id	sid	tid	token	lemma	pos
ejpam-2741	1	1	european	european	PROPN
ejpam-2741	1	2	journal	journal	PROPN
ejpam-2741	1	3	of	of	ADP
ejpam-2741	1	4	pure	pure	ADJ
ejpam-2741	1	5	and	and	CCONJ
ejpam-2741	1	6	applied	apply	VERB
ejpam-2741	1	7	mathematics	mathematic	NOUN
ejpam-2741	1	8	vol	vol	NOUN
ejpam-2741	1	9	.	.	PUNCT
ejpam-2741	2	1	11	11	NUM
ejpam-2741	2	2	,	,	PUNCT
ejpam-2741	2	3	no	no	INTJ
ejpam-2741	2	4	.	.	NOUN
ejpam-2741	2	5	1	1	NUM
ejpam-2741	2	6	,	,	PUNCT
ejpam-2741	2	7	2018	2018	NUM
ejpam-2741	2	8	,	,	PUNCT
ejpam-2741	2	9	238	238	NUM
ejpam-2741	2	10	-	-	SYM
ejpam-2741	2	11	243	243	NUM
ejpam-2741	2	12	issn	issn	PROPN
ejpam-2741	2	13	1307	1307	NUM
ejpam-2741	2	14	-	-	SYM
ejpam-2741	2	15	5543	5543	NUM
ejpam-2741	2	16	–	–	PUNCT
ejpam-2741	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2741	2	18	published	publish	VERB
ejpam-2741	2	19	by	by	ADP
ejpam-2741	2	20	new	new	PROPN
ejpam-2741	2	21	york	york	PROPN
ejpam-2741	2	22	business	business	PROPN
ejpam-2741	2	23	global	global	ADJ
ejpam-2741	2	24	beta	beta	PROPN
ejpam-2741	2	25	g	g	PROPN
ejpam-2741	2	26	-	-	PUNCT
ejpam-2741	2	27	star	star	NOUN
ejpam-2741	2	28	relation	relation	NOUN
ejpam-2741	2	29	on	on	ADP
ejpam-2741	2	30	modules	module	NOUN
ejpam-2741	2	31	celil	celil	NOUN
ejpam-2741	2	32	nebiyev1,∗	nebiyev1,∗	NOUN
ejpam-2741	2	33	,	,	PUNCT
ejpam-2741	2	34	nurhan	nurhan	ADP
ejpam-2741	2	35	sökmez1	sökmez1	PROPN
ejpam-2741	2	36	1	1	NUM
ejpam-2741	2	37	department	department	NOUN
ejpam-2741	2	38	of	of	ADP
ejpam-2741	2	39	mathematics	mathematic	NOUN
ejpam-2741	2	40	,	,	PUNCT
ejpam-2741	2	41	ondokuz	ondokuz	PROPN
ejpam-2741	2	42	mayıs	mayıs	PROPN
ejpam-2741	2	43	university	university	PROPN
ejpam-2741	2	44	,	,	PUNCT
ejpam-2741	2	45	kurupelit	kurupelit	NOUN
ejpam-2741	2	46	-	-	PUNCT
ejpam-2741	2	47	atakum	atakum	NOUN
ejpam-2741	2	48	,	,	PUNCT
ejpam-2741	2	49	samsun	samsun	NOUN
ejpam-2741	2	50	,	,	PUNCT
ejpam-2741	2	51	turkey	turkey	PROPN
ejpam-2741	2	52	abstract	abstract	NOUN
ejpam-2741	2	53	.	.	PUNCT
ejpam-2741	3	1	in	in	ADP
ejpam-2741	3	2	this	this	DET
ejpam-2741	3	3	work	work	NOUN
ejpam-2741	3	4	,	,	PUNCT
ejpam-2741	3	5	we	we	PRON
ejpam-2741	3	6	say	say	VERB
ejpam-2741	3	7	submodules	submodule	NOUN
ejpam-2741	3	8	x	x	PUNCT
ejpam-2741	3	9	and	and	CCONJ
ejpam-2741	3	10	y	y	PROPN
ejpam-2741	3	11	of	of	ADP
ejpam-2741	3	12	m	m	PROPN
ejpam-2741	3	13	are	be	AUX
ejpam-2741	3	14	β∗	β∗	NOUN
ejpam-2741	3	15	g	g	NOUN
ejpam-2741	3	16	equivalence	equivalence	NOUN
ejpam-2741	3	17	,	,	PUNCT
ejpam-2741	3	18	xβ∗	xβ∗	PROPN
ejpam-2741	3	19	gy	gy	PROPN
ejpam-2741	3	20	,	,	PUNCT
ejpam-2741	3	21	if	if	SCONJ
ejpam-2741	3	22	and	and	CCONJ
ejpam-2741	3	23	only	only	ADV
ejpam-2741	3	24	if	if	SCONJ
ejpam-2741	3	25	y	y	PROPN
ejpam-2741	4	1	+	+	PROPN
ejpam-2741	4	2	k	k	X
ejpam-2741	4	3	=	=	PUNCT
ejpam-2741	4	4	m	m	VERB
ejpam-2741	4	5	for	for	ADP
ejpam-2741	4	6	every	every	DET
ejpam-2741	4	7	k	k	PROPN
ejpam-2741	4	8	e	e	PROPN
ejpam-2741	4	9	m	m	VERB
ejpam-2741	4	10	such	such	ADJ
ejpam-2741	4	11	that	that	SCONJ
ejpam-2741	4	12	x	x	X
ejpam-2741	5	1	+	+	NUM
ejpam-2741	5	2	k	k	X
ejpam-2741	5	3	=	=	VERB
ejpam-2741	5	4	m	m	PROPN
ejpam-2741	5	5	and	and	CCONJ
ejpam-2741	5	6	x	x	X
ejpam-2741	5	7	+	+	NUM
ejpam-2741	5	8	t	t	X
ejpam-2741	5	9	=	=	PUNCT
ejpam-2741	5	10	m	m	VERB
ejpam-2741	5	11	for	for	ADP
ejpam-2741	5	12	every	every	DET
ejpam-2741	5	13	t	t	NOUN
ejpam-2741	5	14	e	e	NOUN
ejpam-2741	5	15	m	m	VERB
ejpam-2741	5	16	such	such	ADJ
ejpam-2741	5	17	that	that	SCONJ
ejpam-2741	5	18	y	y	PROPN
ejpam-2741	5	19	+	+	PROPN
ejpam-2741	5	20	t	t	X
ejpam-2741	5	21	=	=	X
ejpam-2741	5	22	m	m	VERB
ejpam-2741	5	23	.	.	PUNCT
ejpam-2741	6	1	it	it	PRON
ejpam-2741	6	2	is	be	AUX
ejpam-2741	6	3	proved	prove	VERB
ejpam-2741	6	4	that	that	SCONJ
ejpam-2741	6	5	the	the	DET
ejpam-2741	6	6	β∗	β∗	NOUN
ejpam-2741	6	7	g	g	PROPN
ejpam-2741	6	8	relation	relation	NOUN
ejpam-2741	6	9	is	be	AUX
ejpam-2741	6	10	an	an	DET
ejpam-2741	6	11	equivalent	equivalent	ADJ
ejpam-2741	6	12	relation	relation	NOUN
ejpam-2741	6	13	and	and	CCONJ
ejpam-2741	6	14	has	have	VERB
ejpam-2741	6	15	good	good	ADJ
ejpam-2741	6	16	behaviour	behaviour	NOUN
ejpam-2741	6	17	with	with	ADP
ejpam-2741	6	18	respect	respect	NOUN
ejpam-2741	6	19	to	to	ADP
ejpam-2741	6	20	addition	addition	NOUN
ejpam-2741	6	21	of	of	ADP
ejpam-2741	6	22	submodules	submodule	NOUN
ejpam-2741	6	23	and	and	CCONJ
ejpam-2741	6	24	homomorphisms	homomorphism	NOUN
ejpam-2741	6	25	.	.	PUNCT
ejpam-2741	7	1	2010	2010	NUM
ejpam-2741	7	2	mathematics	mathematic	NOUN
ejpam-2741	7	3	subject	subject	NOUN
ejpam-2741	7	4	classifications	classification	NOUN
ejpam-2741	7	5	:	:	PUNCT
ejpam-2741	7	6	16d10	16d10	NUM
ejpam-2741	7	7	,	,	PUNCT
ejpam-2741	7	8	16d70	16d70	NUM
ejpam-2741	7	9	key	key	ADJ
ejpam-2741	7	10	words	word	NOUN
ejpam-2741	7	11	and	and	CCONJ
ejpam-2741	7	12	phrases	phrase	NOUN
ejpam-2741	7	13	:	:	PUNCT
ejpam-2741	7	14	small	small	ADJ
ejpam-2741	7	15	submodules	submodule	NOUN
ejpam-2741	7	16	,	,	PUNCT
ejpam-2741	7	17	generalized	generalize	VERB
ejpam-2741	7	18	small	small	ADJ
ejpam-2741	7	19	submodules	submodule	NOUN
ejpam-2741	7	20	,	,	PUNCT
ejpam-2741	7	21	supplemented	supplement	VERB
ejpam-2741	7	22	modules	module	NOUN
ejpam-2741	7	23	,	,	PUNCT
ejpam-2741	7	24	g	g	NOUN
ejpam-2741	7	25	-	-	PUNCT
ejpam-2741	7	26	supplemented	supplement	VERB
ejpam-2741	7	27	modules	module	NOUN
ejpam-2741	7	28	1	1	NUM
ejpam-2741	7	29	.	.	PUNCT
ejpam-2741	8	1	introduction	introduction	NOUN
ejpam-2741	8	2	throughout	throughout	ADP
ejpam-2741	8	3	this	this	DET
ejpam-2741	8	4	paper	paper	NOUN
ejpam-2741	8	5	all	all	DET
ejpam-2741	8	6	rings	ring	NOUN
ejpam-2741	8	7	will	will	AUX
ejpam-2741	8	8	be	be	AUX
ejpam-2741	8	9	associative	associative	ADJ
ejpam-2741	8	10	with	with	ADP
ejpam-2741	8	11	identity	identity	NOUN
ejpam-2741	8	12	and	and	CCONJ
ejpam-2741	8	13	all	all	DET
ejpam-2741	8	14	modules	module	NOUN
ejpam-2741	8	15	will	will	AUX
ejpam-2741	8	16	be	be	AUX
ejpam-2741	8	17	unital	unital	ADJ
ejpam-2741	8	18	left	leave	VERB
ejpam-2741	8	19	modules	module	NOUN
ejpam-2741	8	20	.	.	PUNCT
ejpam-2741	9	1	let	let	VERB
ejpam-2741	9	2	r	r	PRON
ejpam-2741	9	3	be	be	AUX
ejpam-2741	9	4	a	a	DET
ejpam-2741	9	5	ring	ring	NOUN
ejpam-2741	9	6	and	and	CCONJ
ejpam-2741	9	7	m	m	AUX
ejpam-2741	9	8	be	be	AUX
ejpam-2741	9	9	an	an	DET
ejpam-2741	9	10	r−module	r−module	PROPN
ejpam-2741	9	11	.	.	PUNCT
ejpam-2741	10	1	we	we	PRON
ejpam-2741	10	2	will	will	AUX
ejpam-2741	10	3	denote	denote	VERB
ejpam-2741	10	4	a	a	DET
ejpam-2741	10	5	submodule	submodule	NOUN
ejpam-2741	10	6	n	n	PROPN
ejpam-2741	10	7	of	of	ADP
ejpam-2741	10	8	m	m	PRON
ejpam-2741	10	9	by	by	ADP
ejpam-2741	10	10	n	n	PRON
ejpam-2741	10	11	≤	≤	NOUN
ejpam-2741	10	12	m	m	VERB
ejpam-2741	10	13	.	.	PUNCT
ejpam-2741	11	1	let	let	VERB
ejpam-2741	11	2	m	m	PRON
ejpam-2741	11	3	be	be	AUX
ejpam-2741	11	4	an	an	DET
ejpam-2741	11	5	r−module	r−module	NOUN
ejpam-2741	11	6	and	and	CCONJ
ejpam-2741	11	7	n	n	PRON
ejpam-2741	11	8	≤	≤	NOUN
ejpam-2741	11	9	m	m	VERB
ejpam-2741	11	10	.	.	PUNCT
ejpam-2741	12	1	if	if	SCONJ
ejpam-2741	12	2	l	l	PROPN
ejpam-2741	12	3	=	=	VERB
ejpam-2741	12	4	m	m	VERB
ejpam-2741	12	5	for	for	ADP
ejpam-2741	12	6	every	every	DET
ejpam-2741	12	7	submodule	submodule	NOUN
ejpam-2741	12	8	l	l	NOUN
ejpam-2741	12	9	of	of	ADP
ejpam-2741	12	10	m	m	PRON
ejpam-2741	12	11	such	such	ADJ
ejpam-2741	12	12	that	that	SCONJ
ejpam-2741	12	13	m	m	VERB
ejpam-2741	12	14	=	=	SYM
ejpam-2741	12	15	n	n	PROPN
ejpam-2741	12	16	+	+	NOUN
ejpam-2741	12	17	l	l	NOUN
ejpam-2741	12	18	,	,	PUNCT
ejpam-2741	12	19	then	then	ADV
ejpam-2741	12	20	n	n	CCONJ
ejpam-2741	12	21	is	be	AUX
ejpam-2741	12	22	called	call	VERB
ejpam-2741	12	23	a	a	DET
ejpam-2741	12	24	small	small	ADJ
ejpam-2741	12	25	submodule	submodule	NOUN
ejpam-2741	12	26	of	of	ADP
ejpam-2741	12	27	m	m	PRON
ejpam-2741	12	28	and	and	CCONJ
ejpam-2741	12	29	denoted	denote	VERB
ejpam-2741	12	30	by	by	ADP
ejpam-2741	12	31	n	n	DET
ejpam-2741	12	32	�	�	PROPN
ejpam-2741	12	33	m	m	PROPN
ejpam-2741	12	34	.	.	PUNCT
ejpam-2741	13	1	let	let	VERB
ejpam-2741	13	2	m	m	PRON
ejpam-2741	13	3	be	be	AUX
ejpam-2741	13	4	an	an	DET
ejpam-2741	13	5	r−module	r−module	NOUN
ejpam-2741	13	6	and	and	CCONJ
ejpam-2741	13	7	n	n	PRON
ejpam-2741	13	8	≤m	≤m	NOUN
ejpam-2741	13	9	.	.	PUNCT
ejpam-2741	14	1	n	n	PROPN
ejpam-2741	14	2	is	be	AUX
ejpam-2741	14	3	called	call	VERB
ejpam-2741	14	4	essential	essential	ADJ
ejpam-2741	14	5	submodule	submodule	NOUN
ejpam-2741	14	6	of	of	ADP
ejpam-2741	14	7	m	m	PRON
ejpam-2741	14	8	and	and	CCONJ
ejpam-2741	14	9	denoted	denote	VERB
ejpam-2741	14	10	by	by	ADP
ejpam-2741	14	11	n	n	PRON
ejpam-2741	14	12	em	em	PRON
ejpam-2741	14	13	in	in	ADP
ejpam-2741	14	14	case	case	NOUN
ejpam-2741	14	15	k∩n	k∩n	PROPN
ejpam-2741	14	16	6=	6=	PRON
ejpam-2741	14	17	0	0	NUM
ejpam-2741	14	18	for	for	ADP
ejpam-2741	14	19	every	every	DET
ejpam-2741	14	20	submodule	submodule	PROPN
ejpam-2741	14	21	k	k	PROPN
ejpam-2741	14	22	6=	6=	PROPN
ejpam-2741	14	23	0	0	X
ejpam-2741	14	24	.	.	PUNCT
ejpam-2741	15	1	let	let	VERB
ejpam-2741	15	2	m	m	PRON
ejpam-2741	15	3	be	be	AUX
ejpam-2741	15	4	an	an	DET
ejpam-2741	15	5	r−module	r−module	NOUN
ejpam-2741	15	6	and	and	CCONJ
ejpam-2741	15	7	k	k	PROPN
ejpam-2741	15	8	be	be	AUX
ejpam-2741	15	9	a	a	DET
ejpam-2741	15	10	submodule	submodule	NOUN
ejpam-2741	15	11	of	of	ADP
ejpam-2741	15	12	m	m	PROPN
ejpam-2741	15	13	.	.	PUNCT
ejpam-2741	16	1	k	k	PROPN
ejpam-2741	16	2	is	be	AUX
ejpam-2741	16	3	called	call	VERB
ejpam-2741	16	4	a	a	DET
ejpam-2741	16	5	generalized	generalized	ADJ
ejpam-2741	16	6	small	small	ADJ
ejpam-2741	16	7	(	(	PUNCT
ejpam-2741	16	8	briefly	briefly	ADV
ejpam-2741	16	9	,	,	PUNCT
ejpam-2741	16	10	g	g	NOUN
ejpam-2741	16	11	-	-	PUNCT
ejpam-2741	16	12	small	small	ADJ
ejpam-2741	16	13	)	)	PUNCT
ejpam-2741	16	14	submodule	submodule	NOUN
ejpam-2741	16	15	of	of	ADP
ejpam-2741	16	16	m	m	PROPN
ejpam-2741	16	17	if	if	SCONJ
ejpam-2741	16	18	for	for	ADP
ejpam-2741	16	19	every	every	DET
ejpam-2741	16	20	essential	essential	ADJ
ejpam-2741	16	21	submodule	submodule	PROPN
ejpam-2741	16	22	t	t	PROPN
ejpam-2741	16	23	of	of	ADP
ejpam-2741	16	24	m	m	PROPN
ejpam-2741	16	25	with	with	ADP
ejpam-2741	16	26	the	the	DET
ejpam-2741	16	27	property	property	NOUN
ejpam-2741	16	28	m	m	NOUN
ejpam-2741	16	29	=	=	SYM
ejpam-2741	17	1	k	k	X
ejpam-2741	18	1	+	+	PROPN
ejpam-2741	18	2	t	t	PROPN
ejpam-2741	18	3	implies	imply	VERB
ejpam-2741	18	4	that	that	SCONJ
ejpam-2741	18	5	t	t	NOUN
ejpam-2741	18	6	=	=	PUNCT
ejpam-2741	18	7	m	m	PROPN
ejpam-2741	18	8	,	,	PUNCT
ejpam-2741	18	9	then	then	ADV
ejpam-2741	18	10	we	we	PRON
ejpam-2741	18	11	write	write	VERB
ejpam-2741	18	12	k	k	PROPN
ejpam-2741	18	13	�	�	PROPN
ejpam-2741	18	14	g	g	PROPN
ejpam-2741	18	15	m	m	PROPN
ejpam-2741	18	16	.	.	PUNCT
ejpam-2741	19	1	(	(	PUNCT
ejpam-2741	19	2	in	in	ADP
ejpam-2741	19	3	[	[	PUNCT
ejpam-2741	19	4	11	11	NUM
ejpam-2741	19	5	]	]	PUNCT
ejpam-2741	19	6	,	,	PUNCT
ejpam-2741	19	7	it	it	PRON
ejpam-2741	19	8	is	be	AUX
ejpam-2741	19	9	called	call	VERB
ejpam-2741	19	10	an	an	DET
ejpam-2741	19	11	e	e	ADJ
ejpam-2741	19	12	-	-	ADJ
ejpam-2741	19	13	small	small	ADJ
ejpam-2741	19	14	submodule	submodule	NOUN
ejpam-2741	19	15	of	of	ADP
ejpam-2741	19	16	m	m	PROPN
ejpam-2741	19	17	and	and	CCONJ
ejpam-2741	19	18	denoted	denote	VERB
ejpam-2741	19	19	by	by	ADP
ejpam-2741	19	20	k	k	PROPN
ejpam-2741	19	21	�	�	PROPN
ejpam-2741	19	22	e	e	PROPN
ejpam-2741	19	23	m	m	PROPN
ejpam-2741	19	24	)	)	PUNCT
ejpam-2741	19	25	.	.	PUNCT
ejpam-2741	20	1	it	it	PRON
ejpam-2741	20	2	is	be	AUX
ejpam-2741	20	3	clear	clear	ADJ
ejpam-2741	20	4	that	that	SCONJ
ejpam-2741	20	5	every	every	DET
ejpam-2741	20	6	small	small	ADJ
ejpam-2741	20	7	submodule	submodule	NOUN
ejpam-2741	20	8	is	be	AUX
ejpam-2741	20	9	a	a	DET
ejpam-2741	20	10	generalized	generalized	ADJ
ejpam-2741	20	11	small	small	ADJ
ejpam-2741	20	12	submodule	submodule	NOUN
ejpam-2741	20	13	but	but	CCONJ
ejpam-2741	20	14	the	the	DET
ejpam-2741	20	15	converse	converse	NOUN
ejpam-2741	20	16	is	be	AUX
ejpam-2741	20	17	not	not	PART
ejpam-2741	20	18	true	true	ADJ
ejpam-2741	20	19	generally	generally	ADV
ejpam-2741	20	20	.	.	PUNCT
ejpam-2741	21	1	m	m	PROPN
ejpam-2741	21	2	is	be	AUX
ejpam-2741	21	3	called	call	VERB
ejpam-2741	21	4	a	a	DET
ejpam-2741	21	5	(	(	PUNCT
ejpam-2741	21	6	generalized	generalized	ADJ
ejpam-2741	21	7	)	)	PUNCT
ejpam-2741	21	8	hollow	hollow	ADJ
ejpam-2741	21	9	module	module	NOUN
ejpam-2741	21	10	if	if	SCONJ
ejpam-2741	21	11	every	every	DET
ejpam-2741	21	12	proper	proper	ADJ
ejpam-2741	21	13	submodule	submodule	NOUN
ejpam-2741	21	14	of	of	ADP
ejpam-2741	21	15	m	m	PROPN
ejpam-2741	21	16	is	be	AUX
ejpam-2741	21	17	(	(	PUNCT
ejpam-2741	21	18	generalized	generalized	ADJ
ejpam-2741	21	19	)	)	PUNCT
ejpam-2741	21	20	small	small	ADJ
ejpam-2741	21	21	in	in	ADP
ejpam-2741	21	22	m	m	PROPN
ejpam-2741	21	23	.	.	PUNCT
ejpam-2741	22	1	here	here	ADV
ejpam-2741	22	2	it	it	PRON
ejpam-2741	22	3	is	be	AUX
ejpam-2741	22	4	clear	clear	ADJ
ejpam-2741	22	5	that	that	SCONJ
ejpam-2741	22	6	every	every	DET
ejpam-2741	22	7	hollow	hollow	ADJ
ejpam-2741	22	8	module	module	NOUN
ejpam-2741	22	9	is	be	AUX
ejpam-2741	22	10	generalized	generalize	VERB
ejpam-2741	22	11	hollow	hollow	ADJ
ejpam-2741	22	12	module	module	NOUN
ejpam-2741	22	13	.	.	PUNCT
ejpam-2741	23	1	let	let	VERB
ejpam-2741	23	2	m	m	PRON
ejpam-2741	23	3	be	be	AUX
ejpam-2741	23	4	an	an	DET
ejpam-2741	23	5	r−module	r−module	NOUN
ejpam-2741	23	6	and	and	CCONJ
ejpam-2741	23	7	u	u	NOUN
ejpam-2741	23	8	,	,	PUNCT
ejpam-2741	23	9	v	v	NOUN
ejpam-2741	23	10	≤	≤	NOUN
ejpam-2741	23	11	m	m	NOUN
ejpam-2741	23	12	.	.	PUNCT
ejpam-2741	24	1	if	if	SCONJ
ejpam-2741	24	2	m	m	NOUN
ejpam-2741	24	3	=	=	VERB
ejpam-2741	24	4	u	u	NOUN
ejpam-2741	24	5	+	+	X
ejpam-2741	24	6	v	v	NOUN
ejpam-2741	24	7	and	and	CCONJ
ejpam-2741	24	8	v	v	NOUN
ejpam-2741	24	9	is	be	AUX
ejpam-2741	24	10	minimal	minimal	ADJ
ejpam-2741	24	11	with	with	ADP
ejpam-2741	24	12	respect	respect	NOUN
ejpam-2741	24	13	to	to	ADP
ejpam-2741	24	14	this	this	DET
ejpam-2741	24	15	property	property	NOUN
ejpam-2741	24	16	,	,	PUNCT
ejpam-2741	24	17	or	or	CCONJ
ejpam-2741	24	18	equivalently	equivalently	ADV
ejpam-2741	24	19	,	,	PUNCT
ejpam-2741	24	20	m	m	VERB
ejpam-2741	24	21	=	=	SYM
ejpam-2741	24	22	u	u	NOUN
ejpam-2741	24	23	+	+	X
ejpam-2741	24	24	v	v	NOUN
ejpam-2741	24	25	and	and	CCONJ
ejpam-2741	24	26	u	u	NOUN
ejpam-2741	24	27	∩	∩	PROPN
ejpam-2741	24	28	v	v	ADP
ejpam-2741	24	29	�	�	PROPN
ejpam-2741	24	30	v	v	NOUN
ejpam-2741	24	31	,	,	PUNCT
ejpam-2741	24	32	then	then	ADV
ejpam-2741	24	33	v	v	NOUN
ejpam-2741	24	34	is	be	AUX
ejpam-2741	24	35	called	call	VERB
ejpam-2741	24	36	a	a	DET
ejpam-2741	24	37	supplement	supplement	NOUN
ejpam-2741	24	38	of	of	ADP
ejpam-2741	24	39	u	u	NOUN
ejpam-2741	24	40	in	in	ADP
ejpam-2741	24	41	m	m	PROPN
ejpam-2741	24	42	.	.	PUNCT
ejpam-2741	25	1	m	m	PROPN
ejpam-2741	25	2	is	be	AUX
ejpam-2741	25	3	called	call	VERB
ejpam-2741	25	4	a	a	DET
ejpam-2741	25	5	supplemented	supplement	VERB
ejpam-2741	25	6	module	module	NOUN
ejpam-2741	25	7	if	if	SCONJ
ejpam-2741	25	8	every	every	DET
ejpam-2741	25	9	submodule	submodule	NOUN
ejpam-2741	25	10	of	of	ADP
ejpam-2741	25	11	m	m	PROPN
ejpam-2741	25	12	has	have	VERB
ejpam-2741	25	13	a	a	DET
ejpam-2741	25	14	supplement	supplement	NOUN
ejpam-2741	25	15	in	in	ADP
ejpam-2741	25	16	m	m	PROPN
ejpam-2741	25	17	.	.	PUNCT
ejpam-2741	26	1	let	let	VERB
ejpam-2741	26	2	m	m	PRON
ejpam-2741	26	3	be	be	AUX
ejpam-2741	26	4	an	an	DET
ejpam-2741	26	5	r−module	r−module	NOUN
ejpam-2741	26	6	and	and	CCONJ
ejpam-2741	26	7	u	u	NOUN
ejpam-2741	26	8	,	,	PUNCT
ejpam-2741	26	9	v	v	NOUN
ejpam-2741	26	10	≤	≤	NOUN
ejpam-2741	26	11	m	m	NOUN
ejpam-2741	26	12	.	.	PUNCT
ejpam-2741	27	1	if	if	SCONJ
ejpam-2741	27	2	m	m	VERB
ejpam-2741	27	3	=	=	VERB
ejpam-2741	27	4	u	u	NOUN
ejpam-2741	27	5	+	+	NOUN
ejpam-2741	27	6	v	v	NOUN
ejpam-2741	27	7	and	and	CCONJ
ejpam-2741	27	8	m	m	PROPN
ejpam-2741	27	9	=	=	SYM
ejpam-2741	27	10	u	u	PROPN
ejpam-2741	27	11	+	+	PROPN
ejpam-2741	27	12	t	t	NOUN
ejpam-2741	27	13	with	with	ADP
ejpam-2741	27	14	t	t	PROPN
ejpam-2741	27	15	e	e	X
ejpam-2741	27	16	v	v	PROPN
ejpam-2741	27	17	implies	imply	VERB
ejpam-2741	27	18	that	that	SCONJ
ejpam-2741	27	19	t	t	NOUN
ejpam-2741	27	20	=	=	SYM
ejpam-2741	27	21	v	v	NOUN
ejpam-2741	27	22	,	,	PUNCT
ejpam-2741	27	23	or	or	CCONJ
ejpam-2741	27	24	equivalently	equivalently	ADV
ejpam-2741	27	25	,	,	PUNCT
ejpam-2741	27	26	m	m	VERB
ejpam-2741	27	27	=	=	VERB
ejpam-2741	27	28	u	u	NOUN
ejpam-2741	27	29	+	+	NOUN
ejpam-2741	27	30	v	v	NOUN
ejpam-2741	27	31	and	and	CCONJ
ejpam-2741	27	32	u	u	NOUN
ejpam-2741	27	33	∩v	∩v	PROPN
ejpam-2741	27	34	�	�	PROPN
ejpam-2741	27	35	g	g	NOUN
ejpam-2741	27	36	v	v	NUM
ejpam-2741	27	37	,	,	PUNCT
ejpam-2741	27	38	then	then	ADV
ejpam-2741	27	39	v	v	NOUN
ejpam-2741	27	40	is	be	AUX
ejpam-2741	27	41	called	call	VERB
ejpam-2741	27	42	a	a	DET
ejpam-2741	27	43	g	g	NOUN
ejpam-2741	27	44	-	-	PUNCT
ejpam-2741	27	45	supplement	supplement	NOUN
ejpam-2741	27	46	of	of	ADP
ejpam-2741	27	47	u	u	NOUN
ejpam-2741	27	48	in	in	ADP
ejpam-2741	27	49	m	m	PROPN
ejpam-2741	27	50	.	.	PUNCT
ejpam-2741	28	1	m	m	PROPN
ejpam-2741	28	2	is	be	AUX
ejpam-2741	28	3	called	call	VERB
ejpam-2741	28	4	g	g	NOUN
ejpam-2741	28	5	-	-	PUNCT
ejpam-2741	28	6	supplemented	supplement	VERB
ejpam-2741	28	7	∗corresponding	∗corresponde	VERB
ejpam-2741	28	8	author	author	NOUN
ejpam-2741	28	9	.	.	PUNCT
ejpam-2741	29	1	email	email	NOUN
ejpam-2741	29	2	addresses	address	NOUN
ejpam-2741	29	3	:	:	PUNCT
ejpam-2741	29	4	cnebiyev@omu.edu.tr	cnebiyev@omu.edu.tr	PROPN
ejpam-2741	29	5	(	(	PUNCT
ejpam-2741	29	6	celil	celil	NOUN
ejpam-2741	29	7	nebiyev	nebiyev	ADV
ejpam-2741	29	8	)	)	PUNCT
ejpam-2741	29	9	,	,	PUNCT
ejpam-2741	29	10	nozkan@omu.edu.tr	nozkan@omu.edu.tr	X
ejpam-2741	29	11	(	(	PUNCT
ejpam-2741	29	12	nurhan	nurhan	ADJ
ejpam-2741	29	13	sökmez	sökmez	NOUN
ejpam-2741	29	14	)	)	PUNCT
ejpam-2741	29	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2741	30	1	238	238	NUM
ejpam-2741	30	2	c	c	X
ejpam-2741	30	3	©	©	PROPN
ejpam-2741	30	4	2018	2018	NUM
ejpam-2741	30	5	ejpam	ejpam	VERB
ejpam-2741	30	6	all	all	DET
ejpam-2741	30	7	rights	right	NOUN
ejpam-2741	30	8	reserved	reserve	VERB
ejpam-2741	30	9	.	.	PUNCT
ejpam-2741	31	1	c.	c.	PROPN
ejpam-2741	31	2	nebiyev	nebiyev	PROPN
ejpam-2741	31	3	,	,	PUNCT
ejpam-2741	31	4	n.	n.	PROPN
ejpam-2741	31	5	sökmez	sökmez	PROPN
ejpam-2741	31	6	/	/	SYM
ejpam-2741	31	7	eur	eur	PROPN
ejpam-2741	31	8	.	.	PUNCT
ejpam-2741	32	1	j.	j.	PROPN
ejpam-2741	32	2	pure	pure	PROPN
ejpam-2741	32	3	appl	appl	PROPN
ejpam-2741	32	4	.	.	PROPN
ejpam-2741	32	5	math	math	PROPN
ejpam-2741	32	6	,	,	PUNCT
ejpam-2741	32	7	11	11	NUM
ejpam-2741	32	8	(	(	PUNCT
ejpam-2741	32	9	1	1	NUM
ejpam-2741	32	10	)	)	PUNCT
ejpam-2741	32	11	(	(	PUNCT
ejpam-2741	32	12	2018	2018	NUM
ejpam-2741	32	13	)	)	PUNCT
ejpam-2741	32	14	,	,	PUNCT
ejpam-2741	32	15	238	238	NUM
ejpam-2741	32	16	-	-	SYM
ejpam-2741	32	17	243	243	NUM
ejpam-2741	32	18	239	239	NUM
ejpam-2741	32	19	if	if	SCONJ
ejpam-2741	32	20	every	every	DET
ejpam-2741	32	21	submodule	submodule	NOUN
ejpam-2741	32	22	of	of	ADP
ejpam-2741	32	23	m	m	PROPN
ejpam-2741	32	24	has	have	VERB
ejpam-2741	32	25	a	a	DET
ejpam-2741	32	26	g	g	NOUN
ejpam-2741	32	27	-	-	PUNCT
ejpam-2741	32	28	supplement	supplement	NOUN
ejpam-2741	32	29	in	in	ADP
ejpam-2741	32	30	m	m	PROPN
ejpam-2741	32	31	.	.	PUNCT
ejpam-2741	33	1	let	let	VERB
ejpam-2741	33	2	m	m	PRON
ejpam-2741	33	3	be	be	AUX
ejpam-2741	33	4	an	an	DET
ejpam-2741	33	5	r−module	r−module	NOUN
ejpam-2741	33	6	and	and	CCONJ
ejpam-2741	33	7	u	u	PROPN
ejpam-2741	33	8	≤m	≤m	NOUN
ejpam-2741	33	9	.	.	PUNCT
ejpam-2741	34	1	if	if	SCONJ
ejpam-2741	34	2	for	for	ADP
ejpam-2741	34	3	every	every	DET
ejpam-2741	34	4	v	v	NOUN
ejpam-2741	34	5	≤m	≤m	NOUN
ejpam-2741	34	6	such	such	ADJ
ejpam-2741	34	7	that	that	SCONJ
ejpam-2741	34	8	m	m	AUX
ejpam-2741	34	9	=	=	SYM
ejpam-2741	34	10	u	u	PROPN
ejpam-2741	34	11	+	+	X
ejpam-2741	34	12	v	v	NUM
ejpam-2741	34	13	,	,	PUNCT
ejpam-2741	34	14	u	u	NOUN
ejpam-2741	34	15	has	have	VERB
ejpam-2741	34	16	a	a	DET
ejpam-2741	34	17	g	g	NOUN
ejpam-2741	34	18	-	-	PUNCT
ejpam-2741	34	19	supplement	supplement	NOUN
ejpam-2741	34	20	v	v	NOUN
ejpam-2741	34	21	′	′	NOUN
ejpam-2741	34	22	with	with	ADP
ejpam-2741	34	23	v	v	NOUN
ejpam-2741	34	24	′	′	NUM
ejpam-2741	34	25	≤	≤	NUM
ejpam-2741	34	26	v	v	NOUN
ejpam-2741	34	27	,	,	PUNCT
ejpam-2741	34	28	we	we	PRON
ejpam-2741	34	29	say	say	VERB
ejpam-2741	34	30	u	u	NOUN
ejpam-2741	34	31	has	have	VERB
ejpam-2741	34	32	ample	ample	ADJ
ejpam-2741	34	33	g	g	NOUN
ejpam-2741	34	34	-	-	PUNCT
ejpam-2741	34	35	supplements	supplement	NOUN
ejpam-2741	34	36	in	in	ADP
ejpam-2741	34	37	m	m	PROPN
ejpam-2741	34	38	.	.	PUNCT
ejpam-2741	35	1	if	if	SCONJ
ejpam-2741	35	2	every	every	DET
ejpam-2741	35	3	submodule	submodule	NOUN
ejpam-2741	35	4	of	of	ADP
ejpam-2741	35	5	m	m	PROPN
ejpam-2741	35	6	has	have	VERB
ejpam-2741	35	7	ample	ample	ADJ
ejpam-2741	35	8	g	g	NOUN
ejpam-2741	35	9	-	-	PUNCT
ejpam-2741	35	10	supplements	supplement	NOUN
ejpam-2741	35	11	in	in	ADP
ejpam-2741	35	12	m	m	PROPN
ejpam-2741	35	13	,	,	PUNCT
ejpam-2741	35	14	then	then	ADV
ejpam-2741	35	15	m	m	VERB
ejpam-2741	35	16	is	be	AUX
ejpam-2741	35	17	called	call	VERB
ejpam-2741	35	18	an	an	DET
ejpam-2741	35	19	amply	amply	NOUN
ejpam-2741	35	20	g	g	NOUN
ejpam-2741	35	21	-	-	PUNCT
ejpam-2741	35	22	supplemented	supplement	VERB
ejpam-2741	35	23	module	module	NOUN
ejpam-2741	35	24	.	.	PUNCT
ejpam-2741	36	1	socm	socm	PROPN
ejpam-2741	36	2	indicates	indicate	VERB
ejpam-2741	36	3	the	the	DET
ejpam-2741	36	4	socle	socle	NOUN
ejpam-2741	36	5	of	of	ADP
ejpam-2741	36	6	m	m	PROPN
ejpam-2741	36	7	(	(	PUNCT
ejpam-2741	36	8	the	the	DET
ejpam-2741	36	9	sum	sum	NOUN
ejpam-2741	36	10	of	of	ADP
ejpam-2741	36	11	all	all	DET
ejpam-2741	36	12	simple	simple	ADJ
ejpam-2741	36	13	submodules	submodule	NOUN
ejpam-2741	36	14	of	of	ADP
ejpam-2741	36	15	m	m	PROPN
ejpam-2741	36	16	)	)	PUNCT
ejpam-2741	36	17	.	.	PUNCT
ejpam-2741	37	1	lemma	lemma	PROPN
ejpam-2741	37	2	1	1	X
ejpam-2741	37	3	.	.	PUNCT
ejpam-2741	38	1	let	let	VERB
ejpam-2741	38	2	m	m	VERB
ejpam-2741	38	3	=	=	VERB
ejpam-2741	38	4	u	u	PROPN
ejpam-2741	38	5	+	+	X
ejpam-2741	38	6	v	v	NOUN
ejpam-2741	38	7	and	and	CCONJ
ejpam-2741	38	8	m	m	PROPN
ejpam-2741	38	9	=	=	SYM
ejpam-2741	38	10	u	u	NOUN
ejpam-2741	38	11	∩	∩	NOUN
ejpam-2741	38	12	v	v	ADP
ejpam-2741	38	13	+	+	X
ejpam-2741	38	14	t	t	PROPN
ejpam-2741	38	15	.	.	PUNCT
ejpam-2741	39	1	then	then	ADV
ejpam-2741	39	2	m	m	VERB
ejpam-2741	39	3	=	=	SYM
ejpam-2741	39	4	u	u	PROPN
ejpam-2741	39	5	+	+	NOUN
ejpam-2741	39	6	v	v	NOUN
ejpam-2741	39	7	∩	∩	NOUN
ejpam-2741	39	8	t	t	NOUN
ejpam-2741	39	9	=	=	SYM
ejpam-2741	39	10	v	v	PROPN
ejpam-2741	39	11	+	+	CCONJ
ejpam-2741	39	12	u	u	NOUN
ejpam-2741	39	13	∩	∩	NOUN
ejpam-2741	39	14	t	t	NOUN
ejpam-2741	39	15	.	.	PUNCT
ejpam-2741	40	1	proof	proof	NOUN
ejpam-2741	40	2	.	.	PUNCT
ejpam-2741	41	1	see	see	VERB
ejpam-2741	41	2	[	[	X
ejpam-2741	41	3	4	4	NUM
ejpam-2741	41	4	,	,	PUNCT
ejpam-2741	41	5	lemma	lemma	PROPN
ejpam-2741	41	6	1.24	1.24	NUM
ejpam-2741	41	7	]	]	PUNCT
ejpam-2741	41	8	.	.	PUNCT
ejpam-2741	42	1	2	2	X
ejpam-2741	42	2	.	.	X
ejpam-2741	42	3	the	the	DET
ejpam-2741	42	4	β∗g	β∗g	NUM
ejpam-2741	42	5	relation	relation	NOUN
ejpam-2741	42	6	definition	definition	NOUN
ejpam-2741	42	7	1	1	X
ejpam-2741	42	8	.	.	PUNCT
ejpam-2741	43	1	we	we	PRON
ejpam-2741	43	2	define	define	VERB
ejpam-2741	43	3	the	the	DET
ejpam-2741	43	4	relation	relation	NOUN
ejpam-2741	43	5	′β∗g	′β∗g	NOUN
ejpam-2741	43	6	′	′	NOUN
ejpam-2741	43	7	on	on	ADP
ejpam-2741	43	8	the	the	DET
ejpam-2741	43	9	set	set	NOUN
ejpam-2741	43	10	of	of	ADP
ejpam-2741	43	11	submodules	submodule	NOUN
ejpam-2741	43	12	of	of	ADP
ejpam-2741	43	13	an	an	DET
ejpam-2741	43	14	r−module	r−module	NOUN
ejpam-2741	43	15	m	m	VERB
ejpam-2741	43	16	by	by	ADP
ejpam-2741	43	17	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	44	1	if	if	SCONJ
ejpam-2741	44	2	and	and	CCONJ
ejpam-2741	44	3	only	only	ADV
ejpam-2741	44	4	if	if	SCONJ
ejpam-2741	44	5	y	y	PROPN
ejpam-2741	44	6	+	+	PROPN
ejpam-2741	44	7	k	k	PROPN
ejpam-2741	44	8	=	=	X
ejpam-2741	44	9	m	m	VERB
ejpam-2741	44	10	for	for	ADP
ejpam-2741	44	11	every	every	DET
ejpam-2741	44	12	k	k	NOUN
ejpam-2741	44	13	em	em	PRON
ejpam-2741	44	14	such	such	ADJ
ejpam-2741	44	15	that	that	SCONJ
ejpam-2741	44	16	x+k	x+k	NUM
ejpam-2741	44	17	=	=	SYM
ejpam-2741	44	18	m	m	PROPN
ejpam-2741	44	19	and	and	CCONJ
ejpam-2741	44	20	x+t	x+t	NUM
ejpam-2741	44	21	=	=	PUNCT
ejpam-2741	44	22	m	m	VERB
ejpam-2741	44	23	for	for	ADP
ejpam-2741	44	24	every	every	DET
ejpam-2741	44	25	t	t	NOUN
ejpam-2741	44	26	em	em	PRON
ejpam-2741	44	27	such	such	ADJ
ejpam-2741	44	28	that	that	SCONJ
ejpam-2741	44	29	y	y	PROPN
ejpam-2741	44	30	+	+	PROPN
ejpam-2741	44	31	t	t	PROPN
ejpam-2741	44	32	=	=	SYM
ejpam-2741	44	33	m	m	NOUN
ejpam-2741	44	34	.	.	PUNCT
ejpam-2741	45	1	proposition	proposition	NOUN
ejpam-2741	45	2	1	1	NUM
ejpam-2741	45	3	.	.	PUNCT
ejpam-2741	46	1	let	let	VERB
ejpam-2741	46	2	m	m	PRON
ejpam-2741	46	3	be	be	AUX
ejpam-2741	46	4	an	an	DET
ejpam-2741	46	5	r−module	r−module	PROPN
ejpam-2741	46	6	and	and	CCONJ
ejpam-2741	46	7	x	x	X
ejpam-2741	46	8	,	,	PUNCT
ejpam-2741	46	9	y	y	PROPN
ejpam-2741	46	10	≤m	≤m	PROPN
ejpam-2741	46	11	.	.	PUNCT
ejpam-2741	47	1	if	if	SCONJ
ejpam-2741	47	2	xβ∗y	xβ∗y	PROPN
ejpam-2741	47	3	,	,	PUNCT
ejpam-2741	47	4	then	then	ADV
ejpam-2741	47	5	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	47	6	.	.	PUNCT
ejpam-2741	48	1	proof	proof	NOUN
ejpam-2741	48	2	.	.	PUNCT
ejpam-2741	49	1	clear	clear	ADJ
ejpam-2741	49	2	from	from	ADP
ejpam-2741	49	3	definitions	definition	NOUN
ejpam-2741	49	4	.	.	PUNCT
ejpam-2741	50	1	(	(	PUNCT
ejpam-2741	50	2	see	see	VERB
ejpam-2741	50	3	[	[	X
ejpam-2741	50	4	2	2	NUM
ejpam-2741	50	5	]	]	NUM
ejpam-2741	50	6	)	)	PUNCT
ejpam-2741	50	7	.	.	PUNCT
ejpam-2741	51	1	lemma	lemma	PROPN
ejpam-2741	51	2	2	2	X
ejpam-2741	51	3	.	.	PUNCT
ejpam-2741	52	1	the	the	DET
ejpam-2741	52	2	β∗g	β∗g	NUM
ejpam-2741	52	3	relation	relation	NOUN
ejpam-2741	52	4	is	be	AUX
ejpam-2741	52	5	an	an	DET
ejpam-2741	52	6	equivalence	equivalence	NOUN
ejpam-2741	52	7	relation	relation	NOUN
ejpam-2741	52	8	.	.	PUNCT
ejpam-2741	53	1	proof	proof	NOUN
ejpam-2741	53	2	.	.	PUNCT
ejpam-2741	54	1	the	the	DET
ejpam-2741	54	2	reflective	reflective	ADJ
ejpam-2741	54	3	and	and	CCONJ
ejpam-2741	54	4	symmetric	symmetric	ADJ
ejpam-2741	54	5	properties	property	NOUN
ejpam-2741	54	6	are	be	AUX
ejpam-2741	54	7	clear	clear	ADJ
ejpam-2741	54	8	.	.	PUNCT
ejpam-2741	55	1	for	for	ADP
ejpam-2741	55	2	transitive	transitive	ADJ
ejpam-2741	55	3	property	property	NOUN
ejpam-2741	55	4	,	,	PUNCT
ejpam-2741	55	5	assume	assume	VERB
ejpam-2741	55	6	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	55	7	and	and	CCONJ
ejpam-2741	55	8	y	y	PROPN
ejpam-2741	55	9	β∗gz	β∗gz	PROPN
ejpam-2741	55	10	.	.	PUNCT
ejpam-2741	56	1	let	let	VERB
ejpam-2741	56	2	k	k	PROPN
ejpam-2741	56	3	em	em	PRON
ejpam-2741	56	4	and	and	CCONJ
ejpam-2741	56	5	x	x	X
ejpam-2741	57	1	+	+	NOUN
ejpam-2741	57	2	k	k	X
ejpam-2741	57	3	=	=	NOUN
ejpam-2741	57	4	m	m	VERB
ejpam-2741	57	5	.	.	PUNCT
ejpam-2741	58	1	since	since	SCONJ
ejpam-2741	58	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	58	3	,	,	PUNCT
ejpam-2741	58	4	then	then	ADV
ejpam-2741	58	5	y	y	PROPN
ejpam-2741	59	1	+	+	PROPN
ejpam-2741	59	2	k	k	PROPN
ejpam-2741	59	3	=	=	X
ejpam-2741	59	4	m	m	PROPN
ejpam-2741	59	5	,	,	PUNCT
ejpam-2741	59	6	and	and	CCONJ
ejpam-2741	59	7	since	since	SCONJ
ejpam-2741	59	8	y	y	PROPN
ejpam-2741	59	9	β∗gz	β∗gz	PROPN
ejpam-2741	59	10	,	,	PUNCT
ejpam-2741	59	11	then	then	ADV
ejpam-2741	59	12	z	z	PROPN
ejpam-2741	60	1	+	+	NUM
ejpam-2741	60	2	k	k	X
ejpam-2741	60	3	=	=	NOUN
ejpam-2741	60	4	m	m	VERB
ejpam-2741	60	5	.	.	PUNCT
ejpam-2741	61	1	let	let	VERB
ejpam-2741	61	2	t	t	PROPN
ejpam-2741	61	3	e	e	NOUN
ejpam-2741	61	4	m	m	PROPN
ejpam-2741	61	5	and	and	CCONJ
ejpam-2741	61	6	z	z	PROPN
ejpam-2741	62	1	+	+	NUM
ejpam-2741	62	2	t	t	X
ejpam-2741	62	3	=	=	NOUN
ejpam-2741	62	4	m	m	PROPN
ejpam-2741	62	5	.	.	PUNCT
ejpam-2741	63	1	since	since	SCONJ
ejpam-2741	63	2	y	y	PROPN
ejpam-2741	63	3	β∗gz	β∗gz	PROPN
ejpam-2741	63	4	,	,	PUNCT
ejpam-2741	63	5	then	then	ADV
ejpam-2741	63	6	y	y	PROPN
ejpam-2741	63	7	+	+	PROPN
ejpam-2741	63	8	t	t	PROPN
ejpam-2741	63	9	=	=	SYM
ejpam-2741	63	10	m	m	PROPN
ejpam-2741	63	11	,	,	PUNCT
ejpam-2741	63	12	and	and	CCONJ
ejpam-2741	63	13	since	since	SCONJ
ejpam-2741	63	14	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	63	15	,	,	PUNCT
ejpam-2741	63	16	then	then	ADV
ejpam-2741	63	17	x	x	X
ejpam-2741	64	1	+	+	NUM
ejpam-2741	64	2	t	t	X
ejpam-2741	64	3	=	=	SYM
ejpam-2741	64	4	m	m	NOUN
ejpam-2741	64	5	.	.	PUNCT
ejpam-2741	65	1	hence	hence	ADV
ejpam-2741	65	2	xβ∗gz	xβ∗gz	PROPN
ejpam-2741	65	3	.	.	PUNCT
ejpam-2741	66	1	lemma	lemma	PROPN
ejpam-2741	66	2	3	3	X
ejpam-2741	66	3	.	.	PUNCT
ejpam-2741	67	1	let	let	VERB
ejpam-2741	67	2	x	x	PRON
ejpam-2741	67	3	,	,	PUNCT
ejpam-2741	67	4	y	y	PROPN
ejpam-2741	67	5	≤m	≤m	PROPN
ejpam-2741	67	6	.	.	PUNCT
ejpam-2741	68	1	the	the	DET
ejpam-2741	68	2	following	follow	VERB
ejpam-2741	68	3	statements	statement	NOUN
ejpam-2741	68	4	are	be	AUX
ejpam-2741	68	5	equivalent	equivalent	ADJ
ejpam-2741	68	6	.	.	PUNCT
ejpam-2741	69	1	(	(	PUNCT
ejpam-2741	69	2	i	i	NOUN
ejpam-2741	69	3	)	)	PUNCT
ejpam-2741	69	4	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	69	5	.	.	PUNCT
ejpam-2741	70	1	(	(	PUNCT
ejpam-2741	70	2	ii	ii	NOUN
ejpam-2741	70	3	)	)	PUNCT
ejpam-2741	70	4	for	for	ADP
ejpam-2741	70	5	every	every	DET
ejpam-2741	70	6	t	t	NOUN
ejpam-2741	70	7	em	em	PRON
ejpam-2741	70	8	such	such	ADJ
ejpam-2741	70	9	that	that	SCONJ
ejpam-2741	70	10	x	x	X
ejpam-2741	71	1	+	+	NUM
ejpam-2741	71	2	y	y	PROPN
ejpam-2741	72	1	+	+	NOUN
ejpam-2741	72	2	t	t	NOUN
ejpam-2741	72	3	=	=	SYM
ejpam-2741	72	4	m	m	PROPN
ejpam-2741	72	5	,	,	PUNCT
ejpam-2741	72	6	x	x	PROPN
ejpam-2741	73	1	+	+	NUM
ejpam-2741	73	2	t	t	X
ejpam-2741	73	3	=	=	SYM
ejpam-2741	73	4	m	m	PROPN
ejpam-2741	73	5	and	and	CCONJ
ejpam-2741	73	6	y	y	PROPN
ejpam-2741	73	7	+	+	PROPN
ejpam-2741	73	8	t	t	PROPN
ejpam-2741	73	9	=	=	NOUN
ejpam-2741	73	10	m	m	NOUN
ejpam-2741	73	11	.	.	PUNCT
ejpam-2741	74	1	proof	proof	NOUN
ejpam-2741	74	2	.	.	PUNCT
ejpam-2741	75	1	(	(	PUNCT
ejpam-2741	75	2	i	i	NOUN
ejpam-2741	75	3	)	)	PUNCT
ejpam-2741	76	1	=	=	NOUN
ejpam-2741	76	2	⇒	⇒	NOUN
ejpam-2741	76	3	(	(	PUNCT
ejpam-2741	76	4	ii	ii	NOUN
ejpam-2741	76	5	)	)	PUNCT
ejpam-2741	76	6	let	let	VERB
ejpam-2741	76	7	t	t	PROPN
ejpam-2741	76	8	e	e	NOUN
ejpam-2741	76	9	m	m	PROPN
ejpam-2741	76	10	and	and	CCONJ
ejpam-2741	76	11	x	x	X
ejpam-2741	77	1	+	+	CCONJ
ejpam-2741	77	2	y	y	PROPN
ejpam-2741	77	3	+	+	NOUN
ejpam-2741	77	4	t	t	X
ejpam-2741	77	5	=	=	NOUN
ejpam-2741	77	6	m	m	PROPN
ejpam-2741	77	7	.	.	PUNCT
ejpam-2741	78	1	since	since	SCONJ
ejpam-2741	78	2	t	t	PROPN
ejpam-2741	78	3	e	e	PROPN
ejpam-2741	78	4	m	m	PROPN
ejpam-2741	78	5	,	,	PUNCT
ejpam-2741	78	6	then	then	ADV
ejpam-2741	78	7	y	y	PROPN
ejpam-2741	79	1	+	+	PROPN
ejpam-2741	79	2	t	t	PROPN
ejpam-2741	79	3	e	e	NOUN
ejpam-2741	79	4	m	m	PROPN
ejpam-2741	79	5	and	and	CCONJ
ejpam-2741	79	6	x	x	X
ejpam-2741	80	1	+	+	NUM
ejpam-2741	80	2	t	t	X
ejpam-2741	80	3	e	e	X
ejpam-2741	80	4	m	m	NOUN
ejpam-2741	80	5	.	.	PUNCT
ejpam-2741	81	1	then	then	ADV
ejpam-2741	81	2	by	by	ADP
ejpam-2741	81	3	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	81	4	,	,	PUNCT
ejpam-2741	81	5	m	m	VERB
ejpam-2741	81	6	=	=	PUNCT
ejpam-2741	81	7	x	x	PUNCT
ejpam-2741	82	1	+	+	NUM
ejpam-2741	82	2	y	y	PROPN
ejpam-2741	83	1	+	+	NOUN
ejpam-2741	83	2	t	t	NOUN
ejpam-2741	83	3	=	=	PUNCT
ejpam-2741	84	1	x	x	PUNCT
ejpam-2741	85	1	+	+	PUNCT
ejpam-2741	85	2	x	x	X
ejpam-2741	86	1	+	+	NUM
ejpam-2741	86	2	t	t	NOUN
ejpam-2741	86	3	=	=	PUNCT
ejpam-2741	86	4	x	x	PROPN
ejpam-2741	87	1	+	+	NUM
ejpam-2741	87	2	t	t	PROPN
ejpam-2741	87	3	and	and	CCONJ
ejpam-2741	87	4	m	m	PROPN
ejpam-2741	87	5	=	=	NOUN
ejpam-2741	87	6	x	x	PUNCT
ejpam-2741	88	1	+	+	NUM
ejpam-2741	88	2	y	y	PROPN
ejpam-2741	89	1	+	+	PROPN
ejpam-2741	89	2	t	t	NOUN
ejpam-2741	89	3	=	=	SYM
ejpam-2741	89	4	y	y	PROPN
ejpam-2741	89	5	+	+	NUM
ejpam-2741	89	6	y	y	PROPN
ejpam-2741	89	7	+	+	CCONJ
ejpam-2741	89	8	t	t	NOUN
ejpam-2741	89	9	=	=	SYM
ejpam-2741	89	10	y	y	PROPN
ejpam-2741	89	11	+	+	PROPN
ejpam-2741	89	12	t	t	PROPN
ejpam-2741	89	13	.	.	PUNCT
ejpam-2741	90	1	(	(	PUNCT
ejpam-2741	90	2	ii	ii	NOUN
ejpam-2741	90	3	)	)	PUNCT
ejpam-2741	90	4	=	=	NOUN
ejpam-2741	90	5	⇒	⇒	NOUN
ejpam-2741	90	6	(	(	PUNCT
ejpam-2741	90	7	i	i	NOUN
ejpam-2741	90	8	)	)	PUNCT
ejpam-2741	90	9	let	let	VERB
ejpam-2741	90	10	k	k	PROPN
ejpam-2741	90	11	em	em	PRON
ejpam-2741	90	12	and	and	CCONJ
ejpam-2741	90	13	x	x	X
ejpam-2741	91	1	+	+	NOUN
ejpam-2741	91	2	k	k	X
ejpam-2741	91	3	=	=	NOUN
ejpam-2741	91	4	m	m	PROPN
ejpam-2741	91	5	.	.	PUNCT
ejpam-2741	92	1	then	then	ADV
ejpam-2741	92	2	x	x	X
ejpam-2741	93	1	+	+	PUNCT
ejpam-2741	93	2	y	y	PROPN
ejpam-2741	93	3	+	+	PROPN
ejpam-2741	93	4	k	k	NOUN
ejpam-2741	93	5	=	=	VERB
ejpam-2741	93	6	m	m	ADJ
ejpam-2741	93	7	and	and	CCONJ
ejpam-2741	93	8	by	by	ADP
ejpam-2741	93	9	hypothesis	hypothesis	NOUN
ejpam-2741	93	10	,	,	PUNCT
ejpam-2741	93	11	y	y	PROPN
ejpam-2741	94	1	+	+	PROPN
ejpam-2741	94	2	k	k	X
ejpam-2741	94	3	=	=	NOUN
ejpam-2741	94	4	m	m	VERB
ejpam-2741	94	5	.	.	PUNCT
ejpam-2741	95	1	similarly	similarly	ADV
ejpam-2741	95	2	we	we	PRON
ejpam-2741	95	3	prove	prove	VERB
ejpam-2741	95	4	that	that	SCONJ
ejpam-2741	95	5	for	for	ADP
ejpam-2741	95	6	every	every	DET
ejpam-2741	95	7	t	t	NOUN
ejpam-2741	95	8	em	em	PRON
ejpam-2741	95	9	such	such	ADJ
ejpam-2741	95	10	that	that	SCONJ
ejpam-2741	95	11	y	y	PROPN
ejpam-2741	95	12	+	+	PROPN
ejpam-2741	95	13	t	t	PROPN
ejpam-2741	95	14	=	=	SYM
ejpam-2741	95	15	m	m	PROPN
ejpam-2741	95	16	,	,	PUNCT
ejpam-2741	95	17	x+t	x+t	PUNCT
ejpam-2741	96	1	=	=	PUNCT
ejpam-2741	96	2	m	m	NOUN
ejpam-2741	96	3	.	.	PUNCT
ejpam-2741	97	1	proposition	proposition	NOUN
ejpam-2741	97	2	2	2	NUM
ejpam-2741	97	3	.	.	PUNCT
ejpam-2741	98	1	let	let	VERB
ejpam-2741	98	2	x	x	PRON
ejpam-2741	98	3	,	,	PUNCT
ejpam-2741	98	4	y	y	PROPN
ejpam-2741	98	5	≤m	≤m	PROPN
ejpam-2741	98	6	.	.	PUNCT
ejpam-2741	99	1	if	if	SCONJ
ejpam-2741	99	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	99	3	,	,	PUNCT
ejpam-2741	99	4	then	then	ADV
ejpam-2741	99	5	x+y	x+y	NUM
ejpam-2741	99	6	x	x	SYM
ejpam-2741	99	7	�	�	PROPN
ejpam-2741	99	8	g	g	PROPN
ejpam-2741	99	9	m	m	PROPN
ejpam-2741	99	10	x	x	NOUN
ejpam-2741	99	11	and	and	CCONJ
ejpam-2741	99	12	x+y	x+y	NUM
ejpam-2741	99	13	y	y	PROPN
ejpam-2741	99	14	�	�	PROPN
ejpam-2741	99	15	g	g	PROPN
ejpam-2741	99	16	m	m	PROPN
ejpam-2741	99	17	y	y	NOUN
ejpam-2741	99	18	.	.	PUNCT
ejpam-2741	100	1	proof	proof	NOUN
ejpam-2741	100	2	.	.	PUNCT
ejpam-2741	101	1	let	let	VERB
ejpam-2741	101	2	x+y	x+y	NUM
ejpam-2741	101	3	x	x	SYM
ejpam-2741	102	1	+	+	NUM
ejpam-2741	102	2	t	t	NOUN
ejpam-2741	102	3	x	x	X
ejpam-2741	103	1	=	=	PUNCT
ejpam-2741	103	2	m	m	PUNCT
ejpam-2741	103	3	x	x	PUNCT
ejpam-2741	103	4	for	for	ADP
ejpam-2741	103	5	t	t	NOUN
ejpam-2741	103	6	x	x	PUNCT
ejpam-2741	103	7	e	e	NOUN
ejpam-2741	103	8	m	m	NOUN
ejpam-2741	103	9	x	x	INTJ
ejpam-2741	103	10	.	.	PUNCT
ejpam-2741	104	1	clearly	clearly	ADV
ejpam-2741	104	2	,	,	PUNCT
ejpam-2741	104	3	we	we	PRON
ejpam-2741	104	4	can	can	AUX
ejpam-2741	104	5	see	see	VERB
ejpam-2741	104	6	that	that	PRON
ejpam-2741	104	7	t	t	PROPN
ejpam-2741	104	8	e	e	X
ejpam-2741	104	9	m	m	PROPN
ejpam-2741	104	10	.	.	PUNCT
ejpam-2741	105	1	since	since	SCONJ
ejpam-2741	105	2	x+y	x+y	PROPN
ejpam-2741	105	3	x	x	SYM
ejpam-2741	105	4	+	+	NUM
ejpam-2741	105	5	t	t	NOUN
ejpam-2741	105	6	x	x	X
ejpam-2741	106	1	=	=	VERB
ejpam-2741	106	2	m	m	VERB
ejpam-2741	106	3	x	x	INTJ
ejpam-2741	106	4	,	,	PUNCT
ejpam-2741	106	5	then	then	ADV
ejpam-2741	106	6	m	m	VERB
ejpam-2741	106	7	x	x	X
ejpam-2741	106	8	=	=	SYM
ejpam-2741	106	9	x+y	x+y	PUNCT
ejpam-2741	106	10	x	x	X
ejpam-2741	107	1	+	+	NUM
ejpam-2741	107	2	t	t	NOUN
ejpam-2741	107	3	x	x	PUNCT
ejpam-2741	107	4	=	=	PUNCT
ejpam-2741	107	5	y+t	y+t	PROPN
ejpam-2741	107	6	x	x	PUNCT
ejpam-2741	107	7	and	and	CCONJ
ejpam-2741	107	8	y	y	PROPN
ejpam-2741	107	9	+	+	PROPN
ejpam-2741	107	10	t	t	PROPN
ejpam-2741	107	11	=	=	NOUN
ejpam-2741	107	12	m	m	PROPN
ejpam-2741	107	13	.	.	PUNCT
ejpam-2741	108	1	then	then	ADV
ejpam-2741	108	2	by	by	ADP
ejpam-2741	108	3	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	108	4	,	,	PUNCT
ejpam-2741	108	5	x	x	PROPN
ejpam-2741	108	6	+	+	NUM
ejpam-2741	108	7	t	t	X
ejpam-2741	108	8	=	=	SYM
ejpam-2741	108	9	m	m	PROPN
ejpam-2741	108	10	,	,	PUNCT
ejpam-2741	108	11	and	and	CCONJ
ejpam-2741	108	12	since	since	SCONJ
ejpam-2741	108	13	x	x	PROPN
ejpam-2741	108	14	≤	≤	PROPN
ejpam-2741	108	15	t	t	PROPN
ejpam-2741	108	16	,	,	PUNCT
ejpam-2741	108	17	t	t	PROPN
ejpam-2741	108	18	=	=	PUNCT
ejpam-2741	108	19	m	m	NOUN
ejpam-2741	108	20	.	.	PUNCT
ejpam-2741	109	1	hence	hence	ADV
ejpam-2741	109	2	x+y	x+y	NUM
ejpam-2741	109	3	x	x	SYM
ejpam-2741	109	4	�	�	PROPN
ejpam-2741	109	5	g	g	PROPN
ejpam-2741	109	6	m	m	PROPN
ejpam-2741	109	7	x	x	X
ejpam-2741	109	8	.	.	PUNCT
ejpam-2741	110	1	similarly	similarly	ADV
ejpam-2741	110	2	,	,	PUNCT
ejpam-2741	110	3	we	we	PRON
ejpam-2741	110	4	can	can	AUX
ejpam-2741	110	5	prove	prove	VERB
ejpam-2741	110	6	that	that	SCONJ
ejpam-2741	110	7	x+y	x+y	PROPN
ejpam-2741	110	8	y	y	PROPN
ejpam-2741	110	9	�	�	PROPN
ejpam-2741	110	10	g	g	PROPN
ejpam-2741	110	11	m	m	PROPN
ejpam-2741	110	12	y	y	PROPN
ejpam-2741	110	13	.	.	PUNCT
ejpam-2741	111	1	c.	c.	PROPN
ejpam-2741	111	2	nebiyev	nebiyev	PROPN
ejpam-2741	111	3	,	,	PUNCT
ejpam-2741	111	4	n.	n.	PROPN
ejpam-2741	111	5	sökmez	sökmez	PROPN
ejpam-2741	111	6	/	/	SYM
ejpam-2741	111	7	eur	eur	PROPN
ejpam-2741	111	8	.	.	PUNCT
ejpam-2741	112	1	j.	j.	PROPN
ejpam-2741	112	2	pure	pure	PROPN
ejpam-2741	112	3	appl	appl	PROPN
ejpam-2741	112	4	.	.	PROPN
ejpam-2741	112	5	math	math	PROPN
ejpam-2741	112	6	,	,	PUNCT
ejpam-2741	112	7	11	11	NUM
ejpam-2741	112	8	(	(	PUNCT
ejpam-2741	112	9	1	1	NUM
ejpam-2741	112	10	)	)	PUNCT
ejpam-2741	112	11	(	(	PUNCT
ejpam-2741	112	12	2018	2018	NUM
ejpam-2741	112	13	)	)	PUNCT
ejpam-2741	112	14	,	,	PUNCT
ejpam-2741	112	15	238	238	NUM
ejpam-2741	112	16	-	-	SYM
ejpam-2741	112	17	243	243	NUM
ejpam-2741	112	18	240	240	NUM
ejpam-2741	112	19	remark	remark	NOUN
ejpam-2741	112	20	1	1	NUM
ejpam-2741	112	21	.	.	PUNCT
ejpam-2741	113	1	the	the	DET
ejpam-2741	113	2	converse	converse	NOUN
ejpam-2741	113	3	of	of	ADP
ejpam-2741	113	4	the	the	DET
ejpam-2741	113	5	proposition	proposition	NOUN
ejpam-2741	113	6	2	2	NUM
ejpam-2741	113	7	is	be	AUX
ejpam-2741	113	8	not	not	PART
ejpam-2741	113	9	true	true	ADJ
ejpam-2741	113	10	in	in	ADP
ejpam-2741	113	11	general	general	ADJ
ejpam-2741	113	12	.	.	PUNCT
ejpam-2741	114	1	for	for	ADP
ejpam-2741	114	2	example	example	NOUN
ejpam-2741	114	3	,	,	PUNCT
ejpam-2741	114	4	consider	consider	VERB
ejpam-2741	114	5	the	the	DET
ejpam-2741	114	6	z	z	NOUN
ejpam-2741	114	7	-	-	PUNCT
ejpam-2741	114	8	module	module	NOUN
ejpam-2741	114	9	zz	zz	PROPN
ejpam-2741	114	10	and	and	CCONJ
ejpam-2741	114	11	let	let	VERB
ejpam-2741	114	12	p	p	NOUN
ejpam-2741	114	13	and	and	CCONJ
ejpam-2741	114	14	q	q	ADJ
ejpam-2741	114	15	be	be	AUX
ejpam-2741	114	16	primes	prime	NOUN
ejpam-2741	114	17	with	with	ADP
ejpam-2741	114	18	p	p	PROPN
ejpam-2741	114	19	6=	6=	ADP
ejpam-2741	114	20	q.	q.	NOUN
ejpam-2741	114	21	since	since	SCONJ
ejpam-2741	114	22	z	z	PROPN
ejpam-2741	114	23	zp	zp	PROPN
ejpam-2741	114	24	and	and	CCONJ
ejpam-2741	114	25	z	z	PROPN
ejpam-2741	114	26	zq	zq	PROPN
ejpam-2741	114	27	are	be	AUX
ejpam-2741	114	28	simple	simple	ADJ
ejpam-2741	114	29	,	,	PUNCT
ejpam-2741	114	30	zp+zq	zp+zq	PROPN
ejpam-2741	114	31	zp	zp	NOUN
ejpam-2741	115	1	=	=	PUNCT
ejpam-2741	115	2	z	z	PROPN
ejpam-2741	115	3	zp	zp	PROPN
ejpam-2741	115	4	�	�	PROPN
ejpam-2741	115	5	g	g	PROPN
ejpam-2741	115	6	z	z	PROPN
ejpam-2741	115	7	zp	zp	PROPN
ejpam-2741	115	8	and	and	CCONJ
ejpam-2741	115	9	zp+zq	zp+zq	NUM
ejpam-2741	115	10	zq	zq	PROPN
ejpam-2741	115	11	=	=	SYM
ejpam-2741	115	12	z	z	PROPN
ejpam-2741	115	13	zq	zq	PROPN
ejpam-2741	115	14	�	�	PROPN
ejpam-2741	115	15	g	g	PROPN
ejpam-2741	115	16	z	z	PROPN
ejpam-2741	115	17	zq	zq	PROPN
ejpam-2741	115	18	.	.	PUNCT
ejpam-2741	116	1	but	but	CCONJ
ejpam-2741	116	2	zpβ∗gzq	zpβ∗gzq	PROPN
ejpam-2741	116	3	is	be	AUX
ejpam-2741	116	4	not	not	PART
ejpam-2741	116	5	true	true	ADJ
ejpam-2741	116	6	.	.	PUNCT
ejpam-2741	117	1	theorem	theorem	NOUN
ejpam-2741	117	2	1	1	X
ejpam-2741	117	3	.	.	PUNCT
ejpam-2741	118	1	let	let	VERB
ejpam-2741	118	2	x	x	PRON
ejpam-2741	118	3	,	,	PUNCT
ejpam-2741	118	4	y	y	PROPN
ejpam-2741	118	5	≤	≤	NUM
ejpam-2741	118	6	m	m	VERB
ejpam-2741	118	7	such	such	ADJ
ejpam-2741	118	8	that	that	SCONJ
ejpam-2741	118	9	x	x	X
ejpam-2741	118	10	≤	≤	ADJ
ejpam-2741	118	11	y	y	NOUN
ejpam-2741	118	12	+	+	CCONJ
ejpam-2741	118	13	a	a	PRON
ejpam-2741	118	14	and	and	CCONJ
ejpam-2741	118	15	y	y	PROPN
ejpam-2741	118	16	≤	≤	NUM
ejpam-2741	118	17	x	x	PUNCT
ejpam-2741	119	1	+	+	NUM
ejpam-2741	119	2	b	b	NOUN
ejpam-2741	119	3	,	,	PUNCT
ejpam-2741	119	4	where	where	SCONJ
ejpam-2741	119	5	a	a	DET
ejpam-2741	119	6	,	,	PUNCT
ejpam-2741	119	7	b	b	PROPN
ejpam-2741	119	8	�	�	PROPN
ejpam-2741	119	9	g	g	NOUN
ejpam-2741	119	10	m	m	NOUN
ejpam-2741	119	11	.	.	PUNCT
ejpam-2741	120	1	then	then	ADV
ejpam-2741	120	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	120	3	.	.	PUNCT
ejpam-2741	121	1	proof	proof	NOUN
ejpam-2741	121	2	.	.	PUNCT
ejpam-2741	122	1	let	let	VERB
ejpam-2741	122	2	t	t	VERB
ejpam-2741	122	3	em	em	PRON
ejpam-2741	122	4	and	and	CCONJ
ejpam-2741	122	5	x+y	x+y	NUM
ejpam-2741	123	1	+	+	NOUN
ejpam-2741	123	2	t	t	X
ejpam-2741	123	3	=	=	X
ejpam-2741	123	4	m	m	PROPN
ejpam-2741	123	5	.	.	PUNCT
ejpam-2741	124	1	then	then	ADV
ejpam-2741	124	2	(	(	PUNCT
ejpam-2741	124	3	y	y	PROPN
ejpam-2741	124	4	+	+	PROPN
ejpam-2741	124	5	a)+y	a)+y	PROPN
ejpam-2741	124	6	+	+	PROPN
ejpam-2741	124	7	t	t	NOUN
ejpam-2741	124	8	=	=	SYM
ejpam-2741	124	9	m	m	NOUN
ejpam-2741	124	10	and	and	CCONJ
ejpam-2741	124	11	a+y	a+y	NUM
ejpam-2741	124	12	+	+	PROPN
ejpam-2741	124	13	t	t	NOUN
ejpam-2741	124	14	=	=	SYM
ejpam-2741	124	15	m.	m.	NOUN
ejpam-2741	124	16	since	since	SCONJ
ejpam-2741	124	17	t	t	PROPN
ejpam-2741	124	18	e	e	PROPN
ejpam-2741	124	19	m	m	PROPN
ejpam-2741	124	20	,	,	PUNCT
ejpam-2741	124	21	then	then	ADV
ejpam-2741	124	22	y	y	PROPN
ejpam-2741	125	1	+	+	PROPN
ejpam-2741	125	2	t	t	PROPN
ejpam-2741	125	3	e	e	X
ejpam-2741	125	4	m	m	NOUN
ejpam-2741	125	5	.	.	PUNCT
ejpam-2741	126	1	then	then	ADV
ejpam-2741	126	2	,	,	PUNCT
ejpam-2741	126	3	by	by	ADP
ejpam-2741	126	4	a	a	DET
ejpam-2741	126	5	�	�	PROPN
ejpam-2741	126	6	g	g	NOUN
ejpam-2741	126	7	m	m	PROPN
ejpam-2741	126	8	,	,	PUNCT
ejpam-2741	126	9	y	y	PROPN
ejpam-2741	126	10	+	+	PROPN
ejpam-2741	126	11	t	t	PROPN
ejpam-2741	126	12	=	=	NOUN
ejpam-2741	126	13	m	m	VERB
ejpam-2741	126	14	.	.	PUNCT
ejpam-2741	127	1	similarly	similarly	ADV
ejpam-2741	127	2	,	,	PUNCT
ejpam-2741	127	3	we	we	PRON
ejpam-2741	127	4	can	can	AUX
ejpam-2741	127	5	see	see	VERB
ejpam-2741	127	6	that	that	PRON
ejpam-2741	127	7	x	x	PROPN
ejpam-2741	128	1	+	+	NUM
ejpam-2741	128	2	t	t	X
ejpam-2741	128	3	=	=	SYM
ejpam-2741	128	4	m	m	PROPN
ejpam-2741	128	5	.	.	PUNCT
ejpam-2741	129	1	lemma	lemma	PROPN
ejpam-2741	129	2	4	4	X
ejpam-2741	129	3	.	.	PUNCT
ejpam-2741	130	1	let	let	VERB
ejpam-2741	130	2	x	x	PROPN
ejpam-2741	130	3	≤m	≤m	NOUN
ejpam-2741	130	4	.	.	PUNCT
ejpam-2741	131	1	x	x	X
ejpam-2741	131	2	�	�	PROPN
ejpam-2741	131	3	g	g	NOUN
ejpam-2741	131	4	m	m	NOUN
ejpam-2741	131	5	if	if	SCONJ
ejpam-2741	131	6	and	and	CCONJ
ejpam-2741	131	7	only	only	ADV
ejpam-2741	131	8	if	if	SCONJ
ejpam-2741	131	9	xβ∗g0	xβ∗g0	NUM
ejpam-2741	131	10	.	.	PUNCT
ejpam-2741	132	1	proof	proof	NOUN
ejpam-2741	132	2	.	.	PUNCT
ejpam-2741	133	1	(=	(=	X
ejpam-2741	133	2	⇒	⇒	NOUN
ejpam-2741	133	3	)	)	PUNCT
ejpam-2741	133	4	let	let	VERB
ejpam-2741	133	5	x	x	PART
ejpam-2741	133	6	�	�	VERB
ejpam-2741	133	7	g	g	NOUN
ejpam-2741	133	8	m	m	PROPN
ejpam-2741	133	9	and	and	CCONJ
ejpam-2741	133	10	let	let	VERB
ejpam-2741	133	11	x	x	PUNCT
ejpam-2741	134	1	+	+	CCONJ
ejpam-2741	134	2	0	0	NUM
ejpam-2741	135	1	+	+	NUM
ejpam-2741	135	2	t	t	NOUN
ejpam-2741	135	3	=	=	PUNCT
ejpam-2741	135	4	x	x	PROPN
ejpam-2741	136	1	+	+	NUM
ejpam-2741	136	2	t	t	X
ejpam-2741	136	3	=	=	SYM
ejpam-2741	136	4	m	m	VERB
ejpam-2741	136	5	for	for	ADP
ejpam-2741	136	6	t	t	PROPN
ejpam-2741	136	7	e	e	X
ejpam-2741	136	8	m	m	PROPN
ejpam-2741	136	9	.	.	PUNCT
ejpam-2741	137	1	since	since	SCONJ
ejpam-2741	137	2	x	x	PROPN
ejpam-2741	137	3	�	�	PROPN
ejpam-2741	137	4	g	g	NOUN
ejpam-2741	137	5	m	m	PROPN
ejpam-2741	137	6	and	and	CCONJ
ejpam-2741	137	7	x	x	X
ejpam-2741	137	8	+	+	NUM
ejpam-2741	137	9	t	t	X
ejpam-2741	137	10	=	=	SYM
ejpam-2741	137	11	m	m	PROPN
ejpam-2741	137	12	,	,	PUNCT
ejpam-2741	137	13	then	then	ADV
ejpam-2741	137	14	0	0	NUM
ejpam-2741	137	15	+	+	NUM
ejpam-2741	137	16	t	t	X
ejpam-2741	137	17	=	=	SYM
ejpam-2741	137	18	t	t	PROPN
ejpam-2741	137	19	=	=	NOUN
ejpam-2741	137	20	m	m	PROPN
ejpam-2741	137	21	.	.	PUNCT
ejpam-2741	138	1	then	then	ADV
ejpam-2741	138	2	,	,	PUNCT
ejpam-2741	138	3	by	by	ADP
ejpam-2741	138	4	lemma	lemma	PROPN
ejpam-2741	138	5	3	3	NUM
ejpam-2741	138	6	xβ∗g0	xβ∗g0	PROPN
ejpam-2741	138	7	.	.	PUNCT
ejpam-2741	139	1	(	(	PUNCT
ejpam-2741	139	2	⇐	⇐	ADJ
ejpam-2741	139	3	=)	=)	PROPN
ejpam-2741	139	4	let	let	VERB
ejpam-2741	139	5	xβ∗g0	xβ∗g0	VERB
ejpam-2741	139	6	.	.	PUNCT
ejpam-2741	140	1	let	let	VERB
ejpam-2741	140	2	x	x	PUNCT
ejpam-2741	141	1	+	+	NUM
ejpam-2741	141	2	t	t	X
ejpam-2741	141	3	=	=	SYM
ejpam-2741	141	4	m	m	VERB
ejpam-2741	141	5	for	for	ADP
ejpam-2741	141	6	t	t	PROPN
ejpam-2741	141	7	e	e	X
ejpam-2741	141	8	m	m	PROPN
ejpam-2741	141	9	.	.	PUNCT
ejpam-2741	142	1	since	since	SCONJ
ejpam-2741	142	2	xβ∗g0	xβ∗g0	PROPN
ejpam-2741	142	3	,	,	PUNCT
ejpam-2741	142	4	then	then	ADV
ejpam-2741	142	5	t	t	PROPN
ejpam-2741	142	6	=	=	PUNCT
ejpam-2741	142	7	0	0	PUNCT
ejpam-2741	143	1	+	+	NUM
ejpam-2741	143	2	t	t	X
ejpam-2741	143	3	=	=	NOUN
ejpam-2741	143	4	m	m	NOUN
ejpam-2741	143	5	.	.	PUNCT
ejpam-2741	144	1	hence	hence	ADV
ejpam-2741	144	2	x	x	X
ejpam-2741	144	3	�	�	PROPN
ejpam-2741	144	4	g	g	NOUN
ejpam-2741	144	5	m	m	PROPN
ejpam-2741	144	6	.	.	PUNCT
ejpam-2741	145	1	corollary	corollary	ADJ
ejpam-2741	145	2	1	1	NUM
ejpam-2741	145	3	.	.	PUNCT
ejpam-2741	146	1	let	let	VERB
ejpam-2741	146	2	x	x	PRON
ejpam-2741	146	3	,	,	PUNCT
ejpam-2741	146	4	y	y	PROPN
ejpam-2741	146	5	≤m	≤m	PROPN
ejpam-2741	146	6	and	and	CCONJ
ejpam-2741	146	7	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	146	8	.	.	PUNCT
ejpam-2741	147	1	if	if	SCONJ
ejpam-2741	147	2	x	x	PROPN
ejpam-2741	147	3	�	�	VERB
ejpam-2741	147	4	g	g	NOUN
ejpam-2741	147	5	m	m	PROPN
ejpam-2741	147	6	,	,	PUNCT
ejpam-2741	147	7	then	then	ADV
ejpam-2741	147	8	y	y	PROPN
ejpam-2741	147	9	�	�	PROPN
ejpam-2741	147	10	g	g	PROPN
ejpam-2741	147	11	m	m	NOUN
ejpam-2741	147	12	.	.	PUNCT
ejpam-2741	148	1	proof	proof	NOUN
ejpam-2741	148	2	.	.	PUNCT
ejpam-2741	149	1	since	since	SCONJ
ejpam-2741	149	2	x	x	PROPN
ejpam-2741	149	3	�	�	PROPN
ejpam-2741	149	4	g	g	NOUN
ejpam-2741	149	5	m	m	PROPN
ejpam-2741	149	6	,	,	PUNCT
ejpam-2741	149	7	then	then	ADV
ejpam-2741	149	8	by	by	ADP
ejpam-2741	149	9	lemma	lemma	PROPN
ejpam-2741	149	10	4	4	NUM
ejpam-2741	149	11	,	,	PUNCT
ejpam-2741	149	12	xβ∗g0	xβ∗g0	PROPN
ejpam-2741	149	13	,	,	PUNCT
ejpam-2741	149	14	and	and	CCONJ
ejpam-2741	149	15	since	since	SCONJ
ejpam-2741	149	16	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	149	17	,	,	PUNCT
ejpam-2741	149	18	then	then	ADV
ejpam-2741	149	19	by	by	ADP
ejpam-2741	149	20	lemma	lemma	PROPN
ejpam-2741	149	21	2	2	NUM
ejpam-2741	149	22	,	,	PUNCT
ejpam-2741	149	23	y	y	PROPN
ejpam-2741	149	24	β∗g0	β∗g0	PROPN
ejpam-2741	149	25	.	.	PUNCT
ejpam-2741	150	1	then	then	ADV
ejpam-2741	150	2	,	,	PUNCT
ejpam-2741	150	3	by	by	ADP
ejpam-2741	150	4	lemma	lemma	PROPN
ejpam-2741	150	5	4	4	NUM
ejpam-2741	150	6	,	,	PUNCT
ejpam-2741	150	7	y	y	PROPN
ejpam-2741	150	8	�	�	PROPN
ejpam-2741	150	9	g	g	PROPN
ejpam-2741	150	10	m	m	PROPN
ejpam-2741	150	11	.	.	PUNCT
ejpam-2741	151	1	corollary	corollary	ADJ
ejpam-2741	151	2	2	2	NUM
ejpam-2741	151	3	.	.	PUNCT
ejpam-2741	152	1	let	let	VERB
ejpam-2741	152	2	m	m	PRON
ejpam-2741	152	3	be	be	AUX
ejpam-2741	152	4	an	an	DET
ejpam-2741	152	5	r−module	r−module	PROPN
ejpam-2741	152	6	.	.	PUNCT
ejpam-2741	153	1	then	then	ADV
ejpam-2741	153	2	m	m	VERB
ejpam-2741	153	3	is	be	AUX
ejpam-2741	153	4	generalized	generalize	VERB
ejpam-2741	153	5	hollow	hollow	ADJ
ejpam-2741	153	6	if	if	SCONJ
ejpam-2741	154	1	and	and	CCONJ
ejpam-2741	154	2	only	only	ADV
ejpam-2741	154	3	if	if	SCONJ
ejpam-2741	154	4	xβ∗g0	xβ∗g0	PROPN
ejpam-2741	154	5	for	for	ADP
ejpam-2741	154	6	every	every	DET
ejpam-2741	154	7	proper	proper	ADJ
ejpam-2741	154	8	submodule	submodule	NOUN
ejpam-2741	154	9	x	x	PROPN
ejpam-2741	154	10	of	of	ADP
ejpam-2741	154	11	m	m	PROPN
ejpam-2741	154	12	.	.	PUNCT
ejpam-2741	155	1	proof	proof	NOUN
ejpam-2741	155	2	.	.	PUNCT
ejpam-2741	156	1	clear	clear	ADJ
ejpam-2741	156	2	from	from	ADP
ejpam-2741	156	3	lemma	lemma	PROPN
ejpam-2741	156	4	4	4	NUM
ejpam-2741	156	5	.	.	PUNCT
ejpam-2741	156	6	corollary	corollary	ADJ
ejpam-2741	156	7	3	3	X
ejpam-2741	156	8	.	.	PUNCT
ejpam-2741	157	1	let	let	VERB
ejpam-2741	157	2	m	m	PRON
ejpam-2741	157	3	be	be	AUX
ejpam-2741	157	4	an	an	DET
ejpam-2741	157	5	r−module	r−module	PROPN
ejpam-2741	157	6	.	.	PUNCT
ejpam-2741	158	1	then	then	ADV
ejpam-2741	158	2	m	m	VERB
ejpam-2741	158	3	is	be	AUX
ejpam-2741	158	4	generalized	generalize	VERB
ejpam-2741	158	5	hollow	hollow	ADJ
ejpam-2741	158	6	if	if	SCONJ
ejpam-2741	159	1	and	and	CCONJ
ejpam-2741	159	2	only	only	ADV
ejpam-2741	159	3	if	if	SCONJ
ejpam-2741	159	4	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	159	5	for	for	ADP
ejpam-2741	159	6	every	every	DET
ejpam-2741	159	7	proper	proper	ADJ
ejpam-2741	159	8	submodules	submodule	NOUN
ejpam-2741	159	9	x	x	X
ejpam-2741	159	10	,	,	PUNCT
ejpam-2741	159	11	y	y	PROPN
ejpam-2741	159	12	of	of	ADP
ejpam-2741	159	13	m	m	PROPN
ejpam-2741	159	14	.	.	PUNCT
ejpam-2741	160	1	proof	proof	NOUN
ejpam-2741	160	2	.	.	PUNCT
ejpam-2741	161	1	clear	clear	ADJ
ejpam-2741	161	2	from	from	ADP
ejpam-2741	161	3	lemma	lemma	PROPN
ejpam-2741	161	4	4	4	NUM
ejpam-2741	161	5	.	.	NOUN
ejpam-2741	161	6	remark	remark	NOUN
ejpam-2741	161	7	2	2	NUM
ejpam-2741	161	8	.	.	PUNCT
ejpam-2741	162	1	let	let	VERB
ejpam-2741	162	2	m	m	PRON
ejpam-2741	162	3	be	be	AUX
ejpam-2741	162	4	a	a	DET
ejpam-2741	162	5	nonzero	nonzero	ADJ
ejpam-2741	162	6	semisimple	semisimple	PROPN
ejpam-2741	162	7	r−module	r−module	PROPN
ejpam-2741	162	8	.	.	PUNCT
ejpam-2741	163	1	since	since	SCONJ
ejpam-2741	163	2	m	m	PRON
ejpam-2741	163	3	have	have	VERB
ejpam-2741	163	4	no	no	DET
ejpam-2741	163	5	proper	proper	ADJ
ejpam-2741	163	6	essential	essential	ADJ
ejpam-2741	163	7	submodules	submodule	NOUN
ejpam-2741	163	8	,	,	PUNCT
ejpam-2741	163	9	m	m	VERB
ejpam-2741	163	10	�	�	PROPN
ejpam-2741	163	11	g	g	NOUN
ejpam-2741	163	12	m	m	PROPN
ejpam-2741	163	13	and	and	CCONJ
ejpam-2741	163	14	by	by	ADP
ejpam-2741	163	15	lemma	lemma	PROPN
ejpam-2741	163	16	4	4	NUM
ejpam-2741	163	17	,	,	PUNCT
ejpam-2741	163	18	mβ∗g0	mβ∗g0	PROPN
ejpam-2741	163	19	.	.	PUNCT
ejpam-2741	164	1	but	but	CCONJ
ejpam-2741	164	2	mβ∗0	mβ∗0	NOUN
ejpam-2741	164	3	is	be	AUX
ejpam-2741	164	4	not	not	PART
ejpam-2741	164	5	true	true	ADJ
ejpam-2741	164	6	.	.	PUNCT
ejpam-2741	165	1	corollary	corollary	ADJ
ejpam-2741	165	2	4	4	NUM
ejpam-2741	165	3	.	.	PUNCT
ejpam-2741	166	1	let	let	VERB
ejpam-2741	166	2	m	m	PRON
ejpam-2741	166	3	be	be	AUX
ejpam-2741	166	4	an	an	DET
ejpam-2741	166	5	r−module	r−module	PROPN
ejpam-2741	166	6	.	.	PUNCT
ejpam-2741	167	1	then	then	ADV
ejpam-2741	167	2	socmβ∗g0	socmβ∗g0	VERB
ejpam-2741	167	3	.	.	PUNCT
ejpam-2741	168	1	lemma	lemma	PROPN
ejpam-2741	168	2	5	5	NUM
ejpam-2741	168	3	.	.	PUNCT
ejpam-2741	169	1	let	let	VERB
ejpam-2741	169	2	x1	x1	NUM
ejpam-2741	169	3	,	,	PUNCT
ejpam-2741	169	4	x2	x2	PROPN
ejpam-2741	169	5	,	,	PUNCT
ejpam-2741	169	6	y1	y1	NOUN
ejpam-2741	169	7	,	,	PUNCT
ejpam-2741	169	8	y2	y2	NOUN
ejpam-2741	169	9	≤m	≤m	NOUN
ejpam-2741	169	10	such	such	ADJ
ejpam-2741	169	11	that	that	SCONJ
ejpam-2741	169	12	x1β	x1β	PROPN
ejpam-2741	169	13	∗	∗	NOUN
ejpam-2741	169	14	gy1	gy1	PROPN
ejpam-2741	169	15	and	and	CCONJ
ejpam-2741	169	16	x2β	x2β	PROPN
ejpam-2741	169	17	∗	∗	NOUN
ejpam-2741	169	18	gy2	gy2	PROPN
ejpam-2741	169	19	.	.	PUNCT
ejpam-2741	170	1	then	then	ADV
ejpam-2741	170	2	(	(	PUNCT
ejpam-2741	170	3	x1	x1	PROPN
ejpam-2741	170	4	+	+	PROPN
ejpam-2741	170	5	x2)β	x2)β	ADJ
ejpam-2741	170	6	∗	∗	NOUN
ejpam-2741	170	7	g	g	PROPN
ejpam-2741	170	8	(	(	PUNCT
ejpam-2741	170	9	y1	y1	INTJ
ejpam-2741	170	10	+	+	NUM
ejpam-2741	170	11	y2	y2	NOUN
ejpam-2741	170	12	)	)	PUNCT
ejpam-2741	170	13	.	.	PUNCT
ejpam-2741	171	1	proof	proof	NOUN
ejpam-2741	171	2	.	.	PUNCT
ejpam-2741	172	1	let	let	VERB
ejpam-2741	172	2	x1	x1	PROPN
ejpam-2741	173	1	+	+	ADJ
ejpam-2741	173	2	x2	x2	PROPN
ejpam-2741	174	1	+	+	PROPN
ejpam-2741	174	2	k	k	X
ejpam-2741	174	3	=	=	X
ejpam-2741	174	4	m	m	VERB
ejpam-2741	174	5	for	for	ADP
ejpam-2741	174	6	k	k	PROPN
ejpam-2741	174	7	e	e	PROPN
ejpam-2741	174	8	m	m	PROPN
ejpam-2741	174	9	.	.	PUNCT
ejpam-2741	175	1	since	since	SCONJ
ejpam-2741	175	2	k	k	PROPN
ejpam-2741	175	3	e	e	PROPN
ejpam-2741	175	4	m	m	PROPN
ejpam-2741	175	5	,	,	PUNCT
ejpam-2741	175	6	then	then	ADV
ejpam-2741	175	7	x2	x2	PROPN
ejpam-2741	176	1	+	+	PROPN
ejpam-2741	176	2	k	k	X
ejpam-2741	176	3	e	e	ADJ
ejpam-2741	176	4	m	m	PROPN
ejpam-2741	176	5	.	.	PUNCT
ejpam-2741	177	1	then	then	ADV
ejpam-2741	177	2	,	,	PUNCT
ejpam-2741	177	3	by	by	ADP
ejpam-2741	177	4	x1β	x1β	PROPN
ejpam-2741	177	5	∗	∗	NOUN
ejpam-2741	177	6	gy1	gy1	PROPN
ejpam-2741	177	7	,	,	PUNCT
ejpam-2741	177	8	y1	y1	PROPN
ejpam-2741	178	1	+	+	CCONJ
ejpam-2741	178	2	x2	x2	PROPN
ejpam-2741	179	1	+	+	CCONJ
ejpam-2741	179	2	k	k	X
ejpam-2741	179	3	=	=	NOUN
ejpam-2741	179	4	m	m	VERB
ejpam-2741	179	5	.	.	PUNCT
ejpam-2741	180	1	since	since	SCONJ
ejpam-2741	180	2	k	k	PROPN
ejpam-2741	180	3	e	e	PROPN
ejpam-2741	180	4	m	m	PROPN
ejpam-2741	180	5	,	,	PUNCT
ejpam-2741	180	6	then	then	ADV
ejpam-2741	180	7	y1	y1	INTJ
ejpam-2741	180	8	+	+	CCONJ
ejpam-2741	180	9	k	k	PROPN
ejpam-2741	180	10	e	e	PROPN
ejpam-2741	180	11	m	m	PROPN
ejpam-2741	180	12	.	.	PUNCT
ejpam-2741	181	1	then	then	ADV
ejpam-2741	181	2	,	,	PUNCT
ejpam-2741	181	3	by	by	ADP
ejpam-2741	181	4	x2β	x2β	PROPN
ejpam-2741	181	5	∗	∗	NOUN
ejpam-2741	181	6	gy2	gy2	PROPN
ejpam-2741	181	7	,	,	PUNCT
ejpam-2741	181	8	y1	y1	NOUN
ejpam-2741	181	9	+	+	NOUN
ejpam-2741	182	1	y2	y2	PROPN
ejpam-2741	183	1	+	+	CCONJ
ejpam-2741	183	2	k	k	X
ejpam-2741	183	3	=	=	NOUN
ejpam-2741	183	4	m	m	VERB
ejpam-2741	183	5	.	.	PUNCT
ejpam-2741	184	1	similarly	similarly	ADV
ejpam-2741	184	2	,	,	PUNCT
ejpam-2741	184	3	we	we	PRON
ejpam-2741	184	4	can	can	AUX
ejpam-2741	184	5	see	see	VERB
ejpam-2741	184	6	that	that	SCONJ
ejpam-2741	184	7	x1	x1	PROPN
ejpam-2741	185	1	+	+	NUM
ejpam-2741	185	2	x2	x2	PROPN
ejpam-2741	186	1	+	+	NUM
ejpam-2741	186	2	t	t	NOUN
ejpam-2741	186	3	=	=	PUNCT
ejpam-2741	186	4	m	m	VERB
ejpam-2741	186	5	for	for	ADP
ejpam-2741	186	6	every	every	DET
ejpam-2741	186	7	t	t	NOUN
ejpam-2741	186	8	e	e	NOUN
ejpam-2741	186	9	m	m	VERB
ejpam-2741	186	10	such	such	ADJ
ejpam-2741	186	11	that	that	SCONJ
ejpam-2741	186	12	y1	y1	NOUN
ejpam-2741	187	1	+	+	NOUN
ejpam-2741	187	2	y2	y2	PROPN
ejpam-2741	187	3	+	+	NUM
ejpam-2741	187	4	t	t	NOUN
ejpam-2741	187	5	=	=	SYM
ejpam-2741	187	6	m	m	NOUN
ejpam-2741	187	7	.	.	PUNCT
ejpam-2741	188	1	corollary	corollary	ADJ
ejpam-2741	188	2	5	5	NUM
ejpam-2741	188	3	.	.	PUNCT
ejpam-2741	189	1	let	let	VERB
ejpam-2741	189	2	x1	x1	NUM
ejpam-2741	189	3	,	,	PUNCT
ejpam-2741	189	4	x2	x2	PROPN
ejpam-2741	189	5	,	,	PUNCT
ejpam-2741	189	6	...	...	PUNCT
ejpam-2741	189	7	,	,	PUNCT
ejpam-2741	189	8	xn	xn	PROPN
ejpam-2741	189	9	,	,	PUNCT
ejpam-2741	189	10	y1	y1	NOUN
ejpam-2741	189	11	,	,	PUNCT
ejpam-2741	189	12	y2	y2	PROPN
ejpam-2741	189	13	,	,	PUNCT
ejpam-2741	189	14	...	...	PUNCT
ejpam-2741	189	15	,	,	PUNCT
ejpam-2741	189	16	yn	yn	PROPN
ejpam-2741	189	17	≤	≤	PUNCT
ejpam-2741	189	18	m	m	VERB
ejpam-2741	189	19	and	and	CCONJ
ejpam-2741	189	20	xiβ	xiβ	PROPN
ejpam-2741	189	21	∗	∗	PROPN
ejpam-2741	189	22	gyi	gyi	PROPN
ejpam-2741	189	23	for	for	ADP
ejpam-2741	189	24	every	every	DET
ejpam-2741	189	25	i	i	NOUN
ejpam-2741	189	26	=	=	NOUN
ejpam-2741	189	27	1	1	NUM
ejpam-2741	189	28	,	,	PUNCT
ejpam-2741	189	29	2	2	NUM
ejpam-2741	189	30	,	,	PUNCT
ejpam-2741	189	31	...	...	PUNCT
ejpam-2741	189	32	,	,	PUNCT
ejpam-2741	189	33	n.	n.	PROPN
ejpam-2741	189	34	then	then	ADV
ejpam-2741	190	1	x1	x1	PROPN
ejpam-2741	191	1	+	+	PROPN
ejpam-2741	191	2	x2	x2	PROPN
ejpam-2741	191	3	+	+	X
ejpam-2741	191	4	...	...	PUNCT
ejpam-2741	192	1	+	+	ADJ
ejpam-2741	192	2	xnβ	xnβ	ADJ
ejpam-2741	192	3	∗	∗	X
ejpam-2741	192	4	gy1	gy1	NOUN
ejpam-2741	192	5	+	+	CCONJ
ejpam-2741	192	6	y2	y2	PROPN
ejpam-2741	192	7	+	+	CCONJ
ejpam-2741	192	8	...	...	PUNCT
ejpam-2741	193	1	+	+	CCONJ
ejpam-2741	193	2	yn	yn	PROPN
ejpam-2741	193	3	.	.	PUNCT
ejpam-2741	193	4	c.	c.	PROPN
ejpam-2741	193	5	nebiyev	nebiyev	PROPN
ejpam-2741	193	6	,	,	PUNCT
ejpam-2741	193	7	n.	n.	PROPN
ejpam-2741	193	8	sökmez	sökmez	PROPN
ejpam-2741	193	9	/	/	SYM
ejpam-2741	193	10	eur	eur	PROPN
ejpam-2741	193	11	.	.	PUNCT
ejpam-2741	194	1	j.	j.	PROPN
ejpam-2741	194	2	pure	pure	PROPN
ejpam-2741	194	3	appl	appl	PROPN
ejpam-2741	194	4	.	.	PROPN
ejpam-2741	194	5	math	math	PROPN
ejpam-2741	194	6	,	,	PUNCT
ejpam-2741	194	7	11	11	NUM
ejpam-2741	194	8	(	(	PUNCT
ejpam-2741	194	9	1	1	NUM
ejpam-2741	194	10	)	)	PUNCT
ejpam-2741	194	11	(	(	PUNCT
ejpam-2741	194	12	2018	2018	NUM
ejpam-2741	194	13	)	)	PUNCT
ejpam-2741	194	14	,	,	PUNCT
ejpam-2741	194	15	238	238	NUM
ejpam-2741	194	16	-	-	SYM
ejpam-2741	194	17	243	243	NUM
ejpam-2741	194	18	241	241	NUM
ejpam-2741	194	19	proof	proof	NOUN
ejpam-2741	194	20	.	.	PUNCT
ejpam-2741	195	1	clear	clear	ADJ
ejpam-2741	195	2	from	from	ADP
ejpam-2741	195	3	lemma	lemma	PROPN
ejpam-2741	195	4	5	5	NUM
ejpam-2741	195	5	.	.	PUNCT
ejpam-2741	195	6	corollary	corollary	ADJ
ejpam-2741	195	7	6	6	NUM
ejpam-2741	195	8	.	.	PUNCT
ejpam-2741	196	1	let	let	VERB
ejpam-2741	196	2	x1	x1	NUM
ejpam-2741	196	3	,	,	PUNCT
ejpam-2741	196	4	x2	x2	PROPN
ejpam-2741	196	5	,	,	PUNCT
ejpam-2741	196	6	...	...	PUNCT
ejpam-2741	196	7	,	,	PUNCT
ejpam-2741	196	8	xn	xn	PROPN
ejpam-2741	196	9	,	,	PUNCT
ejpam-2741	196	10	y	y	PROPN
ejpam-2741	196	11	≤	≤	PROPN
ejpam-2741	196	12	m	m	VERB
ejpam-2741	196	13	and	and	CCONJ
ejpam-2741	196	14	xiβ	xiβ	PROPN
ejpam-2741	196	15	∗	∗	NOUN
ejpam-2741	196	16	gy	gy	PROPN
ejpam-2741	196	17	for	for	ADP
ejpam-2741	196	18	every	every	DET
ejpam-2741	196	19	i	i	NOUN
ejpam-2741	196	20	=	=	NOUN
ejpam-2741	196	21	1	1	NUM
ejpam-2741	196	22	,	,	PUNCT
ejpam-2741	196	23	2	2	NUM
ejpam-2741	196	24	,	,	PUNCT
ejpam-2741	196	25	...	...	PUNCT
ejpam-2741	196	26	,	,	PUNCT
ejpam-2741	196	27	n.	n.	PROPN
ejpam-2741	196	28	then	then	ADV
ejpam-2741	197	1	x1	x1	PROPN
ejpam-2741	198	1	+	+	PROPN
ejpam-2741	198	2	x2	x2	PROPN
ejpam-2741	198	3	+	+	X
ejpam-2741	198	4	...	...	PUNCT
ejpam-2741	199	1	+	+	ADJ
ejpam-2741	199	2	xnβ	xnβ	ADJ
ejpam-2741	199	3	∗	∗	NOUN
ejpam-2741	199	4	gy	gy	NOUN
ejpam-2741	199	5	.	.	PUNCT
ejpam-2741	200	1	proof	proof	NOUN
ejpam-2741	200	2	.	.	PUNCT
ejpam-2741	201	1	clear	clear	ADJ
ejpam-2741	201	2	from	from	ADP
ejpam-2741	201	3	lemma	lemma	PROPN
ejpam-2741	201	4	5	5	NUM
ejpam-2741	201	5	.	.	PUNCT
ejpam-2741	202	1	lemma	lemma	PROPN
ejpam-2741	202	2	6	6	NUM
ejpam-2741	202	3	.	.	PUNCT
ejpam-2741	203	1	let	let	VERB
ejpam-2741	203	2	f	f	NOUN
ejpam-2741	203	3	:	:	PUNCT
ejpam-2741	203	4	m	m	VERB
ejpam-2741	203	5	−→	−→	ADJ
ejpam-2741	203	6	n	n	AUX
ejpam-2741	203	7	be	be	AUX
ejpam-2741	203	8	an	an	DET
ejpam-2741	203	9	r−module	r−module	NOUN
ejpam-2741	203	10	epimorphism	epimorphism	NOUN
ejpam-2741	203	11	and	and	CCONJ
ejpam-2741	203	12	x	x	NOUN
ejpam-2741	203	13	,	,	PUNCT
ejpam-2741	203	14	y	y	PROPN
ejpam-2741	203	15	≤	≤	NOUN
ejpam-2741	203	16	m	m	VERB
ejpam-2741	203	17	.	.	PUNCT
ejpam-2741	204	1	if	if	SCONJ
ejpam-2741	204	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	204	3	,	,	PUNCT
ejpam-2741	204	4	then	then	ADV
ejpam-2741	204	5	f	f	X
ejpam-2741	204	6	(	(	PUNCT
ejpam-2741	204	7	x)β∗gf	x)β∗gf	NUM
ejpam-2741	204	8	(	(	PUNCT
ejpam-2741	204	9	y	y	PROPN
ejpam-2741	204	10	)	)	PUNCT
ejpam-2741	204	11	.	.	PUNCT
ejpam-2741	205	1	proof	proof	NOUN
ejpam-2741	205	2	.	.	PUNCT
ejpam-2741	206	1	let	let	VERB
ejpam-2741	206	2	f	f	PROPN
ejpam-2741	206	3	(	(	PUNCT
ejpam-2741	206	4	x	x	X
ejpam-2741	206	5	)	)	PUNCT
ejpam-2741	207	1	+	+	NUM
ejpam-2741	207	2	f	f	X
ejpam-2741	207	3	(	(	PUNCT
ejpam-2741	207	4	y	y	PROPN
ejpam-2741	207	5	)	)	PUNCT
ejpam-2741	208	1	+	+	NUM
ejpam-2741	208	2	t	t	X
ejpam-2741	208	3	=	=	SYM
ejpam-2741	208	4	n	n	PROPN
ejpam-2741	208	5	for	for	ADP
ejpam-2741	208	6	t	t	NOUN
ejpam-2741	208	7	e	e	X
ejpam-2741	208	8	n	n	PROPN
ejpam-2741	208	9	.	.	PUNCT
ejpam-2741	209	1	then	then	ADV
ejpam-2741	209	2	x	x	X
ejpam-2741	209	3	+	+	CCONJ
ejpam-2741	209	4	y	y	PROPN
ejpam-2741	209	5	+	+	CCONJ
ejpam-2741	209	6	f−1	f−1	PROPN
ejpam-2741	209	7	(	(	PUNCT
ejpam-2741	209	8	t	t	PROPN
ejpam-2741	209	9	)	)	PUNCT
ejpam-2741	209	10	=	=	PUNCT
ejpam-2741	210	1	m	m	VERB
ejpam-2741	210	2	.	.	PUNCT
ejpam-2741	211	1	since	since	SCONJ
ejpam-2741	211	2	t	t	PROPN
ejpam-2741	211	3	e	e	PROPN
ejpam-2741	211	4	n	n	PROPN
ejpam-2741	211	5	,	,	PUNCT
ejpam-2741	211	6	then	then	ADV
ejpam-2741	211	7	we	we	PRON
ejpam-2741	211	8	can	can	AUX
ejpam-2741	211	9	see	see	VERB
ejpam-2741	211	10	that	that	SCONJ
ejpam-2741	211	11	f−1	f−1	PROPN
ejpam-2741	211	12	(	(	PUNCT
ejpam-2741	211	13	t	t	PROPN
ejpam-2741	211	14	)	)	PUNCT
ejpam-2741	211	15	e	e	PROPN
ejpam-2741	211	16	m	m	PROPN
ejpam-2741	211	17	.	.	PUNCT
ejpam-2741	212	1	then	then	ADV
ejpam-2741	212	2	,	,	PUNCT
ejpam-2741	212	3	by	by	ADP
ejpam-2741	212	4	lemma	lemma	PROPN
ejpam-2741	212	5	3	3	NUM
ejpam-2741	212	6	,	,	PUNCT
ejpam-2741	212	7	x	x	PUNCT
ejpam-2741	212	8	+	+	NUM
ejpam-2741	212	9	f−1	f−1	PROPN
ejpam-2741	212	10	(	(	PUNCT
ejpam-2741	212	11	t	t	PROPN
ejpam-2741	212	12	)	)	PUNCT
ejpam-2741	212	13	=	=	PUNCT
ejpam-2741	213	1	m	m	PROPN
ejpam-2741	213	2	and	and	CCONJ
ejpam-2741	213	3	y	y	PROPN
ejpam-2741	213	4	+	+	CCONJ
ejpam-2741	213	5	f−1	f−1	PROPN
ejpam-2741	213	6	(	(	PUNCT
ejpam-2741	213	7	t	t	PROPN
ejpam-2741	213	8	)	)	PUNCT
ejpam-2741	214	1	=	=	PUNCT
ejpam-2741	215	1	m	m	VERB
ejpam-2741	215	2	.	.	PUNCT
ejpam-2741	216	1	since	since	SCONJ
ejpam-2741	216	2	x	x	PROPN
ejpam-2741	216	3	+	+	NUM
ejpam-2741	216	4	f−1	f−1	PROPN
ejpam-2741	216	5	(	(	PUNCT
ejpam-2741	216	6	t	t	PROPN
ejpam-2741	216	7	)	)	PUNCT
ejpam-2741	216	8	=	=	PUNCT
ejpam-2741	216	9	m	m	PROPN
ejpam-2741	216	10	and	and	CCONJ
ejpam-2741	216	11	y	y	PROPN
ejpam-2741	216	12	+	+	CCONJ
ejpam-2741	216	13	f−1	f−1	PROPN
ejpam-2741	216	14	(	(	PUNCT
ejpam-2741	216	15	t	t	PROPN
ejpam-2741	216	16	)	)	PUNCT
ejpam-2741	217	1	=	=	PUNCT
ejpam-2741	218	1	m	m	PROPN
ejpam-2741	218	2	,	,	PUNCT
ejpam-2741	218	3	then	then	ADV
ejpam-2741	218	4	f	f	X
ejpam-2741	218	5	(	(	PUNCT
ejpam-2741	218	6	x	x	X
ejpam-2741	218	7	)	)	PUNCT
ejpam-2741	218	8	+	+	NUM
ejpam-2741	218	9	t	t	NOUN
ejpam-2741	218	10	=	=	SYM
ejpam-2741	218	11	n	n	PROPN
ejpam-2741	218	12	and	and	CCONJ
ejpam-2741	218	13	f	f	PROPN
ejpam-2741	218	14	(	(	PUNCT
ejpam-2741	218	15	y	y	PROPN
ejpam-2741	218	16	)	)	PUNCT
ejpam-2741	219	1	+	+	NUM
ejpam-2741	219	2	t	t	X
ejpam-2741	219	3	=	=	SYM
ejpam-2741	219	4	n	n	PROPN
ejpam-2741	219	5	.	.	PUNCT
ejpam-2741	220	1	hence	hence	ADV
ejpam-2741	220	2	,	,	PUNCT
ejpam-2741	220	3	by	by	ADP
ejpam-2741	220	4	lemma	lemma	PROPN
ejpam-2741	220	5	3	3	NUM
ejpam-2741	220	6	,	,	PUNCT
ejpam-2741	220	7	f	f	PROPN
ejpam-2741	220	8	(	(	PUNCT
ejpam-2741	220	9	x)β∗gf	x)β∗gf	NUM
ejpam-2741	220	10	(	(	PUNCT
ejpam-2741	220	11	y	y	PROPN
ejpam-2741	220	12	)	)	PUNCT
ejpam-2741	220	13	.	.	PUNCT
ejpam-2741	221	1	corollary	corollary	ADJ
ejpam-2741	221	2	7	7	NUM
ejpam-2741	221	3	.	.	PUNCT
ejpam-2741	222	1	let	let	VERB
ejpam-2741	222	2	x	x	PRON
ejpam-2741	222	3	,	,	PUNCT
ejpam-2741	222	4	y	y	PROPN
ejpam-2741	222	5	,	,	PUNCT
ejpam-2741	222	6	z	z	NOUN
ejpam-2741	222	7	≤m	≤m	NOUN
ejpam-2741	222	8	.	.	PUNCT
ejpam-2741	223	1	if	if	SCONJ
ejpam-2741	223	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	223	3	,	,	PUNCT
ejpam-2741	223	4	then	then	ADV
ejpam-2741	223	5	x+z	x+z	PUNCT
ejpam-2741	223	6	z	z	NOUN
ejpam-2741	223	7	β∗g	β∗g	PUNCT
ejpam-2741	223	8	y+z	y+z	PROPN
ejpam-2741	223	9	z	z	NOUN
ejpam-2741	223	10	.	.	PUNCT
ejpam-2741	224	1	proof	proof	NOUN
ejpam-2741	224	2	.	.	PUNCT
ejpam-2741	225	1	clear	clear	ADJ
ejpam-2741	225	2	from	from	ADP
ejpam-2741	225	3	lemma	lemma	PROPN
ejpam-2741	225	4	6	6	NUM
ejpam-2741	225	5	.	.	PUNCT
ejpam-2741	225	6	corollary	corollary	ADJ
ejpam-2741	225	7	8	8	NUM
ejpam-2741	225	8	.	.	PUNCT
ejpam-2741	226	1	let	let	VERB
ejpam-2741	226	2	m	m	PRON
ejpam-2741	226	3	be	be	AUX
ejpam-2741	226	4	an	an	DET
ejpam-2741	226	5	r−module	r−module	PROPN
ejpam-2741	226	6	,	,	PUNCT
ejpam-2741	226	7	a	a	PRON
ejpam-2741	226	8	be	be	AUX
ejpam-2741	226	9	a	a	DET
ejpam-2741	226	10	direct	direct	ADJ
ejpam-2741	226	11	summand	summand	NOUN
ejpam-2741	226	12	of	of	ADP
ejpam-2741	226	13	m	m	PROPN
ejpam-2741	226	14	and	and	CCONJ
ejpam-2741	226	15	x	x	X
ejpam-2741	226	16	,	,	PUNCT
ejpam-2741	226	17	y	y	PROPN
ejpam-2741	226	18	≤	≤	NUM
ejpam-2741	226	19	a.	a.	NOUN
ejpam-2741	227	1	if	if	SCONJ
ejpam-2741	227	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	227	3	in	in	ADP
ejpam-2741	227	4	m	m	PROPN
ejpam-2741	227	5	,	,	PUNCT
ejpam-2741	227	6	then	then	ADV
ejpam-2741	227	7	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	227	8	in	in	ADP
ejpam-2741	227	9	a	a	DET
ejpam-2741	227	10	also	also	ADV
ejpam-2741	227	11	holds	hold	VERB
ejpam-2741	227	12	.	.	PUNCT
ejpam-2741	228	1	proof	proof	NOUN
ejpam-2741	228	2	.	.	PUNCT
ejpam-2741	229	1	clear	clear	ADJ
ejpam-2741	229	2	from	from	ADP
ejpam-2741	229	3	lemma	lemma	PROPN
ejpam-2741	229	4	6	6	NUM
ejpam-2741	229	5	.	.	PUNCT
ejpam-2741	229	6	proposition	proposition	NOUN
ejpam-2741	229	7	3	3	X
ejpam-2741	229	8	.	.	PUNCT
ejpam-2741	230	1	let	let	VERB
ejpam-2741	230	2	x	x	PRON
ejpam-2741	230	3	,	,	PUNCT
ejpam-2741	230	4	y	y	PROPN
ejpam-2741	230	5	≤	≤	NOUN
ejpam-2741	230	6	m	m	VERB
ejpam-2741	230	7	.	.	PUNCT
ejpam-2741	231	1	if	if	SCONJ
ejpam-2741	231	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	231	3	and	and	CCONJ
ejpam-2741	231	4	y	y	PROPN
ejpam-2741	231	5	is	be	AUX
ejpam-2741	231	6	an	an	DET
ejpam-2741	231	7	essential	essential	ADJ
ejpam-2741	231	8	maximal	maximal	ADJ
ejpam-2741	231	9	submodule	submodule	NOUN
ejpam-2741	231	10	of	of	ADP
ejpam-2741	231	11	m	m	PROPN
ejpam-2741	231	12	,	,	PUNCT
ejpam-2741	231	13	then	then	ADV
ejpam-2741	231	14	x	x	X
ejpam-2741	231	15	≤	≤	ADJ
ejpam-2741	231	16	y	y	NOUN
ejpam-2741	231	17	.	.	PUNCT
ejpam-2741	232	1	proof	proof	NOUN
ejpam-2741	232	2	.	.	PUNCT
ejpam-2741	233	1	assume	assume	VERB
ejpam-2741	233	2	x	x	PUNCT
ejpam-2741	233	3	�	�	PROPN
ejpam-2741	233	4	y	y	PROPN
ejpam-2741	233	5	.	.	PUNCT
ejpam-2741	234	1	then	then	ADV
ejpam-2741	234	2	,	,	PUNCT
ejpam-2741	234	3	because	because	SCONJ
ejpam-2741	234	4	y	y	PROPN
ejpam-2741	234	5	is	be	AUX
ejpam-2741	234	6	an	an	DET
ejpam-2741	234	7	essential	essential	ADJ
ejpam-2741	234	8	maximal	maximal	ADJ
ejpam-2741	234	9	submodule	submodule	NOUN
ejpam-2741	234	10	of	of	ADP
ejpam-2741	234	11	m	m	PRON
ejpam-2741	234	12	,	,	PUNCT
ejpam-2741	234	13	x	x	PROPN
ejpam-2741	235	1	+	+	PUNCT
ejpam-2741	235	2	y	y	NOUN
ejpam-2741	235	3	=	=	SYM
ejpam-2741	235	4	m	m	PROPN
ejpam-2741	235	5	and	and	CCONJ
ejpam-2741	235	6	since	since	SCONJ
ejpam-2741	235	7	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	235	8	,	,	PUNCT
ejpam-2741	235	9	y	y	PROPN
ejpam-2741	235	10	=	=	SYM
ejpam-2741	235	11	y	y	PROPN
ejpam-2741	236	1	+	+	NUM
ejpam-2741	236	2	y	y	PROPN
ejpam-2741	236	3	=	=	NOUN
ejpam-2741	236	4	m	m	PROPN
ejpam-2741	236	5	.	.	PUNCT
ejpam-2741	237	1	this	this	PRON
ejpam-2741	237	2	contradicts	contradict	VERB
ejpam-2741	237	3	maximality	maximality	NOUN
ejpam-2741	237	4	of	of	ADP
ejpam-2741	237	5	y	y	PROPN
ejpam-2741	237	6	.	.	PUNCT
ejpam-2741	238	1	definition	definition	NOUN
ejpam-2741	238	2	2	2	NUM
ejpam-2741	238	3	.	.	PUNCT
ejpam-2741	239	1	let	let	VERB
ejpam-2741	239	2	m	m	PRON
ejpam-2741	239	3	be	be	AUX
ejpam-2741	239	4	an	an	DET
ejpam-2741	239	5	r−module	r−module	NOUN
ejpam-2741	239	6	and	and	CCONJ
ejpam-2741	239	7	u	u	NOUN
ejpam-2741	239	8	,	,	PUNCT
ejpam-2741	239	9	v	v	NOUN
ejpam-2741	239	10	≤m	≤m	NOUN
ejpam-2741	239	11	.	.	PUNCT
ejpam-2741	240	1	if	if	SCONJ
ejpam-2741	240	2	u	u	PROPN
ejpam-2741	240	3	+	+	X
ejpam-2741	240	4	v	v	NOUN
ejpam-2741	240	5	=	=	SYM
ejpam-2741	240	6	m	m	ADJ
ejpam-2741	240	7	and	and	CCONJ
ejpam-2741	240	8	u	u	PROPN
ejpam-2741	240	9	∩	∩	PROPN
ejpam-2741	240	10	v	v	ADP
ejpam-2741	240	11	�	�	PROPN
ejpam-2741	240	12	g	g	NOUN
ejpam-2741	240	13	m	m	PROPN
ejpam-2741	240	14	,	,	PUNCT
ejpam-2741	240	15	then	then	ADV
ejpam-2741	240	16	v	v	NOUN
ejpam-2741	240	17	is	be	AUX
ejpam-2741	240	18	called	call	VERB
ejpam-2741	240	19	a	a	DET
ejpam-2741	240	20	weak	weak	ADJ
ejpam-2741	240	21	g	g	NOUN
ejpam-2741	240	22	-	-	PUNCT
ejpam-2741	240	23	supplement	supplement	NOUN
ejpam-2741	240	24	of	of	ADP
ejpam-2741	240	25	u	u	NOUN
ejpam-2741	240	26	in	in	ADP
ejpam-2741	240	27	m	m	PROPN
ejpam-2741	240	28	.	.	PUNCT
ejpam-2741	241	1	if	if	SCONJ
ejpam-2741	241	2	every	every	DET
ejpam-2741	241	3	submodule	submodule	NOUN
ejpam-2741	241	4	of	of	ADP
ejpam-2741	241	5	m	m	PROPN
ejpam-2741	241	6	has	have	VERB
ejpam-2741	241	7	a	a	DET
ejpam-2741	241	8	weak	weak	ADJ
ejpam-2741	241	9	g	g	NOUN
ejpam-2741	241	10	-	-	PUNCT
ejpam-2741	241	11	supplement	supplement	NOUN
ejpam-2741	241	12	in	in	ADP
ejpam-2741	241	13	m	m	PROPN
ejpam-2741	241	14	,	,	PUNCT
ejpam-2741	241	15	then	then	ADV
ejpam-2741	241	16	m	m	VERB
ejpam-2741	241	17	is	be	AUX
ejpam-2741	241	18	called	call	VERB
ejpam-2741	241	19	a	a	DET
ejpam-2741	241	20	weakly	weakly	ADJ
ejpam-2741	241	21	g	g	NOUN
ejpam-2741	241	22	-	-	PUNCT
ejpam-2741	241	23	supplemented	supplement	VERB
ejpam-2741	241	24	module	module	NOUN
ejpam-2741	241	25	.	.	PUNCT
ejpam-2741	242	1	(	(	PUNCT
ejpam-2741	242	2	see	see	VERB
ejpam-2741	242	3	[	[	X
ejpam-2741	242	4	8	8	NUM
ejpam-2741	242	5	]	]	SYM
ejpam-2741	242	6	)	)	PUNCT
ejpam-2741	242	7	proposition	proposition	NOUN
ejpam-2741	242	8	4	4	NUM
ejpam-2741	242	9	.	.	PUNCT
ejpam-2741	243	1	let	let	VERB
ejpam-2741	243	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	243	3	in	in	ADP
ejpam-2741	243	4	m	m	PROPN
ejpam-2741	243	5	.	.	PUNCT
ejpam-2741	244	1	(	(	PUNCT
ejpam-2741	244	2	i	i	NOUN
ejpam-2741	244	3	)	)	PUNCT
ejpam-2741	244	4	if	if	SCONJ
ejpam-2741	244	5	x	x	PRON
ejpam-2741	244	6	has	have	VERB
ejpam-2741	244	7	an	an	DET
ejpam-2741	244	8	essential	essential	ADJ
ejpam-2741	244	9	g	g	NOUN
ejpam-2741	244	10	-	-	PUNCT
ejpam-2741	244	11	supplement	supplement	NOUN
ejpam-2741	244	12	v	v	NOUN
ejpam-2741	244	13	in	in	ADP
ejpam-2741	244	14	m	m	PROPN
ejpam-2741	244	15	,	,	PUNCT
ejpam-2741	244	16	then	then	ADV
ejpam-2741	244	17	v	v	NOUN
ejpam-2741	244	18	is	be	AUX
ejpam-2741	244	19	also	also	ADV
ejpam-2741	244	20	a	a	DET
ejpam-2741	244	21	g	g	NOUN
ejpam-2741	244	22	-	-	PUNCT
ejpam-2741	244	23	supplement	supplement	NOUN
ejpam-2741	244	24	of	of	ADP
ejpam-2741	244	25	y	y	PROPN
ejpam-2741	244	26	in	in	ADP
ejpam-2741	244	27	m	m	PROPN
ejpam-2741	244	28	.	.	PUNCT
ejpam-2741	245	1	(	(	PUNCT
ejpam-2741	245	2	ii	ii	NOUN
ejpam-2741	245	3	)	)	PUNCT
ejpam-2741	245	4	if	if	SCONJ
ejpam-2741	245	5	x	x	PRON
ejpam-2741	245	6	has	have	VERB
ejpam-2741	245	7	an	an	DET
ejpam-2741	245	8	essential	essential	ADJ
ejpam-2741	245	9	weak	weak	ADJ
ejpam-2741	245	10	g	g	NOUN
ejpam-2741	245	11	-	-	PUNCT
ejpam-2741	245	12	supplement	supplement	NOUN
ejpam-2741	245	13	v	v	NOUN
ejpam-2741	245	14	in	in	ADP
ejpam-2741	245	15	m	m	PROPN
ejpam-2741	245	16	,	,	PUNCT
ejpam-2741	245	17	then	then	ADV
ejpam-2741	245	18	v	v	NOUN
ejpam-2741	245	19	is	be	AUX
ejpam-2741	245	20	also	also	ADV
ejpam-2741	245	21	a	a	DET
ejpam-2741	245	22	weak	weak	ADJ
ejpam-2741	245	23	gsupplement	gsupplement	NOUN
ejpam-2741	245	24	of	of	ADP
ejpam-2741	245	25	y	y	PROPN
ejpam-2741	245	26	in	in	ADP
ejpam-2741	245	27	m	m	PROPN
ejpam-2741	245	28	.	.	PUNCT
ejpam-2741	246	1	proof	proof	NOUN
ejpam-2741	246	2	.	.	PUNCT
ejpam-2741	247	1	(	(	PUNCT
ejpam-2741	247	2	i	i	NOUN
ejpam-2741	247	3	)	)	PUNCT
ejpam-2741	247	4	since	since	SCONJ
ejpam-2741	247	5	m	m	PROPN
ejpam-2741	247	6	=	=	SYM
ejpam-2741	247	7	x+v	x+v	PROPN
ejpam-2741	247	8	and	and	CCONJ
ejpam-2741	247	9	v	v	ADP
ejpam-2741	247	10	em	em	PRON
ejpam-2741	247	11	,	,	PUNCT
ejpam-2741	247	12	then	then	ADV
ejpam-2741	247	13	by	by	ADP
ejpam-2741	247	14	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	247	15	,	,	PUNCT
ejpam-2741	247	16	y	y	PROPN
ejpam-2741	248	1	+	+	PROPN
ejpam-2741	248	2	v	v	NOUN
ejpam-2741	248	3	=	=	NOUN
ejpam-2741	248	4	m	m	VERB
ejpam-2741	248	5	.	.	PUNCT
ejpam-2741	249	1	let	let	VERB
ejpam-2741	249	2	m	m	VERB
ejpam-2741	249	3	=	=	VERB
ejpam-2741	249	4	y	y	PROPN
ejpam-2741	249	5	+	+	PROPN
ejpam-2741	249	6	t	t	PROPN
ejpam-2741	249	7	with	with	ADP
ejpam-2741	249	8	t	t	PROPN
ejpam-2741	249	9	e	e	NOUN
ejpam-2741	249	10	v	v	NOUN
ejpam-2741	249	11	.	.	PUNCT
ejpam-2741	250	1	since	since	SCONJ
ejpam-2741	250	2	t	t	PROPN
ejpam-2741	250	3	e	e	PROPN
ejpam-2741	250	4	v	v	NOUN
ejpam-2741	250	5	and	and	CCONJ
ejpam-2741	250	6	v	v	NOUN
ejpam-2741	250	7	e	e	NOUN
ejpam-2741	250	8	m	m	PROPN
ejpam-2741	250	9	,	,	PUNCT
ejpam-2741	250	10	then	then	ADV
ejpam-2741	250	11	we	we	PRON
ejpam-2741	250	12	can	can	AUX
ejpam-2741	250	13	see	see	VERB
ejpam-2741	250	14	that	that	PRON
ejpam-2741	250	15	t	t	PROPN
ejpam-2741	250	16	e	e	NOUN
ejpam-2741	250	17	m	m	PROPN
ejpam-2741	250	18	.	.	PUNCT
ejpam-2741	251	1	then	then	ADV
ejpam-2741	251	2	by	by	ADP
ejpam-2741	251	3	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	251	4	,	,	PUNCT
ejpam-2741	251	5	x	x	PROPN
ejpam-2741	251	6	+	+	NUM
ejpam-2741	251	7	t	t	X
ejpam-2741	251	8	=	=	SYM
ejpam-2741	251	9	m	m	VERB
ejpam-2741	251	10	.	.	PUNCT
ejpam-2741	252	1	since	since	SCONJ
ejpam-2741	252	2	x	x	PROPN
ejpam-2741	252	3	+	+	NUM
ejpam-2741	252	4	t	t	NOUN
ejpam-2741	252	5	=	=	SYM
ejpam-2741	252	6	m	m	PROPN
ejpam-2741	252	7	and	and	CCONJ
ejpam-2741	252	8	t	t	PROPN
ejpam-2741	252	9	e	e	NOUN
ejpam-2741	252	10	v	v	X
ejpam-2741	252	11	,	,	PUNCT
ejpam-2741	252	12	then	then	ADV
ejpam-2741	252	13	t	t	PROPN
ejpam-2741	252	14	=	=	SYM
ejpam-2741	252	15	v	v	PROPN
ejpam-2741	252	16	.	.	PUNCT
ejpam-2741	253	1	hence	hence	ADV
ejpam-2741	253	2	v	v	NOUN
ejpam-2741	253	3	is	be	AUX
ejpam-2741	253	4	a	a	DET
ejpam-2741	253	5	g	g	NOUN
ejpam-2741	253	6	-	-	PUNCT
ejpam-2741	253	7	supplement	supplement	NOUN
ejpam-2741	253	8	of	of	ADP
ejpam-2741	253	9	y	y	PROPN
ejpam-2741	253	10	in	in	ADP
ejpam-2741	253	11	m	m	PROPN
ejpam-2741	253	12	.	.	PUNCT
ejpam-2741	254	1	(	(	PUNCT
ejpam-2741	254	2	ii	ii	NOUN
ejpam-2741	254	3	)	)	PUNCT
ejpam-2741	254	4	since	since	SCONJ
ejpam-2741	254	5	m	m	NOUN
ejpam-2741	254	6	=	=	SYM
ejpam-2741	254	7	x	x	SYM
ejpam-2741	255	1	+	+	NUM
ejpam-2741	255	2	v	v	NOUN
ejpam-2741	255	3	and	and	CCONJ
ejpam-2741	255	4	v	v	ADP
ejpam-2741	255	5	em	em	PRON
ejpam-2741	255	6	,	,	PUNCT
ejpam-2741	255	7	then	then	ADV
ejpam-2741	255	8	by	by	ADP
ejpam-2741	255	9	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	255	10	,	,	PUNCT
ejpam-2741	255	11	y	y	PROPN
ejpam-2741	255	12	+	+	PROPN
ejpam-2741	255	13	v	v	NOUN
ejpam-2741	255	14	=	=	NOUN
ejpam-2741	255	15	m	m	VERB
ejpam-2741	255	16	.	.	PUNCT
ejpam-2741	256	1	let	let	VERB
ejpam-2741	256	2	y	y	PROPN
ejpam-2741	256	3	∩	∩	PROPN
ejpam-2741	256	4	v	v	ADP
ejpam-2741	256	5	+	+	X
ejpam-2741	256	6	t	t	NOUN
ejpam-2741	256	7	=	=	SYM
ejpam-2741	256	8	m	m	VERB
ejpam-2741	256	9	with	with	ADP
ejpam-2741	256	10	t	t	PROPN
ejpam-2741	256	11	em	em	PRON
ejpam-2741	256	12	.	.	PUNCT
ejpam-2741	257	1	since	since	SCONJ
ejpam-2741	257	2	m	m	PROPN
ejpam-2741	257	3	=	=	SYM
ejpam-2741	257	4	y	y	PROPN
ejpam-2741	257	5	+	+	PROPN
ejpam-2741	257	6	v	v	NOUN
ejpam-2741	257	7	and	and	CCONJ
ejpam-2741	257	8	m	m	VERB
ejpam-2741	257	9	=	=	ADJ
ejpam-2741	257	10	y	y	PROPN
ejpam-2741	258	1	∩v	∩v	PUNCT
ejpam-2741	259	1	+	+	CCONJ
ejpam-2741	259	2	t	t	PROPN
ejpam-2741	259	3	,	,	PUNCT
ejpam-2741	259	4	then	then	ADV
ejpam-2741	259	5	by	by	ADP
ejpam-2741	259	6	lemma	lemma	PROPN
ejpam-2741	259	7	1	1	NUM
ejpam-2741	259	8	,	,	PUNCT
ejpam-2741	259	9	m	m	VERB
ejpam-2741	259	10	=	=	SYM
ejpam-2741	259	11	y	y	PROPN
ejpam-2741	259	12	+	+	PROPN
ejpam-2741	259	13	v	v	NOUN
ejpam-2741	259	14	∩t	∩t	NOUN
ejpam-2741	259	15	.	.	PUNCT
ejpam-2741	260	1	c.	c.	PROPN
ejpam-2741	260	2	nebiyev	nebiyev	PROPN
ejpam-2741	260	3	,	,	PUNCT
ejpam-2741	260	4	n.	n.	PROPN
ejpam-2741	260	5	sökmez	sökmez	PROPN
ejpam-2741	260	6	/	/	SYM
ejpam-2741	260	7	eur	eur	PROPN
ejpam-2741	260	8	.	.	PUNCT
ejpam-2741	261	1	j.	j.	PROPN
ejpam-2741	261	2	pure	pure	PROPN
ejpam-2741	261	3	appl	appl	PROPN
ejpam-2741	261	4	.	.	PROPN
ejpam-2741	261	5	math	math	PROPN
ejpam-2741	261	6	,	,	PUNCT
ejpam-2741	261	7	11	11	NUM
ejpam-2741	261	8	(	(	PUNCT
ejpam-2741	261	9	1	1	NUM
ejpam-2741	261	10	)	)	PUNCT
ejpam-2741	261	11	(	(	PUNCT
ejpam-2741	261	12	2018	2018	NUM
ejpam-2741	261	13	)	)	PUNCT
ejpam-2741	261	14	,	,	PUNCT
ejpam-2741	261	15	238	238	NUM
ejpam-2741	261	16	-	-	SYM
ejpam-2741	261	17	243	243	NUM
ejpam-2741	261	18	242	242	NUM
ejpam-2741	261	19	since	since	SCONJ
ejpam-2741	261	20	v	v	NUM
ejpam-2741	261	21	e	e	NOUN
ejpam-2741	261	22	m	m	NOUN
ejpam-2741	261	23	and	and	CCONJ
ejpam-2741	261	24	t	t	PROPN
ejpam-2741	261	25	e	e	X
ejpam-2741	261	26	m	m	PROPN
ejpam-2741	261	27	,	,	PUNCT
ejpam-2741	261	28	then	then	ADV
ejpam-2741	261	29	v	v	ADP
ejpam-2741	261	30	∩	∩	NOUN
ejpam-2741	261	31	t	t	NOUN
ejpam-2741	261	32	e	e	NOUN
ejpam-2741	261	33	m	m	PROPN
ejpam-2741	261	34	.	.	PUNCT
ejpam-2741	262	1	then	then	ADV
ejpam-2741	262	2	by	by	ADP
ejpam-2741	262	3	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	262	4	,	,	PUNCT
ejpam-2741	262	5	x	x	PUNCT
ejpam-2741	262	6	+	+	SYM
ejpam-2741	262	7	v	v	NUM
ejpam-2741	262	8	∩	∩	NOUN
ejpam-2741	262	9	t	t	NOUN
ejpam-2741	262	10	=	=	SYM
ejpam-2741	262	11	m	m	PROPN
ejpam-2741	262	12	.	.	PUNCT
ejpam-2741	263	1	since	since	SCONJ
ejpam-2741	263	2	m	m	PROPN
ejpam-2741	263	3	=	=	SYM
ejpam-2741	263	4	v	v	PROPN
ejpam-2741	263	5	+	+	X
ejpam-2741	263	6	t	t	PROPN
ejpam-2741	263	7	and	and	CCONJ
ejpam-2741	263	8	m	m	PROPN
ejpam-2741	263	9	=	=	NOUN
ejpam-2741	263	10	x	x	SYM
ejpam-2741	263	11	+	+	NUM
ejpam-2741	263	12	v	v	NUM
ejpam-2741	263	13	∩	∩	ADJ
ejpam-2741	263	14	t	t	NOUN
ejpam-2741	263	15	,	,	PUNCT
ejpam-2741	263	16	then	then	ADV
ejpam-2741	263	17	by	by	ADP
ejpam-2741	263	18	lemma	lemma	PROPN
ejpam-2741	263	19	1	1	NUM
ejpam-2741	263	20	,	,	PUNCT
ejpam-2741	263	21	x	x	PRON
ejpam-2741	263	22	∩	∩	NOUN
ejpam-2741	263	23	v	v	ADP
ejpam-2741	263	24	+	+	X
ejpam-2741	263	25	t	t	NOUN
ejpam-2741	263	26	=	=	SYM
ejpam-2741	263	27	m.	m.	NOUN
ejpam-2741	263	28	because	because	SCONJ
ejpam-2741	263	29	x	x	PROPN
ejpam-2741	263	30	∩	∩	PROPN
ejpam-2741	263	31	v	v	ADP
ejpam-2741	263	32	+	+	X
ejpam-2741	263	33	t	t	NOUN
ejpam-2741	263	34	=	=	SYM
ejpam-2741	263	35	m	m	PROPN
ejpam-2741	263	36	and	and	CCONJ
ejpam-2741	263	37	t	t	VERB
ejpam-2741	263	38	em	em	PRON
ejpam-2741	263	39	and	and	CCONJ
ejpam-2741	263	40	x	x	PROPN
ejpam-2741	263	41	∩	∩	PROPN
ejpam-2741	263	42	v	v	ADP
ejpam-2741	263	43	�	�	PROPN
ejpam-2741	263	44	g	g	NOUN
ejpam-2741	263	45	m	m	PROPN
ejpam-2741	263	46	,	,	PUNCT
ejpam-2741	263	47	then	then	ADV
ejpam-2741	263	48	t	t	PROPN
ejpam-2741	263	49	=	=	PUNCT
ejpam-2741	263	50	m	m	NOUN
ejpam-2741	263	51	.	.	PUNCT
ejpam-2741	264	1	hence	hence	ADV
ejpam-2741	264	2	y	y	PROPN
ejpam-2741	264	3	∩	∩	PROPN
ejpam-2741	264	4	v	v	ADP
ejpam-2741	264	5	�	�	PROPN
ejpam-2741	264	6	g	g	NOUN
ejpam-2741	264	7	m	m	PROPN
ejpam-2741	264	8	and	and	CCONJ
ejpam-2741	264	9	v	v	NOUN
ejpam-2741	264	10	is	be	AUX
ejpam-2741	264	11	a	a	DET
ejpam-2741	264	12	weak	weak	ADJ
ejpam-2741	264	13	g	g	NOUN
ejpam-2741	264	14	-	-	PUNCT
ejpam-2741	264	15	supplement	supplement	NOUN
ejpam-2741	264	16	of	of	ADP
ejpam-2741	264	17	y	y	PROPN
ejpam-2741	264	18	in	in	ADP
ejpam-2741	264	19	m	m	PROPN
ejpam-2741	264	20	.	.	PUNCT
ejpam-2741	265	1	proposition	proposition	NOUN
ejpam-2741	265	2	5	5	NUM
ejpam-2741	265	3	.	.	PUNCT
ejpam-2741	266	1	let	let	VERB
ejpam-2741	266	2	m	m	PRON
ejpam-2741	266	3	be	be	AUX
ejpam-2741	266	4	an	an	DET
ejpam-2741	266	5	amply	amply	NOUN
ejpam-2741	266	6	g	g	NOUN
ejpam-2741	266	7	-	-	PUNCT
ejpam-2741	266	8	supplemented	supplement	VERB
ejpam-2741	266	9	module	module	NOUN
ejpam-2741	266	10	and	and	CCONJ
ejpam-2741	266	11	x	x	NOUN
ejpam-2741	266	12	,	,	PUNCT
ejpam-2741	266	13	y	y	PROPN
ejpam-2741	266	14	≤	≤	NOUN
ejpam-2741	266	15	m	m	VERB
ejpam-2741	266	16	.	.	PUNCT
ejpam-2741	267	1	if	if	SCONJ
ejpam-2741	267	2	gsupplements	gsupplement	NOUN
ejpam-2741	267	3	of	of	ADP
ejpam-2741	267	4	x	x	X
ejpam-2741	267	5	and	and	CCONJ
ejpam-2741	267	6	y	y	PROPN
ejpam-2741	267	7	in	in	ADP
ejpam-2741	267	8	m	m	PROPN
ejpam-2741	267	9	is	be	AUX
ejpam-2741	267	10	the	the	DET
ejpam-2741	267	11	same	same	ADJ
ejpam-2741	267	12	,	,	PUNCT
ejpam-2741	267	13	then	then	ADV
ejpam-2741	267	14	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	267	15	.	.	PUNCT
ejpam-2741	268	1	proof	proof	NOUN
ejpam-2741	268	2	.	.	PUNCT
ejpam-2741	269	1	let	let	VERB
ejpam-2741	269	2	x	x	PUNCT
ejpam-2741	270	1	+	+	NOUN
ejpam-2741	270	2	k	k	X
ejpam-2741	270	3	=	=	X
ejpam-2741	270	4	m	m	VERB
ejpam-2741	270	5	with	with	ADP
ejpam-2741	270	6	k	k	PROPN
ejpam-2741	270	7	em	em	PRON
ejpam-2741	270	8	.	.	PUNCT
ejpam-2741	271	1	since	since	SCONJ
ejpam-2741	271	2	m	m	PROPN
ejpam-2741	271	3	is	be	AUX
ejpam-2741	271	4	amply	amply	ADV
ejpam-2741	271	5	g	g	NOUN
ejpam-2741	271	6	-	-	PUNCT
ejpam-2741	271	7	supplemented	supplement	VERB
ejpam-2741	271	8	,	,	PUNCT
ejpam-2741	271	9	there	there	PRON
ejpam-2741	271	10	exists	exist	VERB
ejpam-2741	271	11	a	a	DET
ejpam-2741	271	12	g	g	NOUN
ejpam-2741	271	13	-	-	PUNCT
ejpam-2741	271	14	supplement	supplement	NOUN
ejpam-2741	271	15	k	k	NOUN
ejpam-2741	271	16	′	′	NUM
ejpam-2741	271	17	of	of	ADP
ejpam-2741	271	18	x	x	PUNCT
ejpam-2741	271	19	with	with	ADP
ejpam-2741	271	20	k	k	PROPN
ejpam-2741	271	21	′	′	ADJ
ejpam-2741	271	22	≤	≤	PROPN
ejpam-2741	271	23	k.	k.	INTJ
ejpam-2741	272	1	by	by	ADP
ejpam-2741	272	2	hypothesis	hypothesis	NOUN
ejpam-2741	272	3	,	,	PUNCT
ejpam-2741	272	4	k	k	PROPN
ejpam-2741	272	5	′	′	NOUN
ejpam-2741	272	6	is	be	AUX
ejpam-2741	272	7	a	a	DET
ejpam-2741	272	8	g	g	NOUN
ejpam-2741	272	9	-	-	PUNCT
ejpam-2741	272	10	supplement	supplement	NOUN
ejpam-2741	272	11	of	of	ADP
ejpam-2741	272	12	y	y	PROPN
ejpam-2741	272	13	in	in	ADP
ejpam-2741	272	14	m	m	PROPN
ejpam-2741	272	15	.	.	PUNCT
ejpam-2741	273	1	then	then	ADV
ejpam-2741	273	2	y	y	PROPN
ejpam-2741	273	3	+	+	PROPN
ejpam-2741	273	4	k	k	PROPN
ejpam-2741	273	5	′	′	NOUN
ejpam-2741	274	1	=	=	VERB
ejpam-2741	274	2	m	m	VERB
ejpam-2741	274	3	and	and	CCONJ
ejpam-2741	274	4	since	since	SCONJ
ejpam-2741	274	5	k	k	PROPN
ejpam-2741	274	6	′	′	NOUN
ejpam-2741	274	7	≤	≤	PROPN
ejpam-2741	275	1	k	k	PROPN
ejpam-2741	275	2	,	,	PUNCT
ejpam-2741	275	3	y	y	PROPN
ejpam-2741	276	1	+	+	PROPN
ejpam-2741	276	2	k	k	X
ejpam-2741	276	3	=	=	X
ejpam-2741	276	4	m	m	VERB
ejpam-2741	276	5	.	.	PUNCT
ejpam-2741	277	1	similarly	similarly	ADV
ejpam-2741	277	2	,	,	PUNCT
ejpam-2741	277	3	we	we	PRON
ejpam-2741	277	4	can	can	AUX
ejpam-2741	277	5	see	see	VERB
ejpam-2741	277	6	that	that	PRON
ejpam-2741	277	7	x+t	x+t	PUNCT
ejpam-2741	278	1	=	=	PUNCT
ejpam-2741	278	2	m	m	VERB
ejpam-2741	278	3	for	for	ADP
ejpam-2741	278	4	every	every	DET
ejpam-2741	278	5	t	t	NOUN
ejpam-2741	278	6	em	em	PRON
ejpam-2741	278	7	such	such	ADJ
ejpam-2741	278	8	that	that	SCONJ
ejpam-2741	278	9	y	y	PROPN
ejpam-2741	279	1	+	+	PROPN
ejpam-2741	279	2	t	t	PROPN
ejpam-2741	279	3	=	=	SYM
ejpam-2741	279	4	m	m	NOUN
ejpam-2741	279	5	.	.	PUNCT
ejpam-2741	280	1	proposition	proposition	NOUN
ejpam-2741	280	2	6	6	NUM
ejpam-2741	280	3	.	.	PUNCT
ejpam-2741	281	1	let	let	VERB
ejpam-2741	281	2	m	m	PRON
ejpam-2741	281	3	be	be	AUX
ejpam-2741	281	4	weakly	weakly	ADV
ejpam-2741	281	5	g	g	NOUN
ejpam-2741	281	6	-	-	PUNCT
ejpam-2741	281	7	supplemented	supplement	VERB
ejpam-2741	281	8	module	module	NOUN
ejpam-2741	281	9	and	and	CCONJ
ejpam-2741	281	10	x	x	NOUN
ejpam-2741	281	11	,	,	PUNCT
ejpam-2741	281	12	y	y	PROPN
ejpam-2741	281	13	≤	≤	NOUN
ejpam-2741	281	14	m	m	VERB
ejpam-2741	281	15	.	.	PUNCT
ejpam-2741	282	1	if	if	SCONJ
ejpam-2741	282	2	weak	weak	ADJ
ejpam-2741	282	3	gsupplements	gsupplement	NOUN
ejpam-2741	282	4	of	of	ADP
ejpam-2741	282	5	x	x	X
ejpam-2741	282	6	and	and	CCONJ
ejpam-2741	282	7	y	y	PROPN
ejpam-2741	282	8	in	in	ADP
ejpam-2741	282	9	m	m	PROPN
ejpam-2741	282	10	is	be	AUX
ejpam-2741	282	11	the	the	DET
ejpam-2741	282	12	same	same	ADJ
ejpam-2741	282	13	,	,	PUNCT
ejpam-2741	282	14	then	then	ADV
ejpam-2741	282	15	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	282	16	.	.	PUNCT
ejpam-2741	283	1	proof	proof	NOUN
ejpam-2741	283	2	.	.	PUNCT
ejpam-2741	284	1	let	let	VERB
ejpam-2741	284	2	x	x	PUNCT
ejpam-2741	285	1	+	+	CCONJ
ejpam-2741	285	2	k	k	X
ejpam-2741	285	3	=	=	NOUN
ejpam-2741	285	4	m	m	VERB
ejpam-2741	285	5	with	with	ADP
ejpam-2741	285	6	k	k	PROPN
ejpam-2741	285	7	e	e	PROPN
ejpam-2741	285	8	m	m	PROPN
ejpam-2741	285	9	.	.	PUNCT
ejpam-2741	286	1	since	since	SCONJ
ejpam-2741	286	2	m	m	PROPN
ejpam-2741	286	3	is	be	AUX
ejpam-2741	286	4	weakly	weakly	ADJ
ejpam-2741	286	5	g	g	NOUN
ejpam-2741	286	6	-	-	PUNCT
ejpam-2741	286	7	supplemented	supplement	VERB
ejpam-2741	286	8	,	,	PUNCT
ejpam-2741	286	9	by	by	ADP
ejpam-2741	286	10	[	[	X
ejpam-2741	286	11	8	8	NUM
ejpam-2741	286	12	,	,	PUNCT
ejpam-2741	286	13	proposition	proposition	NOUN
ejpam-2741	286	14	1	1	NUM
ejpam-2741	286	15	]	]	PUNCT
ejpam-2741	286	16	there	there	PRON
ejpam-2741	286	17	exists	exist	VERB
ejpam-2741	286	18	a	a	DET
ejpam-2741	286	19	weak	weak	ADJ
ejpam-2741	286	20	g	g	NOUN
ejpam-2741	286	21	-	-	PUNCT
ejpam-2741	286	22	supplement	supplement	NOUN
ejpam-2741	286	23	k	k	NOUN
ejpam-2741	286	24	′	′	NUM
ejpam-2741	286	25	of	of	ADP
ejpam-2741	286	26	x	x	PUNCT
ejpam-2741	286	27	with	with	ADP
ejpam-2741	286	28	k	k	PROPN
ejpam-2741	286	29	′	′	ADJ
ejpam-2741	286	30	≤	≤	PROPN
ejpam-2741	286	31	k.	k.	INTJ
ejpam-2741	287	1	by	by	ADP
ejpam-2741	287	2	hypothesis	hypothesis	NOUN
ejpam-2741	287	3	,	,	PUNCT
ejpam-2741	287	4	k	k	PROPN
ejpam-2741	287	5	′	′	NOUN
ejpam-2741	287	6	is	be	AUX
ejpam-2741	287	7	a	a	DET
ejpam-2741	287	8	weak	weak	ADJ
ejpam-2741	287	9	g	g	NOUN
ejpam-2741	287	10	-	-	PUNCT
ejpam-2741	287	11	supplement	supplement	NOUN
ejpam-2741	287	12	of	of	ADP
ejpam-2741	287	13	y	y	PROPN
ejpam-2741	287	14	in	in	ADP
ejpam-2741	287	15	m	m	PROPN
ejpam-2741	287	16	.	.	PUNCT
ejpam-2741	288	1	then	then	ADV
ejpam-2741	288	2	y	y	PROPN
ejpam-2741	288	3	+	+	PROPN
ejpam-2741	288	4	k	k	PROPN
ejpam-2741	288	5	′	′	NOUN
ejpam-2741	289	1	=	=	VERB
ejpam-2741	289	2	m	m	VERB
ejpam-2741	289	3	and	and	CCONJ
ejpam-2741	289	4	since	since	SCONJ
ejpam-2741	289	5	k	k	PROPN
ejpam-2741	289	6	′	′	NOUN
ejpam-2741	289	7	≤	≤	PROPN
ejpam-2741	290	1	k	k	PROPN
ejpam-2741	290	2	,	,	PUNCT
ejpam-2741	290	3	y	y	PROPN
ejpam-2741	291	1	+	+	PROPN
ejpam-2741	291	2	k	k	X
ejpam-2741	291	3	=	=	X
ejpam-2741	291	4	m	m	VERB
ejpam-2741	291	5	.	.	PUNCT
ejpam-2741	292	1	similarly	similarly	ADV
ejpam-2741	292	2	,	,	PUNCT
ejpam-2741	292	3	we	we	PRON
ejpam-2741	292	4	can	can	AUX
ejpam-2741	292	5	see	see	VERB
ejpam-2741	292	6	that	that	PRON
ejpam-2741	292	7	x	x	PROPN
ejpam-2741	293	1	+	+	NUM
ejpam-2741	293	2	t	t	X
ejpam-2741	293	3	=	=	PUNCT
ejpam-2741	293	4	m	m	VERB
ejpam-2741	293	5	for	for	ADP
ejpam-2741	293	6	every	every	DET
ejpam-2741	293	7	t	t	NOUN
ejpam-2741	293	8	em	em	PRON
ejpam-2741	293	9	such	such	ADJ
ejpam-2741	293	10	that	that	SCONJ
ejpam-2741	293	11	y	y	PROPN
ejpam-2741	293	12	+	+	PROPN
ejpam-2741	293	13	t	t	PROPN
ejpam-2741	293	14	=	=	SYM
ejpam-2741	293	15	m	m	NOUN
ejpam-2741	293	16	.	.	PUNCT
ejpam-2741	294	1	proposition	proposition	NOUN
ejpam-2741	294	2	7	7	NUM
ejpam-2741	294	3	.	.	PUNCT
ejpam-2741	295	1	let	let	VERB
ejpam-2741	295	2	m	m	PRON
ejpam-2741	295	3	be	be	AUX
ejpam-2741	295	4	an	an	DET
ejpam-2741	295	5	r−module	r−module	PROPN
ejpam-2741	295	6	,	,	PUNCT
ejpam-2741	295	7	x	x	PUNCT
ejpam-2741	295	8	≤	≤	NUM
ejpam-2741	295	9	y	y	PROPN
ejpam-2741	295	10	≤	≤	NUM
ejpam-2741	295	11	m	m	PROPN
ejpam-2741	295	12	and	and	CCONJ
ejpam-2741	295	13	c	c	PROPN
ejpam-2741	295	14	be	be	AUX
ejpam-2741	295	15	an	an	DET
ejpam-2741	295	16	essential	essential	ADJ
ejpam-2741	295	17	weak	weak	ADJ
ejpam-2741	295	18	g	g	NOUN
ejpam-2741	295	19	-	-	PUNCT
ejpam-2741	295	20	supplement	supplement	NOUN
ejpam-2741	295	21	of	of	ADP
ejpam-2741	295	22	x	x	PUNCT
ejpam-2741	295	23	in	in	ADP
ejpam-2741	295	24	m	m	PROPN
ejpam-2741	295	25	.	.	PUNCT
ejpam-2741	296	1	if	if	SCONJ
ejpam-2741	296	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	296	3	,	,	PUNCT
ejpam-2741	296	4	then	then	ADV
ejpam-2741	296	5	y	y	PROPN
ejpam-2741	296	6	∩	∩	PROPN
ejpam-2741	296	7	c	c	PROPN
ejpam-2741	296	8	�	�	PROPN
ejpam-2741	296	9	g	g	PROPN
ejpam-2741	296	10	m	m	NOUN
ejpam-2741	296	11	.	.	PUNCT
ejpam-2741	297	1	proof	proof	NOUN
ejpam-2741	297	2	.	.	PUNCT
ejpam-2741	298	1	since	since	SCONJ
ejpam-2741	298	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	298	3	and	and	CCONJ
ejpam-2741	298	4	c	c	PROPN
ejpam-2741	298	5	is	be	AUX
ejpam-2741	298	6	an	an	DET
ejpam-2741	298	7	essential	essential	ADJ
ejpam-2741	298	8	weak	weak	ADJ
ejpam-2741	298	9	g	g	NOUN
ejpam-2741	298	10	-	-	PUNCT
ejpam-2741	298	11	supplement	supplement	NOUN
ejpam-2741	298	12	of	of	ADP
ejpam-2741	298	13	x	x	PUNCT
ejpam-2741	298	14	in	in	ADP
ejpam-2741	298	15	m	m	PROPN
ejpam-2741	298	16	,	,	PUNCT
ejpam-2741	298	17	then	then	ADV
ejpam-2741	298	18	by	by	ADP
ejpam-2741	298	19	proposition	proposition	NOUN
ejpam-2741	298	20	4	4	NUM
ejpam-2741	298	21	,	,	PUNCT
ejpam-2741	298	22	c	c	PROPN
ejpam-2741	298	23	is	be	AUX
ejpam-2741	298	24	also	also	ADV
ejpam-2741	298	25	a	a	DET
ejpam-2741	298	26	weak	weak	ADJ
ejpam-2741	298	27	g	g	NOUN
ejpam-2741	298	28	-	-	PUNCT
ejpam-2741	298	29	supplement	supplement	NOUN
ejpam-2741	298	30	of	of	ADP
ejpam-2741	298	31	y	y	PROPN
ejpam-2741	298	32	in	in	ADP
ejpam-2741	298	33	m	m	PROPN
ejpam-2741	298	34	.	.	PUNCT
ejpam-2741	299	1	hence	hence	ADV
ejpam-2741	299	2	y	y	PROPN
ejpam-2741	299	3	∩	∩	PROPN
ejpam-2741	299	4	c	c	PROPN
ejpam-2741	299	5	�	�	PROPN
ejpam-2741	299	6	g	g	PROPN
ejpam-2741	299	7	m	m	PROPN
ejpam-2741	299	8	.	.	PUNCT
ejpam-2741	300	1	lemma	lemma	PROPN
ejpam-2741	300	2	7	7	X
ejpam-2741	300	3	.	.	PUNCT
ejpam-2741	301	1	let	let	VERB
ejpam-2741	301	2	m	m	PRON
ejpam-2741	301	3	be	be	AUX
ejpam-2741	301	4	an	an	DET
ejpam-2741	301	5	r−module	r−module	PROPN
ejpam-2741	301	6	,	,	PUNCT
ejpam-2741	301	7	x	x	PUNCT
ejpam-2741	301	8	≤	≤	PROPN
ejpam-2741	301	9	y	y	PROPN
ejpam-2741	301	10	≤m	≤m	PROPN
ejpam-2741	301	11	and	and	CCONJ
ejpam-2741	301	12	c	c	PROPN
ejpam-2741	301	13	be	be	AUX
ejpam-2741	301	14	a	a	DET
ejpam-2741	301	15	weak	weak	ADJ
ejpam-2741	301	16	g	g	NOUN
ejpam-2741	301	17	-	-	PUNCT
ejpam-2741	301	18	supplement	supplement	NOUN
ejpam-2741	301	19	of	of	ADP
ejpam-2741	301	20	x	x	PUNCT
ejpam-2741	301	21	in	in	ADP
ejpam-2741	301	22	m	m	PROPN
ejpam-2741	301	23	.	.	PUNCT
ejpam-2741	302	1	if	if	SCONJ
ejpam-2741	302	2	y	y	PROPN
ejpam-2741	302	3	∩	∩	PROPN
ejpam-2741	302	4	c	c	PROPN
ejpam-2741	302	5	�	�	PROPN
ejpam-2741	302	6	g	g	PROPN
ejpam-2741	302	7	m	m	PROPN
ejpam-2741	302	8	,	,	PUNCT
ejpam-2741	302	9	then	then	ADV
ejpam-2741	302	10	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	302	11	.	.	PUNCT
ejpam-2741	303	1	proof	proof	NOUN
ejpam-2741	303	2	.	.	PUNCT
ejpam-2741	304	1	let	let	VERB
ejpam-2741	304	2	y	y	NOUN
ejpam-2741	304	3	+	+	NOUN
ejpam-2741	304	4	t	t	X
ejpam-2741	304	5	=	=	PUNCT
ejpam-2741	304	6	m	m	VERB
ejpam-2741	304	7	with	with	ADP
ejpam-2741	304	8	t	t	PROPN
ejpam-2741	304	9	e	e	X
ejpam-2741	304	10	m	m	PROPN
ejpam-2741	304	11	.	.	PUNCT
ejpam-2741	305	1	since	since	SCONJ
ejpam-2741	305	2	c	c	PROPN
ejpam-2741	305	3	is	be	AUX
ejpam-2741	305	4	a	a	DET
ejpam-2741	305	5	weak	weak	ADJ
ejpam-2741	305	6	g	g	NOUN
ejpam-2741	305	7	-	-	PUNCT
ejpam-2741	305	8	supplement	supplement	NOUN
ejpam-2741	305	9	of	of	ADP
ejpam-2741	305	10	x	x	PUNCT
ejpam-2741	305	11	in	in	ADP
ejpam-2741	305	12	m	m	PROPN
ejpam-2741	305	13	,	,	PUNCT
ejpam-2741	305	14	c	c	PROPN
ejpam-2741	305	15	+	+	NOUN
ejpam-2741	305	16	x	x	X
ejpam-2741	305	17	=	=	VERB
ejpam-2741	305	18	m	m	VERB
ejpam-2741	305	19	.	.	PUNCT
ejpam-2741	306	1	since	since	SCONJ
ejpam-2741	306	2	x	x	PROPN
ejpam-2741	306	3	≤	≤	NUM
ejpam-2741	306	4	y	y	PROPN
ejpam-2741	306	5	,	,	PUNCT
ejpam-2741	306	6	by	by	ADP
ejpam-2741	306	7	modular	modular	ADJ
ejpam-2741	306	8	law	law	NOUN
ejpam-2741	306	9	,	,	PUNCT
ejpam-2741	306	10	y	y	PROPN
ejpam-2741	306	11	=	=	PUNCT
ejpam-2741	306	12	y	y	PROPN
ejpam-2741	306	13	∩m	∩m	PROPN
ejpam-2741	306	14	=	=	PUNCT
ejpam-2741	306	15	y	y	PROPN
ejpam-2741	306	16	∩	∩	NOUN
ejpam-2741	306	17	(	(	PUNCT
ejpam-2741	306	18	c	c	NOUN
ejpam-2741	306	19	+	+	NOUN
ejpam-2741	306	20	x	x	X
ejpam-2741	306	21	)	)	PUNCT
ejpam-2741	306	22	=	=	SYM
ejpam-2741	306	23	y	y	PROPN
ejpam-2741	306	24	∩	∩	NOUN
ejpam-2741	306	25	c	c	PROPN
ejpam-2741	306	26	+	+	PROPN
ejpam-2741	306	27	x.	x.	NOUN
ejpam-2741	306	28	then	then	ADV
ejpam-2741	306	29	m	m	VERB
ejpam-2741	306	30	=	=	PUNCT
ejpam-2741	306	31	y	y	PROPN
ejpam-2741	306	32	+	+	PROPN
ejpam-2741	306	33	t	t	NOUN
ejpam-2741	306	34	=	=	SYM
ejpam-2741	306	35	y	y	PROPN
ejpam-2741	306	36	∩c+x	∩c+x	PROPN
ejpam-2741	306	37	+	+	CCONJ
ejpam-2741	306	38	t	t	PROPN
ejpam-2741	306	39	and	and	CCONJ
ejpam-2741	306	40	since	since	SCONJ
ejpam-2741	306	41	y	y	PROPN
ejpam-2741	306	42	∩c	∩c	PROPN
ejpam-2741	306	43	�	�	PROPN
ejpam-2741	306	44	g	g	PROPN
ejpam-2741	306	45	m	m	PROPN
ejpam-2741	306	46	and	and	CCONJ
ejpam-2741	306	47	x	x	X
ejpam-2741	306	48	+	+	NOUN
ejpam-2741	306	49	t	t	X
ejpam-2741	306	50	em	em	PRON
ejpam-2741	306	51	,	,	PUNCT
ejpam-2741	306	52	x	x	PROPN
ejpam-2741	306	53	+	+	NOUN
ejpam-2741	306	54	t	t	X
ejpam-2741	306	55	=	=	SYM
ejpam-2741	306	56	m	m	VERB
ejpam-2741	306	57	.	.	PUNCT
ejpam-2741	307	1	if	if	SCONJ
ejpam-2741	307	2	x	x	X
ejpam-2741	307	3	+	+	NOUN
ejpam-2741	307	4	k	k	X
ejpam-2741	307	5	=	=	X
ejpam-2741	307	6	m	m	VERB
ejpam-2741	307	7	with	with	ADP
ejpam-2741	307	8	k	k	PROPN
ejpam-2741	307	9	em	em	PRON
ejpam-2741	307	10	,	,	PUNCT
ejpam-2741	307	11	y	y	PROPN
ejpam-2741	307	12	+	+	PROPN
ejpam-2741	307	13	k	k	X
ejpam-2741	307	14	=	=	NOUN
ejpam-2741	307	15	m	m	VERB
ejpam-2741	307	16	also	also	ADV
ejpam-2741	307	17	holds	hold	VERB
ejpam-2741	307	18	since	since	SCONJ
ejpam-2741	307	19	x	x	PROPN
ejpam-2741	307	20	≤	≤	NUM
ejpam-2741	307	21	y	y	NOUN
ejpam-2741	307	22	.	.	PUNCT
ejpam-2741	308	1	hence	hence	ADV
ejpam-2741	308	2	xβ∗gy	xβ∗gy	PROPN
ejpam-2741	308	3	.	.	PUNCT
ejpam-2741	309	1	proposition	proposition	NOUN
ejpam-2741	309	2	8	8	NUM
ejpam-2741	309	3	.	.	PUNCT
ejpam-2741	310	1	let	let	VERB
ejpam-2741	310	2	m	m	NOUN
ejpam-2741	310	3	=	=	SYM
ejpam-2741	310	4	m1	m1	PROPN
ejpam-2741	310	5	⊕m2	⊕m2	NUM
ejpam-2741	310	6	and	and	CCONJ
ejpam-2741	310	7	m1	m1	PROPN
ejpam-2741	310	8	≤	≤	NUM
ejpam-2741	310	9	x	x	PUNCT
ejpam-2741	310	10	≤m	≤m	NOUN
ejpam-2741	310	11	.	.	PUNCT
ejpam-2741	311	1	if	if	SCONJ
ejpam-2741	311	2	x	x	PRON
ejpam-2741	311	3	∩m2	∩m2	PROPN
ejpam-2741	311	4	�	�	PROPN
ejpam-2741	311	5	g	g	NOUN
ejpam-2741	311	6	m	m	PROPN
ejpam-2741	311	7	,	,	PUNCT
ejpam-2741	311	8	then	then	ADV
ejpam-2741	311	9	xβ∗gm1	xβ∗gm1	PROPN
ejpam-2741	311	10	.	.	PUNCT
ejpam-2741	312	1	proof	proof	NOUN
ejpam-2741	312	2	.	.	PUNCT
ejpam-2741	313	1	clear	clear	ADJ
ejpam-2741	313	2	from	from	ADP
ejpam-2741	313	3	lemma	lemma	PROPN
ejpam-2741	313	4	7	7	NUM
ejpam-2741	313	5	.	.	PUNCT
ejpam-2741	313	6	proposition	proposition	NOUN
ejpam-2741	313	7	9	9	NUM
ejpam-2741	313	8	.	.	PUNCT
ejpam-2741	314	1	let	let	VERB
ejpam-2741	314	2	m	m	PRON
ejpam-2741	314	3	be	be	AUX
ejpam-2741	314	4	an	an	DET
ejpam-2741	314	5	r−module	r−module	NOUN
ejpam-2741	314	6	.	.	PUNCT
ejpam-2741	315	1	if	if	SCONJ
ejpam-2741	315	2	every	every	DET
ejpam-2741	315	3	submodule	submodule	NOUN
ejpam-2741	315	4	of	of	ADP
ejpam-2741	315	5	m	m	PROPN
ejpam-2741	315	6	equivalent	equivalent	ADJ
ejpam-2741	315	7	to	to	ADP
ejpam-2741	315	8	an	an	DET
ejpam-2741	315	9	essential	essential	ADJ
ejpam-2741	315	10	weak	weak	ADJ
ejpam-2741	315	11	g	g	NOUN
ejpam-2741	315	12	-	-	PUNCT
ejpam-2741	315	13	supplement	supplement	NOUN
ejpam-2741	315	14	in	in	ADP
ejpam-2741	315	15	m	m	PROPN
ejpam-2741	315	16	by	by	ADP
ejpam-2741	315	17	β∗g	β∗g	NUM
ejpam-2741	315	18	relation	relation	NOUN
ejpam-2741	315	19	,	,	PUNCT
ejpam-2741	315	20	then	then	ADV
ejpam-2741	315	21	m	m	VERB
ejpam-2741	315	22	is	be	AUX
ejpam-2741	315	23	weakly	weakly	ADJ
ejpam-2741	315	24	g	g	NOUN
ejpam-2741	315	25	-	-	PUNCT
ejpam-2741	315	26	supplemented	supplement	VERB
ejpam-2741	315	27	.	.	PUNCT
ejpam-2741	316	1	proof	proof	NOUN
ejpam-2741	316	2	.	.	PUNCT
ejpam-2741	317	1	let	let	VERB
ejpam-2741	317	2	x	x	SYM
ejpam-2741	317	3	≤	≤	ADV
ejpam-2741	317	4	m	m	VERB
ejpam-2741	317	5	.	.	PUNCT
ejpam-2741	318	1	by	by	ADP
ejpam-2741	318	2	hypothesis	hypothesis	NOUN
ejpam-2741	318	3	,	,	PUNCT
ejpam-2741	318	4	there	there	PRON
ejpam-2741	318	5	exists	exist	VERB
ejpam-2741	318	6	an	an	DET
ejpam-2741	318	7	essential	essential	ADJ
ejpam-2741	318	8	weak	weak	ADJ
ejpam-2741	318	9	g	g	NOUN
ejpam-2741	318	10	-	-	PUNCT
ejpam-2741	318	11	supplement	supplement	NOUN
ejpam-2741	318	12	v	v	NOUN
ejpam-2741	318	13	in	in	ADP
ejpam-2741	318	14	m	m	PROPN
ejpam-2741	318	15	such	such	ADJ
ejpam-2741	318	16	that	that	SCONJ
ejpam-2741	318	17	xβ∗gv	xβ∗gv	PROPN
ejpam-2741	318	18	.	.	PUNCT
ejpam-2741	319	1	let	let	VERB
ejpam-2741	319	2	v	v	PART
ejpam-2741	319	3	be	be	AUX
ejpam-2741	319	4	a	a	DET
ejpam-2741	319	5	weak	weak	ADJ
ejpam-2741	319	6	g	g	NOUN
ejpam-2741	319	7	-	-	PUNCT
ejpam-2741	319	8	supplement	supplement	NOUN
ejpam-2741	319	9	of	of	ADP
ejpam-2741	319	10	u	u	NOUN
ejpam-2741	319	11	in	in	ADP
ejpam-2741	319	12	m	m	PROPN
ejpam-2741	319	13	.	.	PUNCT
ejpam-2741	320	1	by	by	ADP
ejpam-2741	320	2	hypothesis	hypothesis	NOUN
ejpam-2741	320	3	,	,	PUNCT
ejpam-2741	320	4	there	there	PRON
ejpam-2741	320	5	exists	exist	VERB
ejpam-2741	320	6	an	an	DET
ejpam-2741	320	7	essential	essential	ADJ
ejpam-2741	320	8	weak	weak	ADJ
ejpam-2741	320	9	g	g	NOUN
ejpam-2741	320	10	-	-	PUNCT
ejpam-2741	320	11	supplement	supplement	NOUN
ejpam-2741	320	12	y	y	NOUN
ejpam-2741	320	13	in	in	ADP
ejpam-2741	320	14	m	m	PRON
ejpam-2741	320	15	such	such	ADJ
ejpam-2741	320	16	that	that	SCONJ
ejpam-2741	320	17	uβ∗gy	uβ∗gy	PROPN
ejpam-2741	320	18	.	.	PUNCT
ejpam-2741	321	1	since	since	SCONJ
ejpam-2741	321	2	v	v	NOUN
ejpam-2741	321	3	is	be	AUX
ejpam-2741	321	4	an	an	DET
ejpam-2741	321	5	essential	essential	ADJ
ejpam-2741	321	6	weak	weak	ADJ
ejpam-2741	321	7	g	g	NOUN
ejpam-2741	321	8	-	-	PUNCT
ejpam-2741	321	9	supplement	supplement	NOUN
ejpam-2741	321	10	of	of	ADP
ejpam-2741	321	11	u	u	NOUN
ejpam-2741	321	12	in	in	ADP
ejpam-2741	321	13	m	m	PROPN
ejpam-2741	321	14	,	,	PUNCT
ejpam-2741	321	15	by	by	ADP
ejpam-2741	321	16	proposition	proposition	NOUN
ejpam-2741	321	17	4	4	NUM
ejpam-2741	321	18	,	,	PUNCT
ejpam-2741	321	19	v	v	NOUN
ejpam-2741	321	20	is	be	AUX
ejpam-2741	321	21	a	a	DET
ejpam-2741	321	22	weak	weak	ADJ
ejpam-2741	321	23	g	g	NOUN
ejpam-2741	321	24	-	-	PUNCT
ejpam-2741	321	25	supplement	supplement	NOUN
ejpam-2741	321	26	of	of	ADP
ejpam-2741	321	27	y	y	PROPN
ejpam-2741	321	28	in	in	ADP
ejpam-2741	321	29	m	m	PROPN
ejpam-2741	321	30	.	.	PUNCT
ejpam-2741	322	1	then	then	ADV
ejpam-2741	322	2	y	y	PROPN
ejpam-2741	322	3	is	be	AUX
ejpam-2741	322	4	an	an	DET
ejpam-2741	322	5	essential	essential	ADJ
ejpam-2741	322	6	weak	weak	ADJ
ejpam-2741	322	7	g	g	NOUN
ejpam-2741	322	8	-	-	PUNCT
ejpam-2741	322	9	supplement	supplement	NOUN
ejpam-2741	322	10	of	of	ADP
ejpam-2741	322	11	v	v	NOUN
ejpam-2741	322	12	in	in	ADP
ejpam-2741	322	13	m	m	PROPN
ejpam-2741	322	14	and	and	CCONJ
ejpam-2741	322	15	since	since	SCONJ
ejpam-2741	322	16	xβ∗gv	xβ∗gv	PROPN
ejpam-2741	322	17	,	,	PUNCT
ejpam-2741	322	18	by	by	ADP
ejpam-2741	322	19	proposition	proposition	NOUN
ejpam-2741	322	20	4	4	NUM
ejpam-2741	322	21	,	,	PUNCT
ejpam-2741	322	22	y	y	PROPN
ejpam-2741	322	23	is	be	AUX
ejpam-2741	322	24	a	a	DET
ejpam-2741	322	25	weak	weak	ADJ
ejpam-2741	322	26	g	g	NOUN
ejpam-2741	322	27	-	-	PUNCT
ejpam-2741	322	28	supplement	supplement	NOUN
ejpam-2741	322	29	of	of	ADP
ejpam-2741	322	30	x	x	PUNCT
ejpam-2741	322	31	in	in	ADP
ejpam-2741	322	32	m	m	PROPN
ejpam-2741	322	33	.	.	PUNCT
ejpam-2741	323	1	hence	hence	ADV
ejpam-2741	323	2	m	m	PROPN
ejpam-2741	323	3	is	be	AUX
ejpam-2741	323	4	weakly	weakly	ADJ
ejpam-2741	323	5	g	g	NOUN
ejpam-2741	323	6	-	-	PUNCT
ejpam-2741	323	7	supplemented	supplement	VERB
ejpam-2741	323	8	.	.	PUNCT
ejpam-2741	324	1	references	reference	NOUN
ejpam-2741	324	2	243	243	NUM
ejpam-2741	324	3	references	reference	NOUN
ejpam-2741	324	4	[	[	X
ejpam-2741	324	5	1	1	NUM
ejpam-2741	324	6	]	]	PUNCT
ejpam-2741	324	7	frank	frank	PROPN
ejpam-2741	324	8	w.	w.	PROPN
ejpam-2741	324	9	anderson	anderson	PROPN
ejpam-2741	324	10	and	and	CCONJ
ejpam-2741	324	11	kent	kent	PROPN
ejpam-2741	324	12	r.	r.	PROPN
ejpam-2741	324	13	fuller	fuller	PROPN
ejpam-2741	324	14	.	.	PUNCT
ejpam-2741	325	1	rings	ring	NOUN
ejpam-2741	325	2	and	and	CCONJ
ejpam-2741	325	3	categories	category	NOUN
ejpam-2741	325	4	of	of	ADP
ejpam-2741	325	5	modules	module	NOUN
ejpam-2741	325	6	(	(	PUNCT
ejpam-2741	325	7	graduate	graduate	NOUN
ejpam-2741	325	8	texts	text	NOUN
ejpam-2741	325	9	in	in	ADP
ejpam-2741	325	10	mathematics	mathematic	NOUN
ejpam-2741	325	11	)	)	PUNCT
ejpam-2741	325	12	.	.	PUNCT
ejpam-2741	326	1	springer	springer	NOUN
ejpam-2741	326	2	,	,	PUNCT
ejpam-2741	326	3	1998	1998	NUM
ejpam-2741	326	4	.	.	PUNCT
ejpam-2741	327	1	[	[	X
ejpam-2741	327	2	2	2	X
ejpam-2741	327	3	]	]	X
ejpam-2741	327	4	g.	g.	PROPN
ejpam-2741	327	5	f.	f.	PROPN
ejpam-2741	327	6	birkenmeier	birkenmeier	PROPN
ejpam-2741	327	7	,	,	PUNCT
ejpam-2741	327	8	f.	f.	PROPN
ejpam-2741	327	9	t.	t.	PROPN
ejpam-2741	327	10	mutlu	mutlu	PROPN
ejpam-2741	327	11	,	,	PUNCT
ejpam-2741	327	12	c.	c.	PROPN
ejpam-2741	327	13	nebiyev	nebiyev	PROPN
ejpam-2741	327	14	,	,	PUNCT
ejpam-2741	327	15	n.	n.	NOUN
ejpam-2741	327	16	sokmez	sokmez	NOUN
ejpam-2741	327	17	,	,	PUNCT
ejpam-2741	327	18	and	and	CCONJ
ejpam-2741	327	19	a.	a.	NOUN
ejpam-2741	327	20	tercan	tercan	PROPN
ejpam-2741	327	21	.	.	PUNCT
ejpam-2741	328	1	goldie*supplemented	goldie*supplemente	VERB
ejpam-2741	328	2	modules	module	NOUN
ejpam-2741	328	3	.	.	PUNCT
ejpam-2741	329	1	glasgow	glasgow	PROPN
ejpam-2741	329	2	mathematical	mathematical	ADJ
ejpam-2741	329	3	journal	journal	NOUN
ejpam-2741	329	4	,	,	PUNCT
ejpam-2741	329	5	52a:41–52	52a:41–52	NUM
ejpam-2741	329	6	,	,	PUNCT
ejpam-2741	329	7	2010	2010	NUM
ejpam-2741	329	8	.	.	PUNCT
ejpam-2741	330	1	[	[	X
ejpam-2741	330	2	3	3	X
ejpam-2741	330	3	]	]	X
ejpam-2741	330	4	f.	f.	PROPN
ejpam-2741	330	5	çallıalp	çallıalp	PROPN
ejpam-2741	330	6	and	and	CCONJ
ejpam-2741	330	7	ü.	ü.	PROPN
ejpam-2741	330	8	tekir	tekir	NOUN
ejpam-2741	330	9	.	.	PUNCT
ejpam-2741	331	1	degişmeli	degişmeli	INTJ
ejpam-2741	331	2	halkalar	halkalar	INTJ
ejpam-2741	331	3	ve	ve	VERB
ejpam-2741	331	4	modüller	modüller	NUM
ejpam-2741	331	5	.	.	PROPN
ejpam-2741	331	6	birsen	birsen	PROPN
ejpam-2741	331	7	yayınevi	yayınevi	PROPN
ejpam-2741	331	8	,	,	PUNCT
ejpam-2741	331	9	i̇stanbul	i̇stanbul	INTJ
ejpam-2741	331	10	,	,	PUNCT
ejpam-2741	331	11	2009	2009	NUM
ejpam-2741	331	12	.	.	PUNCT
ejpam-2741	332	1	[	[	X
ejpam-2741	332	2	4	4	X
ejpam-2741	332	3	]	]	X
ejpam-2741	332	4	john	john	PROPN
ejpam-2741	332	5	clark	clark	PROPN
ejpam-2741	332	6	,	,	PUNCT
ejpam-2741	332	7	christian	christian	PROPN
ejpam-2741	332	8	lomp	lomp	NOUN
ejpam-2741	332	9	,	,	PUNCT
ejpam-2741	332	10	n.	n.	NOUN
ejpam-2741	332	11	vanaja	vanaja	PROPN
ejpam-2741	332	12	,	,	PUNCT
ejpam-2741	332	13	and	and	CCONJ
ejpam-2741	332	14	robert	robert	PROPN
ejpam-2741	332	15	wisbauer	wisbauer	PROPN
ejpam-2741	332	16	.	.	PUNCT
ejpam-2741	333	1	lifting	lift	VERB
ejpam-2741	333	2	modules	module	NOUN
ejpam-2741	333	3	:	:	PUNCT
ejpam-2741	333	4	supplements	supplement	NOUN
ejpam-2741	333	5	and	and	CCONJ
ejpam-2741	333	6	projectivity	projectivity	NOUN
ejpam-2741	333	7	in	in	ADP
ejpam-2741	333	8	module	module	NOUN
ejpam-2741	333	9	theory	theory	NOUN
ejpam-2741	333	10	(	(	PUNCT
ejpam-2741	333	11	frontiers	frontier	NOUN
ejpam-2741	333	12	in	in	ADP
ejpam-2741	333	13	mathematics	mathematic	NOUN
ejpam-2741	333	14	)	)	PUNCT
ejpam-2741	333	15	.	.	PUNCT
ejpam-2741	334	1	birkhäuser	birkhäuser	NOUN
ejpam-2741	334	2	,	,	PUNCT
ejpam-2741	334	3	basel	basel	PROPN
ejpam-2741	334	4	,	,	PUNCT
ejpam-2741	334	5	2006	2006	NUM
ejpam-2741	334	6	edition	edition	NOUN
ejpam-2741	334	7	,	,	PUNCT
ejpam-2741	334	8	8	8	NUM
ejpam-2741	334	9	2006	2006	NUM
ejpam-2741	334	10	.	.	PUNCT
ejpam-2741	335	1	[	[	X
ejpam-2741	335	2	5	5	X
ejpam-2741	335	3	]	]	PUNCT
ejpam-2741	335	4	f.	f.	PROPN
ejpam-2741	335	5	kasch	kasch	PROPN
ejpam-2741	335	6	.	.	PUNCT
ejpam-2741	335	7	modules	module	NOUN
ejpam-2741	335	8	and	and	CCONJ
ejpam-2741	335	9	rings	ring	NOUN
ejpam-2741	335	10	.	.	PUNCT
ejpam-2741	336	1	academic	academic	ADJ
ejpam-2741	336	2	press	press	NOUN
ejpam-2741	336	3	,	,	PUNCT
ejpam-2741	336	4	new	new	PROPN
ejpam-2741	336	5	york	york	PROPN
ejpam-2741	336	6	,	,	PUNCT
ejpam-2741	336	7	1982	1982	NUM
ejpam-2741	336	8	.	.	PUNCT
ejpam-2741	337	1	[	[	X
ejpam-2741	337	2	6	6	NUM
ejpam-2741	337	3	]	]	PUNCT
ejpam-2741	337	4	b.	b.	PROPN
ejpam-2741	337	5	koşar	koşar	PROPN
ejpam-2741	337	6	,	,	PUNCT
ejpam-2741	337	7	c.	c.	PROPN
ejpam-2741	337	8	nebiyev	nebiyev	PROPN
ejpam-2741	337	9	,	,	PUNCT
ejpam-2741	337	10	and	and	CCONJ
ejpam-2741	337	11	n.	n.	PROPN
ejpam-2741	337	12	sökmez	sökmez	NOUN
ejpam-2741	337	13	.	.	PUNCT
ejpam-2741	338	1	g	g	NOUN
ejpam-2741	338	2	-	-	PUNCT
ejpam-2741	338	3	supplemented	supplement	VERB
ejpam-2741	338	4	modules	module	NOUN
ejpam-2741	338	5	.	.	PUNCT
ejpam-2741	339	1	ukrainian	ukrainian	ADJ
ejpam-2741	339	2	mathematical	mathematical	ADJ
ejpam-2741	339	3	journal	journal	NOUN
ejpam-2741	339	4	,	,	PUNCT
ejpam-2741	339	5	67(6):861–864	67(6):861–864	PROPN
ejpam-2741	339	6	,	,	PUNCT
ejpam-2741	339	7	2015	2015	NUM
ejpam-2741	339	8	.	.	PUNCT
ejpam-2741	340	1	[	[	X
ejpam-2741	340	2	7	7	X
ejpam-2741	340	3	]	]	X
ejpam-2741	340	4	c.	c.	NOUN
ejpam-2741	340	5	nebiyev	nebiyev	PROPN
ejpam-2741	340	6	and	and	CCONJ
ejpam-2741	340	7	h.	h.	PROPN
ejpam-2741	340	8	h.	h.	PROPN
ejpam-2741	340	9	ökten	ökten	PROPN
ejpam-2741	340	10	.	.	PUNCT
ejpam-2741	341	1	beta	beta	PROPN
ejpam-2741	341	2	star	star	PROPN
ejpam-2741	341	3	relation	relation	NOUN
ejpam-2741	341	4	on	on	ADP
ejpam-2741	341	5	lattices	lattice	NOUN
ejpam-2741	341	6	.	.	PUNCT
ejpam-2741	342	1	miskolc	miskolc	ADJ
ejpam-2741	342	2	mathematical	mathematical	ADJ
ejpam-2741	342	3	notes	note	NOUN
ejpam-2741	342	4	,	,	PUNCT
ejpam-2741	342	5	2017	2017	NUM
ejpam-2741	342	6	.	.	PUNCT
ejpam-2741	343	1	(	(	PUNCT
ejpam-2741	343	2	accepted	accept	VERB
ejpam-2741	343	3	)	)	PUNCT
ejpam-2741	343	4	.	.	PUNCT
ejpam-2741	344	1	[	[	X
ejpam-2741	344	2	8	8	X
ejpam-2741	344	3	]	]	X
ejpam-2741	344	4	c.	c.	NOUN
ejpam-2741	344	5	nebiyev	nebiyev	PROPN
ejpam-2741	344	6	and	and	CCONJ
ejpam-2741	344	7	h.	h.	PROPN
ejpam-2741	344	8	h.	h.	PROPN
ejpam-2741	344	9	ökten	ökten	PROPN
ejpam-2741	344	10	.	.	PUNCT
ejpam-2741	345	1	weakly	weakly	ADJ
ejpam-2741	345	2	g	g	NOUN
ejpam-2741	345	3	-	-	PUNCT
ejpam-2741	345	4	supplemented	supplement	VERB
ejpam-2741	345	5	modules	module	NOUN
ejpam-2741	345	6	.	.	PUNCT
ejpam-2741	346	1	european	european	ADJ
ejpam-2741	346	2	journal	journal	PROPN
ejpam-2741	346	3	of	of	ADP
ejpam-2741	346	4	pure	pure	ADJ
ejpam-2741	346	5	and	and	CCONJ
ejpam-2741	346	6	applied	applied	ADJ
ejpam-2741	346	7	mathematics	mathematic	NOUN
ejpam-2741	346	8	,	,	PUNCT
ejpam-2741	346	9	10(3):521–528	10(3):521–528	PROPN
ejpam-2741	346	10	,	,	PUNCT
ejpam-2741	346	11	2017	2017	NUM
ejpam-2741	346	12	.	.	PUNCT
ejpam-2741	347	1	[	[	X
ejpam-2741	347	2	9	9	NUM
ejpam-2741	347	3	]	]	X
ejpam-2741	347	4	n.	n.	NOUN
ejpam-2741	347	5	sökmez	sökmez	PROPN
ejpam-2741	347	6	,	,	PUNCT
ejpam-2741	347	7	b.	b.	PROPN
ejpam-2741	347	8	koşar	koşar	PROPN
ejpam-2741	347	9	,	,	PUNCT
ejpam-2741	347	10	and	and	CCONJ
ejpam-2741	347	11	c.	c.	PROPN
ejpam-2741	347	12	nebiyev	nebiyev	PROPN
ejpam-2741	347	13	.	.	PUNCT
ejpam-2741	348	1	genelleştirilmiş	genelleştirilmiş	PROPN
ejpam-2741	348	2	küçük	küçük	PROPN
ejpam-2741	348	3	alt	alt	VERB
ejpam-2741	348	4	modüller	modüller	PROPN
ejpam-2741	348	5	.	.	PUNCT
ejpam-2741	349	1	in	in	ADP
ejpam-2741	349	2	xiii	xiii	PROPN
ejpam-2741	349	3	.	.	PUNCT
ejpam-2741	350	1	ulusal	ulusal	PROPN
ejpam-2741	350	2	matematik	matematik	PROPN
ejpam-2741	350	3	sempozyumu	sempozyumu	PROPN
ejpam-2741	350	4	,	,	PUNCT
ejpam-2741	350	5	kayseri	kayseri	PROPN
ejpam-2741	350	6	,	,	PUNCT
ejpam-2741	350	7	2010	2010	NUM
ejpam-2741	350	8	.	.	PUNCT
ejpam-2741	351	1	erciyes	erciye	NOUN
ejpam-2741	351	2	üniversitesi	üniversitesi	PROPN
ejpam-2741	351	3	.	.	PUNCT
ejpam-2741	352	1	[	[	X
ejpam-2741	352	2	10	10	NUM
ejpam-2741	352	3	]	]	X
ejpam-2741	352	4	r.	r.	PROPN
ejpam-2741	352	5	wisbauer	wisbauer	NOUN
ejpam-2741	352	6	.	.	PUNCT
ejpam-2741	353	1	foundations	foundation	NOUN
ejpam-2741	353	2	of	of	ADP
ejpam-2741	353	3	module	module	NOUN
ejpam-2741	353	4	and	and	CCONJ
ejpam-2741	353	5	ring	ring	NOUN
ejpam-2741	353	6	theory	theory	NOUN
ejpam-2741	353	7	.	.	PUNCT
ejpam-2741	354	1	gordon	gordon	PROPN
ejpam-2741	354	2	and	and	CCONJ
ejpam-2741	354	3	breach	breach	PROPN
ejpam-2741	354	4	,	,	PUNCT
ejpam-2741	354	5	philadelphia	philadelphia	PROPN
ejpam-2741	354	6	,	,	PUNCT
ejpam-2741	354	7	1991	1991	NUM
ejpam-2741	354	8	.	.	PUNCT
ejpam-2741	355	1	[	[	X
ejpam-2741	355	2	11	11	NUM
ejpam-2741	355	3	]	]	PUNCT
ejpam-2741	355	4	d.	d.	PROPN
ejpam-2741	355	5	x.	x.	PROPN
ejpam-2741	355	6	zhou	zhou	PROPN
ejpam-2741	355	7	and	and	CCONJ
ejpam-2741	355	8	x.	x.	PROPN
ejpam-2741	355	9	r.	r.	PROPN
ejpam-2741	355	10	zhang	zhang	PROPN
ejpam-2741	355	11	.	.	PUNCT
ejpam-2741	356	1	small	small	ADJ
ejpam-2741	356	2	-	-	PUNCT
ejpam-2741	356	3	essential	essential	ADJ
ejpam-2741	356	4	submodules	submodule	NOUN
ejpam-2741	356	5	and	and	CCONJ
ejpam-2741	356	6	morita	morita	PROPN
ejpam-2741	356	7	duality	duality	PROPN
ejpam-2741	356	8	.	.	PUNCT
ejpam-2741	357	1	southeast	southeast	ADJ
ejpam-2741	357	2	asian	asian	ADJ
ejpam-2741	357	3	bulletin	bulletin	NOUN
ejpam-2741	357	4	of	of	ADP
ejpam-2741	357	5	mathematics	mathematic	NOUN
ejpam-2741	357	6	,	,	PUNCT
ejpam-2741	357	7	35:1051–1062	35:1051–1062	PROPN
ejpam-2741	357	8	,	,	PUNCT
ejpam-2741	357	9	2011	2011	NUM
ejpam-2741	357	10	.	.	PUNCT