id	sid	tid	token	lemma	pos
ejpam-2095	1	1	main.dvi	main.dvi	PROPN
ejpam-2095	1	2	european	european	PROPN
ejpam-2095	1	3	journal	journal	PROPN
ejpam-2095	1	4	of	of	ADP
ejpam-2095	1	5	pure	pure	ADJ
ejpam-2095	1	6	and	and	CCONJ
ejpam-2095	1	7	applied	apply	VERB
ejpam-2095	1	8	mathematics	mathematic	NOUN
ejpam-2095	1	9	vol	vol	NOUN
ejpam-2095	1	10	.	.	PUNCT
ejpam-2095	2	1	7	7	NUM
ejpam-2095	2	2	,	,	PUNCT
ejpam-2095	2	3	no	no	INTJ
ejpam-2095	2	4	.	.	NOUN
ejpam-2095	2	5	2	2	NUM
ejpam-2095	2	6	,	,	PUNCT
ejpam-2095	2	7	2014	2014	NUM
ejpam-2095	2	8	,	,	PUNCT
ejpam-2095	2	9	201	201	NUM
ejpam-2095	2	10	-	-	SYM
ejpam-2095	2	11	209	209	NUM
ejpam-2095	2	12	issn	issn	PROPN
ejpam-2095	2	13	1307	1307	NUM
ejpam-2095	2	14	-	-	SYM
ejpam-2095	2	15	5543	5543	NUM
ejpam-2095	2	16	–	–	PUNCT
ejpam-2095	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2095	2	18	primary	primary	ADJ
ejpam-2095	2	19	decomposition	decomposition	NOUN
ejpam-2095	2	20	in	in	ADP
ejpam-2095	2	21	lattice	lattice	NOUN
ejpam-2095	2	22	modules	module	NOUN
ejpam-2095	3	1	c	c	PROPN
ejpam-2095	3	2	s	s	PART
ejpam-2095	3	3	manjarekar	manjarekar	NOUN
ejpam-2095	3	4	1	1	NUM
ejpam-2095	3	5	,	,	PUNCT
ejpam-2095	3	6	u	u	NOUN
ejpam-2095	3	7	n	n	NOUN
ejpam-2095	3	8	kandale2,∗	kandale2,∗	VERB
ejpam-2095	3	9	1	1	NUM
ejpam-2095	3	10	department	department	NOUN
ejpam-2095	3	11	of	of	ADP
ejpam-2095	3	12	mathematics	mathematics	PROPN
ejpam-2095	3	13	,	,	PUNCT
ejpam-2095	3	14	shivaji	shivaji	PROPN
ejpam-2095	3	15	university	university	PROPN
ejpam-2095	3	16	,	,	PUNCT
ejpam-2095	3	17	kolhapur	kolhapur	PROPN
ejpam-2095	3	18	,	,	PUNCT
ejpam-2095	3	19	maharashtra	maharashtra	PROPN
ejpam-2095	3	20	,	,	PUNCT
ejpam-2095	3	21	india	india	PROPN
ejpam-2095	3	22	2	2	NUM
ejpam-2095	3	23	general	general	ADJ
ejpam-2095	3	24	engineering	engineering	NOUN
ejpam-2095	3	25	department	department	PROPN
ejpam-2095	3	26	,	,	PUNCT
ejpam-2095	3	27	sharad	sharad	PROPN
ejpam-2095	3	28	institute	institute	PROPN
ejpam-2095	3	29	of	of	ADP
ejpam-2095	3	30	technology	technology	PROPN
ejpam-2095	3	31	college	college	PROPN
ejpam-2095	3	32	of	of	ADP
ejpam-2095	3	33	engineering	engineering	PROPN
ejpam-2095	3	34	,	,	PUNCT
ejpam-2095	3	35	shivaji	shivaji	PROPN
ejpam-2095	3	36	university	university	PROPN
ejpam-2095	3	37	,	,	PUNCT
ejpam-2095	3	38	kolhapur	kolhapur	PROPN
ejpam-2095	3	39	,	,	PUNCT
ejpam-2095	3	40	india	india	PROPN
ejpam-2095	3	41	abstract	abstract	NOUN
ejpam-2095	3	42	.	.	PUNCT
ejpam-2095	4	1	in	in	ADP
ejpam-2095	4	2	this	this	DET
ejpam-2095	4	3	paper	paper	NOUN
ejpam-2095	4	4	,	,	PUNCT
ejpam-2095	4	5	we	we	PRON
ejpam-2095	4	6	study	study	VERB
ejpam-2095	4	7	primary	primary	ADJ
ejpam-2095	4	8	decomposition	decomposition	NOUN
ejpam-2095	4	9	of	of	ADP
ejpam-2095	4	10	elements	element	NOUN
ejpam-2095	4	11	in	in	ADP
ejpam-2095	4	12	lattice	lattice	NOUN
ejpam-2095	4	13	modules	module	NOUN
ejpam-2095	4	14	.	.	PUNCT
ejpam-2095	5	1	a	a	DET
ejpam-2095	5	2	necessary	necessary	ADJ
ejpam-2095	5	3	and	and	CCONJ
ejpam-2095	5	4	sufficient	sufficient	ADJ
ejpam-2095	5	5	condition	condition	NOUN
ejpam-2095	5	6	for	for	ADP
ejpam-2095	5	7	a	a	DET
ejpam-2095	5	8	prime	prime	ADJ
ejpam-2095	5	9	element	element	NOUN
ejpam-2095	5	10	p	p	NOUN
ejpam-2095	5	11	of	of	ADP
ejpam-2095	5	12	a	a	DET
ejpam-2095	5	13	multiplicative	multiplicative	ADJ
ejpam-2095	5	14	lattice	lattice	NOUN
ejpam-2095	5	15	l	l	NOUN
ejpam-2095	5	16	to	to	PART
ejpam-2095	5	17	be	be	AUX
ejpam-2095	5	18	equal	equal	ADJ
ejpam-2095	5	19	to	to	ADP
ejpam-2095	5	20	some	some	DET
ejpam-2095	5	21	associated	associated	ADJ
ejpam-2095	5	22	prime	prime	NOUN
ejpam-2095	5	23	of	of	ADP
ejpam-2095	5	24	an	an	DET
ejpam-2095	5	25	element	element	NOUN
ejpam-2095	5	26	in	in	ADP
ejpam-2095	5	27	a	a	DET
ejpam-2095	5	28	lattice	lattice	NOUN
ejpam-2095	5	29	module	module	NOUN
ejpam-2095	5	30	having	have	VERB
ejpam-2095	5	31	primary	primary	ADJ
ejpam-2095	5	32	decomposition	decomposition	NOUN
ejpam-2095	5	33	is	be	AUX
ejpam-2095	5	34	obtained	obtain	VERB
ejpam-2095	5	35	.	.	PUNCT
ejpam-2095	6	1	2010	2010	NUM
ejpam-2095	6	2	mathematics	mathematic	NOUN
ejpam-2095	6	3	subject	subject	NOUN
ejpam-2095	6	4	classifications	classification	NOUN
ejpam-2095	6	5	:	:	PUNCT
ejpam-2095	6	6	13a99	13a99	NUM
ejpam-2095	6	7	key	key	ADJ
ejpam-2095	6	8	words	word	NOUN
ejpam-2095	6	9	and	and	CCONJ
ejpam-2095	6	10	phrases	phrase	NOUN
ejpam-2095	6	11	:	:	PUNCT
ejpam-2095	6	12	prime	prime	ADJ
ejpam-2095	6	13	element	element	NOUN
ejpam-2095	6	14	,	,	PUNCT
ejpam-2095	6	15	primary	primary	ADJ
ejpam-2095	6	16	element	element	NOUN
ejpam-2095	6	17	,	,	PUNCT
ejpam-2095	6	18	lattice	lattice	NOUN
ejpam-2095	6	19	modules	module	NOUN
ejpam-2095	6	20	,	,	PUNCT
ejpam-2095	6	21	primary	primary	ADJ
ejpam-2095	6	22	decomposition	decomposition	NOUN
ejpam-2095	6	23	1	1	NUM
ejpam-2095	6	24	.	.	PUNCT
ejpam-2095	7	1	introduction	introduction	NOUN
ejpam-2095	7	2	a	a	DET
ejpam-2095	7	3	multiplicative	multiplicative	ADJ
ejpam-2095	7	4	lattice	lattice	NOUN
ejpam-2095	7	5	l	l	NOUN
ejpam-2095	7	6	is	be	AUX
ejpam-2095	7	7	a	a	DET
ejpam-2095	7	8	complete	complete	ADJ
ejpam-2095	7	9	lattice	lattice	NOUN
ejpam-2095	7	10	provided	provide	VERB
ejpam-2095	7	11	with	with	ADP
ejpam-2095	7	12	commutative	commutative	ADJ
ejpam-2095	7	13	,	,	PUNCT
ejpam-2095	7	14	associative	associative	ADJ
ejpam-2095	7	15	and	and	CCONJ
ejpam-2095	7	16	join	join	VERB
ejpam-2095	7	17	distributive	distributive	ADJ
ejpam-2095	7	18	multiplication	multiplication	NOUN
ejpam-2095	7	19	in	in	ADP
ejpam-2095	7	20	which	which	PRON
ejpam-2095	7	21	the	the	DET
ejpam-2095	7	22	largest	large	ADJ
ejpam-2095	7	23	element	element	NOUN
ejpam-2095	7	24	1	1	NUM
ejpam-2095	7	25	acts	act	NOUN
ejpam-2095	7	26	as	as	ADP
ejpam-2095	7	27	a	a	DET
ejpam-2095	7	28	multiplicative	multiplicative	ADJ
ejpam-2095	7	29	identity	identity	NOUN
ejpam-2095	7	30	.	.	PUNCT
ejpam-2095	8	1	an	an	DET
ejpam-2095	8	2	element	element	NOUN
ejpam-2095	8	3	a	a	DET
ejpam-2095	8	4	∈	∈	PROPN
ejpam-2095	8	5	l	l	NOUN
ejpam-2095	8	6	is	be	AUX
ejpam-2095	8	7	called	call	VERB
ejpam-2095	8	8	proper	proper	ADJ
ejpam-2095	8	9	if	if	SCONJ
ejpam-2095	8	10	a	a	DET
ejpam-2095	8	11	<	<	X
ejpam-2095	8	12	1	1	NUM
ejpam-2095	8	13	.	.	PUNCT
ejpam-2095	9	1	a	a	DET
ejpam-2095	9	2	proper	proper	ADJ
ejpam-2095	9	3	element	element	NOUN
ejpam-2095	9	4	p	p	NOUN
ejpam-2095	9	5	of	of	ADP
ejpam-2095	9	6	l	l	NOUN
ejpam-2095	9	7	is	be	AUX
ejpam-2095	9	8	said	say	VERB
ejpam-2095	9	9	to	to	PART
ejpam-2095	9	10	be	be	AUX
ejpam-2095	9	11	prime	prime	ADJ
ejpam-2095	9	12	if	if	SCONJ
ejpam-2095	9	13	ab	ab	PROPN
ejpam-2095	9	14	≤	≤	PROPN
ejpam-2095	9	15	p	p	PROPN
ejpam-2095	9	16	implies	imply	VERB
ejpam-2095	9	17	a	a	DET
ejpam-2095	9	18	≤	≤	NUM
ejpam-2095	9	19	p	p	NOUN
ejpam-2095	9	20	or	or	CCONJ
ejpam-2095	9	21	b	b	NOUN
ejpam-2095	9	22	≤	≤	NOUN
ejpam-2095	10	1	p.	p.	NOUN
ejpam-2095	10	2	if	if	SCONJ
ejpam-2095	10	3	a	a	DET
ejpam-2095	10	4	∈	∈	PROPN
ejpam-2095	10	5	l	l	NOUN
ejpam-2095	10	6	,	,	PUNCT
ejpam-2095	10	7	b	b	PROPN
ejpam-2095	10	8	∈	∈	PROPN
ejpam-2095	10	9	l	l	NOUN
ejpam-2095	10	10	,	,	PUNCT
ejpam-2095	10	11	(	(	PUNCT
ejpam-2095	10	12	a	a	DET
ejpam-2095	10	13	:	:	PUNCT
ejpam-2095	10	14	b	b	X
ejpam-2095	10	15	)	)	PUNCT
ejpam-2095	10	16	is	be	AUX
ejpam-2095	10	17	the	the	DET
ejpam-2095	10	18	join	join	NOUN
ejpam-2095	10	19	of	of	ADP
ejpam-2095	10	20	all	all	DET
ejpam-2095	10	21	elements	element	NOUN
ejpam-2095	10	22	c	c	NOUN
ejpam-2095	10	23	in	in	ADP
ejpam-2095	10	24	l	l	NOUN
ejpam-2095	10	25	such	such	ADJ
ejpam-2095	10	26	that	that	SCONJ
ejpam-2095	10	27	cb	cb	PROPN
ejpam-2095	10	28	≤	≤	PROPN
ejpam-2095	10	29	a.	a.	NOUN
ejpam-2095	10	30	a	a	DET
ejpam-2095	10	31	proper	proper	ADJ
ejpam-2095	10	32	element	element	NOUN
ejpam-2095	10	33	p	p	NOUN
ejpam-2095	10	34	of	of	ADP
ejpam-2095	10	35	l	l	NOUN
ejpam-2095	10	36	is	be	AUX
ejpam-2095	10	37	said	say	VERB
ejpam-2095	10	38	to	to	PART
ejpam-2095	10	39	be	be	AUX
ejpam-2095	10	40	primary	primary	ADJ
ejpam-2095	10	41	if	if	SCONJ
ejpam-2095	10	42	ab	ab	PROPN
ejpam-2095	10	43	≤	≤	PROPN
ejpam-2095	10	44	p	p	PROPN
ejpam-2095	10	45	implies	imply	VERB
ejpam-2095	10	46	a	a	DET
ejpam-2095	10	47	≤	≤	NUM
ejpam-2095	10	48	p	p	NOUN
ejpam-2095	10	49	or	or	CCONJ
ejpam-2095	10	50	bn	bn	NOUN
ejpam-2095	10	51	≤	≤	NOUN
ejpam-2095	10	52	p	p	NOUN
ejpam-2095	10	53	for	for	ADP
ejpam-2095	10	54	some	some	DET
ejpam-2095	10	55	positive	positive	ADJ
ejpam-2095	10	56	integer	integer	NOUN
ejpam-2095	10	57	n.	n.	NOUN
ejpam-2095	10	58	if	if	SCONJ
ejpam-2095	10	59	a	a	DET
ejpam-2095	10	60	∈	∈	PROPN
ejpam-2095	10	61	l	l	NOUN
ejpam-2095	10	62	,	,	PUNCT
ejpam-2095	10	63	the	the	DET
ejpam-2095	10	64	radical	radical	NOUN
ejpam-2095	10	65	of	of	ADP
ejpam-2095	10	66	a	a	DET
ejpam-2095	10	67	denoted	denote	VERB
ejpam-2095	10	68	by	by	ADP
ejpam-2095	10	69	p	p	PROPN
ejpam-2095	11	1	a	a	DET
ejpam-2095	11	2	=	=	SYM
ejpam-2095	11	3	∨{x	∨{x	NOUN
ejpam-2095	11	4	∈	∈	NOUN
ejpam-2095	11	5	l	l	NOUN
ejpam-2095	12	1	|	|	ADV
ejpam-2095	12	2	xn	xn	PROPN
ejpam-2095	12	3	¶	¶	PROPN
ejpam-2095	12	4	a	a	PROPN
ejpam-2095	12	5	,	,	PUNCT
ejpam-2095	12	6	n	n	NOUN
ejpam-2095	12	7	∈	∈	NOUN
ejpam-2095	12	8	z+	z+	NUM
ejpam-2095	12	9	}	}	PUNCT
ejpam-2095	12	10	.	.	PUNCT
ejpam-2095	13	1	an	an	DET
ejpam-2095	13	2	element	element	NOUN
ejpam-2095	13	3	a	a	DET
ejpam-2095	13	4	∈	∈	PROPN
ejpam-2095	13	5	l	l	NOUN
ejpam-2095	13	6	is	be	AUX
ejpam-2095	13	7	called	call	VERB
ejpam-2095	13	8	compact	compact	ADJ
ejpam-2095	13	9	if	if	SCONJ
ejpam-2095	13	10	a	a	DET
ejpam-2095	13	11	¶	¶	PROPN
ejpam-2095	13	12	∨	∨	NOUN
ejpam-2095	13	13	α	α	PROPN
ejpam-2095	14	1	b	b	PROPN
ejpam-2095	14	2	α	α	PROPN
ejpam-2095	14	3	implies	imply	VERB
ejpam-2095	14	4	a	a	DET
ejpam-2095	14	5	¶	¶	PROPN
ejpam-2095	14	6	b	b	PROPN
ejpam-2095	14	7	α1	α1	PROPN
ejpam-2095	14	8	∨	∨	NUM
ejpam-2095	14	9	b	b	PRON
ejpam-2095	14	10	α2	α2	ADJ
ejpam-2095	14	11	∨	∨	NOUN
ejpam-2095	14	12	.	.	PUNCT
ejpam-2095	14	13	.	.	PUNCT
ejpam-2095	14	14	.	.	PUNCT
ejpam-2095	15	1	∨	∨	NUM
ejpam-2095	15	2	b	b	SYM
ejpam-2095	15	3	αn	αn	NOUN
ejpam-2095	15	4	for	for	ADP
ejpam-2095	15	5	some	some	DET
ejpam-2095	15	6	finite	finite	NOUN
ejpam-2095	15	7	subset	subset	NOUN
ejpam-2095	15	8	{	{	PUNCT
ejpam-2095	15	9	α1,α2	α1,α2	PROPN
ejpam-2095	15	10	,	,	PUNCT
ejpam-2095	15	11	.	.	PUNCT
ejpam-2095	15	12	.	.	PUNCT
ejpam-2095	16	1	.	.	PUNCT
ejpam-2095	17	1	,	,	PUNCT
ejpam-2095	17	2	αn	αn	NOUN
ejpam-2095	17	3	}	}	PUNCT
ejpam-2095	17	4	.	.	PUNCT
ejpam-2095	18	1	throughout	throughout	ADP
ejpam-2095	18	2	this	this	DET
ejpam-2095	18	3	paper	paper	NOUN
ejpam-2095	18	4	,	,	PUNCT
ejpam-2095	18	5	l	l	NOUN
ejpam-2095	18	6	denotes	denote	VERB
ejpam-2095	18	7	a	a	DET
ejpam-2095	18	8	multiplicative	multiplicative	ADJ
ejpam-2095	18	9	lattice	lattice	NOUN
ejpam-2095	18	10	which	which	PRON
ejpam-2095	18	11	satisfies	satisfy	VERB
ejpam-2095	18	12	the	the	DET
ejpam-2095	18	13	acc	acc	NOUN
ejpam-2095	18	14	so	so	SCONJ
ejpam-2095	18	15	that	that	SCONJ
ejpam-2095	18	16	each	each	DET
ejpam-2095	18	17	element	element	NOUN
ejpam-2095	18	18	of	of	ADP
ejpam-2095	18	19	l	l	NOUN
ejpam-2095	18	20	is	be	AUX
ejpam-2095	18	21	compact	compact	ADJ
ejpam-2095	18	22	.	.	PUNCT
ejpam-2095	19	1	if	if	SCONJ
ejpam-2095	19	2	q	q	NOUN
ejpam-2095	19	3	is	be	AUX
ejpam-2095	19	4	a	a	DET
ejpam-2095	19	5	primary	primary	ADJ
ejpam-2095	19	6	element	element	NOUN
ejpam-2095	19	7	of	of	ADP
ejpam-2095	19	8	l	l	PROPN
ejpam-2095	19	9	then	then	ADV
ejpam-2095	20	1	pq	pq	PROPN
ejpam-2095	20	2	=	=	SYM
ejpam-2095	20	3	∨{x	∨{x	PROPN
ejpam-2095	20	4	∈	∈	PROPN
ejpam-2095	20	5	l	l	NOUN
ejpam-2095	21	1	|	|	ADV
ejpam-2095	21	2	xn	xn	PROPN
ejpam-2095	21	3	¶	¶	PROPN
ejpam-2095	21	4	q	q	PROPN
ejpam-2095	21	5	,	,	PUNCT
ejpam-2095	21	6	for	for	ADP
ejpam-2095	21	7	some	some	DET
ejpam-2095	21	8	integer	integer	NOUN
ejpam-2095	21	9	n	n	CCONJ
ejpam-2095	21	10	}	}	PUNCT
ejpam-2095	21	11	is	be	AUX
ejpam-2095	21	12	a	a	DET
ejpam-2095	21	13	prime	prime	ADJ
ejpam-2095	21	14	element	element	NOUN
ejpam-2095	21	15	containing	contain	VERB
ejpam-2095	21	16	q.	q.	NOUN
ejpam-2095	21	17	it	it	PRON
ejpam-2095	21	18	is	be	AUX
ejpam-2095	21	19	easily	easily	ADV
ejpam-2095	21	20	verified	verify	VERB
ejpam-2095	21	21	that	that	SCONJ
ejpam-2095	21	22	,	,	PUNCT
ejpam-2095	21	23	pq	pq	INTJ
ejpam-2095	21	24	is	be	AUX
ejpam-2095	21	25	a	a	DET
ejpam-2095	21	26	minimal	minimal	ADJ
ejpam-2095	21	27	prime	prime	NOUN
ejpam-2095	21	28	containing	contain	VERB
ejpam-2095	21	29	q	q	NOUN
ejpam-2095	22	1	[	[	X
ejpam-2095	22	2	2	2	NUM
ejpam-2095	22	3	]	]	PUNCT
ejpam-2095	22	4	.	.	PUNCT
ejpam-2095	23	1	the	the	DET
ejpam-2095	23	2	prime	prime	PROPN
ejpam-2095	23	3	pq	pq	PROPN
ejpam-2095	23	4	which	which	PRON
ejpam-2095	23	5	is	be	AUX
ejpam-2095	23	6	same	same	ADJ
ejpam-2095	23	7	as	as	SCONJ
ejpam-2095	23	8	p	p	PROPN
ejpam-2095	23	9	q	q	PROPN
ejpam-2095	23	10	is	be	AUX
ejpam-2095	23	11	called	call	VERB
ejpam-2095	23	12	the	the	DET
ejpam-2095	23	13	prime	prime	NOUN
ejpam-2095	23	14	associated	associate	VERB
ejpam-2095	23	15	with	with	ADP
ejpam-2095	23	16	q	q	PROPN
ejpam-2095	23	17	and	and	CCONJ
ejpam-2095	23	18	has	have	VERB
ejpam-2095	23	19	the	the	DET
ejpam-2095	23	20	properties	property	NOUN
ejpam-2095	23	21	,	,	PUNCT
ejpam-2095	23	22	pk	pk	NOUN
ejpam-2095	23	23	q	q	PROPN
ejpam-2095	23	24	¶	¶	PROPN
ejpam-2095	23	25	q	q	PROPN
ejpam-2095	23	26	¶	¶	PROPN
ejpam-2095	23	27	pq	pq	NOUN
ejpam-2095	23	28	for	for	ADP
ejpam-2095	23	29	some	some	DET
ejpam-2095	23	30	integer	integer	NOUN
ejpam-2095	23	31	k	k	PROPN
ejpam-2095	23	32	and	and	CCONJ
ejpam-2095	23	33	ab	ab	PROPN
ejpam-2095	23	34	¶	¶	PROPN
ejpam-2095	23	35	q	q	PROPN
ejpam-2095	23	36	implies	imply	VERB
ejpam-2095	23	37	a	a	DET
ejpam-2095	23	38	¶	¶	PROPN
ejpam-2095	23	39	q	q	NOUN
ejpam-2095	23	40	or	or	CCONJ
ejpam-2095	23	41	b	b	NOUN
ejpam-2095	23	42	¶	¶	PROPN
ejpam-2095	23	43	pq	pq	PROPN
ejpam-2095	23	44	.	.	PUNCT
ejpam-2095	24	1	an	an	DET
ejpam-2095	24	2	element	element	NOUN
ejpam-2095	24	3	a	a	PRON
ejpam-2095	24	4	is	be	AUX
ejpam-2095	24	5	said	say	VERB
ejpam-2095	24	6	to	to	PART
ejpam-2095	24	7	have	have	VERB
ejpam-2095	24	8	a	a	DET
ejpam-2095	24	9	primary	primary	ADJ
ejpam-2095	24	10	decomposition	decomposition	NOUN
ejpam-2095	24	11	if	if	SCONJ
ejpam-2095	24	12	there	there	PRON
ejpam-2095	24	13	exist	exist	VERB
ejpam-2095	24	14	primary	primary	ADJ
ejpam-2095	24	15	elements	element	NOUN
ejpam-2095	24	16	q1,q2	q1,q2	PROPN
ejpam-2095	24	17	,	,	PUNCT
ejpam-2095	24	18	.	.	PUNCT
ejpam-2095	24	19	.	.	PUNCT
ejpam-2095	24	20	.	.	PUNCT
ejpam-2095	25	1	,	,	PUNCT
ejpam-2095	25	2	qn	qn	INTJ
ejpam-2095	25	3	such	such	ADJ
ejpam-2095	25	4	that	that	SCONJ
ejpam-2095	25	5	a	a	DET
ejpam-2095	25	6	=	=	PROPN
ejpam-2095	25	7	q1	q1	NOUN
ejpam-2095	25	8	∧	∧	PROPN
ejpam-2095	25	9	q2	q2	PROPN
ejpam-2095	25	10	∧	∧	PROPN
ejpam-2095	25	11	.	.	PUNCT
ejpam-2095	25	12	.	.	PUNCT
ejpam-2095	25	13	.	.	PUNCT
ejpam-2095	26	1	∧	∧	NOUN
ejpam-2095	26	2	qn	qn	INTJ
ejpam-2095	26	3	.	.	PUNCT
ejpam-2095	27	1	if	if	SCONJ
ejpam-2095	27	2	some	some	DET
ejpam-2095	27	3	qi	qi	NOUN
ejpam-2095	27	4	contains	contain	VERB
ejpam-2095	27	5	the	the	DET
ejpam-2095	27	6	meet	meet	NOUN
ejpam-2095	27	7	of	of	ADP
ejpam-2095	27	8	remaining	remain	VERB
ejpam-2095	27	9	ones	one	NOUN
ejpam-2095	27	10	∗corresponding	∗corresponde	VERB
ejpam-2095	27	11	author	author	NOUN
ejpam-2095	27	12	.	.	PUNCT
ejpam-2095	28	1	email	email	NOUN
ejpam-2095	28	2	addresses	address	NOUN
ejpam-2095	28	3	:	:	PUNCT
ejpam-2095	28	4	smanjrekar	smanjrekar	PROPN
ejpam-2095	28	5	�	�	PROPN
ejpam-2095	28	6	yahoo	yahoo	PROPN
ejpam-2095	28	7	.	.	PUNCT
ejpam-2095	29	1	o.in	o.in	PROPN
ejpam-2095	29	2	(	(	PUNCT
ejpam-2095	29	3	c.	c.	PROPN
ejpam-2095	29	4	manjarekar	manjarekar	PROPN
ejpam-2095	29	5	)	)	PUNCT
ejpam-2095	29	6	,	,	PUNCT
ejpam-2095	29	7	ujwalabiraje	ujwalabiraje	ADJ
ejpam-2095	29	8	�	�	PROPN
ejpam-2095	29	9	gmail	gmail	NOUN
ejpam-2095	29	10	.	.	PUNCT
ejpam-2095	30	1	om	om	PROPN
ejpam-2095	30	2	(	(	PUNCT
ejpam-2095	30	3	u.	u.	PROPN
ejpam-2095	30	4	kandale	kandale	PROPN
ejpam-2095	30	5	)	)	PUNCT
ejpam-2095	30	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2095	31	1	201	201	NUM
ejpam-2095	31	2	c	c	X
ejpam-2095	31	3	©	©	PROPN
ejpam-2095	31	4	2014	2014	NUM
ejpam-2095	31	5	ejpam	ejpam	NOUN
ejpam-2095	31	6	all	all	DET
ejpam-2095	31	7	rights	right	NOUN
ejpam-2095	31	8	reserved	reserve	VERB
ejpam-2095	31	9	.	.	PUNCT
ejpam-2095	32	1	c.	c.	PROPN
ejpam-2095	32	2	manjarekar	manjarekar	PROPN
ejpam-2095	32	3	,	,	PUNCT
ejpam-2095	32	4	u.	u.	PROPN
ejpam-2095	32	5	kandale	kandale	PROPN
ejpam-2095	32	6	/	/	SYM
ejpam-2095	32	7	eur	eur	PROPN
ejpam-2095	32	8	.	.	PUNCT
ejpam-2095	33	1	j.	j.	PROPN
ejpam-2095	33	2	pure	pure	PROPN
ejpam-2095	33	3	appl	appl	PROPN
ejpam-2095	33	4	.	.	PROPN
ejpam-2095	33	5	math	math	PROPN
ejpam-2095	33	6	,	,	PUNCT
ejpam-2095	33	7	7	7	NUM
ejpam-2095	33	8	(	(	PUNCT
ejpam-2095	33	9	2014	2014	NUM
ejpam-2095	33	10	)	)	PUNCT
ejpam-2095	33	11	,	,	PUNCT
ejpam-2095	33	12	201	201	NUM
ejpam-2095	33	13	-	-	SYM
ejpam-2095	33	14	209	209	NUM
ejpam-2095	33	15	202	202	NUM
ejpam-2095	33	16	then	then	ADV
ejpam-2095	33	17	this	this	DET
ejpam-2095	33	18	qi	qi	NOUN
ejpam-2095	33	19	can	can	AUX
ejpam-2095	33	20	be	be	AUX
ejpam-2095	33	21	dropped	drop	VERB
ejpam-2095	33	22	from	from	ADP
ejpam-2095	33	23	the	the	DET
ejpam-2095	33	24	primary	primary	ADJ
ejpam-2095	33	25	decomposition	decomposition	NOUN
ejpam-2095	33	26	.	.	PUNCT
ejpam-2095	34	1	after	after	ADP
ejpam-2095	34	2	deleting	delete	VERB
ejpam-2095	34	3	such	such	ADJ
ejpam-2095	34	4	primary	primary	ADJ
ejpam-2095	34	5	components	component	NOUN
ejpam-2095	34	6	and	and	CCONJ
ejpam-2095	34	7	combining	combine	VERB
ejpam-2095	34	8	the	the	DET
ejpam-2095	34	9	primary	primary	ADJ
ejpam-2095	34	10	components	component	NOUN
ejpam-2095	34	11	with	with	ADP
ejpam-2095	34	12	same	same	ADJ
ejpam-2095	34	13	associated	associate	VERB
ejpam-2095	34	14	prime	prime	NOUN
ejpam-2095	34	15	we	we	PRON
ejpam-2095	34	16	get	get	VERB
ejpam-2095	34	17	a	a	DET
ejpam-2095	34	18	reduced	reduce	VERB
ejpam-2095	34	19	primary	primary	ADJ
ejpam-2095	34	20	decomposition	decomposition	NOUN
ejpam-2095	34	21	in	in	ADP
ejpam-2095	34	22	which	which	PRON
ejpam-2095	34	23	distinct	distinct	ADJ
ejpam-2095	34	24	primaries	primary	NOUN
ejpam-2095	34	25	are	be	AUX
ejpam-2095	34	26	associated	associate	VERB
ejpam-2095	34	27	with	with	ADP
ejpam-2095	34	28	distinct	distinct	ADJ
ejpam-2095	34	29	primes.such	primes.such	NOUN
ejpam-2095	34	30	a	a	DET
ejpam-2095	34	31	primary	primary	ADJ
ejpam-2095	34	32	decomposition	decomposition	NOUN
ejpam-2095	34	33	is	be	AUX
ejpam-2095	34	34	also	also	ADV
ejpam-2095	34	35	called	call	VERB
ejpam-2095	34	36	an	an	DET
ejpam-2095	34	37	irredundant	irredundant	ADJ
ejpam-2095	34	38	primary	primary	ADJ
ejpam-2095	34	39	decomposition	decomposition	NOUN
ejpam-2095	34	40	,	,	PUNCT
ejpam-2095	34	41	reduced	reduce	VERB
ejpam-2095	34	42	primary	primary	ADJ
ejpam-2095	34	43	decomposition	decomposition	NOUN
ejpam-2095	34	44	or	or	CCONJ
ejpam-2095	34	45	normal	normal	ADJ
ejpam-2095	34	46	primary	primary	ADJ
ejpam-2095	34	47	decomposition	decomposition	NOUN
ejpam-2095	34	48	.	.	PUNCT
ejpam-2095	35	1	let	let	VERB
ejpam-2095	35	2	a	a	PRON
ejpam-2095	35	3	=	=	PUNCT
ejpam-2095	35	4	q1∧q2	q1∧q2	NOUN
ejpam-2095	35	5	.	.	PUNCT
ejpam-2095	35	6	.	.	PUNCT
ejpam-2095	36	1	.∧qn	.∧qn	PROPN
ejpam-2095	36	2	be	be	AUX
ejpam-2095	36	3	a	a	DET
ejpam-2095	36	4	reduced	reduce	VERB
ejpam-2095	36	5	primary	primary	ADJ
ejpam-2095	36	6	decomposition	decomposition	NOUN
ejpam-2095	36	7	of	of	ADP
ejpam-2095	36	8	a	a	DET
ejpam-2095	36	9	and	and	CCONJ
ejpam-2095	36	10	let	let	VERB
ejpam-2095	36	11	p1	p1	NOUN
ejpam-2095	36	12	,	,	PUNCT
ejpam-2095	36	13	p2	p2	NOUN
ejpam-2095	36	14	,	,	PUNCT
ejpam-2095	36	15	.	.	PUNCT
ejpam-2095	36	16	.	.	PUNCT
ejpam-2095	37	1	.	.	PUNCT
ejpam-2095	38	1	,	,	PUNCT
ejpam-2095	38	2	pn	pn	PROPN
ejpam-2095	38	3	denotes	denote	VERB
ejpam-2095	38	4	the	the	DET
ejpam-2095	38	5	associated	associated	ADJ
ejpam-2095	38	6	primes	prime	NOUN
ejpam-2095	38	7	of	of	ADP
ejpam-2095	38	8	q1,q2	q1,q2	PROPN
ejpam-2095	38	9	,	,	PUNCT
ejpam-2095	38	10	.	.	PUNCT
ejpam-2095	38	11	.	.	PUNCT
ejpam-2095	39	1	.	.	PUNCT
ejpam-2095	40	1	,	,	PUNCT
ejpam-2095	40	2	qn	qn	PROPN
ejpam-2095	40	3	respectively	respectively	ADV
ejpam-2095	40	4	,	,	PUNCT
ejpam-2095	40	5	which	which	PRON
ejpam-2095	40	6	are	be	AUX
ejpam-2095	40	7	also	also	ADV
ejpam-2095	40	8	called	call	VERB
ejpam-2095	40	9	associated	associated	ADJ
ejpam-2095	40	10	primes	prime	NOUN
ejpam-2095	40	11	of	of	ADP
ejpam-2095	40	12	a.	a.	NOUN
ejpam-2095	40	13	a	a	DET
ejpam-2095	40	14	subset	subset	NOUN
ejpam-2095	40	15	c	c	PROPN
ejpam-2095	40	16	of	of	ADP
ejpam-2095	40	17	{	{	PUNCT
ejpam-2095	40	18	p1	p1	NOUN
ejpam-2095	40	19	,	,	PUNCT
ejpam-2095	40	20	p2	p2	NOUN
ejpam-2095	40	21	,	,	PUNCT
ejpam-2095	40	22	.	.	PUNCT
ejpam-2095	40	23	.	.	PUNCT
ejpam-2095	41	1	.	.	PUNCT
ejpam-2095	42	1	,	,	PUNCT
ejpam-2095	42	2	pn	pn	PROPN
ejpam-2095	42	3	}	}	PUNCT
ejpam-2095	42	4	is	be	AUX
ejpam-2095	42	5	called	call	VERB
ejpam-2095	42	6	isolated	isolated	ADJ
ejpam-2095	42	7	if	if	SCONJ
ejpam-2095	42	8	pi	pi	PROPN
ejpam-2095	42	9	∈	∈	PROPN
ejpam-2095	42	10	c	c	PROPN
ejpam-2095	42	11	implies	imply	VERB
ejpam-2095	42	12	p	p	X
ejpam-2095	42	13	j	j	PROPN
ejpam-2095	42	14	∈	∈	PROPN
ejpam-2095	42	15	c	c	NOUN
ejpam-2095	42	16	whenever	whenever	SCONJ
ejpam-2095	42	17	p	p	PROPN
ejpam-2095	42	18	j	j	PROPN
ejpam-2095	42	19	¶	¶	PROPN
ejpam-2095	42	20	pi	pi	NOUN
ejpam-2095	42	21	.	.	PUNCT
ejpam-2095	43	1	let	let	VERB
ejpam-2095	43	2	m	m	PRON
ejpam-2095	43	3	be	be	AUX
ejpam-2095	43	4	a	a	DET
ejpam-2095	43	5	complete	complete	ADJ
ejpam-2095	43	6	lattice	lattice	NOUN
ejpam-2095	43	7	and	and	CCONJ
ejpam-2095	43	8	l	l	NOUN
ejpam-2095	43	9	be	be	AUX
ejpam-2095	43	10	a	a	DET
ejpam-2095	43	11	multiplicative	multiplicative	ADJ
ejpam-2095	43	12	lattice	lattice	NOUN
ejpam-2095	43	13	.	.	PUNCT
ejpam-2095	44	1	then	then	ADV
ejpam-2095	44	2	m	m	PROPN
ejpam-2095	44	3	is	be	AUX
ejpam-2095	44	4	called	call	VERB
ejpam-2095	44	5	l	l	NOUN
ejpam-2095	44	6	-	-	NOUN
ejpam-2095	44	7	module	module	NOUN
ejpam-2095	44	8	or	or	CCONJ
ejpam-2095	44	9	module	module	NOUN
ejpam-2095	44	10	over	over	ADP
ejpam-2095	44	11	l	l	NOUN
ejpam-2095	44	12	if	if	SCONJ
ejpam-2095	44	13	there	there	PRON
ejpam-2095	44	14	is	be	VERB
ejpam-2095	44	15	a	a	DET
ejpam-2095	44	16	multiplication	multiplication	NOUN
ejpam-2095	44	17	between	between	ADP
ejpam-2095	44	18	elements	element	NOUN
ejpam-2095	44	19	of	of	ADP
ejpam-2095	44	20	l	l	NOUN
ejpam-2095	44	21	and	and	CCONJ
ejpam-2095	44	22	m	m	AUX
ejpam-2095	44	23	written	write	VERB
ejpam-2095	44	24	as	as	ADP
ejpam-2095	44	25	ab	ab	NUM
ejpam-2095	44	26	where	where	SCONJ
ejpam-2095	44	27	a	a	DET
ejpam-2095	44	28	∈	∈	PROPN
ejpam-2095	44	29	l	l	NOUN
ejpam-2095	44	30	and	and	CCONJ
ejpam-2095	44	31	b	b	X
ejpam-2095	44	32	∈	∈	NOUN
ejpam-2095	44	33	m	m	VERB
ejpam-2095	44	34	which	which	PRON
ejpam-2095	44	35	satisfies	satisfy	VERB
ejpam-2095	44	36	the	the	DET
ejpam-2095	44	37	following	follow	VERB
ejpam-2095	44	38	properties	property	NOUN
ejpam-2095	44	39	,	,	PUNCT
ejpam-2095	44	40	i	i	NOUN
ejpam-2095	44	41	)	)	PUNCT
ejpam-2095	44	42	(	(	PUNCT
ejpam-2095	44	43	∨	∨	NOUN
ejpam-2095	44	44	α	α	PROPN
ejpam-2095	44	45	a	a	DET
ejpam-2095	44	46	α	α	NOUN
ejpam-2095	44	47	)	)	PUNCT
ejpam-2095	44	48	a=	a=	PROPN
ejpam-2095	45	1	∨	∨	NUM
ejpam-2095	45	2	α	α	PROPN
ejpam-2095	45	3	a	a	DET
ejpam-2095	45	4	α	α	NOUN
ejpam-2095	45	5	a	a	DET
ejpam-2095	45	6	∀a	∀a	NOUN
ejpam-2095	45	7	α	α	X
ejpam-2095	45	8	∈	∈	PROPN
ejpam-2095	45	9	l	l	NOUN
ejpam-2095	45	10	,	,	PUNCT
ejpam-2095	45	11	a∈	a∈	PROPN
ejpam-2095	45	12	m	m	PROPN
ejpam-2095	45	13	ii	ii	PROPN
ejpam-2095	45	14	)	)	PUNCT
ejpam-2095	45	15	a(∨	a(∨	X
ejpam-2095	46	1	α	α	NOUN
ejpam-2095	46	2	a	a	DET
ejpam-2095	46	3	α	α	NOUN
ejpam-2095	46	4	)	)	PUNCT
ejpam-2095	47	1	=	=	SYM
ejpam-2095	47	2	∨	∨	NUM
ejpam-2095	47	3	α	α	NOUN
ejpam-2095	48	1	aa	aa	NOUN
ejpam-2095	48	2	α	α	NOUN
ejpam-2095	48	3	∀a	∀a	NOUN
ejpam-2095	48	4	∈	∈	PROPN
ejpam-2095	48	5	l	l	NOUN
ejpam-2095	48	6	,	,	PUNCT
ejpam-2095	48	7	a	a	DET
ejpam-2095	48	8	α	α	PROPN
ejpam-2095	48	9	∈	∈	PROPN
ejpam-2095	48	10	m	m	PROPN
ejpam-2095	48	11	iii	iii	NOUN
ejpam-2095	48	12	)	)	PUNCT
ejpam-2095	48	13	(	(	PUNCT
ejpam-2095	48	14	ab)a=	ab)a=	ADP
ejpam-2095	48	15	a(ba	a(ba	NOUN
ejpam-2095	48	16	)	)	PUNCT
ejpam-2095	48	17	∀a	∀a	NOUN
ejpam-2095	48	18	,	,	PUNCT
ejpam-2095	48	19	b	b	X
ejpam-2095	48	20	∈	∈	PROPN
ejpam-2095	48	21	l	l	NOUN
ejpam-2095	48	22	,	,	PUNCT
ejpam-2095	48	23	a∈	a∈	PROPN
ejpam-2095	48	24	m	m	PROPN
ejpam-2095	48	25	iv	iv	X
ejpam-2095	48	26	)	)	PUNCT
ejpam-2095	48	27	ib	ib	NOUN
ejpam-2095	49	1	=	=	SYM
ejpam-2095	49	2	b	b	PROPN
ejpam-2095	49	3	v	v	NOUN
ejpam-2095	49	4	)	)	PUNCT
ejpam-2095	49	5	0b	0b	NOUN
ejpam-2095	49	6	=	=	SYM
ejpam-2095	49	7	0	0	NUM
ejpam-2095	49	8	m	m	VERB
ejpam-2095	49	9	,	,	PUNCT
ejpam-2095	49	10	for	for	ADP
ejpam-2095	49	11	all	all	DET
ejpam-2095	49	12	a	a	DET
ejpam-2095	49	13	,	,	PUNCT
ejpam-2095	49	14	a	a	DET
ejpam-2095	49	15	α	α	NOUN
ejpam-2095	49	16	,	,	PUNCT
ejpam-2095	49	17	b	b	PROPN
ejpam-2095	49	18	∈	∈	PROPN
ejpam-2095	49	19	l	l	NOUN
ejpam-2095	49	20	and	and	CCONJ
ejpam-2095	49	21	a	a	PRON
ejpam-2095	49	22	,	,	PUNCT
ejpam-2095	49	23	a	a	DET
ejpam-2095	49	24	α	α	NOUN
ejpam-2095	49	25	∈	∈	NOUN
ejpam-2095	49	26	m	m	NOUN
ejpam-2095	49	27	,	,	PUNCT
ejpam-2095	49	28	where	where	SCONJ
ejpam-2095	49	29	i	i	PRON
ejpam-2095	49	30	is	be	AUX
ejpam-2095	49	31	the	the	DET
ejpam-2095	49	32	supremum	supremum	NOUN
ejpam-2095	49	33	of	of	ADP
ejpam-2095	49	34	l	l	NOUN
ejpam-2095	49	35	and	and	CCONJ
ejpam-2095	49	36	0	0	NUM
ejpam-2095	49	37	is	be	AUX
ejpam-2095	49	38	the	the	DET
ejpam-2095	49	39	infimum	infimum	NOUN
ejpam-2095	49	40	of	of	ADP
ejpam-2095	49	41	l.	l.	NOUN
ejpam-2095	49	42	we	we	PRON
ejpam-2095	49	43	denote	denote	VERB
ejpam-2095	49	44	by	by	ADP
ejpam-2095	49	45	0	0	NUM
ejpam-2095	49	46	m	m	VERB
ejpam-2095	49	47	and	and	CCONJ
ejpam-2095	49	48	i	i	PRON
ejpam-2095	49	49	m	m	VERB
ejpam-2095	49	50	the	the	DET
ejpam-2095	49	51	least	least	ADJ
ejpam-2095	49	52	element	element	NOUN
ejpam-2095	49	53	and	and	CCONJ
ejpam-2095	49	54	the	the	DET
ejpam-2095	49	55	greatest	great	ADJ
ejpam-2095	49	56	element	element	NOUN
ejpam-2095	49	57	of	of	ADP
ejpam-2095	49	58	m	m	PROPN
ejpam-2095	49	59	.	.	PUNCT
ejpam-2095	50	1	the	the	DET
ejpam-2095	50	2	elements	element	NOUN
ejpam-2095	50	3	of	of	ADP
ejpam-2095	50	4	l	l	NOUN
ejpam-2095	50	5	will	will	AUX
ejpam-2095	50	6	generally	generally	ADV
ejpam-2095	50	7	be	be	AUX
ejpam-2095	50	8	denoted	denote	VERB
ejpam-2095	50	9	by	by	ADP
ejpam-2095	50	10	a	a	DET
ejpam-2095	50	11	,	,	PUNCT
ejpam-2095	50	12	b	b	PROPN
ejpam-2095	50	13	,	,	PUNCT
ejpam-2095	50	14	c	c	NOUN
ejpam-2095	50	15	,	,	PUNCT
ejpam-2095	50	16	.	.	PUNCT
ejpam-2095	50	17	.	.	PUNCT
ejpam-2095	51	1	.	.	PUNCT
ejpam-2095	52	1	and	and	CCONJ
ejpam-2095	52	2	elements	element	NOUN
ejpam-2095	52	3	of	of	ADP
ejpam-2095	52	4	m	m	PROPN
ejpam-2095	52	5	will	will	AUX
ejpam-2095	52	6	generally	generally	ADV
ejpam-2095	52	7	be	be	AUX
ejpam-2095	52	8	denoted	denote	VERB
ejpam-2095	52	9	by	by	ADP
ejpam-2095	52	10	a	a	DET
ejpam-2095	52	11	,	,	PUNCT
ejpam-2095	52	12	b	b	NOUN
ejpam-2095	52	13	,	,	PUNCT
ejpam-2095	52	14	c	c	NOUN
ejpam-2095	52	15	.	.	PUNCT
ejpam-2095	52	16	.	.	PUNCT
ejpam-2095	53	1	..	..	PUNCT
ejpam-2095	53	2	let	let	VERB
ejpam-2095	53	3	m	m	PRON
ejpam-2095	53	4	be	be	AUX
ejpam-2095	53	5	a	a	DET
ejpam-2095	53	6	l	l	NOUN
ejpam-2095	53	7	-	-	NOUN
ejpam-2095	53	8	module	module	NOUN
ejpam-2095	53	9	.	.	PUNCT
ejpam-2095	54	1	if	if	SCONJ
ejpam-2095	54	2	n	n	PRON
ejpam-2095	54	3	∈	∈	VERB
ejpam-2095	54	4	m	m	VERB
ejpam-2095	54	5	and	and	CCONJ
ejpam-2095	54	6	a	a	DET
ejpam-2095	54	7	∈	∈	NOUN
ejpam-2095	54	8	l	l	NOUN
ejpam-2095	54	9	then	then	ADV
ejpam-2095	54	10	(	(	PUNCT
ejpam-2095	54	11	n	n	X
ejpam-2095	54	12	:	:	PUNCT
ejpam-2095	54	13	a	a	X
ejpam-2095	54	14	)	)	PUNCT
ejpam-2095	55	1	=	=	SYM
ejpam-2095	55	2	∨{x	∨{x	PROPN
ejpam-2095	56	1	∈	∈	NOUN
ejpam-2095	56	2	m	m	VERB
ejpam-2095	56	3	|	|	ADV
ejpam-2095	56	4	ax	ax	NOUN
ejpam-2095	56	5	¶	¶	PROPN
ejpam-2095	56	6	n	n	CCONJ
ejpam-2095	56	7	}	}	PUNCT
ejpam-2095	56	8	.	.	PUNCT
ejpam-2095	57	1	if	if	SCONJ
ejpam-2095	57	2	a	a	PRON
ejpam-2095	57	3	,	,	PUNCT
ejpam-2095	57	4	b	b	X
ejpam-2095	57	5	∈	∈	ADV
ejpam-2095	57	6	m	m	NOUN
ejpam-2095	57	7	,	,	PUNCT
ejpam-2095	57	8	then	then	ADV
ejpam-2095	57	9	(	(	PUNCT
ejpam-2095	57	10	a	a	DET
ejpam-2095	57	11	:	:	PUNCT
ejpam-2095	57	12	b	b	X
ejpam-2095	57	13	)	)	PUNCT
ejpam-2095	57	14	=	=	SYM
ejpam-2095	58	1	∨{x	∨{x	NOUN
ejpam-2095	58	2	∈	∈	NOUN
ejpam-2095	58	3	l	l	NOUN
ejpam-2095	59	1	|	|	ADV
ejpam-2095	59	2	xb	xb	PROPN
ejpam-2095	59	3	¶	¶	PROPN
ejpam-2095	59	4	a	a	PRON
ejpam-2095	59	5	}	}	PUNCT
ejpam-2095	59	6	.	.	PUNCT
ejpam-2095	60	1	an	an	DET
ejpam-2095	60	2	l	l	NOUN
ejpam-2095	60	3	-	-	NOUN
ejpam-2095	60	4	module	module	NOUN
ejpam-2095	60	5	m	m	NOUN
ejpam-2095	60	6	is	be	AUX
ejpam-2095	60	7	called	call	VERB
ejpam-2095	60	8	a	a	DET
ejpam-2095	60	9	multiplication	multiplication	NOUN
ejpam-2095	60	10	l	l	NOUN
ejpam-2095	60	11	-	-	NOUN
ejpam-2095	60	12	module	module	NOUN
ejpam-2095	60	13	if	if	SCONJ
ejpam-2095	60	14	for	for	ADP
ejpam-2095	60	15	every	every	DET
ejpam-2095	60	16	element	element	NOUN
ejpam-2095	60	17	n	n	PRON
ejpam-2095	60	18	∈	∈	NOUN
ejpam-2095	61	1	m	m	AUX
ejpam-2095	61	2	there	there	PRON
ejpam-2095	61	3	exists	exist	VERB
ejpam-2095	61	4	an	an	DET
ejpam-2095	61	5	element	element	NOUN
ejpam-2095	61	6	a	a	DET
ejpam-2095	61	7	∈	∈	NOUN
ejpam-2095	61	8	l	l	NOUN
ejpam-2095	61	9	such	such	ADJ
ejpam-2095	61	10	that	that	SCONJ
ejpam-2095	61	11	n	n	NOUN
ejpam-2095	61	12	=	=	PRON
ejpam-2095	61	13	aim	aim	VERB
ejpam-2095	61	14	[	[	X
ejpam-2095	61	15	4	4	NUM
ejpam-2095	61	16	]	]	PUNCT
ejpam-2095	61	17	.	.	PUNCT
ejpam-2095	62	1	a	a	DET
ejpam-2095	62	2	proper	proper	ADJ
ejpam-2095	62	3	element	element	NOUN
ejpam-2095	62	4	n	n	PROPN
ejpam-2095	62	5	of	of	ADP
ejpam-2095	62	6	m	m	PROPN
ejpam-2095	62	7	is	be	AUX
ejpam-2095	62	8	said	say	VERB
ejpam-2095	62	9	to	to	PART
ejpam-2095	62	10	be	be	AUX
ejpam-2095	62	11	prime	prime	ADJ
ejpam-2095	62	12	if	if	SCONJ
ejpam-2095	62	13	ax	ax	NOUN
ejpam-2095	62	14	¶	¶	PROPN
ejpam-2095	62	15	n	n	PRON
ejpam-2095	62	16	implies	imply	VERB
ejpam-2095	62	17	x	x	PROPN
ejpam-2095	62	18	¶	¶	NOUN
ejpam-2095	62	19	n	n	PRON
ejpam-2095	62	20	or	or	CCONJ
ejpam-2095	62	21	aim	aim	VERB
ejpam-2095	62	22	¶	¶	PROPN
ejpam-2095	62	23	n	n	CCONJ
ejpam-2095	62	24	that	that	PRON
ejpam-2095	62	25	is	be	AUX
ejpam-2095	62	26	a	a	DET
ejpam-2095	62	27	¶	¶	NOUN
ejpam-2095	62	28	(	(	PUNCT
ejpam-2095	62	29	n	n	NUM
ejpam-2095	62	30	:	:	PUNCT
ejpam-2095	62	31	i	i	PRON
ejpam-2095	62	32	m	m	VERB
ejpam-2095	62	33	)	)	PUNCT
ejpam-2095	62	34	for	for	ADP
ejpam-2095	62	35	every	every	DET
ejpam-2095	62	36	a	a	DET
ejpam-2095	62	37	∈	∈	PROPN
ejpam-2095	62	38	l	l	NOUN
ejpam-2095	62	39	,	,	PUNCT
ejpam-2095	62	40	x	x	SYM
ejpam-2095	62	41	∈	∈	PROPN
ejpam-2095	62	42	m	m	VERB
ejpam-2095	62	43	.	.	PUNCT
ejpam-2095	63	1	an	an	DET
ejpam-2095	63	2	element	element	NOUN
ejpam-2095	63	3	n	n	CCONJ
ejpam-2095	63	4	<	<	X
ejpam-2095	63	5	i	i	X
ejpam-2095	63	6	m	m	VERB
ejpam-2095	63	7	in	in	ADP
ejpam-2095	63	8	m	m	PROPN
ejpam-2095	63	9	is	be	AUX
ejpam-2095	63	10	said	say	VERB
ejpam-2095	63	11	to	to	PART
ejpam-2095	63	12	be	be	AUX
ejpam-2095	63	13	primary	primary	ADJ
ejpam-2095	63	14	if	if	SCONJ
ejpam-2095	63	15	ax	ax	NOUN
ejpam-2095	63	16	¶	¶	PROPN
ejpam-2095	63	17	n	n	PRON
ejpam-2095	63	18	implies	imply	VERB
ejpam-2095	63	19	x	x	PROPN
ejpam-2095	63	20	¶	¶	NOUN
ejpam-2095	63	21	n	n	ADP
ejpam-2095	63	22	or	or	CCONJ
ejpam-2095	63	23	an	an	DET
ejpam-2095	63	24	i	i	NOUN
ejpam-2095	63	25	m	m	PROPN
ejpam-2095	63	26	¶	¶	PROPN
ejpam-2095	63	27	n	n	CCONJ
ejpam-2095	63	28	that	that	PRON
ejpam-2095	63	29	is	be	AUX
ejpam-2095	63	30	an	an	DET
ejpam-2095	63	31	¶	¶	NOUN
ejpam-2095	63	32	(	(	PUNCT
ejpam-2095	63	33	n	n	NUM
ejpam-2095	63	34	:	:	PUNCT
ejpam-2095	63	35	i	i	PRON
ejpam-2095	63	36	m	m	VERB
ejpam-2095	63	37	)	)	PUNCT
ejpam-2095	63	38	for	for	ADP
ejpam-2095	63	39	some	some	DET
ejpam-2095	63	40	integer	integer	NOUN
ejpam-2095	63	41	n.	n.	NOUN
ejpam-2095	63	42	an	an	DET
ejpam-2095	63	43	element	element	NOUN
ejpam-2095	63	44	n	n	PROPN
ejpam-2095	63	45	of	of	ADP
ejpam-2095	63	46	m	m	PROPN
ejpam-2095	63	47	is	be	AUX
ejpam-2095	63	48	called	call	VERB
ejpam-2095	63	49	a	a	DET
ejpam-2095	63	50	radical	radical	ADJ
ejpam-2095	63	51	element	element	NOUN
ejpam-2095	63	52	if	if	SCONJ
ejpam-2095	63	53	(	(	PUNCT
ejpam-2095	63	54	n	n	X
ejpam-2095	63	55	:	:	PUNCT
ejpam-2095	63	56	i	i	PRON
ejpam-2095	63	57	m	m	VERB
ejpam-2095	63	58	)	)	PUNCT
ejpam-2095	64	1	=	=	SYM
ejpam-2095	64	2	p	p	X
ejpam-2095	64	3	(	(	PUNCT
ejpam-2095	64	4	n	n	NOUN
ejpam-2095	64	5	:	:	PUNCT
ejpam-2095	64	6	i	i	PRON
ejpam-2095	64	7	m	m	PROPN
ejpam-2095	64	8	)	)	PUNCT
ejpam-2095	64	9	.	.	PUNCT
ejpam-2095	65	1	noether	noether	PROPN
ejpam-2095	65	2	lattice	lattice	PROPN
ejpam-2095	65	3	is	be	AUX
ejpam-2095	65	4	a	a	DET
ejpam-2095	65	5	modular	modular	ADJ
ejpam-2095	65	6	multiplicative	multiplicative	ADJ
ejpam-2095	65	7	lattice	lattice	NOUN
ejpam-2095	65	8	satisfying	satisfy	VERB
ejpam-2095	65	9	ascending	ascend	VERB
ejpam-2095	65	10	chain	chain	NOUN
ejpam-2095	65	11	condition	condition	NOUN
ejpam-2095	65	12	in	in	ADP
ejpam-2095	65	13	which	which	PRON
ejpam-2095	65	14	every	every	DET
ejpam-2095	65	15	element	element	NOUN
ejpam-2095	65	16	is	be	AUX
ejpam-2095	65	17	the	the	DET
ejpam-2095	65	18	join	join	NOUN
ejpam-2095	65	19	of	of	ADP
ejpam-2095	65	20	principal	principal	ADJ
ejpam-2095	65	21	elements	element	NOUN
ejpam-2095	65	22	.	.	PUNCT
ejpam-2095	66	1	let	let	VERB
ejpam-2095	66	2	n	n	PRON
ejpam-2095	66	3	be	be	AUX
ejpam-2095	66	4	an	an	DET
ejpam-2095	66	5	element	element	NOUN
ejpam-2095	66	6	of	of	ADP
ejpam-2095	66	7	a	a	DET
ejpam-2095	66	8	lattice	lattice	NOUN
ejpam-2095	66	9	module	module	NOUN
ejpam-2095	66	10	m	m	PROPN
ejpam-2095	66	11	.	.	PUNCT
ejpam-2095	67	1	then	then	ADV
ejpam-2095	67	2	n	n	PRON
ejpam-2095	67	3	is	be	AUX
ejpam-2095	67	4	said	say	VERB
ejpam-2095	67	5	to	to	PART
ejpam-2095	67	6	have	have	VERB
ejpam-2095	67	7	a	a	DET
ejpam-2095	67	8	primary	primary	ADJ
ejpam-2095	67	9	decomposition	decomposition	NOUN
ejpam-2095	67	10	if	if	SCONJ
ejpam-2095	67	11	there	there	PRON
ejpam-2095	67	12	exist	exist	VERB
ejpam-2095	67	13	primary	primary	ADJ
ejpam-2095	67	14	element	element	NOUN
ejpam-2095	67	15	q1,q2	q1,q2	PROPN
ejpam-2095	67	16	,	,	PUNCT
ejpam-2095	67	17	.	.	PUNCT
ejpam-2095	67	18	.	.	PUNCT
ejpam-2095	67	19	.	.	PUNCT
ejpam-2095	68	1	,	,	PUNCT
ejpam-2095	68	2	qn	qn	INTJ
ejpam-2095	68	3	such	such	ADJ
ejpam-2095	68	4	that	that	SCONJ
ejpam-2095	68	5	n	n	PROPN
ejpam-2095	68	6	=	=	PROPN
ejpam-2095	68	7	q1	q1	PROPN
ejpam-2095	68	8	∧q2	∧q2	VERB
ejpam-2095	68	9	∧	∧	PROPN
ejpam-2095	68	10	.	.	PUNCT
ejpam-2095	68	11	.	.	PUNCT
ejpam-2095	69	1	.qn	.qn	PUNCT
ejpam-2095	69	2	.	.	PUNCT
ejpam-2095	70	1	if	if	SCONJ
ejpam-2095	70	2	some	some	PRON
ejpam-2095	70	3	q	q	NOUN
ejpam-2095	70	4	i	i	PRON
ejpam-2095	70	5	contains	contain	VERB
ejpam-2095	70	6	the	the	DET
ejpam-2095	70	7	meet	meet	NOUN
ejpam-2095	70	8	of	of	ADP
ejpam-2095	70	9	remaining	remain	VERB
ejpam-2095	70	10	ones	one	NOUN
ejpam-2095	70	11	then	then	ADV
ejpam-2095	70	12	this	this	DET
ejpam-2095	70	13	q	q	NOUN
ejpam-2095	70	14	i	i	PRON
ejpam-2095	70	15	can	can	AUX
ejpam-2095	70	16	be	be	AUX
ejpam-2095	70	17	dropped	drop	VERB
ejpam-2095	70	18	from	from	ADP
ejpam-2095	70	19	the	the	DET
ejpam-2095	70	20	primary	primary	ADJ
ejpam-2095	70	21	decomposition	decomposition	NOUN
ejpam-2095	70	22	.	.	PUNCT
ejpam-2095	71	1	similarly	similarly	ADV
ejpam-2095	71	2	,	,	PUNCT
ejpam-2095	71	3	any	any	DET
ejpam-2095	71	4	other	other	ADJ
ejpam-2095	71	5	primary	primary	ADJ
ejpam-2095	71	6	components	component	NOUN
ejpam-2095	71	7	which	which	PRON
ejpam-2095	71	8	contains	contain	VERB
ejpam-2095	71	9	the	the	DET
ejpam-2095	71	10	meet	meet	NOUN
ejpam-2095	71	11	of	of	ADP
ejpam-2095	71	12	remaining	remain	VERB
ejpam-2095	71	13	ones	one	NOUN
ejpam-2095	71	14	can	can	AUX
ejpam-2095	71	15	be	be	AUX
ejpam-2095	71	16	dropped	drop	VERB
ejpam-2095	71	17	from	from	ADP
ejpam-2095	71	18	the	the	DET
ejpam-2095	71	19	primary	primary	ADJ
ejpam-2095	71	20	decomposition	decomposition	NOUN
ejpam-2095	71	21	.	.	PUNCT
ejpam-2095	72	1	if	if	SCONJ
ejpam-2095	72	2	no	no	DET
ejpam-2095	72	3	q	q	NOUN
ejpam-2095	72	4	i	i	PRON
ejpam-2095	72	5	can	can	AUX
ejpam-2095	72	6	be	be	AUX
ejpam-2095	72	7	dropped	drop	VERB
ejpam-2095	72	8	further	far	ADV
ejpam-2095	72	9	we	we	PRON
ejpam-2095	72	10	get	get	VERB
ejpam-2095	72	11	a	a	DET
ejpam-2095	72	12	reduced	reduced	ADJ
ejpam-2095	72	13	primary	primary	ADJ
ejpam-2095	72	14	decomposition	decomposition	NOUN
ejpam-2095	72	15	of	of	ADP
ejpam-2095	72	16	n	n	PROPN
ejpam-2095	72	17	.	.	PUNCT
ejpam-2095	73	1	such	such	DET
ejpam-2095	73	2	a	a	DET
ejpam-2095	73	3	primary	primary	ADJ
ejpam-2095	73	4	decomposition	decomposition	NOUN
ejpam-2095	73	5	is	be	AUX
ejpam-2095	73	6	also	also	ADV
ejpam-2095	73	7	called	call	VERB
ejpam-2095	73	8	an	an	DET
ejpam-2095	73	9	irredundant	irredundant	ADJ
ejpam-2095	73	10	primary	primary	ADJ
ejpam-2095	73	11	decomposition	decomposition	NOUN
ejpam-2095	73	12	.	.	PUNCT
ejpam-2095	74	1	if	if	SCONJ
ejpam-2095	74	2	q	q	NOUN
ejpam-2095	74	3	is	be	AUX
ejpam-2095	74	4	primary	primary	ADJ
ejpam-2095	74	5	then	then	ADV
ejpam-2095	74	6	p	p	X
ejpam-2095	74	7	q	q	PROPN
ejpam-2095	74	8	=	=	X
ejpam-2095	74	9	p	p	X
ejpam-2095	74	10	(	(	PUNCT
ejpam-2095	74	11	q	q	NOUN
ejpam-2095	74	12	:	:	PUNCT
ejpam-2095	74	13	i	i	PRON
ejpam-2095	74	14	m	m	PROPN
ejpam-2095	74	15	)	)	PUNCT
ejpam-2095	74	16	is	be	AUX
ejpam-2095	74	17	prime	prime	ADJ
ejpam-2095	74	18	.	.	PUNCT
ejpam-2095	75	1	we	we	PRON
ejpam-2095	75	2	note	note	VERB
ejpam-2095	75	3	that	that	SCONJ
ejpam-2095	75	4	,	,	PUNCT
ejpam-2095	75	5	p	p	X
ejpam-2095	75	6	(	(	PUNCT
ejpam-2095	75	7	n	n	NOUN
ejpam-2095	75	8	:	:	PUNCT
ejpam-2095	75	9	i	i	PRON
ejpam-2095	75	10	m	m	PROPN
ejpam-2095	75	11	)	)	PUNCT
ejpam-2095	75	12	may	may	AUX
ejpam-2095	75	13	also	also	ADV
ejpam-2095	75	14	be	be	AUX
ejpam-2095	75	15	denoted	denote	VERB
ejpam-2095	75	16	by	by	ADP
ejpam-2095	75	17	p	p	PROPN
ejpam-2095	75	18	n	n	NOUN
ejpam-2095	75	19	.	.	PUNCT
ejpam-2095	76	1	2	2	X
ejpam-2095	76	2	.	.	X
ejpam-2095	76	3	primary	primary	ADJ
ejpam-2095	76	4	decomposition	decomposition	NOUN
ejpam-2095	76	5	of	of	ADP
ejpam-2095	76	6	elements	element	NOUN
ejpam-2095	76	7	the	the	DET
ejpam-2095	76	8	following	following	ADJ
ejpam-2095	76	9	result	result	NOUN
ejpam-2095	76	10	gives	give	VERB
ejpam-2095	76	11	the	the	DET
ejpam-2095	76	12	relation	relation	NOUN
ejpam-2095	76	13	between	between	ADP
ejpam-2095	76	14	a	a	DET
ejpam-2095	76	15	primary	primary	ADJ
ejpam-2095	76	16	element	element	NOUN
ejpam-2095	76	17	q	q	PROPN
ejpam-2095	77	1	and	and	CCONJ
ejpam-2095	77	2	p	p	X
ejpam-2095	77	3	(	(	PUNCT
ejpam-2095	77	4	q	q	NOUN
ejpam-2095	77	5	:	:	PUNCT
ejpam-2095	77	6	i	i	PRON
ejpam-2095	77	7	m	m	PROPN
ejpam-2095	77	8	)	)	PUNCT
ejpam-2095	77	9	.	.	PUNCT
ejpam-2095	78	1	c.	c.	PROPN
ejpam-2095	78	2	manjarekar	manjarekar	PROPN
ejpam-2095	78	3	,	,	PUNCT
ejpam-2095	78	4	u.	u.	PROPN
ejpam-2095	78	5	kandale	kandale	PROPN
ejpam-2095	78	6	/	/	SYM
ejpam-2095	78	7	eur	eur	PROPN
ejpam-2095	78	8	.	.	PUNCT
ejpam-2095	79	1	j.	j.	PROPN
ejpam-2095	79	2	pure	pure	PROPN
ejpam-2095	79	3	appl	appl	PROPN
ejpam-2095	79	4	.	.	PROPN
ejpam-2095	79	5	math	math	PROPN
ejpam-2095	79	6	,	,	PUNCT
ejpam-2095	79	7	7	7	NUM
ejpam-2095	79	8	(	(	PUNCT
ejpam-2095	79	9	2014	2014	NUM
ejpam-2095	79	10	)	)	PUNCT
ejpam-2095	79	11	,	,	PUNCT
ejpam-2095	79	12	201	201	NUM
ejpam-2095	79	13	-	-	SYM
ejpam-2095	79	14	209	209	NUM
ejpam-2095	79	15	203	203	NUM
ejpam-2095	79	16	theorem	theorem	NOUN
ejpam-2095	79	17	1	1	NUM
ejpam-2095	79	18	.	.	PUNCT
ejpam-2095	80	1	if	if	SCONJ
ejpam-2095	80	2	q	q	NOUN
ejpam-2095	80	3	is	be	AUX
ejpam-2095	80	4	a	a	DET
ejpam-2095	80	5	primary	primary	ADJ
ejpam-2095	80	6	element	element	NOUN
ejpam-2095	80	7	of	of	ADP
ejpam-2095	80	8	a	a	DET
ejpam-2095	80	9	lattice	lattice	NOUN
ejpam-2095	80	10	module	module	NOUN
ejpam-2095	80	11	m	m	VERB
ejpam-2095	80	12	then	then	ADV
ejpam-2095	80	13	p	p	X
ejpam-2095	80	14	(	(	PUNCT
ejpam-2095	80	15	q	q	NOUN
ejpam-2095	80	16	:	:	PUNCT
ejpam-2095	80	17	i	i	PRON
ejpam-2095	80	18	m	m	PROPN
ejpam-2095	80	19	)	)	PUNCT
ejpam-2095	80	20	is	be	AUX
ejpam-2095	80	21	a	a	DET
ejpam-2095	80	22	prime	prime	ADJ
ejpam-2095	80	23	element	element	NOUN
ejpam-2095	80	24	of	of	ADP
ejpam-2095	80	25	l.	l.	PROPN
ejpam-2095	80	26	if	if	SCONJ
ejpam-2095	80	27	a	a	PRON
ejpam-2095	80	28	is	be	AUX
ejpam-2095	80	29	an	an	DET
ejpam-2095	80	30	element	element	NOUN
ejpam-2095	80	31	of	of	ADP
ejpam-2095	80	32	l	l	NOUN
ejpam-2095	80	33	and	and	CCONJ
ejpam-2095	80	34	if	if	SCONJ
ejpam-2095	80	35	a	a	DET
ejpam-2095	80	36	¶	¶	NOUN
ejpam-2095	80	37	p	p	NOUN
ejpam-2095	80	38	where	where	SCONJ
ejpam-2095	80	39	p	p	NOUN
ejpam-2095	80	40	is	be	AUX
ejpam-2095	80	41	a	a	DET
ejpam-2095	80	42	prime	prime	ADJ
ejpam-2095	80	43	element	element	NOUN
ejpam-2095	80	44	of	of	ADP
ejpam-2095	80	45	l	l	PROPN
ejpam-2095	80	46	then	then	ADV
ejpam-2095	80	47	p	p	X
ejpam-2095	80	48	a	a	DET
ejpam-2095	80	49	¶	¶	NOUN
ejpam-2095	80	50	p.	p.	NOUN
ejpam-2095	80	51	proof	proof	NOUN
ejpam-2095	80	52	.	.	PUNCT
ejpam-2095	81	1	let	let	VERB
ejpam-2095	81	2	ab	ab	PROPN
ejpam-2095	81	3	¶	¶	PROPN
ejpam-2095	81	4	p	p	PROPN
ejpam-2095	81	5	(	(	PUNCT
ejpam-2095	81	6	q	q	NOUN
ejpam-2095	81	7	:	:	PUNCT
ejpam-2095	81	8	i	i	PRON
ejpam-2095	81	9	m	m	PROPN
ejpam-2095	81	10	)	)	PUNCT
ejpam-2095	81	11	and	and	CCONJ
ejpam-2095	81	12	suppose	suppose	VERB
ejpam-2095	81	13	b	b	X
ejpam-2095	81	14	p	p	X
ejpam-2095	81	15	(	(	PUNCT
ejpam-2095	81	16	q	q	NOUN
ejpam-2095	81	17	:	:	PUNCT
ejpam-2095	81	18	i	i	PRON
ejpam-2095	81	19	m	m	PROPN
ejpam-2095	81	20	)	)	PUNCT
ejpam-2095	81	21	.	.	PUNCT
ejpam-2095	82	1	then	then	ADV
ejpam-2095	82	2	(	(	PUNCT
ejpam-2095	82	3	ab)n	ab)n	PROPN
ejpam-2095	82	4	=	=	PUNCT
ejpam-2095	82	5	an	an	DET
ejpam-2095	82	6	bn	bn	PROPN
ejpam-2095	82	7	¶	¶	NOUN
ejpam-2095	82	8	(	(	PUNCT
ejpam-2095	82	9	q	q	NOUN
ejpam-2095	82	10	:	:	PUNCT
ejpam-2095	82	11	i	i	PRON
ejpam-2095	82	12	m	m	VERB
ejpam-2095	82	13	)	)	PUNCT
ejpam-2095	82	14	for	for	ADP
ejpam-2095	82	15	some	some	DET
ejpam-2095	82	16	positive	positive	ADJ
ejpam-2095	82	17	integer	integer	NOUN
ejpam-2095	82	18	n.	n.	NOUN
ejpam-2095	82	19	now	now	ADV
ejpam-2095	82	20	b	b	X
ejpam-2095	82	21	p	p	X
ejpam-2095	82	22	(	(	PUNCT
ejpam-2095	82	23	q	q	NOUN
ejpam-2095	82	24	:	:	PUNCT
ejpam-2095	82	25	i	i	PRON
ejpam-2095	82	26	m	m	PROPN
ejpam-2095	82	27	)	)	PUNCT
ejpam-2095	82	28	implies	imply	VERB
ejpam-2095	82	29	bm	bm	PROPN
ejpam-2095	82	30	(	(	PUNCT
ejpam-2095	82	31	q	q	NOUN
ejpam-2095	82	32	:	:	PUNCT
ejpam-2095	82	33	i	i	PRON
ejpam-2095	82	34	m	m	VERB
ejpam-2095	82	35	)	)	PUNCT
ejpam-2095	82	36	for	for	ADP
ejpam-2095	82	37	any	any	DET
ejpam-2095	82	38	positive	positive	ADJ
ejpam-2095	82	39	integer	integer	NOUN
ejpam-2095	82	40	m.	m.	NOUN
ejpam-2095	82	41	in	in	ADP
ejpam-2095	82	42	particular	particular	ADJ
ejpam-2095	82	43	bn	bn	INTJ
ejpam-2095	82	44	(	(	PUNCT
ejpam-2095	82	45	q	q	NOUN
ejpam-2095	82	46	:	:	PUNCT
ejpam-2095	82	47	i	i	PRON
ejpam-2095	82	48	m	m	PROPN
ejpam-2095	82	49	)	)	PUNCT
ejpam-2095	82	50	.	.	PUNCT
ejpam-2095	83	1	as	as	SCONJ
ejpam-2095	83	2	q	q	PROPN
ejpam-2095	83	3	is	be	AUX
ejpam-2095	83	4	primary	primary	ADJ
ejpam-2095	83	5	,	,	PUNCT
ejpam-2095	83	6	(	(	PUNCT
ejpam-2095	83	7	an)k	an)k	PROPN
ejpam-2095	83	8	¶	¶	PROPN
ejpam-2095	83	9	(	(	PUNCT
ejpam-2095	83	10	q	q	NOUN
ejpam-2095	83	11	:	:	PUNCT
ejpam-2095	83	12	i	i	PRON
ejpam-2095	83	13	m	m	VERB
ejpam-2095	83	14	)	)	PUNCT
ejpam-2095	83	15	for	for	ADP
ejpam-2095	83	16	some	some	DET
ejpam-2095	83	17	positive	positive	ADJ
ejpam-2095	83	18	integer	integer	NOUN
ejpam-2095	83	19	k.	k.	PROPN
ejpam-2095	84	1	that	that	PRON
ejpam-2095	84	2	is	be	AUX
ejpam-2095	84	3	at	at	ADP
ejpam-2095	84	4	¶	¶	PROPN
ejpam-2095	84	5	(	(	PUNCT
ejpam-2095	84	6	q	q	NOUN
ejpam-2095	84	7	:	:	PUNCT
ejpam-2095	84	8	i	i	PRON
ejpam-2095	84	9	m	m	VERB
ejpam-2095	84	10	)	)	PUNCT
ejpam-2095	84	11	for	for	ADP
ejpam-2095	84	12	some	some	DET
ejpam-2095	84	13	positive	positive	ADJ
ejpam-2095	84	14	integer	integer	NOUN
ejpam-2095	84	15	t	t	PROPN
ejpam-2095	84	16	and	and	CCONJ
ejpam-2095	84	17	a	a	DET
ejpam-2095	84	18	¶	¶	NOUN
ejpam-2095	84	19	p	p	NOUN
ejpam-2095	84	20	(	(	PUNCT
ejpam-2095	84	21	q	q	NOUN
ejpam-2095	84	22	:	:	PUNCT
ejpam-2095	84	23	i	i	PRON
ejpam-2095	84	24	m	m	PROPN
ejpam-2095	84	25	)	)	PUNCT
ejpam-2095	84	26	.	.	PUNCT
ejpam-2095	85	1	therefore	therefore	ADV
ejpam-2095	85	2	p	p	X
ejpam-2095	85	3	(	(	PUNCT
ejpam-2095	85	4	q	q	NOUN
ejpam-2095	85	5	:	:	PUNCT
ejpam-2095	85	6	i	i	PRON
ejpam-2095	85	7	m	m	PROPN
ejpam-2095	85	8	)	)	PUNCT
ejpam-2095	85	9	is	be	AUX
ejpam-2095	85	10	prime	prime	ADJ
ejpam-2095	85	11	.	.	PUNCT
ejpam-2095	86	1	let	let	VERB
ejpam-2095	86	2	a	a	DET
ejpam-2095	86	3	¶	¶	PROPN
ejpam-2095	86	4	p.	p.	NOUN
ejpam-2095	86	5	take	take	VERB
ejpam-2095	86	6	any	any	DET
ejpam-2095	86	7	x	x	SYM
ejpam-2095	86	8	¶	¶	PROPN
ejpam-2095	86	9	p	p	NOUN
ejpam-2095	86	10	a.	a.	NOUN
ejpam-2095	86	11	then	then	ADV
ejpam-2095	86	12	xn	xn	PROPN
ejpam-2095	86	13	¶	¶	PROPN
ejpam-2095	86	14	a	a	DET
ejpam-2095	86	15	¶	¶	PROPN
ejpam-2095	86	16	p	p	NOUN
ejpam-2095	86	17	for	for	ADP
ejpam-2095	86	18	some	some	DET
ejpam-2095	86	19	positive	positive	ADJ
ejpam-2095	86	20	integer	integer	NOUN
ejpam-2095	86	21	n.	n.	NOUN
ejpam-2095	86	22	as	as	SCONJ
ejpam-2095	86	23	p	p	PROPN
ejpam-2095	86	24	is	be	AUX
ejpam-2095	86	25	prime	prime	ADJ
ejpam-2095	86	26	,	,	PUNCT
ejpam-2095	86	27	x	x	PROPN
ejpam-2095	87	1	¶	¶	NOUN
ejpam-2095	87	2	p	p	NOUN
ejpam-2095	87	3	and	and	CCONJ
ejpam-2095	87	4	hence	hence	ADV
ejpam-2095	87	5	p	p	X
ejpam-2095	87	6	a	a	DET
ejpam-2095	87	7	¶	¶	PROPN
ejpam-2095	87	8	p.	p.	NOUN
ejpam-2095	87	9	the	the	DET
ejpam-2095	87	10	following	follow	VERB
ejpam-2095	87	11	theorem	theorem	NOUN
ejpam-2095	87	12	gives	give	VERB
ejpam-2095	87	13	the	the	DET
ejpam-2095	87	14	relation	relation	NOUN
ejpam-2095	87	15	between	between	ADP
ejpam-2095	87	16	meet	meet	NOUN
ejpam-2095	87	17	of	of	ADP
ejpam-2095	87	18	primary	primary	ADJ
ejpam-2095	87	19	elements	element	NOUN
ejpam-2095	87	20	and	and	CCONJ
ejpam-2095	87	21	their	their	PRON
ejpam-2095	87	22	equal	equal	ADJ
ejpam-2095	87	23	associated	associated	ADJ
ejpam-2095	87	24	primes	prime	NOUN
ejpam-2095	87	25	.	.	PUNCT
ejpam-2095	88	1	theorem	theorem	NOUN
ejpam-2095	88	2	2	2	NUM
ejpam-2095	88	3	.	.	PUNCT
ejpam-2095	89	1	if	if	SCONJ
ejpam-2095	89	2	q1,q2	q1,q2	PROPN
ejpam-2095	89	3	,	,	PUNCT
ejpam-2095	89	4	.	.	PUNCT
ejpam-2095	89	5	.	.	PUNCT
ejpam-2095	90	1	.	.	PUNCT
ejpam-2095	91	1	,	,	PUNCT
ejpam-2095	91	2	qk	qk	NOUN
ejpam-2095	91	3	are	be	AUX
ejpam-2095	91	4	p	p	ADJ
ejpam-2095	91	5	-	-	PUNCT
ejpam-2095	91	6	primary	primary	ADJ
ejpam-2095	91	7	elements	element	NOUN
ejpam-2095	91	8	of	of	ADP
ejpam-2095	91	9	a	a	DET
ejpam-2095	91	10	lattice	lattice	NOUN
ejpam-2095	91	11	module	module	NOUN
ejpam-2095	91	12	m	m	PROPN
ejpam-2095	91	13	then	then	ADV
ejpam-2095	91	14	q1∧q2∧.	q1∧q2∧.	PROPN
ejpam-2095	91	15	.	.	PUNCT
ejpam-2095	91	16	.∧qk	.∧qk	PROPN
ejpam-2095	92	1	is	be	AUX
ejpam-2095	92	2	p	p	NOUN
ejpam-2095	92	3	-	-	PUNCT
ejpam-2095	92	4	primary	primary	ADJ
ejpam-2095	92	5	.	.	PUNCT
ejpam-2095	93	1	proof	proof	NOUN
ejpam-2095	93	2	.	.	PUNCT
ejpam-2095	94	1	by	by	ADP
ejpam-2095	94	2	hypothesis	hypothesis	NOUN
ejpam-2095	94	3	p	p	NOUN
ejpam-2095	94	4	(	(	PUNCT
ejpam-2095	94	5	q	q	NOUN
ejpam-2095	94	6	i	i	PRON
ejpam-2095	94	7	:	:	PUNCT
ejpam-2095	94	8	i	i	PRON
ejpam-2095	94	9	m	m	VERB
ejpam-2095	94	10	)	)	PUNCT
ejpam-2095	95	1	=	=	SYM
ejpam-2095	96	1	p	p	X
ejpam-2095	96	2	,	,	PUNCT
ejpam-2095	96	3	i	i	NOUN
ejpam-2095	96	4	=	=	NOUN
ejpam-2095	96	5	1,2	1,2	NUM
ejpam-2095	96	6	,	,	PUNCT
ejpam-2095	96	7	.	.	PUNCT
ejpam-2095	96	8	.	.	PUNCT
ejpam-2095	96	9	.	.	PUNCT
ejpam-2095	97	1	,	,	PUNCT
ejpam-2095	97	2	k.	k.	PROPN
ejpam-2095	97	3	let	let	VERB
ejpam-2095	97	4	q	q	NOUN
ejpam-2095	97	5	=	=	PUNCT
ejpam-2095	97	6	q1	q1	PROPN
ejpam-2095	97	7	∧q2	∧q2	VERB
ejpam-2095	97	8	∧	∧	PROPN
ejpam-2095	97	9	.	.	PUNCT
ejpam-2095	97	10	.	.	PUNCT
ejpam-2095	97	11	.	.	PUNCT
ejpam-2095	98	1	∧qk	∧qk	PROPN
ejpam-2095	98	2	.	.	PUNCT
ejpam-2095	99	1	we	we	PRON
ejpam-2095	99	2	have	have	VERB
ejpam-2095	99	3	,	,	PUNCT
ejpam-2095	99	4	p	p	X
ejpam-2095	99	5	∧q	∧q	PROPN
ejpam-2095	99	6	i	i	PRON
ejpam-2095	99	7	=	=	NOUN
ejpam-2095	99	8	p	p	X
ejpam-2095	99	9	(	(	PUNCT
ejpam-2095	99	10	(	(	PUNCT
ejpam-2095	99	11	∧q	∧q	PROPN
ejpam-2095	99	12	i	i	PROPN
ejpam-2095	99	13	)	)	PUNCT
ejpam-2095	99	14	:	:	PUNCT
ejpam-2095	100	1	i	i	PRON
ejpam-2095	100	2	m	m	VERB
ejpam-2095	100	3	)	)	PUNCT
ejpam-2095	101	1	=	=	SYM
ejpam-2095	101	2	p	p	X
ejpam-2095	101	3	(	(	PUNCT
ejpam-2095	101	4	q1	q1	PROPN
ejpam-2095	101	5	:	:	PUNCT
ejpam-2095	101	6	i	i	PRON
ejpam-2095	101	7	m	m	VERB
ejpam-2095	101	8	)	)	PUNCT
ejpam-2095	102	1	∧	∧	PROPN
ejpam-2095	102	2	p	p	NOUN
ejpam-2095	102	3	(	(	PUNCT
ejpam-2095	102	4	q2	q2	NOUN
ejpam-2095	102	5	:	:	PUNCT
ejpam-2095	102	6	i	i	PRON
ejpam-2095	102	7	m	m	VERB
ejpam-2095	102	8	)	)	PUNCT
ejpam-2095	102	9	∧	∧	PROPN
ejpam-2095	102	10	.	.	PUNCT
ejpam-2095	102	11	.	.	PUNCT
ejpam-2095	102	12	.	.	PUNCT
ejpam-2095	103	1	∧	∧	NOUN
ejpam-2095	103	2	p	p	NOUN
ejpam-2095	103	3	(	(	PUNCT
ejpam-2095	103	4	qk	qk	NOUN
ejpam-2095	103	5	:	:	PUNCT
ejpam-2095	103	6	i	i	PRON
ejpam-2095	103	7	m	m	VERB
ejpam-2095	103	8	)	)	PUNCT
ejpam-2095	104	1	=	=	PUNCT
ejpam-2095	105	1	p.	p.	NOUN
ejpam-2095	105	2	we	we	PRON
ejpam-2095	105	3	show	show	VERB
ejpam-2095	105	4	that	that	SCONJ
ejpam-2095	105	5	∧q	∧q	PROPN
ejpam-2095	105	6	i	i	PRON
ejpam-2095	105	7	is	be	AUX
ejpam-2095	105	8	primary	primary	ADJ
ejpam-2095	105	9	,	,	PUNCT
ejpam-2095	105	10	where	where	SCONJ
ejpam-2095	105	11	i	i	PRON
ejpam-2095	105	12	=	=	SYM
ejpam-2095	105	13	1,2	1,2	NUM
ejpam-2095	105	14	,	,	PUNCT
ejpam-2095	105	15	.	.	PUNCT
ejpam-2095	105	16	.	.	PUNCT
ejpam-2095	105	17	.	.	PUNCT
ejpam-2095	106	1	,	,	PUNCT
ejpam-2095	106	2	k.	k.	PROPN
ejpam-2095	106	3	let	let	VERB
ejpam-2095	106	4	ax	ax	NOUN
ejpam-2095	106	5	¶	¶	PROPN
ejpam-2095	106	6	q	q	PROPN
ejpam-2095	107	1	=	=	SYM
ejpam-2095	107	2	∧q	∧q	PROPN
ejpam-2095	107	3	i	i	PRON
ejpam-2095	107	4	where	where	SCONJ
ejpam-2095	107	5	a	a	DET
ejpam-2095	107	6	∈	∈	PROPN
ejpam-2095	107	7	l	l	NOUN
ejpam-2095	107	8	,	,	PUNCT
ejpam-2095	107	9	x	x	SYM
ejpam-2095	107	10	∈	∈	PROPN
ejpam-2095	107	11	m	m	VERB
ejpam-2095	107	12	.	.	PUNCT
ejpam-2095	107	13	suppose	suppose	VERB
ejpam-2095	107	14	,	,	PUNCT
ejpam-2095	107	15	x	x	PUNCT
ejpam-2095	107	16	q.	q.	PROPN
ejpam-2095	107	17	then	then	ADV
ejpam-2095	107	18	x	x	X
ejpam-2095	107	19	q	q	PROPN
ejpam-2095	107	20	i	i	PRON
ejpam-2095	107	21	for	for	ADP
ejpam-2095	107	22	some	some	DET
ejpam-2095	107	23	i	i	PRON
ejpam-2095	107	24	(	(	PUNCT
ejpam-2095	107	25	1	1	NUM
ejpam-2095	107	26	¶	¶	NUM
ejpam-2095	107	27	i	i	PRON
ejpam-2095	107	28	¶	¶	PROPN
ejpam-2095	107	29	k	k	PROPN
ejpam-2095	107	30	)	)	PUNCT
ejpam-2095	107	31	.	.	PUNCT
ejpam-2095	108	1	as	as	SCONJ
ejpam-2095	108	2	q	q	PROPN
ejpam-2095	108	3	i	i	PRON
ejpam-2095	108	4	is	be	AUX
ejpam-2095	108	5	primary	primary	ADJ
ejpam-2095	108	6	,	,	PUNCT
ejpam-2095	108	7	ax	ax	PROPN
ejpam-2095	108	8	¶	¶	PROPN
ejpam-2095	108	9	q	q	PROPN
ejpam-2095	109	1	i	i	PROPN
ejpam-2095	109	2	and	and	CCONJ
ejpam-2095	109	3	x	x	SYM
ejpam-2095	109	4	q	q	NOUN
ejpam-2095	109	5	i	i	PRON
ejpam-2095	109	6	implies	imply	VERB
ejpam-2095	109	7	a	a	DET
ejpam-2095	109	8	¶	¶	PROPN
ejpam-2095	109	9	p	p	NOUN
ejpam-2095	109	10	(	(	PUNCT
ejpam-2095	109	11	q	q	NOUN
ejpam-2095	109	12	i	i	X
ejpam-2095	109	13	:	:	PUNCT
ejpam-2095	109	14	i	i	PRON
ejpam-2095	109	15	m	m	VERB
ejpam-2095	109	16	)	)	PUNCT
ejpam-2095	110	1	=	=	PUNCT
ejpam-2095	111	1	p.	p.	NOUN
ejpam-2095	111	2	that	that	PRON
ejpam-2095	111	3	is	be	AUX
ejpam-2095	111	4	a	a	DET
ejpam-2095	111	5	¶	¶	PROPN
ejpam-2095	111	6	p	p	NOUN
ejpam-2095	111	7	(	(	PUNCT
ejpam-2095	111	8	q	q	NOUN
ejpam-2095	111	9	:	:	PUNCT
ejpam-2095	111	10	i	i	PRON
ejpam-2095	111	11	m	m	PROPN
ejpam-2095	111	12	)	)	PUNCT
ejpam-2095	111	13	.	.	PUNCT
ejpam-2095	112	1	therefore	therefore	ADV
ejpam-2095	112	2	,	,	PUNCT
ejpam-2095	112	3	q	q	PROPN
ejpam-2095	112	4	is	be	AUX
ejpam-2095	112	5	primary	primary	ADJ
ejpam-2095	112	6	.	.	PUNCT
ejpam-2095	113	1	it	it	PRON
ejpam-2095	113	2	is	be	AUX
ejpam-2095	113	3	shown	show	VERB
ejpam-2095	113	4	by	by	ADP
ejpam-2095	113	5	thakare	thakare	NOUN
ejpam-2095	113	6	and	and	CCONJ
ejpam-2095	113	7	manjarekar	manjarekar	NOUN
ejpam-2095	114	1	[	[	X
ejpam-2095	114	2	6	6	NUM
ejpam-2095	114	3	]	]	PUNCT
ejpam-2095	114	4	that	that	SCONJ
ejpam-2095	114	5	the	the	DET
ejpam-2095	114	6	radical	radical	NOUN
ejpam-2095	114	7	of	of	ADP
ejpam-2095	114	8	any	any	DET
ejpam-2095	114	9	element	element	NOUN
ejpam-2095	114	10	a	a	PRON
ejpam-2095	114	11	of	of	ADP
ejpam-2095	114	12	a	a	DET
ejpam-2095	114	13	multiplicative	multiplicative	ADJ
ejpam-2095	114	14	lattice	lattice	NOUN
ejpam-2095	114	15	satisfing	satisfe	VERB
ejpam-2095	114	16	the	the	DET
ejpam-2095	114	17	acc	acc	PROPN
ejpam-2095	114	18	can	can	AUX
ejpam-2095	114	19	be	be	AUX
ejpam-2095	114	20	written	write	VERB
ejpam-2095	114	21	as	as	ADP
ejpam-2095	114	22	the	the	DET
ejpam-2095	114	23	meet	meet	NOUN
ejpam-2095	114	24	of	of	ADP
ejpam-2095	114	25	minimal	minimal	ADJ
ejpam-2095	114	26	prime	prime	ADJ
ejpam-2095	114	27	divisors	divisor	NOUN
ejpam-2095	114	28	of	of	ADP
ejpam-2095	114	29	a.	a.	NOUN
ejpam-2095	114	30	hence	hence	ADV
ejpam-2095	114	31	,	,	PUNCT
ejpam-2095	114	32	we	we	PRON
ejpam-2095	114	33	have	have	VERB
ejpam-2095	114	34	the	the	DET
ejpam-2095	114	35	following	follow	VERB
ejpam-2095	114	36	result	result	NOUN
ejpam-2095	114	37	.	.	PUNCT
ejpam-2095	115	1	theorem	theorem	NOUN
ejpam-2095	115	2	3	3	X
ejpam-2095	115	3	.	.	PUNCT
ejpam-2095	116	1	let	let	VERB
ejpam-2095	116	2	l	l	NOUN
ejpam-2095	116	3	be	be	AUX
ejpam-2095	116	4	a	a	DET
ejpam-2095	116	5	multiplicative	multiplicative	ADJ
ejpam-2095	116	6	lattice	lattice	NOUN
ejpam-2095	116	7	satisfing	satisfe	VERB
ejpam-2095	116	8	the	the	DET
ejpam-2095	116	9	acc	acc	PROPN
ejpam-2095	116	10	and	and	CCONJ
ejpam-2095	116	11	n	n	CCONJ
ejpam-2095	116	12	be	be	VERB
ejpam-2095	116	13	an	an	DET
ejpam-2095	116	14	element	element	NOUN
ejpam-2095	116	15	of	of	ADP
ejpam-2095	116	16	m	m	PRON
ejpam-2095	116	17	then	then	ADV
ejpam-2095	116	18	p	p	X
ejpam-2095	116	19	(	(	PUNCT
ejpam-2095	116	20	n	n	NUM
ejpam-2095	116	21	:	:	PUNCT
ejpam-2095	116	22	i	i	PRON
ejpam-2095	116	23	m	m	VERB
ejpam-2095	116	24	)	)	PUNCT
ejpam-2095	117	1	=	=	PUNCT
ejpam-2095	118	1	∧{p	∧{p	PROPN
ejpam-2095	119	1	|	|	ADV
ejpam-2095	119	2	p	p	NOUN
ejpam-2095	119	3	is	be	AUX
ejpam-2095	119	4	minimal	minimal	ADJ
ejpam-2095	119	5	prime	prime	ADJ
ejpam-2095	119	6	containing	contain	VERB
ejpam-2095	119	7	(	(	PUNCT
ejpam-2095	119	8	n	n	NUM
ejpam-2095	119	9	:	:	PUNCT
ejpam-2095	119	10	i	i	PRON
ejpam-2095	119	11	m	m	VERB
ejpam-2095	119	12	)	)	PUNCT
ejpam-2095	119	13	}	}	PUNCT
ejpam-2095	119	14	.	.	PUNCT
ejpam-2095	120	1	an	an	DET
ejpam-2095	120	2	element	element	NOUN
ejpam-2095	120	3	n	n	PROPN
ejpam-2095	120	4	of	of	ADP
ejpam-2095	120	5	a	a	DET
ejpam-2095	120	6	lattice	lattice	NOUN
ejpam-2095	120	7	module	module	NOUN
ejpam-2095	120	8	m	m	NOUN
ejpam-2095	120	9	is	be	AUX
ejpam-2095	120	10	said	say	VERB
ejpam-2095	120	11	to	to	PART
ejpam-2095	120	12	be	be	AUX
ejpam-2095	120	13	meet	meet	VERB
ejpam-2095	120	14	irreducible	irreducible	ADJ
ejpam-2095	120	15	if	if	SCONJ
ejpam-2095	120	16	for	for	ADP
ejpam-2095	120	17	any	any	DET
ejpam-2095	120	18	two	two	NUM
ejpam-2095	120	19	elements	element	NOUN
ejpam-2095	120	20	a1	a1	NOUN
ejpam-2095	120	21	and	and	CCONJ
ejpam-2095	120	22	a2	a2	PROPN
ejpam-2095	120	23	of	of	ADP
ejpam-2095	120	24	m	m	PROPN
ejpam-2095	120	25	,	,	PUNCT
ejpam-2095	120	26	n	n	NOUN
ejpam-2095	120	27	=	=	NOUN
ejpam-2095	120	28	a1	a1	NOUN
ejpam-2095	120	29	∧	∧	PROPN
ejpam-2095	120	30	a2	a2	PROPN
ejpam-2095	120	31	implies	imply	VERB
ejpam-2095	120	32	either	either	CCONJ
ejpam-2095	120	33	a1	a1	NOUN
ejpam-2095	120	34	=	=	SYM
ejpam-2095	120	35	n	n	NOUN
ejpam-2095	120	36	or	or	CCONJ
ejpam-2095	120	37	a2	a2	PROPN
ejpam-2095	120	38	=	=	SYM
ejpam-2095	120	39	n	n	X
ejpam-2095	120	40	.	.	PUNCT
ejpam-2095	121	1	theorem	theorem	ADJ
ejpam-2095	121	2	4	4	NUM
ejpam-2095	121	3	.	.	PUNCT
ejpam-2095	122	1	if	if	SCONJ
ejpam-2095	122	2	a	a	DET
ejpam-2095	122	3	lattice	lattice	NOUN
ejpam-2095	122	4	module	module	NOUN
ejpam-2095	122	5	m	m	NOUN
ejpam-2095	122	6	satisfies	satisfy	VERB
ejpam-2095	122	7	acc	acc	PROPN
ejpam-2095	122	8	the	the	DET
ejpam-2095	122	9	chain	chain	NOUN
ejpam-2095	122	10	a1	a1	PROPN
ejpam-2095	122	11	¶	¶	PROPN
ejpam-2095	122	12	a2	a2	PROPN
ejpam-2095	122	13	¶	¶	PROPN
ejpam-2095	122	14	.	.	PUNCT
ejpam-2095	122	15	.	.	PUNCT
ejpam-2095	123	1	.	.	PUNCT
ejpam-2095	124	1	implies	imply	VERB
ejpam-2095	124	2	there	there	PRON
ejpam-2095	124	3	exist	exist	VERB
ejpam-2095	124	4	positive	positive	ADJ
ejpam-2095	124	5	integer	integer	NOUN
ejpam-2095	124	6	m	m	VERB
ejpam-2095	124	7	such	such	ADJ
ejpam-2095	124	8	that	that	SCONJ
ejpam-2095	124	9	an	an	DET
ejpam-2095	124	10	=	=	NOUN
ejpam-2095	124	11	am	be	AUX
ejpam-2095	124	12	for	for	ADP
ejpam-2095	124	13	all	all	PRON
ejpam-2095	124	14	n	n	DET
ejpam-2095	124	15	≥	≥	NOUN
ejpam-2095	124	16	m.	m.	NOUN
ejpam-2095	124	17	then	then	ADV
ejpam-2095	124	18	every	every	DET
ejpam-2095	124	19	element	element	NOUN
ejpam-2095	124	20	of	of	ADP
ejpam-2095	124	21	m	m	PROPN
ejpam-2095	124	22	can	can	AUX
ejpam-2095	124	23	be	be	AUX
ejpam-2095	124	24	written	write	VERB
ejpam-2095	124	25	as	as	ADP
ejpam-2095	124	26	the	the	DET
ejpam-2095	124	27	meet	meet	NOUN
ejpam-2095	124	28	of	of	ADP
ejpam-2095	124	29	a	a	DET
ejpam-2095	124	30	finite	finite	ADJ
ejpam-2095	124	31	number	number	NOUN
ejpam-2095	124	32	of	of	ADP
ejpam-2095	124	33	meet	meet	VERB
ejpam-2095	124	34	irreducible	irreducible	ADJ
ejpam-2095	124	35	elements	element	NOUN
ejpam-2095	124	36	of	of	ADP
ejpam-2095	124	37	m.	m.	NOUN
ejpam-2095	124	38	proof	proof	NOUN
ejpam-2095	124	39	.	.	PUNCT
ejpam-2095	125	1	let	let	VERB
ejpam-2095	125	2	τ	τ	PROPN
ejpam-2095	125	3	be	be	AUX
ejpam-2095	125	4	the	the	DET
ejpam-2095	125	5	set	set	NOUN
ejpam-2095	125	6	of	of	ADP
ejpam-2095	125	7	all	all	DET
ejpam-2095	125	8	elements	element	NOUN
ejpam-2095	125	9	of	of	ADP
ejpam-2095	125	10	m	m	PRON
ejpam-2095	125	11	which	which	PRON
ejpam-2095	125	12	can	can	AUX
ejpam-2095	125	13	not	not	PART
ejpam-2095	125	14	be	be	AUX
ejpam-2095	125	15	written	write	VERB
ejpam-2095	125	16	as	as	ADP
ejpam-2095	125	17	a	a	DET
ejpam-2095	125	18	meet	meet	NOUN
ejpam-2095	125	19	of	of	ADP
ejpam-2095	125	20	a	a	DET
ejpam-2095	125	21	finite	finite	ADJ
ejpam-2095	125	22	number	number	NOUN
ejpam-2095	125	23	of	of	ADP
ejpam-2095	125	24	meet	meet	VERB
ejpam-2095	125	25	irreducible	irreducible	ADJ
ejpam-2095	125	26	elements	element	NOUN
ejpam-2095	125	27	of	of	ADP
ejpam-2095	125	28	m	m	PROPN
ejpam-2095	125	29	.	.	PUNCT
ejpam-2095	126	1	if	if	SCONJ
ejpam-2095	126	2	τ	τ	PROPN
ejpam-2095	126	3	is	be	AUX
ejpam-2095	126	4	empty	empty	ADJ
ejpam-2095	126	5	we	we	PRON
ejpam-2095	126	6	have	have	VERB
ejpam-2095	126	7	nothing	nothing	PRON
ejpam-2095	126	8	to	to	PART
ejpam-2095	126	9	prove	prove	VERB
ejpam-2095	126	10	.	.	PUNCT
ejpam-2095	127	1	suppose	suppose	VERB
ejpam-2095	127	2	τ	τ	PROPN
ejpam-2095	127	3	is	be	AUX
ejpam-2095	127	4	not	not	PART
ejpam-2095	127	5	empty	empty	ADJ
ejpam-2095	127	6	.	.	PUNCT
ejpam-2095	128	1	as	as	ADP
ejpam-2095	128	2	m	m	PROPN
ejpam-2095	128	3	satifies	satifies	PROPN
ejpam-2095	128	4	acc	acc	PROPN
ejpam-2095	128	5	,	,	PUNCT
ejpam-2095	128	6	τ	τ	PROPN
ejpam-2095	128	7	has	have	VERB
ejpam-2095	128	8	a	a	DET
ejpam-2095	128	9	maximal	maximal	ADJ
ejpam-2095	128	10	element	element	NOUN
ejpam-2095	128	11	say	say	VERB
ejpam-2095	128	12	n	n	ADV
ejpam-2095	128	13	.	.	PUNCT
ejpam-2095	129	1	as	as	SCONJ
ejpam-2095	129	2	n	n	PROPN
ejpam-2095	129	3	∈	∈	PROPN
ejpam-2095	129	4	τ	τ	PROPN
ejpam-2095	129	5	,	,	PUNCT
ejpam-2095	129	6	n	n	PRON
ejpam-2095	129	7	is	be	AUX
ejpam-2095	129	8	not	not	PART
ejpam-2095	129	9	irreducible	irreducible	ADJ
ejpam-2095	129	10	.	.	PUNCT
ejpam-2095	130	1	so	so	ADV
ejpam-2095	130	2	there	there	PRON
ejpam-2095	130	3	exists	exist	VERB
ejpam-2095	130	4	elements	element	NOUN
ejpam-2095	130	5	a1	a1	NOUN
ejpam-2095	130	6	and	and	CCONJ
ejpam-2095	130	7	a2	a2	PROPN
ejpam-2095	130	8	of	of	ADP
ejpam-2095	130	9	m	m	PROPN
ejpam-2095	130	10	such	such	ADJ
ejpam-2095	130	11	that	that	SCONJ
ejpam-2095	130	12	n	n	NOUN
ejpam-2095	130	13	=	=	NOUN
ejpam-2095	130	14	a1	a1	NOUN
ejpam-2095	130	15	∧	∧	PROPN
ejpam-2095	130	16	a2	a2	PROPN
ejpam-2095	130	17	where	where	SCONJ
ejpam-2095	130	18	n	n	X
ejpam-2095	130	19	6=	6=	ADP
ejpam-2095	130	20	a1	a1	NOUN
ejpam-2095	130	21	,	,	PUNCT
ejpam-2095	130	22	n	n	CCONJ
ejpam-2095	130	23	6=	6=	NUM
ejpam-2095	130	24	a2	a2	PROPN
ejpam-2095	130	25	.	.	PUNCT
ejpam-2095	131	1	so	so	ADV
ejpam-2095	131	2	,	,	PUNCT
ejpam-2095	131	3	n	n	CCONJ
ejpam-2095	131	4	<	<	X
ejpam-2095	131	5	a1	a1	NOUN
ejpam-2095	131	6	,	,	PUNCT
ejpam-2095	131	7	n	n	CCONJ
ejpam-2095	131	8	<	<	X
ejpam-2095	131	9	a2	a2	PROPN
ejpam-2095	131	10	.	.	PUNCT
ejpam-2095	132	1	this	this	PRON
ejpam-2095	132	2	shows	show	VERB
ejpam-2095	132	3	that	that	SCONJ
ejpam-2095	132	4	both	both	DET
ejpam-2095	132	5	a1	a1	NOUN
ejpam-2095	132	6	and	and	CCONJ
ejpam-2095	132	7	a2	a2	PROPN
ejpam-2095	132	8	can	can	AUX
ejpam-2095	132	9	be	be	AUX
ejpam-2095	132	10	written	write	VERB
ejpam-2095	132	11	as	as	ADP
ejpam-2095	132	12	the	the	DET
ejpam-2095	132	13	meet	meet	NOUN
ejpam-2095	132	14	of	of	ADP
ejpam-2095	132	15	a	a	DET
ejpam-2095	132	16	finite	finite	ADJ
ejpam-2095	132	17	number	number	NOUN
ejpam-2095	132	18	of	of	ADP
ejpam-2095	132	19	meet	meet	VERB
ejpam-2095	132	20	irreducible	irreducible	ADJ
ejpam-2095	132	21	elements	element	NOUN
ejpam-2095	132	22	of	of	ADP
ejpam-2095	132	23	m	m	PROPN
ejpam-2095	132	24	.	.	PUNCT
ejpam-2095	133	1	so	so	ADV
ejpam-2095	133	2	there	there	PRON
ejpam-2095	133	3	are	be	VERB
ejpam-2095	133	4	irreducible	irreducible	ADJ
ejpam-2095	133	5	elements	element	NOUN
ejpam-2095	133	6	k1	k1	NOUN
ejpam-2095	133	7	,	,	PUNCT
ejpam-2095	133	8	k2	k2	NOUN
ejpam-2095	133	9	,	,	PUNCT
ejpam-2095	133	10	.	.	PUNCT
ejpam-2095	133	11	.	.	PUNCT
ejpam-2095	134	1	.	.	PUNCT
ejpam-2095	135	1	,	,	PUNCT
ejpam-2095	135	2	km	km	NOUN
ejpam-2095	135	3	and	and	CCONJ
ejpam-2095	135	4	k	k	PROPN
ejpam-2095	136	1	′	′	NUM
ejpam-2095	136	2	1	1	NUM
ejpam-2095	136	3	,	,	PUNCT
ejpam-2095	136	4	k	k	PROPN
ejpam-2095	136	5	′	′	NUM
ejpam-2095	136	6	2	2	NUM
ejpam-2095	136	7	,	,	PUNCT
ejpam-2095	136	8	.	.	PUNCT
ejpam-2095	136	9	.	.	PUNCT
ejpam-2095	136	10	.	.	PUNCT
ejpam-2095	137	1	,	,	PUNCT
ejpam-2095	138	1	k	k	PROPN
ejpam-2095	138	2	′	′	NUM
ejpam-2095	138	3	n	n	PROPN
ejpam-2095	138	4	of	of	ADP
ejpam-2095	138	5	m	m	PRON
ejpam-2095	138	6	such	such	ADJ
ejpam-2095	138	7	that	that	DET
ejpam-2095	138	8	a1	a1	NOUN
ejpam-2095	138	9	=	=	SYM
ejpam-2095	138	10	k1	k1	NOUN
ejpam-2095	138	11	∧	∧	PROPN
ejpam-2095	138	12	k2	k2	PROPN
ejpam-2095	138	13	∧	∧	PROPN
ejpam-2095	138	14	.	.	PUNCT
ejpam-2095	138	15	.	.	PUNCT
ejpam-2095	139	1	.∧	.∧	PUNCT
ejpam-2095	140	1	km	km	NOUN
ejpam-2095	140	2	and	and	CCONJ
ejpam-2095	140	3	a2	a2	PROPN
ejpam-2095	140	4	=	=	SYM
ejpam-2095	141	1	k	k	PROPN
ejpam-2095	142	1	′	′	NOUN
ejpam-2095	142	2	1	1	NUM
ejpam-2095	142	3	∧k	∧k	NUM
ejpam-2095	142	4	′	′	NUM
ejpam-2095	142	5	2	2	NUM
ejpam-2095	142	6	∧	∧	NOUN
ejpam-2095	142	7	.	.	PUNCT
ejpam-2095	142	8	.	.	PUNCT
ejpam-2095	142	9	.∧	.∧	PUNCT
ejpam-2095	143	1	k	k	X
ejpam-2095	144	1	′	′	NUM
ejpam-2095	144	2	n.	n.	NOUN
ejpam-2095	145	1	but	but	CCONJ
ejpam-2095	145	2	then	then	ADV
ejpam-2095	145	3	n	n	PROPN
ejpam-2095	145	4	=	=	SYM
ejpam-2095	145	5	k1	k1	PROPN
ejpam-2095	145	6	∧k2	∧k2	NOUN
ejpam-2095	145	7	.	.	PUNCT
ejpam-2095	145	8	.	.	PUNCT
ejpam-2095	146	1	.∧	.∧	PUNCT
ejpam-2095	147	1	km	km	NOUN
ejpam-2095	147	2	∧k	∧k	NUM
ejpam-2095	147	3	′	′	NUM
ejpam-2095	147	4	1	1	NUM
ejpam-2095	148	1	∧	∧	PROPN
ejpam-2095	148	2	k	k	NOUN
ejpam-2095	148	3	′	′	NOUN
ejpam-2095	148	4	2	2	NUM
ejpam-2095	148	5	∧	∧	NOUN
ejpam-2095	148	6	.	.	PUNCT
ejpam-2095	148	7	.	.	PUNCT
ejpam-2095	148	8	.	.	PUNCT
ejpam-2095	149	1	k	k	NOUN
ejpam-2095	150	1	′	′	NUM
ejpam-2095	150	2	n.	n.	NOUN
ejpam-2095	150	3	that	that	PRON
ejpam-2095	150	4	is	be	AUX
ejpam-2095	150	5	n	n	PRON
ejpam-2095	150	6	is	be	AUX
ejpam-2095	150	7	the	the	DET
ejpam-2095	150	8	meet	meet	NOUN
ejpam-2095	150	9	of	of	ADP
ejpam-2095	150	10	a	a	DET
ejpam-2095	150	11	finite	finite	ADJ
ejpam-2095	150	12	number	number	NOUN
ejpam-2095	150	13	of	of	ADP
ejpam-2095	150	14	meet	meet	ADJ
ejpam-2095	150	15	irreducible	irreducible	ADJ
ejpam-2095	150	16	elements	element	NOUN
ejpam-2095	150	17	.	.	PUNCT
ejpam-2095	151	1	this	this	PRON
ejpam-2095	151	2	contradicts	contradict	VERB
ejpam-2095	151	3	the	the	DET
ejpam-2095	151	4	fact	fact	NOUN
ejpam-2095	151	5	that	that	SCONJ
ejpam-2095	151	6	n	n	PRON
ejpam-2095	151	7	∈	∈	PROPN
ejpam-2095	151	8	τ	τ	PROPN
ejpam-2095	151	9	.	.	PUNCT
ejpam-2095	152	1	hence	hence	ADV
ejpam-2095	152	2	,	,	PUNCT
ejpam-2095	152	3	τ	τ	PROPN
ejpam-2095	152	4	is	be	AUX
ejpam-2095	152	5	empty	empty	ADJ
ejpam-2095	152	6	.	.	PUNCT
ejpam-2095	153	1	c.	c.	PROPN
ejpam-2095	153	2	manjarekar	manjarekar	PROPN
ejpam-2095	153	3	,	,	PUNCT
ejpam-2095	153	4	u.	u.	PROPN
ejpam-2095	153	5	kandale	kandale	PROPN
ejpam-2095	153	6	/	/	SYM
ejpam-2095	153	7	eur	eur	PROPN
ejpam-2095	153	8	.	.	PUNCT
ejpam-2095	154	1	j.	j.	PROPN
ejpam-2095	154	2	pure	pure	PROPN
ejpam-2095	154	3	appl	appl	PROPN
ejpam-2095	154	4	.	.	PROPN
ejpam-2095	154	5	math	math	PROPN
ejpam-2095	154	6	,	,	PUNCT
ejpam-2095	154	7	7	7	NUM
ejpam-2095	154	8	(	(	PUNCT
ejpam-2095	154	9	2014	2014	NUM
ejpam-2095	154	10	)	)	PUNCT
ejpam-2095	154	11	,	,	PUNCT
ejpam-2095	154	12	201	201	NUM
ejpam-2095	154	13	-	-	SYM
ejpam-2095	154	14	209	209	NUM
ejpam-2095	154	15	204	204	NUM
ejpam-2095	154	16	the	the	DET
ejpam-2095	154	17	study	study	NOUN
ejpam-2095	154	18	of	of	ADP
ejpam-2095	154	19	primary	primary	ADJ
ejpam-2095	154	20	elements	element	NOUN
ejpam-2095	154	21	and	and	CCONJ
ejpam-2095	154	22	their	their	PRON
ejpam-2095	154	23	associated	associate	VERB
ejpam-2095	154	24	primes	prime	NOUN
ejpam-2095	154	25	for	for	ADP
ejpam-2095	154	26	modules	module	NOUN
ejpam-2095	154	27	is	be	AUX
ejpam-2095	154	28	carried	carry	VERB
ejpam-2095	154	29	out	out	ADP
ejpam-2095	154	30	by	by	ADP
ejpam-2095	154	31	p	p	PROPN
ejpam-2095	154	32	j	j	PROPN
ejpam-2095	154	33	mc	mc	PROPN
ejpam-2095	154	34	carthy	carthy	PROPN
ejpam-2095	154	35	and	and	CCONJ
ejpam-2095	154	36	larsen	larsen	PROPN
ejpam-2095	155	1	[	[	X
ejpam-2095	155	2	5	5	NUM
ejpam-2095	155	3	]	]	PUNCT
ejpam-2095	155	4	.	.	PUNCT
ejpam-2095	156	1	we	we	PRON
ejpam-2095	156	2	give	give	VERB
ejpam-2095	156	3	eqivalent	eqivalent	NOUN
ejpam-2095	156	4	formulation	formulation	NOUN
ejpam-2095	156	5	in	in	ADP
ejpam-2095	156	6	the	the	DET
ejpam-2095	156	7	next	next	ADJ
ejpam-2095	156	8	theorems	theorem	NOUN
ejpam-2095	156	9	for	for	ADP
ejpam-2095	156	10	lattice	lattice	NOUN
ejpam-2095	156	11	modules	module	NOUN
ejpam-2095	156	12	.	.	PUNCT
ejpam-2095	157	1	theorem	theorem	NOUN
ejpam-2095	157	2	5	5	NUM
ejpam-2095	157	3	.	.	PUNCT
ejpam-2095	158	1	let	let	VERB
ejpam-2095	158	2	q	q	PART
ejpam-2095	158	3	be	be	AUX
ejpam-2095	158	4	a	a	DET
ejpam-2095	158	5	p	p	NOUN
ejpam-2095	158	6	-	-	PUNCT
ejpam-2095	158	7	primary	primary	ADJ
ejpam-2095	158	8	element	element	NOUN
ejpam-2095	158	9	of	of	ADP
ejpam-2095	158	10	lattice	lattice	PROPN
ejpam-2095	158	11	module	module	NOUN
ejpam-2095	158	12	m	m	PROPN
ejpam-2095	158	13	and	and	CCONJ
ejpam-2095	158	14	n	n	ADV
ejpam-2095	158	15	be	be	VERB
ejpam-2095	158	16	an	an	DET
ejpam-2095	158	17	element	element	NOUN
ejpam-2095	158	18	of	of	ADP
ejpam-2095	158	19	m.	m.	NOUN
ejpam-2095	158	20	if	if	SCONJ
ejpam-2095	158	21	n	n	ADV
ejpam-2095	158	22	q	q	NOUN
ejpam-2095	159	1	then	then	ADV
ejpam-2095	159	2	(	(	PUNCT
ejpam-2095	159	3	q	q	NOUN
ejpam-2095	159	4	:	:	PUNCT
ejpam-2095	159	5	n	n	CCONJ
ejpam-2095	159	6	)	)	PUNCT
ejpam-2095	159	7	is	be	AUX
ejpam-2095	159	8	a	a	DET
ejpam-2095	159	9	p	p	NOUN
ejpam-2095	159	10	-	-	PUNCT
ejpam-2095	159	11	primary	primary	ADJ
ejpam-2095	159	12	element	element	NOUN
ejpam-2095	159	13	.	.	PUNCT
ejpam-2095	160	1	proof	proof	NOUN
ejpam-2095	160	2	.	.	PUNCT
ejpam-2095	161	1	first	first	ADV
ejpam-2095	161	2	we	we	PRON
ejpam-2095	161	3	show	show	VERB
ejpam-2095	161	4	that	that	SCONJ
ejpam-2095	161	5	(	(	PUNCT
ejpam-2095	161	6	q	q	NOUN
ejpam-2095	161	7	:	:	PUNCT
ejpam-2095	161	8	n	n	CCONJ
ejpam-2095	161	9	)	)	PUNCT
ejpam-2095	161	10	is	be	AUX
ejpam-2095	161	11	a	a	DET
ejpam-2095	161	12	p	p	NOUN
ejpam-2095	161	13	-	-	PUNCT
ejpam-2095	161	14	primary	primary	ADJ
ejpam-2095	161	15	element	element	NOUN
ejpam-2095	161	16	.	.	PUNCT
ejpam-2095	162	1	let	let	VERB
ejpam-2095	162	2	a	a	DET
ejpam-2095	162	3	,	,	PUNCT
ejpam-2095	162	4	b	b	PROPN
ejpam-2095	162	5	∈	∈	PROPN
ejpam-2095	162	6	l	l	NOUN
ejpam-2095	162	7	,	,	PUNCT
ejpam-2095	162	8	ab	ab	PROPN
ejpam-2095	162	9	¶	¶	PROPN
ejpam-2095	162	10	(	(	PUNCT
ejpam-2095	162	11	q	q	NOUN
ejpam-2095	162	12	:	:	PUNCT
ejpam-2095	162	13	n	n	CCONJ
ejpam-2095	162	14	)	)	PUNCT
ejpam-2095	162	15	and	and	CCONJ
ejpam-2095	162	16	suppose	suppose	VERB
ejpam-2095	162	17	,	,	PUNCT
ejpam-2095	162	18	a	a	DET
ejpam-2095	162	19	(	(	PUNCT
ejpam-2095	162	20	q	q	NOUN
ejpam-2095	162	21	:	:	PUNCT
ejpam-2095	162	22	n	n	CCONJ
ejpam-2095	162	23	)	)	PUNCT
ejpam-2095	162	24	.	.	PUNCT
ejpam-2095	163	1	as	as	ADP
ejpam-2095	163	2	a	a	DET
ejpam-2095	163	3	(	(	PUNCT
ejpam-2095	163	4	q	q	NOUN
ejpam-2095	163	5	:	:	PUNCT
ejpam-2095	163	6	n	n	CCONJ
ejpam-2095	163	7	)	)	PUNCT
ejpam-2095	163	8	,	,	PUNCT
ejpam-2095	163	9	an	an	DET
ejpam-2095	163	10	q.	q.	NOUN
ejpam-2095	163	11	also	also	ADV
ejpam-2095	163	12	as	as	ADP
ejpam-2095	163	13	ab	ab	PROPN
ejpam-2095	163	14	¶	¶	PROPN
ejpam-2095	163	15	(	(	PUNCT
ejpam-2095	163	16	q	q	NOUN
ejpam-2095	163	17	:	:	PUNCT
ejpam-2095	163	18	n	n	CCONJ
ejpam-2095	163	19	)	)	PUNCT
ejpam-2095	163	20	,	,	PUNCT
ejpam-2095	163	21	abn	abn	PROPN
ejpam-2095	163	22	¶	¶	PROPN
ejpam-2095	163	23	q.	q.	PROPN
ejpam-2095	163	24	but	but	CCONJ
ejpam-2095	163	25	an	an	DET
ejpam-2095	163	26	q	q	NOUN
ejpam-2095	163	27	and	and	CCONJ
ejpam-2095	163	28	q	q	NOUN
ejpam-2095	163	29	is	be	AUX
ejpam-2095	163	30	a	a	DET
ejpam-2095	163	31	primary	primary	ADJ
ejpam-2095	163	32	element	element	NOUN
ejpam-2095	163	33	implies	imply	VERB
ejpam-2095	163	34	that	that	SCONJ
ejpam-2095	164	1	bn	bn	PROPN
ejpam-2095	164	2	¶	¶	INTJ
ejpam-2095	164	3	(	(	PUNCT
ejpam-2095	164	4	q	q	NOUN
ejpam-2095	164	5	:	:	PUNCT
ejpam-2095	164	6	i	i	PRON
ejpam-2095	164	7	m	m	VERB
ejpam-2095	164	8	)	)	PUNCT
ejpam-2095	164	9	for	for	ADP
ejpam-2095	164	10	some	some	DET
ejpam-2095	164	11	integer	integer	NOUN
ejpam-2095	164	12	n.	n.	NOUN
ejpam-2095	164	13	but	but	CCONJ
ejpam-2095	164	14	bn	bn	INTJ
ejpam-2095	164	15	i	i	NOUN
ejpam-2095	164	16	m	m	VERB
ejpam-2095	164	17	¶	¶	PROPN
ejpam-2095	164	18	q	q	PROPN
ejpam-2095	164	19	implies	imply	VERB
ejpam-2095	164	20	bnn	bnn	PROPN
ejpam-2095	164	21	¶	¶	PROPN
ejpam-2095	164	22	q.	q.	PROPN
ejpam-2095	164	23	hence	hence	ADV
ejpam-2095	164	24	,	,	PUNCT
ejpam-2095	164	25	b	b	PROPN
ejpam-2095	164	26	¶	¶	NUM
ejpam-2095	164	27	p	p	NOUN
ejpam-2095	164	28	(	(	PUNCT
ejpam-2095	164	29	q	q	NOUN
ejpam-2095	164	30	:	:	PUNCT
ejpam-2095	164	31	n	n	CCONJ
ejpam-2095	164	32	)	)	PUNCT
ejpam-2095	164	33	.	.	PUNCT
ejpam-2095	165	1	therefore	therefore	ADV
ejpam-2095	165	2	,	,	PUNCT
ejpam-2095	165	3	(	(	PUNCT
ejpam-2095	165	4	q	q	NOUN
ejpam-2095	165	5	:	:	PUNCT
ejpam-2095	165	6	n	n	CCONJ
ejpam-2095	165	7	)	)	PUNCT
ejpam-2095	165	8	is	be	AUX
ejpam-2095	165	9	a	a	DET
ejpam-2095	165	10	primary	primary	ADJ
ejpam-2095	165	11	element	element	NOUN
ejpam-2095	165	12	of	of	ADP
ejpam-2095	165	13	l.	l.	PROPN
ejpam-2095	165	14	now	now	ADV
ejpam-2095	165	15	since	since	SCONJ
ejpam-2095	165	16	n	n	PROPN
ejpam-2095	165	17	q	q	NOUN
ejpam-2095	165	18	,	,	PUNCT
ejpam-2095	165	19	there	there	PRON
ejpam-2095	165	20	exists	exist	VERB
ejpam-2095	165	21	a∈	a∈	PROPN
ejpam-2095	165	22	m	m	PROPN
ejpam-2095	165	23	and	and	CCONJ
ejpam-2095	165	24	a¶	a¶	X
ejpam-2095	165	25	n	n	PRON
ejpam-2095	165	26	such	such	ADJ
ejpam-2095	165	27	that	that	SCONJ
ejpam-2095	165	28	a	a	DET
ejpam-2095	165	29	q.	q.	NOUN
ejpam-2095	165	30	let	let	VERB
ejpam-2095	165	31	a	a	DET
ejpam-2095	165	32	¶	¶	PROPN
ejpam-2095	165	33	p	p	NOUN
ejpam-2095	165	34	(	(	PUNCT
ejpam-2095	165	35	q	q	NOUN
ejpam-2095	165	36	:	:	PUNCT
ejpam-2095	165	37	n	n	CCONJ
ejpam-2095	165	38	)	)	PUNCT
ejpam-2095	165	39	.	.	PUNCT
ejpam-2095	166	1	then	then	ADV
ejpam-2095	166	2	ann	ann	PROPN
ejpam-2095	166	3	¶	¶	PROPN
ejpam-2095	166	4	q.	q.	PROPN
ejpam-2095	166	5	hence	hence	ADV
ejpam-2095	166	6	,	,	PUNCT
ejpam-2095	166	7	ana	ana	PROPN
ejpam-2095	166	8	¶	¶	PROPN
ejpam-2095	166	9	q.	q.	PROPN
ejpam-2095	167	1	but	but	CCONJ
ejpam-2095	167	2	a	a	DET
ejpam-2095	167	3	q	q	NOUN
ejpam-2095	167	4	and	and	CCONJ
ejpam-2095	167	5	q	q	NOUN
ejpam-2095	167	6	is	be	AUX
ejpam-2095	167	7	primary	primary	ADJ
ejpam-2095	167	8	implies	imply	VERB
ejpam-2095	167	9	that	that	SCONJ
ejpam-2095	167	10	(	(	PUNCT
ejpam-2095	167	11	an)k	an)k	PROPN
ejpam-2095	167	12	=	=	PRON
ejpam-2095	167	13	am	be	AUX
ejpam-2095	167	14	¶	¶	NOUN
ejpam-2095	167	15	(	(	PUNCT
ejpam-2095	167	16	q	q	NOUN
ejpam-2095	167	17	:	:	PUNCT
ejpam-2095	167	18	i	i	PRON
ejpam-2095	167	19	m	m	VERB
ejpam-2095	167	20	)	)	PUNCT
ejpam-2095	167	21	for	for	ADP
ejpam-2095	167	22	some	some	DET
ejpam-2095	167	23	integer	integer	NOUN
ejpam-2095	167	24	m.	m.	NOUN
ejpam-2095	167	25	that	that	PRON
ejpam-2095	167	26	is	be	AUX
ejpam-2095	167	27	a	a	DET
ejpam-2095	167	28	¶	¶	PROPN
ejpam-2095	167	29	p	p	NOUN
ejpam-2095	167	30	(	(	PUNCT
ejpam-2095	167	31	q	q	NOUN
ejpam-2095	167	32	:	:	PUNCT
ejpam-2095	167	33	i	i	PRON
ejpam-2095	167	34	m	m	VERB
ejpam-2095	167	35	)	)	PUNCT
ejpam-2095	168	1	=	=	SYM
ejpam-2095	169	1	p	p	NOUN
ejpam-2095	169	2	and	and	CCONJ
ejpam-2095	169	3	p	p	X
ejpam-2095	169	4	(	(	PUNCT
ejpam-2095	169	5	q	q	NOUN
ejpam-2095	169	6	:	:	PUNCT
ejpam-2095	169	7	n	n	NUM
ejpam-2095	169	8	)	)	PUNCT
ejpam-2095	169	9	¶	¶	PROPN
ejpam-2095	169	10	p.	p.	NOUN
ejpam-2095	170	1	conversely	conversely	ADV
ejpam-2095	170	2	,	,	PUNCT
ejpam-2095	170	3	let	let	VERB
ejpam-2095	170	4	a	a	DET
ejpam-2095	170	5	¶	¶	PROPN
ejpam-2095	170	6	p	p	NOUN
ejpam-2095	170	7	(	(	PUNCT
ejpam-2095	170	8	q	q	NOUN
ejpam-2095	170	9	:	:	PUNCT
ejpam-2095	170	10	i	i	PRON
ejpam-2095	170	11	m	m	VERB
ejpam-2095	170	12	)	)	PUNCT
ejpam-2095	171	1	=	=	SYM
ejpam-2095	172	1	p.	p.	NOUN
ejpam-2095	172	2	hence	hence	ADV
ejpam-2095	172	3	,	,	PUNCT
ejpam-2095	172	4	anim	anim	PROPN
ejpam-2095	172	5	¶	¶	PROPN
ejpam-2095	172	6	q	q	PROPN
ejpam-2095	172	7	for	for	ADP
ejpam-2095	172	8	some	some	DET
ejpam-2095	172	9	integer	integer	NOUN
ejpam-2095	172	10	n.	n.	NOUN
ejpam-2095	172	11	so	so	ADV
ejpam-2095	172	12	ann	ann	PROPN
ejpam-2095	172	13	¶	¶	PROPN
ejpam-2095	172	14	q	q	PROPN
ejpam-2095	172	15	for	for	ADP
ejpam-2095	172	16	some	some	DET
ejpam-2095	172	17	integer	integer	NOUN
ejpam-2095	172	18	n.	n.	NOUN
ejpam-2095	172	19	thus	thus	ADV
ejpam-2095	172	20	an	an	DET
ejpam-2095	172	21	¶	¶	NOUN
ejpam-2095	172	22	(	(	PUNCT
ejpam-2095	172	23	q	q	NOUN
ejpam-2095	172	24	:	:	PUNCT
ejpam-2095	172	25	n	n	CCONJ
ejpam-2095	172	26	)	)	PUNCT
ejpam-2095	172	27	and	and	CCONJ
ejpam-2095	172	28	a	a	DET
ejpam-2095	172	29	¶	¶	NOUN
ejpam-2095	172	30	p	p	NOUN
ejpam-2095	172	31	(	(	PUNCT
ejpam-2095	172	32	q	q	NOUN
ejpam-2095	172	33	:	:	PUNCT
ejpam-2095	172	34	n	n	CCONJ
ejpam-2095	172	35	)	)	PUNCT
ejpam-2095	172	36	.	.	PUNCT
ejpam-2095	173	1	this	this	PRON
ejpam-2095	173	2	shows	show	VERB
ejpam-2095	173	3	that	that	SCONJ
ejpam-2095	173	4	p	p	PROPN
ejpam-2095	173	5	¶	¶	PROPN
ejpam-2095	173	6	p	p	NOUN
ejpam-2095	173	7	(	(	PUNCT
ejpam-2095	173	8	q	q	NOUN
ejpam-2095	173	9	:	:	PUNCT
ejpam-2095	173	10	n	n	CCONJ
ejpam-2095	173	11	)	)	PUNCT
ejpam-2095	173	12	and	and	CCONJ
ejpam-2095	173	13	we	we	PRON
ejpam-2095	173	14	have	have	VERB
ejpam-2095	173	15	p	p	NOUN
ejpam-2095	173	16	(	(	PUNCT
ejpam-2095	173	17	q	q	NOUN
ejpam-2095	173	18	:	:	PUNCT
ejpam-2095	173	19	n	n	X
ejpam-2095	173	20	)	)	PUNCT
ejpam-2095	173	21	=	=	VERB
ejpam-2095	174	1	p.	p.	NOUN
ejpam-2095	174	2	therefore,(q	therefore,(q	INTJ
ejpam-2095	174	3	:	:	PUNCT
ejpam-2095	174	4	n	n	X
ejpam-2095	174	5	)	)	PUNCT
ejpam-2095	174	6	is	be	AUX
ejpam-2095	174	7	a	a	DET
ejpam-2095	174	8	p	p	NOUN
ejpam-2095	174	9	-	-	PUNCT
ejpam-2095	174	10	primary	primary	ADJ
ejpam-2095	174	11	element	element	NOUN
ejpam-2095	174	12	.	.	PUNCT
ejpam-2095	175	1	theorem	theorem	VERB
ejpam-2095	175	2	6	6	NUM
ejpam-2095	175	3	.	.	PUNCT
ejpam-2095	176	1	let	let	VERB
ejpam-2095	176	2	m	m	PRON
ejpam-2095	176	3	be	be	AUX
ejpam-2095	176	4	a	a	DET
ejpam-2095	176	5	lattice	lattice	NOUN
ejpam-2095	176	6	module	module	NOUN
ejpam-2095	176	7	and	and	CCONJ
ejpam-2095	176	8	a	a	DET
ejpam-2095	176	9	be	be	AUX
ejpam-2095	176	10	an	an	DET
ejpam-2095	176	11	element	element	NOUN
ejpam-2095	176	12	of	of	ADP
ejpam-2095	176	13	l	l	NOUN
ejpam-2095	176	14	,	,	PUNCT
ejpam-2095	176	15	p	p	PROPN
ejpam-2095	176	16	be	be	AUX
ejpam-2095	176	17	a	a	DET
ejpam-2095	176	18	prime	prime	ADJ
ejpam-2095	176	19	element	element	NOUN
ejpam-2095	176	20	of	of	ADP
ejpam-2095	176	21	l	l	PROPN
ejpam-2095	176	22	and	and	CCONJ
ejpam-2095	176	23	q	q	AUX
ejpam-2095	176	24	be	be	AUX
ejpam-2095	176	25	p	p	NOUN
ejpam-2095	176	26	-	-	PUNCT
ejpam-2095	176	27	primary	primary	ADJ
ejpam-2095	176	28	element	element	NOUN
ejpam-2095	176	29	of	of	ADP
ejpam-2095	176	30	m.	m.	NOUN
ejpam-2095	176	31	if	if	SCONJ
ejpam-2095	176	32	a	a	DET
ejpam-2095	176	33	p	p	NOUN
ejpam-2095	176	34	then	then	ADV
ejpam-2095	176	35	(	(	PUNCT
ejpam-2095	176	36	q	q	NOUN
ejpam-2095	176	37	:	:	PUNCT
ejpam-2095	176	38	a	a	X
ejpam-2095	176	39	)	)	PUNCT
ejpam-2095	176	40	=	=	SYM
ejpam-2095	176	41	q.	q.	NOUN
ejpam-2095	176	42	proof	proof	NOUN
ejpam-2095	176	43	.	.	PUNCT
ejpam-2095	177	1	suppose	suppose	VERB
ejpam-2095	177	2	a	a	DET
ejpam-2095	177	3	p	p	NOUN
ejpam-2095	177	4	where	where	SCONJ
ejpam-2095	177	5	p	p	NOUN
ejpam-2095	177	6	=	=	X
ejpam-2095	177	7	p	p	X
ejpam-2095	177	8	(	(	PUNCT
ejpam-2095	177	9	q	q	NOUN
ejpam-2095	177	10	:	:	PUNCT
ejpam-2095	177	11	i	i	PRON
ejpam-2095	177	12	m	m	PROPN
ejpam-2095	177	13	)	)	PUNCT
ejpam-2095	177	14	.	.	PUNCT
ejpam-2095	178	1	since	since	SCONJ
ejpam-2095	178	2	,	,	PUNCT
ejpam-2095	178	3	a	a	DET
ejpam-2095	178	4	p	p	NOUN
ejpam-2095	178	5	there	there	PRON
ejpam-2095	178	6	is	be	VERB
ejpam-2095	178	7	some	some	DET
ejpam-2095	178	8	b	b	NOUN
ejpam-2095	178	9	¶	¶	NOUN
ejpam-2095	178	10	a	a	PRON
ejpam-2095	178	11	such	such	ADJ
ejpam-2095	178	12	that	that	DET
ejpam-2095	178	13	b	b	NOUN
ejpam-2095	179	1	p.	p.	NOUN
ejpam-2095	179	2	let	let	VERB
ejpam-2095	179	3	x	x	SYM
ejpam-2095	179	4	¶	¶	VERB
ejpam-2095	179	5	(	(	PUNCT
ejpam-2095	179	6	q	q	NOUN
ejpam-2095	179	7	:	:	PUNCT
ejpam-2095	179	8	a	a	X
ejpam-2095	179	9	)	)	PUNCT
ejpam-2095	179	10	.	.	PUNCT
ejpam-2095	180	1	then	then	ADV
ejpam-2095	180	2	ax	ax	VERB
ejpam-2095	180	3	¶	¶	PROPN
ejpam-2095	180	4	q	q	PROPN
ejpam-2095	180	5	and	and	CCONJ
ejpam-2095	180	6	hence	hence	ADV
ejpam-2095	180	7	bx	bx	PROPN
ejpam-2095	180	8	¶	¶	PROPN
ejpam-2095	180	9	q	q	PROPN
ejpam-2095	180	10	where	where	SCONJ
ejpam-2095	180	11	b	b	X
ejpam-2095	180	12	p	p	X
ejpam-2095	180	13	(	(	PUNCT
ejpam-2095	180	14	q	q	NOUN
ejpam-2095	180	15	:	:	PUNCT
ejpam-2095	180	16	i	i	PRON
ejpam-2095	180	17	m	m	VERB
ejpam-2095	180	18	)	)	PUNCT
ejpam-2095	181	1	=	=	SYM
ejpam-2095	182	1	p.	p.	NOUN
ejpam-2095	182	2	as	as	SCONJ
ejpam-2095	182	3	q	q	PROPN
ejpam-2095	182	4	is	be	AUX
ejpam-2095	182	5	a	a	DET
ejpam-2095	182	6	p	p	NOUN
ejpam-2095	182	7	-	-	PUNCT
ejpam-2095	182	8	primary	primary	NOUN
ejpam-2095	182	9	,	,	PUNCT
ejpam-2095	182	10	x	x	PROPN
ejpam-2095	182	11	¶	¶	PROPN
ejpam-2095	182	12	q.	q.	PROPN
ejpam-2095	182	13	hence	hence	ADV
ejpam-2095	182	14	,	,	PUNCT
ejpam-2095	182	15	(	(	PUNCT
ejpam-2095	182	16	q	q	NOUN
ejpam-2095	182	17	:	:	PUNCT
ejpam-2095	182	18	a	a	X
ejpam-2095	182	19	)	)	PUNCT
ejpam-2095	182	20	¶	¶	PROPN
ejpam-2095	182	21	q.	q.	PROPN
ejpam-2095	182	22	conversely	conversely	ADV
ejpam-2095	182	23	let	let	VERB
ejpam-2095	182	24	x	x	X
ejpam-2095	182	25	¶	¶	PROPN
ejpam-2095	182	26	q.	q.	PROPN
ejpam-2095	182	27	since	since	SCONJ
ejpam-2095	182	28	a	a	DET
ejpam-2095	182	29	¶	¶	PROPN
ejpam-2095	182	30	1	1	NUM
ejpam-2095	182	31	,	,	PUNCT
ejpam-2095	182	32	ax	ax	NOUN
ejpam-2095	182	33	¶	¶	PROPN
ejpam-2095	182	34	q.	q.	PROPN
ejpam-2095	183	1	so	so	ADV
ejpam-2095	183	2	x	x	SYM
ejpam-2095	183	3	¶	¶	PROPN
ejpam-2095	183	4	(	(	PUNCT
ejpam-2095	183	5	q	q	NOUN
ejpam-2095	183	6	:	:	PUNCT
ejpam-2095	183	7	a	a	X
ejpam-2095	183	8	)	)	PUNCT
ejpam-2095	183	9	and	and	CCONJ
ejpam-2095	183	10	hence	hence	ADV
ejpam-2095	183	11	q	q	PROPN
ejpam-2095	183	12	¶	¶	PROPN
ejpam-2095	183	13	(	(	PUNCT
ejpam-2095	183	14	q	q	NOUN
ejpam-2095	183	15	:	:	PUNCT
ejpam-2095	183	16	a	a	X
ejpam-2095	183	17	)	)	PUNCT
ejpam-2095	183	18	.	.	PUNCT
ejpam-2095	184	1	therefore	therefore	ADV
ejpam-2095	184	2	,	,	PUNCT
ejpam-2095	184	3	q	q	X
ejpam-2095	184	4	=	=	X
ejpam-2095	184	5	(	(	PUNCT
ejpam-2095	184	6	q	q	NOUN
ejpam-2095	184	7	:	:	PUNCT
ejpam-2095	184	8	a	a	X
ejpam-2095	184	9	)	)	PUNCT
ejpam-2095	184	10	.	.	PUNCT
ejpam-2095	185	1	the	the	DET
ejpam-2095	185	2	following	follow	VERB
ejpam-2095	185	3	theorem	theorem	NOUN
ejpam-2095	185	4	gives	give	VERB
ejpam-2095	185	5	the	the	DET
ejpam-2095	185	6	characterisation	characterisation	NOUN
ejpam-2095	185	7	of	of	ADP
ejpam-2095	185	8	a	a	DET
ejpam-2095	185	9	prime	prime	ADJ
ejpam-2095	185	10	element	element	NOUN
ejpam-2095	185	11	p	p	NOUN
ejpam-2095	185	12	of	of	ADP
ejpam-2095	185	13	l	l	NOUN
ejpam-2095	185	14	to	to	PART
ejpam-2095	185	15	be	be	AUX
ejpam-2095	185	16	equal	equal	ADJ
ejpam-2095	185	17	to	to	ADP
ejpam-2095	185	18	some	some	DET
ejpam-2095	185	19	associated	associated	ADJ
ejpam-2095	185	20	prime	prime	NOUN
ejpam-2095	185	21	of	of	ADP
ejpam-2095	185	22	an	an	DET
ejpam-2095	185	23	element	element	NOUN
ejpam-2095	185	24	which	which	PRON
ejpam-2095	185	25	has	have	VERB
ejpam-2095	185	26	a	a	DET
ejpam-2095	185	27	primary	primary	ADJ
ejpam-2095	185	28	decomposition	decomposition	NOUN
ejpam-2095	185	29	.	.	PUNCT
ejpam-2095	186	1	theorem	theorem	VERB
ejpam-2095	186	2	7	7	NUM
ejpam-2095	186	3	.	.	PUNCT
ejpam-2095	187	1	let	let	VERB
ejpam-2095	187	2	n	n	PRON
ejpam-2095	187	3	6=	6=	NUM
ejpam-2095	188	1	i	i	PRON
ejpam-2095	188	2	m	m	AUX
ejpam-2095	188	3	be	be	VERB
ejpam-2095	188	4	an	an	DET
ejpam-2095	188	5	element	element	NOUN
ejpam-2095	188	6	of	of	ADP
ejpam-2095	188	7	a	a	DET
ejpam-2095	188	8	lattice	lattice	NOUN
ejpam-2095	188	9	module	module	NOUN
ejpam-2095	188	10	m	m	PROPN
ejpam-2095	188	11	and	and	CCONJ
ejpam-2095	188	12	assume	assume	VERB
ejpam-2095	188	13	that	that	SCONJ
ejpam-2095	188	14	n	n	PRON
ejpam-2095	188	15	has	have	VERB
ejpam-2095	188	16	a	a	DET
ejpam-2095	188	17	primary	primary	ADJ
ejpam-2095	188	18	decomposition	decomposition	NOUN
ejpam-2095	188	19	.	.	PUNCT
ejpam-2095	189	1	let	let	VERB
ejpam-2095	189	2	n	n	PRON
ejpam-2095	189	3	=	=	PROPN
ejpam-2095	189	4	q1	q1	PROPN
ejpam-2095	189	5	∧q2	∧q2	VERB
ejpam-2095	189	6	∧	∧	PROPN
ejpam-2095	189	7	.	.	PUNCT
ejpam-2095	189	8	.	.	PUNCT
ejpam-2095	189	9	.	.	PUNCT
ejpam-2095	190	1	∧qk	∧qk	PROPN
ejpam-2095	190	2	be	be	AUX
ejpam-2095	190	3	a	a	DET
ejpam-2095	190	4	reduced	reduce	VERB
ejpam-2095	190	5	primary	primary	ADJ
ejpam-2095	190	6	decomposition	decomposition	NOUN
ejpam-2095	190	7	of	of	ADP
ejpam-2095	190	8	n	n	PROPN
ejpam-2095	190	9	and	and	CCONJ
ejpam-2095	190	10	p	p	NOUN
ejpam-2095	190	11	be	be	AUX
ejpam-2095	190	12	prime	prime	ADJ
ejpam-2095	190	13	element	element	NOUN
ejpam-2095	190	14	of	of	ADP
ejpam-2095	190	15	l.	l.	PROPN
ejpam-2095	190	16	then	then	ADV
ejpam-2095	190	17	p	p	PROPN
ejpam-2095	191	1	=	=	PUNCT
ejpam-2095	191	2	p	p	X
ejpam-2095	191	3	q	q	X
ejpam-2095	191	4	i	i	NOUN
ejpam-2095	191	5	for	for	ADP
ejpam-2095	191	6	some	some	DET
ejpam-2095	191	7	i	i	PRON
ejpam-2095	191	8	if	if	VERB
ejpam-2095	192	1	and	and	CCONJ
ejpam-2095	192	2	only	only	ADV
ejpam-2095	192	3	if	if	SCONJ
ejpam-2095	192	4	(	(	PUNCT
ejpam-2095	192	5	n	n	NUM
ejpam-2095	192	6	:	:	PUNCT
ejpam-2095	192	7	x	x	X
ejpam-2095	192	8	)	)	PUNCT
ejpam-2095	192	9	is	be	AUX
ejpam-2095	192	10	a	a	DET
ejpam-2095	192	11	p	p	NOUN
ejpam-2095	192	12	-	-	PUNCT
ejpam-2095	192	13	primary	primary	ADJ
ejpam-2095	192	14	element	element	NOUN
ejpam-2095	192	15	of	of	ADP
ejpam-2095	192	16	l	l	NOUN
ejpam-2095	192	17	for	for	ADP
ejpam-2095	192	18	some	some	DET
ejpam-2095	192	19	x	x	SYM
ejpam-2095	192	20	n.	n.	NOUN
ejpam-2095	192	21	proof	proof	NOUN
ejpam-2095	192	22	.	.	PUNCT
ejpam-2095	193	1	let	let	VERB
ejpam-2095	193	2	n	n	PRON
ejpam-2095	193	3	=	=	PROPN
ejpam-2095	193	4	q1	q1	PROPN
ejpam-2095	193	5	∧q2	∧q2	VERB
ejpam-2095	193	6	∧	∧	PROPN
ejpam-2095	193	7	.	.	PUNCT
ejpam-2095	193	8	.	.	PUNCT
ejpam-2095	194	1	.∧qk	.∧qk	PUNCT
ejpam-2095	194	2	be	be	AUX
ejpam-2095	194	3	a	a	DET
ejpam-2095	194	4	reduced	reduce	VERB
ejpam-2095	194	5	primary	primary	ADJ
ejpam-2095	194	6	decomposition	decomposition	NOUN
ejpam-2095	194	7	of	of	ADP
ejpam-2095	194	8	n	n	PROPN
ejpam-2095	194	9	.	.	PUNCT
ejpam-2095	195	1	first	first	ADV
ejpam-2095	195	2	suppose	suppose	VERB
ejpam-2095	195	3	that	that	SCONJ
ejpam-2095	195	4	,	,	PUNCT
ejpam-2095	195	5	p	p	X
ejpam-2095	195	6	=	=	NOUN
ejpam-2095	195	7	p	p	X
ejpam-2095	195	8	q	q	X
ejpam-2095	195	9	i	i	NOUN
ejpam-2095	195	10	for	for	ADP
ejpam-2095	195	11	some	some	DET
ejpam-2095	195	12	i.	i.	NOUN
ejpam-2095	195	13	without	without	ADP
ejpam-2095	195	14	loss	loss	NOUN
ejpam-2095	195	15	of	of	ADP
ejpam-2095	195	16	generality	generality	NOUN
ejpam-2095	195	17	we	we	PRON
ejpam-2095	195	18	can	can	AUX
ejpam-2095	195	19	assume	assume	VERB
ejpam-2095	195	20	that	that	SCONJ
ejpam-2095	195	21	p	p	NOUN
ejpam-2095	195	22	=	=	X
ejpam-2095	195	23	p	p	X
ejpam-2095	195	24	(	(	PUNCT
ejpam-2095	195	25	q1	q1	PROPN
ejpam-2095	195	26	:	:	PUNCT
ejpam-2095	195	27	i	i	PRON
ejpam-2095	195	28	m	m	VERB
ejpam-2095	195	29	)	)	PUNCT
ejpam-2095	195	30	where	where	SCONJ
ejpam-2095	195	31	pi	pi	NOUN
ejpam-2095	195	32	=	=	PUNCT
ejpam-2095	195	33	p	p	X
ejpam-2095	195	34	(	(	PUNCT
ejpam-2095	195	35	q	q	NOUN
ejpam-2095	196	1	i	i	PRON
ejpam-2095	196	2	:	:	PUNCT
ejpam-2095	196	3	i	i	PRON
ejpam-2095	196	4	m	m	VERB
ejpam-2095	196	5	)	)	PUNCT
ejpam-2095	197	1	i	i	NOUN
ejpam-2095	197	2	=	=	SYM
ejpam-2095	197	3	1,2	1,2	NUM
ejpam-2095	197	4	,	,	PUNCT
ejpam-2095	197	5	.	.	PUNCT
ejpam-2095	197	6	.	.	PUNCT
ejpam-2095	198	1	.	.	PUNCT
ejpam-2095	199	1	,	,	PUNCT
ejpam-2095	199	2	k.	k.	PROPN
ejpam-2095	199	3	we	we	PRON
ejpam-2095	199	4	prove	prove	VERB
ejpam-2095	199	5	that	that	SCONJ
ejpam-2095	199	6	,	,	PUNCT
ejpam-2095	199	7	(	(	PUNCT
ejpam-2095	199	8	n	n	X
ejpam-2095	199	9	:	:	PUNCT
ejpam-2095	199	10	x	x	X
ejpam-2095	199	11	)	)	PUNCT
ejpam-2095	199	12	is	be	AUX
ejpam-2095	199	13	a	a	DET
ejpam-2095	199	14	p	p	NOUN
ejpam-2095	199	15	-	-	PUNCT
ejpam-2095	199	16	primary	primary	ADJ
ejpam-2095	199	17	element	element	NOUN
ejpam-2095	199	18	of	of	ADP
ejpam-2095	199	19	l	l	NOUN
ejpam-2095	199	20	for	for	ADP
ejpam-2095	199	21	some	some	PRON
ejpam-2095	199	22	x	x	SYM
ejpam-2095	199	23	n	n	NOUN
ejpam-2095	199	24	.	.	PUNCT
ejpam-2095	200	1	since	since	SCONJ
ejpam-2095	200	2	the	the	DET
ejpam-2095	200	3	decomposition	decomposition	NOUN
ejpam-2095	200	4	is	be	AUX
ejpam-2095	200	5	reduced	reduce	VERB
ejpam-2095	200	6	q	q	PROPN
ejpam-2095	200	7	i	i	PROPN
ejpam-2095	200	8	�	�	PROPN
ejpam-2095	200	9	q1	q1	PROPN
ejpam-2095	200	10	∧q2	∧q2	VERB
ejpam-2095	200	11	∧	∧	PROPN
ejpam-2095	200	12	.	.	PUNCT
ejpam-2095	200	13	.	.	PUNCT
ejpam-2095	200	14	.	.	PUNCT
ejpam-2095	201	1	∧q	∧q	PROPN
ejpam-2095	202	1	i−1	i−1	PROPN
ejpam-2095	202	2	∧q	∧q	PROPN
ejpam-2095	202	3	i+1	i+1	ADJ
ejpam-2095	202	4	∧	∧	PROPN
ejpam-2095	202	5	.	.	PUNCT
ejpam-2095	202	6	.	.	PUNCT
ejpam-2095	202	7	.	.	PUNCT
ejpam-2095	203	1	∧qk	∧qk	NOUN
ejpam-2095	203	2	for	for	ADP
ejpam-2095	203	3	i	i	X
ejpam-2095	203	4	=	=	NOUN
ejpam-2095	203	5	1,2	1,2	NUM
ejpam-2095	203	6	,	,	PUNCT
ejpam-2095	203	7	.	.	PUNCT
ejpam-2095	203	8	.	.	PUNCT
ejpam-2095	204	1	.	.	PUNCT
ejpam-2095	205	1	,	,	PUNCT
ejpam-2095	205	2	k.	k.	PROPN
ejpam-2095	205	3	in	in	ADP
ejpam-2095	205	4	particular	particular	ADJ
ejpam-2095	205	5	,	,	PUNCT
ejpam-2095	205	6	q1	q1	PROPN
ejpam-2095	205	7	�	�	PROPN
ejpam-2095	205	8	q2	q2	PROPN
ejpam-2095	205	9	∧q3	∧q3	PROPN
ejpam-2095	205	10	∧	∧	PROPN
ejpam-2095	205	11	.	.	PUNCT
ejpam-2095	205	12	.	.	PUNCT
ejpam-2095	206	1	.∧qk	.∧qk	PROPN
ejpam-2095	206	2	.	.	PUNCT
ejpam-2095	207	1	so	so	ADV
ejpam-2095	207	2	there	there	PRON
ejpam-2095	207	3	exists	exist	VERB
ejpam-2095	207	4	x	x	PROPN
ejpam-2095	207	5	¶	¶	PROPN
ejpam-2095	207	6	q2	q2	PROPN
ejpam-2095	207	7	∧q3	∧q3	PROPN
ejpam-2095	207	8	∧	∧	PROPN
ejpam-2095	207	9	.	.	PUNCT
ejpam-2095	207	10	.	.	PUNCT
ejpam-2095	208	1	.∧qk	.∧qk	PUNCT
ejpam-2095	209	1	such	such	ADJ
ejpam-2095	209	2	that	that	SCONJ
ejpam-2095	209	3	x	x	SYM
ejpam-2095	209	4	q1	q1	PROPN
ejpam-2095	209	5	and	and	CCONJ
ejpam-2095	209	6	hence	hence	ADV
ejpam-2095	209	7	x	x	ADP
ejpam-2095	209	8	n	n	CCONJ
ejpam-2095	209	9	=	=	NUM
ejpam-2095	209	10	q1	q1	PROPN
ejpam-2095	209	11	∧q2	∧q2	VERB
ejpam-2095	209	12	∧	∧	PROPN
ejpam-2095	209	13	.	.	PUNCT
ejpam-2095	209	14	.	.	PUNCT
ejpam-2095	209	15	.	.	PUNCT
ejpam-2095	210	1	∧qk	∧qk	PROPN
ejpam-2095	210	2	.	.	PUNCT
ejpam-2095	211	1	also	also	ADV
ejpam-2095	211	2	(	(	PUNCT
ejpam-2095	211	3	n	n	X
ejpam-2095	211	4	:	:	PUNCT
ejpam-2095	211	5	x	x	X
ejpam-2095	211	6	)	)	PUNCT
ejpam-2095	211	7	=	=	SYM
ejpam-2095	211	8	(	(	PUNCT
ejpam-2095	211	9	q1	q1	PROPN
ejpam-2095	211	10	∧q2	∧q2	VERB
ejpam-2095	211	11	∧	∧	PROPN
ejpam-2095	211	12	.	.	PUNCT
ejpam-2095	211	13	.	.	PUNCT
ejpam-2095	212	1	.∧qk	.∧qk	NUM
ejpam-2095	212	2	)	)	PUNCT
ejpam-2095	213	1	:	:	PUNCT
ejpam-2095	213	2	x	x	X
ejpam-2095	213	3	=	=	SYM
ejpam-2095	213	4	(	(	PUNCT
ejpam-2095	213	5	q1	q1	INTJ
ejpam-2095	213	6	:	:	PUNCT
ejpam-2095	213	7	x	x	X
ejpam-2095	213	8	)	)	PUNCT
ejpam-2095	213	9	∧	∧	PROPN
ejpam-2095	213	10	(	(	PUNCT
ejpam-2095	213	11	q2	q2	NOUN
ejpam-2095	213	12	:	:	PUNCT
ejpam-2095	213	13	x	x	X
ejpam-2095	213	14	)	)	PUNCT
ejpam-2095	213	15	∧	∧	NOUN
ejpam-2095	213	16	.	.	PUNCT
ejpam-2095	213	17	.	.	PUNCT
ejpam-2095	213	18	.∧	.∧	PUNCT
ejpam-2095	214	1	(	(	PUNCT
ejpam-2095	214	2	qk	qk	INTJ
ejpam-2095	214	3	:	:	PUNCT
ejpam-2095	214	4	x	x	X
ejpam-2095	214	5	)	)	PUNCT
ejpam-2095	214	6	.	.	PUNCT
ejpam-2095	215	1	for	for	ADP
ejpam-2095	215	2	i	i	PRON
ejpam-2095	215	3	=	=	NOUN
ejpam-2095	215	4	2,3	2,3	NUM
ejpam-2095	215	5	,	,	PUNCT
ejpam-2095	215	6	.	.	PUNCT
ejpam-2095	215	7	.	.	PUNCT
ejpam-2095	215	8	.	.	PUNCT
ejpam-2095	216	1	,	,	PUNCT
ejpam-2095	216	2	k	k	PROPN
ejpam-2095	216	3	we	we	PRON
ejpam-2095	216	4	show	show	VERB
ejpam-2095	216	5	that	that	SCONJ
ejpam-2095	216	6	(	(	PUNCT
ejpam-2095	216	7	q	q	NOUN
ejpam-2095	216	8	i	i	PRON
ejpam-2095	216	9	:	:	PUNCT
ejpam-2095	216	10	x	x	X
ejpam-2095	216	11	)	)	PUNCT
ejpam-2095	217	1	=	=	SYM
ejpam-2095	217	2	1	1	X
ejpam-2095	217	3	.	.	PUNCT
ejpam-2095	218	1	since	since	SCONJ
ejpam-2095	218	2	x	x	PROPN
ejpam-2095	218	3	¶	¶	PROPN
ejpam-2095	218	4	q2∧q3	q2∧q3	PROPN
ejpam-2095	218	5	.	.	PUNCT
ejpam-2095	218	6	.	.	PUNCT
ejpam-2095	219	1	.∧qk	.∧qk	PROPN
ejpam-2095	219	2	,	,	PUNCT
ejpam-2095	219	3	we	we	PRON
ejpam-2095	219	4	have	have	VERB
ejpam-2095	219	5	x	x	X
ejpam-2095	219	6	¶	¶	PROPN
ejpam-2095	219	7	q	q	PROPN
ejpam-2095	219	8	i	i	PROPN
ejpam-2095	219	9	for	for	ADP
ejpam-2095	219	10	all	all	DET
ejpam-2095	219	11	i	i	PRON
ejpam-2095	219	12	=	=	NOUN
ejpam-2095	219	13	2	2	NUM
ejpam-2095	219	14	,	,	PUNCT
ejpam-2095	219	15	.	.	PUNCT
ejpam-2095	219	16	.	.	PUNCT
ejpam-2095	220	1	.	.	PUNCT
ejpam-2095	221	1	,	,	PUNCT
ejpam-2095	221	2	k.	k.	PROPN
ejpam-2095	221	3	then	then	ADV
ejpam-2095	221	4	ax	ax	VERB
ejpam-2095	221	5	¶	¶	PROPN
ejpam-2095	221	6	q	q	PROPN
ejpam-2095	221	7	i	i	PROPN
ejpam-2095	221	8	for	for	ADP
ejpam-2095	221	9	all	all	DET
ejpam-2095	221	10	a	a	DET
ejpam-2095	221	11	∈	∈	ADJ
ejpam-2095	221	12	l	l	NOUN
ejpam-2095	221	13	and	and	CCONJ
ejpam-2095	221	14	for	for	ADP
ejpam-2095	221	15	all	all	DET
ejpam-2095	221	16	i	i	NOUN
ejpam-2095	221	17	=	=	NOUN
ejpam-2095	221	18	2,3	2,3	NUM
ejpam-2095	221	19	,	,	PUNCT
ejpam-2095	221	20	.	.	PUNCT
ejpam-2095	221	21	.	.	PUNCT
ejpam-2095	221	22	.	.	PUNCT
ejpam-2095	222	1	,	,	PUNCT
ejpam-2095	222	2	k.	k.	PROPN
ejpam-2095	223	1	that	that	PRON
ejpam-2095	223	2	is	be	AUX
ejpam-2095	223	3	a	a	DET
ejpam-2095	223	4	¶	¶	NOUN
ejpam-2095	223	5	(	(	PUNCT
ejpam-2095	223	6	q	q	NOUN
ejpam-2095	223	7	i	i	PRON
ejpam-2095	223	8	:	:	PUNCT
ejpam-2095	223	9	x	x	X
ejpam-2095	223	10	)	)	PUNCT
ejpam-2095	223	11	for	for	ADP
ejpam-2095	223	12	all	all	DET
ejpam-2095	223	13	c.	c.	PROPN
ejpam-2095	223	14	manjarekar	manjarekar	PROPN
ejpam-2095	223	15	,	,	PUNCT
ejpam-2095	223	16	u.	u.	PROPN
ejpam-2095	223	17	kandale	kandale	PROPN
ejpam-2095	223	18	/	/	SYM
ejpam-2095	223	19	eur	eur	PROPN
ejpam-2095	223	20	.	.	PUNCT
ejpam-2095	224	1	j.	j.	PROPN
ejpam-2095	224	2	pure	pure	PROPN
ejpam-2095	224	3	appl	appl	PROPN
ejpam-2095	224	4	.	.	PROPN
ejpam-2095	224	5	math	math	PROPN
ejpam-2095	224	6	,	,	PUNCT
ejpam-2095	224	7	7	7	NUM
ejpam-2095	224	8	(	(	PUNCT
ejpam-2095	224	9	2014	2014	NUM
ejpam-2095	224	10	)	)	PUNCT
ejpam-2095	224	11	,	,	PUNCT
ejpam-2095	224	12	201	201	NUM
ejpam-2095	224	13	-	-	SYM
ejpam-2095	224	14	209	209	NUM
ejpam-2095	224	15	205	205	NUM
ejpam-2095	224	16	i	i	NOUN
ejpam-2095	224	17	=	=	PUNCT
ejpam-2095	224	18	2,3	2,3	NUM
ejpam-2095	224	19	,	,	PUNCT
ejpam-2095	224	20	.	.	PUNCT
ejpam-2095	224	21	.	.	PUNCT
ejpam-2095	224	22	.	.	PUNCT
ejpam-2095	225	1	,	,	PUNCT
ejpam-2095	225	2	k.	k.	PROPN
ejpam-2095	226	1	so	so	ADV
ejpam-2095	226	2	1¶	1¶	PROPN
ejpam-2095	226	3	(	(	PUNCT
ejpam-2095	226	4	q	q	NOUN
ejpam-2095	226	5	i	i	PRON
ejpam-2095	226	6	:	:	PUNCT
ejpam-2095	226	7	x	x	X
ejpam-2095	226	8	)	)	PUNCT
ejpam-2095	226	9	.	.	PUNCT
ejpam-2095	227	1	but	but	CCONJ
ejpam-2095	227	2	(	(	PUNCT
ejpam-2095	227	3	q	q	NOUN
ejpam-2095	227	4	i	i	PRON
ejpam-2095	227	5	:	:	PUNCT
ejpam-2095	227	6	x	x	X
ejpam-2095	227	7	)	)	PUNCT
ejpam-2095	227	8	¶	¶	NOUN
ejpam-2095	227	9	1	1	NUM
ejpam-2095	227	10	implies	imply	VERB
ejpam-2095	227	11	(	(	PUNCT
ejpam-2095	227	12	q	q	NOUN
ejpam-2095	227	13	i	i	PRON
ejpam-2095	227	14	:	:	PUNCT
ejpam-2095	227	15	x	x	X
ejpam-2095	227	16	)	)	PUNCT
ejpam-2095	227	17	=	=	SYM
ejpam-2095	227	18	1	1	NUM
ejpam-2095	227	19	for	for	ADP
ejpam-2095	227	20	i	i	PRON
ejpam-2095	227	21	=	=	NOUN
ejpam-2095	227	22	2,3	2,3	NUM
ejpam-2095	227	23	,	,	PUNCT
ejpam-2095	227	24	.	.	PUNCT
ejpam-2095	227	25	.	.	PUNCT
ejpam-2095	227	26	.	.	PUNCT
ejpam-2095	228	1	,	,	PUNCT
ejpam-2095	228	2	k.	k.	PROPN
ejpam-2095	228	3	hence	hence	ADV
ejpam-2095	228	4	,	,	PUNCT
ejpam-2095	228	5	(	(	PUNCT
ejpam-2095	228	6	n	n	CCONJ
ejpam-2095	228	7	:	:	PUNCT
ejpam-2095	228	8	x	x	X
ejpam-2095	228	9	)	)	PUNCT
ejpam-2095	228	10	=	=	SYM
ejpam-2095	228	11	(	(	PUNCT
ejpam-2095	228	12	q1	q1	INTJ
ejpam-2095	228	13	:	:	PUNCT
ejpam-2095	228	14	x	x	X
ejpam-2095	228	15	)	)	PUNCT
ejpam-2095	229	1	∧	∧	NOUN
ejpam-2095	229	2	1∧	1∧	NUM
ejpam-2095	229	3	.	.	PUNCT
ejpam-2095	229	4	.	.	PUNCT
ejpam-2095	229	5	.	.	PUNCT
ejpam-2095	230	1	∧	∧	NOUN
ejpam-2095	230	2	1	1	NUM
ejpam-2095	230	3	=	=	SYM
ejpam-2095	230	4	(	(	PUNCT
ejpam-2095	230	5	q1	q1	INTJ
ejpam-2095	230	6	:	:	PUNCT
ejpam-2095	230	7	x	x	X
ejpam-2095	230	8	)	)	PUNCT
ejpam-2095	230	9	.	.	PUNCT
ejpam-2095	231	1	so	so	ADV
ejpam-2095	231	2	by	by	ADP
ejpam-2095	231	3	above	above	ADP
ejpam-2095	231	4	result	result	NOUN
ejpam-2095	231	5	,	,	PUNCT
ejpam-2095	231	6	(	(	PUNCT
ejpam-2095	231	7	q1	q1	INTJ
ejpam-2095	231	8	:	:	PUNCT
ejpam-2095	231	9	x	x	X
ejpam-2095	231	10	)	)	PUNCT
ejpam-2095	231	11	is	be	AUX
ejpam-2095	231	12	p	p	ADJ
ejpam-2095	231	13	-	-	PUNCT
ejpam-2095	231	14	primary	primary	ADJ
ejpam-2095	231	15	element	element	NOUN
ejpam-2095	231	16	implies	imply	VERB
ejpam-2095	231	17	(	(	PUNCT
ejpam-2095	231	18	n	n	X
ejpam-2095	231	19	:	:	PUNCT
ejpam-2095	231	20	x	x	X
ejpam-2095	231	21	)	)	PUNCT
ejpam-2095	231	22	is	be	AUX
ejpam-2095	231	23	a	a	DET
ejpam-2095	231	24	p	p	NOUN
ejpam-2095	231	25	-	-	PUNCT
ejpam-2095	231	26	primary	primary	ADJ
ejpam-2095	231	27	element	element	NOUN
ejpam-2095	231	28	of	of	ADP
ejpam-2095	231	29	l	l	NOUN
ejpam-2095	231	30	where	where	SCONJ
ejpam-2095	231	31	x	x	X
ejpam-2095	231	32	n	n	X
ejpam-2095	231	33	.	.	PUNCT
ejpam-2095	232	1	conversely	conversely	ADV
ejpam-2095	232	2	assume	assume	VERB
ejpam-2095	232	3	that	that	SCONJ
ejpam-2095	232	4	(	(	PUNCT
ejpam-2095	232	5	n	n	X
ejpam-2095	232	6	:	:	PUNCT
ejpam-2095	232	7	x	x	X
ejpam-2095	232	8	)	)	PUNCT
ejpam-2095	232	9	is	be	AUX
ejpam-2095	232	10	a	a	DET
ejpam-2095	232	11	p	p	NOUN
ejpam-2095	232	12	-	-	PUNCT
ejpam-2095	232	13	primary	primary	ADJ
ejpam-2095	232	14	element	element	NOUN
ejpam-2095	232	15	of	of	ADP
ejpam-2095	232	16	l	l	NOUN
ejpam-2095	232	17	for	for	ADP
ejpam-2095	232	18	some	some	DET
ejpam-2095	232	19	x	x	SYM
ejpam-2095	232	20	n	n	NOUN
ejpam-2095	232	21	,	,	PUNCT
ejpam-2095	232	22	x	x	PUNCT
ejpam-2095	232	23	∈	∈	NOUN
ejpam-2095	232	24	m	m	VERB
ejpam-2095	232	25	.	.	PUNCT
ejpam-2095	233	1	we	we	PRON
ejpam-2095	233	2	prove	prove	VERB
ejpam-2095	233	3	that	that	SCONJ
ejpam-2095	233	4	p	p	NOUN
ejpam-2095	233	5	q	q	X
ejpam-2095	234	1	i	i	PRON
ejpam-2095	234	2	=	=	PUNCT
ejpam-2095	234	3	p	p	NOUN
ejpam-2095	234	4	for	for	ADP
ejpam-2095	234	5	some	some	DET
ejpam-2095	234	6	i.	i.	NOUN
ejpam-2095	234	7	we	we	PRON
ejpam-2095	234	8	have	have	VERB
ejpam-2095	234	9	,	,	PUNCT
ejpam-2095	234	10	p	p	X
ejpam-2095	234	11	=	=	PUNCT
ejpam-2095	234	12	p	p	X
ejpam-2095	234	13	(	(	PUNCT
ejpam-2095	234	14	n	n	NOUN
ejpam-2095	234	15	:	:	PUNCT
ejpam-2095	234	16	x	x	X
ejpam-2095	234	17	)	)	PUNCT
ejpam-2095	235	1	=	=	PUNCT
ejpam-2095	236	1	p	p	X
ejpam-2095	236	2	[	[	X
ejpam-2095	236	3	(	(	PUNCT
ejpam-2095	236	4	q1	q1	PROPN
ejpam-2095	236	5	∧q2	∧q2	VERB
ejpam-2095	236	6	∧	∧	PROPN
ejpam-2095	236	7	.	.	PUNCT
ejpam-2095	236	8	.	.	PUNCT
ejpam-2095	236	9	.	.	PUNCT
ejpam-2095	237	1	∧qk	∧qk	NOUN
ejpam-2095	237	2	)	)	PUNCT
ejpam-2095	237	3	:	:	PUNCT
ejpam-2095	238	1	x	x	X
ejpam-2095	238	2	]	]	PUNCT
ejpam-2095	238	3	=	=	SYM
ejpam-2095	238	4	p	p	X
ejpam-2095	238	5	(	(	PUNCT
ejpam-2095	238	6	q1	q1	PROPN
ejpam-2095	238	7	:	:	PUNCT
ejpam-2095	238	8	x	x	X
ejpam-2095	238	9	)	)	PUNCT
ejpam-2095	238	10	∧	∧	NOUN
ejpam-2095	238	11	p	p	NOUN
ejpam-2095	238	12	(	(	PUNCT
ejpam-2095	238	13	q2	q2	NOUN
ejpam-2095	238	14	:	:	PUNCT
ejpam-2095	238	15	x	x	X
ejpam-2095	238	16	)	)	PUNCT
ejpam-2095	238	17	∧	∧	NOUN
ejpam-2095	238	18	.	.	PUNCT
ejpam-2095	238	19	.	.	PUNCT
ejpam-2095	238	20	.	.	PUNCT
ejpam-2095	239	1	∧	∧	NOUN
ejpam-2095	239	2	p	p	NOUN
ejpam-2095	239	3	(	(	PUNCT
ejpam-2095	239	4	qk	qk	NOUN
ejpam-2095	239	5	:	:	PUNCT
ejpam-2095	239	6	x	x	X
ejpam-2095	239	7	)	)	PUNCT
ejpam-2095	239	8	.	.	PUNCT
ejpam-2095	240	1	we	we	PRON
ejpam-2095	240	2	claim	claim	VERB
ejpam-2095	240	3	that	that	SCONJ
ejpam-2095	240	4	for	for	ADP
ejpam-2095	240	5	each	each	DET
ejpam-2095	240	6	i	i	PRON
ejpam-2095	240	7	,	,	PUNCT
ejpam-2095	240	8	p	p	X
ejpam-2095	240	9	(	(	PUNCT
ejpam-2095	240	10	q	q	NOUN
ejpam-2095	240	11	i	i	PRON
ejpam-2095	240	12	:	:	PUNCT
ejpam-2095	240	13	x	x	X
ejpam-2095	240	14	)	)	PUNCT
ejpam-2095	240	15	=	=	SYM
ejpam-2095	240	16	pi	pi	NOUN
ejpam-2095	240	17	or	or	CCONJ
ejpam-2095	240	18	1	1	NUM
ejpam-2095	240	19	and	and	CCONJ
ejpam-2095	240	20	equal	equal	ADJ
ejpam-2095	240	21	to	to	AUX
ejpam-2095	240	22	pi	pi	VERB
ejpam-2095	240	23	for	for	ADP
ejpam-2095	240	24	at	at	ADV
ejpam-2095	240	25	least	least	ADV
ejpam-2095	240	26	one	one	NUM
ejpam-2095	240	27	i.	i.	NOUN
ejpam-2095	240	28	we	we	PRON
ejpam-2095	240	29	have	have	VERB
ejpam-2095	240	30	x	x	PROPN
ejpam-2095	240	31	n	n	PROPN
ejpam-2095	240	32	=	=	PROPN
ejpam-2095	240	33	q1	q1	PROPN
ejpam-2095	240	34	∧q2	∧q2	VERB
ejpam-2095	240	35	∧	∧	PROPN
ejpam-2095	240	36	.	.	PUNCT
ejpam-2095	240	37	.	.	PUNCT
ejpam-2095	241	1	.∧qk	.∧qk	PROPN
ejpam-2095	241	2	implies	imply	VERB
ejpam-2095	241	3	x	x	PUNCT
ejpam-2095	241	4	q	q	X
ejpam-2095	241	5	i	i	PRON
ejpam-2095	241	6	for	for	ADP
ejpam-2095	241	7	at	at	ADV
ejpam-2095	241	8	least	least	ADV
ejpam-2095	241	9	one	one	NUM
ejpam-2095	241	10	i	i	PRON
ejpam-2095	241	11	(	(	PUNCT
ejpam-2095	241	12	1¶	1¶	NOUN
ejpam-2095	241	13	i	i	NOUN
ejpam-2095	241	14	¶	¶	PROPN
ejpam-2095	241	15	k	k	NOUN
ejpam-2095	241	16	)	)	PUNCT
ejpam-2095	241	17	.	.	PUNCT
ejpam-2095	242	1	suppose	suppose	VERB
ejpam-2095	242	2	x	x	SYM
ejpam-2095	242	3	qr	qr	X
ejpam-2095	242	4	(	(	PUNCT
ejpam-2095	242	5	1¶	1¶	NOUN
ejpam-2095	242	6	r	r	NOUN
ejpam-2095	242	7	¶	¶	PROPN
ejpam-2095	242	8	k	k	NOUN
ejpam-2095	242	9	)	)	PUNCT
ejpam-2095	242	10	and	and	CCONJ
ejpam-2095	242	11	x	x	SYM
ejpam-2095	242	12	¶	¶	NOUN
ejpam-2095	242	13	q1∧q2∧	q1∧q2∧	NOUN
ejpam-2095	242	14	.	.	PUNCT
ejpam-2095	242	15	.	.	PUNCT
ejpam-2095	243	1	.∧qr−1∧qr+1∧	.∧qr−1∧qr+1∧	PROPN
ejpam-2095	243	2	.	.	PUNCT
ejpam-2095	244	1	.	.	PUNCT
ejpam-2095	245	1	.qk	.qk	PUNCT
ejpam-2095	246	1	that	that	PRON
ejpam-2095	246	2	is	is	ADV
ejpam-2095	246	3	x	x	X
ejpam-2095	246	4	¶	¶	PROPN
ejpam-2095	246	5	∧q	∧q	PROPN
ejpam-2095	247	1	i	i	PRON
ejpam-2095	247	2	,	,	PUNCT
ejpam-2095	247	3	where	where	SCONJ
ejpam-2095	247	4	(	(	PUNCT
ejpam-2095	247	5	i	i	NOUN
ejpam-2095	247	6	6=	6=	NOUN
ejpam-2095	247	7	r	r	NOUN
ejpam-2095	247	8	)	)	PUNCT
ejpam-2095	247	9	.	.	PUNCT
ejpam-2095	248	1	we	we	PRON
ejpam-2095	248	2	have	have	VERB
ejpam-2095	248	3	,	,	PUNCT
ejpam-2095	248	4	ax	ax	NOUN
ejpam-2095	248	5	¶	¶	PROPN
ejpam-2095	248	6	q	q	PROPN
ejpam-2095	248	7	i	i	PROPN
ejpam-2095	248	8	for	for	ADP
ejpam-2095	248	9	all	all	DET
ejpam-2095	248	10	i	i	PRON
ejpam-2095	248	11	6=	6=	NOUN
ejpam-2095	248	12	r	r	NOUN
ejpam-2095	248	13	and	and	CCONJ
ejpam-2095	248	14	a	a	DET
ejpam-2095	248	15	∈	∈	PROPN
ejpam-2095	248	16	l.	l.	NOUN
ejpam-2095	248	17	hence	hence	ADV
ejpam-2095	248	18	,	,	PUNCT
ejpam-2095	248	19	a	a	DET
ejpam-2095	248	20	¶	¶	NOUN
ejpam-2095	248	21	p	p	NOUN
ejpam-2095	248	22	(	(	PUNCT
ejpam-2095	248	23	q	q	NOUN
ejpam-2095	248	24	i	i	PRON
ejpam-2095	248	25	:	:	PUNCT
ejpam-2095	248	26	x	x	X
ejpam-2095	248	27	)	)	PUNCT
ejpam-2095	248	28	for	for	ADP
ejpam-2095	248	29	all	all	DET
ejpam-2095	248	30	a	a	DET
ejpam-2095	248	31	∈	∈	PROPN
ejpam-2095	248	32	l.	l.	NOUN
ejpam-2095	248	33	in	in	ADP
ejpam-2095	248	34	particular	particular	ADJ
ejpam-2095	248	35	,	,	PUNCT
ejpam-2095	248	36	1	1	NUM
ejpam-2095	248	37	¶	¶	NOUN
ejpam-2095	248	38	p	p	NOUN
ejpam-2095	248	39	(	(	PUNCT
ejpam-2095	248	40	q	q	NOUN
ejpam-2095	248	41	i	i	PRON
ejpam-2095	248	42	:	:	PUNCT
ejpam-2095	248	43	x	x	X
ejpam-2095	248	44	)	)	PUNCT
ejpam-2095	248	45	for	for	ADP
ejpam-2095	248	46	all	all	DET
ejpam-2095	248	47	i	i	PRON
ejpam-2095	248	48	6=	6=	PROPN
ejpam-2095	248	49	r.	r.	PROPN
ejpam-2095	249	1	but	but	CCONJ
ejpam-2095	249	2	,	,	PUNCT
ejpam-2095	250	1	p	p	X
ejpam-2095	250	2	(	(	PUNCT
ejpam-2095	250	3	q	q	NOUN
ejpam-2095	250	4	i	i	PRON
ejpam-2095	250	5	:	:	PUNCT
ejpam-2095	250	6	x	x	X
ejpam-2095	250	7	)	)	PUNCT
ejpam-2095	250	8	¶	¶	NOUN
ejpam-2095	250	9	1	1	NUM
ejpam-2095	250	10	for	for	ADP
ejpam-2095	250	11	all	all	DET
ejpam-2095	250	12	i	i	PRON
ejpam-2095	250	13	6=	6=	PROPN
ejpam-2095	250	14	r.	r.	PROPN
ejpam-2095	250	15	therefore	therefore	ADV
ejpam-2095	250	16	,	,	PUNCT
ejpam-2095	250	17	p	p	X
ejpam-2095	250	18	(	(	PUNCT
ejpam-2095	250	19	q	q	NOUN
ejpam-2095	250	20	i	i	PRON
ejpam-2095	250	21	:	:	PUNCT
ejpam-2095	250	22	x	x	X
ejpam-2095	250	23	)	)	PUNCT
ejpam-2095	250	24	=	=	SYM
ejpam-2095	250	25	1	1	NUM
ejpam-2095	250	26	for	for	ADP
ejpam-2095	250	27	all	all	DET
ejpam-2095	250	28	i	i	PRON
ejpam-2095	250	29	6=	6=	PROPN
ejpam-2095	250	30	r.	r.	NOUN
ejpam-2095	250	31	for	for	ADP
ejpam-2095	250	32	i	i	PROPN
ejpam-2095	250	33	=	=	SYM
ejpam-2095	250	34	r	r	NOUN
ejpam-2095	250	35	,	,	PUNCT
ejpam-2095	250	36	x	x	NOUN
ejpam-2095	250	37	qr	qr	NOUN
ejpam-2095	250	38	.	.	PUNCT
ejpam-2095	251	1	let	let	VERB
ejpam-2095	251	2	a	a	DET
ejpam-2095	251	3	¶	¶	PROPN
ejpam-2095	251	4	p	p	NOUN
ejpam-2095	251	5	(	(	PUNCT
ejpam-2095	251	6	qr	qr	NOUN
ejpam-2095	251	7	:	:	PUNCT
ejpam-2095	251	8	x	x	X
ejpam-2095	251	9	)	)	PUNCT
ejpam-2095	251	10	.	.	PUNCT
ejpam-2095	252	1	hence	hence	ADV
ejpam-2095	252	2	,	,	PUNCT
ejpam-2095	252	3	anx	anx	PROPN
ejpam-2095	252	4	¶	¶	PROPN
ejpam-2095	252	5	qr	qr	PROPN
ejpam-2095	252	6	,	,	PUNCT
ejpam-2095	252	7	for	for	ADP
ejpam-2095	252	8	some	some	DET
ejpam-2095	252	9	positive	positive	ADJ
ejpam-2095	252	10	integer	integer	NOUN
ejpam-2095	252	11	n	n	CCONJ
ejpam-2095	252	12	,	,	PUNCT
ejpam-2095	252	13	where	where	SCONJ
ejpam-2095	252	14	x	x	PROPN
ejpam-2095	252	15	qr	qr	INTJ
ejpam-2095	252	16	.	.	PUNCT
ejpam-2095	253	1	as	as	SCONJ
ejpam-2095	253	2	qr	qr	PROPN
ejpam-2095	253	3	is	be	AUX
ejpam-2095	253	4	primary	primary	ADJ
ejpam-2095	253	5	,	,	PUNCT
ejpam-2095	253	6	an	an	DET
ejpam-2095	253	7	¶	¶	PROPN
ejpam-2095	253	8	p	p	NOUN
ejpam-2095	253	9	(	(	PUNCT
ejpam-2095	253	10	qr	qr	INTJ
ejpam-2095	253	11	:	:	PUNCT
ejpam-2095	253	12	i	i	PRON
ejpam-2095	253	13	m	m	VERB
ejpam-2095	253	14	)	)	PUNCT
ejpam-2095	254	1	=	=	NOUN
ejpam-2095	254	2	pr	pr	NOUN
ejpam-2095	254	3	.	.	PUNCT
ejpam-2095	255	1	thus	thus	ADV
ejpam-2095	255	2	,	,	PUNCT
ejpam-2095	255	3	a	a	DET
ejpam-2095	255	4	¶	¶	NOUN
ejpam-2095	255	5	pr	pr	NOUN
ejpam-2095	255	6	,	,	PUNCT
ejpam-2095	255	7	since	since	SCONJ
ejpam-2095	255	8	pr	pr	NOUN
ejpam-2095	255	9	is	be	AUX
ejpam-2095	255	10	prime	prime	ADJ
ejpam-2095	255	11	and	and	CCONJ
ejpam-2095	255	12	we	we	PRON
ejpam-2095	255	13	have	have	VERB
ejpam-2095	255	14	,	,	PUNCT
ejpam-2095	255	15	p	p	X
ejpam-2095	255	16	(	(	PUNCT
ejpam-2095	255	17	qr	qr	NOUN
ejpam-2095	255	18	:	:	PUNCT
ejpam-2095	255	19	x	x	SYM
ejpam-2095	255	20	)	)	PUNCT
ejpam-2095	255	21	¶	¶	INTJ
ejpam-2095	255	22	pr	pr	NOUN
ejpam-2095	255	23	.	.	PUNCT
ejpam-2095	256	1	on	on	ADP
ejpam-2095	256	2	the	the	DET
ejpam-2095	256	3	other	other	ADJ
ejpam-2095	256	4	hand	hand	NOUN
ejpam-2095	256	5	,	,	PUNCT
ejpam-2095	256	6	let	let	VERB
ejpam-2095	256	7	a	a	DET
ejpam-2095	256	8	¶	¶	NOUN
ejpam-2095	256	9	pr	pr	NOUN
ejpam-2095	257	1	=	=	NOUN
ejpam-2095	257	2	p	p	X
ejpam-2095	257	3	qr	qr	NOUN
ejpam-2095	257	4	=	=	SYM
ejpam-2095	257	5	p	p	X
ejpam-2095	257	6	(	(	PUNCT
ejpam-2095	257	7	qr	qr	NOUN
ejpam-2095	257	8	:	:	PUNCT
ejpam-2095	257	9	i	i	PRON
ejpam-2095	257	10	m	m	PROPN
ejpam-2095	257	11	)	)	PUNCT
ejpam-2095	257	12	.	.	PUNCT
ejpam-2095	258	1	hence	hence	ADV
ejpam-2095	258	2	,	,	PUNCT
ejpam-2095	258	3	an	an	DET
ejpam-2095	258	4	¶	¶	PROPN
ejpam-2095	258	5	(	(	PUNCT
ejpam-2095	258	6	qr	qr	NOUN
ejpam-2095	258	7	:	:	PUNCT
ejpam-2095	258	8	i	i	PRON
ejpam-2095	258	9	m	m	VERB
ejpam-2095	258	10	)	)	PUNCT
ejpam-2095	258	11	for	for	ADP
ejpam-2095	258	12	some	some	DET
ejpam-2095	258	13	positive	positive	ADJ
ejpam-2095	258	14	integer	integer	NOUN
ejpam-2095	258	15	n.	n.	NOUN
ejpam-2095	258	16	that	that	PRON
ejpam-2095	258	17	is	be	AUX
ejpam-2095	258	18	an	an	DET
ejpam-2095	258	19	i	i	NOUN
ejpam-2095	258	20	m	m	NOUN
ejpam-2095	258	21	¶	¶	PROPN
ejpam-2095	258	22	qr	qr	NOUN
ejpam-2095	258	23	and	and	CCONJ
ejpam-2095	258	24	therefore	therefore	ADV
ejpam-2095	258	25	,	,	PUNCT
ejpam-2095	258	26	anx	anx	PROPN
ejpam-2095	258	27	¶	¶	PROPN
ejpam-2095	258	28	qr	qr	PROPN
ejpam-2095	258	29	,	,	PUNCT
ejpam-2095	258	30	for	for	ADP
ejpam-2095	258	31	some	some	DET
ejpam-2095	258	32	positive	positive	ADJ
ejpam-2095	258	33	integer	integer	NOUN
ejpam-2095	258	34	n.	n.	NOUN
ejpam-2095	258	35	consequently	consequently	ADV
ejpam-2095	258	36	,	,	PUNCT
ejpam-2095	258	37	an	an	DET
ejpam-2095	258	38	¶	¶	PROPN
ejpam-2095	258	39	(	(	PUNCT
ejpam-2095	258	40	qr	qr	NOUN
ejpam-2095	258	41	:	:	PUNCT
ejpam-2095	258	42	x	x	X
ejpam-2095	258	43	)	)	PUNCT
ejpam-2095	258	44	and	and	CCONJ
ejpam-2095	258	45	hence	hence	ADV
ejpam-2095	258	46	a	a	DET
ejpam-2095	258	47	¶	¶	NOUN
ejpam-2095	258	48	p	p	NOUN
ejpam-2095	258	49	(	(	PUNCT
ejpam-2095	258	50	qr	qr	NOUN
ejpam-2095	258	51	:	:	PUNCT
ejpam-2095	258	52	x	x	X
ejpam-2095	258	53	)	)	PUNCT
ejpam-2095	258	54	.	.	PUNCT
ejpam-2095	259	1	this	this	PRON
ejpam-2095	259	2	gives	give	VERB
ejpam-2095	259	3	pr	pr	NOUN
ejpam-2095	259	4	¶	¶	PROPN
ejpam-2095	259	5	p	p	NOUN
ejpam-2095	259	6	(	(	PUNCT
ejpam-2095	259	7	qr	qr	NOUN
ejpam-2095	259	8	:	:	PUNCT
ejpam-2095	259	9	x	x	X
ejpam-2095	259	10	)	)	PUNCT
ejpam-2095	259	11	.	.	PUNCT
ejpam-2095	260	1	hence	hence	ADV
ejpam-2095	260	2	,	,	PUNCT
ejpam-2095	260	3	p	p	X
ejpam-2095	260	4	(	(	PUNCT
ejpam-2095	260	5	qr	qr	NOUN
ejpam-2095	260	6	:	:	PUNCT
ejpam-2095	260	7	x	x	X
ejpam-2095	260	8	)	)	PUNCT
ejpam-2095	260	9	=	=	SYM
ejpam-2095	260	10	pr	pr	X
ejpam-2095	260	11	where	where	SCONJ
ejpam-2095	260	12	x	x	PRON
ejpam-2095	260	13	qr	qr	INTJ
ejpam-2095	260	14	.	.	PUNCT
ejpam-2095	261	1	we	we	PRON
ejpam-2095	261	2	have	have	AUX
ejpam-2095	261	3	shown	show	VERB
ejpam-2095	261	4	that	that	SCONJ
ejpam-2095	261	5	for	for	ADP
ejpam-2095	261	6	each	each	DET
ejpam-2095	261	7	i	i	PRON
ejpam-2095	261	8	,	,	PUNCT
ejpam-2095	261	9	p	p	X
ejpam-2095	261	10	(	(	PUNCT
ejpam-2095	261	11	q	q	NOUN
ejpam-2095	261	12	i	i	PRON
ejpam-2095	261	13	:	:	PUNCT
ejpam-2095	261	14	x	x	X
ejpam-2095	261	15	)	)	PUNCT
ejpam-2095	262	1	=	=	SYM
ejpam-2095	262	2	pi	pi	NOUN
ejpam-2095	262	3	or	or	CCONJ
ejpam-2095	262	4	1	1	NUM
ejpam-2095	262	5	and	and	CCONJ
ejpam-2095	262	6	is	be	AUX
ejpam-2095	262	7	equal	equal	ADJ
ejpam-2095	262	8	to	to	PART
ejpam-2095	262	9	pi	pi	VERB
ejpam-2095	262	10	for	for	ADP
ejpam-2095	262	11	at	at	ADV
ejpam-2095	262	12	least	least	ADV
ejpam-2095	262	13	one	one	NUM
ejpam-2095	262	14	i	i	PRON
ejpam-2095	262	15	,	,	PUNCT
ejpam-2095	262	16	since	since	SCONJ
ejpam-2095	262	17	x	x	PROPN
ejpam-2095	262	18	n	n	CCONJ
ejpam-2095	262	19	.	.	PUNCT
ejpam-2095	263	1	then	then	ADV
ejpam-2095	263	2	,	,	PUNCT
ejpam-2095	263	3	p	p	X
ejpam-2095	263	4	=	=	PUNCT
ejpam-2095	263	5	p	p	X
ejpam-2095	263	6	(	(	PUNCT
ejpam-2095	263	7	n	n	NOUN
ejpam-2095	263	8	:	:	PUNCT
ejpam-2095	263	9	x	x	X
ejpam-2095	263	10	)	)	PUNCT
ejpam-2095	264	1	=	=	SYM
ejpam-2095	264	2	p	p	X
ejpam-2095	264	3	(	(	PUNCT
ejpam-2095	264	4	q1	q1	PROPN
ejpam-2095	264	5	:	:	PUNCT
ejpam-2095	264	6	x	x	X
ejpam-2095	264	7	)	)	PUNCT
ejpam-2095	264	8	∧	∧	NOUN
ejpam-2095	264	9	.	.	PUNCT
ejpam-2095	264	10	.	.	PUNCT
ejpam-2095	265	1	.∧	.∧	PUNCT
ejpam-2095	266	1	p	p	X
ejpam-2095	266	2	(	(	PUNCT
ejpam-2095	266	3	qk	qk	NOUN
ejpam-2095	266	4	:	:	PUNCT
ejpam-2095	266	5	x	x	X
ejpam-2095	266	6	)	)	PUNCT
ejpam-2095	266	7	is	be	AUX
ejpam-2095	266	8	the	the	DET
ejpam-2095	266	9	meet	meet	NOUN
ejpam-2095	266	10	of	of	ADP
ejpam-2095	266	11	some	some	PRON
ejpam-2095	266	12	of	of	ADP
ejpam-2095	266	13	the	the	DET
ejpam-2095	266	14	prime	prime	ADJ
ejpam-2095	266	15	elements	element	NOUN
ejpam-2095	266	16	p1	p1	NOUN
ejpam-2095	266	17	,	,	PUNCT
ejpam-2095	266	18	p2	p2	NOUN
ejpam-2095	266	19	,	,	PUNCT
ejpam-2095	266	20	.	.	PUNCT
ejpam-2095	266	21	.	.	PUNCT
ejpam-2095	267	1	.	.	PUNCT
ejpam-2095	268	1	,	,	PUNCT
ejpam-2095	268	2	pl	pl	PROPN
ejpam-2095	268	3	(	(	PUNCT
ejpam-2095	268	4	1¶	1¶	PROPN
ejpam-2095	268	5	l	l	PROPN
ejpam-2095	268	6	¶	¶	PROPN
ejpam-2095	268	7	k	k	NOUN
ejpam-2095	268	8	)	)	PUNCT
ejpam-2095	268	9	.	.	PUNCT
ejpam-2095	269	1	that	that	PRON
ejpam-2095	269	2	is	be	AUX
ejpam-2095	269	3	p	p	NOUN
ejpam-2095	269	4	=	=	PUNCT
ejpam-2095	269	5	p	p	X
ejpam-2095	269	6	(	(	PUNCT
ejpam-2095	269	7	n	n	NOUN
ejpam-2095	269	8	:	:	PUNCT
ejpam-2095	269	9	x	x	X
ejpam-2095	269	10	)	)	PUNCT
ejpam-2095	270	1	=	=	SYM
ejpam-2095	270	2	p1	p1	NOUN
ejpam-2095	270	3	∧	∧	PROPN
ejpam-2095	270	4	p2	p2	NOUN
ejpam-2095	270	5	∧	∧	PROPN
ejpam-2095	270	6	.	.	PUNCT
ejpam-2095	270	7	.	.	PUNCT
ejpam-2095	270	8	.∧	.∧	PUNCT
ejpam-2095	271	1	pl	pl	X
ejpam-2095	271	2	.	.	PUNCT
ejpam-2095	272	1	we	we	PRON
ejpam-2095	272	2	show	show	VERB
ejpam-2095	272	3	that	that	SCONJ
ejpam-2095	272	4	p	p	X
ejpam-2095	272	5	=	=	PUNCT
ejpam-2095	272	6	pi	pi	NOUN
ejpam-2095	272	7	for	for	ADP
ejpam-2095	272	8	some	some	DET
ejpam-2095	272	9	i.	i.	NOUN
ejpam-2095	272	10	we	we	PRON
ejpam-2095	272	11	have	have	VERB
ejpam-2095	272	12	,	,	PUNCT
ejpam-2095	272	13	p	p	PROPN
ejpam-2095	272	14	¶	¶	NUM
ejpam-2095	272	15	pi	pi	NOUN
ejpam-2095	272	16	i	i	NOUN
ejpam-2095	272	17	=	=	NOUN
ejpam-2095	272	18	1,2	1,2	NUM
ejpam-2095	272	19	,	,	PUNCT
ejpam-2095	272	20	.	.	PUNCT
ejpam-2095	272	21	.	.	PUNCT
ejpam-2095	273	1	.	.	PUNCT
ejpam-2095	274	1	,	,	PUNCT
ejpam-2095	274	2	l.	l.	PROPN
ejpam-2095	274	3	if	if	SCONJ
ejpam-2095	274	4	for	for	ADP
ejpam-2095	274	5	each	each	DET
ejpam-2095	274	6	i	i	NOUN
ejpam-2095	274	7	,	,	PUNCT
ejpam-2095	274	8	p	p	PROPN
ejpam-2095	274	9	6=	6=	NUM
ejpam-2095	274	10	pi	pi	NOUN
ejpam-2095	274	11	then	then	ADV
ejpam-2095	274	12	pi	pi	NOUN
ejpam-2095	274	13	p	p	PROPN
ejpam-2095	274	14	for	for	ADP
ejpam-2095	274	15	all	all	DET
ejpam-2095	274	16	i	i	NOUN
ejpam-2095	274	17	=	=	SYM
ejpam-2095	274	18	1,2	1,2	NUM
ejpam-2095	274	19	,	,	PUNCT
ejpam-2095	274	20	.	.	PUNCT
ejpam-2095	274	21	.	.	PUNCT
ejpam-2095	275	1	.	.	PUNCT
ejpam-2095	276	1	,	,	PUNCT
ejpam-2095	276	2	l.	l.	PROPN
ejpam-2095	276	3	this	this	PRON
ejpam-2095	276	4	implies	imply	VERB
ejpam-2095	276	5	that	that	SCONJ
ejpam-2095	276	6	there	there	PRON
ejpam-2095	276	7	exist	exist	VERB
ejpam-2095	276	8	x	x	PUNCT
ejpam-2095	277	1	i	i	PRON
ejpam-2095	277	2	¶	¶	VERB
ejpam-2095	277	3	pi	pi	NOUN
ejpam-2095	277	4	such	such	ADJ
ejpam-2095	277	5	that	that	SCONJ
ejpam-2095	277	6	x	x	X
ejpam-2095	278	1	i	i	PRON
ejpam-2095	278	2	p	p	NOUN
ejpam-2095	278	3	for	for	ADP
ejpam-2095	278	4	all	all	DET
ejpam-2095	278	5	i	i	NOUN
ejpam-2095	278	6	=	=	SYM
ejpam-2095	278	7	1,2	1,2	NUM
ejpam-2095	278	8	,	,	PUNCT
ejpam-2095	278	9	.	.	PUNCT
ejpam-2095	278	10	.	.	PUNCT
ejpam-2095	278	11	.	.	PUNCT
ejpam-2095	279	1	,	,	PUNCT
ejpam-2095	280	1	l	l	NOUN
ejpam-2095	280	2	then	then	ADV
ejpam-2095	280	3	,	,	PUNCT
ejpam-2095	280	4	x1	x1	PROPN
ejpam-2095	280	5	x2	x2	PROPN
ejpam-2095	280	6	.	.	PUNCT
ejpam-2095	280	7	.	.	PUNCT
ejpam-2095	280	8	.	.	PUNCT
ejpam-2095	281	1	x	x	X
ejpam-2095	281	2	l	l	NOUN
ejpam-2095	281	3	¶	¶	PROPN
ejpam-2095	281	4	p1	p1	PROPN
ejpam-2095	281	5	∧	∧	PROPN
ejpam-2095	281	6	p2	p2	NOUN
ejpam-2095	281	7	∧	∧	PROPN
ejpam-2095	281	8	.	.	PUNCT
ejpam-2095	281	9	.	.	PUNCT
ejpam-2095	281	10	.	.	PUNCT
ejpam-2095	282	1	∧	∧	NOUN
ejpam-2095	282	2	pl	pl	PROPN
ejpam-2095	282	3	=	=	SYM
ejpam-2095	282	4	p.	p.	NOUN
ejpam-2095	282	5	this	this	PRON
ejpam-2095	282	6	shows	show	VERB
ejpam-2095	282	7	that	that	SCONJ
ejpam-2095	282	8	x	x	PUNCT
ejpam-2095	283	1	i	i	PRON
ejpam-2095	283	2	¶	¶	VERB
ejpam-2095	283	3	p	p	NOUN
ejpam-2095	283	4	for	for	ADP
ejpam-2095	283	5	at	at	ADV
ejpam-2095	283	6	least	least	ADV
ejpam-2095	283	7	one	one	NUM
ejpam-2095	283	8	i	i	PRON
ejpam-2095	283	9	(	(	PUNCT
ejpam-2095	283	10	1¶	1¶	NOUN
ejpam-2095	283	11	i	i	NOUN
ejpam-2095	283	12	¶	¶	PROPN
ejpam-2095	283	13	k	k	NOUN
ejpam-2095	283	14	)	)	PUNCT
ejpam-2095	283	15	a	a	DET
ejpam-2095	283	16	contradiction	contradiction	NOUN
ejpam-2095	283	17	.	.	PUNCT
ejpam-2095	284	1	hence	hence	ADV
ejpam-2095	284	2	,	,	PUNCT
ejpam-2095	284	3	p	p	NOUN
ejpam-2095	284	4	=	=	NOUN
ejpam-2095	284	5	pi	pi	NOUN
ejpam-2095	284	6	for	for	ADP
ejpam-2095	284	7	at	at	ADV
ejpam-2095	284	8	least	least	ADV
ejpam-2095	284	9	one	one	NUM
ejpam-2095	284	10	i.	i.	NOUN
ejpam-2095	284	11	this	this	PRON
ejpam-2095	284	12	leads	lead	VERB
ejpam-2095	284	13	us	we	PRON
ejpam-2095	284	14	to	to	ADP
ejpam-2095	284	15	the	the	DET
ejpam-2095	284	16	following	following	ADJ
ejpam-2095	284	17	result	result	NOUN
ejpam-2095	284	18	.	.	PUNCT
ejpam-2095	285	1	theorem	theorem	ADJ
ejpam-2095	285	2	8	8	NUM
ejpam-2095	285	3	.	.	PUNCT
ejpam-2095	286	1	let	let	VERB
ejpam-2095	286	2	n	n	PRON
ejpam-2095	286	3	6=	6=	NUM
ejpam-2095	287	1	i	i	PRON
ejpam-2095	287	2	m	m	AUX
ejpam-2095	287	3	be	be	VERB
ejpam-2095	287	4	an	an	DET
ejpam-2095	287	5	element	element	NOUN
ejpam-2095	287	6	of	of	ADP
ejpam-2095	287	7	a	a	DET
ejpam-2095	287	8	lattice	lattice	NOUN
ejpam-2095	287	9	module	module	NOUN
ejpam-2095	287	10	m	m	PROPN
ejpam-2095	287	11	and	and	CCONJ
ejpam-2095	287	12	assume	assume	VERB
ejpam-2095	287	13	that	that	SCONJ
ejpam-2095	287	14	n	n	PRON
ejpam-2095	287	15	has	have	VERB
ejpam-2095	287	16	a	a	DET
ejpam-2095	287	17	primary	primary	ADJ
ejpam-2095	287	18	decomposition	decomposition	NOUN
ejpam-2095	287	19	.	.	PUNCT
ejpam-2095	288	1	if	if	SCONJ
ejpam-2095	288	2	n	n	NUM
ejpam-2095	288	3	=	=	PROPN
ejpam-2095	288	4	q1	q1	PROPN
ejpam-2095	288	5	∧q2	∧q2	VERB
ejpam-2095	288	6	∧	∧	PROPN
ejpam-2095	288	7	.	.	PUNCT
ejpam-2095	288	8	.	.	PUNCT
ejpam-2095	288	9	.	.	PUNCT
ejpam-2095	289	1	∧qm	∧qm	PROPN
ejpam-2095	289	2	=	=	SYM
ejpam-2095	289	3	s1	s1	PROPN
ejpam-2095	289	4	∧	∧	PROPN
ejpam-2095	289	5	s2	s2	NOUN
ejpam-2095	289	6	∧	∧	PROPN
ejpam-2095	289	7	.	.	PUNCT
ejpam-2095	289	8	.	.	PUNCT
ejpam-2095	289	9	.	.	PUNCT
ejpam-2095	290	1	∧	∧	PROPN
ejpam-2095	290	2	sn	sn	PROPN
ejpam-2095	290	3	are	be	AUX
ejpam-2095	290	4	two	two	NUM
ejpam-2095	290	5	reduced	reduce	VERB
ejpam-2095	290	6	primary	primary	ADJ
ejpam-2095	290	7	decompositions	decomposition	NOUN
ejpam-2095	290	8	of	of	ADP
ejpam-2095	290	9	n	n	PRON
ejpam-2095	290	10	then	then	ADV
ejpam-2095	290	11	n	n	PROPN
ejpam-2095	290	12	=	=	SYM
ejpam-2095	290	13	m	m	PROPN
ejpam-2095	290	14	and	and	CCONJ
ejpam-2095	291	1	the	the	DET
ejpam-2095	291	2	q	q	X
ejpam-2095	291	3	i	i	PRON
ejpam-2095	291	4	and	and	CCONJ
ejpam-2095	291	5	si	si	PROPN
ejpam-2095	291	6	can	can	AUX
ejpam-2095	291	7	be	be	AUX
ejpam-2095	291	8	so	so	ADV
ejpam-2095	291	9	numbered	number	VERB
ejpam-2095	291	10	that	that	SCONJ
ejpam-2095	291	11	p	p	NOUN
ejpam-2095	291	12	(	(	PUNCT
ejpam-2095	291	13	q	q	NOUN
ejpam-2095	291	14	i	i	PRON
ejpam-2095	291	15	:	:	PUNCT
ejpam-2095	291	16	i	i	PRON
ejpam-2095	291	17	m	m	VERB
ejpam-2095	291	18	)	)	PUNCT
ejpam-2095	292	1	=	=	SYM
ejpam-2095	292	2	p	p	X
ejpam-2095	292	3	(	(	PUNCT
ejpam-2095	292	4	si	si	X
ejpam-2095	292	5	:	:	PUNCT
ejpam-2095	292	6	i	i	PRON
ejpam-2095	292	7	m	m	VERB
ejpam-2095	292	8	)	)	PUNCT
ejpam-2095	292	9	for	for	ADP
ejpam-2095	292	10	i	i	X
ejpam-2095	292	11	=	=	SYM
ejpam-2095	292	12	1,2	1,2	NUM
ejpam-2095	292	13	,	,	PUNCT
ejpam-2095	292	14	.	.	PUNCT
ejpam-2095	292	15	.	.	PUNCT
ejpam-2095	292	16	.	.	PUNCT
ejpam-2095	293	1	,	,	PUNCT
ejpam-2095	293	2	n.	n.	VERB
ejpam-2095	293	3	the	the	DET
ejpam-2095	293	4	above	above	ADJ
ejpam-2095	293	5	theorem	theorem	NOUN
ejpam-2095	293	6	proves	prove	VERB
ejpam-2095	293	7	the	the	DET
ejpam-2095	293	8	uniqueness	uniqueness	NOUN
ejpam-2095	293	9	of	of	ADP
ejpam-2095	293	10	associated	associated	ADJ
ejpam-2095	293	11	primes	prime	NOUN
ejpam-2095	293	12	in	in	ADP
ejpam-2095	293	13	reduced	reduced	ADJ
ejpam-2095	293	14	primary	primary	ADJ
ejpam-2095	293	15	decomposition	decomposition	NOUN
ejpam-2095	293	16	.	.	PUNCT
ejpam-2095	294	1	the	the	DET
ejpam-2095	294	2	next	next	ADJ
ejpam-2095	294	3	result	result	NOUN
ejpam-2095	294	4	gives	give	VERB
ejpam-2095	294	5	the	the	DET
ejpam-2095	294	6	relation	relation	NOUN
ejpam-2095	294	7	between	between	ADP
ejpam-2095	294	8	zero	zero	NUM
ejpam-2095	294	9	divisors	divisor	NOUN
ejpam-2095	294	10	of	of	ADP
ejpam-2095	294	11	l	l	NOUN
ejpam-2095	294	12	and	and	CCONJ
ejpam-2095	294	13	associated	associated	ADJ
ejpam-2095	294	14	primes	prime	NOUN
ejpam-2095	294	15	of	of	ADP
ejpam-2095	294	16	zero	zero	NUM
ejpam-2095	294	17	.	.	PUNCT
ejpam-2095	295	1	theorem	theorem	NOUN
ejpam-2095	295	2	9	9	NUM
ejpam-2095	295	3	.	.	PUNCT
ejpam-2095	296	1	let	let	VERB
ejpam-2095	296	2	l	l	NOUN
ejpam-2095	296	3	be	be	AUX
ejpam-2095	296	4	a	a	DET
ejpam-2095	296	5	noetherian	noetherian	ADJ
ejpam-2095	296	6	lattice	lattice	NOUN
ejpam-2095	296	7	and	and	CCONJ
ejpam-2095	296	8	p1	p1	NOUN
ejpam-2095	296	9	,	,	PUNCT
ejpam-2095	296	10	p2	p2	NOUN
ejpam-2095	296	11	,	,	PUNCT
ejpam-2095	296	12	.	.	PUNCT
ejpam-2095	296	13	.	.	PUNCT
ejpam-2095	297	1	.	.	PUNCT
ejpam-2095	298	1	,	,	PUNCT
ejpam-2095	298	2	pk	pk	NOUN
ejpam-2095	298	3	be	be	AUX
ejpam-2095	298	4	the	the	DET
ejpam-2095	298	5	prime	prime	ADJ
ejpam-2095	298	6	divisors	divisor	NOUN
ejpam-2095	298	7	of	of	ADP
ejpam-2095	298	8	the	the	DET
ejpam-2095	298	9	element	element	NOUN
ejpam-2095	298	10	0	0	NUM
ejpam-2095	299	1	that	that	PRON
ejpam-2095	299	2	is	be	AUX
ejpam-2095	299	3	associated	associate	VERB
ejpam-2095	299	4	prime	prime	ADJ
ejpam-2095	299	5	elements	element	NOUN
ejpam-2095	299	6	of	of	ADP
ejpam-2095	299	7	element	element	NOUN
ejpam-2095	299	8	0	0	NUM
ejpam-2095	299	9	.	.	PUNCT
ejpam-2095	300	1	then	then	ADV
ejpam-2095	300	2	every	every	DET
ejpam-2095	300	3	zero	zero	NUM
ejpam-2095	300	4	divisors	divisor	NOUN
ejpam-2095	300	5	of	of	ADP
ejpam-2095	300	6	l	l	NOUN
ejpam-2095	300	7	is	be	AUX
ejpam-2095	300	8	contained	contain	VERB
ejpam-2095	300	9	in	in	ADP
ejpam-2095	300	10	p1	p1	PROPN
ejpam-2095	300	11	∨	∨	NUM
ejpam-2095	300	12	p2	p2	PROPN
ejpam-2095	300	13	∨	∨	NOUN
ejpam-2095	300	14	.	.	PUNCT
ejpam-2095	300	15	.	.	PUNCT
ejpam-2095	301	1	.∨	.∨	PROPN
ejpam-2095	302	1	pk	pk	PROPN
ejpam-2095	302	2	.	.	PROPN
ejpam-2095	302	3	c.	c.	PROPN
ejpam-2095	302	4	manjarekar	manjarekar	PROPN
ejpam-2095	302	5	,	,	PUNCT
ejpam-2095	302	6	u.	u.	PROPN
ejpam-2095	302	7	kandale	kandale	PROPN
ejpam-2095	302	8	/	/	SYM
ejpam-2095	302	9	eur	eur	PROPN
ejpam-2095	302	10	.	.	PUNCT
ejpam-2095	303	1	j.	j.	PROPN
ejpam-2095	303	2	pure	pure	PROPN
ejpam-2095	303	3	appl	appl	PROPN
ejpam-2095	303	4	.	.	PROPN
ejpam-2095	303	5	math	math	PROPN
ejpam-2095	303	6	,	,	PUNCT
ejpam-2095	303	7	7	7	NUM
ejpam-2095	303	8	(	(	PUNCT
ejpam-2095	303	9	2014	2014	NUM
ejpam-2095	303	10	)	)	PUNCT
ejpam-2095	303	11	,	,	PUNCT
ejpam-2095	303	12	201	201	NUM
ejpam-2095	303	13	-	-	SYM
ejpam-2095	303	14	209	209	NUM
ejpam-2095	303	15	206	206	NUM
ejpam-2095	303	16	proof	proof	NOUN
ejpam-2095	303	17	.	.	PUNCT
ejpam-2095	304	1	let	let	VERB
ejpam-2095	304	2	0	0	NUM
ejpam-2095	304	3	=	=	SYM
ejpam-2095	304	4	q1	q1	NOUN
ejpam-2095	304	5	∧	∧	PROPN
ejpam-2095	304	6	q2	q2	PROPN
ejpam-2095	304	7	∧	∧	PROPN
ejpam-2095	304	8	.	.	PUNCT
ejpam-2095	304	9	.	.	PUNCT
ejpam-2095	304	10	.	.	PUNCT
ejpam-2095	305	1	∧	∧	NOUN
ejpam-2095	305	2	qk	qk	AUX
ejpam-2095	305	3	be	be	AUX
ejpam-2095	305	4	a	a	DET
ejpam-2095	305	5	reduced	reduce	VERB
ejpam-2095	305	6	primary	primary	ADJ
ejpam-2095	305	7	decomposition	decomposition	NOUN
ejpam-2095	305	8	of	of	ADP
ejpam-2095	305	9	0	0	NUM
ejpam-2095	305	10	and	and	CCONJ
ejpam-2095	305	11	pi	pi	NOUN
ejpam-2095	306	1	=	=	SYM
ejpam-2095	306	2	p	p	PROPN
ejpam-2095	306	3	qi	qi	PROPN
ejpam-2095	306	4	,	,	PUNCT
ejpam-2095	306	5	i	i	NOUN
ejpam-2095	306	6	=	=	NOUN
ejpam-2095	306	7	1,2	1,2	NUM
ejpam-2095	306	8	,	,	PUNCT
ejpam-2095	306	9	.	.	PUNCT
ejpam-2095	306	10	.	.	PUNCT
ejpam-2095	306	11	.	.	PUNCT
ejpam-2095	307	1	,	,	PUNCT
ejpam-2095	307	2	k.	k.	PROPN
ejpam-2095	307	3	suppose	suppose	VERB
ejpam-2095	307	4	a	a	PRON
ejpam-2095	307	5	is	be	AUX
ejpam-2095	307	6	a	a	DET
ejpam-2095	307	7	zero	zero	NUM
ejpam-2095	307	8	divisor	divisor	NOUN
ejpam-2095	307	9	.	.	PUNCT
ejpam-2095	308	1	then	then	ADV
ejpam-2095	308	2	if	if	SCONJ
ejpam-2095	308	3	a	a	DET
ejpam-2095	308	4	=	=	NOUN
ejpam-2095	308	5	0	0	NUM
ejpam-2095	308	6	obviously	obviously	ADV
ejpam-2095	308	7	a	a	DET
ejpam-2095	308	8	¶	¶	PROPN
ejpam-2095	308	9	p1	p1	PROPN
ejpam-2095	308	10	∨	∨	NUM
ejpam-2095	308	11	p2	p2	PROPN
ejpam-2095	308	12	∨	∨	NOUN
ejpam-2095	308	13	.	.	PUNCT
ejpam-2095	308	14	.	.	PUNCT
ejpam-2095	308	15	.	.	PUNCT
ejpam-2095	309	1	∨	∨	NUM
ejpam-2095	309	2	pk	pk	PROPN
ejpam-2095	309	3	.	.	PROPN
ejpam-2095	309	4	suppose	suppose	VERB
ejpam-2095	309	5	,	,	PUNCT
ejpam-2095	309	6	a	a	PRON
ejpam-2095	309	7	is	be	AUX
ejpam-2095	309	8	a	a	DET
ejpam-2095	309	9	proper	proper	ADJ
ejpam-2095	309	10	zero	zero	NUM
ejpam-2095	309	11	divisor	divisor	NOUN
ejpam-2095	309	12	that	that	PRON
ejpam-2095	309	13	is	be	AUX
ejpam-2095	309	14	a	a	DET
ejpam-2095	309	15	6=	6=	NOUN
ejpam-2095	309	16	0	0	NUM
ejpam-2095	309	17	and	and	CCONJ
ejpam-2095	309	18	let	let	VERB
ejpam-2095	309	19	ab	ab	PROPN
ejpam-2095	309	20	=	=	PUNCT
ejpam-2095	309	21	0	0	NUM
ejpam-2095	310	1	where	where	SCONJ
ejpam-2095	310	2	b	b	X
ejpam-2095	310	3	6=	6=	NUM
ejpam-2095	310	4	0	0	NUM
ejpam-2095	310	5	.	.	PUNCT
ejpam-2095	310	6	now	now	ADV
ejpam-2095	310	7	,	,	PUNCT
ejpam-2095	310	8	ab	ab	PROPN
ejpam-2095	310	9	=	=	NOUN
ejpam-2095	310	10	0¶	0¶	NOUN
ejpam-2095	310	11	q1∧q2∧	q1∧q2∧	NOUN
ejpam-2095	310	12	.	.	PUNCT
ejpam-2095	310	13	.	.	PUNCT
ejpam-2095	310	14	.∧qk	.∧qk	PUNCT
ejpam-2095	311	1	=	=	PUNCT
ejpam-2095	311	2	{	{	PUNCT
ejpam-2095	311	3	0	0	NUM
ejpam-2095	311	4	}	}	PUNCT
ejpam-2095	311	5	.	.	PUNCT
ejpam-2095	312	1	hence	hence	ADV
ejpam-2095	312	2	,	,	PUNCT
ejpam-2095	312	3	ab	ab	PROPN
ejpam-2095	312	4	¶	¶	PROPN
ejpam-2095	312	5	qi	qi	PROPN
ejpam-2095	312	6	for	for	ADP
ejpam-2095	312	7	all	all	DET
ejpam-2095	312	8	i	i	PRON
ejpam-2095	312	9	and	and	CCONJ
ejpam-2095	312	10	b	b	PROPN
ejpam-2095	312	11	qi	qi	NOUN
ejpam-2095	312	12	for	for	ADP
ejpam-2095	312	13	at	at	ADV
ejpam-2095	312	14	least	least	ADV
ejpam-2095	312	15	one	one	NUM
ejpam-2095	312	16	i.	i.	NOUN
ejpam-2095	312	17	because	because	SCONJ
ejpam-2095	312	18	,	,	PUNCT
ejpam-2095	312	19	b	b	PROPN
ejpam-2095	312	20	¶	¶	PROPN
ejpam-2095	312	21	qi	qi	PROPN
ejpam-2095	312	22	for	for	ADP
ejpam-2095	312	23	all	all	PRON
ejpam-2095	312	24	i	i	PRON
ejpam-2095	312	25	implies	imply	VERB
ejpam-2095	312	26	b	b	PROPN
ejpam-2095	312	27	¶	¶	PROPN
ejpam-2095	312	28	q1	q1	PROPN
ejpam-2095	312	29	∧	∧	PROPN
ejpam-2095	312	30	q2	q2	PROPN
ejpam-2095	312	31	∧	∧	PROPN
ejpam-2095	312	32	.	.	PUNCT
ejpam-2095	312	33	.	.	PUNCT
ejpam-2095	312	34	.	.	PUNCT
ejpam-2095	313	1	∧	∧	NOUN
ejpam-2095	313	2	qk	qk	NOUN
ejpam-2095	313	3	=	=	PUNCT
ejpam-2095	313	4	{	{	PUNCT
ejpam-2095	313	5	0	0	NUM
ejpam-2095	313	6	}	}	PUNCT
ejpam-2095	313	7	and	and	CCONJ
ejpam-2095	313	8	hence	hence	ADV
ejpam-2095	313	9	b	b	X
ejpam-2095	313	10	=	=	SYM
ejpam-2095	313	11	0	0	PROPN
ejpam-2095	313	12	,	,	PUNCT
ejpam-2095	313	13	a	a	DET
ejpam-2095	313	14	contradiction	contradiction	NOUN
ejpam-2095	313	15	.	.	PUNCT
ejpam-2095	314	1	let	let	VERB
ejpam-2095	314	2	b	b	NOUN
ejpam-2095	314	3	q	q	PROPN
ejpam-2095	314	4	j	j	PROPN
ejpam-2095	314	5	.	.	PUNCT
ejpam-2095	315	1	then	then	ADV
ejpam-2095	315	2	,	,	PUNCT
ejpam-2095	315	3	ab	ab	PROPN
ejpam-2095	315	4	¶	¶	PROPN
ejpam-2095	315	5	q	q	PROPN
ejpam-2095	315	6	j	j	PROPN
ejpam-2095	315	7	,	,	PUNCT
ejpam-2095	315	8	b	b	PROPN
ejpam-2095	315	9	q	q	X
ejpam-2095	315	10	j	j	PROPN
ejpam-2095	315	11	and	and	CCONJ
ejpam-2095	315	12	q	q	PROPN
ejpam-2095	315	13	j	j	PROPN
ejpam-2095	315	14	is	be	AUX
ejpam-2095	315	15	a	a	DET
ejpam-2095	315	16	primary	primary	ADJ
ejpam-2095	315	17	element	element	NOUN
ejpam-2095	315	18	.	.	PUNCT
ejpam-2095	316	1	therefore	therefore	ADV
ejpam-2095	316	2	,	,	PUNCT
ejpam-2095	316	3	a	a	DET
ejpam-2095	316	4	¶	¶	NOUN
ejpam-2095	316	5	p	p	NOUN
ejpam-2095	316	6	q	q	PROPN
ejpam-2095	316	7	j	j	PROPN
ejpam-2095	316	8	=	=	PUNCT
ejpam-2095	316	9	p	p	PROPN
ejpam-2095	316	10	j	j	PROPN
ejpam-2095	316	11	,	,	PUNCT
ejpam-2095	316	12	which	which	PRON
ejpam-2095	316	13	shows	show	VERB
ejpam-2095	316	14	that	that	SCONJ
ejpam-2095	316	15	a	a	DET
ejpam-2095	316	16	¶	¶	PROPN
ejpam-2095	316	17	p1	p1	PROPN
ejpam-2095	316	18	∨	∨	NUM
ejpam-2095	316	19	p2	p2	PROPN
ejpam-2095	316	20	∨	∨	NOUN
ejpam-2095	316	21	.	.	PUNCT
ejpam-2095	316	22	.	.	PUNCT
ejpam-2095	317	1	.∨	.∨	PUNCT
ejpam-2095	318	1	pk	pk	PROPN
ejpam-2095	318	2	.	.	PROPN
ejpam-2095	318	3	theorem	theorem	PROPN
ejpam-2095	318	4	10	10	NUM
ejpam-2095	318	5	.	.	PUNCT
ejpam-2095	319	1	let	let	VERB
ejpam-2095	319	2	m	m	PRON
ejpam-2095	319	3	be	be	AUX
ejpam-2095	319	4	a	a	DET
ejpam-2095	319	5	lattice	lattice	NOUN
ejpam-2095	319	6	module	module	NOUN
ejpam-2095	319	7	and	and	CCONJ
ejpam-2095	319	8	n	n	NOUN
ejpam-2095	319	9	6=	6=	PROPN
ejpam-2095	320	1	i	i	PRON
ejpam-2095	320	2	m	m	AUX
ejpam-2095	320	3	be	be	VERB
ejpam-2095	320	4	an	an	DET
ejpam-2095	320	5	element	element	NOUN
ejpam-2095	320	6	of	of	ADP
ejpam-2095	320	7	m	m	PRON
ejpam-2095	320	8	which	which	PRON
ejpam-2095	320	9	has	have	VERB
ejpam-2095	320	10	a	a	DET
ejpam-2095	320	11	reduced	reduce	VERB
ejpam-2095	320	12	primary	primary	ADJ
ejpam-2095	320	13	decomposition	decomposition	NOUN
ejpam-2095	320	14	n	n	PROPN
ejpam-2095	320	15	=	=	PROPN
ejpam-2095	320	16	q1	q1	PROPN
ejpam-2095	320	17	∧q2	∧q2	VERB
ejpam-2095	320	18	∧	∧	PROPN
ejpam-2095	320	19	.	.	PUNCT
ejpam-2095	320	20	.	.	PUNCT
ejpam-2095	320	21	.	.	PUNCT
ejpam-2095	321	1	∧qm	∧qm	PROPN
ejpam-2095	321	2	.	.	PUNCT
ejpam-2095	322	1	if	if	SCONJ
ejpam-2095	322	2	every	every	DET
ejpam-2095	322	3	q	q	X
ejpam-2095	322	4	i	i	PRON
ejpam-2095	322	5	(	(	PUNCT
ejpam-2095	322	6	1¶	1¶	NOUN
ejpam-2095	322	7	i	i	PROPN
ejpam-2095	322	8	¶	¶	PROPN
ejpam-2095	322	9	m	m	VERB
ejpam-2095	322	10	)	)	PUNCT
ejpam-2095	322	11	is	be	AUX
ejpam-2095	322	12	a	a	DET
ejpam-2095	322	13	prime	prime	ADJ
ejpam-2095	322	14	element	element	NOUN
ejpam-2095	322	15	then	then	ADV
ejpam-2095	322	16	(	(	PUNCT
ejpam-2095	322	17	n	n	X
ejpam-2095	322	18	:	:	PUNCT
ejpam-2095	322	19	i	i	PRON
ejpam-2095	322	20	m	m	VERB
ejpam-2095	322	21	)	)	PUNCT
ejpam-2095	323	1	=	=	SYM
ejpam-2095	323	2	p	p	X
ejpam-2095	323	3	(	(	PUNCT
ejpam-2095	323	4	n	n	NOUN
ejpam-2095	323	5	:	:	PUNCT
ejpam-2095	323	6	i	i	PRON
ejpam-2095	323	7	m	m	PROPN
ejpam-2095	323	8	)	)	PUNCT
ejpam-2095	323	9	and	and	CCONJ
ejpam-2095	323	10	the	the	DET
ejpam-2095	323	11	converse	converse	NOUN
ejpam-2095	323	12	holds	hold	VERB
ejpam-2095	323	13	if	if	SCONJ
ejpam-2095	323	14	(	(	PUNCT
ejpam-2095	323	15	q	q	NOUN
ejpam-2095	323	16	i	i	X
ejpam-2095	323	17	:	:	PUNCT
ejpam-2095	323	18	i	i	PRON
ejpam-2095	323	19	m	m	VERB
ejpam-2095	323	20	)	)	PUNCT
ejpam-2095	323	21	are	be	AUX
ejpam-2095	323	22	prime	prime	ADJ
ejpam-2095	323	23	elements	element	NOUN
ejpam-2095	323	24	.	.	PUNCT
ejpam-2095	324	1	proof	proof	NOUN
ejpam-2095	324	2	.	.	PUNCT
ejpam-2095	325	1	suppose	suppose	VERB
ejpam-2095	325	2	each	each	DET
ejpam-2095	325	3	q	q	NOUN
ejpam-2095	325	4	i	i	PRON
ejpam-2095	325	5	is	be	AUX
ejpam-2095	325	6	a	a	DET
ejpam-2095	325	7	prime	prime	ADJ
ejpam-2095	325	8	element	element	NOUN
ejpam-2095	325	9	.	.	PUNCT
ejpam-2095	326	1	let	let	VERB
ejpam-2095	326	2	a	a	DET
ejpam-2095	326	3	¶	¶	PROPN
ejpam-2095	326	4	p	p	NOUN
ejpam-2095	326	5	(	(	PUNCT
ejpam-2095	326	6	n	n	NUM
ejpam-2095	326	7	:	:	PUNCT
ejpam-2095	326	8	i	i	PRON
ejpam-2095	326	9	m	m	PROPN
ejpam-2095	326	10	)	)	PUNCT
ejpam-2095	326	11	.	.	PUNCT
ejpam-2095	327	1	then	then	ADV
ejpam-2095	327	2	an	an	DET
ejpam-2095	327	3	i	i	NOUN
ejpam-2095	327	4	m	m	VERB
ejpam-2095	327	5	¶	¶	PROPN
ejpam-2095	327	6	n	n	PROPN
ejpam-2095	327	7	=	=	PROPN
ejpam-2095	327	8	q1	q1	PROPN
ejpam-2095	327	9	∧q2	∧q2	VERB
ejpam-2095	327	10	∧	∧	PROPN
ejpam-2095	327	11	.	.	PUNCT
ejpam-2095	327	12	.	.	PUNCT
ejpam-2095	328	1	.∧qm	.∧qm	PUNCT
ejpam-2095	329	1	for	for	ADP
ejpam-2095	329	2	some	some	DET
ejpam-2095	329	3	positive	positive	ADJ
ejpam-2095	329	4	integer	integer	NOUN
ejpam-2095	329	5	n.	n.	NOUN
ejpam-2095	329	6	this	this	PRON
ejpam-2095	329	7	implies	imply	VERB
ejpam-2095	329	8	that	that	SCONJ
ejpam-2095	329	9	,	,	PUNCT
ejpam-2095	329	10	an	an	DET
ejpam-2095	329	11	i	i	NOUN
ejpam-2095	329	12	m	m	VERB
ejpam-2095	329	13	¶	¶	PROPN
ejpam-2095	329	14	q	q	PROPN
ejpam-2095	329	15	i	i	PROPN
ejpam-2095	329	16	for	for	ADP
ejpam-2095	329	17	each	each	DET
ejpam-2095	329	18	i.	i.	NOUN
ejpam-2095	329	19	as	as	SCONJ
ejpam-2095	329	20	q	q	PROPN
ejpam-2095	329	21	i	i	PRON
ejpam-2095	329	22	is	be	AUX
ejpam-2095	329	23	a	a	DET
ejpam-2095	329	24	prime	prime	ADJ
ejpam-2095	329	25	element	element	NOUN
ejpam-2095	329	26	,	,	PUNCT
ejpam-2095	329	27	aim	aim	VERB
ejpam-2095	329	28	¶	¶	PROPN
ejpam-2095	329	29	q	q	PROPN
ejpam-2095	330	1	i	i	PROPN
ejpam-2095	330	2	or	or	CCONJ
ejpam-2095	330	3	an−1	an−1	PROPN
ejpam-2095	330	4	i	i	PRON
ejpam-2095	330	5	m	m	VERB
ejpam-2095	330	6	¶	¶	NOUN
ejpam-2095	330	7	q	q	PROPN
ejpam-2095	330	8	i	i	INTJ
ejpam-2095	330	9	.	.	PUNCT
ejpam-2095	331	1	if	if	SCONJ
ejpam-2095	331	2	aim	aim	VERB
ejpam-2095	331	3	¶	¶	PROPN
ejpam-2095	331	4	q	q	PROPN
ejpam-2095	332	1	i	i	PRON
ejpam-2095	332	2	then	then	ADV
ejpam-2095	332	3	a	a	DET
ejpam-2095	332	4	¶	¶	NOUN
ejpam-2095	332	5	(	(	PUNCT
ejpam-2095	332	6	q	q	NOUN
ejpam-2095	333	1	i	i	PRON
ejpam-2095	333	2	:	:	PUNCT
ejpam-2095	333	3	i	i	PRON
ejpam-2095	333	4	m	m	PROPN
ejpam-2095	333	5	)	)	PUNCT
ejpam-2095	333	6	.	.	PUNCT
ejpam-2095	334	1	otherwise	otherwise	ADV
ejpam-2095	334	2	an−1	an−1	PROPN
ejpam-2095	335	1	i	i	PRON
ejpam-2095	335	2	m	m	VERB
ejpam-2095	335	3	¶	¶	NOUN
ejpam-2095	335	4	q	q	PUNCT
ejpam-2095	336	1	i	i	PRON
ejpam-2095	336	2	implies	imply	VERB
ejpam-2095	336	3	aim	aim	NOUN
ejpam-2095	336	4	¶	¶	PROPN
ejpam-2095	336	5	q	q	PROPN
ejpam-2095	336	6	i	i	PROPN
ejpam-2095	336	7	or	or	CCONJ
ejpam-2095	336	8	an−2	an−2	PROPN
ejpam-2095	336	9	i	i	PROPN
ejpam-2095	336	10	m	m	VERB
ejpam-2095	336	11	¶	¶	PROPN
ejpam-2095	336	12	q	q	PROPN
ejpam-2095	337	1	i.	i.	PROPN
ejpam-2095	337	2	continuing	continue	VERB
ejpam-2095	337	3	in	in	ADP
ejpam-2095	337	4	this	this	DET
ejpam-2095	337	5	way	way	NOUN
ejpam-2095	337	6	we	we	PRON
ejpam-2095	337	7	obtain	obtain	VERB
ejpam-2095	337	8	,	,	PUNCT
ejpam-2095	337	9	a	a	DET
ejpam-2095	337	10	¶	¶	NOUN
ejpam-2095	337	11	(	(	PUNCT
ejpam-2095	337	12	q	q	NOUN
ejpam-2095	337	13	i	i	PRON
ejpam-2095	337	14	:	:	PUNCT
ejpam-2095	337	15	i	i	PRON
ejpam-2095	337	16	m	m	VERB
ejpam-2095	337	17	)	)	PUNCT
ejpam-2095	337	18	for	for	ADP
ejpam-2095	337	19	each	each	DET
ejpam-2095	337	20	i.	i.	NOUN
ejpam-2095	337	21	therefore	therefore	ADV
ejpam-2095	337	22	a	a	DET
ejpam-2095	337	23	¶	¶	PROPN
ejpam-2095	337	24	(	(	PUNCT
ejpam-2095	337	25	q1	q1	PROPN
ejpam-2095	337	26	:	:	PUNCT
ejpam-2095	337	27	i	i	PRON
ejpam-2095	337	28	m	m	VERB
ejpam-2095	337	29	)	)	PUNCT
ejpam-2095	337	30	∧	∧	PROPN
ejpam-2095	337	31	(	(	PUNCT
ejpam-2095	337	32	q2	q2	NOUN
ejpam-2095	337	33	:	:	PUNCT
ejpam-2095	337	34	i	i	PRON
ejpam-2095	337	35	m	m	VERB
ejpam-2095	337	36	)	)	PUNCT
ejpam-2095	337	37	∧	∧	PROPN
ejpam-2095	337	38	.	.	PUNCT
ejpam-2095	337	39	.	.	PUNCT
ejpam-2095	337	40	.	.	PUNCT
ejpam-2095	338	1	∧	∧	NOUN
ejpam-2095	338	2	(	(	PUNCT
ejpam-2095	338	3	qm	qm	NOUN
ejpam-2095	338	4	:	:	PUNCT
ejpam-2095	338	5	i	i	PRON
ejpam-2095	338	6	m	m	PROPN
ejpam-2095	338	7	)	)	PUNCT
ejpam-2095	338	8	.	.	PUNCT
ejpam-2095	339	1	that	that	PRON
ejpam-2095	339	2	is	be	AUX
ejpam-2095	339	3	a	a	DET
ejpam-2095	339	4	¶	¶	NOUN
ejpam-2095	339	5	(	(	PUNCT
ejpam-2095	339	6	n	n	NUM
ejpam-2095	339	7	:	:	PUNCT
ejpam-2095	339	8	i	i	PRON
ejpam-2095	339	9	m	m	PROPN
ejpam-2095	339	10	)	)	PUNCT
ejpam-2095	339	11	and	and	CCONJ
ejpam-2095	339	12	hence	hence	ADV
ejpam-2095	339	13	(	(	PUNCT
ejpam-2095	339	14	n	n	X
ejpam-2095	339	15	:	:	PUNCT
ejpam-2095	339	16	i	i	PRON
ejpam-2095	339	17	m	m	VERB
ejpam-2095	339	18	)	)	PUNCT
ejpam-2095	340	1	=	=	SYM
ejpam-2095	340	2	p	p	X
ejpam-2095	340	3	(	(	PUNCT
ejpam-2095	340	4	n	n	NOUN
ejpam-2095	340	5	:	:	PUNCT
ejpam-2095	340	6	i	i	PRON
ejpam-2095	340	7	m	m	PROPN
ejpam-2095	340	8	)	)	PUNCT
ejpam-2095	340	9	.	.	PUNCT
ejpam-2095	341	1	conversely	conversely	ADV
ejpam-2095	341	2	assume	assume	VERB
ejpam-2095	341	3	that	that	SCONJ
ejpam-2095	341	4	,	,	PUNCT
ejpam-2095	341	5	(	(	PUNCT
ejpam-2095	341	6	n	n	X
ejpam-2095	341	7	:	:	PUNCT
ejpam-2095	341	8	i	i	PRON
ejpam-2095	341	9	m	m	VERB
ejpam-2095	341	10	)	)	PUNCT
ejpam-2095	342	1	=	=	SYM
ejpam-2095	342	2	p	p	X
ejpam-2095	342	3	(	(	PUNCT
ejpam-2095	342	4	n	n	NOUN
ejpam-2095	342	5	:	:	PUNCT
ejpam-2095	342	6	i	i	PRON
ejpam-2095	342	7	m	m	PROPN
ejpam-2095	342	8	)	)	PUNCT
ejpam-2095	342	9	.	.	PUNCT
ejpam-2095	343	1	we	we	PRON
ejpam-2095	343	2	show	show	VERB
ejpam-2095	343	3	that	that	SCONJ
ejpam-2095	343	4	(	(	PUNCT
ejpam-2095	343	5	q	q	NOUN
ejpam-2095	343	6	i	i	X
ejpam-2095	343	7	:	:	PUNCT
ejpam-2095	343	8	i	i	PRON
ejpam-2095	343	9	m	m	VERB
ejpam-2095	343	10	)	)	PUNCT
ejpam-2095	344	1	=	=	SYM
ejpam-2095	345	1	pi	pi	NOUN
ejpam-2095	345	2	.	.	PUNCT
ejpam-2095	346	1	let	let	VERB
ejpam-2095	346	2	y	y	PROPN
ejpam-2095	346	3	¶	¶	PROPN
ejpam-2095	346	4	pi	pi	NOUN
ejpam-2095	347	1	=	=	X
ejpam-2095	347	2	p	p	X
ejpam-2095	347	3	(	(	PUNCT
ejpam-2095	347	4	q	q	NOUN
ejpam-2095	347	5	i	i	PRON
ejpam-2095	347	6	:	:	PUNCT
ejpam-2095	347	7	i	i	PRON
ejpam-2095	347	8	m	m	PROPN
ejpam-2095	347	9	)	)	PUNCT
ejpam-2095	347	10	.	.	PUNCT
ejpam-2095	348	1	as	as	SCONJ
ejpam-2095	348	2	m∧	m∧	PROPN
ejpam-2095	348	3	i=1	i=1	PROPN
ejpam-2095	348	4	pi	pi	PROPN
ejpam-2095	348	5	is	be	AUX
ejpam-2095	348	6	irredundant(reduced	irredundant(reduce	VERB
ejpam-2095	348	7	)	)	PUNCT
ejpam-2095	348	8	there	there	PRON
ejpam-2095	348	9	exists	exist	VERB
ejpam-2095	348	10	z	z	PROPN
ejpam-2095	348	11	¶	¶	PROPN
ejpam-2095	348	12	∧p	∧p	PROPN
ejpam-2095	348	13	j	j	PROPN
ejpam-2095	349	1	such	such	ADJ
ejpam-2095	349	2	that	that	SCONJ
ejpam-2095	349	3	z	z	PROPN
ejpam-2095	349	4	�	�	PROPN
ejpam-2095	349	5	pi	pi	PROPN
ejpam-2095	349	6	in	in	ADP
ejpam-2095	349	7	l.	l.	PROPN
ejpam-2095	349	8	now	now	ADV
ejpam-2095	349	9	yz	yz	PROPN
ejpam-2095	349	10	¶	¶	PROPN
ejpam-2095	349	11	m∧	m∧	PROPN
ejpam-2095	349	12	i=1	i=1	PROPN
ejpam-2095	349	13	pi	pi	PROPN
ejpam-2095	349	14	=	=	SYM
ejpam-2095	349	15	m∧	m∧	PROPN
ejpam-2095	349	16	i=1	i=1	PROPN
ejpam-2095	350	1	p	p	X
ejpam-2095	351	1	(	(	PUNCT
ejpam-2095	351	2	q	q	NOUN
ejpam-2095	351	3	i	i	PRON
ejpam-2095	351	4	:	:	PUNCT
ejpam-2095	351	5	i	i	PRON
ejpam-2095	351	6	m	m	VERB
ejpam-2095	351	7	)	)	PUNCT
ejpam-2095	352	1	=	=	PUNCT
ejpam-2095	352	2	m∧	m∧	PROPN
ejpam-2095	352	3	i=1	i=1	X
ejpam-2095	353	1	(	(	PUNCT
ejpam-2095	353	2	q	q	NOUN
ejpam-2095	353	3	i	i	X
ejpam-2095	353	4	:	:	PUNCT
ejpam-2095	353	5	i	i	PRON
ejpam-2095	353	6	m	m	PROPN
ejpam-2095	353	7	)	)	PUNCT
ejpam-2095	353	8	implies	imply	VERB
ejpam-2095	353	9	yzim	yzim	PROPN
ejpam-2095	353	10	¶	¶	PROPN
ejpam-2095	353	11	q	q	PROPN
ejpam-2095	354	1	i	i	PROPN
ejpam-2095	354	2	for	for	ADP
ejpam-2095	354	3	each	each	DET
ejpam-2095	354	4	i.	i.	NOUN
ejpam-2095	354	5	since	since	SCONJ
ejpam-2095	354	6	q	q	PROPN
ejpam-2095	354	7	i	i	PRON
ejpam-2095	354	8	is	be	AUX
ejpam-2095	354	9	primary	primary	ADJ
ejpam-2095	354	10	,	,	PUNCT
ejpam-2095	354	11	z	z	PROPN
ejpam-2095	354	12	�	�	PROPN
ejpam-2095	354	13	pi	pi	PROPN
ejpam-2095	354	14	gives	give	VERB
ejpam-2095	354	15	y	y	PROPN
ejpam-2095	354	16	¶	¶	PROPN
ejpam-2095	354	17	(	(	PUNCT
ejpam-2095	354	18	q	q	NOUN
ejpam-2095	354	19	i	i	X
ejpam-2095	354	20	:	:	PUNCT
ejpam-2095	354	21	i	i	PRON
ejpam-2095	354	22	m	m	PROPN
ejpam-2095	354	23	)	)	PUNCT
ejpam-2095	354	24	.	.	PUNCT
ejpam-2095	355	1	hence	hence	ADV
ejpam-2095	355	2	,	,	PUNCT
ejpam-2095	355	3	pi	pi	PROPN
ejpam-2095	355	4	¶	¶	PROPN
ejpam-2095	355	5	(	(	PUNCT
ejpam-2095	355	6	q	q	NOUN
ejpam-2095	355	7	i	i	X
ejpam-2095	355	8	:	:	PUNCT
ejpam-2095	355	9	i	i	PRON
ejpam-2095	355	10	m	m	PROPN
ejpam-2095	355	11	)	)	PUNCT
ejpam-2095	355	12	.	.	PUNCT
ejpam-2095	356	1	consequently	consequently	ADV
ejpam-2095	356	2	,	,	PUNCT
ejpam-2095	356	3	pi	pi	NOUN
ejpam-2095	356	4	=	=	PUNCT
ejpam-2095	356	5	(	(	PUNCT
ejpam-2095	356	6	q	q	NOUN
ejpam-2095	356	7	i	i	X
ejpam-2095	356	8	:	:	PUNCT
ejpam-2095	356	9	i	i	PRON
ejpam-2095	356	10	m	m	PROPN
ejpam-2095	356	11	)	)	PUNCT
ejpam-2095	356	12	.	.	PUNCT
ejpam-2095	357	1	now	now	ADV
ejpam-2095	357	2	we	we	PRON
ejpam-2095	357	3	obtain	obtain	VERB
ejpam-2095	357	4	a	a	DET
ejpam-2095	357	5	characterization	characterization	NOUN
ejpam-2095	357	6	of	of	ADP
ejpam-2095	357	7	a	a	DET
ejpam-2095	357	8	prime	prime	ADJ
ejpam-2095	357	9	element	element	NOUN
ejpam-2095	357	10	p	p	NOUN
ejpam-2095	357	11	of	of	ADP
ejpam-2095	357	12	l	l	NOUN
ejpam-2095	357	13	containing	contain	VERB
ejpam-2095	357	14	some	some	DET
ejpam-2095	357	15	associated	associate	VERB
ejpam-2095	357	16	prime	prime	ADJ
ejpam-2095	357	17	pi	pi	NOUN
ejpam-2095	357	18	of	of	ADP
ejpam-2095	357	19	n	n	PROPN
ejpam-2095	357	20	6=	6=	PROPN
ejpam-2095	358	1	i	i	PRON
ejpam-2095	358	2	m	m	VERB
ejpam-2095	358	3	in	in	ADP
ejpam-2095	358	4	a	a	DET
ejpam-2095	358	5	lattice	lattice	NOUN
ejpam-2095	358	6	module	module	NOUN
ejpam-2095	358	7	m	m	PROPN
ejpam-2095	358	8	.	.	PUNCT
ejpam-2095	359	1	theorem	theorem	ADJ
ejpam-2095	359	2	11	11	NUM
ejpam-2095	359	3	.	.	PUNCT
ejpam-2095	360	1	let	let	VERB
ejpam-2095	360	2	n	n	PRON
ejpam-2095	360	3	6=	6=	NUM
ejpam-2095	361	1	i	i	PRON
ejpam-2095	361	2	m	m	AUX
ejpam-2095	361	3	have	have	VERB
ejpam-2095	361	4	a	a	DET
ejpam-2095	361	5	reduced	reduce	VERB
ejpam-2095	361	6	primary	primary	ADJ
ejpam-2095	361	7	decomposition	decomposition	NOUN
ejpam-2095	361	8	n	n	PROPN
ejpam-2095	361	9	=	=	PROPN
ejpam-2095	361	10	q1	q1	PROPN
ejpam-2095	361	11	∧q2	∧q2	VERB
ejpam-2095	361	12	∧	∧	PROPN
ejpam-2095	361	13	.	.	PUNCT
ejpam-2095	361	14	.	.	PUNCT
ejpam-2095	361	15	.	.	PUNCT
ejpam-2095	362	1	∧qm	∧qm	PROPN
ejpam-2095	362	2	and	and	CCONJ
ejpam-2095	362	3	pi	pi	NOUN
ejpam-2095	362	4	=	=	PUNCT
ejpam-2095	362	5	p	p	X
ejpam-2095	362	6	(	(	PUNCT
ejpam-2095	362	7	q	q	NOUN
ejpam-2095	363	1	i	i	PRON
ejpam-2095	363	2	:	:	PUNCT
ejpam-2095	363	3	i	i	PRON
ejpam-2095	363	4	m	m	VERB
ejpam-2095	363	5	)	)	PUNCT
ejpam-2095	363	6	be	be	VERB
ejpam-2095	363	7	the	the	DET
ejpam-2095	363	8	associated	associated	ADJ
ejpam-2095	363	9	primes	prime	NOUN
ejpam-2095	363	10	of	of	ADP
ejpam-2095	363	11	n.	n.	NOUN
ejpam-2095	363	12	for	for	ADP
ejpam-2095	363	13	a	a	DET
ejpam-2095	363	14	prime	prime	ADJ
ejpam-2095	363	15	element	element	NOUN
ejpam-2095	363	16	p	p	NOUN
ejpam-2095	363	17	of	of	ADP
ejpam-2095	363	18	l	l	NOUN
ejpam-2095	363	19	to	to	PART
ejpam-2095	363	20	contain	contain	VERB
ejpam-2095	363	21	(	(	PUNCT
ejpam-2095	363	22	n	n	NUM
ejpam-2095	363	23	:	:	PUNCT
ejpam-2095	364	1	i	i	PRON
ejpam-2095	364	2	m	m	VERB
ejpam-2095	364	3	)	)	PUNCT
ejpam-2095	365	1	it	it	PRON
ejpam-2095	365	2	is	be	AUX
ejpam-2095	365	3	necessary	necessary	ADJ
ejpam-2095	365	4	and	and	CCONJ
ejpam-2095	365	5	sufficient	sufficient	ADJ
ejpam-2095	365	6	that	that	SCONJ
ejpam-2095	365	7	p	p	NOUN
ejpam-2095	365	8	contains	contain	VERB
ejpam-2095	365	9	pi	pi	NOUN
ejpam-2095	365	10	for	for	ADP
ejpam-2095	365	11	some	some	DET
ejpam-2095	365	12	i.	i.	NOUN
ejpam-2095	365	13	proof	proof	NOUN
ejpam-2095	365	14	.	.	PUNCT
ejpam-2095	366	1	suppose	suppose	VERB
ejpam-2095	366	2	pi	pi	PROPN
ejpam-2095	366	3	¶	¶	PROPN
ejpam-2095	366	4	p	p	NOUN
ejpam-2095	366	5	for	for	ADP
ejpam-2095	366	6	some	some	DET
ejpam-2095	366	7	i.	i.	NOUN
ejpam-2095	366	8	then	then	ADV
ejpam-2095	366	9	(	(	PUNCT
ejpam-2095	366	10	n	n	X
ejpam-2095	366	11	:	:	PUNCT
ejpam-2095	366	12	i	i	PRON
ejpam-2095	366	13	m	m	VERB
ejpam-2095	366	14	)	)	PUNCT
ejpam-2095	367	1	=	=	PUNCT
ejpam-2095	367	2	m∧	m∧	PROPN
ejpam-2095	367	3	i=1	i=1	X
ejpam-2095	368	1	(	(	PUNCT
ejpam-2095	368	2	q	q	NOUN
ejpam-2095	368	3	i	i	X
ejpam-2095	368	4	:	:	PUNCT
ejpam-2095	368	5	i	i	PRON
ejpam-2095	368	6	m	m	PROPN
ejpam-2095	368	7	)	)	PUNCT
ejpam-2095	368	8	implies	imply	VERB
ejpam-2095	368	9	(	(	PUNCT
ejpam-2095	368	10	n	n	X
ejpam-2095	368	11	:	:	PUNCT
ejpam-2095	368	12	i	i	PRON
ejpam-2095	368	13	m	m	VERB
ejpam-2095	368	14	)	)	PUNCT
ejpam-2095	369	1	¶	¶	PROPN
ejpam-2095	369	2	p.	p.	NOUN
ejpam-2095	369	3	conversely	conversely	ADV
ejpam-2095	369	4	assume	assume	VERB
ejpam-2095	369	5	that	that	SCONJ
ejpam-2095	369	6	(	(	PUNCT
ejpam-2095	369	7	n	n	X
ejpam-2095	369	8	:	:	PUNCT
ejpam-2095	369	9	i	i	PRON
ejpam-2095	369	10	m	m	VERB
ejpam-2095	369	11	)	)	PUNCT
ejpam-2095	369	12	¶	¶	PROPN
ejpam-2095	370	1	p.	p.	NOUN
ejpam-2095	370	2	then	then	ADV
ejpam-2095	370	3	(	(	PUNCT
ejpam-2095	370	4	q1	q1	INTJ
ejpam-2095	370	5	:	:	PUNCT
ejpam-2095	370	6	i	i	PRON
ejpam-2095	370	7	m	m	VERB
ejpam-2095	370	8	)	)	PUNCT
ejpam-2095	370	9	∧	∧	PROPN
ejpam-2095	370	10	(	(	PUNCT
ejpam-2095	370	11	q2	q2	NOUN
ejpam-2095	370	12	:	:	PUNCT
ejpam-2095	370	13	i	i	PRON
ejpam-2095	370	14	m	m	VERB
ejpam-2095	370	15	)	)	PUNCT
ejpam-2095	370	16	∧	∧	PROPN
ejpam-2095	370	17	.	.	PUNCT
ejpam-2095	370	18	.	.	PUNCT
ejpam-2095	371	1	.	.	PUNCT
ejpam-2095	372	1	∧	∧	NOUN
ejpam-2095	372	2	(	(	PUNCT
ejpam-2095	372	3	qm	qm	NOUN
ejpam-2095	372	4	:	:	PUNCT
ejpam-2095	372	5	i	i	PROPN
ejpam-2095	372	6	m	m	VERB
ejpam-2095	372	7	)	)	PUNCT
ejpam-2095	372	8	¶	¶	PROPN
ejpam-2095	372	9	p	p	PROPN
ejpam-2095	372	10	implies	imply	VERB
ejpam-2095	372	11	(	(	PUNCT
ejpam-2095	372	12	q	q	NOUN
ejpam-2095	372	13	i	i	X
ejpam-2095	372	14	:	:	PUNCT
ejpam-2095	372	15	i	i	PRON
ejpam-2095	372	16	m	m	VERB
ejpam-2095	372	17	)	)	PUNCT
ejpam-2095	373	1	¶	¶	PROPN
ejpam-2095	373	2	p	p	NOUN
ejpam-2095	373	3	for	for	ADP
ejpam-2095	373	4	some	some	DET
ejpam-2095	373	5	i.	i.	NOUN
ejpam-2095	374	1	but	but	CCONJ
ejpam-2095	374	2	p	p	X
ejpam-2095	374	3	(	(	PUNCT
ejpam-2095	374	4	q	q	NOUN
ejpam-2095	374	5	i	i	X
ejpam-2095	374	6	:	:	PUNCT
ejpam-2095	374	7	i	i	PRON
ejpam-2095	374	8	m	m	VERB
ejpam-2095	374	9	)	)	PUNCT
ejpam-2095	375	1	=	=	PRON
ejpam-2095	376	1	pi	pi	NOUN
ejpam-2095	376	2	is	be	AUX
ejpam-2095	376	3	the	the	DET
ejpam-2095	376	4	smallest	small	ADJ
ejpam-2095	376	5	prime	prime	NOUN
ejpam-2095	376	6	containing	contain	VERB
ejpam-2095	376	7	(	(	PUNCT
ejpam-2095	376	8	q	q	NOUN
ejpam-2095	376	9	i	i	PRON
ejpam-2095	376	10	:	:	PUNCT
ejpam-2095	376	11	i	i	PRON
ejpam-2095	376	12	m	m	PROPN
ejpam-2095	376	13	)	)	PUNCT
ejpam-2095	376	14	.	.	PUNCT
ejpam-2095	377	1	hence	hence	ADV
ejpam-2095	377	2	,	,	PUNCT
ejpam-2095	377	3	pi	pi	PROPN
ejpam-2095	377	4	¶	¶	PROPN
ejpam-2095	377	5	p	p	NOUN
ejpam-2095	377	6	for	for	ADP
ejpam-2095	377	7	some	some	DET
ejpam-2095	377	8	i.	i.	NOUN
ejpam-2095	377	9	in	in	ADP
ejpam-2095	377	10	our	our	PRON
ejpam-2095	377	11	next	next	ADJ
ejpam-2095	377	12	result	result	NOUN
ejpam-2095	377	13	we	we	PRON
ejpam-2095	377	14	show	show	VERB
ejpam-2095	377	15	that	that	SCONJ
ejpam-2095	377	16	those	those	PRON
ejpam-2095	377	17	q	q	X
ejpam-2095	378	1	′s	′s	PROPN
ejpam-2095	378	2	i	i	PRON
ejpam-2095	378	3	can	can	AUX
ejpam-2095	378	4	be	be	AUX
ejpam-2095	378	5	uniquely	uniquely	ADV
ejpam-2095	378	6	determined	determine	VERB
ejpam-2095	378	7	which	which	PRON
ejpam-2095	378	8	are	be	AUX
ejpam-2095	378	9	isolated	isolate	VERB
ejpam-2095	378	10	primary	primary	ADJ
ejpam-2095	378	11	components	component	NOUN
ejpam-2095	378	12	of	of	ADP
ejpam-2095	378	13	n	n	PROPN
ejpam-2095	378	14	6=	6=	PROPN
ejpam-2095	379	1	i	i	PRON
ejpam-2095	379	2	m	m	PROPN
ejpam-2095	379	3	.	.	PUNCT
ejpam-2095	380	1	theorem	theorem	NOUN
ejpam-2095	380	2	12	12	NUM
ejpam-2095	380	3	.	.	PUNCT
ejpam-2095	381	1	let	let	VERB
ejpam-2095	381	2	n	n	PRON
ejpam-2095	381	3	6=	6=	NUM
ejpam-2095	382	1	i	i	PRON
ejpam-2095	382	2	m	m	AUX
ejpam-2095	382	3	have	have	VERB
ejpam-2095	382	4	a	a	DET
ejpam-2095	382	5	reduced	reduce	VERB
ejpam-2095	382	6	primary	primary	ADJ
ejpam-2095	382	7	decomposition	decomposition	NOUN
ejpam-2095	382	8	n	n	PROPN
ejpam-2095	382	9	=	=	PROPN
ejpam-2095	382	10	q1	q1	PROPN
ejpam-2095	382	11	∧q2	∧q2	VERB
ejpam-2095	382	12	∧	∧	PROPN
ejpam-2095	382	13	.	.	PUNCT
ejpam-2095	382	14	.	.	PUNCT
ejpam-2095	382	15	.	.	PUNCT
ejpam-2095	383	1	∧qm	∧qm	PROPN
ejpam-2095	383	2	and	and	CCONJ
ejpam-2095	383	3	p1	p1	PROPN
ejpam-2095	383	4	,	,	PUNCT
ejpam-2095	383	5	p2	p2	NOUN
ejpam-2095	383	6	,	,	PUNCT
ejpam-2095	383	7	.	.	PUNCT
ejpam-2095	383	8	.	.	PUNCT
ejpam-2095	384	1	.	.	PUNCT
ejpam-2095	385	1	,	,	PUNCT
ejpam-2095	385	2	pm	pm	NOUN
ejpam-2095	385	3	be	be	AUX
ejpam-2095	385	4	the	the	DET
ejpam-2095	385	5	associated	associated	ADJ
ejpam-2095	385	6	primes	prime	NOUN
ejpam-2095	385	7	of	of	ADP
ejpam-2095	385	8	q1,q2	q1,q2	PROPN
ejpam-2095	385	9	,	,	PUNCT
ejpam-2095	385	10	.	.	PUNCT
ejpam-2095	385	11	.	.	PUNCT
ejpam-2095	386	1	.	.	PUNCT
ejpam-2095	387	1	,	,	PUNCT
ejpam-2095	387	2	qm	qm	PROPN
ejpam-2095	387	3	respectively	respectively	ADV
ejpam-2095	387	4	.	.	PUNCT
ejpam-2095	388	1	the	the	DET
ejpam-2095	388	2	element	element	NOUN
ejpam-2095	388	3	q	q	PROPN
ejpam-2095	388	4	′	′	NUM
ejpam-2095	389	1	i	i	NOUN
ejpam-2095	389	2	=	=	PUNCT
ejpam-2095	389	3	∨{x	∨{x	PROPN
ejpam-2095	390	1	∈	∈	PROPN
ejpam-2095	390	2	m	m	VERB
ejpam-2095	390	3	|	|	ADV
ejpam-2095	390	4	(	(	PUNCT
ejpam-2095	390	5	n	n	NOUN
ejpam-2095	390	6	:	:	PUNCT
ejpam-2095	390	7	x	x	X
ejpam-2095	390	8	)	)	PUNCT
ejpam-2095	390	9	�	�	PROPN
ejpam-2095	390	10	pi	pi	NOUN
ejpam-2095	390	11	}	}	PUNCT
ejpam-2095	390	12	c.	c.	PROPN
ejpam-2095	390	13	manjarekar	manjarekar	PROPN
ejpam-2095	390	14	,	,	PUNCT
ejpam-2095	390	15	u.	u.	PROPN
ejpam-2095	390	16	kandale	kandale	PROPN
ejpam-2095	390	17	/	/	SYM
ejpam-2095	390	18	eur	eur	PROPN
ejpam-2095	390	19	.	.	PUNCT
ejpam-2095	391	1	j.	j.	PROPN
ejpam-2095	391	2	pure	pure	PROPN
ejpam-2095	391	3	appl	appl	PROPN
ejpam-2095	391	4	.	.	PROPN
ejpam-2095	391	5	math	math	PROPN
ejpam-2095	391	6	,	,	PUNCT
ejpam-2095	391	7	7	7	NUM
ejpam-2095	391	8	(	(	PUNCT
ejpam-2095	391	9	2014	2014	NUM
ejpam-2095	391	10	)	)	PUNCT
ejpam-2095	391	11	,	,	PUNCT
ejpam-2095	391	12	201	201	NUM
ejpam-2095	391	13	-	-	SYM
ejpam-2095	391	14	209	209	NUM
ejpam-2095	391	15	207	207	NUM
ejpam-2095	391	16	is	be	AUX
ejpam-2095	391	17	an	an	DET
ejpam-2095	391	18	element	element	NOUN
ejpam-2095	391	19	of	of	ADP
ejpam-2095	391	20	m	m	PRON
ejpam-2095	391	21	which	which	PRON
ejpam-2095	391	22	is	be	AUX
ejpam-2095	391	23	contained	contain	VERB
ejpam-2095	391	24	in	in	ADP
ejpam-2095	391	25	q	q	PROPN
ejpam-2095	392	1	i	i	PRON
ejpam-2095	392	2	.	.	PUNCT
ejpam-2095	393	1	if	if	SCONJ
ejpam-2095	393	2	q	q	X
ejpam-2095	393	3	i	i	PRON
ejpam-2095	393	4	is	be	AUX
ejpam-2095	393	5	an	an	DET
ejpam-2095	393	6	isolated	isolated	ADJ
ejpam-2095	393	7	primary	primary	ADJ
ejpam-2095	393	8	component	component	NOUN
ejpam-2095	393	9	of	of	ADP
ejpam-2095	393	10	n	n	PRON
ejpam-2095	393	11	then	then	ADV
ejpam-2095	393	12	q	q	PROPN
ejpam-2095	393	13	i	i	NOUN
ejpam-2095	393	14	=	=	PUNCT
ejpam-2095	394	1	q	q	PROPN
ejpam-2095	395	1	′	′	INTJ
ejpam-2095	395	2	i	i	PRON
ejpam-2095	395	3	.	.	PUNCT
ejpam-2095	396	1	proof	proof	NOUN
ejpam-2095	396	2	.	.	PUNCT
ejpam-2095	397	1	take	take	VERB
ejpam-2095	397	2	any	any	DET
ejpam-2095	397	3	element	element	NOUN
ejpam-2095	397	4	a	a	DET
ejpam-2095	397	5	∈	∈	NOUN
ejpam-2095	397	6	{	{	PUNCT
ejpam-2095	397	7	x	x	SYM
ejpam-2095	397	8	∈	∈	PROPN
ejpam-2095	397	9	m	m	VERB
ejpam-2095	397	10	|	|	NOUN
ejpam-2095	397	11	(	(	PUNCT
ejpam-2095	397	12	n	n	NOUN
ejpam-2095	397	13	:	:	PUNCT
ejpam-2095	397	14	x	x	X
ejpam-2095	397	15	)	)	PUNCT
ejpam-2095	397	16	�	�	PROPN
ejpam-2095	397	17	pi	pi	NOUN
ejpam-2095	397	18	}	}	PUNCT
ejpam-2095	397	19	.	.	PUNCT
ejpam-2095	398	1	then	then	ADV
ejpam-2095	398	2	(	(	PUNCT
ejpam-2095	398	3	n	n	X
ejpam-2095	398	4	:	:	PUNCT
ejpam-2095	398	5	a	a	X
ejpam-2095	398	6	)	)	PUNCT
ejpam-2095	398	7	�	�	PROPN
ejpam-2095	398	8	pi	pi	NOUN
ejpam-2095	398	9	.	.	PUNCT
ejpam-2095	399	1	so	so	ADV
ejpam-2095	399	2	there	there	PRON
ejpam-2095	399	3	exists	exist	VERB
ejpam-2095	399	4	a	a	DET
ejpam-2095	399	5	∈	∈	NOUN
ejpam-2095	399	6	l	l	NOUN
ejpam-2095	399	7	such	such	ADJ
ejpam-2095	399	8	that	that	SCONJ
ejpam-2095	399	9	aa¶	aa¶	NOUN
ejpam-2095	399	10	n	n	PRON
ejpam-2095	399	11	and	and	CCONJ
ejpam-2095	399	12	a	a	DET
ejpam-2095	399	13	�	�	PROPN
ejpam-2095	399	14	pi	pi	NOUN
ejpam-2095	399	15	=	=	SYM
ejpam-2095	399	16	p	p	X
ejpam-2095	399	17	(	(	PUNCT
ejpam-2095	399	18	q	q	NOUN
ejpam-2095	399	19	i	i	PRON
ejpam-2095	399	20	:	:	PUNCT
ejpam-2095	399	21	i	i	PRON
ejpam-2095	399	22	m	m	PROPN
ejpam-2095	399	23	)	)	PUNCT
ejpam-2095	399	24	.	.	PUNCT
ejpam-2095	400	1	hence	hence	ADV
ejpam-2095	400	2	,	,	PUNCT
ejpam-2095	400	3	an	an	DET
ejpam-2095	400	4	i	i	PROPN
ejpam-2095	400	5	m	m	VERB
ejpam-2095	400	6	�	�	PROPN
ejpam-2095	400	7	q	q	PROPN
ejpam-2095	400	8	i	i	PROPN
ejpam-2095	400	9	for	for	ADP
ejpam-2095	400	10	any	any	DET
ejpam-2095	400	11	integer	integer	NOUN
ejpam-2095	400	12	n.	n.	NOUN
ejpam-2095	400	13	now	now	ADV
ejpam-2095	400	14	aa¶	aa¶	ADV
ejpam-2095	400	15	q	q	PROPN
ejpam-2095	400	16	i	i	PROPN
ejpam-2095	400	17	,	,	PUNCT
ejpam-2095	400	18	anim	anim	PROPN
ejpam-2095	400	19	�	�	PROPN
ejpam-2095	400	20	q	q	PROPN
ejpam-2095	401	1	i	i	PROPN
ejpam-2095	401	2	and	and	CCONJ
ejpam-2095	401	3	q	q	PROPN
ejpam-2095	401	4	i	i	PRON
ejpam-2095	401	5	is	be	AUX
ejpam-2095	401	6	primary	primary	ADJ
ejpam-2095	401	7	gives	give	NOUN
ejpam-2095	402	1	a¶q	a¶q	ADV
ejpam-2095	402	2	i	i	PRON
ejpam-2095	402	3	.	.	PUNCT
ejpam-2095	403	1	hence	hence	ADV
ejpam-2095	403	2	q	q	X
ejpam-2095	404	1	i	i	PRON
ejpam-2095	404	2	′	′	NUM
ejpam-2095	405	1	¶	¶	PROPN
ejpam-2095	405	2	q	q	PROPN
ejpam-2095	406	1	i	i	PROPN
ejpam-2095	406	2	and	and	CCONJ
ejpam-2095	406	3	the	the	DET
ejpam-2095	406	4	first	first	ADJ
ejpam-2095	406	5	part	part	NOUN
ejpam-2095	406	6	is	be	AUX
ejpam-2095	406	7	proved	prove	VERB
ejpam-2095	406	8	.	.	PUNCT
ejpam-2095	407	1	if	if	SCONJ
ejpam-2095	407	2	pi	pi	NOUN
ejpam-2095	407	3	is	be	AUX
ejpam-2095	407	4	a	a	DET
ejpam-2095	407	5	minimal	minimal	ADJ
ejpam-2095	407	6	associated	associate	VERB
ejpam-2095	407	7	primes	prime	NOUN
ejpam-2095	407	8	of	of	ADP
ejpam-2095	407	9	n	n	PRON
ejpam-2095	407	10	it	it	PRON
ejpam-2095	407	11	follows	follow	VERB
ejpam-2095	407	12	that	that	SCONJ
ejpam-2095	407	13	p	p	PROPN
ejpam-2095	407	14	j	j	PROPN
ejpam-2095	407	15	�	�	PROPN
ejpam-2095	407	16	pi	pi	NOUN
ejpam-2095	407	17	for	for	ADP
ejpam-2095	407	18	i	i	PROPN
ejpam-2095	407	19	6=	6=	PROPN
ejpam-2095	407	20	j.	j.	PROPN
ejpam-2095	407	21	then	then	ADV
ejpam-2095	407	22	there	there	PRON
ejpam-2095	407	23	exists	exist	VERB
ejpam-2095	407	24	b	b	PROPN
ejpam-2095	407	25	j	j	PROPN
ejpam-2095	407	26	¶	¶	PROPN
ejpam-2095	407	27	p	p	PROPN
ejpam-2095	407	28	j	j	PROPN
ejpam-2095	407	29	in	in	ADP
ejpam-2095	407	30	l	l	PROPN
ejpam-2095	408	1	such	such	ADJ
ejpam-2095	408	2	that	that	PRON
ejpam-2095	408	3	b	b	PROPN
ejpam-2095	408	4	j	j	PROPN
ejpam-2095	408	5	�	�	PROPN
ejpam-2095	408	6	pi	pi	PROPN
ejpam-2095	408	7	.	.	PUNCT
ejpam-2095	409	1	we	we	PRON
ejpam-2095	409	2	have	have	VERB
ejpam-2095	409	3	b	b	PROPN
ejpam-2095	409	4	j	j	PROPN
ejpam-2095	409	5	¶	¶	PROPN
ejpam-2095	409	6	p	p	PROPN
ejpam-2095	409	7	j	j	PROPN
ejpam-2095	409	8	=	=	PUNCT
ejpam-2095	409	9	p	p	X
ejpam-2095	409	10	(	(	PUNCT
ejpam-2095	409	11	q	q	PROPN
ejpam-2095	409	12	j	j	NOUN
ejpam-2095	409	13	:	:	PUNCT
ejpam-2095	409	14	i	i	PRON
ejpam-2095	409	15	m	m	VERB
ejpam-2095	409	16	)	)	PUNCT
ejpam-2095	410	1	=	=	PUNCT
ejpam-2095	411	1	∨{a	∨{a	PROPN
ejpam-2095	411	2	j	j	PROPN
ejpam-2095	411	3	∈	∈	PROPN
ejpam-2095	411	4	l	l	NOUN
ejpam-2095	412	1	|	|	ADV
ejpam-2095	412	2	as	as	SCONJ
ejpam-2095	412	3	j	j	PROPN
ejpam-2095	412	4	j	j	PROPN
ejpam-2095	412	5	i	i	PROPN
ejpam-2095	412	6	m	m	VERB
ejpam-2095	412	7	¶	¶	PROPN
ejpam-2095	412	8	q	q	PROPN
ejpam-2095	412	9	j	j	PROPN
ejpam-2095	412	10	for	for	ADP
ejpam-2095	412	11	some	some	DET
ejpam-2095	412	12	integer	integer	NOUN
ejpam-2095	412	13	s	s	PROPN
ejpam-2095	412	14	j	j	NOUN
ejpam-2095	412	15	}	}	PUNCT
ejpam-2095	412	16	.	.	PUNCT
ejpam-2095	413	1	since	since	SCONJ
ejpam-2095	413	2	each	each	DET
ejpam-2095	413	3	element	element	NOUN
ejpam-2095	413	4	of	of	ADP
ejpam-2095	413	5	l	l	NOUN
ejpam-2095	413	6	is	be	AUX
ejpam-2095	413	7	compact	compact	ADJ
ejpam-2095	413	8	,	,	PUNCT
ejpam-2095	413	9	we	we	PRON
ejpam-2095	413	10	have	have	VERB
ejpam-2095	413	11	b	b	PROPN
ejpam-2095	413	12	j	j	PROPN
ejpam-2095	413	13	¶	¶	PROPN
ejpam-2095	413	14	p	p	PROPN
ejpam-2095	413	15	j	j	PROPN
ejpam-2095	413	16	=	=	SYM
ejpam-2095	413	17	n∨	n∨	PROPN
ejpam-2095	413	18	j=1	j=1	PROPN
ejpam-2095	413	19	{	{	PUNCT
ejpam-2095	413	20	a	a	DET
ejpam-2095	413	21	j	j	PROPN
ejpam-2095	413	22	|	|	ADV
ejpam-2095	413	23	as	as	SCONJ
ejpam-2095	413	24	j	j	PROPN
ejpam-2095	413	25	j	j	PROPN
ejpam-2095	413	26	i	i	PROPN
ejpam-2095	413	27	m	m	VERB
ejpam-2095	413	28	¶	¶	PROPN
ejpam-2095	413	29	q	q	PROPN
ejpam-2095	413	30	j	j	PROPN
ejpam-2095	413	31	for	for	ADP
ejpam-2095	413	32	some	some	DET
ejpam-2095	413	33	integer	integer	NOUN
ejpam-2095	413	34	s	s	PROPN
ejpam-2095	413	35	j	j	NOUN
ejpam-2095	413	36	}	}	PUNCT
ejpam-2095	413	37	.	.	PUNCT
ejpam-2095	414	1	put	put	VERB
ejpam-2095	414	2	s1	s1	NOUN
ejpam-2095	414	3	+	+	CCONJ
ejpam-2095	414	4	s2	s2	NOUN
ejpam-2095	414	5	+	+	NOUN
ejpam-2095	414	6	.	.	PUNCT
ejpam-2095	414	7	.	.	PUNCT
ejpam-2095	415	1	.+	.+	NOUN
ejpam-2095	415	2	sn	sn	NOUN
ejpam-2095	416	1	=	=	SYM
ejpam-2095	417	1	k	k	PROPN
ejpam-2095	418	1	(	(	PUNCT
ejpam-2095	418	2	j	j	PROPN
ejpam-2095	418	3	)	)	PUNCT
ejpam-2095	418	4	.	.	PUNCT
ejpam-2095	419	1	then	then	ADV
ejpam-2095	419	2	b	b	X
ejpam-2095	419	3	j	j	PROPN
ejpam-2095	419	4	k	k	PROPN
ejpam-2095	419	5	(	(	PUNCT
ejpam-2095	419	6	j)im	j)im	PROPN
ejpam-2095	419	7	¶	¶	PROPN
ejpam-2095	419	8	(	(	PUNCT
ejpam-2095	419	9	a1	a1	PROPN
ejpam-2095	419	10	∨	∨	PROPN
ejpam-2095	419	11	a2	a2	PROPN
ejpam-2095	419	12	∨	∨	PROPN
ejpam-2095	419	13	.	.	PUNCT
ejpam-2095	419	14	.	.	PUNCT
ejpam-2095	420	1	.∨	.∨	PROPN
ejpam-2095	421	1	an	an	PRON
ejpam-2095	421	2	)	)	PUNCT
ejpam-2095	421	3	k	k	NOUN
ejpam-2095	421	4	(	(	PUNCT
ejpam-2095	421	5	j)im	j)im	PROPN
ejpam-2095	421	6	¶	¶	PROPN
ejpam-2095	421	7	q	q	PROPN
ejpam-2095	422	1	j.	j.	PROPN
ejpam-2095	422	2	clearly	clearly	ADV
ejpam-2095	422	3	b	b	X
ejpam-2095	423	1	=	=	SYM
ejpam-2095	423	2	π	π	X
ejpam-2095	423	3	j	j	PROPN
ejpam-2095	423	4	6	6	NUM
ejpam-2095	423	5	=	=	NOUN
ejpam-2095	423	6	i	i	NOUN
ejpam-2095	423	7	b	b	PROPN
ejpam-2095	423	8	j	j	PROPN
ejpam-2095	423	9	k	k	X
ejpam-2095	423	10	(	(	PUNCT
ejpam-2095	423	11	j	j	PROPN
ejpam-2095	423	12	)	)	PUNCT
ejpam-2095	423	13	�	�	PROPN
ejpam-2095	423	14	pi	pi	NOUN
ejpam-2095	423	15	as	as	SCONJ
ejpam-2095	423	16	pi	pi	PROPN
ejpam-2095	423	17	is	be	AUX
ejpam-2095	423	18	prime	prime	ADJ
ejpam-2095	423	19	.	.	PUNCT
ejpam-2095	424	1	however	however	ADV
ejpam-2095	424	2	,	,	PUNCT
ejpam-2095	424	3	bim	bim	PROPN
ejpam-2095	424	4	¶	¶	PROPN
ejpam-2095	424	5	∧	∧	PROPN
ejpam-2095	424	6	j	j	PROPN
ejpam-2095	424	7	6	6	NUM
ejpam-2095	425	1	=	=	NOUN
ejpam-2095	425	2	i	i	PRON
ejpam-2095	425	3	q	q	PROPN
ejpam-2095	425	4	j.	j.	PROPN
ejpam-2095	425	5	next	next	ADV
ejpam-2095	425	6	take	take	VERB
ejpam-2095	425	7	any	any	DET
ejpam-2095	425	8	x	x	PROPN
ejpam-2095	425	9	¶	¶	PROPN
ejpam-2095	425	10	q	q	PROPN
ejpam-2095	425	11	i.	i.	PROPN
ejpam-2095	425	12	then	then	ADV
ejpam-2095	425	13	x	x	SYM
ejpam-2095	425	14	bim	bim	PROPN
ejpam-2095	425	15	¶	¶	PROPN
ejpam-2095	425	16	m∧	m∧	PROPN
ejpam-2095	426	1	i=1	i=1	PROPN
ejpam-2095	426	2	q	q	PROPN
ejpam-2095	427	1	i	i	NOUN
ejpam-2095	427	2	=	=	SYM
ejpam-2095	427	3	n	n	PROPN
ejpam-2095	427	4	.	.	PUNCT
ejpam-2095	428	1	so	so	ADV
ejpam-2095	428	2	b	b	PROPN
ejpam-2095	428	3	¶	¶	PROPN
ejpam-2095	428	4	(	(	PUNCT
ejpam-2095	428	5	n	n	NOUN
ejpam-2095	428	6	:	:	PUNCT
ejpam-2095	428	7	x	x	X
ejpam-2095	428	8	)	)	PUNCT
ejpam-2095	428	9	�	�	PROPN
ejpam-2095	428	10	pi	pi	NOUN
ejpam-2095	428	11	.	.	PUNCT
ejpam-2095	429	1	this	this	PRON
ejpam-2095	429	2	implies	imply	VERB
ejpam-2095	429	3	that	that	SCONJ
ejpam-2095	429	4	x	x	SYM
ejpam-2095	429	5	∈	∈	NOUN
ejpam-2095	429	6	{	{	PUNCT
ejpam-2095	429	7	x	x	SYM
ejpam-2095	429	8	∈	∈	PROPN
ejpam-2095	429	9	m	m	VERB
ejpam-2095	429	10	|	|	NOUN
ejpam-2095	429	11	(	(	PUNCT
ejpam-2095	429	12	n	n	NOUN
ejpam-2095	429	13	:	:	PUNCT
ejpam-2095	429	14	x	x	X
ejpam-2095	429	15	)	)	PUNCT
ejpam-2095	429	16	6=	6=	NUM
ejpam-2095	429	17	pi	pi	NOUN
ejpam-2095	429	18	}	}	PUNCT
ejpam-2095	429	19	.	.	PUNCT
ejpam-2095	430	1	hence	hence	ADV
ejpam-2095	430	2	x	x	SYM
ejpam-2095	430	3	¶	¶	NUM
ejpam-2095	430	4	∨{x	∨{x	PROPN
ejpam-2095	430	5	∈	∈	PROPN
ejpam-2095	430	6	m	m	VERB
ejpam-2095	430	7	|	|	ADV
ejpam-2095	430	8	(	(	PUNCT
ejpam-2095	430	9	n	n	NOUN
ejpam-2095	430	10	:	:	PUNCT
ejpam-2095	430	11	x	x	X
ejpam-2095	430	12	)	)	PUNCT
ejpam-2095	430	13	�	�	PROPN
ejpam-2095	430	14	pi}=	pi}=	PROPN
ejpam-2095	431	1	q	q	PROPN
ejpam-2095	432	1	i	i	PRON
ejpam-2095	432	2	′	′	VERB
ejpam-2095	433	1	and	and	CCONJ
ejpam-2095	433	2	we	we	PRON
ejpam-2095	433	3	have	have	VERB
ejpam-2095	433	4	q	q	PROPN
ejpam-2095	434	1	i	i	PRON
ejpam-2095	434	2	¶	¶	PROPN
ejpam-2095	434	3	q	q	PROPN
ejpam-2095	435	1	i	i	PRON
ejpam-2095	435	2	′	′	VERB
ejpam-2095	435	3	.	.	PUNCT
ejpam-2095	436	1	consequently	consequently	ADV
ejpam-2095	436	2	,	,	PUNCT
ejpam-2095	436	3	q	q	PROPN
ejpam-2095	436	4	i	i	NOUN
ejpam-2095	436	5	=	=	PUNCT
ejpam-2095	436	6	q	q	PROPN
ejpam-2095	437	1	i	i	PRON
ejpam-2095	437	2	′	′	VERB
ejpam-2095	437	3	.	.	PUNCT
ejpam-2095	438	1	we	we	PRON
ejpam-2095	438	2	now	now	ADV
ejpam-2095	438	3	relate	relate	VERB
ejpam-2095	438	4	the	the	DET
ejpam-2095	438	5	radical	radical	NOUN
ejpam-2095	438	6	of	of	ADP
ejpam-2095	438	7	n	n	NOUN
ejpam-2095	438	8	with	with	ADP
ejpam-2095	438	9	the	the	DET
ejpam-2095	438	10	isolated	isolated	ADJ
ejpam-2095	438	11	primes	prime	NOUN
ejpam-2095	438	12	of	of	ADP
ejpam-2095	438	13	n	n	NOUN
ejpam-2095	438	14	∈	∈	NOUN
ejpam-2095	438	15	m	m	NOUN
ejpam-2095	438	16	.	.	PUNCT
ejpam-2095	439	1	in	in	ADP
ejpam-2095	439	2	that	that	DET
ejpam-2095	439	3	direction	direction	NOUN
ejpam-2095	439	4	we	we	PRON
ejpam-2095	439	5	have	have	AUX
ejpam-2095	439	6	:	:	PUNCT
ejpam-2095	439	7	theorem	theorem	VERB
ejpam-2095	439	8	13	13	NUM
ejpam-2095	439	9	.	.	PUNCT
ejpam-2095	440	1	let	let	VERB
ejpam-2095	440	2	m	m	PRON
ejpam-2095	440	3	be	be	AUX
ejpam-2095	440	4	a	a	DET
ejpam-2095	440	5	lattice	lattice	NOUN
ejpam-2095	440	6	module	module	NOUN
ejpam-2095	440	7	and	and	CCONJ
ejpam-2095	440	8	n	n	NOUN
ejpam-2095	440	9	6=	6=	PROPN
ejpam-2095	441	1	i	i	PRON
ejpam-2095	441	2	m	m	AUX
ejpam-2095	441	3	have	have	VERB
ejpam-2095	441	4	an	an	DET
ejpam-2095	441	5	irredudent(reduced	irredudent(reduce	VERB
ejpam-2095	441	6	)	)	PUNCT
ejpam-2095	441	7	primary	primary	ADJ
ejpam-2095	441	8	decomposition	decomposition	NOUN
ejpam-2095	441	9	n	n	PROPN
ejpam-2095	441	10	=	=	PROPN
ejpam-2095	441	11	q1	q1	PROPN
ejpam-2095	441	12	∧q2	∧q2	VERB
ejpam-2095	441	13	∧	∧	PROPN
ejpam-2095	441	14	.	.	PUNCT
ejpam-2095	441	15	.	.	PUNCT
ejpam-2095	441	16	.	.	PUNCT
ejpam-2095	442	1	∧qn	∧qn	PROPN
ejpam-2095	442	2	then	then	ADV
ejpam-2095	442	3	p	p	X
ejpam-2095	442	4	(	(	PUNCT
ejpam-2095	442	5	n	n	NUM
ejpam-2095	442	6	:	:	PUNCT
ejpam-2095	442	7	i	i	PRON
ejpam-2095	442	8	m	m	PROPN
ejpam-2095	442	9	)	)	PUNCT
ejpam-2095	442	10	is	be	AUX
ejpam-2095	442	11	the	the	DET
ejpam-2095	442	12	meet	meet	NOUN
ejpam-2095	442	13	of	of	ADP
ejpam-2095	442	14	isolated	isolated	ADJ
ejpam-2095	442	15	prime	prime	ADJ
ejpam-2095	442	16	elements	element	NOUN
ejpam-2095	442	17	of	of	ADP
ejpam-2095	442	18	n.	n.	NOUN
ejpam-2095	442	19	proof	proof	NOUN
ejpam-2095	442	20	.	.	PUNCT
ejpam-2095	443	1	we	we	PRON
ejpam-2095	443	2	have	have	VERB
ejpam-2095	443	3	p	p	NOUN
ejpam-2095	443	4	(	(	PUNCT
ejpam-2095	443	5	n	n	NUM
ejpam-2095	443	6	:	:	PUNCT
ejpam-2095	443	7	i	i	PRON
ejpam-2095	443	8	m	m	VERB
ejpam-2095	443	9	)	)	PUNCT
ejpam-2095	444	1	=	=	SYM
ejpam-2095	444	2	p	p	X
ejpam-2095	444	3	(	(	PUNCT
ejpam-2095	444	4	q1	q1	PROPN
ejpam-2095	444	5	∧q2	∧q2	VERB
ejpam-2095	444	6	∧	∧	PROPN
ejpam-2095	444	7	.	.	PUNCT
ejpam-2095	444	8	.	.	PUNCT
ejpam-2095	444	9	.qn	.qn	PUNCT
ejpam-2095	444	10	)	)	PUNCT
ejpam-2095	444	11	:	:	PUNCT
ejpam-2095	445	1	i	i	PRON
ejpam-2095	445	2	m	m	VERB
ejpam-2095	445	3	=	=	ADJ
ejpam-2095	445	4	p	p	X
ejpam-2095	445	5	(	(	PUNCT
ejpam-2095	445	6	q1	q1	PROPN
ejpam-2095	445	7	:	:	PUNCT
ejpam-2095	445	8	i	i	PRON
ejpam-2095	445	9	m	m	VERB
ejpam-2095	445	10	)	)	PUNCT
ejpam-2095	445	11	∧	∧	NOUN
ejpam-2095	445	12	p	p	NOUN
ejpam-2095	445	13	(	(	PUNCT
ejpam-2095	445	14	q2	q2	NOUN
ejpam-2095	445	15	:	:	PUNCT
ejpam-2095	445	16	i	i	PRON
ejpam-2095	445	17	m	m	VERB
ejpam-2095	445	18	)	)	PUNCT
ejpam-2095	445	19	.	.	PUNCT
ejpam-2095	445	20	.	.	PUNCT
ejpam-2095	445	21	.	.	PUNCT
ejpam-2095	446	1	∧	∧	NOUN
ejpam-2095	446	2	p	p	NOUN
ejpam-2095	446	3	(	(	PUNCT
ejpam-2095	446	4	qn	qn	NOUN
ejpam-2095	446	5	:	:	PUNCT
ejpam-2095	446	6	i	i	PRON
ejpam-2095	446	7	m	m	VERB
ejpam-2095	446	8	)	)	PUNCT
ejpam-2095	447	1	=	=	NOUN
ejpam-2095	447	2	p1	p1	NOUN
ejpam-2095	447	3	∧	∧	NOUN
ejpam-2095	447	4	p2	p2	NOUN
ejpam-2095	447	5	∧	∧	PROPN
ejpam-2095	447	6	.	.	PUNCT
ejpam-2095	447	7	.	.	PUNCT
ejpam-2095	447	8	.∧	.∧	PUNCT
ejpam-2095	448	1	pn	pn	INTJ
ejpam-2095	448	2	,	,	PUNCT
ejpam-2095	448	3	where	where	SCONJ
ejpam-2095	448	4	pi	pi	NOUN
ejpam-2095	448	5	=	=	PROPN
ejpam-2095	448	6	p	p	X
ejpam-2095	448	7	(	(	PUNCT
ejpam-2095	448	8	q	q	NOUN
ejpam-2095	449	1	i	i	PRON
ejpam-2095	449	2	:	:	PUNCT
ejpam-2095	449	3	i	i	PRON
ejpam-2095	449	4	m	m	VERB
ejpam-2095	449	5	)	)	PUNCT
ejpam-2095	449	6	are	be	AUX
ejpam-2095	449	7	associated	associate	VERB
ejpam-2095	449	8	primes	prime	NOUN
ejpam-2095	449	9	of	of	ADP
ejpam-2095	449	10	n	n	NOUN
ejpam-2095	449	11	.	.	PUNCT
ejpam-2095	450	1	if	if	SCONJ
ejpam-2095	450	2	some	some	DET
ejpam-2095	450	3	pk	pk	NOUN
ejpam-2095	450	4	is	be	AUX
ejpam-2095	450	5	not	not	PART
ejpam-2095	450	6	isolated	isolate	VERB
ejpam-2095	450	7	then	then	ADV
ejpam-2095	450	8	pk	pk	NOUN
ejpam-2095	450	9	≥	≥	NOUN
ejpam-2095	450	10	pi	pi	NOUN
ejpam-2095	450	11	for	for	ADP
ejpam-2095	450	12	some	some	DET
ejpam-2095	450	13	pi	pi	NOUN
ejpam-2095	450	14	and	and	CCONJ
ejpam-2095	450	15	hence	hence	ADV
ejpam-2095	450	16	we	we	PRON
ejpam-2095	450	17	can	can	AUX
ejpam-2095	450	18	delete	delete	VERB
ejpam-2095	450	19	such	such	ADJ
ejpam-2095	450	20	elements	element	NOUN
ejpam-2095	450	21	from	from	ADP
ejpam-2095	450	22	the	the	DET
ejpam-2095	450	23	above	above	ADJ
ejpam-2095	450	24	primary	primary	ADJ
ejpam-2095	450	25	decomposition	decomposition	NOUN
ejpam-2095	450	26	and	and	CCONJ
ejpam-2095	450	27	we	we	PRON
ejpam-2095	450	28	are	be	AUX
ejpam-2095	450	29	through	through	ADP
ejpam-2095	450	30	.	.	PUNCT
ejpam-2095	451	1	we	we	PRON
ejpam-2095	451	2	note	note	VERB
ejpam-2095	451	3	that	that	SCONJ
ejpam-2095	451	4	an	an	DET
ejpam-2095	451	5	element	element	NOUN
ejpam-2095	451	6	a=	a=	NOUN
ejpam-2095	451	7	aim	aim	NOUN
ejpam-2095	451	8	of	of	ADP
ejpam-2095	451	9	m	m	PRON
ejpam-2095	451	10	where	where	SCONJ
ejpam-2095	451	11	a	a	DET
ejpam-2095	451	12	∈	∈	PROPN
ejpam-2095	451	13	l	l	NOUN
ejpam-2095	451	14	is	be	AUX
ejpam-2095	451	15	said	say	VERB
ejpam-2095	451	16	to	to	PART
ejpam-2095	451	17	be	be	AUX
ejpam-2095	451	18	nilpotent	nilpotent	ADJ
ejpam-2095	451	19	if	if	SCONJ
ejpam-2095	451	20	an	an	PRON
ejpam-2095	451	21	i	i	NOUN
ejpam-2095	451	22	m	m	VERB
ejpam-2095	451	23	=	=	ADJ
ejpam-2095	451	24	0	0	NUM
ejpam-2095	451	25	m	m	VERB
ejpam-2095	451	26	for	for	ADP
ejpam-2095	451	27	some	some	DET
ejpam-2095	451	28	positive	positive	ADJ
ejpam-2095	451	29	integer	integer	NOUN
ejpam-2095	451	30	n.	n.	NOUN
ejpam-2095	451	31	if	if	SCONJ
ejpam-2095	451	32	a	a	DET
ejpam-2095	451	33	lattice	lattice	NOUN
ejpam-2095	451	34	module	module	NOUN
ejpam-2095	451	35	m	m	VERB
ejpam-2095	451	36	satisfies	satisfy	VERB
ejpam-2095	451	37	the	the	DET
ejpam-2095	451	38	acc	acc	NOUN
ejpam-2095	451	39	and	and	CCONJ
ejpam-2095	451	40	if	if	SCONJ
ejpam-2095	451	41	every	every	DET
ejpam-2095	451	42	element	element	NOUN
ejpam-2095	451	43	of	of	ADP
ejpam-2095	451	44	l	l	NOUN
ejpam-2095	451	45	is	be	AUX
ejpam-2095	451	46	the	the	DET
ejpam-2095	451	47	join	join	NOUN
ejpam-2095	451	48	of	of	ADP
ejpam-2095	451	49	meet	meet	VERB
ejpam-2095	451	50	prncipal	prncipal	ADJ
ejpam-2095	451	51	elements	element	NOUN
ejpam-2095	451	52	then	then	ADV
ejpam-2095	451	53	every	every	DET
ejpam-2095	451	54	element	element	NOUN
ejpam-2095	451	55	of	of	ADP
ejpam-2095	451	56	m	m	PROPN
ejpam-2095	451	57	can	can	AUX
ejpam-2095	451	58	be	be	AUX
ejpam-2095	451	59	written	write	VERB
ejpam-2095	451	60	as	as	ADP
ejpam-2095	451	61	a	a	DET
ejpam-2095	451	62	meet	meet	NOUN
ejpam-2095	451	63	of	of	ADP
ejpam-2095	451	64	finite	finite	ADJ
ejpam-2095	451	65	number	number	NOUN
ejpam-2095	451	66	of	of	ADP
ejpam-2095	451	67	primary	primary	ADJ
ejpam-2095	451	68	elements	element	NOUN
ejpam-2095	451	69	[	[	X
ejpam-2095	451	70	1	1	NUM
ejpam-2095	451	71	]	]	PUNCT
ejpam-2095	451	72	.	.	PUNCT
ejpam-2095	452	1	theorem	theorem	PROPN
ejpam-2095	452	2	14	14	NUM
ejpam-2095	452	3	.	.	PUNCT
ejpam-2095	453	1	let	let	VERB
ejpam-2095	453	2	m	m	PRON
ejpam-2095	453	3	be	be	AUX
ejpam-2095	453	4	a	a	DET
ejpam-2095	453	5	lattice	lattice	NOUN
ejpam-2095	453	6	module	module	NOUN
ejpam-2095	453	7	satisfying	satisfy	VERB
ejpam-2095	453	8	the	the	DET
ejpam-2095	453	9	acc	acc	PROPN
ejpam-2095	453	10	over	over	ADP
ejpam-2095	453	11	a	a	DET
ejpam-2095	453	12	multiplicative	multiplicative	ADJ
ejpam-2095	453	13	lattice	lattice	NOUN
ejpam-2095	453	14	l	l	NOUN
ejpam-2095	453	15	in	in	ADP
ejpam-2095	453	16	which	which	PRON
ejpam-2095	453	17	every	every	DET
ejpam-2095	453	18	element	element	NOUN
ejpam-2095	453	19	is	be	AUX
ejpam-2095	453	20	the	the	DET
ejpam-2095	453	21	join	join	NOUN
ejpam-2095	453	22	of	of	ADP
ejpam-2095	453	23	meet	meet	VERB
ejpam-2095	453	24	principal	principal	ADJ
ejpam-2095	453	25	elements	element	NOUN
ejpam-2095	453	26	.	.	PUNCT
ejpam-2095	454	1	then	then	ADV
ejpam-2095	454	2	the	the	DET
ejpam-2095	454	3	join	join	NOUN
ejpam-2095	454	4	of	of	ADP
ejpam-2095	454	5	the	the	DET
ejpam-2095	454	6	set	set	NOUN
ejpam-2095	454	7	of	of	ADP
ejpam-2095	454	8	all	all	DET
ejpam-2095	454	9	elements	element	NOUN
ejpam-2095	454	10	a	a	DET
ejpam-2095	454	11	∈	∈	NOUN
ejpam-2095	454	12	l	l	NOUN
ejpam-2095	454	13	such	such	ADJ
ejpam-2095	454	14	that	that	SCONJ
ejpam-2095	454	15	aim	aim	NOUN
ejpam-2095	454	16	is	be	AUX
ejpam-2095	454	17	nilpotent	nilpotent	ADJ
ejpam-2095	454	18	is	be	AUX
ejpam-2095	454	19	the	the	DET
ejpam-2095	454	20	meet	meet	NOUN
ejpam-2095	454	21	of	of	ADP
ejpam-2095	454	22	the	the	DET
ejpam-2095	454	23	isolated	isolated	ADJ
ejpam-2095	454	24	primes	prime	NOUN
ejpam-2095	454	25	of	of	ADP
ejpam-2095	454	26	0	0	NUM
ejpam-2095	454	27	m	m	NOUN
ejpam-2095	454	28	.	.	PUNCT
ejpam-2095	455	1	proof	proof	NOUN
ejpam-2095	455	2	.	.	PUNCT
ejpam-2095	456	1	let	let	VERB
ejpam-2095	456	2	0	0	NUM
ejpam-2095	456	3	m	m	VERB
ejpam-2095	456	4	=	=	ADJ
ejpam-2095	456	5	n∧	n∧	NUM
ejpam-2095	457	1	i=1	i=1	PROPN
ejpam-2095	458	1	q	q	X
ejpam-2095	459	1	i	i	PRON
ejpam-2095	459	2	be	be	VERB
ejpam-2095	459	3	a	a	DET
ejpam-2095	459	4	reduced	reduce	VERB
ejpam-2095	459	5	primary	primary	ADJ
ejpam-2095	459	6	decomposition	decomposition	NOUN
ejpam-2095	459	7	of	of	ADP
ejpam-2095	459	8	0	0	NUM
ejpam-2095	459	9	m	m	NOUN
ejpam-2095	459	10	and	and	CCONJ
ejpam-2095	459	11	pi	pi	NOUN
ejpam-2095	460	1	=	=	SYM
ejpam-2095	460	2	p	p	X
ejpam-2095	460	3	(	(	PUNCT
ejpam-2095	460	4	q	q	NOUN
ejpam-2095	460	5	i	i	PRON
ejpam-2095	460	6	:	:	PUNCT
ejpam-2095	461	1	i	i	PRON
ejpam-2095	461	2	m	m	VERB
ejpam-2095	461	3	)	)	PUNCT
ejpam-2095	461	4	be	be	VERB
ejpam-2095	461	5	an	an	DET
ejpam-2095	461	6	associated	associated	ADJ
ejpam-2095	461	7	prime	prime	NOUN
ejpam-2095	461	8	of	of	ADP
ejpam-2095	461	9	q	q	PROPN
ejpam-2095	462	1	i	i	PROPN
ejpam-2095	462	2	,	,	PUNCT
ejpam-2095	462	3	i	i	NOUN
ejpam-2095	462	4	=	=	NOUN
ejpam-2095	462	5	1,2	1,2	NUM
ejpam-2095	462	6	,	,	PUNCT
ejpam-2095	462	7	.	.	PUNCT
ejpam-2095	462	8	.	.	PUNCT
ejpam-2095	462	9	.	.	PUNCT
ejpam-2095	463	1	n.	n.	INTJ
ejpam-2095	463	2	we	we	PRON
ejpam-2095	463	3	have	have	VERB
ejpam-2095	463	4	p	p	NOUN
ejpam-2095	463	5	(	(	PUNCT
ejpam-2095	463	6	0	0	NUM
ejpam-2095	463	7	m	m	VERB
ejpam-2095	463	8	:	:	PUNCT
ejpam-2095	463	9	i	i	PRON
ejpam-2095	463	10	m	m	VERB
ejpam-2095	463	11	)	)	PUNCT
ejpam-2095	464	1	=	=	PUNCT
ejpam-2095	465	1	∨{a	∨{a	PROPN
ejpam-2095	465	2	∈	∈	PROPN
ejpam-2095	465	3	l	l	NOUN
ejpam-2095	466	1	|	|	ADV
ejpam-2095	466	2	anim	anim	NOUN
ejpam-2095	466	3	=	=	PUNCT
ejpam-2095	466	4	0	0	NUM
ejpam-2095	466	5	m	m	VERB
ejpam-2095	466	6	for	for	ADP
ejpam-2095	466	7	some	some	DET
ejpam-2095	466	8	positive	positive	ADJ
ejpam-2095	466	9	integer	integer	NOUN
ejpam-2095	466	10	n	n	CCONJ
ejpam-2095	466	11	}	}	PUNCT
ejpam-2095	466	12	c.	c.	PROPN
ejpam-2095	466	13	manjarekar	manjarekar	PROPN
ejpam-2095	466	14	,	,	PUNCT
ejpam-2095	466	15	u.	u.	PROPN
ejpam-2095	466	16	kandale	kandale	PROPN
ejpam-2095	466	17	/	/	SYM
ejpam-2095	466	18	eur	eur	PROPN
ejpam-2095	466	19	.	.	PUNCT
ejpam-2095	467	1	j.	j.	PROPN
ejpam-2095	467	2	pure	pure	PROPN
ejpam-2095	467	3	appl	appl	PROPN
ejpam-2095	467	4	.	.	PROPN
ejpam-2095	467	5	math	math	PROPN
ejpam-2095	467	6	,	,	PUNCT
ejpam-2095	467	7	7	7	NUM
ejpam-2095	467	8	(	(	PUNCT
ejpam-2095	467	9	2014	2014	NUM
ejpam-2095	467	10	)	)	PUNCT
ejpam-2095	467	11	,	,	PUNCT
ejpam-2095	467	12	201	201	NUM
ejpam-2095	467	13	-	-	SYM
ejpam-2095	467	14	209	209	NUM
ejpam-2095	467	15	208	208	NUM
ejpam-2095	467	16	and	and	CCONJ
ejpam-2095	467	17	p	p	X
ejpam-2095	467	18	(	(	PUNCT
ejpam-2095	467	19	0	0	NUM
ejpam-2095	467	20	m	m	VERB
ejpam-2095	467	21	:	:	PUNCT
ejpam-2095	468	1	i	i	PRON
ejpam-2095	468	2	m	m	VERB
ejpam-2095	468	3	)	)	PUNCT
ejpam-2095	469	1	=	=	SYM
ejpam-2095	469	2	p1∧	p1∧	ADJ
ejpam-2095	469	3	p2∧	p2∧	NOUN
ejpam-2095	469	4	.	.	PUNCT
ejpam-2095	469	5	.	.	PUNCT
ejpam-2095	469	6	.∧	.∧	PUNCT
ejpam-2095	470	1	pk	pk	NOUN
ejpam-2095	470	2	where	where	SCONJ
ejpam-2095	470	3	p1	p1	NOUN
ejpam-2095	470	4	,	,	PUNCT
ejpam-2095	470	5	p2	p2	NOUN
ejpam-2095	470	6	,	,	PUNCT
ejpam-2095	470	7	.	.	PUNCT
ejpam-2095	470	8	.	.	PUNCT
ejpam-2095	470	9	.	.	PUNCT
ejpam-2095	471	1	,	,	PUNCT
ejpam-2095	471	2	pk	pk	NOUN
ejpam-2095	471	3	are	be	AUX
ejpam-2095	471	4	the	the	DET
ejpam-2095	471	5	isolated	isolated	ADJ
ejpam-2095	471	6	primes	prime	NOUN
ejpam-2095	471	7	of	of	ADP
ejpam-2095	471	8	0	0	NUM
ejpam-2095	471	9	m	m	NOUN
ejpam-2095	471	10	.	.	PUNCT
ejpam-2095	472	1	hence	hence	ADV
ejpam-2095	472	2	,	,	PUNCT
ejpam-2095	472	3	p	p	X
ejpam-2095	472	4	(	(	PUNCT
ejpam-2095	472	5	0	0	NUM
ejpam-2095	472	6	m	m	VERB
ejpam-2095	472	7	:	:	PUNCT
ejpam-2095	472	8	i	i	PRON
ejpam-2095	472	9	m	m	PROPN
ejpam-2095	472	10	)	)	PUNCT
ejpam-2095	472	11	is	be	AUX
ejpam-2095	472	12	the	the	DET
ejpam-2095	472	13	meet	meet	NOUN
ejpam-2095	472	14	of	of	ADP
ejpam-2095	472	15	isolated	isolated	ADJ
ejpam-2095	472	16	primes	prime	NOUN
ejpam-2095	472	17	of	of	ADP
ejpam-2095	472	18	0	0	NUM
ejpam-2095	472	19	m	m	NOUN
ejpam-2095	472	20	.	.	PUNCT
ejpam-2095	473	1	the	the	DET
ejpam-2095	473	2	primeness	primeness	NOUN
ejpam-2095	473	3	of	of	ADP
ejpam-2095	473	4	the	the	DET
ejpam-2095	473	5	radical	radical	NOUN
ejpam-2095	473	6	of	of	ADP
ejpam-2095	473	7	a∈	a∈	PROPN
ejpam-2095	473	8	m	m	VERB
ejpam-2095	473	9	is	be	AUX
ejpam-2095	473	10	characterized	characterize	VERB
ejpam-2095	473	11	in	in	ADP
ejpam-2095	473	12	the	the	DET
ejpam-2095	473	13	following	follow	VERB
ejpam-2095	473	14	result	result	NOUN
ejpam-2095	473	15	.	.	PUNCT
ejpam-2095	474	1	theorem	theorem	ADJ
ejpam-2095	474	2	15	15	NUM
ejpam-2095	474	3	.	.	PUNCT
ejpam-2095	475	1	for	for	ADP
ejpam-2095	475	2	a∈	a∈	PROPN
ejpam-2095	475	3	m	m	PROPN
ejpam-2095	475	4	,	,	PUNCT
ejpam-2095	475	5	p	p	X
ejpam-2095	475	6	(	(	PUNCT
ejpam-2095	475	7	a	a	NOUN
ejpam-2095	475	8	:	:	PUNCT
ejpam-2095	475	9	i	i	PRON
ejpam-2095	475	10	m	m	PROPN
ejpam-2095	475	11	)	)	PUNCT
ejpam-2095	475	12	is	be	AUX
ejpam-2095	475	13	prime	prime	ADJ
ejpam-2095	475	14	if	if	SCONJ
ejpam-2095	476	1	and	and	CCONJ
ejpam-2095	476	2	only	only	ADV
ejpam-2095	476	3	if	if	SCONJ
ejpam-2095	476	4	a	a	PRON
ejpam-2095	476	5	has	have	VERB
ejpam-2095	476	6	a	a	DET
ejpam-2095	476	7	single	single	ADJ
ejpam-2095	476	8	isolated	isolated	ADJ
ejpam-2095	476	9	prime	prime	ADJ
ejpam-2095	476	10	element	element	NOUN
ejpam-2095	476	11	.	.	PUNCT
ejpam-2095	477	1	proof	proof	NOUN
ejpam-2095	477	2	.	.	PUNCT
ejpam-2095	478	1	let	let	VERB
ejpam-2095	478	2	a	a	DET
ejpam-2095	478	3	=	=	PROPN
ejpam-2095	478	4	q1	q1	PROPN
ejpam-2095	478	5	∧	∧	PROPN
ejpam-2095	478	6	q2	q2	PROPN
ejpam-2095	478	7	∧	∧	PROPN
ejpam-2095	478	8	.	.	PUNCT
ejpam-2095	478	9	.	.	PUNCT
ejpam-2095	478	10	.	.	PUNCT
ejpam-2095	479	1	∧qn	∧qn	PROPN
ejpam-2095	479	2	be	be	VERB
ejpam-2095	479	3	primary	primary	ADJ
ejpam-2095	479	4	decomposition	decomposition	NOUN
ejpam-2095	479	5	of	of	ADP
ejpam-2095	479	6	a.	a.	NOUN
ejpam-2095	479	7	if	if	SCONJ
ejpam-2095	479	8	a	a	PRON
ejpam-2095	479	9	has	have	VERB
ejpam-2095	479	10	a	a	DET
ejpam-2095	479	11	single	single	ADJ
ejpam-2095	479	12	isolated	isolate	VERB
ejpam-2095	479	13	prime	prime	ADJ
ejpam-2095	479	14	element	element	NOUN
ejpam-2095	479	15	p	p	NOUN
ejpam-2095	479	16	then	then	ADV
ejpam-2095	479	17	p	p	X
ejpam-2095	479	18	(	(	PUNCT
ejpam-2095	479	19	a	a	NOUN
ejpam-2095	479	20	:	:	PUNCT
ejpam-2095	479	21	i	i	PRON
ejpam-2095	479	22	m	m	VERB
ejpam-2095	479	23	)	)	PUNCT
ejpam-2095	480	1	=	=	VERB
ejpam-2095	481	1	p.	p.	NOUN
ejpam-2095	481	2	conversely	conversely	ADV
ejpam-2095	481	3	,	,	PUNCT
ejpam-2095	481	4	assume	assume	VERB
ejpam-2095	481	5	that	that	SCONJ
ejpam-2095	481	6	p	p	X
ejpam-2095	481	7	(	(	PUNCT
ejpam-2095	481	8	a	a	NOUN
ejpam-2095	481	9	:	:	PUNCT
ejpam-2095	481	10	i	i	PRON
ejpam-2095	481	11	m	m	PROPN
ejpam-2095	481	12	)	)	PUNCT
ejpam-2095	481	13	is	be	AUX
ejpam-2095	481	14	prime	prime	ADJ
ejpam-2095	481	15	and	and	CCONJ
ejpam-2095	481	16	p	p	X
ejpam-2095	481	17	(	(	PUNCT
ejpam-2095	481	18	a	a	NOUN
ejpam-2095	481	19	:	:	PUNCT
ejpam-2095	481	20	i	i	PRON
ejpam-2095	481	21	m	m	VERB
ejpam-2095	481	22	)	)	PUNCT
ejpam-2095	482	1	=	=	SYM
ejpam-2095	482	2	p1	p1	NOUN
ejpam-2095	482	3	∧	∧	PROPN
ejpam-2095	482	4	p2	p2	NOUN
ejpam-2095	482	5	where	where	SCONJ
ejpam-2095	482	6	p1	p1	NOUN
ejpam-2095	482	7	,	,	PUNCT
ejpam-2095	482	8	p2	p2	PROPN
ejpam-2095	482	9	are	be	AUX
ejpam-2095	482	10	isolated	isolate	VERB
ejpam-2095	482	11	primes	prime	NOUN
ejpam-2095	482	12	of	of	ADP
ejpam-2095	482	13	a.	a.	NOUN
ejpam-2095	482	14	then	then	ADV
ejpam-2095	482	15	there	there	PRON
ejpam-2095	482	16	are	be	VERB
ejpam-2095	482	17	x	x	X
ejpam-2095	482	18	,	,	PUNCT
ejpam-2095	482	19	y	y	PROPN
ejpam-2095	482	20	∈	∈	PROPN
ejpam-2095	482	21	l	l	NOUN
ejpam-2095	482	22	such	such	ADJ
ejpam-2095	482	23	that	that	SCONJ
ejpam-2095	482	24	x	x	PROPN
ejpam-2095	482	25	¶	¶	PROPN
ejpam-2095	482	26	p1	p1	PROPN
ejpam-2095	482	27	,	,	PUNCT
ejpam-2095	482	28	x	x	PROPN
ejpam-2095	482	29	�	�	PROPN
ejpam-2095	482	30	p2	p2	PROPN
ejpam-2095	482	31	and	and	CCONJ
ejpam-2095	482	32	y	y	PROPN
ejpam-2095	482	33	¶	¶	PROPN
ejpam-2095	482	34	p2	p2	PROPN
ejpam-2095	482	35	,	,	PUNCT
ejpam-2095	482	36	y	y	PROPN
ejpam-2095	482	37	�	�	PROPN
ejpam-2095	482	38	p1	p1	PROPN
ejpam-2095	482	39	.	.	PUNCT
ejpam-2095	483	1	hence	hence	ADV
ejpam-2095	483	2	x	x	PUNCT
ejpam-2095	483	3	y	y	PROPN
ejpam-2095	483	4	¶	¶	PROPN
ejpam-2095	483	5	p	p	PROPN
ejpam-2095	483	6	(	(	PUNCT
ejpam-2095	483	7	a	a	PRON
ejpam-2095	483	8	:	:	PUNCT
ejpam-2095	483	9	i	i	PRON
ejpam-2095	483	10	m	m	VERB
ejpam-2095	483	11	)	)	PUNCT
ejpam-2095	483	12	which	which	PRON
ejpam-2095	483	13	is	be	AUX
ejpam-2095	483	14	prime	prime	ADJ
ejpam-2095	483	15	.	.	PUNCT
ejpam-2095	484	1	but	but	CCONJ
ejpam-2095	484	2	then	then	ADV
ejpam-2095	484	3	x	x	SYM
ejpam-2095	484	4	¶	¶	PROPN
ejpam-2095	484	5	p2	p2	PROPN
ejpam-2095	484	6	or	or	CCONJ
ejpam-2095	484	7	y	y	PROPN
ejpam-2095	484	8	¶	¶	PROPN
ejpam-2095	484	9	p1	p1	PROPN
ejpam-2095	484	10	which	which	PRON
ejpam-2095	484	11	is	be	AUX
ejpam-2095	484	12	a	a	DET
ejpam-2095	484	13	contradiction	contradiction	NOUN
ejpam-2095	484	14	.	.	PUNCT
ejpam-2095	485	1	thus	thus	ADV
ejpam-2095	485	2	a	a	PRON
ejpam-2095	485	3	has	have	VERB
ejpam-2095	485	4	a	a	DET
ejpam-2095	485	5	single	single	ADJ
ejpam-2095	485	6	isolated	isolated	ADJ
ejpam-2095	485	7	prime	prime	ADJ
ejpam-2095	485	8	element	element	NOUN
ejpam-2095	485	9	.	.	PUNCT
ejpam-2095	486	1	in	in	ADP
ejpam-2095	486	2	the	the	DET
ejpam-2095	486	3	remaining	remain	VERB
ejpam-2095	486	4	part	part	NOUN
ejpam-2095	486	5	we	we	PRON
ejpam-2095	486	6	assume	assume	VERB
ejpam-2095	486	7	that	that	SCONJ
ejpam-2095	486	8	a	a	DET
ejpam-2095	486	9	lattice	lattice	NOUN
ejpam-2095	486	10	module	module	NOUN
ejpam-2095	486	11	m	m	VERB
ejpam-2095	486	12	satisfies	satisfy	VERB
ejpam-2095	486	13	the	the	DET
ejpam-2095	486	14	acc	acc	PROPN
ejpam-2095	486	15	over	over	ADP
ejpam-2095	486	16	a	a	DET
ejpam-2095	486	17	multiplicative	multiplicative	ADJ
ejpam-2095	486	18	lattice	lattice	NOUN
ejpam-2095	486	19	l	l	NOUN
ejpam-2095	486	20	in	in	ADP
ejpam-2095	486	21	which	which	PRON
ejpam-2095	486	22	every	every	DET
ejpam-2095	486	23	element	element	NOUN
ejpam-2095	486	24	is	be	AUX
ejpam-2095	486	25	the	the	DET
ejpam-2095	486	26	join	join	NOUN
ejpam-2095	486	27	of	of	ADP
ejpam-2095	486	28	meet	meet	VERB
ejpam-2095	486	29	principal	principal	ADJ
ejpam-2095	486	30	elements	element	NOUN
ejpam-2095	486	31	.	.	PUNCT
ejpam-2095	487	1	this	this	DET
ejpam-2095	487	2	condition	condition	NOUN
ejpam-2095	487	3	assures	assure	VERB
ejpam-2095	487	4	that	that	SCONJ
ejpam-2095	487	5	any	any	DET
ejpam-2095	487	6	element	element	NOUN
ejpam-2095	487	7	m	m	VERB
ejpam-2095	487	8	has	have	VERB
ejpam-2095	487	9	a	a	DET
ejpam-2095	487	10	reduced	reduce	VERB
ejpam-2095	487	11	primary	primary	ADJ
ejpam-2095	487	12	decomposition	decomposition	NOUN
ejpam-2095	487	13	.	.	PUNCT
ejpam-2095	488	1	theorem	theorem	VERB
ejpam-2095	488	2	16	16	NUM
ejpam-2095	488	3	.	.	PUNCT
ejpam-2095	489	1	let	let	VERB
ejpam-2095	489	2	a	a	DET
ejpam-2095	489	3	be	be	AUX
ejpam-2095	489	4	any	any	DET
ejpam-2095	489	5	element	element	NOUN
ejpam-2095	489	6	of	of	ADP
ejpam-2095	489	7	m	m	PROPN
ejpam-2095	489	8	and	and	CCONJ
ejpam-2095	489	9	b	b	X
ejpam-2095	489	10	∈	∈	PROPN
ejpam-2095	489	11	l	l	NOUN
ejpam-2095	489	12	be	be	AUX
ejpam-2095	489	13	such	such	ADJ
ejpam-2095	489	14	that	that	SCONJ
ejpam-2095	489	15	a	a	DET
ejpam-2095	489	16	6=	6=	NOUN
ejpam-2095	489	17	i	i	PRON
ejpam-2095	489	18	m	m	PROPN
ejpam-2095	489	19	.	.	PUNCT
ejpam-2095	490	1	then	then	ADV
ejpam-2095	490	2	a=	a=	VERB
ejpam-2095	490	3	(	(	PUNCT
ejpam-2095	490	4	a	a	DET
ejpam-2095	490	5	:	:	PUNCT
ejpam-2095	490	6	b	b	X
ejpam-2095	490	7	)	)	PUNCT
ejpam-2095	490	8	if	if	SCONJ
ejpam-2095	491	1	and	and	CCONJ
ejpam-2095	491	2	only	only	ADV
ejpam-2095	491	3	if	if	SCONJ
ejpam-2095	491	4	b	b	NOUN
ejpam-2095	491	5	is	be	AUX
ejpam-2095	491	6	contained	contain	VERB
ejpam-2095	491	7	in	in	ADP
ejpam-2095	491	8	no	no	DET
ejpam-2095	491	9	associated	associated	ADJ
ejpam-2095	491	10	prime	prime	ADJ
ejpam-2095	491	11	element	element	NOUN
ejpam-2095	491	12	of	of	ADP
ejpam-2095	491	13	a.	a.	NOUN
ejpam-2095	491	14	proof	proof	NOUN
ejpam-2095	491	15	.	.	PUNCT
ejpam-2095	492	1	let	let	VERB
ejpam-2095	492	2	a	a	DET
ejpam-2095	492	3	=	=	PROPN
ejpam-2095	492	4	q1	q1	PROPN
ejpam-2095	492	5	∧q2	∧q2	VERB
ejpam-2095	492	6	∧	∧	PROPN
ejpam-2095	492	7	.	.	PUNCT
ejpam-2095	492	8	.	.	PUNCT
ejpam-2095	492	9	.	.	PUNCT
ejpam-2095	493	1	∧qm	∧qm	PROPN
ejpam-2095	493	2	be	be	AUX
ejpam-2095	493	3	an	an	DET
ejpam-2095	493	4	irredundant	irredundant	ADJ
ejpam-2095	493	5	primary	primary	ADJ
ejpam-2095	493	6	decomposition	decomposition	NOUN
ejpam-2095	493	7	of	of	ADP
ejpam-2095	493	8	a	a	PRON
ejpam-2095	493	9	and	and	CCONJ
ejpam-2095	493	10	let	let	VERB
ejpam-2095	493	11	pi	pi	NOUN
ejpam-2095	493	12	=	=	PUNCT
ejpam-2095	493	13	p	p	X
ejpam-2095	493	14	(	(	PUNCT
ejpam-2095	493	15	q	q	NOUN
ejpam-2095	493	16	i	i	PRON
ejpam-2095	493	17	:	:	PUNCT
ejpam-2095	493	18	i	i	PRON
ejpam-2095	493	19	m	m	PROPN
ejpam-2095	493	20	)	)	PUNCT
ejpam-2095	493	21	.	.	PUNCT
ejpam-2095	494	1	suppose	suppose	VERB
ejpam-2095	494	2	b	b	X
ejpam-2095	494	3	�	�	PROPN
ejpam-2095	494	4	pi	pi	NOUN
ejpam-2095	494	5	for	for	ADP
ejpam-2095	494	6	any	any	DET
ejpam-2095	494	7	i	i	NOUN
ejpam-2095	494	8	=	=	NOUN
ejpam-2095	494	9	1,2	1,2	NUM
ejpam-2095	494	10	,	,	PUNCT
ejpam-2095	494	11	.	.	PUNCT
ejpam-2095	494	12	.	.	PUNCT
ejpam-2095	495	1	.	.	PUNCT
ejpam-2095	496	1	,	,	PUNCT
ejpam-2095	496	2	m.	m.	NOUN
ejpam-2095	496	3	this	this	PRON
ejpam-2095	496	4	leads	lead	VERB
ejpam-2095	496	5	us	we	PRON
ejpam-2095	496	6	to	to	ADP
ejpam-2095	496	7	the	the	DET
ejpam-2095	496	8	fact	fact	NOUN
ejpam-2095	496	9	bn	bn	NUM
ejpam-2095	496	10	�	�	PROPN
ejpam-2095	496	11	pi	pi	NOUN
ejpam-2095	496	12	for	for	ADP
ejpam-2095	496	13	any	any	DET
ejpam-2095	496	14	positive	positive	ADJ
ejpam-2095	496	15	integer	integer	NOUN
ejpam-2095	496	16	n.	n.	NOUN
ejpam-2095	496	17	we	we	PRON
ejpam-2095	496	18	know	know	VERB
ejpam-2095	497	1	that	that	SCONJ
ejpam-2095	497	2	(	(	PUNCT
ejpam-2095	497	3	a	a	DET
ejpam-2095	497	4	:	:	PUNCT
ejpam-2095	497	5	b)b	b)b	X
ejpam-2095	497	6	¶	¶	X
ejpam-2095	497	7	a	a	PRON
ejpam-2095	498	1	[	[	X
ejpam-2095	498	2	3	3	NUM
ejpam-2095	498	3	]	]	PUNCT
ejpam-2095	498	4	and	and	CCONJ
ejpam-2095	498	5	thus	thus	ADV
ejpam-2095	498	6	(	(	PUNCT
ejpam-2095	498	7	a	a	PRON
ejpam-2095	498	8	:	:	PUNCT
ejpam-2095	498	9	b)b	b)b	X
ejpam-2095	498	10	¶	¶	PROPN
ejpam-2095	498	11	q	q	PROPN
ejpam-2095	498	12	i	i	PROPN
ejpam-2095	498	13	for	for	ADP
ejpam-2095	498	14	all	all	DET
ejpam-2095	498	15	i.	i.	NOUN
ejpam-2095	498	16	since	since	SCONJ
ejpam-2095	498	17	q	q	PROPN
ejpam-2095	498	18	i	i	PRON
ejpam-2095	498	19	is	be	AUX
ejpam-2095	498	20	primary	primary	ADJ
ejpam-2095	498	21	and	and	CCONJ
ejpam-2095	498	22	b	b	PROPN
ejpam-2095	498	23	�	�	PROPN
ejpam-2095	498	24	pi	pi	NOUN
ejpam-2095	498	25	we	we	PRON
ejpam-2095	498	26	have	have	VERB
ejpam-2095	498	27	(	(	PUNCT
ejpam-2095	498	28	a	a	DET
ejpam-2095	498	29	:	:	PUNCT
ejpam-2095	498	30	b	b	X
ejpam-2095	498	31	)	)	PUNCT
ejpam-2095	498	32	¶	¶	PROPN
ejpam-2095	498	33	q	q	PROPN
ejpam-2095	499	1	i	i	INTJ
ejpam-2095	499	2	.	.	PUNCT
ejpam-2095	500	1	that	that	PRON
ejpam-2095	500	2	is	is	ADV
ejpam-2095	500	3	(	(	PUNCT
ejpam-2095	500	4	a	a	DET
ejpam-2095	500	5	:	:	PUNCT
ejpam-2095	500	6	b	b	X
ejpam-2095	500	7	)	)	PUNCT
ejpam-2095	500	8	¶	¶	PROPN
ejpam-2095	500	9	m∧	m∧	PROPN
ejpam-2095	500	10	i=1	i=1	PROPN
ejpam-2095	501	1	q	q	PROPN
ejpam-2095	502	1	i	i	NOUN
ejpam-2095	502	2	=	=	PUNCT
ejpam-2095	502	3	a.	a.	NOUN
ejpam-2095	502	4	but	but	CCONJ
ejpam-2095	502	5	a	a	DET
ejpam-2095	502	6	¶	¶	PROPN
ejpam-2095	502	7	(	(	PUNCT
ejpam-2095	502	8	a	a	DET
ejpam-2095	502	9	:	:	PUNCT
ejpam-2095	502	10	b	b	X
ejpam-2095	502	11	)	)	PUNCT
ejpam-2095	502	12	gives	give	VERB
ejpam-2095	502	13	(	(	PUNCT
ejpam-2095	502	14	a	a	DET
ejpam-2095	502	15	:	:	PUNCT
ejpam-2095	502	16	b	b	X
ejpam-2095	502	17	)	)	PUNCT
ejpam-2095	502	18	=	=	VERB
ejpam-2095	502	19	a.	a.	NOUN
ejpam-2095	502	20	conversely	conversely	ADV
ejpam-2095	502	21	,	,	PUNCT
ejpam-2095	502	22	suppose	suppose	VERB
ejpam-2095	502	23	(	(	PUNCT
ejpam-2095	502	24	a	a	DET
ejpam-2095	502	25	:	:	PUNCT
ejpam-2095	502	26	b	b	X
ejpam-2095	502	27	)	)	PUNCT
ejpam-2095	502	28	=	=	PUNCT
ejpam-2095	502	29	a	a	PROPN
ejpam-2095	502	30	and	and	CCONJ
ejpam-2095	502	31	if	if	SCONJ
ejpam-2095	502	32	possible	possible	ADJ
ejpam-2095	502	33	without	without	ADP
ejpam-2095	502	34	loss	loss	NOUN
ejpam-2095	502	35	of	of	ADP
ejpam-2095	502	36	generality	generality	NOUN
ejpam-2095	502	37	assume	assume	VERB
ejpam-2095	502	38	that	that	SCONJ
ejpam-2095	502	39	b	b	PROPN
ejpam-2095	502	40	¶	¶	PROPN
ejpam-2095	502	41	p1	p1	PROPN
ejpam-2095	502	42	.	.	PUNCT
ejpam-2095	503	1	then	then	ADV
ejpam-2095	503	2	(	(	PUNCT
ejpam-2095	503	3	q	q	NOUN
ejpam-2095	503	4	:	:	PUNCT
ejpam-2095	503	5	bs	bs	NOUN
ejpam-2095	503	6	)	)	PUNCT
ejpam-2095	503	7	=	=	PUNCT
ejpam-2095	504	1	i	i	PRON
ejpam-2095	504	2	m	m	VERB
ejpam-2095	504	3	for	for	ADP
ejpam-2095	504	4	some	some	DET
ejpam-2095	504	5	integer	integer	NOUN
ejpam-2095	504	6	s.	s.	PROPN
ejpam-2095	504	7	we	we	PRON
ejpam-2095	504	8	have	have	VERB
ejpam-2095	504	9	(	(	PUNCT
ejpam-2095	504	10	a	a	DET
ejpam-2095	504	11	:	:	PUNCT
ejpam-2095	504	12	b	b	X
ejpam-2095	504	13	)	)	PUNCT
ejpam-2095	504	14	:	:	PUNCT
ejpam-2095	505	1	b	b	X
ejpam-2095	505	2	=	=	SYM
ejpam-2095	505	3	a	a	DET
ejpam-2095	505	4	:	:	PUNCT
ejpam-2095	505	5	b2	b2	NOUN
ejpam-2095	505	6	[	[	X
ejpam-2095	505	7	3	3	NUM
ejpam-2095	505	8	]	]	PUNCT
ejpam-2095	505	9	.	.	PUNCT
ejpam-2095	506	1	continuing	continue	VERB
ejpam-2095	506	2	in	in	ADP
ejpam-2095	506	3	this	this	DET
ejpam-2095	506	4	way	way	NOUN
ejpam-2095	506	5	a	a	DET
ejpam-2095	506	6	:	:	PUNCT
ejpam-2095	506	7	b	b	X
ejpam-2095	506	8	=	=	SYM
ejpam-2095	506	9	a	a	PRON
ejpam-2095	506	10	:	:	PUNCT
ejpam-2095	506	11	bs	bs	NOUN
ejpam-2095	506	12	.	.	PUNCT
ejpam-2095	507	1	but	but	CCONJ
ejpam-2095	507	2	a	a	DET
ejpam-2095	507	3	:	:	PUNCT
ejpam-2095	507	4	b	b	X
ejpam-2095	507	5	=	=	PUNCT
ejpam-2095	507	6	a	a	PRON
ejpam-2095	507	7	implies	imply	VERB
ejpam-2095	507	8	a	a	DET
ejpam-2095	507	9	:	:	PUNCT
ejpam-2095	507	10	bs	bs	NOUN
ejpam-2095	507	11	=	=	PUNCT
ejpam-2095	507	12	a.	a.	NOUN
ejpam-2095	507	13	finally	finally	ADV
ejpam-2095	507	14	,	,	PUNCT
ejpam-2095	507	15	a=(a	a=(a	NOUN
ejpam-2095	507	16	:	:	PUNCT
ejpam-2095	507	17	bs	bs	NOUN
ejpam-2095	507	18	)	)	PUNCT
ejpam-2095	507	19	=	=	SYM
ejpam-2095	507	20	(	(	PUNCT
ejpam-2095	507	21	(	(	PUNCT
ejpam-2095	507	22	q1	q1	PROPN
ejpam-2095	507	23	∧q2	∧q2	VERB
ejpam-2095	507	24	∧	∧	PROPN
ejpam-2095	507	25	.	.	PUNCT
ejpam-2095	507	26	.	.	PUNCT
ejpam-2095	507	27	.∧qm	.∧qm	PUNCT
ejpam-2095	507	28	)	)	PUNCT
ejpam-2095	508	1	:	:	PUNCT
ejpam-2095	508	2	bs	bs	X
ejpam-2095	508	3	)	)	PUNCT
ejpam-2095	508	4	=	=	SYM
ejpam-2095	508	5	(	(	PUNCT
ejpam-2095	508	6	(	(	PUNCT
ejpam-2095	508	7	q1	q1	NOUN
ejpam-2095	508	8	:	:	PUNCT
ejpam-2095	508	9	bs)∧	bs)∧	NOUN
ejpam-2095	508	10	(	(	PUNCT
ejpam-2095	508	11	q2	q2	NOUN
ejpam-2095	508	12	:	:	PUNCT
ejpam-2095	508	13	bs)∧	bs)∧	NOUN
ejpam-2095	508	14	.	.	PUNCT
ejpam-2095	508	15	.	.	PUNCT
ejpam-2095	508	16	.	.	PUNCT
ejpam-2095	509	1	∧	∧	NOUN
ejpam-2095	509	2	(	(	PUNCT
ejpam-2095	509	3	qm	qm	NOUN
ejpam-2095	509	4	:	:	PUNCT
ejpam-2095	509	5	bs	bs	NOUN
ejpam-2095	509	6	)	)	PUNCT
ejpam-2095	509	7	=	=	PUNCT
ejpam-2095	510	1	∧	∧	PROPN
ejpam-2095	510	2	j	j	PROPN
ejpam-2095	510	3	6=1	6=1	NUM
ejpam-2095	510	4	(	(	PUNCT
ejpam-2095	510	5	q	q	PROPN
ejpam-2095	510	6	j	j	NOUN
ejpam-2095	510	7	:	:	PUNCT
ejpam-2095	510	8	bs	bs	PROPN
ejpam-2095	510	9	)	)	PUNCT
ejpam-2095	510	10	≥	≥	NOUN
ejpam-2095	510	11	∧	∧	PROPN
ejpam-2095	510	12	j	j	PROPN
ejpam-2095	510	13	6=1	6=1	NUM
ejpam-2095	510	14	q	q	X
ejpam-2095	510	15	j	j	PROPN
ejpam-2095	510	16	≥	≥	X
ejpam-2095	510	17	a.	a.	NOUN
ejpam-2095	510	18	that	that	PRON
ejpam-2095	510	19	is	be	AUX
ejpam-2095	510	20	a=	a=	ADV
ejpam-2095	510	21	∧	∧	PROPN
ejpam-2095	510	22	j	j	PROPN
ejpam-2095	510	23	6=1	6=1	NUM
ejpam-2095	510	24	q	q	NOUN
ejpam-2095	510	25	j.	j.	PROPN
ejpam-2095	511	1	this	this	PRON
ejpam-2095	511	2	contradicts	contradict	VERB
ejpam-2095	511	3	the	the	DET
ejpam-2095	511	4	fact	fact	NOUN
ejpam-2095	511	5	that	that	SCONJ
ejpam-2095	511	6	a=	a=	ADV
ejpam-2095	511	7	m∧	m∧	VERB
ejpam-2095	511	8	i=1	i=1	PROPN
ejpam-2095	512	1	q	q	X
ejpam-2095	513	1	i	i	PRON
ejpam-2095	513	2	is	be	AUX
ejpam-2095	513	3	a	a	DET
ejpam-2095	513	4	reduced	reduce	VERB
ejpam-2095	513	5	primary	primary	ADJ
ejpam-2095	513	6	decomposition	decomposition	NOUN
ejpam-2095	513	7	of	of	ADP
ejpam-2095	513	8	a.	a.	NOUN
ejpam-2095	513	9	the	the	DET
ejpam-2095	513	10	above	above	ADJ
ejpam-2095	513	11	theorem	theorem	NOUN
ejpam-2095	513	12	can	can	AUX
ejpam-2095	513	13	be	be	AUX
ejpam-2095	513	14	restated	restate	VERB
ejpam-2095	513	15	in	in	ADP
ejpam-2095	513	16	the	the	DET
ejpam-2095	513	17	following	follow	VERB
ejpam-2095	513	18	form	form	NOUN
ejpam-2095	513	19	.	.	PUNCT
ejpam-2095	514	1	theorem	theorem	NOUN
ejpam-2095	514	2	17	17	NUM
ejpam-2095	514	3	.	.	PUNCT
ejpam-2095	515	1	let	let	VERB
ejpam-2095	515	2	n	n	PRON
ejpam-2095	515	3	6=	6=	NUM
ejpam-2095	516	1	i	i	PRON
ejpam-2095	516	2	m	m	AUX
ejpam-2095	516	3	have	have	VERB
ejpam-2095	516	4	a	a	DET
ejpam-2095	516	5	reduced	reduce	VERB
ejpam-2095	516	6	primary	primary	ADJ
ejpam-2095	516	7	decomposition	decomposition	NOUN
ejpam-2095	516	8	q1	q1	PROPN
ejpam-2095	516	9	∧	∧	PROPN
ejpam-2095	516	10	q2	q2	PROPN
ejpam-2095	516	11	∧	∧	PROPN
ejpam-2095	516	12	.	.	PUNCT
ejpam-2095	516	13	.	.	PUNCT
ejpam-2095	516	14	.	.	PUNCT
ejpam-2095	517	1	∧	∧	PROPN
ejpam-2095	517	2	qm	qm	PROPN
ejpam-2095	517	3	and	and	CCONJ
ejpam-2095	517	4	p1	p1	PROPN
ejpam-2095	517	5	,	,	PUNCT
ejpam-2095	517	6	p2	p2	NOUN
ejpam-2095	517	7	,	,	PUNCT
ejpam-2095	517	8	.	.	PUNCT
ejpam-2095	517	9	.	.	PUNCT
ejpam-2095	518	1	.	.	PUNCT
ejpam-2095	519	1	,	,	PUNCT
ejpam-2095	519	2	pm	pm	NOUN
ejpam-2095	519	3	be	be	AUX
ejpam-2095	519	4	the	the	DET
ejpam-2095	519	5	associated	associated	ADJ
ejpam-2095	519	6	primes	prime	NOUN
ejpam-2095	519	7	of	of	ADP
ejpam-2095	519	8	q	q	PROPN
ejpam-2095	519	9	′s	′s	PROPN
ejpam-2095	519	10	i	i	PRON
ejpam-2095	519	11	.	.	PUNCT
ejpam-2095	520	1	for	for	ADP
ejpam-2095	520	2	an	an	DET
ejpam-2095	520	3	element	element	NOUN
ejpam-2095	520	4	b	b	PROPN
ejpam-2095	520	5	of	of	ADP
ejpam-2095	520	6	l	l	NOUN
ejpam-2095	520	7	to	to	PART
ejpam-2095	520	8	be	be	AUX
ejpam-2095	520	9	contained	contain	VERB
ejpam-2095	520	10	in	in	ADP
ejpam-2095	520	11	some	some	DET
ejpam-2095	520	12	associated	associate	VERB
ejpam-2095	520	13	prime	prime	ADJ
ejpam-2095	520	14	element	element	NOUN
ejpam-2095	520	15	of	of	ADP
ejpam-2095	520	16	n	n	PRON
ejpam-2095	520	17	it	it	PRON
ejpam-2095	520	18	is	be	AUX
ejpam-2095	520	19	necessary	necessary	ADJ
ejpam-2095	520	20	and	and	CCONJ
ejpam-2095	520	21	sufficient	sufficient	ADJ
ejpam-2095	521	1	that	that	SCONJ
ejpam-2095	521	2	(	(	PUNCT
ejpam-2095	521	3	n	n	NUM
ejpam-2095	521	4	:	:	PUNCT
ejpam-2095	521	5	b	b	X
ejpam-2095	521	6	)	)	PUNCT
ejpam-2095	521	7	6=	6=	PUNCT
ejpam-2095	521	8	n.	n.	PROPN
ejpam-2095	521	9	direct	direct	ADJ
ejpam-2095	521	10	application	application	NOUN
ejpam-2095	521	11	of	of	ADP
ejpam-2095	521	12	the	the	DET
ejpam-2095	521	13	above	above	ADJ
ejpam-2095	521	14	theorem	theorem	NOUN
ejpam-2095	521	15	gives	give	VERB
ejpam-2095	521	16	the	the	DET
ejpam-2095	521	17	following	follow	VERB
ejpam-2095	521	18	result	result	NOUN
ejpam-2095	521	19	.	.	PUNCT
ejpam-2095	522	1	theorem	theorem	ADJ
ejpam-2095	522	2	18	18	NUM
ejpam-2095	522	3	.	.	PUNCT
ejpam-2095	523	1	for	for	ADP
ejpam-2095	523	2	an	an	DET
ejpam-2095	523	3	element	element	NOUN
ejpam-2095	523	4	b	b	PROPN
ejpam-2095	523	5	of	of	ADP
ejpam-2095	523	6	l	l	NOUN
ejpam-2095	523	7	to	to	PART
ejpam-2095	523	8	be	be	AUX
ejpam-2095	523	9	contained	contain	VERB
ejpam-2095	523	10	in	in	ADP
ejpam-2095	523	11	some	some	DET
ejpam-2095	523	12	associated	associate	VERB
ejpam-2095	523	13	prime	prime	ADJ
ejpam-2095	523	14	element	element	NOUN
ejpam-2095	523	15	of	of	ADP
ejpam-2095	523	16	n	n	CCONJ
ejpam-2095	523	17	,	,	PUNCT
ejpam-2095	523	18	it	it	PRON
ejpam-2095	523	19	is	be	AUX
ejpam-2095	523	20	necessary	necessary	ADJ
ejpam-2095	523	21	and	and	CCONJ
ejpam-2095	523	22	sufficient	sufficient	ADJ
ejpam-2095	523	23	that	that	SCONJ
ejpam-2095	523	24	there	there	PRON
ejpam-2095	523	25	is	be	VERB
ejpam-2095	523	26	an	an	DET
ejpam-2095	523	27	element	element	NOUN
ejpam-2095	523	28	y	y	PROPN
ejpam-2095	523	29	�	�	PROPN
ejpam-2095	523	30	n	n	CCONJ
ejpam-2095	523	31	such	such	ADJ
ejpam-2095	523	32	that	that	SCONJ
ejpam-2095	523	33	by	by	ADP
ejpam-2095	523	34	¶	¶	PROPN
ejpam-2095	523	35	n.	n.	PROPN
ejpam-2095	523	36	references	reference	VERB
ejpam-2095	523	37	209	209	NUM
ejpam-2095	523	38	an	an	DET
ejpam-2095	523	39	element	element	NOUN
ejpam-2095	523	40	x	x	SYM
ejpam-2095	523	41	∈	∈	NOUN
ejpam-2095	523	42	m	m	VERB
ejpam-2095	523	43	is	be	AUX
ejpam-2095	523	44	called	call	VERB
ejpam-2095	523	45	a	a	DET
ejpam-2095	523	46	zero	zero	NUM
ejpam-2095	523	47	divisor	divisor	NOUN
ejpam-2095	523	48	if	if	SCONJ
ejpam-2095	523	49	(	(	PUNCT
ejpam-2095	523	50	0	0	NUM
ejpam-2095	523	51	m	m	VERB
ejpam-2095	523	52	:	:	PUNCT
ejpam-2095	523	53	x	x	X
ejpam-2095	523	54	)	)	PUNCT
ejpam-2095	524	1	6=	6=	ADP
ejpam-2095	524	2	0	0	NUM
ejpam-2095	525	1	so	so	ADV
ejpam-2095	525	2	there	there	PRON
ejpam-2095	525	3	exists	exist	VERB
ejpam-2095	525	4	a	a	DET
ejpam-2095	525	5	6=	6=	NOUN
ejpam-2095	525	6	0	0	NUM
ejpam-2095	525	7	in	in	ADP
ejpam-2095	525	8	l	l	NOUN
ejpam-2095	525	9	such	such	ADJ
ejpam-2095	525	10	that	that	DET
ejpam-2095	525	11	ax	ax	NOUN
ejpam-2095	525	12	=	=	PUNCT
ejpam-2095	525	13	0	0	NUM
ejpam-2095	525	14	m	m	NOUN
ejpam-2095	525	15	.	.	PUNCT
ejpam-2095	526	1	theorem	theorem	NOUN
ejpam-2095	526	2	19	19	NUM
ejpam-2095	526	3	.	.	PUNCT
ejpam-2095	527	1	let	let	VERB
ejpam-2095	527	2	m	m	PRON
ejpam-2095	527	3	be	be	AUX
ejpam-2095	527	4	a	a	DET
ejpam-2095	527	5	lattice	lattice	NOUN
ejpam-2095	527	6	module	module	NOUN
ejpam-2095	527	7	where	where	SCONJ
ejpam-2095	527	8	m	m	PROPN
ejpam-2095	527	9	satisfies	satisfy	VERB
ejpam-2095	527	10	the	the	DET
ejpam-2095	527	11	acc	acc	NOUN
ejpam-2095	527	12	and	and	CCONJ
ejpam-2095	527	13	every	every	DET
ejpam-2095	527	14	element	element	NOUN
ejpam-2095	527	15	of	of	ADP
ejpam-2095	527	16	m	m	PROPN
ejpam-2095	527	17	is	be	AUX
ejpam-2095	527	18	the	the	DET
ejpam-2095	527	19	join	join	NOUN
ejpam-2095	527	20	of	of	ADP
ejpam-2095	527	21	meet	meet	VERB
ejpam-2095	527	22	principal	principal	ADJ
ejpam-2095	527	23	elements	element	NOUN
ejpam-2095	527	24	.	.	PUNCT
ejpam-2095	528	1	if	if	SCONJ
ejpam-2095	528	2	x	x	SYM
ejpam-2095	528	3	∈	∈	PROPN
ejpam-2095	528	4	m	m	VERB
ejpam-2095	528	5	then	then	ADV
ejpam-2095	528	6	the	the	DET
ejpam-2095	528	7	join	join	NOUN
ejpam-2095	528	8	of	of	ADP
ejpam-2095	528	9	all	all	DET
ejpam-2095	528	10	a	a	DET
ejpam-2095	528	11	∈	∈	NOUN
ejpam-2095	528	12	l	l	NOUN
ejpam-2095	528	13	such	such	ADJ
ejpam-2095	528	14	that	that	SCONJ
ejpam-2095	528	15	a	a	DET
ejpam-2095	528	16	6=	6=	NUM
ejpam-2095	528	17	0	0	NUM
ejpam-2095	528	18	and	and	CCONJ
ejpam-2095	528	19	ax	ax	NOUN
ejpam-2095	528	20	=	=	NOUN
ejpam-2095	528	21	0	0	NUM
ejpam-2095	528	22	m	m	VERB
ejpam-2095	528	23	is	be	AUX
ejpam-2095	528	24	contained	contain	VERB
ejpam-2095	528	25	in	in	ADP
ejpam-2095	528	26	the	the	DET
ejpam-2095	528	27	join	join	NOUN
ejpam-2095	528	28	of	of	ADP
ejpam-2095	528	29	all	all	DET
ejpam-2095	528	30	associated	associate	VERB
ejpam-2095	528	31	prime	prime	ADJ
ejpam-2095	528	32	elements	element	NOUN
ejpam-2095	528	33	of	of	ADP
ejpam-2095	528	34	0	0	NUM
ejpam-2095	528	35	m	m	NOUN
ejpam-2095	528	36	.	.	PUNCT
ejpam-2095	529	1	proof	proof	NOUN
ejpam-2095	529	2	.	.	PUNCT
ejpam-2095	530	1	let	let	VERB
ejpam-2095	530	2	x	x	PRON
ejpam-2095	530	3	be	be	AUX
ejpam-2095	530	4	a	a	DET
ejpam-2095	530	5	zero	zero	NUM
ejpam-2095	530	6	divisor	divisor	NOUN
ejpam-2095	530	7	of	of	ADP
ejpam-2095	530	8	m	m	PROPN
ejpam-2095	530	9	.	.	PUNCT
ejpam-2095	531	1	then	then	ADV
ejpam-2095	531	2	0	0	NUM
ejpam-2095	531	3	m	m	VERB
ejpam-2095	531	4	:	:	PUNCT
ejpam-2095	531	5	x	x	SYM
ejpam-2095	531	6	6=	6=	ADP
ejpam-2095	531	7	0	0	NUM
ejpam-2095	531	8	that	that	PRON
ejpam-2095	531	9	is	be	AUX
ejpam-2095	531	10	there	there	PRON
ejpam-2095	531	11	exists	exist	VERB
ejpam-2095	531	12	a	a	DET
ejpam-2095	531	13	6=	6=	NOUN
ejpam-2095	531	14	0	0	NUM
ejpam-2095	531	15	in	in	ADP
ejpam-2095	531	16	l	l	NOUN
ejpam-2095	531	17	such	such	ADJ
ejpam-2095	531	18	that	that	DET
ejpam-2095	531	19	ax	ax	NOUN
ejpam-2095	531	20	=	=	PUNCT
ejpam-2095	531	21	0	0	NUM
ejpam-2095	531	22	m	m	VERB
ejpam-2095	531	23	.	.	PUNCT
ejpam-2095	532	1	we	we	PRON
ejpam-2095	532	2	know	know	VERB
ejpam-2095	532	3	that	that	PRON
ejpam-2095	532	4	for	for	SCONJ
ejpam-2095	532	5	an	an	DET
ejpam-2095	532	6	element	element	NOUN
ejpam-2095	532	7	b	b	PROPN
ejpam-2095	532	8	of	of	ADP
ejpam-2095	532	9	l	l	PROPN
ejpam-2095	532	10	(	(	PUNCT
ejpam-2095	532	11	b	b	PROPN
ejpam-2095	532	12	6=	6=	NUM
ejpam-2095	532	13	0	0	NUM
ejpam-2095	532	14	,	,	PUNCT
ejpam-2095	532	15	bx	bx	NOUN
ejpam-2095	532	16	=	=	PUNCT
ejpam-2095	532	17	0	0	NUM
ejpam-2095	532	18	m	m	VERB
ejpam-2095	532	19	)	)	PUNCT
ejpam-2095	532	20	to	to	PART
ejpam-2095	532	21	be	be	AUX
ejpam-2095	532	22	contained	contain	VERB
ejpam-2095	532	23	in	in	ADP
ejpam-2095	532	24	some	some	DET
ejpam-2095	532	25	associated	associated	ADJ
ejpam-2095	532	26	prime	prime	NOUN
ejpam-2095	532	27	of	of	ADP
ejpam-2095	532	28	0	0	NUM
ejpam-2095	532	29	m	m	VERB
ejpam-2095	532	30	it	it	PRON
ejpam-2095	532	31	is	be	AUX
ejpam-2095	532	32	necessary	necessary	ADJ
ejpam-2095	532	33	and	and	CCONJ
ejpam-2095	532	34	sufficient	sufficient	ADJ
ejpam-2095	532	35	that	that	SCONJ
ejpam-2095	532	36	(	(	PUNCT
ejpam-2095	532	37	0	0	NUM
ejpam-2095	532	38	m	m	VERB
ejpam-2095	532	39	:	:	PUNCT
ejpam-2095	532	40	b	b	X
ejpam-2095	532	41	)	)	PUNCT
ejpam-2095	532	42	=	=	SYM
ejpam-2095	533	1	∨{x	∨{x	PROPN
ejpam-2095	533	2	∈	∈	PROPN
ejpam-2095	533	3	m	m	VERB
ejpam-2095	533	4	|	|	ADV
ejpam-2095	533	5	bx	bx	VERB
ejpam-2095	533	6	=	=	NOUN
ejpam-2095	533	7	0	0	NUM
ejpam-2095	533	8	m	m	NOUN
ejpam-2095	533	9	}	}	PUNCT
ejpam-2095	533	10	6=	6=	ADP
ejpam-2095	533	11	0	0	NUM
ejpam-2095	533	12	m	m	NOUN
ejpam-2095	533	13	.	.	PUNCT
ejpam-2095	534	1	hence	hence	ADV
ejpam-2095	534	2	the	the	DET
ejpam-2095	534	3	join	join	NOUN
ejpam-2095	534	4	of	of	ADP
ejpam-2095	534	5	all	all	DET
ejpam-2095	534	6	elements	element	NOUN
ejpam-2095	534	7	a	a	PRON
ejpam-2095	534	8	of	of	ADP
ejpam-2095	534	9	l	l	NOUN
ejpam-2095	534	10	such	such	ADJ
ejpam-2095	534	11	that	that	SCONJ
ejpam-2095	534	12	a	a	DET
ejpam-2095	534	13	6=	6=	NUM
ejpam-2095	534	14	0	0	NUM
ejpam-2095	534	15	and	and	CCONJ
ejpam-2095	534	16	ax	ax	NOUN
ejpam-2095	534	17	=	=	NOUN
ejpam-2095	534	18	0	0	NUM
ejpam-2095	534	19	m	m	VERB
ejpam-2095	534	20	is	be	AUX
ejpam-2095	534	21	contained	contain	VERB
ejpam-2095	534	22	in	in	ADP
ejpam-2095	534	23	the	the	DET
ejpam-2095	534	24	join	join	NOUN
ejpam-2095	534	25	of	of	ADP
ejpam-2095	534	26	all	all	DET
ejpam-2095	534	27	associated	associate	VERB
ejpam-2095	534	28	prime	prime	ADJ
ejpam-2095	534	29	elements	element	NOUN
ejpam-2095	534	30	of	of	ADP
ejpam-2095	534	31	0	0	NUM
ejpam-2095	534	32	m	m	VERB
ejpam-2095	534	33	,	,	PUNCT
ejpam-2095	534	34	by	by	ADP
ejpam-2095	534	35	theorem	theorem	NOUN
ejpam-2095	534	36	16	16	NUM
ejpam-2095	534	37	.	.	PUNCT
ejpam-2095	535	1	acknowledgements	acknowledgement	NOUN
ejpam-2095	535	2	the	the	DET
ejpam-2095	535	3	authors	author	NOUN
ejpam-2095	535	4	thank	thank	VERB
ejpam-2095	535	5	the	the	DET
ejpam-2095	535	6	readers	reader	NOUN
ejpam-2095	535	7	of	of	ADP
ejpam-2095	535	8	european	european	PROPN
ejpam-2095	535	9	journal	journal	PROPN
ejpam-2095	535	10	of	of	ADP
ejpam-2095	535	11	pure	pure	ADJ
ejpam-2095	535	12	and	and	CCONJ
ejpam-2095	535	13	applied	applied	ADJ
ejpam-2095	535	14	mathematics	mathematic	NOUN
ejpam-2095	535	15	,	,	PUNCT
ejpam-2095	535	16	for	for	ADP
ejpam-2095	535	17	making	make	VERB
ejpam-2095	535	18	our	our	PRON
ejpam-2095	535	19	journal	journal	NOUN
ejpam-2095	535	20	successful	successful	ADJ
ejpam-2095	535	21	.	.	PUNCT
ejpam-2095	536	1	references	reference	NOUN
ejpam-2095	536	2	[	[	X
ejpam-2095	536	3	1	1	NUM
ejpam-2095	536	4	]	]	X
ejpam-2095	536	5	d.d	d.d	PROPN
ejpam-2095	536	6	.	.	PROPN
ejpam-2095	536	7	anderson	anderson	PROPN
ejpam-2095	536	8	.	.	PUNCT
ejpam-2095	537	1	multiplicative	multiplicative	PROPN
ejpam-2095	537	2	latices	latex	NOUN
ejpam-2095	537	3	.	.	PUNCT
ejpam-2095	538	1	ph.d	ph.d	PROPN
ejpam-2095	538	2	.	.	PUNCT
ejpam-2095	539	1	thesis	thesis	PROPN
ejpam-2095	539	2	,	,	PUNCT
ejpam-2095	539	3	chicago	chicago	PROPN
ejpam-2095	539	4	university	university	NOUN
ejpam-2095	539	5	,	,	PUNCT
ejpam-2095	539	6	1974	1974	NUM
ejpam-2095	539	7	.	.	PUNCT
ejpam-2095	540	1	[	[	X
ejpam-2095	540	2	2	2	NUM
ejpam-2095	540	3	]	]	X
ejpam-2095	540	4	r.p	r.p	PROPN
ejpam-2095	540	5	.	.	PROPN
ejpam-2095	540	6	dilworth	dilworth	PROPN
ejpam-2095	540	7	.	.	PUNCT
ejpam-2095	541	1	abstract	abstract	ADJ
ejpam-2095	541	2	commutative	commutative	ADJ
ejpam-2095	541	3	ideal	ideal	PROPN
ejpam-2095	541	4	theory	theory	NOUN
ejpam-2095	541	5	,	,	PUNCT
ejpam-2095	541	6	pacific	pacific	PROPN
ejpam-2095	541	7	journal	journal	PROPN
ejpam-2095	541	8	of	of	ADP
ejpam-2095	541	9	mathematics	mathematic	NOUN
ejpam-2095	541	10	,	,	PUNCT
ejpam-2095	541	11	12	12	NUM
ejpam-2095	541	12	,	,	PUNCT
ejpam-2095	541	13	481	481	NUM
ejpam-2095	541	14	-	-	SYM
ejpam-2095	541	15	498	498	NUM
ejpam-2095	541	16	.	.	PUNCT
ejpam-2095	542	1	1962	1962	NUM
ejpam-2095	542	2	.	.	PUNCT
ejpam-2095	543	1	[	[	X
ejpam-2095	543	2	3	3	X
ejpam-2095	543	3	]	]	X
ejpam-2095	543	4	j.a	j.a	PROPN
ejpam-2095	543	5	.	.	PROPN
ejpam-2095	543	6	johnson	johnson	PROPN
ejpam-2095	543	7	.	.	PUNCT
ejpam-2095	544	1	a-adic.completions	a-adic.completion	NOUN
ejpam-2095	544	2	of	of	ADP
ejpam-2095	544	3	noetherian	noetherian	ADJ
ejpam-2095	544	4	lattice	lattice	NOUN
ejpam-2095	544	5	modules	module	NOUN
ejpam-2095	544	6	,	,	PUNCT
ejpam-2095	544	7	fundamenta	fundamenta	PROPN
ejpam-2095	544	8	mathematicae	mathematicae	PROPN
ejpam-2095	544	9	,	,	PUNCT
ejpam-2095	544	10	66	66	NUM
ejpam-2095	544	11	,	,	PUNCT
ejpam-2095	544	12	341	341	NUM
ejpam-2095	544	13	-	-	SYM
ejpam-2095	544	14	371	371	NUM
ejpam-2095	544	15	.	.	PUNCT
ejpam-2095	544	16	1970	1970	NUM
ejpam-2095	544	17	.	.	PUNCT
ejpam-2095	545	1	[	[	X
ejpam-2095	545	2	4	4	X
ejpam-2095	545	3	]	]	PUNCT
ejpam-2095	545	4	f.	f.	PROPN
ejpam-2095	545	5	callialp	callialp	PROPN
ejpam-2095	545	6	and	and	CCONJ
ejpam-2095	545	7	u.	u.	PROPN
ejpam-2095	545	8	tekir	tekir	PROPN
ejpam-2095	545	9	.	.	PUNCT
ejpam-2095	546	1	multiplication	multiplication	NOUN
ejpam-2095	546	2	lattice	lattice	PROPN
ejpam-2095	546	3	modules	module	NOUN
ejpam-2095	546	4	,	,	PUNCT
ejpam-2095	546	5	iranian	iranian	ADJ
ejpam-2095	546	6	journal	journal	PROPN
ejpam-2095	546	7	of	of	ADP
ejpam-2095	546	8	science	science	NOUN
ejpam-2095	546	9	and	and	CCONJ
ejpam-2095	546	10	technology	technology	NOUN
ejpam-2095	546	11	,	,	PUNCT
ejpam-2095	546	12	309	309	NUM
ejpam-2095	546	13	-	-	SYM
ejpam-2095	546	14	313	313	NUM
ejpam-2095	546	15	.	.	PUNCT
ejpam-2095	546	16	2011	2011	NUM
ejpam-2095	546	17	.	.	PUNCT
ejpam-2095	547	1	[	[	X
ejpam-2095	547	2	5	5	NUM
ejpam-2095	547	3	]	]	X
ejpam-2095	547	4	m.d	m.d	PROPN
ejpam-2095	547	5	.	.	PROPN
ejpam-2095	547	6	larsen	larsen	PROPN
ejpam-2095	547	7	and	and	CCONJ
ejpam-2095	547	8	p.j	p.j	PROPN
ejpam-2095	547	9	.	.	PROPN
ejpam-2095	547	10	mccarthy	mccarthy	PROPN
ejpam-2095	547	11	.	.	PUNCT
ejpam-2095	548	1	multiplicative	multiplicative	ADJ
ejpam-2095	548	2	theory	theory	NOUN
ejpam-2095	548	3	of	of	ADP
ejpam-2095	548	4	ideals	ideal	NOUN
ejpam-2095	548	5	,	,	PUNCT
ejpam-2095	548	6	academic	academic	ADJ
ejpam-2095	548	7	press	press	NOUN
ejpam-2095	548	8	,	,	PUNCT
ejpam-2095	548	9	new	new	PROPN
ejpam-2095	548	10	york	york	PROPN
ejpam-2095	548	11	,	,	PUNCT
ejpam-2095	548	12	usa	usa	PROPN
ejpam-2095	548	13	.	.	PROPN
ejpam-2095	548	14	1970	1970	NUM
ejpam-2095	548	15	.	.	PUNCT
ejpam-2095	549	1	[	[	X
ejpam-2095	549	2	6	6	NUM
ejpam-2095	549	3	]	]	X
ejpam-2095	549	4	n.k	n.k	PROPN
ejpam-2095	549	5	.	.	PROPN
ejpam-2095	549	6	thakare	thakare	PROPN
ejpam-2095	549	7	and	and	CCONJ
ejpam-2095	549	8	c.s	c.s	PROPN
ejpam-2095	549	9	.	.	PROPN
ejpam-2095	549	10	manjarekar	manjarekar	PROPN
ejpam-2095	549	11	.	.	PUNCT
ejpam-2095	550	1	radicals	radical	NOUN
ejpam-2095	550	2	and	and	CCONJ
ejpam-2095	550	3	uniqueness	uniqueness	NOUN
ejpam-2095	550	4	theorem	theorem	VERB
ejpam-2095	550	5	in	in	ADP
ejpam-2095	550	6	multiplicative	multiplicative	ADJ
ejpam-2095	550	7	lattices	lattice	NOUN
ejpam-2095	550	8	with	with	ADP
ejpam-2095	550	9	chain	chain	NOUN
ejpam-2095	550	10	conditions	condition	NOUN
ejpam-2095	550	11	,	,	PUNCT
ejpam-2095	550	12	studia	studia	PROPN
ejpam-2095	550	13	scientifica	scientifica	PROPN
ejpam-2095	550	14	mathematicarum	mathematicarum	PROPN
ejpam-2095	550	15	hungarica	hungarica	PROPN
ejpam-2095	550	16	18	18	NUM
ejpam-2095	550	17	,	,	PUNCT
ejpam-2095	550	18	13	13	NUM
ejpam-2095	550	19	-	-	SYM
ejpam-2095	550	20	19	19	NUM
ejpam-2095	550	21	.	.	NOUN
ejpam-2095	550	22	1983	1983	NUM