id	sid	tid	token	lemma	pos
ejpam-709	1	1	9_709_rao.dvi	9_709_rao.dvi	NUM
ejpam-709	1	2	european	european	ADJ
ejpam-709	1	3	journal	journal	PROPN
ejpam-709	1	4	of	of	ADP
ejpam-709	1	5	pure	pure	ADJ
ejpam-709	1	6	and	and	CCONJ
ejpam-709	1	7	applied	apply	VERB
ejpam-709	1	8	mathematics	mathematic	NOUN
ejpam-709	1	9	vol	vol	NOUN
ejpam-709	1	10	.	.	PUNCT
ejpam-709	2	1	3	3	NUM
ejpam-709	2	2	,	,	PUNCT
ejpam-709	2	3	no	no	INTJ
ejpam-709	2	4	.	.	NOUN
ejpam-709	2	5	4	4	NUM
ejpam-709	2	6	,	,	PUNCT
ejpam-709	2	7	2010	2010	NUM
ejpam-709	2	8	,	,	PUNCT
ejpam-709	2	9	704	704	NUM
ejpam-709	2	10	-	-	SYM
ejpam-709	2	11	716	716	NUM
ejpam-709	2	12	issn	issn	PROPN
ejpam-709	2	13	1307	1307	NUM
ejpam-709	2	14	-	-	SYM
ejpam-709	2	15	5543	5543	NUM
ejpam-709	2	16	–	–	PUNCT
ejpam-709	3	1	www.ejpam.com	www.ejpam.com	X
ejpam-709	3	2	s−linear	s−linear	VERB
ejpam-709	3	3	almost	almost	ADV
ejpam-709	3	4	distributive	distributive	ADJ
ejpam-709	3	5	lattices	lattice	NOUN
ejpam-709	3	6	g.c	g.c	PROPN
ejpam-709	3	7	.	.	PROPN
ejpam-709	3	8	rao1,∗	rao1,∗	PROPN
ejpam-709	3	9	,	,	PUNCT
ejpam-709	3	10	n.	n.	PROPN
ejpam-709	3	11	rafi1	rafi1	PROPN
ejpam-709	3	12	,	,	PUNCT
ejpam-709	3	13	ravi	ravi	PROPN
ejpam-709	3	14	kumar	kumar	PROPN
ejpam-709	3	15	bandaru2	bandaru2	PROPN
ejpam-709	3	16	1	1	NUM
ejpam-709	3	17	department	department	NOUN
ejpam-709	3	18	of	of	ADP
ejpam-709	3	19	mathematics	mathematics	PROPN
ejpam-709	3	20	,	,	PUNCT
ejpam-709	3	21	andhra	andhra	PROPN
ejpam-709	3	22	university	university	PROPN
ejpam-709	3	23	,	,	PUNCT
ejpam-709	3	24	visakhapatnam	visakhapatnam	PROPN
ejpam-709	3	25	,	,	PUNCT
ejpam-709	3	26	andhra	andhra	PROPN
ejpam-709	3	27	pradesh	pradesh	PROPN
ejpam-709	3	28	,	,	PUNCT
ejpam-709	3	29	india	india	PROPN
ejpam-709	3	30	530003	530003	NUM
ejpam-709	3	31	.	.	PUNCT
ejpam-709	4	1	2	2	NUM
ejpam-709	4	2	department	department	NOUN
ejpam-709	4	3	of	of	ADP
ejpam-709	4	4	engineering	engineering	NOUN
ejpam-709	4	5	mathematics	mathematic	NOUN
ejpam-709	4	6	,	,	PUNCT
ejpam-709	4	7	gitam	gitam	NOUN
ejpam-709	4	8	university	university	NOUN
ejpam-709	4	9	,	,	PUNCT
ejpam-709	4	10	hyderabad	hyderabad	PROPN
ejpam-709	4	11	campus	campus	PROPN
ejpam-709	4	12	,	,	PUNCT
ejpam-709	4	13	andhra	andhra	PROPN
ejpam-709	4	14	pradesh	pradesh	PROPN
ejpam-709	4	15	,	,	PUNCT
ejpam-709	4	16	india	india	PROPN
ejpam-709	4	17	abstract	abstract	PROPN
ejpam-709	4	18	.	.	PUNCT
ejpam-709	5	1	the	the	DET
ejpam-709	5	2	concept	concept	NOUN
ejpam-709	5	3	of	of	ADP
ejpam-709	5	4	an	an	DET
ejpam-709	5	5	s−linear	s−linear	X
ejpam-709	5	6	adl	adl	PROPN
ejpam-709	5	7	is	be	AUX
ejpam-709	5	8	defined	define	VERB
ejpam-709	5	9	and	and	CCONJ
ejpam-709	5	10	characterized	characterize	VERB
ejpam-709	5	11	in	in	ADP
ejpam-709	5	12	terms	term	NOUN
ejpam-709	5	13	of	of	ADP
ejpam-709	5	14	the	the	DET
ejpam-709	5	15	s−prime	s−prime	NOUN
ejpam-709	5	16	ideals	ideal	NOUN
ejpam-709	5	17	and	and	CCONJ
ejpam-709	5	18	s−prime	s−prime	NOUN
ejpam-709	5	19	filters	filter	NOUN
ejpam-709	5	20	.	.	PUNCT
ejpam-709	6	1	equivalent	equivalent	ADJ
ejpam-709	6	2	condition	condition	NOUN
ejpam-709	6	3	for	for	ADP
ejpam-709	6	4	an	an	DET
ejpam-709	6	5	adl	adl	NOUN
ejpam-709	6	6	r	r	NOUN
ejpam-709	6	7	to	to	PART
ejpam-709	6	8	become	become	VERB
ejpam-709	6	9	a	a	DET
ejpam-709	6	10	(	(	PUNCT
ejpam-709	6	11	dually)b−relatively	dually)b−relatively	ADV
ejpam-709	6	12	normal	normal	ADJ
ejpam-709	6	13	adl	adl	NOUN
ejpam-709	6	14	in	in	ADP
ejpam-709	6	15	terms	term	NOUN
ejpam-709	6	16	of	of	ADP
ejpam-709	6	17	minimal	minimal	ADJ
ejpam-709	6	18	prime	prime	ADJ
ejpam-709	6	19	ideals(filters	ideals(filter	NOUN
ejpam-709	6	20	)	)	PUNCT
ejpam-709	6	21	and	and	CCONJ
ejpam-709	6	22	b−maximal	b−maximal	NOUN
ejpam-709	6	23	ideals(filters	ideals(filter	NOUN
ejpam-709	6	24	)	)	PUNCT
ejpam-709	6	25	is	be	AUX
ejpam-709	6	26	obtained	obtain	VERB
ejpam-709	6	27	,	,	PUNCT
ejpam-709	6	28	where	where	SCONJ
ejpam-709	6	29	b	b	NOUN
ejpam-709	6	30	is	be	AUX
ejpam-709	6	31	the	the	DET
ejpam-709	6	32	birkhoff	birkhoff	NOUN
ejpam-709	6	33	centre	centre	NOUN
ejpam-709	6	34	of	of	ADP
ejpam-709	6	35	r.	r.	PROPN
ejpam-709	6	36	2000	2000	NUM
ejpam-709	6	37	mathematics	mathematics	PROPN
ejpam-709	6	38	subject	subject	NOUN
ejpam-709	6	39	classifications	classification	NOUN
ejpam-709	6	40	:	:	PUNCT
ejpam-709	6	41	06d99	06d99	NUM
ejpam-709	6	42	key	key	ADJ
ejpam-709	6	43	words	word	NOUN
ejpam-709	6	44	and	and	CCONJ
ejpam-709	6	45	phrases	phrase	NOUN
ejpam-709	6	46	:	:	PUNCT
ejpam-709	6	47	almost	almost	ADV
ejpam-709	6	48	distributive	distributive	ADJ
ejpam-709	6	49	lattice	lattice	NOUN
ejpam-709	6	50	(	(	PUNCT
ejpam-709	6	51	adl	adl	PROPN
ejpam-709	6	52	)	)	PUNCT
ejpam-709	6	53	,	,	PUNCT
ejpam-709	6	54	s−normal	s−normal	X
ejpam-709	6	55	adl	adl	NOUN
ejpam-709	6	56	,	,	PUNCT
ejpam-709	6	57	s−relatively	s−relatively	ADV
ejpam-709	6	58	normal	normal	ADJ
ejpam-709	6	59	adl	adl	PROPN
ejpam-709	6	60	,	,	PUNCT
ejpam-709	6	61	s−linear	s−linear	CCONJ
ejpam-709	6	62	adl	adl	PROPN
ejpam-709	6	63	,	,	PUNCT
ejpam-709	6	64	uni	uni	PROPN
ejpam-709	6	65	subadl	subadl	NOUN
ejpam-709	6	66	,	,	PUNCT
ejpam-709	6	67	birkhoff	birkhoff	NOUN
ejpam-709	6	68	centre	centre	NOUN
ejpam-709	6	69	,	,	PUNCT
ejpam-709	6	70	s−ideal	s−ideal	ADJ
ejpam-709	6	71	,	,	PUNCT
ejpam-709	6	72	s−filter	s−filter	ADV
ejpam-709	6	73	,	,	PUNCT
ejpam-709	6	74	prime	prime	ADJ
ejpam-709	6	75	ideal	ideal	NOUN
ejpam-709	6	76	,	,	PUNCT
ejpam-709	6	77	prime	prime	ADJ
ejpam-709	6	78	filter	filter	NOUN
ejpam-709	6	79	.	.	PUNCT
ejpam-709	7	1	1	1	X
ejpam-709	7	2	.	.	X
ejpam-709	7	3	introduction	introduction	NOUN
ejpam-709	7	4	the	the	DET
ejpam-709	7	5	concepts	concept	NOUN
ejpam-709	7	6	of	of	ADP
ejpam-709	7	7	s−completely	s−completely	ADV
ejpam-709	7	8	normal	normal	ADJ
ejpam-709	7	9	lattice	lattice	NOUN
ejpam-709	7	10	and	and	CCONJ
ejpam-709	7	11	dually	dually	ADV
ejpam-709	7	12	s−completely	s−completely	ADV
ejpam-709	7	13	normal	normal	ADJ
ejpam-709	7	14	lattice	lattice	NOUN
ejpam-709	7	15	were	be	AUX
ejpam-709	7	16	given	give	VERB
ejpam-709	7	17	by	by	ADP
ejpam-709	7	18	cignoli	cignoli	NOUN
ejpam-709	8	1	[	[	X
ejpam-709	8	2	2	2	NUM
ejpam-709	8	3	]	]	PUNCT
ejpam-709	8	4	.	.	PUNCT
ejpam-709	9	1	the	the	DET
ejpam-709	9	2	concept	concept	NOUN
ejpam-709	9	3	of	of	ADP
ejpam-709	9	4	an	an	DET
ejpam-709	9	5	almost	almost	ADV
ejpam-709	9	6	distributive	distributive	ADJ
ejpam-709	9	7	lattice	lattice	NOUN
ejpam-709	9	8	(	(	PUNCT
ejpam-709	9	9	adl	adl	PROPN
ejpam-709	9	10	)	)	PUNCT
ejpam-709	9	11	was	be	AUX
ejpam-709	9	12	introduced	introduce	VERB
ejpam-709	9	13	by	by	ADP
ejpam-709	9	14	swamy	swamy	NOUN
ejpam-709	9	15	and	and	CCONJ
ejpam-709	9	16	rao	rao	NOUN
ejpam-709	10	1	[	[	X
ejpam-709	10	2	9	9	NUM
ejpam-709	10	3	]	]	PUNCT
ejpam-709	10	4	as	as	ADP
ejpam-709	10	5	a	a	DET
ejpam-709	10	6	common	common	ADJ
ejpam-709	10	7	abstraction	abstraction	NOUN
ejpam-709	10	8	of	of	ADP
ejpam-709	10	9	the	the	DET
ejpam-709	10	10	existing	exist	VERB
ejpam-709	10	11	ring	ring	NOUN
ejpam-709	10	12	theoretic	theoretic	NOUN
ejpam-709	10	13	and	and	CCONJ
ejpam-709	10	14	lattice	lattice	ADJ
ejpam-709	10	15	theoretic	theoretic	ADJ
ejpam-709	10	16	generalizations	generalization	NOUN
ejpam-709	10	17	of	of	ADP
ejpam-709	10	18	a	a	DET
ejpam-709	10	19	boolean	boolean	ADJ
ejpam-709	10	20	algebra	algebra	NOUN
ejpam-709	10	21	.	.	PUNCT
ejpam-709	11	1	the	the	DET
ejpam-709	11	2	concept	concept	NOUN
ejpam-709	11	3	of	of	ADP
ejpam-709	11	4	an	an	DET
ejpam-709	11	5	ideal	ideal	NOUN
ejpam-709	11	6	in	in	ADP
ejpam-709	11	7	an	an	DET
ejpam-709	11	8	adl	adl	NOUN
ejpam-709	11	9	was	be	AUX
ejpam-709	11	10	introduced	introduce	VERB
ejpam-709	11	11	in	in	ADP
ejpam-709	11	12	[	[	PUNCT
ejpam-709	11	13	9	9	NUM
ejpam-709	11	14	]	]	PUNCT
ejpam-709	11	15	analogous	analogous	ADJ
ejpam-709	11	16	to	to	ADP
ejpam-709	11	17	that	that	PRON
ejpam-709	11	18	in	in	ADP
ejpam-709	11	19	a	a	DET
ejpam-709	11	20	distributive	distributive	ADJ
ejpam-709	11	21	lattice	lattice	NOUN
ejpam-709	11	22	and	and	CCONJ
ejpam-709	11	23	it	it	PRON
ejpam-709	11	24	was	be	AUX
ejpam-709	11	25	observed	observe	VERB
ejpam-709	11	26	that	that	SCONJ
ejpam-709	11	27	the	the	DET
ejpam-709	11	28	set	set	NOUN
ejpam-709	11	29	pi(r	pi(r	NOUN
ejpam-709	11	30	)	)	PUNCT
ejpam-709	11	31	of	of	ADP
ejpam-709	11	32	all	all	DET
ejpam-709	11	33	principal	principal	ADJ
ejpam-709	11	34	ideals	ideal	NOUN
ejpam-709	11	35	of	of	ADP
ejpam-709	11	36	r	r	NOUN
ejpam-709	11	37	forms	form	VERB
ejpam-709	11	38	a	a	DET
ejpam-709	11	39	distributive	distributive	ADJ
ejpam-709	11	40	lattice	lattice	NOUN
ejpam-709	11	41	.	.	PUNCT
ejpam-709	12	1	this	this	PRON
ejpam-709	12	2	enables	enable	VERB
ejpam-709	12	3	us	we	PRON
ejpam-709	12	4	to	to	PART
ejpam-709	12	5	extend	extend	VERB
ejpam-709	12	6	many	many	ADJ
ejpam-709	12	7	existing	exist	VERB
ejpam-709	12	8	concepts	concept	NOUN
ejpam-709	12	9	from	from	ADP
ejpam-709	12	10	the	the	DET
ejpam-709	12	11	class	class	NOUN
ejpam-709	12	12	of	of	ADP
ejpam-709	12	13	distributive	distributive	ADJ
ejpam-709	12	14	lattices	lattice	NOUN
ejpam-709	12	15	to	to	ADP
ejpam-709	12	16	the	the	DET
ejpam-709	12	17	class	class	NOUN
ejpam-709	12	18	of	of	ADP
ejpam-709	12	19	adls	adls	PROPN
ejpam-709	12	20	.	.	PUNCT
ejpam-709	13	1	in	in	ADP
ejpam-709	13	2	our	our	PRON
ejpam-709	13	3	paper	paper	NOUN
ejpam-709	13	4	[	[	X
ejpam-709	13	5	5	5	NUM
ejpam-709	13	6	]	]	PUNCT
ejpam-709	13	7	,	,	PUNCT
ejpam-709	13	8	we	we	PRON
ejpam-709	13	9	introduced	introduce	VERB
ejpam-709	13	10	the	the	DET
ejpam-709	13	11	concept	concept	NOUN
ejpam-709	13	12	of	of	ADP
ejpam-709	13	13	an	an	DET
ejpam-709	13	14	s−normal	s−normal	PROPN
ejpam-709	13	15	adl	adl	NOUN
ejpam-709	13	16	r	r	NOUN
ejpam-709	13	17	,	,	PUNCT
ejpam-709	13	18	where	where	SCONJ
ejpam-709	13	19	s	s	NOUN
ejpam-709	13	20	is	be	AUX
ejpam-709	13	21	a	a	DET
ejpam-709	13	22	uni	uni	ADJ
ejpam-709	13	23	subadl	subadl	NOUN
ejpam-709	13	24	of	of	ADP
ejpam-709	13	25	r	r	NOUN
ejpam-709	13	26	and	and	CCONJ
ejpam-709	13	27	obtained	obtain	VERB
ejpam-709	13	28	necessary	necessary	ADJ
ejpam-709	13	29	and	and	CCONJ
ejpam-709	13	30	sufficient	sufficient	ADJ
ejpam-709	13	31	conditions	condition	NOUN
ejpam-709	13	32	for	for	ADP
ejpam-709	13	33	an	an	DET
ejpam-709	13	34	adl	adl	NOUN
ejpam-709	13	35	r	r	NOUN
ejpam-709	13	36	to	to	PART
ejpam-709	13	37	become	become	VERB
ejpam-709	13	38	an	an	DET
ejpam-709	13	39	s−normal	s−normal	ADJ
ejpam-709	13	40	adl	adl	NOUN
ejpam-709	13	41	in	in	ADP
ejpam-709	13	42	terms	term	NOUN
ejpam-709	13	43	of	of	ADP
ejpam-709	13	44	s−prime	s−prime	NOUN
ejpam-709	13	45	filters	filter	NOUN
ejpam-709	13	46	,	,	PUNCT
ejpam-709	13	47	s−maximal	s−maximal	NOUN
ejpam-709	13	48	filters	filter	NOUN
ejpam-709	13	49	.	.	PUNCT
ejpam-709	14	1	b−normal	b−normal	PROPN
ejpam-709	15	1	adls	adls	NOUN
ejpam-709	15	2	were	be	AUX
ejpam-709	15	3	also	also	ADV
ejpam-709	15	4	studied	study	VERB
ejpam-709	15	5	,	,	PUNCT
ejpam-709	15	6	where	where	SCONJ
ejpam-709	15	7	b	b	NOUN
ejpam-709	15	8	is	be	AUX
ejpam-709	15	9	the	the	DET
ejpam-709	15	10	birkhoff	birkhoff	NOUN
ejpam-709	15	11	centre	centre	PROPN
ejpam-709	15	12	of	of	ADP
ejpam-709	15	13	r.	r.	PROPN
ejpam-709	15	14	in	in	ADP
ejpam-709	15	15	this	this	DET
ejpam-709	15	16	paper	paper	NOUN
ejpam-709	15	17	,	,	PUNCT
ejpam-709	15	18	we	we	PRON
ejpam-709	15	19	define	define	VERB
ejpam-709	15	20	the	the	DET
ejpam-709	15	21	concept	concept	NOUN
ejpam-709	15	22	of	of	ADP
ejpam-709	15	23	an	an	DET
ejpam-709	15	24	s−relative	s−relative	ADJ
ejpam-709	15	25	annihilator	annihilator	NOUN
ejpam-709	15	26	of	of	ADP
ejpam-709	15	27	any	any	DET
ejpam-709	15	28	two	two	NUM
ejpam-709	15	29	elements	element	NOUN
ejpam-709	15	30	of	of	ADP
ejpam-709	15	31	r	r	NOUN
ejpam-709	15	32	and	and	CCONJ
ejpam-709	15	33	characterize	characterize	VERB
ejpam-709	15	34	an	an	DET
ejpam-709	15	35	s−normal	s−normal	ADJ
ejpam-709	15	36	adl	adl	NOUN
ejpam-709	15	37	in	in	ADP
ejpam-709	15	38	terms	term	NOUN
ejpam-709	15	39	of	of	ADP
ejpam-709	15	40	s−relative	s−relative	ADJ
ejpam-709	15	41	annihilators	annihilator	NOUN
ejpam-709	15	42	.	.	PUNCT
ejpam-709	16	1	∗corresponding	∗corresponde	VERB
ejpam-709	16	2	author	author	NOUN
ejpam-709	16	3	.	.	PUNCT
ejpam-709	17	1	email	email	NOUN
ejpam-709	17	2	addresses	address	NOUN
ejpam-709	17	3	:	:	PUNCT
ejpam-709	17	4	g	g	PROPN
ejpam-709	17	5	raomaths	raomaths	PROPN
ejpam-709	17	6	�	�	PROPN
ejpam-709	17	7	yahoo	yahoo	PROPN
ejpam-709	17	8	.	.	PUNCT
ejpam-709	18	1	o.in	o.in	PROPN
ejpam-709	18	2	(	(	PUNCT
ejpam-709	18	3	g.	g.	PROPN
ejpam-709	18	4	rao	rao	PROPN
ejpam-709	18	5	)	)	PUNCT
ejpam-709	18	6	,	,	PUNCT
ejpam-709	18	7	rafimaths	rafimath	NOUN
ejpam-709	18	8	�	�	NOUN
ejpam-709	18	9	gmail	gmail	NOUN
ejpam-709	18	10	.	.	PUNCT
ejpam-709	19	1	om	om	PROPN
ejpam-709	19	2	(	(	PUNCT
ejpam-709	19	3	n.	n.	PROPN
ejpam-709	19	4	rafi),ravimaths83	rafi),ravimaths83	NOUN
ejpam-709	19	5	�	�	NOUN
ejpam-709	19	6	gmail	gmail	NOUN
ejpam-709	19	7	.	.	PUNCT
ejpam-709	20	1	om	om	PROPN
ejpam-709	20	2	(	(	PUNCT
ejpam-709	20	3	b.	b.	PROPN
ejpam-709	20	4	kumar	kumar	PROPN
ejpam-709	20	5	)	)	PUNCT
ejpam-709	20	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-709	21	1	704	704	NUM
ejpam-709	21	2	c	c	NOUN
ejpam-709	21	3	©	©	PROPN
ejpam-709	21	4	2010	2010	NUM
ejpam-709	21	5	ejpam	ejpam	NOUN
ejpam-709	21	6	all	all	DET
ejpam-709	21	7	rights	right	NOUN
ejpam-709	21	8	reserved	reserve	VERB
ejpam-709	21	9	.	.	PUNCT
ejpam-709	22	1	g.	g.	PROPN
ejpam-709	22	2	rao	rao	PROPN
ejpam-709	22	3	,	,	PUNCT
ejpam-709	22	4	n.	n.	PROPN
ejpam-709	22	5	rafi	rafi	PROPN
ejpam-709	22	6	and	and	CCONJ
ejpam-709	22	7	b.	b.	PROPN
ejpam-709	22	8	kumar	kumar	PROPN
ejpam-709	22	9	/	/	SYM
ejpam-709	22	10	eur	eur	PROPN
ejpam-709	22	11	.	.	PUNCT
ejpam-709	23	1	j.	j.	PROPN
ejpam-709	23	2	pure	pure	PROPN
ejpam-709	23	3	appl	appl	PROPN
ejpam-709	23	4	.	.	PROPN
ejpam-709	23	5	math	math	PROPN
ejpam-709	23	6	,	,	PUNCT
ejpam-709	23	7	3	3	NUM
ejpam-709	23	8	(	(	PUNCT
ejpam-709	23	9	2010	2010	NUM
ejpam-709	23	10	)	)	PUNCT
ejpam-709	23	11	,	,	PUNCT
ejpam-709	23	12	704	704	NUM
ejpam-709	23	13	-	-	SYM
ejpam-709	23	14	716	716	NUM
ejpam-709	23	15	705	705	NUM
ejpam-709	23	16	we	we	PRON
ejpam-709	23	17	introduce	introduce	VERB
ejpam-709	23	18	the	the	DET
ejpam-709	23	19	concepts	concept	NOUN
ejpam-709	23	20	of	of	ADP
ejpam-709	23	21	s−relatively	s−relatively	ADV
ejpam-709	23	22	normal	normal	ADJ
ejpam-709	23	23	adl	adl	NOUN
ejpam-709	23	24	and	and	CCONJ
ejpam-709	23	25	dually	dually	ADV
ejpam-709	23	26	s−relatively	s−relatively	ADV
ejpam-709	23	27	normal	normal	ADJ
ejpam-709	23	28	adl	adl	NOUN
ejpam-709	23	29	.	.	PUNCT
ejpam-709	24	1	we	we	PRON
ejpam-709	24	2	characterize	characterize	VERB
ejpam-709	24	3	the	the	DET
ejpam-709	24	4	(	(	PUNCT
ejpam-709	24	5	dually)s−relatively	dually)s−relatively	ADV
ejpam-709	24	6	normal	normal	ADJ
ejpam-709	24	7	adl	adl	NOUN
ejpam-709	24	8	in	in	ADP
ejpam-709	24	9	terms	term	NOUN
ejpam-709	24	10	of	of	ADP
ejpam-709	24	11	s−prime	s−prime	NOUN
ejpam-709	24	12	filters(ideals	filters(ideal	NOUN
ejpam-709	24	13	)	)	PUNCT
ejpam-709	24	14	.	.	PUNCT
ejpam-709	25	1	if	if	SCONJ
ejpam-709	25	2	b	b	PROPN
ejpam-709	25	3	is	be	AUX
ejpam-709	25	4	the	the	DET
ejpam-709	25	5	birkhoff	birkhoff	NOUN
ejpam-709	25	6	centre	centre	NOUN
ejpam-709	25	7	of	of	ADP
ejpam-709	25	8	r	r	NOUN
ejpam-709	25	9	,	,	PUNCT
ejpam-709	25	10	then	then	ADV
ejpam-709	25	11	we	we	PRON
ejpam-709	25	12	define	define	VERB
ejpam-709	25	13	the	the	DET
ejpam-709	25	14	concept	concept	NOUN
ejpam-709	25	15	of	of	ADP
ejpam-709	25	16	(	(	PUNCT
ejpam-709	25	17	dually)b−relatively	dually)b−relatively	ADV
ejpam-709	25	18	normal	normal	ADJ
ejpam-709	25	19	adl	adl	NOUN
ejpam-709	25	20	and	and	CCONJ
ejpam-709	25	21	characterize	characterize	VERB
ejpam-709	25	22	it	it	PRON
ejpam-709	25	23	in	in	ADP
ejpam-709	25	24	terms	term	NOUN
ejpam-709	25	25	of	of	ADP
ejpam-709	25	26	minimal	minimal	ADJ
ejpam-709	25	27	prime	prime	ADJ
ejpam-709	25	28	ideals(filters	ideals(filter	NOUN
ejpam-709	25	29	)	)	PUNCT
ejpam-709	25	30	and	and	CCONJ
ejpam-709	25	31	b−maximal	b−maximal	X
ejpam-709	25	32	ideals(filters)of	ideals(filters)of	PROPN
ejpam-709	25	33	r.	r.	PROPN
ejpam-709	25	34	2	2	NUM
ejpam-709	25	35	.	.	PUNCT
ejpam-709	25	36	preliminaries	preliminary	NOUN
ejpam-709	25	37	definition	definition	NOUN
ejpam-709	25	38	1	1	NUM
ejpam-709	25	39	(	(	PUNCT
ejpam-709	25	40	[	[	X
ejpam-709	25	41	9	9	NUM
ejpam-709	25	42	]	]	NUM
ejpam-709	25	43	)	)	PUNCT
ejpam-709	25	44	.	.	PUNCT
ejpam-709	26	1	an	an	DET
ejpam-709	26	2	almost	almost	ADV
ejpam-709	26	3	distributive	distributive	ADJ
ejpam-709	26	4	lattice	lattice	NOUN
ejpam-709	26	5	with	with	ADP
ejpam-709	26	6	zero	zero	NUM
ejpam-709	26	7	or	or	CCONJ
ejpam-709	26	8	simply	simply	ADV
ejpam-709	26	9	adl	adl	PROPN
ejpam-709	26	10	is	be	AUX
ejpam-709	26	11	an	an	DET
ejpam-709	26	12	algebra	algebra	NOUN
ejpam-709	26	13	(	(	PUNCT
ejpam-709	26	14	r,∨,∧	r,∨,∧	NUM
ejpam-709	26	15	,	,	PUNCT
ejpam-709	26	16	0	0	NUM
ejpam-709	26	17	)	)	PUNCT
ejpam-709	26	18	of	of	ADP
ejpam-709	26	19	type	type	NOUN
ejpam-709	26	20	(	(	PUNCT
ejpam-709	26	21	2,2,0	2,2,0	NOUN
ejpam-709	26	22	)	)	PUNCT
ejpam-709	26	23	satisfying	satisfying	NOUN
ejpam-709	26	24	:	:	PUNCT
ejpam-709	26	25	1	1	X
ejpam-709	26	26	.	.	X
ejpam-709	26	27	(	(	PUNCT
ejpam-709	27	1	x	x	PROPN
ejpam-709	27	2	∨	∨	NUM
ejpam-709	27	3	y)∧	y)∧	PROPN
ejpam-709	27	4	z	z	NOUN
ejpam-709	27	5	=	=	SYM
ejpam-709	27	6	(	(	PUNCT
ejpam-709	27	7	x	x	PART
ejpam-709	27	8	∧	∧	PROPN
ejpam-709	27	9	z)∨	z)∨	PROPN
ejpam-709	27	10	(	(	PUNCT
ejpam-709	27	11	y	y	PROPN
ejpam-709	27	12	∧	∧	PROPN
ejpam-709	27	13	z	z	PROPN
ejpam-709	27	14	)	)	PUNCT
ejpam-709	27	15	2	2	NUM
ejpam-709	27	16	.	.	X
ejpam-709	27	17	x	x	X
ejpam-709	28	1	∧	∧	PROPN
ejpam-709	28	2	(	(	PUNCT
ejpam-709	28	3	y	y	PROPN
ejpam-709	28	4	∨	∨	PROPN
ejpam-709	28	5	z	z	PROPN
ejpam-709	28	6	)	)	PUNCT
ejpam-709	28	7	=	=	SYM
ejpam-709	28	8	(	(	PUNCT
ejpam-709	28	9	x	x	PUNCT
ejpam-709	28	10	∧	∧	PROPN
ejpam-709	28	11	y)∨	y)∨	PROPN
ejpam-709	28	12	(	(	PUNCT
ejpam-709	28	13	x	x	PART
ejpam-709	28	14	∧	∧	PROPN
ejpam-709	28	15	z	z	PROPN
ejpam-709	28	16	)	)	PUNCT
ejpam-709	28	17	3	3	NUM
ejpam-709	28	18	.	.	PUNCT
ejpam-709	28	19	(	(	PUNCT
ejpam-709	29	1	x	x	PROPN
ejpam-709	29	2	∨	∨	NUM
ejpam-709	29	3	y)∧	y)∧	NOUN
ejpam-709	29	4	y	y	PROPN
ejpam-709	29	5	=	=	SYM
ejpam-709	29	6	y	y	PROPN
ejpam-709	29	7	4	4	NUM
ejpam-709	29	8	.	.	PUNCT
ejpam-709	30	1	(	(	PUNCT
ejpam-709	30	2	x	x	PROPN
ejpam-709	30	3	∨	∨	NUM
ejpam-709	30	4	y)∧	y)∧	NOUN
ejpam-709	30	5	x	x	SYM
ejpam-709	30	6	=	=	PUNCT
ejpam-709	30	7	x	x	SYM
ejpam-709	30	8	5	5	X
ejpam-709	30	9	.	.	NUM
ejpam-709	30	10	x	x	PUNCT
ejpam-709	30	11	∨	∨	NOUN
ejpam-709	30	12	(	(	PUNCT
ejpam-709	30	13	x	x	PROPN
ejpam-709	30	14	∧	∧	PROPN
ejpam-709	30	15	y	y	NOUN
ejpam-709	30	16	)	)	PUNCT
ejpam-709	30	17	=	=	PUNCT
ejpam-709	31	1	x	x	SYM
ejpam-709	31	2	6	6	NUM
ejpam-709	31	3	.	.	NUM
ejpam-709	31	4	0∧	0∧	NOUN
ejpam-709	31	5	x	x	NOUN
ejpam-709	32	1	=	=	NOUN
ejpam-709	32	2	0	0	NUM
ejpam-709	32	3	7	7	NUM
ejpam-709	32	4	.	.	PUNCT
ejpam-709	32	5	x	x	PUNCT
ejpam-709	33	1	∨	∨	NUM
ejpam-709	33	2	0=	0=	NOUN
ejpam-709	34	1	x	x	X
ejpam-709	34	2	.	.	PUNCT
ejpam-709	35	1	every	every	DET
ejpam-709	35	2	non	non	ADJ
ejpam-709	35	3	-	-	ADJ
ejpam-709	35	4	empty	empty	ADJ
ejpam-709	35	5	set	set	NOUN
ejpam-709	35	6	x	x	PUNCT
ejpam-709	35	7	can	can	AUX
ejpam-709	35	8	be	be	AUX
ejpam-709	35	9	regarded	regard	VERB
ejpam-709	35	10	as	as	ADP
ejpam-709	35	11	an	an	DET
ejpam-709	35	12	adl	adl	NOUN
ejpam-709	35	13	as	as	SCONJ
ejpam-709	35	14	follows	follow	VERB
ejpam-709	35	15	.	.	PUNCT
ejpam-709	36	1	let	let	VERB
ejpam-709	36	2	x0	x0	PROPN
ejpam-709	36	3	∈	∈	PROPN
ejpam-709	37	1	x	x	X
ejpam-709	37	2	.	.	PUNCT
ejpam-709	38	1	define	define	VERB
ejpam-709	38	2	the	the	DET
ejpam-709	38	3	binary	binary	PROPN
ejpam-709	38	4	operations	operation	NOUN
ejpam-709	38	5	∨,∧	∨,∧	NOUN
ejpam-709	38	6	on	on	ADP
ejpam-709	38	7	x	x	PUNCT
ejpam-709	38	8	by	by	ADP
ejpam-709	38	9	x	x	PROPN
ejpam-709	38	10	∨	∨	NUM
ejpam-709	38	11	y	y	NOUN
ejpam-709	38	12	=	=	PUNCT
ejpam-709	39	1	(	(	PUNCT
ejpam-709	39	2	x	x	X
ejpam-709	39	3	if	if	SCONJ
ejpam-709	39	4	x	x	PRON
ejpam-709	39	5	6=	6=	NUM
ejpam-709	39	6	x0	x0	PROPN
ejpam-709	39	7	y	y	PROPN
ejpam-709	39	8	if	if	SCONJ
ejpam-709	39	9	x	x	PRON
ejpam-709	39	10	=	=	SYM
ejpam-709	39	11	x0	x0	PROPN
ejpam-709	39	12	x	x	PUNCT
ejpam-709	39	13	∧	∧	NOUN
ejpam-709	39	14	y	y	NOUN
ejpam-709	39	15	=	=	PRON
ejpam-709	39	16	(	(	PUNCT
ejpam-709	39	17	y	y	NOUN
ejpam-709	39	18	if	if	SCONJ
ejpam-709	39	19	x	x	PRON
ejpam-709	39	20	6=	6=	NUM
ejpam-709	39	21	x0	x0	PROPN
ejpam-709	39	22	x0	x0	PROPN
ejpam-709	40	1	if	if	SCONJ
ejpam-709	40	2	x	x	X
ejpam-709	40	3	=	=	SYM
ejpam-709	40	4	x0	x0	PROPN
ejpam-709	40	5	.	.	PUNCT
ejpam-709	41	1	then	then	ADV
ejpam-709	41	2	(	(	PUNCT
ejpam-709	41	3	x	x	X
ejpam-709	41	4	,	,	PUNCT
ejpam-709	41	5	∨,∧	∨,∧	PROPN
ejpam-709	41	6	,	,	PUNCT
ejpam-709	41	7	x0	x0	PROPN
ejpam-709	41	8	)	)	PUNCT
ejpam-709	41	9	is	be	AUX
ejpam-709	41	10	an	an	DET
ejpam-709	41	11	adl	adl	NOUN
ejpam-709	41	12	(	(	PUNCT
ejpam-709	41	13	where	where	SCONJ
ejpam-709	41	14	x0	x0	PROPN
ejpam-709	41	15	is	be	AUX
ejpam-709	41	16	the	the	DET
ejpam-709	41	17	zero	zero	NUM
ejpam-709	41	18	)	)	PUNCT
ejpam-709	41	19	and	and	CCONJ
ejpam-709	41	20	is	be	AUX
ejpam-709	41	21	called	call	VERB
ejpam-709	41	22	a	a	DET
ejpam-709	41	23	discrete	discrete	ADJ
ejpam-709	41	24	adl	adl	NOUN
ejpam-709	41	25	.	.	PUNCT
ejpam-709	42	1	if	if	SCONJ
ejpam-709	42	2	(	(	PUNCT
ejpam-709	42	3	r,∨,∧	r,∨,∧	NUM
ejpam-709	42	4	,	,	PUNCT
ejpam-709	42	5	0	0	NUM
ejpam-709	42	6	)	)	PUNCT
ejpam-709	42	7	is	be	AUX
ejpam-709	42	8	an	an	DET
ejpam-709	42	9	adl	adl	NOUN
ejpam-709	42	10	,	,	PUNCT
ejpam-709	42	11	for	for	ADP
ejpam-709	42	12	any	any	DET
ejpam-709	42	13	a	a	NOUN
ejpam-709	42	14	,	,	PUNCT
ejpam-709	42	15	b	b	X
ejpam-709	42	16	∈	∈	PROPN
ejpam-709	42	17	r	r	NOUN
ejpam-709	42	18	,	,	PUNCT
ejpam-709	42	19	define	define	VERB
ejpam-709	42	20	a	a	DET
ejpam-709	42	21	≤	≤	NUM
ejpam-709	42	22	b	b	NOUN
ejpam-709	43	1	if	if	SCONJ
ejpam-709	44	1	and	and	CCONJ
ejpam-709	44	2	only	only	ADV
ejpam-709	44	3	if	if	SCONJ
ejpam-709	44	4	a	a	PRON
ejpam-709	44	5	=	=	X
ejpam-709	44	6	a	a	DET
ejpam-709	44	7	∧	∧	PROPN
ejpam-709	44	8	b	b	PROPN
ejpam-709	44	9	(	(	PUNCT
ejpam-709	44	10	or	or	CCONJ
ejpam-709	44	11	equivalently	equivalently	ADV
ejpam-709	44	12	,	,	PUNCT
ejpam-709	44	13	a	a	DET
ejpam-709	44	14	∨	∨	NUM
ejpam-709	44	15	b	b	NOUN
ejpam-709	44	16	=	=	SYM
ejpam-709	44	17	b	b	NOUN
ejpam-709	44	18	)	)	PUNCT
ejpam-709	44	19	,	,	PUNCT
ejpam-709	44	20	then	then	ADV
ejpam-709	44	21	≤	≤	PROPN
ejpam-709	44	22	is	be	AUX
ejpam-709	44	23	a	a	DET
ejpam-709	44	24	partial	partial	ADJ
ejpam-709	44	25	ordering	ordering	NOUN
ejpam-709	44	26	on	on	ADP
ejpam-709	44	27	r.	r.	PROPN
ejpam-709	44	28	theorem	theorem	NOUN
ejpam-709	44	29	1	1	NUM
ejpam-709	44	30	(	(	PUNCT
ejpam-709	44	31	[	[	X
ejpam-709	44	32	9	9	NUM
ejpam-709	44	33	]	]	PUNCT
ejpam-709	44	34	)	)	PUNCT
ejpam-709	44	35	.	.	PUNCT
ejpam-709	45	1	if	if	SCONJ
ejpam-709	45	2	(	(	PUNCT
ejpam-709	45	3	r,∨,∧	r,∨,∧	NUM
ejpam-709	45	4	,	,	PUNCT
ejpam-709	45	5	0	0	NUM
ejpam-709	45	6	)	)	PUNCT
ejpam-709	45	7	is	be	AUX
ejpam-709	45	8	an	an	DET
ejpam-709	45	9	adl	adl	NOUN
ejpam-709	45	10	,	,	PUNCT
ejpam-709	45	11	for	for	ADP
ejpam-709	45	12	any	any	DET
ejpam-709	45	13	a	a	DET
ejpam-709	45	14	,	,	PUNCT
ejpam-709	45	15	b	b	NOUN
ejpam-709	45	16	,	,	PUNCT
ejpam-709	45	17	c	c	PROPN
ejpam-709	45	18	∈	∈	PROPN
ejpam-709	45	19	r	r	NOUN
ejpam-709	45	20	,	,	PUNCT
ejpam-709	45	21	we	we	PRON
ejpam-709	45	22	have	have	VERB
ejpam-709	45	23	the	the	DET
ejpam-709	45	24	following	following	NOUN
ejpam-709	45	25	:	:	PUNCT
ejpam-709	45	26	1	1	X
ejpam-709	45	27	.	.	X
ejpam-709	45	28	a	a	DET
ejpam-709	45	29	∨	∨	NUM
ejpam-709	45	30	b	b	NOUN
ejpam-709	45	31	=	=	NOUN
ejpam-709	45	32	a⇔	a⇔	NOUN
ejpam-709	45	33	a	a	DET
ejpam-709	45	34	∧	∧	PROPN
ejpam-709	45	35	b	b	PROPN
ejpam-709	45	36	=	=	SYM
ejpam-709	45	37	b	b	PROPN
ejpam-709	45	38	2	2	NUM
ejpam-709	45	39	.	.	PUNCT
ejpam-709	45	40	a	a	DET
ejpam-709	45	41	∨	∨	PROPN
ejpam-709	45	42	b	b	NOUN
ejpam-709	45	43	=	=	NOUN
ejpam-709	45	44	b⇔	b⇔	PROPN
ejpam-709	45	45	a	a	DET
ejpam-709	45	46	∧	∧	PROPN
ejpam-709	45	47	b	b	PROPN
ejpam-709	45	48	=	=	PUNCT
ejpam-709	45	49	a	a	DET
ejpam-709	45	50	3	3	NUM
ejpam-709	45	51	.	.	X
ejpam-709	46	1	∧	∧	NOUN
ejpam-709	46	2	is	be	AUX
ejpam-709	46	3	associative	associative	ADJ
ejpam-709	46	4	in	in	ADP
ejpam-709	46	5	r	r	NOUN
ejpam-709	46	6	4	4	NUM
ejpam-709	46	7	.	.	PUNCT
ejpam-709	47	1	a	a	DET
ejpam-709	47	2	∧	∧	PROPN
ejpam-709	47	3	b	b	PROPN
ejpam-709	47	4	∧	∧	PROPN
ejpam-709	47	5	c	c	NOUN
ejpam-709	47	6	=	=	SYM
ejpam-709	47	7	b	b	PROPN
ejpam-709	47	8	∧	∧	PROPN
ejpam-709	47	9	a	a	DET
ejpam-709	47	10	∧	∧	PROPN
ejpam-709	47	11	c	c	NOUN
ejpam-709	47	12	5	5	NUM
ejpam-709	47	13	.	.	PUNCT
ejpam-709	48	1	(	(	PUNCT
ejpam-709	48	2	a	a	DET
ejpam-709	48	3	∨	∨	NUM
ejpam-709	48	4	b)∧	b)∧	PROPN
ejpam-709	48	5	c	c	NOUN
ejpam-709	48	6	=	=	SYM
ejpam-709	48	7	(	(	PUNCT
ejpam-709	48	8	b	b	PROPN
ejpam-709	48	9	∨	∨	NUM
ejpam-709	48	10	a)∧	a)∧	PROPN
ejpam-709	48	11	c	c	PROPN
ejpam-709	48	12	6	6	NUM
ejpam-709	48	13	.	.	PUNCT
ejpam-709	49	1	a	a	DET
ejpam-709	49	2	∧	∧	PROPN
ejpam-709	49	3	b	b	NOUN
ejpam-709	49	4	=	=	SYM
ejpam-709	49	5	0⇔	0⇔	NOUN
ejpam-709	49	6	b	b	X
ejpam-709	49	7	∧	∧	NOUN
ejpam-709	49	8	a	a	DET
ejpam-709	49	9	=	=	SYM
ejpam-709	49	10	0	0	PROPN
ejpam-709	49	11	g.	g.	PROPN
ejpam-709	49	12	rao	rao	PROPN
ejpam-709	49	13	,	,	PUNCT
ejpam-709	49	14	n.	n.	PROPN
ejpam-709	49	15	rafi	rafi	PROPN
ejpam-709	49	16	and	and	CCONJ
ejpam-709	49	17	b.	b.	PROPN
ejpam-709	49	18	kumar	kumar	PROPN
ejpam-709	49	19	/	/	SYM
ejpam-709	49	20	eur	eur	PROPN
ejpam-709	49	21	.	.	PUNCT
ejpam-709	50	1	j.	j.	PROPN
ejpam-709	50	2	pure	pure	PROPN
ejpam-709	50	3	appl	appl	PROPN
ejpam-709	50	4	.	.	PROPN
ejpam-709	50	5	math	math	PROPN
ejpam-709	50	6	,	,	PUNCT
ejpam-709	50	7	3	3	NUM
ejpam-709	50	8	(	(	PUNCT
ejpam-709	50	9	2010	2010	NUM
ejpam-709	50	10	)	)	PUNCT
ejpam-709	50	11	,	,	PUNCT
ejpam-709	50	12	704	704	NUM
ejpam-709	50	13	-	-	SYM
ejpam-709	50	14	716	716	NUM
ejpam-709	50	15	706	706	NUM
ejpam-709	50	16	7	7	NUM
ejpam-709	50	17	.	.	PUNCT
ejpam-709	51	1	a	a	DET
ejpam-709	51	2	∨	∨	NOUN
ejpam-709	51	3	(	(	PUNCT
ejpam-709	51	4	b	b	PROPN
ejpam-709	51	5	∧	∧	PROPN
ejpam-709	51	6	c	c	NOUN
ejpam-709	51	7	)	)	PUNCT
ejpam-709	51	8	=	=	NOUN
ejpam-709	51	9	(	(	PUNCT
ejpam-709	51	10	a	a	DET
ejpam-709	51	11	∨	∨	NUM
ejpam-709	51	12	b)∧	b)∧	PROPN
ejpam-709	51	13	(	(	PUNCT
ejpam-709	51	14	a	a	DET
ejpam-709	51	15	∨	∨	NUM
ejpam-709	51	16	c	c	NOUN
ejpam-709	51	17	)	)	PUNCT
ejpam-709	51	18	8	8	NUM
ejpam-709	51	19	.	.	PUNCT
ejpam-709	52	1	a	a	DET
ejpam-709	52	2	∧	∧	PROPN
ejpam-709	52	3	(	(	PUNCT
ejpam-709	52	4	a	a	DET
ejpam-709	52	5	∨	∨	NUM
ejpam-709	52	6	b	b	NOUN
ejpam-709	52	7	)	)	PUNCT
ejpam-709	52	8	=	=	SYM
ejpam-709	52	9	a	a	PRON
ejpam-709	52	10	,	,	PUNCT
ejpam-709	52	11	(	(	PUNCT
ejpam-709	52	12	a	a	DET
ejpam-709	52	13	∧	∧	PROPN
ejpam-709	52	14	b)∨	b)∨	PROPN
ejpam-709	52	15	b	b	PROPN
ejpam-709	52	16	=	=	SYM
ejpam-709	52	17	b	b	PROPN
ejpam-709	52	18	and	and	CCONJ
ejpam-709	52	19	a	a	DET
ejpam-709	52	20	∨	∨	NOUN
ejpam-709	52	21	(	(	PUNCT
ejpam-709	52	22	b	b	PROPN
ejpam-709	52	23	∧	∧	PROPN
ejpam-709	52	24	a	a	NOUN
ejpam-709	52	25	)	)	PUNCT
ejpam-709	52	26	=	=	PUNCT
ejpam-709	52	27	a	a	DET
ejpam-709	52	28	9	9	NUM
ejpam-709	52	29	.	.	PUNCT
ejpam-709	53	1	a	a	DET
ejpam-709	53	2	≤	≤	NUM
ejpam-709	53	3	a	a	DET
ejpam-709	53	4	∨	∨	NOUN
ejpam-709	53	5	b	b	NOUN
ejpam-709	53	6	and	and	CCONJ
ejpam-709	53	7	a	a	DET
ejpam-709	53	8	∧	∧	PROPN
ejpam-709	53	9	b	b	PROPN
ejpam-709	53	10	≤	≤	NUM
ejpam-709	53	11	b	b	PROPN
ejpam-709	53	12	10	10	NUM
ejpam-709	53	13	.	.	PUNCT
ejpam-709	54	1	a	a	DET
ejpam-709	54	2	∧	∧	PROPN
ejpam-709	54	3	a	a	DET
ejpam-709	54	4	=	=	PUNCT
ejpam-709	54	5	a	a	NOUN
ejpam-709	54	6	and	and	CCONJ
ejpam-709	54	7	a	a	DET
ejpam-709	54	8	∨	∨	NOUN
ejpam-709	54	9	a	a	DET
ejpam-709	54	10	=	=	NOUN
ejpam-709	54	11	a	a	DET
ejpam-709	54	12	11	11	NUM
ejpam-709	54	13	.	.	PUNCT
ejpam-709	55	1	0∨	0∨	NOUN
ejpam-709	55	2	a	a	DET
ejpam-709	55	3	=	=	X
ejpam-709	55	4	a	a	NOUN
ejpam-709	55	5	and	and	CCONJ
ejpam-709	55	6	a	a	DET
ejpam-709	55	7	∧	∧	PROPN
ejpam-709	55	8	0=	0=	NOUN
ejpam-709	55	9	0	0	NUM
ejpam-709	56	1	12	12	NUM
ejpam-709	56	2	.	.	PUNCT
ejpam-709	57	1	if	if	SCONJ
ejpam-709	57	2	a	a	DET
ejpam-709	57	3	≤	≤	NUM
ejpam-709	57	4	c	c	NOUN
ejpam-709	57	5	,	,	PUNCT
ejpam-709	57	6	b	b	PROPN
ejpam-709	57	7	≤	≤	NUM
ejpam-709	57	8	c	c	NOUN
ejpam-709	57	9	then	then	ADV
ejpam-709	57	10	a	a	DET
ejpam-709	57	11	∧	∧	PROPN
ejpam-709	57	12	b	b	PROPN
ejpam-709	57	13	=	=	SYM
ejpam-709	57	14	b	b	PROPN
ejpam-709	57	15	∧	∧	PROPN
ejpam-709	57	16	a	a	PRON
ejpam-709	57	17	and	and	CCONJ
ejpam-709	57	18	a	a	DET
ejpam-709	57	19	∨	∨	NUM
ejpam-709	57	20	b	b	X
ejpam-709	57	21	=	=	SYM
ejpam-709	57	22	b	b	PROPN
ejpam-709	57	23	∨	∨	NUM
ejpam-709	57	24	a	a	DET
ejpam-709	57	25	13	13	NUM
ejpam-709	57	26	.	.	PUNCT
ejpam-709	58	1	a	a	DET
ejpam-709	58	2	∨	∨	PROPN
ejpam-709	58	3	b	b	X
ejpam-709	58	4	=	=	PUNCT
ejpam-709	58	5	(	(	PUNCT
ejpam-709	58	6	a	a	DET
ejpam-709	58	7	∨	∨	PROPN
ejpam-709	58	8	b)∨	b)∨	PROPN
ejpam-709	58	9	a.	a.	NOUN
ejpam-709	58	10	it	it	PRON
ejpam-709	58	11	can	can	AUX
ejpam-709	58	12	be	be	AUX
ejpam-709	58	13	observed	observe	VERB
ejpam-709	58	14	that	that	SCONJ
ejpam-709	58	15	an	an	DET
ejpam-709	58	16	adl	adl	NOUN
ejpam-709	58	17	r	r	NOUN
ejpam-709	58	18	satisfies	satisfie	NOUN
ejpam-709	58	19	almost	almost	ADV
ejpam-709	58	20	all	all	DET
ejpam-709	58	21	the	the	DET
ejpam-709	58	22	properties	property	NOUN
ejpam-709	58	23	of	of	ADP
ejpam-709	58	24	a	a	DET
ejpam-709	58	25	distributive	distributive	ADJ
ejpam-709	58	26	lattice	lattice	NOUN
ejpam-709	58	27	except	except	SCONJ
ejpam-709	58	28	the	the	DET
ejpam-709	58	29	right	right	ADJ
ejpam-709	58	30	distributivity	distributivity	NOUN
ejpam-709	58	31	of	of	ADP
ejpam-709	58	32	∨	∨	NUM
ejpam-709	58	33	over	over	ADP
ejpam-709	58	34	∧	∧	PROPN
ejpam-709	58	35	,	,	PUNCT
ejpam-709	58	36	commutativity	commutativity	NOUN
ejpam-709	58	37	of	of	ADP
ejpam-709	58	38	∨	∨	NOUN
ejpam-709	58	39	,	,	PUNCT
ejpam-709	58	40	commutativity	commutativity	NOUN
ejpam-709	58	41	of	of	ADP
ejpam-709	58	42	∧.	∧.	PROPN
ejpam-709	58	43	any	any	DET
ejpam-709	58	44	one	one	NUM
ejpam-709	58	45	of	of	ADP
ejpam-709	58	46	these	these	DET
ejpam-709	58	47	properties	property	NOUN
ejpam-709	58	48	make	make	VERB
ejpam-709	58	49	an	an	DET
ejpam-709	58	50	adl	adl	NOUN
ejpam-709	58	51	r	r	NOUN
ejpam-709	58	52	a	a	DET
ejpam-709	58	53	distributive	distributive	ADJ
ejpam-709	58	54	lattice	lattice	NOUN
ejpam-709	58	55	.	.	PUNCT
ejpam-709	59	1	that	that	PRON
ejpam-709	59	2	is	be	AUX
ejpam-709	59	3	theorem	theorem	VERB
ejpam-709	59	4	2	2	NUM
ejpam-709	59	5	(	(	PUNCT
ejpam-709	59	6	[	[	X
ejpam-709	59	7	9	9	NUM
ejpam-709	59	8	]	]	PUNCT
ejpam-709	59	9	)	)	PUNCT
ejpam-709	59	10	.	.	PUNCT
ejpam-709	60	1	let	let	VERB
ejpam-709	60	2	(	(	PUNCT
ejpam-709	60	3	r,∨,∧	r,∨,∧	NUM
ejpam-709	60	4	,	,	PUNCT
ejpam-709	60	5	0	0	NUM
ejpam-709	60	6	)	)	PUNCT
ejpam-709	60	7	be	be	AUX
ejpam-709	60	8	an	an	DET
ejpam-709	60	9	adl	adl	NOUN
ejpam-709	60	10	with	with	ADP
ejpam-709	60	11	0	0	NUM
ejpam-709	60	12	.	.	PUNCT
ejpam-709	61	1	then	then	ADV
ejpam-709	61	2	the	the	DET
ejpam-709	61	3	following	following	NOUN
ejpam-709	61	4	are	be	AUX
ejpam-709	61	5	equivalent	equivalent	ADJ
ejpam-709	61	6	:	:	PUNCT
ejpam-709	61	7	1	1	X
ejpam-709	61	8	.	.	PUNCT
ejpam-709	61	9	(	(	PUNCT
ejpam-709	61	10	r,∨,∧	r,∨,∧	NUM
ejpam-709	61	11	,	,	PUNCT
ejpam-709	61	12	0	0	NUM
ejpam-709	61	13	)	)	PUNCT
ejpam-709	61	14	is	be	AUX
ejpam-709	61	15	a	a	DET
ejpam-709	61	16	distributive	distributive	ADJ
ejpam-709	61	17	lattice	lattice	NOUN
ejpam-709	61	18	2	2	NUM
ejpam-709	61	19	.	.	PUNCT
ejpam-709	61	20	a	a	DET
ejpam-709	61	21	∨	∨	PROPN
ejpam-709	61	22	b	b	X
ejpam-709	61	23	=	=	SYM
ejpam-709	61	24	b	b	PROPN
ejpam-709	61	25	∨	∨	NUM
ejpam-709	61	26	a	a	NOUN
ejpam-709	61	27	,	,	PUNCT
ejpam-709	61	28	for	for	ADP
ejpam-709	61	29	all	all	DET
ejpam-709	61	30	a	a	PRON
ejpam-709	61	31	,	,	PUNCT
ejpam-709	61	32	b	b	X
ejpam-709	61	33	∈	∈	PROPN
ejpam-709	61	34	r	r	NOUN
ejpam-709	61	35	3	3	NUM
ejpam-709	61	36	.	.	PUNCT
ejpam-709	62	1	a	a	DET
ejpam-709	62	2	∧	∧	PROPN
ejpam-709	62	3	b	b	PROPN
ejpam-709	62	4	=	=	SYM
ejpam-709	62	5	b	b	PROPN
ejpam-709	62	6	∧	∧	PROPN
ejpam-709	62	7	a	a	PROPN
ejpam-709	62	8	,	,	PUNCT
ejpam-709	62	9	for	for	ADP
ejpam-709	62	10	all	all	DET
ejpam-709	62	11	a	a	PRON
ejpam-709	62	12	,	,	PUNCT
ejpam-709	62	13	b	b	X
ejpam-709	62	14	∈	∈	PROPN
ejpam-709	62	15	r	r	NOUN
ejpam-709	62	16	4	4	NUM
ejpam-709	62	17	.	.	PUNCT
ejpam-709	63	1	(	(	PUNCT
ejpam-709	63	2	a	a	DET
ejpam-709	63	3	∧	∧	PROPN
ejpam-709	63	4	b)∨	b)∨	PROPN
ejpam-709	63	5	c	c	NOUN
ejpam-709	63	6	=	=	PUNCT
ejpam-709	63	7	(	(	PUNCT
ejpam-709	63	8	a	a	DET
ejpam-709	63	9	∨	∨	NUM
ejpam-709	63	10	c)∧	c)∧	PROPN
ejpam-709	63	11	(	(	PUNCT
ejpam-709	63	12	b	b	PROPN
ejpam-709	63	13	∨	∨	NUM
ejpam-709	63	14	c	c	NOUN
ejpam-709	63	15	)	)	PUNCT
ejpam-709	63	16	,	,	PUNCT
ejpam-709	63	17	for	for	ADP
ejpam-709	63	18	all	all	DET
ejpam-709	63	19	a	a	DET
ejpam-709	63	20	,	,	PUNCT
ejpam-709	63	21	b	b	NOUN
ejpam-709	63	22	,	,	PUNCT
ejpam-709	63	23	c	c	PROPN
ejpam-709	63	24	∈	∈	PROPN
ejpam-709	63	25	r.	r.	PROPN
ejpam-709	63	26	as	as	ADP
ejpam-709	63	27	usual	usual	ADJ
ejpam-709	63	28	,	,	PUNCT
ejpam-709	63	29	an	an	DET
ejpam-709	63	30	element	element	NOUN
ejpam-709	63	31	m	m	NOUN
ejpam-709	63	32	∈	∈	NOUN
ejpam-709	63	33	r	r	NOUN
ejpam-709	63	34	is	be	AUX
ejpam-709	63	35	called	call	VERB
ejpam-709	63	36	maximal	maximal	ADJ
ejpam-709	63	37	if	if	SCONJ
ejpam-709	63	38	it	it	PRON
ejpam-709	63	39	is	be	AUX
ejpam-709	63	40	a	a	DET
ejpam-709	63	41	maximal	maximal	ADJ
ejpam-709	63	42	element	element	NOUN
ejpam-709	63	43	in	in	ADP
ejpam-709	63	44	the	the	DET
ejpam-709	63	45	partially	partially	ADV
ejpam-709	63	46	ordered	order	VERB
ejpam-709	63	47	set	set	NOUN
ejpam-709	63	48	(	(	PUNCT
ejpam-709	63	49	r,≤	r,≤	PROPN
ejpam-709	63	50	)	)	PUNCT
ejpam-709	63	51	.	.	PUNCT
ejpam-709	64	1	that	that	PRON
ejpam-709	64	2	is	be	AUX
ejpam-709	64	3	,	,	PUNCT
ejpam-709	64	4	for	for	ADP
ejpam-709	64	5	any	any	DET
ejpam-709	64	6	a	a	DET
ejpam-709	64	7	∈	∈	PROPN
ejpam-709	64	8	r	r	NOUN
ejpam-709	64	9	,	,	PUNCT
ejpam-709	64	10	m≤	m≤	VERB
ejpam-709	64	11	a⇒	a⇒	NOUN
ejpam-709	64	12	m	m	NOUN
ejpam-709	64	13	=	=	NOUN
ejpam-709	64	14	a.	a.	NOUN
ejpam-709	64	15	theorem	theorem	NOUN
ejpam-709	64	16	3	3	NUM
ejpam-709	64	17	(	(	PUNCT
ejpam-709	64	18	[	[	X
ejpam-709	64	19	9	9	NUM
ejpam-709	64	20	]	]	PUNCT
ejpam-709	64	21	)	)	PUNCT
ejpam-709	64	22	.	.	PUNCT
ejpam-709	65	1	let	let	VERB
ejpam-709	65	2	r	r	PRON
ejpam-709	65	3	be	be	AUX
ejpam-709	65	4	an	an	DET
ejpam-709	65	5	adl	adl	NOUN
ejpam-709	65	6	and	and	CCONJ
ejpam-709	65	7	m	m	PROPN
ejpam-709	65	8	∈	∈	PROPN
ejpam-709	65	9	r.	r.	NOUN
ejpam-709	65	10	then	then	ADV
ejpam-709	65	11	the	the	DET
ejpam-709	65	12	following	follow	VERB
ejpam-709	65	13	are	be	AUX
ejpam-709	65	14	equivalent	equivalent	ADJ
ejpam-709	65	15	:	:	PUNCT
ejpam-709	65	16	1	1	X
ejpam-709	65	17	.	.	X
ejpam-709	65	18	m	m	PROPN
ejpam-709	65	19	is	be	AUX
ejpam-709	65	20	maximal	maximal	ADJ
ejpam-709	65	21	with	with	ADP
ejpam-709	65	22	respect	respect	NOUN
ejpam-709	65	23	to	to	ADP
ejpam-709	65	24	≤	≤	NOUN
ejpam-709	65	25	2	2	NUM
ejpam-709	65	26	.	.	PUNCT
ejpam-709	66	1	m∨	m∨	NOUN
ejpam-709	66	2	a	a	DET
ejpam-709	66	3	=	=	NOUN
ejpam-709	66	4	m	m	PROPN
ejpam-709	66	5	,	,	PUNCT
ejpam-709	66	6	for	for	ADP
ejpam-709	66	7	all	all	DET
ejpam-709	66	8	a	a	DET
ejpam-709	66	9	∈	∈	NOUN
ejpam-709	66	10	r	r	NOUN
ejpam-709	66	11	3	3	NUM
ejpam-709	66	12	.	.	PUNCT
ejpam-709	66	13	m∧	m∧	NOUN
ejpam-709	66	14	a	a	DET
ejpam-709	66	15	=	=	X
ejpam-709	66	16	a	a	NOUN
ejpam-709	66	17	,	,	PUNCT
ejpam-709	66	18	for	for	ADP
ejpam-709	66	19	all	all	DET
ejpam-709	66	20	a	a	DET
ejpam-709	66	21	∈	∈	NOUN
ejpam-709	66	22	r	r	NOUN
ejpam-709	66	23	4	4	NUM
ejpam-709	66	24	.	.	PUNCT
ejpam-709	67	1	a	a	DET
ejpam-709	67	2	∨m	∨m	NOUN
ejpam-709	67	3	is	be	AUX
ejpam-709	67	4	maximal	maximal	ADJ
ejpam-709	67	5	,	,	PUNCT
ejpam-709	67	6	for	for	ADP
ejpam-709	67	7	all	all	DET
ejpam-709	67	8	a	a	DET
ejpam-709	67	9	∈	∈	PROPN
ejpam-709	67	10	r.	r.	NOUN
ejpam-709	67	11	as	as	ADP
ejpam-709	67	12	in	in	ADP
ejpam-709	67	13	distributive	distributive	ADJ
ejpam-709	67	14	lattices	lattice	NOUN
ejpam-709	67	15	[	[	X
ejpam-709	67	16	1	1	NUM
ejpam-709	67	17	,	,	PUNCT
ejpam-709	67	18	3	3	NUM
ejpam-709	67	19	]	]	PUNCT
ejpam-709	67	20	,	,	PUNCT
ejpam-709	67	21	a	a	DET
ejpam-709	67	22	non	non	ADJ
ejpam-709	67	23	-	-	ADJ
ejpam-709	67	24	empty	empty	ADJ
ejpam-709	67	25	sub	sub	NOUN
ejpam-709	67	26	set	set	NOUN
ejpam-709	67	27	i	i	PRON
ejpam-709	67	28	of	of	ADP
ejpam-709	67	29	an	an	DET
ejpam-709	67	30	adl	adl	PROPN
ejpam-709	67	31	r	r	NOUN
ejpam-709	67	32	is	be	AUX
ejpam-709	67	33	called	call	VERB
ejpam-709	67	34	an	an	DET
ejpam-709	67	35	ideal	ideal	NOUN
ejpam-709	67	36	of	of	ADP
ejpam-709	67	37	r	r	NOUN
ejpam-709	67	38	if	if	SCONJ
ejpam-709	67	39	a∨	a∨	PROPN
ejpam-709	67	40	b	b	PROPN
ejpam-709	67	41	∈	∈	PROPN
ejpam-709	68	1	i	i	PRON
ejpam-709	68	2	and	and	CCONJ
ejpam-709	68	3	a∧	a∧	NOUN
ejpam-709	68	4	x	x	SYM
ejpam-709	68	5	∈	∈	PROPN
ejpam-709	68	6	i	i	PRON
ejpam-709	68	7	for	for	ADP
ejpam-709	68	8	any	any	DET
ejpam-709	68	9	a	a	NOUN
ejpam-709	68	10	,	,	PUNCT
ejpam-709	68	11	b	b	X
ejpam-709	68	12	∈	∈	NOUN
ejpam-709	68	13	i	i	PRON
ejpam-709	68	14	and	and	CCONJ
ejpam-709	68	15	x	x	PROPN
ejpam-709	68	16	∈	∈	PROPN
ejpam-709	68	17	r.	r.	PROPN
ejpam-709	68	18	also	also	ADV
ejpam-709	68	19	,	,	PUNCT
ejpam-709	68	20	a	a	DET
ejpam-709	68	21	non	non	ADJ
ejpam-709	68	22	-	-	ADJ
ejpam-709	68	23	empty	empty	ADJ
ejpam-709	68	24	subset	subset	NOUN
ejpam-709	68	25	f	f	NOUN
ejpam-709	68	26	of	of	ADP
ejpam-709	68	27	r	r	NOUN
ejpam-709	68	28	is	be	AUX
ejpam-709	68	29	said	say	VERB
ejpam-709	68	30	to	to	PART
ejpam-709	68	31	be	be	AUX
ejpam-709	68	32	a	a	DET
ejpam-709	68	33	filter	filter	NOUN
ejpam-709	68	34	of	of	ADP
ejpam-709	68	35	r	r	NOUN
ejpam-709	68	36	if	if	SCONJ
ejpam-709	68	37	a	a	DET
ejpam-709	68	38	∧	∧	PROPN
ejpam-709	68	39	b	b	PROPN
ejpam-709	68	40	∈	∈	PROPN
ejpam-709	68	41	f	f	PROPN
ejpam-709	68	42	and	and	CCONJ
ejpam-709	68	43	x	x	PROPN
ejpam-709	68	44	∨	∨	NOUN
ejpam-709	68	45	a	a	DET
ejpam-709	68	46	∈	∈	PROPN
ejpam-709	68	47	f	f	NOUN
ejpam-709	68	48	for	for	ADP
ejpam-709	68	49	a	a	DET
ejpam-709	68	50	,	,	PUNCT
ejpam-709	68	51	b	b	PROPN
ejpam-709	68	52	∈	∈	PROPN
ejpam-709	68	53	f	f	PROPN
ejpam-709	68	54	and	and	CCONJ
ejpam-709	68	55	x	x	PROPN
ejpam-709	68	56	∈	∈	PROPN
ejpam-709	68	57	r.	r.	NOUN
ejpam-709	68	58	the	the	DET
ejpam-709	68	59	set	set	PROPN
ejpam-709	68	60	i(r	i(r	PROPN
ejpam-709	68	61	)	)	PUNCT
ejpam-709	68	62	of	of	ADP
ejpam-709	68	63	all	all	DET
ejpam-709	68	64	ideals	ideal	NOUN
ejpam-709	68	65	of	of	ADP
ejpam-709	68	66	r	r	NOUN
ejpam-709	68	67	is	be	AUX
ejpam-709	68	68	a	a	DET
ejpam-709	68	69	bounded	bounded	ADJ
ejpam-709	68	70	distributive	distributive	ADJ
ejpam-709	68	71	lattice	lattice	NOUN
ejpam-709	68	72	with	with	ADP
ejpam-709	68	73	least	least	ADJ
ejpam-709	68	74	element	element	ADJ
ejpam-709	68	75	{	{	PUNCT
ejpam-709	68	76	0	0	NUM
ejpam-709	68	77	}	}	PUNCT
ejpam-709	68	78	and	and	CCONJ
ejpam-709	68	79	greatest	great	ADJ
ejpam-709	68	80	element	element	NOUN
ejpam-709	68	81	r	r	NOUN
ejpam-709	68	82	under	under	ADP
ejpam-709	68	83	set	set	NOUN
ejpam-709	68	84	inclusion	inclusion	NOUN
ejpam-709	68	85	in	in	ADP
ejpam-709	68	86	which	which	PRON
ejpam-709	68	87	,	,	PUNCT
ejpam-709	68	88	for	for	ADP
ejpam-709	68	89	any	any	DET
ejpam-709	68	90	i	i	NOUN
ejpam-709	68	91	,	,	PUNCT
ejpam-709	68	92	j	j	PROPN
ejpam-709	68	93	∈	∈	PROPN
ejpam-709	68	94	i(r	i(r	PROPN
ejpam-709	68	95	)	)	PUNCT
ejpam-709	68	96	,	,	PUNCT
ejpam-709	68	97	i	i	PROPN
ejpam-709	68	98	∩	∩	VERB
ejpam-709	68	99	j	j	PROPN
ejpam-709	68	100	is	be	AUX
ejpam-709	68	101	the	the	DET
ejpam-709	68	102	infimum	infimum	NOUN
ejpam-709	68	103	of	of	ADP
ejpam-709	68	104	i	i	PRON
ejpam-709	68	105	and	and	CCONJ
ejpam-709	68	106	j	j	PROPN
ejpam-709	68	107	while	while	SCONJ
ejpam-709	68	108	the	the	DET
ejpam-709	68	109	supremum	supremum	NOUN
ejpam-709	68	110	is	be	AUX
ejpam-709	68	111	given	give	VERB
ejpam-709	68	112	by	by	ADP
ejpam-709	68	113	i	i	PROPN
ejpam-709	68	114	∨	∨	PROPN
ejpam-709	68	115	j	j	PROPN
ejpam-709	68	116	:	:	PUNCT
ejpam-709	68	117	=	=	X
ejpam-709	68	118	{	{	PUNCT
ejpam-709	68	119	a	a	DET
ejpam-709	68	120	∨	∨	NUM
ejpam-709	68	121	b	b	NOUN
ejpam-709	68	122	|	|	NOUN
ejpam-709	68	123	a	a	DET
ejpam-709	68	124	∈	∈	NOUN
ejpam-709	69	1	i	i	PRON
ejpam-709	69	2	,	,	PUNCT
ejpam-709	69	3	b	b	PROPN
ejpam-709	69	4	∈	∈	PROPN
ejpam-709	69	5	j	j	PROPN
ejpam-709	69	6	}	}	PUNCT
ejpam-709	69	7	.	.	PUNCT
ejpam-709	70	1	a	a	DET
ejpam-709	70	2	proper	proper	ADJ
ejpam-709	70	3	ideal	ideal	NOUN
ejpam-709	70	4	p	p	NOUN
ejpam-709	70	5	of	of	ADP
ejpam-709	70	6	r	r	NOUN
ejpam-709	70	7	is	be	AUX
ejpam-709	70	8	called	call	VERB
ejpam-709	70	9	a	a	DET
ejpam-709	70	10	prime	prime	ADJ
ejpam-709	70	11	ideal	ideal	NOUN
ejpam-709	70	12	if	if	SCONJ
ejpam-709	70	13	,	,	PUNCT
ejpam-709	70	14	for	for	ADP
ejpam-709	70	15	any	any	DET
ejpam-709	70	16	x	x	NOUN
ejpam-709	70	17	,	,	PUNCT
ejpam-709	70	18	y	y	PROPN
ejpam-709	70	19	∈	∈	PROPN
ejpam-709	70	20	r	r	NOUN
ejpam-709	70	21	,	,	PUNCT
ejpam-709	70	22	x	x	PUNCT
ejpam-709	70	23	∧	∧	NOUN
ejpam-709	70	24	y	y	PROPN
ejpam-709	70	25	∈	∈	PROPN
ejpam-709	70	26	p	p	PROPN
ejpam-709	70	27	⇒	⇒	X
ejpam-709	70	28	x	x	PUNCT
ejpam-709	70	29	∈	∈	PROPN
ejpam-709	70	30	p	p	NOUN
ejpam-709	70	31	or	or	CCONJ
ejpam-709	70	32	y	y	PROPN
ejpam-709	70	33	∈	∈	PROPN
ejpam-709	70	34	p.	p.	NOUN
ejpam-709	70	35	a	a	DET
ejpam-709	70	36	proper	proper	ADJ
ejpam-709	70	37	ideal	ideal	NOUN
ejpam-709	70	38	m	m	NOUN
ejpam-709	70	39	of	of	ADP
ejpam-709	70	40	r	r	NOUN
ejpam-709	70	41	is	be	AUX
ejpam-709	70	42	said	say	VERB
ejpam-709	70	43	to	to	PART
ejpam-709	70	44	be	be	AUX
ejpam-709	70	45	maximal	maximal	ADJ
ejpam-709	70	46	if	if	SCONJ
ejpam-709	70	47	it	it	PRON
ejpam-709	70	48	is	be	AUX
ejpam-709	70	49	not	not	PART
ejpam-709	70	50	properly	properly	ADV
ejpam-709	70	51	contained	contain	VERB
ejpam-709	70	52	in	in	ADP
ejpam-709	70	53	any	any	DET
ejpam-709	70	54	proper	proper	ADJ
ejpam-709	70	55	ideal	ideal	NOUN
ejpam-709	70	56	of	of	ADP
ejpam-709	70	57	r.	r.	PROPN
ejpam-709	70	58	it	it	PRON
ejpam-709	70	59	can	can	AUX
ejpam-709	70	60	be	be	AUX
ejpam-709	70	61	g.	g.	PROPN
ejpam-709	70	62	rao	rao	PROPN
ejpam-709	70	63	,	,	PUNCT
ejpam-709	70	64	n.	n.	PROPN
ejpam-709	70	65	rafi	rafi	PROPN
ejpam-709	70	66	and	and	CCONJ
ejpam-709	70	67	b.	b.	PROPN
ejpam-709	70	68	kumar	kumar	PROPN
ejpam-709	70	69	/	/	SYM
ejpam-709	70	70	eur	eur	PROPN
ejpam-709	70	71	.	.	PUNCT
ejpam-709	71	1	j.	j.	PROPN
ejpam-709	71	2	pure	pure	PROPN
ejpam-709	71	3	appl	appl	PROPN
ejpam-709	71	4	.	.	PROPN
ejpam-709	71	5	math	math	PROPN
ejpam-709	71	6	,	,	PUNCT
ejpam-709	71	7	3	3	NUM
ejpam-709	71	8	(	(	PUNCT
ejpam-709	71	9	2010	2010	NUM
ejpam-709	71	10	)	)	PUNCT
ejpam-709	71	11	,	,	PUNCT
ejpam-709	71	12	704	704	NUM
ejpam-709	71	13	-	-	SYM
ejpam-709	71	14	716	716	NUM
ejpam-709	71	15	707	707	NUM
ejpam-709	71	16	observed	observe	VERB
ejpam-709	71	17	that	that	SCONJ
ejpam-709	71	18	every	every	DET
ejpam-709	71	19	maximal	maximal	ADJ
ejpam-709	71	20	ideal	ideal	NOUN
ejpam-709	71	21	of	of	ADP
ejpam-709	71	22	r	r	NOUN
ejpam-709	71	23	is	be	AUX
ejpam-709	71	24	a	a	DET
ejpam-709	71	25	prime	prime	ADJ
ejpam-709	71	26	ideal	ideal	NOUN
ejpam-709	71	27	.	.	PUNCT
ejpam-709	72	1	every	every	DET
ejpam-709	72	2	proper	proper	ADJ
ejpam-709	72	3	ideal	ideal	NOUN
ejpam-709	72	4	of	of	ADP
ejpam-709	72	5	r	r	NOUN
ejpam-709	72	6	is	be	AUX
ejpam-709	72	7	contained	contain	VERB
ejpam-709	72	8	in	in	ADP
ejpam-709	72	9	a	a	DET
ejpam-709	72	10	maximal	maximal	ADJ
ejpam-709	72	11	ideal	ideal	NOUN
ejpam-709	72	12	.	.	PUNCT
ejpam-709	73	1	for	for	ADP
ejpam-709	73	2	any	any	DET
ejpam-709	73	3	subset	subset	NOUN
ejpam-709	73	4	s	s	NOUN
ejpam-709	73	5	of	of	ADP
ejpam-709	73	6	r	r	NOUN
ejpam-709	73	7	the	the	DET
ejpam-709	73	8	smallest	small	ADJ
ejpam-709	73	9	ideal	ideal	NOUN
ejpam-709	73	10	containing	contain	VERB
ejpam-709	73	11	s	s	NOUN
ejpam-709	73	12	is	be	AUX
ejpam-709	73	13	given	give	VERB
ejpam-709	73	14	by	by	ADP
ejpam-709	73	15	(	(	PUNCT
ejpam-709	73	16	s	s	X
ejpam-709	73	17	]	]	X
ejpam-709	73	18	:	:	PUNCT
ejpam-709	73	19	=	=	SYM
ejpam-709	73	20	{	{	PUNCT
ejpam-709	73	21	(	(	PUNCT
ejpam-709	73	22	n	n	NUM
ejpam-709	73	23	∨	∨	NOUN
ejpam-709	73	24	i=1	i=1	PROPN
ejpam-709	73	25	si)∧	si)∧	NOUN
ejpam-709	73	26	x	x	PUNCT
ejpam-709	73	27	|	|	ADV
ejpam-709	73	28	si	si	PROPN
ejpam-709	73	29	∈	∈	PROPN
ejpam-709	73	30	s	s	PROPN
ejpam-709	73	31	,	,	PUNCT
ejpam-709	73	32	x	x	SYM
ejpam-709	73	33	∈	∈	NOUN
ejpam-709	73	34	r	r	NOUN
ejpam-709	73	35	and	and	CCONJ
ejpam-709	73	36	n	n	CCONJ
ejpam-709	73	37	∈	∈	PROPN
ejpam-709	73	38	n	n	CCONJ
ejpam-709	73	39	}	}	PUNCT
ejpam-709	73	40	.	.	PUNCT
ejpam-709	74	1	if	if	SCONJ
ejpam-709	74	2	s	s	VERB
ejpam-709	74	3	=	=	X
ejpam-709	74	4	{	{	PUNCT
ejpam-709	74	5	s	s	PROPN
ejpam-709	74	6	}	}	PUNCT
ejpam-709	74	7	,	,	PUNCT
ejpam-709	74	8	we	we	PRON
ejpam-709	74	9	write	write	VERB
ejpam-709	74	10	(	(	PUNCT
ejpam-709	74	11	s	s	X
ejpam-709	74	12	]	]	X
ejpam-709	74	13	instead	instead	ADV
ejpam-709	74	14	of	of	ADP
ejpam-709	74	15	(	(	PUNCT
ejpam-709	74	16	s	s	X
ejpam-709	74	17	]	]	X
ejpam-709	74	18	.	.	PUNCT
ejpam-709	75	1	similarly	similarly	ADV
ejpam-709	75	2	,	,	PUNCT
ejpam-709	75	3	for	for	ADP
ejpam-709	75	4	any	any	DET
ejpam-709	75	5	s	s	NOUN
ejpam-709	75	6	⊆	⊆	NUM
ejpam-709	75	7	r	r	NOUN
ejpam-709	75	8	,	,	PUNCT
ejpam-709	75	9	[	[	X
ejpam-709	75	10	s	s	X
ejpam-709	75	11	)	)	PUNCT
ejpam-709	75	12	:	:	PUNCT
ejpam-709	75	13	=	=	SYM
ejpam-709	75	14	{	{	PUNCT
ejpam-709	75	15	x	x	PROPN
ejpam-709	75	16	∨	∨	X
ejpam-709	75	17	(	(	PUNCT
ejpam-709	75	18	n	n	CCONJ
ejpam-709	75	19	∧	∧	PROPN
ejpam-709	75	20	i=1	i=1	PROPN
ejpam-709	75	21	si	si	NOUN
ejpam-709	75	22	)	)	PUNCT
ejpam-709	75	23	|	|	ADV
ejpam-709	75	24	si	si	PROPN
ejpam-709	75	25	∈	∈	PROPN
ejpam-709	75	26	s	s	PROPN
ejpam-709	75	27	,	,	PUNCT
ejpam-709	75	28	x	x	SYM
ejpam-709	75	29	∈	∈	NOUN
ejpam-709	75	30	r	r	NOUN
ejpam-709	75	31	and	and	CCONJ
ejpam-709	75	32	n	n	CCONJ
ejpam-709	75	33	∈	∈	PROPN
ejpam-709	75	34	n	n	CCONJ
ejpam-709	75	35	}	}	PUNCT
ejpam-709	75	36	.	.	PUNCT
ejpam-709	76	1	if	if	SCONJ
ejpam-709	76	2	s	s	VERB
ejpam-709	76	3	=	=	X
ejpam-709	76	4	{	{	PUNCT
ejpam-709	76	5	s	s	PROPN
ejpam-709	76	6	}	}	PUNCT
ejpam-709	76	7	,	,	PUNCT
ejpam-709	76	8	we	we	PRON
ejpam-709	76	9	write	write	VERB
ejpam-709	76	10	[	[	X
ejpam-709	76	11	s	s	X
ejpam-709	76	12	)	)	PUNCT
ejpam-709	76	13	instead	instead	ADV
ejpam-709	76	14	of	of	ADP
ejpam-709	76	15	[	[	X
ejpam-709	76	16	s	s	X
ejpam-709	76	17	)	)	PUNCT
ejpam-709	76	18	.	.	PUNCT
ejpam-709	77	1	theorem	theorem	ADJ
ejpam-709	77	2	4	4	NUM
ejpam-709	77	3	(	(	PUNCT
ejpam-709	77	4	[	[	X
ejpam-709	77	5	9	9	NUM
ejpam-709	77	6	]	]	NUM
ejpam-709	77	7	)	)	PUNCT
ejpam-709	77	8	.	.	PUNCT
ejpam-709	78	1	for	for	ADP
ejpam-709	78	2	any	any	DET
ejpam-709	78	3	x	x	NOUN
ejpam-709	78	4	,	,	PUNCT
ejpam-709	78	5	y	y	PROPN
ejpam-709	78	6	in	in	ADP
ejpam-709	78	7	r	r	NOUN
ejpam-709	78	8	the	the	DET
ejpam-709	78	9	following	following	NOUN
ejpam-709	78	10	are	be	AUX
ejpam-709	78	11	equivalent	equivalent	ADJ
ejpam-709	78	12	:	:	PUNCT
ejpam-709	78	13	1	1	X
ejpam-709	78	14	.	.	X
ejpam-709	78	15	(	(	PUNCT
ejpam-709	78	16	x]⊆	x]⊆	PROPN
ejpam-709	78	17	(	(	PUNCT
ejpam-709	78	18	y	y	NOUN
ejpam-709	78	19	]	]	X
ejpam-709	78	20	2	2	X
ejpam-709	78	21	.	.	PUNCT
ejpam-709	78	22	y	y	PROPN
ejpam-709	78	23	∧	∧	PROPN
ejpam-709	78	24	x	x	X
ejpam-709	79	1	=	=	PUNCT
ejpam-709	79	2	x	x	SYM
ejpam-709	79	3	3	3	X
ejpam-709	79	4	.	.	X
ejpam-709	79	5	y	y	PROPN
ejpam-709	79	6	∨	∨	NUM
ejpam-709	79	7	x	x	X
ejpam-709	79	8	=	=	SYM
ejpam-709	79	9	y	y	PROPN
ejpam-709	79	10	4	4	NUM
ejpam-709	79	11	.	.	PUNCT
ejpam-709	80	1	[	[	X
ejpam-709	80	2	y	y	X
ejpam-709	80	3	)	)	PUNCT
ejpam-709	80	4	⊆	⊆	NUM
ejpam-709	81	1	[	[	X
ejpam-709	81	2	x	x	X
ejpam-709	81	3	)	)	PUNCT
ejpam-709	81	4	.	.	PUNCT
ejpam-709	82	1	for	for	ADP
ejpam-709	82	2	any	any	DET
ejpam-709	82	3	x	x	SYM
ejpam-709	82	4	,	,	PUNCT
ejpam-709	82	5	y	y	PROPN
ejpam-709	82	6	∈	∈	PROPN
ejpam-709	82	7	r	r	NOUN
ejpam-709	82	8	,	,	PUNCT
ejpam-709	82	9	it	it	PRON
ejpam-709	82	10	can	can	AUX
ejpam-709	82	11	be	be	AUX
ejpam-709	82	12	verified	verify	VERB
ejpam-709	82	13	that	that	SCONJ
ejpam-709	82	14	(	(	PUNCT
ejpam-709	82	15	x]∨	x]∨	INTJ
ejpam-709	82	16	(	(	PUNCT
ejpam-709	82	17	y	y	NOUN
ejpam-709	82	18	]	]	X
ejpam-709	82	19	=	=	SYM
ejpam-709	82	20	(	(	PUNCT
ejpam-709	82	21	x	x	PROPN
ejpam-709	82	22	∨	∨	NUM
ejpam-709	82	23	y	y	PROPN
ejpam-709	82	24	]	]	PUNCT
ejpam-709	82	25	and	and	CCONJ
ejpam-709	82	26	(	(	PUNCT
ejpam-709	82	27	x]∧	x]∧	X
ejpam-709	82	28	(	(	PUNCT
ejpam-709	82	29	y	y	NOUN
ejpam-709	82	30	]	]	X
ejpam-709	82	31	=	=	SYM
ejpam-709	82	32	(	(	PUNCT
ejpam-709	82	33	x	x	PUNCT
ejpam-709	82	34	∧	∧	NOUN
ejpam-709	82	35	y	y	PROPN
ejpam-709	82	36	]	]	PUNCT
ejpam-709	82	37	.	.	PUNCT
ejpam-709	83	1	hence	hence	ADV
ejpam-709	83	2	the	the	DET
ejpam-709	83	3	set	set	NOUN
ejpam-709	83	4	pi(r	pi(r	NOUN
ejpam-709	83	5	)	)	PUNCT
ejpam-709	83	6	of	of	ADP
ejpam-709	83	7	all	all	DET
ejpam-709	83	8	principal	principal	ADJ
ejpam-709	83	9	ideals	ideal	NOUN
ejpam-709	83	10	of	of	ADP
ejpam-709	83	11	r	r	NOUN
ejpam-709	83	12	is	be	AUX
ejpam-709	83	13	a	a	DET
ejpam-709	83	14	sublattice	sublattice	NOUN
ejpam-709	83	15	of	of	ADP
ejpam-709	83	16	the	the	DET
ejpam-709	83	17	distributive	distributive	ADJ
ejpam-709	83	18	lattice	lattice	NOUN
ejpam-709	83	19	i(r	i(r	PROPN
ejpam-709	83	20	)	)	PUNCT
ejpam-709	83	21	of	of	ADP
ejpam-709	83	22	ideals	ideal	NOUN
ejpam-709	83	23	of	of	ADP
ejpam-709	83	24	r.	r.	PROPN
ejpam-709	83	25	3	3	NUM
ejpam-709	83	26	.	.	PUNCT
ejpam-709	84	1	s−relatively	s−relatively	ADV
ejpam-709	84	2	normal	normal	ADJ
ejpam-709	84	3	adls	adls	NOUN
ejpam-709	84	4	if	if	SCONJ
ejpam-709	84	5	r	r	NOUN
ejpam-709	84	6	is	be	AUX
ejpam-709	84	7	an	an	DET
ejpam-709	84	8	adl	adl	NOUN
ejpam-709	84	9	and	and	CCONJ
ejpam-709	84	10	s	s	NOUN
ejpam-709	84	11	is	be	AUX
ejpam-709	84	12	a	a	DET
ejpam-709	84	13	subadl	subadl	NOUN
ejpam-709	84	14	with	with	ADP
ejpam-709	84	15	0	0	NUM
ejpam-709	84	16	,	,	PUNCT
ejpam-709	84	17	then	then	ADV
ejpam-709	84	18	the	the	DET
ejpam-709	84	19	concept	concept	NOUN
ejpam-709	84	20	of	of	ADP
ejpam-709	84	21	s−normality	s−normality	NOUN
ejpam-709	84	22	in	in	ADP
ejpam-709	84	23	r	r	NOUN
ejpam-709	84	24	introduced	introduce	VERB
ejpam-709	84	25	in	in	ADP
ejpam-709	84	26	[	[	X
ejpam-709	84	27	5	5	NUM
ejpam-709	84	28	]	]	PUNCT
ejpam-709	84	29	and	and	CCONJ
ejpam-709	84	30	its	its	PRON
ejpam-709	84	31	properties	property	NOUN
ejpam-709	84	32	were	be	AUX
ejpam-709	84	33	discussed	discuss	VERB
ejpam-709	84	34	.	.	PUNCT
ejpam-709	85	1	r.	r.	PROPN
ejpam-709	85	2	cignoli	cignoli	PROPN
ejpam-709	86	1	[	[	X
ejpam-709	86	2	2	2	NUM
ejpam-709	86	3	]	]	PUNCT
ejpam-709	86	4	gave	give	VERB
ejpam-709	86	5	the	the	DET
ejpam-709	86	6	concept	concept	NOUN
ejpam-709	86	7	of	of	ADP
ejpam-709	86	8	s−completely	s−completely	ADV
ejpam-709	86	9	normal	normal	ADJ
ejpam-709	86	10	lattice	lattice	NOUN
ejpam-709	86	11	.	.	PUNCT
ejpam-709	87	1	in	in	ADP
ejpam-709	87	2	this	this	DET
ejpam-709	87	3	section	section	NOUN
ejpam-709	87	4	we	we	PRON
ejpam-709	87	5	define	define	VERB
ejpam-709	87	6	the	the	DET
ejpam-709	87	7	concept	concept	NOUN
ejpam-709	87	8	of	of	ADP
ejpam-709	87	9	s−relative	s−relative	ADJ
ejpam-709	87	10	normality	normality	NOUN
ejpam-709	87	11	in	in	ADP
ejpam-709	87	12	an	an	DET
ejpam-709	87	13	adl	adl	NOUN
ejpam-709	87	14	r	r	NOUN
ejpam-709	87	15	through	through	ADP
ejpam-709	87	16	its	its	PRON
ejpam-709	87	17	principal	principal	ADJ
ejpam-709	87	18	ideal	ideal	NOUN
ejpam-709	87	19	lattice	lattice	PROPN
ejpam-709	87	20	pi(r	pi(r	NOUN
ejpam-709	87	21	)	)	PUNCT
ejpam-709	87	22	.	.	PUNCT
ejpam-709	88	1	a	a	DET
ejpam-709	88	2	subadl	subadl	NOUN
ejpam-709	88	3	of	of	ADP
ejpam-709	88	4	an	an	DET
ejpam-709	88	5	adl	adl	NOUN
ejpam-709	88	6	with	with	ADP
ejpam-709	88	7	0	0	NUM
ejpam-709	88	8	carries	carry	VERB
ejpam-709	88	9	the	the	DET
ejpam-709	88	10	usual	usual	ADJ
ejpam-709	88	11	meaning	meaning	NOUN
ejpam-709	88	12	where	where	SCONJ
ejpam-709	88	13	0	0	NUM
ejpam-709	88	14	is	be	AUX
ejpam-709	88	15	treated	treat	VERB
ejpam-709	88	16	as	as	ADP
ejpam-709	88	17	a	a	DET
ejpam-709	88	18	nullary	nullary	ADJ
ejpam-709	88	19	operation	operation	NOUN
ejpam-709	88	20	.	.	PUNCT
ejpam-709	89	1	through	through	ADP
ejpam-709	89	2	out	out	ADP
ejpam-709	89	3	this	this	DET
ejpam-709	89	4	paper	paper	NOUN
ejpam-709	89	5	r	r	NOUN
ejpam-709	89	6	represents	represent	VERB
ejpam-709	89	7	an	an	DET
ejpam-709	89	8	adl	adl	PROPN
ejpam-709	89	9	and	and	CCONJ
ejpam-709	89	10	s	s	NOUN
ejpam-709	89	11	stands	stand	NOUN
ejpam-709	89	12	for	for	ADP
ejpam-709	89	13	a	a	DET
ejpam-709	89	14	subadl	subadl	NOUN
ejpam-709	89	15	of	of	ADP
ejpam-709	89	16	r	r	NOUN
ejpam-709	89	17	with	with	ADP
ejpam-709	89	18	0	0	NUM
ejpam-709	89	19	.	.	PUNCT
ejpam-709	90	1	by	by	ADP
ejpam-709	90	2	a	a	DET
ejpam-709	90	3	uni	uni	PROPN
ejpam-709	90	4	subadl	subadl	NOUN
ejpam-709	90	5	of	of	ADP
ejpam-709	90	6	r	r	PRON
ejpam-709	90	7	we	we	PRON
ejpam-709	90	8	mean	mean	VERB
ejpam-709	90	9	a	a	DET
ejpam-709	90	10	subadl	subadl	NOUN
ejpam-709	90	11	of	of	ADP
ejpam-709	90	12	r	r	NOUN
ejpam-709	90	13	containing	contain	VERB
ejpam-709	90	14	all	all	DET
ejpam-709	90	15	maximal	maximal	ADJ
ejpam-709	90	16	elements	element	NOUN
ejpam-709	90	17	of	of	ADP
ejpam-709	90	18	r.	r.	PROPN
ejpam-709	90	19	in	in	ADP
ejpam-709	90	20	[	[	X
ejpam-709	90	21	8	8	NUM
ejpam-709	90	22	]	]	PUNCT
ejpam-709	90	23	,	,	PUNCT
ejpam-709	90	24	the	the	DET
ejpam-709	90	25	concept	concept	NOUN
ejpam-709	90	26	of	of	ADP
ejpam-709	90	27	relative	relative	ADJ
ejpam-709	90	28	annihilator	annihilator	NOUN
ejpam-709	90	29	in	in	ADP
ejpam-709	90	30	an	an	DET
ejpam-709	90	31	adl	adl	NOUN
ejpam-709	90	32	was	be	AUX
ejpam-709	90	33	given	give	VERB
ejpam-709	90	34	.	.	PUNCT
ejpam-709	91	1	if	if	SCONJ
ejpam-709	91	2	x	x	PRON
ejpam-709	91	3	,	,	PUNCT
ejpam-709	91	4	y	y	PROPN
ejpam-709	91	5	∈	∈	PROPN
ejpam-709	91	6	r	r	NOUN
ejpam-709	91	7	,	,	PUNCT
ejpam-709	91	8	then	then	ADV
ejpam-709	91	9	⌊x	⌊x	PROPN
ejpam-709	91	10	,	,	PUNCT
ejpam-709	91	11	y⌋	y⌋	PROPN
ejpam-709	91	12	=	=	PRON
ejpam-709	91	13	{	{	PUNCT
ejpam-709	91	14	a	a	PRON
ejpam-709	91	15	∈	∈	PROPN
ejpam-709	91	16	r	r	NOUN
ejpam-709	91	17	|	|	NOUN
ejpam-709	91	18	y	y	PROPN
ejpam-709	91	19	∧	∧	PROPN
ejpam-709	91	20	a	a	DET
ejpam-709	91	21	∧	∧	PROPN
ejpam-709	91	22	x	x	X
ejpam-709	91	23	=	=	PUNCT
ejpam-709	91	24	a	a	DET
ejpam-709	91	25	∧	∧	PROPN
ejpam-709	91	26	x	x	VERB
ejpam-709	91	27	}	}	PUNCT
ejpam-709	91	28	is	be	AUX
ejpam-709	91	29	called	call	VERB
ejpam-709	91	30	a	a	DET
ejpam-709	91	31	relative	relative	ADJ
ejpam-709	91	32	annihilator	annihilator	NOUN
ejpam-709	91	33	in	in	ADP
ejpam-709	91	34	r	r	PROPN
ejpam-709	91	35	and	and	CCONJ
ejpam-709	91	36	⌊x	⌊x	NOUN
ejpam-709	91	37	,	,	PUNCT
ejpam-709	91	38	0⌋	0⌋	NOUN
ejpam-709	91	39	=	=	PUNCT
ejpam-709	91	40	(	(	PUNCT
ejpam-709	91	41	x)∗	x)∗	PROPN
ejpam-709	91	42	is	be	AUX
ejpam-709	91	43	the	the	DET
ejpam-709	91	44	annihilator	annihilator	NOUN
ejpam-709	91	45	of	of	ADP
ejpam-709	91	46	x	x	PROPN
ejpam-709	91	47	in	in	ADP
ejpam-709	91	48	r.	r.	PROPN
ejpam-709	91	49	now	now	ADV
ejpam-709	91	50	we	we	PRON
ejpam-709	91	51	define	define	VERB
ejpam-709	91	52	the	the	DET
ejpam-709	91	53	concept	concept	NOUN
ejpam-709	91	54	of	of	ADP
ejpam-709	91	55	an	an	DET
ejpam-709	91	56	s−relative	s−relative	ADJ
ejpam-709	91	57	annihilator	annihilator	NOUN
ejpam-709	91	58	in	in	ADP
ejpam-709	91	59	r	r	NOUN
ejpam-709	91	60	as	as	SCONJ
ejpam-709	91	61	follows	follow	VERB
ejpam-709	91	62	.	.	PUNCT
ejpam-709	92	1	definition	definition	NOUN
ejpam-709	92	2	2	2	NUM
ejpam-709	92	3	.	.	PUNCT
ejpam-709	93	1	let	let	VERB
ejpam-709	93	2	x	x	PRON
ejpam-709	93	3	,	,	PUNCT
ejpam-709	93	4	y	y	PROPN
ejpam-709	93	5	∈	∈	PROPN
ejpam-709	93	6	r.	r.	PROPN
ejpam-709	93	7	define	define	VERB
ejpam-709	93	8	⌊x	⌊x	PROPN
ejpam-709	93	9	,	,	PUNCT
ejpam-709	93	10	y⌋s	y⌋s	PROPN
ejpam-709	94	1	=	=	PRON
ejpam-709	94	2	{	{	PUNCT
ejpam-709	94	3	a	a	DET
ejpam-709	94	4	∈	∈	NOUN
ejpam-709	94	5	s	s	VERB
ejpam-709	94	6	|	|	ADV
ejpam-709	94	7	y	y	PROPN
ejpam-709	94	8	∧	∧	PROPN
ejpam-709	94	9	a	a	DET
ejpam-709	94	10	∧	∧	PROPN
ejpam-709	94	11	x	x	X
ejpam-709	94	12	=	=	PUNCT
ejpam-709	94	13	a	a	DET
ejpam-709	94	14	∧	∧	PROPN
ejpam-709	94	15	x	x	X
ejpam-709	94	16	}	}	PUNCT
ejpam-709	94	17	.	.	PUNCT
ejpam-709	95	1	we	we	PRON
ejpam-709	95	2	call	call	VERB
ejpam-709	95	3	⌊x	⌊x	PROPN
ejpam-709	95	4	,	,	PUNCT
ejpam-709	95	5	y⌋s	y⌋s	PROPN
ejpam-709	95	6	an	an	DET
ejpam-709	95	7	s−relative	s−relative	PROPN
ejpam-709	95	8	annihilator	annihilator	NOUN
ejpam-709	95	9	.	.	PUNCT
ejpam-709	96	1	it	it	PRON
ejpam-709	96	2	can	can	AUX
ejpam-709	96	3	be	be	AUX
ejpam-709	96	4	observed	observe	VERB
ejpam-709	96	5	that	that	SCONJ
ejpam-709	96	6	a	a	DET
ejpam-709	96	7	∈	∈	PROPN
ejpam-709	96	8	⌊x	⌊x	NOUN
ejpam-709	96	9	,	,	PUNCT
ejpam-709	96	10	y⌋s	y⌋s	PROPN
ejpam-709	97	1	iff	iff	PROPN
ejpam-709	97	2	y	y	PROPN
ejpam-709	97	3	=	=	SYM
ejpam-709	97	4	y	y	PROPN
ejpam-709	97	5	∨	∨	NOUN
ejpam-709	97	6	(	(	PUNCT
ejpam-709	97	7	a	a	DET
ejpam-709	97	8	∧	∧	PROPN
ejpam-709	97	9	x	x	NOUN
ejpam-709	97	10	)	)	PUNCT
ejpam-709	97	11	.	.	PUNCT
ejpam-709	98	1	clearly	clearly	ADV
ejpam-709	98	2	⌊x	⌊x	PROPN
ejpam-709	98	3	,	,	PUNCT
ejpam-709	98	4	y⌋s	y⌋s	PROPN
ejpam-709	98	5	is	be	AUX
ejpam-709	98	6	an	an	DET
ejpam-709	98	7	ideal	ideal	NOUN
ejpam-709	98	8	of	of	ADP
ejpam-709	98	9	s.	s.	PROPN
ejpam-709	98	10	the	the	DET
ejpam-709	98	11	following	following	ADJ
ejpam-709	98	12	result	result	NOUN
ejpam-709	98	13	can	can	AUX
ejpam-709	98	14	be	be	AUX
ejpam-709	98	15	verified	verify	VERB
ejpam-709	98	16	easily	easily	ADV
ejpam-709	98	17	.	.	PUNCT
ejpam-709	99	1	lemma	lemma	PROPN
ejpam-709	99	2	1	1	X
ejpam-709	99	3	.	.	PUNCT
ejpam-709	100	1	let	let	VERB
ejpam-709	100	2	x	x	PRON
ejpam-709	100	3	,	,	PUNCT
ejpam-709	100	4	y	y	PROPN
ejpam-709	100	5	∈	∈	PROPN
ejpam-709	100	6	r.	r.	PROPN
ejpam-709	100	7	then	then	ADV
ejpam-709	100	8	for	for	ADP
ejpam-709	100	9	any	any	DET
ejpam-709	100	10	a	a	DET
ejpam-709	100	11	∈	∈	ADJ
ejpam-709	100	12	s	s	NOUN
ejpam-709	100	13	,	,	PUNCT
ejpam-709	100	14	a	a	DET
ejpam-709	100	15	∈	∈	PROPN
ejpam-709	100	16	⌊x	⌊x	NOUN
ejpam-709	100	17	,	,	PUNCT
ejpam-709	101	1	y⌋s	y⌋s	PROPN
ejpam-709	101	2	iff	iff	PROPN
ejpam-709	101	3	x	x	X
ejpam-709	101	4	∧	∧	PROPN
ejpam-709	101	5	a	a	DET
ejpam-709	101	6	≤	≤	NUM
ejpam-709	101	7	y	y	PROPN
ejpam-709	101	8	∧	∧	PROPN
ejpam-709	101	9	a.	a.	NOUN
ejpam-709	101	10	the	the	DET
ejpam-709	101	11	following	follow	VERB
ejpam-709	101	12	definition	definition	NOUN
ejpam-709	101	13	is	be	AUX
ejpam-709	101	14	taken	take	VERB
ejpam-709	101	15	from	from	ADP
ejpam-709	101	16	[	[	X
ejpam-709	101	17	5	5	NUM
ejpam-709	101	18	]	]	PUNCT
ejpam-709	101	19	.	.	PUNCT
ejpam-709	102	1	g.	g.	PROPN
ejpam-709	102	2	rao	rao	PROPN
ejpam-709	102	3	,	,	PUNCT
ejpam-709	102	4	n.	n.	PROPN
ejpam-709	102	5	rafi	rafi	PROPN
ejpam-709	102	6	and	and	CCONJ
ejpam-709	102	7	b.	b.	PROPN
ejpam-709	102	8	kumar	kumar	PROPN
ejpam-709	102	9	/	/	SYM
ejpam-709	102	10	eur	eur	PROPN
ejpam-709	102	11	.	.	PUNCT
ejpam-709	103	1	j.	j.	PROPN
ejpam-709	103	2	pure	pure	PROPN
ejpam-709	103	3	appl	appl	PROPN
ejpam-709	103	4	.	.	PROPN
ejpam-709	103	5	math	math	PROPN
ejpam-709	103	6	,	,	PUNCT
ejpam-709	103	7	3	3	NUM
ejpam-709	103	8	(	(	PUNCT
ejpam-709	103	9	2010	2010	NUM
ejpam-709	103	10	)	)	PUNCT
ejpam-709	103	11	,	,	PUNCT
ejpam-709	103	12	704	704	NUM
ejpam-709	103	13	-	-	SYM
ejpam-709	103	14	716	716	NUM
ejpam-709	103	15	708	708	NUM
ejpam-709	103	16	definition	definition	NOUN
ejpam-709	103	17	3	3	NUM
ejpam-709	103	18	.	.	PUNCT
ejpam-709	104	1	let	let	VERB
ejpam-709	104	2	s	s	PRON
ejpam-709	104	3	be	be	AUX
ejpam-709	104	4	a	a	DET
ejpam-709	104	5	subadl	subadl	NOUN
ejpam-709	104	6	of	of	ADP
ejpam-709	104	7	r.	r.	PROPN
ejpam-709	104	8	an	an	DET
ejpam-709	104	9	ideal	ideal	NOUN
ejpam-709	104	10	i	i	PRON
ejpam-709	104	11	of	of	ADP
ejpam-709	104	12	r	r	NOUN
ejpam-709	104	13	is	be	AUX
ejpam-709	104	14	called	call	VERB
ejpam-709	104	15	an	an	DET
ejpam-709	104	16	s	s	NOUN
ejpam-709	104	17	-	-	NOUN
ejpam-709	104	18	ideal	ideal	NOUN
ejpam-709	104	19	of	of	ADP
ejpam-709	104	20	r	r	NOUN
ejpam-709	104	21	if	if	SCONJ
ejpam-709	104	22	i	i	PRON
ejpam-709	104	23	is	be	AUX
ejpam-709	104	24	generated	generate	VERB
ejpam-709	104	25	by	by	ADP
ejpam-709	104	26	the	the	DET
ejpam-709	104	27	set	set	NOUN
ejpam-709	104	28	i	i	PRON
ejpam-709	104	29	∩	∩	NOUN
ejpam-709	104	30	s(i	s(i	PROPN
ejpam-709	104	31	=	=	PUNCT
ejpam-709	105	1	(	(	PUNCT
ejpam-709	105	2	i	i	PROPN
ejpam-709	105	3	∩	∩	X
ejpam-709	105	4	s	s	PART
ejpam-709	105	5	]	]	X
ejpam-709	105	6	)	)	PUNCT
ejpam-709	105	7	.	.	PUNCT
ejpam-709	106	1	an	an	DET
ejpam-709	106	2	s−ideal	s−ideal	ADJ
ejpam-709	106	3	i	i	PRON
ejpam-709	106	4	is	be	AUX
ejpam-709	106	5	called	call	VERB
ejpam-709	106	6	an	an	DET
ejpam-709	106	7	s−prime	s−prime	NOUN
ejpam-709	106	8	ideal	ideal	NOUN
ejpam-709	106	9	of	of	ADP
ejpam-709	106	10	r	r	NOUN
ejpam-709	106	11	if	if	SCONJ
ejpam-709	106	12	i	i	PRON
ejpam-709	106	13	∩	∩	VERB
ejpam-709	106	14	s	s	PART
ejpam-709	106	15	is	be	AUX
ejpam-709	106	16	a	a	DET
ejpam-709	106	17	prime	prime	ADJ
ejpam-709	106	18	ideal	ideal	NOUN
ejpam-709	106	19	of	of	ADP
ejpam-709	106	20	s	s	PRON
ejpam-709	106	21	and	and	CCONJ
ejpam-709	106	22	s−maximal	s−maximal	NOUN
ejpam-709	106	23	ideal	ideal	ADJ
ejpam-709	106	24	if	if	SCONJ
ejpam-709	106	25	i	i	PRON
ejpam-709	106	26	∩	∩	VERB
ejpam-709	106	27	s	s	PART
ejpam-709	106	28	is	be	AUX
ejpam-709	106	29	a	a	DET
ejpam-709	106	30	maximal	maximal	ADJ
ejpam-709	106	31	ideal	ideal	NOUN
ejpam-709	106	32	of	of	ADP
ejpam-709	106	33	s.	s.	PROPN
ejpam-709	106	34	it	it	PRON
ejpam-709	106	35	can	can	AUX
ejpam-709	106	36	be	be	AUX
ejpam-709	106	37	observed	observe	VERB
ejpam-709	106	38	that	that	SCONJ
ejpam-709	106	39	every	every	DET
ejpam-709	106	40	s−maximal	s−maximal	PROPN
ejpam-709	106	41	ideal	ideal	NOUN
ejpam-709	106	42	of	of	ADP
ejpam-709	106	43	r	r	NOUN
ejpam-709	106	44	is	be	AUX
ejpam-709	106	45	an	an	DET
ejpam-709	106	46	s−prime	s−prime	NOUN
ejpam-709	106	47	ideal	ideal	ADJ
ejpam-709	106	48	.	.	PUNCT
ejpam-709	107	1	the	the	DET
ejpam-709	107	2	concepts	concept	NOUN
ejpam-709	107	3	of	of	ADP
ejpam-709	107	4	s−	s−	PROPN
ejpam-709	107	5	filters	filter	NOUN
ejpam-709	107	6	,	,	PUNCT
ejpam-709	107	7	s−prime	s−prime	NOUN
ejpam-709	107	8	filters	filter	NOUN
ejpam-709	107	9	and	and	CCONJ
ejpam-709	107	10	s−maximal	s−maximal	PRON
ejpam-709	107	11	filters	filter	NOUN
ejpam-709	107	12	are	be	AUX
ejpam-709	107	13	defined	define	VERB
ejpam-709	107	14	analogously	analogously	ADV
ejpam-709	107	15	.	.	PUNCT
ejpam-709	108	1	now	now	ADV
ejpam-709	108	2	,	,	PUNCT
ejpam-709	108	3	the	the	DET
ejpam-709	108	4	following	follow	VERB
ejpam-709	108	5	lemma	lemma	PROPN
ejpam-709	108	6	can	can	AUX
ejpam-709	108	7	be	be	AUX
ejpam-709	108	8	verified	verify	VERB
ejpam-709	108	9	easily	easily	ADV
ejpam-709	108	10	.	.	PUNCT
ejpam-709	109	1	lemma	lemma	PROPN
ejpam-709	109	2	2	2	X
ejpam-709	109	3	.	.	PUNCT
ejpam-709	110	1	let	let	VERB
ejpam-709	110	2	r	r	PRON
ejpam-709	110	3	be	be	AUX
ejpam-709	110	4	an	an	DET
ejpam-709	110	5	adl	adl	NOUN
ejpam-709	110	6	,	,	PUNCT
ejpam-709	110	7	s	s	VERB
ejpam-709	110	8	a	a	DET
ejpam-709	110	9	subadl	subadl	NOUN
ejpam-709	110	10	of	of	ADP
ejpam-709	110	11	r	r	NOUN
ejpam-709	110	12	and	and	CCONJ
ejpam-709	110	13	f1	f1	NOUN
ejpam-709	110	14	a	a	DET
ejpam-709	110	15	filter	filter	NOUN
ejpam-709	110	16	of	of	ADP
ejpam-709	110	17	s.	s.	PROPN
ejpam-709	110	18	then	then	ADV
ejpam-709	110	19	the	the	DET
ejpam-709	110	20	filter	filter	NOUN
ejpam-709	110	21	f	f	PROPN
ejpam-709	110	22	of	of	ADP
ejpam-709	110	23	r	r	NOUN
ejpam-709	110	24	is	be	AUX
ejpam-709	110	25	generated	generate	VERB
ejpam-709	110	26	by	by	ADP
ejpam-709	110	27	f1	f1	PROPN
ejpam-709	110	28	is	be	AUX
ejpam-709	110	29	s−filter	s−filt	ADJ
ejpam-709	110	30	of	of	ADP
ejpam-709	110	31	r	r	NOUN
ejpam-709	110	32	and	and	CCONJ
ejpam-709	110	33	f1	f1	NOUN
ejpam-709	110	34	=	=	SYM
ejpam-709	110	35	f	f	PROPN
ejpam-709	110	36	∩	∩	PROPN
ejpam-709	110	37	s.	s.	PROPN
ejpam-709	110	38	we	we	PRON
ejpam-709	110	39	recall	recall	VERB
ejpam-709	110	40	the	the	DET
ejpam-709	110	41	following	following	NOUN
ejpam-709	110	42	from	from	ADP
ejpam-709	110	43	[	[	X
ejpam-709	110	44	5	5	NUM
ejpam-709	110	45	]	]	PUNCT
ejpam-709	110	46	.	.	PUNCT
ejpam-709	111	1	definition	definition	NOUN
ejpam-709	111	2	4	4	NUM
ejpam-709	111	3	.	.	PUNCT
ejpam-709	112	1	let	let	VERB
ejpam-709	112	2	r	r	PRON
ejpam-709	112	3	be	be	AUX
ejpam-709	112	4	an	an	DET
ejpam-709	112	5	adl	adl	NOUN
ejpam-709	112	6	with	with	ADP
ejpam-709	112	7	maximal	maximal	ADJ
ejpam-709	112	8	elements	element	NOUN
ejpam-709	112	9	and	and	CCONJ
ejpam-709	112	10	s	s	VERB
ejpam-709	112	11	a	a	DET
ejpam-709	112	12	uni	uni	PROPN
ejpam-709	112	13	subadl	subadl	NOUN
ejpam-709	112	14	of	of	ADP
ejpam-709	112	15	r.	r.	PROPN
ejpam-709	112	16	r	r	PROPN
ejpam-709	112	17	is	be	AUX
ejpam-709	112	18	called	call	VERB
ejpam-709	112	19	s−normal	s−normal	PROPN
ejpam-709	112	20	if	if	SCONJ
ejpam-709	112	21	for	for	ADP
ejpam-709	112	22	any	any	DET
ejpam-709	112	23	x	x	NOUN
ejpam-709	112	24	,	,	PUNCT
ejpam-709	112	25	y	y	PROPN
ejpam-709	112	26	∈	∈	PROPN
ejpam-709	112	27	r	r	NOUN
ejpam-709	113	1	such	such	ADJ
ejpam-709	113	2	that	that	SCONJ
ejpam-709	113	3	x	x	SYM
ejpam-709	113	4	∧	∧	NOUN
ejpam-709	113	5	y	y	NOUN
ejpam-709	113	6	=	=	NOUN
ejpam-709	113	7	0	0	PROPN
ejpam-709	114	1	then	then	ADV
ejpam-709	114	2	there	there	PRON
ejpam-709	114	3	exist	exist	VERB
ejpam-709	114	4	elements	element	NOUN
ejpam-709	114	5	a	a	PRON
ejpam-709	114	6	,	,	PUNCT
ejpam-709	114	7	b	b	X
ejpam-709	114	8	∈	∈	NOUN
ejpam-709	114	9	s	s	VERB
ejpam-709	114	10	such	such	ADJ
ejpam-709	114	11	that	that	SCONJ
ejpam-709	114	12	x	x	SYM
ejpam-709	114	13	∧	∧	NOUN
ejpam-709	114	14	a	a	X
ejpam-709	114	15	=	=	X
ejpam-709	114	16	0=	0=	NUM
ejpam-709	114	17	y	y	PROPN
ejpam-709	114	18	∧	∧	PROPN
ejpam-709	114	19	b	b	PROPN
ejpam-709	114	20	and	and	CCONJ
ejpam-709	114	21	a	a	DET
ejpam-709	114	22	∨	∨	PROPN
ejpam-709	114	23	b	b	NOUN
ejpam-709	114	24	is	be	AUX
ejpam-709	114	25	a	a	DET
ejpam-709	114	26	maximal	maximal	ADJ
ejpam-709	114	27	element	element	NOUN
ejpam-709	114	28	.	.	PUNCT
ejpam-709	115	1	in	in	ADP
ejpam-709	115	2	the	the	DET
ejpam-709	115	3	following	following	NOUN
ejpam-709	115	4	theorem	theorem	NOUN
ejpam-709	115	5	,	,	PUNCT
ejpam-709	115	6	we	we	PRON
ejpam-709	115	7	characterize	characterize	VERB
ejpam-709	115	8	the	the	DET
ejpam-709	115	9	s−normal	s−normal	ADJ
ejpam-709	115	10	adl	adl	NOUN
ejpam-709	115	11	in	in	ADP
ejpam-709	115	12	terms	term	NOUN
ejpam-709	115	13	of	of	ADP
ejpam-709	115	14	s−relative	s−relative	ADJ
ejpam-709	115	15	annihilators	annihilator	NOUN
ejpam-709	115	16	.	.	PUNCT
ejpam-709	116	1	theorem	theorem	NOUN
ejpam-709	116	2	5	5	NUM
ejpam-709	116	3	.	.	PUNCT
ejpam-709	117	1	let	let	VERB
ejpam-709	117	2	r	r	PRON
ejpam-709	117	3	be	be	AUX
ejpam-709	117	4	an	an	DET
ejpam-709	117	5	adl	adl	NOUN
ejpam-709	117	6	with	with	ADP
ejpam-709	117	7	maximal	maximal	ADJ
ejpam-709	117	8	elements	element	NOUN
ejpam-709	117	9	and	and	CCONJ
ejpam-709	117	10	s	s	VERB
ejpam-709	117	11	a	a	DET
ejpam-709	117	12	uni	uni	PROPN
ejpam-709	117	13	subadl	subadl	NOUN
ejpam-709	117	14	of	of	ADP
ejpam-709	117	15	r.	r.	PROPN
ejpam-709	117	16	then	then	ADV
ejpam-709	117	17	the	the	DET
ejpam-709	117	18	following	follow	VERB
ejpam-709	117	19	conditions	condition	NOUN
ejpam-709	117	20	are	be	AUX
ejpam-709	117	21	equivalent	equivalent	ADJ
ejpam-709	117	22	:	:	PUNCT
ejpam-709	118	1	1	1	X
ejpam-709	118	2	.	.	X
ejpam-709	118	3	r	r	NOUN
ejpam-709	118	4	is	be	AUX
ejpam-709	118	5	s−normal	s−normal	ADJ
ejpam-709	118	6	2	2	NUM
ejpam-709	118	7	.	.	PUNCT
ejpam-709	118	8	⌊x	⌊x	PROPN
ejpam-709	118	9	,	,	PUNCT
ejpam-709	118	10	y⌋s	y⌋s	PROPN
ejpam-709	118	11	∨	∨	NUM
ejpam-709	118	12	⌊y	⌊y	PROPN
ejpam-709	118	13	,	,	PUNCT
ejpam-709	118	14	x⌋s	x⌋s	PROPN
ejpam-709	119	1	=	=	SYM
ejpam-709	119	2	s	s	PROPN
ejpam-709	119	3	,	,	PUNCT
ejpam-709	119	4	for	for	ADP
ejpam-709	119	5	any	any	DET
ejpam-709	119	6	x	x	NOUN
ejpam-709	119	7	,	,	PUNCT
ejpam-709	119	8	y	y	PROPN
ejpam-709	119	9	∈	∈	PROPN
ejpam-709	119	10	r	r	NOUN
ejpam-709	119	11	with	with	ADP
ejpam-709	119	12	x	x	PUNCT
ejpam-709	119	13	∧	∧	NOUN
ejpam-709	119	14	y	y	NOUN
ejpam-709	119	15	=	=	SYM
ejpam-709	119	16	0	0	PROPN
ejpam-709	119	17	3	3	X
ejpam-709	119	18	.	.	X
ejpam-709	120	1	for	for	ADP
ejpam-709	120	2	any	any	DET
ejpam-709	120	3	prime	prime	ADJ
ejpam-709	120	4	filter	filter	NOUN
ejpam-709	120	5	f	f	PROPN
ejpam-709	120	6	of	of	ADP
ejpam-709	120	7	s	s	PRON
ejpam-709	120	8	and	and	CCONJ
ejpam-709	120	9	for	for	ADP
ejpam-709	120	10	any	any	DET
ejpam-709	120	11	x	x	NOUN
ejpam-709	120	12	,	,	PUNCT
ejpam-709	120	13	y	y	PROPN
ejpam-709	120	14	∈	∈	PROPN
ejpam-709	120	15	r	r	NOUN
ejpam-709	120	16	with	with	ADP
ejpam-709	120	17	x	x	PUNCT
ejpam-709	120	18	∧	∧	PROPN
ejpam-709	120	19	y	y	PROPN
ejpam-709	120	20	=	=	SYM
ejpam-709	120	21	0	0	NUM
ejpam-709	120	22	,	,	PUNCT
ejpam-709	120	23	there	there	PRON
ejpam-709	120	24	exists	exist	VERB
ejpam-709	120	25	a	a	DET
ejpam-709	120	26	∈	∈	NOUN
ejpam-709	120	27	f	f	NOUN
ejpam-709	120	28	such	such	ADJ
ejpam-709	120	29	that	that	SCONJ
ejpam-709	120	30	x	x	SYM
ejpam-709	120	31	∧	∧	PROPN
ejpam-709	120	32	a	a	PROPN
ejpam-709	120	33	and	and	CCONJ
ejpam-709	120	34	y	y	PROPN
ejpam-709	120	35	∧	∧	PROPN
ejpam-709	120	36	a	a	PRON
ejpam-709	120	37	are	be	AUX
ejpam-709	120	38	comparable	comparable	ADJ
ejpam-709	120	39	.	.	PUNCT
ejpam-709	121	1	proof	proof	NOUN
ejpam-709	121	2	.	.	PUNCT
ejpam-709	122	1	(	(	PUNCT
ejpam-709	122	2	1)⇒	1)⇒	NUM
ejpam-709	122	3	(	(	PUNCT
ejpam-709	122	4	2	2	NUM
ejpam-709	122	5	)	)	PUNCT
ejpam-709	122	6	:	:	PUNCT
ejpam-709	122	7	assume	assume	VERB
ejpam-709	122	8	that	that	SCONJ
ejpam-709	122	9	r	r	NOUN
ejpam-709	122	10	is	be	AUX
ejpam-709	122	11	an	an	DET
ejpam-709	122	12	s−normal	s−normal	PROPN
ejpam-709	122	13	adl	adl	NOUN
ejpam-709	122	14	.	.	PUNCT
ejpam-709	123	1	let	let	VERB
ejpam-709	123	2	x	x	PRON
ejpam-709	123	3	,	,	PUNCT
ejpam-709	123	4	y	y	PROPN
ejpam-709	123	5	∈	∈	PROPN
ejpam-709	123	6	r	r	NOUN
ejpam-709	123	7	such	such	ADJ
ejpam-709	123	8	that	that	SCONJ
ejpam-709	123	9	x	x	SYM
ejpam-709	123	10	∧	∧	NOUN
ejpam-709	123	11	y	y	NOUN
ejpam-709	123	12	=	=	NOUN
ejpam-709	123	13	0	0	PROPN
ejpam-709	123	14	.	.	PUNCT
ejpam-709	124	1	then	then	ADV
ejpam-709	124	2	there	there	PRON
ejpam-709	124	3	exist	exist	VERB
ejpam-709	124	4	a	a	DET
ejpam-709	124	5	,	,	PUNCT
ejpam-709	124	6	b	b	X
ejpam-709	124	7	∈	∈	NOUN
ejpam-709	124	8	s	s	VERB
ejpam-709	124	9	such	such	ADJ
ejpam-709	124	10	that	that	SCONJ
ejpam-709	124	11	a	a	DET
ejpam-709	124	12	∧	∧	PROPN
ejpam-709	124	13	x	x	PUNCT
ejpam-709	124	14	=	=	SYM
ejpam-709	124	15	0	0	PUNCT
ejpam-709	125	1	=	=	SYM
ejpam-709	125	2	b	b	PROPN
ejpam-709	125	3	∧	∧	PROPN
ejpam-709	125	4	y	y	PROPN
ejpam-709	125	5	and	and	CCONJ
ejpam-709	125	6	a	a	DET
ejpam-709	125	7	∨	∨	PROPN
ejpam-709	125	8	b	b	NOUN
ejpam-709	125	9	is	be	AUX
ejpam-709	125	10	a	a	DET
ejpam-709	125	11	maximal	maximal	ADJ
ejpam-709	125	12	element	element	NOUN
ejpam-709	125	13	.	.	PUNCT
ejpam-709	126	1	that	that	PRON
ejpam-709	126	2	implies	imply	VERB
ejpam-709	126	3	y	y	PROPN
ejpam-709	126	4	∧	∧	PROPN
ejpam-709	126	5	a	a	DET
ejpam-709	126	6	∧	∧	PROPN
ejpam-709	126	7	x	x	X
ejpam-709	126	8	=	=	PUNCT
ejpam-709	126	9	a	a	DET
ejpam-709	126	10	∧	∧	NOUN
ejpam-709	126	11	x	x	X
ejpam-709	127	1	=	=	PUNCT
ejpam-709	127	2	0=	0=	PUNCT
ejpam-709	128	1	x	x	PUNCT
ejpam-709	128	2	∧	∧	NOUN
ejpam-709	128	3	b	b	PROPN
ejpam-709	128	4	∧	∧	PROPN
ejpam-709	128	5	y	y	PROPN
ejpam-709	128	6	=	=	SYM
ejpam-709	128	7	b	b	PROPN
ejpam-709	128	8	∧	∧	PROPN
ejpam-709	128	9	y.	y.	PROPN
ejpam-709	128	10	therefore	therefore	ADV
ejpam-709	128	11	⌊x	⌊x	PROPN
ejpam-709	128	12	,	,	PUNCT
ejpam-709	128	13	y⌋s	y⌋s	PROPN
ejpam-709	128	14	∨	∨	NUM
ejpam-709	128	15	⌊y	⌊y	PROPN
ejpam-709	128	16	,	,	PUNCT
ejpam-709	128	17	x⌋s	x⌋s	PROPN
ejpam-709	129	1	=	=	SYM
ejpam-709	129	2	s.	s.	PROPN
ejpam-709	129	3	(	(	PUNCT
ejpam-709	129	4	2	2	NUM
ejpam-709	129	5	)	)	PUNCT
ejpam-709	129	6	⇒	⇒	NOUN
ejpam-709	129	7	(	(	PUNCT
ejpam-709	129	8	3	3	NUM
ejpam-709	129	9	)	)	PUNCT
ejpam-709	129	10	:	:	PUNCT
ejpam-709	129	11	let	let	VERB
ejpam-709	129	12	f	f	PRON
ejpam-709	129	13	be	be	AUX
ejpam-709	129	14	any	any	DET
ejpam-709	129	15	prime	prime	ADJ
ejpam-709	129	16	filter	filter	NOUN
ejpam-709	129	17	of	of	ADP
ejpam-709	129	18	s	s	PRON
ejpam-709	129	19	and	and	CCONJ
ejpam-709	129	20	x	x	INTJ
ejpam-709	129	21	,	,	PUNCT
ejpam-709	129	22	y	y	PROPN
ejpam-709	129	23	∈	∈	PROPN
ejpam-709	129	24	r	r	NOUN
ejpam-709	129	25	such	such	ADJ
ejpam-709	129	26	that	that	SCONJ
ejpam-709	129	27	x	x	SYM
ejpam-709	129	28	∧	∧	NOUN
ejpam-709	129	29	y	y	NOUN
ejpam-709	129	30	=	=	NOUN
ejpam-709	129	31	0	0	PROPN
ejpam-709	129	32	.	.	PUNCT
ejpam-709	130	1	then	then	ADV
ejpam-709	130	2	⌊x	⌊x	PROPN
ejpam-709	130	3	,	,	PUNCT
ejpam-709	130	4	y⌋s	y⌋s	PROPN
ejpam-709	130	5	∨	∨	NUM
ejpam-709	130	6	⌊y	⌊y	PROPN
ejpam-709	130	7	,	,	PUNCT
ejpam-709	130	8	x⌋s	x⌋s	PROPN
ejpam-709	130	9	=	=	SYM
ejpam-709	130	10	s.	s.	PROPN
ejpam-709	130	11	let	let	VERB
ejpam-709	130	12	m	m	PRON
ejpam-709	130	13	be	be	AUX
ejpam-709	130	14	any	any	DET
ejpam-709	130	15	maximal	maximal	ADJ
ejpam-709	130	16	element	element	NOUN
ejpam-709	130	17	in	in	ADP
ejpam-709	130	18	s.	s.	PROPN
ejpam-709	130	19	then	then	ADV
ejpam-709	130	20	m	m	VERB
ejpam-709	130	21	=	=	SYM
ejpam-709	130	22	a	a	DET
ejpam-709	130	23	∨	∨	NUM
ejpam-709	130	24	b	b	NOUN
ejpam-709	130	25	,	,	PUNCT
ejpam-709	130	26	for	for	ADP
ejpam-709	130	27	some	some	DET
ejpam-709	130	28	a	a	DET
ejpam-709	130	29	∈	∈	ADJ
ejpam-709	130	30	⌊x	⌊x	NOUN
ejpam-709	130	31	,	,	PUNCT
ejpam-709	130	32	y⌋s	y⌋s	PROPN
ejpam-709	130	33	and	and	CCONJ
ejpam-709	130	34	b	b	X
ejpam-709	130	35	∈	∈	PROPN
ejpam-709	130	36	⌊y	⌊y	NOUN
ejpam-709	130	37	,	,	PUNCT
ejpam-709	130	38	x⌋s	x⌋s	PROPN
ejpam-709	130	39	.	.	PUNCT
ejpam-709	131	1	that	that	PRON
ejpam-709	131	2	implies	imply	VERB
ejpam-709	131	3	a∧	a∧	NOUN
ejpam-709	131	4	x	x	PUNCT
ejpam-709	131	5	=	=	PUNCT
ejpam-709	131	6	y∧a∧	y∧a∧	PUNCT
ejpam-709	131	7	x	x	PUNCT
ejpam-709	131	8	=	=	SYM
ejpam-709	131	9	0	0	NUM
ejpam-709	131	10	and	and	CCONJ
ejpam-709	131	11	b∧	b∧	ADJ
ejpam-709	131	12	y	y	NOUN
ejpam-709	131	13	=	=	SYM
ejpam-709	131	14	x∧	x∧	PROPN
ejpam-709	131	15	b∧	b∧	NOUN
ejpam-709	131	16	y	y	PROPN
ejpam-709	131	17	=	=	PUNCT
ejpam-709	131	18	0	0	PROPN
ejpam-709	131	19	.	.	PUNCT
ejpam-709	132	1	since	since	SCONJ
ejpam-709	132	2	a	a	DET
ejpam-709	132	3	∨	∨	NUM
ejpam-709	132	4	b	b	NOUN
ejpam-709	132	5	∈	∈	ADJ
ejpam-709	132	6	f	f	X
ejpam-709	132	7	,	,	PUNCT
ejpam-709	132	8	we	we	PRON
ejpam-709	132	9	get	get	VERB
ejpam-709	132	10	either	either	CCONJ
ejpam-709	132	11	a	a	DET
ejpam-709	132	12	∈	∈	ADJ
ejpam-709	132	13	f	f	NOUN
ejpam-709	132	14	or	or	CCONJ
ejpam-709	132	15	b	b	PROPN
ejpam-709	132	16	∈	∈	PROPN
ejpam-709	132	17	f.	f.	PROPN
ejpam-709	132	18	suppose	suppose	VERB
ejpam-709	132	19	a	a	DET
ejpam-709	132	20	∈	∈	PROPN
ejpam-709	132	21	f.	f.	NOUN
ejpam-709	132	22	since	since	SCONJ
ejpam-709	132	23	a	a	DET
ejpam-709	132	24	∈	∈	PROPN
ejpam-709	132	25	⌊x	⌊x	NOUN
ejpam-709	132	26	,	,	PUNCT
ejpam-709	132	27	y⌋s	y⌋s	NUM
ejpam-709	132	28	,	,	PUNCT
ejpam-709	132	29	we	we	PRON
ejpam-709	132	30	get	get	VERB
ejpam-709	132	31	x	x	PUNCT
ejpam-709	132	32	∧	∧	PROPN
ejpam-709	132	33	a	a	DET
ejpam-709	132	34	≤	≤	NUM
ejpam-709	132	35	y	y	PROPN
ejpam-709	132	36	∧	∧	PROPN
ejpam-709	132	37	a.	a.	NOUN
ejpam-709	132	38	thus	thus	ADV
ejpam-709	132	39	there	there	PRON
ejpam-709	132	40	is	be	VERB
ejpam-709	132	41	an	an	DET
ejpam-709	132	42	element	element	NOUN
ejpam-709	132	43	a	a	DET
ejpam-709	132	44	∈	∈	NOUN
ejpam-709	132	45	f	f	NOUN
ejpam-709	132	46	such	such	ADJ
ejpam-709	132	47	that	that	SCONJ
ejpam-709	132	48	x	x	SYM
ejpam-709	132	49	∧	∧	PROPN
ejpam-709	132	50	a	a	PROPN
ejpam-709	132	51	and	and	CCONJ
ejpam-709	132	52	y	y	PROPN
ejpam-709	132	53	∧	∧	PROPN
ejpam-709	132	54	a	a	PRON
ejpam-709	132	55	are	be	AUX
ejpam-709	132	56	comparable	comparable	ADJ
ejpam-709	132	57	.	.	PUNCT
ejpam-709	133	1	similarly	similarly	ADV
ejpam-709	133	2	,	,	PUNCT
ejpam-709	133	3	we	we	PRON
ejpam-709	133	4	get	get	VERB
ejpam-709	133	5	x	x	PUNCT
ejpam-709	133	6	∧	∧	PROPN
ejpam-709	133	7	b	b	PROPN
ejpam-709	133	8	and	and	CCONJ
ejpam-709	133	9	y	y	PROPN
ejpam-709	133	10	∧	∧	PROPN
ejpam-709	133	11	b	b	PROPN
ejpam-709	133	12	are	be	AUX
ejpam-709	133	13	comparable	comparable	ADJ
ejpam-709	133	14	,	,	PUNCT
ejpam-709	133	15	if	if	SCONJ
ejpam-709	133	16	b	b	PROPN
ejpam-709	133	17	∈	∈	PROPN
ejpam-709	133	18	f.	f.	PROPN
ejpam-709	133	19	(	(	PUNCT
ejpam-709	133	20	3)⇒	3)⇒	NUM
ejpam-709	133	21	(	(	PUNCT
ejpam-709	133	22	1	1	NUM
ejpam-709	133	23	)	)	PUNCT
ejpam-709	133	24	:	:	PUNCT
ejpam-709	133	25	let	let	VERB
ejpam-709	133	26	x	x	PRON
ejpam-709	133	27	,	,	PUNCT
ejpam-709	133	28	y	y	PROPN
ejpam-709	133	29	∈	∈	PROPN
ejpam-709	133	30	r	r	NOUN
ejpam-709	133	31	such	such	ADJ
ejpam-709	133	32	that	that	SCONJ
ejpam-709	133	33	x	x	SYM
ejpam-709	133	34	∧	∧	NOUN
ejpam-709	133	35	y	y	NOUN
ejpam-709	133	36	=	=	SYM
ejpam-709	133	37	0	0	PROPN
ejpam-709	133	38	.	.	PUNCT
ejpam-709	133	39	suppose	suppose	VERB
ejpam-709	133	40	that	that	SCONJ
ejpam-709	133	41	(	(	PUNCT
ejpam-709	133	42	(	(	PUNCT
ejpam-709	133	43	x)∗	x)∗	PROPN
ejpam-709	133	44	∩	∩	NOUN
ejpam-709	133	45	s	s	PART
ejpam-709	133	46	)	)	PUNCT
ejpam-709	133	47	∨	∨	NUM
ejpam-709	133	48	(	(	PUNCT
ejpam-709	133	49	(	(	PUNCT
ejpam-709	133	50	y)∗	y)∗	PROPN
ejpam-709	133	51	∩	∩	NOUN
ejpam-709	133	52	s	s	PART
ejpam-709	133	53	)	)	PUNCT
ejpam-709	133	54	6=	6=	NUM
ejpam-709	133	55	s.	s.	PROPN
ejpam-709	133	56	then	then	ADV
ejpam-709	133	57	there	there	PRON
ejpam-709	133	58	exists	exist	VERB
ejpam-709	133	59	a	a	DET
ejpam-709	133	60	maximal	maximal	ADJ
ejpam-709	133	61	ideal	ideal	NOUN
ejpam-709	133	62	m	m	NOUN
ejpam-709	133	63	of	of	ADP
ejpam-709	133	64	s	s	PRON
ejpam-709	133	65	such	such	ADJ
ejpam-709	133	66	that	that	SCONJ
ejpam-709	133	67	(	(	PUNCT
ejpam-709	133	68	(	(	PUNCT
ejpam-709	133	69	x)∗	x)∗	PROPN
ejpam-709	133	70	∩	∩	NOUN
ejpam-709	133	71	s	s	PART
ejpam-709	133	72	)	)	PUNCT
ejpam-709	133	73	∨	∨	NUM
ejpam-709	133	74	(	(	PUNCT
ejpam-709	133	75	(	(	PUNCT
ejpam-709	133	76	y)∗	y)∗	NOUN
ejpam-709	133	77	∩	∩	NOUN
ejpam-709	133	78	s	s	PART
ejpam-709	133	79	)	)	PUNCT
ejpam-709	133	80	⊆	⊆	NUM
ejpam-709	133	81	m	m	NOUN
ejpam-709	133	82	.	.	PUNCT
ejpam-709	134	1	that	that	PRON
ejpam-709	134	2	implies	imply	VERB
ejpam-709	134	3	s	s	VERB
ejpam-709	134	4	\	\	PROPN
ejpam-709	134	5	m	m	VERB
ejpam-709	134	6	is	be	AUX
ejpam-709	134	7	a	a	DET
ejpam-709	134	8	prime	prime	ADJ
ejpam-709	134	9	filter	filter	NOUN
ejpam-709	134	10	of	of	ADP
ejpam-709	134	11	s.	s.	PROPN
ejpam-709	134	12	by	by	ADP
ejpam-709	134	13	(	(	PUNCT
ejpam-709	134	14	3	3	NUM
ejpam-709	134	15	)	)	PUNCT
ejpam-709	134	16	,	,	PUNCT
ejpam-709	134	17	there	there	PRON
ejpam-709	134	18	exists	exist	VERB
ejpam-709	134	19	x	x	X
ejpam-709	134	20	∈	∈	PROPN
ejpam-709	134	21	s	s	PART
ejpam-709	134	22	\	\	NOUN
ejpam-709	134	23	m	m	VERB
ejpam-709	134	24	such	such	ADJ
ejpam-709	134	25	that	that	SCONJ
ejpam-709	134	26	x	x	SYM
ejpam-709	134	27	∧	∧	PROPN
ejpam-709	134	28	a	a	PROPN
ejpam-709	134	29	and	and	CCONJ
ejpam-709	134	30	y	y	PROPN
ejpam-709	134	31	∧	∧	PROPN
ejpam-709	134	32	a	a	PRON
ejpam-709	134	33	are	be	AUX
ejpam-709	134	34	comparable	comparable	ADJ
ejpam-709	134	35	.	.	PUNCT
ejpam-709	135	1	suppose	suppose	VERB
ejpam-709	135	2	x	x	PUNCT
ejpam-709	136	1	∧	∧	PROPN
ejpam-709	136	2	a	a	DET
ejpam-709	136	3	≤	≤	NUM
ejpam-709	136	4	y	y	PROPN
ejpam-709	136	5	∧	∧	PROPN
ejpam-709	136	6	a.	a.	NOUN
ejpam-709	136	7	then	then	ADV
ejpam-709	136	8	x	x	PART
ejpam-709	136	9	∧	∧	NOUN
ejpam-709	136	10	a	a	NOUN
ejpam-709	136	11	=	=	NOUN
ejpam-709	136	12	x	x	SYM
ejpam-709	136	13	∧	∧	PROPN
ejpam-709	136	14	a	a	DET
ejpam-709	136	15	∧	∧	PROPN
ejpam-709	136	16	y	y	PROPN
ejpam-709	136	17	∧	∧	PROPN
ejpam-709	136	18	a	a	PRON
ejpam-709	136	19	=	=	NOUN
ejpam-709	136	20	0	0	PROPN
ejpam-709	136	21	.	.	PUNCT
ejpam-709	137	1	then	then	ADV
ejpam-709	137	2	a	a	DET
ejpam-709	137	3	∈	∈	PROPN
ejpam-709	137	4	⌊x	⌊x	NOUN
ejpam-709	137	5	,	,	PUNCT
ejpam-709	137	6	y⌋s	y⌋s	PROPN
ejpam-709	137	7	∩	∩	NOUN
ejpam-709	137	8	(	(	PUNCT
ejpam-709	137	9	s	s	NOUN
ejpam-709	137	10	\	\	PROPN
ejpam-709	137	11	m	m	PROPN
ejpam-709	137	12	)	)	PUNCT
ejpam-709	137	13	,	,	PUNCT
ejpam-709	137	14	which	which	PRON
ejpam-709	137	15	is	be	AUX
ejpam-709	137	16	a	a	DET
ejpam-709	137	17	contradiction	contradiction	NOUN
ejpam-709	137	18	.	.	PUNCT
ejpam-709	138	1	therefore	therefore	ADV
ejpam-709	138	2	(	(	PUNCT
ejpam-709	138	3	(	(	PUNCT
ejpam-709	138	4	x)∗	x)∗	PROPN
ejpam-709	138	5	∩	∩	NOUN
ejpam-709	138	6	s	s	PART
ejpam-709	138	7	)	)	PUNCT
ejpam-709	138	8	∨	∨	NUM
ejpam-709	138	9	(	(	PUNCT
ejpam-709	138	10	(	(	PUNCT
ejpam-709	138	11	y)∗	y)∗	NOUN
ejpam-709	138	12	∩	∩	NOUN
ejpam-709	138	13	s	s	PART
ejpam-709	138	14	)	)	PUNCT
ejpam-709	138	15	=	=	VERB
ejpam-709	138	16	s.	s.	PROPN
ejpam-709	138	17	hence	hence	ADV
ejpam-709	138	18	r	r	NOUN
ejpam-709	138	19	is	be	AUX
ejpam-709	138	20	s−normal	s−normal	X
ejpam-709	138	21	.	.	PUNCT
ejpam-709	139	1	g.	g.	PROPN
ejpam-709	139	2	rao	rao	PROPN
ejpam-709	139	3	,	,	PUNCT
ejpam-709	139	4	n.	n.	PROPN
ejpam-709	139	5	rafi	rafi	PROPN
ejpam-709	139	6	and	and	CCONJ
ejpam-709	139	7	b.	b.	PROPN
ejpam-709	139	8	kumar	kumar	PROPN
ejpam-709	139	9	/	/	SYM
ejpam-709	139	10	eur	eur	PROPN
ejpam-709	139	11	.	.	PUNCT
ejpam-709	140	1	j.	j.	PROPN
ejpam-709	140	2	pure	pure	PROPN
ejpam-709	140	3	appl	appl	PROPN
ejpam-709	140	4	.	.	PROPN
ejpam-709	140	5	math	math	PROPN
ejpam-709	140	6	,	,	PUNCT
ejpam-709	140	7	3	3	NUM
ejpam-709	140	8	(	(	PUNCT
ejpam-709	140	9	2010	2010	NUM
ejpam-709	140	10	)	)	PUNCT
ejpam-709	140	11	,	,	PUNCT
ejpam-709	140	12	704	704	NUM
ejpam-709	140	13	-	-	SYM
ejpam-709	140	14	716	716	NUM
ejpam-709	140	15	709	709	NUM
ejpam-709	140	16	in	in	ADP
ejpam-709	140	17	[	[	X
ejpam-709	140	18	7	7	NUM
ejpam-709	140	19	]	]	PUNCT
ejpam-709	140	20	,	,	PUNCT
ejpam-709	140	21	the	the	DET
ejpam-709	140	22	concept	concept	NOUN
ejpam-709	140	23	of	of	ADP
ejpam-709	140	24	relatively	relatively	ADV
ejpam-709	140	25	normal	normal	ADJ
ejpam-709	140	26	adl	adl	NOUN
ejpam-709	140	27	was	be	AUX
ejpam-709	140	28	given	give	VERB
ejpam-709	140	29	as	as	SCONJ
ejpam-709	140	30	follows	follow	VERB
ejpam-709	140	31	.	.	PUNCT
ejpam-709	141	1	definition	definition	NOUN
ejpam-709	141	2	5	5	NUM
ejpam-709	141	3	.	.	PUNCT
ejpam-709	142	1	let	let	VERB
ejpam-709	142	2	r	r	PRON
ejpam-709	142	3	be	be	AUX
ejpam-709	142	4	an	an	DET
ejpam-709	142	5	adl	adl	NOUN
ejpam-709	142	6	with	with	ADP
ejpam-709	142	7	maximal	maximal	ADJ
ejpam-709	142	8	elements	element	NOUN
ejpam-709	142	9	.	.	PUNCT
ejpam-709	143	1	then	then	ADV
ejpam-709	143	2	r	r	NOUN
ejpam-709	143	3	is	be	AUX
ejpam-709	143	4	called	call	VERB
ejpam-709	143	5	relatively	relatively	ADV
ejpam-709	143	6	normal	normal	ADJ
ejpam-709	143	7	if	if	SCONJ
ejpam-709	143	8	for	for	ADP
ejpam-709	143	9	any	any	DET
ejpam-709	143	10	x	x	NOUN
ejpam-709	143	11	,	,	PUNCT
ejpam-709	143	12	y	y	PROPN
ejpam-709	143	13	∈	∈	PROPN
ejpam-709	143	14	r	r	NOUN
ejpam-709	143	15	,	,	PUNCT
ejpam-709	143	16	there	there	PRON
ejpam-709	143	17	exist	exist	VERB
ejpam-709	143	18	a	a	DET
ejpam-709	143	19	,	,	PUNCT
ejpam-709	143	20	b	b	X
ejpam-709	143	21	∈	∈	NOUN
ejpam-709	143	22	r	r	NOUN
ejpam-709	143	23	such	such	ADJ
ejpam-709	143	24	that	that	SCONJ
ejpam-709	143	25	y	y	PROPN
ejpam-709	143	26	∧	∧	PROPN
ejpam-709	143	27	a	a	DET
ejpam-709	143	28	∧	∧	PROPN
ejpam-709	143	29	x	x	X
ejpam-709	143	30	=	=	PUNCT
ejpam-709	143	31	a	a	DET
ejpam-709	143	32	∧	∧	PROPN
ejpam-709	143	33	x	x	X
ejpam-709	143	34	,	,	PUNCT
ejpam-709	143	35	x	x	PUNCT
ejpam-709	144	1	∧	∧	NOUN
ejpam-709	144	2	b	b	PROPN
ejpam-709	144	3	∧	∧	PROPN
ejpam-709	144	4	y	y	PROPN
ejpam-709	144	5	=	=	SYM
ejpam-709	144	6	b	b	PROPN
ejpam-709	144	7	∧	∧	PROPN
ejpam-709	144	8	y	y	PROPN
ejpam-709	144	9	and	and	CCONJ
ejpam-709	144	10	a	a	DET
ejpam-709	144	11	∨	∨	PROPN
ejpam-709	144	12	b	b	NOUN
ejpam-709	144	13	is	be	AUX
ejpam-709	144	14	a	a	DET
ejpam-709	144	15	maximal	maximal	ADJ
ejpam-709	144	16	element	element	NOUN
ejpam-709	144	17	.	.	PUNCT
ejpam-709	145	1	the	the	DET
ejpam-709	145	2	following	follow	VERB
ejpam-709	145	3	definition	definition	NOUN
ejpam-709	145	4	is	be	AUX
ejpam-709	145	5	taken	take	VERB
ejpam-709	145	6	from	from	ADP
ejpam-709	145	7	cignoli	cignoli	NOUN
ejpam-709	145	8	[	[	X
ejpam-709	145	9	2	2	NUM
ejpam-709	145	10	]	]	PUNCT
ejpam-709	145	11	.	.	PUNCT
ejpam-709	146	1	definition	definition	NOUN
ejpam-709	146	2	6	6	NUM
ejpam-709	146	3	.	.	PUNCT
ejpam-709	147	1	let	let	VERB
ejpam-709	147	2	(	(	PUNCT
ejpam-709	147	3	l,∨,∧	l,∨,∧	NOUN
ejpam-709	147	4	,	,	PUNCT
ejpam-709	147	5	0,1	0,1	NUM
ejpam-709	147	6	)	)	PUNCT
ejpam-709	147	7	be	be	VERB
ejpam-709	147	8	a	a	DET
ejpam-709	147	9	bounded	bounded	ADJ
ejpam-709	147	10	distributive	distributive	ADJ
ejpam-709	147	11	lattice	lattice	NOUN
ejpam-709	147	12	and	and	CCONJ
ejpam-709	147	13	s	s	VERB
ejpam-709	147	14	a	a	DET
ejpam-709	147	15	sublattice	sublattice	NOUN
ejpam-709	147	16	of	of	ADP
ejpam-709	147	17	l	l	NOUN
ejpam-709	147	18	containing	contain	VERB
ejpam-709	147	19	0	0	NUM
ejpam-709	147	20	and	and	CCONJ
ejpam-709	147	21	1	1	NUM
ejpam-709	147	22	.	.	PUNCT
ejpam-709	148	1	then	then	ADV
ejpam-709	148	2	l	l	PROPN
ejpam-709	148	3	is	be	AUX
ejpam-709	148	4	called	call	VERB
ejpam-709	148	5	s−completely	s−completely	ADV
ejpam-709	148	6	normal	normal	ADJ
ejpam-709	148	7	,	,	PUNCT
ejpam-709	148	8	if	if	SCONJ
ejpam-709	148	9	for	for	ADP
ejpam-709	148	10	any	any	DET
ejpam-709	148	11	x	x	NOUN
ejpam-709	148	12	,	,	PUNCT
ejpam-709	148	13	y	y	PROPN
ejpam-709	148	14	∈	∈	PROPN
ejpam-709	148	15	l	l	NOUN
ejpam-709	148	16	,	,	PUNCT
ejpam-709	148	17	there	there	PRON
ejpam-709	148	18	exist	exist	VERB
ejpam-709	148	19	a	a	DET
ejpam-709	148	20	,	,	PUNCT
ejpam-709	148	21	b	b	X
ejpam-709	148	22	∈	∈	NOUN
ejpam-709	148	23	s	s	VERB
ejpam-709	148	24	such	such	ADJ
ejpam-709	148	25	that	that	SCONJ
ejpam-709	148	26	x	x	SYM
ejpam-709	148	27	∧	∧	PROPN
ejpam-709	148	28	a	a	DET
ejpam-709	148	29	≤	≤	NUM
ejpam-709	148	30	y	y	PROPN
ejpam-709	148	31	,	,	PUNCT
ejpam-709	148	32	y	y	PROPN
ejpam-709	148	33	∧	∧	PROPN
ejpam-709	148	34	b	b	PROPN
ejpam-709	148	35	≤	≤	NUM
ejpam-709	148	36	x	x	PUNCT
ejpam-709	148	37	and	and	CCONJ
ejpam-709	148	38	a	a	DET
ejpam-709	148	39	∨	∨	NUM
ejpam-709	148	40	b	b	NOUN
ejpam-709	148	41	=	=	SYM
ejpam-709	148	42	1	1	X
ejpam-709	148	43	.	.	PUNCT
ejpam-709	149	1	now	now	ADV
ejpam-709	149	2	we	we	PRON
ejpam-709	149	3	define	define	VERB
ejpam-709	149	4	the	the	DET
ejpam-709	149	5	concept	concept	NOUN
ejpam-709	149	6	of	of	ADP
ejpam-709	149	7	an	an	DET
ejpam-709	149	8	s−relatively	s−relatively	ADV
ejpam-709	149	9	normal	normal	ADJ
ejpam-709	149	10	adl	adl	NOUN
ejpam-709	149	11	in	in	ADP
ejpam-709	149	12	the	the	DET
ejpam-709	149	13	following	following	NOUN
ejpam-709	149	14	.	.	PUNCT
ejpam-709	150	1	definition	definition	NOUN
ejpam-709	150	2	7	7	NUM
ejpam-709	150	3	.	.	PUNCT
ejpam-709	151	1	let	let	VERB
ejpam-709	151	2	r	r	PRON
ejpam-709	151	3	be	be	AUX
ejpam-709	151	4	an	an	DET
ejpam-709	151	5	adl	adl	NOUN
ejpam-709	151	6	with	with	ADP
ejpam-709	151	7	maximal	maximal	ADJ
ejpam-709	151	8	elements	element	NOUN
ejpam-709	151	9	and	and	CCONJ
ejpam-709	151	10	s	s	VERB
ejpam-709	151	11	a	a	DET
ejpam-709	151	12	uni	uni	PROPN
ejpam-709	151	13	subadl	subadl	NOUN
ejpam-709	151	14	of	of	ADP
ejpam-709	151	15	r.	r.	PROPN
ejpam-709	151	16	r	r	PROPN
ejpam-709	151	17	is	be	AUX
ejpam-709	151	18	called	call	VERB
ejpam-709	151	19	s−relatively	s−relatively	ADV
ejpam-709	151	20	normal	normal	ADJ
ejpam-709	151	21	if	if	SCONJ
ejpam-709	151	22	p	p	PROPN
ejpam-709	151	23	i(r	i(r	PROPN
ejpam-709	151	24	)	)	PUNCT
ejpam-709	151	25	is	be	AUX
ejpam-709	151	26	p	p	X
ejpam-709	151	27	i(s)−completely	i(s)−completely	ADV
ejpam-709	151	28	normal	normal	ADJ
ejpam-709	151	29	lattice	lattice	NOUN
ejpam-709	151	30	.	.	PUNCT
ejpam-709	152	1	the	the	DET
ejpam-709	152	2	following	follow	VERB
ejpam-709	152	3	lemma	lemma	PROPN
ejpam-709	152	4	can	can	AUX
ejpam-709	152	5	be	be	AUX
ejpam-709	152	6	verified	verify	VERB
ejpam-709	152	7	directly	directly	ADV
ejpam-709	152	8	.	.	PUNCT
ejpam-709	153	1	lemma	lemma	PROPN
ejpam-709	153	2	3	3	X
ejpam-709	153	3	.	.	PUNCT
ejpam-709	154	1	let	let	VERB
ejpam-709	154	2	r	r	PRON
ejpam-709	154	3	be	be	AUX
ejpam-709	154	4	an	an	DET
ejpam-709	154	5	adl	adl	NOUN
ejpam-709	154	6	with	with	ADP
ejpam-709	154	7	maximal	maximal	ADJ
ejpam-709	154	8	elements	element	NOUN
ejpam-709	154	9	and	and	CCONJ
ejpam-709	154	10	s	s	VERB
ejpam-709	154	11	a	a	DET
ejpam-709	154	12	uni	uni	PROPN
ejpam-709	154	13	subadl	subadl	NOUN
ejpam-709	154	14	of	of	ADP
ejpam-709	154	15	r.	r.	PROPN
ejpam-709	154	16	then	then	ADV
ejpam-709	154	17	r	r	NOUN
ejpam-709	154	18	is	be	AUX
ejpam-709	154	19	s−relatively	s−relatively	ADV
ejpam-709	154	20	normal	normal	ADJ
ejpam-709	154	21	if	if	SCONJ
ejpam-709	154	22	and	and	CCONJ
ejpam-709	154	23	if	if	SCONJ
ejpam-709	154	24	only	only	ADV
ejpam-709	154	25	for	for	ADP
ejpam-709	154	26	any	any	DET
ejpam-709	154	27	x	x	NOUN
ejpam-709	154	28	,	,	PUNCT
ejpam-709	154	29	y	y	PROPN
ejpam-709	154	30	∈	∈	PROPN
ejpam-709	154	31	r	r	NOUN
ejpam-709	154	32	,	,	PUNCT
ejpam-709	154	33	there	there	PRON
ejpam-709	154	34	exist	exist	VERB
ejpam-709	154	35	a	a	DET
ejpam-709	154	36	,	,	PUNCT
ejpam-709	154	37	b	b	X
ejpam-709	154	38	∈	∈	NOUN
ejpam-709	154	39	s	s	VERB
ejpam-709	154	40	such	such	ADJ
ejpam-709	154	41	that	that	SCONJ
ejpam-709	154	42	y	y	PROPN
ejpam-709	154	43	∧	∧	PROPN
ejpam-709	154	44	a	a	DET
ejpam-709	154	45	∧	∧	PROPN
ejpam-709	154	46	x	x	X
ejpam-709	154	47	=	=	PUNCT
ejpam-709	154	48	a	a	DET
ejpam-709	154	49	∧	∧	PROPN
ejpam-709	154	50	x	x	X
ejpam-709	154	51	,	,	PUNCT
ejpam-709	154	52	x	x	PUNCT
ejpam-709	155	1	∧	∧	NOUN
ejpam-709	155	2	b	b	PROPN
ejpam-709	155	3	∧	∧	PROPN
ejpam-709	155	4	y	y	PROPN
ejpam-709	155	5	=	=	SYM
ejpam-709	155	6	b	b	PROPN
ejpam-709	155	7	∧	∧	PROPN
ejpam-709	155	8	y	y	PROPN
ejpam-709	155	9	and	and	CCONJ
ejpam-709	155	10	a	a	DET
ejpam-709	155	11	∨	∨	PROPN
ejpam-709	155	12	b	b	NOUN
ejpam-709	155	13	is	be	AUX
ejpam-709	155	14	a	a	DET
ejpam-709	155	15	maximal	maximal	ADJ
ejpam-709	155	16	element	element	NOUN
ejpam-709	155	17	.	.	PUNCT
ejpam-709	155	18	example	example	NOUN
ejpam-709	156	1	1	1	NUM
ejpam-709	156	2	.	.	PUNCT
ejpam-709	156	3	let	let	VERB
ejpam-709	156	4	a	a	DET
ejpam-709	156	5	be	be	AUX
ejpam-709	156	6	a	a	DET
ejpam-709	156	7	discrete	discrete	ADJ
ejpam-709	156	8	adl	adl	NOUN
ejpam-709	156	9	and	and	CCONJ
ejpam-709	156	10	b	b	PROPN
ejpam-709	156	11	a	a	DET
ejpam-709	156	12	boolean	boolean	ADJ
ejpam-709	156	13	algebra	algebra	NOUN
ejpam-709	156	14	.	.	PUNCT
ejpam-709	157	1	then	then	ADV
ejpam-709	157	2	r	r	NOUN
ejpam-709	157	3	=	=	PUNCT
ejpam-709	157	4	a×	a×	PUNCT
ejpam-709	157	5	b	b	NOUN
ejpam-709	157	6	is	be	AUX
ejpam-709	157	7	an	an	DET
ejpam-709	157	8	adl	adl	NOUN
ejpam-709	157	9	.	.	PUNCT
ejpam-709	158	1	let	let	VERB
ejpam-709	158	2	d	d	PRON
ejpam-709	158	3	be	be	AUX
ejpam-709	158	4	a	a	DET
ejpam-709	158	5	subadl	subadl	NOUN
ejpam-709	158	6	of	of	ADP
ejpam-709	158	7	a	a	DET
ejpam-709	158	8	containing	contain	VERB
ejpam-709	158	9	at	at	ADV
ejpam-709	158	10	least	least	ADV
ejpam-709	158	11	two	two	NUM
ejpam-709	158	12	elements	element	NOUN
ejpam-709	158	13	.	.	PUNCT
ejpam-709	159	1	then	then	ADV
ejpam-709	159	2	s	s	VERB
ejpam-709	159	3	=	=	SYM
ejpam-709	159	4	d×b	d×b	PROPN
ejpam-709	159	5	is	be	AUX
ejpam-709	159	6	a	a	DET
ejpam-709	159	7	subadl	subadl	NOUN
ejpam-709	159	8	of	of	ADP
ejpam-709	159	9	r.	r.	PROPN
ejpam-709	159	10	let	let	VERB
ejpam-709	159	11	x	x	PRON
ejpam-709	159	12	,	,	PUNCT
ejpam-709	159	13	y	y	PROPN
ejpam-709	159	14	∈	∈	PROPN
ejpam-709	159	15	r.	r.	NOUN
ejpam-709	159	16	then	then	ADV
ejpam-709	159	17	x	x	PUNCT
ejpam-709	159	18	=	=	PRON
ejpam-709	159	19	(	(	PUNCT
ejpam-709	159	20	x1	x1	PROPN
ejpam-709	159	21	,	,	PUNCT
ejpam-709	159	22	x2	x2	PROPN
ejpam-709	159	23	)	)	PUNCT
ejpam-709	159	24	and	and	CCONJ
ejpam-709	159	25	y	y	PROPN
ejpam-709	159	26	=	=	SYM
ejpam-709	159	27	(	(	PUNCT
ejpam-709	159	28	y1	y1	INTJ
ejpam-709	159	29	,	,	PUNCT
ejpam-709	159	30	y2	y2	PROPN
ejpam-709	159	31	)	)	PUNCT
ejpam-709	159	32	.	.	PUNCT
ejpam-709	160	1	let	let	VERB
ejpam-709	160	2	t	t	NOUN
ejpam-709	160	3	be	be	AUX
ejpam-709	160	4	any	any	DET
ejpam-709	160	5	non	non	ADJ
ejpam-709	160	6	-	-	ADJ
ejpam-709	160	7	zero	zero	NUM
ejpam-709	160	8	element	element	NOUN
ejpam-709	160	9	of	of	ADP
ejpam-709	160	10	d.	d.	PROPN
ejpam-709	160	11	suppose	suppose	VERB
ejpam-709	160	12	x1	x1	PROPN
ejpam-709	160	13	,	,	PUNCT
ejpam-709	160	14	y1	y1	PROPN
ejpam-709	160	15	6=	6=	ADP
ejpam-709	160	16	0	0	NUM
ejpam-709	160	17	.	.	PUNCT
ejpam-709	161	1	write	write	VERB
ejpam-709	161	2	a	a	DET
ejpam-709	161	3	=	=	X
ejpam-709	161	4	(	(	PUNCT
ejpam-709	161	5	t	t	PROPN
ejpam-709	161	6	,	,	PUNCT
ejpam-709	161	7	y2∨x	y2∨x	ADJ
ejpam-709	161	8	′2	′2	NOUN
ejpam-709	161	9	)	)	PUNCT
ejpam-709	161	10	and	and	CCONJ
ejpam-709	161	11	b	b	X
ejpam-709	161	12	=	=	SYM
ejpam-709	161	13	(	(	PUNCT
ejpam-709	161	14	t	t	PROPN
ejpam-709	161	15	,	,	PUNCT
ejpam-709	161	16	x2∨y	x2∨y	PROPN
ejpam-709	161	17	′2	′2	NOUN
ejpam-709	161	18	)	)	PUNCT
ejpam-709	161	19	.	.	PUNCT
ejpam-709	162	1	now	now	ADV
ejpam-709	162	2	,	,	PUNCT
ejpam-709	162	3	y∧a∧x	y∧a∧x	PROPN
ejpam-709	162	4	=	=	SYM
ejpam-709	162	5	(	(	PUNCT
ejpam-709	162	6	y1	y1	PROPN
ejpam-709	162	7	,	,	PUNCT
ejpam-709	162	8	y2)∧(t	y2)∧(t	PROPN
ejpam-709	162	9	,	,	PUNCT
ejpam-709	162	10	y2∨x	y2∨x	PROPN
ejpam-709	162	11	′2)∧(x1	′2)∧(x1	NOUN
ejpam-709	162	12	,	,	PUNCT
ejpam-709	162	13	x2	x2	PROPN
ejpam-709	162	14	)	)	PUNCT
ejpam-709	162	15	=	=	SYM
ejpam-709	162	16	(	(	PUNCT
ejpam-709	162	17	y1∧	y1∧	PROPN
ejpam-709	162	18	t∧x1	t∧x1	PROPN
ejpam-709	162	19	,	,	PUNCT
ejpam-709	162	20	y2∧(y2∨x	y2∧(y2∨x	PROPN
ejpam-709	162	21	′2)∧x2	′2)∧x2	NOUN
ejpam-709	162	22	)	)	PUNCT
ejpam-709	162	23	=	=	SYM
ejpam-709	162	24	(	(	PUNCT
ejpam-709	162	25	x1	x1	PROPN
ejpam-709	162	26	,	,	PUNCT
ejpam-709	162	27	y2∧x2	y2∧x2	PROPN
ejpam-709	162	28	)	)	PUNCT
ejpam-709	162	29	=	=	SYM
ejpam-709	162	30	a∧x	a∧x	NOUN
ejpam-709	162	31	and	and	CCONJ
ejpam-709	162	32	x∧b∧	x∧b∧	PROPN
ejpam-709	162	33	y	y	PROPN
ejpam-709	162	34	=	=	SYM
ejpam-709	162	35	(	(	PUNCT
ejpam-709	162	36	x1∧	x1∧	NOUN
ejpam-709	162	37	t∧	t∧	PROPN
ejpam-709	162	38	y1	y1	PROPN
ejpam-709	162	39	,	,	PUNCT
ejpam-709	162	40	x2∧(x2∨	x2∧(x2∨	PROPN
ejpam-709	162	41	y	y	PROPN
ejpam-709	162	42	′2)∧	′2)∧	NOUN
ejpam-709	162	43	y2	y2	PROPN
ejpam-709	162	44	)	)	PUNCT
ejpam-709	163	1	=	=	PRON
ejpam-709	163	2	(	(	PUNCT
ejpam-709	163	3	y1	y1	INTJ
ejpam-709	163	4	,	,	PUNCT
ejpam-709	163	5	x2∧	x2∧	PROPN
ejpam-709	163	6	y2	y2	PROPN
ejpam-709	163	7	)	)	PUNCT
ejpam-709	163	8	=	=	PUNCT
ejpam-709	164	1	b∧	b∧	NUM
ejpam-709	164	2	y.	y.	NOUN
ejpam-709	164	3	also	also	ADV
ejpam-709	164	4	a∨	a∨	PROPN
ejpam-709	164	5	b	b	PROPN
ejpam-709	164	6	=	=	SYM
ejpam-709	164	7	(	(	PUNCT
ejpam-709	164	8	t	t	PROPN
ejpam-709	164	9	,	,	PUNCT
ejpam-709	164	10	1	1	NUM
ejpam-709	164	11	)	)	PUNCT
ejpam-709	164	12	.	.	PUNCT
ejpam-709	165	1	now	now	ADV
ejpam-709	165	2	,	,	PUNCT
ejpam-709	165	3	suppose	suppose	VERB
ejpam-709	165	4	x1	x1	PROPN
ejpam-709	165	5	=	=	SYM
ejpam-709	165	6	0	0	NUM
ejpam-709	165	7	and	and	CCONJ
ejpam-709	165	8	y1	y1	VERB
ejpam-709	165	9	6=	6=	ADP
ejpam-709	165	10	0	0	NUM
ejpam-709	165	11	.	.	PUNCT
ejpam-709	166	1	take	take	VERB
ejpam-709	166	2	a	a	DET
ejpam-709	166	3	=	=	SYM
ejpam-709	166	4	(	(	PUNCT
ejpam-709	166	5	t	t	PROPN
ejpam-709	166	6	,	,	PUNCT
ejpam-709	166	7	y2	y2	PROPN
ejpam-709	166	8	∨	∨	NOUN
ejpam-709	166	9	x	x	SYM
ejpam-709	166	10	′2	′2	NOUN
ejpam-709	166	11	)	)	PUNCT
ejpam-709	166	12	and	and	CCONJ
ejpam-709	166	13	b	b	X
ejpam-709	166	14	=	=	SYM
ejpam-709	166	15	(	(	PUNCT
ejpam-709	166	16	0	0	NUM
ejpam-709	166	17	,	,	PUNCT
ejpam-709	166	18	x2	x2	PROPN
ejpam-709	166	19	∨	∨	NUM
ejpam-709	166	20	y	y	PROPN
ejpam-709	166	21	′2	′2	PROPN
ejpam-709	166	22	)	)	PUNCT
ejpam-709	166	23	.	.	PUNCT
ejpam-709	167	1	now	now	ADV
ejpam-709	167	2	,	,	PUNCT
ejpam-709	167	3	y	y	PROPN
ejpam-709	167	4	∧	∧	PROPN
ejpam-709	167	5	a	a	DET
ejpam-709	167	6	∧	∧	PROPN
ejpam-709	167	7	x	x	X
ejpam-709	167	8	=	=	SYM
ejpam-709	167	9	(	(	PUNCT
ejpam-709	167	10	y1	y1	INTJ
ejpam-709	167	11	∧	∧	PROPN
ejpam-709	167	12	t	t	PROPN
ejpam-709	167	13	∧	∧	PROPN
ejpam-709	167	14	0	0	NUM
ejpam-709	167	15	,	,	PUNCT
ejpam-709	167	16	y2	y2	INTJ
ejpam-709	167	17	∧	∧	PROPN
ejpam-709	167	18	(	(	PUNCT
ejpam-709	168	1	y2	y2	PROPN
ejpam-709	168	2	∨	∨	NUM
ejpam-709	168	3	x	x	SYM
ejpam-709	168	4	′2)∧	′2)∧	NOUN
ejpam-709	168	5	x2	x2	NUM
ejpam-709	168	6	)	)	PUNCT
ejpam-709	169	1	=	=	SYM
ejpam-709	169	2	(	(	PUNCT
ejpam-709	169	3	0	0	NUM
ejpam-709	169	4	,	,	PUNCT
ejpam-709	169	5	y2	y2	INTJ
ejpam-709	169	6	∧	∧	NOUN
ejpam-709	169	7	x2	x2	PROPN
ejpam-709	169	8	)	)	PUNCT
ejpam-709	169	9	=	=	PUNCT
ejpam-709	170	1	a	a	DET
ejpam-709	170	2	∧	∧	PROPN
ejpam-709	170	3	x	x	X
ejpam-709	170	4	and	and	CCONJ
ejpam-709	170	5	x	x	PART
ejpam-709	170	6	∧	∧	NOUN
ejpam-709	170	7	b	b	PROPN
ejpam-709	170	8	∧	∧	PROPN
ejpam-709	170	9	y	y	PROPN
ejpam-709	170	10	=	=	PUNCT
ejpam-709	170	11	(	(	PUNCT
ejpam-709	170	12	0∧	0∧	NOUN
ejpam-709	170	13	0∧	0∧	NOUN
ejpam-709	170	14	y1	y1	NOUN
ejpam-709	170	15	,	,	PUNCT
ejpam-709	170	16	x2	x2	PROPN
ejpam-709	170	17	∧	∧	PROPN
ejpam-709	170	18	(	(	PUNCT
ejpam-709	170	19	x2	x2	PROPN
ejpam-709	170	20	∨	∨	NUM
ejpam-709	170	21	y	y	PROPN
ejpam-709	170	22	′2)∧	′2)∧	NOUN
ejpam-709	170	23	y2	y2	NOUN
ejpam-709	170	24	)	)	PUNCT
ejpam-709	170	25	=	=	SYM
ejpam-709	170	26	(	(	PUNCT
ejpam-709	170	27	0	0	NUM
ejpam-709	170	28	,	,	PUNCT
ejpam-709	170	29	x2	x2	PROPN
ejpam-709	170	30	∧	∧	PROPN
ejpam-709	170	31	y2	y2	PROPN
ejpam-709	170	32	)	)	PUNCT
ejpam-709	170	33	=	=	SYM
ejpam-709	171	1	b	b	X
ejpam-709	171	2	∧	∧	PROPN
ejpam-709	171	3	y.	y.	PROPN
ejpam-709	171	4	clearly	clearly	ADV
ejpam-709	171	5	a	a	DET
ejpam-709	171	6	∨	∨	NOUN
ejpam-709	171	7	b	b	NOUN
ejpam-709	171	8	=	=	SYM
ejpam-709	171	9	(	(	PUNCT
ejpam-709	171	10	t	t	PROPN
ejpam-709	171	11	,	,	PUNCT
ejpam-709	171	12	1	1	NUM
ejpam-709	171	13	)	)	PUNCT
ejpam-709	171	14	.	.	PUNCT
ejpam-709	172	1	thus	thus	ADV
ejpam-709	172	2	r	r	NOUN
ejpam-709	172	3	is	be	AUX
ejpam-709	172	4	an	an	DET
ejpam-709	172	5	s−relatively	s−relatively	ADV
ejpam-709	172	6	normal	normal	ADJ
ejpam-709	172	7	adl	adl	PROPN
ejpam-709	172	8	.	.	PUNCT
ejpam-709	173	1	lemma	lemma	PROPN
ejpam-709	173	2	4	4	X
ejpam-709	173	3	.	.	PUNCT
ejpam-709	174	1	let	let	VERB
ejpam-709	174	2	r	r	PRON
ejpam-709	174	3	be	be	AUX
ejpam-709	174	4	an	an	DET
ejpam-709	174	5	adl	adl	NOUN
ejpam-709	174	6	with	with	ADP
ejpam-709	174	7	maximal	maximal	ADJ
ejpam-709	174	8	elements	element	NOUN
ejpam-709	174	9	and	and	CCONJ
ejpam-709	174	10	s	s	VERB
ejpam-709	174	11	a	a	DET
ejpam-709	174	12	uni	uni	PROPN
ejpam-709	174	13	subadl	subadl	NOUN
ejpam-709	174	14	of	of	ADP
ejpam-709	174	15	r.	r.	PROPN
ejpam-709	174	16	if	if	SCONJ
ejpam-709	174	17	r	r	NOUN
ejpam-709	174	18	is	be	AUX
ejpam-709	174	19	s−relatively	s−relatively	ADV
ejpam-709	174	20	normal	normal	ADJ
ejpam-709	174	21	,	,	PUNCT
ejpam-709	174	22	then	then	ADV
ejpam-709	174	23	,	,	PUNCT
ejpam-709	174	24	for	for	SCONJ
ejpam-709	174	25	each	each	DET
ejpam-709	174	26	pair	pair	NOUN
ejpam-709	174	27	a	a	PRON
ejpam-709	174	28	,	,	PUNCT
ejpam-709	174	29	b	b	X
ejpam-709	174	30	∈	∈	NOUN
ejpam-709	174	31	s	s	VERB
ejpam-709	174	32	such	such	ADJ
ejpam-709	174	33	that	that	SCONJ
ejpam-709	174	34	a	a	DET
ejpam-709	174	35	<	<	X
ejpam-709	174	36	b	b	PROPN
ejpam-709	174	37	,	,	PUNCT
ejpam-709	174	38	the	the	DET
ejpam-709	174	39	segment	segment	NOUN
ejpam-709	174	40	[	[	X
ejpam-709	174	41	a	a	X
ejpam-709	174	42	,	,	PUNCT
ejpam-709	174	43	b	b	X
ejpam-709	174	44	]	]	X
ejpam-709	174	45	is	be	AUX
ejpam-709	174	46	an	an	DET
ejpam-709	174	47	s	s	NOUN
ejpam-709	174	48	∩	∩	NOUN
ejpam-709	174	49	[	[	X
ejpam-709	174	50	a	a	DET
ejpam-709	174	51	,	,	PUNCT
ejpam-709	174	52	b]−normal	b]−normal	ADJ
ejpam-709	174	53	lattice	lattice	NOUN
ejpam-709	174	54	.	.	PUNCT
ejpam-709	175	1	proof	proof	NOUN
ejpam-709	175	2	.	.	PUNCT
ejpam-709	176	1	let	let	VERB
ejpam-709	176	2	x	x	PRON
ejpam-709	176	3	,	,	PUNCT
ejpam-709	176	4	y	y	PROPN
ejpam-709	176	5	∈	∈	PROPN
ejpam-709	177	1	[	[	X
ejpam-709	177	2	a	a	X
ejpam-709	177	3	,	,	PUNCT
ejpam-709	177	4	b	b	NOUN
ejpam-709	177	5	]	]	X
ejpam-709	177	6	such	such	ADJ
ejpam-709	177	7	that	that	SCONJ
ejpam-709	177	8	x	x	PUNCT
ejpam-709	177	9	∧	∧	NOUN
ejpam-709	177	10	y	y	NOUN
ejpam-709	177	11	=	=	NOUN
ejpam-709	177	12	a.	a.	NOUN
ejpam-709	177	13	since	since	SCONJ
ejpam-709	177	14	r	r	NOUN
ejpam-709	177	15	is	be	AUX
ejpam-709	177	16	s−relatively	s−relatively	ADV
ejpam-709	177	17	normal	normal	ADJ
ejpam-709	177	18	,	,	PUNCT
ejpam-709	177	19	there	there	PRON
ejpam-709	177	20	exist	exist	VERB
ejpam-709	177	21	c	c	NOUN
ejpam-709	177	22	,	,	PUNCT
ejpam-709	177	23	d	d	PROPN
ejpam-709	177	24	∈	∈	PROPN
ejpam-709	177	25	s	s	VERB
ejpam-709	177	26	such	such	ADJ
ejpam-709	177	27	that	that	SCONJ
ejpam-709	177	28	y	y	PROPN
ejpam-709	177	29	∧	∧	PROPN
ejpam-709	177	30	c	c	NOUN
ejpam-709	177	31	∧	∧	NOUN
ejpam-709	177	32	x	x	X
ejpam-709	178	1	=	=	PUNCT
ejpam-709	178	2	c	c	X
ejpam-709	178	3	∧	∧	PROPN
ejpam-709	178	4	x	x	INTJ
ejpam-709	178	5	,	,	PUNCT
ejpam-709	178	6	x	x	PUNCT
ejpam-709	178	7	∧	∧	NOUN
ejpam-709	179	1	d	d	NOUN
ejpam-709	179	2	∧	∧	NOUN
ejpam-709	179	3	y	y	PROPN
ejpam-709	179	4	=	=	SYM
ejpam-709	179	5	d	d	PROPN
ejpam-709	179	6	∧	∧	PROPN
ejpam-709	179	7	y	y	PROPN
ejpam-709	179	8	and	and	CCONJ
ejpam-709	179	9	c	c	PROPN
ejpam-709	179	10	∨	∨	PROPN
ejpam-709	179	11	d	d	PROPN
ejpam-709	179	12	is	be	AUX
ejpam-709	179	13	a	a	DET
ejpam-709	179	14	maximal	maximal	ADJ
ejpam-709	179	15	element	element	NOUN
ejpam-709	179	16	.	.	PUNCT
ejpam-709	180	1	now	now	ADV
ejpam-709	180	2	,	,	PUNCT
ejpam-709	180	3	take	take	VERB
ejpam-709	180	4	c1	c1	NOUN
ejpam-709	180	5	=	=	PUNCT
ejpam-709	180	6	a	a	DET
ejpam-709	180	7	∨	∨	NOUN
ejpam-709	180	8	(	(	PUNCT
ejpam-709	180	9	c	c	PROPN
ejpam-709	180	10	∧	∧	PROPN
ejpam-709	180	11	b	b	PROPN
ejpam-709	180	12	)	)	PUNCT
ejpam-709	180	13	and	and	CCONJ
ejpam-709	180	14	d1	d1	PROPN
ejpam-709	180	15	=	=	PUNCT
ejpam-709	180	16	a	a	DET
ejpam-709	180	17	∨	∨	NOUN
ejpam-709	180	18	(	(	PUNCT
ejpam-709	180	19	d	d	PROPN
ejpam-709	180	20	∧	∧	PROPN
ejpam-709	180	21	b	b	PROPN
ejpam-709	180	22	)	)	PUNCT
ejpam-709	180	23	.	.	PUNCT
ejpam-709	181	1	clearly	clearly	ADV
ejpam-709	181	2	c1	c1	PROPN
ejpam-709	181	3	,	,	PUNCT
ejpam-709	181	4	d1	d1	PROPN
ejpam-709	181	5	∈	∈	PROPN
ejpam-709	182	1	[	[	X
ejpam-709	182	2	a	a	X
ejpam-709	182	3	,	,	PUNCT
ejpam-709	182	4	b	b	NOUN
ejpam-709	182	5	]	]	PUNCT
ejpam-709	182	6	∩	∩	PROPN
ejpam-709	182	7	s.	s.	PROPN
ejpam-709	182	8	now	now	ADV
ejpam-709	182	9	,	,	PUNCT
ejpam-709	182	10	c1	c1	PROPN
ejpam-709	182	11	∧	∧	PROPN
ejpam-709	182	12	x	x	INTJ
ejpam-709	182	13	=	=	PRON
ejpam-709	182	14	(	(	PUNCT
ejpam-709	182	15	a	a	DET
ejpam-709	182	16	∨	∨	NOUN
ejpam-709	182	17	(	(	PUNCT
ejpam-709	182	18	c	c	NOUN
ejpam-709	182	19	∧	∧	PROPN
ejpam-709	182	20	b))∧	b))∧	NOUN
ejpam-709	182	21	x	x	SYM
ejpam-709	182	22	=	=	SYM
ejpam-709	182	23	(	(	PUNCT
ejpam-709	182	24	a	a	DET
ejpam-709	182	25	∧	∧	PROPN
ejpam-709	182	26	x)∨	x)∨	PROPN
ejpam-709	182	27	(	(	PUNCT
ejpam-709	182	28	c	c	PROPN
ejpam-709	182	29	∧	∧	PROPN
ejpam-709	182	30	b	b	PROPN
ejpam-709	182	31	∧	∧	PROPN
ejpam-709	182	32	x	x	X
ejpam-709	182	33	)	)	PUNCT
ejpam-709	182	34	=	=	PUNCT
ejpam-709	182	35	a	a	DET
ejpam-709	182	36	∨	∨	NOUN
ejpam-709	182	37	(	(	PUNCT
ejpam-709	182	38	c	c	PROPN
ejpam-709	182	39	∧	∧	PROPN
ejpam-709	182	40	x	x	NOUN
ejpam-709	182	41	)	)	PUNCT
ejpam-709	182	42	=	=	PUNCT
ejpam-709	182	43	a	a	DET
ejpam-709	182	44	∨	∨	NOUN
ejpam-709	182	45	(	(	PUNCT
ejpam-709	182	46	c	c	PROPN
ejpam-709	182	47	∧	∧	PROPN
ejpam-709	182	48	y	y	PROPN
ejpam-709	182	49	∧	∧	PROPN
ejpam-709	182	50	x	x	X
ejpam-709	182	51	)	)	PUNCT
ejpam-709	182	52	=	=	PUNCT
ejpam-709	182	53	a	a	DET
ejpam-709	182	54	∨	∨	NOUN
ejpam-709	182	55	(	(	PUNCT
ejpam-709	182	56	c	c	PROPN
ejpam-709	182	57	∧	∧	PROPN
ejpam-709	182	58	a	a	NOUN
ejpam-709	182	59	)	)	PUNCT
ejpam-709	182	60	=	=	SYM
ejpam-709	182	61	a	a	PRON
ejpam-709	182	62	and	and	CCONJ
ejpam-709	182	63	d1∧	d1∧	ADJ
ejpam-709	182	64	y	y	PROPN
ejpam-709	182	65	=	=	PRON
ejpam-709	182	66	(	(	PUNCT
ejpam-709	182	67	a∨(d∧	a∨(d∧	X
ejpam-709	182	68	b))∧	b))∧	X
ejpam-709	182	69	y	y	X
ejpam-709	182	70	=	=	SYM
ejpam-709	182	71	(	(	PUNCT
ejpam-709	182	72	a∧	a∧	NOUN
ejpam-709	182	73	y)∨(d∧	y)∨(d∧	PROPN
ejpam-709	182	74	b∧	b∧	PROPN
ejpam-709	182	75	y	y	NOUN
ejpam-709	182	76	)	)	PUNCT
ejpam-709	182	77	=	=	PUNCT
ejpam-709	183	1	a∨(d∧	a∨(d∧	PUNCT
ejpam-709	183	2	y	y	X
ejpam-709	183	3	)	)	PUNCT
ejpam-709	183	4	=	=	PUNCT
ejpam-709	184	1	a∨(d∧	a∨(d∧	DET
ejpam-709	184	2	x∧	x∧	PROPN
ejpam-709	184	3	y	y	PROPN
ejpam-709	184	4	)	)	PUNCT
ejpam-709	184	5	=	=	PUNCT
ejpam-709	184	6	a∨(d∧a	a∨(d∧a	X
ejpam-709	184	7	)	)	PUNCT
ejpam-709	184	8	=	=	VERB
ejpam-709	184	9	a.	a.	NOUN
ejpam-709	184	10	clearly	clearly	ADV
ejpam-709	184	11	c1	c1	PROPN
ejpam-709	184	12	∨	∨	NUM
ejpam-709	184	13	d1	d1	PROPN
ejpam-709	184	14	=	=	SYM
ejpam-709	184	15	b.	b.	PROPN
ejpam-709	185	1	therefore	therefore	ADV
ejpam-709	185	2	[	[	X
ejpam-709	185	3	a	a	DET
ejpam-709	185	4	,	,	PUNCT
ejpam-709	185	5	b	b	X
ejpam-709	185	6	]	]	X
ejpam-709	185	7	is	be	AUX
ejpam-709	185	8	s	s	NOUN
ejpam-709	185	9	∩	∩	ADJ
ejpam-709	185	10	[	[	X
ejpam-709	185	11	a	a	DET
ejpam-709	185	12	,	,	PUNCT
ejpam-709	185	13	b]−normal	b]−normal	ADJ
ejpam-709	185	14	lattice	lattice	NOUN
ejpam-709	185	15	.	.	PUNCT
ejpam-709	186	1	the	the	DET
ejpam-709	186	2	following	follow	VERB
ejpam-709	186	3	two	two	NUM
ejpam-709	186	4	results	result	NOUN
ejpam-709	186	5	can	can	AUX
ejpam-709	186	6	be	be	AUX
ejpam-709	186	7	verified	verify	VERB
ejpam-709	186	8	easily	easily	ADV
ejpam-709	186	9	.	.	PUNCT
ejpam-709	187	1	lemma	lemma	PROPN
ejpam-709	187	2	5	5	X
ejpam-709	187	3	.	.	PUNCT
ejpam-709	188	1	let	let	VERB
ejpam-709	188	2	r	r	PRON
ejpam-709	188	3	be	be	AUX
ejpam-709	188	4	an	an	DET
ejpam-709	188	5	adl	adl	NOUN
ejpam-709	188	6	with	with	ADP
ejpam-709	188	7	maximal	maximal	ADJ
ejpam-709	188	8	elements	element	NOUN
ejpam-709	188	9	and	and	CCONJ
ejpam-709	188	10	s	s	VERB
ejpam-709	188	11	a	a	DET
ejpam-709	188	12	uni	uni	PROPN
ejpam-709	188	13	subadl	subadl	NOUN
ejpam-709	188	14	of	of	ADP
ejpam-709	188	15	r.	r.	PROPN
ejpam-709	188	16	then	then	ADV
ejpam-709	188	17	r	r	NOUN
ejpam-709	188	18	is	be	AUX
ejpam-709	188	19	s−relatively	s−relatively	ADV
ejpam-709	188	20	normal	normal	ADJ
ejpam-709	188	21	if	if	SCONJ
ejpam-709	188	22	and	and	CCONJ
ejpam-709	188	23	only	only	ADV
ejpam-709	188	24	if	if	SCONJ
ejpam-709	188	25	for	for	ADP
ejpam-709	188	26	any	any	DET
ejpam-709	188	27	x	x	NOUN
ejpam-709	188	28	,	,	PUNCT
ejpam-709	188	29	y	y	PROPN
ejpam-709	188	30	∈	∈	PROPN
ejpam-709	188	31	r	r	PROPN
ejpam-709	188	32	,	,	PUNCT
ejpam-709	188	33	⌊x	⌊x	NOUN
ejpam-709	188	34	,	,	PUNCT
ejpam-709	188	35	y⌋s	y⌋s	PROPN
ejpam-709	188	36	∨	∨	NUM
ejpam-709	188	37	⌊y	⌊y	PROPN
ejpam-709	188	38	,	,	PUNCT
ejpam-709	188	39	x⌋s	x⌋s	PROPN
ejpam-709	188	40	=	=	SYM
ejpam-709	188	41	s.	s.	PROPN
ejpam-709	188	42	g.	g.	PROPN
ejpam-709	188	43	rao	rao	PROPN
ejpam-709	188	44	,	,	PUNCT
ejpam-709	188	45	n.	n.	PROPN
ejpam-709	188	46	rafi	rafi	PROPN
ejpam-709	188	47	and	and	CCONJ
ejpam-709	188	48	b.	b.	PROPN
ejpam-709	188	49	kumar	kumar	PROPN
ejpam-709	188	50	/	/	SYM
ejpam-709	188	51	eur	eur	PROPN
ejpam-709	188	52	.	.	PUNCT
ejpam-709	189	1	j.	j.	PROPN
ejpam-709	189	2	pure	pure	PROPN
ejpam-709	189	3	appl	appl	PROPN
ejpam-709	189	4	.	.	PROPN
ejpam-709	189	5	math	math	PROPN
ejpam-709	189	6	,	,	PUNCT
ejpam-709	189	7	3	3	NUM
ejpam-709	189	8	(	(	PUNCT
ejpam-709	189	9	2010	2010	NUM
ejpam-709	189	10	)	)	PUNCT
ejpam-709	189	11	,	,	PUNCT
ejpam-709	189	12	704	704	NUM
ejpam-709	189	13	-	-	SYM
ejpam-709	189	14	716	716	NUM
ejpam-709	189	15	710	710	NUM
ejpam-709	189	16	lemma	lemma	PROPN
ejpam-709	189	17	6	6	NUM
ejpam-709	189	18	.	.	PUNCT
ejpam-709	190	1	let	let	VERB
ejpam-709	190	2	r	r	PRON
ejpam-709	190	3	be	be	AUX
ejpam-709	190	4	an	an	DET
ejpam-709	190	5	adl	adl	NOUN
ejpam-709	190	6	with	with	ADP
ejpam-709	190	7	maximal	maximal	ADJ
ejpam-709	190	8	elements	element	NOUN
ejpam-709	190	9	and	and	CCONJ
ejpam-709	190	10	s	s	VERB
ejpam-709	190	11	a	a	DET
ejpam-709	190	12	uni	uni	PROPN
ejpam-709	190	13	subadl	subadl	NOUN
ejpam-709	190	14	of	of	ADP
ejpam-709	190	15	r.	r.	PROPN
ejpam-709	190	16	then	then	ADV
ejpam-709	190	17	r	r	NOUN
ejpam-709	190	18	is	be	AUX
ejpam-709	190	19	s−relatively	s−relatively	ADV
ejpam-709	190	20	normal	normal	ADJ
ejpam-709	190	21	if	if	SCONJ
ejpam-709	190	22	and	and	CCONJ
ejpam-709	190	23	only	only	ADV
ejpam-709	190	24	if	if	SCONJ
ejpam-709	190	25	for	for	ADP
ejpam-709	190	26	any	any	DET
ejpam-709	190	27	prime	prime	ADJ
ejpam-709	190	28	filter	filter	NOUN
ejpam-709	190	29	f	f	PROPN
ejpam-709	190	30	of	of	ADP
ejpam-709	190	31	s	s	PRON
ejpam-709	190	32	and	and	CCONJ
ejpam-709	190	33	for	for	ADP
ejpam-709	190	34	any	any	DET
ejpam-709	190	35	x	x	NOUN
ejpam-709	190	36	,	,	PUNCT
ejpam-709	190	37	y	y	PROPN
ejpam-709	190	38	∈	∈	PROPN
ejpam-709	190	39	r	r	NOUN
ejpam-709	190	40	,	,	PUNCT
ejpam-709	190	41	there	there	PRON
ejpam-709	190	42	exists	exist	VERB
ejpam-709	190	43	a	a	DET
ejpam-709	190	44	∈	∈	NOUN
ejpam-709	190	45	f	f	NOUN
ejpam-709	190	46	such	such	ADJ
ejpam-709	190	47	that	that	SCONJ
ejpam-709	190	48	x	x	SYM
ejpam-709	190	49	∧	∧	PROPN
ejpam-709	190	50	a	a	PROPN
ejpam-709	190	51	and	and	CCONJ
ejpam-709	190	52	y	y	PROPN
ejpam-709	190	53	∧	∧	PROPN
ejpam-709	190	54	a	a	PRON
ejpam-709	190	55	are	be	AUX
ejpam-709	190	56	comparable	comparable	ADJ
ejpam-709	190	57	.	.	PUNCT
ejpam-709	191	1	theorem	theorem	NOUN
ejpam-709	191	2	6	6	NUM
ejpam-709	191	3	.	.	PUNCT
ejpam-709	192	1	let	let	VERB
ejpam-709	192	2	r	r	PRON
ejpam-709	192	3	be	be	AUX
ejpam-709	192	4	an	an	DET
ejpam-709	192	5	adl	adl	NOUN
ejpam-709	192	6	with	with	ADP
ejpam-709	192	7	maximal	maximal	ADJ
ejpam-709	192	8	elements	element	NOUN
ejpam-709	192	9	,	,	PUNCT
ejpam-709	192	10	s	s	VERB
ejpam-709	192	11	a	a	DET
ejpam-709	192	12	uni	uni	PROPN
ejpam-709	192	13	subadl	subadl	NOUN
ejpam-709	192	14	of	of	ADP
ejpam-709	192	15	r	r	PROPN
ejpam-709	192	16	,	,	PUNCT
ejpam-709	192	17	f	f	PROPN
ejpam-709	192	18	an	an	DET
ejpam-709	192	19	s−filter	s−filter	NOUN
ejpam-709	192	20	of	of	ADP
ejpam-709	192	21	r	r	NOUN
ejpam-709	192	22	and	and	CCONJ
ejpam-709	192	23	k	k	PROPN
ejpam-709	192	24	a	a	DET
ejpam-709	192	25	non	non	ADJ
ejpam-709	192	26	-	-	ADJ
ejpam-709	192	27	empty	empty	ADJ
ejpam-709	192	28	subset	subset	NOUN
ejpam-709	192	29	of	of	ADP
ejpam-709	192	30	r	r	NOUN
ejpam-709	192	31	,	,	PUNCT
ejpam-709	192	32	which	which	PRON
ejpam-709	192	33	is	be	AUX
ejpam-709	192	34	closed	close	VERB
ejpam-709	192	35	under	under	ADP
ejpam-709	192	36	the	the	DET
ejpam-709	192	37	operation	operation	NOUN
ejpam-709	192	38	join	join	VERB
ejpam-709	192	39	such	such	ADJ
ejpam-709	192	40	that	that	SCONJ
ejpam-709	192	41	f	f	PROPN
ejpam-709	192	42	∩	∩	PROPN
ejpam-709	192	43	k	k	PROPN
ejpam-709	192	44	=	=	PUNCT
ejpam-709	192	45	;	;	PUNCT
ejpam-709	192	46	.	.	PUNCT
ejpam-709	193	1	then	then	ADV
ejpam-709	193	2	there	there	PRON
ejpam-709	193	3	exists	exist	VERB
ejpam-709	193	4	an	an	DET
ejpam-709	193	5	s−prime	s−prime	NOUN
ejpam-709	193	6	filter	filter	NOUN
ejpam-709	193	7	p	p	NOUN
ejpam-709	193	8	of	of	ADP
ejpam-709	193	9	r	r	NOUN
ejpam-709	194	1	such	such	ADJ
ejpam-709	194	2	that	that	SCONJ
ejpam-709	194	3	f	f	PROPN
ejpam-709	194	4	⊆	⊆	NUM
ejpam-709	194	5	p	p	NOUN
ejpam-709	194	6	and	and	CCONJ
ejpam-709	194	7	p	p	NOUN
ejpam-709	194	8	∩	∩	NOUN
ejpam-709	194	9	k	k	PROPN
ejpam-709	194	10	=	=	PUNCT
ejpam-709	194	11	;	;	PUNCT
ejpam-709	194	12	.	.	PUNCT
ejpam-709	195	1	theorem	theorem	ADJ
ejpam-709	195	2	7	7	NUM
ejpam-709	195	3	.	.	PUNCT
ejpam-709	196	1	let	let	VERB
ejpam-709	196	2	r	r	PRON
ejpam-709	196	3	be	be	AUX
ejpam-709	196	4	an	an	DET
ejpam-709	196	5	adl	adl	NOUN
ejpam-709	196	6	with	with	ADP
ejpam-709	196	7	maximal	maximal	ADJ
ejpam-709	196	8	elements	element	NOUN
ejpam-709	196	9	and	and	CCONJ
ejpam-709	196	10	s	s	VERB
ejpam-709	196	11	a	a	DET
ejpam-709	196	12	uni	uni	PROPN
ejpam-709	196	13	subadl	subadl	NOUN
ejpam-709	196	14	of	of	ADP
ejpam-709	196	15	r.	r.	PROPN
ejpam-709	196	16	then	then	ADV
ejpam-709	196	17	the	the	DET
ejpam-709	196	18	following	follow	VERB
ejpam-709	196	19	conditions	condition	NOUN
ejpam-709	196	20	are	be	AUX
ejpam-709	196	21	equivalent	equivalent	ADJ
ejpam-709	196	22	:	:	PUNCT
ejpam-709	197	1	1	1	X
ejpam-709	197	2	.	.	X
ejpam-709	197	3	r	r	NOUN
ejpam-709	197	4	is	be	AUX
ejpam-709	197	5	s−relatively	s−relatively	ADV
ejpam-709	197	6	normal	normal	ADJ
ejpam-709	197	7	2	2	NUM
ejpam-709	197	8	.	.	PUNCT
ejpam-709	197	9	for	for	ADP
ejpam-709	197	10	each	each	DET
ejpam-709	197	11	pair	pair	NOUN
ejpam-709	197	12	x	x	X
ejpam-709	197	13	,	,	PUNCT
ejpam-709	197	14	y	y	PROPN
ejpam-709	197	15	∈	∈	PROPN
ejpam-709	197	16	r	r	NOUN
ejpam-709	197	17	,	,	PUNCT
ejpam-709	197	18	there	there	PRON
ejpam-709	197	19	is	be	VERB
ejpam-709	197	20	no	no	DET
ejpam-709	197	21	proper	proper	ADJ
ejpam-709	197	22	ideal	ideal	NOUN
ejpam-709	197	23	of	of	ADP
ejpam-709	197	24	s	s	PRON
ejpam-709	197	25	contain	contain	VERB
ejpam-709	197	26	both	both	DET
ejpam-709	197	27	⌊x	⌊x	PROPN
ejpam-709	197	28	,	,	PUNCT
ejpam-709	197	29	y⌋s	y⌋s	PROPN
ejpam-709	197	30	and	and	CCONJ
ejpam-709	197	31	⌊y	⌊y	PROPN
ejpam-709	197	32	,	,	PUNCT
ejpam-709	197	33	x⌋s	x⌋s	PROPN
ejpam-709	197	34	3	3	NUM
ejpam-709	197	35	.	.	PUNCT
ejpam-709	198	1	the	the	DET
ejpam-709	198	2	set	set	NOUN
ejpam-709	198	3	of	of	ADP
ejpam-709	198	4	all	all	DET
ejpam-709	198	5	filters	filter	NOUN
ejpam-709	198	6	of	of	ADP
ejpam-709	198	7	r	r	NOUN
ejpam-709	198	8	that	that	PRON
ejpam-709	198	9	contain	contain	VERB
ejpam-709	198	10	a	a	DET
ejpam-709	198	11	given	give	VERB
ejpam-709	198	12	s−prime	s−prime	NOUN
ejpam-709	198	13	filter	filter	NOUN
ejpam-709	198	14	of	of	ADP
ejpam-709	198	15	r	r	NOUN
ejpam-709	198	16	form	form	NOUN
ejpam-709	198	17	a	a	DET
ejpam-709	198	18	chain	chain	NOUN
ejpam-709	198	19	4	4	NUM
ejpam-709	198	20	.	.	PUNCT
ejpam-709	199	1	the	the	DET
ejpam-709	199	2	set	set	NOUN
ejpam-709	199	3	of	of	ADP
ejpam-709	199	4	all	all	DET
ejpam-709	199	5	prime	prime	ADJ
ejpam-709	199	6	filters	filter	NOUN
ejpam-709	199	7	of	of	ADP
ejpam-709	199	8	r	r	NOUN
ejpam-709	199	9	that	that	PRON
ejpam-709	199	10	contain	contain	VERB
ejpam-709	199	11	a	a	DET
ejpam-709	199	12	given	give	VERB
ejpam-709	199	13	s−prime	s−prime	NOUN
ejpam-709	199	14	filter	filter	NOUN
ejpam-709	199	15	of	of	ADP
ejpam-709	199	16	r	r	NOUN
ejpam-709	199	17	form	form	NOUN
ejpam-709	199	18	a	a	DET
ejpam-709	199	19	chain	chain	NOUN
ejpam-709	199	20	5	5	NUM
ejpam-709	199	21	.	.	PUNCT
ejpam-709	200	1	any	any	DET
ejpam-709	200	2	proper	proper	ADJ
ejpam-709	200	3	filter	filter	NOUN
ejpam-709	200	4	of	of	ADP
ejpam-709	200	5	r	r	NOUN
ejpam-709	200	6	that	that	PRON
ejpam-709	200	7	contain	contain	VERB
ejpam-709	200	8	a	a	DET
ejpam-709	200	9	given	give	VERB
ejpam-709	200	10	s−prime	s−prime	NOUN
ejpam-709	200	11	filter	filter	NOUN
ejpam-709	200	12	of	of	ADP
ejpam-709	200	13	r	r	NOUN
ejpam-709	200	14	is	be	AUX
ejpam-709	200	15	prime	prime	ADJ
ejpam-709	200	16	.	.	PUNCT
ejpam-709	201	1	proof	proof	NOUN
ejpam-709	201	2	.	.	PUNCT
ejpam-709	202	1	(	(	PUNCT
ejpam-709	202	2	1)⇒	1)⇒	NUM
ejpam-709	202	3	(	(	PUNCT
ejpam-709	202	4	2	2	NUM
ejpam-709	202	5	)	)	PUNCT
ejpam-709	202	6	:	:	PUNCT
ejpam-709	202	7	it	it	PRON
ejpam-709	202	8	follows	follow	VERB
ejpam-709	202	9	from	from	ADP
ejpam-709	202	10	lemma	lemma	PROPN
ejpam-709	202	11	5	5	NUM
ejpam-709	202	12	.	.	PUNCT
ejpam-709	203	1	(	(	PUNCT
ejpam-709	203	2	2)⇒	2)⇒	NUM
ejpam-709	203	3	(	(	PUNCT
ejpam-709	203	4	3	3	NUM
ejpam-709	203	5	)	)	PUNCT
ejpam-709	203	6	:	:	PUNCT
ejpam-709	203	7	assume	assume	VERB
ejpam-709	203	8	(	(	PUNCT
ejpam-709	203	9	2	2	NUM
ejpam-709	203	10	)	)	PUNCT
ejpam-709	203	11	.	.	PUNCT
ejpam-709	204	1	suppose	suppose	VERB
ejpam-709	204	2	p	p	NOUN
ejpam-709	204	3	is	be	AUX
ejpam-709	204	4	an	an	DET
ejpam-709	204	5	s−prime	s−prime	NOUN
ejpam-709	204	6	filter	filter	NOUN
ejpam-709	204	7	of	of	ADP
ejpam-709	204	8	r	r	NOUN
ejpam-709	204	9	and	and	CCONJ
ejpam-709	204	10	f1	f1	NOUN
ejpam-709	204	11	,	,	PUNCT
ejpam-709	204	12	f2	f2	PROPN
ejpam-709	204	13	are	be	AUX
ejpam-709	204	14	two	two	NUM
ejpam-709	204	15	filters	filter	NOUN
ejpam-709	204	16	of	of	ADP
ejpam-709	204	17	r	r	NOUN
ejpam-709	204	18	such	such	ADJ
ejpam-709	204	19	that	that	SCONJ
ejpam-709	204	20	p	p	PROPN
ejpam-709	204	21	⊆	⊆	NUM
ejpam-709	204	22	f1	f1	NOUN
ejpam-709	204	23	and	and	CCONJ
ejpam-709	204	24	p	p	NOUN
ejpam-709	204	25	⊆	⊆	NUM
ejpam-709	204	26	f2	f2	PROPN
ejpam-709	204	27	.	.	PUNCT
ejpam-709	205	1	suppose	suppose	VERB
ejpam-709	205	2	f1	f1	PROPN
ejpam-709	205	3	*	*	PUNCT
ejpam-709	205	4	f2	f2	PROPN
ejpam-709	205	5	and	and	CCONJ
ejpam-709	205	6	f2	f2	PROPN
ejpam-709	205	7	*	*	PUNCT
ejpam-709	205	8	f1	f1	PROPN
ejpam-709	205	9	.	.	PUNCT
ejpam-709	206	1	choose	choose	VERB
ejpam-709	206	2	x	x	PUNCT
ejpam-709	206	3	∈	∈	PROPN
ejpam-709	206	4	f1	f1	NOUN
ejpam-709	206	5	\	\	PROPN
ejpam-709	206	6	f2	f2	PROPN
ejpam-709	206	7	and	and	CCONJ
ejpam-709	206	8	y	y	PROPN
ejpam-709	206	9	∈	∈	PROPN
ejpam-709	206	10	f2	f2	PROPN
ejpam-709	206	11	\	\	PROPN
ejpam-709	206	12	f1	f1	NOUN
ejpam-709	206	13	.	.	PUNCT
ejpam-709	207	1	let	let	VERB
ejpam-709	207	2	a	a	DET
ejpam-709	207	3	∈	∈	ADJ
ejpam-709	207	4	⌊x	⌊x	NOUN
ejpam-709	207	5	,	,	PUNCT
ejpam-709	207	6	y⌋s	y⌋s	PROPN
ejpam-709	207	7	.	.	PUNCT
ejpam-709	208	1	then	then	ADV
ejpam-709	208	2	y	y	PROPN
ejpam-709	208	3	∧	∧	PROPN
ejpam-709	208	4	a	a	DET
ejpam-709	208	5	∧	∧	PROPN
ejpam-709	208	6	x	x	X
ejpam-709	208	7	=	=	PUNCT
ejpam-709	208	8	a	a	DET
ejpam-709	208	9	∧	∧	PROPN
ejpam-709	208	10	x	x	X
ejpam-709	208	11	.	.	PUNCT
ejpam-709	208	12	suppose	suppose	VERB
ejpam-709	208	13	a	a	DET
ejpam-709	208	14	/∈	/∈	PRON
ejpam-709	208	15	s	s	NOUN
ejpam-709	208	16	\	\	PUNCT
ejpam-709	209	1	(	(	PUNCT
ejpam-709	209	2	p	p	X
ejpam-709	209	3	∩	∩	X
ejpam-709	209	4	s	s	PART
ejpam-709	209	5	)	)	PUNCT
ejpam-709	209	6	.	.	PUNCT
ejpam-709	210	1	then	then	ADV
ejpam-709	210	2	a	a	DET
ejpam-709	210	3	∈	∈	PROPN
ejpam-709	210	4	p	p	NOUN
ejpam-709	210	5	∩	∩	ADJ
ejpam-709	210	6	s.	s.	PROPN
ejpam-709	210	7	that	that	PRON
ejpam-709	210	8	implies	imply	VERB
ejpam-709	210	9	a	a	DET
ejpam-709	210	10	∈	∈	PROPN
ejpam-709	210	11	f1	f1	NOUN
ejpam-709	210	12	and	and	CCONJ
ejpam-709	210	13	x	x	PART
ejpam-709	210	14	∈	∈	PROPN
ejpam-709	210	15	f1	f1	NOUN
ejpam-709	210	16	.	.	PUNCT
ejpam-709	211	1	hence	hence	ADV
ejpam-709	211	2	a∧	a∧	NOUN
ejpam-709	211	3	x	x	SYM
ejpam-709	211	4	∈	∈	PROPN
ejpam-709	211	5	f1	f1	NOUN
ejpam-709	211	6	.	.	PUNCT
ejpam-709	212	1	thus	thus	ADV
ejpam-709	212	2	y	y	PROPN
ejpam-709	212	3	∨	∨	NOUN
ejpam-709	212	4	(	(	PUNCT
ejpam-709	212	5	a∧	a∧	NOUN
ejpam-709	212	6	x	x	NOUN
ejpam-709	212	7	)	)	PUNCT
ejpam-709	212	8	=	=	SYM
ejpam-709	212	9	y	y	PROPN
ejpam-709	212	10	∈	∈	PROPN
ejpam-709	212	11	f1	f1	NOUN
ejpam-709	212	12	,	,	PUNCT
ejpam-709	212	13	which	which	PRON
ejpam-709	212	14	is	be	AUX
ejpam-709	212	15	a	a	DET
ejpam-709	212	16	contradiction	contradiction	NOUN
ejpam-709	212	17	.	.	PUNCT
ejpam-709	213	1	therefore	therefore	ADV
ejpam-709	213	2	a	a	DET
ejpam-709	213	3	∈	∈	NOUN
ejpam-709	213	4	s	s	PART
ejpam-709	213	5	\	\	PUNCT
ejpam-709	213	6	(	(	PUNCT
ejpam-709	213	7	p	p	X
ejpam-709	213	8	∩	∩	X
ejpam-709	213	9	s	s	PART
ejpam-709	213	10	)	)	PUNCT
ejpam-709	213	11	.	.	PUNCT
ejpam-709	214	1	hence	hence	ADV
ejpam-709	214	2	⌊x	⌊x	PROPN
ejpam-709	214	3	,	,	PUNCT
ejpam-709	214	4	y⌋s	y⌋s	PROPN
ejpam-709	214	5	⊆	⊆	NUM
ejpam-709	214	6	s	s	NOUN
ejpam-709	214	7	\	\	PUNCT
ejpam-709	214	8	(	(	PUNCT
ejpam-709	214	9	p	p	X
ejpam-709	214	10	∩	∩	X
ejpam-709	214	11	s	s	PART
ejpam-709	214	12	)	)	PUNCT
ejpam-709	214	13	and	and	CCONJ
ejpam-709	214	14	similarly	similarly	ADV
ejpam-709	214	15	,	,	PUNCT
ejpam-709	214	16	we	we	PRON
ejpam-709	214	17	have	have	VERB
ejpam-709	214	18	⌊y	⌊y	NOUN
ejpam-709	214	19	,	,	PUNCT
ejpam-709	214	20	x⌋s	x⌋s	PROPN
ejpam-709	215	1	⊆	⊆	NUM
ejpam-709	215	2	s	s	NOUN
ejpam-709	215	3	\	\	PUNCT
ejpam-709	215	4	(	(	PUNCT
ejpam-709	215	5	p	p	X
ejpam-709	215	6	∩	∩	X
ejpam-709	215	7	s	s	PART
ejpam-709	215	8	)	)	PUNCT
ejpam-709	215	9	.	.	PUNCT
ejpam-709	216	1	since	since	SCONJ
ejpam-709	216	2	s	s	PRON
ejpam-709	216	3	\	\	PROPN
ejpam-709	216	4	(	(	PUNCT
ejpam-709	216	5	p	p	X
ejpam-709	216	6	∩	∩	NOUN
ejpam-709	216	7	s	s	PART
ejpam-709	216	8	)	)	PUNCT
ejpam-709	216	9	is	be	AUX
ejpam-709	216	10	a	a	DET
ejpam-709	216	11	prime	prime	ADJ
ejpam-709	216	12	ideal	ideal	NOUN
ejpam-709	216	13	of	of	ADP
ejpam-709	216	14	s	s	PROPN
ejpam-709	216	15	,	,	PUNCT
ejpam-709	216	16	this	this	PRON
ejpam-709	216	17	is	be	AUX
ejpam-709	216	18	a	a	DET
ejpam-709	216	19	contradiction	contradiction	NOUN
ejpam-709	216	20	.	.	PUNCT
ejpam-709	217	1	(	(	PUNCT
ejpam-709	217	2	3)⇒	3)⇒	NUM
ejpam-709	217	3	(	(	PUNCT
ejpam-709	217	4	4	4	NUM
ejpam-709	217	5	)	)	PUNCT
ejpam-709	217	6	:	:	PUNCT
ejpam-709	217	7	clear	clear	ADJ
ejpam-709	217	8	.	.	PUNCT
ejpam-709	218	1	(	(	PUNCT
ejpam-709	218	2	4)⇒	4)⇒	X
ejpam-709	218	3	(	(	PUNCT
ejpam-709	218	4	5	5	NUM
ejpam-709	218	5	)	)	PUNCT
ejpam-709	218	6	:	:	PUNCT
ejpam-709	218	7	assume	assume	VERB
ejpam-709	218	8	(	(	PUNCT
ejpam-709	218	9	4	4	NUM
ejpam-709	218	10	)	)	PUNCT
ejpam-709	218	11	.	.	PUNCT
ejpam-709	219	1	let	let	VERB
ejpam-709	219	2	p	p	PRON
ejpam-709	219	3	be	be	AUX
ejpam-709	219	4	an	an	DET
ejpam-709	219	5	s−prime	s−prime	NOUN
ejpam-709	219	6	filter	filter	NOUN
ejpam-709	219	7	of	of	ADP
ejpam-709	219	8	r	r	NOUN
ejpam-709	219	9	and	and	CCONJ
ejpam-709	219	10	f	f	PROPN
ejpam-709	219	11	a	a	DET
ejpam-709	219	12	proper	proper	ADJ
ejpam-709	219	13	filter	filter	NOUN
ejpam-709	219	14	of	of	ADP
ejpam-709	219	15	r	r	NOUN
ejpam-709	219	16	such	such	ADJ
ejpam-709	219	17	that	that	SCONJ
ejpam-709	219	18	p	p	PROPN
ejpam-709	219	19	⊆	⊆	NUM
ejpam-709	219	20	f.	f.	NOUN
ejpam-709	219	21	suppose	suppose	VERB
ejpam-709	219	22	f	f	PROPN
ejpam-709	219	23	is	be	AUX
ejpam-709	219	24	not	not	PART
ejpam-709	219	25	prime	prime	ADJ
ejpam-709	219	26	filter	filter	NOUN
ejpam-709	219	27	of	of	ADP
ejpam-709	219	28	r.	r.	PROPN
ejpam-709	219	29	then	then	ADV
ejpam-709	219	30	there	there	PRON
ejpam-709	219	31	exist	exist	VERB
ejpam-709	219	32	a	a	DET
ejpam-709	219	33	,	,	PUNCT
ejpam-709	219	34	b	b	X
ejpam-709	219	35	∈	∈	NOUN
ejpam-709	219	36	r	r	NOUN
ejpam-709	220	1	such	such	DET
ejpam-709	220	2	that	that	SCONJ
ejpam-709	220	3	a	a	PRON
ejpam-709	220	4	/∈	/∈	NOUN
ejpam-709	221	1	f	f	NOUN
ejpam-709	221	2	,	,	PUNCT
ejpam-709	221	3	b	b	PROPN
ejpam-709	221	4	/∈	/∈	PUNCT
ejpam-709	221	5	f	f	PROPN
ejpam-709	221	6	and	and	CCONJ
ejpam-709	221	7	a	a	DET
ejpam-709	221	8	∨	∨	NUM
ejpam-709	221	9	b	b	PROPN
ejpam-709	221	10	∈	∈	PROPN
ejpam-709	221	11	f.	f.	NOUN
ejpam-709	222	1	then	then	ADV
ejpam-709	222	2	there	there	PRON
ejpam-709	222	3	exist	exist	VERB
ejpam-709	222	4	prime	prime	ADJ
ejpam-709	222	5	filters	filter	NOUN
ejpam-709	222	6	pa	pa	PROPN
ejpam-709	222	7	,	,	PUNCT
ejpam-709	222	8	pb	pb	ADP
ejpam-709	222	9	of	of	ADP
ejpam-709	222	10	r	r	NOUN
ejpam-709	223	1	such	such	ADJ
ejpam-709	223	2	that	that	SCONJ
ejpam-709	223	3	a	a	PRON
ejpam-709	223	4	/∈	/∈	PUNCT
ejpam-709	223	5	pa	pa	PROPN
ejpam-709	223	6	,	,	PUNCT
ejpam-709	223	7	b	b	PROPN
ejpam-709	223	8	/∈	/∈	PUNCT
ejpam-709	223	9	pb	pb	X
ejpam-709	223	10	and	and	CCONJ
ejpam-709	223	11	f	f	PROPN
ejpam-709	223	12	⊆	⊆	NUM
ejpam-709	223	13	pa	pa	PROPN
ejpam-709	223	14	∩	∩	PROPN
ejpam-709	223	15	pb	pb	PROPN
ejpam-709	223	16	.	.	PUNCT
ejpam-709	224	1	since	since	SCONJ
ejpam-709	224	2	a∨	a∨	PROPN
ejpam-709	224	3	b	b	PROPN
ejpam-709	224	4	∈	∈	PROPN
ejpam-709	224	5	pa	pa	PROPN
ejpam-709	224	6	∩	∩	PROPN
ejpam-709	224	7	pb	pb	ADP
ejpam-709	224	8	,	,	PUNCT
ejpam-709	224	9	we	we	PRON
ejpam-709	224	10	get	get	VERB
ejpam-709	224	11	b	b	DET
ejpam-709	224	12	∈	∈	PROPN
ejpam-709	224	13	pa	pa	PROPN
ejpam-709	224	14	and	and	CCONJ
ejpam-709	224	15	a	a	DET
ejpam-709	224	16	∈	∈	NOUN
ejpam-709	224	17	pb	pb	ADP
ejpam-709	224	18	.	.	PUNCT
ejpam-709	225	1	therefore	therefore	ADV
ejpam-709	225	2	pa	pa	PROPN
ejpam-709	225	3	*	*	PUNCT
ejpam-709	225	4	pb	pb	ADP
ejpam-709	225	5	and	and	CCONJ
ejpam-709	225	6	pb	pb	X
ejpam-709	225	7	*	*	PUNCT
ejpam-709	225	8	pa	pa	PROPN
ejpam-709	225	9	,	,	PUNCT
ejpam-709	225	10	which	which	PRON
ejpam-709	225	11	is	be	AUX
ejpam-709	225	12	a	a	DET
ejpam-709	225	13	contradiction	contradiction	NOUN
ejpam-709	225	14	.	.	PUNCT
ejpam-709	226	1	hence	hence	ADV
ejpam-709	226	2	f	f	PROPN
ejpam-709	226	3	is	be	AUX
ejpam-709	226	4	a	a	DET
ejpam-709	226	5	prime	prime	ADJ
ejpam-709	226	6	filter	filter	NOUN
ejpam-709	226	7	of	of	ADP
ejpam-709	226	8	r.	r.	PROPN
ejpam-709	226	9	(	(	PUNCT
ejpam-709	226	10	5	5	NUM
ejpam-709	226	11	)	)	PUNCT
ejpam-709	226	12	⇒	⇒	NOUN
ejpam-709	226	13	(	(	PUNCT
ejpam-709	226	14	1	1	NUM
ejpam-709	226	15	)	)	PUNCT
ejpam-709	226	16	:	:	PUNCT
ejpam-709	226	17	assume	assume	VERB
ejpam-709	226	18	(	(	PUNCT
ejpam-709	226	19	5	5	NUM
ejpam-709	226	20	)	)	PUNCT
ejpam-709	226	21	.	.	PUNCT
ejpam-709	227	1	let	let	VERB
ejpam-709	227	2	x	x	PRON
ejpam-709	227	3	,	,	PUNCT
ejpam-709	227	4	y	y	PROPN
ejpam-709	227	5	∈	∈	PROPN
ejpam-709	227	6	r.	r.	PROPN
ejpam-709	227	7	suppose	suppose	VERB
ejpam-709	227	8	⌊x	⌊x	PROPN
ejpam-709	227	9	,	,	PUNCT
ejpam-709	227	10	y⌋s	y⌋s	PROPN
ejpam-709	227	11	∨	∨	NUM
ejpam-709	227	12	⌊y	⌊y	PROPN
ejpam-709	227	13	,	,	PUNCT
ejpam-709	227	14	x⌋s	x⌋s	PROPN
ejpam-709	227	15	6=	6=	PROPN
ejpam-709	228	1	s.	s.	PROPN
ejpam-709	228	2	let	let	VERB
ejpam-709	228	3	m	m	PRON
ejpam-709	228	4	be	be	AUX
ejpam-709	228	5	any	any	DET
ejpam-709	228	6	maximal	maximal	ADJ
ejpam-709	228	7	element	element	NOUN
ejpam-709	228	8	in	in	ADP
ejpam-709	228	9	r.	r.	PROPN
ejpam-709	228	10	then	then	ADV
ejpam-709	228	11	m	m	PROPN
ejpam-709	228	12	/∈	/∈	PROPN
ejpam-709	229	1	⌊x	⌊x	PROPN
ejpam-709	229	2	,	,	PUNCT
ejpam-709	229	3	y⌋s	y⌋s	PROPN
ejpam-709	229	4	∨⌊y	∨⌊y	PROPN
ejpam-709	229	5	,	,	PUNCT
ejpam-709	229	6	x⌋s	x⌋s	PROPN
ejpam-709	229	7	and	and	CCONJ
ejpam-709	229	8	hence	hence	ADV
ejpam-709	229	9	there	there	PRON
ejpam-709	229	10	exists	exist	VERB
ejpam-709	229	11	an	an	DET
ejpam-709	229	12	prime	prime	ADJ
ejpam-709	229	13	filter	filter	NOUN
ejpam-709	229	14	p	p	NOUN
ejpam-709	229	15	′	′	NUM
ejpam-709	229	16	of	of	ADP
ejpam-709	229	17	s	s	PRON
ejpam-709	230	1	such	such	ADJ
ejpam-709	230	2	that	that	PRON
ejpam-709	230	3	(	(	PUNCT
ejpam-709	230	4	⌊x	⌊x	NOUN
ejpam-709	230	5	,	,	PUNCT
ejpam-709	230	6	y⌋s∨⌊y	y⌋s∨⌊y	PROPN
ejpam-709	230	7	,	,	PUNCT
ejpam-709	230	8	x⌋s)∩p	x⌋s)∩p	NUM
ejpam-709	230	9	′	′	NUM
ejpam-709	230	10	=	=	PUNCT
ejpam-709	230	11	;	;	PUNCT
ejpam-709	230	12	.	.	PUNCT
ejpam-709	231	1	so	so	ADV
ejpam-709	231	2	that	that	DET
ejpam-709	231	3	⌊x	⌊x	PROPN
ejpam-709	231	4	,	,	PUNCT
ejpam-709	231	5	y⌋s∩p	y⌋s∩p	PROPN
ejpam-709	231	6	′	′	NUM
ejpam-709	231	7	=	=	PUNCT
ejpam-709	231	8	;	;	PUNCT
ejpam-709	231	9	and	and	CCONJ
ejpam-709	231	10	⌊y	⌊y	PRON
ejpam-709	231	11	,	,	PUNCT
ejpam-709	231	12	x⌋s∩p	x⌋s∩p	PROPN
ejpam-709	231	13	′	′	NUM
ejpam-709	232	1	=	=	PUNCT
ejpam-709	233	1	;	;	PUNCT
ejpam-709	233	2	.	.	PUNCT
ejpam-709	234	1	let	let	VERB
ejpam-709	234	2	p	p	PRON
ejpam-709	234	3	be	be	AUX
ejpam-709	234	4	the	the	DET
ejpam-709	234	5	filter	filter	NOUN
ejpam-709	234	6	of	of	ADP
ejpam-709	234	7	r	r	NOUN
ejpam-709	234	8	generated	generate	VERB
ejpam-709	234	9	by	by	ADP
ejpam-709	234	10	p	p	PROPN
ejpam-709	234	11	′.	′.	NOUN
ejpam-709	234	12	by	by	ADP
ejpam-709	234	13	the	the	DET
ejpam-709	234	14	lemma	lemma	PROPN
ejpam-709	234	15	2	2	NUM
ejpam-709	234	16	,	,	PUNCT
ejpam-709	234	17	we	we	PRON
ejpam-709	234	18	get	get	VERB
ejpam-709	234	19	that	that	PRON
ejpam-709	234	20	p	p	NOUN
ejpam-709	234	21	is	be	AUX
ejpam-709	234	22	an	an	DET
ejpam-709	234	23	s−prime	s−prime	NOUN
ejpam-709	234	24	filter	filter	NOUN
ejpam-709	234	25	of	of	ADP
ejpam-709	234	26	r	r	NOUN
ejpam-709	234	27	and	and	CCONJ
ejpam-709	234	28	p	p	NOUN
ejpam-709	235	1	′	′	NOUN
ejpam-709	235	2	=	=	SYM
ejpam-709	235	3	p	p	NOUN
ejpam-709	235	4	∩	∩	PROPN
ejpam-709	235	5	s.	s.	PROPN
ejpam-709	235	6	if	if	SCONJ
ejpam-709	235	7	0	0	NUM
ejpam-709	235	8	∈	∈	PROPN
ejpam-709	235	9	p	p	NOUN
ejpam-709	235	10	∨	∨	NOUN
ejpam-709	235	11	[	[	X
ejpam-709	235	12	x	x	X
ejpam-709	235	13	∨	∨	NUM
ejpam-709	235	14	y	y	PROPN
ejpam-709	235	15	)	)	PUNCT
ejpam-709	235	16	,	,	PUNCT
ejpam-709	235	17	then	then	ADV
ejpam-709	235	18	0	0	X
ejpam-709	236	1	=	=	SYM
ejpam-709	236	2	p	p	PROPN
ejpam-709	236	3	∧	∧	PROPN
ejpam-709	236	4	(	(	PUNCT
ejpam-709	236	5	x	x	PROPN
ejpam-709	236	6	∨	∨	NUM
ejpam-709	236	7	y	y	PROPN
ejpam-709	236	8	)	)	PUNCT
ejpam-709	236	9	and	and	CCONJ
ejpam-709	236	10	hence	hence	ADV
ejpam-709	236	11	p	p	X
ejpam-709	236	12	∧	∧	PROPN
ejpam-709	236	13	x	x	X
ejpam-709	236	14	=	=	SYM
ejpam-709	236	15	0	0	NUM
ejpam-709	236	16	and	and	CCONJ
ejpam-709	236	17	p	p	X
ejpam-709	236	18	∧	∧	PROPN
ejpam-709	236	19	y	y	PROPN
ejpam-709	236	20	=	=	NOUN
ejpam-709	236	21	0	0	PROPN
ejpam-709	236	22	.	.	PUNCT
ejpam-709	237	1	since	since	SCONJ
ejpam-709	237	2	p	p	PROPN
ejpam-709	237	3	∈	∈	PROPN
ejpam-709	237	4	p	p	X
ejpam-709	237	5	,	,	PUNCT
ejpam-709	237	6	there	there	PRON
ejpam-709	237	7	exists	exist	VERB
ejpam-709	237	8	s	s	PROPN
ejpam-709	237	9	∈	∈	PROPN
ejpam-709	237	10	p	p	NOUN
ejpam-709	237	11	∩	∩	NOUN
ejpam-709	237	12	s	s	PART
ejpam-709	237	13	=	=	X
ejpam-709	237	14	p	p	NOUN
ejpam-709	237	15	′	′	NUM
ejpam-709	237	16	such	such	ADJ
ejpam-709	237	17	that	that	SCONJ
ejpam-709	237	18	p	p	PROPN
ejpam-709	237	19	∨	∨	NUM
ejpam-709	237	20	s	s	PART
ejpam-709	238	1	=	=	NOUN
ejpam-709	238	2	p.	p.	NOUN
ejpam-709	238	3	now	now	ADV
ejpam-709	238	4	,	,	PUNCT
ejpam-709	238	5	we	we	PRON
ejpam-709	238	6	prove	prove	VERB
ejpam-709	238	7	that	that	SCONJ
ejpam-709	238	8	the	the	DET
ejpam-709	238	9	filter	filter	NOUN
ejpam-709	238	10	p	p	NOUN
ejpam-709	238	11	∨	∨	NOUN
ejpam-709	238	12	[	[	X
ejpam-709	238	13	x	x	PROPN
ejpam-709	238	14	∨	∨	NUM
ejpam-709	238	15	y	y	NOUN
ejpam-709	238	16	)	)	PUNCT
ejpam-709	238	17	is	be	AUX
ejpam-709	238	18	a	a	DET
ejpam-709	238	19	proper	proper	ADJ
ejpam-709	238	20	filter	filter	NOUN
ejpam-709	238	21	of	of	ADP
ejpam-709	238	22	r.	r.	PROPN
ejpam-709	238	23	now	now	ADV
ejpam-709	238	24	,	,	PUNCT
ejpam-709	238	25	s	s	VERB
ejpam-709	238	26	∧	∧	NOUN
ejpam-709	238	27	x	x	X
ejpam-709	238	28	=	=	PUNCT
ejpam-709	238	29	p	p	PROPN
ejpam-709	238	30	∧	∧	PROPN
ejpam-709	238	31	s	s	PART
ejpam-709	238	32	∧	∧	PROPN
ejpam-709	238	33	x	x	PUNCT
ejpam-709	238	34	=	=	NOUN
ejpam-709	238	35	0	0	PROPN
ejpam-709	238	36	.	.	PUNCT
ejpam-709	239	1	so	so	ADV
ejpam-709	239	2	g.	g.	PROPN
ejpam-709	239	3	rao	rao	PROPN
ejpam-709	239	4	,	,	PUNCT
ejpam-709	239	5	n.	n.	PROPN
ejpam-709	239	6	rafi	rafi	PROPN
ejpam-709	239	7	and	and	CCONJ
ejpam-709	239	8	b.	b.	PROPN
ejpam-709	239	9	kumar	kumar	PROPN
ejpam-709	239	10	/	/	SYM
ejpam-709	239	11	eur	eur	PROPN
ejpam-709	239	12	.	.	PUNCT
ejpam-709	240	1	j.	j.	PROPN
ejpam-709	240	2	pure	pure	PROPN
ejpam-709	240	3	appl	appl	PROPN
ejpam-709	240	4	.	.	PROPN
ejpam-709	240	5	math	math	PROPN
ejpam-709	240	6	,	,	PUNCT
ejpam-709	240	7	3	3	NUM
ejpam-709	240	8	(	(	PUNCT
ejpam-709	240	9	2010	2010	NUM
ejpam-709	240	10	)	)	PUNCT
ejpam-709	240	11	,	,	PUNCT
ejpam-709	240	12	704	704	NUM
ejpam-709	240	13	-	-	SYM
ejpam-709	240	14	716	716	NUM
ejpam-709	240	15	711	711	NUM
ejpam-709	240	16	that	that	PRON
ejpam-709	240	17	s	s	VERB
ejpam-709	240	18	∈	∈	X
ejpam-709	240	19	⌊x	⌊x	NOUN
ejpam-709	240	20	,	,	PUNCT
ejpam-709	240	21	y⌋s	y⌋s	PROPN
ejpam-709	240	22	∩	∩	NOUN
ejpam-709	240	23	p	p	PROPN
ejpam-709	240	24	′	′	NOUN
ejpam-709	240	25	,	,	PUNCT
ejpam-709	240	26	which	which	PRON
ejpam-709	240	27	is	be	AUX
ejpam-709	240	28	a	a	DET
ejpam-709	240	29	contradiction	contradiction	NOUN
ejpam-709	240	30	.	.	PUNCT
ejpam-709	241	1	therefore	therefore	ADV
ejpam-709	241	2	p	p	X
ejpam-709	241	3	∨	∨	PROPN
ejpam-709	241	4	[	[	X
ejpam-709	241	5	x	x	PROPN
ejpam-709	241	6	∨	∨	NUM
ejpam-709	241	7	y	y	NOUN
ejpam-709	241	8	)	)	PUNCT
ejpam-709	241	9	is	be	AUX
ejpam-709	241	10	a	a	DET
ejpam-709	241	11	proper	proper	ADJ
ejpam-709	241	12	filter	filter	NOUN
ejpam-709	241	13	of	of	ADP
ejpam-709	241	14	r	r	NOUN
ejpam-709	241	15	containing	contain	VERB
ejpam-709	241	16	p.	p.	NOUN
ejpam-709	241	17	by	by	ADP
ejpam-709	241	18	our	our	PRON
ejpam-709	241	19	assumption	assumption	NOUN
ejpam-709	241	20	,	,	PUNCT
ejpam-709	241	21	p	p	PROPN
ejpam-709	241	22	∨	∨	NOUN
ejpam-709	241	23	[	[	X
ejpam-709	241	24	x	x	PROPN
ejpam-709	241	25	∨	∨	NUM
ejpam-709	241	26	y	y	NOUN
ejpam-709	241	27	)	)	PUNCT
ejpam-709	242	1	is	be	AUX
ejpam-709	242	2	a	a	DET
ejpam-709	242	3	prime	prime	ADJ
ejpam-709	242	4	filter	filter	NOUN
ejpam-709	242	5	of	of	ADP
ejpam-709	242	6	r.	r.	PROPN
ejpam-709	242	7	without	without	ADP
ejpam-709	242	8	loss	loss	NOUN
ejpam-709	242	9	of	of	ADP
ejpam-709	242	10	generality	generality	NOUN
ejpam-709	242	11	,	,	PUNCT
ejpam-709	242	12	suppose	suppose	VERB
ejpam-709	242	13	x	x	X
ejpam-709	242	14	∈	∈	PROPN
ejpam-709	242	15	p	p	NOUN
ejpam-709	242	16	∨	∨	NOUN
ejpam-709	243	1	[	[	X
ejpam-709	243	2	x	x	X
ejpam-709	243	3	∨	∨	NUM
ejpam-709	243	4	y	y	PROPN
ejpam-709	243	5	)	)	PUNCT
ejpam-709	243	6	.	.	PUNCT
ejpam-709	244	1	then	then	ADV
ejpam-709	244	2	x	x	X
ejpam-709	244	3	=	=	SYM
ejpam-709	244	4	t	t	PROPN
ejpam-709	244	5	∧	∧	PROPN
ejpam-709	244	6	(	(	PUNCT
ejpam-709	244	7	x	x	PROPN
ejpam-709	244	8	∨	∨	NUM
ejpam-709	244	9	y	y	PROPN
ejpam-709	244	10	)	)	PUNCT
ejpam-709	244	11	,	,	PUNCT
ejpam-709	244	12	for	for	ADP
ejpam-709	244	13	some	some	DET
ejpam-709	244	14	t	t	NOUN
ejpam-709	244	15	∈	∈	PROPN
ejpam-709	244	16	p.	p.	NOUN
ejpam-709	244	17	since	since	SCONJ
ejpam-709	244	18	t	t	PROPN
ejpam-709	244	19	∈	∈	PROPN
ejpam-709	244	20	p	p	X
ejpam-709	244	21	,	,	PUNCT
ejpam-709	244	22	there	there	PRON
ejpam-709	244	23	exists	exist	VERB
ejpam-709	244	24	s1	s1	PROPN
ejpam-709	244	25	∈	∈	PROPN
ejpam-709	244	26	p∩s	p∩s	NOUN
ejpam-709	244	27	such	such	ADJ
ejpam-709	244	28	that	that	SCONJ
ejpam-709	244	29	t∨s1	t∨s1	NUM
ejpam-709	244	30	=	=	PUNCT
ejpam-709	245	1	t.	t.	NOUN
ejpam-709	245	2	now	now	ADV
ejpam-709	245	3	,	,	PUNCT
ejpam-709	245	4	s1∧	s1∧	VERB
ejpam-709	245	5	x	x	PUNCT
ejpam-709	246	1	=	=	PUNCT
ejpam-709	246	2	s1∧	s1∧	PROPN
ejpam-709	246	3	t∧(x∨	t∧(x∨	PROPN
ejpam-709	246	4	y	y	PROPN
ejpam-709	246	5	)	)	PUNCT
ejpam-709	246	6	=	=	PUNCT
ejpam-709	247	1	(	(	PUNCT
ejpam-709	247	2	s1∧	s1∧	VERB
ejpam-709	247	3	x)∨(s1∧	x)∨(s1∧	PROPN
ejpam-709	247	4	y	y	PROPN
ejpam-709	247	5	)	)	PUNCT
ejpam-709	247	6	and	and	CCONJ
ejpam-709	247	7	hence	hence	ADV
ejpam-709	247	8	s1	s1	PROPN
ejpam-709	247	9	∧	∧	PROPN
ejpam-709	247	10	y	y	PROPN
ejpam-709	247	11	=	=	SYM
ejpam-709	247	12	s1	s1	PROPN
ejpam-709	247	13	∧	∧	PROPN
ejpam-709	247	14	x	x	SYM
ejpam-709	247	15	∧	∧	PROPN
ejpam-709	247	16	s1	s1	NOUN
ejpam-709	247	17	∧	∧	PROPN
ejpam-709	247	18	y	y	PROPN
ejpam-709	247	19	=	=	PUNCT
ejpam-709	247	20	x	x	SYM
ejpam-709	247	21	∧	∧	NOUN
ejpam-709	247	22	s1	s1	PROPN
ejpam-709	247	23	∧	∧	PROPN
ejpam-709	247	24	y.	y.	PROPN
ejpam-709	247	25	that	that	PRON
ejpam-709	247	26	implies	imply	VERB
ejpam-709	247	27	s1	s1	PROPN
ejpam-709	247	28	∈	∈	PROPN
ejpam-709	247	29	⌊y	⌊y	NOUN
ejpam-709	247	30	,	,	PUNCT
ejpam-709	247	31	x⌋s	x⌋s	PROPN
ejpam-709	247	32	∩	∩	PROPN
ejpam-709	247	33	p	p	X
ejpam-709	247	34	,	,	PUNCT
ejpam-709	247	35	which	which	PRON
ejpam-709	247	36	is	be	AUX
ejpam-709	247	37	a	a	DET
ejpam-709	247	38	contradiction	contradiction	NOUN
ejpam-709	247	39	.	.	PUNCT
ejpam-709	248	1	therefore	therefore	ADV
ejpam-709	248	2	⌊x	⌊x	PROPN
ejpam-709	248	3	,	,	PUNCT
ejpam-709	248	4	y⌋s	y⌋s	PROPN
ejpam-709	248	5	∨	∨	NUM
ejpam-709	248	6	⌊y	⌊y	PROPN
ejpam-709	248	7	,	,	PUNCT
ejpam-709	248	8	x⌋s	x⌋s	PROPN
ejpam-709	248	9	=	=	PROPN
ejpam-709	249	1	s.	s.	PROPN
ejpam-709	249	2	corollary	corollary	PROPN
ejpam-709	249	3	1	1	X
ejpam-709	249	4	.	.	PUNCT
ejpam-709	250	1	let	let	VERB
ejpam-709	250	2	r	r	PRON
ejpam-709	250	3	be	be	AUX
ejpam-709	250	4	an	an	DET
ejpam-709	250	5	adl	adl	NOUN
ejpam-709	250	6	with	with	ADP
ejpam-709	250	7	maximal	maximal	ADJ
ejpam-709	250	8	elements	element	NOUN
ejpam-709	250	9	and	and	CCONJ
ejpam-709	250	10	s1,s2	s1,s2	PROPN
ejpam-709	250	11	uni	uni	INTJ
ejpam-709	250	12	subadls	subadls	NOUN
ejpam-709	250	13	of	of	ADP
ejpam-709	250	14	r	r	NOUN
ejpam-709	250	15	such	such	ADJ
ejpam-709	250	16	that	that	SCONJ
ejpam-709	250	17	s1	s1	PROPN
ejpam-709	250	18	⊆	⊆	NUM
ejpam-709	250	19	s2	s2	PROPN
ejpam-709	250	20	.	.	PUNCT
ejpam-709	251	1	then	then	ADV
ejpam-709	251	2	the	the	DET
ejpam-709	251	3	following	follow	VERB
ejpam-709	251	4	conditions	condition	NOUN
ejpam-709	251	5	are	be	AUX
ejpam-709	251	6	equivalent	equivalent	ADJ
ejpam-709	251	7	:	:	PUNCT
ejpam-709	251	8	1	1	X
ejpam-709	251	9	.	.	X
ejpam-709	251	10	r	r	NOUN
ejpam-709	251	11	is	be	AUX
ejpam-709	251	12	s1−relatively	s1−relatively	ADV
ejpam-709	251	13	normal	normal	ADJ
ejpam-709	251	14	2	2	NUM
ejpam-709	251	15	.	.	PUNCT
ejpam-709	252	1	r	r	NOUN
ejpam-709	252	2	is	be	AUX
ejpam-709	252	3	s2−relatively	s2−relatively	ADV
ejpam-709	252	4	normal	normal	ADJ
ejpam-709	252	5	and	and	CCONJ
ejpam-709	252	6	the	the	DET
ejpam-709	252	7	filters	filter	NOUN
ejpam-709	252	8	generated	generate	VERB
ejpam-709	252	9	in	in	ADP
ejpam-709	252	10	s2	s2	PROPN
ejpam-709	252	11	by	by	ADP
ejpam-709	252	12	prime	prime	ADJ
ejpam-709	252	13	filters	filter	NOUN
ejpam-709	252	14	of	of	ADP
ejpam-709	252	15	s1	s1	NOUN
ejpam-709	252	16	are	be	AUX
ejpam-709	252	17	prime	prime	ADJ
ejpam-709	252	18	.	.	PUNCT
ejpam-709	253	1	proof	proof	NOUN
ejpam-709	253	2	.	.	PUNCT
ejpam-709	254	1	(	(	PUNCT
ejpam-709	254	2	1	1	X
ejpam-709	254	3	)	)	PUNCT
ejpam-709	254	4	⇒	⇒	NOUN
ejpam-709	254	5	(	(	PUNCT
ejpam-709	254	6	2	2	NUM
ejpam-709	254	7	)	)	PUNCT
ejpam-709	254	8	:	:	PUNCT
ejpam-709	254	9	assume	assume	VERB
ejpam-709	254	10	that	that	SCONJ
ejpam-709	254	11	r	r	NOUN
ejpam-709	254	12	is	be	AUX
ejpam-709	254	13	s1−relatively	s1−relatively	ADV
ejpam-709	254	14	normal	normal	ADJ
ejpam-709	254	15	.	.	PUNCT
ejpam-709	255	1	clearly	clearly	ADV
ejpam-709	255	2	r	r	NOUN
ejpam-709	255	3	is	be	AUX
ejpam-709	255	4	s2−relatively	s2−relatively	ADV
ejpam-709	255	5	normal	normal	ADJ
ejpam-709	255	6	and	and	CCONJ
ejpam-709	255	7	s2	s2	NOUN
ejpam-709	255	8	is	be	AUX
ejpam-709	255	9	s1−relatively	s1−relatively	ADV
ejpam-709	255	10	normal	normal	ADJ
ejpam-709	255	11	.	.	PUNCT
ejpam-709	256	1	let	let	VERB
ejpam-709	256	2	p	p	PRON
ejpam-709	256	3	be	be	AUX
ejpam-709	256	4	a	a	DET
ejpam-709	256	5	prime	prime	ADJ
ejpam-709	256	6	filter	filter	NOUN
ejpam-709	256	7	of	of	ADP
ejpam-709	256	8	s1	s1	NOUN
ejpam-709	256	9	.	.	PUNCT
ejpam-709	257	1	we	we	PRON
ejpam-709	257	2	have	have	VERB
ejpam-709	257	3	to	to	PART
ejpam-709	257	4	prove	prove	VERB
ejpam-709	257	5	that	that	SCONJ
ejpam-709	257	6	[	[	X
ejpam-709	257	7	p	p	X
ejpam-709	257	8	)	)	PUNCT
ejpam-709	257	9	is	be	AUX
ejpam-709	257	10	an	an	DET
ejpam-709	257	11	s1−prime	s1−prime	ADJ
ejpam-709	257	12	filter	filter	NOUN
ejpam-709	257	13	of	of	ADP
ejpam-709	257	14	s2	s2	PROPN
ejpam-709	257	15	,	,	PUNCT
ejpam-709	257	16	where	where	SCONJ
ejpam-709	257	17	[	[	X
ejpam-709	257	18	p	p	X
ejpam-709	257	19	)	)	PUNCT
ejpam-709	257	20	=	=	PUNCT
ejpam-709	257	21	{	{	PUNCT
ejpam-709	257	22	s	s	PROPN
ejpam-709	257	23	∨	∨	NOUN
ejpam-709	257	24	a	a	DET
ejpam-709	257	25	|	|	NOUN
ejpam-709	257	26	s	s	NOUN
ejpam-709	257	27	∈	∈	PROPN
ejpam-709	257	28	s2	s2	NOUN
ejpam-709	257	29	and	and	CCONJ
ejpam-709	257	30	a	a	DET
ejpam-709	257	31	∈	∈	NOUN
ejpam-709	257	32	p	p	X
ejpam-709	257	33	}	}	PUNCT
ejpam-709	257	34	.	.	PUNCT
ejpam-709	258	1	let	let	VERB
ejpam-709	258	2	x	x	PRON
ejpam-709	258	3	,	,	PUNCT
ejpam-709	258	4	y	y	PROPN
ejpam-709	258	5	∈	∈	PROPN
ejpam-709	259	1	[	[	X
ejpam-709	259	2	p	p	X
ejpam-709	259	3	)	)	PUNCT
ejpam-709	259	4	.	.	PUNCT
ejpam-709	260	1	then	then	ADV
ejpam-709	260	2	x	x	X
ejpam-709	260	3	=	=	SYM
ejpam-709	260	4	s1	s1	PROPN
ejpam-709	260	5	∨	∨	NUM
ejpam-709	260	6	a1	a1	NOUN
ejpam-709	260	7	and	and	CCONJ
ejpam-709	260	8	y	y	PROPN
ejpam-709	260	9	=	=	PROPN
ejpam-709	260	10	s2	s2	PROPN
ejpam-709	260	11	∨	∨	NUM
ejpam-709	260	12	a2	a2	PROPN
ejpam-709	260	13	,	,	PUNCT
ejpam-709	260	14	for	for	ADP
ejpam-709	260	15	some	some	DET
ejpam-709	260	16	s1	s1	NOUN
ejpam-709	260	17	,	,	PUNCT
ejpam-709	260	18	s2	s2	NOUN
ejpam-709	260	19	∈	∈	PROPN
ejpam-709	260	20	s2	s2	NOUN
ejpam-709	260	21	and	and	CCONJ
ejpam-709	260	22	a1	a1	NOUN
ejpam-709	260	23	,	,	PUNCT
ejpam-709	260	24	a2	a2	PROPN
ejpam-709	260	25	∈	∈	PROPN
ejpam-709	261	1	p.	p.	NOUN
ejpam-709	261	2	now	now	ADV
ejpam-709	261	3	,	,	PUNCT
ejpam-709	261	4	x	x	PUNCT
ejpam-709	261	5	∧	∧	NOUN
ejpam-709	261	6	y	y	PROPN
ejpam-709	261	7	=	=	SYM
ejpam-709	261	8	(	(	PUNCT
ejpam-709	261	9	s1∨a1)∧(s2∨a2	s1∨a1)∧(s2∨a2	PROPN
ejpam-709	261	10	)	)	PUNCT
ejpam-709	261	11	=	=	SYM
ejpam-709	261	12	(	(	PUNCT
ejpam-709	261	13	s1∧(s2∨a2))∨(a1∧(s2∨a2	s1∧(s2∨a2))∨(a1∧(s2∨a2	NOUN
ejpam-709	261	14	)	)	PUNCT
ejpam-709	261	15	)	)	PUNCT
ejpam-709	262	1	=	=	PRON
ejpam-709	262	2	(	(	PUNCT
ejpam-709	262	3	s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2	s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2	PROPN
ejpam-709	262	4	)	)	PUNCT
ejpam-709	262	5	)	)	PUNCT
ejpam-709	262	6	and	and	CCONJ
ejpam-709	262	7	hence	hence	ADV
ejpam-709	262	8	(	(	PUNCT
ejpam-709	262	9	x∧	x∧	PROPN
ejpam-709	262	10	y)∧(a1∧a2	y)∧(a1∧a2	NUM
ejpam-709	262	11	)	)	PUNCT
ejpam-709	262	12	=	=	SYM
ejpam-709	262	13	(	(	PUNCT
ejpam-709	262	14	(	(	PUNCT
ejpam-709	262	15	s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2)))∧(a1∧a2	s1∧(s2∨a2))∨((a1∧s2)∨(a1∧a2)))∧(a1∧a2	NOUN
ejpam-709	262	16	)	)	PUNCT
ejpam-709	262	17	=	=	SYM
ejpam-709	262	18	(	(	PUNCT
ejpam-709	262	19	a1∧a2	a1∧a2	PROPN
ejpam-709	262	20	)	)	PUNCT
ejpam-709	262	21	.	.	PUNCT
ejpam-709	263	1	thus	thus	ADV
ejpam-709	263	2	x	x	X
ejpam-709	263	3	∧	∧	NOUN
ejpam-709	263	4	y	y	NOUN
ejpam-709	263	5	=	=	SYM
ejpam-709	263	6	(	(	PUNCT
ejpam-709	263	7	x	x	PUNCT
ejpam-709	263	8	∧	∧	PROPN
ejpam-709	263	9	y	y	PROPN
ejpam-709	263	10	)	)	PUNCT
ejpam-709	263	11	∨	∨	PROPN
ejpam-709	263	12	(	(	PUNCT
ejpam-709	263	13	a1	a1	NOUN
ejpam-709	263	14	∧	∧	PROPN
ejpam-709	263	15	a2	a2	PROPN
ejpam-709	263	16	)	)	PUNCT
ejpam-709	263	17	and	and	CCONJ
ejpam-709	263	18	hence	hence	ADV
ejpam-709	263	19	x	x	PART
ejpam-709	263	20	∧	∧	NOUN
ejpam-709	263	21	y	y	PROPN
ejpam-709	263	22	∈	∈	PROPN
ejpam-709	264	1	[	[	X
ejpam-709	264	2	p	p	X
ejpam-709	264	3	)	)	PUNCT
ejpam-709	264	4	.	.	PUNCT
ejpam-709	265	1	let	let	VERB
ejpam-709	265	2	x	x	PUNCT
ejpam-709	265	3	∈	∈	PROPN
ejpam-709	265	4	[	[	X
ejpam-709	265	5	p	p	X
ejpam-709	265	6	)	)	PUNCT
ejpam-709	265	7	and	and	CCONJ
ejpam-709	265	8	r	r	NOUN
ejpam-709	265	9	∈	∈	PROPN
ejpam-709	265	10	s2	s2	PROPN
ejpam-709	265	11	.	.	PUNCT
ejpam-709	266	1	then	then	ADV
ejpam-709	266	2	x	x	X
ejpam-709	266	3	=	=	SYM
ejpam-709	266	4	s	s	PART
ejpam-709	266	5	∨	∨	NOUN
ejpam-709	266	6	a	a	NOUN
ejpam-709	266	7	,	,	PUNCT
ejpam-709	266	8	for	for	ADP
ejpam-709	266	9	some	some	DET
ejpam-709	266	10	s	s	X
ejpam-709	266	11	∈	∈	PROPN
ejpam-709	266	12	s2	s2	NOUN
ejpam-709	266	13	and	and	CCONJ
ejpam-709	266	14	a	a	DET
ejpam-709	266	15	∈	∈	PROPN
ejpam-709	266	16	p.	p.	NOUN
ejpam-709	266	17	now	now	ADV
ejpam-709	266	18	,	,	PUNCT
ejpam-709	266	19	(	(	PUNCT
ejpam-709	266	20	r	r	NOUN
ejpam-709	266	21	∨	∨	PROPN
ejpam-709	266	22	x)∧	x)∧	PUNCT
ejpam-709	266	23	a	a	X
ejpam-709	266	24	=	=	X
ejpam-709	266	25	(	(	PUNCT
ejpam-709	266	26	r	r	NOUN
ejpam-709	266	27	∨	∨	NUM
ejpam-709	266	28	(	(	PUNCT
ejpam-709	266	29	s	s	PROPN
ejpam-709	266	30	∨	∨	NOUN
ejpam-709	266	31	a))∧	a))∧	PUNCT
ejpam-709	266	32	a	a	DET
ejpam-709	266	33	=	=	X
ejpam-709	266	34	a	a	NOUN
ejpam-709	266	35	and	and	CCONJ
ejpam-709	266	36	hence	hence	ADV
ejpam-709	266	37	r	r	NOUN
ejpam-709	266	38	∨	∨	NOUN
ejpam-709	266	39	x	x	X
ejpam-709	266	40	=	=	SYM
ejpam-709	266	41	(	(	PUNCT
ejpam-709	266	42	r	r	NOUN
ejpam-709	266	43	∨	∨	PROPN
ejpam-709	266	44	x)∨	x)∨	PROPN
ejpam-709	266	45	a.	a.	NOUN
ejpam-709	266	46	therefore	therefore	ADV
ejpam-709	266	47	r	r	NOUN
ejpam-709	266	48	∨	∨	NUM
ejpam-709	266	49	x	x	SYM
ejpam-709	266	50	∈	∈	PROPN
ejpam-709	267	1	[	[	X
ejpam-709	267	2	p	p	X
ejpam-709	267	3	)	)	PUNCT
ejpam-709	267	4	.	.	PUNCT
ejpam-709	268	1	hence	hence	ADV
ejpam-709	268	2	[	[	X
ejpam-709	268	3	p	p	X
ejpam-709	268	4	)	)	PUNCT
ejpam-709	268	5	is	be	AUX
ejpam-709	268	6	a	a	DET
ejpam-709	268	7	filter	filter	NOUN
ejpam-709	268	8	of	of	ADP
ejpam-709	268	9	s2	s2	PROPN
ejpam-709	268	10	.	.	PUNCT
ejpam-709	269	1	let	let	VERB
ejpam-709	269	2	x	x	PUNCT
ejpam-709	269	3	∈	∈	PROPN
ejpam-709	270	1	[	[	X
ejpam-709	270	2	p	p	X
ejpam-709	270	3	)	)	PUNCT
ejpam-709	270	4	.	.	PUNCT
ejpam-709	271	1	then	then	ADV
ejpam-709	271	2	x	x	X
ejpam-709	271	3	=	=	SYM
ejpam-709	271	4	s	s	PART
ejpam-709	271	5	∨	∨	NOUN
ejpam-709	271	6	a	a	NOUN
ejpam-709	271	7	,	,	PUNCT
ejpam-709	271	8	for	for	ADP
ejpam-709	271	9	some	some	DET
ejpam-709	271	10	s	s	X
ejpam-709	271	11	∈	∈	PROPN
ejpam-709	271	12	s2	s2	NOUN
ejpam-709	271	13	and	and	CCONJ
ejpam-709	271	14	a	a	DET
ejpam-709	271	15	∈	∈	PROPN
ejpam-709	271	16	p.	p.	NOUN
ejpam-709	272	1	now	now	ADV
ejpam-709	272	2	,	,	PUNCT
ejpam-709	272	3	x	x	PROPN
ejpam-709	272	4	∨	∨	NUM
ejpam-709	272	5	a	a	X
ejpam-709	272	6	=	=	X
ejpam-709	272	7	(	(	PUNCT
ejpam-709	272	8	s	s	NOUN
ejpam-709	272	9	∨	∨	NUM
ejpam-709	272	10	a)∨	a)∨	PROPN
ejpam-709	272	11	a	a	DET
ejpam-709	272	12	=	=	SYM
ejpam-709	272	13	s	s	PART
ejpam-709	272	14	∨	∨	NOUN
ejpam-709	272	15	a	a	DET
ejpam-709	272	16	=	=	X
ejpam-709	272	17	x	x	X
ejpam-709	272	18	.	.	PUNCT
ejpam-709	273	1	hence	hence	ADV
ejpam-709	273	2	[	[	X
ejpam-709	273	3	p	p	X
ejpam-709	273	4	)	)	PUNCT
ejpam-709	273	5	is	be	AUX
ejpam-709	273	6	an	an	DET
ejpam-709	273	7	s1−filter	s1−filter	NOUN
ejpam-709	273	8	of	of	ADP
ejpam-709	273	9	r.	r.	PROPN
ejpam-709	273	10	let	let	VERB
ejpam-709	273	11	a	a	DET
ejpam-709	273	12	,	,	PUNCT
ejpam-709	273	13	b	b	PROPN
ejpam-709	273	14	∈	∈	PROPN
ejpam-709	273	15	s1	s1	NOUN
ejpam-709	273	16	such	such	ADJ
ejpam-709	273	17	that	that	SCONJ
ejpam-709	273	18	a	a	DET
ejpam-709	273	19	∨	∨	NUM
ejpam-709	273	20	b	b	X
ejpam-709	273	21	∈	∈	PROPN
ejpam-709	273	22	[	[	X
ejpam-709	273	23	p)∩	p)∩	NOUN
ejpam-709	273	24	s1	s1	NOUN
ejpam-709	273	25	.	.	PUNCT
ejpam-709	274	1	then	then	ADV
ejpam-709	274	2	a	a	DET
ejpam-709	274	3	∨	∨	PROPN
ejpam-709	274	4	b	b	X
ejpam-709	274	5	=	=	SYM
ejpam-709	274	6	s	s	PART
ejpam-709	274	7	∨	∨	NOUN
ejpam-709	274	8	x	x	X
ejpam-709	274	9	,	,	PUNCT
ejpam-709	274	10	for	for	ADP
ejpam-709	274	11	some	some	DET
ejpam-709	274	12	s	s	X
ejpam-709	274	13	∈	∈	PROPN
ejpam-709	274	14	s2	s2	NOUN
ejpam-709	274	15	and	and	CCONJ
ejpam-709	274	16	x	x	PUNCT
ejpam-709	274	17	∈	∈	PROPN
ejpam-709	275	1	p.	p.	NOUN
ejpam-709	275	2	now	now	ADV
ejpam-709	275	3	,	,	PUNCT
ejpam-709	275	4	x	x	SYM
ejpam-709	275	5	=	=	SYM
ejpam-709	275	6	(	(	PUNCT
ejpam-709	275	7	a	a	DET
ejpam-709	275	8	∨	∨	NUM
ejpam-709	275	9	b	b	NOUN
ejpam-709	275	10	)	)	PUNCT
ejpam-709	275	11	∧	∧	NOUN
ejpam-709	275	12	x	x	X
ejpam-709	275	13	=	=	PUNCT
ejpam-709	275	14	(	(	PUNCT
ejpam-709	275	15	a	a	DET
ejpam-709	275	16	∧	∧	PROPN
ejpam-709	275	17	x	x	NOUN
ejpam-709	275	18	)	)	PUNCT
ejpam-709	275	19	∨	∨	PROPN
ejpam-709	275	20	(	(	PUNCT
ejpam-709	275	21	b	b	PROPN
ejpam-709	275	22	∧	∧	PROPN
ejpam-709	275	23	x	x	NOUN
ejpam-709	275	24	)	)	PUNCT
ejpam-709	275	25	∈	∈	PROPN
ejpam-709	276	1	p	p	NOUN
ejpam-709	276	2	(	(	PUNCT
ejpam-709	276	3	since	since	SCONJ
ejpam-709	276	4	x	x	PROPN
ejpam-709	276	5	∈	∈	PROPN
ejpam-709	276	6	p	p	NOUN
ejpam-709	276	7	)	)	PUNCT
ejpam-709	276	8	.	.	PUNCT
ejpam-709	277	1	that	that	PRON
ejpam-709	277	2	implies	imply	VERB
ejpam-709	277	3	either	either	CCONJ
ejpam-709	277	4	a	a	DET
ejpam-709	277	5	∧	∧	PROPN
ejpam-709	277	6	x	x	SYM
ejpam-709	277	7	∈	∈	PROPN
ejpam-709	277	8	p	p	NOUN
ejpam-709	277	9	or	or	CCONJ
ejpam-709	277	10	b	b	NOUN
ejpam-709	277	11	∧	∧	PROPN
ejpam-709	277	12	x	x	SYM
ejpam-709	277	13	∈	∈	PROPN
ejpam-709	277	14	p.	p.	NOUN
ejpam-709	277	15	suppose	suppose	VERB
ejpam-709	278	1	a	a	DET
ejpam-709	278	2	∧	∧	PROPN
ejpam-709	278	3	x	x	SYM
ejpam-709	278	4	∈	∈	PROPN
ejpam-709	278	5	p.	p.	NOUN
ejpam-709	278	6	then	then	ADV
ejpam-709	278	7	a	a	DET
ejpam-709	278	8	∧	∧	PROPN
ejpam-709	278	9	x	x	SYM
ejpam-709	278	10	∈	∈	PROPN
ejpam-709	279	1	[	[	X
ejpam-709	279	2	p	p	X
ejpam-709	279	3	)	)	PUNCT
ejpam-709	279	4	.	.	PUNCT
ejpam-709	280	1	that	that	PRON
ejpam-709	280	2	implies	imply	VERB
ejpam-709	280	3	a	a	DET
ejpam-709	280	4	∨	∨	NOUN
ejpam-709	280	5	(	(	PUNCT
ejpam-709	280	6	a	a	DET
ejpam-709	280	7	∧	∧	PROPN
ejpam-709	280	8	x	x	NOUN
ejpam-709	280	9	)	)	PUNCT
ejpam-709	280	10	∈	∈	PROPN
ejpam-709	280	11	[	[	X
ejpam-709	280	12	p)∩	p)∩	NOUN
ejpam-709	280	13	s1	s1	NOUN
ejpam-709	280	14	.	.	PUNCT
ejpam-709	281	1	hence	hence	ADV
ejpam-709	281	2	a	a	DET
ejpam-709	281	3	∈	∈	NOUN
ejpam-709	281	4	[	[	X
ejpam-709	281	5	p)∩	p)∩	NOUN
ejpam-709	281	6	s1	s1	NOUN
ejpam-709	281	7	.	.	PUNCT
ejpam-709	282	1	thus	thus	ADV
ejpam-709	282	2	[	[	X
ejpam-709	282	3	p	p	X
ejpam-709	282	4	)	)	PUNCT
ejpam-709	282	5	is	be	AUX
ejpam-709	282	6	an	an	DET
ejpam-709	282	7	s1−	s1−	PROPN
ejpam-709	282	8	prime	prime	ADJ
ejpam-709	282	9	filter	filter	NOUN
ejpam-709	282	10	of	of	ADP
ejpam-709	282	11	s2	s2	PROPN
ejpam-709	282	12	.	.	PUNCT
ejpam-709	283	1	since	since	SCONJ
ejpam-709	283	2	s2	s2	PROPN
ejpam-709	283	3	is	be	AUX
ejpam-709	283	4	s1−relatively	s1−relatively	ADV
ejpam-709	283	5	normal	normal	ADJ
ejpam-709	283	6	,	,	PUNCT
ejpam-709	283	7	[	[	X
ejpam-709	283	8	p	p	X
ejpam-709	283	9	)	)	PUNCT
ejpam-709	283	10	is	be	AUX
ejpam-709	283	11	a	a	DET
ejpam-709	283	12	prime	prime	ADJ
ejpam-709	283	13	filter	filter	NOUN
ejpam-709	283	14	of	of	ADP
ejpam-709	283	15	s2	s2	PROPN
ejpam-709	283	16	.	.	PUNCT
ejpam-709	284	1	(	(	PUNCT
ejpam-709	284	2	2	2	X
ejpam-709	284	3	)	)	PUNCT
ejpam-709	284	4	⇒	⇒	NOUN
ejpam-709	284	5	(	(	PUNCT
ejpam-709	284	6	1	1	NUM
ejpam-709	284	7	)	)	PUNCT
ejpam-709	284	8	:	:	PUNCT
ejpam-709	284	9	assume	assume	VERB
ejpam-709	284	10	that	that	SCONJ
ejpam-709	284	11	r	r	NOUN
ejpam-709	284	12	is	be	AUX
ejpam-709	284	13	s2−relatively	s2−relatively	ADV
ejpam-709	284	14	normal	normal	ADJ
ejpam-709	284	15	and	and	CCONJ
ejpam-709	284	16	the	the	DET
ejpam-709	284	17	filters	filter	NOUN
ejpam-709	284	18	generated	generate	VERB
ejpam-709	284	19	in	in	ADP
ejpam-709	284	20	s2	s2	PROPN
ejpam-709	284	21	by	by	ADP
ejpam-709	284	22	prime	prime	ADJ
ejpam-709	284	23	filters	filter	NOUN
ejpam-709	284	24	of	of	ADP
ejpam-709	284	25	s1	s1	NOUN
ejpam-709	284	26	are	be	AUX
ejpam-709	284	27	prime	prime	ADJ
ejpam-709	284	28	.	.	PUNCT
ejpam-709	285	1	let	let	VERB
ejpam-709	285	2	p	p	PRON
ejpam-709	285	3	be	be	AUX
ejpam-709	285	4	an	an	DET
ejpam-709	285	5	s1−prime	s1−prime	ADJ
ejpam-709	285	6	filter	filter	NOUN
ejpam-709	285	7	of	of	ADP
ejpam-709	285	8	r.	r.	PROPN
ejpam-709	285	9	let	let	VERB
ejpam-709	285	10	f	f	PRON
ejpam-709	285	11	be	be	AUX
ejpam-709	285	12	a	a	DET
ejpam-709	285	13	proper	proper	ADJ
ejpam-709	285	14	filter	filter	NOUN
ejpam-709	285	15	of	of	ADP
ejpam-709	285	16	r	r	NOUN
ejpam-709	285	17	such	such	ADJ
ejpam-709	285	18	that	that	SCONJ
ejpam-709	285	19	p	p	PROPN
ejpam-709	285	20	⊆	⊆	NUM
ejpam-709	285	21	f.	f.	NOUN
ejpam-709	285	22	clearly	clearly	ADV
ejpam-709	285	23	p	p	PROPN
ejpam-709	285	24	is	be	AUX
ejpam-709	285	25	an	an	DET
ejpam-709	285	26	s2−filter	s2−filter	NOUN
ejpam-709	285	27	of	of	ADP
ejpam-709	285	28	r.	r.	PROPN
ejpam-709	285	29	we	we	PRON
ejpam-709	285	30	have	have	VERB
ejpam-709	285	31	to	to	PART
ejpam-709	285	32	prove	prove	VERB
ejpam-709	285	33	that	that	SCONJ
ejpam-709	285	34	[	[	X
ejpam-709	285	35	p∩s1	p∩s1	NOUN
ejpam-709	285	36	)	)	PUNCT
ejpam-709	285	37	=	=	SYM
ejpam-709	285	38	p∩s2	p∩s2	PROPN
ejpam-709	285	39	.	.	PUNCT
ejpam-709	286	1	let	let	VERB
ejpam-709	286	2	a	a	DET
ejpam-709	286	3	∈	∈	NOUN
ejpam-709	286	4	[	[	X
ejpam-709	286	5	p	p	NOUN
ejpam-709	286	6	∩s1	∩s1	NOUN
ejpam-709	286	7	)	)	PUNCT
ejpam-709	286	8	.	.	PUNCT
ejpam-709	287	1	then	then	ADV
ejpam-709	287	2	a	a	DET
ejpam-709	287	3	=	=	PUNCT
ejpam-709	287	4	s∨	s∨	PROPN
ejpam-709	287	5	x	x	SYM
ejpam-709	287	6	,	,	PUNCT
ejpam-709	287	7	for	for	ADP
ejpam-709	287	8	some	some	DET
ejpam-709	287	9	s	s	X
ejpam-709	287	10	∈	∈	PROPN
ejpam-709	287	11	s2	s2	NOUN
ejpam-709	287	12	and	and	CCONJ
ejpam-709	287	13	x	x	PUNCT
ejpam-709	287	14	∈	∈	PROPN
ejpam-709	287	15	p	p	NOUN
ejpam-709	287	16	∩s1	∩s1	PROPN
ejpam-709	287	17	.	.	PUNCT
ejpam-709	288	1	that	that	PRON
ejpam-709	288	2	implies	imply	VERB
ejpam-709	288	3	a	a	DET
ejpam-709	288	4	∈	∈	PROPN
ejpam-709	288	5	p	p	NOUN
ejpam-709	288	6	∩s2	∩s2	PROPN
ejpam-709	288	7	.	.	PUNCT
ejpam-709	289	1	therefore	therefore	ADV
ejpam-709	289	2	[	[	X
ejpam-709	289	3	p	p	NOUN
ejpam-709	289	4	∩	∩	ADJ
ejpam-709	289	5	s1	s1	NOUN
ejpam-709	289	6	)	)	PUNCT
ejpam-709	289	7	⊆	⊆	NUM
ejpam-709	289	8	p	p	PROPN
ejpam-709	289	9	∩	∩	ADJ
ejpam-709	289	10	s2	s2	NOUN
ejpam-709	289	11	.	.	PUNCT
ejpam-709	290	1	let	let	VERB
ejpam-709	290	2	a	a	DET
ejpam-709	290	3	∈	∈	PROPN
ejpam-709	290	4	p	p	NOUN
ejpam-709	290	5	∩	∩	ADJ
ejpam-709	290	6	s2	s2	PROPN
ejpam-709	290	7	.	.	PUNCT
ejpam-709	291	1	then	then	ADV
ejpam-709	291	2	there	there	PRON
ejpam-709	291	3	exists	exist	VERB
ejpam-709	291	4	s	s	PROPN
ejpam-709	291	5	∈	∈	PROPN
ejpam-709	291	6	p	p	NOUN
ejpam-709	291	7	∩	∩	ADJ
ejpam-709	291	8	s1	s1	NOUN
ejpam-709	291	9	such	such	ADJ
ejpam-709	291	10	that	that	SCONJ
ejpam-709	291	11	a	a	DET
ejpam-709	291	12	∨	∨	NUM
ejpam-709	291	13	s	s	PART
ejpam-709	291	14	=	=	NOUN
ejpam-709	291	15	a.	a.	NOUN
ejpam-709	291	16	that	that	PRON
ejpam-709	291	17	implies	imply	VERB
ejpam-709	291	18	a	a	DET
ejpam-709	291	19	∈	∈	NOUN
ejpam-709	291	20	[	[	X
ejpam-709	291	21	p	p	NOUN
ejpam-709	291	22	∩	∩	ADJ
ejpam-709	291	23	s1	s1	NOUN
ejpam-709	291	24	)	)	PUNCT
ejpam-709	291	25	.	.	PUNCT
ejpam-709	292	1	hence	hence	ADV
ejpam-709	292	2	p	p	NOUN
ejpam-709	292	3	∩	∩	ADJ
ejpam-709	292	4	s2	s2	NOUN
ejpam-709	292	5	is	be	AUX
ejpam-709	292	6	a	a	DET
ejpam-709	292	7	prime	prime	ADJ
ejpam-709	292	8	filter	filter	NOUN
ejpam-709	292	9	of	of	ADP
ejpam-709	292	10	s2	s2	PROPN
ejpam-709	292	11	.	.	PUNCT
ejpam-709	293	1	that	that	PRON
ejpam-709	293	2	implies	imply	VERB
ejpam-709	293	3	p	p	NOUN
ejpam-709	293	4	is	be	AUX
ejpam-709	293	5	an	an	DET
ejpam-709	293	6	s2−prime	s2−prime	NOUN
ejpam-709	293	7	filter	filter	NOUN
ejpam-709	293	8	of	of	ADP
ejpam-709	293	9	r.	r.	PROPN
ejpam-709	293	10	therefore	therefore	ADV
ejpam-709	293	11	f	f	PROPN
ejpam-709	293	12	is	be	AUX
ejpam-709	293	13	prime	prime	ADJ
ejpam-709	293	14	filter	filter	NOUN
ejpam-709	293	15	of	of	ADP
ejpam-709	293	16	r.	r.	PROPN
ejpam-709	294	1	thus	thus	ADV
ejpam-709	294	2	r	r	NOUN
ejpam-709	294	3	is	be	AUX
ejpam-709	294	4	an	an	DET
ejpam-709	294	5	s1−relatively	s1−relatively	ADV
ejpam-709	294	6	normal	normal	ADJ
ejpam-709	294	7	adl	adl	PROPN
ejpam-709	294	8	.	.	PUNCT
ejpam-709	294	9	corollary	corollary	ADJ
ejpam-709	294	10	2	2	NUM
ejpam-709	294	11	.	.	PUNCT
ejpam-709	295	1	let	let	VERB
ejpam-709	295	2	r	r	PRON
ejpam-709	295	3	be	be	AUX
ejpam-709	295	4	an	an	DET
ejpam-709	295	5	adl	adl	NOUN
ejpam-709	295	6	with	with	ADP
ejpam-709	295	7	maximal	maximal	ADJ
ejpam-709	295	8	elements	element	NOUN
ejpam-709	295	9	and	and	CCONJ
ejpam-709	295	10	s	s	VERB
ejpam-709	295	11	a	a	DET
ejpam-709	295	12	uni	uni	PROPN
ejpam-709	295	13	subadl	subadl	NOUN
ejpam-709	295	14	of	of	ADP
ejpam-709	295	15	r.	r.	PROPN
ejpam-709	295	16	then	then	ADV
ejpam-709	295	17	r	r	NOUN
ejpam-709	295	18	is	be	AUX
ejpam-709	295	19	s−relatively	s−relatively	ADV
ejpam-709	295	20	normal	normal	ADJ
ejpam-709	295	21	if	if	SCONJ
ejpam-709	295	22	and	and	CCONJ
ejpam-709	295	23	only	only	ADV
ejpam-709	295	24	if	if	SCONJ
ejpam-709	295	25	r	r	NOUN
ejpam-709	295	26	is	be	AUX
ejpam-709	295	27	relatively	relatively	ADV
ejpam-709	295	28	normal	normal	ADJ
ejpam-709	295	29	and	and	CCONJ
ejpam-709	295	30	the	the	DET
ejpam-709	295	31	s−prime	s−prime	NOUN
ejpam-709	295	32	filters	filter	NOUN
ejpam-709	295	33	of	of	ADP
ejpam-709	295	34	r	r	NOUN
ejpam-709	295	35	are	be	AUX
ejpam-709	295	36	prime	prime	ADJ
ejpam-709	295	37	.	.	PUNCT
ejpam-709	296	1	g.	g.	PROPN
ejpam-709	296	2	rao	rao	PROPN
ejpam-709	296	3	,	,	PUNCT
ejpam-709	296	4	n.	n.	PROPN
ejpam-709	296	5	rafi	rafi	PROPN
ejpam-709	296	6	and	and	CCONJ
ejpam-709	296	7	b.	b.	PROPN
ejpam-709	296	8	kumar	kumar	PROPN
ejpam-709	296	9	/	/	SYM
ejpam-709	296	10	eur	eur	PROPN
ejpam-709	296	11	.	.	PUNCT
ejpam-709	297	1	j.	j.	PROPN
ejpam-709	297	2	pure	pure	PROPN
ejpam-709	297	3	appl	appl	PROPN
ejpam-709	297	4	.	.	PROPN
ejpam-709	297	5	math	math	PROPN
ejpam-709	297	6	,	,	PUNCT
ejpam-709	297	7	3	3	NUM
ejpam-709	297	8	(	(	PUNCT
ejpam-709	297	9	2010	2010	NUM
ejpam-709	297	10	)	)	PUNCT
ejpam-709	297	11	,	,	PUNCT
ejpam-709	297	12	704	704	NUM
ejpam-709	297	13	-	-	SYM
ejpam-709	297	14	716	716	NUM
ejpam-709	297	15	712	712	NUM
ejpam-709	297	16	proof	proof	NOUN
ejpam-709	297	17	.	.	PUNCT
ejpam-709	298	1	take	take	VERB
ejpam-709	298	2	s1	s1	NOUN
ejpam-709	298	3	=	=	SYM
ejpam-709	298	4	s	s	PROPN
ejpam-709	298	5	and	and	CCONJ
ejpam-709	298	6	s2	s2	NOUN
ejpam-709	298	7	=	=	PUNCT
ejpam-709	298	8	r	r	NOUN
ejpam-709	298	9	in	in	ADP
ejpam-709	298	10	the	the	DET
ejpam-709	298	11	above	above	ADJ
ejpam-709	298	12	corollary	corollary	NOUN
ejpam-709	298	13	.	.	PUNCT
ejpam-709	299	1	let	let	VERB
ejpam-709	299	2	r	r	PRON
ejpam-709	299	3	be	be	AUX
ejpam-709	299	4	an	an	DET
ejpam-709	299	5	adl	adl	PROPN
ejpam-709	299	6	and	and	CCONJ
ejpam-709	299	7	f	f	PROPN
ejpam-709	299	8	a	a	DET
ejpam-709	299	9	filter	filter	NOUN
ejpam-709	299	10	in	in	ADP
ejpam-709	299	11	r.	r.	PROPN
ejpam-709	299	12	then	then	ADV
ejpam-709	299	13	the	the	DET
ejpam-709	299	14	relation	relation	NOUN
ejpam-709	299	15	ψ(f	ψ(f	NOUN
ejpam-709	299	16	)	)	PUNCT
ejpam-709	300	1	=	=	PRON
ejpam-709	300	2	{	{	PUNCT
ejpam-709	300	3	(	(	PUNCT
ejpam-709	300	4	x	x	INTJ
ejpam-709	300	5	,	,	PUNCT
ejpam-709	300	6	y	y	PROPN
ejpam-709	300	7	)	)	PUNCT
ejpam-709	300	8	∈	∈	PROPN
ejpam-709	301	1	r×r	r×r	PROPN
ejpam-709	301	2	|	|	NOUN
ejpam-709	301	3	x∧	x∧	PROPN
ejpam-709	301	4	t	t	PROPN
ejpam-709	301	5	=	=	SYM
ejpam-709	301	6	y∧	y∧	PROPN
ejpam-709	301	7	t	t	PROPN
ejpam-709	301	8	,	,	PUNCT
ejpam-709	301	9	for	for	ADP
ejpam-709	301	10	some	some	DET
ejpam-709	301	11	t	t	NOUN
ejpam-709	301	12	∈	∈	NOUN
ejpam-709	301	13	f	f	X
ejpam-709	301	14	}	}	PUNCT
ejpam-709	301	15	is	be	AUX
ejpam-709	301	16	a	a	DET
ejpam-709	301	17	congruence	congruence	NOUN
ejpam-709	301	18	relation	relation	NOUN
ejpam-709	301	19	on	on	ADP
ejpam-709	301	20	r	r	NOUN
ejpam-709	301	21	and	and	CCONJ
ejpam-709	301	22	the	the	DET
ejpam-709	301	23	set	set	ADJ
ejpam-709	301	24	r	r	NOUN
ejpam-709	301	25	/	/	SYM
ejpam-709	301	26	ψ(f	ψ(f	NOUN
ejpam-709	301	27	)	)	PUNCT
ejpam-709	302	1	=	=	PRON
ejpam-709	302	2	{	{	PUNCT
ejpam-709	302	3	x	x	X
ejpam-709	302	4	/	/	SYM
ejpam-709	302	5	ψ(f	ψ(f	NOUN
ejpam-709	302	6	)	)	PUNCT
ejpam-709	303	1	|	|	ADV
ejpam-709	303	2	x	x	SYM
ejpam-709	303	3	∈	∈	NOUN
ejpam-709	303	4	r	r	NOUN
ejpam-709	303	5	}	}	PUNCT
ejpam-709	303	6	is	be	AUX
ejpam-709	303	7	an	an	DET
ejpam-709	303	8	adl	adl	PROPN
ejpam-709	303	9	.	.	PUNCT
ejpam-709	304	1	let	let	VERB
ejpam-709	304	2	∏	∏	PRON
ejpam-709	304	3	be	be	AUX
ejpam-709	304	4	the	the	DET
ejpam-709	304	5	natural	natural	ADJ
ejpam-709	304	6	homomorphism	homomorphism	NOUN
ejpam-709	304	7	from	from	ADP
ejpam-709	304	8	r	r	NOUN
ejpam-709	304	9	onto	onto	ADP
ejpam-709	304	10	r	r	NOUN
ejpam-709	304	11	/	/	SYM
ejpam-709	304	12	ψ(f	ψ(f	NOUN
ejpam-709	304	13	)	)	PUNCT
ejpam-709	304	14	defined	define	VERB
ejpam-709	304	15	by	by	ADP
ejpam-709	304	16	∏	∏	PROPN
ejpam-709	304	17	(	(	PUNCT
ejpam-709	304	18	x	x	NOUN
ejpam-709	304	19	)	)	PUNCT
ejpam-709	304	20	=	=	SYM
ejpam-709	305	1	x	x	X
ejpam-709	305	2	/	/	SYM
ejpam-709	305	3	ψ(f	ψ(f	NOUN
ejpam-709	305	4	)	)	PUNCT
ejpam-709	305	5	for	for	ADP
ejpam-709	305	6	all	all	DET
ejpam-709	305	7	x	x	PROPN
ejpam-709	305	8	∈	∈	PROPN
ejpam-709	305	9	r.	r.	PROPN
ejpam-709	305	10	theorem	theorem	VERB
ejpam-709	305	11	8	8	NUM
ejpam-709	305	12	.	.	PUNCT
ejpam-709	306	1	let	let	VERB
ejpam-709	306	2	r	r	PRON
ejpam-709	306	3	be	be	AUX
ejpam-709	306	4	an	an	DET
ejpam-709	306	5	adl	adl	NOUN
ejpam-709	306	6	with	with	ADP
ejpam-709	306	7	maximal	maximal	ADJ
ejpam-709	306	8	elements	element	NOUN
ejpam-709	306	9	and	and	CCONJ
ejpam-709	306	10	s	s	VERB
ejpam-709	306	11	a	a	DET
ejpam-709	306	12	uni	uni	PROPN
ejpam-709	306	13	subadl	subadl	NOUN
ejpam-709	306	14	of	of	ADP
ejpam-709	306	15	r.	r.	PROPN
ejpam-709	306	16	then	then	ADV
ejpam-709	306	17	r	r	NOUN
ejpam-709	306	18	is	be	AUX
ejpam-709	306	19	s−relatively	s−relatively	ADV
ejpam-709	306	20	normal	normal	ADJ
ejpam-709	306	21	if	if	SCONJ
ejpam-709	306	22	and	and	CCONJ
ejpam-709	306	23	only	only	ADV
ejpam-709	306	24	if	if	SCONJ
ejpam-709	306	25	r	r	NOUN
ejpam-709	306	26	/	/	SYM
ejpam-709	306	27	ψ(f	ψ(f	NOUN
ejpam-709	306	28	)	)	PUNCT
ejpam-709	306	29	is	be	AUX
ejpam-709	306	30	a	a	DET
ejpam-709	306	31	chain	chain	NOUN
ejpam-709	306	32	,	,	PUNCT
ejpam-709	306	33	for	for	ADP
ejpam-709	306	34	each	each	DET
ejpam-709	306	35	prime	prime	ADJ
ejpam-709	306	36	filter	filter	NOUN
ejpam-709	306	37	f	f	PROPN
ejpam-709	306	38	of	of	ADP
ejpam-709	306	39	s.	s.	PROPN
ejpam-709	306	40	proof	proof	PROPN
ejpam-709	306	41	.	.	PUNCT
ejpam-709	307	1	assume	assume	VERB
ejpam-709	307	2	that	that	SCONJ
ejpam-709	307	3	r	r	NOUN
ejpam-709	307	4	is	be	AUX
ejpam-709	307	5	s−relatively	s−relatively	ADV
ejpam-709	307	6	normal	normal	ADJ
ejpam-709	307	7	.	.	PUNCT
ejpam-709	308	1	let	let	VERB
ejpam-709	308	2	x	x	X
ejpam-709	308	3	/	/	SYM
ejpam-709	308	4	ψ(f	ψ(f	NOUN
ejpam-709	308	5	)	)	PUNCT
ejpam-709	308	6	,	,	PUNCT
ejpam-709	308	7	y	y	PROPN
ejpam-709	308	8	/	/	SYM
ejpam-709	308	9	ψ(f	ψ(f	NOUN
ejpam-709	308	10	)	)	PUNCT
ejpam-709	308	11	∈	∈	PROPN
ejpam-709	308	12	r	r	NOUN
ejpam-709	308	13	/	/	SYM
ejpam-709	308	14	ψ(f	ψ(f	NOUN
ejpam-709	308	15	)	)	PUNCT
ejpam-709	308	16	.	.	PUNCT
ejpam-709	309	1	since	since	SCONJ
ejpam-709	309	2	x	x	X
ejpam-709	309	3	,	,	PUNCT
ejpam-709	309	4	y	y	PROPN
ejpam-709	309	5	∈	∈	PROPN
ejpam-709	309	6	r	r	NOUN
ejpam-709	309	7	,	,	PUNCT
ejpam-709	309	8	by	by	ADP
ejpam-709	309	9	theorem	theorem	NOUN
ejpam-709	309	10	6	6	NUM
ejpam-709	309	11	,	,	PUNCT
ejpam-709	309	12	there	there	PRON
ejpam-709	309	13	exists	exist	VERB
ejpam-709	309	14	a	a	DET
ejpam-709	309	15	∈	∈	NOUN
ejpam-709	309	16	f	f	NOUN
ejpam-709	309	17	such	such	ADJ
ejpam-709	309	18	that	that	SCONJ
ejpam-709	309	19	x	x	SYM
ejpam-709	309	20	∧	∧	PROPN
ejpam-709	309	21	a	a	PROPN
ejpam-709	309	22	and	and	CCONJ
ejpam-709	309	23	y	y	PROPN
ejpam-709	309	24	∧	∧	PROPN
ejpam-709	309	25	a	a	PRON
ejpam-709	309	26	are	be	AUX
ejpam-709	309	27	comparable	comparable	ADJ
ejpam-709	309	28	.	.	PUNCT
ejpam-709	310	1	with	with	ADP
ejpam-709	310	2	out	out	ADP
ejpam-709	310	3	loss	loss	NOUN
ejpam-709	310	4	of	of	ADP
ejpam-709	310	5	generality	generality	NOUN
ejpam-709	310	6	,	,	PUNCT
ejpam-709	310	7	suppose	suppose	VERB
ejpam-709	310	8	x∧a	x∧a	PROPN
ejpam-709	310	9	≤	≤	PROPN
ejpam-709	310	10	y∧a	y∧a	PROPN
ejpam-709	310	11	.	.	PUNCT
ejpam-709	311	1	then	then	ADV
ejpam-709	311	2	x∧a	x∧a	X
ejpam-709	312	1	=	=	PUNCT
ejpam-709	312	2	x∧a∧	x∧a∧	PROPN
ejpam-709	312	3	y∧a	y∧a	X
ejpam-709	312	4	=	=	SYM
ejpam-709	312	5	x∧	x∧	PROPN
ejpam-709	312	6	y∧a	y∧a	PROPN
ejpam-709	312	7	.	.	PUNCT
ejpam-709	313	1	that	that	PRON
ejpam-709	313	2	implies	imply	VERB
ejpam-709	313	3	(	(	PUNCT
ejpam-709	313	4	x	x	X
ejpam-709	313	5	,	,	PUNCT
ejpam-709	313	6	x∧	x∧	PROPN
ejpam-709	313	7	y	y	PROPN
ejpam-709	313	8	)	)	PUNCT
ejpam-709	313	9	∈	∈	PROPN
ejpam-709	313	10	ψ(f	ψ(f	NOUN
ejpam-709	313	11	)	)	PUNCT
ejpam-709	313	12	and	and	CCONJ
ejpam-709	313	13	hence	hence	ADV
ejpam-709	313	14	x	x	NOUN
ejpam-709	313	15	/	/	SYM
ejpam-709	313	16	ψ(f	ψ(f	NOUN
ejpam-709	313	17	)	)	PUNCT
ejpam-709	313	18	=	=	PUNCT
ejpam-709	314	1	(	(	PUNCT
ejpam-709	314	2	x	x	PUNCT
ejpam-709	314	3	∧	∧	PROPN
ejpam-709	314	4	y)/ψ(f	y)/ψ(f	PROPN
ejpam-709	314	5	)	)	PUNCT
ejpam-709	314	6	=	=	PUNCT
ejpam-709	315	1	x	x	X
ejpam-709	315	2	/	/	SYM
ejpam-709	315	3	ψ(f)∧	ψ(f)∧	PROPN
ejpam-709	315	4	y	y	PROPN
ejpam-709	315	5	/	/	SYM
ejpam-709	315	6	ψ(f	ψ(f	NOUN
ejpam-709	315	7	)	)	PUNCT
ejpam-709	315	8	.	.	PUNCT
ejpam-709	316	1	therefore	therefore	ADV
ejpam-709	316	2	x	x	X
ejpam-709	316	3	/	/	SYM
ejpam-709	316	4	ψ(f	ψ(f	NOUN
ejpam-709	316	5	)	)	PUNCT
ejpam-709	316	6	≤	≤	NOUN
ejpam-709	316	7	y	y	X
ejpam-709	316	8	/	/	SYM
ejpam-709	316	9	ψ(f	ψ(f	NOUN
ejpam-709	316	10	)	)	PUNCT
ejpam-709	316	11	.	.	PUNCT
ejpam-709	317	1	hence	hence	ADV
ejpam-709	317	2	r	r	NOUN
ejpam-709	317	3	/	/	SYM
ejpam-709	317	4	ψ(f	ψ(f	NOUN
ejpam-709	317	5	)	)	PUNCT
ejpam-709	317	6	is	be	AUX
ejpam-709	317	7	a	a	DET
ejpam-709	317	8	chain	chain	NOUN
ejpam-709	317	9	.	.	PUNCT
ejpam-709	318	1	conversely	conversely	ADV
ejpam-709	318	2	,	,	PUNCT
ejpam-709	318	3	assume	assume	VERB
ejpam-709	318	4	that	that	SCONJ
ejpam-709	318	5	r	r	NOUN
ejpam-709	318	6	/	/	SYM
ejpam-709	318	7	ψ(f	ψ(f	NOUN
ejpam-709	318	8	)	)	PUNCT
ejpam-709	318	9	is	be	AUX
ejpam-709	318	10	a	a	DET
ejpam-709	318	11	chain	chain	NOUN
ejpam-709	318	12	.	.	PUNCT
ejpam-709	319	1	let	let	VERB
ejpam-709	319	2	x	x	PRON
ejpam-709	319	3	,	,	PUNCT
ejpam-709	319	4	y	y	PROPN
ejpam-709	319	5	∈	∈	PROPN
ejpam-709	319	6	r.	r.	PROPN
ejpam-709	319	7	then	then	ADV
ejpam-709	319	8	x	x	X
ejpam-709	319	9	/	/	SYM
ejpam-709	319	10	ψ(f	ψ(f	NOUN
ejpam-709	319	11	)	)	PUNCT
ejpam-709	319	12	,	,	PUNCT
ejpam-709	319	13	y	y	PROPN
ejpam-709	319	14	/	/	SYM
ejpam-709	319	15	ψ(f	ψ(f	NOUN
ejpam-709	319	16	)	)	PUNCT
ejpam-709	319	17	∈	∈	PROPN
ejpam-709	319	18	r	r	NOUN
ejpam-709	319	19	/	/	SYM
ejpam-709	319	20	ψ(f	ψ(f	NOUN
ejpam-709	319	21	)	)	PUNCT
ejpam-709	319	22	.	.	PUNCT
ejpam-709	320	1	since	since	SCONJ
ejpam-709	320	2	r	r	NOUN
ejpam-709	320	3	/	/	SYM
ejpam-709	320	4	ψ(f	ψ(f	NOUN
ejpam-709	320	5	)	)	PUNCT
ejpam-709	320	6	is	be	AUX
ejpam-709	320	7	a	a	DET
ejpam-709	320	8	chain	chain	NOUN
ejpam-709	320	9	,	,	PUNCT
ejpam-709	320	10	x	x	X
ejpam-709	320	11	/	/	SYM
ejpam-709	320	12	ψ(f	ψ(f	NOUN
ejpam-709	320	13	)	)	PUNCT
ejpam-709	320	14	,	,	PUNCT
ejpam-709	320	15	y	y	PROPN
ejpam-709	320	16	/	/	SYM
ejpam-709	320	17	ψ(f	ψ(f	NOUN
ejpam-709	320	18	)	)	PUNCT
ejpam-709	320	19	are	be	AUX
ejpam-709	320	20	comparable	comparable	ADJ
ejpam-709	320	21	.	.	PUNCT
ejpam-709	321	1	with	with	ADP
ejpam-709	321	2	out	out	ADP
ejpam-709	321	3	loss	loss	NOUN
ejpam-709	321	4	of	of	ADP
ejpam-709	321	5	generality	generality	NOUN
ejpam-709	321	6	,	,	PUNCT
ejpam-709	321	7	suppose	suppose	VERB
ejpam-709	321	8	x	x	X
ejpam-709	321	9	/	/	SYM
ejpam-709	321	10	ψ(f	ψ(f	NOUN
ejpam-709	321	11	)	)	PUNCT
ejpam-709	321	12	≤	≤	NOUN
ejpam-709	322	1	y	y	X
ejpam-709	322	2	/	/	SYM
ejpam-709	322	3	ψ(f	ψ(f	NOUN
ejpam-709	322	4	)	)	PUNCT
ejpam-709	322	5	.	.	PUNCT
ejpam-709	323	1	then	then	ADV
ejpam-709	323	2	x	x	X
ejpam-709	323	3	/	/	SYM
ejpam-709	323	4	ψ(f	ψ(f	NOUN
ejpam-709	323	5	)	)	PUNCT
ejpam-709	323	6	=	=	SYM
ejpam-709	324	1	x	x	X
ejpam-709	324	2	/	/	SYM
ejpam-709	324	3	ψ(f	ψ(f	NOUN
ejpam-709	324	4	)	)	PUNCT
ejpam-709	324	5	∧	∧	PROPN
ejpam-709	324	6	y	y	PROPN
ejpam-709	324	7	/	/	SYM
ejpam-709	324	8	ψ(f	ψ(f	NOUN
ejpam-709	324	9	)	)	PUNCT
ejpam-709	324	10	.	.	PUNCT
ejpam-709	325	1	that	that	PRON
ejpam-709	325	2	implies	imply	VERB
ejpam-709	325	3	(	(	PUNCT
ejpam-709	325	4	x	x	X
ejpam-709	325	5	,	,	PUNCT
ejpam-709	325	6	x	x	PUNCT
ejpam-709	325	7	∧	∧	PROPN
ejpam-709	325	8	y	y	PROPN
ejpam-709	325	9	)	)	PUNCT
ejpam-709	325	10	∈	∈	PROPN
ejpam-709	325	11	ψ(f	ψ(f	NOUN
ejpam-709	325	12	)	)	PUNCT
ejpam-709	325	13	.	.	PUNCT
ejpam-709	326	1	then	then	ADV
ejpam-709	326	2	x	x	X
ejpam-709	326	3	∧	∧	NOUN
ejpam-709	326	4	a	a	NOUN
ejpam-709	326	5	=	=	NOUN
ejpam-709	326	6	x	x	SYM
ejpam-709	326	7	∧	∧	NOUN
ejpam-709	326	8	y	y	PROPN
ejpam-709	326	9	∧	∧	PROPN
ejpam-709	326	10	a	a	PROPN
ejpam-709	326	11	,	,	PUNCT
ejpam-709	326	12	for	for	ADP
ejpam-709	326	13	some	some	DET
ejpam-709	326	14	a	a	DET
ejpam-709	326	15	∈	∈	PROPN
ejpam-709	326	16	f.	f.	NOUN
ejpam-709	326	17	therefore	therefore	ADV
ejpam-709	326	18	x	x	X
ejpam-709	326	19	∧	∧	PROPN
ejpam-709	326	20	a	a	DET
ejpam-709	326	21	≤	≤	NUM
ejpam-709	326	22	y	y	PROPN
ejpam-709	326	23	∧	∧	PROPN
ejpam-709	326	24	a.	a.	NOUN
ejpam-709	326	25	thus	thus	ADV
ejpam-709	326	26	r	r	NOUN
ejpam-709	326	27	is	be	AUX
ejpam-709	326	28	an	an	DET
ejpam-709	326	29	s−relatively	s−relatively	ADV
ejpam-709	326	30	normal	normal	ADJ
ejpam-709	326	31	.	.	PUNCT
ejpam-709	327	1	the	the	DET
ejpam-709	327	2	following	following	ADJ
ejpam-709	327	3	result	result	NOUN
ejpam-709	327	4	follows	follow	VERB
ejpam-709	327	5	directly	directly	ADV
ejpam-709	327	6	from	from	ADP
ejpam-709	327	7	the	the	DET
ejpam-709	327	8	above	above	ADJ
ejpam-709	327	9	theorem	theorem	PROPN
ejpam-709	327	10	.	.	PUNCT
ejpam-709	327	11	theorem	theorem	NOUN
ejpam-709	327	12	9	9	NUM
ejpam-709	327	13	.	.	PUNCT
ejpam-709	328	1	each	each	DET
ejpam-709	328	2	s−relatively	s−relatively	ADV
ejpam-709	328	3	normal	normal	ADJ
ejpam-709	328	4	adl	adl	PROPN
ejpam-709	328	5	is	be	AUX
ejpam-709	328	6	a	a	DET
ejpam-709	328	7	subdirect	subdirect	NOUN
ejpam-709	328	8	product	product	NOUN
ejpam-709	328	9	of	of	ADP
ejpam-709	328	10	the	the	DET
ejpam-709	328	11	bounded	bound	VERB
ejpam-709	328	12	chains	chain	NOUN
ejpam-709	328	13	r	r	NOUN
ejpam-709	328	14	\	\	NOUN
ejpam-709	328	15	p	p	NOUN
ejpam-709	328	16	,	,	PUNCT
ejpam-709	328	17	where	where	SCONJ
ejpam-709	328	18	p	p	NOUN
ejpam-709	328	19	runs	run	VERB
ejpam-709	328	20	through	through	ADP
ejpam-709	328	21	the	the	DET
ejpam-709	328	22	set	set	NOUN
ejpam-709	328	23	of	of	ADP
ejpam-709	328	24	all	all	DET
ejpam-709	328	25	prime	prime	ADJ
ejpam-709	328	26	ideals	ideal	NOUN
ejpam-709	328	27	of	of	ADP
ejpam-709	328	28	s.	s.	PROPN
ejpam-709	328	29	4	4	NUM
ejpam-709	328	30	.	.	PUNCT
ejpam-709	328	31	dually	dually	ADV
ejpam-709	328	32	s−relatively	s−relatively	ADV
ejpam-709	328	33	normal	normal	ADJ
ejpam-709	328	34	adls	adls	NOUN
ejpam-709	328	35	the	the	DET
ejpam-709	328	36	concept	concept	NOUN
ejpam-709	328	37	of	of	ADP
ejpam-709	328	38	a	a	DET
ejpam-709	328	39	dually	dually	NOUN
ejpam-709	328	40	s−completely	s−completely	ADV
ejpam-709	328	41	normal	normal	ADJ
ejpam-709	328	42	lattices	lattice	NOUN
ejpam-709	328	43	was	be	AUX
ejpam-709	328	44	given	give	VERB
ejpam-709	328	45	by	by	ADP
ejpam-709	328	46	cignoli	cignoli	NOUN
ejpam-709	328	47	[	[	X
ejpam-709	328	48	2	2	NUM
ejpam-709	328	49	]	]	PUNCT
ejpam-709	328	50	.	.	PUNCT
ejpam-709	329	1	in	in	ADP
ejpam-709	329	2	this	this	DET
ejpam-709	329	3	section	section	NOUN
ejpam-709	329	4	we	we	PRON
ejpam-709	329	5	define	define	VERB
ejpam-709	329	6	the	the	DET
ejpam-709	329	7	concept	concept	NOUN
ejpam-709	329	8	of	of	ADP
ejpam-709	329	9	dually	dually	ADJ
ejpam-709	329	10	s−relative	s−relative	ADJ
ejpam-709	329	11	normality	normality	NOUN
ejpam-709	329	12	in	in	ADP
ejpam-709	329	13	an	an	DET
ejpam-709	329	14	adl	adl	NOUN
ejpam-709	329	15	r	r	NOUN
ejpam-709	329	16	through	through	ADP
ejpam-709	329	17	its	its	PRON
ejpam-709	329	18	principal	principal	ADJ
ejpam-709	329	19	filter	filter	NOUN
ejpam-709	329	20	lattice	lattice	NOUN
ejpam-709	329	21	pf(r	pf(r	NOUN
ejpam-709	329	22	)	)	PUNCT
ejpam-709	329	23	.	.	PUNCT
ejpam-709	330	1	we	we	PRON
ejpam-709	330	2	begin	begin	VERB
ejpam-709	330	3	with	with	ADP
ejpam-709	330	4	the	the	DET
ejpam-709	330	5	following	following	NOUN
ejpam-709	330	6	.	.	PUNCT
ejpam-709	331	1	definition	definition	NOUN
ejpam-709	331	2	8	8	NUM
ejpam-709	331	3	.	.	PUNCT
ejpam-709	332	1	let	let	VERB
ejpam-709	332	2	r	r	PRON
ejpam-709	332	3	be	be	AUX
ejpam-709	332	4	an	an	DET
ejpam-709	332	5	adl	adl	NOUN
ejpam-709	332	6	,	,	PUNCT
ejpam-709	332	7	s	s	VERB
ejpam-709	332	8	a	a	DET
ejpam-709	332	9	uni	uni	PROPN
ejpam-709	332	10	subadl	subadl	NOUN
ejpam-709	332	11	of	of	ADP
ejpam-709	332	12	r	r	NOUN
ejpam-709	332	13	and	and	CCONJ
ejpam-709	332	14	x	x	NOUN
ejpam-709	332	15	,	,	PUNCT
ejpam-709	332	16	y	y	PROPN
ejpam-709	332	17	∈	∈	PROPN
ejpam-709	332	18	r.	r.	NOUN
ejpam-709	332	19	we	we	PRON
ejpam-709	332	20	define	define	VERB
ejpam-709	332	21	⌈x	⌈x	NOUN
ejpam-709	332	22	,	,	PUNCT
ejpam-709	333	1	y⌉s	y⌉s	PROPN
ejpam-709	333	2	=	=	X
ejpam-709	333	3	{	{	PUNCT
ejpam-709	333	4	a	a	DET
ejpam-709	333	5	∈	∈	NOUN
ejpam-709	334	1	s	s	PART
ejpam-709	334	2	|	|	NOUN
ejpam-709	334	3	(	(	PUNCT
ejpam-709	334	4	x	x	PROPN
ejpam-709	334	5	∨	∨	NUM
ejpam-709	334	6	a)∨	a)∨	PROPN
ejpam-709	334	7	y	y	NOUN
ejpam-709	335	1	=	=	PUNCT
ejpam-709	336	1	x	x	PROPN
ejpam-709	337	1	∨	∨	NUM
ejpam-709	337	2	a	a	DET
ejpam-709	337	3	}	}	PUNCT
ejpam-709	337	4	.	.	PUNCT
ejpam-709	338	1	we	we	PRON
ejpam-709	338	2	call	call	VERB
ejpam-709	338	3	⌈x	⌈x	NOUN
ejpam-709	338	4	,	,	PUNCT
ejpam-709	338	5	y⌉s	y⌉s	ADV
ejpam-709	338	6	an	an	DET
ejpam-709	338	7	s−relative	s−relative	ADJ
ejpam-709	338	8	dual	dual	ADJ
ejpam-709	338	9	annihilator	annihilator	NOUN
ejpam-709	338	10	.	.	PUNCT
ejpam-709	339	1	it	it	PRON
ejpam-709	339	2	can	can	AUX
ejpam-709	339	3	be	be	AUX
ejpam-709	339	4	observed	observe	VERB
ejpam-709	339	5	that	that	SCONJ
ejpam-709	339	6	a	a	DET
ejpam-709	339	7	∈	∈	PROPN
ejpam-709	339	8	⌈x	⌈x	NOUN
ejpam-709	339	9	,	,	PUNCT
ejpam-709	339	10	y⌉s	y⌉s	PROPN
ejpam-709	339	11	iff	iff	PROPN
ejpam-709	339	12	y	y	PROPN
ejpam-709	339	13	=	=	PRON
ejpam-709	339	14	(	(	PUNCT
ejpam-709	339	15	x	x	SYM
ejpam-709	339	16	∨	∨	NOUN
ejpam-709	339	17	a)∧	a)∧	NOUN
ejpam-709	339	18	y.	y.	PROPN
ejpam-709	339	19	clearly	clearly	ADV
ejpam-709	339	20	⌈x	⌈x	NOUN
ejpam-709	339	21	,	,	PUNCT
ejpam-709	339	22	y⌉s	y⌉s	ADV
ejpam-709	339	23	is	be	AUX
ejpam-709	339	24	a	a	DET
ejpam-709	339	25	filter	filter	NOUN
ejpam-709	339	26	of	of	ADP
ejpam-709	339	27	s.	s.	PROPN
ejpam-709	339	28	the	the	DET
ejpam-709	339	29	usual	usual	ADJ
ejpam-709	339	30	lattice	lattice	NOUN
ejpam-709	339	31	theoretic	theoretic	ADJ
ejpam-709	339	32	duality	duality	NOUN
ejpam-709	339	33	principle	principle	NOUN
ejpam-709	339	34	does	do	AUX
ejpam-709	339	35	n’t	not	PART
ejpam-709	339	36	hold	hold	VERB
ejpam-709	339	37	in	in	ADP
ejpam-709	339	38	adls	adls	PROPN
ejpam-709	339	39	.	.	PUNCT
ejpam-709	340	1	for	for	ADP
ejpam-709	340	2	example	example	NOUN
ejpam-709	340	3	,	,	PUNCT
ejpam-709	340	4	in	in	ADP
ejpam-709	340	5	an	an	DET
ejpam-709	340	6	adl	adl	NOUN
ejpam-709	340	7	r	r	NOUN
ejpam-709	340	8	,	,	PUNCT
ejpam-709	340	9	∧	∧	PROPN
ejpam-709	340	10	is	be	AUX
ejpam-709	340	11	right	right	ADV
ejpam-709	340	12	distributive	distributive	ADJ
ejpam-709	340	13	over	over	ADP
ejpam-709	340	14	∨	∨	NUM
ejpam-709	340	15	but	but	CCONJ
ejpam-709	340	16	∨	∨	NOUN
ejpam-709	340	17	is	be	AUX
ejpam-709	340	18	not	not	PART
ejpam-709	340	19	right	right	ADV
ejpam-709	340	20	distributive	distributive	ADJ
ejpam-709	340	21	over	over	ADP
ejpam-709	340	22	∧.	∧.	PROPN
ejpam-709	340	23	however	however	ADV
ejpam-709	340	24	,	,	PUNCT
ejpam-709	340	25	we	we	PRON
ejpam-709	340	26	get	get	VERB
ejpam-709	340	27	that	that	SCONJ
ejpam-709	340	28	the	the	DET
ejpam-709	340	29	dual	dual	ADJ
ejpam-709	340	30	of	of	ADP
ejpam-709	340	31	many	many	ADJ
ejpam-709	340	32	results	result	NOUN
ejpam-709	340	33	of	of	ADP
ejpam-709	340	34	section	section	NOUN
ejpam-709	340	35	3	3	NUM
ejpam-709	340	36	,	,	PUNCT
ejpam-709	340	37	hold	hold	VERB
ejpam-709	340	38	good	good	ADJ
ejpam-709	340	39	in	in	ADP
ejpam-709	340	40	dually	dually	ADJ
ejpam-709	340	41	s−	s−	PROPN
ejpam-709	340	42	relatively	relatively	ADV
ejpam-709	340	43	normal	normal	ADJ
ejpam-709	340	44	adls	adls	NOUN
ejpam-709	340	45	.	.	PUNCT
ejpam-709	341	1	for	for	ADP
ejpam-709	341	2	this	this	DET
ejpam-709	341	3	reason	reason	NOUN
ejpam-709	341	4	we	we	PRON
ejpam-709	341	5	give	give	VERB
ejpam-709	341	6	only	only	ADJ
ejpam-709	341	7	statements	statement	NOUN
ejpam-709	341	8	of	of	ADP
ejpam-709	341	9	these	these	DET
ejpam-709	341	10	results	result	NOUN
ejpam-709	341	11	.	.	PUNCT
ejpam-709	342	1	lemma	lemma	PROPN
ejpam-709	342	2	7	7	X
ejpam-709	342	3	.	.	PUNCT
ejpam-709	343	1	let	let	VERB
ejpam-709	343	2	r	r	PRON
ejpam-709	343	3	be	be	AUX
ejpam-709	343	4	an	an	DET
ejpam-709	343	5	adl	adl	NOUN
ejpam-709	343	6	with	with	ADP
ejpam-709	343	7	maximal	maximal	ADJ
ejpam-709	343	8	elements	element	NOUN
ejpam-709	343	9	and	and	CCONJ
ejpam-709	343	10	s	s	VERB
ejpam-709	343	11	a	a	DET
ejpam-709	343	12	uni	uni	PROPN
ejpam-709	343	13	subadl	subadl	NOUN
ejpam-709	343	14	of	of	ADP
ejpam-709	343	15	r.	r.	PROPN
ejpam-709	343	16	if	if	SCONJ
ejpam-709	343	17	m1	m1	PROPN
ejpam-709	343	18	,	,	PUNCT
ejpam-709	343	19	m2	m2	PROPN
ejpam-709	343	20	are	be	AUX
ejpam-709	343	21	two	two	NUM
ejpam-709	343	22	maximal	maximal	ADJ
ejpam-709	343	23	elements	element	NOUN
ejpam-709	343	24	in	in	ADP
ejpam-709	343	25	r	r	NOUN
ejpam-709	343	26	,	,	PUNCT
ejpam-709	343	27	then	then	ADV
ejpam-709	343	28	for	for	ADP
ejpam-709	343	29	any	any	DET
ejpam-709	343	30	x	x	SYM
ejpam-709	343	31	∈	∈	PROPN
ejpam-709	343	32	r	r	NOUN
ejpam-709	343	33	,	,	PUNCT
ejpam-709	343	34	⌈x	⌈x	NOUN
ejpam-709	343	35	,	,	PUNCT
ejpam-709	343	36	m1⌉s	m1⌉s	PROPN
ejpam-709	343	37	=	=	SYM
ejpam-709	343	38	⌈x	⌈x	PROPN
ejpam-709	343	39	,	,	PUNCT
ejpam-709	343	40	m2⌉s	m2⌉s	PROPN
ejpam-709	343	41	.	.	PUNCT
ejpam-709	344	1	lemma	lemma	PROPN
ejpam-709	344	2	8	8	NUM
ejpam-709	344	3	.	.	PUNCT
ejpam-709	345	1	let	let	VERB
ejpam-709	345	2	p	p	PRON
ejpam-709	345	3	be	be	AUX
ejpam-709	345	4	any	any	DET
ejpam-709	345	5	prime	prime	ADJ
ejpam-709	345	6	ideal	ideal	NOUN
ejpam-709	345	7	of	of	ADP
ejpam-709	345	8	s.	s.	PROPN
ejpam-709	345	9	for	for	ADP
ejpam-709	345	10	any	any	DET
ejpam-709	345	11	x	x	NOUN
ejpam-709	345	12	,	,	PUNCT
ejpam-709	345	13	y	y	PROPN
ejpam-709	345	14	∈	∈	PROPN
ejpam-709	345	15	r	r	NOUN
ejpam-709	345	16	,	,	PUNCT
ejpam-709	345	17	if	if	SCONJ
ejpam-709	345	18	y	y	PROPN
ejpam-709	345	19	∈	∈	PROPN
ejpam-709	345	20	p	p	PROPN
ejpam-709	345	21	∨	∨	X
ejpam-709	345	22	(	(	PUNCT
ejpam-709	345	23	x	x	X
ejpam-709	345	24	]	]	X
ejpam-709	345	25	,	,	PUNCT
ejpam-709	345	26	then	then	ADV
ejpam-709	345	27	p	p	NOUN
ejpam-709	345	28	∩	∩	ADJ
ejpam-709	345	29	⌈x	⌈x	NOUN
ejpam-709	345	30	,	,	PUNCT
ejpam-709	345	31	y⌉s	y⌉s	PROPN
ejpam-709	345	32	is	be	AUX
ejpam-709	345	33	non	non	ADJ
ejpam-709	345	34	-	-	ADJ
ejpam-709	345	35	empty	empty	ADJ
ejpam-709	345	36	.	.	PUNCT
ejpam-709	346	1	g.	g.	PROPN
ejpam-709	346	2	rao	rao	PROPN
ejpam-709	346	3	,	,	PUNCT
ejpam-709	346	4	n.	n.	PROPN
ejpam-709	346	5	rafi	rafi	PROPN
ejpam-709	346	6	and	and	CCONJ
ejpam-709	346	7	b.	b.	PROPN
ejpam-709	346	8	kumar	kumar	PROPN
ejpam-709	346	9	/	/	SYM
ejpam-709	346	10	eur	eur	PROPN
ejpam-709	346	11	.	.	PUNCT
ejpam-709	347	1	j.	j.	PROPN
ejpam-709	347	2	pure	pure	PROPN
ejpam-709	347	3	appl	appl	PROPN
ejpam-709	347	4	.	.	PROPN
ejpam-709	347	5	math	math	PROPN
ejpam-709	347	6	,	,	PUNCT
ejpam-709	347	7	3	3	NUM
ejpam-709	347	8	(	(	PUNCT
ejpam-709	347	9	2010	2010	NUM
ejpam-709	347	10	)	)	PUNCT
ejpam-709	347	11	,	,	PUNCT
ejpam-709	347	12	704	704	NUM
ejpam-709	347	13	-	-	SYM
ejpam-709	347	14	716	716	NUM
ejpam-709	347	15	713	713	NUM
ejpam-709	347	16	definition	definition	NOUN
ejpam-709	347	17	9	9	NUM
ejpam-709	347	18	.	.	PUNCT
ejpam-709	348	1	let	let	VERB
ejpam-709	348	2	r	r	PRON
ejpam-709	348	3	be	be	AUX
ejpam-709	348	4	an	an	DET
ejpam-709	348	5	adl	adl	NOUN
ejpam-709	348	6	with	with	ADP
ejpam-709	348	7	maximal	maximal	ADJ
ejpam-709	348	8	elements	element	NOUN
ejpam-709	348	9	and	and	CCONJ
ejpam-709	348	10	s	s	VERB
ejpam-709	348	11	a	a	DET
ejpam-709	348	12	uni	uni	PROPN
ejpam-709	348	13	subadl	subadl	NOUN
ejpam-709	348	14	of	of	ADP
ejpam-709	348	15	r.	r.	PROPN
ejpam-709	348	16	r	r	PROPN
ejpam-709	348	17	is	be	AUX
ejpam-709	348	18	called	call	VERB
ejpam-709	348	19	dually	dually	ADV
ejpam-709	348	20	s−normal	s−normal	PROPN
ejpam-709	348	21	if	if	SCONJ
ejpam-709	348	22	for	for	ADP
ejpam-709	348	23	any	any	DET
ejpam-709	348	24	x	x	NOUN
ejpam-709	348	25	,	,	PUNCT
ejpam-709	348	26	y	y	PROPN
ejpam-709	348	27	∈	∈	PROPN
ejpam-709	348	28	r	r	NOUN
ejpam-709	348	29	with	with	ADP
ejpam-709	348	30	x	x	PROPN
ejpam-709	348	31	∨	∨	PROPN
ejpam-709	348	32	y	y	PROPN
ejpam-709	348	33	is	be	AUX
ejpam-709	348	34	a	a	DET
ejpam-709	348	35	maximal	maximal	ADJ
ejpam-709	348	36	element	element	NOUN
ejpam-709	348	37	in	in	ADP
ejpam-709	348	38	r	r	NOUN
ejpam-709	348	39	,	,	PUNCT
ejpam-709	348	40	then	then	ADV
ejpam-709	348	41	there	there	PRON
ejpam-709	348	42	exist	exist	VERB
ejpam-709	348	43	a	a	DET
ejpam-709	348	44	,	,	PUNCT
ejpam-709	348	45	b	b	X
ejpam-709	348	46	∈	∈	NOUN
ejpam-709	348	47	r	r	NOUN
ejpam-709	348	48	such	such	ADJ
ejpam-709	348	49	that	that	SCONJ
ejpam-709	348	50	x	x	PROPN
ejpam-709	348	51	∨	∨	NUM
ejpam-709	348	52	a	a	X
ejpam-709	348	53	,	,	PUNCT
ejpam-709	348	54	y	y	PROPN
ejpam-709	348	55	∨	∨	PROPN
ejpam-709	348	56	b	b	PROPN
ejpam-709	348	57	are	be	AUX
ejpam-709	348	58	maximal	maximal	ADJ
ejpam-709	348	59	elements	element	NOUN
ejpam-709	348	60	and	and	CCONJ
ejpam-709	348	61	a	a	DET
ejpam-709	348	62	∧	∧	PROPN
ejpam-709	348	63	b	b	PROPN
ejpam-709	348	64	=	=	SYM
ejpam-709	348	65	0	0	PROPN
ejpam-709	348	66	.	.	PUNCT
ejpam-709	349	1	theorem	theorem	NOUN
ejpam-709	349	2	10	10	NUM
ejpam-709	349	3	.	.	PUNCT
ejpam-709	350	1	let	let	VERB
ejpam-709	350	2	r	r	PRON
ejpam-709	350	3	be	be	AUX
ejpam-709	350	4	an	an	DET
ejpam-709	350	5	adl	adl	NOUN
ejpam-709	350	6	with	with	ADP
ejpam-709	350	7	maximal	maximal	ADJ
ejpam-709	350	8	elements	element	NOUN
ejpam-709	350	9	and	and	CCONJ
ejpam-709	350	10	s	s	VERB
ejpam-709	350	11	a	a	DET
ejpam-709	350	12	uni	uni	PROPN
ejpam-709	350	13	subadl	subadl	NOUN
ejpam-709	350	14	of	of	ADP
ejpam-709	350	15	r.	r.	PROPN
ejpam-709	350	16	then	then	ADV
ejpam-709	350	17	the	the	DET
ejpam-709	350	18	following	following	NOUN
ejpam-709	350	19	are	be	AUX
ejpam-709	350	20	equivalent	equivalent	ADJ
ejpam-709	350	21	:	:	PUNCT
ejpam-709	350	22	1	1	X
ejpam-709	350	23	.	.	X
ejpam-709	350	24	r	r	NOUN
ejpam-709	350	25	is	be	AUX
ejpam-709	350	26	dually	dually	ADV
ejpam-709	350	27	s−normal	s−normal	ADJ
ejpam-709	350	28	2	2	NUM
ejpam-709	350	29	.	.	X
ejpam-709	350	30	⌈x	⌈x	NOUN
ejpam-709	350	31	,	,	PUNCT
ejpam-709	350	32	y⌉s	y⌉s	PROPN
ejpam-709	350	33	∨	∨	NUM
ejpam-709	350	34	⌈y	⌈y	PROPN
ejpam-709	350	35	,	,	PUNCT
ejpam-709	350	36	x⌉s	x⌉s	PROPN
ejpam-709	351	1	=	=	SYM
ejpam-709	351	2	s	s	PROPN
ejpam-709	351	3	,	,	PUNCT
ejpam-709	351	4	for	for	ADP
ejpam-709	351	5	any	any	DET
ejpam-709	351	6	x	x	NOUN
ejpam-709	351	7	,	,	PUNCT
ejpam-709	351	8	y	y	PROPN
ejpam-709	351	9	∈	∈	PROPN
ejpam-709	351	10	r	r	NOUN
ejpam-709	351	11	with	with	ADP
ejpam-709	351	12	x	x	PROPN
ejpam-709	351	13	∨	∨	PROPN
ejpam-709	351	14	y	y	PROPN
ejpam-709	351	15	is	be	AUX
ejpam-709	351	16	a	a	DET
ejpam-709	351	17	maximal	maximal	ADJ
ejpam-709	351	18	element	element	NOUN
ejpam-709	351	19	.	.	PUNCT
ejpam-709	352	1	the	the	DET
ejpam-709	352	2	following	follow	VERB
ejpam-709	352	3	definition	definition	NOUN
ejpam-709	352	4	is	be	AUX
ejpam-709	352	5	taken	take	VERB
ejpam-709	352	6	from	from	ADP
ejpam-709	352	7	[	[	X
ejpam-709	352	8	8	8	NUM
ejpam-709	352	9	]	]	PUNCT
ejpam-709	352	10	.	.	PUNCT
ejpam-709	353	1	definition	definition	NOUN
ejpam-709	353	2	10	10	NUM
ejpam-709	353	3	.	.	PUNCT
ejpam-709	354	1	let	let	VERB
ejpam-709	354	2	r	r	PRON
ejpam-709	354	3	be	be	AUX
ejpam-709	354	4	an	an	DET
ejpam-709	354	5	adl	adl	NOUN
ejpam-709	354	6	with	with	ADP
ejpam-709	354	7	maximal	maximal	ADJ
ejpam-709	354	8	elements	element	NOUN
ejpam-709	354	9	.	.	PUNCT
ejpam-709	355	1	then	then	ADV
ejpam-709	355	2	r	r	NOUN
ejpam-709	355	3	is	be	AUX
ejpam-709	355	4	called	call	VERB
ejpam-709	355	5	dually	dually	ADV
ejpam-709	355	6	relatively	relatively	ADV
ejpam-709	355	7	normal	normal	ADJ
ejpam-709	355	8	if	if	SCONJ
ejpam-709	355	9	for	for	ADP
ejpam-709	355	10	any	any	DET
ejpam-709	355	11	x	x	NOUN
ejpam-709	355	12	,	,	PUNCT
ejpam-709	355	13	y	y	PROPN
ejpam-709	355	14	∈	∈	PROPN
ejpam-709	355	15	r	r	NOUN
ejpam-709	355	16	there	there	PRON
ejpam-709	355	17	exist	exist	VERB
ejpam-709	355	18	a	a	DET
ejpam-709	355	19	,	,	PUNCT
ejpam-709	355	20	b	b	X
ejpam-709	355	21	∈	∈	NOUN
ejpam-709	355	22	r	r	NOUN
ejpam-709	356	1	such	such	ADJ
ejpam-709	356	2	that	that	PRON
ejpam-709	356	3	(	(	PUNCT
ejpam-709	356	4	x	x	PROPN
ejpam-709	356	5	∨	∨	NUM
ejpam-709	356	6	a	a	PRON
ejpam-709	356	7	)	)	PUNCT
ejpam-709	356	8	∨	∨	NOUN
ejpam-709	356	9	y	y	NOUN
ejpam-709	356	10	=	=	PUNCT
ejpam-709	356	11	x	x	PROPN
ejpam-709	356	12	∨	∨	NUM
ejpam-709	356	13	a	a	PRON
ejpam-709	356	14	,	,	PUNCT
ejpam-709	356	15	(	(	PUNCT
ejpam-709	356	16	y	y	PROPN
ejpam-709	356	17	∨	∨	NUM
ejpam-709	356	18	b	b	PROPN
ejpam-709	356	19	)	)	PUNCT
ejpam-709	356	20	∨	∨	NOUN
ejpam-709	356	21	x	x	X
ejpam-709	356	22	=	=	SYM
ejpam-709	356	23	y	y	PROPN
ejpam-709	356	24	∨	∨	NUM
ejpam-709	356	25	b	b	PROPN
ejpam-709	356	26	and	and	CCONJ
ejpam-709	356	27	a	a	DET
ejpam-709	356	28	∧	∧	PROPN
ejpam-709	356	29	b	b	PROPN
ejpam-709	356	30	=	=	SYM
ejpam-709	356	31	0	0	PROPN
ejpam-709	356	32	.	.	PUNCT
ejpam-709	357	1	the	the	DET
ejpam-709	357	2	following	follow	VERB
ejpam-709	357	3	definition	definition	NOUN
ejpam-709	357	4	is	be	AUX
ejpam-709	357	5	taken	take	VERB
ejpam-709	357	6	from	from	ADP
ejpam-709	357	7	cignoli	cignoli	NOUN
ejpam-709	357	8	[	[	X
ejpam-709	357	9	2	2	NUM
ejpam-709	357	10	]	]	PUNCT
ejpam-709	357	11	.	.	PUNCT
ejpam-709	358	1	definition	definition	NOUN
ejpam-709	358	2	11	11	NUM
ejpam-709	358	3	.	.	PUNCT
ejpam-709	359	1	let	let	VERB
ejpam-709	359	2	(	(	PUNCT
ejpam-709	359	3	l,∨,∧	l,∨,∧	NOUN
ejpam-709	359	4	,	,	PUNCT
ejpam-709	359	5	0,1	0,1	NUM
ejpam-709	359	6	)	)	PUNCT
ejpam-709	359	7	be	be	VERB
ejpam-709	359	8	a	a	DET
ejpam-709	359	9	bounded	bounded	ADJ
ejpam-709	359	10	distributive	distributive	ADJ
ejpam-709	359	11	lattice	lattice	NOUN
ejpam-709	359	12	and	and	CCONJ
ejpam-709	359	13	s	s	VERB
ejpam-709	359	14	a	a	DET
ejpam-709	359	15	sublattice	sublattice	NOUN
ejpam-709	359	16	of	of	ADP
ejpam-709	359	17	l	l	NOUN
ejpam-709	359	18	containing	contain	VERB
ejpam-709	359	19	0	0	NUM
ejpam-709	359	20	and	and	CCONJ
ejpam-709	359	21	1	1	NUM
ejpam-709	359	22	.	.	PUNCT
ejpam-709	360	1	then	then	ADV
ejpam-709	360	2	l	l	PROPN
ejpam-709	360	3	is	be	AUX
ejpam-709	360	4	called	call	VERB
ejpam-709	360	5	dually	dually	ADV
ejpam-709	360	6	s−completely	s−completely	ADV
ejpam-709	360	7	normal	normal	ADJ
ejpam-709	360	8	,	,	PUNCT
ejpam-709	360	9	if	if	SCONJ
ejpam-709	360	10	for	for	ADP
ejpam-709	360	11	any	any	DET
ejpam-709	360	12	x	x	NOUN
ejpam-709	360	13	,	,	PUNCT
ejpam-709	360	14	y	y	PROPN
ejpam-709	360	15	∈	∈	PROPN
ejpam-709	360	16	l	l	NOUN
ejpam-709	360	17	,	,	PUNCT
ejpam-709	360	18	there	there	PRON
ejpam-709	360	19	exist	exist	VERB
ejpam-709	360	20	a	a	DET
ejpam-709	360	21	,	,	PUNCT
ejpam-709	360	22	b	b	X
ejpam-709	360	23	∈	∈	NOUN
ejpam-709	360	24	s	s	VERB
ejpam-709	360	25	such	such	ADJ
ejpam-709	360	26	that	that	SCONJ
ejpam-709	360	27	x	x	PROPN
ejpam-709	360	28	∨	∨	NUM
ejpam-709	360	29	a	a	DET
ejpam-709	360	30	≥	≥	NOUN
ejpam-709	360	31	y	y	NOUN
ejpam-709	360	32	,	,	PUNCT
ejpam-709	360	33	y	y	PROPN
ejpam-709	360	34	∨	∨	NUM
ejpam-709	360	35	b	b	PROPN
ejpam-709	360	36	≥	≥	NOUN
ejpam-709	360	37	x	x	X
ejpam-709	360	38	and	and	CCONJ
ejpam-709	360	39	a	a	DET
ejpam-709	360	40	∧	∧	PROPN
ejpam-709	360	41	b	b	PROPN
ejpam-709	360	42	=	=	SYM
ejpam-709	360	43	0	0	PROPN
ejpam-709	360	44	.	.	PUNCT
ejpam-709	361	1	now	now	ADV
ejpam-709	361	2	we	we	PRON
ejpam-709	361	3	define	define	VERB
ejpam-709	361	4	the	the	DET
ejpam-709	361	5	concept	concept	NOUN
ejpam-709	361	6	of	of	ADP
ejpam-709	361	7	dually	dually	ADV
ejpam-709	361	8	s−relatively	s−relatively	ADV
ejpam-709	361	9	normal	normal	ADJ
ejpam-709	361	10	adl	adl	NOUN
ejpam-709	361	11	in	in	ADP
ejpam-709	361	12	the	the	DET
ejpam-709	361	13	following	following	NOUN
ejpam-709	361	14	.	.	PUNCT
ejpam-709	362	1	definition	definition	NOUN
ejpam-709	362	2	12	12	NUM
ejpam-709	362	3	.	.	PUNCT
ejpam-709	363	1	let	let	VERB
ejpam-709	363	2	r	r	PRON
ejpam-709	363	3	be	be	AUX
ejpam-709	363	4	an	an	DET
ejpam-709	363	5	adl	adl	NOUN
ejpam-709	363	6	with	with	ADP
ejpam-709	363	7	maximal	maximal	ADJ
ejpam-709	363	8	elements	element	NOUN
ejpam-709	363	9	and	and	CCONJ
ejpam-709	363	10	s	s	VERB
ejpam-709	363	11	a	a	DET
ejpam-709	363	12	uni	uni	PROPN
ejpam-709	363	13	subadl	subadl	NOUN
ejpam-709	363	14	of	of	ADP
ejpam-709	363	15	r.	r.	PROPN
ejpam-709	363	16	r	r	PROPN
ejpam-709	363	17	is	be	AUX
ejpam-709	363	18	called	call	VERB
ejpam-709	363	19	dually	dually	ADV
ejpam-709	363	20	s−relatively	s−relatively	ADV
ejpam-709	363	21	normal	normal	ADJ
ejpam-709	363	22	if	if	SCONJ
ejpam-709	363	23	pf(r	pf(r	NOUN
ejpam-709	363	24	)	)	PUNCT
ejpam-709	363	25	is	be	AUX
ejpam-709	363	26	dually	dually	ADV
ejpam-709	363	27	pf(s)−completely	pf(s)−completely	ADV
ejpam-709	363	28	normal	normal	ADJ
ejpam-709	363	29	lattice	lattice	NOUN
ejpam-709	363	30	.	.	PUNCT
ejpam-709	364	1	lemma	lemma	PROPN
ejpam-709	364	2	9	9	NUM
ejpam-709	364	3	.	.	PUNCT
ejpam-709	365	1	let	let	VERB
ejpam-709	365	2	r	r	PRON
ejpam-709	365	3	be	be	AUX
ejpam-709	365	4	an	an	DET
ejpam-709	365	5	adl	adl	NOUN
ejpam-709	365	6	with	with	ADP
ejpam-709	365	7	maximal	maximal	ADJ
ejpam-709	365	8	elements	element	NOUN
ejpam-709	365	9	and	and	CCONJ
ejpam-709	365	10	s	s	VERB
ejpam-709	365	11	a	a	DET
ejpam-709	365	12	uni	uni	PROPN
ejpam-709	365	13	subadl	subadl	NOUN
ejpam-709	365	14	of	of	ADP
ejpam-709	365	15	r.	r.	PROPN
ejpam-709	365	16	then	then	ADV
ejpam-709	365	17	r	r	NOUN
ejpam-709	365	18	is	be	AUX
ejpam-709	365	19	dually	dually	ADV
ejpam-709	365	20	s−relatively	s−relatively	ADV
ejpam-709	365	21	normal	normal	ADJ
ejpam-709	365	22	if	if	SCONJ
ejpam-709	365	23	and	and	CCONJ
ejpam-709	365	24	only	only	ADV
ejpam-709	365	25	if	if	SCONJ
ejpam-709	365	26	for	for	ADP
ejpam-709	365	27	any	any	DET
ejpam-709	365	28	x	x	NOUN
ejpam-709	365	29	,	,	PUNCT
ejpam-709	365	30	y	y	PROPN
ejpam-709	365	31	∈	∈	PROPN
ejpam-709	365	32	r	r	NOUN
ejpam-709	365	33	,	,	PUNCT
ejpam-709	365	34	there	there	PRON
ejpam-709	365	35	exist	exist	VERB
ejpam-709	365	36	a	a	DET
ejpam-709	365	37	,	,	PUNCT
ejpam-709	365	38	b	b	X
ejpam-709	365	39	∈	∈	NOUN
ejpam-709	365	40	r	r	NOUN
ejpam-709	366	1	such	such	ADJ
ejpam-709	366	2	that	that	PRON
ejpam-709	366	3	(	(	PUNCT
ejpam-709	366	4	x	x	PROPN
ejpam-709	366	5	∨	∨	NUM
ejpam-709	366	6	a)∨	a)∨	PROPN
ejpam-709	366	7	y	y	NOUN
ejpam-709	366	8	=	=	PUNCT
ejpam-709	366	9	x	x	PROPN
ejpam-709	366	10	∨	∨	NUM
ejpam-709	366	11	a	a	PRON
ejpam-709	366	12	,	,	PUNCT
ejpam-709	366	13	(	(	PUNCT
ejpam-709	366	14	y	y	PROPN
ejpam-709	366	15	∨	∨	PROPN
ejpam-709	366	16	b)∨	b)∨	PROPN
ejpam-709	366	17	x	x	SYM
ejpam-709	366	18	=	=	SYM
ejpam-709	366	19	y	y	PROPN
ejpam-709	366	20	∨	∨	NUM
ejpam-709	366	21	b	b	PROPN
ejpam-709	366	22	and	and	CCONJ
ejpam-709	366	23	a	a	DET
ejpam-709	366	24	∧	∧	PROPN
ejpam-709	366	25	b	b	PROPN
ejpam-709	366	26	=	=	SYM
ejpam-709	366	27	0	0	PROPN
ejpam-709	366	28	.	.	PUNCT
ejpam-709	367	1	lemma	lemma	PROPN
ejpam-709	367	2	10	10	NUM
ejpam-709	367	3	.	.	PUNCT
ejpam-709	368	1	let	let	VERB
ejpam-709	368	2	r	r	PRON
ejpam-709	368	3	be	be	AUX
ejpam-709	368	4	an	an	DET
ejpam-709	368	5	adl	adl	NOUN
ejpam-709	368	6	with	with	ADP
ejpam-709	368	7	maximal	maximal	ADJ
ejpam-709	368	8	elements	element	NOUN
ejpam-709	368	9	and	and	CCONJ
ejpam-709	368	10	s	s	VERB
ejpam-709	368	11	a	a	DET
ejpam-709	368	12	uni	uni	PROPN
ejpam-709	368	13	subadl	subadl	NOUN
ejpam-709	368	14	of	of	ADP
ejpam-709	368	15	r.	r.	PROPN
ejpam-709	368	16	then	then	ADV
ejpam-709	368	17	r	r	NOUN
ejpam-709	368	18	is	be	AUX
ejpam-709	368	19	dually	dually	ADV
ejpam-709	368	20	s−relatively	s−relatively	ADV
ejpam-709	368	21	normal	normal	ADJ
ejpam-709	368	22	if	if	SCONJ
ejpam-709	368	23	and	and	CCONJ
ejpam-709	368	24	only	only	ADV
ejpam-709	368	25	if	if	SCONJ
ejpam-709	368	26	for	for	ADP
ejpam-709	368	27	any	any	DET
ejpam-709	368	28	x	x	NOUN
ejpam-709	368	29	,	,	PUNCT
ejpam-709	368	30	y	y	PROPN
ejpam-709	368	31	∈	∈	PROPN
ejpam-709	368	32	r	r	PROPN
ejpam-709	368	33	,	,	PUNCT
ejpam-709	368	34	⌈x	⌈x	NOUN
ejpam-709	368	35	,	,	PUNCT
ejpam-709	368	36	y⌉s	y⌉s	PROPN
ejpam-709	368	37	∨	∨	NUM
ejpam-709	368	38	⌈y	⌈y	PROPN
ejpam-709	368	39	,	,	PUNCT
ejpam-709	368	40	x⌉s	x⌉s	PROPN
ejpam-709	369	1	=	=	PUNCT
ejpam-709	369	2	s.	s.	PROPN
ejpam-709	369	3	theorem	theorem	VERB
ejpam-709	369	4	11	11	NUM
ejpam-709	369	5	.	.	PUNCT
ejpam-709	370	1	let	let	VERB
ejpam-709	370	2	r	r	PRON
ejpam-709	370	3	be	be	AUX
ejpam-709	370	4	an	an	DET
ejpam-709	370	5	adl	adl	NOUN
ejpam-709	370	6	with	with	ADP
ejpam-709	370	7	maximal	maximal	ADJ
ejpam-709	370	8	elements	element	NOUN
ejpam-709	370	9	and	and	CCONJ
ejpam-709	370	10	s	s	VERB
ejpam-709	370	11	a	a	DET
ejpam-709	370	12	uni	uni	PROPN
ejpam-709	370	13	subadl	subadl	NOUN
ejpam-709	370	14	of	of	ADP
ejpam-709	370	15	r.	r.	PROPN
ejpam-709	370	16	then	then	ADV
ejpam-709	370	17	the	the	DET
ejpam-709	370	18	following	follow	VERB
ejpam-709	370	19	conditions	condition	NOUN
ejpam-709	370	20	are	be	AUX
ejpam-709	370	21	equivalent	equivalent	ADJ
ejpam-709	370	22	:	:	PUNCT
ejpam-709	370	23	1	1	X
ejpam-709	370	24	.	.	X
ejpam-709	370	25	r	r	NOUN
ejpam-709	370	26	is	be	AUX
ejpam-709	370	27	dually	dually	ADV
ejpam-709	370	28	s−relatively	s−relatively	ADV
ejpam-709	370	29	normal	normal	ADJ
ejpam-709	370	30	2	2	NUM
ejpam-709	370	31	.	.	PUNCT
ejpam-709	370	32	for	for	ADP
ejpam-709	370	33	each	each	DET
ejpam-709	370	34	pair	pair	NOUN
ejpam-709	370	35	x	x	X
ejpam-709	370	36	,	,	PUNCT
ejpam-709	370	37	y	y	PROPN
ejpam-709	370	38	∈	∈	PROPN
ejpam-709	370	39	r	r	NOUN
ejpam-709	370	40	,	,	PUNCT
ejpam-709	370	41	there	there	PRON
ejpam-709	370	42	is	be	VERB
ejpam-709	370	43	no	no	DET
ejpam-709	370	44	proper	proper	ADJ
ejpam-709	370	45	filter	filter	NOUN
ejpam-709	370	46	of	of	ADP
ejpam-709	370	47	s	s	AUX
ejpam-709	370	48	containing	contain	VERB
ejpam-709	370	49	both	both	DET
ejpam-709	370	50	⌈x	⌈x	NOUN
ejpam-709	370	51	,	,	PUNCT
ejpam-709	370	52	y⌉s	y⌉s	PROPN
ejpam-709	370	53	and	and	CCONJ
ejpam-709	370	54	⌈y	⌈y	PROPN
ejpam-709	370	55	,	,	PUNCT
ejpam-709	370	56	x⌉s	x⌉s	PROPN
ejpam-709	371	1	3	3	X
ejpam-709	371	2	.	.	PUNCT
ejpam-709	372	1	the	the	DET
ejpam-709	372	2	set	set	NOUN
ejpam-709	372	3	of	of	ADP
ejpam-709	372	4	all	all	DET
ejpam-709	372	5	ideals	ideal	NOUN
ejpam-709	372	6	of	of	ADP
ejpam-709	372	7	r	r	NOUN
ejpam-709	372	8	that	that	PRON
ejpam-709	372	9	contain	contain	VERB
ejpam-709	372	10	a	a	DET
ejpam-709	372	11	given	give	VERB
ejpam-709	372	12	s−prime	s−prime	NOUN
ejpam-709	372	13	ideal	ideal	NOUN
ejpam-709	372	14	of	of	ADP
ejpam-709	372	15	r	r	NOUN
ejpam-709	372	16	form	form	NOUN
ejpam-709	372	17	a	a	DET
ejpam-709	372	18	chain	chain	NOUN
ejpam-709	372	19	4	4	NUM
ejpam-709	372	20	.	.	PUNCT
ejpam-709	373	1	the	the	DET
ejpam-709	373	2	set	set	NOUN
ejpam-709	373	3	of	of	ADP
ejpam-709	373	4	all	all	DET
ejpam-709	373	5	prime	prime	ADJ
ejpam-709	373	6	ideals	ideal	NOUN
ejpam-709	373	7	of	of	ADP
ejpam-709	373	8	r	r	NOUN
ejpam-709	373	9	that	that	PRON
ejpam-709	373	10	contain	contain	VERB
ejpam-709	373	11	a	a	DET
ejpam-709	373	12	given	give	VERB
ejpam-709	373	13	s−prime	s−prime	NOUN
ejpam-709	373	14	ideal	ideal	NOUN
ejpam-709	373	15	of	of	ADP
ejpam-709	373	16	r	r	NOUN
ejpam-709	373	17	form	form	NOUN
ejpam-709	373	18	a	a	DET
ejpam-709	373	19	chain	chain	NOUN
ejpam-709	373	20	5	5	NUM
ejpam-709	373	21	.	.	PUNCT
ejpam-709	374	1	any	any	DET
ejpam-709	374	2	proper	proper	ADJ
ejpam-709	374	3	ideal	ideal	NOUN
ejpam-709	374	4	of	of	ADP
ejpam-709	374	5	r	r	NOUN
ejpam-709	374	6	that	that	PRON
ejpam-709	374	7	contain	contain	VERB
ejpam-709	374	8	a	a	DET
ejpam-709	374	9	given	give	VERB
ejpam-709	374	10	s−prime	s−prime	NOUN
ejpam-709	374	11	ideal	ideal	NOUN
ejpam-709	374	12	of	of	ADP
ejpam-709	374	13	r	r	NOUN
ejpam-709	374	14	is	be	AUX
ejpam-709	374	15	a	a	DET
ejpam-709	374	16	prime	prime	NOUN
ejpam-709	374	17	.	.	PUNCT
ejpam-709	375	1	corollary	corollary	ADJ
ejpam-709	375	2	3	3	X
ejpam-709	375	3	.	.	PUNCT
ejpam-709	376	1	let	let	VERB
ejpam-709	376	2	r	r	PRON
ejpam-709	376	3	be	be	AUX
ejpam-709	376	4	an	an	DET
ejpam-709	376	5	adl	adl	NOUN
ejpam-709	376	6	with	with	ADP
ejpam-709	376	7	maximal	maximal	ADJ
ejpam-709	376	8	elements	element	NOUN
ejpam-709	376	9	and	and	CCONJ
ejpam-709	376	10	s1,s2	s1,s2	PROPN
ejpam-709	376	11	uni	uni	INTJ
ejpam-709	376	12	subadls	subadls	NOUN
ejpam-709	376	13	of	of	ADP
ejpam-709	376	14	r	r	NOUN
ejpam-709	376	15	such	such	ADJ
ejpam-709	376	16	that	that	SCONJ
ejpam-709	376	17	s1	s1	PROPN
ejpam-709	376	18	⊆	⊆	NUM
ejpam-709	376	19	s2	s2	PROPN
ejpam-709	376	20	.	.	PUNCT
ejpam-709	377	1	then	then	ADV
ejpam-709	377	2	the	the	DET
ejpam-709	377	3	following	follow	VERB
ejpam-709	377	4	conditions	condition	NOUN
ejpam-709	377	5	are	be	AUX
ejpam-709	377	6	equivalent	equivalent	ADJ
ejpam-709	377	7	:	:	PUNCT
ejpam-709	377	8	g.	g.	PROPN
ejpam-709	377	9	rao	rao	PROPN
ejpam-709	377	10	,	,	PUNCT
ejpam-709	377	11	n.	n.	PROPN
ejpam-709	377	12	rafi	rafi	PROPN
ejpam-709	377	13	and	and	CCONJ
ejpam-709	377	14	b.	b.	PROPN
ejpam-709	377	15	kumar	kumar	PROPN
ejpam-709	377	16	/	/	SYM
ejpam-709	377	17	eur	eur	PROPN
ejpam-709	377	18	.	.	PUNCT
ejpam-709	378	1	j.	j.	PROPN
ejpam-709	378	2	pure	pure	PROPN
ejpam-709	378	3	appl	appl	PROPN
ejpam-709	378	4	.	.	PROPN
ejpam-709	378	5	math	math	PROPN
ejpam-709	378	6	,	,	PUNCT
ejpam-709	378	7	3	3	NUM
ejpam-709	378	8	(	(	PUNCT
ejpam-709	378	9	2010	2010	NUM
ejpam-709	378	10	)	)	PUNCT
ejpam-709	378	11	,	,	PUNCT
ejpam-709	378	12	704	704	NUM
ejpam-709	378	13	-	-	SYM
ejpam-709	378	14	716	716	NUM
ejpam-709	378	15	714	714	NUM
ejpam-709	378	16	1	1	NUM
ejpam-709	378	17	.	.	PUNCT
ejpam-709	379	1	r	r	NOUN
ejpam-709	379	2	is	be	AUX
ejpam-709	379	3	dually	dually	ADV
ejpam-709	379	4	s1−relatively	s1−relatively	ADV
ejpam-709	379	5	normal	normal	ADJ
ejpam-709	379	6	2	2	NUM
ejpam-709	379	7	.	.	PUNCT
ejpam-709	380	1	r	r	NOUN
ejpam-709	380	2	is	be	AUX
ejpam-709	380	3	dually	dually	ADV
ejpam-709	380	4	s2−relatively	s2−relatively	ADV
ejpam-709	380	5	normal	normal	ADJ
ejpam-709	380	6	and	and	CCONJ
ejpam-709	380	7	the	the	DET
ejpam-709	380	8	ideals	ideal	NOUN
ejpam-709	380	9	generated	generate	VERB
ejpam-709	380	10	in	in	ADP
ejpam-709	380	11	s2	s2	PROPN
ejpam-709	380	12	by	by	ADP
ejpam-709	380	13	prime	prime	ADJ
ejpam-709	380	14	ideals	ideal	NOUN
ejpam-709	380	15	of	of	ADP
ejpam-709	380	16	s1	s1	NOUN
ejpam-709	380	17	are	be	AUX
ejpam-709	380	18	prime	prime	ADJ
ejpam-709	380	19	.	.	PUNCT
ejpam-709	381	1	corollary	corollary	ADJ
ejpam-709	381	2	4	4	NUM
ejpam-709	381	3	.	.	PUNCT
ejpam-709	382	1	let	let	VERB
ejpam-709	382	2	r	r	PRON
ejpam-709	382	3	be	be	AUX
ejpam-709	382	4	an	an	DET
ejpam-709	382	5	adl	adl	NOUN
ejpam-709	382	6	with	with	ADP
ejpam-709	382	7	maximal	maximal	ADJ
ejpam-709	382	8	elements	element	NOUN
ejpam-709	382	9	and	and	CCONJ
ejpam-709	382	10	s	s	VERB
ejpam-709	382	11	a	a	DET
ejpam-709	382	12	uni	uni	PROPN
ejpam-709	382	13	subadl	subadl	NOUN
ejpam-709	382	14	of	of	ADP
ejpam-709	382	15	r.	r.	PROPN
ejpam-709	382	16	then	then	ADV
ejpam-709	382	17	r	r	NOUN
ejpam-709	382	18	is	be	AUX
ejpam-709	382	19	dually	dually	ADV
ejpam-709	382	20	s−relatively	s−relatively	ADV
ejpam-709	382	21	normal	normal	ADJ
ejpam-709	382	22	if	if	SCONJ
ejpam-709	382	23	and	and	CCONJ
ejpam-709	382	24	only	only	ADV
ejpam-709	382	25	if	if	SCONJ
ejpam-709	382	26	r	r	NOUN
ejpam-709	382	27	is	be	AUX
ejpam-709	382	28	dually	dually	ADV
ejpam-709	382	29	relatively	relatively	ADV
ejpam-709	382	30	normal	normal	ADJ
ejpam-709	382	31	and	and	CCONJ
ejpam-709	382	32	the	the	DET
ejpam-709	382	33	s−prime	s−prime	NOUN
ejpam-709	382	34	ideals	ideal	NOUN
ejpam-709	382	35	of	of	ADP
ejpam-709	382	36	r	r	NOUN
ejpam-709	382	37	are	be	AUX
ejpam-709	382	38	prime	prime	ADJ
ejpam-709	382	39	.	.	PUNCT
ejpam-709	383	1	proof	proof	NOUN
ejpam-709	383	2	.	.	PUNCT
ejpam-709	384	1	take	take	VERB
ejpam-709	384	2	s1	s1	NOUN
ejpam-709	384	3	=	=	SYM
ejpam-709	384	4	s	s	PROPN
ejpam-709	384	5	and	and	CCONJ
ejpam-709	384	6	s2	s2	NOUN
ejpam-709	384	7	=	=	PUNCT
ejpam-709	384	8	r	r	NOUN
ejpam-709	384	9	in	in	ADP
ejpam-709	384	10	the	the	DET
ejpam-709	384	11	above	above	ADJ
ejpam-709	384	12	corollary	corollary	NOUN
ejpam-709	384	13	.	.	PUNCT
ejpam-709	385	1	definition	definition	NOUN
ejpam-709	385	2	13	13	NUM
ejpam-709	385	3	.	.	PUNCT
ejpam-709	386	1	let	let	VERB
ejpam-709	386	2	r	r	PRON
ejpam-709	386	3	be	be	AUX
ejpam-709	386	4	an	an	DET
ejpam-709	386	5	adl	adl	NOUN
ejpam-709	386	6	with	with	ADP
ejpam-709	386	7	maximal	maximal	ADJ
ejpam-709	386	8	elements	element	NOUN
ejpam-709	386	9	.	.	PUNCT
ejpam-709	387	1	r	r	NOUN
ejpam-709	387	2	is	be	AUX
ejpam-709	387	3	called	call	VERB
ejpam-709	387	4	relatively	relatively	ADV
ejpam-709	387	5	normal	normal	ADJ
ejpam-709	387	6	if	if	SCONJ
ejpam-709	387	7	for	for	ADP
ejpam-709	387	8	any	any	DET
ejpam-709	387	9	x	x	NOUN
ejpam-709	387	10	,	,	PUNCT
ejpam-709	387	11	y	y	PROPN
ejpam-709	387	12	∈	∈	PROPN
ejpam-709	387	13	r	r	NOUN
ejpam-709	387	14	,	,	PUNCT
ejpam-709	387	15	there	there	PRON
ejpam-709	387	16	exist	exist	VERB
ejpam-709	387	17	a	a	DET
ejpam-709	387	18	,	,	PUNCT
ejpam-709	387	19	b	b	X
ejpam-709	387	20	∈	∈	NOUN
ejpam-709	387	21	r	r	NOUN
ejpam-709	387	22	such	such	ADJ
ejpam-709	387	23	that	that	SCONJ
ejpam-709	387	24	y	y	PROPN
ejpam-709	387	25	∧	∧	PROPN
ejpam-709	387	26	a	a	DET
ejpam-709	387	27	∧	∧	PROPN
ejpam-709	387	28	x	x	X
ejpam-709	387	29	=	=	PUNCT
ejpam-709	387	30	a	a	DET
ejpam-709	387	31	∧	∧	PROPN
ejpam-709	387	32	x	x	X
ejpam-709	387	33	,	,	PUNCT
ejpam-709	387	34	x	x	PUNCT
ejpam-709	388	1	∧	∧	NOUN
ejpam-709	388	2	b	b	PROPN
ejpam-709	388	3	∧	∧	PROPN
ejpam-709	388	4	y	y	PROPN
ejpam-709	388	5	=	=	SYM
ejpam-709	388	6	b	b	PROPN
ejpam-709	388	7	∧	∧	PROPN
ejpam-709	388	8	y	y	PROPN
ejpam-709	388	9	and	and	CCONJ
ejpam-709	388	10	a	a	DET
ejpam-709	388	11	∨	∨	PROPN
ejpam-709	388	12	b	b	NOUN
ejpam-709	388	13	is	be	AUX
ejpam-709	388	14	a	a	DET
ejpam-709	388	15	maximal	maximal	ADJ
ejpam-709	388	16	element	element	NOUN
ejpam-709	388	17	.	.	PUNCT
ejpam-709	389	1	r	r	NOUN
ejpam-709	389	2	is	be	AUX
ejpam-709	389	3	called	call	VERB
ejpam-709	389	4	dually	dually	ADV
ejpam-709	389	5	relatively	relatively	ADV
ejpam-709	389	6	normal	normal	ADJ
ejpam-709	389	7	if	if	SCONJ
ejpam-709	389	8	for	for	ADP
ejpam-709	389	9	any	any	DET
ejpam-709	389	10	x	x	NOUN
ejpam-709	389	11	,	,	PUNCT
ejpam-709	389	12	y	y	PROPN
ejpam-709	389	13	∈	∈	PROPN
ejpam-709	389	14	r	r	NOUN
ejpam-709	389	15	,	,	PUNCT
ejpam-709	389	16	there	there	PRON
ejpam-709	389	17	exist	exist	VERB
ejpam-709	389	18	a	a	DET
ejpam-709	389	19	,	,	PUNCT
ejpam-709	389	20	b	b	X
ejpam-709	389	21	∈	∈	NOUN
ejpam-709	389	22	r	r	NOUN
ejpam-709	390	1	such	such	ADJ
ejpam-709	390	2	that	that	PRON
ejpam-709	390	3	(	(	PUNCT
ejpam-709	390	4	x	x	PROPN
ejpam-709	390	5	∨	∨	NUM
ejpam-709	390	6	a)∨	a)∨	PROPN
ejpam-709	390	7	y	y	NOUN
ejpam-709	390	8	=	=	PUNCT
ejpam-709	390	9	x	x	PROPN
ejpam-709	390	10	∨	∨	NUM
ejpam-709	390	11	a	a	PRON
ejpam-709	390	12	,	,	PUNCT
ejpam-709	390	13	(	(	PUNCT
ejpam-709	390	14	y	y	PROPN
ejpam-709	390	15	∨	∨	PROPN
ejpam-709	390	16	b)∨	b)∨	PROPN
ejpam-709	390	17	x	x	SYM
ejpam-709	390	18	=	=	SYM
ejpam-709	390	19	y	y	PROPN
ejpam-709	390	20	∨	∨	NUM
ejpam-709	390	21	b	b	PROPN
ejpam-709	390	22	and	and	CCONJ
ejpam-709	390	23	a	a	DET
ejpam-709	390	24	∧	∧	PROPN
ejpam-709	390	25	b	b	PROPN
ejpam-709	390	26	=	=	SYM
ejpam-709	390	27	0	0	PROPN
ejpam-709	390	28	.	.	PUNCT
ejpam-709	391	1	definition	definition	NOUN
ejpam-709	391	2	14	14	NUM
ejpam-709	391	3	.	.	PUNCT
ejpam-709	392	1	let	let	VERB
ejpam-709	392	2	r	r	PRON
ejpam-709	392	3	be	be	AUX
ejpam-709	392	4	an	an	DET
ejpam-709	392	5	adl	adl	NOUN
ejpam-709	392	6	with	with	ADP
ejpam-709	392	7	maximal	maximal	ADJ
ejpam-709	392	8	elements	element	NOUN
ejpam-709	392	9	.	.	PUNCT
ejpam-709	393	1	then	then	ADV
ejpam-709	393	2	r	r	NOUN
ejpam-709	393	3	is	be	AUX
ejpam-709	393	4	called	call	VERB
ejpam-709	393	5	a	a	DET
ejpam-709	393	6	linear	linear	ADJ
ejpam-709	393	7	adl	adl	NOUN
ejpam-709	393	8	if	if	SCONJ
ejpam-709	393	9	r	r	NOUN
ejpam-709	393	10	is	be	AUX
ejpam-709	393	11	both	both	PRON
ejpam-709	393	12	relatively	relatively	ADV
ejpam-709	393	13	normal	normal	ADJ
ejpam-709	393	14	and	and	CCONJ
ejpam-709	393	15	dually	dually	ADV
ejpam-709	393	16	relatively	relatively	ADV
ejpam-709	393	17	normal	normal	ADJ
ejpam-709	393	18	.	.	PUNCT
ejpam-709	394	1	if	if	SCONJ
ejpam-709	394	2	s	s	PROPN
ejpam-709	394	3	is	be	AUX
ejpam-709	394	4	a	a	DET
ejpam-709	394	5	uni	uni	ADJ
ejpam-709	394	6	subadl	subadl	NOUN
ejpam-709	394	7	of	of	ADP
ejpam-709	394	8	r	r	PROPN
ejpam-709	394	9	,	,	PUNCT
ejpam-709	394	10	then	then	ADV
ejpam-709	394	11	r	r	NOUN
ejpam-709	394	12	is	be	AUX
ejpam-709	394	13	called	call	VERB
ejpam-709	394	14	an	an	DET
ejpam-709	394	15	s−linear	s−linear	NOUN
ejpam-709	394	16	adl	adl	NOUN
ejpam-709	394	17	if	if	SCONJ
ejpam-709	394	18	r	r	NOUN
ejpam-709	394	19	is	be	AUX
ejpam-709	394	20	both	both	PRON
ejpam-709	394	21	s−relatively	s−relatively	ADV
ejpam-709	394	22	normal	normal	ADJ
ejpam-709	394	23	and	and	CCONJ
ejpam-709	394	24	dually	dually	ADV
ejpam-709	394	25	s−relatively	s−relatively	ADV
ejpam-709	394	26	normal	normal	ADJ
ejpam-709	394	27	.	.	PUNCT
ejpam-709	395	1	the	the	DET
ejpam-709	395	2	following	follow	VERB
ejpam-709	395	3	theorem	theorem	NOUN
ejpam-709	395	4	can	can	AUX
ejpam-709	395	5	be	be	AUX
ejpam-709	395	6	verified	verify	VERB
ejpam-709	395	7	easily	easily	ADV
ejpam-709	395	8	.	.	PUNCT
ejpam-709	396	1	theorem	theorem	NOUN
ejpam-709	396	2	12	12	NUM
ejpam-709	396	3	.	.	PUNCT
ejpam-709	397	1	let	let	VERB
ejpam-709	397	2	r	r	PRON
ejpam-709	397	3	be	be	AUX
ejpam-709	397	4	an	an	DET
ejpam-709	397	5	adl	adl	NOUN
ejpam-709	397	6	with	with	ADP
ejpam-709	397	7	maximal	maximal	ADJ
ejpam-709	397	8	elements	element	NOUN
ejpam-709	397	9	and	and	CCONJ
ejpam-709	397	10	s	s	VERB
ejpam-709	397	11	a	a	DET
ejpam-709	397	12	uni	uni	PROPN
ejpam-709	397	13	subadl	subadl	NOUN
ejpam-709	397	14	of	of	ADP
ejpam-709	397	15	r.	r.	PROPN
ejpam-709	397	16	then	then	ADV
ejpam-709	397	17	r	r	NOUN
ejpam-709	397	18	is	be	AUX
ejpam-709	397	19	s−linear	s−linear	PUNCT
ejpam-709	397	20	if	if	SCONJ
ejpam-709	398	1	and	and	CCONJ
ejpam-709	398	2	only	only	ADV
ejpam-709	398	3	if	if	SCONJ
ejpam-709	398	4	1	1	NUM
ejpam-709	398	5	.	.	X
ejpam-709	399	1	r	r	NOUN
ejpam-709	399	2	is	be	AUX
ejpam-709	399	3	a	a	DET
ejpam-709	399	4	linear	linear	ADJ
ejpam-709	399	5	adl	adl	NOUN
ejpam-709	399	6	2	2	NUM
ejpam-709	399	7	.	.	PUNCT
ejpam-709	400	1	the	the	DET
ejpam-709	400	2	s−prime	s−prime	NOUN
ejpam-709	400	3	filters	filter	NOUN
ejpam-709	400	4	of	of	ADP
ejpam-709	400	5	r	r	NOUN
ejpam-709	400	6	are	be	AUX
ejpam-709	400	7	prime	prime	ADJ
ejpam-709	400	8	in	in	ADP
ejpam-709	400	9	r	r	NOUN
ejpam-709	400	10	3	3	NUM
ejpam-709	400	11	.	.	PUNCT
ejpam-709	401	1	the	the	DET
ejpam-709	401	2	s−prime	s−prime	NOUN
ejpam-709	401	3	ideals	ideal	NOUN
ejpam-709	401	4	of	of	ADP
ejpam-709	401	5	r	r	NOUN
ejpam-709	401	6	are	be	AUX
ejpam-709	401	7	prime	prime	ADJ
ejpam-709	401	8	in	in	ADP
ejpam-709	401	9	r.	r.	PROPN
ejpam-709	401	10	definition	definition	NOUN
ejpam-709	401	11	15	15	NUM
ejpam-709	401	12	.	.	PUNCT
ejpam-709	402	1	let	let	VERB
ejpam-709	402	2	r	r	PRON
ejpam-709	402	3	be	be	AUX
ejpam-709	402	4	an	an	DET
ejpam-709	402	5	adl	adl	NOUN
ejpam-709	402	6	with	with	ADP
ejpam-709	402	7	maximal	maximal	ADJ
ejpam-709	402	8	elements	element	NOUN
ejpam-709	402	9	.	.	PUNCT
ejpam-709	403	1	then	then	ADV
ejpam-709	403	2	b	b	X
ejpam-709	403	3	=	=	PRON
ejpam-709	403	4	{	{	PUNCT
ejpam-709	403	5	a	a	DET
ejpam-709	403	6	∈	∈	NOUN
ejpam-709	403	7	r	r	NOUN
ejpam-709	404	1	|	|	ADV
ejpam-709	404	2	there	there	PRON
ejpam-709	404	3	exists	exist	VERB
ejpam-709	404	4	b	b	PROPN
ejpam-709	404	5	∈	∈	NOUN
ejpam-709	404	6	r	r	NOUN
ejpam-709	404	7	such	such	DET
ejpam-709	404	8	that	that	SCONJ
ejpam-709	404	9	a	a	DET
ejpam-709	404	10	∧	∧	PROPN
ejpam-709	404	11	b	b	NOUN
ejpam-709	404	12	=	=	SYM
ejpam-709	404	13	0	0	PROPN
ejpam-709	404	14	and	and	CCONJ
ejpam-709	404	15	a	a	DET
ejpam-709	404	16	∨	∨	NOUN
ejpam-709	404	17	b	b	NOUN
ejpam-709	404	18	is	be	AUX
ejpam-709	404	19	maximal	maximal	ADJ
ejpam-709	404	20	}	}	PUNCT
ejpam-709	404	21	is	be	AUX
ejpam-709	404	22	called	call	VERB
ejpam-709	404	23	the	the	DET
ejpam-709	404	24	birkhoff	birkhoff	NOUN
ejpam-709	404	25	centre	centre	NOUN
ejpam-709	404	26	of	of	ADP
ejpam-709	404	27	r	r	NOUN
ejpam-709	404	28	and	and	CCONJ
ejpam-709	404	29	(	(	PUNCT
ejpam-709	404	30	b	b	PROPN
ejpam-709	404	31	,	,	PUNCT
ejpam-709	404	32	∨	∨	NOUN
ejpam-709	404	33	,	,	PUNCT
ejpam-709	404	34	∧	∧	PROPN
ejpam-709	404	35	)	)	PUNCT
ejpam-709	404	36	is	be	AUX
ejpam-709	404	37	a	a	DET
ejpam-709	404	38	uni	uni	PROPN
ejpam-709	404	39	sub	sub	NOUN
ejpam-709	404	40	adl	adl	PROPN
ejpam-709	404	41	of	of	ADP
ejpam-709	404	42	r	r	NOUN
ejpam-709	404	43	which	which	PRON
ejpam-709	404	44	is	be	AUX
ejpam-709	404	45	also	also	ADV
ejpam-709	404	46	a	a	DET
ejpam-709	404	47	relatively	relatively	ADV
ejpam-709	404	48	complemented	complement	VERB
ejpam-709	404	49	adl	adl	NOUN
ejpam-709	405	1	[	[	X
ejpam-709	405	2	10	10	NUM
ejpam-709	405	3	]	]	PUNCT
ejpam-709	405	4	.	.	PUNCT
ejpam-709	406	1	if	if	SCONJ
ejpam-709	406	2	a	a	DET
ejpam-709	406	3	∈	∈	PROPN
ejpam-709	406	4	b	b	NOUN
ejpam-709	406	5	,	,	PUNCT
ejpam-709	406	6	then	then	ADV
ejpam-709	406	7	an	an	DET
ejpam-709	406	8	element	element	NOUN
ejpam-709	406	9	b	b	PROPN
ejpam-709	406	10	∈	∈	NOUN
ejpam-709	406	11	r	r	NOUN
ejpam-709	406	12	with	with	ADP
ejpam-709	406	13	the	the	DET
ejpam-709	406	14	property	property	NOUN
ejpam-709	406	15	a	a	DET
ejpam-709	406	16	∧	∧	PROPN
ejpam-709	406	17	b	b	NOUN
ejpam-709	406	18	=	=	SYM
ejpam-709	406	19	0	0	PROPN
ejpam-709	406	20	and	and	CCONJ
ejpam-709	406	21	a	a	DET
ejpam-709	406	22	∨	∨	NOUN
ejpam-709	406	23	b	b	NOUN
ejpam-709	406	24	is	be	AUX
ejpam-709	406	25	maximal	maximal	ADJ
ejpam-709	406	26	is	be	AUX
ejpam-709	406	27	called	call	VERB
ejpam-709	406	28	a	a	DET
ejpam-709	406	29	complement	complement	NOUN
ejpam-709	406	30	of	of	ADP
ejpam-709	406	31	a	a	PRON
ejpam-709	406	32	in	in	ADP
ejpam-709	406	33	b.	b.	PROPN
ejpam-709	406	34	it	it	PRON
ejpam-709	406	35	was	be	AUX
ejpam-709	406	36	observed	observe	VERB
ejpam-709	406	37	in	in	ADP
ejpam-709	406	38	[	[	X
ejpam-709	406	39	7	7	X
ejpam-709	406	40	]	]	PUNCT
ejpam-709	406	41	that	that	SCONJ
ejpam-709	406	42	every	every	DET
ejpam-709	406	43	relatively	relatively	ADV
ejpam-709	406	44	complemented	complement	VERB
ejpam-709	406	45	adl	adl	NOUN
ejpam-709	406	46	is	be	AUX
ejpam-709	406	47	a	a	DET
ejpam-709	406	48	normal	normal	ADJ
ejpam-709	406	49	adl	adl	NOUN
ejpam-709	406	50	and	and	CCONJ
ejpam-709	406	51	hence	hence	ADV
ejpam-709	406	52	b	b	NOUN
ejpam-709	406	53	is	be	AUX
ejpam-709	406	54	normal	normal	ADJ
ejpam-709	406	55	.	.	PUNCT
ejpam-709	407	1	we	we	PRON
ejpam-709	407	2	conclude	conclude	VERB
ejpam-709	407	3	this	this	DET
ejpam-709	407	4	paper	paper	NOUN
ejpam-709	407	5	with	with	ADP
ejpam-709	407	6	the	the	DET
ejpam-709	407	7	following	follow	VERB
ejpam-709	407	8	characterization	characterization	NOUN
ejpam-709	407	9	theorem	theorem	VERB
ejpam-709	407	10	.	.	PUNCT
ejpam-709	407	11	theorem	theorem	NOUN
ejpam-709	407	12	13	13	NUM
ejpam-709	407	13	.	.	PUNCT
ejpam-709	408	1	let	let	VERB
ejpam-709	408	2	r	r	PRON
ejpam-709	408	3	be	be	AUX
ejpam-709	408	4	an	an	DET
ejpam-709	408	5	adl	adl	NOUN
ejpam-709	408	6	with	with	ADP
ejpam-709	408	7	maximal	maximal	ADJ
ejpam-709	408	8	elements	element	NOUN
ejpam-709	408	9	and	and	CCONJ
ejpam-709	408	10	b	b	X
ejpam-709	408	11	the	the	DET
ejpam-709	408	12	birkhoff	birkhoff	NOUN
ejpam-709	408	13	centre	centre	PROPN
ejpam-709	408	14	of	of	ADP
ejpam-709	408	15	r.	r.	PROPN
ejpam-709	408	16	then	then	ADV
ejpam-709	408	17	the	the	DET
ejpam-709	408	18	following	follow	VERB
ejpam-709	408	19	conditions	condition	NOUN
ejpam-709	408	20	are	be	AUX
ejpam-709	408	21	equivalent	equivalent	ADJ
ejpam-709	408	22	:	:	PUNCT
ejpam-709	408	23	1	1	X
ejpam-709	408	24	.	.	X
ejpam-709	408	25	r	r	NOUN
ejpam-709	408	26	is	be	AUX
ejpam-709	408	27	b−relatively	b−relatively	ADV
ejpam-709	408	28	normal	normal	ADJ
ejpam-709	408	29	references	reference	NOUN
ejpam-709	408	30	715	715	NUM
ejpam-709	408	31	1.′	1.′	NUM
ejpam-709	408	32	r	r	NOUN
ejpam-709	408	33	is	be	AUX
ejpam-709	408	34	dually	dually	ADV
ejpam-709	408	35	b−relatively	b−relatively	ADV
ejpam-709	408	36	normal	normal	ADJ
ejpam-709	408	37	2	2	NUM
ejpam-709	408	38	.	.	PUNCT
ejpam-709	409	1	given	give	VERB
ejpam-709	409	2	x	x	PROPN
ejpam-709	409	3	,	,	PUNCT
ejpam-709	409	4	y	y	PROPN
ejpam-709	409	5	∈	∈	PROPN
ejpam-709	409	6	r	r	NOUN
ejpam-709	409	7	,	,	PUNCT
ejpam-709	409	8	there	there	PRON
ejpam-709	409	9	is	be	VERB
ejpam-709	409	10	a	a	DET
ejpam-709	409	11	∈	∈	PROPN
ejpam-709	409	12	b	b	NOUN
ejpam-709	409	13	and	and	CCONJ
ejpam-709	409	14	a	a	DET
ejpam-709	409	15	complement	complement	NOUN
ejpam-709	409	16	a′	a′	NOUN
ejpam-709	409	17	of	of	ADP
ejpam-709	409	18	a	a	DET
ejpam-709	409	19	such	such	ADJ
ejpam-709	409	20	that	that	SCONJ
ejpam-709	409	21	y	y	PROPN
ejpam-709	409	22	∧	∧	PROPN
ejpam-709	409	23	a	a	DET
ejpam-709	409	24	∧	∧	PROPN
ejpam-709	409	25	x	x	X
ejpam-709	409	26	=	=	PUNCT
ejpam-709	409	27	a	a	DET
ejpam-709	409	28	∧	∧	PROPN
ejpam-709	409	29	x	x	X
ejpam-709	409	30	and	and	CCONJ
ejpam-709	409	31	x	x	PROPN
ejpam-709	409	32	∧	∧	PROPN
ejpam-709	409	33	a′	a′	VERB
ejpam-709	409	34	∧	∧	PROPN
ejpam-709	409	35	y	y	PROPN
ejpam-709	409	36	=	=	NOUN
ejpam-709	409	37	a′	a′	PROPN
ejpam-709	409	38	∧	∧	PROPN
ejpam-709	409	39	y	y	PROPN
ejpam-709	409	40	2.′	2.′	PROPN
ejpam-709	409	41	given	give	VERB
ejpam-709	409	42	x	x	PUNCT
ejpam-709	409	43	,	,	PUNCT
ejpam-709	409	44	y	y	PROPN
ejpam-709	409	45	∈	∈	PROPN
ejpam-709	409	46	r	r	NOUN
ejpam-709	409	47	,	,	PUNCT
ejpam-709	409	48	there	there	PRON
ejpam-709	409	49	is	be	VERB
ejpam-709	409	50	a	a	DET
ejpam-709	409	51	∈	∈	PROPN
ejpam-709	409	52	b	b	NOUN
ejpam-709	409	53	a	a	DET
ejpam-709	409	54	complement	complement	NOUN
ejpam-709	409	55	a′	a′	NOUN
ejpam-709	409	56	of	of	ADP
ejpam-709	409	57	a	a	DET
ejpam-709	409	58	such	such	ADJ
ejpam-709	409	59	that	that	SCONJ
ejpam-709	409	60	x	x	PROPN
ejpam-709	409	61	∨	∨	NUM
ejpam-709	409	62	a	a	DET
ejpam-709	409	63	∨	∨	NOUN
ejpam-709	409	64	y	y	NOUN
ejpam-709	409	65	=	=	PUNCT
ejpam-709	409	66	x	x	PROPN
ejpam-709	409	67	∨	∨	NUM
ejpam-709	409	68	a	a	PRON
ejpam-709	409	69	and	and	CCONJ
ejpam-709	409	70	y	y	PROPN
ejpam-709	409	71	∨	∨	PROPN
ejpam-709	409	72	a′	a′	PROPN
ejpam-709	409	73	∨	∨	NOUN
ejpam-709	409	74	x	x	X
ejpam-709	409	75	=	=	SYM
ejpam-709	409	76	y	y	PROPN
ejpam-709	409	77	∨	∨	NUM
ejpam-709	409	78	a′	a′	PROPN
ejpam-709	409	79	3	3	NUM
ejpam-709	409	80	.	.	PUNCT
ejpam-709	410	1	r	r	NOUN
ejpam-709	410	2	is	be	AUX
ejpam-709	410	3	a	a	DET
ejpam-709	410	4	linear	linear	ADJ
ejpam-709	410	5	adl	adl	NOUN
ejpam-709	410	6	and	and	CCONJ
ejpam-709	410	7	the	the	DET
ejpam-709	410	8	minimal	minimal	ADJ
ejpam-709	410	9	prime	prime	ADJ
ejpam-709	410	10	ideals	ideal	NOUN
ejpam-709	410	11	of	of	ADP
ejpam-709	410	12	r	r	NOUN
ejpam-709	410	13	are	be	AUX
ejpam-709	410	14	b−maximal	b−maximal	ADJ
ejpam-709	410	15	ideals	ideal	NOUN
ejpam-709	410	16	of	of	ADP
ejpam-709	410	17	r	r	NOUN
ejpam-709	410	18	3.′	3.′	NUM
ejpam-709	410	19	r	r	NOUN
ejpam-709	410	20	is	be	AUX
ejpam-709	410	21	linear	linear	PROPN
ejpam-709	410	22	adl	adl	PROPN
ejpam-709	410	23	and	and	CCONJ
ejpam-709	410	24	the	the	DET
ejpam-709	410	25	minimal	minimal	ADJ
ejpam-709	410	26	prime	prime	ADJ
ejpam-709	410	27	filters	filter	NOUN
ejpam-709	410	28	of	of	ADP
ejpam-709	410	29	r	r	NOUN
ejpam-709	410	30	are	be	AUX
ejpam-709	410	31	b−maximal	b−maximal	NOUN
ejpam-709	410	32	filters	filter	NOUN
ejpam-709	410	33	of	of	ADP
ejpam-709	410	34	r.	r.	PROPN
ejpam-709	410	35	proof	proof	NOUN
ejpam-709	410	36	.	.	PUNCT
ejpam-709	411	1	(	(	PUNCT
ejpam-709	411	2	1)⇒	1)⇒	NUM
ejpam-709	411	3	(	(	PUNCT
ejpam-709	411	4	2	2	NUM
ejpam-709	411	5	)	)	PUNCT
ejpam-709	411	6	:	:	PUNCT
ejpam-709	411	7	assume	assume	VERB
ejpam-709	411	8	(	(	PUNCT
ejpam-709	411	9	1	1	NUM
ejpam-709	411	10	)	)	PUNCT
ejpam-709	411	11	.	.	PUNCT
ejpam-709	412	1	let	let	VERB
ejpam-709	412	2	x	x	PRON
ejpam-709	412	3	,	,	PUNCT
ejpam-709	412	4	y	y	PROPN
ejpam-709	412	5	∈	∈	PROPN
ejpam-709	412	6	r.	r.	PROPN
ejpam-709	412	7	then	then	ADV
ejpam-709	412	8	there	there	PRON
ejpam-709	412	9	exist	exist	VERB
ejpam-709	412	10	a	a	DET
ejpam-709	412	11	,	,	PUNCT
ejpam-709	412	12	b	b	PROPN
ejpam-709	412	13	∈	∈	PROPN
ejpam-709	412	14	b	b	NOUN
ejpam-709	412	15	such	such	ADJ
ejpam-709	412	16	that	that	SCONJ
ejpam-709	412	17	y	y	PROPN
ejpam-709	412	18	∧	∧	PROPN
ejpam-709	412	19	a	a	DET
ejpam-709	412	20	∧	∧	PROPN
ejpam-709	412	21	x	x	PUNCT
ejpam-709	412	22	=	=	PRON
ejpam-709	412	23	a∧x	a∧x	NOUN
ejpam-709	412	24	,	,	PUNCT
ejpam-709	412	25	x∧b∧y	x∧b∧y	PROPN
ejpam-709	412	26	=	=	PUNCT
ejpam-709	412	27	b∧y	b∧y	PROPN
ejpam-709	412	28	and	and	CCONJ
ejpam-709	412	29	a∨b	a∨b	PROPN
ejpam-709	412	30	is	be	AUX
ejpam-709	412	31	a	a	DET
ejpam-709	412	32	maximal	maximal	ADJ
ejpam-709	412	33	element	element	NOUN
ejpam-709	412	34	.	.	PUNCT
ejpam-709	413	1	since	since	SCONJ
ejpam-709	413	2	a	a	DET
ejpam-709	413	3	∈	∈	PROPN
ejpam-709	413	4	b	b	NOUN
ejpam-709	413	5	,	,	PUNCT
ejpam-709	413	6	there	there	PRON
ejpam-709	413	7	exists	exist	VERB
ejpam-709	413	8	c	c	NOUN
ejpam-709	413	9	∈	∈	PROPN
ejpam-709	413	10	r	r	NOUN
ejpam-709	413	11	such	such	DET
ejpam-709	413	12	that	that	DET
ejpam-709	413	13	a∧	a∧	NOUN
ejpam-709	413	14	c	c	NOUN
ejpam-709	413	15	=	=	SYM
ejpam-709	413	16	0	0	PROPN
ejpam-709	413	17	and	and	CCONJ
ejpam-709	413	18	a∨	a∨	PROPN
ejpam-709	413	19	c	c	PROPN
ejpam-709	413	20	is	be	AUX
ejpam-709	413	21	a	a	DET
ejpam-709	413	22	maximal	maximal	ADJ
ejpam-709	413	23	element	element	NOUN
ejpam-709	413	24	.	.	PUNCT
ejpam-709	414	1	now	now	ADV
ejpam-709	414	2	,	,	PUNCT
ejpam-709	414	3	a∨	a∨	PROPN
ejpam-709	414	4	(	(	PUNCT
ejpam-709	414	5	c∧	c∧	PROPN
ejpam-709	414	6	b	b	NOUN
ejpam-709	414	7	)	)	PUNCT
ejpam-709	414	8	=	=	PUNCT
ejpam-709	414	9	(	(	PUNCT
ejpam-709	414	10	a∨	a∨	PROPN
ejpam-709	414	11	c)∧	c)∧	PROPN
ejpam-709	414	12	(	(	PUNCT
ejpam-709	414	13	a∨	a∨	PROPN
ejpam-709	414	14	b	b	PROPN
ejpam-709	414	15	)	)	PUNCT
ejpam-709	414	16	=	=	SYM
ejpam-709	414	17	a∨	a∨	PROPN
ejpam-709	414	18	b	b	PROPN
ejpam-709	414	19	and	and	CCONJ
ejpam-709	414	20	a	a	DET
ejpam-709	414	21	∧	∧	PROPN
ejpam-709	414	22	c	c	PROPN
ejpam-709	414	23	∧	∧	PROPN
ejpam-709	414	24	b	b	PROPN
ejpam-709	414	25	=	=	NOUN
ejpam-709	414	26	0	0	PROPN
ejpam-709	414	27	.	.	PUNCT
ejpam-709	415	1	so	so	ADV
ejpam-709	415	2	that	that	SCONJ
ejpam-709	415	3	c	c	PROPN
ejpam-709	415	4	∧	∧	PROPN
ejpam-709	415	5	b(=	b(=	PROPN
ejpam-709	415	6	a′	a′	PROPN
ejpam-709	415	7	say	say	VERB
ejpam-709	415	8	)	)	PUNCT
ejpam-709	415	9	is	be	AUX
ejpam-709	415	10	a	a	DET
ejpam-709	415	11	complement	complement	NOUN
ejpam-709	415	12	of	of	ADP
ejpam-709	415	13	a	a	PRON
ejpam-709	415	14	in	in	ADP
ejpam-709	415	15	b	b	NOUN
ejpam-709	415	16	and	and	CCONJ
ejpam-709	415	17	x	x	PROPN
ejpam-709	415	18	∧	∧	PROPN
ejpam-709	415	19	a′	a′	VERB
ejpam-709	415	20	∧	∧	PROPN
ejpam-709	415	21	y	y	NOUN
ejpam-709	415	22	=	=	PUNCT
ejpam-709	415	23	x	x	SYM
ejpam-709	415	24	∧	∧	NOUN
ejpam-709	415	25	c	c	PROPN
ejpam-709	415	26	∧	∧	PROPN
ejpam-709	415	27	b	b	PROPN
ejpam-709	415	28	∧	∧	PROPN
ejpam-709	415	29	y	y	PROPN
ejpam-709	416	1	=	=	SYM
ejpam-709	416	2	c	c	PROPN
ejpam-709	416	3	∧	∧	NOUN
ejpam-709	416	4	x	x	PUNCT
ejpam-709	416	5	∧	∧	PROPN
ejpam-709	416	6	b	b	PROPN
ejpam-709	416	7	∧	∧	PROPN
ejpam-709	416	8	y	y	PROPN
ejpam-709	416	9	=	=	SYM
ejpam-709	416	10	c	c	PROPN
ejpam-709	416	11	∧	∧	PROPN
ejpam-709	416	12	b	b	PROPN
ejpam-709	416	13	∧	∧	PROPN
ejpam-709	416	14	y	y	PROPN
ejpam-709	416	15	=	=	PUNCT
ejpam-709	416	16	a′	a′	PROPN
ejpam-709	416	17	∧	∧	PROPN
ejpam-709	416	18	y.	y.	PROPN
ejpam-709	416	19	(	(	PUNCT
ejpam-709	416	20	2)⇒	2)⇒	NUM
ejpam-709	416	21	(	(	PUNCT
ejpam-709	416	22	2′	2′	NUM
ejpam-709	416	23	)	)	PUNCT
ejpam-709	416	24	:	:	PUNCT
ejpam-709	416	25	assume	assume	VERB
ejpam-709	416	26	(	(	PUNCT
ejpam-709	416	27	2	2	NUM
ejpam-709	416	28	)	)	PUNCT
ejpam-709	416	29	.	.	PUNCT
ejpam-709	417	1	let	let	VERB
ejpam-709	417	2	x	x	PRON
ejpam-709	417	3	,	,	PUNCT
ejpam-709	417	4	y	y	PROPN
ejpam-709	417	5	∈	∈	PROPN
ejpam-709	417	6	r.	r.	PROPN
ejpam-709	417	7	then	then	ADV
ejpam-709	417	8	by	by	ADP
ejpam-709	417	9	our	our	PRON
ejpam-709	417	10	assumption	assumption	NOUN
ejpam-709	417	11	there	there	PRON
ejpam-709	417	12	exists	exist	VERB
ejpam-709	417	13	a	a	DET
ejpam-709	417	14	∈	∈	PROPN
ejpam-709	417	15	b	b	NOUN
ejpam-709	417	16	and	and	CCONJ
ejpam-709	417	17	a	a	DET
ejpam-709	417	18	complement	complement	NOUN
ejpam-709	417	19	a′	a′	NOUN
ejpam-709	417	20	of	of	ADP
ejpam-709	417	21	a	a	PRON
ejpam-709	417	22	in	in	ADP
ejpam-709	417	23	b	b	NOUN
ejpam-709	417	24	such	such	ADJ
ejpam-709	417	25	that	that	SCONJ
ejpam-709	417	26	y	y	PROPN
ejpam-709	417	27	∧	∧	PROPN
ejpam-709	417	28	a	a	DET
ejpam-709	417	29	∧	∧	PROPN
ejpam-709	417	30	x	x	X
ejpam-709	417	31	=	=	PUNCT
ejpam-709	417	32	a	a	DET
ejpam-709	417	33	∧	∧	PROPN
ejpam-709	417	34	x	x	X
ejpam-709	417	35	and	and	CCONJ
ejpam-709	417	36	x	x	PROPN
ejpam-709	417	37	∧	∧	PROPN
ejpam-709	417	38	a′	a′	VERB
ejpam-709	417	39	∧	∧	PROPN
ejpam-709	417	40	y	y	PROPN
ejpam-709	417	41	=	=	PROPN
ejpam-709	417	42	a′	a′	PROPN
ejpam-709	417	43	∧	∧	PROPN
ejpam-709	417	44	y.	y.	PROPN
ejpam-709	417	45	now	now	ADV
ejpam-709	417	46	,	,	PUNCT
ejpam-709	417	47	(	(	PUNCT
ejpam-709	417	48	x	x	X
ejpam-709	417	49	∨	∨	NUM
ejpam-709	417	50	a)∧	a)∧	PROPN
ejpam-709	417	51	y	y	NOUN
ejpam-709	417	52	=	=	PUNCT
ejpam-709	417	53	(	(	PUNCT
ejpam-709	417	54	(	(	PUNCT
ejpam-709	417	55	x	x	SYM
ejpam-709	417	56	∨	∨	NOUN
ejpam-709	417	57	(	(	PUNCT
ejpam-709	417	58	a′	a′	PROPN
ejpam-709	417	59	∧	∧	PROPN
ejpam-709	417	60	y)∨	y)∨	PROPN
ejpam-709	417	61	a)∧	a)∧	PROPN
ejpam-709	417	62	y	y	PROPN
ejpam-709	417	63	=	=	SYM
ejpam-709	417	64	(	(	PUNCT
ejpam-709	417	65	x	x	X
ejpam-709	417	66	∨	∨	NUM
ejpam-709	417	67	a	a	DET
ejpam-709	417	68	∨	∨	NOUN
ejpam-709	417	69	a′)∧	a′)∧	PROPN
ejpam-709	417	70	(	(	PUNCT
ejpam-709	417	71	x	x	PROPN
ejpam-709	417	72	∨	∨	NUM
ejpam-709	417	73	a	a	DET
ejpam-709	417	74	∨	∨	NUM
ejpam-709	417	75	y)∧	y)∧	NOUN
ejpam-709	417	76	y	y	PROPN
ejpam-709	417	77	=	=	PRON
ejpam-709	417	78	(	(	PUNCT
ejpam-709	417	79	a	a	DET
ejpam-709	417	80	∨	∨	NUM
ejpam-709	418	1	a′)∧	a′)∧	PROPN
ejpam-709	418	2	y	y	PROPN
ejpam-709	418	3	=	=	PUNCT
ejpam-709	418	4	y.	y.	PROPN
ejpam-709	418	5	therefore	therefore	ADV
ejpam-709	418	6	(	(	PUNCT
ejpam-709	418	7	x	x	PROPN
ejpam-709	418	8	∨	∨	NUM
ejpam-709	419	1	a)∨	a)∨	PROPN
ejpam-709	419	2	y	y	NOUN
ejpam-709	419	3	=	=	PUNCT
ejpam-709	419	4	x	x	SYM
ejpam-709	419	5	∨	∨	NUM
ejpam-709	419	6	a.	a.	NOUN
ejpam-709	419	7	similarly	similarly	ADV
ejpam-709	419	8	,	,	PUNCT
ejpam-709	419	9	(	(	PUNCT
ejpam-709	419	10	y	y	PROPN
ejpam-709	419	11	∨	∨	NUM
ejpam-709	419	12	a′)∨	a′)∨	PROPN
ejpam-709	419	13	x	x	X
ejpam-709	419	14	=	=	SYM
ejpam-709	419	15	y	y	PROPN
ejpam-709	419	16	∨	∨	NOUN
ejpam-709	419	17	a′.	a′.	NOUN
ejpam-709	419	18	(	(	PUNCT
ejpam-709	419	19	2′	2′	NUM
ejpam-709	419	20	)	)	PUNCT
ejpam-709	419	21	⇒	⇒	NOUN
ejpam-709	419	22	(	(	PUNCT
ejpam-709	419	23	2	2	NUM
ejpam-709	419	24	)	)	PUNCT
ejpam-709	419	25	:	:	PUNCT
ejpam-709	419	26	assume	assume	VERB
ejpam-709	419	27	(	(	PUNCT
ejpam-709	419	28	2′	2′	NUM
ejpam-709	419	29	)	)	PUNCT
ejpam-709	419	30	.	.	PUNCT
ejpam-709	420	1	let	let	VERB
ejpam-709	420	2	x	x	PRON
ejpam-709	420	3	,	,	PUNCT
ejpam-709	420	4	y	y	PROPN
ejpam-709	420	5	∈	∈	PROPN
ejpam-709	420	6	r.	r.	PROPN
ejpam-709	420	7	then	then	ADV
ejpam-709	420	8	there	there	PRON
ejpam-709	420	9	exists	exist	VERB
ejpam-709	420	10	a	a	DET
ejpam-709	420	11	∈	∈	PROPN
ejpam-709	420	12	b	b	NOUN
ejpam-709	420	13	and	and	CCONJ
ejpam-709	420	14	a	a	DET
ejpam-709	420	15	complement	complement	NOUN
ejpam-709	420	16	a′	a′	NOUN
ejpam-709	420	17	of	of	ADP
ejpam-709	420	18	a	a	PRON
ejpam-709	420	19	in	in	ADP
ejpam-709	420	20	b	b	NOUN
ejpam-709	420	21	such	such	ADJ
ejpam-709	420	22	that	that	SCONJ
ejpam-709	420	23	(	(	PUNCT
ejpam-709	420	24	x	x	PROPN
ejpam-709	420	25	∨	∨	NUM
ejpam-709	420	26	a	a	PRON
ejpam-709	420	27	)	)	PUNCT
ejpam-709	420	28	∨	∨	NOUN
ejpam-709	420	29	y	y	NOUN
ejpam-709	420	30	=	=	PUNCT
ejpam-709	420	31	x	x	PROPN
ejpam-709	420	32	∨	∨	NUM
ejpam-709	420	33	a	a	PRON
ejpam-709	420	34	and	and	CCONJ
ejpam-709	420	35	(	(	PUNCT
ejpam-709	420	36	y	y	PROPN
ejpam-709	420	37	∨	∨	NUM
ejpam-709	420	38	a′	a′	PROPN
ejpam-709	420	39	)	)	PUNCT
ejpam-709	420	40	∨	∨	NOUN
ejpam-709	420	41	x	x	X
ejpam-709	420	42	=	=	SYM
ejpam-709	420	43	y	y	PROPN
ejpam-709	420	44	∨	∨	NOUN
ejpam-709	420	45	a′.	a′.	PROPN
ejpam-709	420	46	now	now	ADV
ejpam-709	420	47	,	,	PUNCT
ejpam-709	420	48	y	y	PROPN
ejpam-709	420	49	∨	∨	NUM
ejpam-709	420	50	(	(	PUNCT
ejpam-709	420	51	a	a	DET
ejpam-709	420	52	∧	∧	PROPN
ejpam-709	420	53	x	x	NOUN
ejpam-709	420	54	)	)	PUNCT
ejpam-709	420	55	=	=	SYM
ejpam-709	420	56	y	y	PROPN
ejpam-709	420	57	∨	∨	NOUN
ejpam-709	420	58	(	(	PUNCT
ejpam-709	420	59	a	a	DET
ejpam-709	420	60	∧	∧	PROPN
ejpam-709	420	61	(	(	PUNCT
ejpam-709	420	62	y	y	PROPN
ejpam-709	420	63	∨	∨	NUM
ejpam-709	420	64	a′)∧	a′)∧	PROPN
ejpam-709	420	65	x	x	SYM
ejpam-709	420	66	)	)	PUNCT
ejpam-709	420	67	=	=	SYM
ejpam-709	420	68	y	y	PROPN
ejpam-709	420	69	∨	∨	NOUN
ejpam-709	420	70	(	(	PUNCT
ejpam-709	420	71	(	(	PUNCT
ejpam-709	420	72	a∧	a∧	NOUN
ejpam-709	420	73	y	y	PROPN
ejpam-709	420	74	∧	∧	PROPN
ejpam-709	420	75	x)∨	x)∨	PROPN
ejpam-709	420	76	(	(	PUNCT
ejpam-709	420	77	a∧	a∧	NOUN
ejpam-709	420	78	a′	a′	PROPN
ejpam-709	420	79	∧	∧	PROPN
ejpam-709	420	80	x	x	NOUN
ejpam-709	420	81	)	)	PUNCT
ejpam-709	420	82	)	)	PUNCT
ejpam-709	421	1	=	=	SYM
ejpam-709	421	2	y	y	PROPN
ejpam-709	421	3	∨	∨	NOUN
ejpam-709	421	4	(	(	PUNCT
ejpam-709	421	5	a	a	DET
ejpam-709	421	6	∧	∧	PROPN
ejpam-709	421	7	y	y	PROPN
ejpam-709	421	8	∧	∧	PROPN
ejpam-709	421	9	x	x	X
ejpam-709	421	10	)	)	PUNCT
ejpam-709	421	11	=	=	VERB
ejpam-709	422	1	y.	y.	NOUN
ejpam-709	422	2	therefore	therefore	ADV
ejpam-709	422	3	y	y	PROPN
ejpam-709	422	4	∧	∧	PROPN
ejpam-709	422	5	(	(	PUNCT
ejpam-709	422	6	a	a	DET
ejpam-709	422	7	∧	∧	PROPN
ejpam-709	422	8	x	x	NOUN
ejpam-709	422	9	)	)	PUNCT
ejpam-709	422	10	=	=	PUNCT
ejpam-709	422	11	a	a	DET
ejpam-709	422	12	∧	∧	PROPN
ejpam-709	422	13	x	x	X
ejpam-709	422	14	.	.	PUNCT
ejpam-709	423	1	similarly	similarly	ADV
ejpam-709	423	2	,	,	PUNCT
ejpam-709	423	3	x	x	PART
ejpam-709	423	4	∧	∧	PROPN
ejpam-709	423	5	(	(	PUNCT
ejpam-709	423	6	a′	a′	PROPN
ejpam-709	423	7	∧	∧	PROPN
ejpam-709	423	8	y	y	PROPN
ejpam-709	423	9	)	)	PUNCT
ejpam-709	423	10	=	=	SYM
ejpam-709	423	11	a′	a′	PROPN
ejpam-709	423	12	∧	∧	PROPN
ejpam-709	423	13	y.	y.	PROPN
ejpam-709	423	14	(	(	PUNCT
ejpam-709	423	15	2)⇒	2)⇒	NUM
ejpam-709	423	16	(	(	PUNCT
ejpam-709	423	17	3	3	NUM
ejpam-709	423	18	)	)	PUNCT
ejpam-709	423	19	:	:	PUNCT
ejpam-709	423	20	assume	assume	VERB
ejpam-709	423	21	(	(	PUNCT
ejpam-709	423	22	2	2	NUM
ejpam-709	423	23	)	)	PUNCT
ejpam-709	423	24	.	.	PUNCT
ejpam-709	424	1	since	since	SCONJ
ejpam-709	424	2	(	(	PUNCT
ejpam-709	424	3	2	2	NUM
ejpam-709	424	4	)	)	PUNCT
ejpam-709	424	5	and	and	CCONJ
ejpam-709	424	6	(	(	PUNCT
ejpam-709	424	7	2′	2′	NUM
ejpam-709	424	8	)	)	PUNCT
ejpam-709	424	9	are	be	AUX
ejpam-709	424	10	equivalent	equivalent	ADJ
ejpam-709	424	11	,	,	PUNCT
ejpam-709	424	12	r	r	NOUN
ejpam-709	424	13	is	be	AUX
ejpam-709	424	14	a	a	DET
ejpam-709	424	15	linear	linear	PROPN
ejpam-709	424	16	adl	adl	PROPN
ejpam-709	424	17	.	.	PUNCT
ejpam-709	425	1	let	let	VERB
ejpam-709	425	2	p	p	PRON
ejpam-709	425	3	be	be	AUX
ejpam-709	425	4	a	a	DET
ejpam-709	425	5	minimal	minimal	ADJ
ejpam-709	425	6	prime	prime	ADJ
ejpam-709	425	7	ideal	ideal	NOUN
ejpam-709	425	8	of	of	ADP
ejpam-709	425	9	r	r	NOUN
ejpam-709	425	10	and	and	CCONJ
ejpam-709	425	11	x	x	PUNCT
ejpam-709	425	12	∈	∈	PROPN
ejpam-709	426	1	p.	p.	NOUN
ejpam-709	426	2	then	then	ADV
ejpam-709	426	3	there	there	PRON
ejpam-709	426	4	exists	exist	VERB
ejpam-709	426	5	y	y	PROPN
ejpam-709	426	6	∈	∈	PROPN
ejpam-709	426	7	r	r	NOUN
ejpam-709	426	8	\	\	NOUN
ejpam-709	426	9	p	p	NOUN
ejpam-709	426	10	such	such	ADJ
ejpam-709	426	11	that	that	SCONJ
ejpam-709	426	12	x	x	SYM
ejpam-709	426	13	∧	∧	NOUN
ejpam-709	426	14	y	y	NOUN
ejpam-709	426	15	=	=	NOUN
ejpam-709	426	16	0	0	X
ejpam-709	426	17	.	.	PUNCT
ejpam-709	427	1	by	by	ADP
ejpam-709	427	2	(	(	PUNCT
ejpam-709	427	3	2	2	NUM
ejpam-709	427	4	)	)	PUNCT
ejpam-709	427	5	,	,	PUNCT
ejpam-709	427	6	there	there	PRON
ejpam-709	427	7	exists	exist	VERB
ejpam-709	427	8	a	a	DET
ejpam-709	427	9	∈	∈	PROPN
ejpam-709	427	10	b	b	NOUN
ejpam-709	427	11	and	and	CCONJ
ejpam-709	427	12	a	a	DET
ejpam-709	427	13	complement	complement	NOUN
ejpam-709	427	14	a′	a′	NOUN
ejpam-709	427	15	of	of	ADP
ejpam-709	427	16	a	a	DET
ejpam-709	427	17	such	such	ADJ
ejpam-709	427	18	that	that	SCONJ
ejpam-709	427	19	a	a	DET
ejpam-709	427	20	∧	∧	PROPN
ejpam-709	427	21	x	x	X
ejpam-709	427	22	=	=	PUNCT
ejpam-709	427	23	y	y	PROPN
ejpam-709	427	24	∧	∧	PROPN
ejpam-709	427	25	(	(	PUNCT
ejpam-709	427	26	a	a	DET
ejpam-709	427	27	∧	∧	PROPN
ejpam-709	427	28	x	x	NOUN
ejpam-709	427	29	)	)	PUNCT
ejpam-709	427	30	=	=	SYM
ejpam-709	427	31	0	0	NUM
ejpam-709	427	32	and	and	CCONJ
ejpam-709	427	33	a′	a′	PROPN
ejpam-709	427	34	∧	∧	PROPN
ejpam-709	427	35	y	y	PROPN
ejpam-709	427	36	=	=	PUNCT
ejpam-709	427	37	x	x	SYM
ejpam-709	427	38	∧	∧	PROPN
ejpam-709	427	39	(	(	PUNCT
ejpam-709	427	40	a′	a′	PROPN
ejpam-709	427	41	∧	∧	PROPN
ejpam-709	427	42	y	y	PROPN
ejpam-709	427	43	)	)	PUNCT
ejpam-709	427	44	=	=	SYM
ejpam-709	428	1	0	0	X
ejpam-709	428	2	.	.	PUNCT
ejpam-709	429	1	so	so	SCONJ
ejpam-709	429	2	that	that	SCONJ
ejpam-709	429	3	a′	a′	PROPN
ejpam-709	429	4	∧	∧	PROPN
ejpam-709	429	5	y	y	PROPN
ejpam-709	429	6	∈	∈	PROPN
ejpam-709	429	7	p	p	NOUN
ejpam-709	429	8	and	and	CCONJ
ejpam-709	429	9	hence	hence	ADV
ejpam-709	429	10	a′	a′	PROPN
ejpam-709	429	11	∈	∈	PROPN
ejpam-709	429	12	p.	p.	NOUN
ejpam-709	429	13	now	now	ADV
ejpam-709	429	14	,	,	PUNCT
ejpam-709	429	15	x	x	SYM
ejpam-709	429	16	=	=	SYM
ejpam-709	429	17	(	(	PUNCT
ejpam-709	429	18	a	a	DET
ejpam-709	429	19	∨	∨	NUM
ejpam-709	429	20	a′)∧	a′)∧	PROPN
ejpam-709	429	21	x	x	PUNCT
ejpam-709	430	1	=	=	PRON
ejpam-709	430	2	(	(	PUNCT
ejpam-709	430	3	a	a	DET
ejpam-709	430	4	∧	∧	PROPN
ejpam-709	430	5	x	x	NOUN
ejpam-709	430	6	)	)	PUNCT
ejpam-709	430	7	∨	∨	PROPN
ejpam-709	430	8	(	(	PUNCT
ejpam-709	430	9	a′	a′	PROPN
ejpam-709	430	10	∧	∧	PROPN
ejpam-709	430	11	x	x	NOUN
ejpam-709	430	12	)	)	PUNCT
ejpam-709	430	13	=	=	SYM
ejpam-709	430	14	a′	a′	PROPN
ejpam-709	430	15	∧	∧	PROPN
ejpam-709	430	16	x	x	X
ejpam-709	430	17	.	.	PUNCT
ejpam-709	431	1	therefore	therefore	ADV
ejpam-709	431	2	p	p	PROPN
ejpam-709	431	3	is	be	AUX
ejpam-709	431	4	b−ideal	b−ideal	NOUN
ejpam-709	431	5	of	of	ADP
ejpam-709	431	6	r	r	NOUN
ejpam-709	431	7	and	and	CCONJ
ejpam-709	431	8	hence	hence	ADV
ejpam-709	431	9	p	p	PRON
ejpam-709	431	10	is	be	AUX
ejpam-709	431	11	a	a	DET
ejpam-709	431	12	b−maximal	b−maximal	NOUN
ejpam-709	431	13	ideal	ideal	NOUN
ejpam-709	431	14	of	of	ADP
ejpam-709	431	15	r	r	NOUN
ejpam-709	431	16	,	,	PUNCT
ejpam-709	431	17	since	since	SCONJ
ejpam-709	431	18	b	b	NOUN
ejpam-709	431	19	is	be	AUX
ejpam-709	431	20	a	a	DET
ejpam-709	431	21	relatively	relatively	ADV
ejpam-709	431	22	complemented	complement	VERB
ejpam-709	431	23	adl	adl	PROPN
ejpam-709	431	24	.	.	PUNCT
ejpam-709	432	1	similarly	similarly	ADV
ejpam-709	432	2	,	,	PUNCT
ejpam-709	432	3	we	we	PRON
ejpam-709	432	4	get	get	VERB
ejpam-709	432	5	that	that	PRON
ejpam-709	432	6	(	(	PUNCT
ejpam-709	432	7	2′)⇒(3′	2′)⇒(3′	NOUN
ejpam-709	432	8	)	)	PUNCT
ejpam-709	432	9	.	.	PUNCT
ejpam-709	433	1	references	reference	NOUN
ejpam-709	433	2	[	[	X
ejpam-709	433	3	1	1	NUM
ejpam-709	433	4	]	]	PUNCT
ejpam-709	433	5	birkhoff	birkhoff	NOUN
ejpam-709	433	6	,	,	PUNCT
ejpam-709	433	7	g.	g.	PROPN
ejpam-709	433	8	:	:	PUNCT
ejpam-709	433	9	lattice	lattice	PROPN
ejpam-709	433	10	theory	theory	NOUN
ejpam-709	433	11	.	.	PUNCT
ejpam-709	434	1	amer	amer	PROPN
ejpam-709	434	2	.	.	PUNCT
ejpam-709	434	3	math	math	PROPN
ejpam-709	434	4	.	.	PUNCT
ejpam-709	435	1	soc	soc	PROPN
ejpam-709	435	2	.	.	PUNCT
ejpam-709	436	1	colloq	colloq	PROPN
ejpam-709	436	2	.	.	PUNCT
ejpam-709	437	1	publ	publ	PROPN
ejpam-709	437	2	.	.	PUNCT
ejpam-709	438	1	xxv	xxv	PROPN
ejpam-709	438	2	,	,	PUNCT
ejpam-709	438	3	providence	providence	NOUN
ejpam-709	438	4	.	.	PUNCT
ejpam-709	439	1	1967	1967	NUM
ejpam-709	439	2	.	.	PUNCT
ejpam-709	440	1	[	[	X
ejpam-709	440	2	2	2	NUM
ejpam-709	440	3	]	]	PUNCT
ejpam-709	440	4	cignoli	cignoli	NOUN
ejpam-709	440	5	,	,	PUNCT
ejpam-709	440	6	r.	r.	PROPN
ejpam-709	440	7	:	:	PUNCT
ejpam-709	440	8	the	the	DET
ejpam-709	440	9	lattice	lattice	NOUN
ejpam-709	440	10	of	of	ADP
ejpam-709	440	11	global	global	ADJ
ejpam-709	440	12	sections	section	NOUN
ejpam-709	440	13	of	of	ADP
ejpam-709	440	14	sheaves	sheaf	NOUN
ejpam-709	440	15	of	of	ADP
ejpam-709	440	16	chains	chain	NOUN
ejpam-709	440	17	over	over	ADP
ejpam-709	440	18	boolean	boolean	ADJ
ejpam-709	440	19	spaces	space	NOUN
ejpam-709	440	20	.	.	PUNCT
ejpam-709	441	1	algebra	algebra	NOUN
ejpam-709	441	2	univesalis	univesali	NOUN
ejpam-709	441	3	,	,	PUNCT
ejpam-709	441	4	8	8	NUM
ejpam-709	441	5	,	,	PUNCT
ejpam-709	441	6	357	357	NUM
ejpam-709	441	7	-	-	SYM
ejpam-709	441	8	373	373	NUM
ejpam-709	441	9	.	.	PUNCT
ejpam-709	441	10	1978	1978	NUM
ejpam-709	441	11	.	.	PUNCT
ejpam-709	442	1	[	[	X
ejpam-709	442	2	3	3	NUM
ejpam-709	442	3	]	]	X
ejpam-709	442	4	gratzer	gratzer	NOUN
ejpam-709	442	5	,	,	PUNCT
ejpam-709	442	6	g.	g.	PROPN
ejpam-709	442	7	:	:	PUNCT
ejpam-709	442	8	general	general	ADJ
ejpam-709	442	9	lattice	lattice	PROPN
ejpam-709	442	10	theory	theory	NOUN
ejpam-709	442	11	.	.	PUNCT
ejpam-709	443	1	academic	academic	ADJ
ejpam-709	443	2	press	press	NOUN
ejpam-709	443	3	,	,	PUNCT
ejpam-709	443	4	new	new	PROPN
ejpam-709	443	5	york	york	PROPN
ejpam-709	443	6	,	,	PUNCT
ejpam-709	443	7	sanfransisco	sanfransisco	PROPN
ejpam-709	443	8	.	.	PROPN
ejpam-709	443	9	1978	1978	NUM
ejpam-709	443	10	.	.	PUNCT
ejpam-709	443	11	references	reference	NOUN
ejpam-709	443	12	716	716	NUM
ejpam-709	443	13	[	[	SYM
ejpam-709	443	14	4	4	NUM
ejpam-709	443	15	]	]	X
ejpam-709	443	16	rao	rao	PROPN
ejpam-709	443	17	,	,	PUNCT
ejpam-709	443	18	g.c	g.c	PROPN
ejpam-709	443	19	.	.	PROPN
ejpam-709	443	20	:	:	PUNCT
ejpam-709	443	21	almost	almost	ADV
ejpam-709	443	22	distributive	distributive	ADJ
ejpam-709	443	23	lattices	lattice	NOUN
ejpam-709	443	24	.	.	PUNCT
ejpam-709	444	1	doctoral	doctoral	ADJ
ejpam-709	444	2	thesis	thesis	NOUN
ejpam-709	444	3	,	,	PUNCT
ejpam-709	444	4	dept	dept	NOUN
ejpam-709	444	5	.	.	PROPN
ejpam-709	444	6	of	of	ADP
ejpam-709	444	7	mathematics	mathematics	PROPN
ejpam-709	444	8	,	,	PUNCT
ejpam-709	444	9	andhra	andhra	PROPN
ejpam-709	444	10	university	university	PROPN
ejpam-709	444	11	,	,	PUNCT
ejpam-709	444	12	visakhapatnam	visakhapatnam	PROPN
ejpam-709	444	13	.	.	PUNCT
ejpam-709	444	14	1980	1980	NUM
ejpam-709	444	15	.	.	PUNCT
ejpam-709	445	1	[	[	X
ejpam-709	445	2	5	5	NUM
ejpam-709	445	3	]	]	SYM
ejpam-709	445	4	rao	rao	PROPN
ejpam-709	445	5	,	,	PUNCT
ejpam-709	445	6	g.c	g.c	PROPN
ejpam-709	445	7	.	.	PROPN
ejpam-709	445	8	rafi	rafi	PROPN
ejpam-709	445	9	,	,	PUNCT
ejpam-709	445	10	n.	n.	PROPN
ejpam-709	445	11	and	and	CCONJ
ejpam-709	445	12	ravi	ravi	PROPN
ejpam-709	445	13	kumar	kumar	PROPN
ejpam-709	445	14	bandaru	bandaru	PROPN
ejpam-709	445	15	.	.	PUNCT
ejpam-709	445	16	:	:	PUNCT
ejpam-709	445	17	s−ideals	s−ideal	NOUN
ejpam-709	445	18	in	in	ADP
ejpam-709	445	19	almost	almost	ADV
ejpam-709	445	20	distributive	distributive	ADJ
ejpam-709	445	21	lattices	lattice	NOUN
ejpam-709	445	22	.	.	PUNCT
ejpam-709	446	1	accepted	accept	VERB
ejpam-709	446	2	for	for	ADP
ejpam-709	446	3	publication	publication	NOUN
ejpam-709	446	4	in	in	ADP
ejpam-709	446	5	southeast	southeast	ADJ
ejpam-709	446	6	asian	asian	ADJ
ejpam-709	446	7	bulletin	bulletin	NOUN
ejpam-709	446	8	of	of	ADP
ejpam-709	446	9	mathematics	mathematic	NOUN
ejpam-709	446	10	.	.	PUNCT
ejpam-709	447	1	[	[	X
ejpam-709	447	2	6	6	NUM
ejpam-709	447	3	]	]	X
ejpam-709	447	4	rao	rao	PROPN
ejpam-709	447	5	,	,	PUNCT
ejpam-709	447	6	g.c	g.c	PROPN
ejpam-709	447	7	.	.	PROPN
ejpam-709	447	8	and	and	CCONJ
ejpam-709	447	9	ravi	ravi	PROPN
ejpam-709	447	10	kumar	kumar	PROPN
ejpam-709	447	11	,	,	PUNCT
ejpam-709	447	12	s.	s.	PROPN
ejpam-709	447	13	:	:	PUNCT
ejpam-709	447	14	minimal	minimal	ADJ
ejpam-709	447	15	prime	prime	ADJ
ejpam-709	447	16	ideals	ideal	NOUN
ejpam-709	447	17	in	in	ADP
ejpam-709	447	18	an	an	DET
ejpam-709	447	19	adl	adl	PROPN
ejpam-709	447	20	.	.	PUNCT
ejpam-709	447	21	int	int	PROPN
ejpam-709	447	22	.	.	PUNCT
ejpam-709	448	1	j.	j.	PROPN
ejpam-709	448	2	contemp	contemp	PROPN
ejpam-709	448	3	.	.	PUNCT
ejpam-709	449	1	sciences	science	NOUN
ejpam-709	449	2	,	,	PUNCT
ejpam-709	449	3	4	4	NUM
ejpam-709	449	4	,	,	PUNCT
ejpam-709	449	5	475	475	NUM
ejpam-709	449	6	-	-	SYM
ejpam-709	449	7	484	484	NUM
ejpam-709	449	8	.	.	PUNCT
ejpam-709	449	9	2009	2009	NUM
ejpam-709	449	10	.	.	PUNCT
ejpam-709	450	1	[	[	X
ejpam-709	450	2	7	7	NUM
ejpam-709	450	3	]	]	X
ejpam-709	450	4	rao	rao	PROPN
ejpam-709	450	5	,	,	PUNCT
ejpam-709	450	6	g.c	g.c	PROPN
ejpam-709	450	7	.	.	PROPN
ejpam-709	450	8	and	and	CCONJ
ejpam-709	450	9	ravi	ravi	PROPN
ejpam-709	450	10	kumar	kumar	PROPN
ejpam-709	450	11	,	,	PUNCT
ejpam-709	450	12	s.	s.	PROPN
ejpam-709	450	13	:	:	PUNCT
ejpam-709	450	14	normal	normal	ADJ
ejpam-709	450	15	almost	almost	ADV
ejpam-709	450	16	distributive	distributive	ADJ
ejpam-709	450	17	lattices	lattice	NOUN
ejpam-709	450	18	.	.	PUNCT
ejpam-709	451	1	southeast	southeast	ADJ
ejpam-709	451	2	asian	asian	ADJ
ejpam-709	451	3	bullettin	bullettin	NOUN
ejpam-709	451	4	of	of	ADP
ejpam-709	451	5	mathematics	mathematic	NOUN
ejpam-709	451	6	,	,	PUNCT
ejpam-709	451	7	32	32	NUM
ejpam-709	451	8	,	,	PUNCT
ejpam-709	451	9	831	831	NUM
ejpam-709	451	10	-	-	SYM
ejpam-709	451	11	841	841	NUM
ejpam-709	451	12	.	.	PUNCT
ejpam-709	451	13	2008	2008	NUM
ejpam-709	451	14	.	.	PUNCT
ejpam-709	452	1	[	[	X
ejpam-709	452	2	8	8	NUM
ejpam-709	452	3	]	]	X
ejpam-709	452	4	ravi	ravi	PROPN
ejpam-709	452	5	kumar	kumar	PROPN
ejpam-709	452	6	,	,	PUNCT
ejpam-709	452	7	s.	s.	PROPN
ejpam-709	452	8	:	:	PUNCT
ejpam-709	452	9	normal	normal	ADJ
ejpam-709	452	10	almost	almost	ADV
ejpam-709	452	11	distributive	distributive	ADJ
ejpam-709	452	12	lattices	lattice	NOUN
ejpam-709	452	13	.	.	PUNCT
ejpam-709	453	1	doctoral	doctoral	ADJ
ejpam-709	453	2	thesis	thesis	NOUN
ejpam-709	453	3	,	,	PUNCT
ejpam-709	453	4	dept	dept	NOUN
ejpam-709	453	5	.	.	PROPN
ejpam-709	453	6	of	of	ADP
ejpam-709	453	7	mathematics	mathematics	PROPN
ejpam-709	453	8	,	,	PUNCT
ejpam-709	453	9	andhra	andhra	PROPN
ejpam-709	453	10	university	university	PROPN
ejpam-709	453	11	,	,	PUNCT
ejpam-709	453	12	visakhapatnam	visakhapatnam	PROPN
ejpam-709	453	13	.	.	PUNCT
ejpam-709	453	14	2009	2009	NUM
ejpam-709	453	15	.	.	PUNCT
ejpam-709	454	1	[	[	X
ejpam-709	454	2	9	9	NUM
ejpam-709	454	3	]	]	SYM
ejpam-709	454	4	swamy	swamy	NOUN
ejpam-709	454	5	,	,	PUNCT
ejpam-709	454	6	u.m	u.m	PROPN
ejpam-709	454	7	.	.	PROPN
ejpam-709	454	8	and	and	CCONJ
ejpam-709	454	9	rao	rao	PROPN
ejpam-709	454	10	,	,	PUNCT
ejpam-709	454	11	g.c	g.c	PROPN
ejpam-709	454	12	.	.	PROPN
ejpam-709	454	13	:	:	PUNCT
ejpam-709	454	14	almost	almost	ADV
ejpam-709	454	15	distributive	distributive	ADJ
ejpam-709	454	16	lattices	lattice	NOUN
ejpam-709	454	17	.	.	PUNCT
ejpam-709	455	1	j.	j.	PROPN
ejpam-709	455	2	aust	aust	PROPN
ejpam-709	455	3	.	.	PUNCT
ejpam-709	456	1	math	math	PROPN
ejpam-709	456	2	.	.	PUNCT
ejpam-709	457	1	soc	soc	PROPN
ejpam-709	457	2	.	.	PUNCT
ejpam-709	458	1	(	(	PUNCT
ejpam-709	458	2	series	series	PROPN
ejpam-709	458	3	a	a	PROPN
ejpam-709	458	4	)	)	PUNCT
ejpam-709	458	5	,	,	PUNCT
ejpam-709	458	6	31	31	NUM
ejpam-709	458	7	,	,	PUNCT
ejpam-709	458	8	77	77	NUM
ejpam-709	458	9	-	-	SYM
ejpam-709	458	10	91	91	NUM
ejpam-709	458	11	.	.	PUNCT
ejpam-709	458	12	1981	1981	NUM
ejpam-709	458	13	.	.	PUNCT
ejpam-709	459	1	[	[	X
ejpam-709	459	2	10	10	NUM
ejpam-709	459	3	]	]	X
ejpam-709	459	4	swamy	swamy	NOUN
ejpam-709	459	5	,	,	PUNCT
ejpam-709	459	6	u.m	u.m	PROPN
ejpam-709	459	7	.	.	PROPN
ejpam-709	459	8	and	and	CCONJ
ejpam-709	459	9	ramesh	ramesh	PROPN
ejpam-709	459	10	,	,	PUNCT
ejpam-709	459	11	s.	s.	PROPN
ejpam-709	459	12	:	:	PUNCT
ejpam-709	459	13	birkhoff	birkhoff	PROPN
ejpam-709	459	14	centre	centre	PROPN
ejpam-709	459	15	of	of	ADP
ejpam-709	459	16	adl	adl	PROPN
ejpam-709	459	17	.	.	PUNCT
ejpam-709	459	18	int	int	PROPN
ejpam-709	459	19	.	.	PUNCT
ejpam-709	460	1	j.	j.	PROPN
ejpam-709	460	2	algebra	algebra	PROPN
ejpam-709	460	3	,	,	PUNCT
ejpam-709	460	4	3	3	NUM
ejpam-709	460	5	,	,	PUNCT
ejpam-709	460	6	539	539	NUM
ejpam-709	460	7	-	-	SYM
ejpam-709	460	8	546	546	NUM
ejpam-709	460	9	.	.	PUNCT
ejpam-709	460	10	2009	2009	NUM
ejpam-709	460	11	.	.	PUNCT