id	sid	tid	token	lemma	pos
ejpam-239	1	1	3_rao.dvi	3_rao.dvi	PROPN
ejpam-239	1	2	european	european	PROPN
ejpam-239	1	3	journal	journal	PROPN
ejpam-239	1	4	of	of	ADP
ejpam-239	1	5	pure	pure	ADJ
ejpam-239	1	6	and	and	CCONJ
ejpam-239	1	7	applied	apply	VERB
ejpam-239	1	8	mathematics	mathematic	NOUN
ejpam-239	1	9	vol	vol	NOUN
ejpam-239	1	10	.	.	PROPN
ejpam-239	2	1	2	2	NUM
ejpam-239	2	2	,	,	PUNCT
ejpam-239	2	3	no	no	INTJ
ejpam-239	2	4	.	.	NOUN
ejpam-239	2	5	1	1	NUM
ejpam-239	2	6	,	,	PUNCT
ejpam-239	2	7	2009	2009	NUM
ejpam-239	2	8	,	,	PUNCT
ejpam-239	2	9	(	(	PUNCT
ejpam-239	2	10	58	58	NUM
ejpam-239	2	11	-	-	SYM
ejpam-239	2	12	72	72	NUM
ejpam-239	2	13	)	)	PUNCT
ejpam-239	2	14	issn	issn	PROPN
ejpam-239	2	15	1307	1307	NUM
ejpam-239	2	16	-	-	SYM
ejpam-239	2	17	5543	5543	NUM
ejpam-239	2	18	–	–	PUNCT
ejpam-239	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-239	2	20	annulets	annulet	VERB
ejpam-239	2	21	in	in	ADP
ejpam-239	2	22	almost	almost	ADV
ejpam-239	2	23	distributive	distributive	ADJ
ejpam-239	2	24	lattices	lattice	NOUN
ejpam-239	2	25	g.	g.	PROPN
ejpam-239	2	26	c.	c.	PROPN
ejpam-239	2	27	rao1∗	rao1∗	PROPN
ejpam-239	2	28	and	and	CCONJ
ejpam-239	2	29	m.	m.	PROPN
ejpam-239	2	30	sambasiva	sambasiva	PROPN
ejpam-239	2	31	rao2	rao2	PROPN
ejpam-239	2	32	1	1	NUM
ejpam-239	2	33	department	department	NOUN
ejpam-239	2	34	of	of	ADP
ejpam-239	2	35	mathematics	mathematics	PROPN
ejpam-239	2	36	,	,	PUNCT
ejpam-239	2	37	andhra	andhra	PROPN
ejpam-239	2	38	university	university	PROPN
ejpam-239	2	39	visakhapatnam	visakhapatnam	PROPN
ejpam-239	2	40	,	,	PUNCT
ejpam-239	2	41	andhra	andhra	PROPN
ejpam-239	2	42	pradesh	pradesh	PROPN
ejpam-239	2	43	,	,	PUNCT
ejpam-239	2	44	india-530003	india-530003	ADJ
ejpam-239	2	45	2	2	NUM
ejpam-239	2	46	department	department	NOUN
ejpam-239	2	47	of	of	ADP
ejpam-239	2	48	mathematics	mathematic	NOUN
ejpam-239	2	49	,	,	PUNCT
ejpam-239	2	50	m.v.g.r.college	m.v.g.r.college	NOUN
ejpam-239	2	51	of	of	ADP
ejpam-239	2	52	engineering	engineering	NOUN
ejpam-239	2	53	chintalavalasa	chintalavalasa	PROPN
ejpam-239	2	54	,	,	PUNCT
ejpam-239	2	55	vizianagaram	vizianagaram	PROPN
ejpam-239	2	56	,	,	PUNCT
ejpam-239	2	57	andhra	andhra	PROPN
ejpam-239	2	58	pradesh	pradesh	PROPN
ejpam-239	2	59	,	,	PUNCT
ejpam-239	2	60	india-535003	india-535003	NOUN
ejpam-239	2	61	abstract	abstract	NOUN
ejpam-239	2	62	.	.	PUNCT
ejpam-239	3	1	we	we	PRON
ejpam-239	3	2	introduce	introduce	VERB
ejpam-239	3	3	the	the	DET
ejpam-239	3	4	concept	concept	NOUN
ejpam-239	3	5	of	of	ADP
ejpam-239	3	6	annulets	annulet	NOUN
ejpam-239	3	7	in	in	ADP
ejpam-239	3	8	an	an	DET
ejpam-239	3	9	almost	almost	ADV
ejpam-239	3	10	distributive	distributive	ADJ
ejpam-239	3	11	lattice(adl	lattice(adl	NOUN
ejpam-239	3	12	)	)	PUNCT
ejpam-239	3	13	r	r	NOUN
ejpam-239	3	14	with	with	ADP
ejpam-239	3	15	0	0	NUM
ejpam-239	3	16	.	.	PUNCT
ejpam-239	4	1	we	we	PRON
ejpam-239	4	2	characterize	characterize	VERB
ejpam-239	4	3	both	both	DET
ejpam-239	4	4	generalized	generalized	ADJ
ejpam-239	4	5	stone	stone	NOUN
ejpam-239	4	6	adl	adl	NOUN
ejpam-239	4	7	and	and	CCONJ
ejpam-239	4	8	normal	normal	ADJ
ejpam-239	4	9	adl	adl	NOUN
ejpam-239	4	10	in	in	ADP
ejpam-239	4	11	terms	term	NOUN
ejpam-239	4	12	of	of	ADP
ejpam-239	4	13	their	their	PRON
ejpam-239	4	14	annulets	annulet	NOUN
ejpam-239	4	15	.	.	PUNCT
ejpam-239	5	1	we	we	PRON
ejpam-239	5	2	characterize	characterize	VERB
ejpam-239	5	3	⋆-adls	⋆-adl	NOUN
ejpam-239	5	4	by	by	ADP
ejpam-239	5	5	means	mean	NOUN
ejpam-239	5	6	of	of	ADP
ejpam-239	5	7	their	their	PRON
ejpam-239	5	8	annulets	annulet	NOUN
ejpam-239	5	9	.	.	PUNCT
ejpam-239	6	1	it	it	PRON
ejpam-239	6	2	is	be	AUX
ejpam-239	6	3	proved	prove	VERB
ejpam-239	6	4	that	that	SCONJ
ejpam-239	6	5	the	the	DET
ejpam-239	6	6	lattice	lattice	PROPN
ejpam-239	6	7	a0(r	a0(r	PROPN
ejpam-239	6	8	)	)	PUNCT
ejpam-239	6	9	of	of	ADP
ejpam-239	6	10	all	all	DET
ejpam-239	6	11	annulets	annulet	NOUN
ejpam-239	6	12	of	of	ADP
ejpam-239	6	13	a	a	DET
ejpam-239	6	14	generalized	generalized	ADJ
ejpam-239	6	15	stone	stone	NOUN
ejpam-239	6	16	adl	adl	NOUN
ejpam-239	6	17	r	r	NOUN
ejpam-239	6	18	is	be	AUX
ejpam-239	6	19	a	a	DET
ejpam-239	6	20	relatively	relatively	ADV
ejpam-239	6	21	complemented	complemented	ADJ
ejpam-239	6	22	sublattice	sublattice	NOUN
ejpam-239	6	23	of	of	ADP
ejpam-239	6	24	the	the	DET
ejpam-239	6	25	lattice	lattice	NOUN
ejpam-239	7	1	i	i	PRON
ejpam-239	7	2	(	(	PUNCT
ejpam-239	7	3	r	r	NOUN
ejpam-239	7	4	)	)	PUNCT
ejpam-239	7	5	of	of	ADP
ejpam-239	7	6	all	all	DET
ejpam-239	7	7	ideals	ideal	NOUN
ejpam-239	7	8	of	of	ADP
ejpam-239	7	9	r.	r.	PROPN
ejpam-239	7	10	finally	finally	ADV
ejpam-239	7	11	,	,	PUNCT
ejpam-239	7	12	it	it	PRON
ejpam-239	7	13	is	be	AUX
ejpam-239	7	14	proved	prove	VERB
ejpam-239	7	15	thata0(r	thata0(r	ADV
ejpam-239	7	16	)	)	PUNCT
ejpam-239	7	17	is	be	AUX
ejpam-239	7	18	relatively	relatively	ADV
ejpam-239	7	19	complemented	complement	VERB
ejpam-239	7	20	iff	iff	PROPN
ejpam-239	7	21	r	r	NOUN
ejpam-239	7	22	is	be	AUX
ejpam-239	7	23	sectionally	sectionally	ADV
ejpam-239	7	24	⋆-adl	⋆-adl	PROPN
ejpam-239	7	25	.	.	PUNCT
ejpam-239	8	1	ams	am	NOUN
ejpam-239	8	2	subject	subject	ADJ
ejpam-239	8	3	classifications	classification	NOUN
ejpam-239	8	4	:	:	PUNCT
ejpam-239	8	5	06d99	06d99	NUM
ejpam-239	8	6	,	,	PUNCT
ejpam-239	8	7	06d15	06d15	X
ejpam-239	8	8	.	.	PUNCT
ejpam-239	9	1	key	key	ADJ
ejpam-239	9	2	words	word	NOUN
ejpam-239	9	3	:	:	PUNCT
ejpam-239	9	4	almost	almost	ADV
ejpam-239	9	5	distributive	distributive	ADJ
ejpam-239	9	6	lattice(adl	lattice(adl	NOUN
ejpam-239	9	7	)	)	PUNCT
ejpam-239	9	8	,	,	PUNCT
ejpam-239	9	9	boolean	boolean	ADJ
ejpam-239	9	10	algebra	algebra	NOUN
ejpam-239	9	11	,	,	PUNCT
ejpam-239	9	12	dense	dense	ADJ
ejpam-239	9	13	elements	element	NOUN
ejpam-239	9	14	,	,	PUNCT
ejpam-239	9	15	maximal	maximal	ADJ
ejpam-239	9	16	element	element	NOUN
ejpam-239	9	17	,	,	PUNCT
ejpam-239	9	18	annihilator	annihilator	PROPN
ejpam-239	9	19	ideal	ideal	NOUN
ejpam-239	9	20	,	,	PUNCT
ejpam-239	9	21	annulet	annulet	NOUN
ejpam-239	9	22	,	,	PUNCT
ejpam-239	9	23	normal	normal	ADJ
ejpam-239	9	24	adl	adl	PROPN
ejpam-239	9	25	,	,	PUNCT
ejpam-239	9	26	⋆-adl	⋆-adl	PROPN
ejpam-239	9	27	,	,	PUNCT
ejpam-239	9	28	generalized	generalized	ADJ
ejpam-239	9	29	stone	stone	NOUN
ejpam-239	9	30	adl	adl	PROPN
ejpam-239	9	31	,	,	PUNCT
ejpam-239	9	32	disjunctive	disjunctive	ADJ
ejpam-239	9	33	adl	adl	PROPN
ejpam-239	9	34	.	.	PROPN
ejpam-239	9	35	1	1	NUM
ejpam-239	9	36	.	.	X
ejpam-239	9	37	introduction	introduction	NOUN
ejpam-239	9	38	the	the	DET
ejpam-239	9	39	concept	concept	NOUN
ejpam-239	9	40	of	of	ADP
ejpam-239	9	41	an	an	DET
ejpam-239	9	42	almost	almost	ADV
ejpam-239	9	43	distributive	distributive	ADJ
ejpam-239	9	44	lattice(adl	lattice(adl	NOUN
ejpam-239	9	45	)	)	PUNCT
ejpam-239	9	46	was	be	AUX
ejpam-239	9	47	introduced	introduce	VERB
ejpam-239	9	48	by	by	ADP
ejpam-239	9	49	swamy	swamy	NOUN
ejpam-239	9	50	.	.	PUNCT
ejpam-239	10	1	u.m	u.m	PROPN
ejpam-239	10	2	.	.	PROPN
ejpam-239	11	1	and	and	CCONJ
ejpam-239	11	2	rao.g.c	rao.g.c	NOUN
ejpam-239	12	1	[	[	X
ejpam-239	12	2	8	8	NUM
ejpam-239	12	3	]	]	PUNCT
ejpam-239	12	4	as	as	ADP
ejpam-239	12	5	a	a	DET
ejpam-239	12	6	common	common	ADJ
ejpam-239	12	7	abstraction	abstraction	NOUN
ejpam-239	12	8	to	to	ADP
ejpam-239	12	9	most	most	ADJ
ejpam-239	12	10	of	of	ADP
ejpam-239	12	11	the	the	DET
ejpam-239	12	12	existing	exist	VERB
ejpam-239	12	13	ring	ring	NOUN
ejpam-239	12	14	theoretic	theoretic	NOUN
ejpam-239	12	15	and	and	CCONJ
ejpam-239	12	16	lattice	lattice	ADJ
ejpam-239	12	17	theoretic	theoretic	ADJ
ejpam-239	12	18	generalizations	generalization	NOUN
ejpam-239	12	19	of	of	ADP
ejpam-239	12	20	a	a	DET
ejpam-239	12	21	boolean	boolean	ADJ
ejpam-239	12	22	algebra	algebra	NOUN
ejpam-239	12	23	.	.	PUNCT
ejpam-239	13	1	later	later	ADV
ejpam-239	13	2	a	a	DET
ejpam-239	13	3	more	more	ADV
ejpam-239	13	4	general	general	ADJ
ejpam-239	13	5	class	class	NOUN
ejpam-239	13	6	called	call	VERB
ejpam-239	13	7	⋆-adls	⋆-adl	NOUN
ejpam-239	13	8	was	be	AUX
ejpam-239	13	9	introduced	introduce	VERB
ejpam-239	13	10	in	in	ADP
ejpam-239	13	11	the	the	DET
ejpam-239	13	12	paper	paper	NOUN
ejpam-239	13	13	[	[	X
ejpam-239	13	14	10	10	NUM
ejpam-239	13	15	]	]	PUNCT
ejpam-239	13	16	.	.	PUNCT
ejpam-239	14	1	the	the	DET
ejpam-239	14	2	characterization	characterization	NOUN
ejpam-239	14	3	of	of	ADP
ejpam-239	14	4	⋆-adl	⋆-adl	PROPN
ejpam-239	14	5	by	by	ADP
ejpam-239	14	6	means	mean	NOUN
ejpam-239	14	7	of	of	ADP
ejpam-239	14	8	it	it	PRON
ejpam-239	14	9	’s	’	VERB
ejpam-239	14	10	dense	dense	ADJ
ejpam-239	14	11	elements	element	NOUN
ejpam-239	14	12	was	be	AUX
ejpam-239	14	13	studied	study	VERB
ejpam-239	14	14	in	in	ADP
ejpam-239	14	15	[	[	X
ejpam-239	14	16	11	11	NUM
ejpam-239	14	17	]	]	PUNCT
ejpam-239	14	18	.	.	PUNCT
ejpam-239	15	1	in	in	ADP
ejpam-239	15	2	[	[	X
ejpam-239	15	3	5	5	NUM
ejpam-239	15	4	]	]	PUNCT
ejpam-239	15	5	,	,	PUNCT
ejpam-239	15	6	mandelker	mandelker	NOUN
ejpam-239	15	7	studied	study	VERB
ejpam-239	15	8	the	the	DET
ejpam-239	15	9	properties	property	NOUN
ejpam-239	15	10	of	of	ADP
ejpam-239	15	11	relative	relative	ADJ
ejpam-239	15	12	annihilators	annihilator	NOUN
ejpam-239	15	13	and	and	CCONJ
ejpam-239	15	14	∗corresponding	∗corresponde	VERB
ejpam-239	15	15	author	author	NOUN
ejpam-239	15	16	.	.	PUNCT
ejpam-239	16	1	email	email	NOUN
ejpam-239	16	2	addresses	address	NOUN
ejpam-239	16	3	:	:	PUNCT
ejpam-239	16	4	g	g	PROPN
ejpam-239	16	5	raomaths	raomaths	PROPN
ejpam-239	16	6	�	�	PROPN
ejpam-239	16	7	yahoo	yahoo	PROPN
ejpam-239	16	8	.	.	PUNCT
ejpam-239	17	1	o.in	o.in	PROPN
ejpam-239	17	2	(	(	PUNCT
ejpam-239	17	3	g.	g.	PROPN
ejpam-239	17	4	rao),mssraomaths35	rao),mssraomaths35	PROPN
ejpam-239	17	5	�	�	PROPN
ejpam-239	17	6	rediffmail	rediffmail	NOUN
ejpam-239	17	7	.	.	PUNCT
ejpam-239	18	1	om	om	PROPN
ejpam-239	18	2	(	(	PUNCT
ejpam-239	18	3	m.	m.	NOUN
ejpam-239	18	4	rao	rao	PROPN
ejpam-239	18	5	)	)	PUNCT
ejpam-239	18	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-239	19	1	58	58	NUM
ejpam-239	19	2	c	c	X
ejpam-239	19	3	©	©	PROPN
ejpam-239	19	4	2009	2009	NUM
ejpam-239	19	5	ejpam	ejpam	NOUN
ejpam-239	19	6	all	all	DET
ejpam-239	19	7	rights	right	NOUN
ejpam-239	19	8	reserved	reserve	VERB
ejpam-239	19	9	.	.	PUNCT
ejpam-239	20	1	g.	g.	PROPN
ejpam-239	20	2	c.	c.	PROPN
ejpam-239	20	3	rao	rao	PROPN
ejpam-239	20	4	and	and	CCONJ
ejpam-239	20	5	m.	m.	PROPN
ejpam-239	20	6	sambasiva	sambasiva	PROPN
ejpam-239	20	7	rao	rao	PROPN
ejpam-239	20	8	/	/	SYM
ejpam-239	20	9	eur	eur	PROPN
ejpam-239	20	10	.	.	PUNCT
ejpam-239	21	1	j.	j.	PROPN
ejpam-239	21	2	pure	pure	PROPN
ejpam-239	21	3	appl	appl	PROPN
ejpam-239	21	4	.	.	PROPN
ejpam-239	21	5	math	math	PROPN
ejpam-239	21	6	,	,	PUNCT
ejpam-239	21	7	2	2	NUM
ejpam-239	21	8	(	(	PUNCT
ejpam-239	21	9	2009	2009	NUM
ejpam-239	21	10	)	)	PUNCT
ejpam-239	21	11	,	,	PUNCT
ejpam-239	21	12	(	(	PUNCT
ejpam-239	21	13	58	58	NUM
ejpam-239	21	14	-	-	SYM
ejpam-239	21	15	72	72	NUM
ejpam-239	21	16	)	)	PUNCT
ejpam-239	21	17	59	59	NUM
ejpam-239	21	18	characterized	characterize	VERB
ejpam-239	21	19	the	the	DET
ejpam-239	21	20	distributive	distributive	ADJ
ejpam-239	21	21	lattice	lattice	NOUN
ejpam-239	21	22	in	in	ADP
ejpam-239	21	23	terms	term	NOUN
ejpam-239	21	24	of	of	ADP
ejpam-239	21	25	relative	relative	ADJ
ejpam-239	21	26	annihilators	annihilator	NOUN
ejpam-239	21	27	.	.	PUNCT
ejpam-239	22	1	in	in	ADP
ejpam-239	22	2	this	this	DET
ejpam-239	22	3	paper	paper	NOUN
ejpam-239	22	4	the	the	DET
ejpam-239	22	5	concept	concept	NOUN
ejpam-239	22	6	of	of	ADP
ejpam-239	22	7	annulet	annulet	NOUN
ejpam-239	22	8	as	as	ADP
ejpam-239	22	9	an	an	DET
ejpam-239	22	10	ideal	ideal	NOUN
ejpam-239	22	11	of	of	ADP
ejpam-239	22	12	the	the	DET
ejpam-239	22	13	form	form	NOUN
ejpam-239	22	14	(	(	PUNCT
ejpam-239	22	15	x]∗	x]∗	PROPN
ejpam-239	22	16	=	=	PRON
ejpam-239	22	17	{	{	PUNCT
ejpam-239	22	18	a	a	DET
ejpam-239	22	19	∈	∈	NOUN
ejpam-239	22	20	r	r	NOUN
ejpam-239	22	21	|	|	NOUN
ejpam-239	22	22	x	x	PUNCT
ejpam-239	22	23	∧	∧	PROPN
ejpam-239	22	24	a	a	PRON
ejpam-239	22	25	=	=	NOUN
ejpam-239	22	26	0	0	NUM
ejpam-239	22	27	}	}	PUNCT
ejpam-239	22	28	in	in	ADP
ejpam-239	22	29	an	an	DET
ejpam-239	22	30	adl	adl	NOUN
ejpam-239	22	31	r	r	NOUN
ejpam-239	22	32	with	with	ADP
ejpam-239	22	33	0	0	NUM
ejpam-239	22	34	is	be	AUX
ejpam-239	22	35	introduced	introduce	VERB
ejpam-239	22	36	,	,	PUNCT
ejpam-239	22	37	analogous	analogous	ADJ
ejpam-239	22	38	to	to	ADP
ejpam-239	22	39	that	that	PRON
ejpam-239	22	40	in	in	ADP
ejpam-239	22	41	a	a	DET
ejpam-239	22	42	distributive	distributive	ADJ
ejpam-239	22	43	lattice[4	lattice[4	NOUN
ejpam-239	22	44	]	]	PUNCT
ejpam-239	22	45	.	.	PUNCT
ejpam-239	23	1	it	it	PRON
ejpam-239	23	2	is	be	AUX
ejpam-239	23	3	proved	prove	VERB
ejpam-239	23	4	that	that	SCONJ
ejpam-239	23	5	the	the	DET
ejpam-239	23	6	set	set	PROPN
ejpam-239	23	7	a0(r	a0(r	NOUN
ejpam-239	23	8	)	)	PUNCT
ejpam-239	23	9	of	of	ADP
ejpam-239	23	10	all	all	DET
ejpam-239	23	11	annulets	annulet	NOUN
ejpam-239	23	12	of	of	ADP
ejpam-239	23	13	an	an	DET
ejpam-239	23	14	adl	adl	NOUN
ejpam-239	23	15	r	r	NOUN
ejpam-239	23	16	with	with	ADP
ejpam-239	23	17	0	0	NUM
ejpam-239	23	18	can	can	AUX
ejpam-239	23	19	be	be	AUX
ejpam-239	23	20	made	make	VERB
ejpam-239	23	21	into	into	ADP
ejpam-239	23	22	a	a	DET
ejpam-239	23	23	distributive	distributive	ADJ
ejpam-239	23	24	lattice	lattice	NOUN
ejpam-239	23	25	and	and	CCONJ
ejpam-239	23	26	sublattice	sublattice	NOUN
ejpam-239	23	27	of	of	ADP
ejpam-239	23	28	the	the	DET
ejpam-239	23	29	boolean	boolean	ADJ
ejpam-239	23	30	algebraa	algebraa	NOUN
ejpam-239	23	31	(	(	PUNCT
ejpam-239	23	32	r	r	NOUN
ejpam-239	23	33	)	)	PUNCT
ejpam-239	23	34	of	of	ADP
ejpam-239	23	35	all	all	DET
ejpam-239	23	36	annihilator	annihilator	PROPN
ejpam-239	23	37	ideals	ideal	NOUN
ejpam-239	23	38	of	of	ADP
ejpam-239	23	39	r.	r.	NOUN
ejpam-239	23	40	we	we	PRON
ejpam-239	23	41	characterize	characterize	VERB
ejpam-239	23	42	the	the	DET
ejpam-239	23	43	generalized	generalized	ADJ
ejpam-239	23	44	stone	stone	NOUN
ejpam-239	23	45	adl	adl	NOUN
ejpam-239	23	46	and	and	CCONJ
ejpam-239	23	47	normal	normal	ADJ
ejpam-239	23	48	adl	adl	NOUN
ejpam-239	23	49	in	in	ADP
ejpam-239	23	50	terms	term	NOUN
ejpam-239	23	51	of	of	ADP
ejpam-239	23	52	their	their	PRON
ejpam-239	23	53	annulets	annulet	NOUN
ejpam-239	23	54	.	.	PUNCT
ejpam-239	24	1	we	we	PRON
ejpam-239	24	2	introduce	introduce	VERB
ejpam-239	24	3	a	a	DET
ejpam-239	24	4	more	more	ADV
ejpam-239	24	5	general	general	ADJ
ejpam-239	24	6	class	class	NOUN
ejpam-239	24	7	of	of	ADP
ejpam-239	24	8	adls	adls	PROPN
ejpam-239	24	9	called	call	VERB
ejpam-239	24	10	disjunctive	disjunctive	ADJ
ejpam-239	24	11	adls	adls	PROPN
ejpam-239	24	12	with	with	ADP
ejpam-239	24	13	suitable	suitable	ADJ
ejpam-239	24	14	examples	example	NOUN
ejpam-239	24	15	and	and	CCONJ
ejpam-239	24	16	prove	prove	VERB
ejpam-239	24	17	that	that	SCONJ
ejpam-239	24	18	a	a	DET
ejpam-239	24	19	disjunctive	disjunctive	ADJ
ejpam-239	24	20	normal	normal	ADJ
ejpam-239	24	21	adl	adl	PROPN
ejpam-239	24	22	is	be	AUX
ejpam-239	24	23	dually	dually	ADV
ejpam-239	24	24	isomorphic	isomorphic	ADJ
ejpam-239	24	25	to	to	ADP
ejpam-239	24	26	the	the	DET
ejpam-239	24	27	latticea0(r	latticea0(r	NOUN
ejpam-239	24	28	)	)	PUNCT
ejpam-239	24	29	.	.	PUNCT
ejpam-239	25	1	we	we	PRON
ejpam-239	25	2	characterize	characterize	VERB
ejpam-239	25	3	⋆-adls	⋆-adl	NOUN
ejpam-239	25	4	by	by	ADP
ejpam-239	25	5	means	mean	NOUN
ejpam-239	25	6	of	of	ADP
ejpam-239	25	7	their	their	PRON
ejpam-239	25	8	annulets	annulet	NOUN
ejpam-239	25	9	.	.	PUNCT
ejpam-239	26	1	if	if	SCONJ
ejpam-239	26	2	r	r	NOUN
ejpam-239	26	3	is	be	AUX
ejpam-239	26	4	a	a	DET
ejpam-239	26	5	generalized	generalized	ADJ
ejpam-239	26	6	stone	stone	NOUN
ejpam-239	26	7	adl	adl	NOUN
ejpam-239	26	8	,	,	PUNCT
ejpam-239	26	9	then	then	ADV
ejpam-239	26	10	it	it	PRON
ejpam-239	26	11	is	be	AUX
ejpam-239	26	12	proved	prove	VERB
ejpam-239	26	13	that	that	SCONJ
ejpam-239	26	14	the	the	DET
ejpam-239	26	15	lattice	lattice	PROPN
ejpam-239	26	16	a0(r	a0(r	PROPN
ejpam-239	26	17	)	)	PUNCT
ejpam-239	26	18	is	be	AUX
ejpam-239	26	19	a	a	DET
ejpam-239	26	20	relatively	relatively	ADV
ejpam-239	26	21	complemented	complemented	ADJ
ejpam-239	26	22	sublattice	sublattice	NOUN
ejpam-239	26	23	of	of	ADP
ejpam-239	26	24	the	the	DET
ejpam-239	26	25	lattice	lattice	NOUN
ejpam-239	27	1	i	i	PRON
ejpam-239	27	2	(	(	PUNCT
ejpam-239	27	3	r	r	NOUN
ejpam-239	27	4	)	)	PUNCT
ejpam-239	27	5	of	of	ADP
ejpam-239	27	6	all	all	DET
ejpam-239	27	7	ideals	ideal	NOUN
ejpam-239	27	8	of	of	ADP
ejpam-239	27	9	r.	r.	PROPN
ejpam-239	27	10	finally	finally	ADV
ejpam-239	27	11	,	,	PUNCT
ejpam-239	27	12	it	it	PRON
ejpam-239	27	13	is	be	AUX
ejpam-239	27	14	proved	prove	VERB
ejpam-239	27	15	thata0(r	thata0(r	ADV
ejpam-239	27	16	)	)	PUNCT
ejpam-239	27	17	is	be	AUX
ejpam-239	27	18	relatively	relatively	ADV
ejpam-239	27	19	complemented	complement	VERB
ejpam-239	27	20	iff	iff	PROPN
ejpam-239	27	21	r	r	NOUN
ejpam-239	27	22	is	be	AUX
ejpam-239	27	23	sectionally	sectionally	ADV
ejpam-239	27	24	⋆-adl	⋆-adl	NUM
ejpam-239	27	25	.	.	PUNCT
ejpam-239	28	1	2	2	X
ejpam-239	28	2	.	.	NOUN
ejpam-239	28	3	preliminaries	preliminary	NOUN
ejpam-239	28	4	an	an	DET
ejpam-239	28	5	almost	almost	ADV
ejpam-239	28	6	distributive	distributive	ADJ
ejpam-239	28	7	lattice	lattice	NOUN
ejpam-239	28	8	(	(	PUNCT
ejpam-239	28	9	adl	adl	PROPN
ejpam-239	28	10	)	)	PUNCT
ejpam-239	28	11	is	be	AUX
ejpam-239	28	12	an	an	DET
ejpam-239	28	13	algebra	algebra	NOUN
ejpam-239	28	14	(	(	PUNCT
ejpam-239	28	15	r,∨,∧	r,∨,∧	NUM
ejpam-239	28	16	)	)	PUNCT
ejpam-239	28	17	of	of	ADP
ejpam-239	28	18	type	type	NOUN
ejpam-239	28	19	(	(	PUNCT
ejpam-239	28	20	2,2	2,2	NOUN
ejpam-239	28	21	)	)	PUNCT
ejpam-239	28	22	satisfying	satisfy	VERB
ejpam-239	28	23	1	1	NUM
ejpam-239	28	24	.	.	PUNCT
ejpam-239	29	1	(	(	PUNCT
ejpam-239	29	2	x	x	PROPN
ejpam-239	29	3	∨	∨	NUM
ejpam-239	29	4	y)∧	y)∧	PROPN
ejpam-239	29	5	z	z	NOUN
ejpam-239	29	6	=	=	SYM
ejpam-239	29	7	(	(	PUNCT
ejpam-239	29	8	x	x	PART
ejpam-239	29	9	∧	∧	PROPN
ejpam-239	29	10	z)∨	z)∨	PROPN
ejpam-239	29	11	(	(	PUNCT
ejpam-239	29	12	y	y	PROPN
ejpam-239	29	13	∧	∧	PROPN
ejpam-239	29	14	z	z	PROPN
ejpam-239	29	15	)	)	PUNCT
ejpam-239	29	16	2	2	NUM
ejpam-239	29	17	.	.	X
ejpam-239	30	1	x	x	X
ejpam-239	30	2	∧	∧	PROPN
ejpam-239	30	3	(	(	PUNCT
ejpam-239	30	4	y	y	PROPN
ejpam-239	30	5	∨	∨	PROPN
ejpam-239	30	6	z	z	PROPN
ejpam-239	30	7	)	)	PUNCT
ejpam-239	30	8	=	=	SYM
ejpam-239	30	9	(	(	PUNCT
ejpam-239	30	10	x	x	PUNCT
ejpam-239	30	11	∧	∧	PROPN
ejpam-239	30	12	y)∨	y)∨	PROPN
ejpam-239	30	13	(	(	PUNCT
ejpam-239	30	14	x	x	PART
ejpam-239	30	15	∧	∧	PROPN
ejpam-239	30	16	z	z	PROPN
ejpam-239	30	17	)	)	PUNCT
ejpam-239	30	18	3	3	NUM
ejpam-239	30	19	.	.	PUNCT
ejpam-239	30	20	(	(	PUNCT
ejpam-239	31	1	x	x	PROPN
ejpam-239	31	2	∨	∨	NUM
ejpam-239	31	3	y)∧	y)∧	NOUN
ejpam-239	31	4	y	y	PROPN
ejpam-239	31	5	=	=	SYM
ejpam-239	31	6	y	y	PROPN
ejpam-239	31	7	4	4	NUM
ejpam-239	31	8	.	.	PUNCT
ejpam-239	32	1	(	(	PUNCT
ejpam-239	32	2	x	x	PROPN
ejpam-239	32	3	∨	∨	NUM
ejpam-239	32	4	y)∧	y)∧	NOUN
ejpam-239	32	5	x	x	SYM
ejpam-239	32	6	=	=	PUNCT
ejpam-239	32	7	x	x	SYM
ejpam-239	32	8	5	5	X
ejpam-239	32	9	.	.	NUM
ejpam-239	32	10	x	x	PUNCT
ejpam-239	32	11	∨	∨	NOUN
ejpam-239	32	12	(	(	PUNCT
ejpam-239	32	13	x	x	PROPN
ejpam-239	32	14	∧	∧	PROPN
ejpam-239	32	15	y	y	NOUN
ejpam-239	32	16	)	)	PUNCT
ejpam-239	32	17	=	=	PUNCT
ejpam-239	33	1	x	x	X
ejpam-239	33	2	.	.	PUNCT
ejpam-239	34	1	for	for	ADP
ejpam-239	34	2	any	any	DET
ejpam-239	34	3	x	x	SYM
ejpam-239	34	4	,	,	PUNCT
ejpam-239	34	5	y	y	PROPN
ejpam-239	34	6	,	,	PUNCT
ejpam-239	34	7	z	z	PROPN
ejpam-239	34	8	∈	∈	PROPN
ejpam-239	34	9	r.	r.	NOUN
ejpam-239	34	10	if	if	SCONJ
ejpam-239	34	11	r	r	NOUN
ejpam-239	34	12	has	have	VERB
ejpam-239	34	13	an	an	DET
ejpam-239	34	14	element	element	NOUN
ejpam-239	34	15	0	0	PUNCT
ejpam-239	34	16	and	and	CCONJ
ejpam-239	34	17	satisfies	satisfy	VERB
ejpam-239	34	18	0∧	0∧	NOUN
ejpam-239	34	19	x	x	NOUN
ejpam-239	34	20	=	=	SYM
ejpam-239	34	21	0	0	NUM
ejpam-239	34	22	and	and	CCONJ
ejpam-239	34	23	x∨0=	x∨0=	NOUN
ejpam-239	34	24	x	x	PUNCT
ejpam-239	34	25	along	along	ADP
ejpam-239	34	26	with	with	ADP
ejpam-239	34	27	the	the	DET
ejpam-239	34	28	above	above	ADJ
ejpam-239	34	29	properties	property	NOUN
ejpam-239	34	30	,	,	PUNCT
ejpam-239	34	31	then	then	ADV
ejpam-239	34	32	r	r	NOUN
ejpam-239	34	33	is	be	AUX
ejpam-239	34	34	called	call	VERB
ejpam-239	34	35	an	an	DET
ejpam-239	34	36	adl	adl	NOUN
ejpam-239	34	37	with	with	ADP
ejpam-239	34	38	0	0	NUM
ejpam-239	34	39	.	.	PUNCT
ejpam-239	35	1	every	every	DET
ejpam-239	35	2	non	non	ADJ
ejpam-239	35	3	-	-	ADJ
ejpam-239	35	4	empty	empty	ADJ
ejpam-239	35	5	set	set	NOUN
ejpam-239	35	6	x	x	PUNCT
ejpam-239	35	7	can	can	AUX
ejpam-239	35	8	be	be	AUX
ejpam-239	35	9	regarded	regard	VERB
ejpam-239	35	10	as	as	ADP
ejpam-239	35	11	an	an	DET
ejpam-239	35	12	adl	adl	NOUN
ejpam-239	35	13	as	as	SCONJ
ejpam-239	35	14	follows	follow	VERB
ejpam-239	35	15	.	.	PUNCT
ejpam-239	36	1	let	let	VERB
ejpam-239	36	2	x0	x0	PROPN
ejpam-239	36	3	∈	∈	PROPN
ejpam-239	37	1	x	x	X
ejpam-239	37	2	.	.	PUNCT
ejpam-239	38	1	define	define	VERB
ejpam-239	38	2	two	two	NUM
ejpam-239	38	3	g.	g.	PROPN
ejpam-239	38	4	c.	c.	PROPN
ejpam-239	38	5	rao	rao	PROPN
ejpam-239	38	6	and	and	CCONJ
ejpam-239	38	7	m.	m.	PROPN
ejpam-239	38	8	sambasiva	sambasiva	PROPN
ejpam-239	38	9	rao	rao	PROPN
ejpam-239	38	10	/	/	SYM
ejpam-239	38	11	eur	eur	PROPN
ejpam-239	38	12	.	.	PUNCT
ejpam-239	39	1	j.	j.	PROPN
ejpam-239	39	2	pure	pure	PROPN
ejpam-239	39	3	appl	appl	PROPN
ejpam-239	39	4	.	.	PROPN
ejpam-239	39	5	math	math	PROPN
ejpam-239	39	6	,	,	PUNCT
ejpam-239	39	7	2	2	NUM
ejpam-239	39	8	(	(	PUNCT
ejpam-239	39	9	2009	2009	NUM
ejpam-239	39	10	)	)	PUNCT
ejpam-239	39	11	,	,	PUNCT
ejpam-239	39	12	(	(	PUNCT
ejpam-239	39	13	58	58	NUM
ejpam-239	39	14	-	-	SYM
ejpam-239	39	15	72	72	NUM
ejpam-239	39	16	)	)	PUNCT
ejpam-239	39	17	60	60	NUM
ejpam-239	39	18	binary	binary	NOUN
ejpam-239	39	19	operations	operation	NOUN
ejpam-239	39	20	∨,∧	∨,∧	NOUN
ejpam-239	39	21	on	on	ADP
ejpam-239	39	22	x	x	PUNCT
ejpam-239	39	23	by	by	ADP
ejpam-239	39	24	x	x	PROPN
ejpam-239	39	25	∨	∨	NUM
ejpam-239	39	26	y	y	NOUN
ejpam-239	39	27	=	=	PUNCT
ejpam-239	39	28			PROPN
ejpam-239	39	29			PRON
ejpam-239	39	30			NOUN
ejpam-239	39	31	x	x	X
ejpam-239	39	32	if	if	SCONJ
ejpam-239	39	33	x	x	PROPN
ejpam-239	39	34	6=	6=	NUM
ejpam-239	39	35	x0	x0	PROPN
ejpam-239	39	36	y	y	PROPN
ejpam-239	40	1	if	if	SCONJ
ejpam-239	40	2	x	x	PRON
ejpam-239	40	3	=	=	SYM
ejpam-239	41	1	x0	x0	PROPN
ejpam-239	41	2	x	x	PUNCT
ejpam-239	41	3	∧	∧	NOUN
ejpam-239	41	4	y	y	NOUN
ejpam-239	41	5	=	=	PUNCT
ejpam-239	41	6			PROPN
ejpam-239	41	7			VERB
ejpam-239	41	8			NOUN
ejpam-239	41	9	y	y	PROPN
ejpam-239	41	10	if	if	SCONJ
ejpam-239	41	11	x	x	PROPN
ejpam-239	41	12	6=	6=	NUM
ejpam-239	41	13	x0	x0	PROPN
ejpam-239	41	14	x0	x0	PROPN
ejpam-239	41	15	if	if	SCONJ
ejpam-239	41	16	x	x	PRON
ejpam-239	41	17	=	=	SYM
ejpam-239	41	18	x0	x0	PROPN
ejpam-239	41	19	then	then	ADV
ejpam-239	41	20	(	(	PUNCT
ejpam-239	41	21	x	x	X
ejpam-239	41	22	,	,	PUNCT
ejpam-239	41	23	∨,∧	∨,∧	PROPN
ejpam-239	41	24	,	,	PUNCT
ejpam-239	41	25	x0	x0	PROPN
ejpam-239	41	26	)	)	PUNCT
ejpam-239	41	27	is	be	AUX
ejpam-239	41	28	an	an	DET
ejpam-239	41	29	adl	adl	NOUN
ejpam-239	41	30	with	with	ADP
ejpam-239	41	31	x0	x0	PROPN
ejpam-239	41	32	as	as	ADP
ejpam-239	41	33	zero	zero	NUM
ejpam-239	41	34	element	element	NOUN
ejpam-239	41	35	and	and	CCONJ
ejpam-239	41	36	is	be	AUX
ejpam-239	41	37	called	call	VERB
ejpam-239	41	38	a	a	DET
ejpam-239	41	39	discrete	discrete	ADJ
ejpam-239	41	40	adl	adl	NOUN
ejpam-239	41	41	.	.	PUNCT
ejpam-239	42	1	if	if	SCONJ
ejpam-239	42	2	(	(	PUNCT
ejpam-239	42	3	r,∨,∧	r,∨,∧	NUM
ejpam-239	42	4	,	,	PUNCT
ejpam-239	42	5	0	0	NUM
ejpam-239	42	6	)	)	PUNCT
ejpam-239	42	7	is	be	AUX
ejpam-239	42	8	an	an	DET
ejpam-239	42	9	adl	adl	NOUN
ejpam-239	42	10	,	,	PUNCT
ejpam-239	42	11	for	for	ADP
ejpam-239	42	12	any	any	DET
ejpam-239	42	13	a	a	NOUN
ejpam-239	42	14	,	,	PUNCT
ejpam-239	42	15	b	b	X
ejpam-239	42	16	∈	∈	PROPN
ejpam-239	42	17	r	r	NOUN
ejpam-239	42	18	,	,	PUNCT
ejpam-239	42	19	define	define	VERB
ejpam-239	42	20	a	a	DET
ejpam-239	42	21	≤	≤	NUM
ejpam-239	42	22	b	b	NOUN
ejpam-239	43	1	if	if	SCONJ
ejpam-239	44	1	and	and	CCONJ
ejpam-239	44	2	only	only	ADV
ejpam-239	44	3	if	if	SCONJ
ejpam-239	44	4	a	a	PRON
ejpam-239	44	5	=	=	X
ejpam-239	44	6	a	a	DET
ejpam-239	44	7	∧	∧	PROPN
ejpam-239	44	8	b	b	PROPN
ejpam-239	44	9	(	(	PUNCT
ejpam-239	44	10	or	or	CCONJ
ejpam-239	44	11	equivalently	equivalently	ADV
ejpam-239	44	12	,	,	PUNCT
ejpam-239	44	13	a	a	DET
ejpam-239	44	14	∨	∨	NUM
ejpam-239	44	15	b	b	NOUN
ejpam-239	44	16	=	=	SYM
ejpam-239	44	17	b	b	PROPN
ejpam-239	44	18	)	)	PUNCT
ejpam-239	44	19	,	,	PUNCT
ejpam-239	44	20	then	then	ADV
ejpam-239	44	21	≤	≤	PROPN
ejpam-239	44	22	is	be	AUX
ejpam-239	44	23	a	a	DET
ejpam-239	44	24	partial	partial	ADJ
ejpam-239	44	25	ordering	ordering	NOUN
ejpam-239	44	26	on	on	ADP
ejpam-239	44	27	r.	r.	PROPN
ejpam-239	44	28	theorem	theorem	VERB
ejpam-239	44	29	2.1	2.1	NUM
ejpam-239	44	30	.	.	PUNCT
ejpam-239	45	1	for	for	ADP
ejpam-239	45	2	any	any	DET
ejpam-239	45	3	a	a	DET
ejpam-239	45	4	,	,	PUNCT
ejpam-239	45	5	b	b	NOUN
ejpam-239	45	6	,	,	PUNCT
ejpam-239	45	7	c	c	PROPN
ejpam-239	45	8	∈	∈	PROPN
ejpam-239	45	9	r	r	NOUN
ejpam-239	45	10	,	,	PUNCT
ejpam-239	45	11	we	we	PRON
ejpam-239	45	12	have	have	VERB
ejpam-239	45	13	the	the	DET
ejpam-239	45	14	following	following	NOUN
ejpam-239	45	15	:	:	PUNCT
ejpam-239	45	16	1	1	X
ejpam-239	45	17	.	.	X
ejpam-239	45	18	a	a	DET
ejpam-239	45	19	∨	∨	NUM
ejpam-239	45	20	b	b	NOUN
ejpam-239	45	21	=	=	NOUN
ejpam-239	45	22	a⇔	a⇔	NOUN
ejpam-239	45	23	a	a	DET
ejpam-239	45	24	∧	∧	PROPN
ejpam-239	45	25	b	b	PROPN
ejpam-239	45	26	=	=	SYM
ejpam-239	45	27	b	b	PROPN
ejpam-239	45	28	2	2	NUM
ejpam-239	45	29	.	.	PUNCT
ejpam-239	45	30	a	a	DET
ejpam-239	45	31	∨	∨	PROPN
ejpam-239	45	32	b	b	NOUN
ejpam-239	45	33	=	=	NOUN
ejpam-239	45	34	b⇔	b⇔	PROPN
ejpam-239	45	35	a	a	DET
ejpam-239	45	36	∧	∧	PROPN
ejpam-239	45	37	b	b	PROPN
ejpam-239	45	38	=	=	PUNCT
ejpam-239	45	39	a	a	DET
ejpam-239	45	40	3	3	NUM
ejpam-239	45	41	.	.	PUNCT
ejpam-239	46	1	a	a	DET
ejpam-239	46	2	∧	∧	PROPN
ejpam-239	46	3	b	b	PROPN
ejpam-239	46	4	=	=	SYM
ejpam-239	46	5	b	b	PROPN
ejpam-239	46	6	∧	∧	PROPN
ejpam-239	46	7	a	a	PRON
ejpam-239	46	8	whenever	whenever	SCONJ
ejpam-239	46	9	a	a	DET
ejpam-239	46	10	≤	≤	NUM
ejpam-239	46	11	b	b	NOUN
ejpam-239	46	12	4	4	NUM
ejpam-239	46	13	.	.	X
ejpam-239	47	1	∧	∧	NOUN
ejpam-239	47	2	is	be	AUX
ejpam-239	47	3	associative	associative	ADJ
ejpam-239	47	4	in	in	ADP
ejpam-239	47	5	r	r	NOUN
ejpam-239	47	6	5	5	NUM
ejpam-239	47	7	.	.	PUNCT
ejpam-239	48	1	a	a	DET
ejpam-239	48	2	∧	∧	PROPN
ejpam-239	48	3	b	b	PROPN
ejpam-239	48	4	∧	∧	PROPN
ejpam-239	48	5	c	c	NOUN
ejpam-239	48	6	=	=	SYM
ejpam-239	48	7	b	b	PROPN
ejpam-239	48	8	∧	∧	PROPN
ejpam-239	48	9	a	a	DET
ejpam-239	48	10	∧	∧	PROPN
ejpam-239	48	11	c	c	NOUN
ejpam-239	48	12	6	6	NUM
ejpam-239	48	13	.	.	PUNCT
ejpam-239	49	1	(	(	PUNCT
ejpam-239	49	2	a	a	DET
ejpam-239	49	3	∨	∨	NUM
ejpam-239	49	4	b)∧	b)∧	PROPN
ejpam-239	49	5	c	c	NOUN
ejpam-239	49	6	=	=	SYM
ejpam-239	49	7	(	(	PUNCT
ejpam-239	49	8	b	b	PROPN
ejpam-239	49	9	∨	∨	NUM
ejpam-239	49	10	a)∧	a)∧	PROPN
ejpam-239	49	11	c	c	PROPN
ejpam-239	49	12	7	7	NUM
ejpam-239	49	13	.	.	PUNCT
ejpam-239	50	1	a	a	DET
ejpam-239	50	2	∧	∧	PROPN
ejpam-239	50	3	b	b	NOUN
ejpam-239	50	4	=	=	SYM
ejpam-239	50	5	0⇔	0⇔	NOUN
ejpam-239	50	6	b	b	X
ejpam-239	50	7	∧	∧	PROPN
ejpam-239	50	8	a	a	PRON
ejpam-239	50	9	=	=	SYM
ejpam-239	50	10	0	0	NUM
ejpam-239	50	11	8	8	NUM
ejpam-239	50	12	.	.	PUNCT
ejpam-239	51	1	a	a	DET
ejpam-239	51	2	∨	∨	NUM
ejpam-239	51	3	b	b	X
ejpam-239	51	4	=	=	SYM
ejpam-239	51	5	b	b	PROPN
ejpam-239	51	6	∨	∨	X
ejpam-239	51	7	a	a	PRON
ejpam-239	51	8	whenever	whenever	SCONJ
ejpam-239	51	9	a	a	DET
ejpam-239	51	10	∧	∧	PROPN
ejpam-239	51	11	b	b	NOUN
ejpam-239	51	12	=	=	SYM
ejpam-239	51	13	0	0	NUM
ejpam-239	51	14	9	9	NUM
ejpam-239	51	15	.	.	PUNCT
ejpam-239	52	1	a	a	DET
ejpam-239	52	2	∨	∨	NOUN
ejpam-239	52	3	(	(	PUNCT
ejpam-239	52	4	b	b	PROPN
ejpam-239	52	5	∧	∧	PROPN
ejpam-239	52	6	c	c	NOUN
ejpam-239	52	7	)	)	PUNCT
ejpam-239	52	8	=	=	NOUN
ejpam-239	52	9	(	(	PUNCT
ejpam-239	52	10	a	a	DET
ejpam-239	52	11	∨	∨	NUM
ejpam-239	52	12	b)∧	b)∧	PROPN
ejpam-239	52	13	(	(	PUNCT
ejpam-239	52	14	a	a	DET
ejpam-239	52	15	∨	∨	NUM
ejpam-239	52	16	c	c	NOUN
ejpam-239	52	17	)	)	PUNCT
ejpam-239	52	18	10	10	NUM
ejpam-239	52	19	.	.	PUNCT
ejpam-239	53	1	a	a	DET
ejpam-239	53	2	∧	∧	PROPN
ejpam-239	53	3	(	(	PUNCT
ejpam-239	53	4	a	a	DET
ejpam-239	53	5	∨	∨	NUM
ejpam-239	53	6	b	b	NOUN
ejpam-239	53	7	)	)	PUNCT
ejpam-239	53	8	=	=	SYM
ejpam-239	53	9	a	a	PRON
ejpam-239	53	10	,	,	PUNCT
ejpam-239	53	11	(	(	PUNCT
ejpam-239	53	12	a	a	DET
ejpam-239	53	13	∧	∧	PROPN
ejpam-239	53	14	b)∨	b)∨	PROPN
ejpam-239	53	15	b	b	PROPN
ejpam-239	53	16	=	=	SYM
ejpam-239	53	17	b	b	PROPN
ejpam-239	53	18	,	,	PUNCT
ejpam-239	53	19	and	and	CCONJ
ejpam-239	53	20	a	a	DET
ejpam-239	53	21	∨	∨	NOUN
ejpam-239	53	22	(	(	PUNCT
ejpam-239	53	23	b	b	PROPN
ejpam-239	53	24	∧	∧	PROPN
ejpam-239	53	25	a	a	NOUN
ejpam-239	53	26	)	)	PUNCT
ejpam-239	53	27	=	=	PUNCT
ejpam-239	53	28	a	a	DET
ejpam-239	53	29	11	11	NUM
ejpam-239	53	30	.	.	PUNCT
ejpam-239	54	1	a	a	DET
ejpam-239	54	2	≤	≤	NUM
ejpam-239	54	3	a	a	DET
ejpam-239	54	4	∨	∨	NOUN
ejpam-239	54	5	b	b	NOUN
ejpam-239	54	6	and	and	CCONJ
ejpam-239	54	7	a	a	DET
ejpam-239	54	8	∧	∧	PROPN
ejpam-239	54	9	b	b	PROPN
ejpam-239	54	10	≤	≤	NUM
ejpam-239	54	11	b	b	NOUN
ejpam-239	54	12	12	12	NUM
ejpam-239	54	13	.	.	PUNCT
ejpam-239	55	1	a	a	DET
ejpam-239	55	2	∧	∧	PROPN
ejpam-239	55	3	a	a	DET
ejpam-239	55	4	=	=	PUNCT
ejpam-239	55	5	a	a	NOUN
ejpam-239	55	6	and	and	CCONJ
ejpam-239	55	7	a	a	DET
ejpam-239	55	8	∨	∨	NOUN
ejpam-239	55	9	a	a	DET
ejpam-239	55	10	=	=	NOUN
ejpam-239	55	11	a	a	DET
ejpam-239	55	12	13	13	NUM
ejpam-239	55	13	.	.	PUNCT
ejpam-239	56	1	0∨	0∨	NOUN
ejpam-239	56	2	a	a	DET
ejpam-239	56	3	=	=	X
ejpam-239	56	4	a	a	NOUN
ejpam-239	56	5	and	and	CCONJ
ejpam-239	56	6	a	a	DET
ejpam-239	56	7	∧	∧	PROPN
ejpam-239	56	8	0=	0=	NOUN
ejpam-239	56	9	0	0	NUM
ejpam-239	57	1	14	14	NUM
ejpam-239	57	2	.	.	PUNCT
ejpam-239	58	1	if	if	SCONJ
ejpam-239	58	2	a	a	DET
ejpam-239	58	3	≤	≤	NUM
ejpam-239	58	4	c	c	NOUN
ejpam-239	58	5	and	and	CCONJ
ejpam-239	58	6	b	b	NOUN
ejpam-239	58	7	≤	≤	NOUN
ejpam-239	58	8	c	c	NOUN
ejpam-239	58	9	then	then	ADV
ejpam-239	58	10	a	a	DET
ejpam-239	58	11	∧	∧	PROPN
ejpam-239	58	12	b	b	PROPN
ejpam-239	58	13	=	=	SYM
ejpam-239	58	14	b	b	PROPN
ejpam-239	58	15	∧	∧	PROPN
ejpam-239	58	16	a	a	PRON
ejpam-239	58	17	and	and	CCONJ
ejpam-239	58	18	a	a	DET
ejpam-239	58	19	∨	∨	NUM
ejpam-239	58	20	b	b	X
ejpam-239	58	21	=	=	SYM
ejpam-239	58	22	b	b	PROPN
ejpam-239	58	23	∨	∨	NUM
ejpam-239	58	24	a	a	DET
ejpam-239	58	25	15	15	NUM
ejpam-239	58	26	.	.	PUNCT
ejpam-239	59	1	a	a	DET
ejpam-239	59	2	∨	∨	NUM
ejpam-239	59	3	b	b	X
ejpam-239	59	4	=	=	PUNCT
ejpam-239	59	5	a	a	DET
ejpam-239	59	6	∨	∨	NUM
ejpam-239	59	7	b	b	PROPN
ejpam-239	59	8	∨	∨	NUM
ejpam-239	59	9	a.	a.	NOUN
ejpam-239	59	10	an	an	DET
ejpam-239	59	11	element	element	NOUN
ejpam-239	59	12	m	m	NOUN
ejpam-239	59	13	∈	∈	NOUN
ejpam-239	59	14	r	r	NOUN
ejpam-239	59	15	is	be	AUX
ejpam-239	59	16	called	call	VERB
ejpam-239	59	17	maximal	maximal	ADJ
ejpam-239	59	18	if	if	SCONJ
ejpam-239	59	19	it	it	PRON
ejpam-239	59	20	is	be	AUX
ejpam-239	59	21	maximal	maximal	ADJ
ejpam-239	59	22	in	in	ADP
ejpam-239	59	23	the	the	DET
ejpam-239	59	24	partial	partial	ADJ
ejpam-239	59	25	ordered	order	VERB
ejpam-239	59	26	set	set	NOUN
ejpam-239	59	27	(	(	PUNCT
ejpam-239	59	28	r,≤	r,≤	PROPN
ejpam-239	59	29	)	)	PUNCT
ejpam-239	59	30	.	.	PUNCT
ejpam-239	60	1	that	that	PRON
ejpam-239	60	2	is	be	AUX
ejpam-239	60	3	,	,	PUNCT
ejpam-239	60	4	for	for	ADP
ejpam-239	60	5	any	any	DET
ejpam-239	60	6	x	x	SYM
ejpam-239	60	7	∈	∈	PROPN
ejpam-239	60	8	r	r	NOUN
ejpam-239	60	9	,	,	PUNCT
ejpam-239	60	10	m	m	VERB
ejpam-239	60	11	≤	≤	NOUN
ejpam-239	60	12	x	x	X
ejpam-239	60	13	⇒	⇒	NOUN
ejpam-239	60	14	m	m	VERB
ejpam-239	60	15	=	=	ADJ
ejpam-239	60	16	x	x	X
ejpam-239	60	17	.	.	PUNCT
ejpam-239	61	1	theorem	theorem	VERB
ejpam-239	61	2	2.2	2.2	NUM
ejpam-239	61	3	.	.	PUNCT
ejpam-239	62	1	let	let	VERB
ejpam-239	62	2	r	r	PRON
ejpam-239	62	3	be	be	AUX
ejpam-239	62	4	an	an	DET
ejpam-239	62	5	adl	adl	NOUN
ejpam-239	62	6	and	and	CCONJ
ejpam-239	62	7	m	m	PROPN
ejpam-239	62	8	∈	∈	PROPN
ejpam-239	62	9	r.	r.	NOUN
ejpam-239	62	10	then	then	ADV
ejpam-239	62	11	the	the	DET
ejpam-239	62	12	following	follow	VERB
ejpam-239	62	13	are	be	AUX
ejpam-239	62	14	equivalent	equivalent	ADJ
ejpam-239	62	15	:	:	PUNCT
ejpam-239	62	16	g.	g.	PROPN
ejpam-239	62	17	c.	c.	PROPN
ejpam-239	62	18	rao	rao	PROPN
ejpam-239	62	19	and	and	CCONJ
ejpam-239	62	20	m.	m.	PROPN
ejpam-239	62	21	sambasiva	sambasiva	PROPN
ejpam-239	62	22	rao	rao	PROPN
ejpam-239	62	23	/	/	SYM
ejpam-239	62	24	eur	eur	PROPN
ejpam-239	62	25	.	.	PUNCT
ejpam-239	63	1	j.	j.	PROPN
ejpam-239	63	2	pure	pure	PROPN
ejpam-239	63	3	appl	appl	PROPN
ejpam-239	63	4	.	.	PROPN
ejpam-239	63	5	math	math	PROPN
ejpam-239	63	6	,	,	PUNCT
ejpam-239	63	7	2	2	NUM
ejpam-239	63	8	(	(	PUNCT
ejpam-239	63	9	2009	2009	NUM
ejpam-239	63	10	)	)	PUNCT
ejpam-239	63	11	,	,	PUNCT
ejpam-239	63	12	(	(	PUNCT
ejpam-239	63	13	58	58	NUM
ejpam-239	63	14	-	-	SYM
ejpam-239	63	15	72	72	NUM
ejpam-239	63	16	)	)	PUNCT
ejpam-239	63	17	61	61	NUM
ejpam-239	63	18	1	1	NUM
ejpam-239	63	19	.	.	PUNCT
ejpam-239	64	1	m	m	PROPN
ejpam-239	64	2	is	be	AUX
ejpam-239	64	3	a	a	DET
ejpam-239	64	4	maximal	maximal	ADJ
ejpam-239	64	5	element	element	NOUN
ejpam-239	64	6	with	with	ADP
ejpam-239	64	7	respect	respect	NOUN
ejpam-239	64	8	to	to	ADP
ejpam-239	64	9	≤	≤	NOUN
ejpam-239	64	10	2	2	NUM
ejpam-239	64	11	.	.	PUNCT
ejpam-239	65	1	m∨	m∨	NOUN
ejpam-239	65	2	x	x	PUNCT
ejpam-239	66	1	=	=	PUNCT
ejpam-239	66	2	m	m	PROPN
ejpam-239	66	3	,	,	PUNCT
ejpam-239	66	4	for	for	ADP
ejpam-239	66	5	all	all	PRON
ejpam-239	66	6	x	x	SYM
ejpam-239	66	7	∈	∈	NOUN
ejpam-239	66	8	r	r	NOUN
ejpam-239	66	9	3	3	NUM
ejpam-239	66	10	.	.	PUNCT
ejpam-239	66	11	m∧	m∧	NOUN
ejpam-239	66	12	x	x	X
ejpam-239	66	13	=	=	SYM
ejpam-239	66	14	x	x	X
ejpam-239	66	15	,	,	PUNCT
ejpam-239	66	16	for	for	ADP
ejpam-239	66	17	all	all	PRON
ejpam-239	66	18	x	x	SYM
ejpam-239	66	19	∈	∈	NOUN
ejpam-239	66	20	r	r	NOUN
ejpam-239	66	21	4	4	NUM
ejpam-239	66	22	.	.	PUNCT
ejpam-239	67	1	x	x	SYM
ejpam-239	67	2	∨m	∨m	NOUN
ejpam-239	67	3	is	be	AUX
ejpam-239	67	4	maximal	maximal	ADJ
ejpam-239	67	5	for	for	ADP
ejpam-239	67	6	all	all	DET
ejpam-239	67	7	x	x	PROPN
ejpam-239	67	8	∈	∈	PROPN
ejpam-239	67	9	r.	r.	NOUN
ejpam-239	67	10	a	a	DET
ejpam-239	67	11	non	non	ADJ
ejpam-239	67	12	-	-	ADJ
ejpam-239	67	13	empty	empty	ADJ
ejpam-239	67	14	subset	subset	NOUN
ejpam-239	68	1	i	i	PRON
ejpam-239	68	2	of	of	ADP
ejpam-239	68	3	r	r	NOUN
ejpam-239	68	4	is	be	AUX
ejpam-239	68	5	called	call	VERB
ejpam-239	68	6	an	an	DET
ejpam-239	68	7	ideal(filter)of	ideal(filter)of	NOUN
ejpam-239	68	8	r	r	NOUN
ejpam-239	68	9	if	if	SCONJ
ejpam-239	68	10	a	a	DET
ejpam-239	68	11	∨	∨	NUM
ejpam-239	68	12	b	b	X
ejpam-239	68	13	∈	∈	PROPN
ejpam-239	68	14	i(a	i(a	PROPN
ejpam-239	68	15	∧	∧	PROPN
ejpam-239	68	16	b	b	PROPN
ejpam-239	68	17	∈	∈	PROPN
ejpam-239	68	18	i	i	PROPN
ejpam-239	68	19	)	)	PUNCT
ejpam-239	68	20	and	and	CCONJ
ejpam-239	68	21	a	a	DET
ejpam-239	68	22	∧	∧	PROPN
ejpam-239	68	23	x	x	SYM
ejpam-239	68	24	∈	∈	PROPN
ejpam-239	68	25	i(x	i(x	PROPN
ejpam-239	68	26	∨	∨	NOUN
ejpam-239	68	27	a	a	DET
ejpam-239	68	28	∈	∈	PROPN
ejpam-239	68	29	i	i	NOUN
ejpam-239	68	30	)	)	PUNCT
ejpam-239	68	31	whenever	whenever	SCONJ
ejpam-239	68	32	a	a	PRON
ejpam-239	68	33	,	,	PUNCT
ejpam-239	68	34	b	b	X
ejpam-239	68	35	∈	∈	NOUN
ejpam-239	68	36	i	i	PRON
ejpam-239	68	37	and	and	CCONJ
ejpam-239	68	38	x	x	PROPN
ejpam-239	68	39	∈	∈	PROPN
ejpam-239	68	40	r.	r.	NOUN
ejpam-239	68	41	if	if	SCONJ
ejpam-239	68	42	i	i	PRON
ejpam-239	68	43	is	be	AUX
ejpam-239	68	44	an	an	DET
ejpam-239	68	45	ideal	ideal	NOUN
ejpam-239	68	46	of	of	ADP
ejpam-239	68	47	r	r	NOUN
ejpam-239	68	48	and	and	CCONJ
ejpam-239	68	49	a	a	DET
ejpam-239	68	50	,	,	PUNCT
ejpam-239	68	51	b	b	X
ejpam-239	68	52	∈	∈	PROPN
ejpam-239	68	53	r	r	NOUN
ejpam-239	68	54	,	,	PUNCT
ejpam-239	68	55	then	then	ADV
ejpam-239	69	1	a	a	DET
ejpam-239	69	2	∧	∧	PROPN
ejpam-239	69	3	b	b	PROPN
ejpam-239	69	4	∈	∈	PROPN
ejpam-239	69	5	i	i	PRON
ejpam-239	69	6	⇔	⇔	PROPN
ejpam-239	69	7	b	b	PROPN
ejpam-239	69	8	∧	∧	PROPN
ejpam-239	69	9	a	a	DET
ejpam-239	69	10	∈	∈	NOUN
ejpam-239	69	11	i	i	PRON
ejpam-239	69	12	.	.	PUNCT
ejpam-239	70	1	the	the	DET
ejpam-239	70	2	set	set	NOUN
ejpam-239	70	3	i	i	PRON
ejpam-239	70	4	(	(	PUNCT
ejpam-239	70	5	r	r	NOUN
ejpam-239	70	6	)	)	PUNCT
ejpam-239	70	7	of	of	ADP
ejpam-239	70	8	all	all	DET
ejpam-239	70	9	ideals	ideal	NOUN
ejpam-239	70	10	of	of	ADP
ejpam-239	70	11	r	r	NOUN
ejpam-239	70	12	is	be	AUX
ejpam-239	70	13	a	a	DET
ejpam-239	70	14	complete	complete	ADJ
ejpam-239	70	15	distributive	distributive	ADJ
ejpam-239	70	16	lattice	lattice	NOUN
ejpam-239	70	17	with	with	ADP
ejpam-239	70	18	least	least	ADJ
ejpam-239	70	19	element	element	ADJ
ejpam-239	70	20	{	{	PUNCT
ejpam-239	70	21	0	0	NUM
ejpam-239	70	22	}	}	PUNCT
ejpam-239	70	23	and	and	CCONJ
ejpam-239	70	24	the	the	DET
ejpam-239	70	25	greatest	great	ADJ
ejpam-239	70	26	element	element	NOUN
ejpam-239	70	27	r	r	NOUN
ejpam-239	70	28	under	under	ADP
ejpam-239	70	29	set	set	NOUN
ejpam-239	70	30	inclusion	inclusion	NOUN
ejpam-239	70	31	in	in	ADP
ejpam-239	70	32	which	which	PRON
ejpam-239	70	33	,	,	PUNCT
ejpam-239	70	34	for	for	ADP
ejpam-239	70	35	any	any	DET
ejpam-239	70	36	i	i	NOUN
ejpam-239	70	37	,	,	PUNCT
ejpam-239	70	38	j	j	PROPN
ejpam-239	70	39	∈	∈	PROPN
ejpam-239	70	40	i	i	PRON
ejpam-239	70	41	(	(	PUNCT
ejpam-239	70	42	r	r	NOUN
ejpam-239	70	43	)	)	PUNCT
ejpam-239	70	44	,	,	PUNCT
ejpam-239	70	45	i	i	PROPN
ejpam-239	70	46	∩	∩	VERB
ejpam-239	70	47	j	j	PROPN
ejpam-239	70	48	is	be	AUX
ejpam-239	70	49	the	the	DET
ejpam-239	70	50	infimum	infimum	NOUN
ejpam-239	70	51	of	of	ADP
ejpam-239	70	52	i	i	PRON
ejpam-239	70	53	,	,	PUNCT
ejpam-239	70	54	j	j	PROPN
ejpam-239	70	55	and	and	CCONJ
ejpam-239	70	56	the	the	DET
ejpam-239	70	57	supremum	supremum	NOUN
ejpam-239	70	58	is	be	AUX
ejpam-239	70	59	given	give	VERB
ejpam-239	70	60	by	by	ADP
ejpam-239	70	61	i	i	PROPN
ejpam-239	70	62	∨	∨	NOUN
ejpam-239	70	63	j	j	PROPN
ejpam-239	71	1	=	=	PRON
ejpam-239	71	2	{	{	PUNCT
ejpam-239	71	3	i	i	PROPN
ejpam-239	71	4	∨	∨	PROPN
ejpam-239	71	5	j	j	PROPN
ejpam-239	72	1	|	|	ADV
ejpam-239	72	2	i	i	PRON
ejpam-239	72	3	∈	∈	VERB
ejpam-239	73	1	i	i	PRON
ejpam-239	73	2	,	,	PUNCT
ejpam-239	74	1	j	j	PROPN
ejpam-239	74	2	∈	∈	PROPN
ejpam-239	74	3	j	j	PROPN
ejpam-239	74	4	}	}	PUNCT
ejpam-239	74	5	.	.	PUNCT
ejpam-239	75	1	for	for	ADP
ejpam-239	75	2	any	any	DET
ejpam-239	75	3	a	a	DET
ejpam-239	75	4	∈	∈	PROPN
ejpam-239	75	5	r	r	NOUN
ejpam-239	75	6	,	,	PUNCT
ejpam-239	75	7	(	(	PUNCT
ejpam-239	75	8	a	a	X
ejpam-239	75	9	]	]	X
ejpam-239	75	10	=	=	X
ejpam-239	75	11	{	{	PUNCT
ejpam-239	75	12	a	a	DET
ejpam-239	75	13	∧	∧	PROPN
ejpam-239	75	14	x	x	SYM
ejpam-239	75	15	|	|	ADV
ejpam-239	75	16	x	x	SYM
ejpam-239	75	17	∈	∈	NOUN
ejpam-239	75	18	r	r	NOUN
ejpam-239	75	19	}	}	PUNCT
ejpam-239	75	20	is	be	AUX
ejpam-239	75	21	the	the	DET
ejpam-239	75	22	principal	principal	ADJ
ejpam-239	75	23	ideal	ideal	NOUN
ejpam-239	75	24	generated	generate	VERB
ejpam-239	75	25	by	by	ADP
ejpam-239	75	26	a.	a.	NOUN
ejpam-239	75	27	similarly	similarly	ADV
ejpam-239	75	28	,	,	PUNCT
ejpam-239	75	29	for	for	SCONJ
ejpam-239	75	30	any	any	DET
ejpam-239	75	31	a	a	DET
ejpam-239	75	32	∈	∈	NOUN
ejpam-239	75	33	r	r	NOUN
ejpam-239	75	34	,	,	PUNCT
ejpam-239	75	35	[	[	X
ejpam-239	75	36	a	a	X
ejpam-239	75	37	)	)	PUNCT
ejpam-239	75	38	=	=	SYM
ejpam-239	75	39	{	{	PUNCT
ejpam-239	76	1	x	x	SYM
ejpam-239	76	2	∨	∨	NUM
ejpam-239	76	3	a	a	DET
ejpam-239	76	4	|	|	NOUN
ejpam-239	76	5	x	x	SYM
ejpam-239	76	6	∈	∈	NOUN
ejpam-239	76	7	r	r	NOUN
ejpam-239	76	8	}	}	PUNCT
ejpam-239	76	9	is	be	AUX
ejpam-239	76	10	the	the	DET
ejpam-239	76	11	filter	filter	NOUN
ejpam-239	76	12	generated	generate	VERB
ejpam-239	76	13	by	by	ADP
ejpam-239	76	14	a.	a.	NOUN
ejpam-239	76	15	an	an	DET
ejpam-239	76	16	ideal	ideal	NOUN
ejpam-239	76	17	i	i	PRON
ejpam-239	76	18	of	of	ADP
ejpam-239	76	19	r	r	NOUN
ejpam-239	76	20	is	be	AUX
ejpam-239	76	21	called	call	VERB
ejpam-239	76	22	a	a	DET
ejpam-239	76	23	direct	direct	ADJ
ejpam-239	76	24	summand	summand	NOUN
ejpam-239	76	25	of	of	ADP
ejpam-239	76	26	r	r	NOUN
ejpam-239	76	27	if	if	SCONJ
ejpam-239	76	28	there	there	PRON
ejpam-239	76	29	exists	exist	VERB
ejpam-239	76	30	an	an	DET
ejpam-239	76	31	ideal	ideal	ADJ
ejpam-239	76	32	j	j	NOUN
ejpam-239	76	33	in	in	ADP
ejpam-239	76	34	r	r	NOUN
ejpam-239	76	35	such	such	ADJ
ejpam-239	76	36	that	that	SCONJ
ejpam-239	76	37	i	i	PROPN
ejpam-239	76	38	∩	∩	NOUN
ejpam-239	76	39	j	j	PROPN
ejpam-239	76	40	=	=	PUNCT
ejpam-239	76	41	(	(	PUNCT
ejpam-239	76	42	0	0	NUM
ejpam-239	76	43	]	]	PUNCT
ejpam-239	76	44	and	and	CCONJ
ejpam-239	76	45	i	i	PRON
ejpam-239	76	46	∨	∨	PROPN
ejpam-239	76	47	j	j	PROPN
ejpam-239	76	48	=	=	SYM
ejpam-239	76	49	r.	r.	PROPN
ejpam-239	76	50	theorem	theorem	VERB
ejpam-239	76	51	2.3	2.3	NUM
ejpam-239	76	52	.	.	PUNCT
ejpam-239	77	1	for	for	ADP
ejpam-239	77	2	any	any	DET
ejpam-239	77	3	a	a	PRON
ejpam-239	77	4	,	,	PUNCT
ejpam-239	77	5	b	b	X
ejpam-239	77	6	∈	∈	PROPN
ejpam-239	77	7	r	r	NOUN
ejpam-239	77	8	,	,	PUNCT
ejpam-239	77	9	we	we	PRON
ejpam-239	77	10	have	have	VERB
ejpam-239	77	11	the	the	DET
ejpam-239	77	12	following	following	NOUN
ejpam-239	77	13	:	:	PUNCT
ejpam-239	77	14	1	1	X
ejpam-239	77	15	.	.	PUNCT
ejpam-239	77	16	(	(	PUNCT
ejpam-239	77	17	a]∨	a]∨	PROPN
ejpam-239	77	18	(	(	PUNCT
ejpam-239	77	19	b	b	NOUN
ejpam-239	77	20	]	]	X
ejpam-239	77	21	=	=	X
ejpam-239	77	22	(	(	PUNCT
ejpam-239	77	23	a	a	DET
ejpam-239	77	24	∨	∨	NUM
ejpam-239	77	25	b	b	NOUN
ejpam-239	77	26	]	]	X
ejpam-239	77	27	=	=	SYM
ejpam-239	77	28	(	(	PUNCT
ejpam-239	77	29	b	b	PROPN
ejpam-239	77	30	∨	∨	NUM
ejpam-239	77	31	a	a	PRON
ejpam-239	77	32	]	]	X
ejpam-239	77	33	2	2	NUM
ejpam-239	77	34	.	.	PUNCT
ejpam-239	77	35	(	(	PUNCT
ejpam-239	77	36	a]∩	a]∩	X
ejpam-239	77	37	(	(	PUNCT
ejpam-239	77	38	b	b	X
ejpam-239	77	39	]	]	X
ejpam-239	77	40	=	=	X
ejpam-239	77	41	(	(	PUNCT
ejpam-239	77	42	a	a	DET
ejpam-239	77	43	∧	∧	PROPN
ejpam-239	77	44	b	b	PROPN
ejpam-239	77	45	]	]	X
ejpam-239	77	46	=	=	SYM
ejpam-239	77	47	(	(	PUNCT
ejpam-239	77	48	b	b	X
ejpam-239	77	49	∧	∧	PROPN
ejpam-239	77	50	a	a	X
ejpam-239	77	51	]	]	X
ejpam-239	77	52	3	3	NUM
ejpam-239	77	53	.	.	PUNCT
ejpam-239	78	1	[	[	X
ejpam-239	78	2	a)∨	a)∨	PROPN
ejpam-239	78	3	[	[	X
ejpam-239	78	4	b	b	NOUN
ejpam-239	78	5	)	)	PUNCT
ejpam-239	78	6	=	=	PUNCT
ejpam-239	79	1	[	[	X
ejpam-239	79	2	a	a	DET
ejpam-239	79	3	∧	∧	PROPN
ejpam-239	79	4	b	b	NOUN
ejpam-239	79	5	)	)	PUNCT
ejpam-239	79	6	=	=	PUNCT
ejpam-239	80	1	[	[	X
ejpam-239	80	2	b	b	X
ejpam-239	80	3	∧	∧	PROPN
ejpam-239	80	4	a	a	PRON
ejpam-239	80	5	)	)	PUNCT
ejpam-239	80	6	4	4	NUM
ejpam-239	80	7	.	.	PUNCT
ejpam-239	81	1	[	[	X
ejpam-239	81	2	a)∩	a)∩	X
ejpam-239	81	3	[	[	X
ejpam-239	81	4	b	b	X
ejpam-239	81	5	)	)	PUNCT
ejpam-239	81	6	=	=	PUNCT
ejpam-239	82	1	[	[	X
ejpam-239	82	2	a	a	DET
ejpam-239	82	3	∨	∨	NUM
ejpam-239	82	4	b	b	NOUN
ejpam-239	82	5	)	)	PUNCT
ejpam-239	82	6	=	=	PUNCT
ejpam-239	83	1	[	[	X
ejpam-239	83	2	b	b	X
ejpam-239	83	3	∨	∨	NUM
ejpam-239	83	4	a	a	NOUN
ejpam-239	83	5	)	)	PUNCT
ejpam-239	83	6	thus	thus	ADV
ejpam-239	83	7	the	the	DET
ejpam-239	83	8	set	set	NOUN
ejpam-239	83	9	p	p	X
ejpam-239	83	10	i	i	PRON
ejpam-239	83	11	(	(	PUNCT
ejpam-239	83	12	r	r	NOUN
ejpam-239	83	13	)	)	PUNCT
ejpam-239	83	14	of	of	ADP
ejpam-239	83	15	all	all	DET
ejpam-239	83	16	principal	principal	ADJ
ejpam-239	83	17	ideals	ideal	NOUN
ejpam-239	83	18	of	of	ADP
ejpam-239	83	19	r	r	NOUN
ejpam-239	83	20	is	be	AUX
ejpam-239	83	21	a	a	DET
ejpam-239	83	22	sublattice	sublattice	NOUN
ejpam-239	83	23	of	of	ADP
ejpam-239	83	24	the	the	DET
ejpam-239	83	25	distributive	distributive	ADJ
ejpam-239	83	26	lattice	lattice	NOUN
ejpam-239	84	1	i	i	PRON
ejpam-239	84	2	(	(	PUNCT
ejpam-239	84	3	r	r	NOUN
ejpam-239	84	4	)	)	PUNCT
ejpam-239	84	5	of	of	ADP
ejpam-239	84	6	ideals	ideal	NOUN
ejpam-239	84	7	of	of	ADP
ejpam-239	84	8	r.	r.	PROPN
ejpam-239	84	9	a	a	DET
ejpam-239	84	10	proper	proper	ADJ
ejpam-239	84	11	ideal	ideal	NOUN
ejpam-239	84	12	p	p	NOUN
ejpam-239	84	13	of	of	ADP
ejpam-239	84	14	r	r	NOUN
ejpam-239	84	15	is	be	AUX
ejpam-239	84	16	said	say	VERB
ejpam-239	84	17	to	to	PART
ejpam-239	84	18	be	be	AUX
ejpam-239	84	19	prime	prime	ADJ
ejpam-239	84	20	if	if	SCONJ
ejpam-239	84	21	for	for	SCONJ
ejpam-239	84	22	any	any	DET
ejpam-239	84	23	x	x	NOUN
ejpam-239	84	24	,	,	PUNCT
ejpam-239	84	25	y	y	PROPN
ejpam-239	84	26	∈	∈	PROPN
ejpam-239	84	27	r	r	NOUN
ejpam-239	84	28	,	,	PUNCT
ejpam-239	84	29	x	x	PUNCT
ejpam-239	84	30	∧	∧	NOUN
ejpam-239	84	31	y	y	PROPN
ejpam-239	84	32	∈	∈	PROPN
ejpam-239	84	33	p	p	PROPN
ejpam-239	84	34	⇒	⇒	NOUN
ejpam-239	84	35	either	either	CCONJ
ejpam-239	84	36	x	x	SYM
ejpam-239	84	37	∈	∈	PROPN
ejpam-239	84	38	p	p	NOUN
ejpam-239	84	39	or	or	CCONJ
ejpam-239	84	40	y	y	PROPN
ejpam-239	84	41	∈	∈	PROPN
ejpam-239	85	1	p.	p.	NOUN
ejpam-239	86	1	it	it	PRON
ejpam-239	86	2	is	be	AUX
ejpam-239	86	3	clear	clear	ADJ
ejpam-239	86	4	that	that	SCONJ
ejpam-239	86	5	a	a	DET
ejpam-239	86	6	subset	subset	NOUN
ejpam-239	86	7	p	p	NOUN
ejpam-239	86	8	of	of	ADP
ejpam-239	86	9	r	r	NOUN
ejpam-239	86	10	is	be	AUX
ejpam-239	86	11	a	a	DET
ejpam-239	86	12	prime	prime	ADJ
ejpam-239	86	13	ideal	ideal	NOUN
ejpam-239	86	14	iff	iff	PROPN
ejpam-239	86	15	r−	r−	PROPN
ejpam-239	86	16	p	p	PROPN
ejpam-239	86	17	is	be	AUX
ejpam-239	86	18	a	a	DET
ejpam-239	86	19	prime	prime	ADJ
ejpam-239	86	20	filter	filter	NOUN
ejpam-239	86	21	.	.	PUNCT
ejpam-239	87	1	for	for	ADP
ejpam-239	87	2	any	any	DET
ejpam-239	87	3	a⊆	a⊆	PROPN
ejpam-239	87	4	r	r	NOUN
ejpam-239	87	5	,	,	PUNCT
ejpam-239	87	6	a∗	a∗	NOUN
ejpam-239	87	7	=	=	SYM
ejpam-239	87	8	{	{	PUNCT
ejpam-239	87	9	x	x	PUNCT
ejpam-239	87	10	∈	∈	NOUN
ejpam-239	87	11	r	r	NOUN
ejpam-239	87	12	|	|	ADV
ejpam-239	87	13	a	a	DET
ejpam-239	87	14	∧	∧	NOUN
ejpam-239	87	15	x	x	PUNCT
ejpam-239	87	16	=	=	SYM
ejpam-239	87	17	0	0	NUM
ejpam-239	87	18	for	for	SCONJ
ejpam-239	87	19	all	all	DET
ejpam-239	87	20	a	a	DET
ejpam-239	87	21	∈	∈	PROPN
ejpam-239	87	22	a	a	PRON
ejpam-239	87	23	}	}	PUNCT
ejpam-239	87	24	is	be	AUX
ejpam-239	87	25	an	an	DET
ejpam-239	87	26	ideal	ideal	NOUN
ejpam-239	87	27	of	of	ADP
ejpam-239	87	28	r.	r.	PROPN
ejpam-239	87	29	we	we	PRON
ejpam-239	87	30	write	write	VERB
ejpam-239	87	31	(	(	PUNCT
ejpam-239	87	32	a]∗	a]∗	NOUN
ejpam-239	87	33	for	for	ADP
ejpam-239	87	34	{	{	PUNCT
ejpam-239	87	35	a}∗.	a}∗.	VERB
ejpam-239	87	36	then	then	ADV
ejpam-239	87	37	clearly	clearly	ADV
ejpam-239	87	38	(	(	PUNCT
ejpam-239	87	39	0]∗	0]∗	NOUN
ejpam-239	87	40	=	=	SYM
ejpam-239	87	41	r	r	NOUN
ejpam-239	87	42	and	and	CCONJ
ejpam-239	87	43	r∗	r∗	PROPN
ejpam-239	87	44	=	=	SYM
ejpam-239	87	45	(	(	PUNCT
ejpam-239	87	46	0	0	NUM
ejpam-239	87	47	]	]	PUNCT
ejpam-239	87	48	.	.	PUNCT
ejpam-239	88	1	an	an	DET
ejpam-239	88	2	element	element	NOUN
ejpam-239	88	3	a	a	DET
ejpam-239	88	4	∈	∈	NOUN
ejpam-239	88	5	r	r	NOUN
ejpam-239	88	6	is	be	AUX
ejpam-239	88	7	called	call	VERB
ejpam-239	88	8	dense	dense	ADJ
ejpam-239	88	9	if	if	SCONJ
ejpam-239	88	10	(	(	PUNCT
ejpam-239	88	11	a]∗	a]∗	PROPN
ejpam-239	88	12	=	=	SYM
ejpam-239	88	13	(	(	PUNCT
ejpam-239	88	14	0	0	NUM
ejpam-239	88	15	]	]	PUNCT
ejpam-239	88	16	.	.	PUNCT
ejpam-239	89	1	the	the	DET
ejpam-239	89	2	set	set	NOUN
ejpam-239	89	3	of	of	ADP
ejpam-239	89	4	all	all	DET
ejpam-239	89	5	dense	dense	ADJ
ejpam-239	89	6	elements	element	NOUN
ejpam-239	89	7	of	of	ADP
ejpam-239	89	8	r	r	NOUN
ejpam-239	89	9	is	be	AUX
ejpam-239	89	10	denoted	denote	VERB
ejpam-239	89	11	by	by	ADP
ejpam-239	89	12	d.	d.	PROPN
ejpam-239	89	13	an	an	DET
ejpam-239	89	14	ideal	ideal	NOUN
ejpam-239	89	15	i	i	PRON
ejpam-239	89	16	of	of	ADP
ejpam-239	89	17	r	r	NOUN
ejpam-239	89	18	is	be	AUX
ejpam-239	89	19	called	call	VERB
ejpam-239	89	20	dense	dense	ADJ
ejpam-239	89	21	if	if	SCONJ
ejpam-239	89	22	i∗	i∗	NOUN
ejpam-239	89	23	=	=	SYM
ejpam-239	89	24	(	(	PUNCT
ejpam-239	89	25	0	0	NUM
ejpam-239	89	26	]	]	PUNCT
ejpam-239	89	27	.	.	PUNCT
ejpam-239	90	1	an	an	DET
ejpam-239	90	2	adl	adl	PROPN
ejpam-239	90	3	r	r	NOUN
ejpam-239	90	4	with	with	ADP
ejpam-239	90	5	0	0	NUM
ejpam-239	90	6	is	be	AUX
ejpam-239	90	7	called	call	VERB
ejpam-239	90	8	a	a	DET
ejpam-239	90	9	⋆-adl	⋆-adl	PROPN
ejpam-239	90	10	[	[	X
ejpam-239	90	11	10	10	NUM
ejpam-239	90	12	]	]	PUNCT
ejpam-239	90	13	,	,	PUNCT
ejpam-239	90	14	if	if	SCONJ
ejpam-239	90	15	for	for	ADP
ejpam-239	90	16	each	each	DET
ejpam-239	90	17	x	x	SYM
ejpam-239	90	18	∈	∈	PROPN
ejpam-239	90	19	r	r	NOUN
ejpam-239	90	20	,	,	PUNCT
ejpam-239	90	21	there	there	PRON
ejpam-239	90	22	exists	exist	VERB
ejpam-239	90	23	an	an	DET
ejpam-239	90	24	element	element	NOUN
ejpam-239	90	25	x	x	NOUN
ejpam-239	90	26	′	′	NUM
ejpam-239	90	27	∈	∈	NOUN
ejpam-239	90	28	r	r	NOUN
ejpam-239	90	29	such	such	ADJ
ejpam-239	90	30	that	that	PRON
ejpam-239	90	31	(	(	PUNCT
ejpam-239	90	32	x]∗∗	x]∗∗	NOUN
ejpam-239	90	33	=	=	SYM
ejpam-239	90	34	(	(	PUNCT
ejpam-239	90	35	x	x	SYM
ejpam-239	90	36	′]∗.	′]∗.	PROPN
ejpam-239	90	37	r	r	NOUN
ejpam-239	90	38	is	be	AUX
ejpam-239	90	39	a	a	DET
ejpam-239	90	40	⋆-adl	⋆-adl	PROPN
ejpam-239	90	41	iff	iff	NOUN
ejpam-239	90	42	to	to	ADP
ejpam-239	90	43	each	each	DET
ejpam-239	90	44	x	x	SYM
ejpam-239	90	45	∈	∈	PROPN
ejpam-239	90	46	r	r	NOUN
ejpam-239	90	47	,	,	PUNCT
ejpam-239	90	48	there	there	PRON
ejpam-239	90	49	exists	exist	VERB
ejpam-239	90	50	x	x	PUNCT
ejpam-239	90	51	′	′	NUM
ejpam-239	90	52	∈	∈	NOUN
ejpam-239	90	53	r	r	NOUN
ejpam-239	90	54	such	such	ADJ
ejpam-239	90	55	that	that	SCONJ
ejpam-239	90	56	x	x	X
ejpam-239	90	57	∧	∧	NOUN
ejpam-239	90	58	x	x	NOUN
ejpam-239	91	1	′	′	NUM
ejpam-239	91	2	=	=	SYM
ejpam-239	91	3	0	0	NUM
ejpam-239	92	1	and	and	CCONJ
ejpam-239	92	2	x	x	SYM
ejpam-239	92	3	∨	∨	NOUN
ejpam-239	92	4	x	x	SYM
ejpam-239	92	5	′	′	NOUN
ejpam-239	92	6	is	be	AUX
ejpam-239	92	7	dense	dense	ADJ
ejpam-239	92	8	.	.	PUNCT
ejpam-239	93	1	every	every	DET
ejpam-239	93	2	⋆-adl	⋆-adl	PROPN
ejpam-239	93	3	possesses	possess	VERB
ejpam-239	93	4	a	a	DET
ejpam-239	93	5	dense	dense	ADJ
ejpam-239	93	6	element	element	NOUN
ejpam-239	93	7	.	.	PUNCT
ejpam-239	94	1	an	an	DET
ejpam-239	94	2	adl	adl	PROPN
ejpam-239	94	3	r	r	NOUN
ejpam-239	94	4	with	with	ADP
ejpam-239	94	5	0	0	NUM
ejpam-239	94	6	is	be	AUX
ejpam-239	94	7	called	call	VERB
ejpam-239	94	8	relatively	relatively	ADV
ejpam-239	94	9	g.	g.	PROPN
ejpam-239	94	10	c.	c.	PROPN
ejpam-239	94	11	rao	rao	PROPN
ejpam-239	94	12	and	and	CCONJ
ejpam-239	94	13	m.	m.	PROPN
ejpam-239	94	14	sambasiva	sambasiva	PROPN
ejpam-239	94	15	rao	rao	PROPN
ejpam-239	94	16	/	/	SYM
ejpam-239	94	17	eur	eur	PROPN
ejpam-239	94	18	.	.	PUNCT
ejpam-239	95	1	j.	j.	PROPN
ejpam-239	95	2	pure	pure	PROPN
ejpam-239	95	3	appl	appl	PROPN
ejpam-239	95	4	.	.	PROPN
ejpam-239	95	5	math	math	PROPN
ejpam-239	95	6	,	,	PUNCT
ejpam-239	95	7	2	2	NUM
ejpam-239	95	8	(	(	PUNCT
ejpam-239	95	9	2009	2009	NUM
ejpam-239	95	10	)	)	PUNCT
ejpam-239	95	11	,	,	PUNCT
ejpam-239	95	12	(	(	PUNCT
ejpam-239	95	13	58	58	NUM
ejpam-239	95	14	-	-	SYM
ejpam-239	95	15	72	72	NUM
ejpam-239	95	16	)	)	PUNCT
ejpam-239	95	17	62	62	NUM
ejpam-239	95	18	complemented	complement	VERB
ejpam-239	95	19	if	if	SCONJ
ejpam-239	95	20	each	each	DET
ejpam-239	95	21	interval	interval	NOUN
ejpam-239	95	22	[	[	X
ejpam-239	95	23	a	a	X
ejpam-239	95	24	,	,	PUNCT
ejpam-239	95	25	b	b	NOUN
ejpam-239	95	26	]	]	X
ejpam-239	95	27	,	,	PUNCT
ejpam-239	95	28	a	a	DET
ejpam-239	95	29	≤	≤	PROPN
ejpam-239	95	30	b	b	NOUN
ejpam-239	95	31	,	,	PUNCT
ejpam-239	95	32	in	in	ADP
ejpam-239	95	33	r	r	NOUN
ejpam-239	95	34	is	be	AUX
ejpam-239	95	35	a	a	DET
ejpam-239	95	36	complemented	complemented	ADJ
ejpam-239	95	37	lattice	lattice	NOUN
ejpam-239	95	38	.	.	PUNCT
ejpam-239	96	1	an	an	DET
ejpam-239	96	2	ideal	ideal	ADJ
ejpam-239	96	3	i	i	PRON
ejpam-239	96	4	of	of	ADP
ejpam-239	96	5	r	r	NOUN
ejpam-239	96	6	is	be	AUX
ejpam-239	96	7	called	call	VERB
ejpam-239	96	8	an	an	DET
ejpam-239	96	9	annihilator	annihilator	NOUN
ejpam-239	96	10	ideal	ideal	NOUN
ejpam-239	96	11	if	if	SCONJ
ejpam-239	96	12	i	i	PRON
ejpam-239	96	13	=	=	SYM
ejpam-239	96	14	i∗∗	i∗∗	PROPN
ejpam-239	96	15	,	,	PUNCT
ejpam-239	96	16	or	or	CCONJ
ejpam-239	96	17	equivalently	equivalently	ADV
ejpam-239	96	18	,	,	PUNCT
ejpam-239	96	19	i	i	PRON
ejpam-239	96	20	=	=	PUNCT
ejpam-239	96	21	s∗	s∗	PROPN
ejpam-239	96	22	=	=	SYM
ejpam-239	96	23	{	{	PUNCT
ejpam-239	96	24	y	y	PROPN
ejpam-239	96	25	∈	∈	PROPN
ejpam-239	96	26	r	r	NOUN
ejpam-239	96	27	|	|	NOUN
ejpam-239	96	28	y	y	PROPN
ejpam-239	96	29	∧	∧	PROPN
ejpam-239	96	30	s	s	PART
ejpam-239	96	31	=	=	NOUN
ejpam-239	96	32	0	0	NUM
ejpam-239	96	33	for	for	ADP
ejpam-239	96	34	all	all	DET
ejpam-239	96	35	s	s	PART
ejpam-239	96	36	∈	∈	NOUN
ejpam-239	96	37	s	s	PART
ejpam-239	96	38	}	}	PUNCT
ejpam-239	96	39	for	for	ADP
ejpam-239	96	40	some	some	DET
ejpam-239	96	41	non	non	ADJ
ejpam-239	96	42	-	-	ADJ
ejpam-239	96	43	empty	empty	ADJ
ejpam-239	96	44	subset	subset	NOUN
ejpam-239	96	45	s	s	PROPN
ejpam-239	96	46	of	of	ADP
ejpam-239	96	47	r.	r.	PROPN
ejpam-239	96	48	we	we	PRON
ejpam-239	96	49	denote	denote	VERB
ejpam-239	96	50	the	the	DET
ejpam-239	96	51	set	set	NOUN
ejpam-239	96	52	of	of	ADP
ejpam-239	96	53	all	all	DET
ejpam-239	96	54	annihilator	annihilator	NOUN
ejpam-239	96	55	ideals	ideal	NOUN
ejpam-239	96	56	of	of	ADP
ejpam-239	96	57	r	r	NOUN
ejpam-239	96	58	by	by	ADP
ejpam-239	96	59	a	a	DET
ejpam-239	96	60	(	(	PUNCT
ejpam-239	96	61	r	r	NOUN
ejpam-239	96	62	)	)	PUNCT
ejpam-239	96	63	.	.	PUNCT
ejpam-239	97	1	the	the	DET
ejpam-239	97	2	set	set	NOUN
ejpam-239	97	3	a	a	DET
ejpam-239	97	4	(	(	PUNCT
ejpam-239	97	5	r	r	NOUN
ejpam-239	97	6	)	)	PUNCT
ejpam-239	97	7	forms	form	VERB
ejpam-239	97	8	a	a	DET
ejpam-239	97	9	complete	complete	ADJ
ejpam-239	97	10	boolean	boolean	ADJ
ejpam-239	97	11	algebra	algebra	NOUN
ejpam-239	97	12	with	with	ADP
ejpam-239	97	13	bounds	bound	NOUN
ejpam-239	97	14	{	{	PUNCT
ejpam-239	97	15	0},r	0},r	NOUN
ejpam-239	97	16	and	and	CCONJ
ejpam-239	97	17	the	the	DET
ejpam-239	97	18	complement	complement	NOUN
ejpam-239	97	19	of	of	ADP
ejpam-239	97	20	any	any	PRON
ejpam-239	97	21	i	i	PROPN
ejpam-239	97	22	∈a	∈a	X
ejpam-239	97	23	(	(	PUNCT
ejpam-239	97	24	r	r	NOUN
ejpam-239	97	25	)	)	PUNCT
ejpam-239	97	26	is	be	AUX
ejpam-239	97	27	i∗	i∗	NOUN
ejpam-239	97	28	with	with	ADP
ejpam-239	97	29	respect	respect	NOUN
ejpam-239	97	30	to	to	ADP
ejpam-239	97	31	the	the	DET
ejpam-239	97	32	operations	operation	NOUN
ejpam-239	97	33	∧	∧	PROPN
ejpam-239	97	34	and	and	CCONJ
ejpam-239	97	35	∨	∨	NOUN
ejpam-239	97	36	given	give	VERB
ejpam-239	97	37	by	by	ADP
ejpam-239	97	38	i	i	PROPN
ejpam-239	98	1	∧	∧	PROPN
ejpam-239	98	2	j	j	PROPN
ejpam-239	98	3	=	=	SYM
ejpam-239	98	4	i	i	PROPN
ejpam-239	98	5	∩	∩	PROPN
ejpam-239	98	6	j	j	PROPN
ejpam-239	98	7	and	and	CCONJ
ejpam-239	98	8	i	i	PROPN
ejpam-239	98	9	∨	∨	PROPN
ejpam-239	98	10	j	j	PROPN
ejpam-239	99	1	=	=	PRON
ejpam-239	99	2	(	(	PUNCT
ejpam-239	99	3	i∗	i∗	NOUN
ejpam-239	99	4	∩	∩	NOUN
ejpam-239	99	5	j∗)∗.	j∗)∗.	PROPN
ejpam-239	99	6	3	3	X
ejpam-239	99	7	.	.	PUNCT
ejpam-239	99	8	annulets	annulet	NOUN
ejpam-239	99	9	in	in	ADP
ejpam-239	99	10	this	this	DET
ejpam-239	99	11	section	section	NOUN
ejpam-239	99	12	,	,	PUNCT
ejpam-239	99	13	we	we	PRON
ejpam-239	99	14	introduce	introduce	VERB
ejpam-239	99	15	the	the	DET
ejpam-239	99	16	concept	concept	NOUN
ejpam-239	99	17	of	of	ADP
ejpam-239	99	18	annulets	annulet	NOUN
ejpam-239	99	19	in	in	ADP
ejpam-239	99	20	r	r	NOUN
ejpam-239	99	21	and	and	CCONJ
ejpam-239	99	22	study	study	VERB
ejpam-239	99	23	some	some	DET
ejpam-239	99	24	basic	basic	ADJ
ejpam-239	99	25	properties	property	NOUN
ejpam-239	99	26	of	of	ADP
ejpam-239	99	27	these	these	DET
ejpam-239	99	28	annulets	annulet	NOUN
ejpam-239	99	29	.	.	PUNCT
ejpam-239	100	1	we	we	PRON
ejpam-239	100	2	prove	prove	VERB
ejpam-239	100	3	charactarization	charactarization	NOUN
ejpam-239	100	4	theorems	theorem	NOUN
ejpam-239	100	5	of	of	ADP
ejpam-239	100	6	a	a	DET
ejpam-239	100	7	few	few	ADJ
ejpam-239	100	8	algebraic	algebraic	ADJ
ejpam-239	100	9	structures	structure	NOUN
ejpam-239	100	10	with	with	ADP
ejpam-239	100	11	the	the	DET
ejpam-239	100	12	help	help	NOUN
ejpam-239	100	13	of	of	ADP
ejpam-239	100	14	their	their	PRON
ejpam-239	100	15	annulets	annulet	NOUN
ejpam-239	100	16	.	.	PUNCT
ejpam-239	101	1	we	we	PRON
ejpam-239	101	2	begin	begin	VERB
ejpam-239	101	3	with	with	ADP
ejpam-239	101	4	the	the	DET
ejpam-239	101	5	following	follow	VERB
ejpam-239	101	6	definition	definition	NOUN
ejpam-239	101	7	.	.	PUNCT
ejpam-239	102	1	definition	definition	NOUN
ejpam-239	102	2	3.1	3.1	NUM
ejpam-239	102	3	.	.	PUNCT
ejpam-239	103	1	let	let	VERB
ejpam-239	103	2	r	r	PRON
ejpam-239	103	3	be	be	AUX
ejpam-239	103	4	an	an	DET
ejpam-239	103	5	adl	adl	NOUN
ejpam-239	103	6	with	with	ADP
ejpam-239	103	7	0	0	NUM
ejpam-239	103	8	and	and	CCONJ
ejpam-239	103	9	x	x	SYM
ejpam-239	103	10	∈	∈	PROPN
ejpam-239	103	11	r.	r.	PROPN
ejpam-239	103	12	then	then	ADV
ejpam-239	103	13	define	define	VERB
ejpam-239	103	14	the	the	DET
ejpam-239	103	15	annulet	annulet	NOUN
ejpam-239	103	16	(	(	PUNCT
ejpam-239	103	17	x]∗	x]∗	PROPN
ejpam-239	103	18	as	as	SCONJ
ejpam-239	103	19	follows	follow	VERB
ejpam-239	103	20	:	:	PUNCT
ejpam-239	103	21	(	(	PUNCT
ejpam-239	104	1	x]∗	x]∗	PROPN
ejpam-239	104	2	=	=	PRON
ejpam-239	104	3	{	{	PUNCT
ejpam-239	104	4	y	y	PROPN
ejpam-239	104	5	∈	∈	PROPN
ejpam-239	104	6	r	r	NOUN
ejpam-239	104	7	|	|	NOUN
ejpam-239	104	8	x	x	NOUN
ejpam-239	104	9	∧	∧	NOUN
ejpam-239	104	10	y	y	NOUN
ejpam-239	104	11	=	=	NOUN
ejpam-239	104	12	0	0	PUNCT
ejpam-239	104	13	}	}	PUNCT
ejpam-239	104	14	clearly	clearly	ADV
ejpam-239	104	15	(	(	PUNCT
ejpam-239	104	16	x]∗	x]∗	PROPN
ejpam-239	104	17	is	be	AUX
ejpam-239	104	18	an	an	DET
ejpam-239	104	19	ideal	ideal	NOUN
ejpam-239	104	20	in	in	ADP
ejpam-239	104	21	r	r	NOUN
ejpam-239	104	22	and	and	CCONJ
ejpam-239	104	23	hence	hence	ADV
ejpam-239	104	24	an	an	DET
ejpam-239	104	25	annihilator	annihilator	PROPN
ejpam-239	104	26	ideal	ideal	NOUN
ejpam-239	104	27	.	.	PUNCT
ejpam-239	105	1	let	let	VERB
ejpam-239	105	2	us	we	PRON
ejpam-239	105	3	denotea0(r	denotea0(r	VERB
ejpam-239	105	4	)	)	PUNCT
ejpam-239	105	5	=	=	PRON
ejpam-239	105	6	{	{	PUNCT
ejpam-239	105	7	(	(	PUNCT
ejpam-239	105	8	x	x	SYM
ejpam-239	105	9	]	]	X
ejpam-239	105	10	∗	∗	NOUN
ejpam-239	106	1	|	|	NOUN
ejpam-239	106	2	x	x	SYM
ejpam-239	106	3	∈	∈	NOUN
ejpam-239	106	4	r	r	NOUN
ejpam-239	106	5	}	}	PUNCT
ejpam-239	106	6	.	.	PUNCT
ejpam-239	107	1	annulets	annulet	NOUN
ejpam-239	107	2	have	have	VERB
ejpam-239	107	3	many	many	ADJ
ejpam-239	107	4	important	important	ADJ
ejpam-239	107	5	properties	property	NOUN
ejpam-239	107	6	.	.	PUNCT
ejpam-239	108	1	we	we	PRON
ejpam-239	108	2	give	give	VERB
ejpam-239	108	3	some	some	PRON
ejpam-239	108	4	of	of	ADP
ejpam-239	108	5	them	they	PRON
ejpam-239	108	6	in	in	ADP
ejpam-239	108	7	the	the	DET
ejpam-239	108	8	following	follow	VERB
ejpam-239	108	9	lemma	lemma	PROPN
ejpam-239	108	10	which	which	PRON
ejpam-239	108	11	can	can	AUX
ejpam-239	108	12	be	be	AUX
ejpam-239	108	13	proved	prove	VERB
ejpam-239	108	14	directly	directly	ADV
ejpam-239	108	15	.	.	PUNCT
ejpam-239	109	1	lemma	lemma	PROPN
ejpam-239	109	2	3.2	3.2	NUM
ejpam-239	109	3	.	.	PUNCT
ejpam-239	110	1	let	let	VERB
ejpam-239	110	2	r	r	PRON
ejpam-239	110	3	be	be	AUX
ejpam-239	110	4	an	an	DET
ejpam-239	110	5	adl	adl	NOUN
ejpam-239	110	6	with	with	ADP
ejpam-239	110	7	0	0	NUM
ejpam-239	110	8	and	and	CCONJ
ejpam-239	110	9	x	x	NOUN
ejpam-239	110	10	,	,	PUNCT
ejpam-239	110	11	y	y	PROPN
ejpam-239	110	12	∈	∈	PROPN
ejpam-239	110	13	r.	r.	PROPN
ejpam-239	110	14	then	then	ADV
ejpam-239	110	15	we	we	PRON
ejpam-239	110	16	have	have	VERB
ejpam-239	110	17	:	:	PUNCT
ejpam-239	110	18	1	1	NUM
ejpam-239	110	19	.	.	NUM
ejpam-239	110	20	x	x	SYM
ejpam-239	110	21	≤	≤	NUM
ejpam-239	110	22	y	y	PROPN
ejpam-239	110	23	⇒	⇒	NOUN
ejpam-239	110	24	(	(	PUNCT
ejpam-239	110	25	y]∗	y]∗	PROPN
ejpam-239	110	26	⊆	⊆	NUM
ejpam-239	110	27	(	(	PUNCT
ejpam-239	110	28	x]∗	x]∗	PROPN
ejpam-239	110	29	2	2	NUM
ejpam-239	110	30	.	.	PUNCT
ejpam-239	111	1	(	(	PUNCT
ejpam-239	111	2	x	x	PUNCT
ejpam-239	111	3	∧	∧	NOUN
ejpam-239	111	4	y]∗	y]∗	PROPN
ejpam-239	111	5	=	=	PUNCT
ejpam-239	111	6	(	(	PUNCT
ejpam-239	111	7	y	y	PROPN
ejpam-239	111	8	∧	∧	PROPN
ejpam-239	111	9	x]∗	x]∗	PROPN
ejpam-239	111	10	3	3	X
ejpam-239	111	11	.	.	PUNCT
ejpam-239	112	1	(	(	PUNCT
ejpam-239	112	2	x	x	X
ejpam-239	112	3	∨	∨	ADP
ejpam-239	112	4	y]∗	y]∗	PROPN
ejpam-239	112	5	=	=	SYM
ejpam-239	112	6	(	(	PUNCT
ejpam-239	112	7	y	y	PROPN
ejpam-239	112	8	∨	∨	NUM
ejpam-239	112	9	x]∗	x]∗	PROPN
ejpam-239	112	10	4	4	NUM
ejpam-239	112	11	.	.	PUNCT
ejpam-239	113	1	(	(	PUNCT
ejpam-239	113	2	x	x	X
ejpam-239	113	3	∨	∨	ADP
ejpam-239	113	4	y]∗	y]∗	PROPN
ejpam-239	113	5	=	=	SYM
ejpam-239	113	6	(	(	PUNCT
ejpam-239	113	7	x]∗	x]∗	PROPN
ejpam-239	113	8	∩	∩	NOUN
ejpam-239	113	9	(	(	PUNCT
ejpam-239	113	10	y]∗	y]∗	PROPN
ejpam-239	113	11	5	5	NUM
ejpam-239	113	12	.	.	PUNCT
ejpam-239	114	1	(	(	PUNCT
ejpam-239	114	2	x]∗	x]∗	PROPN
ejpam-239	114	3	∨	∨	PROPN
ejpam-239	114	4	(	(	PUNCT
ejpam-239	114	5	y]∗	y]∗	PROPN
ejpam-239	114	6	⊆	⊆	NUM
ejpam-239	114	7	(	(	PUNCT
ejpam-239	114	8	x	x	SYM
ejpam-239	114	9	∧	∧	PROPN
ejpam-239	114	10	y]∗.	y]∗.	PROPN
ejpam-239	114	11	note	note	VERB
ejpam-239	114	12	:	:	PUNCT
ejpam-239	114	13	since	since	SCONJ
ejpam-239	114	14	each	each	DET
ejpam-239	114	15	annulet	annulet	NOUN
ejpam-239	114	16	is	be	AUX
ejpam-239	114	17	an	an	DET
ejpam-239	114	18	annihilator	annihilator	PROPN
ejpam-239	114	19	ideal	ideal	NOUN
ejpam-239	114	20	,	,	PUNCT
ejpam-239	114	21	we	we	PRON
ejpam-239	114	22	can	can	AUX
ejpam-239	114	23	have	have	VERB
ejpam-239	114	24	the	the	DET
ejpam-239	114	25	following	following	NOUN
ejpam-239	114	26	:	:	PUNCT
ejpam-239	114	27	(	(	PUNCT
ejpam-239	114	28	x]∗∨(y]∗	x]∗∨(y]∗	PROPN
ejpam-239	114	29	=	=	SYM
ejpam-239	114	30	�	�	PROPN
ejpam-239	114	31	(	(	PUNCT
ejpam-239	114	32	x]∗∗	x]∗∗	PROPN
ejpam-239	114	33	∩	∩	NOUN
ejpam-239	114	34	(	(	PUNCT
ejpam-239	114	35	y]∗∗	y]∗∗	NOUN
ejpam-239	114	36	�	�	NOUN
ejpam-239	114	37	∗	∗	NOUN
ejpam-239	114	38	=	=	SYM
ejpam-239	114	39	�	�	PROPN
ejpam-239	114	40	(	(	PUNCT
ejpam-239	114	41	x	x	PROPN
ejpam-239	114	42	∧	∧	PROPN
ejpam-239	114	43	y]∗∗	y]∗∗	NOUN
ejpam-239	114	44	�	�	NOUN
ejpam-239	114	45	∗	∗	NOUN
ejpam-239	114	46	=	=	SYM
ejpam-239	114	47	(	(	PUNCT
ejpam-239	114	48	x	x	PART
ejpam-239	114	49	∧	∧	PROPN
ejpam-239	114	50	y]∗	y]∗	PROPN
ejpam-239	114	51	g.	g.	PROPN
ejpam-239	114	52	c.	c.	PROPN
ejpam-239	114	53	rao	rao	PROPN
ejpam-239	114	54	and	and	CCONJ
ejpam-239	114	55	m.	m.	PROPN
ejpam-239	114	56	sambasiva	sambasiva	PROPN
ejpam-239	114	57	rao	rao	PROPN
ejpam-239	114	58	/	/	SYM
ejpam-239	114	59	eur	eur	PROPN
ejpam-239	114	60	.	.	PUNCT
ejpam-239	115	1	j.	j.	PROPN
ejpam-239	115	2	pure	pure	PROPN
ejpam-239	115	3	appl	appl	PROPN
ejpam-239	115	4	.	.	PROPN
ejpam-239	115	5	math	math	PROPN
ejpam-239	115	6	,	,	PUNCT
ejpam-239	115	7	2	2	NUM
ejpam-239	115	8	(	(	PUNCT
ejpam-239	115	9	2009	2009	NUM
ejpam-239	115	10	)	)	PUNCT
ejpam-239	115	11	,	,	PUNCT
ejpam-239	115	12	(	(	PUNCT
ejpam-239	115	13	58	58	NUM
ejpam-239	115	14	-	-	SYM
ejpam-239	115	15	72	72	NUM
ejpam-239	115	16	)	)	PUNCT
ejpam-239	115	17	63	63	NUM
ejpam-239	115	18	(	(	PUNCT
ejpam-239	115	19	x]∗	x]∗	PROPN
ejpam-239	115	20	∧	∧	PROPN
ejpam-239	115	21	(	(	PUNCT
ejpam-239	115	22	y]∗	y]∗	PROPN
ejpam-239	115	23	=	=	SYM
ejpam-239	115	24	(	(	PUNCT
ejpam-239	115	25	x]∗	x]∗	PROPN
ejpam-239	115	26	∩	∩	NOUN
ejpam-239	115	27	(	(	PUNCT
ejpam-239	115	28	y]∗	y]∗	PROPN
ejpam-239	115	29	=	=	SYM
ejpam-239	115	30	(	(	PUNCT
ejpam-239	115	31	x	x	PROPN
ejpam-239	115	32	∨	∨	NUM
ejpam-239	115	33	y]∗.	y]∗.	NUM
ejpam-239	115	34	now	now	ADV
ejpam-239	115	35	we	we	PRON
ejpam-239	115	36	prove	prove	VERB
ejpam-239	115	37	in	in	ADP
ejpam-239	115	38	the	the	DET
ejpam-239	115	39	following	following	NOUN
ejpam-239	115	40	theorem	theorem	NOUN
ejpam-239	115	41	that	that	SCONJ
ejpam-239	115	42	the	the	DET
ejpam-239	115	43	set	set	PROPN
ejpam-239	115	44	a0(r	a0(r	NOUN
ejpam-239	115	45	)	)	PUNCT
ejpam-239	115	46	of	of	ADP
ejpam-239	115	47	all	all	DET
ejpam-239	115	48	annulets	annulet	NOUN
ejpam-239	115	49	of	of	ADP
ejpam-239	115	50	an	an	DET
ejpam-239	115	51	adl	adl	NOUN
ejpam-239	115	52	r	r	NOUN
ejpam-239	115	53	forms	form	VERB
ejpam-239	115	54	a	a	DET
ejpam-239	115	55	distributive	distributive	ADJ
ejpam-239	115	56	lattice	lattice	NOUN
ejpam-239	115	57	.	.	PUNCT
ejpam-239	116	1	theorem	theorem	VERB
ejpam-239	116	2	3.3	3.3	NUM
ejpam-239	116	3	.	.	PUNCT
ejpam-239	117	1	let	let	VERB
ejpam-239	117	2	r	r	PRON
ejpam-239	117	3	be	be	AUX
ejpam-239	117	4	an	an	DET
ejpam-239	117	5	adl	adl	NOUN
ejpam-239	117	6	with	with	ADP
ejpam-239	117	7	0	0	NUM
ejpam-239	117	8	.	.	PUNCT
ejpam-239	118	1	then	then	ADV
ejpam-239	118	2	(	(	PUNCT
ejpam-239	118	3	a0(r),∩,∨	a0(r),∩,∨	NOUN
ejpam-239	118	4	)	)	PUNCT
ejpam-239	118	5	is	be	AUX
ejpam-239	118	6	a	a	DET
ejpam-239	118	7	distributive	distributive	ADJ
ejpam-239	118	8	lattice	lattice	NOUN
ejpam-239	118	9	and	and	CCONJ
ejpam-239	118	10	a	a	DET
ejpam-239	118	11	sublattice	sublattice	NOUN
ejpam-239	118	12	of	of	ADP
ejpam-239	118	13	the	the	DET
ejpam-239	118	14	boolean	boolean	ADJ
ejpam-239	118	15	algebra	algebra	NOUN
ejpam-239	118	16	a	a	DET
ejpam-239	118	17	(	(	PUNCT
ejpam-239	118	18	r),∩,∨,∗	r),∩,∨,∗	PROPN
ejpam-239	118	19	,	,	PUNCT
ejpam-239	118	20	(	(	PUNCT
ejpam-239	118	21	0],r	0],r	X
ejpam-239	118	22	�	�	PROPN
ejpam-239	118	23	of	of	ADP
ejpam-239	118	24	annihilator	annihilator	PROPN
ejpam-239	118	25	ideals	ideal	NOUN
ejpam-239	118	26	of	of	ADP
ejpam-239	118	27	r.	r.	PROPN
ejpam-239	118	28	a0(r	a0(r	PROPN
ejpam-239	118	29	)	)	PUNCT
ejpam-239	118	30	has	have	VERB
ejpam-239	118	31	the	the	DET
ejpam-239	118	32	same	same	ADJ
ejpam-239	118	33	greatest	great	ADJ
ejpam-239	118	34	element	element	NOUN
ejpam-239	118	35	r	r	NOUN
ejpam-239	118	36	=	=	PUNCT
ejpam-239	118	37	(	(	PUNCT
ejpam-239	118	38	0]∗	0]∗	PROPN
ejpam-239	118	39	as	as	ADP
ejpam-239	118	40	a	a	DET
ejpam-239	118	41	(	(	PUNCT
ejpam-239	118	42	r	r	NOUN
ejpam-239	118	43	)	)	PUNCT
ejpam-239	118	44	while	while	SCONJ
ejpam-239	118	45	a0(r	a0(r	PROPN
ejpam-239	118	46	)	)	PUNCT
ejpam-239	118	47	has	have	VERB
ejpam-239	118	48	the	the	DET
ejpam-239	118	49	smallest	small	ADJ
ejpam-239	118	50	element	element	NOUN
ejpam-239	118	51	iff	iff	NOUN
ejpam-239	118	52	r	r	NOUN
ejpam-239	118	53	possesses	possess	VERB
ejpam-239	118	54	a	a	DET
ejpam-239	118	55	dense	dense	ADJ
ejpam-239	118	56	element	element	NOUN
ejpam-239	118	57	.	.	PUNCT
ejpam-239	119	1	proof	proof	NOUN
ejpam-239	119	2	:	:	PUNCT
ejpam-239	119	3	let	let	VERB
ejpam-239	119	4	(	(	PUNCT
ejpam-239	119	5	x]∗	x]∗	PROPN
ejpam-239	119	6	,	,	PUNCT
ejpam-239	119	7	(	(	PUNCT
ejpam-239	119	8	y]∗	y]∗	PROPN
ejpam-239	119	9	∈a0(r	∈a0(r	NOUN
ejpam-239	119	10	)	)	PUNCT
ejpam-239	119	11	,	,	PUNCT
ejpam-239	119	12	where	where	SCONJ
ejpam-239	119	13	x	x	X
ejpam-239	119	14	,	,	PUNCT
ejpam-239	119	15	y	y	PROPN
ejpam-239	119	16	∈	∈	PROPN
ejpam-239	119	17	r.	r.	PROPN
ejpam-239	119	18	then	then	ADV
ejpam-239	119	19	1	1	X
ejpam-239	119	20	.	.	PUNCT
ejpam-239	120	1	(	(	PUNCT
ejpam-239	120	2	x]∗	x]∗	PROPN
ejpam-239	120	3	∧	∧	PROPN
ejpam-239	120	4	(	(	PUNCT
ejpam-239	120	5	y]∗	y]∗	PROPN
ejpam-239	120	6	=	=	SYM
ejpam-239	120	7	(	(	PUNCT
ejpam-239	120	8	x]∗	x]∗	PROPN
ejpam-239	120	9	∩	∩	NOUN
ejpam-239	120	10	(	(	PUNCT
ejpam-239	120	11	y]∗	y]∗	PROPN
ejpam-239	120	12	=	=	SYM
ejpam-239	120	13	(	(	PUNCT
ejpam-239	120	14	x	x	X
ejpam-239	120	15	∨	∨	ADP
ejpam-239	120	16	y]∗	y]∗	PROPN
ejpam-239	120	17	∈a0(r	∈a0(r	NOUN
ejpam-239	120	18	)	)	PUNCT
ejpam-239	120	19	and	and	CCONJ
ejpam-239	120	20	2	2	X
ejpam-239	120	21	.	.	PUNCT
ejpam-239	121	1	(	(	PUNCT
ejpam-239	121	2	x]∗∨(y]∗	x]∗∨(y]∗	X
ejpam-239	121	3	=	=	SYM
ejpam-239	121	4	(	(	PUNCT
ejpam-239	121	5	x	x	PART
ejpam-239	121	6	∧	∧	PROPN
ejpam-239	121	7	y]∗	y]∗	PROPN
ejpam-239	121	8	∈a0(r	∈a0(r	NOUN
ejpam-239	121	9	)	)	PUNCT
ejpam-239	121	10	.	.	PUNCT
ejpam-239	122	1	hence	hence	ADV
ejpam-239	122	2	a0(r	a0(r	PROPN
ejpam-239	122	3	)	)	PUNCT
ejpam-239	122	4	is	be	AUX
ejpam-239	122	5	a	a	DET
ejpam-239	122	6	sublattice	sublattice	NOUN
ejpam-239	122	7	of	of	ADP
ejpam-239	122	8	a	a	DET
ejpam-239	122	9	(	(	PUNCT
ejpam-239	122	10	r	r	NOUN
ejpam-239	122	11	)	)	PUNCT
ejpam-239	122	12	.	.	PUNCT
ejpam-239	123	1	since	since	SCONJ
ejpam-239	123	2	a	a	DET
ejpam-239	123	3	(	(	PUNCT
ejpam-239	123	4	r	r	NOUN
ejpam-239	123	5	)	)	PUNCT
ejpam-239	123	6	is	be	AUX
ejpam-239	123	7	distributive	distributive	ADJ
ejpam-239	123	8	,	,	PUNCT
ejpam-239	123	9	we	we	PRON
ejpam-239	123	10	have	have	VERB
ejpam-239	123	11	that	that	PRON
ejpam-239	123	12	a0(r	a0(r	NOUN
ejpam-239	123	13	)	)	PUNCT
ejpam-239	123	14	is	be	AUX
ejpam-239	123	15	also	also	ADV
ejpam-239	123	16	distributive	distributive	ADJ
ejpam-239	123	17	.	.	PUNCT
ejpam-239	124	1	clearly	clearly	ADV
ejpam-239	124	2	(	(	PUNCT
ejpam-239	124	3	0]∗	0]∗	PRON
ejpam-239	124	4	is	be	AUX
ejpam-239	124	5	the	the	DET
ejpam-239	124	6	greatest	great	ADJ
ejpam-239	124	7	element	element	NOUN
ejpam-239	124	8	of	of	ADP
ejpam-239	124	9	a	a	DET
ejpam-239	124	10	(	(	PUNCT
ejpam-239	124	11	r	r	NOUN
ejpam-239	124	12	)	)	PUNCT
ejpam-239	124	13	.	.	PUNCT
ejpam-239	125	1	now	now	ADV
ejpam-239	125	2	for	for	ADP
ejpam-239	125	3	any	any	DET
ejpam-239	125	4	(	(	PUNCT
ejpam-239	125	5	x]∗	x]∗	PROPN
ejpam-239	125	6	∈	∈	PROPN
ejpam-239	125	7	a0(r	a0(r	PROPN
ejpam-239	125	8	)	)	PUNCT
ejpam-239	125	9	,	,	PUNCT
ejpam-239	125	10	we	we	PRON
ejpam-239	125	11	get	get	VERB
ejpam-239	125	12	(	(	PUNCT
ejpam-239	125	13	x]∗∩(0]∗	x]∗∩(0]∗	PROPN
ejpam-239	125	14	=	=	SYM
ejpam-239	126	1	(	(	PUNCT
ejpam-239	126	2	x∨0]∗	x∨0]∗	PROPN
ejpam-239	126	3	=	=	PRON
ejpam-239	126	4	(	(	PUNCT
ejpam-239	126	5	x]∗	x]∗	PROPN
ejpam-239	126	6	and	and	CCONJ
ejpam-239	126	7	(	(	PUNCT
ejpam-239	126	8	x]∗∨(0]∗	x]∗∨(0]∗	PROPN
ejpam-239	126	9	=	=	PRON
ejpam-239	126	10	(	(	PUNCT
ejpam-239	126	11	x∧0]∗	x∧0]∗	PROPN
ejpam-239	126	12	=	=	SYM
ejpam-239	127	1	(	(	PUNCT
ejpam-239	127	2	0]∗.	0]∗.	NUM
ejpam-239	127	3	it	it	PRON
ejpam-239	127	4	shows	show	VERB
ejpam-239	127	5	that	that	SCONJ
ejpam-239	127	6	(	(	PUNCT
ejpam-239	127	7	0]∗	0]∗	PRON
ejpam-239	127	8	is	be	AUX
ejpam-239	127	9	the	the	DET
ejpam-239	127	10	greatest	great	ADJ
ejpam-239	127	11	element	element	NOUN
ejpam-239	127	12	ina0(r	ina0(r	NOUN
ejpam-239	127	13	)	)	PUNCT
ejpam-239	127	14	.	.	PUNCT
ejpam-239	128	1	now	now	ADV
ejpam-239	128	2	,	,	PUNCT
ejpam-239	128	3	it	it	PRON
ejpam-239	128	4	remains	remain	VERB
ejpam-239	128	5	to	to	PART
ejpam-239	128	6	prove	prove	VERB
ejpam-239	128	7	the	the	DET
ejpam-239	128	8	final	final	ADJ
ejpam-239	128	9	condition	condition	NOUN
ejpam-239	128	10	of	of	ADP
ejpam-239	128	11	the	the	DET
ejpam-239	128	12	theorem	theorem	NOUN
ejpam-239	128	13	.	.	PROPN
ejpam-239	128	14	assumea0(r	assumea0(r	NOUN
ejpam-239	128	15	)	)	PUNCT
ejpam-239	128	16	has	have	VERB
ejpam-239	128	17	the	the	DET
ejpam-239	128	18	smallest	small	ADJ
ejpam-239	128	19	element	element	NOUN
ejpam-239	128	20	,	,	PUNCT
ejpam-239	128	21	say	say	VERB
ejpam-239	128	22	(	(	PUNCT
ejpam-239	128	23	d]∗	d]∗	INTJ
ejpam-239	128	24	where	where	SCONJ
ejpam-239	128	25	d	d	PROPN
ejpam-239	128	26	∈	∈	PROPN
ejpam-239	128	27	r.	r.	PROPN
ejpam-239	128	28	suppose	suppose	VERB
ejpam-239	128	29	x	x	X
ejpam-239	128	30	∈	∈	PROPN
ejpam-239	128	31	(	(	PUNCT
ejpam-239	128	32	d]∗.	d]∗.	NOUN
ejpam-239	128	33	then	then	ADV
ejpam-239	128	34	x	x	PART
ejpam-239	128	35	∧	∧	PROPN
ejpam-239	128	36	d	d	NOUN
ejpam-239	128	37	=	=	NOUN
ejpam-239	128	38	0	0	PROPN
ejpam-239	128	39	.	.	PUNCT
ejpam-239	129	1	since	since	SCONJ
ejpam-239	129	2	(	(	PUNCT
ejpam-239	129	3	d]∗	d]∗	PROPN
ejpam-239	129	4	is	be	AUX
ejpam-239	129	5	the	the	DET
ejpam-239	129	6	least	least	ADJ
ejpam-239	129	7	element	element	NOUN
ejpam-239	129	8	,	,	PUNCT
ejpam-239	129	9	we	we	PRON
ejpam-239	129	10	get	get	VERB
ejpam-239	129	11	(	(	PUNCT
ejpam-239	129	12	x]∗	x]∗	PROPN
ejpam-239	129	13	=	=	PROPN
ejpam-239	129	14	(	(	PUNCT
ejpam-239	129	15	x]∗∨(d]∗	x]∗∨(d]∗	PROPN
ejpam-239	129	16	=	=	SYM
ejpam-239	129	17	(	(	PUNCT
ejpam-239	129	18	x	x	PUNCT
ejpam-239	129	19	∧	∧	NOUN
ejpam-239	129	20	d]∗	d]∗	NOUN
ejpam-239	129	21	=	=	PUNCT
ejpam-239	129	22	(	(	PUNCT
ejpam-239	129	23	0]∗	0]∗	X
ejpam-239	129	24	=	=	PUNCT
ejpam-239	129	25	r.	r.	PROPN
ejpam-239	129	26	hence	hence	ADV
ejpam-239	129	27	x	x	PUNCT
ejpam-239	130	1	=	=	SYM
ejpam-239	130	2	0	0	NUM
ejpam-239	130	3	.	.	PUNCT
ejpam-239	131	1	thus	thus	ADV
ejpam-239	131	2	(	(	PUNCT
ejpam-239	131	3	d]∗	d]∗	NOUN
ejpam-239	131	4	=	=	SYM
ejpam-239	131	5	(	(	PUNCT
ejpam-239	131	6	0	0	NUM
ejpam-239	131	7	]	]	PUNCT
ejpam-239	131	8	.	.	PUNCT
ejpam-239	132	1	therefore	therefore	ADV
ejpam-239	132	2	d	d	X
ejpam-239	132	3	is	be	AUX
ejpam-239	132	4	a	a	DET
ejpam-239	132	5	dense	dense	ADJ
ejpam-239	132	6	element	element	NOUN
ejpam-239	132	7	in	in	ADP
ejpam-239	132	8	r.	r.	PROPN
ejpam-239	132	9	conversely	conversely	ADV
ejpam-239	132	10	,	,	PUNCT
ejpam-239	132	11	suppose	suppose	VERB
ejpam-239	132	12	that	that	SCONJ
ejpam-239	132	13	r	r	NOUN
ejpam-239	132	14	possesses	possess	VERB
ejpam-239	132	15	a	a	DET
ejpam-239	132	16	dense	dense	ADJ
ejpam-239	132	17	element	element	NOUN
ejpam-239	132	18	,	,	PUNCT
ejpam-239	132	19	say	say	VERB
ejpam-239	132	20	d	d	INTJ
ejpam-239	132	21	.	.	PUNCT
ejpam-239	133	1	so	so	ADV
ejpam-239	133	2	(	(	PUNCT
ejpam-239	133	3	d]∗	d]∗	PROPN
ejpam-239	133	4	=	=	SYM
ejpam-239	133	5	(	(	PUNCT
ejpam-239	133	6	0	0	NUM
ejpam-239	133	7	]	]	PUNCT
ejpam-239	133	8	.	.	PUNCT
ejpam-239	134	1	clearly	clearly	ADV
ejpam-239	134	2	(	(	PUNCT
ejpam-239	134	3	d]∗	d]∗	PROPN
ejpam-239	134	4	∈	∈	PROPN
ejpam-239	134	5	a0(r	a0(r	PROPN
ejpam-239	134	6	)	)	PUNCT
ejpam-239	134	7	.	.	PUNCT
ejpam-239	135	1	now	now	ADV
ejpam-239	135	2	for	for	SCONJ
ejpam-239	135	3	any	any	DET
ejpam-239	135	4	x	x	SYM
ejpam-239	135	5	∈	∈	PROPN
ejpam-239	135	6	r	r	NOUN
ejpam-239	135	7	,	,	PUNCT
ejpam-239	135	8	consider	consider	VERB
ejpam-239	135	9	(	(	PUNCT
ejpam-239	135	10	x]∗	x]∗	PROPN
ejpam-239	135	11	∩	∩	PROPN
ejpam-239	135	12	(	(	PUNCT
ejpam-239	135	13	d]∗	d]∗	PROPN
ejpam-239	135	14	=	=	SYM
ejpam-239	135	15	(	(	PUNCT
ejpam-239	135	16	x]∗	x]∗	PROPN
ejpam-239	135	17	∩	∩	PROPN
ejpam-239	135	18	(	(	PUNCT
ejpam-239	135	19	0	0	NUM
ejpam-239	135	20	]	]	X
ejpam-239	135	21	=	=	SYM
ejpam-239	135	22	(	(	PUNCT
ejpam-239	135	23	0	0	NUM
ejpam-239	135	24	]	]	PUNCT
ejpam-239	135	25	.	.	PUNCT
ejpam-239	136	1	also	also	ADV
ejpam-239	136	2	(	(	PUNCT
ejpam-239	136	3	x]∗∨(d]∗	x]∗∨(d]∗	PROPN
ejpam-239	136	4	=	=	PUNCT
ejpam-239	137	1	[	[	X
ejpam-239	137	2	(	(	PUNCT
ejpam-239	137	3	x]∗∗	x]∗∗	NOUN
ejpam-239	137	4	∩	∩	NOUN
ejpam-239	137	5	(	(	PUNCT
ejpam-239	137	6	d]∗∗]∗	d]∗∗]∗	NOUN
ejpam-239	137	7	=	=	SYM
ejpam-239	138	1	[	[	X
ejpam-239	138	2	(	(	PUNCT
ejpam-239	138	3	x]∗∗	x]∗∗	NOUN
ejpam-239	138	4	∩	∩	NOUN
ejpam-239	138	5	(	(	PUNCT
ejpam-239	138	6	0]∗]∗	0]∗]∗	X
ejpam-239	138	7	=	=	SYM
ejpam-239	139	1	[	[	X
ejpam-239	139	2	(	(	PUNCT
ejpam-239	139	3	x]∗∗	x]∗∗	NOUN
ejpam-239	139	4	∩	∩	ADJ
ejpam-239	139	5	r	r	NOUN
ejpam-239	139	6	]	]	X
ejpam-239	139	7	∗	∗	NOUN
ejpam-239	139	8	=	=	SYM
ejpam-239	139	9	(	(	PUNCT
ejpam-239	139	10	x]∗∗∗	x]∗∗∗	PROPN
ejpam-239	139	11	=	=	SYM
ejpam-239	139	12	(	(	PUNCT
ejpam-239	139	13	x]∗.	x]∗.	PROPN
ejpam-239	139	14	hence	hence	ADV
ejpam-239	139	15	(	(	PUNCT
ejpam-239	139	16	d]∗	d]∗	PROPN
ejpam-239	139	17	is	be	AUX
ejpam-239	139	18	the	the	DET
ejpam-239	139	19	smallest	small	ADJ
ejpam-239	139	20	element	element	NOUN
ejpam-239	139	21	ina0(r	ina0(r	NOUN
ejpam-239	139	22	)	)	PUNCT
ejpam-239	139	23	.	.	PUNCT
ejpam-239	140	1	�	�	PROPN
ejpam-239	141	1	the	the	DET
ejpam-239	141	2	following	follow	VERB
ejpam-239	141	3	definition	definition	NOUN
ejpam-239	141	4	of	of	ADP
ejpam-239	141	5	a	a	DET
ejpam-239	141	6	normal	normal	ADJ
ejpam-239	141	7	adl	adl	NOUN
ejpam-239	141	8	is	be	AUX
ejpam-239	141	9	taken	take	VERB
ejpam-239	141	10	from	from	ADP
ejpam-239	141	11	[	[	X
ejpam-239	141	12	7	7	NUM
ejpam-239	141	13	]	]	PUNCT
ejpam-239	141	14	.	.	PUNCT
ejpam-239	142	1	definition	definition	NOUN
ejpam-239	142	2	3.4	3.4	NUM
ejpam-239	142	3	.	.	PUNCT
ejpam-239	143	1	an	an	DET
ejpam-239	143	2	adl	adl	PROPN
ejpam-239	143	3	r	r	NOUN
ejpam-239	143	4	with	with	ADP
ejpam-239	143	5	0	0	NUM
ejpam-239	143	6	is	be	AUX
ejpam-239	143	7	called	call	VERB
ejpam-239	143	8	normal	normal	ADJ
ejpam-239	143	9	adl	adl	PROPN
ejpam-239	143	10	iff	iff	PROPN
ejpam-239	143	11	for	for	ADP
ejpam-239	143	12	all	all	DET
ejpam-239	143	13	x	x	SYM
ejpam-239	143	14	,	,	PUNCT
ejpam-239	143	15	y	y	PROPN
ejpam-239	143	16	∈	∈	PROPN
ejpam-239	143	17	r	r	NOUN
ejpam-239	143	18	(	(	PUNCT
ejpam-239	143	19	x]∗	x]∗	PROPN
ejpam-239	143	20	∨	∨	PROPN
ejpam-239	143	21	(	(	PUNCT
ejpam-239	143	22	y]∗	y]∗	PROPN
ejpam-239	143	23	=	=	SYM
ejpam-239	143	24	(	(	PUNCT
ejpam-239	143	25	x	x	PUNCT
ejpam-239	143	26	∧	∧	NOUN
ejpam-239	143	27	y]∗.	y]∗.	PROPN
ejpam-239	143	28	swamy.u.m	swamy.u.m	NOUN
ejpam-239	143	29	.	.	PUNCT
ejpam-239	143	30	,	,	PUNCT
ejpam-239	143	31	rao.g.c	rao.g.c	PROPN
ejpam-239	143	32	.	.	PUNCT
ejpam-239	143	33	,	,	PUNCT
ejpam-239	143	34	nanaji	nanaji	ADJ
ejpam-239	143	35	rao.g.[9	rao.g.[9	NOUN
ejpam-239	143	36	]	]	PUNCT
ejpam-239	143	37	and	and	CCONJ
ejpam-239	144	1	[	[	X
ejpam-239	144	2	10	10	NUM
ejpam-239	144	3	]	]	PUNCT
ejpam-239	144	4	,	,	PUNCT
ejpam-239	144	5	have	have	AUX
ejpam-239	144	6	studied	study	VERB
ejpam-239	144	7	the	the	DET
ejpam-239	144	8	properties	property	NOUN
ejpam-239	144	9	of	of	ADP
ejpam-239	144	10	a	a	DET
ejpam-239	144	11	psuedocomplemented	psuedocomplemente	VERB
ejpam-239	144	12	adl	adl	NOUN
ejpam-239	144	13	and	and	CCONJ
ejpam-239	144	14	later	later	ADV
ejpam-239	144	15	introduced	introduce	VERB
ejpam-239	144	16	the	the	DET
ejpam-239	144	17	concept	concept	NOUN
ejpam-239	144	18	of	of	ADP
ejpam-239	144	19	stone	stone	NOUN
ejpam-239	144	20	adl	adl	PROPN
ejpam-239	145	1	[	[	X
ejpam-239	145	2	10	10	NUM
ejpam-239	145	3	]	]	PUNCT
ejpam-239	145	4	as	as	ADP
ejpam-239	145	5	a	a	DET
ejpam-239	145	6	psuedo	psuedo	ADV
ejpam-239	145	7	-	-	PUNCT
ejpam-239	145	8	complemented	complement	VERB
ejpam-239	145	9	g.	g.	PROPN
ejpam-239	145	10	c.	c.	PROPN
ejpam-239	145	11	rao	rao	PROPN
ejpam-239	145	12	and	and	CCONJ
ejpam-239	145	13	m.	m.	PROPN
ejpam-239	145	14	sambasiva	sambasiva	PROPN
ejpam-239	145	15	rao	rao	PROPN
ejpam-239	145	16	/	/	SYM
ejpam-239	145	17	eur	eur	PROPN
ejpam-239	145	18	.	.	PUNCT
ejpam-239	146	1	j.	j.	PROPN
ejpam-239	146	2	pure	pure	PROPN
ejpam-239	146	3	appl	appl	PROPN
ejpam-239	146	4	.	.	PROPN
ejpam-239	146	5	math	math	PROPN
ejpam-239	146	6	,	,	PUNCT
ejpam-239	146	7	2	2	NUM
ejpam-239	146	8	(	(	PUNCT
ejpam-239	146	9	2009	2009	NUM
ejpam-239	146	10	)	)	PUNCT
ejpam-239	146	11	,	,	PUNCT
ejpam-239	146	12	(	(	PUNCT
ejpam-239	146	13	58	58	NUM
ejpam-239	146	14	-	-	SYM
ejpam-239	146	15	72	72	NUM
ejpam-239	146	16	)	)	PUNCT
ejpam-239	146	17	64	64	NUM
ejpam-239	146	18	adl	adl	NOUN
ejpam-239	146	19	r	r	NOUN
ejpam-239	146	20	with	with	ADP
ejpam-239	146	21	0	0	NUM
ejpam-239	146	22	,	,	PUNCT
ejpam-239	146	23	in	in	ADP
ejpam-239	146	24	which	which	PRON
ejpam-239	146	25	x∗∨x∗∗	x∗∨x∗∗	PUNCT
ejpam-239	146	26	=	=	SYM
ejpam-239	146	27	0∗	0∗	PROPN
ejpam-239	146	28	,	,	PUNCT
ejpam-239	146	29	for	for	ADP
ejpam-239	146	30	all	all	DET
ejpam-239	146	31	x	x	SYM
ejpam-239	146	32	∈	∈	PROPN
ejpam-239	146	33	r.	r.	NOUN
ejpam-239	146	34	now	now	ADV
ejpam-239	146	35	we	we	PRON
ejpam-239	146	36	give	give	VERB
ejpam-239	146	37	the	the	DET
ejpam-239	146	38	definition	definition	NOUN
ejpam-239	146	39	of	of	ADP
ejpam-239	146	40	a	a	DET
ejpam-239	146	41	generalized	generalized	ADJ
ejpam-239	146	42	stone	stone	NOUN
ejpam-239	146	43	adl	adl	NOUN
ejpam-239	146	44	in	in	ADP
ejpam-239	146	45	the	the	DET
ejpam-239	146	46	following	following	NOUN
ejpam-239	146	47	.	.	PUNCT
ejpam-239	147	1	definition	definition	NOUN
ejpam-239	147	2	3.5	3.5	NUM
ejpam-239	147	3	.	.	PUNCT
ejpam-239	148	1	an	an	DET
ejpam-239	148	2	adl	adl	PROPN
ejpam-239	148	3	r	r	NOUN
ejpam-239	148	4	with	with	ADP
ejpam-239	148	5	0	0	NUM
ejpam-239	148	6	is	be	AUX
ejpam-239	148	7	called	call	VERB
ejpam-239	148	8	a	a	DET
ejpam-239	148	9	generalized	generalize	VERB
ejpam-239	148	10	stone	stone	NOUN
ejpam-239	148	11	adl	adl	PROPN
ejpam-239	148	12	iff	iff	PROPN
ejpam-239	148	13	(	(	PUNCT
ejpam-239	148	14	x]∗	x]∗	PROPN
ejpam-239	148	15	∨	∨	PROPN
ejpam-239	148	16	(	(	PUNCT
ejpam-239	148	17	x]∗∗	x]∗∗	PROPN
ejpam-239	148	18	=	=	SYM
ejpam-239	148	19	r	r	NOUN
ejpam-239	148	20	for	for	ADP
ejpam-239	148	21	each	each	DET
ejpam-239	148	22	x	x	PROPN
ejpam-239	148	23	∈	∈	PROPN
ejpam-239	148	24	r.	r.	PROPN
ejpam-239	148	25	example	example	NOUN
ejpam-239	148	26	3.6	3.6	NUM
ejpam-239	148	27	.	.	PUNCT
ejpam-239	149	1	let	let	VERB
ejpam-239	149	2	a	a	PRON
ejpam-239	149	3	=	=	PUNCT
ejpam-239	149	4	{	{	PUNCT
ejpam-239	149	5	0	0	NUM
ejpam-239	149	6	,	,	PUNCT
ejpam-239	149	7	a	a	PRON
ejpam-239	149	8	}	}	PUNCT
ejpam-239	149	9	and	and	CCONJ
ejpam-239	149	10	b	b	X
ejpam-239	149	11	=	=	SYM
ejpam-239	149	12	{	{	PUNCT
ejpam-239	149	13	0	0	PROPN
ejpam-239	149	14	,	,	PUNCT
ejpam-239	149	15	b1	b1	NOUN
ejpam-239	149	16	,	,	PUNCT
ejpam-239	149	17	b2	b2	NOUN
ejpam-239	149	18	}	}	PUNCT
ejpam-239	149	19	be	be	VERB
ejpam-239	149	20	two	two	NUM
ejpam-239	149	21	discrete	discrete	ADJ
ejpam-239	149	22	adls	adls	PROPN
ejpam-239	149	23	.	.	PUNCT
ejpam-239	150	1	write	write	VERB
ejpam-239	150	2	r	r	NOUN
ejpam-239	150	3	=	=	PUNCT
ejpam-239	150	4	a×	a×	PUNCT
ejpam-239	150	5	b	b	X
ejpam-239	150	6	=	=	PRON
ejpam-239	150	7	{	{	PUNCT
ejpam-239	150	8	(	(	PUNCT
ejpam-239	150	9	0,0	0,0	NOUN
ejpam-239	150	10	)	)	PUNCT
ejpam-239	150	11	,	,	PUNCT
ejpam-239	150	12	(	(	PUNCT
ejpam-239	150	13	0	0	NUM
ejpam-239	150	14	,	,	PUNCT
ejpam-239	150	15	b1	b1	NOUN
ejpam-239	150	16	)	)	PUNCT
ejpam-239	150	17	,	,	PUNCT
ejpam-239	150	18	(	(	PUNCT
ejpam-239	150	19	0	0	NUM
ejpam-239	150	20	,	,	PUNCT
ejpam-239	150	21	b2	b2	NOUN
ejpam-239	150	22	)	)	PUNCT
ejpam-239	150	23	,	,	PUNCT
ejpam-239	150	24	(	(	PUNCT
ejpam-239	150	25	a	a	PRON
ejpam-239	150	26	,	,	PUNCT
ejpam-239	150	27	0	0	NUM
ejpam-239	150	28	)	)	PUNCT
ejpam-239	150	29	,	,	PUNCT
ejpam-239	150	30	(	(	PUNCT
ejpam-239	150	31	a	a	DET
ejpam-239	150	32	,	,	PUNCT
ejpam-239	150	33	b1	b1	NOUN
ejpam-239	150	34	)	)	PUNCT
ejpam-239	150	35	,	,	PUNCT
ejpam-239	150	36	(	(	PUNCT
ejpam-239	150	37	a	a	DET
ejpam-239	150	38	,	,	PUNCT
ejpam-239	150	39	b2	b2	NOUN
ejpam-239	150	40	)	)	PUNCT
ejpam-239	150	41	}	}	PUNCT
ejpam-239	150	42	.	.	PUNCT
ejpam-239	151	1	then	then	ADV
ejpam-239	151	2	(	(	PUNCT
ejpam-239	151	3	r,∨,∧	r,∨,∧	NUM
ejpam-239	151	4	,	,	PUNCT
ejpam-239	151	5	0′	0′	NUM
ejpam-239	151	6	)	)	PUNCT
ejpam-239	151	7	is	be	AUX
ejpam-239	151	8	an	an	DET
ejpam-239	151	9	adl	adl	NOUN
ejpam-239	151	10	where	where	SCONJ
ejpam-239	151	11	0′	0′	NUM
ejpam-239	152	1	=	=	SYM
ejpam-239	152	2	(	(	PUNCT
ejpam-239	152	3	0,0	0,0	NOUN
ejpam-239	152	4	)	)	PUNCT
ejpam-239	152	5	,	,	PUNCT
ejpam-239	152	6	under	under	ADP
ejpam-239	152	7	point	point	NOUN
ejpam-239	152	8	-	-	PUNCT
ejpam-239	152	9	wise	wise	ADJ
ejpam-239	152	10	operations	operation	NOUN
ejpam-239	152	11	.	.	PUNCT
ejpam-239	153	1	now	now	ADV
ejpam-239	153	2	(	(	PUNCT
ejpam-239	153	3	(	(	PUNCT
ejpam-239	153	4	a	a	X
ejpam-239	153	5	,	,	PUNCT
ejpam-239	153	6	0)]∗	0)]∗	PROPN
ejpam-239	153	7	∨	∨	NUM
ejpam-239	153	8	(	(	PUNCT
ejpam-239	153	9	(	(	PUNCT
ejpam-239	153	10	a	a	PRON
ejpam-239	153	11	,	,	PUNCT
ejpam-239	153	12	0)]∗∗	0)]∗∗	NUM
ejpam-239	153	13	=	=	SYM
ejpam-239	153	14	{	{	PUNCT
ejpam-239	153	15	(	(	PUNCT
ejpam-239	153	16	0,0	0,0	NOUN
ejpam-239	153	17	)	)	PUNCT
ejpam-239	153	18	,	,	PUNCT
ejpam-239	153	19	(	(	PUNCT
ejpam-239	153	20	0	0	NUM
ejpam-239	153	21	,	,	PUNCT
ejpam-239	153	22	b1	b1	NOUN
ejpam-239	153	23	)	)	PUNCT
ejpam-239	153	24	,	,	PUNCT
ejpam-239	153	25	(	(	PUNCT
ejpam-239	153	26	0	0	NUM
ejpam-239	153	27	,	,	PUNCT
ejpam-239	153	28	b2	b2	NOUN
ejpam-239	153	29	)	)	PUNCT
ejpam-239	153	30	}	}	PUNCT
ejpam-239	153	31	∨	∨	X
ejpam-239	153	32	{	{	PUNCT
ejpam-239	153	33	(	(	PUNCT
ejpam-239	153	34	0,0	0,0	NOUN
ejpam-239	153	35	)	)	PUNCT
ejpam-239	153	36	,	,	PUNCT
ejpam-239	153	37	(	(	PUNCT
ejpam-239	153	38	a	a	X
ejpam-239	153	39	,	,	PUNCT
ejpam-239	153	40	0)}=	0)}=	PROPN
ejpam-239	153	41	r.	r.	PROPN
ejpam-239	153	42	(	(	PUNCT
ejpam-239	153	43	(	(	PUNCT
ejpam-239	153	44	0	0	NUM
ejpam-239	153	45	,	,	PUNCT
ejpam-239	153	46	b1	b1	NOUN
ejpam-239	153	47	)	)	PUNCT
ejpam-239	153	48	]	]	PUNCT
ejpam-239	153	49	∗	∗	X
ejpam-239	153	50	∨	∨	X
ejpam-239	153	51	(	(	PUNCT
ejpam-239	153	52	(	(	PUNCT
ejpam-239	153	53	0	0	NUM
ejpam-239	153	54	,	,	PUNCT
ejpam-239	153	55	b1	b1	NOUN
ejpam-239	153	56	)	)	PUNCT
ejpam-239	153	57	]	]	PUNCT
ejpam-239	154	1	∗∗	∗∗	X
ejpam-239	154	2	=	=	SYM
ejpam-239	154	3	{	{	PUNCT
ejpam-239	154	4	(	(	PUNCT
ejpam-239	154	5	0,0	0,0	NOUN
ejpam-239	154	6	)	)	PUNCT
ejpam-239	154	7	,	,	PUNCT
ejpam-239	154	8	(	(	PUNCT
ejpam-239	154	9	a	a	PRON
ejpam-239	154	10	,	,	PUNCT
ejpam-239	154	11	0	0	NUM
ejpam-239	154	12	)	)	PUNCT
ejpam-239	154	13	}	}	PUNCT
ejpam-239	154	14	∨	∨	X
ejpam-239	154	15	{	{	PUNCT
ejpam-239	154	16	(	(	PUNCT
ejpam-239	154	17	0,0	0,0	NOUN
ejpam-239	154	18	)	)	PUNCT
ejpam-239	154	19	,	,	PUNCT
ejpam-239	154	20	(	(	PUNCT
ejpam-239	154	21	0	0	NUM
ejpam-239	154	22	,	,	PUNCT
ejpam-239	154	23	b1	b1	NOUN
ejpam-239	154	24	)	)	PUNCT
ejpam-239	154	25	,	,	PUNCT
ejpam-239	154	26	(	(	PUNCT
ejpam-239	154	27	0	0	NUM
ejpam-239	154	28	,	,	PUNCT
ejpam-239	154	29	b2)}=	b2)}=	NOUN
ejpam-239	154	30	r.	r.	PROPN
ejpam-239	154	31	also	also	ADV
ejpam-239	154	32	(	(	PUNCT
ejpam-239	154	33	(	(	PUNCT
ejpam-239	154	34	a	a	PRON
ejpam-239	154	35	,	,	PUNCT
ejpam-239	154	36	b1	b1	NOUN
ejpam-239	154	37	)	)	PUNCT
ejpam-239	154	38	]	]	PUNCT
ejpam-239	155	1	∗	∗	X
ejpam-239	155	2	∨	∨	X
ejpam-239	155	3	(	(	PUNCT
ejpam-239	155	4	(	(	PUNCT
ejpam-239	155	5	a	a	PRON
ejpam-239	155	6	,	,	PUNCT
ejpam-239	155	7	b1	b1	NOUN
ejpam-239	155	8	)	)	PUNCT
ejpam-239	155	9	]	]	PUNCT
ejpam-239	156	1	∗∗	∗∗	X
ejpam-239	156	2	=	=	SYM
ejpam-239	156	3	{	{	PUNCT
ejpam-239	156	4	(	(	PUNCT
ejpam-239	156	5	0,0	0,0	NOUN
ejpam-239	156	6	)	)	PUNCT
ejpam-239	156	7	}	}	PUNCT
ejpam-239	156	8	∨	∨	NUM
ejpam-239	156	9	r=	r=	PROPN
ejpam-239	156	10	r.	r.	PROPN
ejpam-239	156	11	hence	hence	ADV
ejpam-239	156	12	(	(	PUNCT
ejpam-239	156	13	r,∨,∧	r,∨,∧	NUM
ejpam-239	156	14	,	,	PUNCT
ejpam-239	156	15	0′	0′	NUM
ejpam-239	156	16	)	)	PUNCT
ejpam-239	156	17	is	be	AUX
ejpam-239	156	18	a	a	DET
ejpam-239	156	19	generalized	generalized	ADJ
ejpam-239	156	20	stone	stone	NOUN
ejpam-239	156	21	adl	adl	PROPN
ejpam-239	156	22	.	.	PUNCT
ejpam-239	157	1	we	we	PRON
ejpam-239	157	2	now	now	ADV
ejpam-239	157	3	characterize	characterize	VERB
ejpam-239	157	4	normal	normal	ADJ
ejpam-239	157	5	adl	adl	NOUN
ejpam-239	157	6	and	and	CCONJ
ejpam-239	157	7	the	the	DET
ejpam-239	157	8	generalized	generalize	VERB
ejpam-239	157	9	stone	stone	NOUN
ejpam-239	157	10	adl	adl	NOUN
ejpam-239	157	11	in	in	ADP
ejpam-239	157	12	terms	term	NOUN
ejpam-239	157	13	of	of	ADP
ejpam-239	157	14	annulets	annulet	NOUN
ejpam-239	157	15	.	.	PUNCT
ejpam-239	158	1	theorem	theorem	VERB
ejpam-239	158	2	3.7	3.7	NUM
ejpam-239	158	3	.	.	PUNCT
ejpam-239	159	1	let	let	VERB
ejpam-239	159	2	r	r	PRON
ejpam-239	159	3	be	be	AUX
ejpam-239	159	4	an	an	DET
ejpam-239	159	5	adl	adl	NOUN
ejpam-239	159	6	with	with	ADP
ejpam-239	159	7	0	0	NUM
ejpam-239	159	8	.	.	PUNCT
ejpam-239	160	1	consider	consider	VERB
ejpam-239	160	2	the	the	DET
ejpam-239	160	3	following	follow	VERB
ejpam-239	160	4	conditions	condition	NOUN
ejpam-239	160	5	:	:	PUNCT
ejpam-239	160	6	(	(	PUNCT
ejpam-239	160	7	1	1	NUM
ejpam-239	160	8	)	)	PUNCT
ejpam-239	160	9	.	.	PUNCT
ejpam-239	161	1	each	each	DET
ejpam-239	161	2	annulet	annulet	NOUN
ejpam-239	161	3	is	be	AUX
ejpam-239	161	4	a	a	DET
ejpam-239	161	5	direct	direct	ADJ
ejpam-239	161	6	summand	summand	NOUN
ejpam-239	161	7	of	of	ADP
ejpam-239	161	8	r	r	NOUN
ejpam-239	161	9	(	(	PUNCT
ejpam-239	161	10	2	2	NUM
ejpam-239	161	11	)	)	PUNCT
ejpam-239	161	12	.	.	PUNCT
ejpam-239	162	1	r	r	NOUN
ejpam-239	162	2	is	be	AUX
ejpam-239	162	3	a	a	DET
ejpam-239	162	4	generalized	generalized	ADJ
ejpam-239	162	5	stone	stone	NOUN
ejpam-239	162	6	adl	adl	NOUN
ejpam-239	162	7	(	(	PUNCT
ejpam-239	162	8	3	3	NUM
ejpam-239	162	9	)	)	PUNCT
ejpam-239	162	10	.	.	PUNCT
ejpam-239	163	1	r	r	NOUN
ejpam-239	163	2	is	be	AUX
ejpam-239	163	3	normal	normal	ADJ
ejpam-239	163	4	(	(	PUNCT
ejpam-239	163	5	4	4	NUM
ejpam-239	163	6	)	)	PUNCT
ejpam-239	163	7	.	.	PUNCT
ejpam-239	164	1	a0(r	a0(r	NOUN
ejpam-239	164	2	)	)	PUNCT
ejpam-239	164	3	is	be	AUX
ejpam-239	164	4	a	a	DET
ejpam-239	164	5	sublattice	sublattice	NOUN
ejpam-239	164	6	of	of	ADP
ejpam-239	164	7	the	the	DET
ejpam-239	164	8	lattice	lattice	NOUN
ejpam-239	164	9	i	i	PRON
ejpam-239	164	10	(	(	PUNCT
ejpam-239	164	11	r	r	NOUN
ejpam-239	164	12	)	)	PUNCT
ejpam-239	164	13	of	of	ADP
ejpam-239	164	14	all	all	DET
ejpam-239	164	15	ideals	ideal	NOUN
ejpam-239	164	16	of	of	ADP
ejpam-239	164	17	r.	r.	PROPN
ejpam-239	164	18	then	then	ADV
ejpam-239	164	19	(	(	PUNCT
ejpam-239	164	20	1	1	X
ejpam-239	164	21	)	)	PUNCT
ejpam-239	164	22	is	be	AUX
ejpam-239	164	23	equivalent	equivalent	ADJ
ejpam-239	164	24	to	to	ADP
ejpam-239	164	25	(	(	PUNCT
ejpam-239	164	26	2	2	NUM
ejpam-239	164	27	)	)	PUNCT
ejpam-239	164	28	,	,	PUNCT
ejpam-239	164	29	(	(	PUNCT
ejpam-239	164	30	3	3	X
ejpam-239	164	31	)	)	PUNCT
ejpam-239	164	32	is	be	AUX
ejpam-239	164	33	equivalent	equivalent	ADJ
ejpam-239	164	34	to	to	ADP
ejpam-239	164	35	(	(	PUNCT
ejpam-239	164	36	4	4	NUM
ejpam-239	164	37	)	)	PUNCT
ejpam-239	164	38	,	,	PUNCT
ejpam-239	164	39	and	and	CCONJ
ejpam-239	164	40	(	(	PUNCT
ejpam-239	164	41	2	2	X
ejpam-239	164	42	)	)	PUNCT
ejpam-239	164	43	implies	imply	VERB
ejpam-239	164	44	(	(	PUNCT
ejpam-239	164	45	3	3	NUM
ejpam-239	164	46	)	)	PUNCT
ejpam-239	164	47	.	.	PUNCT
ejpam-239	165	1	if	if	SCONJ
ejpam-239	165	2	r	r	NOUN
ejpam-239	165	3	is	be	AUX
ejpam-239	165	4	a	a	DET
ejpam-239	165	5	⋆-adl	⋆-adl	PROPN
ejpam-239	165	6	,	,	PUNCT
ejpam-239	165	7	then	then	ADV
ejpam-239	165	8	(	(	PUNCT
ejpam-239	165	9	4	4	X
ejpam-239	165	10	)	)	PUNCT
ejpam-239	165	11	implies	imply	VERB
ejpam-239	165	12	(	(	PUNCT
ejpam-239	165	13	1	1	NUM
ejpam-239	165	14	)	)	PUNCT
ejpam-239	165	15	.	.	PUNCT
ejpam-239	166	1	proof	proof	NOUN
ejpam-239	166	2	:	:	PUNCT
ejpam-239	166	3	(	(	PUNCT
ejpam-239	166	4	1)⇒	1)⇒	NUM
ejpam-239	166	5	(	(	PUNCT
ejpam-239	166	6	2	2	NUM
ejpam-239	166	7	):	):	PUNCT
ejpam-239	166	8	let	let	VERB
ejpam-239	166	9	x	x	PROPN
ejpam-239	166	10	∈	∈	PROPN
ejpam-239	166	11	r.	r.	PROPN
ejpam-239	166	12	then	then	ADV
ejpam-239	166	13	by	by	ADP
ejpam-239	166	14	(	(	PUNCT
ejpam-239	166	15	1	1	NUM
ejpam-239	166	16	)	)	PUNCT
ejpam-239	166	17	,	,	PUNCT
ejpam-239	166	18	there	there	PRON
ejpam-239	166	19	exists	exist	VERB
ejpam-239	166	20	an	an	DET
ejpam-239	166	21	ideal	ideal	ADJ
ejpam-239	166	22	j	j	NOUN
ejpam-239	166	23	of	of	ADP
ejpam-239	166	24	r	r	NOUN
ejpam-239	166	25	such	such	ADJ
ejpam-239	166	26	that	that	PRON
ejpam-239	166	27	(	(	PUNCT
ejpam-239	166	28	x]∗	x]∗	PROPN
ejpam-239	166	29	∩	∩	PROPN
ejpam-239	166	30	j	j	PROPN
ejpam-239	166	31	=	=	PUNCT
ejpam-239	166	32	(	(	PUNCT
ejpam-239	166	33	0	0	NUM
ejpam-239	166	34	]	]	PUNCT
ejpam-239	166	35	and	and	CCONJ
ejpam-239	166	36	(	(	PUNCT
ejpam-239	166	37	x]∗∨j	x]∗∨j	X
ejpam-239	166	38	=	=	SYM
ejpam-239	166	39	r.	r.	PROPN
ejpam-239	166	40	now	now	ADV
ejpam-239	166	41	(	(	PUNCT
ejpam-239	166	42	x]∗∩j	x]∗∩j	PROPN
ejpam-239	166	43	=	=	SYM
ejpam-239	166	44	(	(	PUNCT
ejpam-239	166	45	0	0	NUM
ejpam-239	166	46	]	]	PUNCT
ejpam-239	166	47	implies	imply	VERB
ejpam-239	166	48	that	that	SCONJ
ejpam-239	166	49	j	j	PROPN
ejpam-239	166	50	⊆	⊆	NUM
ejpam-239	166	51	(	(	PUNCT
ejpam-239	166	52	x]∗∗.	x]∗∗.	PROPN
ejpam-239	166	53	hence	hence	ADV
ejpam-239	166	54	r=	r=	ADJ
ejpam-239	166	55	(	(	PUNCT
ejpam-239	166	56	x]∗∨j	x]∗∨j	NOUN
ejpam-239	166	57	⊆	⊆	NUM
ejpam-239	166	58	(	(	PUNCT
ejpam-239	166	59	x]∗∨(x]∗∗.	x]∗∨(x]∗∗.	X
ejpam-239	166	60	thus	thus	ADV
ejpam-239	166	61	r=	r=	ADJ
ejpam-239	166	62	(	(	PUNCT
ejpam-239	166	63	x]∗	x]∗	PROPN
ejpam-239	166	64	∨	∨	PROPN
ejpam-239	166	65	(	(	PUNCT
ejpam-239	166	66	x]∗∗	x]∗∗	X
ejpam-239	166	67	∀	∀	X
ejpam-239	167	1	x	x	X
ejpam-239	167	2	∈	∈	PROPN
ejpam-239	167	3	r.	r.	PROPN
ejpam-239	167	4	(	(	PUNCT
ejpam-239	167	5	2)⇒	2)⇒	NUM
ejpam-239	167	6	(	(	PUNCT
ejpam-239	167	7	1	1	NUM
ejpam-239	167	8	):	):	PUNCT
ejpam-239	167	9	assume	assume	VERB
ejpam-239	167	10	that	that	SCONJ
ejpam-239	167	11	r	r	NOUN
ejpam-239	167	12	is	be	AUX
ejpam-239	167	13	a	a	DET
ejpam-239	167	14	generalized	generalized	ADJ
ejpam-239	167	15	stone	stone	NOUN
ejpam-239	167	16	adl	adl	PROPN
ejpam-239	167	17	.	.	PUNCT
ejpam-239	168	1	let	let	VERB
ejpam-239	168	2	x	x	SYM
ejpam-239	168	3	∈	∈	PROPN
ejpam-239	168	4	r.	r.	PROPN
ejpam-239	168	5	we	we	PRON
ejpam-239	168	6	have	have	VERB
ejpam-239	168	7	always	always	ADV
ejpam-239	168	8	(	(	PUNCT
ejpam-239	168	9	x]∗	x]∗	PROPN
ejpam-239	168	10	∩	∩	PROPN
ejpam-239	168	11	(	(	PUNCT
ejpam-239	168	12	x]∗∗	x]∗∗	PROPN
ejpam-239	168	13	=	=	SYM
ejpam-239	168	14	(	(	PUNCT
ejpam-239	168	15	0	0	NUM
ejpam-239	168	16	]	]	PUNCT
ejpam-239	168	17	.	.	PUNCT
ejpam-239	169	1	by	by	ADP
ejpam-239	169	2	(	(	PUNCT
ejpam-239	169	3	2	2	NUM
ejpam-239	169	4	)	)	PUNCT
ejpam-239	169	5	,	,	PUNCT
ejpam-239	169	6	we	we	PRON
ejpam-239	169	7	get	get	VERB
ejpam-239	169	8	(	(	PUNCT
ejpam-239	169	9	x]∗	x]∗	PROPN
ejpam-239	169	10	∨	∨	PROPN
ejpam-239	169	11	(	(	PUNCT
ejpam-239	169	12	x]∗∗	x]∗∗	PROPN
ejpam-239	169	13	=	=	SYM
ejpam-239	169	14	r.	r.	PROPN
ejpam-239	169	15	(	(	PUNCT
ejpam-239	169	16	2	2	NUM
ejpam-239	169	17	)	)	PUNCT
ejpam-239	169	18	⇒	⇒	NOUN
ejpam-239	169	19	(	(	PUNCT
ejpam-239	169	20	3	3	NUM
ejpam-239	169	21	):	):	PUNCT
ejpam-239	169	22	assume	assume	VERB
ejpam-239	169	23	that	that	SCONJ
ejpam-239	169	24	r	r	NOUN
ejpam-239	169	25	is	be	AUX
ejpam-239	169	26	a	a	DET
ejpam-239	169	27	generalized	generalized	ADJ
ejpam-239	169	28	stone	stone	NOUN
ejpam-239	169	29	adl	adl	PROPN
ejpam-239	169	30	.	.	PUNCT
ejpam-239	170	1	let	let	VERB
ejpam-239	170	2	x	x	PRON
ejpam-239	170	3	,	,	PUNCT
ejpam-239	170	4	y	y	PROPN
ejpam-239	170	5	∈	∈	PROPN
ejpam-239	170	6	r.	r.	NOUN
ejpam-239	170	7	always	always	ADV
ejpam-239	170	8	we	we	PRON
ejpam-239	170	9	have	have	VERB
ejpam-239	170	10	(	(	PUNCT
ejpam-239	170	11	x]∗	x]∗	PROPN
ejpam-239	170	12	∨	∨	PROPN
ejpam-239	170	13	(	(	PUNCT
ejpam-239	170	14	y]∗	y]∗	PROPN
ejpam-239	170	15	⊆	⊆	NUM
ejpam-239	170	16	(	(	PUNCT
ejpam-239	170	17	x	x	SYM
ejpam-239	170	18	∧	∧	PROPN
ejpam-239	170	19	y]∗.	y]∗.	PROPN
ejpam-239	170	20	let	let	VERB
ejpam-239	170	21	a	a	DET
ejpam-239	170	22	∈	∈	NOUN
ejpam-239	170	23	(	(	PUNCT
ejpam-239	170	24	x	x	PART
ejpam-239	170	25	∧	∧	PROPN
ejpam-239	170	26	y]∗.	y]∗.	PROPN
ejpam-239	170	27	then	then	ADV
ejpam-239	170	28	a	a	DET
ejpam-239	170	29	∧	∧	PROPN
ejpam-239	170	30	x	x	PUNCT
ejpam-239	170	31	∧	∧	NOUN
ejpam-239	170	32	y	y	PROPN
ejpam-239	170	33	=	=	SYM
ejpam-239	170	34	0	0	PROPN
ejpam-239	170	35	.	.	PUNCT
ejpam-239	171	1	⇒	⇒	PROPN
ejpam-239	171	2	(	(	PUNCT
ejpam-239	171	3	a	a	DET
ejpam-239	171	4	∧	∧	PROPN
ejpam-239	171	5	x	x	PUNCT
ejpam-239	171	6	∧	∧	PROPN
ejpam-239	171	7	y	y	PROPN
ejpam-239	171	8	]	]	X
ejpam-239	171	9	=	=	PUNCT
ejpam-239	171	10	(	(	PUNCT
ejpam-239	171	11	0	0	NUM
ejpam-239	171	12	]	]	X
ejpam-239	171	13	g.	g.	PROPN
ejpam-239	171	14	c.	c.	PROPN
ejpam-239	171	15	rao	rao	PROPN
ejpam-239	171	16	and	and	CCONJ
ejpam-239	171	17	m.	m.	PROPN
ejpam-239	171	18	sambasiva	sambasiva	PROPN
ejpam-239	171	19	rao	rao	PROPN
ejpam-239	171	20	/	/	SYM
ejpam-239	171	21	eur	eur	PROPN
ejpam-239	171	22	.	.	PUNCT
ejpam-239	172	1	j.	j.	PROPN
ejpam-239	172	2	pure	pure	PROPN
ejpam-239	172	3	appl	appl	PROPN
ejpam-239	172	4	.	.	PROPN
ejpam-239	172	5	math	math	PROPN
ejpam-239	172	6	,	,	PUNCT
ejpam-239	172	7	2	2	NUM
ejpam-239	172	8	(	(	PUNCT
ejpam-239	172	9	2009	2009	NUM
ejpam-239	172	10	)	)	PUNCT
ejpam-239	172	11	,	,	PUNCT
ejpam-239	172	12	(	(	PUNCT
ejpam-239	172	13	58	58	NUM
ejpam-239	172	14	-	-	SYM
ejpam-239	172	15	72	72	NUM
ejpam-239	172	16	)	)	PUNCT
ejpam-239	172	17	65	65	NUM
ejpam-239	172	18	⇒	⇒	NOUN
ejpam-239	172	19	(	(	PUNCT
ejpam-239	172	20	x]∩	x]∩	X
ejpam-239	172	21	(	(	PUNCT
ejpam-239	172	22	a	a	DET
ejpam-239	172	23	∧	∧	PROPN
ejpam-239	172	24	y	y	PROPN
ejpam-239	172	25	]	]	X
ejpam-239	172	26	=	=	PUNCT
ejpam-239	172	27	(	(	PUNCT
ejpam-239	172	28	0	0	NUM
ejpam-239	172	29	]	]	PUNCT
ejpam-239	172	30	⇒	⇒	NOUN
ejpam-239	172	31	(	(	PUNCT
ejpam-239	172	32	a	a	DET
ejpam-239	172	33	∧	∧	PROPN
ejpam-239	172	34	y	y	PROPN
ejpam-239	172	35	]	]	X
ejpam-239	172	36	⊆	⊆	NUM
ejpam-239	172	37	(	(	PUNCT
ejpam-239	172	38	x]∗	x]∗	PROPN
ejpam-239	172	39	⇒	⇒	PROPN
ejpam-239	172	40	(	(	PUNCT
ejpam-239	172	41	x]∗∗	x]∗∗	PROPN
ejpam-239	172	42	⊆	⊆	NUM
ejpam-239	172	43	(	(	PUNCT
ejpam-239	172	44	a	a	DET
ejpam-239	172	45	∧	∧	PROPN
ejpam-239	172	46	y]∗	y]∗	PROPN
ejpam-239	172	47	⇒	⇒	PROPN
ejpam-239	172	48	(	(	PUNCT
ejpam-239	172	49	x]∗∗	x]∗∗	PROPN
ejpam-239	172	50	∩	∩	X
ejpam-239	172	51	(	(	PUNCT
ejpam-239	172	52	a	a	DET
ejpam-239	172	53	∧	∧	PROPN
ejpam-239	172	54	y	y	PROPN
ejpam-239	172	55	]	]	X
ejpam-239	172	56	=	=	PUNCT
ejpam-239	172	57	(	(	PUNCT
ejpam-239	172	58	0	0	NUM
ejpam-239	172	59	]	]	PUNCT
ejpam-239	172	60	⇒	⇒	NOUN
ejpam-239	172	61	(	(	PUNCT
ejpam-239	172	62	x]∗∗	x]∗∗	PROPN
ejpam-239	172	63	∩	∩	X
ejpam-239	172	64	{	{	PUNCT
ejpam-239	172	65	(	(	PUNCT
ejpam-239	172	66	a]∩	a]∩	X
ejpam-239	172	67	(	(	PUNCT
ejpam-239	172	68	y	y	NOUN
ejpam-239	172	69	]	]	X
ejpam-239	172	70	}	}	PUNCT
ejpam-239	172	71	=	=	SYM
ejpam-239	172	72	(	(	PUNCT
ejpam-239	172	73	0	0	NUM
ejpam-239	172	74	]	]	PUNCT
ejpam-239	172	75	⇒	⇒	NOUN
ejpam-239	172	76	{	{	PUNCT
ejpam-239	172	77	(	(	PUNCT
ejpam-239	172	78	x]∗∗	x]∗∗	NOUN
ejpam-239	172	79	∩	∩	X
ejpam-239	172	80	(	(	PUNCT
ejpam-239	172	81	a	a	DET
ejpam-239	172	82	]	]	X
ejpam-239	172	83	}	}	PUNCT
ejpam-239	172	84	∩	∩	NOUN
ejpam-239	172	85	(	(	PUNCT
ejpam-239	172	86	y	y	X
ejpam-239	172	87	]	]	X
ejpam-239	172	88	=	=	PUNCT
ejpam-239	172	89	(	(	PUNCT
ejpam-239	172	90	0	0	NUM
ejpam-239	172	91	]	]	PUNCT
ejpam-239	172	92	⇒	⇒	NOUN
ejpam-239	172	93	(	(	PUNCT
ejpam-239	172	94	x]∗∗	x]∗∗	PROPN
ejpam-239	172	95	∩	∩	X
ejpam-239	172	96	(	(	PUNCT
ejpam-239	172	97	a]⊆	a]⊆	PROPN
ejpam-239	172	98	(	(	PUNCT
ejpam-239	172	99	y]∗	y]∗	ADP
ejpam-239	172	100	it	it	PRON
ejpam-239	172	101	is	be	AUX
ejpam-239	172	102	clear	clear	ADJ
ejpam-239	172	103	that	that	SCONJ
ejpam-239	172	104	(	(	PUNCT
ejpam-239	172	105	x]∗	x]∗	PROPN
ejpam-239	172	106	∩	∩	PROPN
ejpam-239	172	107	(	(	PUNCT
ejpam-239	172	108	a]⊆	a]⊆	PROPN
ejpam-239	172	109	(	(	PUNCT
ejpam-239	172	110	x]∗	x]∗	PROPN
ejpam-239	172	111	thus	thus	ADV
ejpam-239	172	112	we	we	PRON
ejpam-239	172	113	get	get	VERB
ejpam-239	172	114	that	that	PRON
ejpam-239	172	115	{	{	PUNCT
ejpam-239	172	116	(	(	PUNCT
ejpam-239	172	117	x]∗	x]∗	PROPN
ejpam-239	172	118	∩	∩	PROPN
ejpam-239	172	119	(	(	PUNCT
ejpam-239	172	120	a	a	DET
ejpam-239	172	121	]	]	X
ejpam-239	172	122	}	}	PUNCT
ejpam-239	172	123	∨	∨	NUM
ejpam-239	172	124	{	{	PUNCT
ejpam-239	172	125	(	(	PUNCT
ejpam-239	173	1	x]∗∗	x]∗∗	NOUN
ejpam-239	173	2	∩	∩	X
ejpam-239	173	3	(	(	PUNCT
ejpam-239	173	4	a	a	PRON
ejpam-239	173	5	]	]	X
ejpam-239	173	6	}	}	NUM
ejpam-239	173	7	⊆	⊆	NUM
ejpam-239	173	8	(	(	PUNCT
ejpam-239	173	9	x]∗	x]∗	PROPN
ejpam-239	173	10	∨	∨	PROPN
ejpam-239	173	11	(	(	PUNCT
ejpam-239	173	12	y]∗	y]∗	PROPN
ejpam-239	173	13	⇒	⇒	PROPN
ejpam-239	173	14	{	{	PUNCT
ejpam-239	173	15	(	(	PUNCT
ejpam-239	173	16	x]∗	x]∗	PROPN
ejpam-239	173	17	∨	∨	PROPN
ejpam-239	173	18	(	(	PUNCT
ejpam-239	173	19	x]∗∗	x]∗∗	NOUN
ejpam-239	173	20	}	}	PUNCT
ejpam-239	173	21	∩	∩	NOUN
ejpam-239	173	22	(	(	PUNCT
ejpam-239	173	23	a]⊆	a]⊆	PROPN
ejpam-239	173	24	(	(	PUNCT
ejpam-239	173	25	x]∗	x]∗	PROPN
ejpam-239	173	26	∨	∨	PROPN
ejpam-239	173	27	(	(	PUNCT
ejpam-239	173	28	y]∗	y]∗	PROPN
ejpam-239	173	29	⇒	⇒	NOUN
ejpam-239	173	30	r∩	r∩	PROPN
ejpam-239	173	31	(	(	PUNCT
ejpam-239	173	32	a]⊆	a]⊆	PROPN
ejpam-239	173	33	(	(	PUNCT
ejpam-239	173	34	x]∗	x]∗	PROPN
ejpam-239	173	35	∨	∨	PROPN
ejpam-239	173	36	(	(	PUNCT
ejpam-239	173	37	y]∗	y]∗	PROPN
ejpam-239	173	38	(	(	PUNCT
ejpam-239	173	39	since	since	SCONJ
ejpam-239	173	40	r	r	NOUN
ejpam-239	173	41	is	be	AUX
ejpam-239	173	42	a	a	DET
ejpam-239	173	43	generalized	generalized	ADJ
ejpam-239	173	44	stone	stone	NOUN
ejpam-239	173	45	adl	adl	NOUN
ejpam-239	173	46	)	)	PUNCT
ejpam-239	173	47	⇒	⇒	NOUN
ejpam-239	173	48	(	(	PUNCT
ejpam-239	173	49	a]⊆	a]⊆	PROPN
ejpam-239	173	50	(	(	PUNCT
ejpam-239	173	51	x]∗	x]∗	PROPN
ejpam-239	173	52	∨	∨	PROPN
ejpam-239	173	53	(	(	PUNCT
ejpam-239	173	54	y]∗	y]∗	PROPN
ejpam-239	173	55	⇒	⇒	VERB
ejpam-239	173	56	a	a	DET
ejpam-239	173	57	∈	∈	PROPN
ejpam-239	173	58	(	(	PUNCT
ejpam-239	173	59	x]∗	x]∗	PROPN
ejpam-239	173	60	∨	∨	PROPN
ejpam-239	173	61	(	(	PUNCT
ejpam-239	173	62	y]∗	y]∗	PROPN
ejpam-239	173	63	hence	hence	ADV
ejpam-239	173	64	(	(	PUNCT
ejpam-239	173	65	x	x	PUNCT
ejpam-239	173	66	∧	∧	PROPN
ejpam-239	173	67	y]∗	y]∗	PROPN
ejpam-239	173	68	⊆	⊆	NUM
ejpam-239	173	69	(	(	PUNCT
ejpam-239	173	70	x]∗	x]∗	PROPN
ejpam-239	173	71	∨	∨	PROPN
ejpam-239	173	72	(	(	PUNCT
ejpam-239	173	73	y]∗.	y]∗.	PRON
ejpam-239	173	74	thus	thus	ADV
ejpam-239	173	75	(	(	PUNCT
ejpam-239	173	76	x	x	SYM
ejpam-239	173	77	∧	∧	NOUN
ejpam-239	173	78	y]∗	y]∗	PROPN
ejpam-239	173	79	=	=	PUNCT
ejpam-239	173	80	(	(	PUNCT
ejpam-239	173	81	x]∗	x]∗	PROPN
ejpam-239	173	82	∨	∨	PROPN
ejpam-239	173	83	(	(	PUNCT
ejpam-239	173	84	y]∗.	y]∗.	PRON
ejpam-239	173	85	therefore	therefore	ADV
ejpam-239	173	86	r	r	NOUN
ejpam-239	173	87	is	be	AUX
ejpam-239	173	88	normal	normal	ADJ
ejpam-239	173	89	.	.	PUNCT
ejpam-239	174	1	now	now	ADV
ejpam-239	174	2	we	we	PRON
ejpam-239	174	3	prove	prove	VERB
ejpam-239	174	4	the	the	DET
ejpam-239	174	5	equivalency	equivalency	NOUN
ejpam-239	174	6	of	of	ADP
ejpam-239	174	7	(	(	PUNCT
ejpam-239	174	8	3	3	NUM
ejpam-239	174	9	)	)	PUNCT
ejpam-239	174	10	and	and	CCONJ
ejpam-239	174	11	(	(	PUNCT
ejpam-239	174	12	4	4	NUM
ejpam-239	174	13	)	)	PUNCT
ejpam-239	174	14	.	.	PUNCT
ejpam-239	175	1	(	(	PUNCT
ejpam-239	175	2	3)⇒	3)⇒	NUM
ejpam-239	175	3	(	(	PUNCT
ejpam-239	175	4	4	4	NUM
ejpam-239	175	5	):	):	PUNCT
ejpam-239	175	6	assume	assume	VERB
ejpam-239	175	7	that	that	SCONJ
ejpam-239	175	8	r	r	NOUN
ejpam-239	175	9	is	be	AUX
ejpam-239	175	10	normal	normal	ADJ
ejpam-239	175	11	.	.	PUNCT
ejpam-239	176	1	let	let	VERB
ejpam-239	176	2	x	x	PRON
ejpam-239	176	3	,	,	PUNCT
ejpam-239	176	4	y	y	PROPN
ejpam-239	176	5	∈	∈	PROPN
ejpam-239	176	6	r.	r.	NOUN
ejpam-239	176	7	we	we	PRON
ejpam-239	176	8	have	have	VERB
ejpam-239	176	9	always	always	ADV
ejpam-239	176	10	(	(	PUNCT
ejpam-239	176	11	x]∗	x]∗	PROPN
ejpam-239	176	12	∩	∩	NOUN
ejpam-239	176	13	(	(	PUNCT
ejpam-239	176	14	y]∗	y]∗	PROPN
ejpam-239	176	15	=	=	SYM
ejpam-239	176	16	(	(	PUNCT
ejpam-239	176	17	x	x	X
ejpam-239	176	18	∨	∨	ADP
ejpam-239	176	19	y]∗	y]∗	PROPN
ejpam-239	176	20	∈	∈	PROPN
ejpam-239	176	21	a0(r	a0(r	PROPN
ejpam-239	176	22	)	)	PUNCT
ejpam-239	176	23	.	.	PUNCT
ejpam-239	177	1	since	since	SCONJ
ejpam-239	177	2	r	r	NOUN
ejpam-239	177	3	is	be	AUX
ejpam-239	177	4	normal	normal	ADJ
ejpam-239	177	5	,	,	PUNCT
ejpam-239	177	6	we	we	PRON
ejpam-239	177	7	get	get	VERB
ejpam-239	177	8	(	(	PUNCT
ejpam-239	177	9	x]∗	x]∗	PROPN
ejpam-239	177	10	∨	∨	PROPN
ejpam-239	177	11	(	(	PUNCT
ejpam-239	177	12	y]∗	y]∗	PROPN
ejpam-239	177	13	=	=	SYM
ejpam-239	177	14	(	(	PUNCT
ejpam-239	177	15	x	x	PART
ejpam-239	177	16	∧	∧	PROPN
ejpam-239	177	17	y]∗	y]∗	PROPN
ejpam-239	177	18	∈	∈	PROPN
ejpam-239	177	19	a0(r	a0(r	PROPN
ejpam-239	177	20	)	)	PUNCT
ejpam-239	177	21	.	.	PUNCT
ejpam-239	178	1	therefore	therefore	ADV
ejpam-239	178	2	a0(r	a0(r	PROPN
ejpam-239	178	3	)	)	PUNCT
ejpam-239	178	4	is	be	AUX
ejpam-239	178	5	a	a	DET
ejpam-239	178	6	sublattice	sublattice	NOUN
ejpam-239	178	7	of	of	ADP
ejpam-239	178	8	i	i	PRON
ejpam-239	178	9	(	(	PUNCT
ejpam-239	178	10	r	r	NOUN
ejpam-239	178	11	)	)	PUNCT
ejpam-239	178	12	.	.	PUNCT
ejpam-239	179	1	(	(	PUNCT
ejpam-239	179	2	4)⇒	4)⇒	X
ejpam-239	179	3	(	(	PUNCT
ejpam-239	179	4	3	3	NUM
ejpam-239	179	5	):	):	PUNCT
ejpam-239	179	6	assume	assume	VERB
ejpam-239	179	7	the	the	DET
ejpam-239	179	8	condition	condition	NOUN
ejpam-239	179	9	(	(	PUNCT
ejpam-239	179	10	4	4	NUM
ejpam-239	179	11	)	)	PUNCT
ejpam-239	179	12	.	.	PUNCT
ejpam-239	180	1	let	let	VERB
ejpam-239	180	2	x	x	PRON
ejpam-239	180	3	,	,	PUNCT
ejpam-239	180	4	y	y	PROPN
ejpam-239	180	5	∈	∈	PROPN
ejpam-239	180	6	r.	r.	PROPN
ejpam-239	180	7	then	then	ADV
ejpam-239	180	8	by	by	ADP
ejpam-239	180	9	(	(	PUNCT
ejpam-239	180	10	4	4	NUM
ejpam-239	180	11	)	)	PUNCT
ejpam-239	180	12	,	,	PUNCT
ejpam-239	180	13	(	(	PUNCT
ejpam-239	180	14	x]∗	x]∗	PROPN
ejpam-239	180	15	∨	∨	PROPN
ejpam-239	180	16	(	(	PUNCT
ejpam-239	180	17	y]∗	y]∗	PROPN
ejpam-239	180	18	=	=	SYM
ejpam-239	180	19	(	(	PUNCT
ejpam-239	180	20	z]∗	z]∗	PROPN
ejpam-239	180	21	,	,	PUNCT
ejpam-239	180	22	for	for	ADP
ejpam-239	180	23	some	some	DET
ejpam-239	180	24	z	z	PROPN
ejpam-239	180	25	∈	∈	PROPN
ejpam-239	180	26	r.	r.	PROPN
ejpam-239	180	27	now	now	ADV
ejpam-239	180	28	(	(	PUNCT
ejpam-239	180	29	z]∗∗	z]∗∗	NOUN
ejpam-239	180	30	=	=	SYM
ejpam-239	180	31	{	{	PUNCT
ejpam-239	180	32	(	(	PUNCT
ejpam-239	180	33	x]∗	x]∗	PROPN
ejpam-239	180	34	∨	∨	PROPN
ejpam-239	180	35	(	(	PUNCT
ejpam-239	180	36	y]∗}∗	y]∗}∗	NOUN
ejpam-239	180	37	=	=	SYM
ejpam-239	180	38	(	(	PUNCT
ejpam-239	180	39	x]∗∗	x]∗∗	PROPN
ejpam-239	180	40	∩	∩	X
ejpam-239	180	41	(	(	PUNCT
ejpam-239	180	42	y]∗∗	y]∗∗	X
ejpam-239	180	43	=	=	SYM
ejpam-239	180	44	(	(	PUNCT
ejpam-239	180	45	x	x	PUNCT
ejpam-239	180	46	∧	∧	NOUN
ejpam-239	180	47	y]∗∗.	y]∗∗.	PROPN
ejpam-239	180	48	hence	hence	ADV
ejpam-239	180	49	(	(	PUNCT
ejpam-239	180	50	x]∗	x]∗	PROPN
ejpam-239	180	51	∨	∨	PROPN
ejpam-239	180	52	(	(	PUNCT
ejpam-239	180	53	y]∗	y]∗	PROPN
ejpam-239	180	54	=	=	SYM
ejpam-239	180	55	(	(	PUNCT
ejpam-239	180	56	x	x	PART
ejpam-239	180	57	∧	∧	PROPN
ejpam-239	180	58	y]∗.	y]∗.	PROPN
ejpam-239	180	59	therefore	therefore	ADV
ejpam-239	180	60	r	r	NOUN
ejpam-239	180	61	is	be	AUX
ejpam-239	180	62	normal	normal	ADJ
ejpam-239	180	63	.	.	PUNCT
ejpam-239	181	1	(	(	PUNCT
ejpam-239	181	2	4	4	X
ejpam-239	181	3	)	)	PUNCT
ejpam-239	181	4	⇒	⇒	NOUN
ejpam-239	181	5	(	(	PUNCT
ejpam-239	181	6	1	1	NUM
ejpam-239	181	7	):	):	PUNCT
ejpam-239	181	8	suppose	suppose	VERB
ejpam-239	181	9	r	r	NOUN
ejpam-239	181	10	is	be	AUX
ejpam-239	181	11	a	a	DET
ejpam-239	181	12	⋆−	⋆−	NOUN
ejpam-239	181	13	adl	adl	NOUN
ejpam-239	181	14	.	.	PROPN
ejpam-239	181	15	assume	assume	VERB
ejpam-239	181	16	that	that	SCONJ
ejpam-239	181	17	a0(r	a0(r	PROPN
ejpam-239	181	18	)	)	PUNCT
ejpam-239	181	19	is	be	AUX
ejpam-239	181	20	a	a	DET
ejpam-239	181	21	sublattice	sublattice	NOUN
ejpam-239	181	22	of	of	ADP
ejpam-239	181	23	i	i	PRON
ejpam-239	181	24	(	(	PUNCT
ejpam-239	181	25	r	r	NOUN
ejpam-239	181	26	)	)	PUNCT
ejpam-239	181	27	.	.	PUNCT
ejpam-239	182	1	let	let	VERB
ejpam-239	182	2	x	x	SYM
ejpam-239	182	3	∈	∈	PROPN
ejpam-239	182	4	r.	r.	PROPN
ejpam-239	182	5	then	then	ADV
ejpam-239	182	6	there	there	PRON
ejpam-239	182	7	exists	exist	VERB
ejpam-239	182	8	x	x	PUNCT
ejpam-239	182	9	′	′	NUM
ejpam-239	182	10	∈	∈	NOUN
ejpam-239	182	11	r	r	NOUN
ejpam-239	182	12	such	such	ADJ
ejpam-239	182	13	that	that	PRON
ejpam-239	182	14	(	(	PUNCT
ejpam-239	182	15	x]∗∗	x]∗∗	NOUN
ejpam-239	182	16	=	=	SYM
ejpam-239	182	17	(	(	PUNCT
ejpam-239	182	18	x	x	X
ejpam-239	182	19	′]∗.	′]∗.	PROPN
ejpam-239	182	20	we	we	PRON
ejpam-239	182	21	have	have	VERB
ejpam-239	182	22	always	always	ADV
ejpam-239	182	23	(	(	PUNCT
ejpam-239	182	24	x]∗	x]∗	PROPN
ejpam-239	182	25	∩	∩	PROPN
ejpam-239	182	26	(	(	PUNCT
ejpam-239	182	27	x]∗∗	x]∗∗	PROPN
ejpam-239	182	28	=	=	SYM
ejpam-239	182	29	(	(	PUNCT
ejpam-239	182	30	0	0	NUM
ejpam-239	182	31	]	]	PUNCT
ejpam-239	182	32	.	.	PUNCT
ejpam-239	183	1	now	now	ADV
ejpam-239	183	2	(	(	PUNCT
ejpam-239	183	3	x]∗	x]∗	PROPN
ejpam-239	183	4	∨	∨	PROPN
ejpam-239	183	5	(	(	PUNCT
ejpam-239	183	6	x]∗∗	x]∗∗	PROPN
ejpam-239	183	7	=	=	SYM
ejpam-239	183	8	(	(	PUNCT
ejpam-239	183	9	x]∗	x]∗	PROPN
ejpam-239	183	10	∨	∨	PROPN
ejpam-239	183	11	(	(	PUNCT
ejpam-239	183	12	x	x	PROPN
ejpam-239	183	13	′]∗	′]∗	PROPN
ejpam-239	183	14	=	=	SYM
ejpam-239	183	15	(	(	PUNCT
ejpam-239	183	16	z]∗	z]∗	PROPN
ejpam-239	183	17	,	,	PUNCT
ejpam-239	183	18	for	for	ADP
ejpam-239	183	19	some	some	DET
ejpam-239	183	20	z	z	PROPN
ejpam-239	183	21	∈	∈	PROPN
ejpam-239	183	22	r(by	r(by	NOUN
ejpam-239	183	23	condition	condition	NOUN
ejpam-239	183	24	(	(	PUNCT
ejpam-239	183	25	4	4	NUM
ejpam-239	183	26	)	)	PUNCT
ejpam-239	183	27	)	)	PUNCT
ejpam-239	183	28	.	.	PUNCT
ejpam-239	184	1	hence	hence	ADV
ejpam-239	184	2	(	(	PUNCT
ejpam-239	184	3	z]∗∗	z]∗∗	NOUN
ejpam-239	184	4	=	=	SYM
ejpam-239	184	5	{	{	PUNCT
ejpam-239	184	6	(	(	PUNCT
ejpam-239	184	7	x]∗	x]∗	PROPN
ejpam-239	184	8	∨	∨	PROPN
ejpam-239	184	9	(	(	PUNCT
ejpam-239	184	10	x	x	X
ejpam-239	184	11	′]∗}∗	′]∗}∗	PROPN
ejpam-239	184	12	=	=	SYM
ejpam-239	184	13	(	(	PUNCT
ejpam-239	184	14	x]∗∗	x]∗∗	PROPN
ejpam-239	184	15	∩	∩	X
ejpam-239	184	16	(	(	PUNCT
ejpam-239	184	17	x	x	X
ejpam-239	184	18	′]∗∗	′]∗∗	NOUN
ejpam-239	184	19	=	=	SYM
ejpam-239	184	20	(	(	PUNCT
ejpam-239	184	21	x]∗∗	x]∗∗	PROPN
ejpam-239	184	22	∩	∩	NOUN
ejpam-239	184	23	(	(	PUNCT
ejpam-239	184	24	x]∗∗∗	x]∗∗∗	PROPN
ejpam-239	184	25	=	=	SYM
ejpam-239	184	26	(	(	PUNCT
ejpam-239	184	27	0	0	NUM
ejpam-239	184	28	]	]	PUNCT
ejpam-239	184	29	.	.	PUNCT
ejpam-239	185	1	thus	thus	ADV
ejpam-239	185	2	(	(	PUNCT
ejpam-239	185	3	x]∗	x]∗	PROPN
ejpam-239	185	4	∨	∨	PROPN
ejpam-239	185	5	(	(	PUNCT
ejpam-239	185	6	x]∗∗	x]∗∗	PROPN
ejpam-239	185	7	=	=	SYM
ejpam-239	185	8	(	(	PUNCT
ejpam-239	185	9	z]∗	z]∗	NUM
ejpam-239	185	10	=	=	SYM
ejpam-239	185	11	(	(	PUNCT
ejpam-239	185	12	0]∗	0]∗	X
ejpam-239	185	13	=	=	PUNCT
ejpam-239	185	14	r.	r.	PROPN
ejpam-239	185	15	thus	thus	ADV
ejpam-239	185	16	(	(	PUNCT
ejpam-239	185	17	x]∗	x]∗	PROPN
ejpam-239	185	18	is	be	AUX
ejpam-239	185	19	a	a	DET
ejpam-239	185	20	direct	direct	ADJ
ejpam-239	185	21	summand	summand	NOUN
ejpam-239	185	22	of	of	ADP
ejpam-239	185	23	r.	r.	PROPN
ejpam-239	185	24	�	�	PROPN
ejpam-239	185	25	definition	definition	NOUN
ejpam-239	185	26	3.8	3.8	NUM
ejpam-239	185	27	.	.	PUNCT
ejpam-239	186	1	an	an	DET
ejpam-239	186	2	adl	adl	NOUN
ejpam-239	186	3	r	r	NOUN
ejpam-239	186	4	with	with	ADP
ejpam-239	186	5	0	0	NUM
ejpam-239	186	6	,	,	PUNCT
ejpam-239	186	7	is	be	AUX
ejpam-239	186	8	called	call	VERB
ejpam-239	186	9	disjunctive	disjunctive	ADJ
ejpam-239	186	10	iff	iff	PROPN
ejpam-239	186	11	for	for	ADP
ejpam-239	186	12	all	all	DET
ejpam-239	186	13	a	a	PRON
ejpam-239	186	14	,	,	PUNCT
ejpam-239	186	15	b	b	X
ejpam-239	186	16	∈	∈	PROPN
ejpam-239	186	17	r	r	NOUN
ejpam-239	186	18	,	,	PUNCT
ejpam-239	186	19	(	(	PUNCT
ejpam-239	186	20	a]∗	a]∗	PROPN
ejpam-239	186	21	=	=	SYM
ejpam-239	186	22	(	(	PUNCT
ejpam-239	186	23	b]∗	b]∗	PROPN
ejpam-239	186	24	implies	imply	VERB
ejpam-239	186	25	a	a	DET
ejpam-239	186	26	=	=	PROPN
ejpam-239	186	27	b.	b.	PROPN
ejpam-239	186	28	g.	g.	PROPN
ejpam-239	186	29	c.	c.	PROPN
ejpam-239	186	30	rao	rao	PROPN
ejpam-239	186	31	and	and	CCONJ
ejpam-239	186	32	m.	m.	PROPN
ejpam-239	186	33	sambasiva	sambasiva	PROPN
ejpam-239	186	34	rao	rao	PROPN
ejpam-239	186	35	/	/	SYM
ejpam-239	186	36	eur	eur	PROPN
ejpam-239	186	37	.	.	PUNCT
ejpam-239	187	1	j.	j.	PROPN
ejpam-239	187	2	pure	pure	PROPN
ejpam-239	187	3	appl	appl	PROPN
ejpam-239	187	4	.	.	PROPN
ejpam-239	187	5	math	math	PROPN
ejpam-239	187	6	,	,	PUNCT
ejpam-239	187	7	2	2	NUM
ejpam-239	187	8	(	(	PUNCT
ejpam-239	187	9	2009	2009	NUM
ejpam-239	187	10	)	)	PUNCT
ejpam-239	187	11	,	,	PUNCT
ejpam-239	187	12	(	(	PUNCT
ejpam-239	187	13	58	58	NUM
ejpam-239	187	14	-	-	SYM
ejpam-239	187	15	72	72	NUM
ejpam-239	187	16	)	)	PUNCT
ejpam-239	187	17	66	66	NUM
ejpam-239	187	18	example	example	NOUN
ejpam-239	187	19	3.9	3.9	NUM
ejpam-239	187	20	.	.	PUNCT
ejpam-239	188	1	let	let	VERB
ejpam-239	188	2	r=	r=	ADJ
ejpam-239	188	3	{	{	PUNCT
ejpam-239	188	4	0	0	NUM
ejpam-239	188	5	,	,	PUNCT
ejpam-239	188	6	a	a	DET
ejpam-239	188	7	,	,	PUNCT
ejpam-239	188	8	b	b	NOUN
ejpam-239	188	9	,	,	PUNCT
ejpam-239	188	10	c	c	AUX
ejpam-239	188	11	}	}	PUNCT
ejpam-239	188	12	be	be	AUX
ejpam-239	188	13	a	a	DET
ejpam-239	188	14	set	set	NOUN
ejpam-239	188	15	.	.	PUNCT
ejpam-239	189	1	define	define	VERB
ejpam-239	189	2	∨	∨	NOUN
ejpam-239	189	3	and	and	CCONJ
ejpam-239	189	4	∧	∧	NOUN
ejpam-239	189	5	on	on	ADP
ejpam-239	189	6	r	r	NOUN
ejpam-239	189	7	as	as	SCONJ
ejpam-239	189	8	follows	follow	VERB
ejpam-239	189	9	:	:	PUNCT
ejpam-239	189	10	∨	∨	NUM
ejpam-239	189	11	0	0	NUM
ejpam-239	189	12	a	a	DET
ejpam-239	189	13	b	b	NOUN
ejpam-239	189	14	c	c	NOUN
ejpam-239	189	15	0	0	NUM
ejpam-239	189	16	0	0	NUM
ejpam-239	189	17	a	a	DET
ejpam-239	189	18	b	b	NOUN
ejpam-239	189	19	c	c	ADP
ejpam-239	189	20	a	a	DET
ejpam-239	189	21	a	a	DET
ejpam-239	189	22	a	a	DET
ejpam-239	189	23	a	a	DET
ejpam-239	189	24	a	a	DET
ejpam-239	189	25	b	b	PROPN
ejpam-239	189	26	b	b	PROPN
ejpam-239	189	27	a	a	DET
ejpam-239	189	28	b	b	NOUN
ejpam-239	189	29	a	a	DET
ejpam-239	189	30	c	c	NOUN
ejpam-239	189	31	c	c	NOUN
ejpam-239	189	32	a	a	DET
ejpam-239	189	33	a	a	DET
ejpam-239	189	34	c	c	NOUN
ejpam-239	189	35	∧	∧	PROPN
ejpam-239	189	36	0	0	NUM
ejpam-239	189	37	a	a	DET
ejpam-239	189	38	b	b	X
ejpam-239	189	39	c	c	NOUN
ejpam-239	189	40	0	0	NUM
ejpam-239	189	41	0	0	NUM
ejpam-239	189	42	0	0	NUM
ejpam-239	189	43	0	0	NUM
ejpam-239	189	44	0	0	NUM
ejpam-239	189	45	a	a	DET
ejpam-239	189	46	0	0	NUM
ejpam-239	189	47	a	a	DET
ejpam-239	189	48	b	b	NOUN
ejpam-239	189	49	c	c	NOUN
ejpam-239	189	50	b	b	PROPN
ejpam-239	189	51	0	0	NUM
ejpam-239	189	52	b	b	PROPN
ejpam-239	189	53	b	b	PROPN
ejpam-239	189	54	0	0	NUM
ejpam-239	190	1	c	c	NOUN
ejpam-239	190	2	0	0	PUNCT
ejpam-239	191	1	c	c	NOUN
ejpam-239	191	2	0	0	PUNCT
ejpam-239	192	1	c	c	NOUN
ejpam-239	192	2	then	then	ADV
ejpam-239	192	3	clearly	clearly	ADV
ejpam-239	192	4	(	(	PUNCT
ejpam-239	192	5	r,∨,∧	r,∨,∧	NUM
ejpam-239	192	6	,	,	PUNCT
ejpam-239	192	7	0	0	NUM
ejpam-239	192	8	)	)	PUNCT
ejpam-239	192	9	is	be	AUX
ejpam-239	192	10	an	an	DET
ejpam-239	192	11	adl	adl	NOUN
ejpam-239	192	12	with	with	ADP
ejpam-239	192	13	0	0	NUM
ejpam-239	192	14	.	.	PUNCT
ejpam-239	193	1	now	now	ADV
ejpam-239	193	2	,	,	PUNCT
ejpam-239	193	3	(	(	PUNCT
ejpam-239	193	4	a]∗	a]∗	PROPN
ejpam-239	193	5	=	=	SYM
ejpam-239	193	6	(	(	PUNCT
ejpam-239	193	7	0	0	NUM
ejpam-239	193	8	]	]	PUNCT
ejpam-239	193	9	,	,	PUNCT
ejpam-239	193	10	(	(	PUNCT
ejpam-239	193	11	b]∗	b]∗	PROPN
ejpam-239	193	12	=	=	NUM
ejpam-239	193	13	{	{	PUNCT
ejpam-239	193	14	0	0	NUM
ejpam-239	193	15	,	,	PUNCT
ejpam-239	193	16	c	c	NOUN
ejpam-239	193	17	}	}	PUNCT
ejpam-239	193	18	and	and	CCONJ
ejpam-239	193	19	(	(	PUNCT
ejpam-239	193	20	c]∗	c]∗	PROPN
ejpam-239	193	21	=	=	PUNCT
ejpam-239	193	22	{	{	PUNCT
ejpam-239	193	23	0	0	NUM
ejpam-239	193	24	,	,	PUNCT
ejpam-239	193	25	b	b	NOUN
ejpam-239	193	26	}	}	PUNCT
ejpam-239	193	27	.	.	PUNCT
ejpam-239	194	1	thus	thus	ADV
ejpam-239	194	2	x	x	X
ejpam-239	194	3	6=	6=	NUM
ejpam-239	194	4	y	y	PROPN
ejpam-239	194	5	implies	imply	VERB
ejpam-239	194	6	that	that	SCONJ
ejpam-239	194	7	(	(	PUNCT
ejpam-239	194	8	x]∗	x]∗	PROPN
ejpam-239	194	9	6=	6=	PROPN
ejpam-239	194	10	(	(	PUNCT
ejpam-239	194	11	y]∗	y]∗	PROPN
ejpam-239	194	12	for	for	ADP
ejpam-239	194	13	all	all	DET
ejpam-239	194	14	x	x	SYM
ejpam-239	194	15	,	,	PUNCT
ejpam-239	194	16	y	y	PROPN
ejpam-239	194	17	∈	∈	PROPN
ejpam-239	194	18	r.	r.	PROPN
ejpam-239	194	19	hence	hence	ADV
ejpam-239	194	20	r	r	NOUN
ejpam-239	194	21	is	be	AUX
ejpam-239	194	22	disjunctive	disjunctive	ADJ
ejpam-239	194	23	.	.	PUNCT
ejpam-239	195	1	theorem	theorem	VERB
ejpam-239	195	2	3.10	3.10	NUM
ejpam-239	195	3	.	.	PUNCT
ejpam-239	196	1	a	a	DET
ejpam-239	196	2	disjunctive	disjunctive	ADJ
ejpam-239	196	3	adl	adl	NOUN
ejpam-239	196	4	r	r	NOUN
ejpam-239	196	5	is	be	AUX
ejpam-239	196	6	dually	dually	ADV
ejpam-239	196	7	isomorphic	isomorphic	ADJ
ejpam-239	196	8	toa0(r	toa0(r	NOUN
ejpam-239	196	9	)	)	PUNCT
ejpam-239	196	10	.	.	PUNCT
ejpam-239	197	1	proof	proof	NOUN
ejpam-239	197	2	:	:	PUNCT
ejpam-239	197	3	let	let	VERB
ejpam-239	197	4	r	r	PRON
ejpam-239	197	5	be	be	AUX
ejpam-239	197	6	a	a	DET
ejpam-239	197	7	disjunctive	disjunctive	ADJ
ejpam-239	197	8	adl	adl	PROPN
ejpam-239	197	9	.	.	PROPN
ejpam-239	197	10	define	define	VERB
ejpam-239	197	11	a	a	DET
ejpam-239	197	12	mapping	mapping	NOUN
ejpam-239	197	13	φ	φ	NOUN
ejpam-239	197	14	:	:	PUNCT
ejpam-239	197	15	r	r	NOUN
ejpam-239	197	16	−→a0(r	−→a0(r	NOUN
ejpam-239	197	17	)	)	PUNCT
ejpam-239	197	18	by	by	ADP
ejpam-239	197	19	φ(x	φ(x	NOUN
ejpam-239	197	20	)	)	PUNCT
ejpam-239	197	21	=	=	SYM
ejpam-239	197	22	(	(	PUNCT
ejpam-239	197	23	x]∗	x]∗	PROPN
ejpam-239	197	24	,	,	PUNCT
ejpam-239	197	25	for	for	SCONJ
ejpam-239	197	26	all	all	PRON
ejpam-239	197	27	x	x	PROPN
ejpam-239	197	28	∈	∈	PROPN
ejpam-239	197	29	r.	r.	PROPN
ejpam-239	197	30	clearly	clearly	ADV
ejpam-239	197	31	φ	φ	PROPN
ejpam-239	197	32	is	be	AUX
ejpam-239	197	33	well	well	ADV
ejpam-239	197	34	-	-	PUNCT
ejpam-239	197	35	defined	define	VERB
ejpam-239	197	36	.	.	PUNCT
ejpam-239	198	1	(	(	PUNCT
ejpam-239	198	2	i	i	NOUN
ejpam-239	198	3	)	)	PUNCT
ejpam-239	198	4	.	.	PUNCT
ejpam-239	199	1	let	let	VERB
ejpam-239	199	2	x	x	PRON
ejpam-239	199	3	,	,	PUNCT
ejpam-239	199	4	y	y	PROPN
ejpam-239	199	5	∈	∈	PROPN
ejpam-239	199	6	r	r	NOUN
ejpam-239	199	7	be	be	VERB
ejpam-239	199	8	such	such	ADJ
ejpam-239	199	9	that	that	SCONJ
ejpam-239	199	10	φ(x	φ(x	NOUN
ejpam-239	199	11	)	)	PUNCT
ejpam-239	199	12	=	=	SYM
ejpam-239	199	13	φ(y	φ(y	NOUN
ejpam-239	199	14	)	)	PUNCT
ejpam-239	199	15	.	.	PUNCT
ejpam-239	200	1	then	then	ADV
ejpam-239	200	2	(	(	PUNCT
ejpam-239	200	3	x]∗	x]∗	PROPN
ejpam-239	200	4	=	=	PROPN
ejpam-239	200	5	(	(	PUNCT
ejpam-239	200	6	y]∗.	y]∗.	PROPN
ejpam-239	200	7	since	since	SCONJ
ejpam-239	200	8	r	r	NOUN
ejpam-239	200	9	is	be	AUX
ejpam-239	200	10	disjunctive	disjunctive	ADJ
ejpam-239	200	11	,	,	PUNCT
ejpam-239	200	12	we	we	PRON
ejpam-239	200	13	get	get	VERB
ejpam-239	200	14	that	that	PRON
ejpam-239	200	15	x	x	PUNCT
ejpam-239	201	1	=	=	SYM
ejpam-239	201	2	y.	y.	PROPN
ejpam-239	201	3	therefore	therefore	ADV
ejpam-239	201	4	φ	φ	PROPN
ejpam-239	201	5	is	be	AUX
ejpam-239	201	6	one	one	NUM
ejpam-239	201	7	-	-	PUNCT
ejpam-239	201	8	one	one	NUM
ejpam-239	201	9	.	.	PUNCT
ejpam-239	202	1	(	(	PUNCT
ejpam-239	202	2	ii	ii	NOUN
ejpam-239	202	3	)	)	PUNCT
ejpam-239	202	4	.	.	PUNCT
ejpam-239	203	1	let	let	VERB
ejpam-239	203	2	y	y	PROPN
ejpam-239	203	3	∈	∈	PROPN
ejpam-239	203	4	a0(r	a0(r	PROPN
ejpam-239	203	5	)	)	PUNCT
ejpam-239	203	6	.	.	PUNCT
ejpam-239	204	1	then	then	ADV
ejpam-239	204	2	y	y	PROPN
ejpam-239	204	3	=	=	SYM
ejpam-239	204	4	(	(	PUNCT
ejpam-239	204	5	x]∗	x]∗	PROPN
ejpam-239	204	6	,	,	PUNCT
ejpam-239	204	7	for	for	ADP
ejpam-239	204	8	some	some	DET
ejpam-239	204	9	x	x	SYM
ejpam-239	204	10	∈	∈	PROPN
ejpam-239	204	11	r.	r.	NOUN
ejpam-239	204	12	now	now	ADV
ejpam-239	204	13	for	for	ADP
ejpam-239	204	14	this	this	PRON
ejpam-239	204	15	x	x	SYM
ejpam-239	204	16	,	,	PUNCT
ejpam-239	204	17	φ(x	φ(x	PROPN
ejpam-239	204	18	)	)	PUNCT
ejpam-239	204	19	=	=	PUNCT
ejpam-239	204	20	(	(	PUNCT
ejpam-239	204	21	x]∗	x]∗	PROPN
ejpam-239	204	22	=	=	SYM
ejpam-239	204	23	y.	y.	PROPN
ejpam-239	204	24	therefore	therefore	ADV
ejpam-239	204	25	φ	φ	PROPN
ejpam-239	204	26	is	be	AUX
ejpam-239	204	27	onto	onto	ADP
ejpam-239	204	28	.	.	PUNCT
ejpam-239	205	1	(	(	PUNCT
ejpam-239	205	2	iii	iii	NOUN
ejpam-239	205	3	)	)	PUNCT
ejpam-239	205	4	.	.	PUNCT
ejpam-239	206	1	let(x]∗	let(x]∗	PROPN
ejpam-239	206	2	,	,	PUNCT
ejpam-239	206	3	(	(	PUNCT
ejpam-239	206	4	y]∗	y]∗	PROPN
ejpam-239	206	5	∈a0(r	∈a0(r	NOUN
ejpam-239	206	6	)	)	PUNCT
ejpam-239	206	7	,	,	PUNCT
ejpam-239	206	8	where	where	SCONJ
ejpam-239	206	9	x	x	X
ejpam-239	206	10	,	,	PUNCT
ejpam-239	206	11	y	y	PROPN
ejpam-239	206	12	∈	∈	PROPN
ejpam-239	206	13	r.	r.	PROPN
ejpam-239	206	14	then	then	ADV
ejpam-239	206	15	φ(x	φ(x	PROPN
ejpam-239	206	16	∧	∧	PROPN
ejpam-239	206	17	y	y	PROPN
ejpam-239	206	18	)	)	PUNCT
ejpam-239	206	19	=	=	PRON
ejpam-239	207	1	(	(	PUNCT
ejpam-239	207	2	x	x	PART
ejpam-239	207	3	∧	∧	NOUN
ejpam-239	207	4	y]∗	y]∗	PROPN
ejpam-239	207	5	=	=	SYM
ejpam-239	207	6	(	(	PUNCT
ejpam-239	207	7	x]∗∨(y]∗	x]∗∨(y]∗	PROPN
ejpam-239	207	8	=	=	SYM
ejpam-239	207	9	φ(x)∨φ(y	φ(x)∨φ(y	PROPN
ejpam-239	207	10	)	)	PUNCT
ejpam-239	207	11	.	.	PUNCT
ejpam-239	208	1	again	again	ADV
ejpam-239	208	2	φ(x	φ(x	PROPN
ejpam-239	208	3	∨	∨	NUM
ejpam-239	208	4	y	y	PROPN
ejpam-239	208	5	)	)	PUNCT
ejpam-239	208	6	=	=	PRON
ejpam-239	208	7	(	(	PUNCT
ejpam-239	208	8	x	x	X
ejpam-239	208	9	∨	∨	ADP
ejpam-239	208	10	y]∗	y]∗	PROPN
ejpam-239	208	11	=	=	SYM
ejpam-239	208	12	(	(	PUNCT
ejpam-239	208	13	x]∗	x]∗	PROPN
ejpam-239	208	14	∩	∩	NOUN
ejpam-239	208	15	(	(	PUNCT
ejpam-239	208	16	y]∗	y]∗	PROPN
ejpam-239	208	17	=	=	SYM
ejpam-239	208	18	φ(x)∧φ(y	φ(x)∧φ(y	NUM
ejpam-239	208	19	)	)	PUNCT
ejpam-239	208	20	.	.	PUNCT
ejpam-239	209	1	hence	hence	ADV
ejpam-239	209	2	φ	φ	PROPN
ejpam-239	209	3	is	be	AUX
ejpam-239	209	4	a	a	DET
ejpam-239	209	5	dual	dual	ADJ
ejpam-239	209	6	isomorphism	isomorphism	NOUN
ejpam-239	209	7	.	.	PUNCT
ejpam-239	210	1	�	�	PROPN
ejpam-239	210	2	in	in	ADP
ejpam-239	210	3	an	an	DET
ejpam-239	210	4	adl	adl	NOUN
ejpam-239	210	5	r	r	NOUN
ejpam-239	210	6	with	with	ADP
ejpam-239	210	7	0	0	NUM
ejpam-239	210	8	,	,	PUNCT
ejpam-239	210	9	we	we	PRON
ejpam-239	210	10	know	know	VERB
ejpam-239	210	11	that	that	SCONJ
ejpam-239	210	12	a	a	DET
ejpam-239	210	13	maximal	maximal	ADJ
ejpam-239	210	14	element	element	NOUN
ejpam-239	210	15	is	be	AUX
ejpam-239	210	16	always	always	ADV
ejpam-239	210	17	a	a	DET
ejpam-239	210	18	dense	dense	ADJ
ejpam-239	210	19	element	element	NOUN
ejpam-239	210	20	.	.	PUNCT
ejpam-239	211	1	now	now	ADV
ejpam-239	211	2	we	we	PRON
ejpam-239	211	3	prove	prove	VERB
ejpam-239	211	4	the	the	DET
ejpam-239	211	5	converse	converse	NOUN
ejpam-239	211	6	in	in	ADP
ejpam-239	211	7	disjunctive	disjunctive	ADJ
ejpam-239	211	8	adl	adl	PROPN
ejpam-239	211	9	.	.	PUNCT
ejpam-239	211	10	theorem	theorem	PROPN
ejpam-239	211	11	3.11	3.11	NUM
ejpam-239	211	12	.	.	PUNCT
ejpam-239	212	1	if	if	SCONJ
ejpam-239	212	2	r	r	NOUN
ejpam-239	212	3	is	be	AUX
ejpam-239	212	4	a	a	DET
ejpam-239	212	5	disjunctive	disjunctive	ADJ
ejpam-239	212	6	adl	adl	NOUN
ejpam-239	212	7	,	,	PUNCT
ejpam-239	212	8	then	then	ADV
ejpam-239	212	9	every	every	DET
ejpam-239	212	10	dense	dense	ADJ
ejpam-239	212	11	element	element	NOUN
ejpam-239	212	12	of	of	ADP
ejpam-239	212	13	r	r	NOUN
ejpam-239	212	14	is	be	AUX
ejpam-239	212	15	a	a	DET
ejpam-239	212	16	maximal	maximal	ADJ
ejpam-239	212	17	element	element	NOUN
ejpam-239	212	18	.	.	PUNCT
ejpam-239	213	1	proof	proof	NOUN
ejpam-239	213	2	:	:	PUNCT
ejpam-239	213	3	assume	assume	VERB
ejpam-239	213	4	that	that	SCONJ
ejpam-239	213	5	r	r	NOUN
ejpam-239	213	6	is	be	AUX
ejpam-239	213	7	disjunctive	disjunctive	ADJ
ejpam-239	213	8	.	.	PUNCT
ejpam-239	214	1	let	let	VERB
ejpam-239	214	2	m	m	PRON
ejpam-239	214	3	be	be	AUX
ejpam-239	214	4	a	a	DET
ejpam-239	214	5	dense	dense	ADJ
ejpam-239	214	6	element	element	NOUN
ejpam-239	214	7	of	of	ADP
ejpam-239	214	8	r.	r.	PROPN
ejpam-239	214	9	that	that	PRON
ejpam-239	214	10	is	be	AUX
ejpam-239	214	11	(	(	PUNCT
ejpam-239	214	12	m]∗	m]∗	NOUN
ejpam-239	214	13	=	=	SYM
ejpam-239	214	14	(	(	PUNCT
ejpam-239	214	15	0	0	NUM
ejpam-239	214	16	]	]	PUNCT
ejpam-239	214	17	.	.	PUNCT
ejpam-239	215	1	for	for	ADP
ejpam-239	215	2	any	any	DET
ejpam-239	215	3	x	x	SYM
ejpam-239	215	4	∈	∈	PROPN
ejpam-239	215	5	r	r	NOUN
ejpam-239	215	6	,	,	PUNCT
ejpam-239	215	7	(	(	PUNCT
ejpam-239	215	8	m∨	m∨	PROPN
ejpam-239	215	9	x]∗	x]∗	PROPN
ejpam-239	215	10	=	=	PROPN
ejpam-239	215	11	(	(	PUNCT
ejpam-239	215	12	m]∗∩(x]∗	m]∗∩(x]∗	NOUN
ejpam-239	215	13	=	=	SYM
ejpam-239	215	14	(	(	PUNCT
ejpam-239	215	15	0]∩(x]∗	0]∩(x]∗	NUM
ejpam-239	215	16	=	=	SYM
ejpam-239	215	17	(	(	PUNCT
ejpam-239	215	18	0	0	NUM
ejpam-239	215	19	]	]	X
ejpam-239	215	20	=	=	X
ejpam-239	215	21	(	(	PUNCT
ejpam-239	215	22	m]∗.	m]∗.	NOUN
ejpam-239	215	23	since	since	SCONJ
ejpam-239	215	24	r	r	NOUN
ejpam-239	215	25	is	be	AUX
ejpam-239	215	26	disjunctive	disjunctive	ADJ
ejpam-239	215	27	,	,	PUNCT
ejpam-239	215	28	we	we	PRON
ejpam-239	215	29	get	get	VERB
ejpam-239	215	30	that	that	DET
ejpam-239	215	31	m∨	m∨	PROPN
ejpam-239	215	32	g.	g.	PROPN
ejpam-239	215	33	c.	c.	PROPN
ejpam-239	215	34	rao	rao	PROPN
ejpam-239	215	35	and	and	CCONJ
ejpam-239	215	36	m.	m.	PROPN
ejpam-239	215	37	sambasiva	sambasiva	PROPN
ejpam-239	215	38	rao	rao	PROPN
ejpam-239	215	39	/	/	SYM
ejpam-239	215	40	eur	eur	PROPN
ejpam-239	215	41	.	.	PUNCT
ejpam-239	216	1	j.	j.	PROPN
ejpam-239	216	2	pure	pure	PROPN
ejpam-239	216	3	appl	appl	PROPN
ejpam-239	216	4	.	.	PROPN
ejpam-239	216	5	math	math	PROPN
ejpam-239	216	6	,	,	PUNCT
ejpam-239	216	7	2	2	NUM
ejpam-239	216	8	(	(	PUNCT
ejpam-239	216	9	2009	2009	NUM
ejpam-239	216	10	)	)	PUNCT
ejpam-239	216	11	,	,	PUNCT
ejpam-239	216	12	(	(	PUNCT
ejpam-239	216	13	58	58	NUM
ejpam-239	216	14	-	-	SYM
ejpam-239	216	15	72	72	NUM
ejpam-239	216	16	)	)	PUNCT
ejpam-239	216	17	67	67	NUM
ejpam-239	216	18	x	x	X
ejpam-239	216	19	=	=	PUNCT
ejpam-239	216	20	m.	m.	NOUN
ejpam-239	217	1	therefore	therefore	ADV
ejpam-239	217	2	m	m	VERB
ejpam-239	217	3	is	be	AUX
ejpam-239	217	4	a	a	DET
ejpam-239	217	5	maximal	maximal	ADJ
ejpam-239	217	6	element	element	NOUN
ejpam-239	217	7	of	of	ADP
ejpam-239	217	8	r.	r.	PROPN
ejpam-239	217	9	�	�	PROPN
ejpam-239	217	10	we	we	PRON
ejpam-239	217	11	now	now	ADV
ejpam-239	217	12	characterize	characterize	VERB
ejpam-239	217	13	a	a	DET
ejpam-239	217	14	⋆-adl	⋆-adl	PROPN
ejpam-239	217	15	in	in	ADP
ejpam-239	217	16	terms	term	NOUN
ejpam-239	217	17	of	of	ADP
ejpam-239	217	18	it	it	PRON
ejpam-239	217	19	’s	’	VERB
ejpam-239	217	20	lattice	lattice	NOUN
ejpam-239	217	21	of	of	ADP
ejpam-239	217	22	annulets	annulet	NOUN
ejpam-239	217	23	in	in	ADP
ejpam-239	217	24	the	the	DET
ejpam-239	217	25	following	follow	VERB
ejpam-239	217	26	theorem	theorem	PROPN
ejpam-239	217	27	.	.	PUNCT
ejpam-239	217	28	theorem	theorem	PROPN
ejpam-239	217	29	3.12	3.12	NUM
ejpam-239	217	30	.	.	PUNCT
ejpam-239	218	1	let	let	VERB
ejpam-239	218	2	r	r	PRON
ejpam-239	218	3	be	be	AUX
ejpam-239	218	4	an	an	DET
ejpam-239	218	5	adl	adl	NOUN
ejpam-239	218	6	with	with	ADP
ejpam-239	218	7	0	0	NUM
ejpam-239	218	8	.	.	PUNCT
ejpam-239	219	1	then	then	ADV
ejpam-239	219	2	r	r	NOUN
ejpam-239	219	3	is	be	AUX
ejpam-239	219	4	a	a	DET
ejpam-239	219	5	⋆-adl	⋆-adl	PROPN
ejpam-239	219	6	iffa0(r	iffa0(r	NOUN
ejpam-239	219	7	)	)	PUNCT
ejpam-239	219	8	is	be	AUX
ejpam-239	219	9	a	a	DET
ejpam-239	219	10	boolean	boolean	ADJ
ejpam-239	219	11	subalgebra	subalgebra	NOUN
ejpam-239	219	12	ofa	ofa	PROPN
ejpam-239	219	13	(	(	PUNCT
ejpam-239	219	14	r	r	NOUN
ejpam-239	219	15	)	)	PUNCT
ejpam-239	219	16	.	.	PUNCT
ejpam-239	220	1	proof	proof	NOUN
ejpam-239	220	2	:	:	PUNCT
ejpam-239	220	3	assume	assume	VERB
ejpam-239	220	4	that	that	SCONJ
ejpam-239	220	5	r	r	NOUN
ejpam-239	220	6	is	be	AUX
ejpam-239	220	7	a	a	DET
ejpam-239	220	8	⋆-adl	⋆-adl	PROPN
ejpam-239	220	9	.	.	PUNCT
ejpam-239	221	1	then	then	ADV
ejpam-239	221	2	r	r	NOUN
ejpam-239	221	3	has	have	VERB
ejpam-239	221	4	a	a	DET
ejpam-239	221	5	dense	dense	ADJ
ejpam-239	221	6	element	element	NOUN
ejpam-239	221	7	,	,	PUNCT
ejpam-239	221	8	say	say	VERB
ejpam-239	221	9	d	d	INTJ
ejpam-239	221	10	.	.	PUNCT
ejpam-239	222	1	then	then	ADV
ejpam-239	222	2	(	(	PUNCT
ejpam-239	222	3	d]∗	d]∗	NOUN
ejpam-239	222	4	=	=	SYM
ejpam-239	222	5	(	(	PUNCT
ejpam-239	222	6	0	0	NUM
ejpam-239	222	7	]	]	PUNCT
ejpam-239	222	8	is	be	AUX
ejpam-239	222	9	the	the	DET
ejpam-239	222	10	least	least	ADJ
ejpam-239	222	11	element	element	ADJ
ejpam-239	222	12	and(0]∗	and(0]∗	NOUN
ejpam-239	222	13	is	be	AUX
ejpam-239	222	14	the	the	DET
ejpam-239	222	15	greatest	great	ADJ
ejpam-239	222	16	element	element	NOUN
ejpam-239	222	17	of	of	ADP
ejpam-239	222	18	the	the	DET
ejpam-239	222	19	sublatticea0(r	sublatticea0(r	NOUN
ejpam-239	222	20	)	)	PUNCT
ejpam-239	222	21	ofa	ofa	PROPN
ejpam-239	222	22	(	(	PUNCT
ejpam-239	222	23	r	r	NOUN
ejpam-239	222	24	)	)	PUNCT
ejpam-239	222	25	.	.	PUNCT
ejpam-239	223	1	let	let	VERB
ejpam-239	223	2	x	x	SYM
ejpam-239	223	3	∈	∈	PROPN
ejpam-239	223	4	r.	r.	PROPN
ejpam-239	223	5	since	since	SCONJ
ejpam-239	223	6	r	r	NOUN
ejpam-239	223	7	is	be	AUX
ejpam-239	223	8	a	a	DET
ejpam-239	223	9	⋆-adl	⋆-adl	PROPN
ejpam-239	223	10	,	,	PUNCT
ejpam-239	223	11	there	there	PRON
ejpam-239	223	12	exists	exist	VERB
ejpam-239	223	13	x	x	PUNCT
ejpam-239	223	14	′	′	NUM
ejpam-239	223	15	∈	∈	NOUN
ejpam-239	223	16	r	r	NOUN
ejpam-239	223	17	such	such	ADJ
ejpam-239	223	18	that	that	PRON
ejpam-239	223	19	(	(	PUNCT
ejpam-239	223	20	x]∗∗	x]∗∗	NOUN
ejpam-239	223	21	=	=	SYM
ejpam-239	223	22	(	(	PUNCT
ejpam-239	223	23	x	x	PUNCT
ejpam-239	223	24	′]∗.	′]∗.	PROPN
ejpam-239	223	25	we	we	PRON
ejpam-239	223	26	now	now	ADV
ejpam-239	223	27	show	show	VERB
ejpam-239	223	28	that	that	SCONJ
ejpam-239	223	29	(	(	PUNCT
ejpam-239	223	30	x	x	SYM
ejpam-239	223	31	′]∗	′]∗	PROPN
ejpam-239	223	32	is	be	AUX
ejpam-239	223	33	the	the	DET
ejpam-239	223	34	complement	complement	NOUN
ejpam-239	223	35	of	of	ADP
ejpam-239	223	36	(	(	PUNCT
ejpam-239	223	37	x]∗	x]∗	PROPN
ejpam-239	223	38	ina0(r	ina0(r	NOUN
ejpam-239	223	39	)	)	PUNCT
ejpam-239	223	40	,	,	PUNCT
ejpam-239	223	41	for	for	ADP
ejpam-239	223	42	each	each	DET
ejpam-239	223	43	x	x	PROPN
ejpam-239	223	44	∈	∈	PROPN
ejpam-239	223	45	r.	r.	PROPN
ejpam-239	223	46	now	now	ADV
ejpam-239	223	47	(	(	PUNCT
ejpam-239	223	48	x]∗∩(x	x]∗∩(x	PROPN
ejpam-239	223	49	′]∗	′]∗	PROPN
ejpam-239	223	50	=	=	PUNCT
ejpam-239	223	51	(	(	PUNCT
ejpam-239	223	52	x]∗∩(x]∗∗	x]∗∩(x]∗∗	PROPN
ejpam-239	223	53	=	=	PUNCT
ejpam-239	223	54	(	(	PUNCT
ejpam-239	223	55	0	0	NUM
ejpam-239	223	56	]	]	PUNCT
ejpam-239	223	57	and	and	CCONJ
ejpam-239	223	58	(	(	PUNCT
ejpam-239	223	59	x]∗∨(x	x]∗∨(x	PROPN
ejpam-239	223	60	′]∗	′]∗	PROPN
ejpam-239	223	61	=	=	SYM
ejpam-239	223	62	�	�	PROPN
ejpam-239	223	63	(	(	PUNCT
ejpam-239	223	64	x]∗∗	x]∗∗	PROPN
ejpam-239	223	65	∩	∩	X
ejpam-239	223	66	(	(	PUNCT
ejpam-239	223	67	x	x	SYM
ejpam-239	223	68	′]∗∗	′]∗∗	PROPN
ejpam-239	223	69	�	�	PROPN
ejpam-239	223	70	∗	∗	NOUN
ejpam-239	223	71	=	=	PUNCT
ejpam-239	224	1	[	[	X
ejpam-239	224	2	(	(	PUNCT
ejpam-239	224	3	x]∗∗	x]∗∗	NOUN
ejpam-239	224	4	∩	∩	X
ejpam-239	224	5	(	(	PUNCT
ejpam-239	224	6	x]∗∗∗]∗	x]∗∗∗]∗	X
ejpam-239	224	7	=	=	PUNCT
ejpam-239	224	8	[	[	X
ejpam-239	224	9	(	(	PUNCT
ejpam-239	224	10	x]∗∗	x]∗∗	NOUN
ejpam-239	224	11	∩	∩	NOUN
ejpam-239	224	12	(	(	PUNCT
ejpam-239	224	13	x]∗]∗	x]∗]∗	PROPN
ejpam-239	224	14	=	=	SYM
ejpam-239	224	15	(	(	PUNCT
ejpam-239	224	16	0]∗.	0]∗.	NUM
ejpam-239	224	17	thusa0(r	thusa0(r	NUM
ejpam-239	224	18	)	)	PUNCT
ejpam-239	224	19	is	be	AUX
ejpam-239	224	20	a	a	DET
ejpam-239	224	21	boolean	boolean	ADJ
ejpam-239	224	22	subalgebra	subalgebra	NOUN
ejpam-239	224	23	ofa	ofa	PROPN
ejpam-239	224	24	(	(	PUNCT
ejpam-239	224	25	r	r	NOUN
ejpam-239	224	26	)	)	PUNCT
ejpam-239	224	27	.	.	PUNCT
ejpam-239	225	1	conversely	conversely	ADV
ejpam-239	225	2	assume	assume	VERB
ejpam-239	225	3	that	that	SCONJ
ejpam-239	225	4	a0(r	a0(r	PROPN
ejpam-239	225	5	)	)	PUNCT
ejpam-239	225	6	is	be	AUX
ejpam-239	225	7	a	a	DET
ejpam-239	225	8	boolean	boolean	ADJ
ejpam-239	225	9	subalgebra	subalgebra	NOUN
ejpam-239	225	10	ofa	ofa	PROPN
ejpam-239	225	11	(	(	PUNCT
ejpam-239	225	12	r	r	NOUN
ejpam-239	225	13	)	)	PUNCT
ejpam-239	225	14	.	.	PUNCT
ejpam-239	226	1	let	let	VERB
ejpam-239	226	2	x	x	SYM
ejpam-239	226	3	∈	∈	PROPN
ejpam-239	226	4	r.	r.	PROPN
ejpam-239	226	5	then	then	ADV
ejpam-239	226	6	(	(	PUNCT
ejpam-239	226	7	x]∗	x]∗	PROPN
ejpam-239	226	8	∈	∈	PROPN
ejpam-239	226	9	a0(r	a0(r	PROPN
ejpam-239	226	10	)	)	PUNCT
ejpam-239	226	11	.	.	PUNCT
ejpam-239	227	1	since	since	SCONJ
ejpam-239	227	2	a0(r	a0(r	PROPN
ejpam-239	227	3	)	)	PUNCT
ejpam-239	227	4	is	be	AUX
ejpam-239	227	5	a	a	DET
ejpam-239	227	6	subalgebra	subalgebra	NOUN
ejpam-239	227	7	of	of	ADP
ejpam-239	227	8	a	a	DET
ejpam-239	227	9	(	(	PUNCT
ejpam-239	227	10	r	r	NOUN
ejpam-239	227	11	)	)	PUNCT
ejpam-239	227	12	,	,	PUNCT
ejpam-239	227	13	there	there	PRON
ejpam-239	227	14	exists	exist	VERB
ejpam-239	227	15	(	(	PUNCT
ejpam-239	227	16	y]∗	y]∗	PROPN
ejpam-239	227	17	∈	∈	PROPN
ejpam-239	227	18	a0(r	a0(r	PROPN
ejpam-239	227	19	)	)	PUNCT
ejpam-239	227	20	,	,	PUNCT
ejpam-239	227	21	with	with	ADP
ejpam-239	227	22	y	y	PROPN
ejpam-239	227	23	∈	∈	PROPN
ejpam-239	227	24	r	r	NOUN
ejpam-239	227	25	such	such	ADJ
ejpam-239	227	26	that	that	PRON
ejpam-239	227	27	(	(	PUNCT
ejpam-239	227	28	x]∗	x]∗	PROPN
ejpam-239	227	29	∩	∩	NOUN
ejpam-239	227	30	(	(	PUNCT
ejpam-239	227	31	y]∗	y]∗	PROPN
ejpam-239	227	32	=	=	SYM
ejpam-239	227	33	(	(	PUNCT
ejpam-239	227	34	0	0	NUM
ejpam-239	227	35	]	]	PUNCT
ejpam-239	227	36	and	and	CCONJ
ejpam-239	227	37	(	(	PUNCT
ejpam-239	227	38	x]∗	x]∗	PROPN
ejpam-239	227	39	∨	∨	PROPN
ejpam-239	227	40	(	(	PUNCT
ejpam-239	227	41	y]∗	y]∗	PROPN
ejpam-239	227	42	=	=	SYM
ejpam-239	227	43	(	(	PUNCT
ejpam-239	227	44	0]∗.	0]∗.	NUM
ejpam-239	227	45	now	now	ADV
ejpam-239	227	46	(	(	PUNCT
ejpam-239	227	47	x]∗	x]∗	PROPN
ejpam-239	227	48	∨	∨	PROPN
ejpam-239	227	49	(	(	PUNCT
ejpam-239	227	50	y]∗	y]∗	PROPN
ejpam-239	227	51	=	=	SYM
ejpam-239	227	52	(	(	PUNCT
ejpam-239	227	53	0]∗	0]∗	X
ejpam-239	227	54	⇒	⇒	NOUN
ejpam-239	227	55	(	(	PUNCT
ejpam-239	227	56	x	x	PUNCT
ejpam-239	227	57	∧	∧	NOUN
ejpam-239	227	58	y]∗	y]∗	PROPN
ejpam-239	227	59	=	=	PUNCT
ejpam-239	227	60	(	(	PUNCT
ejpam-239	227	61	0]∗	0]∗	X
ejpam-239	227	62	=	=	PUNCT
ejpam-239	227	63	r	r	NOUN
ejpam-239	227	64	⇒	⇒	NOUN
ejpam-239	227	65	x	x	PUNCT
ejpam-239	227	66	∧	∧	NOUN
ejpam-239	227	67	y	y	PROPN
ejpam-239	227	68	=	=	NOUN
ejpam-239	227	69	0	0	X
ejpam-239	227	70	.	.	PUNCT
ejpam-239	228	1	again	again	ADV
ejpam-239	228	2	,	,	PUNCT
ejpam-239	228	3	(	(	PUNCT
ejpam-239	228	4	x]∗	x]∗	PROPN
ejpam-239	228	5	∩	∩	PROPN
ejpam-239	228	6	(	(	PUNCT
ejpam-239	228	7	y]∗	y]∗	PROPN
ejpam-239	228	8	=	=	SYM
ejpam-239	228	9	(	(	PUNCT
ejpam-239	228	10	0	0	NUM
ejpam-239	228	11	]	]	PUNCT
ejpam-239	228	12	⇒	⇒	NOUN
ejpam-239	228	13	(	(	PUNCT
ejpam-239	228	14	x	x	X
ejpam-239	228	15	∨	∨	ADP
ejpam-239	228	16	y]∗	y]∗	PROPN
ejpam-239	228	17	=	=	SYM
ejpam-239	228	18	(	(	PUNCT
ejpam-239	228	19	0	0	X
ejpam-239	228	20	]	]	PUNCT
ejpam-239	228	21	⇒	⇒	NOUN
ejpam-239	228	22	x	x	PUNCT
ejpam-239	228	23	∨	∨	NUM
ejpam-239	228	24	y	y	PROPN
ejpam-239	228	25	is	be	AUX
ejpam-239	228	26	a	a	DET
ejpam-239	228	27	dense	dense	ADJ
ejpam-239	228	28	element	element	NOUN
ejpam-239	228	29	.	.	PUNCT
ejpam-239	229	1	thus	thus	ADV
ejpam-239	229	2	we	we	PRON
ejpam-239	229	3	proved	prove	VERB
ejpam-239	229	4	that	that	SCONJ
ejpam-239	229	5	for	for	ADP
ejpam-239	229	6	each	each	DET
ejpam-239	229	7	x	x	SYM
ejpam-239	229	8	∈	∈	PROPN
ejpam-239	229	9	r	r	NOUN
ejpam-239	229	10	,	,	PUNCT
ejpam-239	229	11	there	there	PRON
ejpam-239	229	12	exists	exist	VERB
ejpam-239	229	13	y	y	PROPN
ejpam-239	229	14	∈	∈	PROPN
ejpam-239	229	15	r	r	NOUN
ejpam-239	229	16	such	such	ADJ
ejpam-239	229	17	that	that	SCONJ
ejpam-239	229	18	x	x	SYM
ejpam-239	229	19	∧	∧	NOUN
ejpam-239	229	20	y	y	NOUN
ejpam-239	229	21	=	=	SYM
ejpam-239	229	22	0	0	PUNCT
ejpam-239	230	1	and	and	CCONJ
ejpam-239	230	2	x	x	SYM
ejpam-239	230	3	∨	∨	NOUN
ejpam-239	230	4	y	y	PROPN
ejpam-239	230	5	is	be	AUX
ejpam-239	230	6	a	a	DET
ejpam-239	230	7	dense	dense	ADJ
ejpam-239	230	8	element	element	NOUN
ejpam-239	230	9	.	.	PUNCT
ejpam-239	231	1	therefore	therefore	ADV
ejpam-239	231	2	r	r	NOUN
ejpam-239	231	3	is	be	AUX
ejpam-239	231	4	a	a	DET
ejpam-239	231	5	⋆-adl	⋆-adl	PROPN
ejpam-239	231	6	.	.	PUNCT
ejpam-239	232	1	�	�	PROPN
ejpam-239	232	2	definition	definition	NOUN
ejpam-239	232	3	3.13	3.13	NUM
ejpam-239	232	4	.	.	PUNCT
ejpam-239	233	1	an	an	DET
ejpam-239	233	2	adl	adl	PROPN
ejpam-239	233	3	r	r	NOUN
ejpam-239	233	4	with	with	ADP
ejpam-239	233	5	0	0	NUM
ejpam-239	233	6	is	be	AUX
ejpam-239	233	7	called	call	VERB
ejpam-239	233	8	sectionally	sectionally	ADV
ejpam-239	233	9	⋆-adl	⋆-adl	PROPN
ejpam-239	233	10	iff	iff	PROPN
ejpam-239	233	11	for	for	ADP
ejpam-239	233	12	any	any	DET
ejpam-239	233	13	x	x	X
ejpam-239	233	14	(	(	PUNCT
ejpam-239	233	15	6=	6=	NOUN
ejpam-239	233	16	0	0	NUM
ejpam-239	233	17	)	)	PUNCT
ejpam-239	233	18	∈	∈	PROPN
ejpam-239	233	19	r	r	NOUN
ejpam-239	233	20	,	,	PUNCT
ejpam-239	233	21	the	the	DET
ejpam-239	233	22	interval	interval	NOUN
ejpam-239	233	23	[	[	X
ejpam-239	233	24	0	0	NUM
ejpam-239	233	25	,	,	PUNCT
ejpam-239	233	26	x	x	X
ejpam-239	233	27	]	]	X
ejpam-239	233	28	is	be	AUX
ejpam-239	233	29	a	a	DET
ejpam-239	233	30	⋆-adl	⋆-adl	PROPN
ejpam-239	233	31	.	.	PUNCT
ejpam-239	234	1	before	before	ADP
ejpam-239	234	2	proving	prove	VERB
ejpam-239	234	3	the	the	DET
ejpam-239	234	4	next	next	ADJ
ejpam-239	234	5	theorem	theorem	NOUN
ejpam-239	234	6	,	,	PUNCT
ejpam-239	234	7	we	we	PRON
ejpam-239	234	8	need	need	VERB
ejpam-239	234	9	the	the	DET
ejpam-239	234	10	following	follow	VERB
ejpam-239	234	11	lemma	lemma	PROPN
ejpam-239	234	12	.	.	PUNCT
ejpam-239	235	1	lemma	lemma	PROPN
ejpam-239	235	2	3.14	3.14	NUM
ejpam-239	235	3	.	.	PUNCT
ejpam-239	236	1	let	let	VERB
ejpam-239	236	2	i	i	PRON
ejpam-239	236	3	,	,	PUNCT
ejpam-239	236	4	j	j	PROPN
ejpam-239	236	5	be	be	VERB
ejpam-239	236	6	two	two	NUM
ejpam-239	236	7	ideals	ideal	NOUN
ejpam-239	236	8	in	in	ADP
ejpam-239	236	9	an	an	DET
ejpam-239	236	10	adl	adl	PROPN
ejpam-239	236	11	r.	r.	PROPN
ejpam-239	236	12	if	if	SCONJ
ejpam-239	236	13	i	i	PRON
ejpam-239	236	14	∩	∩	VERB
ejpam-239	236	15	j	j	PROPN
ejpam-239	236	16	and	and	CCONJ
ejpam-239	236	17	i	i	PROPN
ejpam-239	236	18	∨	∨	PROPN
ejpam-239	236	19	j	j	PROPN
ejpam-239	236	20	(	(	PUNCT
ejpam-239	236	21	i.e.	i.e.	X
ejpam-239	236	22	the	the	DET
ejpam-239	236	23	infimum	infimum	NOUN
ejpam-239	236	24	and	and	CCONJ
ejpam-239	236	25	the	the	DET
ejpam-239	236	26	supremum	supremum	NOUN
ejpam-239	236	27	of	of	ADP
ejpam-239	236	28	i	i	PRON
ejpam-239	236	29	,	,	PUNCT
ejpam-239	236	30	j	j	PROPN
ejpam-239	236	31	in	in	ADP
ejpam-239	236	32	the	the	DET
ejpam-239	236	33	distributive	distributive	ADJ
ejpam-239	236	34	lattice	lattice	NOUN
ejpam-239	237	1	i	i	PRON
ejpam-239	237	2	(	(	PUNCT
ejpam-239	237	3	r	r	NOUN
ejpam-239	237	4	)	)	PUNCT
ejpam-239	237	5	)	)	PUNCT
ejpam-239	237	6	are	be	AUX
ejpam-239	237	7	both	both	PRON
ejpam-239	237	8	principal	principal	ADJ
ejpam-239	237	9	ideals	ideal	NOUN
ejpam-239	237	10	,	,	PUNCT
ejpam-239	237	11	then	then	ADV
ejpam-239	237	12	i	i	PRON
ejpam-239	237	13	,	,	PUNCT
ejpam-239	237	14	j	j	PROPN
ejpam-239	237	15	are	be	AUX
ejpam-239	237	16	also	also	ADV
ejpam-239	237	17	principal	principal	ADJ
ejpam-239	237	18	ideals	ideal	NOUN
ejpam-239	237	19	.	.	PUNCT
ejpam-239	238	1	proof	proof	NOUN
ejpam-239	238	2	:	:	PUNCT
ejpam-239	238	3	suppose	suppose	VERB
ejpam-239	238	4	i	i	PRON
ejpam-239	238	5	∨	∨	PROPN
ejpam-239	238	6	j	j	PROPN
ejpam-239	238	7	=	=	PRON
ejpam-239	238	8	(	(	PUNCT
ejpam-239	238	9	a	a	X
ejpam-239	238	10	]	]	X
ejpam-239	238	11	and	and	CCONJ
ejpam-239	238	12	i	i	PROPN
ejpam-239	238	13	∩	∩	ADJ
ejpam-239	238	14	j	j	PROPN
ejpam-239	238	15	=	=	PRON
ejpam-239	238	16	(	(	PUNCT
ejpam-239	238	17	b	b	NOUN
ejpam-239	238	18	]	]	X
ejpam-239	238	19	,	,	PUNCT
ejpam-239	238	20	for	for	ADP
ejpam-239	238	21	some	some	PRON
ejpam-239	238	22	a	a	PRON
ejpam-239	238	23	,	,	PUNCT
ejpam-239	238	24	b	b	PROPN
ejpam-239	238	25	∈	∈	PROPN
ejpam-239	238	26	r.	r.	NOUN
ejpam-239	238	27	now	now	ADV
ejpam-239	238	28	a	a	DET
ejpam-239	238	29	∈	∈	NOUN
ejpam-239	238	30	i	i	PRON
ejpam-239	238	31	∨	∨	PROPN
ejpam-239	238	32	j	j	PROPN
ejpam-239	238	33	⇒	⇒	VERB
ejpam-239	238	34	a	a	DET
ejpam-239	238	35	=	=	SYM
ejpam-239	238	36	c	c	X
ejpam-239	238	37	∨	∨	NUM
ejpam-239	238	38	d	d	NOUN
ejpam-239	238	39	for	for	ADP
ejpam-239	238	40	some	some	PRON
ejpam-239	238	41	c	c	NOUN
ejpam-239	238	42	∈	∈	PROPN
ejpam-239	239	1	i	i	PRON
ejpam-239	239	2	and	and	CCONJ
ejpam-239	239	3	d	d	PROPN
ejpam-239	239	4	∈	∈	PROPN
ejpam-239	239	5	j	j	PROPN
ejpam-239	239	6	.	.	PUNCT
ejpam-239	240	1	then	then	ADV
ejpam-239	240	2	c	c	PROPN
ejpam-239	240	3	∨	∨	X
ejpam-239	240	4	(	(	PUNCT
ejpam-239	240	5	b	b	PROPN
ejpam-239	240	6	∧	∧	PROPN
ejpam-239	240	7	d	d	PROPN
ejpam-239	240	8	)	)	PUNCT
ejpam-239	240	9	∈	∈	PROPN
ejpam-239	241	1	i	i	PRON
ejpam-239	241	2	.	.	PUNCT
ejpam-239	242	1	so	so	ADV
ejpam-239	242	2	that	that	SCONJ
ejpam-239	242	3	g.	g.	PROPN
ejpam-239	242	4	c.	c.	PROPN
ejpam-239	242	5	rao	rao	PROPN
ejpam-239	242	6	and	and	CCONJ
ejpam-239	242	7	m.	m.	PROPN
ejpam-239	242	8	sambasiva	sambasiva	PROPN
ejpam-239	242	9	rao	rao	PROPN
ejpam-239	242	10	/	/	SYM
ejpam-239	242	11	eur	eur	PROPN
ejpam-239	242	12	.	.	PUNCT
ejpam-239	243	1	j.	j.	PROPN
ejpam-239	243	2	pure	pure	PROPN
ejpam-239	243	3	appl	appl	PROPN
ejpam-239	243	4	.	.	PROPN
ejpam-239	243	5	math	math	PROPN
ejpam-239	243	6	,	,	PUNCT
ejpam-239	243	7	2	2	NUM
ejpam-239	243	8	(	(	PUNCT
ejpam-239	243	9	2009	2009	NUM
ejpam-239	243	10	)	)	PUNCT
ejpam-239	243	11	,	,	PUNCT
ejpam-239	243	12	(	(	PUNCT
ejpam-239	243	13	58	58	NUM
ejpam-239	243	14	-	-	SYM
ejpam-239	243	15	72	72	NUM
ejpam-239	243	16	)	)	PUNCT
ejpam-239	243	17	68	68	NUM
ejpam-239	243	18	(	(	PUNCT
ejpam-239	243	19	c	c	NOUN
ejpam-239	243	20	∨	∨	X
ejpam-239	243	21	(	(	PUNCT
ejpam-239	243	22	b	b	PROPN
ejpam-239	243	23	∧	∧	PROPN
ejpam-239	243	24	d	d	PROPN
ejpam-239	243	25	)	)	PUNCT
ejpam-239	243	26	]	]	PUNCT
ejpam-239	243	27	⊆	⊆	NUM
ejpam-239	243	28	i	i	PRON
ejpam-239	243	29	.	.	PUNCT
ejpam-239	244	1	we	we	PRON
ejpam-239	244	2	now	now	ADV
ejpam-239	244	3	prove	prove	VERB
ejpam-239	244	4	that	that	SCONJ
ejpam-239	244	5	i	i	PRON
ejpam-239	244	6	=	=	PUNCT
ejpam-239	244	7	(	(	PUNCT
ejpam-239	244	8	c	c	PROPN
ejpam-239	244	9	∨	∨	X
ejpam-239	244	10	(	(	PUNCT
ejpam-239	244	11	b	b	PROPN
ejpam-239	244	12	∧	∧	PROPN
ejpam-239	244	13	d	d	PROPN
ejpam-239	244	14	)	)	PUNCT
ejpam-239	244	15	]	]	PUNCT
ejpam-239	244	16	.	.	PUNCT
ejpam-239	245	1	let	let	VERB
ejpam-239	245	2	x	x	SYM
ejpam-239	246	1	∈	∈	PROPN
ejpam-239	247	1	i	i	PRON
ejpam-239	247	2	.	.	PUNCT
ejpam-239	248	1	then	then	ADV
ejpam-239	248	2	x	x	SYM
ejpam-239	248	3	∈	∈	PROPN
ejpam-239	248	4	i	i	PRON
ejpam-239	248	5	∨	∨	NOUN
ejpam-239	248	6	j	j	PROPN
ejpam-239	248	7	=	=	PRON
ejpam-239	248	8	(	(	PUNCT
ejpam-239	248	9	a	a	X
ejpam-239	248	10	]	]	X
ejpam-239	248	11	.	.	PUNCT
ejpam-239	249	1	so	so	ADV
ejpam-239	249	2	x	x	X
ejpam-239	249	3	=	=	PUNCT
ejpam-239	249	4	a	a	DET
ejpam-239	249	5	∧	∧	NOUN
ejpam-239	249	6	x	x	X
ejpam-239	249	7	=	=	PUNCT
ejpam-239	249	8	(	(	PUNCT
ejpam-239	249	9	c	c	NOUN
ejpam-239	249	10	∨	∨	NOUN
ejpam-239	249	11	d)∧	d)∧	PROPN
ejpam-239	249	12	x	x	X
ejpam-239	249	13	=	=	PUNCT
ejpam-239	249	14	(	(	PUNCT
ejpam-239	249	15	c	c	X
ejpam-239	249	16	∧	∧	PROPN
ejpam-239	249	17	x)∨	x)∨	PROPN
ejpam-239	249	18	(	(	PUNCT
ejpam-239	249	19	d	d	PROPN
ejpam-239	249	20	∧	∧	PROPN
ejpam-239	249	21	x	x	NOUN
ejpam-239	249	22	)	)	PUNCT
ejpam-239	249	23	−→	−→	NOUN
ejpam-239	249	24	(	(	PUNCT
ejpam-239	249	25	1	1	NUM
ejpam-239	249	26	)	)	PUNCT
ejpam-239	249	27	.	.	PUNCT
ejpam-239	250	1	now	now	ADV
ejpam-239	250	2	x	x	SYM
ejpam-239	250	3	∈	∈	PROPN
ejpam-239	250	4	i	i	PRON
ejpam-239	250	5	and	and	CCONJ
ejpam-239	250	6	d	d	PROPN
ejpam-239	250	7	∈	∈	PROPN
ejpam-239	250	8	j	j	PROPN
ejpam-239	250	9	⇒	⇒	VERB
ejpam-239	250	10	x	x	X
ejpam-239	250	11	∧	∧	NOUN
ejpam-239	250	12	d	d	X
ejpam-239	250	13	∈	∈	PROPN
ejpam-239	250	14	i	i	PRON
ejpam-239	250	15	∩	∩	NOUN
ejpam-239	250	16	j	j	PROPN
ejpam-239	251	1	=	=	PRON
ejpam-239	252	1	(	(	PUNCT
ejpam-239	252	2	b]⇒	b]⇒	PROPN
ejpam-239	252	3	d	d	PROPN
ejpam-239	252	4	∧	∧	PROPN
ejpam-239	252	5	x	x	SYM
ejpam-239	252	6	∈	∈	PROPN
ejpam-239	252	7	(	(	PUNCT
ejpam-239	252	8	b	b	NOUN
ejpam-239	252	9	]	]	X
ejpam-239	252	10	.	.	PUNCT
ejpam-239	253	1	hence	hence	ADV
ejpam-239	253	2	d	d	X
ejpam-239	253	3	∧	∧	NOUN
ejpam-239	253	4	x	x	X
ejpam-239	253	5	=	=	SYM
ejpam-239	253	6	b	b	PROPN
ejpam-239	253	7	∧	∧	PROPN
ejpam-239	253	8	d	d	PROPN
ejpam-239	253	9	∧	∧	PROPN
ejpam-239	253	10	x	x	PUNCT
ejpam-239	253	11	−→	−→	NOUN
ejpam-239	253	12	(	(	PUNCT
ejpam-239	253	13	2	2	NUM
ejpam-239	253	14	)	)	PUNCT
ejpam-239	253	15	.	.	PUNCT
ejpam-239	254	1	from	from	ADP
ejpam-239	254	2	(	(	PUNCT
ejpam-239	254	3	1	1	NUM
ejpam-239	254	4	)	)	PUNCT
ejpam-239	254	5	and	and	CCONJ
ejpam-239	254	6	(	(	PUNCT
ejpam-239	254	7	2	2	NUM
ejpam-239	254	8	)	)	PUNCT
ejpam-239	254	9	,	,	PUNCT
ejpam-239	254	10	we	we	PRON
ejpam-239	254	11	can	can	AUX
ejpam-239	254	12	obtain	obtain	VERB
ejpam-239	254	13	x	x	X
ejpam-239	254	14	=	=	SYM
ejpam-239	254	15	(	(	PUNCT
ejpam-239	254	16	c	c	PROPN
ejpam-239	254	17	∧	∧	PROPN
ejpam-239	254	18	x	x	X
ejpam-239	254	19	)	)	PUNCT
ejpam-239	254	20	∨	∨	PROPN
ejpam-239	254	21	(	(	PUNCT
ejpam-239	254	22	b	b	PROPN
ejpam-239	254	23	∧	∧	PROPN
ejpam-239	254	24	d	d	PROPN
ejpam-239	254	25	∧	∧	PROPN
ejpam-239	254	26	x	x	X
ejpam-239	254	27	)	)	PUNCT
ejpam-239	254	28	=	=	PUNCT
ejpam-239	255	1	[	[	X
ejpam-239	255	2	c	c	X
ejpam-239	255	3	∨	∨	X
ejpam-239	255	4	(	(	PUNCT
ejpam-239	255	5	b	b	PROPN
ejpam-239	255	6	∧	∧	PROPN
ejpam-239	255	7	d	d	PROPN
ejpam-239	255	8	)	)	PUNCT
ejpam-239	255	9	]	]	PUNCT
ejpam-239	256	1	∧	∧	NOUN
ejpam-239	256	2	x	x	INTJ
ejpam-239	256	3	.	.	PUNCT
ejpam-239	257	1	hence	hence	ADV
ejpam-239	257	2	x	x	X
ejpam-239	257	3	∈	∈	PROPN
ejpam-239	257	4	(	(	PUNCT
ejpam-239	257	5	c	c	NOUN
ejpam-239	257	6	∨	∨	X
ejpam-239	257	7	(	(	PUNCT
ejpam-239	257	8	b	b	PROPN
ejpam-239	257	9	∧	∧	PROPN
ejpam-239	257	10	d	d	PROPN
ejpam-239	257	11	)	)	PUNCT
ejpam-239	257	12	]	]	PUNCT
ejpam-239	257	13	.	.	PUNCT
ejpam-239	258	1	therefore	therefore	ADV
ejpam-239	258	2	i	i	PRON
ejpam-239	258	3	⊆	⊆	NUM
ejpam-239	258	4	(	(	PUNCT
ejpam-239	258	5	c	c	PROPN
ejpam-239	258	6	∨	∨	X
ejpam-239	258	7	(	(	PUNCT
ejpam-239	258	8	b	b	PROPN
ejpam-239	258	9	∧	∧	PROPN
ejpam-239	258	10	d	d	PROPN
ejpam-239	258	11	)	)	PUNCT
ejpam-239	258	12	]	]	PUNCT
ejpam-239	258	13	.	.	PUNCT
ejpam-239	259	1	by	by	ADP
ejpam-239	259	2	symmetry	symmetry	NOUN
ejpam-239	259	3	,	,	PUNCT
ejpam-239	259	4	we	we	PRON
ejpam-239	259	5	get	get	VERB
ejpam-239	259	6	that	that	PRON
ejpam-239	259	7	j	j	PROPN
ejpam-239	259	8	is	be	AUX
ejpam-239	259	9	also	also	ADV
ejpam-239	259	10	a	a	DET
ejpam-239	259	11	principal	principal	ADJ
ejpam-239	259	12	ideal	ideal	NOUN
ejpam-239	259	13	.	.	PUNCT
ejpam-239	260	1	�	�	PROPN
ejpam-239	260	2	theorem	theorem	VERB
ejpam-239	260	3	3.15	3.15	NUM
ejpam-239	260	4	.	.	PUNCT
ejpam-239	261	1	let	let	VERB
ejpam-239	261	2	r	r	PRON
ejpam-239	261	3	be	be	AUX
ejpam-239	261	4	a	a	DET
ejpam-239	261	5	generalized	generalized	ADJ
ejpam-239	261	6	stone	stone	NOUN
ejpam-239	261	7	adl	adl	PROPN
ejpam-239	261	8	.	.	PUNCT
ejpam-239	262	1	then	then	ADV
ejpam-239	262	2	a0(r	a0(r	PROPN
ejpam-239	262	3	)	)	PUNCT
ejpam-239	262	4	is	be	AUX
ejpam-239	262	5	a	a	DET
ejpam-239	262	6	relatively	relatively	ADV
ejpam-239	262	7	complemented	complemented	ADJ
ejpam-239	262	8	sublattice	sublattice	NOUN
ejpam-239	262	9	of	of	ADP
ejpam-239	262	10	the	the	DET
ejpam-239	262	11	lattice	lattice	NOUN
ejpam-239	263	1	i	i	PRON
ejpam-239	263	2	(	(	PUNCT
ejpam-239	263	3	r	r	NOUN
ejpam-239	263	4	)	)	PUNCT
ejpam-239	263	5	of	of	ADP
ejpam-239	263	6	all	all	DET
ejpam-239	263	7	ideals	ideal	NOUN
ejpam-239	263	8	of	of	ADP
ejpam-239	263	9	r.	r.	PROPN
ejpam-239	263	10	proof	proof	NOUN
ejpam-239	263	11	:	:	PUNCT
ejpam-239	263	12	let	let	VERB
ejpam-239	263	13	r	r	PRON
ejpam-239	263	14	be	be	AUX
ejpam-239	263	15	a	a	DET
ejpam-239	263	16	generalized	generalized	ADJ
ejpam-239	263	17	stone	stone	NOUN
ejpam-239	263	18	adl	adl	PROPN
ejpam-239	263	19	.	.	PUNCT
ejpam-239	264	1	by	by	ADP
ejpam-239	264	2	theorem	theorem	NOUN
ejpam-239	264	3	3.7	3.7	NUM
ejpam-239	264	4	,	,	PUNCT
ejpam-239	264	5	a0(r	a0(r	PROPN
ejpam-239	264	6	)	)	PUNCT
ejpam-239	264	7	is	be	AUX
ejpam-239	264	8	a	a	DET
ejpam-239	264	9	sublattice	sublattice	NOUN
ejpam-239	264	10	of	of	ADP
ejpam-239	264	11	i	i	PRON
ejpam-239	264	12	(	(	PUNCT
ejpam-239	264	13	r	r	NOUN
ejpam-239	264	14	)	)	PUNCT
ejpam-239	264	15	.	.	PUNCT
ejpam-239	265	1	so	so	ADV
ejpam-239	265	2	we	we	PRON
ejpam-239	265	3	can	can	AUX
ejpam-239	265	4	treate	treate	VERB
ejpam-239	265	5	∨	∨	NOUN
ejpam-239	265	6	as	as	ADP
ejpam-239	265	7	∨.	∨.	NOUN
ejpam-239	265	8	sincea0(r	sincea0(r	NOUN
ejpam-239	265	9	)	)	PUNCT
ejpam-239	265	10	is	be	AUX
ejpam-239	265	11	a	a	DET
ejpam-239	265	12	distributive	distributive	ADJ
ejpam-239	265	13	lattice	lattice	NOUN
ejpam-239	265	14	with	with	ADP
ejpam-239	265	15	the	the	DET
ejpam-239	265	16	greatest	great	ADJ
ejpam-239	265	17	element	element	NOUN
ejpam-239	265	18	(	(	PUNCT
ejpam-239	265	19	0]∗	0]∗	NOUN
ejpam-239	265	20	=	=	SYM
ejpam-239	265	21	r	r	NOUN
ejpam-239	265	22	,	,	PUNCT
ejpam-239	265	23	it	it	PRON
ejpam-239	265	24	is	be	AUX
ejpam-239	265	25	enough	enough	ADJ
ejpam-239	265	26	to	to	PART
ejpam-239	265	27	prove	prove	VERB
ejpam-239	265	28	that	that	SCONJ
ejpam-239	265	29	each	each	DET
ejpam-239	265	30	interval	interval	NOUN
ejpam-239	265	31	of	of	ADP
ejpam-239	265	32	the	the	DET
ejpam-239	265	33	form	form	NOUN
ejpam-239	266	1	[	[	X
ejpam-239	266	2	i	i	PRON
ejpam-239	266	3	,	,	PUNCT
ejpam-239	266	4	r	r	NOUN
ejpam-239	266	5	]	]	X
ejpam-239	266	6	,	,	PUNCT
ejpam-239	266	7	where	where	SCONJ
ejpam-239	266	8	i	i	PRON
ejpam-239	266	9	∈	∈	PROPN
ejpam-239	266	10	a0(r	a0(r	PROPN
ejpam-239	266	11	)	)	PUNCT
ejpam-239	266	12	,	,	PUNCT
ejpam-239	266	13	is	be	AUX
ejpam-239	266	14	complemented	complement	VERB
ejpam-239	266	15	.	.	PUNCT
ejpam-239	267	1	let	let	VERB
ejpam-239	267	2	j	j	PROPN
ejpam-239	268	1	=	=	PUNCT
ejpam-239	269	1	[	[	X
ejpam-239	269	2	(	(	PUNCT
ejpam-239	269	3	x]∗,r	x]∗,r	NOUN
ejpam-239	269	4	]	]	X
ejpam-239	269	5	be	be	VERB
ejpam-239	269	6	an	an	DET
ejpam-239	269	7	interval	interval	NOUN
ejpam-239	269	8	ina0(r	ina0(r	NOUN
ejpam-239	269	9	)	)	PUNCT
ejpam-239	269	10	and	and	CCONJ
ejpam-239	269	11	(	(	PUNCT
ejpam-239	269	12	y]∗	y]∗	PROPN
ejpam-239	269	13	∈	∈	PROPN
ejpam-239	269	14	j	j	PROPN
ejpam-239	269	15	.	.	PUNCT
ejpam-239	270	1	we	we	PRON
ejpam-239	270	2	have	have	VERB
ejpam-239	270	3	clearly	clearly	ADV
ejpam-239	270	4	(	(	PUNCT
ejpam-239	270	5	y]∗	y]∗	PROPN
ejpam-239	270	6	∩	∩	NOUN
ejpam-239	270	7	(	(	PUNCT
ejpam-239	270	8	y]∗∗	y]∗∗	X
ejpam-239	270	9	=	=	SYM
ejpam-239	270	10	(	(	PUNCT
ejpam-239	270	11	0	0	NUM
ejpam-239	270	12	]	]	PUNCT
ejpam-239	270	13	.	.	PUNCT
ejpam-239	271	1	since	since	SCONJ
ejpam-239	271	2	r	r	NOUN
ejpam-239	271	3	is	be	AUX
ejpam-239	271	4	generalized	generalize	VERB
ejpam-239	271	5	stone	stone	NOUN
ejpam-239	271	6	adl	adl	PROPN
ejpam-239	271	7	,	,	PUNCT
ejpam-239	271	8	we	we	PRON
ejpam-239	271	9	have	have	VERB
ejpam-239	271	10	(	(	PUNCT
ejpam-239	271	11	y]∗	y]∗	PROPN
ejpam-239	271	12	∨	∨	NUM
ejpam-239	271	13	(	(	PUNCT
ejpam-239	271	14	y]∗∗	y]∗∗	NOUN
ejpam-239	271	15	=	=	SYM
ejpam-239	271	16	r	r	NOUN
ejpam-239	271	17	for	for	ADP
ejpam-239	271	18	all	all	DET
ejpam-239	271	19	y	y	PROPN
ejpam-239	271	20	∈	∈	PROPN
ejpam-239	271	21	r.	r.	PROPN
ejpam-239	271	22	now	now	ADV
ejpam-239	271	23	�	�	PROPN
ejpam-239	271	24	(	(	PUNCT
ejpam-239	271	25	x]∩	x]∩	X
ejpam-239	271	26	(	(	PUNCT
ejpam-239	271	27	y]∗	y]∗	PROPN
ejpam-239	271	28	∨	∨	NUM
ejpam-239	271	29	�	�	PROPN
ejpam-239	271	30	(	(	PUNCT
ejpam-239	271	31	x]∩	x]∩	X
ejpam-239	271	32	(	(	PUNCT
ejpam-239	271	33	y]∗∗	y]∗∗	X
ejpam-239	271	34	=	=	SYM
ejpam-239	271	35	(	(	PUNCT
ejpam-239	271	36	x]∩	x]∩	ADP
ejpam-239	271	37	�	�	PROPN
ejpam-239	271	38	(	(	PUNCT
ejpam-239	271	39	y]∗	y]∗	PROPN
ejpam-239	271	40	∨	∨	NUM
ejpam-239	271	41	(	(	PUNCT
ejpam-239	271	42	y]∗∗	y]∗∗	X
ejpam-239	271	43	=	=	SYM
ejpam-239	271	44	(	(	PUNCT
ejpam-239	271	45	x]∩	x]∩	X
ejpam-239	271	46	r=	r=	PROPN
ejpam-239	271	47	(	(	PUNCT
ejpam-239	271	48	x	x	X
ejpam-239	271	49	]	]	X
ejpam-239	271	50	.	.	PUNCT
ejpam-239	272	1	also	also	ADV
ejpam-239	272	2	�	�	PROPN
ejpam-239	272	3	(	(	PUNCT
ejpam-239	272	4	x]∩	x]∩	X
ejpam-239	272	5	(	(	PUNCT
ejpam-239	272	6	y]∗	y]∗	PROPN
ejpam-239	272	7	∩	∩	ADJ
ejpam-239	272	8	�	�	PROPN
ejpam-239	272	9	(	(	PUNCT
ejpam-239	272	10	x]∩	x]∩	X
ejpam-239	272	11	(	(	PUNCT
ejpam-239	272	12	y]∗∗	y]∗∗	X
ejpam-239	272	13	=	=	SYM
ejpam-239	272	14	(	(	PUNCT
ejpam-239	272	15	x]∩	x]∩	ADP
ejpam-239	272	16	�	�	PROPN
ejpam-239	272	17	(	(	PUNCT
ejpam-239	272	18	y]∗	y]∗	PROPN
ejpam-239	272	19	∩	∩	NOUN
ejpam-239	272	20	(	(	PUNCT
ejpam-239	272	21	y]∗∗	y]∗∗	X
ejpam-239	272	22	=	=	SYM
ejpam-239	272	23	(	(	PUNCT
ejpam-239	272	24	x]∩	x]∩	X
ejpam-239	272	25	(	(	PUNCT
ejpam-239	272	26	0	0	NUM
ejpam-239	272	27	]	]	X
ejpam-239	272	28	=	=	SYM
ejpam-239	272	29	(	(	PUNCT
ejpam-239	272	30	0	0	NUM
ejpam-239	272	31	]	]	PUNCT
ejpam-239	272	32	.	.	PUNCT
ejpam-239	273	1	thus	thus	ADV
ejpam-239	273	2	we	we	PRON
ejpam-239	273	3	have	have	VERB
ejpam-239	273	4	that	that	SCONJ
ejpam-239	273	5	the	the	DET
ejpam-239	273	6	infimum	infimum	NOUN
ejpam-239	273	7	and	and	CCONJ
ejpam-239	273	8	the	the	DET
ejpam-239	273	9	supremum	supremum	NOUN
ejpam-239	273	10	of	of	ADP
ejpam-239	273	11	the	the	DET
ejpam-239	273	12	ideals	ideal	NOUN
ejpam-239	273	13	(	(	PUNCT
ejpam-239	273	14	x]∩	x]∩	X
ejpam-239	273	15	(	(	PUNCT
ejpam-239	273	16	y]∗	y]∗	PROPN
ejpam-239	273	17	and	and	CCONJ
ejpam-239	273	18	(	(	PUNCT
ejpam-239	273	19	x]∩	x]∩	X
ejpam-239	273	20	(	(	PUNCT
ejpam-239	273	21	y]∗∗	y]∗∗	NOUN
ejpam-239	273	22	are	be	AUX
ejpam-239	273	23	the	the	DET
ejpam-239	273	24	principal	principal	ADJ
ejpam-239	273	25	ideals	ideal	NOUN
ejpam-239	273	26	(	(	PUNCT
ejpam-239	273	27	0	0	NUM
ejpam-239	273	28	]	]	PUNCT
ejpam-239	273	29	and	and	CCONJ
ejpam-239	273	30	(	(	PUNCT
ejpam-239	273	31	x	x	X
ejpam-239	273	32	]	]	X
ejpam-239	273	33	.	.	PUNCT
ejpam-239	274	1	therefore	therefore	ADV
ejpam-239	274	2	,	,	PUNCT
ejpam-239	274	3	by	by	ADP
ejpam-239	274	4	the	the	DET
ejpam-239	274	5	above	above	ADJ
ejpam-239	274	6	lemma	lemma	PROPN
ejpam-239	274	7	,	,	PUNCT
ejpam-239	274	8	(	(	PUNCT
ejpam-239	274	9	x	x	X
ejpam-239	274	10	]	]	X
ejpam-239	274	11	∩	∩	NOUN
ejpam-239	274	12	(	(	PUNCT
ejpam-239	274	13	y]∗	y]∗	PROPN
ejpam-239	274	14	and	and	CCONJ
ejpam-239	274	15	(	(	PUNCT
ejpam-239	274	16	x	x	NOUN
ejpam-239	274	17	]	]	X
ejpam-239	274	18	∩	∩	NOUN
ejpam-239	274	19	(	(	PUNCT
ejpam-239	274	20	y]∗∗	y]∗∗	NOUN
ejpam-239	274	21	must	must	AUX
ejpam-239	274	22	be	be	AUX
ejpam-239	274	23	the	the	DET
ejpam-239	274	24	principal	principal	ADJ
ejpam-239	274	25	ideals	ideal	NOUN
ejpam-239	274	26	.	.	PUNCT
ejpam-239	275	1	suppose	suppose	VERB
ejpam-239	275	2	(	(	PUNCT
ejpam-239	275	3	x]∩	x]∩	X
ejpam-239	275	4	(	(	PUNCT
ejpam-239	275	5	y]∗	y]∗	PROPN
ejpam-239	275	6	=	=	SYM
ejpam-239	275	7	(	(	PUNCT
ejpam-239	275	8	a	a	X
ejpam-239	275	9	]	]	PUNCT
ejpam-239	275	10	and	and	CCONJ
ejpam-239	275	11	(	(	PUNCT
ejpam-239	275	12	x]∩	x]∩	X
ejpam-239	275	13	(	(	PUNCT
ejpam-239	275	14	y]∗∗	y]∗∗	X
ejpam-239	275	15	=	=	SYM
ejpam-239	275	16	(	(	PUNCT
ejpam-239	275	17	b	b	X
ejpam-239	275	18	]	]	X
ejpam-239	275	19	for	for	ADP
ejpam-239	275	20	some	some	PRON
ejpam-239	275	21	a	a	PRON
ejpam-239	275	22	,	,	PUNCT
ejpam-239	275	23	b	b	PROPN
ejpam-239	275	24	∈	∈	PROPN
ejpam-239	275	25	r.	r.	NOUN
ejpam-239	275	26	now	now	ADV
ejpam-239	275	27	a	a	DET
ejpam-239	275	28	∈	∈	NOUN
ejpam-239	275	29	(	(	PUNCT
ejpam-239	275	30	x]∩	x]∩	X
ejpam-239	275	31	(	(	PUNCT
ejpam-239	275	32	y]∗⇒	y]∗⇒	NUM
ejpam-239	275	33	(	(	PUNCT
ejpam-239	275	34	a]⊆	a]⊆	PROPN
ejpam-239	275	35	(	(	PUNCT
ejpam-239	275	36	x]⇒	x]⇒	PROPN
ejpam-239	275	37	(	(	PUNCT
ejpam-239	275	38	x]∗	x]∗	PROPN
ejpam-239	275	39	⊆	⊆	NUM
ejpam-239	275	40	(	(	PUNCT
ejpam-239	275	41	a]∗.therefore	a]∗.therefore	ADV
ejpam-239	275	42	(	(	PUNCT
ejpam-239	275	43	a]∗	a]∗	PROPN
ejpam-239	275	44	∈	∈	PROPN
ejpam-239	275	45	j	j	PROPN
ejpam-239	275	46	.	.	PUNCT
ejpam-239	276	1	also	also	ADV
ejpam-239	276	2	(	(	PUNCT
ejpam-239	276	3	a	a	X
ejpam-239	276	4	]	]	X
ejpam-239	276	5	=	=	SYM
ejpam-239	276	6	(	(	PUNCT
ejpam-239	276	7	x	x	SYM
ejpam-239	276	8	]	]	X
ejpam-239	276	9	∩	∩	NOUN
ejpam-239	276	10	(	(	PUNCT
ejpam-239	276	11	y]∗	y]∗	PROPN
ejpam-239	276	12	⊆	⊆	NUM
ejpam-239	276	13	(	(	PUNCT
ejpam-239	276	14	y]∗	y]∗	PROPN
ejpam-239	276	15	⇒	⇒	PROPN
ejpam-239	276	16	(	(	PUNCT
ejpam-239	276	17	y]∗∗	y]∗∗	NOUN
ejpam-239	276	18	⊆	⊆	NUM
ejpam-239	276	19	(	(	PUNCT
ejpam-239	276	20	a]∗.	a]∗.	NOUN
ejpam-239	276	21	hence	hence	ADV
ejpam-239	276	22	(	(	PUNCT
ejpam-239	276	23	y]∗	y]∗	PROPN
ejpam-239	276	24	∨	∨	NUM
ejpam-239	276	25	(	(	PUNCT
ejpam-239	276	26	y]∗∗	y]∗∗	NOUN
ejpam-239	276	27	⊆	⊆	NUM
ejpam-239	276	28	(	(	PUNCT
ejpam-239	276	29	y]∗	y]∗	PROPN
ejpam-239	276	30	∨	∨	NUM
ejpam-239	276	31	(	(	PUNCT
ejpam-239	276	32	a]∗	a]∗	NOUN
ejpam-239	276	33	⇒	⇒	VERB
ejpam-239	276	34	r	r	NOUN
ejpam-239	276	35	⊆	⊆	NUM
ejpam-239	276	36	(	(	PUNCT
ejpam-239	276	37	a]∗	a]∗	PROPN
ejpam-239	276	38	∨	∨	PROPN
ejpam-239	276	39	(	(	PUNCT
ejpam-239	276	40	y]∗.	y]∗.	PRON
ejpam-239	276	41	thus	thus	ADV
ejpam-239	276	42	r=	r=	ADJ
ejpam-239	276	43	(	(	PUNCT
ejpam-239	276	44	a]∗	a]∗	PROPN
ejpam-239	276	45	∨	∨	PROPN
ejpam-239	276	46	(	(	PUNCT
ejpam-239	276	47	y]∗	y]∗	PROPN
ejpam-239	276	48	−→	−→	NOUN
ejpam-239	276	49	(	(	PUNCT
ejpam-239	276	50	1	1	NUM
ejpam-239	276	51	)	)	PUNCT
ejpam-239	276	52	again	again	ADV
ejpam-239	276	53	(	(	PUNCT
ejpam-239	276	54	a]∗	a]∗	PROPN
ejpam-239	276	55	∩	∩	NOUN
ejpam-239	276	56	(	(	PUNCT
ejpam-239	276	57	y]∗	y]∗	PROPN
ejpam-239	276	58	∩	∩	NOUN
ejpam-239	276	59	(	(	PUNCT
ejpam-239	276	60	x	x	X
ejpam-239	276	61	]	]	X
ejpam-239	276	62	=	=	SYM
ejpam-239	276	63	(	(	PUNCT
ejpam-239	276	64	a]∗	a]∗	PROPN
ejpam-239	276	65	∩	∩	NOUN
ejpam-239	276	66	(	(	PUNCT
ejpam-239	276	67	a	a	X
ejpam-239	276	68	]	]	X
ejpam-239	276	69	=	=	SYM
ejpam-239	276	70	(	(	PUNCT
ejpam-239	276	71	0	0	NUM
ejpam-239	276	72	]	]	PUNCT
ejpam-239	276	73	.	.	PUNCT
ejpam-239	277	1	hence	hence	ADV
ejpam-239	277	2	(	(	PUNCT
ejpam-239	277	3	a]∗	a]∗	PROPN
ejpam-239	277	4	∩	∩	NOUN
ejpam-239	277	5	(	(	PUNCT
ejpam-239	277	6	y]∗	y]∗	PROPN
ejpam-239	277	7	⊆	⊆	NUM
ejpam-239	277	8	(	(	PUNCT
ejpam-239	277	9	x]∗.	x]∗.	PROPN
ejpam-239	277	10	but	but	CCONJ
ejpam-239	277	11	(	(	PUNCT
ejpam-239	277	12	x]∗	x]∗	PROPN
ejpam-239	277	13	⊆	⊆	NUM
ejpam-239	277	14	(	(	PUNCT
ejpam-239	277	15	y]∗	y]∗	PROPN
ejpam-239	277	16	and	and	CCONJ
ejpam-239	277	17	(	(	PUNCT
ejpam-239	277	18	x]∗	x]∗	PROPN
ejpam-239	277	19	⊆	⊆	NUM
ejpam-239	277	20	(	(	PUNCT
ejpam-239	277	21	a]∗	a]∗	NOUN
ejpam-239	277	22	imply	imply	VERB
ejpam-239	277	23	that	that	SCONJ
ejpam-239	277	24	(	(	PUNCT
ejpam-239	277	25	x]∗	x]∗	PROPN
ejpam-239	277	26	⊆	⊆	NUM
ejpam-239	277	27	(	(	PUNCT
ejpam-239	277	28	a]∗	a]∗	PROPN
ejpam-239	277	29	∩	∩	NOUN
ejpam-239	277	30	(	(	PUNCT
ejpam-239	277	31	y]∗.	y]∗.	PROPN
ejpam-239	277	32	hence	hence	ADV
ejpam-239	277	33	(	(	PUNCT
ejpam-239	277	34	a]∗	a]∗	PROPN
ejpam-239	277	35	∩	∩	NOUN
ejpam-239	277	36	(	(	PUNCT
ejpam-239	277	37	y]∗	y]∗	PROPN
ejpam-239	277	38	=	=	SYM
ejpam-239	277	39	(	(	PUNCT
ejpam-239	277	40	x]∗	x]∗	PROPN
ejpam-239	277	41	−→	−→	PROPN
ejpam-239	277	42	(	(	PUNCT
ejpam-239	277	43	2	2	NUM
ejpam-239	277	44	)	)	PUNCT
ejpam-239	277	45	from	from	ADP
ejpam-239	277	46	(	(	PUNCT
ejpam-239	277	47	1	1	NUM
ejpam-239	277	48	)	)	PUNCT
ejpam-239	277	49	and	and	CCONJ
ejpam-239	277	50	(	(	PUNCT
ejpam-239	277	51	2	2	NUM
ejpam-239	277	52	)	)	PUNCT
ejpam-239	277	53	,	,	PUNCT
ejpam-239	277	54	(	(	PUNCT
ejpam-239	277	55	a]∗	a]∗	PROPN
ejpam-239	277	56	is	be	AUX
ejpam-239	277	57	the	the	DET
ejpam-239	277	58	required	require	VERB
ejpam-239	277	59	complement	complement	NOUN
ejpam-239	277	60	of	of	ADP
ejpam-239	277	61	(	(	PUNCT
ejpam-239	277	62	y]∗	y]∗	PROPN
ejpam-239	277	63	in	in	ADP
ejpam-239	277	64	j	j	PROPN
ejpam-239	277	65	.	.	PUNCT
ejpam-239	278	1	g.	g.	PROPN
ejpam-239	278	2	c.	c.	PROPN
ejpam-239	278	3	rao	rao	PROPN
ejpam-239	278	4	and	and	CCONJ
ejpam-239	278	5	m.	m.	PROPN
ejpam-239	278	6	sambasiva	sambasiva	PROPN
ejpam-239	278	7	rao	rao	PROPN
ejpam-239	278	8	/	/	SYM
ejpam-239	278	9	eur	eur	PROPN
ejpam-239	278	10	.	.	PUNCT
ejpam-239	279	1	j.	j.	PROPN
ejpam-239	279	2	pure	pure	PROPN
ejpam-239	279	3	appl	appl	PROPN
ejpam-239	279	4	.	.	PROPN
ejpam-239	279	5	math	math	PROPN
ejpam-239	279	6	,	,	PUNCT
ejpam-239	279	7	2	2	NUM
ejpam-239	279	8	(	(	PUNCT
ejpam-239	279	9	2009	2009	NUM
ejpam-239	279	10	)	)	PUNCT
ejpam-239	279	11	,	,	PUNCT
ejpam-239	279	12	(	(	PUNCT
ejpam-239	279	13	58	58	NUM
ejpam-239	279	14	-	-	SYM
ejpam-239	279	15	72	72	NUM
ejpam-239	279	16	)	)	PUNCT
ejpam-239	279	17	69	69	NUM
ejpam-239	279	18	hencea0(r	hencea0(r	NOUN
ejpam-239	279	19	)	)	PUNCT
ejpam-239	279	20	is	be	AUX
ejpam-239	279	21	a	a	DET
ejpam-239	279	22	relatively	relatively	ADV
ejpam-239	279	23	complemented	complemented	ADJ
ejpam-239	279	24	sublattice	sublattice	NOUN
ejpam-239	279	25	of	of	ADP
ejpam-239	279	26	i	i	PRON
ejpam-239	279	27	(	(	PUNCT
ejpam-239	279	28	r	r	NOUN
ejpam-239	279	29	)	)	PUNCT
ejpam-239	279	30	.	.	PUNCT
ejpam-239	280	1	�	�	PROPN
ejpam-239	280	2	definition	definition	NOUN
ejpam-239	280	3	3.16	3.16	NUM
ejpam-239	280	4	.	.	PUNCT
ejpam-239	281	1	let	let	VERB
ejpam-239	281	2	i	i	PRON
ejpam-239	281	3	=	=	PUNCT
ejpam-239	282	1	[	[	X
ejpam-239	282	2	0	0	NUM
ejpam-239	282	3	,	,	PUNCT
ejpam-239	282	4	x	x	X
ejpam-239	282	5	]	]	X
ejpam-239	282	6	,	,	PUNCT
ejpam-239	282	7	0	0	PUNCT
ejpam-239	282	8	<	<	X
ejpam-239	282	9	x	x	X
ejpam-239	282	10	,	,	PUNCT
ejpam-239	282	11	be	be	AUX
ejpam-239	282	12	an	an	DET
ejpam-239	282	13	interval	interval	NOUN
ejpam-239	282	14	in	in	ADP
ejpam-239	282	15	an	an	DET
ejpam-239	282	16	adl	adl	NOUN
ejpam-239	282	17	r	r	NOUN
ejpam-239	282	18	with	with	ADP
ejpam-239	282	19	0	0	NUM
ejpam-239	282	20	.	.	PUNCT
ejpam-239	283	1	for	for	ADP
ejpam-239	283	2	a	a	DET
ejpam-239	283	3	∈	∈	NOUN
ejpam-239	283	4	i	i	PRON
ejpam-239	283	5	,	,	PUNCT
ejpam-239	283	6	define	define	VERB
ejpam-239	283	7	the	the	DET
ejpam-239	283	8	annihilator	annihilator	NOUN
ejpam-239	283	9	(	(	PUNCT
ejpam-239	283	10	a]+	a]+	PROPN
ejpam-239	283	11	of	of	ADP
ejpam-239	283	12	a	a	PRON
ejpam-239	283	13	with	with	ADP
ejpam-239	283	14	respect	respect	NOUN
ejpam-239	283	15	to	to	ADP
ejpam-239	283	16	i	i	PRON
ejpam-239	283	17	as	as	SCONJ
ejpam-239	283	18	follows	follow	VERB
ejpam-239	283	19	:	:	PUNCT
ejpam-239	283	20	(	(	PUNCT
ejpam-239	283	21	a]+	a]+	NOUN
ejpam-239	283	22	=	=	PRON
ejpam-239	283	23	{	{	PUNCT
ejpam-239	283	24	y	y	PROPN
ejpam-239	283	25	∈	∈	PROPN
ejpam-239	284	1	i	i	PRON
ejpam-239	284	2	|	|	ADV
ejpam-239	284	3	y	y	PROPN
ejpam-239	284	4	∧	∧	PROPN
ejpam-239	284	5	a	a	PRON
ejpam-239	284	6	=	=	NOUN
ejpam-239	284	7	0	0	NUM
ejpam-239	284	8	}	}	PUNCT
ejpam-239	284	9	.	.	PUNCT
ejpam-239	285	1	observe	observe	VERB
ejpam-239	285	2	that	that	SCONJ
ejpam-239	285	3	(	(	PUNCT
ejpam-239	285	4	a]∗	a]∗	NOUN
ejpam-239	285	5	∩	∩	NOUN
ejpam-239	285	6	i	i	PRON
ejpam-239	285	7	=	=	SYM
ejpam-239	285	8	(	(	PUNCT
ejpam-239	285	9	a]+	a]+	PROPN
ejpam-239	285	10	.	.	PUNCT
ejpam-239	286	1	lemma	lemma	PROPN
ejpam-239	286	2	3.17	3.17	NUM
ejpam-239	286	3	.	.	PUNCT
ejpam-239	287	1	for	for	ADP
ejpam-239	287	2	a	a	DET
ejpam-239	287	3	∈	∈	NOUN
ejpam-239	287	4	i	i	PRON
ejpam-239	287	5	,	,	PUNCT
ejpam-239	287	6	the	the	DET
ejpam-239	287	7	annihilator	annihilator	PROPN
ejpam-239	287	8	(	(	PUNCT
ejpam-239	287	9	a]+	a]+	PROPN
ejpam-239	287	10	is	be	AUX
ejpam-239	287	11	an	an	DET
ejpam-239	287	12	ideal	ideal	NOUN
ejpam-239	287	13	in	in	ADP
ejpam-239	287	14	i.	i.	NOUN
ejpam-239	287	15	proof	proof	NOUN
ejpam-239	287	16	:	:	PUNCT
ejpam-239	287	17	since	since	SCONJ
ejpam-239	287	18	0	0	NUM
ejpam-239	287	19	∈	∈	PROPN
ejpam-239	287	20	i	i	PRON
ejpam-239	287	21	and	and	CCONJ
ejpam-239	287	22	0	0	NUM
ejpam-239	287	23	∧	∧	NOUN
ejpam-239	287	24	a	a	DET
ejpam-239	287	25	=	=	SYM
ejpam-239	287	26	0	0	NUM
ejpam-239	287	27	,	,	PUNCT
ejpam-239	287	28	we	we	PRON
ejpam-239	287	29	get	get	VERB
ejpam-239	287	30	that	that	PRON
ejpam-239	287	31	0	0	NUM
ejpam-239	287	32	∈	∈	NOUN
ejpam-239	287	33	(	(	PUNCT
ejpam-239	287	34	a]+	a]+	PROPN
ejpam-239	287	35	.	.	PUNCT
ejpam-239	288	1	let	let	VERB
ejpam-239	288	2	r	r	NOUN
ejpam-239	288	3	,	,	PUNCT
ejpam-239	288	4	s	s	NOUN
ejpam-239	288	5	∈	∈	PROPN
ejpam-239	288	6	(	(	PUNCT
ejpam-239	288	7	a]+	a]+	PROPN
ejpam-239	288	8	.	.	PUNCT
ejpam-239	289	1	then	then	ADV
ejpam-239	289	2	r	r	NOUN
ejpam-239	289	3	,	,	PUNCT
ejpam-239	289	4	s	s	PART
ejpam-239	289	5	∈	∈	NOUN
ejpam-239	290	1	i	i	PRON
ejpam-239	290	2	and	and	CCONJ
ejpam-239	290	3	r	r	NOUN
ejpam-239	290	4	∧	∧	PROPN
ejpam-239	290	5	a	a	DET
ejpam-239	290	6	=	=	SYM
ejpam-239	290	7	s	s	NOUN
ejpam-239	290	8	∧	∧	NOUN
ejpam-239	290	9	a	a	PRON
ejpam-239	290	10	=	=	NOUN
ejpam-239	290	11	0	0	NUM
ejpam-239	290	12	.	.	PUNCT
ejpam-239	291	1	since	since	SCONJ
ejpam-239	291	2	r	r	NOUN
ejpam-239	291	3	,	,	PUNCT
ejpam-239	291	4	s	s	PART
ejpam-239	291	5	∈	∈	NOUN
ejpam-239	292	1	i	i	PRON
ejpam-239	292	2	,	,	PUNCT
ejpam-239	292	3	we	we	PRON
ejpam-239	292	4	get	get	VERB
ejpam-239	292	5	r	r	NOUN
ejpam-239	292	6	∨	∨	NUM
ejpam-239	292	7	s	s	PART
ejpam-239	292	8	∈	∈	NOUN
ejpam-239	292	9	i	i	PRON
ejpam-239	292	10	,	,	PUNCT
ejpam-239	292	11	and	and	CCONJ
ejpam-239	292	12	(	(	PUNCT
ejpam-239	292	13	r	r	PROPN
ejpam-239	292	14	∨	∨	NUM
ejpam-239	292	15	s)∧	s)∧	PROPN
ejpam-239	292	16	a	a	X
ejpam-239	292	17	=	=	X
ejpam-239	292	18	(	(	PUNCT
ejpam-239	292	19	r	r	NOUN
ejpam-239	292	20	∧	∧	PROPN
ejpam-239	292	21	a)∨	a)∨	PROPN
ejpam-239	292	22	(	(	PUNCT
ejpam-239	292	23	s	s	VERB
ejpam-239	292	24	∧	∧	PROPN
ejpam-239	292	25	a	a	NOUN
ejpam-239	292	26	)	)	PUNCT
ejpam-239	292	27	=	=	SYM
ejpam-239	293	1	0∨	0∨	NUM
ejpam-239	293	2	0=	0=	NOUN
ejpam-239	293	3	0	0	X
ejpam-239	293	4	.	.	PUNCT
ejpam-239	294	1	hence	hence	ADV
ejpam-239	294	2	r	r	NOUN
ejpam-239	294	3	∨	∨	NUM
ejpam-239	294	4	s	s	NOUN
ejpam-239	294	5	∈	∈	PROPN
ejpam-239	294	6	(	(	PUNCT
ejpam-239	294	7	a]+	a]+	PROPN
ejpam-239	294	8	.	.	PUNCT
ejpam-239	295	1	let	let	VERB
ejpam-239	295	2	y	y	PROPN
ejpam-239	295	3	∈	∈	PROPN
ejpam-239	295	4	(	(	PUNCT
ejpam-239	295	5	a]+	a]+	PROPN
ejpam-239	295	6	and	and	CCONJ
ejpam-239	295	7	t	t	PROPN
ejpam-239	295	8	∈	∈	PROPN
ejpam-239	295	9	i	i	PRON
ejpam-239	295	10	.	.	PUNCT
ejpam-239	296	1	then	then	ADV
ejpam-239	296	2	y	y	PROPN
ejpam-239	296	3	∈	∈	PROPN
ejpam-239	297	1	i	i	PRON
ejpam-239	297	2	and	and	CCONJ
ejpam-239	297	3	y	y	PROPN
ejpam-239	297	4	∧	∧	PROPN
ejpam-239	297	5	a	a	PRON
ejpam-239	297	6	=	=	NOUN
ejpam-239	297	7	0	0	NUM
ejpam-239	297	8	.	.	PUNCT
ejpam-239	298	1	hence	hence	ADV
ejpam-239	298	2	y	y	PROPN
ejpam-239	298	3	∧	∧	PROPN
ejpam-239	298	4	t	t	PROPN
ejpam-239	298	5	∈	∈	PROPN
ejpam-239	299	1	i	i	PRON
ejpam-239	299	2	.	.	PUNCT
ejpam-239	300	1	now	now	ADV
ejpam-239	300	2	(	(	PUNCT
ejpam-239	300	3	y	y	PROPN
ejpam-239	300	4	∧	∧	PROPN
ejpam-239	300	5	t)∧	t)∧	VERB
ejpam-239	300	6	a	a	DET
ejpam-239	300	7	=	=	SYM
ejpam-239	300	8	t	t	NOUN
ejpam-239	300	9	∧	∧	PROPN
ejpam-239	300	10	y	y	PROPN
ejpam-239	300	11	∧	∧	PROPN
ejpam-239	300	12	a	a	DET
ejpam-239	300	13	=	=	X
ejpam-239	300	14	t	t	NOUN
ejpam-239	300	15	∧0=	∧0=	NOUN
ejpam-239	300	16	0	0	NUM
ejpam-239	300	17	,	,	PUNCT
ejpam-239	300	18	which	which	PRON
ejpam-239	300	19	implies	imply	VERB
ejpam-239	300	20	that	that	SCONJ
ejpam-239	300	21	y	y	PROPN
ejpam-239	300	22	∧	∧	PROPN
ejpam-239	300	23	t	t	PROPN
ejpam-239	300	24	∈	∈	PROPN
ejpam-239	300	25	(	(	PUNCT
ejpam-239	300	26	a]+	a]+	PROPN
ejpam-239	300	27	.	.	PUNCT
ejpam-239	301	1	thus	thus	ADV
ejpam-239	301	2	(	(	PUNCT
ejpam-239	301	3	a]+	a]+	PROPN
ejpam-239	301	4	is	be	AUX
ejpam-239	301	5	an	an	DET
ejpam-239	301	6	ideal	ideal	NOUN
ejpam-239	301	7	of	of	ADP
ejpam-239	301	8	i	i	PRON
ejpam-239	301	9	.	.	PUNCT
ejpam-239	302	1	�	�	PROPN
ejpam-239	302	2	lemma	lemma	PROPN
ejpam-239	302	3	3.18	3.18	NUM
ejpam-239	302	4	.	.	PUNCT
ejpam-239	303	1	let	let	VERB
ejpam-239	303	2	i	i	PRON
ejpam-239	303	3	=	=	PUNCT
ejpam-239	304	1	[	[	X
ejpam-239	304	2	0	0	NUM
ejpam-239	304	3	,	,	PUNCT
ejpam-239	304	4	x	x	X
ejpam-239	304	5	]	]	X
ejpam-239	304	6	,	,	PUNCT
ejpam-239	304	7	0	0	PUNCT
ejpam-239	304	8	<	<	X
ejpam-239	304	9	x	x	X
ejpam-239	304	10	,	,	PUNCT
ejpam-239	304	11	be	be	AUX
ejpam-239	304	12	an	an	DET
ejpam-239	304	13	interval	interval	NOUN
ejpam-239	304	14	in	in	ADP
ejpam-239	304	15	an	an	DET
ejpam-239	304	16	adl	adl	NOUN
ejpam-239	304	17	r	r	NOUN
ejpam-239	304	18	with	with	ADP
ejpam-239	304	19	0	0	NUM
ejpam-239	304	20	.	.	PUNCT
ejpam-239	305	1	then	then	ADV
ejpam-239	305	2	we	we	PRON
ejpam-239	305	3	have	have	VERB
ejpam-239	305	4	the	the	DET
ejpam-239	305	5	following	following	NOUN
ejpam-239	305	6	:	:	PUNCT
ejpam-239	305	7	(	(	PUNCT
ejpam-239	305	8	i	i	NOUN
ejpam-239	305	9	)	)	PUNCT
ejpam-239	305	10	.	.	PUNCT
ejpam-239	306	1	for	for	ADP
ejpam-239	306	2	a	a	DET
ejpam-239	306	3	,	,	PUNCT
ejpam-239	306	4	b	b	X
ejpam-239	306	5	∈	∈	NOUN
ejpam-239	306	6	i	i	PRON
ejpam-239	306	7	,	,	PUNCT
ejpam-239	306	8	(	(	PUNCT
ejpam-239	306	9	a]+	a]+	PROPN
ejpam-239	306	10	⊆	⊆	NUM
ejpam-239	306	11	(	(	PUNCT
ejpam-239	306	12	b]+	b]+	ADJ
ejpam-239	306	13	implies	imply	VERB
ejpam-239	306	14	(	(	PUNCT
ejpam-239	306	15	a]∗	a]∗	PROPN
ejpam-239	306	16	⊆	⊆	NUM
ejpam-239	306	17	(	(	PUNCT
ejpam-239	306	18	b]∗.	b]∗.	PROPN
ejpam-239	306	19	(	(	PUNCT
ejpam-239	306	20	ii	ii	NOUN
ejpam-239	306	21	)	)	PUNCT
ejpam-239	306	22	.	.	PUNCT
ejpam-239	307	1	if	if	SCONJ
ejpam-239	307	2	z	z	NOUN
ejpam-239	307	3	∈	∈	PROPN
ejpam-239	307	4	r	r	NOUN
ejpam-239	307	5	,	,	PUNCT
ejpam-239	307	6	then	then	ADV
ejpam-239	307	7	(	(	PUNCT
ejpam-239	307	8	z]∗	z]∗	X
ejpam-239	307	9	∩	∩	NOUN
ejpam-239	307	10	i	i	PRON
ejpam-239	307	11	=	=	SYM
ejpam-239	307	12	(	(	PUNCT
ejpam-239	307	13	z	z	PROPN
ejpam-239	307	14	∧	∧	PROPN
ejpam-239	307	15	x]+	x]+	PROPN
ejpam-239	307	16	.	.	PUNCT
ejpam-239	308	1	proof	proof	NOUN
ejpam-239	308	2	:	:	PUNCT
ejpam-239	308	3	(	(	PUNCT
ejpam-239	308	4	i	i	NOUN
ejpam-239	308	5	)	)	PUNCT
ejpam-239	308	6	.	.	PUNCT
ejpam-239	309	1	let	let	VERB
ejpam-239	309	2	a	a	DET
ejpam-239	309	3	,	,	PUNCT
ejpam-239	309	4	b	b	X
ejpam-239	309	5	∈	∈	NOUN
ejpam-239	309	6	i	i	PRON
ejpam-239	309	7	and	and	CCONJ
ejpam-239	309	8	suppose	suppose	VERB
ejpam-239	309	9	(	(	PUNCT
ejpam-239	309	10	a]+	a]+	PROPN
ejpam-239	309	11	⊆	⊆	NUM
ejpam-239	309	12	(	(	PUNCT
ejpam-239	309	13	b]+	b]+	X
ejpam-239	309	14	.	.	PUNCT
ejpam-239	310	1	let	let	VERB
ejpam-239	310	2	t	t	PROPN
ejpam-239	310	3	∈	∈	PROPN
ejpam-239	310	4	(	(	PUNCT
ejpam-239	310	5	a]∗.	a]∗.	NOUN
ejpam-239	310	6	then	then	ADV
ejpam-239	310	7	t	t	PROPN
ejpam-239	310	8	∧	∧	PROPN
ejpam-239	310	9	a	a	DET
ejpam-239	310	10	=	=	SYM
ejpam-239	310	11	0	0	NUM
ejpam-239	311	1	and	and	CCONJ
ejpam-239	311	2	t	t	PROPN
ejpam-239	311	3	∈	∈	PROPN
ejpam-239	311	4	r	r	NOUN
ejpam-239	311	5	⇒	⇒	NOUN
ejpam-239	311	6	t	t	PROPN
ejpam-239	311	7	∧	∧	NOUN
ejpam-239	311	8	x	x	X
ejpam-239	311	9	∧	∧	PROPN
ejpam-239	311	10	a	a	PRON
ejpam-239	311	11	=	=	SYM
ejpam-239	311	12	0	0	NUM
ejpam-239	312	1	and	and	CCONJ
ejpam-239	312	2	t	t	PROPN
ejpam-239	312	3	∧	∧	PROPN
ejpam-239	312	4	x	x	PUNCT
ejpam-239	312	5	∈	∈	PROPN
ejpam-239	313	1	i	i	PRON
ejpam-239	313	2	,	,	PUNCT
ejpam-239	313	3	since	since	SCONJ
ejpam-239	313	4	x	x	PROPN
ejpam-239	313	5	∈	∈	PROPN
ejpam-239	313	6	i	i	PRON
ejpam-239	313	7	.	.	PUNCT
ejpam-239	314	1	which	which	PRON
ejpam-239	314	2	implies	imply	VERB
ejpam-239	314	3	t	t	PROPN
ejpam-239	314	4	∧	∧	PROPN
ejpam-239	314	5	x	x	SYM
ejpam-239	314	6	∈	∈	PROPN
ejpam-239	314	7	(	(	PUNCT
ejpam-239	314	8	a]+	a]+	PROPN
ejpam-239	314	9	⊆	⊆	NUM
ejpam-239	314	10	(	(	PUNCT
ejpam-239	314	11	b]+	b]+	ADJ
ejpam-239	314	12	⇒	⇒	NOUN
ejpam-239	314	13	t	t	PROPN
ejpam-239	315	1	∧	∧	NOUN
ejpam-239	315	2	x	x	PUNCT
ejpam-239	315	3	∧	∧	PROPN
ejpam-239	315	4	b	b	NOUN
ejpam-239	315	5	=	=	SYM
ejpam-239	315	6	0⇒	0⇒	PROPN
ejpam-239	315	7	t	t	PROPN
ejpam-239	315	8	∧	∧	PROPN
ejpam-239	315	9	b	b	PROPN
ejpam-239	315	10	=	=	SYM
ejpam-239	315	11	0	0	NUM
ejpam-239	315	12	,	,	PUNCT
ejpam-239	315	13	since	since	SCONJ
ejpam-239	315	14	t	t	PROPN
ejpam-239	315	15	∈	∈	PROPN
ejpam-239	316	1	i	i	PRON
ejpam-239	316	2	=	=	PUNCT
ejpam-239	317	1	[	[	X
ejpam-239	317	2	0	0	NUM
ejpam-239	317	3	,	,	PUNCT
ejpam-239	317	4	x	x	NOUN
ejpam-239	317	5	]	]	X
ejpam-239	317	6	.	.	PUNCT
ejpam-239	318	1	hence	hence	ADV
ejpam-239	318	2	t	t	PROPN
ejpam-239	318	3	∈	∈	PROPN
ejpam-239	318	4	(	(	PUNCT
ejpam-239	318	5	b]∗.	b]∗.	PROPN
ejpam-239	318	6	(	(	PUNCT
ejpam-239	318	7	ii	ii	NOUN
ejpam-239	318	8	)	)	PUNCT
ejpam-239	318	9	.	.	PUNCT
ejpam-239	319	1	let	let	VERB
ejpam-239	319	2	t	t	PROPN
ejpam-239	319	3	∈	∈	PROPN
ejpam-239	319	4	(	(	PUNCT
ejpam-239	319	5	z]∗	z]∗	X
ejpam-239	319	6	∩	∩	NOUN
ejpam-239	319	7	i	i	PRON
ejpam-239	319	8	.	.	PUNCT
ejpam-239	320	1	then	then	ADV
ejpam-239	320	2	t	t	PROPN
ejpam-239	320	3	∈	∈	PROPN
ejpam-239	320	4	(	(	PUNCT
ejpam-239	320	5	z]∗	z]∗	PROPN
ejpam-239	320	6	and	and	CCONJ
ejpam-239	320	7	t	t	PROPN
ejpam-239	320	8	∈	∈	PROPN
ejpam-239	320	9	i	i	PRON
ejpam-239	320	10	.	.	PUNCT
ejpam-239	321	1	hence	hence	ADV
ejpam-239	321	2	t	t	X
ejpam-239	321	3	∧	∧	PROPN
ejpam-239	321	4	z	z	PROPN
ejpam-239	321	5	=	=	SYM
ejpam-239	321	6	0	0	NUM
ejpam-239	322	1	and	and	CCONJ
ejpam-239	322	2	t	t	NOUN
ejpam-239	322	3	∈	∈	PROPN
ejpam-239	323	1	i	i	PRON
ejpam-239	323	2	.	.	PUNCT
ejpam-239	324	1	thus	thus	ADV
ejpam-239	324	2	t	t	X
ejpam-239	324	3	∧	∧	PROPN
ejpam-239	324	4	z	z	PROPN
ejpam-239	324	5	∧	∧	NOUN
ejpam-239	324	6	x	x	X
ejpam-239	324	7	=	=	SYM
ejpam-239	324	8	0	0	NUM
ejpam-239	324	9	and	and	CCONJ
ejpam-239	324	10	t	t	PROPN
ejpam-239	324	11	∈	∈	PROPN
ejpam-239	324	12	i	i	PRON
ejpam-239	324	13	⇒	⇒	VERB
ejpam-239	324	14	t	t	PROPN
ejpam-239	324	15	∧	∧	PROPN
ejpam-239	324	16	(	(	PUNCT
ejpam-239	324	17	z	z	NOUN
ejpam-239	324	18	∧	∧	PROPN
ejpam-239	324	19	x	x	NOUN
ejpam-239	324	20	)	)	PUNCT
ejpam-239	324	21	=	=	SYM
ejpam-239	324	22	0	0	NUM
ejpam-239	324	23	and	and	CCONJ
ejpam-239	324	24	t	t	PROPN
ejpam-239	324	25	∈	∈	PROPN
ejpam-239	324	26	i	i	PRON
ejpam-239	324	27	⇒	⇒	VERB
ejpam-239	324	28	t	t	PROPN
ejpam-239	324	29	∈	∈	PROPN
ejpam-239	324	30	(	(	PUNCT
ejpam-239	324	31	z	z	NOUN
ejpam-239	324	32	∧	∧	PROPN
ejpam-239	324	33	x]+	x]+	PROPN
ejpam-239	324	34	.	.	PUNCT
ejpam-239	325	1	therefore	therefore	ADV
ejpam-239	325	2	(	(	PUNCT
ejpam-239	325	3	z]∗∩	z]∗∩	NOUN
ejpam-239	325	4	i	i	PRON
ejpam-239	325	5	⊆	⊆	NUM
ejpam-239	325	6	(	(	PUNCT
ejpam-239	325	7	z∧	z∧	PROPN
ejpam-239	325	8	x]+	x]+	PROPN
ejpam-239	325	9	.	.	PUNCT
ejpam-239	325	10	again	again	ADV
ejpam-239	325	11	,	,	PUNCT
ejpam-239	325	12	let	let	VERB
ejpam-239	325	13	t	t	PROPN
ejpam-239	325	14	∈	∈	PROPN
ejpam-239	325	15	(	(	PUNCT
ejpam-239	325	16	z∧	z∧	PROPN
ejpam-239	325	17	x]+	x]+	PROPN
ejpam-239	325	18	,	,	PUNCT
ejpam-239	325	19	then	then	ADV
ejpam-239	325	20	t∧z∧	t∧z∧	NOUN
ejpam-239	325	21	x	x	PUNCT
ejpam-239	325	22	=	=	SYM
ejpam-239	325	23	0	0	NUM
ejpam-239	325	24	and	and	CCONJ
ejpam-239	325	25	t	t	PROPN
ejpam-239	325	26	∈	∈	PROPN
ejpam-239	325	27	i	i	PRON
ejpam-239	325	28	⇒	⇒	VERB
ejpam-239	325	29	z∧	z∧	PROPN
ejpam-239	325	30	t∧	t∧	NUM
ejpam-239	325	31	x	x	PROPN
ejpam-239	325	32	=	=	SYM
ejpam-239	325	33	0	0	NUM
ejpam-239	326	1	and	and	CCONJ
ejpam-239	326	2	t	t	PROPN
ejpam-239	326	3	∈	∈	PROPN
ejpam-239	326	4	i	i	PRON
ejpam-239	326	5	⇒	⇒	VERB
ejpam-239	326	6	z∧t	z∧t	PROPN
ejpam-239	327	1	=	=	SYM
ejpam-239	327	2	0	0	NUM
ejpam-239	328	1	and	and	CCONJ
ejpam-239	328	2	t	t	PROPN
ejpam-239	328	3	∈	∈	PROPN
ejpam-239	328	4	i	i	PRON
ejpam-239	328	5	⇒	⇒	VERB
ejpam-239	328	6	t	t	PROPN
ejpam-239	328	7	∈	∈	PROPN
ejpam-239	328	8	(	(	PUNCT
ejpam-239	328	9	z]∗	z]∗	PROPN
ejpam-239	328	10	and	and	CCONJ
ejpam-239	328	11	t	t	PROPN
ejpam-239	328	12	∈	∈	PROPN
ejpam-239	329	1	i	i	PRON
ejpam-239	329	2	.	.	PUNCT
ejpam-239	330	1	hence	hence	ADV
ejpam-239	330	2	t	t	PROPN
ejpam-239	330	3	∈	∈	PROPN
ejpam-239	330	4	(	(	PUNCT
ejpam-239	330	5	z]∗∩i	z]∗∩i	PROPN
ejpam-239	330	6	.	.	PUNCT
ejpam-239	331	1	thus	thus	ADV
ejpam-239	331	2	(	(	PUNCT
ejpam-239	331	3	z∧x]+	z∧x]+	NOUN
ejpam-239	331	4	⊆	⊆	NUM
ejpam-239	331	5	(	(	PUNCT
ejpam-239	331	6	z]∗∩i	z]∗∩i	PROPN
ejpam-239	331	7	.	.	PUNCT
ejpam-239	332	1	therefore	therefore	ADV
ejpam-239	332	2	(	(	PUNCT
ejpam-239	332	3	z]∗	z]∗	X
ejpam-239	332	4	∩	∩	NOUN
ejpam-239	332	5	i	i	PRON
ejpam-239	332	6	=	=	SYM
ejpam-239	332	7	(	(	PUNCT
ejpam-239	332	8	z	z	PROPN
ejpam-239	332	9	∧	∧	PROPN
ejpam-239	332	10	x]+	x]+	PROPN
ejpam-239	332	11	.	.	PUNCT
ejpam-239	333	1	�	�	PROPN
ejpam-239	333	2	we	we	PRON
ejpam-239	333	3	now	now	ADV
ejpam-239	333	4	prove	prove	VERB
ejpam-239	333	5	the	the	DET
ejpam-239	333	6	characterization	characterization	NOUN
ejpam-239	333	7	theorem	theorem	NOUN
ejpam-239	333	8	of	of	ADP
ejpam-239	333	9	a	a	DET
ejpam-239	333	10	sectionally	sectionally	ADV
ejpam-239	333	11	⋆-adl	⋆-adl	PROPN
ejpam-239	333	12	in	in	ADP
ejpam-239	333	13	terms	term	NOUN
ejpam-239	333	14	of	of	ADP
ejpam-239	333	15	it	it	PRON
ejpam-239	333	16	’s	’	VERB
ejpam-239	333	17	annulets	annulet	NOUN
ejpam-239	333	18	.	.	PUNCT
ejpam-239	334	1	before	before	ADP
ejpam-239	334	2	proving	prove	VERB
ejpam-239	334	3	it	it	PRON
ejpam-239	334	4	,	,	PUNCT
ejpam-239	334	5	we	we	PRON
ejpam-239	334	6	can	can	AUX
ejpam-239	334	7	observe	observe	VERB
ejpam-239	334	8	that	that	SCONJ
ejpam-239	334	9	if	if	SCONJ
ejpam-239	334	10	r	r	NOUN
ejpam-239	334	11	is	be	AUX
ejpam-239	334	12	an	an	DET
ejpam-239	334	13	adl	adl	NOUN
ejpam-239	334	14	with	with	ADP
ejpam-239	334	15	0	0	NUM
ejpam-239	335	1	and	and	CCONJ
ejpam-239	335	2	i	i	PRON
ejpam-239	335	3	=	=	PUNCT
ejpam-239	336	1	[	[	X
ejpam-239	336	2	0	0	NUM
ejpam-239	336	3	,	,	PUNCT
ejpam-239	336	4	x	x	NOUN
ejpam-239	336	5	]	]	X
ejpam-239	336	6	,	,	PUNCT
ejpam-239	336	7	0	0	PUNCT
ejpam-239	336	8	<	<	X
ejpam-239	336	9	x	x	X
ejpam-239	336	10	for	for	ADP
ejpam-239	336	11	some	some	DET
ejpam-239	336	12	x	x	SYM
ejpam-239	336	13	∈	∈	PROPN
ejpam-239	336	14	r	r	NOUN
ejpam-239	336	15	,	,	PUNCT
ejpam-239	336	16	then	then	ADV
ejpam-239	336	17	a0(i	a0(i	X
ejpam-239	336	18	)	)	PUNCT
ejpam-239	336	19	is	be	AUX
ejpam-239	336	20	a	a	DET
ejpam-239	336	21	bounded	bounded	ADJ
ejpam-239	336	22	distributive	distributive	ADJ
ejpam-239	336	23	lattice	lattice	NOUN
ejpam-239	336	24	(	(	PUNCT
ejpam-239	336	25	with	with	ADP
ejpam-239	336	26	respect	respect	NOUN
ejpam-239	336	27	to	to	ADP
ejpam-239	336	28	the	the	DET
ejpam-239	336	29	operations	operation	NOUN
ejpam-239	336	30	given	give	VERB
ejpam-239	336	31	in	in	ADP
ejpam-239	336	32	the	the	DET
ejpam-239	336	33	theorem	theorem	ADJ
ejpam-239	336	34	3.3	3.3	NUM
ejpam-239	336	35	)	)	PUNCT
ejpam-239	336	36	with	with	ADP
ejpam-239	336	37	the	the	DET
ejpam-239	336	38	greatest	great	ADJ
ejpam-239	336	39	element	element	NOUN
ejpam-239	336	40	i	i	NOUN
ejpam-239	336	41	=	=	PUNCT
ejpam-239	337	1	(	(	PUNCT
ejpam-239	337	2	0]+	0]+	NUM
ejpam-239	337	3	and	and	CCONJ
ejpam-239	337	4	the	the	DET
ejpam-239	337	5	least	least	ADJ
ejpam-239	337	6	element	element	NOUN
ejpam-239	337	7	(	(	PUNCT
ejpam-239	337	8	x]+	x]+	PROPN
ejpam-239	337	9	.	.	PUNCT
ejpam-239	338	1	g.	g.	PROPN
ejpam-239	338	2	c.	c.	PROPN
ejpam-239	338	3	rao	rao	PROPN
ejpam-239	338	4	and	and	CCONJ
ejpam-239	338	5	m.	m.	PROPN
ejpam-239	338	6	sambasiva	sambasiva	PROPN
ejpam-239	338	7	rao	rao	PROPN
ejpam-239	338	8	/	/	SYM
ejpam-239	338	9	eur	eur	PROPN
ejpam-239	338	10	.	.	PUNCT
ejpam-239	339	1	j.	j.	PROPN
ejpam-239	339	2	pure	pure	PROPN
ejpam-239	339	3	appl	appl	PROPN
ejpam-239	339	4	.	.	PROPN
ejpam-239	339	5	math	math	PROPN
ejpam-239	339	6	,	,	PUNCT
ejpam-239	339	7	2	2	NUM
ejpam-239	339	8	(	(	PUNCT
ejpam-239	339	9	2009	2009	NUM
ejpam-239	339	10	)	)	PUNCT
ejpam-239	339	11	,	,	PUNCT
ejpam-239	339	12	(	(	PUNCT
ejpam-239	339	13	58	58	NUM
ejpam-239	339	14	-	-	SYM
ejpam-239	339	15	72	72	NUM
ejpam-239	339	16	)	)	PUNCT
ejpam-239	339	17	70	70	NUM
ejpam-239	339	18	theorem	theorem	VERB
ejpam-239	339	19	3.19	3.19	NUM
ejpam-239	339	20	.	.	PUNCT
ejpam-239	340	1	let	let	VERB
ejpam-239	340	2	r	r	PRON
ejpam-239	340	3	be	be	AUX
ejpam-239	340	4	an	an	DET
ejpam-239	340	5	adl	adl	NOUN
ejpam-239	340	6	with	with	ADP
ejpam-239	340	7	0	0	NUM
ejpam-239	340	8	.	.	PUNCT
ejpam-239	340	9	thena0(r	thena0(r	NOUN
ejpam-239	340	10	)	)	PUNCT
ejpam-239	340	11	is	be	AUX
ejpam-239	340	12	relatively	relatively	ADV
ejpam-239	340	13	complemented	complemented	ADJ
ejpam-239	340	14	if	if	SCONJ
ejpam-239	341	1	and	and	CCONJ
ejpam-239	341	2	only	only	ADV
ejpam-239	341	3	if	if	SCONJ
ejpam-239	341	4	r	r	NOUN
ejpam-239	341	5	is	be	AUX
ejpam-239	341	6	sectionally	sectionally	ADV
ejpam-239	341	7	⋆-adl	⋆-adl	PROPN
ejpam-239	341	8	.	.	PUNCT
ejpam-239	342	1	proof	proof	NOUN
ejpam-239	342	2	:	:	PUNCT
ejpam-239	342	3	assume	assume	VERB
ejpam-239	342	4	thata0(r	thata0(r	NUM
ejpam-239	342	5	)	)	PUNCT
ejpam-239	342	6	is	be	AUX
ejpam-239	342	7	relatively	relatively	ADV
ejpam-239	342	8	complemented	complemented	ADJ
ejpam-239	342	9	.	.	PUNCT
ejpam-239	343	1	we	we	PRON
ejpam-239	343	2	have	have	VERB
ejpam-239	343	3	to	to	PART
ejpam-239	343	4	prove	prove	VERB
ejpam-239	343	5	that	that	SCONJ
ejpam-239	343	6	each	each	DET
ejpam-239	343	7	interval	interval	NOUN
ejpam-239	344	1	i	i	PRON
ejpam-239	345	1	=	=	PUNCT
ejpam-239	346	1	[	[	X
ejpam-239	346	2	0	0	NUM
ejpam-239	346	3	,	,	PUNCT
ejpam-239	346	4	x	x	X
ejpam-239	346	5	]	]	X
ejpam-239	346	6	in	in	ADP
ejpam-239	346	7	r	r	NOUN
ejpam-239	346	8	is	be	AUX
ejpam-239	346	9	a	a	DET
ejpam-239	346	10	⋆−adl	⋆−adl	NOUN
ejpam-239	346	11	.	.	PUNCT
ejpam-239	347	1	by	by	ADP
ejpam-239	347	2	theorem	theorem	NOUN
ejpam-239	347	3	3.12	3.12	NUM
ejpam-239	347	4	,	,	PUNCT
ejpam-239	347	5	it	it	PRON
ejpam-239	347	6	is	be	AUX
ejpam-239	347	7	enough	enough	ADJ
ejpam-239	347	8	to	to	PART
ejpam-239	347	9	prove	prove	VERB
ejpam-239	347	10	thata0(i	thata0(i	VERB
ejpam-239	347	11	)	)	PUNCT
ejpam-239	348	1	is	be	AUX
ejpam-239	348	2	relatively	relatively	ADV
ejpam-239	348	3	complemented	complemented	ADJ
ejpam-239	348	4	.	.	PUNCT
ejpam-239	349	1	since	since	SCONJ
ejpam-239	349	2	a0(i	a0(i	SYM
ejpam-239	349	3	)	)	PUNCT
ejpam-239	349	4	is	be	AUX
ejpam-239	349	5	a	a	DET
ejpam-239	349	6	distributive	distributive	ADJ
ejpam-239	349	7	lattice	lattice	NOUN
ejpam-239	349	8	with	with	ADP
ejpam-239	349	9	the	the	DET
ejpam-239	349	10	greatest	great	ADJ
ejpam-239	349	11	element	element	NOUN
ejpam-239	349	12	i	i	NOUN
ejpam-239	349	13	=	=	PUNCT
ejpam-239	349	14	(	(	PUNCT
ejpam-239	349	15	0]+	0]+	NOUN
ejpam-239	349	16	,	,	PUNCT
ejpam-239	349	17	it	it	PRON
ejpam-239	349	18	is	be	AUX
ejpam-239	349	19	enough	enough	ADJ
ejpam-239	349	20	to	to	PART
ejpam-239	349	21	prove	prove	VERB
ejpam-239	350	1	that	that	SCONJ
ejpam-239	350	2	each	each	DET
ejpam-239	350	3	interval	interval	NOUN
ejpam-239	351	1	[	[	X
ejpam-239	351	2	j	j	X
ejpam-239	351	3	,	,	PUNCT
ejpam-239	351	4	i	i	PROPN
ejpam-239	351	5	]	]	PUNCT
ejpam-239	351	6	,	,	PUNCT
ejpam-239	351	7	j	j	PROPN
ejpam-239	351	8	∈a0(i	∈a0(i	PROPN
ejpam-239	351	9	)	)	PUNCT
ejpam-239	351	10	is	be	AUX
ejpam-239	351	11	complemented	complement	VERB
ejpam-239	351	12	.	.	PUNCT
ejpam-239	352	1	choose	choose	VERB
ejpam-239	352	2	a	a	DET
ejpam-239	352	3	,	,	PUNCT
ejpam-239	352	4	b	b	X
ejpam-239	352	5	∈	∈	NOUN
ejpam-239	352	6	i	i	PRON
ejpam-239	352	7	such	such	ADJ
ejpam-239	352	8	that	that	SCONJ
ejpam-239	352	9	(	(	PUNCT
ejpam-239	352	10	b]+	b]+	PROPN
ejpam-239	352	11	∈	∈	PROPN
ejpam-239	352	12	�	�	PROPN
ejpam-239	352	13	(	(	PUNCT
ejpam-239	352	14	a]+	a]+	PROPN
ejpam-239	352	15	,	,	PUNCT
ejpam-239	352	16	i	i	PRON
ejpam-239	352	17	�	�	PROPN
ejpam-239	352	18	⊆a0(i	⊆a0(i	NUM
ejpam-239	352	19	)	)	PUNCT
ejpam-239	352	20	.	.	PUNCT
ejpam-239	353	1	then	then	ADV
ejpam-239	353	2	(	(	PUNCT
ejpam-239	353	3	a]+	a]+	PROPN
ejpam-239	353	4	⊆	⊆	NUM
ejpam-239	353	5	(	(	PUNCT
ejpam-239	353	6	b]+	b]+	NOUN
ejpam-239	353	7	⊆	⊆	NUM
ejpam-239	353	8	i	i	PRON
ejpam-239	353	9	.	.	PUNCT
ejpam-239	354	1	by	by	ADP
ejpam-239	354	2	lemma	lemma	PROPN
ejpam-239	354	3	3.18(i	3.18(i	PROPN
ejpam-239	354	4	)	)	PUNCT
ejpam-239	354	5	,	,	PUNCT
ejpam-239	354	6	(	(	PUNCT
ejpam-239	354	7	a]∗	a]∗	PROPN
ejpam-239	354	8	⊆	⊆	NUM
ejpam-239	354	9	(	(	PUNCT
ejpam-239	354	10	b]∗	b]∗	PROPN
ejpam-239	354	11	⊆	⊆	NUM
ejpam-239	354	12	r.	r.	NOUN
ejpam-239	354	13	since	since	SCONJ
ejpam-239	354	14	a0(r	a0(r	PROPN
ejpam-239	354	15	)	)	PUNCT
ejpam-239	354	16	is	be	AUX
ejpam-239	354	17	relatively	relatively	ADV
ejpam-239	354	18	complemented	complemented	ADJ
ejpam-239	354	19	and	and	CCONJ
ejpam-239	354	20	(	(	PUNCT
ejpam-239	354	21	b]∗	b]∗	PROPN
ejpam-239	354	22	∈	∈	PROPN
ejpam-239	355	1	[	[	X
ejpam-239	355	2	(	(	PUNCT
ejpam-239	355	3	a]∗,r	a]∗,r	NOUN
ejpam-239	355	4	]	]	PUNCT
ejpam-239	355	5	,	,	PUNCT
ejpam-239	355	6	there	there	PRON
ejpam-239	355	7	exists	exist	VERB
ejpam-239	355	8	an	an	DET
ejpam-239	355	9	element	element	NOUN
ejpam-239	355	10	c	c	PROPN
ejpam-239	355	11	∈	∈	NOUN
ejpam-239	355	12	r	r	NOUN
ejpam-239	355	13	such	such	ADJ
ejpam-239	355	14	that	that	PRON
ejpam-239	355	15	(	(	PUNCT
ejpam-239	355	16	c]∗	c]∗	PROPN
ejpam-239	355	17	∈	∈	PROPN
ejpam-239	356	1	[	[	X
ejpam-239	356	2	(	(	PUNCT
ejpam-239	356	3	a]∗,r	a]∗,r	NOUN
ejpam-239	356	4	]	]	PUNCT
ejpam-239	356	5	and	and	CCONJ
ejpam-239	356	6	(	(	PUNCT
ejpam-239	356	7	b]∗	b]∗	PROPN
ejpam-239	356	8	∩	∩	NOUN
ejpam-239	356	9	(	(	PUNCT
ejpam-239	356	10	c]∗	c]∗	PROPN
ejpam-239	356	11	=	=	SYM
ejpam-239	356	12	(	(	PUNCT
ejpam-239	356	13	a]∗	a]∗	PROPN
ejpam-239	356	14	and	and	CCONJ
ejpam-239	356	15	(	(	PUNCT
ejpam-239	356	16	b]∗∨(c]∗	b]∗∨(c]∗	PROPN
ejpam-239	356	17	=	=	SYM
ejpam-239	356	18	r.	r.	PROPN
ejpam-239	356	19	now	now	ADV
ejpam-239	356	20	(	(	PUNCT
ejpam-239	356	21	b]∗	b]∗	PROPN
ejpam-239	356	22	∩	∩	NOUN
ejpam-239	356	23	(	(	PUNCT
ejpam-239	356	24	c]∗	c]∗	PROPN
ejpam-239	356	25	=	=	SYM
ejpam-239	356	26	(	(	PUNCT
ejpam-239	356	27	a]∗	a]∗	PROPN
ejpam-239	356	28	⇒	⇒	PROPN
ejpam-239	356	29	(	(	PUNCT
ejpam-239	356	30	b]∗	b]∗	PROPN
ejpam-239	356	31	∩	∩	NOUN
ejpam-239	356	32	(	(	PUNCT
ejpam-239	356	33	c]∗	c]∗	PROPN
ejpam-239	356	34	∩	∩	PROPN
ejpam-239	356	35	i	i	PRON
ejpam-239	356	36	=	=	SYM
ejpam-239	356	37	(	(	PUNCT
ejpam-239	356	38	a]∗	a]∗	PROPN
ejpam-239	356	39	∩	∩	NOUN
ejpam-239	356	40	i	i	PRON
ejpam-239	356	41	⇒	⇒	VERB
ejpam-239	356	42	[	[	X
ejpam-239	356	43	(	(	PUNCT
ejpam-239	356	44	b]∗	b]∗	PROPN
ejpam-239	356	45	∩	∩	ADJ
ejpam-239	356	46	i	i	X
ejpam-239	356	47	]	]	X
ejpam-239	356	48	∩	∩	NOUN
ejpam-239	356	49	[	[	X
ejpam-239	356	50	(	(	PUNCT
ejpam-239	356	51	c]∗	c]∗	PROPN
ejpam-239	356	52	∩	∩	PROPN
ejpam-239	356	53	i	i	PRON
ejpam-239	356	54	]	]	X
ejpam-239	356	55	=	=	SYM
ejpam-239	356	56	(	(	PUNCT
ejpam-239	356	57	a]∗	a]∗	PROPN
ejpam-239	356	58	∩	∩	NOUN
ejpam-239	356	59	i	i	PRON
ejpam-239	356	60	⇒	⇒	VERB
ejpam-239	356	61	(	(	PUNCT
ejpam-239	356	62	b]+	b]+	INTJ
ejpam-239	356	63	∩	∩	NOUN
ejpam-239	356	64	(	(	PUNCT
ejpam-239	356	65	c]+	c]+	X
ejpam-239	356	66	=	=	SYM
ejpam-239	356	67	(	(	PUNCT
ejpam-239	356	68	a]+	a]+	PROPN
ejpam-239	356	69	−→	−→	NOUN
ejpam-239	356	70	(	(	PUNCT
ejpam-239	356	71	1	1	NUM
ejpam-239	356	72	)	)	PUNCT
ejpam-239	356	73	secondly	secondly	ADV
ejpam-239	356	74	,	,	PUNCT
ejpam-239	356	75	(	(	PUNCT
ejpam-239	356	76	b]∗∨(c]∗	b]∗∨(c]∗	PROPN
ejpam-239	356	77	=	=	SYM
ejpam-239	356	78	r⇒	r⇒	DET
ejpam-239	356	79	�	�	PROPN
ejpam-239	356	80	(	(	PUNCT
ejpam-239	356	81	b]∗∨(c]∗	b]∗∨(c]∗	PROPN
ejpam-239	356	82	�	�	PROPN
ejpam-239	356	83	∩i	∩i	PROPN
ejpam-239	356	84	=	=	SYM
ejpam-239	356	85	r∩i	r∩i	NOUN
ejpam-239	356	86	⇒	⇒	NOUN
ejpam-239	356	87	[	[	X
ejpam-239	356	88	(	(	PUNCT
ejpam-239	356	89	b]∗	b]∗	PROPN
ejpam-239	356	90	∩	∩	NOUN
ejpam-239	356	91	i]∨	i]∨	X
ejpam-239	356	92	[	[	X
ejpam-239	356	93	(	(	PUNCT
ejpam-239	356	94	c]∗	c]∗	PROPN
ejpam-239	356	95	∩	∩	PROPN
ejpam-239	356	96	i	i	PRON
ejpam-239	356	97	]	]	X
ejpam-239	356	98	=	=	PUNCT
ejpam-239	356	99	i	i	PRON
ejpam-239	356	100	⇒	⇒	VERB
ejpam-239	356	101	(	(	PUNCT
ejpam-239	356	102	b]+∨(c]+	b]+∨(c]+	PROPN
ejpam-239	356	103	=	=	PUNCT
ejpam-239	356	104	i	i	PRON
ejpam-239	356	105	−→	−→	VERB
ejpam-239	356	106	(	(	PUNCT
ejpam-239	356	107	2	2	NUM
ejpam-239	356	108	)	)	PUNCT
ejpam-239	356	109	from	from	ADP
ejpam-239	356	110	(	(	PUNCT
ejpam-239	356	111	1	1	NUM
ejpam-239	356	112	)	)	PUNCT
ejpam-239	356	113	and	and	CCONJ
ejpam-239	356	114	(	(	PUNCT
ejpam-239	356	115	2	2	NUM
ejpam-239	356	116	)	)	PUNCT
ejpam-239	356	117	,	,	PUNCT
ejpam-239	356	118	we	we	PRON
ejpam-239	356	119	get	get	VERB
ejpam-239	356	120	that	that	PRON
ejpam-239	356	121	(	(	PUNCT
ejpam-239	356	122	c]+	c]+	INTJ
ejpam-239	356	123	is	be	AUX
ejpam-239	356	124	the	the	DET
ejpam-239	356	125	complement	complement	NOUN
ejpam-239	356	126	of	of	ADP
ejpam-239	356	127	(	(	PUNCT
ejpam-239	356	128	b]+	b]+	ADJ
ejpam-239	356	129	in	in	ADP
ejpam-239	356	130	�	�	PROPN
ejpam-239	356	131	(	(	PUNCT
ejpam-239	356	132	a]+	a]+	PROPN
ejpam-239	356	133	,	,	PUNCT
ejpam-239	356	134	i	i	PRON
ejpam-239	356	135	�	�	PROPN
ejpam-239	356	136	.	.	PUNCT
ejpam-239	357	1	hence	hence	ADV
ejpam-239	357	2	�	�	PROPN
ejpam-239	357	3	(	(	PUNCT
ejpam-239	357	4	a]+	a]+	PROPN
ejpam-239	357	5	,	,	PUNCT
ejpam-239	357	6	i	i	PRON
ejpam-239	357	7	�	�	PROPN
ejpam-239	357	8	is	be	AUX
ejpam-239	357	9	relatively	relatively	ADV
ejpam-239	357	10	complemented	complemented	ADJ
ejpam-239	357	11	.	.	PUNCT
ejpam-239	358	1	conversely	conversely	ADV
ejpam-239	358	2	assume	assume	VERB
ejpam-239	358	3	that	that	SCONJ
ejpam-239	358	4	r	r	NOUN
ejpam-239	358	5	is	be	AUX
ejpam-239	358	6	sectionally	sectionally	ADV
ejpam-239	358	7	⋆-adl	⋆-adl	PROPN
ejpam-239	358	8	.	.	PUNCT
ejpam-239	359	1	since	since	SCONJ
ejpam-239	359	2	a0(r	a0(r	PROPN
ejpam-239	359	3	)	)	PUNCT
ejpam-239	359	4	is	be	AUX
ejpam-239	359	5	a	a	DET
ejpam-239	359	6	distributive	distributive	ADJ
ejpam-239	359	7	lattice	lattice	NOUN
ejpam-239	359	8	with	with	ADP
ejpam-239	359	9	the	the	DET
ejpam-239	359	10	greatest	great	ADJ
ejpam-239	359	11	element	element	NOUN
ejpam-239	359	12	r	r	NOUN
ejpam-239	359	13	,	,	PUNCT
ejpam-239	359	14	it	it	PRON
ejpam-239	359	15	is	be	AUX
ejpam-239	359	16	enough	enough	ADJ
ejpam-239	359	17	to	to	PART
ejpam-239	359	18	prove	prove	VERB
ejpam-239	359	19	that	that	SCONJ
ejpam-239	359	20	each	each	DET
ejpam-239	359	21	interval	interval	NOUN
ejpam-239	359	22	[	[	X
ejpam-239	359	23	(	(	PUNCT
ejpam-239	359	24	a]∗,r	a]∗,r	NOUN
ejpam-239	359	25	]	]	PUNCT
ejpam-239	359	26	,	,	PUNCT
ejpam-239	359	27	(	(	PUNCT
ejpam-239	359	28	a]∗	a]∗	PROPN
ejpam-239	359	29	∈a0(r	∈a0(r	NOUN
ejpam-239	359	30	)	)	PUNCT
ejpam-239	359	31	is	be	AUX
ejpam-239	359	32	complemented	complement	VERB
ejpam-239	359	33	.	.	PUNCT
ejpam-239	360	1	let	let	VERB
ejpam-239	360	2	(	(	PUNCT
ejpam-239	360	3	b]∗	b]∗	PROPN
ejpam-239	360	4	∈	∈	PROPN
ejpam-239	361	1	[	[	X
ejpam-239	361	2	(	(	PUNCT
ejpam-239	361	3	a]∗,r	a]∗,r	NOUN
ejpam-239	361	4	]	]	X
ejpam-239	361	5	.	.	PUNCT
ejpam-239	362	1	therefore	therefore	ADV
ejpam-239	362	2	(	(	PUNCT
ejpam-239	362	3	a]∗	a]∗	PROPN
ejpam-239	362	4	⊆	⊆	NUM
ejpam-239	362	5	(	(	PUNCT
ejpam-239	362	6	b]∗	b]∗	PROPN
ejpam-239	362	7	⊆	⊆	NUM
ejpam-239	362	8	r.	r.	NOUN
ejpam-239	362	9	consider	consider	VERB
ejpam-239	362	10	the	the	DET
ejpam-239	362	11	interval	interval	NOUN
ejpam-239	363	1	i	i	NOUN
ejpam-239	363	2	=	=	PUNCT
ejpam-239	364	1	[	[	X
ejpam-239	364	2	0	0	NUM
ejpam-239	364	3	,	,	PUNCT
ejpam-239	364	4	b	b	PROPN
ejpam-239	364	5	∨	∨	NUM
ejpam-239	364	6	a	a	DET
ejpam-239	364	7	]	]	X
ejpam-239	364	8	.	.	PUNCT
ejpam-239	365	1	then	then	ADV
ejpam-239	365	2	by	by	ADP
ejpam-239	365	3	the	the	DET
ejpam-239	365	4	hypothesis	hypothesis	NOUN
ejpam-239	365	5	,	,	PUNCT
ejpam-239	365	6	i	i	PRON
ejpam-239	365	7	is	be	AUX
ejpam-239	365	8	a	a	DET
ejpam-239	365	9	⋆−	⋆−	NOUN
ejpam-239	365	10	adl	adl	NOUN
ejpam-239	365	11	.	.	PUNCT
ejpam-239	366	1	so	so	ADV
ejpam-239	366	2	by	by	ADP
ejpam-239	366	3	theorem	theorem	NOUN
ejpam-239	366	4	3.12,a0(i	3.12,a0(i	NUM
ejpam-239	366	5	)	)	PUNCT
ejpam-239	366	6	is	be	AUX
ejpam-239	366	7	complemented	complement	VERB
ejpam-239	366	8	.	.	PUNCT
ejpam-239	367	1	hence	hence	ADV
ejpam-239	367	2	each	each	DET
ejpam-239	367	3	interval	interval	NOUN
ejpam-239	367	4	�	�	PROPN
ejpam-239	367	5	(	(	PUNCT
ejpam-239	367	6	a]+	a]+	PROPN
ejpam-239	367	7	,	,	PUNCT
ejpam-239	367	8	i	i	PRON
ejpam-239	367	9	�	�	PROPN
ejpam-239	367	10	,	,	PUNCT
ejpam-239	367	11	(	(	PUNCT
ejpam-239	367	12	a]+	a]+	PROPN
ejpam-239	367	13	∈a0(i	∈a0(i	NOUN
ejpam-239	367	14	)	)	PUNCT
ejpam-239	367	15	,	,	PUNCT
ejpam-239	367	16	where	where	SCONJ
ejpam-239	367	17	a	a	DET
ejpam-239	367	18	∈	∈	NOUN
ejpam-239	367	19	i	i	PRON
ejpam-239	367	20	is	be	AUX
ejpam-239	367	21	complemented	complement	VERB
ejpam-239	367	22	.	.	PUNCT
ejpam-239	368	1	we	we	PRON
ejpam-239	368	2	have	have	VERB
ejpam-239	368	3	by	by	ADP
ejpam-239	368	4	the	the	DET
ejpam-239	368	5	lemma	lemma	PROPN
ejpam-239	368	6	3.18(ii	3.18(ii	PROPN
ejpam-239	368	7	)	)	PUNCT
ejpam-239	368	8	,	,	PUNCT
ejpam-239	368	9	(	(	PUNCT
ejpam-239	368	10	a]∗	a]∗	PROPN
ejpam-239	368	11	∩	∩	NOUN
ejpam-239	368	12	i	i	PRON
ejpam-239	368	13	=	=	SYM
ejpam-239	368	14	(	(	PUNCT
ejpam-239	368	15	a	a	DET
ejpam-239	368	16	∧	∧	PROPN
ejpam-239	368	17	(	(	PUNCT
ejpam-239	368	18	b	b	PROPN
ejpam-239	368	19	∨	∨	NUM
ejpam-239	368	20	a)]+	a)]+	PROPN
ejpam-239	368	21	and	and	CCONJ
ejpam-239	368	22	(	(	PUNCT
ejpam-239	368	23	b]∗	b]∗	PROPN
ejpam-239	368	24	∩	∩	NOUN
ejpam-239	368	25	i	i	PRON
ejpam-239	368	26	=	=	PUNCT
ejpam-239	368	27	(	(	PUNCT
ejpam-239	368	28	b	b	PROPN
ejpam-239	368	29	∧	∧	PROPN
ejpam-239	368	30	(	(	PUNCT
ejpam-239	368	31	b	b	PROPN
ejpam-239	368	32	∨	∨	NOUN
ejpam-239	368	33	a)]+	a)]+	NOUN
ejpam-239	368	34	=	=	SYM
ejpam-239	368	35	(	(	PUNCT
ejpam-239	368	36	b]+	b]+	NOUN
ejpam-239	368	37	⊆	⊆	NUM
ejpam-239	368	38	i	i	PRON
ejpam-239	368	39	,	,	PUNCT
ejpam-239	368	40	that	that	ADV
ejpam-239	368	41	is	is	ADV
ejpam-239	368	42	(	(	PUNCT
ejpam-239	368	43	b]+	b]+	PROPN
ejpam-239	368	44	∈	∈	PROPN
ejpam-239	368	45	�	�	PROPN
ejpam-239	368	46	(	(	PUNCT
ejpam-239	368	47	a	a	DET
ejpam-239	368	48	∧	∧	PROPN
ejpam-239	368	49	(	(	PUNCT
ejpam-239	368	50	b	b	PROPN
ejpam-239	368	51	∨	∨	NUM
ejpam-239	368	52	a)]+	a)]+	PROPN
ejpam-239	368	53	,	,	PUNCT
ejpam-239	368	54	i	i	PRON
ejpam-239	368	55	�	�	PROPN
ejpam-239	368	56	.	.	PUNCT
ejpam-239	368	57	sincea0(i	sincea0(i	PUNCT
ejpam-239	368	58	)	)	PUNCT
ejpam-239	368	59	is	be	AUX
ejpam-239	368	60	complemented	complement	VERB
ejpam-239	368	61	,	,	PUNCT
ejpam-239	368	62	there	there	PRON
ejpam-239	368	63	exists	exist	VERB
ejpam-239	368	64	an	an	DET
ejpam-239	368	65	element	element	NOUN
ejpam-239	368	66	c	c	NOUN
ejpam-239	368	67	∈	∈	PROPN
ejpam-239	368	68	i	i	PRON
ejpam-239	368	69	such	such	ADJ
ejpam-239	368	70	that	that	SCONJ
ejpam-239	368	71	(	(	PUNCT
ejpam-239	368	72	b]+	b]+	ADJ
ejpam-239	368	73	∩	∩	NOUN
ejpam-239	368	74	(	(	PUNCT
ejpam-239	368	75	c]+	c]+	X
ejpam-239	368	76	=	=	SYM
ejpam-239	368	77	(	(	PUNCT
ejpam-239	368	78	a	a	DET
ejpam-239	368	79	∧	∧	PROPN
ejpam-239	368	80	(	(	PUNCT
ejpam-239	368	81	b	b	PROPN
ejpam-239	368	82	∨	∨	NOUN
ejpam-239	368	83	a)]+	a)]+	PROPN
ejpam-239	368	84	and	and	CCONJ
ejpam-239	368	85	(	(	PUNCT
ejpam-239	368	86	b]+∨(c]+	b]+∨(c]+	NOUN
ejpam-239	368	87	=	=	PUNCT
ejpam-239	369	1	i	i	PRON
ejpam-239	369	2	−→	−→	VERB
ejpam-239	369	3	(	(	PUNCT
ejpam-239	369	4	3	3	NUM
ejpam-239	369	5	)	)	PUNCT
ejpam-239	369	6	now	now	ADV
ejpam-239	369	7	our	our	PRON
ejpam-239	369	8	claim	claim	NOUN
ejpam-239	369	9	is	be	AUX
ejpam-239	369	10	(	(	PUNCT
ejpam-239	369	11	b]∗	b]∗	PROPN
ejpam-239	369	12	∩	∩	NOUN
ejpam-239	369	13	(	(	PUNCT
ejpam-239	369	14	c]∗	c]∗	PROPN
ejpam-239	369	15	=	=	SYM
ejpam-239	369	16	(	(	PUNCT
ejpam-239	369	17	a]∗	a]∗	PROPN
ejpam-239	369	18	and	and	CCONJ
ejpam-239	369	19	(	(	PUNCT
ejpam-239	369	20	b]∗∨(c]∗	b]∗∨(c]∗	PROPN
ejpam-239	369	21	=	=	SYM
ejpam-239	369	22	r.	r.	PROPN
ejpam-239	369	23	g.	g.	PROPN
ejpam-239	369	24	c.	c.	PROPN
ejpam-239	369	25	rao	rao	PROPN
ejpam-239	369	26	and	and	CCONJ
ejpam-239	369	27	m.	m.	PROPN
ejpam-239	369	28	sambasiva	sambasiva	PROPN
ejpam-239	369	29	rao	rao	PROPN
ejpam-239	369	30	/	/	SYM
ejpam-239	369	31	eur	eur	PROPN
ejpam-239	369	32	.	.	PUNCT
ejpam-239	370	1	j.	j.	PROPN
ejpam-239	370	2	pure	pure	PROPN
ejpam-239	370	3	appl	appl	PROPN
ejpam-239	370	4	.	.	PROPN
ejpam-239	370	5	math	math	PROPN
ejpam-239	370	6	,	,	PUNCT
ejpam-239	370	7	2	2	NUM
ejpam-239	370	8	(	(	PUNCT
ejpam-239	370	9	2009	2009	NUM
ejpam-239	370	10	)	)	PUNCT
ejpam-239	370	11	,	,	PUNCT
ejpam-239	370	12	(	(	PUNCT
ejpam-239	370	13	58	58	NUM
ejpam-239	370	14	-	-	SYM
ejpam-239	370	15	72	72	NUM
ejpam-239	370	16	)	)	PUNCT
ejpam-239	370	17	71	71	NUM
ejpam-239	370	18	let	let	VERB
ejpam-239	370	19	x	x	X
ejpam-239	370	20	∈	∈	PROPN
ejpam-239	370	21	(	(	PUNCT
ejpam-239	370	22	b]∗	b]∗	PROPN
ejpam-239	370	23	∩	∩	NOUN
ejpam-239	370	24	(	(	PUNCT
ejpam-239	370	25	c]∗.	c]∗.	NOUN
ejpam-239	370	26	then	then	ADV
ejpam-239	370	27	x	x	SYM
ejpam-239	370	28	∈	∈	PROPN
ejpam-239	370	29	(	(	PUNCT
ejpam-239	370	30	b]∗	b]∗	PROPN
ejpam-239	370	31	and	and	CCONJ
ejpam-239	370	32	x	x	SYM
ejpam-239	370	33	∈	∈	PROPN
ejpam-239	370	34	(	(	PUNCT
ejpam-239	370	35	c]∗	c]∗	PROPN
ejpam-239	370	36	,	,	PUNCT
ejpam-239	370	37	implies	imply	VERB
ejpam-239	370	38	b	b	X
ejpam-239	370	39	∧	∧	NOUN
ejpam-239	370	40	x	x	X
ejpam-239	370	41	=	=	SYM
ejpam-239	370	42	0	0	NUM
ejpam-239	370	43	and	and	CCONJ
ejpam-239	370	44	c	c	X
ejpam-239	370	45	∧	∧	NOUN
ejpam-239	370	46	x	x	PUNCT
ejpam-239	370	47	=	=	SYM
ejpam-239	370	48	0	0	NUM
ejpam-239	370	49	⇒	⇒	NOUN
ejpam-239	370	50	x	x	X
ejpam-239	371	1	∧	∧	PROPN
ejpam-239	371	2	(	(	PUNCT
ejpam-239	371	3	b∨	b∨	PROPN
ejpam-239	371	4	a)∧	a)∧	PROPN
ejpam-239	371	5	b	b	PROPN
ejpam-239	371	6	=	=	SYM
ejpam-239	371	7	0	0	PROPN
ejpam-239	371	8	and	and	CCONJ
ejpam-239	371	9	x	x	PROPN
ejpam-239	371	10	∧	∧	PROPN
ejpam-239	371	11	(	(	PUNCT
ejpam-239	371	12	b	b	PROPN
ejpam-239	371	13	∨	∨	NUM
ejpam-239	371	14	a)∧	a)∧	PROPN
ejpam-239	371	15	c	c	NOUN
ejpam-239	371	16	=	=	SYM
ejpam-239	371	17	0	0	X
ejpam-239	371	18	.	.	PUNCT
ejpam-239	371	19	⇒	⇒	NOUN
ejpam-239	371	20	x	x	X
ejpam-239	372	1	∧	∧	PROPN
ejpam-239	372	2	(	(	PUNCT
ejpam-239	372	3	b∨	b∨	PROPN
ejpam-239	372	4	a	a	DET
ejpam-239	372	5	)	)	PUNCT
ejpam-239	372	6	∈	∈	PROPN
ejpam-239	372	7	(	(	PUNCT
ejpam-239	372	8	b]+	b]+	ADJ
ejpam-239	372	9	and	and	CCONJ
ejpam-239	372	10	x	x	PART
ejpam-239	372	11	∧	∧	PROPN
ejpam-239	372	12	(	(	PUNCT
ejpam-239	372	13	b	b	PROPN
ejpam-239	372	14	∨	∨	NUM
ejpam-239	372	15	a	a	PRON
ejpam-239	372	16	)	)	PUNCT
ejpam-239	372	17	∈	∈	PROPN
ejpam-239	372	18	(	(	PUNCT
ejpam-239	372	19	c]+	c]+	PROPN
ejpam-239	372	20	,	,	PUNCT
ejpam-239	372	21	since	since	SCONJ
ejpam-239	372	22	x	x	PROPN
ejpam-239	372	23	∧	∧	PROPN
ejpam-239	372	24	(	(	PUNCT
ejpam-239	372	25	b	b	PROPN
ejpam-239	372	26	∨	∨	NUM
ejpam-239	372	27	a	a	PRON
ejpam-239	372	28	)	)	PUNCT
ejpam-239	372	29	∈	∈	PROPN
ejpam-239	372	30	i	i	PRON
ejpam-239	372	31	.	.	PUNCT
ejpam-239	373	1	⇒	⇒	PROPN
ejpam-239	373	2	x	x	X
ejpam-239	374	1	∧	∧	PROPN
ejpam-239	374	2	(	(	PUNCT
ejpam-239	374	3	b	b	PROPN
ejpam-239	374	4	∨	∨	NUM
ejpam-239	374	5	a	a	PRON
ejpam-239	374	6	)	)	PUNCT
ejpam-239	374	7	∈	∈	PROPN
ejpam-239	374	8	(	(	PUNCT
ejpam-239	374	9	b]+	b]+	ADJ
ejpam-239	374	10	∩	∩	NOUN
ejpam-239	374	11	(	(	PUNCT
ejpam-239	374	12	c]+	c]+	PROPN
ejpam-239	374	13	⇒	⇒	NOUN
ejpam-239	374	14	x	x	PUNCT
ejpam-239	374	15	∧	∧	PROPN
ejpam-239	374	16	(	(	PUNCT
ejpam-239	374	17	b	b	PROPN
ejpam-239	374	18	∨	∨	NUM
ejpam-239	374	19	a	a	PRON
ejpam-239	374	20	)	)	PUNCT
ejpam-239	374	21	∈	∈	PROPN
ejpam-239	374	22	(	(	PUNCT
ejpam-239	374	23	a	a	DET
ejpam-239	374	24	∧	∧	PROPN
ejpam-239	374	25	(	(	PUNCT
ejpam-239	374	26	b	b	PROPN
ejpam-239	374	27	∨	∨	NUM
ejpam-239	374	28	a)]+	a)]+	PROPN
ejpam-239	374	29	,	,	PUNCT
ejpam-239	374	30	by	by	ADP
ejpam-239	374	31	(	(	PUNCT
ejpam-239	374	32	3	3	X
ejpam-239	374	33	)	)	PUNCT
ejpam-239	374	34	⇒	⇒	NOUN
ejpam-239	374	35	x	x	PUNCT
ejpam-239	374	36	∧	∧	PROPN
ejpam-239	374	37	(	(	PUNCT
ejpam-239	374	38	b	b	PROPN
ejpam-239	374	39	∨	∨	NUM
ejpam-239	374	40	a)∧	a)∧	PROPN
ejpam-239	374	41	a	a	DET
ejpam-239	374	42	∧	∧	PROPN
ejpam-239	374	43	(	(	PUNCT
ejpam-239	374	44	b	b	PROPN
ejpam-239	374	45	∨	∨	NUM
ejpam-239	374	46	a	a	PRON
ejpam-239	374	47	)	)	PUNCT
ejpam-239	374	48	=	=	SYM
ejpam-239	374	49	0	0	NUM
ejpam-239	374	50	⇒	⇒	NOUN
ejpam-239	374	51	x	x	PUNCT
ejpam-239	374	52	∧	∧	PROPN
ejpam-239	374	53	a	a	DET
ejpam-239	374	54	∧	∧	PROPN
ejpam-239	374	55	(	(	PUNCT
ejpam-239	374	56	b	b	PROPN
ejpam-239	374	57	∨	∨	NUM
ejpam-239	374	58	a)∧	a)∧	PROPN
ejpam-239	374	59	(	(	PUNCT
ejpam-239	374	60	b	b	PROPN
ejpam-239	374	61	∨	∨	NUM
ejpam-239	374	62	a	a	PRON
ejpam-239	374	63	)	)	PUNCT
ejpam-239	374	64	=	=	SYM
ejpam-239	374	65	0	0	NUM
ejpam-239	374	66	⇒	⇒	NOUN
ejpam-239	374	67	(	(	PUNCT
ejpam-239	374	68	x	x	SYM
ejpam-239	374	69	∧	∧	NOUN
ejpam-239	374	70	a)∧	a)∧	PROPN
ejpam-239	374	71	(	(	PUNCT
ejpam-239	374	72	b	b	PROPN
ejpam-239	374	73	∨	∨	NUM
ejpam-239	374	74	a	a	PRON
ejpam-239	374	75	)	)	PUNCT
ejpam-239	374	76	=	=	SYM
ejpam-239	374	77	0	0	NUM
ejpam-239	374	78	⇒	⇒	NOUN
ejpam-239	374	79	(	(	PUNCT
ejpam-239	374	80	b	b	PROPN
ejpam-239	374	81	∨	∨	NUM
ejpam-239	374	82	a)∧	a)∧	PROPN
ejpam-239	374	83	(	(	PUNCT
ejpam-239	374	84	x	x	X
ejpam-239	374	85	∧	∧	NOUN
ejpam-239	374	86	a	a	NOUN
ejpam-239	374	87	)	)	PUNCT
ejpam-239	374	88	=	=	SYM
ejpam-239	374	89	0	0	NUM
ejpam-239	374	90	⇒	⇒	NOUN
ejpam-239	374	91	x	x	X
ejpam-239	374	92	∧	∧	PROPN
ejpam-239	374	93	(	(	PUNCT
ejpam-239	374	94	b	b	PROPN
ejpam-239	374	95	∨	∨	NUM
ejpam-239	374	96	a)∧	a)∧	PROPN
ejpam-239	374	97	a	a	DET
ejpam-239	374	98	=	=	SYM
ejpam-239	374	99	0	0	NUM
ejpam-239	374	100	⇒	⇒	NOUN
ejpam-239	374	101	x	x	PUNCT
ejpam-239	375	1	∧	∧	NOUN
ejpam-239	375	2	a	a	DET
ejpam-239	375	3	=	=	SYM
ejpam-239	375	4	0	0	NUM
ejpam-239	375	5	⇒	⇒	NOUN
ejpam-239	375	6	x	x	X
ejpam-239	375	7	∈	∈	PROPN
ejpam-239	375	8	(	(	PUNCT
ejpam-239	375	9	a]∗	a]∗	PROPN
ejpam-239	375	10	hence	hence	ADV
ejpam-239	375	11	(	(	PUNCT
ejpam-239	375	12	b]∗	b]∗	PROPN
ejpam-239	375	13	∩	∩	NOUN
ejpam-239	375	14	(	(	PUNCT
ejpam-239	375	15	c]∗	c]∗	PROPN
ejpam-239	375	16	⊆	⊆	NUM
ejpam-239	375	17	(	(	PUNCT
ejpam-239	375	18	a]∗	a]∗	NOUN
ejpam-239	375	19	−→	−→	NOUN
ejpam-239	375	20	(	(	PUNCT
ejpam-239	375	21	4	4	NUM
ejpam-239	375	22	)	)	PUNCT
ejpam-239	375	23	conversely	conversely	ADV
ejpam-239	375	24	,	,	PUNCT
ejpam-239	375	25	let	let	VERB
ejpam-239	375	26	x	x	X
ejpam-239	375	27	∈	∈	PROPN
ejpam-239	375	28	(	(	PUNCT
ejpam-239	375	29	a]∗.	a]∗.	NOUN
ejpam-239	375	30	then	then	ADV
ejpam-239	375	31	x	x	PART
ejpam-239	375	32	∧	∧	PROPN
ejpam-239	375	33	a	a	DET
ejpam-239	375	34	=	=	SYM
ejpam-239	375	35	0	0	NUM
ejpam-239	375	36	⇒	⇒	NOUN
ejpam-239	375	37	x	x	PUNCT
ejpam-239	375	38	∧	∧	PROPN
ejpam-239	375	39	a	a	DET
ejpam-239	375	40	∧	∧	PROPN
ejpam-239	375	41	(	(	PUNCT
ejpam-239	375	42	b	b	PROPN
ejpam-239	375	43	∨	∨	NUM
ejpam-239	375	44	a)∧	a)∧	PROPN
ejpam-239	375	45	(	(	PUNCT
ejpam-239	375	46	b	b	PROPN
ejpam-239	375	47	∨	∨	NUM
ejpam-239	375	48	a	a	PRON
ejpam-239	375	49	)	)	PUNCT
ejpam-239	375	50	=	=	SYM
ejpam-239	375	51	0	0	NUM
ejpam-239	375	52	⇒	⇒	NOUN
ejpam-239	375	53	x	x	X
ejpam-239	376	1	∧	∧	PROPN
ejpam-239	376	2	(	(	PUNCT
ejpam-239	376	3	b	b	PROPN
ejpam-239	376	4	∨	∨	NUM
ejpam-239	376	5	a)∧	a)∧	PROPN
ejpam-239	376	6	a	a	DET
ejpam-239	376	7	∧	∧	PROPN
ejpam-239	376	8	(	(	PUNCT
ejpam-239	376	9	b	b	PROPN
ejpam-239	376	10	∨	∨	NUM
ejpam-239	376	11	a	a	PRON
ejpam-239	376	12	)	)	PUNCT
ejpam-239	376	13	=	=	SYM
ejpam-239	376	14	0	0	NUM
ejpam-239	376	15	⇒	⇒	NOUN
ejpam-239	376	16	x	x	X
ejpam-239	376	17	∧	∧	PROPN
ejpam-239	376	18	(	(	PUNCT
ejpam-239	376	19	b	b	PROPN
ejpam-239	376	20	∨	∨	NUM
ejpam-239	376	21	a	a	PRON
ejpam-239	376	22	)	)	PUNCT
ejpam-239	376	23	∈	∈	PROPN
ejpam-239	376	24	(	(	PUNCT
ejpam-239	376	25	a	a	DET
ejpam-239	376	26	∧	∧	PROPN
ejpam-239	376	27	(	(	PUNCT
ejpam-239	376	28	b	b	PROPN
ejpam-239	376	29	∨	∨	NUM
ejpam-239	376	30	a)]+	a)]+	PROPN
ejpam-239	376	31	,	,	PUNCT
ejpam-239	376	32	since	since	SCONJ
ejpam-239	376	33	x	x	PROPN
ejpam-239	376	34	∧	∧	PROPN
ejpam-239	376	35	(	(	PUNCT
ejpam-239	376	36	b	b	PROPN
ejpam-239	376	37	∨	∨	NUM
ejpam-239	376	38	a	a	PRON
ejpam-239	376	39	)	)	PUNCT
ejpam-239	376	40	∈	∈	PROPN
ejpam-239	376	41	i	i	PRON
ejpam-239	376	42	.	.	PUNCT
ejpam-239	377	1	⇒	⇒	PROPN
ejpam-239	377	2	x	x	X
ejpam-239	378	1	∧	∧	PROPN
ejpam-239	378	2	(	(	PUNCT
ejpam-239	378	3	b	b	PROPN
ejpam-239	378	4	∨	∨	NUM
ejpam-239	378	5	a	a	PRON
ejpam-239	378	6	)	)	PUNCT
ejpam-239	378	7	∈	∈	PROPN
ejpam-239	378	8	(	(	PUNCT
ejpam-239	378	9	b]+	b]+	ADJ
ejpam-239	378	10	∩	∩	NOUN
ejpam-239	378	11	(	(	PUNCT
ejpam-239	378	12	c]+	c]+	PROPN
ejpam-239	378	13	,	,	PUNCT
ejpam-239	378	14	by	by	ADP
ejpam-239	378	15	(	(	PUNCT
ejpam-239	378	16	3	3	X
ejpam-239	378	17	)	)	PUNCT
ejpam-239	378	18	⇒	⇒	NOUN
ejpam-239	378	19	x	x	PUNCT
ejpam-239	378	20	∧	∧	PROPN
ejpam-239	378	21	(	(	PUNCT
ejpam-239	378	22	b	b	PROPN
ejpam-239	378	23	∨	∨	NUM
ejpam-239	378	24	a	a	PRON
ejpam-239	378	25	)	)	PUNCT
ejpam-239	378	26	∈	∈	NOUN
ejpam-239	378	27	(	(	PUNCT
ejpam-239	378	28	b]+	b]+	ADJ
ejpam-239	378	29	and	and	CCONJ
ejpam-239	378	30	x	x	PART
ejpam-239	378	31	∧	∧	PROPN
ejpam-239	378	32	(	(	PUNCT
ejpam-239	378	33	b	b	PROPN
ejpam-239	378	34	∨	∨	NUM
ejpam-239	378	35	a	a	PRON
ejpam-239	378	36	)	)	PUNCT
ejpam-239	378	37	∈	∈	PROPN
ejpam-239	378	38	(	(	PUNCT
ejpam-239	378	39	c]+	c]+	INTJ
ejpam-239	378	40	⇒	⇒	VERB
ejpam-239	378	41	x	x	PUNCT
ejpam-239	378	42	∧	∧	NOUN
ejpam-239	378	43	(	(	PUNCT
ejpam-239	378	44	b	b	PROPN
ejpam-239	378	45	∨	∨	NUM
ejpam-239	378	46	a)∧	a)∧	PROPN
ejpam-239	378	47	b	b	PROPN
ejpam-239	378	48	=	=	SYM
ejpam-239	378	49	0	0	PROPN
ejpam-239	378	50	and	and	CCONJ
ejpam-239	378	51	x	x	PROPN
ejpam-239	378	52	∧	∧	PROPN
ejpam-239	378	53	(	(	PUNCT
ejpam-239	378	54	b	b	PROPN
ejpam-239	378	55	∨	∨	NUM
ejpam-239	378	56	a)∧	a)∧	PROPN
ejpam-239	378	57	c	c	NOUN
ejpam-239	378	58	=	=	SYM
ejpam-239	378	59	0	0	X
ejpam-239	378	60	.	.	PUNCT
ejpam-239	378	61	⇒	⇒	NOUN
ejpam-239	378	62	x	x	X
ejpam-239	379	1	∧	∧	NOUN
ejpam-239	379	2	b	b	NOUN
ejpam-239	379	3	=	=	SYM
ejpam-239	379	4	0	0	PROPN
ejpam-239	379	5	and	and	CCONJ
ejpam-239	379	6	x	x	PART
ejpam-239	379	7	∧	∧	NOUN
ejpam-239	379	8	c	c	NOUN
ejpam-239	379	9	=	=	SYM
ejpam-239	379	10	0	0	NUM
ejpam-239	379	11	,	,	PUNCT
ejpam-239	379	12	since	since	SCONJ
ejpam-239	379	13	c	c	PROPN
ejpam-239	379	14	∈	∈	PROPN
ejpam-239	380	1	i	i	PRON
ejpam-239	380	2	=	=	PUNCT
ejpam-239	381	1	[	[	X
ejpam-239	381	2	0	0	NUM
ejpam-239	381	3	,	,	PUNCT
ejpam-239	381	4	b	b	PROPN
ejpam-239	381	5	∨	∨	NUM
ejpam-239	381	6	a	a	DET
ejpam-239	381	7	]	]	X
ejpam-239	381	8	.	.	PUNCT
ejpam-239	382	1	⇒	⇒	PROPN
ejpam-239	382	2	x	x	PUNCT
ejpam-239	382	3	∈	∈	PROPN
ejpam-239	382	4	(	(	PUNCT
ejpam-239	382	5	b]∗	b]∗	PROPN
ejpam-239	382	6	and	and	CCONJ
ejpam-239	382	7	x	x	SYM
ejpam-239	382	8	∈	∈	PROPN
ejpam-239	382	9	(	(	PUNCT
ejpam-239	382	10	c]∗	c]∗	PROPN
ejpam-239	382	11	⇒	⇒	VERB
ejpam-239	382	12	x	x	PUNCT
ejpam-239	382	13	∈	∈	PROPN
ejpam-239	382	14	(	(	PUNCT
ejpam-239	382	15	b]∗	b]∗	PROPN
ejpam-239	382	16	∩	∩	NOUN
ejpam-239	382	17	(	(	PUNCT
ejpam-239	382	18	c]∗	c]∗	PROPN
ejpam-239	382	19	hence	hence	ADV
ejpam-239	382	20	(	(	PUNCT
ejpam-239	382	21	a]∗	a]∗	PROPN
ejpam-239	382	22	⊆	⊆	NUM
ejpam-239	382	23	(	(	PUNCT
ejpam-239	382	24	b]∗	b]∗	PROPN
ejpam-239	382	25	∩	∩	NOUN
ejpam-239	382	26	(	(	PUNCT
ejpam-239	382	27	c]∗.	c]∗.	NOUN
ejpam-239	382	28	−→	−→	NOUN
ejpam-239	382	29	(	(	PUNCT
ejpam-239	382	30	5	5	NUM
ejpam-239	382	31	)	)	PUNCT
ejpam-239	382	32	from	from	ADP
ejpam-239	382	33	(	(	PUNCT
ejpam-239	382	34	4	4	NUM
ejpam-239	382	35	)	)	PUNCT
ejpam-239	382	36	and	and	CCONJ
ejpam-239	382	37	(	(	PUNCT
ejpam-239	382	38	5	5	NUM
ejpam-239	382	39	)	)	PUNCT
ejpam-239	382	40	,	,	PUNCT
ejpam-239	382	41	we	we	PRON
ejpam-239	382	42	can	can	AUX
ejpam-239	382	43	obtain	obtain	VERB
ejpam-239	382	44	(	(	PUNCT
ejpam-239	382	45	b]∗	b]∗	PROPN
ejpam-239	382	46	∩	∩	NOUN
ejpam-239	382	47	(	(	PUNCT
ejpam-239	382	48	c]∗	c]∗	PROPN
ejpam-239	382	49	=	=	SYM
ejpam-239	382	50	(	(	PUNCT
ejpam-239	382	51	a]∗.	a]∗.	NOUN
ejpam-239	382	52	again	again	ADV
ejpam-239	382	53	from	from	ADP
ejpam-239	382	54	(	(	PUNCT
ejpam-239	382	55	3	3	NUM
ejpam-239	382	56	)	)	PUNCT
ejpam-239	382	57	,	,	PUNCT
ejpam-239	382	58	we	we	PRON
ejpam-239	382	59	have	have	VERB
ejpam-239	382	60	(	(	PUNCT
ejpam-239	382	61	b]+∨(c]+	b]+∨(c]+	NOUN
ejpam-239	382	62	=	=	PUNCT
ejpam-239	382	63	i	i	PRON
ejpam-239	382	64	⇒	⇒	VERB
ejpam-239	382	65	(	(	PUNCT
ejpam-239	382	66	b	b	X
ejpam-239	382	67	∧	∧	NOUN
ejpam-239	382	68	c]+	c]+	PROPN
ejpam-239	382	69	=	=	PUNCT
ejpam-239	382	70	(	(	PUNCT
ejpam-239	382	71	b]+∨(c]+	b]+∨(c]+	PROPN
ejpam-239	382	72	=	=	PUNCT
ejpam-239	382	73	i	i	PRON
ejpam-239	382	74	⇒	⇒	VERB
ejpam-239	382	75	(	(	PUNCT
ejpam-239	382	76	b	b	X
ejpam-239	382	77	∧	∧	NOUN
ejpam-239	382	78	c]+	c]+	PROPN
ejpam-239	382	79	=	=	PRON
ejpam-239	383	1	i	i	PRON
ejpam-239	383	2	⇒	⇒	VERB
ejpam-239	383	3	b	b	X
ejpam-239	383	4	∧	∧	NOUN
ejpam-239	383	5	c	c	NOUN
ejpam-239	383	6	=	=	SYM
ejpam-239	383	7	0	0	NUM
ejpam-239	383	8	references	reference	NOUN
ejpam-239	383	9	72	72	NUM
ejpam-239	383	10	⇒	⇒	NOUN
ejpam-239	383	11	(	(	PUNCT
ejpam-239	383	12	b	b	NOUN
ejpam-239	383	13	∧	∧	PROPN
ejpam-239	383	14	c]∗	c]∗	PROPN
ejpam-239	383	15	=	=	PUNCT
ejpam-239	383	16	(	(	PUNCT
ejpam-239	383	17	0]∗	0]∗	X
ejpam-239	383	18	=	=	PUNCT
ejpam-239	383	19	r	r	NOUN
ejpam-239	383	20	⇒	⇒	NOUN
ejpam-239	383	21	(	(	PUNCT
ejpam-239	384	1	b]∗∨(c]∗	b]∗∨(c]∗	PROPN
ejpam-239	384	2	=	=	SYM
ejpam-239	384	3	r	r	NOUN
ejpam-239	384	4	hence	hence	ADV
ejpam-239	384	5	(	(	PUNCT
ejpam-239	384	6	c]∗	c]∗	PROPN
ejpam-239	384	7	is	be	AUX
ejpam-239	384	8	the	the	DET
ejpam-239	384	9	complement	complement	NOUN
ejpam-239	384	10	of	of	ADP
ejpam-239	384	11	(	(	PUNCT
ejpam-239	384	12	b]∗	b]∗	PROPN
ejpam-239	384	13	in	in	ADP
ejpam-239	384	14	[	[	X
ejpam-239	384	15	(	(	PUNCT
ejpam-239	384	16	a]∗,r	a]∗,r	NOUN
ejpam-239	384	17	]	]	PUNCT
ejpam-239	384	18	.	.	PUNCT
ejpam-239	385	1	thusa0(r	thusa0(r	CCONJ
ejpam-239	385	2	)	)	PUNCT
ejpam-239	385	3	is	be	AUX
ejpam-239	385	4	relatively	relatively	ADV
ejpam-239	385	5	complemented	complemented	ADJ
ejpam-239	385	6	.	.	PUNCT
ejpam-239	386	1	�	�	PROPN
ejpam-239	386	2	acknowledgements	acknowledgement	NOUN
ejpam-239	386	3	.	.	PUNCT
ejpam-239	387	1	the	the	DET
ejpam-239	387	2	authors	author	NOUN
ejpam-239	387	3	would	would	AUX
ejpam-239	387	4	like	like	VERB
ejpam-239	387	5	to	to	PART
ejpam-239	387	6	thank	thank	VERB
ejpam-239	387	7	the	the	DET
ejpam-239	387	8	referee	referee	NOUN
ejpam-239	387	9	for	for	ADP
ejpam-239	387	10	his	his	PRON
ejpam-239	387	11	comments	comment	NOUN
ejpam-239	387	12	and	and	CCONJ
ejpam-239	387	13	valuable	valuable	ADJ
ejpam-239	387	14	suggestions	suggestion	NOUN
ejpam-239	387	15	.	.	PUNCT
ejpam-239	388	1	references	reference	NOUN
ejpam-239	388	2	[	[	X
ejpam-239	388	3	1	1	NUM
ejpam-239	388	4	]	]	PUNCT
ejpam-239	388	5	birkhoff	birkhoff	NOUN
ejpam-239	388	6	.	.	PUNCT
ejpam-239	389	1	g.	g.	PROPN
ejpam-239	389	2	:	:	PUNCT
ejpam-239	390	1	lattice	lattice	PROPN
ejpam-239	390	2	theory	theory	NOUN
ejpam-239	390	3	,	,	PUNCT
ejpam-239	390	4	amer.math.soc.colloq	amer.math.soc.colloq	PROPN
ejpam-239	390	5	.	.	PUNCT
ejpam-239	391	1	xxv	xxv	PROPN
ejpam-239	391	2	,	,	PUNCT
ejpam-239	391	3	providence	providence	NOUN
ejpam-239	391	4	,	,	PUNCT
ejpam-239	391	5	(	(	PUNCT
ejpam-239	391	6	1967	1967	NUM
ejpam-239	391	7	)	)	PUNCT
ejpam-239	391	8	,	,	PUNCT
ejpam-239	391	9	u.s.a	u.s.a	NOUN
ejpam-239	391	10	.	.	PUNCT
ejpam-239	392	1	[	[	X
ejpam-239	392	2	2	2	NUM
ejpam-239	392	3	]	]	PUNCT
ejpam-239	392	4	burris	burris	PROPN
ejpam-239	392	5	.	.	PUNCT
ejpam-239	393	1	s.	s.	PROPN
ejpam-239	393	2	,	,	PUNCT
ejpam-239	393	3	sankappanavar	sankappanavar	PROPN
ejpam-239	393	4	,	,	PUNCT
ejpam-239	393	5	h.p	h.p	PROPN
ejpam-239	393	6	.	.	PUNCT
ejpam-239	393	7	:	:	PUNCT
ejpam-239	393	8	a	a	DET
ejpam-239	393	9	cource	cource	NOUN
ejpam-239	393	10	in	in	ADP
ejpam-239	393	11	univerasal	univerasal	ADJ
ejpam-239	393	12	algebra	algebra	PROPN
ejpam-239	393	13	,	,	PUNCT
ejpam-239	393	14	springer	springer	NOUN
ejpam-239	393	15	verlag	verlag	NOUN
ejpam-239	393	16	,	,	PUNCT
ejpam-239	393	17	(	(	PUNCT
ejpam-239	393	18	1981	1981	NUM
ejpam-239	393	19	)	)	PUNCT
ejpam-239	393	20	.	.	PUNCT
ejpam-239	394	1	[	[	X
ejpam-239	394	2	3	3	NUM
ejpam-239	394	3	]	]	SYM
ejpam-239	394	4	cornish.w.h	cornish.w.h	NUM
ejpam-239	394	5	.	.	PUNCT
ejpam-239	395	1	:	:	PUNCT
ejpam-239	395	2	normal	normal	ADJ
ejpam-239	395	3	lattices	lattice	NOUN
ejpam-239	395	4	,	,	PUNCT
ejpam-239	395	5	j.austral.math.soc	j.austral.math.soc	PROPN
ejpam-239	395	6	.	.	PROPN
ejpam-239	395	7	,	,	PUNCT
ejpam-239	395	8	14	14	NUM
ejpam-239	395	9	(	(	PUNCT
ejpam-239	395	10	1972	1972	NUM
ejpam-239	395	11	)	)	PUNCT
ejpam-239	395	12	,	,	PUNCT
ejpam-239	395	13	200	200	NUM
ejpam-239	395	14	-	-	SYM
ejpam-239	395	15	215	215	NUM
ejpam-239	395	16	.	.	PUNCT
ejpam-239	396	1	[	[	X
ejpam-239	396	2	4	4	NUM
ejpam-239	396	3	]	]	SYM
ejpam-239	396	4	cornish.w.h	cornish.w.h	NUM
ejpam-239	396	5	.	.	PUNCT
ejpam-239	396	6	:	:	PUNCT
ejpam-239	397	1	annulets	annulet	NOUN
ejpam-239	397	2	and	and	CCONJ
ejpam-239	397	3	αideals	αideal	NOUN
ejpam-239	397	4	in	in	ADP
ejpam-239	397	5	distributive	distributive	ADJ
ejpam-239	397	6	lattices	lattice	NOUN
ejpam-239	397	7	,	,	PUNCT
ejpam-239	397	8	j.austral.math.soc	j.austral.math.soc	PROPN
ejpam-239	397	9	.	.	PROPN
ejpam-239	397	10	,	,	PUNCT
ejpam-239	397	11	15	15	NUM
ejpam-239	397	12	(	(	PUNCT
ejpam-239	397	13	1973	1973	NUM
ejpam-239	397	14	)	)	PUNCT
ejpam-239	397	15	,	,	PUNCT
ejpam-239	397	16	70	70	NUM
ejpam-239	397	17	-	-	SYM
ejpam-239	397	18	77	77	NUM
ejpam-239	397	19	.	.	PUNCT
ejpam-239	398	1	[	[	X
ejpam-239	398	2	5	5	NUM
ejpam-239	398	3	]	]	PUNCT
ejpam-239	398	4	mandelker	mandelker	NOUN
ejpam-239	398	5	.	.	PUNCT
ejpam-239	399	1	m	m	VERB
ejpam-239	399	2	:	:	PUNCT
ejpam-239	399	3	relative	relative	ADJ
ejpam-239	399	4	annihilators	annihilator	NOUN
ejpam-239	399	5	in	in	ADP
ejpam-239	399	6	lattices	lattice	NOUN
ejpam-239	399	7	,	,	PUNCT
ejpam-239	399	8	duke	duke	PROPN
ejpam-239	399	9	math	math	PROPN
ejpam-239	399	10	.	.	PUNCT
ejpam-239	400	1	j	j	PROPN
ejpam-239	400	2	,	,	PUNCT
ejpam-239	400	3	37	37	NUM
ejpam-239	400	4	(	(	PUNCT
ejpam-239	400	5	1970	1970	NUM
ejpam-239	400	6	)	)	PUNCT
ejpam-239	400	7	,	,	PUNCT
ejpam-239	400	8	377	377	NUM
ejpam-239	400	9	-	-	SYM
ejpam-239	400	10	386	386	NUM
ejpam-239	400	11	.	.	PUNCT
ejpam-239	401	1	[	[	X
ejpam-239	401	2	6	6	NUM
ejpam-239	401	3	]	]	SYM
ejpam-239	401	4	rao.g.c	rao.g.c	NOUN
ejpam-239	401	5	.	.	PUNCT
ejpam-239	402	1	:	:	PUNCT
ejpam-239	402	2	almost	almost	ADV
ejpam-239	402	3	distributive	distributive	ADJ
ejpam-239	402	4	lattices	lattice	NOUN
ejpam-239	402	5	,	,	PUNCT
ejpam-239	402	6	doctoral	doctoral	ADJ
ejpam-239	402	7	thesis	thesis	NOUN
ejpam-239	402	8	,	,	PUNCT
ejpam-239	402	9	andhra	andhra	PROPN
ejpam-239	402	10	university	university	PROPN
ejpam-239	402	11	,	,	PUNCT
ejpam-239	402	12	waltair	waltair	NOUN
ejpam-239	402	13	,	,	PUNCT
ejpam-239	402	14	(	(	PUNCT
ejpam-239	402	15	1980	1980	NUM
ejpam-239	402	16	)	)	PUNCT
ejpam-239	402	17	.	.	PUNCT
ejpam-239	403	1	[	[	X
ejpam-239	403	2	7	7	X
ejpam-239	403	3	]	]	X
ejpam-239	403	4	rao.g.c.and	rao.g.c.and	PROPN
ejpam-239	403	5	ravikumar.s	ravikumar.s	PROPN
ejpam-239	403	6	:	:	PUNCT
ejpam-239	403	7	normal	normal	ADJ
ejpam-239	403	8	almost	almost	ADV
ejpam-239	403	9	distributive	distributive	ADJ
ejpam-239	403	10	lattices	lattice	NOUN
ejpam-239	403	11	,	,	PUNCT
ejpam-239	403	12	southeast	southeast	ADJ
ejpam-239	403	13	asian	asian	ADJ
ejpam-239	403	14	bulletin	bulletin	NOUN
ejpam-239	403	15	of	of	ADP
ejpam-239	403	16	mathematics,(to	mathematics,(to	NOUN
ejpam-239	403	17	appear	appear	VERB
ejpam-239	403	18	)	)	PUNCT
ejpam-239	403	19	.	.	PUNCT
ejpam-239	404	1	[	[	X
ejpam-239	404	2	8	8	NUM
ejpam-239	404	3	]	]	SYM
ejpam-239	404	4	swamy.u.m	swamy.u.m	NOUN
ejpam-239	404	5	.	.	PROPN
ejpam-239	404	6	,	,	PUNCT
ejpam-239	404	7	rao.g.c	rao.g.c	PROPN
ejpam-239	404	8	.	.	PUNCT
ejpam-239	405	1	:	:	PUNCT
ejpam-239	405	2	almost	almost	ADV
ejpam-239	405	3	distributive	distributive	ADJ
ejpam-239	405	4	lattices	lattice	NOUN
ejpam-239	405	5	,	,	PUNCT
ejpam-239	405	6	j.austral.math.soc	j.austral.math.soc	PROPN
ejpam-239	405	7	.	.	PUNCT
ejpam-239	406	1	(	(	PUNCT
ejpam-239	406	2	series	series	PROPN
ejpam-239	406	3	a	a	PROPN
ejpam-239	406	4	)	)	PUNCT
ejpam-239	406	5	,	,	PUNCT
ejpam-239	406	6	31	31	NUM
ejpam-239	406	7	(	(	PUNCT
ejpam-239	406	8	1981	1981	NUM
ejpam-239	406	9	)	)	PUNCT
ejpam-239	406	10	,	,	PUNCT
ejpam-239	406	11	77	77	NUM
ejpam-239	406	12	-	-	SYM
ejpam-239	406	13	91	91	NUM
ejpam-239	406	14	.	.	PUNCT
ejpam-239	407	1	[	[	X
ejpam-239	407	2	9	9	NUM
ejpam-239	407	3	]	]	SYM
ejpam-239	407	4	swamy.u.m	swamy.u.m	NOUN
ejpam-239	407	5	.	.	PROPN
ejpam-239	407	6	,	,	PUNCT
ejpam-239	407	7	rao.g.c	rao.g.c	PROPN
ejpam-239	407	8	.	.	PUNCT
ejpam-239	407	9	,	,	PUNCT
ejpam-239	407	10	nanaji	nanaji	PROPN
ejpam-239	407	11	rao.g	rao.g	PROPN
ejpam-239	407	12	.	.	PUNCT
ejpam-239	407	13	:	:	PUNCT
ejpam-239	408	1	pseudo	pseudo	NOUN
ejpam-239	408	2	-	-	NOUN
ejpam-239	408	3	complementation	complementation	NOUN
ejpam-239	408	4	on	on	ADP
ejpam-239	408	5	almost	almost	ADV
ejpam-239	408	6	distributive	distributive	ADJ
ejpam-239	408	7	lattices	lattice	NOUN
ejpam-239	408	8	,	,	PUNCT
ejpam-239	408	9	southeast	southeast	ADJ
ejpam-239	408	10	asian	asian	ADJ
ejpam-239	408	11	bulletin	bulletin	NOUN
ejpam-239	408	12	of	of	ADP
ejpam-239	408	13	mathematics	mathematic	NOUN
ejpam-239	408	14	,	,	PUNCT
ejpam-239	408	15	24	24	NUM
ejpam-239	408	16	(	(	PUNCT
ejpam-239	408	17	2000	2000	NUM
ejpam-239	408	18	)	)	PUNCT
ejpam-239	408	19	,	,	PUNCT
ejpam-239	408	20	95	95	NUM
ejpam-239	408	21	-	-	SYM
ejpam-239	408	22	104	104	NUM
ejpam-239	408	23	.	.	PUNCT
ejpam-239	409	1	[	[	X
ejpam-239	409	2	10	10	NUM
ejpam-239	409	3	]	]	SYM
ejpam-239	409	4	swamy.u.m	swamy.u.m	PROPN
ejpam-239	409	5	.	.	PROPN
ejpam-239	409	6	,	,	PUNCT
ejpam-239	409	7	rao.g.c	rao.g.c	PROPN
ejpam-239	409	8	.	.	PUNCT
ejpam-239	409	9	,	,	PUNCT
ejpam-239	409	10	nanaji	nanaji	PROPN
ejpam-239	409	11	rao.g	rao.g	PROPN
ejpam-239	409	12	.	.	PUNCT
ejpam-239	409	13	:	:	PUNCT
ejpam-239	409	14	stone	stone	NOUN
ejpam-239	409	15	almost	almost	ADV
ejpam-239	409	16	distributive	distributive	ADJ
ejpam-239	409	17	lattices	lattice	NOUN
ejpam-239	409	18	,	,	PUNCT
ejpam-239	409	19	southeast	southeast	ADJ
ejpam-239	409	20	asian	asian	ADJ
ejpam-239	409	21	bulletin	bulletin	NOUN
ejpam-239	409	22	of	of	ADP
ejpam-239	409	23	mathematics	mathematic	NOUN
ejpam-239	409	24	,	,	PUNCT
ejpam-239	409	25	27	27	NUM
ejpam-239	409	26	(	(	PUNCT
ejpam-239	409	27	2003	2003	NUM
ejpam-239	409	28	)	)	PUNCT
ejpam-239	409	29	,	,	PUNCT
ejpam-239	409	30	513	513	NUM
ejpam-239	409	31	-	-	SYM
ejpam-239	409	32	526	526	NUM
ejpam-239	409	33	.	.	PUNCT
ejpam-239	410	1	[	[	X
ejpam-239	410	2	11	11	NUM
ejpam-239	410	3	]	]	SYM
ejpam-239	410	4	swamy.u.m	swamy.u.m	PROPN
ejpam-239	410	5	.	.	PROPN
ejpam-239	410	6	,	,	PUNCT
ejpam-239	410	7	rao.g.c	rao.g.c	PROPN
ejpam-239	410	8	.	.	PUNCT
ejpam-239	410	9	,	,	PUNCT
ejpam-239	410	10	nanaji	nanaji	PROPN
ejpam-239	410	11	rao.g	rao.g	PROPN
ejpam-239	410	12	.	.	PUNCT
ejpam-239	410	13	:	:	PUNCT
ejpam-239	410	14	dense	dense	ADJ
ejpam-239	410	15	elements	element	NOUN
ejpam-239	410	16	in	in	ADP
ejpam-239	410	17	almost	almost	ADV
ejpam-239	410	18	distributive	distributive	ADJ
ejpam-239	410	19	lattices	lattice	NOUN
ejpam-239	410	20	,	,	PUNCT
ejpam-239	410	21	southeast	southeast	ADJ
ejpam-239	410	22	asian	asian	ADJ
ejpam-239	410	23	bulletin	bulletin	NOUN
ejpam-239	410	24	of	of	ADP
ejpam-239	410	25	mathematics	mathematic	NOUN
ejpam-239	410	26	,	,	PUNCT
ejpam-239	410	27	27	27	NUM
ejpam-239	410	28	(	(	PUNCT
ejpam-239	410	29	2004	2004	NUM
ejpam-239	410	30	)	)	PUNCT
ejpam-239	410	31	,	,	PUNCT
ejpam-239	410	32	1081	1081	NUM
ejpam-239	410	33	-	-	SYM
ejpam-239	410	34	1088	1088	NUM
ejpam-239	410	35	.	.	PUNCT