id	sid	tid	token	lemma	pos
ejpam-1781	1	1	10_shaikh.dvi	10_shaikh.dvi	NUM
ejpam-1781	1	2	european	european	PROPN
ejpam-1781	1	3	journal	journal	PROPN
ejpam-1781	1	4	of	of	ADP
ejpam-1781	1	5	pure	pure	ADJ
ejpam-1781	1	6	and	and	CCONJ
ejpam-1781	1	7	applied	apply	VERB
ejpam-1781	1	8	mathematics	mathematic	NOUN
ejpam-1781	1	9	vol	vol	NOUN
ejpam-1781	1	10	.	.	PROPN
ejpam-1781	2	1	6	6	NUM
ejpam-1781	2	2	,	,	PUNCT
ejpam-1781	2	3	no	no	INTJ
ejpam-1781	2	4	.	.	NOUN
ejpam-1781	2	5	1	1	NUM
ejpam-1781	2	6	,	,	PUNCT
ejpam-1781	2	7	2013	2013	NUM
ejpam-1781	2	8	,	,	PUNCT
ejpam-1781	2	9	107	107	NUM
ejpam-1781	2	10	-	-	SYM
ejpam-1781	2	11	118	118	NUM
ejpam-1781	2	12	issn	issn	PROPN
ejpam-1781	2	13	1307	1307	NUM
ejpam-1781	2	14	-	-	SYM
ejpam-1781	2	15	5543	5543	NUM
ejpam-1781	2	16	–	–	PUNCT
ejpam-1781	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1781	2	18	on	on	ADP
ejpam-1781	2	19	prime	prime	ADJ
ejpam-1781	2	20	,	,	PUNCT
ejpam-1781	2	21	minimal	minimal	ADJ
ejpam-1781	2	22	prime	prime	NOUN
ejpam-1781	2	23	and	and	CCONJ
ejpam-1781	2	24	annihilator	annihilator	NOUN
ejpam-1781	2	25	ideals	ideal	NOUN
ejpam-1781	2	26	in	in	ADP
ejpam-1781	2	27	an	an	DET
ejpam-1781	2	28	almost	almost	ADV
ejpam-1781	2	29	distributive	distributive	ADJ
ejpam-1781	2	30	lattice	lattice	NOUN
ejpam-1781	2	31	y.	y.	PROPN
ejpam-1781	2	32	s.	s.	PROPN
ejpam-1781	2	33	pawar1	pawar1	PROPN
ejpam-1781	2	34	,	,	PUNCT
ejpam-1781	2	35	i.	i.	PROPN
ejpam-1781	2	36	a.	a.	PROPN
ejpam-1781	2	37	shaikh2,∗	shaikh2,∗	PROPN
ejpam-1781	2	38	1	1	NUM
ejpam-1781	2	39	department	department	NOUN
ejpam-1781	2	40	of	of	ADP
ejpam-1781	2	41	mathematics	mathematic	NOUN
ejpam-1781	2	42	,	,	PUNCT
ejpam-1781	2	43	spspmšs	spspmšs	PROPN
ejpam-1781	2	44	skn	skn	PROPN
ejpam-1781	2	45	sinhgad	sinhgad	VERB
ejpam-1781	2	46	college	college	NOUN
ejpam-1781	2	47	of	of	ADP
ejpam-1781	2	48	engineering	engineering	NOUN
ejpam-1781	2	49	,	,	PUNCT
ejpam-1781	2	50	at	at	ADP
ejpam-1781	2	51	post	post	PROPN
ejpam-1781	2	52	korti	korti	PROPN
ejpam-1781	2	53	,	,	PUNCT
ejpam-1781	2	54	pandharpur-413304	pandharpur-413304	PROPN
ejpam-1781	2	55	,	,	PUNCT
ejpam-1781	2	56	india	india	PROPN
ejpam-1781	2	57	2	2	NUM
ejpam-1781	2	58	department	department	NOUN
ejpam-1781	2	59	of	of	ADP
ejpam-1781	2	60	mathematics	mathematic	NOUN
ejpam-1781	2	61	,	,	PUNCT
ejpam-1781	2	62	nagesh	nagesh	PROPN
ejpam-1781	2	63	karajagi	karajagi	PROPN
ejpam-1781	2	64	orchid	orchid	PROPN
ejpam-1781	2	65	college	college	PROPN
ejpam-1781	2	66	of	of	ADP
ejpam-1781	2	67	engineering	engineering	NOUN
ejpam-1781	2	68	and	and	CCONJ
ejpam-1781	2	69	technology	technology	NOUN
ejpam-1781	2	70	,	,	PUNCT
ejpam-1781	2	71	solapur-413002,(m.s	solapur-413002,(m.s	PROPN
ejpam-1781	2	72	.	.	PUNCT
ejpam-1781	2	73	)	)	PUNCT
ejpam-1781	2	74	,	,	PUNCT
ejpam-1781	2	75	india	india	PROPN
ejpam-1781	2	76	abstract	abstract	NOUN
ejpam-1781	2	77	.	.	PUNCT
ejpam-1781	3	1	necessary	necessary	ADJ
ejpam-1781	3	2	and	and	CCONJ
ejpam-1781	3	3	sufficient	sufficient	ADJ
ejpam-1781	3	4	conditions	condition	NOUN
ejpam-1781	3	5	for	for	ADP
ejpam-1781	3	6	a	a	DET
ejpam-1781	3	7	prime	prime	ADJ
ejpam-1781	3	8	ideal	ideal	NOUN
ejpam-1781	3	9	to	to	PART
ejpam-1781	3	10	be	be	AUX
ejpam-1781	3	11	a	a	DET
ejpam-1781	3	12	minimal	minimal	ADJ
ejpam-1781	3	13	prime	prime	ADJ
ejpam-1781	3	14	ideal	ideal	NOUN
ejpam-1781	3	15	and	and	CCONJ
ejpam-1781	3	16	prime	prime	ADJ
ejpam-1781	3	17	ideal	ideal	NOUN
ejpam-1781	3	18	to	to	PART
ejpam-1781	3	19	be	be	AUX
ejpam-1781	3	20	a	a	DET
ejpam-1781	3	21	principal	principal	ADJ
ejpam-1781	3	22	ideal	ideal	NOUN
ejpam-1781	3	23	in	in	ADP
ejpam-1781	3	24	an	an	DET
ejpam-1781	3	25	adl	adl	NOUN
ejpam-1781	3	26	are	be	AUX
ejpam-1781	3	27	furnished	furnish	VERB
ejpam-1781	3	28	.	.	PUNCT
ejpam-1781	4	1	and	and	CCONJ
ejpam-1781	4	2	some	some	DET
ejpam-1781	4	3	properties	property	NOUN
ejpam-1781	4	4	of	of	ADP
ejpam-1781	4	5	the	the	DET
ejpam-1781	4	6	special	special	ADJ
ejpam-1781	4	7	subsets	subset	NOUN
ejpam-1781	4	8	of	of	ADP
ejpam-1781	4	9	the	the	DET
ejpam-1781	4	10	set	set	NOUN
ejpam-1781	4	11	of	of	ADP
ejpam-1781	4	12	all	all	DET
ejpam-1781	4	13	prime	prime	ADJ
ejpam-1781	4	14	ideals	ideal	NOUN
ejpam-1781	4	15	in	in	ADP
ejpam-1781	4	16	an	an	DET
ejpam-1781	4	17	adl	adl	NOUN
ejpam-1781	4	18	are	be	AUX
ejpam-1781	4	19	studied	study	VERB
ejpam-1781	4	20	.	.	PUNCT
ejpam-1781	5	1	2010	2010	NUM
ejpam-1781	5	2	mathematics	mathematic	NOUN
ejpam-1781	5	3	subject	subject	NOUN
ejpam-1781	5	4	classifications	classification	NOUN
ejpam-1781	5	5	:	:	PUNCT
ejpam-1781	5	6	06d99	06d99	NUM
ejpam-1781	5	7	key	key	ADJ
ejpam-1781	5	8	words	word	NOUN
ejpam-1781	5	9	and	and	CCONJ
ejpam-1781	5	10	phrases	phrase	NOUN
ejpam-1781	5	11	:	:	PUNCT
ejpam-1781	5	12	almost	almost	ADV
ejpam-1781	5	13	distributive	distributive	ADJ
ejpam-1781	5	14	lattice	lattice	NOUN
ejpam-1781	5	15	(	(	PUNCT
ejpam-1781	5	16	adl	adl	PROPN
ejpam-1781	5	17	)	)	PUNCT
ejpam-1781	5	18	,	,	PUNCT
ejpam-1781	5	19	prime	prime	ADJ
ejpam-1781	5	20	ideal	ideal	NOUN
ejpam-1781	5	21	,	,	PUNCT
ejpam-1781	5	22	minimal	minimal	ADJ
ejpam-1781	5	23	prime	prime	ADJ
ejpam-1781	5	24	ideal	ideal	ADJ
ejpam-1781	5	25	,	,	PUNCT
ejpam-1781	5	26	maximal	maximal	ADJ
ejpam-1781	5	27	ideal	ideal	NOUN
ejpam-1781	5	28	,	,	PUNCT
ejpam-1781	5	29	semi	semi	ADV
ejpam-1781	5	30	filter	filter	NOUN
ejpam-1781	5	31	1	1	NUM
ejpam-1781	5	32	.	.	PUNCT
ejpam-1781	6	1	introduction	introduction	NOUN
ejpam-1781	6	2	an	an	DET
ejpam-1781	6	3	almost	almost	ADV
ejpam-1781	6	4	distributed	distribute	VERB
ejpam-1781	6	5	lattice(adl	lattice(adl	NOUN
ejpam-1781	6	6	)	)	PUNCT
ejpam-1781	6	7	was	be	AUX
ejpam-1781	6	8	introduced	introduce	VERB
ejpam-1781	6	9	by	by	ADP
ejpam-1781	6	10	u.	u.	PROPN
ejpam-1781	6	11	m.	m.	PROPN
ejpam-1781	6	12	swamy	swamy	PROPN
ejpam-1781	6	13	and	and	CCONJ
ejpam-1781	6	14	rao	rao	PROPN
ejpam-1781	6	15	.	.	PUNCT
ejpam-1781	7	1	g	g	PROPN
ejpam-1781	7	2	.c	.c	PUNCT
ejpam-1781	8	1	[	[	X
ejpam-1781	8	2	3	3	NUM
ejpam-1781	8	3	]	]	PUNCT
ejpam-1781	8	4	.	.	PUNCT
ejpam-1781	9	1	after	after	ADP
ejpam-1781	9	2	the	the	DET
ejpam-1781	9	3	boole	boole	PROPN
ejpam-1781	9	4	’s	’s	PART
ejpam-1781	9	5	axiomatization	axiomatization	NOUN
ejpam-1781	9	6	of	of	ADP
ejpam-1781	9	7	the	the	DET
ejpam-1781	9	8	two	two	NUM
ejpam-1781	9	9	valued	value	VERB
ejpam-1781	9	10	propositional	propositional	ADJ
ejpam-1781	9	11	calculus	calculus	NOUN
ejpam-1781	9	12	as	as	ADP
ejpam-1781	9	13	the	the	DET
ejpam-1781	9	14	boolean	boolean	ADJ
ejpam-1781	9	15	algebra	algebra	NOUN
ejpam-1781	9	16	many	many	ADJ
ejpam-1781	9	17	generalizations	generalization	NOUN
ejpam-1781	9	18	of	of	ADP
ejpam-1781	9	19	a	a	DET
ejpam-1781	9	20	boolean	boolean	ADJ
ejpam-1781	9	21	algebra	algebra	NOUN
ejpam-1781	9	22	both	both	DET
ejpam-1781	9	23	ring	ring	NOUN
ejpam-1781	9	24	theoretically	theoretically	ADV
ejpam-1781	9	25	and	and	CCONJ
ejpam-1781	9	26	lattice	lattice	VERB
ejpam-1781	9	27	theoretically	theoretically	ADV
ejpam-1781	9	28	,	,	PUNCT
ejpam-1781	9	29	have	have	AUX
ejpam-1781	9	30	come	come	VERB
ejpam-1781	9	31	into	into	ADP
ejpam-1781	9	32	being	be	AUX
ejpam-1781	9	33	.	.	PUNCT
ejpam-1781	10	1	with	with	ADP
ejpam-1781	10	2	an	an	DET
ejpam-1781	10	3	idea	idea	NOUN
ejpam-1781	10	4	of	of	ADP
ejpam-1781	10	5	bringing	bring	VERB
ejpam-1781	10	6	common	common	ADJ
ejpam-1781	10	7	abstraction	abstraction	NOUN
ejpam-1781	10	8	to	to	ADP
ejpam-1781	10	9	most	most	ADJ
ejpam-1781	10	10	of	of	ADP
ejpam-1781	10	11	the	the	DET
ejpam-1781	10	12	existing	exist	VERB
ejpam-1781	10	13	ring	ring	NOUN
ejpam-1781	10	14	theoretic	theoretic	NOUN
ejpam-1781	10	15	and	and	CCONJ
ejpam-1781	10	16	lattice	lattice	ADJ
ejpam-1781	10	17	theoretic	theoretic	ADJ
ejpam-1781	10	18	generalizations	generalization	NOUN
ejpam-1781	10	19	of	of	ADP
ejpam-1781	10	20	a	a	DET
ejpam-1781	10	21	boolean	boolean	ADJ
ejpam-1781	10	22	algebra	algebra	NOUN
ejpam-1781	10	23	,	,	PUNCT
ejpam-1781	10	24	the	the	DET
ejpam-1781	10	25	concept	concept	NOUN
ejpam-1781	10	26	of	of	ADP
ejpam-1781	10	27	an	an	DET
ejpam-1781	10	28	"	"	PUNCT
ejpam-1781	10	29	almost	almost	ADV
ejpam-1781	10	30	distributive	distributive	ADJ
ejpam-1781	10	31	lattice(adl	lattice(adl	NOUN
ejpam-1781	10	32	)	)	PUNCT
ejpam-1781	10	33	"	"	PUNCT
ejpam-1781	10	34	was	be	AUX
ejpam-1781	10	35	introduced	introduce	VERB
ejpam-1781	10	36	.	.	PUNCT
ejpam-1781	11	1	an	an	DET
ejpam-1781	11	2	adl	adl	NOUN
ejpam-1781	11	3	is	be	AUX
ejpam-1781	11	4	an	an	DET
ejpam-1781	11	5	algebra	algebra	NOUN
ejpam-1781	11	6	(	(	PUNCT
ejpam-1781	11	7	r,∨,∧	r,∨,∧	NUM
ejpam-1781	11	8	)	)	PUNCT
ejpam-1781	11	9	of	of	ADP
ejpam-1781	11	10	type	type	NOUN
ejpam-1781	11	11	(	(	PUNCT
ejpam-1781	11	12	2,2	2,2	NUM
ejpam-1781	11	13	)	)	PUNCT
ejpam-1781	11	14	which	which	PRON
ejpam-1781	11	15	satisfies	satisfy	VERB
ejpam-1781	11	16	almost	almost	ADV
ejpam-1781	11	17	all	all	DET
ejpam-1781	11	18	the	the	DET
ejpam-1781	11	19	properties	property	NOUN
ejpam-1781	11	20	of	of	ADP
ejpam-1781	11	21	a	a	DET
ejpam-1781	11	22	distributive	distributive	ADJ
ejpam-1781	11	23	lattice	lattice	NOUN
ejpam-1781	11	24	except	except	SCONJ
ejpam-1781	11	25	possibly	possibly	ADV
ejpam-1781	11	26	the	the	DET
ejpam-1781	11	27	commutative	commutative	ADJ
ejpam-1781	11	28	of	of	ADP
ejpam-1781	11	29	∨	∨	NUM
ejpam-1781	11	30	,	,	PUNCT
ejpam-1781	11	31	the	the	DET
ejpam-1781	11	32	commutative	commutative	ADJ
ejpam-1781	11	33	of	of	ADP
ejpam-1781	11	34	∧	∧	PROPN
ejpam-1781	11	35	and	and	CCONJ
ejpam-1781	11	36	the	the	DET
ejpam-1781	11	37	right	right	ADJ
ejpam-1781	11	38	distributivity	distributivity	NOUN
ejpam-1781	11	39	of	of	ADP
ejpam-1781	11	40	∨	∨	NUM
ejpam-1781	11	41	over	over	ADP
ejpam-1781	11	42	∧.	∧.	PROPN
ejpam-1781	11	43	it	it	PRON
ejpam-1781	11	44	was	be	AUX
ejpam-1781	11	45	also	also	ADV
ejpam-1781	11	46	observed	observe	VERB
ejpam-1781	11	47	that	that	SCONJ
ejpam-1781	11	48	any	any	DET
ejpam-1781	11	49	one	one	NUM
ejpam-1781	11	50	of	of	ADP
ejpam-1781	11	51	these	these	DET
ejpam-1781	11	52	three	three	NUM
ejpam-1781	11	53	properties	property	NOUN
ejpam-1781	11	54	converts	convert	VERB
ejpam-1781	11	55	an	an	DET
ejpam-1781	11	56	adl	adl	NOUN
ejpam-1781	11	57	into	into	ADP
ejpam-1781	11	58	a	a	DET
ejpam-1781	11	59	distributive	distributive	ADJ
ejpam-1781	11	60	lattice	lattice	NOUN
ejpam-1781	11	61	.	.	PUNCT
ejpam-1781	12	1	the	the	DET
ejpam-1781	12	2	concept	concept	NOUN
ejpam-1781	12	3	of	of	ADP
ejpam-1781	12	4	an	an	DET
ejpam-1781	12	5	ideal	ideal	NOUN
ejpam-1781	12	6	was	be	AUX
ejpam-1781	12	7	introduced	introduce	VERB
ejpam-1781	12	8	in	in	ADP
ejpam-1781	12	9	an	an	DET
ejpam-1781	12	10	adl	adl	NOUN
ejpam-1781	12	11	analogous	analogous	ADJ
ejpam-1781	12	12	to	to	ADP
ejpam-1781	12	13	that	that	PRON
ejpam-1781	12	14	in	in	ADP
ejpam-1781	12	15	a	a	DET
ejpam-1781	12	16	distributive	distributive	ADJ
ejpam-1781	12	17	lattice	lattice	NOUN
ejpam-1781	13	1	[	[	X
ejpam-1781	13	2	3	3	NUM
ejpam-1781	13	3	]	]	PUNCT
ejpam-1781	13	4	.	.	PUNCT
ejpam-1781	14	1	if	if	SCONJ
ejpam-1781	14	2	r	r	NOUN
ejpam-1781	14	3	is	be	AUX
ejpam-1781	14	4	an	an	DET
ejpam-1781	14	5	adl	adl	NOUN
ejpam-1781	14	6	,	,	PUNCT
ejpam-1781	14	7	then	then	ADV
ejpam-1781	14	8	the	the	DET
ejpam-1781	14	9	set	set	NOUN
ejpam-1781	14	10	pi(r	pi(r	CCONJ
ejpam-1781	14	11	)	)	PUNCT
ejpam-1781	14	12	of	of	ADP
ejpam-1781	14	13	all	all	DET
ejpam-1781	14	14	principal	principal	ADJ
ejpam-1781	14	15	ideals	ideal	NOUN
ejpam-1781	14	16	of	of	ADP
ejpam-1781	14	17	r	r	NOUN
ejpam-1781	14	18	form	form	NOUN
ejpam-1781	14	19	a	a	DET
ejpam-1781	14	20	distributive	distributive	ADJ
ejpam-1781	14	21	lattice	lattice	NOUN
ejpam-1781	14	22	.	.	PUNCT
ejpam-1781	15	1	this	this	PRON
ejpam-1781	15	2	enables	enable	VERB
ejpam-1781	15	3	to	to	PART
ejpam-1781	15	4	extend	extend	VERB
ejpam-1781	15	5	many	many	ADJ
ejpam-1781	15	6	existing	exist	VERB
ejpam-1781	15	7	concepts	concept	NOUN
ejpam-1781	15	8	in	in	ADP
ejpam-1781	15	9	distributive	distributive	ADJ
ejpam-1781	15	10	lattices	lattice	NOUN
ejpam-1781	15	11	to	to	ADP
ejpam-1781	15	12	the	the	DET
ejpam-1781	15	13	class	class	NOUN
ejpam-1781	15	14	of	of	ADP
ejpam-1781	15	15	adls	adls	PROPN
ejpam-1781	15	16	.	.	PUNCT
ejpam-1781	16	1	almost	almost	ADV
ejpam-1781	16	2	distributive	distributive	ADJ
ejpam-1781	16	3	lattice	lattice	NOUN
ejpam-1781	16	4	arise	arise	NOUN
ejpam-1781	16	5	as	as	ADP
ejpam-1781	16	6	a	a	DET
ejpam-1781	16	7	natural	natural	ADJ
ejpam-1781	16	8	generalization	generalization	NOUN
ejpam-1781	16	9	of	of	ADP
ejpam-1781	16	10	a	a	DET
ejpam-1781	16	11	distributive	distributive	ADJ
ejpam-1781	16	12	lattices	lattice	NOUN
ejpam-1781	16	13	and	and	CCONJ
ejpam-1781	16	14	hence	hence	ADV
ejpam-1781	16	15	it	it	PRON
ejpam-1781	16	16	is	be	AUX
ejpam-1781	16	17	natural	natural	ADJ
ejpam-1781	16	18	to	to	PART
ejpam-1781	16	19	consider	consider	VERB
ejpam-1781	16	20	the	the	DET
ejpam-1781	16	21	properties	property	NOUN
ejpam-1781	16	22	of	of	ADP
ejpam-1781	16	23	prime	prime	ADJ
ejpam-1781	16	24	ideals	ideal	NOUN
ejpam-1781	16	25	in	in	ADP
ejpam-1781	16	26	an	an	DET
ejpam-1781	16	27	almost	almost	ADV
ejpam-1781	16	28	distributive	distributive	ADJ
ejpam-1781	16	29	lattice	lattice	NOUN
ejpam-1781	16	30	.	.	PUNCT
ejpam-1781	17	1	it	it	PRON
ejpam-1781	17	2	is	be	AUX
ejpam-1781	17	3	∗corresponding	∗corresponde	VERB
ejpam-1781	17	4	author	author	NOUN
ejpam-1781	17	5	.	.	PUNCT
ejpam-1781	18	1	email	email	NOUN
ejpam-1781	18	2	addresses	address	NOUN
ejpam-1781	18	3	:	:	PUNCT
ejpam-1781	18	4	pawar_y_s	pawar_y_s	PROPN
ejpam-1781	18	5	�	�	NOUN
ejpam-1781	18	6	yahoo	yahoo	PROPN
ejpam-1781	18	7	.	.	PUNCT
ejpam-1781	19	1	om	om	PROPN
ejpam-1781	19	2	(	(	PUNCT
ejpam-1781	19	3	y.	y.	PROPN
ejpam-1781	19	4	pawar	pawar	PROPN
ejpam-1781	19	5	)	)	PUNCT
ejpam-1781	19	6	,	,	PUNCT
ejpam-1781	19	7	shaikh_i_a	shaikh_i_a	PROPN
ejpam-1781	19	8	�	�	PROPN
ejpam-1781	19	9	yahoo	yahoo	PROPN
ejpam-1781	19	10	.	.	PUNCT
ejpam-1781	20	1	om	om	PROPN
ejpam-1781	20	2	(	(	PUNCT
ejpam-1781	20	3	i.	i.	PROPN
ejpam-1781	20	4	shaikh	shaikh	PROPN
ejpam-1781	20	5	)	)	PUNCT
ejpam-1781	20	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1781	21	1	107	107	NUM
ejpam-1781	22	1	c	c	X
ejpam-1781	22	2	©	©	PROPN
ejpam-1781	22	3	2013	2013	NUM
ejpam-1781	22	4	ejpam	ejpam	NOUN
ejpam-1781	22	5	all	all	DET
ejpam-1781	22	6	rights	right	NOUN
ejpam-1781	22	7	reserved	reserve	VERB
ejpam-1781	22	8	.	.	PUNCT
ejpam-1781	23	1	y.	y.	PROPN
ejpam-1781	23	2	pawar	pawar	PROPN
ejpam-1781	23	3	,	,	PUNCT
ejpam-1781	23	4	i.	i.	PROPN
ejpam-1781	23	5	shaikh	shaikh	PROPN
ejpam-1781	23	6	/	/	SYM
ejpam-1781	23	7	eur	eur	PROPN
ejpam-1781	23	8	.	.	PUNCT
ejpam-1781	24	1	j.	j.	PROPN
ejpam-1781	24	2	pure	pure	PROPN
ejpam-1781	24	3	appl	appl	PROPN
ejpam-1781	24	4	.	.	PROPN
ejpam-1781	24	5	math	math	PROPN
ejpam-1781	24	6	,	,	PUNCT
ejpam-1781	24	7	6	6	NUM
ejpam-1781	24	8	(	(	PUNCT
ejpam-1781	24	9	2013	2013	NUM
ejpam-1781	24	10	)	)	PUNCT
ejpam-1781	24	11	,	,	PUNCT
ejpam-1781	24	12	107	107	NUM
ejpam-1781	24	13	-	-	SYM
ejpam-1781	24	14	118	118	NUM
ejpam-1781	24	15	108	108	NUM
ejpam-1781	24	16	interesting	interesting	ADJ
ejpam-1781	24	17	to	to	PART
ejpam-1781	24	18	note	note	VERB
ejpam-1781	24	19	that	that	SCONJ
ejpam-1781	24	20	the	the	DET
ejpam-1781	24	21	results	result	NOUN
ejpam-1781	24	22	which	which	PRON
ejpam-1781	24	23	are	be	AUX
ejpam-1781	24	24	valid	valid	ADJ
ejpam-1781	24	25	for	for	ADP
ejpam-1781	24	26	distributive	distributive	ADJ
ejpam-1781	24	27	lattices	lattice	NOUN
ejpam-1781	24	28	in	in	ADP
ejpam-1781	24	29	verbatim	verbatim	ADJ
ejpam-1781	24	30	,	,	PUNCT
ejpam-1781	24	31	for	for	ADP
ejpam-1781	24	32	adls	adls	PROPN
ejpam-1781	24	33	,	,	PUNCT
ejpam-1781	24	34	even	even	ADV
ejpam-1781	24	35	though	though	SCONJ
ejpam-1781	24	36	the	the	DET
ejpam-1781	24	37	techniques	technique	NOUN
ejpam-1781	24	38	of	of	ADP
ejpam-1781	24	39	the	the	DET
ejpam-1781	24	40	proofs	proof	NOUN
ejpam-1781	24	41	in	in	ADP
ejpam-1781	24	42	the	the	DET
ejpam-1781	24	43	case	case	NOUN
ejpam-1781	24	44	of	of	ADP
ejpam-1781	24	45	adls	adls	PROPN
ejpam-1781	24	46	are	be	AUX
ejpam-1781	24	47	slightly	slightly	ADV
ejpam-1781	24	48	different	different	ADJ
ejpam-1781	24	49	,	,	PUNCT
ejpam-1781	24	50	for	for	ADP
ejpam-1781	24	51	the	the	DET
ejpam-1781	24	52	reason	reason	NOUN
ejpam-1781	24	53	that	that	SCONJ
ejpam-1781	24	54	the	the	DET
ejpam-1781	24	55	operations	operation	NOUN
ejpam-1781	24	56	∨	∨	NOUN
ejpam-1781	24	57	and	and	CCONJ
ejpam-1781	24	58	∧	∧	NOUN
ejpam-1781	24	59	are	be	AUX
ejpam-1781	24	60	not	not	PART
ejpam-1781	24	61	commutative	commutative	ADJ
ejpam-1781	24	62	.	.	PUNCT
ejpam-1781	25	1	if	if	SCONJ
ejpam-1781	25	2	i	i	PRON
ejpam-1781	25	3	is	be	AUX
ejpam-1781	25	4	an	an	DET
ejpam-1781	25	5	ideal	ideal	NOUN
ejpam-1781	25	6	of	of	ADP
ejpam-1781	25	7	r	r	NOUN
ejpam-1781	25	8	,	,	PUNCT
ejpam-1781	25	9	the	the	DET
ejpam-1781	25	10	concept	concept	NOUN
ejpam-1781	25	11	of	of	ADP
ejpam-1781	25	12	the	the	DET
ejpam-1781	25	13	minimal	minimal	ADJ
ejpam-1781	25	14	prime	prime	ADJ
ejpam-1781	25	15	ideal	ideal	NOUN
ejpam-1781	25	16	belonging	belong	VERB
ejpam-1781	25	17	to	to	ADP
ejpam-1781	25	18	i	i	PRON
ejpam-1781	25	19	is	be	AUX
ejpam-1781	25	20	defined	define	VERB
ejpam-1781	25	21	in	in	ADP
ejpam-1781	25	22	[	[	X
ejpam-1781	25	23	1	1	NUM
ejpam-1781	25	24	]	]	PUNCT
ejpam-1781	25	25	.	.	PUNCT
ejpam-1781	26	1	in	in	ADP
ejpam-1781	26	2	[	[	X
ejpam-1781	26	3	2	2	X
ejpam-1781	26	4	]	]	PUNCT
ejpam-1781	26	5	the	the	DET
ejpam-1781	26	6	concept	concept	NOUN
ejpam-1781	26	7	of	of	ADP
ejpam-1781	26	8	annihilator	annihilator	PROPN
ejpam-1781	26	9	ideals	ideal	NOUN
ejpam-1781	26	10	in	in	ADP
ejpam-1781	26	11	an	an	DET
ejpam-1781	26	12	adl	adl	NOUN
ejpam-1781	26	13	is	be	AUX
ejpam-1781	26	14	introduced	introduce	VERB
ejpam-1781	26	15	with	with	ADP
ejpam-1781	26	16	suitable	suitable	ADJ
ejpam-1781	26	17	examples	example	NOUN
ejpam-1781	26	18	and	and	CCONJ
ejpam-1781	26	19	proved	prove	VERB
ejpam-1781	26	20	some	some	DET
ejpam-1781	26	21	basic	basic	ADJ
ejpam-1781	26	22	properties	property	NOUN
ejpam-1781	26	23	of	of	ADP
ejpam-1781	26	24	the	the	DET
ejpam-1781	26	25	annihilator	annihilator	PROPN
ejpam-1781	26	26	ideals	ideal	NOUN
ejpam-1781	26	27	,	,	PUNCT
ejpam-1781	26	28	analogous	analogous	ADJ
ejpam-1781	26	29	to	to	ADP
ejpam-1781	26	30	that	that	PRON
ejpam-1781	26	31	in	in	ADP
ejpam-1781	26	32	a	a	DET
ejpam-1781	26	33	distributive	distributive	ADJ
ejpam-1781	26	34	lattice	lattice	NOUN
ejpam-1781	26	35	.	.	PUNCT
ejpam-1781	27	1	it	it	PRON
ejpam-1781	27	2	is	be	AUX
ejpam-1781	27	3	proved	prove	VERB
ejpam-1781	27	4	that	that	SCONJ
ejpam-1781	27	5	the	the	DET
ejpam-1781	27	6	set	set	NOUN
ejpam-1781	27	7	a(r	a(r	NOUN
ejpam-1781	27	8	)	)	PUNCT
ejpam-1781	27	9	of	of	ADP
ejpam-1781	27	10	all	all	DET
ejpam-1781	27	11	annihilator	annihilator	NOUN
ejpam-1781	27	12	ideals	ideal	NOUN
ejpam-1781	27	13	of	of	ADP
ejpam-1781	27	14	an	an	DET
ejpam-1781	27	15	adl	adl	NOUN
ejpam-1781	27	16	r	r	NOUN
ejpam-1781	27	17	with	with	ADP
ejpam-1781	27	18	0	0	NUM
ejpam-1781	27	19	can	can	AUX
ejpam-1781	27	20	be	be	AUX
ejpam-1781	27	21	made	make	VERB
ejpam-1781	27	22	into	into	ADP
ejpam-1781	27	23	a	a	DET
ejpam-1781	27	24	complete	complete	ADJ
ejpam-1781	27	25	boolean	boolean	ADJ
ejpam-1781	27	26	algebra	algebra	NOUN
ejpam-1781	27	27	.	.	PUNCT
ejpam-1781	28	1	the	the	DET
ejpam-1781	28	2	aim	aim	NOUN
ejpam-1781	28	3	of	of	ADP
ejpam-1781	28	4	this	this	DET
ejpam-1781	28	5	paper	paper	NOUN
ejpam-1781	28	6	is	be	AUX
ejpam-1781	28	7	to	to	PART
ejpam-1781	28	8	study	study	VERB
ejpam-1781	28	9	some	some	DET
ejpam-1781	28	10	additional	additional	ADJ
ejpam-1781	28	11	properties	property	NOUN
ejpam-1781	28	12	of	of	ADP
ejpam-1781	28	13	prime	prime	ADJ
ejpam-1781	28	14	,	,	PUNCT
ejpam-1781	28	15	minimal	minimal	ADJ
ejpam-1781	28	16	prime	prime	NOUN
ejpam-1781	28	17	and	and	CCONJ
ejpam-1781	28	18	annihilators	annihilators	PROPN
ejpam-1781	28	19	ideals	ideal	NOUN
ejpam-1781	28	20	in	in	ADP
ejpam-1781	28	21	an	an	DET
ejpam-1781	28	22	adl	adl	NOUN
ejpam-1781	28	23	.	.	PUNCT
ejpam-1781	29	1	this	this	DET
ejpam-1781	29	2	paper	paper	NOUN
ejpam-1781	29	3	consists	consist	VERB
ejpam-1781	29	4	of	of	ADP
ejpam-1781	29	5	four	four	NUM
ejpam-1781	29	6	sections	section	NOUN
ejpam-1781	29	7	.	.	PUNCT
ejpam-1781	30	1	in	in	ADP
ejpam-1781	30	2	the	the	DET
ejpam-1781	30	3	second	second	ADJ
ejpam-1781	30	4	section	section	NOUN
ejpam-1781	30	5	we	we	PRON
ejpam-1781	30	6	recall	recall	VERB
ejpam-1781	30	7	some	some	DET
ejpam-1781	30	8	basic	basic	ADJ
ejpam-1781	30	9	definitions	definition	NOUN
ejpam-1781	30	10	and	and	CCONJ
ejpam-1781	30	11	results	result	NOUN
ejpam-1781	30	12	.	.	PUNCT
ejpam-1781	31	1	third	third	ADJ
ejpam-1781	31	2	section	section	NOUN
ejpam-1781	31	3	is	be	AUX
ejpam-1781	31	4	devoted	devote	VERB
ejpam-1781	31	5	to	to	PART
ejpam-1781	31	6	prove	prove	VERB
ejpam-1781	31	7	several	several	ADJ
ejpam-1781	31	8	necessary	necessary	ADJ
ejpam-1781	31	9	and	and	CCONJ
ejpam-1781	31	10	sufficient	sufficient	ADJ
ejpam-1781	31	11	conditions	condition	NOUN
ejpam-1781	31	12	for	for	ADP
ejpam-1781	31	13	a	a	DET
ejpam-1781	31	14	prime	prime	ADJ
ejpam-1781	31	15	ideal	ideal	NOUN
ejpam-1781	31	16	to	to	PART
ejpam-1781	31	17	be	be	AUX
ejpam-1781	31	18	a	a	DET
ejpam-1781	31	19	minimal	minimal	ADJ
ejpam-1781	31	20	prime	prime	ADJ
ejpam-1781	31	21	ideal	ideal	NOUN
ejpam-1781	31	22	and	and	CCONJ
ejpam-1781	31	23	prime	prime	ADJ
ejpam-1781	31	24	ideal	ideal	NOUN
ejpam-1781	31	25	to	to	PART
ejpam-1781	31	26	be	be	AUX
ejpam-1781	31	27	a	a	DET
ejpam-1781	31	28	principal	principal	ADJ
ejpam-1781	31	29	ideals	ideal	NOUN
ejpam-1781	31	30	in	in	ADP
ejpam-1781	31	31	an	an	DET
ejpam-1781	31	32	adl	adl	PROPN
ejpam-1781	31	33	.	.	PUNCT
ejpam-1781	32	1	fourth	fourth	ADJ
ejpam-1781	32	2	section	section	NOUN
ejpam-1781	32	3	deals	deal	VERB
ejpam-1781	32	4	with	with	ADP
ejpam-1781	32	5	some	some	DET
ejpam-1781	32	6	properties	property	NOUN
ejpam-1781	32	7	of	of	ADP
ejpam-1781	32	8	the	the	DET
ejpam-1781	32	9	special	special	ADJ
ejpam-1781	32	10	subsets	subset	NOUN
ejpam-1781	32	11	of	of	ADP
ejpam-1781	32	12	the	the	DET
ejpam-1781	32	13	set	set	NOUN
ejpam-1781	32	14	of	of	ADP
ejpam-1781	32	15	all	all	DET
ejpam-1781	32	16	prime	prime	ADJ
ejpam-1781	32	17	ideals	ideal	NOUN
ejpam-1781	32	18	in	in	ADP
ejpam-1781	32	19	an	an	DET
ejpam-1781	32	20	adl	adl	NOUN
ejpam-1781	32	21	.	.	PROPN
ejpam-1781	32	22	2	2	NUM
ejpam-1781	32	23	.	.	X
ejpam-1781	32	24	preliminaries	preliminary	NOUN
ejpam-1781	32	25	in	in	ADP
ejpam-1781	32	26	this	this	DET
ejpam-1781	32	27	article	article	NOUN
ejpam-1781	32	28	we	we	PRON
ejpam-1781	32	29	recall	recall	VERB
ejpam-1781	32	30	certain	certain	ADJ
ejpam-1781	32	31	definitions	definition	NOUN
ejpam-1781	32	32	and	and	CCONJ
ejpam-1781	32	33	important	important	ADJ
ejpam-1781	32	34	results	result	NOUN
ejpam-1781	32	35	mostly	mostly	ADV
ejpam-1781	32	36	from	from	ADP
ejpam-1781	32	37	[	[	X
ejpam-1781	32	38	1	1	NUM
ejpam-1781	32	39	]	]	PUNCT
ejpam-1781	32	40	and	and	CCONJ
ejpam-1781	32	41	[	[	X
ejpam-1781	32	42	2	2	NUM
ejpam-1781	32	43	]	]	PUNCT
ejpam-1781	32	44	,	,	PUNCT
ejpam-1781	32	45	that	that	SCONJ
ejpam-1781	32	46	we	we	PRON
ejpam-1781	32	47	need	need	VERB
ejpam-1781	32	48	in	in	ADP
ejpam-1781	32	49	sequel	sequel	NOUN
ejpam-1781	32	50	.	.	PUNCT
ejpam-1781	33	1	an	an	DET
ejpam-1781	33	2	almost	almost	ADV
ejpam-1781	33	3	distributive	distributive	ADJ
ejpam-1781	33	4	lattice	lattice	NOUN
ejpam-1781	33	5	(	(	PUNCT
ejpam-1781	33	6	adl	adl	PROPN
ejpam-1781	33	7	)	)	PUNCT
ejpam-1781	33	8	is	be	AUX
ejpam-1781	33	9	an	an	DET
ejpam-1781	33	10	algebra	algebra	NOUN
ejpam-1781	33	11	(	(	PUNCT
ejpam-1781	33	12	r,∨,∧	r,∨,∧	NUM
ejpam-1781	33	13	,	,	PUNCT
ejpam-1781	33	14	0	0	NUM
ejpam-1781	33	15	)	)	PUNCT
ejpam-1781	33	16	of	of	ADP
ejpam-1781	33	17	type	type	NOUN
ejpam-1781	33	18	(	(	PUNCT
ejpam-1781	33	19	2,2,0	2,2,0	NUM
ejpam-1781	33	20	)	)	PUNCT
ejpam-1781	33	21	satisfying	satisfy	VERB
ejpam-1781	33	22	the	the	DET
ejpam-1781	33	23	following	follow	VERB
ejpam-1781	33	24	axioms	axiom	NOUN
ejpam-1781	33	25	.	.	PUNCT
ejpam-1781	34	1	1	1	X
ejpam-1781	34	2	.	.	X
ejpam-1781	34	3	a	a	DET
ejpam-1781	34	4	∨	∨	NOUN
ejpam-1781	34	5	0=	0=	NOUN
ejpam-1781	34	6	a	a	DET
ejpam-1781	34	7	,	,	PUNCT
ejpam-1781	34	8	2	2	NUM
ejpam-1781	34	9	.	.	NOUN
ejpam-1781	34	10	0∧	0∧	NOUN
ejpam-1781	34	11	a	a	DET
ejpam-1781	34	12	=	=	SYM
ejpam-1781	34	13	0	0	NUM
ejpam-1781	34	14	,	,	PUNCT
ejpam-1781	34	15	3	3	NUM
ejpam-1781	34	16	.	.	PUNCT
ejpam-1781	35	1	(	(	PUNCT
ejpam-1781	35	2	a	a	DET
ejpam-1781	35	3	∨	∨	NUM
ejpam-1781	35	4	b)∧	b)∧	PROPN
ejpam-1781	35	5	c	c	NOUN
ejpam-1781	35	6	=	=	PUNCT
ejpam-1781	35	7	(	(	PUNCT
ejpam-1781	35	8	a	a	DET
ejpam-1781	35	9	∧	∧	NOUN
ejpam-1781	35	10	c)∨	c)∨	NOUN
ejpam-1781	35	11	(	(	PUNCT
ejpam-1781	35	12	b	b	PROPN
ejpam-1781	35	13	∧	∧	PROPN
ejpam-1781	35	14	c	c	NOUN
ejpam-1781	35	15	)	)	PUNCT
ejpam-1781	35	16	,	,	PUNCT
ejpam-1781	35	17	4	4	X
ejpam-1781	35	18	.	.	PUNCT
ejpam-1781	36	1	a	a	DET
ejpam-1781	36	2	∧	∧	PROPN
ejpam-1781	36	3	(	(	PUNCT
ejpam-1781	36	4	b	b	PROPN
ejpam-1781	36	5	∨	∨	NUM
ejpam-1781	36	6	c	c	NOUN
ejpam-1781	36	7	)	)	PUNCT
ejpam-1781	36	8	=	=	NOUN
ejpam-1781	36	9	(	(	PUNCT
ejpam-1781	36	10	a	a	DET
ejpam-1781	36	11	∧	∧	PROPN
ejpam-1781	36	12	b)∨	b)∨	PROPN
ejpam-1781	36	13	(	(	PUNCT
ejpam-1781	36	14	a	a	DET
ejpam-1781	36	15	∧	∧	PROPN
ejpam-1781	36	16	c	c	NOUN
ejpam-1781	36	17	)	)	PUNCT
ejpam-1781	36	18	,	,	PUNCT
ejpam-1781	36	19	5	5	X
ejpam-1781	36	20	.	.	PUNCT
ejpam-1781	36	21	a	a	DET
ejpam-1781	36	22	∨	∨	NOUN
ejpam-1781	36	23	(	(	PUNCT
ejpam-1781	36	24	b	b	PROPN
ejpam-1781	36	25	∧	∧	PROPN
ejpam-1781	36	26	c	c	NOUN
ejpam-1781	36	27	)	)	PUNCT
ejpam-1781	36	28	=	=	NOUN
ejpam-1781	36	29	(	(	PUNCT
ejpam-1781	36	30	a	a	DET
ejpam-1781	36	31	∨	∨	NUM
ejpam-1781	36	32	b)∧	b)∧	PROPN
ejpam-1781	36	33	(	(	PUNCT
ejpam-1781	36	34	a	a	DET
ejpam-1781	36	35	∨	∨	NUM
ejpam-1781	36	36	c	c	NOUN
ejpam-1781	36	37	)	)	PUNCT
ejpam-1781	36	38	,	,	PUNCT
ejpam-1781	36	39	6	6	NUM
ejpam-1781	36	40	.	.	PUNCT
ejpam-1781	37	1	(	(	PUNCT
ejpam-1781	37	2	a	a	DET
ejpam-1781	37	3	∨	∨	NUM
ejpam-1781	37	4	b)∧	b)∧	PROPN
ejpam-1781	37	5	b	b	PROPN
ejpam-1781	37	6	=	=	SYM
ejpam-1781	37	7	b	b	PROPN
ejpam-1781	37	8	,	,	PUNCT
ejpam-1781	37	9	for	for	ADP
ejpam-1781	37	10	all	all	DET
ejpam-1781	37	11	a	a	DET
ejpam-1781	37	12	,	,	PUNCT
ejpam-1781	37	13	b	b	NOUN
ejpam-1781	37	14	,	,	PUNCT
ejpam-1781	37	15	c	c	PROPN
ejpam-1781	37	16	∈	∈	PROPN
ejpam-1781	37	17	r.	r.	PROPN
ejpam-1781	37	18	throughout	throughout	ADP
ejpam-1781	37	19	this	this	DET
ejpam-1781	37	20	paper	paper	NOUN
ejpam-1781	37	21	,	,	PUNCT
ejpam-1781	37	22	r	r	NOUN
ejpam-1781	37	23	stands	stand	VERB
ejpam-1781	37	24	for	for	ADP
ejpam-1781	37	25	an	an	DET
ejpam-1781	37	26	adl	adl	PROPN
ejpam-1781	37	27	(	(	PUNCT
ejpam-1781	37	28	r,∧,∨	r,∧,∨	ADJ
ejpam-1781	37	29	,	,	PUNCT
ejpam-1781	37	30	0	0	NUM
ejpam-1781	37	31	)	)	PUNCT
ejpam-1781	37	32	with	with	ADP
ejpam-1781	37	33	zero	zero	NUM
ejpam-1781	37	34	unless	unless	SCONJ
ejpam-1781	37	35	otherwise	otherwise	ADV
ejpam-1781	37	36	mentioned	mention	VERB
ejpam-1781	37	37	.	.	PUNCT
ejpam-1781	38	1	for	for	ADP
ejpam-1781	38	2	any	any	DET
ejpam-1781	38	3	a	a	PRON
ejpam-1781	38	4	,	,	PUNCT
ejpam-1781	38	5	b	b	X
ejpam-1781	38	6	∈	∈	PROPN
ejpam-1781	38	7	r	r	NOUN
ejpam-1781	38	8	,	,	PUNCT
ejpam-1781	38	9	define	define	VERB
ejpam-1781	38	10	a	a	DET
ejpam-1781	38	11	≤	≤	NUM
ejpam-1781	38	12	b	b	NOUN
ejpam-1781	39	1	if	if	SCONJ
ejpam-1781	40	1	and	and	CCONJ
ejpam-1781	40	2	only	only	ADV
ejpam-1781	40	3	if	if	SCONJ
ejpam-1781	40	4	a	a	PRON
ejpam-1781	40	5	=	=	X
ejpam-1781	40	6	a	a	DET
ejpam-1781	40	7	∧	∧	PROPN
ejpam-1781	40	8	b	b	PROPN
ejpam-1781	40	9	or	or	CCONJ
ejpam-1781	40	10	,	,	PUNCT
ejpam-1781	40	11	equivalently	equivalently	ADV
ejpam-1781	40	12	,	,	PUNCT
ejpam-1781	40	13	a	a	DET
ejpam-1781	40	14	∨	∨	NUM
ejpam-1781	40	15	b	b	X
ejpam-1781	40	16	=	=	SYM
ejpam-1781	40	17	b	b	PROPN
ejpam-1781	40	18	,	,	PUNCT
ejpam-1781	40	19	then	then	ADV
ejpam-1781	40	20	≤	≤	PROPN
ejpam-1781	40	21	is	be	AUX
ejpam-1781	40	22	a	a	DET
ejpam-1781	40	23	partial	partial	ADJ
ejpam-1781	40	24	ordering	ordering	NOUN
ejpam-1781	40	25	on	on	ADP
ejpam-1781	40	26	r.	r.	PROPN
ejpam-1781	40	27	an	an	DET
ejpam-1781	40	28	element	element	NOUN
ejpam-1781	40	29	m	m	PROPN
ejpam-1781	40	30	∈	∈	NOUN
ejpam-1781	40	31	r	r	NOUN
ejpam-1781	40	32	is	be	AUX
ejpam-1781	40	33	called	call	VERB
ejpam-1781	40	34	maximal	maximal	ADJ
ejpam-1781	40	35	element	element	NOUN
ejpam-1781	40	36	in	in	ADP
ejpam-1781	40	37	the	the	DET
ejpam-1781	40	38	poset	poset	NOUN
ejpam-1781	40	39	(	(	PUNCT
ejpam-1781	40	40	r,≤	r,≤	PROPN
ejpam-1781	40	41	)	)	PUNCT
ejpam-1781	40	42	.	.	PUNCT
ejpam-1781	41	1	that	that	PRON
ejpam-1781	41	2	is	be	AUX
ejpam-1781	41	3	for	for	ADP
ejpam-1781	41	4	any	any	DET
ejpam-1781	41	5	a	a	DET
ejpam-1781	41	6	∈	∈	PROPN
ejpam-1781	41	7	r	r	NOUN
ejpam-1781	41	8	,	,	PUNCT
ejpam-1781	41	9	m≤	m≤	VERB
ejpam-1781	41	10	a⇒	a⇒	NOUN
ejpam-1781	41	11	m	m	ADJ
ejpam-1781	41	12	=	=	VERB
ejpam-1781	41	13	a.	a.	NOUN
ejpam-1781	41	14	a	a	DET
ejpam-1781	41	15	non	non	X
ejpam-1781	41	16	empty	empty	ADJ
ejpam-1781	41	17	subset	subset	NOUN
ejpam-1781	41	18	i	i	PRON
ejpam-1781	41	19	of	of	ADP
ejpam-1781	41	20	r	r	NOUN
ejpam-1781	41	21	is	be	AUX
ejpam-1781	41	22	said	say	VERB
ejpam-1781	41	23	to	to	PART
ejpam-1781	41	24	be	be	AUX
ejpam-1781	41	25	an	an	DET
ejpam-1781	41	26	ideal	ideal	ADJ
ejpam-1781	41	27	(	(	PUNCT
ejpam-1781	41	28	filter	filter	NOUN
ejpam-1781	41	29	)	)	PUNCT
ejpam-1781	41	30	of	of	ADP
ejpam-1781	41	31	r	r	NOUN
ejpam-1781	41	32	,	,	PUNCT
ejpam-1781	41	33	if	if	SCONJ
ejpam-1781	41	34	a	a	DET
ejpam-1781	41	35	∨	∨	NUM
ejpam-1781	41	36	b	b	X
ejpam-1781	41	37	∈	∈	PROPN
ejpam-1781	41	38	i(a	i(a	PROPN
ejpam-1781	41	39	∧	∧	PROPN
ejpam-1781	41	40	b	b	PROPN
ejpam-1781	41	41	∈	∈	PROPN
ejpam-1781	41	42	i	i	PROPN
ejpam-1781	41	43	)	)	PUNCT
ejpam-1781	41	44	and	and	CCONJ
ejpam-1781	41	45	a	a	DET
ejpam-1781	41	46	∧	∧	PROPN
ejpam-1781	41	47	x	x	SYM
ejpam-1781	41	48	∈	∈	PROPN
ejpam-1781	42	1	i	i	PRON
ejpam-1781	42	2	(	(	PUNCT
ejpam-1781	42	3	x	x	PROPN
ejpam-1781	42	4	∨	∨	NUM
ejpam-1781	42	5	a	a	DET
ejpam-1781	42	6	∈	∈	PROPN
ejpam-1781	42	7	i	i	NOUN
ejpam-1781	42	8	)	)	PUNCT
ejpam-1781	42	9	whenever	whenever	SCONJ
ejpam-1781	42	10	a	a	PRON
ejpam-1781	42	11	,	,	PUNCT
ejpam-1781	42	12	b	b	X
ejpam-1781	42	13	∈	∈	NOUN
ejpam-1781	42	14	i	i	PRON
ejpam-1781	42	15	and	and	CCONJ
ejpam-1781	42	16	x	x	PROPN
ejpam-1781	42	17	∈	∈	PROPN
ejpam-1781	42	18	r.	r.	NOUN
ejpam-1781	42	19	if	if	SCONJ
ejpam-1781	42	20	i	i	PRON
ejpam-1781	42	21	is	be	AUX
ejpam-1781	42	22	an	an	DET
ejpam-1781	42	23	ideal	ideal	NOUN
ejpam-1781	42	24	of	of	ADP
ejpam-1781	42	25	r	r	NOUN
ejpam-1781	42	26	and	and	CCONJ
ejpam-1781	42	27	a	a	PRON
ejpam-1781	42	28	,	,	PUNCT
ejpam-1781	42	29	b	b	X
ejpam-1781	42	30	∈	∈	PROPN
ejpam-1781	42	31	r	r	NOUN
ejpam-1781	42	32	,	,	PUNCT
ejpam-1781	42	33	then	then	ADV
ejpam-1781	42	34	a	a	DET
ejpam-1781	42	35	∧	∧	PROPN
ejpam-1781	42	36	b	b	PROPN
ejpam-1781	42	37	∈	∈	PROPN
ejpam-1781	42	38	i⇔	i⇔	PROPN
ejpam-1781	42	39	b	b	PROPN
ejpam-1781	42	40	∧	∧	PROPN
ejpam-1781	42	41	a	a	DET
ejpam-1781	42	42	∈	∈	NOUN
ejpam-1781	42	43	i	i	PRON
ejpam-1781	42	44	.	.	PUNCT
ejpam-1781	43	1	a	a	DET
ejpam-1781	43	2	proper	proper	ADJ
ejpam-1781	43	3	ideal	ideal	NOUN
ejpam-1781	43	4	p	p	NOUN
ejpam-1781	43	5	of	of	ADP
ejpam-1781	43	6	r	r	NOUN
ejpam-1781	43	7	is	be	AUX
ejpam-1781	43	8	said	say	VERB
ejpam-1781	43	9	to	to	PART
ejpam-1781	43	10	be	be	AUX
ejpam-1781	43	11	prime	prime	ADJ
ejpam-1781	43	12	if	if	SCONJ
ejpam-1781	43	13	for	for	ADP
ejpam-1781	43	14	any	any	DET
ejpam-1781	43	15	x	x	NOUN
ejpam-1781	43	16	,	,	PUNCT
ejpam-1781	43	17	y	y	PROPN
ejpam-1781	43	18	∈	∈	PROPN
ejpam-1781	43	19	r	r	NOUN
ejpam-1781	43	20	,	,	PUNCT
ejpam-1781	43	21	x	x	PUNCT
ejpam-1781	43	22	∧	∧	NOUN
ejpam-1781	43	23	y	y	PROPN
ejpam-1781	43	24	∈	∈	PROPN
ejpam-1781	43	25	p	p	PROPN
ejpam-1781	43	26	implies	imply	VERB
ejpam-1781	43	27	either	either	CCONJ
ejpam-1781	43	28	x	x	SYM
ejpam-1781	43	29	∈	∈	PROPN
ejpam-1781	43	30	p	p	NOUN
ejpam-1781	43	31	or	or	CCONJ
ejpam-1781	43	32	y	y	PROPN
ejpam-1781	43	33	∈	∈	PROPN
ejpam-1781	43	34	p.	p.	NOUN
ejpam-1781	43	35	a	a	DET
ejpam-1781	43	36	prime	prime	ADJ
ejpam-1781	43	37	ideal	ideal	NOUN
ejpam-1781	43	38	of	of	ADP
ejpam-1781	43	39	p	p	NOUN
ejpam-1781	43	40	of	of	ADP
ejpam-1781	43	41	r	r	NOUN
ejpam-1781	43	42	is	be	AUX
ejpam-1781	43	43	said	say	VERB
ejpam-1781	43	44	to	to	PART
ejpam-1781	43	45	be	be	AUX
ejpam-1781	43	46	minimal	minimal	ADJ
ejpam-1781	43	47	if	if	SCONJ
ejpam-1781	43	48	there	there	PRON
ejpam-1781	43	49	exists	exist	VERB
ejpam-1781	43	50	no	no	DET
ejpam-1781	43	51	prime	prime	ADJ
ejpam-1781	43	52	ideal	ideal	NOUN
ejpam-1781	43	53	q	q	PROPN
ejpam-1781	43	54	of	of	ADP
ejpam-1781	43	55	r	r	NOUN
ejpam-1781	44	1	such	such	ADJ
ejpam-1781	44	2	that	that	PRON
ejpam-1781	44	3	q	q	PROPN
ejpam-1781	44	4	⊂	⊂	PROPN
ejpam-1781	44	5	r.	r.	PROPN
ejpam-1781	44	6	a	a	DET
ejpam-1781	44	7	proper	proper	ADJ
ejpam-1781	44	8	ideal	ideal	NOUN
ejpam-1781	44	9	p	p	NOUN
ejpam-1781	44	10	of	of	ADP
ejpam-1781	44	11	r	r	NOUN
ejpam-1781	44	12	is	be	AUX
ejpam-1781	44	13	said	say	VERB
ejpam-1781	44	14	to	to	PART
ejpam-1781	44	15	be	be	AUX
ejpam-1781	44	16	maximal	maximal	ADJ
ejpam-1781	44	17	if	if	SCONJ
ejpam-1781	44	18	,	,	PUNCT
ejpam-1781	44	19	there	there	PRON
ejpam-1781	44	20	is	be	VERB
ejpam-1781	44	21	no	no	DET
ejpam-1781	44	22	proper	proper	ADJ
ejpam-1781	44	23	ideal	ideal	NOUN
ejpam-1781	44	24	q	q	NOUN
ejpam-1781	44	25	of	of	ADP
ejpam-1781	44	26	r	r	NOUN
ejpam-1781	44	27	such	such	ADJ
ejpam-1781	44	28	that	that	SCONJ
ejpam-1781	44	29	p	p	PROPN
ejpam-1781	44	30	⊆	⊆	NUM
ejpam-1781	44	31	q.	q.	NOUN
ejpam-1781	44	32	note	note	NOUN
ejpam-1781	44	33	that	that	SCONJ
ejpam-1781	44	34	every	every	DET
ejpam-1781	44	35	maximal	maximal	ADJ
ejpam-1781	44	36	ideal	ideal	NOUN
ejpam-1781	44	37	of	of	ADP
ejpam-1781	44	38	r	r	NOUN
ejpam-1781	44	39	is	be	AUX
ejpam-1781	44	40	prime	prime	ADJ
ejpam-1781	44	41	.	.	PUNCT
ejpam-1781	45	1	dually	dually	ADV
ejpam-1781	45	2	we	we	PRON
ejpam-1781	45	3	can	can	AUX
ejpam-1781	45	4	define	define	VERB
ejpam-1781	45	5	prime	prime	ADJ
ejpam-1781	45	6	filter	filter	NOUN
ejpam-1781	45	7	,	,	PUNCT
ejpam-1781	45	8	minimal	minimal	ADJ
ejpam-1781	45	9	prime	prime	ADJ
ejpam-1781	45	10	filter	filter	NOUN
ejpam-1781	45	11	and	and	CCONJ
ejpam-1781	45	12	maximal	maximal	ADJ
ejpam-1781	45	13	filter	filter	NOUN
ejpam-1781	45	14	.	.	PUNCT
ejpam-1781	46	1	for	for	ADP
ejpam-1781	46	2	any	any	DET
ejpam-1781	46	3	non	non	ADJ
ejpam-1781	46	4	-	-	ADJ
ejpam-1781	46	5	empty	empty	ADJ
ejpam-1781	46	6	subset	subset	VERB
ejpam-1781	46	7	a	a	PRON
ejpam-1781	46	8	of	of	ADP
ejpam-1781	46	9	an	an	DET
ejpam-1781	46	10	adl	adl	PROPN
ejpam-1781	46	11	r	r	NOUN
ejpam-1781	46	12	,	,	PUNCT
ejpam-1781	46	13	define	define	VERB
ejpam-1781	46	14	a∗	a∗	NOUN
ejpam-1781	46	15	=	=	SYM
ejpam-1781	46	16	{	{	PUNCT
ejpam-1781	46	17	x	x	SYM
ejpam-1781	46	18	∈	∈	NOUN
ejpam-1781	46	19	r	r	NOUN
ejpam-1781	46	20	|	|	ADV
ejpam-1781	46	21	a	a	DET
ejpam-1781	46	22	∧	∧	NOUN
ejpam-1781	46	23	x	x	PUNCT
ejpam-1781	46	24	=	=	SYM
ejpam-1781	46	25	0	0	NUM
ejpam-1781	46	26	,	,	PUNCT
ejpam-1781	46	27	for	for	ADP
ejpam-1781	46	28	all	all	DET
ejpam-1781	46	29	a	a	DET
ejpam-1781	46	30	∈	∈	PROPN
ejpam-1781	46	31	a	a	PRON
ejpam-1781	46	32	}	}	PUNCT
ejpam-1781	46	33	and	and	CCONJ
ejpam-1781	46	34	is	be	AUX
ejpam-1781	46	35	called	call	VERB
ejpam-1781	46	36	an	an	DET
ejpam-1781	46	37	annihilator	annihilator	PROPN
ejpam-1781	46	38	ideal	ideal	NOUN
ejpam-1781	46	39	of	of	ADP
ejpam-1781	46	40	y.	y.	PROPN
ejpam-1781	46	41	pawar	pawar	PROPN
ejpam-1781	46	42	,	,	PUNCT
ejpam-1781	46	43	i.	i.	PROPN
ejpam-1781	46	44	shaikh	shaikh	PROPN
ejpam-1781	46	45	/	/	SYM
ejpam-1781	46	46	eur	eur	PROPN
ejpam-1781	46	47	.	.	PUNCT
ejpam-1781	47	1	j.	j.	PROPN
ejpam-1781	47	2	pure	pure	PROPN
ejpam-1781	47	3	appl	appl	PROPN
ejpam-1781	47	4	.	.	PROPN
ejpam-1781	47	5	math	math	PROPN
ejpam-1781	47	6	,	,	PUNCT
ejpam-1781	47	7	6	6	NUM
ejpam-1781	47	8	(	(	PUNCT
ejpam-1781	47	9	2013	2013	NUM
ejpam-1781	47	10	)	)	PUNCT
ejpam-1781	47	11	,	,	PUNCT
ejpam-1781	47	12	107	107	NUM
ejpam-1781	47	13	-	-	SYM
ejpam-1781	47	14	118	118	NUM
ejpam-1781	47	15	109	109	NUM
ejpam-1781	47	16	a.	a.	NOUN
ejpam-1781	47	17	for	for	ADP
ejpam-1781	47	18	x	x	PROPN
ejpam-1781	47	19	∈	∈	PROPN
ejpam-1781	47	20	r	r	NOUN
ejpam-1781	47	21	,	,	PUNCT
ejpam-1781	47	22	the	the	DET
ejpam-1781	47	23	annulate	annulate	NOUN
ejpam-1781	47	24	(	(	PUNCT
ejpam-1781	47	25	x]∗	x]∗	PROPN
ejpam-1781	47	26	of	of	ADP
ejpam-1781	47	27	x	x	PROPN
ejpam-1781	47	28	is	be	AUX
ejpam-1781	47	29	defined	define	VERB
ejpam-1781	47	30	as	as	ADP
ejpam-1781	47	31	(	(	PUNCT
ejpam-1781	47	32	x]∗	x]∗	PROPN
ejpam-1781	47	33	=	=	PROPN
ejpam-1781	47	34	{	{	PUNCT
ejpam-1781	47	35	y	y	PROPN
ejpam-1781	47	36	∈	∈	PROPN
ejpam-1781	47	37	r	r	NOUN
ejpam-1781	48	1	|	|	NOUN
ejpam-1781	48	2	x	x	NOUN
ejpam-1781	48	3	∧	∧	NOUN
ejpam-1781	48	4	y	y	NOUN
ejpam-1781	48	5	=	=	PROPN
ejpam-1781	48	6	0	0	NUM
ejpam-1781	48	7	}	}	PUNCT
ejpam-1781	48	8	.	.	PUNCT
ejpam-1781	49	1	let	let	VERB
ejpam-1781	49	2	℘	℘	PROPN
ejpam-1781	49	3	,	,	PUNCT
ejpam-1781	49	4	σ	σ	PROPN
ejpam-1781	49	5	and	and	CCONJ
ejpam-1781	49	6	m	m	PROPN
ejpam-1781	49	7	denote	denote	VERB
ejpam-1781	49	8	the	the	DET
ejpam-1781	49	9	set	set	NOUN
ejpam-1781	49	10	of	of	ADP
ejpam-1781	49	11	all	all	DET
ejpam-1781	49	12	prime	prime	ADJ
ejpam-1781	49	13	ideals	ideal	NOUN
ejpam-1781	49	14	,	,	PUNCT
ejpam-1781	49	15	maximal	maximal	ADJ
ejpam-1781	49	16	ideals	ideal	NOUN
ejpam-1781	49	17	and	and	CCONJ
ejpam-1781	49	18	minimal	minimal	ADJ
ejpam-1781	49	19	prime	prime	ADJ
ejpam-1781	49	20	ideals	ideal	NOUN
ejpam-1781	49	21	in	in	ADP
ejpam-1781	49	22	r	r	NOUN
ejpam-1781	49	23	respectively	respectively	ADV
ejpam-1781	49	24	.	.	PUNCT
ejpam-1781	50	1	now	now	ADV
ejpam-1781	50	2	we	we	PRON
ejpam-1781	50	3	quote	quote	VERB
ejpam-1781	50	4	some	some	DET
ejpam-1781	50	5	results	result	NOUN
ejpam-1781	50	6	result	result	VERB
ejpam-1781	50	7	1	1	NUM
ejpam-1781	50	8	.	.	PUNCT
ejpam-1781	51	1	let	let	VERB
ejpam-1781	51	2	i	i	PRON
ejpam-1781	51	3	be	be	AUX
ejpam-1781	51	4	an	an	DET
ejpam-1781	51	5	ideal	ideal	NOUN
ejpam-1781	51	6	of	of	ADP
ejpam-1781	51	7	r	r	NOUN
ejpam-1781	51	8	and	and	CCONJ
ejpam-1781	51	9	a	a	DET
ejpam-1781	51	10	∈	∈	NOUN
ejpam-1781	51	11	r	r	NOUN
ejpam-1781	51	12	such	such	DET
ejpam-1781	51	13	that	that	SCONJ
ejpam-1781	51	14	a	a	PRON
ejpam-1781	51	15	/∈	/∈	INTJ
ejpam-1781	52	1	i	i	INTJ
ejpam-1781	52	2	.	.	PUNCT
ejpam-1781	53	1	then	then	ADV
ejpam-1781	53	2	there	there	PRON
ejpam-1781	53	3	exists	exist	VERB
ejpam-1781	53	4	a	a	DET
ejpam-1781	53	5	prime	prime	ADJ
ejpam-1781	53	6	ideal	ideal	NOUN
ejpam-1781	53	7	p	p	NOUN
ejpam-1781	53	8	of	of	ADP
ejpam-1781	53	9	r	r	NOUN
ejpam-1781	53	10	such	such	ADJ
ejpam-1781	53	11	that	that	SCONJ
ejpam-1781	53	12	i	i	PRON
ejpam-1781	53	13	⊆	⊆	NUM
ejpam-1781	53	14	p	p	NOUN
ejpam-1781	53	15	and	and	CCONJ
ejpam-1781	53	16	a	a	PRON
ejpam-1781	53	17	/∈	/∈	NOUN
ejpam-1781	54	1	p	p	NOUN
ejpam-1781	54	2	result	result	NOUN
ejpam-1781	54	3	2	2	NUM
ejpam-1781	54	4	.	.	PUNCT
ejpam-1781	55	1	a	a	DET
ejpam-1781	55	2	prime	prime	ADJ
ejpam-1781	55	3	ideal	ideal	NOUN
ejpam-1781	55	4	of	of	ADP
ejpam-1781	55	5	an	an	DET
ejpam-1781	55	6	adl	adl	PROPN
ejpam-1781	55	7	r	r	NOUN
ejpam-1781	55	8	is	be	AUX
ejpam-1781	55	9	minimal	minimal	ADJ
ejpam-1781	55	10	if	if	SCONJ
ejpam-1781	55	11	and	and	CCONJ
ejpam-1781	55	12	only	only	ADV
ejpam-1781	55	13	if	if	SCONJ
ejpam-1781	55	14	a	a	DET
ejpam-1781	55	15	∈	∈	PROPN
ejpam-1781	55	16	p	p	NOUN
ejpam-1781	55	17	⇒	⇒	NOUN
ejpam-1781	55	18	(	(	PUNCT
ejpam-1781	55	19	a]∗	a]∗	PROPN
ejpam-1781	55	20	/∈	/∈	PUNCT
ejpam-1781	56	1	p.	p.	NOUN
ejpam-1781	56	2	result	result	NOUN
ejpam-1781	56	3	3	3	X
ejpam-1781	56	4	.	.	PUNCT
ejpam-1781	57	1	every	every	DET
ejpam-1781	57	2	prime	prime	ADJ
ejpam-1781	57	3	ideal	ideal	NOUN
ejpam-1781	57	4	of	of	ADP
ejpam-1781	57	5	r	r	NOUN
ejpam-1781	57	6	contains	contain	VERB
ejpam-1781	57	7	a	a	DET
ejpam-1781	57	8	minimal	minimal	ADJ
ejpam-1781	57	9	prime	prime	ADJ
ejpam-1781	57	10	ideal	ideal	NOUN
ejpam-1781	57	11	.	.	PUNCT
ejpam-1781	58	1	result	result	VERB
ejpam-1781	58	2	4	4	NUM
ejpam-1781	58	3	.	.	PUNCT
ejpam-1781	59	1	the	the	DET
ejpam-1781	59	2	set	set	NOUN
ejpam-1781	59	3	i	i	PRON
ejpam-1781	59	4	(	(	PUNCT
ejpam-1781	59	5	r	r	NOUN
ejpam-1781	59	6	)	)	PUNCT
ejpam-1781	59	7	of	of	ADP
ejpam-1781	59	8	all	all	DET
ejpam-1781	59	9	ideals	ideal	NOUN
ejpam-1781	59	10	of	of	ADP
ejpam-1781	59	11	r	r	NOUN
ejpam-1781	59	12	is	be	AUX
ejpam-1781	59	13	a	a	DET
ejpam-1781	59	14	complete	complete	ADJ
ejpam-1781	59	15	distributive	distributive	ADJ
ejpam-1781	59	16	lattice	lattice	NOUN
ejpam-1781	59	17	with	with	ADP
ejpam-1781	59	18	the	the	DET
ejpam-1781	59	19	least	least	ADJ
ejpam-1781	59	20	element	element	NOUN
ejpam-1781	59	21	{	{	PUNCT
ejpam-1781	59	22	0	0	NUM
ejpam-1781	59	23	}	}	PUNCT
ejpam-1781	59	24	and	and	CCONJ
ejpam-1781	59	25	the	the	DET
ejpam-1781	59	26	greatest	great	ADJ
ejpam-1781	59	27	element	element	NOUN
ejpam-1781	59	28	r	r	NOUN
ejpam-1781	59	29	under	under	ADP
ejpam-1781	59	30	set	set	NOUN
ejpam-1781	59	31	inclusion	inclusion	NOUN
ejpam-1781	59	32	in	in	ADP
ejpam-1781	59	33	which	which	PRON
ejpam-1781	59	34	,	,	PUNCT
ejpam-1781	59	35	for	for	ADP
ejpam-1781	59	36	any	any	DET
ejpam-1781	59	37	i	i	NOUN
ejpam-1781	59	38	,	,	PUNCT
ejpam-1781	59	39	j	j	PROPN
ejpam-1781	59	40	∈	∈	PROPN
ejpam-1781	59	41	i	i	PRON
ejpam-1781	59	42	(	(	PUNCT
ejpam-1781	59	43	r	r	NOUN
ejpam-1781	59	44	)	)	PUNCT
ejpam-1781	59	45	,	,	PUNCT
ejpam-1781	59	46	i	i	PROPN
ejpam-1781	59	47	∩	∩	VERB
ejpam-1781	59	48	j	j	PROPN
ejpam-1781	59	49	is	be	AUX
ejpam-1781	59	50	the	the	DET
ejpam-1781	59	51	infimum	infimum	NOUN
ejpam-1781	59	52	of	of	ADP
ejpam-1781	59	53	i	i	PRON
ejpam-1781	59	54	,	,	PUNCT
ejpam-1781	59	55	j	j	PROPN
ejpam-1781	59	56	and	and	CCONJ
ejpam-1781	59	57	the	the	DET
ejpam-1781	59	58	supremum	supremum	NOUN
ejpam-1781	59	59	is	be	AUX
ejpam-1781	59	60	given	give	VERB
ejpam-1781	59	61	by	by	ADP
ejpam-1781	59	62	i	i	PROPN
ejpam-1781	59	63	∨	∨	PROPN
ejpam-1781	59	64	j	j	PROPN
ejpam-1781	59	65	=	=	SYM
ejpam-1781	59	66	�	�	PROPN
ejpam-1781	60	1	i	i	PRON
ejpam-1781	60	2	∨	∨	PROPN
ejpam-1781	61	1	j	j	PROPN
ejpam-1781	62	1	|	|	ADV
ejpam-1781	62	2	i	i	PRON
ejpam-1781	62	3	∈	∈	VERB
ejpam-1781	63	1	i	i	PRON
ejpam-1781	63	2	,	,	PUNCT
ejpam-1781	63	3	j	j	PROPN
ejpam-1781	63	4	∈	∈	PROPN
ejpam-1781	63	5	j	j	PROPN
ejpam-1781	63	6	.	.	PUNCT
ejpam-1781	64	1	result	result	VERB
ejpam-1781	64	2	5	5	NUM
ejpam-1781	64	3	.	.	PUNCT
ejpam-1781	65	1	p	p	NOUN
ejpam-1781	65	2	is	be	AUX
ejpam-1781	65	3	a	a	DET
ejpam-1781	65	4	minimal	minimal	ADJ
ejpam-1781	65	5	prime	prime	ADJ
ejpam-1781	65	6	ideal	ideal	NOUN
ejpam-1781	65	7	of	of	ADP
ejpam-1781	65	8	r	r	NOUN
ejpam-1781	65	9	if	if	SCONJ
ejpam-1781	66	1	and	and	CCONJ
ejpam-1781	66	2	only	only	ADV
ejpam-1781	66	3	if	if	SCONJ
ejpam-1781	66	4	r	r	NOUN
ejpam-1781	66	5	\	\	PROPN
ejpam-1781	66	6	p	p	NOUN
ejpam-1781	66	7	is	be	AUX
ejpam-1781	66	8	a	a	DET
ejpam-1781	66	9	maximal	maximal	ADJ
ejpam-1781	66	10	filter	filter	NOUN
ejpam-1781	66	11	of	of	ADP
ejpam-1781	66	12	r.	r.	PROPN
ejpam-1781	66	13	result	result	PROPN
ejpam-1781	66	14	6	6	NUM
ejpam-1781	66	15	.	.	X
ejpam-1781	67	1	for	for	ADP
ejpam-1781	67	2	any	any	DET
ejpam-1781	67	3	a	a	NOUN
ejpam-1781	67	4	,	,	PUNCT
ejpam-1781	67	5	b	b	X
ejpam-1781	67	6	∈	∈	PROPN
ejpam-1781	67	7	r	r	NOUN
ejpam-1781	67	8	,	,	PUNCT
ejpam-1781	67	9	we	we	PRON
ejpam-1781	67	10	have	have	VERB
ejpam-1781	67	11	the	the	DET
ejpam-1781	67	12	following	follow	VERB
ejpam-1781	67	13	:	:	PUNCT
ejpam-1781	67	14	1	1	X
ejpam-1781	67	15	.	.	PUNCT
ejpam-1781	67	16	(	(	PUNCT
ejpam-1781	67	17	a]∨	a]∨	PROPN
ejpam-1781	67	18	(	(	PUNCT
ejpam-1781	67	19	b	b	NOUN
ejpam-1781	67	20	]	]	X
ejpam-1781	67	21	=	=	X
ejpam-1781	67	22	(	(	PUNCT
ejpam-1781	67	23	a	a	DET
ejpam-1781	67	24	∨	∨	NUM
ejpam-1781	67	25	b	b	NOUN
ejpam-1781	67	26	]	]	X
ejpam-1781	68	1	=	=	SYM
ejpam-1781	68	2	(	(	PUNCT
ejpam-1781	68	3	b	b	PROPN
ejpam-1781	68	4	∨	∨	NUM
ejpam-1781	68	5	a	a	PRON
ejpam-1781	68	6	]	]	X
ejpam-1781	68	7	2	2	NUM
ejpam-1781	68	8	.	.	PUNCT
ejpam-1781	68	9	(	(	PUNCT
ejpam-1781	68	10	a]∧	a]∧	X
ejpam-1781	68	11	(	(	PUNCT
ejpam-1781	68	12	b	b	X
ejpam-1781	68	13	]	]	X
ejpam-1781	68	14	=	=	X
ejpam-1781	68	15	(	(	PUNCT
ejpam-1781	68	16	a	a	DET
ejpam-1781	68	17	∧	∧	PROPN
ejpam-1781	68	18	b	b	PROPN
ejpam-1781	68	19	]	]	X
ejpam-1781	68	20	=	=	SYM
ejpam-1781	68	21	(	(	PUNCT
ejpam-1781	68	22	b	b	X
ejpam-1781	68	23	∧	∧	PROPN
ejpam-1781	68	24	a	a	X
ejpam-1781	68	25	]	]	X
ejpam-1781	68	26	.	.	PUNCT
ejpam-1781	69	1	result	result	VERB
ejpam-1781	69	2	7	7	NUM
ejpam-1781	69	3	.	.	X
ejpam-1781	69	4	intersection	intersection	NOUN
ejpam-1781	69	5	of	of	ADP
ejpam-1781	69	6	all	all	DET
ejpam-1781	69	7	minimal	minimal	ADJ
ejpam-1781	69	8	prime	prime	ADJ
ejpam-1781	69	9	ideals	ideal	NOUN
ejpam-1781	69	10	of	of	ADP
ejpam-1781	69	11	r	r	NOUN
ejpam-1781	69	12	is	be	AUX
ejpam-1781	69	13	{	{	PUNCT
ejpam-1781	69	14	0	0	NUM
ejpam-1781	69	15	}	}	PUNCT
ejpam-1781	69	16	.	.	PUNCT
ejpam-1781	70	1	result	result	VERB
ejpam-1781	70	2	8	8	NUM
ejpam-1781	70	3	.	.	PUNCT
ejpam-1781	71	1	for	for	ADP
ejpam-1781	71	2	any	any	DET
ejpam-1781	71	3	non	non	ADJ
ejpam-1781	71	4	-	-	ADJ
ejpam-1781	71	5	empty	empty	ADJ
ejpam-1781	71	6	subset	subset	NOUN
ejpam-1781	71	7	a	a	PRON
ejpam-1781	71	8	of	of	ADP
ejpam-1781	71	9	r	r	NOUN
ejpam-1781	71	10	,	,	PUNCT
ejpam-1781	71	11	a∗	a∗	PROPN
ejpam-1781	71	12	is	be	AUX
ejpam-1781	71	13	an	an	DET
ejpam-1781	71	14	ideal	ideal	NOUN
ejpam-1781	71	15	of	of	ADP
ejpam-1781	71	16	r.	r.	PROPN
ejpam-1781	71	17	result	result	PROPN
ejpam-1781	71	18	9	9	NUM
ejpam-1781	71	19	.	.	X
ejpam-1781	72	1	for	for	ADP
ejpam-1781	72	2	any	any	DET
ejpam-1781	72	3	non	non	ADJ
ejpam-1781	72	4	-	-	ADJ
ejpam-1781	72	5	empty	empty	ADJ
ejpam-1781	72	6	subset	subset	NOUN
ejpam-1781	72	7	a	a	PRON
ejpam-1781	72	8	of	of	ADP
ejpam-1781	72	9	r	r	NOUN
ejpam-1781	72	10	,	,	PUNCT
ejpam-1781	72	11	a∗	a∗	ADJ
ejpam-1781	72	12	∩	∩	NOUN
ejpam-1781	72	13	a=	a=	NOUN
ejpam-1781	72	14	;	;	PUNCT
ejpam-1781	72	15	.	.	PUNCT
ejpam-1781	73	1	result	result	VERB
ejpam-1781	73	2	10	10	NUM
ejpam-1781	73	3	.	.	PUNCT
ejpam-1781	74	1	for	for	ADP
ejpam-1781	74	2	any	any	DET
ejpam-1781	74	3	non	non	ADJ
ejpam-1781	74	4	-	-	ADJ
ejpam-1781	74	5	empty	empty	ADJ
ejpam-1781	74	6	subset	subset	NOUN
ejpam-1781	74	7	a	a	PRON
ejpam-1781	74	8	of	of	ADP
ejpam-1781	74	9	r	r	NOUN
ejpam-1781	74	10	,	,	PUNCT
ejpam-1781	74	11	a∗	a∗	NOUN
ejpam-1781	74	12	=	=	PUNCT
ejpam-1781	74	13	∩{m	∩{m	PROPN
ejpam-1781	74	14	∈m	∈m	VERB
ejpam-1781	74	15	|a	|a	NOUN
ejpam-1781	74	16	*	*	SYM
ejpam-1781	74	17	m	m	NOUN
ejpam-1781	74	18	}	}	PUNCT
ejpam-1781	74	19	.	.	PUNCT
ejpam-1781	75	1	result	result	VERB
ejpam-1781	75	2	11	11	NUM
ejpam-1781	75	3	.	.	PUNCT
ejpam-1781	76	1	let	let	VERB
ejpam-1781	76	2	i	i	PRON
ejpam-1781	76	3	be	be	AUX
ejpam-1781	76	4	an	an	DET
ejpam-1781	76	5	ideal	ideal	NOUN
ejpam-1781	76	6	and	and	CCONJ
ejpam-1781	76	7	s	s	AUX
ejpam-1781	76	8	be	be	AUX
ejpam-1781	76	9	a	a	DET
ejpam-1781	76	10	multiplicatively	multiplicatively	ADV
ejpam-1781	76	11	closed	close	VERB
ejpam-1781	76	12	subset	subset	NOUN
ejpam-1781	76	13	of	of	ADP
ejpam-1781	76	14	r	r	NOUN
ejpam-1781	76	15	such	such	ADJ
ejpam-1781	76	16	that	that	SCONJ
ejpam-1781	76	17	i	i	PRON
ejpam-1781	76	18	∩	∩	NOUN
ejpam-1781	76	19	s	s	PART
ejpam-1781	76	20	=	=	X
ejpam-1781	76	21	;	;	PUNCT
ejpam-1781	76	22	.	.	PUNCT
ejpam-1781	77	1	then	then	ADV
ejpam-1781	77	2	there	there	PRON
ejpam-1781	77	3	is	be	VERB
ejpam-1781	77	4	a	a	DET
ejpam-1781	77	5	prime	prime	ADJ
ejpam-1781	77	6	ideal	ideal	NOUN
ejpam-1781	77	7	m	m	VERB
ejpam-1781	77	8	of	of	ADP
ejpam-1781	77	9	r	r	NOUN
ejpam-1781	77	10	such	such	ADJ
ejpam-1781	77	11	that	that	SCONJ
ejpam-1781	77	12	i	i	PRON
ejpam-1781	77	13	⊆	⊆	NUM
ejpam-1781	77	14	m	m	VERB
ejpam-1781	77	15	and	and	CCONJ
ejpam-1781	77	16	m	m	NOUN
ejpam-1781	77	17	∩	∩	NOUN
ejpam-1781	77	18	s	s	PART
ejpam-1781	77	19	=	=	PUNCT
ejpam-1781	77	20	;	;	PUNCT
ejpam-1781	77	21	.	.	PUNCT
ejpam-1781	78	1	result	result	PROPN
ejpam-1781	78	2	12	12	NUM
ejpam-1781	78	3	.	.	PUNCT
ejpam-1781	79	1	for	for	ADP
ejpam-1781	79	2	any	any	DET
ejpam-1781	79	3	ideal	ideal	NOUN
ejpam-1781	79	4	i	i	PRON
ejpam-1781	79	5	of	of	ADP
ejpam-1781	79	6	r	r	PROPN
ejpam-1781	79	7	,	,	PUNCT
ejpam-1781	79	8	we	we	PRON
ejpam-1781	79	9	have	have	VERB
ejpam-1781	79	10	,	,	PUNCT
ejpam-1781	79	11	i	i	PRON
ejpam-1781	79	12	=	=	VERB
ejpam-1781	79	13	∩{p	∩{p	PROPN
ejpam-1781	80	1	|	|	ADV
ejpam-1781	80	2	p	p	NOUN
ejpam-1781	80	3	is	be	AUX
ejpam-1781	80	4	a	a	DET
ejpam-1781	80	5	prime	prime	ADJ
ejpam-1781	80	6	ideal	ideal	NOUN
ejpam-1781	80	7	of	of	ADP
ejpam-1781	80	8	r	r	NOUN
ejpam-1781	80	9	,	,	PUNCT
ejpam-1781	80	10	i	i	NOUN
ejpam-1781	80	11	⊆	⊆	NUM
ejpam-1781	80	12	p	p	X
ejpam-1781	80	13	}	}	PUNCT
ejpam-1781	80	14	.	.	PUNCT
ejpam-1781	81	1	result	result	VERB
ejpam-1781	81	2	13	13	NUM
ejpam-1781	81	3	.	.	PUNCT
ejpam-1781	82	1	every	every	DET
ejpam-1781	82	2	maximal	maximal	ADJ
ejpam-1781	82	3	ideal	ideal	NOUN
ejpam-1781	82	4	is	be	AUX
ejpam-1781	82	5	prime	prime	ADJ
ejpam-1781	82	6	in	in	ADP
ejpam-1781	82	7	r.	r.	PROPN
ejpam-1781	82	8	3	3	NUM
ejpam-1781	82	9	.	.	PUNCT
ejpam-1781	83	1	prime	prime	ADJ
ejpam-1781	83	2	ideals	ideal	NOUN
ejpam-1781	83	3	addition	addition	NOUN
ejpam-1781	83	4	to	to	ADP
ejpam-1781	83	5	the	the	DET
ejpam-1781	83	6	properties	property	NOUN
ejpam-1781	83	7	of	of	ADP
ejpam-1781	83	8	prime	prime	ADJ
ejpam-1781	83	9	,	,	PUNCT
ejpam-1781	83	10	minimal	minimal	ADJ
ejpam-1781	83	11	prime	prime	NOUN
ejpam-1781	83	12	and	and	CCONJ
ejpam-1781	83	13	annihilator	annihilator	NOUN
ejpam-1781	83	14	ideals	ideal	NOUN
ejpam-1781	83	15	in	in	ADP
ejpam-1781	83	16	an	an	DET
ejpam-1781	83	17	adl	adl	NOUN
ejpam-1781	83	18	carried	carry	VERB
ejpam-1781	83	19	out	out	ADP
ejpam-1781	83	20	in	in	ADP
ejpam-1781	83	21	[	[	X
ejpam-1781	83	22	1	1	NUM
ejpam-1781	83	23	]	]	PUNCT
ejpam-1781	83	24	and	and	CCONJ
ejpam-1781	83	25	[	[	X
ejpam-1781	83	26	2	2	NUM
ejpam-1781	83	27	]	]	PUNCT
ejpam-1781	83	28	,	,	PUNCT
ejpam-1781	83	29	we	we	PRON
ejpam-1781	83	30	study	study	VERB
ejpam-1781	83	31	some	some	DET
ejpam-1781	83	32	more	more	ADJ
ejpam-1781	83	33	properties	property	NOUN
ejpam-1781	83	34	of	of	ADP
ejpam-1781	83	35	these	these	DET
ejpam-1781	83	36	ideals	ideal	NOUN
ejpam-1781	83	37	in	in	ADP
ejpam-1781	83	38	an	an	DET
ejpam-1781	83	39	adl	adl	NOUN
ejpam-1781	83	40	in	in	ADP
ejpam-1781	83	41	this	this	DET
ejpam-1781	83	42	article	article	NOUN
ejpam-1781	83	43	.	.	PUNCT
ejpam-1781	84	1	two	two	NUM
ejpam-1781	84	2	ideals	ideal	NOUN
ejpam-1781	84	3	i	i	PRON
ejpam-1781	84	4	and	and	CCONJ
ejpam-1781	84	5	j	j	PROPN
ejpam-1781	84	6	of	of	ADP
ejpam-1781	84	7	r	r	NOUN
ejpam-1781	84	8	are	be	AUX
ejpam-1781	84	9	said	say	VERB
ejpam-1781	84	10	to	to	PART
ejpam-1781	84	11	be	be	AUX
ejpam-1781	84	12	co	co	VERB
ejpam-1781	84	13	-	-	ADJ
ejpam-1781	84	14	maximal	maximal	ADJ
ejpam-1781	84	15	if	if	SCONJ
ejpam-1781	84	16	i	i	PRON
ejpam-1781	84	17	∨	∨	VERB
ejpam-1781	84	18	j	j	PROPN
ejpam-1781	84	19	=	=	SYM
ejpam-1781	84	20	r.	r.	PROPN
ejpam-1781	84	21	if	if	SCONJ
ejpam-1781	84	22	the	the	DET
ejpam-1781	84	23	ideals	ideal	NOUN
ejpam-1781	84	24	(	(	PUNCT
ejpam-1781	84	25	a	a	X
ejpam-1781	84	26	]	]	X
ejpam-1781	84	27	are	be	AUX
ejpam-1781	84	28	(	(	PUNCT
ejpam-1781	84	29	a]∗	a]∗	NOUN
ejpam-1781	84	30	are	be	AUX
ejpam-1781	84	31	comaximal	comaximal	ADJ
ejpam-1781	84	32	,	,	PUNCT
ejpam-1781	84	33	then	then	ADV
ejpam-1781	84	34	we	we	PRON
ejpam-1781	84	35	have	have	AUX
ejpam-1781	84	36	theorem	theorem	VERB
ejpam-1781	84	37	1	1	NUM
ejpam-1781	84	38	.	.	PUNCT
ejpam-1781	85	1	for	for	ADP
ejpam-1781	85	2	any	any	DET
ejpam-1781	85	3	a	a	DET
ejpam-1781	85	4	∈	∈	NOUN
ejpam-1781	85	5	r	r	NOUN
ejpam-1781	85	6	if	if	SCONJ
ejpam-1781	85	7	the	the	DET
ejpam-1781	85	8	ideals	ideal	NOUN
ejpam-1781	85	9	(	(	PUNCT
ejpam-1781	85	10	a	a	X
ejpam-1781	85	11	]	]	X
ejpam-1781	85	12	are	be	AUX
ejpam-1781	85	13	(	(	PUNCT
ejpam-1781	85	14	a]∗	a]∗	NOUN
ejpam-1781	85	15	are	be	AUX
ejpam-1781	85	16	comaximal	comaximal	ADJ
ejpam-1781	85	17	,	,	PUNCT
ejpam-1781	85	18	then	then	ADV
ejpam-1781	85	19	(	(	PUNCT
ejpam-1781	85	20	a	a	X
ejpam-1781	85	21	]	]	X
ejpam-1781	85	22	=	=	SYM
ejpam-1781	85	23	(	(	PUNCT
ejpam-1781	85	24	a]∗∗	a]∗∗	PUNCT
ejpam-1781	85	25	and	and	CCONJ
ejpam-1781	85	26	the	the	DET
ejpam-1781	85	27	ideals	ideal	NOUN
ejpam-1781	85	28	(	(	PUNCT
ejpam-1781	85	29	a]∗	a]∗	PROPN
ejpam-1781	85	30	and	and	CCONJ
ejpam-1781	85	31	(	(	PUNCT
ejpam-1781	85	32	a]∗∗	a]∗∗	NOUN
ejpam-1781	85	33	are	be	AUX
ejpam-1781	85	34	comaximal	comaximal	ADJ
ejpam-1781	85	35	and	and	CCONJ
ejpam-1781	85	36	conversely	conversely	ADV
ejpam-1781	85	37	.	.	PUNCT
ejpam-1781	86	1	y.	y.	PROPN
ejpam-1781	86	2	pawar	pawar	PROPN
ejpam-1781	86	3	,	,	PUNCT
ejpam-1781	86	4	i.	i.	PROPN
ejpam-1781	86	5	shaikh	shaikh	PROPN
ejpam-1781	86	6	/	/	SYM
ejpam-1781	86	7	eur	eur	PROPN
ejpam-1781	86	8	.	.	PUNCT
ejpam-1781	87	1	j.	j.	PROPN
ejpam-1781	87	2	pure	pure	PROPN
ejpam-1781	87	3	appl	appl	PROPN
ejpam-1781	87	4	.	.	PROPN
ejpam-1781	87	5	math	math	PROPN
ejpam-1781	87	6	,	,	PUNCT
ejpam-1781	87	7	6	6	NUM
ejpam-1781	87	8	(	(	PUNCT
ejpam-1781	87	9	2013	2013	NUM
ejpam-1781	87	10	)	)	PUNCT
ejpam-1781	87	11	,	,	PUNCT
ejpam-1781	87	12	107	107	NUM
ejpam-1781	87	13	-	-	SYM
ejpam-1781	87	14	118	118	NUM
ejpam-1781	87	15	110	110	NUM
ejpam-1781	87	16	proof	proof	NOUN
ejpam-1781	87	17	.	.	PUNCT
ejpam-1781	88	1	for	for	ADP
ejpam-1781	88	2	a	a	DET
ejpam-1781	88	3	∈	∈	NOUN
ejpam-1781	88	4	r	r	NOUN
ejpam-1781	88	5	let	let	VERB
ejpam-1781	88	6	r=	r=	ADJ
ejpam-1781	88	7	(	(	PUNCT
ejpam-1781	88	8	a]∨	a]∨	INTJ
ejpam-1781	88	9	(	(	PUNCT
ejpam-1781	88	10	a]∗.	a]∗.	NOUN
ejpam-1781	88	11	we	we	PRON
ejpam-1781	88	12	have	have	VERB
ejpam-1781	88	13	(	(	PUNCT
ejpam-1781	88	14	a]∗∗	a]∗∗	X
ejpam-1781	88	15	=	=	SYM
ejpam-1781	88	16	(	(	PUNCT
ejpam-1781	88	17	a]∗∗	a]∗∗	X
ejpam-1781	88	18	∩	∩	X
ejpam-1781	88	19	r=	r=	ADJ
ejpam-1781	88	20	(	(	PUNCT
ejpam-1781	88	21	a]∗∗	a]∗∗	X
ejpam-1781	88	22	∩	∩	X
ejpam-1781	88	23	�	�	PROPN
ejpam-1781	88	24	(	(	PUNCT
ejpam-1781	88	25	a]∨	a]∨	PROPN
ejpam-1781	88	26	(	(	PUNCT
ejpam-1781	88	27	a]∗	a]∗	PROPN
ejpam-1781	88	28	�	�	PROPN
ejpam-1781	88	29	=	=	SYM
ejpam-1781	88	30	�	�	PROPN
ejpam-1781	88	31	(	(	PUNCT
ejpam-1781	88	32	a]∗∗	a]∗∗	NOUN
ejpam-1781	88	33	∩	∩	X
ejpam-1781	88	34	(	(	PUNCT
ejpam-1781	88	35	a	a	PRON
ejpam-1781	88	36	]	]	X
ejpam-1781	88	37	�	�	PROPN
ejpam-1781	88	38	∨	∨	NUM
ejpam-1781	88	39	�	�	PROPN
ejpam-1781	88	40	(	(	PUNCT
ejpam-1781	88	41	a]∗∗	a]∗∗	NOUN
ejpam-1781	88	42	∩	∩	NOUN
ejpam-1781	88	43	(	(	PUNCT
ejpam-1781	88	44	a]∗	a]∗	PROPN
ejpam-1781	88	45	�	�	PROPN
ejpam-1781	88	46	=	=	PUNCT
ejpam-1781	88	47	(	(	PUNCT
ejpam-1781	88	48	a	a	X
ejpam-1781	88	49	]	]	X
ejpam-1781	88	50	since	since	SCONJ
ejpam-1781	88	51	(	(	PUNCT
ejpam-1781	88	52	a	a	X
ejpam-1781	88	53	]	]	X
ejpam-1781	88	54	⊆	⊆	NUM
ejpam-1781	88	55	(	(	PUNCT
ejpam-1781	88	56	a]∗∗	a]∗∗	NOUN
ejpam-1781	88	57	)	)	PUNCT
ejpam-1781	88	58	.	.	PUNCT
ejpam-1781	89	1	hence	hence	ADV
ejpam-1781	89	2	in	in	ADP
ejpam-1781	89	3	this	this	DET
ejpam-1781	89	4	case	case	NOUN
ejpam-1781	89	5	r	r	NOUN
ejpam-1781	89	6	=	=	SYM
ejpam-1781	89	7	(	(	PUNCT
ejpam-1781	89	8	a	a	X
ejpam-1781	89	9	]	]	X
ejpam-1781	89	10	∨	∨	X
ejpam-1781	89	11	(	(	PUNCT
ejpam-1781	89	12	a]∗	a]∗	PROPN
ejpam-1781	89	13	=	=	SYM
ejpam-1781	89	14	(	(	PUNCT
ejpam-1781	89	15	a]∗	a]∗	PROPN
ejpam-1781	89	16	∨	∨	PROPN
ejpam-1781	89	17	(	(	PUNCT
ejpam-1781	89	18	a]∗∗.	a]∗∗.	NOUN
ejpam-1781	89	19	proof	proof	NOUN
ejpam-1781	89	20	of	of	ADP
ejpam-1781	89	21	converse	converse	NOUN
ejpam-1781	89	22	is	be	AUX
ejpam-1781	89	23	obvious	obvious	ADJ
ejpam-1781	89	24	.	.	PUNCT
ejpam-1781	90	1	now	now	ADV
ejpam-1781	90	2	we	we	PRON
ejpam-1781	90	3	prove	prove	VERB
ejpam-1781	90	4	necessary	necessary	ADJ
ejpam-1781	90	5	and	and	CCONJ
ejpam-1781	90	6	sufficient	sufficient	ADJ
ejpam-1781	90	7	conditions	condition	NOUN
ejpam-1781	90	8	for	for	SCONJ
ejpam-1781	90	9	every	every	DET
ejpam-1781	90	10	prime	prime	ADJ
ejpam-1781	90	11	ideal	ideal	NOUN
ejpam-1781	90	12	to	to	PART
ejpam-1781	90	13	be	be	AUX
ejpam-1781	90	14	minimal	minimal	ADJ
ejpam-1781	90	15	in	in	ADP
ejpam-1781	90	16	r.	r.	PROPN
ejpam-1781	90	17	theorem	theorem	PROPN
ejpam-1781	90	18	2	2	NUM
ejpam-1781	90	19	.	.	PUNCT
ejpam-1781	91	1	every	every	DET
ejpam-1781	91	2	prime	prime	ADJ
ejpam-1781	91	3	ideal	ideal	NOUN
ejpam-1781	91	4	in	in	ADP
ejpam-1781	91	5	r	r	NOUN
ejpam-1781	91	6	is	be	AUX
ejpam-1781	91	7	minimal	minimal	ADJ
ejpam-1781	91	8	prime	prime	ADJ
ejpam-1781	91	9	if	if	SCONJ
ejpam-1781	92	1	and	and	CCONJ
ejpam-1781	92	2	only	only	ADV
ejpam-1781	92	3	if	if	SCONJ
ejpam-1781	92	4	the	the	DET
ejpam-1781	92	5	ideals	ideal	NOUN
ejpam-1781	92	6	(	(	PUNCT
ejpam-1781	92	7	a	a	X
ejpam-1781	92	8	]	]	X
ejpam-1781	92	9	are	be	AUX
ejpam-1781	92	10	(	(	PUNCT
ejpam-1781	92	11	a]∗	a]∗	NOUN
ejpam-1781	92	12	are	be	AUX
ejpam-1781	92	13	co	co	ADJ
ejpam-1781	92	14	-	-	ADJ
ejpam-1781	92	15	maximal	maximal	ADJ
ejpam-1781	92	16	for	for	ADP
ejpam-1781	92	17	each	each	DET
ejpam-1781	92	18	a	a	DET
ejpam-1781	92	19	∈	∈	PROPN
ejpam-1781	92	20	r.	r.	NOUN
ejpam-1781	92	21	proof	proof	NOUN
ejpam-1781	92	22	.	.	PUNCT
ejpam-1781	93	1	let	let	VERB
ejpam-1781	93	2	every	every	DET
ejpam-1781	93	3	prime	prime	ADJ
ejpam-1781	93	4	ideal	ideal	NOUN
ejpam-1781	93	5	in	in	ADP
ejpam-1781	93	6	r	r	NOUN
ejpam-1781	93	7	be	be	AUX
ejpam-1781	93	8	minimal	minimal	ADJ
ejpam-1781	93	9	prime	prime	ADJ
ejpam-1781	93	10	.	.	PUNCT
ejpam-1781	94	1	let	let	VERB
ejpam-1781	94	2	if	if	SCONJ
ejpam-1781	94	3	possible	possible	ADJ
ejpam-1781	94	4	there	there	PRON
ejpam-1781	94	5	exists	exist	VERB
ejpam-1781	94	6	a	a	DET
ejpam-1781	94	7	∈	∈	NOUN
ejpam-1781	94	8	r	r	NOUN
ejpam-1781	94	9	such	such	ADJ
ejpam-1781	94	10	that	that	SCONJ
ejpam-1781	94	11	(	(	PUNCT
ejpam-1781	94	12	a	a	X
ejpam-1781	94	13	]	]	X
ejpam-1781	94	14	∨	∨	X
ejpam-1781	94	15	(	(	PUNCT
ejpam-1781	94	16	a]∗	a]∗	PROPN
ejpam-1781	94	17	⊂	⊂	PROPN
ejpam-1781	94	18	r.	r.	AUX
ejpam-1781	94	19	select	select	VERB
ejpam-1781	94	20	x	x	PUNCT
ejpam-1781	94	21	∈	∈	NOUN
ejpam-1781	94	22	r	r	NOUN
ejpam-1781	94	23	such	such	ADJ
ejpam-1781	94	24	that	that	PRON
ejpam-1781	94	25	x	x	SYM
ejpam-1781	94	26	/∈	/∈	PUNCT
ejpam-1781	95	1	(	(	PUNCT
ejpam-1781	95	2	a	a	X
ejpam-1781	95	3	]	]	X
ejpam-1781	95	4	∨	∨	NOUN
ejpam-1781	95	5	(	(	PUNCT
ejpam-1781	95	6	a]∗.	a]∗.	NOUN
ejpam-1781	95	7	hence	hence	ADV
ejpam-1781	95	8	by	by	ADP
ejpam-1781	95	9	result	result	NOUN
ejpam-1781	95	10	1	1	NUM
ejpam-1781	95	11	,	,	PUNCT
ejpam-1781	95	12	there	there	PRON
ejpam-1781	95	13	exists	exist	VERB
ejpam-1781	95	14	a	a	DET
ejpam-1781	95	15	prime	prime	ADJ
ejpam-1781	95	16	ideal	ideal	NOUN
ejpam-1781	95	17	,	,	PUNCT
ejpam-1781	95	18	say	say	VERB
ejpam-1781	95	19	p	p	X
ejpam-1781	95	20	,	,	PUNCT
ejpam-1781	95	21	in	in	ADP
ejpam-1781	95	22	r	r	NOUN
ejpam-1781	95	23	such	such	ADJ
ejpam-1781	95	24	that	that	SCONJ
ejpam-1781	96	1	[	[	X
ejpam-1781	96	2	(	(	PUNCT
ejpam-1781	96	3	a	a	X
ejpam-1781	96	4	]	]	X
ejpam-1781	96	5	∨	∨	X
ejpam-1781	96	6	(	(	PUNCT
ejpam-1781	96	7	a]∗	a]∗	PROPN
ejpam-1781	96	8	]	]	PUNCT
ejpam-1781	96	9	⊆	⊆	NUM
ejpam-1781	96	10	p	p	NOUN
ejpam-1781	96	11	and	and	CCONJ
ejpam-1781	96	12	p	p	NOUN
ejpam-1781	96	13	∩	∩	ADJ
ejpam-1781	96	14	[	[	X
ejpam-1781	96	15	x	x	X
ejpam-1781	96	16	)	)	PUNCT
ejpam-1781	96	17	=	=	SYM
ejpam-1781	96	18	;	;	PUNCT
ejpam-1781	96	19	.	.	PUNCT
ejpam-1781	97	1	p	p	X
ejpam-1781	97	2	being	be	AUX
ejpam-1781	97	3	minimal	minimal	ADJ
ejpam-1781	97	4	by	by	ADP
ejpam-1781	97	5	assumption	assumption	NOUN
ejpam-1781	97	6	,	,	PUNCT
ejpam-1781	97	7	(	(	PUNCT
ejpam-1781	97	8	a	a	X
ejpam-1781	97	9	]	]	X
ejpam-1781	97	10	and	and	CCONJ
ejpam-1781	97	11	(	(	PUNCT
ejpam-1781	97	12	a]∗	a]∗	PROPN
ejpam-1781	97	13	can	can	AUX
ejpam-1781	97	14	not	not	PART
ejpam-1781	97	15	be	be	AUX
ejpam-1781	97	16	contained	contain	VERB
ejpam-1781	97	17	in	in	ADP
ejpam-1781	97	18	p	p	NOUN
ejpam-1781	97	19	simultaneously	simultaneously	ADV
ejpam-1781	97	20	(	(	PUNCT
ejpam-1781	97	21	see	see	VERB
ejpam-1781	97	22	result	result	NOUN
ejpam-1781	97	23	2	2	NUM
ejpam-1781	97	24	)	)	PUNCT
ejpam-1781	97	25	.	.	PUNCT
ejpam-1781	98	1	this	this	PRON
ejpam-1781	98	2	in	in	ADP
ejpam-1781	98	3	turn	turn	NOUN
ejpam-1781	98	4	shows	show	VERB
ejpam-1781	98	5	that	that	SCONJ
ejpam-1781	98	6	r=	r=	ADJ
ejpam-1781	98	7	(	(	PUNCT
ejpam-1781	98	8	a]∨	a]∨	PROPN
ejpam-1781	98	9	(	(	PUNCT
ejpam-1781	98	10	a]∗	a]∗	NOUN
ejpam-1781	98	11	for	for	ADP
ejpam-1781	98	12	each	each	DET
ejpam-1781	98	13	a	a	DET
ejpam-1781	98	14	∈	∈	PROPN
ejpam-1781	98	15	r.	r.	NOUN
ejpam-1781	98	16	conversely	conversely	ADV
ejpam-1781	98	17	,	,	PUNCT
ejpam-1781	98	18	let	let	VERB
ejpam-1781	98	19	r	r	NOUN
ejpam-1781	98	20	=	=	SYM
ejpam-1781	98	21	(	(	PUNCT
ejpam-1781	98	22	a]∨	a]∨	PROPN
ejpam-1781	98	23	(	(	PUNCT
ejpam-1781	98	24	a]∗	a]∗	NOUN
ejpam-1781	98	25	for	for	ADP
ejpam-1781	98	26	each	each	DET
ejpam-1781	98	27	a	a	DET
ejpam-1781	98	28	∈	∈	PROPN
ejpam-1781	98	29	r.	r.	NOUN
ejpam-1781	98	30	let	let	VERB
ejpam-1781	98	31	if	if	SCONJ
ejpam-1781	98	32	possible	possible	ADJ
ejpam-1781	98	33	,	,	PUNCT
ejpam-1781	98	34	there	there	PRON
ejpam-1781	98	35	exists	exist	VERB
ejpam-1781	98	36	a	a	DET
ejpam-1781	98	37	prime	prime	ADJ
ejpam-1781	98	38	ideal	ideal	NOUN
ejpam-1781	98	39	p	p	NOUN
ejpam-1781	98	40	in	in	ADP
ejpam-1781	98	41	r	r	NOUN
ejpam-1781	98	42	which	which	PRON
ejpam-1781	98	43	is	be	AUX
ejpam-1781	98	44	not	not	PART
ejpam-1781	98	45	minimal	minimal	ADJ
ejpam-1781	98	46	.	.	PUNCT
ejpam-1781	99	1	by	by	ADP
ejpam-1781	99	2	result	result	NOUN
ejpam-1781	99	3	3	3	NUM
ejpam-1781	99	4	,	,	PUNCT
ejpam-1781	99	5	there	there	PRON
ejpam-1781	99	6	exists	exist	VERB
ejpam-1781	99	7	a	a	DET
ejpam-1781	99	8	minimal	minimal	ADJ
ejpam-1781	99	9	prime	prime	ADJ
ejpam-1781	99	10	ideal	ideal	NOUN
ejpam-1781	99	11	,	,	PUNCT
ejpam-1781	99	12	say	say	VERB
ejpam-1781	99	13	m	m	PRON
ejpam-1781	99	14	,	,	PUNCT
ejpam-1781	99	15	in	in	ADP
ejpam-1781	99	16	r	r	NOUN
ejpam-1781	99	17	such	such	ADJ
ejpam-1781	99	18	that	that	SCONJ
ejpam-1781	99	19	m	m	VERB
ejpam-1781	99	20	⊂	⊂	PROPN
ejpam-1781	99	21	p.	p.	NOUN
ejpam-1781	99	22	select	select	ADJ
ejpam-1781	99	23	x	x	PUNCT
ejpam-1781	99	24	∈	∈	PROPN
ejpam-1781	99	25	p	p	PROPN
ejpam-1781	99	26	\m	\m	NOUN
ejpam-1781	99	27	.	.	PUNCT
ejpam-1781	100	1	as	as	ADP
ejpam-1781	100	2	x	x	PROPN
ejpam-1781	100	3	/∈	/∈	PROPN
ejpam-1781	100	4	m	m	VERB
ejpam-1781	100	5	,	,	PUNCT
ejpam-1781	100	6	(	(	PUNCT
ejpam-1781	100	7	x]∗	x]∗	PROPN
ejpam-1781	100	8	⊆	⊆	NUM
ejpam-1781	100	9	m	m	PROPN
ejpam-1781	100	10	(	(	PUNCT
ejpam-1781	100	11	see	see	VERB
ejpam-1781	100	12	result	result	NOUN
ejpam-1781	100	13	2	2	NUM
ejpam-1781	100	14	)	)	PUNCT
ejpam-1781	100	15	.	.	PUNCT
ejpam-1781	101	1	but	but	CCONJ
ejpam-1781	101	2	then	then	ADV
ejpam-1781	101	3	(	(	PUNCT
ejpam-1781	101	4	x]⊆	x]⊆	PROPN
ejpam-1781	101	5	p	p	PROPN
ejpam-1781	101	6	and	and	CCONJ
ejpam-1781	101	7	(	(	PUNCT
ejpam-1781	101	8	x]∗	x]∗	PROPN
ejpam-1781	101	9	⊆	⊆	NUM
ejpam-1781	101	10	p	p	PROPN
ejpam-1781	101	11	will	will	AUX
ejpam-1781	101	12	give	give	VERB
ejpam-1781	101	13	r	r	NOUN
ejpam-1781	101	14	=	=	PUNCT
ejpam-1781	101	15	(	(	PUNCT
ejpam-1781	101	16	x]∨	x]∨	X
ejpam-1781	101	17	(	(	PUNCT
ejpam-1781	101	18	x]∗	x]∗	PROPN
ejpam-1781	101	19	⊆	⊆	NUM
ejpam-1781	101	20	p	p	X
ejpam-1781	101	21	;	;	PUNCT
ejpam-1781	101	22	a	a	DET
ejpam-1781	101	23	contradiction	contradiction	NOUN
ejpam-1781	101	24	.	.	PUNCT
ejpam-1781	102	1	hence	hence	ADV
ejpam-1781	102	2	every	every	DET
ejpam-1781	102	3	prime	prime	ADJ
ejpam-1781	102	4	ideal	ideal	NOUN
ejpam-1781	102	5	in	in	ADP
ejpam-1781	102	6	r	r	NOUN
ejpam-1781	102	7	must	must	AUX
ejpam-1781	102	8	be	be	AUX
ejpam-1781	102	9	minimal	minimal	ADJ
ejpam-1781	102	10	prime	prime	NOUN
ejpam-1781	102	11	.	.	PUNCT
ejpam-1781	103	1	using	use	VERB
ejpam-1781	103	2	theorem	theorem	ADJ
ejpam-1781	103	3	1	1	NUM
ejpam-1781	103	4	and	and	CCONJ
ejpam-1781	103	5	theorem	theorem	VERB
ejpam-1781	103	6	2	2	NUM
ejpam-1781	103	7	,	,	PUNCT
ejpam-1781	103	8	we	we	PRON
ejpam-1781	103	9	have	have	VERB
ejpam-1781	103	10	corollary	corollary	ADJ
ejpam-1781	103	11	1	1	NUM
ejpam-1781	103	12	.	.	PUNCT
ejpam-1781	104	1	following	follow	VERB
ejpam-1781	104	2	statements	statement	NOUN
ejpam-1781	104	3	are	be	AUX
ejpam-1781	104	4	equivalent	equivalent	ADJ
ejpam-1781	104	5	in	in	ADP
ejpam-1781	104	6	r	r	NOUN
ejpam-1781	104	7	1	1	NUM
ejpam-1781	104	8	.	.	PUNCT
ejpam-1781	105	1	every	every	DET
ejpam-1781	105	2	prime	prime	ADJ
ejpam-1781	105	3	ideal	ideal	NOUN
ejpam-1781	105	4	in	in	ADP
ejpam-1781	105	5	r	r	NOUN
ejpam-1781	105	6	is	be	AUX
ejpam-1781	105	7	minimal	minimal	ADJ
ejpam-1781	105	8	prime	prime	NOUN
ejpam-1781	105	9	.	.	PUNCT
ejpam-1781	106	1	2	2	X
ejpam-1781	106	2	.	.	X
ejpam-1781	106	3	the	the	DET
ejpam-1781	106	4	ideals	ideal	NOUN
ejpam-1781	106	5	(	(	PUNCT
ejpam-1781	106	6	a	a	X
ejpam-1781	106	7	]	]	PUNCT
ejpam-1781	106	8	and	and	CCONJ
ejpam-1781	106	9	(	(	PUNCT
ejpam-1781	106	10	a]∗	a]∗	PROPN
ejpam-1781	106	11	are	be	AUX
ejpam-1781	106	12	co	co	ADJ
ejpam-1781	106	13	-	-	ADJ
ejpam-1781	106	14	maximal	maximal	ADJ
ejpam-1781	106	15	,	,	PUNCT
ejpam-1781	106	16	for	for	ADP
ejpam-1781	106	17	each	each	DET
ejpam-1781	106	18	a	a	DET
ejpam-1781	106	19	∈	∈	PROPN
ejpam-1781	106	20	r.	r.	PROPN
ejpam-1781	106	21	3	3	NUM
ejpam-1781	106	22	.	.	PUNCT
ejpam-1781	107	1	(	(	PUNCT
ejpam-1781	107	2	a	a	X
ejpam-1781	107	3	]	]	X
ejpam-1781	107	4	=	=	SYM
ejpam-1781	107	5	(	(	PUNCT
ejpam-1781	107	6	a]∗∗	a]∗∗	PUNCT
ejpam-1781	107	7	and	and	CCONJ
ejpam-1781	107	8	the	the	DET
ejpam-1781	107	9	ideals	ideal	NOUN
ejpam-1781	107	10	(	(	PUNCT
ejpam-1781	107	11	a]∗	a]∗	PROPN
ejpam-1781	107	12	and	and	CCONJ
ejpam-1781	107	13	(	(	PUNCT
ejpam-1781	107	14	a]∗∗	a]∗∗	NOUN
ejpam-1781	107	15	are	be	AUX
ejpam-1781	107	16	co	co	ADJ
ejpam-1781	107	17	-	-	ADJ
ejpam-1781	107	18	maximal	maximal	ADJ
ejpam-1781	107	19	,	,	PUNCT
ejpam-1781	107	20	for	for	ADP
ejpam-1781	107	21	each	each	DET
ejpam-1781	107	22	a	a	DET
ejpam-1781	107	23	∈	∈	PROPN
ejpam-1781	107	24	r.	r.	NOUN
ejpam-1781	107	25	we	we	PRON
ejpam-1781	107	26	know	know	VERB
ejpam-1781	107	27	that	that	SCONJ
ejpam-1781	107	28	p	p	NOUN
ejpam-1781	107	29	is	be	AUX
ejpam-1781	107	30	a	a	DET
ejpam-1781	107	31	minimal	minimal	ADJ
ejpam-1781	107	32	prime	prime	ADJ
ejpam-1781	107	33	ideal	ideal	NOUN
ejpam-1781	107	34	of	of	ADP
ejpam-1781	107	35	r	r	NOUN
ejpam-1781	107	36	if	if	SCONJ
ejpam-1781	108	1	and	and	CCONJ
ejpam-1781	108	2	only	only	ADV
ejpam-1781	108	3	if	if	SCONJ
ejpam-1781	108	4	r	r	NOUN
ejpam-1781	108	5	\	\	PROPN
ejpam-1781	108	6	p	p	NOUN
ejpam-1781	108	7	is	be	AUX
ejpam-1781	108	8	a	a	DET
ejpam-1781	108	9	maximal	maximal	ADJ
ejpam-1781	108	10	filter	filter	NOUN
ejpam-1781	108	11	of	of	ADP
ejpam-1781	108	12	r	r	NOUN
ejpam-1781	108	13	(	(	PUNCT
ejpam-1781	108	14	see	see	VERB
ejpam-1781	108	15	result	result	NOUN
ejpam-1781	108	16	5	5	NUM
ejpam-1781	108	17	)	)	PUNCT
ejpam-1781	108	18	.	.	PUNCT
ejpam-1781	109	1	using	use	VERB
ejpam-1781	109	2	this	this	DET
ejpam-1781	109	3	relation	relation	NOUN
ejpam-1781	109	4	between	between	ADP
ejpam-1781	109	5	minimal	minimal	ADJ
ejpam-1781	109	6	prime	prime	ADJ
ejpam-1781	109	7	ideals	ideal	NOUN
ejpam-1781	109	8	and	and	CCONJ
ejpam-1781	109	9	maximal	maximal	ADJ
ejpam-1781	109	10	filters	filter	NOUN
ejpam-1781	109	11	of	of	ADP
ejpam-1781	109	12	r	r	NOUN
ejpam-1781	109	13	,	,	PUNCT
ejpam-1781	109	14	we	we	PRON
ejpam-1781	109	15	get	get	VERB
ejpam-1781	109	16	corollary	corollary	ADJ
ejpam-1781	109	17	2	2	NUM
ejpam-1781	109	18	.	.	PUNCT
ejpam-1781	109	19	following	follow	VERB
ejpam-1781	109	20	statements	statement	NOUN
ejpam-1781	109	21	are	be	AUX
ejpam-1781	109	22	equivalent	equivalent	ADJ
ejpam-1781	109	23	in	in	ADP
ejpam-1781	109	24	r	r	NOUN
ejpam-1781	109	25	with	with	ADP
ejpam-1781	109	26	maximal	maximal	ADJ
ejpam-1781	109	27	elements	element	NOUN
ejpam-1781	109	28	.	.	PUNCT
ejpam-1781	110	1	1	1	X
ejpam-1781	110	2	.	.	X
ejpam-1781	110	3	every	every	DET
ejpam-1781	110	4	prime	prime	ADJ
ejpam-1781	110	5	ideal	ideal	NOUN
ejpam-1781	110	6	in	in	ADP
ejpam-1781	110	7	r	r	NOUN
ejpam-1781	110	8	is	be	AUX
ejpam-1781	110	9	minimal	minimal	ADJ
ejpam-1781	110	10	prime	prime	NOUN
ejpam-1781	110	11	.	.	PUNCT
ejpam-1781	111	1	2	2	X
ejpam-1781	111	2	.	.	X
ejpam-1781	111	3	every	every	DET
ejpam-1781	111	4	prime	prime	ADJ
ejpam-1781	111	5	filter	filter	NOUN
ejpam-1781	111	6	in	in	ADP
ejpam-1781	111	7	r	r	NOUN
ejpam-1781	111	8	is	be	AUX
ejpam-1781	111	9	maximal	maximal	ADJ
ejpam-1781	111	10	.	.	PUNCT
ejpam-1781	112	1	3	3	X
ejpam-1781	112	2	.	.	X
ejpam-1781	112	3	every	every	DET
ejpam-1781	112	4	prime	prime	ADJ
ejpam-1781	112	5	filter	filter	NOUN
ejpam-1781	112	6	in	in	ADP
ejpam-1781	112	7	r	r	NOUN
ejpam-1781	112	8	is	be	AUX
ejpam-1781	112	9	minimal	minimal	ADJ
ejpam-1781	112	10	prime	prime	NOUN
ejpam-1781	112	11	.	.	PUNCT
ejpam-1781	113	1	it	it	PRON
ejpam-1781	113	2	is	be	AUX
ejpam-1781	113	3	well	well	ADV
ejpam-1781	113	4	known	know	VERB
ejpam-1781	113	5	that	that	SCONJ
ejpam-1781	113	6	a	a	DET
ejpam-1781	113	7	proper	proper	ADJ
ejpam-1781	113	8	ideal	ideal	NOUN
ejpam-1781	113	9	in	in	ADP
ejpam-1781	113	10	r	r	NOUN
ejpam-1781	113	11	need	need	AUX
ejpam-1781	113	12	not	not	PART
ejpam-1781	113	13	be	be	AUX
ejpam-1781	113	14	prime	prime	ADJ
ejpam-1781	113	15	.	.	PUNCT
ejpam-1781	114	1	a	a	DET
ejpam-1781	114	2	sufficient	sufficient	ADJ
ejpam-1781	114	3	condition	condition	NOUN
ejpam-1781	114	4	for	for	ADP
ejpam-1781	114	5	a	a	DET
ejpam-1781	114	6	proper	proper	ADJ
ejpam-1781	114	7	ideal	ideal	NOUN
ejpam-1781	114	8	in	in	ADP
ejpam-1781	114	9	r	r	NOUN
ejpam-1781	114	10	to	to	PART
ejpam-1781	114	11	be	be	AUX
ejpam-1781	114	12	prime	prime	ADJ
ejpam-1781	114	13	is	be	AUX
ejpam-1781	114	14	proved	prove	VERB
ejpam-1781	114	15	in	in	ADP
ejpam-1781	114	16	the	the	DET
ejpam-1781	114	17	following	following	NOUN
ejpam-1781	114	18	theorem	theorem	VERB
ejpam-1781	114	19	.	.	PUNCT
ejpam-1781	115	1	y.	y.	PROPN
ejpam-1781	115	2	pawar	pawar	PROPN
ejpam-1781	115	3	,	,	PUNCT
ejpam-1781	115	4	i.	i.	PROPN
ejpam-1781	115	5	shaikh	shaikh	PROPN
ejpam-1781	115	6	/	/	SYM
ejpam-1781	115	7	eur	eur	PROPN
ejpam-1781	115	8	.	.	PUNCT
ejpam-1781	116	1	j.	j.	PROPN
ejpam-1781	116	2	pure	pure	PROPN
ejpam-1781	116	3	appl	appl	PROPN
ejpam-1781	116	4	.	.	PROPN
ejpam-1781	116	5	math	math	PROPN
ejpam-1781	116	6	,	,	PUNCT
ejpam-1781	116	7	6	6	NUM
ejpam-1781	116	8	(	(	PUNCT
ejpam-1781	116	9	2013	2013	NUM
ejpam-1781	116	10	)	)	PUNCT
ejpam-1781	116	11	,	,	PUNCT
ejpam-1781	116	12	107	107	NUM
ejpam-1781	116	13	-	-	SYM
ejpam-1781	116	14	118	118	NUM
ejpam-1781	116	15	111	111	NUM
ejpam-1781	116	16	theorem	theorem	NOUN
ejpam-1781	116	17	3	3	NUM
ejpam-1781	116	18	.	.	PUNCT
ejpam-1781	116	19	a	a	DET
ejpam-1781	116	20	proper	proper	ADJ
ejpam-1781	116	21	ideal	ideal	NOUN
ejpam-1781	116	22	p	p	NOUN
ejpam-1781	116	23	in	in	ADP
ejpam-1781	116	24	r	r	NOUN
ejpam-1781	116	25	is	be	AUX
ejpam-1781	116	26	prime	prime	ADJ
ejpam-1781	116	27	,	,	PUNCT
ejpam-1781	116	28	if	if	SCONJ
ejpam-1781	116	29	the	the	DET
ejpam-1781	116	30	set	set	NOUN
ejpam-1781	116	31	{	{	PUNCT
ejpam-1781	116	32	i	i	NOUN
ejpam-1781	116	33	∈	∈	PROPN
ejpam-1781	116	34	i	i	PRON
ejpam-1781	116	35	(	(	PUNCT
ejpam-1781	116	36	r	r	NOUN
ejpam-1781	116	37	)	)	PUNCT
ejpam-1781	117	1	|	|	ADV
ejpam-1781	117	2	p	p	NOUN
ejpam-1781	117	3	⊆	⊆	NUM
ejpam-1781	117	4	i	i	PRON
ejpam-1781	117	5	}	}	PUNCT
ejpam-1781	117	6	is	be	AUX
ejpam-1781	117	7	a	a	DET
ejpam-1781	117	8	totally	totally	ADV
ejpam-1781	117	9	ordered	order	VERB
ejpam-1781	117	10	subset	subset	NOUN
ejpam-1781	117	11	of	of	ADP
ejpam-1781	117	12	i	i	PRON
ejpam-1781	117	13	(	(	PUNCT
ejpam-1781	117	14	r	r	NOUN
ejpam-1781	117	15	)	)	PUNCT
ejpam-1781	117	16	.	.	PUNCT
ejpam-1781	118	1	proof	proof	NOUN
ejpam-1781	118	2	.	.	PUNCT
ejpam-1781	119	1	let	let	VERB
ejpam-1781	119	2	a	a	DET
ejpam-1781	119	3	proper	proper	ADJ
ejpam-1781	119	4	ideal	ideal	NOUN
ejpam-1781	119	5	p	p	NOUN
ejpam-1781	119	6	in	in	ADP
ejpam-1781	119	7	r	r	NOUN
ejpam-1781	119	8	be	be	VERB
ejpam-1781	119	9	such	such	ADJ
ejpam-1781	119	10	that	that	SCONJ
ejpam-1781	119	11	the	the	DET
ejpam-1781	119	12	set	set	NOUN
ejpam-1781	119	13	{	{	PUNCT
ejpam-1781	119	14	i	i	NOUN
ejpam-1781	119	15	∈	∈	PROPN
ejpam-1781	119	16	i	i	PRON
ejpam-1781	119	17	(	(	PUNCT
ejpam-1781	119	18	r	r	NOUN
ejpam-1781	119	19	)	)	PUNCT
ejpam-1781	120	1	|	|	ADV
ejpam-1781	120	2	p	p	NOUN
ejpam-1781	120	3	⊆	⊆	NUM
ejpam-1781	120	4	i	i	PRON
ejpam-1781	120	5	}	}	PUNCT
ejpam-1781	120	6	is	be	AUX
ejpam-1781	120	7	a	a	DET
ejpam-1781	120	8	totally	totally	ADV
ejpam-1781	120	9	ordered	order	VERB
ejpam-1781	120	10	subset	subset	NOUN
ejpam-1781	120	11	of	of	ADP
ejpam-1781	120	12	i	i	PRON
ejpam-1781	120	13	(	(	PUNCT
ejpam-1781	120	14	r	r	NOUN
ejpam-1781	120	15	)	)	PUNCT
ejpam-1781	120	16	.	.	PUNCT
ejpam-1781	121	1	let	let	VERB
ejpam-1781	121	2	p	p	PRON
ejpam-1781	121	3	be	be	AUX
ejpam-1781	121	4	not	not	PART
ejpam-1781	121	5	prime	prime	ADJ
ejpam-1781	121	6	.	.	PUNCT
ejpam-1781	122	1	then	then	ADV
ejpam-1781	122	2	there	there	PRON
ejpam-1781	122	3	exists	exist	VERB
ejpam-1781	122	4	a	a	DET
ejpam-1781	122	5	,	,	PUNCT
ejpam-1781	122	6	b	b	X
ejpam-1781	122	7	∈	∈	NOUN
ejpam-1781	122	8	r	r	NOUN
ejpam-1781	122	9	such	such	ADJ
ejpam-1781	122	10	that	that	SCONJ
ejpam-1781	122	11	a	a	DET
ejpam-1781	122	12	∧	∧	PROPN
ejpam-1781	122	13	b	b	PROPN
ejpam-1781	122	14	∈	∈	PROPN
ejpam-1781	122	15	p	p	NOUN
ejpam-1781	122	16	with	with	ADP
ejpam-1781	122	17	a	a	DET
ejpam-1781	122	18	/∈	/∈	PUNCT
ejpam-1781	122	19	p	p	NOUN
ejpam-1781	122	20	and	and	CCONJ
ejpam-1781	122	21	b	b	NOUN
ejpam-1781	122	22	/∈	/∈	PROPN
ejpam-1781	122	23	p	p	X
ejpam-1781	122	24	.	.	PUNCT
ejpam-1781	123	1	as	as	ADP
ejpam-1781	123	2	p∨(a]⊃	p∨(a]⊃	PROPN
ejpam-1781	123	3	p	p	PROPN
ejpam-1781	123	4	and	and	CCONJ
ejpam-1781	123	5	p∨(b	p∨(b	PROPN
ejpam-1781	123	6	]	]	X
ejpam-1781	123	7	⊃	⊃	X
ejpam-1781	123	8	p	p	VERB
ejpam-1781	123	9	by	by	ADP
ejpam-1781	123	10	assumption	assumption	NOUN
ejpam-1781	123	11	p∨(a]⊆	p∨(a]⊆	PROPN
ejpam-1781	123	12	p∨(b	p∨(b	PROPN
ejpam-1781	123	13	]	]	PUNCT
ejpam-1781	123	14	or	or	CCONJ
ejpam-1781	123	15	p∨(b]⊆	p∨(b]⊆	PROPN
ejpam-1781	123	16	p∨(a	p∨(a	NOUN
ejpam-1781	123	17	]	]	PUNCT
ejpam-1781	123	18	.	.	PUNCT
ejpam-1781	124	1	let	let	VERB
ejpam-1781	124	2	us	we	PRON
ejpam-1781	124	3	assume	assume	VERB
ejpam-1781	124	4	without	without	ADP
ejpam-1781	124	5	loss	loss	NOUN
ejpam-1781	124	6	of	of	ADP
ejpam-1781	124	7	generality	generality	NOUN
ejpam-1781	124	8	p	p	PROPN
ejpam-1781	124	9	∨	∨	X
ejpam-1781	124	10	(	(	PUNCT
ejpam-1781	124	11	a]⊆	a]⊆	PROPN
ejpam-1781	124	12	p	p	NOUN
ejpam-1781	124	13	∨	∨	NOUN
ejpam-1781	124	14	(	(	PUNCT
ejpam-1781	124	15	b	b	NOUN
ejpam-1781	124	16	]	]	X
ejpam-1781	124	17	.	.	PUNCT
ejpam-1781	125	1	as	as	ADP
ejpam-1781	125	2	a	a	DET
ejpam-1781	125	3	∧	∧	PROPN
ejpam-1781	125	4	b	b	PROPN
ejpam-1781	125	5	∈	∈	PROPN
ejpam-1781	125	6	p	p	NOUN
ejpam-1781	125	7	,	,	PUNCT
ejpam-1781	125	8	we	we	PRON
ejpam-1781	125	9	get	get	VERB
ejpam-1781	125	10	p	p	NOUN
ejpam-1781	125	11	=	=	NOUN
ejpam-1781	125	12	p	p	NOUN
ejpam-1781	125	13	∨	∨	NOUN
ejpam-1781	125	14	(	(	PUNCT
ejpam-1781	125	15	a	a	DET
ejpam-1781	125	16	∧	∧	PROPN
ejpam-1781	125	17	b	b	PROPN
ejpam-1781	125	18	]	]	X
ejpam-1781	125	19	=	=	PUNCT
ejpam-1781	126	1	p	p	X
ejpam-1781	126	2	∨	∨	NUM
ejpam-1781	126	3	[	[	X
ejpam-1781	126	4	(	(	PUNCT
ejpam-1781	126	5	a]∧	a]∧	X
ejpam-1781	126	6	(	(	PUNCT
ejpam-1781	126	7	b	b	NOUN
ejpam-1781	126	8	]	]	X
ejpam-1781	126	9	]	]	X
ejpam-1781	126	10	(	(	PUNCT
ejpam-1781	126	11	by	by	ADP
ejpam-1781	126	12	result	result	NOUN
ejpam-1781	126	13	6	6	NUM
ejpam-1781	126	14	)	)	PUNCT
ejpam-1781	126	15	=[	=[	NOUN
ejpam-1781	126	16	p	p	NOUN
ejpam-1781	126	17	∨	∨	NOUN
ejpam-1781	126	18	(	(	PUNCT
ejpam-1781	126	19	a]]∧	a]]∧	VERB
ejpam-1781	126	20	[	[	X
ejpam-1781	126	21	p	p	X
ejpam-1781	126	22	∨	∨	X
ejpam-1781	126	23	(	(	PUNCT
ejpam-1781	126	24	b	b	NOUN
ejpam-1781	126	25	]	]	X
ejpam-1781	126	26	]	]	X
ejpam-1781	126	27	=	=	PUNCT
ejpam-1781	126	28	p	p	X
ejpam-1781	126	29	∨	∨	NOUN
ejpam-1781	126	30	(	(	PUNCT
ejpam-1781	126	31	a	a	X
ejpam-1781	126	32	]	]	X
ejpam-1781	126	33	(	(	PUNCT
ejpam-1781	126	34	since	since	SCONJ
ejpam-1781	126	35	p	p	ADJ
ejpam-1781	126	36	∨	∨	NOUN
ejpam-1781	126	37	(	(	PUNCT
ejpam-1781	126	38	a]⊆	a]⊆	PROPN
ejpam-1781	126	39	p	p	NOUN
ejpam-1781	126	40	∨	∨	NOUN
ejpam-1781	126	41	(	(	PUNCT
ejpam-1781	126	42	b	b	NOUN
ejpam-1781	126	43	]	]	X
ejpam-1781	126	44	)	)	PUNCT
ejpam-1781	126	45	this	this	PRON
ejpam-1781	126	46	shows	show	VERB
ejpam-1781	126	47	that	that	SCONJ
ejpam-1781	126	48	a	a	DET
ejpam-1781	126	49	∈	∈	PROPN
ejpam-1781	126	50	p	p	X
ejpam-1781	126	51	;	;	PUNCT
ejpam-1781	126	52	a	a	DET
ejpam-1781	126	53	contradiction	contradiction	NOUN
ejpam-1781	126	54	.	.	PUNCT
ejpam-1781	127	1	hence	hence	ADV
ejpam-1781	127	2	p	p	PRON
ejpam-1781	127	3	must	must	AUX
ejpam-1781	127	4	be	be	AUX
ejpam-1781	127	5	a	a	DET
ejpam-1781	127	6	prime	prime	ADJ
ejpam-1781	127	7	ideal	ideal	NOUN
ejpam-1781	127	8	.	.	PUNCT
ejpam-1781	128	1	for	for	ADP
ejpam-1781	128	2	a	a	DET
ejpam-1781	128	3	special	special	ADJ
ejpam-1781	128	4	subset	subset	NOUN
ejpam-1781	128	5	of	of	ADP
ejpam-1781	128	6	the	the	DET
ejpam-1781	128	7	set	set	PROPN
ejpam-1781	128	8	a0(r	a0(r	NOUN
ejpam-1781	128	9	)	)	PUNCT
ejpam-1781	128	10	of	of	ADP
ejpam-1781	128	11	all	all	DET
ejpam-1781	128	12	annulets	annulet	NOUN
ejpam-1781	128	13	of	of	ADP
ejpam-1781	128	14	r	r	NOUN
ejpam-1781	128	15	,	,	PUNCT
ejpam-1781	128	16	we	we	PRON
ejpam-1781	128	17	have	have	AUX
ejpam-1781	128	18	theorem	theorem	VERB
ejpam-1781	128	19	4	4	NUM
ejpam-1781	128	20	.	.	PUNCT
ejpam-1781	129	1	let	let	VERB
ejpam-1781	129	2	x	x	PRON
ejpam-1781	129	3	be	be	AUX
ejpam-1781	129	4	a	a	DET
ejpam-1781	129	5	non	non	ADJ
ejpam-1781	129	6	-	-	ADJ
ejpam-1781	129	7	empty	empty	ADJ
ejpam-1781	129	8	subset	subset	NOUN
ejpam-1781	129	9	of	of	ADP
ejpam-1781	129	10	r	r	NOUN
ejpam-1781	129	11	such	such	ADJ
ejpam-1781	129	12	that	that	DET
ejpam-1781	129	13	0	0	NUM
ejpam-1781	129	14	/∈	/∈	NUM
ejpam-1781	130	1	x	x	X
ejpam-1781	130	2	.	.	PUNCT
ejpam-1781	131	1	then	then	ADV
ejpam-1781	131	2	⋃	⋃	PROPN
ejpam-1781	131	3	�	�	PROPN
ejpam-1781	131	4	{	{	PUNCT
ejpam-1781	131	5	a}∗	a}∗	PROPN
ejpam-1781	131	6	|	|	ADV
ejpam-1781	131	7	a	a	DET
ejpam-1781	131	8	∈	∈	NOUN
ejpam-1781	131	9	x	x	X
ejpam-1781	131	10	=	=	SYM
ejpam-1781	131	11	⋂	⋂	PROPN
ejpam-1781	131	12	{	{	PUNCT
ejpam-1781	131	13	m	m	NOUN
ejpam-1781	131	14	∈m	∈m	NOUN
ejpam-1781	131	15	|m	|m	NOUN
ejpam-1781	131	16	∩	∩	NOUN
ejpam-1781	131	17	x	x	X
ejpam-1781	131	18	=	=	PUNCT
ejpam-1781	131	19	;	;	PUNCT
ejpam-1781	131	20	}	}	PUNCT
ejpam-1781	131	21	=	=	SYM
ejpam-1781	131	22	⋂	⋂	PROPN
ejpam-1781	131	23	�	�	PROPN
ejpam-1781	131	24	p	p	NOUN
ejpam-1781	131	25	∈	∈	PROPN
ejpam-1781	131	26	℘	℘	PROPN
ejpam-1781	131	27	|	|	ADV
ejpam-1781	131	28	p	p	NOUN
ejpam-1781	131	29	∩	∩	NOUN
ejpam-1781	131	30	x	x	X
ejpam-1781	131	31	=	=	PRON
ejpam-1781	131	32	;	;	PUNCT
ejpam-1781	131	33	.	.	PUNCT
ejpam-1781	132	1	proof	proof	NOUN
ejpam-1781	132	2	.	.	PUNCT
ejpam-1781	133	1	let	let	VERB
ejpam-1781	133	2	x	x	SYM
ejpam-1781	133	3	∈	∈	PROPN
ejpam-1781	133	4	⋃	⋃	PROPN
ejpam-1781	133	5	{	{	PUNCT
ejpam-1781	133	6	{	{	PUNCT
ejpam-1781	133	7	a}∗	a}∗	PROPN
ejpam-1781	133	8	|	|	ADV
ejpam-1781	133	9	a	a	DET
ejpam-1781	133	10	∈	∈	NOUN
ejpam-1781	133	11	x	x	PUNCT
ejpam-1781	133	12	}	}	PUNCT
ejpam-1781	133	13	.	.	PUNCT
ejpam-1781	134	1	then	then	ADV
ejpam-1781	134	2	x	x	X
ejpam-1781	134	3	∧	∧	NOUN
ejpam-1781	134	4	a	a	DET
ejpam-1781	134	5	=	=	X
ejpam-1781	134	6	0	0	NUM
ejpam-1781	134	7	for	for	ADP
ejpam-1781	134	8	some	some	DET
ejpam-1781	134	9	a	a	DET
ejpam-1781	134	10	∈	∈	NOUN
ejpam-1781	134	11	x	x	X
ejpam-1781	134	12	.	.	PUNCT
ejpam-1781	135	1	now	now	ADV
ejpam-1781	135	2	,	,	PUNCT
ejpam-1781	135	3	0	0	NUM
ejpam-1781	136	1	=	=	SYM
ejpam-1781	136	2	x	x	SYM
ejpam-1781	136	3	∧	∧	PROPN
ejpam-1781	136	4	a	a	DET
ejpam-1781	136	5	∈	∈	PROPN
ejpam-1781	136	6	m	m	VERB
ejpam-1781	136	7	for	for	ADP
ejpam-1781	136	8	m	m	NOUN
ejpam-1781	136	9	∈m	∈m	NOUN
ejpam-1781	136	10	with	with	ADP
ejpam-1781	136	11	m	m	NOUN
ejpam-1781	136	12	∩x	∩x	X
ejpam-1781	136	13	=	=	PUNCT
ejpam-1781	136	14	;	;	PUNCT
ejpam-1781	136	15	implies	imply	VERB
ejpam-1781	136	16	x	x	PUNCT
ejpam-1781	136	17	∈	∈	ADV
ejpam-1781	136	18	m	m	VERB
ejpam-1781	136	19	.	.	PUNCT
ejpam-1781	137	1	hence	hence	ADV
ejpam-1781	137	2	we	we	PRON
ejpam-1781	137	3	get	get	VERB
ejpam-1781	137	4	x	x	PUNCT
ejpam-1781	137	5	∈	∈	PROPN
ejpam-1781	137	6	⋂	⋂	PROPN
ejpam-1781	137	7	{	{	PUNCT
ejpam-1781	137	8	m	m	PROPN
ejpam-1781	137	9	∈m	∈m	NOUN
ejpam-1781	137	10	|m	|m	NOUN
ejpam-1781	137	11	∩	∩	NOUN
ejpam-1781	137	12	x	x	X
ejpam-1781	137	13	=	=	PUNCT
ejpam-1781	137	14	;	;	PUNCT
ejpam-1781	137	15	}	}	PUNCT
ejpam-1781	137	16	.	.	PUNCT
ejpam-1781	138	1	this	this	DET
ejpam-1781	138	2	shows⋃	shows⋃	ADJ
ejpam-1781	138	3	{	{	PUNCT
ejpam-1781	138	4	{	{	PUNCT
ejpam-1781	138	5	a}∗	a}∗	PROPN
ejpam-1781	138	6	|	|	ADV
ejpam-1781	138	7	a	a	DET
ejpam-1781	138	8	∈	∈	NOUN
ejpam-1781	138	9	x	x	SYM
ejpam-1781	138	10	}	}	PUNCT
ejpam-1781	138	11	⊆	⊆	NUM
ejpam-1781	138	12	⋂	⋂	PROPN
ejpam-1781	138	13	{	{	PUNCT
ejpam-1781	138	14	m	m	NOUN
ejpam-1781	138	15	∈m	∈m	NOUN
ejpam-1781	138	16	|m	|m	NOUN
ejpam-1781	138	17	∩	∩	NOUN
ejpam-1781	138	18	x	x	X
ejpam-1781	138	19	=	=	PUNCT
ejpam-1781	138	20	;	;	PUNCT
ejpam-1781	138	21	}	}	PUNCT
ejpam-1781	138	22	.	.	PUNCT
ejpam-1781	139	1	conversely	conversely	ADV
ejpam-1781	139	2	,	,	PUNCT
ejpam-1781	139	3	let	let	VERB
ejpam-1781	139	4	if	if	SCONJ
ejpam-1781	139	5	possible	possible	ADJ
ejpam-1781	139	6	,	,	PUNCT
ejpam-1781	139	7	there	there	PRON
ejpam-1781	139	8	exists	exist	VERB
ejpam-1781	139	9	x	x	X
ejpam-1781	139	10	∈	∈	PROPN
ejpam-1781	139	11	⋂	⋂	PROPN
ejpam-1781	139	12	{	{	PUNCT
ejpam-1781	139	13	m	m	PROPN
ejpam-1781	139	14	∈m	∈m	NOUN
ejpam-1781	139	15	|m	|m	NOUN
ejpam-1781	139	16	∩	∩	NOUN
ejpam-1781	139	17	x	x	X
ejpam-1781	139	18	=	=	PUNCT
ejpam-1781	139	19	;	;	PUNCT
ejpam-1781	139	20	}	}	PUNCT
ejpam-1781	139	21	such	such	ADJ
ejpam-1781	139	22	that	that	SCONJ
ejpam-1781	139	23	x	x	SYM
ejpam-1781	139	24	/∈	/∈	PUNCT
ejpam-1781	139	25	⋃	⋃	ADV
ejpam-1781	139	26	{	{	PUNCT
ejpam-1781	139	27	{	{	PUNCT
ejpam-1781	139	28	a}∗	a}∗	PROPN
ejpam-1781	139	29	|	|	ADV
ejpam-1781	139	30	a	a	DET
ejpam-1781	139	31	∈	∈	NOUN
ejpam-1781	139	32	x	x	PUNCT
ejpam-1781	139	33	}	}	PUNCT
ejpam-1781	139	34	.	.	PUNCT
ejpam-1781	140	1	then	then	ADV
ejpam-1781	140	2	x	x	X
ejpam-1781	140	3	∧	∧	PROPN
ejpam-1781	140	4	a	a	PRON
ejpam-1781	140	5	6=	6=	NOUN
ejpam-1781	140	6	0	0	NUM
ejpam-1781	140	7	for	for	ADP
ejpam-1781	140	8	each	each	DET
ejpam-1781	140	9	a	a	DET
ejpam-1781	140	10	∈	∈	NOUN
ejpam-1781	140	11	x	x	X
ejpam-1781	140	12	.	.	PUNCT
ejpam-1781	141	1	let	let	VERB
ejpam-1781	141	2	x	x	PUNCT
ejpam-1781	141	3	=	=	PRON
ejpam-1781	141	4	{	{	PUNCT
ejpam-1781	141	5	x	x	PART
ejpam-1781	141	6	∧	∧	NOUN
ejpam-1781	141	7	a	a	DET
ejpam-1781	141	8	|	|	NOUN
ejpam-1781	141	9	a	a	DET
ejpam-1781	141	10	∈	∈	NOUN
ejpam-1781	141	11	x	x	NOUN
ejpam-1781	141	12	}	}	PUNCT
ejpam-1781	141	13	.	.	PUNCT
ejpam-1781	142	1	then	then	ADV
ejpam-1781	142	2	as	as	SCONJ
ejpam-1781	142	3	0	0	NUM
ejpam-1781	142	4	/∈	/∈	NUM
ejpam-1781	142	5	x	x	X
ejpam-1781	142	6	,	,	PUNCT
ejpam-1781	142	7	[	[	X
ejpam-1781	142	8	x	x	X
ejpam-1781	142	9	)	)	PUNCT
ejpam-1781	142	10	is	be	AUX
ejpam-1781	142	11	a	a	DET
ejpam-1781	142	12	proper	proper	ADJ
ejpam-1781	142	13	filter	filter	NOUN
ejpam-1781	142	14	of	of	ADP
ejpam-1781	142	15	r.	r.	PROPN
ejpam-1781	142	16	hence	hence	ADV
ejpam-1781	142	17	it	it	PRON
ejpam-1781	142	18	must	must	AUX
ejpam-1781	142	19	be	be	AUX
ejpam-1781	142	20	contained	contain	VERB
ejpam-1781	142	21	in	in	ADP
ejpam-1781	142	22	some	some	DET
ejpam-1781	142	23	maximal	maximal	ADJ
ejpam-1781	142	24	filter	filter	NOUN
ejpam-1781	142	25	say	say	VERB
ejpam-1781	142	26	f	f	PROPN
ejpam-1781	142	27	in	in	ADP
ejpam-1781	142	28	r	r	NOUN
ejpam-1781	142	29	.	.	PUNCT
ejpam-1781	143	1	define	define	VERB
ejpam-1781	143	2	m	m	NOUN
ejpam-1781	143	3	=	=	SYM
ejpam-1781	143	4	r	r	NOUN
ejpam-1781	143	5	\	\	PROPN
ejpam-1781	143	6	f	f	NOUN
ejpam-1781	143	7	.	.	PUNCT
ejpam-1781	144	1	then	then	ADV
ejpam-1781	144	2	m	m	VERB
ejpam-1781	144	3	∈m	∈m	NOUN
ejpam-1781	144	4	(	(	PUNCT
ejpam-1781	144	5	see	see	VERB
ejpam-1781	144	6	result	result	NOUN
ejpam-1781	144	7	5	5	NUM
ejpam-1781	144	8	)	)	PUNCT
ejpam-1781	144	9	and	and	CCONJ
ejpam-1781	144	10	x	x	PUNCT
ejpam-1781	144	11	∩m	∩m	PROPN
ejpam-1781	144	12	=	=	X
ejpam-1781	144	13	;	;	PUNCT
ejpam-1781	144	14	.	.	PUNCT
ejpam-1781	145	1	thus	thus	ADV
ejpam-1781	145	2	m	m	PROPN
ejpam-1781	145	3	∈	∈	PROPN
ejpam-1781	145	4	⋂	⋂	PROPN
ejpam-1781	145	5	{	{	PUNCT
ejpam-1781	145	6	m	m	PROPN
ejpam-1781	145	7	∈m	∈m	NOUN
ejpam-1781	145	8	|m	|m	NOUN
ejpam-1781	145	9	∩	∩	NOUN
ejpam-1781	145	10	x	x	X
ejpam-1781	145	11	=	=	PUNCT
ejpam-1781	145	12	;	;	PUNCT
ejpam-1781	145	13	}	}	PUNCT
ejpam-1781	145	14	.	.	PUNCT
ejpam-1781	146	1	hence	hence	ADV
ejpam-1781	146	2	by	by	ADP
ejpam-1781	146	3	the	the	DET
ejpam-1781	146	4	choice	choice	NOUN
ejpam-1781	146	5	of	of	ADP
ejpam-1781	146	6	x	x	X
ejpam-1781	146	7	,	,	PUNCT
ejpam-1781	146	8	x	x	SYM
ejpam-1781	146	9	∈	∈	NOUN
ejpam-1781	146	10	m	m	NOUN
ejpam-1781	146	11	;	;	PUNCT
ejpam-1781	146	12	a	a	DET
ejpam-1781	146	13	contradiction	contradiction	NOUN
ejpam-1781	146	14	.	.	PUNCT
ejpam-1781	147	1	this	this	PRON
ejpam-1781	147	2	shows	show	VERB
ejpam-1781	147	3	that	that	SCONJ
ejpam-1781	147	4	⋂	⋂	PROPN
ejpam-1781	147	5	{	{	PUNCT
ejpam-1781	147	6	m	m	PROPN
ejpam-1781	147	7	∈m	∈m	NOUN
ejpam-1781	147	8	|m	|m	NOUN
ejpam-1781	147	9	∩	∩	NOUN
ejpam-1781	147	10	x	x	X
ejpam-1781	147	11	=	=	PUNCT
ejpam-1781	147	12	;	;	PUNCT
ejpam-1781	147	13	}	}	PUNCT
ejpam-1781	147	14	⊆	⊆	NUM
ejpam-1781	147	15	⋃	⋃	NOUN
ejpam-1781	147	16	{	{	PUNCT
ejpam-1781	147	17	{	{	PUNCT
ejpam-1781	147	18	a}∗	a}∗	PROPN
ejpam-1781	147	19	|	|	ADV
ejpam-1781	147	20	a	a	DET
ejpam-1781	147	21	∈	∈	NOUN
ejpam-1781	147	22	x	x	PUNCT
ejpam-1781	147	23	}	}	PUNCT
ejpam-1781	147	24	.	.	PUNCT
ejpam-1781	148	1	combining	combine	VERB
ejpam-1781	148	2	both	both	DET
ejpam-1781	148	3	the	the	DET
ejpam-1781	148	4	inclusions	inclusion	NOUN
ejpam-1781	148	5	,	,	PUNCT
ejpam-1781	148	6	we	we	PRON
ejpam-1781	148	7	get	get	VERB
ejpam-1781	148	8	⋃	⋃	NOUN
ejpam-1781	148	9	{	{	PUNCT
ejpam-1781	148	10	{	{	PUNCT
ejpam-1781	148	11	a}∗	a}∗	PROPN
ejpam-1781	148	12	|	|	ADV
ejpam-1781	148	13	a	a	DET
ejpam-1781	148	14	∈	∈	NOUN
ejpam-1781	148	15	x	x	PUNCT
ejpam-1781	148	16	}	}	PUNCT
ejpam-1781	148	17	=	=	SYM
ejpam-1781	148	18	⋂	⋂	PROPN
ejpam-1781	148	19	{	{	PUNCT
ejpam-1781	148	20	m	m	NOUN
ejpam-1781	148	21	∈m	∈m	NOUN
ejpam-1781	148	22	|m	|m	NOUN
ejpam-1781	148	23	∩	∩	NOUN
ejpam-1781	148	24	x	x	X
ejpam-1781	148	25	=	=	PUNCT
ejpam-1781	148	26	;	;	PUNCT
ejpam-1781	148	27	}	}	PUNCT
ejpam-1781	148	28	.	.	PUNCT
ejpam-1781	149	1	as⋂	as⋂	PROPN
ejpam-1781	149	2	{	{	PUNCT
ejpam-1781	149	3	m	m	VERB
ejpam-1781	149	4	∈m	∈m	ADJ
ejpam-1781	149	5	|m	|m	NOUN
ejpam-1781	149	6	∩	∩	NOUN
ejpam-1781	149	7	x	x	X
ejpam-1781	149	8	=	=	PUNCT
ejpam-1781	149	9	;	;	PUNCT
ejpam-1781	149	10	}	}	PUNCT
ejpam-1781	149	11	=	=	SYM
ejpam-1781	149	12	⋂	⋂	PROPN
ejpam-1781	149	13	�	�	PROPN
ejpam-1781	149	14	p	p	NOUN
ejpam-1781	149	15	∈	∈	PROPN
ejpam-1781	149	16	℘	℘	PROPN
ejpam-1781	149	17	|	|	ADV
ejpam-1781	149	18	p	p	NOUN
ejpam-1781	149	19	∩	∩	NOUN
ejpam-1781	149	20	x	x	X
ejpam-1781	149	21	=	=	PUNCT
ejpam-1781	149	22	;	;	PUNCT
ejpam-1781	149	23	holds	hold	VERB
ejpam-1781	149	24	always	always	ADV
ejpam-1781	149	25	,	,	PUNCT
ejpam-1781	149	26	the	the	DET
ejpam-1781	149	27	result	result	NOUN
ejpam-1781	149	28	follows	follow	VERB
ejpam-1781	149	29	.	.	PUNCT
ejpam-1781	150	1	recall	recall	VERB
ejpam-1781	150	2	that	that	SCONJ
ejpam-1781	150	3	an	an	DET
ejpam-1781	150	4	ideal	ideal	NOUN
ejpam-1781	150	5	i	i	PRON
ejpam-1781	150	6	of	of	ADP
ejpam-1781	150	7	r	r	NOUN
ejpam-1781	150	8	is	be	AUX
ejpam-1781	150	9	said	say	VERB
ejpam-1781	150	10	to	to	PART
ejpam-1781	150	11	be	be	AUX
ejpam-1781	150	12	an	an	DET
ejpam-1781	150	13	annihilator	annihilator	NOUN
ejpam-1781	150	14	ideal	ideal	NOUN
ejpam-1781	150	15	if	if	SCONJ
ejpam-1781	150	16	i	i	PRON
ejpam-1781	150	17	=	=	VERB
ejpam-1781	150	18	i∗∗.	i∗∗.	VERB
ejpam-1781	150	19	the	the	DET
ejpam-1781	150	20	set	set	NOUN
ejpam-1781	150	21	of	of	ADP
ejpam-1781	150	22	all	all	DET
ejpam-1781	150	23	annihilator	annihilator	NOUN
ejpam-1781	150	24	ideals	ideal	NOUN
ejpam-1781	150	25	in	in	ADP
ejpam-1781	150	26	r	r	NOUN
ejpam-1781	150	27	is	be	AUX
ejpam-1781	150	28	denoted	denote	VERB
ejpam-1781	150	29	by	by	ADP
ejpam-1781	150	30	a(r	a(r	NOUN
ejpam-1781	150	31	)	)	PUNCT
ejpam-1781	150	32	.	.	PUNCT
ejpam-1781	151	1	further	far	ADV
ejpam-1781	151	2	we	we	PRON
ejpam-1781	151	3	have	have	AUX
ejpam-1781	151	4	theorem	theorem	VERB
ejpam-1781	151	5	5	5	NUM
ejpam-1781	151	6	.	.	X
ejpam-1781	151	7	for	for	ADP
ejpam-1781	151	8	any	any	DET
ejpam-1781	151	9	prime	prime	ADJ
ejpam-1781	151	10	ideal	ideal	NOUN
ejpam-1781	151	11	p	p	NOUN
ejpam-1781	151	12	in	in	ADP
ejpam-1781	151	13	r	r	NOUN
ejpam-1781	151	14	,	,	PUNCT
ejpam-1781	151	15	consider	consider	VERB
ejpam-1781	151	16	the	the	DET
ejpam-1781	151	17	following	follow	VERB
ejpam-1781	151	18	statements	statement	NOUN
ejpam-1781	151	19	.	.	PUNCT
ejpam-1781	152	1	1	1	X
ejpam-1781	152	2	.	.	X
ejpam-1781	152	3	for	for	ADP
ejpam-1781	152	4	any	any	DET
ejpam-1781	152	5	i	i	PRON
ejpam-1781	152	6	∈	∈	PROPN
ejpam-1781	152	7	i	i	PRON
ejpam-1781	152	8	(	(	PUNCT
ejpam-1781	152	9	r	r	NOUN
ejpam-1781	152	10	)	)	PUNCT
ejpam-1781	152	11	,	,	PUNCT
ejpam-1781	152	12	i	i	PRON
ejpam-1781	152	13	and	and	CCONJ
ejpam-1781	152	14	p	p	NOUN
ejpam-1781	152	15	are	be	AUX
ejpam-1781	152	16	comparable	comparable	ADJ
ejpam-1781	152	17	.	.	PUNCT
ejpam-1781	153	1	2	2	X
ejpam-1781	153	2	.	.	X
ejpam-1781	153	3	for	for	ADP
ejpam-1781	153	4	any	any	DET
ejpam-1781	153	5	n	n	NOUN
ejpam-1781	153	6	6=	6=	NOUN
ejpam-1781	153	7	r	r	NOUN
ejpam-1781	153	8	in	in	ADP
ejpam-1781	153	9	a(r	a(r	NOUN
ejpam-1781	153	10	)	)	PUNCT
ejpam-1781	153	11	,	,	PUNCT
ejpam-1781	153	12	n	n	PROPN
ejpam-1781	153	13	⊆	⊆	NUM
ejpam-1781	153	14	p.	p.	NOUN
ejpam-1781	153	15	3	3	NUM
ejpam-1781	153	16	.	.	PUNCT
ejpam-1781	154	1	for	for	ADP
ejpam-1781	154	2	any	any	DET
ejpam-1781	154	3	m	m	NOUN
ejpam-1781	154	4	∈m	∈m	NOUN
ejpam-1781	154	5	,	,	PUNCT
ejpam-1781	154	6	m	m	VERB
ejpam-1781	154	7	⊆	⊆	NUM
ejpam-1781	154	8	p.	p.	NOUN
ejpam-1781	154	9	4	4	NUM
ejpam-1781	154	10	.	.	PUNCT
ejpam-1781	155	1	for	for	ADP
ejpam-1781	155	2	any	any	PRON
ejpam-1781	155	3	x	x	NOUN
ejpam-1781	155	4	/∈	/∈	PUNCT
ejpam-1781	156	1	p	p	X
ejpam-1781	156	2	,	,	PUNCT
ejpam-1781	156	3	{	{	PUNCT
ejpam-1781	156	4	x}∗	x}∗	NOUN
ejpam-1781	156	5	=	=	SYM
ejpam-1781	156	6	{	{	PUNCT
ejpam-1781	156	7	0	0	NUM
ejpam-1781	156	8	}	}	PUNCT
ejpam-1781	156	9	.	.	PUNCT
ejpam-1781	157	1	then	then	ADV
ejpam-1781	157	2	1⇒	1⇒	PROPN
ejpam-1781	157	3	2⇒	2⇒	NUM
ejpam-1781	157	4	3⇒	3⇒	NUM
ejpam-1781	157	5	4	4	NUM
ejpam-1781	157	6	.	.	PUNCT
ejpam-1781	158	1	y.	y.	PROPN
ejpam-1781	158	2	pawar	pawar	PROPN
ejpam-1781	158	3	,	,	PUNCT
ejpam-1781	158	4	i.	i.	PROPN
ejpam-1781	158	5	shaikh	shaikh	PROPN
ejpam-1781	158	6	/	/	SYM
ejpam-1781	158	7	eur	eur	PROPN
ejpam-1781	158	8	.	.	PUNCT
ejpam-1781	159	1	j.	j.	PROPN
ejpam-1781	159	2	pure	pure	PROPN
ejpam-1781	159	3	appl	appl	PROPN
ejpam-1781	159	4	.	.	PROPN
ejpam-1781	159	5	math	math	PROPN
ejpam-1781	159	6	,	,	PUNCT
ejpam-1781	159	7	6	6	NUM
ejpam-1781	159	8	(	(	PUNCT
ejpam-1781	159	9	2013	2013	NUM
ejpam-1781	159	10	)	)	PUNCT
ejpam-1781	159	11	,	,	PUNCT
ejpam-1781	159	12	107	107	NUM
ejpam-1781	159	13	-	-	SYM
ejpam-1781	159	14	118	118	NUM
ejpam-1781	159	15	112	112	NUM
ejpam-1781	159	16	proof	proof	NOUN
ejpam-1781	159	17	.	.	PUNCT
ejpam-1781	160	1	1⇒	1⇒	NOUN
ejpam-1781	160	2	2	2	NUM
ejpam-1781	160	3	let	let	VERB
ejpam-1781	160	4	if	if	SCONJ
ejpam-1781	160	5	possible	possible	ADJ
ejpam-1781	160	6	,	,	PUNCT
ejpam-1781	160	7	there	there	PRON
ejpam-1781	160	8	exist	exist	VERB
ejpam-1781	160	9	n	n	PRON
ejpam-1781	160	10	6=	6=	NUM
ejpam-1781	160	11	r	r	NOUN
ejpam-1781	160	12	in	in	ADP
ejpam-1781	160	13	a(r	a(r	NOUN
ejpam-1781	160	14	)	)	PUNCT
ejpam-1781	160	15	such	such	ADJ
ejpam-1781	160	16	that	that	SCONJ
ejpam-1781	160	17	n	n	PROPN
ejpam-1781	160	18	*	*	PUNCT
ejpam-1781	160	19	p.	p.	NOUN
ejpam-1781	160	20	hence	hence	ADV
ejpam-1781	160	21	by	by	ADP
ejpam-1781	160	22	(	(	PUNCT
ejpam-1781	160	23	1	1	X
ejpam-1781	160	24	)	)	PUNCT
ejpam-1781	160	25	p	p	X
ejpam-1781	160	26	⊂	⊂	PROPN
ejpam-1781	160	27	n	n	PROPN
ejpam-1781	160	28	.	.	PUNCT
ejpam-1781	161	1	select	select	ADJ
ejpam-1781	161	2	x	x	SYM
ejpam-1781	161	3	∈	∈	PROPN
ejpam-1781	161	4	n	n	PRON
ejpam-1781	161	5	\	\	NOUN
ejpam-1781	161	6	p.	p.	NOUN
ejpam-1781	161	7	as	as	SCONJ
ejpam-1781	161	8	p	p	PROPN
ejpam-1781	161	9	is	be	AUX
ejpam-1781	161	10	a	a	DET
ejpam-1781	161	11	prime	prime	ADJ
ejpam-1781	161	12	ideal	ideal	NOUN
ejpam-1781	161	13	and	and	CCONJ
ejpam-1781	161	14	x	x	NOUN
ejpam-1781	161	15	/∈	/∈	PUNCT
ejpam-1781	162	1	p	p	X
ejpam-1781	162	2	,	,	PUNCT
ejpam-1781	162	3	we	we	PRON
ejpam-1781	162	4	get	get	VERB
ejpam-1781	163	1	n	n	DET
ejpam-1781	163	2	∗	∗	NOUN
ejpam-1781	163	3	⊆	⊆	NUM
ejpam-1781	163	4	{	{	PUNCT
ejpam-1781	163	5	x}∗	x}∗	NOUN
ejpam-1781	163	6	⊆	⊆	NUM
ejpam-1781	163	7	p	p	NOUN
ejpam-1781	163	8	⊆	⊆	NUM
ejpam-1781	163	9	n	n	NOUN
ejpam-1781	163	10	.	.	PUNCT
ejpam-1781	164	1	this	this	PRON
ejpam-1781	164	2	in	in	ADP
ejpam-1781	164	3	turn	turn	NOUN
ejpam-1781	164	4	implies	imply	VERB
ejpam-1781	164	5	that	that	SCONJ
ejpam-1781	164	6	n	n	NOUN
ejpam-1781	164	7	∗	∗	NOUN
ejpam-1781	164	8	=	=	SYM
ejpam-1781	164	9	{	{	PUNCT
ejpam-1781	164	10	0	0	NUM
ejpam-1781	164	11	}	}	PUNCT
ejpam-1781	164	12	;	;	PUNCT
ejpam-1781	164	13	and	and	CCONJ
ejpam-1781	164	14	hence	hence	ADV
ejpam-1781	164	15	n	n	NOUN
ejpam-1781	164	16	=	=	SYM
ejpam-1781	164	17	n	n	PRON
ejpam-1781	164	18	∗∗	∗∗	NOUN
ejpam-1781	164	19	=	=	SYM
ejpam-1781	164	20	r	r	NOUN
ejpam-1781	164	21	contradicting	contradict	VERB
ejpam-1781	164	22	the	the	DET
ejpam-1781	164	23	fact	fact	NOUN
ejpam-1781	164	24	that	that	SCONJ
ejpam-1781	164	25	n	n	PRON
ejpam-1781	164	26	6=	6=	ADP
ejpam-1781	164	27	r.	r.	NOUN
ejpam-1781	164	28	hence	hence	ADV
ejpam-1781	164	29	n	n	NOUN
ejpam-1781	164	30	⊆	⊆	NUM
ejpam-1781	164	31	p	p	NOUN
ejpam-1781	164	32	for	for	ADP
ejpam-1781	164	33	each	each	DET
ejpam-1781	164	34	n	n	NOUN
ejpam-1781	164	35	6=	6=	NOUN
ejpam-1781	164	36	r	r	NOUN
ejpam-1781	164	37	in	in	ADP
ejpam-1781	164	38	n(r	n(r	NOUN
ejpam-1781	164	39	)	)	PUNCT
ejpam-1781	164	40	.	.	PUNCT
ejpam-1781	165	1	2⇒	2⇒	NOUN
ejpam-1781	165	2	3	3	NUM
ejpam-1781	165	3	let	let	VERB
ejpam-1781	165	4	m	m	PRON
ejpam-1781	165	5	∈m	∈m	NOUN
ejpam-1781	165	6	.	.	PUNCT
ejpam-1781	166	1	define	define	VERB
ejpam-1781	166	2	x	x	X
ejpam-1781	166	3	=	=	SYM
ejpam-1781	166	4	l	l	NOUN
ejpam-1781	166	5	\m	\m	NOUN
ejpam-1781	166	6	.	.	PUNCT
ejpam-1781	167	1	then	then	ADV
ejpam-1781	167	2	by	by	ADP
ejpam-1781	167	3	theorem	theorem	NOUN
ejpam-1781	167	4	4	4	NUM
ejpam-1781	167	5	we	we	PRON
ejpam-1781	167	6	have	have	VERB
ejpam-1781	167	7	⋃	⋃	ADP
ejpam-1781	167	8	�	�	PROPN
ejpam-1781	167	9	{	{	PUNCT
ejpam-1781	167	10	a}∗	a}∗	PROPN
ejpam-1781	167	11	|	|	ADV
ejpam-1781	167	12	a	a	DET
ejpam-1781	167	13	∈	∈	NOUN
ejpam-1781	167	14	x	x	X
ejpam-1781	167	15	=	=	SYM
ejpam-1781	167	16	⋂	⋂	PROPN
ejpam-1781	167	17	{	{	PUNCT
ejpam-1781	167	18	m	m	NOUN
ejpam-1781	167	19	∈m	∈m	NOUN
ejpam-1781	167	20	|m	|m	NOUN
ejpam-1781	167	21	∩	∩	NOUN
ejpam-1781	167	22	x	x	X
ejpam-1781	167	23	=	=	PUNCT
ejpam-1781	167	24	;	;	PUNCT
ejpam-1781	167	25	}	}	PUNCT
ejpam-1781	167	26	.	.	PUNCT
ejpam-1781	168	1	hence	hence	ADV
ejpam-1781	168	2	⋃	⋃	PROPN
ejpam-1781	168	3	{	{	PUNCT
ejpam-1781	168	4	{	{	PUNCT
ejpam-1781	168	5	a}∗	a}∗	INTJ
ejpam-1781	168	6	|	|	ADV
ejpam-1781	168	7	a	a	PRON
ejpam-1781	168	8	/∈	/∈	NOUN
ejpam-1781	168	9	m	m	VERB
ejpam-1781	168	10	}	}	PUNCT
ejpam-1781	168	11	=	=	ADJ
ejpam-1781	168	12	m	m	NOUN
ejpam-1781	168	13	.	.	PUNCT
ejpam-1781	169	1	now	now	ADV
ejpam-1781	169	2	a	a	DET
ejpam-1781	169	3	/∈	/∈	NOUN
ejpam-1781	169	4	m	m	AUX
ejpam-1781	169	5	⇒	⇒	NOUN
ejpam-1781	169	6	{	{	PUNCT
ejpam-1781	169	7	a}∗	a}∗	PROPN
ejpam-1781	169	8	6=	6=	PROPN
ejpam-1781	169	9	r.	r.	NOUN
ejpam-1781	169	10	hence	hence	ADV
ejpam-1781	169	11	by	by	ADP
ejpam-1781	169	12	(	(	PUNCT
ejpam-1781	169	13	2	2	NUM
ejpam-1781	169	14	)	)	PUNCT
ejpam-1781	169	15	,	,	PUNCT
ejpam-1781	169	16	{	{	PUNCT
ejpam-1781	169	17	a}∗	a}∗	PROPN
ejpam-1781	169	18	⊆	⊆	NUM
ejpam-1781	169	19	p	p	NOUN
ejpam-1781	169	20	for	for	ADP
ejpam-1781	169	21	each	each	DET
ejpam-1781	169	22	a	a	PRON
ejpam-1781	169	23	/∈	/∈	INTJ
ejpam-1781	169	24	m	m	VERB
ejpam-1781	169	25	.	.	PUNCT
ejpam-1781	170	1	this	this	PRON
ejpam-1781	170	2	gives	give	VERB
ejpam-1781	170	3	m	m	PRON
ejpam-1781	170	4	=	=	PUNCT
ejpam-1781	170	5	⋃	⋃	NOUN
ejpam-1781	170	6	{	{	PUNCT
ejpam-1781	170	7	{	{	PUNCT
ejpam-1781	170	8	a}∗	a}∗	INTJ
ejpam-1781	170	9	|	|	ADV
ejpam-1781	170	10	a	a	DET
ejpam-1781	170	11	/∈	/∈	NOUN
ejpam-1781	170	12	m	m	VERB
ejpam-1781	170	13	}	}	PUNCT
ejpam-1781	170	14	⊆	⊆	NUM
ejpam-1781	170	15	p	p	NOUN
ejpam-1781	170	16	and	and	CCONJ
ejpam-1781	170	17	the	the	DET
ejpam-1781	170	18	implication	implication	NOUN
ejpam-1781	170	19	follows	follow	VERB
ejpam-1781	170	20	.	.	PUNCT
ejpam-1781	171	1	3⇒	3⇒	NUM
ejpam-1781	171	2	4	4	NUM
ejpam-1781	171	3	let	let	VERB
ejpam-1781	171	4	a	a	DET
ejpam-1781	171	5	/∈	/∈	PUNCT
ejpam-1781	171	6	p.	p.	NOUN
ejpam-1781	171	7	by	by	ADP
ejpam-1781	171	8	assumption	assumption	NOUN
ejpam-1781	171	9	7	7	NUM
ejpam-1781	171	10	,	,	PUNCT
ejpam-1781	171	11	m	m	VERB
ejpam-1781	171	12	⊆	⊆	NUM
ejpam-1781	171	13	p	p	NOUN
ejpam-1781	171	14	for	for	ADP
ejpam-1781	171	15	each	each	DET
ejpam-1781	171	16	m	m	NOUN
ejpam-1781	171	17	∈m	∈m	NOUN
ejpam-1781	171	18	.	.	PUNCT
ejpam-1781	172	1	hence	hence	ADV
ejpam-1781	172	2	a	a	DET
ejpam-1781	172	3	/∈	/∈	NOUN
ejpam-1781	172	4	m	m	VERB
ejpam-1781	172	5	for	for	ADP
ejpam-1781	172	6	each	each	DET
ejpam-1781	172	7	m	m	NOUN
ejpam-1781	172	8	∈m	∈m	NOUN
ejpam-1781	172	9	.	.	PUNCT
ejpam-1781	173	1	but	but	CCONJ
ejpam-1781	173	2	then	then	ADV
ejpam-1781	173	3	{	{	PUNCT
ejpam-1781	173	4	a}∗	a}∗	PROPN
ejpam-1781	173	5	⊆	⊆	NUM
ejpam-1781	173	6	m	m	NOUN
ejpam-1781	173	7	for	for	ADP
ejpam-1781	173	8	each	each	DET
ejpam-1781	173	9	m	m	NOUN
ejpam-1781	173	10	∈m	∈m	NOUN
ejpam-1781	173	11	(	(	PUNCT
ejpam-1781	173	12	see	see	VERB
ejpam-1781	173	13	result	result	NOUN
ejpam-1781	173	14	2	2	NUM
ejpam-1781	173	15	)	)	PUNCT
ejpam-1781	173	16	will	will	AUX
ejpam-1781	173	17	give	give	VERB
ejpam-1781	173	18	{	{	PUNCT
ejpam-1781	173	19	a}∗	a}∗	PROPN
ejpam-1781	173	20	∩	∩	NOUN
ejpam-1781	173	21	{	{	PUNCT
ejpam-1781	173	22	m	m	PROPN
ejpam-1781	173	23	|m	|m	NOUN
ejpam-1781	173	24	∈m	∈m	NOUN
ejpam-1781	173	25	}	}	PUNCT
ejpam-1781	173	26	=	=	PUNCT
ejpam-1781	173	27	{	{	PUNCT
ejpam-1781	173	28	0	0	NUM
ejpam-1781	173	29	}	}	PUNCT
ejpam-1781	173	30	(	(	PUNCT
ejpam-1781	173	31	see	see	VERB
ejpam-1781	173	32	result	result	NOUN
ejpam-1781	173	33	7	7	NUM
ejpam-1781	173	34	)	)	PUNCT
ejpam-1781	173	35	.	.	PUNCT
ejpam-1781	174	1	thus	thus	ADV
ejpam-1781	174	2	we	we	PRON
ejpam-1781	174	3	get	get	VERB
ejpam-1781	174	4	1⇒	1⇒	PROPN
ejpam-1781	174	5	2⇒	2⇒	NUM
ejpam-1781	174	6	3⇒	3⇒	NUM
ejpam-1781	174	7	4	4	NUM
ejpam-1781	174	8	.	.	PUNCT
ejpam-1781	174	9	theorem	theorem	VERB
ejpam-1781	174	10	6	6	NUM
ejpam-1781	174	11	.	.	PUNCT
ejpam-1781	175	1	the	the	DET
ejpam-1781	175	2	statements	statement	NOUN
ejpam-1781	175	3	of	of	ADP
ejpam-1781	175	4	theorem	theorem	ADJ
ejpam-1781	175	5	5	5	NUM
ejpam-1781	175	6	are	be	AUX
ejpam-1781	175	7	equivalent	equivalent	ADJ
ejpam-1781	175	8	if	if	SCONJ
ejpam-1781	175	9	r	r	NOUN
ejpam-1781	175	10	satisfies	satisfie	NOUN
ejpam-1781	175	11	following	follow	VERB
ejpam-1781	175	12	condition	condition	NOUN
ejpam-1781	175	13	(	(	PUNCT
ejpam-1781	175	14	*	*	NOUN
ejpam-1781	175	15	)	)	PUNCT
ejpam-1781	175	16	.	.	PUNCT
ejpam-1781	176	1	(	(	PUNCT
ejpam-1781	176	2	*	*	PUNCT
ejpam-1781	176	3	)	)	PUNCT
ejpam-1781	176	4	for	for	ADP
ejpam-1781	176	5	any	any	PRON
ejpam-1781	176	6	i	i	PROPN
ejpam-1781	176	7	ver	ver	NOUN
ejpam-1781	176	8	tj	tj	NOUN
ejpam-1781	176	9	,	,	PUNCT
ejpam-1781	176	10	i	i	PRON
ejpam-1781	176	11	,	,	PUNCT
ejpam-1781	176	12	j	j	PROPN
ejpam-1781	176	13	∈	∈	PROPN
ejpam-1781	177	1	i	i	PRON
ejpam-1781	177	2	(	(	PUNCT
ejpam-1781	177	3	r	r	NOUN
ejpam-1781	177	4	)	)	PUNCT
ejpam-1781	177	5	,	,	PUNCT
ejpam-1781	177	6	there	there	PRON
ejpam-1781	177	7	exists	exist	VERB
ejpam-1781	177	8	x	x	X
ejpam-1781	177	9	∈	∈	PROPN
ejpam-1781	178	1	i	i	NOUN
ejpam-1781	178	2	\	\	NOUN
ejpam-1781	178	3	j	j	PROPN
ejpam-1781	178	4	and	and	CCONJ
ejpam-1781	178	5	y	y	PROPN
ejpam-1781	178	6	∈	∈	PROPN
ejpam-1781	179	1	j	j	PROPN
ejpam-1781	179	2	\	\	PROPN
ejpam-1781	180	1	i	i	PRON
ejpam-1781	180	2	such	such	ADJ
ejpam-1781	180	3	that	that	SCONJ
ejpam-1781	180	4	x	x	PUNCT
ejpam-1781	180	5	∧	∧	NOUN
ejpam-1781	180	6	y	y	NOUN
ejpam-1781	180	7	=	=	SYM
ejpam-1781	180	8	0	0	PROPN
ejpam-1781	180	9	.	.	PUNCT
ejpam-1781	181	1	proof	proof	NOUN
ejpam-1781	181	2	.	.	PUNCT
ejpam-1781	182	1	to	to	PART
ejpam-1781	182	2	prove	prove	VERB
ejpam-1781	182	3	that	that	SCONJ
ejpam-1781	182	4	conditions	condition	NOUN
ejpam-1781	182	5	are	be	AUX
ejpam-1781	182	6	equivalent	equivalent	ADJ
ejpam-1781	182	7	in	in	ADP
ejpam-1781	182	8	r	r	NOUN
ejpam-1781	182	9	,	,	PUNCT
ejpam-1781	182	10	it	it	PRON
ejpam-1781	182	11	is	be	AUX
ejpam-1781	182	12	enough	enough	ADJ
ejpam-1781	182	13	to	to	PART
ejpam-1781	182	14	prove	prove	VERB
ejpam-1781	182	15	that	that	SCONJ
ejpam-1781	182	16	4⇒	4⇒	NUM
ejpam-1781	182	17	1	1	NUM
ejpam-1781	182	18	under	under	ADP
ejpam-1781	182	19	the	the	DET
ejpam-1781	182	20	condition	condition	NOUN
ejpam-1781	182	21	(	(	PUNCT
ejpam-1781	182	22	*	*	NOUN
ejpam-1781	182	23	)	)	PUNCT
ejpam-1781	182	24	.	.	PUNCT
ejpam-1781	183	1	let	let	VERB
ejpam-1781	183	2	there	there	PRON
ejpam-1781	183	3	exist	exist	VERB
ejpam-1781	183	4	an	an	DET
ejpam-1781	183	5	ideal	ideal	NOUN
ejpam-1781	184	1	i	i	PRON
ejpam-1781	184	2	∈	∈	PROPN
ejpam-1781	185	1	i	i	PRON
ejpam-1781	185	2	(	(	PUNCT
ejpam-1781	185	3	r	r	NOUN
ejpam-1781	185	4	)	)	PUNCT
ejpam-1781	185	5	such	such	ADJ
ejpam-1781	185	6	that	that	SCONJ
ejpam-1781	185	7	i	i	PRON
ejpam-1781	185	8	ver	ver	VERB
ejpam-1781	185	9	tp	tp	PART
ejpam-1781	185	10	,	,	PUNCT
ejpam-1781	185	11	by	by	ADP
ejpam-1781	185	12	condition	condition	NOUN
ejpam-1781	185	13	(	(	PUNCT
ejpam-1781	185	14	*	*	NOUN
ejpam-1781	185	15	)	)	PUNCT
ejpam-1781	185	16	select	select	ADJ
ejpam-1781	185	17	x	x	PUNCT
ejpam-1781	185	18	∈	∈	PROPN
ejpam-1781	186	1	i	i	NOUN
ejpam-1781	186	2	\	\	PROPN
ejpam-1781	187	1	p	p	PROPN
ejpam-1781	187	2	and	and	CCONJ
ejpam-1781	187	3	y	y	PROPN
ejpam-1781	187	4	∈	∈	PROPN
ejpam-1781	188	1	p	p	X
ejpam-1781	188	2	\	\	PROPN
ejpam-1781	189	1	i	i	PRON
ejpam-1781	189	2	such	such	ADJ
ejpam-1781	189	3	that	that	SCONJ
ejpam-1781	189	4	x	x	PUNCT
ejpam-1781	189	5	∧	∧	NOUN
ejpam-1781	189	6	y	y	NOUN
ejpam-1781	189	7	=	=	NOUN
ejpam-1781	189	8	0	0	PROPN
ejpam-1781	189	9	.	.	PUNCT
ejpam-1781	190	1	then	then	ADV
ejpam-1781	190	2	x	x	X
ejpam-1781	190	3	>	>	X
ejpam-1781	190	4	0	0	PROPN
ejpam-1781	190	5	,	,	PUNCT
ejpam-1781	190	6	y	y	PROPN
ejpam-1781	190	7	>	>	X
ejpam-1781	190	8	0	0	PUNCT
ejpam-1781	191	1	and	and	CCONJ
ejpam-1781	191	2	y	y	PROPN
ejpam-1781	191	3	∈	∈	PROPN
ejpam-1781	191	4	{	{	PUNCT
ejpam-1781	191	5	x}∗.	x}∗.	VERB
ejpam-1781	191	6	again	again	ADV
ejpam-1781	191	7	by	by	ADP
ejpam-1781	191	8	assumption	assumption	NOUN
ejpam-1781	191	9	,	,	PUNCT
ejpam-1781	191	10	x	x	PUNCT
ejpam-1781	191	11	∈	∈	PROPN
ejpam-1781	191	12	p	p	NOUN
ejpam-1781	191	13	implies	imply	VERB
ejpam-1781	191	14	{	{	PUNCT
ejpam-1781	191	15	x}∗	x}∗	NOUN
ejpam-1781	191	16	=	=	SYM
ejpam-1781	191	17	{	{	PUNCT
ejpam-1781	191	18	0	0	NUM
ejpam-1781	191	19	}	}	PUNCT
ejpam-1781	191	20	;	;	PUNCT
ejpam-1781	191	21	a	a	DET
ejpam-1781	191	22	contradiction	contradiction	NOUN
ejpam-1781	191	23	.	.	PUNCT
ejpam-1781	192	1	hence	hence	ADV
ejpam-1781	192	2	i	i	PRON
ejpam-1781	192	3	and	and	CCONJ
ejpam-1781	192	4	p	p	NOUN
ejpam-1781	192	5	must	must	AUX
ejpam-1781	192	6	be	be	AUX
ejpam-1781	192	7	comparable	comparable	ADJ
ejpam-1781	192	8	for	for	ADP
ejpam-1781	192	9	each	each	DET
ejpam-1781	192	10	i	i	PRON
ejpam-1781	192	11	∈	∈	PROPN
ejpam-1781	193	1	i	i	PRON
ejpam-1781	193	2	(	(	PUNCT
ejpam-1781	193	3	r	r	NOUN
ejpam-1781	193	4	)	)	PUNCT
ejpam-1781	193	5	.	.	PUNCT
ejpam-1781	194	1	theorem	theorem	VERB
ejpam-1781	194	2	7	7	NUM
ejpam-1781	194	3	.	.	PUNCT
ejpam-1781	195	1	in	in	ADP
ejpam-1781	195	2	an	an	DET
ejpam-1781	195	3	adl	adl	NOUN
ejpam-1781	195	4	r	r	NOUN
ejpam-1781	195	5	,	,	PUNCT
ejpam-1781	195	6	if	if	SCONJ
ejpam-1781	195	7	an	an	DET
ejpam-1781	195	8	ideal	ideal	NOUN
ejpam-1781	195	9	i	i	PRON
ejpam-1781	195	10	6=	6=	NUM
ejpam-1781	195	11	{	{	PUNCT
ejpam-1781	195	12	0	0	NUM
ejpam-1781	195	13	}	}	PUNCT
ejpam-1781	195	14	is	be	AUX
ejpam-1781	195	15	a	a	DET
ejpam-1781	195	16	totally	totally	ADV
ejpam-1781	195	17	ordered	order	VERB
ejpam-1781	195	18	subset	subset	NOUN
ejpam-1781	195	19	of	of	ADP
ejpam-1781	195	20	r	r	NOUN
ejpam-1781	195	21	,	,	PUNCT
ejpam-1781	195	22	then	then	ADV
ejpam-1781	195	23	i∗	i∗	NOUN
ejpam-1781	195	24	is	be	AUX
ejpam-1781	195	25	a	a	DET
ejpam-1781	195	26	minimal	minimal	ADJ
ejpam-1781	195	27	prime	prime	ADJ
ejpam-1781	195	28	ideal	ideal	NOUN
ejpam-1781	195	29	in	in	ADP
ejpam-1781	195	30	r.	r.	PROPN
ejpam-1781	195	31	proof	proof	NOUN
ejpam-1781	195	32	.	.	PUNCT
ejpam-1781	196	1	claim	claim	VERB
ejpam-1781	196	2	1	1	NUM
ejpam-1781	196	3	:	:	PUNCT
ejpam-1781	196	4	i∗	i∗	NOUN
ejpam-1781	196	5	=	=	SYM
ejpam-1781	196	6	{	{	PUNCT
ejpam-1781	196	7	a}∗	a}∗	PROPN
ejpam-1781	196	8	for	for	ADP
ejpam-1781	196	9	any	any	DET
ejpam-1781	196	10	0	0	NUM
ejpam-1781	196	11	<	<	X
ejpam-1781	196	12	a	a	DET
ejpam-1781	196	13	∈	∈	NOUN
ejpam-1781	197	1	i	i	PRON
ejpam-1781	197	2	.	.	PUNCT
ejpam-1781	198	1	let	let	VERB
ejpam-1781	198	2	0	0	PUNCT
ejpam-1781	198	3	<	<	X
ejpam-1781	199	1	a	a	PRON
ejpam-1781	199	2	∈	∈	ADJ
ejpam-1781	200	1	i	i	PRON
ejpam-1781	200	2	.	.	PUNCT
ejpam-1781	201	1	then	then	ADV
ejpam-1781	201	2	i∗	i∗	VERB
ejpam-1781	201	3	⊆	⊆	NUM
ejpam-1781	201	4	{	{	PUNCT
ejpam-1781	201	5	a∗	a∗	NOUN
ejpam-1781	201	6	}	}	PUNCT
ejpam-1781	201	7	always	always	ADV
ejpam-1781	201	8	.	.	PUNCT
ejpam-1781	202	1	let	let	VERB
ejpam-1781	202	2	if	if	SCONJ
ejpam-1781	202	3	possible	possible	ADJ
ejpam-1781	202	4	,	,	PUNCT
ejpam-1781	202	5	i∗	i∗	PROPN
ejpam-1781	202	6	⊂	⊂	X
ejpam-1781	202	7	{	{	PUNCT
ejpam-1781	202	8	a}∗.	a}∗.	ADJ
ejpam-1781	202	9	select	select	ADJ
ejpam-1781	202	10	x	x	X
ejpam-1781	202	11	∈	∈	PROPN
ejpam-1781	202	12	{	{	PUNCT
ejpam-1781	202	13	a}∗	a}∗	NOUN
ejpam-1781	202	14	\	\	PROPN
ejpam-1781	202	15	i∗.	i∗.	PROPN
ejpam-1781	202	16	then	then	ADV
ejpam-1781	202	17	x	x	X
ejpam-1781	202	18	>	>	X
ejpam-1781	202	19	0	0	NUM
ejpam-1781	202	20	,	,	PUNCT
ejpam-1781	202	21	x	x	PUNCT
ejpam-1781	202	22	∧	∧	NOUN
ejpam-1781	202	23	a	a	PRON
ejpam-1781	202	24	=	=	SYM
ejpam-1781	202	25	0	0	NUM
ejpam-1781	203	1	and	and	CCONJ
ejpam-1781	203	2	x	x	PART
ejpam-1781	203	3	∧	∧	PROPN
ejpam-1781	203	4	b	b	PROPN
ejpam-1781	203	5	6=	6=	PROPN
ejpam-1781	203	6	0	0	NUM
ejpam-1781	204	1	for	for	ADP
ejpam-1781	204	2	some	some	DET
ejpam-1781	204	3	b	b	NOUN
ejpam-1781	204	4	∈	∈	NOUN
ejpam-1781	204	5	i	i	PRON
ejpam-1781	204	6	.	.	PUNCT
ejpam-1781	205	1	as	as	SCONJ
ejpam-1781	205	2	i	i	PRON
ejpam-1781	205	3	is	be	AUX
ejpam-1781	205	4	totally	totally	ADV
ejpam-1781	205	5	ordered	order	VERB
ejpam-1781	205	6	,	,	PUNCT
ejpam-1781	205	7	either	either	CCONJ
ejpam-1781	205	8	x	x	PART
ejpam-1781	205	9	∧	∧	PROPN
ejpam-1781	205	10	b	b	PROPN
ejpam-1781	205	11	≤	≤	NUM
ejpam-1781	205	12	a	a	PRON
ejpam-1781	205	13	or	or	CCONJ
ejpam-1781	205	14	a	a	DET
ejpam-1781	205	15	≤	≤	ADJ
ejpam-1781	205	16	x∧b	x∧b	NOUN
ejpam-1781	205	17	.	.	PUNCT
ejpam-1781	206	1	if	if	SCONJ
ejpam-1781	206	2	x∧b	x∧b	PROPN
ejpam-1781	206	3	≤	≤	NOUN
ejpam-1781	206	4	a	a	PRON
ejpam-1781	206	5	,	,	PUNCT
ejpam-1781	206	6	then	then	ADV
ejpam-1781	206	7	x∧b	x∧b	PUNCT
ejpam-1781	207	1	=	=	PRON
ejpam-1781	208	1	(	(	PUNCT
ejpam-1781	208	2	x∧b)∧a=	x∧b)∧a=	PROPN
ejpam-1781	208	3	x∧(b∧a	x∧(b∧a	PROPN
ejpam-1781	208	4	)	)	PUNCT
ejpam-1781	208	5	=	=	SYM
ejpam-1781	208	6	x∧0	x∧0	PROPN
ejpam-1781	208	7	(	(	PUNCT
ejpam-1781	208	8	since	since	SCONJ
ejpam-1781	208	9	a∧b	a∧b	NOUN
ejpam-1781	209	1	=	=	SYM
ejpam-1781	209	2	0⇒	0⇒	PROPN
ejpam-1781	209	3	b∧a	b∧a	ADV
ejpam-1781	209	4	=	=	NOUN
ejpam-1781	209	5	0	0	NUM
ejpam-1781	209	6	)	)	PUNCT
ejpam-1781	209	7	.	.	PUNCT
ejpam-1781	210	1	hence	hence	ADV
ejpam-1781	210	2	x	x	PUNCT
ejpam-1781	210	3	∧	∧	NOUN
ejpam-1781	210	4	b	b	PROPN
ejpam-1781	210	5	=	=	SYM
ejpam-1781	210	6	0	0	NUM
ejpam-1781	210	7	;	;	PUNCT
ejpam-1781	210	8	a	a	DET
ejpam-1781	210	9	contradiction	contradiction	NOUN
ejpam-1781	210	10	.	.	PUNCT
ejpam-1781	211	1	if	if	SCONJ
ejpam-1781	211	2	a	a	DET
ejpam-1781	211	3	≤	≤	ADV
ejpam-1781	211	4	x	x	PUNCT
ejpam-1781	211	5	∧	∧	PROPN
ejpam-1781	211	6	b	b	PROPN
ejpam-1781	211	7	,	,	PUNCT
ejpam-1781	211	8	then	then	ADV
ejpam-1781	211	9	a	a	DET
ejpam-1781	211	10	=	=	NOUN
ejpam-1781	211	11	a∧	a∧	NOUN
ejpam-1781	211	12	(	(	PUNCT
ejpam-1781	211	13	x	x	PROPN
ejpam-1781	211	14	∧	∧	PROPN
ejpam-1781	211	15	b	b	NOUN
ejpam-1781	211	16	)	)	PUNCT
ejpam-1781	211	17	=	=	SYM
ejpam-1781	211	18	x	x	SYM
ejpam-1781	211	19	∧	∧	PROPN
ejpam-1781	211	20	(	(	PUNCT
ejpam-1781	211	21	a∧	a∧	NOUN
ejpam-1781	211	22	b	b	NOUN
ejpam-1781	211	23	)	)	PUNCT
ejpam-1781	211	24	=	=	SYM
ejpam-1781	211	25	x	x	SYM
ejpam-1781	211	26	∧0=	∧0=	NOUN
ejpam-1781	211	27	0	0	NUM
ejpam-1781	211	28	;	;	PUNCT
ejpam-1781	211	29	a	a	DET
ejpam-1781	211	30	contradiction	contradiction	NOUN
ejpam-1781	211	31	.	.	PUNCT
ejpam-1781	212	1	thus	thus	ADV
ejpam-1781	212	2	i∗	i∗	NOUN
ejpam-1781	212	3	=	=	SYM
ejpam-1781	212	4	{	{	PUNCT
ejpam-1781	212	5	a}∗	a}∗	PROPN
ejpam-1781	212	6	for	for	ADP
ejpam-1781	212	7	any	any	DET
ejpam-1781	212	8	a	a	DET
ejpam-1781	212	9	∈	∈	PROPN
ejpam-1781	212	10	r.	r.	NOUN
ejpam-1781	212	11	claim	claim	NOUN
ejpam-1781	212	12	2	2	NUM
ejpam-1781	212	13	:	:	PUNCT
ejpam-1781	212	14	i∗	i∗	NOUN
ejpam-1781	212	15	is	be	AUX
ejpam-1781	212	16	a	a	DET
ejpam-1781	212	17	prime	prime	ADJ
ejpam-1781	212	18	ideal	ideal	NOUN
ejpam-1781	212	19	in	in	ADP
ejpam-1781	212	20	r.	r.	PROPN
ejpam-1781	212	21	i∗	i∗	PROPN
ejpam-1781	212	22	is	be	AUX
ejpam-1781	212	23	an	an	DET
ejpam-1781	212	24	ideal	ideal	NOUN
ejpam-1781	212	25	in	in	ADP
ejpam-1781	212	26	r	r	NOUN
ejpam-1781	212	27	(	(	PUNCT
ejpam-1781	212	28	see	see	VERB
ejpam-1781	212	29	result	result	NOUN
ejpam-1781	212	30	8)	8)	NUM
ejpam-1781	212	31	.	.	PUNCT
ejpam-1781	213	1	let	let	VERB
ejpam-1781	213	2	there	there	PRON
ejpam-1781	213	3	exist	exist	VERB
ejpam-1781	213	4	a	a	DET
ejpam-1781	213	5	,	,	PUNCT
ejpam-1781	213	6	b	b	X
ejpam-1781	213	7	∈	∈	NOUN
ejpam-1781	213	8	r	r	NOUN
ejpam-1781	213	9	such	such	ADJ
ejpam-1781	213	10	that	that	SCONJ
ejpam-1781	213	11	a	a	DET
ejpam-1781	213	12	∧	∧	PROPN
ejpam-1781	213	13	b	b	PROPN
ejpam-1781	213	14	∈	∈	PROPN
ejpam-1781	213	15	i∗	i∗	NOUN
ejpam-1781	213	16	with	with	ADP
ejpam-1781	213	17	a	a	DET
ejpam-1781	213	18	/∈	/∈	INTJ
ejpam-1781	213	19	i∗	i∗	NOUN
ejpam-1781	213	20	and	and	CCONJ
ejpam-1781	213	21	b	b	NOUN
ejpam-1781	213	22	/∈	/∈	PUNCT
ejpam-1781	214	1	i∗.	i∗.	PROPN
ejpam-1781	215	1	but	but	CCONJ
ejpam-1781	215	2	then	then	ADV
ejpam-1781	215	3	a	a	DET
ejpam-1781	215	4	∧	∧	PROPN
ejpam-1781	215	5	x	x	X
ejpam-1781	215	6	>	>	X
ejpam-1781	215	7	0	0	NUM
ejpam-1781	215	8	and	and	CCONJ
ejpam-1781	215	9	b	b	PROPN
ejpam-1781	215	10	∧	∧	PROPN
ejpam-1781	215	11	y	y	PROPN
ejpam-1781	215	12	>	>	X
ejpam-1781	215	13	0	0	PUNCT
ejpam-1781	216	1	for	for	ADP
ejpam-1781	216	2	some	some	DET
ejpam-1781	216	3	x	x	NOUN
ejpam-1781	216	4	,	,	PUNCT
ejpam-1781	216	5	y	y	PROPN
ejpam-1781	216	6	∈	∈	PROPN
ejpam-1781	216	7	i	i	PRON
ejpam-1781	216	8	.	.	PUNCT
ejpam-1781	217	1	now	now	ADV
ejpam-1781	217	2	,	,	PUNCT
ejpam-1781	217	3	(	(	PUNCT
ejpam-1781	217	4	a	a	DET
ejpam-1781	217	5	∧	∧	PROPN
ejpam-1781	217	6	x)∧	x)∧	PUNCT
ejpam-1781	217	7	(	(	PUNCT
ejpam-1781	217	8	b	b	PROPN
ejpam-1781	217	9	∧	∧	PROPN
ejpam-1781	217	10	y	y	PROPN
ejpam-1781	217	11	)	)	PUNCT
ejpam-1781	217	12	=	=	PUNCT
ejpam-1781	218	1	a	a	DET
ejpam-1781	218	2	∧	∧	PROPN
ejpam-1781	219	1	[	[	X
ejpam-1781	219	2	x	x	X
ejpam-1781	219	3	∧	∧	PROPN
ejpam-1781	219	4	b	b	PROPN
ejpam-1781	219	5	∧	∧	PROPN
ejpam-1781	219	6	y	y	PROPN
ejpam-1781	219	7	]	]	X
ejpam-1781	219	8	=	=	PUNCT
ejpam-1781	219	9	a	a	DET
ejpam-1781	219	10	∧	∧	PROPN
ejpam-1781	220	1	[	[	X
ejpam-1781	220	2	b	b	X
ejpam-1781	220	3	∧	∧	PROPN
ejpam-1781	220	4	x	x	PUNCT
ejpam-1781	220	5	∧	∧	PROPN
ejpam-1781	220	6	y	y	PROPN
ejpam-1781	220	7	]	]	X
ejpam-1781	220	8	=	=	X
ejpam-1781	220	9	(	(	PUNCT
ejpam-1781	220	10	a	a	DET
ejpam-1781	220	11	∧	∧	PROPN
ejpam-1781	220	12	b)∧	b)∧	PROPN
ejpam-1781	220	13	(	(	PUNCT
ejpam-1781	220	14	x	x	PROPN
ejpam-1781	220	15	∧	∧	PROPN
ejpam-1781	220	16	y	y	PROPN
ejpam-1781	220	17	)	)	PUNCT
ejpam-1781	220	18	=	=	SYM
ejpam-1781	220	19	0	0	PUNCT
ejpam-1781	221	1	(	(	PUNCT
ejpam-1781	221	2	as	as	ADP
ejpam-1781	221	3	x	x	PROPN
ejpam-1781	221	4	∧	∧	NOUN
ejpam-1781	221	5	y	y	PROPN
ejpam-1781	221	6	∈	∈	PROPN
ejpam-1781	221	7	i	i	PRON
ejpam-1781	221	8	and	and	CCONJ
ejpam-1781	221	9	a	a	DET
ejpam-1781	221	10	∧	∧	PROPN
ejpam-1781	221	11	b	b	PROPN
ejpam-1781	221	12	∈	∈	PROPN
ejpam-1781	221	13	i∗	i∗	NOUN
ejpam-1781	221	14	)	)	PUNCT
ejpam-1781	221	15	.	.	PUNCT
ejpam-1781	222	1	but	but	CCONJ
ejpam-1781	222	2	this	this	PRON
ejpam-1781	222	3	shows	show	VERB
ejpam-1781	222	4	that	that	SCONJ
ejpam-1781	222	5	b	b	X
ejpam-1781	222	6	∧	∧	PROPN
ejpam-1781	222	7	y	y	PROPN
ejpam-1781	222	8	∈	∈	PROPN
ejpam-1781	222	9	{	{	PUNCT
ejpam-1781	222	10	a	a	DET
ejpam-1781	222	11	∧	∧	PROPN
ejpam-1781	222	12	x}∗.	x}∗.	PUNCT
ejpam-1781	222	13	thus	thus	ADV
ejpam-1781	222	14	b	b	X
ejpam-1781	222	15	∧	∧	PROPN
ejpam-1781	222	16	y	y	PROPN
ejpam-1781	222	17	∈	∈	PROPN
ejpam-1781	222	18	i	i	PRON
ejpam-1781	222	19	∩	∩	VERB
ejpam-1781	222	20	i∗	i∗	NOUN
ejpam-1781	222	21	=	=	PUNCT
ejpam-1781	222	22	{	{	PUNCT
ejpam-1781	222	23	0	0	NUM
ejpam-1781	222	24	}	}	PUNCT
ejpam-1781	222	25	(	(	PUNCT
ejpam-1781	222	26	see	see	VERB
ejpam-1781	222	27	result	result	NOUN
ejpam-1781	222	28	9	9	NUM
ejpam-1781	222	29	)	)	PUNCT
ejpam-1781	222	30	.	.	PUNCT
ejpam-1781	223	1	hence	hence	ADV
ejpam-1781	223	2	b	b	X
ejpam-1781	223	3	∧	∧	NOUN
ejpam-1781	223	4	y	y	PROPN
ejpam-1781	223	5	=	=	PROPN
ejpam-1781	223	6	0	0	NUM
ejpam-1781	223	7	;	;	PUNCT
ejpam-1781	223	8	a	a	DET
ejpam-1781	223	9	contradiction	contradiction	NOUN
ejpam-1781	223	10	.	.	PUNCT
ejpam-1781	224	1	hence	hence	ADV
ejpam-1781	224	2	i∗	i∗	NOUN
ejpam-1781	224	3	is	be	AUX
ejpam-1781	224	4	a	a	DET
ejpam-1781	224	5	prime	prime	ADJ
ejpam-1781	224	6	ideal	ideal	NOUN
ejpam-1781	224	7	.	.	PUNCT
ejpam-1781	225	1	y.	y.	PROPN
ejpam-1781	225	2	pawar	pawar	PROPN
ejpam-1781	225	3	,	,	PUNCT
ejpam-1781	225	4	i.	i.	PROPN
ejpam-1781	225	5	shaikh	shaikh	PROPN
ejpam-1781	225	6	/	/	SYM
ejpam-1781	225	7	eur	eur	PROPN
ejpam-1781	225	8	.	.	PUNCT
ejpam-1781	226	1	j.	j.	PROPN
ejpam-1781	226	2	pure	pure	PROPN
ejpam-1781	226	3	appl	appl	PROPN
ejpam-1781	226	4	.	.	PROPN
ejpam-1781	226	5	math	math	PROPN
ejpam-1781	226	6	,	,	PUNCT
ejpam-1781	226	7	6	6	NUM
ejpam-1781	226	8	(	(	PUNCT
ejpam-1781	226	9	2013	2013	NUM
ejpam-1781	226	10	)	)	PUNCT
ejpam-1781	226	11	,	,	PUNCT
ejpam-1781	226	12	107	107	NUM
ejpam-1781	226	13	-	-	SYM
ejpam-1781	226	14	118	118	NUM
ejpam-1781	226	15	113	113	NUM
ejpam-1781	226	16	claim	claim	NOUN
ejpam-1781	226	17	3	3	NUM
ejpam-1781	226	18	:	:	PUNCT
ejpam-1781	226	19	i∗	i∗	NOUN
ejpam-1781	226	20	is	be	AUX
ejpam-1781	226	21	a	a	DET
ejpam-1781	226	22	minimal	minimal	ADJ
ejpam-1781	226	23	prime	prime	ADJ
ejpam-1781	226	24	ideal	ideal	NOUN
ejpam-1781	226	25	in	in	ADP
ejpam-1781	226	26	r.	r.	PROPN
ejpam-1781	226	27	we	we	PRON
ejpam-1781	226	28	know	know	VERB
ejpam-1781	226	29	that	that	DET
ejpam-1781	226	30	i∗	i∗	NOUN
ejpam-1781	226	31	=	=	SYM
ejpam-1781	226	32	⋂	⋂	PROPN
ejpam-1781	226	33	{	{	PUNCT
ejpam-1781	226	34	m	m	PROPN
ejpam-1781	226	35	∈	∈	NOUN
ejpam-1781	227	1	m	m	VERB
ejpam-1781	227	2	|	|	ADV
ejpam-1781	228	1	i	i	PRON
ejpam-1781	228	2	*	*	PUNCT
ejpam-1781	228	3	m	m	VERB
ejpam-1781	228	4	}	}	PUNCT
ejpam-1781	228	5	(	(	PUNCT
ejpam-1781	228	6	by	by	ADP
ejpam-1781	228	7	result	result	NOUN
ejpam-1781	228	8	10	10	NUM
ejpam-1781	228	9	)	)	PUNCT
ejpam-1781	228	10	.	.	PUNCT
ejpam-1781	228	11	.	.	PUNCT
ejpam-1781	228	12	.	.	PUNCT
ejpam-1781	229	1	(	(	PUNCT
ejpam-1781	229	2	i	i	NOUN
ejpam-1781	229	3	)	)	PUNCT
ejpam-1781	229	4	.	.	PUNCT
ejpam-1781	230	1	let	let	VERB
ejpam-1781	230	2	if	if	SCONJ
ejpam-1781	230	3	possible	possible	ADJ
ejpam-1781	230	4	there	there	PRON
ejpam-1781	230	5	exists	exist	VERB
ejpam-1781	230	6	a	a	DET
ejpam-1781	230	7	minimal	minimal	ADJ
ejpam-1781	230	8	prime	prime	ADJ
ejpam-1781	230	9	ideal	ideal	NOUN
ejpam-1781	230	10	m	m	VERB
ejpam-1781	230	11	in	in	ADP
ejpam-1781	230	12	r	r	NOUN
ejpam-1781	230	13	such	such	ADJ
ejpam-1781	230	14	that	that	SCONJ
ejpam-1781	230	15	m	m	VERB
ejpam-1781	230	16	⊂	⊂	ADJ
ejpam-1781	230	17	i∗	i∗	NOUN
ejpam-1781	230	18	(	(	PUNCT
ejpam-1781	230	19	see	see	VERB
ejpam-1781	230	20	result	result	NOUN
ejpam-1781	230	21	3	3	NUM
ejpam-1781	230	22	)	)	PUNCT
ejpam-1781	230	23	.	.	PUNCT
ejpam-1781	231	1	if	if	SCONJ
ejpam-1781	231	2	i	i	PRON
ejpam-1781	231	3	*	*	VERB
ejpam-1781	231	4	m	m	VERB
ejpam-1781	231	5	,	,	PUNCT
ejpam-1781	231	6	then	then	ADV
ejpam-1781	231	7	there	there	PRON
ejpam-1781	231	8	exists	exist	VERB
ejpam-1781	231	9	x	x	X
ejpam-1781	231	10	∈	∈	PROPN
ejpam-1781	231	11	i	i	PRON
ejpam-1781	231	12	such	such	VERB
ejpam-1781	231	13	that	that	PRON
ejpam-1781	231	14	x	x	PROPN
ejpam-1781	231	15	/∈	/∈	INTJ
ejpam-1781	232	1	m	m	INTJ
ejpam-1781	232	2	.	.	PUNCT
ejpam-1781	233	1	x	x	X
ejpam-1781	233	2	∈	∈	PROPN
ejpam-1781	234	1	i	i	PRON
ejpam-1781	234	2	and	and	CCONJ
ejpam-1781	234	3	x	x	NOUN
ejpam-1781	234	4	/∈	/∈	PUNCT
ejpam-1781	235	1	m	m	AUX
ejpam-1781	235	2	⇒	⇒	NOUN
ejpam-1781	235	3	x	x	PUNCT
ejpam-1781	235	4	>	>	X
ejpam-1781	235	5	0	0	NUM
ejpam-1781	235	6	.	.	PUNCT
ejpam-1781	236	1	x	x	X
ejpam-1781	236	2	∈	∈	PROPN
ejpam-1781	236	3	i	i	PRON
ejpam-1781	236	4	implies	imply	VERB
ejpam-1781	236	5	i	i	PRON
ejpam-1781	236	6	=	=	PUNCT
ejpam-1781	236	7	{	{	PUNCT
ejpam-1781	236	8	x}∗	x}∗	NOUN
ejpam-1781	236	9	by	by	ADP
ejpam-1781	236	10	claim	claim	NOUN
ejpam-1781	236	11	1	1	NUM
ejpam-1781	236	12	x	x	NOUN
ejpam-1781	236	13	/∈	/∈	PUNCT
ejpam-1781	236	14	m	m	VERB
ejpam-1781	236	15	,	,	PUNCT
ejpam-1781	236	16	m	m	VERB
ejpam-1781	236	17	is	be	AUX
ejpam-1781	236	18	minimal	minimal	ADJ
ejpam-1781	236	19	⇒	⇒	NOUN
ejpam-1781	236	20	{	{	PUNCT
ejpam-1781	236	21	x}∗	x}∗	NOUN
ejpam-1781	236	22	⊆	⊆	NUM
ejpam-1781	236	23	m	m	NOUN
ejpam-1781	236	24	⇒i	⇒i	VERB
ejpam-1781	236	25	⊆	⊆	NUM
ejpam-1781	236	26	m	m	NOUN
ejpam-1781	236	27	⇒i	⇒i	VERB
ejpam-1781	236	28	=	=	PUNCT
ejpam-1781	237	1	i	i	PRON
ejpam-1781	237	2	∩m	∩m	PROPN
ejpam-1781	238	1	⊆	⊆	NUM
ejpam-1781	238	2	i	i	PRON
ejpam-1781	238	3	∩	∩	ADJ
ejpam-1781	238	4	i∗	i∗	NOUN
ejpam-1781	238	5	=	=	SYM
ejpam-1781	238	6	{	{	PUNCT
ejpam-1781	238	7	0	0	NUM
ejpam-1781	238	8	}	}	PUNCT
ejpam-1781	238	9	⇒i	⇒i	VERB
ejpam-1781	238	10	=	=	X
ejpam-1781	238	11	{	{	PUNCT
ejpam-1781	238	12	0	0	NUM
ejpam-1781	238	13	}	}	PUNCT
ejpam-1781	238	14	,	,	PUNCT
ejpam-1781	238	15	a	a	DET
ejpam-1781	238	16	contradiction	contradiction	NOUN
ejpam-1781	238	17	(	(	PUNCT
ejpam-1781	238	18	i∗	i∗	NOUN
ejpam-1781	238	19	⊆	⊆	NUM
ejpam-1781	238	20	m	m	NOUN
ejpam-1781	238	21	by	by	ADP
ejpam-1781	238	22	result	result	NOUN
ejpam-1781	238	23	9	9	NUM
ejpam-1781	238	24	)	)	PUNCT
ejpam-1781	238	25	hence	hence	ADV
ejpam-1781	238	26	i∗	i∗	NOUN
ejpam-1781	238	27	∈m	∈m	NOUN
ejpam-1781	238	28	necessary	necessary	ADJ
ejpam-1781	238	29	and	and	CCONJ
ejpam-1781	238	30	sufficient	sufficient	ADJ
ejpam-1781	238	31	conditions	condition	NOUN
ejpam-1781	238	32	for	for	ADP
ejpam-1781	238	33	any	any	DET
ejpam-1781	238	34	prime	prime	ADJ
ejpam-1781	238	35	ideal	ideal	NOUN
ejpam-1781	238	36	in	in	ADP
ejpam-1781	238	37	r	r	NOUN
ejpam-1781	238	38	to	to	PART
ejpam-1781	238	39	be	be	AUX
ejpam-1781	238	40	a	a	DET
ejpam-1781	238	41	principal	principal	ADJ
ejpam-1781	238	42	ideal	ideal	NOUN
ejpam-1781	238	43	are	be	AUX
ejpam-1781	238	44	given	give	VERB
ejpam-1781	238	45	in	in	ADP
ejpam-1781	238	46	the	the	DET
ejpam-1781	238	47	following	follow	VERB
ejpam-1781	238	48	theorem	theorem	NOUN
ejpam-1781	238	49	.	.	PUNCT
ejpam-1781	238	50	theorem	theorem	NOUN
ejpam-1781	238	51	8	8	NUM
ejpam-1781	238	52	.	.	PUNCT
ejpam-1781	239	1	let	let	VERB
ejpam-1781	239	2	{	{	PUNCT
ejpam-1781	239	3	pα	pα	INTJ
ejpam-1781	239	4	|α	|α	NOUN
ejpam-1781	239	5	∈	∈	NOUN
ejpam-1781	239	6	∆	∆	PROPN
ejpam-1781	239	7	}	}	PUNCT
ejpam-1781	239	8	(	(	PUNCT
ejpam-1781	239	9	∆	∆	X
ejpam-1781	239	10	any	any	DET
ejpam-1781	239	11	indexing	indexing	NOUN
ejpam-1781	239	12	set	set	NOUN
ejpam-1781	239	13	)	)	PUNCT
ejpam-1781	239	14	be	be	VERB
ejpam-1781	239	15	any	any	DET
ejpam-1781	239	16	family	family	NOUN
ejpam-1781	239	17	prime	prime	ADJ
ejpam-1781	239	18	ideals	ideal	NOUN
ejpam-1781	239	19	in	in	ADP
ejpam-1781	239	20	r.	r.	PROPN
ejpam-1781	239	21	then	then	ADV
ejpam-1781	239	22	following	follow	VERB
ejpam-1781	239	23	statements	statement	NOUN
ejpam-1781	239	24	are	be	AUX
ejpam-1781	239	25	equivalent	equivalent	ADJ
ejpam-1781	239	26	:	:	PUNCT
ejpam-1781	239	27	1	1	X
ejpam-1781	239	28	.	.	X
ejpam-1781	240	1	for	for	ADP
ejpam-1781	240	2	any	any	DET
ejpam-1781	240	3	ideal	ideal	NOUN
ejpam-1781	240	4	i	i	PRON
ejpam-1781	240	5	in	in	ADP
ejpam-1781	240	6	r	r	NOUN
ejpam-1781	240	7	if	if	SCONJ
ejpam-1781	240	8	i	i	PRON
ejpam-1781	240	9	⊆	⊆	NUM
ejpam-1781	240	10	⋃	⋃	ADP
ejpam-1781	240	11	α∈∆	α∈∆	PRON
ejpam-1781	240	12	pα	pα	NOUN
ejpam-1781	240	13	,	,	PUNCT
ejpam-1781	240	14	then	then	ADV
ejpam-1781	240	15	i	i	PRON
ejpam-1781	240	16	⊆	⊆	NUM
ejpam-1781	240	17	pα	pα	NOUN
ejpam-1781	240	18	for	for	ADP
ejpam-1781	240	19	some	some	DET
ejpam-1781	240	20	α	α	NOUN
ejpam-1781	240	21	∈∆.	∈∆.	PROPN
ejpam-1781	240	22	2	2	NUM
ejpam-1781	240	23	.	.	PUNCT
ejpam-1781	241	1	for	for	ADP
ejpam-1781	241	2	any	any	DET
ejpam-1781	241	3	prime	prime	ADJ
ejpam-1781	241	4	ideal	ideal	NOUN
ejpam-1781	241	5	i	i	PRON
ejpam-1781	241	6	in	in	ADP
ejpam-1781	241	7	r	r	NOUN
ejpam-1781	241	8	if	if	SCONJ
ejpam-1781	241	9	p	p	PRON
ejpam-1781	241	10	⊆	⊆	NUM
ejpam-1781	241	11	⋃	⋃	NOUN
ejpam-1781	241	12	α∈∆	α∈∆	PRON
ejpam-1781	241	13	pα	pα	NOUN
ejpam-1781	241	14	,	,	PUNCT
ejpam-1781	241	15	then	then	ADV
ejpam-1781	241	16	p	p	NOUN
ejpam-1781	241	17	⊆	⊆	NUM
ejpam-1781	241	18	pα	pα	NOUN
ejpam-1781	241	19	for	for	ADP
ejpam-1781	241	20	some	some	DET
ejpam-1781	241	21	α	α	NOUN
ejpam-1781	241	22	∈∆.	∈∆.	PROPN
ejpam-1781	241	23	3	3	NUM
ejpam-1781	241	24	.	.	PUNCT
ejpam-1781	242	1	every	every	DET
ejpam-1781	242	2	(	(	PUNCT
ejpam-1781	242	3	proper	proper	ADJ
ejpam-1781	242	4	)	)	PUNCT
ejpam-1781	242	5	ideal	ideal	NOUN
ejpam-1781	242	6	in	in	ADP
ejpam-1781	242	7	r	r	NOUN
ejpam-1781	242	8	is	be	AUX
ejpam-1781	242	9	a	a	DET
ejpam-1781	242	10	principal	principal	NOUN
ejpam-1781	242	11	.	.	PUNCT
ejpam-1781	243	1	4	4	X
ejpam-1781	243	2	.	.	X
ejpam-1781	243	3	every	every	DET
ejpam-1781	243	4	prime	prime	ADJ
ejpam-1781	243	5	ideal	ideal	NOUN
ejpam-1781	243	6	in	in	ADP
ejpam-1781	243	7	r	r	NOUN
ejpam-1781	243	8	is	be	AUX
ejpam-1781	243	9	a	a	DET
ejpam-1781	243	10	principal	principal	NOUN
ejpam-1781	243	11	.	.	PUNCT
ejpam-1781	244	1	proof	proof	NOUN
ejpam-1781	244	2	.	.	PUNCT
ejpam-1781	245	1	this	this	DET
ejpam-1781	245	2	implications	implication	NOUN
ejpam-1781	245	3	(	(	PUNCT
ejpam-1781	245	4	1)⇒	1)⇒	NUM
ejpam-1781	245	5	(	(	PUNCT
ejpam-1781	245	6	2	2	NUM
ejpam-1781	245	7	)	)	PUNCT
ejpam-1781	245	8	and	and	CCONJ
ejpam-1781	245	9	(	(	PUNCT
ejpam-1781	245	10	3)⇒	3)⇒	NUM
ejpam-1781	245	11	(	(	PUNCT
ejpam-1781	245	12	4	4	NUM
ejpam-1781	245	13	)	)	PUNCT
ejpam-1781	245	14	are	be	AUX
ejpam-1781	245	15	obviously	obviously	ADV
ejpam-1781	245	16	true	true	ADJ
ejpam-1781	245	17	.	.	PUNCT
ejpam-1781	246	1	(	(	PUNCT
ejpam-1781	246	2	2)⇒	2)⇒	NUM
ejpam-1781	246	3	(	(	PUNCT
ejpam-1781	246	4	1	1	X
ejpam-1781	246	5	)	)	PUNCT
ejpam-1781	246	6	let	let	VERB
ejpam-1781	246	7	i	i	PRON
ejpam-1781	246	8	⊆	⊆	X
ejpam-1781	246	9	⋃	⋃	ADP
ejpam-1781	246	10	α∈∆	α∈∆	PRON
ejpam-1781	246	11	pα	pα	VERB
ejpam-1781	246	12	,	,	PUNCT
ejpam-1781	246	13	i	i	PRON
ejpam-1781	246	14	an	an	DET
ejpam-1781	246	15	ideal	ideal	NOUN
ejpam-1781	246	16	on	on	ADP
ejpam-1781	246	17	r.	r.	PROPN
ejpam-1781	246	18	define	define	VERB
ejpam-1781	246	19	m	m	PROPN
ejpam-1781	247	1	=	=	SYM
ejpam-1781	247	2	r	r	NOUN
ejpam-1781	247	3	\	\	NOUN
ejpam-1781	247	4	⋃	⋃	SCONJ
ejpam-1781	247	5	α∈∆	α∈∆	PRON
ejpam-1781	247	6	pα	pα	NOUN
ejpam-1781	247	7	.	.	PUNCT
ejpam-1781	248	1	then	then	ADV
ejpam-1781	248	2	m	m	VERB
ejpam-1781	248	3	6=	6=	NUM
ejpam-1781	248	4	;	;	PUNCT
ejpam-1781	248	5	.	.	PUNCT
ejpam-1781	249	1	let	let	VERB
ejpam-1781	249	2	x	x	PRON
ejpam-1781	249	3	,	,	PUNCT
ejpam-1781	249	4	y	y	PROPN
ejpam-1781	249	5	∈	∈	PROPN
ejpam-1781	249	6	m	m	VERB
ejpam-1781	249	7	.	.	PUNCT
ejpam-1781	250	1	then	then	ADV
ejpam-1781	250	2	x	x	X
ejpam-1781	250	3	,	,	PUNCT
ejpam-1781	250	4	y	y	PROPN
ejpam-1781	250	5	/∈	/∈	PUNCT
ejpam-1781	250	6	⋃	⋃	VERB
ejpam-1781	250	7	α∈∆	α∈∆	PRON
ejpam-1781	250	8	pα	pα	VERB
ejpam-1781	250	9	imply	imply	ADV
ejpam-1781	250	10	x	x	INTJ
ejpam-1781	250	11	/∈	/∈	PUNCT
ejpam-1781	251	1	pα	pα	INTJ
ejpam-1781	251	2	and	and	CCONJ
ejpam-1781	251	3	y	y	PROPN
ejpam-1781	251	4	/∈	/∈	PUNCT
ejpam-1781	252	1	pα	pα	INTJ
ejpam-1781	252	2	for	for	ADP
ejpam-1781	252	3	each	each	DET
ejpam-1781	252	4	α	α	NOUN
ejpam-1781	252	5	∈	∈	PROPN
ejpam-1781	253	1	∆.	∆.	NOUN
ejpam-1781	253	2	pα	pα	AUX
ejpam-1781	253	3	being	be	AUX
ejpam-1781	253	4	a	a	DET
ejpam-1781	253	5	prime	prime	ADJ
ejpam-1781	253	6	ideal	ideal	NOUN
ejpam-1781	253	7	,	,	PUNCT
ejpam-1781	253	8	x	x	PUNCT
ejpam-1781	253	9	∧	∧	NOUN
ejpam-1781	253	10	y	y	NOUN
ejpam-1781	253	11	/∈	/∈	PUNCT
ejpam-1781	254	1	pα	pα	INTJ
ejpam-1781	254	2	for	for	ADP
ejpam-1781	254	3	each	each	DET
ejpam-1781	254	4	α	α	NOUN
ejpam-1781	254	5	∈	∈	PROPN
ejpam-1781	255	1	∆.	∆.	NOUN
ejpam-1781	256	1	but	but	CCONJ
ejpam-1781	256	2	then	then	ADV
ejpam-1781	256	3	x	x	PART
ejpam-1781	256	4	∧	∧	NOUN
ejpam-1781	256	5	y	y	PROPN
ejpam-1781	256	6	∈	∈	PROPN
ejpam-1781	256	7	m	m	VERB
ejpam-1781	256	8	.	.	PUNCT
ejpam-1781	257	1	this	this	PRON
ejpam-1781	257	2	shows	show	VERB
ejpam-1781	257	3	that	that	SCONJ
ejpam-1781	257	4	m	m	NOUN
ejpam-1781	257	5	is	be	AUX
ejpam-1781	257	6	closed	close	VERB
ejpam-1781	257	7	for	for	ADP
ejpam-1781	257	8	∧.	∧.	PROPN
ejpam-1781	257	9	further	far	ADV
ejpam-1781	257	10	i	i	PRON
ejpam-1781	257	11	∩	∩	ADJ
ejpam-1781	257	12	m	m	VERB
ejpam-1781	257	13	=	=	PUNCT
ejpam-1781	257	14	;	;	PUNCT
ejpam-1781	257	15	.	.	PUNCT
ejpam-1781	258	1	hence	hence	ADV
ejpam-1781	258	2	by	by	ADP
ejpam-1781	258	3	result	result	NOUN
ejpam-1781	258	4	11	11	NUM
ejpam-1781	258	5	there	there	PRON
ejpam-1781	258	6	exist	exist	VERB
ejpam-1781	258	7	a	a	DET
ejpam-1781	258	8	prime	prime	ADJ
ejpam-1781	258	9	ideal	ideal	NOUN
ejpam-1781	258	10	p	p	NOUN
ejpam-1781	258	11	in	in	ADP
ejpam-1781	258	12	r	r	NOUN
ejpam-1781	258	13	such	such	ADJ
ejpam-1781	258	14	that	that	SCONJ
ejpam-1781	258	15	i	i	PRON
ejpam-1781	258	16	⊆	⊆	NUM
ejpam-1781	258	17	p	p	NOUN
ejpam-1781	258	18	and	and	CCONJ
ejpam-1781	258	19	p	p	NOUN
ejpam-1781	258	20	∩m	∩m	PROPN
ejpam-1781	258	21	=	=	X
ejpam-1781	258	22	;	;	PUNCT
ejpam-1781	258	23	.	.	PUNCT
ejpam-1781	259	1	but	but	CCONJ
ejpam-1781	259	2	then	then	ADV
ejpam-1781	259	3	p	p	X
ejpam-1781	259	4	⊆	⊆	NUM
ejpam-1781	259	5	⋃	⋃	NOUN
ejpam-1781	259	6	α∈∆	α∈∆	NOUN
ejpam-1781	259	7	pα	pα	NOUN
ejpam-1781	259	8	will	will	AUX
ejpam-1781	259	9	imply	imply	VERB
ejpam-1781	259	10	p	p	NOUN
ejpam-1781	259	11	⊆	⊆	NUM
ejpam-1781	259	12	pα	pα	NOUN
ejpam-1781	259	13	for	for	ADP
ejpam-1781	259	14	some	some	DET
ejpam-1781	259	15	α	α	NOUN
ejpam-1781	259	16	∈∆	∈∆	NOUN
ejpam-1781	259	17	,	,	PUNCT
ejpam-1781	259	18	by	by	ADP
ejpam-1781	259	19	the	the	DET
ejpam-1781	259	20	condition	condition	NOUN
ejpam-1781	259	21	(	(	PUNCT
ejpam-1781	259	22	2	2	NUM
ejpam-1781	259	23	)	)	PUNCT
ejpam-1781	259	24	.	.	PUNCT
ejpam-1781	260	1	as	as	SCONJ
ejpam-1781	260	2	i	i	PRON
ejpam-1781	260	3	⊆	⊆	NUM
ejpam-1781	260	4	p	p	NOUN
ejpam-1781	260	5	we	we	PRON
ejpam-1781	260	6	get	get	VERB
ejpam-1781	260	7	i	i	PRON
ejpam-1781	260	8	⊆	⊆	NUM
ejpam-1781	260	9	pα	pα	NOUN
ejpam-1781	260	10	and	and	CCONJ
ejpam-1781	260	11	the	the	DET
ejpam-1781	260	12	implication	implication	NOUN
ejpam-1781	260	13	follows	follow	VERB
ejpam-1781	260	14	.	.	PUNCT
ejpam-1781	261	1	(	(	PUNCT
ejpam-1781	261	2	2)⇒	2)⇒	NUM
ejpam-1781	261	3	(	(	PUNCT
ejpam-1781	261	4	3	3	X
ejpam-1781	261	5	)	)	PUNCT
ejpam-1781	261	6	suppose	suppose	VERB
ejpam-1781	261	7	r	r	NOUN
ejpam-1781	261	8	satisfies	satisfy	VERB
ejpam-1781	261	9	the	the	DET
ejpam-1781	261	10	condition	condition	NOUN
ejpam-1781	261	11	(	(	PUNCT
ejpam-1781	261	12	2	2	NUM
ejpam-1781	261	13	)	)	PUNCT
ejpam-1781	261	14	.	.	PUNCT
ejpam-1781	262	1	assume	assume	VERB
ejpam-1781	262	2	that	that	SCONJ
ejpam-1781	262	3	there	there	PRON
ejpam-1781	262	4	exists	exist	VERB
ejpam-1781	262	5	a	a	DET
ejpam-1781	262	6	prime	prime	ADJ
ejpam-1781	262	7	ideal	ideal	NOUN
ejpam-1781	262	8	p	p	NOUN
ejpam-1781	262	9	in	in	ADP
ejpam-1781	262	10	r	r	NOUN
ejpam-1781	262	11	which	which	PRON
ejpam-1781	262	12	is	be	AUX
ejpam-1781	262	13	not	not	PART
ejpam-1781	262	14	principal	principal	ADJ
ejpam-1781	262	15	.	.	PUNCT
ejpam-1781	263	1	hence	hence	ADV
ejpam-1781	263	2	p	p	X
ejpam-1781	263	3	6=	6=	PROPN
ejpam-1781	263	4	(	(	PUNCT
ejpam-1781	263	5	y	y	NOUN
ejpam-1781	263	6	]	]	X
ejpam-1781	263	7	for	for	ADP
ejpam-1781	263	8	any	any	DET
ejpam-1781	263	9	y	y	PROPN
ejpam-1781	263	10	/∈	/∈	PUNCT
ejpam-1781	264	1	p.	p.	NOUN
ejpam-1781	264	2	as	as	ADP
ejpam-1781	264	3	(	(	PUNCT
ejpam-1781	264	4	y	y	NOUN
ejpam-1781	264	5	]	]	X
ejpam-1781	264	6	=	=	SYM
ejpam-1781	264	7	⋂	⋂	PROPN
ejpam-1781	264	8	{	{	PUNCT
ejpam-1781	264	9	p	p	NOUN
ejpam-1781	265	1	|	|	ADV
ejpam-1781	265	2	p	p	PROPN
ejpam-1781	265	3	is	be	AUX
ejpam-1781	265	4	a	a	DET
ejpam-1781	265	5	prime	prime	ADJ
ejpam-1781	265	6	ideal	ideal	NOUN
ejpam-1781	265	7	and	and	CCONJ
ejpam-1781	265	8	y	y	PROPN
ejpam-1781	265	9	∈	∈	PROPN
ejpam-1781	266	1	p	p	X
ejpam-1781	266	2	}	}	PUNCT
ejpam-1781	266	3	(	(	PUNCT
ejpam-1781	266	4	see	see	VERB
ejpam-1781	266	5	result	result	NOUN
ejpam-1781	266	6	12	12	NUM
ejpam-1781	266	7	)	)	PUNCT
ejpam-1781	266	8	we	we	PRON
ejpam-1781	266	9	get	get	VERB
ejpam-1781	266	10	p	p	NOUN
ejpam-1781	266	11	6=	6=	ADP
ejpam-1781	266	12	⋂	⋂	PROPN
ejpam-1781	266	13	{	{	PUNCT
ejpam-1781	266	14	py	py	INTJ
ejpam-1781	267	1	|	|	ADV
ejpam-1781	267	2	py	py	PROPN
ejpam-1781	267	3	is	be	AUX
ejpam-1781	267	4	a	a	DET
ejpam-1781	267	5	prime	prime	ADJ
ejpam-1781	267	6	ideal	ideal	NOUN
ejpam-1781	267	7	in	in	ADP
ejpam-1781	267	8	r	r	NOUN
ejpam-1781	267	9	containing	contain	VERB
ejpam-1781	267	10	y	y	NOUN
ejpam-1781	267	11	}	}	PUNCT
ejpam-1781	267	12	.	.	PUNCT
ejpam-1781	268	1	if	if	SCONJ
ejpam-1781	268	2	p	p	PROPN
ejpam-1781	268	3	⊆	⊆	NUM
ejpam-1781	268	4	py	py	PROPN
ejpam-1781	268	5	for	for	ADP
ejpam-1781	268	6	each	each	DET
ejpam-1781	268	7	prime	prime	ADJ
ejpam-1781	268	8	ideal	ideal	NOUN
ejpam-1781	268	9	py	py	PROPN
ejpam-1781	268	10	containing	contain	VERB
ejpam-1781	268	11	y	y	NOUN
ejpam-1781	268	12	,	,	PUNCT
ejpam-1781	268	13	then	then	ADV
ejpam-1781	268	14	(	(	PUNCT
ejpam-1781	268	15	y	y	X
ejpam-1781	268	16	]	]	X
ejpam-1781	268	17	⊆	⊆	NUM
ejpam-1781	268	18	{	{	PUNCT
ejpam-1781	268	19	py	py	INTJ
ejpam-1781	268	20	|	|	INTJ
ejpam-1781	268	21	y	y	PROPN
ejpam-1781	268	22	∈	∈	PROPN
ejpam-1781	268	23	p	p	X
ejpam-1781	268	24	}	}	PUNCT
ejpam-1781	268	25	=	=	SYM
ejpam-1781	268	26	(	(	PUNCT
ejpam-1781	268	27	y	y	NOUN
ejpam-1781	268	28	]	]	X
ejpam-1781	268	29	will	will	AUX
ejpam-1781	268	30	imply	imply	VERB
ejpam-1781	268	31	p	p	PROPN
ejpam-1781	268	32	=	=	PUNCT
ejpam-1781	268	33	(	(	PUNCT
ejpam-1781	268	34	y	y	PROPN
ejpam-1781	268	35	]	]	X
ejpam-1781	268	36	;	;	PUNCT
ejpam-1781	268	37	a	a	DET
ejpam-1781	268	38	contradiction	contradiction	NOUN
ejpam-1781	268	39	.	.	PUNCT
ejpam-1781	269	1	hence	hence	ADV
ejpam-1781	269	2	for	for	ADP
ejpam-1781	269	3	each	each	DET
ejpam-1781	269	4	y	y	PROPN
ejpam-1781	269	5	∈	∈	PROPN
ejpam-1781	269	6	r	r	NOUN
ejpam-1781	269	7	there	there	PRON
ejpam-1781	269	8	exists	exist	VERB
ejpam-1781	269	9	a	a	DET
ejpam-1781	269	10	prime	prime	ADJ
ejpam-1781	269	11	ideal	ideal	NOUN
ejpam-1781	269	12	py	py	PROPN
ejpam-1781	269	13	in	in	ADP
ejpam-1781	269	14	r	r	NOUN
ejpam-1781	269	15	such	such	ADJ
ejpam-1781	269	16	that	that	SCONJ
ejpam-1781	269	17	y	y	PROPN
ejpam-1781	269	18	∈	∈	PROPN
ejpam-1781	269	19	py	py	INTJ
ejpam-1781	269	20	and	and	CCONJ
ejpam-1781	269	21	p	p	PROPN
ejpam-1781	269	22	*	*	PUNCT
ejpam-1781	269	23	py	py	INTJ
ejpam-1781	269	24	.	.	PUNCT
ejpam-1781	270	1	again	again	ADV
ejpam-1781	270	2	p	p	ADP
ejpam-1781	270	3	⊆	⊆	NUM
ejpam-1781	270	4	⋃	⋃	NOUN
ejpam-1781	270	5	{	{	PUNCT
ejpam-1781	270	6	py	py	NOUN
ejpam-1781	270	7	|	|	INTJ
ejpam-1781	270	8	y	y	PROPN
ejpam-1781	270	9	∈	∈	PROPN
ejpam-1781	270	10	p	p	X
ejpam-1781	270	11	}	}	PUNCT
ejpam-1781	270	12	implies	imply	VERB
ejpam-1781	270	13	p	p	PROPN
ejpam-1781	270	14	⊆	⊆	NUM
ejpam-1781	270	15	py	py	NOUN
ejpam-1781	270	16	for	for	ADP
ejpam-1781	270	17	some	some	DET
ejpam-1781	270	18	y	y	PROPN
ejpam-1781	270	19	∈	∈	PROPN
ejpam-1781	270	20	p	p	X
ejpam-1781	270	21	;	;	PUNCT
ejpam-1781	270	22	a	a	DET
ejpam-1781	270	23	contradiction	contradiction	NOUN
ejpam-1781	270	24	.	.	PUNCT
ejpam-1781	271	1	hence	hence	ADV
ejpam-1781	271	2	our	our	PRON
ejpam-1781	271	3	assumption	assumption	NOUN
ejpam-1781	271	4	is	be	AUX
ejpam-1781	271	5	wrong	wrong	ADJ
ejpam-1781	271	6	.	.	PUNCT
ejpam-1781	272	1	therefore	therefore	ADV
ejpam-1781	272	2	every	every	DET
ejpam-1781	272	3	prime	prime	ADJ
ejpam-1781	272	4	ideal	ideal	NOUN
ejpam-1781	272	5	in	in	ADP
ejpam-1781	272	6	r	r	NOUN
ejpam-1781	272	7	is	be	AUX
ejpam-1781	272	8	principal	principal	ADJ
ejpam-1781	272	9	.	.	PUNCT
ejpam-1781	273	1	(	(	PUNCT
ejpam-1781	273	2	3)⇒	3)⇒	NUM
ejpam-1781	273	3	(	(	PUNCT
ejpam-1781	273	4	4	4	NUM
ejpam-1781	273	5	)	)	PUNCT
ejpam-1781	273	6	y.	y.	PROPN
ejpam-1781	273	7	pawar	pawar	PROPN
ejpam-1781	273	8	,	,	PUNCT
ejpam-1781	273	9	i.	i.	PROPN
ejpam-1781	273	10	shaikh	shaikh	PROPN
ejpam-1781	273	11	/	/	SYM
ejpam-1781	273	12	eur	eur	PROPN
ejpam-1781	273	13	.	.	PUNCT
ejpam-1781	274	1	j.	j.	PROPN
ejpam-1781	274	2	pure	pure	PROPN
ejpam-1781	274	3	appl	appl	PROPN
ejpam-1781	274	4	.	.	PROPN
ejpam-1781	274	5	math	math	PROPN
ejpam-1781	274	6	,	,	PUNCT
ejpam-1781	274	7	6	6	NUM
ejpam-1781	274	8	(	(	PUNCT
ejpam-1781	274	9	2013	2013	NUM
ejpam-1781	274	10	)	)	PUNCT
ejpam-1781	274	11	,	,	PUNCT
ejpam-1781	274	12	107	107	NUM
ejpam-1781	274	13	-	-	SYM
ejpam-1781	274	14	118	118	NUM
ejpam-1781	274	15	114	114	NUM
ejpam-1781	274	16	let	let	VERB
ejpam-1781	274	17	a	a	DET
ejpam-1781	274	18	prime	prime	ADJ
ejpam-1781	274	19	ideal	ideal	NOUN
ejpam-1781	274	20	p	p	NOUN
ejpam-1781	274	21	⊆	⊆	NUM
ejpam-1781	274	22	⋃	⋃	NOUN
ejpam-1781	274	23	α∈∆	α∈∆	PRON
ejpam-1781	274	24	pα	pα	VERB
ejpam-1781	274	25	,	,	PUNCT
ejpam-1781	274	26	(	(	PUNCT
ejpam-1781	274	27	∆	∆	X
ejpam-1781	275	1	any	any	DET
ejpam-1781	275	2	indexing	indexing	NOUN
ejpam-1781	275	3	set	set	NOUN
ejpam-1781	275	4	)	)	PUNCT
ejpam-1781	275	5	.	.	PUNCT
ejpam-1781	276	1	by	by	ADP
ejpam-1781	276	2	assumption	assumption	NOUN
ejpam-1781	276	3	,	,	PUNCT
ejpam-1781	276	4	p	p	NOUN
ejpam-1781	276	5	=	=	X
ejpam-1781	276	6	(	(	PUNCT
ejpam-1781	276	7	x	x	X
ejpam-1781	276	8	]	]	X
ejpam-1781	276	9	for	for	ADP
ejpam-1781	276	10	some	some	DET
ejpam-1781	276	11	x	x	SYM
ejpam-1781	276	12	∈	∈	PROPN
ejpam-1781	276	13	r.	r.	NOUN
ejpam-1781	276	14	then	then	ADV
ejpam-1781	276	15	(	(	PUNCT
ejpam-1781	276	16	x	x	X
ejpam-1781	276	17	]	]	X
ejpam-1781	276	18	⊆	⊆	NUM
ejpam-1781	276	19	⋃	⋃	ADP
ejpam-1781	276	20	α∈∆	α∈∆	NOUN
ejpam-1781	276	21	pα	pα	NOUN
ejpam-1781	276	22	implies	imply	VERB
ejpam-1781	276	23	x	x	X
ejpam-1781	276	24	∈	∈	NOUN
ejpam-1781	276	25	pα	pα	VERB
ejpam-1781	276	26	for	for	ADP
ejpam-1781	276	27	some	some	DET
ejpam-1781	276	28	α	α	NOUN
ejpam-1781	276	29	∈	∈	PROPN
ejpam-1781	277	1	∆.	∆.	NOUN
ejpam-1781	278	1	but	but	CCONJ
ejpam-1781	278	2	then	then	ADV
ejpam-1781	278	3	p	p	X
ejpam-1781	278	4	=	=	SYM
ejpam-1781	278	5	(	(	PUNCT
ejpam-1781	278	6	x	x	X
ejpam-1781	278	7	]	]	X
ejpam-1781	278	8	⊆	⊆	NUM
ejpam-1781	278	9	pα	pα	NOUN
ejpam-1781	278	10	and	and	CCONJ
ejpam-1781	278	11	we	we	PRON
ejpam-1781	278	12	are	be	AUX
ejpam-1781	278	13	through	through	ADP
ejpam-1781	278	14	.	.	PUNCT
ejpam-1781	279	1	(	(	PUNCT
ejpam-1781	279	2	4)⇒	4)⇒	X
ejpam-1781	279	3	(	(	PUNCT
ejpam-1781	279	4	3	3	NUM
ejpam-1781	279	5	)	)	PUNCT
ejpam-1781	279	6	suppose	suppose	VERB
ejpam-1781	279	7	the	the	DET
ejpam-1781	279	8	statement	statement	NOUN
ejpam-1781	279	9	(	(	PUNCT
ejpam-1781	279	10	3	3	X
ejpam-1781	279	11	)	)	PUNCT
ejpam-1781	279	12	is	be	AUX
ejpam-1781	279	13	false	false	ADJ
ejpam-1781	279	14	.	.	PUNCT
ejpam-1781	280	1	then	then	ADV
ejpam-1781	280	2	there	there	PRON
ejpam-1781	280	3	exists	exist	VERB
ejpam-1781	280	4	a	a	DET
ejpam-1781	280	5	proper	proper	ADJ
ejpam-1781	280	6	non	non	ADJ
ejpam-1781	280	7	principal	principal	NOUN
ejpam-1781	280	8	ideal	ideal	NOUN
ejpam-1781	280	9	in	in	ADP
ejpam-1781	280	10	r.	r.	PROPN
ejpam-1781	280	11	let	let	VERB
ejpam-1781	280	12	a	a	DET
ejpam-1781	280	13	denote	denote	NOUN
ejpam-1781	280	14	the	the	DET
ejpam-1781	280	15	non	non	ADJ
ejpam-1781	280	16	-	-	ADJ
ejpam-1781	280	17	empty	empty	ADJ
ejpam-1781	280	18	collection	collection	NOUN
ejpam-1781	280	19	of	of	ADP
ejpam-1781	280	20	non	non	ADJ
ejpam-1781	280	21	-	-	ADJ
ejpam-1781	280	22	principal	principal	ADJ
ejpam-1781	280	23	proper	proper	ADJ
ejpam-1781	280	24	ideals	ideal	NOUN
ejpam-1781	280	25	of	of	ADP
ejpam-1781	280	26	r.	r.	PROPN
ejpam-1781	280	27	it	it	PRON
ejpam-1781	280	28	is	be	AUX
ejpam-1781	280	29	clear	clear	ADJ
ejpam-1781	280	30	that	that	SCONJ
ejpam-1781	280	31	a	a	PRON
ejpam-1781	280	32	is	be	AUX
ejpam-1781	280	33	closed	close	VERB
ejpam-1781	280	34	under	under	ADP
ejpam-1781	280	35	the	the	DET
ejpam-1781	280	36	formation	formation	NOUN
ejpam-1781	280	37	of	of	ADP
ejpam-1781	280	38	unions	union	NOUN
ejpam-1781	280	39	of	of	ADP
ejpam-1781	280	40	chains	chain	NOUN
ejpam-1781	280	41	in	in	ADP
ejpam-1781	280	42	a.	a.	NOUN
ejpam-1781	280	43	so	so	ADV
ejpam-1781	280	44	,	,	PUNCT
ejpam-1781	280	45	by	by	ADP
ejpam-1781	280	46	zorn	zorn	PROPN
ejpam-1781	280	47	’s	’s	PART
ejpam-1781	280	48	lemma	lemma	PROPN
ejpam-1781	280	49	we	we	PRON
ejpam-1781	280	50	get	get	VERB
ejpam-1781	280	51	a	a	DET
ejpam-1781	280	52	maximal	maximal	ADJ
ejpam-1781	280	53	element	element	NOUN
ejpam-1781	280	54	m	m	NOUN
ejpam-1781	280	55	in	in	ADP
ejpam-1781	280	56	a	a	PRON
ejpam-1781	280	57	which	which	PRON
ejpam-1781	280	58	is	be	AUX
ejpam-1781	280	59	not	not	PART
ejpam-1781	280	60	principal	principal	ADJ
ejpam-1781	280	61	.	.	PUNCT
ejpam-1781	281	1	since	since	SCONJ
ejpam-1781	281	2	m	m	PROPN
ejpam-1781	281	3	is	be	AUX
ejpam-1781	281	4	proper	proper	ADJ
ejpam-1781	281	5	,	,	PUNCT
ejpam-1781	281	6	r	r	NOUN
ejpam-1781	281	7	6=	6=	NOUN
ejpam-1781	281	8	m	m	NOUN
ejpam-1781	281	9	.	.	PUNCT
ejpam-1781	282	1	as	as	SCONJ
ejpam-1781	282	2	m	m	PROPN
ejpam-1781	282	3	is	be	AUX
ejpam-1781	282	4	not	not	PART
ejpam-1781	282	5	prime	prime	ADJ
ejpam-1781	282	6	,	,	PUNCT
ejpam-1781	282	7	there	there	PRON
ejpam-1781	282	8	exist	exist	VERB
ejpam-1781	282	9	elements	element	NOUN
ejpam-1781	282	10	a	a	PRON
ejpam-1781	282	11	,	,	PUNCT
ejpam-1781	282	12	b	b	X
ejpam-1781	282	13	∈	∈	PROPN
ejpam-1781	282	14	m	m	VERB
ejpam-1781	282	15	such	such	ADJ
ejpam-1781	282	16	that	that	SCONJ
ejpam-1781	282	17	a	a	DET
ejpam-1781	282	18	∧	∧	PROPN
ejpam-1781	282	19	b	b	PROPN
ejpam-1781	282	20	∈	∈	PROPN
ejpam-1781	282	21	m	m	VERB
ejpam-1781	282	22	.	.	PUNCT
ejpam-1781	283	1	now	now	ADV
ejpam-1781	283	2	as	as	SCONJ
ejpam-1781	283	3	m	m	PROPN
ejpam-1781	283	4	is	be	AUX
ejpam-1781	283	5	a	a	DET
ejpam-1781	283	6	maximal	maximal	ADJ
ejpam-1781	283	7	element	element	NOUN
ejpam-1781	283	8	in	in	ADP
ejpam-1781	283	9	a	a	DET
ejpam-1781	283	10	,	,	PUNCT
ejpam-1781	283	11	m	m	VERB
ejpam-1781	283	12	∨	∨	NOUN
ejpam-1781	283	13	(	(	PUNCT
ejpam-1781	283	14	a	a	PRON
ejpam-1781	283	15	]	]	X
ejpam-1781	283	16	and	and	CCONJ
ejpam-1781	283	17	m	m	PROPN
ejpam-1781	283	18	∨	∨	NOUN
ejpam-1781	283	19	(	(	PUNCT
ejpam-1781	283	20	b	b	X
ejpam-1781	283	21	]	]	X
ejpam-1781	283	22	are	be	AUX
ejpam-1781	283	23	principal	principal	ADJ
ejpam-1781	283	24	ideals	ideal	NOUN
ejpam-1781	283	25	.	.	PUNCT
ejpam-1781	284	1	let	let	VERB
ejpam-1781	284	2	(	(	PUNCT
ejpam-1781	284	3	m	m	VERB
ejpam-1781	284	4	∨	∨	NOUN
ejpam-1781	284	5	(	(	PUNCT
ejpam-1781	284	6	a	a	PRON
ejpam-1781	284	7	]	]	X
ejpam-1781	284	8	)	)	PUNCT
ejpam-1781	284	9	=	=	SYM
ejpam-1781	284	10	(	(	PUNCT
ejpam-1781	284	11	x	x	SYM
ejpam-1781	284	12	]	]	X
ejpam-1781	284	13	and	and	CCONJ
ejpam-1781	284	14	(	(	PUNCT
ejpam-1781	284	15	m	m	VERB
ejpam-1781	284	16	∨	∨	NOUN
ejpam-1781	284	17	(	(	PUNCT
ejpam-1781	284	18	b	b	NOUN
ejpam-1781	284	19	]	]	X
ejpam-1781	284	20	)	)	PUNCT
ejpam-1781	284	21	=	=	SYM
ejpam-1781	285	1	(	(	PUNCT
ejpam-1781	285	2	y	y	NOUN
ejpam-1781	285	3	]	]	X
ejpam-1781	285	4	.	.	PUNCT
ejpam-1781	286	1	hence	hence	ADV
ejpam-1781	286	2	m	m	VERB
ejpam-1781	286	3	=	=	PUNCT
ejpam-1781	286	4	(	(	PUNCT
ejpam-1781	286	5	m	m	PROPN
ejpam-1781	286	6	∨	∨	NOUN
ejpam-1781	286	7	(	(	PUNCT
ejpam-1781	287	1	a	a	PRON
ejpam-1781	287	2	]	]	X
ejpam-1781	287	3	)	)	PUNCT
ejpam-1781	287	4	∧	∧	PROPN
ejpam-1781	287	5	(	(	PUNCT
ejpam-1781	287	6	m	m	PROPN
ejpam-1781	287	7	∨	∨	NOUN
ejpam-1781	287	8	(	(	PUNCT
ejpam-1781	287	9	b	b	NOUN
ejpam-1781	287	10	]	]	X
ejpam-1781	287	11	)	)	PUNCT
ejpam-1781	287	12	=	=	SYM
ejpam-1781	287	13	(	(	PUNCT
ejpam-1781	287	14	x	x	SYM
ejpam-1781	287	15	]	]	X
ejpam-1781	287	16	∧	∧	PROPN
ejpam-1781	287	17	(	(	PUNCT
ejpam-1781	287	18	y	y	NOUN
ejpam-1781	287	19	]	]	X
ejpam-1781	287	20	=	=	SYM
ejpam-1781	287	21	(	(	PUNCT
ejpam-1781	287	22	x	x	PUNCT
ejpam-1781	287	23	∧	∧	PROPN
ejpam-1781	287	24	y	y	PROPN
ejpam-1781	287	25	]	]	X
ejpam-1781	287	26	(	(	PUNCT
ejpam-1781	287	27	by	by	ADP
ejpam-1781	287	28	result	result	NOUN
ejpam-1781	287	29	6	6	NUM
ejpam-1781	287	30	)	)	PUNCT
ejpam-1781	287	31	;	;	PUNCT
ejpam-1781	287	32	contradicting	contradict	VERB
ejpam-1781	287	33	that	that	SCONJ
ejpam-1781	287	34	m	m	NOUN
ejpam-1781	287	35	is	be	AUX
ejpam-1781	287	36	not	not	PART
ejpam-1781	287	37	principal	principal	ADJ
ejpam-1781	287	38	.	.	PUNCT
ejpam-1781	288	1	hence	hence	ADV
ejpam-1781	288	2	the	the	DET
ejpam-1781	288	3	implication	implication	NOUN
ejpam-1781	288	4	.	.	PUNCT
ejpam-1781	289	1	thus	thus	ADV
ejpam-1781	289	2	as	as	SCONJ
ejpam-1781	289	3	(	(	PUNCT
ejpam-1781	289	4	1)⇔	1)⇔	NUM
ejpam-1781	289	5	(	(	PUNCT
ejpam-1781	289	6	2)⇔	2)⇔	NUM
ejpam-1781	289	7	(	(	PUNCT
ejpam-1781	289	8	3)⇔	3)⇔	NUM
ejpam-1781	289	9	(	(	PUNCT
ejpam-1781	289	10	4	4	NUM
ejpam-1781	289	11	)	)	PUNCT
ejpam-1781	289	12	result	result	NOUN
ejpam-1781	289	13	follows	follow	VERB
ejpam-1781	289	14	.	.	PUNCT
ejpam-1781	290	1	for	for	ADP
ejpam-1781	290	2	a	a	DET
ejpam-1781	290	3	∈	∈	PROPN
ejpam-1781	290	4	r	r	NOUN
ejpam-1781	290	5	,	,	PUNCT
ejpam-1781	290	6	an	an	DET
ejpam-1781	290	7	ideal	ideal	NOUN
ejpam-1781	290	8	in	in	ADP
ejpam-1781	290	9	r	r	NOUN
ejpam-1781	290	10	which	which	PRON
ejpam-1781	290	11	is	be	AUX
ejpam-1781	290	12	maximal	maximal	ADJ
ejpam-1781	290	13	w.r.t	w.r.t	NOUN
ejpam-1781	290	14	not	not	PART
ejpam-1781	290	15	containing	contain	VERB
ejpam-1781	290	16	the	the	DET
ejpam-1781	290	17	element	element	NOUN
ejpam-1781	290	18	a	a	PRON
ejpam-1781	290	19	is	be	AUX
ejpam-1781	290	20	called	call	VERB
ejpam-1781	290	21	amaximal	amaximal	ADJ
ejpam-1781	290	22	ideal	ideal	NOUN
ejpam-1781	290	23	.	.	PUNCT
ejpam-1781	291	1	interestingly	interestingly	ADV
ejpam-1781	291	2	,	,	PUNCT
ejpam-1781	291	3	we	we	PRON
ejpam-1781	291	4	have	have	AUX
ejpam-1781	291	5	theorem	theorem	VERB
ejpam-1781	291	6	9	9	NUM
ejpam-1781	291	7	.	.	PUNCT
ejpam-1781	292	1	any	any	DET
ejpam-1781	292	2	a	a	PRON
ejpam-1781	292	3	-	-	PUNCT
ejpam-1781	292	4	maximal	maximal	ADJ
ejpam-1781	292	5	ideal	ideal	NOUN
ejpam-1781	292	6	in	in	ADP
ejpam-1781	292	7	r	r	NOUN
ejpam-1781	292	8	is	be	AUX
ejpam-1781	292	9	prime	prime	ADJ
ejpam-1781	292	10	.	.	PUNCT
ejpam-1781	293	1	proof	proof	NOUN
ejpam-1781	293	2	.	.	PUNCT
ejpam-1781	294	1	let	let	VERB
ejpam-1781	294	2	m	m	PRON
ejpam-1781	294	3	be	be	AUX
ejpam-1781	294	4	a	a	DET
ejpam-1781	294	5	-	-	PUNCT
ejpam-1781	294	6	maximal	maximal	ADJ
ejpam-1781	294	7	ideal	ideal	NOUN
ejpam-1781	294	8	in	in	ADP
ejpam-1781	294	9	r.	r.	PROPN
ejpam-1781	294	10	then	then	ADV
ejpam-1781	294	11	a	a	PRON
ejpam-1781	294	12	/∈	/∈	INTJ
ejpam-1781	294	13	m	m	VERB
ejpam-1781	294	14	.	.	PUNCT
ejpam-1781	295	1	suppose	suppose	VERB
ejpam-1781	295	2	there	there	PRON
ejpam-1781	295	3	exist	exist	VERB
ejpam-1781	295	4	x	x	PRON
ejpam-1781	295	5	,	,	PUNCT
ejpam-1781	295	6	y	y	PROPN
ejpam-1781	295	7	∈	∈	PROPN
ejpam-1781	295	8	r	r	NOUN
ejpam-1781	295	9	such	such	ADJ
ejpam-1781	295	10	that	that	SCONJ
ejpam-1781	295	11	x	x	PUNCT
ejpam-1781	295	12	∧	∧	NOUN
ejpam-1781	295	13	y	y	PROPN
ejpam-1781	295	14	∈	∈	PROPN
ejpam-1781	295	15	m	m	VERB
ejpam-1781	295	16	with	with	ADP
ejpam-1781	295	17	x	x	PROPN
ejpam-1781	295	18	/∈	/∈	PUNCT
ejpam-1781	295	19	m	m	VERB
ejpam-1781	295	20	and	and	CCONJ
ejpam-1781	295	21	y	y	PROPN
ejpam-1781	295	22	/∈	/∈	PUNCT
ejpam-1781	296	1	m	m	INTJ
ejpam-1781	296	2	.	.	PUNCT
ejpam-1781	297	1	but	but	CCONJ
ejpam-1781	297	2	then	then	ADV
ejpam-1781	297	3	a	a	DET
ejpam-1781	297	4	∈	∈	NOUN
ejpam-1781	297	5	m	m	VERB
ejpam-1781	297	6	∨	∨	NOUN
ejpam-1781	297	7	(	(	PUNCT
ejpam-1781	297	8	x	x	X
ejpam-1781	297	9	]	]	X
ejpam-1781	297	10	and	and	CCONJ
ejpam-1781	297	11	y	y	PROPN
ejpam-1781	297	12	∨	∨	PROPN
ejpam-1781	297	13	m	m	PROPN
ejpam-1781	297	14	∨	∨	NOUN
ejpam-1781	297	15	(	(	PUNCT
ejpam-1781	297	16	y	y	NOUN
ejpam-1781	297	17	]	]	X
ejpam-1781	297	18	will	will	AUX
ejpam-1781	297	19	imply	imply	VERB
ejpam-1781	297	20	a	a	DET
ejpam-1781	297	21	=	=	X
ejpam-1781	297	22	m1	m1	PROPN
ejpam-1781	297	23	∨	∨	X
ejpam-1781	297	24	(	(	PUNCT
ejpam-1781	297	25	t	t	PROPN
ejpam-1781	297	26	∧	∧	PROPN
ejpam-1781	297	27	x	x	NOUN
ejpam-1781	297	28	)	)	PUNCT
ejpam-1781	297	29	and	and	CCONJ
ejpam-1781	297	30	y	y	PROPN
ejpam-1781	297	31	=	=	SYM
ejpam-1781	297	32	m2	m2	PROPN
ejpam-1781	297	33	∨	∨	PROPN
ejpam-1781	297	34	(	(	PUNCT
ejpam-1781	297	35	s	s	AUX
ejpam-1781	297	36	∧	∧	PROPN
ejpam-1781	297	37	y	y	PROPN
ejpam-1781	297	38	)	)	PUNCT
ejpam-1781	297	39	,	,	PUNCT
ejpam-1781	297	40	for	for	ADP
ejpam-1781	297	41	some	some	DET
ejpam-1781	297	42	t	t	PROPN
ejpam-1781	297	43	,	,	PUNCT
ejpam-1781	297	44	s	s	PART
ejpam-1781	297	45	∈	∈	PROPN
ejpam-1781	297	46	r	r	NOUN
ejpam-1781	297	47	.	.	PUNCT
ejpam-1781	298	1	thus	thus	ADV
ejpam-1781	298	2	a	a	DET
ejpam-1781	298	3	=	=	SYM
ejpam-1781	298	4	�	�	PROPN
ejpam-1781	298	5	m1	m1	PROPN
ejpam-1781	298	6	∨	∨	PROPN
ejpam-1781	298	7	(	(	PUNCT
ejpam-1781	298	8	t	t	PROPN
ejpam-1781	298	9	∧	∧	PROPN
ejpam-1781	298	10	x	x	NOUN
ejpam-1781	298	11	)	)	PUNCT
ejpam-1781	298	12	�	�	PROPN
ejpam-1781	298	13	∧	∧	PROPN
ejpam-1781	298	14	�	�	PROPN
ejpam-1781	298	15	m2	m2	PROPN
ejpam-1781	298	16	∨	∨	PROPN
ejpam-1781	298	17	(	(	PUNCT
ejpam-1781	298	18	s	s	VERB
ejpam-1781	298	19	∧	∧	PROPN
ejpam-1781	298	20	y	y	PROPN
ejpam-1781	298	21	)	)	PUNCT
ejpam-1781	298	22	�	�	PROPN
ejpam-1781	299	1	=	=	SYM
ejpam-1781	299	2	m1	m1	PROPN
ejpam-1781	299	3	∧	∧	PROPN
ejpam-1781	299	4	�	�	PROPN
ejpam-1781	299	5	m2	m2	PROPN
ejpam-1781	299	6	∨	∨	PROPN
ejpam-1781	299	7	(	(	PUNCT
ejpam-1781	299	8	s	s	VERB
ejpam-1781	299	9	∧	∧	PROPN
ejpam-1781	299	10	y	y	PROPN
ejpam-1781	299	11	)	)	PUNCT
ejpam-1781	299	12	�	�	PROPN
ejpam-1781	299	13	∨	∨	PROPN
ejpam-1781	299	14	(	(	PUNCT
ejpam-1781	299	15	t	t	PROPN
ejpam-1781	299	16	∧	∧	PROPN
ejpam-1781	299	17	x)∧	x)∧	PUNCT
ejpam-1781	299	18	�	�	PROPN
ejpam-1781	299	19	m2	m2	PROPN
ejpam-1781	299	20	∨	∨	PROPN
ejpam-1781	299	21	(	(	PUNCT
ejpam-1781	299	22	s	s	VERB
ejpam-1781	299	23	∧	∧	PROPN
ejpam-1781	299	24	y	y	PROPN
ejpam-1781	299	25	)	)	PUNCT
ejpam-1781	299	26	�	�	PROPN
ejpam-1781	299	27	=(	=(	PROPN
ejpam-1781	299	28	m1	m1	PROPN
ejpam-1781	299	29	∧m2)∨	∧m2)∨	PROPN
ejpam-1781	299	30	�	�	PROPN
ejpam-1781	299	31	m1	m1	PROPN
ejpam-1781	299	32	∧	∧	PROPN
ejpam-1781	299	33	(	(	PUNCT
ejpam-1781	299	34	s	s	PROPN
ejpam-1781	299	35	∧	∧	PROPN
ejpam-1781	299	36	y	y	PROPN
ejpam-1781	299	37	)	)	PUNCT
ejpam-1781	299	38	�	�	PROPN
ejpam-1781	299	39	∨	∨	PROPN
ejpam-1781	299	40	(	(	PUNCT
ejpam-1781	299	41	t	t	PROPN
ejpam-1781	299	42	∧	∧	PROPN
ejpam-1781	299	43	x)∧m2)∨	x)∧m2)∨	INTJ
ejpam-1781	299	44	(	(	PUNCT
ejpam-1781	299	45	t	t	PROPN
ejpam-1781	299	46	∧	∧	PROPN
ejpam-1781	299	47	x)∧	x)∧	PUNCT
ejpam-1781	299	48	(	(	PUNCT
ejpam-1781	299	49	s	s	PROPN
ejpam-1781	299	50	∧	∧	PROPN
ejpam-1781	299	51	y	y	PROPN
ejpam-1781	299	52	)	)	PUNCT
ejpam-1781	299	53	.	.	PUNCT
ejpam-1781	299	54	.	.	PUNCT
ejpam-1781	299	55	.	.	PUNCT
ejpam-1781	300	1	(	(	PUNCT
ejpam-1781	300	2	1	1	X
ejpam-1781	300	3	)	)	PUNCT
ejpam-1781	300	4	m1	m1	NOUN
ejpam-1781	300	5	∧m2	∧m2	NOUN
ejpam-1781	300	6	∈	∈	PROPN
ejpam-1781	300	7	m	m	NOUN
ejpam-1781	300	8	and	and	CCONJ
ejpam-1781	300	9	m1	m1	PROPN
ejpam-1781	300	10	∧	∧	PROPN
ejpam-1781	300	11	(	(	PUNCT
ejpam-1781	300	12	s	s	PROPN
ejpam-1781	300	13	∧	∧	PROPN
ejpam-1781	300	14	y	y	PROPN
ejpam-1781	300	15	)	)	PUNCT
ejpam-1781	300	16	∈	∈	PROPN
ejpam-1781	300	17	m	m	X
ejpam-1781	300	18	(	(	PUNCT
ejpam-1781	300	19	since	since	SCONJ
ejpam-1781	300	20	m1	m1	PROPN
ejpam-1781	300	21	∈	∈	PROPN
ejpam-1781	300	22	m	m	PROPN
ejpam-1781	300	23	)	)	PUNCT
ejpam-1781	300	24	.	.	PUNCT
ejpam-1781	301	1	further	far	ADV
ejpam-1781	301	2	m2	m2	PROPN
ejpam-1781	301	3	∈	∈	PROPN
ejpam-1781	301	4	m	m	VERB
ejpam-1781	301	5	⇒	⇒	NOUN
ejpam-1781	301	6	m2	m2	PROPN
ejpam-1781	301	7	∧	∧	PROPN
ejpam-1781	301	8	(	(	PUNCT
ejpam-1781	301	9	t	t	PROPN
ejpam-1781	301	10	∧	∧	PROPN
ejpam-1781	301	11	x	x	NOUN
ejpam-1781	301	12	)	)	PUNCT
ejpam-1781	301	13	∈	∈	PROPN
ejpam-1781	301	14	m	m	VERB
ejpam-1781	301	15	⇒	⇒	NOUN
ejpam-1781	301	16	(	(	PUNCT
ejpam-1781	301	17	t	t	PROPN
ejpam-1781	301	18	∧	∧	PROPN
ejpam-1781	301	19	x)∧m2	x)∧m2	PROPN
ejpam-1781	301	20	∈	∈	PROPN
ejpam-1781	301	21	m	m	X
ejpam-1781	301	22	(	(	PUNCT
ejpam-1781	301	23	since	since	SCONJ
ejpam-1781	301	24	m	m	PROPN
ejpam-1781	301	25	is	be	AUX
ejpam-1781	301	26	an	an	DET
ejpam-1781	301	27	ideal	ideal	NOUN
ejpam-1781	301	28	)	)	PUNCT
ejpam-1781	301	29	.	.	PUNCT
ejpam-1781	302	1	again	again	ADV
ejpam-1781	302	2	(	(	PUNCT
ejpam-1781	302	3	t∧	t∧	NOUN
ejpam-1781	302	4	x)∧(s∧	x)∧(s∧	PROPN
ejpam-1781	302	5	y	y	X
ejpam-1781	302	6	)	)	PUNCT
ejpam-1781	303	1	=	=	PRON
ejpam-1781	303	2	t∧(x∧	t∧(x∧	VERB
ejpam-1781	303	3	s∧	s∧	PROPN
ejpam-1781	303	4	y	y	NOUN
ejpam-1781	303	5	)	)	PUNCT
ejpam-1781	303	6	=	=	SYM
ejpam-1781	304	1	t∧(s∧	t∧(s∧	PROPN
ejpam-1781	304	2	x∧	x∧	PROPN
ejpam-1781	304	3	y	y	PROPN
ejpam-1781	304	4	)	)	PUNCT
ejpam-1781	304	5	=	=	PUNCT
ejpam-1781	305	1	(	(	PUNCT
ejpam-1781	305	2	t∧	t∧	NOUN
ejpam-1781	305	3	s)∧(x∧	s)∧(x∧	PROPN
ejpam-1781	305	4	y	y	PROPN
ejpam-1781	305	5	)	)	PUNCT
ejpam-1781	305	6	.	.	PUNCT
ejpam-1781	306	1	as	as	SCONJ
ejpam-1781	306	2	x	x	X
ejpam-1781	306	3	∧	∧	PROPN
ejpam-1781	306	4	y	y	PROPN
ejpam-1781	306	5	∈	∈	PROPN
ejpam-1781	306	6	m	m	VERB
ejpam-1781	306	7	,	,	PUNCT
ejpam-1781	306	8	we	we	PRON
ejpam-1781	306	9	get	get	VERB
ejpam-1781	306	10	(	(	PUNCT
ejpam-1781	306	11	x	x	PUNCT
ejpam-1781	306	12	∧	∧	PROPN
ejpam-1781	306	13	y	y	NOUN
ejpam-1781	306	14	)	)	PUNCT
ejpam-1781	306	15	∧	∧	PROPN
ejpam-1781	306	16	(	(	PUNCT
ejpam-1781	306	17	t	t	PROPN
ejpam-1781	306	18	∧	∧	PROPN
ejpam-1781	306	19	s	s	PART
ejpam-1781	306	20	)	)	PUNCT
ejpam-1781	306	21	∈	∈	PROPN
ejpam-1781	306	22	m	m	NOUN
ejpam-1781	306	23	and	and	CCONJ
ejpam-1781	306	24	hence	hence	ADV
ejpam-1781	306	25	(	(	PUNCT
ejpam-1781	306	26	t	t	PROPN
ejpam-1781	306	27	∧	∧	PROPN
ejpam-1781	306	28	s)∧	s)∧	PROPN
ejpam-1781	306	29	(	(	PUNCT
ejpam-1781	306	30	x	x	PROPN
ejpam-1781	306	31	∧	∧	PROPN
ejpam-1781	306	32	y	y	PROPN
ejpam-1781	306	33	)	)	PUNCT
ejpam-1781	306	34	∈	∈	PROPN
ejpam-1781	306	35	m	m	NOUN
ejpam-1781	306	36	.	.	PUNCT
ejpam-1781	307	1	but	but	CCONJ
ejpam-1781	307	2	then	then	ADV
ejpam-1781	307	3	by	by	ADP
ejpam-1781	307	4	(	(	PUNCT
ejpam-1781	307	5	1	1	NUM
ejpam-1781	307	6	)	)	PUNCT
ejpam-1781	307	7	,	,	PUNCT
ejpam-1781	307	8	we	we	PRON
ejpam-1781	307	9	have	have	VERB
ejpam-1781	307	10	a	a	DET
ejpam-1781	307	11	∈	∈	NOUN
ejpam-1781	307	12	m	m	NOUN
ejpam-1781	307	13	;	;	PUNCT
ejpam-1781	307	14	a	a	DET
ejpam-1781	307	15	contradiction	contradiction	NOUN
ejpam-1781	307	16	.	.	PUNCT
ejpam-1781	308	1	hence	hence	ADV
ejpam-1781	308	2	m	m	PROPN
ejpam-1781	308	3	is	be	AUX
ejpam-1781	308	4	prime	prime	ADJ
ejpam-1781	308	5	.	.	PUNCT
ejpam-1781	309	1	an	an	DET
ejpam-1781	309	2	ideal	ideal	ADJ
ejpam-1781	309	3	j	j	PROPN
ejpam-1781	309	4	in	in	ADP
ejpam-1781	309	5	r	r	NOUN
ejpam-1781	309	6	is	be	AUX
ejpam-1781	309	7	meet	meet	VERB
ejpam-1781	309	8	irreducible	irreducible	ADJ
ejpam-1781	309	9	if	if	SCONJ
ejpam-1781	309	10	j	j	PROPN
ejpam-1781	309	11	=	=	SYM
ejpam-1781	309	12	⋂	⋂	PROPN
ejpam-1781	310	1	λ∈∆	λ∈∆	X
ejpam-1781	310	2	iλ	iλ	PROPN
ejpam-1781	310	3	where	where	SCONJ
ejpam-1781	310	4	{	{	PUNCT
ejpam-1781	310	5	iλ}λ∈∆	iλ}λ∈∆	PROPN
ejpam-1781	310	6	is	be	AUX
ejpam-1781	310	7	a	a	DET
ejpam-1781	310	8	family	family	NOUN
ejpam-1781	310	9	of	of	ADP
ejpam-1781	310	10	ideals	ideal	NOUN
ejpam-1781	310	11	in	in	ADP
ejpam-1781	310	12	r	r	NOUN
ejpam-1781	310	13	(	(	PUNCT
ejpam-1781	310	14	∆	∆	PROPN
ejpam-1781	310	15	is	be	AUX
ejpam-1781	310	16	any	any	DET
ejpam-1781	310	17	indexing	indexing	NOUN
ejpam-1781	310	18	family	family	NOUN
ejpam-1781	310	19	)	)	PUNCT
ejpam-1781	310	20	,	,	PUNCT
ejpam-1781	310	21	then	then	ADV
ejpam-1781	310	22	j	j	PROPN
ejpam-1781	310	23	=	=	PUNCT
ejpam-1781	310	24	iλ	iλ	PROPN
ejpam-1781	310	25	for	for	ADP
ejpam-1781	310	26	some	some	DET
ejpam-1781	310	27	λ	λ	NOUN
ejpam-1781	310	28	∈∆.	∈∆.	PROPN
ejpam-1781	310	29	in	in	ADP
ejpam-1781	310	30	the	the	DET
ejpam-1781	310	31	following	follow	VERB
ejpam-1781	310	32	theorem	theorem	NOUN
ejpam-1781	310	33	we	we	PRON
ejpam-1781	310	34	furnish	furnish	VERB
ejpam-1781	310	35	some	some	DET
ejpam-1781	310	36	characterizations	characterization	NOUN
ejpam-1781	310	37	of	of	ADP
ejpam-1781	310	38	a	a	DET
ejpam-1781	310	39	-	-	PUNCT
ejpam-1781	310	40	maximal	maximal	ADJ
ejpam-1781	310	41	ideals	ideal	NOUN
ejpam-1781	310	42	in	in	ADP
ejpam-1781	310	43	r.	r.	PROPN
ejpam-1781	310	44	theorem	theorem	PROPN
ejpam-1781	310	45	10	10	NUM
ejpam-1781	310	46	.	.	PUNCT
ejpam-1781	311	1	following	follow	VERB
ejpam-1781	311	2	statements	statement	NOUN
ejpam-1781	311	3	are	be	AUX
ejpam-1781	311	4	equivalent	equivalent	ADJ
ejpam-1781	311	5	in	in	ADP
ejpam-1781	311	6	r.	r.	PROPN
ejpam-1781	311	7	1	1	NUM
ejpam-1781	311	8	.	.	PUNCT
ejpam-1781	312	1	m	m	PROPN
ejpam-1781	312	2	is	be	AUX
ejpam-1781	312	3	a	a	PRON
ejpam-1781	312	4	-	-	PUNCT
ejpam-1781	312	5	maximal	maximal	ADJ
ejpam-1781	312	6	ideal	ideal	NOUN
ejpam-1781	312	7	for	for	ADP
ejpam-1781	312	8	some	some	DET
ejpam-1781	312	9	a	a	DET
ejpam-1781	312	10	∈	∈	PROPN
ejpam-1781	312	11	r.	r.	PROPN
ejpam-1781	312	12	2	2	NUM
ejpam-1781	312	13	.	.	PUNCT
ejpam-1781	313	1	m	m	PROPN
ejpam-1781	313	2	is	be	AUX
ejpam-1781	313	3	meet	meet	VERB
ejpam-1781	313	4	irreducible	irreducible	ADJ
ejpam-1781	313	5	3	3	NUM
ejpam-1781	313	6	.	.	PUNCT
ejpam-1781	313	7	m	m	VERB
ejpam-1781	314	1	⊂	⊂	NOUN
ejpam-1781	314	2	m	m	VERB
ejpam-1781	314	3	′	′	NUM
ejpam-1781	315	1	=	=	PUNCT
ejpam-1781	315	2	∩{i	∩{i	X
ejpam-1781	315	3	∈	∈	NOUN
ejpam-1781	316	1	i	i	PRON
ejpam-1781	316	2	(	(	PUNCT
ejpam-1781	316	3	r	r	NOUN
ejpam-1781	316	4	)	)	PUNCT
ejpam-1781	317	1	|	|	ADV
ejpam-1781	317	2	i	i	PRON
ejpam-1781	317	3	⊃	⊃	NOUN
ejpam-1781	317	4	m	m	VERB
ejpam-1781	317	5	}	}	PUNCT
ejpam-1781	317	6	4	4	NUM
ejpam-1781	317	7	.	.	PUNCT
ejpam-1781	317	8	m	m	PROPN
ejpam-1781	317	9	is	be	AUX
ejpam-1781	317	10	x	x	X
ejpam-1781	317	11	-	-	ADJ
ejpam-1781	317	12	maximal	maximal	ADJ
ejpam-1781	317	13	for	for	ADP
ejpam-1781	317	14	some	some	DET
ejpam-1781	317	15	x	x	SYM
ejpam-1781	317	16	∈	∈	PROPN
ejpam-1781	317	17	m	m	NOUN
ejpam-1781	317	18	′	′	NOUN
ejpam-1781	317	19	\m	\m	NOUN
ejpam-1781	317	20	.	.	PUNCT
ejpam-1781	318	1	y.	y.	PROPN
ejpam-1781	318	2	pawar	pawar	PROPN
ejpam-1781	318	3	,	,	PUNCT
ejpam-1781	318	4	i.	i.	PROPN
ejpam-1781	318	5	shaikh	shaikh	PROPN
ejpam-1781	318	6	/	/	SYM
ejpam-1781	318	7	eur	eur	PROPN
ejpam-1781	318	8	.	.	PUNCT
ejpam-1781	319	1	j.	j.	PROPN
ejpam-1781	319	2	pure	pure	PROPN
ejpam-1781	319	3	appl	appl	PROPN
ejpam-1781	319	4	.	.	PROPN
ejpam-1781	319	5	math	math	PROPN
ejpam-1781	319	6	,	,	PUNCT
ejpam-1781	319	7	6	6	NUM
ejpam-1781	319	8	(	(	PUNCT
ejpam-1781	319	9	2013	2013	NUM
ejpam-1781	319	10	)	)	PUNCT
ejpam-1781	319	11	,	,	PUNCT
ejpam-1781	319	12	107	107	NUM
ejpam-1781	319	13	-	-	SYM
ejpam-1781	319	14	118	118	NUM
ejpam-1781	319	15	115	115	NUM
ejpam-1781	319	16	proof	proof	NOUN
ejpam-1781	319	17	.	.	PUNCT
ejpam-1781	320	1	(	(	PUNCT
ejpam-1781	320	2	1)⇒	1)⇒	NUM
ejpam-1781	320	3	(	(	PUNCT
ejpam-1781	320	4	2	2	NUM
ejpam-1781	320	5	)	)	PUNCT
ejpam-1781	320	6	let	let	VERB
ejpam-1781	320	7	m	m	NOUN
ejpam-1781	320	8	=	=	SYM
ejpam-1781	320	9	⋂	⋂	PROPN
ejpam-1781	321	1	λ∈∆	λ∈∆	X
ejpam-1781	321	2	iλ	iλ	PROPN
ejpam-1781	321	3	where	where	SCONJ
ejpam-1781	321	4	{	{	PUNCT
ejpam-1781	321	5	iλ}λ∈∆	iλ}λ∈∆	PROPN
ejpam-1781	321	6	is	be	AUX
ejpam-1781	321	7	a	a	DET
ejpam-1781	321	8	family	family	NOUN
ejpam-1781	321	9	of	of	ADP
ejpam-1781	321	10	ideals	ideal	NOUN
ejpam-1781	321	11	in	in	ADP
ejpam-1781	321	12	r	r	NOUN
ejpam-1781	321	13	and	and	CCONJ
ejpam-1781	321	14	∆	∆	PROPN
ejpam-1781	321	15	is	be	AUX
ejpam-1781	321	16	any	any	DET
ejpam-1781	321	17	indexing	indexing	NOUN
ejpam-1781	321	18	family	family	NOUN
ejpam-1781	321	19	.	.	PUNCT
ejpam-1781	322	1	m	m	PROPN
ejpam-1781	322	2	is	be	AUX
ejpam-1781	322	3	a	a	DET
ejpam-1781	322	4	-	-	PUNCT
ejpam-1781	322	5	maximal⇒	maximal⇒	PROPN
ejpam-1781	322	6	a	a	NOUN
ejpam-1781	322	7	/∈	/∈	NOUN
ejpam-1781	322	8	m	m	AUX
ejpam-1781	322	9	⇒	⇒	NOUN
ejpam-1781	322	10	a	a	DET
ejpam-1781	322	11	/∈	/∈	INTJ
ejpam-1781	322	12	iλ0	iλ0	NOUN
ejpam-1781	322	13	for	for	ADP
ejpam-1781	322	14	some	some	DET
ejpam-1781	322	15	λ0	λ0	NOUN
ejpam-1781	322	16	∈	∈	NOUN
ejpam-1781	322	17	∆.	∆.	X
ejpam-1781	322	18	m	m	AUX
ejpam-1781	322	19	being	be	AUX
ejpam-1781	322	20	amaximal	amaximal	ADJ
ejpam-1781	322	21	ideal	ideal	NOUN
ejpam-1781	322	22	,	,	PUNCT
ejpam-1781	322	23	we	we	PRON
ejpam-1781	322	24	get	get	VERB
ejpam-1781	322	25	iλ0	iλ0	ADJ
ejpam-1781	322	26	=	=	VERB
ejpam-1781	322	27	m	m	VERB
ejpam-1781	322	28	as	as	ADP
ejpam-1781	322	29	m	m	PROPN
ejpam-1781	322	30	⊆	⊆	NUM
ejpam-1781	322	31	iλ0	iλ0	NOUN
ejpam-1781	322	32	.	.	PUNCT
ejpam-1781	323	1	(	(	PUNCT
ejpam-1781	323	2	2)⇒	2)⇒	NUM
ejpam-1781	323	3	(	(	PUNCT
ejpam-1781	323	4	3	3	X
ejpam-1781	323	5	)	)	PUNCT
ejpam-1781	323	6	let	let	VERB
ejpam-1781	323	7	if	if	SCONJ
ejpam-1781	323	8	possible	possible	ADJ
ejpam-1781	323	9	m	m	VERB
ejpam-1781	323	10	⊂	⊂	NOUN
ejpam-1781	323	11	m	m	VERB
ejpam-1781	323	12	′	′	NUM
ejpam-1781	324	1	=	=	PUNCT
ejpam-1781	324	2	∩{i	∩{i	X
ejpam-1781	324	3	∈	∈	NOUN
ejpam-1781	325	1	i	i	PRON
ejpam-1781	325	2	(	(	PUNCT
ejpam-1781	325	3	r	r	NOUN
ejpam-1781	325	4	)	)	PUNCT
ejpam-1781	326	1	|	|	ADV
ejpam-1781	326	2	i	i	PRON
ejpam-1781	326	3	⊃	⊃	NOUN
ejpam-1781	326	4	m	m	VERB
ejpam-1781	326	5	}	}	PUNCT
ejpam-1781	326	6	.	.	PUNCT
ejpam-1781	327	1	by	by	ADP
ejpam-1781	327	2	assumption	assumption	NOUN
ejpam-1781	327	3	(	(	PUNCT
ejpam-1781	327	4	2	2	NUM
ejpam-1781	327	5	)	)	PUNCT
ejpam-1781	327	6	,	,	PUNCT
ejpam-1781	327	7	m	m	VERB
ejpam-1781	327	8	=	=	VERB
ejpam-1781	328	1	i	i	PRON
ejpam-1781	328	2	for	for	ADP
ejpam-1781	328	3	some	some	PRON
ejpam-1781	328	4	i	i	PRON
ejpam-1781	328	5	⊃	⊃	VERB
ejpam-1781	328	6	m	m	VERB
ejpam-1781	328	7	,	,	PUNCT
ejpam-1781	328	8	a	a	DET
ejpam-1781	328	9	contradiction	contradiction	NOUN
ejpam-1781	328	10	.	.	PUNCT
ejpam-1781	329	1	hence	hence	ADV
ejpam-1781	329	2	m	m	VERB
ejpam-1781	329	3	⊂	⊂	PROPN
ejpam-1781	329	4	m	m	VERB
ejpam-1781	329	5	′.	′.	NOUN
ejpam-1781	329	6	(	(	PUNCT
ejpam-1781	329	7	3)⇒	3)⇒	NUM
ejpam-1781	329	8	(	(	PUNCT
ejpam-1781	329	9	4	4	NUM
ejpam-1781	329	10	)	)	PUNCT
ejpam-1781	329	11	by	by	ADP
ejpam-1781	329	12	(	(	PUNCT
ejpam-1781	329	13	3	3	NUM
ejpam-1781	329	14	)	)	PUNCT
ejpam-1781	329	15	,	,	PUNCT
ejpam-1781	329	16	m	m	VERB
ejpam-1781	329	17	⊂	⊂	NOUN
ejpam-1781	329	18	m	m	VERB
ejpam-1781	329	19	′	′	NUM
ejpam-1781	330	1	=	=	PUNCT
ejpam-1781	330	2	∩{i	∩{i	X
ejpam-1781	330	3	∈	∈	NOUN
ejpam-1781	331	1	i	i	PRON
ejpam-1781	331	2	(	(	PUNCT
ejpam-1781	331	3	r	r	NOUN
ejpam-1781	331	4	)	)	PUNCT
ejpam-1781	332	1	|	|	ADV
ejpam-1781	332	2	i	i	PRON
ejpam-1781	332	3	⊃	⊃	NOUN
ejpam-1781	332	4	m	m	ADV
ejpam-1781	332	5	}	}	PUNCT
ejpam-1781	332	6	.	.	PUNCT
ejpam-1781	333	1	select	select	ADJ
ejpam-1781	333	2	x	x	PUNCT
ejpam-1781	333	3	∈	∈	PROPN
ejpam-1781	333	4	m	m	NOUN
ejpam-1781	333	5	′	′	NOUN
ejpam-1781	333	6	\m	\m	NOUN
ejpam-1781	333	7	.	.	PUNCT
ejpam-1781	334	1	thus	thus	ADV
ejpam-1781	334	2	x	x	X
ejpam-1781	334	3	∈	∈	NOUN
ejpam-1781	334	4	i	i	PRON
ejpam-1781	334	5	for	for	ADP
ejpam-1781	334	6	each	each	DET
ejpam-1781	334	7	i	i	PRON
ejpam-1781	334	8	∈	∈	PROPN
ejpam-1781	334	9	i(r	i(r	PROPN
ejpam-1781	334	10	)	)	PUNCT
ejpam-1781	334	11	with	with	ADP
ejpam-1781	334	12	i	i	PROPN
ejpam-1781	334	13	⊃	⊃	NOUN
ejpam-1781	334	14	m	m	VERB
ejpam-1781	334	15	.	.	PUNCT
ejpam-1781	335	1	if	if	SCONJ
ejpam-1781	335	2	m	m	NOUN
ejpam-1781	335	3	is	be	AUX
ejpam-1781	335	4	not	not	PART
ejpam-1781	335	5	x	x	SYM
ejpam-1781	335	6	-maximal	-maximal	ADJ
ejpam-1781	335	7	,	,	PUNCT
ejpam-1781	335	8	then	then	ADV
ejpam-1781	335	9	there	there	PRON
ejpam-1781	335	10	exists	exist	VERB
ejpam-1781	335	11	an	an	DET
ejpam-1781	335	12	ideal	ideal	NOUN
ejpam-1781	335	13	say	say	VERB
ejpam-1781	335	14	j	j	PROPN
ejpam-1781	335	15	properly	properly	ADV
ejpam-1781	335	16	containing	contain	VERB
ejpam-1781	335	17	m	m	PRON
ejpam-1781	335	18	and	and	CCONJ
ejpam-1781	335	19	not	not	PART
ejpam-1781	335	20	containing	contain	VERB
ejpam-1781	335	21	x	x	PUNCT
ejpam-1781	335	22	(	(	PUNCT
ejpam-1781	335	23	see	see	VERB
ejpam-1781	335	24	result	result	NOUN
ejpam-1781	335	25	1	1	NUM
ejpam-1781	335	26	)	)	PUNCT
ejpam-1781	335	27	.	.	PUNCT
ejpam-1781	336	1	but	but	CCONJ
ejpam-1781	336	2	then	then	ADV
ejpam-1781	336	3	j	j	PROPN
ejpam-1781	336	4	∈	∈	PROPN
ejpam-1781	336	5	{	{	PUNCT
ejpam-1781	336	6	i	i	NOUN
ejpam-1781	336	7	∈	∈	PROPN
ejpam-1781	337	1	i	i	PRON
ejpam-1781	337	2	(	(	PUNCT
ejpam-1781	337	3	r	r	NOUN
ejpam-1781	337	4	)	)	PUNCT
ejpam-1781	338	1	|	|	ADV
ejpam-1781	338	2	i	i	PRON
ejpam-1781	338	3	⊃	⊃	NOUN
ejpam-1781	338	4	m	m	VERB
ejpam-1781	338	5	}	}	PUNCT
ejpam-1781	338	6	.	.	PUNCT
ejpam-1781	339	1	hence	hence	ADV
ejpam-1781	339	2	x	x	SYM
ejpam-1781	339	3	∈	∈	PROPN
ejpam-1781	339	4	j	j	PROPN
ejpam-1781	339	5	;	;	PUNCT
ejpam-1781	339	6	a	a	DET
ejpam-1781	339	7	contradiction	contradiction	NOUN
ejpam-1781	339	8	.	.	PUNCT
ejpam-1781	340	1	therefore	therefore	ADV
ejpam-1781	340	2	m	m	PROPN
ejpam-1781	340	3	is	be	AUX
ejpam-1781	340	4	x	x	SYM
ejpam-1781	340	5	-maximal	-maximal	PROPN
ejpam-1781	340	6	for	for	ADP
ejpam-1781	340	7	any	any	DET
ejpam-1781	340	8	x	x	SYM
ejpam-1781	340	9	∈	∈	PROPN
ejpam-1781	340	10	m	m	NOUN
ejpam-1781	340	11	′	′	NOUN
ejpam-1781	340	12	\m	\m	NOUN
ejpam-1781	340	13	.	.	PUNCT
ejpam-1781	341	1	(	(	PUNCT
ejpam-1781	341	2	4)⇒	4)⇒	X
ejpam-1781	341	3	(	(	PUNCT
ejpam-1781	341	4	1	1	NUM
ejpam-1781	341	5	)	)	PUNCT
ejpam-1781	341	6	being	be	AUX
ejpam-1781	341	7	obviously	obviously	ADV
ejpam-1781	341	8	true	true	ADJ
ejpam-1781	341	9	,	,	PUNCT
ejpam-1781	341	10	all	all	DET
ejpam-1781	341	11	the	the	DET
ejpam-1781	341	12	statements	statement	NOUN
ejpam-1781	341	13	are	be	AUX
ejpam-1781	341	14	equivalent	equivalent	ADJ
ejpam-1781	341	15	.	.	PUNCT
ejpam-1781	342	1	similar	similar	ADJ
ejpam-1781	342	2	to	to	ADP
ejpam-1781	342	3	the	the	DET
ejpam-1781	342	4	result	result	NOUN
ejpam-1781	342	5	in	in	ADP
ejpam-1781	342	6	ring	ring	NOUN
ejpam-1781	342	7	-	-	PUNCT
ejpam-1781	342	8	theory	theory	NOUN
ejpam-1781	342	9	,	,	PUNCT
ejpam-1781	342	10	we	we	PRON
ejpam-1781	342	11	have	have	AUX
ejpam-1781	342	12	(	(	PUNCT
ejpam-1781	342	13	see	see	VERB
ejpam-1781	342	14	[	[	X
ejpam-1781	342	15	reticulated	reticulate	VERB
ejpam-1781	342	16	rings	ring	NOUN
ejpam-1781	342	17	]	]	PUNCT
ejpam-1781	342	18	)	)	PUNCT
ejpam-1781	342	19	theorem	theorem	VERB
ejpam-1781	342	20	11	11	NUM
ejpam-1781	342	21	.	.	PUNCT
ejpam-1781	343	1	for	for	ADP
ejpam-1781	343	2	each	each	DET
ejpam-1781	343	3	element	element	NOUN
ejpam-1781	343	4	x	x	PUNCT
ejpam-1781	343	5	and	and	CCONJ
ejpam-1781	343	6	a	a	DET
ejpam-1781	343	7	prime	prime	ADJ
ejpam-1781	343	8	ideal	ideal	NOUN
ejpam-1781	343	9	p	p	NOUN
ejpam-1781	343	10	of	of	ADP
ejpam-1781	343	11	r	r	NOUN
ejpam-1781	343	12	,	,	PUNCT
ejpam-1781	343	13	the	the	DET
ejpam-1781	343	14	following	follow	VERB
ejpam-1781	343	15	are	be	AUX
ejpam-1781	343	16	equivalent	equivalent	ADJ
ejpam-1781	343	17	(	(	PUNCT
ejpam-1781	343	18	i	i	NOUN
ejpam-1781	343	19	)	)	PUNCT
ejpam-1781	343	20	{	{	PUNCT
ejpam-1781	343	21	x}∗	x}∗	NOUN
ejpam-1781	343	22	⊆	⊆	NUM
ejpam-1781	343	23	p	p	PROPN
ejpam-1781	343	24	(	(	PUNCT
ejpam-1781	343	25	ii	ii	NOUN
ejpam-1781	343	26	)	)	PUNCT
ejpam-1781	343	27	there	there	PRON
ejpam-1781	343	28	is	be	VERB
ejpam-1781	343	29	some	some	DET
ejpam-1781	343	30	m	m	NOUN
ejpam-1781	343	31	∈m	∈m	NOUN
ejpam-1781	343	32	with	with	ADP
ejpam-1781	343	33	m	m	PROPN
ejpam-1781	343	34	⊆	⊆	NUM
ejpam-1781	343	35	p	p	NOUN
ejpam-1781	343	36	and	and	CCONJ
ejpam-1781	343	37	x	x	NOUN
ejpam-1781	343	38	/∈	/∈	PUNCT
ejpam-1781	343	39	m.	m.	NOUN
ejpam-1781	343	40	proof	proof	NOUN
ejpam-1781	343	41	.	.	PUNCT
ejpam-1781	344	1	(	(	PUNCT
ejpam-1781	344	2	i)⇒	i)⇒	PROPN
ejpam-1781	344	3	(	(	PUNCT
ejpam-1781	344	4	ii	ii	NOUN
ejpam-1781	344	5	)	)	PUNCT
ejpam-1781	344	6	define	define	VERB
ejpam-1781	344	7	s	s	NOUN
ejpam-1781	344	8	=	=	PUNCT
ejpam-1781	344	9	{	{	PUNCT
ejpam-1781	344	10	a	a	DET
ejpam-1781	344	11	∧	∧	PROPN
ejpam-1781	344	12	x	x	PUNCT
ejpam-1781	344	13	|	|	ADV
ejpam-1781	344	14	a	a	PRON
ejpam-1781	344	15	/∈	/∈	NOUN
ejpam-1781	344	16	p	p	X
ejpam-1781	344	17	}	}	PUNCT
ejpam-1781	344	18	.	.	PUNCT
ejpam-1781	345	1	if	if	SCONJ
ejpam-1781	345	2	0	0	NUM
ejpam-1781	345	3	∈	∈	PROPN
ejpam-1781	345	4	s	s	NOUN
ejpam-1781	345	5	,	,	PUNCT
ejpam-1781	345	6	then	then	ADV
ejpam-1781	345	7	a	a	DET
ejpam-1781	345	8	∧	∧	PROPN
ejpam-1781	345	9	x	x	X
ejpam-1781	345	10	=	=	SYM
ejpam-1781	345	11	0	0	NUM
ejpam-1781	345	12	for	for	ADP
ejpam-1781	345	13	some	some	DET
ejpam-1781	345	14	a	a	PRON
ejpam-1781	345	15	/∈	/∈	PUNCT
ejpam-1781	346	1	p	p	NOUN
ejpam-1781	346	2	will	will	AUX
ejpam-1781	346	3	imply	imply	VERB
ejpam-1781	346	4	a	a	DET
ejpam-1781	346	5	∈	∈	NOUN
ejpam-1781	346	6	{	{	PUNCT
ejpam-1781	346	7	x}∗	x}∗	NOUN
ejpam-1781	346	8	⊆	⊆	NUM
ejpam-1781	346	9	p	p	NOUN
ejpam-1781	346	10	(	(	PUNCT
ejpam-1781	346	11	by	by	ADP
ejpam-1781	346	12	(	(	PUNCT
ejpam-1781	346	13	i	i	NOUN
ejpam-1781	346	14	)	)	PUNCT
ejpam-1781	346	15	)	)	PUNCT
ejpam-1781	346	16	;	;	PUNCT
ejpam-1781	346	17	a	a	DET
ejpam-1781	346	18	contradiction	contradiction	NOUN
ejpam-1781	346	19	.	.	PUNCT
ejpam-1781	347	1	hence	hence	ADV
ejpam-1781	347	2	0	0	NUM
ejpam-1781	347	3	/∈	/∈	PUNCT
ejpam-1781	348	1	s.	s.	PROPN
ejpam-1781	348	2	again	again	ADV
ejpam-1781	348	3	for	for	ADP
ejpam-1781	348	4	a	a	DET
ejpam-1781	348	5	maximal	maximal	ADJ
ejpam-1781	348	6	element	element	NOUN
ejpam-1781	348	7	m	m	VERB
ejpam-1781	348	8	in	in	ADP
ejpam-1781	348	9	r	r	NOUN
ejpam-1781	348	10	,	,	PUNCT
ejpam-1781	348	11	m	m	VERB
ejpam-1781	348	12	∧	∧	NOUN
ejpam-1781	348	13	x	x	X
ejpam-1781	348	14	=	=	PUNCT
ejpam-1781	348	15	x	x	X
ejpam-1781	348	16	and	and	CCONJ
ejpam-1781	348	17	m	m	PROPN
ejpam-1781	348	18	/∈	/∈	PUNCT
ejpam-1781	349	1	p	p	NOUN
ejpam-1781	349	2	will	will	AUX
ejpam-1781	349	3	give	give	VERB
ejpam-1781	349	4	x	x	SYM
ejpam-1781	349	5	∈	∈	PROPN
ejpam-1781	349	6	s.	s.	PROPN
ejpam-1781	349	7	hence	hence	ADV
ejpam-1781	349	8	s	s	VERB
ejpam-1781	349	9	is	be	AUX
ejpam-1781	349	10	non	non	ADJ
ejpam-1781	349	11	-	-	ADJ
ejpam-1781	349	12	empty	empty	ADJ
ejpam-1781	349	13	.	.	PUNCT
ejpam-1781	350	1	further	far	ADV
ejpam-1781	350	2	a	a	DET
ejpam-1781	350	3	∧	∧	PROPN
ejpam-1781	350	4	x	x	SYM
ejpam-1781	350	5	,	,	PUNCT
ejpam-1781	350	6	b	b	X
ejpam-1781	350	7	∧	∧	PROPN
ejpam-1781	350	8	y	y	PROPN
ejpam-1781	350	9	∈	∈	PROPN
ejpam-1781	350	10	s	s	VERB
ejpam-1781	350	11	for	for	ADP
ejpam-1781	350	12	a	a	PRON
ejpam-1781	350	13	,	,	PUNCT
ejpam-1781	350	14	b	b	X
ejpam-1781	350	15	∈	∈	PROPN
ejpam-1781	350	16	p	p	NOUN
ejpam-1781	350	17	implies	imply	VERB
ejpam-1781	350	18	(	(	PUNCT
ejpam-1781	350	19	a∧	a∧	NOUN
ejpam-1781	350	20	x)∧(b∧	x)∧(b∧	PROPN
ejpam-1781	350	21	y	y	X
ejpam-1781	350	22	)	)	PUNCT
ejpam-1781	350	23	=	=	PUNCT
ejpam-1781	351	1	a∧(x	a∧(x	NOUN
ejpam-1781	351	2	∧	∧	NOUN
ejpam-1781	351	3	b∧	b∧	PROPN
ejpam-1781	351	4	y	y	NOUN
ejpam-1781	351	5	)	)	PUNCT
ejpam-1781	352	1	=	=	SYM
ejpam-1781	352	2	a∧(b∧	a∧(b∧	NOUN
ejpam-1781	352	3	x	x	PUNCT
ejpam-1781	352	4	∧	∧	PROPN
ejpam-1781	352	5	y	y	PROPN
ejpam-1781	352	6	)	)	PUNCT
ejpam-1781	352	7	=	=	SYM
ejpam-1781	353	1	(	(	PUNCT
ejpam-1781	353	2	a∧	a∧	NOUN
ejpam-1781	353	3	b)∧(x	b)∧(x	X
ejpam-1781	353	4	∧	∧	PROPN
ejpam-1781	353	5	y	y	PROPN
ejpam-1781	353	6	)	)	PUNCT
ejpam-1781	353	7	∈	∈	PROPN
ejpam-1781	353	8	s	s	NOUN
ejpam-1781	353	9	as	as	ADP
ejpam-1781	353	10	a∧	a∧	PROPN
ejpam-1781	353	11	b	b	PROPN
ejpam-1781	353	12	/∈	/∈	PUNCT
ejpam-1781	354	1	p	p	X
ejpam-1781	354	2	(	(	PUNCT
ejpam-1781	354	3	p	p	X
ejpam-1781	354	4	being	be	AUX
ejpam-1781	354	5	a	a	DET
ejpam-1781	354	6	prime	prime	ADJ
ejpam-1781	354	7	ideal	ideal	NOUN
ejpam-1781	354	8	in	in	ADP
ejpam-1781	354	9	r	r	NOUN
ejpam-1781	354	10	)	)	PUNCT
ejpam-1781	354	11	.	.	PUNCT
ejpam-1781	355	1	thus	thus	ADV
ejpam-1781	355	2	s	s	X
ejpam-1781	355	3	is	be	AUX
ejpam-1781	355	4	a	a	DET
ejpam-1781	355	5	multiplicatively	multiplicatively	ADV
ejpam-1781	355	6	closed	close	VERB
ejpam-1781	355	7	subset	subset	NOUN
ejpam-1781	355	8	of	of	ADP
ejpam-1781	355	9	r	r	NOUN
ejpam-1781	355	10	not	not	PART
ejpam-1781	355	11	containing	contain	VERB
ejpam-1781	355	12	0	0	NUM
ejpam-1781	355	13	.	.	PUNCT
ejpam-1781	356	1	hence	hence	ADV
ejpam-1781	356	2	by	by	ADP
ejpam-1781	356	3	result	result	NOUN
ejpam-1781	356	4	12	12	NUM
ejpam-1781	356	5	there	there	PRON
ejpam-1781	356	6	exists	exist	VERB
ejpam-1781	356	7	a	a	DET
ejpam-1781	356	8	prime	prime	ADJ
ejpam-1781	356	9	ideal	ideal	NOUN
ejpam-1781	356	10	p	p	NOUN
ejpam-1781	356	11	in	in	ADP
ejpam-1781	356	12	r	r	NOUN
ejpam-1781	356	13	with	with	ADP
ejpam-1781	356	14	p	p	NOUN
ejpam-1781	356	15	∩	∩	NOUN
ejpam-1781	356	16	s	s	PART
ejpam-1781	356	17	=	=	X
ejpam-1781	356	18	;	;	PUNCT
ejpam-1781	356	19	.	.	PUNCT
ejpam-1781	357	1	hence	hence	ADV
ejpam-1781	357	2	x	x	PUNCT
ejpam-1781	357	3	/∈	/∈	PUNCT
ejpam-1781	358	1	p.	p.	NOUN
ejpam-1781	358	2	as	as	ADP
ejpam-1781	358	3	every	every	DET
ejpam-1781	358	4	prime	prime	ADJ
ejpam-1781	358	5	ideal	ideal	NOUN
ejpam-1781	358	6	in	in	ADP
ejpam-1781	358	7	r	r	NOUN
ejpam-1781	358	8	contains	contain	VERB
ejpam-1781	358	9	a	a	DET
ejpam-1781	358	10	minimal	minimal	ADJ
ejpam-1781	358	11	prime	prime	ADJ
ejpam-1781	358	12	ideal	ideal	NOUN
ejpam-1781	358	13	,	,	PUNCT
ejpam-1781	358	14	select	select	ADJ
ejpam-1781	358	15	m	m	NOUN
ejpam-1781	358	16	∈m	∈m	NOUN
ejpam-1781	358	17	such	such	ADJ
ejpam-1781	358	18	that	that	SCONJ
ejpam-1781	358	19	m	m	PROPN
ejpam-1781	358	20	⊆	⊆	NUM
ejpam-1781	358	21	p	p	NOUN
ejpam-1781	358	22	,	,	PUNCT
ejpam-1781	358	23	and	and	CCONJ
ejpam-1781	358	24	the	the	DET
ejpam-1781	358	25	implication	implication	NOUN
ejpam-1781	358	26	follows	follow	VERB
ejpam-1781	358	27	.	.	PUNCT
ejpam-1781	359	1	(	(	PUNCT
ejpam-1781	359	2	ii)⇒	ii)⇒	PROPN
ejpam-1781	359	3	(	(	PUNCT
ejpam-1781	359	4	i	i	NOUN
ejpam-1781	359	5	)	)	PUNCT
ejpam-1781	359	6	follows	follow	VERB
ejpam-1781	359	7	by	by	ADP
ejpam-1781	359	8	result	result	NOUN
ejpam-1781	359	9	2	2	NUM
ejpam-1781	359	10	.	.	PUNCT
ejpam-1781	360	1	4	4	X
ejpam-1781	360	2	.	.	X
ejpam-1781	361	1	the	the	DET
ejpam-1781	361	2	properties	property	NOUN
ejpam-1781	361	3	of	of	ADP
ejpam-1781	361	4	the	the	DET
ejpam-1781	361	5	set	set	NOUN
ejpam-1781	361	6	u(i	u(i	NOUN
ejpam-1781	361	7	)	)	PUNCT
ejpam-1781	361	8	for	for	ADP
ejpam-1781	361	9	any	any	DET
ejpam-1781	361	10	ideal	ideal	NOUN
ejpam-1781	361	11	i	i	PRON
ejpam-1781	361	12	in	in	ADP
ejpam-1781	361	13	r	r	NOUN
ejpam-1781	361	14	,	,	PUNCT
ejpam-1781	361	15	define	define	VERB
ejpam-1781	361	16	u(i	u(i	NOUN
ejpam-1781	361	17	)	)	PUNCT
ejpam-1781	361	18	=	=	PUNCT
ejpam-1781	361	19	{	{	PUNCT
ejpam-1781	361	20	p	p	NOUN
ejpam-1781	361	21	∈	∈	PROPN
ejpam-1781	361	22	℘	℘	NOUN
ejpam-1781	361	23	|	|	ADV
ejpam-1781	361	24	i	i	PRON
ejpam-1781	361	25	*	*	PUNCT
ejpam-1781	362	1	p	p	X
ejpam-1781	362	2	}	}	PUNCT
ejpam-1781	362	3	and	and	CCONJ
ejpam-1781	362	4	for	for	ADP
ejpam-1781	362	5	any	any	DET
ejpam-1781	362	6	a	a	DET
ejpam-1781	362	7	∈	∈	NOUN
ejpam-1781	362	8	r	r	NOUN
ejpam-1781	362	9	define	define	NOUN
ejpam-1781	362	10	u(a	u(a	NOUN
ejpam-1781	362	11	)	)	PUNCT
ejpam-1781	362	12	=	=	PUNCT
ejpam-1781	362	13	{	{	PUNCT
ejpam-1781	362	14	p	p	NOUN
ejpam-1781	362	15	∈	∈	PROPN
ejpam-1781	362	16	℘	℘	NOUN
ejpam-1781	362	17	|	|	ADV
ejpam-1781	362	18	a	a	PRON
ejpam-1781	362	19	/∈	/∈	NOUN
ejpam-1781	363	1	p	p	NOUN
ejpam-1781	363	2	}	}	PUNCT
ejpam-1781	363	3	.	.	PUNCT
ejpam-1781	364	1	note	note	VERB
ejpam-1781	364	2	that	that	SCONJ
ejpam-1781	364	3	u(a	u(a	PROPN
ejpam-1781	364	4	)	)	PUNCT
ejpam-1781	364	5	=	=	PUNCT
ejpam-1781	364	6	u((a	u((a	PROPN
ejpam-1781	364	7	]	]	X
ejpam-1781	364	8	)	)	PUNCT
ejpam-1781	364	9	for	for	ADP
ejpam-1781	364	10	any	any	DET
ejpam-1781	364	11	a	a	DET
ejpam-1781	364	12	∈	∈	PROPN
ejpam-1781	364	13	r.	r.	NOUN
ejpam-1781	364	14	the	the	DET
ejpam-1781	364	15	aim	aim	NOUN
ejpam-1781	364	16	of	of	ADP
ejpam-1781	364	17	this	this	DET
ejpam-1781	364	18	article	article	NOUN
ejpam-1781	364	19	is	be	AUX
ejpam-1781	364	20	to	to	PART
ejpam-1781	364	21	study	study	VERB
ejpam-1781	364	22	some	some	DET
ejpam-1781	364	23	properties	property	NOUN
ejpam-1781	364	24	of	of	ADP
ejpam-1781	364	25	the	the	DET
ejpam-1781	364	26	sets	set	NOUN
ejpam-1781	364	27	u(i	u(i	NOUN
ejpam-1781	364	28	)	)	PUNCT
ejpam-1781	364	29	.	.	PUNCT
ejpam-1781	365	1	for	for	ADP
ejpam-1781	365	2	any	any	DET
ejpam-1781	365	3	prime	prime	ADJ
ejpam-1781	365	4	ideal	ideal	NOUN
ejpam-1781	365	5	p	p	NOUN
ejpam-1781	365	6	in	in	ADP
ejpam-1781	365	7	r	r	NOUN
ejpam-1781	365	8	we	we	PRON
ejpam-1781	365	9	define	define	VERB
ejpam-1781	365	10	sp	sp	ADP
ejpam-1781	365	11	=	=	SYM
ejpam-1781	365	12	⋂	⋂	PROPN
ejpam-1781	365	13	{	{	PUNCT
ejpam-1781	365	14	m	m	NOUN
ejpam-1781	365	15	∈	∈	NOUN
ejpam-1781	365	16	℘	℘	NOUN
ejpam-1781	365	17	|m	|m	NOUN
ejpam-1781	365	18	⊆	⊆	NUM
ejpam-1781	365	19	p	p	NOUN
ejpam-1781	365	20	}	}	PUNCT
ejpam-1781	365	21	.	.	PUNCT
ejpam-1781	366	1	theorem	theorem	NOUN
ejpam-1781	366	2	12	12	NUM
ejpam-1781	366	3	.	.	PUNCT
ejpam-1781	367	1	for	for	ADP
ejpam-1781	367	2	a	a	DET
ejpam-1781	367	3	prime	prime	ADJ
ejpam-1781	367	4	ideal	ideal	NOUN
ejpam-1781	367	5	p	p	NOUN
ejpam-1781	367	6	in	in	ADP
ejpam-1781	367	7	r	r	NOUN
ejpam-1781	367	8	,	,	PUNCT
ejpam-1781	367	9	sp	sp	ADP
ejpam-1781	367	10	=	=	PUNCT
ejpam-1781	367	11	{	{	PUNCT
ejpam-1781	367	12	a	a	PRON
ejpam-1781	367	13	∈	∈	NOUN
ejpam-1781	367	14	r	r	NOUN
ejpam-1781	368	1	|	|	NOUN
ejpam-1781	368	2	a	a	DET
ejpam-1781	368	3	=	=	NOUN
ejpam-1781	368	4	0	0	NUM
ejpam-1781	368	5	or	or	CCONJ
ejpam-1781	368	6	p‖q	p‖q	NOUN
ejpam-1781	368	7	for	for	ADP
ejpam-1781	368	8	each	each	DET
ejpam-1781	368	9	q	q	PROPN
ejpam-1781	368	10	∈	∈	PROPN
ejpam-1781	368	11	u(a	u(a	PROPN
ejpam-1781	368	12	)	)	PUNCT
ejpam-1781	368	13	}	}	PUNCT
ejpam-1781	368	14	proof	proof	NOUN
ejpam-1781	368	15	.	.	PUNCT
ejpam-1781	369	1	let	let	VERB
ejpam-1781	369	2	k	k	PROPN
ejpam-1781	370	1	=	=	PRON
ejpam-1781	370	2	{	{	PUNCT
ejpam-1781	370	3	a	a	PRON
ejpam-1781	370	4	∈	∈	NOUN
ejpam-1781	370	5	r	r	NOUN
ejpam-1781	370	6	|	|	NOUN
ejpam-1781	370	7	a	a	DET
ejpam-1781	370	8	=	=	NOUN
ejpam-1781	370	9	0	0	NUM
ejpam-1781	370	10	or	or	CCONJ
ejpam-1781	370	11	p‖q	p‖q	NOUN
ejpam-1781	370	12	for	for	ADP
ejpam-1781	370	13	each	each	DET
ejpam-1781	370	14	q	q	PROPN
ejpam-1781	370	15	∈	∈	PROPN
ejpam-1781	370	16	u(a	u(a	PROPN
ejpam-1781	370	17	)	)	PUNCT
ejpam-1781	370	18	}	}	PUNCT
ejpam-1781	370	19	.	.	PUNCT
ejpam-1781	371	1	to	to	PART
ejpam-1781	371	2	prove	prove	VERB
ejpam-1781	371	3	that	that	PRON
ejpam-1781	371	4	sp	sp	ADP
ejpam-1781	371	5	=	=	SYM
ejpam-1781	371	6	k	k	PROPN
ejpam-1781	371	7	.	.	PUNCT
ejpam-1781	372	1	let	let	VERB
ejpam-1781	372	2	sp	sp	ADP
ejpam-1781	372	3	6=	6=	PROPN
ejpam-1781	372	4	k	k	X
ejpam-1781	372	5	.	.	PUNCT
ejpam-1781	373	1	select	select	ADJ
ejpam-1781	373	2	0	0	PUNCT
ejpam-1781	373	3	<	<	X
ejpam-1781	373	4	a	a	DET
ejpam-1781	373	5	∈	∈	NOUN
ejpam-1781	373	6	sp	sp	ADP
ejpam-1781	373	7	such	such	DET
ejpam-1781	373	8	that	that	SCONJ
ejpam-1781	373	9	a	a	PRON
ejpam-1781	373	10	/∈	/∈	PUNCT
ejpam-1781	374	1	k	k	X
ejpam-1781	374	2	.	.	PUNCT
ejpam-1781	375	1	by	by	ADP
ejpam-1781	375	2	the	the	DET
ejpam-1781	375	3	definition	definition	NOUN
ejpam-1781	375	4	of	of	ADP
ejpam-1781	375	5	k	k	PROPN
ejpam-1781	375	6	,	,	PUNCT
ejpam-1781	375	7	there	there	PRON
ejpam-1781	375	8	exist	exist	VERB
ejpam-1781	375	9	q	q	PROPN
ejpam-1781	375	10	∈	∈	PROPN
ejpam-1781	375	11	u(a	u(a	PROPN
ejpam-1781	375	12	)	)	PUNCT
ejpam-1781	375	13	such	such	ADJ
ejpam-1781	375	14	that	that	SCONJ
ejpam-1781	375	15	p	p	PROPN
ejpam-1781	375	16	and	and	CCONJ
ejpam-1781	375	17	q	q	NOUN
ejpam-1781	375	18	are	be	AUX
ejpam-1781	375	19	comparable	comparable	ADJ
ejpam-1781	375	20	.	.	PUNCT
ejpam-1781	376	1	if	if	SCONJ
ejpam-1781	376	2	p	p	PROPN
ejpam-1781	376	3	⊆	⊆	NUM
ejpam-1781	376	4	q	q	NOUN
ejpam-1781	376	5	,	,	PUNCT
ejpam-1781	376	6	then	then	ADV
ejpam-1781	376	7	sp	sp	ADP
ejpam-1781	376	8	⊆	⊆	NUM
ejpam-1781	376	9	p	p	NOUN
ejpam-1781	376	10	⇒	⇒	NOUN
ejpam-1781	376	11	sp	sp	ADP
ejpam-1781	376	12	⊆	⊆	NUM
ejpam-1781	376	13	q.	q.	NOUN
ejpam-1781	376	14	if	if	SCONJ
ejpam-1781	376	15	q	q	PROPN
ejpam-1781	376	16	⊆	⊆	NUM
ejpam-1781	376	17	p	p	NOUN
ejpam-1781	376	18	,	,	PUNCT
ejpam-1781	376	19	then	then	ADV
ejpam-1781	376	20	sp	sp	ADP
ejpam-1781	376	21	⊆	⊆	NUM
ejpam-1781	376	22	sq	sq	PROPN
ejpam-1781	376	23	y.	y.	PROPN
ejpam-1781	376	24	pawar	pawar	PROPN
ejpam-1781	376	25	,	,	PUNCT
ejpam-1781	376	26	i.	i.	PROPN
ejpam-1781	376	27	shaikh	shaikh	PROPN
ejpam-1781	376	28	/	/	SYM
ejpam-1781	376	29	eur	eur	PROPN
ejpam-1781	376	30	.	.	PUNCT
ejpam-1781	377	1	j.	j.	PROPN
ejpam-1781	377	2	pure	pure	PROPN
ejpam-1781	377	3	appl	appl	PROPN
ejpam-1781	377	4	.	.	PROPN
ejpam-1781	377	5	math	math	PROPN
ejpam-1781	377	6	,	,	PUNCT
ejpam-1781	377	7	6	6	NUM
ejpam-1781	377	8	(	(	PUNCT
ejpam-1781	377	9	2013	2013	NUM
ejpam-1781	377	10	)	)	PUNCT
ejpam-1781	377	11	,	,	PUNCT
ejpam-1781	377	12	107	107	NUM
ejpam-1781	377	13	-	-	SYM
ejpam-1781	377	14	118	118	NUM
ejpam-1781	377	15	116	116	NUM
ejpam-1781	377	16	and	and	CCONJ
ejpam-1781	377	17	sq	sq	NUM
ejpam-1781	377	18	⊆	⊆	NUM
ejpam-1781	377	19	q	q	NOUN
ejpam-1781	377	20	imply	imply	NOUN
ejpam-1781	377	21	sp	sp	ADP
ejpam-1781	377	22	⊆	⊆	NUM
ejpam-1781	377	23	q.	q.	NOUN
ejpam-1781	377	24	thus	thus	ADV
ejpam-1781	377	25	in	in	ADP
ejpam-1781	377	26	either	either	CCONJ
ejpam-1781	377	27	the	the	DET
ejpam-1781	377	28	case	case	NOUN
ejpam-1781	377	29	sp	sp	ADP
ejpam-1781	377	30	⊆	⊆	NUM
ejpam-1781	377	31	q.	q.	NOUN
ejpam-1781	377	32	but	but	CCONJ
ejpam-1781	377	33	then	then	ADV
ejpam-1781	377	34	a	a	DET
ejpam-1781	377	35	∈	∈	PROPN
ejpam-1781	377	36	q	q	NOUN
ejpam-1781	377	37	;	;	PUNCT
ejpam-1781	377	38	a	a	DET
ejpam-1781	377	39	contradiction	contradiction	NOUN
ejpam-1781	377	40	.	.	PUNCT
ejpam-1781	378	1	therefore	therefore	ADV
ejpam-1781	378	2	sp	sp	ADP
ejpam-1781	378	3	⊆k	⊆k	NOUN
ejpam-1781	378	4	.	.	PUNCT
ejpam-1781	378	5	.	.	PUNCT
ejpam-1781	378	6	.	.	PUNCT
ejpam-1781	379	1	(	(	PUNCT
ejpam-1781	379	2	i	i	NOUN
ejpam-1781	379	3	)	)	PUNCT
ejpam-1781	379	4	let	let	VERB
ejpam-1781	379	5	if	if	SCONJ
ejpam-1781	379	6	possible	possible	ADJ
ejpam-1781	379	7	k	k	PROPN
ejpam-1781	379	8	*	*	PUNCT
ejpam-1781	379	9	sp	sp	PROPN
ejpam-1781	379	10	.	.	PUNCT
ejpam-1781	380	1	pick	pick	VERB
ejpam-1781	380	2	0	0	NUM
ejpam-1781	381	1	<	<	X
ejpam-1781	381	2	b	b	X
ejpam-1781	381	3	∈	∈	PROPN
ejpam-1781	381	4	k	k	X
ejpam-1781	381	5	\	\	PROPN
ejpam-1781	381	6	sp	sp	PROPN
ejpam-1781	381	7	.	.	PUNCT
ejpam-1781	382	1	as	as	ADP
ejpam-1781	382	2	b	b	PROPN
ejpam-1781	382	3	/∈	/∈	PUNCT
ejpam-1781	382	4	sp	sp	ADP
ejpam-1781	382	5	there	there	ADV
ejpam-1781	382	6	exists	exist	VERB
ejpam-1781	382	7	m	m	VERB
ejpam-1781	382	8	∈m	∈m	NOUN
ejpam-1781	382	9	such	such	ADJ
ejpam-1781	382	10	that	that	SCONJ
ejpam-1781	382	11	m	m	PROPN
ejpam-1781	382	12	⊆	⊆	NUM
ejpam-1781	382	13	p	p	NOUN
ejpam-1781	382	14	but	but	CCONJ
ejpam-1781	382	15	b	b	NOUN
ejpam-1781	382	16	/∈	/∈	INTJ
ejpam-1781	383	1	m	m	INTJ
ejpam-1781	383	2	.	.	PUNCT
ejpam-1781	384	1	but	but	CCONJ
ejpam-1781	384	2	then	then	ADV
ejpam-1781	384	3	m	m	VERB
ejpam-1781	384	4	∈	∈	ADJ
ejpam-1781	384	5	x	x	SYM
ejpam-1781	384	6	(	(	PUNCT
ejpam-1781	384	7	b	b	NOUN
ejpam-1781	384	8	)	)	PUNCT
ejpam-1781	384	9	.	.	PUNCT
ejpam-1781	385	1	as	as	ADP
ejpam-1781	385	2	b	b	PROPN
ejpam-1781	385	3	∈k	∈k	NOUN
ejpam-1781	385	4	and	and	CCONJ
ejpam-1781	385	5	b	b	NOUN
ejpam-1781	385	6	>	>	X
ejpam-1781	385	7	0	0	NUM
ejpam-1781	385	8	,	,	PUNCT
ejpam-1781	385	9	we	we	PRON
ejpam-1781	385	10	get	get	VERB
ejpam-1781	385	11	m	m	PROPN
ejpam-1781	385	12	ver	ver	NOUN
ejpam-1781	385	13	tp	tp	NOUN
ejpam-1781	385	14	;	;	PUNCT
ejpam-1781	385	15	a	a	DET
ejpam-1781	385	16	contradiction	contradiction	NOUN
ejpam-1781	385	17	.	.	PUNCT
ejpam-1781	386	1	hence	hence	ADV
ejpam-1781	386	2	k	k	PROPN
ejpam-1781	386	3	⊆	⊆	NUM
ejpam-1781	386	4	sp	sp	NOUN
ejpam-1781	386	5	.	.	PUNCT
ejpam-1781	386	6	.	.	PUNCT
ejpam-1781	386	7	.	.	PUNCT
ejpam-1781	387	1	(	(	PUNCT
ejpam-1781	387	2	ii	ii	NOUN
ejpam-1781	387	3	)	)	PUNCT
ejpam-1781	387	4	from	from	ADP
ejpam-1781	387	5	(	(	PUNCT
ejpam-1781	387	6	i	i	NOUN
ejpam-1781	387	7	)	)	PUNCT
ejpam-1781	387	8	and	and	CCONJ
ejpam-1781	387	9	(	(	PUNCT
ejpam-1781	387	10	ii	ii	NOUN
ejpam-1781	387	11	)	)	PUNCT
ejpam-1781	387	12	we	we	PRON
ejpam-1781	387	13	get	get	VERB
ejpam-1781	387	14	k	k	NOUN
ejpam-1781	387	15	=	=	PUNCT
ejpam-1781	387	16	sp	sp	PROPN
ejpam-1781	387	17	and	and	CCONJ
ejpam-1781	387	18	the	the	DET
ejpam-1781	387	19	result	result	NOUN
ejpam-1781	387	20	follows	follow	VERB
ejpam-1781	387	21	.	.	PUNCT
ejpam-1781	388	1	let	let	VERB
ejpam-1781	388	2	<	<	X
ejpam-1781	388	3	p,≤	p,≤	PUNCT
ejpam-1781	388	4	>	>	X
ejpam-1781	388	5	be	be	AUX
ejpam-1781	388	6	a	a	DET
ejpam-1781	388	7	bounded	bounded	ADJ
ejpam-1781	388	8	poset	poset	NOUN
ejpam-1781	388	9	.	.	PUNCT
ejpam-1781	389	1	a	a	DET
ejpam-1781	389	2	non	non	X
ejpam-1781	389	3	empty	empty	ADJ
ejpam-1781	389	4	subset	subset	NOUN
ejpam-1781	389	5	f	f	PROPN
ejpam-1781	389	6	of	of	ADP
ejpam-1781	389	7	p	p	PROPN
ejpam-1781	389	8	is	be	AUX
ejpam-1781	389	9	a	a	DET
ejpam-1781	389	10	semi	semi	ADJ
ejpam-1781	389	11	filter	filter	NOUN
ejpam-1781	389	12	(	(	PUNCT
ejpam-1781	389	13	or	or	CCONJ
ejpam-1781	389	14	dual	dual	ADJ
ejpam-1781	389	15	semi	semi	ADV
ejpam-1781	389	16	ideal	ideal	ADJ
ejpam-1781	389	17	)	)	PUNCT
ejpam-1781	389	18	in	in	ADP
ejpam-1781	389	19	p	p	PRON
ejpam-1781	389	20	if	if	SCONJ
ejpam-1781	389	21	a	a	DET
ejpam-1781	389	22	∈	∈	PROPN
ejpam-1781	389	23	f	f	X
ejpam-1781	389	24	,	,	PUNCT
ejpam-1781	389	25	b	b	PROPN
ejpam-1781	389	26	∈	∈	PROPN
ejpam-1781	389	27	p	p	NOUN
ejpam-1781	389	28	and	and	CCONJ
ejpam-1781	389	29	a	a	DET
ejpam-1781	389	30	≤	≤	NUM
ejpam-1781	389	31	b	b	NOUN
ejpam-1781	389	32	imply	imply	NOUN
ejpam-1781	389	33	b	b	NOUN
ejpam-1781	389	34	∈	∈	NOUN
ejpam-1781	390	1	f	f	X
ejpam-1781	391	1	[	[	X
ejpam-1781	391	2	see	see	VERB
ejpam-1781	391	3	4	4	NUM
ejpam-1781	391	4	]	]	PUNCT
ejpam-1781	391	5	.	.	PUNCT
ejpam-1781	392	1	for	for	ADP
ejpam-1781	392	2	m	m	PROPN
ejpam-1781	392	3	∈	∈	PROPN
ejpam-1781	392	4	σ	σ	NOUN
ejpam-1781	392	5	,	,	PUNCT
ejpam-1781	392	6	we	we	PRON
ejpam-1781	392	7	define	define	VERB
ejpam-1781	392	8	dwm	dwm	NOUN
ejpam-1781	392	9	=	=	PUNCT
ejpam-1781	392	10	{	{	PUNCT
ejpam-1781	392	11	p	p	NOUN
ejpam-1781	392	12	∈	∈	PROPN
ejpam-1781	392	13	℘	℘	NOUN
ejpam-1781	392	14	|	|	ADV
ejpam-1781	392	15	p	p	NOUN
ejpam-1781	392	16	⊆	⊆	NUM
ejpam-1781	392	17	m	m	NOUN
ejpam-1781	392	18	}	}	PUNCT
ejpam-1781	392	19	and	and	CCONJ
ejpam-1781	392	20	wm	wm	X
ejpam-1781	392	21	=	=	SYM
ejpam-1781	392	22	⋂dwm	⋂dwm	NOUN
ejpam-1781	392	23	using	use	VERB
ejpam-1781	392	24	the	the	DET
ejpam-1781	392	25	concept	concept	NOUN
ejpam-1781	392	26	of	of	ADP
ejpam-1781	392	27	semi	semi	ADJ
ejpam-1781	392	28	filter	filter	NOUN
ejpam-1781	392	29	in	in	ADP
ejpam-1781	392	30	the	the	DET
ejpam-1781	392	31	poset	poset	NOUN
ejpam-1781	392	32	of	of	ADP
ejpam-1781	392	33	prime	prime	ADJ
ejpam-1781	392	34	idals	idal	NOUN
ejpam-1781	392	35	(	(	PUNCT
ejpam-1781	392	36	℘,⊆	℘,⊆	NOUN
ejpam-1781	392	37	)	)	PUNCT
ejpam-1781	392	38	in	in	ADP
ejpam-1781	392	39	r	r	NOUN
ejpam-1781	392	40	,	,	PUNCT
ejpam-1781	392	41	we	we	PRON
ejpam-1781	392	42	have	have	AUX
ejpam-1781	392	43	theorem	theorem	VERB
ejpam-1781	392	44	13	13	NUM
ejpam-1781	392	45	.	.	PUNCT
ejpam-1781	393	1	let	let	VERB
ejpam-1781	393	2	i	i	PRON
ejpam-1781	393	3	∈	∈	VERB
ejpam-1781	393	4	i	i	PRON
ejpam-1781	393	5	(	(	PUNCT
ejpam-1781	393	6	r	r	NOUN
ejpam-1781	393	7	)	)	PUNCT
ejpam-1781	393	8	be	be	AUX
ejpam-1781	393	9	such	such	ADJ
ejpam-1781	393	10	that	that	SCONJ
ejpam-1781	393	11	u(i	u(i	NOUN
ejpam-1781	393	12	)	)	PUNCT
ejpam-1781	393	13	is	be	AUX
ejpam-1781	393	14	a	a	DET
ejpam-1781	393	15	semi	semi	ADJ
ejpam-1781	393	16	filter	filter	NOUN
ejpam-1781	393	17	in	in	ADP
ejpam-1781	393	18	the	the	DET
ejpam-1781	393	19	poset	poset	NOUN
ejpam-1781	393	20	(	(	PUNCT
ejpam-1781	393	21	℘,⊆	℘,⊆	NOUN
ejpam-1781	393	22	)	)	PUNCT
ejpam-1781	393	23	.	.	PUNCT
ejpam-1781	394	1	then	then	ADV
ejpam-1781	394	2	following	follow	VERB
ejpam-1781	394	3	properties	property	NOUN
ejpam-1781	394	4	hold	hold	VERB
ejpam-1781	394	5	in	in	ADP
ejpam-1781	394	6	r.	r.	PROPN
ejpam-1781	394	7	1	1	NUM
ejpam-1781	394	8	.	.	PUNCT
ejpam-1781	395	1	if	if	SCONJ
ejpam-1781	395	2	i	i	PRON
ejpam-1781	395	3	⊆	⊆	NUM
ejpam-1781	395	4	m	m	NOUN
ejpam-1781	395	5	∈	∈	PROPN
ejpam-1781	395	6	σ	σ	NOUN
ejpam-1781	395	7	and	and	CCONJ
ejpam-1781	395	8	p	p	NOUN
ejpam-1781	395	9	∈	∈	PROPN
ejpam-1781	395	10	℘	℘	PROPN
ejpam-1781	395	11	,	,	PUNCT
ejpam-1781	395	12	p	p	NOUN
ejpam-1781	395	13	⊆	⊆	NUM
ejpam-1781	395	14	m.	m.	NOUN
ejpam-1781	395	15	then	then	ADV
ejpam-1781	395	16	i	i	PRON
ejpam-1781	395	17	⊆	⊆	NUM
ejpam-1781	395	18	p	p	PROPN
ejpam-1781	395	19	2	2	NUM
ejpam-1781	395	20	.	.	PUNCT
ejpam-1781	396	1	if	if	SCONJ
ejpam-1781	396	2	i	i	PRON
ejpam-1781	396	3	⊆	⊆	NUM
ejpam-1781	396	4	m	m	NOUN
ejpam-1781	396	5	∈	∈	NOUN
ejpam-1781	396	6	σ	σ	NOUN
ejpam-1781	396	7	imply	imply	VERB
ejpam-1781	396	8	i	i	PRON
ejpam-1781	396	9	⊆wm	⊆wm	NUM
ejpam-1781	397	1	3	3	X
ejpam-1781	397	2	.	.	PUNCT
ejpam-1781	398	1	i	i	PRON
ejpam-1781	398	2	=	=	SYM
ejpam-1781	399	1	⋂	⋂	PROPN
ejpam-1781	399	2	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	400	1	|	|	ADV
ejpam-1781	400	2	i	i	PRON
ejpam-1781	400	3	⊆wm	⊆wm	NOUN
ejpam-1781	400	4	}	}	PUNCT
ejpam-1781	400	5	4	4	NUM
ejpam-1781	400	6	.	.	X
ejpam-1781	401	1	u(i	u(i	NOUN
ejpam-1781	401	2	)	)	PUNCT
ejpam-1781	401	3	=	=	SYM
ejpam-1781	401	4	℘	℘	X
ejpam-1781	401	5	\	\	NOUN
ejpam-1781	401	6	⋃	⋃	NOUN
ejpam-1781	401	7	m∈k	m∈k	NOUN
ejpam-1781	401	8	dwm	dwm	NOUN
ejpam-1781	401	9	=	=	SYM
ejpam-1781	401	10	℘	℘	PROPN
ejpam-1781	401	11	\	\	NOUN
ejpam-1781	401	12	⋃	⋃	NOUN
ejpam-1781	401	13	m∈k	m∈k	NOUN
ejpam-1781	401	14	{	{	PUNCT
ejpam-1781	401	15	p	p	NOUN
ejpam-1781	401	16	∈	∈	PROPN
ejpam-1781	401	17	℘	℘	NOUN
ejpam-1781	401	18	|	|	ADV
ejpam-1781	401	19	p	p	ADJ
ejpam-1781	401	20	⊆	⊆	NUM
ejpam-1781	401	21	m	m	NOUN
ejpam-1781	401	22	}	}	PUNCT
ejpam-1781	401	23	5	5	NUM
ejpam-1781	401	24	.	.	PUNCT
ejpam-1781	402	1	if	if	SCONJ
ejpam-1781	402	2	a	a	DET
ejpam-1781	402	3	∈	∈	X
ejpam-1781	402	4	i	i	PRON
ejpam-1781	402	5	,	,	PUNCT
ejpam-1781	402	6	we	we	PRON
ejpam-1781	402	7	have	have	VERB
ejpam-1781	402	8	(	(	PUNCT
ejpam-1781	402	9	a]∗	a]∗	PROPN
ejpam-1781	402	10	∨	∨	NUM
ejpam-1781	402	11	i	i	NOUN
ejpam-1781	402	12	=	=	SYM
ejpam-1781	402	13	r.	r.	NOUN
ejpam-1781	402	14	proof	proof	NOUN
ejpam-1781	402	15	.	.	PUNCT
ejpam-1781	403	1	1	1	X
ejpam-1781	403	2	.	.	X
ejpam-1781	403	3	suppose	suppose	VERB
ejpam-1781	404	1	that	that	SCONJ
ejpam-1781	404	2	i	i	PRON
ejpam-1781	404	3	*	*	PUNCT
ejpam-1781	405	1	p.	p.	NOUN
ejpam-1781	405	2	then	then	ADV
ejpam-1781	405	3	p	p	PROPN
ejpam-1781	405	4	∈	∈	PROPN
ejpam-1781	405	5	u(i	u(i	NOUN
ejpam-1781	405	6	)	)	PUNCT
ejpam-1781	405	7	.	.	PUNCT
ejpam-1781	406	1	but	but	CCONJ
ejpam-1781	406	2	as	as	ADP
ejpam-1781	406	3	u(i	u(i	NOUN
ejpam-1781	406	4	)	)	PUNCT
ejpam-1781	406	5	is	be	AUX
ejpam-1781	406	6	a	a	DET
ejpam-1781	406	7	semi	semi	ADJ
ejpam-1781	406	8	filter	filter	NOUN
ejpam-1781	406	9	and	and	CCONJ
ejpam-1781	406	10	p	p	NOUN
ejpam-1781	406	11	⊆	⊆	NUM
ejpam-1781	406	12	m	m	VERB
ejpam-1781	406	13	we	we	PRON
ejpam-1781	406	14	get	get	VERB
ejpam-1781	406	15	m	m	VERB
ejpam-1781	406	16	∈	∈	NOUN
ejpam-1781	406	17	u(i	u(i	NOUN
ejpam-1781	406	18	)	)	PUNCT
ejpam-1781	406	19	.	.	PUNCT
ejpam-1781	407	1	so	so	ADV
ejpam-1781	407	2	i	i	PRON
ejpam-1781	407	3	*	*	VERB
ejpam-1781	407	4	m	m	VERB
ejpam-1781	407	5	,	,	PUNCT
ejpam-1781	407	6	which	which	PRON
ejpam-1781	407	7	is	be	AUX
ejpam-1781	407	8	absurd	absurd	ADJ
ejpam-1781	407	9	.	.	PUNCT
ejpam-1781	408	1	therefore	therefore	ADV
ejpam-1781	408	2	i	i	PRON
ejpam-1781	408	3	⊆	⊆	NUM
ejpam-1781	408	4	p.	p.	NOUN
ejpam-1781	408	5	2	2	NUM
ejpam-1781	408	6	.	.	X
ejpam-1781	408	7	select	select	VERB
ejpam-1781	408	8	p	p	PROPN
ejpam-1781	408	9	∈	∈	PROPN
ejpam-1781	408	10	℘	℘	NOUN
ejpam-1781	408	11	such	such	ADJ
ejpam-1781	408	12	that	that	SCONJ
ejpam-1781	408	13	p	p	PROPN
ejpam-1781	408	14	⊆	⊆	NUM
ejpam-1781	408	15	m	m	NOUN
ejpam-1781	408	16	.	.	PUNCT
ejpam-1781	409	1	if	if	SCONJ
ejpam-1781	409	2	i	i	PRON
ejpam-1781	409	3	*	*	PUNCT
ejpam-1781	410	1	p	p	X
ejpam-1781	410	2	,	,	PUNCT
ejpam-1781	410	3	then	then	ADV
ejpam-1781	410	4	p	p	PROPN
ejpam-1781	410	5	∈	∈	PROPN
ejpam-1781	410	6	u(i	u(i	PROPN
ejpam-1781	410	7	)	)	PUNCT
ejpam-1781	410	8	.	.	PUNCT
ejpam-1781	411	1	as	as	ADP
ejpam-1781	411	2	u(i	u(i	NOUN
ejpam-1781	411	3	)	)	PUNCT
ejpam-1781	411	4	is	be	AUX
ejpam-1781	411	5	a	a	DET
ejpam-1781	411	6	semi	semi	ADJ
ejpam-1781	411	7	filter	filter	NOUN
ejpam-1781	411	8	in	in	ADP
ejpam-1781	411	9	(	(	PUNCT
ejpam-1781	411	10	℘,⊆	℘,⊆	NOUN
ejpam-1781	411	11	)	)	PUNCT
ejpam-1781	411	12	and	and	CCONJ
ejpam-1781	411	13	p	p	X
ejpam-1781	411	14	⊆	⊆	NUM
ejpam-1781	411	15	m	m	VERB
ejpam-1781	411	16	we	we	PRON
ejpam-1781	411	17	get	get	VERB
ejpam-1781	411	18	m	m	VERB
ejpam-1781	411	19	∈	∈	NOUN
ejpam-1781	411	20	u(i	u(i	NOUN
ejpam-1781	411	21	)	)	PUNCT
ejpam-1781	411	22	(	(	PUNCT
ejpam-1781	411	23	since	since	SCONJ
ejpam-1781	411	24	m	m	PROPN
ejpam-1781	411	25	∈	∈	PROPN
ejpam-1781	411	26	℘	℘	PROPN
ejpam-1781	411	27	)	)	PUNCT
ejpam-1781	411	28	.	.	PUNCT
ejpam-1781	412	1	but	but	CCONJ
ejpam-1781	412	2	then	then	ADV
ejpam-1781	412	3	i	i	PRON
ejpam-1781	412	4	*	*	VERB
ejpam-1781	412	5	m	m	VERB
ejpam-1781	412	6	;	;	PUNCT
ejpam-1781	412	7	a	a	DET
ejpam-1781	412	8	contradiction	contradiction	NOUN
ejpam-1781	412	9	.	.	PUNCT
ejpam-1781	413	1	hence	hence	ADV
ejpam-1781	413	2	i	i	PRON
ejpam-1781	413	3	⊆	⊆	NUM
ejpam-1781	413	4	p.	p.	NOUN
ejpam-1781	413	5	this	this	PRON
ejpam-1781	413	6	shows	show	VERB
ejpam-1781	413	7	that	that	SCONJ
ejpam-1781	413	8	i	i	PRON
ejpam-1781	413	9	⊆	⊆	NUM
ejpam-1781	413	10	p	p	NOUN
ejpam-1781	413	11	for	for	ADP
ejpam-1781	413	12	each	each	DET
ejpam-1781	413	13	p	p	NOUN
ejpam-1781	413	14	∈	∈	PROPN
ejpam-1781	413	15	℘	℘	PROPN
ejpam-1781	413	16	with	with	ADP
ejpam-1781	413	17	p	p	PROPN
ejpam-1781	413	18	⊆	⊆	NUM
ejpam-1781	413	19	m	m	NOUN
ejpam-1781	413	20	.	.	PUNCT
ejpam-1781	414	1	hence	hence	ADV
ejpam-1781	414	2	i	i	PRON
ejpam-1781	414	3	⊆	⊆	PROPN
ejpam-1781	414	4	⋂	⋂	PROPN
ejpam-1781	414	5	{	{	PUNCT
ejpam-1781	414	6	p	p	NOUN
ejpam-1781	414	7	∈	∈	PROPN
ejpam-1781	414	8	℘	℘	NOUN
ejpam-1781	414	9	|	|	ADV
ejpam-1781	414	10	p	p	ADJ
ejpam-1781	414	11	⊆	⊆	NUM
ejpam-1781	414	12	m	m	PRON
ejpam-1781	414	13	}	}	PUNCT
ejpam-1781	414	14	=	=	PROPN
ejpam-1781	414	15	wm	wm	NOUN
ejpam-1781	414	16	.	.	PUNCT
ejpam-1781	415	1	3	3	X
ejpam-1781	415	2	.	.	X
ejpam-1781	415	3	obviously	obviously	ADV
ejpam-1781	415	4	,	,	PUNCT
ejpam-1781	415	5	i	i	PROPN
ejpam-1781	415	6	⊆	⊆	NUM
ejpam-1781	415	7	⋂	⋂	PROPN
ejpam-1781	415	8	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	416	1	|	|	ADV
ejpam-1781	416	2	i	i	PRON
ejpam-1781	416	3	⊆	⊆	NUM
ejpam-1781	416	4	wm	wm	PROPN
ejpam-1781	416	5	}	}	PUNCT
ejpam-1781	416	6	.	.	PUNCT
ejpam-1781	417	1	hence	hence	ADV
ejpam-1781	417	2	to	to	PART
ejpam-1781	417	3	prove	prove	VERB
ejpam-1781	417	4	that	that	SCONJ
ejpam-1781	417	5	⋂	⋂	PROPN
ejpam-1781	417	6	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	418	1	|	|	ADV
ejpam-1781	418	2	i	i	PRON
ejpam-1781	418	3	⊆	⊆	NUM
ejpam-1781	418	4	wm	wm	PROPN
ejpam-1781	418	5	}	}	PUNCT
ejpam-1781	418	6	⊆	⊆	NUM
ejpam-1781	418	7	i	i	PRON
ejpam-1781	418	8	.	.	PUNCT
ejpam-1781	419	1	let	let	VERB
ejpam-1781	419	2	if	if	SCONJ
ejpam-1781	419	3	possible	possible	ADJ
ejpam-1781	419	4	⋂	⋂	PROPN
ejpam-1781	419	5	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	420	1	|	|	ADV
ejpam-1781	420	2	i	i	PRON
ejpam-1781	420	3	⊆	⊆	NUM
ejpam-1781	420	4	wm	wm	PROPN
ejpam-1781	420	5	}	}	PUNCT
ejpam-1781	420	6	*	*	PROPN
ejpam-1781	421	1	i	i	PRON
ejpam-1781	421	2	.	.	PUNCT
ejpam-1781	422	1	select	select	ADJ
ejpam-1781	422	2	x	x	PUNCT
ejpam-1781	422	3	∈	∈	PROPN
ejpam-1781	422	4	⋂	⋂	PROPN
ejpam-1781	422	5	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	423	1	|	|	ADV
ejpam-1781	423	2	i	i	PRON
ejpam-1781	423	3	⊆	⊆	NUM
ejpam-1781	423	4	wm	wm	PROPN
ejpam-1781	423	5	}	}	PUNCT
ejpam-1781	423	6	such	such	ADJ
ejpam-1781	423	7	that	that	SCONJ
ejpam-1781	423	8	x	x	X
ejpam-1781	423	9	/∈	/∈	INTJ
ejpam-1781	423	10	i	i	INTJ
ejpam-1781	423	11	.	.	PUNCT
ejpam-1781	424	1	by	by	ADP
ejpam-1781	424	2	result	result	NOUN
ejpam-1781	424	3	11	11	NUM
ejpam-1781	424	4	,	,	PUNCT
ejpam-1781	424	5	there	there	PRON
ejpam-1781	424	6	exists	exist	VERB
ejpam-1781	424	7	a	a	DET
ejpam-1781	424	8	prime	prime	ADJ
ejpam-1781	424	9	ideal	ideal	NOUN
ejpam-1781	424	10	q	q	NOUN
ejpam-1781	424	11	in	in	ADP
ejpam-1781	424	12	r	r	NOUN
ejpam-1781	425	1	such	such	ADJ
ejpam-1781	425	2	that	that	SCONJ
ejpam-1781	425	3	i	i	PRON
ejpam-1781	425	4	⊆	⊆	NUM
ejpam-1781	425	5	q	q	NOUN
ejpam-1781	425	6	and	and	CCONJ
ejpam-1781	425	7	x	x	SYM
ejpam-1781	425	8	/∈	/∈	PUNCT
ejpam-1781	425	9	q.	q.	PROPN
ejpam-1781	425	10	as	as	SCONJ
ejpam-1781	425	11	q	q	PROPN
ejpam-1781	425	12	is	be	AUX
ejpam-1781	425	13	a	a	DET
ejpam-1781	425	14	proper	proper	ADJ
ejpam-1781	425	15	ideal	ideal	NOUN
ejpam-1781	425	16	,	,	PUNCT
ejpam-1781	425	17	q	q	PROPN
ejpam-1781	425	18	must	must	AUX
ejpam-1781	425	19	be	be	AUX
ejpam-1781	425	20	contained	contain	VERB
ejpam-1781	425	21	in	in	ADP
ejpam-1781	425	22	some	some	DET
ejpam-1781	425	23	maximal	maximal	ADJ
ejpam-1781	425	24	ideal	ideal	NOUN
ejpam-1781	425	25	say	say	VERB
ejpam-1781	425	26	m	m	VERB
ejpam-1781	425	27	in	in	ADP
ejpam-1781	425	28	r	r	NOUN
ejpam-1781	425	29	(	(	PUNCT
ejpam-1781	425	30	by	by	ADP
ejpam-1781	425	31	result	result	NOUN
ejpam-1781	425	32	3	3	NUM
ejpam-1781	425	33	)	)	PUNCT
ejpam-1781	425	34	.	.	PUNCT
ejpam-1781	426	1	but	but	CCONJ
ejpam-1781	426	2	then	then	ADV
ejpam-1781	426	3	i	i	PRON
ejpam-1781	426	4	⊆	⊆	NUM
ejpam-1781	426	5	m	m	VERB
ejpam-1781	426	6	will	will	AUX
ejpam-1781	426	7	imply	imply	VERB
ejpam-1781	426	8	i	i	PRON
ejpam-1781	426	9	⊆	⊆	NUM
ejpam-1781	426	10	wm	wm	PROPN
ejpam-1781	426	11	(	(	PUNCT
ejpam-1781	426	12	by	by	ADP
ejpam-1781	426	13	property	property	NOUN
ejpam-1781	426	14	1	1	NUM
ejpam-1781	426	15	)	)	PUNCT
ejpam-1781	426	16	.	.	PUNCT
ejpam-1781	427	1	therefore	therefore	ADV
ejpam-1781	427	2	x	x	X
ejpam-1781	427	3	∈	∈	PROPN
ejpam-1781	427	4	wm	wm	PROPN
ejpam-1781	427	5	.	.	PUNCT
ejpam-1781	428	1	as	as	ADP
ejpam-1781	428	2	wm	wm	PROPN
ejpam-1781	428	3	=	=	PUNCT
ejpam-1781	428	4	{	{	PUNCT
ejpam-1781	428	5	p	p	NOUN
ejpam-1781	428	6	∈	∈	PROPN
ejpam-1781	428	7	℘	℘	NOUN
ejpam-1781	428	8	|	|	ADV
ejpam-1781	428	9	p	p	NOUN
ejpam-1781	428	10	⊆	⊆	NUM
ejpam-1781	428	11	m	m	PRON
ejpam-1781	428	12	}	}	PUNCT
ejpam-1781	428	13	,	,	PUNCT
ejpam-1781	428	14	we	we	PRON
ejpam-1781	428	15	get	get	VERB
ejpam-1781	428	16	wm	wm	PROPN
ejpam-1781	428	17	⊆	⊆	NUM
ejpam-1781	428	18	q.	q.	NOUN
ejpam-1781	429	1	but	but	CCONJ
ejpam-1781	429	2	then	then	ADV
ejpam-1781	429	3	x	x	PART
ejpam-1781	429	4	∈	∈	PROPN
ejpam-1781	429	5	q	q	NOUN
ejpam-1781	429	6	;	;	PUNCT
ejpam-1781	429	7	which	which	PRON
ejpam-1781	429	8	is	be	AUX
ejpam-1781	429	9	absurd	absurd	ADJ
ejpam-1781	429	10	.	.	PUNCT
ejpam-1781	430	1	hence	hence	ADV
ejpam-1781	430	2	⋂	⋂	PROPN
ejpam-1781	430	3	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	431	1	|	|	ADV
ejpam-1781	431	2	i	i	PRON
ejpam-1781	431	3	⊆wm	⊆wm	VERB
ejpam-1781	431	4	}	}	PUNCT
ejpam-1781	431	5	⊆	⊆	NUM
ejpam-1781	431	6	i	i	PRON
ejpam-1781	431	7	.	.	PUNCT
ejpam-1781	432	1	combining	combine	VERB
ejpam-1781	432	2	both	both	DET
ejpam-1781	432	3	the	the	DET
ejpam-1781	432	4	inclusions	inclusion	NOUN
ejpam-1781	432	5	,	,	PUNCT
ejpam-1781	432	6	we	we	PRON
ejpam-1781	432	7	get	get	VERB
ejpam-1781	432	8	i	i	PRON
ejpam-1781	432	9	=	=	SYM
ejpam-1781	433	1	⋂	⋂	NUM
ejpam-1781	433	2	m∈σ{wm	m∈σ{wm	NOUN
ejpam-1781	434	1	|	|	ADV
ejpam-1781	434	2	i	i	PRON
ejpam-1781	434	3	⊆wm	⊆wm	VERB
ejpam-1781	434	4	}	}	PUNCT
ejpam-1781	434	5	.	.	PUNCT
ejpam-1781	435	1	4	4	X
ejpam-1781	435	2	.	.	X
ejpam-1781	435	3	let	let	VERB
ejpam-1781	435	4	q	q	NOUN
ejpam-1781	435	5	∈	∈	PROPN
ejpam-1781	435	6	⋃	⋃	NOUN
ejpam-1781	435	7	m∈k	m∈k	NOUN
ejpam-1781	435	8	{	{	PUNCT
ejpam-1781	435	9	p	p	NOUN
ejpam-1781	435	10	∈	∈	PROPN
ejpam-1781	435	11	℘	℘	NOUN
ejpam-1781	435	12	|	|	ADV
ejpam-1781	435	13	p	p	NOUN
ejpam-1781	435	14	⊆	⊆	NUM
ejpam-1781	435	15	m	m	NOUN
ejpam-1781	435	16	}	}	PUNCT
ejpam-1781	435	17	.	.	PUNCT
ejpam-1781	436	1	then	then	ADV
ejpam-1781	436	2	q	q	PROPN
ejpam-1781	436	3	∈	∈	PROPN
ejpam-1781	436	4	℘	℘	PROPN
ejpam-1781	436	5	,	,	PUNCT
ejpam-1781	436	6	q	q	NOUN
ejpam-1781	436	7	⊆	⊆	NUM
ejpam-1781	436	8	m	m	NOUN
ejpam-1781	436	9	and	and	CCONJ
ejpam-1781	436	10	i	i	PRON
ejpam-1781	436	11	⊆	⊆	NUM
ejpam-1781	436	12	m	m	NOUN
ejpam-1781	436	13	.	.	PUNCT
ejpam-1781	437	1	hence	hence	ADV
ejpam-1781	437	2	i	i	PRON
ejpam-1781	437	3	⊆	⊆	NUM
ejpam-1781	437	4	q	q	NOUN
ejpam-1781	437	5	(	(	PUNCT
ejpam-1781	437	6	see	see	VERB
ejpam-1781	437	7	property	property	NOUN
ejpam-1781	437	8	1	1	NUM
ejpam-1781	437	9	)	)	PUNCT
ejpam-1781	437	10	.	.	PUNCT
ejpam-1781	438	1	therefore	therefore	ADV
ejpam-1781	438	2	q	q	X
ejpam-1781	438	3	/∈	/∈	PUNCT
ejpam-1781	438	4	u(i	u(i	NOUN
ejpam-1781	438	5	)	)	PUNCT
ejpam-1781	438	6	.	.	PUNCT
ejpam-1781	439	1	this	this	PRON
ejpam-1781	439	2	shows	show	VERB
ejpam-1781	439	3	that	that	SCONJ
ejpam-1781	439	4	u(i	u(i	NOUN
ejpam-1781	439	5	)	)	PUNCT
ejpam-1781	439	6	⊆	⊆	NUM
ejpam-1781	439	7	℘\	℘\	PROPN
ejpam-1781	439	8	⋃	⋃	PROPN
ejpam-1781	439	9	m∈k	m∈k	NOUN
ejpam-1781	439	10	{	{	PUNCT
ejpam-1781	439	11	p	p	NOUN
ejpam-1781	439	12	∈	∈	PROPN
ejpam-1781	439	13	℘	℘	NOUN
ejpam-1781	439	14	|	|	ADV
ejpam-1781	439	15	p	p	NOUN
ejpam-1781	439	16	⊆	⊆	NUM
ejpam-1781	439	17	m	m	NOUN
ejpam-1781	439	18	}	}	PUNCT
ejpam-1781	439	19	.	.	PUNCT
ejpam-1781	440	1	now	now	ADV
ejpam-1781	440	2	,	,	PUNCT
ejpam-1781	440	3	let	let	VERB
ejpam-1781	440	4	q	q	X
ejpam-1781	440	5	/∈	/∈	VERB
ejpam-1781	441	1	u(i	u(i	NOUN
ejpam-1781	441	2	)	)	PUNCT
ejpam-1781	441	3	.	.	PUNCT
ejpam-1781	442	1	then	then	ADV
ejpam-1781	442	2	i	i	PRON
ejpam-1781	442	3	⊆	⊆	NUM
ejpam-1781	442	4	q.	q.	PROPN
ejpam-1781	442	5	let	let	VERB
ejpam-1781	442	6	m	m	PRON
ejpam-1781	442	7	denote	denote	VERB
ejpam-1781	442	8	a	a	DET
ejpam-1781	442	9	maximal	maximal	ADJ
ejpam-1781	442	10	ideal	ideal	NOUN
ejpam-1781	442	11	containing	contain	VERB
ejpam-1781	442	12	q	q	NOUN
ejpam-1781	442	13	(	(	PUNCT
ejpam-1781	442	14	by	by	ADP
ejpam-1781	442	15	result	result	NOUN
ejpam-1781	442	16	3	3	NUM
ejpam-1781	442	17	)	)	PUNCT
ejpam-1781	442	18	.	.	PUNCT
ejpam-1781	443	1	as	as	SCONJ
ejpam-1781	443	2	i	i	PRON
ejpam-1781	443	3	⊆	⊆	NUM
ejpam-1781	443	4	m	m	NOUN
ejpam-1781	443	5	,	,	PUNCT
ejpam-1781	443	6	m	m	VERB
ejpam-1781	443	7	∈k	∈k	ADV
ejpam-1781	443	8	.	.	PUNCT
ejpam-1781	444	1	this	this	PRON
ejpam-1781	444	2	shows	show	VERB
ejpam-1781	444	3	that	that	SCONJ
ejpam-1781	444	4	q	q	PUNCT
ejpam-1781	444	5	∈	∈	PROPN
ejpam-1781	444	6	⋃	⋃	NOUN
ejpam-1781	444	7	m∈k	m∈k	NOUN
ejpam-1781	444	8	{	{	PUNCT
ejpam-1781	444	9	p	p	NOUN
ejpam-1781	444	10	∈	∈	PROPN
ejpam-1781	444	11	℘	℘	NOUN
ejpam-1781	444	12	|	|	ADV
ejpam-1781	444	13	p	p	NOUN
ejpam-1781	444	14	⊆	⊆	NUM
ejpam-1781	444	15	m	m	NOUN
ejpam-1781	444	16	}	}	PUNCT
ejpam-1781	444	17	.	.	PUNCT
ejpam-1781	445	1	therefore	therefore	ADV
ejpam-1781	445	2	q	q	PROPN
ejpam-1781	445	3	/∈	/∈	PUNCT
ejpam-1781	446	1	℘\	℘\	PROPN
ejpam-1781	446	2	⋃	⋃	NOUN
ejpam-1781	446	3	m∈k	m∈k	NOUN
ejpam-1781	446	4	{	{	PUNCT
ejpam-1781	446	5	p	p	NOUN
ejpam-1781	446	6	∈	∈	PROPN
ejpam-1781	446	7	℘	℘	NOUN
ejpam-1781	446	8	|	|	ADV
ejpam-1781	446	9	p	p	NOUN
ejpam-1781	446	10	⊆	⊆	NUM
ejpam-1781	446	11	m	m	NOUN
ejpam-1781	446	12	}	}	PUNCT
ejpam-1781	446	13	.	.	PUNCT
ejpam-1781	447	1	this	this	PRON
ejpam-1781	447	2	shows	show	VERB
ejpam-1781	447	3	that	that	SCONJ
ejpam-1781	447	4	℘\	℘\	PROPN
ejpam-1781	447	5	⋃	⋃	PROPN
ejpam-1781	447	6	m∈k	m∈k	NOUN
ejpam-1781	447	7	{	{	PUNCT
ejpam-1781	447	8	p	p	NOUN
ejpam-1781	447	9	∈	∈	PROPN
ejpam-1781	447	10	℘	℘	NOUN
ejpam-1781	447	11	|	|	ADV
ejpam-1781	447	12	p	p	ADJ
ejpam-1781	447	13	⊆	⊆	NUM
ejpam-1781	447	14	m	m	NOUN
ejpam-1781	447	15	}	}	PUNCT
ejpam-1781	447	16	⊆	⊆	NUM
ejpam-1781	447	17	u(i	u(i	NOUN
ejpam-1781	447	18	)	)	PUNCT
ejpam-1781	447	19	.	.	PUNCT
ejpam-1781	448	1	combining	combine	VERB
ejpam-1781	448	2	both	both	DET
ejpam-1781	448	3	the	the	DET
ejpam-1781	448	4	inclusions	inclusion	NOUN
ejpam-1781	448	5	u(i	u(i	NOUN
ejpam-1781	448	6	)	)	PUNCT
ejpam-1781	448	7	=	=	SYM
ejpam-1781	448	8	℘	℘	X
ejpam-1781	448	9	\	\	NOUN
ejpam-1781	448	10	⋃	⋃	NOUN
ejpam-1781	448	11	m∈k	m∈k	NOUN
ejpam-1781	448	12	{	{	PUNCT
ejpam-1781	448	13	p	p	NOUN
ejpam-1781	448	14	∈	∈	PROPN
ejpam-1781	448	15	℘	℘	NOUN
ejpam-1781	448	16	|	|	ADV
ejpam-1781	448	17	p	p	NOUN
ejpam-1781	448	18	⊆	⊆	NUM
ejpam-1781	448	19	m	m	NOUN
ejpam-1781	448	20	}	}	PUNCT
ejpam-1781	448	21	.	.	PUNCT
ejpam-1781	449	1	5	5	X
ejpam-1781	449	2	.	.	X
ejpam-1781	449	3	let	let	VERB
ejpam-1781	449	4	i	i	PRON
ejpam-1781	449	5	∈	∈	VERB
ejpam-1781	449	6	i	i	PRON
ejpam-1781	449	7	(	(	PUNCT
ejpam-1781	449	8	r	r	NOUN
ejpam-1781	449	9	)	)	PUNCT
ejpam-1781	449	10	be	be	AUX
ejpam-1781	449	11	such	such	ADJ
ejpam-1781	449	12	that	that	SCONJ
ejpam-1781	449	13	u(i	u(i	NOUN
ejpam-1781	449	14	)	)	PUNCT
ejpam-1781	449	15	is	be	AUX
ejpam-1781	449	16	a	a	DET
ejpam-1781	449	17	semi	semi	ADJ
ejpam-1781	449	18	filter	filter	NOUN
ejpam-1781	449	19	in	in	ADP
ejpam-1781	449	20	(	(	PUNCT
ejpam-1781	449	21	℘,⊆	℘,⊆	NOUN
ejpam-1781	449	22	)	)	PUNCT
ejpam-1781	449	23	but	but	CCONJ
ejpam-1781	449	24	(	(	PUNCT
ejpam-1781	449	25	a]∗	a]∗	PROPN
ejpam-1781	449	26	∨	∨	NUM
ejpam-1781	449	27	i	i	PROPN
ejpam-1781	450	1	6=	6=	PROPN
ejpam-1781	450	2	r	r	NOUN
ejpam-1781	450	3	for	for	ADP
ejpam-1781	450	4	some	some	DET
ejpam-1781	450	5	a	a	DET
ejpam-1781	450	6	∈	∈	NOUN
ejpam-1781	450	7	i	i	PRON
ejpam-1781	450	8	,	,	PUNCT
ejpam-1781	450	9	i.e.	i.e.	X
ejpam-1781	450	10	(	(	PUNCT
ejpam-1781	450	11	a]∗	a]∗	PROPN
ejpam-1781	450	12	∨	∨	NUM
ejpam-1781	450	13	i	i	PROPN
ejpam-1781	450	14	⊂	⊂	PROPN
ejpam-1781	450	15	r.	r.	PROPN
ejpam-1781	450	16	then	then	ADV
ejpam-1781	450	17	(	(	PUNCT
ejpam-1781	450	18	a]∗	a]∗	PROPN
ejpam-1781	450	19	∨	∨	PROPN
ejpam-1781	450	20	i	i	PRON
ejpam-1781	450	21	is	be	AUX
ejpam-1781	450	22	a	a	DET
ejpam-1781	450	23	proper	proper	ADJ
ejpam-1781	450	24	ideal	ideal	NOUN
ejpam-1781	450	25	of	of	ADP
ejpam-1781	450	26	r.	r.	PROPN
ejpam-1781	450	27	then	then	ADV
ejpam-1781	450	28	by	by	ADP
ejpam-1781	450	29	theorem	theorem	NOUN
ejpam-1781	450	30	11	11	NUM
ejpam-1781	450	31	there	there	PRON
ejpam-1781	450	32	exists	exist	VERB
ejpam-1781	450	33	a	a	DET
ejpam-1781	450	34	maximal	maximal	ADJ
ejpam-1781	450	35	ideal	ideal	NOUN
ejpam-1781	450	36	m	m	NOUN
ejpam-1781	450	37	of	of	ADP
ejpam-1781	450	38	r	r	NOUN
ejpam-1781	450	39	such	such	ADJ
ejpam-1781	450	40	that	that	SCONJ
ejpam-1781	450	41	(	(	PUNCT
ejpam-1781	450	42	a]∗	a]∗	PROPN
ejpam-1781	450	43	∨	∨	PROPN
ejpam-1781	450	44	i	i	PRON
ejpam-1781	450	45	⊆	⊆	NUM
ejpam-1781	450	46	m	m	VERB
ejpam-1781	450	47	(	(	PUNCT
ejpam-1781	450	48	see	see	VERB
ejpam-1781	450	49	result	result	NOUN
ejpam-1781	450	50	5	5	NUM
ejpam-1781	450	51	)	)	PUNCT
ejpam-1781	450	52	.	.	PUNCT
ejpam-1781	451	1	then	then	ADV
ejpam-1781	451	2	we	we	PRON
ejpam-1781	451	3	have	have	VERB
ejpam-1781	451	4	(	(	PUNCT
ejpam-1781	451	5	a]∗	a]∗	PROPN
ejpam-1781	451	6	⊆	⊆	NUM
ejpam-1781	451	7	m	m	NOUN
ejpam-1781	451	8	.	.	PUNCT
ejpam-1781	452	1	then	then	ADV
ejpam-1781	452	2	y.	y.	PROPN
ejpam-1781	452	3	pawar	pawar	PROPN
ejpam-1781	452	4	,	,	PUNCT
ejpam-1781	452	5	i.	i.	PROPN
ejpam-1781	452	6	shaikh	shaikh	PROPN
ejpam-1781	452	7	/	/	SYM
ejpam-1781	452	8	eur	eur	PROPN
ejpam-1781	452	9	.	.	PUNCT
ejpam-1781	453	1	j.	j.	PROPN
ejpam-1781	453	2	pure	pure	PROPN
ejpam-1781	453	3	appl	appl	PROPN
ejpam-1781	453	4	.	.	PROPN
ejpam-1781	453	5	math	math	PROPN
ejpam-1781	453	6	,	,	PUNCT
ejpam-1781	453	7	6	6	NUM
ejpam-1781	453	8	(	(	PUNCT
ejpam-1781	453	9	2013	2013	NUM
ejpam-1781	453	10	)	)	PUNCT
ejpam-1781	453	11	,	,	PUNCT
ejpam-1781	453	12	107	107	NUM
ejpam-1781	453	13	-	-	SYM
ejpam-1781	453	14	118	118	NUM
ejpam-1781	453	15	117	117	NUM
ejpam-1781	453	16	by	by	ADP
ejpam-1781	453	17	result	result	NOUN
ejpam-1781	453	18	13	13	NUM
ejpam-1781	453	19	there	there	PRON
ejpam-1781	453	20	exists	exist	VERB
ejpam-1781	453	21	a	a	DET
ejpam-1781	453	22	minimal	minimal	ADJ
ejpam-1781	453	23	prime	prime	ADJ
ejpam-1781	453	24	ideal	ideal	NOUN
ejpam-1781	453	25	q	q	PROPN
ejpam-1781	453	26	⊆	⊆	NUM
ejpam-1781	453	27	m	m	NOUN
ejpam-1781	453	28	and	and	CCONJ
ejpam-1781	453	29	a	a	DET
ejpam-1781	453	30	/∈	/∈	NOUN
ejpam-1781	453	31	q.	q.	NOUN
ejpam-1781	453	32	as	as	ADP
ejpam-1781	453	33	a	a	DET
ejpam-1781	453	34	∈	∈	NOUN
ejpam-1781	453	35	i	i	PRON
ejpam-1781	453	36	and	and	CCONJ
ejpam-1781	453	37	a	a	DET
ejpam-1781	453	38	/∈	/∈	NOUN
ejpam-1781	453	39	q	q	NOUN
ejpam-1781	454	1	we	we	PRON
ejpam-1781	454	2	have	have	VERB
ejpam-1781	454	3	i	i	PRON
ejpam-1781	454	4	*	*	PUNCT
ejpam-1781	454	5	q	q	X
ejpam-1781	454	6	,	,	PUNCT
ejpam-1781	454	7	which	which	PRON
ejpam-1781	454	8	means	mean	VERB
ejpam-1781	454	9	that	that	SCONJ
ejpam-1781	454	10	q	q	PUNCT
ejpam-1781	454	11	∈	∈	PROPN
ejpam-1781	454	12	u(i	u(i	NOUN
ejpam-1781	454	13	)	)	PUNCT
ejpam-1781	454	14	.	.	PUNCT
ejpam-1781	455	1	since	since	SCONJ
ejpam-1781	455	2	q	q	PROPN
ejpam-1781	455	3	∈	∈	PROPN
ejpam-1781	455	4	u(i	u(i	NOUN
ejpam-1781	455	5	)	)	PUNCT
ejpam-1781	455	6	and	and	CCONJ
ejpam-1781	455	7	q	q	PRON
ejpam-1781	455	8	⊆	⊆	NUM
ejpam-1781	455	9	m	m	NOUN
ejpam-1781	455	10	and	and	CCONJ
ejpam-1781	455	11	u(i	u(i	NOUN
ejpam-1781	455	12	)	)	PUNCT
ejpam-1781	455	13	is	be	AUX
ejpam-1781	455	14	semi	semi	ADV
ejpam-1781	455	15	filter	filter	NOUN
ejpam-1781	455	16	,	,	PUNCT
ejpam-1781	455	17	we	we	PRON
ejpam-1781	455	18	conclude	conclude	VERB
ejpam-1781	455	19	that	that	SCONJ
ejpam-1781	455	20	m	m	VERB
ejpam-1781	455	21	∈	∈	PROPN
ejpam-1781	455	22	u(i	u(i	NOUN
ejpam-1781	455	23	)	)	PUNCT
ejpam-1781	455	24	,	,	PUNCT
ejpam-1781	456	1	i.e.	i.e.	X
ejpam-1781	456	2	i	i	NOUN
ejpam-1781	456	3	*	*	VERB
ejpam-1781	456	4	m	m	VERB
ejpam-1781	456	5	.	.	PUNCT
ejpam-1781	457	1	this	this	PRON
ejpam-1781	457	2	is	be	AUX
ejpam-1781	457	3	a	a	DET
ejpam-1781	457	4	contradiction	contradiction	NOUN
ejpam-1781	457	5	since	since	SCONJ
ejpam-1781	457	6	i	i	PRON
ejpam-1781	457	7	⊆	⊆	NUM
ejpam-1781	457	8	(	(	PUNCT
ejpam-1781	457	9	a]∗	a]∗	PROPN
ejpam-1781	457	10	∨	∨	PROPN
ejpam-1781	457	11	i	i	PRON
ejpam-1781	457	12	⊆	⊆	NUM
ejpam-1781	457	13	m	m	NOUN
ejpam-1781	457	14	.	.	PUNCT
ejpam-1781	458	1	hence	hence	ADV
ejpam-1781	458	2	(	(	PUNCT
ejpam-1781	458	3	a]∗	a]∗	PROPN
ejpam-1781	458	4	∨	∨	NUM
ejpam-1781	458	5	i	i	PRON
ejpam-1781	458	6	=	=	SYM
ejpam-1781	458	7	r.	r.	PROPN
ejpam-1781	458	8	in	in	ADP
ejpam-1781	458	9	the	the	DET
ejpam-1781	458	10	following	follow	VERB
ejpam-1781	458	11	theorem	theorem	NOUN
ejpam-1781	458	12	we	we	PRON
ejpam-1781	458	13	prove	prove	VERB
ejpam-1781	458	14	sufficient	sufficient	ADJ
ejpam-1781	458	15	condition	condition	NOUN
ejpam-1781	458	16	on	on	ADP
ejpam-1781	458	17	an	an	DET
ejpam-1781	458	18	ideal	ideal	ADJ
ejpam-1781	458	19	i	i	PRON
ejpam-1781	458	20	for	for	ADP
ejpam-1781	458	21	u(i	u(i	NOUN
ejpam-1781	458	22	)	)	PUNCT
ejpam-1781	458	23	to	to	PART
ejpam-1781	458	24	be	be	AUX
ejpam-1781	458	25	a	a	DET
ejpam-1781	458	26	semi	semi	ADJ
ejpam-1781	458	27	filter	filter	NOUN
ejpam-1781	458	28	in	in	ADP
ejpam-1781	458	29	(	(	PUNCT
ejpam-1781	458	30	℘,⊆	℘,⊆	NOUN
ejpam-1781	458	31	)	)	PUNCT
ejpam-1781	458	32	.	.	PUNCT
ejpam-1781	459	1	theorem	theorem	ADJ
ejpam-1781	459	2	14	14	NUM
ejpam-1781	459	3	.	.	PUNCT
ejpam-1781	460	1	let	let	VERB
ejpam-1781	460	2	m1	m1	PROPN
ejpam-1781	460	3	,	,	PUNCT
ejpam-1781	460	4	m2	m2	PROPN
ejpam-1781	460	5	,	,	PUNCT
ejpam-1781	460	6	.	.	PUNCT
ejpam-1781	460	7	.	.	PUNCT
ejpam-1781	461	1	.	.	PUNCT
ejpam-1781	462	1	mn	mn	PROPN
ejpam-1781	462	2	(	(	PUNCT
ejpam-1781	462	3	n	n	CCONJ
ejpam-1781	462	4	-	-	PUNCT
ejpam-1781	462	5	finite	finite	NOUN
ejpam-1781	462	6	)	)	PUNCT
ejpam-1781	462	7	be	be	VERB
ejpam-1781	462	8	maximal	maximal	ADJ
ejpam-1781	462	9	ideals	ideal	NOUN
ejpam-1781	462	10	in	in	ADP
ejpam-1781	462	11	r.	r.	PROPN
ejpam-1781	462	12	let	let	VERB
ejpam-1781	462	13	i	i	PRON
ejpam-1781	462	14	=	=	NOUN
ejpam-1781	462	15	⋂n	⋂n	PROPN
ejpam-1781	462	16	i=1	i=1	PROPN
ejpam-1781	462	17	wmi	wmi	PROPN
ejpam-1781	462	18	.	.	PUNCT
ejpam-1781	463	1	then	then	ADV
ejpam-1781	463	2	u(i	u(i	NOUN
ejpam-1781	463	3	)	)	PUNCT
ejpam-1781	463	4	is	be	AUX
ejpam-1781	463	5	a	a	DET
ejpam-1781	463	6	semi	semi	ADJ
ejpam-1781	463	7	filter	filter	NOUN
ejpam-1781	463	8	in	in	ADP
ejpam-1781	463	9	poset	poset	NOUN
ejpam-1781	463	10	(	(	PUNCT
ejpam-1781	463	11	℘,⊆	℘,⊆	NOUN
ejpam-1781	463	12	)	)	PUNCT
ejpam-1781	463	13	.	.	PUNCT
ejpam-1781	464	1	proof	proof	NOUN
ejpam-1781	464	2	.	.	PUNCT
ejpam-1781	465	1	claim	claim	VERB
ejpam-1781	465	2	1	1	NUM
ejpam-1781	465	3	:	:	PUNCT
ejpam-1781	465	4	u(i	u(i	NOUN
ejpam-1781	465	5	)	)	PUNCT
ejpam-1781	465	6	=	=	SYM
ejpam-1781	465	7	℘℘	℘℘	ADJ
ejpam-1781	465	8	\	\	PROPN
ejpam-1781	465	9	⋃n	⋃n	PROPN
ejpam-1781	465	10	i=1{p	i=1{p	PROPN
ejpam-1781	465	11	∈	∈	PROPN
ejpam-1781	465	12	℘	℘	NOUN
ejpam-1781	465	13	|	|	ADV
ejpam-1781	465	14	p	p	NOUN
ejpam-1781	465	15	⊆	⊆	NUM
ejpam-1781	465	16	mi	mi	NOUN
ejpam-1781	465	17	}	}	PUNCT
ejpam-1781	465	18	.	.	PUNCT
ejpam-1781	466	1	selectp	selectp	PROPN
ejpam-1781	466	2	∈	∈	PROPN
ejpam-1781	467	1	u(i)⇒	u(i)⇒	ADV
ejpam-1781	467	2	i	i	X
ejpam-1781	467	3	*	*	PUNCT
ejpam-1781	468	1	p	p	X
ejpam-1781	468	2	,	,	PUNCT
ejpam-1781	468	3	p	p	NOUN
ejpam-1781	468	4	∈	∈	PROPN
ejpam-1781	468	5	℘	℘	PROPN
ejpam-1781	468	6	⇒	⇒	NOUN
ejpam-1781	468	7	n⋃	n⋃	VERB
ejpam-1781	468	8	i=1	i=1	PROPN
ejpam-1781	468	9	wi	wi	PROPN
ejpam-1781	468	10	*	*	PUNCT
ejpam-1781	469	1	p	p	X
ejpam-1781	469	2	,	,	PUNCT
ejpam-1781	469	3	p	p	NOUN
ejpam-1781	469	4	∈	∈	PROPN
ejpam-1781	469	5	℘	℘	NOUN
ejpam-1781	469	6	⇒wmi	⇒wmi	ADJ
ejpam-1781	469	7	*	*	PUNCT
ejpam-1781	470	1	p	p	X
ejpam-1781	470	2	,	,	PUNCT
ejpam-1781	470	3	p	p	NOUN
ejpam-1781	470	4	∈	∈	PROPN
ejpam-1781	470	5	℘	℘	PROPN
ejpam-1781	470	6	for	for	ADP
ejpam-1781	470	7	all	all	DET
ejpam-1781	470	8	i	i	PRON
ejpam-1781	470	9	,	,	PUNCT
ejpam-1781	470	10	1	1	NUM
ejpam-1781	470	11	≤	≤	NUM
ejpam-1781	470	12	i	i	PRON
ejpam-1781	470	13	≤	≤	ADJ
ejpam-1781	471	1	n.	n.	NOUN
ejpam-1781	472	1	but	but	CCONJ
ejpam-1781	472	2	this	this	PRON
ejpam-1781	472	3	gives	give	VERB
ejpam-1781	472	4	⋂	⋂	PROPN
ejpam-1781	472	5	{	{	PUNCT
ejpam-1781	472	6	p	p	NOUN
ejpam-1781	472	7	∈	∈	PROPN
ejpam-1781	472	8	℘	℘	NOUN
ejpam-1781	472	9	|	|	ADV
ejpam-1781	472	10	p	p	NOUN
ejpam-1781	472	11	⊆	⊆	NUM
ejpam-1781	472	12	mi	mi	NOUN
ejpam-1781	472	13	}	}	PUNCT
ejpam-1781	472	14	*	*	PROPN
ejpam-1781	473	1	p	p	X
ejpam-1781	473	2	,	,	PUNCT
ejpam-1781	473	3	for	for	ADP
ejpam-1781	473	4	all	all	DET
ejpam-1781	473	5	i	i	PRON
ejpam-1781	473	6	,	,	PUNCT
ejpam-1781	473	7	1	1	NUM
ejpam-1781	473	8	≤	≤	NUM
ejpam-1781	473	9	i	i	PRON
ejpam-1781	473	10	≤	≤	NOUN
ejpam-1781	473	11	n	n	CCONJ
ejpam-1781	473	12	thus	thus	ADV
ejpam-1781	473	13	p	p	X
ejpam-1781	473	14	/∈	/∈	PUNCT
ejpam-1781	473	15	{	{	PUNCT
ejpam-1781	473	16	p	p	NOUN
ejpam-1781	473	17	∈	∈	PROPN
ejpam-1781	473	18	℘	℘	NOUN
ejpam-1781	473	19	|	|	ADV
ejpam-1781	473	20	p	p	NOUN
ejpam-1781	473	21	⊆	⊆	NUM
ejpam-1781	473	22	m	m	NOUN
ejpam-1781	473	23	}	}	PUNCT
ejpam-1781	473	24	for	for	ADP
ejpam-1781	473	25	all	all	DET
ejpam-1781	473	26	i	i	PRON
ejpam-1781	473	27	,	,	PUNCT
ejpam-1781	473	28	1	1	NUM
ejpam-1781	473	29	≤	≤	NUM
ejpam-1781	473	30	i	i	PRON
ejpam-1781	473	31	≤	≤	ADJ
ejpam-1781	473	32	n.	n.	NOUN
ejpam-1781	473	33	hence	hence	ADV
ejpam-1781	473	34	p	p	X
ejpam-1781	473	35	/∈	/∈	PUNCT
ejpam-1781	473	36	⋃n	⋃n	PROPN
ejpam-1781	474	1	i=1{p	i=1{p	PROPN
ejpam-1781	474	2	∈	∈	PROPN
ejpam-1781	474	3	℘	℘	NOUN
ejpam-1781	474	4	|	|	ADV
ejpam-1781	474	5	p	p	NOUN
ejpam-1781	474	6	⊆	⊆	NUM
ejpam-1781	474	7	mi	mi	NOUN
ejpam-1781	474	8	}	}	PUNCT
ejpam-1781	474	9	.	.	PUNCT
ejpam-1781	475	1	i.e.	i.e.	X
ejpam-1781	475	2	p	p	X
ejpam-1781	475	3	∈	∈	PROPN
ejpam-1781	475	4	℘	℘	PROPN
ejpam-1781	475	5	\	\	NOUN
ejpam-1781	475	6	⋃n	⋃n	PROPN
ejpam-1781	475	7	i=1{p	i=1{p	PROPN
ejpam-1781	475	8	∈	∈	PROPN
ejpam-1781	475	9	℘	℘	NOUN
ejpam-1781	475	10	|	|	ADV
ejpam-1781	475	11	p	p	NOUN
ejpam-1781	475	12	⊆	⊆	NUM
ejpam-1781	475	13	mi	mi	NOUN
ejpam-1781	475	14	}	}	PUNCT
ejpam-1781	475	15	.	.	PUNCT
ejpam-1781	476	1	thus	thus	ADV
ejpam-1781	476	2	u(i	u(i	NOUN
ejpam-1781	476	3	)	)	PUNCT
ejpam-1781	476	4	⊆	⊆	NUM
ejpam-1781	476	5	℘	℘	PROPN
ejpam-1781	476	6	\	\	NOUN
ejpam-1781	476	7	⋃n	⋃n	PROPN
ejpam-1781	476	8	i=1{p	i=1{p	PROPN
ejpam-1781	476	9	∈	∈	PROPN
ejpam-1781	476	10	℘	℘	NOUN
ejpam-1781	476	11	|	|	ADV
ejpam-1781	476	12	p	p	NOUN
ejpam-1781	476	13	⊆	⊆	NUM
ejpam-1781	476	14	mi	mi	NOUN
ejpam-1781	476	15	}	}	PUNCT
ejpam-1781	476	16	.	.	PUNCT
ejpam-1781	476	17	.	.	PUNCT
ejpam-1781	476	18	.	.	PUNCT
ejpam-1781	477	1	(	(	PUNCT
ejpam-1781	477	2	i	i	NOUN
ejpam-1781	477	3	)	)	PUNCT
ejpam-1781	477	4	now	now	ADV
ejpam-1781	477	5	,	,	PUNCT
ejpam-1781	477	6	select	select	VERB
ejpam-1781	477	7	p	p	PRON
ejpam-1781	477	8	∈	∈	PROPN
ejpam-1781	477	9	℘	℘	NOUN
ejpam-1781	477	10	such	such	ADJ
ejpam-1781	477	11	that	that	SCONJ
ejpam-1781	477	12	p	p	PROPN
ejpam-1781	477	13	/∈	/∈	PUNCT
ejpam-1781	477	14	⋃n	⋃n	PROPN
ejpam-1781	477	15	i=1{p	i=1{p	PROPN
ejpam-1781	477	16	∈	∈	PROPN
ejpam-1781	477	17	℘	℘	NOUN
ejpam-1781	477	18	|	|	ADV
ejpam-1781	477	19	p	p	NOUN
ejpam-1781	477	20	⊆	⊆	NUM
ejpam-1781	477	21	mi	mi	NOUN
ejpam-1781	477	22	}	}	PUNCT
ejpam-1781	477	23	,	,	PUNCT
ejpam-1781	477	24	p	p	PROPN
ejpam-1781	477	25	∈	∈	PROPN
ejpam-1781	477	26	℘.	℘.	PROPN
ejpam-1781	477	27	but	but	CCONJ
ejpam-1781	477	28	then	then	ADV
ejpam-1781	477	29	⋂	⋂	PROPN
ejpam-1781	477	30	{	{	PUNCT
ejpam-1781	477	31	p	p	NOUN
ejpam-1781	477	32	∈	∈	PROPN
ejpam-1781	477	33	℘	℘	NOUN
ejpam-1781	477	34	|	|	ADV
ejpam-1781	477	35	p	p	NOUN
ejpam-1781	477	36	⊆	⊆	NUM
ejpam-1781	477	37	mi	mi	NOUN
ejpam-1781	477	38	}	}	PUNCT
ejpam-1781	477	39	*	*	PUNCT
ejpam-1781	478	1	p	p	NOUN
ejpam-1781	478	2	for	for	ADP
ejpam-1781	478	3	each	each	DET
ejpam-1781	478	4	i	i	PROPN
ejpam-1781	478	5	,	,	PUNCT
ejpam-1781	478	6	1≤	1≤	INTJ
ejpam-1781	478	7	i	i	PROPN
ejpam-1781	478	8	≤	≤	PROPN
ejpam-1781	478	9	n	n	CCONJ
ejpam-1781	478	10	,	,	PUNCT
ejpam-1781	478	11	p	p	PROPN
ejpam-1781	478	12	∈	∈	PROPN
ejpam-1781	478	13	℘	℘	PROPN
ejpam-1781	478	14	implies	imply	VERB
ejpam-1781	478	15	i	i	PRON
ejpam-1781	478	16	*	*	PUNCT
ejpam-1781	479	1	p	p	X
ejpam-1781	479	2	p	p	X
ejpam-1781	479	3	∈	∈	PROPN
ejpam-1781	479	4	℘.	℘.	PROPN
ejpam-1781	479	5	hence	hence	ADV
ejpam-1781	479	6	p	p	X
ejpam-1781	479	7	∈	∈	PROPN
ejpam-1781	479	8	u(i	u(i	NOUN
ejpam-1781	479	9	)	)	PUNCT
ejpam-1781	479	10	.	.	PUNCT
ejpam-1781	480	1	thus	thus	ADV
ejpam-1781	480	2	℘	℘	VERB
ejpam-1781	480	3	\	\	NOUN
ejpam-1781	480	4	⋃n	⋃n	PROPN
ejpam-1781	480	5	i=1{p	i=1{p	PROPN
ejpam-1781	480	6	∈	∈	PROPN
ejpam-1781	480	7	℘	℘	NOUN
ejpam-1781	480	8	|	|	ADV
ejpam-1781	480	9	p	p	NOUN
ejpam-1781	480	10	⊆	⊆	NUM
ejpam-1781	480	11	mi	mi	NOUN
ejpam-1781	480	12	}	}	PUNCT
ejpam-1781	480	13	⊆	⊆	NUM
ejpam-1781	480	14	u(i	u(i	NOUN
ejpam-1781	480	15	)	)	PUNCT
ejpam-1781	480	16	.	.	PUNCT
ejpam-1781	480	17	.	.	PUNCT
ejpam-1781	480	18	.	.	PUNCT
ejpam-1781	481	1	(	(	PUNCT
ejpam-1781	481	2	ii	ii	X
ejpam-1781	481	3	)	)	PUNCT
ejpam-1781	481	4	combining	combine	VERB
ejpam-1781	481	5	both	both	DET
ejpam-1781	481	6	the	the	DET
ejpam-1781	481	7	inclusions	inclusion	NOUN
ejpam-1781	481	8	we	we	PRON
ejpam-1781	481	9	get	get	VERB
ejpam-1781	481	10	u(i	u(i	NOUN
ejpam-1781	481	11	)	)	PUNCT
ejpam-1781	481	12	=	=	SYM
ejpam-1781	481	13	℘	℘	PROPN
ejpam-1781	481	14	\	\	NOUN
ejpam-1781	481	15	⋃n	⋃n	PROPN
ejpam-1781	481	16	i=1{p	i=1{p	PROPN
ejpam-1781	481	17	∈	∈	PROPN
ejpam-1781	481	18	℘	℘	NOUN
ejpam-1781	481	19	|	|	ADV
ejpam-1781	481	20	p	p	NOUN
ejpam-1781	481	21	⊆	⊆	NUM
ejpam-1781	481	22	mi	mi	NOUN
ejpam-1781	481	23	}	}	PUNCT
ejpam-1781	481	24	claim	claim	NOUN
ejpam-1781	481	25	2	2	NUM
ejpam-1781	481	26	:	:	PUNCT
ejpam-1781	481	27	u(i	u(i	NOUN
ejpam-1781	481	28	)	)	PUNCT
ejpam-1781	481	29	is	be	AUX
ejpam-1781	481	30	a	a	DET
ejpam-1781	481	31	semi	semi	ADJ
ejpam-1781	481	32	filter	filter	NOUN
ejpam-1781	481	33	in	in	ADP
ejpam-1781	481	34	(	(	PUNCT
ejpam-1781	481	35	℘,⊆	℘,⊆	NOUN
ejpam-1781	481	36	)	)	PUNCT
ejpam-1781	481	37	let	let	VERB
ejpam-1781	481	38	p	p	PRON
ejpam-1781	481	39	,	,	PUNCT
ejpam-1781	481	40	q	q	NOUN
ejpam-1781	481	41	∈	∈	NOUN
ejpam-1781	481	42	℘	℘	PROPN
ejpam-1781	481	43	with	with	ADP
ejpam-1781	481	44	p	p	NOUN
ejpam-1781	481	45	⊆	⊆	NUM
ejpam-1781	481	46	q	q	NOUN
ejpam-1781	481	47	and	and	CCONJ
ejpam-1781	481	48	p	p	NOUN
ejpam-1781	481	49	∈	∈	PROPN
ejpam-1781	481	50	u(i	u(i	NOUN
ejpam-1781	481	51	)	)	PUNCT
ejpam-1781	481	52	.	.	PUNCT
ejpam-1781	482	1	p	p	X
ejpam-1781	482	2	∈	∈	PROPN
ejpam-1781	482	3	u(i	u(i	NOUN
ejpam-1781	482	4	)	)	PUNCT
ejpam-1781	482	5	gives	give	VERB
ejpam-1781	482	6	p	p	PRON
ejpam-1781	482	7	/∈	/∈	PUNCT
ejpam-1781	482	8	⋃n	⋃n	PROPN
ejpam-1781	482	9	i=1{p	i=1{p	PROPN
ejpam-1781	482	10	∈	∈	PROPN
ejpam-1781	482	11	℘	℘	NOUN
ejpam-1781	482	12	|	|	ADV
ejpam-1781	482	13	p	p	NOUN
ejpam-1781	482	14	⊆	⊆	NUM
ejpam-1781	482	15	mi	mi	NOUN
ejpam-1781	482	16	}	}	PUNCT
ejpam-1781	482	17	,	,	PUNCT
ejpam-1781	482	18	(	(	PUNCT
ejpam-1781	482	19	by	by	ADP
ejpam-1781	482	20	claim	claim	NOUN
ejpam-1781	482	21	1	1	NUM
ejpam-1781	482	22	)	)	PUNCT
ejpam-1781	482	23	.	.	PUNCT
ejpam-1781	483	1	as	as	SCONJ
ejpam-1781	483	2	p	p	DET
ejpam-1781	483	3	⊆	⊆	NUM
ejpam-1781	483	4	q	q	NOUN
ejpam-1781	483	5	,	,	PUNCT
ejpam-1781	483	6	obviously	obviously	ADV
ejpam-1781	483	7	,	,	PUNCT
ejpam-1781	483	8	q	q	PROPN
ejpam-1781	483	9	/∈	/∈	PUNCT
ejpam-1781	483	10	⋃n	⋃n	PROPN
ejpam-1781	483	11	i=1{p	i=1{p	PROPN
ejpam-1781	483	12	∈	∈	PROPN
ejpam-1781	483	13	℘	℘	NOUN
ejpam-1781	483	14	|	|	ADV
ejpam-1781	483	15	p	p	NOUN
ejpam-1781	483	16	⊆	⊆	NUM
ejpam-1781	483	17	mi	mi	NOUN
ejpam-1781	483	18	}	}	PUNCT
ejpam-1781	483	19	.	.	PUNCT
ejpam-1781	484	1	but	but	CCONJ
ejpam-1781	484	2	then	then	ADV
ejpam-1781	484	3	q	q	PROPN
ejpam-1781	484	4	∈	∈	PROPN
ejpam-1781	484	5	℘	℘	PROPN
ejpam-1781	484	6	\	\	NOUN
ejpam-1781	484	7	⋃n	⋃n	PROPN
ejpam-1781	484	8	i=1{p	i=1{p	PROPN
ejpam-1781	484	9	∈	∈	PROPN
ejpam-1781	484	10	℘	℘	NOUN
ejpam-1781	484	11	|	|	ADV
ejpam-1781	484	12	p	p	NOUN
ejpam-1781	484	13	⊆	⊆	NUM
ejpam-1781	484	14	mi	mi	NOUN
ejpam-1781	484	15	}	}	PUNCT
ejpam-1781	484	16	=	=	SYM
ejpam-1781	484	17	u(i	u(i	NOUN
ejpam-1781	484	18	)	)	PUNCT
ejpam-1781	484	19	.	.	PUNCT
ejpam-1781	485	1	this	this	PRON
ejpam-1781	485	2	shows	show	VERB
ejpam-1781	485	3	that	that	SCONJ
ejpam-1781	485	4	u(i	u(i	NOUN
ejpam-1781	485	5	)	)	PUNCT
ejpam-1781	485	6	is	be	AUX
ejpam-1781	485	7	a	a	DET
ejpam-1781	485	8	semi	semi	ADJ
ejpam-1781	485	9	filter	filter	NOUN
ejpam-1781	485	10	in	in	ADP
ejpam-1781	485	11	(	(	PUNCT
ejpam-1781	485	12	℘,⊆	℘,⊆	NOUN
ejpam-1781	485	13	)	)	PUNCT
ejpam-1781	485	14	and	and	CCONJ
ejpam-1781	485	15	the	the	DET
ejpam-1781	485	16	result	result	NOUN
ejpam-1781	485	17	follows	follow	VERB
ejpam-1781	485	18	.	.	PUNCT
ejpam-1781	486	1	theorem	theorem	ADJ
ejpam-1781	486	2	15	15	NUM
ejpam-1781	486	3	.	.	PUNCT
ejpam-1781	487	1	let	let	VERB
ejpam-1781	487	2	i	i	PRON
ejpam-1781	487	3	be	be	AUX
ejpam-1781	487	4	an	an	DET
ejpam-1781	487	5	ideal	ideal	NOUN
ejpam-1781	487	6	of	of	ADP
ejpam-1781	487	7	r	r	NOUN
ejpam-1781	487	8	such	such	ADJ
ejpam-1781	487	9	that	that	PRON
ejpam-1781	487	10	(	(	PUNCT
ejpam-1781	487	11	x]∗	x]∗	PROPN
ejpam-1781	487	12	and	and	CCONJ
ejpam-1781	487	13	i	i	PRON
ejpam-1781	487	14	are	be	AUX
ejpam-1781	487	15	co	co	ADJ
ejpam-1781	487	16	-	-	ADJ
ejpam-1781	487	17	maximal	maximal	ADJ
ejpam-1781	487	18	ideals	ideal	NOUN
ejpam-1781	487	19	for	for	ADP
ejpam-1781	487	20	each	each	DET
ejpam-1781	487	21	x	x	PROPN
ejpam-1781	487	22	∈	∈	PROPN
ejpam-1781	487	23	r.	r.	NOUN
ejpam-1781	487	24	then	then	ADV
ejpam-1781	487	25	u(i	u(i	NOUN
ejpam-1781	487	26	)	)	PUNCT
ejpam-1781	487	27	is	be	AUX
ejpam-1781	487	28	a	a	DET
ejpam-1781	487	29	semi	semi	ADJ
ejpam-1781	487	30	filter	filter	NOUN
ejpam-1781	487	31	in	in	ADP
ejpam-1781	487	32	(	(	PUNCT
ejpam-1781	487	33	℘,⊆	℘,⊆	NOUN
ejpam-1781	487	34	)	)	PUNCT
ejpam-1781	487	35	.	.	PUNCT
ejpam-1781	488	1	proof	proof	NOUN
ejpam-1781	488	2	.	.	PUNCT
ejpam-1781	489	1	let	let	VERB
ejpam-1781	489	2	p	p	PRON
ejpam-1781	489	3	∈	∈	PROPN
ejpam-1781	489	4	u(i	u(i	NOUN
ejpam-1781	489	5	)	)	PUNCT
ejpam-1781	489	6	,	,	PUNCT
ejpam-1781	489	7	p	p	ADP
ejpam-1781	489	8	⊆	⊆	NUM
ejpam-1781	489	9	q	q	NOUN
ejpam-1781	489	10	and	and	CCONJ
ejpam-1781	489	11	q	q	PROPN
ejpam-1781	489	12	∈	∈	PROPN
ejpam-1781	489	13	℘.	℘.	PROPN
ejpam-1781	489	14	we	we	PRON
ejpam-1781	489	15	must	must	AUX
ejpam-1781	489	16	show	show	VERB
ejpam-1781	489	17	that	that	SCONJ
ejpam-1781	489	18	q	q	PUNCT
ejpam-1781	489	19	∈	∈	PROPN
ejpam-1781	489	20	u(i	u(i	NOUN
ejpam-1781	489	21	)	)	PUNCT
ejpam-1781	489	22	.	.	PUNCT
ejpam-1781	490	1	assume	assume	VERB
ejpam-1781	490	2	on	on	ADP
ejpam-1781	490	3	the	the	DET
ejpam-1781	490	4	contrary	contrary	NOUN
ejpam-1781	490	5	that	that	DET
ejpam-1781	490	6	q	q	NOUN
ejpam-1781	490	7	/∈	/∈	PUNCT
ejpam-1781	490	8	u(i	u(i	NOUN
ejpam-1781	490	9	)	)	PUNCT
ejpam-1781	490	10	.	.	PUNCT
ejpam-1781	491	1	this	this	PRON
ejpam-1781	491	2	implies	imply	VERB
ejpam-1781	491	3	i	i	PRON
ejpam-1781	491	4	⊆	⊆	NUM
ejpam-1781	491	5	q.	q.	NOUN
ejpam-1781	491	6	as	as	SCONJ
ejpam-1781	491	7	p	p	PROPN
ejpam-1781	491	8	is	be	AUX
ejpam-1781	491	9	prime	prime	ADJ
ejpam-1781	491	10	ideal	ideal	NOUN
ejpam-1781	491	11	we	we	PRON
ejpam-1781	491	12	have	have	VERB
ejpam-1781	491	13	some	some	DET
ejpam-1781	491	14	j	j	NOUN
ejpam-1781	491	15	∈m	∈m	NOUN
ejpam-1781	491	16	such	such	ADJ
ejpam-1781	491	17	that	that	SCONJ
ejpam-1781	491	18	j	j	PROPN
ejpam-1781	491	19	⊆	⊆	NUM
ejpam-1781	491	20	p	p	X
ejpam-1781	491	21	(	(	PUNCT
ejpam-1781	491	22	by	by	ADP
ejpam-1781	491	23	result	result	NOUN
ejpam-1781	491	24	2	2	NUM
ejpam-1781	491	25	)	)	PUNCT
ejpam-1781	491	26	.	.	PUNCT
ejpam-1781	492	1	if	if	SCONJ
ejpam-1781	492	2	i	i	PRON
ejpam-1781	492	3	⊆	⊆	NUM
ejpam-1781	492	4	j	j	NOUN
ejpam-1781	492	5	⊆	⊆	NUM
ejpam-1781	492	6	p	p	NOUN
ejpam-1781	492	7	then	then	ADV
ejpam-1781	492	8	p	p	X
ejpam-1781	492	9	/∈	/∈	PUNCT
ejpam-1781	492	10	u(i	u(i	NOUN
ejpam-1781	492	11	)	)	PUNCT
ejpam-1781	492	12	,	,	PUNCT
ejpam-1781	492	13	which	which	PRON
ejpam-1781	492	14	is	be	AUX
ejpam-1781	492	15	a	a	DET
ejpam-1781	492	16	contradiction	contradiction	NOUN
ejpam-1781	492	17	.	.	PUNCT
ejpam-1781	493	1	hence	hence	ADV
ejpam-1781	493	2	i	i	PRON
ejpam-1781	493	3	*	*	PUNCT
ejpam-1781	494	1	j	j	PROPN
ejpam-1781	494	2	.	.	PUNCT
ejpam-1781	495	1	select	select	ADJ
ejpam-1781	495	2	x	x	PUNCT
ejpam-1781	495	3	∈	∈	PROPN
ejpam-1781	495	4	i	i	PRON
ejpam-1781	495	5	such	such	VERB
ejpam-1781	495	6	that	that	PRON
ejpam-1781	495	7	x	x	PROPN
ejpam-1781	495	8	/∈	/∈	PROPN
ejpam-1781	496	1	j	j	PROPN
ejpam-1781	496	2	.	.	PUNCT
ejpam-1781	497	1	if	if	SCONJ
ejpam-1781	497	2	(	(	PUNCT
ejpam-1781	497	3	x]∗	x]∗	PROPN
ejpam-1781	497	4	*	*	PUNCT
ejpam-1781	497	5	j	j	PROPN
ejpam-1781	497	6	,	,	PUNCT
ejpam-1781	497	7	then	then	ADV
ejpam-1781	497	8	there	there	PRON
ejpam-1781	497	9	exists	exist	VERB
ejpam-1781	497	10	t	t	PROPN
ejpam-1781	497	11	∈	∈	PROPN
ejpam-1781	497	12	(	(	PUNCT
ejpam-1781	497	13	x]∗	x]∗	PROPN
ejpam-1781	497	14	,	,	PUNCT
ejpam-1781	497	15	t	t	PROPN
ejpam-1781	497	16	/∈	/∈	PUNCT
ejpam-1781	498	1	j	j	PROPN
ejpam-1781	498	2	.	.	PUNCT
ejpam-1781	499	1	as	as	ADP
ejpam-1781	499	2	t	t	PROPN
ejpam-1781	499	3	∈	∈	PROPN
ejpam-1781	499	4	(	(	PUNCT
ejpam-1781	499	5	x]∗	x]∗	PROPN
ejpam-1781	499	6	gives	give	VERB
ejpam-1781	499	7	that	that	DET
ejpam-1781	499	8	t	t	NOUN
ejpam-1781	499	9	∧	∧	PROPN
ejpam-1781	499	10	x	x	PUNCT
ejpam-1781	499	11	=	=	SYM
ejpam-1781	499	12	0	0	NUM
ejpam-1781	499	13	∈	∈	PROPN
ejpam-1781	499	14	j	j	PROPN
ejpam-1781	499	15	.	.	PUNCT
ejpam-1781	500	1	but	but	CCONJ
ejpam-1781	500	2	as	as	ADP
ejpam-1781	500	3	x	x	PROPN
ejpam-1781	500	4	/∈	/∈	PROPN
ejpam-1781	500	5	j	j	PROPN
ejpam-1781	500	6	and	and	CCONJ
ejpam-1781	500	7	t	t	PROPN
ejpam-1781	500	8	/∈	/∈	PUNCT
ejpam-1781	501	1	j	j	PROPN
ejpam-1781	501	2	will	will	AUX
ejpam-1781	501	3	give	give	VERB
ejpam-1781	501	4	t	t	PROPN
ejpam-1781	501	5	∧	∧	PROPN
ejpam-1781	501	6	x	x	NOUN
ejpam-1781	501	7	/∈	/∈	PUNCT
ejpam-1781	502	1	j	j	PROPN
ejpam-1781	502	2	i.e.	i.e.	X
ejpam-1781	502	3	0	0	X
ejpam-1781	502	4	/∈	/∈	SYM
ejpam-1781	502	5	j	j	PROPN
ejpam-1781	502	6	which	which	PRON
ejpam-1781	502	7	is	be	AUX
ejpam-1781	502	8	impossible	impossible	ADJ
ejpam-1781	502	9	.	.	PUNCT
ejpam-1781	503	1	hence	hence	ADV
ejpam-1781	503	2	(	(	PUNCT
ejpam-1781	503	3	x]∗	x]∗	PROPN
ejpam-1781	503	4	⊆	⊆	NUM
ejpam-1781	503	5	j	j	PROPN
ejpam-1781	503	6	.	.	PUNCT
ejpam-1781	504	1	but	but	CCONJ
ejpam-1781	504	2	j	j	PROPN
ejpam-1781	505	1	⊆	⊆	NUM
ejpam-1781	505	2	p	p	PRON
ejpam-1781	505	3	⊆	⊆	NUM
ejpam-1781	505	4	q	q	NOUN
ejpam-1781	505	5	implies	imply	VERB
ejpam-1781	505	6	(	(	PUNCT
ejpam-1781	505	7	x]∗	x]∗	PROPN
ejpam-1781	505	8	⊆	⊆	NUM
ejpam-1781	505	9	q.	q.	NOUN
ejpam-1781	505	10	since	since	SCONJ
ejpam-1781	505	11	i	i	PRON
ejpam-1781	505	12	⊆	⊆	NUM
ejpam-1781	505	13	q	q	NOUN
ejpam-1781	505	14	,	,	PUNCT
ejpam-1781	505	15	(	(	PUNCT
ejpam-1781	505	16	x]∗	x]∗	PROPN
ejpam-1781	505	17	∨	∨	PROPN
ejpam-1781	505	18	i	i	PRON
ejpam-1781	505	19	⊆	⊆	NUM
ejpam-1781	505	20	q.	q.	NOUN
ejpam-1781	505	21	thus	thus	ADV
ejpam-1781	505	22	r	r	NOUN
ejpam-1781	505	23	=	=	SYM
ejpam-1781	505	24	q	q	X
ejpam-1781	505	25	which	which	PRON
ejpam-1781	505	26	is	be	AUX
ejpam-1781	505	27	absurd	absurd	ADJ
ejpam-1781	505	28	.	.	PUNCT
ejpam-1781	506	1	hence	hence	ADV
ejpam-1781	506	2	we	we	PRON
ejpam-1781	506	3	conclude	conclude	VERB
ejpam-1781	506	4	that	that	DET
ejpam-1781	506	5	q	q	PUNCT
ejpam-1781	506	6	∈	∈	PROPN
ejpam-1781	506	7	u(i	u(i	NOUN
ejpam-1781	506	8	)	)	PUNCT
ejpam-1781	506	9	.	.	PUNCT
ejpam-1781	507	1	from	from	ADP
ejpam-1781	507	2	this	this	PRON
ejpam-1781	507	3	it	it	PRON
ejpam-1781	507	4	follows	follow	VERB
ejpam-1781	507	5	that	that	SCONJ
ejpam-1781	507	6	u(i	u(i	NOUN
ejpam-1781	507	7	)	)	PUNCT
ejpam-1781	507	8	is	be	AUX
ejpam-1781	507	9	a	a	DET
ejpam-1781	507	10	semi	semi	ADJ
ejpam-1781	507	11	filter	filter	NOUN
ejpam-1781	507	12	.	.	PUNCT
ejpam-1781	507	13	necessary	necessary	ADJ
ejpam-1781	507	14	and	and	CCONJ
ejpam-1781	507	15	sufficient	sufficient	ADJ
ejpam-1781	507	16	for	for	ADP
ejpam-1781	507	17	u(i	u(i	NOUN
ejpam-1781	507	18	)	)	PUNCT
ejpam-1781	507	19	to	to	PART
ejpam-1781	507	20	be	be	AUX
ejpam-1781	507	21	a	a	DET
ejpam-1781	507	22	semi	semi	ADJ
ejpam-1781	507	23	filter	filter	NOUN
ejpam-1781	507	24	(	(	PUNCT
ejpam-1781	508	1	i	i	PRON
ejpam-1781	509	1	∈	∈	PROPN
ejpam-1781	510	1	i	i	PRON
ejpam-1781	510	2	(	(	PUNCT
ejpam-1781	510	3	r	r	NOUN
ejpam-1781	510	4	)	)	PUNCT
ejpam-1781	510	5	)	)	PUNCT
ejpam-1781	510	6	in	in	ADP
ejpam-1781	510	7	(	(	PUNCT
ejpam-1781	510	8	℘,⊆	℘,⊆	NOUN
ejpam-1781	510	9	)	)	PUNCT
ejpam-1781	510	10	is	be	AUX
ejpam-1781	510	11	proved	prove	VERB
ejpam-1781	510	12	in	in	ADP
ejpam-1781	510	13	the	the	DET
ejpam-1781	510	14	following	follow	VERB
ejpam-1781	510	15	theorem	theorem	VERB
ejpam-1781	510	16	.	.	PROPN
ejpam-1781	511	1	references	reference	NOUN
ejpam-1781	511	2	118	118	NUM
ejpam-1781	511	3	theorem	theorem	VERB
ejpam-1781	511	4	16	16	NUM
ejpam-1781	511	5	.	.	PUNCT
ejpam-1781	512	1	for	for	ADP
ejpam-1781	512	2	i	i	PRON
ejpam-1781	512	3	∈	∈	PROPN
ejpam-1781	512	4	i	i	PRON
ejpam-1781	512	5	(	(	PUNCT
ejpam-1781	512	6	r	r	NOUN
ejpam-1781	512	7	)	)	PUNCT
ejpam-1781	512	8	,	,	PUNCT
ejpam-1781	512	9	u(i	u(i	NOUN
ejpam-1781	512	10	)	)	PUNCT
ejpam-1781	512	11	is	be	AUX
ejpam-1781	512	12	a	a	DET
ejpam-1781	512	13	semi	semi	ADJ
ejpam-1781	512	14	filter	filter	NOUN
ejpam-1781	512	15	in	in	ADP
ejpam-1781	512	16	(	(	PUNCT
ejpam-1781	512	17	℘,⊆	℘,⊆	NOUN
ejpam-1781	512	18	)	)	PUNCT
ejpam-1781	512	19	if	if	SCONJ
ejpam-1781	513	1	and	and	CCONJ
ejpam-1781	513	2	only	only	ADV
ejpam-1781	513	3	if	if	SCONJ
ejpam-1781	513	4	u(i	u(i	ADJ
ejpam-1781	513	5	)	)	PUNCT
ejpam-1781	513	6	=	=	PUNCT
ejpam-1781	513	7	⋃	⋃	ADP
ejpam-1781	513	8	a∈i	a∈i	NOUN
ejpam-1781	513	9	v	v	NOUN
ejpam-1781	513	10	(	(	PUNCT
ejpam-1781	513	11	(	(	PUNCT
ejpam-1781	513	12	a]∗	a]∗	PROPN
ejpam-1781	513	13	)	)	PUNCT
ejpam-1781	513	14	.	.	PUNCT
ejpam-1781	514	1	proof	proof	NOUN
ejpam-1781	514	2	.	.	PUNCT
ejpam-1781	515	1	let	let	VERB
ejpam-1781	515	2	u(i	u(i	PRON
ejpam-1781	515	3	)	)	PUNCT
ejpam-1781	515	4	=	=	PUNCT
ejpam-1781	516	1	⋃	⋃	ADP
ejpam-1781	516	2	a∈i	a∈i	NOUN
ejpam-1781	516	3	v	v	NOUN
ejpam-1781	516	4	(	(	PUNCT
ejpam-1781	516	5	(	(	PUNCT
ejpam-1781	516	6	a]∗	a]∗	PROPN
ejpam-1781	516	7	)	)	PUNCT
ejpam-1781	516	8	,	,	PUNCT
ejpam-1781	516	9	p	p	PROPN
ejpam-1781	516	10	∈	∈	PROPN
ejpam-1781	516	11	u(i),q	u(i),q	PROPN
ejpam-1781	516	12	∈	∈	PROPN
ejpam-1781	516	13	℘	℘	PROPN
ejpam-1781	516	14	and	and	CCONJ
ejpam-1781	516	15	p	p	NOUN
ejpam-1781	516	16	⊆	⊆	NUM
ejpam-1781	516	17	q.	q.	NOUN
ejpam-1781	517	1	then	then	ADV
ejpam-1781	517	2	i	i	PRON
ejpam-1781	517	3	*	*	PUNCT
ejpam-1781	518	1	p.	p.	NOUN
ejpam-1781	518	2	therefore	therefore	ADV
ejpam-1781	518	3	there	there	PRON
ejpam-1781	518	4	exists	exist	VERB
ejpam-1781	518	5	x1	x1	PROPN
ejpam-1781	518	6	∈	∈	PROPN
ejpam-1781	519	1	i	i	PRON
ejpam-1781	519	2	\	\	PUNCT
ejpam-1781	520	1	p.	p.	NOUN
ejpam-1781	521	1	but	but	CCONJ
ejpam-1781	521	2	then	then	ADV
ejpam-1781	521	3	(	(	PUNCT
ejpam-1781	521	4	x1	x1	PROPN
ejpam-1781	521	5	]	]	PUNCT
ejpam-1781	521	6	∗	∗	VERB
ejpam-1781	521	7	⊆	⊆	NUM
ejpam-1781	521	8	p	p	PRON
ejpam-1781	521	9	⊆	⊆	NUM
ejpam-1781	521	10	q	q	NOUN
ejpam-1781	521	11	shows	show	VERB
ejpam-1781	521	12	that	that	SCONJ
ejpam-1781	521	13	q	q	PROPN
ejpam-1781	521	14	∈	∈	PROPN
ejpam-1781	521	15	v	v	ADP
ejpam-1781	521	16	�	�	PROPN
ejpam-1781	521	17	(	(	PUNCT
ejpam-1781	521	18	x1	x1	PROPN
ejpam-1781	521	19	]	]	PUNCT
ejpam-1781	521	20	∗	∗	NOUN
ejpam-1781	521	21	�	�	PROPN
ejpam-1781	521	22	⊆	⊆	NUM
ejpam-1781	521	23	⋃	⋃	NOUN
ejpam-1781	521	24	a∈i	a∈i	NOUN
ejpam-1781	521	25	v	v	NOUN
ejpam-1781	521	26	(	(	PUNCT
ejpam-1781	521	27	(	(	PUNCT
ejpam-1781	521	28	a]∗	a]∗	PROPN
ejpam-1781	521	29	)	)	PUNCT
ejpam-1781	521	30	=	=	SYM
ejpam-1781	521	31	u(i	u(i	NOUN
ejpam-1781	521	32	)	)	PUNCT
ejpam-1781	521	33	.	.	PUNCT
ejpam-1781	522	1	this	this	PRON
ejpam-1781	522	2	in	in	ADP
ejpam-1781	522	3	turn	turn	NOUN
ejpam-1781	522	4	shows	show	VERB
ejpam-1781	522	5	that	that	SCONJ
ejpam-1781	522	6	u(i	u(i	NOUN
ejpam-1781	522	7	)	)	PUNCT
ejpam-1781	522	8	is	be	AUX
ejpam-1781	522	9	a	a	DET
ejpam-1781	522	10	semi	semi	ADJ
ejpam-1781	522	11	filter	filter	NOUN
ejpam-1781	522	12	.	.	PUNCT
ejpam-1781	523	1	now	now	ADV
ejpam-1781	523	2	u(i	u(i	NOUN
ejpam-1781	523	3	)	)	PUNCT
ejpam-1781	523	4	is	be	AUX
ejpam-1781	523	5	a	a	DET
ejpam-1781	523	6	semi	semi	ADJ
ejpam-1781	523	7	filter	filter	NOUN
ejpam-1781	523	8	in	in	ADP
ejpam-1781	523	9	(	(	PUNCT
ejpam-1781	523	10	℘,⊆	℘,⊆	NOUN
ejpam-1781	523	11	)	)	PUNCT
ejpam-1781	523	12	.	.	PUNCT
ejpam-1781	524	1	to	to	PART
ejpam-1781	524	2	prove	prove	VERB
ejpam-1781	524	3	that	that	SCONJ
ejpam-1781	524	4	u(i	u(i	NOUN
ejpam-1781	524	5	)	)	PUNCT
ejpam-1781	524	6	=	=	PUNCT
ejpam-1781	524	7	⋃	⋃	ADP
ejpam-1781	524	8	a∈i	a∈i	NOUN
ejpam-1781	524	9	v	v	NOUN
ejpam-1781	524	10	(	(	PUNCT
ejpam-1781	524	11	(	(	PUNCT
ejpam-1781	524	12	a]∗	a]∗	PROPN
ejpam-1781	524	13	)	)	PUNCT
ejpam-1781	524	14	.	.	PUNCT
ejpam-1781	525	1	let	let	VERB
ejpam-1781	525	2	p	p	PRON
ejpam-1781	525	3	∈	∈	PROPN
ejpam-1781	525	4	u(i	u(i	NOUN
ejpam-1781	525	5	)	)	PUNCT
ejpam-1781	525	6	.	.	PUNCT
ejpam-1781	526	1	but	but	CCONJ
ejpam-1781	526	2	then	then	ADV
ejpam-1781	527	1	i	i	PRON
ejpam-1781	527	2	*	*	PUNCT
ejpam-1781	528	1	p.	p.	NOUN
ejpam-1781	528	2	therefore	therefore	ADV
ejpam-1781	528	3	there	there	PRON
ejpam-1781	528	4	exists	exist	VERB
ejpam-1781	528	5	an	an	DET
ejpam-1781	528	6	element	element	NOUN
ejpam-1781	528	7	x1	x1	PROPN
ejpam-1781	528	8	∈	∈	PROPN
ejpam-1781	528	9	i\p	i\p	NOUN
ejpam-1781	528	10	.	.	PUNCT
ejpam-1781	529	1	but	but	CCONJ
ejpam-1781	529	2	then	then	ADV
ejpam-1781	529	3	(	(	PUNCT
ejpam-1781	529	4	x1	x1	PROPN
ejpam-1781	529	5	]	]	PUNCT
ejpam-1781	529	6	∗	∗	NOUN
ejpam-1781	529	7	⊆	⊆	NUM
ejpam-1781	530	1	p.	p.	NOUN
ejpam-1781	530	2	i.e	i.e	PRON
ejpam-1781	531	1	p	p	X
ejpam-1781	531	2	∈	∈	PROPN
ejpam-1781	531	3	v	v	ADP
ejpam-1781	531	4	�	�	PROPN
ejpam-1781	531	5	(	(	PUNCT
ejpam-1781	531	6	x1	x1	PROPN
ejpam-1781	531	7	]	]	PUNCT
ejpam-1781	531	8	∗	∗	NOUN
ejpam-1781	531	9	�	�	PROPN
ejpam-1781	531	10	⊆	⊆	NUM
ejpam-1781	531	11	⋃	⋃	NOUN
ejpam-1781	531	12	a∈i	a∈i	NOUN
ejpam-1781	531	13	v	v	NOUN
ejpam-1781	531	14	(	(	PUNCT
ejpam-1781	531	15	(	(	PUNCT
ejpam-1781	531	16	a]∗	a]∗	PROPN
ejpam-1781	531	17	)	)	PUNCT
ejpam-1781	531	18	.	.	PUNCT
ejpam-1781	532	1	hence	hence	ADV
ejpam-1781	532	2	u(i	u(i	NOUN
ejpam-1781	532	3	)	)	PUNCT
ejpam-1781	532	4	⊆	⊆	NUM
ejpam-1781	532	5	⋃	⋃	NOUN
ejpam-1781	532	6	a∈i	a∈i	NOUN
ejpam-1781	532	7	v	v	NOUN
ejpam-1781	532	8	(	(	PUNCT
ejpam-1781	532	9	(	(	PUNCT
ejpam-1781	532	10	a]∗	a]∗	PROPN
ejpam-1781	532	11	)	)	PUNCT
ejpam-1781	532	12	.	.	PUNCT
ejpam-1781	533	1	let	let	VERB
ejpam-1781	533	2	p	p	PRON
ejpam-1781	533	3	∈	∈	PROPN
ejpam-1781	533	4	⋃	⋃	NOUN
ejpam-1781	533	5	a∈i	a∈i	NOUN
ejpam-1781	533	6	v	v	NOUN
ejpam-1781	533	7	(	(	PUNCT
ejpam-1781	533	8	(	(	PUNCT
ejpam-1781	533	9	a]∗	a]∗	PROPN
ejpam-1781	533	10	)	)	PUNCT
ejpam-1781	533	11	then	then	ADV
ejpam-1781	533	12	there	there	PRON
ejpam-1781	533	13	exists	exist	VERB
ejpam-1781	533	14	an	an	DET
ejpam-1781	533	15	element	element	NOUN
ejpam-1781	533	16	y	y	PROPN
ejpam-1781	533	17	∈	∈	PROPN
ejpam-1781	534	1	i	i	PRON
ejpam-1781	534	2	such	such	ADJ
ejpam-1781	534	3	that	that	SCONJ
ejpam-1781	534	4	p	p	PROPN
ejpam-1781	534	5	∈	∈	PROPN
ejpam-1781	534	6	v	v	NOUN
ejpam-1781	534	7	(	(	PUNCT
ejpam-1781	534	8	y]∗	y]∗	PROPN
ejpam-1781	534	9	but	but	CCONJ
ejpam-1781	534	10	then	then	ADV
ejpam-1781	534	11	(	(	PUNCT
ejpam-1781	534	12	y]∗	y]∗	PROPN
ejpam-1781	534	13	⊆	⊆	NUM
ejpam-1781	534	14	p.	p.	NOUN
ejpam-1781	534	15	if	if	SCONJ
ejpam-1781	534	16	i	i	PRON
ejpam-1781	534	17	⊆	⊆	NUM
ejpam-1781	534	18	p	p	NOUN
ejpam-1781	534	19	,	,	PUNCT
ejpam-1781	534	20	then	then	ADV
ejpam-1781	534	21	(	(	PUNCT
ejpam-1781	534	22	y]∗	y]∗	PROPN
ejpam-1781	534	23	∨	∨	NUM
ejpam-1781	534	24	i	i	PRON
ejpam-1781	534	25	⊆	⊆	NUM
ejpam-1781	534	26	p.	p.	NOUN
ejpam-1781	534	27	hence	hence	ADV
ejpam-1781	534	28	by	by	ADP
ejpam-1781	534	29	theorem	theorem	NOUN
ejpam-1781	534	30	13	13	NUM
ejpam-1781	534	31	r	r	NOUN
ejpam-1781	534	32	⊆	⊆	NUM
ejpam-1781	534	33	p	p	ADP
ejpam-1781	534	34	a	a	DET
ejpam-1781	534	35	contradiction	contradiction	NOUN
ejpam-1781	534	36	.	.	PUNCT
ejpam-1781	535	1	therefore	therefore	ADV
ejpam-1781	535	2	we	we	PRON
ejpam-1781	535	3	must	must	AUX
ejpam-1781	535	4	we	we	PRON
ejpam-1781	535	5	have	have	VERB
ejpam-1781	535	6	i	i	PRON
ejpam-1781	535	7	*	*	PUNCT
ejpam-1781	536	1	p.	p.	NOUN
ejpam-1781	536	2	therefore	therefore	ADV
ejpam-1781	536	3	p	p	X
ejpam-1781	536	4	∈	∈	PROPN
ejpam-1781	536	5	u(i	u(i	PROPN
ejpam-1781	536	6	)	)	PUNCT
ejpam-1781	536	7	.	.	PUNCT
ejpam-1781	537	1	thus	thus	ADV
ejpam-1781	537	2	⋃	⋃	ADP
ejpam-1781	537	3	a∈i	a∈i	ADJ
ejpam-1781	537	4	v	v	NOUN
ejpam-1781	537	5	(	(	PUNCT
ejpam-1781	537	6	(	(	PUNCT
ejpam-1781	537	7	a]∗)⊆	a]∗)⊆	NOUN
ejpam-1781	537	8	u(i	u(i	NOUN
ejpam-1781	537	9	)	)	PUNCT
ejpam-1781	537	10	.	.	PUNCT
ejpam-1781	538	1	combining	combine	VERB
ejpam-1781	538	2	both	both	DET
ejpam-1781	538	3	the	the	DET
ejpam-1781	538	4	inclusions	inclusion	NOUN
ejpam-1781	538	5	we	we	PRON
ejpam-1781	538	6	get	get	VERB
ejpam-1781	538	7	u(i	u(i	NOUN
ejpam-1781	538	8	)	)	PUNCT
ejpam-1781	538	9	=	=	PUNCT
ejpam-1781	538	10	⋃	⋃	ADP
ejpam-1781	538	11	a∈i	a∈i	NOUN
ejpam-1781	538	12	v	v	NOUN
ejpam-1781	538	13	(	(	PUNCT
ejpam-1781	538	14	(	(	PUNCT
ejpam-1781	538	15	a]∗	a]∗	PROPN
ejpam-1781	538	16	)	)	PUNCT
ejpam-1781	538	17	.	.	PUNCT
ejpam-1781	539	1	references	reference	NOUN
ejpam-1781	539	2	[	[	X
ejpam-1781	539	3	1	1	X
ejpam-1781	539	4	]	]	X
ejpam-1781	539	5	g	g	PROPN
ejpam-1781	539	6	c	c	PROPN
ejpam-1781	539	7	rao	rao	NOUN
ejpam-1781	539	8	and	and	CCONJ
ejpam-1781	539	9	s	s	NOUN
ejpam-1781	539	10	ravikumar	ravikumar	NOUN
ejpam-1781	539	11	.	.	PUNCT
ejpam-1781	540	1	minimal	minimal	ADJ
ejpam-1781	540	2	prime	prime	ADJ
ejpam-1781	540	3	ideals	ideal	NOUN
ejpam-1781	540	4	in	in	ADP
ejpam-1781	540	5	almost	almost	ADV
ejpam-1781	540	6	distributive	distributive	ADJ
ejpam-1781	540	7	lattices	lattice	NOUN
ejpam-1781	540	8	.	.	PUNCT
ejpam-1781	541	1	international	international	ADJ
ejpam-1781	541	2	journal	journal	PROPN
ejpam-1781	541	3	of	of	ADP
ejpam-1781	541	4	contemporary	contemporary	PROPN
ejpam-1781	541	5	mathematical	mathematical	PROPN
ejpam-1781	541	6	sciences	sciences	PROPN
ejpam-1781	541	7	.	.	PUNCT
ejpam-1781	542	1	vol	vol	NOUN
ejpam-1781	542	2	4	4	NUM
ejpam-1781	542	3	,	,	PUNCT
ejpam-1781	542	4	10	10	NUM
ejpam-1781	542	5	.	.	PUNCT
ejpam-1781	542	6	475	475	NUM
ejpam-1781	542	7	-	-	SYM
ejpam-1781	542	8	484	484	NUM
ejpam-1781	542	9	.	.	PUNCT
ejpam-1781	542	10	2009	2009	NUM
ejpam-1781	542	11	.	.	PUNCT
ejpam-1781	543	1	[	[	X
ejpam-1781	543	2	2	2	X
ejpam-1781	543	3	]	]	PUNCT
ejpam-1781	543	4	g	g	PROPN
ejpam-1781	543	5	c	c	PROPN
ejpam-1781	543	6	rao	rao	PROPN
ejpam-1781	543	7	and	and	CCONJ
ejpam-1781	543	8	s	s	NOUN
ejpam-1781	543	9	m	m	PROPN
ejpam-1781	543	10	rao	rao	NOUN
ejpam-1781	543	11	.	.	PUNCT
ejpam-1781	544	1	annihilator	annihilator	PROPN
ejpam-1781	544	2	ideals	ideal	NOUN
ejpam-1781	544	3	in	in	ADP
ejpam-1781	544	4	almost	almost	ADV
ejpam-1781	544	5	distributive	distributive	ADJ
ejpam-1781	544	6	lattices	lattice	NOUN
ejpam-1781	544	7	.	.	PUNCT
ejpam-1781	545	1	international	international	ADJ
ejpam-1781	545	2	mathematical	mathematical	PROPN
ejpam-1781	545	3	forum	forum	PROPN
ejpam-1781	545	4	,	,	PUNCT
ejpam-1781	545	5	vol	vol	NOUN
ejpam-1781	545	6	4	4	NUM
ejpam-1781	545	7	,	,	PUNCT
ejpam-1781	545	8	733	733	NUM
ejpam-1781	545	9	-	-	SYM
ejpam-1781	545	10	746	746	NUM
ejpam-1781	545	11	.	.	PUNCT
ejpam-1781	545	12	2009	2009	NUM
ejpam-1781	545	13	.	.	PUNCT
ejpam-1781	546	1	[	[	X
ejpam-1781	546	2	3	3	NUM
ejpam-1781	546	3	]	]	X
ejpam-1781	546	4	u	u	NOUN
ejpam-1781	546	5	m	m	NOUN
ejpam-1781	546	6	swamy	swamy	NOUN
ejpam-1781	546	7	and	and	CCONJ
ejpam-1781	546	8	g	g	PROPN
ejpam-1781	546	9	c	c	PROPN
ejpam-1781	546	10	rao	rao	PROPN
ejpam-1781	546	11	.	.	PUNCT
ejpam-1781	547	1	almost	almost	ADV
ejpam-1781	547	2	distributive	distributive	ADJ
ejpam-1781	547	3	lattices	lattice	NOUN
ejpam-1781	547	4	.	.	PUNCT
ejpam-1781	548	1	journal	journal	NOUN
ejpam-1781	548	2	of	of	ADP
ejpam-1781	548	3	the	the	DET
ejpam-1781	548	4	australian	australian	ADJ
ejpam-1781	548	5	mathematical	mathematical	ADJ
ejpam-1781	548	6	society,(series	society,(serie	NOUN
ejpam-1781	548	7	a	a	PRON
ejpam-1781	548	8	)	)	PUNCT
ejpam-1781	548	9	,	,	PUNCT
ejpam-1781	548	10	vol	vol	NOUN
ejpam-1781	548	11	31	31	NUM
ejpam-1781	548	12	,	,	PUNCT
ejpam-1781	548	13	77	77	NUM
ejpam-1781	548	14	-	-	SYM
ejpam-1781	548	15	91	91	NUM
ejpam-1781	548	16	.	.	PUNCT
ejpam-1781	548	17	1981	1981	NUM
ejpam-1781	548	18	.	.	PUNCT
ejpam-1781	549	1	[	[	X
ejpam-1781	549	2	4	4	X
ejpam-1781	549	3	]	]	X
ejpam-1781	549	4	p	p	X
ejpam-1781	549	5	v	v	X
ejpam-1781	549	6	venkatanarasimhan	venkatanarasimhan	ADJ
ejpam-1781	549	7	.	.	PUNCT
ejpam-1781	550	1	stone	stone	NOUN
ejpam-1781	550	2	’s	’s	PART
ejpam-1781	550	3	topology	topology	NOUN
ejpam-1781	550	4	for	for	ADP
ejpam-1781	550	5	pseudocomplemented	pseudocomplemented	ADJ
ejpam-1781	550	6	and	and	CCONJ
ejpam-1781	550	7	bicomplemented	bicomplemente	VERB
ejpam-1781	550	8	lattices	lattice	NOUN
ejpam-1781	550	9	,	,	PUNCT
ejpam-1781	550	10	transactions	transaction	NOUN
ejpam-1781	550	11	of	of	ADP
ejpam-1781	550	12	the	the	DET
ejpam-1781	550	13	american	american	PROPN
ejpam-1781	550	14	mathematical	mathematical	PROPN
ejpam-1781	550	15	society	society	NOUN
ejpam-1781	550	16	,	,	PUNCT
ejpam-1781	550	17	vol	vol	NOUN
ejpam-1781	550	18	170	170	NUM
ejpam-1781	550	19	,	,	PUNCT
ejpam-1781	550	20	57	57	NUM
ejpam-1781	550	21	-	-	SYM
ejpam-1781	550	22	70	70	NUM
ejpam-1781	550	23	.	.	PUNCT
ejpam-1781	550	24	1972	1972	NUM
ejpam-1781	550	25	.	.	PUNCT