id	sid	tid	token	lemma	pos
ejpam-5127	1	1	european	european	PROPN
ejpam-5127	1	2	journal	journal	PROPN
ejpam-5127	1	3	of	of	ADP
ejpam-5127	1	4	pure	pure	ADJ
ejpam-5127	1	5	and	and	CCONJ
ejpam-5127	1	6	applied	apply	VERB
ejpam-5127	1	7	mathematics	mathematic	NOUN
ejpam-5127	1	8	vol	vol	NOUN
ejpam-5127	1	9	.	.	PROPN
ejpam-5127	2	1	17	17	NUM
ejpam-5127	2	2	,	,	PUNCT
ejpam-5127	2	3	no	no	INTJ
ejpam-5127	2	4	.	.	NOUN
ejpam-5127	2	5	2	2	NUM
ejpam-5127	2	6	,	,	PUNCT
ejpam-5127	2	7	2024	2024	NUM
ejpam-5127	2	8	,	,	PUNCT
ejpam-5127	2	9	1306	1306	NUM
ejpam-5127	2	10	-	-	SYM
ejpam-5127	2	11	1320	1320	NUM
ejpam-5127	2	12	issn	issn	PROPN
ejpam-5127	2	13	1307	1307	NUM
ejpam-5127	2	14	-	-	SYM
ejpam-5127	2	15	5543	5543	NUM
ejpam-5127	2	16	–	–	PUNCT
ejpam-5127	3	1	ejpam.com	ejpam.com	X
ejpam-5127	3	2	published	publish	VERB
ejpam-5127	3	3	by	by	ADP
ejpam-5127	3	4	new	new	PROPN
ejpam-5127	3	5	york	york	PROPN
ejpam-5127	3	6	business	business	PROPN
ejpam-5127	3	7	global	global	ADJ
ejpam-5127	3	8	normal	normal	ADJ
ejpam-5127	3	9	paradistributive	paradistributive	ADJ
ejpam-5127	3	10	latticoids	latticoids	PROPN
ejpam-5127	3	11	ravikumar	ravikumar	PROPN
ejpam-5127	3	12	bandaru1	bandaru1	PROPN
ejpam-5127	3	13	,	,	PUNCT
ejpam-5127	3	14	prashant	prashant	PROPN
ejpam-5127	3	15	patel2	patel2	PROPN
ejpam-5127	3	16	,	,	PUNCT
ejpam-5127	3	17	noorbhasha	noorbhasha	PROPN
ejpam-5127	3	18	rafi3	rafi3	ADJ
ejpam-5127	3	19	,	,	PUNCT
ejpam-5127	3	20	rahul	rahul	PROPN
ejpam-5127	3	21	shukla4,∗	shukla4,∗	NOUN
ejpam-5127	3	22	,	,	PUNCT
ejpam-5127	3	23	suryavardhani	suryavardhani	ADJ
ejpam-5127	3	24	ajjarapu5	ajjarapu5	NOUN
ejpam-5127	3	25	1,2	1,2	NUM
ejpam-5127	3	26	department	department	NOUN
ejpam-5127	3	27	of	of	ADP
ejpam-5127	3	28	mathematics	mathematic	NOUN
ejpam-5127	3	29	,	,	PUNCT
ejpam-5127	3	30	school	school	NOUN
ejpam-5127	3	31	of	of	ADP
ejpam-5127	3	32	advanced	advanced	ADJ
ejpam-5127	3	33	sciences	science	NOUN
ejpam-5127	3	34	,	,	PUNCT
ejpam-5127	3	35	vit	vit	PROPN
ejpam-5127	3	36	-	-	PUNCT
ejpam-5127	3	37	ap	ap	PROPN
ejpam-5127	3	38	university	university	PROPN
ejpam-5127	3	39	,	,	PUNCT
ejpam-5127	3	40	andhra	andhra	PROPN
ejpam-5127	3	41	pradesh-522237	pradesh-522237	NOUN
ejpam-5127	3	42	,	,	PUNCT
ejpam-5127	3	43	india	india	PROPN
ejpam-5127	3	44	3	3	NUM
ejpam-5127	3	45	department	department	NOUN
ejpam-5127	3	46	of	of	ADP
ejpam-5127	3	47	mathematics	mathematic	NOUN
ejpam-5127	3	48	,	,	PUNCT
ejpam-5127	3	49	bapatla	bapatla	VERB
ejpam-5127	3	50	engineering	engineering	NOUN
ejpam-5127	3	51	college	college	NOUN
ejpam-5127	3	52	,	,	PUNCT
ejpam-5127	3	53	bapatla	bapatla	NOUN
ejpam-5127	3	54	,	,	PUNCT
ejpam-5127	3	55	andhra	andhra	PROPN
ejpam-5127	3	56	pradesh-522101	pradesh-522101	NOUN
ejpam-5127	3	57	,	,	PUNCT
ejpam-5127	3	58	india	india	PROPN
ejpam-5127	3	59	4	4	NUM
ejpam-5127	3	60	department	department	PROPN
ejpam-5127	3	61	of	of	ADP
ejpam-5127	3	62	mathematical	mathematical	ADJ
ejpam-5127	3	63	sciences	sciences	PROPN
ejpam-5127	3	64	and	and	CCONJ
ejpam-5127	3	65	computing	computing	NOUN
ejpam-5127	3	66	,	,	PUNCT
ejpam-5127	3	67	walter	walter	PROPN
ejpam-5127	3	68	sisulu	sisulu	PROPN
ejpam-5127	3	69	university	university	PROPN
ejpam-5127	3	70	,	,	PUNCT
ejpam-5127	3	71	mthatha	mthatha	NOUN
ejpam-5127	3	72	5117	5117	NUM
ejpam-5127	3	73	,	,	PUNCT
ejpam-5127	3	74	south	south	PROPN
ejpam-5127	3	75	africa	africa	PROPN
ejpam-5127	3	76	5	5	NUM
ejpam-5127	3	77	department	department	NOUN
ejpam-5127	3	78	of	of	ADP
ejpam-5127	3	79	mathematics	mathematic	NOUN
ejpam-5127	3	80	,	,	PUNCT
ejpam-5127	3	81	gitam	gitam	NOUN
ejpam-5127	3	82	deemed	deem	VERB
ejpam-5127	3	83	to	to	PART
ejpam-5127	3	84	be	be	AUX
ejpam-5127	3	85	university	university	NOUN
ejpam-5127	3	86	,	,	PUNCT
ejpam-5127	3	87	hyderabad	hyderabad	PROPN
ejpam-5127	3	88	campus	campus	NOUN
ejpam-5127	3	89	,	,	PUNCT
ejpam-5127	3	90	telangana-502329	telangana-502329	ADJ
ejpam-5127	3	91	,	,	PUNCT
ejpam-5127	3	92	india	india	PROPN
ejpam-5127	3	93	abstract	abstract	NOUN
ejpam-5127	3	94	.	.	PUNCT
ejpam-5127	4	1	for	for	ADP
ejpam-5127	4	2	any	any	DET
ejpam-5127	4	3	filter	filter	NOUN
ejpam-5127	4	4	p	p	NOUN
ejpam-5127	4	5	of	of	ADP
ejpam-5127	4	6	a	a	DET
ejpam-5127	4	7	paradistributive	paradistributive	ADJ
ejpam-5127	4	8	latticoid	latticoid	NOUN
ejpam-5127	4	9	,	,	PUNCT
ejpam-5127	4	10	o(p	o(p	PROPN
ejpam-5127	4	11	)	)	PUNCT
ejpam-5127	4	12	is	be	AUX
ejpam-5127	4	13	defined	define	VERB
ejpam-5127	4	14	and	and	CCONJ
ejpam-5127	4	15	it	it	PRON
ejpam-5127	4	16	is	be	AUX
ejpam-5127	4	17	proved	prove	VERB
ejpam-5127	4	18	that	that	SCONJ
ejpam-5127	4	19	o(p	o(p	PROPN
ejpam-5127	4	20	)	)	PUNCT
ejpam-5127	4	21	is	be	AUX
ejpam-5127	4	22	a	a	DET
ejpam-5127	4	23	filter	filter	NOUN
ejpam-5127	4	24	if	if	SCONJ
ejpam-5127	4	25	p	p	NOUN
ejpam-5127	4	26	is	be	AUX
ejpam-5127	4	27	prime	prime	ADJ
ejpam-5127	4	28	.	.	PUNCT
ejpam-5127	5	1	it	it	PRON
ejpam-5127	5	2	is	be	AUX
ejpam-5127	5	3	also	also	ADV
ejpam-5127	5	4	proved	prove	VERB
ejpam-5127	5	5	that	that	SCONJ
ejpam-5127	5	6	each	each	DET
ejpam-5127	5	7	minimal	minimal	ADJ
ejpam-5127	5	8	prime	prime	ADJ
ejpam-5127	5	9	filter	filter	NOUN
ejpam-5127	5	10	belonging	belong	VERB
ejpam-5127	5	11	to	to	ADP
ejpam-5127	5	12	o(p	o(p	NUM
ejpam-5127	5	13	)	)	PUNCT
ejpam-5127	5	14	is	be	AUX
ejpam-5127	5	15	contained	contain	VERB
ejpam-5127	5	16	in	in	ADP
ejpam-5127	5	17	p	p	NOUN
ejpam-5127	5	18	,	,	PUNCT
ejpam-5127	5	19	and	and	CCONJ
ejpam-5127	5	20	o(p	o(p	NUM
ejpam-5127	5	21	)	)	PUNCT
ejpam-5127	5	22	is	be	AUX
ejpam-5127	5	23	the	the	DET
ejpam-5127	5	24	intersection	intersection	NOUN
ejpam-5127	5	25	of	of	ADP
ejpam-5127	5	26	all	all	DET
ejpam-5127	5	27	the	the	DET
ejpam-5127	5	28	minimal	minimal	ADJ
ejpam-5127	5	29	prime	prime	ADJ
ejpam-5127	5	30	filters	filter	NOUN
ejpam-5127	5	31	contained	contain	VERB
ejpam-5127	5	32	in	in	ADP
ejpam-5127	5	33	p.	p.	NOUN
ejpam-5127	5	34	the	the	DET
ejpam-5127	5	35	concept	concept	NOUN
ejpam-5127	5	36	of	of	ADP
ejpam-5127	5	37	a	a	DET
ejpam-5127	5	38	normal	normal	ADJ
ejpam-5127	5	39	paradistributive	paradistributive	ADJ
ejpam-5127	5	40	latticoid	latticoid	NOUN
ejpam-5127	5	41	is	be	AUX
ejpam-5127	5	42	introduced	introduce	VERB
ejpam-5127	5	43	and	and	CCONJ
ejpam-5127	5	44	characterized	characterize	VERB
ejpam-5127	5	45	in	in	ADP
ejpam-5127	5	46	terms	term	NOUN
ejpam-5127	5	47	of	of	ADP
ejpam-5127	5	48	the	the	DET
ejpam-5127	5	49	prime	prime	ADJ
ejpam-5127	5	50	filters	filter	NOUN
ejpam-5127	5	51	and	and	CCONJ
ejpam-5127	5	52	minimal	minimal	ADJ
ejpam-5127	5	53	prime	prime	ADJ
ejpam-5127	5	54	filters	filter	NOUN
ejpam-5127	5	55	.	.	PUNCT
ejpam-5127	6	1	we	we	PRON
ejpam-5127	6	2	proved	prove	VERB
ejpam-5127	6	3	that	that	SCONJ
ejpam-5127	6	4	every	every	DET
ejpam-5127	6	5	relatively	relatively	ADV
ejpam-5127	6	6	complemented	complemented	ADJ
ejpam-5127	6	7	paradistributive	paradistributive	ADJ
ejpam-5127	6	8	latticoid	latticoid	NOUN
ejpam-5127	6	9	is	be	AUX
ejpam-5127	6	10	normal	normal	ADJ
ejpam-5127	6	11	.	.	PUNCT
ejpam-5127	7	1	2020	2020	NUM
ejpam-5127	7	2	mathematics	mathematic	NOUN
ejpam-5127	7	3	subject	subject	NOUN
ejpam-5127	7	4	classifications	classification	NOUN
ejpam-5127	7	5	:	:	PUNCT
ejpam-5127	7	6	06d99	06d99	NUM
ejpam-5127	7	7	key	key	ADJ
ejpam-5127	7	8	words	word	NOUN
ejpam-5127	7	9	and	and	CCONJ
ejpam-5127	7	10	phrases	phrase	NOUN
ejpam-5127	7	11	:	:	PUNCT
ejpam-5127	7	12	paradistributive	paradistributive	ADJ
ejpam-5127	7	13	latticoid(pdl	latticoid(pdl	NOUN
ejpam-5127	7	14	)	)	PUNCT
ejpam-5127	7	15	,	,	PUNCT
ejpam-5127	7	16	minimal	minimal	ADJ
ejpam-5127	7	17	element	element	NOUN
ejpam-5127	7	18	,	,	PUNCT
ejpam-5127	7	19	filter	filter	NOUN
ejpam-5127	7	20	.	.	PUNCT
ejpam-5127	8	1	1	1	X
ejpam-5127	8	2	.	.	X
ejpam-5127	8	3	introduction	introduction	NOUN
ejpam-5127	8	4	in	in	ADP
ejpam-5127	8	5	1972	1972	NUM
ejpam-5127	8	6	,	,	PUNCT
ejpam-5127	8	7	cornish[3	cornish[3	PROPN
ejpam-5127	8	8	]	]	PUNCT
ejpam-5127	8	9	coined	coin	VERB
ejpam-5127	8	10	the	the	DET
ejpam-5127	8	11	term	term	NOUN
ejpam-5127	8	12	“	"	PUNCT
ejpam-5127	8	13	normal	normal	ADJ
ejpam-5127	8	14	lattices	lattice	NOUN
ejpam-5127	8	15	”	"	PUNCT
ejpam-5127	8	16	for	for	ADP
ejpam-5127	8	17	distributive	distributive	ADJ
ejpam-5127	8	18	lattices	lattice	NOUN
ejpam-5127	8	19	with	with	ADP
ejpam-5127	8	20	0	0	NUM
ejpam-5127	8	21	,	,	PUNCT
ejpam-5127	8	22	where	where	SCONJ
ejpam-5127	8	23	each	each	DET
ejpam-5127	8	24	prime	prime	ADJ
ejpam-5127	8	25	ideal	ideal	NOUN
ejpam-5127	8	26	contains	contain	VERB
ejpam-5127	8	27	a	a	DET
ejpam-5127	8	28	unique	unique	ADJ
ejpam-5127	8	29	minimal	minimal	ADJ
ejpam-5127	8	30	prime	prime	ADJ
ejpam-5127	8	31	ideal	ideal	NOUN
ejpam-5127	8	32	.	.	PUNCT
ejpam-5127	9	1	he	he	PRON
ejpam-5127	9	2	established	establish	VERB
ejpam-5127	9	3	that	that	SCONJ
ejpam-5127	9	4	a	a	DET
ejpam-5127	9	5	distributive	distributive	ADJ
ejpam-5127	9	6	lattice	lattice	NOUN
ejpam-5127	9	7	a	a	PRON
ejpam-5127	9	8	with	with	ADP
ejpam-5127	9	9	0	0	NUM
ejpam-5127	9	10	is	be	AUX
ejpam-5127	9	11	normal	normal	ADJ
ejpam-5127	9	12	iff	iff	NOUN
ejpam-5127	9	13	for	for	ADP
ejpam-5127	9	14	all	all	DET
ejpam-5127	9	15	x	x	NOUN
ejpam-5127	9	16	,	,	PUNCT
ejpam-5127	9	17	y	y	PROPN
ejpam-5127	9	18	∈	∈	PROPN
ejpam-5127	9	19	a	a	PRON
ejpam-5127	9	20	,	,	PUNCT
ejpam-5127	9	21	x	x	PUNCT
ejpam-5127	9	22	∧	∧	NOUN
ejpam-5127	9	23	y	y	NOUN
ejpam-5127	9	24	=	=	SYM
ejpam-5127	9	25	0	0	NUM
ejpam-5127	9	26	implies	imply	VERB
ejpam-5127	9	27	x⊥	x⊥	PROPN
ejpam-5127	9	28	and	and	CCONJ
ejpam-5127	9	29	y⊥	y⊥	NOUN
ejpam-5127	9	30	are	be	AUX
ejpam-5127	9	31	comaximal	comaximal	ADJ
ejpam-5127	9	32	.	.	PUNCT
ejpam-5127	10	1	pawar[4	pawar[4	NOUN
ejpam-5127	10	2	]	]	PUNCT
ejpam-5127	10	3	characterized	characterize	VERB
ejpam-5127	10	4	normal	normal	ADJ
ejpam-5127	10	5	lattices	lattice	NOUN
ejpam-5127	10	6	using	use	VERB
ejpam-5127	10	7	the	the	DET
ejpam-5127	10	8	properties	property	NOUN
ejpam-5127	10	9	of	of	ADP
ejpam-5127	10	10	stone	stone	NOUN
ejpam-5127	10	11	space	space	NOUN
ejpam-5127	10	12	of	of	ADP
ejpam-5127	10	13	prime	prime	ADJ
ejpam-5127	10	14	filters	filter	NOUN
ejpam-5127	10	15	and	and	CCONJ
ejpam-5127	10	16	the	the	DET
ejpam-5127	10	17	stone	stone	NOUN
ejpam-5127	10	18	space	space	NOUN
ejpam-5127	10	19	of	of	ADP
ejpam-5127	10	20	maximal	maximal	ADJ
ejpam-5127	10	21	filters	filter	NOUN
ejpam-5127	10	22	of	of	ADP
ejpam-5127	10	23	a	a	DET
ejpam-5127	10	24	bounded	bound	VERB
ejpam-5127	10	25	distributive	distributive	ADJ
ejpam-5127	10	26	lattice	lattice	NOUN
ejpam-5127	10	27	.	.	PUNCT
ejpam-5127	11	1	in	in	ADP
ejpam-5127	11	2	1977	1977	NUM
ejpam-5127	11	3	,	,	PUNCT
ejpam-5127	11	4	pawar	pawar	PROPN
ejpam-5127	11	5	and	and	CCONJ
ejpam-5127	11	6	thakare[5	thakare[5	PROPN
ejpam-5127	11	7	]	]	PUNCT
ejpam-5127	11	8	introduced	introduce	VERB
ejpam-5127	11	9	pm	pm	NOUN
ejpam-5127	11	10	-	-	PUNCT
ejpam-5127	11	11	lattices	lattice	NOUN
ejpam-5127	11	12	as	as	ADP
ejpam-5127	11	13	bounded	bound	VERB
ejpam-5127	11	14	distributive	distributive	ADJ
ejpam-5127	11	15	lattices	lattice	NOUN
ejpam-5127	11	16	where	where	SCONJ
ejpam-5127	11	17	each	each	DET
ejpam-5127	11	18	prime	prime	ADJ
ejpam-5127	11	19	ideal	ideal	NOUN
ejpam-5127	11	20	is	be	AUX
ejpam-5127	11	21	uniquely	uniquely	ADV
ejpam-5127	11	22	contained	contain	VERB
ejpam-5127	11	23	within	within	ADP
ejpam-5127	11	24	a	a	DET
ejpam-5127	11	25	maximal	maximal	ADJ
ejpam-5127	11	26	ideal	ideal	NOUN
ejpam-5127	11	27	.	.	PUNCT
ejpam-5127	12	1	in	in	ADP
ejpam-5127	12	2	1980	1980	NUM
ejpam-5127	12	3	,	,	PUNCT
ejpam-5127	12	4	simmons[9	simmons[9	PROPN
ejpam-5127	12	5	]	]	PUNCT
ejpam-5127	12	6	demonstrated	demonstrate	VERB
ejpam-5127	12	7	the	the	DET
ejpam-5127	12	8	equivalence	equivalence	NOUN
ejpam-5127	12	9	between	between	ADP
ejpam-5127	12	10	pm	pm	NOUN
ejpam-5127	12	11	-	-	PUNCT
ejpam-5127	12	12	lattices	lattice	NOUN
ejpam-5127	12	13	and	and	CCONJ
ejpam-5127	12	14	normal	normal	ADJ
ejpam-5127	12	15	lattices	lattice	NOUN
ejpam-5127	12	16	,	,	PUNCT
ejpam-5127	12	17	revealing	reveal	VERB
ejpam-5127	12	18	that	that	SCONJ
ejpam-5127	12	19	a	a	DET
ejpam-5127	12	20	bounded	bound	VERB
ejpam-5127	12	21	distributive	distributive	ADJ
ejpam-5127	12	22	lattice	lattice	NOUN
ejpam-5127	12	23	is	be	AUX
ejpam-5127	12	24	a	a	DET
ejpam-5127	12	25	pm	pm	NOUN
ejpam-5127	12	26	-	-	PUNCT
ejpam-5127	12	27	lattice	lattice	NOUN
ejpam-5127	12	28	precisely	precisely	ADV
ejpam-5127	12	29	when	when	SCONJ
ejpam-5127	12	30	it	it	PRON
ejpam-5127	12	31	is	be	AUX
ejpam-5127	12	32	normal	normal	ADJ
ejpam-5127	12	33	.	.	PUNCT
ejpam-5127	13	1	in	in	ADP
ejpam-5127	13	2	pm	pm	NOUN
ejpam-5127	13	3	-	-	PUNCT
ejpam-5127	13	4	lattices	lattice	NOUN
ejpam-5127	13	5	,	,	PUNCT
ejpam-5127	13	6	∗corresponding	∗corresponde	VERB
ejpam-5127	13	7	author	author	NOUN
ejpam-5127	13	8	.	.	PUNCT
ejpam-5127	14	1	doi	doi	NOUN
ejpam-5127	14	2	:	:	PUNCT
ejpam-5127	14	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5127	https://doi.org/10.29020/nybg.ejpam.v17i2.5127	NOUN
ejpam-5127	14	4	email	email	NOUN
ejpam-5127	14	5	addresses	address	VERB
ejpam-5127	14	6	:	:	PUNCT
ejpam-5127	15	1	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-5127	15	2	(	(	PUNCT
ejpam-5127	15	3	r.	r.	PROPN
ejpam-5127	15	4	bandaru	bandaru	PROPN
ejpam-5127	15	5	)	)	PUNCT
ejpam-5127	15	6	,	,	PUNCT
ejpam-5127	15	7	prashant.patel9999@gmail.com	prashant.patel9999@gmail.com	PROPN
ejpam-5127	15	8	(	(	PUNCT
ejpam-5127	15	9	p.	p.	NOUN
ejpam-5127	15	10	patel	patel	PROPN
ejpam-5127	15	11	)	)	PUNCT
ejpam-5127	15	12	,	,	PUNCT
ejpam-5127	15	13	rafimaths@gmail.com	rafimaths@gmail.com	X
ejpam-5127	15	14	(	(	PUNCT
ejpam-5127	15	15	r.	r.	PROPN
ejpam-5127	15	16	noorbhasha	noorbhasha	PROPN
ejpam-5127	15	17	)	)	PUNCT
ejpam-5127	15	18	,	,	PUNCT
ejpam-5127	15	19	rshukla@wsu.ac.za	rshukla@wsu.ac.za	NOUN
ejpam-5127	15	20	(	(	PUNCT
ejpam-5127	15	21	r.	r.	NOUN
ejpam-5127	15	22	shukla	shukla	PROPN
ejpam-5127	15	23	)	)	PUNCT
ejpam-5127	15	24	,	,	PUNCT
ejpam-5127	15	25	syerrapr@gitam.in	syerrapr@gitam.in	PROPN
ejpam-5127	15	26	(	(	PUNCT
ejpam-5127	15	27	s.	s.	PROPN
ejpam-5127	15	28	ajjarapu	ajjarapu	PROPN
ejpam-5127	15	29	)	)	PUNCT
ejpam-5127	15	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5127	15	31	1306	1306	NUM
ejpam-5127	16	1	©	©	ADP
ejpam-5127	16	2	2024	2024	NUM
ejpam-5127	16	3	ejpam	ejpam	NOUN
ejpam-5127	16	4	all	all	DET
ejpam-5127	16	5	rights	right	NOUN
ejpam-5127	16	6	reserved	reserve	VERB
ejpam-5127	16	7	.	.	PUNCT
ejpam-5127	17	1	r.	r.	PROPN
ejpam-5127	17	2	shukla	shukla	PROPN
ejpam-5127	17	3	et	et	PROPN
ejpam-5127	17	4	al	al	PROPN
ejpam-5127	17	5	.	.	PUNCT
ejpam-5127	17	6	/	/	SYM
ejpam-5127	17	7	eur	eur	PROPN
ejpam-5127	17	8	.	.	PUNCT
ejpam-5127	18	1	j.	j.	PROPN
ejpam-5127	18	2	pure	pure	PROPN
ejpam-5127	18	3	appl	appl	PROPN
ejpam-5127	18	4	.	.	PROPN
ejpam-5127	18	5	math	math	PROPN
ejpam-5127	18	6	,	,	PUNCT
ejpam-5127	18	7	17	17	NUM
ejpam-5127	18	8	(	(	PUNCT
ejpam-5127	18	9	2	2	NUM
ejpam-5127	18	10	)	)	PUNCT
ejpam-5127	18	11	(	(	PUNCT
ejpam-5127	18	12	2024	2024	NUM
ejpam-5127	18	13	)	)	PUNCT
ejpam-5127	18	14	,	,	PUNCT
ejpam-5127	18	15	1306	1306	NUM
ejpam-5127	18	16	-	-	SYM
ejpam-5127	18	17	1320	1320	NUM
ejpam-5127	18	18	1307	1307	NUM
ejpam-5127	18	19	the	the	DET
ejpam-5127	18	20	map	map	NOUN
ejpam-5127	18	21	from	from	ADP
ejpam-5127	18	22	prime	prime	ADJ
ejpam-5127	18	23	ideals	ideal	NOUN
ejpam-5127	18	24	to	to	ADP
ejpam-5127	18	25	their	their	PRON
ejpam-5127	18	26	unique	unique	ADJ
ejpam-5127	18	27	maximal	maximal	ADJ
ejpam-5127	18	28	ideals	ideal	NOUN
ejpam-5127	18	29	is	be	AUX
ejpam-5127	18	30	continuous	continuous	ADJ
ejpam-5127	18	31	,	,	PUNCT
ejpam-5127	18	32	serving	serve	VERB
ejpam-5127	18	33	as	as	ADP
ejpam-5127	18	34	the	the	DET
ejpam-5127	18	35	exclusive	exclusive	ADJ
ejpam-5127	18	36	retraction	retraction	NOUN
ejpam-5127	18	37	from	from	ADP
ejpam-5127	18	38	prime	prime	ADJ
ejpam-5127	18	39	to	to	ADP
ejpam-5127	18	40	maximal	maximal	ADJ
ejpam-5127	18	41	spectrum	spectrum	NOUN
ejpam-5127	18	42	.	.	PUNCT
ejpam-5127	19	1	moreover	moreover	ADV
ejpam-5127	19	2	,	,	PUNCT
ejpam-5127	19	3	the	the	DET
ejpam-5127	19	4	pm	pm	NOUN
ejpam-5127	19	5	-	-	PUNCT
ejpam-5127	19	6	property	property	NOUN
ejpam-5127	19	7	signifies	signifie	NOUN
ejpam-5127	19	8	prime	prime	ADJ
ejpam-5127	19	9	spectrum	spectrum	NOUN
ejpam-5127	19	10	normality	normality	NOUN
ejpam-5127	19	11	and	and	CCONJ
ejpam-5127	19	12	ensures	ensure	VERB
ejpam-5127	19	13	the	the	DET
ejpam-5127	19	14	maximal	maximal	ADJ
ejpam-5127	19	15	spectrum	spectrum	NOUN
ejpam-5127	19	16	satisfies	satisfy	VERB
ejpam-5127	19	17	the	the	DET
ejpam-5127	19	18	t2	t2	PROPN
ejpam-5127	19	19	separation	separation	NOUN
ejpam-5127	19	20	axiom	axiom	NOUN
ejpam-5127	19	21	.	.	PUNCT
ejpam-5127	20	1	in	in	ADP
ejpam-5127	20	2	2012	2012	NUM
ejpam-5127	20	3	,	,	PUNCT
ejpam-5127	20	4	borumand	borumand	PROPN
ejpam-5127	20	5	saeid	saeid	PROPN
ejpam-5127	20	6	and	and	CCONJ
ejpam-5127	20	7	mohtashamnia[8	mohtashamnia[8	PROPN
ejpam-5127	20	8	]	]	PUNCT
ejpam-5127	20	9	introduced	introduce	VERB
ejpam-5127	20	10	the	the	DET
ejpam-5127	20	11	notion	notion	NOUN
ejpam-5127	20	12	of	of	ADP
ejpam-5127	20	13	(	(	PUNCT
ejpam-5127	20	14	right	right	ADJ
ejpam-5127	20	15	and	and	CCONJ
ejpam-5127	20	16	left	left	ADJ
ejpam-5127	20	17	)	)	PUNCT
ejpam-5127	20	18	stabilizer	stabilizer	NOUN
ejpam-5127	20	19	in	in	ADP
ejpam-5127	20	20	residuated	residuate	VERB
ejpam-5127	20	21	lattices	lattice	NOUN
ejpam-5127	20	22	and	and	CCONJ
ejpam-5127	20	23	proved	prove	VERB
ejpam-5127	20	24	some	some	DET
ejpam-5127	20	25	theorems	theorem	NOUN
ejpam-5127	20	26	which	which	PRON
ejpam-5127	20	27	gives	give	VERB
ejpam-5127	20	28	the	the	DET
ejpam-5127	20	29	relationship	relationship	NOUN
ejpam-5127	20	30	between	between	ADP
ejpam-5127	20	31	this	this	DET
ejpam-5127	20	32	notion	notion	NOUN
ejpam-5127	20	33	and	and	CCONJ
ejpam-5127	20	34	all	all	DET
ejpam-5127	20	35	types	type	NOUN
ejpam-5127	20	36	of	of	ADP
ejpam-5127	20	37	filters	filter	NOUN
ejpam-5127	20	38	in	in	ADP
ejpam-5127	20	39	residuated	residuate	VERB
ejpam-5127	20	40	lattices	lattice	NOUN
ejpam-5127	20	41	.	.	PUNCT
ejpam-5127	21	1	after	after	ADP
ejpam-5127	21	2	that	that	PRON
ejpam-5127	21	3	they	they	PRON
ejpam-5127	21	4	constructed	construct	VERB
ejpam-5127	21	5	quotient	quotient	NOUN
ejpam-5127	21	6	of	of	ADP
ejpam-5127	21	7	residuated	residuate	VERB
ejpam-5127	21	8	lattices	lattice	NOUN
ejpam-5127	21	9	via	via	ADP
ejpam-5127	21	10	stabilizer	stabilizer	NOUN
ejpam-5127	21	11	and	and	CCONJ
ejpam-5127	21	12	studied	study	VERB
ejpam-5127	21	13	its	its	PRON
ejpam-5127	21	14	properties	property	NOUN
ejpam-5127	21	15	.	.	PUNCT
ejpam-5127	22	1	rasouli	rasouli	PROPN
ejpam-5127	22	2	and	and	CCONJ
ejpam-5127	22	3	dehghani[6	dehghani[6	PROPN
ejpam-5127	22	4	]	]	PUNCT
ejpam-5127	22	5	investigated	investigate	VERB
ejpam-5127	22	6	the	the	DET
ejpam-5127	22	7	notion	notion	NOUN
ejpam-5127	22	8	of	of	ADP
ejpam-5127	22	9	an	an	DET
ejpam-5127	22	10	mp	mp	NOUN
ejpam-5127	22	11	-	-	PUNCT
ejpam-5127	22	12	residuated	residuate	VERB
ejpam-5127	22	13	lattice	lattice	NOUN
ejpam-5127	22	14	,	,	PUNCT
ejpam-5127	22	15	and	and	CCONJ
ejpam-5127	22	16	extracted	extract	VERB
ejpam-5127	22	17	their	their	PRON
ejpam-5127	22	18	topological	topological	ADJ
ejpam-5127	22	19	characterizations	characterization	NOUN
ejpam-5127	22	20	.	.	PUNCT
ejpam-5127	23	1	rasouli	rasouli	PROPN
ejpam-5127	23	2	and	and	CCONJ
ejpam-5127	23	3	kondo[7	kondo[7	PROPN
ejpam-5127	23	4	]	]	PUNCT
ejpam-5127	23	5	,	,	PUNCT
ejpam-5127	23	6	introduced	introduce	VERB
ejpam-5127	23	7	and	and	CCONJ
ejpam-5127	23	8	investigated	investigate	VERB
ejpam-5127	23	9	the	the	DET
ejpam-5127	23	10	notion	notion	NOUN
ejpam-5127	23	11	of	of	ADP
ejpam-5127	23	12	n	n	CCONJ
ejpam-5127	23	13	-	-	PUNCT
ejpam-5127	23	14	normal	normal	ADJ
ejpam-5127	23	15	residuated	residuate	VERB
ejpam-5127	23	16	lattice	lattice	NOUN
ejpam-5127	23	17	,	,	PUNCT
ejpam-5127	23	18	as	as	ADP
ejpam-5127	23	19	a	a	DET
ejpam-5127	23	20	subclass	subclass	NOUN
ejpam-5127	23	21	of	of	ADP
ejpam-5127	23	22	residuated	residuate	VERB
ejpam-5127	23	23	lattices	lattice	NOUN
ejpam-5127	23	24	in	in	ADP
ejpam-5127	23	25	which	which	PRON
ejpam-5127	23	26	every	every	DET
ejpam-5127	23	27	prime	prime	ADJ
ejpam-5127	23	28	filter	filter	NOUN
ejpam-5127	23	29	contains	contain	VERB
ejpam-5127	23	30	at	at	ADP
ejpam-5127	23	31	most	most	ADV
ejpam-5127	23	32	n	n	PRON
ejpam-5127	23	33	minimal	minimal	ADJ
ejpam-5127	23	34	prime	prime	ADJ
ejpam-5127	23	35	filters	filter	NOUN
ejpam-5127	23	36	.	.	PUNCT
ejpam-5127	24	1	bandaru	bandaru	PROPN
ejpam-5127	24	2	et	et	PROPN
ejpam-5127	24	3	al.[2	al.[2	PROPN
ejpam-5127	24	4	]	]	X
ejpam-5127	24	5	,	,	PUNCT
ejpam-5127	24	6	generalized	generalize	VERB
ejpam-5127	24	7	the	the	DET
ejpam-5127	24	8	the	the	DET
ejpam-5127	24	9	concept	concept	NOUN
ejpam-5127	24	10	of	of	ADP
ejpam-5127	24	11	distributive	distributive	ADJ
ejpam-5127	24	12	lattice	lattice	NOUN
ejpam-5127	24	13	and	and	CCONJ
ejpam-5127	24	14	introduced	introduce	VERB
ejpam-5127	24	15	the	the	DET
ejpam-5127	24	16	concept	concept	NOUN
ejpam-5127	24	17	of	of	ADP
ejpam-5127	24	18	paradistributive	paradistributive	ADJ
ejpam-5127	24	19	latticoid(pdl	latticoid(pdl	NOUN
ejpam-5127	24	20	)	)	PUNCT
ejpam-5127	24	21	and	and	CCONJ
ejpam-5127	24	22	investigated	investigate	VERB
ejpam-5127	24	23	its	its	PRON
ejpam-5127	24	24	properties	property	NOUN
ejpam-5127	24	25	.	.	PUNCT
ejpam-5127	25	1	they	they	PRON
ejpam-5127	25	2	also	also	ADV
ejpam-5127	25	3	obtained	obtain	VERB
ejpam-5127	25	4	subdirect	subdirect	NOUN
ejpam-5127	25	5	representation	representation	NOUN
ejpam-5127	25	6	of	of	ADP
ejpam-5127	25	7	a	a	DET
ejpam-5127	25	8	paradistributive	paradistributive	ADJ
ejpam-5127	25	9	latticoid	latticoid	NOUN
ejpam-5127	25	10	.	.	PUNCT
ejpam-5127	26	1	recently	recently	ADV
ejpam-5127	26	2	,	,	PUNCT
ejpam-5127	26	3	bandaru	bandaru	NOUN
ejpam-5127	26	4	et	et	NOUN
ejpam-5127	26	5	al.[1	al.[1	PROPN
ejpam-5127	26	6	]	]	PUNCT
ejpam-5127	26	7	,	,	PUNCT
ejpam-5127	26	8	introduced	introduce	VERB
ejpam-5127	26	9	the	the	DET
ejpam-5127	26	10	concept	concept	NOUN
ejpam-5127	26	11	of	of	ADP
ejpam-5127	26	12	a	a	DET
ejpam-5127	26	13	parapseudo	parapseudo	NOUN
ejpam-5127	26	14	-	-	NOUN
ejpam-5127	26	15	complementation	complementation	NOUN
ejpam-5127	26	16	in	in	ADP
ejpam-5127	26	17	a	a	DET
ejpam-5127	26	18	paradistributive	paradistributive	ADJ
ejpam-5127	26	19	latticoid(pdl	latticoid(pdl	NOUN
ejpam-5127	26	20	)	)	PUNCT
ejpam-5127	26	21	and	and	CCONJ
ejpam-5127	26	22	investigated	investigate	VERB
ejpam-5127	26	23	its	its	PRON
ejpam-5127	26	24	elementary	elementary	ADJ
ejpam-5127	26	25	properties	property	NOUN
ejpam-5127	26	26	.	.	PUNCT
ejpam-5127	27	1	additionally	additionally	ADV
ejpam-5127	27	2	,	,	PUNCT
ejpam-5127	27	3	authors	author	NOUN
ejpam-5127	27	4	established	establish	VERB
ejpam-5127	27	5	necessary	necessary	ADJ
ejpam-5127	27	6	conditions	condition	NOUN
ejpam-5127	27	7	for	for	ADP
ejpam-5127	27	8	a	a	DET
ejpam-5127	27	9	pdl	pdl	NOUN
ejpam-5127	27	10	with	with	ADP
ejpam-5127	27	11	a	a	DET
ejpam-5127	27	12	minimal	minimal	ADJ
ejpam-5127	27	13	element	element	NOUN
ejpam-5127	27	14	to	to	PART
ejpam-5127	27	15	be	be	AUX
ejpam-5127	27	16	parapseudo	parapseudo	NOUN
ejpam-5127	27	17	-	-	VERB
ejpam-5127	27	18	complemented	complement	VERB
ejpam-5127	27	19	and	and	CCONJ
ejpam-5127	27	20	explored	explore	VERB
ejpam-5127	27	21	the	the	DET
ejpam-5127	27	22	properties	property	NOUN
ejpam-5127	27	23	required	require	VERB
ejpam-5127	27	24	for	for	ADP
ejpam-5127	27	25	parapseudo	parapseudo	NOUN
ejpam-5127	27	26	-	-	NOUN
ejpam-5127	27	27	complementation	complementation	NOUN
ejpam-5127	27	28	to	to	PART
ejpam-5127	27	29	be	be	AUX
ejpam-5127	27	30	equationally	equationally	ADV
ejpam-5127	27	31	definable	definable	ADJ
ejpam-5127	27	32	.	.	PUNCT
ejpam-5127	28	1	moreover	moreover	ADV
ejpam-5127	28	2	,	,	PUNCT
ejpam-5127	28	3	they	they	PRON
ejpam-5127	28	4	established	establish	VERB
ejpam-5127	28	5	a	a	DET
ejpam-5127	28	6	one	one	NUM
ejpam-5127	28	7	-	-	PUNCT
ejpam-5127	28	8	to	to	ADP
ejpam-5127	28	9	-	-	PUNCT
ejpam-5127	28	10	one	one	NUM
ejpam-5127	28	11	correspondence	correspondence	NOUN
ejpam-5127	28	12	between	between	ADP
ejpam-5127	28	13	the	the	DET
ejpam-5127	28	14	set	set	NOUN
ejpam-5127	28	15	of	of	ADP
ejpam-5127	28	16	all	all	DET
ejpam-5127	28	17	minimal	minimal	ADJ
ejpam-5127	28	18	elements	element	NOUN
ejpam-5127	28	19	and	and	CCONJ
ejpam-5127	28	20	the	the	DET
ejpam-5127	28	21	set	set	NOUN
ejpam-5127	28	22	of	of	ADP
ejpam-5127	28	23	all	all	DET
ejpam-5127	28	24	parapseudo	parapseudo	NOUN
ejpam-5127	28	25	-	-	PUNCT
ejpam-5127	28	26	complementations	complementation	NOUN
ejpam-5127	28	27	in	in	ADP
ejpam-5127	28	28	this	this	DET
ejpam-5127	28	29	paper	paper	NOUN
ejpam-5127	28	30	,	,	PUNCT
ejpam-5127	28	31	we	we	PRON
ejpam-5127	28	32	introduce	introduce	VERB
ejpam-5127	28	33	the	the	DET
ejpam-5127	28	34	concept	concept	NOUN
ejpam-5127	28	35	of	of	ADP
ejpam-5127	28	36	annihilator	annihilator	PROPN
ejpam-5127	28	37	filters	filter	NOUN
ejpam-5127	28	38	and	and	CCONJ
ejpam-5127	28	39	study	study	VERB
ejpam-5127	28	40	their	their	PRON
ejpam-5127	28	41	properties	property	NOUN
ejpam-5127	28	42	.	.	PUNCT
ejpam-5127	29	1	we	we	PRON
ejpam-5127	29	2	define	define	VERB
ejpam-5127	29	3	the	the	DET
ejpam-5127	29	4	set	set	NOUN
ejpam-5127	29	5	o(p	o(p	PROPN
ejpam-5127	29	6	)	)	PUNCT
ejpam-5127	29	7	in	in	ADP
ejpam-5127	29	8	a	a	DET
ejpam-5127	29	9	pdl	pdl	NOUN
ejpam-5127	29	10	and	and	CCONJ
ejpam-5127	29	11	derive	derive	VERB
ejpam-5127	29	12	that	that	SCONJ
ejpam-5127	29	13	o(p	o(p	PROPN
ejpam-5127	29	14	)	)	PUNCT
ejpam-5127	29	15	is	be	AUX
ejpam-5127	29	16	intersection	intersection	NOUN
ejpam-5127	29	17	of	of	ADP
ejpam-5127	29	18	all	all	DET
ejpam-5127	29	19	minimal	minimal	ADJ
ejpam-5127	29	20	primr	primr	NOUN
ejpam-5127	29	21	filters	filter	NOUN
ejpam-5127	29	22	of	of	ADP
ejpam-5127	29	23	v	v	NOUN
ejpam-5127	29	24	contained	contain	VERB
ejpam-5127	29	25	in	in	ADP
ejpam-5127	29	26	p	p	PRON
ejpam-5127	29	27	,	,	PUNCT
ejpam-5127	29	28	in	in	ADP
ejpam-5127	29	29	case	case	NOUN
ejpam-5127	29	30	of	of	ADP
ejpam-5127	29	31	p	p	PROPN
ejpam-5127	29	32	is	be	AUX
ejpam-5127	29	33	a	a	DET
ejpam-5127	29	34	prime	prime	ADJ
ejpam-5127	29	35	filter	filter	NOUN
ejpam-5127	29	36	.	.	PUNCT
ejpam-5127	30	1	also	also	ADV
ejpam-5127	30	2	,	,	PUNCT
ejpam-5127	30	3	observe	observe	VERB
ejpam-5127	30	4	the	the	DET
ejpam-5127	30	5	relations	relation	NOUN
ejpam-5127	30	6	between	between	ADP
ejpam-5127	30	7	prime	prime	ADJ
ejpam-5127	30	8	ideals	ideal	NOUN
ejpam-5127	30	9	and	and	CCONJ
ejpam-5127	30	10	prime	prime	ADJ
ejpam-5127	30	11	filters	filter	NOUN
ejpam-5127	30	12	.	.	PUNCT
ejpam-5127	31	1	we	we	PRON
ejpam-5127	31	2	also	also	ADV
ejpam-5127	31	3	derive	derive	VERB
ejpam-5127	31	4	that	that	SCONJ
ejpam-5127	31	5	,	,	PUNCT
ejpam-5127	31	6	every	every	DET
ejpam-5127	31	7	maximal	maximal	ADJ
ejpam-5127	31	8	ideal	ideal	NOUN
ejpam-5127	31	9	is	be	AUX
ejpam-5127	31	10	prime	prime	ADJ
ejpam-5127	31	11	ideal	ideal	NOUN
ejpam-5127	31	12	and	and	CCONJ
ejpam-5127	31	13	converse	converse	NOUN
ejpam-5127	31	14	holds	hold	VERB
ejpam-5127	31	15	true	true	ADJ
ejpam-5127	31	16	if	if	SCONJ
ejpam-5127	31	17	v	v	NOUN
ejpam-5127	31	18	is	be	AUX
ejpam-5127	31	19	relatively	relatively	ADV
ejpam-5127	31	20	complemented	complemented	ADJ
ejpam-5127	31	21	pdl	pdl	NOUN
ejpam-5127	31	22	.	.	PUNCT
ejpam-5127	32	1	also	also	ADV
ejpam-5127	32	2	,	,	PUNCT
ejpam-5127	32	3	we	we	PRON
ejpam-5127	32	4	introduce	introduce	VERB
ejpam-5127	32	5	the	the	DET
ejpam-5127	32	6	concept	concept	NOUN
ejpam-5127	32	7	of	of	ADP
ejpam-5127	32	8	normal	normal	ADJ
ejpam-5127	32	9	pdl	pdl	NOUN
ejpam-5127	32	10	in	in	ADP
ejpam-5127	32	11	the	the	DET
ejpam-5127	32	12	terms	term	NOUN
ejpam-5127	32	13	of	of	ADP
ejpam-5127	32	14	prime	prime	ADJ
ejpam-5127	32	15	filters	filter	NOUN
ejpam-5127	32	16	and	and	CCONJ
ejpam-5127	32	17	characterize	characterize	VERB
ejpam-5127	32	18	it	it	PRON
ejpam-5127	32	19	,	,	PUNCT
ejpam-5127	32	20	in	in	ADP
ejpam-5127	32	21	element	element	NOUN
ejpam-5127	32	22	wise	wise	ADJ
ejpam-5127	32	23	and	and	CCONJ
ejpam-5127	32	24	its	its	PRON
ejpam-5127	32	25	annihilators	annihilator	NOUN
ejpam-5127	32	26	.	.	PUNCT
ejpam-5127	33	1	2	2	X
ejpam-5127	33	2	.	.	X
ejpam-5127	33	3	preliminaries	preliminary	NOUN
ejpam-5127	33	4	first	first	ADV
ejpam-5127	33	5	we	we	PRON
ejpam-5127	33	6	recall	recall	VERB
ejpam-5127	33	7	the	the	DET
ejpam-5127	33	8	necessary	necessary	ADJ
ejpam-5127	33	9	definitions	definition	NOUN
ejpam-5127	33	10	and	and	CCONJ
ejpam-5127	33	11	results	result	NOUN
ejpam-5127	33	12	from	from	ADP
ejpam-5127	33	13	[	[	X
ejpam-5127	33	14	2	2	NUM
ejpam-5127	33	15	]	]	PUNCT
ejpam-5127	33	16	.	.	PUNCT
ejpam-5127	34	1	definition	definition	NOUN
ejpam-5127	34	2	1	1	NUM
ejpam-5127	34	3	.	.	PUNCT
ejpam-5127	35	1	an	an	DET
ejpam-5127	35	2	algebra	algebra	NOUN
ejpam-5127	35	3	(	(	PUNCT
ejpam-5127	35	4	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	35	5	,	,	PUNCT
ejpam-5127	35	6	1	1	NUM
ejpam-5127	35	7	)	)	PUNCT
ejpam-5127	35	8	of	of	ADP
ejpam-5127	35	9	type	type	NOUN
ejpam-5127	35	10	(	(	PUNCT
ejpam-5127	35	11	2,2,0	2,2,0	NOUN
ejpam-5127	35	12	)	)	PUNCT
ejpam-5127	35	13	is	be	AUX
ejpam-5127	35	14	called	call	VERB
ejpam-5127	35	15	a	a	DET
ejpam-5127	35	16	paradistributive	paradistributive	ADJ
ejpam-5127	35	17	latticoid	latticoid	NOUN
ejpam-5127	35	18	,	,	PUNCT
ejpam-5127	35	19	abbreviated	abbreviate	VERB
ejpam-5127	35	20	as	as	ADP
ejpam-5127	35	21	pdl	pdl	NOUN
ejpam-5127	35	22	,	,	PUNCT
ejpam-5127	35	23	if	if	SCONJ
ejpam-5127	35	24	it	it	PRON
ejpam-5127	35	25	assures	assure	VERB
ejpam-5127	35	26	the	the	DET
ejpam-5127	35	27	subsequent	subsequent	ADJ
ejpam-5127	35	28	axioms	axiom	NOUN
ejpam-5127	35	29	:	:	PUNCT
ejpam-5127	35	30	(	(	PUNCT
ejpam-5127	35	31	ld∨	ld∨	NOUN
ejpam-5127	35	32	)	)	PUNCT
ejpam-5127	35	33	p	p	X
ejpam-5127	35	34	∨	∨	NOUN
ejpam-5127	35	35	(	(	PUNCT
ejpam-5127	35	36	q	q	NOUN
ejpam-5127	35	37	∧	∧	PROPN
ejpam-5127	35	38	r	r	NOUN
ejpam-5127	35	39	)	)	PUNCT
ejpam-5127	35	40	=	=	SYM
ejpam-5127	35	41	(	(	PUNCT
ejpam-5127	35	42	p	p	PROPN
ejpam-5127	35	43	∨	∨	NUM
ejpam-5127	35	44	q	q	NOUN
ejpam-5127	35	45	)	)	PUNCT
ejpam-5127	35	46	∧	∧	NOUN
ejpam-5127	35	47	(	(	PUNCT
ejpam-5127	35	48	p	p	NOUN
ejpam-5127	35	49	∨	∨	NUM
ejpam-5127	35	50	r	r	NOUN
ejpam-5127	35	51	)	)	PUNCT
ejpam-5127	35	52	,	,	PUNCT
ejpam-5127	35	53	(	(	PUNCT
ejpam-5127	35	54	rd∨	rd∨	X
ejpam-5127	35	55	)	)	PUNCT
ejpam-5127	35	56	(	(	PUNCT
ejpam-5127	35	57	p	p	PROPN
ejpam-5127	35	58	∧	∧	PROPN
ejpam-5127	35	59	q	q	NOUN
ejpam-5127	35	60	)	)	PUNCT
ejpam-5127	35	61	∨	∨	NOUN
ejpam-5127	35	62	r	r	NOUN
ejpam-5127	35	63	=	=	PUNCT
ejpam-5127	35	64	(	(	PUNCT
ejpam-5127	35	65	p	p	NOUN
ejpam-5127	35	66	∨	∨	NUM
ejpam-5127	35	67	r	r	NOUN
ejpam-5127	35	68	)	)	PUNCT
ejpam-5127	35	69	∧	∧	NOUN
ejpam-5127	35	70	(	(	PUNCT
ejpam-5127	35	71	q	q	NOUN
ejpam-5127	35	72	∨	∨	NUM
ejpam-5127	35	73	r	r	NOUN
ejpam-5127	35	74	)	)	PUNCT
ejpam-5127	35	75	,	,	PUNCT
ejpam-5127	35	76	(	(	PUNCT
ejpam-5127	35	77	l1	l1	PROPN
ejpam-5127	35	78	)	)	PUNCT
ejpam-5127	35	79	(	(	PUNCT
ejpam-5127	35	80	p	p	PROPN
ejpam-5127	35	81	∨	∨	NUM
ejpam-5127	35	82	q	q	NOUN
ejpam-5127	35	83	)	)	PUNCT
ejpam-5127	35	84	∧	∧	NOUN
ejpam-5127	35	85	q	q	NOUN
ejpam-5127	36	1	=	=	SYM
ejpam-5127	36	2	q	q	NOUN
ejpam-5127	36	3	,	,	PUNCT
ejpam-5127	36	4	(	(	PUNCT
ejpam-5127	36	5	l2	l2	NOUN
ejpam-5127	36	6	)	)	PUNCT
ejpam-5127	36	7	(	(	PUNCT
ejpam-5127	36	8	p	p	PROPN
ejpam-5127	36	9	∨	∨	NUM
ejpam-5127	36	10	q	q	NOUN
ejpam-5127	36	11	)	)	PUNCT
ejpam-5127	36	12	∧	∧	NOUN
ejpam-5127	36	13	p	p	NOUN
ejpam-5127	36	14	=	=	SYM
ejpam-5127	36	15	p	p	X
ejpam-5127	36	16	,	,	PUNCT
ejpam-5127	36	17	(	(	PUNCT
ejpam-5127	36	18	l3	l3	NOUN
ejpam-5127	36	19	)	)	PUNCT
ejpam-5127	36	20	p	p	X
ejpam-5127	36	21	∨	∨	NOUN
ejpam-5127	36	22	(	(	PUNCT
ejpam-5127	36	23	p	p	PROPN
ejpam-5127	36	24	∧	∧	PROPN
ejpam-5127	36	25	q	q	NOUN
ejpam-5127	36	26	)	)	PUNCT
ejpam-5127	36	27	=	=	SYM
ejpam-5127	37	1	p	p	X
ejpam-5127	37	2	,	,	PUNCT
ejpam-5127	37	3	(	(	PUNCT
ejpam-5127	37	4	i1	i1	PROPN
ejpam-5127	37	5	)	)	PUNCT
ejpam-5127	37	6	p	p	X
ejpam-5127	37	7	∨	∨	NUM
ejpam-5127	37	8	1	1	NUM
ejpam-5127	37	9	=	=	SYM
ejpam-5127	37	10	1	1	NUM
ejpam-5127	37	11	,	,	PUNCT
ejpam-5127	37	12	for	for	ADP
ejpam-5127	37	13	any	any	DET
ejpam-5127	37	14	p	p	X
ejpam-5127	37	15	,	,	PUNCT
ejpam-5127	37	16	q	q	ADJ
ejpam-5127	37	17	,	,	PUNCT
ejpam-5127	37	18	r	r	NOUN
ejpam-5127	37	19	∈	∈	PROPN
ejpam-5127	37	20	v	v	NOUN
ejpam-5127	37	21	.	.	PUNCT
ejpam-5127	38	1	for	for	ADP
ejpam-5127	38	2	any	any	DET
ejpam-5127	38	3	p	p	NOUN
ejpam-5127	38	4	,	,	PUNCT
ejpam-5127	38	5	q	q	PROPN
ejpam-5127	38	6	∈	∈	PROPN
ejpam-5127	38	7	v	v	NOUN
ejpam-5127	38	8	,	,	PUNCT
ejpam-5127	38	9	we	we	PRON
ejpam-5127	38	10	say	say	VERB
ejpam-5127	38	11	that	that	SCONJ
ejpam-5127	38	12	p	p	NOUN
ejpam-5127	38	13	is	be	AUX
ejpam-5127	38	14	less	less	ADJ
ejpam-5127	38	15	than	than	ADP
ejpam-5127	38	16	or	or	CCONJ
ejpam-5127	38	17	equal	equal	ADJ
ejpam-5127	38	18	to	to	ADP
ejpam-5127	38	19	q	q	PUNCT
ejpam-5127	38	20	and	and	CCONJ
ejpam-5127	38	21	write	write	VERB
ejpam-5127	38	22	p	p	PROPN
ejpam-5127	38	23	≤	≤	ADJ
ejpam-5127	38	24	q	q	NOUN
ejpam-5127	39	1	if	if	SCONJ
ejpam-5127	39	2	p	p	PROPN
ejpam-5127	39	3	∧	∧	NOUN
ejpam-5127	39	4	q	q	X
ejpam-5127	40	1	=	=	PUNCT
ejpam-5127	40	2	p	p	NOUN
ejpam-5127	40	3	or	or	CCONJ
ejpam-5127	40	4	equivalently	equivalently	ADV
ejpam-5127	40	5	p∨	p∨	PROPN
ejpam-5127	40	6	q	q	NOUN
ejpam-5127	41	1	=	=	PUNCT
ejpam-5127	41	2	q	q	NOUN
ejpam-5127	42	1	and	and	CCONJ
ejpam-5127	42	2	it	it	PRON
ejpam-5127	42	3	can	can	AUX
ejpam-5127	42	4	be	be	AUX
ejpam-5127	42	5	easily	easily	ADV
ejpam-5127	42	6	observed	observe	VERB
ejpam-5127	42	7	that	that	SCONJ
ejpam-5127	42	8	≤	≤	NUM
ejpam-5127	42	9	is	be	AUX
ejpam-5127	42	10	a	a	DET
ejpam-5127	42	11	partial	partial	ADJ
ejpam-5127	42	12	order	order	NOUN
ejpam-5127	42	13	on	on	ADP
ejpam-5127	42	14	v	v	NOUN
ejpam-5127	42	15	.	.	PUNCT
ejpam-5127	43	1	the	the	DET
ejpam-5127	43	2	r.	r.	PROPN
ejpam-5127	43	3	shukla	shukla	PROPN
ejpam-5127	43	4	et	et	PROPN
ejpam-5127	43	5	al	al	PROPN
ejpam-5127	43	6	.	.	PUNCT
ejpam-5127	43	7	/	/	SYM
ejpam-5127	43	8	eur	eur	PROPN
ejpam-5127	43	9	.	.	PUNCT
ejpam-5127	44	1	j.	j.	PROPN
ejpam-5127	44	2	pure	pure	PROPN
ejpam-5127	44	3	appl	appl	PROPN
ejpam-5127	44	4	.	.	PROPN
ejpam-5127	44	5	math	math	PROPN
ejpam-5127	44	6	,	,	PUNCT
ejpam-5127	44	7	17	17	NUM
ejpam-5127	44	8	(	(	PUNCT
ejpam-5127	44	9	2	2	NUM
ejpam-5127	44	10	)	)	PUNCT
ejpam-5127	44	11	(	(	PUNCT
ejpam-5127	44	12	2024	2024	NUM
ejpam-5127	44	13	)	)	PUNCT
ejpam-5127	44	14	,	,	PUNCT
ejpam-5127	44	15	1306	1306	NUM
ejpam-5127	44	16	-	-	SYM
ejpam-5127	44	17	1320	1320	NUM
ejpam-5127	44	18	1308	1308	NUM
ejpam-5127	44	19	element	element	NOUN
ejpam-5127	44	20	1	1	NUM
ejpam-5127	44	21	,	,	PUNCT
ejpam-5127	44	22	in	in	ADP
ejpam-5127	44	23	definition	definition	NOUN
ejpam-5127	44	24	1	1	NUM
ejpam-5127	44	25	,	,	PUNCT
ejpam-5127	44	26	is	be	AUX
ejpam-5127	44	27	called	call	VERB
ejpam-5127	44	28	the	the	DET
ejpam-5127	44	29	greatest	great	ADJ
ejpam-5127	44	30	element	element	NOUN
ejpam-5127	44	31	.	.	PUNCT
ejpam-5127	44	32	example	example	NOUN
ejpam-5127	45	1	1	1	NUM
ejpam-5127	45	2	.	.	PUNCT
ejpam-5127	45	3	let	let	VERB
ejpam-5127	45	4	v	v	PART
ejpam-5127	45	5	be	be	AUX
ejpam-5127	45	6	a	a	DET
ejpam-5127	45	7	non	non	ADJ
ejpam-5127	45	8	-	-	ADJ
ejpam-5127	45	9	empty	empty	ADJ
ejpam-5127	45	10	set	set	NOUN
ejpam-5127	45	11	.	.	PUNCT
ejpam-5127	46	1	fix	fix	VERB
ejpam-5127	46	2	some	some	DET
ejpam-5127	46	3	element	element	NOUN
ejpam-5127	46	4	y0	y0	PROPN
ejpam-5127	46	5	∈	∈	NOUN
ejpam-5127	46	6	v	v	NOUN
ejpam-5127	46	7	.	.	PUNCT
ejpam-5127	47	1	then	then	ADV
ejpam-5127	47	2	,	,	PUNCT
ejpam-5127	47	3	for	for	ADP
ejpam-5127	47	4	any	any	DET
ejpam-5127	47	5	x	x	NOUN
ejpam-5127	47	6	,	,	PUNCT
ejpam-5127	47	7	y	y	PROPN
ejpam-5127	47	8	∈	∈	PROPN
ejpam-5127	47	9	v	v	PART
ejpam-5127	47	10	define	define	VERB
ejpam-5127	47	11	∨	∨	NOUN
ejpam-5127	47	12	and	and	CCONJ
ejpam-5127	47	13	∧	∧	NOUN
ejpam-5127	47	14	on	on	ADP
ejpam-5127	47	15	v	v	NUM
ejpam-5127	47	16	by	by	ADP
ejpam-5127	47	17	x	x	PROPN
ejpam-5127	47	18	∨	∨	NOUN
ejpam-5127	47	19	y	y	NOUN
ejpam-5127	47	20	=	=	PRON
ejpam-5127	47	21	{	{	PUNCT
ejpam-5127	47	22	x	x	PUNCT
ejpam-5127	47	23	y	y	PROPN
ejpam-5127	47	24	̸=	̸=	PROPN
ejpam-5127	47	25	y0	y0	NOUN
ejpam-5127	47	26	y0	y0	NOUN
ejpam-5127	47	27	y	y	NOUN
ejpam-5127	47	28	=	=	SYM
ejpam-5127	47	29	y0	y0	PROPN
ejpam-5127	47	30	and	and	CCONJ
ejpam-5127	47	31	x	x	PART
ejpam-5127	47	32	∧	∧	NOUN
ejpam-5127	47	33	y	y	NOUN
ejpam-5127	48	1	=	=	PRON
ejpam-5127	48	2	{	{	PUNCT
ejpam-5127	48	3	y	y	NOUN
ejpam-5127	48	4	y	y	NOUN
ejpam-5127	48	5	̸=	̸=	PROPN
ejpam-5127	48	6	y0	y0	PROPN
ejpam-5127	48	7	x	x	SYM
ejpam-5127	48	8	y	y	NOUN
ejpam-5127	48	9	=	=	SYM
ejpam-5127	48	10	y0	y0	PROPN
ejpam-5127	48	11	then	then	ADV
ejpam-5127	48	12	(	(	PUNCT
ejpam-5127	48	13	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	48	14	,	,	PUNCT
ejpam-5127	48	15	y0	y0	PROPN
ejpam-5127	48	16	)	)	PUNCT
ejpam-5127	48	17	is	be	AUX
ejpam-5127	48	18	a	a	DET
ejpam-5127	48	19	disconnected	disconnected	ADJ
ejpam-5127	48	20	pdl	pdl	NOUN
ejpam-5127	48	21	with	with	ADP
ejpam-5127	48	22	y0	y0	PROPN
ejpam-5127	48	23	as	as	ADP
ejpam-5127	48	24	its	its	PRON
ejpam-5127	48	25	greatest	great	ADJ
ejpam-5127	48	26	element	element	NOUN
ejpam-5127	48	27	.	.	PUNCT
ejpam-5127	49	1	lemma	lemma	PROPN
ejpam-5127	49	2	1	1	X
ejpam-5127	49	3	.	.	PUNCT
ejpam-5127	50	1	let	let	AUX
ejpam-5127	50	2	(	(	PUNCT
ejpam-5127	50	3	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	50	4	,	,	PUNCT
ejpam-5127	50	5	1	1	NUM
ejpam-5127	50	6	)	)	PUNCT
ejpam-5127	50	7	be	be	AUX
ejpam-5127	50	8	a	a	DET
ejpam-5127	50	9	pdl	pdl	NOUN
ejpam-5127	50	10	.	.	PUNCT
ejpam-5127	51	1	then	then	ADV
ejpam-5127	51	2	for	for	ADP
ejpam-5127	51	3	any	any	DET
ejpam-5127	51	4	p	p	X
ejpam-5127	51	5	,	,	PUNCT
ejpam-5127	51	6	q	q	ADJ
ejpam-5127	51	7	,	,	PUNCT
ejpam-5127	51	8	r	r	NOUN
ejpam-5127	51	9	,	,	PUNCT
ejpam-5127	51	10	s	s	NOUN
ejpam-5127	51	11	∈	∈	NOUN
ejpam-5127	51	12	v	v	NOUN
ejpam-5127	51	13	,	,	PUNCT
ejpam-5127	51	14	we	we	PRON
ejpam-5127	51	15	have	have	VERB
ejpam-5127	51	16	the	the	DET
ejpam-5127	51	17	following	following	NOUN
ejpam-5127	51	18	:	:	PUNCT
ejpam-5127	51	19	(	(	PUNCT
ejpam-5127	51	20	1	1	X
ejpam-5127	51	21	)	)	SYM
ejpam-5127	51	22	1	1	NUM
ejpam-5127	51	23	∧	∧	NOUN
ejpam-5127	51	24	p	p	NOUN
ejpam-5127	51	25	=	=	SYM
ejpam-5127	51	26	p	p	NOUN
ejpam-5127	51	27	,	,	PUNCT
ejpam-5127	51	28	(	(	PUNCT
ejpam-5127	51	29	2	2	X
ejpam-5127	51	30	)	)	PUNCT
ejpam-5127	51	31	p	p	NOUN
ejpam-5127	51	32	∧	∧	PROPN
ejpam-5127	51	33	1	1	NUM
ejpam-5127	51	34	=	=	SYM
ejpam-5127	51	35	p	p	NOUN
ejpam-5127	51	36	,	,	PUNCT
ejpam-5127	51	37	(	(	PUNCT
ejpam-5127	51	38	3	3	NUM
ejpam-5127	51	39	)	)	PUNCT
ejpam-5127	51	40	1	1	NUM
ejpam-5127	51	41	∨	∨	NOUN
ejpam-5127	51	42	p	p	NOUN
ejpam-5127	51	43	=	=	NOUN
ejpam-5127	51	44	1	1	NUM
ejpam-5127	51	45	,	,	PUNCT
ejpam-5127	51	46	(	(	PUNCT
ejpam-5127	51	47	4	4	NUM
ejpam-5127	51	48	)	)	PUNCT
ejpam-5127	51	49	(	(	PUNCT
ejpam-5127	51	50	p	p	PROPN
ejpam-5127	51	51	∨	∨	NUM
ejpam-5127	51	52	q	q	NOUN
ejpam-5127	51	53	)	)	PUNCT
ejpam-5127	51	54	∧	∧	NOUN
ejpam-5127	51	55	r	r	NOUN
ejpam-5127	51	56	=	=	PUNCT
ejpam-5127	51	57	(	(	PUNCT
ejpam-5127	51	58	p	p	NOUN
ejpam-5127	51	59	∧	∧	PROPN
ejpam-5127	51	60	r	r	NOUN
ejpam-5127	51	61	)	)	PUNCT
ejpam-5127	51	62	∨	∨	NOUN
ejpam-5127	51	63	(	(	PUNCT
ejpam-5127	51	64	q	q	NOUN
ejpam-5127	51	65	∧	∧	PROPN
ejpam-5127	51	66	r	r	NOUN
ejpam-5127	51	67	)	)	PUNCT
ejpam-5127	51	68	,	,	PUNCT
ejpam-5127	51	69	(	(	PUNCT
ejpam-5127	51	70	5	5	X
ejpam-5127	51	71	)	)	PUNCT
ejpam-5127	51	72	p	p	NOUN
ejpam-5127	51	73	∨	∨	NOUN
ejpam-5127	51	74	(	(	PUNCT
ejpam-5127	51	75	q	q	NOUN
ejpam-5127	51	76	∧	∧	PROPN
ejpam-5127	51	77	r	r	NOUN
ejpam-5127	51	78	)	)	PUNCT
ejpam-5127	51	79	=	=	SYM
ejpam-5127	52	1	p	p	NOUN
ejpam-5127	52	2	∨	∨	NOUN
ejpam-5127	52	3	(	(	PUNCT
ejpam-5127	52	4	r	r	NOUN
ejpam-5127	52	5	∧	∧	PROPN
ejpam-5127	52	6	q	q	NOUN
ejpam-5127	52	7	)	)	PUNCT
ejpam-5127	52	8	,	,	PUNCT
ejpam-5127	52	9	(	(	PUNCT
ejpam-5127	52	10	6	6	X
ejpam-5127	52	11	)	)	PUNCT
ejpam-5127	52	12	the	the	DET
ejpam-5127	52	13	operation	operation	NOUN
ejpam-5127	52	14	∨	∨	NOUN
ejpam-5127	52	15	is	be	AUX
ejpam-5127	52	16	associative	associative	ADJ
ejpam-5127	52	17	in	in	ADP
ejpam-5127	52	18	v	v	NOUN
ejpam-5127	52	19	i.e.	i.e.	X
ejpam-5127	52	20	,	,	PUNCT
ejpam-5127	52	21	p	p	ADJ
ejpam-5127	52	22	∨	∨	NOUN
ejpam-5127	52	23	(	(	PUNCT
ejpam-5127	52	24	q	q	PROPN
ejpam-5127	52	25	∨	∨	NUM
ejpam-5127	52	26	r	r	NOUN
ejpam-5127	52	27	)	)	PUNCT
ejpam-5127	52	28	=	=	PUNCT
ejpam-5127	52	29	(	(	PUNCT
ejpam-5127	52	30	p	p	PROPN
ejpam-5127	52	31	∨	∨	NUM
ejpam-5127	52	32	q	q	NOUN
ejpam-5127	52	33	)	)	PUNCT
ejpam-5127	52	34	∨	∨	NOUN
ejpam-5127	52	35	r	r	NOUN
ejpam-5127	52	36	,	,	PUNCT
ejpam-5127	52	37	(	(	PUNCT
ejpam-5127	52	38	7	7	X
ejpam-5127	52	39	)	)	PUNCT
ejpam-5127	52	40	the	the	DET
ejpam-5127	52	41	set	set	NOUN
ejpam-5127	52	42	va	va	NOUN
ejpam-5127	52	43	=	=	PUNCT
ejpam-5127	52	44	{	{	PUNCT
ejpam-5127	52	45	p	p	X
ejpam-5127	52	46	∈	∈	PROPN
ejpam-5127	52	47	v	v	ADP
ejpam-5127	52	48	|	|	ADV
ejpam-5127	52	49	a	a	DET
ejpam-5127	52	50	≤	≤	NOUN
ejpam-5127	52	51	p	p	X
ejpam-5127	52	52	}	}	PUNCT
ejpam-5127	52	53	=	=	SYM
ejpam-5127	52	54	{	{	PUNCT
ejpam-5127	52	55	a	a	DET
ejpam-5127	52	56	∨	∨	NUM
ejpam-5127	52	57	p	p	NOUN
ejpam-5127	53	1	|	|	NOUN
ejpam-5127	53	2	p	p	NOUN
ejpam-5127	53	3	∈	∈	PROPN
ejpam-5127	53	4	v	v	NOUN
ejpam-5127	53	5	}	}	PUNCT
ejpam-5127	53	6	is	be	AUX
ejpam-5127	53	7	a	a	DET
ejpam-5127	53	8	distributive	distributive	ADJ
ejpam-5127	53	9	lattice	lattice	NOUN
ejpam-5127	53	10	under	under	ADP
ejpam-5127	53	11	induced	induced	ADJ
ejpam-5127	53	12	operations	operation	NOUN
ejpam-5127	53	13	∨	∨	NOUN
ejpam-5127	53	14	and	and	CCONJ
ejpam-5127	53	15	∧	∧	NOUN
ejpam-5127	53	16	with	with	ADP
ejpam-5127	53	17	a	a	DET
ejpam-5127	53	18	as	as	ADP
ejpam-5127	53	19	its	its	PRON
ejpam-5127	53	20	least	least	ADJ
ejpam-5127	53	21	element	element	NOUN
ejpam-5127	53	22	,	,	PUNCT
ejpam-5127	53	23	(	(	PUNCT
ejpam-5127	53	24	8)	8)	NUM
ejpam-5127	53	25	s	s	NOUN
ejpam-5127	53	26	∨	∨	NOUN
ejpam-5127	53	27	{	{	PUNCT
ejpam-5127	53	28	p	p	PROPN
ejpam-5127	53	29	∧	∧	PROPN
ejpam-5127	53	30	(	(	PUNCT
ejpam-5127	53	31	q	q	NOUN
ejpam-5127	53	32	∧	∧	PROPN
ejpam-5127	53	33	r	r	NOUN
ejpam-5127	53	34	)	)	PUNCT
ejpam-5127	53	35	}	}	PUNCT
ejpam-5127	53	36	=	=	SYM
ejpam-5127	53	37	s	s	PROPN
ejpam-5127	53	38	∨	∨	NOUN
ejpam-5127	53	39	{	{	PUNCT
ejpam-5127	53	40	(	(	PUNCT
ejpam-5127	53	41	p	p	PROPN
ejpam-5127	53	42	∧	∧	PROPN
ejpam-5127	53	43	q	q	NOUN
ejpam-5127	53	44	)	)	PUNCT
ejpam-5127	53	45	∧	∧	PROPN
ejpam-5127	53	46	r	r	NOUN
ejpam-5127	53	47	}	}	PUNCT
ejpam-5127	53	48	,	,	PUNCT
ejpam-5127	53	49	(	(	PUNCT
ejpam-5127	53	50	9	9	X
ejpam-5127	53	51	)	)	PUNCT
ejpam-5127	53	52	p	p	NOUN
ejpam-5127	53	53	∨	∨	NOUN
ejpam-5127	53	54	(	(	PUNCT
ejpam-5127	53	55	q	q	PROPN
ejpam-5127	53	56	∨	∨	NUM
ejpam-5127	53	57	r	r	NOUN
ejpam-5127	53	58	)	)	PUNCT
ejpam-5127	53	59	=	=	SYM
ejpam-5127	53	60	p	p	NOUN
ejpam-5127	53	61	∨	∨	NOUN
ejpam-5127	53	62	(	(	PUNCT
ejpam-5127	53	63	r	r	PROPN
ejpam-5127	53	64	∨	∨	NUM
ejpam-5127	53	65	q	q	NOUN
ejpam-5127	53	66	)	)	PUNCT
ejpam-5127	53	67	,	,	PUNCT
ejpam-5127	53	68	(	(	PUNCT
ejpam-5127	53	69	10	10	NUM
ejpam-5127	53	70	)	)	PUNCT
ejpam-5127	53	71	p	p	NOUN
ejpam-5127	53	72	∨	∨	NUM
ejpam-5127	53	73	q	q	NOUN
ejpam-5127	53	74	=	=	SYM
ejpam-5127	53	75	1	1	NUM
ejpam-5127	53	76	if	if	SCONJ
ejpam-5127	53	77	and	and	CCONJ
ejpam-5127	53	78	only	only	ADV
ejpam-5127	53	79	if	if	SCONJ
ejpam-5127	53	80	q	q	PROPN
ejpam-5127	53	81	∨	∨	NUM
ejpam-5127	53	82	p	p	NOUN
ejpam-5127	53	83	=	=	NOUN
ejpam-5127	53	84	1	1	NUM
ejpam-5127	53	85	,	,	PUNCT
ejpam-5127	53	86	(	(	PUNCT
ejpam-5127	53	87	11	11	NUM
ejpam-5127	53	88	)	)	PUNCT
ejpam-5127	53	89	p	p	NOUN
ejpam-5127	53	90	∧	∧	PROPN
ejpam-5127	53	91	q	q	NOUN
ejpam-5127	53	92	=	=	PUNCT
ejpam-5127	53	93	q	q	NOUN
ejpam-5127	53	94	∧	∧	PROPN
ejpam-5127	53	95	p	p	NOUN
ejpam-5127	53	96	whenever	whenever	SCONJ
ejpam-5127	53	97	p	p	NOUN
ejpam-5127	53	98	∨	∨	PROPN
ejpam-5127	53	99	q	q	NOUN
ejpam-5127	53	100	=	=	ADJ
ejpam-5127	53	101	1	1	X
ejpam-5127	53	102	.	.	PUNCT
ejpam-5127	53	103	theorem	theorem	NOUN
ejpam-5127	53	104	1	1	NUM
ejpam-5127	53	105	.	.	PUNCT
ejpam-5127	54	1	an	an	DET
ejpam-5127	54	2	algebra	algebra	NOUN
ejpam-5127	54	3	(	(	PUNCT
ejpam-5127	54	4	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	54	5	,	,	PUNCT
ejpam-5127	54	6	1	1	NUM
ejpam-5127	54	7	)	)	PUNCT
ejpam-5127	54	8	of	of	ADP
ejpam-5127	54	9	type	type	NOUN
ejpam-5127	54	10	(	(	PUNCT
ejpam-5127	54	11	2	2	NUM
ejpam-5127	54	12	,	,	PUNCT
ejpam-5127	54	13	2	2	NUM
ejpam-5127	54	14	,	,	PUNCT
ejpam-5127	54	15	0	0	NUM
ejpam-5127	54	16	)	)	PUNCT
ejpam-5127	54	17	is	be	AUX
ejpam-5127	54	18	a	a	DET
ejpam-5127	54	19	pdl	pdl	NOUN
ejpam-5127	54	20	if	if	SCONJ
ejpam-5127	55	1	and	and	CCONJ
ejpam-5127	55	2	only	only	ADV
ejpam-5127	55	3	if	if	SCONJ
ejpam-5127	55	4	it	it	PRON
ejpam-5127	55	5	satisfies	satisfy	VERB
ejpam-5127	55	6	the	the	DET
ejpam-5127	55	7	following	following	NOUN
ejpam-5127	55	8	:	:	PUNCT
ejpam-5127	55	9	(	(	PUNCT
ejpam-5127	55	10	ld∨	ld∨	NOUN
ejpam-5127	55	11	)	)	PUNCT
ejpam-5127	55	12	p	p	X
ejpam-5127	55	13	∨	∨	NOUN
ejpam-5127	55	14	(	(	PUNCT
ejpam-5127	55	15	q	q	NOUN
ejpam-5127	55	16	∧	∧	PROPN
ejpam-5127	55	17	r	r	NOUN
ejpam-5127	55	18	)	)	PUNCT
ejpam-5127	55	19	=	=	SYM
ejpam-5127	55	20	(	(	PUNCT
ejpam-5127	55	21	p	p	PROPN
ejpam-5127	55	22	∨	∨	NUM
ejpam-5127	55	23	q	q	NOUN
ejpam-5127	55	24	)	)	PUNCT
ejpam-5127	55	25	∧	∧	NOUN
ejpam-5127	55	26	(	(	PUNCT
ejpam-5127	55	27	p	p	NOUN
ejpam-5127	55	28	∨	∨	NUM
ejpam-5127	55	29	r	r	NOUN
ejpam-5127	55	30	)	)	PUNCT
ejpam-5127	55	31	,	,	PUNCT
ejpam-5127	55	32	(	(	PUNCT
ejpam-5127	55	33	rd∨	rd∨	X
ejpam-5127	55	34	)	)	PUNCT
ejpam-5127	55	35	(	(	PUNCT
ejpam-5127	55	36	p	p	PROPN
ejpam-5127	55	37	∧	∧	PROPN
ejpam-5127	55	38	q	q	NOUN
ejpam-5127	55	39	)	)	PUNCT
ejpam-5127	55	40	∨	∨	NOUN
ejpam-5127	55	41	r	r	NOUN
ejpam-5127	55	42	=	=	PUNCT
ejpam-5127	55	43	(	(	PUNCT
ejpam-5127	55	44	p	p	NOUN
ejpam-5127	55	45	∨	∨	NUM
ejpam-5127	55	46	r	r	NOUN
ejpam-5127	55	47	)	)	PUNCT
ejpam-5127	55	48	∧	∧	NOUN
ejpam-5127	55	49	(	(	PUNCT
ejpam-5127	55	50	q	q	NOUN
ejpam-5127	55	51	∨	∨	NUM
ejpam-5127	55	52	r	r	NOUN
ejpam-5127	55	53	)	)	PUNCT
ejpam-5127	55	54	,	,	PUNCT
ejpam-5127	55	55	(	(	PUNCT
ejpam-5127	55	56	rd∧	rd∧	PROPN
ejpam-5127	55	57	)	)	PUNCT
ejpam-5127	55	58	(	(	PUNCT
ejpam-5127	55	59	p	p	PROPN
ejpam-5127	55	60	∨	∨	NUM
ejpam-5127	55	61	q	q	NOUN
ejpam-5127	55	62	)	)	PUNCT
ejpam-5127	55	63	∧	∧	NOUN
ejpam-5127	55	64	r	r	NOUN
ejpam-5127	55	65	=	=	PUNCT
ejpam-5127	55	66	(	(	PUNCT
ejpam-5127	55	67	p	p	NOUN
ejpam-5127	55	68	∧	∧	PROPN
ejpam-5127	55	69	r	r	NOUN
ejpam-5127	55	70	)	)	PUNCT
ejpam-5127	55	71	∨	∨	NOUN
ejpam-5127	55	72	(	(	PUNCT
ejpam-5127	55	73	q	q	NOUN
ejpam-5127	55	74	∧	∧	PROPN
ejpam-5127	55	75	r	r	NOUN
ejpam-5127	55	76	)	)	PUNCT
ejpam-5127	55	77	,	,	PUNCT
ejpam-5127	55	78	(	(	PUNCT
ejpam-5127	55	79	l1	l1	PROPN
ejpam-5127	55	80	)	)	PUNCT
ejpam-5127	55	81	(	(	PUNCT
ejpam-5127	55	82	p	p	PROPN
ejpam-5127	55	83	∨	∨	NUM
ejpam-5127	55	84	q	q	NOUN
ejpam-5127	55	85	)	)	PUNCT
ejpam-5127	55	86	∧	∧	NOUN
ejpam-5127	55	87	q	q	NOUN
ejpam-5127	56	1	=	=	SYM
ejpam-5127	56	2	q	q	NOUN
ejpam-5127	56	3	,	,	PUNCT
ejpam-5127	56	4	(	(	PUNCT
ejpam-5127	56	5	l3	l3	NOUN
ejpam-5127	56	6	)	)	PUNCT
ejpam-5127	56	7	p	p	X
ejpam-5127	56	8	∨	∨	NOUN
ejpam-5127	56	9	(	(	PUNCT
ejpam-5127	56	10	p	p	PROPN
ejpam-5127	56	11	∧	∧	PROPN
ejpam-5127	56	12	q	q	NOUN
ejpam-5127	56	13	)	)	PUNCT
ejpam-5127	56	14	=	=	SYM
ejpam-5127	57	1	p	p	X
ejpam-5127	57	2	,	,	PUNCT
ejpam-5127	57	3	(	(	PUNCT
ejpam-5127	57	4	i1	i1	PROPN
ejpam-5127	57	5	)	)	PUNCT
ejpam-5127	57	6	p	p	X
ejpam-5127	57	7	∨	∨	NUM
ejpam-5127	57	8	1	1	NUM
ejpam-5127	57	9	=	=	SYM
ejpam-5127	57	10	1	1	NUM
ejpam-5127	57	11	,	,	PUNCT
ejpam-5127	57	12	(	(	PUNCT
ejpam-5127	57	13	i2	i2	PROPN
ejpam-5127	57	14	)	)	PUNCT
ejpam-5127	57	15	1	1	NUM
ejpam-5127	57	16	∧	∧	NOUN
ejpam-5127	57	17	p	p	NOUN
ejpam-5127	57	18	=	=	SYM
ejpam-5127	57	19	p	p	X
ejpam-5127	57	20	,	,	PUNCT
ejpam-5127	57	21	for	for	ADP
ejpam-5127	57	22	all	all	DET
ejpam-5127	57	23	p	p	NOUN
ejpam-5127	57	24	,	,	PUNCT
ejpam-5127	57	25	q	q	ADJ
ejpam-5127	57	26	,	,	PUNCT
ejpam-5127	57	27	r	r	NOUN
ejpam-5127	57	28	∈	∈	PROPN
ejpam-5127	57	29	v	v	NOUN
ejpam-5127	57	30	.	.	PUNCT
ejpam-5127	58	1	definition	definition	NOUN
ejpam-5127	58	2	2	2	NUM
ejpam-5127	58	3	.	.	PUNCT
ejpam-5127	59	1	a	a	DET
ejpam-5127	59	2	paradistributive	paradistributive	ADJ
ejpam-5127	59	3	latticoid	latticoid	NOUN
ejpam-5127	59	4	(	(	PUNCT
ejpam-5127	59	5	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	59	6	,	,	PUNCT
ejpam-5127	59	7	1	1	NUM
ejpam-5127	59	8	)	)	PUNCT
ejpam-5127	59	9	is	be	AUX
ejpam-5127	59	10	said	say	VERB
ejpam-5127	59	11	to	to	PART
ejpam-5127	59	12	be	be	AUX
ejpam-5127	59	13	associative	associative	ADJ
ejpam-5127	59	14	if	if	SCONJ
ejpam-5127	59	15	it	it	PRON
ejpam-5127	59	16	satisfies	satisfy	VERB
ejpam-5127	59	17	the	the	DET
ejpam-5127	59	18	following	follow	VERB
ejpam-5127	59	19	condition	condition	NOUN
ejpam-5127	59	20	p	p	X
ejpam-5127	59	21	∧	∧	PROPN
ejpam-5127	59	22	(	(	PUNCT
ejpam-5127	59	23	q	q	NOUN
ejpam-5127	59	24	∧	∧	PROPN
ejpam-5127	59	25	r	r	NOUN
ejpam-5127	59	26	)	)	PUNCT
ejpam-5127	59	27	=	=	NOUN
ejpam-5127	59	28	(	(	PUNCT
ejpam-5127	59	29	p	p	X
ejpam-5127	59	30	∧	∧	PROPN
ejpam-5127	59	31	q	q	NOUN
ejpam-5127	59	32	)	)	PUNCT
ejpam-5127	59	33	∧	∧	NOUN
ejpam-5127	59	34	r	r	NOUN
ejpam-5127	59	35	for	for	ADP
ejpam-5127	59	36	all	all	DET
ejpam-5127	59	37	p	p	NOUN
ejpam-5127	59	38	,	,	PUNCT
ejpam-5127	59	39	q	q	ADJ
ejpam-5127	59	40	,	,	PUNCT
ejpam-5127	59	41	r	r	NOUN
ejpam-5127	59	42	∈	∈	PROPN
ejpam-5127	59	43	v.	v.	ADP
ejpam-5127	59	44	definition	definition	NOUN
ejpam-5127	60	1	3	3	X
ejpam-5127	60	2	.	.	PUNCT
ejpam-5127	61	1	let	let	VERB
ejpam-5127	61	2	v	v	PART
ejpam-5127	61	3	be	be	AUX
ejpam-5127	61	4	a	a	DET
ejpam-5127	61	5	pdl	pdl	NOUN
ejpam-5127	61	6	.	.	PUNCT
ejpam-5127	62	1	then	then	ADV
ejpam-5127	62	2	,	,	PUNCT
ejpam-5127	62	3	an	an	DET
ejpam-5127	62	4	element	element	NOUN
ejpam-5127	62	5	a	a	DET
ejpam-5127	62	6	∈	∈	PROPN
ejpam-5127	62	7	v	v	NOUN
ejpam-5127	62	8	is	be	AUX
ejpam-5127	62	9	said	say	VERB
ejpam-5127	62	10	to	to	PART
ejpam-5127	62	11	be	be	AUX
ejpam-5127	62	12	a	a	DET
ejpam-5127	62	13	minimal	minimal	ADJ
ejpam-5127	62	14	element	element	NOUN
ejpam-5127	62	15	if	if	SCONJ
ejpam-5127	62	16	for	for	ADP
ejpam-5127	62	17	any	any	DET
ejpam-5127	62	18	u	u	PROPN
ejpam-5127	62	19	∈	∈	PROPN
ejpam-5127	62	20	v	v	NOUN
ejpam-5127	62	21	,	,	PUNCT
ejpam-5127	62	22	u	u	NOUN
ejpam-5127	62	23	≤	≤	VERB
ejpam-5127	62	24	a	a	DET
ejpam-5127	62	25	⇒	⇒	NOUN
ejpam-5127	62	26	u	u	NOUN
ejpam-5127	62	27	=	=	PROPN
ejpam-5127	62	28	a.	a.	PROPN
ejpam-5127	62	29	r.	r.	PROPN
ejpam-5127	62	30	shukla	shukla	PROPN
ejpam-5127	62	31	et	et	PROPN
ejpam-5127	62	32	al	al	PROPN
ejpam-5127	62	33	.	.	PUNCT
ejpam-5127	62	34	/	/	SYM
ejpam-5127	62	35	eur	eur	PROPN
ejpam-5127	62	36	.	.	PUNCT
ejpam-5127	63	1	j.	j.	PROPN
ejpam-5127	63	2	pure	pure	PROPN
ejpam-5127	63	3	appl	appl	PROPN
ejpam-5127	63	4	.	.	PROPN
ejpam-5127	63	5	math	math	PROPN
ejpam-5127	63	6	,	,	PUNCT
ejpam-5127	63	7	17	17	NUM
ejpam-5127	63	8	(	(	PUNCT
ejpam-5127	63	9	2	2	NUM
ejpam-5127	63	10	)	)	PUNCT
ejpam-5127	63	11	(	(	PUNCT
ejpam-5127	63	12	2024	2024	NUM
ejpam-5127	63	13	)	)	PUNCT
ejpam-5127	63	14	,	,	PUNCT
ejpam-5127	63	15	1306	1306	NUM
ejpam-5127	63	16	-	-	SYM
ejpam-5127	63	17	1320	1320	NUM
ejpam-5127	63	18	1309	1309	NUM
ejpam-5127	63	19	lemma	lemma	PROPN
ejpam-5127	63	20	2	2	X
ejpam-5127	63	21	.	.	PUNCT
ejpam-5127	64	1	let	let	VERB
ejpam-5127	64	2	v	v	PART
ejpam-5127	64	3	be	be	AUX
ejpam-5127	64	4	a	a	DET
ejpam-5127	64	5	pdl	pdl	NOUN
ejpam-5127	64	6	.	.	PUNCT
ejpam-5127	65	1	then	then	ADV
ejpam-5127	65	2	,	,	PUNCT
ejpam-5127	65	3	for	for	ADP
ejpam-5127	65	4	any	any	DET
ejpam-5127	65	5	a	a	DET
ejpam-5127	65	6	∈	∈	PROPN
ejpam-5127	65	7	v	v	NOUN
ejpam-5127	65	8	,	,	PUNCT
ejpam-5127	65	9	the	the	DET
ejpam-5127	65	10	following	follow	VERB
ejpam-5127	65	11	are	be	AUX
ejpam-5127	65	12	equivalent	equivalent	ADJ
ejpam-5127	65	13	:	:	PUNCT
ejpam-5127	65	14	(	(	PUNCT
ejpam-5127	65	15	1	1	X
ejpam-5127	65	16	)	)	PUNCT
ejpam-5127	65	17	a	a	PRON
ejpam-5127	65	18	is	be	AUX
ejpam-5127	65	19	minimal	minimal	ADJ
ejpam-5127	65	20	,	,	PUNCT
ejpam-5127	65	21	(	(	PUNCT
ejpam-5127	65	22	2	2	X
ejpam-5127	65	23	)	)	PUNCT
ejpam-5127	65	24	p	p	NOUN
ejpam-5127	65	25	∧	∧	PROPN
ejpam-5127	65	26	a	a	DET
ejpam-5127	65	27	=	=	X
ejpam-5127	65	28	a	a	NOUN
ejpam-5127	65	29	for	for	ADP
ejpam-5127	65	30	all	all	DET
ejpam-5127	65	31	p	p	NOUN
ejpam-5127	65	32	∈	∈	PROPN
ejpam-5127	65	33	v	v	NOUN
ejpam-5127	65	34	,	,	PUNCT
ejpam-5127	65	35	(	(	PUNCT
ejpam-5127	65	36	3	3	X
ejpam-5127	65	37	)	)	PUNCT
ejpam-5127	65	38	p	p	NOUN
ejpam-5127	65	39	∨	∨	NUM
ejpam-5127	65	40	a	a	DET
ejpam-5127	65	41	=	=	X
ejpam-5127	65	42	p	p	NOUN
ejpam-5127	65	43	for	for	ADP
ejpam-5127	65	44	all	all	DET
ejpam-5127	65	45	p	p	NOUN
ejpam-5127	65	46	∈	∈	PROPN
ejpam-5127	65	47	v	v	NOUN
ejpam-5127	65	48	.	.	PUNCT
ejpam-5127	66	1	definition	definition	NOUN
ejpam-5127	66	2	4	4	NUM
ejpam-5127	66	3	.	.	PUNCT
ejpam-5127	67	1	a	a	DET
ejpam-5127	67	2	non	non	ADJ
ejpam-5127	67	3	-	-	ADJ
ejpam-5127	67	4	empty	empty	ADJ
ejpam-5127	67	5	subset	subset	NOUN
ejpam-5127	67	6	f	f	PROPN
ejpam-5127	67	7	of	of	ADP
ejpam-5127	67	8	a	a	DET
ejpam-5127	67	9	pdl	pdl	NOUN
ejpam-5127	67	10	v	v	NOUN
ejpam-5127	67	11	is	be	AUX
ejpam-5127	67	12	said	say	VERB
ejpam-5127	67	13	to	to	PART
ejpam-5127	67	14	be	be	AUX
ejpam-5127	67	15	a	a	DET
ejpam-5127	67	16	filter	filter	NOUN
ejpam-5127	67	17	if	if	SCONJ
ejpam-5127	67	18	it	it	PRON
ejpam-5127	67	19	satisfies	satisfy	VERB
ejpam-5127	67	20	the	the	DET
ejpam-5127	67	21	following	follow	VERB
ejpam-5127	67	22	:	:	PUNCT
ejpam-5127	67	23	p	p	X
ejpam-5127	67	24	,	,	PUNCT
ejpam-5127	67	25	q	q	PROPN
ejpam-5127	67	26	∈	∈	PROPN
ejpam-5127	67	27	f	f	PROPN
ejpam-5127	67	28	⇒	⇒	VERB
ejpam-5127	67	29	p	p	PROPN
ejpam-5127	67	30	∧	∧	PROPN
ejpam-5127	67	31	q	q	PROPN
ejpam-5127	67	32	∈	∈	PROPN
ejpam-5127	67	33	f	f	NOUN
ejpam-5127	67	34	,	,	PUNCT
ejpam-5127	67	35	p	p	PROPN
ejpam-5127	67	36	∈	∈	PROPN
ejpam-5127	67	37	f	f	PROPN
ejpam-5127	67	38	,	,	PUNCT
ejpam-5127	67	39	a	a	DET
ejpam-5127	67	40	∈	∈	PROPN
ejpam-5127	67	41	v	v	ADP
ejpam-5127	67	42	⇒	⇒	NOUN
ejpam-5127	67	43	a	a	DET
ejpam-5127	67	44	∨	∨	PROPN
ejpam-5127	67	45	p	p	NOUN
ejpam-5127	67	46	∈	∈	PROPN
ejpam-5127	67	47	f	f	X
ejpam-5127	67	48	.	.	PUNCT
ejpam-5127	68	1	theorem	theorem	NOUN
ejpam-5127	68	2	2	2	NUM
ejpam-5127	68	3	.	.	PUNCT
ejpam-5127	69	1	let	let	VERB
ejpam-5127	69	2	s	s	PRON
ejpam-5127	69	3	be	be	AUX
ejpam-5127	69	4	a	a	DET
ejpam-5127	69	5	non	non	ADJ
ejpam-5127	69	6	-	-	ADJ
ejpam-5127	69	7	empty	empty	ADJ
ejpam-5127	69	8	subset	subset	NOUN
ejpam-5127	69	9	of	of	ADP
ejpam-5127	69	10	v	v	NOUN
ejpam-5127	69	11	.	.	PUNCT
ejpam-5127	70	1	then	then	ADV
ejpam-5127	70	2	[	[	X
ejpam-5127	70	3	s	s	X
ejpam-5127	70	4	)	)	PUNCT
ejpam-5127	70	5	=	=	SYM
ejpam-5127	70	6	{	{	PUNCT
ejpam-5127	70	7	p	p	NOUN
ejpam-5127	70	8	∨	∨	NOUN
ejpam-5127	70	9	(	(	PUNCT
ejpam-5127	70	10	n	n	CCONJ
ejpam-5127	70	11	∧	∧	PROPN
ejpam-5127	70	12	i=1	i=1	PROPN
ejpam-5127	70	13	si	si	NOUN
ejpam-5127	70	14	)	)	PUNCT
ejpam-5127	70	15	|	|	ADV
ejpam-5127	70	16	si	si	PROPN
ejpam-5127	70	17	∈	∈	PROPN
ejpam-5127	70	18	s	s	PROPN
ejpam-5127	70	19	,	,	PUNCT
ejpam-5127	70	20	p	p	PROPN
ejpam-5127	70	21	∈	∈	PROPN
ejpam-5127	70	22	v	v	NOUN
ejpam-5127	70	23	,	,	PUNCT
ejpam-5127	70	24	1	1	NUM
ejpam-5127	70	25	≤	≤	NUM
ejpam-5127	70	26	i	i	PRON
ejpam-5127	70	27	≤	≤	ADJ
ejpam-5127	70	28	n	n	CCONJ
ejpam-5127	70	29	and	and	CCONJ
ejpam-5127	70	30	n	n	PROPN
ejpam-5127	70	31	is	be	AUX
ejpam-5127	70	32	a	a	DET
ejpam-5127	70	33	positive	positive	ADJ
ejpam-5127	70	34	integer	integer	NOUN
ejpam-5127	70	35	}	}	PUNCT
ejpam-5127	70	36	is	be	AUX
ejpam-5127	70	37	the	the	DET
ejpam-5127	70	38	smallest	small	ADJ
ejpam-5127	70	39	filter	filter	NOUN
ejpam-5127	70	40	of	of	ADP
ejpam-5127	70	41	v	v	NOUN
ejpam-5127	70	42	containing	contain	VERB
ejpam-5127	70	43	s.	s.	PROPN
ejpam-5127	70	44	lemma	lemma	PROPN
ejpam-5127	70	45	3	3	X
ejpam-5127	70	46	.	.	PUNCT
ejpam-5127	71	1	let	let	VERB
ejpam-5127	71	2	v	v	PART
ejpam-5127	71	3	be	be	AUX
ejpam-5127	71	4	a	a	DET
ejpam-5127	71	5	pdl	pdl	NOUN
ejpam-5127	72	1	and	and	CCONJ
ejpam-5127	72	2	f	f	PROPN
ejpam-5127	72	3	be	be	AUX
ejpam-5127	72	4	a	a	DET
ejpam-5127	72	5	filter	filter	NOUN
ejpam-5127	72	6	of	of	ADP
ejpam-5127	72	7	v	v	NOUN
ejpam-5127	72	8	.	.	PUNCT
ejpam-5127	73	1	then	then	ADV
ejpam-5127	73	2	for	for	ADP
ejpam-5127	73	3	any	any	DET
ejpam-5127	73	4	p	p	NOUN
ejpam-5127	73	5	,	,	PUNCT
ejpam-5127	73	6	q	q	PROPN
ejpam-5127	73	7	∈	∈	PROPN
ejpam-5127	73	8	v	v	NOUN
ejpam-5127	73	9	,	,	PUNCT
ejpam-5127	73	10	we	we	PRON
ejpam-5127	73	11	have	have	VERB
ejpam-5127	73	12	the	the	DET
ejpam-5127	73	13	following	following	NOUN
ejpam-5127	73	14	:	:	PUNCT
ejpam-5127	73	15	(	(	PUNCT
ejpam-5127	73	16	1	1	X
ejpam-5127	73	17	)	)	PUNCT
ejpam-5127	74	1	[	[	X
ejpam-5127	74	2	p	p	X
ejpam-5127	74	3	)	)	PUNCT
ejpam-5127	74	4	=	=	SYM
ejpam-5127	74	5	{	{	PUNCT
ejpam-5127	74	6	x	x	PROPN
ejpam-5127	74	7	∨	∨	NUM
ejpam-5127	74	8	p	p	NOUN
ejpam-5127	74	9	|	|	NOUN
ejpam-5127	74	10	x	x	SYM
ejpam-5127	74	11	∈	∈	NOUN
ejpam-5127	74	12	v	v	ADP
ejpam-5127	74	13	}	}	PUNCT
ejpam-5127	74	14	,	,	PUNCT
ejpam-5127	74	15	(	(	PUNCT
ejpam-5127	74	16	2	2	X
ejpam-5127	74	17	)	)	PUNCT
ejpam-5127	74	18	p	p	NOUN
ejpam-5127	74	19	∈	∈	PROPN
ejpam-5127	75	1	[	[	X
ejpam-5127	75	2	q	q	X
ejpam-5127	75	3	)	)	PUNCT
ejpam-5127	75	4	if	if	SCONJ
ejpam-5127	75	5	and	and	CCONJ
ejpam-5127	75	6	only	only	ADV
ejpam-5127	75	7	if	if	SCONJ
ejpam-5127	75	8	p	p	X
ejpam-5127	75	9	=	=	PUNCT
ejpam-5127	75	10	p	p	X
ejpam-5127	75	11	∨	∨	NUM
ejpam-5127	75	12	q	q	NOUN
ejpam-5127	75	13	for	for	ADP
ejpam-5127	75	14	all	all	DET
ejpam-5127	75	15	p	p	NOUN
ejpam-5127	75	16	,	,	PUNCT
ejpam-5127	75	17	q	q	PROPN
ejpam-5127	75	18	∈	∈	PROPN
ejpam-5127	75	19	v	v	NOUN
ejpam-5127	75	20	,	,	PUNCT
ejpam-5127	75	21	(	(	PUNCT
ejpam-5127	75	22	3	3	X
ejpam-5127	75	23	)	)	PUNCT
ejpam-5127	75	24	p	p	NOUN
ejpam-5127	75	25	∨	∨	NUM
ejpam-5127	75	26	q	q	PROPN
ejpam-5127	75	27	∈	∈	PROPN
ejpam-5127	75	28	f	f	NOUN
ejpam-5127	76	1	if	if	SCONJ
ejpam-5127	76	2	and	and	CCONJ
ejpam-5127	76	3	only	only	ADV
ejpam-5127	76	4	if	if	SCONJ
ejpam-5127	76	5	q	q	PROPN
ejpam-5127	76	6	∨	∨	NUM
ejpam-5127	76	7	p	p	X
ejpam-5127	76	8	∈	∈	PROPN
ejpam-5127	76	9	f	f	X
ejpam-5127	76	10	,	,	PUNCT
ejpam-5127	76	11	(	(	PUNCT
ejpam-5127	76	12	4	4	X
ejpam-5127	76	13	)	)	PUNCT
ejpam-5127	77	1	[	[	X
ejpam-5127	77	2	p	p	X
ejpam-5127	77	3	∨	∨	NUM
ejpam-5127	77	4	q	q	NOUN
ejpam-5127	77	5	)	)	PUNCT
ejpam-5127	77	6	=	=	PUNCT
ejpam-5127	78	1	[	[	X
ejpam-5127	78	2	q	q	X
ejpam-5127	78	3	∨	∨	NUM
ejpam-5127	78	4	p	p	NOUN
ejpam-5127	78	5	)	)	PUNCT
ejpam-5127	78	6	,	,	PUNCT
ejpam-5127	78	7	(	(	PUNCT
ejpam-5127	78	8	5	5	X
ejpam-5127	78	9	)	)	PUNCT
ejpam-5127	78	10	[	[	X
ejpam-5127	78	11	p	p	X
ejpam-5127	78	12	∧	∧	PROPN
ejpam-5127	78	13	q	q	NOUN
ejpam-5127	78	14	)	)	PUNCT
ejpam-5127	78	15	=	=	PUNCT
ejpam-5127	79	1	[	[	X
ejpam-5127	79	2	q	q	X
ejpam-5127	79	3	∧	∧	NOUN
ejpam-5127	79	4	p	p	NOUN
ejpam-5127	79	5	)	)	PUNCT
ejpam-5127	79	6	=	=	PUNCT
ejpam-5127	80	1	[	[	X
ejpam-5127	80	2	p	p	X
ejpam-5127	80	3	)	)	PUNCT
ejpam-5127	80	4	∨	∨	NOUN
ejpam-5127	81	1	[	[	X
ejpam-5127	81	2	q	q	X
ejpam-5127	81	3	)	)	PUNCT
ejpam-5127	81	4	.	.	PUNCT
ejpam-5127	82	1	theorem	theorem	NOUN
ejpam-5127	82	2	3	3	NUM
ejpam-5127	82	3	.	.	PUNCT
ejpam-5127	83	1	the	the	DET
ejpam-5127	83	2	collection	collection	NOUN
ejpam-5127	83	3	f(v	f(v	PROPN
ejpam-5127	83	4	)	)	PUNCT
ejpam-5127	83	5	of	of	ADP
ejpam-5127	83	6	all	all	DET
ejpam-5127	83	7	filters	filter	NOUN
ejpam-5127	83	8	of	of	ADP
ejpam-5127	83	9	a	a	DET
ejpam-5127	83	10	pdl	pdl	NOUN
ejpam-5127	83	11	v	v	NOUN
ejpam-5127	83	12	forms	form	NOUN
ejpam-5127	83	13	a	a	DET
ejpam-5127	83	14	distributive	distributive	ADJ
ejpam-5127	83	15	lattice	lattice	NOUN
ejpam-5127	83	16	under	under	ADP
ejpam-5127	83	17	set	set	ADJ
ejpam-5127	83	18	inclusion	inclusion	NOUN
ejpam-5127	83	19	,	,	PUNCT
ejpam-5127	83	20	in	in	ADP
ejpam-5127	83	21	which	which	PRON
ejpam-5127	83	22	,	,	PUNCT
ejpam-5127	83	23	the	the	DET
ejpam-5127	83	24	glb	glb	NOUN
ejpam-5127	83	25	and	and	CCONJ
ejpam-5127	83	26	lub	lub	NOUN
ejpam-5127	83	27	of	of	ADP
ejpam-5127	83	28	any	any	DET
ejpam-5127	83	29	tow	tow	NOUN
ejpam-5127	83	30	filters	filter	NOUN
ejpam-5127	83	31	f	f	PROPN
ejpam-5127	83	32	and	and	CCONJ
ejpam-5127	83	33	g	g	PROPN
ejpam-5127	83	34	are	be	AUX
ejpam-5127	83	35	given	give	VERB
ejpam-5127	83	36	by	by	ADP
ejpam-5127	83	37	f	f	PROPN
ejpam-5127	83	38	∧	∧	PROPN
ejpam-5127	83	39	g	g	PROPN
ejpam-5127	83	40	=	=	SYM
ejpam-5127	83	41	f	f	PROPN
ejpam-5127	83	42	∩	∩	PROPN
ejpam-5127	83	43	g	g	PROPN
ejpam-5127	83	44	and	and	CCONJ
ejpam-5127	83	45	f	f	PROPN
ejpam-5127	83	46	∨	∨	NOUN
ejpam-5127	83	47	g	g	PROPN
ejpam-5127	83	48	=	=	PUNCT
ejpam-5127	83	49	{	{	PUNCT
ejpam-5127	83	50	p	p	X
ejpam-5127	83	51	∧	∧	PROPN
ejpam-5127	83	52	q	q	NOUN
ejpam-5127	83	53	|	|	ADV
ejpam-5127	83	54	p	p	NOUN
ejpam-5127	83	55	∈	∈	PROPN
ejpam-5127	83	56	f	f	NOUN
ejpam-5127	83	57	and	and	CCONJ
ejpam-5127	83	58	q	q	PROPN
ejpam-5127	83	59	∈	∈	PROPN
ejpam-5127	83	60	g	g	NOUN
ejpam-5127	83	61	}	}	PUNCT
ejpam-5127	83	62	,	,	PUNCT
ejpam-5127	83	63	respectively	respectively	ADV
ejpam-5127	83	64	.	.	PUNCT
ejpam-5127	84	1	definition	definition	NOUN
ejpam-5127	84	2	5	5	NUM
ejpam-5127	84	3	.	.	PUNCT
ejpam-5127	85	1	a	a	DET
ejpam-5127	85	2	non	non	ADJ
ejpam-5127	85	3	-	-	ADJ
ejpam-5127	85	4	empty	empty	ADJ
ejpam-5127	85	5	subset	subset	NOUN
ejpam-5127	85	6	i	i	PRON
ejpam-5127	85	7	of	of	ADP
ejpam-5127	85	8	a	a	DET
ejpam-5127	85	9	pdl	pdl	NOUN
ejpam-5127	85	10	v	v	NOUN
ejpam-5127	85	11	is	be	AUX
ejpam-5127	85	12	said	say	VERB
ejpam-5127	85	13	to	to	PART
ejpam-5127	85	14	be	be	AUX
ejpam-5127	85	15	an	an	DET
ejpam-5127	85	16	ideal	ideal	NOUN
ejpam-5127	85	17	if	if	SCONJ
ejpam-5127	85	18	it	it	PRON
ejpam-5127	85	19	satisfies	satisfy	VERB
ejpam-5127	85	20	the	the	DET
ejpam-5127	85	21	following	follow	VERB
ejpam-5127	85	22	:	:	PUNCT
ejpam-5127	86	1	p	p	X
ejpam-5127	86	2	,	,	PUNCT
ejpam-5127	86	3	q	q	PROPN
ejpam-5127	86	4	∈	∈	NOUN
ejpam-5127	86	5	i	i	PRON
ejpam-5127	86	6	⇒	⇒	VERB
ejpam-5127	86	7	p	p	PROPN
ejpam-5127	86	8	∨	∨	PROPN
ejpam-5127	86	9	q	q	PROPN
ejpam-5127	86	10	∈	∈	PROPN
ejpam-5127	86	11	i	i	PRON
ejpam-5127	86	12	,	,	PUNCT
ejpam-5127	86	13	p	p	PROPN
ejpam-5127	86	14	∈	∈	PROPN
ejpam-5127	87	1	i	i	PRON
ejpam-5127	87	2	,	,	PUNCT
ejpam-5127	87	3	a	a	DET
ejpam-5127	87	4	∈	∈	NOUN
ejpam-5127	87	5	v	v	ADP
ejpam-5127	87	6	⇒	⇒	NOUN
ejpam-5127	87	7	p	p	PROPN
ejpam-5127	87	8	∧	∧	PROPN
ejpam-5127	87	9	a	a	DET
ejpam-5127	87	10	∈	∈	PROPN
ejpam-5127	87	11	i.	i.	NOUN
ejpam-5127	87	12	theorem	theorem	VERB
ejpam-5127	87	13	4	4	X
ejpam-5127	87	14	.	.	PUNCT
ejpam-5127	88	1	let	let	VERB
ejpam-5127	88	2	s	s	PRON
ejpam-5127	88	3	be	be	AUX
ejpam-5127	88	4	a	a	DET
ejpam-5127	88	5	non	non	ADJ
ejpam-5127	88	6	-	-	ADJ
ejpam-5127	88	7	empty	empty	ADJ
ejpam-5127	88	8	subset	subset	NOUN
ejpam-5127	88	9	of	of	ADP
ejpam-5127	88	10	v	v	NOUN
ejpam-5127	88	11	.	.	PUNCT
ejpam-5127	89	1	then	then	ADV
ejpam-5127	89	2	(	(	PUNCT
ejpam-5127	89	3	s	s	X
ejpam-5127	89	4	]	]	X
ejpam-5127	89	5	=	=	X
ejpam-5127	89	6	{	{	PUNCT
ejpam-5127	89	7	(	(	PUNCT
ejpam-5127	89	8	n	n	NUM
ejpam-5127	89	9	∨	∨	NUM
ejpam-5127	89	10	i=1	i=1	PROPN
ejpam-5127	89	11	si	si	ADJ
ejpam-5127	89	12	)	)	PUNCT
ejpam-5127	89	13	∧	∧	NOUN
ejpam-5127	89	14	p	p	NOUN
ejpam-5127	89	15	|	|	ADV
ejpam-5127	89	16	si	si	PROPN
ejpam-5127	89	17	∈	∈	PROPN
ejpam-5127	89	18	s	s	PROPN
ejpam-5127	89	19	,	,	PUNCT
ejpam-5127	89	20	p	p	PROPN
ejpam-5127	89	21	∈	∈	PROPN
ejpam-5127	89	22	v	v	NOUN
ejpam-5127	89	23	,	,	PUNCT
ejpam-5127	89	24	1	1	NUM
ejpam-5127	89	25	≤	≤	NUM
ejpam-5127	89	26	i	i	PRON
ejpam-5127	89	27	≤	≤	ADJ
ejpam-5127	89	28	n	n	CCONJ
ejpam-5127	89	29	and	and	CCONJ
ejpam-5127	89	30	n	n	PROPN
ejpam-5127	89	31	is	be	AUX
ejpam-5127	89	32	a	a	DET
ejpam-5127	89	33	positive	positive	ADJ
ejpam-5127	89	34	integer	integer	NOUN
ejpam-5127	89	35	}	}	PUNCT
ejpam-5127	89	36	is	be	AUX
ejpam-5127	89	37	the	the	DET
ejpam-5127	89	38	smallest	small	ADJ
ejpam-5127	89	39	ideal	ideal	NOUN
ejpam-5127	89	40	of	of	ADP
ejpam-5127	89	41	v	v	NOUN
ejpam-5127	89	42	containing	contain	VERB
ejpam-5127	89	43	s.	s.	PROPN
ejpam-5127	89	44	lemma	lemma	PROPN
ejpam-5127	89	45	4	4	X
ejpam-5127	89	46	.	.	PUNCT
ejpam-5127	90	1	let	let	VERB
ejpam-5127	90	2	v	v	PART
ejpam-5127	90	3	be	be	AUX
ejpam-5127	90	4	a	a	DET
ejpam-5127	90	5	pdl	pdl	NOUN
ejpam-5127	91	1	and	and	CCONJ
ejpam-5127	91	2	i	i	PRON
ejpam-5127	91	3	be	be	VERB
ejpam-5127	91	4	an	an	DET
ejpam-5127	91	5	ideal	ideal	NOUN
ejpam-5127	91	6	of	of	ADP
ejpam-5127	91	7	v	v	NOUN
ejpam-5127	91	8	.	.	PUNCT
ejpam-5127	92	1	then	then	ADV
ejpam-5127	92	2	,	,	PUNCT
ejpam-5127	92	3	for	for	ADP
ejpam-5127	92	4	any	any	DET
ejpam-5127	92	5	p	p	NOUN
ejpam-5127	92	6	,	,	PUNCT
ejpam-5127	92	7	q	q	PROPN
ejpam-5127	92	8	∈	∈	PROPN
ejpam-5127	92	9	v	v	NOUN
ejpam-5127	92	10	,	,	PUNCT
ejpam-5127	92	11	we	we	PRON
ejpam-5127	92	12	have	have	VERB
ejpam-5127	92	13	the	the	DET
ejpam-5127	92	14	following	following	NOUN
ejpam-5127	92	15	:	:	PUNCT
ejpam-5127	92	16	(	(	PUNCT
ejpam-5127	92	17	1	1	X
ejpam-5127	92	18	)	)	PUNCT
ejpam-5127	92	19	(	(	PUNCT
ejpam-5127	93	1	p	p	X
ejpam-5127	93	2	]	]	X
ejpam-5127	93	3	=	=	PUNCT
ejpam-5127	93	4	{	{	PUNCT
ejpam-5127	93	5	p	p	X
ejpam-5127	93	6	∧	∧	PROPN
ejpam-5127	93	7	x	x	PUNCT
ejpam-5127	93	8	|	|	ADV
ejpam-5127	93	9	x	x	SYM
ejpam-5127	93	10	∈	∈	NOUN
ejpam-5127	93	11	v	v	ADP
ejpam-5127	93	12	}	}	PUNCT
ejpam-5127	93	13	,	,	PUNCT
ejpam-5127	93	14	(	(	PUNCT
ejpam-5127	93	15	2	2	X
ejpam-5127	93	16	)	)	PUNCT
ejpam-5127	93	17	p	p	NOUN
ejpam-5127	93	18	∈	∈	PROPN
ejpam-5127	93	19	(	(	PUNCT
ejpam-5127	93	20	q	q	X
ejpam-5127	93	21	]	]	X
ejpam-5127	93	22	if	if	SCONJ
ejpam-5127	93	23	and	and	CCONJ
ejpam-5127	93	24	only	only	ADV
ejpam-5127	93	25	if	if	SCONJ
ejpam-5127	93	26	p	p	X
ejpam-5127	93	27	=	=	X
ejpam-5127	93	28	q	q	NOUN
ejpam-5127	93	29	∧	∧	PROPN
ejpam-5127	93	30	p	p	NOUN
ejpam-5127	93	31	,	,	PUNCT
ejpam-5127	93	32	(	(	PUNCT
ejpam-5127	93	33	3	3	X
ejpam-5127	93	34	)	)	PUNCT
ejpam-5127	93	35	p	p	NOUN
ejpam-5127	93	36	∧	∧	PROPN
ejpam-5127	93	37	q	q	PUNCT
ejpam-5127	93	38	∈	∈	PROPN
ejpam-5127	94	1	i	i	PRON
ejpam-5127	94	2	if	if	SCONJ
ejpam-5127	94	3	and	and	CCONJ
ejpam-5127	94	4	only	only	ADV
ejpam-5127	94	5	if	if	SCONJ
ejpam-5127	94	6	q	q	PROPN
ejpam-5127	94	7	∧	∧	PROPN
ejpam-5127	94	8	p	p	X
ejpam-5127	94	9	∈	∈	PROPN
ejpam-5127	95	1	i	i	PRON
ejpam-5127	95	2	,	,	PUNCT
ejpam-5127	95	3	(	(	PUNCT
ejpam-5127	95	4	4	4	NUM
ejpam-5127	95	5	)	)	PUNCT
ejpam-5127	95	6	(	(	PUNCT
ejpam-5127	95	7	p	p	X
ejpam-5127	95	8	∧	∧	PROPN
ejpam-5127	95	9	q	q	X
ejpam-5127	95	10	]	]	X
ejpam-5127	95	11	=	=	SYM
ejpam-5127	95	12	(	(	PUNCT
ejpam-5127	95	13	q	q	PUNCT
ejpam-5127	95	14	∧	∧	PROPN
ejpam-5127	95	15	p	p	X
ejpam-5127	95	16	]	]	X
ejpam-5127	95	17	,	,	PUNCT
ejpam-5127	95	18	(	(	PUNCT
ejpam-5127	95	19	5	5	NUM
ejpam-5127	95	20	)	)	PUNCT
ejpam-5127	95	21	(	(	PUNCT
ejpam-5127	95	22	p	p	X
ejpam-5127	95	23	∧	∧	PROPN
ejpam-5127	95	24	q	q	X
ejpam-5127	95	25	]	]	X
ejpam-5127	95	26	=	=	SYM
ejpam-5127	95	27	(	(	PUNCT
ejpam-5127	95	28	q	q	PUNCT
ejpam-5127	95	29	∧	∧	PROPN
ejpam-5127	95	30	p	p	X
ejpam-5127	95	31	]	]	X
ejpam-5127	95	32	=	=	PUNCT
ejpam-5127	95	33	(	(	PUNCT
ejpam-5127	95	34	p	p	X
ejpam-5127	95	35	]	]	X
ejpam-5127	95	36	∧	∧	PROPN
ejpam-5127	95	37	(	(	PUNCT
ejpam-5127	95	38	q	q	X
ejpam-5127	95	39	]	]	X
ejpam-5127	95	40	.	.	PUNCT
ejpam-5127	96	1	r.	r.	PROPN
ejpam-5127	96	2	shukla	shukla	PROPN
ejpam-5127	96	3	et	et	PROPN
ejpam-5127	96	4	al	al	PROPN
ejpam-5127	96	5	.	.	PUNCT
ejpam-5127	96	6	/	/	SYM
ejpam-5127	96	7	eur	eur	PROPN
ejpam-5127	96	8	.	.	PUNCT
ejpam-5127	97	1	j.	j.	PROPN
ejpam-5127	97	2	pure	pure	PROPN
ejpam-5127	97	3	appl	appl	PROPN
ejpam-5127	97	4	.	.	PROPN
ejpam-5127	97	5	math	math	PROPN
ejpam-5127	97	6	,	,	PUNCT
ejpam-5127	97	7	17	17	NUM
ejpam-5127	97	8	(	(	PUNCT
ejpam-5127	97	9	2	2	NUM
ejpam-5127	97	10	)	)	PUNCT
ejpam-5127	97	11	(	(	PUNCT
ejpam-5127	97	12	2024	2024	NUM
ejpam-5127	97	13	)	)	PUNCT
ejpam-5127	97	14	,	,	PUNCT
ejpam-5127	97	15	1306	1306	NUM
ejpam-5127	97	16	-	-	SYM
ejpam-5127	97	17	1320	1320	NUM
ejpam-5127	97	18	1310	1310	NUM
ejpam-5127	97	19	theorem	theorem	NOUN
ejpam-5127	97	20	5	5	NUM
ejpam-5127	97	21	.	.	PUNCT
ejpam-5127	98	1	the	the	DET
ejpam-5127	98	2	collection	collection	NOUN
ejpam-5127	98	3	i(v	i(v	NOUN
ejpam-5127	98	4	)	)	PUNCT
ejpam-5127	98	5	of	of	ADP
ejpam-5127	98	6	all	all	DET
ejpam-5127	98	7	ideals	ideal	NOUN
ejpam-5127	98	8	of	of	ADP
ejpam-5127	98	9	a	a	DET
ejpam-5127	98	10	pdl	pdl	NOUN
ejpam-5127	98	11	v	v	NOUN
ejpam-5127	98	12	forms	form	NOUN
ejpam-5127	98	13	a	a	DET
ejpam-5127	98	14	distributive	distributive	ADJ
ejpam-5127	98	15	lattice	lattice	NOUN
ejpam-5127	98	16	under	under	ADP
ejpam-5127	98	17	set	set	ADJ
ejpam-5127	98	18	inclusion	inclusion	NOUN
ejpam-5127	98	19	,	,	PUNCT
ejpam-5127	98	20	in	in	ADP
ejpam-5127	98	21	which	which	PRON
ejpam-5127	98	22	,	,	PUNCT
ejpam-5127	98	23	the	the	DET
ejpam-5127	98	24	glb	glb	NOUN
ejpam-5127	98	25	and	and	CCONJ
ejpam-5127	98	26	lub	lub	NOUN
ejpam-5127	98	27	of	of	ADP
ejpam-5127	98	28	any	any	DET
ejpam-5127	98	29	two	two	NUM
ejpam-5127	98	30	ideals	ideal	NOUN
ejpam-5127	98	31	i	i	PRON
ejpam-5127	98	32	and	and	CCONJ
ejpam-5127	98	33	j	j	PROPN
ejpam-5127	98	34	are	be	AUX
ejpam-5127	98	35	given	give	VERB
ejpam-5127	98	36	by	by	ADP
ejpam-5127	98	37	i	i	PROPN
ejpam-5127	98	38	∧	∧	PROPN
ejpam-5127	98	39	j	j	PROPN
ejpam-5127	99	1	=	=	SYM
ejpam-5127	99	2	i	i	PROPN
ejpam-5127	99	3	∩	∩	PROPN
ejpam-5127	99	4	j	j	PROPN
ejpam-5127	99	5	and	and	CCONJ
ejpam-5127	99	6	i	i	PROPN
ejpam-5127	99	7	∨	∨	PROPN
ejpam-5127	99	8	j	j	PROPN
ejpam-5127	100	1	=	=	PRON
ejpam-5127	100	2	{	{	PUNCT
ejpam-5127	100	3	p	p	PROPN
ejpam-5127	100	4	∨	∨	NUM
ejpam-5127	100	5	q	q	NOUN
ejpam-5127	101	1	|	|	ADV
ejpam-5127	101	2	p	p	NOUN
ejpam-5127	101	3	∈	∈	PROPN
ejpam-5127	102	1	i	i	PRON
ejpam-5127	102	2	and	and	CCONJ
ejpam-5127	102	3	q	q	PROPN
ejpam-5127	102	4	∈	∈	PROPN
ejpam-5127	102	5	j	j	PROPN
ejpam-5127	102	6	}	}	PUNCT
ejpam-5127	102	7	,	,	PUNCT
ejpam-5127	102	8	respectively	respectively	ADV
ejpam-5127	102	9	.	.	PUNCT
ejpam-5127	103	1	a	a	DET
ejpam-5127	103	2	proper	proper	ADJ
ejpam-5127	103	3	filter(ideal	filter(ideal	NOUN
ejpam-5127	103	4	)	)	PUNCT
ejpam-5127	104	1	p	p	NOUN
ejpam-5127	104	2	of	of	ADP
ejpam-5127	104	3	v	v	NOUN
ejpam-5127	104	4	is	be	AUX
ejpam-5127	104	5	said	say	VERB
ejpam-5127	104	6	to	to	PART
ejpam-5127	104	7	be	be	AUX
ejpam-5127	104	8	a	a	DET
ejpam-5127	104	9	prime	prime	ADJ
ejpam-5127	104	10	filter(ideal	filter(ideal	NOUN
ejpam-5127	104	11	)	)	PUNCT
ejpam-5127	104	12	if	if	SCONJ
ejpam-5127	104	13	for	for	ADP
ejpam-5127	104	14	any	any	DET
ejpam-5127	104	15	x	x	NOUN
ejpam-5127	104	16	,	,	PUNCT
ejpam-5127	104	17	y	y	PROPN
ejpam-5127	104	18	∈	∈	PROPN
ejpam-5127	104	19	v	v	NOUN
ejpam-5127	104	20	,	,	PUNCT
ejpam-5127	104	21	x	x	PROPN
ejpam-5127	104	22	∨	∨	NUM
ejpam-5127	104	23	y	y	PROPN
ejpam-5127	104	24	∈	∈	PROPN
ejpam-5127	104	25	p(x	p(x	PROPN
ejpam-5127	104	26	∧	∧	PROPN
ejpam-5127	104	27	y	y	PROPN
ejpam-5127	104	28	∈	∈	PROPN
ejpam-5127	104	29	p	p	NOUN
ejpam-5127	104	30	)	)	PUNCT
ejpam-5127	104	31	⇒	⇒	NOUN
ejpam-5127	104	32	x	x	X
ejpam-5127	104	33	∈	∈	PROPN
ejpam-5127	104	34	p	p	NOUN
ejpam-5127	104	35	or	or	CCONJ
ejpam-5127	104	36	y	y	PROPN
ejpam-5127	104	37	∈	∈	PROPN
ejpam-5127	104	38	p.	p.	NOUN
ejpam-5127	104	39	a	a	DET
ejpam-5127	104	40	proper	proper	ADJ
ejpam-5127	104	41	filter(ideal	filter(ideal	NOUN
ejpam-5127	104	42	)	)	PUNCT
ejpam-5127	105	1	m	m	PROPN
ejpam-5127	105	2	of	of	ADP
ejpam-5127	105	3	v	v	NOUN
ejpam-5127	105	4	is	be	AUX
ejpam-5127	105	5	said	say	VERB
ejpam-5127	105	6	to	to	PART
ejpam-5127	105	7	be	be	AUX
ejpam-5127	105	8	maximal	maximal	ADJ
ejpam-5127	105	9	if	if	SCONJ
ejpam-5127	105	10	it	it	PRON
ejpam-5127	105	11	is	be	AUX
ejpam-5127	105	12	not	not	PART
ejpam-5127	105	13	properly	properly	ADV
ejpam-5127	105	14	contained	contain	VERB
ejpam-5127	105	15	in	in	ADP
ejpam-5127	105	16	any	any	DET
ejpam-5127	105	17	proper	proper	ADJ
ejpam-5127	105	18	filter(ideal	filter(ideal	NOUN
ejpam-5127	105	19	)	)	PUNCT
ejpam-5127	105	20	of	of	ADP
ejpam-5127	105	21	v	v	NOUN
ejpam-5127	105	22	.	.	PUNCT
ejpam-5127	106	1	a	a	DET
ejpam-5127	106	2	prime	prime	ADJ
ejpam-5127	106	3	filter	filter	NOUN
ejpam-5127	106	4	p	p	NOUN
ejpam-5127	106	5	of	of	ADP
ejpam-5127	106	6	v	v	NOUN
ejpam-5127	106	7	is	be	AUX
ejpam-5127	106	8	said	say	VERB
ejpam-5127	106	9	to	to	PART
ejpam-5127	106	10	be	be	AUX
ejpam-5127	106	11	minimal	minimal	ADJ
ejpam-5127	106	12	,	,	PUNCT
ejpam-5127	106	13	if	if	SCONJ
ejpam-5127	106	14	it	it	PRON
ejpam-5127	106	15	is	be	AUX
ejpam-5127	106	16	minimal	minimal	ADJ
ejpam-5127	106	17	among	among	ADP
ejpam-5127	106	18	all	all	DET
ejpam-5127	106	19	the	the	DET
ejpam-5127	106	20	prime	prime	ADJ
ejpam-5127	106	21	filters	filter	NOUN
ejpam-5127	106	22	of	of	ADP
ejpam-5127	106	23	v	v	NOUN
ejpam-5127	106	24	.	.	PUNCT
ejpam-5127	107	1	a	a	DET
ejpam-5127	107	2	prime	prime	ADJ
ejpam-5127	107	3	filter	filter	NOUN
ejpam-5127	107	4	p	p	NOUN
ejpam-5127	107	5	is	be	AUX
ejpam-5127	107	6	said	say	VERB
ejpam-5127	107	7	to	to	PART
ejpam-5127	107	8	be	be	AUX
ejpam-5127	107	9	a	a	DET
ejpam-5127	107	10	minimal	minimal	ADJ
ejpam-5127	107	11	prime	prime	ADJ
ejpam-5127	107	12	filter	filter	NOUN
ejpam-5127	107	13	belonging	belong	VERB
ejpam-5127	107	14	to	to	ADP
ejpam-5127	107	15	a	a	DET
ejpam-5127	107	16	filter	filter	NOUN
ejpam-5127	108	1	i	i	PRON
ejpam-5127	108	2	,	,	PUNCT
ejpam-5127	108	3	if	if	SCONJ
ejpam-5127	108	4	it	it	PRON
ejpam-5127	108	5	is	be	AUX
ejpam-5127	108	6	minimal	minimal	ADJ
ejpam-5127	108	7	among	among	ADP
ejpam-5127	108	8	all	all	DET
ejpam-5127	108	9	the	the	DET
ejpam-5127	108	10	prime	prime	ADJ
ejpam-5127	108	11	filters	filter	NOUN
ejpam-5127	108	12	of	of	ADP
ejpam-5127	108	13	v	v	NOUN
ejpam-5127	108	14	containing	contain	VERB
ejpam-5127	108	15	i.	i.	NOUN
ejpam-5127	108	16	a	a	DET
ejpam-5127	108	17	prime	prime	ADJ
ejpam-5127	108	18	filer	filer	NOUN
ejpam-5127	108	19	p	p	PROPN
ejpam-5127	108	20	of	of	ADP
ejpam-5127	108	21	v	v	NOUN
ejpam-5127	108	22	is	be	AUX
ejpam-5127	108	23	a	a	DET
ejpam-5127	108	24	minimal	minimal	ADJ
ejpam-5127	108	25	prime	prime	ADJ
ejpam-5127	108	26	filter	filter	NOUN
ejpam-5127	108	27	if	if	SCONJ
ejpam-5127	108	28	and	and	CCONJ
ejpam-5127	108	29	only	only	ADV
ejpam-5127	108	30	if	if	SCONJ
ejpam-5127	108	31	for	for	ADP
ejpam-5127	109	1	each	each	DET
ejpam-5127	109	2	x	x	SYM
ejpam-5127	109	3	∈	∈	PROPN
ejpam-5127	109	4	p	p	NOUN
ejpam-5127	109	5	,	,	PUNCT
ejpam-5127	109	6	there	there	PRON
ejpam-5127	109	7	exists	exist	VERB
ejpam-5127	109	8	y	y	PROPN
ejpam-5127	109	9	/∈	/∈	PUNCT
ejpam-5127	110	1	p	p	X
ejpam-5127	110	2	such	such	ADJ
ejpam-5127	110	3	that	that	SCONJ
ejpam-5127	110	4	x	x	PROPN
ejpam-5127	110	5	∨	∨	NUM
ejpam-5127	110	6	y	y	NOUN
ejpam-5127	110	7	=	=	SYM
ejpam-5127	110	8	1	1	X
ejpam-5127	110	9	.	.	PUNCT
ejpam-5127	110	10	definition	definition	NOUN
ejpam-5127	110	11	6	6	NUM
ejpam-5127	110	12	.	.	PUNCT
ejpam-5127	111	1	by	by	ADP
ejpam-5127	111	2	a	a	DET
ejpam-5127	111	3	homomorphism	homomorphism	NOUN
ejpam-5127	111	4	of	of	ADP
ejpam-5127	111	5	a	a	DET
ejpam-5127	111	6	pdl	pdl	PROPN
ejpam-5127	111	7	(	(	PUNCT
ejpam-5127	111	8	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	111	9	,	,	PUNCT
ejpam-5127	111	10	1	1	NUM
ejpam-5127	111	11	)	)	PUNCT
ejpam-5127	111	12	into	into	ADP
ejpam-5127	111	13	a	a	DET
ejpam-5127	111	14	pdl	pdl	NOUN
ejpam-5127	111	15	(	(	PUNCT
ejpam-5127	111	16	v	v	NOUN
ejpam-5127	111	17	′,∨′,∧′	′,∨′,∧′	NOUN
ejpam-5127	111	18	,	,	PUNCT
ejpam-5127	111	19	1′	1′	NUM
ejpam-5127	111	20	)	)	PUNCT
ejpam-5127	111	21	,	,	PUNCT
ejpam-5127	111	22	we	we	PRON
ejpam-5127	111	23	mean	mean	VERB
ejpam-5127	111	24	,	,	PUNCT
ejpam-5127	111	25	a	a	DET
ejpam-5127	111	26	mapping	mapping	NOUN
ejpam-5127	111	27	f	f	NOUN
ejpam-5127	111	28	:	:	PUNCT
ejpam-5127	111	29	v	v	X
ejpam-5127	111	30	→	→	SYM
ejpam-5127	111	31	v	v	NOUN
ejpam-5127	111	32	′	′	NOUN
ejpam-5127	111	33	satisfying	satisfy	VERB
ejpam-5127	111	34	the	the	DET
ejpam-5127	111	35	following	following	NOUN
ejpam-5127	111	36	:	:	PUNCT
ejpam-5127	111	37	(	(	PUNCT
ejpam-5127	111	38	1	1	X
ejpam-5127	111	39	)	)	PUNCT
ejpam-5127	111	40	f(a	f(a	NOUN
ejpam-5127	111	41	∨	∨	NUM
ejpam-5127	111	42	b	b	NOUN
ejpam-5127	111	43	)	)	PUNCT
ejpam-5127	111	44	=	=	SYM
ejpam-5127	111	45	f(a	f(a	PROPN
ejpam-5127	111	46	)	)	PUNCT
ejpam-5127	112	1	∨′	∨′	PROPN
ejpam-5127	112	2	f(b	f(b	PROPN
ejpam-5127	112	3	)	)	PUNCT
ejpam-5127	112	4	,	,	PUNCT
ejpam-5127	112	5	(	(	PUNCT
ejpam-5127	112	6	2	2	X
ejpam-5127	112	7	)	)	PUNCT
ejpam-5127	112	8	f(a	f(a	NOUN
ejpam-5127	112	9	∧	∧	PROPN
ejpam-5127	112	10	b	b	PROPN
ejpam-5127	112	11	)	)	PUNCT
ejpam-5127	112	12	=	=	SYM
ejpam-5127	112	13	f(a	f(a	NOUN
ejpam-5127	112	14	)	)	PUNCT
ejpam-5127	112	15	∧′	∧′	PROPN
ejpam-5127	112	16	f(b	f(b	NOUN
ejpam-5127	112	17	)	)	PUNCT
ejpam-5127	112	18	,	,	PUNCT
ejpam-5127	112	19	(	(	PUNCT
ejpam-5127	112	20	3	3	X
ejpam-5127	112	21	)	)	PUNCT
ejpam-5127	112	22	f(1	f(1	PROPN
ejpam-5127	112	23	)	)	PUNCT
ejpam-5127	112	24	=	=	SYM
ejpam-5127	112	25	f(1′	f(1′	PROPN
ejpam-5127	112	26	)	)	PUNCT
ejpam-5127	112	27	.	.	PUNCT
ejpam-5127	113	1	3	3	X
ejpam-5127	113	2	.	.	X
ejpam-5127	113	3	normal	normal	ADJ
ejpam-5127	113	4	pdl	pdl	NOUN
ejpam-5127	113	5	in	in	ADP
ejpam-5127	113	6	this	this	DET
ejpam-5127	113	7	section	section	NOUN
ejpam-5127	113	8	,	,	PUNCT
ejpam-5127	113	9	we	we	PRON
ejpam-5127	113	10	introduce	introduce	VERB
ejpam-5127	113	11	the	the	DET
ejpam-5127	113	12	notion	notion	NOUN
ejpam-5127	113	13	of	of	ADP
ejpam-5127	113	14	a	a	DET
ejpam-5127	113	15	normal	normal	ADJ
ejpam-5127	113	16	paradistributive	paradistributive	ADJ
ejpam-5127	113	17	latticoid	latticoid	NOUN
ejpam-5127	113	18	and	and	CCONJ
ejpam-5127	113	19	characterize	characterize	VERB
ejpam-5127	113	20	interms	interm	NOUN
ejpam-5127	113	21	of	of	ADP
ejpam-5127	113	22	prime	prime	ADJ
ejpam-5127	113	23	filters	filter	NOUN
ejpam-5127	113	24	and	and	CCONJ
ejpam-5127	113	25	minimal	minimal	ADJ
ejpam-5127	113	26	prime	prime	ADJ
ejpam-5127	113	27	filters	filter	NOUN
ejpam-5127	113	28	.	.	PUNCT
ejpam-5127	114	1	throughout	throughout	ADP
ejpam-5127	114	2	this	this	DET
ejpam-5127	114	3	section	section	NOUN
ejpam-5127	114	4	,	,	PUNCT
ejpam-5127	114	5	a	a	DET
ejpam-5127	114	6	pdl	pdl	PROPN
ejpam-5127	114	7	v	v	NOUN
ejpam-5127	114	8	means	mean	VERB
ejpam-5127	114	9	a	a	DET
ejpam-5127	114	10	paradistributive	paradistributive	ADJ
ejpam-5127	114	11	latticoid	latticoid	NOUN
ejpam-5127	114	12	(	(	PUNCT
ejpam-5127	114	13	v,∨,∧	v,∨,∧	NOUN
ejpam-5127	114	14	,	,	PUNCT
ejpam-5127	114	15	1	1	NUM
ejpam-5127	114	16	)	)	PUNCT
ejpam-5127	114	17	with	with	ADP
ejpam-5127	114	18	minimal	minimal	ADJ
ejpam-5127	114	19	elements	element	NOUN
ejpam-5127	114	20	.	.	PUNCT
ejpam-5127	115	1	first	first	ADV
ejpam-5127	115	2	,	,	PUNCT
ejpam-5127	115	3	we	we	PRON
ejpam-5127	115	4	give	give	VERB
ejpam-5127	115	5	the	the	DET
ejpam-5127	115	6	following	following	NOUN
ejpam-5127	115	7	:	:	PUNCT
ejpam-5127	115	8	definition	definition	NOUN
ejpam-5127	115	9	7	7	NUM
ejpam-5127	115	10	.	.	PUNCT
ejpam-5127	116	1	for	for	ADP
ejpam-5127	116	2	any	any	DET
ejpam-5127	116	3	non	non	ADJ
ejpam-5127	116	4	-	-	ADJ
ejpam-5127	116	5	empty	empty	ADJ
ejpam-5127	116	6	subset	subset	NOUN
ejpam-5127	116	7	s	s	NOUN
ejpam-5127	116	8	of	of	ADP
ejpam-5127	116	9	a	a	DET
ejpam-5127	116	10	pdl	pdl	NOUN
ejpam-5127	116	11	v	v	NOUN
ejpam-5127	116	12	,	,	PUNCT
ejpam-5127	116	13	write	write	VERB
ejpam-5127	116	14	(	(	PUNCT
ejpam-5127	116	15	s)•	s)•	PROPN
ejpam-5127	116	16	=	=	SYM
ejpam-5127	116	17	{	{	PUNCT
ejpam-5127	116	18	a	a	DET
ejpam-5127	116	19	∈	∈	PROPN
ejpam-5127	116	20	v	v	ADP
ejpam-5127	116	21	|s	|s	PROPN
ejpam-5127	116	22	∨	∨	NUM
ejpam-5127	116	23	a	a	DET
ejpam-5127	116	24	=	=	SYM
ejpam-5127	116	25	1	1	NUM
ejpam-5127	116	26	for	for	ADP
ejpam-5127	116	27	all	all	DET
ejpam-5127	116	28	s	s	PART
ejpam-5127	116	29	∈	∈	NOUN
ejpam-5127	116	30	s	s	PART
ejpam-5127	116	31	}	}	PUNCT
ejpam-5127	116	32	.	.	PUNCT
ejpam-5127	117	1	then	then	ADV
ejpam-5127	117	2	(	(	PUNCT
ejpam-5127	117	3	s)•	s)•	X
ejpam-5127	117	4	is	be	AUX
ejpam-5127	117	5	a	a	DET
ejpam-5127	117	6	filter	filter	NOUN
ejpam-5127	117	7	of	of	ADP
ejpam-5127	117	8	v	v	NOUN
ejpam-5127	117	9	,	,	PUNCT
ejpam-5127	117	10	and	and	CCONJ
ejpam-5127	117	11	is	be	AUX
ejpam-5127	117	12	called	call	VERB
ejpam-5127	117	13	the	the	DET
ejpam-5127	117	14	annihilator	annihilator	NOUN
ejpam-5127	117	15	of	of	ADP
ejpam-5127	117	16	s	s	PRON
ejpam-5127	117	17	in	in	ADP
ejpam-5127	117	18	v	v	NOUN
ejpam-5127	117	19	.	.	PUNCT
ejpam-5127	118	1	if	if	SCONJ
ejpam-5127	118	2	s	s	VERB
ejpam-5127	118	3	=	=	X
ejpam-5127	118	4	{	{	PUNCT
ejpam-5127	118	5	s	s	PROPN
ejpam-5127	118	6	}	}	PUNCT
ejpam-5127	118	7	,	,	PUNCT
ejpam-5127	118	8	we	we	PRON
ejpam-5127	118	9	write	write	VERB
ejpam-5127	118	10	(	(	PUNCT
ejpam-5127	118	11	s)•	s)•	X
ejpam-5127	118	12	for	for	ADP
ejpam-5127	118	13	(	(	PUNCT
ejpam-5127	118	14	{	{	PUNCT
ejpam-5127	118	15	s})•.	s})•.	PUNCT
ejpam-5127	118	16	the	the	DET
ejpam-5127	118	17	following	follow	VERB
ejpam-5127	118	18	lemma	lemma	PROPN
ejpam-5127	118	19	can	can	AUX
ejpam-5127	118	20	be	be	AUX
ejpam-5127	118	21	verified	verify	VERB
ejpam-5127	118	22	routinely	routinely	ADV
ejpam-5127	118	23	.	.	PUNCT
ejpam-5127	119	1	lemma	lemma	PROPN
ejpam-5127	119	2	5	5	NUM
ejpam-5127	119	3	.	.	PUNCT
ejpam-5127	120	1	for	for	ADP
ejpam-5127	120	2	any	any	DET
ejpam-5127	120	3	a	a	NOUN
ejpam-5127	120	4	,	,	PUNCT
ejpam-5127	120	5	b	b	PROPN
ejpam-5127	120	6	∈	∈	PROPN
ejpam-5127	120	7	v	v	NOUN
ejpam-5127	120	8	,	,	PUNCT
ejpam-5127	120	9	(	(	PUNCT
ejpam-5127	120	10	1	1	X
ejpam-5127	120	11	)	)	PUNCT
ejpam-5127	120	12	a	a	DET
ejpam-5127	120	13	≤	≤	PROPN
ejpam-5127	120	14	b	b	X
ejpam-5127	120	15	=	=	NOUN
ejpam-5127	120	16	⇒	⇒	NOUN
ejpam-5127	120	17	(	(	PUNCT
ejpam-5127	120	18	a)•	a)•	NUM
ejpam-5127	120	19	⊆	⊆	NUM
ejpam-5127	120	20	(	(	PUNCT
ejpam-5127	120	21	b)•	b)•	NUM
ejpam-5127	120	22	,	,	PUNCT
ejpam-5127	120	23	(	(	PUNCT
ejpam-5127	120	24	2	2	NUM
ejpam-5127	120	25	)	)	PUNCT
ejpam-5127	120	26	(	(	PUNCT
ejpam-5127	120	27	a	a	DET
ejpam-5127	120	28	∨	∨	NUM
ejpam-5127	120	29	b)•	b)•	PRON
ejpam-5127	120	30	=	=	SYM
ejpam-5127	120	31	(	(	PUNCT
ejpam-5127	120	32	b	b	PROPN
ejpam-5127	120	33	∨	∨	NUM
ejpam-5127	120	34	a)•	a)•	PROPN
ejpam-5127	120	35	,	,	PUNCT
ejpam-5127	120	36	(	(	PUNCT
ejpam-5127	120	37	3	3	NUM
ejpam-5127	120	38	)	)	PUNCT
ejpam-5127	120	39	(	(	PUNCT
ejpam-5127	120	40	a	a	DET
ejpam-5127	120	41	∧	∧	PROPN
ejpam-5127	120	42	b)•	b)•	PRON
ejpam-5127	120	43	=	=	PUNCT
ejpam-5127	120	44	(	(	PUNCT
ejpam-5127	120	45	b	b	PROPN
ejpam-5127	120	46	∧	∧	PROPN
ejpam-5127	120	47	a)•	a)•	PROPN
ejpam-5127	120	48	,	,	PUNCT
ejpam-5127	120	49	(	(	PUNCT
ejpam-5127	120	50	4	4	NUM
ejpam-5127	120	51	)	)	PUNCT
ejpam-5127	120	52	(	(	PUNCT
ejpam-5127	120	53	a	a	DET
ejpam-5127	120	54	∧	∧	PROPN
ejpam-5127	120	55	b)•	b)•	PRON
ejpam-5127	120	56	=	=	PUNCT
ejpam-5127	120	57	(	(	PUNCT
ejpam-5127	120	58	a)•	a)•	NUM
ejpam-5127	120	59	∩	∩	NOUN
ejpam-5127	120	60	(	(	PUNCT
ejpam-5127	120	61	b)•	b)•	NUM
ejpam-5127	120	62	,	,	PUNCT
ejpam-5127	120	63	(	(	PUNCT
ejpam-5127	120	64	5	5	NUM
ejpam-5127	120	65	)	)	PUNCT
ejpam-5127	120	66	(	(	PUNCT
ejpam-5127	120	67	a)•	a)•	NUM
ejpam-5127	120	68	∨	∨	NUM
ejpam-5127	120	69	(	(	PUNCT
ejpam-5127	120	70	b)•	b)•	PRON
ejpam-5127	120	71	⊆	⊆	NUM
ejpam-5127	120	72	(	(	PUNCT
ejpam-5127	120	73	a	a	DET
ejpam-5127	120	74	∨	∨	NUM
ejpam-5127	120	75	b)•	b)•	NUM
ejpam-5127	120	76	,	,	PUNCT
ejpam-5127	120	77	(	(	PUNCT
ejpam-5127	120	78	6	6	NUM
ejpam-5127	120	79	)	)	PUNCT
ejpam-5127	120	80	a	a	DET
ejpam-5127	120	81	∈	∈	NOUN
ejpam-5127	120	82	(	(	PUNCT
ejpam-5127	120	83	x)•	x)•	PROPN
ejpam-5127	120	84	=	=	AUX
ejpam-5127	120	85	⇒	⇒	PROPN
ejpam-5127	120	86	(	(	PUNCT
ejpam-5127	120	87	x)••	x)••	PROPN
ejpam-5127	120	88	⊆	⊆	NUM
ejpam-5127	120	89	(	(	PUNCT
ejpam-5127	120	90	a)•	a)•	NUM
ejpam-5127	120	91	,	,	PUNCT
ejpam-5127	120	92	(	(	PUNCT
ejpam-5127	120	93	7	7	X
ejpam-5127	120	94	)	)	PUNCT
ejpam-5127	120	95	a	a	DET
ejpam-5127	120	96	∈	∈	NOUN
ejpam-5127	121	1	[	[	X
ejpam-5127	121	2	b	b	X
ejpam-5127	121	3	)	)	PUNCT
ejpam-5127	121	4	=	=	NOUN
ejpam-5127	121	5	⇒	⇒	NOUN
ejpam-5127	121	6	(	(	PUNCT
ejpam-5127	121	7	b)•	b)•	NUM
ejpam-5127	121	8	⊆	⊆	NUM
ejpam-5127	121	9	(	(	PUNCT
ejpam-5127	121	10	a)•	a)•	NUM
ejpam-5127	121	11	,	,	PUNCT
ejpam-5127	121	12	(	(	PUNCT
ejpam-5127	121	13	8)	8)	NUM
ejpam-5127	121	14	[	[	X
ejpam-5127	121	15	a	a	X
ejpam-5127	121	16	)	)	PUNCT
ejpam-5127	121	17	⊆	⊆	NUM
ejpam-5127	121	18	[	[	X
ejpam-5127	121	19	b	b	X
ejpam-5127	121	20	)	)	PUNCT
ejpam-5127	122	1	=	=	NOUN
ejpam-5127	122	2	⇒	⇒	NOUN
ejpam-5127	122	3	(	(	PUNCT
ejpam-5127	122	4	b)•	b)•	NUM
ejpam-5127	122	5	⊆	⊆	NUM
ejpam-5127	122	6	(	(	PUNCT
ejpam-5127	122	7	a)•.	a)•.	PROPN
ejpam-5127	122	8	r.	r.	PROPN
ejpam-5127	122	9	shukla	shukla	PROPN
ejpam-5127	122	10	et	et	PROPN
ejpam-5127	122	11	al	al	PROPN
ejpam-5127	122	12	.	.	PUNCT
ejpam-5127	122	13	/	/	SYM
ejpam-5127	122	14	eur	eur	PROPN
ejpam-5127	122	15	.	.	PUNCT
ejpam-5127	123	1	j.	j.	PROPN
ejpam-5127	123	2	pure	pure	PROPN
ejpam-5127	123	3	appl	appl	PROPN
ejpam-5127	123	4	.	.	PROPN
ejpam-5127	123	5	math	math	PROPN
ejpam-5127	123	6	,	,	PUNCT
ejpam-5127	123	7	17	17	NUM
ejpam-5127	123	8	(	(	PUNCT
ejpam-5127	123	9	2	2	NUM
ejpam-5127	123	10	)	)	PUNCT
ejpam-5127	123	11	(	(	PUNCT
ejpam-5127	123	12	2024	2024	NUM
ejpam-5127	123	13	)	)	PUNCT
ejpam-5127	123	14	,	,	PUNCT
ejpam-5127	123	15	1306	1306	NUM
ejpam-5127	123	16	-	-	SYM
ejpam-5127	123	17	1320	1320	NUM
ejpam-5127	123	18	1311	1311	NUM
ejpam-5127	123	19	definition	definition	NOUN
ejpam-5127	123	20	8	8	NUM
ejpam-5127	123	21	.	.	PUNCT
ejpam-5127	124	1	let	let	VERB
ejpam-5127	124	2	s	s	PRON
ejpam-5127	124	3	be	be	AUX
ejpam-5127	124	4	a	a	DET
ejpam-5127	124	5	subset	subset	NOUN
ejpam-5127	124	6	of	of	ADP
ejpam-5127	124	7	a	a	DET
ejpam-5127	124	8	pdl	pdl	NOUN
ejpam-5127	124	9	v	v	NOUN
ejpam-5127	124	10	.	.	PUNCT
ejpam-5127	125	1	then	then	ADV
ejpam-5127	125	2	we	we	PRON
ejpam-5127	125	3	define	define	VERB
ejpam-5127	125	4	o(s	o(s	PROPN
ejpam-5127	125	5	)	)	PUNCT
ejpam-5127	125	6	=	=	PRON
ejpam-5127	126	1	{	{	PUNCT
ejpam-5127	126	2	x	x	PUNCT
ejpam-5127	126	3	∈	∈	PROPN
ejpam-5127	126	4	v	v	NOUN
ejpam-5127	126	5	|a	|a	VERB
ejpam-5127	126	6	∨	∨	NOUN
ejpam-5127	126	7	x	x	SYM
ejpam-5127	126	8	=	=	SYM
ejpam-5127	126	9	1	1	NUM
ejpam-5127	126	10	for	for	ADP
ejpam-5127	126	11	some	some	PRON
ejpam-5127	126	12	a	a	DET
ejpam-5127	126	13	∈	∈	NOUN
ejpam-5127	126	14	v	v	ADP
ejpam-5127	126	15	\s	\s	NOUN
ejpam-5127	126	16	}	}	PUNCT
ejpam-5127	126	17	.	.	PUNCT
ejpam-5127	127	1	observe	observe	VERB
ejpam-5127	127	2	that	that	SCONJ
ejpam-5127	127	3	for	for	ADP
ejpam-5127	127	4	any	any	DET
ejpam-5127	127	5	non	non	ADJ
ejpam-5127	127	6	-	-	ADJ
ejpam-5127	127	7	empty	empty	ADJ
ejpam-5127	127	8	subset	subset	NOUN
ejpam-5127	127	9	s	s	NOUN
ejpam-5127	127	10	of	of	ADP
ejpam-5127	127	11	v	v	NOUN
ejpam-5127	127	12	,	,	PUNCT
ejpam-5127	127	13	o(s	o(s	PROPN
ejpam-5127	127	14	)	)	PUNCT
ejpam-5127	127	15	=	=	PUNCT
ejpam-5127	127	16	∪	∪	ADP
ejpam-5127	127	17	a∈v	a∈v	NOUN
ejpam-5127	127	18	\s	\s	PROPN
ejpam-5127	127	19	(	(	PUNCT
ejpam-5127	127	20	a)•.	a)•.	PROPN
ejpam-5127	127	21	the	the	DET
ejpam-5127	127	22	following	follow	VERB
ejpam-5127	127	23	is	be	AUX
ejpam-5127	127	24	an	an	DET
ejpam-5127	127	25	important	important	ADJ
ejpam-5127	127	26	characterization	characterization	NOUN
ejpam-5127	127	27	of	of	ADP
ejpam-5127	127	28	o(s	o(s	PROPN
ejpam-5127	127	29	)	)	PUNCT
ejpam-5127	127	30	.	.	PUNCT
ejpam-5127	128	1	lemma	lemma	PROPN
ejpam-5127	128	2	6	6	NUM
ejpam-5127	128	3	.	.	PUNCT
ejpam-5127	129	1	if	if	SCONJ
ejpam-5127	129	2	s	s	NOUN
ejpam-5127	129	3	is	be	AUX
ejpam-5127	129	4	any	any	DET
ejpam-5127	129	5	non	non	ADJ
ejpam-5127	129	6	-	-	ADJ
ejpam-5127	129	7	empty	empty	ADJ
ejpam-5127	129	8	subset	subset	NOUN
ejpam-5127	129	9	of	of	ADP
ejpam-5127	129	10	a	a	DET
ejpam-5127	129	11	pdl	pdl	NOUN
ejpam-5127	129	12	v	v	NOUN
ejpam-5127	129	13	and	and	CCONJ
ejpam-5127	129	14	x	x	ADP
ejpam-5127	129	15	∈	∈	NOUN
ejpam-5127	129	16	v	v	NOUN
ejpam-5127	129	17	,	,	PUNCT
ejpam-5127	129	18	then	then	ADV
ejpam-5127	129	19	x	x	SYM
ejpam-5127	129	20	∈	∈	PROPN
ejpam-5127	129	21	o(s	o(s	PROPN
ejpam-5127	129	22	)	)	PUNCT
ejpam-5127	129	23	if	if	SCONJ
ejpam-5127	129	24	and	and	CCONJ
ejpam-5127	129	25	only	only	ADV
ejpam-5127	129	26	if	if	SCONJ
ejpam-5127	129	27	(	(	PUNCT
ejpam-5127	129	28	x)•	x)•	PROPN
ejpam-5127	129	29	⊈	⊈	PROPN
ejpam-5127	129	30	s.	s.	PROPN
ejpam-5127	129	31	proof	proof	PROPN
ejpam-5127	129	32	.	.	PUNCT
ejpam-5127	130	1	let	let	VERB
ejpam-5127	130	2	s	s	PRON
ejpam-5127	130	3	be	be	AUX
ejpam-5127	130	4	any	any	DET
ejpam-5127	130	5	non	non	ADJ
ejpam-5127	130	6	-	-	ADJ
ejpam-5127	130	7	empty	empty	ADJ
ejpam-5127	130	8	subset	subset	NOUN
ejpam-5127	130	9	of	of	ADP
ejpam-5127	130	10	a	a	DET
ejpam-5127	130	11	pdl	pdl	NOUN
ejpam-5127	130	12	v	v	NOUN
ejpam-5127	130	13	and	and	CCONJ
ejpam-5127	130	14	x	x	NOUN
ejpam-5127	130	15	∈	∈	NOUN
ejpam-5127	130	16	v	v	NOUN
ejpam-5127	130	17	.	.	PUNCT
ejpam-5127	131	1	now	now	ADV
ejpam-5127	131	2	x	x	X
ejpam-5127	131	3	∈	∈	PROPN
ejpam-5127	131	4	o(s	o(s	PROPN
ejpam-5127	131	5	)	)	PUNCT
ejpam-5127	131	6	implies	imply	VERB
ejpam-5127	131	7	that	that	SCONJ
ejpam-5127	131	8	y	y	PROPN
ejpam-5127	131	9	∨	∨	NUM
ejpam-5127	131	10	x	x	X
ejpam-5127	132	1	=	=	SYM
ejpam-5127	132	2	1	1	NUM
ejpam-5127	132	3	for	for	ADP
ejpam-5127	132	4	some	some	DET
ejpam-5127	132	5	y	y	PROPN
ejpam-5127	132	6	/∈	/∈	PUNCT
ejpam-5127	132	7	s.	s.	PROPN
ejpam-5127	132	8	hence	hence	ADV
ejpam-5127	132	9	y	y	PROPN
ejpam-5127	132	10	∈	∈	PROPN
ejpam-5127	132	11	(	(	PUNCT
ejpam-5127	132	12	x)•	x)•	PROPN
ejpam-5127	132	13	for	for	ADP
ejpam-5127	132	14	some	some	DET
ejpam-5127	132	15	y	y	PROPN
ejpam-5127	132	16	/∈	/∈	PUNCT
ejpam-5127	132	17	s.	s.	PROPN
ejpam-5127	133	1	therefore	therefore	ADV
ejpam-5127	133	2	(	(	PUNCT
ejpam-5127	133	3	x)•	x)•	PROPN
ejpam-5127	133	4	⊈	⊈	PROPN
ejpam-5127	133	5	s.	s.	PROPN
ejpam-5127	133	6	conversely	conversely	ADV
ejpam-5127	133	7	,	,	PUNCT
ejpam-5127	133	8	(	(	PUNCT
ejpam-5127	133	9	x)•	x)•	PROPN
ejpam-5127	133	10	⊈	⊈	PROPN
ejpam-5127	133	11	s	s	PART
ejpam-5127	133	12	implies	imply	VERB
ejpam-5127	133	13	that	that	SCONJ
ejpam-5127	133	14	there	there	PRON
ejpam-5127	133	15	is	be	VERB
ejpam-5127	133	16	y	y	PROPN
ejpam-5127	133	17	∈	∈	PROPN
ejpam-5127	133	18	(	(	PUNCT
ejpam-5127	133	19	x)•	x)•	PROPN
ejpam-5127	133	20	such	such	ADJ
ejpam-5127	133	21	that	that	SCONJ
ejpam-5127	133	22	y	y	PROPN
ejpam-5127	133	23	/∈	/∈	PUNCT
ejpam-5127	133	24	s.	s.	PROPN
ejpam-5127	133	25	hence	hence	ADV
ejpam-5127	133	26	y	y	PROPN
ejpam-5127	133	27	∨	∨	NUM
ejpam-5127	133	28	x	x	SYM
ejpam-5127	133	29	=	=	SYM
ejpam-5127	133	30	1	1	NUM
ejpam-5127	133	31	and	and	CCONJ
ejpam-5127	133	32	y	y	PROPN
ejpam-5127	133	33	/∈	/∈	PUNCT
ejpam-5127	133	34	s.	s.	PROPN
ejpam-5127	133	35	therefore	therefore	ADV
ejpam-5127	133	36	,	,	PUNCT
ejpam-5127	133	37	x	x	PROPN
ejpam-5127	133	38	∈	∈	PROPN
ejpam-5127	133	39	o(s	o(s	PROPN
ejpam-5127	133	40	)	)	PUNCT
ejpam-5127	133	41	.	.	PUNCT
ejpam-5127	134	1	the	the	DET
ejpam-5127	134	2	following	follow	VERB
ejpam-5127	134	3	lemma	lemma	PROPN
ejpam-5127	134	4	can	can	AUX
ejpam-5127	134	5	be	be	AUX
ejpam-5127	134	6	proved	prove	VERB
ejpam-5127	134	7	easily	easily	ADV
ejpam-5127	134	8	.	.	PUNCT
ejpam-5127	135	1	lemma	lemma	PROPN
ejpam-5127	135	2	7	7	NUM
ejpam-5127	135	3	.	.	X
ejpam-5127	135	4	for	for	ADP
ejpam-5127	135	5	any	any	DET
ejpam-5127	135	6	prime	prime	ADJ
ejpam-5127	135	7	filter	filter	NOUN
ejpam-5127	135	8	p	p	NOUN
ejpam-5127	135	9	of	of	ADP
ejpam-5127	135	10	a	a	DET
ejpam-5127	135	11	pdl	pdl	NOUN
ejpam-5127	135	12	v	v	NOUN
ejpam-5127	135	13	,	,	PUNCT
ejpam-5127	135	14	o(p	o(p	PROPN
ejpam-5127	135	15	)	)	PUNCT
ejpam-5127	135	16	is	be	AUX
ejpam-5127	135	17	a	a	DET
ejpam-5127	135	18	filter	filter	NOUN
ejpam-5127	135	19	of	of	ADP
ejpam-5127	135	20	v	v	NOUN
ejpam-5127	135	21	and	and	CCONJ
ejpam-5127	135	22	o(p	o(p	NOUN
ejpam-5127	135	23	)	)	PUNCT
ejpam-5127	136	1	⊆	⊆	NUM
ejpam-5127	136	2	p.	p.	NOUN
ejpam-5127	136	3	lemma	lemma	PROPN
ejpam-5127	136	4	8	8	NUM
ejpam-5127	136	5	.	.	PUNCT
ejpam-5127	137	1	if	if	SCONJ
ejpam-5127	137	2	p	p	NOUN
ejpam-5127	137	3	is	be	AUX
ejpam-5127	137	4	a	a	DET
ejpam-5127	137	5	prime	prime	ADJ
ejpam-5127	137	6	filter	filter	NOUN
ejpam-5127	137	7	of	of	ADP
ejpam-5127	137	8	a	a	DET
ejpam-5127	137	9	pdl	pdl	NOUN
ejpam-5127	137	10	v	v	NOUN
ejpam-5127	137	11	,	,	PUNCT
ejpam-5127	137	12	then	then	ADV
ejpam-5127	137	13	each	each	DET
ejpam-5127	137	14	minimal	minimal	ADJ
ejpam-5127	137	15	prime	prime	ADJ
ejpam-5127	137	16	filter	filter	NOUN
ejpam-5127	137	17	belonging	belong	VERB
ejpam-5127	137	18	to	to	ADP
ejpam-5127	137	19	o(p	o(p	NUM
ejpam-5127	137	20	)	)	PUNCT
ejpam-5127	137	21	is	be	AUX
ejpam-5127	137	22	contained	contain	VERB
ejpam-5127	137	23	in	in	ADP
ejpam-5127	137	24	p.	p.	NOUN
ejpam-5127	137	25	proof	proof	NOUN
ejpam-5127	137	26	.	.	PUNCT
ejpam-5127	138	1	let	let	VERB
ejpam-5127	138	2	p	p	PRON
ejpam-5127	138	3	be	be	AUX
ejpam-5127	138	4	a	a	DET
ejpam-5127	138	5	prime	prime	ADJ
ejpam-5127	138	6	filter	filter	NOUN
ejpam-5127	138	7	of	of	ADP
ejpam-5127	138	8	a	a	DET
ejpam-5127	138	9	pdl	pdl	NOUN
ejpam-5127	138	10	v	v	NOUN
ejpam-5127	138	11	and	and	CCONJ
ejpam-5127	138	12	q	q	NOUN
ejpam-5127	138	13	be	be	AUX
ejpam-5127	138	14	any	any	DET
ejpam-5127	138	15	minimal	minimal	ADJ
ejpam-5127	138	16	prime	prime	ADJ
ejpam-5127	138	17	filter	filter	NOUN
ejpam-5127	138	18	belonging	belong	VERB
ejpam-5127	138	19	to	to	ADP
ejpam-5127	138	20	o(p	o(p	NUM
ejpam-5127	138	21	)	)	PUNCT
ejpam-5127	138	22	.	.	PUNCT
ejpam-5127	139	1	we	we	PRON
ejpam-5127	139	2	have	have	VERB
ejpam-5127	139	3	to	to	PART
ejpam-5127	139	4	prove	prove	VERB
ejpam-5127	139	5	that	that	SCONJ
ejpam-5127	139	6	q	q	PROPN
ejpam-5127	139	7	⊆	⊆	NUM
ejpam-5127	139	8	p.	p.	NOUN
ejpam-5127	139	9	suppose	suppose	VERB
ejpam-5127	139	10	that	that	SCONJ
ejpam-5127	139	11	q	q	PROPN
ejpam-5127	139	12	⊈	⊈	PROPN
ejpam-5127	139	13	p.	p.	NOUN
ejpam-5127	139	14	then	then	ADV
ejpam-5127	139	15	there	there	PRON
ejpam-5127	139	16	exists	exist	VERB
ejpam-5127	139	17	an	an	DET
ejpam-5127	139	18	element	element	NOUN
ejpam-5127	139	19	x	x	SYM
ejpam-5127	139	20	∈	∈	PROPN
ejpam-5127	139	21	q	q	NOUN
ejpam-5127	139	22	such	such	ADJ
ejpam-5127	139	23	that	that	PRON
ejpam-5127	139	24	x	x	PUNCT
ejpam-5127	139	25	/∈	/∈	PUNCT
ejpam-5127	140	1	p.	p.	NOUN
ejpam-5127	140	2	since	since	SCONJ
ejpam-5127	140	3	q	q	PROPN
ejpam-5127	140	4	is	be	AUX
ejpam-5127	140	5	a	a	DET
ejpam-5127	140	6	minimal	minimal	ADJ
ejpam-5127	140	7	prime	prime	ADJ
ejpam-5127	140	8	filter	filter	NOUN
ejpam-5127	140	9	,	,	PUNCT
ejpam-5127	140	10	there	there	PRON
ejpam-5127	140	11	is	be	VERB
ejpam-5127	140	12	an	an	DET
ejpam-5127	140	13	element	element	NOUN
ejpam-5127	140	14	y	y	NOUN
ejpam-5127	140	15	/∈	/∈	PUNCT
ejpam-5127	141	1	q	q	PROPN
ejpam-5127	141	2	such	such	ADJ
ejpam-5127	141	3	that	that	SCONJ
ejpam-5127	141	4	x	x	PROPN
ejpam-5127	141	5	∨	∨	NUM
ejpam-5127	141	6	y	y	NOUN
ejpam-5127	141	7	=	=	SYM
ejpam-5127	141	8	1	1	NUM
ejpam-5127	142	1	and	and	CCONJ
ejpam-5127	142	2	hence	hence	ADV
ejpam-5127	142	3	y	y	PROPN
ejpam-5127	142	4	∈	∈	PROPN
ejpam-5127	142	5	o(p	o(p	PROPN
ejpam-5127	142	6	)	)	PUNCT
ejpam-5127	142	7	.	.	PUNCT
ejpam-5127	143	1	since	since	SCONJ
ejpam-5127	143	2	o(p	o(p	PROPN
ejpam-5127	143	3	)	)	PUNCT
ejpam-5127	143	4	⊆	⊆	NUM
ejpam-5127	143	5	q	q	NOUN
ejpam-5127	143	6	,	,	PUNCT
ejpam-5127	143	7	we	we	PRON
ejpam-5127	143	8	get	get	VERB
ejpam-5127	143	9	y	y	PROPN
ejpam-5127	143	10	∈	∈	PROPN
ejpam-5127	143	11	q.	q.	NOUN
ejpam-5127	143	12	this	this	PRON
ejpam-5127	143	13	is	be	AUX
ejpam-5127	143	14	a	a	DET
ejpam-5127	143	15	contradiction	contradiction	NOUN
ejpam-5127	143	16	.	.	PUNCT
ejpam-5127	144	1	therefore	therefore	ADV
ejpam-5127	144	2	,	,	PUNCT
ejpam-5127	144	3	we	we	PRON
ejpam-5127	144	4	get	get	VERB
ejpam-5127	144	5	q	q	NOUN
ejpam-5127	144	6	⊆	⊆	NUM
ejpam-5127	144	7	p.	p.	NOUN
ejpam-5127	144	8	thus	thus	ADV
ejpam-5127	144	9	each	each	DET
ejpam-5127	144	10	minimal	minimal	ADJ
ejpam-5127	144	11	prime	prime	ADJ
ejpam-5127	144	12	filter	filter	NOUN
ejpam-5127	144	13	belonging	belong	VERB
ejpam-5127	144	14	to	to	ADP
ejpam-5127	144	15	o(p	o(p	NUM
ejpam-5127	144	16	)	)	PUNCT
ejpam-5127	144	17	is	be	AUX
ejpam-5127	144	18	contained	contain	VERB
ejpam-5127	144	19	in	in	ADP
ejpam-5127	144	20	p.	p.	PROPN
ejpam-5127	144	21	lemma	lemma	PROPN
ejpam-5127	145	1	9	9	X
ejpam-5127	145	2	.	.	PUNCT
ejpam-5127	145	3	let	let	VERB
ejpam-5127	145	4	p	p	PRON
ejpam-5127	145	5	be	be	AUX
ejpam-5127	145	6	a	a	DET
ejpam-5127	145	7	prime	prime	ADJ
ejpam-5127	145	8	filter	filter	NOUN
ejpam-5127	145	9	of	of	ADP
ejpam-5127	145	10	a	a	DET
ejpam-5127	145	11	pdl	pdl	NOUN
ejpam-5127	145	12	v	v	NOUN
ejpam-5127	145	13	.	.	PUNCT
ejpam-5127	146	1	then	then	ADV
ejpam-5127	146	2	x	x	SYM
ejpam-5127	146	3	∈	∈	NOUN
ejpam-5127	146	4	v	v	ADP
ejpam-5127	146	5	\p	\p	ADV
ejpam-5127	146	6	=	=	NOUN
ejpam-5127	146	7	⇒	⇒	NOUN
ejpam-5127	146	8	(	(	PUNCT
ejpam-5127	146	9	x)•	x)•	PROPN
ejpam-5127	146	10	⊆	⊆	NUM
ejpam-5127	146	11	p.	p.	NOUN
ejpam-5127	146	12	corollary	corollary	NOUN
ejpam-5127	146	13	1	1	NUM
ejpam-5127	146	14	.	.	PUNCT
ejpam-5127	147	1	let	let	VERB
ejpam-5127	147	2	p	p	PRON
ejpam-5127	147	3	be	be	AUX
ejpam-5127	147	4	a	a	DET
ejpam-5127	147	5	prime	prime	ADJ
ejpam-5127	147	6	filter	filter	NOUN
ejpam-5127	147	7	of	of	ADP
ejpam-5127	147	8	a	a	DET
ejpam-5127	147	9	pdl	pdl	NOUN
ejpam-5127	147	10	v	v	NOUN
ejpam-5127	147	11	.	.	PUNCT
ejpam-5127	148	1	then	then	ADV
ejpam-5127	148	2	x	x	SYM
ejpam-5127	148	3	∈	∈	PROPN
ejpam-5127	148	4	v	v	NUM
ejpam-5127	148	5	\o(p	\o(p	NOUN
ejpam-5127	148	6	)	)	PUNCT
ejpam-5127	148	7	if	if	SCONJ
ejpam-5127	148	8	and	and	CCONJ
ejpam-5127	148	9	only	only	ADV
ejpam-5127	148	10	if	if	SCONJ
ejpam-5127	148	11	(	(	PUNCT
ejpam-5127	148	12	x)•	x)•	PROPN
ejpam-5127	148	13	⊆	⊆	NUM
ejpam-5127	148	14	p.	p.	PROPN
ejpam-5127	148	15	lemma	lemma	PROPN
ejpam-5127	148	16	10	10	NUM
ejpam-5127	148	17	.	.	PUNCT
ejpam-5127	149	1	if	if	SCONJ
ejpam-5127	149	2	v	v	NOUN
ejpam-5127	149	3	is	be	AUX
ejpam-5127	149	4	a	a	DET
ejpam-5127	149	5	pdl	pdl	NOUN
ejpam-5127	149	6	then	then	ADV
ejpam-5127	149	7	every	every	DET
ejpam-5127	149	8	proper	proper	ADJ
ejpam-5127	149	9	filter	filter	NOUN
ejpam-5127	149	10	of	of	ADP
ejpam-5127	149	11	v	v	NOUN
ejpam-5127	149	12	is	be	AUX
ejpam-5127	149	13	contained	contain	VERB
ejpam-5127	149	14	in	in	ADP
ejpam-5127	149	15	a	a	DET
ejpam-5127	149	16	maximal	maximal	ADJ
ejpam-5127	149	17	filter	filter	NOUN
ejpam-5127	149	18	.	.	PUNCT
ejpam-5127	150	1	proof	proof	NOUN
ejpam-5127	150	2	.	.	PUNCT
ejpam-5127	151	1	let	let	VERB
ejpam-5127	151	2	f	f	PRON
ejpam-5127	151	3	be	be	AUX
ejpam-5127	151	4	a	a	DET
ejpam-5127	151	5	proper	proper	ADJ
ejpam-5127	151	6	filter	filter	NOUN
ejpam-5127	151	7	of	of	ADP
ejpam-5127	151	8	a	a	DET
ejpam-5127	151	9	pdl	pdl	NOUN
ejpam-5127	151	10	v	v	NOUN
ejpam-5127	151	11	and	and	CCONJ
ejpam-5127	151	12	s	s	NOUN
ejpam-5127	151	13	=	=	PUNCT
ejpam-5127	151	14	{	{	PUNCT
ejpam-5127	151	15	g|g	g|g	NOUN
ejpam-5127	151	16	is	be	AUX
ejpam-5127	151	17	a	a	DET
ejpam-5127	151	18	filter	filter	NOUN
ejpam-5127	151	19	of	of	ADP
ejpam-5127	151	20	v	v	NOUN
ejpam-5127	151	21	containing	contain	VERB
ejpam-5127	151	22	f	f	NOUN
ejpam-5127	151	23	}	}	PUNCT
ejpam-5127	151	24	.	.	PUNCT
ejpam-5127	152	1	clearly	clearly	ADV
ejpam-5127	152	2	f	f	PROPN
ejpam-5127	152	3	∈	∈	PROPN
ejpam-5127	152	4	s.	s.	PROPN
ejpam-5127	152	5	therefore	therefore	ADV
ejpam-5127	152	6	s	s	VERB
ejpam-5127	152	7	=	=	PUNCT
ejpam-5127	152	8	̸	̸	ADV
ejpam-5127	152	9	∅.	∅.	ADV
ejpam-5127	152	10	let	let	VERB
ejpam-5127	152	11	{	{	PUNCT
ejpam-5127	152	12	gα|α	gα|α	ADJ
ejpam-5127	152	13	∈	∈	NOUN
ejpam-5127	152	14	∆	∆	PROPN
ejpam-5127	152	15	}	}	PUNCT
ejpam-5127	152	16	be	be	AUX
ejpam-5127	152	17	a	a	DET
ejpam-5127	152	18	chain	chain	NOUN
ejpam-5127	152	19	in	in	ADP
ejpam-5127	152	20	s.	s.	PROPN
ejpam-5127	152	21	now	now	ADV
ejpam-5127	152	22	we	we	PRON
ejpam-5127	152	23	prove	prove	VERB
ejpam-5127	152	24	that	that	SCONJ
ejpam-5127	152	25	∪	∪	ADJ
ejpam-5127	152	26	α∈∆	α∈∆	PRON
ejpam-5127	152	27	gα	gα	NOUN
ejpam-5127	152	28	is	be	AUX
ejpam-5127	152	29	an	an	DET
ejpam-5127	152	30	upper	upper	ADJ
ejpam-5127	152	31	bound	bind	VERB
ejpam-5127	152	32	of	of	ADP
ejpam-5127	152	33	{	{	PUNCT
ejpam-5127	152	34	gα|α	gα|α	ADJ
ejpam-5127	152	35	∈	∈	NOUN
ejpam-5127	152	36	∆	∆	X
ejpam-5127	152	37	}	}	PUNCT
ejpam-5127	152	38	.	.	PUNCT
ejpam-5127	153	1	let	let	VERB
ejpam-5127	153	2	x	x	PRON
ejpam-5127	153	3	,	,	PUNCT
ejpam-5127	153	4	y	y	PROPN
ejpam-5127	153	5	∈	∈	PROPN
ejpam-5127	153	6	∪	∪	ADP
ejpam-5127	153	7	α∈∆	α∈∆	PRON
ejpam-5127	153	8	gα	gα	NOUN
ejpam-5127	153	9	.	.	PUNCT
ejpam-5127	154	1	then	then	ADV
ejpam-5127	154	2	x	x	SYM
ejpam-5127	154	3	∈	∈	NOUN
ejpam-5127	154	4	gi	gi	NOUN
ejpam-5127	154	5	and	and	CCONJ
ejpam-5127	154	6	y	y	PROPN
ejpam-5127	154	7	∈	∈	PROPN
ejpam-5127	154	8	gj	gj	NOUN
ejpam-5127	154	9	for	for	ADP
ejpam-5127	154	10	some	some	DET
ejpam-5127	154	11	i	i	PROPN
ejpam-5127	154	12	,	,	PUNCT
ejpam-5127	154	13	j	j	PROPN
ejpam-5127	154	14	∈	∈	PROPN
ejpam-5127	155	1	∆.	∆.	X
ejpam-5127	155	2	since	since	SCONJ
ejpam-5127	155	3	{	{	PUNCT
ejpam-5127	155	4	gα|α	gα|α	ADJ
ejpam-5127	155	5	∈	∈	NOUN
ejpam-5127	155	6	∆	∆	X
ejpam-5127	155	7	}	}	PUNCT
ejpam-5127	155	8	is	be	AUX
ejpam-5127	155	9	a	a	DET
ejpam-5127	155	10	chain	chain	NOUN
ejpam-5127	155	11	,	,	PUNCT
ejpam-5127	155	12	take	take	VERB
ejpam-5127	155	13	gi	gi	NOUN
ejpam-5127	155	14	⊆	⊆	NUM
ejpam-5127	155	15	gj	gj	NOUN
ejpam-5127	155	16	.	.	PUNCT
ejpam-5127	156	1	therefore	therefore	ADV
ejpam-5127	156	2	x	x	X
ejpam-5127	156	3	,	,	PUNCT
ejpam-5127	156	4	y	y	PROPN
ejpam-5127	156	5	∈	∈	PROPN
ejpam-5127	156	6	gj	gj	NOUN
ejpam-5127	156	7	.	.	PUNCT
ejpam-5127	157	1	since	since	SCONJ
ejpam-5127	157	2	gj	gj	PROPN
ejpam-5127	157	3	is	be	AUX
ejpam-5127	157	4	a	a	DET
ejpam-5127	157	5	filter	filter	NOUN
ejpam-5127	157	6	of	of	ADP
ejpam-5127	157	7	v	v	NOUN
ejpam-5127	157	8	,	,	PUNCT
ejpam-5127	157	9	x∨y	x∨y	PROPN
ejpam-5127	157	10	∈	∈	PROPN
ejpam-5127	157	11	gj	gj	PROPN
ejpam-5127	157	12	⊆	⊆	NOUN
ejpam-5127	157	13	∪	∪	ADP
ejpam-5127	157	14	α∈∆	α∈∆	PRON
ejpam-5127	157	15	gα	gα	NOUN
ejpam-5127	157	16	.	.	NOUN
ejpam-5127	157	17	again	again	ADV
ejpam-5127	157	18	,	,	PUNCT
ejpam-5127	157	19	let	let	VERB
ejpam-5127	157	20	x	x	PUNCT
ejpam-5127	157	21	∈	∈	PROPN
ejpam-5127	157	22	∪	∪	ADP
ejpam-5127	157	23	α∈∆	α∈∆	NOUN
ejpam-5127	157	24	gα	gα	NOUN
ejpam-5127	157	25	and	and	CCONJ
ejpam-5127	157	26	r	r	NOUN
ejpam-5127	157	27	∈	∈	PROPN
ejpam-5127	157	28	v	v	NOUN
ejpam-5127	157	29	.	.	PUNCT
ejpam-5127	158	1	then	then	ADV
ejpam-5127	158	2	x	x	SYM
ejpam-5127	158	3	∈	∈	NOUN
ejpam-5127	158	4	gα	gα	ADP
ejpam-5127	158	5	for	for	ADP
ejpam-5127	158	6	some	some	DET
ejpam-5127	158	7	r.	r.	PROPN
ejpam-5127	158	8	shukla	shukla	PROPN
ejpam-5127	158	9	et	et	PROPN
ejpam-5127	158	10	al	al	PROPN
ejpam-5127	158	11	.	.	PUNCT
ejpam-5127	158	12	/	/	SYM
ejpam-5127	158	13	eur	eur	PROPN
ejpam-5127	158	14	.	.	PUNCT
ejpam-5127	159	1	j.	j.	PROPN
ejpam-5127	159	2	pure	pure	PROPN
ejpam-5127	159	3	appl	appl	PROPN
ejpam-5127	159	4	.	.	PROPN
ejpam-5127	159	5	math	math	PROPN
ejpam-5127	159	6	,	,	PUNCT
ejpam-5127	159	7	17	17	NUM
ejpam-5127	159	8	(	(	PUNCT
ejpam-5127	159	9	2	2	NUM
ejpam-5127	159	10	)	)	PUNCT
ejpam-5127	159	11	(	(	PUNCT
ejpam-5127	159	12	2024	2024	NUM
ejpam-5127	159	13	)	)	PUNCT
ejpam-5127	159	14	,	,	PUNCT
ejpam-5127	159	15	1306	1306	NUM
ejpam-5127	159	16	-	-	SYM
ejpam-5127	159	17	1320	1320	NUM
ejpam-5127	159	18	1312	1312	NUM
ejpam-5127	159	19	α	α	NOUN
ejpam-5127	159	20	∈	∈	PROPN
ejpam-5127	160	1	∆.	∆.	X
ejpam-5127	160	2	therefore	therefore	ADV
ejpam-5127	160	3	,	,	PUNCT
ejpam-5127	160	4	x	x	PUNCT
ejpam-5127	160	5	∧	∧	NOUN
ejpam-5127	160	6	r	r	NOUN
ejpam-5127	160	7	∈	∈	NOUN
ejpam-5127	160	8	gα	gα	ADP
ejpam-5127	160	9	⊆	⊆	NUM
ejpam-5127	160	10	∪	∪	ADP
ejpam-5127	160	11	α∈∆	α∈∆	PRON
ejpam-5127	160	12	gα	gα	NOUN
ejpam-5127	160	13	.	.	PUNCT
ejpam-5127	161	1	since	since	SCONJ
ejpam-5127	161	2	f	f	PROPN
ejpam-5127	161	3	⊆	⊆	NUM
ejpam-5127	161	4	gα	gα	NOUN
ejpam-5127	161	5	for	for	ADP
ejpam-5127	161	6	each	each	DET
ejpam-5127	161	7	α	α	NOUN
ejpam-5127	161	8	,	,	PUNCT
ejpam-5127	161	9	we	we	PRON
ejpam-5127	161	10	get	get	VERB
ejpam-5127	161	11	f	f	NOUN
ejpam-5127	161	12	⊆	⊆	NOUN
ejpam-5127	161	13	∪	∪	ADP
ejpam-5127	161	14	α∈∆	α∈∆	PRON
ejpam-5127	161	15	gα	gα	NOUN
ejpam-5127	161	16	.	.	PUNCT
ejpam-5127	162	1	therefore	therefore	ADV
ejpam-5127	162	2	∪α∈∆	∪α∈∆	NOUN
ejpam-5127	162	3	gα	gα	NOUN
ejpam-5127	162	4	is	be	AUX
ejpam-5127	162	5	an	an	DET
ejpam-5127	162	6	upper	upper	ADJ
ejpam-5127	162	7	bound	bind	VERB
ejpam-5127	162	8	of	of	ADP
ejpam-5127	162	9	{	{	PUNCT
ejpam-5127	162	10	gα|α	gα|α	ADJ
ejpam-5127	162	11	∈	∈	NOUN
ejpam-5127	162	12	∆	∆	NOUN
ejpam-5127	162	13	}	}	PUNCT
ejpam-5127	162	14	in	in	ADP
ejpam-5127	162	15	s.	s.	PROPN
ejpam-5127	162	16	thus	thus	ADV
ejpam-5127	162	17	by	by	ADP
ejpam-5127	162	18	zorn	zorn	PROPN
ejpam-5127	162	19	’s	’s	PART
ejpam-5127	162	20	lemma	lemma	PROPN
ejpam-5127	162	21	s	s	PROPN
ejpam-5127	162	22	has	have	VERB
ejpam-5127	162	23	a	a	DET
ejpam-5127	162	24	maximal	maximal	ADJ
ejpam-5127	162	25	element	element	NOUN
ejpam-5127	162	26	.	.	PUNCT
ejpam-5127	163	1	in	in	ADP
ejpam-5127	163	2	the	the	DET
ejpam-5127	163	3	lattice	lattice	PROPN
ejpam-5127	163	4	theory	theory	NOUN
ejpam-5127	163	5	,	,	PUNCT
ejpam-5127	163	6	due	due	ADP
ejpam-5127	163	7	to	to	PART
ejpam-5127	163	8	lattice	lattice	VERB
ejpam-5127	163	9	theoretic	theoretic	ADJ
ejpam-5127	163	10	duality	duality	NOUN
ejpam-5127	163	11	principle	principle	NOUN
ejpam-5127	163	12	,	,	PUNCT
ejpam-5127	163	13	those	those	DET
ejpam-5127	163	14	results	result	NOUN
ejpam-5127	163	15	which	which	PRON
ejpam-5127	163	16	are	be	AUX
ejpam-5127	163	17	valid	valid	ADJ
ejpam-5127	163	18	for	for	ADP
ejpam-5127	163	19	filters	filter	NOUN
ejpam-5127	163	20	are	be	AUX
ejpam-5127	163	21	also	also	ADV
ejpam-5127	163	22	valid	valid	ADJ
ejpam-5127	163	23	for	for	ADP
ejpam-5127	163	24	ideals	ideal	NOUN
ejpam-5127	163	25	.	.	PUNCT
ejpam-5127	164	1	but	but	CCONJ
ejpam-5127	164	2	the	the	DET
ejpam-5127	164	3	dual	dual	ADJ
ejpam-5127	164	4	of	of	ADP
ejpam-5127	164	5	a	a	DET
ejpam-5127	164	6	pdl	pdl	NOUN
ejpam-5127	164	7	need	need	NOUN
ejpam-5127	164	8	not	not	PART
ejpam-5127	164	9	be	be	AUX
ejpam-5127	164	10	a	a	DET
ejpam-5127	164	11	pdl	pdl	NOUN
ejpam-5127	164	12	.	.	PUNCT
ejpam-5127	165	1	for	for	ADP
ejpam-5127	165	2	this	this	DET
ejpam-5127	165	3	reason	reason	NOUN
ejpam-5127	165	4	,	,	PUNCT
ejpam-5127	165	5	it	it	PRON
ejpam-5127	165	6	is	be	AUX
ejpam-5127	165	7	necessary	necessary	ADJ
ejpam-5127	165	8	to	to	PART
ejpam-5127	165	9	provide	provide	VERB
ejpam-5127	165	10	proofs	proof	NOUN
ejpam-5127	165	11	for	for	ADP
ejpam-5127	165	12	similar	similar	ADJ
ejpam-5127	165	13	results	result	NOUN
ejpam-5127	165	14	on	on	ADP
ejpam-5127	165	15	ideals	ideal	NOUN
ejpam-5127	165	16	.	.	PUNCT
ejpam-5127	166	1	lemma	lemma	PROPN
ejpam-5127	166	2	11	11	NUM
ejpam-5127	166	3	.	.	PUNCT
ejpam-5127	167	1	a	a	DET
ejpam-5127	167	2	subset	subset	NOUN
ejpam-5127	167	3	p	p	NOUN
ejpam-5127	167	4	of	of	ADP
ejpam-5127	167	5	a	a	DET
ejpam-5127	167	6	pdl	pdl	NOUN
ejpam-5127	167	7	v	v	NOUN
ejpam-5127	167	8	is	be	AUX
ejpam-5127	167	9	a	a	DET
ejpam-5127	167	10	prime	prime	ADJ
ejpam-5127	167	11	filter	filter	NOUN
ejpam-5127	167	12	if	if	SCONJ
ejpam-5127	167	13	and	and	CCONJ
ejpam-5127	167	14	only	only	ADV
ejpam-5127	167	15	if	if	SCONJ
ejpam-5127	167	16	v	v	NOUN
ejpam-5127	167	17	\p	\p	ADV
ejpam-5127	167	18	is	be	AUX
ejpam-5127	167	19	a	a	DET
ejpam-5127	167	20	prime	prime	ADJ
ejpam-5127	167	21	ideal	ideal	NOUN
ejpam-5127	167	22	.	.	PUNCT
ejpam-5127	168	1	proof	proof	NOUN
ejpam-5127	168	2	.	.	PUNCT
ejpam-5127	169	1	let	let	VERB
ejpam-5127	169	2	v	v	PART
ejpam-5127	169	3	be	be	AUX
ejpam-5127	169	4	a	a	DET
ejpam-5127	169	5	pdl	pdl	NOUN
ejpam-5127	169	6	and	and	CCONJ
ejpam-5127	169	7	p	p	NOUN
ejpam-5127	169	8	⊆	⊆	NUM
ejpam-5127	169	9	v	v	NOUN
ejpam-5127	169	10	.	.	PUNCT
ejpam-5127	170	1	assume	assume	VERB
ejpam-5127	170	2	that	that	SCONJ
ejpam-5127	170	3	p	p	NOUN
ejpam-5127	170	4	is	be	AUX
ejpam-5127	170	5	a	a	DET
ejpam-5127	170	6	prime	prime	ADJ
ejpam-5127	170	7	filter	filter	NOUN
ejpam-5127	170	8	of	of	ADP
ejpam-5127	170	9	v	v	NOUN
ejpam-5127	170	10	.	.	PUNCT
ejpam-5127	171	1	we	we	PRON
ejpam-5127	171	2	have	have	VERB
ejpam-5127	171	3	to	to	PART
ejpam-5127	171	4	prove	prove	VERB
ejpam-5127	171	5	that	that	SCONJ
ejpam-5127	171	6	v	v	NOUN
ejpam-5127	171	7	\p	\p	ADV
ejpam-5127	171	8	is	be	AUX
ejpam-5127	171	9	a	a	DET
ejpam-5127	171	10	prime	prime	ADJ
ejpam-5127	171	11	ideal	ideal	NOUN
ejpam-5127	171	12	of	of	ADP
ejpam-5127	171	13	v	v	NOUN
ejpam-5127	171	14	.	.	PUNCT
ejpam-5127	172	1	let	let	VERB
ejpam-5127	172	2	x	x	PRON
ejpam-5127	172	3	,	,	PUNCT
ejpam-5127	172	4	y	y	PROPN
ejpam-5127	172	5	∈	∈	PROPN
ejpam-5127	172	6	v	v	ADP
ejpam-5127	172	7	\p	\p	ADV
ejpam-5127	172	8	.	.	PUNCT
ejpam-5127	173	1	then	then	ADV
ejpam-5127	173	2	x	x	X
ejpam-5127	173	3	/∈	/∈	PUNCT
ejpam-5127	174	1	p	p	NOUN
ejpam-5127	174	2	and	and	CCONJ
ejpam-5127	174	3	y	y	PROPN
ejpam-5127	174	4	/∈	/∈	PUNCT
ejpam-5127	175	1	p.	p.	NOUN
ejpam-5127	175	2	since	since	SCONJ
ejpam-5127	175	3	p	p	PROPN
ejpam-5127	175	4	is	be	AUX
ejpam-5127	175	5	prime	prime	ADJ
ejpam-5127	175	6	,	,	PUNCT
ejpam-5127	175	7	we	we	PRON
ejpam-5127	175	8	get	get	VERB
ejpam-5127	175	9	x	x	PUNCT
ejpam-5127	175	10	∨	∨	NUM
ejpam-5127	175	11	y	y	PROPN
ejpam-5127	175	12	/∈	/∈	PUNCT
ejpam-5127	176	1	p.	p.	NOUN
ejpam-5127	176	2	therefore	therefore	ADV
ejpam-5127	176	3	,	,	PUNCT
ejpam-5127	176	4	x	x	PROPN
ejpam-5127	176	5	∨	∨	NUM
ejpam-5127	176	6	y	y	PROPN
ejpam-5127	176	7	∈	∈	PROPN
ejpam-5127	176	8	v	v	ADP
ejpam-5127	176	9	\p	\p	ADV
ejpam-5127	176	10	.	.	PUNCT
ejpam-5127	177	1	again	again	ADV
ejpam-5127	177	2	,	,	PUNCT
ejpam-5127	177	3	let	let	VERB
ejpam-5127	177	4	a	a	DET
ejpam-5127	177	5	∈	∈	NOUN
ejpam-5127	177	6	v	v	NOUN
ejpam-5127	177	7	and	and	CCONJ
ejpam-5127	177	8	x	x	ADP
ejpam-5127	177	9	∈	∈	PROPN
ejpam-5127	177	10	v	v	ADP
ejpam-5127	177	11	\p	\p	ADV
ejpam-5127	177	12	.	.	PUNCT
ejpam-5127	178	1	if	if	SCONJ
ejpam-5127	178	2	x	x	PUNCT
ejpam-5127	178	3	∧	∧	NOUN
ejpam-5127	178	4	a	a	DET
ejpam-5127	178	5	∈	∈	NOUN
ejpam-5127	179	1	p	p	NOUN
ejpam-5127	179	2	then	then	ADV
ejpam-5127	179	3	x	x	X
ejpam-5127	179	4	∨	∨	NOUN
ejpam-5127	179	5	(	(	PUNCT
ejpam-5127	179	6	x	x	PROPN
ejpam-5127	179	7	∧	∧	NOUN
ejpam-5127	179	8	a	a	DET
ejpam-5127	179	9	)	)	PUNCT
ejpam-5127	179	10	∈	∈	PROPN
ejpam-5127	179	11	p	p	NOUN
ejpam-5127	179	12	and	and	CCONJ
ejpam-5127	179	13	hence	hence	ADV
ejpam-5127	179	14	x	x	PUNCT
ejpam-5127	179	15	∈	∈	NOUN
ejpam-5127	180	1	p.	p.	NOUN
ejpam-5127	181	1	but	but	CCONJ
ejpam-5127	181	2	x	x	X
ejpam-5127	181	3	/∈	/∈	PUNCT
ejpam-5127	182	1	p.	p.	NOUN
ejpam-5127	182	2	therefore	therefore	ADV
ejpam-5127	182	3	,	,	PUNCT
ejpam-5127	182	4	x	x	PROPN
ejpam-5127	182	5	∧	∧	PROPN
ejpam-5127	182	6	a	a	PRON
ejpam-5127	182	7	/∈	/∈	NOUN
ejpam-5127	182	8	p	p	NOUN
ejpam-5127	182	9	and	and	CCONJ
ejpam-5127	182	10	hence	hence	ADV
ejpam-5127	182	11	a	a	DET
ejpam-5127	182	12	∧	∧	PROPN
ejpam-5127	182	13	x	x	SYM
ejpam-5127	182	14	∈	∈	PROPN
ejpam-5127	182	15	v	v	ADP
ejpam-5127	182	16	\p	\p	ADV
ejpam-5127	182	17	.	.	PUNCT
ejpam-5127	183	1	thus	thus	ADV
ejpam-5127	183	2	v	v	NOUN
ejpam-5127	183	3	\p	\p	ADV
ejpam-5127	183	4	is	be	AUX
ejpam-5127	183	5	an	an	DET
ejpam-5127	183	6	ideal	ideal	NOUN
ejpam-5127	183	7	.	.	PUNCT
ejpam-5127	184	1	now	now	ADV
ejpam-5127	184	2	we	we	PRON
ejpam-5127	184	3	prove	prove	VERB
ejpam-5127	184	4	that	that	SCONJ
ejpam-5127	184	5	v	v	NOUN
ejpam-5127	184	6	\p	\p	ADV
ejpam-5127	184	7	is	be	AUX
ejpam-5127	184	8	a	a	DET
ejpam-5127	184	9	prime	prime	ADJ
ejpam-5127	184	10	ideal	ideal	NOUN
ejpam-5127	184	11	.	.	PUNCT
ejpam-5127	185	1	let	let	VERB
ejpam-5127	185	2	x	x	PRON
ejpam-5127	185	3	,	,	PUNCT
ejpam-5127	185	4	y	y	PROPN
ejpam-5127	185	5	∈	∈	PROPN
ejpam-5127	185	6	v	v	ADP
ejpam-5127	185	7	such	such	ADJ
ejpam-5127	185	8	that	that	SCONJ
ejpam-5127	185	9	x	x	PUNCT
ejpam-5127	185	10	∧	∧	NOUN
ejpam-5127	185	11	y	y	PROPN
ejpam-5127	185	12	∈	∈	PROPN
ejpam-5127	185	13	v	v	ADP
ejpam-5127	185	14	\p	\p	ADV
ejpam-5127	185	15	.	.	PUNCT
ejpam-5127	186	1	suppose	suppose	VERB
ejpam-5127	186	2	that	that	SCONJ
ejpam-5127	186	3	x	x	X
ejpam-5127	186	4	/∈	/∈	PUNCT
ejpam-5127	187	1	v	v	INTJ
ejpam-5127	187	2	\p	\p	ADV
ejpam-5127	187	3	and	and	CCONJ
ejpam-5127	187	4	y	y	PROPN
ejpam-5127	187	5	/∈	/∈	PROPN
ejpam-5127	188	1	v	v	ADP
ejpam-5127	188	2	\p	\p	ADV
ejpam-5127	188	3	.	.	PUNCT
ejpam-5127	189	1	then	then	ADV
ejpam-5127	189	2	x	x	SYM
ejpam-5127	189	3	∈	∈	PROPN
ejpam-5127	189	4	p	p	NOUN
ejpam-5127	189	5	and	and	CCONJ
ejpam-5127	189	6	y	y	PROPN
ejpam-5127	189	7	∈	∈	PROPN
ejpam-5127	189	8	p.	p.	NOUN
ejpam-5127	189	9	since	since	SCONJ
ejpam-5127	189	10	p	p	PROPN
ejpam-5127	189	11	is	be	AUX
ejpam-5127	189	12	a	a	DET
ejpam-5127	189	13	filter	filter	NOUN
ejpam-5127	189	14	,	,	PUNCT
ejpam-5127	189	15	x	x	PUNCT
ejpam-5127	189	16	∧	∧	NOUN
ejpam-5127	189	17	y	y	PROPN
ejpam-5127	189	18	∈	∈	PROPN
ejpam-5127	189	19	p	p	NOUN
ejpam-5127	189	20	and	and	CCONJ
ejpam-5127	189	21	hence	hence	ADV
ejpam-5127	189	22	x	x	PART
ejpam-5127	189	23	∧	∧	NOUN
ejpam-5127	189	24	y	y	PROPN
ejpam-5127	189	25	/∈	/∈	PROPN
ejpam-5127	189	26	v	v	ADP
ejpam-5127	189	27	\p	\p	ADV
ejpam-5127	189	28	.	.	PUNCT
ejpam-5127	190	1	this	this	PRON
ejpam-5127	190	2	is	be	AUX
ejpam-5127	190	3	a	a	DET
ejpam-5127	190	4	contradiction	contradiction	NOUN
ejpam-5127	190	5	.	.	PUNCT
ejpam-5127	191	1	therefore	therefore	ADV
ejpam-5127	191	2	,	,	PUNCT
ejpam-5127	191	3	we	we	PRON
ejpam-5127	191	4	get	get	VERB
ejpam-5127	191	5	either	either	CCONJ
ejpam-5127	191	6	x	x	SYM
ejpam-5127	191	7	∈	∈	PROPN
ejpam-5127	191	8	v	v	ADP
ejpam-5127	191	9	\p	\p	ADV
ejpam-5127	191	10	or	or	CCONJ
ejpam-5127	191	11	y	y	PROPN
ejpam-5127	191	12	∈	∈	PROPN
ejpam-5127	191	13	v	v	ADP
ejpam-5127	191	14	\p	\p	NOUN
ejpam-5127	191	15	.	.	PUNCT
ejpam-5127	192	1	hence	hence	ADV
ejpam-5127	192	2	v	v	NOUN
ejpam-5127	192	3	\p	\p	ADV
ejpam-5127	192	4	is	be	AUX
ejpam-5127	192	5	a	a	DET
ejpam-5127	192	6	prime	prime	ADJ
ejpam-5127	192	7	ideal	ideal	NOUN
ejpam-5127	192	8	of	of	ADP
ejpam-5127	192	9	v	v	NOUN
ejpam-5127	192	10	.	.	PUNCT
ejpam-5127	193	1	conversely	conversely	ADV
ejpam-5127	193	2	,	,	PUNCT
ejpam-5127	193	3	assume	assume	VERB
ejpam-5127	193	4	v	v	ADP
ejpam-5127	193	5	\p	\p	ADV
ejpam-5127	193	6	is	be	AUX
ejpam-5127	193	7	a	a	DET
ejpam-5127	193	8	prime	prime	ADJ
ejpam-5127	193	9	ideal	ideal	NOUN
ejpam-5127	193	10	of	of	ADP
ejpam-5127	193	11	v	v	NOUN
ejpam-5127	193	12	.	.	PUNCT
ejpam-5127	194	1	first	first	ADV
ejpam-5127	194	2	we	we	PRON
ejpam-5127	194	3	prove	prove	VERB
ejpam-5127	194	4	that	that	SCONJ
ejpam-5127	194	5	p	p	NOUN
ejpam-5127	194	6	is	be	AUX
ejpam-5127	194	7	a	a	DET
ejpam-5127	194	8	filter	filter	NOUN
ejpam-5127	194	9	of	of	ADP
ejpam-5127	194	10	v	v	NOUN
ejpam-5127	194	11	.	.	PUNCT
ejpam-5127	195	1	let	let	VERB
ejpam-5127	195	2	x	x	PRON
ejpam-5127	195	3	,	,	PUNCT
ejpam-5127	195	4	y	y	PROPN
ejpam-5127	195	5	∈	∈	PROPN
ejpam-5127	196	1	p.	p.	NOUN
ejpam-5127	196	2	then	then	ADV
ejpam-5127	197	1	x	x	X
ejpam-5127	197	2	/∈	/∈	PUNCT
ejpam-5127	198	1	v	v	INTJ
ejpam-5127	198	2	\p	\p	ADV
ejpam-5127	198	3	and	and	CCONJ
ejpam-5127	198	4	y	y	PROPN
ejpam-5127	198	5	/∈	/∈	PROPN
ejpam-5127	198	6	v	v	ADP
ejpam-5127	198	7	\p	\p	ADV
ejpam-5127	198	8	.	.	PUNCT
ejpam-5127	199	1	since	since	SCONJ
ejpam-5127	199	2	v	v	NOUN
ejpam-5127	199	3	\p	\p	ADV
ejpam-5127	199	4	is	be	AUX
ejpam-5127	199	5	a	a	DET
ejpam-5127	199	6	prime	prime	ADJ
ejpam-5127	199	7	ideal	ideal	NOUN
ejpam-5127	199	8	,	,	PUNCT
ejpam-5127	199	9	x	x	PUNCT
ejpam-5127	199	10	∧	∧	NOUN
ejpam-5127	199	11	y	y	PROPN
ejpam-5127	199	12	/∈	/∈	PROPN
ejpam-5127	199	13	v	v	ADP
ejpam-5127	199	14	\p	\p	ADV
ejpam-5127	199	15	.	.	PUNCT
ejpam-5127	200	1	therefore	therefore	ADV
ejpam-5127	200	2	,	,	PUNCT
ejpam-5127	200	3	x	x	PUNCT
ejpam-5127	200	4	∧	∧	NOUN
ejpam-5127	200	5	y	y	PROPN
ejpam-5127	200	6	∈	∈	PROPN
ejpam-5127	200	7	p.	p.	NOUN
ejpam-5127	200	8	again	again	ADV
ejpam-5127	200	9	,	,	PUNCT
ejpam-5127	200	10	let	let	VERB
ejpam-5127	200	11	a	a	DET
ejpam-5127	200	12	∈	∈	NOUN
ejpam-5127	200	13	v	v	NOUN
ejpam-5127	200	14	and	and	CCONJ
ejpam-5127	200	15	x	x	PUNCT
ejpam-5127	200	16	∈	∈	PROPN
ejpam-5127	200	17	p.	p.	NOUN
ejpam-5127	200	18	suppose	suppose	VERB
ejpam-5127	200	19	a	a	DET
ejpam-5127	200	20	∨	∨	NOUN
ejpam-5127	200	21	x	x	SYM
ejpam-5127	200	22	∈	∈	PROPN
ejpam-5127	200	23	v	v	ADP
ejpam-5127	200	24	\p	\p	ADV
ejpam-5127	200	25	.	.	PUNCT
ejpam-5127	201	1	then	then	ADV
ejpam-5127	201	2	(	(	PUNCT
ejpam-5127	201	3	a	a	DET
ejpam-5127	201	4	∨	∨	NUM
ejpam-5127	201	5	x	x	NOUN
ejpam-5127	201	6	)	)	PUNCT
ejpam-5127	201	7	∧	∧	NOUN
ejpam-5127	201	8	x	x	SYM
ejpam-5127	201	9	∈	∈	PROPN
ejpam-5127	201	10	v	v	ADP
ejpam-5127	201	11	\p	\p	ADV
ejpam-5127	201	12	and	and	CCONJ
ejpam-5127	201	13	hence	hence	ADV
ejpam-5127	201	14	x	x	PART
ejpam-5127	201	15	∈	∈	NOUN
ejpam-5127	201	16	v	v	ADP
ejpam-5127	201	17	\p	\p	ADV
ejpam-5127	201	18	.	.	PUNCT
ejpam-5127	202	1	but	but	CCONJ
ejpam-5127	202	2	x	x	X
ejpam-5127	202	3	/∈	/∈	PUNCT
ejpam-5127	203	1	v	v	ADP
ejpam-5127	203	2	\p	\p	ADV
ejpam-5127	203	3	.	.	PUNCT
ejpam-5127	204	1	therefore	therefore	ADV
ejpam-5127	204	2	,	,	PUNCT
ejpam-5127	204	3	a	a	DET
ejpam-5127	204	4	∨	∨	NOUN
ejpam-5127	204	5	x	x	SYM
ejpam-5127	204	6	/∈	/∈	NOUN
ejpam-5127	204	7	v	v	NOUN
ejpam-5127	204	8	\p	\p	ADV
ejpam-5127	204	9	.	.	PUNCT
ejpam-5127	205	1	thus	thus	ADV
ejpam-5127	205	2	a	a	DET
ejpam-5127	205	3	∨	∨	NOUN
ejpam-5127	205	4	x	x	SYM
ejpam-5127	205	5	∈	∈	PROPN
ejpam-5127	205	6	p.	p.	NOUN
ejpam-5127	205	7	therefore	therefore	ADV
ejpam-5127	205	8	,	,	PUNCT
ejpam-5127	205	9	p	p	PROPN
ejpam-5127	205	10	is	be	AUX
ejpam-5127	205	11	a	a	DET
ejpam-5127	205	12	filter	filter	NOUN
ejpam-5127	205	13	of	of	ADP
ejpam-5127	205	14	v	v	NOUN
ejpam-5127	205	15	.	.	PUNCT
ejpam-5127	206	1	now	now	ADV
ejpam-5127	206	2	,	,	PUNCT
ejpam-5127	206	3	we	we	PRON
ejpam-5127	206	4	prove	prove	VERB
ejpam-5127	206	5	that	that	SCONJ
ejpam-5127	206	6	p	p	NOUN
ejpam-5127	206	7	is	be	AUX
ejpam-5127	206	8	a	a	DET
ejpam-5127	206	9	prime	prime	ADJ
ejpam-5127	206	10	filter	filter	NOUN
ejpam-5127	206	11	of	of	ADP
ejpam-5127	206	12	v	v	NOUN
ejpam-5127	206	13	.	.	PUNCT
ejpam-5127	207	1	let	let	VERB
ejpam-5127	207	2	x	x	PUNCT
ejpam-5127	207	3	∨	∨	PROPN
ejpam-5127	207	4	y	y	PROPN
ejpam-5127	207	5	∈	∈	PROPN
ejpam-5127	207	6	p.	p.	NOUN
ejpam-5127	207	7	suppose	suppose	VERB
ejpam-5127	208	1	x	x	X
ejpam-5127	208	2	/∈	/∈	PUNCT
ejpam-5127	209	1	p	p	NOUN
ejpam-5127	209	2	and	and	CCONJ
ejpam-5127	209	3	y	y	PROPN
ejpam-5127	209	4	/∈	/∈	PUNCT
ejpam-5127	210	1	p.	p.	NOUN
ejpam-5127	210	2	then	then	ADV
ejpam-5127	210	3	x	x	SYM
ejpam-5127	210	4	∈	∈	PROPN
ejpam-5127	210	5	v	v	ADP
ejpam-5127	210	6	\p	\p	ADV
ejpam-5127	210	7	and	and	CCONJ
ejpam-5127	210	8	y	y	PROPN
ejpam-5127	210	9	∈	∈	PROPN
ejpam-5127	210	10	v	v	ADP
ejpam-5127	210	11	\p	\p	ADV
ejpam-5127	210	12	.	.	PUNCT
ejpam-5127	211	1	since	since	SCONJ
ejpam-5127	211	2	v	v	NOUN
ejpam-5127	211	3	\p	\p	ADV
ejpam-5127	211	4	is	be	AUX
ejpam-5127	211	5	an	an	DET
ejpam-5127	211	6	ideal	ideal	NOUN
ejpam-5127	211	7	,	,	PUNCT
ejpam-5127	211	8	x∨y	x∨y	PROPN
ejpam-5127	211	9	∈	∈	PROPN
ejpam-5127	211	10	v	v	ADP
ejpam-5127	211	11	\p	\p	ADV
ejpam-5127	211	12	and	and	CCONJ
ejpam-5127	211	13	hence	hence	ADV
ejpam-5127	211	14	x∨y	x∨y	PROPN
ejpam-5127	211	15	/∈	/∈	PUNCT
ejpam-5127	212	1	p.	p.	NOUN
ejpam-5127	212	2	this	this	PRON
ejpam-5127	212	3	is	be	AUX
ejpam-5127	212	4	a	a	DET
ejpam-5127	212	5	contradiction	contradiction	NOUN
ejpam-5127	212	6	.	.	PUNCT
ejpam-5127	213	1	thus	thus	ADV
ejpam-5127	213	2	we	we	PRON
ejpam-5127	213	3	get	get	VERB
ejpam-5127	213	4	either	either	CCONJ
ejpam-5127	213	5	x	x	SYM
ejpam-5127	213	6	∈	∈	PROPN
ejpam-5127	213	7	p	p	NOUN
ejpam-5127	213	8	or	or	CCONJ
ejpam-5127	213	9	y	y	PROPN
ejpam-5127	213	10	∈	∈	PROPN
ejpam-5127	214	1	p.	p.	NOUN
ejpam-5127	214	2	therefore	therefore	ADV
ejpam-5127	214	3	p	p	PROPN
ejpam-5127	214	4	is	be	AUX
ejpam-5127	214	5	a	a	DET
ejpam-5127	214	6	prime	prime	ADJ
ejpam-5127	214	7	filter	filter	NOUN
ejpam-5127	214	8	.	.	PUNCT
ejpam-5127	215	1	theorem	theorem	VERB
ejpam-5127	215	2	6	6	NUM
ejpam-5127	215	3	.	.	PUNCT
ejpam-5127	216	1	in	in	ADP
ejpam-5127	216	2	a	a	DET
ejpam-5127	216	3	pdl	pdl	NOUN
ejpam-5127	216	4	,	,	PUNCT
ejpam-5127	216	5	every	every	DET
ejpam-5127	216	6	maximal	maximal	ADJ
ejpam-5127	216	7	ideal	ideal	NOUN
ejpam-5127	216	8	is	be	AUX
ejpam-5127	216	9	a	a	DET
ejpam-5127	216	10	prime	prime	ADJ
ejpam-5127	216	11	ideal	ideal	NOUN
ejpam-5127	216	12	.	.	PUNCT
ejpam-5127	217	1	proof	proof	NOUN
ejpam-5127	217	2	.	.	PUNCT
ejpam-5127	218	1	let	let	VERB
ejpam-5127	218	2	v	v	PART
ejpam-5127	218	3	be	be	AUX
ejpam-5127	218	4	a	a	DET
ejpam-5127	218	5	pdl	pdl	NOUN
ejpam-5127	218	6	and	and	CCONJ
ejpam-5127	218	7	g	g	NOUN
ejpam-5127	218	8	be	be	AUX
ejpam-5127	218	9	a	a	DET
ejpam-5127	218	10	maximal	maximal	ADJ
ejpam-5127	218	11	ideal	ideal	NOUN
ejpam-5127	218	12	of	of	ADP
ejpam-5127	218	13	v	v	NOUN
ejpam-5127	218	14	.	.	PUNCT
ejpam-5127	219	1	we	we	PRON
ejpam-5127	219	2	have	have	VERB
ejpam-5127	219	3	to	to	PART
ejpam-5127	219	4	prove	prove	VERB
ejpam-5127	219	5	that	that	SCONJ
ejpam-5127	219	6	g	g	PROPN
ejpam-5127	219	7	is	be	AUX
ejpam-5127	219	8	a	a	DET
ejpam-5127	219	9	prime	prime	ADJ
ejpam-5127	219	10	ideal	ideal	NOUN
ejpam-5127	219	11	of	of	ADP
ejpam-5127	219	12	v	v	NOUN
ejpam-5127	219	13	.	.	PUNCT
ejpam-5127	220	1	let	let	VERB
ejpam-5127	220	2	x	x	PRON
ejpam-5127	220	3	,	,	PUNCT
ejpam-5127	220	4	y	y	PROPN
ejpam-5127	220	5	∈	∈	PROPN
ejpam-5127	220	6	v	v	NOUN
ejpam-5127	220	7	and	and	CCONJ
ejpam-5127	220	8	x	x	PROPN
ejpam-5127	220	9	∧	∧	PROPN
ejpam-5127	220	10	y	y	PROPN
ejpam-5127	220	11	∈	∈	PROPN
ejpam-5127	220	12	g.	g.	NOUN
ejpam-5127	220	13	we	we	PRON
ejpam-5127	220	14	have	have	VERB
ejpam-5127	220	15	to	to	PART
ejpam-5127	220	16	prove	prove	VERB
ejpam-5127	220	17	that	that	SCONJ
ejpam-5127	220	18	x	x	PUNCT
ejpam-5127	220	19	∈	∈	PROPN
ejpam-5127	220	20	g	g	NOUN
ejpam-5127	220	21	or	or	CCONJ
ejpam-5127	220	22	y	y	PROPN
ejpam-5127	220	23	∈	∈	PROPN
ejpam-5127	220	24	g.	g.	PROPN
ejpam-5127	220	25	suppose	suppose	VERB
ejpam-5127	220	26	x	x	X
ejpam-5127	220	27	/∈	/∈	VERB
ejpam-5127	221	1	g	g	NOUN
ejpam-5127	221	2	,	,	PUNCT
ejpam-5127	221	3	y	y	PROPN
ejpam-5127	221	4	/∈	/∈	PUNCT
ejpam-5127	222	1	g	g	NOUN
ejpam-5127	222	2	,	,	PUNCT
ejpam-5127	222	3	we	we	PRON
ejpam-5127	222	4	get	get	VERB
ejpam-5127	222	5	g	g	NOUN
ejpam-5127	222	6	∨	∨	NOUN
ejpam-5127	222	7	(	(	PUNCT
ejpam-5127	222	8	x	x	X
ejpam-5127	222	9	]	]	X
ejpam-5127	222	10	=	=	SYM
ejpam-5127	222	11	v	v	NOUN
ejpam-5127	222	12	and	and	CCONJ
ejpam-5127	222	13	g	g	PROPN
ejpam-5127	222	14	∨	∨	NUM
ejpam-5127	222	15	(	(	PUNCT
ejpam-5127	222	16	y	y	NOUN
ejpam-5127	222	17	]	]	X
ejpam-5127	222	18	=	=	SYM
ejpam-5127	222	19	v	v	NOUN
ejpam-5127	222	20	.	.	PUNCT
ejpam-5127	223	1	now	now	ADV
ejpam-5127	223	2	1	1	NUM
ejpam-5127	223	3	∈	∈	NOUN
ejpam-5127	223	4	v	v	NOUN
ejpam-5127	223	5	=	=	NOUN
ejpam-5127	223	6	⇒	⇒	ADJ
ejpam-5127	223	7	1	1	NUM
ejpam-5127	223	8	=	=	SYM
ejpam-5127	223	9	s1	s1	PROPN
ejpam-5127	223	10	∨	∨	NUM
ejpam-5127	223	11	t	t	PROPN
ejpam-5127	223	12	for	for	ADP
ejpam-5127	223	13	some	some	DET
ejpam-5127	223	14	s1	s1	PROPN
ejpam-5127	223	15	∈	∈	PROPN
ejpam-5127	223	16	g	g	NOUN
ejpam-5127	223	17	and	and	CCONJ
ejpam-5127	223	18	t	t	PROPN
ejpam-5127	223	19	∈	∈	PROPN
ejpam-5127	223	20	(	(	PUNCT
ejpam-5127	223	21	x	x	X
ejpam-5127	223	22	]	]	X
ejpam-5127	223	23	=	=	SYM
ejpam-5127	223	24	⇒	⇒	NOUN
ejpam-5127	223	25	1	1	NUM
ejpam-5127	223	26	=	=	SYM
ejpam-5127	223	27	s1	s1	PROPN
ejpam-5127	223	28	∨	∨	X
ejpam-5127	223	29	(	(	PUNCT
ejpam-5127	223	30	x	x	PROPN
ejpam-5127	223	31	∧	∧	PROPN
ejpam-5127	223	32	t	t	PROPN
ejpam-5127	223	33	)	)	PUNCT
ejpam-5127	224	1	=	=	VERB
ejpam-5127	224	2	⇒	⇒	NOUN
ejpam-5127	224	3	1	1	NUM
ejpam-5127	224	4	=	=	SYM
ejpam-5127	224	5	(	(	PUNCT
ejpam-5127	224	6	s1	s1	PROPN
ejpam-5127	224	7	∨	∨	NUM
ejpam-5127	224	8	x	x	NOUN
ejpam-5127	224	9	)	)	PUNCT
ejpam-5127	224	10	∧	∧	PROPN
ejpam-5127	224	11	(	(	PUNCT
ejpam-5127	224	12	s1	s1	PROPN
ejpam-5127	224	13	∨	∨	NUM
ejpam-5127	224	14	t	t	PROPN
ejpam-5127	224	15	)	)	PUNCT
ejpam-5127	224	16	=	=	VERB
ejpam-5127	224	17	⇒	⇒	NOUN
ejpam-5127	224	18	1	1	NUM
ejpam-5127	224	19	=	=	SYM
ejpam-5127	224	20	s1	s1	PROPN
ejpam-5127	224	21	∨	∨	NUM
ejpam-5127	224	22	x.	x.	NOUN
ejpam-5127	224	23	similarly	similarly	ADV
ejpam-5127	224	24	,	,	PUNCT
ejpam-5127	224	25	we	we	PRON
ejpam-5127	224	26	can	can	AUX
ejpam-5127	224	27	find	find	VERB
ejpam-5127	224	28	some	some	DET
ejpam-5127	224	29	s2	s2	NOUN
ejpam-5127	224	30	∈	∈	PROPN
ejpam-5127	224	31	g	g	NOUN
ejpam-5127	224	32	such	such	ADJ
ejpam-5127	224	33	that	that	DET
ejpam-5127	224	34	1	1	NUM
ejpam-5127	224	35	=	=	SYM
ejpam-5127	224	36	s2	s2	PROPN
ejpam-5127	224	37	∨	∨	NUM
ejpam-5127	224	38	y.	y.	PROPN
ejpam-5127	224	39	now	now	ADV
ejpam-5127	224	40	(	(	PUNCT
ejpam-5127	224	41	x	x	PROPN
ejpam-5127	224	42	∧	∧	PROPN
ejpam-5127	224	43	y	y	PROPN
ejpam-5127	224	44	)	)	PUNCT
ejpam-5127	224	45	∨	∨	NOUN
ejpam-5127	224	46	s1	s1	PROPN
ejpam-5127	224	47	∨	∨	NUM
ejpam-5127	224	48	s2	s2	NOUN
ejpam-5127	224	49	=	=	SYM
ejpam-5127	224	50	(	(	PUNCT
ejpam-5127	224	51	x	x	PROPN
ejpam-5127	224	52	∨	∨	NUM
ejpam-5127	224	53	s1	s1	PROPN
ejpam-5127	224	54	∨	∨	NUM
ejpam-5127	224	55	s2	s2	PROPN
ejpam-5127	224	56	)	)	PUNCT
ejpam-5127	224	57	∧	∧	PROPN
ejpam-5127	224	58	(	(	PUNCT
ejpam-5127	224	59	y	y	PROPN
ejpam-5127	224	60	∨	∨	NUM
ejpam-5127	224	61	s1	s1	PROPN
ejpam-5127	224	62	∨	∨	NUM
ejpam-5127	224	63	s2	s2	PROPN
ejpam-5127	224	64	)	)	PUNCT
ejpam-5127	224	65	=	=	SYM
ejpam-5127	225	1	1	1	X
ejpam-5127	225	2	.	.	PUNCT
ejpam-5127	225	3	since	since	SCONJ
ejpam-5127	225	4	,	,	PUNCT
ejpam-5127	225	5	s1	s1	NOUN
ejpam-5127	225	6	,	,	PUNCT
ejpam-5127	225	7	s2	s2	NOUN
ejpam-5127	225	8	∈	∈	PROPN
ejpam-5127	225	9	g	g	PROPN
ejpam-5127	225	10	and	and	CCONJ
ejpam-5127	225	11	x	x	PROPN
ejpam-5127	225	12	∧	∧	NOUN
ejpam-5127	225	13	y	y	PROPN
ejpam-5127	225	14	∈	∈	PROPN
ejpam-5127	225	15	g	g	PROPN
ejpam-5127	225	16	,	,	PUNCT
ejpam-5127	225	17	we	we	PRON
ejpam-5127	225	18	get	get	VERB
ejpam-5127	225	19	(	(	PUNCT
ejpam-5127	225	20	x	x	SYM
ejpam-5127	225	21	∧	∧	PROPN
ejpam-5127	225	22	y	y	PROPN
ejpam-5127	225	23	)	)	PUNCT
ejpam-5127	225	24	∨	∨	NOUN
ejpam-5127	225	25	s1	s1	PROPN
ejpam-5127	225	26	∨	∨	NUM
ejpam-5127	225	27	s2	s2	NOUN
ejpam-5127	225	28	∈	∈	PROPN
ejpam-5127	225	29	g	g	NOUN
ejpam-5127	225	30	and	and	CCONJ
ejpam-5127	225	31	hence	hence	ADV
ejpam-5127	225	32	1	1	NUM
ejpam-5127	225	33	∈	∈	NOUN
ejpam-5127	225	34	g.	g.	NOUN
ejpam-5127	225	35	this	this	PRON
ejpam-5127	225	36	is	be	AUX
ejpam-5127	225	37	a	a	DET
ejpam-5127	225	38	contradiction	contradiction	NOUN
ejpam-5127	225	39	.	.	PUNCT
ejpam-5127	226	1	therefore	therefore	ADV
ejpam-5127	226	2	x	x	X
ejpam-5127	226	3	∈	∈	PROPN
ejpam-5127	226	4	g	g	NOUN
ejpam-5127	226	5	or	or	CCONJ
ejpam-5127	226	6	y	y	PROPN
ejpam-5127	226	7	∈	∈	PROPN
ejpam-5127	226	8	g.	g.	NOUN
ejpam-5127	226	9	hence	hence	ADV
ejpam-5127	226	10	g	g	PROPN
ejpam-5127	226	11	is	be	AUX
ejpam-5127	226	12	a	a	DET
ejpam-5127	226	13	prime	prime	ADJ
ejpam-5127	226	14	ideal	ideal	NOUN
ejpam-5127	226	15	of	of	ADP
ejpam-5127	226	16	v	v	NOUN
ejpam-5127	226	17	.	.	PUNCT
ejpam-5127	227	1	r.	r.	PROPN
ejpam-5127	227	2	shukla	shukla	PROPN
ejpam-5127	227	3	et	et	PROPN
ejpam-5127	227	4	al	al	PROPN
ejpam-5127	227	5	.	.	PUNCT
ejpam-5127	227	6	/	/	SYM
ejpam-5127	227	7	eur	eur	PROPN
ejpam-5127	227	8	.	.	PUNCT
ejpam-5127	228	1	j.	j.	PROPN
ejpam-5127	228	2	pure	pure	PROPN
ejpam-5127	228	3	appl	appl	PROPN
ejpam-5127	228	4	.	.	PROPN
ejpam-5127	228	5	math	math	PROPN
ejpam-5127	228	6	,	,	PUNCT
ejpam-5127	228	7	17	17	NUM
ejpam-5127	228	8	(	(	PUNCT
ejpam-5127	228	9	2	2	NUM
ejpam-5127	228	10	)	)	PUNCT
ejpam-5127	228	11	(	(	PUNCT
ejpam-5127	228	12	2024	2024	NUM
ejpam-5127	228	13	)	)	PUNCT
ejpam-5127	228	14	,	,	PUNCT
ejpam-5127	228	15	1306	1306	NUM
ejpam-5127	228	16	-	-	SYM
ejpam-5127	228	17	1320	1320	NUM
ejpam-5127	228	18	1313	1313	NUM
ejpam-5127	228	19	theorem	theorem	VERB
ejpam-5127	228	20	7	7	NUM
ejpam-5127	228	21	.	.	PUNCT
ejpam-5127	229	1	in	in	ADP
ejpam-5127	229	2	a	a	DET
ejpam-5127	229	3	relatively	relatively	ADV
ejpam-5127	229	4	complemented	complement	VERB
ejpam-5127	229	5	pdl	pdl	PROPN
ejpam-5127	229	6	v	v	NOUN
ejpam-5127	229	7	,	,	PUNCT
ejpam-5127	229	8	every	every	DET
ejpam-5127	229	9	prime	prime	ADJ
ejpam-5127	229	10	ideal	ideal	NOUN
ejpam-5127	229	11	of	of	ADP
ejpam-5127	229	12	v	v	NOUN
ejpam-5127	229	13	is	be	AUX
ejpam-5127	229	14	a	a	DET
ejpam-5127	229	15	maximal	maximal	ADJ
ejpam-5127	229	16	ideal	ideal	NOUN
ejpam-5127	229	17	.	.	PUNCT
ejpam-5127	230	1	proof	proof	NOUN
ejpam-5127	230	2	.	.	PUNCT
ejpam-5127	231	1	let	let	VERB
ejpam-5127	231	2	v	v	PART
ejpam-5127	231	3	be	be	AUX
ejpam-5127	231	4	relatively	relatively	ADV
ejpam-5127	231	5	complemented	complement	VERB
ejpam-5127	231	6	pdl	pdl	NOUN
ejpam-5127	231	7	and	and	CCONJ
ejpam-5127	231	8	f	f	PROPN
ejpam-5127	231	9	be	be	AUX
ejpam-5127	231	10	any	any	DET
ejpam-5127	231	11	prime	prime	ADJ
ejpam-5127	231	12	ideal	ideal	NOUN
ejpam-5127	231	13	of	of	ADP
ejpam-5127	231	14	v	v	NOUN
ejpam-5127	231	15	.	.	PUNCT
ejpam-5127	232	1	let	let	VERB
ejpam-5127	232	2	g	g	PRON
ejpam-5127	232	3	be	be	AUX
ejpam-5127	232	4	an	an	DET
ejpam-5127	232	5	ideal	ideal	NOUN
ejpam-5127	232	6	of	of	ADP
ejpam-5127	232	7	v	v	NOUN
ejpam-5127	232	8	such	such	ADJ
ejpam-5127	232	9	that	that	SCONJ
ejpam-5127	232	10	f	f	PROPN
ejpam-5127	232	11	⊂	⊂	PROPN
ejpam-5127	232	12	g.	g.	PROPN
ejpam-5127	232	13	since	since	SCONJ
ejpam-5127	232	14	f	f	PROPN
ejpam-5127	232	15	̸=	̸=	PROPN
ejpam-5127	232	16	g	g	PROPN
ejpam-5127	232	17	,	,	PUNCT
ejpam-5127	232	18	there	there	PRON
ejpam-5127	232	19	exists	exist	VERB
ejpam-5127	232	20	an	an	DET
ejpam-5127	232	21	element	element	NOUN
ejpam-5127	232	22	x	x	PUNCT
ejpam-5127	232	23	∈	∈	PROPN
ejpam-5127	232	24	g	g	NOUN
ejpam-5127	232	25	such	such	ADJ
ejpam-5127	232	26	that	that	PRON
ejpam-5127	232	27	x	x	PROPN
ejpam-5127	232	28	/∈	/∈	PROPN
ejpam-5127	233	1	f	f	PROPN
ejpam-5127	233	2	.	.	PUNCT
ejpam-5127	234	1	now	now	ADV
ejpam-5127	234	2	,	,	PUNCT
ejpam-5127	234	3	chose	choose	VERB
ejpam-5127	234	4	t	t	PROPN
ejpam-5127	234	5	∈	∈	PROPN
ejpam-5127	234	6	f	f	PROPN
ejpam-5127	234	7	and	and	CCONJ
ejpam-5127	234	8	consider	consider	VERB
ejpam-5127	234	9	the	the	DET
ejpam-5127	234	10	interval	interval	NOUN
ejpam-5127	235	1	i	i	PRON
ejpam-5127	235	2	=	=	PUNCT
ejpam-5127	236	1	[	[	X
ejpam-5127	236	2	t	t	X
ejpam-5127	236	3	∧	∧	PROPN
ejpam-5127	236	4	x	x	X
ejpam-5127	236	5	,	,	PUNCT
ejpam-5127	236	6	1	1	NUM
ejpam-5127	236	7	]	]	PUNCT
ejpam-5127	236	8	.	.	PUNCT
ejpam-5127	237	1	since	since	SCONJ
ejpam-5127	237	2	v	v	NOUN
ejpam-5127	237	3	is	be	AUX
ejpam-5127	237	4	a	a	DET
ejpam-5127	237	5	relatively	relatively	ADV
ejpam-5127	237	6	complemented	complemented	ADJ
ejpam-5127	237	7	pdl	pdl	NOUN
ejpam-5127	237	8	,	,	PUNCT
ejpam-5127	237	9	i	i	PRON
ejpam-5127	237	10	is	be	AUX
ejpam-5127	237	11	a	a	DET
ejpam-5127	237	12	dually	dually	ADV
ejpam-5127	237	13	complemented	complement	VERB
ejpam-5127	237	14	lattice	lattice	NOUN
ejpam-5127	237	15	and	and	CCONJ
ejpam-5127	237	16	x	x	SYM
ejpam-5127	237	17	∈	∈	PROPN
ejpam-5127	237	18	i.	i.	NOUN
ejpam-5127	237	19	therefore	therefore	ADV
ejpam-5127	237	20	there	there	PRON
ejpam-5127	237	21	exists	exist	VERB
ejpam-5127	237	22	some	some	DET
ejpam-5127	237	23	z	z	NOUN
ejpam-5127	237	24	∈	∈	PROPN
ejpam-5127	237	25	i	i	PRON
ejpam-5127	237	26	such	such	ADJ
ejpam-5127	237	27	that	that	SCONJ
ejpam-5127	237	28	z	z	NOUN
ejpam-5127	237	29	∨	∨	NOUN
ejpam-5127	237	30	x	x	SYM
ejpam-5127	237	31	=	=	SYM
ejpam-5127	237	32	1	1	NUM
ejpam-5127	237	33	and	and	CCONJ
ejpam-5127	237	34	z	z	NOUN
ejpam-5127	237	35	∧	∧	NOUN
ejpam-5127	237	36	x	x	PUNCT
ejpam-5127	238	1	=	=	SYM
ejpam-5127	238	2	t	t	PROPN
ejpam-5127	238	3	∧	∧	PROPN
ejpam-5127	238	4	x.	x.	NOUN
ejpam-5127	238	5	since	since	SCONJ
ejpam-5127	238	6	f	f	PROPN
ejpam-5127	238	7	is	be	AUX
ejpam-5127	238	8	an	an	DET
ejpam-5127	238	9	ideal	ideal	NOUN
ejpam-5127	238	10	and	and	CCONJ
ejpam-5127	238	11	t	t	NOUN
ejpam-5127	238	12	∈	∈	PROPN
ejpam-5127	238	13	f	f	PROPN
ejpam-5127	238	14	,	,	PUNCT
ejpam-5127	238	15	we	we	PRON
ejpam-5127	238	16	get	get	VERB
ejpam-5127	238	17	t	t	NOUN
ejpam-5127	238	18	∧	∧	NOUN
ejpam-5127	238	19	x	x	SYM
ejpam-5127	238	20	∈	∈	PROPN
ejpam-5127	238	21	f	f	NOUN
ejpam-5127	238	22	and	and	CCONJ
ejpam-5127	238	23	hence	hence	ADV
ejpam-5127	238	24	z	z	NOUN
ejpam-5127	238	25	∧	∧	NOUN
ejpam-5127	238	26	x	x	SYM
ejpam-5127	238	27	∈	∈	PROPN
ejpam-5127	238	28	f	f	X
ejpam-5127	238	29	.	.	PUNCT
ejpam-5127	239	1	this	this	PRON
ejpam-5127	239	2	gives	give	VERB
ejpam-5127	239	3	z	z	PROPN
ejpam-5127	239	4	∈	∈	PROPN
ejpam-5127	239	5	f	f	NOUN
ejpam-5127	239	6	or	or	CCONJ
ejpam-5127	239	7	x	x	PROPN
ejpam-5127	239	8	∈	∈	PROPN
ejpam-5127	239	9	f	f	X
ejpam-5127	239	10	.	.	PUNCT
ejpam-5127	240	1	but	but	CCONJ
ejpam-5127	240	2	x	x	X
ejpam-5127	240	3	/∈	/∈	PROPN
ejpam-5127	241	1	f	f	PROPN
ejpam-5127	241	2	.	.	PUNCT
ejpam-5127	242	1	therefore	therefore	ADV
ejpam-5127	242	2	we	we	PRON
ejpam-5127	242	3	get	get	VERB
ejpam-5127	242	4	z	z	NOUN
ejpam-5127	242	5	∈	∈	PROPN
ejpam-5127	242	6	f	f	PROPN
ejpam-5127	242	7	⊂	⊂	PROPN
ejpam-5127	242	8	g.	g.	PROPN
ejpam-5127	243	1	since	since	SCONJ
ejpam-5127	243	2	x	x	PROPN
ejpam-5127	243	3	∈	∈	PROPN
ejpam-5127	243	4	g	g	PROPN
ejpam-5127	243	5	and	and	CCONJ
ejpam-5127	243	6	z	z	PROPN
ejpam-5127	243	7	∈	∈	PROPN
ejpam-5127	243	8	g	g	NOUN
ejpam-5127	243	9	,	,	PUNCT
ejpam-5127	243	10	we	we	PRON
ejpam-5127	243	11	get	get	VERB
ejpam-5127	243	12	z	z	NOUN
ejpam-5127	243	13	∨	∨	NUM
ejpam-5127	243	14	x	x	SYM
ejpam-5127	243	15	∈	∈	PROPN
ejpam-5127	243	16	g.	g.	NOUN
ejpam-5127	243	17	this	this	PRON
ejpam-5127	243	18	gives	give	VERB
ejpam-5127	243	19	1	1	NUM
ejpam-5127	243	20	∈	∈	NOUN
ejpam-5127	243	21	g.	g.	NOUN
ejpam-5127	243	22	since	since	SCONJ
ejpam-5127	243	23	g	g	PROPN
ejpam-5127	243	24	is	be	AUX
ejpam-5127	243	25	an	an	DET
ejpam-5127	243	26	ideal	ideal	ADJ
ejpam-5127	243	27	and	and	CCONJ
ejpam-5127	243	28	1	1	NUM
ejpam-5127	243	29	∈	∈	NOUN
ejpam-5127	243	30	g	g	NOUN
ejpam-5127	243	31	,	,	PUNCT
ejpam-5127	243	32	we	we	PRON
ejpam-5127	243	33	get	get	VERB
ejpam-5127	243	34	g	g	NOUN
ejpam-5127	243	35	=	=	SYM
ejpam-5127	243	36	v	v	NOUN
ejpam-5127	243	37	.	.	PUNCT
ejpam-5127	244	1	therefore	therefore	ADV
ejpam-5127	244	2	f	f	PROPN
ejpam-5127	244	3	is	be	AUX
ejpam-5127	244	4	a	a	DET
ejpam-5127	244	5	maximal	maximal	ADJ
ejpam-5127	244	6	ideal	ideal	NOUN
ejpam-5127	244	7	of	of	ADP
ejpam-5127	244	8	v	v	NUM
ejpam-5127	244	9	.	.	PUNCT
ejpam-5127	245	1	theorem	theorem	ADJ
ejpam-5127	245	2	8	8	NUM
ejpam-5127	245	3	.	.	PUNCT
ejpam-5127	246	1	let	let	VERB
ejpam-5127	246	2	v	v	PART
ejpam-5127	246	3	be	be	AUX
ejpam-5127	246	4	a	a	DET
ejpam-5127	246	5	pdl	pdl	NOUN
ejpam-5127	246	6	and	and	CCONJ
ejpam-5127	246	7	p	p	NOUN
ejpam-5127	246	8	be	be	AUX
ejpam-5127	246	9	a	a	DET
ejpam-5127	246	10	prime	prime	ADJ
ejpam-5127	246	11	filter	filter	NOUN
ejpam-5127	246	12	of	of	ADP
ejpam-5127	246	13	v	v	NOUN
ejpam-5127	246	14	.	.	PUNCT
ejpam-5127	247	1	then	then	ADV
ejpam-5127	247	2	p	p	PROPN
ejpam-5127	247	3	is	be	AUX
ejpam-5127	247	4	a	a	DET
ejpam-5127	247	5	minimal	minimal	ADJ
ejpam-5127	247	6	prime	prime	ADJ
ejpam-5127	247	7	filter	filter	NOUN
ejpam-5127	247	8	of	of	ADP
ejpam-5127	247	9	v	v	NOUN
ejpam-5127	247	10	if	if	SCONJ
ejpam-5127	248	1	and	and	CCONJ
ejpam-5127	248	2	only	only	ADV
ejpam-5127	248	3	if	if	SCONJ
ejpam-5127	248	4	v	v	NOUN
ejpam-5127	248	5	\p	\p	ADV
ejpam-5127	248	6	is	be	AUX
ejpam-5127	248	7	a	a	DET
ejpam-5127	248	8	maximal	maximal	ADJ
ejpam-5127	248	9	prime	prime	ADJ
ejpam-5127	248	10	ideal	ideal	NOUN
ejpam-5127	248	11	of	of	ADP
ejpam-5127	248	12	v	v	NOUN
ejpam-5127	248	13	.	.	PUNCT
ejpam-5127	249	1	proof	proof	NOUN
ejpam-5127	249	2	.	.	PUNCT
ejpam-5127	250	1	assume	assume	VERB
ejpam-5127	250	2	that	that	SCONJ
ejpam-5127	250	3	p	p	NOUN
ejpam-5127	250	4	is	be	AUX
ejpam-5127	250	5	a	a	DET
ejpam-5127	250	6	minimal	minimal	ADJ
ejpam-5127	250	7	prime	prime	ADJ
ejpam-5127	250	8	filter	filter	NOUN
ejpam-5127	250	9	of	of	ADP
ejpam-5127	250	10	v	v	NOUN
ejpam-5127	250	11	.	.	PUNCT
ejpam-5127	251	1	since	since	SCONJ
ejpam-5127	251	2	p	p	NOUN
ejpam-5127	251	3	is	be	AUX
ejpam-5127	251	4	a	a	DET
ejpam-5127	251	5	prime	prime	ADJ
ejpam-5127	251	6	filter	filter	NOUN
ejpam-5127	251	7	of	of	ADP
ejpam-5127	251	8	v	v	NUM
ejpam-5127	251	9	,	,	PUNCT
ejpam-5127	251	10	v	v	NOUN
ejpam-5127	251	11	\p	\p	ADV
ejpam-5127	251	12	is	be	AUX
ejpam-5127	251	13	a	a	DET
ejpam-5127	251	14	prime	prime	ADJ
ejpam-5127	251	15	ideal	ideal	NOUN
ejpam-5127	251	16	of	of	ADP
ejpam-5127	251	17	v	v	NOUN
ejpam-5127	251	18	.	.	PUNCT
ejpam-5127	252	1	now	now	ADV
ejpam-5127	252	2	we	we	PRON
ejpam-5127	252	3	prove	prove	VERB
ejpam-5127	252	4	that	that	SCONJ
ejpam-5127	252	5	v	v	NOUN
ejpam-5127	252	6	\p	\p	ADV
ejpam-5127	252	7	is	be	AUX
ejpam-5127	252	8	maximal	maximal	ADJ
ejpam-5127	252	9	ideal	ideal	NOUN
ejpam-5127	252	10	of	of	ADP
ejpam-5127	252	11	v	v	NOUN
ejpam-5127	252	12	.	.	PUNCT
ejpam-5127	253	1	suppose	suppose	VERB
ejpam-5127	253	2	v	v	NOUN
ejpam-5127	253	3	\p	\p	ADV
ejpam-5127	253	4	is	be	AUX
ejpam-5127	253	5	not	not	PART
ejpam-5127	253	6	a	a	DET
ejpam-5127	253	7	maximal	maximal	ADJ
ejpam-5127	253	8	ideal	ideal	NOUN
ejpam-5127	253	9	of	of	ADP
ejpam-5127	253	10	v	v	NOUN
ejpam-5127	253	11	.	.	PUNCT
ejpam-5127	254	1	then	then	ADV
ejpam-5127	254	2	there	there	PRON
ejpam-5127	254	3	is	be	VERB
ejpam-5127	254	4	a	a	DET
ejpam-5127	254	5	maximal	maximal	ADJ
ejpam-5127	254	6	ideal	ideal	NOUN
ejpam-5127	254	7	(	(	PUNCT
ejpam-5127	254	8	prime	prime	ADJ
ejpam-5127	254	9	ideal	ideal	NOUN
ejpam-5127	254	10	)	)	PUNCT
ejpam-5127	254	11	g	g	NOUN
ejpam-5127	254	12	in	in	ADP
ejpam-5127	254	13	v	v	NUM
ejpam-5127	254	14	such	such	DET
ejpam-5127	254	15	that	that	DET
ejpam-5127	254	16	v	v	NOUN
ejpam-5127	254	17	\p	\p	ADV
ejpam-5127	254	18	⊂	⊂	PROPN
ejpam-5127	254	19	g	g	PROPN
ejpam-5127	254	20	⊆	⊆	NUM
ejpam-5127	254	21	v	v	NOUN
ejpam-5127	254	22	.	.	PUNCT
ejpam-5127	255	1	this	this	PRON
ejpam-5127	255	2	gives	give	VERB
ejpam-5127	255	3	v	v	ADP
ejpam-5127	255	4	\g	\g	NOUN
ejpam-5127	255	5	⊂	⊂	PROPN
ejpam-5127	256	1	p.	p.	NOUN
ejpam-5127	256	2	that	that	PRON
ejpam-5127	256	3	is	be	AUX
ejpam-5127	256	4	v	v	NOUN
ejpam-5127	256	5	\g	\g	NOUN
ejpam-5127	256	6	is	be	AUX
ejpam-5127	256	7	a	a	DET
ejpam-5127	256	8	prime	prime	ADJ
ejpam-5127	256	9	filter	filter	NOUN
ejpam-5127	256	10	of	of	ADP
ejpam-5127	256	11	v	v	NOUN
ejpam-5127	256	12	contained	contain	VERB
ejpam-5127	256	13	in	in	ADP
ejpam-5127	256	14	p.	p.	NOUN
ejpam-5127	256	15	since	since	SCONJ
ejpam-5127	256	16	p	p	NOUN
ejpam-5127	256	17	is	be	AUX
ejpam-5127	256	18	minimal	minimal	ADJ
ejpam-5127	256	19	prime	prime	ADJ
ejpam-5127	256	20	filter	filter	NOUN
ejpam-5127	256	21	of	of	ADP
ejpam-5127	256	22	v	v	NUM
ejpam-5127	256	23	,	,	PUNCT
ejpam-5127	256	24	v	v	NOUN
ejpam-5127	256	25	\g	\g	NOUN
ejpam-5127	257	1	⊂	⊂	PROPN
ejpam-5127	257	2	p	p	X
ejpam-5127	257	3	is	be	AUX
ejpam-5127	257	4	not	not	PART
ejpam-5127	257	5	possible	possible	ADJ
ejpam-5127	257	6	.	.	PUNCT
ejpam-5127	258	1	therefore	therefore	ADV
ejpam-5127	258	2	,	,	PUNCT
ejpam-5127	258	3	v	v	NOUN
ejpam-5127	258	4	\p	\p	ADV
ejpam-5127	258	5	is	be	AUX
ejpam-5127	258	6	a	a	DET
ejpam-5127	258	7	maximal	maximal	ADJ
ejpam-5127	258	8	ideal	ideal	NOUN
ejpam-5127	258	9	of	of	ADP
ejpam-5127	258	10	v	v	NOUN
ejpam-5127	258	11	.	.	PUNCT
ejpam-5127	259	1	conversely	conversely	ADV
ejpam-5127	259	2	,	,	PUNCT
ejpam-5127	259	3	assume	assume	VERB
ejpam-5127	259	4	that	that	SCONJ
ejpam-5127	259	5	v	v	NOUN
ejpam-5127	259	6	\p	\p	ADV
ejpam-5127	259	7	is	be	AUX
ejpam-5127	259	8	a	a	DET
ejpam-5127	259	9	maximal	maximal	ADJ
ejpam-5127	259	10	prime	prime	ADJ
ejpam-5127	259	11	ideal	ideal	NOUN
ejpam-5127	259	12	of	of	ADP
ejpam-5127	259	13	v	v	NOUN
ejpam-5127	259	14	.	.	PUNCT
ejpam-5127	260	1	then	then	ADV
ejpam-5127	260	2	p	p	PROPN
ejpam-5127	260	3	is	be	AUX
ejpam-5127	260	4	a	a	DET
ejpam-5127	260	5	prime	prime	ADJ
ejpam-5127	260	6	filter	filter	NOUN
ejpam-5127	260	7	of	of	ADP
ejpam-5127	260	8	v	v	NOUN
ejpam-5127	260	9	.	.	PUNCT
ejpam-5127	261	1	now	now	ADV
ejpam-5127	261	2	suppose	suppose	VERB
ejpam-5127	261	3	q	q	NOUN
ejpam-5127	261	4	is	be	AUX
ejpam-5127	261	5	any	any	DET
ejpam-5127	261	6	prime	prime	ADJ
ejpam-5127	261	7	filter	filter	NOUN
ejpam-5127	261	8	of	of	ADP
ejpam-5127	261	9	v	v	NOUN
ejpam-5127	261	10	such	such	ADJ
ejpam-5127	261	11	that	that	PRON
ejpam-5127	261	12	q	q	PROPN
ejpam-5127	262	1	⊂	⊂	PROPN
ejpam-5127	262	2	p.	p.	NOUN
ejpam-5127	262	3	then	then	ADV
ejpam-5127	262	4	v	v	ADP
ejpam-5127	262	5	\p	\p	ADV
ejpam-5127	263	1	⊂	⊂	PROPN
ejpam-5127	263	2	v	v	ADP
ejpam-5127	263	3	\q	\q	NOUN
ejpam-5127	263	4	.	.	PUNCT
ejpam-5127	264	1	that	that	PRON
ejpam-5127	264	2	is	be	AUX
ejpam-5127	264	3	,	,	PUNCT
ejpam-5127	264	4	the	the	DET
ejpam-5127	264	5	prime	prime	ADJ
ejpam-5127	264	6	ideal	ideal	NOUN
ejpam-5127	264	7	v	v	ADP
ejpam-5127	264	8	\q	\q	NOUN
ejpam-5127	264	9	is	be	AUX
ejpam-5127	264	10	containing	contain	VERB
ejpam-5127	264	11	the	the	DET
ejpam-5127	264	12	maximal	maximal	ADJ
ejpam-5127	264	13	prime	prime	ADJ
ejpam-5127	264	14	ideal	ideal	NOUN
ejpam-5127	264	15	v	v	ADP
ejpam-5127	264	16	\p	\p	NOUN
ejpam-5127	264	17	of	of	ADP
ejpam-5127	264	18	v	v	NOUN
ejpam-5127	264	19	.	.	PUNCT
ejpam-5127	265	1	this	this	PRON
ejpam-5127	265	2	is	be	AUX
ejpam-5127	265	3	a	a	DET
ejpam-5127	265	4	contradiction	contradiction	NOUN
ejpam-5127	265	5	.	.	PUNCT
ejpam-5127	266	1	therefore	therefore	ADV
ejpam-5127	266	2	p	p	PROPN
ejpam-5127	266	3	is	be	AUX
ejpam-5127	266	4	a	a	DET
ejpam-5127	266	5	minimal	minimal	ADJ
ejpam-5127	266	6	prime	prime	ADJ
ejpam-5127	266	7	filter	filter	NOUN
ejpam-5127	266	8	of	of	ADP
ejpam-5127	266	9	v	v	NUM
ejpam-5127	266	10	.	.	PUNCT
ejpam-5127	267	1	theorem	theorem	VERB
ejpam-5127	267	2	9	9	NUM
ejpam-5127	267	3	.	.	PUNCT
ejpam-5127	268	1	every	every	DET
ejpam-5127	268	2	prime	prime	ADJ
ejpam-5127	268	3	filter	filter	NOUN
ejpam-5127	268	4	of	of	ADP
ejpam-5127	268	5	v	v	PROPN
ejpam-5127	268	6	contains	contain	VERB
ejpam-5127	268	7	a	a	DET
ejpam-5127	268	8	minimal	minimal	ADJ
ejpam-5127	268	9	prime	prime	ADJ
ejpam-5127	268	10	filter	filter	NOUN
ejpam-5127	268	11	of	of	ADP
ejpam-5127	268	12	v	v	NOUN
ejpam-5127	268	13	.	.	PUNCT
ejpam-5127	269	1	proof	proof	NOUN
ejpam-5127	269	2	.	.	PUNCT
ejpam-5127	270	1	let	let	VERB
ejpam-5127	270	2	p	p	PRON
ejpam-5127	270	3	be	be	AUX
ejpam-5127	270	4	a	a	DET
ejpam-5127	270	5	prime	prime	ADJ
ejpam-5127	270	6	filter	filter	NOUN
ejpam-5127	270	7	of	of	ADP
ejpam-5127	270	8	v	v	NOUN
ejpam-5127	270	9	.	.	PUNCT
ejpam-5127	271	1	then	then	ADV
ejpam-5127	271	2	v	v	NOUN
ejpam-5127	271	3	\p	\p	ADV
ejpam-5127	271	4	is	be	AUX
ejpam-5127	271	5	a	a	DET
ejpam-5127	271	6	prime	prime	ADJ
ejpam-5127	271	7	ideal	ideal	NOUN
ejpam-5127	271	8	of	of	ADP
ejpam-5127	271	9	v	v	NOUN
ejpam-5127	271	10	.	.	PUNCT
ejpam-5127	272	1	then	then	ADV
ejpam-5127	272	2	by	by	ADP
ejpam-5127	272	3	zorn	zorn	PROPN
ejpam-5127	272	4	’s	’s	PART
ejpam-5127	272	5	lemma	lemma	PROPN
ejpam-5127	272	6	,	,	PUNCT
ejpam-5127	272	7	there	there	PRON
ejpam-5127	272	8	is	be	VERB
ejpam-5127	272	9	a	a	DET
ejpam-5127	272	10	maximal	maximal	ADJ
ejpam-5127	272	11	prime	prime	ADJ
ejpam-5127	272	12	ideal	ideal	NOUN
ejpam-5127	272	13	g	g	NOUN
ejpam-5127	272	14	in	in	ADP
ejpam-5127	272	15	v	v	ADP
ejpam-5127	272	16	such	such	DET
ejpam-5127	272	17	that	that	DET
ejpam-5127	272	18	v	v	NOUN
ejpam-5127	272	19	\p	\p	ADV
ejpam-5127	273	1	=	=	SYM
ejpam-5127	273	2	f	f	PROPN
ejpam-5127	273	3	⊆	⊆	NUM
ejpam-5127	273	4	g.	g.	NOUN
ejpam-5127	273	5	thus	thus	ADV
ejpam-5127	273	6	v	v	ADP
ejpam-5127	273	7	\g	\g	NOUN
ejpam-5127	273	8	⊆	⊆	NUM
ejpam-5127	273	9	v	v	NOUN
ejpam-5127	273	10	\f	\f	NOUN
ejpam-5127	273	11	=	=	PUNCT
ejpam-5127	274	1	p.	p.	NOUN
ejpam-5127	274	2	therefore	therefore	ADV
ejpam-5127	274	3	,	,	PUNCT
ejpam-5127	274	4	v	v	ADP
ejpam-5127	274	5	\g	\g	ADJ
ejpam-5127	274	6	is	be	AUX
ejpam-5127	274	7	a	a	DET
ejpam-5127	274	8	minimal	minimal	ADJ
ejpam-5127	274	9	prime	prime	ADJ
ejpam-5127	274	10	filter	filter	NOUN
ejpam-5127	274	11	of	of	ADP
ejpam-5127	274	12	v	v	NOUN
ejpam-5127	274	13	contained	contain	VERB
ejpam-5127	274	14	in	in	ADP
ejpam-5127	274	15	the	the	DET
ejpam-5127	274	16	prime	prime	ADJ
ejpam-5127	274	17	filter	filter	NOUN
ejpam-5127	274	18	p.	p.	NOUN
ejpam-5127	274	19	theorem	theorem	VERB
ejpam-5127	274	20	10	10	NUM
ejpam-5127	274	21	.	.	PUNCT
ejpam-5127	275	1	if	if	SCONJ
ejpam-5127	275	2	p	p	NOUN
ejpam-5127	275	3	is	be	AUX
ejpam-5127	275	4	a	a	DET
ejpam-5127	275	5	prime	prime	ADJ
ejpam-5127	275	6	filter	filter	NOUN
ejpam-5127	275	7	in	in	ADP
ejpam-5127	275	8	a	a	DET
ejpam-5127	275	9	pdl	pdl	NOUN
ejpam-5127	275	10	v	v	NOUN
ejpam-5127	275	11	,	,	PUNCT
ejpam-5127	275	12	then	then	ADV
ejpam-5127	275	13	the	the	DET
ejpam-5127	275	14	filter	filter	NOUN
ejpam-5127	275	15	o(p	o(p	PROPN
ejpam-5127	275	16	)	)	PUNCT
ejpam-5127	275	17	is	be	AUX
ejpam-5127	275	18	the	the	DET
ejpam-5127	275	19	intersection	intersection	NOUN
ejpam-5127	275	20	of	of	ADP
ejpam-5127	275	21	all	all	DET
ejpam-5127	275	22	the	the	DET
ejpam-5127	275	23	minimal	minimal	ADJ
ejpam-5127	275	24	prime	prime	ADJ
ejpam-5127	275	25	filters	filter	NOUN
ejpam-5127	275	26	of	of	ADP
ejpam-5127	275	27	v	v	NOUN
ejpam-5127	275	28	contained	contain	VERB
ejpam-5127	275	29	in	in	ADP
ejpam-5127	275	30	p.	p.	NOUN
ejpam-5127	275	31	proof	proof	NOUN
ejpam-5127	275	32	.	.	PUNCT
ejpam-5127	276	1	let	let	VERB
ejpam-5127	276	2	p	p	PRON
ejpam-5127	276	3	be	be	AUX
ejpam-5127	276	4	a	a	DET
ejpam-5127	276	5	prime	prime	ADJ
ejpam-5127	276	6	filter	filter	NOUN
ejpam-5127	276	7	of	of	ADP
ejpam-5127	276	8	v	v	NOUN
ejpam-5127	276	9	.	.	PUNCT
ejpam-5127	277	1	then	then	ADV
ejpam-5127	277	2	p	p	PROPN
ejpam-5127	277	3	contains	contain	VERB
ejpam-5127	277	4	atleast	atleast	ADJ
ejpam-5127	277	5	one	one	NUM
ejpam-5127	277	6	minimal	minimal	ADJ
ejpam-5127	277	7	prime	prime	ADJ
ejpam-5127	277	8	filter	filter	NOUN
ejpam-5127	277	9	of	of	ADP
ejpam-5127	277	10	v	v	NOUN
ejpam-5127	277	11	.	.	PUNCT
ejpam-5127	278	1	let	let	VERB
ejpam-5127	278	2	{	{	PUNCT
ejpam-5127	278	3	qα|α	qα|α	NOUN
ejpam-5127	278	4	∈	∈	NOUN
ejpam-5127	278	5	∆	∆	X
ejpam-5127	278	6	}	}	PUNCT
ejpam-5127	278	7	be	be	VERB
ejpam-5127	278	8	the	the	DET
ejpam-5127	278	9	set	set	NOUN
ejpam-5127	278	10	of	of	ADP
ejpam-5127	278	11	all	all	DET
ejpam-5127	278	12	minimal	minimal	ADJ
ejpam-5127	278	13	prime	prime	ADJ
ejpam-5127	278	14	filters	filter	NOUN
ejpam-5127	278	15	of	of	ADP
ejpam-5127	278	16	v	v	NOUN
ejpam-5127	278	17	contained	contain	VERB
ejpam-5127	278	18	in	in	ADP
ejpam-5127	278	19	p.	p.	NOUN
ejpam-5127	278	20	now	now	ADV
ejpam-5127	278	21	,	,	PUNCT
ejpam-5127	278	22	we	we	PRON
ejpam-5127	278	23	prove	prove	VERB
ejpam-5127	278	24	that	that	SCONJ
ejpam-5127	278	25	o(p	o(p	PROPN
ejpam-5127	278	26	)	)	PUNCT
ejpam-5127	278	27	=	=	NOUN
ejpam-5127	278	28	∩	∩	NOUN
ejpam-5127	278	29	α∈∆	α∈∆	PROPN
ejpam-5127	278	30	qα	qα	PROPN
ejpam-5127	278	31	.	.	PUNCT
ejpam-5127	279	1	let	let	VERB
ejpam-5127	279	2	x	x	SYM
ejpam-5127	279	3	∈	∈	PROPN
ejpam-5127	279	4	o(p	o(p	PROPN
ejpam-5127	279	5	)	)	PUNCT
ejpam-5127	279	6	.	.	PUNCT
ejpam-5127	280	1	then	then	ADV
ejpam-5127	280	2	a	a	DET
ejpam-5127	280	3	∨	∨	NOUN
ejpam-5127	280	4	x	x	SYM
ejpam-5127	280	5	=	=	SYM
ejpam-5127	280	6	1	1	NUM
ejpam-5127	280	7	for	for	ADP
ejpam-5127	280	8	some	some	DET
ejpam-5127	280	9	a	a	DET
ejpam-5127	280	10	/∈	/∈	PUNCT
ejpam-5127	280	11	p.	p.	NOUN
ejpam-5127	280	12	now	now	ADV
ejpam-5127	281	1	a	a	PRON
ejpam-5127	281	2	/∈	/∈	PUNCT
ejpam-5127	281	3	p	p	NOUN
ejpam-5127	281	4	gives	give	VERB
ejpam-5127	281	5	that	that	PRON
ejpam-5127	281	6	a	a	DET
ejpam-5127	281	7	/∈	/∈	INTJ
ejpam-5127	281	8	qα	qα	PROPN
ejpam-5127	281	9	,	,	PUNCT
ejpam-5127	281	10	for	for	ADP
ejpam-5127	281	11	all	all	DET
ejpam-5127	281	12	α	α	NOUN
ejpam-5127	281	13	∈	∈	PROPN
ejpam-5127	282	1	∆.	∆.	NOUN
ejpam-5127	282	2	since	since	SCONJ
ejpam-5127	282	3	qα	qα	PROPN
ejpam-5127	282	4	is	be	AUX
ejpam-5127	282	5	a	a	DET
ejpam-5127	282	6	prime	prime	ADJ
ejpam-5127	282	7	filter	filter	NOUN
ejpam-5127	282	8	for	for	ADP
ejpam-5127	282	9	all	all	DET
ejpam-5127	282	10	α	α	DET
ejpam-5127	282	11	∈	∈	NOUN
ejpam-5127	282	12	∆	∆	PROPN
ejpam-5127	282	13	and	and	CCONJ
ejpam-5127	282	14	1	1	NUM
ejpam-5127	282	15	=	=	SYM
ejpam-5127	282	16	a	a	DET
ejpam-5127	282	17	∨	∨	NOUN
ejpam-5127	282	18	x	x	SYM
ejpam-5127	282	19	∈	∈	PROPN
ejpam-5127	282	20	qα	qα	PROPN
ejpam-5127	282	21	,	,	PUNCT
ejpam-5127	282	22	we	we	PRON
ejpam-5127	282	23	get	get	VERB
ejpam-5127	282	24	either	either	CCONJ
ejpam-5127	282	25	a	a	DET
ejpam-5127	282	26	∈	∈	ADJ
ejpam-5127	282	27	qα	qα	NOUN
ejpam-5127	282	28	or	or	CCONJ
ejpam-5127	282	29	x	x	PROPN
ejpam-5127	282	30	∈	∈	PROPN
ejpam-5127	282	31	qα	qα	PROPN
ejpam-5127	282	32	.	.	PUNCT
ejpam-5127	283	1	but	but	CCONJ
ejpam-5127	283	2	a	a	DET
ejpam-5127	283	3	/∈	/∈	INTJ
ejpam-5127	283	4	qα	qα	PROPN
ejpam-5127	283	5	.	.	PUNCT
ejpam-5127	284	1	therefore	therefore	ADV
ejpam-5127	284	2	,	,	PUNCT
ejpam-5127	284	3	we	we	PRON
ejpam-5127	284	4	get	get	VERB
ejpam-5127	284	5	x	x	X
ejpam-5127	284	6	∈	∈	PROPN
ejpam-5127	284	7	qα	qα	PROPN
ejpam-5127	284	8	for	for	ADP
ejpam-5127	284	9	all	all	PRON
ejpam-5127	284	10	α	α	DET
ejpam-5127	284	11	∈	∈	NOUN
ejpam-5127	284	12	∆	∆	PROPN
ejpam-5127	284	13	and	and	CCONJ
ejpam-5127	284	14	hence	hence	ADV
ejpam-5127	284	15	o(p	o(p	PROPN
ejpam-5127	284	16	)	)	PUNCT
ejpam-5127	284	17	⊆	⊆	NUM
ejpam-5127	284	18	∩	∩	NOUN
ejpam-5127	284	19	α∈∆	α∈∆	PROPN
ejpam-5127	284	20	qα	qα	X
ejpam-5127	284	21	.	.	PUNCT
ejpam-5127	285	1	conversely	conversely	ADV
ejpam-5127	285	2	,	,	PUNCT
ejpam-5127	285	3	suppose	suppose	VERB
ejpam-5127	285	4	x	x	X
ejpam-5127	285	5	/∈	/∈	PUNCT
ejpam-5127	285	6	o(p	o(p	PROPN
ejpam-5127	285	7	)	)	PUNCT
ejpam-5127	285	8	.	.	PUNCT
ejpam-5127	286	1	since	since	SCONJ
ejpam-5127	286	2	p	p	NOUN
ejpam-5127	286	3	is	be	AUX
ejpam-5127	286	4	a	a	DET
ejpam-5127	286	5	prime	prime	ADJ
ejpam-5127	286	6	filter	filter	NOUN
ejpam-5127	286	7	of	of	ADP
ejpam-5127	286	8	v	v	NUM
ejpam-5127	286	9	,	,	PUNCT
ejpam-5127	286	10	v	v	NOUN
ejpam-5127	286	11	\p	\p	ADV
ejpam-5127	286	12	is	be	AUX
ejpam-5127	286	13	a	a	DET
ejpam-5127	286	14	prime	prime	ADJ
ejpam-5127	286	15	ideal	ideal	NOUN
ejpam-5127	286	16	.	.	PUNCT
ejpam-5127	287	1	now	now	ADV
ejpam-5127	287	2	,	,	PUNCT
ejpam-5127	287	3	consider	consider	VERB
ejpam-5127	287	4	the	the	DET
ejpam-5127	287	5	ideal	ideal	NOUN
ejpam-5127	287	6	a	a	DET
ejpam-5127	287	7	=	=	X
ejpam-5127	287	8	(	(	PUNCT
ejpam-5127	287	9	x	x	SYM
ejpam-5127	287	10	]	]	X
ejpam-5127	287	11	∨	∨	X
ejpam-5127	287	12	(	(	PUNCT
ejpam-5127	287	13	v	v	NOUN
ejpam-5127	287	14	\p	\p	NOUN
ejpam-5127	287	15	)	)	PUNCT
ejpam-5127	287	16	.	.	PUNCT
ejpam-5127	288	1	since	since	SCONJ
ejpam-5127	288	2	x	x	PROPN
ejpam-5127	288	3	/∈	/∈	PROPN
ejpam-5127	288	4	o(p	o(p	PROPN
ejpam-5127	288	5	)	)	PUNCT
ejpam-5127	288	6	,	,	PUNCT
ejpam-5127	288	7	we	we	PRON
ejpam-5127	288	8	get	get	VERB
ejpam-5127	288	9	y	y	PROPN
ejpam-5127	288	10	∨	∨	NOUN
ejpam-5127	288	11	x	x	SYM
ejpam-5127	288	12	̸=	̸=	PROPN
ejpam-5127	288	13	1	1	NUM
ejpam-5127	288	14	for	for	ADP
ejpam-5127	288	15	all	all	DET
ejpam-5127	288	16	y	y	PROPN
ejpam-5127	288	17	∈	∈	PROPN
ejpam-5127	288	18	v	v	ADP
ejpam-5127	288	19	\p	\p	ADV
ejpam-5127	288	20	.	.	PUNCT
ejpam-5127	289	1	hence	hence	ADV
ejpam-5127	289	2	1	1	NUM
ejpam-5127	289	3	/∈	/∈	NOUN
ejpam-5127	289	4	a.	a.	NOUN
ejpam-5127	289	5	therefore	therefore	ADV
ejpam-5127	289	6	,	,	PUNCT
ejpam-5127	289	7	a	a	PRON
ejpam-5127	289	8	is	be	AUX
ejpam-5127	289	9	a	a	DET
ejpam-5127	289	10	proper	proper	ADJ
ejpam-5127	289	11	ideal	ideal	NOUN
ejpam-5127	289	12	of	of	ADP
ejpam-5127	289	13	v	v	NOUN
ejpam-5127	289	14	.	.	PUNCT
ejpam-5127	290	1	let	let	VERB
ejpam-5127	290	2	g	g	NOUN
ejpam-5127	290	3	be	be	AUX
ejpam-5127	290	4	any	any	DET
ejpam-5127	290	5	maximal	maximal	ADJ
ejpam-5127	290	6	ideal	ideal	NOUN
ejpam-5127	290	7	of	of	ADP
ejpam-5127	290	8	v	v	NOUN
ejpam-5127	290	9	containing	contain	VERB
ejpam-5127	290	10	a.	a.	NOUN
ejpam-5127	290	11	then	then	ADV
ejpam-5127	290	12	v	v	NOUN
ejpam-5127	290	13	\g	\g	NOUN
ejpam-5127	290	14	is	be	AUX
ejpam-5127	290	15	a	a	DET
ejpam-5127	290	16	minimal	minimal	ADJ
ejpam-5127	290	17	prime	prime	ADJ
ejpam-5127	290	18	filter	filter	NOUN
ejpam-5127	290	19	r.	r.	PROPN
ejpam-5127	290	20	shukla	shukla	PROPN
ejpam-5127	290	21	et	et	PROPN
ejpam-5127	290	22	al	al	PROPN
ejpam-5127	290	23	.	.	PUNCT
ejpam-5127	290	24	/	/	SYM
ejpam-5127	290	25	eur	eur	PROPN
ejpam-5127	290	26	.	.	PUNCT
ejpam-5127	291	1	j.	j.	PROPN
ejpam-5127	291	2	pure	pure	PROPN
ejpam-5127	291	3	appl	appl	PROPN
ejpam-5127	291	4	.	.	PROPN
ejpam-5127	291	5	math	math	PROPN
ejpam-5127	291	6	,	,	PUNCT
ejpam-5127	291	7	17	17	NUM
ejpam-5127	291	8	(	(	PUNCT
ejpam-5127	291	9	2	2	NUM
ejpam-5127	291	10	)	)	PUNCT
ejpam-5127	291	11	(	(	PUNCT
ejpam-5127	291	12	2024	2024	NUM
ejpam-5127	291	13	)	)	PUNCT
ejpam-5127	291	14	,	,	PUNCT
ejpam-5127	291	15	1306	1306	NUM
ejpam-5127	291	16	-	-	SYM
ejpam-5127	291	17	1320	1320	NUM
ejpam-5127	291	18	1314	1314	NUM
ejpam-5127	291	19	of	of	ADP
ejpam-5127	291	20	v	v	NOUN
ejpam-5127	291	21	.	.	PUNCT
ejpam-5127	292	1	since	since	SCONJ
ejpam-5127	292	2	x	x	PROPN
ejpam-5127	292	3	∈	∈	PROPN
ejpam-5127	292	4	a	a	DET
ejpam-5127	292	5	⊆	⊆	NUM
ejpam-5127	292	6	g	g	NOUN
ejpam-5127	292	7	and	and	CCONJ
ejpam-5127	292	8	v	v	NOUN
ejpam-5127	292	9	\p	\p	ADV
ejpam-5127	292	10	⊆	⊆	NUM
ejpam-5127	292	11	a	a	DET
ejpam-5127	292	12	⊆	⊆	NUM
ejpam-5127	292	13	g	g	NOUN
ejpam-5127	292	14	,	,	PUNCT
ejpam-5127	292	15	we	we	PRON
ejpam-5127	292	16	get	get	VERB
ejpam-5127	292	17	q	q	NOUN
ejpam-5127	292	18	=	=	X
ejpam-5127	292	19	v	v	NOUN
ejpam-5127	292	20	\g	\g	NOUN
ejpam-5127	292	21	⊆	⊆	NUM
ejpam-5127	292	22	p	p	NOUN
ejpam-5127	292	23	and	and	CCONJ
ejpam-5127	292	24	x	x	ADJ
ejpam-5127	292	25	/∈	/∈	PUNCT
ejpam-5127	292	26	q.	q.	NOUN
ejpam-5127	293	1	that	that	PRON
ejpam-5127	293	2	is	be	AUX
ejpam-5127	293	3	,	,	PUNCT
ejpam-5127	293	4	there	there	PRON
ejpam-5127	293	5	is	be	VERB
ejpam-5127	293	6	a	a	DET
ejpam-5127	293	7	minimal	minimal	ADJ
ejpam-5127	293	8	prime	prime	ADJ
ejpam-5127	293	9	filter	filter	NOUN
ejpam-5127	293	10	q	q	NOUN
ejpam-5127	293	11	of	of	ADP
ejpam-5127	293	12	v	v	NOUN
ejpam-5127	293	13	contained	contain	VERB
ejpam-5127	293	14	in	in	ADP
ejpam-5127	293	15	p	p	NOUN
ejpam-5127	293	16	and	and	CCONJ
ejpam-5127	293	17	x	x	PROPN
ejpam-5127	293	18	/∈	/∈	PUNCT
ejpam-5127	293	19	q.	q.	PROPN
ejpam-5127	294	1	therefore	therefore	ADV
ejpam-5127	294	2	,	,	PUNCT
ejpam-5127	294	3	x	x	X
ejpam-5127	294	4	/∈	/∈	PUNCT
ejpam-5127	294	5	∩	∩	ADJ
ejpam-5127	294	6	α∈∆	α∈∆	PROPN
ejpam-5127	294	7	qα	qα	PROPN
ejpam-5127	294	8	.	.	PUNCT
ejpam-5127	294	9	thus	thus	ADV
ejpam-5127	294	10	∩	∩	ADJ
ejpam-5127	294	11	α∈∆	α∈∆	PROPN
ejpam-5127	294	12	qα	qα	PROPN
ejpam-5127	294	13	⊆	⊆	NUM
ejpam-5127	294	14	o(p	o(p	PROPN
ejpam-5127	294	15	)	)	PUNCT
ejpam-5127	294	16	.	.	PUNCT
ejpam-5127	295	1	hence	hence	ADV
ejpam-5127	295	2	o(p	o(p	PROPN
ejpam-5127	295	3	)	)	PUNCT
ejpam-5127	296	1	=	=	NOUN
ejpam-5127	296	2	∩	∩	X
ejpam-5127	296	3	α∈∆	α∈∆	PROPN
ejpam-5127	296	4	qα	qα	PROPN
ejpam-5127	296	5	.	.	PUNCT
ejpam-5127	297	1	if	if	SCONJ
ejpam-5127	297	2	f	f	PROPN
ejpam-5127	297	3	:	:	PUNCT
ejpam-5127	297	4	v1	v1	PROPN
ejpam-5127	297	5	→	→	SYM
ejpam-5127	297	6	v2	v2	PROPN
ejpam-5127	297	7	is	be	AUX
ejpam-5127	297	8	a	a	DET
ejpam-5127	297	9	homomorphism	homomorphism	NOUN
ejpam-5127	297	10	and	and	CCONJ
ejpam-5127	297	11	i	i	PRON
ejpam-5127	297	12	is	be	AUX
ejpam-5127	297	13	a	a	DET
ejpam-5127	297	14	filter	filter	NOUN
ejpam-5127	297	15	of	of	ADP
ejpam-5127	297	16	v1	v1	NOUN
ejpam-5127	297	17	then	then	ADV
ejpam-5127	297	18	f(i	f(i	NUM
ejpam-5127	297	19	)	)	PUNCT
ejpam-5127	297	20	need	need	AUX
ejpam-5127	297	21	not	not	PART
ejpam-5127	297	22	be	be	AUX
ejpam-5127	297	23	a	a	DET
ejpam-5127	297	24	filter	filter	NOUN
ejpam-5127	297	25	of	of	ADP
ejpam-5127	297	26	v2	v2	NOUN
ejpam-5127	297	27	.	.	PUNCT
ejpam-5127	298	1	but	but	CCONJ
ejpam-5127	298	2	[	[	X
ejpam-5127	298	3	f(i	f(i	NUM
ejpam-5127	298	4	)	)	PUNCT
ejpam-5127	298	5	)	)	PUNCT
ejpam-5127	298	6	is	be	AUX
ejpam-5127	298	7	the	the	DET
ejpam-5127	298	8	filter	filter	NOUN
ejpam-5127	298	9	generated	generate	VERB
ejpam-5127	298	10	by	by	ADP
ejpam-5127	298	11	f(i	f(i	PROPN
ejpam-5127	298	12	)	)	PUNCT
ejpam-5127	298	13	in	in	ADP
ejpam-5127	298	14	v2	v2	NOUN
ejpam-5127	298	15	.	.	PUNCT
ejpam-5127	299	1	we	we	PRON
ejpam-5127	299	2	denote	denote	VERB
ejpam-5127	299	3	this	this	DET
ejpam-5127	299	4	filter	filter	NOUN
ejpam-5127	299	5	by	by	ADP
ejpam-5127	299	6	ie	ie	PRON
ejpam-5127	299	7	.	.	PUNCT
ejpam-5127	300	1	that	that	PRON
ejpam-5127	300	2	is	be	AUX
ejpam-5127	300	3	ie	ie	X
ejpam-5127	300	4	=	=	PUNCT
ejpam-5127	301	1	[	[	X
ejpam-5127	301	2	f(i	f(i	NUM
ejpam-5127	301	3	)	)	PUNCT
ejpam-5127	301	4	)	)	PUNCT
ejpam-5127	301	5	.	.	PUNCT
ejpam-5127	302	1	in	in	ADP
ejpam-5127	302	2	the	the	DET
ejpam-5127	302	3	following	follow	VERB
ejpam-5127	302	4	lemma	lemma	PROPN
ejpam-5127	302	5	,	,	PUNCT
ejpam-5127	302	6	we	we	PRON
ejpam-5127	302	7	can	can	AUX
ejpam-5127	302	8	observe	observe	VERB
ejpam-5127	302	9	that	that	SCONJ
ejpam-5127	302	10	that	that	DET
ejpam-5127	302	11	f−1(j	f−1(j	NOUN
ejpam-5127	302	12	)	)	PUNCT
ejpam-5127	302	13	is	be	AUX
ejpam-5127	302	14	a	a	DET
ejpam-5127	302	15	filter	filter	NOUN
ejpam-5127	302	16	of	of	ADP
ejpam-5127	302	17	v1	v1	NOUN
ejpam-5127	302	18	if	if	SCONJ
ejpam-5127	302	19	j	j	PROPN
ejpam-5127	302	20	is	be	AUX
ejpam-5127	302	21	a	a	DET
ejpam-5127	302	22	filter	filter	NOUN
ejpam-5127	302	23	of	of	ADP
ejpam-5127	302	24	v2	v2	NOUN
ejpam-5127	302	25	.	.	PUNCT
ejpam-5127	303	1	we	we	PRON
ejpam-5127	303	2	denote	denote	VERB
ejpam-5127	303	3	this	this	DET
ejpam-5127	303	4	filter	filter	NOUN
ejpam-5127	303	5	by	by	ADP
ejpam-5127	303	6	j	j	PROPN
ejpam-5127	303	7	c	c	PROPN
ejpam-5127	303	8	=	=	SYM
ejpam-5127	303	9	f−1(j	f−1(j	PROPN
ejpam-5127	303	10	)	)	PUNCT
ejpam-5127	303	11	.	.	PUNCT
ejpam-5127	304	1	lemma	lemma	PROPN
ejpam-5127	304	2	12	12	NUM
ejpam-5127	304	3	.	.	PUNCT
ejpam-5127	305	1	let	let	VERB
ejpam-5127	305	2	v1	v1	VERB
ejpam-5127	305	3	and	and	CCONJ
ejpam-5127	305	4	v2	v2	NOUN
ejpam-5127	305	5	be	be	AUX
ejpam-5127	305	6	two	two	NUM
ejpam-5127	305	7	pdls	pdl	NOUN
ejpam-5127	305	8	and	and	CCONJ
ejpam-5127	305	9	f	f	NOUN
ejpam-5127	305	10	:	:	PUNCT
ejpam-5127	305	11	v1	v1	PROPN
ejpam-5127	305	12	→	→	SYM
ejpam-5127	305	13	v2	v2	PROPN
ejpam-5127	305	14	be	be	AUX
ejpam-5127	305	15	a	a	DET
ejpam-5127	305	16	homomorphism	homomorphism	NOUN
ejpam-5127	305	17	.	.	PUNCT
ejpam-5127	306	1	let	let	VERB
ejpam-5127	306	2	i	i	PRON
ejpam-5127	306	3	,	,	PUNCT
ejpam-5127	306	4	i1	i1	PROPN
ejpam-5127	306	5	be	be	AUX
ejpam-5127	306	6	filters	filter	NOUN
ejpam-5127	306	7	of	of	ADP
ejpam-5127	306	8	v1	v1	NOUN
ejpam-5127	306	9	and	and	CCONJ
ejpam-5127	306	10	j	j	PROPN
ejpam-5127	306	11	,	,	PUNCT
ejpam-5127	306	12	j1	j1	PROPN
ejpam-5127	306	13	be	be	VERB
ejpam-5127	306	14	filters	filter	NOUN
ejpam-5127	306	15	of	of	ADP
ejpam-5127	306	16	v2	v2	NOUN
ejpam-5127	306	17	.	.	PUNCT
ejpam-5127	307	1	then	then	ADV
ejpam-5127	307	2	(	(	PUNCT
ejpam-5127	307	3	1	1	X
ejpam-5127	307	4	)	)	PUNCT
ejpam-5127	307	5	if	if	SCONJ
ejpam-5127	307	6	j	j	PROPN
ejpam-5127	307	7	is	be	AUX
ejpam-5127	307	8	a	a	DET
ejpam-5127	307	9	filter	filter	NOUN
ejpam-5127	307	10	of	of	ADP
ejpam-5127	307	11	v2	v2	NOUN
ejpam-5127	307	12	,	,	PUNCT
ejpam-5127	307	13	then	then	ADV
ejpam-5127	307	14	j	j	PROPN
ejpam-5127	307	15	c	c	PROPN
ejpam-5127	307	16	=	=	PRON
ejpam-5127	307	17	{	{	PUNCT
ejpam-5127	307	18	x	x	PROPN
ejpam-5127	307	19	∈	∈	PROPN
ejpam-5127	307	20	v1|f(x	v1|f(x	PROPN
ejpam-5127	307	21	)	)	PUNCT
ejpam-5127	307	22	∈	∈	PROPN
ejpam-5127	307	23	j	j	PROPN
ejpam-5127	307	24	⊆	⊆	NUM
ejpam-5127	307	25	v2	v2	PROPN
ejpam-5127	307	26	}	}	PUNCT
ejpam-5127	307	27	is	be	AUX
ejpam-5127	307	28	a	a	DET
ejpam-5127	307	29	filter	filter	NOUN
ejpam-5127	307	30	of	of	ADP
ejpam-5127	307	31	v1	v1	NOUN
ejpam-5127	307	32	.	.	PUNCT
ejpam-5127	308	1	(	(	PUNCT
ejpam-5127	308	2	2	2	X
ejpam-5127	308	3	)	)	PUNCT
ejpam-5127	308	4	i	i	PROPN
ejpam-5127	308	5	⊆	⊆	NUM
ejpam-5127	308	6	i1	i1	PROPN
ejpam-5127	308	7	=	=	AUX
ejpam-5127	308	8	⇒	⇒	VERB
ejpam-5127	308	9	ie	ie	PRON
ejpam-5127	308	10	⊆	⊆	NUM
ejpam-5127	308	11	ie	ie	X
ejpam-5127	308	12	1	1	NUM
ejpam-5127	308	13	.	.	PUNCT
ejpam-5127	309	1	(	(	PUNCT
ejpam-5127	309	2	3	3	X
ejpam-5127	309	3	)	)	PUNCT
ejpam-5127	309	4	j	j	NOUN
ejpam-5127	309	5	⊆	⊆	NUM
ejpam-5127	309	6	j1	j1	PROPN
ejpam-5127	309	7	=	=	SYM
ejpam-5127	309	8	⇒	⇒	PROPN
ejpam-5127	309	9	j	j	PROPN
ejpam-5127	309	10	c	c	PROPN
ejpam-5127	309	11	⊆	⊆	NUM
ejpam-5127	309	12	j	j	PROPN
ejpam-5127	309	13	c	c	SYM
ejpam-5127	309	14	1	1	NUM
ejpam-5127	309	15	.	.	PUNCT
ejpam-5127	310	1	(	(	PUNCT
ejpam-5127	310	2	4	4	NUM
ejpam-5127	310	3	)	)	PUNCT
ejpam-5127	310	4	for	for	ADP
ejpam-5127	310	5	every	every	DET
ejpam-5127	310	6	i	i	PROPN
ejpam-5127	310	7	∈	∈	PROPN
ejpam-5127	310	8	f(v1	f(v1	NOUN
ejpam-5127	310	9	)	)	PUNCT
ejpam-5127	310	10	,	,	PUNCT
ejpam-5127	310	11	i	i	PRON
ejpam-5127	310	12	⊆	⊆	NUM
ejpam-5127	310	13	iec	iec	NOUN
ejpam-5127	310	14	and	and	CCONJ
ejpam-5127	310	15	i	i	NOUN
ejpam-5127	310	16	=	=	NOUN
ejpam-5127	310	17	iec	iec	NOUN
ejpam-5127	310	18	if	if	SCONJ
ejpam-5127	310	19	f	f	PROPN
ejpam-5127	310	20	is	be	AUX
ejpam-5127	310	21	a	a	DET
ejpam-5127	310	22	bijection	bijection	NOUN
ejpam-5127	310	23	.	.	PUNCT
ejpam-5127	311	1	(	(	PUNCT
ejpam-5127	311	2	5	5	NUM
ejpam-5127	311	3	)	)	PUNCT
ejpam-5127	311	4	for	for	ADP
ejpam-5127	311	5	any	any	DET
ejpam-5127	311	6	j	j	PROPN
ejpam-5127	311	7	∈	∈	PROPN
ejpam-5127	311	8	f(v2),j	f(v2),j	PROPN
ejpam-5127	311	9	ce	ce	PROPN
ejpam-5127	312	1	⊆	⊆	NUM
ejpam-5127	312	2	j	j	PROPN
ejpam-5127	312	3	and	and	CCONJ
ejpam-5127	312	4	j	j	PROPN
ejpam-5127	312	5	ce	ce	PROPN
ejpam-5127	313	1	=	=	PROPN
ejpam-5127	313	2	j	j	PROPN
ejpam-5127	313	3	if	if	SCONJ
ejpam-5127	313	4	f	f	PROPN
ejpam-5127	313	5	is	be	AUX
ejpam-5127	313	6	onto	onto	ADP
ejpam-5127	313	7	.	.	PUNCT
ejpam-5127	314	1	since	since	SCONJ
ejpam-5127	314	2	the	the	DET
ejpam-5127	314	3	mapping	mapping	NOUN
ejpam-5127	314	4	x	x	SYM
ejpam-5127	314	5	7→	7→	NUM
ejpam-5127	315	1	[	[	X
ejpam-5127	315	2	x	x	X
ejpam-5127	315	3	)	)	PUNCT
ejpam-5127	315	4	is	be	AUX
ejpam-5127	315	5	a	a	DET
ejpam-5127	315	6	homomorphism	homomorphism	NOUN
ejpam-5127	315	7	of	of	ADP
ejpam-5127	315	8	the	the	DET
ejpam-5127	315	9	pdl	pdl	PROPN
ejpam-5127	315	10	v	v	NOUN
ejpam-5127	315	11	into	into	ADP
ejpam-5127	315	12	the	the	DET
ejpam-5127	315	13	lattice	lattice	NOUN
ejpam-5127	315	14	pf(v	pf(v	NOUN
ejpam-5127	315	15	)	)	PUNCT
ejpam-5127	315	16	of	of	ADP
ejpam-5127	315	17	all	all	DET
ejpam-5127	315	18	principal	principal	ADJ
ejpam-5127	315	19	filters	filter	NOUN
ejpam-5127	315	20	of	of	ADP
ejpam-5127	315	21	v	v	NOUN
ejpam-5127	315	22	,	,	PUNCT
ejpam-5127	315	23	we	we	PRON
ejpam-5127	315	24	have	have	VERB
ejpam-5127	315	25	the	the	DET
ejpam-5127	315	26	following	follow	VERB
ejpam-5127	315	27	theorem	theorem	VERB
ejpam-5127	315	28	.	.	PUNCT
ejpam-5127	315	29	theorem	theorem	PROPN
ejpam-5127	315	30	11	11	NUM
ejpam-5127	315	31	.	.	PUNCT
ejpam-5127	316	1	(	(	PUNCT
ejpam-5127	316	2	1	1	X
ejpam-5127	316	3	)	)	PUNCT
ejpam-5127	316	4	for	for	ADP
ejpam-5127	316	5	any	any	DET
ejpam-5127	316	6	filter	filter	NOUN
ejpam-5127	316	7	i	i	PRON
ejpam-5127	316	8	of	of	ADP
ejpam-5127	316	9	v	v	NOUN
ejpam-5127	316	10	,	,	PUNCT
ejpam-5127	316	11	ie	ie	ADV
ejpam-5127	316	12	=	=	PUNCT
ejpam-5127	316	13	{	{	PUNCT
ejpam-5127	317	1	[	[	X
ejpam-5127	317	2	a)|a	a)|a	X
ejpam-5127	317	3	∈	∈	PROPN
ejpam-5127	317	4	i	i	PRON
ejpam-5127	317	5	}	}	PUNCT
ejpam-5127	317	6	is	be	AUX
ejpam-5127	317	7	a	a	DET
ejpam-5127	317	8	filter	filter	NOUN
ejpam-5127	317	9	of	of	ADP
ejpam-5127	317	10	pf(v	pf(v	NOUN
ejpam-5127	317	11	)	)	PUNCT
ejpam-5127	317	12	.	.	PUNCT
ejpam-5127	318	1	moreover	moreover	ADV
ejpam-5127	318	2	,	,	PUNCT
ejpam-5127	318	3	i	i	PRON
ejpam-5127	318	4	is	be	AUX
ejpam-5127	318	5	prime	prime	ADJ
ejpam-5127	318	6	if	if	SCONJ
ejpam-5127	319	1	and	and	CCONJ
ejpam-5127	319	2	only	only	ADV
ejpam-5127	319	3	if	if	SCONJ
ejpam-5127	319	4	ie	ie	ADV
ejpam-5127	319	5	is	be	AUX
ejpam-5127	319	6	prime	prime	ADJ
ejpam-5127	319	7	.	.	PUNCT
ejpam-5127	320	1	(	(	PUNCT
ejpam-5127	320	2	2	2	X
ejpam-5127	320	3	)	)	PUNCT
ejpam-5127	320	4	for	for	ADP
ejpam-5127	320	5	any	any	DET
ejpam-5127	320	6	filter	filter	NOUN
ejpam-5127	320	7	k	k	NOUN
ejpam-5127	320	8	of	of	ADP
ejpam-5127	320	9	the	the	DET
ejpam-5127	320	10	lattice	lattice	NOUN
ejpam-5127	320	11	pf(v	pf(v	NOUN
ejpam-5127	320	12	)	)	PUNCT
ejpam-5127	320	13	,	,	PUNCT
ejpam-5127	320	14	kc	kc	PROPN
ejpam-5127	320	15	=	=	PRON
ejpam-5127	320	16	{	{	PUNCT
ejpam-5127	320	17	a	a	PRON
ejpam-5127	320	18	∈	∈	PROPN
ejpam-5127	320	19	v	v	ADP
ejpam-5127	320	20	|[a	|[a	NOUN
ejpam-5127	320	21	)	)	PUNCT
ejpam-5127	320	22	∈	∈	PROPN
ejpam-5127	321	1	k	k	X
ejpam-5127	321	2	}	}	PUNCT
ejpam-5127	321	3	is	be	AUX
ejpam-5127	321	4	a	a	DET
ejpam-5127	321	5	filter	filter	NOUN
ejpam-5127	321	6	of	of	ADP
ejpam-5127	321	7	v	v	NOUN
ejpam-5127	321	8	.	.	PUNCT
ejpam-5127	322	1	further	far	ADV
ejpam-5127	322	2	,	,	PUNCT
ejpam-5127	322	3	k	k	PROPN
ejpam-5127	322	4	is	be	AUX
ejpam-5127	322	5	prime	prime	ADJ
ejpam-5127	322	6	if	if	SCONJ
ejpam-5127	323	1	and	and	CCONJ
ejpam-5127	323	2	only	only	ADV
ejpam-5127	323	3	if	if	SCONJ
ejpam-5127	323	4	kc	kc	PROPN
ejpam-5127	323	5	is	be	AUX
ejpam-5127	323	6	prime	prime	ADJ
ejpam-5127	323	7	.	.	PUNCT
ejpam-5127	324	1	(	(	PUNCT
ejpam-5127	324	2	3	3	X
ejpam-5127	324	3	)	)	PUNCT
ejpam-5127	324	4	for	for	ADP
ejpam-5127	324	5	any	any	DET
ejpam-5127	324	6	filters	filter	NOUN
ejpam-5127	324	7	i1	i1	PROPN
ejpam-5127	324	8	,	,	PUNCT
ejpam-5127	324	9	i2	i2	PROPN
ejpam-5127	324	10	of	of	ADP
ejpam-5127	324	11	v	v	NOUN
ejpam-5127	324	12	,	,	PUNCT
ejpam-5127	324	13	i1	i1	PROPN
ejpam-5127	324	14	⊆	⊆	NUM
ejpam-5127	324	15	i2	i2	PROPN
ejpam-5127	324	16	if	if	SCONJ
ejpam-5127	324	17	and	and	CCONJ
ejpam-5127	324	18	only	only	ADV
ejpam-5127	324	19	if	if	SCONJ
ejpam-5127	324	20	ie	ie	ADV
ejpam-5127	324	21	1	1	NUM
ejpam-5127	324	22	⊆	⊆	NUM
ejpam-5127	324	23	ie	ie	ADP
ejpam-5127	324	24	2	2	NUM
ejpam-5127	324	25	.	.	PUNCT
ejpam-5127	325	1	(	(	PUNCT
ejpam-5127	325	2	4	4	NUM
ejpam-5127	325	3	)	)	PUNCT
ejpam-5127	325	4	for	for	ADP
ejpam-5127	325	5	any	any	DET
ejpam-5127	325	6	filters	filter	NOUN
ejpam-5127	325	7	k1,k2	k1,k2	PROPN
ejpam-5127	325	8	of	of	ADP
ejpam-5127	325	9	pf(v	pf(v	NOUN
ejpam-5127	325	10	)	)	PUNCT
ejpam-5127	325	11	,	,	PUNCT
ejpam-5127	325	12	k1	k1	VERB
ejpam-5127	325	13	⊆	⊆	NUM
ejpam-5127	325	14	k2	k2	PROPN
ejpam-5127	325	15	if	if	SCONJ
ejpam-5127	325	16	and	and	CCONJ
ejpam-5127	325	17	only	only	ADV
ejpam-5127	325	18	if	if	SCONJ
ejpam-5127	325	19	kc	kc	PROPN
ejpam-5127	325	20	1	1	NUM
ejpam-5127	325	21	⊆	⊆	NUM
ejpam-5127	325	22	kc	kc	PROPN
ejpam-5127	325	23	2	2	NUM
ejpam-5127	325	24	.	.	PUNCT
ejpam-5127	326	1	(	(	PUNCT
ejpam-5127	326	2	5	5	X
ejpam-5127	326	3	)	)	PUNCT
ejpam-5127	326	4	iec	iec	NOUN
ejpam-5127	326	5	=	=	SYM
ejpam-5127	326	6	i	i	PROPN
ejpam-5127	326	7	,	,	PUNCT
ejpam-5127	326	8	for	for	ADP
ejpam-5127	326	9	all	all	DET
ejpam-5127	326	10	filters	filter	NOUN
ejpam-5127	326	11	i	i	PRON
ejpam-5127	326	12	of	of	ADP
ejpam-5127	326	13	v	v	NOUN
ejpam-5127	326	14	.	.	PUNCT
ejpam-5127	327	1	(	(	PUNCT
ejpam-5127	327	2	6	6	NUM
ejpam-5127	327	3	)	)	PUNCT
ejpam-5127	327	4	kce	kce	NOUN
ejpam-5127	327	5	=	=	SYM
ejpam-5127	327	6	k	k	PROPN
ejpam-5127	327	7	,	,	PUNCT
ejpam-5127	327	8	for	for	ADP
ejpam-5127	327	9	all	all	DET
ejpam-5127	327	10	filters	filter	NOUN
ejpam-5127	327	11	k	k	PROPN
ejpam-5127	327	12	of	of	ADP
ejpam-5127	327	13	pf(v	pf(v	NOUN
ejpam-5127	327	14	)	)	PUNCT
ejpam-5127	327	15	.	.	PUNCT
ejpam-5127	328	1	lemma	lemma	PROPN
ejpam-5127	328	2	13	13	NUM
ejpam-5127	328	3	.	.	PUNCT
ejpam-5127	329	1	let	let	VERB
ejpam-5127	329	2	v	v	PART
ejpam-5127	329	3	be	be	AUX
ejpam-5127	329	4	a	a	DET
ejpam-5127	329	5	pdl	pdl	NOUN
ejpam-5127	329	6	and	and	CCONJ
ejpam-5127	329	7	pf(v	pf(v	NUM
ejpam-5127	329	8	)	)	PUNCT
ejpam-5127	329	9	is	be	AUX
ejpam-5127	329	10	the	the	DET
ejpam-5127	329	11	principle	principle	ADJ
ejpam-5127	329	12	filter	filter	NOUN
ejpam-5127	329	13	lattice	lattice	NOUN
ejpam-5127	329	14	of	of	ADP
ejpam-5127	329	15	v	v	NOUN
ejpam-5127	329	16	.	.	PUNCT
ejpam-5127	330	1	then	then	ADV
ejpam-5127	330	2	p	p	PROPN
ejpam-5127	330	3	is	be	AUX
ejpam-5127	330	4	a	a	DET
ejpam-5127	330	5	minimal	minimal	ADJ
ejpam-5127	330	6	prime	prime	ADJ
ejpam-5127	330	7	filter	filter	NOUN
ejpam-5127	330	8	of	of	ADP
ejpam-5127	330	9	v	v	NOUN
ejpam-5127	330	10	if	if	SCONJ
ejpam-5127	331	1	and	and	CCONJ
ejpam-5127	331	2	only	only	ADV
ejpam-5127	331	3	if	if	SCONJ
ejpam-5127	331	4	pe	pe	PROPN
ejpam-5127	331	5	is	be	AUX
ejpam-5127	331	6	a	a	DET
ejpam-5127	331	7	minimal	minimal	ADJ
ejpam-5127	331	8	prime	prime	ADJ
ejpam-5127	331	9	filter	filter	NOUN
ejpam-5127	331	10	of	of	ADP
ejpam-5127	331	11	pf(v	pf(v	NOUN
ejpam-5127	331	12	)	)	PUNCT
ejpam-5127	331	13	.	.	PUNCT
ejpam-5127	332	1	proof	proof	NOUN
ejpam-5127	332	2	.	.	PUNCT
ejpam-5127	333	1	let	let	VERB
ejpam-5127	333	2	p	p	PRON
ejpam-5127	333	3	be	be	AUX
ejpam-5127	333	4	a	a	DET
ejpam-5127	333	5	minimal	minimal	ADJ
ejpam-5127	333	6	prime	prime	ADJ
ejpam-5127	333	7	filter	filter	NOUN
ejpam-5127	333	8	of	of	ADP
ejpam-5127	333	9	v	v	NOUN
ejpam-5127	333	10	.	.	PUNCT
ejpam-5127	334	1	since	since	SCONJ
ejpam-5127	334	2	p	p	NOUN
ejpam-5127	334	3	is	be	AUX
ejpam-5127	334	4	a	a	DET
ejpam-5127	334	5	prime	prime	ADJ
ejpam-5127	334	6	filter	filter	NOUN
ejpam-5127	334	7	of	of	ADP
ejpam-5127	334	8	v	v	NUM
ejpam-5127	334	9	,	,	PUNCT
ejpam-5127	334	10	pe	pe	PROPN
ejpam-5127	334	11	is	be	AUX
ejpam-5127	334	12	a	a	DET
ejpam-5127	334	13	prime	prime	ADJ
ejpam-5127	334	14	filter	filter	NOUN
ejpam-5127	334	15	of	of	ADP
ejpam-5127	334	16	pf(v	pf(v	NOUN
ejpam-5127	334	17	)	)	PUNCT
ejpam-5127	334	18	.	.	PUNCT
ejpam-5127	335	1	now	now	ADV
ejpam-5127	335	2	we	we	PRON
ejpam-5127	335	3	prove	prove	VERB
ejpam-5127	335	4	that	that	SCONJ
ejpam-5127	335	5	pe	pe	PROPN
ejpam-5127	335	6	is	be	AUX
ejpam-5127	335	7	a	a	DET
ejpam-5127	335	8	minimal	minimal	ADJ
ejpam-5127	335	9	prime	prime	ADJ
ejpam-5127	335	10	filter	filter	NOUN
ejpam-5127	335	11	of	of	ADP
ejpam-5127	335	12	pf(v	pf(v	NOUN
ejpam-5127	335	13	)	)	PUNCT
ejpam-5127	335	14	such	such	ADJ
ejpam-5127	335	15	that	that	DET
ejpam-5127	335	16	q	q	PROPN
ejpam-5127	335	17	⊆	⊆	NUM
ejpam-5127	335	18	pe	pe	NOUN
ejpam-5127	335	19	.	.	PUNCT
ejpam-5127	336	1	then	then	ADV
ejpam-5127	336	2	,	,	PUNCT
ejpam-5127	336	3	qc	qc	PROPN
ejpam-5127	336	4	is	be	AUX
ejpam-5127	336	5	a	a	DET
ejpam-5127	336	6	prime	prime	ADJ
ejpam-5127	336	7	filter	filter	NOUN
ejpam-5127	336	8	of	of	ADP
ejpam-5127	336	9	v	v	NOUN
ejpam-5127	336	10	.	.	PUNCT
ejpam-5127	337	1	also	also	ADV
ejpam-5127	337	2	qc	qc	PROPN
ejpam-5127	337	3	⊆	⊆	NUM
ejpam-5127	337	4	pec	pec	NOUN
ejpam-5127	337	5	=	=	PROPN
ejpam-5127	338	1	p.	p.	NOUN
ejpam-5127	339	1	but	but	CCONJ
ejpam-5127	339	2	p	p	NOUN
ejpam-5127	339	3	is	be	AUX
ejpam-5127	339	4	a	a	DET
ejpam-5127	339	5	minimal	minimal	ADJ
ejpam-5127	339	6	prime	prime	ADJ
ejpam-5127	339	7	filter	filter	NOUN
ejpam-5127	339	8	of	of	ADP
ejpam-5127	339	9	v	v	NOUN
ejpam-5127	339	10	.	.	PUNCT
ejpam-5127	340	1	therefore	therefore	ADV
ejpam-5127	340	2	we	we	PRON
ejpam-5127	340	3	get	get	VERB
ejpam-5127	340	4	qc	qc	PROPN
ejpam-5127	341	1	=	=	SYM
ejpam-5127	341	2	p	p	NOUN
ejpam-5127	341	3	which	which	PRON
ejpam-5127	341	4	gives	give	VERB
ejpam-5127	341	5	qce	qce	NOUN
ejpam-5127	341	6	=	=	PUNCT
ejpam-5127	341	7	pe	pe	X
ejpam-5127	341	8	and	and	CCONJ
ejpam-5127	342	1	hence	hence	ADV
ejpam-5127	342	2	q	q	PROPN
ejpam-5127	342	3	=	=	ADJ
ejpam-5127	342	4	pe	pe	PROPN
ejpam-5127	342	5	.	.	PUNCT
ejpam-5127	343	1	therefore	therefore	ADV
ejpam-5127	343	2	pe	pe	PROPN
ejpam-5127	343	3	is	be	AUX
ejpam-5127	343	4	a	a	DET
ejpam-5127	343	5	minimal	minimal	ADJ
ejpam-5127	343	6	prime	prime	ADJ
ejpam-5127	343	7	filter	filter	NOUN
ejpam-5127	343	8	of	of	ADP
ejpam-5127	343	9	v	v	NOUN
ejpam-5127	343	10	.	.	PUNCT
ejpam-5127	344	1	similarly	similarly	ADV
ejpam-5127	344	2	,	,	PUNCT
ejpam-5127	344	3	we	we	PRON
ejpam-5127	344	4	can	can	AUX
ejpam-5127	344	5	prove	prove	VERB
ejpam-5127	344	6	the	the	DET
ejpam-5127	344	7	converse	converse	NOUN
ejpam-5127	344	8	.	.	PUNCT
ejpam-5127	345	1	now	now	ADV
ejpam-5127	345	2	,	,	PUNCT
ejpam-5127	345	3	we	we	PRON
ejpam-5127	345	4	define	define	VERB
ejpam-5127	345	5	the	the	DET
ejpam-5127	345	6	notion	notion	NOUN
ejpam-5127	345	7	of	of	ADP
ejpam-5127	345	8	a	a	DET
ejpam-5127	345	9	normal	normal	ADJ
ejpam-5127	345	10	paradistributive	paradistributive	ADJ
ejpam-5127	345	11	latticoid	latticoid	NOUN
ejpam-5127	345	12	.	.	PUNCT
ejpam-5127	346	1	r.	r.	PROPN
ejpam-5127	346	2	shukla	shukla	PROPN
ejpam-5127	346	3	et	et	PROPN
ejpam-5127	346	4	al	al	PROPN
ejpam-5127	346	5	.	.	PUNCT
ejpam-5127	346	6	/	/	SYM
ejpam-5127	346	7	eur	eur	PROPN
ejpam-5127	346	8	.	.	PUNCT
ejpam-5127	347	1	j.	j.	PROPN
ejpam-5127	347	2	pure	pure	PROPN
ejpam-5127	347	3	appl	appl	PROPN
ejpam-5127	347	4	.	.	PROPN
ejpam-5127	347	5	math	math	PROPN
ejpam-5127	347	6	,	,	PUNCT
ejpam-5127	347	7	17	17	NUM
ejpam-5127	347	8	(	(	PUNCT
ejpam-5127	347	9	2	2	NUM
ejpam-5127	347	10	)	)	PUNCT
ejpam-5127	347	11	(	(	PUNCT
ejpam-5127	347	12	2024	2024	NUM
ejpam-5127	347	13	)	)	PUNCT
ejpam-5127	347	14	,	,	PUNCT
ejpam-5127	347	15	1306	1306	NUM
ejpam-5127	347	16	-	-	SYM
ejpam-5127	347	17	1320	1320	NUM
ejpam-5127	347	18	1315	1315	NUM
ejpam-5127	347	19	definition	definition	NOUN
ejpam-5127	347	20	9	9	NUM
ejpam-5127	347	21	.	.	PUNCT
ejpam-5127	348	1	a	a	DET
ejpam-5127	348	2	paradistributive	paradistributive	ADJ
ejpam-5127	348	3	latticoid	latticoid	NOUN
ejpam-5127	348	4	v	v	NOUN
ejpam-5127	348	5	is	be	AUX
ejpam-5127	348	6	called	call	VERB
ejpam-5127	348	7	normal	normal	ADJ
ejpam-5127	348	8	if	if	SCONJ
ejpam-5127	348	9	every	every	DET
ejpam-5127	348	10	prime	prime	ADJ
ejpam-5127	348	11	filter	filter	NOUN
ejpam-5127	348	12	of	of	ADP
ejpam-5127	348	13	v	v	PROPN
ejpam-5127	348	14	contains	contain	VERB
ejpam-5127	348	15	a	a	DET
ejpam-5127	348	16	unique	unique	ADJ
ejpam-5127	348	17	minimal	minimal	ADJ
ejpam-5127	348	18	prime	prime	ADJ
ejpam-5127	348	19	filter	filter	NOUN
ejpam-5127	348	20	of	of	ADP
ejpam-5127	348	21	v	v	NOUN
ejpam-5127	348	22	.	.	PUNCT
ejpam-5127	348	23	example	example	NOUN
ejpam-5127	349	1	2	2	NUM
ejpam-5127	349	2	.	.	PUNCT
ejpam-5127	349	3	let	let	VERB
ejpam-5127	349	4	v	v	VERB
ejpam-5127	349	5	=	=	SYM
ejpam-5127	349	6	{	{	PUNCT
ejpam-5127	349	7	0	0	NUM
ejpam-5127	349	8	,	,	PUNCT
ejpam-5127	349	9	1	1	NUM
ejpam-5127	349	10	,	,	PUNCT
ejpam-5127	349	11	2	2	NUM
ejpam-5127	349	12	,	,	PUNCT
ejpam-5127	349	13	3	3	NUM
ejpam-5127	349	14	,	,	PUNCT
ejpam-5127	349	15	4	4	NUM
ejpam-5127	349	16	,	,	PUNCT
ejpam-5127	349	17	5	5	NUM
ejpam-5127	349	18	}	}	PUNCT
ejpam-5127	349	19	be	be	AUX
ejpam-5127	349	20	a	a	DET
ejpam-5127	349	21	set	set	NOUN
ejpam-5127	349	22	with	with	ADP
ejpam-5127	349	23	binary	binary	ADJ
ejpam-5127	349	24	operations	operation	NOUN
ejpam-5127	349	25	∨	∨	NOUN
ejpam-5127	349	26	and	and	CCONJ
ejpam-5127	349	27	∧	∧	NOUN
ejpam-5127	349	28	given	give	VERB
ejpam-5127	349	29	in	in	ADP
ejpam-5127	349	30	the	the	DET
ejpam-5127	349	31	following	following	ADJ
ejpam-5127	349	32	tables	table	NOUN
ejpam-5127	349	33	:	:	PUNCT
ejpam-5127	349	34	∨	∨	NOUN
ejpam-5127	349	35	0	0	NUM
ejpam-5127	349	36	1	1	NUM
ejpam-5127	349	37	2	2	NUM
ejpam-5127	349	38	3	3	NUM
ejpam-5127	349	39	4	4	NUM
ejpam-5127	349	40	5	5	NUM
ejpam-5127	349	41	0	0	NUM
ejpam-5127	349	42	0	0	NUM
ejpam-5127	349	43	1	1	NUM
ejpam-5127	349	44	1	1	NUM
ejpam-5127	349	45	0	0	NUM
ejpam-5127	349	46	0	0	NUM
ejpam-5127	349	47	0	0	NUM
ejpam-5127	349	48	1	1	NUM
ejpam-5127	349	49	1	1	NUM
ejpam-5127	349	50	1	1	NUM
ejpam-5127	349	51	1	1	NUM
ejpam-5127	349	52	1	1	NUM
ejpam-5127	349	53	1	1	NUM
ejpam-5127	349	54	1	1	NUM
ejpam-5127	349	55	2	2	NUM
ejpam-5127	349	56	1	1	NUM
ejpam-5127	349	57	1	1	NUM
ejpam-5127	349	58	2	2	NUM
ejpam-5127	349	59	2	2	NUM
ejpam-5127	349	60	1	1	NUM
ejpam-5127	349	61	2	2	NUM
ejpam-5127	349	62	3	3	NUM
ejpam-5127	349	63	0	0	NUM
ejpam-5127	349	64	1	1	NUM
ejpam-5127	349	65	2	2	NUM
ejpam-5127	349	66	3	3	NUM
ejpam-5127	349	67	0	0	NUM
ejpam-5127	349	68	3	3	NUM
ejpam-5127	349	69	4	4	NUM
ejpam-5127	349	70	4	4	NUM
ejpam-5127	349	71	1	1	NUM
ejpam-5127	349	72	1	1	NUM
ejpam-5127	349	73	4	4	NUM
ejpam-5127	349	74	4	4	NUM
ejpam-5127	349	75	4	4	NUM
ejpam-5127	349	76	5	5	NUM
ejpam-5127	349	77	4	4	NUM
ejpam-5127	349	78	1	1	NUM
ejpam-5127	349	79	2	2	NUM
ejpam-5127	349	80	5	5	NUM
ejpam-5127	349	81	4	4	NUM
ejpam-5127	349	82	5	5	NUM
ejpam-5127	349	83	∧	∧	NOUN
ejpam-5127	349	84	0	0	NUM
ejpam-5127	349	85	1	1	NUM
ejpam-5127	349	86	2	2	NUM
ejpam-5127	349	87	3	3	NUM
ejpam-5127	349	88	4	4	NUM
ejpam-5127	349	89	5	5	NUM
ejpam-5127	349	90	0	0	NUM
ejpam-5127	349	91	0	0	NUM
ejpam-5127	349	92	0	0	NUM
ejpam-5127	349	93	3	3	NUM
ejpam-5127	349	94	3	3	NUM
ejpam-5127	349	95	4	4	NUM
ejpam-5127	349	96	5	5	NUM
ejpam-5127	349	97	1	1	NUM
ejpam-5127	349	98	0	0	NUM
ejpam-5127	349	99	1	1	NUM
ejpam-5127	349	100	2	2	NUM
ejpam-5127	349	101	3	3	NUM
ejpam-5127	349	102	4	4	NUM
ejpam-5127	349	103	5	5	NUM
ejpam-5127	349	104	2	2	NUM
ejpam-5127	349	105	3	3	NUM
ejpam-5127	349	106	2	2	NUM
ejpam-5127	349	107	2	2	NUM
ejpam-5127	349	108	3	3	NUM
ejpam-5127	349	109	5	5	NUM
ejpam-5127	349	110	5	5	NUM
ejpam-5127	349	111	3	3	NUM
ejpam-5127	349	112	3	3	NUM
ejpam-5127	349	113	3	3	NUM
ejpam-5127	349	114	3	3	NUM
ejpam-5127	349	115	3	3	NUM
ejpam-5127	349	116	5	5	NUM
ejpam-5127	349	117	5	5	NUM
ejpam-5127	349	118	4	4	NUM
ejpam-5127	349	119	0	0	NUM
ejpam-5127	349	120	4	4	NUM
ejpam-5127	349	121	5	5	NUM
ejpam-5127	349	122	3	3	NUM
ejpam-5127	349	123	4	4	NUM
ejpam-5127	349	124	5	5	NUM
ejpam-5127	349	125	5	5	NUM
ejpam-5127	349	126	3	3	NUM
ejpam-5127	349	127	5	5	NUM
ejpam-5127	349	128	5	5	NUM
ejpam-5127	349	129	3	3	NUM
ejpam-5127	349	130	5	5	NUM
ejpam-5127	349	131	5	5	NUM
ejpam-5127	349	132	then	then	ADV
ejpam-5127	349	133	(	(	PUNCT
ejpam-5127	349	134	v	v	NOUN
ejpam-5127	349	135	,	,	PUNCT
ejpam-5127	349	136	∨	∨	NUM
ejpam-5127	349	137	∧	∧	PROPN
ejpam-5127	349	138	1	1	NUM
ejpam-5127	349	139	)	)	PUNCT
ejpam-5127	349	140	is	be	AUX
ejpam-5127	349	141	a	a	DET
ejpam-5127	349	142	normal	normal	ADJ
ejpam-5127	349	143	paradistributive	paradistributive	ADJ
ejpam-5127	349	144	latticoid	latticoid	NOUN
ejpam-5127	349	145	.	.	PUNCT
ejpam-5127	350	1	here	here	ADV
ejpam-5127	350	2	{	{	PUNCT
ejpam-5127	350	3	1	1	NUM
ejpam-5127	350	4	,	,	PUNCT
ejpam-5127	350	5	2	2	NUM
ejpam-5127	350	6	}	}	PUNCT
ejpam-5127	350	7	and	and	CCONJ
ejpam-5127	350	8	{	{	PUNCT
ejpam-5127	350	9	1	1	NUM
ejpam-5127	350	10	,	,	PUNCT
ejpam-5127	350	11	0	0	NUM
ejpam-5127	350	12	,	,	PUNCT
ejpam-5127	350	13	4	4	NUM
ejpam-5127	350	14	}	}	PUNCT
ejpam-5127	350	15	are	be	AUX
ejpam-5127	350	16	the	the	DET
ejpam-5127	350	17	only	only	ADJ
ejpam-5127	350	18	prime	prime	ADJ
ejpam-5127	350	19	filters	filter	NOUN
ejpam-5127	350	20	of	of	ADP
ejpam-5127	350	21	v	v	NOUN
ejpam-5127	350	22	.	.	PUNCT
ejpam-5127	350	23	example	example	NOUN
ejpam-5127	351	1	3	3	X
ejpam-5127	351	2	.	.	PUNCT
ejpam-5127	351	3	let	let	VERB
ejpam-5127	351	4	v	v	VERB
ejpam-5127	351	5	=	=	SYM
ejpam-5127	351	6	{	{	PUNCT
ejpam-5127	351	7	0	0	NUM
ejpam-5127	351	8	,	,	PUNCT
ejpam-5127	351	9	1	1	NUM
ejpam-5127	351	10	,	,	PUNCT
ejpam-5127	351	11	2	2	NUM
ejpam-5127	351	12	,	,	PUNCT
ejpam-5127	351	13	3	3	NUM
ejpam-5127	351	14	,	,	PUNCT
ejpam-5127	351	15	4	4	NUM
ejpam-5127	351	16	,	,	PUNCT
ejpam-5127	351	17	5	5	NUM
ejpam-5127	351	18	,	,	PUNCT
ejpam-5127	351	19	6	6	NUM
ejpam-5127	351	20	,	,	PUNCT
ejpam-5127	351	21	7	7	NUM
ejpam-5127	351	22	,	,	PUNCT
ejpam-5127	351	23	8	8	NUM
ejpam-5127	351	24	}	}	PUNCT
ejpam-5127	351	25	be	be	AUX
ejpam-5127	351	26	a	a	DET
ejpam-5127	351	27	set	set	NOUN
ejpam-5127	351	28	with	with	ADP
ejpam-5127	351	29	binary	binary	ADJ
ejpam-5127	351	30	operations	operation	NOUN
ejpam-5127	351	31	∨	∨	NOUN
ejpam-5127	351	32	and	and	CCONJ
ejpam-5127	351	33	∧	∧	NOUN
ejpam-5127	351	34	given	give	VERB
ejpam-5127	351	35	in	in	ADP
ejpam-5127	351	36	the	the	DET
ejpam-5127	351	37	following	following	ADJ
ejpam-5127	351	38	tables	table	NOUN
ejpam-5127	351	39	:	:	PUNCT
ejpam-5127	351	40	∨	∨	NOUN
ejpam-5127	351	41	0	0	NUM
ejpam-5127	351	42	1	1	NUM
ejpam-5127	351	43	2	2	NUM
ejpam-5127	351	44	3	3	NUM
ejpam-5127	351	45	4	4	NUM
ejpam-5127	351	46	5	5	NUM
ejpam-5127	351	47	6	6	NUM
ejpam-5127	351	48	7	7	NUM
ejpam-5127	351	49	8	8	NUM
ejpam-5127	351	50	0	0	NUM
ejpam-5127	351	51	0	0	NUM
ejpam-5127	351	52	1	1	NUM
ejpam-5127	351	53	1	1	NUM
ejpam-5127	351	54	0	0	NUM
ejpam-5127	351	55	0	0	NUM
ejpam-5127	351	56	1	1	NUM
ejpam-5127	351	57	0	0	NUM
ejpam-5127	351	58	0	0	NUM
ejpam-5127	351	59	0	0	NUM
ejpam-5127	351	60	1	1	NUM
ejpam-5127	351	61	1	1	NUM
ejpam-5127	351	62	1	1	NUM
ejpam-5127	351	63	1	1	NUM
ejpam-5127	351	64	1	1	NUM
ejpam-5127	351	65	1	1	NUM
ejpam-5127	351	66	1	1	NUM
ejpam-5127	351	67	1	1	NUM
ejpam-5127	351	68	1	1	NUM
ejpam-5127	351	69	1	1	NUM
ejpam-5127	351	70	2	2	NUM
ejpam-5127	351	71	1	1	NUM
ejpam-5127	351	72	1	1	NUM
ejpam-5127	351	73	2	2	NUM
ejpam-5127	351	74	2	2	NUM
ejpam-5127	351	75	2	2	NUM
ejpam-5127	351	76	2	2	NUM
ejpam-5127	351	77	2	2	NUM
ejpam-5127	351	78	2	2	NUM
ejpam-5127	351	79	2	2	NUM
ejpam-5127	351	80	3	3	NUM
ejpam-5127	351	81	0	0	NUM
ejpam-5127	351	82	1	1	NUM
ejpam-5127	351	83	2	2	NUM
ejpam-5127	351	84	3	3	NUM
ejpam-5127	351	85	3	3	NUM
ejpam-5127	351	86	2	2	NUM
ejpam-5127	351	87	3	3	NUM
ejpam-5127	351	88	3	3	NUM
ejpam-5127	351	89	3	3	NUM
ejpam-5127	351	90	4	4	NUM
ejpam-5127	351	91	0	0	NUM
ejpam-5127	351	92	1	1	NUM
ejpam-5127	351	93	2	2	NUM
ejpam-5127	351	94	3	3	NUM
ejpam-5127	351	95	4	4	NUM
ejpam-5127	351	96	2	2	NUM
ejpam-5127	351	97	3	3	NUM
ejpam-5127	351	98	4	4	NUM
ejpam-5127	351	99	4	4	NUM
ejpam-5127	351	100	5	5	NUM
ejpam-5127	351	101	1	1	NUM
ejpam-5127	351	102	1	1	NUM
ejpam-5127	351	103	5	5	NUM
ejpam-5127	351	104	5	5	NUM
ejpam-5127	351	105	5	5	NUM
ejpam-5127	351	106	5	5	NUM
ejpam-5127	351	107	5	5	NUM
ejpam-5127	351	108	5	5	NUM
ejpam-5127	351	109	5	5	NUM
ejpam-5127	351	110	6	6	NUM
ejpam-5127	351	111	0	0	NUM
ejpam-5127	351	112	1	1	NUM
ejpam-5127	351	113	5	5	NUM
ejpam-5127	351	114	6	6	NUM
ejpam-5127	351	115	6	6	NUM
ejpam-5127	351	116	5	5	NUM
ejpam-5127	351	117	6	6	NUM
ejpam-5127	351	118	6	6	NUM
ejpam-5127	351	119	6	6	NUM
ejpam-5127	351	120	7	7	NUM
ejpam-5127	351	121	0	0	NUM
ejpam-5127	351	122	1	1	NUM
ejpam-5127	351	123	5	5	NUM
ejpam-5127	351	124	6	6	NUM
ejpam-5127	351	125	7	7	NUM
ejpam-5127	351	126	5	5	NUM
ejpam-5127	351	127	6	6	NUM
ejpam-5127	351	128	7	7	NUM
ejpam-5127	351	129	7	7	NUM
ejpam-5127	351	130	8	8	NUM
ejpam-5127	351	131	0	0	NUM
ejpam-5127	351	132	1	1	NUM
ejpam-5127	351	133	5	5	NUM
ejpam-5127	351	134	6	6	NUM
ejpam-5127	351	135	8	8	NUM
ejpam-5127	351	136	5	5	NUM
ejpam-5127	351	137	6	6	NUM
ejpam-5127	351	138	8	8	NUM
ejpam-5127	351	139	8	8	NUM
ejpam-5127	351	140	∧	∧	NOUN
ejpam-5127	351	141	0	0	NUM
ejpam-5127	351	142	1	1	NUM
ejpam-5127	351	143	2	2	NUM
ejpam-5127	351	144	3	3	NUM
ejpam-5127	351	145	4	4	NUM
ejpam-5127	351	146	5	5	NUM
ejpam-5127	351	147	6	6	NUM
ejpam-5127	351	148	7	7	NUM
ejpam-5127	351	149	8	8	NUM
ejpam-5127	351	150	0	0	NUM
ejpam-5127	351	151	0	0	NUM
ejpam-5127	351	152	0	0	NUM
ejpam-5127	351	153	3	3	NUM
ejpam-5127	351	154	3	3	NUM
ejpam-5127	351	155	4	4	NUM
ejpam-5127	351	156	6	6	NUM
ejpam-5127	351	157	6	6	NUM
ejpam-5127	351	158	7	7	NUM
ejpam-5127	351	159	8	8	NUM
ejpam-5127	351	160	1	1	NUM
ejpam-5127	351	161	0	0	NUM
ejpam-5127	351	162	1	1	NUM
ejpam-5127	351	163	2	2	NUM
ejpam-5127	351	164	3	3	NUM
ejpam-5127	351	165	4	4	NUM
ejpam-5127	351	166	5	5	NUM
ejpam-5127	351	167	6	6	NUM
ejpam-5127	351	168	7	7	NUM
ejpam-5127	351	169	8	8	NUM
ejpam-5127	351	170	2	2	NUM
ejpam-5127	351	171	3	3	NUM
ejpam-5127	351	172	2	2	NUM
ejpam-5127	351	173	2	2	NUM
ejpam-5127	351	174	3	3	NUM
ejpam-5127	351	175	4	4	NUM
ejpam-5127	351	176	5	5	NUM
ejpam-5127	351	177	6	6	NUM
ejpam-5127	351	178	7	7	NUM
ejpam-5127	351	179	8	8	NUM
ejpam-5127	351	180	3	3	NUM
ejpam-5127	351	181	3	3	NUM
ejpam-5127	351	182	3	3	NUM
ejpam-5127	351	183	3	3	NUM
ejpam-5127	351	184	3	3	NUM
ejpam-5127	351	185	4	4	NUM
ejpam-5127	351	186	6	6	NUM
ejpam-5127	351	187	6	6	NUM
ejpam-5127	351	188	7	7	NUM
ejpam-5127	351	189	8	8	NUM
ejpam-5127	351	190	4	4	NUM
ejpam-5127	351	191	4	4	NUM
ejpam-5127	351	192	4	4	NUM
ejpam-5127	351	193	4	4	NUM
ejpam-5127	351	194	4	4	NUM
ejpam-5127	351	195	4	4	NUM
ejpam-5127	351	196	7	7	NUM
ejpam-5127	351	197	8	8	NUM
ejpam-5127	351	198	7	7	NUM
ejpam-5127	351	199	8	8	NUM
ejpam-5127	351	200	5	5	NUM
ejpam-5127	351	201	6	6	NUM
ejpam-5127	351	202	5	5	NUM
ejpam-5127	351	203	2	2	NUM
ejpam-5127	351	204	3	3	NUM
ejpam-5127	351	205	4	4	NUM
ejpam-5127	351	206	5	5	NUM
ejpam-5127	351	207	6	6	NUM
ejpam-5127	351	208	7	7	NUM
ejpam-5127	351	209	8	8	NUM
ejpam-5127	351	210	6	6	NUM
ejpam-5127	351	211	6	6	NUM
ejpam-5127	351	212	6	6	NUM
ejpam-5127	351	213	3	3	NUM
ejpam-5127	351	214	3	3	NUM
ejpam-5127	351	215	4	4	NUM
ejpam-5127	351	216	6	6	NUM
ejpam-5127	351	217	6	6	NUM
ejpam-5127	351	218	7	7	NUM
ejpam-5127	351	219	8	8	NUM
ejpam-5127	351	220	7	7	NUM
ejpam-5127	351	221	7	7	NUM
ejpam-5127	351	222	7	7	NUM
ejpam-5127	351	223	4	4	NUM
ejpam-5127	351	224	4	4	NUM
ejpam-5127	351	225	4	4	NUM
ejpam-5127	351	226	7	7	NUM
ejpam-5127	351	227	7	7	NUM
ejpam-5127	351	228	7	7	NUM
ejpam-5127	351	229	8	8	NUM
ejpam-5127	351	230	8	8	NUM
ejpam-5127	351	231	8	8	NUM
ejpam-5127	351	232	8	8	NUM
ejpam-5127	351	233	4	4	NUM
ejpam-5127	351	234	4	4	NUM
ejpam-5127	351	235	4	4	NUM
ejpam-5127	351	236	8	8	NUM
ejpam-5127	351	237	8	8	NUM
ejpam-5127	351	238	7	7	NUM
ejpam-5127	351	239	8	8	NUM
ejpam-5127	351	240	then	then	ADV
ejpam-5127	351	241	(	(	PUNCT
ejpam-5127	351	242	v	v	NOUN
ejpam-5127	351	243	,	,	PUNCT
ejpam-5127	351	244	∨	∨	NUM
ejpam-5127	351	245	∧	∧	PROPN
ejpam-5127	351	246	1	1	NUM
ejpam-5127	351	247	)	)	PUNCT
ejpam-5127	351	248	is	be	AUX
ejpam-5127	351	249	not	not	PART
ejpam-5127	351	250	a	a	DET
ejpam-5127	351	251	normal	normal	ADJ
ejpam-5127	351	252	paradistributive	paradistributive	ADJ
ejpam-5127	351	253	latticoid	latticoid	NOUN
ejpam-5127	351	254	.	.	PUNCT
ejpam-5127	352	1	here	here	ADV
ejpam-5127	352	2	{	{	PUNCT
ejpam-5127	352	3	0	0	NUM
ejpam-5127	352	4	,	,	PUNCT
ejpam-5127	352	5	1	1	NUM
ejpam-5127	352	6	}	}	PUNCT
ejpam-5127	352	7	,	,	PUNCT
ejpam-5127	352	8	{	{	PUNCT
ejpam-5127	352	9	1	1	NUM
ejpam-5127	352	10	,	,	PUNCT
ejpam-5127	352	11	2	2	NUM
ejpam-5127	352	12	,	,	PUNCT
ejpam-5127	352	13	5	5	NUM
ejpam-5127	352	14	}	}	PUNCT
ejpam-5127	352	15	and	and	CCONJ
ejpam-5127	352	16	{	{	PUNCT
ejpam-5127	352	17	0	0	NUM
ejpam-5127	352	18	,	,	PUNCT
ejpam-5127	352	19	1	1	NUM
ejpam-5127	352	20	,	,	PUNCT
ejpam-5127	352	21	2	2	NUM
ejpam-5127	352	22	,	,	PUNCT
ejpam-5127	352	23	3	3	NUM
ejpam-5127	352	24	,	,	PUNCT
ejpam-5127	352	25	5	5	NUM
ejpam-5127	352	26	,	,	PUNCT
ejpam-5127	352	27	6	6	NUM
ejpam-5127	352	28	}	}	PUNCT
ejpam-5127	352	29	are	be	AUX
ejpam-5127	352	30	the	the	DET
ejpam-5127	352	31	only	only	ADJ
ejpam-5127	352	32	prime	prime	ADJ
ejpam-5127	352	33	filters	filter	NOUN
ejpam-5127	352	34	of	of	ADP
ejpam-5127	352	35	v	v	NUM
ejpam-5127	352	36	.	.	PUNCT
ejpam-5127	353	1	remark	remark	NOUN
ejpam-5127	353	2	1	1	NUM
ejpam-5127	353	3	.	.	PUNCT
ejpam-5127	354	1	:	:	PUNCT
ejpam-5127	354	2	if	if	SCONJ
ejpam-5127	354	3	v	v	NOUN
ejpam-5127	354	4	is	be	AUX
ejpam-5127	354	5	a	a	DET
ejpam-5127	354	6	pdl	pdl	NOUN
ejpam-5127	354	7	in	in	ADP
ejpam-5127	354	8	which	which	PRON
ejpam-5127	354	9	a∨	a∨	PROPN
ejpam-5127	354	10	b	b	PROPN
ejpam-5127	354	11	̸=	̸=	PROPN
ejpam-5127	354	12	1	1	NUM
ejpam-5127	354	13	for	for	ADP
ejpam-5127	354	14	all	all	DET
ejpam-5127	354	15	a	a	DET
ejpam-5127	354	16	̸=	̸=	PROPN
ejpam-5127	354	17	1	1	NUM
ejpam-5127	354	18	and	and	CCONJ
ejpam-5127	354	19	b	b	PROPN
ejpam-5127	354	20	̸=	̸=	PROPN
ejpam-5127	354	21	1	1	NUM
ejpam-5127	354	22	,	,	PUNCT
ejpam-5127	354	23	then	then	ADV
ejpam-5127	354	24	v	v	NOUN
ejpam-5127	354	25	is	be	AUX
ejpam-5127	354	26	normal	normal	ADJ
ejpam-5127	354	27	(	(	PUNCT
ejpam-5127	354	28	since	since	SCONJ
ejpam-5127	354	29	{	{	PUNCT
ejpam-5127	354	30	1	1	NUM
ejpam-5127	354	31	}	}	PUNCT
ejpam-5127	354	32	becomes	become	VERB
ejpam-5127	354	33	the	the	DET
ejpam-5127	354	34	unique	unique	ADJ
ejpam-5127	354	35	minimal	minimal	ADJ
ejpam-5127	354	36	prime	prime	ADJ
ejpam-5127	354	37	filter	filter	NOUN
ejpam-5127	354	38	)	)	PUNCT
ejpam-5127	354	39	.	.	PUNCT
ejpam-5127	355	1	proposition	proposition	NOUN
ejpam-5127	355	2	1	1	NUM
ejpam-5127	355	3	.	.	PUNCT
ejpam-5127	356	1	let	let	VERB
ejpam-5127	356	2	v	v	PART
ejpam-5127	356	3	be	be	AUX
ejpam-5127	356	4	a	a	DET
ejpam-5127	356	5	paradistributive	paradistributive	ADJ
ejpam-5127	356	6	latticoid	latticoid	NOUN
ejpam-5127	356	7	with	with	ADP
ejpam-5127	356	8	minimal	minimal	ADJ
ejpam-5127	356	9	elements	element	NOUN
ejpam-5127	356	10	.	.	PUNCT
ejpam-5127	357	1	then	then	ADV
ejpam-5127	357	2	the	the	DET
ejpam-5127	357	3	following	follow	VERB
ejpam-5127	357	4	assertions	assertion	NOUN
ejpam-5127	357	5	are	be	AUX
ejpam-5127	357	6	equivalent	equivalent	ADJ
ejpam-5127	357	7	.	.	PUNCT
ejpam-5127	358	1	(	(	PUNCT
ejpam-5127	358	2	1	1	X
ejpam-5127	358	3	)	)	PUNCT
ejpam-5127	358	4	any	any	DET
ejpam-5127	358	5	two	two	NUM
ejpam-5127	358	6	distinct	distinct	ADJ
ejpam-5127	358	7	minimal	minimal	ADJ
ejpam-5127	358	8	prime	prime	ADJ
ejpam-5127	358	9	filters	filter	NOUN
ejpam-5127	358	10	are	be	AUX
ejpam-5127	358	11	co	co	ADJ
ejpam-5127	358	12	-	-	ADJ
ejpam-5127	358	13	maximal	maximal	ADJ
ejpam-5127	358	14	,	,	PUNCT
ejpam-5127	358	15	(	(	PUNCT
ejpam-5127	358	16	2	2	X
ejpam-5127	358	17	)	)	PUNCT
ejpam-5127	358	18	v	v	NOUN
ejpam-5127	358	19	is	be	AUX
ejpam-5127	358	20	normal	normal	ADJ
ejpam-5127	358	21	,	,	PUNCT
ejpam-5127	358	22	(	(	PUNCT
ejpam-5127	358	23	3	3	X
ejpam-5127	358	24	)	)	PUNCT
ejpam-5127	358	25	for	for	ADP
ejpam-5127	358	26	any	any	DET
ejpam-5127	358	27	prime	prime	ADJ
ejpam-5127	358	28	filter	filter	NOUN
ejpam-5127	358	29	p	p	NOUN
ejpam-5127	358	30	,	,	PUNCT
ejpam-5127	358	31	o(p	o(p	PROPN
ejpam-5127	358	32	)	)	PUNCT
ejpam-5127	358	33	is	be	AUX
ejpam-5127	358	34	a	a	DET
ejpam-5127	358	35	minimal	minimal	ADJ
ejpam-5127	358	36	prime	prime	ADJ
ejpam-5127	358	37	filter	filter	NOUN
ejpam-5127	358	38	,	,	PUNCT
ejpam-5127	358	39	(	(	PUNCT
ejpam-5127	358	40	4	4	NUM
ejpam-5127	358	41	)	)	PUNCT
ejpam-5127	358	42	for	for	ADP
ejpam-5127	358	43	any	any	DET
ejpam-5127	358	44	x	x	NOUN
ejpam-5127	358	45	,	,	PUNCT
ejpam-5127	358	46	y	y	PROPN
ejpam-5127	358	47	∈	∈	PROPN
ejpam-5127	358	48	v	v	NOUN
ejpam-5127	358	49	,	,	PUNCT
ejpam-5127	358	50	x	x	PROPN
ejpam-5127	358	51	∨	∨	NUM
ejpam-5127	358	52	y	y	NOUN
ejpam-5127	358	53	=	=	SYM
ejpam-5127	358	54	1	1	NUM
ejpam-5127	358	55	=	=	NOUN
ejpam-5127	358	56	⇒	⇒	NOUN
ejpam-5127	358	57	(	(	PUNCT
ejpam-5127	358	58	x)•	x)•	PROPN
ejpam-5127	358	59	∨	∨	NUM
ejpam-5127	358	60	(	(	PUNCT
ejpam-5127	358	61	y)•	y)•	NUM
ejpam-5127	358	62	=	=	SYM
ejpam-5127	358	63	v	v	PROPN
ejpam-5127	358	64	,	,	PUNCT
ejpam-5127	358	65	r.	r.	PROPN
ejpam-5127	358	66	shukla	shukla	PROPN
ejpam-5127	358	67	et	et	PROPN
ejpam-5127	358	68	al	al	PROPN
ejpam-5127	358	69	.	.	PUNCT
ejpam-5127	358	70	/	/	SYM
ejpam-5127	358	71	eur	eur	PROPN
ejpam-5127	358	72	.	.	PUNCT
ejpam-5127	359	1	j.	j.	PROPN
ejpam-5127	359	2	pure	pure	PROPN
ejpam-5127	359	3	appl	appl	PROPN
ejpam-5127	359	4	.	.	PROPN
ejpam-5127	359	5	math	math	PROPN
ejpam-5127	359	6	,	,	PUNCT
ejpam-5127	359	7	17	17	NUM
ejpam-5127	359	8	(	(	PUNCT
ejpam-5127	359	9	2	2	NUM
ejpam-5127	359	10	)	)	PUNCT
ejpam-5127	359	11	(	(	PUNCT
ejpam-5127	359	12	2024	2024	NUM
ejpam-5127	359	13	)	)	PUNCT
ejpam-5127	359	14	,	,	PUNCT
ejpam-5127	359	15	1306	1306	NUM
ejpam-5127	359	16	-	-	SYM
ejpam-5127	359	17	1320	1320	NUM
ejpam-5127	359	18	1316	1316	NUM
ejpam-5127	359	19	(	(	PUNCT
ejpam-5127	359	20	5	5	NUM
ejpam-5127	359	21	)	)	PUNCT
ejpam-5127	359	22	for	for	ADP
ejpam-5127	359	23	any	any	DET
ejpam-5127	359	24	x	x	NOUN
ejpam-5127	359	25	,	,	PUNCT
ejpam-5127	359	26	y	y	PROPN
ejpam-5127	359	27	∈	∈	PROPN
ejpam-5127	359	28	v	v	NOUN
ejpam-5127	359	29	,	,	PUNCT
ejpam-5127	359	30	(	(	PUNCT
ejpam-5127	359	31	x	x	X
ejpam-5127	359	32	∨	∨	NOUN
ejpam-5127	359	33	y)•	y)•	NUM
ejpam-5127	359	34	=	=	SYM
ejpam-5127	359	35	(	(	PUNCT
ejpam-5127	359	36	x)•	x)•	PROPN
ejpam-5127	359	37	∨	∨	NUM
ejpam-5127	359	38	(	(	PUNCT
ejpam-5127	359	39	y)•	y)•	NUM
ejpam-5127	359	40	,	,	PUNCT
ejpam-5127	359	41	(	(	PUNCT
ejpam-5127	359	42	6	6	NUM
ejpam-5127	359	43	)	)	PUNCT
ejpam-5127	359	44	for	for	ADP
ejpam-5127	359	45	any	any	DET
ejpam-5127	359	46	x	x	NOUN
ejpam-5127	359	47	,	,	PUNCT
ejpam-5127	359	48	y	y	PROPN
ejpam-5127	359	49	∈	∈	PROPN
ejpam-5127	359	50	v	v	NOUN
ejpam-5127	359	51	,	,	PUNCT
ejpam-5127	359	52	x	x	PROPN
ejpam-5127	359	53	∨	∨	NUM
ejpam-5127	359	54	y	y	NOUN
ejpam-5127	359	55	=	=	SYM
ejpam-5127	359	56	1	1	NUM
ejpam-5127	359	57	implies	imply	VERB
ejpam-5127	359	58	that	that	SCONJ
ejpam-5127	359	59	there	there	PRON
ejpam-5127	359	60	exists	exist	VERB
ejpam-5127	359	61	u	u	PROPN
ejpam-5127	359	62	∈	∈	PROPN
ejpam-5127	359	63	(	(	PUNCT
ejpam-5127	359	64	x)•	x)•	PROPN
ejpam-5127	359	65	and	and	CCONJ
ejpam-5127	359	66	v	v	ADP
ejpam-5127	359	67	∈	∈	PROPN
ejpam-5127	359	68	(	(	PUNCT
ejpam-5127	359	69	y)•	y)•	NUM
ejpam-5127	359	70	such	such	ADJ
ejpam-5127	359	71	that	that	SCONJ
ejpam-5127	359	72	u	u	PROPN
ejpam-5127	359	73	∧	∧	PROPN
ejpam-5127	359	74	v	v	NOUN
ejpam-5127	359	75	is	be	AUX
ejpam-5127	359	76	minimal	minimal	ADJ
ejpam-5127	359	77	.	.	PUNCT
ejpam-5127	360	1	proof	proof	NOUN
ejpam-5127	360	2	.	.	PUNCT
ejpam-5127	361	1	(	(	PUNCT
ejpam-5127	361	2	1	1	X
ejpam-5127	361	3	)	)	PUNCT
ejpam-5127	361	4	⇒	⇒	NOUN
ejpam-5127	361	5	(	(	PUNCT
ejpam-5127	361	6	2	2	X
ejpam-5127	361	7	)	)	PUNCT
ejpam-5127	361	8	assume	assume	VERB
ejpam-5127	361	9	(	(	PUNCT
ejpam-5127	361	10	1	1	NUM
ejpam-5127	361	11	)	)	PUNCT
ejpam-5127	361	12	.	.	PUNCT
ejpam-5127	362	1	let	let	VERB
ejpam-5127	362	2	p	p	PRON
ejpam-5127	362	3	be	be	AUX
ejpam-5127	362	4	a	a	DET
ejpam-5127	362	5	prime	prime	ADJ
ejpam-5127	362	6	filter	filter	NOUN
ejpam-5127	362	7	of	of	ADP
ejpam-5127	362	8	v	v	NOUN
ejpam-5127	362	9	.	.	PUNCT
ejpam-5127	363	1	suppose	suppose	VERB
ejpam-5127	363	2	that	that	SCONJ
ejpam-5127	363	3	q1,q2	q1,q2	PROPN
ejpam-5127	363	4	are	be	AUX
ejpam-5127	363	5	two	two	NUM
ejpam-5127	363	6	distinct	distinct	ADJ
ejpam-5127	363	7	minimal	minimal	ADJ
ejpam-5127	363	8	prime	prime	ADJ
ejpam-5127	363	9	filters	filter	NOUN
ejpam-5127	363	10	of	of	ADP
ejpam-5127	363	11	v	v	NOUN
ejpam-5127	363	12	contained	contain	VERB
ejpam-5127	363	13	in	in	ADP
ejpam-5127	363	14	p.	p.	NOUN
ejpam-5127	363	15	then	then	ADV
ejpam-5127	363	16	q1	q1	PROPN
ejpam-5127	363	17	∨	∨	PROPN
ejpam-5127	363	18	q2	q2	PROPN
ejpam-5127	363	19	⊆	⊆	NUM
ejpam-5127	364	1	p.	p.	NOUN
ejpam-5127	364	2	but	but	CCONJ
ejpam-5127	364	3	from	from	ADP
ejpam-5127	364	4	our	our	PRON
ejpam-5127	364	5	assumption	assumption	NOUN
ejpam-5127	364	6	,	,	PUNCT
ejpam-5127	364	7	we	we	PRON
ejpam-5127	364	8	have	have	VERB
ejpam-5127	364	9	q1∨q2	q1∨q2	NOUN
ejpam-5127	364	10	=	=	SYM
ejpam-5127	364	11	v	v	NOUN
ejpam-5127	364	12	.	.	PUNCT
ejpam-5127	365	1	therefore	therefore	ADV
ejpam-5127	365	2	,	,	PUNCT
ejpam-5127	365	3	we	we	PRON
ejpam-5127	365	4	get	get	VERB
ejpam-5127	365	5	v	v	ADP
ejpam-5127	365	6	⊆	⊆	NUM
ejpam-5127	365	7	p.	p.	NOUN
ejpam-5127	365	8	this	this	PRON
ejpam-5127	365	9	is	be	AUX
ejpam-5127	365	10	a	a	DET
ejpam-5127	365	11	contradiction	contradiction	NOUN
ejpam-5127	365	12	.	.	PUNCT
ejpam-5127	366	1	thus	thus	ADV
ejpam-5127	366	2	p	p	X
ejpam-5127	366	3	contains	contain	VERB
ejpam-5127	366	4	a	a	DET
ejpam-5127	366	5	unique	unique	ADJ
ejpam-5127	366	6	minimal	minimal	ADJ
ejpam-5127	366	7	prime	prime	ADJ
ejpam-5127	366	8	filter	filter	NOUN
ejpam-5127	366	9	of	of	ADP
ejpam-5127	366	10	v	v	NOUN
ejpam-5127	366	11	.	.	PUNCT
ejpam-5127	367	1	hence	hence	ADV
ejpam-5127	367	2	v	v	NOUN
ejpam-5127	367	3	is	be	AUX
ejpam-5127	367	4	normal	normal	ADJ
ejpam-5127	367	5	.	.	PUNCT
ejpam-5127	368	1	(	(	PUNCT
ejpam-5127	368	2	2	2	X
ejpam-5127	368	3	)	)	PUNCT
ejpam-5127	368	4	⇒	⇒	NOUN
ejpam-5127	368	5	(	(	PUNCT
ejpam-5127	368	6	3	3	X
ejpam-5127	368	7	)	)	PUNCT
ejpam-5127	368	8	assume	assume	VERB
ejpam-5127	368	9	(	(	PUNCT
ejpam-5127	368	10	2	2	NUM
ejpam-5127	368	11	)	)	PUNCT
ejpam-5127	368	12	.	.	PUNCT
ejpam-5127	369	1	let	let	VERB
ejpam-5127	369	2	p	p	PRON
ejpam-5127	369	3	be	be	AUX
ejpam-5127	369	4	a	a	DET
ejpam-5127	369	5	prime	prime	ADJ
ejpam-5127	369	6	filter	filter	NOUN
ejpam-5127	369	7	of	of	ADP
ejpam-5127	369	8	v	v	NOUN
ejpam-5127	369	9	.	.	PUNCT
ejpam-5127	370	1	then	then	ADV
ejpam-5127	370	2	o(p	o(p	PROPN
ejpam-5127	370	3	)	)	PUNCT
ejpam-5127	370	4	is	be	AUX
ejpam-5127	370	5	a	a	DET
ejpam-5127	370	6	filter	filter	NOUN
ejpam-5127	370	7	of	of	ADP
ejpam-5127	370	8	v	v	NOUN
ejpam-5127	370	9	contained	contain	VERB
ejpam-5127	370	10	in	in	ADP
ejpam-5127	370	11	p.	p.	NOUN
ejpam-5127	370	12	since	since	SCONJ
ejpam-5127	370	13	v	v	NOUN
ejpam-5127	370	14	is	be	AUX
ejpam-5127	370	15	normal	normal	ADJ
ejpam-5127	370	16	,	,	PUNCT
ejpam-5127	370	17	p	p	NOUN
ejpam-5127	370	18	contains	contain	VERB
ejpam-5127	370	19	a	a	DET
ejpam-5127	370	20	unique	unique	ADJ
ejpam-5127	370	21	minimal	minimal	ADJ
ejpam-5127	370	22	prime	prime	ADJ
ejpam-5127	370	23	filter	filter	NOUN
ejpam-5127	370	24	,	,	PUNCT
ejpam-5127	370	25	say	say	VERB
ejpam-5127	370	26	q.	q.	NOUN
ejpam-5127	370	27	we	we	PRON
ejpam-5127	370	28	know	know	VERB
ejpam-5127	370	29	that	that	SCONJ
ejpam-5127	370	30	o(p	o(p	PROPN
ejpam-5127	370	31	)	)	PUNCT
ejpam-5127	370	32	is	be	AUX
ejpam-5127	370	33	the	the	DET
ejpam-5127	370	34	intersection	intersection	NOUN
ejpam-5127	370	35	of	of	ADP
ejpam-5127	370	36	all	all	DET
ejpam-5127	370	37	the	the	DET
ejpam-5127	370	38	minimal	minimal	ADJ
ejpam-5127	370	39	prime	prime	ADJ
ejpam-5127	370	40	filters	filter	NOUN
ejpam-5127	370	41	of	of	ADP
ejpam-5127	370	42	v	v	NOUN
ejpam-5127	370	43	contained	contain	VERB
ejpam-5127	370	44	in	in	ADP
ejpam-5127	370	45	p.	p.	NOUN
ejpam-5127	370	46	hence	hence	ADV
ejpam-5127	370	47	o(p	o(p	PROPN
ejpam-5127	370	48	)	)	PUNCT
ejpam-5127	371	1	=	=	SYM
ejpam-5127	371	2	q.	q.	PROPN
ejpam-5127	371	3	therefore	therefore	ADV
ejpam-5127	371	4	o(p	o(p	PROPN
ejpam-5127	371	5	)	)	PUNCT
ejpam-5127	371	6	is	be	AUX
ejpam-5127	371	7	the	the	DET
ejpam-5127	371	8	minimal	minimal	ADJ
ejpam-5127	371	9	prime	prime	ADJ
ejpam-5127	371	10	filter	filter	NOUN
ejpam-5127	371	11	of	of	ADP
ejpam-5127	371	12	v	v	NUM
ejpam-5127	371	13	.	.	PUNCT
ejpam-5127	372	1	(	(	PUNCT
ejpam-5127	372	2	3	3	X
ejpam-5127	372	3	)	)	PUNCT
ejpam-5127	372	4	⇒	⇒	NOUN
ejpam-5127	372	5	(	(	PUNCT
ejpam-5127	372	6	4	4	X
ejpam-5127	372	7	)	)	PUNCT
ejpam-5127	372	8	assume	assume	VERB
ejpam-5127	372	9	(	(	PUNCT
ejpam-5127	372	10	3	3	NUM
ejpam-5127	372	11	)	)	PUNCT
ejpam-5127	372	12	.	.	PUNCT
ejpam-5127	373	1	let	let	VERB
ejpam-5127	373	2	x	x	PRON
ejpam-5127	373	3	,	,	PUNCT
ejpam-5127	373	4	y	y	PROPN
ejpam-5127	373	5	∈	∈	PROPN
ejpam-5127	373	6	v	v	NOUN
ejpam-5127	373	7	and	and	CCONJ
ejpam-5127	373	8	x	x	PROPN
ejpam-5127	373	9	∨	∨	NUM
ejpam-5127	374	1	y	y	NOUN
ejpam-5127	374	2	=	=	SYM
ejpam-5127	374	3	1	1	X
ejpam-5127	374	4	.	.	X
ejpam-5127	375	1	we	we	PRON
ejpam-5127	375	2	have	have	VERB
ejpam-5127	375	3	to	to	PART
ejpam-5127	375	4	prove	prove	VERB
ejpam-5127	375	5	that	that	SCONJ
ejpam-5127	375	6	(	(	PUNCT
ejpam-5127	375	7	x)•	x)•	PROPN
ejpam-5127	375	8	∨	∨	NUM
ejpam-5127	375	9	(	(	PUNCT
ejpam-5127	375	10	y)•	y)•	NUM
ejpam-5127	375	11	=	=	PUNCT
ejpam-5127	375	12	v	v	NOUN
ejpam-5127	375	13	.	.	PUNCT
ejpam-5127	376	1	suppose	suppose	VERB
ejpam-5127	376	2	(	(	PUNCT
ejpam-5127	376	3	x)•	x)•	PROPN
ejpam-5127	376	4	∨	∨	NUM
ejpam-5127	376	5	(	(	PUNCT
ejpam-5127	376	6	y)•	y)•	NUM
ejpam-5127	376	7	̸=	̸=	PROPN
ejpam-5127	376	8	v	v	NOUN
ejpam-5127	376	9	.	.	PUNCT
ejpam-5127	377	1	then	then	ADV
ejpam-5127	377	2	there	there	PRON
ejpam-5127	377	3	exists	exist	VERB
ejpam-5127	377	4	a	a	DET
ejpam-5127	377	5	maximal	maximal	ADJ
ejpam-5127	377	6	filter	filter	NOUN
ejpam-5127	377	7	m	m	NOUN
ejpam-5127	377	8	in	in	ADP
ejpam-5127	377	9	v	v	NUM
ejpam-5127	377	10	such	such	ADJ
ejpam-5127	377	11	that	that	PRON
ejpam-5127	377	12	(	(	PUNCT
ejpam-5127	377	13	x)•	x)•	PROPN
ejpam-5127	377	14	∨	∨	NUM
ejpam-5127	377	15	(	(	PUNCT
ejpam-5127	377	16	y)•	y)•	NUM
ejpam-5127	377	17	⊆	⊆	NUM
ejpam-5127	377	18	m.	m.	NOUN
ejpam-5127	377	19	since	since	SCONJ
ejpam-5127	377	20	m	m	PROPN
ejpam-5127	377	21	is	be	AUX
ejpam-5127	377	22	a	a	DET
ejpam-5127	377	23	prime	prime	ADJ
ejpam-5127	377	24	filter	filter	NOUN
ejpam-5127	377	25	of	of	ADP
ejpam-5127	377	26	v	v	NOUN
ejpam-5127	377	27	,	,	PUNCT
ejpam-5127	377	28	from	from	ADP
ejpam-5127	377	29	condition	condition	NOUN
ejpam-5127	377	30	(	(	PUNCT
ejpam-5127	377	31	3	3	NUM
ejpam-5127	377	32	)	)	PUNCT
ejpam-5127	377	33	,	,	PUNCT
ejpam-5127	377	34	we	we	PRON
ejpam-5127	377	35	get	get	VERB
ejpam-5127	377	36	o(m	o(m	NOUN
ejpam-5127	377	37	)	)	PUNCT
ejpam-5127	377	38	is	be	AUX
ejpam-5127	377	39	a	a	DET
ejpam-5127	377	40	prime	prime	ADJ
ejpam-5127	377	41	filter	filter	NOUN
ejpam-5127	377	42	of	of	ADP
ejpam-5127	377	43	v	v	NOUN
ejpam-5127	377	44	.	.	PUNCT
ejpam-5127	378	1	now	now	ADV
ejpam-5127	378	2	x	x	X
ejpam-5127	378	3	∨	∨	NUM
ejpam-5127	378	4	y	y	NOUN
ejpam-5127	378	5	=	=	SYM
ejpam-5127	378	6	1	1	NUM
ejpam-5127	378	7	∈	∈	PROPN
ejpam-5127	378	8	o(m	o(m	NOUN
ejpam-5127	378	9	)	)	PUNCT
ejpam-5127	378	10	implies	imply	VERB
ejpam-5127	378	11	that	that	SCONJ
ejpam-5127	378	12	either	either	CCONJ
ejpam-5127	378	13	x	x	PROPN
ejpam-5127	378	14	∈	∈	PROPN
ejpam-5127	378	15	o(m	o(m	NOUN
ejpam-5127	378	16	)	)	PUNCT
ejpam-5127	378	17	or	or	CCONJ
ejpam-5127	378	18	y	y	PROPN
ejpam-5127	378	19	∈	∈	PROPN
ejpam-5127	378	20	o(m	o(m	PROPN
ejpam-5127	378	21	)	)	PUNCT
ejpam-5127	378	22	.	.	PUNCT
ejpam-5127	379	1	if	if	SCONJ
ejpam-5127	379	2	x	x	PROPN
ejpam-5127	379	3	∈	∈	PROPN
ejpam-5127	379	4	o(m	o(m	PROPN
ejpam-5127	379	5	)	)	PUNCT
ejpam-5127	379	6	then	then	ADV
ejpam-5127	379	7	(	(	PUNCT
ejpam-5127	379	8	x)•	x)•	PROPN
ejpam-5127	379	9	⊈	⊈	PROPN
ejpam-5127	379	10	m.	m.	NOUN
ejpam-5127	379	11	this	this	PRON
ejpam-5127	379	12	is	be	AUX
ejpam-5127	379	13	not	not	PART
ejpam-5127	379	14	possible	possible	ADJ
ejpam-5127	379	15	.	.	PUNCT
ejpam-5127	380	1	therefore	therefore	ADV
ejpam-5127	380	2	x	x	X
ejpam-5127	380	3	/∈	/∈	PUNCT
ejpam-5127	380	4	o(m	o(m	PROPN
ejpam-5127	380	5	)	)	PUNCT
ejpam-5127	380	6	.	.	PUNCT
ejpam-5127	381	1	similarly	similarly	ADV
ejpam-5127	381	2	,	,	PUNCT
ejpam-5127	381	3	we	we	PRON
ejpam-5127	381	4	get	get	VERB
ejpam-5127	381	5	y	y	PROPN
ejpam-5127	381	6	/∈	/∈	PUNCT
ejpam-5127	381	7	o(m	o(m	PROPN
ejpam-5127	381	8	)	)	PUNCT
ejpam-5127	381	9	.	.	PUNCT
ejpam-5127	382	1	this	this	PRON
ejpam-5127	382	2	is	be	AUX
ejpam-5127	382	3	a	a	DET
ejpam-5127	382	4	contradiction	contradiction	NOUN
ejpam-5127	382	5	.	.	PUNCT
ejpam-5127	383	1	therefore	therefore	ADV
ejpam-5127	383	2	,	,	PUNCT
ejpam-5127	383	3	(	(	PUNCT
ejpam-5127	383	4	x)•	x)•	PROPN
ejpam-5127	383	5	∨	∨	NUM
ejpam-5127	383	6	(	(	PUNCT
ejpam-5127	383	7	y)•	y)•	NUM
ejpam-5127	383	8	=	=	PUNCT
ejpam-5127	383	9	v	v	NOUN
ejpam-5127	383	10	.	.	PUNCT
ejpam-5127	384	1	(	(	PUNCT
ejpam-5127	384	2	4	4	X
ejpam-5127	384	3	)	)	PUNCT
ejpam-5127	384	4	⇒	⇒	NOUN
ejpam-5127	384	5	(	(	PUNCT
ejpam-5127	384	6	5	5	X
ejpam-5127	384	7	)	)	PUNCT
ejpam-5127	384	8	assume	assume	VERB
ejpam-5127	384	9	(	(	PUNCT
ejpam-5127	384	10	4	4	NUM
ejpam-5127	384	11	)	)	PUNCT
ejpam-5127	384	12	.	.	PUNCT
ejpam-5127	385	1	let	let	VERB
ejpam-5127	385	2	a	a	DET
ejpam-5127	385	3	∈	∈	NOUN
ejpam-5127	385	4	(	(	PUNCT
ejpam-5127	385	5	x	x	PROPN
ejpam-5127	385	6	∨	∨	NUM
ejpam-5127	385	7	y)•.	y)•.	PROPN
ejpam-5127	385	8	then	then	ADV
ejpam-5127	385	9	a	a	DET
ejpam-5127	385	10	∨	∨	NOUN
ejpam-5127	385	11	x	x	SYM
ejpam-5127	385	12	∨	∨	NUM
ejpam-5127	385	13	y	y	NOUN
ejpam-5127	385	14	=	=	SYM
ejpam-5127	385	15	1	1	X
ejpam-5127	385	16	.	.	PUNCT
ejpam-5127	386	1	this	this	PRON
ejpam-5127	386	2	gives	give	VERB
ejpam-5127	386	3	a	a	DET
ejpam-5127	386	4	∨	∨	NOUN
ejpam-5127	386	5	x	x	SYM
ejpam-5127	386	6	∨	∨	NUM
ejpam-5127	386	7	a	a	DET
ejpam-5127	386	8	∨	∨	NOUN
ejpam-5127	386	9	y	y	NOUN
ejpam-5127	386	10	=	=	SYM
ejpam-5127	386	11	1	1	X
ejpam-5127	386	12	.	.	PUNCT
ejpam-5127	386	13	therefore	therefore	ADV
ejpam-5127	386	14	from	from	ADP
ejpam-5127	386	15	condition	condition	NOUN
ejpam-5127	386	16	(	(	PUNCT
ejpam-5127	386	17	4	4	NUM
ejpam-5127	386	18	)	)	PUNCT
ejpam-5127	386	19	,	,	PUNCT
ejpam-5127	386	20	we	we	PRON
ejpam-5127	386	21	have	have	AUX
ejpam-5127	386	22	(	(	PUNCT
ejpam-5127	386	23	a	a	DET
ejpam-5127	386	24	∨	∨	NUM
ejpam-5127	386	25	x)•	x)•	PROPN
ejpam-5127	386	26	∨	∨	NUM
ejpam-5127	386	27	(	(	PUNCT
ejpam-5127	386	28	a	a	DET
ejpam-5127	386	29	∨	∨	NUM
ejpam-5127	386	30	y)•	y)•	NUM
ejpam-5127	386	31	=	=	SYM
ejpam-5127	386	32	v	v	NOUN
ejpam-5127	386	33	.	.	PUNCT
ejpam-5127	387	1	now	now	ADV
ejpam-5127	387	2	a	a	DET
ejpam-5127	387	3	∈	∈	NOUN
ejpam-5127	387	4	v	v	NOUN
ejpam-5127	387	5	implies	imply	VERB
ejpam-5127	387	6	that	that	SCONJ
ejpam-5127	387	7	a	a	DET
ejpam-5127	387	8	=	=	NOUN
ejpam-5127	387	9	t∧s	t∧s	NOUN
ejpam-5127	387	10	where	where	SCONJ
ejpam-5127	387	11	t	t	PROPN
ejpam-5127	387	12	∈	∈	PROPN
ejpam-5127	387	13	(	(	PUNCT
ejpam-5127	387	14	a∨x)•	a∨x)•	PROPN
ejpam-5127	387	15	and	and	CCONJ
ejpam-5127	387	16	s	s	PROPN
ejpam-5127	387	17	∈	∈	PROPN
ejpam-5127	387	18	(	(	PUNCT
ejpam-5127	387	19	a∨y)•.	a∨y)•.	X
ejpam-5127	387	20	so	so	SCONJ
ejpam-5127	387	21	that	that	SCONJ
ejpam-5127	387	22	a	a	DET
ejpam-5127	387	23	=	=	X
ejpam-5127	387	24	t∧s	t∧s	NOUN
ejpam-5127	387	25	,	,	PUNCT
ejpam-5127	387	26	where	where	SCONJ
ejpam-5127	387	27	t∨a∨x	t∨a∨x	PROPN
ejpam-5127	387	28	=	=	NOUN
ejpam-5127	387	29	1	1	NUM
ejpam-5127	387	30	and	and	CCONJ
ejpam-5127	387	31	s	s	ADJ
ejpam-5127	387	32	∨	∨	NUM
ejpam-5127	387	33	a	a	DET
ejpam-5127	387	34	∨	∨	NOUN
ejpam-5127	387	35	y	y	NOUN
ejpam-5127	387	36	=	=	SYM
ejpam-5127	387	37	1	1	NUM
ejpam-5127	387	38	.	.	PUNCT
ejpam-5127	387	39	hence	hence	ADV
ejpam-5127	387	40	a	a	DET
ejpam-5127	387	41	=	=	SYM
ejpam-5127	387	42	t	t	PROPN
ejpam-5127	387	43	∧	∧	PROPN
ejpam-5127	387	44	s	s	PROPN
ejpam-5127	387	45	,	,	PUNCT
ejpam-5127	387	46	where	where	SCONJ
ejpam-5127	387	47	t	t	PROPN
ejpam-5127	387	48	∨	∨	NUM
ejpam-5127	387	49	a	a	DET
ejpam-5127	387	50	∈	∈	NOUN
ejpam-5127	387	51	(	(	PUNCT
ejpam-5127	387	52	x)•	x)•	PROPN
ejpam-5127	387	53	and	and	CCONJ
ejpam-5127	387	54	s	s	PROPN
ejpam-5127	387	55	∨	∨	NOUN
ejpam-5127	387	56	a	a	DET
ejpam-5127	387	57	∈	∈	NOUN
ejpam-5127	387	58	(	(	PUNCT
ejpam-5127	387	59	y)•.	y)•.	PROPN
ejpam-5127	387	60	therefore	therefore	ADV
ejpam-5127	387	61	a	a	DET
ejpam-5127	387	62	=	=	SYM
ejpam-5127	387	63	t	t	PROPN
ejpam-5127	387	64	∧	∧	PROPN
ejpam-5127	387	65	s	s	PART
ejpam-5127	387	66	=	=	PUNCT
ejpam-5127	387	67	(	(	PUNCT
ejpam-5127	387	68	t	t	PROPN
ejpam-5127	387	69	∧	∧	PROPN
ejpam-5127	387	70	s	s	PART
ejpam-5127	387	71	)	)	PUNCT
ejpam-5127	387	72	∨	∨	NOUN
ejpam-5127	387	73	a	a	PRON
ejpam-5127	387	74	=	=	X
ejpam-5127	387	75	(	(	PUNCT
ejpam-5127	387	76	t	t	PROPN
ejpam-5127	387	77	∨	∨	NUM
ejpam-5127	387	78	a	a	PRON
ejpam-5127	387	79	)	)	PUNCT
ejpam-5127	387	80	∧	∧	NOUN
ejpam-5127	387	81	(	(	PUNCT
ejpam-5127	387	82	s	s	PROPN
ejpam-5127	387	83	∨	∨	NUM
ejpam-5127	387	84	a	a	DET
ejpam-5127	387	85	)	)	PUNCT
ejpam-5127	387	86	∈	∈	PROPN
ejpam-5127	387	87	(	(	PUNCT
ejpam-5127	387	88	x)•	x)•	PROPN
ejpam-5127	387	89	∨	∨	NUM
ejpam-5127	387	90	(	(	PUNCT
ejpam-5127	387	91	y)•.	y)•.	PROPN
ejpam-5127	387	92	thus	thus	ADV
ejpam-5127	387	93	(	(	PUNCT
ejpam-5127	387	94	x	x	PROPN
ejpam-5127	387	95	∨	∨	PROPN
ejpam-5127	387	96	y)•	y)•	NUM
ejpam-5127	387	97	⊆	⊆	NUM
ejpam-5127	387	98	(	(	PUNCT
ejpam-5127	387	99	x)•	x)•	PROPN
ejpam-5127	387	100	∨	∨	NUM
ejpam-5127	387	101	(	(	PUNCT
ejpam-5127	387	102	y)•.	y)•.	PROPN
ejpam-5127	387	103	hence	hence	ADV
ejpam-5127	387	104	(	(	PUNCT
ejpam-5127	387	105	x	x	X
ejpam-5127	387	106	∨	∨	NOUN
ejpam-5127	387	107	y)•	y)•	NUM
ejpam-5127	387	108	=	=	SYM
ejpam-5127	387	109	(	(	PUNCT
ejpam-5127	387	110	x)•	x)•	PROPN
ejpam-5127	387	111	∨	∨	NUM
ejpam-5127	387	112	(	(	PUNCT
ejpam-5127	387	113	y)•.	y)•.	PROPN
ejpam-5127	387	114	(	(	PUNCT
ejpam-5127	387	115	5	5	NUM
ejpam-5127	387	116	)	)	PUNCT
ejpam-5127	387	117	⇒	⇒	NOUN
ejpam-5127	387	118	(	(	PUNCT
ejpam-5127	387	119	6	6	X
ejpam-5127	387	120	)	)	PUNCT
ejpam-5127	387	121	let	let	VERB
ejpam-5127	387	122	x	x	PRON
ejpam-5127	387	123	,	,	PUNCT
ejpam-5127	387	124	y	y	PROPN
ejpam-5127	387	125	∈	∈	PROPN
ejpam-5127	387	126	v	v	NOUN
ejpam-5127	387	127	and	and	CCONJ
ejpam-5127	387	128	x	x	PROPN
ejpam-5127	387	129	∨	∨	NUM
ejpam-5127	388	1	y	y	NOUN
ejpam-5127	388	2	=	=	SYM
ejpam-5127	388	3	1	1	X
ejpam-5127	388	4	.	.	PUNCT
ejpam-5127	389	1	let	let	VERB
ejpam-5127	389	2	m	m	PRON
ejpam-5127	389	3	be	be	AUX
ejpam-5127	389	4	a	a	DET
ejpam-5127	389	5	minimal	minimal	ADJ
ejpam-5127	389	6	element	element	NOUN
ejpam-5127	389	7	.	.	PUNCT
ejpam-5127	390	1	then	then	ADV
ejpam-5127	390	2	(	(	PUNCT
ejpam-5127	390	3	1)•	1)•	NUM
ejpam-5127	390	4	=	=	SYM
ejpam-5127	390	5	(	(	PUNCT
ejpam-5127	390	6	x	x	X
ejpam-5127	390	7	∨	∨	NOUN
ejpam-5127	390	8	y)•	y)•	NUM
ejpam-5127	390	9	=	=	SYM
ejpam-5127	390	10	(	(	PUNCT
ejpam-5127	390	11	x)•	x)•	PROPN
ejpam-5127	390	12	∨	∨	NUM
ejpam-5127	390	13	(	(	PUNCT
ejpam-5127	390	14	y)•.	y)•.	PROPN
ejpam-5127	390	15	this	this	PRON
ejpam-5127	390	16	gives	give	VERB
ejpam-5127	390	17	v	v	NOUN
ejpam-5127	390	18	=	=	PUNCT
ejpam-5127	390	19	(	(	PUNCT
ejpam-5127	390	20	x)•	x)•	PROPN
ejpam-5127	390	21	∨	∨	NUM
ejpam-5127	390	22	(	(	PUNCT
ejpam-5127	390	23	y)•.	y)•.	PROPN
ejpam-5127	390	24	now	now	ADV
ejpam-5127	390	25	m	m	VERB
ejpam-5127	390	26	∈	∈	NOUN
ejpam-5127	390	27	v	v	ADP
ejpam-5127	390	28	=	=	NOUN
ejpam-5127	390	29	⇒	⇒	NOUN
ejpam-5127	390	30	m	m	VERB
ejpam-5127	390	31	∈	∈	PROPN
ejpam-5127	390	32	(	(	PUNCT
ejpam-5127	390	33	x)•	x)•	PROPN
ejpam-5127	390	34	∨	∨	NUM
ejpam-5127	390	35	(	(	PUNCT
ejpam-5127	390	36	y)•	y)•	NUM
ejpam-5127	390	37	=	=	NOUN
ejpam-5127	390	38	⇒	⇒	NOUN
ejpam-5127	390	39	m	m	VERB
ejpam-5127	390	40	=	=	SYM
ejpam-5127	390	41	u	u	PROPN
ejpam-5127	390	42	∧	∧	PROPN
ejpam-5127	390	43	v	v	ADP
ejpam-5127	390	44	where	where	SCONJ
ejpam-5127	390	45	u	u	PROPN
ejpam-5127	390	46	∈	∈	PROPN
ejpam-5127	390	47	(	(	PUNCT
ejpam-5127	390	48	x)•	x)•	PROPN
ejpam-5127	390	49	and	and	CCONJ
ejpam-5127	390	50	v	v	ADP
ejpam-5127	390	51	∈	∈	PROPN
ejpam-5127	390	52	(	(	PUNCT
ejpam-5127	390	53	y)•	y)•	NUM
ejpam-5127	390	54	=	=	NOUN
ejpam-5127	390	55	⇒	⇒	NOUN
ejpam-5127	390	56	m	m	VERB
ejpam-5127	390	57	=	=	SYM
ejpam-5127	390	58	u	u	PROPN
ejpam-5127	390	59	∧	∧	PROPN
ejpam-5127	390	60	v	v	ADP
ejpam-5127	390	61	where	where	SCONJ
ejpam-5127	390	62	u	u	NOUN
ejpam-5127	390	63	∨	∨	NOUN
ejpam-5127	390	64	x	x	SYM
ejpam-5127	390	65	=	=	SYM
ejpam-5127	390	66	1	1	NUM
ejpam-5127	390	67	and	and	CCONJ
ejpam-5127	390	68	v	v	ADP
ejpam-5127	390	69	∨	∨	NUM
ejpam-5127	390	70	y	y	PROPN
ejpam-5127	390	71	=	=	SYM
ejpam-5127	390	72	1	1	X
ejpam-5127	390	73	.	.	PUNCT
ejpam-5127	391	1	therefore	therefore	ADV
ejpam-5127	391	2	,	,	PUNCT
ejpam-5127	391	3	there	there	PRON
ejpam-5127	391	4	exists	exist	VERB
ejpam-5127	391	5	u	u	PROPN
ejpam-5127	391	6	∈	∈	PROPN
ejpam-5127	391	7	(	(	PUNCT
ejpam-5127	391	8	x)•	x)•	PROPN
ejpam-5127	391	9	and	and	CCONJ
ejpam-5127	391	10	v	v	ADP
ejpam-5127	391	11	∈	∈	PROPN
ejpam-5127	391	12	(	(	PUNCT
ejpam-5127	391	13	y)•	y)•	NUM
ejpam-5127	391	14	such	such	ADJ
ejpam-5127	391	15	that	that	SCONJ
ejpam-5127	391	16	u	u	PROPN
ejpam-5127	391	17	∧	∧	PROPN
ejpam-5127	391	18	v	v	NOUN
ejpam-5127	391	19	=	=	NOUN
ejpam-5127	391	20	m.	m.	NOUN
ejpam-5127	391	21	(	(	PUNCT
ejpam-5127	391	22	6	6	NUM
ejpam-5127	391	23	)	)	PUNCT
ejpam-5127	391	24	⇒	⇒	NOUN
ejpam-5127	391	25	(	(	PUNCT
ejpam-5127	391	26	1	1	X
ejpam-5127	391	27	)	)	PUNCT
ejpam-5127	391	28	assume	assume	VERB
ejpam-5127	391	29	(	(	PUNCT
ejpam-5127	391	30	6	6	NUM
ejpam-5127	391	31	)	)	PUNCT
ejpam-5127	391	32	.	.	PUNCT
ejpam-5127	392	1	let	let	VERB
ejpam-5127	392	2	p	p	PRON
ejpam-5127	392	3	,	,	PUNCT
ejpam-5127	392	4	q	q	ADJ
ejpam-5127	392	5	be	be	AUX
ejpam-5127	392	6	two	two	NUM
ejpam-5127	392	7	distinct	distinct	ADJ
ejpam-5127	392	8	minimal	minimal	ADJ
ejpam-5127	392	9	prime	prime	ADJ
ejpam-5127	392	10	filters	filter	NOUN
ejpam-5127	392	11	of	of	ADP
ejpam-5127	392	12	v	v	NOUN
ejpam-5127	392	13	.	.	PUNCT
ejpam-5127	393	1	then	then	ADV
ejpam-5127	393	2	v	v	NOUN
ejpam-5127	393	3	\p	\p	ADV
ejpam-5127	393	4	is	be	AUX
ejpam-5127	393	5	a	a	DET
ejpam-5127	393	6	maximal	maximal	ADJ
ejpam-5127	393	7	ideal	ideal	NOUN
ejpam-5127	393	8	of	of	ADP
ejpam-5127	393	9	v	v	NOUN
ejpam-5127	393	10	.	.	PUNCT
ejpam-5127	394	1	then	then	ADV
ejpam-5127	394	2	v	v	NOUN
ejpam-5127	394	3	\p	\p	ADV
ejpam-5127	394	4	is	be	AUX
ejpam-5127	394	5	a	a	DET
ejpam-5127	394	6	maximal	maximal	ADJ
ejpam-5127	394	7	ideal	ideal	NOUN
ejpam-5127	394	8	of	of	ADP
ejpam-5127	394	9	v	v	NOUN
ejpam-5127	394	10	.	.	PUNCT
ejpam-5127	395	1	since	since	SCONJ
ejpam-5127	395	2	p	p	PRON
ejpam-5127	395	3	,	,	PUNCT
ejpam-5127	395	4	q	q	X
ejpam-5127	395	5	are	be	AUX
ejpam-5127	395	6	distinct	distinct	ADJ
ejpam-5127	395	7	,	,	PUNCT
ejpam-5127	395	8	chose	choose	VERB
ejpam-5127	395	9	x	x	PUNCT
ejpam-5127	395	10	∈	∈	PROPN
ejpam-5127	395	11	p\q	p\q	NOUN
ejpam-5127	395	12	.	.	PUNCT
ejpam-5127	396	1	clearly	clearly	ADV
ejpam-5127	396	2	,	,	PUNCT
ejpam-5127	396	3	x	x	PROPN
ejpam-5127	396	4	/∈	/∈	PUNCT
ejpam-5127	396	5	v	v	ADP
ejpam-5127	396	6	\p	\p	ADV
ejpam-5127	396	7	.	.	PUNCT
ejpam-5127	397	1	since	since	SCONJ
ejpam-5127	397	2	v	v	NOUN
ejpam-5127	397	3	\p	\p	ADV
ejpam-5127	397	4	is	be	AUX
ejpam-5127	397	5	a	a	DET
ejpam-5127	397	6	maximal	maximal	ADJ
ejpam-5127	397	7	ideal	ideal	NOUN
ejpam-5127	397	8	of	of	ADP
ejpam-5127	397	9	v	v	NUM
ejpam-5127	397	10	,	,	PUNCT
ejpam-5127	397	11	we	we	PRON
ejpam-5127	397	12	get	get	VERB
ejpam-5127	397	13	(	(	PUNCT
ejpam-5127	397	14	v	v	NOUN
ejpam-5127	397	15	\p	\p	NOUN
ejpam-5127	397	16	)	)	PUNCT
ejpam-5127	397	17	∨	∨	NOUN
ejpam-5127	397	18	(	(	PUNCT
ejpam-5127	397	19	x	x	X
ejpam-5127	397	20	]	]	X
ejpam-5127	397	21	=	=	SYM
ejpam-5127	397	22	v	v	NOUN
ejpam-5127	397	23	.	.	PUNCT
ejpam-5127	398	1	now	now	ADV
ejpam-5127	398	2	1	1	NUM
ejpam-5127	398	3	∈	∈	NOUN
ejpam-5127	398	4	v	v	NOUN
ejpam-5127	398	5	=	=	NOUN
ejpam-5127	398	6	⇒	⇒	ADJ
ejpam-5127	398	7	1	1	NUM
ejpam-5127	398	8	∈	∈	PROPN
ejpam-5127	398	9	(	(	PUNCT
ejpam-5127	398	10	v	v	NOUN
ejpam-5127	398	11	\p	\p	ADV
ejpam-5127	398	12	)	)	PUNCT
ejpam-5127	398	13	∨	∨	NOUN
ejpam-5127	398	14	(	(	PUNCT
ejpam-5127	398	15	x	x	SYM
ejpam-5127	398	16	]	]	X
ejpam-5127	398	17	r.	r.	PROPN
ejpam-5127	398	18	shukla	shukla	PROPN
ejpam-5127	398	19	et	et	PROPN
ejpam-5127	398	20	al	al	PROPN
ejpam-5127	398	21	.	.	PUNCT
ejpam-5127	398	22	/	/	SYM
ejpam-5127	398	23	eur	eur	PROPN
ejpam-5127	398	24	.	.	PUNCT
ejpam-5127	399	1	j.	j.	PROPN
ejpam-5127	399	2	pure	pure	PROPN
ejpam-5127	399	3	appl	appl	PROPN
ejpam-5127	399	4	.	.	PROPN
ejpam-5127	399	5	math	math	PROPN
ejpam-5127	399	6	,	,	PUNCT
ejpam-5127	399	7	17	17	NUM
ejpam-5127	399	8	(	(	PUNCT
ejpam-5127	399	9	2	2	NUM
ejpam-5127	399	10	)	)	PUNCT
ejpam-5127	399	11	(	(	PUNCT
ejpam-5127	399	12	2024	2024	NUM
ejpam-5127	399	13	)	)	PUNCT
ejpam-5127	399	14	,	,	PUNCT
ejpam-5127	399	15	1306	1306	NUM
ejpam-5127	399	16	-	-	SYM
ejpam-5127	399	17	1320	1320	NUM
ejpam-5127	399	18	1317	1317	NUM
ejpam-5127	399	19	=	=	NOUN
ejpam-5127	399	20	⇒	⇒	NOUN
ejpam-5127	399	21	1	1	NUM
ejpam-5127	399	22	=	=	SYM
ejpam-5127	399	23	a	a	DET
ejpam-5127	399	24	∨	∨	NUM
ejpam-5127	399	25	b	b	NOUN
ejpam-5127	399	26	,	,	PUNCT
ejpam-5127	399	27	where	where	SCONJ
ejpam-5127	399	28	a	a	DET
ejpam-5127	399	29	∈	∈	NOUN
ejpam-5127	399	30	v	v	ADP
ejpam-5127	399	31	\p	\p	NOUN
ejpam-5127	399	32	and	and	CCONJ
ejpam-5127	399	33	b	b	X
ejpam-5127	399	34	∈	∈	PROPN
ejpam-5127	399	35	(	(	PUNCT
ejpam-5127	399	36	x	x	X
ejpam-5127	399	37	]	]	X
ejpam-5127	399	38	=	=	SYM
ejpam-5127	399	39	⇒	⇒	NOUN
ejpam-5127	399	40	1	1	NUM
ejpam-5127	399	41	∨	∨	NUM
ejpam-5127	399	42	x	x	X
ejpam-5127	399	43	=	=	PUNCT
ejpam-5127	399	44	a	a	DET
ejpam-5127	399	45	∨	∨	NUM
ejpam-5127	399	46	b	b	NOUN
ejpam-5127	399	47	∨	∨	NOUN
ejpam-5127	399	48	x	x	X
ejpam-5127	400	1	=	=	NOUN
ejpam-5127	400	2	⇒	⇒	NOUN
ejpam-5127	400	3	1	1	NUM
ejpam-5127	400	4	=	=	SYM
ejpam-5127	400	5	a	a	DET
ejpam-5127	400	6	∨	∨	NOUN
ejpam-5127	400	7	x.	x.	NOUN
ejpam-5127	400	8	thus	thus	ADV
ejpam-5127	400	9	from	from	ADP
ejpam-5127	400	10	condition	condition	NOUN
ejpam-5127	400	11	(	(	PUNCT
ejpam-5127	400	12	6	6	NUM
ejpam-5127	400	13	)	)	PUNCT
ejpam-5127	400	14	,	,	PUNCT
ejpam-5127	400	15	there	there	PRON
ejpam-5127	400	16	exists	exist	VERB
ejpam-5127	400	17	u	u	PROPN
ejpam-5127	400	18	∈	∈	PROPN
ejpam-5127	400	19	(	(	PUNCT
ejpam-5127	400	20	a)•	a)•	NUM
ejpam-5127	400	21	and	and	CCONJ
ejpam-5127	400	22	v	v	ADP
ejpam-5127	400	23	∈	∈	PROPN
ejpam-5127	400	24	(	(	PUNCT
ejpam-5127	400	25	x)•	x)•	PROPN
ejpam-5127	400	26	such	such	ADJ
ejpam-5127	400	27	that	that	SCONJ
ejpam-5127	400	28	u	u	PROPN
ejpam-5127	400	29	∧	∧	PROPN
ejpam-5127	400	30	v	v	NOUN
ejpam-5127	400	31	is	be	AUX
ejpam-5127	400	32	minimal	minimal	ADJ
ejpam-5127	400	33	.	.	PUNCT
ejpam-5127	401	1	hence	hence	ADV
ejpam-5127	401	2	u	u	PROPN
ejpam-5127	401	3	∧	∧	PROPN
ejpam-5127	401	4	v	v	ADP
ejpam-5127	401	5	∈	∈	PROPN
ejpam-5127	401	6	(	(	PUNCT
ejpam-5127	401	7	a)•	a)•	NUM
ejpam-5127	401	8	∨	∨	NUM
ejpam-5127	401	9	(	(	PUNCT
ejpam-5127	401	10	x)•	x)•	PROPN
ejpam-5127	401	11	⊆	⊆	NUM
ejpam-5127	401	12	p	p	PROPN
ejpam-5127	401	13	∨	∨	NUM
ejpam-5127	401	14	q.	q.	PROPN
ejpam-5127	401	15	therefore	therefore	ADV
ejpam-5127	401	16	p	p	PROPN
ejpam-5127	401	17	∨	∨	PROPN
ejpam-5127	401	18	q	q	NOUN
ejpam-5127	401	19	=	=	X
ejpam-5127	401	20	v	v	NOUN
ejpam-5127	401	21	.	.	PUNCT
ejpam-5127	402	1	thus	thus	ADV
ejpam-5127	402	2	any	any	DET
ejpam-5127	402	3	two	two	NUM
ejpam-5127	402	4	distinct	distinct	ADJ
ejpam-5127	402	5	minimal	minimal	ADJ
ejpam-5127	402	6	prime	prime	ADJ
ejpam-5127	402	7	filters	filter	NOUN
ejpam-5127	402	8	of	of	ADP
ejpam-5127	402	9	v	v	NOUN
ejpam-5127	402	10	are	be	AUX
ejpam-5127	402	11	co	co	ADJ
ejpam-5127	402	12	-	-	ADJ
ejpam-5127	402	13	maximal	maximal	ADJ
ejpam-5127	402	14	.	.	PUNCT
ejpam-5127	403	1	let	let	VERB
ejpam-5127	403	2	v	v	PART
ejpam-5127	403	3	be	be	AUX
ejpam-5127	403	4	a	a	DET
ejpam-5127	403	5	bounded	bounded	ADJ
ejpam-5127	403	6	distributive	distributive	ADJ
ejpam-5127	403	7	lattice	lattice	NOUN
ejpam-5127	403	8	.	.	PUNCT
ejpam-5127	404	1	then	then	ADV
ejpam-5127	404	2	v	v	NOUN
ejpam-5127	404	3	is	be	AUX
ejpam-5127	404	4	said	say	VERB
ejpam-5127	404	5	to	to	PART
ejpam-5127	404	6	be	be	AUX
ejpam-5127	404	7	conormal	conormal	ADJ
ejpam-5127	404	8	provided	provide	VERB
ejpam-5127	404	9	that	that	SCONJ
ejpam-5127	404	10	x	x	NOUN
ejpam-5127	404	11	,	,	PUNCT
ejpam-5127	404	12	y	y	PROPN
ejpam-5127	404	13	∈	∈	PROPN
ejpam-5127	404	14	v	v	NOUN
ejpam-5127	404	15	,	,	PUNCT
ejpam-5127	404	16	x	x	PUNCT
ejpam-5127	404	17	∧	∧	NOUN
ejpam-5127	404	18	y	y	NOUN
ejpam-5127	404	19	=	=	SYM
ejpam-5127	404	20	0	0	NUM
ejpam-5127	404	21	implies	imply	VERB
ejpam-5127	404	22	there	there	PRON
ejpam-5127	404	23	exist	exist	VERB
ejpam-5127	404	24	u	u	NOUN
ejpam-5127	404	25	,	,	PUNCT
ejpam-5127	404	26	v	v	PROPN
ejpam-5127	404	27	∈	∈	NOUN
ejpam-5127	404	28	v	v	ADP
ejpam-5127	404	29	such	such	ADJ
ejpam-5127	404	30	that	that	DET
ejpam-5127	404	31	u	u	NOUN
ejpam-5127	404	32	∧	∧	NOUN
ejpam-5127	404	33	x	x	PUNCT
ejpam-5127	404	34	=	=	SYM
ejpam-5127	404	35	v	v	ADP
ejpam-5127	404	36	∧	∧	PROPN
ejpam-5127	404	37	y	y	PROPN
ejpam-5127	404	38	=	=	SYM
ejpam-5127	404	39	0	0	NUM
ejpam-5127	404	40	and	and	CCONJ
ejpam-5127	404	41	u	u	NOUN
ejpam-5127	404	42	∨	∨	NUM
ejpam-5127	404	43	v	v	NOUN
ejpam-5127	404	44	=	=	SYM
ejpam-5127	404	45	1	1	NUM
ejpam-5127	404	46	.	.	PUNCT
ejpam-5127	405	1	this	this	DET
ejpam-5127	405	2	property	property	NOUN
ejpam-5127	405	3	,	,	PUNCT
ejpam-5127	405	4	discussed	discuss	VERB
ejpam-5127	405	5	by	by	ADP
ejpam-5127	405	6	cornish[3	cornish[3	NOUN
ejpam-5127	405	7	]	]	PUNCT
ejpam-5127	405	8	under	under	ADP
ejpam-5127	405	9	the	the	DET
ejpam-5127	405	10	name	name	NOUN
ejpam-5127	405	11	of	of	ADP
ejpam-5127	405	12	a	a	DET
ejpam-5127	405	13	normal	normal	ADJ
ejpam-5127	405	14	lattice	lattice	NOUN
ejpam-5127	405	15	,	,	PUNCT
ejpam-5127	405	16	aligns	align	VERB
ejpam-5127	405	17	with	with	ADP
ejpam-5127	405	18	the	the	DET
ejpam-5127	405	19	version	version	NOUN
ejpam-5127	405	20	of	of	ADP
ejpam-5127	405	21	the	the	DET
ejpam-5127	405	22	definition	definition	NOUN
ejpam-5127	405	23	presented	present	VERB
ejpam-5127	405	24	by	by	ADP
ejpam-5127	405	25	simmons[9	simmons[9	PROPN
ejpam-5127	405	26	]	]	PUNCT
ejpam-5127	405	27	in	in	ADP
ejpam-5127	405	28	1980	1980	NUM
ejpam-5127	405	29	(	(	PUNCT
ejpam-5127	405	30	specifically	specifically	ADV
ejpam-5127	405	31	,	,	PUNCT
ejpam-5127	405	32	definition	definition	NOUN
ejpam-5127	405	33	4.3	4.3	NUM
ejpam-5127	405	34	)	)	PUNCT
ejpam-5127	405	35	.	.	PUNCT
ejpam-5127	406	1	theorem	theorem	NOUN
ejpam-5127	406	2	12	12	NUM
ejpam-5127	406	3	.	.	PUNCT
ejpam-5127	407	1	a	a	DET
ejpam-5127	407	2	pdl	pdl	PROPN
ejpam-5127	407	3	v	v	NOUN
ejpam-5127	407	4	is	be	AUX
ejpam-5127	407	5	normal	normal	ADJ
ejpam-5127	407	6	if	if	SCONJ
ejpam-5127	407	7	and	and	CCONJ
ejpam-5127	407	8	only	only	ADV
ejpam-5127	407	9	if	if	SCONJ
ejpam-5127	407	10	the	the	DET
ejpam-5127	407	11	bounded	bounded	ADJ
ejpam-5127	407	12	distributive	distributive	ADJ
ejpam-5127	407	13	lattice	lattice	NOUN
ejpam-5127	407	14	pf(v	pf(v	NOUN
ejpam-5127	407	15	)	)	PUNCT
ejpam-5127	407	16	is	be	AUX
ejpam-5127	407	17	conormal	conormal	ADJ
ejpam-5127	407	18	.	.	PUNCT
ejpam-5127	408	1	corollary	corollary	ADJ
ejpam-5127	408	2	2	2	NUM
ejpam-5127	408	3	.	.	PUNCT
ejpam-5127	409	1	if	if	SCONJ
ejpam-5127	409	2	v	v	NOUN
ejpam-5127	409	3	is	be	AUX
ejpam-5127	409	4	a	a	DET
ejpam-5127	409	5	pdl	pdl	NOUN
ejpam-5127	409	6	with	with	ADP
ejpam-5127	409	7	minimal	minimal	ADJ
ejpam-5127	409	8	element	element	NOUN
ejpam-5127	409	9	m	m	VERB
ejpam-5127	409	10	then	then	ADV
ejpam-5127	409	11	the	the	DET
ejpam-5127	409	12	following	following	NOUN
ejpam-5127	409	13	are	be	AUX
ejpam-5127	409	14	equivalent	equivalent	ADJ
ejpam-5127	409	15	.	.	PUNCT
ejpam-5127	410	1	(	(	PUNCT
ejpam-5127	410	2	1	1	X
ejpam-5127	410	3	)	)	PUNCT
ejpam-5127	410	4	v	v	NOUN
ejpam-5127	410	5	is	be	AUX
ejpam-5127	410	6	normal	normal	ADJ
ejpam-5127	410	7	.	.	PUNCT
ejpam-5127	411	1	(	(	PUNCT
ejpam-5127	411	2	2	2	X
ejpam-5127	411	3	)	)	PUNCT
ejpam-5127	411	4	each	each	DET
ejpam-5127	411	5	maximal	maximal	ADJ
ejpam-5127	411	6	filter	filter	NOUN
ejpam-5127	411	7	contains	contain	VERB
ejpam-5127	411	8	a	a	DET
ejpam-5127	411	9	unique	unique	ADJ
ejpam-5127	411	10	minimal	minimal	ADJ
ejpam-5127	411	11	prime	prime	ADJ
ejpam-5127	411	12	filter	filter	NOUN
ejpam-5127	411	13	.	.	PUNCT
ejpam-5127	412	1	(	(	PUNCT
ejpam-5127	412	2	3	3	X
ejpam-5127	412	3	)	)	PUNCT
ejpam-5127	412	4	for	for	ADP
ejpam-5127	412	5	each	each	DET
ejpam-5127	412	6	maximal	maximal	ADJ
ejpam-5127	412	7	filter	filter	NOUN
ejpam-5127	412	8	m	m	NOUN
ejpam-5127	412	9	,	,	PUNCT
ejpam-5127	412	10	o(m	o(m	PROPN
ejpam-5127	412	11	)	)	PUNCT
ejpam-5127	412	12	is	be	AUX
ejpam-5127	412	13	a	a	DET
ejpam-5127	412	14	prime	prime	ADJ
ejpam-5127	412	15	filter	filter	NOUN
ejpam-5127	412	16	.	.	PUNCT
ejpam-5127	413	1	(	(	PUNCT
ejpam-5127	413	2	4	4	NUM
ejpam-5127	413	3	)	)	PUNCT
ejpam-5127	413	4	for	for	ADP
ejpam-5127	413	5	any	any	DET
ejpam-5127	413	6	x	x	NOUN
ejpam-5127	413	7	,	,	PUNCT
ejpam-5127	413	8	y	y	PROPN
ejpam-5127	413	9	∈	∈	PROPN
ejpam-5127	413	10	v	v	ADP
ejpam-5127	413	11	if	if	SCONJ
ejpam-5127	413	12	x	x	PROPN
ejpam-5127	413	13	∨	∨	NOUN
ejpam-5127	413	14	y	y	NOUN
ejpam-5127	413	15	=	=	NOUN
ejpam-5127	413	16	1	1	NUM
ejpam-5127	413	17	then	then	ADV
ejpam-5127	413	18	there	there	PRON
ejpam-5127	413	19	exist	exist	VERB
ejpam-5127	413	20	x1	x1	PROPN
ejpam-5127	413	21	,	,	PUNCT
ejpam-5127	413	22	y1	y1	PROPN
ejpam-5127	413	23	∈	∈	PROPN
ejpam-5127	413	24	v	v	ADP
ejpam-5127	413	25	such	such	ADJ
ejpam-5127	413	26	that	that	SCONJ
ejpam-5127	413	27	x1	x1	PROPN
ejpam-5127	413	28	∨	∨	NOUN
ejpam-5127	413	29	x	x	SYM
ejpam-5127	413	30	=	=	SYM
ejpam-5127	413	31	1	1	NUM
ejpam-5127	413	32	,	,	PUNCT
ejpam-5127	413	33	y1	y1	X
ejpam-5127	413	34	∨	∨	NUM
ejpam-5127	413	35	y	y	PROPN
ejpam-5127	413	36	=	=	SYM
ejpam-5127	413	37	1	1	NUM
ejpam-5127	413	38	and	and	CCONJ
ejpam-5127	413	39	x1	x1	DET
ejpam-5127	413	40	∧	∧	PROPN
ejpam-5127	413	41	y1	y1	NOUN
ejpam-5127	413	42	=	=	PUNCT
ejpam-5127	413	43	m.	m.	NOUN
ejpam-5127	413	44	we	we	PRON
ejpam-5127	413	45	know	know	VERB
ejpam-5127	413	46	that	that	SCONJ
ejpam-5127	413	47	every	every	DET
ejpam-5127	413	48	filter	filter	NOUN
ejpam-5127	413	49	of	of	ADP
ejpam-5127	413	50	a	a	DET
ejpam-5127	413	51	pdl	pdl	NOUN
ejpam-5127	413	52	v	v	NOUN
ejpam-5127	413	53	is	be	AUX
ejpam-5127	413	54	a	a	DET
ejpam-5127	413	55	sub	sub	NOUN
ejpam-5127	413	56	pdl	pdl	NOUN
ejpam-5127	413	57	of	of	ADP
ejpam-5127	413	58	v	v	NOUN
ejpam-5127	413	59	and	and	CCONJ
ejpam-5127	413	60	every	every	DET
ejpam-5127	413	61	ideal	ideal	NOUN
ejpam-5127	413	62	is	be	AUX
ejpam-5127	413	63	also	also	ADV
ejpam-5127	413	64	a	a	DET
ejpam-5127	413	65	sub	sub	NOUN
ejpam-5127	413	66	pdl	pdl	NOUN
ejpam-5127	413	67	of	of	ADP
ejpam-5127	413	68	v	v	NOUN
ejpam-5127	413	69	.	.	PUNCT
ejpam-5127	414	1	now	now	ADV
ejpam-5127	414	2	we	we	PRON
ejpam-5127	414	3	prove	prove	VERB
ejpam-5127	414	4	the	the	DET
ejpam-5127	414	5	following	following	NOUN
ejpam-5127	414	6	.	.	PUNCT
ejpam-5127	415	1	lemma	lemma	PROPN
ejpam-5127	415	2	14	14	NUM
ejpam-5127	415	3	.	.	PUNCT
ejpam-5127	416	1	a	a	DET
ejpam-5127	416	2	filter	filter	NOUN
ejpam-5127	416	3	j	j	NOUN
ejpam-5127	416	4	of	of	ADP
ejpam-5127	416	5	a	a	DET
ejpam-5127	416	6	pdl	pdl	NOUN
ejpam-5127	416	7	v	v	NOUN
ejpam-5127	416	8	is	be	AUX
ejpam-5127	416	9	normal	normal	ADJ
ejpam-5127	416	10	as	as	ADP
ejpam-5127	416	11	a	a	DET
ejpam-5127	416	12	sub	sub	NOUN
ejpam-5127	416	13	pdl	pdl	NOUN
ejpam-5127	416	14	of	of	ADP
ejpam-5127	416	15	v	v	NOUN
ejpam-5127	416	16	if	if	SCONJ
ejpam-5127	417	1	and	and	CCONJ
ejpam-5127	417	2	only	only	ADV
ejpam-5127	417	3	if	if	SCONJ
ejpam-5127	417	4	j	j	PROPN
ejpam-5127	417	5	e	e	PROPN
ejpam-5127	417	6	is	be	AUX
ejpam-5127	417	7	normal	normal	ADJ
ejpam-5127	417	8	as	as	ADP
ejpam-5127	417	9	a	a	DET
ejpam-5127	417	10	sub	sub	NOUN
ejpam-5127	417	11	lattice	lattice	NOUN
ejpam-5127	417	12	of	of	ADP
ejpam-5127	417	13	pf(v	pf(v	NOUN
ejpam-5127	417	14	)	)	PUNCT
ejpam-5127	417	15	.	.	PUNCT
ejpam-5127	418	1	proof	proof	NOUN
ejpam-5127	418	2	.	.	PUNCT
ejpam-5127	419	1	let	let	VERB
ejpam-5127	419	2	j	j	PROPN
ejpam-5127	419	3	be	be	AUX
ejpam-5127	419	4	a	a	DET
ejpam-5127	419	5	filter	filter	NOUN
ejpam-5127	419	6	of	of	ADP
ejpam-5127	419	7	v	v	NOUN
ejpam-5127	419	8	.	.	PUNCT
ejpam-5127	420	1	assume	assume	VERB
ejpam-5127	420	2	that	that	SCONJ
ejpam-5127	420	3	j	j	PROPN
ejpam-5127	420	4	is	be	AUX
ejpam-5127	420	5	normal	normal	ADJ
ejpam-5127	420	6	as	as	ADP
ejpam-5127	420	7	a	a	DET
ejpam-5127	420	8	sub	sub	NOUN
ejpam-5127	420	9	pdl	pdl	NOUN
ejpam-5127	420	10	of	of	ADP
ejpam-5127	420	11	v	v	NOUN
ejpam-5127	420	12	.	.	PUNCT
ejpam-5127	421	1	let	let	VERB
ejpam-5127	421	2	p	p	PRON
ejpam-5127	421	3	be	be	AUX
ejpam-5127	421	4	any	any	DET
ejpam-5127	421	5	prime	prime	ADJ
ejpam-5127	421	6	filter	filter	NOUN
ejpam-5127	421	7	of	of	ADP
ejpam-5127	421	8	j	j	PROPN
ejpam-5127	421	9	e.	e.	PROPN
ejpam-5127	421	10	now	now	ADV
ejpam-5127	421	11	we	we	PRON
ejpam-5127	421	12	prove	prove	VERB
ejpam-5127	421	13	that	that	SCONJ
ejpam-5127	421	14	p	p	NOUN
ejpam-5127	421	15	contains	contain	VERB
ejpam-5127	421	16	a	a	DET
ejpam-5127	421	17	unique	unique	ADJ
ejpam-5127	421	18	minimal	minimal	ADJ
ejpam-5127	421	19	prime	prime	ADJ
ejpam-5127	421	20	filter	filter	NOUN
ejpam-5127	421	21	of	of	ADP
ejpam-5127	421	22	j	j	PROPN
ejpam-5127	421	23	e.	e.	PROPN
ejpam-5127	421	24	suppose	suppose	VERB
ejpam-5127	421	25	q1,q2	q1,q2	PROPN
ejpam-5127	421	26	are	be	AUX
ejpam-5127	421	27	two	two	NUM
ejpam-5127	421	28	minimal	minimal	ADJ
ejpam-5127	421	29	prime	prime	ADJ
ejpam-5127	421	30	filters	filter	NOUN
ejpam-5127	421	31	of	of	ADP
ejpam-5127	421	32	j	j	PROPN
ejpam-5127	421	33	e	e	PROPN
ejpam-5127	421	34	such	such	ADJ
ejpam-5127	421	35	that	that	PRON
ejpam-5127	421	36	q1	q1	VERB
ejpam-5127	421	37	⊆	⊆	NUM
ejpam-5127	421	38	p	p	NOUN
ejpam-5127	421	39	and	and	CCONJ
ejpam-5127	421	40	q2	q2	NOUN
ejpam-5127	421	41	⊆	⊆	NUM
ejpam-5127	422	1	p.	p.	NOUN
ejpam-5127	422	2	then	then	ADV
ejpam-5127	422	3	qc	qc	PROPN
ejpam-5127	422	4	1	1	NUM
ejpam-5127	422	5	⊆	⊆	NUM
ejpam-5127	422	6	pc	pc	NOUN
ejpam-5127	422	7	and	and	CCONJ
ejpam-5127	422	8	qc	qc	PROPN
ejpam-5127	422	9	2	2	NUM
ejpam-5127	422	10	⊆	⊆	NUM
ejpam-5127	422	11	pc	pc	NOUN
ejpam-5127	422	12	.	.	PUNCT
ejpam-5127	423	1	since	since	SCONJ
ejpam-5127	423	2	p	p	NOUN
ejpam-5127	423	3	is	be	AUX
ejpam-5127	423	4	a	a	DET
ejpam-5127	423	5	prime	prime	ADJ
ejpam-5127	423	6	filter	filter	NOUN
ejpam-5127	423	7	of	of	ADP
ejpam-5127	423	8	j	j	PROPN
ejpam-5127	423	9	e	e	PROPN
ejpam-5127	423	10	,	,	PUNCT
ejpam-5127	423	11	pc	pc	NOUN
ejpam-5127	423	12	is	be	AUX
ejpam-5127	423	13	a	a	DET
ejpam-5127	423	14	prime	prime	ADJ
ejpam-5127	423	15	filter	filter	NOUN
ejpam-5127	423	16	of	of	ADP
ejpam-5127	423	17	j	j	PROPN
ejpam-5127	423	18	ec	ec	PROPN
ejpam-5127	423	19	=	=	PROPN
ejpam-5127	423	20	j	j	PROPN
ejpam-5127	423	21	.	.	PUNCT
ejpam-5127	424	1	since	since	SCONJ
ejpam-5127	424	2	j	j	PROPN
ejpam-5127	424	3	is	be	AUX
ejpam-5127	424	4	normal	normal	ADJ
ejpam-5127	424	5	,	,	PUNCT
ejpam-5127	424	6	the	the	DET
ejpam-5127	424	7	prime	prime	ADJ
ejpam-5127	424	8	filter	filter	NOUN
ejpam-5127	424	9	pc	pc	NOUN
ejpam-5127	424	10	of	of	ADP
ejpam-5127	424	11	j	j	PROPN
ejpam-5127	424	12	contains	contain	VERB
ejpam-5127	424	13	a	a	DET
ejpam-5127	424	14	unique	unique	ADJ
ejpam-5127	424	15	minimal	minimal	ADJ
ejpam-5127	424	16	prime	prime	ADJ
ejpam-5127	424	17	filter	filter	NOUN
ejpam-5127	424	18	.	.	PUNCT
ejpam-5127	425	1	therefore	therefore	ADV
ejpam-5127	425	2	qc	qc	PROPN
ejpam-5127	425	3	1	1	NUM
ejpam-5127	425	4	=	=	SYM
ejpam-5127	425	5	qc	qc	PROPN
ejpam-5127	425	6	2	2	NUM
ejpam-5127	425	7	.	.	PUNCT
ejpam-5127	425	8	hence	hence	ADV
ejpam-5127	425	9	q1	q1	PROPN
ejpam-5127	425	10	=	=	PUNCT
ejpam-5127	425	11	qce	qce	VERB
ejpam-5127	425	12	1	1	NUM
ejpam-5127	425	13	=	=	NOUN
ejpam-5127	425	14	qce	qce	NOUN
ejpam-5127	425	15	2	2	NUM
ejpam-5127	425	16	=	=	SYM
ejpam-5127	425	17	q2	q2	NOUN
ejpam-5127	425	18	.	.	PUNCT
ejpam-5127	426	1	therefore	therefore	ADV
ejpam-5127	426	2	j	j	PROPN
ejpam-5127	426	3	e	e	PROPN
ejpam-5127	426	4	is	be	AUX
ejpam-5127	426	5	a	a	DET
ejpam-5127	426	6	normal	normal	ADJ
ejpam-5127	426	7	as	as	ADP
ejpam-5127	426	8	a	a	DET
ejpam-5127	426	9	sub	sub	NOUN
ejpam-5127	426	10	lattice	lattice	NOUN
ejpam-5127	426	11	of	of	ADP
ejpam-5127	426	12	pf(v	pf(v	NOUN
ejpam-5127	426	13	)	)	PUNCT
ejpam-5127	426	14	.	.	PUNCT
ejpam-5127	427	1	conversely	conversely	ADV
ejpam-5127	427	2	,	,	PUNCT
ejpam-5127	427	3	assume	assume	VERB
ejpam-5127	427	4	that	that	SCONJ
ejpam-5127	427	5	j	j	PROPN
ejpam-5127	427	6	e	e	PROPN
ejpam-5127	427	7	is	be	AUX
ejpam-5127	427	8	normal	normal	ADJ
ejpam-5127	427	9	as	as	ADP
ejpam-5127	427	10	a	a	DET
ejpam-5127	427	11	sub	sub	NOUN
ejpam-5127	427	12	lattice	lattice	NOUN
ejpam-5127	427	13	of	of	ADP
ejpam-5127	427	14	pf(v	pf(v	NOUN
ejpam-5127	427	15	)	)	PUNCT
ejpam-5127	427	16	.	.	PUNCT
ejpam-5127	428	1	now	now	ADV
ejpam-5127	428	2	,	,	PUNCT
ejpam-5127	428	3	we	we	PRON
ejpam-5127	428	4	prove	prove	VERB
ejpam-5127	428	5	that	that	SCONJ
ejpam-5127	428	6	the	the	DET
ejpam-5127	428	7	filter	filter	NOUN
ejpam-5127	428	8	j	j	PROPN
ejpam-5127	428	9	is	be	AUX
ejpam-5127	428	10	normal	normal	ADJ
ejpam-5127	428	11	as	as	ADP
ejpam-5127	428	12	a	a	DET
ejpam-5127	428	13	sub	sub	NOUN
ejpam-5127	428	14	pdl	pdl	NOUN
ejpam-5127	428	15	of	of	ADP
ejpam-5127	428	16	v	v	NOUN
ejpam-5127	428	17	.	.	PUNCT
ejpam-5127	429	1	let	let	VERB
ejpam-5127	429	2	p	p	PRON
ejpam-5127	429	3	be	be	AUX
ejpam-5127	429	4	a	a	DET
ejpam-5127	429	5	prime	prime	ADJ
ejpam-5127	429	6	filter	filter	NOUN
ejpam-5127	429	7	of	of	ADP
ejpam-5127	429	8	j	j	PROPN
ejpam-5127	429	9	.	.	PUNCT
ejpam-5127	430	1	then	then	ADV
ejpam-5127	430	2	pe	pe	PROPN
ejpam-5127	430	3	is	be	AUX
ejpam-5127	430	4	a	a	DET
ejpam-5127	430	5	prime	prime	ADJ
ejpam-5127	430	6	filter	filter	NOUN
ejpam-5127	430	7	of	of	ADP
ejpam-5127	430	8	j	j	PROPN
ejpam-5127	430	9	e.	e.	PROPN
ejpam-5127	430	10	since	since	SCONJ
ejpam-5127	430	11	j	j	PROPN
ejpam-5127	430	12	e	e	PROPN
ejpam-5127	430	13	is	be	AUX
ejpam-5127	430	14	normal	normal	ADJ
ejpam-5127	430	15	,	,	PUNCT
ejpam-5127	430	16	pe	pe	PROPN
ejpam-5127	430	17	contains	contain	VERB
ejpam-5127	430	18	a	a	DET
ejpam-5127	430	19	unique	unique	ADJ
ejpam-5127	430	20	minimal	minimal	ADJ
ejpam-5127	430	21	prime	prime	ADJ
ejpam-5127	430	22	filter	filter	NOUN
ejpam-5127	430	23	say	say	VERB
ejpam-5127	430	24	q.	q.	PROPN
ejpam-5127	430	25	then	then	ADV
ejpam-5127	430	26	qc	qc	PROPN
ejpam-5127	430	27	is	be	AUX
ejpam-5127	430	28	a	a	DET
ejpam-5127	430	29	minimal	minimal	ADJ
ejpam-5127	430	30	prime	prime	ADJ
ejpam-5127	430	31	filter	filter	NOUN
ejpam-5127	430	32	contained	contain	VERB
ejpam-5127	430	33	in	in	ADP
ejpam-5127	430	34	pec	pec	NOUN
ejpam-5127	431	1	=	=	PROPN
ejpam-5127	432	1	p.	p.	NOUN
ejpam-5127	432	2	now	now	ADV
ejpam-5127	432	3	,	,	PUNCT
ejpam-5127	432	4	we	we	PRON
ejpam-5127	432	5	prove	prove	VERB
ejpam-5127	432	6	that	that	SCONJ
ejpam-5127	432	7	qc	qc	PROPN
ejpam-5127	432	8	is	be	AUX
ejpam-5127	432	9	unique	unique	ADJ
ejpam-5127	432	10	.	.	PUNCT
ejpam-5127	433	1	let	let	VERB
ejpam-5127	433	2	q1	q1	PROPN
ejpam-5127	433	3	be	be	AUX
ejpam-5127	433	4	any	any	DET
ejpam-5127	433	5	other	other	ADJ
ejpam-5127	433	6	minimal	minimal	ADJ
ejpam-5127	433	7	prime	prime	ADJ
ejpam-5127	433	8	filter	filter	NOUN
ejpam-5127	433	9	of	of	ADP
ejpam-5127	433	10	j	j	PROPN
ejpam-5127	433	11	contained	contain	VERB
ejpam-5127	433	12	in	in	ADP
ejpam-5127	433	13	p.	p.	PROPN
ejpam-5127	433	14	qe	qe	PROPN
ejpam-5127	433	15	1	1	NUM
ejpam-5127	433	16	is	be	AUX
ejpam-5127	433	17	a	a	DET
ejpam-5127	433	18	minimal	minimal	ADJ
ejpam-5127	433	19	prime	prime	ADJ
ejpam-5127	433	20	filter	filter	NOUN
ejpam-5127	433	21	r.	r.	PROPN
ejpam-5127	433	22	shukla	shukla	PROPN
ejpam-5127	433	23	et	et	PROPN
ejpam-5127	433	24	al	al	PROPN
ejpam-5127	433	25	.	.	PUNCT
ejpam-5127	433	26	/	/	SYM
ejpam-5127	433	27	eur	eur	PROPN
ejpam-5127	433	28	.	.	PUNCT
ejpam-5127	434	1	j.	j.	PROPN
ejpam-5127	434	2	pure	pure	PROPN
ejpam-5127	434	3	appl	appl	PROPN
ejpam-5127	434	4	.	.	PROPN
ejpam-5127	434	5	math	math	PROPN
ejpam-5127	434	6	,	,	PUNCT
ejpam-5127	434	7	17	17	NUM
ejpam-5127	434	8	(	(	PUNCT
ejpam-5127	434	9	2	2	NUM
ejpam-5127	434	10	)	)	PUNCT
ejpam-5127	434	11	(	(	PUNCT
ejpam-5127	434	12	2024	2024	NUM
ejpam-5127	434	13	)	)	PUNCT
ejpam-5127	434	14	,	,	PUNCT
ejpam-5127	434	15	1306	1306	NUM
ejpam-5127	434	16	-	-	SYM
ejpam-5127	434	17	1320	1320	NUM
ejpam-5127	434	18	1318	1318	NUM
ejpam-5127	434	19	contained	contain	VERB
ejpam-5127	434	20	in	in	ADP
ejpam-5127	434	21	the	the	DET
ejpam-5127	434	22	prime	prime	ADJ
ejpam-5127	434	23	filter	filter	NOUN
ejpam-5127	434	24	pe	pe	NOUN
ejpam-5127	434	25	.	.	PUNCT
ejpam-5127	435	1	since	since	SCONJ
ejpam-5127	435	2	pe	pe	PROPN
ejpam-5127	435	3	contains	contain	VERB
ejpam-5127	435	4	a	a	DET
ejpam-5127	435	5	unique	unique	ADJ
ejpam-5127	435	6	minimal	minimal	ADJ
ejpam-5127	435	7	prime	prime	ADJ
ejpam-5127	435	8	filter	filter	NOUN
ejpam-5127	435	9	,	,	PUNCT
ejpam-5127	435	10	we	we	PRON
ejpam-5127	435	11	get	get	VERB
ejpam-5127	435	12	q	q	ADJ
ejpam-5127	436	1	=	=	PUNCT
ejpam-5127	436	2	qe	qe	PROPN
ejpam-5127	436	3	1	1	NUM
ejpam-5127	436	4	.	.	PUNCT
ejpam-5127	437	1	this	this	PRON
ejpam-5127	437	2	gives	give	VERB
ejpam-5127	437	3	qc	qc	PROPN
ejpam-5127	437	4	=	=	PRON
ejpam-5127	437	5	(	(	PUNCT
ejpam-5127	437	6	qe	qe	PROPN
ejpam-5127	437	7	1	1	NUM
ejpam-5127	437	8	)	)	PUNCT
ejpam-5127	437	9	c	c	NOUN
ejpam-5127	437	10	=	=	SYM
ejpam-5127	437	11	q1	q1	PROPN
ejpam-5127	437	12	.	.	PUNCT
ejpam-5127	438	1	therefore	therefore	ADV
ejpam-5127	438	2	qc	qc	PROPN
ejpam-5127	438	3	is	be	AUX
ejpam-5127	438	4	unique	unique	ADJ
ejpam-5127	438	5	.	.	PUNCT
ejpam-5127	439	1	now	now	ADV
ejpam-5127	439	2	,	,	PUNCT
ejpam-5127	439	3	we	we	PRON
ejpam-5127	439	4	recall	recall	VERB
ejpam-5127	439	5	that	that	SCONJ
ejpam-5127	439	6	a	a	DET
ejpam-5127	439	7	relatively	relatively	ADV
ejpam-5127	439	8	complemented	complemented	ADJ
ejpam-5127	439	9	pdl	pdl	NOUN
ejpam-5127	439	10	is	be	AUX
ejpam-5127	439	11	a	a	DET
ejpam-5127	439	12	pdl	pdl	NOUN
ejpam-5127	439	13	in	in	ADP
ejpam-5127	439	14	which	which	PRON
ejpam-5127	439	15	every	every	DET
ejpam-5127	439	16	closed	closed	ADJ
ejpam-5127	439	17	interval	interval	NOUN
ejpam-5127	439	18	is	be	AUX
ejpam-5127	439	19	a	a	DET
ejpam-5127	439	20	dually	dually	ADV
ejpam-5127	439	21	complemented	complement	VERB
ejpam-5127	439	22	lattice	lattice	NOUN
ejpam-5127	439	23	.	.	PUNCT
ejpam-5127	440	1	every	every	DET
ejpam-5127	440	2	relatively	relatively	ADV
ejpam-5127	440	3	complemented	complemented	ADJ
ejpam-5127	440	4	pdl	pdl	NOUN
ejpam-5127	440	5	is	be	AUX
ejpam-5127	440	6	associative	associative	ADJ
ejpam-5127	440	7	.	.	PUNCT
ejpam-5127	441	1	in	in	ADP
ejpam-5127	441	2	the	the	DET
ejpam-5127	441	3	following	following	ADJ
ejpam-5127	441	4	result	result	NOUN
ejpam-5127	441	5	we	we	PRON
ejpam-5127	441	6	prove	prove	VERB
ejpam-5127	441	7	that	that	SCONJ
ejpam-5127	441	8	every	every	DET
ejpam-5127	441	9	relatively	relatively	ADV
ejpam-5127	441	10	complemented	complemented	ADJ
ejpam-5127	441	11	pdl	pdl	NOUN
ejpam-5127	441	12	is	be	AUX
ejpam-5127	441	13	a	a	DET
ejpam-5127	441	14	normal	normal	ADJ
ejpam-5127	441	15	pdl	pdl	NOUN
ejpam-5127	441	16	.	.	PUNCT
ejpam-5127	441	17	theorem	theorem	PROPN
ejpam-5127	441	18	13	13	NUM
ejpam-5127	441	19	.	.	PUNCT
ejpam-5127	442	1	every	every	DET
ejpam-5127	442	2	relatively	relatively	ADV
ejpam-5127	442	3	complemented	complemented	ADJ
ejpam-5127	442	4	pdl	pdl	NOUN
ejpam-5127	442	5	is	be	AUX
ejpam-5127	442	6	a	a	DET
ejpam-5127	442	7	normal	normal	ADJ
ejpam-5127	442	8	pdl	pdl	NOUN
ejpam-5127	442	9	.	.	PUNCT
ejpam-5127	443	1	proof	proof	NOUN
ejpam-5127	443	2	.	.	PUNCT
ejpam-5127	444	1	let	let	VERB
ejpam-5127	444	2	v	v	PART
ejpam-5127	444	3	be	be	AUX
ejpam-5127	444	4	a	a	DET
ejpam-5127	444	5	relatively	relatively	ADV
ejpam-5127	444	6	complemented	complemented	ADJ
ejpam-5127	444	7	pdl	pdl	NOUN
ejpam-5127	444	8	.	.	PUNCT
ejpam-5127	445	1	let	let	VERB
ejpam-5127	445	2	x	x	PRON
ejpam-5127	445	3	,	,	PUNCT
ejpam-5127	445	4	y	y	PROPN
ejpam-5127	445	5	∈	∈	PROPN
ejpam-5127	445	6	v	v	NOUN
ejpam-5127	445	7	and	and	CCONJ
ejpam-5127	445	8	x	x	PROPN
ejpam-5127	445	9	∨	∨	NUM
ejpam-5127	446	1	y	y	NOUN
ejpam-5127	446	2	=	=	SYM
ejpam-5127	446	3	1	1	X
ejpam-5127	446	4	.	.	PUNCT
ejpam-5127	447	1	let	let	VERB
ejpam-5127	447	2	z	z	NOUN
ejpam-5127	447	3	∈	∈	PROPN
ejpam-5127	447	4	v	v	NOUN
ejpam-5127	447	5	.	.	PUNCT
ejpam-5127	448	1	now	now	ADV
ejpam-5127	448	2	consider	consider	VERB
ejpam-5127	448	3	the	the	DET
ejpam-5127	448	4	interval	interval	NOUN
ejpam-5127	449	1	[	[	X
ejpam-5127	449	2	z	z	X
ejpam-5127	449	3	∧	∧	NOUN
ejpam-5127	449	4	y	y	PROPN
ejpam-5127	449	5	∧	∧	PROPN
ejpam-5127	449	6	x	x	X
ejpam-5127	449	7	,	,	PUNCT
ejpam-5127	449	8	1	1	NUM
ejpam-5127	449	9	]	]	PUNCT
ejpam-5127	449	10	.	.	PUNCT
ejpam-5127	450	1	since	since	SCONJ
ejpam-5127	450	2	x∨	x∨	PROPN
ejpam-5127	450	3	y	y	PROPN
ejpam-5127	450	4	=	=	SYM
ejpam-5127	450	5	1	1	NUM
ejpam-5127	450	6	,	,	PUNCT
ejpam-5127	450	7	we	we	PRON
ejpam-5127	450	8	have	have	VERB
ejpam-5127	450	9	x∧	x∧	PROPN
ejpam-5127	450	10	y	y	PROPN
ejpam-5127	450	11	=	=	SYM
ejpam-5127	450	12	y	y	PROPN
ejpam-5127	450	13	∧	∧	PROPN
ejpam-5127	450	14	x.	x.	NOUN
ejpam-5127	451	1	so	so	SCONJ
ejpam-5127	451	2	that	that	SCONJ
ejpam-5127	451	3	x	x	X
ejpam-5127	451	4	,	,	PUNCT
ejpam-5127	451	5	y	y	PROPN
ejpam-5127	451	6	∈	∈	PROPN
ejpam-5127	452	1	[	[	X
ejpam-5127	452	2	z	z	X
ejpam-5127	452	3	∧	∧	PROPN
ejpam-5127	452	4	y	y	PROPN
ejpam-5127	452	5	∧	∧	PROPN
ejpam-5127	452	6	x	x	X
ejpam-5127	452	7	,	,	PUNCT
ejpam-5127	452	8	1	1	NUM
ejpam-5127	452	9	]	]	PUNCT
ejpam-5127	452	10	.	.	PUNCT
ejpam-5127	453	1	therefore	therefore	ADV
ejpam-5127	453	2	there	there	PRON
ejpam-5127	453	3	exist	exist	VERB
ejpam-5127	453	4	complements	complement	NOUN
ejpam-5127	453	5	x′	x′	NUM
ejpam-5127	453	6	,	,	PUNCT
ejpam-5127	453	7	y′	y′	NOUN
ejpam-5127	453	8	of	of	ADP
ejpam-5127	453	9	x	x	PROPN
ejpam-5127	453	10	,	,	PUNCT
ejpam-5127	453	11	y	y	PROPN
ejpam-5127	453	12	,	,	PUNCT
ejpam-5127	453	13	respectively	respectively	ADV
ejpam-5127	453	14	in	in	ADP
ejpam-5127	453	15	[	[	X
ejpam-5127	453	16	z	z	X
ejpam-5127	453	17	∧	∧	PROPN
ejpam-5127	453	18	y	y	PROPN
ejpam-5127	453	19	∧	∧	PROPN
ejpam-5127	453	20	x	x	X
ejpam-5127	453	21	,	,	PUNCT
ejpam-5127	453	22	1	1	NUM
ejpam-5127	453	23	]	]	PUNCT
ejpam-5127	453	24	.	.	PUNCT
ejpam-5127	454	1	also	also	ADV
ejpam-5127	454	2	,	,	PUNCT
ejpam-5127	454	3	since	since	SCONJ
ejpam-5127	454	4	x	x	PROPN
ejpam-5127	454	5	∨	∨	NUM
ejpam-5127	454	6	y	y	NOUN
ejpam-5127	454	7	=	=	SYM
ejpam-5127	454	8	1	1	NUM
ejpam-5127	454	9	,	,	PUNCT
ejpam-5127	454	10	we	we	PRON
ejpam-5127	454	11	have	have	VERB
ejpam-5127	454	12	x′	x′	PROPN
ejpam-5127	454	13	∧	∧	NOUN
ejpam-5127	454	14	y′	y′	X
ejpam-5127	455	1	=	=	SYM
ejpam-5127	455	2	z	z	PROPN
ejpam-5127	455	3	∧	∧	PROPN
ejpam-5127	455	4	y	y	PROPN
ejpam-5127	455	5	∧	∧	PROPN
ejpam-5127	455	6	x.	x.	NOUN
ejpam-5127	455	7	(	(	PUNCT
ejpam-5127	455	8	z	z	PROPN
ejpam-5127	455	9	∨	∨	NUM
ejpam-5127	455	10	x′	x′	NUM
ejpam-5127	455	11	)	)	PUNCT
ejpam-5127	455	12	∧	∧	PROPN
ejpam-5127	455	13	(	(	PUNCT
ejpam-5127	455	14	z	z	NOUN
ejpam-5127	455	15	∨	∨	NUM
ejpam-5127	455	16	y′	y′	NUM
ejpam-5127	455	17	)	)	PUNCT
ejpam-5127	456	1	=	=	SYM
ejpam-5127	456	2	z	z	NOUN
ejpam-5127	456	3	∨	∨	X
ejpam-5127	456	4	(	(	PUNCT
ejpam-5127	456	5	x′	x′	PROPN
ejpam-5127	456	6	∧	∧	PROPN
ejpam-5127	456	7	y′	y′	NUM
ejpam-5127	456	8	)	)	PUNCT
ejpam-5127	456	9	=	=	SYM
ejpam-5127	456	10	z	z	NOUN
ejpam-5127	456	11	∨	∨	NOUN
ejpam-5127	456	12	(	(	PUNCT
ejpam-5127	456	13	z	z	PROPN
ejpam-5127	456	14	∧	∧	PROPN
ejpam-5127	456	15	y	y	PROPN
ejpam-5127	456	16	∧	∧	PROPN
ejpam-5127	456	17	x	x	X
ejpam-5127	456	18	)	)	PUNCT
ejpam-5127	456	19	=	=	SYM
ejpam-5127	456	20	z.	z.	PROPN
ejpam-5127	456	21	since	since	SCONJ
ejpam-5127	456	22	z	z	PROPN
ejpam-5127	456	23	∨	∨	PROPN
ejpam-5127	456	24	x′	x′	PROPN
ejpam-5127	456	25	∨	∨	NOUN
ejpam-5127	456	26	x	x	SYM
ejpam-5127	456	27	=	=	SYM
ejpam-5127	456	28	1	1	NUM
ejpam-5127	456	29	and	and	CCONJ
ejpam-5127	456	30	z	z	NOUN
ejpam-5127	456	31	∨	∨	NUM
ejpam-5127	456	32	y′	y′	X
ejpam-5127	456	33	∨	∨	NUM
ejpam-5127	456	34	y	y	PROPN
ejpam-5127	456	35	=	=	SYM
ejpam-5127	456	36	1	1	NUM
ejpam-5127	456	37	,	,	PUNCT
ejpam-5127	456	38	we	we	PRON
ejpam-5127	456	39	get	get	VERB
ejpam-5127	456	40	z	z	NOUN
ejpam-5127	456	41	∨	∨	NUM
ejpam-5127	456	42	x′	x′	PROPN
ejpam-5127	456	43	∈	∈	PROPN
ejpam-5127	456	44	(	(	PUNCT
ejpam-5127	456	45	x)•	x)•	PROPN
ejpam-5127	456	46	and	and	CCONJ
ejpam-5127	456	47	z	z	PROPN
ejpam-5127	456	48	∨	∨	NUM
ejpam-5127	456	49	y′	y′	NOUN
ejpam-5127	456	50	∈	∈	PROPN
ejpam-5127	456	51	(	(	PUNCT
ejpam-5127	456	52	y)•.	y)•.	PROPN
ejpam-5127	456	53	therefore	therefore	ADV
ejpam-5127	456	54	we	we	PRON
ejpam-5127	456	55	get	get	VERB
ejpam-5127	456	56	z	z	NOUN
ejpam-5127	456	57	∈	∈	PROPN
ejpam-5127	456	58	(	(	PUNCT
ejpam-5127	456	59	x)•	x)•	PROPN
ejpam-5127	456	60	∨	∨	NUM
ejpam-5127	456	61	(	(	PUNCT
ejpam-5127	456	62	y)•.	y)•.	PROPN
ejpam-5127	456	63	thus	thus	ADV
ejpam-5127	456	64	v	v	ADP
ejpam-5127	456	65	⊆	⊆	NUM
ejpam-5127	456	66	(	(	PUNCT
ejpam-5127	456	67	x)•	x)•	PROPN
ejpam-5127	456	68	∨	∨	NUM
ejpam-5127	456	69	(	(	PUNCT
ejpam-5127	456	70	y)•	y)•	NUM
ejpam-5127	456	71	and	and	CCONJ
ejpam-5127	456	72	hence	hence	ADV
ejpam-5127	456	73	v	v	NOUN
ejpam-5127	456	74	is	be	AUX
ejpam-5127	456	75	a	a	DET
ejpam-5127	456	76	normal	normal	ADJ
ejpam-5127	456	77	pdl	pdl	NOUN
ejpam-5127	456	78	.	.	PUNCT
ejpam-5127	456	79	theorem	theorem	PROPN
ejpam-5127	456	80	14	14	NUM
ejpam-5127	456	81	.	.	PUNCT
ejpam-5127	457	1	let	let	VERB
ejpam-5127	457	2	v	v	PART
ejpam-5127	457	3	be	be	AUX
ejpam-5127	457	4	a	a	DET
ejpam-5127	457	5	pdl	pdl	NOUN
ejpam-5127	457	6	.	.	PUNCT
ejpam-5127	458	1	then	then	ADV
ejpam-5127	458	2	v	v	NOUN
ejpam-5127	458	3	is	be	AUX
ejpam-5127	458	4	normal	normal	ADJ
ejpam-5127	458	5	if	if	SCONJ
ejpam-5127	458	6	and	and	CCONJ
ejpam-5127	458	7	only	only	ADV
ejpam-5127	458	8	if	if	SCONJ
ejpam-5127	458	9	each	each	DET
ejpam-5127	458	10	prime	prime	ADJ
ejpam-5127	458	11	ideal	ideal	NOUN
ejpam-5127	458	12	in	in	ADP
ejpam-5127	458	13	v	v	NOUN
ejpam-5127	458	14	is	be	AUX
ejpam-5127	458	15	contained	contain	VERB
ejpam-5127	458	16	in	in	ADP
ejpam-5127	458	17	a	a	DET
ejpam-5127	458	18	unique	unique	ADJ
ejpam-5127	458	19	maximal	maximal	ADJ
ejpam-5127	458	20	ideal	ideal	NOUN
ejpam-5127	458	21	.	.	PUNCT
ejpam-5127	459	1	proof	proof	NOUN
ejpam-5127	459	2	.	.	PUNCT
ejpam-5127	460	1	assume	assume	VERB
ejpam-5127	460	2	that	that	SCONJ
ejpam-5127	460	3	each	each	DET
ejpam-5127	460	4	prime	prime	ADJ
ejpam-5127	460	5	ideal	ideal	NOUN
ejpam-5127	460	6	in	in	ADP
ejpam-5127	460	7	v	v	NOUN
ejpam-5127	460	8	is	be	AUX
ejpam-5127	460	9	contained	contain	VERB
ejpam-5127	460	10	in	in	ADP
ejpam-5127	460	11	a	a	DET
ejpam-5127	460	12	unique	unique	ADJ
ejpam-5127	460	13	maximal	maximal	ADJ
ejpam-5127	460	14	ideal	ideal	NOUN
ejpam-5127	460	15	.	.	PUNCT
ejpam-5127	461	1	now	now	ADV
ejpam-5127	461	2	we	we	PRON
ejpam-5127	461	3	prove	prove	VERB
ejpam-5127	461	4	that	that	SCONJ
ejpam-5127	461	5	v	v	NOUN
ejpam-5127	461	6	is	be	AUX
ejpam-5127	461	7	a	a	DET
ejpam-5127	461	8	normal	normal	ADJ
ejpam-5127	461	9	pdl	pdl	NOUN
ejpam-5127	461	10	.	.	PUNCT
ejpam-5127	462	1	let	let	VERB
ejpam-5127	462	2	p	p	PRON
ejpam-5127	462	3	be	be	AUX
ejpam-5127	462	4	a	a	DET
ejpam-5127	462	5	prime	prime	ADJ
ejpam-5127	462	6	filter	filter	NOUN
ejpam-5127	462	7	of	of	ADP
ejpam-5127	462	8	v	v	NOUN
ejpam-5127	462	9	.	.	PUNCT
ejpam-5127	463	1	then	then	ADV
ejpam-5127	463	2	it	it	PRON
ejpam-5127	463	3	is	be	AUX
ejpam-5127	463	4	enough	enough	ADJ
ejpam-5127	463	5	to	to	PART
ejpam-5127	463	6	prove	prove	VERB
ejpam-5127	463	7	that	that	SCONJ
ejpam-5127	463	8	p	p	NOUN
ejpam-5127	463	9	contains	contain	VERB
ejpam-5127	463	10	a	a	DET
ejpam-5127	463	11	unique	unique	ADJ
ejpam-5127	463	12	minimal	minimal	ADJ
ejpam-5127	463	13	prime	prime	ADJ
ejpam-5127	463	14	filter	filter	NOUN
ejpam-5127	463	15	of	of	ADP
ejpam-5127	463	16	v	v	NOUN
ejpam-5127	463	17	.	.	PUNCT
ejpam-5127	464	1	since	since	SCONJ
ejpam-5127	464	2	p	p	NOUN
ejpam-5127	464	3	is	be	AUX
ejpam-5127	464	4	a	a	DET
ejpam-5127	464	5	prime	prime	ADJ
ejpam-5127	464	6	filter	filter	NOUN
ejpam-5127	464	7	,	,	PUNCT
ejpam-5127	464	8	v	v	NOUN
ejpam-5127	464	9	\p	\p	ADV
ejpam-5127	464	10	is	be	AUX
ejpam-5127	464	11	a	a	DET
ejpam-5127	464	12	prime	prime	ADJ
ejpam-5127	464	13	ideal	ideal	NOUN
ejpam-5127	464	14	of	of	ADP
ejpam-5127	464	15	v	v	NOUN
ejpam-5127	464	16	.	.	PUNCT
ejpam-5127	465	1	therefore	therefore	ADV
ejpam-5127	465	2	,	,	PUNCT
ejpam-5127	465	3	v	v	NOUN
ejpam-5127	465	4	\p	\p	ADV
ejpam-5127	465	5	is	be	AUX
ejpam-5127	465	6	contained	contain	VERB
ejpam-5127	465	7	in	in	ADP
ejpam-5127	465	8	a	a	DET
ejpam-5127	465	9	unique	unique	ADJ
ejpam-5127	465	10	maximal	maximal	ADJ
ejpam-5127	465	11	ideal	ideal	NOUN
ejpam-5127	465	12	,	,	PUNCT
ejpam-5127	465	13	say	say	VERB
ejpam-5127	465	14	g.	g.	PROPN
ejpam-5127	465	15	now	now	ADV
ejpam-5127	465	16	,	,	PUNCT
ejpam-5127	465	17	(	(	PUNCT
ejpam-5127	465	18	v	v	NOUN
ejpam-5127	465	19	\p	\p	ADV
ejpam-5127	465	20	)	)	PUNCT
ejpam-5127	465	21	⊆	⊆	NUM
ejpam-5127	465	22	g	g	NOUN
ejpam-5127	465	23	gives	give	VERB
ejpam-5127	465	24	that	that	PRON
ejpam-5127	465	25	v	v	NOUN
ejpam-5127	465	26	\g	\g	NOUN
ejpam-5127	465	27	⊆	⊆	NUM
ejpam-5127	466	1	p.	p.	NOUN
ejpam-5127	466	2	clearly	clearly	ADV
ejpam-5127	466	3	,	,	PUNCT
ejpam-5127	466	4	we	we	PRON
ejpam-5127	466	5	have	have	VERB
ejpam-5127	466	6	v	v	NOUN
ejpam-5127	466	7	\g	\g	NOUN
ejpam-5127	466	8	is	be	AUX
ejpam-5127	466	9	a	a	DET
ejpam-5127	466	10	minimal	minimal	ADJ
ejpam-5127	466	11	prime	prime	ADJ
ejpam-5127	466	12	filter	filter	NOUN
ejpam-5127	466	13	of	of	ADP
ejpam-5127	466	14	v	v	NOUN
ejpam-5127	466	15	contained	contain	VERB
ejpam-5127	466	16	in	in	ADP
ejpam-5127	466	17	p.	p.	NOUN
ejpam-5127	466	18	let	let	VERB
ejpam-5127	466	19	q	q	PART
ejpam-5127	466	20	be	be	AUX
ejpam-5127	466	21	a	a	DET
ejpam-5127	466	22	minimal	minimal	ADJ
ejpam-5127	466	23	prime	prime	ADJ
ejpam-5127	466	24	filter	filter	NOUN
ejpam-5127	466	25	of	of	ADP
ejpam-5127	466	26	v	v	NOUN
ejpam-5127	466	27	such	such	ADJ
ejpam-5127	466	28	that	that	DET
ejpam-5127	466	29	q	q	NOUN
ejpam-5127	466	30	⊆	⊆	NUM
ejpam-5127	466	31	p.	p.	NOUN
ejpam-5127	466	32	then	then	ADV
ejpam-5127	466	33	v	v	ADP
ejpam-5127	466	34	\p	\p	ADV
ejpam-5127	466	35	⊆	⊆	NUM
ejpam-5127	466	36	v	v	ADP
ejpam-5127	466	37	\q	\q	NOUN
ejpam-5127	466	38	.	.	PUNCT
ejpam-5127	467	1	since	since	SCONJ
ejpam-5127	467	2	q	q	PROPN
ejpam-5127	467	3	is	be	AUX
ejpam-5127	467	4	a	a	DET
ejpam-5127	467	5	minimal	minimal	ADJ
ejpam-5127	467	6	prime	prime	ADJ
ejpam-5127	467	7	filter	filter	NOUN
ejpam-5127	467	8	,	,	PUNCT
ejpam-5127	467	9	v	v	ADJ
ejpam-5127	467	10	\q	\q	NOUN
ejpam-5127	467	11	is	be	AUX
ejpam-5127	467	12	a	a	DET
ejpam-5127	467	13	maximal	maximal	ADJ
ejpam-5127	467	14	prime	prime	ADJ
ejpam-5127	467	15	ideal	ideal	NOUN
ejpam-5127	467	16	of	of	ADP
ejpam-5127	467	17	v	v	NOUN
ejpam-5127	467	18	.	.	PUNCT
ejpam-5127	468	1	that	that	PRON
ejpam-5127	468	2	is	is	ADV
ejpam-5127	468	3	,	,	PUNCT
ejpam-5127	468	4	v	v	X
ejpam-5127	468	5	\p	\p	ADV
ejpam-5127	468	6	is	be	AUX
ejpam-5127	468	7	a	a	DET
ejpam-5127	468	8	prime	prime	ADJ
ejpam-5127	468	9	ideal	ideal	NOUN
ejpam-5127	468	10	which	which	PRON
ejpam-5127	468	11	is	be	AUX
ejpam-5127	468	12	contained	contain	VERB
ejpam-5127	468	13	in	in	ADP
ejpam-5127	468	14	two	two	NUM
ejpam-5127	468	15	maximal	maximal	ADJ
ejpam-5127	468	16	ideal	ideal	NOUN
ejpam-5127	468	17	v	v	ADP
ejpam-5127	468	18	\q	\q	NOUN
ejpam-5127	468	19	and	and	CCONJ
ejpam-5127	468	20	g.	g.	PROPN
ejpam-5127	468	21	this	this	PRON
ejpam-5127	468	22	is	be	AUX
ejpam-5127	468	23	a	a	DET
ejpam-5127	468	24	contradiction	contradiction	NOUN
ejpam-5127	468	25	to	to	ADP
ejpam-5127	468	26	our	our	PRON
ejpam-5127	468	27	assumption	assumption	NOUN
ejpam-5127	468	28	.	.	PUNCT
ejpam-5127	469	1	therefore	therefore	ADV
ejpam-5127	469	2	v	v	ADJ
ejpam-5127	469	3	\g	\g	NOUN
ejpam-5127	469	4	is	be	AUX
ejpam-5127	469	5	unique	unique	ADJ
ejpam-5127	469	6	.	.	PUNCT
ejpam-5127	470	1	therefore	therefore	ADV
ejpam-5127	470	2	,	,	PUNCT
ejpam-5127	470	3	v	v	NOUN
ejpam-5127	470	4	is	be	AUX
ejpam-5127	470	5	a	a	DET
ejpam-5127	470	6	normal	normal	ADJ
ejpam-5127	470	7	pdl	pdl	NOUN
ejpam-5127	470	8	.	.	PUNCT
ejpam-5127	471	1	similarly	similarly	ADV
ejpam-5127	471	2	,	,	PUNCT
ejpam-5127	471	3	we	we	PRON
ejpam-5127	471	4	can	can	AUX
ejpam-5127	471	5	prove	prove	VERB
ejpam-5127	471	6	the	the	DET
ejpam-5127	471	7	converse	converse	NOUN
ejpam-5127	471	8	.	.	PUNCT
ejpam-5127	472	1	theorem	theorem	VERB
ejpam-5127	472	2	15	15	NUM
ejpam-5127	472	3	.	.	PUNCT
ejpam-5127	473	1	let	let	VERB
ejpam-5127	473	2	v	v	PART
ejpam-5127	473	3	be	be	AUX
ejpam-5127	473	4	a	a	DET
ejpam-5127	473	5	pdl	pdl	NOUN
ejpam-5127	473	6	.	.	PUNCT
ejpam-5127	474	1	then	then	ADV
ejpam-5127	474	2	v	v	NOUN
ejpam-5127	474	3	is	be	AUX
ejpam-5127	474	4	normal	normal	ADJ
ejpam-5127	474	5	if	if	SCONJ
ejpam-5127	474	6	and	and	CCONJ
ejpam-5127	474	7	only	only	ADV
ejpam-5127	474	8	if	if	SCONJ
ejpam-5127	474	9	each	each	DET
ejpam-5127	474	10	minimal	minimal	ADJ
ejpam-5127	474	11	prime	prime	ADJ
ejpam-5127	474	12	ideal	ideal	NOUN
ejpam-5127	474	13	in	in	ADP
ejpam-5127	474	14	v	v	NOUN
ejpam-5127	474	15	is	be	AUX
ejpam-5127	474	16	contained	contain	VERB
ejpam-5127	474	17	in	in	ADP
ejpam-5127	474	18	a	a	DET
ejpam-5127	474	19	unique	unique	ADJ
ejpam-5127	474	20	maximal	maximal	ADJ
ejpam-5127	474	21	ideal	ideal	NOUN
ejpam-5127	474	22	of	of	ADP
ejpam-5127	474	23	v	v	NOUN
ejpam-5127	474	24	.	.	PUNCT
ejpam-5127	475	1	proof	proof	NOUN
ejpam-5127	475	2	.	.	PUNCT
ejpam-5127	476	1	assume	assume	VERB
ejpam-5127	476	2	that	that	SCONJ
ejpam-5127	476	3	v	v	NOUN
ejpam-5127	476	4	is	be	AUX
ejpam-5127	476	5	a	a	DET
ejpam-5127	476	6	normal	normal	ADJ
ejpam-5127	476	7	pdl	pdl	NOUN
ejpam-5127	476	8	.	.	PUNCT
ejpam-5127	477	1	let	let	VERB
ejpam-5127	477	2	f	f	PRON
ejpam-5127	477	3	be	be	AUX
ejpam-5127	477	4	any	any	DET
ejpam-5127	477	5	minimal	minimal	ADJ
ejpam-5127	477	6	prime	prime	ADJ
ejpam-5127	477	7	ideal	ideal	NOUN
ejpam-5127	477	8	in	in	ADP
ejpam-5127	477	9	v	v	NOUN
ejpam-5127	477	10	.	.	PUNCT
ejpam-5127	478	1	then	then	ADV
ejpam-5127	478	2	,	,	PUNCT
ejpam-5127	478	3	the	the	DET
ejpam-5127	478	4	prime	prime	ADJ
ejpam-5127	478	5	ideal	ideal	NOUN
ejpam-5127	478	6	f	f	PROPN
ejpam-5127	478	7	is	be	AUX
ejpam-5127	478	8	contained	contain	VERB
ejpam-5127	478	9	in	in	ADP
ejpam-5127	478	10	a	a	DET
ejpam-5127	478	11	unique	unique	ADJ
ejpam-5127	478	12	maximal	maximal	ADJ
ejpam-5127	478	13	ideal	ideal	NOUN
ejpam-5127	478	14	of	of	ADP
ejpam-5127	478	15	v	v	NOUN
ejpam-5127	478	16	.	.	PUNCT
ejpam-5127	479	1	that	that	PRON
ejpam-5127	479	2	is	is	ADV
ejpam-5127	479	3	,	,	PUNCT
ejpam-5127	479	4	every	every	DET
ejpam-5127	479	5	minimal	minimal	ADJ
ejpam-5127	479	6	prime	prime	ADJ
ejpam-5127	479	7	ideal	ideal	NOUN
ejpam-5127	479	8	of	of	ADP
ejpam-5127	479	9	v	v	NOUN
ejpam-5127	479	10	is	be	AUX
ejpam-5127	479	11	contained	contain	VERB
ejpam-5127	479	12	in	in	ADP
ejpam-5127	479	13	a	a	DET
ejpam-5127	479	14	unique	unique	ADJ
ejpam-5127	479	15	maximal	maximal	ADJ
ejpam-5127	479	16	ideal	ideal	NOUN
ejpam-5127	479	17	of	of	ADP
ejpam-5127	479	18	v	v	NOUN
ejpam-5127	479	19	.	.	PUNCT
ejpam-5127	480	1	conversely	conversely	ADV
ejpam-5127	480	2	,	,	PUNCT
ejpam-5127	480	3	assume	assume	VERB
ejpam-5127	480	4	that	that	SCONJ
ejpam-5127	480	5	each	each	DET
ejpam-5127	480	6	minimal	minimal	ADJ
ejpam-5127	480	7	prime	prime	ADJ
ejpam-5127	480	8	ideal	ideal	NOUN
ejpam-5127	480	9	in	in	ADP
ejpam-5127	480	10	v	v	NOUN
ejpam-5127	480	11	is	be	AUX
ejpam-5127	480	12	contained	contain	VERB
ejpam-5127	480	13	in	in	ADP
ejpam-5127	480	14	a	a	DET
ejpam-5127	480	15	unique	unique	ADJ
ejpam-5127	480	16	maximal	maximal	ADJ
ejpam-5127	480	17	ideal	ideal	NOUN
ejpam-5127	480	18	of	of	ADP
ejpam-5127	480	19	v	v	NOUN
ejpam-5127	480	20	.	.	PUNCT
ejpam-5127	481	1	suppose	suppose	VERB
ejpam-5127	481	2	,	,	PUNCT
ejpam-5127	481	3	a	a	DET
ejpam-5127	481	4	prime	prime	ADJ
ejpam-5127	481	5	filter	filter	NOUN
ejpam-5127	481	6	p	p	NOUN
ejpam-5127	481	7	of	of	ADP
ejpam-5127	481	8	v	v	PROPN
ejpam-5127	481	9	contains	contain	VERB
ejpam-5127	481	10	two	two	NUM
ejpam-5127	481	11	minimal	minimal	ADJ
ejpam-5127	481	12	prime	prime	ADJ
ejpam-5127	481	13	filters	filter	NOUN
ejpam-5127	481	14	q1	q1	PROPN
ejpam-5127	481	15	and	and	CCONJ
ejpam-5127	481	16	q2	q2	NOUN
ejpam-5127	481	17	of	of	ADP
ejpam-5127	481	18	v	v	NOUN
ejpam-5127	481	19	.	.	PUNCT
ejpam-5127	482	1	since	since	SCONJ
ejpam-5127	482	2	p	p	NOUN
ejpam-5127	482	3	is	be	AUX
ejpam-5127	482	4	a	a	DET
ejpam-5127	482	5	proper	proper	ADJ
ejpam-5127	482	6	filter	filter	NOUN
ejpam-5127	482	7	of	of	ADP
ejpam-5127	482	8	v	v	NOUN
ejpam-5127	482	9	,	,	PUNCT
ejpam-5127	482	10	p	p	NOUN
ejpam-5127	482	11	is	be	AUX
ejpam-5127	482	12	contained	contain	VERB
ejpam-5127	482	13	in	in	ADP
ejpam-5127	482	14	a	a	DET
ejpam-5127	482	15	maximal	maximal	ADJ
ejpam-5127	482	16	filter	filter	NOUN
ejpam-5127	482	17	,	,	PUNCT
ejpam-5127	482	18	say	say	VERB
ejpam-5127	482	19	m	m	ADV
ejpam-5127	482	20	of	of	ADP
ejpam-5127	482	21	v	v	NOUN
ejpam-5127	482	22	.	.	PUNCT
ejpam-5127	483	1	then	then	ADV
ejpam-5127	483	2	,	,	PUNCT
ejpam-5127	483	3	we	we	PRON
ejpam-5127	483	4	get	get	VERB
ejpam-5127	483	5	q1	q1	PROPN
ejpam-5127	483	6	⊆	⊆	NUM
ejpam-5127	483	7	p	p	NOUN
ejpam-5127	483	8	⊆	⊆	NUM
ejpam-5127	483	9	m	m	NOUN
ejpam-5127	483	10	and	and	CCONJ
ejpam-5127	483	11	q2	q2	NOUN
ejpam-5127	483	12	⊆	⊆	NUM
ejpam-5127	483	13	p	p	SYM
ejpam-5127	483	14	⊆	⊆	NUM
ejpam-5127	483	15	m.	m.	NOUN
ejpam-5127	483	16	this	this	PRON
ejpam-5127	483	17	gives	give	VERB
ejpam-5127	483	18	v	v	ADP
ejpam-5127	483	19	\m	\m	NOUN
ejpam-5127	483	20	⊆	⊆	NUM
ejpam-5127	483	21	v	v	ADP
ejpam-5127	483	22	\q1	\q1	PROPN
ejpam-5127	483	23	and	and	CCONJ
ejpam-5127	483	24	v	v	ADP
ejpam-5127	483	25	\m	\m	NOUN
ejpam-5127	483	26	⊆	⊆	NUM
ejpam-5127	483	27	v	v	ADP
ejpam-5127	483	28	\q2	\q2	NUM
ejpam-5127	483	29	.	.	PUNCT
ejpam-5127	484	1	that	that	PRON
ejpam-5127	484	2	is	be	AUX
ejpam-5127	484	3	,	,	PUNCT
ejpam-5127	484	4	the	the	DET
ejpam-5127	484	5	minimal	minimal	ADJ
ejpam-5127	484	6	prime	prime	ADJ
ejpam-5127	484	7	ideal	ideal	NOUN
ejpam-5127	484	8	v	v	ADP
ejpam-5127	484	9	\m	\m	NOUN
ejpam-5127	484	10	is	be	AUX
ejpam-5127	484	11	contained	contain	VERB
ejpam-5127	484	12	in	in	ADP
ejpam-5127	484	13	two	two	NUM
ejpam-5127	484	14	maximal	maximal	ADJ
ejpam-5127	484	15	ideals	ideal	NOUN
ejpam-5127	484	16	v	v	ADP
ejpam-5127	484	17	\q1	\q1	PROPN
ejpam-5127	484	18	and	and	CCONJ
ejpam-5127	484	19	v	v	ADP
ejpam-5127	484	20	\q2	\q2	NUM
ejpam-5127	484	21	.	.	PUNCT
ejpam-5127	485	1	this	this	PRON
ejpam-5127	485	2	is	be	AUX
ejpam-5127	485	3	a	a	DET
ejpam-5127	485	4	contradiction	contradiction	NOUN
ejpam-5127	485	5	.	.	PUNCT
ejpam-5127	486	1	therefore	therefore	ADV
ejpam-5127	486	2	each	each	DET
ejpam-5127	486	3	prime	prime	ADJ
ejpam-5127	486	4	filter	filter	NOUN
ejpam-5127	486	5	in	in	ADP
ejpam-5127	486	6	v	v	NOUN
ejpam-5127	486	7	contains	contain	VERB
ejpam-5127	486	8	a	a	DET
ejpam-5127	486	9	unique	unique	ADJ
ejpam-5127	486	10	minimal	minimal	ADJ
ejpam-5127	486	11	prime	prime	ADJ
ejpam-5127	486	12	filter	filter	NOUN
ejpam-5127	486	13	of	of	ADP
ejpam-5127	486	14	v	v	NOUN
ejpam-5127	486	15	.	.	PUNCT
ejpam-5127	487	1	hence	hence	ADV
ejpam-5127	487	2	v	v	NOUN
ejpam-5127	487	3	is	be	AUX
ejpam-5127	487	4	normal	normal	ADJ
ejpam-5127	487	5	.	.	PUNCT
ejpam-5127	488	1	references	reference	NOUN
ejpam-5127	488	2	1319	1319	NUM
ejpam-5127	488	3	4	4	NUM
ejpam-5127	488	4	.	.	PUNCT
ejpam-5127	489	1	conclusions	conclusion	NOUN
ejpam-5127	489	2	in	in	ADP
ejpam-5127	489	3	summary	summary	NOUN
ejpam-5127	489	4	,	,	PUNCT
ejpam-5127	489	5	we	we	PRON
ejpam-5127	489	6	have	have	AUX
ejpam-5127	489	7	defined	define	VERB
ejpam-5127	489	8	o(p	o(p	NOUN
ejpam-5127	489	9	)	)	PUNCT
ejpam-5127	489	10	for	for	ADP
ejpam-5127	489	11	any	any	DET
ejpam-5127	489	12	filter	filter	NOUN
ejpam-5127	489	13	p	p	NOUN
ejpam-5127	489	14	of	of	ADP
ejpam-5127	489	15	a	a	DET
ejpam-5127	489	16	paradistributive	paradistributive	ADJ
ejpam-5127	489	17	latticoid	latticoid	NOUN
ejpam-5127	489	18	and	and	CCONJ
ejpam-5127	489	19	proved	prove	VERB
ejpam-5127	489	20	that	that	SCONJ
ejpam-5127	489	21	o(p	o(p	PROPN
ejpam-5127	489	22	)	)	PUNCT
ejpam-5127	489	23	is	be	AUX
ejpam-5127	489	24	a	a	DET
ejpam-5127	489	25	filter	filter	NOUN
ejpam-5127	489	26	when	when	SCONJ
ejpam-5127	489	27	p	p	NOUN
ejpam-5127	489	28	is	be	AUX
ejpam-5127	489	29	prime	prime	ADJ
ejpam-5127	489	30	.	.	PUNCT
ejpam-5127	490	1	additionally	additionally	ADV
ejpam-5127	490	2	,	,	PUNCT
ejpam-5127	490	3	we	we	PRON
ejpam-5127	490	4	have	have	AUX
ejpam-5127	490	5	established	establish	VERB
ejpam-5127	490	6	that	that	SCONJ
ejpam-5127	490	7	every	every	DET
ejpam-5127	490	8	minimal	minimal	ADJ
ejpam-5127	490	9	prime	prime	ADJ
ejpam-5127	490	10	filter	filter	NOUN
ejpam-5127	490	11	belonging	belong	VERB
ejpam-5127	490	12	to	to	ADP
ejpam-5127	490	13	o(p	o(p	NUM
ejpam-5127	490	14	)	)	PUNCT
ejpam-5127	490	15	is	be	AUX
ejpam-5127	490	16	contained	contain	VERB
ejpam-5127	490	17	in	in	ADP
ejpam-5127	490	18	p	p	NOUN
ejpam-5127	490	19	,	,	PUNCT
ejpam-5127	490	20	and	and	CCONJ
ejpam-5127	490	21	that	that	PRON
ejpam-5127	490	22	o(p	o(p	PROPN
ejpam-5127	490	23	)	)	PUNCT
ejpam-5127	490	24	is	be	AUX
ejpam-5127	490	25	the	the	DET
ejpam-5127	490	26	intersection	intersection	NOUN
ejpam-5127	490	27	of	of	ADP
ejpam-5127	490	28	all	all	DET
ejpam-5127	490	29	minimal	minimal	ADJ
ejpam-5127	490	30	prime	prime	ADJ
ejpam-5127	490	31	filters	filter	NOUN
ejpam-5127	490	32	contained	contain	VERB
ejpam-5127	490	33	in	in	ADP
ejpam-5127	490	34	p.	p.	NOUN
ejpam-5127	490	35	the	the	DET
ejpam-5127	490	36	concept	concept	NOUN
ejpam-5127	490	37	of	of	ADP
ejpam-5127	490	38	a	a	DET
ejpam-5127	490	39	normal	normal	ADJ
ejpam-5127	490	40	paradistributive	paradistributive	ADJ
ejpam-5127	490	41	latticoid	latticoid	NOUN
ejpam-5127	490	42	has	have	AUX
ejpam-5127	490	43	been	be	AUX
ejpam-5127	490	44	introduced	introduce	VERB
ejpam-5127	490	45	and	and	CCONJ
ejpam-5127	490	46	characterized	characterize	VERB
ejpam-5127	490	47	in	in	ADP
ejpam-5127	490	48	terms	term	NOUN
ejpam-5127	490	49	of	of	ADP
ejpam-5127	490	50	prime	prime	ADJ
ejpam-5127	490	51	filters	filter	NOUN
ejpam-5127	490	52	and	and	CCONJ
ejpam-5127	490	53	minimal	minimal	ADJ
ejpam-5127	490	54	prime	prime	ADJ
ejpam-5127	490	55	filters	filter	NOUN
ejpam-5127	490	56	,	,	PUNCT
ejpam-5127	490	57	with	with	ADP
ejpam-5127	490	58	the	the	DET
ejpam-5127	490	59	proof	proof	NOUN
ejpam-5127	490	60	that	that	SCONJ
ejpam-5127	490	61	every	every	DET
ejpam-5127	490	62	relatively	relatively	ADV
ejpam-5127	490	63	complemented	complemented	ADJ
ejpam-5127	490	64	paradistributive	paradistributive	ADJ
ejpam-5127	490	65	latticoid	latticoid	NOUN
ejpam-5127	490	66	is	be	AUX
ejpam-5127	490	67	normal	normal	ADJ
ejpam-5127	490	68	.	.	PUNCT
ejpam-5127	491	1	our	our	PRON
ejpam-5127	491	2	future	future	ADJ
ejpam-5127	491	3	work	work	NOUN
ejpam-5127	491	4	will	will	AUX
ejpam-5127	491	5	delve	delve	VERB
ejpam-5127	491	6	into	into	ADP
ejpam-5127	491	7	the	the	DET
ejpam-5127	491	8	introduction	introduction	NOUN
ejpam-5127	491	9	of	of	ADP
ejpam-5127	491	10	relatively	relatively	ADV
ejpam-5127	491	11	normal	normal	ADJ
ejpam-5127	491	12	paradistributive	paradistributive	ADJ
ejpam-5127	491	13	latticoids	latticoid	NOUN
ejpam-5127	491	14	.	.	PUNCT
ejpam-5127	492	1	we	we	PRON
ejpam-5127	492	2	also	also	ADV
ejpam-5127	492	3	aim	aim	VERB
ejpam-5127	492	4	to	to	PART
ejpam-5127	492	5	explore	explore	VERB
ejpam-5127	492	6	the	the	DET
ejpam-5127	492	7	topological	topological	ADJ
ejpam-5127	492	8	characterization	characterization	NOUN
ejpam-5127	492	9	of	of	ADP
ejpam-5127	492	10	normal	normal	ADJ
ejpam-5127	492	11	paradistributive	paradistributive	ADJ
ejpam-5127	492	12	latticoids	latticoid	NOUN
ejpam-5127	492	13	.	.	PUNCT
ejpam-5127	493	1	furthermore	furthermore	ADV
ejpam-5127	493	2	,	,	PUNCT
ejpam-5127	493	3	we	we	PRON
ejpam-5127	493	4	plan	plan	VERB
ejpam-5127	493	5	to	to	PART
ejpam-5127	493	6	investigate	investigate	VERB
ejpam-5127	493	7	s	s	NOUN
ejpam-5127	493	8	-	-	ADJ
ejpam-5127	493	9	normal	normal	ADJ
ejpam-5127	493	10	paradistributive	paradistributive	ADJ
ejpam-5127	493	11	latticoids	latticoid	NOUN
ejpam-5127	493	12	,	,	PUNCT
ejpam-5127	493	13	where	where	SCONJ
ejpam-5127	493	14	s	s	NOUN
ejpam-5127	493	15	represents	represent	VERB
ejpam-5127	493	16	a	a	DET
ejpam-5127	493	17	sub	sub	NOUN
ejpam-5127	493	18	paradistributive	paradistributive	ADJ
ejpam-5127	493	19	latticoid	latticoid	NOUN
ejpam-5127	493	20	of	of	ADP
ejpam-5127	493	21	a	a	DET
ejpam-5127	493	22	given	give	VERB
ejpam-5127	493	23	paradistributive	paradistributive	ADJ
ejpam-5127	493	24	latticoid	latticoid	NOUN
ejpam-5127	493	25	.	.	PUNCT
ejpam-5127	494	1	conflicts	conflict	NOUN
ejpam-5127	494	2	of	of	ADP
ejpam-5127	494	3	interest	interest	NOUN
ejpam-5127	494	4	or	or	CCONJ
ejpam-5127	494	5	competing	compete	VERB
ejpam-5127	494	6	interests	interest	NOUN
ejpam-5127	494	7	the	the	DET
ejpam-5127	494	8	authors	author	NOUN
ejpam-5127	494	9	declare	declare	VERB
ejpam-5127	494	10	that	that	SCONJ
ejpam-5127	494	11	they	they	PRON
ejpam-5127	494	12	have	have	VERB
ejpam-5127	494	13	no	no	DET
ejpam-5127	494	14	conflicts	conflict	NOUN
ejpam-5127	494	15	of	of	ADP
ejpam-5127	494	16	interest	interest	NOUN
ejpam-5127	494	17	.	.	PUNCT
ejpam-5127	495	1	informed	inform	VERB
ejpam-5127	495	2	consent	consent	VERB
ejpam-5127	495	3	the	the	DET
ejpam-5127	495	4	authors	author	NOUN
ejpam-5127	495	5	are	be	AUX
ejpam-5127	495	6	fully	fully	ADV
ejpam-5127	495	7	aware	aware	ADJ
ejpam-5127	495	8	and	and	CCONJ
ejpam-5127	495	9	satisfied	satisfied	ADJ
ejpam-5127	495	10	with	with	ADP
ejpam-5127	495	11	the	the	DET
ejpam-5127	495	12	contents	content	NOUN
ejpam-5127	495	13	of	of	ADP
ejpam-5127	495	14	the	the	DET
ejpam-5127	495	15	article	article	NOUN
ejpam-5127	495	16	.	.	PUNCT
ejpam-5127	496	1	acknowledgements	acknowledgement	NOUN
ejpam-5127	496	2	the	the	DET
ejpam-5127	496	3	authors	author	NOUN
ejpam-5127	496	4	wish	wish	VERB
ejpam-5127	496	5	to	to	PART
ejpam-5127	496	6	thank	thank	VERB
ejpam-5127	496	7	the	the	DET
ejpam-5127	496	8	anonymous	anonymous	ADJ
ejpam-5127	496	9	reviewers	reviewer	NOUN
ejpam-5127	496	10	for	for	ADP
ejpam-5127	496	11	their	their	PRON
ejpam-5127	496	12	valuable	valuable	ADJ
ejpam-5127	496	13	suggestions	suggestion	NOUN
ejpam-5127	496	14	.	.	PUNCT
ejpam-5127	497	1	references	reference	NOUN
ejpam-5127	497	2	[	[	X
ejpam-5127	497	3	1	1	NUM
ejpam-5127	497	4	]	]	X
ejpam-5127	497	5	s	s	VERB
ejpam-5127	497	6	ajjarapu	ajjarapu	PROPN
ejpam-5127	497	7	,	,	PUNCT
ejpam-5127	497	8	r	r	NOUN
ejpam-5127	497	9	bandaru	bandaru	NOUN
ejpam-5127	497	10	,	,	PUNCT
ejpam-5127	497	11	r	r	NOUN
ejpam-5127	497	12	shukla	shukla	NOUN
ejpam-5127	497	13	,	,	PUNCT
ejpam-5127	497	14	and	and	CCONJ
ejpam-5127	497	15	y	y	PROPN
ejpam-5127	497	16	b	b	PROPN
ejpam-5127	497	17	jun	jun	PROPN
ejpam-5127	497	18	.	.	PUNCT
ejpam-5127	497	19	parapseudo	parapseudo	NOUN
ejpam-5127	497	20	-	-	NOUN
ejpam-5127	497	21	complementation	complementation	NOUN
ejpam-5127	497	22	on	on	ADP
ejpam-5127	497	23	paradistributive	paradistributive	ADJ
ejpam-5127	497	24	latticoids	latticoid	NOUN
ejpam-5127	497	25	.	.	PUNCT
ejpam-5127	498	1	european	european	PROPN
ejpam-5127	498	2	journal	journal	PROPN
ejpam-5127	498	3	of	of	ADP
ejpam-5127	498	4	pure	pure	ADJ
ejpam-5127	498	5	and	and	CCONJ
ejpam-5127	498	6	applied	applied	ADJ
ejpam-5127	498	7	mathematics	mathematic	NOUN
ejpam-5127	498	8	,	,	PUNCT
ejpam-5127	498	9	in	in	ADP
ejpam-5127	498	10	press	press	NOUN
ejpam-5127	498	11	,	,	PUNCT
ejpam-5127	498	12	2024	2024	NUM
ejpam-5127	498	13	.	.	PUNCT
ejpam-5127	499	1	https://doi.org/10.29020/nybg.ejpam.v17i2.5042	https://doi.org/10.29020/nybg.ejpam.v17i2.5042	NUM
ejpam-5127	499	2	.	.	PUNCT
ejpam-5127	500	1	[	[	X
ejpam-5127	500	2	2	2	NUM
ejpam-5127	500	3	]	]	SYM
ejpam-5127	500	4	r	r	NOUN
ejpam-5127	500	5	bandaru	bandaru	NOUN
ejpam-5127	500	6	and	and	CCONJ
ejpam-5127	500	7	s	s	NOUN
ejpam-5127	500	8	ajjarapu	ajjarapu	PROPN
ejpam-5127	500	9	.	.	PUNCT
ejpam-5127	501	1	paradistributive	paradistributive	ADJ
ejpam-5127	501	2	latticoids	latticoids	PROPN
ejpam-5127	501	3	.	.	PUNCT
ejpam-5127	502	1	european	european	PROPN
ejpam-5127	502	2	journal	journal	PROPN
ejpam-5127	502	3	of	of	ADP
ejpam-5127	502	4	pure	pure	ADJ
ejpam-5127	502	5	and	and	CCONJ
ejpam-5127	502	6	applied	applied	ADJ
ejpam-5127	502	7	mathematics	mathematic	NOUN
ejpam-5127	502	8	,	,	PUNCT
ejpam-5127	502	9	in	in	ADP
ejpam-5127	502	10	press	press	NOUN
ejpam-5127	502	11	,	,	PUNCT
ejpam-5127	502	12	2024	2024	NUM
ejpam-5127	502	13	.	.	PUNCT
ejpam-5127	503	1	https://doi.org/10.29020/nybg.ejpam.v17i2.5125	https://doi.org/10.29020/nybg.ejpam.v17i2.5125	X
ejpam-5127	503	2	.	.	PUNCT
ejpam-5127	504	1	[	[	X
ejpam-5127	504	2	3	3	X
ejpam-5127	504	3	]	]	X
ejpam-5127	504	4	w	w	PROPN
ejpam-5127	504	5	h	h	PROPN
ejpam-5127	504	6	cornish	cornish	PROPN
ejpam-5127	504	7	.	.	PUNCT
ejpam-5127	504	8	normal	normal	ADJ
ejpam-5127	504	9	lattices	lattice	NOUN
ejpam-5127	504	10	.	.	PUNCT
ejpam-5127	505	1	journal	journal	NOUN
ejpam-5127	505	2	of	of	ADP
ejpam-5127	505	3	the	the	DET
ejpam-5127	505	4	australian	australian	ADJ
ejpam-5127	505	5	mathematical	mathematical	ADJ
ejpam-5127	505	6	society	society	NOUN
ejpam-5127	505	7	,	,	PUNCT
ejpam-5127	505	8	14:200	14:200	NUM
ejpam-5127	505	9	–	–	PUNCT
ejpam-5127	505	10	215	215	NUM
ejpam-5127	505	11	,	,	PUNCT
ejpam-5127	505	12	1972	1972	NUM
ejpam-5127	505	13	.	.	PUNCT
ejpam-5127	506	1	[	[	X
ejpam-5127	506	2	4	4	NUM
ejpam-5127	506	3	]	]	X
ejpam-5127	506	4	y	y	PROPN
ejpam-5127	506	5	s	s	PROPN
ejpam-5127	506	6	pawar	pawar	PROPN
ejpam-5127	506	7	.	.	PUNCT
ejpam-5127	507	1	characterizations	characterization	NOUN
ejpam-5127	507	2	of	of	ADP
ejpam-5127	507	3	normal	normal	ADJ
ejpam-5127	507	4	lattices	lattice	NOUN
ejpam-5127	507	5	.	.	PUNCT
ejpam-5127	508	1	indian	indian	ADJ
ejpam-5127	508	2	journal	journal	PROPN
ejpam-5127	508	3	of	of	ADP
ejpam-5127	508	4	pure	pure	ADJ
ejpam-5127	508	5	and	and	CCONJ
ejpam-5127	508	6	applied	applied	ADJ
ejpam-5127	508	7	mathematics	mathematic	NOUN
ejpam-5127	508	8	,	,	PUNCT
ejpam-5127	508	9	24(11):651–656	24(11):651–656	NUM
ejpam-5127	508	10	,	,	PUNCT
ejpam-5127	508	11	1993	1993	NUM
ejpam-5127	508	12	.	.	PUNCT
ejpam-5127	509	1	[	[	X
ejpam-5127	509	2	5	5	NUM
ejpam-5127	509	3	]	]	X
ejpam-5127	509	4	y	y	PROPN
ejpam-5127	509	5	s	s	PROPN
ejpam-5127	509	6	pawar	pawar	NOUN
ejpam-5127	509	7	and	and	CCONJ
ejpam-5127	509	8	n	n	PROPN
ejpam-5127	509	9	k	k	PROPN
ejpam-5127	509	10	thakare	thakare	PROPN
ejpam-5127	509	11	.	.	PUNCT
ejpam-5127	510	1	pm	pm	NOUN
ejpam-5127	510	2	-	-	PUNCT
ejpam-5127	510	3	lattices	lattice	NOUN
ejpam-5127	510	4	.	.	PUNCT
ejpam-5127	511	1	algebra	algebra	NOUN
ejpam-5127	511	2	universalis	universali	VERB
ejpam-5127	511	3	,	,	PUNCT
ejpam-5127	511	4	7:259–263	7:259–263	NUM
ejpam-5127	511	5	,	,	PUNCT
ejpam-5127	511	6	1977	1977	NUM
ejpam-5127	511	7	.	.	PUNCT
ejpam-5127	512	1	references	reference	NOUN
ejpam-5127	512	2	1320	1320	NUM
ejpam-5127	512	3	[	[	X
ejpam-5127	512	4	6	6	NUM
ejpam-5127	512	5	]	]	PUNCT
ejpam-5127	512	6	s	s	PART
ejpam-5127	512	7	rasouli	rasouli	NOUN
ejpam-5127	512	8	and	and	CCONJ
ejpam-5127	512	9	a	a	DET
ejpam-5127	512	10	dehghani	dehghani	PROPN
ejpam-5127	512	11	.	.	PUNCT
ejpam-5127	513	1	mpand	mpand	PROPN
ejpam-5127	513	2	purified	purify	VERB
ejpam-5127	513	3	residuated	residuate	VERB
ejpam-5127	513	4	lattices	lattice	NOUN
ejpam-5127	513	5	.	.	PUNCT
ejpam-5127	514	1	soft	soft	ADJ
ejpam-5127	514	2	computing	computing	NOUN
ejpam-5127	514	3	,	,	PUNCT
ejpam-5127	514	4	27:131–148	27:131–148	PROPN
ejpam-5127	514	5	,	,	PUNCT
ejpam-5127	514	6	2023	2023	NUM
ejpam-5127	514	7	.	.	PUNCT
ejpam-5127	515	1	[	[	X
ejpam-5127	515	2	7	7	NUM
ejpam-5127	515	3	]	]	X
ejpam-5127	515	4	s	s	PART
ejpam-5127	515	5	rasouli	rasouli	PROPN
ejpam-5127	515	6	and	and	CCONJ
ejpam-5127	515	7	m	m	PROPN
ejpam-5127	515	8	kondo	kondo	PROPN
ejpam-5127	515	9	.	.	PUNCT
ejpam-5127	516	1	n	n	CCONJ
ejpam-5127	516	2	-	-	PUNCT
ejpam-5127	516	3	normal	normal	ADJ
ejpam-5127	516	4	residuated	residuate	VERB
ejpam-5127	516	5	lattices	lattice	NOUN
ejpam-5127	516	6	.	.	PUNCT
ejpam-5127	517	1	soft	soft	ADJ
ejpam-5127	517	2	computing	computing	NOUN
ejpam-5127	517	3	,	,	PUNCT
ejpam-5127	517	4	24:247–258	24:247–258	PROPN
ejpam-5127	517	5	,	,	PUNCT
ejpam-5127	517	6	2020	2020	NUM
ejpam-5127	517	7	.	.	PUNCT
ejpam-5127	518	1	[	[	X
ejpam-5127	518	2	8	8	NUM
ejpam-5127	518	3	]	]	PUNCT
ejpam-5127	518	4	a	a	DET
ejpam-5127	518	5	borumand	borumand	ADJ
ejpam-5127	518	6	saeid	saeid	PROPN
ejpam-5127	518	7	and	and	CCONJ
ejpam-5127	518	8	n	n	PRON
ejpam-5127	518	9	mohtashamnia	mohtashamnia	NOUN
ejpam-5127	518	10	.	.	PUNCT
ejpam-5127	519	1	stabilizer	stabilizer	NOUN
ejpam-5127	519	2	in	in	ADP
ejpam-5127	519	3	residuated	residuate	VERB
ejpam-5127	519	4	lattices	lattice	NOUN
ejpam-5127	519	5	.	.	PUNCT
ejpam-5127	520	1	u	u	NOUN
ejpam-5127	520	2	p	p	PROPN
ejpam-5127	520	3	b	b	PROPN
ejpam-5127	520	4	scientific	scientific	ADJ
ejpam-5127	520	5	bulletin	bulletin	NOUN
ejpam-5127	520	6	.	.	PUNCT
ejpam-5127	521	1	series	series	PROPN
ejpam-5127	521	2	a.	a.	PROPN
ejpam-5127	521	3	,	,	PUNCT
ejpam-5127	521	4	74(2):65–74	74(2):65–74	NUM
ejpam-5127	521	5	,	,	PUNCT
ejpam-5127	521	6	2012	2012	NUM
ejpam-5127	521	7	.	.	PUNCT
ejpam-5127	522	1	[	[	X
ejpam-5127	522	2	9	9	NUM
ejpam-5127	522	3	]	]	X
ejpam-5127	522	4	h	h	NOUN
ejpam-5127	522	5	simmons	simmon	NOUN
ejpam-5127	522	6	.	.	PUNCT
ejpam-5127	522	7	reticulated	reticulate	VERB
ejpam-5127	522	8	rings	ring	NOUN
ejpam-5127	522	9	.	.	PUNCT
ejpam-5127	523	1	journal	journal	PROPN
ejpam-5127	523	2	of	of	ADP
ejpam-5127	523	3	algebra	algebra	PROPN
ejpam-5127	523	4	,	,	PUNCT
ejpam-5127	523	5	66(1):169–192	66(1):169–192	PROPN
ejpam-5127	523	6	,	,	PUNCT
ejpam-5127	523	7	1980	1980	NUM
ejpam-5127	523	8	.	.	PUNCT