id	sid	tid	token	lemma	pos
ejpam-7135	1	1	european	european	PROPN
ejpam-7135	1	2	journal	journal	PROPN
ejpam-7135	1	3	of	of	ADP
ejpam-7135	1	4	pure	pure	ADJ
ejpam-7135	1	5	and	and	CCONJ
ejpam-7135	1	6	applied	applied	ADJ
ejpam-7135	1	7	mathematics	mathematic	NOUN
ejpam-7135	1	8	2025	2025	NUM
ejpam-7135	1	9	,	,	PUNCT
ejpam-7135	1	10	vol	vol	NOUN
ejpam-7135	1	11	.	.	PROPN
ejpam-7135	1	12	18	18	NUM
ejpam-7135	1	13	,	,	PUNCT
ejpam-7135	1	14	issue	issue	NOUN
ejpam-7135	1	15	4	4	NUM
ejpam-7135	1	16	,	,	PUNCT
ejpam-7135	1	17	article	article	NOUN
ejpam-7135	1	18	number	number	NOUN
ejpam-7135	1	19	7135	7135	NUM
ejpam-7135	1	20	issn	issn	VERB
ejpam-7135	1	21	1307	1307	NUM
ejpam-7135	1	22	-	-	SYM
ejpam-7135	1	23	5543	5543	NUM
ejpam-7135	1	24	–	–	PUNCT
ejpam-7135	1	25	ejpam.com	ejpam.com	X
ejpam-7135	1	26	published	publish	VERB
ejpam-7135	1	27	by	by	ADP
ejpam-7135	1	28	new	new	PROPN
ejpam-7135	1	29	york	york	PROPN
ejpam-7135	1	30	business	business	PROPN
ejpam-7135	1	31	global	global	ADJ
ejpam-7135	1	32	stone	stone	PROPN
ejpam-7135	1	33	paradistributive	paradistributive	PROPN
ejpam-7135	1	34	latticoids	latticoids	PROPN
ejpam-7135	1	35	ravikumar	ravikumar	PROPN
ejpam-7135	1	36	bandaru1	bandaru1	PROPN
ejpam-7135	1	37	,	,	PUNCT
ejpam-7135	1	38	ramesh	ramesh	PROPN
ejpam-7135	1	39	sirisetti2	sirisetti2	PROPN
ejpam-7135	1	40	,	,	PUNCT
ejpam-7135	1	41	satyanarayana	satyanarayana	PROPN
ejpam-7135	1	42	rao	rao	PROPN
ejpam-7135	1	43	kola3	kola3	PROPN
ejpam-7135	1	44	,	,	PUNCT
ejpam-7135	1	45	rafi	rafi	PROPN
ejpam-7135	1	46	noorbhasha4	noorbhasha4	PROPN
ejpam-7135	1	47	,	,	PUNCT
ejpam-7135	1	48	hashem	hashem	PROPN
ejpam-7135	1	49	bordbar5	bordbar5	PROPN
ejpam-7135	1	50	,	,	PUNCT
ejpam-7135	1	51	aiyared	aiyare	VERB
ejpam-7135	1	52	iampan6,∗	iampan6,∗	PROPN
ejpam-7135	1	53	1	1	NUM
ejpam-7135	1	54	department	department	NOUN
ejpam-7135	1	55	of	of	ADP
ejpam-7135	1	56	mathematics	mathematic	NOUN
ejpam-7135	1	57	,	,	PUNCT
ejpam-7135	1	58	school	school	NOUN
ejpam-7135	1	59	of	of	ADP
ejpam-7135	1	60	advanced	advanced	ADJ
ejpam-7135	1	61	sciences	science	NOUN
ejpam-7135	1	62	,	,	PUNCT
ejpam-7135	1	63	vit	vit	PROPN
ejpam-7135	1	64	-	-	PUNCT
ejpam-7135	1	65	ap	ap	PROPN
ejpam-7135	1	66	university	university	PROPN
ejpam-7135	1	67	,	,	PUNCT
ejpam-7135	1	68	andhra	andhra	PROPN
ejpam-7135	1	69	pradesh-522237	pradesh-522237	NOUN
ejpam-7135	1	70	,	,	PUNCT
ejpam-7135	1	71	india	india	PROPN
ejpam-7135	1	72	2	2	NUM
ejpam-7135	1	73	department	department	NOUN
ejpam-7135	1	74	of	of	ADP
ejpam-7135	1	75	mathematics	mathematics	PROPN
ejpam-7135	1	76	,	,	PUNCT
ejpam-7135	1	77	aditya	aditya	PROPN
ejpam-7135	1	78	university	university	PROPN
ejpam-7135	1	79	,	,	PUNCT
ejpam-7135	1	80	surampalem	surampalem	PROPN
ejpam-7135	1	81	,	,	PUNCT
ejpam-7135	1	82	kakinada	kakinada	PROPN
ejpam-7135	1	83	,	,	PUNCT
ejpam-7135	1	84	andhra	andhra	PROPN
ejpam-7135	1	85	pradesh533437	pradesh533437	PROPN
ejpam-7135	1	86	,	,	PUNCT
ejpam-7135	1	87	india	india	PROPN
ejpam-7135	1	88	3	3	NUM
ejpam-7135	1	89	department	department	PROPN
ejpam-7135	1	90	of	of	ADP
ejpam-7135	1	91	business	business	NOUN
ejpam-7135	1	92	mathematics	mathematics	PROPN
ejpam-7135	1	93	&	&	CCONJ
ejpam-7135	1	94	information	information	NOUN
ejpam-7135	1	95	technology	technology	NOUN
ejpam-7135	1	96	,	,	PUNCT
ejpam-7135	1	97	amity	amity	NOUN
ejpam-7135	1	98	global	global	ADJ
ejpam-7135	1	99	business	business	NOUN
ejpam-7135	1	100	school	school	NOUN
ejpam-7135	1	101	,	,	PUNCT
ejpam-7135	1	102	hyderabad-500082	hyderabad-500082	NOUN
ejpam-7135	1	103	,	,	PUNCT
ejpam-7135	1	104	telangana	telangana	PROPN
ejpam-7135	1	105	,	,	PUNCT
ejpam-7135	1	106	india	india	PROPN
ejpam-7135	1	107	4	4	NUM
ejpam-7135	1	108	department	department	NOUN
ejpam-7135	1	109	of	of	ADP
ejpam-7135	1	110	mathematics	mathematic	NOUN
ejpam-7135	1	111	,	,	PUNCT
ejpam-7135	1	112	sri	sri	PROPN
ejpam-7135	1	113	siddhartha	siddhartha	PROPN
ejpam-7135	1	114	academy	academy	PROPN
ejpam-7135	1	115	of	of	ADP
ejpam-7135	1	116	higher	high	ADJ
ejpam-7135	1	117	education	education	NOUN
ejpam-7135	1	118	(	(	PUNCT
ejpam-7135	1	119	deemed	deem	VERB
ejpam-7135	1	120	to	to	PART
ejpam-7135	1	121	be	be	AUX
ejpam-7135	1	122	university	university	NOUN
ejpam-7135	1	123	)	)	PUNCT
ejpam-7135	1	124	,	,	PUNCT
ejpam-7135	1	125	vijayawada-520007	vijayawada-520007	VERB
ejpam-7135	1	126	,	,	PUNCT
ejpam-7135	1	127	andhra	andhra	PROPN
ejpam-7135	1	128	pradesh	pradesh	PROPN
ejpam-7135	1	129	,	,	PUNCT
ejpam-7135	1	130	india	india	PROPN
ejpam-7135	1	131	5	5	NUM
ejpam-7135	1	132	center	center	NOUN
ejpam-7135	1	133	for	for	ADP
ejpam-7135	1	134	information	information	NOUN
ejpam-7135	1	135	technology	technology	NOUN
ejpam-7135	1	136	and	and	CCONJ
ejpam-7135	1	137	applied	apply	VERB
ejpam-7135	1	138	mathematics	mathematic	NOUN
ejpam-7135	1	139	,	,	PUNCT
ejpam-7135	1	140	university	university	PROPN
ejpam-7135	1	141	of	of	ADP
ejpam-7135	1	142	nova	nova	PROPN
ejpam-7135	1	143	gorica	gorica	PROPN
ejpam-7135	1	144	,	,	PUNCT
ejpam-7135	1	145	5000	5000	NUM
ejpam-7135	1	146	nova	nova	PROPN
ejpam-7135	1	147	gorica	gorica	PROPN
ejpam-7135	1	148	,	,	PUNCT
ejpam-7135	1	149	slovenia	slovenia	PROPN
ejpam-7135	1	150	6	6	NUM
ejpam-7135	1	151	department	department	NOUN
ejpam-7135	1	152	of	of	ADP
ejpam-7135	1	153	mathematics	mathematic	NOUN
ejpam-7135	1	154	,	,	PUNCT
ejpam-7135	1	155	school	school	NOUN
ejpam-7135	1	156	of	of	ADP
ejpam-7135	1	157	science	science	NOUN
ejpam-7135	1	158	,	,	PUNCT
ejpam-7135	1	159	university	university	NOUN
ejpam-7135	1	160	of	of	ADP
ejpam-7135	1	161	phayao	phayao	NOUN
ejpam-7135	1	162	,	,	PUNCT
ejpam-7135	1	163	mae	mae	PROPN
ejpam-7135	1	164	ka	ka	PROPN
ejpam-7135	1	165	,	,	PUNCT
ejpam-7135	1	166	mueang	mueang	PROPN
ejpam-7135	1	167	,	,	PUNCT
ejpam-7135	1	168	phayao	phayao	NOUN
ejpam-7135	1	169	56000	56000	NUM
ejpam-7135	1	170	,	,	PUNCT
ejpam-7135	1	171	thailand	thailand	PROPN
ejpam-7135	1	172	abstract	abstract	NOUN
ejpam-7135	1	173	.	.	PUNCT
ejpam-7135	2	1	we	we	PRON
ejpam-7135	2	2	introduce	introduce	VERB
ejpam-7135	2	3	the	the	DET
ejpam-7135	2	4	concept	concept	NOUN
ejpam-7135	2	5	of	of	ADP
ejpam-7135	2	6	stone	stone	NOUN
ejpam-7135	2	7	paradistributive	paradistributive	ADJ
ejpam-7135	2	8	latticoids	latticoids	PROPN
ejpam-7135	2	9	(	(	PUNCT
ejpam-7135	2	10	stone	stone	NOUN
ejpam-7135	2	11	pdls	pdl	NOUN
ejpam-7135	2	12	)	)	PUNCT
ejpam-7135	2	13	as	as	ADP
ejpam-7135	2	14	a	a	DET
ejpam-7135	2	15	natural	natural	ADJ
ejpam-7135	2	16	generalization	generalization	NOUN
ejpam-7135	2	17	of	of	ADP
ejpam-7135	2	18	stone	stone	NOUN
ejpam-7135	2	19	lattices	lattice	NOUN
ejpam-7135	2	20	to	to	ADP
ejpam-7135	2	21	the	the	DET
ejpam-7135	2	22	broader	broad	ADJ
ejpam-7135	2	23	setting	setting	NOUN
ejpam-7135	2	24	of	of	ADP
ejpam-7135	2	25	paradistributive	paradistributive	ADJ
ejpam-7135	2	26	latticoids	latticoid	NOUN
ejpam-7135	2	27	endowed	endow	VERB
ejpam-7135	2	28	with	with	ADP
ejpam-7135	2	29	parapseudo	parapseudo	NOUN
ejpam-7135	2	30	-	-	NOUN
ejpam-7135	2	31	complementation	complementation	NOUN
ejpam-7135	2	32	.	.	PUNCT
ejpam-7135	3	1	we	we	PRON
ejpam-7135	3	2	provide	provide	VERB
ejpam-7135	3	3	multiple	multiple	ADJ
ejpam-7135	3	4	equivalent	equivalent	ADJ
ejpam-7135	3	5	characterizations	characterization	NOUN
ejpam-7135	3	6	of	of	ADP
ejpam-7135	3	7	stone	stone	NOUN
ejpam-7135	3	8	pdls	pdl	NOUN
ejpam-7135	3	9	,	,	PUNCT
ejpam-7135	3	10	both	both	CCONJ
ejpam-7135	3	11	algebraic	algebraic	ADJ
ejpam-7135	3	12	and	and	CCONJ
ejpam-7135	3	13	topological	topological	ADJ
ejpam-7135	3	14	,	,	PUNCT
ejpam-7135	3	15	including	include	VERB
ejpam-7135	3	16	those	those	PRON
ejpam-7135	3	17	based	base	VERB
ejpam-7135	3	18	on	on	ADP
ejpam-7135	3	19	the	the	DET
ejpam-7135	3	20	structure	structure	NOUN
ejpam-7135	3	21	of	of	ADP
ejpam-7135	3	22	principal	principal	ADJ
ejpam-7135	3	23	filters	filter	NOUN
ejpam-7135	3	24	,	,	PUNCT
ejpam-7135	3	25	the	the	DET
ejpam-7135	3	26	comaximality	comaximality	NOUN
ejpam-7135	3	27	condition	condition	NOUN
ejpam-7135	3	28	of	of	ADP
ejpam-7135	3	29	distinct	distinct	ADJ
ejpam-7135	3	30	minimal	minimal	ADJ
ejpam-7135	3	31	prime	prime	ADJ
ejpam-7135	3	32	filters	filter	NOUN
ejpam-7135	3	33	,	,	PUNCT
ejpam-7135	3	34	and	and	CCONJ
ejpam-7135	3	35	the	the	DET
ejpam-7135	3	36	retract	retract	ADJ
ejpam-7135	3	37	properties	property	NOUN
ejpam-7135	3	38	of	of	ADP
ejpam-7135	3	39	the	the	DET
ejpam-7135	3	40	associated	associated	ADJ
ejpam-7135	3	41	spectral	spectral	ADJ
ejpam-7135	3	42	spaces	space	NOUN
ejpam-7135	3	43	.	.	PUNCT
ejpam-7135	4	1	moreover	moreover	ADV
ejpam-7135	4	2	,	,	PUNCT
ejpam-7135	4	3	we	we	PRON
ejpam-7135	4	4	establish	establish	VERB
ejpam-7135	4	5	canonical	canonical	ADJ
ejpam-7135	4	6	correspondences	correspondence	NOUN
ejpam-7135	4	7	between	between	ADP
ejpam-7135	4	8	prime	prime	ADJ
ejpam-7135	4	9	filters	filter	NOUN
ejpam-7135	4	10	of	of	ADP
ejpam-7135	4	11	pdls	pdl	NOUN
ejpam-7135	4	12	and	and	CCONJ
ejpam-7135	4	13	those	those	PRON
ejpam-7135	4	14	of	of	ADP
ejpam-7135	4	15	associated	associated	ADJ
ejpam-7135	4	16	boolean	boolean	ADJ
ejpam-7135	4	17	algebras	algebra	NOUN
ejpam-7135	4	18	,	,	PUNCT
ejpam-7135	4	19	revealing	reveal	VERB
ejpam-7135	4	20	new	new	ADJ
ejpam-7135	4	21	representation	representation	NOUN
ejpam-7135	4	22	theorems	theorem	NOUN
ejpam-7135	4	23	and	and	CCONJ
ejpam-7135	4	24	duality	duality	NOUN
ejpam-7135	4	25	principles	principle	NOUN
ejpam-7135	4	26	.	.	PUNCT
ejpam-7135	5	1	these	these	DET
ejpam-7135	5	2	results	result	NOUN
ejpam-7135	5	3	unify	unify	VERB
ejpam-7135	5	4	and	and	CCONJ
ejpam-7135	5	5	extend	extend	VERB
ejpam-7135	5	6	classical	classical	ADJ
ejpam-7135	5	7	lattice	lattice	NOUN
ejpam-7135	5	8	-	-	PUNCT
ejpam-7135	5	9	theoretic	theoretic	ADJ
ejpam-7135	5	10	frameworks	framework	NOUN
ejpam-7135	5	11	—	—	PUNCT
ejpam-7135	5	12	particularly	particularly	ADV
ejpam-7135	5	13	those	those	PRON
ejpam-7135	5	14	concerning	concern	VERB
ejpam-7135	5	15	stone	stone	NOUN
ejpam-7135	5	16	lattices	lattice	NOUN
ejpam-7135	5	17	—	—	PUNCT
ejpam-7135	5	18	into	into	ADP
ejpam-7135	5	19	a	a	DET
ejpam-7135	5	20	more	more	ADV
ejpam-7135	5	21	general	general	ADJ
ejpam-7135	5	22	algebraic	algebraic	ADJ
ejpam-7135	5	23	logic	logic	NOUN
ejpam-7135	5	24	context	context	NOUN
ejpam-7135	5	25	,	,	PUNCT
ejpam-7135	5	26	laying	lay	VERB
ejpam-7135	5	27	a	a	DET
ejpam-7135	5	28	robust	robust	ADJ
ejpam-7135	5	29	foundation	foundation	NOUN
ejpam-7135	5	30	for	for	ADP
ejpam-7135	5	31	future	future	ADJ
ejpam-7135	5	32	applications	application	NOUN
ejpam-7135	5	33	in	in	ADP
ejpam-7135	5	34	lattice	lattice	PROPN
ejpam-7135	5	35	theory	theory	NOUN
ejpam-7135	5	36	,	,	PUNCT
ejpam-7135	5	37	universal	universal	ADJ
ejpam-7135	5	38	algebra	algebra	NOUN
ejpam-7135	5	39	,	,	PUNCT
ejpam-7135	5	40	and	and	CCONJ
ejpam-7135	5	41	topological	topological	ADJ
ejpam-7135	5	42	duality	duality	NOUN
ejpam-7135	5	43	.	.	PUNCT
ejpam-7135	6	1	2020	2020	NUM
ejpam-7135	6	2	mathematics	mathematic	NOUN
ejpam-7135	6	3	subject	subject	NOUN
ejpam-7135	6	4	classifications	classification	NOUN
ejpam-7135	6	5	:	:	PUNCT
ejpam-7135	6	6	06d99	06d99	NUM
ejpam-7135	6	7	,	,	PUNCT
ejpam-7135	6	8	06b10	06b10	NOUN
ejpam-7135	6	9	key	key	ADJ
ejpam-7135	6	10	words	word	NOUN
ejpam-7135	6	11	and	and	CCONJ
ejpam-7135	6	12	phrases	phrase	NOUN
ejpam-7135	6	13	:	:	PUNCT
ejpam-7135	6	14	paradistributive	paradistributive	ADJ
ejpam-7135	6	15	latticoid	latticoid	NOUN
ejpam-7135	6	16	,	,	PUNCT
ejpam-7135	6	17	stone	stone	NOUN
ejpam-7135	6	18	lattice	lattice	PROPN
ejpam-7135	6	19	,	,	PUNCT
ejpam-7135	6	20	parapseudo	parapseudo	NOUN
ejpam-7135	6	21	-	-	NOUN
ejpam-7135	6	22	complementation	complementation	NOUN
ejpam-7135	6	23	,	,	PUNCT
ejpam-7135	6	24	prime	prime	ADJ
ejpam-7135	6	25	filter	filter	NOUN
ejpam-7135	6	26	,	,	PUNCT
ejpam-7135	6	27	minimal	minimal	ADJ
ejpam-7135	6	28	prime	prime	ADJ
ejpam-7135	6	29	filter	filter	NOUN
ejpam-7135	6	30	∗corresponding	∗corresponde	VERB
ejpam-7135	6	31	author	author	NOUN
ejpam-7135	6	32	.	.	PUNCT
ejpam-7135	7	1	doi	doi	NOUN
ejpam-7135	7	2	:	:	PUNCT
ejpam-7135	7	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7135	https://doi.org/10.29020/nybg.ejpam.v18i4.7135	NOUN
ejpam-7135	7	4	email	email	NOUN
ejpam-7135	7	5	addresses	address	NOUN
ejpam-7135	7	6	:	:	PUNCT
ejpam-7135	8	1	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-7135	8	2	(	(	PUNCT
ejpam-7135	8	3	r.	r.	PROPN
ejpam-7135	8	4	bandaru	bandaru	PROPN
ejpam-7135	8	5	)	)	PUNCT
ejpam-7135	8	6	,	,	PUNCT
ejpam-7135	8	7	ramesh.sirisetti@gmail.com	ramesh.sirisetti@gmail.com	X
ejpam-7135	8	8	(	(	PUNCT
ejpam-7135	8	9	r.	r.	PROPN
ejpam-7135	8	10	sirisetti	sirisetti	PROPN
ejpam-7135	8	11	)	)	PUNCT
ejpam-7135	8	12	,	,	PUNCT
ejpam-7135	8	13	satyakola@gmail.com	satyakola@gmail.com	X
ejpam-7135	8	14	(	(	PUNCT
ejpam-7135	8	15	s.	s.	PROPN
ejpam-7135	8	16	r.	r.	PROPN
ejpam-7135	8	17	kola	kola	PROPN
ejpam-7135	8	18	)	)	PUNCT
ejpam-7135	8	19	,	,	PUNCT
ejpam-7135	8	20	rafimaths@gmail.com	rafimaths@gmail.com	X
ejpam-7135	8	21	(	(	PUNCT
ejpam-7135	8	22	r.	r.	PROPN
ejpam-7135	8	23	noorbhasha	noorbhasha	PROPN
ejpam-7135	8	24	)	)	PUNCT
ejpam-7135	8	25	,	,	PUNCT
ejpam-7135	8	26	hashem.bordbar@ung.si	hashem.bordbar@ung.si	PROPN
ejpam-7135	8	27	(	(	PUNCT
ejpam-7135	8	28	h.	h.	PROPN
ejpam-7135	8	29	bordbar	bordbar	PROPN
ejpam-7135	8	30	)	)	PUNCT
ejpam-7135	8	31	,	,	PUNCT
ejpam-7135	8	32	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-7135	8	33	(	(	PUNCT
ejpam-7135	8	34	a.	a.	NOUN
ejpam-7135	8	35	iampan	iampan	PROPN
ejpam-7135	8	36	)	)	PUNCT
ejpam-7135	8	37	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-7135	9	1	1	1	NUM
ejpam-7135	9	2	copyright	copyright	NOUN
ejpam-7135	9	3	:	:	PUNCT
ejpam-7135	9	4	©	©	PROPN
ejpam-7135	9	5	2025	2025	NUM
ejpam-7135	9	6	the	the	DET
ejpam-7135	9	7	author(s	author(s	NOUN
ejpam-7135	9	8	)	)	PUNCT
ejpam-7135	9	9	.	.	PUNCT
ejpam-7135	10	1	(	(	PUNCT
ejpam-7135	10	2	cc	cc	NOUN
ejpam-7135	10	3	by	by	ADP
ejpam-7135	10	4	-	-	PUNCT
ejpam-7135	10	5	nc	nc	PROPN
ejpam-7135	10	6	4.0	4.0	NUM
ejpam-7135	10	7	)	)	PUNCT
ejpam-7135	10	8	r.	r.	PROPN
ejpam-7135	10	9	bandaru	bandaru	PROPN
ejpam-7135	10	10	et	et	PROPN
ejpam-7135	10	11	al	al	PROPN
ejpam-7135	10	12	.	.	PUNCT
ejpam-7135	10	13	/	/	SYM
ejpam-7135	10	14	eur	eur	PROPN
ejpam-7135	10	15	.	.	PUNCT
ejpam-7135	11	1	j.	j.	PROPN
ejpam-7135	11	2	pure	pure	PROPN
ejpam-7135	11	3	appl	appl	PROPN
ejpam-7135	11	4	.	.	PROPN
ejpam-7135	11	5	math	math	PROPN
ejpam-7135	11	6	,	,	PUNCT
ejpam-7135	11	7	18	18	NUM
ejpam-7135	11	8	(	(	PUNCT
ejpam-7135	11	9	4	4	NUM
ejpam-7135	11	10	)	)	PUNCT
ejpam-7135	11	11	(	(	PUNCT
ejpam-7135	11	12	2025	2025	NUM
ejpam-7135	11	13	)	)	PUNCT
ejpam-7135	11	14	,	,	PUNCT
ejpam-7135	11	15	7135	7135	NUM
ejpam-7135	11	16	2	2	NUM
ejpam-7135	11	17	of	of	ADP
ejpam-7135	11	18	14	14	NUM
ejpam-7135	11	19	1	1	NUM
ejpam-7135	11	20	.	.	PUNCT
ejpam-7135	12	1	introduction	introduction	NOUN
ejpam-7135	12	2	the	the	DET
ejpam-7135	12	3	study	study	NOUN
ejpam-7135	12	4	of	of	ADP
ejpam-7135	12	5	distributive	distributive	ADJ
ejpam-7135	12	6	lattices	lattice	NOUN
ejpam-7135	12	7	has	have	VERB
ejpam-7135	12	8	its	its	PRON
ejpam-7135	12	9	origin	origin	NOUN
ejpam-7135	12	10	in	in	ADP
ejpam-7135	12	11	the	the	DET
ejpam-7135	12	12	classical	classical	ADJ
ejpam-7135	12	13	work	work	NOUN
ejpam-7135	12	14	of	of	ADP
ejpam-7135	12	15	birkhoff	birkhoff	NOUN
ejpam-7135	13	1	[	[	X
ejpam-7135	13	2	1	1	NUM
ejpam-7135	13	3	]	]	PUNCT
ejpam-7135	13	4	,	,	PUNCT
ejpam-7135	13	5	who	who	PRON
ejpam-7135	13	6	established	establish	VERB
ejpam-7135	13	7	fundamental	fundamental	ADJ
ejpam-7135	13	8	results	result	NOUN
ejpam-7135	13	9	that	that	PRON
ejpam-7135	13	10	shaped	shape	VERB
ejpam-7135	13	11	modern	modern	ADJ
ejpam-7135	13	12	lattice	lattice	NOUN
ejpam-7135	13	13	theory	theory	NOUN
ejpam-7135	13	14	.	.	PUNCT
ejpam-7135	14	1	building	build	VERB
ejpam-7135	14	2	on	on	ADP
ejpam-7135	14	3	these	these	DET
ejpam-7135	14	4	foundations	foundation	NOUN
ejpam-7135	14	5	,	,	PUNCT
ejpam-7135	14	6	stone	stone	NOUN
ejpam-7135	14	7	lattices	lattice	NOUN
ejpam-7135	14	8	were	be	AUX
ejpam-7135	14	9	introduced	introduce	VERB
ejpam-7135	14	10	and	and	CCONJ
ejpam-7135	14	11	studied	study	VERB
ejpam-7135	14	12	in	in	ADP
ejpam-7135	14	13	depth	depth	NOUN
ejpam-7135	14	14	by	by	ADP
ejpam-7135	14	15	balbes	balbe	NOUN
ejpam-7135	14	16	and	and	CCONJ
ejpam-7135	14	17	horn	horn	NOUN
ejpam-7135	14	18	[	[	X
ejpam-7135	14	19	2	2	NUM
ejpam-7135	14	20	]	]	PUNCT
ejpam-7135	14	21	,	,	PUNCT
ejpam-7135	14	22	following	follow	VERB
ejpam-7135	14	23	earlier	early	ADJ
ejpam-7135	14	24	contributions	contribution	NOUN
ejpam-7135	14	25	by	by	ADP
ejpam-7135	14	26	bruns	brun	NOUN
ejpam-7135	14	27	[	[	X
ejpam-7135	14	28	3	3	NUM
ejpam-7135	14	29	]	]	PUNCT
ejpam-7135	14	30	,	,	PUNCT
ejpam-7135	14	31	chen	chen	PROPN
ejpam-7135	14	32	and	and	CCONJ
ejpam-7135	14	33	grätzer	grätzer	NOUN
ejpam-7135	15	1	[	[	X
ejpam-7135	15	2	4	4	NUM
ejpam-7135	15	3	]	]	PUNCT
ejpam-7135	15	4	,	,	PUNCT
ejpam-7135	15	5	varlet	varlet	VERB
ejpam-7135	15	6	[	[	X
ejpam-7135	15	7	5	5	NUM
ejpam-7135	15	8	]	]	PUNCT
ejpam-7135	15	9	,	,	PUNCT
ejpam-7135	15	10	and	and	CCONJ
ejpam-7135	15	11	speed	speed	NOUN
ejpam-7135	15	12	[	[	X
ejpam-7135	15	13	6	6	NUM
ejpam-7135	15	14	]	]	PUNCT
ejpam-7135	15	15	.	.	PUNCT
ejpam-7135	16	1	these	these	DET
ejpam-7135	16	2	works	work	NOUN
ejpam-7135	16	3	explored	explore	VERB
ejpam-7135	16	4	ideal	ideal	ADJ
ejpam-7135	16	5	representations	representation	NOUN
ejpam-7135	16	6	,	,	PUNCT
ejpam-7135	16	7	prime	prime	ADJ
ejpam-7135	16	8	ideals	ideal	NOUN
ejpam-7135	16	9	,	,	PUNCT
ejpam-7135	16	10	and	and	CCONJ
ejpam-7135	16	11	structural	structural	ADJ
ejpam-7135	16	12	characterizations	characterization	NOUN
ejpam-7135	16	13	of	of	ADP
ejpam-7135	16	14	distributive	distributive	ADJ
ejpam-7135	16	15	lattices	lattice	NOUN
ejpam-7135	16	16	with	with	ADP
ejpam-7135	16	17	pseudo	pseudo	NOUN
ejpam-7135	16	18	-	-	NOUN
ejpam-7135	16	19	complements	complement	NOUN
ejpam-7135	16	20	.	.	PUNCT
ejpam-7135	17	1	grätzer	grätzer	NOUN
ejpam-7135	18	1	[	[	X
ejpam-7135	18	2	7	7	NUM
ejpam-7135	18	3	]	]	X
ejpam-7135	18	4	further	further	ADJ
ejpam-7135	18	5	generalized	generalize	VERB
ejpam-7135	18	6	stone	stone	NOUN
ejpam-7135	18	7	’s	’s	PART
ejpam-7135	18	8	representation	representation	NOUN
ejpam-7135	18	9	theorem	theorem	NOUN
ejpam-7135	18	10	for	for	ADP
ejpam-7135	18	11	boolean	boolean	ADJ
ejpam-7135	18	12	algebras	algebra	NOUN
ejpam-7135	18	13	,	,	PUNCT
ejpam-7135	18	14	while	while	SCONJ
ejpam-7135	18	15	swamy	swamy	NOUN
ejpam-7135	18	16	and	and	CCONJ
ejpam-7135	18	17	manikyamba	manikyamba	NOUN
ejpam-7135	19	1	[	[	X
ejpam-7135	19	2	8	8	NUM
ejpam-7135	19	3	]	]	PUNCT
ejpam-7135	19	4	provided	provide	VERB
ejpam-7135	19	5	prime	prime	ADJ
ejpam-7135	19	6	ideal	ideal	ADJ
ejpam-7135	19	7	characterizations	characterization	NOUN
ejpam-7135	19	8	of	of	ADP
ejpam-7135	19	9	stone	stone	NOUN
ejpam-7135	19	10	lattices	lattice	NOUN
ejpam-7135	19	11	.	.	PUNCT
ejpam-7135	20	1	frink	frink	PROPN
ejpam-7135	21	1	[	[	X
ejpam-7135	21	2	9	9	NUM
ejpam-7135	21	3	]	]	PUNCT
ejpam-7135	21	4	introduced	introduce	VERB
ejpam-7135	21	5	pseudo	pseudo	NOUN
ejpam-7135	21	6	-	-	NOUN
ejpam-7135	21	7	complements	complement	NOUN
ejpam-7135	21	8	in	in	ADP
ejpam-7135	21	9	semi	semi	NOUN
ejpam-7135	21	10	-	-	NOUN
ejpam-7135	21	11	lattices	lattice	NOUN
ejpam-7135	21	12	,	,	PUNCT
ejpam-7135	21	13	thereby	thereby	ADV
ejpam-7135	21	14	extending	extend	VERB
ejpam-7135	21	15	the	the	DET
ejpam-7135	21	16	algebraic	algebraic	ADJ
ejpam-7135	21	17	toolkit	toolkit	NOUN
ejpam-7135	21	18	available	available	ADJ
ejpam-7135	21	19	for	for	ADP
ejpam-7135	21	20	distributive	distributive	ADJ
ejpam-7135	21	21	lattice	lattice	NOUN
ejpam-7135	21	22	theory	theory	NOUN
ejpam-7135	21	23	.	.	PUNCT
ejpam-7135	22	1	connections	connection	NOUN
ejpam-7135	22	2	with	with	ADP
ejpam-7135	22	3	ring	ring	NOUN
ejpam-7135	22	4	theory	theory	NOUN
ejpam-7135	22	5	were	be	AUX
ejpam-7135	22	6	established	establish	VERB
ejpam-7135	22	7	through	through	ADP
ejpam-7135	22	8	the	the	DET
ejpam-7135	22	9	notion	notion	NOUN
ejpam-7135	22	10	of	of	ADP
ejpam-7135	22	11	regular	regular	ADJ
ejpam-7135	22	12	rings	ring	NOUN
ejpam-7135	22	13	introduced	introduce	VERB
ejpam-7135	22	14	by	by	ADP
ejpam-7135	22	15	von	von	PROPN
ejpam-7135	22	16	neumann	neumann	PROPN
ejpam-7135	23	1	[	[	X
ejpam-7135	23	2	10	10	NUM
ejpam-7135	23	3	]	]	PUNCT
ejpam-7135	23	4	,	,	PUNCT
ejpam-7135	23	5	which	which	PRON
ejpam-7135	23	6	furnished	furnish	VERB
ejpam-7135	23	7	natural	natural	ADJ
ejpam-7135	23	8	examples	example	NOUN
ejpam-7135	23	9	of	of	ADP
ejpam-7135	23	10	algebraic	algebraic	ADJ
ejpam-7135	23	11	systems	system	NOUN
ejpam-7135	23	12	whose	whose	DET
ejpam-7135	23	13	lattice	lattice	NOUN
ejpam-7135	23	14	of	of	ADP
ejpam-7135	23	15	ideals	ideal	NOUN
ejpam-7135	23	16	exhibits	exhibit	VERB
ejpam-7135	23	17	stone	stone	NOUN
ejpam-7135	23	18	-	-	PUNCT
ejpam-7135	23	19	like	like	ADJ
ejpam-7135	23	20	properties	property	NOUN
ejpam-7135	23	21	.	.	PUNCT
ejpam-7135	24	1	at	at	ADP
ejpam-7135	24	2	the	the	DET
ejpam-7135	24	3	same	same	ADJ
ejpam-7135	24	4	time	time	NOUN
ejpam-7135	24	5	,	,	PUNCT
ejpam-7135	24	6	burris	burris	PROPN
ejpam-7135	24	7	and	and	CCONJ
ejpam-7135	24	8	sankappanavar	sankappanavar	NOUN
ejpam-7135	25	1	[	[	X
ejpam-7135	25	2	11	11	NUM
ejpam-7135	25	3	]	]	PUNCT
ejpam-7135	25	4	placed	place	VERB
ejpam-7135	25	5	these	these	DET
ejpam-7135	25	6	developments	development	NOUN
ejpam-7135	25	7	within	within	ADP
ejpam-7135	25	8	the	the	DET
ejpam-7135	25	9	broader	broad	ADJ
ejpam-7135	25	10	scope	scope	NOUN
ejpam-7135	25	11	of	of	ADP
ejpam-7135	25	12	universal	universal	ADJ
ejpam-7135	25	13	algebra	algebra	NOUN
ejpam-7135	25	14	,	,	PUNCT
ejpam-7135	25	15	thereby	thereby	ADV
ejpam-7135	25	16	highlighting	highlight	VERB
ejpam-7135	25	17	the	the	DET
ejpam-7135	25	18	importance	importance	NOUN
ejpam-7135	25	19	of	of	ADP
ejpam-7135	25	20	distributive	distributive	ADJ
ejpam-7135	25	21	lattices	lattice	NOUN
ejpam-7135	25	22	and	and	CCONJ
ejpam-7135	25	23	their	their	PRON
ejpam-7135	25	24	extensions	extension	NOUN
ejpam-7135	25	25	in	in	ADP
ejpam-7135	25	26	a	a	DET
ejpam-7135	25	27	general	general	ADJ
ejpam-7135	25	28	algebraic	algebraic	ADJ
ejpam-7135	25	29	framework	framework	NOUN
ejpam-7135	25	30	.	.	PUNCT
ejpam-7135	26	1	in	in	ADP
ejpam-7135	26	2	more	more	ADV
ejpam-7135	26	3	recent	recent	ADJ
ejpam-7135	26	4	years	year	NOUN
ejpam-7135	26	5	,	,	PUNCT
ejpam-7135	26	6	significant	significant	ADJ
ejpam-7135	26	7	progress	progress	NOUN
ejpam-7135	26	8	has	have	AUX
ejpam-7135	26	9	been	be	AUX
ejpam-7135	26	10	made	make	VERB
ejpam-7135	26	11	in	in	ADP
ejpam-7135	26	12	extending	extend	VERB
ejpam-7135	26	13	classical	classical	ADJ
ejpam-7135	26	14	lattice	lattice	NOUN
ejpam-7135	26	15	theory	theory	NOUN
ejpam-7135	26	16	to	to	ADP
ejpam-7135	26	17	new	new	ADJ
ejpam-7135	26	18	structures	structure	NOUN
ejpam-7135	26	19	.	.	PUNCT
ejpam-7135	27	1	bandaru	bandaru	NOUN
ejpam-7135	27	2	and	and	CCONJ
ejpam-7135	27	3	ajjarapu	ajjarapu	PROPN
ejpam-7135	28	1	[	[	X
ejpam-7135	28	2	12	12	NUM
ejpam-7135	28	3	]	]	PUNCT
ejpam-7135	28	4	introduced	introduce	VERB
ejpam-7135	28	5	the	the	DET
ejpam-7135	28	6	concept	concept	NOUN
ejpam-7135	28	7	of	of	ADP
ejpam-7135	28	8	paradistributive	paradistributive	ADJ
ejpam-7135	28	9	latticoid	latticoid	NOUN
ejpam-7135	28	10	as	as	ADP
ejpam-7135	28	11	a	a	DET
ejpam-7135	28	12	generalization	generalization	NOUN
ejpam-7135	28	13	of	of	ADP
ejpam-7135	28	14	distributive	distributive	ADJ
ejpam-7135	28	15	lattice	lattice	NOUN
ejpam-7135	28	16	.	.	PUNCT
ejpam-7135	29	1	also	also	ADV
ejpam-7135	29	2	,	,	PUNCT
ejpam-7135	29	3	bandaru	bandaru	PROPN
ejpam-7135	29	4	et	et	PROPN
ejpam-7135	29	5	al	al	PROPN
ejpam-7135	29	6	.	.	PUNCT
ejpam-7135	30	1	[	[	X
ejpam-7135	30	2	13	13	NUM
ejpam-7135	30	3	]	]	PUNCT
ejpam-7135	30	4	studied	study	VERB
ejpam-7135	30	5	their	their	PRON
ejpam-7135	30	6	normal	normal	ADJ
ejpam-7135	30	7	forms	form	NOUN
ejpam-7135	30	8	.	.	PUNCT
ejpam-7135	31	1	ajjarapu	ajjarapu	PROPN
ejpam-7135	31	2	et	et	PROPN
ejpam-7135	31	3	al	al	PROPN
ejpam-7135	31	4	.	.	PUNCT
ejpam-7135	32	1	[	[	X
ejpam-7135	32	2	14	14	NUM
ejpam-7135	32	3	]	]	PUNCT
ejpam-7135	32	4	subsequently	subsequently	ADV
ejpam-7135	32	5	developed	develop	VERB
ejpam-7135	32	6	the	the	DET
ejpam-7135	32	7	notion	notion	NOUN
ejpam-7135	32	8	of	of	ADP
ejpam-7135	32	9	parapseudo	parapseudo	NOUN
ejpam-7135	32	10	-	-	NOUN
ejpam-7135	32	11	complementation	complementation	NOUN
ejpam-7135	32	12	on	on	ADP
ejpam-7135	32	13	paradistributive	paradistributive	ADJ
ejpam-7135	32	14	latticoids	latticoid	NOUN
ejpam-7135	32	15	,	,	PUNCT
ejpam-7135	32	16	thereby	thereby	ADV
ejpam-7135	32	17	generalizing	generalize	VERB
ejpam-7135	32	18	the	the	DET
ejpam-7135	32	19	classical	classical	ADJ
ejpam-7135	32	20	ideas	idea	NOUN
ejpam-7135	32	21	of	of	ADP
ejpam-7135	32	22	pseudo	pseudo	NOUN
ejpam-7135	32	23	-	-	NOUN
ejpam-7135	32	24	complementation	complementation	NOUN
ejpam-7135	32	25	to	to	ADP
ejpam-7135	32	26	this	this	DET
ejpam-7135	32	27	broader	broad	ADJ
ejpam-7135	32	28	framework	framework	NOUN
ejpam-7135	32	29	.	.	PUNCT
ejpam-7135	33	1	also	also	ADV
ejpam-7135	33	2	,	,	PUNCT
ejpam-7135	33	3	ajjarapu	ajjarapu	PROPN
ejpam-7135	33	4	et	et	PROPN
ejpam-7135	33	5	al	al	PROPN
ejpam-7135	33	6	.	.	PUNCT
ejpam-7135	34	1	[	[	X
ejpam-7135	34	2	15	15	NUM
ejpam-7135	34	3	]	]	PUNCT
ejpam-7135	34	4	studied	study	VERB
ejpam-7135	34	5	topological	topological	ADJ
ejpam-7135	34	6	properties	property	NOUN
ejpam-7135	34	7	of	of	ADP
ejpam-7135	34	8	prime	prime	ADJ
ejpam-7135	34	9	filters	filter	NOUN
ejpam-7135	34	10	and	and	CCONJ
ejpam-7135	34	11	minimal	minimal	ADJ
ejpam-7135	34	12	prime	prime	ADJ
ejpam-7135	34	13	filters	filter	NOUN
ejpam-7135	34	14	on	on	ADP
ejpam-7135	34	15	a	a	DET
ejpam-7135	34	16	paradistributive	paradistributive	ADJ
ejpam-7135	34	17	latticoid	latticoid	NOUN
ejpam-7135	34	18	.	.	PUNCT
ejpam-7135	35	1	these	these	DET
ejpam-7135	35	2	advances	advance	NOUN
ejpam-7135	35	3	represent	represent	VERB
ejpam-7135	35	4	a	a	DET
ejpam-7135	35	5	natural	natural	ADJ
ejpam-7135	35	6	evolution	evolution	NOUN
ejpam-7135	35	7	of	of	ADP
ejpam-7135	35	8	the	the	DET
ejpam-7135	35	9	classical	classical	ADJ
ejpam-7135	35	10	results	result	NOUN
ejpam-7135	35	11	on	on	ADP
ejpam-7135	35	12	stone	stone	NOUN
ejpam-7135	35	13	lattices	lattice	NOUN
ejpam-7135	35	14	[	[	X
ejpam-7135	35	15	2–9	2–9	NUM
ejpam-7135	35	16	]	]	X
ejpam-7135	35	17	,	,	PUNCT
ejpam-7135	35	18	adapted	adapt	VERB
ejpam-7135	35	19	to	to	ADP
ejpam-7135	35	20	the	the	DET
ejpam-7135	35	21	setting	setting	NOUN
ejpam-7135	35	22	of	of	ADP
ejpam-7135	35	23	paradistributive	paradistributive	ADJ
ejpam-7135	35	24	latticoids	latticoid	NOUN
ejpam-7135	35	25	.	.	PUNCT
ejpam-7135	36	1	the	the	DET
ejpam-7135	36	2	present	present	ADJ
ejpam-7135	36	3	paper	paper	NOUN
ejpam-7135	36	4	continues	continue	VERB
ejpam-7135	36	5	this	this	DET
ejpam-7135	36	6	line	line	NOUN
ejpam-7135	36	7	of	of	ADP
ejpam-7135	36	8	research	research	NOUN
ejpam-7135	36	9	by	by	ADP
ejpam-7135	36	10	introducing	introduce	VERB
ejpam-7135	36	11	and	and	CCONJ
ejpam-7135	36	12	studying	study	VERB
ejpam-7135	36	13	the	the	DET
ejpam-7135	36	14	class	class	NOUN
ejpam-7135	36	15	of	of	ADP
ejpam-7135	36	16	stone	stone	NOUN
ejpam-7135	36	17	pdls	pdl	NOUN
ejpam-7135	36	18	(	(	PUNCT
ejpam-7135	36	19	stone	stone	NOUN
ejpam-7135	36	20	paradistributive	paradistributive	ADJ
ejpam-7135	36	21	latticoids	latticoids	PROPN
ejpam-7135	36	22	)	)	PUNCT
ejpam-7135	36	23	.	.	PUNCT
ejpam-7135	37	1	we	we	PRON
ejpam-7135	37	2	aim	aim	VERB
ejpam-7135	37	3	to	to	PART
ejpam-7135	37	4	provide	provide	VERB
ejpam-7135	37	5	algebraic	algebraic	ADJ
ejpam-7135	37	6	,	,	PUNCT
ejpam-7135	37	7	topological	topological	ADJ
ejpam-7135	37	8	,	,	PUNCT
ejpam-7135	37	9	and	and	CCONJ
ejpam-7135	37	10	prime	prime	ADJ
ejpam-7135	37	11	filter	filter	NOUN
ejpam-7135	37	12	characterizations	characterization	NOUN
ejpam-7135	37	13	of	of	ADP
ejpam-7135	37	14	these	these	DET
ejpam-7135	37	15	structures	structure	NOUN
ejpam-7135	37	16	.	.	PUNCT
ejpam-7135	38	1	our	our	PRON
ejpam-7135	38	2	results	result	NOUN
ejpam-7135	38	3	unify	unify	VERB
ejpam-7135	38	4	and	and	CCONJ
ejpam-7135	38	5	extend	extend	VERB
ejpam-7135	38	6	earlier	early	ADJ
ejpam-7135	38	7	work	work	NOUN
ejpam-7135	38	8	on	on	ADP
ejpam-7135	38	9	stone	stone	NOUN
ejpam-7135	38	10	lattices	lattice	NOUN
ejpam-7135	38	11	and	and	CCONJ
ejpam-7135	38	12	pseudo	pseudo	NOUN
ejpam-7135	38	13	-	-	NOUN
ejpam-7135	38	14	complements	complement	NOUN
ejpam-7135	38	15	into	into	ADP
ejpam-7135	38	16	the	the	DET
ejpam-7135	38	17	framework	framework	NOUN
ejpam-7135	38	18	of	of	ADP
ejpam-7135	38	19	paradistributive	paradistributive	ADJ
ejpam-7135	38	20	latticoids	latticoids	PROPN
ejpam-7135	38	21	[	[	X
ejpam-7135	38	22	12–14	12–14	NUM
ejpam-7135	38	23	]	]	PUNCT
ejpam-7135	38	24	,	,	PUNCT
ejpam-7135	38	25	while	while	SCONJ
ejpam-7135	38	26	also	also	ADV
ejpam-7135	38	27	connecting	connect	VERB
ejpam-7135	38	28	with	with	ADP
ejpam-7135	38	29	lattice	lattice	NOUN
ejpam-7135	38	30	theory	theory	NOUN
ejpam-7135	38	31	[	[	X
ejpam-7135	38	32	1	1	NUM
ejpam-7135	38	33	]	]	PUNCT
ejpam-7135	38	34	,	,	PUNCT
ejpam-7135	38	35	universal	universal	ADJ
ejpam-7135	38	36	algebra	algebra	NOUN
ejpam-7135	38	37	[	[	X
ejpam-7135	38	38	11	11	NUM
ejpam-7135	38	39	]	]	PUNCT
ejpam-7135	38	40	,	,	PUNCT
ejpam-7135	38	41	and	and	CCONJ
ejpam-7135	38	42	regular	regular	ADJ
ejpam-7135	38	43	rings	ring	NOUN
ejpam-7135	38	44	[	[	X
ejpam-7135	38	45	10	10	NUM
ejpam-7135	38	46	]	]	PUNCT
ejpam-7135	38	47	.	.	PUNCT
ejpam-7135	39	1	2	2	X
ejpam-7135	39	2	.	.	X
ejpam-7135	39	3	preliminaries	preliminary	NOUN
ejpam-7135	39	4	first	first	ADV
ejpam-7135	39	5	,	,	PUNCT
ejpam-7135	39	6	we	we	PRON
ejpam-7135	39	7	recall	recall	VERB
ejpam-7135	39	8	the	the	DET
ejpam-7135	39	9	necessary	necessary	ADJ
ejpam-7135	39	10	definitions	definition	NOUN
ejpam-7135	39	11	and	and	CCONJ
ejpam-7135	39	12	results	result	NOUN
ejpam-7135	39	13	from	from	ADP
ejpam-7135	39	14	[	[	X
ejpam-7135	39	15	12	12	NUM
ejpam-7135	39	16	]	]	PUNCT
ejpam-7135	39	17	.	.	PUNCT
ejpam-7135	40	1	definition	definition	NOUN
ejpam-7135	40	2	1	1	NUM
ejpam-7135	40	3	.	.	PUNCT
ejpam-7135	41	1	[	[	X
ejpam-7135	41	2	12	12	NUM
ejpam-7135	41	3	]	]	PUNCT
ejpam-7135	41	4	an	an	DET
ejpam-7135	41	5	algebra	algebra	NOUN
ejpam-7135	41	6	(	(	PUNCT
ejpam-7135	41	7	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	41	8	,	,	PUNCT
ejpam-7135	41	9	1	1	NUM
ejpam-7135	41	10	)	)	PUNCT
ejpam-7135	41	11	of	of	ADP
ejpam-7135	41	12	type	type	NOUN
ejpam-7135	41	13	(	(	PUNCT
ejpam-7135	41	14	2	2	NUM
ejpam-7135	41	15	,	,	PUNCT
ejpam-7135	41	16	2	2	NUM
ejpam-7135	41	17	,	,	PUNCT
ejpam-7135	41	18	0	0	NUM
ejpam-7135	41	19	)	)	PUNCT
ejpam-7135	41	20	is	be	AUX
ejpam-7135	41	21	called	call	VERB
ejpam-7135	41	22	a	a	DET
ejpam-7135	41	23	paradistributive	paradistributive	ADJ
ejpam-7135	41	24	latticoid	latticoid	NOUN
ejpam-7135	41	25	,	,	PUNCT
ejpam-7135	41	26	abbreviated	abbreviate	VERB
ejpam-7135	41	27	as	as	ADP
ejpam-7135	41	28	pdl	pdl	NOUN
ejpam-7135	41	29	,	,	PUNCT
ejpam-7135	41	30	if	if	SCONJ
ejpam-7135	41	31	it	it	PRON
ejpam-7135	41	32	assures	assure	VERB
ejpam-7135	41	33	the	the	DET
ejpam-7135	41	34	subsequent	subsequent	ADJ
ejpam-7135	41	35	axioms	axiom	NOUN
ejpam-7135	41	36	:	:	PUNCT
ejpam-7135	41	37	(	(	PUNCT
ejpam-7135	41	38	ld∨	ld∨	NOUN
ejpam-7135	41	39	)	)	PUNCT
ejpam-7135	41	40	x	x	SYM
ejpam-7135	41	41	∨	∨	NOUN
ejpam-7135	41	42	(	(	PUNCT
ejpam-7135	41	43	y	y	PROPN
ejpam-7135	41	44	∧	∧	PROPN
ejpam-7135	41	45	z	z	PROPN
ejpam-7135	41	46	)	)	PUNCT
ejpam-7135	41	47	=	=	SYM
ejpam-7135	41	48	(	(	PUNCT
ejpam-7135	41	49	x	x	PROPN
ejpam-7135	41	50	∨	∨	NUM
ejpam-7135	41	51	y	y	NOUN
ejpam-7135	41	52	)	)	PUNCT
ejpam-7135	41	53	∧	∧	NOUN
ejpam-7135	41	54	(	(	PUNCT
ejpam-7135	41	55	x	x	PROPN
ejpam-7135	41	56	∨	∨	PROPN
ejpam-7135	41	57	z	z	PROPN
ejpam-7135	41	58	)	)	PUNCT
ejpam-7135	41	59	,	,	PUNCT
ejpam-7135	41	60	(	(	PUNCT
ejpam-7135	41	61	rd∨	rd∨	X
ejpam-7135	41	62	)	)	PUNCT
ejpam-7135	41	63	(	(	PUNCT
ejpam-7135	41	64	x	x	PUNCT
ejpam-7135	41	65	∧	∧	PROPN
ejpam-7135	41	66	y	y	PROPN
ejpam-7135	41	67	)	)	PUNCT
ejpam-7135	41	68	∨	∨	NUM
ejpam-7135	41	69	z	z	NOUN
ejpam-7135	41	70	=	=	SYM
ejpam-7135	41	71	(	(	PUNCT
ejpam-7135	41	72	x	x	PROPN
ejpam-7135	41	73	∨	∨	PROPN
ejpam-7135	41	74	z	z	NOUN
ejpam-7135	41	75	)	)	PUNCT
ejpam-7135	41	76	∧	∧	PROPN
ejpam-7135	41	77	(	(	PUNCT
ejpam-7135	41	78	y	y	PROPN
ejpam-7135	41	79	∨	∨	PROPN
ejpam-7135	41	80	z	z	PROPN
ejpam-7135	41	81	)	)	PUNCT
ejpam-7135	41	82	,	,	PUNCT
ejpam-7135	41	83	(	(	PUNCT
ejpam-7135	41	84	l1	l1	PROPN
ejpam-7135	41	85	)	)	PUNCT
ejpam-7135	41	86	(	(	PUNCT
ejpam-7135	41	87	x	x	PROPN
ejpam-7135	41	88	∨	∨	NUM
ejpam-7135	41	89	y	y	NOUN
ejpam-7135	41	90	)	)	PUNCT
ejpam-7135	41	91	∧	∧	NOUN
ejpam-7135	41	92	y	y	PROPN
ejpam-7135	41	93	=	=	SYM
ejpam-7135	41	94	y	y	PROPN
ejpam-7135	41	95	,	,	PUNCT
ejpam-7135	41	96	(	(	PUNCT
ejpam-7135	41	97	l2	l2	NOUN
ejpam-7135	41	98	)	)	PUNCT
ejpam-7135	41	99	(	(	PUNCT
ejpam-7135	41	100	x	x	PROPN
ejpam-7135	41	101	∨	∨	NUM
ejpam-7135	41	102	y	y	NOUN
ejpam-7135	41	103	)	)	PUNCT
ejpam-7135	41	104	∧	∧	NOUN
ejpam-7135	41	105	x	x	X
ejpam-7135	41	106	=	=	SYM
ejpam-7135	41	107	x	x	PROPN
ejpam-7135	41	108	,	,	PUNCT
ejpam-7135	41	109	r.	r.	PROPN
ejpam-7135	41	110	bandaru	bandaru	PROPN
ejpam-7135	41	111	et	et	PROPN
ejpam-7135	41	112	al	al	PROPN
ejpam-7135	41	113	.	.	PUNCT
ejpam-7135	41	114	/	/	SYM
ejpam-7135	41	115	eur	eur	PROPN
ejpam-7135	41	116	.	.	PUNCT
ejpam-7135	42	1	j.	j.	PROPN
ejpam-7135	42	2	pure	pure	PROPN
ejpam-7135	42	3	appl	appl	PROPN
ejpam-7135	42	4	.	.	PROPN
ejpam-7135	42	5	math	math	PROPN
ejpam-7135	42	6	,	,	PUNCT
ejpam-7135	42	7	18	18	NUM
ejpam-7135	42	8	(	(	PUNCT
ejpam-7135	42	9	4	4	NUM
ejpam-7135	42	10	)	)	PUNCT
ejpam-7135	42	11	(	(	PUNCT
ejpam-7135	42	12	2025	2025	NUM
ejpam-7135	42	13	)	)	PUNCT
ejpam-7135	42	14	,	,	PUNCT
ejpam-7135	42	15	7135	7135	NUM
ejpam-7135	42	16	3	3	NUM
ejpam-7135	42	17	of	of	ADP
ejpam-7135	42	18	14	14	NUM
ejpam-7135	42	19	(	(	PUNCT
ejpam-7135	42	20	l3	l3	PROPN
ejpam-7135	42	21	)	)	PUNCT
ejpam-7135	42	22	x	x	SYM
ejpam-7135	42	23	∨	∨	NOUN
ejpam-7135	42	24	(	(	PUNCT
ejpam-7135	42	25	x	x	PROPN
ejpam-7135	42	26	∧	∧	PROPN
ejpam-7135	42	27	y	y	NOUN
ejpam-7135	42	28	)	)	PUNCT
ejpam-7135	42	29	=	=	SYM
ejpam-7135	43	1	x	x	X
ejpam-7135	43	2	,	,	PUNCT
ejpam-7135	43	3	(	(	PUNCT
ejpam-7135	43	4	i1	i1	PROPN
ejpam-7135	43	5	)	)	PUNCT
ejpam-7135	43	6	x	x	PUNCT
ejpam-7135	43	7	∨	∨	NUM
ejpam-7135	43	8	1	1	NUM
ejpam-7135	43	9	=	=	SYM
ejpam-7135	43	10	1	1	NUM
ejpam-7135	43	11	,	,	PUNCT
ejpam-7135	43	12	for	for	ADP
ejpam-7135	43	13	any	any	DET
ejpam-7135	43	14	x	x	NOUN
ejpam-7135	43	15	,	,	PUNCT
ejpam-7135	43	16	y	y	PROPN
ejpam-7135	43	17	,	,	PUNCT
ejpam-7135	43	18	z	z	PROPN
ejpam-7135	43	19	∈	∈	PROPN
ejpam-7135	43	20	l.	l.	NOUN
ejpam-7135	43	21	for	for	ADP
ejpam-7135	43	22	any	any	DET
ejpam-7135	43	23	x	x	NOUN
ejpam-7135	43	24	,	,	PUNCT
ejpam-7135	43	25	y	y	PROPN
ejpam-7135	43	26	∈	∈	PROPN
ejpam-7135	43	27	l	l	NOUN
ejpam-7135	43	28	,	,	PUNCT
ejpam-7135	43	29	we	we	PRON
ejpam-7135	43	30	say	say	VERB
ejpam-7135	43	31	that	that	SCONJ
ejpam-7135	43	32	x	x	PRON
ejpam-7135	43	33	is	be	AUX
ejpam-7135	43	34	less	less	ADJ
ejpam-7135	43	35	than	than	ADP
ejpam-7135	43	36	or	or	CCONJ
ejpam-7135	43	37	equal	equal	ADJ
ejpam-7135	43	38	to	to	ADP
ejpam-7135	43	39	y	y	PROPN
ejpam-7135	43	40	and	and	CCONJ
ejpam-7135	43	41	write	write	VERB
ejpam-7135	43	42	x	x	PUNCT
ejpam-7135	43	43	≤	≤	NUM
ejpam-7135	43	44	y	y	NOUN
ejpam-7135	43	45	if	if	SCONJ
ejpam-7135	43	46	x	x	X
ejpam-7135	43	47	∧	∧	NOUN
ejpam-7135	43	48	y	y	NOUN
ejpam-7135	43	49	=	=	PUNCT
ejpam-7135	43	50	x	x	X
ejpam-7135	43	51	or	or	CCONJ
ejpam-7135	43	52	equivalently	equivalently	ADV
ejpam-7135	43	53	x∨	x∨	PROPN
ejpam-7135	43	54	y	y	PROPN
ejpam-7135	43	55	=	=	SYM
ejpam-7135	43	56	y	y	PROPN
ejpam-7135	43	57	and	and	CCONJ
ejpam-7135	43	58	it	it	PRON
ejpam-7135	43	59	can	can	AUX
ejpam-7135	43	60	be	be	AUX
ejpam-7135	43	61	easily	easily	ADV
ejpam-7135	43	62	observed	observe	VERB
ejpam-7135	43	63	that	that	SCONJ
ejpam-7135	43	64	≤	≤	NUM
ejpam-7135	43	65	is	be	AUX
ejpam-7135	43	66	a	a	DET
ejpam-7135	43	67	partial	partial	ADJ
ejpam-7135	43	68	order	order	NOUN
ejpam-7135	43	69	on	on	ADP
ejpam-7135	43	70	l.	l.	NOUN
ejpam-7135	43	71	we	we	PRON
ejpam-7135	43	72	can	can	AUX
ejpam-7135	43	73	observe	observe	VERB
ejpam-7135	43	74	that	that	SCONJ
ejpam-7135	43	75	the	the	DET
ejpam-7135	43	76	element	element	NOUN
ejpam-7135	43	77	1	1	NUM
ejpam-7135	43	78	,	,	PUNCT
ejpam-7135	43	79	in	in	ADP
ejpam-7135	43	80	definition	definition	NOUN
ejpam-7135	43	81	1	1	NUM
ejpam-7135	43	82	,	,	PUNCT
ejpam-7135	43	83	is	be	AUX
ejpam-7135	43	84	the	the	DET
ejpam-7135	43	85	greatest	great	ADJ
ejpam-7135	43	86	element	element	NOUN
ejpam-7135	43	87	with	with	ADP
ejpam-7135	43	88	respect	respect	NOUN
ejpam-7135	43	89	to	to	ADP
ejpam-7135	43	90	the	the	DET
ejpam-7135	43	91	partial	partial	ADJ
ejpam-7135	43	92	ordering	ordering	NOUN
ejpam-7135	43	93	≤.	≤.	PROPN
ejpam-7135	43	94	example	example	NOUN
ejpam-7135	44	1	1	1	NUM
ejpam-7135	44	2	.	.	PUNCT
ejpam-7135	45	1	[	[	X
ejpam-7135	45	2	12	12	NUM
ejpam-7135	45	3	]	]	X
ejpam-7135	45	4	let	let	AUX
ejpam-7135	45	5	l	l	NOUN
ejpam-7135	45	6	be	be	AUX
ejpam-7135	45	7	a	a	DET
ejpam-7135	45	8	non	non	ADJ
ejpam-7135	45	9	-	-	ADJ
ejpam-7135	45	10	empty	empty	ADJ
ejpam-7135	45	11	set	set	NOUN
ejpam-7135	45	12	.	.	PUNCT
ejpam-7135	46	1	fix	fix	VERB
ejpam-7135	46	2	some	some	DET
ejpam-7135	46	3	element	element	NOUN
ejpam-7135	46	4	g	g	PROPN
ejpam-7135	46	5	∈	∈	PROPN
ejpam-7135	46	6	l.	l.	NOUN
ejpam-7135	46	7	then	then	ADV
ejpam-7135	46	8	,	,	PUNCT
ejpam-7135	46	9	for	for	ADP
ejpam-7135	46	10	any	any	DET
ejpam-7135	46	11	x	x	NOUN
ejpam-7135	46	12	,	,	PUNCT
ejpam-7135	46	13	y	y	PROPN
ejpam-7135	46	14	∈	∈	PROPN
ejpam-7135	46	15	l	l	NOUN
ejpam-7135	46	16	define	define	VERB
ejpam-7135	46	17	∨	∨	NOUN
ejpam-7135	46	18	and	and	CCONJ
ejpam-7135	46	19	∧	∧	NOUN
ejpam-7135	46	20	on	on	ADP
ejpam-7135	46	21	l	l	NOUN
ejpam-7135	46	22	by	by	ADP
ejpam-7135	46	23	x	x	PROPN
ejpam-7135	46	24	∨	∨	NUM
ejpam-7135	46	25	y	y	NOUN
ejpam-7135	46	26	=	=	PRON
ejpam-7135	46	27	{	{	PUNCT
ejpam-7135	46	28	x	x	PUNCT
ejpam-7135	46	29	y	y	PROPN
ejpam-7135	46	30	̸=	̸=	PROPN
ejpam-7135	46	31	g	g	NOUN
ejpam-7135	46	32	g	g	PROPN
ejpam-7135	46	33	y	y	PROPN
ejpam-7135	46	34	=	=	PROPN
ejpam-7135	46	35	g	g	PROPN
ejpam-7135	46	36	and	and	CCONJ
ejpam-7135	46	37	x	x	PROPN
ejpam-7135	46	38	∧	∧	NOUN
ejpam-7135	46	39	y	y	NOUN
ejpam-7135	46	40	=	=	PRON
ejpam-7135	46	41	{	{	PUNCT
ejpam-7135	46	42	y	y	NOUN
ejpam-7135	46	43	y	y	PROPN
ejpam-7135	46	44	̸=	̸=	PROPN
ejpam-7135	46	45	g	g	PROPN
ejpam-7135	46	46	x	x	SYM
ejpam-7135	46	47	y	y	PROPN
ejpam-7135	46	48	=	=	PUNCT
ejpam-7135	46	49	g	g	PROPN
ejpam-7135	46	50	then	then	ADV
ejpam-7135	46	51	(	(	PUNCT
ejpam-7135	46	52	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	46	53	,	,	PUNCT
ejpam-7135	46	54	g	g	NOUN
ejpam-7135	46	55	)	)	PUNCT
ejpam-7135	46	56	is	be	AUX
ejpam-7135	46	57	a	a	DET
ejpam-7135	46	58	disconnected	disconnected	ADJ
ejpam-7135	46	59	pdl	pdl	NOUN
ejpam-7135	46	60	with	with	ADP
ejpam-7135	46	61	g	g	PROPN
ejpam-7135	46	62	as	as	ADP
ejpam-7135	46	63	its	its	PRON
ejpam-7135	46	64	greatest	great	ADJ
ejpam-7135	46	65	element	element	NOUN
ejpam-7135	46	66	.	.	PUNCT
ejpam-7135	47	1	according	accord	VERB
ejpam-7135	47	2	to	to	ADP
ejpam-7135	47	3	lemma	lemma	PROPN
ejpam-7135	47	4	7	7	NUM
ejpam-7135	47	5	,	,	PUNCT
ejpam-7135	47	6	theorem	theorem	ADJ
ejpam-7135	47	7	1	1	NUM
ejpam-7135	47	8	,	,	PUNCT
ejpam-7135	47	9	lemma	lemma	PROPN
ejpam-7135	47	10	8	8	NUM
ejpam-7135	47	11	,	,	PUNCT
ejpam-7135	47	12	theorem	theorem	VERB
ejpam-7135	47	13	4	4	NUM
ejpam-7135	47	14	,	,	PUNCT
ejpam-7135	47	15	corollary	corollary	ADJ
ejpam-7135	47	16	8	8	NUM
ejpam-7135	47	17	,	,	PUNCT
ejpam-7135	47	18	lemma	lemma	PROPN
ejpam-7135	47	19	9	9	NUM
ejpam-7135	47	20	,	,	PUNCT
ejpam-7135	47	21	and	and	CCONJ
ejpam-7135	47	22	lemma	lemma	PROPN
ejpam-7135	47	23	10	10	NUM
ejpam-7135	47	24	of	of	ADP
ejpam-7135	47	25	[	[	X
ejpam-7135	47	26	12	12	NUM
ejpam-7135	47	27	]	]	PUNCT
ejpam-7135	47	28	,	,	PUNCT
ejpam-7135	47	29	the	the	DET
ejpam-7135	47	30	following	follow	VERB
ejpam-7135	47	31	lemma	lemma	PROPN
ejpam-7135	47	32	holds	hold	VERB
ejpam-7135	47	33	.	.	PUNCT
ejpam-7135	48	1	lemma	lemma	PROPN
ejpam-7135	48	2	1	1	NUM
ejpam-7135	48	3	.	.	PUNCT
ejpam-7135	49	1	[	[	X
ejpam-7135	49	2	12	12	NUM
ejpam-7135	49	3	]	]	X
ejpam-7135	49	4	let	let	NOUN
ejpam-7135	49	5	(	(	PUNCT
ejpam-7135	49	6	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	49	7	,	,	PUNCT
ejpam-7135	49	8	1	1	NUM
ejpam-7135	49	9	)	)	PUNCT
ejpam-7135	49	10	be	be	AUX
ejpam-7135	49	11	a	a	DET
ejpam-7135	49	12	pdl	pdl	NOUN
ejpam-7135	49	13	.	.	PUNCT
ejpam-7135	50	1	then	then	ADV
ejpam-7135	50	2	for	for	ADP
ejpam-7135	50	3	any	any	DET
ejpam-7135	50	4	x	x	NOUN
ejpam-7135	50	5	,	,	PUNCT
ejpam-7135	50	6	y	y	PROPN
ejpam-7135	50	7	,	,	PUNCT
ejpam-7135	50	8	z	z	PROPN
ejpam-7135	50	9	,	,	PUNCT
ejpam-7135	50	10	s	s	PROPN
ejpam-7135	50	11	∈	∈	PROPN
ejpam-7135	50	12	l	l	NOUN
ejpam-7135	50	13	,	,	PUNCT
ejpam-7135	50	14	we	we	PRON
ejpam-7135	50	15	have	have	VERB
ejpam-7135	50	16	the	the	DET
ejpam-7135	50	17	following	following	NOUN
ejpam-7135	50	18	:	:	PUNCT
ejpam-7135	50	19	(	(	PUNCT
ejpam-7135	50	20	1	1	X
ejpam-7135	50	21	)	)	SYM
ejpam-7135	50	22	1	1	NUM
ejpam-7135	50	23	∧	∧	NOUN
ejpam-7135	50	24	x	x	X
ejpam-7135	51	1	=	=	SYM
ejpam-7135	51	2	x	x	NOUN
ejpam-7135	51	3	,	,	PUNCT
ejpam-7135	51	4	(	(	PUNCT
ejpam-7135	51	5	2	2	NUM
ejpam-7135	51	6	)	)	PUNCT
ejpam-7135	51	7	x	x	SYM
ejpam-7135	51	8	∧	∧	NOUN
ejpam-7135	51	9	1	1	NUM
ejpam-7135	51	10	=	=	SYM
ejpam-7135	51	11	x	x	NOUN
ejpam-7135	51	12	,	,	PUNCT
ejpam-7135	51	13	(	(	PUNCT
ejpam-7135	51	14	3	3	NUM
ejpam-7135	51	15	)	)	PUNCT
ejpam-7135	51	16	1	1	NUM
ejpam-7135	51	17	∨	∨	NOUN
ejpam-7135	51	18	x	x	X
ejpam-7135	51	19	=	=	SYM
ejpam-7135	51	20	1	1	NUM
ejpam-7135	51	21	,	,	PUNCT
ejpam-7135	51	22	(	(	PUNCT
ejpam-7135	51	23	4	4	NUM
ejpam-7135	51	24	)	)	PUNCT
ejpam-7135	51	25	(	(	PUNCT
ejpam-7135	51	26	x	x	PROPN
ejpam-7135	51	27	∨	∨	NUM
ejpam-7135	51	28	y	y	NOUN
ejpam-7135	51	29	)	)	PUNCT
ejpam-7135	51	30	∧	∧	NOUN
ejpam-7135	51	31	z	z	NOUN
ejpam-7135	51	32	=	=	SYM
ejpam-7135	51	33	(	(	PUNCT
ejpam-7135	51	34	x	x	PART
ejpam-7135	51	35	∧	∧	PROPN
ejpam-7135	51	36	z	z	PROPN
ejpam-7135	51	37	)	)	PUNCT
ejpam-7135	51	38	∨	∨	PROPN
ejpam-7135	51	39	(	(	PUNCT
ejpam-7135	51	40	y	y	PROPN
ejpam-7135	51	41	∧	∧	PROPN
ejpam-7135	51	42	z	z	PROPN
ejpam-7135	51	43	)	)	PUNCT
ejpam-7135	51	44	,	,	PUNCT
ejpam-7135	51	45	(	(	PUNCT
ejpam-7135	51	46	5	5	X
ejpam-7135	51	47	)	)	PUNCT
ejpam-7135	51	48	x	x	PUNCT
ejpam-7135	51	49	∨	∨	PROPN
ejpam-7135	51	50	(	(	PUNCT
ejpam-7135	51	51	y	y	PROPN
ejpam-7135	51	52	∧	∧	PROPN
ejpam-7135	51	53	z	z	PROPN
ejpam-7135	51	54	)	)	PUNCT
ejpam-7135	51	55	=	=	SYM
ejpam-7135	52	1	x	x	SYM
ejpam-7135	52	2	∨	∨	X
ejpam-7135	52	3	(	(	PUNCT
ejpam-7135	52	4	z	z	PROPN
ejpam-7135	52	5	∧	∧	PROPN
ejpam-7135	52	6	y	y	PROPN
ejpam-7135	52	7	)	)	PUNCT
ejpam-7135	52	8	,	,	PUNCT
ejpam-7135	52	9	(	(	PUNCT
ejpam-7135	52	10	6	6	X
ejpam-7135	52	11	)	)	PUNCT
ejpam-7135	52	12	the	the	DET
ejpam-7135	52	13	operation	operation	NOUN
ejpam-7135	52	14	∨	∨	NOUN
ejpam-7135	52	15	is	be	AUX
ejpam-7135	52	16	associative	associative	ADJ
ejpam-7135	52	17	in	in	ADP
ejpam-7135	52	18	l	l	NOUN
ejpam-7135	52	19	i.e.	i.e.	X
ejpam-7135	52	20	,	,	PUNCT
ejpam-7135	52	21	x	x	X
ejpam-7135	52	22	∨	∨	X
ejpam-7135	52	23	(	(	PUNCT
ejpam-7135	52	24	y	y	PROPN
ejpam-7135	52	25	∨	∨	PROPN
ejpam-7135	52	26	z	z	PROPN
ejpam-7135	52	27	)	)	PUNCT
ejpam-7135	52	28	=	=	SYM
ejpam-7135	52	29	(	(	PUNCT
ejpam-7135	52	30	x	x	PROPN
ejpam-7135	52	31	∨	∨	NUM
ejpam-7135	52	32	y	y	PROPN
ejpam-7135	52	33	)	)	PUNCT
ejpam-7135	52	34	∨	∨	PROPN
ejpam-7135	52	35	z	z	PROPN
ejpam-7135	52	36	,	,	PUNCT
ejpam-7135	52	37	(	(	PUNCT
ejpam-7135	52	38	7	7	X
ejpam-7135	52	39	)	)	PUNCT
ejpam-7135	52	40	the	the	DET
ejpam-7135	52	41	set	set	NOUN
ejpam-7135	52	42	la	la	NOUN
ejpam-7135	52	43	=	=	SYM
ejpam-7135	52	44	{	{	PUNCT
ejpam-7135	52	45	x	x	PUNCT
ejpam-7135	52	46	∈	∈	NOUN
ejpam-7135	52	47	l	l	NOUN
ejpam-7135	53	1	|	|	ADV
ejpam-7135	53	2	a	a	DET
ejpam-7135	53	3	≤	≤	NOUN
ejpam-7135	53	4	x	x	SYM
ejpam-7135	53	5	}	}	PUNCT
ejpam-7135	53	6	=	=	SYM
ejpam-7135	53	7	{	{	PUNCT
ejpam-7135	53	8	a	a	DET
ejpam-7135	53	9	∨	∨	NOUN
ejpam-7135	53	10	x	x	SYM
ejpam-7135	53	11	|	|	NOUN
ejpam-7135	53	12	x	x	SYM
ejpam-7135	53	13	∈	∈	PROPN
ejpam-7135	53	14	l	l	NOUN
ejpam-7135	53	15	}	}	PUNCT
ejpam-7135	53	16	is	be	AUX
ejpam-7135	53	17	a	a	DET
ejpam-7135	53	18	distributive	distributive	ADJ
ejpam-7135	53	19	lattice	lattice	NOUN
ejpam-7135	53	20	under	under	ADP
ejpam-7135	53	21	induced	induced	ADJ
ejpam-7135	53	22	operations	operation	NOUN
ejpam-7135	53	23	∨	∨	NOUN
ejpam-7135	53	24	and	and	CCONJ
ejpam-7135	53	25	∧	∧	NOUN
ejpam-7135	53	26	with	with	ADP
ejpam-7135	53	27	a	a	PRON
ejpam-7135	53	28	as	as	ADP
ejpam-7135	53	29	its	its	PRON
ejpam-7135	53	30	least	least	ADJ
ejpam-7135	53	31	element	element	NOUN
ejpam-7135	53	32	,	,	PUNCT
ejpam-7135	53	33	(	(	PUNCT
ejpam-7135	53	34	8)	8)	NUM
ejpam-7135	53	35	s	s	NOUN
ejpam-7135	53	36	∨	∨	NOUN
ejpam-7135	53	37	{	{	PUNCT
ejpam-7135	53	38	x	x	PROPN
ejpam-7135	53	39	∧	∧	PROPN
ejpam-7135	53	40	(	(	PUNCT
ejpam-7135	53	41	y	y	PROPN
ejpam-7135	53	42	∧	∧	PROPN
ejpam-7135	53	43	z	z	PROPN
ejpam-7135	53	44	)	)	PUNCT
ejpam-7135	53	45	}	}	PUNCT
ejpam-7135	53	46	=	=	SYM
ejpam-7135	53	47	s	s	PROPN
ejpam-7135	53	48	∨	∨	X
ejpam-7135	53	49	{	{	PUNCT
ejpam-7135	53	50	(	(	PUNCT
ejpam-7135	53	51	x	x	PROPN
ejpam-7135	53	52	∧	∧	PROPN
ejpam-7135	53	53	y	y	NOUN
ejpam-7135	53	54	)	)	PUNCT
ejpam-7135	53	55	∧	∧	PROPN
ejpam-7135	53	56	z	z	PROPN
ejpam-7135	53	57	}	}	PUNCT
ejpam-7135	53	58	,	,	PUNCT
ejpam-7135	53	59	(	(	PUNCT
ejpam-7135	53	60	9	9	X
ejpam-7135	53	61	)	)	PUNCT
ejpam-7135	53	62	x	x	PUNCT
ejpam-7135	53	63	∨	∨	NOUN
ejpam-7135	53	64	(	(	PUNCT
ejpam-7135	53	65	y	y	PROPN
ejpam-7135	53	66	∨	∨	PROPN
ejpam-7135	53	67	z	z	PROPN
ejpam-7135	53	68	)	)	PUNCT
ejpam-7135	53	69	=	=	SYM
ejpam-7135	54	1	x	x	SYM
ejpam-7135	54	2	∨	∨	X
ejpam-7135	54	3	(	(	PUNCT
ejpam-7135	54	4	z	z	PROPN
ejpam-7135	54	5	∨	∨	PROPN
ejpam-7135	54	6	y	y	PROPN
ejpam-7135	54	7	)	)	PUNCT
ejpam-7135	54	8	,	,	PUNCT
ejpam-7135	54	9	(	(	PUNCT
ejpam-7135	54	10	10	10	NUM
ejpam-7135	54	11	)	)	PUNCT
ejpam-7135	54	12	x	x	PUNCT
ejpam-7135	54	13	∨	∨	NUM
ejpam-7135	54	14	y	y	NOUN
ejpam-7135	54	15	=	=	SYM
ejpam-7135	54	16	1	1	NUM
ejpam-7135	54	17	if	if	SCONJ
ejpam-7135	54	18	and	and	CCONJ
ejpam-7135	54	19	only	only	ADV
ejpam-7135	54	20	if	if	SCONJ
ejpam-7135	54	21	y	y	PROPN
ejpam-7135	54	22	∨	∨	NUM
ejpam-7135	54	23	x	x	SYM
ejpam-7135	54	24	=	=	SYM
ejpam-7135	54	25	1	1	NUM
ejpam-7135	54	26	,	,	PUNCT
ejpam-7135	54	27	(	(	PUNCT
ejpam-7135	54	28	11	11	NUM
ejpam-7135	54	29	)	)	PUNCT
ejpam-7135	54	30	x	x	X
ejpam-7135	55	1	∧	∧	NOUN
ejpam-7135	55	2	y	y	PROPN
ejpam-7135	55	3	=	=	SYM
ejpam-7135	55	4	y	y	PROPN
ejpam-7135	55	5	∧	∧	PROPN
ejpam-7135	55	6	x	x	INTJ
ejpam-7135	55	7	whenever	whenever	SCONJ
ejpam-7135	55	8	x	x	PROPN
ejpam-7135	55	9	∨	∨	NUM
ejpam-7135	55	10	y	y	NOUN
ejpam-7135	55	11	=	=	SYM
ejpam-7135	55	12	1	1	X
ejpam-7135	55	13	.	.	PUNCT
ejpam-7135	55	14	theorem	theorem	NOUN
ejpam-7135	55	15	1	1	NUM
ejpam-7135	55	16	.	.	PUNCT
ejpam-7135	56	1	[	[	X
ejpam-7135	56	2	12	12	NUM
ejpam-7135	56	3	]	]	PUNCT
ejpam-7135	56	4	an	an	DET
ejpam-7135	56	5	algebra	algebra	NOUN
ejpam-7135	56	6	(	(	PUNCT
ejpam-7135	56	7	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	56	8	,	,	PUNCT
ejpam-7135	56	9	1	1	NUM
ejpam-7135	56	10	)	)	PUNCT
ejpam-7135	56	11	of	of	ADP
ejpam-7135	56	12	type	type	NOUN
ejpam-7135	56	13	(	(	PUNCT
ejpam-7135	56	14	2	2	NUM
ejpam-7135	56	15	,	,	PUNCT
ejpam-7135	56	16	2	2	NUM
ejpam-7135	56	17	,	,	PUNCT
ejpam-7135	56	18	0	0	NUM
ejpam-7135	56	19	)	)	PUNCT
ejpam-7135	56	20	is	be	AUX
ejpam-7135	56	21	a	a	DET
ejpam-7135	56	22	pdl	pdl	NOUN
ejpam-7135	56	23	if	if	SCONJ
ejpam-7135	57	1	and	and	CCONJ
ejpam-7135	57	2	only	only	ADV
ejpam-7135	57	3	if	if	SCONJ
ejpam-7135	57	4	it	it	PRON
ejpam-7135	57	5	satisfies	satisfy	VERB
ejpam-7135	57	6	the	the	DET
ejpam-7135	57	7	following	following	NOUN
ejpam-7135	57	8	:	:	PUNCT
ejpam-7135	57	9	(	(	PUNCT
ejpam-7135	57	10	ld∨	ld∨	NOUN
ejpam-7135	57	11	)	)	PUNCT
ejpam-7135	57	12	x	x	SYM
ejpam-7135	58	1	∨	∨	NOUN
ejpam-7135	58	2	(	(	PUNCT
ejpam-7135	58	3	y	y	PROPN
ejpam-7135	58	4	∧	∧	PROPN
ejpam-7135	58	5	z	z	PROPN
ejpam-7135	58	6	)	)	PUNCT
ejpam-7135	58	7	=	=	SYM
ejpam-7135	58	8	(	(	PUNCT
ejpam-7135	58	9	x	x	PROPN
ejpam-7135	58	10	∨	∨	NUM
ejpam-7135	58	11	y	y	NOUN
ejpam-7135	58	12	)	)	PUNCT
ejpam-7135	58	13	∧	∧	NOUN
ejpam-7135	58	14	(	(	PUNCT
ejpam-7135	58	15	x	x	PROPN
ejpam-7135	58	16	∨	∨	PROPN
ejpam-7135	58	17	z	z	PROPN
ejpam-7135	58	18	)	)	PUNCT
ejpam-7135	58	19	,	,	PUNCT
ejpam-7135	58	20	(	(	PUNCT
ejpam-7135	58	21	rd∨	rd∨	X
ejpam-7135	58	22	)	)	PUNCT
ejpam-7135	58	23	(	(	PUNCT
ejpam-7135	58	24	x	x	PUNCT
ejpam-7135	58	25	∧	∧	PROPN
ejpam-7135	58	26	y	y	PROPN
ejpam-7135	58	27	)	)	PUNCT
ejpam-7135	58	28	∨	∨	NUM
ejpam-7135	58	29	z	z	NOUN
ejpam-7135	58	30	=	=	SYM
ejpam-7135	58	31	(	(	PUNCT
ejpam-7135	58	32	x	x	PROPN
ejpam-7135	58	33	∨	∨	PROPN
ejpam-7135	58	34	z	z	NOUN
ejpam-7135	58	35	)	)	PUNCT
ejpam-7135	58	36	∧	∧	PROPN
ejpam-7135	58	37	(	(	PUNCT
ejpam-7135	58	38	y	y	PROPN
ejpam-7135	58	39	∨	∨	PROPN
ejpam-7135	58	40	z	z	PROPN
ejpam-7135	58	41	)	)	PUNCT
ejpam-7135	58	42	,	,	PUNCT
ejpam-7135	58	43	(	(	PUNCT
ejpam-7135	58	44	rd∧	rd∧	PROPN
ejpam-7135	58	45	)	)	PUNCT
ejpam-7135	58	46	(	(	PUNCT
ejpam-7135	58	47	x	x	PROPN
ejpam-7135	58	48	∨	∨	NUM
ejpam-7135	58	49	y	y	NOUN
ejpam-7135	58	50	)	)	PUNCT
ejpam-7135	58	51	∧	∧	NOUN
ejpam-7135	58	52	z	z	NOUN
ejpam-7135	58	53	=	=	SYM
ejpam-7135	58	54	(	(	PUNCT
ejpam-7135	58	55	x	x	PART
ejpam-7135	58	56	∧	∧	PROPN
ejpam-7135	58	57	z	z	PROPN
ejpam-7135	58	58	)	)	PUNCT
ejpam-7135	58	59	∨	∨	PROPN
ejpam-7135	58	60	(	(	PUNCT
ejpam-7135	58	61	y	y	PROPN
ejpam-7135	58	62	∧	∧	PROPN
ejpam-7135	58	63	z	z	PROPN
ejpam-7135	58	64	)	)	PUNCT
ejpam-7135	58	65	,	,	PUNCT
ejpam-7135	58	66	(	(	PUNCT
ejpam-7135	58	67	l1	l1	PROPN
ejpam-7135	58	68	)	)	PUNCT
ejpam-7135	58	69	(	(	PUNCT
ejpam-7135	58	70	x	x	PROPN
ejpam-7135	58	71	∨	∨	NUM
ejpam-7135	58	72	y	y	NOUN
ejpam-7135	58	73	)	)	PUNCT
ejpam-7135	58	74	∧	∧	NOUN
ejpam-7135	58	75	y	y	PROPN
ejpam-7135	58	76	=	=	SYM
ejpam-7135	58	77	y	y	PROPN
ejpam-7135	58	78	,	,	PUNCT
ejpam-7135	58	79	(	(	PUNCT
ejpam-7135	58	80	l3	l3	NOUN
ejpam-7135	58	81	)	)	PUNCT
ejpam-7135	58	82	x	x	SYM
ejpam-7135	58	83	∨	∨	NOUN
ejpam-7135	58	84	(	(	PUNCT
ejpam-7135	58	85	x	x	PROPN
ejpam-7135	58	86	∧	∧	PROPN
ejpam-7135	58	87	y	y	NOUN
ejpam-7135	58	88	)	)	PUNCT
ejpam-7135	58	89	=	=	SYM
ejpam-7135	59	1	x	x	PROPN
ejpam-7135	59	2	,	,	PUNCT
ejpam-7135	59	3	r.	r.	PROPN
ejpam-7135	59	4	bandaru	bandaru	PROPN
ejpam-7135	59	5	et	et	PROPN
ejpam-7135	59	6	al	al	PROPN
ejpam-7135	59	7	.	.	PUNCT
ejpam-7135	59	8	/	/	SYM
ejpam-7135	59	9	eur	eur	PROPN
ejpam-7135	59	10	.	.	PUNCT
ejpam-7135	60	1	j.	j.	PROPN
ejpam-7135	60	2	pure	pure	PROPN
ejpam-7135	60	3	appl	appl	PROPN
ejpam-7135	60	4	.	.	PROPN
ejpam-7135	60	5	math	math	PROPN
ejpam-7135	60	6	,	,	PUNCT
ejpam-7135	60	7	18	18	NUM
ejpam-7135	60	8	(	(	PUNCT
ejpam-7135	60	9	4	4	NUM
ejpam-7135	60	10	)	)	PUNCT
ejpam-7135	60	11	(	(	PUNCT
ejpam-7135	60	12	2025	2025	NUM
ejpam-7135	60	13	)	)	PUNCT
ejpam-7135	60	14	,	,	PUNCT
ejpam-7135	60	15	7135	7135	NUM
ejpam-7135	60	16	4	4	NUM
ejpam-7135	60	17	of	of	ADP
ejpam-7135	60	18	14	14	NUM
ejpam-7135	60	19	(	(	PUNCT
ejpam-7135	60	20	i1	i1	PROPN
ejpam-7135	60	21	)	)	PUNCT
ejpam-7135	60	22	x	x	PUNCT
ejpam-7135	61	1	∨	∨	NUM
ejpam-7135	61	2	1	1	NUM
ejpam-7135	61	3	=	=	SYM
ejpam-7135	61	4	1	1	NUM
ejpam-7135	61	5	,	,	PUNCT
ejpam-7135	61	6	(	(	PUNCT
ejpam-7135	61	7	i2	i2	PROPN
ejpam-7135	61	8	)	)	PUNCT
ejpam-7135	61	9	1	1	NUM
ejpam-7135	61	10	∧	∧	NOUN
ejpam-7135	61	11	x	x	X
ejpam-7135	61	12	=	=	SYM
ejpam-7135	61	13	x	x	NOUN
ejpam-7135	61	14	,	,	PUNCT
ejpam-7135	61	15	for	for	ADP
ejpam-7135	61	16	all	all	DET
ejpam-7135	61	17	x	x	NOUN
ejpam-7135	61	18	,	,	PUNCT
ejpam-7135	61	19	y	y	PROPN
ejpam-7135	61	20	,	,	PUNCT
ejpam-7135	61	21	z	z	PROPN
ejpam-7135	61	22	∈	∈	PROPN
ejpam-7135	61	23	l.	l.	NOUN
ejpam-7135	61	24	definition	definition	NOUN
ejpam-7135	61	25	2	2	NUM
ejpam-7135	61	26	.	.	PUNCT
ejpam-7135	62	1	[	[	X
ejpam-7135	62	2	12	12	NUM
ejpam-7135	62	3	]	]	PUNCT
ejpam-7135	62	4	a	a	DET
ejpam-7135	62	5	paradistributive	paradistributive	ADJ
ejpam-7135	62	6	latticoid	latticoid	NOUN
ejpam-7135	62	7	(	(	PUNCT
ejpam-7135	62	8	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	62	9	,	,	PUNCT
ejpam-7135	62	10	1	1	NUM
ejpam-7135	62	11	)	)	PUNCT
ejpam-7135	62	12	is	be	AUX
ejpam-7135	62	13	said	say	VERB
ejpam-7135	62	14	to	to	PART
ejpam-7135	62	15	be	be	AUX
ejpam-7135	62	16	associative	associative	ADJ
ejpam-7135	62	17	if	if	SCONJ
ejpam-7135	62	18	it	it	PRON
ejpam-7135	62	19	satisfies	satisfy	VERB
ejpam-7135	62	20	the	the	DET
ejpam-7135	62	21	following	follow	VERB
ejpam-7135	62	22	condition	condition	NOUN
ejpam-7135	62	23	x	x	X
ejpam-7135	62	24	∧	∧	PROPN
ejpam-7135	62	25	(	(	PUNCT
ejpam-7135	62	26	y	y	PROPN
ejpam-7135	62	27	∧	∧	PROPN
ejpam-7135	62	28	z	z	PROPN
ejpam-7135	62	29	)	)	PUNCT
ejpam-7135	62	30	=	=	SYM
ejpam-7135	62	31	(	(	PUNCT
ejpam-7135	62	32	x	x	PUNCT
ejpam-7135	62	33	∧	∧	PROPN
ejpam-7135	62	34	y	y	NOUN
ejpam-7135	62	35	)	)	PUNCT
ejpam-7135	62	36	∧	∧	PROPN
ejpam-7135	62	37	z	z	NOUN
ejpam-7135	62	38	for	for	ADP
ejpam-7135	62	39	all	all	DET
ejpam-7135	62	40	x	x	NOUN
ejpam-7135	62	41	,	,	PUNCT
ejpam-7135	62	42	y	y	PROPN
ejpam-7135	62	43	,	,	PUNCT
ejpam-7135	62	44	z	z	PROPN
ejpam-7135	62	45	∈	∈	PROPN
ejpam-7135	62	46	l.	l.	NOUN
ejpam-7135	62	47	definition	definition	NOUN
ejpam-7135	62	48	3	3	NUM
ejpam-7135	62	49	.	.	PUNCT
ejpam-7135	63	1	[	[	X
ejpam-7135	63	2	12	12	NUM
ejpam-7135	63	3	]	]	X
ejpam-7135	63	4	let	let	AUX
ejpam-7135	63	5	l	l	NOUN
ejpam-7135	63	6	be	be	AUX
ejpam-7135	63	7	a	a	DET
ejpam-7135	63	8	pdl	pdl	NOUN
ejpam-7135	63	9	.	.	PUNCT
ejpam-7135	64	1	then	then	ADV
ejpam-7135	64	2	,	,	PUNCT
ejpam-7135	64	3	an	an	DET
ejpam-7135	64	4	element	element	NOUN
ejpam-7135	64	5	a	a	DET
ejpam-7135	64	6	∈	∈	PROPN
ejpam-7135	64	7	l	l	NOUN
ejpam-7135	64	8	is	be	AUX
ejpam-7135	64	9	said	say	VERB
ejpam-7135	64	10	to	to	PART
ejpam-7135	64	11	be	be	AUX
ejpam-7135	64	12	a	a	DET
ejpam-7135	64	13	minimal	minimal	ADJ
ejpam-7135	64	14	element	element	NOUN
ejpam-7135	64	15	if	if	SCONJ
ejpam-7135	64	16	for	for	ADP
ejpam-7135	64	17	any	any	DET
ejpam-7135	64	18	x	x	SYM
ejpam-7135	64	19	∈	∈	PROPN
ejpam-7135	64	20	l	l	NOUN
ejpam-7135	64	21	,	,	PUNCT
ejpam-7135	64	22	x	x	SYM
ejpam-7135	64	23	≤	≤	NOUN
ejpam-7135	64	24	a	a	DET
ejpam-7135	64	25	⇒	⇒	NOUN
ejpam-7135	64	26	x	x	X
ejpam-7135	64	27	=	=	PUNCT
ejpam-7135	64	28	a.	a.	NOUN
ejpam-7135	64	29	lemma	lemma	PROPN
ejpam-7135	64	30	2	2	X
ejpam-7135	64	31	.	.	PUNCT
ejpam-7135	65	1	[	[	X
ejpam-7135	65	2	12	12	NUM
ejpam-7135	65	3	]	]	X
ejpam-7135	65	4	let	let	AUX
ejpam-7135	65	5	l	l	NOUN
ejpam-7135	65	6	be	be	AUX
ejpam-7135	65	7	a	a	DET
ejpam-7135	65	8	pdl	pdl	NOUN
ejpam-7135	65	9	.	.	PUNCT
ejpam-7135	66	1	then	then	ADV
ejpam-7135	66	2	,	,	PUNCT
ejpam-7135	66	3	for	for	ADP
ejpam-7135	66	4	any	any	DET
ejpam-7135	66	5	a	a	DET
ejpam-7135	66	6	∈	∈	PROPN
ejpam-7135	66	7	l	l	NOUN
ejpam-7135	66	8	,	,	PUNCT
ejpam-7135	66	9	the	the	DET
ejpam-7135	66	10	following	follow	VERB
ejpam-7135	66	11	are	be	AUX
ejpam-7135	66	12	equivalent	equivalent	ADJ
ejpam-7135	66	13	:	:	PUNCT
ejpam-7135	66	14	(	(	PUNCT
ejpam-7135	66	15	1	1	X
ejpam-7135	66	16	)	)	PUNCT
ejpam-7135	66	17	a	a	PRON
ejpam-7135	66	18	is	be	AUX
ejpam-7135	66	19	minimal	minimal	ADJ
ejpam-7135	66	20	,	,	PUNCT
ejpam-7135	66	21	(	(	PUNCT
ejpam-7135	66	22	2	2	NUM
ejpam-7135	66	23	)	)	PUNCT
ejpam-7135	66	24	x	x	X
ejpam-7135	67	1	∧	∧	NOUN
ejpam-7135	67	2	a	a	PRON
ejpam-7135	67	3	=	=	X
ejpam-7135	67	4	a	a	NOUN
ejpam-7135	67	5	for	for	ADP
ejpam-7135	67	6	all	all	DET
ejpam-7135	67	7	x	x	SYM
ejpam-7135	67	8	∈	∈	PROPN
ejpam-7135	67	9	l	l	NOUN
ejpam-7135	67	10	,	,	PUNCT
ejpam-7135	67	11	(	(	PUNCT
ejpam-7135	67	12	3	3	X
ejpam-7135	67	13	)	)	PUNCT
ejpam-7135	67	14	x	x	SYM
ejpam-7135	67	15	∨	∨	NOUN
ejpam-7135	67	16	a	a	X
ejpam-7135	67	17	=	=	X
ejpam-7135	67	18	x	x	PROPN
ejpam-7135	67	19	for	for	ADP
ejpam-7135	67	20	all	all	DET
ejpam-7135	67	21	x	x	SYM
ejpam-7135	67	22	∈	∈	PROPN
ejpam-7135	67	23	l.	l.	NOUN
ejpam-7135	67	24	definition	definition	NOUN
ejpam-7135	67	25	4	4	NUM
ejpam-7135	67	26	.	.	PUNCT
ejpam-7135	68	1	[	[	X
ejpam-7135	68	2	12	12	NUM
ejpam-7135	68	3	]	]	PUNCT
ejpam-7135	68	4	a	a	DET
ejpam-7135	68	5	non	non	ADJ
ejpam-7135	68	6	-	-	ADJ
ejpam-7135	68	7	empty	empty	ADJ
ejpam-7135	68	8	subset	subset	NOUN
ejpam-7135	68	9	f	f	PROPN
ejpam-7135	68	10	of	of	ADP
ejpam-7135	68	11	a	a	DET
ejpam-7135	68	12	pdl	pdl	NOUN
ejpam-7135	68	13	l	l	NOUN
ejpam-7135	68	14	is	be	AUX
ejpam-7135	68	15	said	say	VERB
ejpam-7135	68	16	to	to	PART
ejpam-7135	68	17	be	be	AUX
ejpam-7135	68	18	a	a	DET
ejpam-7135	68	19	filter	filter	NOUN
ejpam-7135	68	20	if	if	SCONJ
ejpam-7135	68	21	it	it	PRON
ejpam-7135	68	22	satisfies	satisfy	VERB
ejpam-7135	68	23	the	the	DET
ejpam-7135	68	24	following	follow	VERB
ejpam-7135	68	25	:	:	PUNCT
ejpam-7135	68	26	x	x	X
ejpam-7135	68	27	,	,	PUNCT
ejpam-7135	68	28	y	y	PROPN
ejpam-7135	68	29	∈	∈	PROPN
ejpam-7135	68	30	f	f	PROPN
ejpam-7135	68	31	⇒	⇒	VERB
ejpam-7135	68	32	x	x	X
ejpam-7135	69	1	∧	∧	NOUN
ejpam-7135	69	2	y	y	PROPN
ejpam-7135	69	3	∈	∈	PROPN
ejpam-7135	69	4	f	f	AUX
ejpam-7135	69	5	,	,	PUNCT
ejpam-7135	69	6	x	x	SYM
ejpam-7135	69	7	∈	∈	PROPN
ejpam-7135	69	8	f	f	PROPN
ejpam-7135	69	9	,	,	PUNCT
ejpam-7135	69	10	a	a	DET
ejpam-7135	69	11	∈	∈	PROPN
ejpam-7135	69	12	l	l	NOUN
ejpam-7135	69	13	⇒	⇒	NOUN
ejpam-7135	69	14	a	a	DET
ejpam-7135	69	15	∨	∨	NOUN
ejpam-7135	69	16	x	x	SYM
ejpam-7135	69	17	∈	∈	PROPN
ejpam-7135	69	18	f.	f.	PROPN
ejpam-7135	69	19	theorem	theorem	VERB
ejpam-7135	69	20	2	2	NUM
ejpam-7135	69	21	.	.	PUNCT
ejpam-7135	70	1	[	[	X
ejpam-7135	70	2	12	12	NUM
ejpam-7135	70	3	]	]	X
ejpam-7135	70	4	let	let	VERB
ejpam-7135	70	5	s	s	PRON
ejpam-7135	70	6	be	be	AUX
ejpam-7135	70	7	a	a	DET
ejpam-7135	70	8	non	non	ADJ
ejpam-7135	70	9	-	-	ADJ
ejpam-7135	70	10	empty	empty	ADJ
ejpam-7135	70	11	subset	subset	NOUN
ejpam-7135	70	12	of	of	ADP
ejpam-7135	70	13	l.	l.	PROPN
ejpam-7135	70	14	then	then	ADV
ejpam-7135	70	15	[	[	X
ejpam-7135	70	16	s	s	X
ejpam-7135	70	17	)	)	PUNCT
ejpam-7135	70	18	=	=	SYM
ejpam-7135	70	19	{	{	PUNCT
ejpam-7135	70	20	x	x	PROPN
ejpam-7135	70	21	∨	∨	X
ejpam-7135	70	22	(	(	PUNCT
ejpam-7135	70	23	n	n	CCONJ
ejpam-7135	70	24	∧	∧	PROPN
ejpam-7135	70	25	i=1	i=1	PROPN
ejpam-7135	70	26	si	si	NOUN
ejpam-7135	70	27	)	)	PUNCT
ejpam-7135	71	1	|	|	ADV
ejpam-7135	71	2	si	si	PROPN
ejpam-7135	71	3	∈	∈	PROPN
ejpam-7135	71	4	s	s	PROPN
ejpam-7135	71	5	,	,	PUNCT
ejpam-7135	71	6	x	x	SYM
ejpam-7135	71	7	∈	∈	PROPN
ejpam-7135	71	8	l	l	NOUN
ejpam-7135	71	9	,	,	PUNCT
ejpam-7135	71	10	n	n	X
ejpam-7135	71	11	is	be	AUX
ejpam-7135	71	12	a	a	DET
ejpam-7135	71	13	positive	positive	ADJ
ejpam-7135	71	14	integer	integer	NOUN
ejpam-7135	71	15	}	}	PUNCT
ejpam-7135	71	16	is	be	AUX
ejpam-7135	71	17	the	the	DET
ejpam-7135	71	18	smallest	small	ADJ
ejpam-7135	71	19	filter	filter	NOUN
ejpam-7135	71	20	of	of	ADP
ejpam-7135	71	21	l	l	NOUN
ejpam-7135	71	22	containing	contain	VERB
ejpam-7135	71	23	s.	s.	PROPN
ejpam-7135	71	24	note	note	VERB
ejpam-7135	71	25	that	that	SCONJ
ejpam-7135	71	26	if	if	SCONJ
ejpam-7135	71	27	s	s	VERB
ejpam-7135	71	28	=	=	X
ejpam-7135	71	29	{	{	PUNCT
ejpam-7135	71	30	x	x	NOUN
ejpam-7135	71	31	}	}	PUNCT
ejpam-7135	71	32	,	,	PUNCT
ejpam-7135	71	33	then	then	ADV
ejpam-7135	71	34	we	we	PRON
ejpam-7135	71	35	write	write	VERB
ejpam-7135	71	36	[	[	X
ejpam-7135	71	37	s	s	X
ejpam-7135	71	38	)	)	PUNCT
ejpam-7135	71	39	=	=	PUNCT
ejpam-7135	72	1	[	[	X
ejpam-7135	72	2	x	x	X
ejpam-7135	72	3	)	)	PUNCT
ejpam-7135	72	4	,	,	PUNCT
ejpam-7135	72	5	the	the	DET
ejpam-7135	72	6	principal	principal	ADJ
ejpam-7135	72	7	ideal	ideal	NOUN
ejpam-7135	72	8	of	of	ADP
ejpam-7135	72	9	l	l	NOUN
ejpam-7135	72	10	generated	generate	VERB
ejpam-7135	72	11	by	by	ADP
ejpam-7135	72	12	x.	x.	PROPN
ejpam-7135	72	13	hence	hence	ADV
ejpam-7135	72	14	,	,	PUNCT
ejpam-7135	72	15	[	[	X
ejpam-7135	72	16	x	x	X
ejpam-7135	72	17	)	)	PUNCT
ejpam-7135	72	18	=	=	SYM
ejpam-7135	72	19	{	{	PUNCT
ejpam-7135	72	20	a	a	DET
ejpam-7135	72	21	∨	∨	NOUN
ejpam-7135	72	22	x	x	SYM
ejpam-7135	72	23	|	|	ADV
ejpam-7135	72	24	a	a	DET
ejpam-7135	72	25	∈	∈	NOUN
ejpam-7135	72	26	l	l	NOUN
ejpam-7135	72	27	}	}	PUNCT
ejpam-7135	72	28	.	.	PUNCT
ejpam-7135	73	1	according	accord	VERB
ejpam-7135	73	2	to	to	ADP
ejpam-7135	73	3	corollary	corollary	ADJ
ejpam-7135	73	4	8	8	NUM
ejpam-7135	73	5	and	and	CCONJ
ejpam-7135	73	6	lemma	lemma	PROPN
ejpam-7135	73	7	12	12	NUM
ejpam-7135	73	8	of	of	ADP
ejpam-7135	73	9	[	[	X
ejpam-7135	73	10	12	12	NUM
ejpam-7135	73	11	]	]	PUNCT
ejpam-7135	73	12	,	,	PUNCT
ejpam-7135	73	13	the	the	DET
ejpam-7135	73	14	following	follow	VERB
ejpam-7135	73	15	lemma	lemma	PROPN
ejpam-7135	73	16	holds	hold	VERB
ejpam-7135	73	17	.	.	PUNCT
ejpam-7135	74	1	lemma	lemma	PROPN
ejpam-7135	74	2	3	3	NUM
ejpam-7135	74	3	.	.	PUNCT
ejpam-7135	75	1	[	[	X
ejpam-7135	75	2	12	12	NUM
ejpam-7135	75	3	]	]	X
ejpam-7135	75	4	let	let	AUX
ejpam-7135	75	5	l	l	NOUN
ejpam-7135	75	6	be	be	AUX
ejpam-7135	75	7	a	a	DET
ejpam-7135	75	8	pdl	pdl	NOUN
ejpam-7135	75	9	and	and	CCONJ
ejpam-7135	75	10	f	f	PROPN
ejpam-7135	75	11	be	be	AUX
ejpam-7135	75	12	a	a	DET
ejpam-7135	75	13	filter	filter	NOUN
ejpam-7135	75	14	of	of	ADP
ejpam-7135	75	15	l.	l.	PROPN
ejpam-7135	75	16	then	then	ADV
ejpam-7135	75	17	for	for	ADP
ejpam-7135	75	18	any	any	DET
ejpam-7135	75	19	x	x	NOUN
ejpam-7135	75	20	,	,	PUNCT
ejpam-7135	75	21	y	y	PROPN
ejpam-7135	75	22	∈	∈	PROPN
ejpam-7135	75	23	l	l	NOUN
ejpam-7135	75	24	,	,	PUNCT
ejpam-7135	75	25	we	we	PRON
ejpam-7135	75	26	have	have	VERB
ejpam-7135	75	27	the	the	DET
ejpam-7135	75	28	following	following	NOUN
ejpam-7135	75	29	:	:	PUNCT
ejpam-7135	75	30	(	(	PUNCT
ejpam-7135	75	31	1	1	X
ejpam-7135	75	32	)	)	PUNCT
ejpam-7135	75	33	x	x	SYM
ejpam-7135	75	34	∈	∈	PROPN
ejpam-7135	76	1	[	[	X
ejpam-7135	76	2	y	y	NOUN
ejpam-7135	76	3	)	)	PUNCT
ejpam-7135	76	4	if	if	SCONJ
ejpam-7135	76	5	and	and	CCONJ
ejpam-7135	76	6	only	only	ADV
ejpam-7135	76	7	if	if	SCONJ
ejpam-7135	76	8	x	x	NOUN
ejpam-7135	76	9	=	=	SYM
ejpam-7135	76	10	x	x	SYM
ejpam-7135	76	11	∨	∨	NUM
ejpam-7135	76	12	y	y	PROPN
ejpam-7135	76	13	for	for	ADP
ejpam-7135	76	14	all	all	DET
ejpam-7135	76	15	x	x	NOUN
ejpam-7135	76	16	,	,	PUNCT
ejpam-7135	76	17	y	y	PROPN
ejpam-7135	76	18	∈	∈	PROPN
ejpam-7135	76	19	l	l	PROPN
ejpam-7135	76	20	,	,	PUNCT
ejpam-7135	76	21	(	(	PUNCT
ejpam-7135	76	22	2	2	NUM
ejpam-7135	76	23	)	)	PUNCT
ejpam-7135	76	24	x	x	PUNCT
ejpam-7135	76	25	∨	∨	NUM
ejpam-7135	76	26	y	y	PROPN
ejpam-7135	76	27	∈	∈	PROPN
ejpam-7135	76	28	f	f	PROPN
ejpam-7135	77	1	if	if	SCONJ
ejpam-7135	77	2	and	and	CCONJ
ejpam-7135	77	3	only	only	ADV
ejpam-7135	77	4	if	if	SCONJ
ejpam-7135	77	5	y	y	PROPN
ejpam-7135	77	6	∨	∨	NUM
ejpam-7135	77	7	x	x	SYM
ejpam-7135	77	8	∈	∈	PROPN
ejpam-7135	77	9	f	f	X
ejpam-7135	77	10	,	,	PUNCT
ejpam-7135	77	11	(	(	PUNCT
ejpam-7135	77	12	3	3	X
ejpam-7135	77	13	)	)	PUNCT
ejpam-7135	78	1	[	[	X
ejpam-7135	78	2	x	x	X
ejpam-7135	78	3	∨	∨	NUM
ejpam-7135	78	4	y	y	PROPN
ejpam-7135	78	5	)	)	PUNCT
ejpam-7135	78	6	=	=	PUNCT
ejpam-7135	79	1	[	[	X
ejpam-7135	79	2	y	y	PROPN
ejpam-7135	79	3	∨	∨	NUM
ejpam-7135	79	4	x	x	X
ejpam-7135	79	5	)	)	PUNCT
ejpam-7135	79	6	,	,	PUNCT
ejpam-7135	79	7	(	(	PUNCT
ejpam-7135	79	8	4	4	X
ejpam-7135	79	9	)	)	PUNCT
ejpam-7135	79	10	[	[	X
ejpam-7135	79	11	x	x	X
ejpam-7135	79	12	∧	∧	PROPN
ejpam-7135	79	13	y	y	NOUN
ejpam-7135	79	14	)	)	PUNCT
ejpam-7135	79	15	=	=	PUNCT
ejpam-7135	80	1	[	[	X
ejpam-7135	80	2	y	y	PROPN
ejpam-7135	80	3	∧	∧	PROPN
ejpam-7135	80	4	x	x	X
ejpam-7135	80	5	)	)	PUNCT
ejpam-7135	80	6	=	=	PUNCT
ejpam-7135	81	1	[	[	X
ejpam-7135	81	2	x	x	X
ejpam-7135	81	3	)	)	PUNCT
ejpam-7135	81	4	∨	∨	NOUN
ejpam-7135	81	5	[	[	X
ejpam-7135	81	6	y	y	NOUN
ejpam-7135	81	7	)	)	PUNCT
ejpam-7135	81	8	.	.	PUNCT
ejpam-7135	82	1	theorem	theorem	NOUN
ejpam-7135	82	2	3	3	NUM
ejpam-7135	82	3	.	.	PUNCT
ejpam-7135	83	1	[	[	X
ejpam-7135	83	2	12	12	NUM
ejpam-7135	83	3	]	]	PUNCT
ejpam-7135	83	4	the	the	DET
ejpam-7135	83	5	collection	collection	NOUN
ejpam-7135	83	6	f	f	X
ejpam-7135	83	7	(	(	PUNCT
ejpam-7135	83	8	l	l	NOUN
ejpam-7135	83	9	)	)	PUNCT
ejpam-7135	83	10	of	of	ADP
ejpam-7135	83	11	all	all	DET
ejpam-7135	83	12	filters	filter	NOUN
ejpam-7135	83	13	of	of	ADP
ejpam-7135	83	14	a	a	DET
ejpam-7135	83	15	pdl	pdl	NOUN
ejpam-7135	83	16	l	l	NOUN
ejpam-7135	83	17	forms	form	NOUN
ejpam-7135	83	18	a	a	DET
ejpam-7135	83	19	distributive	distributive	ADJ
ejpam-7135	83	20	lattice	lattice	NOUN
ejpam-7135	83	21	under	under	ADP
ejpam-7135	83	22	set	set	ADJ
ejpam-7135	83	23	inclusion	inclusion	NOUN
ejpam-7135	83	24	,	,	PUNCT
ejpam-7135	83	25	in	in	ADP
ejpam-7135	83	26	which	which	PRON
ejpam-7135	83	27	,	,	PUNCT
ejpam-7135	83	28	the	the	DET
ejpam-7135	83	29	glb	glb	NOUN
ejpam-7135	83	30	and	and	CCONJ
ejpam-7135	83	31	lub	lub	NOUN
ejpam-7135	83	32	of	of	ADP
ejpam-7135	83	33	any	any	DET
ejpam-7135	83	34	two	two	NUM
ejpam-7135	83	35	filters	filter	NOUN
ejpam-7135	83	36	f	f	PROPN
ejpam-7135	83	37	and	and	CCONJ
ejpam-7135	83	38	g	g	PROPN
ejpam-7135	83	39	are	be	AUX
ejpam-7135	83	40	given	give	VERB
ejpam-7135	83	41	by	by	ADP
ejpam-7135	83	42	f	f	PROPN
ejpam-7135	83	43	∧g	∧g	PROPN
ejpam-7135	83	44	=	=	SYM
ejpam-7135	83	45	f	f	PROPN
ejpam-7135	83	46	∩g	∩g	NOUN
ejpam-7135	83	47	and	and	CCONJ
ejpam-7135	83	48	f	f	PROPN
ejpam-7135	83	49	∨g	∨g	PROPN
ejpam-7135	83	50	=	=	SYM
ejpam-7135	83	51	{	{	PUNCT
ejpam-7135	83	52	x	x	PROPN
ejpam-7135	83	53	∧	∧	NOUN
ejpam-7135	83	54	y	y	NOUN
ejpam-7135	84	1	|	|	ADV
ejpam-7135	84	2	x	x	SYM
ejpam-7135	84	3	∈	∈	PROPN
ejpam-7135	84	4	f	f	PROPN
ejpam-7135	84	5	and	and	CCONJ
ejpam-7135	84	6	y	y	PROPN
ejpam-7135	84	7	∈	∈	PROPN
ejpam-7135	84	8	g	g	PROPN
ejpam-7135	84	9	}	}	PUNCT
ejpam-7135	84	10	,	,	PUNCT
ejpam-7135	84	11	respectively	respectively	ADV
ejpam-7135	84	12	.	.	PUNCT
ejpam-7135	85	1	definition	definition	NOUN
ejpam-7135	85	2	5	5	NUM
ejpam-7135	85	3	.	.	PUNCT
ejpam-7135	86	1	[	[	X
ejpam-7135	86	2	12	12	NUM
ejpam-7135	86	3	]	]	PUNCT
ejpam-7135	86	4	a	a	DET
ejpam-7135	86	5	non	non	ADJ
ejpam-7135	86	6	-	-	ADJ
ejpam-7135	86	7	empty	empty	ADJ
ejpam-7135	86	8	subset	subset	NOUN
ejpam-7135	86	9	i	i	PRON
ejpam-7135	86	10	of	of	ADP
ejpam-7135	86	11	a	a	DET
ejpam-7135	86	12	pdl	pdl	NOUN
ejpam-7135	86	13	l	l	NOUN
ejpam-7135	86	14	is	be	AUX
ejpam-7135	86	15	said	say	VERB
ejpam-7135	86	16	to	to	PART
ejpam-7135	86	17	be	be	AUX
ejpam-7135	86	18	an	an	DET
ejpam-7135	86	19	ideal	ideal	NOUN
ejpam-7135	86	20	if	if	SCONJ
ejpam-7135	86	21	it	it	PRON
ejpam-7135	86	22	satisfies	satisfy	VERB
ejpam-7135	86	23	the	the	DET
ejpam-7135	86	24	following	follow	VERB
ejpam-7135	86	25	:	:	PUNCT
ejpam-7135	86	26	x	x	X
ejpam-7135	86	27	,	,	PUNCT
ejpam-7135	86	28	y	y	PROPN
ejpam-7135	86	29	∈	∈	PROPN
ejpam-7135	86	30	i	i	PRON
ejpam-7135	86	31	⇒	⇒	VERB
ejpam-7135	86	32	x	x	PUNCT
ejpam-7135	86	33	∨	∨	NUM
ejpam-7135	86	34	y	y	PROPN
ejpam-7135	86	35	∈	∈	PROPN
ejpam-7135	87	1	i	i	PRON
ejpam-7135	87	2	,	,	PUNCT
ejpam-7135	87	3	x	x	PUNCT
ejpam-7135	87	4	∈	∈	PROPN
ejpam-7135	87	5	i	i	PRON
ejpam-7135	87	6	,	,	PUNCT
ejpam-7135	87	7	a	a	DET
ejpam-7135	87	8	∈	∈	PROPN
ejpam-7135	87	9	l	l	NOUN
ejpam-7135	87	10	⇒	⇒	NOUN
ejpam-7135	87	11	x	x	PUNCT
ejpam-7135	87	12	∧	∧	PROPN
ejpam-7135	87	13	a	a	DET
ejpam-7135	87	14	∈	∈	PROPN
ejpam-7135	87	15	i.	i.	PROPN
ejpam-7135	87	16	r.	r.	PROPN
ejpam-7135	87	17	bandaru	bandaru	PROPN
ejpam-7135	87	18	et	et	PROPN
ejpam-7135	87	19	al	al	PROPN
ejpam-7135	87	20	.	.	PUNCT
ejpam-7135	87	21	/	/	SYM
ejpam-7135	87	22	eur	eur	PROPN
ejpam-7135	87	23	.	.	PUNCT
ejpam-7135	88	1	j.	j.	PROPN
ejpam-7135	88	2	pure	pure	PROPN
ejpam-7135	88	3	appl	appl	PROPN
ejpam-7135	88	4	.	.	PROPN
ejpam-7135	88	5	math	math	PROPN
ejpam-7135	88	6	,	,	PUNCT
ejpam-7135	88	7	18	18	NUM
ejpam-7135	88	8	(	(	PUNCT
ejpam-7135	88	9	4	4	NUM
ejpam-7135	88	10	)	)	PUNCT
ejpam-7135	88	11	(	(	PUNCT
ejpam-7135	88	12	2025	2025	NUM
ejpam-7135	88	13	)	)	PUNCT
ejpam-7135	88	14	,	,	PUNCT
ejpam-7135	88	15	7135	7135	NUM
ejpam-7135	88	16	5	5	NUM
ejpam-7135	88	17	of	of	ADP
ejpam-7135	88	18	14	14	NUM
ejpam-7135	88	19	theorem	theorem	NOUN
ejpam-7135	88	20	4	4	NUM
ejpam-7135	88	21	.	.	PUNCT
ejpam-7135	89	1	[	[	X
ejpam-7135	89	2	12	12	NUM
ejpam-7135	89	3	]	]	PUNCT
ejpam-7135	89	4	let	let	VERB
ejpam-7135	89	5	s	s	PRON
ejpam-7135	89	6	be	be	AUX
ejpam-7135	89	7	a	a	DET
ejpam-7135	89	8	non	non	ADJ
ejpam-7135	89	9	-	-	ADJ
ejpam-7135	89	10	empty	empty	ADJ
ejpam-7135	89	11	subset	subset	NOUN
ejpam-7135	89	12	of	of	ADP
ejpam-7135	89	13	l.	l.	PROPN
ejpam-7135	89	14	then	then	ADV
ejpam-7135	89	15	(	(	PUNCT
ejpam-7135	89	16	s	s	X
ejpam-7135	89	17	]	]	X
ejpam-7135	89	18	=	=	X
ejpam-7135	89	19	{	{	PUNCT
ejpam-7135	89	20	(	(	PUNCT
ejpam-7135	89	21	n	n	NUM
ejpam-7135	89	22	∨	∨	NUM
ejpam-7135	89	23	i=1	i=1	PROPN
ejpam-7135	89	24	si	si	ADJ
ejpam-7135	89	25	)	)	PUNCT
ejpam-7135	89	26	∧	∧	NOUN
ejpam-7135	89	27	x	x	INTJ
ejpam-7135	89	28	|	|	ADV
ejpam-7135	89	29	si	si	PROPN
ejpam-7135	89	30	∈	∈	PROPN
ejpam-7135	89	31	s	s	PROPN
ejpam-7135	89	32	,	,	PUNCT
ejpam-7135	89	33	x	x	SYM
ejpam-7135	89	34	∈	∈	PROPN
ejpam-7135	89	35	l	l	NOUN
ejpam-7135	89	36	,	,	PUNCT
ejpam-7135	89	37	n	n	X
ejpam-7135	89	38	is	be	AUX
ejpam-7135	89	39	a	a	DET
ejpam-7135	89	40	positive	positive	ADJ
ejpam-7135	89	41	integer	integer	NOUN
ejpam-7135	89	42	}	}	PUNCT
ejpam-7135	89	43	is	be	AUX
ejpam-7135	89	44	the	the	DET
ejpam-7135	89	45	smallest	small	ADJ
ejpam-7135	89	46	ideal	ideal	NOUN
ejpam-7135	89	47	of	of	ADP
ejpam-7135	89	48	l	l	NOUN
ejpam-7135	89	49	containing	contain	VERB
ejpam-7135	89	50	s.	s.	PROPN
ejpam-7135	89	51	note	note	VERB
ejpam-7135	89	52	that	that	SCONJ
ejpam-7135	89	53	if	if	SCONJ
ejpam-7135	89	54	s	s	VERB
ejpam-7135	89	55	=	=	X
ejpam-7135	89	56	{	{	PUNCT
ejpam-7135	89	57	x	x	NOUN
ejpam-7135	89	58	}	}	PUNCT
ejpam-7135	89	59	,	,	PUNCT
ejpam-7135	89	60	then	then	ADV
ejpam-7135	89	61	we	we	PRON
ejpam-7135	89	62	write	write	VERB
ejpam-7135	89	63	(	(	PUNCT
ejpam-7135	89	64	s	s	X
ejpam-7135	89	65	]	]	X
ejpam-7135	89	66	=	=	SYM
ejpam-7135	89	67	(	(	PUNCT
ejpam-7135	89	68	x	x	X
ejpam-7135	89	69	]	]	X
ejpam-7135	89	70	,	,	PUNCT
ejpam-7135	89	71	the	the	DET
ejpam-7135	89	72	principal	principal	ADJ
ejpam-7135	89	73	ideal	ideal	NOUN
ejpam-7135	89	74	of	of	ADP
ejpam-7135	89	75	l	l	NOUN
ejpam-7135	89	76	generated	generate	VERB
ejpam-7135	89	77	by	by	ADP
ejpam-7135	89	78	x.	x.	PROPN
ejpam-7135	89	79	hence	hence	ADV
ejpam-7135	89	80	,	,	PUNCT
ejpam-7135	89	81	(	(	PUNCT
ejpam-7135	89	82	x	x	X
ejpam-7135	89	83	]	]	X
ejpam-7135	89	84	=	=	SYM
ejpam-7135	89	85	{	{	PUNCT
ejpam-7135	89	86	x	x	PUNCT
ejpam-7135	89	87	∧	∧	NOUN
ejpam-7135	89	88	x	x	PUNCT
ejpam-7135	90	1	|	|	ADV
ejpam-7135	90	2	x	x	X
ejpam-7135	90	3	∈	∈	PROPN
ejpam-7135	90	4	l	l	NOUN
ejpam-7135	90	5	}	}	PUNCT
ejpam-7135	90	6	.	.	PUNCT
ejpam-7135	91	1	according	accord	VERB
ejpam-7135	91	2	to	to	ADP
ejpam-7135	91	3	corollary	corollary	ADJ
ejpam-7135	91	4	5	5	NUM
ejpam-7135	91	5	,	,	PUNCT
ejpam-7135	91	6	lemma	lemma	PROPN
ejpam-7135	91	7	11	11	NUM
ejpam-7135	91	8	,	,	PUNCT
ejpam-7135	91	9	and	and	CCONJ
ejpam-7135	91	10	corollary	corollary	ADJ
ejpam-7135	91	11	6	6	NUM
ejpam-7135	91	12	of	of	ADP
ejpam-7135	91	13	[	[	X
ejpam-7135	91	14	12	12	NUM
ejpam-7135	91	15	]	]	PUNCT
ejpam-7135	91	16	,	,	PUNCT
ejpam-7135	91	17	the	the	DET
ejpam-7135	91	18	following	follow	VERB
ejpam-7135	91	19	lemma	lemma	PROPN
ejpam-7135	91	20	holds	hold	VERB
ejpam-7135	91	21	.	.	PUNCT
ejpam-7135	92	1	lemma	lemma	PROPN
ejpam-7135	92	2	4	4	NUM
ejpam-7135	92	3	.	.	PUNCT
ejpam-7135	93	1	[	[	X
ejpam-7135	93	2	12	12	NUM
ejpam-7135	93	3	]	]	X
ejpam-7135	93	4	let	let	AUX
ejpam-7135	93	5	l	l	NOUN
ejpam-7135	93	6	be	be	AUX
ejpam-7135	93	7	a	a	DET
ejpam-7135	93	8	pdl	pdl	NOUN
ejpam-7135	94	1	and	and	CCONJ
ejpam-7135	94	2	i	i	PRON
ejpam-7135	94	3	be	be	VERB
ejpam-7135	94	4	an	an	DET
ejpam-7135	94	5	ideal	ideal	NOUN
ejpam-7135	94	6	of	of	ADP
ejpam-7135	94	7	l.	l.	PROPN
ejpam-7135	94	8	then	then	ADV
ejpam-7135	94	9	,	,	PUNCT
ejpam-7135	94	10	for	for	ADP
ejpam-7135	94	11	any	any	DET
ejpam-7135	94	12	x	x	NOUN
ejpam-7135	94	13	,	,	PUNCT
ejpam-7135	94	14	y	y	PROPN
ejpam-7135	94	15	∈	∈	PROPN
ejpam-7135	94	16	l	l	NOUN
ejpam-7135	94	17	,	,	PUNCT
ejpam-7135	94	18	we	we	PRON
ejpam-7135	94	19	have	have	VERB
ejpam-7135	94	20	the	the	DET
ejpam-7135	94	21	following	following	NOUN
ejpam-7135	94	22	:	:	PUNCT
ejpam-7135	94	23	(	(	PUNCT
ejpam-7135	94	24	1	1	X
ejpam-7135	94	25	)	)	PUNCT
ejpam-7135	94	26	x	x	SYM
ejpam-7135	94	27	∈	∈	PROPN
ejpam-7135	94	28	(	(	PUNCT
ejpam-7135	94	29	y	y	NOUN
ejpam-7135	94	30	]	]	X
ejpam-7135	94	31	if	if	SCONJ
ejpam-7135	94	32	and	and	CCONJ
ejpam-7135	94	33	only	only	ADV
ejpam-7135	94	34	if	if	SCONJ
ejpam-7135	94	35	x	x	NOUN
ejpam-7135	94	36	=	=	VERB
ejpam-7135	94	37	y	y	PROPN
ejpam-7135	94	38	∧	∧	PROPN
ejpam-7135	94	39	x	x	X
ejpam-7135	94	40	,	,	PUNCT
ejpam-7135	94	41	(	(	PUNCT
ejpam-7135	94	42	2	2	NUM
ejpam-7135	94	43	)	)	PUNCT
ejpam-7135	94	44	x	x	X
ejpam-7135	95	1	∧	∧	NOUN
ejpam-7135	95	2	y	y	PROPN
ejpam-7135	95	3	∈	∈	PROPN
ejpam-7135	96	1	i	i	PRON
ejpam-7135	96	2	if	if	SCONJ
ejpam-7135	96	3	and	and	CCONJ
ejpam-7135	96	4	only	only	ADV
ejpam-7135	96	5	if	if	SCONJ
ejpam-7135	96	6	y	y	PROPN
ejpam-7135	96	7	∧	∧	PROPN
ejpam-7135	96	8	x	x	SYM
ejpam-7135	96	9	∈	∈	PROPN
ejpam-7135	97	1	i	i	PRON
ejpam-7135	97	2	,	,	PUNCT
ejpam-7135	97	3	(	(	PUNCT
ejpam-7135	97	4	3	3	X
ejpam-7135	97	5	)	)	PUNCT
ejpam-7135	97	6	(	(	PUNCT
ejpam-7135	97	7	x	x	PUNCT
ejpam-7135	97	8	∧	∧	NOUN
ejpam-7135	97	9	y	y	PROPN
ejpam-7135	97	10	]	]	X
ejpam-7135	97	11	=	=	SYM
ejpam-7135	97	12	(	(	PUNCT
ejpam-7135	97	13	y	y	PROPN
ejpam-7135	97	14	∧	∧	PROPN
ejpam-7135	97	15	x	x	X
ejpam-7135	97	16	]	]	X
ejpam-7135	97	17	=	=	SYM
ejpam-7135	97	18	(	(	PUNCT
ejpam-7135	97	19	x	x	SYM
ejpam-7135	97	20	]	]	X
ejpam-7135	97	21	∧	∧	PROPN
ejpam-7135	97	22	(	(	PUNCT
ejpam-7135	97	23	y	y	NOUN
ejpam-7135	97	24	]	]	PUNCT
ejpam-7135	97	25	.	.	PUNCT
ejpam-7135	97	26	theorem	theorem	ADJ
ejpam-7135	97	27	5	5	NUM
ejpam-7135	97	28	.	.	PUNCT
ejpam-7135	98	1	[	[	X
ejpam-7135	98	2	12	12	NUM
ejpam-7135	98	3	]	]	PUNCT
ejpam-7135	98	4	the	the	DET
ejpam-7135	98	5	collection	collection	NOUN
ejpam-7135	98	6	i(l	i(l	PROPN
ejpam-7135	98	7	)	)	PUNCT
ejpam-7135	98	8	of	of	ADP
ejpam-7135	98	9	all	all	DET
ejpam-7135	98	10	ideals	ideal	NOUN
ejpam-7135	98	11	of	of	ADP
ejpam-7135	98	12	a	a	DET
ejpam-7135	98	13	pdl	pdl	NOUN
ejpam-7135	98	14	l	l	NOUN
ejpam-7135	98	15	forms	form	NOUN
ejpam-7135	98	16	a	a	DET
ejpam-7135	98	17	distributive	distributive	ADJ
ejpam-7135	98	18	lattice	lattice	NOUN
ejpam-7135	98	19	under	under	ADP
ejpam-7135	98	20	set	set	ADJ
ejpam-7135	98	21	inclusion	inclusion	NOUN
ejpam-7135	98	22	,	,	PUNCT
ejpam-7135	98	23	in	in	ADP
ejpam-7135	98	24	which	which	PRON
ejpam-7135	98	25	,	,	PUNCT
ejpam-7135	98	26	the	the	DET
ejpam-7135	98	27	glb	glb	NOUN
ejpam-7135	98	28	and	and	CCONJ
ejpam-7135	98	29	lub	lub	NOUN
ejpam-7135	98	30	of	of	ADP
ejpam-7135	98	31	any	any	DET
ejpam-7135	98	32	two	two	NUM
ejpam-7135	98	33	ideals	ideal	NOUN
ejpam-7135	98	34	i	i	PRON
ejpam-7135	98	35	and	and	CCONJ
ejpam-7135	98	36	j	j	PROPN
ejpam-7135	98	37	are	be	AUX
ejpam-7135	98	38	given	give	VERB
ejpam-7135	98	39	by	by	ADP
ejpam-7135	98	40	i	i	PROPN
ejpam-7135	98	41	∧	∧	PROPN
ejpam-7135	98	42	j	j	PROPN
ejpam-7135	99	1	=	=	SYM
ejpam-7135	99	2	i	i	PROPN
ejpam-7135	99	3	∩	∩	PROPN
ejpam-7135	99	4	j	j	PROPN
ejpam-7135	99	5	and	and	CCONJ
ejpam-7135	99	6	i	i	PROPN
ejpam-7135	99	7	∨	∨	PROPN
ejpam-7135	99	8	j	j	PROPN
ejpam-7135	99	9	=	=	PRON
ejpam-7135	99	10	{	{	PUNCT
ejpam-7135	99	11	x	x	PROPN
ejpam-7135	99	12	∨	∨	NUM
ejpam-7135	99	13	y	y	NOUN
ejpam-7135	100	1	|	|	ADV
ejpam-7135	100	2	x	x	SYM
ejpam-7135	100	3	∈	∈	PROPN
ejpam-7135	101	1	i	i	PRON
ejpam-7135	101	2	and	and	CCONJ
ejpam-7135	101	3	y	y	PROPN
ejpam-7135	101	4	∈	∈	PROPN
ejpam-7135	101	5	j	j	PROPN
ejpam-7135	101	6	}	}	PUNCT
ejpam-7135	101	7	,	,	PUNCT
ejpam-7135	101	8	respectively	respectively	ADV
ejpam-7135	101	9	.	.	PUNCT
ejpam-7135	102	1	a	a	DET
ejpam-7135	102	2	proper	proper	ADJ
ejpam-7135	102	3	filter(ideal	filter(ideal	NOUN
ejpam-7135	102	4	)	)	PUNCT
ejpam-7135	103	1	p	p	NOUN
ejpam-7135	103	2	of	of	ADP
ejpam-7135	103	3	l	l	NOUN
ejpam-7135	103	4	is	be	AUX
ejpam-7135	103	5	said	say	VERB
ejpam-7135	103	6	to	to	PART
ejpam-7135	103	7	be	be	AUX
ejpam-7135	103	8	a	a	DET
ejpam-7135	103	9	prime	prime	ADJ
ejpam-7135	103	10	filter(ideal	filter(ideal	NOUN
ejpam-7135	103	11	)	)	PUNCT
ejpam-7135	103	12	if	if	SCONJ
ejpam-7135	103	13	for	for	ADP
ejpam-7135	103	14	any	any	DET
ejpam-7135	103	15	x	x	NOUN
ejpam-7135	103	16	,	,	PUNCT
ejpam-7135	103	17	y	y	PROPN
ejpam-7135	103	18	∈	∈	PROPN
ejpam-7135	103	19	l	l	NOUN
ejpam-7135	103	20	,	,	PUNCT
ejpam-7135	103	21	x	x	PROPN
ejpam-7135	103	22	∨	∨	NUM
ejpam-7135	103	23	y	y	PROPN
ejpam-7135	103	24	∈	∈	PROPN
ejpam-7135	103	25	p	p	X
ejpam-7135	103	26	(	(	PUNCT
ejpam-7135	103	27	x	x	PROPN
ejpam-7135	103	28	∧	∧	NOUN
ejpam-7135	103	29	y	y	PROPN
ejpam-7135	103	30	∈	∈	PROPN
ejpam-7135	103	31	p	p	NOUN
ejpam-7135	103	32	)	)	PUNCT
ejpam-7135	103	33	⇒	⇒	NOUN
ejpam-7135	103	34	x	x	X
ejpam-7135	103	35	∈	∈	PROPN
ejpam-7135	103	36	p	p	NOUN
ejpam-7135	103	37	or	or	CCONJ
ejpam-7135	103	38	y	y	PROPN
ejpam-7135	103	39	∈	∈	PROPN
ejpam-7135	103	40	p	p	PROPN
ejpam-7135	103	41	.	.	PUNCT
ejpam-7135	104	1	a	a	DET
ejpam-7135	104	2	proper	proper	ADJ
ejpam-7135	104	3	filter(ideal	filter(ideal	NOUN
ejpam-7135	104	4	)	)	PUNCT
ejpam-7135	104	5	m	m	PROPN
ejpam-7135	104	6	of	of	ADP
ejpam-7135	104	7	l	l	NOUN
ejpam-7135	104	8	is	be	AUX
ejpam-7135	104	9	said	say	VERB
ejpam-7135	104	10	to	to	PART
ejpam-7135	104	11	be	be	AUX
ejpam-7135	104	12	maximal	maximal	ADJ
ejpam-7135	104	13	if	if	SCONJ
ejpam-7135	104	14	it	it	PRON
ejpam-7135	104	15	is	be	AUX
ejpam-7135	104	16	not	not	PART
ejpam-7135	104	17	properly	properly	ADV
ejpam-7135	104	18	contained	contain	VERB
ejpam-7135	104	19	in	in	ADP
ejpam-7135	104	20	any	any	DET
ejpam-7135	104	21	proper	proper	ADJ
ejpam-7135	104	22	filter(ideal	filter(ideal	NOUN
ejpam-7135	104	23	)	)	PUNCT
ejpam-7135	104	24	of	of	ADP
ejpam-7135	104	25	l.	l.	PROPN
ejpam-7135	104	26	a	a	DET
ejpam-7135	104	27	prime	prime	ADJ
ejpam-7135	104	28	filter	filter	NOUN
ejpam-7135	104	29	p	p	NOUN
ejpam-7135	104	30	of	of	ADP
ejpam-7135	104	31	l	l	NOUN
ejpam-7135	104	32	is	be	AUX
ejpam-7135	104	33	said	say	VERB
ejpam-7135	104	34	to	to	PART
ejpam-7135	104	35	be	be	AUX
ejpam-7135	104	36	minimal	minimal	ADJ
ejpam-7135	104	37	if	if	SCONJ
ejpam-7135	104	38	it	it	PRON
ejpam-7135	104	39	is	be	AUX
ejpam-7135	104	40	minimal	minimal	ADJ
ejpam-7135	104	41	among	among	ADP
ejpam-7135	104	42	all	all	DET
ejpam-7135	104	43	the	the	DET
ejpam-7135	104	44	prime	prime	ADJ
ejpam-7135	104	45	filters	filter	NOUN
ejpam-7135	104	46	of	of	ADP
ejpam-7135	104	47	l.	l.	PROPN
ejpam-7135	104	48	a	a	DET
ejpam-7135	104	49	prime	prime	ADJ
ejpam-7135	104	50	filter	filter	NOUN
ejpam-7135	104	51	p	p	NOUN
ejpam-7135	104	52	is	be	AUX
ejpam-7135	104	53	said	say	VERB
ejpam-7135	104	54	to	to	PART
ejpam-7135	104	55	be	be	AUX
ejpam-7135	104	56	a	a	DET
ejpam-7135	104	57	minimal	minimal	ADJ
ejpam-7135	104	58	prime	prime	ADJ
ejpam-7135	104	59	filter	filter	NOUN
ejpam-7135	104	60	belonging	belong	VERB
ejpam-7135	104	61	to	to	ADP
ejpam-7135	104	62	a	a	DET
ejpam-7135	104	63	filter	filter	NOUN
ejpam-7135	105	1	i	i	PRON
ejpam-7135	105	2	if	if	SCONJ
ejpam-7135	105	3	it	it	PRON
ejpam-7135	105	4	is	be	AUX
ejpam-7135	105	5	minimal	minimal	ADJ
ejpam-7135	105	6	among	among	ADP
ejpam-7135	105	7	all	all	DET
ejpam-7135	105	8	the	the	DET
ejpam-7135	105	9	prime	prime	ADJ
ejpam-7135	105	10	filters	filter	NOUN
ejpam-7135	105	11	of	of	ADP
ejpam-7135	105	12	l	l	NOUN
ejpam-7135	105	13	containing	contain	VERB
ejpam-7135	105	14	i.	i.	PROPN
ejpam-7135	105	15	a	a	DET
ejpam-7135	105	16	prime	prime	ADJ
ejpam-7135	105	17	filter	filter	NOUN
ejpam-7135	105	18	p	p	NOUN
ejpam-7135	105	19	of	of	ADP
ejpam-7135	105	20	l	l	NOUN
ejpam-7135	105	21	is	be	AUX
ejpam-7135	105	22	a	a	DET
ejpam-7135	105	23	minimal	minimal	ADJ
ejpam-7135	105	24	prime	prime	ADJ
ejpam-7135	105	25	filter	filter	NOUN
ejpam-7135	105	26	if	if	SCONJ
ejpam-7135	105	27	and	and	CCONJ
ejpam-7135	105	28	only	only	ADV
ejpam-7135	105	29	if	if	SCONJ
ejpam-7135	105	30	for	for	ADP
ejpam-7135	105	31	each	each	DET
ejpam-7135	105	32	x	x	SYM
ejpam-7135	105	33	∈	∈	PROPN
ejpam-7135	105	34	p	p	NOUN
ejpam-7135	105	35	,	,	PUNCT
ejpam-7135	105	36	there	there	PRON
ejpam-7135	105	37	exists	exist	VERB
ejpam-7135	105	38	y	y	PROPN
ejpam-7135	105	39	/∈	/∈	PUNCT
ejpam-7135	106	1	p	p	X
ejpam-7135	107	1	such	such	ADJ
ejpam-7135	107	2	that	that	SCONJ
ejpam-7135	107	3	x	x	PROPN
ejpam-7135	107	4	∨	∨	NUM
ejpam-7135	107	5	y	y	NOUN
ejpam-7135	107	6	=	=	SYM
ejpam-7135	107	7	1	1	X
ejpam-7135	107	8	.	.	PUNCT
ejpam-7135	107	9	lemma	lemma	PROPN
ejpam-7135	107	10	5	5	NUM
ejpam-7135	107	11	.	.	PUNCT
ejpam-7135	108	1	[	[	X
ejpam-7135	108	2	14	14	NUM
ejpam-7135	108	3	]	]	PUNCT
ejpam-7135	108	4	let	let	AUX
ejpam-7135	108	5	l	l	NOUN
ejpam-7135	108	6	be	be	AUX
ejpam-7135	108	7	a	a	DET
ejpam-7135	108	8	pdl	pdl	NOUN
ejpam-7135	108	9	and	and	CCONJ
ejpam-7135	108	10	a	a	DET
ejpam-7135	108	11	⊆	⊆	NUM
ejpam-7135	108	12	l.	l.	NOUN
ejpam-7135	108	13	then	then	ADV
ejpam-7135	108	14	(	(	PUNCT
ejpam-7135	108	15	1	1	X
ejpam-7135	108	16	)	)	PUNCT
ejpam-7135	108	17	a•	a•	NOUN
ejpam-7135	108	18	=	=	PUNCT
ejpam-7135	108	19	{	{	PUNCT
ejpam-7135	108	20	t	t	NOUN
ejpam-7135	108	21	∈	∈	PROPN
ejpam-7135	108	22	l	l	NOUN
ejpam-7135	109	1	|	|	NOUN
ejpam-7135	109	2	t	t	X
ejpam-7135	109	3	∨	∨	NOUN
ejpam-7135	109	4	x	x	X
ejpam-7135	109	5	=	=	SYM
ejpam-7135	109	6	1	1	NUM
ejpam-7135	109	7	for	for	ADP
ejpam-7135	109	8	all	all	DET
ejpam-7135	109	9	x	x	SYM
ejpam-7135	109	10	∈	∈	PROPN
ejpam-7135	109	11	a	a	PRON
ejpam-7135	109	12	}	}	PUNCT
ejpam-7135	109	13	is	be	AUX
ejpam-7135	109	14	a	a	DET
ejpam-7135	109	15	filter	filter	NOUN
ejpam-7135	109	16	of	of	ADP
ejpam-7135	109	17	l.	l.	PROPN
ejpam-7135	109	18	(	(	PUNCT
ejpam-7135	109	19	2	2	NUM
ejpam-7135	109	20	)	)	PUNCT
ejpam-7135	109	21	for	for	ADP
ejpam-7135	109	22	any	any	DET
ejpam-7135	109	23	x	x	NOUN
ejpam-7135	109	24	,	,	PUNCT
ejpam-7135	109	25	y	y	PROPN
ejpam-7135	109	26	∈	∈	PROPN
ejpam-7135	109	27	l	l	NOUN
ejpam-7135	109	28	,	,	PUNCT
ejpam-7135	110	1	[	[	X
ejpam-7135	110	2	x	x	X
ejpam-7135	110	3	∧	∧	NOUN
ejpam-7135	110	4	y]•	y]•	NOUN
ejpam-7135	110	5	=	=	PUNCT
ejpam-7135	111	1	[	[	X
ejpam-7135	111	2	x]•	x]•	X
ejpam-7135	111	3	∩	∩	PROPN
ejpam-7135	111	4	[	[	X
ejpam-7135	111	5	y]•	y]•	NOUN
ejpam-7135	111	6	,	,	PUNCT
ejpam-7135	111	7	where	where	SCONJ
ejpam-7135	111	8	[	[	X
ejpam-7135	111	9	x	x	X
ejpam-7135	111	10	∧	∧	NOUN
ejpam-7135	111	11	y]•	y]•	NOUN
ejpam-7135	111	12	=	=	PROPN
ejpam-7135	111	13	{	{	PUNCT
ejpam-7135	111	14	t	t	NOUN
ejpam-7135	111	15	∈	∈	PROPN
ejpam-7135	111	16	l	l	NOUN
ejpam-7135	112	1	|	|	NOUN
ejpam-7135	112	2	t	t	PROPN
ejpam-7135	112	3	∨	∨	NUM
ejpam-7135	112	4	(	(	PUNCT
ejpam-7135	112	5	x	x	PROPN
ejpam-7135	112	6	∧	∧	PROPN
ejpam-7135	112	7	y	y	NOUN
ejpam-7135	112	8	)	)	PUNCT
ejpam-7135	112	9	=	=	PUNCT
ejpam-7135	112	10	1	1	NUM
ejpam-7135	112	11	}	}	PUNCT
ejpam-7135	112	12	.	.	PUNCT
ejpam-7135	113	1	(	(	PUNCT
ejpam-7135	113	2	3	3	X
ejpam-7135	113	3	)	)	PUNCT
ejpam-7135	113	4	for	for	ADP
ejpam-7135	113	5	any	any	DET
ejpam-7135	113	6	x	x	NOUN
ejpam-7135	113	7	,	,	PUNCT
ejpam-7135	113	8	y	y	PROPN
ejpam-7135	113	9	∈	∈	PROPN
ejpam-7135	113	10	l	l	NOUN
ejpam-7135	113	11	,	,	PUNCT
ejpam-7135	113	12	[	[	X
ejpam-7135	113	13	x	x	X
ejpam-7135	113	14	∨	∨	NUM
ejpam-7135	113	15	y]•	y]•	NOUN
ejpam-7135	113	16	•	•	NOUN
ejpam-7135	113	17	=	=	PUNCT
ejpam-7135	114	1	[	[	X
ejpam-7135	114	2	x]•	x]•	X
ejpam-7135	114	3	•	•	NOUN
ejpam-7135	114	4	∩	∩	NOUN
ejpam-7135	115	1	[	[	X
ejpam-7135	116	1	y]•	y]•	NOUN
ejpam-7135	116	2	•	•	NOUN
ejpam-7135	116	3	,	,	PUNCT
ejpam-7135	116	4	where	where	SCONJ
ejpam-7135	116	5	[	[	X
ejpam-7135	116	6	x	x	X
ejpam-7135	116	7	∧	∧	NOUN
ejpam-7135	116	8	y]•	y]•	NOUN
ejpam-7135	116	9	•	•	NOUN
ejpam-7135	116	10	=	=	PUNCT
ejpam-7135	116	11	{	{	PUNCT
ejpam-7135	116	12	t	t	NOUN
ejpam-7135	116	13	∈	∈	PROPN
ejpam-7135	116	14	l	l	NOUN
ejpam-7135	117	1	|	|	NOUN
ejpam-7135	117	2	t	t	X
ejpam-7135	117	3	∨	∨	NOUN
ejpam-7135	117	4	z	z	X
ejpam-7135	117	5	=	=	SYM
ejpam-7135	117	6	1	1	NUM
ejpam-7135	117	7	for	for	ADP
ejpam-7135	117	8	all	all	DET
ejpam-7135	117	9	z	z	NOUN
ejpam-7135	117	10	∈	∈	PROPN
ejpam-7135	118	1	[	[	X
ejpam-7135	118	2	x	x	X
ejpam-7135	118	3	∧	∧	PROPN
ejpam-7135	118	4	y]•	y]•	PROPN
ejpam-7135	118	5	}	}	PUNCT
ejpam-7135	118	6	.	.	PUNCT
ejpam-7135	119	1	definition	definition	NOUN
ejpam-7135	119	2	6	6	NUM
ejpam-7135	119	3	.	.	PUNCT
ejpam-7135	120	1	[	[	X
ejpam-7135	120	2	14	14	NUM
ejpam-7135	120	3	]	]	X
ejpam-7135	120	4	let	let	NOUN
ejpam-7135	120	5	(	(	PUNCT
ejpam-7135	120	6	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	120	7	,	,	PUNCT
ejpam-7135	120	8	1	1	NUM
ejpam-7135	120	9	)	)	PUNCT
ejpam-7135	120	10	be	be	AUX
ejpam-7135	120	11	a	a	DET
ejpam-7135	120	12	paradistributive	paradistributive	ADJ
ejpam-7135	120	13	latticoid	latticoid	NOUN
ejpam-7135	120	14	(	(	PUNCT
ejpam-7135	120	15	pdl	pdl	NOUN
ejpam-7135	120	16	)	)	PUNCT
ejpam-7135	120	17	and	and	CCONJ
ejpam-7135	120	18	consider	consider	VERB
ejpam-7135	120	19	a	a	DET
ejpam-7135	120	20	unary	unary	ADJ
ejpam-7135	120	21	operation	operation	NOUN
ejpam-7135	120	22	denoted	denote	VERB
ejpam-7135	120	23	as	as	ADP
ejpam-7135	120	24	x	x	PROPN
ejpam-7135	120	25	7→	7→	NUM
ejpam-7135	120	26	x	x	SYM
ejpam-7135	120	27	♦	♦	PROPN
ejpam-7135	120	28	on	on	ADP
ejpam-7135	120	29	l.	l.	PROPN
ejpam-7135	120	30	this	this	DET
ejpam-7135	120	31	operation	operation	NOUN
ejpam-7135	120	32	is	be	AUX
ejpam-7135	120	33	called	call	VERB
ejpam-7135	120	34	a	a	DET
ejpam-7135	120	35	parapseudocomplementation	parapseudocomplementation	NOUN
ejpam-7135	120	36	on	on	ADP
ejpam-7135	120	37	l	l	NOUN
ejpam-7135	120	38	if	if	SCONJ
ejpam-7135	120	39	it	it	PRON
ejpam-7135	120	40	satisfies	satisfy	VERB
ejpam-7135	120	41	the	the	DET
ejpam-7135	120	42	following	follow	VERB
ejpam-7135	120	43	conditions	condition	NOUN
ejpam-7135	120	44	:	:	PUNCT
ejpam-7135	120	45	(	(	PUNCT
ejpam-7135	120	46	ppc1	ppc1	PROPN
ejpam-7135	120	47	)	)	PUNCT
ejpam-7135	121	1	if	if	SCONJ
ejpam-7135	121	2	x	x	PROPN
ejpam-7135	121	3	∨	∨	NUM
ejpam-7135	121	4	y	y	NOUN
ejpam-7135	121	5	=	=	SYM
ejpam-7135	121	6	1	1	NUM
ejpam-7135	121	7	,	,	PUNCT
ejpam-7135	121	8	then	then	ADV
ejpam-7135	121	9	x	x	PROPN
ejpam-7135	121	10	∨	∨	PROPN
ejpam-7135	121	11	y	y	PROPN
ejpam-7135	121	12	♦	♦	PROPN
ejpam-7135	121	13	=	=	PROPN
ejpam-7135	121	14	x.	x.	PROPN
ejpam-7135	121	15	(	(	PUNCT
ejpam-7135	121	16	ppc2	ppc2	PROPN
ejpam-7135	121	17	)	)	PUNCT
ejpam-7135	121	18	x	x	PUNCT
ejpam-7135	121	19	∨	∨	PROPN
ejpam-7135	121	20	x	x	SYM
ejpam-7135	121	21	♦	♦	PROPN
ejpam-7135	121	22	=	=	PROPN
ejpam-7135	121	23	1	1	PROPN
ejpam-7135	121	24	.	.	PUNCT
ejpam-7135	122	1	(	(	PUNCT
ejpam-7135	122	2	ppc3	ppc3	ADJ
ejpam-7135	122	3	)	)	PUNCT
ejpam-7135	122	4	(	(	PUNCT
ejpam-7135	122	5	x	x	PUNCT
ejpam-7135	122	6	∧	∧	PROPN
ejpam-7135	122	7	y	y	PROPN
ejpam-7135	122	8	)	)	PUNCT
ejpam-7135	122	9	♦	♦	PROPN
ejpam-7135	122	10	=	=	PROPN
ejpam-7135	123	1	x	x	PROPN
ejpam-7135	123	2	♦	♦	PROPN
ejpam-7135	123	3	∨	∨	PROPN
ejpam-7135	123	4	y	y	PROPN
ejpam-7135	123	5	♦	♦	PROPN
ejpam-7135	123	6	.	.	PUNCT
ejpam-7135	124	1	r.	r.	PROPN
ejpam-7135	124	2	bandaru	bandaru	PROPN
ejpam-7135	124	3	et	et	PROPN
ejpam-7135	124	4	al	al	PROPN
ejpam-7135	124	5	.	.	PUNCT
ejpam-7135	124	6	/	/	SYM
ejpam-7135	124	7	eur	eur	PROPN
ejpam-7135	124	8	.	.	PUNCT
ejpam-7135	125	1	j.	j.	PROPN
ejpam-7135	125	2	pure	pure	PROPN
ejpam-7135	125	3	appl	appl	PROPN
ejpam-7135	125	4	.	.	PROPN
ejpam-7135	125	5	math	math	PROPN
ejpam-7135	125	6	,	,	PUNCT
ejpam-7135	125	7	18	18	NUM
ejpam-7135	125	8	(	(	PUNCT
ejpam-7135	125	9	4	4	NUM
ejpam-7135	125	10	)	)	PUNCT
ejpam-7135	125	11	(	(	PUNCT
ejpam-7135	125	12	2025	2025	NUM
ejpam-7135	125	13	)	)	PUNCT
ejpam-7135	125	14	,	,	PUNCT
ejpam-7135	125	15	7135	7135	NUM
ejpam-7135	125	16	6	6	NUM
ejpam-7135	125	17	of	of	ADP
ejpam-7135	125	18	14	14	NUM
ejpam-7135	125	19	definition	definition	NOUN
ejpam-7135	125	20	7	7	NUM
ejpam-7135	125	21	.	.	PUNCT
ejpam-7135	126	1	[	[	X
ejpam-7135	126	2	14	14	NUM
ejpam-7135	126	3	]	]	PUNCT
ejpam-7135	126	4	by	by	ADP
ejpam-7135	126	5	a	a	DET
ejpam-7135	126	6	homomorphism	homomorphism	NOUN
ejpam-7135	126	7	of	of	ADP
ejpam-7135	126	8	a	a	DET
ejpam-7135	126	9	pdl	pdl	NOUN
ejpam-7135	126	10	(	(	PUNCT
ejpam-7135	126	11	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	126	12	,	,	PUNCT
ejpam-7135	126	13	1	1	NUM
ejpam-7135	126	14	)	)	PUNCT
ejpam-7135	126	15	into	into	ADP
ejpam-7135	126	16	a	a	DET
ejpam-7135	126	17	pdl	pdl	NOUN
ejpam-7135	126	18	(	(	PUNCT
ejpam-7135	126	19	l′,∨′,∧′	l′,∨′,∧′	NUM
ejpam-7135	126	20	,	,	PUNCT
ejpam-7135	126	21	1′	1′	NUM
ejpam-7135	126	22	)	)	PUNCT
ejpam-7135	126	23	,	,	PUNCT
ejpam-7135	126	24	we	we	PRON
ejpam-7135	126	25	mean	mean	VERB
ejpam-7135	126	26	,	,	PUNCT
ejpam-7135	126	27	a	a	DET
ejpam-7135	126	28	mapping	mapping	NOUN
ejpam-7135	126	29	f	f	NOUN
ejpam-7135	126	30	:	:	PUNCT
ejpam-7135	127	1	l	l	X
ejpam-7135	127	2	→	→	SYM
ejpam-7135	127	3	l′	l′	AUX
ejpam-7135	127	4	satisfying	satisfy	VERB
ejpam-7135	127	5	the	the	DET
ejpam-7135	127	6	following	following	NOUN
ejpam-7135	127	7	:	:	PUNCT
ejpam-7135	127	8	(	(	PUNCT
ejpam-7135	127	9	1	1	X
ejpam-7135	127	10	)	)	PUNCT
ejpam-7135	127	11	f(a	f(a	NOUN
ejpam-7135	127	12	∨	∨	NUM
ejpam-7135	127	13	b	b	NOUN
ejpam-7135	127	14	)	)	PUNCT
ejpam-7135	127	15	=	=	SYM
ejpam-7135	127	16	f(a	f(a	PROPN
ejpam-7135	127	17	)	)	PUNCT
ejpam-7135	127	18	∨′	∨′	PROPN
ejpam-7135	127	19	f(b	f(b	PROPN
ejpam-7135	127	20	)	)	PUNCT
ejpam-7135	127	21	,	,	PUNCT
ejpam-7135	127	22	(	(	PUNCT
ejpam-7135	127	23	2	2	X
ejpam-7135	127	24	)	)	PUNCT
ejpam-7135	127	25	f(a	f(a	NOUN
ejpam-7135	127	26	∧	∧	PROPN
ejpam-7135	127	27	b	b	PROPN
ejpam-7135	127	28	)	)	PUNCT
ejpam-7135	127	29	=	=	SYM
ejpam-7135	127	30	f(a	f(a	NOUN
ejpam-7135	127	31	)	)	PUNCT
ejpam-7135	127	32	∧′	∧′	PROPN
ejpam-7135	127	33	f(b	f(b	NOUN
ejpam-7135	127	34	)	)	PUNCT
ejpam-7135	127	35	,	,	PUNCT
ejpam-7135	127	36	(	(	PUNCT
ejpam-7135	127	37	3	3	X
ejpam-7135	127	38	)	)	PUNCT
ejpam-7135	127	39	f(1	f(1	PROPN
ejpam-7135	127	40	)	)	PUNCT
ejpam-7135	127	41	=	=	SYM
ejpam-7135	127	42	f(1′	f(1′	PROPN
ejpam-7135	127	43	)	)	PUNCT
ejpam-7135	127	44	.	.	PUNCT
ejpam-7135	128	1	3	3	X
ejpam-7135	128	2	.	.	X
ejpam-7135	128	3	stone	stone	NOUN
ejpam-7135	128	4	pdl	pdl	PROPN
ejpam-7135	128	5	let	let	VERB
ejpam-7135	128	6	us	we	PRON
ejpam-7135	128	7	consider	consider	VERB
ejpam-7135	128	8	that	that	PRON
ejpam-7135	128	9	by	by	ADP
ejpam-7135	128	10	l	l	NOUN
ejpam-7135	128	11	we	we	PRON
ejpam-7135	128	12	mean	mean	VERB
ejpam-7135	128	13	a	a	DET
ejpam-7135	128	14	paradistributive	paradistributive	ADJ
ejpam-7135	128	15	latticoid	latticoid	NOUN
ejpam-7135	128	16	(	(	PUNCT
ejpam-7135	128	17	l,∧,∨,m	l,∧,∨,m	ADJ
ejpam-7135	128	18	,	,	PUNCT
ejpam-7135	128	19	1	1	NUM
ejpam-7135	128	20	)	)	PUNCT
ejpam-7135	128	21	with	with	ADP
ejpam-7135	128	22	a	a	DET
ejpam-7135	128	23	parapseudo	parapseudo	NOUN
ejpam-7135	128	24	-	-	PUNCT
ejpam-7135	128	25	complementation	complementation	NOUN
ejpam-7135	128	26	♦	♦	NOUN
ejpam-7135	128	27	,	,	PUNCT
ejpam-7135	128	28	1	1	NUM
ejpam-7135	128	29	is	be	AUX
ejpam-7135	128	30	the	the	DET
ejpam-7135	128	31	greatest	great	ADJ
ejpam-7135	128	32	element	element	NOUN
ejpam-7135	128	33	,	,	PUNCT
ejpam-7135	128	34	and	and	CCONJ
ejpam-7135	128	35	m	m	PROPN
ejpam-7135	128	36	is	be	AUX
ejpam-7135	128	37	a	a	DET
ejpam-7135	128	38	minimal	minimal	ADJ
ejpam-7135	128	39	element	element	NOUN
ejpam-7135	128	40	in	in	ADP
ejpam-7135	128	41	l	l	PROPN
ejpam-7135	128	42	,	,	PUNCT
ejpam-7135	128	43	until	until	SCONJ
ejpam-7135	128	44	otherwise	otherwise	ADV
ejpam-7135	128	45	specified	specify	VERB
ejpam-7135	128	46	.	.	PUNCT
ejpam-7135	129	1	definition	definition	NOUN
ejpam-7135	129	2	8	8	NUM
ejpam-7135	129	3	.	.	PUNCT
ejpam-7135	130	1	l	l	NOUN
ejpam-7135	130	2	with	with	ADP
ejpam-7135	130	3	a	a	DET
ejpam-7135	130	4	parapseudo	parapseudo	NOUN
ejpam-7135	130	5	-	-	PUNCT
ejpam-7135	130	6	complementation	complementation	NOUN
ejpam-7135	130	7	♦	♦	PROPN
ejpam-7135	130	8	said	say	VERB
ejpam-7135	130	9	to	to	PART
ejpam-7135	130	10	be	be	AUX
ejpam-7135	130	11	a	a	DET
ejpam-7135	130	12	stone	stone	NOUN
ejpam-7135	130	13	pdl	pdl	NOUN
ejpam-7135	130	14	if	if	SCONJ
ejpam-7135	130	15	x	x	PROPN
ejpam-7135	130	16	♦	♦	PROPN
ejpam-7135	130	17	∧	∧	PROPN
ejpam-7135	130	18	x	x	PROPN
ejpam-7135	130	19	♦	♦	PROPN
ejpam-7135	130	20	♦	♦	PROPN
ejpam-7135	130	21	=	=	PROPN
ejpam-7135	130	22	1	1	NUM
ejpam-7135	130	23	♦	♦	PROPN
ejpam-7135	130	24	for	for	ADP
ejpam-7135	130	25	all	all	DET
ejpam-7135	130	26	x	x	SYM
ejpam-7135	130	27	∈	∈	PROPN
ejpam-7135	130	28	l.	l.	PROPN
ejpam-7135	130	29	example	example	NOUN
ejpam-7135	130	30	2	2	X
ejpam-7135	130	31	.	.	X
ejpam-7135	131	1	let	let	VERB
ejpam-7135	131	2	(	(	PUNCT
ejpam-7135	131	3	l,+	l,+	NOUN
ejpam-7135	131	4	,	,	PUNCT
ejpam-7135	131	5	·	·	PUNCT
ejpam-7135	131	6	,	,	PUNCT
ejpam-7135	131	7	0	0	NUM
ejpam-7135	131	8	,	,	PUNCT
ejpam-7135	131	9	1	1	NUM
ejpam-7135	131	10	)	)	PUNCT
ejpam-7135	131	11	be	be	AUX
ejpam-7135	131	12	a	a	DET
ejpam-7135	131	13	commutative	commutative	ADJ
ejpam-7135	131	14	regular	regular	ADJ
ejpam-7135	131	15	ring	ring	NOUN
ejpam-7135	131	16	with	with	ADP
ejpam-7135	131	17	unity	unity	NOUN
ejpam-7135	131	18	.	.	PUNCT
ejpam-7135	132	1	for	for	ADP
ejpam-7135	132	2	any	any	DET
ejpam-7135	132	3	a	a	PRON
ejpam-7135	132	4	,	,	PUNCT
ejpam-7135	132	5	b	b	PROPN
ejpam-7135	132	6	∈	∈	PROPN
ejpam-7135	132	7	l	l	NOUN
ejpam-7135	132	8	,	,	PUNCT
ejpam-7135	132	9	define	define	VERB
ejpam-7135	132	10	a	a	DET
ejpam-7135	132	11	∨	∨	NOUN
ejpam-7135	132	12	b	b	NOUN
ejpam-7135	132	13	=	=	PUNCT
ejpam-7135	132	14	b0a	b0a	PROPN
ejpam-7135	132	15	,	,	PUNCT
ejpam-7135	132	16	a	a	DET
ejpam-7135	132	17	∧	∧	PROPN
ejpam-7135	132	18	b	b	NOUN
ejpam-7135	132	19	=	=	PRON
ejpam-7135	132	20	a+	a+	PUNCT
ejpam-7135	132	21	b−	b−	PROPN
ejpam-7135	132	22	b0a	b0a	PROPN
ejpam-7135	132	23	,	,	PUNCT
ejpam-7135	132	24	b	b	NOUN
ejpam-7135	132	25	♦	♦	PROPN
ejpam-7135	132	26	=	=	PROPN
ejpam-7135	132	27	1−	1−	NUM
ejpam-7135	132	28	b0	b0	NOUN
ejpam-7135	132	29	.	.	PUNCT
ejpam-7135	133	1	then	then	ADV
ejpam-7135	133	2	(	(	PUNCT
ejpam-7135	133	3	l,∨,∧	l,∨,∧	NOUN
ejpam-7135	133	4	,	,	PUNCT
ejpam-7135	133	5	1	1	NUM
ejpam-7135	133	6	)	)	PUNCT
ejpam-7135	133	7	is	be	AUX
ejpam-7135	133	8	a	a	DET
ejpam-7135	133	9	stone	stone	NOUN
ejpam-7135	133	10	pdl	pdl	NOUN
ejpam-7135	133	11	,	,	PUNCT
ejpam-7135	133	12	where	where	SCONJ
ejpam-7135	133	13	b0	b0	NOUN
ejpam-7135	133	14	is	be	AUX
ejpam-7135	133	15	the	the	DET
ejpam-7135	133	16	unique	unique	ADJ
ejpam-7135	133	17	idempotent	idempotent	ADJ
ejpam-7135	133	18	element	element	NOUN
ejpam-7135	133	19	in	in	ADP
ejpam-7135	133	20	l	l	PROPN
ejpam-7135	133	21	associated	associate	VERB
ejpam-7135	133	22	with	with	ADP
ejpam-7135	133	23	b	b	PROPN
ejpam-7135	133	24	such	such	ADJ
ejpam-7135	133	25	that	that	PRON
ejpam-7135	133	26	bl	bl	PROPN
ejpam-7135	133	27	=	=	SYM
ejpam-7135	133	28	b0l	b0l	PROPN
ejpam-7135	133	29	.	.	PUNCT
ejpam-7135	133	30	example	example	NOUN
ejpam-7135	134	1	3	3	X
ejpam-7135	134	2	.	.	PUNCT
ejpam-7135	134	3	let	let	VERB
ejpam-7135	134	4	a	a	PRON
ejpam-7135	134	5	be	be	AUX
ejpam-7135	134	6	a	a	DET
ejpam-7135	134	7	non	non	ADJ
ejpam-7135	134	8	-	-	ADJ
ejpam-7135	134	9	empty	empty	ADJ
ejpam-7135	134	10	set	set	NOUN
ejpam-7135	134	11	with	with	ADP
ejpam-7135	134	12	at	at	ADV
ejpam-7135	134	13	least	least	ADV
ejpam-7135	134	14	two	two	NUM
ejpam-7135	134	15	elements	element	NOUN
ejpam-7135	134	16	and	and	CCONJ
ejpam-7135	134	17	b	b	NOUN
ejpam-7135	134	18	any	any	DET
ejpam-7135	134	19	set	set	NOUN
ejpam-7135	134	20	.	.	PUNCT
ejpam-7135	135	1	choose	choose	VERB
ejpam-7135	135	2	p	p	NOUN
ejpam-7135	135	3	,	,	PUNCT
ejpam-7135	135	4	p0	p0	PROPN
ejpam-7135	135	5	∈	∈	PROPN
ejpam-7135	135	6	ab	ab	PROPN
ejpam-7135	135	7	such	such	ADJ
ejpam-7135	135	8	that	that	DET
ejpam-7135	135	9	p(t	p(t	NOUN
ejpam-7135	135	10	)	)	PUNCT
ejpam-7135	135	11	̸=	̸=	PROPN
ejpam-7135	135	12	p0(t	p0(t	NUM
ejpam-7135	135	13	)	)	PUNCT
ejpam-7135	135	14	,	,	PUNCT
ejpam-7135	135	15	for	for	ADP
ejpam-7135	135	16	all	all	DET
ejpam-7135	135	17	t	t	PROPN
ejpam-7135	135	18	∈	∈	PROPN
ejpam-7135	135	19	b.	b.	PROPN
ejpam-7135	135	20	for	for	ADP
ejpam-7135	135	21	any	any	DET
ejpam-7135	135	22	a	a	PRON
ejpam-7135	135	23	,	,	PUNCT
ejpam-7135	135	24	b	b	PROPN
ejpam-7135	135	25	∈	∈	PROPN
ejpam-7135	135	26	ab	ab	PROPN
ejpam-7135	135	27	and	and	CCONJ
ejpam-7135	135	28	t	t	PROPN
ejpam-7135	135	29	∈	∈	PROPN
ejpam-7135	135	30	b	b	PROPN
ejpam-7135	135	31	,	,	PUNCT
ejpam-7135	135	32	define	define	VERB
ejpam-7135	135	33	(	(	PUNCT
ejpam-7135	135	34	a	a	DET
ejpam-7135	135	35	∨	∨	NUM
ejpam-7135	135	36	b)(t	b)(t	NOUN
ejpam-7135	135	37	)	)	PUNCT
ejpam-7135	136	1	=	=	PRON
ejpam-7135	136	2	{	{	PUNCT
ejpam-7135	136	3	a(t	a(t	NOUN
ejpam-7135	136	4	)	)	PUNCT
ejpam-7135	136	5	if	if	SCONJ
ejpam-7135	136	6	b(t	b(t	NOUN
ejpam-7135	136	7	)	)	PUNCT
ejpam-7135	136	8	̸=	̸=	PROPN
ejpam-7135	136	9	p0(t	p0(t	PROPN
ejpam-7135	136	10	)	)	PUNCT
ejpam-7135	136	11	p0(t	p0(t	PROPN
ejpam-7135	136	12	)	)	PUNCT
ejpam-7135	136	13	if	if	SCONJ
ejpam-7135	136	14	b(t	b(t	NOUN
ejpam-7135	136	15	)	)	PUNCT
ejpam-7135	136	16	=	=	SYM
ejpam-7135	137	1	p0(t	p0(t	PROPN
ejpam-7135	137	2	)	)	PUNCT
ejpam-7135	137	3	(	(	PUNCT
ejpam-7135	137	4	a	a	DET
ejpam-7135	137	5	∧	∧	PROPN
ejpam-7135	137	6	b)(t	b)(t	NOUN
ejpam-7135	137	7	)	)	PUNCT
ejpam-7135	137	8	=	=	PRON
ejpam-7135	137	9	{	{	PUNCT
ejpam-7135	137	10	b(t	b(t	NOUN
ejpam-7135	137	11	)	)	PUNCT
ejpam-7135	137	12	if	if	SCONJ
ejpam-7135	137	13	b(t	b(t	NOUN
ejpam-7135	137	14	)	)	PUNCT
ejpam-7135	137	15	̸=	̸=	PROPN
ejpam-7135	137	16	p0(t	p0(t	NUM
ejpam-7135	137	17	)	)	PUNCT
ejpam-7135	137	18	a(t	a(t	NOUN
ejpam-7135	137	19	)	)	PUNCT
ejpam-7135	137	20	if	if	SCONJ
ejpam-7135	137	21	b(t	b(t	NOUN
ejpam-7135	137	22	)	)	PUNCT
ejpam-7135	137	23	=	=	SYM
ejpam-7135	138	1	p0(t	p0(t	PROPN
ejpam-7135	138	2	)	)	PUNCT
ejpam-7135	138	3	ap(t	ap(t	NOUN
ejpam-7135	138	4	)	)	PUNCT
ejpam-7135	138	5	=	=	PRON
ejpam-7135	138	6	{	{	PUNCT
ejpam-7135	139	1	p0(t	p0(t	NOUN
ejpam-7135	139	2	)	)	PUNCT
ejpam-7135	139	3	if	if	SCONJ
ejpam-7135	139	4	a(t	a(t	NOUN
ejpam-7135	139	5	)	)	PUNCT
ejpam-7135	139	6	̸=	̸=	PROPN
ejpam-7135	139	7	p0(t	p0(t	PROPN
ejpam-7135	139	8	)	)	PUNCT
ejpam-7135	139	9	p(t	p(t	NOUN
ejpam-7135	139	10	)	)	PUNCT
ejpam-7135	139	11	if	if	SCONJ
ejpam-7135	139	12	a(t	a(t	VERB
ejpam-7135	139	13	)	)	PUNCT
ejpam-7135	139	14	=	=	PUNCT
ejpam-7135	140	1	p0(t	p0(t	PROPN
ejpam-7135	140	2	)	)	PUNCT
ejpam-7135	140	3	then	then	ADV
ejpam-7135	140	4	(	(	PUNCT
ejpam-7135	140	5	ab,∨,∧	ab,∨,∧	PROPN
ejpam-7135	140	6	,	,	PUNCT
ejpam-7135	140	7	p0	p0	NOUN
ejpam-7135	140	8	)	)	PUNCT
ejpam-7135	140	9	is	be	AUX
ejpam-7135	140	10	a	a	DET
ejpam-7135	140	11	stone	stone	NOUN
ejpam-7135	140	12	pdl	pdl	NOUN
ejpam-7135	140	13	in	in	ADP
ejpam-7135	140	14	which	which	PRON
ejpam-7135	140	15	a	a	DET
ejpam-7135	140	16	7→	7→	NUM
ejpam-7135	140	17	ap	ap	PROPN
ejpam-7135	140	18	is	be	AUX
ejpam-7135	140	19	a	a	DET
ejpam-7135	140	20	parapseudo	parapseudo	NOUN
ejpam-7135	140	21	-	-	NOUN
ejpam-7135	140	22	complementation	complementation	NOUN
ejpam-7135	140	23	.	.	PUNCT
ejpam-7135	141	1	theorem	theorem	VERB
ejpam-7135	141	2	6	6	NUM
ejpam-7135	141	3	.	.	PUNCT
ejpam-7135	141	4	l	l	NOUN
ejpam-7135	141	5	with	with	ADP
ejpam-7135	141	6	a	a	DET
ejpam-7135	141	7	parapseudo	parapseudo	NOUN
ejpam-7135	141	8	-	-	PUNCT
ejpam-7135	141	9	complementation	complementation	NOUN
ejpam-7135	141	10	♦	♦	NOUN
ejpam-7135	141	11	,	,	PUNCT
ejpam-7135	141	12	is	be	AUX
ejpam-7135	141	13	a	a	DET
ejpam-7135	141	14	stone	stone	NOUN
ejpam-7135	141	15	pdl	pdl	NOUN
ejpam-7135	142	1	if	if	SCONJ
ejpam-7135	143	1	and	and	CCONJ
ejpam-7135	143	2	only	only	ADV
ejpam-7135	143	3	if	if	SCONJ
ejpam-7135	143	4	pf	pf	PROPN
ejpam-7135	143	5	(	(	PUNCT
ejpam-7135	143	6	l	l	NOUN
ejpam-7135	143	7	)	)	PUNCT
ejpam-7135	143	8	is	be	AUX
ejpam-7135	143	9	a	a	DET
ejpam-7135	143	10	stone	stone	NOUN
ejpam-7135	143	11	lattice	lattice	NOUN
ejpam-7135	143	12	,	,	PUNCT
ejpam-7135	143	13	where	where	SCONJ
ejpam-7135	143	14	pf	pf	PROPN
ejpam-7135	143	15	(	(	PUNCT
ejpam-7135	143	16	l	l	NOUN
ejpam-7135	143	17	)	)	PUNCT
ejpam-7135	143	18	denotes	denote	VERB
ejpam-7135	143	19	the	the	DET
ejpam-7135	143	20	set	set	NOUN
ejpam-7135	143	21	of	of	ADP
ejpam-7135	143	22	principal	principal	ADJ
ejpam-7135	143	23	filters	filter	NOUN
ejpam-7135	143	24	of	of	ADP
ejpam-7135	143	25	l.	l.	PROPN
ejpam-7135	143	26	proof	proof	PROPN
ejpam-7135	143	27	.	.	PUNCT
ejpam-7135	144	1	suppose	suppose	VERB
ejpam-7135	144	2	first	first	ADV
ejpam-7135	144	3	that	that	SCONJ
ejpam-7135	144	4	pf	pf	PROPN
ejpam-7135	144	5	(	(	PUNCT
ejpam-7135	144	6	l	l	NOUN
ejpam-7135	144	7	)	)	PUNCT
ejpam-7135	144	8	is	be	AUX
ejpam-7135	144	9	a	a	DET
ejpam-7135	144	10	stone	stone	NOUN
ejpam-7135	144	11	lattice	lattice	NOUN
ejpam-7135	144	12	with	with	ADP
ejpam-7135	144	13	pseudocomplementation	pseudocomplementation	NOUN
ejpam-7135	144	14	[	[	X
ejpam-7135	144	15	x	x	X
ejpam-7135	144	16	)	)	PUNCT
ejpam-7135	144	17	7→	7→	NOUN
ejpam-7135	145	1	[	[	X
ejpam-7135	145	2	x)	x)	PROPN
ejpam-7135	145	3	♦	♦	PROPN
ejpam-7135	145	4	.	.	PUNCT
ejpam-7135	146	1	for	for	ADP
ejpam-7135	146	2	each	each	DET
ejpam-7135	146	3	x	x	SYM
ejpam-7135	146	4	∈	∈	PROPN
ejpam-7135	146	5	l	l	NOUN
ejpam-7135	146	6	,	,	PUNCT
ejpam-7135	146	7	note	note	VERB
ejpam-7135	146	8	that	that	SCONJ
ejpam-7135	146	9	[	[	X
ejpam-7135	146	10	x	x	X
ejpam-7135	146	11	)	)	PUNCT
ejpam-7135	146	12	♦	♦	PROPN
ejpam-7135	146	13	=	=	PUNCT
ejpam-7135	147	1	[	[	X
ejpam-7135	147	2	m	m	X
ejpam-7135	147	3	∨	∨	NUM
ejpam-7135	147	4	x	x	X
ejpam-7135	147	5	)	)	PUNCT
ejpam-7135	147	6	,	,	PUNCT
ejpam-7135	147	7	where	where	SCONJ
ejpam-7135	147	8	m	m	NOUN
ejpam-7135	147	9	is	be	AUX
ejpam-7135	147	10	the	the	DET
ejpam-7135	147	11	minimal	minimal	ADJ
ejpam-7135	147	12	element	element	NOUN
ejpam-7135	147	13	of	of	ADP
ejpam-7135	147	14	l.	l.	PROPN
ejpam-7135	147	15	let	let	VERB
ejpam-7135	148	1	[	[	X
ejpam-7135	148	2	x)•	x)•	X
ejpam-7135	148	3	=	=	PUNCT
ejpam-7135	149	1	[	[	X
ejpam-7135	149	2	x1	x1	X
ejpam-7135	149	3	)	)	PUNCT
ejpam-7135	149	4	and	and	CCONJ
ejpam-7135	150	1	[	[	X
ejpam-7135	150	2	x)••	x)••	X
ejpam-7135	150	3	=	=	PUNCT
ejpam-7135	151	1	[	[	X
ejpam-7135	151	2	x2	x2	NOUN
ejpam-7135	151	3	)	)	PUNCT
ejpam-7135	151	4	.	.	PUNCT
ejpam-7135	152	1	then	then	ADV
ejpam-7135	152	2	x	x	X
ejpam-7135	152	3	♦	♦	PROPN
ejpam-7135	152	4	=	=	PROPN
ejpam-7135	152	5	m	m	PROPN
ejpam-7135	152	6	∨	∨	NUM
ejpam-7135	152	7	x1	x1	ADJ
ejpam-7135	152	8	and	and	CCONJ
ejpam-7135	152	9	x	x	PROPN
ejpam-7135	152	10	♦	♦	PROPN
ejpam-7135	152	11	♦	♦	PROPN
ejpam-7135	152	12	=	=	PROPN
ejpam-7135	152	13	m	m	PROPN
ejpam-7135	152	14	∨	∨	NOUN
ejpam-7135	152	15	x2	x2	PROPN
ejpam-7135	152	16	.	.	PUNCT
ejpam-7135	153	1	now	now	ADV
ejpam-7135	153	2	consider	consider	VERB
ejpam-7135	153	3	x	x	PRON
ejpam-7135	153	4	♦	♦	PROPN
ejpam-7135	153	5	♦	♦	PROPN
ejpam-7135	153	6	∧	∧	PROPN
ejpam-7135	153	7	x	x	PROPN
ejpam-7135	153	8	♦	♦	PROPN
ejpam-7135	153	9	=	=	PROPN
ejpam-7135	153	10	(	(	PUNCT
ejpam-7135	153	11	m	m	PROPN
ejpam-7135	153	12	∨	∨	ADJ
ejpam-7135	153	13	x2	x2	PROPN
ejpam-7135	153	14	)	)	PUNCT
ejpam-7135	153	15	∧	∧	PROPN
ejpam-7135	153	16	(	(	PUNCT
ejpam-7135	153	17	m	m	PROPN
ejpam-7135	153	18	∨	∨	NUM
ejpam-7135	153	19	x1	x1	PROPN
ejpam-7135	153	20	)	)	PUNCT
ejpam-7135	154	1	=	=	SYM
ejpam-7135	154	2	m	m	VERB
ejpam-7135	154	3	∨	∨	NOUN
ejpam-7135	154	4	(	(	PUNCT
ejpam-7135	154	5	x1	x1	PROPN
ejpam-7135	154	6	∧	∧	PROPN
ejpam-7135	154	7	x2	x2	PROPN
ejpam-7135	154	8	)	)	PUNCT
ejpam-7135	154	9	.	.	PUNCT
ejpam-7135	155	1	since	since	SCONJ
ejpam-7135	155	2	pf	pf	PROPN
ejpam-7135	155	3	(	(	PUNCT
ejpam-7135	155	4	l	l	NOUN
ejpam-7135	155	5	)	)	PUNCT
ejpam-7135	155	6	is	be	AUX
ejpam-7135	155	7	stone	stone	NOUN
ejpam-7135	155	8	,	,	PUNCT
ejpam-7135	155	9	we	we	PRON
ejpam-7135	155	10	have	have	VERB
ejpam-7135	155	11	[	[	X
ejpam-7135	155	12	x1	x1	PROPN
ejpam-7135	155	13	)	)	PUNCT
ejpam-7135	155	14	∨	∨	NOUN
ejpam-7135	156	1	[	[	X
ejpam-7135	156	2	x2	x2	X
ejpam-7135	156	3	)	)	PUNCT
ejpam-7135	156	4	=	=	PUNCT
ejpam-7135	157	1	[	[	X
ejpam-7135	157	2	x1	x1	PROPN
ejpam-7135	157	3	∧	∧	PROPN
ejpam-7135	157	4	x2	x2	PROPN
ejpam-7135	157	5	)	)	PUNCT
ejpam-7135	157	6	is	be	AUX
ejpam-7135	157	7	a	a	DET
ejpam-7135	157	8	minimal	minimal	ADJ
ejpam-7135	157	9	filter	filter	NOUN
ejpam-7135	157	10	,	,	PUNCT
ejpam-7135	157	11	which	which	PRON
ejpam-7135	157	12	implies	imply	VERB
ejpam-7135	157	13	x	x	PROPN
ejpam-7135	157	14	♦	♦	PROPN
ejpam-7135	157	15	♦	♦	PROPN
ejpam-7135	157	16	∧	∧	PROPN
ejpam-7135	157	17	x	x	PROPN
ejpam-7135	157	18	♦	♦	PROPN
ejpam-7135	157	19	=	=	PROPN
ejpam-7135	157	20	m	m	PROPN
ejpam-7135	157	21	=	=	SYM
ejpam-7135	157	22	1	1	NUM
ejpam-7135	157	23	♦	♦	PROPN
ejpam-7135	157	24	.	.	PUNCT
ejpam-7135	158	1	hence	hence	ADV
ejpam-7135	158	2	,	,	PUNCT
ejpam-7135	158	3	l	l	NOUN
ejpam-7135	158	4	satisfies	satisfy	VERB
ejpam-7135	158	5	the	the	DET
ejpam-7135	158	6	stone	stone	NOUN
ejpam-7135	158	7	condition	condition	NOUN
ejpam-7135	158	8	.	.	PUNCT
ejpam-7135	159	1	r.	r.	PROPN
ejpam-7135	159	2	bandaru	bandaru	PROPN
ejpam-7135	159	3	et	et	PROPN
ejpam-7135	159	4	al	al	PROPN
ejpam-7135	159	5	.	.	PUNCT
ejpam-7135	159	6	/	/	SYM
ejpam-7135	159	7	eur	eur	PROPN
ejpam-7135	159	8	.	.	PUNCT
ejpam-7135	160	1	j.	j.	PROPN
ejpam-7135	160	2	pure	pure	PROPN
ejpam-7135	160	3	appl	appl	PROPN
ejpam-7135	160	4	.	.	PROPN
ejpam-7135	160	5	math	math	PROPN
ejpam-7135	160	6	,	,	PUNCT
ejpam-7135	160	7	18	18	NUM
ejpam-7135	160	8	(	(	PUNCT
ejpam-7135	160	9	4	4	NUM
ejpam-7135	160	10	)	)	PUNCT
ejpam-7135	160	11	(	(	PUNCT
ejpam-7135	160	12	2025	2025	NUM
ejpam-7135	160	13	)	)	PUNCT
ejpam-7135	160	14	,	,	PUNCT
ejpam-7135	160	15	7135	7135	NUM
ejpam-7135	160	16	7	7	NUM
ejpam-7135	160	17	of	of	ADP
ejpam-7135	160	18	14	14	NUM
ejpam-7135	160	19	conversely	conversely	ADV
ejpam-7135	160	20	,	,	PUNCT
ejpam-7135	160	21	suppose	suppose	VERB
ejpam-7135	160	22	l	l	NOUN
ejpam-7135	160	23	is	be	AUX
ejpam-7135	160	24	a	a	DET
ejpam-7135	160	25	stone	stone	NOUN
ejpam-7135	160	26	pdl	pdl	NOUN
ejpam-7135	160	27	.	.	PUNCT
ejpam-7135	161	1	then	then	ADV
ejpam-7135	161	2	for	for	ADP
ejpam-7135	161	3	each	each	DET
ejpam-7135	161	4	x	x	SYM
ejpam-7135	161	5	∈	∈	NOUN
ejpam-7135	161	6	l	l	NOUN
ejpam-7135	161	7	the	the	DET
ejpam-7135	161	8	mapping	mapping	NOUN
ejpam-7135	161	9	x	x	SYM
ejpam-7135	161	10	7→	7→	NUM
ejpam-7135	162	1	[	[	X
ejpam-7135	162	2	x	x	X
ejpam-7135	162	3	)	)	PUNCT
ejpam-7135	162	4	defines	define	VERB
ejpam-7135	162	5	a	a	DET
ejpam-7135	162	6	pseudo	pseudo	NOUN
ejpam-7135	162	7	-	-	ADJ
ejpam-7135	162	8	complemented	complement	VERB
ejpam-7135	162	9	distributive	distributive	ADJ
ejpam-7135	162	10	lattice	lattice	NOUN
ejpam-7135	162	11	of	of	ADP
ejpam-7135	162	12	principal	principal	ADJ
ejpam-7135	162	13	filters	filter	NOUN
ejpam-7135	162	14	,	,	PUNCT
ejpam-7135	162	15	and	and	CCONJ
ejpam-7135	162	16	the	the	DET
ejpam-7135	162	17	stone	stone	NOUN
ejpam-7135	162	18	property	property	NOUN
ejpam-7135	162	19	x	x	PROPN
ejpam-7135	162	20	♦	♦	PROPN
ejpam-7135	162	21	♦	♦	PROPN
ejpam-7135	162	22	∧	∧	PROPN
ejpam-7135	162	23	x	x	PROPN
ejpam-7135	162	24	♦	♦	PROPN
ejpam-7135	162	25	=	=	PROPN
ejpam-7135	162	26	1	1	NUM
ejpam-7135	162	27	♦	♦	PROPN
ejpam-7135	162	28	transfers	transfer	NOUN
ejpam-7135	162	29	to	to	ADP
ejpam-7135	162	30	pf	pf	PROPN
ejpam-7135	162	31	(	(	PUNCT
ejpam-7135	162	32	l	l	NOUN
ejpam-7135	162	33	)	)	PUNCT
ejpam-7135	162	34	.	.	PUNCT
ejpam-7135	163	1	thus	thus	ADV
ejpam-7135	163	2	,	,	PUNCT
ejpam-7135	163	3	pf	pf	PROPN
ejpam-7135	163	4	(	(	PUNCT
ejpam-7135	163	5	l	l	NOUN
ejpam-7135	163	6	)	)	PUNCT
ejpam-7135	163	7	is	be	AUX
ejpam-7135	163	8	a	a	DET
ejpam-7135	163	9	stone	stone	NOUN
ejpam-7135	163	10	lattice	lattice	NOUN
ejpam-7135	163	11	.	.	PUNCT
ejpam-7135	164	1	theorem	theorem	VERB
ejpam-7135	164	2	7	7	NUM
ejpam-7135	164	3	.	.	PUNCT
ejpam-7135	164	4	l	l	NOUN
ejpam-7135	164	5	with	with	ADP
ejpam-7135	164	6	a	a	DET
ejpam-7135	164	7	parapseudo	parapseudo	NOUN
ejpam-7135	164	8	-	-	PUNCT
ejpam-7135	164	9	complementation	complementation	NOUN
ejpam-7135	164	10	♦	♦	PROPN
ejpam-7135	164	11	is	be	AUX
ejpam-7135	164	12	a	a	DET
ejpam-7135	164	13	stone	stone	NOUN
ejpam-7135	164	14	pdl	pdl	NOUN
ejpam-7135	165	1	if	if	SCONJ
ejpam-7135	165	2	and	and	CCONJ
ejpam-7135	165	3	only	only	ADV
ejpam-7135	165	4	if	if	SCONJ
ejpam-7135	165	5	[	[	X
ejpam-7135	165	6	x]•	x]•	X
ejpam-7135	165	7	∨	∨	PROPN
ejpam-7135	165	8	[	[	X
ejpam-7135	165	9	x]••	x]••	PUNCT
ejpam-7135	165	10	=	=	SYM
ejpam-7135	165	11	l	l	NOUN
ejpam-7135	165	12	for	for	ADP
ejpam-7135	165	13	all	all	DET
ejpam-7135	165	14	x	x	SYM
ejpam-7135	165	15	∈	∈	PROPN
ejpam-7135	165	16	l.	l.	NOUN
ejpam-7135	165	17	proof	proof	PROPN
ejpam-7135	165	18	.	.	PUNCT
ejpam-7135	166	1	suppose	suppose	VERB
ejpam-7135	167	1	[	[	X
ejpam-7135	167	2	x]•	x]•	PROPN
ejpam-7135	167	3	∨	∨	PROPN
ejpam-7135	167	4	[	[	X
ejpam-7135	167	5	x]••	x]••	PUNCT
ejpam-7135	167	6	=	=	SYM
ejpam-7135	167	7	l	l	NOUN
ejpam-7135	167	8	for	for	ADP
ejpam-7135	167	9	all	all	DET
ejpam-7135	167	10	x	x	SYM
ejpam-7135	167	11	∈	∈	PROPN
ejpam-7135	167	12	l.	l.	NOUN
ejpam-7135	167	13	let	let	VERB
ejpam-7135	167	14	m	m	PRON
ejpam-7135	167	15	denote	denote	VERB
ejpam-7135	167	16	the	the	DET
ejpam-7135	167	17	minimal	minimal	ADJ
ejpam-7135	167	18	element	element	NOUN
ejpam-7135	167	19	of	of	ADP
ejpam-7135	167	20	l.	l.	PROPN
ejpam-7135	167	21	since	since	SCONJ
ejpam-7135	167	22	m	m	PROPN
ejpam-7135	167	23	∈	∈	PROPN
ejpam-7135	167	24	l	l	NOUN
ejpam-7135	167	25	,	,	PUNCT
ejpam-7135	167	26	we	we	PRON
ejpam-7135	167	27	can	can	AUX
ejpam-7135	167	28	write	write	VERB
ejpam-7135	167	29	m	m	PROPN
ejpam-7135	167	30	=	=	PUNCT
ejpam-7135	167	31	a	a	DET
ejpam-7135	167	32	∧	∧	PROPN
ejpam-7135	167	33	b	b	PROPN
ejpam-7135	167	34	with	with	ADP
ejpam-7135	167	35	a	a	DET
ejpam-7135	167	36	∈	∈	PROPN
ejpam-7135	168	1	[	[	X
ejpam-7135	168	2	x]•	x]•	X
ejpam-7135	168	3	and	and	CCONJ
ejpam-7135	168	4	b	b	X
ejpam-7135	168	5	∈	∈	PROPN
ejpam-7135	169	1	[	[	X
ejpam-7135	169	2	x]••.	x]••.	X
ejpam-7135	169	3	define	define	VERB
ejpam-7135	169	4	x	x	PRON
ejpam-7135	169	5	♦	♦	PROPN
ejpam-7135	169	6	:	:	PUNCT
ejpam-7135	169	7	=	=	SYM
ejpam-7135	169	8	m∨	m∨	PROPN
ejpam-7135	169	9	a.	a.	NOUN
ejpam-7135	169	10	then	then	ADV
ejpam-7135	169	11	x	x	X
ejpam-7135	169	12	♦	♦	PROPN
ejpam-7135	169	13	is	be	AUX
ejpam-7135	169	14	a	a	DET
ejpam-7135	169	15	candidate	candidate	NOUN
ejpam-7135	169	16	for	for	ADP
ejpam-7135	169	17	the	the	DET
ejpam-7135	169	18	parapseudo	parapseudo	NOUN
ejpam-7135	169	19	-	-	NOUN
ejpam-7135	169	20	complementation	complementation	NOUN
ejpam-7135	169	21	of	of	ADP
ejpam-7135	169	22	x.	x.	NOUN
ejpam-7135	169	23	we	we	PRON
ejpam-7135	169	24	claim	claim	VERB
ejpam-7135	169	25	that	that	SCONJ
ejpam-7135	169	26	x	x	PROPN
ejpam-7135	169	27	♦	♦	PROPN
ejpam-7135	169	28	♦	♦	PROPN
ejpam-7135	169	29	∧	∧	PROPN
ejpam-7135	169	30	x	x	PROPN
ejpam-7135	169	31	♦	♦	PROPN
ejpam-7135	169	32	=	=	PROPN
ejpam-7135	169	33	1	1	NUM
ejpam-7135	169	34	♦	♦	PROPN
ejpam-7135	169	35	.	.	PUNCT
ejpam-7135	170	1	first	first	ADV
ejpam-7135	170	2	observe	observe	VERB
ejpam-7135	170	3	that	that	SCONJ
ejpam-7135	171	1	[	[	X
ejpam-7135	171	2	x	x	X
ejpam-7135	171	3	♦	♦	PROPN
ejpam-7135	171	4	]•	]•	PUNCT
ejpam-7135	171	5	=	=	PUNCT
ejpam-7135	172	1	[	[	X
ejpam-7135	172	2	b	b	NOUN
ejpam-7135	172	3	)	)	PUNCT
ejpam-7135	172	4	.	.	PUNCT
ejpam-7135	173	1	indeed	indeed	ADV
ejpam-7135	173	2	,	,	PUNCT
ejpam-7135	173	3	since	since	SCONJ
ejpam-7135	173	4	x	x	PROPN
ejpam-7135	173	5	♦	♦	PROPN
ejpam-7135	173	6	∨	∨	PROPN
ejpam-7135	173	7	b	b	PROPN
ejpam-7135	173	8	=	=	X
ejpam-7135	173	9	m	m	PROPN
ejpam-7135	173	10	∨	∨	NOUN
ejpam-7135	173	11	a	a	DET
ejpam-7135	173	12	∨	∨	NUM
ejpam-7135	173	13	b	b	NOUN
ejpam-7135	173	14	=	=	SYM
ejpam-7135	173	15	1	1	NUM
ejpam-7135	173	16	(	(	PUNCT
ejpam-7135	173	17	as	as	ADP
ejpam-7135	173	18	a	a	DET
ejpam-7135	173	19	∨	∨	NUM
ejpam-7135	173	20	b	b	X
ejpam-7135	173	21	∈	∈	PROPN
ejpam-7135	174	1	[	[	X
ejpam-7135	174	2	x]•	x]•	X
ejpam-7135	174	3	∩	∩	NOUN
ejpam-7135	174	4	[	[	X
ejpam-7135	174	5	x]••	x]••	PUNCT
ejpam-7135	174	6	=	=	SYM
ejpam-7135	174	7	{	{	PUNCT
ejpam-7135	174	8	1	1	NUM
ejpam-7135	174	9	}	}	PUNCT
ejpam-7135	174	10	)	)	PUNCT
ejpam-7135	174	11	,	,	PUNCT
ejpam-7135	174	12	we	we	PRON
ejpam-7135	174	13	obtain	obtain	VERB
ejpam-7135	174	14	b	b	PRON
ejpam-7135	174	15	∈	∈	NOUN
ejpam-7135	175	1	[	[	X
ejpam-7135	175	2	x	x	X
ejpam-7135	175	3	♦	♦	PROPN
ejpam-7135	175	4	]•.	]•.	X
ejpam-7135	175	5	on	on	ADP
ejpam-7135	175	6	the	the	DET
ejpam-7135	175	7	other	other	ADJ
ejpam-7135	175	8	hand	hand	NOUN
ejpam-7135	175	9	,	,	PUNCT
ejpam-7135	175	10	if	if	SCONJ
ejpam-7135	175	11	t	t	PROPN
ejpam-7135	175	12	∈	∈	PROPN
ejpam-7135	176	1	[	[	X
ejpam-7135	176	2	x	x	X
ejpam-7135	176	3	♦	♦	PROPN
ejpam-7135	176	4	]•	]•	PROPN
ejpam-7135	176	5	,	,	PUNCT
ejpam-7135	176	6	then	then	ADV
ejpam-7135	176	7	x	x	PROPN
ejpam-7135	176	8	♦	♦	PROPN
ejpam-7135	176	9	∨	∨	PROPN
ejpam-7135	176	10	t	t	PROPN
ejpam-7135	176	11	=	=	SYM
ejpam-7135	176	12	1	1	NUM
ejpam-7135	176	13	,	,	PUNCT
ejpam-7135	176	14	hence	hence	ADV
ejpam-7135	176	15	m	m	NOUN
ejpam-7135	176	16	∨	∨	NUM
ejpam-7135	176	17	a	a	DET
ejpam-7135	176	18	∨	∨	NOUN
ejpam-7135	176	19	t	t	NOUN
ejpam-7135	176	20	=	=	SYM
ejpam-7135	176	21	1	1	X
ejpam-7135	176	22	.	.	PUNCT
ejpam-7135	177	1	thus	thus	ADV
ejpam-7135	177	2	,	,	PUNCT
ejpam-7135	177	3	a	a	DET
ejpam-7135	177	4	∨	∨	NOUN
ejpam-7135	177	5	t	t	NOUN
ejpam-7135	177	6	=	=	SYM
ejpam-7135	177	7	1	1	NUM
ejpam-7135	177	8	,	,	PUNCT
ejpam-7135	177	9	which	which	PRON
ejpam-7135	177	10	implies	imply	VERB
ejpam-7135	177	11	t	t	PROPN
ejpam-7135	177	12	∈	∈	PROPN
ejpam-7135	178	1	[	[	X
ejpam-7135	178	2	b	b	NOUN
ejpam-7135	178	3	)	)	PUNCT
ejpam-7135	178	4	.	.	PUNCT
ejpam-7135	179	1	therefore	therefore	ADV
ejpam-7135	179	2	,	,	PUNCT
ejpam-7135	179	3	[	[	X
ejpam-7135	179	4	x	x	X
ejpam-7135	179	5	♦	♦	PROPN
ejpam-7135	179	6	]•	]•	PUNCT
ejpam-7135	179	7	=	=	PUNCT
ejpam-7135	180	1	[	[	X
ejpam-7135	180	2	b	b	NOUN
ejpam-7135	180	3	)	)	PUNCT
ejpam-7135	180	4	.	.	PUNCT
ejpam-7135	181	1	now	now	ADV
ejpam-7135	181	2	compute	compute	VERB
ejpam-7135	181	3	x	x	PROPN
ejpam-7135	181	4	♦	♦	PROPN
ejpam-7135	181	5	♦	♦	PROPN
ejpam-7135	181	6	∧	∧	PROPN
ejpam-7135	181	7	x	x	PROPN
ejpam-7135	181	8	♦	♦	PROPN
ejpam-7135	181	9	=	=	PROPN
ejpam-7135	181	10	(	(	PUNCT
ejpam-7135	181	11	m	m	PROPN
ejpam-7135	181	12	∨	∨	NUM
ejpam-7135	181	13	b	b	NOUN
ejpam-7135	181	14	)	)	PUNCT
ejpam-7135	181	15	∧	∧	PROPN
ejpam-7135	181	16	(	(	PUNCT
ejpam-7135	181	17	m	m	PROPN
ejpam-7135	181	18	∨	∨	NUM
ejpam-7135	181	19	a	a	PRON
ejpam-7135	181	20	)	)	PUNCT
ejpam-7135	181	21	=	=	SYM
ejpam-7135	182	1	m	m	VERB
ejpam-7135	182	2	∨	∨	NOUN
ejpam-7135	182	3	(	(	PUNCT
ejpam-7135	182	4	a	a	DET
ejpam-7135	182	5	∧	∧	PROPN
ejpam-7135	182	6	b	b	NOUN
ejpam-7135	182	7	)	)	PUNCT
ejpam-7135	182	8	=	=	PUNCT
ejpam-7135	182	9	m	m	NOUN
ejpam-7135	182	10	=	=	SYM
ejpam-7135	182	11	1	1	NUM
ejpam-7135	182	12	♦	♦	PROPN
ejpam-7135	182	13	.	.	PUNCT
ejpam-7135	183	1	hence	hence	ADV
ejpam-7135	183	2	,	,	PUNCT
ejpam-7135	183	3	l	l	NOUN
ejpam-7135	183	4	is	be	AUX
ejpam-7135	183	5	a	a	DET
ejpam-7135	183	6	stone	stone	NOUN
ejpam-7135	183	7	pdl	pdl	NOUN
ejpam-7135	183	8	.	.	PUNCT
ejpam-7135	184	1	conversely	conversely	ADV
ejpam-7135	184	2	,	,	PUNCT
ejpam-7135	184	3	assume	assume	VERB
ejpam-7135	184	4	l	l	NOUN
ejpam-7135	184	5	is	be	AUX
ejpam-7135	184	6	a	a	DET
ejpam-7135	184	7	stone	stone	NOUN
ejpam-7135	184	8	pdl	pdl	NOUN
ejpam-7135	184	9	.	.	PUNCT
ejpam-7135	185	1	then	then	ADV
ejpam-7135	185	2	by	by	ADP
ejpam-7135	185	3	definition	definition	NOUN
ejpam-7135	185	4	,	,	PUNCT
ejpam-7135	185	5	x	x	PROPN
ejpam-7135	185	6	♦	♦	PROPN
ejpam-7135	185	7	♦	♦	PROPN
ejpam-7135	185	8	∧	∧	PROPN
ejpam-7135	185	9	x	x	PROPN
ejpam-7135	185	10	♦	♦	PROPN
ejpam-7135	185	11	=	=	PROPN
ejpam-7135	185	12	1	1	NUM
ejpam-7135	185	13	♦	♦	PROPN
ejpam-7135	185	14	=	=	PUNCT
ejpam-7135	185	15	m.	m.	NOUN
ejpam-7135	185	16	this	this	PRON
ejpam-7135	185	17	implies	imply	VERB
ejpam-7135	185	18	that	that	SCONJ
ejpam-7135	185	19	any	any	DET
ejpam-7135	185	20	t	t	NOUN
ejpam-7135	185	21	∈	∈	PROPN
ejpam-7135	185	22	l	l	NOUN
ejpam-7135	185	23	can	can	AUX
ejpam-7135	185	24	be	be	AUX
ejpam-7135	185	25	expressed	express	VERB
ejpam-7135	185	26	as	as	ADP
ejpam-7135	185	27	a	a	DET
ejpam-7135	185	28	join	join	NOUN
ejpam-7135	185	29	of	of	ADP
ejpam-7135	185	30	elements	element	NOUN
ejpam-7135	185	31	from	from	ADP
ejpam-7135	185	32	[	[	X
ejpam-7135	185	33	x]•	x]•	PROPN
ejpam-7135	185	34	and	and	CCONJ
ejpam-7135	185	35	[	[	X
ejpam-7135	185	36	x]••.	x]••.	X
ejpam-7135	185	37	thus	thus	ADV
ejpam-7135	185	38	,	,	PUNCT
ejpam-7135	185	39	[	[	X
ejpam-7135	185	40	x]•	x]•	PROPN
ejpam-7135	185	41	∨	∨	PROPN
ejpam-7135	185	42	[	[	X
ejpam-7135	185	43	x]••	x]••	PUNCT
ejpam-7135	185	44	=	=	SYM
ejpam-7135	185	45	l.	l.	PROPN
ejpam-7135	185	46	theorem	theorem	VERB
ejpam-7135	185	47	8	8	NUM
ejpam-7135	185	48	.	.	PUNCT
ejpam-7135	185	49	l	l	NOUN
ejpam-7135	185	50	with	with	ADP
ejpam-7135	185	51	a	a	DET
ejpam-7135	185	52	parapseudo	parapseudo	NOUN
ejpam-7135	185	53	-	-	PUNCT
ejpam-7135	185	54	complementation	complementation	NOUN
ejpam-7135	185	55	♦	♦	PROPN
ejpam-7135	185	56	is	be	AUX
ejpam-7135	185	57	a	a	DET
ejpam-7135	185	58	stone	stone	NOUN
ejpam-7135	185	59	pdl	pdl	NOUN
ejpam-7135	185	60	if	if	SCONJ
ejpam-7135	185	61	and	and	CCONJ
ejpam-7135	185	62	only	only	ADV
ejpam-7135	185	63	if	if	SCONJ
ejpam-7135	185	64	[	[	X
ejpam-7135	185	65	x	x	X
ejpam-7135	185	66	∨	∨	NUM
ejpam-7135	185	67	y]•	y]•	NOUN
ejpam-7135	186	1	=	=	PUNCT
ejpam-7135	187	1	[	[	X
ejpam-7135	187	2	x]•	x]•	X
ejpam-7135	187	3	∨	∨	PROPN
ejpam-7135	188	1	[	[	X
ejpam-7135	188	2	y]•	y]•	NOUN
ejpam-7135	188	3	for	for	ADP
ejpam-7135	188	4	all	all	DET
ejpam-7135	188	5	x	x	NOUN
ejpam-7135	188	6	,	,	PUNCT
ejpam-7135	188	7	y	y	PROPN
ejpam-7135	188	8	∈	∈	PROPN
ejpam-7135	188	9	l.	l.	NOUN
ejpam-7135	188	10	proof	proof	PROPN
ejpam-7135	188	11	.	.	PUNCT
ejpam-7135	189	1	suppose	suppose	VERB
ejpam-7135	189	2	l	l	NOUN
ejpam-7135	189	3	is	be	AUX
ejpam-7135	189	4	a	a	DET
ejpam-7135	189	5	stone	stone	NOUN
ejpam-7135	189	6	pdl	pdl	NOUN
ejpam-7135	189	7	.	.	PUNCT
ejpam-7135	190	1	clearly	clearly	ADV
ejpam-7135	190	2	,	,	PUNCT
ejpam-7135	190	3	since	since	SCONJ
ejpam-7135	190	4	x	x	PROPN
ejpam-7135	190	5	≤	≤	NUM
ejpam-7135	190	6	x	x	PUNCT
ejpam-7135	190	7	∨	∨	NUM
ejpam-7135	190	8	y	y	PROPN
ejpam-7135	190	9	and	and	CCONJ
ejpam-7135	190	10	y	y	PROPN
ejpam-7135	190	11	≤	≤	PROPN
ejpam-7135	190	12	y	y	PROPN
ejpam-7135	190	13	∨	∨	NUM
ejpam-7135	190	14	x	x	AUX
ejpam-7135	190	15	,	,	PUNCT
ejpam-7135	190	16	we	we	PRON
ejpam-7135	190	17	have	have	VERB
ejpam-7135	190	18	[	[	X
ejpam-7135	190	19	x]•	x]•	X
ejpam-7135	190	20	,	,	PUNCT
ejpam-7135	191	1	[	[	X
ejpam-7135	191	2	y]•	y]•	NOUN
ejpam-7135	191	3	⊆	⊆	NUM
ejpam-7135	191	4	[	[	X
ejpam-7135	191	5	x	x	X
ejpam-7135	191	6	∨	∨	NUM
ejpam-7135	191	7	y]•	y]•	NOUN
ejpam-7135	191	8	=	=	PUNCT
ejpam-7135	192	1	[	[	X
ejpam-7135	192	2	y	y	PROPN
ejpam-7135	192	3	∨	∨	NUM
ejpam-7135	192	4	x]•	x]•	PROPN
ejpam-7135	192	5	,	,	PUNCT
ejpam-7135	192	6	so	so	SCONJ
ejpam-7135	192	7	that	that	SCONJ
ejpam-7135	192	8	[	[	X
ejpam-7135	192	9	x]•	x]•	X
ejpam-7135	192	10	∨	∨	PROPN
ejpam-7135	193	1	[	[	X
ejpam-7135	193	2	y]•	y]•	NOUN
ejpam-7135	193	3	⊆	⊆	NUM
ejpam-7135	193	4	[	[	X
ejpam-7135	193	5	x	x	X
ejpam-7135	193	6	∨	∨	NUM
ejpam-7135	193	7	y]•.	y]•.	NUM
ejpam-7135	193	8	for	for	ADP
ejpam-7135	193	9	the	the	DET
ejpam-7135	193	10	reverse	reverse	ADJ
ejpam-7135	193	11	inclusion	inclusion	NOUN
ejpam-7135	193	12	,	,	PUNCT
ejpam-7135	193	13	let	let	VERB
ejpam-7135	193	14	t	t	PROPN
ejpam-7135	193	15	∈	∈	PROPN
ejpam-7135	194	1	[	[	X
ejpam-7135	194	2	x∨	x∨	PROPN
ejpam-7135	194	3	y]•.	y]•.	PROPN
ejpam-7135	194	4	then	then	ADV
ejpam-7135	194	5	t∨	t∨	PROPN
ejpam-7135	194	6	(	(	PUNCT
ejpam-7135	194	7	x∨	x∨	PROPN
ejpam-7135	194	8	y	y	PROPN
ejpam-7135	194	9	)	)	PUNCT
ejpam-7135	194	10	=	=	SYM
ejpam-7135	194	11	1	1	X
ejpam-7135	194	12	.	.	PUNCT
ejpam-7135	194	13	by	by	ADP
ejpam-7135	194	14	the	the	DET
ejpam-7135	194	15	stone	stone	NOUN
ejpam-7135	194	16	property	property	NOUN
ejpam-7135	194	17	,	,	PUNCT
ejpam-7135	194	18	we	we	PRON
ejpam-7135	194	19	know	know	VERB
ejpam-7135	194	20	x	x	PRON
ejpam-7135	194	21	♦	♦	PROPN
ejpam-7135	194	22	∧	∧	PROPN
ejpam-7135	194	23	x	x	PROPN
ejpam-7135	194	24	♦	♦	PROPN
ejpam-7135	194	25	♦	♦	PROPN
ejpam-7135	194	26	=	=	PROPN
ejpam-7135	194	27	1	1	NUM
ejpam-7135	194	28	♦	♦	PROPN
ejpam-7135	194	29	.	.	PUNCT
ejpam-7135	195	1	hence	hence	ADV
ejpam-7135	195	2	,	,	PUNCT
ejpam-7135	195	3	1	1	NUM
ejpam-7135	195	4	=	=	SYM
ejpam-7135	195	5	t	t	PROPN
ejpam-7135	195	6	∨	∨	NUM
ejpam-7135	195	7	(	(	PUNCT
ejpam-7135	195	8	x	x	PROPN
ejpam-7135	195	9	∨	∨	NUM
ejpam-7135	195	10	y	y	PROPN
ejpam-7135	195	11	)	)	PUNCT
ejpam-7135	195	12	=	=	PUNCT
ejpam-7135	196	1	(	(	PUNCT
ejpam-7135	196	2	t	t	PROPN
ejpam-7135	196	3	∨	∨	NUM
ejpam-7135	196	4	y	y	PROPN
ejpam-7135	196	5	)	)	PUNCT
ejpam-7135	196	6	∨	∨	NOUN
ejpam-7135	196	7	(	(	PUNCT
ejpam-7135	196	8	x	x	PROPN
ejpam-7135	196	9	∨	∨	PROPN
ejpam-7135	196	10	x	x	PROPN
ejpam-7135	196	11	♦	♦	PROPN
ejpam-7135	196	12	∨	∨	PROPN
ejpam-7135	196	13	x	x	PROPN
ejpam-7135	196	14	♦	♦	PROPN
ejpam-7135	196	15	♦	♦	PROPN
ejpam-7135	196	16	)	)	PUNCT
ejpam-7135	196	17	.	.	PUNCT
ejpam-7135	197	1	this	this	PRON
ejpam-7135	197	2	implies	imply	VERB
ejpam-7135	197	3	t	t	PROPN
ejpam-7135	197	4	∨	∨	NUM
ejpam-7135	197	5	x	x	PROPN
ejpam-7135	197	6	♦	♦	PROPN
ejpam-7135	197	7	∈	∈	PROPN
ejpam-7135	198	1	[	[	X
ejpam-7135	198	2	x]•	x]•	X
ejpam-7135	198	3	and	and	CCONJ
ejpam-7135	198	4	t	t	PROPN
ejpam-7135	198	5	∨	∨	NUM
ejpam-7135	198	6	x	x	PROPN
ejpam-7135	198	7	♦	♦	PROPN
ejpam-7135	198	8	♦	♦	PROPN
ejpam-7135	198	9	∈	∈	PROPN
ejpam-7135	199	1	[	[	X
ejpam-7135	199	2	y]•.	y]•.	X
ejpam-7135	199	3	thus	thus	ADV
ejpam-7135	199	4	,	,	PUNCT
ejpam-7135	199	5	t	t	PROPN
ejpam-7135	199	6	=	=	SYM
ejpam-7135	199	7	(	(	PUNCT
ejpam-7135	199	8	t	t	PROPN
ejpam-7135	199	9	∨	∨	NUM
ejpam-7135	199	10	x	x	SYM
ejpam-7135	199	11	♦	♦	PROPN
ejpam-7135	199	12	)	)	PUNCT
ejpam-7135	199	13	∧	∧	PROPN
ejpam-7135	199	14	(	(	PUNCT
ejpam-7135	199	15	t	t	PROPN
ejpam-7135	199	16	∨	∨	NUM
ejpam-7135	199	17	x	x	PROPN
ejpam-7135	199	18	♦	♦	PROPN
ejpam-7135	199	19	♦	♦	PROPN
ejpam-7135	199	20	)	)	PUNCT
ejpam-7135	199	21	∈	∈	PROPN
ejpam-7135	200	1	[	[	X
ejpam-7135	200	2	x]•	x]•	PROPN
ejpam-7135	200	3	∨	∨	PROPN
ejpam-7135	200	4	[	[	X
ejpam-7135	200	5	y]•.	y]•.	X
ejpam-7135	200	6	therefore	therefore	ADV
ejpam-7135	200	7	,	,	PUNCT
ejpam-7135	200	8	[	[	X
ejpam-7135	200	9	x	x	X
ejpam-7135	200	10	∨	∨	NUM
ejpam-7135	200	11	y]•	y]•	NOUN
ejpam-7135	200	12	⊆	⊆	NUM
ejpam-7135	200	13	[	[	X
ejpam-7135	200	14	x]•	x]•	PROPN
ejpam-7135	200	15	∨	∨	PROPN
ejpam-7135	201	1	[	[	X
ejpam-7135	201	2	y]•	y]•	NOUN
ejpam-7135	201	3	,	,	PUNCT
ejpam-7135	201	4	and	and	CCONJ
ejpam-7135	201	5	equality	equality	NOUN
ejpam-7135	201	6	holds	hold	VERB
ejpam-7135	201	7	.	.	PUNCT
ejpam-7135	202	1	conversely	conversely	ADV
ejpam-7135	202	2	,	,	PUNCT
ejpam-7135	202	3	suppose	suppose	VERB
ejpam-7135	202	4	the	the	DET
ejpam-7135	202	5	equality	equality	NOUN
ejpam-7135	202	6	[	[	X
ejpam-7135	202	7	x	x	X
ejpam-7135	202	8	∨	∨	NUM
ejpam-7135	202	9	y]•	y]•	NOUN
ejpam-7135	202	10	=	=	PUNCT
ejpam-7135	203	1	[	[	X
ejpam-7135	203	2	x]•	x]•	X
ejpam-7135	203	3	∨	∨	PROPN
ejpam-7135	203	4	[	[	X
ejpam-7135	203	5	y]•	y]•	NOUN
ejpam-7135	203	6	holds	hold	VERB
ejpam-7135	203	7	for	for	ADP
ejpam-7135	203	8	all	all	DET
ejpam-7135	203	9	x	x	NOUN
ejpam-7135	203	10	,	,	PUNCT
ejpam-7135	203	11	y	y	PROPN
ejpam-7135	203	12	∈	∈	PROPN
ejpam-7135	203	13	l.	l.	NOUN
ejpam-7135	203	14	take	take	VERB
ejpam-7135	203	15	y	y	NOUN
ejpam-7135	203	16	=	=	PUNCT
ejpam-7135	203	17	x	x	PROPN
ejpam-7135	203	18	♦	♦	PROPN
ejpam-7135	203	19	.	.	PUNCT
ejpam-7135	204	1	then	then	ADV
ejpam-7135	205	1	[	[	X
ejpam-7135	205	2	x]•	x]•	X
ejpam-7135	205	3	∨	∨	PUNCT
ejpam-7135	205	4	[	[	X
ejpam-7135	205	5	x	x	X
ejpam-7135	205	6	♦	♦	PROPN
ejpam-7135	205	7	]•	]•	PUNCT
ejpam-7135	205	8	=	=	PUNCT
ejpam-7135	206	1	[	[	X
ejpam-7135	206	2	x	x	X
ejpam-7135	206	3	∨	∨	PROPN
ejpam-7135	206	4	x	x	SYM
ejpam-7135	206	5	♦	♦	PROPN
ejpam-7135	206	6	]•	]•	PUNCT
ejpam-7135	206	7	=	=	PUNCT
ejpam-7135	207	1	[	[	X
ejpam-7135	207	2	1]•	1]•	NUM
ejpam-7135	207	3	=	=	SYM
ejpam-7135	207	4	l.	l.	NOUN
ejpam-7135	207	5	this	this	PRON
ejpam-7135	207	6	implies	imply	VERB
ejpam-7135	207	7	that	that	SCONJ
ejpam-7135	207	8	x	x	PROPN
ejpam-7135	207	9	♦	♦	PROPN
ejpam-7135	207	10	♦	♦	PROPN
ejpam-7135	207	11	∧	∧	PROPN
ejpam-7135	207	12	x	x	PROPN
ejpam-7135	207	13	♦	♦	PROPN
ejpam-7135	207	14	=	=	PROPN
ejpam-7135	207	15	1	1	NUM
ejpam-7135	207	16	♦	♦	PROPN
ejpam-7135	207	17	,	,	PUNCT
ejpam-7135	207	18	i.e.	i.e.	X
ejpam-7135	207	19	,	,	PUNCT
ejpam-7135	207	20	l	l	NOUN
ejpam-7135	207	21	is	be	AUX
ejpam-7135	207	22	a	a	DET
ejpam-7135	207	23	stone	stone	NOUN
ejpam-7135	207	24	pdl	pdl	NOUN
ejpam-7135	207	25	.	.	PROPN
ejpam-7135	207	26	4	4	NUM
ejpam-7135	207	27	.	.	X
ejpam-7135	207	28	prime	prime	ADJ
ejpam-7135	207	29	filter	filter	NOUN
ejpam-7135	207	30	characterization	characterization	NOUN
ejpam-7135	207	31	of	of	ADP
ejpam-7135	207	32	stone	stone	NOUN
ejpam-7135	207	33	pdl	pdl	NOUN
ejpam-7135	207	34	in	in	ADP
ejpam-7135	207	35	this	this	DET
ejpam-7135	207	36	section	section	NOUN
ejpam-7135	207	37	,	,	PUNCT
ejpam-7135	207	38	we	we	PRON
ejpam-7135	207	39	provide	provide	VERB
ejpam-7135	207	40	necessary	necessary	ADJ
ejpam-7135	207	41	and	and	CCONJ
ejpam-7135	207	42	sufficient	sufficient	ADJ
ejpam-7135	207	43	conditions	condition	NOUN
ejpam-7135	207	44	for	for	ADP
ejpam-7135	207	45	a	a	DET
ejpam-7135	207	46	pdl	pdl	NOUN
ejpam-7135	207	47	l	l	NOUN
ejpam-7135	207	48	with	with	ADP
ejpam-7135	207	49	parapseudo	parapseudo	NOUN
ejpam-7135	207	50	-	-	PUNCT
ejpam-7135	207	51	complementation	complementation	NOUN
ejpam-7135	207	52	♦	♦	PROPN
ejpam-7135	207	53	in	in	ADP
ejpam-7135	207	54	which	which	PRON
ejpam-7135	207	55	m	m	VERB
ejpam-7135	207	56	is	be	AUX
ejpam-7135	207	57	a	a	DET
ejpam-7135	207	58	minimal	minimal	ADJ
ejpam-7135	207	59	element	element	NOUN
ejpam-7135	207	60	,	,	PUNCT
ejpam-7135	207	61	to	to	PART
ejpam-7135	207	62	be	be	AUX
ejpam-7135	207	63	a	a	DET
ejpam-7135	207	64	stone	stone	NOUN
ejpam-7135	207	65	pdl	pdl	NOUN
ejpam-7135	207	66	in	in	ADP
ejpam-7135	207	67	terms	term	NOUN
ejpam-7135	207	68	of	of	ADP
ejpam-7135	207	69	prime	prime	ADJ
ejpam-7135	207	70	filters	filter	NOUN
ejpam-7135	207	71	in	in	ADP
ejpam-7135	207	72	both	both	CCONJ
ejpam-7135	207	73	algebraic	algebraic	ADJ
ejpam-7135	207	74	and	and	CCONJ
ejpam-7135	207	75	topological	topological	ADJ
ejpam-7135	207	76	aspects	aspect	NOUN
ejpam-7135	207	77	.	.	PUNCT
ejpam-7135	208	1	recall	recall	VERB
ejpam-7135	208	2	that	that	SCONJ
ejpam-7135	208	3	if	if	SCONJ
ejpam-7135	208	4	i	i	PRON
ejpam-7135	208	5	is	be	AUX
ejpam-7135	208	6	a	a	DET
ejpam-7135	208	7	non	non	ADJ
ejpam-7135	208	8	-	-	ADJ
ejpam-7135	208	9	empty	empty	ADJ
ejpam-7135	208	10	subset	subset	NOUN
ejpam-7135	208	11	of	of	ADP
ejpam-7135	208	12	l	l	NOUN
ejpam-7135	208	13	which	which	PRON
ejpam-7135	208	14	is	be	AUX
ejpam-7135	208	15	closed	close	VERB
ejpam-7135	208	16	under	under	ADP
ejpam-7135	208	17	∨	∨	NOUN
ejpam-7135	208	18	,	,	PUNCT
ejpam-7135	208	19	then	then	ADV
ejpam-7135	208	20	the	the	DET
ejpam-7135	208	21	relation	relation	NOUN
ejpam-7135	208	22	θi	θi	ADP
ejpam-7135	208	23	=	=	PRON
ejpam-7135	208	24	{	{	PUNCT
ejpam-7135	208	25	(	(	PUNCT
ejpam-7135	208	26	x	x	NOUN
ejpam-7135	208	27	,	,	PUNCT
ejpam-7135	208	28	y	y	NOUN
ejpam-7135	208	29	)	)	PUNCT
ejpam-7135	208	30	∈	∈	PROPN
ejpam-7135	208	31	l	l	NOUN
ejpam-7135	208	32	×	×	NOUN
ejpam-7135	208	33	l	l	NOUN
ejpam-7135	208	34	|	|	NOUN
ejpam-7135	208	35	x	x	SYM
ejpam-7135	208	36	∨	∨	NUM
ejpam-7135	208	37	d	d	X
ejpam-7135	208	38	=	=	SYM
ejpam-7135	208	39	y	y	PROPN
ejpam-7135	208	40	∨	∨	NUM
ejpam-7135	208	41	d	d	NOUN
ejpam-7135	208	42	for	for	ADP
ejpam-7135	208	43	some	some	PRON
ejpam-7135	208	44	d	d	PROPN
ejpam-7135	208	45	∈	∈	PROPN
ejpam-7135	209	1	i	i	PRON
ejpam-7135	209	2	}	}	PUNCT
ejpam-7135	210	1	[	[	X
ejpam-7135	210	2	12	12	NUM
ejpam-7135	210	3	]	]	PUNCT
ejpam-7135	210	4	is	be	AUX
ejpam-7135	210	5	a	a	DET
ejpam-7135	210	6	congruence	congruence	NOUN
ejpam-7135	210	7	relation	relation	NOUN
ejpam-7135	210	8	on	on	ADP
ejpam-7135	210	9	l.	l.	PROPN
ejpam-7135	210	10	now	now	ADV
ejpam-7135	210	11	we	we	PRON
ejpam-7135	210	12	prove	prove	VERB
ejpam-7135	210	13	the	the	DET
ejpam-7135	210	14	following	following	NOUN
ejpam-7135	210	15	.	.	PUNCT
ejpam-7135	211	1	lemma	lemma	PROPN
ejpam-7135	211	2	6	6	NUM
ejpam-7135	211	3	.	.	PUNCT
ejpam-7135	212	1	let	let	VERB
ejpam-7135	212	2	i	i	PRON
ejpam-7135	212	3	be	be	AUX
ejpam-7135	212	4	a	a	DET
ejpam-7135	212	5	non	non	ADJ
ejpam-7135	212	6	-	-	ADJ
ejpam-7135	212	7	empty	empty	ADJ
ejpam-7135	212	8	subset	subset	NOUN
ejpam-7135	212	9	of	of	ADP
ejpam-7135	212	10	l	l	NOUN
ejpam-7135	212	11	that	that	PRON
ejpam-7135	212	12	is	be	AUX
ejpam-7135	212	13	closed	close	VERB
ejpam-7135	212	14	under	under	ADP
ejpam-7135	212	15	∨	∨	NUM
ejpam-7135	212	16	,	,	PUNCT
ejpam-7135	212	17	and	and	CCONJ
ejpam-7135	212	18	let	let	VERB
ejpam-7135	212	19	p	p	PRON
ejpam-7135	212	20	be	be	AUX
ejpam-7135	212	21	a	a	DET
ejpam-7135	212	22	prime	prime	ADJ
ejpam-7135	212	23	filter	filter	NOUN
ejpam-7135	212	24	of	of	ADP
ejpam-7135	212	25	l	l	NOUN
ejpam-7135	212	26	such	such	ADJ
ejpam-7135	212	27	that	that	SCONJ
ejpam-7135	212	28	p	p	VERB
ejpam-7135	212	29	⊆	⊆	NUM
ejpam-7135	212	30	l	l	NOUN
ejpam-7135	212	31	\	\	PROPN
ejpam-7135	212	32	i.	i.	NOUN
ejpam-7135	212	33	define	define	VERB
ejpam-7135	212	34	p	p	X
ejpam-7135	212	35	:	:	PUNCT
ejpam-7135	212	36	=	=	SYM
ejpam-7135	212	37	{	{	PUNCT
ejpam-7135	212	38	x	x	PROPN
ejpam-7135	212	39	/	/	SYM
ejpam-7135	212	40	θi	θi	ADP
ejpam-7135	212	41	∈	∈	PROPN
ejpam-7135	212	42	l	l	PROPN
ejpam-7135	212	43	/	/	SYM
ejpam-7135	212	44	θi	θi	NOUN
ejpam-7135	213	1	|	|	ADV
ejpam-7135	213	2	x	x	SYM
ejpam-7135	213	3	∈	∈	PROPN
ejpam-7135	213	4	p	p	X
ejpam-7135	213	5	}	}	PUNCT
ejpam-7135	213	6	.	.	PUNCT
ejpam-7135	214	1	then	then	ADV
ejpam-7135	214	2	:	:	PUNCT
ejpam-7135	214	3	(	(	PUNCT
ejpam-7135	214	4	1	1	X
ejpam-7135	214	5	)	)	PUNCT
ejpam-7135	214	6	for	for	ADP
ejpam-7135	214	7	any	any	DET
ejpam-7135	214	8	x	x	SYM
ejpam-7135	214	9	∈	∈	PROPN
ejpam-7135	214	10	l	l	NOUN
ejpam-7135	214	11	,	,	PUNCT
ejpam-7135	214	12	x	x	PROPN
ejpam-7135	214	13	/	/	SYM
ejpam-7135	214	14	θi	θi	ADP
ejpam-7135	214	15	∈	∈	PROPN
ejpam-7135	214	16	p	p	NOUN
ejpam-7135	214	17	if	if	SCONJ
ejpam-7135	215	1	and	and	CCONJ
ejpam-7135	215	2	only	only	ADV
ejpam-7135	215	3	if	if	SCONJ
ejpam-7135	215	4	x	x	PROPN
ejpam-7135	215	5	∈	∈	PROPN
ejpam-7135	215	6	p	p	X
ejpam-7135	215	7	.	.	PUNCT
ejpam-7135	216	1	(	(	PUNCT
ejpam-7135	216	2	2	2	X
ejpam-7135	216	3	)	)	PUNCT
ejpam-7135	216	4	p	p	NOUN
ejpam-7135	216	5	is	be	AUX
ejpam-7135	216	6	a	a	DET
ejpam-7135	216	7	prime	prime	ADJ
ejpam-7135	216	8	filter	filter	NOUN
ejpam-7135	216	9	of	of	ADP
ejpam-7135	216	10	l	l	PROPN
ejpam-7135	216	11	/	/	SYM
ejpam-7135	216	12	θi	θi	PROPN
ejpam-7135	216	13	.	.	PUNCT
ejpam-7135	217	1	r.	r.	PROPN
ejpam-7135	217	2	bandaru	bandaru	PROPN
ejpam-7135	217	3	et	et	PROPN
ejpam-7135	217	4	al	al	PROPN
ejpam-7135	217	5	.	.	PUNCT
ejpam-7135	217	6	/	/	SYM
ejpam-7135	217	7	eur	eur	PROPN
ejpam-7135	217	8	.	.	PUNCT
ejpam-7135	218	1	j.	j.	PROPN
ejpam-7135	218	2	pure	pure	PROPN
ejpam-7135	218	3	appl	appl	PROPN
ejpam-7135	218	4	.	.	PROPN
ejpam-7135	218	5	math	math	PROPN
ejpam-7135	218	6	,	,	PUNCT
ejpam-7135	218	7	18	18	NUM
ejpam-7135	218	8	(	(	PUNCT
ejpam-7135	218	9	4	4	NUM
ejpam-7135	218	10	)	)	PUNCT
ejpam-7135	218	11	(	(	PUNCT
ejpam-7135	218	12	2025	2025	NUM
ejpam-7135	218	13	)	)	PUNCT
ejpam-7135	218	14	,	,	PUNCT
ejpam-7135	218	15	7135	7135	NUM
ejpam-7135	218	16	8	8	NUM
ejpam-7135	218	17	of	of	ADP
ejpam-7135	218	18	14	14	NUM
ejpam-7135	218	19	proof	proof	NOUN
ejpam-7135	218	20	.	.	PUNCT
ejpam-7135	219	1	(	(	PUNCT
ejpam-7135	219	2	1	1	X
ejpam-7135	219	3	)	)	PUNCT
ejpam-7135	219	4	suppose	suppose	VERB
ejpam-7135	219	5	x	x	X
ejpam-7135	219	6	/	/	SYM
ejpam-7135	219	7	θi	θi	ADP
ejpam-7135	219	8	∈	∈	PROPN
ejpam-7135	219	9	p	p	PROPN
ejpam-7135	219	10	.	.	PUNCT
ejpam-7135	220	1	then	then	ADV
ejpam-7135	220	2	there	there	PRON
ejpam-7135	220	3	exists	exist	VERB
ejpam-7135	220	4	y	y	PROPN
ejpam-7135	220	5	∈	∈	PROPN
ejpam-7135	220	6	p	p	NOUN
ejpam-7135	220	7	such	such	ADJ
ejpam-7135	220	8	that	that	SCONJ
ejpam-7135	220	9	x	x	PROPN
ejpam-7135	220	10	/	/	SYM
ejpam-7135	220	11	θi	θi	ADP
ejpam-7135	220	12	=	=	SYM
ejpam-7135	220	13	y	y	PROPN
ejpam-7135	220	14	/	/	SYM
ejpam-7135	220	15	θi	θi	PROPN
ejpam-7135	220	16	.	.	PUNCT
ejpam-7135	221	1	hence	hence	ADV
ejpam-7135	221	2	,	,	PUNCT
ejpam-7135	221	3	(	(	PUNCT
ejpam-7135	221	4	x	x	X
ejpam-7135	221	5	,	,	PUNCT
ejpam-7135	221	6	y	y	PROPN
ejpam-7135	221	7	)	)	PUNCT
ejpam-7135	221	8	∈	∈	PROPN
ejpam-7135	221	9	θi	θi	NOUN
ejpam-7135	221	10	,	,	PUNCT
ejpam-7135	221	11	so	so	ADV
ejpam-7135	221	12	x∨d	x∨d	PROPN
ejpam-7135	222	1	=	=	PUNCT
ejpam-7135	222	2	y∨d	y∨d	PROPN
ejpam-7135	222	3	for	for	ADP
ejpam-7135	222	4	some	some	DET
ejpam-7135	222	5	d	d	PROPN
ejpam-7135	222	6	∈	∈	PROPN
ejpam-7135	222	7	i.	i.	NOUN
ejpam-7135	222	8	since	since	SCONJ
ejpam-7135	222	9	y	y	PROPN
ejpam-7135	222	10	∈	∈	PROPN
ejpam-7135	222	11	p	p	PROPN
ejpam-7135	222	12	and	and	CCONJ
ejpam-7135	222	13	p	p	NOUN
ejpam-7135	222	14	is	be	AUX
ejpam-7135	222	15	a	a	DET
ejpam-7135	222	16	filter	filter	NOUN
ejpam-7135	222	17	,	,	PUNCT
ejpam-7135	222	18	we	we	PRON
ejpam-7135	222	19	have	have	VERB
ejpam-7135	222	20	y∨d	y∨d	PROPN
ejpam-7135	222	21	∈	∈	PROPN
ejpam-7135	222	22	p	p	PROPN
ejpam-7135	222	23	.	.	PUNCT
ejpam-7135	223	1	thus	thus	ADV
ejpam-7135	223	2	,	,	PUNCT
ejpam-7135	223	3	x	x	PUNCT
ejpam-7135	223	4	∨	∨	PROPN
ejpam-7135	223	5	d	d	X
ejpam-7135	223	6	∈	∈	PROPN
ejpam-7135	223	7	p	p	NOUN
ejpam-7135	223	8	.	.	PUNCT
ejpam-7135	224	1	as	as	ADP
ejpam-7135	224	2	d	d	PROPN
ejpam-7135	224	3	/∈	/∈	PUNCT
ejpam-7135	225	1	p	p	X
ejpam-7135	225	2	(	(	PUNCT
ejpam-7135	225	3	because	because	SCONJ
ejpam-7135	225	4	p	p	PRON
ejpam-7135	225	5	⊆	⊆	NUM
ejpam-7135	225	6	l	l	NOUN
ejpam-7135	225	7	\	\	PUNCT
ejpam-7135	225	8	i	i	PROPN
ejpam-7135	225	9	)	)	PUNCT
ejpam-7135	225	10	,	,	PUNCT
ejpam-7135	225	11	it	it	PRON
ejpam-7135	225	12	follows	follow	VERB
ejpam-7135	225	13	that	that	SCONJ
ejpam-7135	225	14	x	x	PUNCT
ejpam-7135	225	15	∈	∈	PROPN
ejpam-7135	225	16	p	p	NOUN
ejpam-7135	225	17	.	.	PUNCT
ejpam-7135	226	1	conversely	conversely	ADV
ejpam-7135	226	2	,	,	PUNCT
ejpam-7135	226	3	if	if	SCONJ
ejpam-7135	226	4	x	x	PROPN
ejpam-7135	226	5	∈	∈	PROPN
ejpam-7135	226	6	p	p	NOUN
ejpam-7135	226	7	,	,	PUNCT
ejpam-7135	226	8	then	then	ADV
ejpam-7135	226	9	trivially	trivially	ADV
ejpam-7135	226	10	x	x	PART
ejpam-7135	226	11	/	/	SYM
ejpam-7135	226	12	θi	θi	ADP
ejpam-7135	226	13	∈	∈	PROPN
ejpam-7135	226	14	p	p	X
ejpam-7135	226	15	.	.	PUNCT
ejpam-7135	227	1	(	(	PUNCT
ejpam-7135	227	2	2	2	NUM
ejpam-7135	227	3	)	)	PUNCT
ejpam-7135	227	4	from	from	ADP
ejpam-7135	227	5	(	(	PUNCT
ejpam-7135	227	6	1	1	NUM
ejpam-7135	227	7	)	)	PUNCT
ejpam-7135	227	8	,	,	PUNCT
ejpam-7135	227	9	membership	membership	NOUN
ejpam-7135	227	10	is	be	AUX
ejpam-7135	227	11	preserved	preserve	VERB
ejpam-7135	227	12	under	under	ADP
ejpam-7135	227	13	the	the	DET
ejpam-7135	227	14	mapping	mapping	NOUN
ejpam-7135	227	15	p	p	PROPN
ejpam-7135	227	16	7→	7→	NUM
ejpam-7135	227	17	p	p	NOUN
ejpam-7135	227	18	.	.	PUNCT
ejpam-7135	228	1	as	as	SCONJ
ejpam-7135	228	2	p	p	PRON
ejpam-7135	228	3	is	be	AUX
ejpam-7135	228	4	a	a	DET
ejpam-7135	228	5	prime	prime	ADJ
ejpam-7135	228	6	filter	filter	NOUN
ejpam-7135	228	7	of	of	ADP
ejpam-7135	228	8	l	l	NOUN
ejpam-7135	228	9	,	,	PUNCT
ejpam-7135	228	10	it	it	PRON
ejpam-7135	228	11	follows	follow	VERB
ejpam-7135	228	12	directly	directly	ADV
ejpam-7135	228	13	that	that	SCONJ
ejpam-7135	228	14	p	p	PROPN
ejpam-7135	228	15	is	be	AUX
ejpam-7135	228	16	a	a	DET
ejpam-7135	228	17	prime	prime	ADJ
ejpam-7135	228	18	filter	filter	NOUN
ejpam-7135	228	19	of	of	ADP
ejpam-7135	228	20	l	l	PROPN
ejpam-7135	228	21	/	/	SYM
ejpam-7135	228	22	θi	θi	PROPN
ejpam-7135	228	23	.	.	PUNCT
ejpam-7135	229	1	theorem	theorem	VERB
ejpam-7135	229	2	9	9	NUM
ejpam-7135	229	3	.	.	PUNCT
ejpam-7135	230	1	let	let	VERB
ejpam-7135	230	2	i	i	PRON
ejpam-7135	230	3	be	be	AUX
ejpam-7135	230	4	a	a	DET
ejpam-7135	230	5	non	non	ADJ
ejpam-7135	230	6	-	-	ADJ
ejpam-7135	230	7	empty	empty	ADJ
ejpam-7135	230	8	subset	subset	NOUN
ejpam-7135	230	9	of	of	ADP
ejpam-7135	230	10	l	l	PROPN
ejpam-7135	230	11	closed	close	VERB
ejpam-7135	230	12	under	under	ADP
ejpam-7135	230	13	∨.	∨.	NOUN
ejpam-7135	230	14	let	let	VERB
ejpam-7135	230	15	e	e	PRON
ejpam-7135	230	16	be	be	AUX
ejpam-7135	230	17	the	the	DET
ejpam-7135	230	18	set	set	NOUN
ejpam-7135	230	19	of	of	ADP
ejpam-7135	230	20	prime	prime	ADJ
ejpam-7135	230	21	filters	filter	NOUN
ejpam-7135	230	22	of	of	ADP
ejpam-7135	230	23	l	l	NOUN
ejpam-7135	230	24	contained	contain	VERB
ejpam-7135	230	25	in	in	ADP
ejpam-7135	230	26	l	l	NOUN
ejpam-7135	230	27	\	\	PROPN
ejpam-7135	231	1	i	i	PRON
ejpam-7135	231	2	,	,	PUNCT
ejpam-7135	231	3	and	and	CCONJ
ejpam-7135	231	4	let	let	VERB
ejpam-7135	231	5	g	g	NOUN
ejpam-7135	231	6	be	be	AUX
ejpam-7135	231	7	the	the	DET
ejpam-7135	231	8	set	set	NOUN
ejpam-7135	231	9	of	of	ADP
ejpam-7135	231	10	prime	prime	ADJ
ejpam-7135	231	11	filters	filter	NOUN
ejpam-7135	231	12	of	of	ADP
ejpam-7135	231	13	l	l	PROPN
ejpam-7135	231	14	/	/	SYM
ejpam-7135	231	15	θi	θi	PROPN
ejpam-7135	231	16	.	.	PUNCT
ejpam-7135	232	1	then	then	ADV
ejpam-7135	232	2	the	the	DET
ejpam-7135	232	3	map	map	NOUN
ejpam-7135	232	4	f	f	X
ejpam-7135	232	5	:	:	PUNCT
ejpam-7135	232	6	e	e	X
ejpam-7135	232	7	→	→	SYM
ejpam-7135	232	8	g	g	PROPN
ejpam-7135	232	9	,	,	PUNCT
ejpam-7135	232	10	f(p	f(p	PROPN
ejpam-7135	232	11	)	)	PUNCT
ejpam-7135	233	1	=	=	PUNCT
ejpam-7135	234	1	p	p	NOUN
ejpam-7135	234	2	,	,	PUNCT
ejpam-7135	234	3	is	be	AUX
ejpam-7135	234	4	an	an	DET
ejpam-7135	234	5	order	order	NOUN
ejpam-7135	234	6	isomorphism	isomorphism	NOUN
ejpam-7135	234	7	with	with	ADP
ejpam-7135	234	8	respect	respect	NOUN
ejpam-7135	234	9	to	to	ADP
ejpam-7135	234	10	inclusion	inclusion	NOUN
ejpam-7135	234	11	.	.	PUNCT
ejpam-7135	235	1	proof	proof	NOUN
ejpam-7135	235	2	.	.	PUNCT
ejpam-7135	236	1	by	by	ADP
ejpam-7135	236	2	lemma	lemma	PROPN
ejpam-7135	236	3	6	6	NUM
ejpam-7135	236	4	,	,	PUNCT
ejpam-7135	236	5	f	f	PROPN
ejpam-7135	236	6	is	be	AUX
ejpam-7135	236	7	well	well	ADV
ejpam-7135	236	8	-	-	PUNCT
ejpam-7135	236	9	defined	define	VERB
ejpam-7135	236	10	.	.	PUNCT
ejpam-7135	237	1	if	if	SCONJ
ejpam-7135	237	2	p	p	X
ejpam-7135	237	3	,	,	PUNCT
ejpam-7135	237	4	q	q	NOUN
ejpam-7135	237	5	∈	∈	PROPN
ejpam-7135	237	6	e	e	NOUN
ejpam-7135	237	7	,	,	PUNCT
ejpam-7135	237	8	then	then	ADV
ejpam-7135	237	9	p	p	NOUN
ejpam-7135	237	10	⊆	⊆	NUM
ejpam-7135	237	11	q	q	NOUN
ejpam-7135	237	12	⇐	⇐	ADJ
ejpam-7135	237	13	⇒	⇒	NOUN
ejpam-7135	237	14	p	p	NOUN
ejpam-7135	237	15	⊆	⊆	NUM
ejpam-7135	237	16	q	q	NOUN
ejpam-7135	237	17	,	,	PUNCT
ejpam-7135	237	18	so	so	CCONJ
ejpam-7135	237	19	f	f	PROPN
ejpam-7135	237	20	is	be	AUX
ejpam-7135	237	21	order	order	NOUN
ejpam-7135	237	22	-	-	PUNCT
ejpam-7135	237	23	preserving	preserve	VERB
ejpam-7135	237	24	and	and	CCONJ
ejpam-7135	237	25	order	order	NOUN
ejpam-7135	237	26	-	-	PUNCT
ejpam-7135	237	27	reflecting	reflect	VERB
ejpam-7135	237	28	.	.	PUNCT
ejpam-7135	238	1	to	to	PART
ejpam-7135	238	2	show	show	VERB
ejpam-7135	238	3	surjectivity	surjectivity	NOUN
ejpam-7135	238	4	,	,	PUNCT
ejpam-7135	238	5	let	let	VERB
ejpam-7135	238	6	m	m	PRON
ejpam-7135	238	7	∈	∈	NOUN
ejpam-7135	238	8	g.	g.	NOUN
ejpam-7135	238	9	define	define	VERB
ejpam-7135	238	10	p	p	X
ejpam-7135	238	11	:	:	PUNCT
ejpam-7135	238	12	=	=	SYM
ejpam-7135	238	13	{	{	PUNCT
ejpam-7135	238	14	x	x	PUNCT
ejpam-7135	238	15	∈	∈	NOUN
ejpam-7135	238	16	l	l	NOUN
ejpam-7135	239	1	|	|	NOUN
ejpam-7135	239	2	x	x	X
ejpam-7135	239	3	/	/	SYM
ejpam-7135	239	4	θi	θi	ADP
ejpam-7135	239	5	∈	∈	PROPN
ejpam-7135	239	6	m	m	PROPN
ejpam-7135	239	7	}	}	PUNCT
ejpam-7135	239	8	.	.	PUNCT
ejpam-7135	240	1	then	then	ADV
ejpam-7135	240	2	p	p	NOUN
ejpam-7135	240	3	is	be	AUX
ejpam-7135	240	4	a	a	DET
ejpam-7135	240	5	prime	prime	ADJ
ejpam-7135	240	6	filter	filter	NOUN
ejpam-7135	240	7	of	of	ADP
ejpam-7135	240	8	l.	l.	PROPN
ejpam-7135	240	9	if	if	SCONJ
ejpam-7135	240	10	p	p	NOUN
ejpam-7135	240	11	∩	∩	NOUN
ejpam-7135	240	12	i	i	PRON
ejpam-7135	240	13	̸=	̸=	PROPN
ejpam-7135	240	14	∅	∅	NOUN
ejpam-7135	240	15	,	,	PUNCT
ejpam-7135	240	16	pick	pick	VERB
ejpam-7135	240	17	x	x	PUNCT
ejpam-7135	240	18	∈	∈	PROPN
ejpam-7135	240	19	p	p	PROPN
ejpam-7135	240	20	∩	∩	ADJ
ejpam-7135	240	21	i.	i.	NOUN
ejpam-7135	240	22	then	then	ADV
ejpam-7135	240	23	for	for	ADP
ejpam-7135	240	24	any	any	DET
ejpam-7135	240	25	y	y	PROPN
ejpam-7135	240	26	∈	∈	PROPN
ejpam-7135	240	27	l	l	NOUN
ejpam-7135	240	28	,	,	PUNCT
ejpam-7135	240	29	(	(	PUNCT
ejpam-7135	240	30	y	y	PROPN
ejpam-7135	240	31	∨	∨	NUM
ejpam-7135	240	32	x	x	PROPN
ejpam-7135	240	33	,	,	PUNCT
ejpam-7135	240	34	y	y	PROPN
ejpam-7135	240	35	)	)	PUNCT
ejpam-7135	240	36	∈	∈	PROPN
ejpam-7135	240	37	θi	θi	ADP
ejpam-7135	240	38	,	,	PUNCT
ejpam-7135	240	39	so	so	ADV
ejpam-7135	240	40	y	y	PROPN
ejpam-7135	240	41	/	/	SYM
ejpam-7135	240	42	θi	θi	NOUN
ejpam-7135	240	43	=	=	SYM
ejpam-7135	240	44	(	(	PUNCT
ejpam-7135	240	45	y	y	PROPN
ejpam-7135	240	46	∨	∨	NUM
ejpam-7135	240	47	x)/θi	x)/θi	PROPN
ejpam-7135	241	1	∈	∈	PROPN
ejpam-7135	241	2	m	m	VERB
ejpam-7135	241	3	.	.	PUNCT
ejpam-7135	242	1	thus	thus	ADV
ejpam-7135	242	2	,	,	PUNCT
ejpam-7135	242	3	m	m	VERB
ejpam-7135	242	4	=	=	ADJ
ejpam-7135	242	5	l	l	NOUN
ejpam-7135	242	6	/	/	SYM
ejpam-7135	242	7	θi	θi	PROPN
ejpam-7135	242	8	,	,	PUNCT
ejpam-7135	242	9	a	a	DET
ejpam-7135	242	10	contradiction	contradiction	NOUN
ejpam-7135	242	11	since	since	SCONJ
ejpam-7135	242	12	m	m	PROPN
ejpam-7135	242	13	is	be	AUX
ejpam-7135	242	14	proper	proper	ADJ
ejpam-7135	242	15	.	.	PUNCT
ejpam-7135	243	1	hence	hence	ADV
ejpam-7135	243	2	,	,	PUNCT
ejpam-7135	243	3	p	p	ADJ
ejpam-7135	243	4	⊆	⊆	NUM
ejpam-7135	243	5	l	l	NOUN
ejpam-7135	243	6	\	\	X
ejpam-7135	244	1	i	i	PRON
ejpam-7135	244	2	,	,	PUNCT
ejpam-7135	244	3	i.e.	i.e.	X
ejpam-7135	244	4	,	,	PUNCT
ejpam-7135	244	5	p	p	NOUN
ejpam-7135	244	6	∈	∈	PROPN
ejpam-7135	244	7	e	e	NOUN
ejpam-7135	244	8	,	,	PUNCT
ejpam-7135	244	9	and	and	CCONJ
ejpam-7135	244	10	f(p	f(p	NOUN
ejpam-7135	244	11	)	)	PUNCT
ejpam-7135	245	1	=	=	PUNCT
ejpam-7135	246	1	p	p	NOUN
ejpam-7135	246	2	=	=	NOUN
ejpam-7135	246	3	m	m	PROPN
ejpam-7135	246	4	.	.	PUNCT
ejpam-7135	247	1	therefore	therefore	ADV
ejpam-7135	247	2	,	,	PUNCT
ejpam-7135	247	3	f	f	PROPN
ejpam-7135	247	4	is	be	AUX
ejpam-7135	247	5	a	a	DET
ejpam-7135	247	6	bijection	bijection	NOUN
ejpam-7135	247	7	and	and	CCONJ
ejpam-7135	247	8	an	an	DET
ejpam-7135	247	9	order	order	NOUN
ejpam-7135	247	10	isomorphism	isomorphism	NOUN
ejpam-7135	247	11	.	.	PUNCT
ejpam-7135	248	1	definition	definition	NOUN
ejpam-7135	248	2	9	9	NUM
ejpam-7135	248	3	.	.	PUNCT
ejpam-7135	249	1	two	two	NUM
ejpam-7135	249	2	filters	filter	NOUN
ejpam-7135	249	3	f	f	NOUN
ejpam-7135	249	4	,	,	PUNCT
ejpam-7135	249	5	g	g	PROPN
ejpam-7135	249	6	of	of	ADP
ejpam-7135	249	7	l	l	NOUN
ejpam-7135	249	8	are	be	AUX
ejpam-7135	249	9	said	say	VERB
ejpam-7135	249	10	to	to	PART
ejpam-7135	249	11	be	be	AUX
ejpam-7135	249	12	co	co	VERB
ejpam-7135	249	13	-	-	ADJ
ejpam-7135	249	14	maximal	maximal	ADJ
ejpam-7135	249	15	if	if	SCONJ
ejpam-7135	249	16	f	f	PROPN
ejpam-7135	249	17	∨g	∨g	PROPN
ejpam-7135	249	18	=	=	SYM
ejpam-7135	249	19	l.	l.	PROPN
ejpam-7135	249	20	theorem	theorem	VERB
ejpam-7135	249	21	10	10	NUM
ejpam-7135	249	22	.	.	PUNCT
ejpam-7135	250	1	l	l	NOUN
ejpam-7135	250	2	with	with	ADP
ejpam-7135	250	3	a	a	DET
ejpam-7135	250	4	parapseudo	parapseudo	NOUN
ejpam-7135	250	5	-	-	PUNCT
ejpam-7135	250	6	complementation	complementation	NOUN
ejpam-7135	250	7	♦	♦	PROPN
ejpam-7135	250	8	is	be	AUX
ejpam-7135	250	9	a	a	DET
ejpam-7135	250	10	stone	stone	NOUN
ejpam-7135	250	11	pdl	pdl	NOUN
ejpam-7135	250	12	if	if	SCONJ
ejpam-7135	251	1	and	and	CCONJ
ejpam-7135	251	2	only	only	ADV
ejpam-7135	251	3	if	if	SCONJ
ejpam-7135	251	4	any	any	DET
ejpam-7135	251	5	two	two	NUM
ejpam-7135	251	6	distinct	distinct	ADJ
ejpam-7135	251	7	minimal	minimal	ADJ
ejpam-7135	251	8	prime	prime	ADJ
ejpam-7135	251	9	filters	filter	NOUN
ejpam-7135	251	10	of	of	ADP
ejpam-7135	251	11	l	l	NOUN
ejpam-7135	251	12	are	be	AUX
ejpam-7135	251	13	co	co	ADJ
ejpam-7135	251	14	-	-	ADJ
ejpam-7135	251	15	maximal	maximal	ADJ
ejpam-7135	251	16	.	.	PUNCT
ejpam-7135	252	1	proof	proof	NOUN
ejpam-7135	252	2	.	.	PUNCT
ejpam-7135	253	1	(	(	PUNCT
ejpam-7135	253	2	⇐	⇐	NOUN
ejpam-7135	253	3	)	)	PUNCT
ejpam-7135	253	4	suppose	suppose	VERB
ejpam-7135	253	5	l	l	NOUN
ejpam-7135	253	6	is	be	AUX
ejpam-7135	253	7	not	not	PART
ejpam-7135	253	8	a	a	DET
ejpam-7135	253	9	stone	stone	NOUN
ejpam-7135	253	10	pdl	pdl	NOUN
ejpam-7135	253	11	.	.	PUNCT
ejpam-7135	254	1	then	then	ADV
ejpam-7135	254	2	there	there	PRON
ejpam-7135	254	3	exists	exist	VERB
ejpam-7135	254	4	x	x	X
ejpam-7135	254	5	∈	∈	NOUN
ejpam-7135	254	6	l	l	NOUN
ejpam-7135	254	7	such	such	ADJ
ejpam-7135	254	8	that	that	SCONJ
ejpam-7135	255	1	[	[	X
ejpam-7135	255	2	x]•	x]•	PROPN
ejpam-7135	255	3	∨	∨	PROPN
ejpam-7135	255	4	[	[	X
ejpam-7135	255	5	x]••	x]••	PUNCT
ejpam-7135	255	6	̸=	̸=	PROPN
ejpam-7135	255	7	l.	l.	NOUN
ejpam-7135	255	8	hence	hence	ADV
ejpam-7135	255	9	,	,	PUNCT
ejpam-7135	255	10	there	there	PRON
ejpam-7135	255	11	is	be	VERB
ejpam-7135	255	12	a	a	DET
ejpam-7135	255	13	prime	prime	ADJ
ejpam-7135	255	14	filter	filter	NOUN
ejpam-7135	255	15	r	r	NOUN
ejpam-7135	255	16	of	of	ADP
ejpam-7135	255	17	l	l	NOUN
ejpam-7135	255	18	with	with	ADP
ejpam-7135	255	19	[	[	X
ejpam-7135	255	20	x]•∨	x]•∨	NUM
ejpam-7135	255	21	[	[	X
ejpam-7135	255	22	x]••	x]••	PUNCT
ejpam-7135	255	23	⊆	⊆	NUM
ejpam-7135	255	24	r.	r.	NOUN
ejpam-7135	255	25	since	since	SCONJ
ejpam-7135	255	26	l	l	PROPN
ejpam-7135	255	27	is	be	AUX
ejpam-7135	255	28	not	not	PART
ejpam-7135	255	29	a	a	DET
ejpam-7135	255	30	stone	stone	NOUN
ejpam-7135	255	31	pdl	pdl	NOUN
ejpam-7135	255	32	,	,	PUNCT
ejpam-7135	255	33	there	there	PRON
ejpam-7135	255	34	exists	exist	VERB
ejpam-7135	255	35	x′	x′	PROPN
ejpam-7135	255	36	∈	∈	PROPN
ejpam-7135	255	37	l	l	NOUN
ejpam-7135	255	38	with	with	ADP
ejpam-7135	255	39	[	[	X
ejpam-7135	255	40	x]••	x]••	PUNCT
ejpam-7135	255	41	=	=	PUNCT
ejpam-7135	256	1	[	[	X
ejpam-7135	256	2	x′]•	x′]•	X
ejpam-7135	256	3	,	,	PUNCT
ejpam-7135	256	4	so	so	SCONJ
ejpam-7135	256	5	[	[	X
ejpam-7135	256	6	x]•∨	x]•∨	PRON
ejpam-7135	256	7	[	[	X
ejpam-7135	256	8	x′]•	x′]•	VERB
ejpam-7135	256	9	⊆	⊆	NUM
ejpam-7135	256	10	r.	r.	NOUN
ejpam-7135	256	11	thus	thus	ADV
ejpam-7135	256	12	,	,	PUNCT
ejpam-7135	256	13	x	x	X
ejpam-7135	256	14	,	,	PUNCT
ejpam-7135	256	15	x′	x′	PROPN
ejpam-7135	256	16	∈	∈	PROPN
ejpam-7135	256	17	r	r	NOUN
ejpam-7135	256	18	and	and	CCONJ
ejpam-7135	256	19	neither	neither	PRON
ejpam-7135	256	20	is	be	AUX
ejpam-7135	256	21	equal	equal	ADJ
ejpam-7135	256	22	to	to	ADP
ejpam-7135	256	23	1	1	NUM
ejpam-7135	256	24	.	.	PUNCT
ejpam-7135	257	1	now	now	ADV
ejpam-7135	257	2	set	set	VERB
ejpam-7135	257	3	i	i	NOUN
ejpam-7135	258	1	=	=	PUNCT
ejpam-7135	258	2	l	l	X
ejpam-7135	258	3	\	\	PROPN
ejpam-7135	258	4	r.	r.	PROPN
ejpam-7135	258	5	then	then	ADV
ejpam-7135	258	6	i	i	PRON
ejpam-7135	258	7	is	be	AUX
ejpam-7135	258	8	a	a	DET
ejpam-7135	258	9	prime	prime	ADJ
ejpam-7135	258	10	ideal	ideal	NOUN
ejpam-7135	258	11	,	,	PUNCT
ejpam-7135	258	12	and	and	CCONJ
ejpam-7135	258	13	θi	θi	NOUN
ejpam-7135	258	14	is	be	AUX
ejpam-7135	258	15	a	a	DET
ejpam-7135	258	16	congruence	congruence	NOUN
ejpam-7135	258	17	.	.	PUNCT
ejpam-7135	259	1	since	since	SCONJ
ejpam-7135	259	2	x	x	PROPN
ejpam-7135	259	3	/	/	SYM
ejpam-7135	259	4	θi	θi	ADP
ejpam-7135	259	5	̸=	̸=	PROPN
ejpam-7135	259	6	1	1	NUM
ejpam-7135	259	7	/	/	SYM
ejpam-7135	259	8	θi	θi	NOUN
ejpam-7135	259	9	and	and	CCONJ
ejpam-7135	259	10	x′/θi	x′/θi	VERB
ejpam-7135	259	11	̸=	̸=	PROPN
ejpam-7135	259	12	1	1	NUM
ejpam-7135	259	13	/	/	SYM
ejpam-7135	259	14	θi	θi	NOUN
ejpam-7135	259	15	,	,	PUNCT
ejpam-7135	259	16	there	there	PRON
ejpam-7135	259	17	exist	exist	VERB
ejpam-7135	259	18	prime	prime	ADJ
ejpam-7135	259	19	filters	filter	NOUN
ejpam-7135	259	20	p	p	NOUN
ejpam-7135	259	21	,	,	PUNCT
ejpam-7135	259	22	q	q	NOUN
ejpam-7135	259	23	of	of	ADP
ejpam-7135	259	24	l	l	PROPN
ejpam-7135	259	25	/	/	SYM
ejpam-7135	259	26	θi	θi	NOUN
ejpam-7135	259	27	such	such	ADJ
ejpam-7135	259	28	that	that	SCONJ
ejpam-7135	259	29	x	x	PROPN
ejpam-7135	259	30	/	/	SYM
ejpam-7135	259	31	θi	θi	NOUN
ejpam-7135	259	32	/∈	/∈	PUNCT
ejpam-7135	259	33	p	p	NOUN
ejpam-7135	259	34	and	and	CCONJ
ejpam-7135	259	35	x′/θi	x′/θi	PROPN
ejpam-7135	259	36	/∈	/∈	PUNCT
ejpam-7135	259	37	q.	q.	PROPN
ejpam-7135	259	38	by	by	ADP
ejpam-7135	259	39	theorem	theorem	NOUN
ejpam-7135	259	40	9	9	NUM
ejpam-7135	259	41	,	,	PUNCT
ejpam-7135	259	42	there	there	PRON
ejpam-7135	259	43	exist	exist	VERB
ejpam-7135	259	44	minimal	minimal	ADJ
ejpam-7135	259	45	prime	prime	ADJ
ejpam-7135	259	46	filters	filter	NOUN
ejpam-7135	259	47	p	p	PROPN
ejpam-7135	259	48	′	′	NOUN
ejpam-7135	259	49	,	,	PUNCT
ejpam-7135	259	50	q′	q′	NOUN
ejpam-7135	259	51	of	of	ADP
ejpam-7135	259	52	l	l	PROPN
ejpam-7135	259	53	contained	contain	VERB
ejpam-7135	259	54	in	in	ADP
ejpam-7135	259	55	r	r	NOUN
ejpam-7135	259	56	with	with	ADP
ejpam-7135	259	57	f(p	f(p	PROPN
ejpam-7135	259	58	′	′	NUM
ejpam-7135	259	59	)	)	PUNCT
ejpam-7135	260	1	=	=	SYM
ejpam-7135	260	2	p	p	NOUN
ejpam-7135	260	3	,	,	PUNCT
ejpam-7135	260	4	f(q′	f(q′	PROPN
ejpam-7135	260	5	)	)	PUNCT
ejpam-7135	260	6	=	=	VERB
ejpam-7135	261	1	q.	q.	NOUN
ejpam-7135	261	2	then	then	ADV
ejpam-7135	261	3	p	p	PROPN
ejpam-7135	261	4	′	′	NOUN
ejpam-7135	261	5	,	,	PUNCT
ejpam-7135	261	6	q′	q′	NOUN
ejpam-7135	261	7	are	be	AUX
ejpam-7135	261	8	distinct	distinct	ADJ
ejpam-7135	261	9	minimal	minimal	ADJ
ejpam-7135	261	10	prime	prime	ADJ
ejpam-7135	261	11	filters	filter	NOUN
ejpam-7135	261	12	of	of	ADP
ejpam-7135	261	13	l	l	NOUN
ejpam-7135	262	1	such	such	ADJ
ejpam-7135	262	2	that	that	SCONJ
ejpam-7135	262	3	p	p	NOUN
ejpam-7135	262	4	′	′	NUM
ejpam-7135	262	5	∨q′	∨q′	NOUN
ejpam-7135	262	6	⊆	⊆	NUM
ejpam-7135	262	7	r	r	NOUN
ejpam-7135	262	8	̸=	̸=	PROPN
ejpam-7135	262	9	l	l	NOUN
ejpam-7135	262	10	,	,	PUNCT
ejpam-7135	262	11	so	so	SCONJ
ejpam-7135	262	12	they	they	PRON
ejpam-7135	262	13	are	be	AUX
ejpam-7135	262	14	not	not	PART
ejpam-7135	262	15	co	co	ADJ
ejpam-7135	262	16	-	-	ADJ
ejpam-7135	262	17	maximal	maximal	ADJ
ejpam-7135	262	18	.	.	PUNCT
ejpam-7135	263	1	(	(	PUNCT
ejpam-7135	263	2	⇒	⇒	NOUN
ejpam-7135	263	3	)	)	PUNCT
ejpam-7135	263	4	conversely	conversely	ADV
ejpam-7135	263	5	,	,	PUNCT
ejpam-7135	263	6	assume	assume	VERB
ejpam-7135	263	7	l	l	NOUN
ejpam-7135	263	8	is	be	AUX
ejpam-7135	263	9	a	a	DET
ejpam-7135	263	10	stone	stone	NOUN
ejpam-7135	263	11	pdl	pdl	NOUN
ejpam-7135	263	12	,	,	PUNCT
ejpam-7135	263	13	and	and	CCONJ
ejpam-7135	263	14	let	let	VERB
ejpam-7135	263	15	p	p	PRON
ejpam-7135	263	16	,	,	PUNCT
ejpam-7135	263	17	q	q	PUNCT
ejpam-7135	263	18	be	be	AUX
ejpam-7135	263	19	distinct	distinct	ADJ
ejpam-7135	263	20	minimal	minimal	ADJ
ejpam-7135	263	21	prime	prime	ADJ
ejpam-7135	263	22	filters	filter	NOUN
ejpam-7135	263	23	.	.	PUNCT
ejpam-7135	264	1	pick	pick	VERB
ejpam-7135	264	2	a	a	DET
ejpam-7135	264	3	∈	∈	NOUN
ejpam-7135	264	4	p	p	PROPN
ejpam-7135	264	5	\q	\q	NOUN
ejpam-7135	264	6	.	.	PUNCT
ejpam-7135	265	1	then	then	ADV
ejpam-7135	265	2	a	a	DET
ejpam-7135	265	3	♦	♦	PROPN
ejpam-7135	265	4	∈	∈	PROPN
ejpam-7135	265	5	q.	q.	PROPN
ejpam-7135	265	6	since	since	SCONJ
ejpam-7135	265	7	l	l	PROPN
ejpam-7135	265	8	\	\	PROPN
ejpam-7135	266	1	p	p	NOUN
ejpam-7135	266	2	is	be	AUX
ejpam-7135	266	3	a	a	DET
ejpam-7135	266	4	maximal	maximal	ADJ
ejpam-7135	266	5	prime	prime	ADJ
ejpam-7135	266	6	ideal	ideal	NOUN
ejpam-7135	266	7	,	,	PUNCT
ejpam-7135	266	8	there	there	PRON
ejpam-7135	266	9	exists	exist	VERB
ejpam-7135	266	10	t	t	PROPN
ejpam-7135	266	11	∈	∈	PROPN
ejpam-7135	266	12	l	l	NOUN
ejpam-7135	266	13	\p	\p	ADV
ejpam-7135	266	14	with	with	ADP
ejpam-7135	266	15	a∨	a∨	PROPN
ejpam-7135	266	16	t	t	NOUN
ejpam-7135	266	17	=	=	SYM
ejpam-7135	266	18	1	1	X
ejpam-7135	266	19	.	.	PUNCT
ejpam-7135	267	1	but	but	CCONJ
ejpam-7135	267	2	then	then	ADV
ejpam-7135	267	3	a	a	DET
ejpam-7135	267	4	♦	♦	PROPN
ejpam-7135	267	5	∈	∈	PROPN
ejpam-7135	267	6	l	l	NOUN
ejpam-7135	267	7	\p	\p	ADV
ejpam-7135	267	8	,	,	PUNCT
ejpam-7135	267	9	so	so	ADV
ejpam-7135	267	10	a	a	DET
ejpam-7135	267	11	♦	♦	PROPN
ejpam-7135	267	12	/∈	/∈	PROPN
ejpam-7135	268	1	p	p	X
ejpam-7135	268	2	.	.	PUNCT
ejpam-7135	269	1	as	as	ADP
ejpam-7135	269	2	a	a	DET
ejpam-7135	269	3	♦	♦	PROPN
ejpam-7135	269	4	∨	∨	PROPN
ejpam-7135	269	5	a	a	DET
ejpam-7135	269	6	♦	♦	PROPN
ejpam-7135	269	7	♦	♦	PROPN
ejpam-7135	269	8	=	=	PROPN
ejpam-7135	269	9	1	1	PROPN
ejpam-7135	269	10	,	,	PUNCT
ejpam-7135	269	11	it	it	PRON
ejpam-7135	269	12	follows	follow	VERB
ejpam-7135	269	13	that	that	SCONJ
ejpam-7135	269	14	1	1	NUM
ejpam-7135	269	15	♦	♦	PROPN
ejpam-7135	269	16	∈	∈	PROPN
ejpam-7135	269	17	p	p	PROPN
ejpam-7135	269	18	∨q	∨q	NOUN
ejpam-7135	269	19	,	,	PUNCT
ejpam-7135	269	20	hence	hence	ADV
ejpam-7135	269	21	p	p	NOUN
ejpam-7135	269	22	∨q	∨q	PROPN
ejpam-7135	269	23	=	=	PROPN
ejpam-7135	269	24	l.	l.	PROPN
ejpam-7135	269	25	therefore	therefore	ADV
ejpam-7135	269	26	,	,	PUNCT
ejpam-7135	269	27	distinct	distinct	ADJ
ejpam-7135	269	28	minimal	minimal	ADJ
ejpam-7135	269	29	prime	prime	ADJ
ejpam-7135	269	30	filters	filter	NOUN
ejpam-7135	269	31	are	be	AUX
ejpam-7135	269	32	co	co	ADJ
ejpam-7135	269	33	-	-	ADJ
ejpam-7135	269	34	maximal	maximal	ADJ
ejpam-7135	269	35	.	.	PUNCT
ejpam-7135	270	1	in	in	ADP
ejpam-7135	270	2	the	the	DET
ejpam-7135	270	3	following	follow	VERB
ejpam-7135	270	4	lemma	lemma	PROPN
ejpam-7135	270	5	,	,	PUNCT
ejpam-7135	270	6	we	we	PRON
ejpam-7135	270	7	establish	establish	VERB
ejpam-7135	270	8	some	some	DET
ejpam-7135	270	9	relations	relation	NOUN
ejpam-7135	270	10	between	between	ADP
ejpam-7135	270	11	the	the	DET
ejpam-7135	270	12	prime	prime	ADJ
ejpam-7135	270	13	filters	filter	NOUN
ejpam-7135	270	14	of	of	ADP
ejpam-7135	270	15	l	l	NOUN
ejpam-7135	270	16	and	and	CCONJ
ejpam-7135	270	17	prime	prime	ADJ
ejpam-7135	270	18	filters	filter	NOUN
ejpam-7135	270	19	of	of	ADP
ejpam-7135	270	20	b[m	b[m	ADJ
ejpam-7135	270	21	,	,	PUNCT
ejpam-7135	270	22	1	1	NUM
ejpam-7135	270	23	]	]	PUNCT
ejpam-7135	270	24	(	(	PUNCT
ejpam-7135	270	25	the	the	DET
ejpam-7135	270	26	boolean	boolean	ADJ
ejpam-7135	270	27	algebra	algebra	NOUN
ejpam-7135	270	28	of	of	ADP
ejpam-7135	270	29	all	all	DET
ejpam-7135	270	30	complemented	complemented	ADJ
ejpam-7135	270	31	elements	element	NOUN
ejpam-7135	270	32	of	of	ADP
ejpam-7135	270	33	the	the	DET
ejpam-7135	270	34	bounded	bounded	ADJ
ejpam-7135	270	35	distributive	distributive	ADJ
ejpam-7135	270	36	lattice	lattice	NOUN
ejpam-7135	271	1	[	[	X
ejpam-7135	271	2	m	m	NOUN
ejpam-7135	271	3	,	,	PUNCT
ejpam-7135	271	4	1	1	NUM
ejpam-7135	271	5	]	]	NUM
ejpam-7135	271	6	)	)	PUNCT
ejpam-7135	271	7	,	,	PUNCT
ejpam-7135	271	8	which	which	PRON
ejpam-7135	271	9	leads	lead	VERB
ejpam-7135	271	10	to	to	ADP
ejpam-7135	271	11	another	another	DET
ejpam-7135	271	12	characterization	characterization	NOUN
ejpam-7135	271	13	of	of	ADP
ejpam-7135	271	14	stone	stone	NOUN
ejpam-7135	271	15	pdls	pdl	NOUN
ejpam-7135	271	16	,	,	PUNCT
ejpam-7135	271	17	in	in	ADP
ejpam-7135	271	18	terms	term	NOUN
ejpam-7135	271	19	of	of	ADP
ejpam-7135	271	20	minimal	minimal	ADJ
ejpam-7135	271	21	prime	prime	ADJ
ejpam-7135	271	22	filters	filter	NOUN
ejpam-7135	271	23	.	.	PUNCT
ejpam-7135	272	1	lemma	lemma	PROPN
ejpam-7135	272	2	7	7	X
ejpam-7135	272	3	.	.	PUNCT
ejpam-7135	273	1	let	let	VERB
ejpam-7135	273	2	l	l	NOUN
ejpam-7135	273	3	be	be	AUX
ejpam-7135	273	4	a	a	DET
ejpam-7135	273	5	pdl	pdl	NOUN
ejpam-7135	273	6	with	with	ADP
ejpam-7135	273	7	a	a	DET
ejpam-7135	273	8	minimal	minimal	ADJ
ejpam-7135	273	9	element	element	NOUN
ejpam-7135	273	10	m	m	NOUN
ejpam-7135	273	11	,	,	PUNCT
ejpam-7135	273	12	and	and	CCONJ
ejpam-7135	273	13	let	let	VERB
ejpam-7135	273	14	p	p	PROPN
ejpam-7135	273	15	∈	∈	PROPN
ejpam-7135	273	16	y	y	PROPN
ejpam-7135	273	17	,	,	PUNCT
ejpam-7135	273	18	where	where	SCONJ
ejpam-7135	273	19	y	y	PROPN
ejpam-7135	273	20	is	be	AUX
ejpam-7135	273	21	the	the	DET
ejpam-7135	273	22	set	set	NOUN
ejpam-7135	273	23	of	of	ADP
ejpam-7135	273	24	all	all	DET
ejpam-7135	273	25	prime	prime	ADJ
ejpam-7135	273	26	filters	filter	NOUN
ejpam-7135	273	27	of	of	ADP
ejpam-7135	273	28	l.	l.	NOUN
ejpam-7135	273	29	then	then	ADV
ejpam-7135	274	1	p	p	X
ejpam-7135	274	2	c	c	NOUN
ejpam-7135	274	3	:	:	PUNCT
ejpam-7135	274	4	=	=	SYM
ejpam-7135	274	5	p	p	NOUN
ejpam-7135	274	6	∩	∩	NOUN
ejpam-7135	274	7	b[m	b[m	ADJ
ejpam-7135	274	8	,	,	PUNCT
ejpam-7135	274	9	1	1	NUM
ejpam-7135	274	10	]	]	PUNCT
ejpam-7135	274	11	is	be	AUX
ejpam-7135	274	12	a	a	DET
ejpam-7135	274	13	prime	prime	ADJ
ejpam-7135	274	14	filter	filter	NOUN
ejpam-7135	274	15	of	of	ADP
ejpam-7135	274	16	the	the	DET
ejpam-7135	274	17	boolean	boolean	ADJ
ejpam-7135	274	18	algebra	algebra	PROPN
ejpam-7135	274	19	b[m	b[m	PROPN
ejpam-7135	274	20	,	,	PUNCT
ejpam-7135	274	21	1	1	NUM
ejpam-7135	274	22	]	]	PUNCT
ejpam-7135	274	23	.	.	PUNCT
ejpam-7135	275	1	r.	r.	PROPN
ejpam-7135	275	2	bandaru	bandaru	PROPN
ejpam-7135	275	3	et	et	PROPN
ejpam-7135	275	4	al	al	PROPN
ejpam-7135	275	5	.	.	PUNCT
ejpam-7135	275	6	/	/	SYM
ejpam-7135	275	7	eur	eur	PROPN
ejpam-7135	275	8	.	.	PUNCT
ejpam-7135	276	1	j.	j.	PROPN
ejpam-7135	276	2	pure	pure	PROPN
ejpam-7135	276	3	appl	appl	PROPN
ejpam-7135	276	4	.	.	PROPN
ejpam-7135	276	5	math	math	PROPN
ejpam-7135	276	6	,	,	PUNCT
ejpam-7135	276	7	18	18	NUM
ejpam-7135	276	8	(	(	PUNCT
ejpam-7135	276	9	4	4	NUM
ejpam-7135	276	10	)	)	PUNCT
ejpam-7135	276	11	(	(	PUNCT
ejpam-7135	276	12	2025	2025	NUM
ejpam-7135	276	13	)	)	PUNCT
ejpam-7135	276	14	,	,	PUNCT
ejpam-7135	276	15	7135	7135	NUM
ejpam-7135	276	16	9	9	NUM
ejpam-7135	276	17	of	of	ADP
ejpam-7135	276	18	14	14	NUM
ejpam-7135	276	19	proof	proof	NOUN
ejpam-7135	276	20	.	.	PUNCT
ejpam-7135	277	1	since	since	SCONJ
ejpam-7135	277	2	p	p	NOUN
ejpam-7135	277	3	is	be	AUX
ejpam-7135	277	4	a	a	DET
ejpam-7135	277	5	prime	prime	ADJ
ejpam-7135	277	6	filter	filter	NOUN
ejpam-7135	277	7	of	of	ADP
ejpam-7135	277	8	l	l	NOUN
ejpam-7135	277	9	,	,	PUNCT
ejpam-7135	277	10	it	it	PRON
ejpam-7135	277	11	is	be	AUX
ejpam-7135	277	12	nonempty	nonempty	ADJ
ejpam-7135	277	13	,	,	PUNCT
ejpam-7135	277	14	proper	proper	ADJ
ejpam-7135	277	15	,	,	PUNCT
ejpam-7135	277	16	and	and	CCONJ
ejpam-7135	277	17	closed	close	VERB
ejpam-7135	277	18	under	under	ADP
ejpam-7135	277	19	finite	finite	ADJ
ejpam-7135	277	20	meets	meet	NOUN
ejpam-7135	277	21	.	.	PUNCT
ejpam-7135	278	1	also	also	ADV
ejpam-7135	278	2	,	,	PUNCT
ejpam-7135	278	3	for	for	ADP
ejpam-7135	278	4	any	any	DET
ejpam-7135	278	5	a	a	PRON
ejpam-7135	278	6	,	,	PUNCT
ejpam-7135	278	7	b	b	PROPN
ejpam-7135	278	8	∈	∈	PROPN
ejpam-7135	278	9	l	l	NOUN
ejpam-7135	278	10	,	,	PUNCT
ejpam-7135	278	11	if	if	SCONJ
ejpam-7135	278	12	a	a	DET
ejpam-7135	278	13	∨	∨	NUM
ejpam-7135	278	14	b	b	NOUN
ejpam-7135	278	15	∈	∈	PROPN
ejpam-7135	278	16	p	p	NOUN
ejpam-7135	278	17	,	,	PUNCT
ejpam-7135	278	18	then	then	ADV
ejpam-7135	278	19	a	a	DET
ejpam-7135	278	20	∈	∈	PROPN
ejpam-7135	278	21	p	p	NOUN
ejpam-7135	278	22	or	or	CCONJ
ejpam-7135	278	23	b	b	NOUN
ejpam-7135	278	24	∈	∈	PROPN
ejpam-7135	278	25	p	p	NOUN
ejpam-7135	278	26	.	.	PUNCT
ejpam-7135	279	1	now	now	ADV
ejpam-7135	279	2	,	,	PUNCT
ejpam-7135	279	3	b[m	b[m	ADJ
ejpam-7135	279	4	,	,	PUNCT
ejpam-7135	279	5	1	1	NUM
ejpam-7135	279	6	]	]	PUNCT
ejpam-7135	279	7	is	be	AUX
ejpam-7135	279	8	a	a	DET
ejpam-7135	279	9	boolean	boolean	ADJ
ejpam-7135	279	10	algebra	algebra	NOUN
ejpam-7135	279	11	with	with	ADP
ejpam-7135	279	12	least	least	ADJ
ejpam-7135	279	13	element	element	ADJ
ejpam-7135	279	14	m	m	PROPN
ejpam-7135	279	15	and	and	CCONJ
ejpam-7135	279	16	greatest	great	ADJ
ejpam-7135	279	17	element	element	NOUN
ejpam-7135	279	18	1	1	NUM
ejpam-7135	279	19	.	.	PUNCT
ejpam-7135	280	1	since	since	SCONJ
ejpam-7135	280	2	p	p	NOUN
ejpam-7135	280	3	is	be	AUX
ejpam-7135	280	4	a	a	DET
ejpam-7135	280	5	filter	filter	NOUN
ejpam-7135	280	6	of	of	ADP
ejpam-7135	280	7	l	l	NOUN
ejpam-7135	280	8	,	,	PUNCT
ejpam-7135	280	9	its	its	PRON
ejpam-7135	280	10	intersection	intersection	NOUN
ejpam-7135	280	11	with	with	ADP
ejpam-7135	280	12	b[m	b[m	ADJ
ejpam-7135	280	13	,	,	PUNCT
ejpam-7135	280	14	1	1	NUM
ejpam-7135	280	15	]	]	PUNCT
ejpam-7135	280	16	is	be	AUX
ejpam-7135	280	17	nonempty	nonempty	X
ejpam-7135	280	18	(	(	PUNCT
ejpam-7135	280	19	as	as	ADP
ejpam-7135	280	20	1	1	NUM
ejpam-7135	280	21	∈	∈	PROPN
ejpam-7135	280	22	p	p	NOUN
ejpam-7135	280	23	∩	∩	ADJ
ejpam-7135	280	24	b[m	b[m	ADJ
ejpam-7135	280	25	,	,	PUNCT
ejpam-7135	280	26	1	1	NUM
ejpam-7135	280	27	]	]	PUNCT
ejpam-7135	280	28	)	)	PUNCT
ejpam-7135	280	29	and	and	CCONJ
ejpam-7135	280	30	proper	proper	ADJ
ejpam-7135	280	31	(	(	PUNCT
ejpam-7135	280	32	as	as	ADP
ejpam-7135	280	33	m	m	PROPN
ejpam-7135	280	34	/∈	/∈	PUNCT
ejpam-7135	280	35	p	p	NOUN
ejpam-7135	280	36	)	)	PUNCT
ejpam-7135	280	37	.	.	PUNCT
ejpam-7135	281	1	let	let	VERB
ejpam-7135	281	2	a	a	DET
ejpam-7135	281	3	,	,	PUNCT
ejpam-7135	281	4	b	b	PROPN
ejpam-7135	281	5	∈	∈	PROPN
ejpam-7135	281	6	p	p	PROPN
ejpam-7135	281	7	c.	c.	NOUN
ejpam-7135	281	8	then	then	ADV
ejpam-7135	281	9	a	a	DET
ejpam-7135	281	10	∧	∧	PROPN
ejpam-7135	281	11	b	b	PROPN
ejpam-7135	281	12	∈	∈	PROPN
ejpam-7135	281	13	p	p	NOUN
ejpam-7135	281	14	and	and	CCONJ
ejpam-7135	281	15	a	a	DET
ejpam-7135	281	16	∧	∧	PROPN
ejpam-7135	281	17	b	b	PROPN
ejpam-7135	281	18	∈	∈	PROPN
ejpam-7135	281	19	b[m	b[m	NOUN
ejpam-7135	281	20	,	,	PUNCT
ejpam-7135	281	21	1	1	NUM
ejpam-7135	281	22	]	]	PUNCT
ejpam-7135	281	23	,	,	PUNCT
ejpam-7135	281	24	so	so	ADV
ejpam-7135	281	25	a	a	DET
ejpam-7135	281	26	∧	∧	PROPN
ejpam-7135	281	27	b	b	PROPN
ejpam-7135	281	28	∈	∈	PROPN
ejpam-7135	281	29	p	p	PROPN
ejpam-7135	281	30	c.	c.	NOUN
ejpam-7135	281	31	if	if	SCONJ
ejpam-7135	281	32	a	a	DET
ejpam-7135	281	33	∈	∈	PROPN
ejpam-7135	281	34	p	p	NOUN
ejpam-7135	281	35	c	c	PROPN
ejpam-7135	281	36	and	and	CCONJ
ejpam-7135	281	37	a	a	DET
ejpam-7135	281	38	≤	≤	ADJ
ejpam-7135	281	39	c	c	NOUN
ejpam-7135	281	40	in	in	ADP
ejpam-7135	281	41	b[m	b[m	ADJ
ejpam-7135	281	42	,	,	PUNCT
ejpam-7135	281	43	1	1	NUM
ejpam-7135	281	44	]	]	PUNCT
ejpam-7135	281	45	,	,	PUNCT
ejpam-7135	281	46	then	then	ADV
ejpam-7135	281	47	c	c	PROPN
ejpam-7135	281	48	∈	∈	PROPN
ejpam-7135	281	49	p	p	X
ejpam-7135	281	50	(	(	PUNCT
ejpam-7135	281	51	since	since	SCONJ
ejpam-7135	281	52	p	p	NOUN
ejpam-7135	281	53	is	be	AUX
ejpam-7135	281	54	upward	upward	ADV
ejpam-7135	281	55	closed	closed	ADJ
ejpam-7135	281	56	)	)	PUNCT
ejpam-7135	281	57	and	and	CCONJ
ejpam-7135	281	58	c	c	PROPN
ejpam-7135	281	59	∈	∈	PROPN
ejpam-7135	281	60	b[m	b[m	PROPN
ejpam-7135	281	61	,	,	PUNCT
ejpam-7135	281	62	1	1	NUM
ejpam-7135	281	63	]	]	PUNCT
ejpam-7135	281	64	,	,	PUNCT
ejpam-7135	281	65	so	so	SCONJ
ejpam-7135	281	66	c	c	PROPN
ejpam-7135	281	67	∈	∈	PROPN
ejpam-7135	281	68	p	p	PROPN
ejpam-7135	281	69	c.	c.	PROPN
ejpam-7135	281	70	finally	finally	ADV
ejpam-7135	281	71	,	,	PUNCT
ejpam-7135	281	72	if	if	SCONJ
ejpam-7135	281	73	a	a	DET
ejpam-7135	281	74	∨	∨	NUM
ejpam-7135	281	75	b	b	X
ejpam-7135	281	76	∈	∈	PROPN
ejpam-7135	281	77	p	p	PROPN
ejpam-7135	281	78	c	c	NOUN
ejpam-7135	281	79	for	for	ADP
ejpam-7135	281	80	a	a	DET
ejpam-7135	281	81	,	,	PUNCT
ejpam-7135	281	82	b	b	PROPN
ejpam-7135	281	83	∈	∈	PROPN
ejpam-7135	281	84	b[m	b[m	NOUN
ejpam-7135	281	85	,	,	PUNCT
ejpam-7135	281	86	1	1	NUM
ejpam-7135	281	87	]	]	PUNCT
ejpam-7135	281	88	,	,	PUNCT
ejpam-7135	281	89	then	then	ADV
ejpam-7135	281	90	a	a	DET
ejpam-7135	281	91	∨	∨	PROPN
ejpam-7135	281	92	b	b	PROPN
ejpam-7135	281	93	∈	∈	PROPN
ejpam-7135	281	94	p	p	NOUN
ejpam-7135	281	95	,	,	PUNCT
ejpam-7135	281	96	so	so	ADV
ejpam-7135	281	97	a	a	DET
ejpam-7135	281	98	∈	∈	PROPN
ejpam-7135	281	99	p	p	NOUN
ejpam-7135	281	100	or	or	CCONJ
ejpam-7135	281	101	b	b	NOUN
ejpam-7135	281	102	∈	∈	PROPN
ejpam-7135	281	103	p	p	NOUN
ejpam-7135	281	104	,	,	PUNCT
ejpam-7135	281	105	hence	hence	ADV
ejpam-7135	281	106	a	a	DET
ejpam-7135	281	107	∈	∈	NOUN
ejpam-7135	281	108	p	p	NOUN
ejpam-7135	281	109	c	c	NOUN
ejpam-7135	281	110	or	or	CCONJ
ejpam-7135	281	111	b	b	NOUN
ejpam-7135	281	112	∈	∈	PROPN
ejpam-7135	281	113	p	p	PROPN
ejpam-7135	281	114	c.	c.	PROPN
ejpam-7135	281	115	thus	thus	ADV
ejpam-7135	281	116	,	,	PUNCT
ejpam-7135	281	117	p	p	PROPN
ejpam-7135	281	118	c	c	PROPN
ejpam-7135	281	119	is	be	AUX
ejpam-7135	281	120	a	a	DET
ejpam-7135	281	121	prime	prime	ADJ
ejpam-7135	281	122	filter	filter	NOUN
ejpam-7135	281	123	of	of	ADP
ejpam-7135	281	124	b[m	b[m	ADJ
ejpam-7135	281	125	,	,	PUNCT
ejpam-7135	281	126	1	1	NUM
ejpam-7135	281	127	]	]	PUNCT
ejpam-7135	281	128	.	.	PUNCT
ejpam-7135	282	1	lemma	lemma	PROPN
ejpam-7135	282	2	8	8	NUM
ejpam-7135	282	3	.	.	PUNCT
ejpam-7135	283	1	let	let	VERB
ejpam-7135	283	2	l	l	NOUN
ejpam-7135	283	3	be	be	AUX
ejpam-7135	283	4	a	a	DET
ejpam-7135	283	5	stone	stone	NOUN
ejpam-7135	283	6	pdl	pdl	NOUN
ejpam-7135	283	7	and	and	CCONJ
ejpam-7135	283	8	let	let	VERB
ejpam-7135	283	9	q	q	PUNCT
ejpam-7135	283	10	be	be	AUX
ejpam-7135	283	11	a	a	DET
ejpam-7135	283	12	prime	prime	ADJ
ejpam-7135	283	13	filter	filter	NOUN
ejpam-7135	283	14	of	of	ADP
ejpam-7135	283	15	b[1	b[1	PROPN
ejpam-7135	283	16	♦	♦	PROPN
ejpam-7135	283	17	,	,	PUNCT
ejpam-7135	283	18	1	1	NUM
ejpam-7135	283	19	]	]	PUNCT
ejpam-7135	283	20	.	.	PUNCT
ejpam-7135	284	1	define	define	VERB
ejpam-7135	284	2	qe	qe	X
ejpam-7135	284	3	:	:	PUNCT
ejpam-7135	284	4	=	=	SYM
ejpam-7135	284	5	{	{	PUNCT
ejpam-7135	284	6	x	x	PROPN
ejpam-7135	284	7	∨	∨	NUM
ejpam-7135	284	8	a	a	DET
ejpam-7135	284	9	|	|	NOUN
ejpam-7135	284	10	a	a	DET
ejpam-7135	284	11	∈	∈	ADJ
ejpam-7135	284	12	q	q	NOUN
ejpam-7135	284	13	,	,	PUNCT
ejpam-7135	284	14	x	x	SYM
ejpam-7135	284	15	∈	∈	NOUN
ejpam-7135	284	16	l	l	NOUN
ejpam-7135	284	17	}	}	PUNCT
ejpam-7135	284	18	.	.	PUNCT
ejpam-7135	285	1	then	then	ADV
ejpam-7135	285	2	qe	qe	PROPN
ejpam-7135	285	3	is	be	AUX
ejpam-7135	285	4	a	a	DET
ejpam-7135	285	5	minimal	minimal	ADJ
ejpam-7135	285	6	prime	prime	ADJ
ejpam-7135	285	7	filter	filter	NOUN
ejpam-7135	285	8	of	of	ADP
ejpam-7135	285	9	l.	l.	PROPN
ejpam-7135	285	10	proof	proof	PROPN
ejpam-7135	285	11	.	.	PUNCT
ejpam-7135	286	1	we	we	PRON
ejpam-7135	286	2	first	first	ADV
ejpam-7135	286	3	show	show	VERB
ejpam-7135	286	4	that	that	SCONJ
ejpam-7135	286	5	qe	qe	PROPN
ejpam-7135	286	6	is	be	AUX
ejpam-7135	286	7	a	a	DET
ejpam-7135	286	8	proper	proper	ADJ
ejpam-7135	286	9	filter	filter	NOUN
ejpam-7135	286	10	of	of	ADP
ejpam-7135	286	11	l.	l.	PROPN
ejpam-7135	286	12	properness	properness	PROPN
ejpam-7135	286	13	:	:	PUNCT
ejpam-7135	286	14	if	if	SCONJ
ejpam-7135	286	15	1	1	NUM
ejpam-7135	286	16	♦	♦	PROPN
ejpam-7135	286	17	∈	∈	PROPN
ejpam-7135	286	18	qe	qe	PROPN
ejpam-7135	286	19	,	,	PUNCT
ejpam-7135	286	20	then	then	ADV
ejpam-7135	286	21	1	1	NUM
ejpam-7135	286	22	♦	♦	NOUN
ejpam-7135	286	23	=	=	PROPN
ejpam-7135	286	24	x	x	PROPN
ejpam-7135	286	25	∨	∨	NUM
ejpam-7135	286	26	a	a	PRON
ejpam-7135	286	27	for	for	ADP
ejpam-7135	286	28	some	some	PRON
ejpam-7135	286	29	a	a	DET
ejpam-7135	286	30	∈	∈	ADJ
ejpam-7135	286	31	q	q	NOUN
ejpam-7135	286	32	,	,	PUNCT
ejpam-7135	286	33	x	x	SYM
ejpam-7135	286	34	∈	∈	PROPN
ejpam-7135	286	35	l.	l.	NOUN
ejpam-7135	286	36	but	but	CCONJ
ejpam-7135	286	37	then	then	ADV
ejpam-7135	286	38	1	1	NUM
ejpam-7135	286	39	♦	♦	PROPN
ejpam-7135	286	40	∨	∨	NUM
ejpam-7135	286	41	a	a	PRON
ejpam-7135	286	42	=	=	X
ejpam-7135	286	43	x	x	SYM
ejpam-7135	286	44	∨	∨	PROPN
ejpam-7135	286	45	a	a	DET
ejpam-7135	286	46	∨	∨	NOUN
ejpam-7135	286	47	a	a	DET
ejpam-7135	286	48	=	=	SYM
ejpam-7135	286	49	x	x	SYM
ejpam-7135	286	50	∨	∨	NOUN
ejpam-7135	286	51	a	a	DET
ejpam-7135	286	52	=	=	ADJ
ejpam-7135	286	53	1	1	NUM
ejpam-7135	286	54	♦	♦	PROPN
ejpam-7135	286	55	,	,	PUNCT
ejpam-7135	286	56	implying	imply	VERB
ejpam-7135	286	57	1	1	NUM
ejpam-7135	286	58	♦	♦	PROPN
ejpam-7135	286	59	∈	∈	PROPN
ejpam-7135	286	60	q	q	PROPN
ejpam-7135	286	61	,	,	PUNCT
ejpam-7135	286	62	a	a	DET
ejpam-7135	286	63	contradiction	contradiction	NOUN
ejpam-7135	286	64	since	since	SCONJ
ejpam-7135	286	65	1	1	NUM
ejpam-7135	286	66	♦	♦	PROPN
ejpam-7135	286	67	is	be	AUX
ejpam-7135	286	68	the	the	DET
ejpam-7135	286	69	least	least	ADJ
ejpam-7135	286	70	element	element	NOUN
ejpam-7135	286	71	of	of	ADP
ejpam-7135	286	72	b[1	b[1	PROPN
ejpam-7135	286	73	♦	♦	PROPN
ejpam-7135	286	74	,	,	PUNCT
ejpam-7135	286	75	1	1	NUM
ejpam-7135	286	76	]	]	PUNCT
ejpam-7135	286	77	.	.	PUNCT
ejpam-7135	287	1	hence	hence	ADV
ejpam-7135	287	2	,	,	PUNCT
ejpam-7135	287	3	1	1	NUM
ejpam-7135	287	4	♦	♦	PROPN
ejpam-7135	287	5	/∈	/∈	PUNCT
ejpam-7135	287	6	qe	qe	PROPN
ejpam-7135	287	7	,	,	PUNCT
ejpam-7135	287	8	so	so	ADV
ejpam-7135	287	9	qe	qe	PROPN
ejpam-7135	287	10	is	be	AUX
ejpam-7135	287	11	proper	proper	ADJ
ejpam-7135	287	12	.	.	PUNCT
ejpam-7135	288	1	filter	filter	NOUN
ejpam-7135	288	2	properties	property	NOUN
ejpam-7135	288	3	:	:	PUNCT
ejpam-7135	288	4	•	•	ADP
ejpam-7135	288	5	if	if	SCONJ
ejpam-7135	288	6	x	x	PROPN
ejpam-7135	288	7	∨	∨	NUM
ejpam-7135	288	8	a	a	X
ejpam-7135	288	9	,	,	PUNCT
ejpam-7135	288	10	y	y	PROPN
ejpam-7135	288	11	∨	∨	NUM
ejpam-7135	288	12	b	b	PROPN
ejpam-7135	288	13	∈	∈	PROPN
ejpam-7135	288	14	qe	qe	PROPN
ejpam-7135	288	15	,	,	PUNCT
ejpam-7135	288	16	then	then	ADV
ejpam-7135	288	17	their	their	PRON
ejpam-7135	288	18	meet	meet	NOUN
ejpam-7135	288	19	is	be	AUX
ejpam-7135	288	20	(	(	PUNCT
ejpam-7135	288	21	x	x	PROPN
ejpam-7135	288	22	∨	∨	NUM
ejpam-7135	288	23	a	a	DET
ejpam-7135	288	24	)	)	PUNCT
ejpam-7135	288	25	∧	∧	PROPN
ejpam-7135	288	26	(	(	PUNCT
ejpam-7135	288	27	y	y	PROPN
ejpam-7135	288	28	∨	∨	NUM
ejpam-7135	288	29	b	b	PROPN
ejpam-7135	288	30	)	)	PUNCT
ejpam-7135	288	31	=	=	SYM
ejpam-7135	289	1	(	(	PUNCT
ejpam-7135	289	2	x	x	PART
ejpam-7135	289	3	∧	∧	PROPN
ejpam-7135	289	4	(	(	PUNCT
ejpam-7135	289	5	y	y	PROPN
ejpam-7135	289	6	∨	∨	NUM
ejpam-7135	289	7	b	b	NOUN
ejpam-7135	289	8	)	)	PUNCT
ejpam-7135	289	9	)	)	PUNCT
ejpam-7135	289	10	∨	∨	NUM
ejpam-7135	289	11	(	(	PUNCT
ejpam-7135	289	12	a	a	DET
ejpam-7135	289	13	∧	∧	PROPN
ejpam-7135	289	14	(	(	PUNCT
ejpam-7135	289	15	y	y	PROPN
ejpam-7135	289	16	∨	∨	NUM
ejpam-7135	289	17	b	b	NOUN
ejpam-7135	289	18	)	)	PUNCT
ejpam-7135	289	19	)	)	PUNCT
ejpam-7135	289	20	=	=	SYM
ejpam-7135	289	21	(	(	PUNCT
ejpam-7135	289	22	x	x	PART
ejpam-7135	289	23	∧	∧	PROPN
ejpam-7135	289	24	(	(	PUNCT
ejpam-7135	289	25	y	y	PROPN
ejpam-7135	289	26	∨	∨	NUM
ejpam-7135	289	27	b	b	NOUN
ejpam-7135	289	28	)	)	PUNCT
ejpam-7135	289	29	)	)	PUNCT
ejpam-7135	289	30	∨	∨	NUM
ejpam-7135	289	31	(	(	PUNCT
ejpam-7135	289	32	(	(	PUNCT
ejpam-7135	289	33	y	y	PROPN
ejpam-7135	289	34	∨	∨	NUM
ejpam-7135	289	35	b	b	PROPN
ejpam-7135	289	36	)	)	PUNCT
ejpam-7135	289	37	∧	∧	PROPN
ejpam-7135	289	38	a	a	NOUN
ejpam-7135	289	39	)	)	PUNCT
ejpam-7135	289	40	.	.	PUNCT
ejpam-7135	290	1	since	since	SCONJ
ejpam-7135	290	2	a	a	DET
ejpam-7135	290	3	∧	∧	PROPN
ejpam-7135	290	4	b	b	PROPN
ejpam-7135	290	5	∈	∈	PROPN
ejpam-7135	290	6	q	q	NOUN
ejpam-7135	290	7	,	,	PUNCT
ejpam-7135	290	8	the	the	DET
ejpam-7135	290	9	expression	expression	NOUN
ejpam-7135	290	10	lies	lie	VERB
ejpam-7135	290	11	in	in	ADP
ejpam-7135	290	12	qe	qe	PROPN
ejpam-7135	290	13	.	.	PUNCT
ejpam-7135	291	1	•	•	INTJ
ejpam-7135	292	1	if	if	SCONJ
ejpam-7135	292	2	x	x	PROPN
ejpam-7135	292	3	∨	∨	VERB
ejpam-7135	292	4	a	a	DET
ejpam-7135	292	5	∈	∈	PROPN
ejpam-7135	292	6	qe	qe	NOUN
ejpam-7135	292	7	and	and	CCONJ
ejpam-7135	292	8	t	t	PROPN
ejpam-7135	292	9	∈	∈	PROPN
ejpam-7135	292	10	l	l	NOUN
ejpam-7135	292	11	,	,	PUNCT
ejpam-7135	292	12	then	then	ADV
ejpam-7135	292	13	t	t	PROPN
ejpam-7135	292	14	∨	∨	PROPN
ejpam-7135	292	15	x	x	X
ejpam-7135	292	16	∨	∨	NUM
ejpam-7135	292	17	a	a	DET
ejpam-7135	292	18	∈	∈	PROPN
ejpam-7135	292	19	qe	qe	NOUN
ejpam-7135	292	20	,	,	PUNCT
ejpam-7135	292	21	so	so	ADV
ejpam-7135	292	22	qe	qe	PROPN
ejpam-7135	292	23	is	be	AUX
ejpam-7135	292	24	upward	upward	ADV
ejpam-7135	292	25	closed	closed	ADJ
ejpam-7135	292	26	.	.	PUNCT
ejpam-7135	293	1	primality	primality	NOUN
ejpam-7135	293	2	:	:	PUNCT
ejpam-7135	293	3	suppose	suppose	VERB
ejpam-7135	293	4	x	x	PUNCT
ejpam-7135	293	5	∨	∨	NUM
ejpam-7135	293	6	y	y	PROPN
ejpam-7135	293	7	∈	∈	PROPN
ejpam-7135	293	8	qe	qe	PROPN
ejpam-7135	293	9	.	.	PUNCT
ejpam-7135	294	1	then	then	ADV
ejpam-7135	294	2	x	x	X
ejpam-7135	294	3	∨	∨	NUM
ejpam-7135	294	4	y	y	PROPN
ejpam-7135	294	5	=	=	SYM
ejpam-7135	294	6	t	t	PROPN
ejpam-7135	294	7	∨	∨	NUM
ejpam-7135	294	8	a	a	PRON
ejpam-7135	294	9	for	for	ADP
ejpam-7135	294	10	some	some	PRON
ejpam-7135	294	11	a	a	DET
ejpam-7135	294	12	∈	∈	PROPN
ejpam-7135	294	13	q	q	NOUN
ejpam-7135	294	14	,	,	PUNCT
ejpam-7135	294	15	t	t	PROPN
ejpam-7135	294	16	∈	∈	PROPN
ejpam-7135	294	17	l.	l.	PROPN
ejpam-7135	294	18	then	then	ADV
ejpam-7135	294	19	x	x	PROPN
ejpam-7135	294	20	♦	♦	PROPN
ejpam-7135	294	21	♦	♦	PROPN
ejpam-7135	294	22	∨	∨	PROPN
ejpam-7135	294	23	y	y	PROPN
ejpam-7135	294	24	♦	♦	PROPN
ejpam-7135	294	25	♦	♦	PROPN
ejpam-7135	294	26	=	=	PROPN
ejpam-7135	294	27	(	(	PUNCT
ejpam-7135	294	28	x	x	PROPN
ejpam-7135	294	29	∨	∨	NUM
ejpam-7135	294	30	y	y	NOUN
ejpam-7135	294	31	)	)	PUNCT
ejpam-7135	294	32	♦	♦	PROPN
ejpam-7135	294	33	♦	♦	PROPN
ejpam-7135	294	34	=	=	PROPN
ejpam-7135	294	35	(	(	PUNCT
ejpam-7135	294	36	t	t	PROPN
ejpam-7135	294	37	∨	∨	NUM
ejpam-7135	294	38	a	a	PRON
ejpam-7135	294	39	)	)	PUNCT
ejpam-7135	294	40	♦	♦	PROPN
ejpam-7135	294	41	♦	♦	PROPN
ejpam-7135	294	42	=	=	PROPN
ejpam-7135	294	43	t	t	PROPN
ejpam-7135	294	44	♦	♦	PROPN
ejpam-7135	294	45	♦	♦	PROPN
ejpam-7135	294	46	∨	∨	PROPN
ejpam-7135	294	47	a	a	DET
ejpam-7135	294	48	♦	♦	PROPN
ejpam-7135	294	49	♦	♦	PROPN
ejpam-7135	294	50	=	=	PROPN
ejpam-7135	294	51	t	t	PROPN
ejpam-7135	294	52	♦	♦	PROPN
ejpam-7135	294	53	♦	♦	PROPN
ejpam-7135	294	54	∨	∨	PROPN
ejpam-7135	294	55	a	a	DET
ejpam-7135	294	56	∈	∈	PROPN
ejpam-7135	294	57	q.	q.	NOUN
ejpam-7135	294	58	since	since	SCONJ
ejpam-7135	294	59	x	x	PROPN
ejpam-7135	294	60	♦	♦	PROPN
ejpam-7135	294	61	♦	♦	PROPN
ejpam-7135	294	62	,	,	PUNCT
ejpam-7135	294	63	y	y	PROPN
ejpam-7135	294	64	♦	♦	PROPN
ejpam-7135	294	65	♦	♦	PROPN
ejpam-7135	294	66	∈	∈	PROPN
ejpam-7135	294	67	b[1	b[1	PROPN
ejpam-7135	294	68	♦	♦	PROPN
ejpam-7135	294	69	,	,	PUNCT
ejpam-7135	294	70	1	1	NUM
ejpam-7135	294	71	]	]	PUNCT
ejpam-7135	294	72	,	,	PUNCT
ejpam-7135	294	73	primality	primality	NOUN
ejpam-7135	294	74	of	of	ADP
ejpam-7135	294	75	q	q	PROPN
ejpam-7135	294	76	implies	imply	VERB
ejpam-7135	295	1	x	x	PUNCT
ejpam-7135	295	2	♦	♦	PROPN
ejpam-7135	295	3	♦	♦	PROPN
ejpam-7135	295	4	∈	∈	PROPN
ejpam-7135	295	5	q	q	PROPN
ejpam-7135	295	6	or	or	CCONJ
ejpam-7135	295	7	y	y	PROPN
ejpam-7135	295	8	♦	♦	PROPN
ejpam-7135	295	9	♦	♦	PROPN
ejpam-7135	295	10	∈	∈	PROPN
ejpam-7135	295	11	q	q	X
ejpam-7135	295	12	,	,	PUNCT
ejpam-7135	295	13	so	so	ADV
ejpam-7135	295	14	x	x	SYM
ejpam-7135	295	15	∈	∈	PROPN
ejpam-7135	295	16	qe	qe	PROPN
ejpam-7135	295	17	or	or	CCONJ
ejpam-7135	295	18	y	y	PROPN
ejpam-7135	295	19	∈	∈	PROPN
ejpam-7135	295	20	qe	qe	PROPN
ejpam-7135	295	21	.	.	PUNCT
ejpam-7135	296	1	minimality	minimality	NOUN
ejpam-7135	296	2	:	:	PUNCT
ejpam-7135	296	3	let	let	VERB
ejpam-7135	296	4	x	x	X
ejpam-7135	296	5	∈	∈	PROPN
ejpam-7135	296	6	qe	qe	PROPN
ejpam-7135	296	7	,	,	PUNCT
ejpam-7135	296	8	so	so	ADV
ejpam-7135	296	9	x	x	SYM
ejpam-7135	296	10	=	=	SYM
ejpam-7135	296	11	t∨	t∨	NOUN
ejpam-7135	296	12	a	a	NOUN
ejpam-7135	296	13	for	for	ADP
ejpam-7135	296	14	some	some	PRON
ejpam-7135	296	15	a	a	DET
ejpam-7135	296	16	∈	∈	PROPN
ejpam-7135	296	17	q	q	NOUN
ejpam-7135	296	18	,	,	PUNCT
ejpam-7135	296	19	t	t	PROPN
ejpam-7135	296	20	∈	∈	PROPN
ejpam-7135	296	21	l.	l.	PROPN
ejpam-7135	296	22	let	let	VERB
ejpam-7135	296	23	a′	a′	PROPN
ejpam-7135	296	24	be	be	AUX
ejpam-7135	296	25	the	the	DET
ejpam-7135	296	26	complement	complement	NOUN
ejpam-7135	296	27	of	of	ADP
ejpam-7135	296	28	a	a	PRON
ejpam-7135	296	29	in	in	ADP
ejpam-7135	296	30	b[1	b[1	PROPN
ejpam-7135	296	31	♦	♦	PROPN
ejpam-7135	296	32	,	,	PUNCT
ejpam-7135	296	33	1	1	NUM
ejpam-7135	296	34	]	]	PUNCT
ejpam-7135	296	35	.	.	PUNCT
ejpam-7135	297	1	then	then	ADV
ejpam-7135	297	2	x	x	X
ejpam-7135	297	3	∨	∨	NOUN
ejpam-7135	297	4	a′	a′	PROPN
ejpam-7135	297	5	=	=	SYM
ejpam-7135	297	6	t	t	PROPN
ejpam-7135	297	7	∨	∨	NUM
ejpam-7135	297	8	a	a	DET
ejpam-7135	297	9	∨	∨	NOUN
ejpam-7135	297	10	a′	a′	NOUN
ejpam-7135	297	11	=	=	SYM
ejpam-7135	297	12	1	1	NUM
ejpam-7135	297	13	,	,	PUNCT
ejpam-7135	297	14	so	so	ADV
ejpam-7135	297	15	a′	a′	PROPN
ejpam-7135	297	16	∈	∈	PROPN
ejpam-7135	298	1	[	[	X
ejpam-7135	298	2	x]	x]	PROPN
ejpam-7135	298	3	♦	♦	PROPN
ejpam-7135	298	4	.	.	PUNCT
ejpam-7135	299	1	if	if	SCONJ
ejpam-7135	299	2	a′	a′	PROPN
ejpam-7135	299	3	∈	∈	PROPN
ejpam-7135	299	4	qe	qe	PROPN
ejpam-7135	299	5	,	,	PUNCT
ejpam-7135	299	6	then	then	ADV
ejpam-7135	299	7	a′	a′	PROPN
ejpam-7135	299	8	=	=	SYM
ejpam-7135	299	9	s	s	PART
ejpam-7135	299	10	∨	∨	NUM
ejpam-7135	299	11	b	b	NOUN
ejpam-7135	299	12	for	for	ADP
ejpam-7135	299	13	some	some	DET
ejpam-7135	299	14	b	b	NOUN
ejpam-7135	299	15	∈	∈	ADJ
ejpam-7135	299	16	q	q	NOUN
ejpam-7135	299	17	,	,	PUNCT
ejpam-7135	299	18	s	s	NOUN
ejpam-7135	299	19	∈	∈	PROPN
ejpam-7135	299	20	l	l	NOUN
ejpam-7135	299	21	,	,	PUNCT
ejpam-7135	299	22	and	and	CCONJ
ejpam-7135	299	23	a′	a′	PROPN
ejpam-7135	299	24	∨	∨	NUM
ejpam-7135	299	25	b	b	PROPN
ejpam-7135	299	26	=	=	SYM
ejpam-7135	299	27	s	s	PART
ejpam-7135	299	28	∨	∨	NUM
ejpam-7135	299	29	b	b	PROPN
ejpam-7135	299	30	∨	∨	NUM
ejpam-7135	299	31	b	b	NOUN
ejpam-7135	299	32	=	=	PUNCT
ejpam-7135	299	33	a′	a′	PROPN
ejpam-7135	299	34	,	,	PUNCT
ejpam-7135	299	35	so	so	SCONJ
ejpam-7135	299	36	a′	a′	PROPN
ejpam-7135	299	37	∈	∈	PROPN
ejpam-7135	299	38	q	q	PROPN
ejpam-7135	299	39	,	,	PUNCT
ejpam-7135	299	40	a	a	DET
ejpam-7135	299	41	contradiction	contradiction	NOUN
ejpam-7135	299	42	.	.	PUNCT
ejpam-7135	300	1	hence	hence	ADV
ejpam-7135	300	2	,	,	PUNCT
ejpam-7135	300	3	a′	a′	PROPN
ejpam-7135	300	4	/∈	/∈	PROPN
ejpam-7135	300	5	qe	qe	PROPN
ejpam-7135	300	6	,	,	PUNCT
ejpam-7135	300	7	so	so	SCONJ
ejpam-7135	300	8	[	[	X
ejpam-7135	300	9	x	x	X
ejpam-7135	300	10	]	]	X
ejpam-7135	300	11	♦	♦	PROPN
ejpam-7135	300	12	\qe	\qe	PROPN
ejpam-7135	300	13	̸=	̸=	PROPN
ejpam-7135	300	14	∅	∅	NOUN
ejpam-7135	300	15	,	,	PUNCT
ejpam-7135	300	16	and	and	CCONJ
ejpam-7135	300	17	thus	thus	ADV
ejpam-7135	300	18	qe	qe	PROPN
ejpam-7135	300	19	is	be	AUX
ejpam-7135	300	20	minimal	minimal	ADJ
ejpam-7135	300	21	.	.	PUNCT
ejpam-7135	301	1	lemma	lemma	PROPN
ejpam-7135	301	2	9	9	NUM
ejpam-7135	301	3	.	.	PUNCT
ejpam-7135	302	1	let	let	VERB
ejpam-7135	302	2	q	q	PART
ejpam-7135	302	3	be	be	AUX
ejpam-7135	302	4	a	a	DET
ejpam-7135	302	5	prime	prime	ADJ
ejpam-7135	302	6	filter	filter	NOUN
ejpam-7135	302	7	of	of	ADP
ejpam-7135	302	8	b[1	b[1	PROPN
ejpam-7135	302	9	♦	♦	PROPN
ejpam-7135	302	10	,	,	PUNCT
ejpam-7135	302	11	1	1	NUM
ejpam-7135	302	12	]	]	PUNCT
ejpam-7135	302	13	.	.	PUNCT
ejpam-7135	303	1	then	then	ADV
ejpam-7135	303	2	qe	qe	PROPN
ejpam-7135	303	3	is	be	AUX
ejpam-7135	303	4	the	the	DET
ejpam-7135	303	5	smallest	small	ADJ
ejpam-7135	303	6	filter	filter	NOUN
ejpam-7135	303	7	of	of	ADP
ejpam-7135	303	8	l	l	NOUN
ejpam-7135	303	9	containing	contain	VERB
ejpam-7135	303	10	q.	q.	NOUN
ejpam-7135	303	11	proof	proof	NOUN
ejpam-7135	303	12	.	.	PUNCT
ejpam-7135	304	1	clearly	clearly	ADV
ejpam-7135	304	2	q	q	PROPN
ejpam-7135	304	3	⊆	⊆	NUM
ejpam-7135	304	4	qe	qe	PROPN
ejpam-7135	304	5	,	,	PUNCT
ejpam-7135	304	6	since	since	SCONJ
ejpam-7135	304	7	for	for	ADP
ejpam-7135	304	8	a	a	DET
ejpam-7135	304	9	∈	∈	PROPN
ejpam-7135	304	10	q	q	NOUN
ejpam-7135	304	11	,	,	PUNCT
ejpam-7135	304	12	we	we	PRON
ejpam-7135	304	13	have	have	VERB
ejpam-7135	304	14	a	a	DET
ejpam-7135	304	15	=	=	NOUN
ejpam-7135	304	16	m∨	m∨	NOUN
ejpam-7135	304	17	a	a	DET
ejpam-7135	304	18	∈	∈	PROPN
ejpam-7135	304	19	qe	qe	PROPN
ejpam-7135	304	20	.	.	PUNCT
ejpam-7135	305	1	now	now	ADV
ejpam-7135	305	2	suppose	suppose	VERB
ejpam-7135	305	3	h	h	NOUN
ejpam-7135	305	4	is	be	AUX
ejpam-7135	305	5	a	a	DET
ejpam-7135	305	6	filter	filter	NOUN
ejpam-7135	305	7	of	of	ADP
ejpam-7135	305	8	l	l	NOUN
ejpam-7135	305	9	containing	contain	VERB
ejpam-7135	305	10	q.	q.	NOUN
ejpam-7135	305	11	let	let	VERB
ejpam-7135	305	12	x∨	x∨	PROPN
ejpam-7135	305	13	a	a	DET
ejpam-7135	305	14	∈	∈	PROPN
ejpam-7135	305	15	qe	qe	NOUN
ejpam-7135	305	16	with	with	ADP
ejpam-7135	305	17	a	a	DET
ejpam-7135	305	18	∈	∈	PROPN
ejpam-7135	305	19	q	q	NOUN
ejpam-7135	305	20	,	,	PUNCT
ejpam-7135	305	21	x	x	SYM
ejpam-7135	305	22	∈	∈	PROPN
ejpam-7135	305	23	l.	l.	NOUN
ejpam-7135	305	24	since	since	SCONJ
ejpam-7135	305	25	a	a	DET
ejpam-7135	305	26	∈	∈	PROPN
ejpam-7135	305	27	h	h	NOUN
ejpam-7135	305	28	and	and	CCONJ
ejpam-7135	305	29	h	h	NOUN
ejpam-7135	305	30	is	be	AUX
ejpam-7135	305	31	a	a	DET
ejpam-7135	305	32	filter	filter	NOUN
ejpam-7135	305	33	,	,	PUNCT
ejpam-7135	305	34	a	a	DET
ejpam-7135	305	35	∨	∨	NOUN
ejpam-7135	305	36	x	x	SYM
ejpam-7135	305	37	∈	∈	PROPN
ejpam-7135	305	38	h.	h.	NOUN
ejpam-7135	305	39	hence	hence	ADV
ejpam-7135	305	40	,	,	PUNCT
ejpam-7135	305	41	qe	qe	PROPN
ejpam-7135	305	42	⊆	⊆	NUM
ejpam-7135	305	43	h.	h.	PROPN
ejpam-7135	305	44	therefore	therefore	ADV
ejpam-7135	305	45	,	,	PUNCT
ejpam-7135	305	46	qe	qe	PROPN
ejpam-7135	305	47	is	be	AUX
ejpam-7135	305	48	the	the	DET
ejpam-7135	305	49	smallest	small	ADJ
ejpam-7135	305	50	filter	filter	NOUN
ejpam-7135	305	51	of	of	ADP
ejpam-7135	305	52	l	l	NOUN
ejpam-7135	305	53	containing	contain	VERB
ejpam-7135	305	54	q.	q.	PROPN
ejpam-7135	305	55	remark	remark	NOUN
ejpam-7135	305	56	1	1	NUM
ejpam-7135	305	57	.	.	PUNCT
ejpam-7135	306	1	let	let	VERB
ejpam-7135	306	2	l	l	NOUN
ejpam-7135	306	3	be	be	AUX
ejpam-7135	306	4	a	a	DET
ejpam-7135	306	5	pdl	pdl	NOUN
ejpam-7135	306	6	with	with	ADP
ejpam-7135	306	7	a	a	DET
ejpam-7135	306	8	minimal	minimal	ADJ
ejpam-7135	306	9	element	element	NOUN
ejpam-7135	306	10	m.	m.	NOUN
ejpam-7135	306	11	then	then	ADV
ejpam-7135	306	12	for	for	ADP
ejpam-7135	306	13	any	any	DET
ejpam-7135	306	14	p	p	PROPN
ejpam-7135	306	15	∈	∈	PROPN
ejpam-7135	306	16	y	y	PROPN
ejpam-7135	306	17	,	,	PUNCT
ejpam-7135	306	18	p	p	PROPN
ejpam-7135	306	19	ce	ce	PROPN
ejpam-7135	306	20	⊆	⊆	NUM
ejpam-7135	306	21	p	p	PROPN
ejpam-7135	306	22	.	.	PUNCT
ejpam-7135	307	1	r.	r.	PROPN
ejpam-7135	307	2	bandaru	bandaru	PROPN
ejpam-7135	307	3	et	et	PROPN
ejpam-7135	307	4	al	al	PROPN
ejpam-7135	307	5	.	.	PUNCT
ejpam-7135	307	6	/	/	SYM
ejpam-7135	307	7	eur	eur	PROPN
ejpam-7135	307	8	.	.	PUNCT
ejpam-7135	308	1	j.	j.	PROPN
ejpam-7135	308	2	pure	pure	PROPN
ejpam-7135	308	3	appl	appl	PROPN
ejpam-7135	308	4	.	.	PROPN
ejpam-7135	308	5	math	math	PROPN
ejpam-7135	308	6	,	,	PUNCT
ejpam-7135	308	7	18	18	NUM
ejpam-7135	308	8	(	(	PUNCT
ejpam-7135	308	9	4	4	NUM
ejpam-7135	308	10	)	)	PUNCT
ejpam-7135	308	11	(	(	PUNCT
ejpam-7135	308	12	2025	2025	NUM
ejpam-7135	308	13	)	)	PUNCT
ejpam-7135	308	14	,	,	PUNCT
ejpam-7135	308	15	7135	7135	NUM
ejpam-7135	308	16	10	10	NUM
ejpam-7135	308	17	of	of	ADP
ejpam-7135	308	18	14	14	NUM
ejpam-7135	308	19	theorem	theorem	NOUN
ejpam-7135	308	20	11	11	NUM
ejpam-7135	308	21	.	.	PUNCT
ejpam-7135	309	1	let	let	VERB
ejpam-7135	309	2	l	l	NOUN
ejpam-7135	309	3	be	be	AUX
ejpam-7135	309	4	a	a	DET
ejpam-7135	309	5	pdl	pdl	NOUN
ejpam-7135	309	6	with	with	ADP
ejpam-7135	309	7	a	a	DET
ejpam-7135	309	8	parapseudo	parapseudo	NOUN
ejpam-7135	309	9	-	-	PUNCT
ejpam-7135	309	10	complementation	complementation	NOUN
ejpam-7135	309	11	♦	♦	PROPN
ejpam-7135	309	12	in	in	ADP
ejpam-7135	309	13	which	which	PRON
ejpam-7135	309	14	m	m	VERB
ejpam-7135	309	15	is	be	AUX
ejpam-7135	309	16	a	a	DET
ejpam-7135	309	17	minimal	minimal	ADJ
ejpam-7135	309	18	element	element	NOUN
ejpam-7135	309	19	,	,	PUNCT
ejpam-7135	309	20	and	and	CCONJ
ejpam-7135	309	21	let	let	VERB
ejpam-7135	309	22	m	m	PRON
ejpam-7135	309	23	be	be	AUX
ejpam-7135	309	24	the	the	DET
ejpam-7135	309	25	set	set	NOUN
ejpam-7135	309	26	of	of	ADP
ejpam-7135	309	27	minimal	minimal	ADJ
ejpam-7135	309	28	prime	prime	ADJ
ejpam-7135	309	29	filters	filter	NOUN
ejpam-7135	309	30	of	of	ADP
ejpam-7135	309	31	l.	l.	PROPN
ejpam-7135	309	32	then	then	ADV
ejpam-7135	309	33	l	l	PROPN
ejpam-7135	309	34	is	be	AUX
ejpam-7135	309	35	a	a	DET
ejpam-7135	309	36	stone	stone	NOUN
ejpam-7135	309	37	pdl	pdl	NOUN
ejpam-7135	309	38	if	if	SCONJ
ejpam-7135	310	1	and	and	CCONJ
ejpam-7135	310	2	only	only	ADV
ejpam-7135	310	3	if	if	SCONJ
ejpam-7135	310	4	p	p	X
ejpam-7135	310	5	=	=	X
ejpam-7135	310	6	p	p	X
ejpam-7135	310	7	ce	ce	PROPN
ejpam-7135	310	8	for	for	ADP
ejpam-7135	310	9	all	all	DET
ejpam-7135	310	10	p	p	NOUN
ejpam-7135	310	11	∈	∈	PROPN
ejpam-7135	310	12	m	m	NOUN
ejpam-7135	310	13	,	,	PUNCT
ejpam-7135	310	14	where	where	SCONJ
ejpam-7135	310	15	p	p	NOUN
ejpam-7135	310	16	c	c	NOUN
ejpam-7135	310	17	=	=	SYM
ejpam-7135	310	18	p	p	NOUN
ejpam-7135	310	19	∩	∩	NOUN
ejpam-7135	310	20	b[m	b[m	ADJ
ejpam-7135	310	21	,	,	PUNCT
ejpam-7135	310	22	1	1	NUM
ejpam-7135	310	23	]	]	PUNCT
ejpam-7135	310	24	and	and	CCONJ
ejpam-7135	310	25	p	p	X
ejpam-7135	310	26	ce	ce	PROPN
ejpam-7135	311	1	=	=	SYM
ejpam-7135	311	2	{	{	PUNCT
ejpam-7135	311	3	x	x	PROPN
ejpam-7135	311	4	∨	∨	NUM
ejpam-7135	311	5	a	a	DET
ejpam-7135	311	6	|	|	NOUN
ejpam-7135	311	7	a	a	DET
ejpam-7135	311	8	∈	∈	NOUN
ejpam-7135	311	9	p	p	NOUN
ejpam-7135	311	10	c	c	NOUN
ejpam-7135	311	11	,	,	PUNCT
ejpam-7135	311	12	x	x	SYM
ejpam-7135	311	13	∈	∈	NOUN
ejpam-7135	311	14	l	l	NOUN
ejpam-7135	311	15	}	}	PUNCT
ejpam-7135	311	16	.	.	PUNCT
ejpam-7135	312	1	proof	proof	NOUN
ejpam-7135	312	2	.	.	PUNCT
ejpam-7135	313	1	(	(	PUNCT
ejpam-7135	313	2	⇒	⇒	PROPN
ejpam-7135	313	3	)	)	PUNCT
ejpam-7135	313	4	suppose	suppose	VERB
ejpam-7135	313	5	l	l	NOUN
ejpam-7135	313	6	is	be	AUX
ejpam-7135	313	7	a	a	DET
ejpam-7135	313	8	stone	stone	NOUN
ejpam-7135	313	9	pdl	pdl	NOUN
ejpam-7135	313	10	and	and	CCONJ
ejpam-7135	313	11	let	let	VERB
ejpam-7135	313	12	p	p	PRON
ejpam-7135	313	13	∈	∈	PROPN
ejpam-7135	313	14	m	m	NOUN
ejpam-7135	313	15	.	.	PUNCT
ejpam-7135	314	1	by	by	ADP
ejpam-7135	314	2	lemma	lemma	PROPN
ejpam-7135	314	3	7	7	NUM
ejpam-7135	314	4	,	,	PUNCT
ejpam-7135	314	5	p	p	NOUN
ejpam-7135	314	6	c	c	NOUN
ejpam-7135	314	7	is	be	AUX
ejpam-7135	314	8	a	a	DET
ejpam-7135	314	9	prime	prime	ADJ
ejpam-7135	314	10	filter	filter	NOUN
ejpam-7135	314	11	of	of	ADP
ejpam-7135	314	12	b[m	b[m	ADJ
ejpam-7135	314	13	,	,	PUNCT
ejpam-7135	314	14	1	1	NUM
ejpam-7135	314	15	]	]	PUNCT
ejpam-7135	314	16	.	.	PUNCT
ejpam-7135	315	1	by	by	ADP
ejpam-7135	315	2	lemma	lemma	PROPN
ejpam-7135	315	3	8	8	NUM
ejpam-7135	315	4	,	,	PUNCT
ejpam-7135	315	5	p	p	PROPN
ejpam-7135	315	6	ce	ce	PROPN
ejpam-7135	315	7	is	be	AUX
ejpam-7135	315	8	a	a	DET
ejpam-7135	315	9	minimal	minimal	ADJ
ejpam-7135	315	10	prime	prime	ADJ
ejpam-7135	315	11	filter	filter	NOUN
ejpam-7135	315	12	of	of	ADP
ejpam-7135	315	13	l.	l.	PROPN
ejpam-7135	315	14	since	since	SCONJ
ejpam-7135	315	15	p	p	PROPN
ejpam-7135	315	16	ce	ce	PROPN
ejpam-7135	315	17	⊆	⊆	NUM
ejpam-7135	315	18	p	p	X
ejpam-7135	315	19	(	(	PUNCT
ejpam-7135	315	20	as	as	ADP
ejpam-7135	315	21	a	a	DET
ejpam-7135	315	22	∈	∈	NOUN
ejpam-7135	315	23	p	p	NOUN
ejpam-7135	315	24	c	c	NOUN
ejpam-7135	316	1	⊆	⊆	NUM
ejpam-7135	316	2	p	p	PROPN
ejpam-7135	316	3	implies	imply	VERB
ejpam-7135	316	4	x	x	ADJ
ejpam-7135	316	5	∨	∨	NUM
ejpam-7135	316	6	a	a	DET
ejpam-7135	316	7	∈	∈	PROPN
ejpam-7135	316	8	p	p	NOUN
ejpam-7135	316	9	for	for	ADP
ejpam-7135	316	10	all	all	DET
ejpam-7135	316	11	x	x	SYM
ejpam-7135	316	12	∈	∈	PROPN
ejpam-7135	316	13	l	l	NOUN
ejpam-7135	316	14	)	)	PUNCT
ejpam-7135	316	15	and	and	CCONJ
ejpam-7135	316	16	both	both	PRON
ejpam-7135	316	17	are	be	AUX
ejpam-7135	316	18	minimal	minimal	ADJ
ejpam-7135	316	19	prime	prime	ADJ
ejpam-7135	316	20	filters	filter	NOUN
ejpam-7135	316	21	,	,	PUNCT
ejpam-7135	316	22	we	we	PRON
ejpam-7135	316	23	have	have	VERB
ejpam-7135	316	24	p	p	NOUN
ejpam-7135	316	25	ce	ce	PROPN
ejpam-7135	317	1	=	=	PROPN
ejpam-7135	317	2	p	p	PROPN
ejpam-7135	317	3	.	.	PUNCT
ejpam-7135	318	1	(	(	PUNCT
ejpam-7135	318	2	⇐	⇐	NOUN
ejpam-7135	318	3	)	)	PUNCT
ejpam-7135	318	4	suppose	suppose	VERB
ejpam-7135	318	5	p	p	X
ejpam-7135	318	6	=	=	PUNCT
ejpam-7135	318	7	p	p	X
ejpam-7135	318	8	ce	ce	PROPN
ejpam-7135	318	9	for	for	ADP
ejpam-7135	318	10	all	all	DET
ejpam-7135	318	11	p	p	PROPN
ejpam-7135	318	12	∈	∈	PROPN
ejpam-7135	318	13	m	m	VERB
ejpam-7135	318	14	.	.	PUNCT
ejpam-7135	319	1	let	let	VERB
ejpam-7135	319	2	p	p	PRON
ejpam-7135	319	3	,	,	PUNCT
ejpam-7135	319	4	q	q	PROPN
ejpam-7135	319	5	∈	∈	PROPN
ejpam-7135	319	6	m	m	VERB
ejpam-7135	319	7	with	with	ADP
ejpam-7135	319	8	p	p	PROPN
ejpam-7135	319	9	̸=	̸=	PROPN
ejpam-7135	319	10	q.	q.	NOUN
ejpam-7135	319	11	then	then	ADV
ejpam-7135	319	12	p	p	PROPN
ejpam-7135	319	13	c	c	PROPN
ejpam-7135	319	14	̸=	̸=	PROPN
ejpam-7135	319	15	qc	qc	PROPN
ejpam-7135	319	16	.	.	PROPN
ejpam-7135	319	17	without	without	ADP
ejpam-7135	319	18	loss	loss	NOUN
ejpam-7135	319	19	of	of	ADP
ejpam-7135	319	20	generality	generality	NOUN
ejpam-7135	319	21	,	,	PUNCT
ejpam-7135	319	22	choose	choose	VERB
ejpam-7135	319	23	a	a	DET
ejpam-7135	319	24	∈	∈	PROPN
ejpam-7135	319	25	p	p	NOUN
ejpam-7135	319	26	c	c	PROPN
ejpam-7135	319	27	\qc	\qc	PROPN
ejpam-7135	319	28	.	.	PUNCT
ejpam-7135	320	1	let	let	VERB
ejpam-7135	320	2	a′	a′	NOUN
ejpam-7135	320	3	be	be	AUX
ejpam-7135	320	4	the	the	DET
ejpam-7135	320	5	complement	complement	NOUN
ejpam-7135	320	6	of	of	ADP
ejpam-7135	320	7	a	a	PRON
ejpam-7135	320	8	in	in	ADP
ejpam-7135	320	9	b[m	b[m	ADJ
ejpam-7135	320	10	,	,	PUNCT
ejpam-7135	320	11	1	1	NUM
ejpam-7135	320	12	]	]	PUNCT
ejpam-7135	320	13	.	.	PUNCT
ejpam-7135	321	1	then	then	ADV
ejpam-7135	321	2	a′	a′	PROPN
ejpam-7135	321	3	∈	∈	PROPN
ejpam-7135	321	4	qc	qc	PROPN
ejpam-7135	321	5	,	,	PUNCT
ejpam-7135	321	6	and	and	CCONJ
ejpam-7135	321	7	since	since	SCONJ
ejpam-7135	321	8	a	a	DET
ejpam-7135	321	9	∈	∈	PROPN
ejpam-7135	321	10	p	p	NOUN
ejpam-7135	321	11	and	and	CCONJ
ejpam-7135	321	12	a′	a′	PROPN
ejpam-7135	321	13	∈	∈	PROPN
ejpam-7135	321	14	q	q	NOUN
ejpam-7135	321	15	,	,	PUNCT
ejpam-7135	321	16	we	we	PRON
ejpam-7135	321	17	have	have	VERB
ejpam-7135	321	18	m	m	NOUN
ejpam-7135	321	19	=	=	SYM
ejpam-7135	321	20	a∧a′	a∧a′	PROPN
ejpam-7135	321	21	∈	∈	PROPN
ejpam-7135	321	22	p	p	NOUN
ejpam-7135	321	23	∨q	∨q	NOUN
ejpam-7135	321	24	,	,	PUNCT
ejpam-7135	321	25	hence	hence	ADV
ejpam-7135	321	26	p	p	X
ejpam-7135	321	27	∨q	∨q	NOUN
ejpam-7135	321	28	=	=	X
ejpam-7135	321	29	l.	l.	NOUN
ejpam-7135	321	30	by	by	ADP
ejpam-7135	321	31	theorem	theorem	NOUN
ejpam-7135	321	32	10	10	NUM
ejpam-7135	321	33	,	,	PUNCT
ejpam-7135	321	34	l	l	NOUN
ejpam-7135	321	35	is	be	AUX
ejpam-7135	321	36	a	a	DET
ejpam-7135	321	37	stone	stone	NOUN
ejpam-7135	321	38	pdl	pdl	NOUN
ejpam-7135	321	39	.	.	PUNCT
ejpam-7135	322	1	in	in	ADP
ejpam-7135	322	2	the	the	DET
ejpam-7135	322	3	following	following	NOUN
ejpam-7135	322	4	,	,	PUNCT
ejpam-7135	322	5	we	we	PRON
ejpam-7135	322	6	give	give	VERB
ejpam-7135	322	7	another	another	DET
ejpam-7135	322	8	characterization	characterization	NOUN
ejpam-7135	322	9	of	of	ADP
ejpam-7135	322	10	stone	stone	NOUN
ejpam-7135	322	11	pdls	pdl	NOUN
ejpam-7135	322	12	.	.	PUNCT
ejpam-7135	323	1	first	first	ADV
ejpam-7135	323	2	,	,	PUNCT
ejpam-7135	323	3	we	we	PRON
ejpam-7135	323	4	prove	prove	VERB
ejpam-7135	323	5	the	the	DET
ejpam-7135	323	6	following	following	NOUN
ejpam-7135	323	7	.	.	PUNCT
ejpam-7135	324	1	lemma	lemma	PROPN
ejpam-7135	324	2	10	10	NUM
ejpam-7135	324	3	.	.	PUNCT
ejpam-7135	325	1	let	let	VERB
ejpam-7135	325	2	l	l	NOUN
ejpam-7135	325	3	be	be	AUX
ejpam-7135	325	4	a	a	DET
ejpam-7135	325	5	pdl	pdl	NOUN
ejpam-7135	325	6	with	with	ADP
ejpam-7135	325	7	a	a	DET
ejpam-7135	325	8	minimal	minimal	ADJ
ejpam-7135	325	9	element	element	NOUN
ejpam-7135	325	10	m	m	NOUN
ejpam-7135	325	11	,	,	PUNCT
ejpam-7135	325	12	and	and	CCONJ
ejpam-7135	325	13	let	let	VERB
ejpam-7135	325	14	a	a	DET
ejpam-7135	325	15	∈	∈	PROPN
ejpam-7135	325	16	l.	l.	NOUN
ejpam-7135	325	17	then	then	ADV
ejpam-7135	325	18	ya	ya	PROPN
ejpam-7135	325	19	=	=	PUNCT
ejpam-7135	325	20	y	y	PROPN
ejpam-7135	326	1	if	if	SCONJ
ejpam-7135	326	2	and	and	CCONJ
ejpam-7135	326	3	only	only	ADV
ejpam-7135	326	4	if	if	SCONJ
ejpam-7135	326	5	a	a	PRON
ejpam-7135	326	6	is	be	AUX
ejpam-7135	326	7	a	a	DET
ejpam-7135	326	8	minimal	minimal	ADJ
ejpam-7135	326	9	element	element	NOUN
ejpam-7135	326	10	,	,	PUNCT
ejpam-7135	326	11	where	where	SCONJ
ejpam-7135	326	12	ya	ya	PRON
ejpam-7135	326	13	=	=	PUNCT
ejpam-7135	326	14	{	{	PUNCT
ejpam-7135	326	15	p	p	X
ejpam-7135	326	16	∈	∈	X
ejpam-7135	326	17	y	y	NOUN
ejpam-7135	326	18	|	|	ADV
ejpam-7135	326	19	a	a	PRON
ejpam-7135	326	20	/∈	/∈	NOUN
ejpam-7135	327	1	p	p	NOUN
ejpam-7135	327	2	}	}	PUNCT
ejpam-7135	327	3	and	and	CCONJ
ejpam-7135	327	4	y	y	PROPN
ejpam-7135	327	5	is	be	AUX
ejpam-7135	327	6	the	the	DET
ejpam-7135	327	7	set	set	NOUN
ejpam-7135	327	8	of	of	ADP
ejpam-7135	327	9	all	all	DET
ejpam-7135	327	10	prime	prime	ADJ
ejpam-7135	327	11	filters	filter	NOUN
ejpam-7135	327	12	of	of	ADP
ejpam-7135	327	13	l.	l.	PROPN
ejpam-7135	327	14	proof	proof	PROPN
ejpam-7135	327	15	.	.	PUNCT
ejpam-7135	328	1	if	if	SCONJ
ejpam-7135	328	2	a	a	PRON
ejpam-7135	328	3	is	be	AUX
ejpam-7135	328	4	minimal	minimal	ADJ
ejpam-7135	328	5	,	,	PUNCT
ejpam-7135	328	6	then	then	ADV
ejpam-7135	328	7	ya	ya	PROPN
ejpam-7135	328	8	=	=	SYM
ejpam-7135	328	9	y	y	PROPN
ejpam-7135	328	10	,	,	PUNCT
ejpam-7135	328	11	since	since	SCONJ
ejpam-7135	328	12	m	m	PROPN
ejpam-7135	328	13	/∈	/∈	VERB
ejpam-7135	329	1	any	any	DET
ejpam-7135	329	2	prime	prime	ADJ
ejpam-7135	329	3	filter	filter	NOUN
ejpam-7135	329	4	.	.	PUNCT
ejpam-7135	330	1	conversely	conversely	ADV
ejpam-7135	330	2	,	,	PUNCT
ejpam-7135	330	3	if	if	SCONJ
ejpam-7135	330	4	ya	ya	PROPN
ejpam-7135	330	5	=	=	SYM
ejpam-7135	330	6	y	y	PROPN
ejpam-7135	330	7	,	,	PUNCT
ejpam-7135	330	8	then	then	ADV
ejpam-7135	330	9	a	a	DET
ejpam-7135	330	10	/∈	/∈	NOUN
ejpam-7135	330	11	p	p	NOUN
ejpam-7135	330	12	for	for	ADP
ejpam-7135	330	13	all	all	DET
ejpam-7135	330	14	prime	prime	ADJ
ejpam-7135	330	15	filters	filter	NOUN
ejpam-7135	330	16	p	p	NOUN
ejpam-7135	330	17	.	.	PUNCT
ejpam-7135	331	1	this	this	PRON
ejpam-7135	331	2	implies	imply	VERB
ejpam-7135	331	3	a	a	PRON
ejpam-7135	331	4	is	be	AUX
ejpam-7135	331	5	contained	contain	VERB
ejpam-7135	331	6	in	in	ADP
ejpam-7135	331	7	every	every	DET
ejpam-7135	331	8	prime	prime	ADJ
ejpam-7135	331	9	ideal	ideal	NOUN
ejpam-7135	331	10	,	,	PUNCT
ejpam-7135	331	11	hence	hence	ADV
ejpam-7135	331	12	in	in	ADP
ejpam-7135	331	13	the	the	DET
ejpam-7135	331	14	intersection	intersection	NOUN
ejpam-7135	331	15	of	of	ADP
ejpam-7135	331	16	all	all	DET
ejpam-7135	331	17	prime	prime	ADJ
ejpam-7135	331	18	ideals	ideal	NOUN
ejpam-7135	331	19	,	,	PUNCT
ejpam-7135	331	20	which	which	PRON
ejpam-7135	331	21	is	be	AUX
ejpam-7135	331	22	the	the	DET
ejpam-7135	331	23	set	set	NOUN
ejpam-7135	331	24	of	of	ADP
ejpam-7135	331	25	minimal	minimal	ADJ
ejpam-7135	331	26	elements	element	NOUN
ejpam-7135	331	27	.	.	PUNCT
ejpam-7135	332	1	thus	thus	ADV
ejpam-7135	332	2	,	,	PUNCT
ejpam-7135	332	3	a	a	PRON
ejpam-7135	332	4	is	be	AUX
ejpam-7135	332	5	minimal	minimal	ADJ
ejpam-7135	332	6	.	.	PUNCT
ejpam-7135	333	1	lemma	lemma	PROPN
ejpam-7135	333	2	11	11	NUM
ejpam-7135	333	3	.	.	PUNCT
ejpam-7135	334	1	let	let	VERB
ejpam-7135	334	2	l	l	NOUN
ejpam-7135	334	3	be	be	AUX
ejpam-7135	334	4	a	a	DET
ejpam-7135	334	5	pdl	pdl	NOUN
ejpam-7135	334	6	with	with	ADP
ejpam-7135	334	7	a	a	DET
ejpam-7135	334	8	minimal	minimal	ADJ
ejpam-7135	334	9	element	element	NOUN
ejpam-7135	334	10	m	m	NOUN
ejpam-7135	334	11	,	,	PUNCT
ejpam-7135	334	12	and	and	CCONJ
ejpam-7135	334	13	let	let	VERB
ejpam-7135	334	14	y	y	PRON
ejpam-7135	334	15	be	be	AUX
ejpam-7135	334	16	the	the	DET
ejpam-7135	334	17	set	set	NOUN
ejpam-7135	334	18	of	of	ADP
ejpam-7135	334	19	prime	prime	ADJ
ejpam-7135	334	20	filters	filter	NOUN
ejpam-7135	334	21	of	of	ADP
ejpam-7135	334	22	l	l	NOUN
ejpam-7135	334	23	with	with	ADP
ejpam-7135	334	24	the	the	DET
ejpam-7135	334	25	hull	hull	NOUN
ejpam-7135	334	26	-	-	PUNCT
ejpam-7135	334	27	kernel	kernel	NOUN
ejpam-7135	334	28	topology	topology	NOUN
ejpam-7135	334	29	.	.	PUNCT
ejpam-7135	335	1	a	a	DET
ejpam-7135	335	2	subset	subset	NOUN
ejpam-7135	335	3	u	u	NOUN
ejpam-7135	335	4	⊆	⊆	NUM
ejpam-7135	335	5	y	y	PROPN
ejpam-7135	335	6	is	be	AUX
ejpam-7135	335	7	clopen	clopen	ADJ
ejpam-7135	335	8	if	if	SCONJ
ejpam-7135	335	9	and	and	CCONJ
ejpam-7135	335	10	only	only	ADV
ejpam-7135	335	11	if	if	SCONJ
ejpam-7135	335	12	u	u	PROPN
ejpam-7135	335	13	=	=	PROPN
ejpam-7135	335	14	ya	ya	PROPN
ejpam-7135	335	15	for	for	ADP
ejpam-7135	335	16	some	some	DET
ejpam-7135	335	17	a	a	DET
ejpam-7135	335	18	∈	∈	PROPN
ejpam-7135	335	19	b[m	b[m	NOUN
ejpam-7135	335	20	,	,	PUNCT
ejpam-7135	335	21	1	1	NUM
ejpam-7135	335	22	]	]	PUNCT
ejpam-7135	335	23	.	.	PUNCT
ejpam-7135	336	1	proof	proof	NOUN
ejpam-7135	336	2	.	.	PUNCT
ejpam-7135	337	1	suppose	suppose	VERB
ejpam-7135	337	2	u	u	PRON
ejpam-7135	337	3	is	be	AUX
ejpam-7135	337	4	clopen	clopen	ADJ
ejpam-7135	337	5	in	in	ADP
ejpam-7135	337	6	y	y	PROPN
ejpam-7135	337	7	.	.	PUNCT
ejpam-7135	338	1	then	then	ADV
ejpam-7135	338	2	u	u	PROPN
ejpam-7135	338	3	=	=	PROPN
ejpam-7135	338	4	yx	yx	PROPN
ejpam-7135	338	5	and	and	CCONJ
ejpam-7135	338	6	y	y	PROPN
ejpam-7135	338	7	\	\	NOUN
ejpam-7135	338	8	u	u	PROPN
ejpam-7135	338	9	=	=	PROPN
ejpam-7135	338	10	yy	yy	PROPN
ejpam-7135	338	11	for	for	ADP
ejpam-7135	338	12	some	some	DET
ejpam-7135	338	13	x	x	NOUN
ejpam-7135	338	14	,	,	PUNCT
ejpam-7135	338	15	y	y	PROPN
ejpam-7135	338	16	∈	∈	PROPN
ejpam-7135	338	17	l.	l.	NOUN
ejpam-7135	339	1	then	then	ADV
ejpam-7135	339	2	:	:	PUNCT
ejpam-7135	339	3	•	•	X
ejpam-7135	339	4	yx	yx	NOUN
ejpam-7135	339	5	∩	∩	ADJ
ejpam-7135	339	6	yy	yy	PROPN
ejpam-7135	339	7	=	=	PUNCT
ejpam-7135	339	8	yx∨y	yx∨y	PROPN
ejpam-7135	339	9	=	=	NOUN
ejpam-7135	339	10	∅	∅	NOUN
ejpam-7135	339	11	⇒	⇒	NOUN
ejpam-7135	339	12	x	x	PROPN
ejpam-7135	339	13	∨	∨	NUM
ejpam-7135	339	14	y	y	NOUN
ejpam-7135	339	15	=	=	SYM
ejpam-7135	339	16	1	1	NUM
ejpam-7135	339	17	•	•	NUM
ejpam-7135	339	18	yx	yx	NOUN
ejpam-7135	339	19	∪	∪	VERB
ejpam-7135	339	20	yy	yy	NOUN
ejpam-7135	339	21	=	=	PUNCT
ejpam-7135	339	22	yx∧y	yx∧y	NOUN
ejpam-7135	339	23	=	=	PUNCT
ejpam-7135	339	24	y	y	PROPN
ejpam-7135	339	25	⇒	⇒	VERB
ejpam-7135	339	26	x	x	PUNCT
ejpam-7135	340	1	∧	∧	NOUN
ejpam-7135	340	2	y	y	PROPN
ejpam-7135	340	3	is	be	AUX
ejpam-7135	340	4	minimal	minimal	ADJ
ejpam-7135	340	5	(	(	PUNCT
ejpam-7135	340	6	by	by	ADP
ejpam-7135	340	7	lemma	lemma	PROPN
ejpam-7135	340	8	10	10	NUM
ejpam-7135	340	9	)	)	PUNCT
ejpam-7135	340	10	since	since	SCONJ
ejpam-7135	340	11	x	x	SYM
ejpam-7135	340	12	∧	∧	PROPN
ejpam-7135	340	13	y	y	PROPN
ejpam-7135	340	14	is	be	AUX
ejpam-7135	340	15	minimal	minimal	ADJ
ejpam-7135	340	16	and	and	CCONJ
ejpam-7135	340	17	x	x	SYM
ejpam-7135	340	18	∨	∨	NUM
ejpam-7135	340	19	y	y	NOUN
ejpam-7135	340	20	=	=	SYM
ejpam-7135	340	21	1	1	NUM
ejpam-7135	340	22	,	,	PUNCT
ejpam-7135	340	23	it	it	PRON
ejpam-7135	340	24	follows	follow	VERB
ejpam-7135	340	25	that	that	SCONJ
ejpam-7135	340	26	x	x	X
ejpam-7135	340	27	,	,	PUNCT
ejpam-7135	340	28	y	y	PROPN
ejpam-7135	340	29	are	be	AUX
ejpam-7135	340	30	complements	complement	NOUN
ejpam-7135	340	31	in	in	ADP
ejpam-7135	340	32	b[m	b[m	ADJ
ejpam-7135	340	33	,	,	PUNCT
ejpam-7135	340	34	1	1	NUM
ejpam-7135	340	35	]	]	PUNCT
ejpam-7135	340	36	,	,	PUNCT
ejpam-7135	340	37	so	so	CCONJ
ejpam-7135	340	38	u	u	X
ejpam-7135	340	39	=	=	PROPN
ejpam-7135	340	40	yx	yx	NOUN
ejpam-7135	340	41	with	with	ADP
ejpam-7135	340	42	x	x	PROPN
ejpam-7135	340	43	∈	∈	PROPN
ejpam-7135	340	44	b[m	b[m	NOUN
ejpam-7135	340	45	,	,	PUNCT
ejpam-7135	340	46	1	1	NUM
ejpam-7135	340	47	]	]	PUNCT
ejpam-7135	340	48	.	.	PUNCT
ejpam-7135	341	1	conversely	conversely	ADV
ejpam-7135	341	2	,	,	PUNCT
ejpam-7135	341	3	if	if	SCONJ
ejpam-7135	341	4	a	a	DET
ejpam-7135	341	5	∈	∈	PROPN
ejpam-7135	341	6	b[m	b[m	NOUN
ejpam-7135	341	7	,	,	PUNCT
ejpam-7135	341	8	1	1	NUM
ejpam-7135	341	9	]	]	PUNCT
ejpam-7135	341	10	,	,	PUNCT
ejpam-7135	341	11	then	then	ADV
ejpam-7135	341	12	ya	ya	PRON
ejpam-7135	341	13	is	be	AUX
ejpam-7135	341	14	open	open	ADJ
ejpam-7135	341	15	and	and	CCONJ
ejpam-7135	341	16	its	its	PRON
ejpam-7135	341	17	complement	complement	NOUN
ejpam-7135	341	18	ya′	ya′	PROPN
ejpam-7135	342	1	(	(	PUNCT
ejpam-7135	342	2	where	where	SCONJ
ejpam-7135	342	3	a′	a′	PROPN
ejpam-7135	342	4	is	be	AUX
ejpam-7135	342	5	the	the	DET
ejpam-7135	342	6	complement	complement	NOUN
ejpam-7135	342	7	of	of	ADP
ejpam-7135	342	8	a	a	PRON
ejpam-7135	342	9	in	in	ADP
ejpam-7135	342	10	b[m	b[m	ADJ
ejpam-7135	342	11	,	,	PUNCT
ejpam-7135	342	12	1	1	NUM
ejpam-7135	342	13	]	]	PUNCT
ejpam-7135	342	14	)	)	PUNCT
ejpam-7135	342	15	is	be	AUX
ejpam-7135	342	16	also	also	ADV
ejpam-7135	342	17	open	open	ADJ
ejpam-7135	342	18	,	,	PUNCT
ejpam-7135	342	19	so	so	ADV
ejpam-7135	342	20	ya	ya	PRON
ejpam-7135	342	21	is	be	AUX
ejpam-7135	342	22	clopen	clopen	ADJ
ejpam-7135	342	23	.	.	PUNCT
ejpam-7135	343	1	let	let	VERB
ejpam-7135	343	2	l	l	NOUN
ejpam-7135	343	3	be	be	AUX
ejpam-7135	343	4	a	a	DET
ejpam-7135	343	5	pdl	pdl	NOUN
ejpam-7135	343	6	with	with	ADP
ejpam-7135	343	7	a	a	DET
ejpam-7135	343	8	minimal	minimal	ADJ
ejpam-7135	343	9	elementm	elementm	PROPN
ejpam-7135	343	10	andx	andx	PROPN
ejpam-7135	343	11	denote	denote	VERB
ejpam-7135	343	12	the	the	DET
ejpam-7135	343	13	boolean	boolean	ADJ
ejpam-7135	343	14	space	space	NOUN
ejpam-7135	343	15	of	of	ADP
ejpam-7135	343	16	all	all	DET
ejpam-7135	343	17	prime	prime	ADJ
ejpam-7135	343	18	filters	filter	NOUN
ejpam-7135	343	19	of	of	ADP
ejpam-7135	343	20	the	the	DET
ejpam-7135	343	21	boolean	boolean	ADJ
ejpam-7135	343	22	algebra	algebra	PROPN
ejpam-7135	343	23	b[m	b[m	PROPN
ejpam-7135	343	24	,	,	PUNCT
ejpam-7135	343	25	1	1	NUM
ejpam-7135	343	26	]	]	PUNCT
ejpam-7135	343	27	with	with	ADP
ejpam-7135	343	28	hull	hull	NOUN
ejpam-7135	343	29	-	-	PUNCT
ejpam-7135	343	30	kernel	kernel	NOUN
ejpam-7135	343	31	topology	topology	NOUN
ejpam-7135	343	32	on	on	ADP
ejpam-7135	343	33	x.	x.	NOUN
ejpam-7135	343	34	that	that	PRON
ejpam-7135	343	35	is	be	AUX
ejpam-7135	343	36	the	the	DET
ejpam-7135	343	37	topology	topology	NOUN
ejpam-7135	343	38	for	for	ADP
ejpam-7135	343	39	which	which	PRON
ejpam-7135	343	40	{	{	PUNCT
ejpam-7135	343	41	xa	xa	PROPN
ejpam-7135	343	42	|	|	ADV
ejpam-7135	343	43	a	a	DET
ejpam-7135	343	44	∈	∈	PROPN
ejpam-7135	343	45	b[m	b[m	NOUN
ejpam-7135	343	46	,	,	PUNCT
ejpam-7135	343	47	1	1	NUM
ejpam-7135	343	48	]	]	PUNCT
ejpam-7135	343	49	}	}	PUNCT
ejpam-7135	343	50	is	be	AUX
ejpam-7135	343	51	basis	basis	NOUN
ejpam-7135	343	52	,	,	PUNCT
ejpam-7135	343	53	where	where	SCONJ
ejpam-7135	343	54	for	for	ADP
ejpam-7135	343	55	any	any	DET
ejpam-7135	343	56	a	a	DET
ejpam-7135	343	57	∈	∈	PROPN
ejpam-7135	343	58	b[m	b[m	NOUN
ejpam-7135	343	59	,	,	PUNCT
ejpam-7135	343	60	1	1	NUM
ejpam-7135	343	61	]	]	PUNCT
ejpam-7135	343	62	,	,	PUNCT
ejpam-7135	343	63	xa	xa	PROPN
ejpam-7135	343	64	=	=	PRON
ejpam-7135	343	65	{	{	PUNCT
ejpam-7135	343	66	p	p	X
ejpam-7135	343	67	∈	∈	PROPN
ejpam-7135	343	68	x	x	X
ejpam-7135	343	69	|	|	ADV
ejpam-7135	343	70	a	a	DET
ejpam-7135	343	71	̸∈	̸∈	PROPN
ejpam-7135	343	72	p	p	PROPN
ejpam-7135	343	73	}	}	PUNCT
ejpam-7135	343	74	.	.	PUNCT
ejpam-7135	344	1	we	we	PRON
ejpam-7135	344	2	observed	observe	VERB
ejpam-7135	344	3	that	that	SCONJ
ejpam-7135	344	4	,	,	PUNCT
ejpam-7135	344	5	if	if	SCONJ
ejpam-7135	344	6	p	p	PROPN
ejpam-7135	344	7	∈	∈	X
ejpam-7135	345	1	y	y	NOUN
ejpam-7135	346	1	then	then	ADV
ejpam-7135	346	2	p	p	X
ejpam-7135	346	3	c	c	PROPN
ejpam-7135	347	1	=	=	SYM
ejpam-7135	347	2	p	p	NOUN
ejpam-7135	347	3	∩	∩	NOUN
ejpam-7135	347	4	b[m	b[m	ADJ
ejpam-7135	347	5	,	,	PUNCT
ejpam-7135	347	6	1	1	NUM
ejpam-7135	347	7	]	]	PUNCT
ejpam-7135	347	8	is	be	AUX
ejpam-7135	347	9	a	a	DET
ejpam-7135	347	10	prime	prime	ADJ
ejpam-7135	347	11	filter	filter	NOUN
ejpam-7135	347	12	of	of	ADP
ejpam-7135	347	13	b[m	b[m	ADJ
ejpam-7135	347	14	,	,	PUNCT
ejpam-7135	347	15	1	1	NUM
ejpam-7135	347	16	]	]	PUNCT
ejpam-7135	347	17	and	and	CCONJ
ejpam-7135	347	18	hence	hence	ADV
ejpam-7135	347	19	p	p	X
ejpam-7135	347	20	c	c	PROPN
ejpam-7135	347	21	∈	∈	PROPN
ejpam-7135	347	22	x.	x.	NOUN
ejpam-7135	348	1	now	now	ADV
ejpam-7135	348	2	we	we	PRON
ejpam-7135	348	3	prove	prove	VERB
ejpam-7135	348	4	the	the	DET
ejpam-7135	348	5	following	follow	VERB
ejpam-7135	348	6	lemmas	lemmas	PROPN
ejpam-7135	348	7	.	.	PUNCT
ejpam-7135	349	1	r.	r.	PROPN
ejpam-7135	349	2	bandaru	bandaru	PROPN
ejpam-7135	349	3	et	et	PROPN
ejpam-7135	349	4	al	al	PROPN
ejpam-7135	349	5	.	.	PUNCT
ejpam-7135	349	6	/	/	SYM
ejpam-7135	349	7	eur	eur	PROPN
ejpam-7135	349	8	.	.	PUNCT
ejpam-7135	350	1	j.	j.	PROPN
ejpam-7135	350	2	pure	pure	PROPN
ejpam-7135	350	3	appl	appl	PROPN
ejpam-7135	350	4	.	.	PROPN
ejpam-7135	350	5	math	math	PROPN
ejpam-7135	350	6	,	,	PUNCT
ejpam-7135	350	7	18	18	NUM
ejpam-7135	350	8	(	(	PUNCT
ejpam-7135	350	9	4	4	NUM
ejpam-7135	350	10	)	)	PUNCT
ejpam-7135	350	11	(	(	PUNCT
ejpam-7135	350	12	2025	2025	NUM
ejpam-7135	350	13	)	)	PUNCT
ejpam-7135	350	14	,	,	PUNCT
ejpam-7135	350	15	7135	7135	NUM
ejpam-7135	350	16	11	11	NUM
ejpam-7135	350	17	of	of	ADP
ejpam-7135	350	18	14	14	NUM
ejpam-7135	350	19	lemma	lemma	PROPN
ejpam-7135	350	20	12	12	NUM
ejpam-7135	350	21	.	.	PUNCT
ejpam-7135	351	1	let	let	VERB
ejpam-7135	351	2	l	l	NOUN
ejpam-7135	351	3	be	be	AUX
ejpam-7135	351	4	a	a	DET
ejpam-7135	351	5	pdl	pdl	NOUN
ejpam-7135	351	6	with	with	ADP
ejpam-7135	351	7	a	a	DET
ejpam-7135	351	8	minimal	minimal	ADJ
ejpam-7135	351	9	element	element	NOUN
ejpam-7135	351	10	m	m	PROPN
ejpam-7135	351	11	,	,	PUNCT
ejpam-7135	351	12	y	y	PROPN
ejpam-7135	351	13	the	the	DET
ejpam-7135	351	14	set	set	NOUN
ejpam-7135	351	15	of	of	ADP
ejpam-7135	351	16	prime	prime	ADJ
ejpam-7135	351	17	filters	filter	NOUN
ejpam-7135	351	18	of	of	ADP
ejpam-7135	351	19	l	l	NOUN
ejpam-7135	351	20	,	,	PUNCT
ejpam-7135	351	21	and	and	CCONJ
ejpam-7135	351	22	x	x	PUNCT
ejpam-7135	351	23	the	the	DET
ejpam-7135	351	24	boolean	boolean	ADJ
ejpam-7135	351	25	space	space	NOUN
ejpam-7135	351	26	of	of	ADP
ejpam-7135	351	27	prime	prime	ADJ
ejpam-7135	351	28	filters	filter	NOUN
ejpam-7135	351	29	of	of	ADP
ejpam-7135	351	30	b[m	b[m	ADJ
ejpam-7135	351	31	,	,	PUNCT
ejpam-7135	351	32	1	1	NUM
ejpam-7135	351	33	]	]	PUNCT
ejpam-7135	351	34	.	.	PUNCT
ejpam-7135	352	1	the	the	DET
ejpam-7135	352	2	map	map	NOUN
ejpam-7135	352	3	f	f	X
ejpam-7135	352	4	:	:	PUNCT
ejpam-7135	352	5	y	y	PROPN
ejpam-7135	352	6	→	→	PUNCT
ejpam-7135	352	7	x	x	NOUN
ejpam-7135	352	8	defined	define	VERB
ejpam-7135	352	9	by	by	ADP
ejpam-7135	352	10	f(p	f(p	NOUN
ejpam-7135	352	11	)	)	PUNCT
ejpam-7135	353	1	=	=	PUNCT
ejpam-7135	354	1	p	p	X
ejpam-7135	354	2	c	c	NOUN
ejpam-7135	354	3	=	=	SYM
ejpam-7135	354	4	p	p	NOUN
ejpam-7135	354	5	∩b[m	∩b[m	NOUN
ejpam-7135	354	6	,	,	PUNCT
ejpam-7135	354	7	1	1	NUM
ejpam-7135	354	8	]	]	PUNCT
ejpam-7135	354	9	is	be	AUX
ejpam-7135	354	10	continuous	continuous	ADJ
ejpam-7135	354	11	.	.	PUNCT
ejpam-7135	355	1	proof	proof	NOUN
ejpam-7135	355	2	.	.	PUNCT
ejpam-7135	356	1	let	let	VERB
ejpam-7135	356	2	xa	xa	PRON
ejpam-7135	356	3	be	be	AUX
ejpam-7135	356	4	a	a	DET
ejpam-7135	356	5	basic	basic	ADJ
ejpam-7135	356	6	open	open	ADJ
ejpam-7135	356	7	set	set	NOUN
ejpam-7135	356	8	in	in	ADP
ejpam-7135	356	9	x	x	PUNCT
ejpam-7135	356	10	for	for	ADP
ejpam-7135	356	11	some	some	DET
ejpam-7135	356	12	a	a	DET
ejpam-7135	356	13	∈	∈	PROPN
ejpam-7135	356	14	b[m	b[m	NOUN
ejpam-7135	356	15	,	,	PUNCT
ejpam-7135	356	16	1	1	NUM
ejpam-7135	356	17	]	]	PUNCT
ejpam-7135	356	18	.	.	PUNCT
ejpam-7135	357	1	then	then	ADV
ejpam-7135	357	2	:	:	PUNCT
ejpam-7135	357	3	f−1(xa	f−1(xa	X
ejpam-7135	357	4	)	)	PUNCT
ejpam-7135	357	5	=	=	PRON
ejpam-7135	357	6	{	{	PUNCT
ejpam-7135	357	7	p	p	X
ejpam-7135	357	8	∈	∈	PROPN
ejpam-7135	357	9	y	y	PROPN
ejpam-7135	357	10	|	|	ADV
ejpam-7135	357	11	f(p	f(p	PROPN
ejpam-7135	357	12	)	)	PUNCT
ejpam-7135	358	1	∈	∈	PROPN
ejpam-7135	359	1	xa	xa	PROPN
ejpam-7135	359	2	}	}	PUNCT
ejpam-7135	359	3	=	=	PUNCT
ejpam-7135	359	4	{	{	PUNCT
ejpam-7135	359	5	p	p	X
ejpam-7135	359	6	∈	∈	X
ejpam-7135	359	7	y	y	NOUN
ejpam-7135	359	8	|	|	ADV
ejpam-7135	359	9	a	a	X
ejpam-7135	359	10	/∈	/∈	PUNCT
ejpam-7135	360	1	p	p	NOUN
ejpam-7135	360	2	c	c	NOUN
ejpam-7135	360	3	}	}	PUNCT
ejpam-7135	360	4	=	=	SYM
ejpam-7135	360	5	{	{	PUNCT
ejpam-7135	360	6	p	p	X
ejpam-7135	360	7	∈	∈	X
ejpam-7135	360	8	y	y	NOUN
ejpam-7135	360	9	|	|	ADV
ejpam-7135	360	10	a	a	PRON
ejpam-7135	360	11	/∈	/∈	NOUN
ejpam-7135	361	1	p	p	NOUN
ejpam-7135	361	2	}	}	PUNCT
ejpam-7135	361	3	=	=	PUNCT
ejpam-7135	361	4	ya	ya	PROPN
ejpam-7135	361	5	which	which	PRON
ejpam-7135	361	6	is	be	AUX
ejpam-7135	361	7	open	open	ADJ
ejpam-7135	361	8	in	in	ADP
ejpam-7135	361	9	y	y	PROPN
ejpam-7135	361	10	.	.	PUNCT
ejpam-7135	362	1	hence	hence	ADV
ejpam-7135	362	2	,	,	PUNCT
ejpam-7135	362	3	f	f	PROPN
ejpam-7135	362	4	is	be	AUX
ejpam-7135	362	5	continuous	continuous	ADJ
ejpam-7135	362	6	.	.	PUNCT
ejpam-7135	363	1	lemma	lemma	PROPN
ejpam-7135	363	2	13	13	NUM
ejpam-7135	363	3	.	.	PUNCT
ejpam-7135	364	1	assume	assume	VERB
ejpam-7135	364	2	that	that	SCONJ
ejpam-7135	364	3	for	for	ADP
ejpam-7135	364	4	all	all	DET
ejpam-7135	364	5	p	p	NOUN
ejpam-7135	364	6	∈	∈	PROPN
ejpam-7135	364	7	x	x	NOUN
ejpam-7135	364	8	,	,	PUNCT
ejpam-7135	364	9	p	p	PROPN
ejpam-7135	364	10	e	e	PROPN
ejpam-7135	364	11	∈	∈	PROPN
ejpam-7135	364	12	y	y	PROPN
ejpam-7135	364	13	(	(	PUNCT
ejpam-7135	364	14	i.e.	i.e.	X
ejpam-7135	364	15	,	,	PUNCT
ejpam-7135	364	16	p	p	PRON
ejpam-7135	364	17	e	e	NOUN
ejpam-7135	364	18	is	be	AUX
ejpam-7135	364	19	a	a	DET
ejpam-7135	364	20	prime	prime	ADJ
ejpam-7135	364	21	filter	filter	NOUN
ejpam-7135	364	22	of	of	ADP
ejpam-7135	364	23	l	l	NOUN
ejpam-7135	364	24	)	)	PUNCT
ejpam-7135	364	25	.	.	PUNCT
ejpam-7135	365	1	define	define	VERB
ejpam-7135	365	2	g	g	NOUN
ejpam-7135	365	3	:	:	PUNCT
ejpam-7135	365	4	x	x	SYM
ejpam-7135	365	5	→	→	SYM
ejpam-7135	365	6	y	y	PROPN
ejpam-7135	365	7	by	by	ADP
ejpam-7135	365	8	g(p	g(p	PROPN
ejpam-7135	365	9	)	)	PUNCT
ejpam-7135	366	1	=	=	PUNCT
ejpam-7135	367	1	p	p	PROPN
ejpam-7135	367	2	e.	e.	PROPN
ejpam-7135	367	3	then	then	ADV
ejpam-7135	367	4	:	:	PUNCT
ejpam-7135	367	5	(	(	PUNCT
ejpam-7135	367	6	1	1	X
ejpam-7135	367	7	)	)	PUNCT
ejpam-7135	367	8	f	f	NOUN
ejpam-7135	367	9	◦	◦	NOUN
ejpam-7135	367	10	g	g	NOUN
ejpam-7135	367	11	=	=	PUNCT
ejpam-7135	367	12	idx	idx	NOUN
ejpam-7135	367	13	(	(	PUNCT
ejpam-7135	367	14	2	2	NUM
ejpam-7135	367	15	)	)	PUNCT
ejpam-7135	367	16	for	for	ADP
ejpam-7135	367	17	all	all	DET
ejpam-7135	367	18	x	x	SYM
ejpam-7135	367	19	∈	∈	PROPN
ejpam-7135	367	20	l	l	NOUN
ejpam-7135	367	21	,	,	PUNCT
ejpam-7135	367	22	g−1(yx	g−1(yx	NOUN
ejpam-7135	367	23	)	)	PUNCT
ejpam-7135	367	24	is	be	AUX
ejpam-7135	367	25	closed	close	VERB
ejpam-7135	367	26	in	in	ADP
ejpam-7135	367	27	x.	x.	NOUN
ejpam-7135	367	28	proof	proof	NOUN
ejpam-7135	367	29	.	.	PUNCT
ejpam-7135	368	1	(	(	PUNCT
ejpam-7135	368	2	1	1	X
ejpam-7135	368	3	)	)	PUNCT
ejpam-7135	368	4	let	let	VERB
ejpam-7135	368	5	p	p	PRON
ejpam-7135	368	6	∈	∈	PROPN
ejpam-7135	368	7	x.	x.	NOUN
ejpam-7135	368	8	then	then	ADV
ejpam-7135	368	9	:	:	PUNCT
ejpam-7135	368	10	(	(	PUNCT
ejpam-7135	368	11	f	f	X
ejpam-7135	368	12	◦	◦	NOUN
ejpam-7135	368	13	g)(p	g)(p	PUNCT
ejpam-7135	368	14	)	)	PUNCT
ejpam-7135	369	1	=	=	PUNCT
ejpam-7135	369	2	f(p	f(p	PROPN
ejpam-7135	369	3	e	e	X
ejpam-7135	369	4	)	)	PUNCT
ejpam-7135	369	5	=	=	SYM
ejpam-7135	369	6	p	p	X
ejpam-7135	369	7	e	e	X
ejpam-7135	369	8	∩	∩	X
ejpam-7135	369	9	b[m	b[m	PROPN
ejpam-7135	369	10	,	,	PUNCT
ejpam-7135	369	11	1	1	NUM
ejpam-7135	369	12	]	]	PUNCT
ejpam-7135	369	13	if	if	SCONJ
ejpam-7135	369	14	a	a	DET
ejpam-7135	369	15	∈	∈	PROPN
ejpam-7135	369	16	p	p	X
ejpam-7135	369	17	e	e	PROPN
ejpam-7135	369	18	∩	∩	X
ejpam-7135	369	19	b[m	b[m	PROPN
ejpam-7135	369	20	,	,	PUNCT
ejpam-7135	369	21	1	1	NUM
ejpam-7135	369	22	]	]	PUNCT
ejpam-7135	369	23	,	,	PUNCT
ejpam-7135	369	24	then	then	ADV
ejpam-7135	369	25	a	a	DET
ejpam-7135	369	26	=	=	PROPN
ejpam-7135	369	27	b∨	b∨	PROPN
ejpam-7135	369	28	t	t	PROPN
ejpam-7135	369	29	for	for	ADP
ejpam-7135	369	30	some	some	DET
ejpam-7135	369	31	b	b	NOUN
ejpam-7135	369	32	∈	∈	PROPN
ejpam-7135	369	33	p	p	NOUN
ejpam-7135	369	34	,	,	PUNCT
ejpam-7135	369	35	t	t	PROPN
ejpam-7135	369	36	∈	∈	PROPN
ejpam-7135	369	37	l.	l.	NOUN
ejpam-7135	369	38	but	but	CCONJ
ejpam-7135	369	39	since	since	SCONJ
ejpam-7135	369	40	a	a	DET
ejpam-7135	369	41	∈	∈	PROPN
ejpam-7135	369	42	b[m	b[m	NOUN
ejpam-7135	369	43	,	,	PUNCT
ejpam-7135	369	44	1	1	NUM
ejpam-7135	369	45	]	]	PUNCT
ejpam-7135	369	46	,	,	PUNCT
ejpam-7135	369	47	we	we	PRON
ejpam-7135	369	48	have	have	VERB
ejpam-7135	369	49	a	a	DET
ejpam-7135	369	50	=	=	PUNCT
ejpam-7135	369	51	b∨	b∨	PROPN
ejpam-7135	369	52	a	a	DET
ejpam-7135	369	53	∈	∈	PROPN
ejpam-7135	369	54	p	p	NOUN
ejpam-7135	369	55	.	.	PUNCT
ejpam-7135	370	1	thus	thus	ADV
ejpam-7135	370	2	,	,	PUNCT
ejpam-7135	370	3	p	p	PROPN
ejpam-7135	370	4	e	e	PROPN
ejpam-7135	370	5	∩b[m	∩b[m	NOUN
ejpam-7135	370	6	,	,	PUNCT
ejpam-7135	370	7	1	1	NUM
ejpam-7135	370	8	]	]	SYM
ejpam-7135	370	9	⊆	⊆	NUM
ejpam-7135	370	10	p	p	NOUN
ejpam-7135	370	11	.	.	PUNCT
ejpam-7135	371	1	the	the	DET
ejpam-7135	371	2	reverse	reverse	ADJ
ejpam-7135	371	3	inclusion	inclusion	NOUN
ejpam-7135	371	4	is	be	AUX
ejpam-7135	371	5	clear	clear	ADJ
ejpam-7135	371	6	since	since	SCONJ
ejpam-7135	371	7	p	p	PROPN
ejpam-7135	371	8	⊆	⊆	NUM
ejpam-7135	371	9	p	p	X
ejpam-7135	371	10	e.	e.	PROPN
ejpam-7135	371	11	hence	hence	PROPN
ejpam-7135	371	12	,	,	PUNCT
ejpam-7135	371	13	f	f	PROPN
ejpam-7135	371	14	◦	◦	NOUN
ejpam-7135	371	15	g	g	NOUN
ejpam-7135	371	16	=	=	NOUN
ejpam-7135	371	17	idx	idx	NOUN
ejpam-7135	371	18	.	.	PUNCT
ejpam-7135	372	1	(	(	PUNCT
ejpam-7135	372	2	2	2	X
ejpam-7135	372	3	)	)	PUNCT
ejpam-7135	372	4	let	let	VERB
ejpam-7135	372	5	x	x	PUNCT
ejpam-7135	372	6	∈	∈	NOUN
ejpam-7135	372	7	l	l	NOUN
ejpam-7135	372	8	and	and	CCONJ
ejpam-7135	372	9	p	p	NOUN
ejpam-7135	372	10	∈	∈	PROPN
ejpam-7135	372	11	x	x	PUNCT
ejpam-7135	372	12	\	\	ADJ
ejpam-7135	372	13	g−1(yx	g−1(yx	NOUN
ejpam-7135	372	14	)	)	PUNCT
ejpam-7135	372	15	.	.	PUNCT
ejpam-7135	373	1	then	then	ADV
ejpam-7135	373	2	x	x	X
ejpam-7135	373	3	∈	∈	PROPN
ejpam-7135	373	4	p	p	NOUN
ejpam-7135	373	5	e	e	NOUN
ejpam-7135	373	6	,	,	PUNCT
ejpam-7135	373	7	so	so	ADV
ejpam-7135	373	8	x	x	SYM
ejpam-7135	373	9	=	=	SYM
ejpam-7135	373	10	t	t	PROPN
ejpam-7135	373	11	∨	∨	NUM
ejpam-7135	373	12	a	a	PRON
ejpam-7135	373	13	for	for	ADP
ejpam-7135	373	14	some	some	PRON
ejpam-7135	373	15	a	a	DET
ejpam-7135	373	16	∈	∈	PROPN
ejpam-7135	373	17	p	p	NOUN
ejpam-7135	373	18	,	,	PUNCT
ejpam-7135	373	19	t	t	PROPN
ejpam-7135	373	20	∈	∈	PROPN
ejpam-7135	373	21	l.	l.	PROPN
ejpam-7135	373	22	let	let	VERB
ejpam-7135	373	23	a′	a′	PROPN
ejpam-7135	373	24	be	be	AUX
ejpam-7135	373	25	the	the	DET
ejpam-7135	373	26	complement	complement	NOUN
ejpam-7135	373	27	of	of	ADP
ejpam-7135	373	28	a	a	PRON
ejpam-7135	373	29	in	in	ADP
ejpam-7135	373	30	b[m	b[m	ADJ
ejpam-7135	373	31	,	,	PUNCT
ejpam-7135	373	32	1	1	NUM
ejpam-7135	373	33	]	]	PUNCT
ejpam-7135	373	34	.	.	PUNCT
ejpam-7135	374	1	then	then	ADV
ejpam-7135	374	2	a′	a′	PROPN
ejpam-7135	374	3	/∈	/∈	PROPN
ejpam-7135	375	1	p	p	NOUN
ejpam-7135	375	2	,	,	PUNCT
ejpam-7135	375	3	so	so	ADV
ejpam-7135	375	4	p	p	ADP
ejpam-7135	375	5	∈	∈	PROPN
ejpam-7135	375	6	xa′	xa′	PROPN
ejpam-7135	375	7	.	.	PUNCT
ejpam-7135	376	1	for	for	ADP
ejpam-7135	376	2	any	any	DET
ejpam-7135	376	3	q	q	NOUN
ejpam-7135	376	4	∈	∈	PROPN
ejpam-7135	376	5	xa′	xa′	NOUN
ejpam-7135	376	6	,	,	PUNCT
ejpam-7135	376	7	we	we	PRON
ejpam-7135	376	8	have	have	VERB
ejpam-7135	376	9	a′	a′	NOUN
ejpam-7135	376	10	/∈	/∈	PUNCT
ejpam-7135	377	1	q	q	INTJ
ejpam-7135	377	2	,	,	PUNCT
ejpam-7135	377	3	so	so	SCONJ
ejpam-7135	377	4	a	a	DET
ejpam-7135	377	5	∈	∈	ADJ
ejpam-7135	377	6	q	q	NOUN
ejpam-7135	377	7	,	,	PUNCT
ejpam-7135	377	8	hence	hence	ADV
ejpam-7135	377	9	x	x	PUNCT
ejpam-7135	377	10	=	=	SYM
ejpam-7135	377	11	t	t	PROPN
ejpam-7135	377	12	∨	∨	NUM
ejpam-7135	377	13	a	a	DET
ejpam-7135	377	14	∈	∈	PROPN
ejpam-7135	377	15	qe	qe	NOUN
ejpam-7135	377	16	,	,	PUNCT
ejpam-7135	377	17	so	so	ADV
ejpam-7135	377	18	q	q	NOUN
ejpam-7135	377	19	/∈	/∈	PUNCT
ejpam-7135	377	20	g−1(yx	g−1(yx	NOUN
ejpam-7135	377	21	)	)	PUNCT
ejpam-7135	377	22	.	.	PUNCT
ejpam-7135	378	1	thus	thus	ADV
ejpam-7135	378	2	,	,	PUNCT
ejpam-7135	378	3	xa′	xa′	PROPN
ejpam-7135	378	4	⊆	⊆	NUM
ejpam-7135	378	5	x	x	SYM
ejpam-7135	378	6	\	\	ADJ
ejpam-7135	378	7	g−1(yx	g−1(yx	NOUN
ejpam-7135	378	8	)	)	PUNCT
ejpam-7135	378	9	,	,	PUNCT
ejpam-7135	378	10	showing	show	VERB
ejpam-7135	378	11	that	that	SCONJ
ejpam-7135	378	12	g−1(yx	g−1(yx	NOUN
ejpam-7135	378	13	)	)	PUNCT
ejpam-7135	378	14	is	be	AUX
ejpam-7135	378	15	closed	close	VERB
ejpam-7135	378	16	.	.	PUNCT
ejpam-7135	379	1	theorem	theorem	NOUN
ejpam-7135	379	2	12	12	NUM
ejpam-7135	379	3	.	.	PUNCT
ejpam-7135	380	1	l	l	NOUN
ejpam-7135	380	2	with	with	ADP
ejpam-7135	380	3	a	a	DET
ejpam-7135	380	4	parapseudo	parapseudo	NOUN
ejpam-7135	380	5	-	-	PUNCT
ejpam-7135	380	6	complementation	complementation	NOUN
ejpam-7135	380	7	♦	♦	PROPN
ejpam-7135	380	8	in	in	ADP
ejpam-7135	380	9	which	which	PRON
ejpam-7135	380	10	m	m	VERB
ejpam-7135	380	11	is	be	AUX
ejpam-7135	380	12	a	a	DET
ejpam-7135	380	13	minimal	minimal	ADJ
ejpam-7135	380	14	element	element	NOUN
ejpam-7135	380	15	,	,	PUNCT
ejpam-7135	380	16	and	and	CCONJ
ejpam-7135	380	17	assume	assume	VERB
ejpam-7135	380	18	that	that	SCONJ
ejpam-7135	380	19	p	p	PROPN
ejpam-7135	380	20	e	e	X
ejpam-7135	380	21	∈	∈	PROPN
ejpam-7135	380	22	y	y	PROPN
ejpam-7135	380	23	for	for	ADP
ejpam-7135	380	24	all	all	DET
ejpam-7135	380	25	p	p	NOUN
ejpam-7135	380	26	∈	∈	PROPN
ejpam-7135	380	27	x	x	NOUN
ejpam-7135	380	28	,	,	PUNCT
ejpam-7135	380	29	where	where	SCONJ
ejpam-7135	380	30	x	x	PRON
ejpam-7135	380	31	is	be	AUX
ejpam-7135	380	32	the	the	DET
ejpam-7135	380	33	boolean	boolean	ADJ
ejpam-7135	380	34	space	space	NOUN
ejpam-7135	380	35	of	of	ADP
ejpam-7135	380	36	prime	prime	ADJ
ejpam-7135	380	37	filters	filter	NOUN
ejpam-7135	380	38	of	of	ADP
ejpam-7135	380	39	b[m	b[m	ADJ
ejpam-7135	380	40	,	,	PUNCT
ejpam-7135	380	41	1	1	NUM
ejpam-7135	380	42	]	]	PUNCT
ejpam-7135	380	43	and	and	CCONJ
ejpam-7135	380	44	y	y	PROPN
ejpam-7135	380	45	is	be	AUX
ejpam-7135	380	46	the	the	DET
ejpam-7135	380	47	set	set	NOUN
ejpam-7135	380	48	of	of	ADP
ejpam-7135	380	49	prime	prime	ADJ
ejpam-7135	380	50	filters	filter	NOUN
ejpam-7135	380	51	of	of	ADP
ejpam-7135	380	52	l.	l.	PROPN
ejpam-7135	380	53	then	then	ADV
ejpam-7135	380	54	the	the	DET
ejpam-7135	380	55	following	follow	VERB
ejpam-7135	380	56	are	be	AUX
ejpam-7135	380	57	equivalent	equivalent	ADJ
ejpam-7135	380	58	:	:	PUNCT
ejpam-7135	380	59	(	(	PUNCT
ejpam-7135	380	60	1	1	X
ejpam-7135	380	61	)	)	PUNCT
ejpam-7135	380	62	l	l	NOUN
ejpam-7135	380	63	is	be	AUX
ejpam-7135	380	64	a	a	DET
ejpam-7135	380	65	stone	stone	NOUN
ejpam-7135	380	66	pdl	pdl	NOUN
ejpam-7135	380	67	(	(	PUNCT
ejpam-7135	380	68	2	2	NUM
ejpam-7135	380	69	)	)	PUNCT
ejpam-7135	380	70	for	for	ADP
ejpam-7135	380	71	every	every	DET
ejpam-7135	380	72	x	x	SYM
ejpam-7135	380	73	∈	∈	PROPN
ejpam-7135	380	74	l	l	NOUN
ejpam-7135	380	75	,	,	PUNCT
ejpam-7135	380	76	there	there	PRON
ejpam-7135	380	77	exists	exist	VERB
ejpam-7135	380	78	a	a	DET
ejpam-7135	380	79	least	least	ADJ
ejpam-7135	380	80	element	element	NOUN
ejpam-7135	380	81	a	a	DET
ejpam-7135	380	82	∈	∈	PROPN
ejpam-7135	380	83	b[m	b[m	NOUN
ejpam-7135	380	84	,	,	PUNCT
ejpam-7135	380	85	1	1	NUM
ejpam-7135	380	86	]	]	PUNCT
ejpam-7135	380	87	such	such	ADJ
ejpam-7135	380	88	that	that	SCONJ
ejpam-7135	380	89	x	x	PROPN
ejpam-7135	380	90	∨	∨	NUM
ejpam-7135	380	91	a	a	DET
ejpam-7135	380	92	=	=	SYM
ejpam-7135	380	93	1	1	NUM
ejpam-7135	380	94	(	(	PUNCT
ejpam-7135	380	95	3	3	NUM
ejpam-7135	380	96	)	)	PUNCT
ejpam-7135	380	97	the	the	DET
ejpam-7135	380	98	map	map	NOUN
ejpam-7135	380	99	g	g	NOUN
ejpam-7135	380	100	:	:	PUNCT
ejpam-7135	380	101	x	x	SYM
ejpam-7135	380	102	→	→	SYM
ejpam-7135	380	103	y	y	PROPN
ejpam-7135	380	104	defined	define	VERB
ejpam-7135	380	105	by	by	ADP
ejpam-7135	380	106	g(p	g(p	NOUN
ejpam-7135	380	107	)	)	PUNCT
ejpam-7135	381	1	=	=	PUNCT
ejpam-7135	382	1	p	p	X
ejpam-7135	382	2	e	e	NOUN
ejpam-7135	382	3	is	be	AUX
ejpam-7135	382	4	continuous	continuous	ADJ
ejpam-7135	382	5	.	.	PUNCT
ejpam-7135	383	1	proof	proof	NOUN
ejpam-7135	383	2	.	.	PUNCT
ejpam-7135	384	1	(	(	PUNCT
ejpam-7135	384	2	1	1	X
ejpam-7135	384	3	)	)	PUNCT
ejpam-7135	384	4	⇒	⇒	NOUN
ejpam-7135	384	5	(	(	PUNCT
ejpam-7135	384	6	2	2	NUM
ejpam-7135	384	7	):	):	PUNCT
ejpam-7135	384	8	assume	assume	PROPN
ejpam-7135	384	9	l	l	NOUN
ejpam-7135	384	10	is	be	AUX
ejpam-7135	384	11	a	a	DET
ejpam-7135	384	12	stone	stone	NOUN
ejpam-7135	384	13	pdl	pdl	NOUN
ejpam-7135	384	14	.	.	PROPN
ejpam-7135	385	1	for	for	ADP
ejpam-7135	385	2	any	any	DET
ejpam-7135	385	3	x	x	SYM
ejpam-7135	385	4	∈	∈	PROPN
ejpam-7135	385	5	l	l	NOUN
ejpam-7135	385	6	,	,	PUNCT
ejpam-7135	385	7	we	we	PRON
ejpam-7135	385	8	have	have	VERB
ejpam-7135	385	9	x∨x	x∨x	PROPN
ejpam-7135	385	10	♦	♦	PROPN
ejpam-7135	385	11	=	=	PROPN
ejpam-7135	385	12	1	1	X
ejpam-7135	385	13	.	.	PUNCT
ejpam-7135	386	1	since	since	SCONJ
ejpam-7135	386	2	l	l	NOUN
ejpam-7135	386	3	is	be	AUX
ejpam-7135	386	4	stone	stone	NOUN
ejpam-7135	386	5	,	,	PUNCT
ejpam-7135	386	6	x	x	NOUN
ejpam-7135	386	7	♦	♦	PROPN
ejpam-7135	386	8	∧	∧	PROPN
ejpam-7135	386	9	x	x	PROPN
ejpam-7135	386	10	♦	♦	PROPN
ejpam-7135	386	11	♦	♦	PROPN
ejpam-7135	386	12	=	=	PROPN
ejpam-7135	386	13	1	1	NUM
ejpam-7135	386	14	♦	♦	PROPN
ejpam-7135	386	15	,	,	PUNCT
ejpam-7135	386	16	and	and	CCONJ
ejpam-7135	386	17	x	x	X
ejpam-7135	386	18	♦	♦	PROPN
ejpam-7135	386	19	∈	∈	PROPN
ejpam-7135	386	20	b[1	b[1	PROPN
ejpam-7135	386	21	♦	♦	PROPN
ejpam-7135	386	22	,	,	PUNCT
ejpam-7135	386	23	1	1	NUM
ejpam-7135	386	24	]	]	SYM
ejpam-7135	386	25	⊆	⊆	NUM
ejpam-7135	386	26	b[m	b[m	ADJ
ejpam-7135	386	27	,	,	PUNCT
ejpam-7135	386	28	1	1	NUM
ejpam-7135	386	29	]	]	PUNCT
ejpam-7135	386	30	.	.	PUNCT
ejpam-7135	387	1	if	if	SCONJ
ejpam-7135	387	2	a	a	DET
ejpam-7135	387	3	∈	∈	PROPN
ejpam-7135	387	4	b[m	b[m	NOUN
ejpam-7135	387	5	,	,	PUNCT
ejpam-7135	387	6	1	1	NUM
ejpam-7135	387	7	]	]	PUNCT
ejpam-7135	387	8	satisfies	satisfie	NOUN
ejpam-7135	387	9	x	x	X
ejpam-7135	387	10	∨	∨	NUM
ejpam-7135	387	11	a	a	DET
ejpam-7135	387	12	=	=	SYM
ejpam-7135	387	13	1	1	NUM
ejpam-7135	387	14	,	,	PUNCT
ejpam-7135	387	15	then	then	ADV
ejpam-7135	387	16	x	x	PROPN
ejpam-7135	387	17	♦	♦	PROPN
ejpam-7135	387	18	∨	∨	VERB
ejpam-7135	387	19	a	a	DET
ejpam-7135	387	20	=	=	SYM
ejpam-7135	387	21	a	a	NOUN
ejpam-7135	387	22	,	,	PUNCT
ejpam-7135	387	23	so	so	SCONJ
ejpam-7135	387	24	a	a	DET
ejpam-7135	387	25	≤	≤	PROPN
ejpam-7135	387	26	x	x	SYM
ejpam-7135	387	27	♦	♦	PROPN
ejpam-7135	387	28	.	.	PUNCT
ejpam-7135	388	1	thus	thus	ADV
ejpam-7135	388	2	,	,	PUNCT
ejpam-7135	388	3	x	x	PRON
ejpam-7135	388	4	♦	♦	PROPN
ejpam-7135	388	5	is	be	AUX
ejpam-7135	388	6	the	the	DET
ejpam-7135	388	7	least	least	ADJ
ejpam-7135	388	8	such	such	ADJ
ejpam-7135	388	9	element	element	NOUN
ejpam-7135	388	10	in	in	ADP
ejpam-7135	388	11	b[m	b[m	ADJ
ejpam-7135	388	12	,	,	PUNCT
ejpam-7135	388	13	1	1	NUM
ejpam-7135	388	14	]	]	PUNCT
ejpam-7135	388	15	.	.	PUNCT
ejpam-7135	389	1	(	(	PUNCT
ejpam-7135	389	2	2	2	X
ejpam-7135	389	3	)	)	PUNCT
ejpam-7135	389	4	⇒	⇒	NOUN
ejpam-7135	389	5	(	(	PUNCT
ejpam-7135	389	6	3	3	NUM
ejpam-7135	389	7	):	):	PUNCT
ejpam-7135	389	8	assume	assume	VERB
ejpam-7135	389	9	(	(	PUNCT
ejpam-7135	389	10	2	2	NUM
ejpam-7135	389	11	)	)	PUNCT
ejpam-7135	389	12	holds	hold	VERB
ejpam-7135	389	13	.	.	PUNCT
ejpam-7135	390	1	let	let	VERB
ejpam-7135	390	2	x	x	PUNCT
ejpam-7135	390	3	∈	∈	PROPN
ejpam-7135	390	4	l	l	NOUN
ejpam-7135	390	5	and	and	CCONJ
ejpam-7135	390	6	p	p	NOUN
ejpam-7135	390	7	∈	∈	PROPN
ejpam-7135	390	8	g−1(yx	g−1(yx	NOUN
ejpam-7135	390	9	)	)	PUNCT
ejpam-7135	390	10	.	.	PUNCT
ejpam-7135	391	1	then	then	ADV
ejpam-7135	391	2	x	x	X
ejpam-7135	391	3	/∈	/∈	PUNCT
ejpam-7135	391	4	g(p	g(p	PROPN
ejpam-7135	391	5	)	)	PUNCT
ejpam-7135	392	1	=	=	PUNCT
ejpam-7135	393	1	p	p	X
ejpam-7135	393	2	e.	e.	PROPN
ejpam-7135	393	3	by	by	ADP
ejpam-7135	393	4	(	(	PUNCT
ejpam-7135	393	5	2	2	NUM
ejpam-7135	393	6	)	)	PUNCT
ejpam-7135	393	7	,	,	PUNCT
ejpam-7135	393	8	there	there	PRON
ejpam-7135	393	9	exists	exist	VERB
ejpam-7135	393	10	a	a	DET
ejpam-7135	393	11	least	least	ADJ
ejpam-7135	393	12	element	element	ADJ
ejpam-7135	393	13	ax	ax	NOUN
ejpam-7135	393	14	∈	∈	PROPN
ejpam-7135	393	15	b[m	b[m	NOUN
ejpam-7135	393	16	,	,	PUNCT
ejpam-7135	393	17	1	1	NUM
ejpam-7135	393	18	]	]	PUNCT
ejpam-7135	393	19	such	such	ADJ
ejpam-7135	393	20	that	that	SCONJ
ejpam-7135	393	21	x∨	x∨	PROPN
ejpam-7135	393	22	ax	ax	NOUN
ejpam-7135	393	23	=	=	NOUN
ejpam-7135	393	24	1	1	X
ejpam-7135	393	25	.	.	PUNCT
ejpam-7135	394	1	since	since	SCONJ
ejpam-7135	394	2	x	x	PROPN
ejpam-7135	394	3	/∈	/∈	PUNCT
ejpam-7135	394	4	p	p	NOUN
ejpam-7135	394	5	e	e	NOUN
ejpam-7135	394	6	,	,	PUNCT
ejpam-7135	394	7	we	we	PRON
ejpam-7135	394	8	must	must	AUX
ejpam-7135	394	9	have	have	VERB
ejpam-7135	394	10	ax	ax	NOUN
ejpam-7135	394	11	∈	∈	PROPN
ejpam-7135	394	12	p	p	PROPN
ejpam-7135	394	13	e	e	NOUN
ejpam-7135	394	14	,	,	PUNCT
ejpam-7135	394	15	hence	hence	ADV
ejpam-7135	394	16	ax	ax	NOUN
ejpam-7135	394	17	∈	∈	PROPN
ejpam-7135	394	18	p	p	PROPN
ejpam-7135	394	19	e	e	PROPN
ejpam-7135	394	20	∩	∩	X
ejpam-7135	394	21	b[m	b[m	ADJ
ejpam-7135	394	22	,	,	PUNCT
ejpam-7135	394	23	1	1	NUM
ejpam-7135	394	24	]	]	PUNCT
ejpam-7135	394	25	=	=	PUNCT
ejpam-7135	394	26	p	p	NOUN
ejpam-7135	394	27	.	.	PUNCT
ejpam-7135	395	1	let	let	VERB
ejpam-7135	395	2	a′x	a′x	PRON
ejpam-7135	395	3	be	be	AUX
ejpam-7135	395	4	the	the	DET
ejpam-7135	395	5	complement	complement	NOUN
ejpam-7135	395	6	of	of	ADP
ejpam-7135	395	7	ax	ax	NOUN
ejpam-7135	395	8	in	in	ADP
ejpam-7135	395	9	b[m	b[m	ADJ
ejpam-7135	395	10	,	,	PUNCT
ejpam-7135	395	11	1	1	NUM
ejpam-7135	395	12	]	]	PUNCT
ejpam-7135	395	13	.	.	PUNCT
ejpam-7135	396	1	then	then	ADV
ejpam-7135	396	2	a′x	a′x	PROPN
ejpam-7135	396	3	/∈	/∈	PUNCT
ejpam-7135	397	1	p	p	X
ejpam-7135	397	2	,	,	PUNCT
ejpam-7135	397	3	so	so	ADV
ejpam-7135	397	4	p	p	X
ejpam-7135	397	5	∈	∈	PROPN
ejpam-7135	397	6	xa′x	xa′x	PROPN
ejpam-7135	397	7	.	.	PUNCT
ejpam-7135	398	1	now	now	ADV
ejpam-7135	398	2	,	,	PUNCT
ejpam-7135	398	3	if	if	SCONJ
ejpam-7135	398	4	q	q	X
ejpam-7135	398	5	∈	∈	PROPN
ejpam-7135	398	6	xa′x	xa′x	NOUN
ejpam-7135	398	7	,	,	PUNCT
ejpam-7135	398	8	then	then	ADV
ejpam-7135	398	9	a′x	a′x	NOUN
ejpam-7135	398	10	/∈	/∈	PUNCT
ejpam-7135	399	1	q	q	INTJ
ejpam-7135	399	2	,	,	PUNCT
ejpam-7135	399	3	so	so	ADV
ejpam-7135	399	4	ax	ax	NOUN
ejpam-7135	399	5	∈	∈	PROPN
ejpam-7135	399	6	q.	q.	NOUN
ejpam-7135	399	7	if	if	SCONJ
ejpam-7135	399	8	x	x	PROPN
ejpam-7135	399	9	∈	∈	PROPN
ejpam-7135	399	10	qe	qe	PROPN
ejpam-7135	399	11	,	,	PUNCT
ejpam-7135	399	12	then	then	ADV
ejpam-7135	399	13	x	x	X
ejpam-7135	399	14	=	=	SYM
ejpam-7135	399	15	t	t	PROPN
ejpam-7135	399	16	∨	∨	NUM
ejpam-7135	399	17	b	b	PROPN
ejpam-7135	399	18	for	for	ADP
ejpam-7135	399	19	some	some	DET
ejpam-7135	399	20	b	b	NOUN
ejpam-7135	399	21	∈	∈	ADJ
ejpam-7135	399	22	q	q	NOUN
ejpam-7135	399	23	,	,	PUNCT
ejpam-7135	399	24	t	t	PROPN
ejpam-7135	399	25	∈	∈	PROPN
ejpam-7135	399	26	l	l	NOUN
ejpam-7135	399	27	,	,	PUNCT
ejpam-7135	399	28	and	and	CCONJ
ejpam-7135	399	29	x	x	X
ejpam-7135	399	30	∨	∨	NOUN
ejpam-7135	399	31	b′	b′	NUM
ejpam-7135	399	32	=	=	NOUN
ejpam-7135	399	33	1	1	NUM
ejpam-7135	399	34	where	where	SCONJ
ejpam-7135	399	35	b′	b′	NOUN
ejpam-7135	399	36	is	be	AUX
ejpam-7135	399	37	the	the	DET
ejpam-7135	399	38	complement	complement	NOUN
ejpam-7135	399	39	of	of	ADP
ejpam-7135	399	40	b.	b.	PROPN
ejpam-7135	399	41	then	then	ADV
ejpam-7135	399	42	b′	b′	NUM
ejpam-7135	399	43	≤	≤	NUM
ejpam-7135	399	44	ax	ax	NOUN
ejpam-7135	399	45	,	,	PUNCT
ejpam-7135	399	46	so	so	ADV
ejpam-7135	399	47	ax	ax	NOUN
ejpam-7135	399	48	/∈	/∈	PUNCT
ejpam-7135	399	49	q	q	X
ejpam-7135	400	1	(	(	PUNCT
ejpam-7135	400	2	since	since	SCONJ
ejpam-7135	400	3	b	b	PROPN
ejpam-7135	400	4	∈	∈	PROPN
ejpam-7135	400	5	q	q	NOUN
ejpam-7135	400	6	)	)	PUNCT
ejpam-7135	400	7	,	,	PUNCT
ejpam-7135	400	8	a	a	DET
ejpam-7135	400	9	contradiction	contradiction	NOUN
ejpam-7135	400	10	.	.	PUNCT
ejpam-7135	401	1	hence	hence	ADV
ejpam-7135	401	2	,	,	PUNCT
ejpam-7135	401	3	x	x	PROPN
ejpam-7135	401	4	/∈	/∈	PUNCT
ejpam-7135	401	5	qe	qe	PROPN
ejpam-7135	401	6	,	,	PUNCT
ejpam-7135	401	7	so	so	ADV
ejpam-7135	401	8	q	q	PUNCT
ejpam-7135	401	9	∈	∈	PROPN
ejpam-7135	401	10	g−1(yx	g−1(yx	NOUN
ejpam-7135	401	11	)	)	PUNCT
ejpam-7135	401	12	.	.	PUNCT
ejpam-7135	402	1	thus	thus	ADV
ejpam-7135	402	2	,	,	PUNCT
ejpam-7135	402	3	xa′x	xa′x	NOUN
ejpam-7135	402	4	⊆	⊆	NUM
ejpam-7135	402	5	g−1(yx	g−1(yx	NOUN
ejpam-7135	402	6	)	)	PUNCT
ejpam-7135	402	7	,	,	PUNCT
ejpam-7135	402	8	showing	show	VERB
ejpam-7135	402	9	g−1(yx	g−1(yx	NOUN
ejpam-7135	402	10	)	)	PUNCT
ejpam-7135	402	11	is	be	AUX
ejpam-7135	402	12	open	open	ADJ
ejpam-7135	402	13	.	.	PUNCT
ejpam-7135	403	1	r.	r.	PROPN
ejpam-7135	403	2	bandaru	bandaru	PROPN
ejpam-7135	403	3	et	et	PROPN
ejpam-7135	403	4	al	al	PROPN
ejpam-7135	403	5	.	.	PUNCT
ejpam-7135	403	6	/	/	SYM
ejpam-7135	403	7	eur	eur	PROPN
ejpam-7135	403	8	.	.	PUNCT
ejpam-7135	404	1	j.	j.	PROPN
ejpam-7135	404	2	pure	pure	PROPN
ejpam-7135	404	3	appl	appl	PROPN
ejpam-7135	404	4	.	.	PROPN
ejpam-7135	404	5	math	math	PROPN
ejpam-7135	404	6	,	,	PUNCT
ejpam-7135	404	7	18	18	NUM
ejpam-7135	404	8	(	(	PUNCT
ejpam-7135	404	9	4	4	NUM
ejpam-7135	404	10	)	)	PUNCT
ejpam-7135	404	11	(	(	PUNCT
ejpam-7135	404	12	2025	2025	NUM
ejpam-7135	404	13	)	)	PUNCT
ejpam-7135	404	14	,	,	PUNCT
ejpam-7135	404	15	7135	7135	NUM
ejpam-7135	404	16	12	12	NUM
ejpam-7135	404	17	of	of	ADP
ejpam-7135	404	18	14	14	NUM
ejpam-7135	404	19	(	(	PUNCT
ejpam-7135	404	20	3	3	NUM
ejpam-7135	404	21	)	)	PUNCT
ejpam-7135	404	22	⇒	⇒	NOUN
ejpam-7135	404	23	(	(	PUNCT
ejpam-7135	404	24	1	1	NUM
ejpam-7135	404	25	):	):	PUNCT
ejpam-7135	404	26	assume	assume	VERB
ejpam-7135	404	27	g	g	PROPN
ejpam-7135	404	28	is	be	AUX
ejpam-7135	404	29	continuous	continuous	ADJ
ejpam-7135	404	30	.	.	PUNCT
ejpam-7135	405	1	for	for	ADP
ejpam-7135	405	2	each	each	DET
ejpam-7135	405	3	x	x	SYM
ejpam-7135	405	4	∈	∈	PROPN
ejpam-7135	405	5	l	l	NOUN
ejpam-7135	405	6	,	,	PUNCT
ejpam-7135	405	7	g−1(yx	g−1(yx	NOUN
ejpam-7135	405	8	)	)	PUNCT
ejpam-7135	405	9	is	be	AUX
ejpam-7135	405	10	clopen	clopen	ADJ
ejpam-7135	405	11	in	in	ADP
ejpam-7135	405	12	x	x	PUNCT
ejpam-7135	405	13	(	(	PUNCT
ejpam-7135	405	14	open	open	ADJ
ejpam-7135	405	15	by	by	ADP
ejpam-7135	405	16	continuity	continuity	NOUN
ejpam-7135	405	17	,	,	PUNCT
ejpam-7135	405	18	closed	close	VERB
ejpam-7135	405	19	by	by	ADP
ejpam-7135	405	20	lemma	lemma	PROPN
ejpam-7135	405	21	13	13	NUM
ejpam-7135	405	22	)	)	PUNCT
ejpam-7135	405	23	.	.	PUNCT
ejpam-7135	406	1	by	by	ADP
ejpam-7135	406	2	lemma	lemma	PROPN
ejpam-7135	406	3	11	11	NUM
ejpam-7135	406	4	,	,	PUNCT
ejpam-7135	406	5	there	there	PRON
ejpam-7135	406	6	exists	exist	VERB
ejpam-7135	406	7	a	a	DET
ejpam-7135	406	8	unique	unique	ADJ
ejpam-7135	406	9	ax	ax	NOUN
ejpam-7135	406	10	∈	∈	PROPN
ejpam-7135	406	11	b[m	b[m	NOUN
ejpam-7135	406	12	,	,	PUNCT
ejpam-7135	406	13	1	1	NUM
ejpam-7135	406	14	]	]	PUNCT
ejpam-7135	406	15	such	such	ADJ
ejpam-7135	406	16	that	that	SCONJ
ejpam-7135	406	17	g−1(yx	g−1(yx	NOUN
ejpam-7135	406	18	)	)	PUNCT
ejpam-7135	406	19	=	=	SYM
ejpam-7135	406	20	xax	xax	PROPN
ejpam-7135	406	21	.	.	PUNCT
ejpam-7135	407	1	define	define	VERB
ejpam-7135	407	2	x	x	PRON
ejpam-7135	407	3	♦	♦	PROPN
ejpam-7135	407	4	=	=	PUNCT
ejpam-7135	407	5	a′x	a′x	PROPN
ejpam-7135	407	6	,	,	PUNCT
ejpam-7135	407	7	where	where	SCONJ
ejpam-7135	407	8	a′x	a′x	PROPN
ejpam-7135	407	9	is	be	AUX
ejpam-7135	407	10	the	the	DET
ejpam-7135	407	11	complement	complement	NOUN
ejpam-7135	407	12	of	of	ADP
ejpam-7135	407	13	ax	ax	NOUN
ejpam-7135	407	14	in	in	ADP
ejpam-7135	407	15	b[m	b[m	ADJ
ejpam-7135	407	16	,	,	PUNCT
ejpam-7135	407	17	1	1	NUM
ejpam-7135	407	18	]	]	PUNCT
ejpam-7135	407	19	.	.	PUNCT
ejpam-7135	408	1	we	we	PRON
ejpam-7135	408	2	verify	verify	VERB
ejpam-7135	408	3	that	that	SCONJ
ejpam-7135	408	4	♦	♦	PROPN
ejpam-7135	408	5	is	be	AUX
ejpam-7135	408	6	a	a	DET
ejpam-7135	408	7	parapseudo	parapseudo	NOUN
ejpam-7135	408	8	-	-	NOUN
ejpam-7135	408	9	complementation	complementation	NOUN
ejpam-7135	408	10	:	:	PUNCT
ejpam-7135	408	11	•	•	NUM
ejpam-7135	408	12	x	x	SYM
ejpam-7135	408	13	∨	∨	NUM
ejpam-7135	408	14	x	x	NOUN
ejpam-7135	408	15	♦	♦	PROPN
ejpam-7135	408	16	=	=	PROPN
ejpam-7135	408	17	1	1	NUM
ejpam-7135	408	18	:	:	PUNCT
ejpam-7135	408	19	if	if	SCONJ
ejpam-7135	408	20	not	not	PART
ejpam-7135	408	21	,	,	PUNCT
ejpam-7135	408	22	there	there	PRON
ejpam-7135	408	23	exists	exist	VERB
ejpam-7135	408	24	p	p	PROPN
ejpam-7135	408	25	∈	∈	PROPN
ejpam-7135	408	26	y	y	PROPN
ejpam-7135	408	27	with	with	ADP
ejpam-7135	408	28	x	x	PUNCT
ejpam-7135	408	29	∨	∨	NUM
ejpam-7135	408	30	a′x	a′x	PROPN
ejpam-7135	408	31	/∈	/∈	PUNCT
ejpam-7135	409	1	p	p	NOUN
ejpam-7135	409	2	,	,	PUNCT
ejpam-7135	409	3	leading	lead	VERB
ejpam-7135	409	4	to	to	ADP
ejpam-7135	409	5	a	a	DET
ejpam-7135	409	6	contradiction	contradiction	NOUN
ejpam-7135	409	7	.	.	PUNCT
ejpam-7135	410	1	•	•	INTJ
ejpam-7135	410	2	if	if	SCONJ
ejpam-7135	410	3	x	x	PROPN
ejpam-7135	410	4	∨	∨	NUM
ejpam-7135	410	5	y	y	NOUN
ejpam-7135	410	6	=	=	SYM
ejpam-7135	410	7	1	1	NUM
ejpam-7135	410	8	,	,	PUNCT
ejpam-7135	410	9	then	then	ADV
ejpam-7135	410	10	x	x	PROPN
ejpam-7135	410	11	♦	♦	PROPN
ejpam-7135	410	12	∨	∨	PROPN
ejpam-7135	410	13	y	y	PROPN
ejpam-7135	410	14	=	=	SYM
ejpam-7135	410	15	y	y	PROPN
ejpam-7135	410	16	:	:	PUNCT
ejpam-7135	410	17	follows	follow	VERB
ejpam-7135	410	18	from	from	ADP
ejpam-7135	410	19	the	the	DET
ejpam-7135	410	20	minimality	minimality	NOUN
ejpam-7135	410	21	of	of	ADP
ejpam-7135	410	22	ax	ax	NOUN
ejpam-7135	410	23	.	.	PUNCT
ejpam-7135	411	1	•	•	NUM
ejpam-7135	411	2	(	(	PUNCT
ejpam-7135	411	3	x	x	PUNCT
ejpam-7135	411	4	∧	∧	PROPN
ejpam-7135	411	5	y	y	PROPN
ejpam-7135	411	6	)	)	PUNCT
ejpam-7135	411	7	♦	♦	PROPN
ejpam-7135	411	8	=	=	PROPN
ejpam-7135	412	1	x	x	PROPN
ejpam-7135	412	2	♦	♦	PROPN
ejpam-7135	412	3	∨	∨	PROPN
ejpam-7135	412	4	y	y	PROPN
ejpam-7135	412	5	♦	♦	PROPN
ejpam-7135	412	6	:	:	PUNCT
ejpam-7135	412	7	since	since	SCONJ
ejpam-7135	412	8	g−1(yx∧y	g−1(yx∧y	PROPN
ejpam-7135	412	9	)	)	PUNCT
ejpam-7135	412	10	=	=	SYM
ejpam-7135	412	11	g−1(yx	g−1(yx	NOUN
ejpam-7135	412	12	)	)	PUNCT
ejpam-7135	412	13	∪	∪	ADP
ejpam-7135	412	14	g−1(yy	g−1(yy	NOUN
ejpam-7135	412	15	)	)	PUNCT
ejpam-7135	412	16	=	=	SYM
ejpam-7135	412	17	xax	xax	PROPN
ejpam-7135	412	18	∪xay	∪xay	PROPN
ejpam-7135	412	19	=	=	PUNCT
ejpam-7135	412	20	xax∧ay	xax∧ay	PROPN
ejpam-7135	412	21	.	.	PUNCT
ejpam-7135	413	1	finally	finally	ADV
ejpam-7135	413	2	,	,	PUNCT
ejpam-7135	413	3	x	x	X
ejpam-7135	413	4	♦	♦	PROPN
ejpam-7135	413	5	∧	∧	PROPN
ejpam-7135	413	6	x	x	PROPN
ejpam-7135	413	7	♦	♦	PROPN
ejpam-7135	414	1	♦	♦	PROPN
ejpam-7135	414	2	=	=	PROPN
ejpam-7135	414	3	a′x	a′x	PROPN
ejpam-7135	414	4	∧	∧	PROPN
ejpam-7135	414	5	(	(	PUNCT
ejpam-7135	414	6	a′x	a′x	ADJ
ejpam-7135	414	7	)	)	PUNCT
ejpam-7135	414	8	♦	♦	PROPN
ejpam-7135	414	9	=	=	PROPN
ejpam-7135	414	10	a′x	a′x	PROPN
ejpam-7135	414	11	∧	∧	NOUN
ejpam-7135	414	12	ax	ax	NOUN
ejpam-7135	414	13	=	=	NOUN
ejpam-7135	414	14	m	m	NOUN
ejpam-7135	414	15	=	=	SYM
ejpam-7135	414	16	1	1	NUM
ejpam-7135	414	17	♦	♦	PROPN
ejpam-7135	414	18	,	,	PUNCT
ejpam-7135	414	19	so	so	ADV
ejpam-7135	414	20	l	l	NOUN
ejpam-7135	414	21	is	be	AUX
ejpam-7135	414	22	a	a	DET
ejpam-7135	414	23	stone	stone	NOUN
ejpam-7135	414	24	pdl	pdl	PROPN
ejpam-7135	414	25	.	.	PUNCT
ejpam-7135	415	1	lemma	lemma	PROPN
ejpam-7135	415	2	14	14	NUM
ejpam-7135	415	3	.	.	PUNCT
ejpam-7135	416	1	let	let	VERB
ejpam-7135	416	2	l	l	NOUN
ejpam-7135	416	3	be	be	AUX
ejpam-7135	416	4	a	a	DET
ejpam-7135	416	5	pdl	pdl	NOUN
ejpam-7135	416	6	with	with	ADP
ejpam-7135	416	7	minimal	minimal	ADJ
ejpam-7135	416	8	element	element	NOUN
ejpam-7135	416	9	m	m	NOUN
ejpam-7135	416	10	,	,	PUNCT
ejpam-7135	416	11	x	x	X
ejpam-7135	416	12	∈	∈	PROPN
ejpam-7135	416	13	l	l	NOUN
ejpam-7135	416	14	,	,	PUNCT
ejpam-7135	416	15	and	and	CCONJ
ejpam-7135	416	16	a	a	DET
ejpam-7135	416	17	∈	∈	PROPN
ejpam-7135	416	18	b[m	b[m	NOUN
ejpam-7135	416	19	,	,	PUNCT
ejpam-7135	416	20	1	1	NUM
ejpam-7135	416	21	]	]	PUNCT
ejpam-7135	416	22	.	.	PUNCT
ejpam-7135	417	1	let	let	VERB
ejpam-7135	417	2	mx	mx	VERB
ejpam-7135	417	3	=	=	PRON
ejpam-7135	417	4	{	{	PUNCT
ejpam-7135	417	5	p	p	X
ejpam-7135	417	6	∈	∈	PROPN
ejpam-7135	417	7	m	m	VERB
ejpam-7135	417	8	|	|	NOUN
ejpam-7135	417	9	x	x	X
ejpam-7135	417	10	/∈	/∈	PUNCT
ejpam-7135	418	1	p	p	X
ejpam-7135	418	2	}	}	PUNCT
ejpam-7135	418	3	,	,	PUNCT
ejpam-7135	418	4	where	where	SCONJ
ejpam-7135	418	5	m	m	NOUN
ejpam-7135	418	6	is	be	AUX
ejpam-7135	418	7	the	the	DET
ejpam-7135	418	8	set	set	NOUN
ejpam-7135	418	9	of	of	ADP
ejpam-7135	418	10	minimal	minimal	ADJ
ejpam-7135	418	11	prime	prime	ADJ
ejpam-7135	418	12	filters	filter	NOUN
ejpam-7135	418	13	of	of	ADP
ejpam-7135	418	14	l.	l.	NOUN
ejpam-7135	418	15	then	then	ADV
ejpam-7135	418	16	a∨x	a∨x	PROPN
ejpam-7135	419	1	=	=	PUNCT
ejpam-7135	420	1	x	x	SYM
ejpam-7135	420	2	if	if	SCONJ
ejpam-7135	420	3	and	and	CCONJ
ejpam-7135	420	4	only	only	ADV
ejpam-7135	420	5	if	if	SCONJ
ejpam-7135	420	6	mx	mx	PROPN
ejpam-7135	420	7	⊆	⊆	NUM
ejpam-7135	420	8	ma	ma	PROPN
ejpam-7135	420	9	.	.	PROPN
ejpam-7135	420	10	proof	proof	PROPN
ejpam-7135	420	11	.	.	PUNCT
ejpam-7135	421	1	(	(	PUNCT
ejpam-7135	421	2	⇒	⇒	NOUN
ejpam-7135	421	3	):	):	PUNCT
ejpam-7135	421	4	suppose	suppose	VERB
ejpam-7135	421	5	a	a	DET
ejpam-7135	421	6	∨	∨	NOUN
ejpam-7135	421	7	x	x	X
ejpam-7135	421	8	=	=	PUNCT
ejpam-7135	421	9	x.	x.	NOUN
ejpam-7135	421	10	then	then	ADV
ejpam-7135	421	11	a	a	DET
ejpam-7135	421	12	≤	≤	NOUN
ejpam-7135	421	13	x.	x.	NOUN
ejpam-7135	422	1	if	if	SCONJ
ejpam-7135	422	2	p	p	PROPN
ejpam-7135	422	3	∈	∈	PROPN
ejpam-7135	422	4	mx	mx	PROPN
ejpam-7135	422	5	,	,	PUNCT
ejpam-7135	422	6	then	then	ADV
ejpam-7135	422	7	x	x	X
ejpam-7135	422	8	/∈	/∈	PUNCT
ejpam-7135	423	1	p	p	X
ejpam-7135	423	2	,	,	PUNCT
ejpam-7135	423	3	so	so	ADV
ejpam-7135	423	4	a	a	DET
ejpam-7135	423	5	/∈	/∈	INTJ
ejpam-7135	423	6	p	p	NOUN
ejpam-7135	423	7	(	(	PUNCT
ejpam-7135	423	8	since	since	SCONJ
ejpam-7135	423	9	p	p	NOUN
ejpam-7135	423	10	is	be	AUX
ejpam-7135	423	11	upward	upward	ADV
ejpam-7135	423	12	closed	closed	ADJ
ejpam-7135	423	13	)	)	PUNCT
ejpam-7135	423	14	,	,	PUNCT
ejpam-7135	423	15	hence	hence	ADV
ejpam-7135	423	16	p	p	PROPN
ejpam-7135	423	17	∈	∈	PROPN
ejpam-7135	423	18	ma	ma	PROPN
ejpam-7135	423	19	.	.	PROPN
ejpam-7135	424	1	thus	thus	ADV
ejpam-7135	424	2	,	,	PUNCT
ejpam-7135	424	3	mx	mx	PROPN
ejpam-7135	424	4	⊆	⊆	NUM
ejpam-7135	424	5	ma	ma	PROPN
ejpam-7135	424	6	.	.	PROPN
ejpam-7135	425	1	(	(	PUNCT
ejpam-7135	425	2	⇐	⇐	PROPN
ejpam-7135	425	3	):	):	PUNCT
ejpam-7135	425	4	suppose	suppose	VERB
ejpam-7135	425	5	mx	mx	PROPN
ejpam-7135	425	6	⊆	⊆	NUM
ejpam-7135	425	7	ma	ma	PROPN
ejpam-7135	425	8	.	.	PUNCT
ejpam-7135	426	1	then	then	ADV
ejpam-7135	426	2	every	every	DET
ejpam-7135	426	3	minimal	minimal	ADJ
ejpam-7135	426	4	prime	prime	ADJ
ejpam-7135	426	5	filter	filter	NOUN
ejpam-7135	426	6	containing	contain	VERB
ejpam-7135	426	7	x	x	PUNCT
ejpam-7135	426	8	also	also	ADV
ejpam-7135	426	9	contains	contain	VERB
ejpam-7135	426	10	a	a	PRON
ejpam-7135	426	11	,	,	PUNCT
ejpam-7135	426	12	which	which	PRON
ejpam-7135	426	13	implies	imply	VERB
ejpam-7135	426	14	a	a	DET
ejpam-7135	426	15	≤	≤	NOUN
ejpam-7135	426	16	x	x	PUNCT
ejpam-7135	426	17	in	in	ADP
ejpam-7135	426	18	the	the	DET
ejpam-7135	426	19	order	order	NOUN
ejpam-7135	426	20	of	of	ADP
ejpam-7135	426	21	l.	l.	PROPN
ejpam-7135	426	22	hence	hence	PROPN
ejpam-7135	426	23	,	,	PUNCT
ejpam-7135	426	24	a	a	DET
ejpam-7135	426	25	∨	∨	NOUN
ejpam-7135	426	26	x	x	X
ejpam-7135	426	27	=	=	PUNCT
ejpam-7135	426	28	x.	x.	NOUN
ejpam-7135	426	29	definition	definition	NOUN
ejpam-7135	426	30	10	10	NUM
ejpam-7135	426	31	.	.	PUNCT
ejpam-7135	427	1	a	a	DET
ejpam-7135	427	2	subspace	subspace	NOUN
ejpam-7135	427	3	s	s	X
ejpam-7135	427	4	of	of	ADP
ejpam-7135	427	5	a	a	DET
ejpam-7135	427	6	topological	topological	ADJ
ejpam-7135	427	7	space	space	NOUN
ejpam-7135	427	8	t	t	PROPN
ejpam-7135	427	9	is	be	AUX
ejpam-7135	427	10	said	say	VERB
ejpam-7135	427	11	to	to	PART
ejpam-7135	427	12	be	be	AUX
ejpam-7135	427	13	a	a	DET
ejpam-7135	427	14	retract	retract	NOUN
ejpam-7135	427	15	of	of	ADP
ejpam-7135	427	16	t	t	PROPN
ejpam-7135	427	17	if	if	SCONJ
ejpam-7135	427	18	there	there	PRON
ejpam-7135	427	19	exists	exist	VERB
ejpam-7135	427	20	a	a	DET
ejpam-7135	427	21	continuous	continuous	ADJ
ejpam-7135	427	22	map	map	NOUN
ejpam-7135	427	23	θ	θ	NOUN
ejpam-7135	427	24	:	:	PUNCT
ejpam-7135	427	25	t	t	PROPN
ejpam-7135	427	26	→	→	SYM
ejpam-7135	427	27	s	s	VERB
ejpam-7135	427	28	such	such	ADJ
ejpam-7135	427	29	that	that	SCONJ
ejpam-7135	427	30	θ(a	θ(a	PROPN
ejpam-7135	427	31	)	)	PUNCT
ejpam-7135	428	1	=	=	PUNCT
ejpam-7135	428	2	a	a	PRON
ejpam-7135	428	3	for	for	ADP
ejpam-7135	428	4	all	all	DET
ejpam-7135	428	5	a	a	DET
ejpam-7135	428	6	∈	∈	PROPN
ejpam-7135	428	7	s.	s.	PROPN
ejpam-7135	428	8	recall	recall	VERB
ejpam-7135	428	9	that	that	SCONJ
ejpam-7135	428	10	m	m	PROPN
ejpam-7135	428	11	is	be	AUX
ejpam-7135	428	12	a	a	DET
ejpam-7135	428	13	subspace	subspace	NOUN
ejpam-7135	428	14	of	of	ADP
ejpam-7135	428	15	y	y	PROPN
ejpam-7135	428	16	under	under	ADP
ejpam-7135	428	17	the	the	DET
ejpam-7135	428	18	induced	induced	ADJ
ejpam-7135	428	19	topology	topology	NOUN
ejpam-7135	428	20	y	y	PROPN
ejpam-7135	428	21	.	.	PUNCT
ejpam-7135	429	1	in	in	ADP
ejpam-7135	429	2	this	this	PRON
ejpam-7135	429	3	,	,	PUNCT
ejpam-7135	429	4	basic	basic	ADJ
ejpam-7135	429	5	open	open	ADJ
ejpam-7135	429	6	sets	set	NOUN
ejpam-7135	429	7	are	be	AUX
ejpam-7135	429	8	{	{	PUNCT
ejpam-7135	429	9	ma|a	ma|a	NOUN
ejpam-7135	429	10	∈	∈	PROPN
ejpam-7135	429	11	l	l	NOUN
ejpam-7135	429	12	}	}	PUNCT
ejpam-7135	429	13	where	where	SCONJ
ejpam-7135	429	14	,	,	PUNCT
ejpam-7135	429	15	for	for	ADP
ejpam-7135	429	16	any	any	DET
ejpam-7135	429	17	a	a	DET
ejpam-7135	429	18	∈	∈	PROPN
ejpam-7135	429	19	l	l	NOUN
ejpam-7135	429	20	,	,	PUNCT
ejpam-7135	429	21	ma	ma	PROPN
ejpam-7135	429	22	=	=	PROPN
ejpam-7135	429	23	m	m	PROPN
ejpam-7135	429	24	∩	∩	PROPN
ejpam-7135	429	25	ya	ya	PROPN
ejpam-7135	429	26	.	.	PUNCT
ejpam-7135	430	1	finally	finally	ADV
ejpam-7135	430	2	,	,	PUNCT
ejpam-7135	430	3	we	we	PRON
ejpam-7135	430	4	conclude	conclude	VERB
ejpam-7135	430	5	with	with	ADP
ejpam-7135	430	6	the	the	DET
ejpam-7135	430	7	following	follow	VERB
ejpam-7135	430	8	theorem	theorem	NOUN
ejpam-7135	430	9	,	,	PUNCT
ejpam-7135	430	10	which	which	PRON
ejpam-7135	430	11	is	be	AUX
ejpam-7135	430	12	another	another	DET
ejpam-7135	430	13	characterization	characterization	NOUN
ejpam-7135	430	14	of	of	ADP
ejpam-7135	430	15	stone	stone	NOUN
ejpam-7135	430	16	pdls	pdl	NOUN
ejpam-7135	430	17	in	in	ADP
ejpam-7135	430	18	terms	term	NOUN
ejpam-7135	430	19	of	of	ADP
ejpam-7135	430	20	minimal	minimal	ADJ
ejpam-7135	430	21	prime	prime	ADJ
ejpam-7135	430	22	filters	filter	NOUN
ejpam-7135	430	23	.	.	PUNCT
ejpam-7135	431	1	theorem	theorem	VERB
ejpam-7135	431	2	13	13	NUM
ejpam-7135	431	3	.	.	PUNCT
ejpam-7135	432	1	let	let	VERB
ejpam-7135	432	2	l	l	NOUN
ejpam-7135	432	3	be	be	AUX
ejpam-7135	432	4	a	a	DET
ejpam-7135	432	5	pdl	pdl	NOUN
ejpam-7135	432	6	with	with	ADP
ejpam-7135	432	7	a	a	DET
ejpam-7135	432	8	parapseudo	parapseudo	NOUN
ejpam-7135	432	9	-	-	PUNCT
ejpam-7135	432	10	complementation	complementation	NOUN
ejpam-7135	432	11	♦	♦	PROPN
ejpam-7135	432	12	in	in	ADP
ejpam-7135	432	13	which	which	PRON
ejpam-7135	432	14	m	m	VERB
ejpam-7135	432	15	is	be	AUX
ejpam-7135	432	16	a	a	DET
ejpam-7135	432	17	minimal	minimal	ADJ
ejpam-7135	432	18	element	element	NOUN
ejpam-7135	432	19	,	,	PUNCT
ejpam-7135	432	20	y	y	PROPN
ejpam-7135	432	21	the	the	DET
ejpam-7135	432	22	set	set	NOUN
ejpam-7135	432	23	of	of	ADP
ejpam-7135	432	24	prime	prime	ADJ
ejpam-7135	432	25	filters	filter	NOUN
ejpam-7135	432	26	of	of	ADP
ejpam-7135	432	27	l	l	NOUN
ejpam-7135	432	28	,	,	PUNCT
ejpam-7135	432	29	and	and	CCONJ
ejpam-7135	432	30	m	m	PROPN
ejpam-7135	432	31	⊆	⊆	NUM
ejpam-7135	432	32	y	y	PROPN
ejpam-7135	432	33	the	the	DET
ejpam-7135	432	34	set	set	NOUN
ejpam-7135	432	35	of	of	ADP
ejpam-7135	432	36	minimal	minimal	ADJ
ejpam-7135	432	37	prime	prime	ADJ
ejpam-7135	432	38	filters	filter	NOUN
ejpam-7135	432	39	.	.	PUNCT
ejpam-7135	433	1	then	then	ADV
ejpam-7135	433	2	the	the	DET
ejpam-7135	433	3	following	follow	VERB
ejpam-7135	433	4	are	be	AUX
ejpam-7135	433	5	equivalent	equivalent	ADJ
ejpam-7135	433	6	:	:	PUNCT
ejpam-7135	433	7	(	(	PUNCT
ejpam-7135	433	8	1	1	X
ejpam-7135	433	9	)	)	PUNCT
ejpam-7135	433	10	l	l	NOUN
ejpam-7135	433	11	is	be	AUX
ejpam-7135	433	12	a	a	DET
ejpam-7135	433	13	stone	stone	NOUN
ejpam-7135	433	14	pdl	pdl	NOUN
ejpam-7135	433	15	(	(	PUNCT
ejpam-7135	433	16	2	2	NUM
ejpam-7135	433	17	)	)	PUNCT
ejpam-7135	433	18	m	m	VERB
ejpam-7135	433	19	is	be	AUX
ejpam-7135	433	20	a	a	DET
ejpam-7135	433	21	retract	retract	NOUN
ejpam-7135	433	22	of	of	ADP
ejpam-7135	433	23	y	y	PROPN
ejpam-7135	433	24	(	(	PUNCT
ejpam-7135	433	25	3	3	X
ejpam-7135	433	26	)	)	PUNCT
ejpam-7135	433	27	the	the	DET
ejpam-7135	433	28	restriction	restriction	NOUN
ejpam-7135	433	29	f	f	NOUN
ejpam-7135	433	30	|m	|m	NOUN
ejpam-7135	433	31	:	:	PUNCT
ejpam-7135	433	32	m	m	VERB
ejpam-7135	433	33	→	→	PUNCT
ejpam-7135	433	34	x	x	X
ejpam-7135	433	35	is	be	AUX
ejpam-7135	433	36	a	a	DET
ejpam-7135	433	37	homeomorphism	homeomorphism	NOUN
ejpam-7135	433	38	.	.	PUNCT
ejpam-7135	434	1	proof	proof	NOUN
ejpam-7135	434	2	.	.	PUNCT
ejpam-7135	435	1	(	(	PUNCT
ejpam-7135	435	2	1	1	X
ejpam-7135	435	3	)	)	PUNCT
ejpam-7135	435	4	⇒	⇒	NOUN
ejpam-7135	435	5	(	(	PUNCT
ejpam-7135	435	6	2	2	NUM
ejpam-7135	435	7	):	):	PUNCT
ejpam-7135	435	8	assume	assume	PROPN
ejpam-7135	435	9	l	l	NOUN
ejpam-7135	435	10	is	be	AUX
ejpam-7135	435	11	a	a	DET
ejpam-7135	435	12	stone	stone	NOUN
ejpam-7135	435	13	pdl	pdl	NOUN
ejpam-7135	435	14	.	.	PUNCT
ejpam-7135	435	15	by	by	ADP
ejpam-7135	435	16	theorem	theorem	NOUN
ejpam-7135	435	17	10	10	NUM
ejpam-7135	435	18	,	,	PUNCT
ejpam-7135	435	19	every	every	DET
ejpam-7135	435	20	prime	prime	ADJ
ejpam-7135	435	21	filter	filter	NOUN
ejpam-7135	435	22	contains	contain	VERB
ejpam-7135	435	23	a	a	DET
ejpam-7135	435	24	unique	unique	ADJ
ejpam-7135	435	25	minimal	minimal	ADJ
ejpam-7135	435	26	prime	prime	ADJ
ejpam-7135	435	27	filter	filter	NOUN
ejpam-7135	435	28	.	.	PUNCT
ejpam-7135	436	1	define	define	VERB
ejpam-7135	436	2	θ	θ	PROPN
ejpam-7135	436	3	:	:	PUNCT
ejpam-7135	437	1	y	y	PROPN
ejpam-7135	437	2	→	→	SYM
ejpam-7135	437	3	m	m	VERB
ejpam-7135	437	4	by	by	ADP
ejpam-7135	437	5	θ(p	θ(p	NOUN
ejpam-7135	437	6	)	)	PUNCT
ejpam-7135	438	1	=	=	PUNCT
ejpam-7135	438	2	the	the	DET
ejpam-7135	438	3	unique	unique	ADJ
ejpam-7135	438	4	minimal	minimal	ADJ
ejpam-7135	438	5	prime	prime	ADJ
ejpam-7135	438	6	filter	filter	NOUN
ejpam-7135	438	7	contained	contain	VERB
ejpam-7135	438	8	in	in	ADP
ejpam-7135	438	9	p	p	NOUN
ejpam-7135	438	10	.	.	PUNCT
ejpam-7135	439	1	for	for	ADP
ejpam-7135	439	2	p	p	PROPN
ejpam-7135	439	3	∈	∈	PROPN
ejpam-7135	439	4	m	m	NOUN
ejpam-7135	439	5	,	,	PUNCT
ejpam-7135	439	6	θ(p	θ(p	PROPN
ejpam-7135	439	7	)	)	PUNCT
ejpam-7135	439	8	=	=	PUNCT
ejpam-7135	440	1	p	p	NOUN
ejpam-7135	440	2	.	.	PUNCT
ejpam-7135	441	1	to	to	PART
ejpam-7135	441	2	show	show	VERB
ejpam-7135	441	3	continuity	continuity	NOUN
ejpam-7135	441	4	,	,	PUNCT
ejpam-7135	441	5	let	let	VERB
ejpam-7135	441	6	x	x	PUNCT
ejpam-7135	441	7	∈	∈	NOUN
ejpam-7135	441	8	l	l	NOUN
ejpam-7135	441	9	and	and	CCONJ
ejpam-7135	441	10	consider	consider	VERB
ejpam-7135	441	11	mx	mx	NOUN
ejpam-7135	441	12	=	=	PRON
ejpam-7135	441	13	{	{	PUNCT
ejpam-7135	441	14	q	q	PROPN
ejpam-7135	441	15	∈	∈	PROPN
ejpam-7135	441	16	m	m	VERB
ejpam-7135	441	17	|	|	NOUN
ejpam-7135	441	18	x	x	X
ejpam-7135	441	19	/∈	/∈	PUNCT
ejpam-7135	442	1	q	q	NOUN
ejpam-7135	442	2	}	}	PUNCT
ejpam-7135	442	3	.	.	PUNCT
ejpam-7135	443	1	then	then	ADV
ejpam-7135	443	2	:	:	PUNCT
ejpam-7135	443	3	θ−1(mx	θ−1(mx	PROPN
ejpam-7135	443	4	)	)	PUNCT
ejpam-7135	443	5	=	=	PRON
ejpam-7135	443	6	{	{	PUNCT
ejpam-7135	443	7	p	p	X
ejpam-7135	443	8	∈	∈	X
ejpam-7135	443	9	y	y	NOUN
ejpam-7135	443	10	|	|	ADV
ejpam-7135	443	11	x	x	PROPN
ejpam-7135	443	12	/∈	/∈	PUNCT
ejpam-7135	443	13	θ(p	θ(p	NOUN
ejpam-7135	443	14	)	)	PUNCT
ejpam-7135	443	15	}	}	PUNCT
ejpam-7135	443	16	=	=	PUNCT
ejpam-7135	443	17	{	{	PUNCT
ejpam-7135	443	18	p	p	X
ejpam-7135	443	19	∈	∈	PROPN
ejpam-7135	443	20	y	y	PROPN
ejpam-7135	443	21	|	|	ADV
ejpam-7135	443	22	x	x	PROPN
ejpam-7135	443	23	♦	♦	PROPN
ejpam-7135	443	24	∈	∈	PROPN
ejpam-7135	443	25	θ(p	θ(p	PROPN
ejpam-7135	443	26	)	)	PUNCT
ejpam-7135	443	27	}	}	PUNCT
ejpam-7135	443	28	=	=	SYM
ejpam-7135	443	29	yx	yx	PROPN
ejpam-7135	443	30	♦	♦	PROPN
ejpam-7135	443	31	♦	♦	PROPN
ejpam-7135	443	32	which	which	PRON
ejpam-7135	443	33	is	be	AUX
ejpam-7135	443	34	open	open	ADJ
ejpam-7135	443	35	in	in	ADP
ejpam-7135	443	36	y	y	PROPN
ejpam-7135	443	37	.	.	PUNCT
ejpam-7135	444	1	hence	hence	ADV
ejpam-7135	444	2	,	,	PUNCT
ejpam-7135	444	3	θ	θ	PROPN
ejpam-7135	444	4	is	be	AUX
ejpam-7135	444	5	continuous	continuous	ADJ
ejpam-7135	444	6	,	,	PUNCT
ejpam-7135	444	7	and	and	CCONJ
ejpam-7135	444	8	m	m	PROPN
ejpam-7135	444	9	is	be	AUX
ejpam-7135	444	10	a	a	DET
ejpam-7135	444	11	retract	retract	NOUN
ejpam-7135	444	12	of	of	ADP
ejpam-7135	444	13	y	y	PROPN
ejpam-7135	444	14	.	.	PUNCT
ejpam-7135	445	1	(	(	PUNCT
ejpam-7135	445	2	2	2	X
ejpam-7135	445	3	)	)	PUNCT
ejpam-7135	445	4	⇒	⇒	NOUN
ejpam-7135	445	5	(	(	PUNCT
ejpam-7135	445	6	3	3	NUM
ejpam-7135	445	7	):	):	PUNCT
ejpam-7135	445	8	assume	assume	VERB
ejpam-7135	445	9	there	there	PRON
ejpam-7135	445	10	exists	exist	VERB
ejpam-7135	445	11	a	a	DET
ejpam-7135	445	12	continuous	continuous	ADJ
ejpam-7135	445	13	retraction	retraction	NOUN
ejpam-7135	445	14	θ	θ	NOUN
ejpam-7135	445	15	:	:	PUNCT
ejpam-7135	445	16	y	y	PROPN
ejpam-7135	445	17	→	→	SYM
ejpam-7135	445	18	m	m	PROPN
ejpam-7135	445	19	.	.	PUNCT
ejpam-7135	446	1	the	the	DET
ejpam-7135	446	2	map	map	NOUN
ejpam-7135	446	3	f	f	NOUN
ejpam-7135	446	4	|m	|m	NOUN
ejpam-7135	446	5	:	:	PUNCT
ejpam-7135	446	6	m	m	VERB
ejpam-7135	446	7	→	→	NOUN
ejpam-7135	446	8	x	x	X
ejpam-7135	446	9	is	be	AUX
ejpam-7135	446	10	continuous	continuous	ADJ
ejpam-7135	446	11	since	since	SCONJ
ejpam-7135	446	12	(	(	PUNCT
ejpam-7135	446	13	f	f	PROPN
ejpam-7135	446	14	|m	|m	NOUN
ejpam-7135	446	15	)	)	PUNCT
ejpam-7135	447	1	−1(xa	−1(xa	NOUN
ejpam-7135	447	2	)	)	PUNCT
ejpam-7135	447	3	=	=	SYM
ejpam-7135	447	4	ma	ma	PROPN
ejpam-7135	447	5	.	.	PROPN
ejpam-7135	447	6	r.	r.	PROPN
ejpam-7135	447	7	bandaru	bandaru	PROPN
ejpam-7135	447	8	et	et	PROPN
ejpam-7135	447	9	al	al	PROPN
ejpam-7135	447	10	.	.	PUNCT
ejpam-7135	447	11	/	/	SYM
ejpam-7135	447	12	eur	eur	PROPN
ejpam-7135	447	13	.	.	PUNCT
ejpam-7135	448	1	j.	j.	PROPN
ejpam-7135	448	2	pure	pure	PROPN
ejpam-7135	448	3	appl	appl	PROPN
ejpam-7135	448	4	.	.	PROPN
ejpam-7135	448	5	math	math	PROPN
ejpam-7135	448	6	,	,	PUNCT
ejpam-7135	448	7	18	18	NUM
ejpam-7135	448	8	(	(	PUNCT
ejpam-7135	448	9	4	4	NUM
ejpam-7135	448	10	)	)	PUNCT
ejpam-7135	448	11	(	(	PUNCT
ejpam-7135	448	12	2025	2025	NUM
ejpam-7135	448	13	)	)	PUNCT
ejpam-7135	448	14	,	,	PUNCT
ejpam-7135	448	15	7135	7135	NUM
ejpam-7135	448	16	13	13	NUM
ejpam-7135	448	17	of	of	ADP
ejpam-7135	448	18	14	14	NUM
ejpam-7135	448	19	to	to	PART
ejpam-7135	448	20	show	show	VERB
ejpam-7135	448	21	injectivity	injectivity	NOUN
ejpam-7135	448	22	,	,	PUNCT
ejpam-7135	448	23	let	let	VERB
ejpam-7135	448	24	p1	p1	NOUN
ejpam-7135	448	25	,	,	PUNCT
ejpam-7135	448	26	p2	p2	PROPN
ejpam-7135	448	27	∈	∈	PROPN
ejpam-7135	448	28	m	m	VERB
ejpam-7135	448	29	with	with	ADP
ejpam-7135	448	30	f(p1	f(p1	NOUN
ejpam-7135	448	31	)	)	PUNCT
ejpam-7135	448	32	=	=	SYM
ejpam-7135	448	33	f(p2	f(p2	NOUN
ejpam-7135	448	34	)	)	PUNCT
ejpam-7135	448	35	.	.	PUNCT
ejpam-7135	449	1	for	for	ADP
ejpam-7135	449	2	any	any	DET
ejpam-7135	449	3	x	x	SYM
ejpam-7135	449	4	∈	∈	PROPN
ejpam-7135	449	5	p1	p1	NOUN
ejpam-7135	449	6	,	,	PUNCT
ejpam-7135	449	7	mx	mx	PROPN
ejpam-7135	449	8	is	be	AUX
ejpam-7135	449	9	clopen	clopen	ADJ
ejpam-7135	449	10	in	in	ADP
ejpam-7135	449	11	m	m	PROPN
ejpam-7135	449	12	,	,	PUNCT
ejpam-7135	449	13	so	so	ADV
ejpam-7135	449	14	θ−1(mx	θ−1(mx	PROPN
ejpam-7135	449	15	)	)	PUNCT
ejpam-7135	450	1	=	=	SYM
ejpam-7135	450	2	ya	ya	PROPN
ejpam-7135	450	3	for	for	ADP
ejpam-7135	450	4	some	some	DET
ejpam-7135	450	5	a	a	DET
ejpam-7135	450	6	∈	∈	PROPN
ejpam-7135	450	7	b[m	b[m	NOUN
ejpam-7135	450	8	,	,	PUNCT
ejpam-7135	450	9	1	1	NUM
ejpam-7135	450	10	]	]	PUNCT
ejpam-7135	450	11	by	by	ADP
ejpam-7135	450	12	lemma	lemma	PROPN
ejpam-7135	450	13	11	11	NUM
ejpam-7135	450	14	.	.	PUNCT
ejpam-7135	451	1	since	since	SCONJ
ejpam-7135	451	2	p1	p1	PROPN
ejpam-7135	451	3	∈	∈	PROPN
ejpam-7135	451	4	mx	mx	PROPN
ejpam-7135	451	5	,	,	PUNCT
ejpam-7135	451	6	we	we	PRON
ejpam-7135	451	7	have	have	VERB
ejpam-7135	451	8	a	a	DET
ejpam-7135	451	9	∈	∈	PROPN
ejpam-7135	451	10	p1	p1	NOUN
ejpam-7135	451	11	,	,	PUNCT
ejpam-7135	451	12	hence	hence	ADV
ejpam-7135	451	13	a	a	DET
ejpam-7135	451	14	∈	∈	PROPN
ejpam-7135	451	15	f(p1	f(p1	NOUN
ejpam-7135	451	16	)	)	PUNCT
ejpam-7135	451	17	=	=	SYM
ejpam-7135	451	18	f(p2	f(p2	NOUN
ejpam-7135	451	19	)	)	PUNCT
ejpam-7135	451	20	,	,	PUNCT
ejpam-7135	451	21	so	so	CCONJ
ejpam-7135	451	22	a	a	DET
ejpam-7135	451	23	∈	∈	NOUN
ejpam-7135	451	24	p2	p2	NOUN
ejpam-7135	451	25	,	,	PUNCT
ejpam-7135	451	26	thus	thus	ADV
ejpam-7135	451	27	p2	p2	PROPN
ejpam-7135	451	28	/∈	/∈	PUNCT
ejpam-7135	452	1	ya	ya	PROPN
ejpam-7135	452	2	,	,	PUNCT
ejpam-7135	452	3	meaning	mean	VERB
ejpam-7135	452	4	p2	p2	PROPN
ejpam-7135	452	5	∈	∈	PROPN
ejpam-7135	452	6	mx	mx	PROPN
ejpam-7135	452	7	,	,	PUNCT
ejpam-7135	452	8	so	so	ADV
ejpam-7135	452	9	x	x	SYM
ejpam-7135	452	10	∈	∈	NOUN
ejpam-7135	452	11	p2	p2	NOUN
ejpam-7135	452	12	.	.	PUNCT
ejpam-7135	453	1	by	by	ADP
ejpam-7135	453	2	symmetry	symmetry	NOUN
ejpam-7135	453	3	,	,	PUNCT
ejpam-7135	453	4	p1	p1	NOUN
ejpam-7135	453	5	=	=	PUNCT
ejpam-7135	453	6	p2	p2	PROPN
ejpam-7135	453	7	.	.	PUNCT
ejpam-7135	454	1	to	to	PART
ejpam-7135	454	2	show	show	VERB
ejpam-7135	454	3	surjectivity	surjectivity	NOUN
ejpam-7135	454	4	,	,	PUNCT
ejpam-7135	454	5	for	for	ADP
ejpam-7135	454	6	p	p	PROPN
ejpam-7135	454	7	∈	∈	PROPN
ejpam-7135	454	8	x	x	SYM
ejpam-7135	454	9	,	,	PUNCT
ejpam-7135	454	10	p	p	ADJ
ejpam-7135	454	11	e	e	NOUN
ejpam-7135	454	12	is	be	AUX
ejpam-7135	454	13	a	a	DET
ejpam-7135	454	14	minimal	minimal	ADJ
ejpam-7135	454	15	prime	prime	ADJ
ejpam-7135	454	16	filter	filter	NOUN
ejpam-7135	454	17	(	(	PUNCT
ejpam-7135	454	18	by	by	ADP
ejpam-7135	454	19	previous	previous	ADJ
ejpam-7135	454	20	results	result	NOUN
ejpam-7135	454	21	)	)	PUNCT
ejpam-7135	454	22	,	,	PUNCT
ejpam-7135	454	23	and	and	CCONJ
ejpam-7135	454	24	f(p	f(p	PROPN
ejpam-7135	454	25	e	e	X
ejpam-7135	454	26	)	)	PUNCT
ejpam-7135	454	27	=	=	SYM
ejpam-7135	454	28	p	p	NOUN
ejpam-7135	454	29	.	.	PUNCT
ejpam-7135	455	1	thus	thus	ADV
ejpam-7135	455	2	,	,	PUNCT
ejpam-7135	455	3	f	f	PROPN
ejpam-7135	455	4	|m	|m	NOUN
ejpam-7135	455	5	is	be	AUX
ejpam-7135	455	6	bijective	bijective	ADJ
ejpam-7135	455	7	.	.	PUNCT
ejpam-7135	456	1	openness	openness	NOUN
ejpam-7135	456	2	follows	follow	VERB
ejpam-7135	456	3	from	from	ADP
ejpam-7135	456	4	showing	show	VERB
ejpam-7135	456	5	f(mx	f(mx	NOUN
ejpam-7135	456	6	)	)	PUNCT
ejpam-7135	457	1	=	=	SYM
ejpam-7135	457	2	xax	xax	PROPN
ejpam-7135	457	3	for	for	ADP
ejpam-7135	457	4	some	some	DET
ejpam-7135	457	5	ax	ax	NOUN
ejpam-7135	457	6	∈	∈	PROPN
ejpam-7135	457	7	b[m	b[m	NOUN
ejpam-7135	457	8	,	,	PUNCT
ejpam-7135	457	9	1	1	NUM
ejpam-7135	457	10	]	]	PUNCT
ejpam-7135	457	11	,	,	PUNCT
ejpam-7135	457	12	making	make	VERB
ejpam-7135	457	13	f	f	PROPN
ejpam-7135	457	14	|m	|m	NOUN
ejpam-7135	457	15	a	a	DET
ejpam-7135	457	16	homeomorphism	homeomorphism	X
ejpam-7135	457	17	.	.	PUNCT
ejpam-7135	458	1	(	(	PUNCT
ejpam-7135	458	2	3	3	X
ejpam-7135	458	3	)	)	PUNCT
ejpam-7135	458	4	⇒	⇒	NOUN
ejpam-7135	458	5	(	(	PUNCT
ejpam-7135	458	6	1	1	NUM
ejpam-7135	458	7	):	):	PUNCT
ejpam-7135	458	8	assume	assume	PROPN
ejpam-7135	458	9	f	f	PROPN
ejpam-7135	458	10	|m	|m	NOUN
ejpam-7135	458	11	:	:	PUNCT
ejpam-7135	458	12	m	m	VERB
ejpam-7135	458	13	→	→	PUNCT
ejpam-7135	458	14	x	x	X
ejpam-7135	458	15	is	be	AUX
ejpam-7135	458	16	a	a	DET
ejpam-7135	458	17	homeomorphism	homeomorphism	NOUN
ejpam-7135	458	18	.	.	PUNCT
ejpam-7135	459	1	for	for	ADP
ejpam-7135	459	2	x	x	PROPN
ejpam-7135	459	3	∈	∈	PROPN
ejpam-7135	459	4	l	l	NOUN
ejpam-7135	459	5	,	,	PUNCT
ejpam-7135	459	6	f(mx	f(mx	PROPN
ejpam-7135	459	7	)	)	PUNCT
ejpam-7135	459	8	=	=	SYM
ejpam-7135	459	9	xax	xax	PROPN
ejpam-7135	459	10	for	for	ADP
ejpam-7135	459	11	some	some	DET
ejpam-7135	459	12	unique	unique	ADJ
ejpam-7135	459	13	ax	ax	NOUN
ejpam-7135	459	14	∈	∈	PROPN
ejpam-7135	459	15	b[m	b[m	NOUN
ejpam-7135	459	16	,	,	PUNCT
ejpam-7135	459	17	1	1	NUM
ejpam-7135	459	18	]	]	PUNCT
ejpam-7135	459	19	.	.	PUNCT
ejpam-7135	460	1	define	define	VERB
ejpam-7135	460	2	x	x	PRON
ejpam-7135	460	3	♦	♦	PROPN
ejpam-7135	460	4	=	=	PUNCT
ejpam-7135	460	5	a′x	a′x	PROPN
ejpam-7135	460	6	.	.	PUNCT
ejpam-7135	461	1	one	one	NUM
ejpam-7135	461	2	verifies	verifie	NOUN
ejpam-7135	461	3	that	that	PRON
ejpam-7135	461	4	♦	♦	PROPN
ejpam-7135	461	5	is	be	AUX
ejpam-7135	461	6	a	a	DET
ejpam-7135	461	7	parapseudocomplementation	parapseudocomplementation	NOUN
ejpam-7135	461	8	satisfying	satisfy	VERB
ejpam-7135	461	9	the	the	DET
ejpam-7135	461	10	stone	stone	NOUN
ejpam-7135	461	11	condition	condition	NOUN
ejpam-7135	461	12	x	x	NOUN
ejpam-7135	461	13	♦	♦	PROPN
ejpam-7135	461	14	∧	∧	PROPN
ejpam-7135	461	15	x	x	PROPN
ejpam-7135	461	16	♦	♦	PROPN
ejpam-7135	461	17	♦	♦	PROPN
ejpam-7135	461	18	=	=	PROPN
ejpam-7135	461	19	1	1	NUM
ejpam-7135	461	20	♦	♦	PROPN
ejpam-7135	461	21	.	.	PUNCT
ejpam-7135	462	1	hence	hence	ADV
ejpam-7135	462	2	,	,	PUNCT
ejpam-7135	462	3	l	l	NOUN
ejpam-7135	462	4	is	be	AUX
ejpam-7135	462	5	a	a	DET
ejpam-7135	462	6	stone	stone	NOUN
ejpam-7135	462	7	pdl	pdl	NOUN
ejpam-7135	462	8	.	.	PROPN
ejpam-7135	462	9	5	5	NUM
ejpam-7135	462	10	.	.	X
ejpam-7135	462	11	conclusion	conclusion	NOUN
ejpam-7135	462	12	in	in	ADP
ejpam-7135	462	13	this	this	DET
ejpam-7135	462	14	work	work	NOUN
ejpam-7135	462	15	,	,	PUNCT
ejpam-7135	462	16	we	we	PRON
ejpam-7135	462	17	have	have	AUX
ejpam-7135	462	18	introduced	introduce	VERB
ejpam-7135	462	19	and	and	CCONJ
ejpam-7135	462	20	systematically	systematically	ADV
ejpam-7135	462	21	investigated	investigate	VERB
ejpam-7135	462	22	the	the	DET
ejpam-7135	462	23	class	class	NOUN
ejpam-7135	462	24	of	of	ADP
ejpam-7135	462	25	stone	stone	NOUN
ejpam-7135	462	26	paradistributive	paradistributive	ADJ
ejpam-7135	462	27	latticoids	latticoids	PROPN
ejpam-7135	462	28	(	(	PUNCT
ejpam-7135	462	29	stone	stone	NOUN
ejpam-7135	462	30	pdls	pdl	NOUN
ejpam-7135	462	31	)	)	PUNCT
ejpam-7135	462	32	as	as	ADP
ejpam-7135	462	33	a	a	DET
ejpam-7135	462	34	unifying	unifying	ADJ
ejpam-7135	462	35	extension	extension	NOUN
ejpam-7135	462	36	of	of	ADP
ejpam-7135	462	37	stone	stone	NOUN
ejpam-7135	462	38	lattices	lattice	NOUN
ejpam-7135	462	39	within	within	ADP
ejpam-7135	462	40	the	the	DET
ejpam-7135	462	41	framework	framework	NOUN
ejpam-7135	462	42	of	of	ADP
ejpam-7135	462	43	paradistributive	paradistributive	ADJ
ejpam-7135	462	44	latticoids	latticoid	NOUN
ejpam-7135	462	45	equipped	equip	VERB
ejpam-7135	462	46	with	with	ADP
ejpam-7135	462	47	parapseudo	parapseudo	NOUN
ejpam-7135	462	48	-	-	NOUN
ejpam-7135	462	49	complementation	complementation	NOUN
ejpam-7135	462	50	.	.	PUNCT
ejpam-7135	463	1	through	through	ADP
ejpam-7135	463	2	a	a	DET
ejpam-7135	463	3	series	series	NOUN
ejpam-7135	463	4	of	of	ADP
ejpam-7135	463	5	equivalent	equivalent	ADJ
ejpam-7135	463	6	algebraic	algebraic	ADJ
ejpam-7135	463	7	,	,	PUNCT
ejpam-7135	463	8	filter	filter	NOUN
ejpam-7135	463	9	-	-	NOUN
ejpam-7135	463	10	theoretic	theoretic	ADJ
ejpam-7135	463	11	,	,	PUNCT
ejpam-7135	463	12	and	and	CCONJ
ejpam-7135	463	13	topological	topological	ADJ
ejpam-7135	463	14	characterizations	characterization	NOUN
ejpam-7135	463	15	—	—	PUNCT
ejpam-7135	463	16	including	include	VERB
ejpam-7135	463	17	conditions	condition	NOUN
ejpam-7135	463	18	on	on	ADP
ejpam-7135	463	19	principal	principal	ADJ
ejpam-7135	463	20	filters	filter	NOUN
ejpam-7135	463	21	,	,	PUNCT
ejpam-7135	463	22	co	co	NOUN
ejpam-7135	463	23	-	-	NOUN
ejpam-7135	463	24	maximality	maximality	NOUN
ejpam-7135	463	25	of	of	ADP
ejpam-7135	463	26	minimal	minimal	ADJ
ejpam-7135	463	27	prime	prime	ADJ
ejpam-7135	463	28	filters	filter	NOUN
ejpam-7135	463	29	,	,	PUNCT
ejpam-7135	463	30	and	and	CCONJ
ejpam-7135	463	31	retract	retract	VERB
ejpam-7135	463	32	properties	property	NOUN
ejpam-7135	463	33	of	of	ADP
ejpam-7135	463	34	prime	prime	ADJ
ejpam-7135	463	35	filter	filter	NOUN
ejpam-7135	463	36	spaces	space	NOUN
ejpam-7135	463	37	—	—	PUNCT
ejpam-7135	463	38	we	we	PRON
ejpam-7135	463	39	demonstrated	demonstrate	VERB
ejpam-7135	463	40	how	how	SCONJ
ejpam-7135	463	41	classical	classical	ADJ
ejpam-7135	463	42	stone	stone	NOUN
ejpam-7135	463	43	-	-	PUNCT
ejpam-7135	463	44	type	type	NOUN
ejpam-7135	463	45	representations	representation	NOUN
ejpam-7135	463	46	can	can	AUX
ejpam-7135	463	47	be	be	AUX
ejpam-7135	463	48	lifted	lift	VERB
ejpam-7135	463	49	to	to	ADP
ejpam-7135	463	50	this	this	DET
ejpam-7135	463	51	more	more	ADV
ejpam-7135	463	52	general	general	ADJ
ejpam-7135	463	53	setting	setting	NOUN
ejpam-7135	463	54	.	.	PUNCT
ejpam-7135	464	1	these	these	DET
ejpam-7135	464	2	findings	finding	NOUN
ejpam-7135	464	3	establish	establish	VERB
ejpam-7135	464	4	a	a	DET
ejpam-7135	464	5	coherent	coherent	ADJ
ejpam-7135	464	6	bridge	bridge	NOUN
ejpam-7135	464	7	between	between	ADP
ejpam-7135	464	8	lattice	lattice	NOUN
ejpam-7135	464	9	theory	theory	NOUN
ejpam-7135	464	10	and	and	CCONJ
ejpam-7135	464	11	algebraic	algebraic	ADJ
ejpam-7135	464	12	topology	topology	NOUN
ejpam-7135	464	13	,	,	PUNCT
ejpam-7135	464	14	offering	offer	VERB
ejpam-7135	464	15	a	a	DET
ejpam-7135	464	16	robust	robust	ADJ
ejpam-7135	464	17	platform	platform	NOUN
ejpam-7135	464	18	for	for	ADP
ejpam-7135	464	19	future	future	ADJ
ejpam-7135	464	20	developments	development	NOUN
ejpam-7135	464	21	.	.	PUNCT
ejpam-7135	465	1	directions	direction	NOUN
ejpam-7135	465	2	for	for	ADP
ejpam-7135	465	3	further	further	ADJ
ejpam-7135	465	4	research	research	NOUN
ejpam-7135	465	5	include	include	VERB
ejpam-7135	465	6	the	the	DET
ejpam-7135	465	7	study	study	NOUN
ejpam-7135	465	8	of	of	ADP
ejpam-7135	465	9	categorical	categorical	ADJ
ejpam-7135	465	10	dualities	duality	NOUN
ejpam-7135	465	11	,	,	PUNCT
ejpam-7135	465	12	refinement	refinement	NOUN
ejpam-7135	465	13	of	of	ADP
ejpam-7135	465	14	spectral	spectral	ADJ
ejpam-7135	465	15	representations	representation	NOUN
ejpam-7135	465	16	,	,	PUNCT
ejpam-7135	465	17	and	and	CCONJ
ejpam-7135	465	18	applications	application	NOUN
ejpam-7135	465	19	to	to	ADP
ejpam-7135	465	20	algebraic	algebraic	ADJ
ejpam-7135	465	21	logic	logic	NOUN
ejpam-7135	465	22	,	,	PUNCT
ejpam-7135	465	23	regular	regular	ADJ
ejpam-7135	465	24	rings	ring	NOUN
ejpam-7135	465	25	,	,	PUNCT
ejpam-7135	465	26	and	and	CCONJ
ejpam-7135	465	27	beyond	beyond	ADP
ejpam-7135	465	28	.	.	PUNCT
ejpam-7135	466	1	acknowledgements	acknowledgement	NOUN
ejpam-7135	466	2	this	this	DET
ejpam-7135	466	3	research	research	NOUN
ejpam-7135	466	4	was	be	AUX
ejpam-7135	466	5	supported	support	VERB
ejpam-7135	466	6	by	by	ADP
ejpam-7135	466	7	university	university	NOUN
ejpam-7135	466	8	of	of	ADP
ejpam-7135	466	9	phayao	phayao	NOUN
ejpam-7135	466	10	and	and	CCONJ
ejpam-7135	466	11	thailand	thailand	PROPN
ejpam-7135	466	12	science	science	PROPN
ejpam-7135	466	13	research	research	PROPN
ejpam-7135	466	14	and	and	CCONJ
ejpam-7135	466	15	innovation	innovation	NOUN
ejpam-7135	466	16	fund	fund	NOUN
ejpam-7135	466	17	(	(	PUNCT
ejpam-7135	466	18	fundamental	fundamental	ADJ
ejpam-7135	466	19	fund	fund	NOUN
ejpam-7135	466	20	2026	2026	NUM
ejpam-7135	466	21	,	,	PUNCT
ejpam-7135	466	22	grant	grant	VERB
ejpam-7135	466	23	no	no	NOUN
ejpam-7135	466	24	.	.	PUNCT
ejpam-7135	466	25	2252/2568	2252/2568	NUM
ejpam-7135	466	26	)	)	PUNCT
ejpam-7135	466	27	.	.	PUNCT
ejpam-7135	467	1	references	reference	NOUN
ejpam-7135	467	2	[	[	X
ejpam-7135	467	3	1	1	NUM
ejpam-7135	467	4	]	]	PUNCT
ejpam-7135	467	5	g.	g.	NOUN
ejpam-7135	467	6	birkhoff	birkhoff	PROPN
ejpam-7135	467	7	.	.	PUNCT
ejpam-7135	468	1	lattice	lattice	PROPN
ejpam-7135	468	2	theory	theory	PROPN
ejpam-7135	468	3	.	.	PUNCT
ejpam-7135	469	1	amer	amer	PROPN
ejpam-7135	469	2	.	.	PUNCT
ejpam-7135	469	3	math	math	PROPN
ejpam-7135	469	4	.	.	PUNCT
ejpam-7135	470	1	soc	soc	PROPN
ejpam-7135	470	2	.	.	PUNCT
ejpam-7135	471	1	colloq	colloq	PROPN
ejpam-7135	471	2	.	.	PUNCT
ejpam-7135	472	1	publ	publ	PROPN
ejpam-7135	472	2	.	.	PUNCT
ejpam-7135	473	1	xxv	xxv	PROPN
ejpam-7135	473	2	,	,	PUNCT
ejpam-7135	473	3	providence	providence	NOUN
ejpam-7135	473	4	,	,	PUNCT
ejpam-7135	473	5	u.s.a	u.s.a	NOUN
ejpam-7135	473	6	.	.	PROPN
ejpam-7135	473	7	,	,	PUNCT
ejpam-7135	473	8	1948	1948	NUM
ejpam-7135	473	9	.	.	PUNCT
ejpam-7135	474	1	[	[	X
ejpam-7135	474	2	2	2	NUM
ejpam-7135	474	3	]	]	X
ejpam-7135	474	4	r.	r.	NOUN
ejpam-7135	474	5	balbes	balbe	NOUN
ejpam-7135	474	6	and	and	CCONJ
ejpam-7135	474	7	a.	a.	NOUN
ejpam-7135	474	8	horn	horn	NOUN
ejpam-7135	474	9	.	.	PUNCT
ejpam-7135	475	1	stone	stone	NOUN
ejpam-7135	475	2	lattices	lattice	NOUN
ejpam-7135	475	3	.	.	PUNCT
ejpam-7135	476	1	duke	duke	PROPN
ejpam-7135	476	2	math	math	PROPN
ejpam-7135	476	3	.	.	PUNCT
ejpam-7135	477	1	j.	j.	PROPN
ejpam-7135	477	2	,	,	PUNCT
ejpam-7135	477	3	37:537–545	37:537–545	PROPN
ejpam-7135	477	4	,	,	PUNCT
ejpam-7135	477	5	1970	1970	NUM
ejpam-7135	477	6	.	.	PUNCT
ejpam-7135	478	1	[	[	X
ejpam-7135	478	2	3	3	X
ejpam-7135	478	3	]	]	X
ejpam-7135	478	4	g.	g.	PROPN
ejpam-7135	478	5	bruns	bruns	PROPN
ejpam-7135	478	6	.	.	PUNCT
ejpam-7135	479	1	ideal	ideal	ADJ
ejpam-7135	479	2	-	-	PUNCT
ejpam-7135	479	3	representations	representation	NOUN
ejpam-7135	479	4	of	of	ADP
ejpam-7135	479	5	stone	stone	NOUN
ejpam-7135	479	6	lattices	lattice	NOUN
ejpam-7135	479	7	.	.	PUNCT
ejpam-7135	480	1	duke	duke	PROPN
ejpam-7135	480	2	math	math	PROPN
ejpam-7135	480	3	.	.	PUNCT
ejpam-7135	481	1	j.	j.	PROPN
ejpam-7135	481	2	,	,	PUNCT
ejpam-7135	481	3	32(3):555–556	32(3):555–556	PROPN
ejpam-7135	481	4	,	,	PUNCT
ejpam-7135	481	5	1965	1965	NUM
ejpam-7135	481	6	.	.	PUNCT
ejpam-7135	482	1	[	[	X
ejpam-7135	482	2	4	4	NUM
ejpam-7135	482	3	]	]	X
ejpam-7135	482	4	c.	c.	PROPN
ejpam-7135	482	5	c.	c.	PROPN
ejpam-7135	482	6	chen	chen	PROPN
ejpam-7135	482	7	and	and	CCONJ
ejpam-7135	482	8	g.	g.	PROPN
ejpam-7135	482	9	grätzer	grätzer	PROPN
ejpam-7135	482	10	.	.	PUNCT
ejpam-7135	483	1	stone	stone	PROPN
ejpam-7135	483	2	lattices	lattices	PROPN
ejpam-7135	483	3	i.	i.	PROPN
ejpam-7135	483	4	canad	canad	PROPN
ejpam-7135	483	5	.	.	PUNCT
ejpam-7135	484	1	j.	j.	PROPN
ejpam-7135	484	2	math	math	PROPN
ejpam-7135	484	3	.	.	PROPN
ejpam-7135	484	4	,	,	PUNCT
ejpam-7135	485	1	21:884–894	21:884–894	NUM
ejpam-7135	485	2	,	,	PUNCT
ejpam-7135	485	3	1969	1969	NUM
ejpam-7135	485	4	.	.	PUNCT
ejpam-7135	486	1	[	[	X
ejpam-7135	486	2	5	5	X
ejpam-7135	486	3	]	]	PUNCT
ejpam-7135	486	4	j.	j.	PROPN
ejpam-7135	486	5	varlet	varlet	PROPN
ejpam-7135	486	6	.	.	PUNCT
ejpam-7135	487	1	on	on	ADP
ejpam-7135	487	2	the	the	DET
ejpam-7135	487	3	characterization	characterization	NOUN
ejpam-7135	487	4	of	of	ADP
ejpam-7135	487	5	stone	stone	NOUN
ejpam-7135	487	6	lattices	lattice	NOUN
ejpam-7135	487	7	.	.	PUNCT
ejpam-7135	488	1	acta	acta	PROPN
ejpam-7135	488	2	sci	sci	PROPN
ejpam-7135	488	3	.	.	PROPN
ejpam-7135	488	4	math	math	PROPN
ejpam-7135	488	5	.	.	PUNCT
ejpam-7135	489	1	(	(	PUNCT
ejpam-7135	489	2	szeged	szeged	PROPN
ejpam-7135	489	3	)	)	PUNCT
ejpam-7135	489	4	,	,	PUNCT
ejpam-7135	489	5	27:81	27:81	NUM
ejpam-7135	489	6	–	–	PUNCT
ejpam-7135	489	7	84	84	NUM
ejpam-7135	489	8	,	,	PUNCT
ejpam-7135	489	9	1966	1966	NUM
ejpam-7135	489	10	.	.	PUNCT
ejpam-7135	490	1	[	[	X
ejpam-7135	490	2	6	6	NUM
ejpam-7135	490	3	]	]	PUNCT
ejpam-7135	490	4	t.	t.	NOUN
ejpam-7135	490	5	p.	p.	NOUN
ejpam-7135	490	6	speed	speed	NOUN
ejpam-7135	490	7	.	.	PUNCT
ejpam-7135	491	1	on	on	ADP
ejpam-7135	491	2	stone	stone	NOUN
ejpam-7135	491	3	lattices	lattice	NOUN
ejpam-7135	491	4	.	.	PUNCT
ejpam-7135	492	1	j.	j.	PROPN
ejpam-7135	492	2	aust	aust	PROPN
ejpam-7135	492	3	.	.	PUNCT
ejpam-7135	493	1	math	math	PROPN
ejpam-7135	493	2	.	.	PUNCT
ejpam-7135	494	1	soc	soc	PROPN
ejpam-7135	494	2	.	.	PUNCT
ejpam-7135	494	3	,	,	PUNCT
ejpam-7135	494	4	9:297–307	9:297–307	NUM
ejpam-7135	494	5	,	,	PUNCT
ejpam-7135	494	6	1969	1969	NUM
ejpam-7135	494	7	.	.	PUNCT
ejpam-7135	495	1	r.	r.	PROPN
ejpam-7135	495	2	bandaru	bandaru	PROPN
ejpam-7135	495	3	et	et	PROPN
ejpam-7135	495	4	al	al	PROPN
ejpam-7135	495	5	.	.	PUNCT
ejpam-7135	495	6	/	/	SYM
ejpam-7135	495	7	eur	eur	PROPN
ejpam-7135	495	8	.	.	PUNCT
ejpam-7135	496	1	j.	j.	PROPN
ejpam-7135	496	2	pure	pure	PROPN
ejpam-7135	496	3	appl	appl	PROPN
ejpam-7135	496	4	.	.	PROPN
ejpam-7135	496	5	math	math	PROPN
ejpam-7135	496	6	,	,	PUNCT
ejpam-7135	496	7	18	18	NUM
ejpam-7135	496	8	(	(	PUNCT
ejpam-7135	496	9	4	4	NUM
ejpam-7135	496	10	)	)	PUNCT
ejpam-7135	496	11	(	(	PUNCT
ejpam-7135	496	12	2025	2025	NUM
ejpam-7135	496	13	)	)	PUNCT
ejpam-7135	496	14	,	,	PUNCT
ejpam-7135	496	15	7135	7135	NUM
ejpam-7135	496	16	14	14	NUM
ejpam-7135	496	17	of	of	ADP
ejpam-7135	496	18	14	14	NUM
ejpam-7135	496	19	[	[	X
ejpam-7135	496	20	7	7	NUM
ejpam-7135	496	21	]	]	X
ejpam-7135	496	22	g.	g.	PROPN
ejpam-7135	496	23	grätzer	grätzer	PROPN
ejpam-7135	496	24	.	.	PUNCT
ejpam-7135	497	1	a	a	DET
ejpam-7135	497	2	generalization	generalization	NOUN
ejpam-7135	497	3	of	of	ADP
ejpam-7135	497	4	stone	stone	NOUN
ejpam-7135	497	5	’s	’s	PART
ejpam-7135	497	6	representation	representation	NOUN
ejpam-7135	497	7	theorem	theorem	NOUN
ejpam-7135	497	8	for	for	ADP
ejpam-7135	497	9	boolean	boolean	ADJ
ejpam-7135	497	10	algebras	algebra	NOUN
ejpam-7135	497	11	.	.	PUNCT
ejpam-7135	498	1	duke	duke	PROPN
ejpam-7135	498	2	math	math	PROPN
ejpam-7135	498	3	.	.	PUNCT
ejpam-7135	499	1	j.	j.	PROPN
ejpam-7135	499	2	,	,	PUNCT
ejpam-7135	499	3	30:469–474	30:469–474	PROPN
ejpam-7135	499	4	,	,	PUNCT
ejpam-7135	499	5	1963	1963	NUM
ejpam-7135	499	6	.	.	PUNCT
ejpam-7135	500	1	[	[	X
ejpam-7135	500	2	8	8	NUM
ejpam-7135	500	3	]	]	X
ejpam-7135	500	4	u.	u.	PROPN
ejpam-7135	500	5	m.	m.	PROPN
ejpam-7135	500	6	swamy	swamy	PROPN
ejpam-7135	500	7	and	and	CCONJ
ejpam-7135	500	8	p.	p.	PROPN
ejpam-7135	500	9	manikyamba	manikyamba	PROPN
ejpam-7135	500	10	.	.	PUNCT
ejpam-7135	501	1	prime	prime	ADJ
ejpam-7135	501	2	ideal	ideal	ADJ
ejpam-7135	501	3	characterization	characterization	NOUN
ejpam-7135	501	4	of	of	ADP
ejpam-7135	501	5	stone	stone	NOUN
ejpam-7135	501	6	lattices	lattice	NOUN
ejpam-7135	501	7	.	.	PUNCT
ejpam-7135	502	1	math	math	NOUN
ejpam-7135	502	2	.	.	PUNCT
ejpam-7135	503	1	semin	semin	PROPN
ejpam-7135	503	2	.	.	PUNCT
ejpam-7135	504	1	notes	notes	PROPN
ejpam-7135	504	2	,	,	PUNCT
ejpam-7135	504	3	kobe	kobe	PROPN
ejpam-7135	504	4	univ	univ	PROPN
ejpam-7135	504	5	.	.	PROPN
ejpam-7135	504	6	,	,	PUNCT
ejpam-7135	504	7	7:25–31	7:25–31	NUM
ejpam-7135	504	8	,	,	PUNCT
ejpam-7135	504	9	1979	1979	NUM
ejpam-7135	504	10	.	.	PUNCT
ejpam-7135	505	1	[	[	X
ejpam-7135	505	2	9	9	NUM
ejpam-7135	505	3	]	]	X
ejpam-7135	505	4	o.	o.	PROPN
ejpam-7135	505	5	frink	frink	PROPN
ejpam-7135	505	6	.	.	PUNCT
ejpam-7135	506	1	pseudo	pseudo	NOUN
ejpam-7135	506	2	-	-	NOUN
ejpam-7135	506	3	complements	complement	NOUN
ejpam-7135	506	4	in	in	ADP
ejpam-7135	506	5	semi	semi	NOUN
ejpam-7135	506	6	-	-	NOUN
ejpam-7135	506	7	lattices	lattice	NOUN
ejpam-7135	506	8	.	.	PUNCT
ejpam-7135	507	1	duke	duke	PROPN
ejpam-7135	507	2	math	math	PROPN
ejpam-7135	507	3	.	.	PUNCT
ejpam-7135	508	1	j.	j.	PROPN
ejpam-7135	508	2	,	,	PUNCT
ejpam-7135	508	3	29:505–514	29:505–514	PROPN
ejpam-7135	508	4	,	,	PUNCT
ejpam-7135	508	5	1962	1962	NUM
ejpam-7135	508	6	.	.	PUNCT
ejpam-7135	509	1	[	[	X
ejpam-7135	509	2	10	10	NUM
ejpam-7135	509	3	]	]	PUNCT
ejpam-7135	509	4	j.	j.	PROPN
ejpam-7135	509	5	von	von	PROPN
ejpam-7135	509	6	neumann	neumann	PROPN
ejpam-7135	509	7	.	.	PUNCT
ejpam-7135	510	1	on	on	ADP
ejpam-7135	510	2	regular	regular	ADJ
ejpam-7135	510	3	rings	ring	NOUN
ejpam-7135	510	4	.	.	PUNCT
ejpam-7135	511	1	proc	proc	PROPN
ejpam-7135	511	2	.	.	PUNCT
ejpam-7135	512	1	nat	nat	PROPN
ejpam-7135	512	2	.	.	PUNCT
ejpam-7135	513	1	acad	acad	PROPN
ejpam-7135	513	2	.	.	PUNCT
ejpam-7135	514	1	sci	sci	PROPN
ejpam-7135	514	2	.	.	PROPN
ejpam-7135	514	3	,	,	PUNCT
ejpam-7135	514	4	u.s.a	u.s.a	PROPN
ejpam-7135	514	5	.	.	PROPN
ejpam-7135	514	6	,	,	PUNCT
ejpam-7135	514	7	22:707–713	22:707–713	NUM
ejpam-7135	514	8	,	,	PUNCT
ejpam-7135	514	9	1936	1936	NUM
ejpam-7135	514	10	.	.	PUNCT
ejpam-7135	515	1	[	[	X
ejpam-7135	515	2	11	11	NUM
ejpam-7135	515	3	]	]	X
ejpam-7135	515	4	s.	s.	PROPN
ejpam-7135	515	5	burris	burris	PROPN
ejpam-7135	515	6	and	and	CCONJ
ejpam-7135	515	7	h.	h.	PROPN
ejpam-7135	515	8	p.	p.	PROPN
ejpam-7135	515	9	sankappanavar	sankappanavar	NOUN
ejpam-7135	515	10	.	.	PUNCT
ejpam-7135	516	1	a	a	DET
ejpam-7135	516	2	course	course	NOUN
ejpam-7135	516	3	in	in	ADP
ejpam-7135	516	4	universal	universal	ADJ
ejpam-7135	516	5	algebra	algebra	NOUN
ejpam-7135	516	6	.	.	PUNCT
ejpam-7135	517	1	springer	springer	NOUN
ejpam-7135	517	2	-	-	PUNCT
ejpam-7135	517	3	verlag	verlag	PROPN
ejpam-7135	517	4	,	,	PUNCT
ejpam-7135	517	5	1981	1981	NUM
ejpam-7135	517	6	.	.	PUNCT
ejpam-7135	518	1	[	[	X
ejpam-7135	518	2	12	12	NUM
ejpam-7135	518	3	]	]	X
ejpam-7135	518	4	r.	r.	PROPN
ejpam-7135	518	5	bandaru	bandaru	PROPN
ejpam-7135	518	6	and	and	CCONJ
ejpam-7135	518	7	s.	s.	PROPN
ejpam-7135	518	8	ajjarapu	ajjarapu	PROPN
ejpam-7135	518	9	.	.	PUNCT
ejpam-7135	519	1	paradistributive	paradistributive	PROPN
ejpam-7135	519	2	latticoids	latticoids	PROPN
ejpam-7135	519	3	.	.	PUNCT
ejpam-7135	520	1	eur	eur	PROPN
ejpam-7135	520	2	.	.	PUNCT
ejpam-7135	521	1	j.	j.	PROPN
ejpam-7135	521	2	pure	pure	PROPN
ejpam-7135	521	3	appl	appl	PROPN
ejpam-7135	521	4	.	.	PUNCT
ejpam-7135	521	5	math	math	PROPN
ejpam-7135	521	6	.	.	PUNCT
ejpam-7135	521	7	,	,	PUNCT
ejpam-7135	521	8	17(2):819–834	17(2):819–834	NUM
ejpam-7135	521	9	,	,	PUNCT
ejpam-7135	521	10	2024	2024	NUM
ejpam-7135	521	11	.	.	PUNCT
ejpam-7135	522	1	[	[	X
ejpam-7135	522	2	13	13	NUM
ejpam-7135	522	3	]	]	X
ejpam-7135	522	4	r.	r.	PROPN
ejpam-7135	522	5	bandaru	bandaru	PROPN
ejpam-7135	522	6	,	,	PUNCT
ejpam-7135	522	7	p.	p.	NOUN
ejpam-7135	522	8	patel	patel	PROPN
ejpam-7135	522	9	,	,	PUNCT
ejpam-7135	522	10	n.	n.	PROPN
ejpam-7135	522	11	rafi	rafi	PROPN
ejpam-7135	522	12	,	,	PUNCT
ejpam-7135	522	13	r.	r.	PROPN
ejpam-7135	522	14	shukla	shukla	PROPN
ejpam-7135	522	15	,	,	PUNCT
ejpam-7135	522	16	and	and	CCONJ
ejpam-7135	522	17	s.	s.	PROPN
ejpam-7135	522	18	ajjarapu	ajjarapu	PROPN
ejpam-7135	522	19	.	.	PUNCT
ejpam-7135	523	1	normal	normal	ADJ
ejpam-7135	523	2	paradistributive	paradistributive	ADJ
ejpam-7135	523	3	latticoids	latticoid	NOUN
ejpam-7135	523	4	.	.	PUNCT
ejpam-7135	524	1	eur	eur	PROPN
ejpam-7135	524	2	.	.	PUNCT
ejpam-7135	525	1	j.	j.	PROPN
ejpam-7135	525	2	pure	pure	PROPN
ejpam-7135	525	3	appl	appl	PROPN
ejpam-7135	525	4	.	.	PUNCT
ejpam-7135	525	5	math	math	PROPN
ejpam-7135	525	6	.	.	PUNCT
ejpam-7135	525	7	,	,	PUNCT
ejpam-7135	525	8	17(2):1306–1320	17(2):1306–1320	NUM
ejpam-7135	525	9	,	,	PUNCT
ejpam-7135	525	10	2024	2024	NUM
ejpam-7135	525	11	.	.	PUNCT
ejpam-7135	526	1	[	[	X
ejpam-7135	526	2	14	14	NUM
ejpam-7135	526	3	]	]	X
ejpam-7135	526	4	s.	s.	PROPN
ejpam-7135	526	5	ajjarapu	ajjarapu	PROPN
ejpam-7135	526	6	,	,	PUNCT
ejpam-7135	526	7	r.	r.	PROPN
ejpam-7135	526	8	bandaru	bandaru	PROPN
ejpam-7135	526	9	,	,	PUNCT
ejpam-7135	526	10	r.	r.	PROPN
ejpam-7135	526	11	shukla	shukla	PROPN
ejpam-7135	526	12	,	,	PUNCT
ejpam-7135	526	13	and	and	CCONJ
ejpam-7135	526	14	y.	y.	PROPN
ejpam-7135	526	15	b.	b.	PROPN
ejpam-7135	526	16	jun	jun	PROPN
ejpam-7135	526	17	.	.	PUNCT
ejpam-7135	527	1	parapseudo	parapseudo	NOUN
ejpam-7135	527	2	-	-	NOUN
ejpam-7135	527	3	complementation	complementation	NOUN
ejpam-7135	527	4	on	on	ADP
ejpam-7135	527	5	paradistributive	paradistributive	ADJ
ejpam-7135	527	6	latticoids	latticoid	NOUN
ejpam-7135	527	7	.	.	PUNCT
ejpam-7135	528	1	eur	eur	PROPN
ejpam-7135	528	2	.	.	PUNCT
ejpam-7135	529	1	j.	j.	PROPN
ejpam-7135	529	2	pure	pure	PROPN
ejpam-7135	529	3	appl	appl	PROPN
ejpam-7135	529	4	.	.	PUNCT
ejpam-7135	529	5	math	math	PROPN
ejpam-7135	529	6	.	.	PUNCT
ejpam-7135	529	7	,	,	PUNCT
ejpam-7135	529	8	17(2):1129–1145	17(2):1129–1145	NUM
ejpam-7135	529	9	,	,	PUNCT
ejpam-7135	529	10	2024	2024	NUM
ejpam-7135	529	11	.	.	PUNCT
ejpam-7135	530	1	[	[	X
ejpam-7135	530	2	15	15	NUM
ejpam-7135	530	3	]	]	X
ejpam-7135	530	4	s.	s.	PROPN
ejpam-7135	530	5	ajjarapu	ajjarapu	PROPN
ejpam-7135	530	6	,	,	PUNCT
ejpam-7135	530	7	r.	r.	PROPN
ejpam-7135	530	8	bandaru	bandaru	PROPN
ejpam-7135	530	9	,	,	PUNCT
ejpam-7135	530	10	r.	r.	PROPN
ejpam-7135	530	11	r.	r.	PROPN
ejpam-7135	530	12	kotha	kotha	PROPN
ejpam-7135	530	13	,	,	PUNCT
ejpam-7135	530	14	and	and	CCONJ
ejpam-7135	530	15	r.	r.	PROPN
ejpam-7135	530	16	shukla	shukla	PROPN
ejpam-7135	530	17	.	.	PUNCT
ejpam-7135	531	1	topological	topological	ADJ
ejpam-7135	531	2	properties	property	NOUN
ejpam-7135	531	3	of	of	ADP
ejpam-7135	531	4	prime	prime	ADJ
ejpam-7135	531	5	filters	filter	NOUN
ejpam-7135	531	6	and	and	CCONJ
ejpam-7135	531	7	minimal	minimal	ADJ
ejpam-7135	531	8	prime	prime	ADJ
ejpam-7135	531	9	filters	filter	NOUN
ejpam-7135	531	10	on	on	ADP
ejpam-7135	531	11	a	a	DET
ejpam-7135	531	12	paradistributive	paradistributive	ADJ
ejpam-7135	531	13	latticoid	latticoid	NOUN
ejpam-7135	531	14	.	.	PUNCT
ejpam-7135	532	1	int	int	NOUN
ejpam-7135	532	2	.	.	PUNCT
ejpam-7135	533	1	j.	j.	PROPN
ejpam-7135	533	2	math	math	PROPN
ejpam-7135	533	3	.	.	PUNCT
ejpam-7135	534	1	math	math	NOUN
ejpam-7135	534	2	.	.	PUNCT
ejpam-7135	535	1	sci	sci	PROPN
ejpam-7135	535	2	.	.	PROPN
ejpam-7135	535	3	,	,	PUNCT
ejpam-7135	535	4	2024	2024	NUM
ejpam-7135	535	5	:	:	PUNCT
ejpam-7135	535	6	article	article	NOUN
ejpam-7135	535	7	i	i	PROPN
ejpam-7135	535	8	d	d	PROPN
ejpam-7135	535	9	1862245	1862245	NUM
ejpam-7135	535	10	,	,	PUNCT
ejpam-7135	535	11	12	12	NUM
ejpam-7135	535	12	pages	page	NOUN
ejpam-7135	535	13	,	,	PUNCT
ejpam-7135	535	14	2024	2024	NUM
ejpam-7135	535	15	.	.	PUNCT