id	sid	tid	token	lemma	pos
ejpam-5535	1	1	european	european	PROPN
ejpam-5535	1	2	journal	journal	PROPN
ejpam-5535	1	3	of	of	ADP
ejpam-5535	1	4	pure	pure	ADJ
ejpam-5535	1	5	and	and	CCONJ
ejpam-5535	1	6	applied	applied	ADJ
ejpam-5535	1	7	mathematics	mathematic	NOUN
ejpam-5535	1	8	2025	2025	NUM
ejpam-5535	1	9	,	,	PUNCT
ejpam-5535	1	10	vol	vol	NOUN
ejpam-5535	1	11	.	.	PROPN
ejpam-5535	1	12	18	18	NUM
ejpam-5535	1	13	,	,	PUNCT
ejpam-5535	1	14	issue	issue	NOUN
ejpam-5535	1	15	1	1	NUM
ejpam-5535	1	16	,	,	PUNCT
ejpam-5535	1	17	article	article	NOUN
ejpam-5535	1	18	number	number	NOUN
ejpam-5535	1	19	5535	5535	NUM
ejpam-5535	1	20	issn	issn	PROPN
ejpam-5535	1	21	1307	1307	NUM
ejpam-5535	1	22	-	-	SYM
ejpam-5535	1	23	5543	5543	NUM
ejpam-5535	1	24	–	–	PUNCT
ejpam-5535	1	25	ejpam.com	ejpam.com	X
ejpam-5535	1	26	published	publish	VERB
ejpam-5535	1	27	by	by	ADP
ejpam-5535	1	28	new	new	PROPN
ejpam-5535	1	29	york	york	PROPN
ejpam-5535	1	30	business	business	PROPN
ejpam-5535	1	31	global	global	PROPN
ejpam-5535	1	32	a	a	DET
ejpam-5535	1	33	certain	certain	ADJ
ejpam-5535	1	34	class	class	NOUN
ejpam-5535	1	35	of	of	ADP
ejpam-5535	1	36	filters	filter	NOUN
ejpam-5535	1	37	in	in	ADP
ejpam-5535	1	38	generalized	generalized	ADJ
ejpam-5535	1	39	complemented	complement	VERB
ejpam-5535	1	40	distributive	distributive	ADJ
ejpam-5535	1	41	lattices	lattice	NOUN
ejpam-5535	1	42	ramesh	ramesh	PROPN
ejpam-5535	1	43	sirisetti1	sirisetti1	PROPN
ejpam-5535	1	44	,	,	PUNCT
ejpam-5535	1	45	jogarao	jogarao	PROPN
ejpam-5535	1	46	gunda2	gunda2	PROPN
ejpam-5535	1	47	,	,	PUNCT
ejpam-5535	1	48	ravikumar	ravikumar	PROPN
ejpam-5535	1	49	bandaru3	bandaru3	PROPN
ejpam-5535	1	50	,	,	PUNCT
ejpam-5535	1	51	rahul	rahul	PROPN
ejpam-5535	1	52	shukla4,∗	shukla4,∗	NOUN
ejpam-5535	1	53	1	1	NUM
ejpam-5535	1	54	department	department	NOUN
ejpam-5535	1	55	of	of	ADP
ejpam-5535	1	56	mathematics	mathematic	NOUN
ejpam-5535	1	57	,	,	PUNCT
ejpam-5535	1	58	gitam	gitam	NOUN
ejpam-5535	1	59	school	school	NOUN
ejpam-5535	1	60	of	of	ADP
ejpam-5535	1	61	science	science	NOUN
ejpam-5535	1	62	,	,	PUNCT
ejpam-5535	1	63	gitam	gitam	NOUN
ejpam-5535	1	64	(	(	PUNCT
ejpam-5535	1	65	deemed	deem	VERB
ejpam-5535	1	66	to	to	PART
ejpam-5535	1	67	be	be	AUX
ejpam-5535	1	68	university	university	NOUN
ejpam-5535	1	69	)	)	PUNCT
ejpam-5535	1	70	,	,	PUNCT
ejpam-5535	1	71	visakhapatnam	visakhapatnam	PROPN
ejpam-5535	1	72	,	,	PUNCT
ejpam-5535	1	73	andhra	andhra	PROPN
ejpam-5535	1	74	pradesh-530045	pradesh-530045	NOUN
ejpam-5535	1	75	,	,	PUNCT
ejpam-5535	1	76	india	india	PROPN
ejpam-5535	1	77	2	2	NUM
ejpam-5535	1	78	department	department	NOUN
ejpam-5535	1	79	of	of	ADP
ejpam-5535	1	80	bs	bs	PROPN
ejpam-5535	1	81	&	&	CCONJ
ejpam-5535	1	82	h	h	PROPN
ejpam-5535	1	83	,	,	PUNCT
ejpam-5535	1	84	aditya	aditya	PROPN
ejpam-5535	1	85	institute	institute	PROPN
ejpam-5535	1	86	of	of	ADP
ejpam-5535	1	87	technology	technology	NOUN
ejpam-5535	1	88	and	and	CCONJ
ejpam-5535	1	89	management	management	NOUN
ejpam-5535	1	90	,	,	PUNCT
ejpam-5535	1	91	tekkali	tekkali	PROPN
ejpam-5535	1	92	,	,	PUNCT
ejpam-5535	1	93	srikakulam	srikakulam	PROPN
ejpam-5535	1	94	,	,	PUNCT
ejpam-5535	1	95	andhra	andhra	PROPN
ejpam-5535	1	96	pradesh-530021	pradesh-530021	PROPN
ejpam-5535	1	97	,	,	PUNCT
ejpam-5535	1	98	india	india	PROPN
ejpam-5535	1	99	3	3	NUM
ejpam-5535	1	100	department	department	NOUN
ejpam-5535	1	101	of	of	ADP
ejpam-5535	1	102	mathematics	mathematic	NOUN
ejpam-5535	1	103	,	,	PUNCT
ejpam-5535	1	104	school	school	NOUN
ejpam-5535	1	105	of	of	ADP
ejpam-5535	1	106	advanced	advanced	ADJ
ejpam-5535	1	107	sciences	science	NOUN
ejpam-5535	1	108	,	,	PUNCT
ejpam-5535	1	109	vit	vit	PROPN
ejpam-5535	1	110	-	-	PUNCT
ejpam-5535	1	111	ap	ap	PROPN
ejpam-5535	1	112	university	university	PROPN
ejpam-5535	1	113	,	,	PUNCT
ejpam-5535	1	114	andhra	andhra	PROPN
ejpam-5535	1	115	pradesh-522237	pradesh-522237	NOUN
ejpam-5535	1	116	,	,	PUNCT
ejpam-5535	1	117	india	india	PROPN
ejpam-5535	1	118	4	4	NUM
ejpam-5535	1	119	department	department	PROPN
ejpam-5535	1	120	of	of	ADP
ejpam-5535	1	121	mathematical	mathematical	ADJ
ejpam-5535	1	122	sciences	sciences	PROPN
ejpam-5535	1	123	and	and	CCONJ
ejpam-5535	1	124	computing	computing	NOUN
ejpam-5535	1	125	,	,	PUNCT
ejpam-5535	1	126	walter	walter	PROPN
ejpam-5535	1	127	sisulu	sisulu	PROPN
ejpam-5535	1	128	university	university	PROPN
ejpam-5535	1	129	,	,	PUNCT
ejpam-5535	1	130	mthatha	mthatha	NOUN
ejpam-5535	1	131	5117	5117	NUM
ejpam-5535	1	132	,	,	PUNCT
ejpam-5535	1	133	south	south	PROPN
ejpam-5535	1	134	africa	africa	PROPN
ejpam-5535	1	135	abstract	abstract	PROPN
ejpam-5535	1	136	.	.	PUNCT
ejpam-5535	2	1	in	in	ADP
ejpam-5535	2	2	this	this	DET
ejpam-5535	2	3	paper	paper	NOUN
ejpam-5535	2	4	,	,	PUNCT
ejpam-5535	2	5	we	we	PRON
ejpam-5535	2	6	introduce	introduce	VERB
ejpam-5535	2	7	kg	kg	NOUN
ejpam-5535	2	8	-	-	PUNCT
ejpam-5535	2	9	filters	filter	NOUN
ejpam-5535	2	10	jointly	jointly	ADV
ejpam-5535	2	11	derived	derive	VERB
ejpam-5535	2	12	from	from	ADP
ejpam-5535	2	13	the	the	DET
ejpam-5535	2	14	class	class	NOUN
ejpam-5535	2	15	of	of	ADP
ejpam-5535	2	16	ideals	ideal	NOUN
ejpam-5535	2	17	and	and	CCONJ
ejpam-5535	2	18	the	the	DET
ejpam-5535	2	19	class	class	NOUN
ejpam-5535	2	20	of	of	ADP
ejpam-5535	2	21	generalized	generalized	ADJ
ejpam-5535	2	22	complementations	complementation	NOUN
ejpam-5535	2	23	in	in	ADP
ejpam-5535	2	24	a	a	DET
ejpam-5535	2	25	generalized	generalize	VERB
ejpam-5535	2	26	complemented	complement	VERB
ejpam-5535	2	27	distributive	distributive	ADJ
ejpam-5535	2	28	lattice	lattice	NOUN
ejpam-5535	2	29	.	.	PUNCT
ejpam-5535	3	1	we	we	PRON
ejpam-5535	3	2	obtain	obtain	VERB
ejpam-5535	3	3	some	some	DET
ejpam-5535	3	4	algebraic	algebraic	ADJ
ejpam-5535	3	5	properties	property	NOUN
ejpam-5535	3	6	on	on	ADP
ejpam-5535	3	7	the	the	DET
ejpam-5535	3	8	obtained	obtain	VERB
ejpam-5535	3	9	class	class	NOUN
ejpam-5535	3	10	,	,	PUNCT
ejpam-5535	3	11	and	and	CCONJ
ejpam-5535	3	12	we	we	PRON
ejpam-5535	3	13	provide	provide	VERB
ejpam-5535	3	14	some	some	DET
ejpam-5535	3	15	counter	counter	NOUN
ejpam-5535	3	16	-	-	NOUN
ejpam-5535	3	17	examples	example	NOUN
ejpam-5535	3	18	.	.	PUNCT
ejpam-5535	4	1	mainly	mainly	ADV
ejpam-5535	4	2	,	,	PUNCT
ejpam-5535	4	3	we	we	PRON
ejpam-5535	4	4	derive	derive	VERB
ejpam-5535	4	5	some	some	DET
ejpam-5535	4	6	boolean	boolean	ADJ
ejpam-5535	4	7	algebras	algebra	NOUN
ejpam-5535	4	8	(	(	PUNCT
ejpam-5535	4	9	distributive	distributive	ADJ
ejpam-5535	4	10	lattices	lattice	NOUN
ejpam-5535	4	11	)	)	PUNCT
ejpam-5535	4	12	through	through	ADP
ejpam-5535	4	13	the	the	DET
ejpam-5535	4	14	class	class	NOUN
ejpam-5535	4	15	of	of	ADP
ejpam-5535	4	16	kg	kg	NOUN
ejpam-5535	4	17	-	-	PUNCT
ejpam-5535	4	18	filters	filter	NOUN
ejpam-5535	4	19	in	in	ADP
ejpam-5535	4	20	a	a	DET
ejpam-5535	4	21	generalized	generalize	VERB
ejpam-5535	4	22	complemented	complement	VERB
ejpam-5535	4	23	distributive	distributive	ADJ
ejpam-5535	4	24	lattice	lattice	NOUN
ejpam-5535	4	25	.	.	PUNCT
ejpam-5535	5	1	finally	finally	ADV
ejpam-5535	5	2	,	,	PUNCT
ejpam-5535	5	3	we	we	PRON
ejpam-5535	5	4	introduce	introduce	VERB
ejpam-5535	5	5	normal	normal	ADJ
ejpam-5535	5	6	kg	kg	NOUN
ejpam-5535	5	7	-	-	PUNCT
ejpam-5535	5	8	filters	filter	NOUN
ejpam-5535	5	9	in	in	ADP
ejpam-5535	5	10	a	a	DET
ejpam-5535	5	11	generalized	generalize	VERB
ejpam-5535	5	12	complemented	complement	VERB
ejpam-5535	5	13	distributive	distributive	ADJ
ejpam-5535	5	14	lattice	lattice	NOUN
ejpam-5535	5	15	and	and	CCONJ
ejpam-5535	5	16	then	then	ADV
ejpam-5535	5	17	prove	prove	VERB
ejpam-5535	5	18	that	that	SCONJ
ejpam-5535	5	19	the	the	DET
ejpam-5535	5	20	class	class	NOUN
ejpam-5535	5	21	of	of	ADP
ejpam-5535	5	22	normal	normal	ADJ
ejpam-5535	5	23	kg	kg	NOUN
ejpam-5535	5	24	-	-	PUNCT
ejpam-5535	5	25	filters	filter	NOUN
ejpam-5535	5	26	is	be	AUX
ejpam-5535	5	27	a	a	DET
ejpam-5535	5	28	boolean	boolean	ADJ
ejpam-5535	5	29	algebra	algebra	NOUN
ejpam-5535	5	30	.	.	PUNCT
ejpam-5535	6	1	2020	2020	NUM
ejpam-5535	6	2	mathematics	mathematic	NOUN
ejpam-5535	6	3	subject	subject	NOUN
ejpam-5535	6	4	classifications	classification	NOUN
ejpam-5535	6	5	:	:	PUNCT
ejpam-5535	6	6	06d05	06d05	NUM
ejpam-5535	6	7	,	,	PUNCT
ejpam-5535	6	8	06d15	06d15	DET
ejpam-5535	6	9	key	key	ADJ
ejpam-5535	6	10	words	word	NOUN
ejpam-5535	6	11	and	and	CCONJ
ejpam-5535	6	12	phrases	phrase	NOUN
ejpam-5535	6	13	:	:	PUNCT
ejpam-5535	6	14	ideals	ideal	NOUN
ejpam-5535	6	15	,	,	PUNCT
ejpam-5535	6	16	filters	filter	NOUN
ejpam-5535	6	17	,	,	PUNCT
ejpam-5535	6	18	distributive	distributive	ADJ
ejpam-5535	6	19	lattices	lattice	NOUN
ejpam-5535	6	20	,	,	PUNCT
ejpam-5535	6	21	generalized	generalize	VERB
ejpam-5535	6	22	complemented	complement	VERB
ejpam-5535	6	23	distributive	distributive	ADJ
ejpam-5535	6	24	lattices	lattice	NOUN
ejpam-5535	6	25	,	,	PUNCT
ejpam-5535	6	26	boolean	boolean	ADJ
ejpam-5535	6	27	algebras	algebra	NOUN
ejpam-5535	6	28	1	1	X
ejpam-5535	6	29	.	.	X
ejpam-5535	6	30	introduction	introduction	NOUN
ejpam-5535	6	31	the	the	DET
ejpam-5535	6	32	concept	concept	NOUN
ejpam-5535	6	33	of	of	ADP
ejpam-5535	6	34	distributive	distributive	ADJ
ejpam-5535	6	35	lattices	lattice	NOUN
ejpam-5535	6	36	[	[	X
ejpam-5535	6	37	2	2	NUM
ejpam-5535	6	38	,	,	PUNCT
ejpam-5535	6	39	3	3	NUM
ejpam-5535	6	40	]	]	PUNCT
ejpam-5535	6	41	has	have	AUX
ejpam-5535	6	42	been	be	AUX
ejpam-5535	6	43	extensively	extensively	ADV
ejpam-5535	6	44	studied	study	VERB
ejpam-5535	6	45	by	by	ADP
ejpam-5535	6	46	several	several	ADJ
ejpam-5535	6	47	authors	author	NOUN
ejpam-5535	6	48	by	by	ADP
ejpam-5535	6	49	taking	take	VERB
ejpam-5535	6	50	a	a	DET
ejpam-5535	6	51	unary	unary	ADJ
ejpam-5535	6	52	operation	operation	NOUN
ejpam-5535	6	53	,	,	PUNCT
ejpam-5535	6	54	like	like	ADP
ejpam-5535	6	55	complementation	complementation	NOUN
ejpam-5535	6	56	[	[	X
ejpam-5535	6	57	7	7	NUM
ejpam-5535	6	58	,	,	PUNCT
ejpam-5535	6	59	12	12	NUM
ejpam-5535	6	60	]	]	PUNCT
ejpam-5535	6	61	,	,	PUNCT
ejpam-5535	6	62	pseudo	pseudo	NOUN
ejpam-5535	6	63	-	-	NOUN
ejpam-5535	6	64	complementation	complementation	NOUN
ejpam-5535	6	65	[	[	X
ejpam-5535	6	66	14	14	NUM
ejpam-5535	6	67	,	,	PUNCT
ejpam-5535	6	68	15	15	NUM
ejpam-5535	6	69	]	]	PUNCT
ejpam-5535	6	70	,	,	PUNCT
ejpam-5535	6	71	quasi	quasi	NOUN
ejpam-5535	6	72	-	-	NOUN
ejpam-5535	6	73	complementation	complementation	NOUN
ejpam-5535	6	74	[	[	X
ejpam-5535	6	75	5	5	NUM
ejpam-5535	6	76	,	,	PUNCT
ejpam-5535	6	77	13	13	NUM
ejpam-5535	6	78	]	]	PUNCT
ejpam-5535	6	79	etc	etc	X
ejpam-5535	6	80	.	.	X
ejpam-5535	6	81	,	,	PUNCT
ejpam-5535	6	82	and	and	CCONJ
ejpam-5535	6	83	also	also	ADV
ejpam-5535	6	84	by	by	ADP
ejpam-5535	6	85	considering	consider	VERB
ejpam-5535	6	86	the	the	DET
ejpam-5535	6	87	class	class	NOUN
ejpam-5535	6	88	of	of	ADP
ejpam-5535	6	89	ideals	ideal	NOUN
ejpam-5535	6	90	(	(	PUNCT
ejpam-5535	6	91	filters	filter	NOUN
ejpam-5535	6	92	)	)	PUNCT
ejpam-5535	6	93	.	.	PUNCT
ejpam-5535	7	1	some	some	PRON
ejpam-5535	7	2	of	of	ADP
ejpam-5535	7	3	authors	author	NOUN
ejpam-5535	7	4	recursively	recursively	ADV
ejpam-5535	7	5	studied	study	VERB
ejpam-5535	7	6	the	the	DET
ejpam-5535	7	7	class	class	NOUN
ejpam-5535	7	8	of	of	ADP
ejpam-5535	7	9	distributive	distributive	ADJ
ejpam-5535	7	10	lattices	lattice	NOUN
ejpam-5535	7	11	by	by	ADP
ejpam-5535	7	12	taking	take	VERB
ejpam-5535	7	13	a	a	DET
ejpam-5535	7	14	binary	binary	ADJ
ejpam-5535	7	15	operations	operation	NOUN
ejpam-5535	7	16	like	like	ADP
ejpam-5535	7	17	generalized	generalized	ADJ
ejpam-5535	7	18	implementation	implementation	NOUN
ejpam-5535	7	19	[	[	X
ejpam-5535	7	20	11	11	NUM
ejpam-5535	7	21	]	]	PUNCT
ejpam-5535	7	22	,	,	PUNCT
ejpam-5535	7	23	by	by	ADP
ejpam-5535	7	24	taking	take	VERB
ejpam-5535	7	25	median	median	ADJ
ejpam-5535	7	26	graphs	graph	NOUN
ejpam-5535	7	27	[	[	X
ejpam-5535	7	28	10	10	NUM
ejpam-5535	7	29	]	]	PUNCT
ejpam-5535	7	30	,	,	PUNCT
ejpam-5535	7	31	by	by	ADP
ejpam-5535	7	32	considering	consider	VERB
ejpam-5535	7	33	canonical	canonical	ADJ
ejpam-5535	7	34	extensions	extension	NOUN
ejpam-5535	7	35	[	[	X
ejpam-5535	7	36	9	9	NUM
ejpam-5535	7	37	]	]	PUNCT
ejpam-5535	7	38	,	,	PUNCT
ejpam-5535	7	39	by	by	ADP
ejpam-5535	7	40	using	use	VERB
ejpam-5535	7	41	subordinations	subordination	NOUN
ejpam-5535	7	42	[	[	X
ejpam-5535	7	43	1	1	NUM
ejpam-5535	7	44	,	,	PUNCT
ejpam-5535	7	45	6	6	NUM
ejpam-5535	7	46	]	]	PUNCT
ejpam-5535	7	47	.	.	PUNCT
ejpam-5535	8	1	the	the	DET
ejpam-5535	8	2	class	class	NOUN
ejpam-5535	8	3	of	of	ADP
ejpam-5535	8	4	ideals	ideal	NOUN
ejpam-5535	8	5	(	(	PUNCT
ejpam-5535	8	6	filters	filter	NOUN
ejpam-5535	8	7	)	)	PUNCT
ejpam-5535	8	8	has	have	VERB
ejpam-5535	8	9	a	a	DET
ejpam-5535	8	10	major	major	ADJ
ejpam-5535	8	11	role	role	NOUN
ejpam-5535	8	12	in	in	ADP
ejpam-5535	8	13	the	the	DET
ejpam-5535	8	14	theory	theory	NOUN
ejpam-5535	8	15	of	of	ADP
ejpam-5535	8	16	lattices	lattice	NOUN
ejpam-5535	8	17	.	.	PUNCT
ejpam-5535	9	1	it	it	PRON
ejpam-5535	9	2	is	be	AUX
ejpam-5535	9	3	known	know	VERB
ejpam-5535	9	4	that	that	SCONJ
ejpam-5535	9	5	there	there	PRON
ejpam-5535	9	6	is	be	VERB
ejpam-5535	9	7	a	a	DET
ejpam-5535	9	8	one	one	NUM
ejpam-5535	9	9	-	-	PUNCT
ejpam-5535	9	10	to	to	ADP
ejpam-5535	9	11	-	-	PUNCT
ejpam-5535	9	12	one	one	NUM
ejpam-5535	9	13	∗corresponding	∗corresponde	VERB
ejpam-5535	9	14	author	author	NOUN
ejpam-5535	9	15	.	.	PUNCT
ejpam-5535	10	1	doi	doi	NOUN
ejpam-5535	10	2	:	:	PUNCT
ejpam-5535	10	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5535	https://doi.org/10.29020/nybg.ejpam.v18i1.5535	NOUN
ejpam-5535	10	4	email	email	NOUN
ejpam-5535	10	5	addresses	address	NOUN
ejpam-5535	10	6	:	:	PUNCT
ejpam-5535	10	7	ramesh.sirisetti@gmail.com	ramesh.sirisetti@gmail.com	X
ejpam-5535	10	8	(	(	PUNCT
ejpam-5535	10	9	r.	r.	PROPN
ejpam-5535	10	10	sirisetti	sirisetti	PROPN
ejpam-5535	10	11	)	)	PUNCT
ejpam-5535	10	12	,	,	PUNCT
ejpam-5535	10	13	jogarao.gunda@gmail.com	jogarao.gunda@gmail.com	PROPN
ejpam-5535	10	14	(	(	PUNCT
ejpam-5535	10	15	j.	j.	PROPN
ejpam-5535	10	16	gunda	gunda	PROPN
ejpam-5535	10	17	)	)	PUNCT
ejpam-5535	10	18	,	,	PUNCT
ejpam-5535	10	19	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-5535	10	20	(	(	PUNCT
ejpam-5535	10	21	r.	r.	PROPN
ejpam-5535	10	22	bandaru	bandaru	PROPN
ejpam-5535	10	23	)	)	PUNCT
ejpam-5535	10	24	,	,	PUNCT
ejpam-5535	10	25	rshukla@wsu.ac.za	rshukla@wsu.ac.za	NOUN
ejpam-5535	10	26	(	(	PUNCT
ejpam-5535	10	27	r.	r.	NOUN
ejpam-5535	10	28	shukla	shukla	PROPN
ejpam-5535	10	29	)	)	PUNCT
ejpam-5535	10	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5535	11	1	1	1	NUM
ejpam-5535	11	2	copyright	copyright	NOUN
ejpam-5535	11	3	:	:	PUNCT
ejpam-5535	11	4	©	©	PROPN
ejpam-5535	11	5	2025	2025	NUM
ejpam-5535	11	6	the	the	DET
ejpam-5535	11	7	author(s	author(s	NOUN
ejpam-5535	11	8	)	)	PUNCT
ejpam-5535	11	9	.	.	PUNCT
ejpam-5535	12	1	(	(	PUNCT
ejpam-5535	12	2	cc	cc	NOUN
ejpam-5535	12	3	by	by	ADP
ejpam-5535	12	4	-	-	PUNCT
ejpam-5535	12	5	nc	nc	PROPN
ejpam-5535	12	6	4.0	4.0	NUM
ejpam-5535	12	7	)	)	PUNCT
ejpam-5535	12	8	r.	r.	PROPN
ejpam-5535	12	9	sirisetti	sirisetti	PROPN
ejpam-5535	12	10	et	et	PROPN
ejpam-5535	12	11	al	al	PROPN
ejpam-5535	12	12	.	.	PUNCT
ejpam-5535	12	13	/	/	SYM
ejpam-5535	12	14	eur	eur	PROPN
ejpam-5535	12	15	.	.	PUNCT
ejpam-5535	13	1	j.	j.	PROPN
ejpam-5535	13	2	pure	pure	PROPN
ejpam-5535	13	3	appl	appl	PROPN
ejpam-5535	13	4	.	.	PROPN
ejpam-5535	13	5	math	math	PROPN
ejpam-5535	13	6	,	,	PUNCT
ejpam-5535	13	7	18	18	NUM
ejpam-5535	13	8	(	(	PUNCT
ejpam-5535	13	9	1	1	NUM
ejpam-5535	13	10	)	)	PUNCT
ejpam-5535	13	11	(	(	PUNCT
ejpam-5535	13	12	2025	2025	NUM
ejpam-5535	13	13	)	)	PUNCT
ejpam-5535	13	14	,	,	PUNCT
ejpam-5535	13	15	5535	5535	NUM
ejpam-5535	13	16	2	2	NUM
ejpam-5535	13	17	of	of	ADP
ejpam-5535	13	18	12	12	NUM
ejpam-5535	13	19	correspondence	correspondence	NOUN
ejpam-5535	13	20	between	between	ADP
ejpam-5535	13	21	the	the	DET
ejpam-5535	13	22	class	class	NOUN
ejpam-5535	13	23	of	of	ADP
ejpam-5535	13	24	prime	prime	ADJ
ejpam-5535	13	25	ideals	ideal	NOUN
ejpam-5535	13	26	and	and	CCONJ
ejpam-5535	13	27	the	the	DET
ejpam-5535	13	28	class	class	NOUN
ejpam-5535	13	29	of	of	ADP
ejpam-5535	13	30	prime	prime	ADJ
ejpam-5535	13	31	filters	filter	NOUN
ejpam-5535	13	32	in	in	ADP
ejpam-5535	13	33	a	a	DET
ejpam-5535	13	34	distributive	distributive	ADJ
ejpam-5535	13	35	lattice	lattice	NOUN
ejpam-5535	13	36	.	.	PUNCT
ejpam-5535	14	1	it	it	PRON
ejpam-5535	14	2	is	be	AUX
ejpam-5535	14	3	easy	easy	ADJ
ejpam-5535	14	4	to	to	PART
ejpam-5535	14	5	observe	observe	VERB
ejpam-5535	14	6	that	that	SCONJ
ejpam-5535	14	7	we	we	PRON
ejpam-5535	14	8	can	can	AUX
ejpam-5535	14	9	obtain	obtain	VERB
ejpam-5535	14	10	a	a	DET
ejpam-5535	14	11	class	class	NOUN
ejpam-5535	14	12	of	of	ADP
ejpam-5535	14	13	ideals	ideal	NOUN
ejpam-5535	14	14	(	(	PUNCT
ejpam-5535	14	15	filter	filter	NOUN
ejpam-5535	14	16	)	)	PUNCT
ejpam-5535	14	17	from	from	ADP
ejpam-5535	14	18	a	a	DET
ejpam-5535	14	19	complementation	complementation	NOUN
ejpam-5535	14	20	on	on	ADP
ejpam-5535	14	21	a	a	DET
ejpam-5535	14	22	distributive	distributive	ADJ
ejpam-5535	14	23	lattice	lattice	NOUN
ejpam-5535	14	24	.	.	PUNCT
ejpam-5535	15	1	the	the	DET
ejpam-5535	15	2	authors	author	NOUN
ejpam-5535	15	3	[	[	X
ejpam-5535	15	4	8	8	NUM
ejpam-5535	15	5	]	]	PUNCT
ejpam-5535	15	6	introduced	introduce	VERB
ejpam-5535	15	7	and	and	CCONJ
ejpam-5535	15	8	studied	study	VERB
ejpam-5535	15	9	a	a	DET
ejpam-5535	15	10	generalized	generalized	ADJ
ejpam-5535	15	11	complementation	complementation	NOUN
ejpam-5535	15	12	(	(	PUNCT
ejpam-5535	15	13	g	g	NOUN
ejpam-5535	15	14	-	-	NOUN
ejpam-5535	15	15	complementation	complementation	NOUN
ejpam-5535	15	16	)	)	PUNCT
ejpam-5535	15	17	on	on	ADP
ejpam-5535	15	18	a	a	DET
ejpam-5535	15	19	distributive	distributive	ADJ
ejpam-5535	15	20	lattice	lattice	NOUN
ejpam-5535	15	21	with	with	ADP
ejpam-5535	15	22	dense	dense	ADJ
ejpam-5535	15	23	elements	element	NOUN
ejpam-5535	15	24	;	;	PUNCT
ejpam-5535	15	25	it	it	PRON
ejpam-5535	15	26	is	be	AUX
ejpam-5535	15	27	a	a	DET
ejpam-5535	15	28	generalization	generalization	NOUN
ejpam-5535	15	29	of	of	ADP
ejpam-5535	15	30	complementation	complementation	NOUN
ejpam-5535	15	31	and	and	CCONJ
ejpam-5535	15	32	dual	dual	ADJ
ejpam-5535	15	33	of	of	ADP
ejpam-5535	15	34	pseudo	pseudo	NOUN
ejpam-5535	15	35	-	-	NOUN
ejpam-5535	15	36	complementation	complementation	NOUN
ejpam-5535	15	37	in	in	ADP
ejpam-5535	15	38	distributive	distributive	ADJ
ejpam-5535	15	39	lattices	lattice	NOUN
ejpam-5535	15	40	.	.	PUNCT
ejpam-5535	16	1	in	in	ADP
ejpam-5535	16	2	this	this	DET
ejpam-5535	16	3	regard	regard	NOUN
ejpam-5535	16	4	,	,	PUNCT
ejpam-5535	16	5	we	we	PRON
ejpam-5535	16	6	derive	derive	VERB
ejpam-5535	16	7	filters	filter	NOUN
ejpam-5535	16	8	from	from	ADP
ejpam-5535	16	9	a	a	DET
ejpam-5535	16	10	generalized	generalized	ADJ
ejpam-5535	16	11	complementation	complementation	NOUN
ejpam-5535	16	12	on	on	ADP
ejpam-5535	16	13	a	a	DET
ejpam-5535	16	14	distributive	distributive	ADJ
ejpam-5535	16	15	lattice	lattice	NOUN
ejpam-5535	16	16	using	use	VERB
ejpam-5535	16	17	dense	dense	ADJ
ejpam-5535	16	18	elements	element	NOUN
ejpam-5535	16	19	.	.	PUNCT
ejpam-5535	17	1	in	in	ADP
ejpam-5535	17	2	this	this	DET
ejpam-5535	17	3	paper	paper	NOUN
ejpam-5535	17	4	,	,	PUNCT
ejpam-5535	17	5	we	we	PRON
ejpam-5535	17	6	introduce	introduce	VERB
ejpam-5535	17	7	a	a	DET
ejpam-5535	17	8	class	class	NOUN
ejpam-5535	17	9	of	of	ADP
ejpam-5535	17	10	filters	filter	NOUN
ejpam-5535	17	11	(	(	PUNCT
ejpam-5535	17	12	kg	kg	NOUN
ejpam-5535	17	13	-	-	PUNCT
ejpam-5535	17	14	filters	filter	NOUN
ejpam-5535	17	15	)	)	PUNCT
ejpam-5535	17	16	,	,	PUNCT
ejpam-5535	17	17	which	which	PRON
ejpam-5535	17	18	are	be	AUX
ejpam-5535	17	19	abstract	abstract	ADV
ejpam-5535	17	20	combined	combine	VERB
ejpam-5535	17	21	from	from	ADP
ejpam-5535	17	22	the	the	DET
ejpam-5535	17	23	class	class	NOUN
ejpam-5535	17	24	of	of	ADP
ejpam-5535	17	25	ideals	ideal	NOUN
ejpam-5535	17	26	and	and	CCONJ
ejpam-5535	17	27	the	the	DET
ejpam-5535	17	28	class	class	NOUN
ejpam-5535	17	29	of	of	ADP
ejpam-5535	17	30	generalized	generalized	ADJ
ejpam-5535	17	31	complementations	complementation	NOUN
ejpam-5535	17	32	on	on	ADP
ejpam-5535	17	33	a	a	DET
ejpam-5535	17	34	distributive	distributive	ADJ
ejpam-5535	17	35	lattice	lattice	NOUN
ejpam-5535	17	36	a	a	PRON
ejpam-5535	17	37	through	through	ADP
ejpam-5535	17	38	dense	dense	ADJ
ejpam-5535	17	39	elements	element	NOUN
ejpam-5535	17	40	and	and	CCONJ
ejpam-5535	17	41	prove	prove	VERB
ejpam-5535	17	42	several	several	ADJ
ejpam-5535	17	43	algebraic	algebraic	ADJ
ejpam-5535	17	44	properties	property	NOUN
ejpam-5535	17	45	on	on	ADP
ejpam-5535	17	46	them	they	PRON
ejpam-5535	17	47	.	.	PUNCT
ejpam-5535	18	1	also	also	ADV
ejpam-5535	18	2	,	,	PUNCT
ejpam-5535	18	3	we	we	PRON
ejpam-5535	18	4	obtain	obtain	VERB
ejpam-5535	18	5	some	some	DET
ejpam-5535	18	6	necessary	necessary	ADJ
ejpam-5535	18	7	and	and	CCONJ
ejpam-5535	18	8	sufficient	sufficient	ADJ
ejpam-5535	18	9	conditions	condition	NOUN
ejpam-5535	18	10	for	for	SCONJ
ejpam-5535	18	11	a	a	DET
ejpam-5535	18	12	filter	filter	NOUN
ejpam-5535	18	13	to	to	PART
ejpam-5535	18	14	become	become	VERB
ejpam-5535	18	15	kg	kg	NOUN
ejpam-5535	18	16	-	-	NOUN
ejpam-5535	18	17	filter	filter	NOUN
ejpam-5535	18	18	in	in	ADP
ejpam-5535	18	19	a.	a.	NOUN
ejpam-5535	18	20	we	we	PRON
ejpam-5535	18	21	obtain	obtain	VERB
ejpam-5535	18	22	some	some	DET
ejpam-5535	18	23	algebraic	algebraic	ADJ
ejpam-5535	18	24	results	result	NOUN
ejpam-5535	18	25	on	on	ADP
ejpam-5535	18	26	the	the	DET
ejpam-5535	18	27	class	class	NOUN
ejpam-5535	18	28	of	of	ADP
ejpam-5535	18	29	kg	kg	NOUN
ejpam-5535	18	30	-	-	PUNCT
ejpam-5535	18	31	filters	filter	NOUN
ejpam-5535	18	32	in	in	ADP
ejpam-5535	18	33	a	a	DET
ejpam-5535	18	34	generalized	generalize	VERB
ejpam-5535	18	35	complemented	complement	VERB
ejpam-5535	18	36	distributive	distributive	ADJ
ejpam-5535	18	37	lattice	lattice	NOUN
ejpam-5535	18	38	.	.	PUNCT
ejpam-5535	19	1	mainly	mainly	ADV
ejpam-5535	19	2	,	,	PUNCT
ejpam-5535	19	3	we	we	PRON
ejpam-5535	19	4	abstract	abstract	VERB
ejpam-5535	19	5	some	some	DET
ejpam-5535	19	6	structures	structure	NOUN
ejpam-5535	19	7	which	which	PRON
ejpam-5535	19	8	are	be	AUX
ejpam-5535	19	9	distributive	distributive	ADJ
ejpam-5535	19	10	lattices	lattice	NOUN
ejpam-5535	19	11	(	(	PUNCT
ejpam-5535	19	12	boolean	boolean	ADJ
ejpam-5535	19	13	algebras	algebra	NOUN
ejpam-5535	19	14	)	)	PUNCT
ejpam-5535	20	1	[	[	X
ejpam-5535	20	2	4	4	NUM
ejpam-5535	20	3	]	]	PUNCT
ejpam-5535	20	4	from	from	ADP
ejpam-5535	20	5	the	the	DET
ejpam-5535	20	6	class	class	NOUN
ejpam-5535	20	7	of	of	ADP
ejpam-5535	20	8	kg	kg	NOUN
ejpam-5535	20	9	-	-	PUNCT
ejpam-5535	20	10	filters	filter	NOUN
ejpam-5535	20	11	.	.	PUNCT
ejpam-5535	21	1	finally	finally	ADV
ejpam-5535	21	2	we	we	PRON
ejpam-5535	21	3	discuss	discuss	VERB
ejpam-5535	21	4	normal	normal	ADJ
ejpam-5535	21	5	kg	kg	NOUN
ejpam-5535	21	6	-	-	PUNCT
ejpam-5535	21	7	filters	filter	NOUN
ejpam-5535	21	8	in	in	ADP
ejpam-5535	21	9	generalized	generalized	ADJ
ejpam-5535	21	10	complemented	complement	VERB
ejpam-5535	21	11	distributive	distributive	ADJ
ejpam-5535	21	12	lattices	lattice	NOUN
ejpam-5535	21	13	and	and	CCONJ
ejpam-5535	21	14	prove	prove	VERB
ejpam-5535	21	15	that	that	SCONJ
ejpam-5535	21	16	the	the	DET
ejpam-5535	21	17	class	class	NOUN
ejpam-5535	21	18	of	of	ADP
ejpam-5535	21	19	normalkg	normalkg	PROPN
ejpam-5535	21	20	-	-	PUNCT
ejpam-5535	21	21	filters	filter	NOUN
ejpam-5535	21	22	forms	form	VERB
ejpam-5535	21	23	a	a	DET
ejpam-5535	21	24	boolean	boolean	ADJ
ejpam-5535	21	25	algebra	algebra	NOUN
ejpam-5535	21	26	.	.	PUNCT
ejpam-5535	22	1	2	2	X
ejpam-5535	22	2	.	.	X
ejpam-5535	22	3	some	some	DET
ejpam-5535	22	4	results	result	NOUN
ejpam-5535	22	5	on	on	ADP
ejpam-5535	22	6	generalized	generalized	ADJ
ejpam-5535	22	7	complemented	complement	VERB
ejpam-5535	22	8	distributive	distributive	ADJ
ejpam-5535	22	9	lattices	lattice	NOUN
ejpam-5535	22	10	in	in	ADP
ejpam-5535	22	11	this	this	DET
ejpam-5535	22	12	section	section	NOUN
ejpam-5535	22	13	,	,	PUNCT
ejpam-5535	22	14	we	we	PRON
ejpam-5535	22	15	introduce	introduce	VERB
ejpam-5535	22	16	a	a	DET
ejpam-5535	22	17	filter	filter	NOUN
ejpam-5535	22	18	corresponding	correspond	VERB
ejpam-5535	22	19	to	to	ADP
ejpam-5535	22	20	an	an	DET
ejpam-5535	22	21	ideal	ideal	NOUN
ejpam-5535	22	22	in	in	ADP
ejpam-5535	22	23	a	a	DET
ejpam-5535	22	24	generalized	generalized	ADJ
ejpam-5535	22	25	distributive	distributive	ADJ
ejpam-5535	22	26	lattice	lattice	NOUN
ejpam-5535	22	27	and	and	CCONJ
ejpam-5535	22	28	obtain	obtain	VERB
ejpam-5535	22	29	some	some	DET
ejpam-5535	22	30	properties	property	NOUN
ejpam-5535	22	31	.	.	PUNCT
ejpam-5535	23	1	also	also	ADV
ejpam-5535	23	2	,	,	PUNCT
ejpam-5535	23	3	we	we	PRON
ejpam-5535	23	4	discuss	discuss	VERB
ejpam-5535	23	5	prime	prime	ADJ
ejpam-5535	23	6	filters	filter	NOUN
ejpam-5535	23	7	corresponding	correspond	VERB
ejpam-5535	23	8	to	to	ADP
ejpam-5535	23	9	an	an	DET
ejpam-5535	23	10	ideal	ideal	NOUN
ejpam-5535	23	11	in	in	ADP
ejpam-5535	23	12	a	a	DET
ejpam-5535	23	13	generalized	generalize	VERB
ejpam-5535	23	14	complemented	complement	VERB
ejpam-5535	23	15	distributive	distributive	ADJ
ejpam-5535	23	16	lattice	lattice	NOUN
ejpam-5535	23	17	.	.	PUNCT
ejpam-5535	24	1	definition	definition	NOUN
ejpam-5535	24	2	2.1	2.1	NUM
ejpam-5535	24	3	.	.	PUNCT
ejpam-5535	25	1	[	[	X
ejpam-5535	25	2	8	8	NUM
ejpam-5535	25	3	]	]	PUNCT
ejpam-5535	25	4	a	a	DET
ejpam-5535	25	5	unary	unary	ADJ
ejpam-5535	25	6	operation	operation	NOUN
ejpam-5535	25	7	g	g	NOUN
ejpam-5535	25	8	on	on	ADP
ejpam-5535	25	9	a	a	DET
ejpam-5535	25	10	distributive	distributive	ADJ
ejpam-5535	25	11	lattice	lattice	NOUN
ejpam-5535	25	12	a	a	PRON
ejpam-5535	25	13	is	be	AUX
ejpam-5535	25	14	said	say	VERB
ejpam-5535	25	15	to	to	PART
ejpam-5535	25	16	be	be	AUX
ejpam-5535	25	17	a	a	DET
ejpam-5535	25	18	gcomplementation	gcomplementation	NOUN
ejpam-5535	25	19	(	(	PUNCT
ejpam-5535	25	20	generalized	generalized	ADJ
ejpam-5535	25	21	complementation	complementation	NOUN
ejpam-5535	25	22	)	)	PUNCT
ejpam-5535	25	23	if	if	SCONJ
ejpam-5535	25	24	for	for	ADP
ejpam-5535	25	25	any	any	DET
ejpam-5535	25	26	v	v	NOUN
ejpam-5535	25	27	,	,	PUNCT
ejpam-5535	25	28	w	w	PROPN
ejpam-5535	25	29	∈	∈	PROPN
ejpam-5535	25	30	a	a	PRON
ejpam-5535	25	31	,	,	PUNCT
ejpam-5535	25	32	v	v	ADP
ejpam-5535	25	33	∨	∨	NOUN
ejpam-5535	25	34	vg	vg	ADP
ejpam-5535	25	35	∈	∈	PROPN
ejpam-5535	25	36	d	d	NOUN
ejpam-5535	25	37	,	,	PUNCT
ejpam-5535	25	38	and	and	CCONJ
ejpam-5535	25	39	v	v	ADP
ejpam-5535	25	40	∨w	∨w	NOUN
ejpam-5535	25	41	∈	∈	PROPN
ejpam-5535	25	42	d	d	NOUN
ejpam-5535	25	43	if	if	SCONJ
ejpam-5535	25	44	and	and	CCONJ
ejpam-5535	25	45	only	only	ADV
ejpam-5535	25	46	if	if	SCONJ
ejpam-5535	25	47	vg	vg	ADP
ejpam-5535	25	48	≤	≤	NOUN
ejpam-5535	25	49	w.	w.	NOUN
ejpam-5535	25	50	in	in	ADP
ejpam-5535	25	51	this	this	DET
ejpam-5535	25	52	case	case	NOUN
ejpam-5535	25	53	,	,	PUNCT
ejpam-5535	25	54	vg	vg	NOUN
ejpam-5535	25	55	is	be	AUX
ejpam-5535	25	56	said	say	VERB
ejpam-5535	25	57	to	to	PART
ejpam-5535	25	58	be	be	AUX
ejpam-5535	25	59	a	a	DET
ejpam-5535	25	60	g	g	NOUN
ejpam-5535	25	61	-	-	NOUN
ejpam-5535	25	62	complement	complement	NOUN
ejpam-5535	25	63	of	of	ADP
ejpam-5535	25	64	v	v	NOUN
ejpam-5535	25	65	,	,	PUNCT
ejpam-5535	25	66	and	and	CCONJ
ejpam-5535	25	67	a	a	PRON
ejpam-5535	25	68	is	be	AUX
ejpam-5535	25	69	called	call	VERB
ejpam-5535	25	70	a	a	DET
ejpam-5535	25	71	g	g	NOUN
ejpam-5535	25	72	-	-	PUNCT
ejpam-5535	25	73	complemented	complement	VERB
ejpam-5535	25	74	distributive	distributive	ADJ
ejpam-5535	25	75	lattice	lattice	NOUN
ejpam-5535	25	76	.	.	PUNCT
ejpam-5535	26	1	let	let	VERB
ejpam-5535	26	2	us	we	PRON
ejpam-5535	26	3	consider	consider	VERB
ejpam-5535	26	4	a	a	DET
ejpam-5535	26	5	means	means	NOUN
ejpam-5535	26	6	that	that	SCONJ
ejpam-5535	26	7	it	it	PRON
ejpam-5535	26	8	is	be	AUX
ejpam-5535	26	9	a	a	DET
ejpam-5535	26	10	distributive	distributive	ADJ
ejpam-5535	26	11	lattice	lattice	NOUN
ejpam-5535	26	12	with	with	ADP
ejpam-5535	26	13	dense	dense	ADJ
ejpam-5535	26	14	elements	element	NOUN
ejpam-5535	26	15	and	and	CCONJ
ejpam-5535	26	16	g	g	NOUN
ejpam-5535	26	17	is	be	AUX
ejpam-5535	26	18	a	a	DET
ejpam-5535	26	19	generalized	generalized	ADJ
ejpam-5535	26	20	complementation	complementation	NOUN
ejpam-5535	26	21	on	on	ADP
ejpam-5535	26	22	a.	a.	NOUN
ejpam-5535	26	23	proposition	proposition	NOUN
ejpam-5535	26	24	2.1	2.1	NUM
ejpam-5535	26	25	.	.	PUNCT
ejpam-5535	27	1	[	[	X
ejpam-5535	27	2	8	8	NUM
ejpam-5535	27	3	]	]	PUNCT
ejpam-5535	27	4	for	for	ADP
ejpam-5535	27	5	any	any	DET
ejpam-5535	27	6	g	g	NOUN
ejpam-5535	27	7	-	-	PUNCT
ejpam-5535	27	8	complementation	complementation	NOUN
ejpam-5535	27	9	g	g	NOUN
ejpam-5535	27	10	on	on	ADP
ejpam-5535	27	11	a	a	DET
ejpam-5535	27	12	,	,	PUNCT
ejpam-5535	27	13	(	(	PUNCT
ejpam-5535	27	14	i	i	NOUN
ejpam-5535	27	15	)	)	PUNCT
ejpam-5535	27	16	0	0	PUNCT
ejpam-5535	27	17	g	g	NOUN
ejpam-5535	27	18	∈	∈	PROPN
ejpam-5535	27	19	d	d	X
ejpam-5535	27	20	(	(	PUNCT
ejpam-5535	27	21	ii	ii	NOUN
ejpam-5535	27	22	)	)	PUNCT
ejpam-5535	27	23	d	d	NOUN
ejpam-5535	27	24	∈	∈	PROPN
ejpam-5535	27	25	d	d	X
ejpam-5535	27	26	⇒	⇒	NOUN
ejpam-5535	27	27	dg	dg	PROPN
ejpam-5535	27	28	=	=	SYM
ejpam-5535	27	29	0	0	NUM
ejpam-5535	27	30	(	(	PUNCT
ejpam-5535	27	31	iii	iii	NOUN
ejpam-5535	27	32	)	)	PUNCT
ejpam-5535	27	33	v	v	NOUN
ejpam-5535	27	34	≤	≤	NUM
ejpam-5535	27	35	w	w	NOUN
ejpam-5535	27	36	⇒	⇒	NOUN
ejpam-5535	27	37	wg	wg	VERB
ejpam-5535	27	38	≤	≤	ADJ
ejpam-5535	27	39	vg	vg	ADV
ejpam-5535	27	40	(	(	PUNCT
ejpam-5535	27	41	iv	iv	X
ejpam-5535	27	42	)	)	PUNCT
ejpam-5535	27	43	vgg	vgg	ADJ
ejpam-5535	27	44	≤	≤	NUM
ejpam-5535	27	45	v	v	ADP
ejpam-5535	27	46	(	(	PUNCT
ejpam-5535	27	47	v	v	NOUN
ejpam-5535	27	48	)	)	PUNCT
ejpam-5535	27	49	vggg	vggg	NOUN
ejpam-5535	27	50	=	=	SYM
ejpam-5535	27	51	vg	vg	NOUN
ejpam-5535	27	52	(	(	PUNCT
ejpam-5535	27	53	vi	vi	NOUN
ejpam-5535	27	54	)	)	PUNCT
ejpam-5535	27	55	0gg	0gg	NOUN
ejpam-5535	28	1	=	=	SYM
ejpam-5535	28	2	0	0	NUM
ejpam-5535	28	3	(	(	PUNCT
ejpam-5535	28	4	vii	vii	PROPN
ejpam-5535	28	5	)	)	PUNCT
ejpam-5535	28	6	v	v	ADP
ejpam-5535	28	7	∈	∈	PROPN
ejpam-5535	28	8	d	d	X
ejpam-5535	28	9	⇐	⇐	ADJ
ejpam-5535	28	10	⇒	⇒	NOUN
ejpam-5535	28	11	vg	vg	NOUN
ejpam-5535	28	12	=	=	SYM
ejpam-5535	28	13	0	0	NUM
ejpam-5535	28	14	⇐	⇐	ADJ
ejpam-5535	28	15	⇒	⇒	PROPN
ejpam-5535	28	16	vgg	vgg	PROPN
ejpam-5535	28	17	∈	∈	PROPN
ejpam-5535	29	1	d	d	X
ejpam-5535	29	2	r.	r.	PROPN
ejpam-5535	29	3	sirisetti	sirisetti	PROPN
ejpam-5535	29	4	et	et	PROPN
ejpam-5535	29	5	al	al	PROPN
ejpam-5535	29	6	.	.	PUNCT
ejpam-5535	29	7	/	/	SYM
ejpam-5535	29	8	eur	eur	PROPN
ejpam-5535	29	9	.	.	PUNCT
ejpam-5535	30	1	j.	j.	PROPN
ejpam-5535	30	2	pure	pure	PROPN
ejpam-5535	30	3	appl	appl	PROPN
ejpam-5535	30	4	.	.	PROPN
ejpam-5535	30	5	math	math	PROPN
ejpam-5535	30	6	,	,	PUNCT
ejpam-5535	30	7	18	18	NUM
ejpam-5535	30	8	(	(	PUNCT
ejpam-5535	30	9	1	1	NUM
ejpam-5535	30	10	)	)	PUNCT
ejpam-5535	30	11	(	(	PUNCT
ejpam-5535	30	12	2025	2025	NUM
ejpam-5535	30	13	)	)	PUNCT
ejpam-5535	30	14	,	,	PUNCT
ejpam-5535	30	15	5535	5535	NUM
ejpam-5535	30	16	3	3	NUM
ejpam-5535	30	17	of	of	ADP
ejpam-5535	30	18	12	12	NUM
ejpam-5535	30	19	(	(	PUNCT
ejpam-5535	30	20	viii	viii	NOUN
ejpam-5535	30	21	)	)	PUNCT
ejpam-5535	30	22	v	v	NOUN
ejpam-5535	30	23	≤	≤	NUM
ejpam-5535	30	24	0	0	NUM
ejpam-5535	30	25	g	g	NOUN
ejpam-5535	30	26	,	,	PUNCT
ejpam-5535	30	27	for	for	ADP
ejpam-5535	30	28	all	all	DET
ejpam-5535	30	29	v	v	NOUN
ejpam-5535	30	30	,	,	PUNCT
ejpam-5535	30	31	w	w	PROPN
ejpam-5535	30	32	∈	∈	PROPN
ejpam-5535	30	33	a.	a.	NOUN
ejpam-5535	30	34	proposition	proposition	NOUN
ejpam-5535	30	35	2.2	2.2	NUM
ejpam-5535	30	36	.	.	PUNCT
ejpam-5535	31	1	[	[	X
ejpam-5535	31	2	8	8	NUM
ejpam-5535	31	3	]	]	PUNCT
ejpam-5535	31	4	for	for	ADP
ejpam-5535	31	5	any	any	DET
ejpam-5535	31	6	g	g	NOUN
ejpam-5535	31	7	-	-	PUNCT
ejpam-5535	31	8	complementation	complementation	NOUN
ejpam-5535	31	9	g	g	NOUN
ejpam-5535	31	10	on	on	ADP
ejpam-5535	31	11	a	a	PRON
ejpam-5535	31	12	and	and	CCONJ
ejpam-5535	31	13	for	for	ADP
ejpam-5535	31	14	any	any	DET
ejpam-5535	31	15	v	v	NOUN
ejpam-5535	31	16	,	,	PUNCT
ejpam-5535	31	17	w	w	PROPN
ejpam-5535	31	18	∈	∈	PROPN
ejpam-5535	31	19	a	a	PRON
ejpam-5535	31	20	,	,	PUNCT
ejpam-5535	31	21	the	the	DET
ejpam-5535	31	22	following	follow	VERB
ejpam-5535	31	23	are	be	AUX
ejpam-5535	31	24	equivalent	equivalent	ADJ
ejpam-5535	31	25	;	;	PUNCT
ejpam-5535	31	26	(	(	PUNCT
ejpam-5535	31	27	i	i	NOUN
ejpam-5535	31	28	)	)	PUNCT
ejpam-5535	31	29	v	v	ADP
ejpam-5535	31	30	∨	∨	PROPN
ejpam-5535	31	31	w	w	PROPN
ejpam-5535	31	32	∈	∈	PROPN
ejpam-5535	31	33	d	d	PROPN
ejpam-5535	31	34	(	(	PUNCT
ejpam-5535	31	35	ii	ii	PROPN
ejpam-5535	31	36	)	)	PUNCT
ejpam-5535	31	37	vgg	vgg	NOUN
ejpam-5535	31	38	∨	∨	NUM
ejpam-5535	31	39	w	w	PROPN
ejpam-5535	31	40	∈	∈	PROPN
ejpam-5535	31	41	d	d	PROPN
ejpam-5535	31	42	(	(	PUNCT
ejpam-5535	31	43	iii	iii	NOUN
ejpam-5535	31	44	)	)	PUNCT
ejpam-5535	31	45	vgg	vgg	NOUN
ejpam-5535	31	46	∨	∨	NUM
ejpam-5535	31	47	wgg	wgg	PROPN
ejpam-5535	32	1	∈	∈	PROPN
ejpam-5535	32	2	d	d	PROPN
ejpam-5535	32	3	(	(	PUNCT
ejpam-5535	32	4	iv	iv	X
ejpam-5535	32	5	)	)	PUNCT
ejpam-5535	32	6	v	v	ADP
ejpam-5535	32	7	∨	∨	PROPN
ejpam-5535	32	8	wgg	wgg	PROPN
ejpam-5535	32	9	∈	∈	PROPN
ejpam-5535	32	10	d.	d.	PROPN
ejpam-5535	32	11	proposition	proposition	NOUN
ejpam-5535	32	12	2.3	2.3	NUM
ejpam-5535	32	13	.	.	PUNCT
ejpam-5535	33	1	[	[	X
ejpam-5535	33	2	8	8	NUM
ejpam-5535	33	3	]	]	PUNCT
ejpam-5535	33	4	for	for	ADP
ejpam-5535	33	5	any	any	DET
ejpam-5535	33	6	v	v	NOUN
ejpam-5535	33	7	,	,	PUNCT
ejpam-5535	33	8	w	w	PROPN
ejpam-5535	33	9	∈	∈	PROPN
ejpam-5535	33	10	a	a	DET
ejpam-5535	33	11	,	,	PUNCT
ejpam-5535	33	12	(	(	PUNCT
ejpam-5535	33	13	i	i	NOUN
ejpam-5535	33	14	)	)	PUNCT
ejpam-5535	33	15	(	(	PUNCT
ejpam-5535	33	16	v	v	NUM
ejpam-5535	33	17	∧	∧	PROPN
ejpam-5535	33	18	w)g	w)g	PUNCT
ejpam-5535	33	19	=	=	PUNCT
ejpam-5535	34	1	vg	vg	ADP
ejpam-5535	34	2	∨	∨	NUM
ejpam-5535	34	3	wg	wg	PROPN
ejpam-5535	34	4	(	(	PUNCT
ejpam-5535	34	5	ii	ii	PROPN
ejpam-5535	34	6	)	)	PUNCT
ejpam-5535	34	7	(	(	PUNCT
ejpam-5535	34	8	v	v	NUM
ejpam-5535	34	9	∨	∨	NOUN
ejpam-5535	34	10	w)g	w)g	X
ejpam-5535	34	11	≤	≤	NOUN
ejpam-5535	34	12	vg	vg	ADP
ejpam-5535	34	13	∧	∧	PROPN
ejpam-5535	34	14	wg	wg	PROPN
ejpam-5535	34	15	(	(	PUNCT
ejpam-5535	34	16	iii	iii	NOUN
ejpam-5535	34	17	)	)	PUNCT
ejpam-5535	34	18	(	(	PUNCT
ejpam-5535	34	19	v	v	NOUN
ejpam-5535	34	20	∨	∨	NUM
ejpam-5535	34	21	w)gg	w)gg	PROPN
ejpam-5535	34	22	=	=	SYM
ejpam-5535	34	23	vgg	vgg	PROPN
ejpam-5535	34	24	∨	∨	NUM
ejpam-5535	34	25	wgg	wgg	NOUN
ejpam-5535	34	26	=	=	SYM
ejpam-5535	34	27	(	(	PUNCT
ejpam-5535	34	28	vg	vg	ADP
ejpam-5535	34	29	∧	∧	PROPN
ejpam-5535	34	30	wg)g	wg)g	PROPN
ejpam-5535	34	31	(	(	PUNCT
ejpam-5535	34	32	iv	iv	X
ejpam-5535	34	33	)	)	PUNCT
ejpam-5535	34	34	(	(	PUNCT
ejpam-5535	34	35	v	v	ADP
ejpam-5535	34	36	∧	∧	PROPN
ejpam-5535	34	37	w)gg	w)gg	PROPN
ejpam-5535	35	1	=	=	SYM
ejpam-5535	35	2	(	(	PUNCT
ejpam-5535	35	3	vg	vg	ADP
ejpam-5535	35	4	∨	∨	NUM
ejpam-5535	35	5	wg)g	wg)g	NOUN
ejpam-5535	35	6	=	=	SYM
ejpam-5535	35	7	(	(	PUNCT
ejpam-5535	35	8	vgg	vgg	INTJ
ejpam-5535	35	9	∧	∧	PROPN
ejpam-5535	35	10	wgg)gg	wgg)gg	NOUN
ejpam-5535	35	11	.	.	PUNCT
ejpam-5535	36	1	given	give	VERB
ejpam-5535	36	2	any	any	DET
ejpam-5535	36	3	ideal	ideal	NOUN
ejpam-5535	36	4	k	k	NOUN
ejpam-5535	36	5	of	of	ADP
ejpam-5535	36	6	a	a	PRON
ejpam-5535	36	7	,	,	PUNCT
ejpam-5535	36	8	consider	consider	VERB
ejpam-5535	36	9	a	a	DET
ejpam-5535	36	10	set	set	NOUN
ejpam-5535	36	11	g(k	g(k	NOUN
ejpam-5535	36	12	)	)	PUNCT
ejpam-5535	36	13	=	=	PRON
ejpam-5535	37	1	{	{	PUNCT
ejpam-5535	37	2	u	u	NOUN
ejpam-5535	37	3	∈	∈	PROPN
ejpam-5535	37	4	a	a	DET
ejpam-5535	37	5	|	|	NOUN
ejpam-5535	37	6	ug	ug	ADP
ejpam-5535	37	7	∈	∈	PROPN
ejpam-5535	37	8	k	k	X
ejpam-5535	37	9	}	}	PUNCT
ejpam-5535	37	10	containing	contain	VERB
ejpam-5535	37	11	d	d	NOUN
ejpam-5535	37	12	(	(	PUNCT
ejpam-5535	37	13	because	because	SCONJ
ejpam-5535	37	14	dg	dg	X
ejpam-5535	37	15	=	=	SYM
ejpam-5535	37	16	0	0	PUNCT
ejpam-5535	37	17	∈	∈	PROPN
ejpam-5535	37	18	k	k	NOUN
ejpam-5535	37	19	,	,	PUNCT
ejpam-5535	37	20	for	for	ADP
ejpam-5535	37	21	all	all	DET
ejpam-5535	37	22	d	d	PROPN
ejpam-5535	37	23	∈	∈	PROPN
ejpam-5535	37	24	d	d	NOUN
ejpam-5535	37	25	)	)	PUNCT
ejpam-5535	37	26	.	.	PUNCT
ejpam-5535	38	1	lemma	lemma	PROPN
ejpam-5535	38	2	2.2	2.2	NUM
ejpam-5535	38	3	.	.	PUNCT
ejpam-5535	39	1	for	for	ADP
ejpam-5535	39	2	any	any	DET
ejpam-5535	39	3	ideal	ideal	NOUN
ejpam-5535	39	4	k	k	PROPN
ejpam-5535	39	5	of	of	ADP
ejpam-5535	39	6	a	a	DET
ejpam-5535	39	7	,	,	PUNCT
ejpam-5535	39	8	g(k	g(k	NOUN
ejpam-5535	39	9	)	)	PUNCT
ejpam-5535	39	10	is	be	AUX
ejpam-5535	39	11	a	a	DET
ejpam-5535	39	12	filter	filter	NOUN
ejpam-5535	39	13	.	.	PUNCT
ejpam-5535	40	1	proof	proof	NOUN
ejpam-5535	40	2	.	.	PUNCT
ejpam-5535	41	1	let	let	VERB
ejpam-5535	41	2	v	v	NOUN
ejpam-5535	41	3	,	,	PUNCT
ejpam-5535	41	4	w	w	PROPN
ejpam-5535	41	5	∈	∈	PROPN
ejpam-5535	41	6	g(k	g(k	NOUN
ejpam-5535	41	7	)	)	PUNCT
ejpam-5535	41	8	.	.	PUNCT
ejpam-5535	42	1	then	then	ADV
ejpam-5535	42	2	vg	vg	VERB
ejpam-5535	42	3	,	,	PUNCT
ejpam-5535	42	4	wg	wg	PROPN
ejpam-5535	42	5	∈	∈	PROPN
ejpam-5535	42	6	k.	k.	PROPN
ejpam-5535	43	1	by	by	ADP
ejpam-5535	43	2	proposition	proposition	NOUN
ejpam-5535	43	3	2.3	2.3	NUM
ejpam-5535	43	4	(	(	PUNCT
ejpam-5535	43	5	i	i	NOUN
ejpam-5535	43	6	)	)	PUNCT
ejpam-5535	43	7	,	,	PUNCT
ejpam-5535	43	8	(	(	PUNCT
ejpam-5535	43	9	v∧w)g	v∧w)g	ADV
ejpam-5535	43	10	=	=	SYM
ejpam-5535	43	11	vg∨wg	vg∨wg	NOUN
ejpam-5535	43	12	∈	∈	ADJ
ejpam-5535	43	13	k	k	X
ejpam-5535	43	14	(	(	PUNCT
ejpam-5535	43	15	since	since	SCONJ
ejpam-5535	43	16	k	k	PROPN
ejpam-5535	43	17	is	be	AUX
ejpam-5535	43	18	ideal	ideal	ADJ
ejpam-5535	43	19	)	)	PUNCT
ejpam-5535	43	20	.	.	PUNCT
ejpam-5535	44	1	therefore	therefore	ADV
ejpam-5535	44	2	v	v	VERB
ejpam-5535	44	3	∧w	∧w	NOUN
ejpam-5535	44	4	∈	∈	PROPN
ejpam-5535	44	5	g(k	g(k	PROPN
ejpam-5535	44	6	)	)	PUNCT
ejpam-5535	44	7	.	.	PUNCT
ejpam-5535	45	1	given	give	VERB
ejpam-5535	45	2	u	u	PRON
ejpam-5535	45	3	∈	∈	PROPN
ejpam-5535	45	4	a	a	PRON
ejpam-5535	45	5	,	,	PUNCT
ejpam-5535	45	6	we	we	PRON
ejpam-5535	45	7	have	have	VERB
ejpam-5535	45	8	v	v	NOUN
ejpam-5535	45	9	,	,	PUNCT
ejpam-5535	45	10	u	u	PROPN
ejpam-5535	45	11	≤	≤	PROPN
ejpam-5535	45	12	v	v	ADP
ejpam-5535	45	13	∨	∨	NUM
ejpam-5535	45	14	u	u	NOUN
ejpam-5535	45	15	,	,	PUNCT
ejpam-5535	45	16	which	which	PRON
ejpam-5535	45	17	implies	imply	VERB
ejpam-5535	45	18	(	(	PUNCT
ejpam-5535	45	19	v	v	NOUN
ejpam-5535	45	20	∨	∨	NOUN
ejpam-5535	45	21	u)g	u)g	NOUN
ejpam-5535	45	22	≤	≤	NOUN
ejpam-5535	45	23	vg	vg	NOUN
ejpam-5535	45	24	,	,	PUNCT
ejpam-5535	45	25	ug	ug	X
ejpam-5535	45	26	(	(	PUNCT
ejpam-5535	45	27	by	by	ADP
ejpam-5535	45	28	proposition	proposition	NOUN
ejpam-5535	45	29	2.1(iii	2.1(iii	NUM
ejpam-5535	45	30	)	)	PUNCT
ejpam-5535	45	31	)	)	PUNCT
ejpam-5535	46	1	and	and	CCONJ
ejpam-5535	46	2	then	then	ADV
ejpam-5535	46	3	(	(	PUNCT
ejpam-5535	46	4	v	v	NOUN
ejpam-5535	46	5	∨	∨	NOUN
ejpam-5535	46	6	u)g	u)g	NOUN
ejpam-5535	46	7	≤	≤	NOUN
ejpam-5535	46	8	vg	vg	ADP
ejpam-5535	46	9	∧	∧	PROPN
ejpam-5535	46	10	ug	ug	ADP
ejpam-5535	46	11	∈	∈	PROPN
ejpam-5535	46	12	k	k	PROPN
ejpam-5535	46	13	(	(	PUNCT
ejpam-5535	46	14	since	since	SCONJ
ejpam-5535	46	15	vg	vg	PROPN
ejpam-5535	46	16	∈	∈	PROPN
ejpam-5535	46	17	k	k	NOUN
ejpam-5535	46	18	,	,	PUNCT
ejpam-5535	46	19	ug	ug	ADP
ejpam-5535	46	20	∈	∈	PROPN
ejpam-5535	46	21	a	a	PRON
ejpam-5535	46	22	,	,	PUNCT
ejpam-5535	46	23	and	and	CCONJ
ejpam-5535	46	24	k	k	PROPN
ejpam-5535	46	25	is	be	AUX
ejpam-5535	46	26	an	an	DET
ejpam-5535	46	27	ideal	ideal	NOUN
ejpam-5535	46	28	)	)	PUNCT
ejpam-5535	46	29	.	.	PUNCT
ejpam-5535	47	1	therefore	therefore	ADV
ejpam-5535	47	2	v	v	X
ejpam-5535	47	3	∨	∨	NUM
ejpam-5535	47	4	u	u	PROPN
ejpam-5535	47	5	∈	∈	PROPN
ejpam-5535	47	6	g(k	g(k	NOUN
ejpam-5535	47	7	)	)	PUNCT
ejpam-5535	47	8	.	.	PUNCT
ejpam-5535	48	1	thus	thus	ADV
ejpam-5535	48	2	g(k	g(k	VERB
ejpam-5535	48	3	)	)	PUNCT
ejpam-5535	48	4	is	be	AUX
ejpam-5535	48	5	filter	filter	NOUN
ejpam-5535	48	6	.	.	PUNCT
ejpam-5535	49	1	remark	remark	VERB
ejpam-5535	49	2	2.3	2.3	NUM
ejpam-5535	49	3	.	.	PUNCT
ejpam-5535	50	1	d	d	X
ejpam-5535	50	2	=	=	SYM
ejpam-5535	50	3	g(0	g(0	PROPN
ejpam-5535	50	4	)	)	PUNCT
ejpam-5535	50	5	=	=	SYM
ejpam-5535	50	6	g((0	g((0	PROPN
ejpam-5535	50	7	]	]	PUNCT
ejpam-5535	50	8	)	)	PUNCT
ejpam-5535	50	9	and	and	CCONJ
ejpam-5535	50	10	g(a	g(a	PROPN
ejpam-5535	50	11	)	)	PUNCT
ejpam-5535	51	1	=	=	PUNCT
ejpam-5535	51	2	a.	a.	NOUN
ejpam-5535	51	3	lemma	lemma	PROPN
ejpam-5535	51	4	2.4	2.4	NUM
ejpam-5535	51	5	.	.	PUNCT
ejpam-5535	52	1	for	for	ADP
ejpam-5535	52	2	any	any	DET
ejpam-5535	52	3	ideals	ideal	NOUN
ejpam-5535	52	4	k	k	NOUN
ejpam-5535	52	5	,	,	PUNCT
ejpam-5535	52	6	h	h	NOUN
ejpam-5535	52	7	of	of	ADP
ejpam-5535	52	8	a	a	PRON
ejpam-5535	52	9	,	,	PUNCT
ejpam-5535	52	10	we	we	PRON
ejpam-5535	52	11	have	have	VERB
ejpam-5535	52	12	;	;	PUNCT
ejpam-5535	52	13	(	(	PUNCT
ejpam-5535	52	14	i	i	NOUN
ejpam-5535	52	15	)	)	PUNCT
ejpam-5535	53	1	k	k	NOUN
ejpam-5535	54	1	⊆	⊆	NUM
ejpam-5535	54	2	h	h	NOUN
ejpam-5535	54	3	implies	imply	VERB
ejpam-5535	54	4	g(k	g(k	NOUN
ejpam-5535	54	5	)	)	PUNCT
ejpam-5535	54	6	⊆	⊆	NUM
ejpam-5535	54	7	g(h	g(h	NUM
ejpam-5535	54	8	)	)	PUNCT
ejpam-5535	54	9	.	.	PUNCT
ejpam-5535	55	1	(	(	PUNCT
ejpam-5535	55	2	ii	ii	NOUN
ejpam-5535	55	3	)	)	PUNCT
ejpam-5535	55	4	g(k	g(k	NOUN
ejpam-5535	55	5	)	)	PUNCT
ejpam-5535	55	6	∩g(h	∩g(h	ADJ
ejpam-5535	55	7	)	)	PUNCT
ejpam-5535	55	8	=	=	PUNCT
ejpam-5535	55	9	g(k	g(k	VERB
ejpam-5535	55	10	∩h	∩h	NOUN
ejpam-5535	55	11	)	)	PUNCT
ejpam-5535	55	12	.	.	PUNCT
ejpam-5535	56	1	(	(	PUNCT
ejpam-5535	56	2	iii	iii	X
ejpam-5535	56	3	)	)	PUNCT
ejpam-5535	56	4	g(k	g(k	NOUN
ejpam-5535	56	5	)	)	PUNCT
ejpam-5535	56	6	∨g(h	∨g(h	PUNCT
ejpam-5535	56	7	)	)	PUNCT
ejpam-5535	56	8	⊆	⊆	NUM
ejpam-5535	56	9	g(k	g(k	NOUN
ejpam-5535	56	10	∨h	∨h	NOUN
ejpam-5535	56	11	)	)	PUNCT
ejpam-5535	56	12	.	.	PUNCT
ejpam-5535	57	1	proof	proof	NOUN
ejpam-5535	57	2	.	.	PUNCT
ejpam-5535	58	1	(	(	PUNCT
ejpam-5535	58	2	i	i	NOUN
ejpam-5535	58	3	)	)	PUNCT
ejpam-5535	58	4	suppose	suppose	VERB
ejpam-5535	58	5	that	that	SCONJ
ejpam-5535	58	6	k	k	PROPN
ejpam-5535	58	7	⊆	⊆	NUM
ejpam-5535	58	8	h	h	NOUN
ejpam-5535	58	9	and	and	CCONJ
ejpam-5535	58	10	u	u	PROPN
ejpam-5535	58	11	∈	∈	PROPN
ejpam-5535	58	12	g(k	g(k	PROPN
ejpam-5535	58	13	)	)	PUNCT
ejpam-5535	58	14	.	.	PUNCT
ejpam-5535	59	1	then	then	ADV
ejpam-5535	59	2	ug	ug	ADP
ejpam-5535	59	3	∈	∈	PROPN
ejpam-5535	59	4	k	k	PROPN
ejpam-5535	59	5	⊆	⊆	X
ejpam-5535	59	6	h.	h.	PROPN
ejpam-5535	59	7	therefore	therefore	ADV
ejpam-5535	59	8	u	u	X
ejpam-5535	59	9	∈	∈	PROPN
ejpam-5535	59	10	g(h	g(h	PROPN
ejpam-5535	59	11	)	)	PUNCT
ejpam-5535	59	12	and	and	CCONJ
ejpam-5535	59	13	hence	hence	ADV
ejpam-5535	59	14	g(k	g(k	VERB
ejpam-5535	59	15	)	)	PUNCT
ejpam-5535	59	16	⊆	⊆	NUM
ejpam-5535	59	17	g(h	g(h	NUM
ejpam-5535	59	18	)	)	PUNCT
ejpam-5535	59	19	.	.	PUNCT
ejpam-5535	60	1	(	(	PUNCT
ejpam-5535	60	2	ii	ii	NOUN
ejpam-5535	60	3	)	)	PUNCT
ejpam-5535	60	4	and	and	CCONJ
ejpam-5535	60	5	(	(	PUNCT
ejpam-5535	60	6	iii	iii	NOUN
ejpam-5535	60	7	)	)	PUNCT
ejpam-5535	60	8	follows	follow	VERB
ejpam-5535	60	9	from	from	ADP
ejpam-5535	60	10	(	(	PUNCT
ejpam-5535	60	11	i	i	NOUN
ejpam-5535	60	12	)	)	PUNCT
ejpam-5535	60	13	.	.	PUNCT
ejpam-5535	61	1	lemma	lemma	PROPN
ejpam-5535	61	2	2.5	2.5	NUM
ejpam-5535	61	3	.	.	PUNCT
ejpam-5535	62	1	for	for	ADP
ejpam-5535	62	2	any	any	DET
ejpam-5535	62	3	ideal	ideal	NOUN
ejpam-5535	62	4	k	k	PROPN
ejpam-5535	62	5	of	of	ADP
ejpam-5535	62	6	a	a	DET
ejpam-5535	62	7	,	,	PUNCT
ejpam-5535	62	8	either	either	CCONJ
ejpam-5535	62	9	g(k	g(k	NOUN
ejpam-5535	62	10	)	)	PUNCT
ejpam-5535	62	11	∩k	∩k	NOUN
ejpam-5535	62	12	=	=	SYM
ejpam-5535	62	13	∅	∅	NOUN
ejpam-5535	62	14	or	or	CCONJ
ejpam-5535	62	15	g(k	g(k	VERB
ejpam-5535	62	16	)	)	PUNCT
ejpam-5535	62	17	=	=	PUNCT
ejpam-5535	62	18	a.	a.	PROPN
ejpam-5535	62	19	r.	r.	PROPN
ejpam-5535	62	20	sirisetti	sirisetti	PROPN
ejpam-5535	62	21	et	et	PROPN
ejpam-5535	62	22	al	al	PROPN
ejpam-5535	62	23	.	.	PUNCT
ejpam-5535	62	24	/	/	SYM
ejpam-5535	62	25	eur	eur	PROPN
ejpam-5535	62	26	.	.	PUNCT
ejpam-5535	63	1	j.	j.	PROPN
ejpam-5535	63	2	pure	pure	PROPN
ejpam-5535	63	3	appl	appl	PROPN
ejpam-5535	63	4	.	.	PROPN
ejpam-5535	63	5	math	math	PROPN
ejpam-5535	63	6	,	,	PUNCT
ejpam-5535	63	7	18	18	NUM
ejpam-5535	63	8	(	(	PUNCT
ejpam-5535	63	9	1	1	NUM
ejpam-5535	63	10	)	)	PUNCT
ejpam-5535	63	11	(	(	PUNCT
ejpam-5535	63	12	2025	2025	NUM
ejpam-5535	63	13	)	)	PUNCT
ejpam-5535	63	14	,	,	PUNCT
ejpam-5535	63	15	5535	5535	NUM
ejpam-5535	63	16	4	4	NUM
ejpam-5535	63	17	of	of	ADP
ejpam-5535	63	18	12	12	NUM
ejpam-5535	63	19	proof	proof	NOUN
ejpam-5535	63	20	.	.	PUNCT
ejpam-5535	63	21	suppose	suppose	VERB
ejpam-5535	63	22	that	that	SCONJ
ejpam-5535	63	23	g(k	g(k	NOUN
ejpam-5535	63	24	)	)	PUNCT
ejpam-5535	63	25	∩k	∩k	NOUN
ejpam-5535	63	26	̸=	̸=	PROPN
ejpam-5535	63	27	ϕ.	ϕ.	NOUN
ejpam-5535	63	28	let	let	VERB
ejpam-5535	63	29	x	x	X
ejpam-5535	63	30	∈	∈	PROPN
ejpam-5535	63	31	g(k	g(k	PROPN
ejpam-5535	63	32	)	)	PUNCT
ejpam-5535	63	33	∩k	∩k	NOUN
ejpam-5535	63	34	.	.	PUNCT
ejpam-5535	64	1	then	then	ADV
ejpam-5535	64	2	xg	xg	PROPN
ejpam-5535	64	3	∈	∈	PROPN
ejpam-5535	64	4	k	k	PROPN
ejpam-5535	64	5	,	,	PUNCT
ejpam-5535	64	6	and	and	CCONJ
ejpam-5535	64	7	x	x	X
ejpam-5535	64	8	∈	∈	PROPN
ejpam-5535	64	9	k.	k.	NOUN
ejpam-5535	65	1	therefore	therefore	ADV
ejpam-5535	65	2	x	x	PROPN
ejpam-5535	65	3	∨	∨	NUM
ejpam-5535	65	4	xg	xg	PROPN
ejpam-5535	65	5	∈	∈	PROPN
ejpam-5535	65	6	d	d	PROPN
ejpam-5535	65	7	∩k	∩k	NOUN
ejpam-5535	65	8	(	(	PUNCT
ejpam-5535	65	9	since	since	SCONJ
ejpam-5535	65	10	k	k	PROPN
ejpam-5535	65	11	is	be	AUX
ejpam-5535	65	12	an	an	DET
ejpam-5535	65	13	ideal	ideal	NOUN
ejpam-5535	65	14	)	)	PUNCT
ejpam-5535	65	15	.	.	PUNCT
ejpam-5535	66	1	since	since	SCONJ
ejpam-5535	66	2	0	0	NUM
ejpam-5535	66	3	g	g	PROPN
ejpam-5535	66	4	≤	≤	NUM
ejpam-5535	66	5	d	d	NOUN
ejpam-5535	66	6	for	for	ADP
ejpam-5535	66	7	all	all	DET
ejpam-5535	66	8	d	d	PROPN
ejpam-5535	66	9	∈	∈	PROPN
ejpam-5535	66	10	d	d	PROPN
ejpam-5535	66	11	,	,	PUNCT
ejpam-5535	66	12	0	0	NUM
ejpam-5535	66	13	g	g	NOUN
ejpam-5535	66	14	≤	≤	NUM
ejpam-5535	66	15	x	x	PUNCT
ejpam-5535	66	16	∨	∨	NUM
ejpam-5535	66	17	xg	xg	PROPN
ejpam-5535	66	18	.	.	PUNCT
ejpam-5535	67	1	therefore	therefore	ADV
ejpam-5535	67	2	0	0	NUM
ejpam-5535	67	3	in	in	ADP
ejpam-5535	67	4	g(k	g(k	NOUN
ejpam-5535	67	5	)	)	PUNCT
ejpam-5535	67	6	and	and	CCONJ
ejpam-5535	67	7	hence	hence	ADV
ejpam-5535	67	8	g(k	g(k	VERB
ejpam-5535	67	9	)	)	PUNCT
ejpam-5535	67	10	=	=	SYM
ejpam-5535	68	1	a	a	PRON
ejpam-5535	68	2	(	(	PUNCT
ejpam-5535	68	3	since	since	SCONJ
ejpam-5535	68	4	g(k	g(k	NOUN
ejpam-5535	68	5	)	)	PUNCT
ejpam-5535	68	6	is	be	AUX
ejpam-5535	68	7	filter	filter	NOUN
ejpam-5535	68	8	)	)	PUNCT
ejpam-5535	68	9	.	.	PUNCT
ejpam-5535	69	1	lemma	lemma	PROPN
ejpam-5535	69	2	2.6	2.6	NUM
ejpam-5535	69	3	.	.	PUNCT
ejpam-5535	70	1	for	for	ADP
ejpam-5535	70	2	any	any	DET
ejpam-5535	70	3	ideal	ideal	NOUN
ejpam-5535	70	4	k	k	PROPN
ejpam-5535	70	5	of	of	ADP
ejpam-5535	70	6	a	a	PRON
ejpam-5535	70	7	and	and	CCONJ
ejpam-5535	70	8	u	u	NOUN
ejpam-5535	70	9	∈	∈	PROPN
ejpam-5535	70	10	a	a	DET
ejpam-5535	70	11	,	,	PUNCT
ejpam-5535	70	12	(	(	PUNCT
ejpam-5535	70	13	i	i	NOUN
ejpam-5535	70	14	)	)	PUNCT
ejpam-5535	70	15	u	u	PROPN
ejpam-5535	70	16	∈	∈	PROPN
ejpam-5535	70	17	g(k	g(k	NOUN
ejpam-5535	70	18	)	)	PUNCT
ejpam-5535	70	19	if	if	SCONJ
ejpam-5535	70	20	and	and	CCONJ
ejpam-5535	70	21	only	only	ADV
ejpam-5535	70	22	if	if	SCONJ
ejpam-5535	70	23	ugg	ugg	PROPN
ejpam-5535	70	24	∈	∈	PROPN
ejpam-5535	70	25	g(k	g(k	PROPN
ejpam-5535	70	26	)	)	PUNCT
ejpam-5535	70	27	(	(	PUNCT
ejpam-5535	70	28	ii	ii	NOUN
ejpam-5535	70	29	)	)	PUNCT
ejpam-5535	70	30	u	u	NOUN
ejpam-5535	70	31	∈	∈	PROPN
ejpam-5535	70	32	k	k	PROPN
ejpam-5535	70	33	implies	imply	VERB
ejpam-5535	70	34	ug	ug	ADP
ejpam-5535	70	35	∈	∈	PROPN
ejpam-5535	70	36	g(k	g(k	NOUN
ejpam-5535	70	37	)	)	PUNCT
ejpam-5535	70	38	.	.	PUNCT
ejpam-5535	71	1	proof	proof	NOUN
ejpam-5535	71	2	.	.	PUNCT
ejpam-5535	72	1	(	(	PUNCT
ejpam-5535	72	2	i	i	NOUN
ejpam-5535	72	3	)	)	PUNCT
ejpam-5535	72	4	follows	follow	VERB
ejpam-5535	72	5	from	from	ADP
ejpam-5535	72	6	proposition	proposition	NOUN
ejpam-5535	72	7	2.1(v	2.1(v	NUM
ejpam-5535	72	8	)	)	PUNCT
ejpam-5535	72	9	.	.	PUNCT
ejpam-5535	73	1	(	(	PUNCT
ejpam-5535	73	2	ii	ii	NOUN
ejpam-5535	73	3	)	)	PUNCT
ejpam-5535	73	4	follows	follow	VERB
ejpam-5535	73	5	from	from	ADP
ejpam-5535	73	6	proposition	proposition	NOUN
ejpam-5535	73	7	2.1(iv	2.1(iv	NUM
ejpam-5535	73	8	)	)	PUNCT
ejpam-5535	73	9	.	.	PUNCT
ejpam-5535	74	1	the	the	DET
ejpam-5535	74	2	converse	converse	NOUN
ejpam-5535	74	3	of	of	ADP
ejpam-5535	74	4	lemma	lemma	PROPN
ejpam-5535	74	5	2.6(ii	2.6(ii	NUM
ejpam-5535	74	6	)	)	PUNCT
ejpam-5535	74	7	need	need	AUX
ejpam-5535	74	8	not	not	PART
ejpam-5535	74	9	be	be	AUX
ejpam-5535	74	10	hold	hold	ADJ
ejpam-5535	74	11	.	.	PUNCT
ejpam-5535	75	1	check	check	VERB
ejpam-5535	75	2	out	out	ADP
ejpam-5535	75	3	the	the	DET
ejpam-5535	75	4	following	follow	VERB
ejpam-5535	75	5	example	example	NOUN
ejpam-5535	75	6	.	.	PUNCT
ejpam-5535	76	1	example	example	NOUN
ejpam-5535	76	2	2.7	2.7	NUM
ejpam-5535	76	3	.	.	PUNCT
ejpam-5535	77	1	consider	consider	VERB
ejpam-5535	77	2	a	a	DET
ejpam-5535	77	3	distributive	distributive	ADJ
ejpam-5535	77	4	lattice	lattice	NOUN
ejpam-5535	77	5	a	a	DET
ejpam-5535	77	6	=	=	PUNCT
ejpam-5535	77	7	{	{	PUNCT
ejpam-5535	77	8	0	0	NUM
ejpam-5535	77	9	,	,	PUNCT
ejpam-5535	77	10	i	i	PRON
ejpam-5535	77	11	,	,	PUNCT
ejpam-5535	77	12	j	j	PROPN
ejpam-5535	77	13	,	,	PUNCT
ejpam-5535	77	14	k	k	PROPN
ejpam-5535	77	15	,	,	PUNCT
ejpam-5535	77	16	1	1	NUM
ejpam-5535	77	17	}	}	PUNCT
ejpam-5535	77	18	,	,	PUNCT
ejpam-5535	77	19	whose	whose	DET
ejpam-5535	77	20	hasse	hasse	NOUN
ejpam-5535	77	21	diagram	diagram	NOUN
ejpam-5535	77	22	is	be	AUX
ejpam-5535	77	23	showing	show	VERB
ejpam-5535	77	24	in	in	ADP
ejpam-5535	77	25	below	below	ADV
ejpam-5535	77	26	;	;	PUNCT
ejpam-5535	78	1	k	k	PROPN
ejpam-5535	78	2	i	i	PROPN
ejpam-5535	78	3	0	0	PUNCT
ejpam-5535	78	4	j	j	PROPN
ejpam-5535	78	5	1	1	NUM
ejpam-5535	78	6	for	for	ADP
ejpam-5535	78	7	the	the	DET
ejpam-5535	78	8	ideal	ideal	NOUN
ejpam-5535	78	9	k	k	PROPN
ejpam-5535	79	1	=	=	PUNCT
ejpam-5535	80	1	{	{	PUNCT
ejpam-5535	80	2	0	0	NUM
ejpam-5535	80	3	,	,	PUNCT
ejpam-5535	80	4	i	i	PRON
ejpam-5535	80	5	}	}	PUNCT
ejpam-5535	80	6	of	of	ADP
ejpam-5535	80	7	a	a	DET
ejpam-5535	80	8	and	and	CCONJ
ejpam-5535	80	9	0	0	NUM
ejpam-5535	80	10	g	g	NOUN
ejpam-5535	80	11	=	=	SYM
ejpam-5535	80	12	k	k	PROPN
ejpam-5535	80	13	,	,	PUNCT
ejpam-5535	80	14	ig	ig	PROPN
ejpam-5535	80	15	=	=	SYM
ejpam-5535	80	16	j	j	PROPN
ejpam-5535	80	17	,	,	PUNCT
ejpam-5535	80	18	jg	jg	PROPN
ejpam-5535	81	1	=	=	SYM
ejpam-5535	81	2	i	i	PROPN
ejpam-5535	81	3	,	,	PUNCT
ejpam-5535	81	4	kg	kg	PROPN
ejpam-5535	81	5	=	=	NOUN
ejpam-5535	81	6	1	1	NUM
ejpam-5535	81	7	g	g	NOUN
ejpam-5535	81	8	=	=	SYM
ejpam-5535	81	9	0	0	NUM
ejpam-5535	81	10	,	,	PUNCT
ejpam-5535	81	11	we	we	PRON
ejpam-5535	81	12	have	have	AUX
ejpam-5535	81	13	g(k	g(k	NOUN
ejpam-5535	81	14	)	)	PUNCT
ejpam-5535	81	15	=	=	PRON
ejpam-5535	81	16	{	{	PUNCT
ejpam-5535	81	17	j	j	PROPN
ejpam-5535	81	18	,	,	PUNCT
ejpam-5535	81	19	k	k	PROPN
ejpam-5535	81	20	,	,	PUNCT
ejpam-5535	81	21	1	1	NUM
ejpam-5535	81	22	}	}	PUNCT
ejpam-5535	81	23	,	,	PUNCT
ejpam-5535	81	24	and	and	CCONJ
ejpam-5535	81	25	kg	kg	X
ejpam-5535	82	1	=	=	SYM
ejpam-5535	82	2	0	0	PUNCT
ejpam-5535	82	3	=	=	SYM
ejpam-5535	82	4	1	1	NUM
ejpam-5535	82	5	g	g	NOUN
ejpam-5535	82	6	∈	∈	PROPN
ejpam-5535	82	7	g(k	g(k	NOUN
ejpam-5535	82	8	)	)	PUNCT
ejpam-5535	82	9	,	,	PUNCT
ejpam-5535	82	10	but	but	CCONJ
ejpam-5535	82	11	k	k	X
ejpam-5535	82	12	,	,	PUNCT
ejpam-5535	82	13	1	1	NUM
ejpam-5535	82	14	/∈	/∈	PUNCT
ejpam-5535	83	1	k.	k.	PROPN
ejpam-5535	83	2	lemma	lemma	PROPN
ejpam-5535	84	1	2.8	2.8	NUM
ejpam-5535	84	2	.	.	PUNCT
ejpam-5535	85	1	for	for	ADP
ejpam-5535	85	2	any	any	DET
ejpam-5535	85	3	ideal	ideal	NOUN
ejpam-5535	85	4	k	k	PROPN
ejpam-5535	85	5	of	of	ADP
ejpam-5535	85	6	a	a	PRON
ejpam-5535	85	7	,	,	PUNCT
ejpam-5535	85	8	k	k	PROPN
ejpam-5535	85	9	∩d	∩d	NOUN
ejpam-5535	85	10	̸=	̸=	PROPN
ejpam-5535	85	11	ϕ	ϕ	NOUN
ejpam-5535	86	1	if	if	SCONJ
ejpam-5535	86	2	and	and	CCONJ
ejpam-5535	86	3	only	only	ADV
ejpam-5535	86	4	if	if	SCONJ
ejpam-5535	86	5	a	a	DET
ejpam-5535	86	6	=	=	X
ejpam-5535	86	7	g(k	g(k	NOUN
ejpam-5535	86	8	)	)	PUNCT
ejpam-5535	86	9	.	.	PUNCT
ejpam-5535	87	1	proof	proof	NOUN
ejpam-5535	87	2	.	.	PUNCT
ejpam-5535	88	1	suppose	suppose	VERB
ejpam-5535	88	2	that	that	SCONJ
ejpam-5535	88	3	k	k	PROPN
ejpam-5535	88	4	∩	∩	PROPN
ejpam-5535	88	5	d	d	PROPN
ejpam-5535	88	6	̸=	̸=	PROPN
ejpam-5535	88	7	ϕ.	ϕ.	NOUN
ejpam-5535	88	8	let	let	VERB
ejpam-5535	88	9	v	v	ADP
ejpam-5535	88	10	∈	∈	PROPN
ejpam-5535	88	11	k	k	PROPN
ejpam-5535	88	12	∩	∩	PROPN
ejpam-5535	88	13	d.	d.	PROPN
ejpam-5535	88	14	then	then	ADV
ejpam-5535	88	15	v	v	ADP
ejpam-5535	88	16	∈	∈	PROPN
ejpam-5535	88	17	k	k	NOUN
ejpam-5535	88	18	and	and	CCONJ
ejpam-5535	88	19	vg	vg	NOUN
ejpam-5535	88	20	=	=	SYM
ejpam-5535	88	21	0	0	NUM
ejpam-5535	88	22	∈	∈	PROPN
ejpam-5535	88	23	k.	k.	NOUN
ejpam-5535	88	24	therefore	therefore	ADV
ejpam-5535	88	25	v	v	ADP
ejpam-5535	88	26	∈	∈	PROPN
ejpam-5535	88	27	k	k	PROPN
ejpam-5535	88	28	∩	∩	ADJ
ejpam-5535	88	29	g(k	g(k	NOUN
ejpam-5535	88	30	)	)	PUNCT
ejpam-5535	88	31	.	.	PUNCT
ejpam-5535	89	1	so	so	ADV
ejpam-5535	89	2	that	that	SCONJ
ejpam-5535	89	3	k	k	PROPN
ejpam-5535	89	4	∩	∩	PROPN
ejpam-5535	89	5	g(k	g(k	NOUN
ejpam-5535	89	6	)	)	PUNCT
ejpam-5535	89	7	is	be	AUX
ejpam-5535	89	8	non	non	ADJ
ejpam-5535	89	9	-	-	ADJ
ejpam-5535	89	10	empty	empty	ADJ
ejpam-5535	89	11	.	.	PUNCT
ejpam-5535	90	1	by	by	ADP
ejpam-5535	90	2	lemma	lemma	PROPN
ejpam-5535	90	3	2.5	2.5	NUM
ejpam-5535	90	4	.	.	PUNCT
ejpam-5535	90	5	,	,	PUNCT
ejpam-5535	90	6	g(k	g(k	NOUN
ejpam-5535	90	7	)	)	PUNCT
ejpam-5535	90	8	=	=	VERB
ejpam-5535	90	9	a.	a.	NOUN
ejpam-5535	90	10	conversely	conversely	ADV
ejpam-5535	90	11	suppose	suppose	VERB
ejpam-5535	90	12	that	that	SCONJ
ejpam-5535	90	13	g(k	g(k	NOUN
ejpam-5535	90	14	)	)	PUNCT
ejpam-5535	90	15	=	=	SYM
ejpam-5535	90	16	a.	a.	NOUN
ejpam-5535	90	17	for	for	ADP
ejpam-5535	90	18	0	0	NUM
ejpam-5535	90	19	∈	∈	PROPN
ejpam-5535	90	20	a	a	DET
ejpam-5535	90	21	=	=	X
ejpam-5535	90	22	g(k	g(k	NOUN
ejpam-5535	90	23	)	)	PUNCT
ejpam-5535	90	24	,	,	PUNCT
ejpam-5535	90	25	we	we	PRON
ejpam-5535	90	26	have	have	VERB
ejpam-5535	90	27	0	0	NUM
ejpam-5535	90	28	g	g	NOUN
ejpam-5535	90	29	∈	∈	NOUN
ejpam-5535	90	30	k	k	X
ejpam-5535	90	31	∩d	∩d	PROPN
ejpam-5535	90	32	.	.	PUNCT
ejpam-5535	91	1	theorem	theorem	VERB
ejpam-5535	91	2	2.9	2.9	NUM
ejpam-5535	91	3	.	.	PUNCT
ejpam-5535	92	1	given	give	VERB
ejpam-5535	92	2	an	an	DET
ejpam-5535	92	3	ideal	ideal	ADJ
ejpam-5535	92	4	k	k	NOUN
ejpam-5535	92	5	of	of	ADP
ejpam-5535	92	6	a	a	DET
ejpam-5535	92	7	,	,	PUNCT
ejpam-5535	92	8	g(k	g(k	NOUN
ejpam-5535	92	9	)	)	PUNCT
ejpam-5535	92	10	is	be	AUX
ejpam-5535	92	11	proper	proper	ADJ
ejpam-5535	92	12	if	if	SCONJ
ejpam-5535	92	13	and	and	CCONJ
ejpam-5535	92	14	only	only	ADV
ejpam-5535	92	15	if	if	SCONJ
ejpam-5535	92	16	either	either	CCONJ
ejpam-5535	92	17	i	i	PRON
ejpam-5535	92	18	/∈	/∈	PUNCT
ejpam-5535	92	19	g(k	g(k	NOUN
ejpam-5535	92	20	)	)	PUNCT
ejpam-5535	92	21	or	or	CCONJ
ejpam-5535	92	22	ig	ig	PROPN
ejpam-5535	92	23	/∈	/∈	PUNCT
ejpam-5535	92	24	g(k	g(k	NOUN
ejpam-5535	92	25	)	)	PUNCT
ejpam-5535	92	26	.	.	PUNCT
ejpam-5535	93	1	that	that	PRON
ejpam-5535	93	2	is	be	AUX
ejpam-5535	93	3	.	.	PUNCT
ejpam-5535	93	4	,	,	PUNCT
ejpam-5535	93	5	for	for	ADP
ejpam-5535	93	6	any	any	DET
ejpam-5535	93	7	i	i	PROPN
ejpam-5535	93	8	∈	∈	PROPN
ejpam-5535	94	1	a	a	PRON
ejpam-5535	94	2	,	,	PUNCT
ejpam-5535	94	3	i	i	PRON
ejpam-5535	94	4	,	,	PUNCT
ejpam-5535	94	5	ig	ig	PROPN
ejpam-5535	94	6	in	in	ADP
ejpam-5535	94	7	g(k	g(k	NOUN
ejpam-5535	94	8	)	)	PUNCT
ejpam-5535	94	9	if	if	SCONJ
ejpam-5535	94	10	and	and	CCONJ
ejpam-5535	94	11	only	only	ADV
ejpam-5535	94	12	if	if	SCONJ
ejpam-5535	94	13	a	a	DET
ejpam-5535	94	14	=	=	X
ejpam-5535	94	15	g(k	g(k	NOUN
ejpam-5535	94	16	)	)	PUNCT
ejpam-5535	94	17	.	.	PUNCT
ejpam-5535	95	1	proof	proof	NOUN
ejpam-5535	95	2	.	.	PUNCT
ejpam-5535	96	1	let	let	VERB
ejpam-5535	96	2	us	we	PRON
ejpam-5535	96	3	consider	consider	VERB
ejpam-5535	96	4	that	that	DET
ejpam-5535	96	5	g(k	g(k	NOUN
ejpam-5535	96	6	)	)	PUNCT
ejpam-5535	96	7	is	be	AUX
ejpam-5535	96	8	a	a	DET
ejpam-5535	96	9	proper	proper	ADJ
ejpam-5535	96	10	filter	filter	NOUN
ejpam-5535	96	11	of	of	ADP
ejpam-5535	96	12	a.	a.	NOUN
ejpam-5535	96	13	let	let	VERB
ejpam-5535	96	14	i	i	PRON
ejpam-5535	96	15	∈	∈	VERB
ejpam-5535	96	16	a.	a.	NOUN
ejpam-5535	96	17	if	if	SCONJ
ejpam-5535	96	18	i	i	PRON
ejpam-5535	96	19	∈	∈	PROPN
ejpam-5535	96	20	g(k	g(k	NOUN
ejpam-5535	96	21	)	)	PUNCT
ejpam-5535	96	22	and	and	CCONJ
ejpam-5535	96	23	ig	ig	PROPN
ejpam-5535	96	24	∈	∈	PROPN
ejpam-5535	96	25	g(k	g(k	PROPN
ejpam-5535	96	26	)	)	PUNCT
ejpam-5535	96	27	,	,	PUNCT
ejpam-5535	96	28	then	then	ADV
ejpam-5535	96	29	ig	ig	PROPN
ejpam-5535	96	30	in	in	ADP
ejpam-5535	96	31	k	k	PROPN
ejpam-5535	96	32	and	and	CCONJ
ejpam-5535	96	33	ig	ig	PROPN
ejpam-5535	96	34	in	in	ADP
ejpam-5535	96	35	g(k	g(k	NOUN
ejpam-5535	96	36	)	)	PUNCT
ejpam-5535	96	37	.	.	PUNCT
ejpam-5535	97	1	therefore	therefore	ADV
ejpam-5535	97	2	g(k	g(k	NOUN
ejpam-5535	97	3	)	)	PUNCT
ejpam-5535	97	4	∩	∩	NOUN
ejpam-5535	97	5	k	k	PROPN
ejpam-5535	97	6	̸=	̸=	PROPN
ejpam-5535	97	7	ϕ.	ϕ.	NOUN
ejpam-5535	97	8	by	by	ADP
ejpam-5535	97	9	theorem	theorem	VERB
ejpam-5535	97	10	2.5	2.5	NUM
ejpam-5535	97	11	.	.	PUNCT
ejpam-5535	97	12	,	,	PUNCT
ejpam-5535	97	13	g(k	g(k	NOUN
ejpam-5535	97	14	)	)	PUNCT
ejpam-5535	97	15	=	=	PUNCT
ejpam-5535	97	16	a.	a.	NOUN
ejpam-5535	97	17	which	which	PRON
ejpam-5535	97	18	is	be	AUX
ejpam-5535	97	19	contradiction	contradiction	NOUN
ejpam-5535	97	20	.	.	PUNCT
ejpam-5535	98	1	here	here	ADV
ejpam-5535	98	2	i	i	PRON
ejpam-5535	98	3	/∈	/∈	PUNCT
ejpam-5535	98	4	g(k	g(k	NOUN
ejpam-5535	98	5	)	)	PUNCT
ejpam-5535	98	6	or	or	CCONJ
ejpam-5535	98	7	ig	ig	PROPN
ejpam-5535	98	8	/∈	/∈	PUNCT
ejpam-5535	98	9	g(k	g(k	NOUN
ejpam-5535	98	10	)	)	PUNCT
ejpam-5535	98	11	.	.	PUNCT
ejpam-5535	99	1	conversely	conversely	ADV
ejpam-5535	99	2	suppose	suppose	VERB
ejpam-5535	99	3	that	that	SCONJ
ejpam-5535	99	4	i	i	PRON
ejpam-5535	99	5	/∈	/∈	PUNCT
ejpam-5535	99	6	g(k	g(k	NOUN
ejpam-5535	99	7	)	)	PUNCT
ejpam-5535	99	8	or	or	CCONJ
ejpam-5535	99	9	ig	ig	PROPN
ejpam-5535	99	10	/∈	/∈	PUNCT
ejpam-5535	99	11	g(k	g(k	NOUN
ejpam-5535	99	12	)	)	PUNCT
ejpam-5535	99	13	.	.	PUNCT
ejpam-5535	100	1	if	if	SCONJ
ejpam-5535	100	2	g(k	g(k	VERB
ejpam-5535	100	3	)	)	PUNCT
ejpam-5535	100	4	=	=	SYM
ejpam-5535	101	1	a	a	PRON
ejpam-5535	101	2	,	,	PUNCT
ejpam-5535	101	3	then	then	ADV
ejpam-5535	101	4	i	i	PRON
ejpam-5535	101	5	,	,	PUNCT
ejpam-5535	101	6	ig	ig	PROPN
ejpam-5535	101	7	∈	∈	PROPN
ejpam-5535	101	8	a	a	DET
ejpam-5535	101	9	=	=	X
ejpam-5535	101	10	g(k	g(k	NOUN
ejpam-5535	101	11	)	)	PUNCT
ejpam-5535	101	12	.	.	PUNCT
ejpam-5535	102	1	which	which	PRON
ejpam-5535	102	2	is	be	AUX
ejpam-5535	102	3	contradiction	contradiction	NOUN
ejpam-5535	102	4	.	.	PUNCT
ejpam-5535	103	1	therefore	therefore	ADV
ejpam-5535	103	2	g(k	g(k	VERB
ejpam-5535	103	3	)	)	PUNCT
ejpam-5535	103	4	̸=	̸=	PROPN
ejpam-5535	103	5	a.	a.	NOUN
ejpam-5535	103	6	hence	hence	ADV
ejpam-5535	103	7	g(k	g(k	NOUN
ejpam-5535	103	8	)	)	PUNCT
ejpam-5535	103	9	is	be	AUX
ejpam-5535	103	10	proper	proper	ADJ
ejpam-5535	103	11	.	.	PUNCT
ejpam-5535	104	1	r.	r.	PROPN
ejpam-5535	104	2	sirisetti	sirisetti	PROPN
ejpam-5535	104	3	et	et	PROPN
ejpam-5535	104	4	al	al	PROPN
ejpam-5535	104	5	.	.	PUNCT
ejpam-5535	104	6	/	/	SYM
ejpam-5535	104	7	eur	eur	PROPN
ejpam-5535	104	8	.	.	PUNCT
ejpam-5535	105	1	j.	j.	PROPN
ejpam-5535	105	2	pure	pure	PROPN
ejpam-5535	105	3	appl	appl	PROPN
ejpam-5535	105	4	.	.	PROPN
ejpam-5535	105	5	math	math	PROPN
ejpam-5535	105	6	,	,	PUNCT
ejpam-5535	105	7	18	18	NUM
ejpam-5535	105	8	(	(	PUNCT
ejpam-5535	105	9	1	1	NUM
ejpam-5535	105	10	)	)	PUNCT
ejpam-5535	105	11	(	(	PUNCT
ejpam-5535	105	12	2025	2025	NUM
ejpam-5535	105	13	)	)	PUNCT
ejpam-5535	105	14	,	,	PUNCT
ejpam-5535	105	15	5535	5535	NUM
ejpam-5535	105	16	5	5	NUM
ejpam-5535	105	17	of	of	ADP
ejpam-5535	105	18	12	12	NUM
ejpam-5535	105	19	theorem	theorem	NOUN
ejpam-5535	105	20	2.10	2.10	NUM
ejpam-5535	105	21	.	.	PUNCT
ejpam-5535	106	1	if	if	SCONJ
ejpam-5535	106	2	a	a	DET
ejpam-5535	106	3	satisfies	satisfie	NOUN
ejpam-5535	106	4	(	(	PUNCT
ejpam-5535	106	5	v	v	NOUN
ejpam-5535	106	6	∨w)g	∨w)g	NOUN
ejpam-5535	106	7	=	=	PUNCT
ejpam-5535	106	8	vg	vg	ADP
ejpam-5535	106	9	∧wg	∧wg	PROPN
ejpam-5535	106	10	,	,	PUNCT
ejpam-5535	106	11	for	for	ADP
ejpam-5535	106	12	all	all	DET
ejpam-5535	106	13	v	v	NOUN
ejpam-5535	106	14	,	,	PUNCT
ejpam-5535	106	15	w	w	PROPN
ejpam-5535	106	16	∈	∈	PROPN
ejpam-5535	106	17	a	a	PRON
ejpam-5535	106	18	and	and	CCONJ
ejpam-5535	106	19	k	k	PROPN
ejpam-5535	106	20	is	be	AUX
ejpam-5535	106	21	a	a	DET
ejpam-5535	106	22	prime	prime	ADJ
ejpam-5535	106	23	ideal	ideal	NOUN
ejpam-5535	106	24	of	of	ADP
ejpam-5535	106	25	a	a	PRON
ejpam-5535	106	26	,	,	PUNCT
ejpam-5535	106	27	then	then	ADV
ejpam-5535	106	28	g(k	g(k	VERB
ejpam-5535	106	29	)	)	PUNCT
ejpam-5535	106	30	is	be	AUX
ejpam-5535	106	31	a	a	DET
ejpam-5535	106	32	prime	prime	ADJ
ejpam-5535	106	33	filter	filter	NOUN
ejpam-5535	106	34	in	in	ADP
ejpam-5535	106	35	a.	a.	NOUN
ejpam-5535	106	36	proof	proof	NOUN
ejpam-5535	106	37	.	.	PUNCT
ejpam-5535	107	1	let	let	VERB
ejpam-5535	107	2	v	v	NOUN
ejpam-5535	107	3	,	,	PUNCT
ejpam-5535	107	4	w	w	PROPN
ejpam-5535	107	5	∈	∈	PROPN
ejpam-5535	107	6	a	a	DET
ejpam-5535	107	7	such	such	ADJ
ejpam-5535	107	8	that	that	DET
ejpam-5535	107	9	v	v	NOUN
ejpam-5535	107	10	∨w	∨w	NOUN
ejpam-5535	107	11	in	in	ADP
ejpam-5535	107	12	g(k	g(k	NOUN
ejpam-5535	107	13	)	)	PUNCT
ejpam-5535	107	14	.	.	PUNCT
ejpam-5535	108	1	then	then	ADV
ejpam-5535	108	2	(	(	PUNCT
ejpam-5535	108	3	v	v	X
ejpam-5535	108	4	∨w)g	∨w)g	NOUN
ejpam-5535	108	5	=	=	PUNCT
ejpam-5535	108	6	vg	vg	ADP
ejpam-5535	108	7	∧wg	∧wg	PROPN
ejpam-5535	108	8	in	in	ADP
ejpam-5535	108	9	k.	k.	PROPN
ejpam-5535	108	10	since	since	SCONJ
ejpam-5535	108	11	k	k	PROPN
ejpam-5535	108	12	is	be	AUX
ejpam-5535	108	13	prime	prime	ADJ
ejpam-5535	108	14	,	,	PUNCT
ejpam-5535	108	15	vg	vg	ADP
ejpam-5535	108	16	∈	∈	PROPN
ejpam-5535	108	17	k	k	NOUN
ejpam-5535	108	18	or	or	CCONJ
ejpam-5535	108	19	wg	wg	PROPN
ejpam-5535	108	20	∈	∈	PROPN
ejpam-5535	108	21	k.	k.	NOUN
ejpam-5535	109	1	therefore	therefore	ADV
ejpam-5535	109	2	v	v	ADP
ejpam-5535	109	3	∈	∈	PROPN
ejpam-5535	109	4	g(k	g(k	NOUN
ejpam-5535	109	5	)	)	PUNCT
ejpam-5535	109	6	or	or	CCONJ
ejpam-5535	109	7	w	w	PROPN
ejpam-5535	109	8	∈	∈	PROPN
ejpam-5535	109	9	g(k	g(k	NOUN
ejpam-5535	109	10	)	)	PUNCT
ejpam-5535	109	11	and	and	CCONJ
ejpam-5535	109	12	hence	hence	ADV
ejpam-5535	109	13	g(k	g(k	VERB
ejpam-5535	109	14	)	)	PUNCT
ejpam-5535	109	15	is	be	AUX
ejpam-5535	109	16	prime	prime	ADJ
ejpam-5535	109	17	.	.	PUNCT
ejpam-5535	110	1	the	the	DET
ejpam-5535	110	2	converse	converse	NOUN
ejpam-5535	110	3	of	of	ADP
ejpam-5535	110	4	theorem	theorem	ADJ
ejpam-5535	110	5	2.10	2.10	NUM
ejpam-5535	110	6	need	need	NOUN
ejpam-5535	110	7	not	not	PART
ejpam-5535	110	8	be	be	AUX
ejpam-5535	110	9	true	true	ADJ
ejpam-5535	110	10	.	.	PUNCT
ejpam-5535	111	1	example	example	NOUN
ejpam-5535	111	2	2.11	2.11	NUM
ejpam-5535	111	3	.	.	PUNCT
ejpam-5535	112	1	consider	consider	VERB
ejpam-5535	112	2	the	the	DET
ejpam-5535	112	3	distributive	distributive	ADJ
ejpam-5535	112	4	lattice	lattice	NOUN
ejpam-5535	112	5	a	a	DET
ejpam-5535	112	6	=	=	X
ejpam-5535	112	7	{	{	PUNCT
ejpam-5535	112	8	0	0	NUM
ejpam-5535	112	9	,	,	PUNCT
ejpam-5535	112	10	i	i	PRON
ejpam-5535	112	11	,	,	PUNCT
ejpam-5535	112	12	j	j	PROPN
ejpam-5535	112	13	,	,	PUNCT
ejpam-5535	112	14	k	k	PROPN
ejpam-5535	112	15	,	,	PUNCT
ejpam-5535	112	16	l	l	NOUN
ejpam-5535	112	17	,	,	PUNCT
ejpam-5535	112	18	1	1	NUM
ejpam-5535	112	19	}	}	PUNCT
ejpam-5535	112	20	,	,	PUNCT
ejpam-5535	112	21	whose	whose	DET
ejpam-5535	112	22	hasse	hasse	NOUN
ejpam-5535	112	23	diagram	diagram	NOUN
ejpam-5535	112	24	is	be	AUX
ejpam-5535	112	25	in	in	ADP
ejpam-5535	112	26	below	below	ADV
ejpam-5535	112	27	;	;	PUNCT
ejpam-5535	112	28	0	0	PUNCT
ejpam-5535	113	1	i	i	PRON
ejpam-5535	113	2	j	j	PROPN
ejpam-5535	113	3	kl	kl	INTJ
ejpam-5535	113	4	1	1	NUM
ejpam-5535	113	5	for	for	ADP
ejpam-5535	113	6	the	the	DET
ejpam-5535	113	7	ideal	ideal	NOUN
ejpam-5535	113	8	k	k	PROPN
ejpam-5535	114	1	=	=	PUNCT
ejpam-5535	114	2	{	{	PUNCT
ejpam-5535	114	3	0	0	NUM
ejpam-5535	114	4	,	,	PUNCT
ejpam-5535	114	5	i	i	PRON
ejpam-5535	114	6	}	}	PUNCT
ejpam-5535	114	7	and	and	CCONJ
ejpam-5535	114	8	0	0	NUM
ejpam-5535	114	9	g	g	NOUN
ejpam-5535	114	10	=	=	SYM
ejpam-5535	114	11	k	k	PROPN
ejpam-5535	114	12	,	,	PUNCT
ejpam-5535	114	13	ig	ig	PROPN
ejpam-5535	114	14	=	=	SYM
ejpam-5535	114	15	j	j	PROPN
ejpam-5535	114	16	,	,	PUNCT
ejpam-5535	114	17	jg	jg	PROPN
ejpam-5535	115	1	=	=	SYM
ejpam-5535	115	2	i	i	PROPN
ejpam-5535	115	3	,	,	PUNCT
ejpam-5535	115	4	kg	kg	PROPN
ejpam-5535	115	5	=	=	NOUN
ejpam-5535	115	6	1	1	NUM
ejpam-5535	115	7	g	g	NOUN
ejpam-5535	115	8	=	=	SYM
ejpam-5535	115	9	0	0	NUM
ejpam-5535	115	10	,	,	PUNCT
ejpam-5535	115	11	g(k	g(k	NOUN
ejpam-5535	115	12	)	)	PUNCT
ejpam-5535	115	13	=	=	SYM
ejpam-5535	115	14	{	{	PUNCT
ejpam-5535	115	15	j	j	PROPN
ejpam-5535	115	16	,	,	PUNCT
ejpam-5535	115	17	k	k	PROPN
ejpam-5535	115	18	,	,	PUNCT
ejpam-5535	115	19	1	1	NUM
ejpam-5535	115	20	}	}	PUNCT
ejpam-5535	115	21	is	be	AUX
ejpam-5535	115	22	a	a	DET
ejpam-5535	115	23	prime	prime	ADJ
ejpam-5535	115	24	filter	filter	NOUN
ejpam-5535	115	25	,	,	PUNCT
ejpam-5535	115	26	but	but	CCONJ
ejpam-5535	115	27	k	k	PROPN
ejpam-5535	115	28	is	be	AUX
ejpam-5535	115	29	not	not	PART
ejpam-5535	115	30	prime	prime	ADJ
ejpam-5535	115	31	.	.	PUNCT
ejpam-5535	116	1	because	because	SCONJ
ejpam-5535	116	2	j	j	PROPN
ejpam-5535	116	3	∧	∧	PROPN
ejpam-5535	116	4	l	l	NOUN
ejpam-5535	116	5	=	=	SYM
ejpam-5535	116	6	0	0	PUNCT
ejpam-5535	116	7	∈	∈	PROPN
ejpam-5535	116	8	k	k	PROPN
ejpam-5535	116	9	,	,	PUNCT
ejpam-5535	116	10	j	j	PROPN
ejpam-5535	116	11	/∈	/∈	PUNCT
ejpam-5535	116	12	k	k	PROPN
ejpam-5535	116	13	and	and	CCONJ
ejpam-5535	116	14	l	l	PROPN
ejpam-5535	116	15	/∈	/∈	PUNCT
ejpam-5535	117	1	k.	k.	PROPN
ejpam-5535	117	2	remark	remark	PROPN
ejpam-5535	117	3	2.12	2.12	NUM
ejpam-5535	117	4	.	.	PUNCT
ejpam-5535	118	1	the	the	DET
ejpam-5535	118	2	converse	converse	NOUN
ejpam-5535	118	3	of	of	ADP
ejpam-5535	118	4	theorem	theorem	ADJ
ejpam-5535	118	5	2.10	2.10	NUM
ejpam-5535	118	6	.	.	PUNCT
ejpam-5535	118	7	is	be	AUX
ejpam-5535	118	8	true	true	ADJ
ejpam-5535	118	9	,	,	PUNCT
ejpam-5535	118	10	provided	provide	VERB
ejpam-5535	118	11	vgg	vgg	PROPN
ejpam-5535	118	12	=	=	SYM
ejpam-5535	118	13	v	v	NOUN
ejpam-5535	118	14	,	,	PUNCT
ejpam-5535	118	15	for	for	ADP
ejpam-5535	118	16	all	all	DET
ejpam-5535	118	17	v	v	NOUN
ejpam-5535	118	18	∈	∈	PROPN
ejpam-5535	118	19	a.	a.	NOUN
ejpam-5535	118	20	for	for	ADP
ejpam-5535	118	21	,	,	PUNCT
ejpam-5535	118	22	suppose	suppose	VERB
ejpam-5535	118	23	that	that	SCONJ
ejpam-5535	118	24	a	a	PRON
ejpam-5535	118	25	does	do	AUX
ejpam-5535	118	26	not	not	PART
ejpam-5535	118	27	have	have	VERB
ejpam-5535	118	28	a	a	DET
ejpam-5535	118	29	prime	prime	ADJ
ejpam-5535	118	30	ideal	ideal	NOUN
ejpam-5535	118	31	k.	k.	PROPN
ejpam-5535	118	32	in	in	ADP
ejpam-5535	118	33	such	such	ADJ
ejpam-5535	118	34	case	case	NOUN
ejpam-5535	118	35	,	,	PUNCT
ejpam-5535	118	36	v	v	ADP
ejpam-5535	118	37	∧	∧	PROPN
ejpam-5535	118	38	w	w	PROPN
ejpam-5535	118	39	∈	∈	PROPN
ejpam-5535	118	40	k	k	PRON
ejpam-5535	118	41	such	such	ADJ
ejpam-5535	118	42	that	that	DET
ejpam-5535	118	43	v	v	NOUN
ejpam-5535	118	44	/∈	/∈	PUNCT
ejpam-5535	119	1	k	k	PROPN
ejpam-5535	119	2	and	and	CCONJ
ejpam-5535	119	3	w	w	PROPN
ejpam-5535	119	4	/∈	/∈	PUNCT
ejpam-5535	120	1	k	k	PROPN
ejpam-5535	120	2	exist	exist	VERB
ejpam-5535	120	3	for	for	ADP
ejpam-5535	120	4	v	v	NOUN
ejpam-5535	120	5	,	,	PUNCT
ejpam-5535	120	6	w	w	PROPN
ejpam-5535	120	7	∈	∈	PROPN
ejpam-5535	120	8	a.	a.	NOUN
ejpam-5535	120	9	(	(	PUNCT
ejpam-5535	120	10	v	v	NUM
ejpam-5535	120	11	∧w)gg	∧w)gg	NOUN
ejpam-5535	120	12	≤	≤	NUM
ejpam-5535	120	13	v	v	NOUN
ejpam-5535	120	14	∧w	∧w	PROPN
ejpam-5535	120	15	∈	∈	PROPN
ejpam-5535	120	16	k	k	NOUN
ejpam-5535	120	17	,	,	PUNCT
ejpam-5535	120	18	by	by	ADP
ejpam-5535	120	19	proposition	proposition	NOUN
ejpam-5535	120	20	2.1	2.1	NUM
ejpam-5535	120	21	(	(	PUNCT
ejpam-5535	120	22	iv	iv	NUM
ejpam-5535	120	23	)	)	PUNCT
ejpam-5535	120	24	.	.	PUNCT
ejpam-5535	121	1	consequently	consequently	ADV
ejpam-5535	121	2	,	,	PUNCT
ejpam-5535	121	3	(	(	PUNCT
ejpam-5535	121	4	v	v	ADP
ejpam-5535	121	5	∧	∧	PROPN
ejpam-5535	121	6	w)gg	w)gg	PROPN
ejpam-5535	121	7	∈	∈	PROPN
ejpam-5535	121	8	k.	k.	PROPN
ejpam-5535	121	9	therefore	therefore	ADV
ejpam-5535	121	10	,	,	PUNCT
ejpam-5535	121	11	(	(	PUNCT
ejpam-5535	121	12	v	v	ADP
ejpam-5535	121	13	∧	∧	PROPN
ejpam-5535	121	14	w)g	w)g	X
ejpam-5535	121	15	∈	∈	PROPN
ejpam-5535	121	16	g(k	g(k	NOUN
ejpam-5535	121	17	)	)	PUNCT
ejpam-5535	121	18	holds	hold	VERB
ejpam-5535	121	19	.	.	PUNCT
ejpam-5535	122	1	(	(	PUNCT
ejpam-5535	122	2	v	v	NUM
ejpam-5535	122	3	∧	∧	PROPN
ejpam-5535	122	4	w)g	w)g	PUNCT
ejpam-5535	122	5	=	=	PUNCT
ejpam-5535	123	1	vg	vg	ADP
ejpam-5535	123	2	∨	∨	NUM
ejpam-5535	123	3	wg	wg	PROPN
ejpam-5535	123	4	∈	∈	PROPN
ejpam-5535	123	5	g(k	g(k	NOUN
ejpam-5535	123	6	)	)	PUNCT
ejpam-5535	123	7	,	,	PUNCT
ejpam-5535	123	8	at	at	ADP
ejpam-5535	123	9	this	this	DET
ejpam-5535	123	10	point	point	NOUN
ejpam-5535	123	11	.	.	PUNCT
ejpam-5535	124	1	vg	vg	ADP
ejpam-5535	124	2	∈	∈	PROPN
ejpam-5535	124	3	g(k	g(k	NOUN
ejpam-5535	124	4	)	)	PUNCT
ejpam-5535	124	5	or	or	CCONJ
ejpam-5535	124	6	wg	wg	PROPN
ejpam-5535	124	7	∈	∈	PROPN
ejpam-5535	124	8	g(k	g(k	NOUN
ejpam-5535	124	9	)	)	PUNCT
ejpam-5535	124	10	if	if	SCONJ
ejpam-5535	124	11	g(k	g(k	NOUN
ejpam-5535	124	12	)	)	PUNCT
ejpam-5535	124	13	is	be	AUX
ejpam-5535	124	14	prime	prime	ADJ
ejpam-5535	124	15	.	.	PUNCT
ejpam-5535	125	1	it	it	PRON
ejpam-5535	125	2	follows	follow	VERB
ejpam-5535	125	3	that	that	SCONJ
ejpam-5535	125	4	either	either	CCONJ
ejpam-5535	125	5	w	w	X
ejpam-5535	125	6	=	=	VERB
ejpam-5535	125	7	wgg	wgg	NOUN
ejpam-5535	125	8	∈	∈	PROPN
ejpam-5535	125	9	k	k	PROPN
ejpam-5535	125	10	or	or	CCONJ
ejpam-5535	125	11	v	v	NOUN
ejpam-5535	125	12	=	=	SYM
ejpam-5535	125	13	vgg	vgg	NOUN
ejpam-5535	126	1	∈	∈	PROPN
ejpam-5535	127	1	k.	k.	NOUN
ejpam-5535	128	1	it	it	PRON
ejpam-5535	128	2	is	be	AUX
ejpam-5535	128	3	contradictory	contradictory	ADJ
ejpam-5535	128	4	.	.	PUNCT
ejpam-5535	129	1	therefore	therefore	ADV
ejpam-5535	129	2	it	it	PRON
ejpam-5535	129	3	is	be	AUX
ejpam-5535	129	4	not	not	PART
ejpam-5535	129	5	prime	prime	ADJ
ejpam-5535	129	6	,	,	PUNCT
ejpam-5535	129	7	g(k	g(k	NOUN
ejpam-5535	129	8	)	)	PUNCT
ejpam-5535	129	9	.	.	PUNCT
ejpam-5535	130	1	theorem	theorem	NOUN
ejpam-5535	130	2	2.13	2.13	NUM
ejpam-5535	130	3	.	.	PUNCT
ejpam-5535	131	1	if	if	SCONJ
ejpam-5535	131	2	vgg	vgg	PROPN
ejpam-5535	131	3	=	=	SYM
ejpam-5535	131	4	v	v	NOUN
ejpam-5535	131	5	,	,	PUNCT
ejpam-5535	131	6	for	for	ADP
ejpam-5535	131	7	all	all	PRON
ejpam-5535	131	8	v	v	ADP
ejpam-5535	131	9	∈	∈	PRON
ejpam-5535	131	10	a	a	PRON
ejpam-5535	131	11	,	,	PUNCT
ejpam-5535	131	12	then	then	ADV
ejpam-5535	131	13	a	a	PRON
ejpam-5535	131	14	has	have	VERB
ejpam-5535	131	15	a	a	DET
ejpam-5535	131	16	unique	unique	ADJ
ejpam-5535	131	17	dense	dense	ADJ
ejpam-5535	131	18	element	element	NOUN
ejpam-5535	131	19	.	.	PUNCT
ejpam-5535	132	1	proof	proof	NOUN
ejpam-5535	132	2	.	.	PUNCT
ejpam-5535	133	1	suppose	suppose	VERB
ejpam-5535	133	2	that	that	SCONJ
ejpam-5535	133	3	d1	d1	PROPN
ejpam-5535	133	4	and	and	CCONJ
ejpam-5535	133	5	d2	d2	PROPN
ejpam-5535	133	6	are	be	AUX
ejpam-5535	133	7	two	two	NUM
ejpam-5535	133	8	dense	dense	ADJ
ejpam-5535	133	9	elements	element	NOUN
ejpam-5535	133	10	ina	ina	PROPN
ejpam-5535	133	11	.	.	PUNCT
ejpam-5535	134	1	now	now	ADV
ejpam-5535	134	2	,	,	PUNCT
ejpam-5535	134	3	d1	d1	PROPN
ejpam-5535	134	4	=	=	SYM
ejpam-5535	134	5	dgg1	dgg1	PROPN
ejpam-5535	134	6	=	=	SYM
ejpam-5535	134	7	(	(	PUNCT
ejpam-5535	134	8	dg1	dg1	PROPN
ejpam-5535	134	9	)	)	PUNCT
ejpam-5535	134	10	g	g	NOUN
ejpam-5535	134	11	=	=	SYM
ejpam-5535	134	12	0	0	NUM
ejpam-5535	134	13	g	g	NOUN
ejpam-5535	134	14	and	and	CCONJ
ejpam-5535	134	15	d2	d2	PROPN
ejpam-5535	134	16	=	=	SYM
ejpam-5535	134	17	dgg2	dgg2	PROPN
ejpam-5535	134	18	=	=	SYM
ejpam-5535	134	19	(	(	PUNCT
ejpam-5535	134	20	dg2	dg2	PROPN
ejpam-5535	134	21	)	)	PUNCT
ejpam-5535	135	1	g	g	NOUN
ejpam-5535	135	2	=	=	SYM
ejpam-5535	135	3	0	0	NUM
ejpam-5535	135	4	g	g	NOUN
ejpam-5535	135	5	(	(	PUNCT
ejpam-5535	135	6	by	by	ADP
ejpam-5535	135	7	proposition	proposition	NOUN
ejpam-5535	135	8	2.1.(ii	2.1.(ii	NUM
ejpam-5535	135	9	)	)	PUNCT
ejpam-5535	135	10	)	)	PUNCT
ejpam-5535	135	11	.	.	PUNCT
ejpam-5535	136	1	therefore	therefore	ADV
ejpam-5535	136	2	d1	d1	PROPN
ejpam-5535	136	3	=	=	SYM
ejpam-5535	136	4	d2	d2	PROPN
ejpam-5535	136	5	=	=	SYM
ejpam-5535	136	6	0	0	NUM
ejpam-5535	136	7	g.	g.	NOUN
ejpam-5535	136	8	hence	hence	ADV
ejpam-5535	136	9	a	a	PRON
ejpam-5535	136	10	has	have	VERB
ejpam-5535	136	11	a	a	DET
ejpam-5535	136	12	unique	unique	ADJ
ejpam-5535	136	13	dense	dense	ADJ
ejpam-5535	136	14	element	element	NOUN
ejpam-5535	136	15	.	.	PUNCT
ejpam-5535	137	1	in	in	ADP
ejpam-5535	137	2	a	a	DET
ejpam-5535	137	3	general	general	ADJ
ejpam-5535	137	4	lattice	lattice	NOUN
ejpam-5535	137	5	,	,	PUNCT
ejpam-5535	137	6	the	the	DET
ejpam-5535	137	7	converse	converse	NOUN
ejpam-5535	137	8	of	of	ADP
ejpam-5535	137	9	theorem	theorem	PROPN
ejpam-5535	137	10	2.13	2.13	NUM
ejpam-5535	137	11	.	.	PUNCT
ejpam-5535	138	1	need	need	AUX
ejpam-5535	138	2	not	not	PART
ejpam-5535	138	3	be	be	AUX
ejpam-5535	138	4	true	true	ADJ
ejpam-5535	138	5	.	.	PUNCT
ejpam-5535	139	1	example	example	NOUN
ejpam-5535	139	2	2.14	2.14	NUM
ejpam-5535	139	3	.	.	PUNCT
ejpam-5535	140	1	consider	consider	VERB
ejpam-5535	140	2	a	a	DET
ejpam-5535	140	3	lattice	lattice	NOUN
ejpam-5535	140	4	a	a	DET
ejpam-5535	140	5	=	=	PUNCT
ejpam-5535	140	6	{	{	PUNCT
ejpam-5535	140	7	0	0	NUM
ejpam-5535	140	8	,	,	PUNCT
ejpam-5535	140	9	i	i	PRON
ejpam-5535	140	10	,	,	PUNCT
ejpam-5535	140	11	j	j	PROPN
ejpam-5535	140	12	,	,	PUNCT
ejpam-5535	140	13	k	k	PROPN
ejpam-5535	140	14	,	,	PUNCT
ejpam-5535	140	15	1	1	NUM
ejpam-5535	140	16	}	}	PUNCT
ejpam-5535	140	17	,	,	PUNCT
ejpam-5535	140	18	whose	whose	DET
ejpam-5535	140	19	hasse	hasse	NOUN
ejpam-5535	140	20	diagram	diagram	NOUN
ejpam-5535	140	21	is	be	AUX
ejpam-5535	140	22	in	in	ADP
ejpam-5535	140	23	below	below	ADV
ejpam-5535	140	24	;	;	PUNCT
ejpam-5535	141	1	r.	r.	PROPN
ejpam-5535	141	2	sirisetti	sirisetti	PROPN
ejpam-5535	141	3	et	et	PROPN
ejpam-5535	141	4	al	al	PROPN
ejpam-5535	141	5	.	.	PUNCT
ejpam-5535	141	6	/	/	SYM
ejpam-5535	141	7	eur	eur	PROPN
ejpam-5535	141	8	.	.	PUNCT
ejpam-5535	142	1	j.	j.	PROPN
ejpam-5535	142	2	pure	pure	PROPN
ejpam-5535	142	3	appl	appl	PROPN
ejpam-5535	142	4	.	.	PROPN
ejpam-5535	142	5	math	math	PROPN
ejpam-5535	142	6	,	,	PUNCT
ejpam-5535	142	7	18	18	NUM
ejpam-5535	142	8	(	(	PUNCT
ejpam-5535	142	9	1	1	NUM
ejpam-5535	142	10	)	)	PUNCT
ejpam-5535	142	11	(	(	PUNCT
ejpam-5535	142	12	2025	2025	NUM
ejpam-5535	142	13	)	)	PUNCT
ejpam-5535	142	14	,	,	PUNCT
ejpam-5535	142	15	5535	5535	NUM
ejpam-5535	142	16	6	6	NUM
ejpam-5535	142	17	of	of	ADP
ejpam-5535	142	18	12	12	NUM
ejpam-5535	142	19	0	0	NUM
ejpam-5535	143	1	i	i	PRON
ejpam-5535	143	2	j	j	PROPN
ejpam-5535	144	1	k	k	NOUN
ejpam-5535	144	2	1	1	NUM
ejpam-5535	144	3	and	and	CCONJ
ejpam-5535	144	4	0	0	NUM
ejpam-5535	144	5	g	g	NOUN
ejpam-5535	144	6	=	=	SYM
ejpam-5535	144	7	1	1	NUM
ejpam-5535	144	8	,	,	PUNCT
ejpam-5535	144	9	1	1	NUM
ejpam-5535	144	10	g	g	NOUN
ejpam-5535	144	11	=	=	SYM
ejpam-5535	144	12	0	0	NUM
ejpam-5535	144	13	,	,	PUNCT
ejpam-5535	144	14	ig	ig	PROPN
ejpam-5535	144	15	=	=	SYM
ejpam-5535	144	16	k	k	PROPN
ejpam-5535	144	17	,	,	PUNCT
ejpam-5535	144	18	jg	jg	PROPN
ejpam-5535	144	19	=	=	SYM
ejpam-5535	144	20	k	k	PROPN
ejpam-5535	144	21	,	,	PUNCT
ejpam-5535	144	22	kg	kg	PROPN
ejpam-5535	144	23	=	=	PUNCT
ejpam-5535	144	24	i.	i.	PROPN
ejpam-5535	145	1	then	then	ADV
ejpam-5535	145	2	a	a	PRON
ejpam-5535	145	3	has	have	VERB
ejpam-5535	145	4	a	a	DET
ejpam-5535	145	5	unique	unique	ADJ
ejpam-5535	145	6	dense	dense	ADJ
ejpam-5535	145	7	element	element	NOUN
ejpam-5535	145	8	,	,	PUNCT
ejpam-5535	145	9	but	but	CCONJ
ejpam-5535	145	10	jgg	jgg	PROPN
ejpam-5535	145	11	=	=	SYM
ejpam-5535	146	1	i	i	PRON
ejpam-5535	146	2	̸=	̸=	PROPN
ejpam-5535	146	3	j.	j.	PROPN
ejpam-5535	146	4	3	3	PROPN
ejpam-5535	146	5	.	.	PUNCT
ejpam-5535	147	1	kg	kg	NOUN
ejpam-5535	147	2	-	-	PUNCT
ejpam-5535	147	3	filters	filter	NOUN
ejpam-5535	147	4	in	in	ADP
ejpam-5535	147	5	g	g	NOUN
ejpam-5535	147	6	-	-	PUNCT
ejpam-5535	147	7	complemented	complement	VERB
ejpam-5535	147	8	distributive	distributive	ADJ
ejpam-5535	147	9	lattices	lattice	NOUN
ejpam-5535	147	10	in	in	ADP
ejpam-5535	147	11	this	this	DET
ejpam-5535	147	12	section	section	NOUN
ejpam-5535	147	13	,	,	PUNCT
ejpam-5535	147	14	we	we	PRON
ejpam-5535	147	15	introduce	introduce	VERB
ejpam-5535	147	16	kg	kg	NOUN
ejpam-5535	147	17	-	-	PUNCT
ejpam-5535	147	18	filters	filter	NOUN
ejpam-5535	147	19	derived	derive	VERB
ejpam-5535	147	20	from	from	ADP
ejpam-5535	147	21	ideals	ideal	NOUN
ejpam-5535	147	22	using	use	VERB
ejpam-5535	147	23	a	a	DET
ejpam-5535	147	24	generalized	generalized	ADJ
ejpam-5535	147	25	complementation	complementation	NOUN
ejpam-5535	147	26	on	on	ADP
ejpam-5535	147	27	a	a	DET
ejpam-5535	147	28	distributive	distributive	ADJ
ejpam-5535	147	29	lattice	lattice	NOUN
ejpam-5535	147	30	with	with	ADP
ejpam-5535	147	31	dense	dense	ADJ
ejpam-5535	147	32	elements	element	NOUN
ejpam-5535	147	33	and	and	CCONJ
ejpam-5535	147	34	study	study	VERB
ejpam-5535	147	35	some	some	DET
ejpam-5535	147	36	algebraic	algebraic	ADJ
ejpam-5535	147	37	properties	property	NOUN
ejpam-5535	147	38	.	.	PUNCT
ejpam-5535	148	1	we	we	PRON
ejpam-5535	148	2	obtain	obtain	VERB
ejpam-5535	148	3	some	some	DET
ejpam-5535	148	4	necessary	necessary	ADJ
ejpam-5535	148	5	conditions	condition	NOUN
ejpam-5535	148	6	for	for	SCONJ
ejpam-5535	148	7	a	a	DET
ejpam-5535	148	8	filter	filter	NOUN
ejpam-5535	148	9	to	to	PART
ejpam-5535	148	10	become	become	VERB
ejpam-5535	148	11	a	a	DET
ejpam-5535	148	12	kg	kg	NOUN
ejpam-5535	148	13	-	-	NOUN
ejpam-5535	148	14	filter	filter	NOUN
ejpam-5535	148	15	.	.	PUNCT
ejpam-5535	149	1	mainly	mainly	ADV
ejpam-5535	149	2	,	,	PUNCT
ejpam-5535	149	3	we	we	PRON
ejpam-5535	149	4	prove	prove	VERB
ejpam-5535	149	5	that	that	SCONJ
ejpam-5535	149	6	the	the	DET
ejpam-5535	149	7	set	set	NOUN
ejpam-5535	149	8	of	of	ADP
ejpam-5535	149	9	kg	kg	NOUN
ejpam-5535	149	10	-	-	PUNCT
ejpam-5535	149	11	filters	filter	NOUN
ejpam-5535	149	12	is	be	AUX
ejpam-5535	149	13	a	a	DET
ejpam-5535	149	14	distributive	distributive	ADJ
ejpam-5535	149	15	lattice	lattice	NOUN
ejpam-5535	149	16	.	.	PUNCT
ejpam-5535	150	1	also	also	ADV
ejpam-5535	150	2	,	,	PUNCT
ejpam-5535	150	3	we	we	PRON
ejpam-5535	150	4	obtain	obtain	VERB
ejpam-5535	150	5	a	a	DET
ejpam-5535	150	6	boolean	boolean	ADJ
ejpam-5535	150	7	algebraic	algebraic	ADJ
ejpam-5535	150	8	structure	structure	NOUN
ejpam-5535	150	9	through	through	ADP
ejpam-5535	150	10	kg	kg	NOUN
ejpam-5535	150	11	-	-	PUNCT
ejpam-5535	150	12	filters	filter	NOUN
ejpam-5535	150	13	in	in	ADP
ejpam-5535	150	14	a	a	DET
ejpam-5535	150	15	g	g	NOUN
ejpam-5535	150	16	-	-	PUNCT
ejpam-5535	150	17	complemented	complement	VERB
ejpam-5535	150	18	distributive	distributive	ADJ
ejpam-5535	150	19	lattice	lattice	NOUN
ejpam-5535	150	20	.	.	PUNCT
ejpam-5535	151	1	finally	finally	ADV
ejpam-5535	151	2	we	we	PRON
ejpam-5535	151	3	introduce	introduce	VERB
ejpam-5535	151	4	normal	normal	ADJ
ejpam-5535	151	5	kg	kg	NOUN
ejpam-5535	151	6	-	-	PUNCT
ejpam-5535	151	7	filters	filter	NOUN
ejpam-5535	151	8	in	in	ADP
ejpam-5535	151	9	a	a	DET
ejpam-5535	151	10	generalized	generalize	VERB
ejpam-5535	151	11	complemented	complement	VERB
ejpam-5535	151	12	distributive	distributive	ADJ
ejpam-5535	151	13	lattice	lattice	NOUN
ejpam-5535	151	14	and	and	CCONJ
ejpam-5535	151	15	prove	prove	VERB
ejpam-5535	151	16	that	that	SCONJ
ejpam-5535	151	17	the	the	DET
ejpam-5535	151	18	set	set	NOUN
ejpam-5535	151	19	of	of	ADP
ejpam-5535	151	20	normal	normal	ADJ
ejpam-5535	151	21	kg	kg	NOUN
ejpam-5535	151	22	-	-	PUNCT
ejpam-5535	151	23	filters	filter	NOUN
ejpam-5535	151	24	is	be	AUX
ejpam-5535	151	25	a	a	DET
ejpam-5535	151	26	boolean	boolean	ADJ
ejpam-5535	151	27	algebra	algebra	NOUN
ejpam-5535	151	28	.	.	PUNCT
ejpam-5535	152	1	definition	definition	NOUN
ejpam-5535	152	2	3.1	3.1	NUM
ejpam-5535	152	3	.	.	PUNCT
ejpam-5535	153	1	a	a	DET
ejpam-5535	153	2	filter	filter	NOUN
ejpam-5535	153	3	n	n	NOUN
ejpam-5535	153	4	of	of	ADP
ejpam-5535	153	5	a	a	PRON
ejpam-5535	153	6	is	be	AUX
ejpam-5535	153	7	called	call	VERB
ejpam-5535	153	8	a	a	DET
ejpam-5535	153	9	kg	kg	NOUN
ejpam-5535	153	10	-	-	NOUN
ejpam-5535	153	11	filter	filter	NOUN
ejpam-5535	153	12	if	if	SCONJ
ejpam-5535	153	13	there	there	PRON
ejpam-5535	153	14	exists	exist	VERB
ejpam-5535	153	15	an	an	DET
ejpam-5535	153	16	ideal	ideal	NOUN
ejpam-5535	153	17	k	k	PROPN
ejpam-5535	153	18	in	in	ADP
ejpam-5535	153	19	a	a	DET
ejpam-5535	153	20	such	such	ADJ
ejpam-5535	153	21	that	that	DET
ejpam-5535	153	22	g(k	g(k	NOUN
ejpam-5535	153	23	)	)	PUNCT
ejpam-5535	153	24	=	=	SYM
ejpam-5535	154	1	n	n	NOUN
ejpam-5535	154	2	.	.	PUNCT
ejpam-5535	155	1	it	it	PRON
ejpam-5535	155	2	is	be	AUX
ejpam-5535	155	3	easy	easy	ADJ
ejpam-5535	155	4	to	to	PART
ejpam-5535	155	5	observe	observe	VERB
ejpam-5535	155	6	that	that	SCONJ
ejpam-5535	155	7	d	d	NOUN
ejpam-5535	155	8	is	be	AUX
ejpam-5535	155	9	a	a	DET
ejpam-5535	155	10	kg	kg	NOUN
ejpam-5535	155	11	-	-	NOUN
ejpam-5535	155	12	filter	filter	NOUN
ejpam-5535	155	13	,	,	PUNCT
ejpam-5535	155	14	because	because	SCONJ
ejpam-5535	155	15	g((0	g((0	PROPN
ejpam-5535	155	16	]	]	PUNCT
ejpam-5535	155	17	)	)	PUNCT
ejpam-5535	155	18	=	=	SYM
ejpam-5535	155	19	d.	d.	PROPN
ejpam-5535	155	20	lemma	lemma	PROPN
ejpam-5535	155	21	3.2	3.2	NUM
ejpam-5535	155	22	.	.	PUNCT
ejpam-5535	156	1	for	for	ADP
ejpam-5535	156	2	any	any	DET
ejpam-5535	156	3	v	v	NOUN
ejpam-5535	156	4	∈	∈	PROPN
ejpam-5535	156	5	a	a	PRON
ejpam-5535	156	6	,	,	PUNCT
ejpam-5535	156	7	the	the	DET
ejpam-5535	156	8	principal	principal	ADJ
ejpam-5535	156	9	filter	filter	NOUN
ejpam-5535	156	10	generated	generate	VERB
ejpam-5535	156	11	by	by	ADP
ejpam-5535	156	12	vg	vg	PROPN
ejpam-5535	156	13	is	be	AUX
ejpam-5535	156	14	a	a	DET
ejpam-5535	156	15	kg	kg	NOUN
ejpam-5535	156	16	-	-	NOUN
ejpam-5535	156	17	filter	filter	NOUN
ejpam-5535	156	18	of	of	ADP
ejpam-5535	156	19	a.	a.	NOUN
ejpam-5535	156	20	moreover	moreover	ADV
ejpam-5535	156	21	[	[	X
ejpam-5535	156	22	vg	vg	NOUN
ejpam-5535	156	23	)	)	PUNCT
ejpam-5535	156	24	=	=	SYM
ejpam-5535	156	25	g((v	g((v	NOUN
ejpam-5535	156	26	]	]	X
ejpam-5535	156	27	)	)	PUNCT
ejpam-5535	156	28	.	.	PUNCT
ejpam-5535	157	1	proof	proof	NOUN
ejpam-5535	157	2	.	.	PUNCT
ejpam-5535	158	1	let	let	VERB
ejpam-5535	158	2	u	u	PRON
ejpam-5535	158	3	∈	∈	PROPN
ejpam-5535	158	4	g((v	g((v	NOUN
ejpam-5535	158	5	]	]	PUNCT
ejpam-5535	158	6	)	)	PUNCT
ejpam-5535	158	7	.	.	PUNCT
ejpam-5535	159	1	then	then	ADV
ejpam-5535	159	2	ug	ug	ADP
ejpam-5535	159	3	∈	∈	PROPN
ejpam-5535	159	4	(	(	PUNCT
ejpam-5535	159	5	v	v	NOUN
ejpam-5535	159	6	]	]	PUNCT
ejpam-5535	159	7	and	and	CCONJ
ejpam-5535	159	8	ug	ug	ADP
ejpam-5535	159	9	≤	≤	PROPN
ejpam-5535	159	10	v.	v.	ADV
ejpam-5535	159	11	since	since	SCONJ
ejpam-5535	159	12	u	u	NOUN
ejpam-5535	159	13	∨	∨	NUM
ejpam-5535	159	14	ug	ug	ADP
ejpam-5535	159	15	∈	∈	PROPN
ejpam-5535	159	16	d	d	PROPN
ejpam-5535	159	17	,	,	PUNCT
ejpam-5535	159	18	u	u	NOUN
ejpam-5535	159	19	∨	∨	NUM
ejpam-5535	159	20	v	v	ADP
ejpam-5535	159	21	∈	∈	PROPN
ejpam-5535	159	22	d.	d.	NOUN
ejpam-5535	159	23	by	by	ADP
ejpam-5535	159	24	definition	definition	NOUN
ejpam-5535	159	25	2.1	2.1	NUM
ejpam-5535	159	26	.	.	PUNCT
ejpam-5535	159	27	,	,	PUNCT
ejpam-5535	159	28	vg	vg	ADP
ejpam-5535	159	29	≤	≤	NOUN
ejpam-5535	159	30	u.	u.	VERB
ejpam-5535	159	31	so	so	SCONJ
ejpam-5535	159	32	that	that	SCONJ
ejpam-5535	159	33	u	u	PRON
ejpam-5535	159	34	∈	∈	PROPN
ejpam-5535	159	35	[	[	X
ejpam-5535	159	36	vg	vg	NOUN
ejpam-5535	159	37	)	)	PUNCT
ejpam-5535	159	38	.	.	PUNCT
ejpam-5535	160	1	therefore	therefore	ADV
ejpam-5535	160	2	g((v	g((v	NOUN
ejpam-5535	160	3	]	]	PUNCT
ejpam-5535	160	4	)	)	PUNCT
ejpam-5535	161	1	⊆	⊆	NUM
ejpam-5535	161	2	[	[	X
ejpam-5535	161	3	vg	vg	NOUN
ejpam-5535	161	4	)	)	PUNCT
ejpam-5535	161	5	.	.	PUNCT
ejpam-5535	162	1	for	for	ADP
ejpam-5535	162	2	any	any	DET
ejpam-5535	162	3	u	u	PROPN
ejpam-5535	162	4	∈	∈	PROPN
ejpam-5535	162	5	a	a	X
ejpam-5535	162	6	,	,	PUNCT
ejpam-5535	162	7	u	u	NOUN
ejpam-5535	162	8	∈	∈	PROPN
ejpam-5535	162	9	[	[	X
ejpam-5535	162	10	vg	vg	NOUN
ejpam-5535	162	11	)	)	PUNCT
ejpam-5535	162	12	⇒	⇒	NOUN
ejpam-5535	162	13	vg	vg	ADP
ejpam-5535	162	14	≤	≤	ADJ
ejpam-5535	162	15	u	u	NOUN
ejpam-5535	162	16	⇒	⇒	NOUN
ejpam-5535	162	17	ug	ug	ADP
ejpam-5535	162	18	≤	≤	NUM
ejpam-5535	162	19	vgg	vgg	NOUN
ejpam-5535	162	20	(	(	PUNCT
ejpam-5535	162	21	proposition	proposition	NOUN
ejpam-5535	162	22	2.1(iii	2.1(iii	NUM
ejpam-5535	162	23	)	)	PUNCT
ejpam-5535	162	24	)	)	PUNCT
ejpam-5535	162	25	⇒	⇒	NOUN
ejpam-5535	162	26	ug	ug	ADP
ejpam-5535	162	27	≤	≤	NUM
ejpam-5535	162	28	v	v	NOUN
ejpam-5535	162	29	(	(	PUNCT
ejpam-5535	162	30	since	since	SCONJ
ejpam-5535	162	31	vgg	vgg	ADJ
ejpam-5535	162	32	≤	≤	NUM
ejpam-5535	162	33	v	v	NOUN
ejpam-5535	162	34	)	)	PUNCT
ejpam-5535	162	35	⇒	⇒	NOUN
ejpam-5535	162	36	ug	ug	ADP
ejpam-5535	162	37	∈	∈	PROPN
ejpam-5535	162	38	(	(	PUNCT
ejpam-5535	162	39	v	v	NOUN
ejpam-5535	162	40	]	]	PUNCT
ejpam-5535	162	41	⇒	⇒	NOUN
ejpam-5535	162	42	u	u	PROPN
ejpam-5535	162	43	∈	∈	PROPN
ejpam-5535	162	44	g((v	g((v	NOUN
ejpam-5535	162	45	]	]	PUNCT
ejpam-5535	162	46	)	)	PUNCT
ejpam-5535	162	47	.	.	PUNCT
ejpam-5535	163	1	therefore	therefore	ADV
ejpam-5535	163	2	[	[	X
ejpam-5535	163	3	vg	vg	NOUN
ejpam-5535	163	4	)	)	PUNCT
ejpam-5535	163	5	⊆	⊆	NUM
ejpam-5535	163	6	g((v	g((v	NOUN
ejpam-5535	163	7	]	]	PUNCT
ejpam-5535	163	8	)	)	PUNCT
ejpam-5535	163	9	and	and	CCONJ
ejpam-5535	163	10	hence	hence	ADV
ejpam-5535	163	11	[	[	X
ejpam-5535	163	12	vg	vg	NOUN
ejpam-5535	163	13	)	)	PUNCT
ejpam-5535	163	14	=	=	SYM
ejpam-5535	163	15	g((v	g((v	NOUN
ejpam-5535	163	16	]	]	X
ejpam-5535	163	17	)	)	PUNCT
ejpam-5535	163	18	.	.	PUNCT
ejpam-5535	164	1	thus	thus	ADV
ejpam-5535	164	2	[	[	X
ejpam-5535	164	3	vg	vg	NOUN
ejpam-5535	164	4	)	)	PUNCT
ejpam-5535	164	5	is	be	AUX
ejpam-5535	164	6	a	a	DET
ejpam-5535	164	7	kg	kg	NOUN
ejpam-5535	164	8	-	-	NOUN
ejpam-5535	164	9	filter	filter	NOUN
ejpam-5535	164	10	.	.	PUNCT
ejpam-5535	165	1	now	now	ADV
ejpam-5535	165	2	,	,	PUNCT
ejpam-5535	165	3	we	we	PRON
ejpam-5535	165	4	provide	provide	VERB
ejpam-5535	165	5	an	an	DET
ejpam-5535	165	6	example	example	NOUN
ejpam-5535	165	7	of	of	ADP
ejpam-5535	165	8	a	a	DET
ejpam-5535	165	9	distributive	distributive	ADJ
ejpam-5535	165	10	lattice	lattice	NOUN
ejpam-5535	165	11	in	in	ADP
ejpam-5535	165	12	which	which	PRON
ejpam-5535	165	13	there	there	PRON
ejpam-5535	165	14	is	be	VERB
ejpam-5535	165	15	a	a	DET
ejpam-5535	165	16	kg	kg	NOUN
ejpam-5535	165	17	-	-	NOUN
ejpam-5535	165	18	filter	filter	NOUN
ejpam-5535	165	19	and	and	CCONJ
ejpam-5535	165	20	there	there	PRON
ejpam-5535	165	21	is	be	VERB
ejpam-5535	165	22	a	a	DET
ejpam-5535	165	23	non	non	ADJ
ejpam-5535	165	24	-	-	ADJ
ejpam-5535	165	25	kg	kg	ADJ
ejpam-5535	165	26	-	-	NOUN
ejpam-5535	165	27	filter	filter	NOUN
ejpam-5535	165	28	.	.	PUNCT
ejpam-5535	166	1	r.	r.	PROPN
ejpam-5535	166	2	sirisetti	sirisetti	PROPN
ejpam-5535	166	3	et	et	PROPN
ejpam-5535	166	4	al	al	PROPN
ejpam-5535	166	5	.	.	PUNCT
ejpam-5535	166	6	/	/	SYM
ejpam-5535	166	7	eur	eur	PROPN
ejpam-5535	166	8	.	.	PUNCT
ejpam-5535	167	1	j.	j.	PROPN
ejpam-5535	167	2	pure	pure	PROPN
ejpam-5535	167	3	appl	appl	PROPN
ejpam-5535	167	4	.	.	PROPN
ejpam-5535	167	5	math	math	PROPN
ejpam-5535	167	6	,	,	PUNCT
ejpam-5535	167	7	18	18	NUM
ejpam-5535	167	8	(	(	PUNCT
ejpam-5535	167	9	1	1	NUM
ejpam-5535	167	10	)	)	PUNCT
ejpam-5535	167	11	(	(	PUNCT
ejpam-5535	167	12	2025	2025	NUM
ejpam-5535	167	13	)	)	PUNCT
ejpam-5535	167	14	,	,	PUNCT
ejpam-5535	167	15	5535	5535	NUM
ejpam-5535	167	16	7	7	NUM
ejpam-5535	167	17	of	of	ADP
ejpam-5535	167	18	12	12	NUM
ejpam-5535	167	19	example	example	NOUN
ejpam-5535	167	20	3.3	3.3	NUM
ejpam-5535	167	21	.	.	PUNCT
ejpam-5535	168	1	consider	consider	VERB
ejpam-5535	168	2	a	a	DET
ejpam-5535	168	3	distributive	distributive	ADJ
ejpam-5535	168	4	lattice	lattice	NOUN
ejpam-5535	168	5	a	a	DET
ejpam-5535	168	6	=	=	PUNCT
ejpam-5535	168	7	{	{	PUNCT
ejpam-5535	168	8	0	0	NUM
ejpam-5535	168	9	,	,	PUNCT
ejpam-5535	168	10	i	i	PRON
ejpam-5535	168	11	,	,	PUNCT
ejpam-5535	168	12	j	j	PROPN
ejpam-5535	168	13	,	,	PUNCT
ejpam-5535	168	14	k	k	PROPN
ejpam-5535	168	15	,	,	PUNCT
ejpam-5535	168	16	1	1	NUM
ejpam-5535	168	17	}	}	PUNCT
ejpam-5535	168	18	whose	whose	DET
ejpam-5535	168	19	hasse	hasse	NOUN
ejpam-5535	168	20	diagram	diagram	NOUN
ejpam-5535	168	21	is	be	AUX
ejpam-5535	168	22	given	give	VERB
ejpam-5535	168	23	below	below	ADV
ejpam-5535	168	24	;	;	PUNCT
ejpam-5535	168	25	0	0	PUNCT
ejpam-5535	169	1	i	i	PRON
ejpam-5535	169	2	j	j	PROPN
ejpam-5535	170	1	k	k	PROPN
ejpam-5535	170	2	1	1	X
ejpam-5535	170	3	then	then	ADV
ejpam-5535	170	4	ka	ka	PROPN
ejpam-5535	171	1	=	=	PUNCT
ejpam-5535	171	2	{	{	PUNCT
ejpam-5535	171	3	0},kb	0},kb	NOUN
ejpam-5535	171	4	=	=	SYM
ejpam-5535	171	5	{	{	PUNCT
ejpam-5535	171	6	0	0	NUM
ejpam-5535	171	7	,	,	PUNCT
ejpam-5535	171	8	i},kc	i},kc	PROPN
ejpam-5535	171	9	=	=	PUNCT
ejpam-5535	171	10	{	{	PUNCT
ejpam-5535	171	11	0	0	NUM
ejpam-5535	171	12	,	,	PUNCT
ejpam-5535	171	13	i	i	PRON
ejpam-5535	171	14	,	,	PUNCT
ejpam-5535	171	15	k},kd	k},kd	PROPN
ejpam-5535	171	16	=	=	SYM
ejpam-5535	171	17	{	{	PUNCT
ejpam-5535	171	18	0	0	NUM
ejpam-5535	171	19	,	,	PUNCT
ejpam-5535	171	20	i	i	PRON
ejpam-5535	171	21	,	,	PUNCT
ejpam-5535	171	22	j},ke	j},ke	PROPN
ejpam-5535	171	23	=	=	PUNCT
ejpam-5535	171	24	a	a	PRON
ejpam-5535	171	25	are	be	AUX
ejpam-5535	171	26	ideals	ideal	NOUN
ejpam-5535	171	27	of	of	ADP
ejpam-5535	171	28	a	a	PRON
ejpam-5535	171	29	and	and	CCONJ
ejpam-5535	171	30	na	na	ADP
ejpam-5535	171	31	=	=	PUNCT
ejpam-5535	171	32	{	{	PUNCT
ejpam-5535	171	33	1	1	NUM
ejpam-5535	171	34	}	}	PUNCT
ejpam-5535	171	35	,	,	PUNCT
ejpam-5535	171	36	nb	nb	INTJ
ejpam-5535	171	37	=	=	SYM
ejpam-5535	171	38	{	{	PUNCT
ejpam-5535	171	39	j	j	PROPN
ejpam-5535	171	40	,	,	PUNCT
ejpam-5535	171	41	1	1	NUM
ejpam-5535	171	42	}	}	PUNCT
ejpam-5535	171	43	,	,	PUNCT
ejpam-5535	171	44	nc	nc	PROPN
ejpam-5535	171	45	=	=	X
ejpam-5535	171	46	{	{	PUNCT
ejpam-5535	171	47	k	k	NOUN
ejpam-5535	171	48	,	,	PUNCT
ejpam-5535	171	49	1	1	NUM
ejpam-5535	171	50	}	}	PUNCT
ejpam-5535	171	51	,	,	PUNCT
ejpam-5535	171	52	nd	nd	NOUN
ejpam-5535	171	53	=	=	SYM
ejpam-5535	171	54	{	{	PUNCT
ejpam-5535	171	55	i	i	PROPN
ejpam-5535	171	56	,	,	PUNCT
ejpam-5535	171	57	j	j	PROPN
ejpam-5535	171	58	,	,	PUNCT
ejpam-5535	171	59	k	k	PROPN
ejpam-5535	171	60	,	,	PUNCT
ejpam-5535	171	61	1	1	NUM
ejpam-5535	171	62	}	}	PUNCT
ejpam-5535	171	63	,	,	PUNCT
ejpam-5535	171	64	ne	ne	X
ejpam-5535	171	65	=	=	PUNCT
ejpam-5535	171	66	a	a	PRON
ejpam-5535	171	67	are	be	AUX
ejpam-5535	171	68	filters	filter	NOUN
ejpam-5535	171	69	of	of	ADP
ejpam-5535	171	70	a.	a.	NOUN
ejpam-5535	172	1	moreover	moreover	ADV
ejpam-5535	172	2	nd	nd	ADV
ejpam-5535	172	3	is	be	AUX
ejpam-5535	172	4	a	a	DET
ejpam-5535	172	5	kg	kg	NOUN
ejpam-5535	172	6	-	-	NOUN
ejpam-5535	172	7	filter	filter	NOUN
ejpam-5535	172	8	but	but	CCONJ
ejpam-5535	172	9	others	other	NOUN
ejpam-5535	172	10	are	be	AUX
ejpam-5535	172	11	not	not	PART
ejpam-5535	172	12	.	.	PUNCT
ejpam-5535	173	1	theorem	theorem	VERB
ejpam-5535	173	2	3.4	3.4	NUM
ejpam-5535	173	3	.	.	PUNCT
ejpam-5535	174	1	let	let	VERB
ejpam-5535	174	2	a	a	PRON
ejpam-5535	174	3	be	be	AUX
ejpam-5535	174	4	a	a	DET
ejpam-5535	174	5	distributive	distributive	ADJ
ejpam-5535	174	6	lattice	lattice	NOUN
ejpam-5535	174	7	with	with	ADP
ejpam-5535	174	8	vgg	vgg	PROPN
ejpam-5535	174	9	=	=	SYM
ejpam-5535	174	10	v	v	NOUN
ejpam-5535	174	11	,	,	PUNCT
ejpam-5535	174	12	for	for	ADP
ejpam-5535	174	13	all	all	DET
ejpam-5535	174	14	v	v	NOUN
ejpam-5535	174	15	∈	∈	NOUN
ejpam-5535	174	16	a.	a.	NOUN
ejpam-5535	174	17	then	then	ADV
ejpam-5535	174	18	for	for	SCONJ
ejpam-5535	174	19	any	any	DET
ejpam-5535	174	20	prime	prime	ADJ
ejpam-5535	174	21	filter	filter	NOUN
ejpam-5535	174	22	containing	contain	VERB
ejpam-5535	174	23	d	d	PROPN
ejpam-5535	174	24	in	in	ADP
ejpam-5535	174	25	a	a	PRON
ejpam-5535	174	26	is	be	AUX
ejpam-5535	174	27	a	a	DET
ejpam-5535	174	28	kg	kg	NOUN
ejpam-5535	174	29	-	-	NOUN
ejpam-5535	174	30	filter	filter	NOUN
ejpam-5535	174	31	.	.	PUNCT
ejpam-5535	175	1	proof	proof	NOUN
ejpam-5535	175	2	.	.	PUNCT
ejpam-5535	176	1	let	let	VERB
ejpam-5535	176	2	n	n	PRON
ejpam-5535	176	3	be	be	AUX
ejpam-5535	176	4	a	a	DET
ejpam-5535	176	5	prime	prime	ADJ
ejpam-5535	176	6	filter	filter	NOUN
ejpam-5535	176	7	containing	contain	VERB
ejpam-5535	176	8	d	d	PROPN
ejpam-5535	176	9	in	in	ADP
ejpam-5535	176	10	a.	a.	NOUN
ejpam-5535	176	11	for	for	ADP
ejpam-5535	176	12	any	any	DET
ejpam-5535	176	13	v	v	NOUN
ejpam-5535	176	14	∈	∈	PROPN
ejpam-5535	176	15	n	n	NOUN
ejpam-5535	176	16	,	,	PUNCT
ejpam-5535	176	17	vg	vg	ADP
ejpam-5535	176	18	∧	∧	PROPN
ejpam-5535	176	19	vgg	vgg	NOUN
ejpam-5535	176	20	=	=	SYM
ejpam-5535	176	21	0	0	SYM
ejpam-5535	176	22	∈	∈	PROPN
ejpam-5535	176	23	a−n	a−n	PROPN
ejpam-5535	176	24	⇒vg	⇒vg	NOUN
ejpam-5535	176	25	∈	∈	PROPN
ejpam-5535	176	26	a−n	a−n	PROPN
ejpam-5535	176	27	or	or	CCONJ
ejpam-5535	176	28	vgg	vgg	PROPN
ejpam-5535	176	29	∈	∈	PROPN
ejpam-5535	176	30	a−n	a−n	PROPN
ejpam-5535	176	31	(	(	PUNCT
ejpam-5535	176	32	since	since	SCONJ
ejpam-5535	176	33	a−n	a−n	PROPN
ejpam-5535	176	34	is	be	AUX
ejpam-5535	176	35	prime	prime	ADJ
ejpam-5535	176	36	ideal	ideal	ADJ
ejpam-5535	176	37	)	)	PUNCT
ejpam-5535	176	38	⇒vg	⇒vg	NOUN
ejpam-5535	176	39	∈	∈	PROPN
ejpam-5535	176	40	a−n	a−n	PROPN
ejpam-5535	176	41	or	or	CCONJ
ejpam-5535	176	42	v	v	ADP
ejpam-5535	176	43	∈	∈	PROPN
ejpam-5535	176	44	a−n	a−n	PROPN
ejpam-5535	176	45	(	(	PUNCT
ejpam-5535	176	46	since	since	SCONJ
ejpam-5535	176	47	vgg	vgg	PROPN
ejpam-5535	176	48	=	=	SYM
ejpam-5535	176	49	v	v	NOUN
ejpam-5535	176	50	)	)	PUNCT
ejpam-5535	176	51	⇒vg	⇒vg	NOUN
ejpam-5535	176	52	∈	∈	PROPN
ejpam-5535	176	53	a−n	a−n	PROPN
ejpam-5535	176	54	(	(	PUNCT
ejpam-5535	176	55	since	since	SCONJ
ejpam-5535	176	56	v	v	NUM
ejpam-5535	176	57	∈	∈	PROPN
ejpam-5535	176	58	n	n	CCONJ
ejpam-5535	176	59	)	)	PUNCT
ejpam-5535	176	60	⇒v	⇒v	NOUN
ejpam-5535	176	61	∈	∈	PROPN
ejpam-5535	176	62	g(a−n	g(a−n	PROPN
ejpam-5535	176	63	)	)	PUNCT
ejpam-5535	176	64	.	.	PUNCT
ejpam-5535	177	1	therefore	therefore	ADV
ejpam-5535	177	2	n	n	PROPN
ejpam-5535	177	3	⊆	⊆	NUM
ejpam-5535	177	4	g(a−n	g(a−n	PROPN
ejpam-5535	177	5	)	)	PUNCT
ejpam-5535	177	6	.	.	PUNCT
ejpam-5535	178	1	let	let	VERB
ejpam-5535	178	2	v	v	NUM
ejpam-5535	178	3	∈	∈	PROPN
ejpam-5535	178	4	g(a−n	g(a−n	PROPN
ejpam-5535	178	5	)	)	PUNCT
ejpam-5535	178	6	.	.	PUNCT
ejpam-5535	179	1	then	then	ADV
ejpam-5535	179	2	vg	vg	ADP
ejpam-5535	179	3	∈	∈	PROPN
ejpam-5535	179	4	a−n	a−n	PROPN
ejpam-5535	179	5	.	.	PUNCT
ejpam-5535	180	1	since	since	SCONJ
ejpam-5535	180	2	d	d	PROPN
ejpam-5535	180	3	⊆	⊆	NUM
ejpam-5535	180	4	n	n	NUM
ejpam-5535	180	5	,	,	PUNCT
ejpam-5535	180	6	v	v	ADP
ejpam-5535	180	7	∨	∨	NOUN
ejpam-5535	180	8	vg	vg	NOUN
ejpam-5535	180	9	in	in	ADP
ejpam-5535	180	10	n	n	PROPN
ejpam-5535	180	11	.	.	PUNCT
ejpam-5535	181	1	then	then	ADV
ejpam-5535	181	2	v	v	NOUN
ejpam-5535	181	3	in	in	ADP
ejpam-5535	181	4	n	n	PRON
ejpam-5535	181	5	or	or	CCONJ
ejpam-5535	181	6	vg	vg	NOUN
ejpam-5535	181	7	in	in	ADP
ejpam-5535	181	8	n	n	PROPN
ejpam-5535	181	9	.	.	PUNCT
ejpam-5535	182	1	therefore	therefore	ADV
ejpam-5535	182	2	v	v	NOUN
ejpam-5535	182	3	in	in	ADP
ejpam-5535	182	4	n(since	n(since	NOUN
ejpam-5535	182	5	vg	vg	NOUN
ejpam-5535	182	6	/∈	/∈	PUNCT
ejpam-5535	182	7	n	n	CCONJ
ejpam-5535	182	8	)	)	PUNCT
ejpam-5535	182	9	.	.	PUNCT
ejpam-5535	183	1	so	so	ADV
ejpam-5535	183	2	that	that	SCONJ
ejpam-5535	183	3	g(a	g(a	PROPN
ejpam-5535	183	4	−	−	PROPN
ejpam-5535	183	5	n	n	CCONJ
ejpam-5535	183	6	)	)	PUNCT
ejpam-5535	183	7	⊆	⊆	NUM
ejpam-5535	183	8	n	n	NOUN
ejpam-5535	183	9	.	.	PUNCT
ejpam-5535	184	1	hence	hence	ADV
ejpam-5535	184	2	g(a−n	g(a−n	PROPN
ejpam-5535	184	3	)	)	PUNCT
ejpam-5535	184	4	=	=	SYM
ejpam-5535	185	1	n	n	NOUN
ejpam-5535	185	2	.	.	PUNCT
ejpam-5535	186	1	thus	thus	ADV
ejpam-5535	186	2	n	n	PRON
ejpam-5535	186	3	is	be	AUX
ejpam-5535	186	4	kg	kg	ADJ
ejpam-5535	186	5	-	-	PUNCT
ejpam-5535	186	6	filter	filter	NOUN
ejpam-5535	186	7	.	.	PUNCT
ejpam-5535	187	1	corollary	corollary	ADJ
ejpam-5535	187	2	3.5	3.5	NUM
ejpam-5535	187	3	.	.	PUNCT
ejpam-5535	188	1	let	let	VERB
ejpam-5535	188	2	a	a	PRON
ejpam-5535	188	3	be	be	AUX
ejpam-5535	188	4	a	a	DET
ejpam-5535	188	5	distributive	distributive	ADJ
ejpam-5535	188	6	lattice	lattice	NOUN
ejpam-5535	188	7	with	with	ADP
ejpam-5535	188	8	vgg	vgg	PROPN
ejpam-5535	188	9	=	=	SYM
ejpam-5535	188	10	v	v	NOUN
ejpam-5535	188	11	,	,	PUNCT
ejpam-5535	188	12	for	for	ADP
ejpam-5535	188	13	all	all	DET
ejpam-5535	188	14	v	v	NOUN
ejpam-5535	188	15	∈	∈	NOUN
ejpam-5535	188	16	a.	a.	NOUN
ejpam-5535	188	17	then	then	ADV
ejpam-5535	188	18	every	every	DET
ejpam-5535	188	19	maximal	maximal	ADJ
ejpam-5535	188	20	filter	filter	NOUN
ejpam-5535	188	21	is	be	AUX
ejpam-5535	188	22	a	a	DET
ejpam-5535	188	23	kg	kg	NOUN
ejpam-5535	188	24	-	-	NOUN
ejpam-5535	188	25	filter	filter	NOUN
ejpam-5535	188	26	.	.	PUNCT
ejpam-5535	189	1	let	let	VERB
ejpam-5535	189	2	us	we	PRON
ejpam-5535	189	3	denote	denote	VERB
ejpam-5535	189	4	the	the	DET
ejpam-5535	189	5	set	set	NOUN
ejpam-5535	189	6	of	of	ADP
ejpam-5535	189	7	kg	kg	NOUN
ejpam-5535	189	8	-	-	PUNCT
ejpam-5535	189	9	filters	filter	NOUN
ejpam-5535	189	10	in	in	ADP
ejpam-5535	189	11	a	a	DET
ejpam-5535	189	12	distributive	distributive	ADJ
ejpam-5535	189	13	lattice	lattice	NOUN
ejpam-5535	189	14	a	a	PRON
ejpam-5535	189	15	with	with	ADP
ejpam-5535	189	16	dense	dense	ADJ
ejpam-5535	189	17	elements	element	NOUN
ejpam-5535	189	18	by	by	ADP
ejpam-5535	189	19	gf(a	gf(a	NOUN
ejpam-5535	189	20	)	)	PUNCT
ejpam-5535	189	21	.	.	PUNCT
ejpam-5535	190	1	now	now	ADV
ejpam-5535	190	2	we	we	PRON
ejpam-5535	190	3	have	have	VERB
ejpam-5535	190	4	the	the	DET
ejpam-5535	190	5	following	follow	VERB
ejpam-5535	190	6	theorem	theorem	VERB
ejpam-5535	190	7	.	.	PUNCT
ejpam-5535	190	8	theorem	theorem	VERB
ejpam-5535	190	9	3.6	3.6	NUM
ejpam-5535	190	10	.	.	PUNCT
ejpam-5535	190	11	gf(a	gf(a	PUNCT
ejpam-5535	190	12	)	)	PUNCT
ejpam-5535	190	13	is	be	AUX
ejpam-5535	190	14	a	a	DET
ejpam-5535	190	15	distributive	distributive	ADJ
ejpam-5535	190	16	lattice	lattice	NOUN
ejpam-5535	190	17	with	with	ADP
ejpam-5535	190	18	the	the	DET
ejpam-5535	190	19	operations	operation	NOUN
ejpam-5535	190	20	g(ka	g(ka	NOUN
ejpam-5535	190	21	)	)	PUNCT
ejpam-5535	190	22	∩	∩	NOUN
ejpam-5535	190	23	g(kb	g(kb	X
ejpam-5535	190	24	)	)	PUNCT
ejpam-5535	190	25	=	=	SYM
ejpam-5535	190	26	g(ka	g(ka	PROPN
ejpam-5535	190	27	∧kb	∧kb	PROPN
ejpam-5535	190	28	)	)	PUNCT
ejpam-5535	190	29	and	and	CCONJ
ejpam-5535	190	30	g(ka	g(ka	NOUN
ejpam-5535	190	31	)	)	PUNCT
ejpam-5535	190	32	⊔g(kb	⊔g(kb	X
ejpam-5535	190	33	)	)	PUNCT
ejpam-5535	191	1	=	=	SYM
ejpam-5535	191	2	g(ka	g(ka	NOUN
ejpam-5535	191	3	∨kb	∨kb	NOUN
ejpam-5535	191	4	)	)	PUNCT
ejpam-5535	191	5	,	,	PUNCT
ejpam-5535	191	6	for	for	ADP
ejpam-5535	191	7	all	all	DET
ejpam-5535	191	8	g(ka	g(ka	NOUN
ejpam-5535	191	9	)	)	PUNCT
ejpam-5535	191	10	,	,	PUNCT
ejpam-5535	191	11	g(kb	g(kb	X
ejpam-5535	191	12	)	)	PUNCT
ejpam-5535	191	13	∈	∈	PROPN
ejpam-5535	191	14	gf(a	gf(a	NOUN
ejpam-5535	191	15	)	)	PUNCT
ejpam-5535	191	16	.	.	PUNCT
ejpam-5535	192	1	r.	r.	PROPN
ejpam-5535	192	2	sirisetti	sirisetti	PROPN
ejpam-5535	192	3	et	et	PROPN
ejpam-5535	192	4	al	al	PROPN
ejpam-5535	192	5	.	.	PUNCT
ejpam-5535	192	6	/	/	SYM
ejpam-5535	192	7	eur	eur	PROPN
ejpam-5535	192	8	.	.	PUNCT
ejpam-5535	193	1	j.	j.	PROPN
ejpam-5535	193	2	pure	pure	PROPN
ejpam-5535	193	3	appl	appl	PROPN
ejpam-5535	193	4	.	.	PROPN
ejpam-5535	193	5	math	math	PROPN
ejpam-5535	193	6	,	,	PUNCT
ejpam-5535	193	7	18	18	NUM
ejpam-5535	193	8	(	(	PUNCT
ejpam-5535	193	9	1	1	NUM
ejpam-5535	193	10	)	)	PUNCT
ejpam-5535	193	11	(	(	PUNCT
ejpam-5535	193	12	2025	2025	NUM
ejpam-5535	193	13	)	)	PUNCT
ejpam-5535	193	14	,	,	PUNCT
ejpam-5535	193	15	5535	5535	NUM
ejpam-5535	193	16	8	8	NUM
ejpam-5535	193	17	of	of	ADP
ejpam-5535	193	18	12	12	NUM
ejpam-5535	193	19	proof	proof	NOUN
ejpam-5535	193	20	.	.	PUNCT
ejpam-5535	194	1	letg(ka	letg(ka	ADJ
ejpam-5535	194	2	)	)	PUNCT
ejpam-5535	194	3	,	,	PUNCT
ejpam-5535	194	4	g(kb	g(kb	X
ejpam-5535	194	5	)	)	PUNCT
ejpam-5535	194	6	∈	∈	PROPN
ejpam-5535	194	7	gf(a	gf(a	NOUN
ejpam-5535	194	8	)	)	PUNCT
ejpam-5535	194	9	,	,	PUNCT
ejpam-5535	194	10	whereka	whereka	PROPN
ejpam-5535	194	11	,	,	PUNCT
ejpam-5535	194	12	kb	kb	PROPN
ejpam-5535	194	13	in	in	ADP
ejpam-5535	194	14	i(a	i(a	PROPN
ejpam-5535	194	15	)	)	PUNCT
ejpam-5535	194	16	.	.	PUNCT
ejpam-5535	195	1	by	by	ADP
ejpam-5535	195	2	lemma	lemma	PROPN
ejpam-5535	195	3	2.4(i	2.4(i	NUM
ejpam-5535	195	4	)	)	PUNCT
ejpam-5535	195	5	,	,	PUNCT
ejpam-5535	195	6	g(ka	g(ka	NOUN
ejpam-5535	195	7	)	)	PUNCT
ejpam-5535	195	8	,	,	PUNCT
ejpam-5535	195	9	g(kb	g(kb	X
ejpam-5535	195	10	)	)	PUNCT
ejpam-5535	195	11	⊆	⊆	NUM
ejpam-5535	195	12	g(ka	g(ka	PROPN
ejpam-5535	195	13	∨	∨	NUM
ejpam-5535	195	14	kb	kb	PROPN
ejpam-5535	195	15	)	)	PUNCT
ejpam-5535	195	16	.	.	PUNCT
ejpam-5535	196	1	therefore	therefore	ADV
ejpam-5535	196	2	g(ka	g(ka	PROPN
ejpam-5535	196	3	∨	∨	NUM
ejpam-5535	196	4	kb	kb	PROPN
ejpam-5535	196	5	)	)	PUNCT
ejpam-5535	196	6	is	be	AUX
ejpam-5535	196	7	an	an	DET
ejpam-5535	196	8	upper	upper	ADJ
ejpam-5535	196	9	bound	bound	NOUN
ejpam-5535	196	10	of	of	ADP
ejpam-5535	196	11	g(ka	g(ka	NOUN
ejpam-5535	196	12	)	)	PUNCT
ejpam-5535	196	13	,	,	PUNCT
ejpam-5535	196	14	g(kb	g(kb	X
ejpam-5535	196	15	)	)	PUNCT
ejpam-5535	196	16	.	.	PUNCT
ejpam-5535	197	1	let	let	VERB
ejpam-5535	197	2	g(kc	g(kc	PRON
ejpam-5535	197	3	)	)	PUNCT
ejpam-5535	197	4	∈	∈	PROPN
ejpam-5535	197	5	gf(a	gf(a	X
ejpam-5535	197	6	)	)	PUNCT
ejpam-5535	197	7	is	be	AUX
ejpam-5535	197	8	an	an	DET
ejpam-5535	197	9	upper	upper	ADJ
ejpam-5535	197	10	bound	bound	NOUN
ejpam-5535	197	11	of	of	ADP
ejpam-5535	197	12	g(ka	g(ka	NOUN
ejpam-5535	197	13	)	)	PUNCT
ejpam-5535	197	14	,	,	PUNCT
ejpam-5535	197	15	g(kb	g(kb	X
ejpam-5535	197	16	)	)	PUNCT
ejpam-5535	197	17	,	,	PUNCT
ejpam-5535	197	18	for	for	ADP
ejpam-5535	197	19	some	some	DET
ejpam-5535	197	20	kc	kc	PROPN
ejpam-5535	197	21	∈	∈	PROPN
ejpam-5535	197	22	i(a	i(a	PROPN
ejpam-5535	197	23	)	)	PUNCT
ejpam-5535	197	24	.	.	PUNCT
ejpam-5535	198	1	for	for	ADP
ejpam-5535	198	2	v	v	NOUN
ejpam-5535	198	3	∈	∈	PROPN
ejpam-5535	198	4	g(ka	g(ka	NOUN
ejpam-5535	198	5	∨kb	∨kb	NOUN
ejpam-5535	198	6	)	)	PUNCT
ejpam-5535	198	7	,	,	PUNCT
ejpam-5535	198	8	vg	vg	PROPN
ejpam-5535	198	9	∈	∈	PROPN
ejpam-5535	198	10	ka	ka	PROPN
ejpam-5535	198	11	∨kb	∨kb	NOUN
ejpam-5535	198	12	⇒vg	⇒vg	NOUN
ejpam-5535	198	13	=	=	SYM
ejpam-5535	199	1	i	i	PROPN
ejpam-5535	199	2	∨	∨	PROPN
ejpam-5535	199	3	j	j	PROPN
ejpam-5535	199	4	for	for	ADP
ejpam-5535	199	5	some	some	DET
ejpam-5535	199	6	i	i	PRON
ejpam-5535	199	7	∈	∈	PROPN
ejpam-5535	199	8	ka	ka	PROPN
ejpam-5535	199	9	and	and	CCONJ
ejpam-5535	199	10	j	j	PROPN
ejpam-5535	199	11	∈	∈	PROPN
ejpam-5535	199	12	kb	kb	PROPN
ejpam-5535	199	13	⇒igg	⇒igg	PROPN
ejpam-5535	199	14	∈	∈	PROPN
ejpam-5535	199	15	ka	ka	PROPN
ejpam-5535	199	16	and	and	CCONJ
ejpam-5535	199	17	jgg	jgg	PROPN
ejpam-5535	199	18	∈	∈	PROPN
ejpam-5535	199	19	kb	kb	PROPN
ejpam-5535	199	20	(	(	PUNCT
ejpam-5535	199	21	since	since	SCONJ
ejpam-5535	199	22	vgg	vgg	PROPN
ejpam-5535	199	23	≤	≤	NUM
ejpam-5535	199	24	v	v	ADP
ejpam-5535	199	25	forall	forall	NOUN
ejpam-5535	199	26	v	v	ADP
ejpam-5535	199	27	∈	∈	PROPN
ejpam-5535	199	28	a	a	DET
ejpam-5535	199	29	)	)	PUNCT
ejpam-5535	199	30	⇒ig	⇒ig	NOUN
ejpam-5535	199	31	∈	∈	PROPN
ejpam-5535	199	32	g(ka	g(ka	NOUN
ejpam-5535	199	33	)	)	PUNCT
ejpam-5535	199	34	and	and	CCONJ
ejpam-5535	199	35	jg	jg	PROPN
ejpam-5535	199	36	∈	∈	PROPN
ejpam-5535	199	37	g(kb	g(kb	PROPN
ejpam-5535	199	38	)	)	PUNCT
ejpam-5535	199	39	⇒ig	⇒ig	PROPN
ejpam-5535	199	40	∈	∈	PROPN
ejpam-5535	199	41	g(kc	g(kc	PROPN
ejpam-5535	199	42	)	)	PUNCT
ejpam-5535	199	43	and	and	CCONJ
ejpam-5535	199	44	jg	jg	PROPN
ejpam-5535	199	45	∈	∈	PROPN
ejpam-5535	199	46	g(kc	g(kc	PROPN
ejpam-5535	199	47	)	)	PUNCT
ejpam-5535	199	48	(	(	PUNCT
ejpam-5535	199	49	since	since	SCONJ
ejpam-5535	199	50	g(ka	g(ka	NOUN
ejpam-5535	199	51	)	)	PUNCT
ejpam-5535	199	52	,	,	PUNCT
ejpam-5535	199	53	g(kb	g(kb	X
ejpam-5535	199	54	)	)	PUNCT
ejpam-5535	199	55	⊆	⊆	NUM
ejpam-5535	199	56	g(kc	g(kc	NUM
ejpam-5535	199	57	)	)	PUNCT
ejpam-5535	199	58	)	)	PUNCT
ejpam-5535	200	1	⇒igg	⇒igg	PROPN
ejpam-5535	200	2	∨	∨	NUM
ejpam-5535	200	3	jgg	jgg	PROPN
ejpam-5535	200	4	∈	∈	PROPN
ejpam-5535	200	5	kc	kc	PROPN
ejpam-5535	200	6	(	(	PUNCT
ejpam-5535	200	7	since	since	SCONJ
ejpam-5535	200	8	kc	kc	PROPN
ejpam-5535	200	9	is	be	AUX
ejpam-5535	200	10	an	an	DET
ejpam-5535	200	11	ideal	ideal	NOUN
ejpam-5535	200	12	)	)	PUNCT
ejpam-5535	200	13	⇒(i	⇒(i	VERB
ejpam-5535	201	1	∨	∨	NUM
ejpam-5535	201	2	j)gg	j)gg	PROPN
ejpam-5535	201	3	∈	∈	PROPN
ejpam-5535	201	4	kc	kc	PROPN
ejpam-5535	201	5	(	(	PUNCT
ejpam-5535	201	6	since	since	SCONJ
ejpam-5535	201	7	proportion	proportion	NOUN
ejpam-5535	201	8	2.3(iii	2.3(iii	NUM
ejpam-5535	201	9	)	)	PUNCT
ejpam-5535	201	10	)	)	PUNCT
ejpam-5535	201	11	⇒(vg)gg	⇒(vg)gg	NOUN
ejpam-5535	201	12	∈	∈	PROPN
ejpam-5535	201	13	kc	kc	PROPN
ejpam-5535	201	14	(	(	PUNCT
ejpam-5535	201	15	since	since	SCONJ
ejpam-5535	201	16	vg	vg	NOUN
ejpam-5535	201	17	=	=	PUNCT
ejpam-5535	201	18	i	i	PROPN
ejpam-5535	201	19	∨	∨	PROPN
ejpam-5535	201	20	j	j	NOUN
ejpam-5535	201	21	)	)	PUNCT
ejpam-5535	201	22	⇒vggg	⇒vggg	NOUN
ejpam-5535	201	23	∈	∈	PROPN
ejpam-5535	201	24	kc	kc	PROPN
ejpam-5535	201	25	⇒vg	⇒vg	PROPN
ejpam-5535	201	26	∈	∈	PROPN
ejpam-5535	201	27	kc	kc	PROPN
ejpam-5535	201	28	(	(	PUNCT
ejpam-5535	201	29	since	since	SCONJ
ejpam-5535	201	30	vggg	vggg	NOUN
ejpam-5535	201	31	=	=	SYM
ejpam-5535	201	32	vg	vg	NOUN
ejpam-5535	201	33	)	)	PUNCT
ejpam-5535	201	34	⇒v	⇒v	NOUN
ejpam-5535	201	35	∈	∈	PROPN
ejpam-5535	201	36	g(kc	g(kc	PROPN
ejpam-5535	201	37	)	)	PUNCT
ejpam-5535	201	38	.	.	PUNCT
ejpam-5535	202	1	therefore	therefore	ADV
ejpam-5535	202	2	g(ka∨kb	g(ka∨kb	X
ejpam-5535	202	3	)	)	PUNCT
ejpam-5535	202	4	⊆	⊆	NUM
ejpam-5535	202	5	g(kc	g(kc	NUM
ejpam-5535	202	6	)	)	PUNCT
ejpam-5535	202	7	and	and	CCONJ
ejpam-5535	202	8	hence	hence	ADV
ejpam-5535	202	9	g(ka∨kb	g(ka∨kb	NUM
ejpam-5535	202	10	)	)	PUNCT
ejpam-5535	202	11	is	be	AUX
ejpam-5535	202	12	the	the	DET
ejpam-5535	202	13	least	least	ADJ
ejpam-5535	202	14	upper	upper	ADJ
ejpam-5535	202	15	bound	bind	VERB
ejpam-5535	202	16	of	of	ADP
ejpam-5535	202	17	g(ka	g(ka	NOUN
ejpam-5535	202	18	)	)	PUNCT
ejpam-5535	202	19	,	,	PUNCT
ejpam-5535	202	20	g(kb	g(kb	X
ejpam-5535	202	21	)	)	PUNCT
ejpam-5535	202	22	.	.	PUNCT
ejpam-5535	203	1	hence	hence	ADV
ejpam-5535	203	2	it	it	PRON
ejpam-5535	203	3	is	be	AUX
ejpam-5535	203	4	denoted	denote	VERB
ejpam-5535	203	5	by	by	ADP
ejpam-5535	203	6	g(ka	g(ka	NOUN
ejpam-5535	203	7	)	)	PUNCT
ejpam-5535	203	8	⊔	⊔	NUM
ejpam-5535	203	9	g(kb	g(kb	NOUN
ejpam-5535	203	10	)	)	PUNCT
ejpam-5535	203	11	.	.	PUNCT
ejpam-5535	204	1	by	by	ADP
ejpam-5535	204	2	lemma	lemma	PROPN
ejpam-5535	204	3	2.4(ii	2.4(ii	NUM
ejpam-5535	204	4	)	)	PUNCT
ejpam-5535	204	5	.	.	PUNCT
ejpam-5535	205	1	,	,	PUNCT
ejpam-5535	205	2	g(ka	g(ka	NOUN
ejpam-5535	205	3	)	)	PUNCT
ejpam-5535	205	4	∧	∧	NOUN
ejpam-5535	205	5	g(kb	g(kb	NOUN
ejpam-5535	205	6	)	)	PUNCT
ejpam-5535	205	7	=	=	SYM
ejpam-5535	205	8	g(ka∧kb	g(ka∧kb	NOUN
ejpam-5535	205	9	)	)	PUNCT
ejpam-5535	205	10	.	.	PUNCT
ejpam-5535	206	1	since	since	SCONJ
ejpam-5535	206	2	i(a	i(a	PROPN
ejpam-5535	206	3	)	)	PUNCT
ejpam-5535	206	4	is	be	AUX
ejpam-5535	206	5	distributive	distributive	ADJ
ejpam-5535	206	6	,	,	PUNCT
ejpam-5535	206	7	gf(a	gf(a	PUNCT
ejpam-5535	206	8	)	)	PUNCT
ejpam-5535	206	9	is	be	AUX
ejpam-5535	206	10	distributive	distributive	ADJ
ejpam-5535	206	11	.	.	PUNCT
ejpam-5535	207	1	hence	hence	ADV
ejpam-5535	207	2	gf(a	gf(a	PUNCT
ejpam-5535	207	3	)	)	PUNCT
ejpam-5535	207	4	is	be	AUX
ejpam-5535	207	5	a	a	DET
ejpam-5535	207	6	distributive	distributive	ADJ
ejpam-5535	207	7	lattice	lattice	NOUN
ejpam-5535	207	8	with	with	ADP
ejpam-5535	207	9	the	the	DET
ejpam-5535	207	10	lease	lease	NOUN
ejpam-5535	207	11	element	element	NOUN
ejpam-5535	207	12	g({0	g({0	NOUN
ejpam-5535	207	13	}	}	PUNCT
ejpam-5535	207	14	)	)	PUNCT
ejpam-5535	207	15	and	and	CCONJ
ejpam-5535	207	16	the	the	DET
ejpam-5535	207	17	greatest	great	ADJ
ejpam-5535	207	18	element	element	NOUN
ejpam-5535	207	19	g(a	g(a	PROPN
ejpam-5535	207	20	)	)	PUNCT
ejpam-5535	207	21	.	.	PUNCT
ejpam-5535	208	1	corollary	corollary	ADJ
ejpam-5535	208	2	3.7	3.7	NUM
ejpam-5535	208	3	.	.	PUNCT
ejpam-5535	209	1	if	if	SCONJ
ejpam-5535	209	2	i(a	i(a	PROPN
ejpam-5535	209	3	)	)	PUNCT
ejpam-5535	209	4	is	be	AUX
ejpam-5535	209	5	a	a	DET
ejpam-5535	209	6	boolean	boolean	ADJ
ejpam-5535	209	7	algebra	algebra	NOUN
ejpam-5535	209	8	,	,	PUNCT
ejpam-5535	209	9	then	then	ADV
ejpam-5535	209	10	gf(a	gf(a	PUNCT
ejpam-5535	209	11	)	)	PUNCT
ejpam-5535	209	12	is	be	AUX
ejpam-5535	209	13	a	a	DET
ejpam-5535	209	14	boolean	boolean	ADJ
ejpam-5535	209	15	algebra	algebra	NOUN
ejpam-5535	209	16	,	,	PUNCT
ejpam-5535	209	17	but	but	CCONJ
ejpam-5535	209	18	not	not	PART
ejpam-5535	209	19	a	a	DET
ejpam-5535	209	20	sub	sub	ADJ
ejpam-5535	209	21	-	-	ADJ
ejpam-5535	209	22	boolean	boolean	ADJ
ejpam-5535	209	23	algebra	algebra	NOUN
ejpam-5535	209	24	of	of	ADP
ejpam-5535	209	25	i(a	i(a	PROPN
ejpam-5535	209	26	)	)	PUNCT
ejpam-5535	209	27	.	.	PUNCT
ejpam-5535	210	1	lemma	lemma	PROPN
ejpam-5535	210	2	3.8	3.8	NUM
ejpam-5535	210	3	.	.	PUNCT
ejpam-5535	211	1	for	for	ADP
ejpam-5535	211	2	any	any	DET
ejpam-5535	211	3	v	v	NOUN
ejpam-5535	211	4	∈	∈	PROPN
ejpam-5535	211	5	a	a	DET
ejpam-5535	211	6	,	,	PUNCT
ejpam-5535	211	7	g((vgg	g((vgg	NOUN
ejpam-5535	211	8	]	]	PUNCT
ejpam-5535	211	9	)	)	PUNCT
ejpam-5535	211	10	=	=	SYM
ejpam-5535	211	11	g((v	g((v	NOUN
ejpam-5535	211	12	]	]	X
ejpam-5535	211	13	)	)	PUNCT
ejpam-5535	211	14	.	.	PUNCT
ejpam-5535	212	1	proof	proof	NOUN
ejpam-5535	212	2	.	.	PUNCT
ejpam-5535	213	1	let	let	VERB
ejpam-5535	213	2	v	v	NUM
ejpam-5535	213	3	∈	∈	NOUN
ejpam-5535	213	4	a.	a.	NOUN
ejpam-5535	213	5	then	then	ADV
ejpam-5535	213	6	vgg	vgg	VERB
ejpam-5535	213	7	≤	≤	PROPN
ejpam-5535	214	1	v.	v.	CCONJ
ejpam-5535	214	2	therefore	therefore	ADV
ejpam-5535	214	3	(	(	PUNCT
ejpam-5535	214	4	vgg	vgg	PROPN
ejpam-5535	214	5	]	]	X
ejpam-5535	214	6	⊆	⊆	NUM
ejpam-5535	214	7	(	(	PUNCT
ejpam-5535	214	8	v	v	NOUN
ejpam-5535	214	9	]	]	PUNCT
ejpam-5535	214	10	.	.	PUNCT
ejpam-5535	215	1	by	by	ADP
ejpam-5535	215	2	lemma	lemma	PROPN
ejpam-5535	215	3	2.4(ii	2.4(ii	NUM
ejpam-5535	215	4	)	)	PUNCT
ejpam-5535	215	5	.	.	PUNCT
ejpam-5535	215	6	,	,	PUNCT
ejpam-5535	215	7	g((vgg	g((vgg	NOUN
ejpam-5535	215	8	]	]	PUNCT
ejpam-5535	215	9	)	)	PUNCT
ejpam-5535	215	10	⊆	⊆	NUM
ejpam-5535	215	11	g((v	g((v	NOUN
ejpam-5535	215	12	]	]	PUNCT
ejpam-5535	215	13	)	)	PUNCT
ejpam-5535	215	14	.	.	PUNCT
ejpam-5535	216	1	let	let	VERB
ejpam-5535	216	2	l	l	NOUN
ejpam-5535	216	3	∈	∈	PROPN
ejpam-5535	216	4	g((v	g((v	NOUN
ejpam-5535	216	5	]	]	PUNCT
ejpam-5535	216	6	)	)	PUNCT
ejpam-5535	216	7	.	.	PUNCT
ejpam-5535	217	1	then	then	ADV
ejpam-5535	217	2	lg	lg	PROPN
ejpam-5535	217	3	∈	∈	PROPN
ejpam-5535	217	4	(	(	PUNCT
ejpam-5535	217	5	v	v	NOUN
ejpam-5535	217	6	]	]	PUNCT
ejpam-5535	217	7	and	and	CCONJ
ejpam-5535	217	8	hence	hence	ADV
ejpam-5535	217	9	lg	lg	NOUN
ejpam-5535	217	10	≤	≤	PROPN
ejpam-5535	217	11	v.	v.	ADP
ejpam-5535	217	12	by	by	ADP
ejpam-5535	217	13	proposition	proposition	NOUN
ejpam-5535	217	14	2.1	2.1	NUM
ejpam-5535	217	15	.	.	PUNCT
ejpam-5535	217	16	,	,	PUNCT
ejpam-5535	217	17	lggg	lggg	NOUN
ejpam-5535	217	18	=	=	PUNCT
ejpam-5535	217	19	lg	lg	PROPN
ejpam-5535	217	20	≤	≤	PROPN
ejpam-5535	217	21	vgg	vgg	ADJ
ejpam-5535	217	22	≤	≤	NUM
ejpam-5535	217	23	v.	v.	CCONJ
ejpam-5535	218	1	so	so	SCONJ
ejpam-5535	218	2	that	that	SCONJ
ejpam-5535	218	3	lg	lg	PROPN
ejpam-5535	218	4	∈	∈	PROPN
ejpam-5535	218	5	(	(	PUNCT
ejpam-5535	218	6	vgg	vgg	NOUN
ejpam-5535	218	7	]	]	X
ejpam-5535	218	8	.	.	PUNCT
ejpam-5535	219	1	hence	hence	ADV
ejpam-5535	219	2	l	l	PROPN
ejpam-5535	219	3	∈	∈	PROPN
ejpam-5535	219	4	g((vgg	g((vgg	NOUN
ejpam-5535	219	5	]	]	PUNCT
ejpam-5535	219	6	)	)	PUNCT
ejpam-5535	219	7	.	.	PUNCT
ejpam-5535	220	1	thus	thus	ADV
ejpam-5535	220	2	g((v	g((v	NOUN
ejpam-5535	220	3	]	]	PUNCT
ejpam-5535	220	4	)	)	PUNCT
ejpam-5535	220	5	=	=	SYM
ejpam-5535	220	6	g((vgg	g((vgg	NOUN
ejpam-5535	220	7	]	]	PUNCT
ejpam-5535	220	8	)	)	PUNCT
ejpam-5535	220	9	.	.	PUNCT
ejpam-5535	221	1	lemma	lemma	PROPN
ejpam-5535	221	2	3.9	3.9	NUM
ejpam-5535	221	3	.	.	PUNCT
ejpam-5535	222	1	for	for	ADP
ejpam-5535	222	2	any	any	DET
ejpam-5535	222	3	v	v	NOUN
ejpam-5535	222	4	,	,	PUNCT
ejpam-5535	222	5	w	w	NOUN
ejpam-5535	222	6	in	in	ADP
ejpam-5535	222	7	a	a	PRON
ejpam-5535	222	8	,	,	PUNCT
ejpam-5535	222	9	the	the	DET
ejpam-5535	222	10	following	follow	VERB
ejpam-5535	222	11	(	(	PUNCT
ejpam-5535	222	12	1	1	NUM
ejpam-5535	222	13	.	.	NUM
ejpam-5535	222	14	)	)	PUNCT
ejpam-5535	222	15	,	,	PUNCT
ejpam-5535	222	16	and	and	CCONJ
ejpam-5535	222	17	(	(	PUNCT
ejpam-5535	222	18	2	2	NUM
ejpam-5535	222	19	.	.	PUNCT
ejpam-5535	222	20	)	)	PUNCT
ejpam-5535	222	21	are	be	AUX
ejpam-5535	222	22	equivalent	equivalent	ADJ
ejpam-5535	222	23	;	;	PUNCT
ejpam-5535	222	24	(	(	PUNCT
ejpam-5535	222	25	1	1	NUM
ejpam-5535	222	26	)	)	PUNCT
ejpam-5535	222	27	.	.	PUNCT
ejpam-5535	223	1	(	(	PUNCT
ejpam-5535	223	2	v	v	X
ejpam-5535	223	3	∧	∧	PROPN
ejpam-5535	223	4	w)gg	w)gg	PROPN
ejpam-5535	223	5	=	=	SYM
ejpam-5535	223	6	vgg	vgg	PROPN
ejpam-5535	223	7	∧	∧	PROPN
ejpam-5535	223	8	wgg	wgg	NOUN
ejpam-5535	223	9	(	(	PUNCT
ejpam-5535	223	10	2	2	NUM
ejpam-5535	223	11	)	)	PUNCT
ejpam-5535	223	12	.	.	PUNCT
ejpam-5535	224	1	(	(	PUNCT
ejpam-5535	224	2	v	v	NOUN
ejpam-5535	224	3	∨	∨	NOUN
ejpam-5535	224	4	w)g	w)g	PUNCT
ejpam-5535	225	1	=	=	PUNCT
ejpam-5535	225	2	vg	vg	ADP
ejpam-5535	225	3	∧	∧	PROPN
ejpam-5535	225	4	wg	wg	PROPN
ejpam-5535	225	5	.	.	PUNCT
ejpam-5535	226	1	proof	proof	NOUN
ejpam-5535	226	2	.	.	PUNCT
ejpam-5535	227	1	(	(	PUNCT
ejpam-5535	227	2	1	1	X
ejpam-5535	227	3	)	)	PUNCT
ejpam-5535	227	4	⇒	⇒	NOUN
ejpam-5535	227	5	(	(	PUNCT
ejpam-5535	227	6	2	2	NUM
ejpam-5535	227	7	)	)	PUNCT
ejpam-5535	227	8	:	:	PUNCT
ejpam-5535	227	9	assume(1	assume(1	X
ejpam-5535	227	10	)	)	PUNCT
ejpam-5535	227	11	,	,	PUNCT
ejpam-5535	227	12	let	let	VERB
ejpam-5535	227	13	v	v	ADP
ejpam-5535	227	14	,	,	PUNCT
ejpam-5535	227	15	w	w	PROPN
ejpam-5535	227	16	∈	∈	PROPN
ejpam-5535	227	17	a.	a.	NOUN
ejpam-5535	227	18	then	then	ADV
ejpam-5535	227	19	(	(	PUNCT
ejpam-5535	227	20	v∨w)g	v∨w)g	NOUN
ejpam-5535	227	21	=	=	PUNCT
ejpam-5535	227	22	(	(	PUNCT
ejpam-5535	227	23	v∨w)ggg	v∨w)ggg	NOUN
ejpam-5535	227	24	=	=	SYM
ejpam-5535	227	25	(	(	PUNCT
ejpam-5535	227	26	vgg∨wgg)g	vgg∨wgg)g	NOUN
ejpam-5535	227	27	=	=	SYM
ejpam-5535	227	28	(	(	PUNCT
ejpam-5535	227	29	vg	vg	ADP
ejpam-5535	227	30	∧	∧	PROPN
ejpam-5535	227	31	wg)gg	wg)gg	NUM
ejpam-5535	227	32	=	=	SYM
ejpam-5535	227	33	vggg	vggg	NOUN
ejpam-5535	227	34	∧	∧	NOUN
ejpam-5535	227	35	wggg	wggg	NOUN
ejpam-5535	227	36	=	=	PUNCT
ejpam-5535	227	37	vg	vg	ADP
ejpam-5535	227	38	∧	∧	PROPN
ejpam-5535	227	39	wg	wg	PROPN
ejpam-5535	227	40	.	.	PUNCT
ejpam-5535	228	1	(	(	PUNCT
ejpam-5535	228	2	2	2	X
ejpam-5535	228	3	)	)	PUNCT
ejpam-5535	228	4	⇒	⇒	NOUN
ejpam-5535	228	5	(	(	PUNCT
ejpam-5535	228	6	1	1	NUM
ejpam-5535	228	7	):	):	PUNCT
ejpam-5535	228	8	assume	assume	VERB
ejpam-5535	228	9	(	(	PUNCT
ejpam-5535	228	10	2	2	NUM
ejpam-5535	228	11	)	)	PUNCT
ejpam-5535	228	12	,	,	PUNCT
ejpam-5535	228	13	let	let	VERB
ejpam-5535	228	14	v	v	ADP
ejpam-5535	228	15	,	,	PUNCT
ejpam-5535	228	16	w	w	PROPN
ejpam-5535	228	17	∈	∈	PROPN
ejpam-5535	228	18	a.	a.	NOUN
ejpam-5535	228	19	then	then	ADV
ejpam-5535	228	20	(	(	PUNCT
ejpam-5535	228	21	v	v	NUM
ejpam-5535	228	22	∧	∧	PROPN
ejpam-5535	228	23	w)gg	w)gg	PROPN
ejpam-5535	229	1	=	=	SYM
ejpam-5535	229	2	(	(	PUNCT
ejpam-5535	229	3	vg	vg	ADP
ejpam-5535	229	4	∨	∨	NUM
ejpam-5535	229	5	wg)g	wg)g	PROPN
ejpam-5535	229	6	=	=	SYM
ejpam-5535	229	7	vgg	vgg	ADJ
ejpam-5535	229	8	∧	∧	PROPN
ejpam-5535	229	9	wgg	wgg	NOUN
ejpam-5535	229	10	.	.	PUNCT
ejpam-5535	230	1	lemma	lemma	PROPN
ejpam-5535	230	2	3.10	3.10	NUM
ejpam-5535	230	3	.	.	PUNCT
ejpam-5535	231	1	for	for	ADP
ejpam-5535	231	2	any	any	DET
ejpam-5535	231	3	v	v	NOUN
ejpam-5535	231	4	,	,	PUNCT
ejpam-5535	231	5	w	w	PROPN
ejpam-5535	231	6	∈	∈	PROPN
ejpam-5535	231	7	a	a	X
ejpam-5535	231	8	,	,	PUNCT
ejpam-5535	231	9	we	we	PRON
ejpam-5535	231	10	have	have	VERB
ejpam-5535	231	11	(	(	PUNCT
ejpam-5535	231	12	i	i	NOUN
ejpam-5535	231	13	)	)	PUNCT
ejpam-5535	231	14	g((v	g((v	NOUN
ejpam-5535	231	15	]	]	X
ejpam-5535	231	16	)	)	PUNCT
ejpam-5535	231	17	=	=	SYM
ejpam-5535	231	18	g((vgg	g((vgg	NOUN
ejpam-5535	231	19	]	]	PUNCT
ejpam-5535	231	20	)	)	PUNCT
ejpam-5535	231	21	(	(	PUNCT
ejpam-5535	231	22	ii	ii	NOUN
ejpam-5535	231	23	)	)	PUNCT
ejpam-5535	231	24	w	w	PROPN
ejpam-5535	231	25	∈	∈	PROPN
ejpam-5535	231	26	n	n	CCONJ
ejpam-5535	231	27	⇔	⇔	X
ejpam-5535	231	28	wgg	wgg	PROPN
ejpam-5535	231	29	∈	∈	PROPN
ejpam-5535	232	1	n	n	ADV
ejpam-5535	232	2	if	if	SCONJ
ejpam-5535	232	3	n	n	ADV
ejpam-5535	232	4	is	be	AUX
ejpam-5535	232	5	kg	kg	NOUN
ejpam-5535	232	6	-	-	NOUN
ejpam-5535	232	7	filter	filter	NOUN
ejpam-5535	232	8	in	in	ADP
ejpam-5535	232	9	a	a	DET
ejpam-5535	232	10	r.	r.	PROPN
ejpam-5535	232	11	sirisetti	sirisetti	PROPN
ejpam-5535	232	12	et	et	PROPN
ejpam-5535	232	13	al	al	PROPN
ejpam-5535	232	14	.	.	PUNCT
ejpam-5535	232	15	/	/	SYM
ejpam-5535	232	16	eur	eur	PROPN
ejpam-5535	232	17	.	.	PUNCT
ejpam-5535	233	1	j.	j.	PROPN
ejpam-5535	233	2	pure	pure	PROPN
ejpam-5535	233	3	appl	appl	PROPN
ejpam-5535	233	4	.	.	PROPN
ejpam-5535	233	5	math	math	PROPN
ejpam-5535	233	6	,	,	PUNCT
ejpam-5535	233	7	18	18	NUM
ejpam-5535	233	8	(	(	PUNCT
ejpam-5535	233	9	1	1	NUM
ejpam-5535	233	10	)	)	PUNCT
ejpam-5535	233	11	(	(	PUNCT
ejpam-5535	233	12	2025	2025	NUM
ejpam-5535	233	13	)	)	PUNCT
ejpam-5535	233	14	,	,	PUNCT
ejpam-5535	233	15	5535	5535	NUM
ejpam-5535	233	16	9	9	NUM
ejpam-5535	233	17	of	of	ADP
ejpam-5535	233	18	12	12	NUM
ejpam-5535	233	19	(	(	PUNCT
ejpam-5535	233	20	iii	iii	NOUN
ejpam-5535	233	21	)	)	PUNCT
ejpam-5535	233	22	if	if	SCONJ
ejpam-5535	233	23	n	n	PRON
ejpam-5535	233	24	is	be	AUX
ejpam-5535	233	25	a	a	DET
ejpam-5535	233	26	kg	kg	NOUN
ejpam-5535	233	27	-	-	NOUN
ejpam-5535	233	28	filter	filter	NOUN
ejpam-5535	233	29	,	,	PUNCT
ejpam-5535	233	30	then	then	ADV
ejpam-5535	233	31	n	n	NOUN
ejpam-5535	233	32	=	=	SYM
ejpam-5535	233	33	⋃	⋃	PROPN
ejpam-5535	233	34	w∈n	w∈n	ADJ
ejpam-5535	233	35	g((wg	g((wg	PROPN
ejpam-5535	233	36	]	]	X
ejpam-5535	233	37	)	)	PUNCT
ejpam-5535	233	38	.	.	PUNCT
ejpam-5535	234	1	proof	proof	NOUN
ejpam-5535	234	2	.	.	PUNCT
ejpam-5535	235	1	(	(	PUNCT
ejpam-5535	235	2	i	i	NOUN
ejpam-5535	235	3	)	)	PUNCT
ejpam-5535	235	4	let	let	VERB
ejpam-5535	235	5	v	v	NUM
ejpam-5535	235	6	∈	∈	PROPN
ejpam-5535	235	7	a.	a.	NOUN
ejpam-5535	235	8	by	by	ADP
ejpam-5535	235	9	lemma	lemma	PROPN
ejpam-5535	235	10	3.2	3.2	NUM
ejpam-5535	235	11	.	.	PUNCT
ejpam-5535	235	12	,	,	PUNCT
ejpam-5535	235	13	g((v	g((v	NOUN
ejpam-5535	235	14	]	]	X
ejpam-5535	235	15	)	)	PUNCT
ejpam-5535	236	1	=	=	PUNCT
ejpam-5535	237	1	[	[	X
ejpam-5535	237	2	vg	vg	NOUN
ejpam-5535	237	3	)	)	PUNCT
ejpam-5535	237	4	=	=	PUNCT
ejpam-5535	238	1	[	[	X
ejpam-5535	238	2	vggg	vggg	NOUN
ejpam-5535	238	3	)	)	PUNCT
ejpam-5535	238	4	=	=	SYM
ejpam-5535	238	5	g((vgg	g((vgg	NOUN
ejpam-5535	238	6	]	]	PUNCT
ejpam-5535	238	7	)	)	PUNCT
ejpam-5535	238	8	.	.	PUNCT
ejpam-5535	239	1	(	(	PUNCT
ejpam-5535	239	2	ii	ii	NOUN
ejpam-5535	239	3	)	)	PUNCT
ejpam-5535	239	4	let	let	VERB
ejpam-5535	239	5	n	n	PRON
ejpam-5535	239	6	be	be	AUX
ejpam-5535	239	7	a	a	DET
ejpam-5535	239	8	kg	kg	NOUN
ejpam-5535	239	9	filter	filter	NOUN
ejpam-5535	239	10	of	of	ADP
ejpam-5535	239	11	a.	a.	NOUN
ejpam-5535	239	12	then	then	ADV
ejpam-5535	239	13	n	n	PROPN
ejpam-5535	239	14	=	=	SYM
ejpam-5535	239	15	g(k	g(k	NOUN
ejpam-5535	239	16	)	)	PUNCT
ejpam-5535	239	17	,	,	PUNCT
ejpam-5535	239	18	for	for	ADP
ejpam-5535	239	19	some	some	DET
ejpam-5535	239	20	ideal	ideal	NOUN
ejpam-5535	239	21	k	k	PROPN
ejpam-5535	239	22	in	in	ADP
ejpam-5535	239	23	a.	a.	NOUN
ejpam-5535	239	24	for	for	ADP
ejpam-5535	239	25	any	any	DET
ejpam-5535	239	26	w	w	PROPN
ejpam-5535	239	27	∈	∈	PROPN
ejpam-5535	239	28	a	a	DET
ejpam-5535	239	29	,	,	PUNCT
ejpam-5535	239	30	w	w	NOUN
ejpam-5535	239	31	in	in	ADP
ejpam-5535	239	32	n	n	PROPN
ejpam-5535	239	33	⇔	⇔	PROPN
ejpam-5535	239	34	w	w	PROPN
ejpam-5535	239	35	∈	∈	PROPN
ejpam-5535	239	36	g(k	g(k	PROPN
ejpam-5535	239	37	)	)	PUNCT
ejpam-5535	239	38	⇔	⇔	NOUN
ejpam-5535	239	39	wg	wg	PROPN
ejpam-5535	239	40	∈	∈	PROPN
ejpam-5535	239	41	k	k	PROPN
ejpam-5535	239	42	⇔	⇔	PROPN
ejpam-5535	239	43	wggg	wggg	PROPN
ejpam-5535	239	44	∈	∈	PROPN
ejpam-5535	239	45	k	k	PROPN
ejpam-5535	239	46	⇔	⇔	PROPN
ejpam-5535	239	47	wgg	wgg	PROPN
ejpam-5535	239	48	∈	∈	PROPN
ejpam-5535	239	49	g(k	g(k	PROPN
ejpam-5535	239	50	)	)	PUNCT
ejpam-5535	239	51	⇔	⇔	PROPN
ejpam-5535	239	52	wgg	wgg	PROPN
ejpam-5535	239	53	∈	∈	PROPN
ejpam-5535	239	54	n.	n.	PROPN
ejpam-5535	239	55	(	(	PUNCT
ejpam-5535	239	56	iii	iii	NOUN
ejpam-5535	239	57	)	)	PUNCT
ejpam-5535	239	58	let	let	VERB
ejpam-5535	239	59	n	n	PRON
ejpam-5535	239	60	be	be	AUX
ejpam-5535	239	61	a	a	DET
ejpam-5535	239	62	kg	kg	NOUN
ejpam-5535	239	63	-	-	NOUN
ejpam-5535	239	64	filter	filter	NOUN
ejpam-5535	239	65	of	of	ADP
ejpam-5535	239	66	a.	a.	NOUN
ejpam-5535	239	67	then	then	ADV
ejpam-5535	239	68	n	n	PROPN
ejpam-5535	239	69	=	=	SYM
ejpam-5535	239	70	g(k	g(k	NOUN
ejpam-5535	239	71	)	)	PUNCT
ejpam-5535	239	72	for	for	ADP
ejpam-5535	239	73	some	some	DET
ejpam-5535	239	74	ideal	ideal	NOUN
ejpam-5535	239	75	k	k	PROPN
ejpam-5535	239	76	in	in	ADP
ejpam-5535	239	77	a.	a.	NOUN
ejpam-5535	239	78	let	let	VERB
ejpam-5535	239	79	v	v	ADP
ejpam-5535	239	80	∈	∈	PROPN
ejpam-5535	239	81	g((wg	g((wg	PROPN
ejpam-5535	239	82	]	]	X
ejpam-5535	239	83	)	)	PUNCT
ejpam-5535	239	84	for	for	ADP
ejpam-5535	239	85	some	some	DET
ejpam-5535	239	86	w	w	PROPN
ejpam-5535	239	87	∈	∈	PROPN
ejpam-5535	239	88	n	n	ADV
ejpam-5535	239	89	.	.	PUNCT
ejpam-5535	240	1	then	then	ADV
ejpam-5535	240	2	vg	vg	PROPN
ejpam-5535	240	3	∈	∈	PROPN
ejpam-5535	240	4	(	(	PUNCT
ejpam-5535	240	5	wg	wg	PROPN
ejpam-5535	240	6	]	]	X
ejpam-5535	240	7	.	.	PUNCT
ejpam-5535	241	1	therefore	therefore	ADV
ejpam-5535	241	2	vg	vg	ADP
ejpam-5535	241	3	≤	≤	NUM
ejpam-5535	241	4	wg	wg	PROPN
ejpam-5535	241	5	⇒	⇒	NOUN
ejpam-5535	241	6	wgg	wgg	VERB
ejpam-5535	241	7	≤	≤	PROPN
ejpam-5535	241	8	vgg	vgg	VERB
ejpam-5535	241	9	.	.	PUNCT
ejpam-5535	242	1	now	now	ADV
ejpam-5535	242	2	,	,	PUNCT
ejpam-5535	242	3	w	w	PROPN
ejpam-5535	242	4	∈	∈	PROPN
ejpam-5535	242	5	n	n	CCONJ
ejpam-5535	242	6	=	=	SYM
ejpam-5535	242	7	g(k	g(k	NOUN
ejpam-5535	242	8	)	)	PUNCT
ejpam-5535	242	9	.	.	PUNCT
ejpam-5535	243	1	then	then	ADV
ejpam-5535	243	2	wg	wg	PROPN
ejpam-5535	243	3	∈	∈	PROPN
ejpam-5535	243	4	k.	k.	NOUN
ejpam-5535	243	5	therefore	therefore	ADV
ejpam-5535	243	6	wggg	wggg	VERB
ejpam-5535	243	7	=	=	PRON
ejpam-5535	243	8	wg	wg	PROPN
ejpam-5535	243	9	∈	∈	PROPN
ejpam-5535	243	10	k	k	PROPN
ejpam-5535	243	11	implies	imply	VERB
ejpam-5535	243	12	wgg	wgg	PROPN
ejpam-5535	243	13	∈	∈	PROPN
ejpam-5535	243	14	g(k	g(k	PROPN
ejpam-5535	243	15	)	)	PUNCT
ejpam-5535	243	16	=	=	SYM
ejpam-5535	243	17	n	n	NOUN
ejpam-5535	243	18	.	.	PUNCT
ejpam-5535	244	1	since	since	SCONJ
ejpam-5535	244	2	n	n	ADV
ejpam-5535	244	3	is	be	AUX
ejpam-5535	244	4	filter	filter	NOUN
ejpam-5535	244	5	,	,	PUNCT
ejpam-5535	244	6	vgg	vgg	PROPN
ejpam-5535	244	7	∈	∈	PROPN
ejpam-5535	244	8	n	n	ADV
ejpam-5535	244	9	.	.	PUNCT
ejpam-5535	245	1	so	so	ADV
ejpam-5535	245	2	that	that	PRON
ejpam-5535	245	3	v	v	ADP
ejpam-5535	245	4	∈	∈	PROPN
ejpam-5535	246	1	n	n	NOUN
ejpam-5535	246	2	.	.	PUNCT
ejpam-5535	247	1	hence	hence	ADV
ejpam-5535	247	2	⋃	⋃	NOUN
ejpam-5535	247	3	w∈n	w∈n	ADJ
ejpam-5535	247	4	g((wg	g((wg	PROPN
ejpam-5535	247	5	]	]	X
ejpam-5535	247	6	)	)	PUNCT
ejpam-5535	247	7	⊆	⊆	NUM
ejpam-5535	247	8	n	n	NOUN
ejpam-5535	247	9	.	.	PUNCT
ejpam-5535	248	1	let	let	VERB
ejpam-5535	248	2	w	w	NOUN
ejpam-5535	248	3	∈	∈	PROPN
ejpam-5535	248	4	n	n	NOUN
ejpam-5535	248	5	.	.	PUNCT
ejpam-5535	249	1	then	then	ADV
ejpam-5535	249	2	wg	wg	PROPN
ejpam-5535	249	3	∈	∈	PROPN
ejpam-5535	249	4	k.	k.	NOUN
ejpam-5535	249	5	therefore	therefore	ADV
ejpam-5535	249	6	wggg	wggg	VERB
ejpam-5535	249	7	=	=	PRON
ejpam-5535	250	1	wg	wg	PROPN
ejpam-5535	250	2	∈	∈	PROPN
ejpam-5535	250	3	(	(	PUNCT
ejpam-5535	250	4	wg	wg	PROPN
ejpam-5535	250	5	]	]	PUNCT
ejpam-5535	250	6	implies	imply	VERB
ejpam-5535	250	7	wgg	wgg	PROPN
ejpam-5535	250	8	∈	∈	PROPN
ejpam-5535	250	9	g((wg	g((wg	PROPN
ejpam-5535	250	10	]	]	X
ejpam-5535	250	11	)	)	PUNCT
ejpam-5535	250	12	.	.	PUNCT
ejpam-5535	251	1	so	so	ADV
ejpam-5535	251	2	that	that	SCONJ
ejpam-5535	251	3	w	w	PROPN
ejpam-5535	251	4	∈	∈	PROPN
ejpam-5535	251	5	g((wg	g((wg	PROPN
ejpam-5535	251	6	]	]	X
ejpam-5535	251	7	)	)	PUNCT
ejpam-5535	251	8	.	.	PUNCT
ejpam-5535	252	1	hence	hence	ADV
ejpam-5535	252	2	n	n	CCONJ
ejpam-5535	252	3	⊆	⊆	NUM
ejpam-5535	252	4	⋃	⋃	NOUN
ejpam-5535	252	5	w∈n	w∈n	ADJ
ejpam-5535	252	6	g((wg	g((wg	PROPN
ejpam-5535	252	7	]	]	X
ejpam-5535	252	8	)	)	PUNCT
ejpam-5535	252	9	.	.	PUNCT
ejpam-5535	253	1	thus	thus	ADV
ejpam-5535	253	2	n	n	ADV
ejpam-5535	253	3	=	=	SYM
ejpam-5535	253	4	⋃	⋃	PROPN
ejpam-5535	253	5	w∈n	w∈n	ADJ
ejpam-5535	253	6	g((wg	g((wg	PROPN
ejpam-5535	253	7	]	]	X
ejpam-5535	253	8	)	)	PUNCT
ejpam-5535	253	9	.	.	PUNCT
ejpam-5535	254	1	let	let	VERB
ejpam-5535	254	2	us	we	PRON
ejpam-5535	254	3	consider	consider	VERB
ejpam-5535	254	4	a	a	DET
ejpam-5535	254	5	set	set	NOUN
ejpam-5535	254	6	d(a	d(a	PROPN
ejpam-5535	254	7	)	)	PUNCT
ejpam-5535	254	8	=	=	PRON
ejpam-5535	254	9	{	{	PUNCT
ejpam-5535	254	10	v	v	NUM
ejpam-5535	254	11	∈	∈	PROPN
ejpam-5535	254	12	a	a	DET
ejpam-5535	254	13	|	|	NOUN
ejpam-5535	254	14	vg	vg	NOUN
ejpam-5535	254	15	∈	∈	NOUN
ejpam-5535	254	16	d	d	NOUN
ejpam-5535	254	17	}	}	PUNCT
ejpam-5535	254	18	.	.	PUNCT
ejpam-5535	255	1	then	then	ADV
ejpam-5535	255	2	it	it	PRON
ejpam-5535	255	3	is	be	AUX
ejpam-5535	255	4	easy	easy	ADJ
ejpam-5535	255	5	to	to	PART
ejpam-5535	255	6	observe	observe	VERB
ejpam-5535	255	7	that	that	SCONJ
ejpam-5535	255	8	d(a	d(a	PROPN
ejpam-5535	255	9	)	)	PUNCT
ejpam-5535	255	10	is	be	AUX
ejpam-5535	255	11	an	an	DET
ejpam-5535	255	12	ideal	ideal	NOUN
ejpam-5535	255	13	of	of	ADP
ejpam-5535	255	14	a	a	DET
ejpam-5535	255	15	and	and	CCONJ
ejpam-5535	255	16	g(d(a	g(d(a	PROPN
ejpam-5535	255	17	)	)	PUNCT
ejpam-5535	255	18	)	)	PUNCT
ejpam-5535	256	1	=	=	PUNCT
ejpam-5535	256	2	d.	d.	PROPN
ejpam-5535	256	3	lemma	lemma	PROPN
ejpam-5535	257	1	3.11	3.11	NUM
ejpam-5535	257	2	.	.	PUNCT
ejpam-5535	258	1	for	for	ADP
ejpam-5535	258	2	any	any	DET
ejpam-5535	258	3	ideal	ideal	ADJ
ejpam-5535	258	4	k	k	PROPN
ejpam-5535	258	5	of	of	ADP
ejpam-5535	258	6	a	a	PRON
ejpam-5535	258	7	,	,	PUNCT
ejpam-5535	258	8	and	and	CCONJ
ejpam-5535	258	9	x	x	PUNCT
ejpam-5535	258	10	∈	∈	PROPN
ejpam-5535	258	11	a	a	X
ejpam-5535	258	12	,	,	PUNCT
ejpam-5535	258	13	we	we	PRON
ejpam-5535	258	14	have	have	VERB
ejpam-5535	258	15	the	the	DET
ejpam-5535	258	16	following	follow	VERB
ejpam-5535	258	17	equivalent	equivalent	ADJ
ejpam-5535	258	18	conditions	condition	NOUN
ejpam-5535	258	19	:	:	PUNCT
ejpam-5535	258	20	(	(	PUNCT
ejpam-5535	258	21	i	i	NOUN
ejpam-5535	258	22	)	)	PUNCT
ejpam-5535	258	23	g(k	g(k	NOUN
ejpam-5535	258	24	)	)	PUNCT
ejpam-5535	258	25	=	=	SYM
ejpam-5535	259	1	d	d	PROPN
ejpam-5535	259	2	(	(	PUNCT
ejpam-5535	259	3	ii	ii	NOUN
ejpam-5535	259	4	)	)	PUNCT
ejpam-5535	259	5	v	v	ADP
ejpam-5535	259	6	∈	∈	PROPN
ejpam-5535	259	7	k	k	PROPN
ejpam-5535	259	8	⇒	⇒	NOUN
ejpam-5535	259	9	vg	vg	PROPN
ejpam-5535	259	10	∈	∈	PROPN
ejpam-5535	259	11	d	d	X
ejpam-5535	259	12	(	(	PUNCT
ejpam-5535	259	13	iii	iii	NOUN
ejpam-5535	259	14	)	)	PUNCT
ejpam-5535	259	15	k	k	PROPN
ejpam-5535	259	16	⊆	⊆	NUM
ejpam-5535	259	17	d(k	d(k	PROPN
ejpam-5535	259	18	)	)	PUNCT
ejpam-5535	259	19	⊆	⊆	NUM
ejpam-5535	259	20	d(a	d(a	PROPN
ejpam-5535	259	21	)	)	PUNCT
ejpam-5535	259	22	proof	proof	NOUN
ejpam-5535	259	23	.	.	PUNCT
ejpam-5535	260	1	(	(	PUNCT
ejpam-5535	260	2	i	i	NOUN
ejpam-5535	260	3	)	)	PUNCT
ejpam-5535	260	4	⇒	⇒	PROPN
ejpam-5535	260	5	(	(	PUNCT
ejpam-5535	260	6	ii	ii	PROPN
ejpam-5535	260	7	)	)	PUNCT
ejpam-5535	260	8	;	;	PUNCT
ejpam-5535	260	9	assume	assume	VERB
ejpam-5535	260	10	(	(	PUNCT
ejpam-5535	260	11	i	i	NOUN
ejpam-5535	260	12	)	)	PUNCT
ejpam-5535	260	13	;	;	PUNCT
ejpam-5535	260	14	let	let	VERB
ejpam-5535	260	15	v	v	ADP
ejpam-5535	260	16	∈	∈	PROPN
ejpam-5535	260	17	k.	k.	NOUN
ejpam-5535	261	1	then	then	ADV
ejpam-5535	261	2	vgg	vgg	PROPN
ejpam-5535	261	3	∈	∈	PROPN
ejpam-5535	261	4	k.	k.	PROPN
ejpam-5535	261	5	therefore	therefore	ADV
ejpam-5535	261	6	vg	vg	ADP
ejpam-5535	261	7	∈	∈	PROPN
ejpam-5535	261	8	g(k	g(k	NOUN
ejpam-5535	261	9	)	)	PUNCT
ejpam-5535	261	10	=	=	SYM
ejpam-5535	261	11	d.	d.	NOUN
ejpam-5535	261	12	hence	hence	ADV
ejpam-5535	261	13	vg	vg	ADP
ejpam-5535	261	14	∈	∈	PROPN
ejpam-5535	261	15	d.	d.	PROPN
ejpam-5535	261	16	(	(	PUNCT
ejpam-5535	261	17	ii	ii	PROPN
ejpam-5535	261	18	)	)	PUNCT
ejpam-5535	261	19	⇒	⇒	NOUN
ejpam-5535	261	20	(	(	PUNCT
ejpam-5535	261	21	iii	iii	NOUN
ejpam-5535	261	22	)	)	PUNCT
ejpam-5535	261	23	;	;	PUNCT
ejpam-5535	261	24	assume	assume	VERB
ejpam-5535	261	25	(	(	PUNCT
ejpam-5535	261	26	ii	ii	NOUN
ejpam-5535	261	27	)	)	PUNCT
ejpam-5535	261	28	;	;	PUNCT
ejpam-5535	261	29	let	let	VERB
ejpam-5535	261	30	v	v	ADP
ejpam-5535	261	31	∈	∈	PROPN
ejpam-5535	262	1	k.	k.	NOUN
ejpam-5535	263	1	then	then	ADV
ejpam-5535	263	2	vg	vg	PROPN
ejpam-5535	263	3	∈	∈	PROPN
ejpam-5535	263	4	d.	d.	PROPN
ejpam-5535	263	5	therefore	therefore	ADV
ejpam-5535	263	6	v	v	ADP
ejpam-5535	263	7	∈	∈	PROPN
ejpam-5535	263	8	d(a	d(a	PROPN
ejpam-5535	263	9	)	)	PUNCT
ejpam-5535	263	10	.	.	PUNCT
ejpam-5535	264	1	since	since	SCONJ
ejpam-5535	264	2	k	k	PROPN
ejpam-5535	264	3	⊆	⊆	PROPN
ejpam-5535	264	4	a	a	PRON
ejpam-5535	264	5	,	,	PUNCT
ejpam-5535	264	6	v	v	PROPN
ejpam-5535	264	7	∈	∈	PROPN
ejpam-5535	264	8	d(k	d(k	PROPN
ejpam-5535	264	9	)	)	PUNCT
ejpam-5535	264	10	.	.	PUNCT
ejpam-5535	265	1	hence	hence	ADV
ejpam-5535	265	2	k	k	PROPN
ejpam-5535	265	3	⊆	⊆	NUM
ejpam-5535	265	4	d(k	d(k	PROPN
ejpam-5535	265	5	)	)	PUNCT
ejpam-5535	265	6	.	.	PUNCT
ejpam-5535	266	1	(	(	PUNCT
ejpam-5535	266	2	iii	iii	X
ejpam-5535	266	3	)	)	PUNCT
ejpam-5535	266	4	⇒	⇒	NOUN
ejpam-5535	266	5	(	(	PUNCT
ejpam-5535	266	6	i	i	NOUN
ejpam-5535	266	7	)	)	PUNCT
ejpam-5535	266	8	;	;	PUNCT
ejpam-5535	266	9	assume	assume	VERB
ejpam-5535	266	10	(	(	PUNCT
ejpam-5535	266	11	iii	iii	NOUN
ejpam-5535	266	12	)	)	PUNCT
ejpam-5535	266	13	;	;	PUNCT
ejpam-5535	266	14	now	now	ADV
ejpam-5535	266	15	,	,	PUNCT
ejpam-5535	266	16	k	k	PROPN
ejpam-5535	266	17	⊆	⊆	NUM
ejpam-5535	266	18	d(k	d(k	PROPN
ejpam-5535	266	19	)	)	PUNCT
ejpam-5535	266	20	⊆	⊆	NUM
ejpam-5535	266	21	d(a	d(a	PROPN
ejpam-5535	266	22	)	)	PUNCT
ejpam-5535	266	23	implies	imply	VERB
ejpam-5535	266	24	g(k	g(k	NOUN
ejpam-5535	266	25	)	)	PUNCT
ejpam-5535	266	26	⊆	⊆	NUM
ejpam-5535	266	27	g(d(k	g(d(k	PROPN
ejpam-5535	266	28	)	)	PUNCT
ejpam-5535	266	29	)	)	PUNCT
ejpam-5535	267	1	⊆	⊆	NUM
ejpam-5535	267	2	g(d(a	g(d(a	PROPN
ejpam-5535	267	3	)	)	PUNCT
ejpam-5535	267	4	)	)	PUNCT
ejpam-5535	267	5	.	.	PUNCT
ejpam-5535	268	1	therefore	therefore	ADV
ejpam-5535	268	2	g(k	g(k	VERB
ejpam-5535	268	3	)	)	PUNCT
ejpam-5535	268	4	⊆	⊆	NUM
ejpam-5535	268	5	g(d(k	g(d(k	PROPN
ejpam-5535	268	6	)	)	PUNCT
ejpam-5535	268	7	)	)	PUNCT
ejpam-5535	269	1	⊆	⊆	NUM
ejpam-5535	269	2	d.	d.	NOUN
ejpam-5535	269	3	since	since	SCONJ
ejpam-5535	269	4	d	d	PROPN
ejpam-5535	269	5	⊆	⊆	NUM
ejpam-5535	269	6	g(k	g(k	NOUN
ejpam-5535	269	7	)	)	PUNCT
ejpam-5535	269	8	,	,	PUNCT
ejpam-5535	269	9	g(k	g(k	NOUN
ejpam-5535	269	10	)	)	PUNCT
ejpam-5535	269	11	=	=	SYM
ejpam-5535	269	12	d.	d.	PROPN
ejpam-5535	269	13	theorem	theorem	VERB
ejpam-5535	269	14	3.12	3.12	NUM
ejpam-5535	269	15	.	.	PUNCT
ejpam-5535	270	1	the	the	DET
ejpam-5535	270	2	set	set	NOUN
ejpam-5535	270	3	bg(a	bg(a	NOUN
ejpam-5535	270	4	)	)	PUNCT
ejpam-5535	270	5	=	=	SYM
ejpam-5535	270	6	{	{	PUNCT
ejpam-5535	270	7	v	v	NUM
ejpam-5535	270	8	∈	∈	PROPN
ejpam-5535	270	9	a	a	DET
ejpam-5535	270	10	|	|	NOUN
ejpam-5535	270	11	vgg	vgg	NOUN
ejpam-5535	270	12	=	=	NOUN
ejpam-5535	270	13	v	v	NOUN
ejpam-5535	270	14	}	}	PUNCT
ejpam-5535	270	15	is	be	AUX
ejpam-5535	270	16	a	a	DET
ejpam-5535	270	17	boolean	boolean	ADJ
ejpam-5535	270	18	algebra	algebra	NOUN
ejpam-5535	270	19	with	with	ADP
ejpam-5535	270	20	the	the	DET
ejpam-5535	270	21	operations	operation	NOUN
ejpam-5535	270	22	∨	∨	ADJ
ejpam-5535	270	23	and	and	CCONJ
ejpam-5535	270	24	v1	v1	VERB
ejpam-5535	270	25	∗	∗	NOUN
ejpam-5535	270	26	v2	v2	NOUN
ejpam-5535	271	1	=	=	SYM
ejpam-5535	271	2	(	(	PUNCT
ejpam-5535	271	3	vg1	vg1	PROPN
ejpam-5535	271	4	∨	∨	NUM
ejpam-5535	271	5	vg2	vg2	PROPN
ejpam-5535	271	6	)	)	PUNCT
ejpam-5535	271	7	g	g	NOUN
ejpam-5535	271	8	,	,	PUNCT
ejpam-5535	271	9	for	for	ADP
ejpam-5535	271	10	all	all	DET
ejpam-5535	271	11	v1	v1	NOUN
ejpam-5535	271	12	,	,	PUNCT
ejpam-5535	271	13	v2	v2	PROPN
ejpam-5535	271	14	∈	∈	PROPN
ejpam-5535	271	15	bg(a	bg(a	NOUN
ejpam-5535	271	16	)	)	PUNCT
ejpam-5535	271	17	.	.	PUNCT
ejpam-5535	272	1	proof	proof	NOUN
ejpam-5535	272	2	.	.	PUNCT
ejpam-5535	273	1	let	let	VERB
ejpam-5535	273	2	v1	v1	NOUN
ejpam-5535	273	3	,	,	PUNCT
ejpam-5535	273	4	v2	v2	PROPN
ejpam-5535	273	5	∈	∈	PROPN
ejpam-5535	273	6	bg(a	bg(a	NOUN
ejpam-5535	273	7	)	)	PUNCT
ejpam-5535	273	8	.	.	PUNCT
ejpam-5535	274	1	now	now	ADV
ejpam-5535	274	2	,	,	PUNCT
ejpam-5535	274	3	(	(	PUNCT
ejpam-5535	274	4	v1	v1	NOUN
ejpam-5535	274	5	∨	∨	NUM
ejpam-5535	274	6	v2	v2	NOUN
ejpam-5535	274	7	)	)	PUNCT
ejpam-5535	274	8	gg	gg	NOUN
ejpam-5535	274	9	=	=	PUNCT
ejpam-5535	274	10	vgg1	vgg1	PROPN
ejpam-5535	274	11	∨	∨	NUM
ejpam-5535	274	12	vgg2	vgg2	NOUN
ejpam-5535	274	13	=	=	SYM
ejpam-5535	274	14	v1	v1	VERB
ejpam-5535	274	15	∨	∨	NUM
ejpam-5535	274	16	v2	v2	NOUN
ejpam-5535	274	17	(	(	PUNCT
ejpam-5535	274	18	by	by	ADP
ejpam-5535	274	19	proposition	proposition	NOUN
ejpam-5535	274	20	2.3(iii	2.3(iii	NUM
ejpam-5535	274	21	)	)	PUNCT
ejpam-5535	274	22	)	)	PUNCT
ejpam-5535	274	23	.	.	PUNCT
ejpam-5535	275	1	therefore	therefore	ADV
ejpam-5535	275	2	v1∨v2	v1∨v2	NOUN
ejpam-5535	275	3	∈	∈	PROPN
ejpam-5535	275	4	bg(a	bg(a	NOUN
ejpam-5535	275	5	)	)	PUNCT
ejpam-5535	275	6	.	.	PUNCT
ejpam-5535	276	1	now	now	ADV
ejpam-5535	276	2	[	[	X
ejpam-5535	276	3	v1∗v2]gg	v1∗v2]gg	X
ejpam-5535	276	4	=	=	SYM
ejpam-5535	277	1	[	[	X
ejpam-5535	277	2	(	(	PUNCT
ejpam-5535	277	3	vg1∨v	vg1∨v	ADP
ejpam-5535	277	4	g	g	NOUN
ejpam-5535	277	5	2	2	NUM
ejpam-5535	277	6	)	)	PUNCT
ejpam-5535	277	7	g]gg	g]gg	NOUN
ejpam-5535	277	8	=	=	PUNCT
ejpam-5535	278	1	(	(	PUNCT
ejpam-5535	278	2	(	(	PUNCT
ejpam-5535	278	3	vg1∨v	vg1∨v	ADV
ejpam-5535	278	4	g	g	NOUN
ejpam-5535	278	5	2	2	NUM
ejpam-5535	278	6	)	)	PUNCT
ejpam-5535	278	7	g	g	NOUN
ejpam-5535	278	8	=	=	SYM
ejpam-5535	278	9	v1∗v2	v1∗v2	SYM
ejpam-5535	278	10	∈	∈	PROPN
ejpam-5535	278	11	bg(a	bg(a	NOUN
ejpam-5535	278	12	)	)	PUNCT
ejpam-5535	278	13	.	.	PUNCT
ejpam-5535	279	1	therefore	therefore	ADV
ejpam-5535	279	2	bg(a	bg(a	PUNCT
ejpam-5535	279	3	)	)	PUNCT
ejpam-5535	279	4	is	be	AUX
ejpam-5535	279	5	closed	close	VERB
ejpam-5535	279	6	under	under	ADP
ejpam-5535	279	7	∨	∨	NOUN
ejpam-5535	279	8	and	and	CCONJ
ejpam-5535	279	9	∗.	∗.	NOUN
ejpam-5535	279	10	now	now	ADV
ejpam-5535	279	11	v1∗v2	v1∗v2	X
ejpam-5535	279	12	=	=	SYM
ejpam-5535	279	13	(	(	PUNCT
ejpam-5535	279	14	vg1∨vg2	vg1∨vg2	NOUN
ejpam-5535	279	15	)	)	PUNCT
ejpam-5535	279	16	g	g	NOUN
ejpam-5535	279	17	=	=	SYM
ejpam-5535	279	18	(	(	PUNCT
ejpam-5535	279	19	v1∧v2	v1∧v2	PROPN
ejpam-5535	279	20	)	)	PUNCT
ejpam-5535	279	21	gg	gg	PROPN
ejpam-5535	279	22	≤	≤	PUNCT
ejpam-5535	279	23	v1∧v2	v1∧v2	PROPN
ejpam-5535	279	24	≤	≤	PROPN
ejpam-5535	279	25	v1	v1	NOUN
ejpam-5535	279	26	,	,	PUNCT
ejpam-5535	279	27	v2	v2	PROPN
ejpam-5535	279	28	.	.	PUNCT
ejpam-5535	280	1	therefore	therefore	ADV
ejpam-5535	280	2	v1∗v2	v1∗v2	PRON
ejpam-5535	280	3	is	be	AUX
ejpam-5535	280	4	a	a	DET
ejpam-5535	280	5	lower	low	ADJ
ejpam-5535	280	6	bound	bind	VERB
ejpam-5535	280	7	of	of	ADP
ejpam-5535	280	8	v1	v1	NOUN
ejpam-5535	280	9	,	,	PUNCT
ejpam-5535	280	10	w2	w2	NOUN
ejpam-5535	280	11	.	.	PUNCT
ejpam-5535	281	1	for	for	ADP
ejpam-5535	281	2	any	any	DET
ejpam-5535	281	3	lower	low	ADJ
ejpam-5535	281	4	bound	bound	ADJ
ejpam-5535	281	5	l	l	NOUN
ejpam-5535	281	6	of	of	ADP
ejpam-5535	281	7	v1	v1	NOUN
ejpam-5535	281	8	,	,	PUNCT
ejpam-5535	281	9	v2	v2	PROPN
ejpam-5535	281	10	in	in	ADP
ejpam-5535	281	11	bg(a	bg(a	NOUN
ejpam-5535	281	12	)	)	PUNCT
ejpam-5535	281	13	,	,	PUNCT
ejpam-5535	281	14	we	we	PRON
ejpam-5535	281	15	have	have	VERB
ejpam-5535	281	16	l	l	NOUN
ejpam-5535	281	17	≤	≤	NUM
ejpam-5535	281	18	v1	v1	NOUN
ejpam-5535	281	19	,	,	PUNCT
ejpam-5535	281	20	v2	v2	PROPN
ejpam-5535	281	21	⇒	⇒	NOUN
ejpam-5535	281	22	vg1	vg1	NOUN
ejpam-5535	281	23	,	,	PUNCT
ejpam-5535	281	24	v	v	ADP
ejpam-5535	281	25	g	g	PROPN
ejpam-5535	281	26	2	2	NUM
ejpam-5535	281	27	≤	≤	NOUN
ejpam-5535	281	28	lg	lg	NOUN
ejpam-5535	281	29	⇒	⇒	PROPN
ejpam-5535	281	30	vg1	vg1	PROPN
ejpam-5535	281	31	∨vg2	∨vg2	PROPN
ejpam-5535	281	32	≤	≤	PROPN
ejpam-5535	281	33	lg	lg	NOUN
ejpam-5535	281	34	⇒	⇒	NOUN
ejpam-5535	281	35	lgg	lgg	VERB
ejpam-5535	281	36	≤	≤	NOUN
ejpam-5535	281	37	(	(	PUNCT
ejpam-5535	281	38	vg1	vg1	PROPN
ejpam-5535	281	39	∨vg2	∨vg2	PROPN
ejpam-5535	281	40	)	)	PUNCT
ejpam-5535	281	41	g	g	PROPN
ejpam-5535	281	42	⇒	⇒	NOUN
ejpam-5535	281	43	l	l	NOUN
ejpam-5535	281	44	≤	≤	PROPN
ejpam-5535	281	45	v1	v1	PROPN
ejpam-5535	281	46	∗v2	∗v2	PROPN
ejpam-5535	281	47	(	(	PUNCT
ejpam-5535	281	48	since	since	SCONJ
ejpam-5535	281	49	l	l	PROPN
ejpam-5535	281	50	∈	∈	PROPN
ejpam-5535	281	51	bg(a	bg(a	NOUN
ejpam-5535	281	52	)	)	PUNCT
ejpam-5535	281	53	)	)	PUNCT
ejpam-5535	281	54	.	.	PUNCT
ejpam-5535	282	1	therefore	therefore	ADV
ejpam-5535	282	2	v1	v1	VERB
ejpam-5535	282	3	∗	∗	NOUN
ejpam-5535	282	4	v2	v2	PROPN
ejpam-5535	282	5	is	be	AUX
ejpam-5535	282	6	the	the	DET
ejpam-5535	282	7	greatest	greatest	ADV
ejpam-5535	282	8	lower	low	ADJ
ejpam-5535	282	9	bound	bind	VERB
ejpam-5535	282	10	of	of	ADP
ejpam-5535	282	11	v1	v1	NOUN
ejpam-5535	282	12	,	,	PUNCT
ejpam-5535	282	13	v2	v2	PROPN
ejpam-5535	282	14	in	in	ADP
ejpam-5535	282	15	bg(a	bg(a	NOUN
ejpam-5535	282	16	)	)	PUNCT
ejpam-5535	282	17	.	.	PUNCT
ejpam-5535	283	1	since	since	SCONJ
ejpam-5535	283	2	a	a	PRON
ejpam-5535	283	3	is	be	AUX
ejpam-5535	283	4	distributive	distributive	ADJ
ejpam-5535	283	5	lattice	lattice	NOUN
ejpam-5535	283	6	,	,	PUNCT
ejpam-5535	283	7	bg(a	bg(a	NOUN
ejpam-5535	283	8	)	)	PUNCT
ejpam-5535	283	9	is	be	AUX
ejpam-5535	283	10	form	form	VERB
ejpam-5535	283	11	a	a	DET
ejpam-5535	283	12	distributive	distributive	ADJ
ejpam-5535	283	13	lattice	lattice	NOUN
ejpam-5535	283	14	.	.	PUNCT
ejpam-5535	284	1	let	let	VERB
ejpam-5535	284	2	b	b	NOUN
ejpam-5535	284	3	∈	∈	PROPN
ejpam-5535	284	4	bg(a	bg(a	NOUN
ejpam-5535	284	5	)	)	PUNCT
ejpam-5535	284	6	.	.	PUNCT
ejpam-5535	285	1	then	then	ADV
ejpam-5535	285	2	there	there	PRON
ejpam-5535	285	3	exists	exist	VERB
ejpam-5535	285	4	bg	bg	PROPN
ejpam-5535	285	5	∈	∈	PROPN
ejpam-5535	285	6	bg(a	bg(a	NOUN
ejpam-5535	285	7	)	)	PUNCT
ejpam-5535	285	8	such	such	ADJ
ejpam-5535	285	9	that	that	DET
ejpam-5535	285	10	b∨bg	b∨bg	NOUN
ejpam-5535	285	11	is	be	AUX
ejpam-5535	285	12	dense	dense	ADJ
ejpam-5535	285	13	and	and	CCONJ
ejpam-5535	285	14	b∨bg	b∨bg	NOUN
ejpam-5535	285	15	=	=	SYM
ejpam-5535	285	16	bgg∨bg	bgg∨bg	PROPN
ejpam-5535	285	17	=	=	SYM
ejpam-5535	285	18	bgg∨(bg)gg	bgg∨(bg)gg	PROPN
ejpam-5535	286	1	=	=	SYM
ejpam-5535	286	2	(	(	PUNCT
ejpam-5535	286	3	b∨bg)gg	b∨bg)gg	PROPN
ejpam-5535	286	4	=	=	SYM
ejpam-5535	286	5	0	0	NUM
ejpam-5535	286	6	g	g	NOUN
ejpam-5535	286	7	is	be	AUX
ejpam-5535	286	8	the	the	DET
ejpam-5535	286	9	greatest	great	ADJ
ejpam-5535	286	10	element	element	NOUN
ejpam-5535	286	11	in	in	ADP
ejpam-5535	286	12	bg(a	bg(a	NOUN
ejpam-5535	286	13	)	)	PUNCT
ejpam-5535	286	14	.	.	PUNCT
ejpam-5535	287	1	now	now	ADV
ejpam-5535	287	2	,	,	PUNCT
ejpam-5535	287	3	b	b	NOUN
ejpam-5535	287	4	∗	∗	X
ejpam-5535	287	5	bg	bg	NOUN
ejpam-5535	287	6	=	=	PUNCT
ejpam-5535	287	7	(	(	PUNCT
ejpam-5535	287	8	bg	bg	PROPN
ejpam-5535	287	9	∨	∨	PROPN
ejpam-5535	287	10	bgg)g	bgg)g	PROPN
ejpam-5535	287	11	.	.	PUNCT
ejpam-5535	288	1	since	since	SCONJ
ejpam-5535	288	2	bg	bg	PROPN
ejpam-5535	288	3	∨	∨	PROPN
ejpam-5535	288	4	bgg	bgg	PROPN
ejpam-5535	288	5	is	be	AUX
ejpam-5535	288	6	dense	dense	ADJ
ejpam-5535	288	7	,	,	PUNCT
ejpam-5535	288	8	(	(	PUNCT
ejpam-5535	288	9	bg	bg	PROPN
ejpam-5535	288	10	∨	∨	NUM
ejpam-5535	288	11	bgg)g	bgg)g	PROPN
ejpam-5535	288	12	=	=	SYM
ejpam-5535	288	13	0	0	PUNCT
ejpam-5535	288	14	=	=	SYM
ejpam-5535	288	15	b	b	PROPN
ejpam-5535	288	16	∗	∗	X
ejpam-5535	288	17	bg	bg	PROPN
ejpam-5535	288	18	.	.	PUNCT
ejpam-5535	288	19	hence	hence	ADV
ejpam-5535	288	20	bg(a	bg(a	PUNCT
ejpam-5535	288	21	)	)	PUNCT
ejpam-5535	288	22	is	be	AUX
ejpam-5535	288	23	a	a	DET
ejpam-5535	288	24	boolean	boolean	ADJ
ejpam-5535	288	25	algebra	algebra	NOUN
ejpam-5535	288	26	.	.	PUNCT
ejpam-5535	289	1	r.	r.	PROPN
ejpam-5535	289	2	sirisetti	sirisetti	PROPN
ejpam-5535	289	3	et	et	PROPN
ejpam-5535	289	4	al	al	PROPN
ejpam-5535	289	5	.	.	PUNCT
ejpam-5535	289	6	/	/	SYM
ejpam-5535	289	7	eur	eur	PROPN
ejpam-5535	289	8	.	.	PUNCT
ejpam-5535	290	1	j.	j.	PROPN
ejpam-5535	290	2	pure	pure	PROPN
ejpam-5535	290	3	appl	appl	PROPN
ejpam-5535	290	4	.	.	PROPN
ejpam-5535	290	5	math	math	PROPN
ejpam-5535	290	6	,	,	PUNCT
ejpam-5535	290	7	18	18	NUM
ejpam-5535	290	8	(	(	PUNCT
ejpam-5535	290	9	1	1	NUM
ejpam-5535	290	10	)	)	PUNCT
ejpam-5535	290	11	(	(	PUNCT
ejpam-5535	290	12	2025	2025	NUM
ejpam-5535	290	13	)	)	PUNCT
ejpam-5535	290	14	,	,	PUNCT
ejpam-5535	290	15	5535	5535	NUM
ejpam-5535	290	16	10	10	NUM
ejpam-5535	290	17	of	of	ADP
ejpam-5535	290	18	12	12	NUM
ejpam-5535	290	19	theorem	theorem	NOUN
ejpam-5535	290	20	3.13	3.13	NUM
ejpam-5535	290	21	.	.	PUNCT
ejpam-5535	291	1	there	there	PRON
ejpam-5535	291	2	is	be	VERB
ejpam-5535	291	3	an	an	PRON
ejpam-5535	291	4	onto	onto	ADP
ejpam-5535	291	5	homomorphism	homomorphism	NOUN
ejpam-5535	291	6	between	between	ADP
ejpam-5535	291	7	i(a	i(a	PROPN
ejpam-5535	291	8	)	)	PUNCT
ejpam-5535	291	9	to	to	PART
ejpam-5535	291	10	gf(a	gf(a	VERB
ejpam-5535	291	11	)	)	PUNCT
ejpam-5535	291	12	.	.	PUNCT
ejpam-5535	292	1	proof	proof	NOUN
ejpam-5535	292	2	.	.	PUNCT
ejpam-5535	293	1	define	define	VERB
ejpam-5535	293	2	a	a	DET
ejpam-5535	293	3	map	map	NOUN
ejpam-5535	293	4	φ	φ	NOUN
ejpam-5535	293	5	from	from	ADP
ejpam-5535	293	6	i(a	i(a	PROPN
ejpam-5535	293	7	)	)	PUNCT
ejpam-5535	293	8	to	to	PART
ejpam-5535	293	9	gf(a	gf(a	VERB
ejpam-5535	293	10	)	)	PUNCT
ejpam-5535	293	11	by	by	ADP
ejpam-5535	293	12	φ(k	φ(k	PROPN
ejpam-5535	293	13	)	)	PUNCT
ejpam-5535	293	14	=	=	SYM
ejpam-5535	293	15	g(k	g(k	NOUN
ejpam-5535	293	16	)	)	PUNCT
ejpam-5535	293	17	for	for	ADP
ejpam-5535	293	18	all	all	DET
ejpam-5535	293	19	k	k	PROPN
ejpam-5535	293	20	∈	∈	PROPN
ejpam-5535	293	21	i(a	i(a	PROPN
ejpam-5535	293	22	)	)	PUNCT
ejpam-5535	293	23	.	.	PUNCT
ejpam-5535	294	1	let	let	VERB
ejpam-5535	294	2	ka	ka	PROPN
ejpam-5535	294	3	,	,	PUNCT
ejpam-5535	294	4	kb	kb	PROPN
ejpam-5535	294	5	∈	∈	PROPN
ejpam-5535	294	6	gf(a	gf(a	PROPN
ejpam-5535	294	7	)	)	PUNCT
ejpam-5535	294	8	.	.	PUNCT
ejpam-5535	295	1	now	now	ADV
ejpam-5535	295	2	,	,	PUNCT
ejpam-5535	295	3	ka	ka	PROPN
ejpam-5535	295	4	=	=	PROPN
ejpam-5535	295	5	kb	kb	PROPN
ejpam-5535	295	6	.	.	PUNCT
ejpam-5535	295	7	then	then	ADV
ejpam-5535	295	8	φ(ka	φ(ka	NOUN
ejpam-5535	295	9	)	)	PUNCT
ejpam-5535	295	10	=	=	PUNCT
ejpam-5535	295	11	φ(kb	φ(kb	NOUN
ejpam-5535	295	12	)	)	PUNCT
ejpam-5535	295	13	.	.	PUNCT
ejpam-5535	296	1	therefore	therefore	ADV
ejpam-5535	296	2	φ	φ	PROPN
ejpam-5535	296	3	is	be	AUX
ejpam-5535	296	4	well	well	ADV
ejpam-5535	296	5	defined	define	VERB
ejpam-5535	296	6	.	.	PUNCT
ejpam-5535	297	1	now	now	ADV
ejpam-5535	297	2	,	,	PUNCT
ejpam-5535	297	3	φ(ka	φ(ka	NOUN
ejpam-5535	297	4	∩	∩	X
ejpam-5535	297	5	kb	kb	NOUN
ejpam-5535	297	6	)	)	PUNCT
ejpam-5535	297	7	=	=	SYM
ejpam-5535	297	8	g(ka	g(ka	NOUN
ejpam-5535	297	9	∩	∩	ADJ
ejpam-5535	297	10	kb	kb	NOUN
ejpam-5535	297	11	)	)	PUNCT
ejpam-5535	297	12	=	=	SYM
ejpam-5535	297	13	g(ka	g(ka	NOUN
ejpam-5535	297	14	)	)	PUNCT
ejpam-5535	297	15	∧	∧	NOUN
ejpam-5535	297	16	g(kb	g(kb	NOUN
ejpam-5535	297	17	)	)	PUNCT
ejpam-5535	297	18	=	=	SYM
ejpam-5535	297	19	φ(ka	φ(ka	NOUN
ejpam-5535	297	20	)	)	PUNCT
ejpam-5535	297	21	∧	∧	PROPN
ejpam-5535	297	22	φ(kb	φ(kb	PROPN
ejpam-5535	297	23	)	)	PUNCT
ejpam-5535	297	24	,	,	PUNCT
ejpam-5535	297	25	φ(ka	φ(ka	ADP
ejpam-5535	297	26	∨	∨	NUM
ejpam-5535	297	27	kb	kb	PROPN
ejpam-5535	297	28	)	)	PUNCT
ejpam-5535	298	1	=	=	SYM
ejpam-5535	298	2	g(ka	g(ka	NOUN
ejpam-5535	298	3	∨kb	∨kb	NOUN
ejpam-5535	298	4	)	)	PUNCT
ejpam-5535	298	5	=	=	SYM
ejpam-5535	298	6	g(ka	g(ka	NOUN
ejpam-5535	298	7	)	)	PUNCT
ejpam-5535	298	8	⊔g(kb	⊔g(kb	X
ejpam-5535	298	9	)	)	PUNCT
ejpam-5535	299	1	=	=	SYM
ejpam-5535	299	2	φ(ka	φ(ka	NOUN
ejpam-5535	299	3	)	)	PUNCT
ejpam-5535	299	4	⊔φ(kb	⊔φ(kb	NUM
ejpam-5535	299	5	)	)	PUNCT
ejpam-5535	299	6	and	and	CCONJ
ejpam-5535	299	7	φ({0	φ({0	NUM
ejpam-5535	299	8	}	}	PUNCT
ejpam-5535	299	9	)	)	PUNCT
ejpam-5535	299	10	=	=	SYM
ejpam-5535	299	11	g({0	g({0	NOUN
ejpam-5535	299	12	}	}	PUNCT
ejpam-5535	299	13	)	)	PUNCT
ejpam-5535	300	1	=	=	SYM
ejpam-5535	300	2	d.	d.	PROPN
ejpam-5535	300	3	therefore	therefore	ADV
ejpam-5535	300	4	φ	φ	PROPN
ejpam-5535	300	5	is	be	AUX
ejpam-5535	300	6	an	an	PRON
ejpam-5535	300	7	onto	onto	ADP
ejpam-5535	300	8	homomorphism	homomorphism	NOUN
ejpam-5535	300	9	.	.	PUNCT
ejpam-5535	301	1	remark	remark	PROPN
ejpam-5535	301	2	3.14	3.14	NUM
ejpam-5535	301	3	.	.	PUNCT
ejpam-5535	302	1	the	the	DET
ejpam-5535	302	2	above	above	ADJ
ejpam-5535	302	3	homomorphism	homomorphism	NOUN
ejpam-5535	302	4	need	need	AUX
ejpam-5535	302	5	not	not	PART
ejpam-5535	302	6	be	be	AUX
ejpam-5535	302	7	one	one	NUM
ejpam-5535	302	8	-	-	PUNCT
ejpam-5535	302	9	one	one	NUM
ejpam-5535	302	10	.	.	PUNCT
ejpam-5535	303	1	in	in	ADP
ejpam-5535	303	2	example	example	NOUN
ejpam-5535	303	3	3.3	3.3	NUM
ejpam-5535	303	4	.	.	PUNCT
ejpam-5535	303	5	,	,	PUNCT
ejpam-5535	303	6	let	let	VERB
ejpam-5535	303	7	kb	kb	PROPN
ejpam-5535	303	8	=	=	PRON
ejpam-5535	303	9	{	{	PUNCT
ejpam-5535	303	10	0	0	NUM
ejpam-5535	303	11	,	,	PUNCT
ejpam-5535	303	12	i},kc	i},kc	PROPN
ejpam-5535	303	13	=	=	PUNCT
ejpam-5535	303	14	{	{	PUNCT
ejpam-5535	303	15	0	0	NUM
ejpam-5535	303	16	,	,	PUNCT
ejpam-5535	303	17	i	i	PRON
ejpam-5535	303	18	,	,	PUNCT
ejpam-5535	303	19	k	k	PROPN
ejpam-5535	303	20	}	}	PUNCT
ejpam-5535	303	21	,	,	PUNCT
ejpam-5535	303	22	then	then	ADV
ejpam-5535	303	23	g(kb	g(kb	X
ejpam-5535	303	24	)	)	PUNCT
ejpam-5535	303	25	=	=	SYM
ejpam-5535	303	26	a	a	PRON
ejpam-5535	303	27	=	=	SYM
ejpam-5535	303	28	g(kc	g(kc	PROPN
ejpam-5535	303	29	)	)	PUNCT
ejpam-5535	303	30	.	.	PUNCT
ejpam-5535	304	1	that	that	PRON
ejpam-5535	304	2	is	be	AUX
ejpam-5535	304	3	φ(kb	φ(kb	NOUN
ejpam-5535	304	4	)	)	PUNCT
ejpam-5535	304	5	=	=	SYM
ejpam-5535	304	6	φ(kc	φ(kc	PROPN
ejpam-5535	304	7	)	)	PUNCT
ejpam-5535	304	8	.	.	PUNCT
ejpam-5535	305	1	but	but	CCONJ
ejpam-5535	305	2	kb	kb	PROPN
ejpam-5535	305	3	̸=	̸=	PROPN
ejpam-5535	305	4	kc	kc	PROPN
ejpam-5535	305	5	.	.	PUNCT
ejpam-5535	306	1	hence	hence	ADV
ejpam-5535	306	2	φ	φ	PROPN
ejpam-5535	306	3	is	be	AUX
ejpam-5535	306	4	not	not	PART
ejpam-5535	306	5	one	one	NUM
ejpam-5535	306	6	-	-	PUNCT
ejpam-5535	306	7	one	one	NUM
ejpam-5535	306	8	.	.	PUNCT
ejpam-5535	307	1	lemma	lemma	PROPN
ejpam-5535	307	2	3.15	3.15	NUM
ejpam-5535	307	3	.	.	PUNCT
ejpam-5535	308	1	let	let	VERB
ejpam-5535	308	2	φ	φ	PROPN
ejpam-5535	308	3	be	be	AUX
ejpam-5535	308	4	a	a	DET
ejpam-5535	308	5	homomorphism	homomorphism	NOUN
ejpam-5535	308	6	between	between	ADP
ejpam-5535	308	7	i(a	i(a	PROPN
ejpam-5535	308	8	)	)	PUNCT
ejpam-5535	308	9	and	and	CCONJ
ejpam-5535	308	10	gf(a	gf(a	NOUN
ejpam-5535	308	11	)	)	PUNCT
ejpam-5535	308	12	.	.	PUNCT
ejpam-5535	309	1	then	then	ADV
ejpam-5535	309	2	kernel	kernel	PROPN
ejpam-5535	309	3	of	of	ADP
ejpam-5535	309	4	φ	φ	PROPN
ejpam-5535	309	5	is	be	AUX
ejpam-5535	309	6	an	an	DET
ejpam-5535	309	7	ideal	ideal	NOUN
ejpam-5535	309	8	of	of	ADP
ejpam-5535	309	9	i(a	i(a	PROPN
ejpam-5535	309	10	)	)	PUNCT
ejpam-5535	309	11	.	.	PUNCT
ejpam-5535	310	1	proof	proof	NOUN
ejpam-5535	310	2	.	.	PUNCT
ejpam-5535	311	1	let	let	VERB
ejpam-5535	311	2	ka	ka	PROPN
ejpam-5535	311	3	,	,	PUNCT
ejpam-5535	311	4	kb	kb	PROPN
ejpam-5535	311	5	∈	∈	PROPN
ejpam-5535	311	6	ker(φ	ker(φ	PROPN
ejpam-5535	311	7	)	)	PUNCT
ejpam-5535	311	8	.	.	PUNCT
ejpam-5535	312	1	then	then	ADV
ejpam-5535	312	2	φ(ka	φ(ka	NOUN
ejpam-5535	312	3	)	)	PUNCT
ejpam-5535	312	4	=	=	SYM
ejpam-5535	313	1	d	d	NOUN
ejpam-5535	313	2	=	=	PUNCT
ejpam-5535	313	3	φ(kb	φ(kb	PROPN
ejpam-5535	313	4	)	)	PUNCT
ejpam-5535	313	5	.	.	PUNCT
ejpam-5535	314	1	therefore	therefore	ADV
ejpam-5535	314	2	g(ka	g(ka	NOUN
ejpam-5535	314	3	)	)	PUNCT
ejpam-5535	314	4	=	=	SYM
ejpam-5535	315	1	d	d	NOUN
ejpam-5535	315	2	=	=	PUNCT
ejpam-5535	315	3	g(kb	g(kb	NUM
ejpam-5535	315	4	)	)	PUNCT
ejpam-5535	315	5	.	.	PUNCT
ejpam-5535	316	1	now	now	ADV
ejpam-5535	316	2	,	,	PUNCT
ejpam-5535	316	3	φ(ka	φ(ka	ADP
ejpam-5535	316	4	∨	∨	NUM
ejpam-5535	316	5	kb	kb	PROPN
ejpam-5535	316	6	)	)	PUNCT
ejpam-5535	316	7	=	=	PUNCT
ejpam-5535	316	8	g(ka	g(ka	PROPN
ejpam-5535	316	9	∨	∨	NUM
ejpam-5535	316	10	kb	kb	PROPN
ejpam-5535	316	11	)	)	PUNCT
ejpam-5535	316	12	=	=	SYM
ejpam-5535	316	13	g(ka	g(ka	NOUN
ejpam-5535	316	14	)	)	PUNCT
ejpam-5535	316	15	⊔	⊔	NUM
ejpam-5535	316	16	g(kb	g(kb	NOUN
ejpam-5535	316	17	)	)	PUNCT
ejpam-5535	316	18	=	=	PUNCT
ejpam-5535	317	1	d	d	X
ejpam-5535	317	2	⊔	⊔	NUM
ejpam-5535	318	1	d	d	X
ejpam-5535	318	2	=	=	SYM
ejpam-5535	318	3	d.	d.	PROPN
ejpam-5535	318	4	therefore	therefore	ADV
ejpam-5535	318	5	ka	ka	PROPN
ejpam-5535	318	6	∨	∨	PROPN
ejpam-5535	318	7	kb	kb	PROPN
ejpam-5535	318	8	∈	∈	PROPN
ejpam-5535	318	9	ker(φ	ker(φ	PROPN
ejpam-5535	318	10	)	)	PUNCT
ejpam-5535	318	11	.	.	PUNCT
ejpam-5535	319	1	for	for	ADP
ejpam-5535	319	2	any	any	DET
ejpam-5535	319	3	k	k	PROPN
ejpam-5535	319	4	∈	∈	PROPN
ejpam-5535	319	5	i(a),φ(k	i(a),φ(k	PROPN
ejpam-5535	319	6	∧	∧	PROPN
ejpam-5535	319	7	ka	ka	PROPN
ejpam-5535	319	8	)	)	PUNCT
ejpam-5535	319	9	=	=	PUNCT
ejpam-5535	319	10	g(k	g(k	VERB
ejpam-5535	319	11	∧	∧	PROPN
ejpam-5535	319	12	ka	ka	PROPN
ejpam-5535	319	13	)	)	PUNCT
ejpam-5535	319	14	=	=	SYM
ejpam-5535	319	15	g(k	g(k	NOUN
ejpam-5535	319	16	)	)	PUNCT
ejpam-5535	319	17	∧	∧	NOUN
ejpam-5535	319	18	g(ka	g(ka	NOUN
ejpam-5535	319	19	)	)	PUNCT
ejpam-5535	319	20	=	=	SYM
ejpam-5535	319	21	g(k	g(k	NOUN
ejpam-5535	319	22	)	)	PUNCT
ejpam-5535	319	23	∧d	∧d	X
ejpam-5535	320	1	=	=	SYM
ejpam-5535	320	2	d.	d.	PROPN
ejpam-5535	320	3	therefore	therefore	ADV
ejpam-5535	320	4	k	k	PROPN
ejpam-5535	320	5	∧ka	∧ka	PROPN
ejpam-5535	320	6	∈	∈	PROPN
ejpam-5535	320	7	ker(φ	ker(φ	PROPN
ejpam-5535	320	8	)	)	PUNCT
ejpam-5535	320	9	.	.	PUNCT
ejpam-5535	321	1	hence	hence	ADV
ejpam-5535	321	2	ker(φ	ker(φ	X
ejpam-5535	321	3	)	)	PUNCT
ejpam-5535	321	4	is	be	AUX
ejpam-5535	321	5	an	an	DET
ejpam-5535	321	6	ideal	ideal	NOUN
ejpam-5535	321	7	of	of	ADP
ejpam-5535	321	8	i(a	i(a	PROPN
ejpam-5535	321	9	)	)	PUNCT
ejpam-5535	321	10	.	.	PUNCT
ejpam-5535	322	1	theorem	theorem	VERB
ejpam-5535	322	2	3.16	3.16	NUM
ejpam-5535	322	3	.	.	PUNCT
ejpam-5535	323	1	if	if	SCONJ
ejpam-5535	323	2	φ	φ	PROPN
ejpam-5535	323	3	is	be	AUX
ejpam-5535	323	4	a	a	DET
ejpam-5535	323	5	homomorphism	homomorphism	NOUN
ejpam-5535	323	6	from	from	ADP
ejpam-5535	323	7	i(a	i(a	PROPN
ejpam-5535	323	8	)	)	PUNCT
ejpam-5535	323	9	to	to	PART
ejpam-5535	323	10	gf(a	gf(a	VERB
ejpam-5535	323	11	)	)	PUNCT
ejpam-5535	323	12	defined	define	VERB
ejpam-5535	323	13	by	by	ADP
ejpam-5535	323	14	φ(k	φ(k	PROPN
ejpam-5535	323	15	)	)	PUNCT
ejpam-5535	323	16	=	=	SYM
ejpam-5535	323	17	g(k	g(k	NOUN
ejpam-5535	323	18	)	)	PUNCT
ejpam-5535	323	19	,	,	PUNCT
ejpam-5535	323	20	for	for	ADP
ejpam-5535	323	21	all	all	DET
ejpam-5535	323	22	k	k	PROPN
ejpam-5535	323	23	∈	∈	PROPN
ejpam-5535	323	24	i(a	i(a	PROPN
ejpam-5535	323	25	)	)	PUNCT
ejpam-5535	323	26	,	,	PUNCT
ejpam-5535	323	27	then	then	ADV
ejpam-5535	323	28	the	the	DET
ejpam-5535	323	29	following	following	NOUN
ejpam-5535	323	30	are	be	AUX
ejpam-5535	323	31	equivalent	equivalent	ADJ
ejpam-5535	323	32	;	;	PUNCT
ejpam-5535	323	33	(	(	PUNCT
ejpam-5535	323	34	i	i	NOUN
ejpam-5535	323	35	)	)	PUNCT
ejpam-5535	323	36	ker(φ	ker(φ	X
ejpam-5535	323	37	)	)	PUNCT
ejpam-5535	323	38	=	=	PRON
ejpam-5535	323	39	{	{	PUNCT
ejpam-5535	323	40	0	0	NUM
ejpam-5535	323	41	}	}	PUNCT
ejpam-5535	323	42	(	(	PUNCT
ejpam-5535	323	43	ii	ii	NOUN
ejpam-5535	323	44	)	)	PUNCT
ejpam-5535	323	45	vgg	vgg	PROPN
ejpam-5535	323	46	=	=	SYM
ejpam-5535	323	47	v	v	NOUN
ejpam-5535	323	48	,	,	PUNCT
ejpam-5535	323	49	for	for	ADP
ejpam-5535	323	50	all	all	PRON
ejpam-5535	323	51	v	v	NOUN
ejpam-5535	323	52	∈	∈	NOUN
ejpam-5535	323	53	a.	a.	NOUN
ejpam-5535	323	54	(	(	PUNCT
ejpam-5535	323	55	iii	iii	NOUN
ejpam-5535	323	56	)	)	PUNCT
ejpam-5535	323	57	φ	φ	PROPN
ejpam-5535	323	58	is	be	AUX
ejpam-5535	323	59	one	one	NUM
ejpam-5535	323	60	-	-	PUNCT
ejpam-5535	323	61	one	one	NUM
ejpam-5535	323	62	.	.	PUNCT
ejpam-5535	324	1	proof	proof	NOUN
ejpam-5535	324	2	.	.	PUNCT
ejpam-5535	325	1	(	(	PUNCT
ejpam-5535	325	2	ii	ii	NOUN
ejpam-5535	325	3	)	)	PUNCT
ejpam-5535	325	4	⇒	⇒	NOUN
ejpam-5535	325	5	(	(	PUNCT
ejpam-5535	325	6	iii	iii	NOUN
ejpam-5535	325	7	)	)	PUNCT
ejpam-5535	325	8	;	;	PUNCT
ejpam-5535	325	9	suppose	suppose	VERB
ejpam-5535	325	10	that	that	SCONJ
ejpam-5535	325	11	vgg	vgg	NOUN
ejpam-5535	325	12	=	=	NOUN
ejpam-5535	325	13	v	v	NOUN
ejpam-5535	325	14	for	for	ADP
ejpam-5535	325	15	all	all	DET
ejpam-5535	325	16	v	v	NOUN
ejpam-5535	325	17	∈	∈	NOUN
ejpam-5535	325	18	a.	a.	NOUN
ejpam-5535	325	19	let	let	VERB
ejpam-5535	325	20	ka	ka	PROPN
ejpam-5535	325	21	,	,	PUNCT
ejpam-5535	325	22	kb	kb	PROPN
ejpam-5535	325	23	∈	∈	PROPN
ejpam-5535	325	24	i(a	i(a	PROPN
ejpam-5535	325	25	)	)	PUNCT
ejpam-5535	325	26	.	.	PUNCT
ejpam-5535	326	1	now	now	ADV
ejpam-5535	326	2	,	,	PUNCT
ejpam-5535	326	3	φ(ka	φ(ka	NOUN
ejpam-5535	326	4	)	)	PUNCT
ejpam-5535	326	5	=	=	PUNCT
ejpam-5535	326	6	φ(kb	φ(kb	NOUN
ejpam-5535	326	7	)	)	PUNCT
ejpam-5535	326	8	.	.	PUNCT
ejpam-5535	327	1	then	then	ADV
ejpam-5535	327	2	g(ka	g(ka	NOUN
ejpam-5535	327	3	)	)	PUNCT
ejpam-5535	327	4	=	=	SYM
ejpam-5535	327	5	g(kb	g(kb	NOUN
ejpam-5535	327	6	)	)	PUNCT
ejpam-5535	327	7	.	.	PUNCT
ejpam-5535	328	1	for	for	ADP
ejpam-5535	328	2	any	any	DET
ejpam-5535	328	3	k	k	PROPN
ejpam-5535	328	4	∈	∈	PROPN
ejpam-5535	328	5	a	a	PRON
ejpam-5535	328	6	,	,	PUNCT
ejpam-5535	328	7	k	k	PROPN
ejpam-5535	328	8	∈	∈	PROPN
ejpam-5535	328	9	ka	ka	PROPN
ejpam-5535	328	10	⇔	⇔	PROPN
ejpam-5535	328	11	kgg	kgg	PROPN
ejpam-5535	328	12	∈	∈	PROPN
ejpam-5535	328	13	ka	ka	PROPN
ejpam-5535	328	14	(	(	PUNCT
ejpam-5535	328	15	since	since	SCONJ
ejpam-5535	328	16	kgg	kgg	PROPN
ejpam-5535	328	17	=	=	SYM
ejpam-5535	328	18	k	k	PROPN
ejpam-5535	328	19	)	)	PUNCT
ejpam-5535	328	20	⇔	⇔	NOUN
ejpam-5535	328	21	kg	kg	PROPN
ejpam-5535	328	22	∈	∈	PROPN
ejpam-5535	328	23	g(ka	g(ka	NOUN
ejpam-5535	328	24	)	)	PUNCT
ejpam-5535	328	25	=	=	SYM
ejpam-5535	328	26	g(kb	g(kb	X
ejpam-5535	328	27	)	)	PUNCT
ejpam-5535	328	28	⇔	⇔	NOUN
ejpam-5535	328	29	kg	kg	X
ejpam-5535	328	30	∈	∈	PROPN
ejpam-5535	328	31	g(kb	g(kb	PROPN
ejpam-5535	328	32	)	)	PUNCT
ejpam-5535	328	33	⇔	⇔	PROPN
ejpam-5535	328	34	kgg	kgg	PROPN
ejpam-5535	328	35	∈	∈	PROPN
ejpam-5535	328	36	kb	kb	PROPN
ejpam-5535	328	37	⇔	⇔	PROPN
ejpam-5535	328	38	k	k	PROPN
ejpam-5535	328	39	∈	∈	PROPN
ejpam-5535	328	40	kb	kb	PROPN
ejpam-5535	328	41	.	.	PUNCT
ejpam-5535	329	1	(	(	PUNCT
ejpam-5535	329	2	since	since	SCONJ
ejpam-5535	329	3	kgg	kgg	PROPN
ejpam-5535	329	4	=	=	SYM
ejpam-5535	329	5	k	k	PROPN
ejpam-5535	329	6	)	)	PUNCT
ejpam-5535	329	7	therefore	therefore	ADV
ejpam-5535	329	8	ka	ka	PROPN
ejpam-5535	329	9	=	=	PROPN
ejpam-5535	329	10	kb	kb	PROPN
ejpam-5535	329	11	.	.	PUNCT
ejpam-5535	330	1	hence	hence	ADV
ejpam-5535	330	2	φ	φ	PROPN
ejpam-5535	330	3	is	be	AUX
ejpam-5535	330	4	one	one	NUM
ejpam-5535	330	5	-	-	PUNCT
ejpam-5535	330	6	one	one	NUM
ejpam-5535	330	7	.	.	PUNCT
ejpam-5535	331	1	(	(	PUNCT
ejpam-5535	331	2	iii	iii	X
ejpam-5535	331	3	)	)	PUNCT
ejpam-5535	331	4	⇒	⇒	NOUN
ejpam-5535	331	5	(	(	PUNCT
ejpam-5535	331	6	i	i	NOUN
ejpam-5535	331	7	)	)	PUNCT
ejpam-5535	331	8	;	;	PUNCT
ejpam-5535	331	9	suppose	suppose	VERB
ejpam-5535	331	10	that	that	SCONJ
ejpam-5535	331	11	φ	φ	PROPN
ejpam-5535	331	12	is	be	AUX
ejpam-5535	331	13	one	one	NUM
ejpam-5535	331	14	-	-	PUNCT
ejpam-5535	331	15	one	one	NUM
ejpam-5535	331	16	.	.	PUNCT
ejpam-5535	332	1	let	let	VERB
ejpam-5535	332	2	k	k	PROPN
ejpam-5535	332	3	∈	∈	PROPN
ejpam-5535	332	4	ker(φ	ker(φ	PROPN
ejpam-5535	332	5	)	)	PUNCT
ejpam-5535	332	6	.	.	PUNCT
ejpam-5535	333	1	then	then	ADV
ejpam-5535	333	2	φ(k	φ(k	PROPN
ejpam-5535	333	3	)	)	PUNCT
ejpam-5535	334	1	=	=	SYM
ejpam-5535	334	2	d.	d.	PROPN
ejpam-5535	334	3	therefore	therefore	ADV
ejpam-5535	334	4	φ(k	φ(k	PROPN
ejpam-5535	334	5	)	)	PUNCT
ejpam-5535	334	6	=	=	SYM
ejpam-5535	334	7	g(k	g(k	NOUN
ejpam-5535	334	8	)	)	PUNCT
ejpam-5535	334	9	=	=	PUNCT
ejpam-5535	335	1	d	d	NOUN
ejpam-5535	335	2	=	=	SYM
ejpam-5535	335	3	g((0	g((0	PROPN
ejpam-5535	335	4	]	]	PUNCT
ejpam-5535	335	5	)	)	PUNCT
ejpam-5535	335	6	=	=	SYM
ejpam-5535	335	7	φ((0	φ((0	PROPN
ejpam-5535	335	8	]	]	PUNCT
ejpam-5535	335	9	)	)	PUNCT
ejpam-5535	335	10	.	.	PUNCT
ejpam-5535	336	1	since	since	SCONJ
ejpam-5535	336	2	φ	φ	PROPN
ejpam-5535	336	3	is	be	AUX
ejpam-5535	336	4	one	one	NUM
ejpam-5535	336	5	-	-	PUNCT
ejpam-5535	336	6	one	one	NUM
ejpam-5535	336	7	,	,	PUNCT
ejpam-5535	336	8	k	k	NOUN
ejpam-5535	336	9	=	=	PUNCT
ejpam-5535	336	10	(	(	PUNCT
ejpam-5535	336	11	0	0	NUM
ejpam-5535	336	12	]	]	PUNCT
ejpam-5535	336	13	.	.	PUNCT
ejpam-5535	337	1	hence	hence	ADV
ejpam-5535	337	2	ker(φ	ker(φ	X
ejpam-5535	337	3	)	)	PUNCT
ejpam-5535	338	1	=	=	PRON
ejpam-5535	338	2	{	{	PUNCT
ejpam-5535	338	3	0	0	NUM
ejpam-5535	338	4	}	}	PUNCT
ejpam-5535	338	5	.	.	PUNCT
ejpam-5535	339	1	(	(	PUNCT
ejpam-5535	339	2	ii	ii	NOUN
ejpam-5535	339	3	)	)	PUNCT
ejpam-5535	339	4	⇒	⇒	NOUN
ejpam-5535	339	5	(	(	PUNCT
ejpam-5535	339	6	i	i	NOUN
ejpam-5535	339	7	)	)	PUNCT
ejpam-5535	339	8	;	;	PUNCT
ejpam-5535	339	9	suppose	suppose	VERB
ejpam-5535	339	10	that	that	SCONJ
ejpam-5535	339	11	vgg	vgg	PROPN
ejpam-5535	339	12	=	=	SYM
ejpam-5535	339	13	v	v	NOUN
ejpam-5535	339	14	,	,	PUNCT
ejpam-5535	339	15	for	for	ADP
ejpam-5535	339	16	all	all	DET
ejpam-5535	339	17	v	v	NOUN
ejpam-5535	339	18	∈	∈	NOUN
ejpam-5535	339	19	a.	a.	NOUN
ejpam-5535	339	20	let	let	VERB
ejpam-5535	339	21	k	k	PROPN
ejpam-5535	339	22	∈	∈	PROPN
ejpam-5535	339	23	ker(φ	ker(φ	PROPN
ejpam-5535	339	24	)	)	PUNCT
ejpam-5535	339	25	.	.	PUNCT
ejpam-5535	340	1	then	then	ADV
ejpam-5535	340	2	φ(k	φ(k	PROPN
ejpam-5535	340	3	)	)	PUNCT
ejpam-5535	341	1	=	=	SYM
ejpam-5535	341	2	d.	d.	PROPN
ejpam-5535	341	3	therefore	therefore	ADV
ejpam-5535	341	4	g(k	g(k	VERB
ejpam-5535	341	5	)	)	PUNCT
ejpam-5535	341	6	=	=	SYM
ejpam-5535	342	1	d.	d.	PROPN
ejpam-5535	342	2	let	let	VERB
ejpam-5535	343	1	k	k	PROPN
ejpam-5535	343	2	∈	∈	PROPN
ejpam-5535	343	3	k.	k.	PROPN
ejpam-5535	344	1	then	then	ADV
ejpam-5535	344	2	kgg	kgg	PROPN
ejpam-5535	345	1	∈	∈	PROPN
ejpam-5535	345	2	k.	k.	PROPN
ejpam-5535	346	1	therefore	therefore	ADV
ejpam-5535	346	2	kg	kg	PROPN
ejpam-5535	346	3	∈	∈	PROPN
ejpam-5535	346	4	g(k	g(k	PROPN
ejpam-5535	346	5	)	)	PUNCT
ejpam-5535	346	6	=	=	SYM
ejpam-5535	347	1	d.	d.	PROPN
ejpam-5535	347	2	for	for	ADP
ejpam-5535	347	3	this	this	DET
ejpam-5535	347	4	kg	kg	NOUN
ejpam-5535	347	5	∈	∈	PROPN
ejpam-5535	347	6	d	d	PROPN
ejpam-5535	347	7	,	,	PUNCT
ejpam-5535	347	8	kgg	kgg	PROPN
ejpam-5535	348	1	=	=	NOUN
ejpam-5535	349	1	0	0	PROPN
ejpam-5535	349	2	.	.	PUNCT
ejpam-5535	350	1	since	since	SCONJ
ejpam-5535	350	2	kgg	kgg	PROPN
ejpam-5535	350	3	=	=	SYM
ejpam-5535	350	4	k	k	PROPN
ejpam-5535	350	5	=	=	PUNCT
ejpam-5535	350	6	0	0	X
ejpam-5535	350	7	.	.	PUNCT
ejpam-5535	350	8	hence	hence	ADV
ejpam-5535	350	9	k	k	PROPN
ejpam-5535	350	10	=	=	PUNCT
ejpam-5535	350	11	{	{	PUNCT
ejpam-5535	350	12	0	0	NUM
ejpam-5535	350	13	}	}	PUNCT
ejpam-5535	350	14	.	.	PUNCT
ejpam-5535	351	1	therefore	therefore	ADV
ejpam-5535	351	2	ker(φ	ker(φ	X
ejpam-5535	351	3	)	)	PUNCT
ejpam-5535	351	4	=	=	PUNCT
ejpam-5535	351	5	{	{	PUNCT
ejpam-5535	351	6	0	0	NUM
ejpam-5535	351	7	}	}	PUNCT
ejpam-5535	351	8	.	.	PUNCT
ejpam-5535	352	1	(	(	PUNCT
ejpam-5535	352	2	iii	iii	X
ejpam-5535	352	3	)	)	PUNCT
ejpam-5535	352	4	⇒	⇒	NOUN
ejpam-5535	352	5	(	(	PUNCT
ejpam-5535	352	6	ii	ii	PROPN
ejpam-5535	352	7	)	)	PUNCT
ejpam-5535	352	8	;	;	PUNCT
ejpam-5535	352	9	suppose	suppose	VERB
ejpam-5535	352	10	that	that	SCONJ
ejpam-5535	352	11	φ	φ	PROPN
ejpam-5535	352	12	is	be	AUX
ejpam-5535	352	13	one	one	NUM
ejpam-5535	352	14	-	-	PUNCT
ejpam-5535	352	15	one	one	NUM
ejpam-5535	352	16	.	.	PUNCT
ejpam-5535	353	1	let	let	VERB
ejpam-5535	353	2	v	v	NUM
ejpam-5535	353	3	∈	∈	VERB
ejpam-5535	353	4	a.	a.	NOUN
ejpam-5535	353	5	by	by	ADP
ejpam-5535	353	6	lemma	lemma	PROPN
ejpam-5535	353	7	3.3	3.3	NUM
ejpam-5535	353	8	.	.	PUNCT
ejpam-5535	353	9	,	,	PUNCT
ejpam-5535	353	10	g((vgg	g((vgg	NOUN
ejpam-5535	353	11	]	]	PUNCT
ejpam-5535	353	12	)	)	PUNCT
ejpam-5535	353	13	=	=	SYM
ejpam-5535	353	14	g((v	g((v	NOUN
ejpam-5535	353	15	]	]	PUNCT
ejpam-5535	353	16	)	)	PUNCT
ejpam-5535	353	17	.	.	PUNCT
ejpam-5535	354	1	then	then	ADV
ejpam-5535	354	2	φ((vgg	φ((vgg	VERB
ejpam-5535	354	3	]	]	PUNCT
ejpam-5535	354	4	)	)	PUNCT
ejpam-5535	354	5	=	=	SYM
ejpam-5535	354	6	φ((v	φ((v	NOUN
ejpam-5535	354	7	]	]	PUNCT
ejpam-5535	354	8	)	)	PUNCT
ejpam-5535	354	9	.	.	PUNCT
ejpam-5535	355	1	therefore	therefore	ADV
ejpam-5535	355	2	(	(	PUNCT
ejpam-5535	355	3	vgg	vgg	NOUN
ejpam-5535	355	4	]	]	X
ejpam-5535	356	1	=	=	PUNCT
ejpam-5535	357	1	(	(	PUNCT
ejpam-5535	357	2	v	v	NOUN
ejpam-5535	357	3	]	]	X
ejpam-5535	357	4	.	.	PUNCT
ejpam-5535	358	1	since	since	SCONJ
ejpam-5535	358	2	vgg	vgg	PROPN
ejpam-5535	358	3	∈	∈	PROPN
ejpam-5535	358	4	(	(	PUNCT
ejpam-5535	358	5	vgg	vgg	NOUN
ejpam-5535	358	6	]	]	X
ejpam-5535	358	7	=	=	PUNCT
ejpam-5535	358	8	(	(	PUNCT
ejpam-5535	358	9	v	v	NOUN
ejpam-5535	358	10	]	]	PUNCT
ejpam-5535	358	11	,	,	PUNCT
ejpam-5535	358	12	(	(	PUNCT
ejpam-5535	358	13	vgg	vgg	NOUN
ejpam-5535	358	14	]	]	X
ejpam-5535	358	15	≤	≤	PROPN
ejpam-5535	358	16	v.	v.	CCONJ
ejpam-5535	358	17	similarly	similarly	ADV
ejpam-5535	358	18	v	v	ADJ
ejpam-5535	358	19	≤	≤	NUM
ejpam-5535	358	20	vgg	vgg	NOUN
ejpam-5535	358	21	.	.	PUNCT
ejpam-5535	359	1	hence	hence	ADV
ejpam-5535	359	2	vgg	vgg	PROPN
ejpam-5535	359	3	=	=	SYM
ejpam-5535	359	4	v	v	NOUN
ejpam-5535	359	5	,	,	PUNCT
ejpam-5535	359	6	for	for	ADP
ejpam-5535	359	7	all	all	DET
ejpam-5535	359	8	v	v	NOUN
ejpam-5535	359	9	∈	∈	PROPN
ejpam-5535	359	10	a.	a.	NOUN
ejpam-5535	359	11	r.	r.	PROPN
ejpam-5535	359	12	sirisetti	sirisetti	PROPN
ejpam-5535	359	13	et	et	PROPN
ejpam-5535	359	14	al	al	PROPN
ejpam-5535	359	15	.	.	PUNCT
ejpam-5535	359	16	/	/	SYM
ejpam-5535	359	17	eur	eur	PROPN
ejpam-5535	359	18	.	.	PUNCT
ejpam-5535	360	1	j.	j.	PROPN
ejpam-5535	360	2	pure	pure	PROPN
ejpam-5535	360	3	appl	appl	PROPN
ejpam-5535	360	4	.	.	PROPN
ejpam-5535	360	5	math	math	PROPN
ejpam-5535	360	6	,	,	PUNCT
ejpam-5535	360	7	18	18	NUM
ejpam-5535	360	8	(	(	PUNCT
ejpam-5535	360	9	1	1	NUM
ejpam-5535	360	10	)	)	PUNCT
ejpam-5535	360	11	(	(	PUNCT
ejpam-5535	360	12	2025	2025	NUM
ejpam-5535	360	13	)	)	PUNCT
ejpam-5535	360	14	,	,	PUNCT
ejpam-5535	360	15	5535	5535	NUM
ejpam-5535	360	16	11	11	NUM
ejpam-5535	360	17	of	of	ADP
ejpam-5535	360	18	12	12	NUM
ejpam-5535	360	19	for	for	ADP
ejpam-5535	360	20	any	any	DET
ejpam-5535	360	21	i	i	PROPN
ejpam-5535	360	22	∈	∈	PROPN
ejpam-5535	360	23	a	a	DET
ejpam-5535	360	24	,	,	PUNCT
ejpam-5535	360	25	g((i	g((i	NOUN
ejpam-5535	360	26	]	]	PUNCT
ejpam-5535	360	27	)	)	PUNCT
ejpam-5535	360	28	is	be	AUX
ejpam-5535	360	29	a	a	DET
ejpam-5535	360	30	kg	kg	NOUN
ejpam-5535	360	31	-	-	NOUN
ejpam-5535	360	32	filter	filter	NOUN
ejpam-5535	360	33	of	of	ADP
ejpam-5535	360	34	a	a	PRON
ejpam-5535	360	35	and	and	CCONJ
ejpam-5535	360	36	it	it	PRON
ejpam-5535	360	37	is	be	AUX
ejpam-5535	360	38	called	call	VERB
ejpam-5535	360	39	a	a	DET
ejpam-5535	360	40	normal	normal	ADJ
ejpam-5535	360	41	kg	kg	NOUN
ejpam-5535	360	42	-	-	NOUN
ejpam-5535	360	43	filter	filter	NOUN
ejpam-5535	360	44	.	.	PUNCT
ejpam-5535	361	1	the	the	DET
ejpam-5535	361	2	set	set	NOUN
ejpam-5535	361	3	of	of	ADP
ejpam-5535	361	4	all	all	DET
ejpam-5535	361	5	normal	normal	ADJ
ejpam-5535	361	6	kg	kg	NOUN
ejpam-5535	361	7	-	-	PUNCT
ejpam-5535	361	8	filters	filter	NOUN
ejpam-5535	361	9	of	of	ADP
ejpam-5535	361	10	a	a	DET
ejpam-5535	361	11	denoted	denote	VERB
ejpam-5535	361	12	by	by	ADP
ejpam-5535	361	13	ng(a	ng(a	NOUN
ejpam-5535	361	14	)	)	PUNCT
ejpam-5535	361	15	=	=	PRON
ejpam-5535	361	16	{	{	PUNCT
ejpam-5535	361	17	g((i	g((i	NOUN
ejpam-5535	361	18	]	]	X
ejpam-5535	361	19	)	)	PUNCT
ejpam-5535	362	1	=	=	PUNCT
ejpam-5535	363	1	[	[	X
ejpam-5535	363	2	ig)|i	ig)|i	X
ejpam-5535	363	3	∈	∈	PROPN
ejpam-5535	363	4	a	a	NOUN
ejpam-5535	363	5	}	}	PUNCT
ejpam-5535	363	6	and	and	CCONJ
ejpam-5535	363	7	it	it	PRON
ejpam-5535	363	8	forms	form	VERB
ejpam-5535	363	9	a	a	DET
ejpam-5535	363	10	sublattice	sublattice	NOUN
ejpam-5535	363	11	of	of	ADP
ejpam-5535	363	12	gf(a	gf(a	NOUN
ejpam-5535	363	13	)	)	PUNCT
ejpam-5535	363	14	.	.	PUNCT
ejpam-5535	364	1	moreover	moreover	ADV
ejpam-5535	364	2	ng(a	ng(a	NUM
ejpam-5535	364	3	)	)	PUNCT
ejpam-5535	364	4	is	be	AUX
ejpam-5535	364	5	a	a	DET
ejpam-5535	364	6	boolean	boolean	ADJ
ejpam-5535	364	7	algebra	algebra	NOUN
ejpam-5535	364	8	.	.	PUNCT
ejpam-5535	365	1	theorem	theorem	VERB
ejpam-5535	365	2	3.17	3.17	NUM
ejpam-5535	365	3	.	.	PUNCT
ejpam-5535	366	1	the	the	DET
ejpam-5535	366	2	set	set	VERB
ejpam-5535	366	3	ng(a	ng(a	NOUN
ejpam-5535	366	4	)	)	PUNCT
ejpam-5535	366	5	forms	form	VERB
ejpam-5535	366	6	a	a	DET
ejpam-5535	366	7	sublattice	sublattice	NOUN
ejpam-5535	366	8	of	of	ADP
ejpam-5535	366	9	gf(a	gf(a	NOUN
ejpam-5535	366	10	)	)	PUNCT
ejpam-5535	366	11	with	with	ADP
ejpam-5535	366	12	the	the	DET
ejpam-5535	366	13	operations	operation	NOUN
ejpam-5535	366	14	g((i])∧	g((i])∧	NOUN
ejpam-5535	366	15	g((j	g((j	NOUN
ejpam-5535	366	16	]	]	X
ejpam-5535	366	17	)	)	PUNCT
ejpam-5535	367	1	=	=	SYM
ejpam-5535	367	2	g((i	g((i	NOUN
ejpam-5535	367	3	∧	∧	PROPN
ejpam-5535	367	4	j	j	PROPN
ejpam-5535	367	5	]	]	X
ejpam-5535	367	6	)	)	PUNCT
ejpam-5535	367	7	and	and	CCONJ
ejpam-5535	367	8	g((i	g((i	NOUN
ejpam-5535	367	9	]	]	X
ejpam-5535	367	10	)	)	PUNCT
ejpam-5535	368	1	⊔	⊔	PROPN
ejpam-5535	368	2	g((j	g((j	NOUN
ejpam-5535	368	3	]	]	PUNCT
ejpam-5535	368	4	)	)	PUNCT
ejpam-5535	368	5	=	=	SYM
ejpam-5535	368	6	g((i	g((i	NOUN
ejpam-5535	368	7	∨	∨	NUM
ejpam-5535	368	8	j	j	PROPN
ejpam-5535	368	9	]	]	X
ejpam-5535	368	10	)	)	PUNCT
ejpam-5535	368	11	,	,	PUNCT
ejpam-5535	368	12	for	for	ADP
ejpam-5535	368	13	all	all	DET
ejpam-5535	368	14	g((i	g((i	NOUN
ejpam-5535	368	15	]	]	X
ejpam-5535	368	16	)	)	PUNCT
ejpam-5535	368	17	,	,	PUNCT
ejpam-5535	368	18	g(j	g(j	PROPN
ejpam-5535	368	19	]	]	PUNCT
ejpam-5535	368	20	)	)	PUNCT
ejpam-5535	368	21	∈	∈	PROPN
ejpam-5535	368	22	ng(a	ng(a	NOUN
ejpam-5535	368	23	)	)	PUNCT
ejpam-5535	368	24	,	,	PUNCT
ejpam-5535	368	25	for	for	ADP
ejpam-5535	368	26	some	some	DET
ejpam-5535	368	27	i	i	PROPN
ejpam-5535	368	28	,	,	PUNCT
ejpam-5535	368	29	j	j	PROPN
ejpam-5535	368	30	∈	∈	PROPN
ejpam-5535	368	31	a.	a.	NOUN
ejpam-5535	368	32	moreover	moreover	ADV
ejpam-5535	368	33	ng(a	ng(a	NUM
ejpam-5535	368	34	)	)	PUNCT
ejpam-5535	368	35	is	be	AUX
ejpam-5535	368	36	a	a	DET
ejpam-5535	368	37	boolean	boolean	ADJ
ejpam-5535	368	38	algebra	algebra	NOUN
ejpam-5535	368	39	.	.	PUNCT
ejpam-5535	369	1	proof	proof	NOUN
ejpam-5535	369	2	.	.	PUNCT
ejpam-5535	370	1	let	let	VERB
ejpam-5535	370	2	g((i	g((i	NOUN
ejpam-5535	370	3	]	]	X
ejpam-5535	370	4	)	)	PUNCT
ejpam-5535	370	5	,	,	PUNCT
ejpam-5535	370	6	g((j	g((j	PROPN
ejpam-5535	370	7	]	]	PUNCT
ejpam-5535	370	8	)	)	PUNCT
ejpam-5535	370	9	∈	∈	PROPN
ejpam-5535	370	10	ng(a	ng(a	NOUN
ejpam-5535	370	11	)	)	PUNCT
ejpam-5535	370	12	.	.	PUNCT
ejpam-5535	371	1	by	by	ADP
ejpam-5535	371	2	lemma	lemma	PROPN
ejpam-5535	371	3	2.4	2.4	NUM
ejpam-5535	371	4	.	.	PUNCT
ejpam-5535	371	5	,	,	PUNCT
ejpam-5535	371	6	g((i	g((i	NOUN
ejpam-5535	371	7	]	]	X
ejpam-5535	371	8	)	)	PUNCT
ejpam-5535	371	9	∧	∧	PROPN
ejpam-5535	371	10	g((j	g((j	NOUN
ejpam-5535	371	11	]	]	X
ejpam-5535	371	12	)	)	PUNCT
ejpam-5535	371	13	=	=	SYM
ejpam-5535	371	14	g((i	g((i	NOUN
ejpam-5535	371	15	]	]	X
ejpam-5535	371	16	∧	∧	PROPN
ejpam-5535	371	17	(	(	PUNCT
ejpam-5535	371	18	j	j	NOUN
ejpam-5535	371	19	]	]	X
ejpam-5535	371	20	)	)	PUNCT
ejpam-5535	371	21	=	=	SYM
ejpam-5535	371	22	g((i	g((i	NOUN
ejpam-5535	371	23	∧	∧	PROPN
ejpam-5535	371	24	j	j	PROPN
ejpam-5535	371	25	]	]	X
ejpam-5535	371	26	)	)	PUNCT
ejpam-5535	371	27	and	and	CCONJ
ejpam-5535	371	28	by	by	ADP
ejpam-5535	371	29	theorem	theorem	NOUN
ejpam-5535	371	30	3.9	3.9	NUM
ejpam-5535	371	31	.	.	PUNCT
ejpam-5535	371	32	,	,	PUNCT
ejpam-5535	371	33	g((i	g((i	NOUN
ejpam-5535	371	34	]	]	X
ejpam-5535	371	35	)	)	PUNCT
ejpam-5535	372	1	⊔	⊔	PROPN
ejpam-5535	372	2	g((j	g((j	NOUN
ejpam-5535	372	3	]	]	PUNCT
ejpam-5535	372	4	)	)	PUNCT
ejpam-5535	372	5	=	=	SYM
ejpam-5535	372	6	g((i	g((i	NOUN
ejpam-5535	372	7	]	]	X
ejpam-5535	372	8	∨	∨	X
ejpam-5535	372	9	(	(	PUNCT
ejpam-5535	372	10	j	j	NOUN
ejpam-5535	372	11	]	]	X
ejpam-5535	372	12	)	)	PUNCT
ejpam-5535	372	13	=	=	SYM
ejpam-5535	372	14	g((i	g((i	NOUN
ejpam-5535	372	15	∨	∨	NUM
ejpam-5535	372	16	j	j	PROPN
ejpam-5535	372	17	]	]	X
ejpam-5535	372	18	)	)	PUNCT
ejpam-5535	372	19	.	.	PUNCT
ejpam-5535	373	1	therefore	therefore	ADV
ejpam-5535	373	2	g((i	g((i	NOUN
ejpam-5535	373	3	]	]	X
ejpam-5535	373	4	)	)	PUNCT
ejpam-5535	374	1	∧	∧	PROPN
ejpam-5535	374	2	g((j	g((j	NOUN
ejpam-5535	374	3	]	]	PUNCT
ejpam-5535	374	4	)	)	PUNCT
ejpam-5535	374	5	∈	∈	PROPN
ejpam-5535	374	6	ng(a	ng(a	NOUN
ejpam-5535	374	7	)	)	PUNCT
ejpam-5535	374	8	and	and	CCONJ
ejpam-5535	374	9	g((i	g((i	NOUN
ejpam-5535	374	10	]	]	X
ejpam-5535	374	11	)	)	PUNCT
ejpam-5535	374	12	⊔	⊔	PROPN
ejpam-5535	374	13	g((j	g((j	NOUN
ejpam-5535	374	14	]	]	PUNCT
ejpam-5535	374	15	)	)	PUNCT
ejpam-5535	374	16	∈	∈	PROPN
ejpam-5535	374	17	ng(a	ng(a	NOUN
ejpam-5535	374	18	)	)	PUNCT
ejpam-5535	374	19	.	.	PUNCT
ejpam-5535	375	1	hence	hence	ADV
ejpam-5535	375	2	ng(a	ng(a	X
ejpam-5535	375	3	)	)	PUNCT
ejpam-5535	375	4	is	be	AUX
ejpam-5535	375	5	a	a	DET
ejpam-5535	375	6	sublattice	sublattice	NOUN
ejpam-5535	375	7	of	of	ADP
ejpam-5535	375	8	distributive	distributive	ADJ
ejpam-5535	375	9	lattice	lattice	NOUN
ejpam-5535	375	10	gf(a	gf(a	PROPN
ejpam-5535	375	11	)	)	PUNCT
ejpam-5535	375	12	with	with	ADP
ejpam-5535	375	13	the	the	DET
ejpam-5535	375	14	least	least	ADJ
ejpam-5535	375	15	element	element	ADJ
ejpam-5535	375	16	g((0	g((0	PROPN
ejpam-5535	375	17	]	]	PUNCT
ejpam-5535	375	18	)	)	PUNCT
ejpam-5535	375	19	and	and	CCONJ
ejpam-5535	375	20	the	the	DET
ejpam-5535	375	21	greatest	great	ADJ
ejpam-5535	375	22	element	element	NOUN
ejpam-5535	375	23	g((d	g((d	NOUN
ejpam-5535	375	24	]	]	PUNCT
ejpam-5535	375	25	)	)	PUNCT
ejpam-5535	375	26	where	where	SCONJ
ejpam-5535	375	27	d	d	NOUN
ejpam-5535	375	28	is	be	AUX
ejpam-5535	375	29	dense	dense	ADJ
ejpam-5535	375	30	in	in	ADP
ejpam-5535	375	31	a.	a.	NOUN
ejpam-5535	375	32	for	for	ADP
ejpam-5535	375	33	any	any	DET
ejpam-5535	375	34	g((i	g((i	NOUN
ejpam-5535	375	35	]	]	PUNCT
ejpam-5535	375	36	)	)	PUNCT
ejpam-5535	375	37	in	in	ADP
ejpam-5535	375	38	ng(a	ng(a	NOUN
ejpam-5535	375	39	)	)	PUNCT
ejpam-5535	375	40	,	,	PUNCT
ejpam-5535	375	41	there	there	PRON
ejpam-5535	375	42	exists	exist	VERB
ejpam-5535	375	43	g((ig	g((ig	NOUN
ejpam-5535	375	44	]	]	PUNCT
ejpam-5535	375	45	)	)	PUNCT
ejpam-5535	375	46	in	in	ADP
ejpam-5535	375	47	ng(a	ng(a	NOUN
ejpam-5535	375	48	)	)	PUNCT
ejpam-5535	375	49	such	such	ADJ
ejpam-5535	375	50	that	that	DET
ejpam-5535	375	51	g((i	g((i	NOUN
ejpam-5535	375	52	]	]	X
ejpam-5535	375	53	)	)	PUNCT
ejpam-5535	375	54	∧	∧	NOUN
ejpam-5535	375	55	g((ig	g((ig	NOUN
ejpam-5535	375	56	]	]	PUNCT
ejpam-5535	375	57	)	)	PUNCT
ejpam-5535	376	1	=	=	SYM
ejpam-5535	376	2	g((i	g((i	NOUN
ejpam-5535	376	3	∧	∧	PROPN
ejpam-5535	376	4	ig	ig	PROPN
ejpam-5535	376	5	]	]	PUNCT
ejpam-5535	376	6	)	)	PUNCT
ejpam-5535	376	7	.	.	PUNCT
ejpam-5535	377	1	let	let	VERB
ejpam-5535	377	2	x	x	PUNCT
ejpam-5535	377	3	∈	∈	NOUN
ejpam-5535	377	4	g((i	g((i	NOUN
ejpam-5535	377	5	∧	∧	PROPN
ejpam-5535	377	6	ig	ig	PROPN
ejpam-5535	377	7	]	]	X
ejpam-5535	377	8	)	)	PUNCT
ejpam-5535	377	9	=	=	SYM
ejpam-5535	377	10	g((i	g((i	NOUN
ejpam-5535	377	11	]	]	X
ejpam-5535	377	12	)	)	PUNCT
ejpam-5535	377	13	∧	∧	NOUN
ejpam-5535	377	14	g((ig	g((ig	NOUN
ejpam-5535	377	15	]	]	PUNCT
ejpam-5535	377	16	)	)	PUNCT
ejpam-5535	377	17	.	.	PUNCT
ejpam-5535	378	1	then	then	ADV
ejpam-5535	378	2	we	we	PRON
ejpam-5535	378	3	have	have	VERB
ejpam-5535	378	4	xg	xg	PROPN
ejpam-5535	378	5	∈	∈	PROPN
ejpam-5535	378	6	(	(	PUNCT
ejpam-5535	378	7	i	i	PRON
ejpam-5535	378	8	∧	∧	PROPN
ejpam-5535	378	9	ig	ig	PROPN
ejpam-5535	378	10	]	]	PUNCT
ejpam-5535	378	11	.	.	PUNCT
ejpam-5535	379	1	so	so	ADV
ejpam-5535	379	2	,	,	PUNCT
ejpam-5535	379	3	xg	xg	PROPN
ejpam-5535	379	4	≤	≤	PROPN
ejpam-5535	380	1	i	i	PRON
ejpam-5535	380	2	∧	∧	PROPN
ejpam-5535	380	3	ig	ig	PROPN
ejpam-5535	380	4	.	.	PUNCT
ejpam-5535	381	1	hence	hence	ADV
ejpam-5535	381	2	(	(	PUNCT
ejpam-5535	381	3	i	i	PRON
ejpam-5535	381	4	∧	∧	NOUN
ejpam-5535	381	5	ig)g	ig)g	PROPN
ejpam-5535	381	6	=	=	PROPN
ejpam-5535	381	7	ig	ig	PROPN
ejpam-5535	381	8	∨	∨	NUM
ejpam-5535	381	9	igg	igg	PROPN
ejpam-5535	381	10	≤	≤	PROPN
ejpam-5535	381	11	xgg	xgg	PROPN
ejpam-5535	381	12	≤	≤	NUM
ejpam-5535	381	13	x.	x.	VERB
ejpam-5535	381	14	since	since	SCONJ
ejpam-5535	381	15	ig	ig	PROPN
ejpam-5535	381	16	∨	∨	NUM
ejpam-5535	381	17	igg	igg	PROPN
ejpam-5535	381	18	is	be	AUX
ejpam-5535	381	19	dense	dense	ADJ
ejpam-5535	381	20	,	,	PUNCT
ejpam-5535	381	21	x	x	PRON
ejpam-5535	381	22	is	be	AUX
ejpam-5535	381	23	dense	dense	ADJ
ejpam-5535	381	24	.	.	PUNCT
ejpam-5535	382	1	then	then	ADV
ejpam-5535	382	2	g((i	g((i	NOUN
ejpam-5535	382	3	]	]	X
ejpam-5535	382	4	∧	∧	PROPN
ejpam-5535	382	5	g(ig	g(ig	PROPN
ejpam-5535	382	6	]	]	PUNCT
ejpam-5535	382	7	)	)	PUNCT
ejpam-5535	383	1	⊆	⊆	NUM
ejpam-5535	383	2	d.	d.	NOUN
ejpam-5535	383	3	hence	hence	ADV
ejpam-5535	383	4	g((i	g((i	NOUN
ejpam-5535	383	5	]	]	X
ejpam-5535	383	6	∧	∧	PROPN
ejpam-5535	383	7	g(ig	g(ig	PROPN
ejpam-5535	383	8	]	]	PUNCT
ejpam-5535	383	9	)	)	PUNCT
ejpam-5535	384	1	=	=	SYM
ejpam-5535	384	2	d.	d.	PROPN
ejpam-5535	384	3	now	now	ADV
ejpam-5535	384	4	,	,	PUNCT
ejpam-5535	384	5	g((i	g((i	NOUN
ejpam-5535	384	6	]	]	X
ejpam-5535	384	7	)	)	PUNCT
ejpam-5535	384	8	⊔g((ig	⊔g((ig	PROPN
ejpam-5535	384	9	]	]	PUNCT
ejpam-5535	384	10	)	)	PUNCT
ejpam-5535	384	11	=	=	SYM
ejpam-5535	384	12	g((i	g((i	NOUN
ejpam-5535	384	13	∨	∨	NUM
ejpam-5535	384	14	ig	ig	PROPN
ejpam-5535	384	15	]	]	X
ejpam-5535	384	16	)	)	PUNCT
ejpam-5535	384	17	=	=	PUNCT
ejpam-5535	385	1	[	[	X
ejpam-5535	385	2	(	(	PUNCT
ejpam-5535	385	3	i	i	PROPN
ejpam-5535	385	4	∨	∨	PROPN
ejpam-5535	385	5	ig)g	ig)g	PROPN
ejpam-5535	385	6	)	)	PUNCT
ejpam-5535	385	7	.	.	PUNCT
ejpam-5535	386	1	since	since	SCONJ
ejpam-5535	386	2	(	(	PUNCT
ejpam-5535	386	3	i	i	PROPN
ejpam-5535	386	4	∨	∨	NUM
ejpam-5535	386	5	ig	ig	PROPN
ejpam-5535	386	6	)	)	PUNCT
ejpam-5535	386	7	is	be	AUX
ejpam-5535	386	8	dense	dense	ADJ
ejpam-5535	386	9	,	,	PUNCT
ejpam-5535	386	10	(	(	PUNCT
ejpam-5535	386	11	i	i	NOUN
ejpam-5535	386	12	∨	∨	VERB
ejpam-5535	386	13	ig)g	ig)g	PROPN
ejpam-5535	386	14	=	=	PROPN
ejpam-5535	386	15	0	0	X
ejpam-5535	386	16	.	.	PUNCT
ejpam-5535	387	1	therefore	therefore	ADV
ejpam-5535	387	2	g((i	g((i	NOUN
ejpam-5535	387	3	∨	∨	NUM
ejpam-5535	387	4	ig	ig	PROPN
ejpam-5535	387	5	]	]	X
ejpam-5535	387	6	)	)	PUNCT
ejpam-5535	388	1	=	=	SYM
ejpam-5535	388	2	a.	a.	NOUN
ejpam-5535	388	3	hence	hence	ADV
ejpam-5535	388	4	g((i	g((i	NOUN
ejpam-5535	388	5	]	]	PUNCT
ejpam-5535	388	6	)	)	PUNCT
ejpam-5535	388	7	⊔g((ig	⊔g((ig	PROPN
ejpam-5535	388	8	]	]	PUNCT
ejpam-5535	388	9	)	)	PUNCT
ejpam-5535	389	1	=	=	SYM
ejpam-5535	389	2	a.	a.	NOUN
ejpam-5535	389	3	thus	thus	ADV
ejpam-5535	389	4	ng(a	ng(a	X
ejpam-5535	389	5	)	)	PUNCT
ejpam-5535	389	6	is	be	AUX
ejpam-5535	389	7	a	a	DET
ejpam-5535	389	8	boolean	boolean	ADJ
ejpam-5535	389	9	algebra	algebra	NOUN
ejpam-5535	389	10	.	.	PUNCT
ejpam-5535	390	1	theorem	theorem	VERB
ejpam-5535	390	2	3.18	3.18	NUM
ejpam-5535	390	3	.	.	PUNCT
ejpam-5535	391	1	for	for	ADP
ejpam-5535	391	2	any	any	DET
ejpam-5535	391	3	normal	normal	ADJ
ejpam-5535	391	4	kg	kg	ADJ
ejpam-5535	391	5	-	-	NOUN
ejpam-5535	391	6	filter	filter	NOUN
ejpam-5535	391	7	n	n	NOUN
ejpam-5535	391	8	of	of	ADP
ejpam-5535	391	9	a	a	PRON
ejpam-5535	391	10	,	,	PUNCT
ejpam-5535	391	11	there	there	PRON
ejpam-5535	391	12	exists	exist	VERB
ejpam-5535	391	13	a	a	DET
ejpam-5535	391	14	prime	prime	ADJ
ejpam-5535	391	15	filter	filter	NOUN
ejpam-5535	391	16	q	q	NOUN
ejpam-5535	391	17	of	of	ADP
ejpam-5535	391	18	a	a	DET
ejpam-5535	391	19	such	such	ADJ
ejpam-5535	391	20	that	that	SCONJ
ejpam-5535	391	21	n	n	NUM
ejpam-5535	391	22	⊆	⊆	NUM
ejpam-5535	391	23	q.	q.	NOUN
ejpam-5535	391	24	proof	proof	NOUN
ejpam-5535	391	25	.	.	PUNCT
ejpam-5535	392	1	let	let	VERB
ejpam-5535	392	2	n	n	PRON
ejpam-5535	392	3	be	be	AUX
ejpam-5535	392	4	a	a	DET
ejpam-5535	392	5	kg	kg	NOUN
ejpam-5535	392	6	-	-	NOUN
ejpam-5535	392	7	filter	filter	NOUN
ejpam-5535	392	8	of	of	ADP
ejpam-5535	392	9	a.	a.	NOUN
ejpam-5535	392	10	then	then	ADV
ejpam-5535	392	11	there	there	PRON
ejpam-5535	392	12	exists	exist	VERB
ejpam-5535	392	13	an	an	DET
ejpam-5535	392	14	ideal	ideal	ADJ
ejpam-5535	392	15	k	k	NOUN
ejpam-5535	392	16	of	of	ADP
ejpam-5535	392	17	a	a	DET
ejpam-5535	392	18	such	such	ADJ
ejpam-5535	392	19	that	that	DET
ejpam-5535	392	20	g(k	g(k	NOUN
ejpam-5535	392	21	)	)	PUNCT
ejpam-5535	392	22	=	=	SYM
ejpam-5535	393	1	n	n	NOUN
ejpam-5535	393	2	.	.	PUNCT
ejpam-5535	394	1	now	now	ADV
ejpam-5535	394	2	,	,	PUNCT
ejpam-5535	394	3	n	n	X
ejpam-5535	394	4	∩	∩	X
ejpam-5535	394	5	k	k	X
ejpam-5535	394	6	=	=	SYM
ejpam-5535	394	7	φ	φ	PROPN
ejpam-5535	394	8	.	.	PUNCT
ejpam-5535	395	1	then	then	ADV
ejpam-5535	395	2	there	there	PRON
ejpam-5535	395	3	exists	exist	VERB
ejpam-5535	395	4	a	a	DET
ejpam-5535	395	5	prime	prime	ADJ
ejpam-5535	395	6	filter	filter	NOUN
ejpam-5535	395	7	q	q	NOUN
ejpam-5535	395	8	of	of	ADP
ejpam-5535	395	9	a	a	DET
ejpam-5535	395	10	such	such	ADJ
ejpam-5535	395	11	that	that	DET
ejpam-5535	395	12	q	q	NOUN
ejpam-5535	395	13	∩k	∩k	NOUN
ejpam-5535	395	14	=	=	PUNCT
ejpam-5535	395	15	φ	φ	PROPN
ejpam-5535	395	16	and	and	CCONJ
ejpam-5535	395	17	n	n	PRON
ejpam-5535	395	18	⊆	⊆	NUM
ejpam-5535	395	19	q.	q.	PROPN
ejpam-5535	395	20	4	4	NUM
ejpam-5535	395	21	.	.	PUNCT
ejpam-5535	395	22	conclusions	conclusion	NOUN
ejpam-5535	395	23	this	this	DET
ejpam-5535	395	24	paper	paper	NOUN
ejpam-5535	395	25	exponentially	exponentially	ADV
ejpam-5535	395	26	enrich	enrich	VERB
ejpam-5535	395	27	algebraic	algebraic	ADJ
ejpam-5535	395	28	properties	property	NOUN
ejpam-5535	395	29	of	of	ADP
ejpam-5535	395	30	a	a	DET
ejpam-5535	395	31	certain	certain	ADJ
ejpam-5535	395	32	class	class	NOUN
ejpam-5535	395	33	of	of	ADP
ejpam-5535	395	34	filters	filter	NOUN
ejpam-5535	395	35	(	(	PUNCT
ejpam-5535	395	36	kgfilters	kgfilter	NOUN
ejpam-5535	395	37	)	)	PUNCT
ejpam-5535	395	38	generated	generate	VERB
ejpam-5535	395	39	by	by	ADP
ejpam-5535	395	40	a	a	DET
ejpam-5535	395	41	generalized	generalized	ADJ
ejpam-5535	395	42	complementation	complementation	NOUN
ejpam-5535	395	43	on	on	ADP
ejpam-5535	395	44	a	a	DET
ejpam-5535	395	45	distributive	distributive	ADJ
ejpam-5535	395	46	lattice	lattice	NOUN
ejpam-5535	395	47	with	with	ADP
ejpam-5535	395	48	dense	dense	ADJ
ejpam-5535	395	49	elements	element	NOUN
ejpam-5535	395	50	.	.	PUNCT
ejpam-5535	396	1	also	also	ADV
ejpam-5535	396	2	,	,	PUNCT
ejpam-5535	396	3	we	we	PRON
ejpam-5535	396	4	prove	prove	VERB
ejpam-5535	396	5	the	the	DET
ejpam-5535	396	6	class	class	NOUN
ejpam-5535	396	7	of	of	ADP
ejpam-5535	396	8	kg	kg	NOUN
ejpam-5535	396	9	-	-	PUNCT
ejpam-5535	396	10	filters	filter	NOUN
ejpam-5535	396	11	forms	form	VERB
ejpam-5535	396	12	a	a	DET
ejpam-5535	396	13	distributive	distributive	ADJ
ejpam-5535	396	14	lattice	lattice	NOUN
ejpam-5535	396	15	which	which	PRON
ejpam-5535	396	16	is	be	AUX
ejpam-5535	396	17	not	not	PART
ejpam-5535	396	18	induced	induce	VERB
ejpam-5535	396	19	.	.	PUNCT
ejpam-5535	397	1	further	far	ADV
ejpam-5535	397	2	,	,	PUNCT
ejpam-5535	397	3	we	we	PRON
ejpam-5535	397	4	derive	derive	VERB
ejpam-5535	397	5	normal	normal	ADJ
ejpam-5535	397	6	kg	kg	NOUN
ejpam-5535	397	7	-	-	PUNCT
ejpam-5535	397	8	filters	filter	NOUN
ejpam-5535	397	9	in	in	ADP
ejpam-5535	397	10	generalized	generalized	ADJ
ejpam-5535	397	11	complemented	complement	VERB
ejpam-5535	397	12	distributive	distributive	ADJ
ejpam-5535	397	13	lattices	lattice	NOUN
ejpam-5535	397	14	and	and	CCONJ
ejpam-5535	397	15	proven	prove	VERB
ejpam-5535	397	16	that	that	SCONJ
ejpam-5535	397	17	the	the	DET
ejpam-5535	397	18	class	class	NOUN
ejpam-5535	397	19	of	of	ADP
ejpam-5535	397	20	normal	normal	ADJ
ejpam-5535	397	21	kg	kg	NOUN
ejpam-5535	397	22	-	-	PUNCT
ejpam-5535	397	23	filters	filter	NOUN
ejpam-5535	397	24	is	be	AUX
ejpam-5535	397	25	a	a	DET
ejpam-5535	397	26	boolean	boolean	ADJ
ejpam-5535	397	27	algebra	algebra	NOUN
ejpam-5535	397	28	which	which	PRON
ejpam-5535	397	29	is	be	AUX
ejpam-5535	397	30	not	not	PART
ejpam-5535	397	31	induced	induce	VERB
ejpam-5535	397	32	.	.	PUNCT
ejpam-5535	398	1	we	we	PRON
ejpam-5535	398	2	can	can	AUX
ejpam-5535	398	3	classify	classify	VERB
ejpam-5535	398	4	kg	kg	NOUN
ejpam-5535	398	5	-	-	PUNCT
ejpam-5535	398	6	filters	filter	NOUN
ejpam-5535	398	7	and	and	CCONJ
ejpam-5535	398	8	normal	normal	ADJ
ejpam-5535	398	9	kg	kg	NOUN
ejpam-5535	398	10	-	-	PUNCT
ejpam-5535	398	11	filters	filter	NOUN
ejpam-5535	398	12	in	in	ADP
ejpam-5535	398	13	different	different	ADJ
ejpam-5535	398	14	class	class	NOUN
ejpam-5535	398	15	of	of	ADP
ejpam-5535	398	16	lattice	lattice	NOUN
ejpam-5535	398	17	.	.	PUNCT
ejpam-5535	399	1	conflict	conflict	NOUN
ejpam-5535	399	2	of	of	ADP
ejpam-5535	399	3	interest	interest	NOUN
ejpam-5535	399	4	the	the	DET
ejpam-5535	399	5	authors	author	NOUN
ejpam-5535	399	6	declare	declare	VERB
ejpam-5535	399	7	that	that	SCONJ
ejpam-5535	399	8	there	there	PRON
ejpam-5535	399	9	are	be	VERB
ejpam-5535	399	10	no	no	DET
ejpam-5535	399	11	conflicts	conflict	NOUN
ejpam-5535	399	12	of	of	ADP
ejpam-5535	399	13	interest	interest	NOUN
ejpam-5535	399	14	regarding	regard	VERB
ejpam-5535	399	15	the	the	DET
ejpam-5535	399	16	publication	publication	NOUN
ejpam-5535	399	17	of	of	ADP
ejpam-5535	399	18	this	this	DET
ejpam-5535	399	19	paper	paper	NOUN
ejpam-5535	399	20	acknowledgements	acknowledgement	NOUN
ejpam-5535	399	21	the	the	DET
ejpam-5535	399	22	authors	author	NOUN
ejpam-5535	399	23	wish	wish	VERB
ejpam-5535	399	24	to	to	PART
ejpam-5535	399	25	thank	thank	VERB
ejpam-5535	399	26	the	the	DET
ejpam-5535	399	27	anonymous	anonymous	ADJ
ejpam-5535	399	28	reviewers	reviewer	NOUN
ejpam-5535	399	29	for	for	ADP
ejpam-5535	399	30	their	their	PRON
ejpam-5535	399	31	valuable	valuable	ADJ
ejpam-5535	399	32	suggestions	suggestion	NOUN
ejpam-5535	399	33	.	.	PUNCT
ejpam-5535	400	1	this	this	DET
ejpam-5535	400	2	work	work	NOUN
ejpam-5535	400	3	was	be	AUX
ejpam-5535	400	4	supported	support	VERB
ejpam-5535	400	5	by	by	ADP
ejpam-5535	400	6	directorate	directorate	NOUN
ejpam-5535	400	7	of	of	ADP
ejpam-5535	400	8	research	research	NOUN
ejpam-5535	400	9	and	and	CCONJ
ejpam-5535	400	10	innovation	innovation	NOUN
ejpam-5535	400	11	,	,	PUNCT
ejpam-5535	400	12	walter	walter	PROPN
ejpam-5535	400	13	sisulu	sisulu	PROPN
ejpam-5535	400	14	university	university	PROPN
ejpam-5535	400	15	,	,	PUNCT
ejpam-5535	400	16	south	south	PROPN
ejpam-5535	400	17	africa	africa	PROPN
ejpam-5535	400	18	.	.	PUNCT
ejpam-5535	401	1	r.	r.	PROPN
ejpam-5535	401	2	sirisetti	sirisetti	PROPN
ejpam-5535	401	3	et	et	PROPN
ejpam-5535	401	4	al	al	PROPN
ejpam-5535	401	5	.	.	PUNCT
ejpam-5535	401	6	/	/	SYM
ejpam-5535	401	7	eur	eur	PROPN
ejpam-5535	401	8	.	.	PUNCT
ejpam-5535	402	1	j.	j.	PROPN
ejpam-5535	402	2	pure	pure	PROPN
ejpam-5535	402	3	appl	appl	PROPN
ejpam-5535	402	4	.	.	PROPN
ejpam-5535	402	5	math	math	PROPN
ejpam-5535	402	6	,	,	PUNCT
ejpam-5535	402	7	18	18	NUM
ejpam-5535	402	8	(	(	PUNCT
ejpam-5535	402	9	1	1	NUM
ejpam-5535	402	10	)	)	PUNCT
ejpam-5535	402	11	(	(	PUNCT
ejpam-5535	402	12	2025	2025	NUM
ejpam-5535	402	13	)	)	PUNCT
ejpam-5535	402	14	,	,	PUNCT
ejpam-5535	402	15	5535	5535	NUM
ejpam-5535	402	16	12	12	NUM
ejpam-5535	402	17	of	of	ADP
ejpam-5535	402	18	12	12	NUM
ejpam-5535	402	19	references	reference	NOUN
ejpam-5535	402	20	[	[	X
ejpam-5535	402	21	1	1	NUM
ejpam-5535	402	22	]	]	PUNCT
ejpam-5535	402	23	sergio	sergio	PROPN
ejpam-5535	402	24	a.	a.	NOUN
ejpam-5535	402	25	subordinations	subordination	NOUN
ejpam-5535	402	26	on	on	ADP
ejpam-5535	402	27	bounded	bounded	ADJ
ejpam-5535	402	28	distributive	distributive	ADJ
ejpam-5535	402	29	lattices	lattice	NOUN
ejpam-5535	402	30	.	.	PUNCT
ejpam-5535	403	1	order	order	NOUN
ejpam-5535	403	2	,	,	PUNCT
ejpam-5535	403	3	40:1–27	40:1–27	NUM
ejpam-5535	403	4	,	,	PUNCT
ejpam-5535	403	5	2023	2023	NUM
ejpam-5535	403	6	.	.	PUNCT
ejpam-5535	404	1	[	[	X
ejpam-5535	404	2	2	2	NUM
ejpam-5535	404	3	]	]	X
ejpam-5535	404	4	g.	g.	NOUN
ejpam-5535	404	5	birkhoff	birkhoff	PROPN
ejpam-5535	404	6	.	.	PUNCT
ejpam-5535	405	1	lattice	lattice	PROPN
ejpam-5535	405	2	theory	theory	PROPN
ejpam-5535	405	3	.	.	PUNCT
ejpam-5535	406	1	amer	amer	PROPN
ejpam-5535	406	2	.	.	PUNCT
ejpam-5535	406	3	math	math	PROPN
ejpam-5535	406	4	.	.	PUNCT
ejpam-5535	407	1	soc	soc	PROPN
ejpam-5535	407	2	.	.	PUNCT
ejpam-5535	408	1	collequium	collequium	NOUN
ejpam-5535	408	2	pub	pub	NOUN
ejpam-5535	408	3	,	,	PUNCT
ejpam-5535	408	4	1967	1967	NUM
ejpam-5535	408	5	.	.	PUNCT
ejpam-5535	409	1	[	[	X
ejpam-5535	409	2	3	3	X
ejpam-5535	409	3	]	]	X
ejpam-5535	409	4	g.	g.	PROPN
ejpam-5535	409	5	boole	boole	PROPN
ejpam-5535	409	6	.	.	PUNCT
ejpam-5535	410	1	an	an	DET
ejpam-5535	410	2	investigation	investigation	NOUN
ejpam-5535	410	3	of	of	ADP
ejpam-5535	410	4	the	the	DET
ejpam-5535	410	5	laws	law	NOUN
ejpam-5535	410	6	of	of	ADP
ejpam-5535	410	7	thought	thought	NOUN
ejpam-5535	410	8	.	.	PUNCT
ejpam-5535	411	1	reprinted	reprint	VERB
ejpam-5535	411	2	by	by	ADP
ejpam-5535	411	3	open	open	ADJ
ejpam-5535	411	4	court	court	PROPN
ejpam-5535	411	5	publishing	publishing	PROPN
ejpam-5535	411	6	co.	co.	PROPN
ejpam-5535	411	7	,	,	PUNCT
ejpam-5535	411	8	chelsea	chelsea	PROPN
ejpam-5535	411	9	,	,	PUNCT
ejpam-5535	411	10	london	london	PROPN
ejpam-5535	411	11	,	,	PUNCT
ejpam-5535	411	12	1940	1940	NUM
ejpam-5535	411	13	.	.	PUNCT
ejpam-5535	412	1	[	[	X
ejpam-5535	412	2	4	4	X
ejpam-5535	412	3	]	]	X
ejpam-5535	412	4	s.	s.	PROPN
ejpam-5535	412	5	burris	burris	PROPN
ejpam-5535	412	6	and	and	CCONJ
ejpam-5535	412	7	h.	h.	PROPN
ejpam-5535	412	8	p.	p.	PROPN
ejpam-5535	412	9	sankappanavar	sankappanavar	NOUN
ejpam-5535	412	10	.	.	PUNCT
ejpam-5535	413	1	a	a	DET
ejpam-5535	413	2	course	course	NOUN
ejpam-5535	413	3	in	in	ADP
ejpam-5535	413	4	universal	universal	ADJ
ejpam-5535	413	5	algebra	algebra	NOUN
ejpam-5535	413	6	.	.	PUNCT
ejpam-5535	414	1	springer	springer	NOUN
ejpam-5535	414	2	-	-	PUNCT
ejpam-5535	414	3	verlag	verlag	PROPN
ejpam-5535	414	4	,	,	PUNCT
ejpam-5535	414	5	1980	1980	NUM
ejpam-5535	414	6	.	.	PUNCT
ejpam-5535	415	1	[	[	X
ejpam-5535	415	2	5	5	X
ejpam-5535	415	3	]	]	PUNCT
ejpam-5535	415	4	w.	w.	PROPN
ejpam-5535	415	5	h.	h.	PROPN
ejpam-5535	415	6	cornish	cornish	PROPN
ejpam-5535	415	7	.	.	PUNCT
ejpam-5535	416	1	quasi	quasi	ADJ
ejpam-5535	416	2	-	-	ADJ
ejpam-5535	416	3	complemented	complemented	ADJ
ejpam-5535	416	4	lattices	lattice	NOUN
ejpam-5535	416	5	.	.	PUNCT
ejpam-5535	417	1	commentationes	commentatione	NOUN
ejpam-5535	417	2	mathematicae	mathematicae	VERB
ejpam-5535	417	3	universitatis	universitatis	PROPN
ejpam-5535	417	4	carolinae	carolinae	PROPN
ejpam-5535	417	5	,	,	PUNCT
ejpam-5535	417	6	15:501–511	15:501–511	NUM
ejpam-5535	417	7	,	,	PUNCT
ejpam-5535	417	8	1974	1974	NUM
ejpam-5535	417	9	.	.	PUNCT
ejpam-5535	418	1	[	[	X
ejpam-5535	418	2	6	6	NUM
ejpam-5535	418	3	]	]	X
ejpam-5535	418	4	g.	g.	PROPN
ejpam-5535	418	5	epstein	epstein	PROPN
ejpam-5535	418	6	and	and	CCONJ
ejpam-5535	418	7	a.	a.	NOUN
ejpam-5535	418	8	horn	horn	NOUN
ejpam-5535	418	9	.	.	PUNCT
ejpam-5535	419	1	chain	chain	NOUN
ejpam-5535	419	2	based	base	VERB
ejpam-5535	419	3	lattices	lattice	NOUN
ejpam-5535	419	4	.	.	PUNCT
ejpam-5535	420	1	pacific	pacific	PROPN
ejpam-5535	420	2	journal	journal	PROPN
ejpam-5535	420	3	of	of	ADP
ejpam-5535	420	4	mathematics	mathematic	NOUN
ejpam-5535	420	5	,	,	PUNCT
ejpam-5535	420	6	55:65–84	55:65–84	NOUN
ejpam-5535	420	7	,	,	PUNCT
ejpam-5535	420	8	1974	1974	NUM
ejpam-5535	420	9	.	.	PUNCT
ejpam-5535	421	1	[	[	X
ejpam-5535	421	2	7	7	X
ejpam-5535	421	3	]	]	X
ejpam-5535	421	4	y.	y.	PROPN
ejpam-5535	421	5	l.	l.	PROPN
ejpam-5535	421	6	ershov	ershov	PROPN
ejpam-5535	421	7	.	.	PUNCT
ejpam-5535	422	1	relatively	relatively	ADV
ejpam-5535	422	2	complemented	complement	VERB
ejpam-5535	422	3	distributive	distributive	ADJ
ejpam-5535	422	4	lattices	lattice	NOUN
ejpam-5535	422	5	.	.	PUNCT
ejpam-5535	423	1	algebra	algebra	NOUN
ejpam-5535	423	2	and	and	CCONJ
ejpam-5535	423	3	logic	logic	NOUN
ejpam-5535	423	4	,	,	PUNCT
ejpam-5535	423	5	18:431–459	18:431–459	PROPN
ejpam-5535	423	6	,	,	PUNCT
ejpam-5535	423	7	1978	1978	NUM
ejpam-5535	423	8	.	.	PUNCT
ejpam-5535	424	1	[	[	X
ejpam-5535	424	2	8	8	NUM
ejpam-5535	424	3	]	]	PUNCT
ejpam-5535	424	4	ravikumar	ravikumar	NOUN
ejpam-5535	424	5	bandaru	bandaru	PROPN
ejpam-5535	424	6	g.	g.	PROPN
ejpam-5535	424	7	jogarao	jogarao	PROPN
ejpam-5535	424	8	,	,	PUNCT
ejpam-5535	424	9	s.ramesh	s.ramesh	ADJ
ejpam-5535	424	10	and	and	CCONJ
ejpam-5535	424	11	rahul	rahul	PROPN
ejpam-5535	424	12	shukla	shukla	NOUN
ejpam-5535	424	13	.	.	PUNCT
ejpam-5535	425	1	g	g	NOUN
ejpam-5535	425	2	-	-	PUNCT
ejpam-5535	425	3	filters	filter	NOUN
ejpam-5535	425	4	and	and	CCONJ
ejpam-5535	425	5	generalized	generalize	VERB
ejpam-5535	425	6	complemented	complemented	ADJ
ejpam-5535	425	7	distributive	distributive	ADJ
ejpam-5535	425	8	lattices	lattice	NOUN
ejpam-5535	425	9	.	.	PUNCT
ejpam-5535	426	1	european	european	ADJ
ejpam-5535	426	2	journal	journal	PROPN
ejpam-5535	426	3	of	of	ADP
ejpam-5535	426	4	pure	pure	ADJ
ejpam-5535	426	5	and	and	CCONJ
ejpam-5535	426	6	applied	applied	ADJ
ejpam-5535	426	7	mathematics	mathematic	NOUN
ejpam-5535	426	8	,	,	PUNCT
ejpam-5535	426	9	in	in	ADP
ejpam-5535	426	10	press:237–249	press:237–249	NUM
ejpam-5535	426	11	,	,	PUNCT
ejpam-5535	426	12	2024	2024	NUM
ejpam-5535	426	13	.	.	PUNCT
ejpam-5535	427	1	[	[	X
ejpam-5535	427	2	9	9	NUM
ejpam-5535	427	3	]	]	X
ejpam-5535	427	4	john	john	PROPN
ejpam-5535	427	5	harding	harding	PROPN
ejpam-5535	427	6	guram	guram	PROPN
ejpam-5535	427	7	bezhanishvili	bezhanishvili	PROPN
ejpam-5535	427	8	and	and	CCONJ
ejpam-5535	427	9	mamuka	mamuka	NOUN
ejpam-5535	427	10	jibladze	jibladze	NOUN
ejpam-5535	427	11	.	.	PUNCT
ejpam-5535	428	1	canonical	canonical	ADJ
ejpam-5535	428	2	extensions	extension	NOUN
ejpam-5535	428	3	,	,	PUNCT
ejpam-5535	428	4	free	free	ADJ
ejpam-5535	428	5	completely	completely	ADV
ejpam-5535	428	6	distributive	distributive	ADJ
ejpam-5535	428	7	lattices	lattice	NOUN
ejpam-5535	428	8	,	,	PUNCT
ejpam-5535	428	9	and	and	CCONJ
ejpam-5535	428	10	complete	complete	ADJ
ejpam-5535	428	11	retracts	retract	NOUN
ejpam-5535	428	12	.	.	PUNCT
ejpam-5535	429	1	algebra	algebra	NOUN
ejpam-5535	429	2	univers	univer	NOUN
ejpam-5535	429	3	.	.	PUNCT
ejpam-5535	429	4	,	,	PUNCT
ejpam-5535	429	5	64:1–6	64:1–6	NOUN
ejpam-5535	429	6	,	,	PUNCT
ejpam-5535	429	7	2021	2021	NUM
ejpam-5535	429	8	.	.	PUNCT
ejpam-5535	430	1	[	[	X
ejpam-5535	430	2	10	10	NUM
ejpam-5535	430	3	]	]	X
ejpam-5535	430	4	alain	alain	PROPN
ejpam-5535	430	5	gélya	gélya	PROPN
ejpam-5535	430	6	,	,	PUNCT
ejpam-5535	430	7	miguel	miguel	PROPN
ejpam-5535	430	8	couceiro	couceiro	PROPN
ejpam-5535	430	9	,	,	PUNCT
ejpam-5535	430	10	laurent	laurent	NOUN
ejpam-5535	430	11	miclet	miclet	NOUN
ejpam-5535	430	12	,	,	PUNCT
ejpam-5535	430	13	and	and	CCONJ
ejpam-5535	430	14	amedeo	amedeo	PROPN
ejpam-5535	430	15	napoli	napoli	PROPN
ejpam-5535	430	16	.	.	PUNCT
ejpam-5535	431	1	a	a	DET
ejpam-5535	431	2	study	study	NOUN
ejpam-5535	431	3	of	of	ADP
ejpam-5535	431	4	algorithms	algorithm	NOUN
ejpam-5535	431	5	relating	relate	VERB
ejpam-5535	431	6	distributive	distributive	ADJ
ejpam-5535	431	7	lattices	lattice	NOUN
ejpam-5535	431	8	,	,	PUNCT
ejpam-5535	431	9	median	median	ADJ
ejpam-5535	431	10	graphs	graph	NOUN
ejpam-5535	431	11	,	,	PUNCT
ejpam-5535	431	12	and	and	CCONJ
ejpam-5535	431	13	formal	formal	ADJ
ejpam-5535	431	14	concept	concept	NOUN
ejpam-5535	431	15	analysis	analysis	NOUN
ejpam-5535	431	16	.	.	PUNCT
ejpam-5535	432	1	international	international	ADJ
ejpam-5535	432	2	journal	journal	PROPN
ejpam-5535	432	3	of	of	ADP
ejpam-5535	432	4	approximate	approximate	ADJ
ejpam-5535	432	5	reasoning	reasoning	NOUN
ejpam-5535	432	6	,	,	PUNCT
ejpam-5535	432	7	142:370–382	142:370–382	NUM
ejpam-5535	432	8	,	,	PUNCT
ejpam-5535	432	9	2022	2022	NUM
ejpam-5535	432	10	.	.	PUNCT
ejpam-5535	433	1	[	[	X
ejpam-5535	433	2	11	11	NUM
ejpam-5535	433	3	]	]	PUNCT
ejpam-5535	433	4	sergio	sergio	PROPN
ejpam-5535	433	5	celani	celani	PROPN
ejpam-5535	433	6	ismael	ismael	PROPN
ejpam-5535	433	7	calomino	calomino	PROPN
ejpam-5535	433	8	,	,	PUNCT
ejpam-5535	433	9	jorge	jorge	PROPN
ejpam-5535	433	10	castro	castro	PROPN
ejpam-5535	433	11	and	and	CCONJ
ejpam-5535	433	12	luciana	luciana	PROPN
ejpam-5535	433	13	valenzuela	valenzuela	PROPN
ejpam-5535	433	14	.	.	PUNCT
ejpam-5535	434	1	a	a	DET
ejpam-5535	434	2	study	study	NOUN
ejpam-5535	434	3	on	on	ADP
ejpam-5535	434	4	some	some	DET
ejpam-5535	434	5	classes	class	NOUN
ejpam-5535	434	6	of	of	ADP
ejpam-5535	434	7	distributive	distributive	ADJ
ejpam-5535	434	8	lattices	lattice	NOUN
ejpam-5535	434	9	with	with	ADP
ejpam-5535	434	10	a	a	DET
ejpam-5535	434	11	generalized	generalized	ADJ
ejpam-5535	434	12	implication	implication	NOUN
ejpam-5535	434	13	.	.	PUNCT
ejpam-5535	435	1	order	order	NOUN
ejpam-5535	435	2	,	,	PUNCT
ejpam-5535	435	3	2023	2023	NUM
ejpam-5535	435	4	.	.	PUNCT
ejpam-5535	436	1	[	[	X
ejpam-5535	436	2	12	12	NUM
ejpam-5535	436	3	]	]	X
ejpam-5535	436	4	c.	c.	PROPN
ejpam-5535	436	5	jayaram	jayaram	PROPN
ejpam-5535	436	6	.	.	PUNCT
ejpam-5535	437	1	weak	weak	ADJ
ejpam-5535	437	2	complemented	complemented	ADJ
ejpam-5535	437	3	and	and	CCONJ
ejpam-5535	437	4	weak	weak	ADJ
ejpam-5535	437	5	invertible	invertible	ADJ
ejpam-5535	437	6	elements	element	NOUN
ejpam-5535	437	7	in	in	ADP
ejpam-5535	437	8	c	c	NOUN
ejpam-5535	437	9	-	-	PUNCT
ejpam-5535	437	10	lattices	lattice	NOUN
ejpam-5535	437	11	.	.	PUNCT
ejpam-5535	438	1	algebra	algebra	NOUN
ejpam-5535	438	2	universalis	universali	VERB
ejpam-5535	438	3	,	,	PUNCT
ejpam-5535	438	4	77:237–249	77:237–249	PROPN
ejpam-5535	438	5	,	,	PUNCT
ejpam-5535	438	6	2017	2017	NUM
ejpam-5535	438	7	.	.	PUNCT
ejpam-5535	439	1	[	[	X
ejpam-5535	439	2	13	13	NUM
ejpam-5535	439	3	]	]	PUNCT
ejpam-5535	439	4	w.	w.	PROPN
ejpam-5535	439	5	b.	b.	PROPN
ejpam-5535	439	6	johnson	johnson	PROPN
ejpam-5535	439	7	.	.	PUNCT
ejpam-5535	440	1	on	on	ADP
ejpam-5535	440	2	quasi	quasi	NOUN
ejpam-5535	440	3	-	-	NOUN
ejpam-5535	440	4	complements	complement	NOUN
ejpam-5535	440	5	.	.	PUNCT
ejpam-5535	441	1	pacific	pacific	PROPN
ejpam-5535	441	2	journal	journal	PROPN
ejpam-5535	441	3	of	of	ADP
ejpam-5535	441	4	mathematics	mathematic	NOUN
ejpam-5535	441	5	,	,	PUNCT
ejpam-5535	441	6	48:113–118	48:113–118	PROPN
ejpam-5535	441	7	,	,	PUNCT
ejpam-5535	441	8	1973	1973	NUM
ejpam-5535	441	9	.	.	PUNCT
ejpam-5535	442	1	[	[	X
ejpam-5535	442	2	14	14	NUM
ejpam-5535	442	3	]	]	X
ejpam-5535	442	4	h.	h.	NOUN
ejpam-5535	442	5	lakser	lakser	PROPN
ejpam-5535	442	6	.	.	PUNCT
ejpam-5535	443	1	the	the	DET
ejpam-5535	443	2	structure	structure	NOUN
ejpam-5535	443	3	of	of	ADP
ejpam-5535	443	4	pseudo	pseudo	NOUN
ejpam-5535	443	5	-	-	ADJ
ejpam-5535	443	6	complemented	complement	VERB
ejpam-5535	443	7	distributive	distributive	ADJ
ejpam-5535	443	8	lattices	lattice	NOUN
ejpam-5535	443	9	-	-	PUNCT
ejpam-5535	443	10	i.	i.	NOUN
ejpam-5535	443	11	transaction	transaction	NOUN
ejpam-5535	443	12	of	of	ADP
ejpam-5535	443	13	the	the	DET
ejpam-5535	443	14	american	american	PROPN
ejpam-5535	443	15	mathematical	mathematical	PROPN
ejpam-5535	443	16	society	society	NOUN
ejpam-5535	443	17	,	,	PUNCT
ejpam-5535	443	18	157:335–342	157:335–342	NUM
ejpam-5535	443	19	,	,	PUNCT
ejpam-5535	443	20	1971	1971	NUM
ejpam-5535	443	21	.	.	PUNCT
ejpam-5535	444	1	[	[	X
ejpam-5535	444	2	15	15	NUM
ejpam-5535	444	3	]	]	X
ejpam-5535	444	4	p.	p.	NOUN
ejpam-5535	444	5	v.	v.	PROPN
ejpam-5535	445	1	venkatanarasimhan	venkatanarasimhan	PROPN
ejpam-5535	445	2	.	.	PUNCT
ejpam-5535	446	1	pseudo	pseudo	NOUN
ejpam-5535	446	2	-	-	NOUN
ejpam-5535	446	3	complements	complement	NOUN
ejpam-5535	446	4	in	in	ADP
ejpam-5535	446	5	posets	poset	NOUN
ejpam-5535	446	6	.	.	PUNCT
ejpam-5535	447	1	american	american	PROPN
ejpam-5535	447	2	mathematical	mathematical	PROPN
ejpam-5535	447	3	society	society	NOUN
ejpam-5535	447	4	,	,	PUNCT
ejpam-5535	447	5	28:9–17	28:9–17	NUM
ejpam-5535	447	6	,	,	PUNCT
ejpam-5535	447	7	1971	1971	NUM
ejpam-5535	447	8	.	.	PUNCT
ejpam-5535	448	1	introduction	introduction	NOUN
ejpam-5535	448	2	some	some	DET
ejpam-5535	448	3	results	result	NOUN
ejpam-5535	448	4	on	on	ADP
ejpam-5535	448	5	generalized	generalized	ADJ
ejpam-5535	448	6	complemented	complement	VERB
ejpam-5535	448	7	distributive	distributive	ADJ
ejpam-5535	448	8	lattices	lattice	NOUN
ejpam-5535	448	9	kg	kg	NOUN
ejpam-5535	448	10	-	-	NOUN
ejpam-5535	448	11	filters	filter	NOUN
ejpam-5535	448	12	in	in	ADP
ejpam-5535	448	13	g	g	NOUN
ejpam-5535	448	14	-	-	PUNCT
ejpam-5535	448	15	complemented	complement	VERB
ejpam-5535	448	16	distributive	distributive	ADJ
ejpam-5535	448	17	lattices	lattice	NOUN
ejpam-5535	448	18	conclusions	conclusion	NOUN