id	sid	tid	token	lemma	pos
ejpam-5125	1	1	european	european	PROPN
ejpam-5125	1	2	journal	journal	PROPN
ejpam-5125	1	3	of	of	ADP
ejpam-5125	1	4	pure	pure	ADJ
ejpam-5125	1	5	and	and	CCONJ
ejpam-5125	1	6	applied	apply	VERB
ejpam-5125	1	7	mathematics	mathematic	NOUN
ejpam-5125	1	8	vol	vol	NOUN
ejpam-5125	1	9	.	.	PROPN
ejpam-5125	2	1	17	17	NUM
ejpam-5125	2	2	,	,	PUNCT
ejpam-5125	2	3	no	no	INTJ
ejpam-5125	2	4	.	.	NOUN
ejpam-5125	2	5	2	2	NUM
ejpam-5125	2	6	,	,	PUNCT
ejpam-5125	2	7	2024	2024	NUM
ejpam-5125	2	8	,	,	PUNCT
ejpam-5125	2	9	819	819	NUM
ejpam-5125	2	10	-	-	SYM
ejpam-5125	2	11	834	834	NUM
ejpam-5125	2	12	issn	issn	PROPN
ejpam-5125	2	13	1307	1307	NUM
ejpam-5125	2	14	-	-	SYM
ejpam-5125	2	15	5543	5543	NUM
ejpam-5125	2	16	–	–	PUNCT
ejpam-5125	3	1	ejpam.com	ejpam.com	X
ejpam-5125	3	2	published	publish	VERB
ejpam-5125	3	3	by	by	ADP
ejpam-5125	3	4	new	new	PROPN
ejpam-5125	3	5	york	york	PROPN
ejpam-5125	3	6	business	business	PROPN
ejpam-5125	3	7	global	global	PROPN
ejpam-5125	3	8	paradistributive	paradistributive	PROPN
ejpam-5125	3	9	latticoids	latticoids	PROPN
ejpam-5125	3	10	ravikumar	ravikumar	PROPN
ejpam-5125	3	11	bandaru1	bandaru1	PROPN
ejpam-5125	3	12	,	,	PUNCT
ejpam-5125	3	13	suryavardhani	suryavardhani	NOUN
ejpam-5125	3	14	ajjarapu2,∗	ajjarapu2,∗	PROPN
ejpam-5125	3	15	1	1	NUM
ejpam-5125	3	16	department	department	NOUN
ejpam-5125	3	17	of	of	ADP
ejpam-5125	3	18	mathematics	mathematic	NOUN
ejpam-5125	3	19	,	,	PUNCT
ejpam-5125	3	20	school	school	NOUN
ejpam-5125	3	21	of	of	ADP
ejpam-5125	3	22	advanced	advanced	ADJ
ejpam-5125	3	23	sciences	science	NOUN
ejpam-5125	3	24	,	,	PUNCT
ejpam-5125	3	25	vit	vit	PROPN
ejpam-5125	3	26	-	-	PUNCT
ejpam-5125	3	27	ap	ap	PROPN
ejpam-5125	3	28	university	university	PROPN
ejpam-5125	3	29	,	,	PUNCT
ejpam-5125	3	30	andhra	andhra	PROPN
ejpam-5125	3	31	pradesh-522237	pradesh-522237	NOUN
ejpam-5125	3	32	,	,	PUNCT
ejpam-5125	3	33	india	india	PROPN
ejpam-5125	3	34	2	2	NUM
ejpam-5125	3	35	department	department	NOUN
ejpam-5125	3	36	of	of	ADP
ejpam-5125	3	37	mathematics	mathematic	NOUN
ejpam-5125	3	38	,	,	PUNCT
ejpam-5125	3	39	gitam	gitam	NOUN
ejpam-5125	3	40	deemed	deem	VERB
ejpam-5125	3	41	to	to	PART
ejpam-5125	3	42	be	be	AUX
ejpam-5125	3	43	university	university	NOUN
ejpam-5125	3	44	,	,	PUNCT
ejpam-5125	3	45	hyderabad	hyderabad	PROPN
ejpam-5125	3	46	campus	campus	NOUN
ejpam-5125	3	47	,	,	PUNCT
ejpam-5125	3	48	telangana-502329	telangana-502329	ADJ
ejpam-5125	3	49	,	,	PUNCT
ejpam-5125	3	50	india	india	PROPN
ejpam-5125	3	51	abstract	abstract	NOUN
ejpam-5125	3	52	.	.	PUNCT
ejpam-5125	4	1	in	in	ADP
ejpam-5125	4	2	this	this	DET
ejpam-5125	4	3	paper	paper	NOUN
ejpam-5125	4	4	,	,	PUNCT
ejpam-5125	4	5	the	the	DET
ejpam-5125	4	6	concept	concept	NOUN
ejpam-5125	4	7	of	of	ADP
ejpam-5125	4	8	paradistributive	paradistributive	ADJ
ejpam-5125	4	9	latticoid	latticoid	NOUN
ejpam-5125	4	10	(	(	PUNCT
ejpam-5125	4	11	pdl	pdl	NOUN
ejpam-5125	4	12	)	)	PUNCT
ejpam-5125	4	13	as	as	ADP
ejpam-5125	4	14	a	a	DET
ejpam-5125	4	15	generalization	generalization	NOUN
ejpam-5125	4	16	of	of	ADP
ejpam-5125	4	17	a	a	DET
ejpam-5125	4	18	distributive	distributive	ADJ
ejpam-5125	4	19	lattice	lattice	NOUN
ejpam-5125	4	20	is	be	AUX
ejpam-5125	4	21	introduced	introduce	VERB
ejpam-5125	4	22	and	and	CCONJ
ejpam-5125	4	23	investigated	investigate	VERB
ejpam-5125	4	24	its	its	PRON
ejpam-5125	4	25	properties	property	NOUN
ejpam-5125	4	26	.	.	PUNCT
ejpam-5125	5	1	a	a	DET
ejpam-5125	5	2	set	set	NOUN
ejpam-5125	5	3	of	of	ADP
ejpam-5125	5	4	equivalent	equivalent	ADJ
ejpam-5125	5	5	conditions	condition	NOUN
ejpam-5125	5	6	for	for	SCONJ
ejpam-5125	5	7	a	a	DET
ejpam-5125	5	8	pdl	pdl	NOUN
ejpam-5125	5	9	to	to	PART
ejpam-5125	5	10	become	become	VERB
ejpam-5125	5	11	a	a	DET
ejpam-5125	5	12	distributive	distributive	ADJ
ejpam-5125	5	13	lattice	lattice	NOUN
ejpam-5125	5	14	are	be	AUX
ejpam-5125	5	15	given	give	VERB
ejpam-5125	5	16	.	.	PUNCT
ejpam-5125	6	1	the	the	DET
ejpam-5125	6	2	notions	notion	NOUN
ejpam-5125	6	3	of	of	ADP
ejpam-5125	6	4	an	an	DET
ejpam-5125	6	5	ideal	ideal	NOUN
ejpam-5125	6	6	and	and	CCONJ
ejpam-5125	6	7	a	a	DET
ejpam-5125	6	8	filter	filter	NOUN
ejpam-5125	6	9	in	in	ADP
ejpam-5125	6	10	a	a	DET
ejpam-5125	6	11	pdl	pdl	NOUN
ejpam-5125	6	12	are	be	AUX
ejpam-5125	6	13	introduced	introduce	VERB
ejpam-5125	6	14	and	and	CCONJ
ejpam-5125	6	15	studied	study	VERB
ejpam-5125	6	16	their	their	PRON
ejpam-5125	6	17	properties	property	NOUN
ejpam-5125	6	18	.	.	PUNCT
ejpam-5125	7	1	subdirect	subdirect	NOUN
ejpam-5125	7	2	representation	representation	NOUN
ejpam-5125	7	3	of	of	ADP
ejpam-5125	7	4	a	a	DET
ejpam-5125	7	5	pdl	pdl	NOUN
ejpam-5125	7	6	is	be	AUX
ejpam-5125	7	7	obtained	obtain	VERB
ejpam-5125	7	8	.	.	PUNCT
ejpam-5125	8	1	2020	2020	NUM
ejpam-5125	8	2	mathematics	mathematic	NOUN
ejpam-5125	8	3	subject	subject	NOUN
ejpam-5125	8	4	classifications	classification	NOUN
ejpam-5125	8	5	:	:	PUNCT
ejpam-5125	8	6	06d99	06d99	NUM
ejpam-5125	8	7	key	key	ADJ
ejpam-5125	8	8	words	word	NOUN
ejpam-5125	8	9	and	and	CCONJ
ejpam-5125	8	10	phrases	phrase	NOUN
ejpam-5125	8	11	:	:	PUNCT
ejpam-5125	8	12	paradistributive	paradistributive	ADJ
ejpam-5125	8	13	latticoid(pdl	latticoid(pdl	NOUN
ejpam-5125	8	14	)	)	PUNCT
ejpam-5125	8	15	,	,	PUNCT
ejpam-5125	8	16	ideal	ideal	ADJ
ejpam-5125	8	17	,	,	PUNCT
ejpam-5125	8	18	filter	filter	NOUN
ejpam-5125	8	19	,	,	PUNCT
ejpam-5125	8	20	congruence	congruence	NOUN
ejpam-5125	8	21	,	,	PUNCT
ejpam-5125	8	22	subdirectly	subdirectly	ADV
ejpam-5125	8	23	irreducible	irreducible	ADJ
ejpam-5125	8	24	.	.	PUNCT
ejpam-5125	9	1	1	1	X
ejpam-5125	9	2	.	.	X
ejpam-5125	9	3	introduction	introduction	NOUN
ejpam-5125	9	4	garret	garret	PROPN
ejpam-5125	9	5	birkhoff	birkhoff	PROPN
ejpam-5125	9	6	’s	’s	PART
ejpam-5125	9	7	effort	effort	NOUN
ejpam-5125	9	8	in	in	ADP
ejpam-5125	9	9	the	the	DET
ejpam-5125	9	10	mid-1930	mid-1930	NOUN
ejpam-5125	9	11	’s	’s	PART
ejpam-5125	9	12	arouse	arouse	VERB
ejpam-5125	9	13	off	off	ADP
ejpam-5125	9	14	with	with	ADP
ejpam-5125	9	15	the	the	DET
ejpam-5125	9	16	overall	overall	ADJ
ejpam-5125	9	17	development	development	NOUN
ejpam-5125	9	18	of	of	ADP
ejpam-5125	9	19	lattice	lattice	PROPN
ejpam-5125	9	20	theory	theory	NOUN
ejpam-5125	9	21	(	(	PUNCT
ejpam-5125	9	22	see	see	VERB
ejpam-5125	9	23	[	[	X
ejpam-5125	9	24	1	1	NUM
ejpam-5125	9	25	]	]	NUM
ejpam-5125	9	26	)	)	PUNCT
ejpam-5125	9	27	.	.	PUNCT
ejpam-5125	10	1	in	in	ADP
ejpam-5125	10	2	a	a	DET
ejpam-5125	10	3	great	great	ADJ
ejpam-5125	10	4	sequence	sequence	NOUN
ejpam-5125	10	5	of	of	ADP
ejpam-5125	10	6	works	work	NOUN
ejpam-5125	10	7	,	,	PUNCT
ejpam-5125	10	8	he	he	PRON
ejpam-5125	10	9	signified	signify	VERB
ejpam-5125	10	10	the	the	DET
ejpam-5125	10	11	circumstances	circumstance	NOUN
ejpam-5125	10	12	of	of	ADP
ejpam-5125	10	13	lattice	lattice	NOUN
ejpam-5125	10	14	theory	theory	NOUN
ejpam-5125	10	15	and	and	CCONJ
ejpam-5125	10	16	demonstrated	demonstrate	VERB
ejpam-5125	10	17	how	how	SCONJ
ejpam-5125	10	18	it	it	PRON
ejpam-5125	10	19	provides	provide	VERB
ejpam-5125	10	20	a	a	DET
ejpam-5125	10	21	conjoining	conjoining	NOUN
ejpam-5125	10	22	for	for	ADP
ejpam-5125	10	23	independent	independent	ADJ
ejpam-5125	10	24	advancements	advancement	NOUN
ejpam-5125	10	25	in	in	ADP
ejpam-5125	10	26	many	many	ADJ
ejpam-5125	10	27	arithmetic	arithmetic	ADJ
ejpam-5125	10	28	disciplines	discipline	NOUN
ejpam-5125	10	29	.	.	PUNCT
ejpam-5125	11	1	birkhoff	birkhoff	NOUN
ejpam-5125	11	2	system	system	NOUN
ejpam-5125	11	3	is	be	AUX
ejpam-5125	11	4	an	an	DET
ejpam-5125	11	5	algebra	algebra	NOUN
ejpam-5125	11	6	in	in	ADP
ejpam-5125	11	7	which	which	PRON
ejpam-5125	11	8	two	two	NUM
ejpam-5125	11	9	binary	binary	ADJ
ejpam-5125	11	10	operations	operation	NOUN
ejpam-5125	11	11	meet	meet	VERB
ejpam-5125	11	12	and	and	CCONJ
ejpam-5125	11	13	join	join	VERB
ejpam-5125	11	14	,	,	PUNCT
ejpam-5125	11	15	each	each	PRON
ejpam-5125	11	16	of	of	ADP
ejpam-5125	11	17	which	which	PRON
ejpam-5125	11	18	is	be	AUX
ejpam-5125	11	19	commutative	commutative	ADJ
ejpam-5125	11	20	,	,	PUNCT
ejpam-5125	11	21	associative	associative	ADJ
ejpam-5125	11	22	,	,	PUNCT
ejpam-5125	11	23	and	and	CCONJ
ejpam-5125	11	24	idempotent	idempotent	NOUN
ejpam-5125	11	25	,	,	PUNCT
ejpam-5125	11	26	and	and	CCONJ
ejpam-5125	11	27	which	which	PRON
ejpam-5125	11	28	,	,	PUNCT
ejpam-5125	11	29	when	when	SCONJ
ejpam-5125	11	30	combined	combine	VERB
ejpam-5125	11	31	,	,	PUNCT
ejpam-5125	11	32	satisfy	satisfy	VERB
ejpam-5125	11	33	the	the	DET
ejpam-5125	11	34	relation	relation	NOUN
ejpam-5125	11	35	x	x	PUNCT
ejpam-5125	11	36	∧	∧	NOUN
ejpam-5125	11	37	(	(	PUNCT
ejpam-5125	11	38	x	x	PROPN
ejpam-5125	11	39	∨	∨	NUM
ejpam-5125	11	40	y	y	PROPN
ejpam-5125	11	41	)	)	PUNCT
ejpam-5125	11	42	=	=	SYM
ejpam-5125	12	1	x	x	SYM
ejpam-5125	12	2	∨	∨	X
ejpam-5125	12	3	(	(	PUNCT
ejpam-5125	12	4	x	x	PROPN
ejpam-5125	12	5	∧	∧	PROPN
ejpam-5125	12	6	y	y	PROPN
ejpam-5125	12	7	)	)	PUNCT
ejpam-5125	12	8	(	(	PUNCT
ejpam-5125	12	9	see	see	VERB
ejpam-5125	12	10	[	[	X
ejpam-5125	12	11	3	3	NUM
ejpam-5125	12	12	,	,	PUNCT
ejpam-5125	12	13	4	4	NUM
ejpam-5125	12	14	]	]	NUM
ejpam-5125	12	15	)	)	PUNCT
ejpam-5125	12	16	.	.	PUNCT
ejpam-5125	13	1	this	this	PRON
ejpam-5125	13	2	is	be	AUX
ejpam-5125	13	3	a	a	DET
ejpam-5125	13	4	weakend	weakend	ADJ
ejpam-5125	13	5	version	version	NOUN
ejpam-5125	13	6	of	of	ADP
ejpam-5125	13	7	the	the	DET
ejpam-5125	13	8	absorption	absorption	NOUN
ejpam-5125	13	9	law	law	NOUN
ejpam-5125	13	10	for	for	ADP
ejpam-5125	13	11	lattices	lattice	NOUN
ejpam-5125	13	12	and	and	CCONJ
ejpam-5125	13	13	was	be	AUX
ejpam-5125	13	14	introduced	introduce	VERB
ejpam-5125	13	15	in	in	ADP
ejpam-5125	13	16	the	the	DET
ejpam-5125	13	17	year	year	NOUN
ejpam-5125	13	18	1948	1948	NUM
ejpam-5125	13	19	.	.	PUNCT
ejpam-5125	14	1	lattices	lattice	NOUN
ejpam-5125	14	2	and	and	CCONJ
ejpam-5125	14	3	quasilattices	quasilattice	NOUN
ejpam-5125	14	4	are	be	AUX
ejpam-5125	14	5	examples	example	NOUN
ejpam-5125	14	6	of	of	ADP
ejpam-5125	14	7	birkhoff	birkhoff	NOUN
ejpam-5125	14	8	systems	system	NOUN
ejpam-5125	14	9	,	,	PUNCT
ejpam-5125	14	10	with	with	ADP
ejpam-5125	14	11	the	the	DET
ejpam-5125	14	12	latter	latter	ADJ
ejpam-5125	14	13	being	be	AUX
ejpam-5125	14	14	the	the	DET
ejpam-5125	14	15	regularization	regularization	NOUN
ejpam-5125	14	16	of	of	ADP
ejpam-5125	14	17	a	a	DET
ejpam-5125	14	18	variety	variety	NOUN
ejpam-5125	14	19	of	of	ADP
ejpam-5125	14	20	lattices	lattice	NOUN
ejpam-5125	14	21	.	.	PUNCT
ejpam-5125	15	1	the	the	DET
ejpam-5125	15	2	types	type	NOUN
ejpam-5125	15	3	of	of	ADP
ejpam-5125	15	4	birkhoff	birkhoff	NOUN
ejpam-5125	15	5	systems	system	NOUN
ejpam-5125	15	6	that	that	PRON
ejpam-5125	15	7	adhere	adhere	VERB
ejpam-5125	15	8	to	to	ADP
ejpam-5125	15	9	one	one	NUM
ejpam-5125	15	10	or	or	CCONJ
ejpam-5125	15	11	both	both	DET
ejpam-5125	15	12	distributive	distributive	ADJ
ejpam-5125	15	13	rules	rule	NOUN
ejpam-5125	15	14	were	be	AUX
ejpam-5125	15	15	studied	study	VERB
ejpam-5125	15	16	.	.	PUNCT
ejpam-5125	16	1	they	they	PRON
ejpam-5125	16	2	were	be	AUX
ejpam-5125	16	3	three	three	NUM
ejpam-5125	16	4	birkhoff	birkhoff	NOUN
ejpam-5125	16	5	systems	system	NOUN
ejpam-5125	16	6	namely	namely	ADV
ejpam-5125	16	7	meet	meet	VERB
ejpam-5125	16	8	-	-	PUNCT
ejpam-5125	16	9	distributive	distributive	ADJ
ejpam-5125	16	10	,	,	PUNCT
ejpam-5125	16	11	join	join	NOUN
ejpam-5125	16	12	-	-	PUNCT
ejpam-5125	16	13	distributive	distributive	ADJ
ejpam-5125	16	14	,	,	PUNCT
ejpam-5125	16	15	and	and	CCONJ
ejpam-5125	16	16	distributive	distributive	ADJ
ejpam-5125	16	17	.	.	PUNCT
ejpam-5125	17	1	the	the	DET
ejpam-5125	17	2	duality	duality	NOUN
ejpam-5125	17	3	between	between	ADP
ejpam-5125	17	4	meet	meet	VERB
ejpam-5125	17	5	and	and	CCONJ
ejpam-5125	17	6	join	join	VERB
ejpam-5125	17	7	operations	operation	NOUN
ejpam-5125	17	8	lead	lead	VERB
ejpam-5125	17	9	to	to	ADP
ejpam-5125	17	10	corresponding	corresponding	ADJ
ejpam-5125	17	11	results	result	NOUN
ejpam-5125	17	12	for	for	ADP
ejpam-5125	17	13	join	join	NOUN
ejpam-5125	17	14	-	-	PUNCT
ejpam-5125	17	15	distributive	distributive	ADJ
ejpam-5125	17	16	birkhoff	birkhoff	NOUN
ejpam-5125	17	17	systems	system	NOUN
ejpam-5125	17	18	.	.	PUNCT
ejpam-5125	18	1	a	a	DET
ejpam-5125	18	2	characterisation	characterisation	NOUN
ejpam-5125	18	3	of	of	ADP
ejpam-5125	18	4	these	these	DET
ejpam-5125	18	5	varieties	variety	NOUN
ejpam-5125	18	6	,	,	PUNCT
ejpam-5125	18	7	subvarieties	subvarietie	NOUN
ejpam-5125	18	8	,	,	PUNCT
ejpam-5125	18	9	a	a	DET
ejpam-5125	18	10	duality	duality	NOUN
ejpam-5125	18	11	thesis	thesis	NOUN
ejpam-5125	18	12	for	for	ADP
ejpam-5125	18	13	distributive	distributive	ADJ
ejpam-5125	18	14	birkhoff	birkhoff	NOUN
ejpam-5125	18	15	systems	system	NOUN
ejpam-5125	18	16	,	,	PUNCT
ejpam-5125	18	17	a	a	DET
ejpam-5125	18	18	structure	structure	NOUN
ejpam-5125	18	19	theory	theory	NOUN
ejpam-5125	18	20	for	for	ADP
ejpam-5125	18	21	meet	meet	ADJ
ejpam-5125	18	22	-	-	PUNCT
ejpam-5125	18	23	distributive	distributive	ADJ
ejpam-5125	18	24	birkhoff	birkhoff	NOUN
ejpam-5125	18	25	systems	system	NOUN
ejpam-5125	18	26	improved	improve	VERB
ejpam-5125	18	27	descriptions	description	NOUN
ejpam-5125	18	28	of	of	ADP
ejpam-5125	18	29	some	some	PRON
ejpam-5125	18	30	of	of	ADP
ejpam-5125	18	31	the	the	DET
ejpam-5125	18	32	subdirectly	subdirectly	ADJ
ejpam-5125	18	33	irreducibles	irreducible	NOUN
ejpam-5125	18	34	.	.	PUNCT
ejpam-5125	19	1	as	as	SCONJ
ejpam-5125	19	2	one	one	NUM
ejpam-5125	19	3	of	of	ADP
ejpam-5125	19	4	the	the	DET
ejpam-5125	19	5	standard	standard	ADJ
ejpam-5125	19	6	application	application	NOUN
ejpam-5125	19	7	of	of	ADP
ejpam-5125	19	8	birkhoff	birkhoff	NOUN
ejpam-5125	19	9	theorem	theorem	ADJ
ejpam-5125	19	10	∗corresponding	∗corresponde	VERB
ejpam-5125	19	11	author	author	NOUN
ejpam-5125	19	12	.	.	PUNCT
ejpam-5125	20	1	doi	doi	NOUN
ejpam-5125	20	2	:	:	PUNCT
ejpam-5125	20	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5125	https://doi.org/10.29020/nybg.ejpam.v17i2.5125	PRON
ejpam-5125	20	4	email	email	NOUN
ejpam-5125	20	5	addresses	address	VERB
ejpam-5125	20	6	:	:	PUNCT
ejpam-5125	21	1	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-5125	21	2	(	(	PUNCT
ejpam-5125	21	3	r.	r.	PROPN
ejpam-5125	21	4	bandaru	bandaru	PROPN
ejpam-5125	21	5	)	)	PUNCT
ejpam-5125	21	6	,	,	PUNCT
ejpam-5125	21	7	syerrapr@gitam.in	syerrapr@gitam.in	PROPN
ejpam-5125	21	8	(	(	PUNCT
ejpam-5125	21	9	s.	s.	PROPN
ejpam-5125	21	10	ajjarapu	ajjarapu	PROPN
ejpam-5125	21	11	)	)	PUNCT
ejpam-5125	21	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5125	22	1	819	819	NUM
ejpam-5125	22	2	©	©	ADP
ejpam-5125	22	3	2024	2024	NUM
ejpam-5125	22	4	ejpam	ejpam	NOUN
ejpam-5125	22	5	all	all	DET
ejpam-5125	22	6	rights	right	NOUN
ejpam-5125	22	7	reserved	reserve	VERB
ejpam-5125	22	8	.	.	PUNCT
ejpam-5125	23	1	r.	r.	PROPN
ejpam-5125	23	2	bandaru	bandaru	PROPN
ejpam-5125	23	3	,	,	PUNCT
ejpam-5125	23	4	s.	s.	PROPN
ejpam-5125	23	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	23	6	/	/	PUNCT
ejpam-5125	23	7	eur	eur	PROPN
ejpam-5125	23	8	.	.	PUNCT
ejpam-5125	24	1	j.	j.	PROPN
ejpam-5125	24	2	pure	pure	PROPN
ejpam-5125	24	3	appl	appl	PROPN
ejpam-5125	24	4	.	.	PROPN
ejpam-5125	24	5	math	math	PROPN
ejpam-5125	24	6	,	,	PUNCT
ejpam-5125	24	7	17	17	NUM
ejpam-5125	24	8	(	(	PUNCT
ejpam-5125	24	9	2	2	NUM
ejpam-5125	24	10	)	)	PUNCT
ejpam-5125	24	11	(	(	PUNCT
ejpam-5125	24	12	2024	2024	NUM
ejpam-5125	24	13	)	)	PUNCT
ejpam-5125	24	14	,	,	PUNCT
ejpam-5125	24	15	819	819	NUM
ejpam-5125	24	16	-	-	SYM
ejpam-5125	24	17	834	834	NUM
ejpam-5125	24	18	820	820	NUM
ejpam-5125	24	19	is	be	AUX
ejpam-5125	24	20	stone	stone	NOUN
ejpam-5125	24	21	’s	’s	PART
ejpam-5125	24	22	theorem	theorem	NOUN
ejpam-5125	24	23	,	,	PUNCT
ejpam-5125	24	24	here	here	ADV
ejpam-5125	24	25	our	our	PRON
ejpam-5125	24	26	aim	aim	NOUN
ejpam-5125	24	27	of	of	ADP
ejpam-5125	24	28	this	this	DET
ejpam-5125	24	29	paper	paper	NOUN
ejpam-5125	24	30	is	be	AUX
ejpam-5125	24	31	to	to	PART
ejpam-5125	24	32	use	use	VERB
ejpam-5125	24	33	birkhoff	birkhoff	NOUN
ejpam-5125	24	34	theorem	theorem	NOUN
ejpam-5125	24	35	for	for	ADP
ejpam-5125	24	36	subdirectly	subdirectly	ADV
ejpam-5125	24	37	irreducible	irreducible	ADJ
ejpam-5125	24	38	pdls	pdl	NOUN
ejpam-5125	24	39	.	.	PUNCT
ejpam-5125	25	1	george	george	PROPN
ejpam-5125	25	2	boole	boole	PROPN
ejpam-5125	25	3	’s	’s	PART
ejpam-5125	25	4	attempt	attempt	NOUN
ejpam-5125	25	5	to	to	PART
ejpam-5125	25	6	formulate	formulate	VERB
ejpam-5125	25	7	propositional	propositional	ADJ
ejpam-5125	25	8	logic	logic	NOUN
ejpam-5125	25	9	in	in	ADP
ejpam-5125	25	10	the	the	DET
ejpam-5125	25	11	first	first	ADJ
ejpam-5125	25	12	part	part	NOUN
ejpam-5125	25	13	of	of	ADP
ejpam-5125	25	14	the	the	DET
ejpam-5125	25	15	nineteenth	nineteenth	ADJ
ejpam-5125	25	16	century	century	NOUN
ejpam-5125	25	17	induced	induce	VERB
ejpam-5125	25	18	the	the	DET
ejpam-5125	25	19	idea	idea	NOUN
ejpam-5125	25	20	of	of	ADP
ejpam-5125	25	21	boolean	boolean	ADJ
ejpam-5125	25	22	algebras	algebra	NOUN
ejpam-5125	25	23	(	(	PUNCT
ejpam-5125	25	24	see	see	VERB
ejpam-5125	25	25	[	[	X
ejpam-5125	25	26	2	2	NUM
ejpam-5125	25	27	]	]	PUNCT
ejpam-5125	25	28	)	)	PUNCT
ejpam-5125	25	29	.	.	PUNCT
ejpam-5125	26	1	at	at	ADP
ejpam-5125	26	2	the	the	DET
ejpam-5125	26	3	end	end	NOUN
ejpam-5125	26	4	of	of	ADP
ejpam-5125	26	5	the	the	DET
ejpam-5125	26	6	course	course	NOUN
ejpam-5125	26	7	,	,	PUNCT
ejpam-5125	26	8	he	he	PRON
ejpam-5125	26	9	looked	look	VERB
ejpam-5125	26	10	into	into	ADP
ejpam-5125	26	11	the	the	DET
ejpam-5125	26	12	axiomatic	axiomatic	NOUN
ejpam-5125	26	13	of	of	ADP
ejpam-5125	26	14	boolean	boolean	ADJ
ejpam-5125	26	15	algebras	algebra	NOUN
ejpam-5125	26	16	.	.	PUNCT
ejpam-5125	27	1	distributive	distributive	ADJ
ejpam-5125	27	2	lattices	lattice	NOUN
ejpam-5125	27	3	had	have	VERB
ejpam-5125	27	4	a	a	DET
ejpam-5125	27	5	main	main	ADJ
ejpam-5125	27	6	role	role	NOUN
ejpam-5125	27	7	in	in	ADP
ejpam-5125	27	8	lattice	lattice	NOUN
ejpam-5125	27	9	theory	theory	NOUN
ejpam-5125	27	10	.	.	PUNCT
ejpam-5125	28	1	as	as	SCONJ
ejpam-5125	28	2	lattice	lattice	NOUN
ejpam-5125	28	3	theory	theory	NOUN
ejpam-5125	28	4	began	begin	VERB
ejpam-5125	28	5	with	with	ADP
ejpam-5125	28	6	boolean	boolean	ADJ
ejpam-5125	28	7	algebra	algebra	NOUN
ejpam-5125	28	8	,	,	PUNCT
ejpam-5125	28	9	the	the	DET
ejpam-5125	28	10	postulation	postulation	NOUN
ejpam-5125	28	11	of	of	ADP
ejpam-5125	28	12	distributive	distributive	ADJ
ejpam-5125	28	13	lattices	lattice	NOUN
ejpam-5125	28	14	is	be	AUX
ejpam-5125	28	15	the	the	DET
ejpam-5125	28	16	most	most	ADV
ejpam-5125	28	17	comprehensive	comprehensive	ADJ
ejpam-5125	28	18	with	with	ADP
ejpam-5125	28	19	fulfilling	fulfil	VERB
ejpam-5125	28	20	chapter	chapter	NOUN
ejpam-5125	28	21	in	in	ADP
ejpam-5125	28	22	the	the	DET
ejpam-5125	28	23	history	history	NOUN
ejpam-5125	28	24	of	of	ADP
ejpam-5125	28	25	lattice	lattice	PROPN
ejpam-5125	28	26	theory	theory	NOUN
ejpam-5125	28	27	.	.	PUNCT
ejpam-5125	29	1	many	many	ADJ
ejpam-5125	29	2	lattice	lattice	ADJ
ejpam-5125	29	3	conditions	condition	NOUN
ejpam-5125	29	4	,	,	PUNCT
ejpam-5125	29	5	as	as	ADV
ejpam-5125	29	6	well	well	ADV
ejpam-5125	29	7	as	as	ADP
ejpam-5125	29	8	lattice	lattice	NOUN
ejpam-5125	29	9	components	component	NOUN
ejpam-5125	29	10	and	and	CCONJ
ejpam-5125	29	11	ideals	ideal	NOUN
ejpam-5125	29	12	,	,	PUNCT
ejpam-5125	29	13	are	be	AUX
ejpam-5125	29	14	debilitate	debilitate	VERB
ejpam-5125	29	15	variants	variant	NOUN
ejpam-5125	29	16	of	of	ADP
ejpam-5125	29	17	distributivity	distributivity	NOUN
ejpam-5125	29	18	.	.	PUNCT
ejpam-5125	30	1	as	as	ADP
ejpam-5125	30	2	a	a	DET
ejpam-5125	30	3	result	result	NOUN
ejpam-5125	30	4	,	,	PUNCT
ejpam-5125	30	5	a	a	DET
ejpam-5125	30	6	detailed	detailed	ADJ
ejpam-5125	30	7	mastery	mastery	NOUN
ejpam-5125	30	8	of	of	ADP
ejpam-5125	30	9	distributive	distributive	ADJ
ejpam-5125	30	10	lattices	lattice	NOUN
ejpam-5125	30	11	is	be	AUX
ejpam-5125	30	12	required	require	VERB
ejpam-5125	30	13	to	to	PART
ejpam-5125	30	14	perform	perform	VERB
ejpam-5125	30	15	in	in	ADP
ejpam-5125	30	16	lattice	lattice	NOUN
ejpam-5125	30	17	theory	theory	NOUN
ejpam-5125	30	18	.	.	PUNCT
ejpam-5125	31	1	distributive	distributive	ADJ
ejpam-5125	31	2	lattices	lattice	NOUN
ejpam-5125	31	3	are	be	AUX
ejpam-5125	31	4	characterized	characterize	VERB
ejpam-5125	31	5	by	by	ADP
ejpam-5125	31	6	their	their	PRON
ejpam-5125	31	7	lattice	lattice	NOUN
ejpam-5125	31	8	of	of	ADP
ejpam-5125	31	9	ideals	ideal	NOUN
ejpam-5125	31	10	.	.	PUNCT
ejpam-5125	32	1	finally	finally	ADV
ejpam-5125	32	2	,	,	PUNCT
ejpam-5125	32	3	in	in	ADP
ejpam-5125	32	4	several	several	ADJ
ejpam-5125	32	5	applications	application	NOUN
ejpam-5125	32	6	,	,	PUNCT
ejpam-5125	32	7	the	the	DET
ejpam-5125	32	8	distributivity	distributivity	NOUN
ejpam-5125	32	9	constraint	constraint	NOUN
ejpam-5125	32	10	is	be	AUX
ejpam-5125	32	11	enforced	enforce	VERB
ejpam-5125	32	12	on	on	ADP
ejpam-5125	32	13	lattices	lattice	NOUN
ejpam-5125	32	14	arising	arise	VERB
ejpam-5125	32	15	in	in	ADP
ejpam-5125	32	16	various	various	ADJ
ejpam-5125	32	17	fields	field	NOUN
ejpam-5125	32	18	of	of	ADP
ejpam-5125	32	19	mathematics	mathematic	NOUN
ejpam-5125	32	20	,	,	PUNCT
ejpam-5125	32	21	particularly	particularly	ADV
ejpam-5125	32	22	algebra	algebra	NOUN
ejpam-5125	32	23	.	.	PUNCT
ejpam-5125	33	1	certain	certain	ADJ
ejpam-5125	33	2	algebras	algebra	NOUN
ejpam-5125	33	3	are	be	AUX
ejpam-5125	33	4	referred	refer	VERB
ejpam-5125	33	5	as	as	ADP
ejpam-5125	33	6	distributive	distributive	ADJ
ejpam-5125	33	7	quasi	quasi	ADJ
ejpam-5125	33	8	lattices	lattice	NOUN
ejpam-5125	33	9	,	,	PUNCT
ejpam-5125	33	10	and	and	CCONJ
ejpam-5125	33	11	they	they	PRON
ejpam-5125	33	12	are	be	AUX
ejpam-5125	33	13	used	use	VERB
ejpam-5125	33	14	to	to	PART
ejpam-5125	33	15	generalize	generalize	VERB
ejpam-5125	33	16	distributive	distributive	ADJ
ejpam-5125	33	17	lattices	lattice	NOUN
ejpam-5125	33	18	.	.	PUNCT
ejpam-5125	34	1	j.	j.	PROPN
ejpam-5125	34	2	a.	a.	PROPN
ejpam-5125	34	3	kalman	kalman	PROPN
ejpam-5125	34	4	given	give	VERB
ejpam-5125	34	5	subdirect	subdirect	NOUN
ejpam-5125	34	6	decomposition	decomposition	NOUN
ejpam-5125	34	7	of	of	ADP
ejpam-5125	34	8	distributive	distributive	ADJ
ejpam-5125	34	9	quasilattices	quasilattice	NOUN
ejpam-5125	34	10	(	(	PUNCT
ejpam-5125	34	11	see	see	VERB
ejpam-5125	34	12	[	[	X
ejpam-5125	34	13	5	5	NUM
ejpam-5125	34	14	]	]	PUNCT
ejpam-5125	34	15	)	)	PUNCT
ejpam-5125	34	16	u.	u.	PROPN
ejpam-5125	34	17	m.	m.	PROPN
ejpam-5125	34	18	swamy	swamy	PROPN
ejpam-5125	34	19	and	and	CCONJ
ejpam-5125	34	20	g.	g.	PROPN
ejpam-5125	34	21	c.	c.	PROPN
ejpam-5125	34	22	rao	rao	PROPN
ejpam-5125	34	23	introduced	introduce	VERB
ejpam-5125	34	24	the	the	DET
ejpam-5125	34	25	concept	concept	NOUN
ejpam-5125	34	26	of	of	ADP
ejpam-5125	34	27	an	an	DET
ejpam-5125	34	28	almost	almost	ADV
ejpam-5125	34	29	distributive	distributive	ADJ
ejpam-5125	34	30	lattice(adl	lattice(adl	NOUN
ejpam-5125	34	31	)	)	PUNCT
ejpam-5125	34	32	(	(	PUNCT
ejpam-5125	34	33	see	see	VERB
ejpam-5125	34	34	[	[	X
ejpam-5125	34	35	8	8	NUM
ejpam-5125	34	36	]	]	NUM
ejpam-5125	34	37	)	)	PUNCT
ejpam-5125	34	38	.	.	PUNCT
ejpam-5125	35	1	this	this	DET
ejpam-5125	35	2	group	group	NOUN
ejpam-5125	35	3	of	of	ADP
ejpam-5125	35	4	adls	adls	PROPN
ejpam-5125	35	5	covers	cover	VERB
ejpam-5125	35	6	nearly	nearly	ADV
ejpam-5125	35	7	all	all	DET
ejpam-5125	35	8	the	the	DET
ejpam-5125	35	9	existing	exist	VERB
ejpam-5125	35	10	ring	ring	NOUN
ejpam-5125	35	11	theoretical	theoretical	ADJ
ejpam-5125	35	12	hypothesis	hypothesis	NOUN
ejpam-5125	35	13	of	of	ADP
ejpam-5125	35	14	a	a	DET
ejpam-5125	35	15	boolean	boolean	ADJ
ejpam-5125	35	16	algebra	algebra	NOUN
ejpam-5125	35	17	.	.	PUNCT
ejpam-5125	36	1	the	the	DET
ejpam-5125	36	2	class	class	NOUN
ejpam-5125	36	3	of	of	ADP
ejpam-5125	36	4	triple	triple	ADJ
ejpam-5125	36	5	systems	system	NOUN
ejpam-5125	36	6	has	have	AUX
ejpam-5125	36	7	been	be	AUX
ejpam-5125	36	8	introduced	introduce	VERB
ejpam-5125	36	9	by	by	ADP
ejpam-5125	36	10	subrahmanyam	subrahmanyam	NOUN
ejpam-5125	36	11	as	as	ADP
ejpam-5125	36	12	a	a	DET
ejpam-5125	36	13	lattice	lattice	NOUN
ejpam-5125	36	14	theoretic	theoretic	ADJ
ejpam-5125	36	15	generalisation	generalisation	NOUN
ejpam-5125	36	16	of	of	ADP
ejpam-5125	36	17	p1	p1	PROPN
ejpam-5125	36	18	-	-	PUNCT
ejpam-5125	36	19	rings	ring	NOUN
ejpam-5125	36	20	(	(	PUNCT
ejpam-5125	36	21	see	see	VERB
ejpam-5125	36	22	[	[	X
ejpam-5125	36	23	6	6	NUM
ejpam-5125	36	24	,	,	PUNCT
ejpam-5125	36	25	7	7	NUM
ejpam-5125	36	26	]	]	NUM
ejpam-5125	36	27	)	)	PUNCT
ejpam-5125	36	28	.	.	PUNCT
ejpam-5125	37	1	for	for	ADP
ejpam-5125	37	2	most	most	ADJ
ejpam-5125	37	3	of	of	ADP
ejpam-5125	37	4	the	the	DET
ejpam-5125	37	5	results	result	NOUN
ejpam-5125	37	6	that	that	PRON
ejpam-5125	37	7	are	be	AUX
ejpam-5125	37	8	valid	valid	ADJ
ejpam-5125	37	9	in	in	ADP
ejpam-5125	37	10	triple	triple	ADJ
ejpam-5125	37	11	systems	system	NOUN
ejpam-5125	37	12	the	the	DET
ejpam-5125	37	13	additive	additive	ADJ
ejpam-5125	37	14	semigroup	semigroup	NOUN
ejpam-5125	37	15	structure	structure	NOUN
ejpam-5125	37	16	in	in	ADP
ejpam-5125	37	17	the	the	DET
ejpam-5125	37	18	triple	triple	ADJ
ejpam-5125	37	19	system	system	NOUN
ejpam-5125	37	20	does	do	AUX
ejpam-5125	37	21	not	not	PART
ejpam-5125	37	22	play	play	VERB
ejpam-5125	37	23	any	any	DET
ejpam-5125	37	24	role	role	NOUN
ejpam-5125	37	25	.	.	PUNCT
ejpam-5125	38	1	this	this	PRON
ejpam-5125	38	2	motivated	motivate	VERB
ejpam-5125	38	3	them	they	PRON
ejpam-5125	38	4	to	to	PART
ejpam-5125	38	5	introduce	introduce	VERB
ejpam-5125	38	6	the	the	DET
ejpam-5125	38	7	class	class	NOUN
ejpam-5125	38	8	of	of	ADP
ejpam-5125	38	9	almost	almost	ADV
ejpam-5125	38	10	distributive	distributive	ADJ
ejpam-5125	38	11	lattices	lattice	NOUN
ejpam-5125	38	12	.	.	PUNCT
ejpam-5125	39	1	an	an	DET
ejpam-5125	39	2	adl	adl	NOUN
ejpam-5125	39	3	is	be	AUX
ejpam-5125	39	4	an	an	DET
ejpam-5125	39	5	algebra	algebra	NOUN
ejpam-5125	39	6	(	(	PUNCT
ejpam-5125	39	7	l,∨,∧	l,∨,∧	NOUN
ejpam-5125	39	8	)	)	PUNCT
ejpam-5125	39	9	which	which	PRON
ejpam-5125	39	10	fulfills	fulfill	VERB
ejpam-5125	39	11	all	all	PRON
ejpam-5125	39	12	of	of	ADP
ejpam-5125	39	13	the	the	DET
ejpam-5125	39	14	distributive	distributive	ADJ
ejpam-5125	39	15	lattice	lattice	NOUN
ejpam-5125	39	16	’s	’s	PART
ejpam-5125	39	17	axioms	axiom	NOUN
ejpam-5125	39	18	with	with	ADP
ejpam-5125	39	19	0	0	NUM
ejpam-5125	39	20	except	except	SCONJ
ejpam-5125	39	21	the	the	DET
ejpam-5125	39	22	possibility	possibility	NOUN
ejpam-5125	39	23	of	of	ADP
ejpam-5125	39	24	commutativity	commutativity	NOUN
ejpam-5125	39	25	with	with	ADP
ejpam-5125	39	26	respect	respect	NOUN
ejpam-5125	39	27	to	to	ADP
ejpam-5125	39	28	∨	∨	NUM
ejpam-5125	39	29	and	and	CCONJ
ejpam-5125	39	30	∧.	∧.	VERB
ejpam-5125	39	31	the	the	DET
ejpam-5125	39	32	idea	idea	NOUN
ejpam-5125	39	33	of	of	ADP
ejpam-5125	39	34	our	our	PRON
ejpam-5125	39	35	paper	paper	NOUN
ejpam-5125	39	36	is	be	AUX
ejpam-5125	39	37	to	to	PART
ejpam-5125	39	38	introduce	introduce	VERB
ejpam-5125	39	39	a	a	DET
ejpam-5125	39	40	new	new	ADJ
ejpam-5125	39	41	algebraic	algebraic	ADJ
ejpam-5125	39	42	structure	structure	NOUN
ejpam-5125	39	43	as	as	ADP
ejpam-5125	39	44	a	a	DET
ejpam-5125	39	45	generalization	generalization	NOUN
ejpam-5125	39	46	of	of	ADP
ejpam-5125	39	47	distributive	distributive	ADJ
ejpam-5125	39	48	lattice	lattice	NOUN
ejpam-5125	39	49	and	and	CCONJ
ejpam-5125	39	50	distributive	distributive	ADJ
ejpam-5125	39	51	birkhoff	birkhoff	NOUN
ejpam-5125	39	52	systems	system	NOUN
ejpam-5125	39	53	.	.	PUNCT
ejpam-5125	40	1	we	we	PRON
ejpam-5125	40	2	introduce	introduce	VERB
ejpam-5125	40	3	paradistributive	paradistributive	ADJ
ejpam-5125	40	4	latticoids	latticoid	NOUN
ejpam-5125	40	5	,	,	PUNCT
ejpam-5125	40	6	which	which	PRON
ejpam-5125	40	7	is	be	AUX
ejpam-5125	40	8	an	an	DET
ejpam-5125	40	9	algebra	algebra	NOUN
ejpam-5125	40	10	of	of	ADP
ejpam-5125	40	11	type	type	NOUN
ejpam-5125	40	12	(	(	PUNCT
ejpam-5125	40	13	2,2,0	2,2,0	NOUN
ejpam-5125	40	14	)	)	PUNCT
ejpam-5125	40	15	that	that	PRON
ejpam-5125	40	16	assures	assure	VERB
ejpam-5125	40	17	all	all	DET
ejpam-5125	40	18	the	the	DET
ejpam-5125	40	19	propositions	proposition	NOUN
ejpam-5125	40	20	of	of	ADP
ejpam-5125	40	21	a	a	DET
ejpam-5125	40	22	distributive	distributive	ADJ
ejpam-5125	40	23	lattice	lattice	NOUN
ejpam-5125	40	24	with	with	ADP
ejpam-5125	40	25	the	the	DET
ejpam-5125	40	26	possible	possible	ADJ
ejpam-5125	40	27	exception	exception	NOUN
ejpam-5125	40	28	of	of	ADP
ejpam-5125	40	29	the	the	DET
ejpam-5125	40	30	left	left	ADJ
ejpam-5125	40	31	distributivity	distributivity	NOUN
ejpam-5125	40	32	of	of	ADP
ejpam-5125	40	33	the	the	DET
ejpam-5125	40	34	operation	operation	NOUN
ejpam-5125	40	35	meet	meet	NOUN
ejpam-5125	40	36	and	and	CCONJ
ejpam-5125	40	37	the	the	DET
ejpam-5125	40	38	commutativity	commutativity	NOUN
ejpam-5125	40	39	of	of	ADP
ejpam-5125	40	40	the	the	DET
ejpam-5125	40	41	operations	operation	NOUN
ejpam-5125	40	42	join	join	VERB
ejpam-5125	40	43	and	and	CCONJ
ejpam-5125	40	44	meet	meet	VERB
ejpam-5125	40	45	.	.	PUNCT
ejpam-5125	41	1	in	in	ADP
ejpam-5125	41	2	§	§	PROPN
ejpam-5125	41	3	2	2	NUM
ejpam-5125	41	4	,	,	PUNCT
ejpam-5125	41	5	we	we	PRON
ejpam-5125	41	6	discuss	discuss	VERB
ejpam-5125	41	7	the	the	DET
ejpam-5125	41	8	basic	basic	ADJ
ejpam-5125	41	9	definition	definition	NOUN
ejpam-5125	41	10	of	of	ADP
ejpam-5125	41	11	a	a	DET
ejpam-5125	41	12	pdl	pdl	NOUN
ejpam-5125	41	13	and	and	CCONJ
ejpam-5125	41	14	some	some	DET
ejpam-5125	41	15	preliminary	preliminary	ADJ
ejpam-5125	41	16	results	result	NOUN
ejpam-5125	41	17	related	relate	VERB
ejpam-5125	41	18	to	to	ADP
ejpam-5125	41	19	pdl	pdl	VERB
ejpam-5125	41	20	and	and	CCONJ
ejpam-5125	41	21	illustrate	illustrate	VERB
ejpam-5125	41	22	few	few	ADJ
ejpam-5125	41	23	examples	example	NOUN
ejpam-5125	41	24	.	.	PUNCT
ejpam-5125	42	1	in	in	ADP
ejpam-5125	42	2	§	§	PROPN
ejpam-5125	42	3	3	3	NUM
ejpam-5125	42	4	,	,	PUNCT
ejpam-5125	42	5	we	we	PRON
ejpam-5125	42	6	introduce	introduce	VERB
ejpam-5125	42	7	the	the	DET
ejpam-5125	42	8	notions	notion	NOUN
ejpam-5125	42	9	of	of	ADP
ejpam-5125	42	10	an	an	DET
ejpam-5125	42	11	ideal	ideal	NOUN
ejpam-5125	42	12	and	and	CCONJ
ejpam-5125	42	13	a	a	DET
ejpam-5125	42	14	filter	filter	NOUN
ejpam-5125	42	15	in	in	ADP
ejpam-5125	42	16	a	a	DET
ejpam-5125	42	17	pdl	pdl	NOUN
ejpam-5125	42	18	and	and	CCONJ
ejpam-5125	42	19	investigate	investigate	VERB
ejpam-5125	42	20	its	its	PRON
ejpam-5125	42	21	properties	property	NOUN
ejpam-5125	42	22	.	.	PUNCT
ejpam-5125	43	1	further	far	ADV
ejpam-5125	43	2	in	in	ADP
ejpam-5125	43	3	§	§	PROPN
ejpam-5125	43	4	4	4	NUM
ejpam-5125	43	5	,	,	PUNCT
ejpam-5125	43	6	we	we	PRON
ejpam-5125	43	7	also	also	ADV
ejpam-5125	43	8	provide	provide	VERB
ejpam-5125	43	9	a	a	DET
ejpam-5125	43	10	subdirect	subdirect	NOUN
ejpam-5125	43	11	representation	representation	NOUN
ejpam-5125	43	12	theorem	theorem	NOUN
ejpam-5125	43	13	for	for	ADP
ejpam-5125	43	14	associative	associative	ADJ
ejpam-5125	43	15	pdls	pdl	NOUN
ejpam-5125	43	16	using	use	VERB
ejpam-5125	43	17	birkhoff	birkhoff	NOUN
ejpam-5125	43	18	theorem	theorem	PROPN
ejpam-5125	43	19	,	,	PUNCT
ejpam-5125	43	20	which	which	PRON
ejpam-5125	43	21	simplifies	simplify	VERB
ejpam-5125	43	22	many	many	ADJ
ejpam-5125	43	23	results	result	NOUN
ejpam-5125	43	24	in	in	ADP
ejpam-5125	43	25	pdls	pdl	NOUN
ejpam-5125	43	26	.	.	PUNCT
ejpam-5125	44	1	2	2	X
ejpam-5125	44	2	.	.	X
ejpam-5125	44	3	paradistributive	paradistributive	ADJ
ejpam-5125	44	4	latticoids	latticoid	NOUN
ejpam-5125	44	5	in	in	ADP
ejpam-5125	44	6	this	this	DET
ejpam-5125	44	7	section	section	NOUN
ejpam-5125	44	8	,	,	PUNCT
ejpam-5125	44	9	we	we	PRON
ejpam-5125	44	10	define	define	VERB
ejpam-5125	44	11	a	a	DET
ejpam-5125	44	12	paradistributive	paradistributive	ADJ
ejpam-5125	44	13	latticoid	latticoid	NOUN
ejpam-5125	44	14	and	and	CCONJ
ejpam-5125	44	15	present	present	VERB
ejpam-5125	44	16	some	some	DET
ejpam-5125	44	17	fundamental	fundamental	ADJ
ejpam-5125	44	18	findings	finding	NOUN
ejpam-5125	44	19	,	,	PUNCT
ejpam-5125	44	20	the	the	DET
ejpam-5125	44	21	most	most	ADJ
ejpam-5125	44	22	of	of	ADP
ejpam-5125	44	23	which	which	PRON
ejpam-5125	44	24	require	require	VERB
ejpam-5125	44	25	just	just	ADV
ejpam-5125	44	26	simple	simple	ADJ
ejpam-5125	44	27	verification	verification	NOUN
ejpam-5125	44	28	.	.	PUNCT
ejpam-5125	45	1	definition	definition	NOUN
ejpam-5125	45	2	1	1	NUM
ejpam-5125	45	3	.	.	PUNCT
ejpam-5125	46	1	an	an	DET
ejpam-5125	46	2	algebra	algebra	NOUN
ejpam-5125	46	3	(	(	PUNCT
ejpam-5125	46	4	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	46	5	,	,	PUNCT
ejpam-5125	46	6	1	1	NUM
ejpam-5125	46	7	)	)	PUNCT
ejpam-5125	46	8	of	of	ADP
ejpam-5125	46	9	type	type	NOUN
ejpam-5125	46	10	(	(	PUNCT
ejpam-5125	46	11	2,2,0	2,2,0	NOUN
ejpam-5125	46	12	)	)	PUNCT
ejpam-5125	46	13	is	be	AUX
ejpam-5125	46	14	called	call	VERB
ejpam-5125	46	15	a	a	DET
ejpam-5125	46	16	paradistributive	paradistributive	ADJ
ejpam-5125	46	17	latticoid	latticoid	NOUN
ejpam-5125	46	18	,	,	PUNCT
ejpam-5125	46	19	if	if	SCONJ
ejpam-5125	46	20	it	it	PRON
ejpam-5125	46	21	assures	assure	VERB
ejpam-5125	46	22	the	the	DET
ejpam-5125	46	23	subsequent	subsequent	ADJ
ejpam-5125	46	24	axioms	axiom	NOUN
ejpam-5125	46	25	:	:	PUNCT
ejpam-5125	46	26	(	(	PUNCT
ejpam-5125	46	27	ld∨	ld∨	NOUN
ejpam-5125	46	28	)	)	PUNCT
ejpam-5125	46	29	p1	p1	NOUN
ejpam-5125	46	30	∨	∨	NUM
ejpam-5125	46	31	(	(	PUNCT
ejpam-5125	46	32	p2	p2	PROPN
ejpam-5125	46	33	∧	∧	PROPN
ejpam-5125	46	34	p3	p3	PROPN
ejpam-5125	46	35	)	)	PUNCT
ejpam-5125	46	36	=	=	PUNCT
ejpam-5125	47	1	(	(	PUNCT
ejpam-5125	47	2	p1	p1	PROPN
ejpam-5125	47	3	∨	∨	NUM
ejpam-5125	47	4	p2	p2	NOUN
ejpam-5125	47	5	)	)	PUNCT
ejpam-5125	47	6	∧	∧	PROPN
ejpam-5125	47	7	(	(	PUNCT
ejpam-5125	47	8	p1	p1	PROPN
ejpam-5125	47	9	∨	∨	NUM
ejpam-5125	47	10	p3	p3	PROPN
ejpam-5125	47	11	)	)	PUNCT
ejpam-5125	47	12	.	.	PUNCT
ejpam-5125	48	1	(	(	PUNCT
ejpam-5125	48	2	rd∨	rd∨	X
ejpam-5125	48	3	)	)	PUNCT
ejpam-5125	48	4	(	(	PUNCT
ejpam-5125	48	5	p1	p1	NOUN
ejpam-5125	48	6	∧	∧	NOUN
ejpam-5125	48	7	p2	p2	NOUN
ejpam-5125	48	8	)	)	PUNCT
ejpam-5125	48	9	∨	∨	NUM
ejpam-5125	48	10	p3	p3	NOUN
ejpam-5125	48	11	=	=	SYM
ejpam-5125	48	12	(	(	PUNCT
ejpam-5125	48	13	p1	p1	PROPN
ejpam-5125	48	14	∨	∨	NUM
ejpam-5125	48	15	p3	p3	PROPN
ejpam-5125	48	16	)	)	PUNCT
ejpam-5125	48	17	∧	∧	PROPN
ejpam-5125	48	18	(	(	PUNCT
ejpam-5125	48	19	p2	p2	PROPN
ejpam-5125	48	20	∨	∨	NUM
ejpam-5125	48	21	p3	p3	PROPN
ejpam-5125	48	22	)	)	PUNCT
ejpam-5125	48	23	.	.	PUNCT
ejpam-5125	49	1	(	(	PUNCT
ejpam-5125	49	2	l1	l1	PROPN
ejpam-5125	49	3	)	)	PUNCT
ejpam-5125	49	4	(	(	PUNCT
ejpam-5125	49	5	p1	p1	PROPN
ejpam-5125	49	6	∨	∨	NUM
ejpam-5125	49	7	p2	p2	NOUN
ejpam-5125	49	8	)	)	PUNCT
ejpam-5125	49	9	∧	∧	NOUN
ejpam-5125	49	10	p2	p2	NOUN
ejpam-5125	49	11	=	=	SYM
ejpam-5125	49	12	p2	p2	X
ejpam-5125	49	13	.	.	PUNCT
ejpam-5125	50	1	(	(	PUNCT
ejpam-5125	50	2	l2	l2	NOUN
ejpam-5125	50	3	)	)	PUNCT
ejpam-5125	50	4	(	(	PUNCT
ejpam-5125	50	5	p1	p1	PROPN
ejpam-5125	50	6	∨	∨	NUM
ejpam-5125	50	7	p2	p2	NOUN
ejpam-5125	50	8	)	)	PUNCT
ejpam-5125	50	9	∧	∧	PROPN
ejpam-5125	50	10	p1	p1	NOUN
ejpam-5125	50	11	=	=	PROPN
ejpam-5125	50	12	p1	p1	PROPN
ejpam-5125	50	13	.	.	PUNCT
ejpam-5125	51	1	r.	r.	PROPN
ejpam-5125	51	2	bandaru	bandaru	PROPN
ejpam-5125	51	3	,	,	PUNCT
ejpam-5125	51	4	s.	s.	PROPN
ejpam-5125	51	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	51	6	/	/	PUNCT
ejpam-5125	51	7	eur	eur	PROPN
ejpam-5125	51	8	.	.	PUNCT
ejpam-5125	52	1	j.	j.	PROPN
ejpam-5125	52	2	pure	pure	PROPN
ejpam-5125	52	3	appl	appl	PROPN
ejpam-5125	52	4	.	.	PROPN
ejpam-5125	52	5	math	math	PROPN
ejpam-5125	52	6	,	,	PUNCT
ejpam-5125	52	7	17	17	NUM
ejpam-5125	52	8	(	(	PUNCT
ejpam-5125	52	9	2	2	NUM
ejpam-5125	52	10	)	)	PUNCT
ejpam-5125	52	11	(	(	PUNCT
ejpam-5125	52	12	2024	2024	NUM
ejpam-5125	52	13	)	)	PUNCT
ejpam-5125	52	14	,	,	PUNCT
ejpam-5125	52	15	819	819	NUM
ejpam-5125	52	16	-	-	SYM
ejpam-5125	52	17	834	834	NUM
ejpam-5125	52	18	821	821	NUM
ejpam-5125	52	19	(	(	PUNCT
ejpam-5125	52	20	l3	l3	NOUN
ejpam-5125	52	21	)	)	PUNCT
ejpam-5125	52	22	p1	p1	PROPN
ejpam-5125	52	23	∨	∨	NUM
ejpam-5125	52	24	(	(	PUNCT
ejpam-5125	52	25	p1	p1	NOUN
ejpam-5125	52	26	∧	∧	NOUN
ejpam-5125	52	27	p2	p2	NOUN
ejpam-5125	52	28	)	)	PUNCT
ejpam-5125	52	29	=	=	SYM
ejpam-5125	52	30	p1	p1	NOUN
ejpam-5125	52	31	.	.	PUNCT
ejpam-5125	53	1	(	(	PUNCT
ejpam-5125	53	2	i1	i1	PROPN
ejpam-5125	53	3	)	)	PUNCT
ejpam-5125	53	4	p1	p1	PROPN
ejpam-5125	53	5	∨	∨	NUM
ejpam-5125	53	6	1	1	NUM
ejpam-5125	53	7	=	=	SYM
ejpam-5125	53	8	1	1	NUM
ejpam-5125	53	9	.	.	PUNCT
ejpam-5125	53	10	for	for	ADP
ejpam-5125	53	11	all	all	DET
ejpam-5125	53	12	p1	p1	NOUN
ejpam-5125	53	13	,	,	PUNCT
ejpam-5125	53	14	p2	p2	NOUN
ejpam-5125	53	15	,	,	PUNCT
ejpam-5125	53	16	p3	p3	PROPN
ejpam-5125	53	17	∈	∈	PROPN
ejpam-5125	53	18	v	v	NOUN
ejpam-5125	53	19	.	.	PUNCT
ejpam-5125	54	1	the	the	DET
ejpam-5125	54	2	independence	independence	NOUN
ejpam-5125	54	3	of	of	ADP
ejpam-5125	54	4	the	the	DET
ejpam-5125	54	5	axioms	axiom	NOUN
ejpam-5125	54	6	mentioned	mention	VERB
ejpam-5125	54	7	in	in	ADP
ejpam-5125	54	8	the	the	DET
ejpam-5125	54	9	above	above	ADJ
ejpam-5125	54	10	definition	definition	NOUN
ejpam-5125	54	11	can	can	AUX
ejpam-5125	54	12	be	be	AUX
ejpam-5125	54	13	verified	verify	VERB
ejpam-5125	54	14	using	use	VERB
ejpam-5125	54	15	non	non	ADJ
ejpam-5125	54	16	-	-	ADJ
ejpam-5125	54	17	trivial	trivial	ADJ
ejpam-5125	54	18	examples	example	NOUN
ejpam-5125	54	19	.	.	PUNCT
ejpam-5125	55	1	the	the	DET
ejpam-5125	55	2	example	example	NOUN
ejpam-5125	55	3	below	below	ADV
ejpam-5125	55	4	demonstrates	demonstrate	VERB
ejpam-5125	55	5	how	how	SCONJ
ejpam-5125	55	6	every	every	DET
ejpam-5125	55	7	non	non	ADJ
ejpam-5125	55	8	-	-	ADJ
ejpam-5125	55	9	empty	empty	ADJ
ejpam-5125	55	10	set	set	NOUN
ejpam-5125	55	11	can	can	AUX
ejpam-5125	55	12	be	be	AUX
ejpam-5125	55	13	transformed	transform	VERB
ejpam-5125	55	14	into	into	ADP
ejpam-5125	55	15	a	a	DET
ejpam-5125	55	16	pdl	pdl	NOUN
ejpam-5125	55	17	with	with	ADP
ejpam-5125	55	18	any	any	DET
ejpam-5125	55	19	member	member	NOUN
ejpam-5125	55	20	that	that	PRON
ejpam-5125	55	21	has	have	AUX
ejpam-5125	55	22	been	be	AUX
ejpam-5125	55	23	arbitrarily	arbitrarily	ADV
ejpam-5125	55	24	preassigned	preassigne	VERB
ejpam-5125	55	25	as	as	ADP
ejpam-5125	55	26	its	its	PRON
ejpam-5125	55	27	unity(greatest	unity(great	ADJ
ejpam-5125	55	28	)	)	PUNCT
ejpam-5125	55	29	element	element	NOUN
ejpam-5125	55	30	.	.	PUNCT
ejpam-5125	56	1	for	for	ADP
ejpam-5125	56	2	any	any	DET
ejpam-5125	56	3	p1	p1	NOUN
ejpam-5125	56	4	,	,	PUNCT
ejpam-5125	56	5	p2	p2	PROPN
ejpam-5125	56	6	∈	∈	PROPN
ejpam-5125	56	7	v	v	NOUN
ejpam-5125	56	8	,	,	PUNCT
ejpam-5125	56	9	we	we	PRON
ejpam-5125	56	10	say	say	VERB
ejpam-5125	56	11	that	that	SCONJ
ejpam-5125	56	12	p1	p1	NOUN
ejpam-5125	56	13	is	be	AUX
ejpam-5125	56	14	less	less	ADJ
ejpam-5125	56	15	than	than	ADP
ejpam-5125	56	16	or	or	CCONJ
ejpam-5125	56	17	equal	equal	ADJ
ejpam-5125	56	18	to	to	ADP
ejpam-5125	56	19	p2	p2	PROPN
ejpam-5125	56	20	and	and	CCONJ
ejpam-5125	56	21	write	write	VERB
ejpam-5125	56	22	p1	p1	NOUN
ejpam-5125	56	23	≤	≤	ADJ
ejpam-5125	56	24	p2	p2	PROPN
ejpam-5125	56	25	if	if	SCONJ
ejpam-5125	56	26	p1	p1	PROPN
ejpam-5125	56	27	∧	∧	PROPN
ejpam-5125	56	28	p2	p2	NOUN
ejpam-5125	56	29	=	=	SYM
ejpam-5125	56	30	p1	p1	PROPN
ejpam-5125	56	31	or	or	CCONJ
ejpam-5125	56	32	equivalently	equivalently	ADV
ejpam-5125	56	33	p1	p1	PROPN
ejpam-5125	56	34	∨	∨	NOUN
ejpam-5125	56	35	p2	p2	X
ejpam-5125	56	36	=	=	SYM
ejpam-5125	56	37	p2	p2	PROPN
ejpam-5125	56	38	and	and	CCONJ
ejpam-5125	56	39	it	it	PRON
ejpam-5125	56	40	can	can	AUX
ejpam-5125	56	41	be	be	AUX
ejpam-5125	56	42	easily	easily	ADV
ejpam-5125	56	43	observed	observe	VERB
ejpam-5125	56	44	that	that	SCONJ
ejpam-5125	56	45	≤	≤	NUM
ejpam-5125	56	46	is	be	AUX
ejpam-5125	56	47	a	a	DET
ejpam-5125	56	48	partial	partial	ADJ
ejpam-5125	56	49	order	order	NOUN
ejpam-5125	56	50	on	on	ADP
ejpam-5125	56	51	v	v	NOUN
ejpam-5125	56	52	.	.	PUNCT
ejpam-5125	57	1	the	the	DET
ejpam-5125	57	2	element	element	NOUN
ejpam-5125	57	3	1	1	NUM
ejpam-5125	57	4	,	,	PUNCT
ejpam-5125	57	5	in	in	ADP
ejpam-5125	57	6	definition	definition	NOUN
ejpam-5125	57	7	1	1	NUM
ejpam-5125	57	8	,	,	PUNCT
ejpam-5125	57	9	is	be	AUX
ejpam-5125	57	10	called	call	VERB
ejpam-5125	57	11	the	the	DET
ejpam-5125	57	12	greatest	great	ADJ
ejpam-5125	57	13	element	element	NOUN
ejpam-5125	57	14	.	.	PUNCT
ejpam-5125	57	15	example	example	NOUN
ejpam-5125	58	1	1	1	NUM
ejpam-5125	58	2	.	.	PUNCT
ejpam-5125	58	3	let	let	VERB
ejpam-5125	58	4	v	v	PART
ejpam-5125	58	5	be	be	AUX
ejpam-5125	58	6	a	a	DET
ejpam-5125	58	7	non	non	ADJ
ejpam-5125	58	8	-	-	ADJ
ejpam-5125	58	9	empty	empty	ADJ
ejpam-5125	58	10	set	set	NOUN
ejpam-5125	58	11	.	.	PUNCT
ejpam-5125	59	1	fix	fix	VERB
ejpam-5125	59	2	some	some	DET
ejpam-5125	59	3	element	element	NOUN
ejpam-5125	59	4	y0	y0	PROPN
ejpam-5125	59	5	∈	∈	NOUN
ejpam-5125	59	6	v	v	NOUN
ejpam-5125	59	7	.	.	PUNCT
ejpam-5125	60	1	then	then	ADV
ejpam-5125	60	2	,	,	PUNCT
ejpam-5125	60	3	for	for	ADP
ejpam-5125	60	4	any	any	DET
ejpam-5125	60	5	x	x	NOUN
ejpam-5125	60	6	,	,	PUNCT
ejpam-5125	60	7	y	y	PROPN
ejpam-5125	60	8	∈	∈	PROPN
ejpam-5125	60	9	v	v	PART
ejpam-5125	60	10	define	define	VERB
ejpam-5125	60	11	∨	∨	NOUN
ejpam-5125	60	12	and	and	CCONJ
ejpam-5125	60	13	∧	∧	NOUN
ejpam-5125	60	14	on	on	ADP
ejpam-5125	60	15	v	v	NUM
ejpam-5125	60	16	by	by	ADP
ejpam-5125	60	17	x	x	PROPN
ejpam-5125	60	18	∨	∨	NOUN
ejpam-5125	60	19	y	y	NOUN
ejpam-5125	60	20	=	=	PRON
ejpam-5125	60	21	{	{	PUNCT
ejpam-5125	60	22	x	x	PUNCT
ejpam-5125	60	23	y	y	PROPN
ejpam-5125	60	24	̸=	̸=	PROPN
ejpam-5125	60	25	y0	y0	NOUN
ejpam-5125	60	26	y0	y0	NOUN
ejpam-5125	60	27	y	y	NOUN
ejpam-5125	60	28	=	=	SYM
ejpam-5125	60	29	y0	y0	PROPN
ejpam-5125	60	30	and	and	CCONJ
ejpam-5125	60	31	x	x	PART
ejpam-5125	60	32	∧	∧	NOUN
ejpam-5125	60	33	y	y	NOUN
ejpam-5125	61	1	=	=	PRON
ejpam-5125	61	2	{	{	PUNCT
ejpam-5125	61	3	y	y	NOUN
ejpam-5125	61	4	y	y	NOUN
ejpam-5125	61	5	̸=	̸=	PROPN
ejpam-5125	61	6	y0	y0	PROPN
ejpam-5125	61	7	x	x	SYM
ejpam-5125	61	8	y	y	NOUN
ejpam-5125	61	9	=	=	SYM
ejpam-5125	61	10	y0	y0	PROPN
ejpam-5125	61	11	then	then	ADV
ejpam-5125	61	12	(	(	PUNCT
ejpam-5125	61	13	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	61	14	,	,	PUNCT
ejpam-5125	61	15	y0	y0	PROPN
ejpam-5125	61	16	)	)	PUNCT
ejpam-5125	61	17	is	be	AUX
ejpam-5125	61	18	a	a	DET
ejpam-5125	61	19	disconnected	disconnected	ADJ
ejpam-5125	61	20	pdl	pdl	NOUN
ejpam-5125	61	21	with	with	ADP
ejpam-5125	61	22	y0	y0	PROPN
ejpam-5125	61	23	as	as	ADP
ejpam-5125	61	24	its	its	PRON
ejpam-5125	61	25	greatest	great	ADJ
ejpam-5125	61	26	element	element	NOUN
ejpam-5125	61	27	.	.	PUNCT
ejpam-5125	61	28	example	example	NOUN
ejpam-5125	62	1	2	2	NUM
ejpam-5125	62	2	.	.	PUNCT
ejpam-5125	62	3	let	let	VERB
ejpam-5125	62	4	v	v	VERB
ejpam-5125	62	5	=	=	SYM
ejpam-5125	62	6	{	{	PUNCT
ejpam-5125	62	7	0	0	NUM
ejpam-5125	62	8	,	,	PUNCT
ejpam-5125	62	9	1	1	NUM
ejpam-5125	62	10	,	,	PUNCT
ejpam-5125	62	11	2	2	NUM
ejpam-5125	62	12	,	,	PUNCT
ejpam-5125	62	13	3	3	NUM
ejpam-5125	62	14	,	,	PUNCT
ejpam-5125	62	15	4	4	NUM
ejpam-5125	62	16	}	}	PUNCT
ejpam-5125	62	17	be	be	AUX
ejpam-5125	62	18	a	a	DET
ejpam-5125	62	19	set	set	NOUN
ejpam-5125	62	20	with	with	ADP
ejpam-5125	62	21	binary	binary	ADJ
ejpam-5125	62	22	operations	operation	NOUN
ejpam-5125	62	23	∨	∨	NOUN
ejpam-5125	62	24	and	and	CCONJ
ejpam-5125	62	25	∧	∧	NOUN
ejpam-5125	62	26	given	give	VERB
ejpam-5125	62	27	in	in	ADP
ejpam-5125	62	28	the	the	DET
ejpam-5125	62	29	following	following	ADJ
ejpam-5125	62	30	tables	table	NOUN
ejpam-5125	62	31	:	:	PUNCT
ejpam-5125	62	32	∨	∨	NOUN
ejpam-5125	62	33	0	0	NUM
ejpam-5125	62	34	1	1	NUM
ejpam-5125	62	35	2	2	NUM
ejpam-5125	62	36	3	3	NUM
ejpam-5125	62	37	4	4	NUM
ejpam-5125	62	38	0	0	NUM
ejpam-5125	62	39	0	0	NUM
ejpam-5125	62	40	1	1	NUM
ejpam-5125	62	41	0	0	NUM
ejpam-5125	62	42	3	3	NUM
ejpam-5125	62	43	3	3	NUM
ejpam-5125	62	44	1	1	NUM
ejpam-5125	62	45	1	1	NUM
ejpam-5125	62	46	1	1	NUM
ejpam-5125	62	47	1	1	NUM
ejpam-5125	62	48	1	1	NUM
ejpam-5125	62	49	1	1	NUM
ejpam-5125	62	50	2	2	NUM
ejpam-5125	62	51	2	2	NUM
ejpam-5125	62	52	1	1	NUM
ejpam-5125	62	53	2	2	NUM
ejpam-5125	62	54	4	4	NUM
ejpam-5125	62	55	4	4	NUM
ejpam-5125	62	56	3	3	NUM
ejpam-5125	62	57	3	3	NUM
ejpam-5125	62	58	1	1	NUM
ejpam-5125	62	59	3	3	NUM
ejpam-5125	62	60	3	3	NUM
ejpam-5125	62	61	3	3	NUM
ejpam-5125	62	62	4	4	NUM
ejpam-5125	62	63	4	4	NUM
ejpam-5125	62	64	1	1	NUM
ejpam-5125	62	65	4	4	NUM
ejpam-5125	62	66	4	4	NUM
ejpam-5125	62	67	4	4	NUM
ejpam-5125	62	68	∧	∧	NOUN
ejpam-5125	62	69	0	0	NUM
ejpam-5125	62	70	1	1	NUM
ejpam-5125	62	71	2	2	NUM
ejpam-5125	62	72	3	3	NUM
ejpam-5125	62	73	4	4	NUM
ejpam-5125	62	74	0	0	NUM
ejpam-5125	62	75	0	0	NUM
ejpam-5125	62	76	0	0	NUM
ejpam-5125	62	77	2	2	NUM
ejpam-5125	62	78	0	0	NUM
ejpam-5125	62	79	2	2	NUM
ejpam-5125	62	80	1	1	NUM
ejpam-5125	62	81	0	0	NUM
ejpam-5125	62	82	1	1	NUM
ejpam-5125	62	83	2	2	NUM
ejpam-5125	62	84	3	3	NUM
ejpam-5125	62	85	4	4	NUM
ejpam-5125	62	86	2	2	NUM
ejpam-5125	62	87	0	0	NUM
ejpam-5125	62	88	2	2	NUM
ejpam-5125	62	89	2	2	NUM
ejpam-5125	62	90	0	0	NUM
ejpam-5125	62	91	2	2	NUM
ejpam-5125	62	92	3	3	NUM
ejpam-5125	62	93	0	0	NUM
ejpam-5125	62	94	3	3	NUM
ejpam-5125	62	95	2	2	NUM
ejpam-5125	62	96	3	3	NUM
ejpam-5125	62	97	4	4	NUM
ejpam-5125	62	98	4	4	NUM
ejpam-5125	62	99	0	0	NUM
ejpam-5125	62	100	4	4	NUM
ejpam-5125	62	101	2	2	NUM
ejpam-5125	62	102	3	3	NUM
ejpam-5125	62	103	4	4	NUM
ejpam-5125	62	104	then	then	ADV
ejpam-5125	62	105	(	(	PUNCT
ejpam-5125	62	106	v	v	NOUN
ejpam-5125	62	107	;	;	PUNCT
ejpam-5125	62	108	∨	∨	NUM
ejpam-5125	62	109	∧	∧	PROPN
ejpam-5125	62	110	1	1	NUM
ejpam-5125	62	111	)	)	PUNCT
ejpam-5125	62	112	is	be	AUX
ejpam-5125	62	113	a	a	DET
ejpam-5125	62	114	paradistributive	paradistributive	ADJ
ejpam-5125	62	115	latticoid	latticoid	NOUN
ejpam-5125	62	116	.	.	PUNCT
ejpam-5125	63	1	but	but	CCONJ
ejpam-5125	63	2	v	v	NOUN
ejpam-5125	63	3	is	be	AUX
ejpam-5125	63	4	not	not	PART
ejpam-5125	63	5	a	a	DET
ejpam-5125	63	6	distributive	distributive	ADJ
ejpam-5125	63	7	lattice	lattice	NOUN
ejpam-5125	63	8	,	,	PUNCT
ejpam-5125	63	9	since	since	SCONJ
ejpam-5125	63	10	0	0	NUM
ejpam-5125	63	11	∧	∧	PROPN
ejpam-5125	63	12	(	(	PUNCT
ejpam-5125	63	13	2	2	NUM
ejpam-5125	63	14	∨	∨	NUM
ejpam-5125	63	15	1	1	NUM
ejpam-5125	63	16	)	)	PUNCT
ejpam-5125	63	17	=	=	SYM
ejpam-5125	63	18	0	0	NUM
ejpam-5125	64	1	∧	∧	NOUN
ejpam-5125	64	2	1	1	NUM
ejpam-5125	64	3	=	=	SYM
ejpam-5125	64	4	0	0	NUM
ejpam-5125	64	5	̸=	̸=	PROPN
ejpam-5125	64	6	2	2	NUM
ejpam-5125	64	7	=	=	SYM
ejpam-5125	64	8	2	2	NUM
ejpam-5125	64	9	∨	∨	NUM
ejpam-5125	64	10	0	0	NUM
ejpam-5125	65	1	=	=	SYM
ejpam-5125	65	2	(	(	PUNCT
ejpam-5125	65	3	0	0	NUM
ejpam-5125	65	4	∧	∧	PROPN
ejpam-5125	65	5	2	2	NUM
ejpam-5125	65	6	)	)	PUNCT
ejpam-5125	65	7	∨	∨	NOUN
ejpam-5125	65	8	(	(	PUNCT
ejpam-5125	65	9	0	0	NUM
ejpam-5125	65	10	∧	∧	PROPN
ejpam-5125	65	11	1	1	NUM
ejpam-5125	65	12	)	)	PUNCT
ejpam-5125	65	13	,	,	PUNCT
ejpam-5125	65	14	0	0	NUM
ejpam-5125	65	15	∧	∧	NOUN
ejpam-5125	65	16	2	2	NUM
ejpam-5125	65	17	=	=	SYM
ejpam-5125	65	18	2	2	NUM
ejpam-5125	65	19	̸=	̸=	PROPN
ejpam-5125	65	20	0	0	NUM
ejpam-5125	66	1	=	=	SYM
ejpam-5125	66	2	2	2	NUM
ejpam-5125	66	3	∧	∧	NOUN
ejpam-5125	66	4	0	0	NUM
ejpam-5125	66	5	and	and	CCONJ
ejpam-5125	66	6	0	0	NUM
ejpam-5125	67	1	∨	∨	NUM
ejpam-5125	67	2	2	2	NUM
ejpam-5125	67	3	=	=	SYM
ejpam-5125	67	4	0	0	NUM
ejpam-5125	67	5	̸=	̸=	PROPN
ejpam-5125	67	6	2	2	NUM
ejpam-5125	67	7	=	=	SYM
ejpam-5125	67	8	2	2	NUM
ejpam-5125	67	9	∨	∨	NUM
ejpam-5125	67	10	0	0	NUM
ejpam-5125	67	11	.	.	PUNCT
ejpam-5125	67	12	example	example	NOUN
ejpam-5125	68	1	3	3	X
ejpam-5125	68	2	.	.	PUNCT
ejpam-5125	69	1	let	let	AUX
ejpam-5125	69	2	(	(	PUNCT
ejpam-5125	69	3	v,+	v,+	NUM
ejpam-5125	69	4	,	,	PUNCT
ejpam-5125	69	5	·	·	PUNCT
ejpam-5125	69	6	,	,	PUNCT
ejpam-5125	69	7	0	0	NUM
ejpam-5125	69	8	,	,	PUNCT
ejpam-5125	69	9	1	1	NUM
ejpam-5125	69	10	)	)	PUNCT
ejpam-5125	69	11	be	be	AUX
ejpam-5125	69	12	a	a	DET
ejpam-5125	69	13	commutative	commutative	ADJ
ejpam-5125	69	14	regular	regular	ADJ
ejpam-5125	69	15	ring	ring	NOUN
ejpam-5125	69	16	with	with	ADP
ejpam-5125	69	17	unity	unity	NOUN
ejpam-5125	69	18	and	and	CCONJ
ejpam-5125	69	19	let	let	VERB
ejpam-5125	69	20	x0	x0	PROPN
ejpam-5125	69	21	be	be	AUX
ejpam-5125	69	22	the	the	DET
ejpam-5125	69	23	unique	unique	ADJ
ejpam-5125	69	24	idempotent	idempotent	ADJ
ejpam-5125	69	25	element	element	NOUN
ejpam-5125	69	26	in	in	ADP
ejpam-5125	69	27	v	v	NOUN
ejpam-5125	69	28	such	such	ADJ
ejpam-5125	69	29	that	that	PRON
ejpam-5125	69	30	xv	xv	VERB
ejpam-5125	70	1	=	=	PUNCT
ejpam-5125	70	2	x0v	x0v	PROPN
ejpam-5125	70	3	.	.	PUNCT
ejpam-5125	71	1	now	now	ADV
ejpam-5125	71	2	,	,	PUNCT
ejpam-5125	71	3	for	for	ADP
ejpam-5125	71	4	any	any	DET
ejpam-5125	71	5	x	x	NOUN
ejpam-5125	71	6	,	,	PUNCT
ejpam-5125	71	7	y	y	PROPN
ejpam-5125	71	8	∈	∈	PROPN
ejpam-5125	71	9	v	v	NOUN
ejpam-5125	71	10	,	,	PUNCT
ejpam-5125	71	11	define	define	VERB
ejpam-5125	71	12	(	(	PUNCT
ejpam-5125	71	13	1	1	NUM
ejpam-5125	71	14	)	)	PUNCT
ejpam-5125	71	15	x	x	PUNCT
ejpam-5125	72	1	∨	∨	NUM
ejpam-5125	72	2	y	y	PROPN
ejpam-5125	72	3	=	=	SYM
ejpam-5125	72	4	y0x	y0x	PROPN
ejpam-5125	72	5	(	(	PUNCT
ejpam-5125	72	6	2	2	NUM
ejpam-5125	72	7	)	)	PUNCT
ejpam-5125	72	8	x	x	X
ejpam-5125	73	1	∧	∧	NOUN
ejpam-5125	73	2	y	y	NOUN
ejpam-5125	73	3	=	=	PUNCT
ejpam-5125	73	4	x+	x+	PROPN
ejpam-5125	73	5	y	y	PROPN
ejpam-5125	73	6	−	−	PROPN
ejpam-5125	73	7	y0x	y0x	PROPN
ejpam-5125	73	8	.	.	PUNCT
ejpam-5125	74	1	then	then	ADV
ejpam-5125	74	2	(	(	PUNCT
ejpam-5125	74	3	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	74	4	,	,	PUNCT
ejpam-5125	74	5	0	0	NUM
ejpam-5125	74	6	)	)	PUNCT
ejpam-5125	74	7	is	be	AUX
ejpam-5125	74	8	a	a	DET
ejpam-5125	74	9	pdl	pdl	NOUN
ejpam-5125	74	10	.	.	PUNCT
ejpam-5125	75	1	we	we	PRON
ejpam-5125	75	2	now	now	ADV
ejpam-5125	75	3	provide	provide	VERB
ejpam-5125	75	4	some	some	DET
ejpam-5125	75	5	essential	essential	ADJ
ejpam-5125	75	6	results	result	NOUN
ejpam-5125	75	7	.	.	PUNCT
ejpam-5125	76	1	r.	r.	PROPN
ejpam-5125	76	2	bandaru	bandaru	PROPN
ejpam-5125	76	3	,	,	PUNCT
ejpam-5125	76	4	s.	s.	PROPN
ejpam-5125	76	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	76	6	/	/	PUNCT
ejpam-5125	76	7	eur	eur	PROPN
ejpam-5125	76	8	.	.	PUNCT
ejpam-5125	77	1	j.	j.	PROPN
ejpam-5125	77	2	pure	pure	PROPN
ejpam-5125	77	3	appl	appl	PROPN
ejpam-5125	77	4	.	.	PROPN
ejpam-5125	77	5	math	math	PROPN
ejpam-5125	77	6	,	,	PUNCT
ejpam-5125	77	7	17	17	NUM
ejpam-5125	77	8	(	(	PUNCT
ejpam-5125	77	9	2	2	NUM
ejpam-5125	77	10	)	)	PUNCT
ejpam-5125	77	11	(	(	PUNCT
ejpam-5125	77	12	2024	2024	NUM
ejpam-5125	77	13	)	)	PUNCT
ejpam-5125	77	14	,	,	PUNCT
ejpam-5125	77	15	819	819	NUM
ejpam-5125	77	16	-	-	SYM
ejpam-5125	77	17	834	834	NUM
ejpam-5125	77	18	822	822	NUM
ejpam-5125	77	19	lemma	lemma	PROPN
ejpam-5125	77	20	1	1	NUM
ejpam-5125	77	21	.	.	PUNCT
ejpam-5125	78	1	for	for	ADP
ejpam-5125	78	2	any	any	DET
ejpam-5125	78	3	p1	p1	NOUN
ejpam-5125	78	4	,	,	PUNCT
ejpam-5125	78	5	p2	p2	PROPN
ejpam-5125	78	6	∈	∈	PROPN
ejpam-5125	78	7	v	v	NOUN
ejpam-5125	78	8	,	,	PUNCT
ejpam-5125	78	9	the	the	DET
ejpam-5125	78	10	following	follow	VERB
ejpam-5125	78	11	holds	hold	VERB
ejpam-5125	78	12	:	:	PUNCT
ejpam-5125	78	13	(	(	PUNCT
ejpam-5125	78	14	1	1	X
ejpam-5125	78	15	)	)	PUNCT
ejpam-5125	78	16	p1	p1	NOUN
ejpam-5125	78	17	∧	∧	PROPN
ejpam-5125	78	18	p1	p1	PROPN
ejpam-5125	78	19	=	=	PROPN
ejpam-5125	78	20	p1	p1	PROPN
ejpam-5125	78	21	.	.	PUNCT
ejpam-5125	79	1	(	(	PUNCT
ejpam-5125	79	2	2	2	X
ejpam-5125	79	3	)	)	PUNCT
ejpam-5125	79	4	p1	p1	NOUN
ejpam-5125	79	5	∨	∨	NUM
ejpam-5125	79	6	p1	p1	PROPN
ejpam-5125	79	7	=	=	PROPN
ejpam-5125	79	8	p1	p1	PROPN
ejpam-5125	79	9	.	.	PUNCT
ejpam-5125	80	1	(	(	PUNCT
ejpam-5125	80	2	l4	l4	PROPN
ejpam-5125	80	3	)	)	PUNCT
ejpam-5125	80	4	(	(	PUNCT
ejpam-5125	80	5	p1	p1	NOUN
ejpam-5125	80	6	∧	∧	NOUN
ejpam-5125	80	7	p2	p2	NOUN
ejpam-5125	80	8	)	)	PUNCT
ejpam-5125	80	9	∨	∨	NOUN
ejpam-5125	80	10	p2	p2	NOUN
ejpam-5125	80	11	=	=	SYM
ejpam-5125	80	12	p2	p2	X
ejpam-5125	80	13	.	.	PUNCT
ejpam-5125	81	1	(	(	PUNCT
ejpam-5125	81	2	l5	l5	PROPN
ejpam-5125	81	3	)	)	PUNCT
ejpam-5125	81	4	p1	p1	NOUN
ejpam-5125	81	5	∨	∨	NUM
ejpam-5125	81	6	(	(	PUNCT
ejpam-5125	81	7	p2	p2	PROPN
ejpam-5125	81	8	∧	∧	PROPN
ejpam-5125	81	9	p1	p1	NOUN
ejpam-5125	81	10	)	)	PUNCT
ejpam-5125	81	11	=	=	SYM
ejpam-5125	81	12	p1	p1	PROPN
ejpam-5125	81	13	.	.	PUNCT
ejpam-5125	82	1	(	(	PUNCT
ejpam-5125	82	2	l6	l6	PROPN
ejpam-5125	82	3	)	)	PUNCT
ejpam-5125	82	4	p1	p1	NOUN
ejpam-5125	82	5	∧	∧	PROPN
ejpam-5125	82	6	(	(	PUNCT
ejpam-5125	82	7	p1	p1	PROPN
ejpam-5125	82	8	∨	∨	NUM
ejpam-5125	82	9	p2	p2	NOUN
ejpam-5125	82	10	)	)	PUNCT
ejpam-5125	82	11	=	=	SYM
ejpam-5125	82	12	p1	p1	NOUN
ejpam-5125	82	13	.	.	PUNCT
ejpam-5125	83	1	proof	proof	NOUN
ejpam-5125	83	2	.	.	PUNCT
ejpam-5125	84	1	proofs	proof	NOUN
ejpam-5125	84	2	of	of	ADP
ejpam-5125	84	3	(	(	PUNCT
ejpam-5125	84	4	1	1	NUM
ejpam-5125	84	5	)	)	PUNCT
ejpam-5125	84	6	,	,	PUNCT
ejpam-5125	84	7	(	(	PUNCT
ejpam-5125	84	8	2	2	X
ejpam-5125	84	9	)	)	PUNCT
ejpam-5125	84	10	are	be	AUX
ejpam-5125	84	11	obvious	obvious	ADJ
ejpam-5125	84	12	.	.	PUNCT
ejpam-5125	85	1	now	now	ADV
ejpam-5125	85	2	(	(	PUNCT
ejpam-5125	85	3	p1	p1	PROPN
ejpam-5125	85	4	∧	∧	PROPN
ejpam-5125	85	5	p2)∨	p2)∨	ADJ
ejpam-5125	85	6	p2	p2	NOUN
ejpam-5125	85	7	=	=	SYM
ejpam-5125	85	8	(	(	PUNCT
ejpam-5125	85	9	p1	p1	PROPN
ejpam-5125	85	10	∨	∨	NUM
ejpam-5125	85	11	p2)∧	p2)∧	NOUN
ejpam-5125	85	12	(	(	PUNCT
ejpam-5125	85	13	p2	p2	PROPN
ejpam-5125	85	14	∨	∨	NUM
ejpam-5125	85	15	p2	p2	NOUN
ejpam-5125	85	16	)	)	PUNCT
ejpam-5125	85	17	=	=	SYM
ejpam-5125	85	18	(	(	PUNCT
ejpam-5125	85	19	p1	p1	PROPN
ejpam-5125	85	20	∨	∨	NUM
ejpam-5125	85	21	p2)∧p2	p2)∧p2	NOUN
ejpam-5125	85	22	=	=	SYM
ejpam-5125	85	23	p2	p2	NOUN
ejpam-5125	85	24	,	,	PUNCT
ejpam-5125	85	25	which	which	PRON
ejpam-5125	85	26	proves	prove	VERB
ejpam-5125	85	27	(	(	PUNCT
ejpam-5125	85	28	l4	l4	PROPN
ejpam-5125	85	29	)	)	PUNCT
ejpam-5125	85	30	.	.	PUNCT
ejpam-5125	86	1	also	also	ADV
ejpam-5125	86	2	p1∨(p2∧p1	p1∨(p2∧p1	ADP
ejpam-5125	86	3	)	)	PUNCT
ejpam-5125	86	4	=	=	SYM
ejpam-5125	86	5	(	(	PUNCT
ejpam-5125	86	6	p1∨p2)∧(p1∧p1	p1∨p2)∧(p1∧p1	PROPN
ejpam-5125	86	7	)	)	PUNCT
ejpam-5125	86	8	=	=	SYM
ejpam-5125	87	1	(	(	PUNCT
ejpam-5125	87	2	p1∨p2)∧p1	p1∨p2)∧p1	NUM
ejpam-5125	87	3	=	=	NOUN
ejpam-5125	87	4	p1	p1	NOUN
ejpam-5125	87	5	which	which	PRON
ejpam-5125	87	6	proves	prove	VERB
ejpam-5125	87	7	(	(	PUNCT
ejpam-5125	87	8	l5	l5	PROPN
ejpam-5125	87	9	)	)	PUNCT
ejpam-5125	87	10	.	.	PUNCT
ejpam-5125	88	1	lastly	lastly	ADV
ejpam-5125	88	2	,	,	PUNCT
ejpam-5125	88	3	(	(	PUNCT
ejpam-5125	88	4	l6	l6	NOUN
ejpam-5125	88	5	)	)	PUNCT
ejpam-5125	88	6	follows	follow	VERB
ejpam-5125	88	7	as	as	ADP
ejpam-5125	88	8	p1∧(p1∨p2	p1∧(p1∨p2	NOUN
ejpam-5125	88	9	)	)	PUNCT
ejpam-5125	88	10	=	=	SYM
ejpam-5125	88	11	(	(	PUNCT
ejpam-5125	88	12	p1∨p1)∧(p1∨p2	p1∨p1)∧(p1∨p2	NOUN
ejpam-5125	88	13	)	)	PUNCT
ejpam-5125	88	14	=	=	SYM
ejpam-5125	88	15	p1∨(p1∧p2	p1∨(p1∧p2	NOUN
ejpam-5125	88	16	)	)	PUNCT
ejpam-5125	89	1	=	=	SYM
ejpam-5125	89	2	p1	p1	PROPN
ejpam-5125	89	3	.	.	PUNCT
ejpam-5125	90	1	lemma	lemma	PROPN
ejpam-5125	90	2	2	2	NUM
ejpam-5125	90	3	.	.	X
ejpam-5125	90	4	for	for	ADP
ejpam-5125	90	5	any	any	DET
ejpam-5125	90	6	p1	p1	NOUN
ejpam-5125	90	7	,	,	PUNCT
ejpam-5125	90	8	p2	p2	PROPN
ejpam-5125	90	9	∈	∈	PROPN
ejpam-5125	90	10	v	v	NOUN
ejpam-5125	90	11	,	,	PUNCT
ejpam-5125	90	12	p1	p1	NOUN
ejpam-5125	90	13	∨	∨	NOUN
ejpam-5125	90	14	p2	p2	X
ejpam-5125	90	15	=	=	SYM
ejpam-5125	90	16	p2	p2	PROPN
ejpam-5125	90	17	∨	∨	NUM
ejpam-5125	90	18	p1	p1	PROPN
ejpam-5125	90	19	whenever	whenever	SCONJ
ejpam-5125	90	20	p1	p1	PROPN
ejpam-5125	90	21	≤	≤	NOUN
ejpam-5125	90	22	p2	p2	NOUN
ejpam-5125	90	23	.	.	PUNCT
ejpam-5125	91	1	proof	proof	NOUN
ejpam-5125	91	2	.	.	PUNCT
ejpam-5125	92	1	let	let	VERB
ejpam-5125	92	2	p1	p1	PROPN
ejpam-5125	92	3	,	,	PUNCT
ejpam-5125	92	4	p2	p2	PROPN
ejpam-5125	92	5	∈	∈	PROPN
ejpam-5125	92	6	v	v	NOUN
ejpam-5125	92	7	and	and	CCONJ
ejpam-5125	92	8	p1	p1	NOUN
ejpam-5125	92	9	≤	≤	NUM
ejpam-5125	92	10	p2	p2	NOUN
ejpam-5125	92	11	.	.	PUNCT
ejpam-5125	93	1	then	then	ADV
ejpam-5125	93	2	p2	p2	PROPN
ejpam-5125	93	3	∨	∨	NUM
ejpam-5125	93	4	p1	p1	PROPN
ejpam-5125	93	5	=	=	PROPN
ejpam-5125	93	6	p1	p1	PROPN
ejpam-5125	93	7	=	=	PROPN
ejpam-5125	93	8	p1	p1	PROPN
ejpam-5125	93	9	∨	∨	NUM
ejpam-5125	93	10	(	(	PUNCT
ejpam-5125	93	11	p2	p2	PROPN
ejpam-5125	93	12	∧	∧	PROPN
ejpam-5125	93	13	p1	p1	NOUN
ejpam-5125	93	14	)	)	PUNCT
ejpam-5125	93	15	=	=	SYM
ejpam-5125	93	16	p1	p1	PROPN
ejpam-5125	93	17	∨	∨	NUM
ejpam-5125	93	18	p2	p2	NOUN
ejpam-5125	93	19	.	.	PUNCT
ejpam-5125	94	1	lemma	lemma	PROPN
ejpam-5125	94	2	3	3	NUM
ejpam-5125	94	3	.	.	PUNCT
ejpam-5125	95	1	the	the	DET
ejpam-5125	95	2	relation	relation	NOUN
ejpam-5125	95	3	≤	≤	PROPN
ejpam-5125	95	4	is	be	AUX
ejpam-5125	95	5	a	a	DET
ejpam-5125	95	6	partial	partial	ADJ
ejpam-5125	95	7	ordering	ordering	NOUN
ejpam-5125	95	8	on	on	ADP
ejpam-5125	95	9	v	v	NOUN
ejpam-5125	95	10	.	.	PUNCT
ejpam-5125	96	1	proof	proof	NOUN
ejpam-5125	96	2	.	.	PUNCT
ejpam-5125	97	1	the	the	DET
ejpam-5125	97	2	reflexivity	reflexivity	NOUN
ejpam-5125	97	3	of	of	ADP
ejpam-5125	97	4	≤	≤	PROPN
ejpam-5125	97	5	follows	follow	VERB
ejpam-5125	97	6	from	from	ADP
ejpam-5125	97	7	lemma	lemma	PROPN
ejpam-5125	97	8	1	1	NUM
ejpam-5125	97	9	.	.	PUNCT
ejpam-5125	98	1	let	let	VERB
ejpam-5125	98	2	p1	p1	PROPN
ejpam-5125	98	3	,	,	PUNCT
ejpam-5125	98	4	p2	p2	PROPN
ejpam-5125	98	5	∈	∈	PROPN
ejpam-5125	98	6	v	v	AUX
ejpam-5125	98	7	be	be	AUX
ejpam-5125	98	8	such	such	ADJ
ejpam-5125	98	9	that	that	SCONJ
ejpam-5125	98	10	p1	p1	PROPN
ejpam-5125	98	11	≤	≤	ADJ
ejpam-5125	98	12	p2	p2	NOUN
ejpam-5125	98	13	and	and	CCONJ
ejpam-5125	98	14	p2	p2	NOUN
ejpam-5125	98	15	≤	≤	NUM
ejpam-5125	98	16	p1	p1	NOUN
ejpam-5125	98	17	.	.	PUNCT
ejpam-5125	99	1	that	that	PRON
ejpam-5125	99	2	is	be	AUX
ejpam-5125	99	3	p1	p1	PROPN
ejpam-5125	99	4	∨	∨	NUM
ejpam-5125	99	5	p2	p2	X
ejpam-5125	99	6	=	=	SYM
ejpam-5125	99	7	p2	p2	PROPN
ejpam-5125	99	8	and	and	CCONJ
ejpam-5125	99	9	p2	p2	PROPN
ejpam-5125	99	10	∨	∨	NUM
ejpam-5125	99	11	p1	p1	PROPN
ejpam-5125	99	12	=	=	PROPN
ejpam-5125	99	13	p1	p1	PROPN
ejpam-5125	99	14	and	and	CCONJ
ejpam-5125	99	15	hence	hence	ADV
ejpam-5125	99	16	by	by	ADP
ejpam-5125	99	17	lemma	lemma	PROPN
ejpam-5125	99	18	2	2	NUM
ejpam-5125	99	19	,	,	PUNCT
ejpam-5125	99	20	we	we	PRON
ejpam-5125	99	21	have	have	VERB
ejpam-5125	99	22	p1	p1	NOUN
ejpam-5125	99	23	=	=	NOUN
ejpam-5125	99	24	p2	p2	NOUN
ejpam-5125	99	25	.	.	PUNCT
ejpam-5125	100	1	thus	thus	ADV
ejpam-5125	100	2	≤	≤	NUM
ejpam-5125	100	3	is	be	AUX
ejpam-5125	100	4	anti	anti	X
ejpam-5125	100	5	symmetric	symmetric	ADJ
ejpam-5125	100	6	.	.	PUNCT
ejpam-5125	101	1	finally	finally	ADV
ejpam-5125	101	2	,	,	PUNCT
ejpam-5125	101	3	suppose	suppose	VERB
ejpam-5125	101	4	p1	p1	NOUN
ejpam-5125	101	5	,	,	PUNCT
ejpam-5125	101	6	p2	p2	NOUN
ejpam-5125	101	7	,	,	PUNCT
ejpam-5125	101	8	p3	p3	PROPN
ejpam-5125	101	9	∈	∈	PROPN
ejpam-5125	101	10	v	v	ADP
ejpam-5125	101	11	such	such	ADJ
ejpam-5125	101	12	that	that	DET
ejpam-5125	101	13	p1	p1	PROPN
ejpam-5125	101	14	≤	≤	PUNCT
ejpam-5125	101	15	p2	p2	NOUN
ejpam-5125	101	16	≤	≤	NUM
ejpam-5125	101	17	p3	p3	PROPN
ejpam-5125	101	18	.	.	PUNCT
ejpam-5125	102	1	then	then	ADV
ejpam-5125	102	2	p1	p1	PROPN
ejpam-5125	102	3	∨	∨	NUM
ejpam-5125	102	4	p3	p3	PROPN
ejpam-5125	102	5	=	=	SYM
ejpam-5125	102	6	(	(	PUNCT
ejpam-5125	102	7	p1	p1	NOUN
ejpam-5125	102	8	∧	∧	NOUN
ejpam-5125	102	9	p2	p2	NOUN
ejpam-5125	102	10	)	)	PUNCT
ejpam-5125	102	11	∨	∨	NUM
ejpam-5125	102	12	p3	p3	NOUN
ejpam-5125	102	13	=	=	SYM
ejpam-5125	102	14	(	(	PUNCT
ejpam-5125	102	15	p1	p1	PROPN
ejpam-5125	102	16	∨	∨	NUM
ejpam-5125	102	17	p3	p3	PROPN
ejpam-5125	102	18	)	)	PUNCT
ejpam-5125	102	19	∧	∧	PROPN
ejpam-5125	102	20	(	(	PUNCT
ejpam-5125	102	21	p2	p2	PROPN
ejpam-5125	102	22	∨	∨	NUM
ejpam-5125	102	23	p3	p3	PROPN
ejpam-5125	102	24	)	)	PUNCT
ejpam-5125	103	1	=	=	SYM
ejpam-5125	103	2	p3	p3	PROPN
ejpam-5125	103	3	∧	∧	PROPN
ejpam-5125	103	4	p3	p3	PROPN
ejpam-5125	103	5	=	=	SYM
ejpam-5125	103	6	p3	p3	PROPN
ejpam-5125	103	7	implies	imply	VERB
ejpam-5125	103	8	p1	p1	PROPN
ejpam-5125	103	9	≤	≤	NUM
ejpam-5125	103	10	p3	p3	PROPN
ejpam-5125	103	11	,	,	PUNCT
ejpam-5125	103	12	hence	hence	ADV
ejpam-5125	103	13	≤	≤	NUM
ejpam-5125	103	14	is	be	AUX
ejpam-5125	103	15	transitive	transitive	ADJ
ejpam-5125	103	16	.	.	PUNCT
ejpam-5125	104	1	lemma	lemma	PROPN
ejpam-5125	104	2	4	4	NUM
ejpam-5125	104	3	.	.	X
ejpam-5125	104	4	for	for	ADP
ejpam-5125	104	5	any	any	DET
ejpam-5125	104	6	p1	p1	NOUN
ejpam-5125	104	7	,	,	PUNCT
ejpam-5125	104	8	p2	p2	PROPN
ejpam-5125	104	9	∈	∈	PROPN
ejpam-5125	104	10	v	v	NOUN
ejpam-5125	104	11	,	,	PUNCT
ejpam-5125	104	12	p1	p1	NOUN
ejpam-5125	104	13	∧	∧	NOUN
ejpam-5125	104	14	p2	p2	PROPN
ejpam-5125	104	15	=	=	SYM
ejpam-5125	104	16	p1	p1	NOUN
ejpam-5125	104	17	if	if	SCONJ
ejpam-5125	104	18	and	and	CCONJ
ejpam-5125	104	19	only	only	ADV
ejpam-5125	104	20	if	if	SCONJ
ejpam-5125	104	21	p1	p1	PROPN
ejpam-5125	104	22	∨	∨	NUM
ejpam-5125	104	23	p2	p2	X
ejpam-5125	104	24	=	=	NOUN
ejpam-5125	104	25	p2	p2	NOUN
ejpam-5125	104	26	.	.	PUNCT
ejpam-5125	105	1	proof	proof	NOUN
ejpam-5125	105	2	.	.	PUNCT
ejpam-5125	106	1	let	let	VERB
ejpam-5125	106	2	p1	p1	PROPN
ejpam-5125	106	3	∧	∧	PROPN
ejpam-5125	106	4	p2	p2	PROPN
ejpam-5125	106	5	=	=	SYM
ejpam-5125	106	6	p1	p1	PROPN
ejpam-5125	106	7	.	.	PUNCT
ejpam-5125	107	1	then	then	ADV
ejpam-5125	107	2	p1	p1	PROPN
ejpam-5125	107	3	∨	∨	NOUN
ejpam-5125	107	4	p2	p2	PROPN
ejpam-5125	107	5	=	=	PUNCT
ejpam-5125	107	6	(	(	PUNCT
ejpam-5125	107	7	p1	p1	NOUN
ejpam-5125	107	8	∧	∧	PROPN
ejpam-5125	107	9	p2)∨	p2)∨	ADJ
ejpam-5125	107	10	p2	p2	NOUN
ejpam-5125	107	11	=	=	SYM
ejpam-5125	107	12	p2	p2	NOUN
ejpam-5125	107	13	.	.	PUNCT
ejpam-5125	108	1	similarly	similarly	ADV
ejpam-5125	108	2	,	,	PUNCT
ejpam-5125	108	3	for	for	ADP
ejpam-5125	108	4	p1	p1	PROPN
ejpam-5125	108	5	∨	∨	NUM
ejpam-5125	108	6	p2	p2	NOUN
ejpam-5125	108	7	=	=	SYM
ejpam-5125	108	8	p2	p2	NOUN
ejpam-5125	108	9	,	,	PUNCT
ejpam-5125	108	10	we	we	PRON
ejpam-5125	108	11	have	have	VERB
ejpam-5125	108	12	,	,	PUNCT
ejpam-5125	108	13	p1	p1	VERB
ejpam-5125	108	14	∧	∧	NOUN
ejpam-5125	108	15	p2	p2	NOUN
ejpam-5125	108	16	=	=	SYM
ejpam-5125	108	17	p1	p1	PROPN
ejpam-5125	108	18	∧	∧	PROPN
ejpam-5125	108	19	(	(	PUNCT
ejpam-5125	108	20	p1	p1	PROPN
ejpam-5125	108	21	∨	∨	NUM
ejpam-5125	108	22	p2	p2	NOUN
ejpam-5125	108	23	)	)	PUNCT
ejpam-5125	108	24	=	=	SYM
ejpam-5125	108	25	p1	p1	PROPN
ejpam-5125	108	26	.	.	PUNCT
ejpam-5125	109	1	lemma	lemma	PROPN
ejpam-5125	109	2	5	5	NUM
ejpam-5125	109	3	.	.	PUNCT
ejpam-5125	109	4	for	for	ADP
ejpam-5125	109	5	any	any	DET
ejpam-5125	109	6	p1	p1	NOUN
ejpam-5125	109	7	,	,	PUNCT
ejpam-5125	109	8	p2	p2	PROPN
ejpam-5125	109	9	∈	∈	PROPN
ejpam-5125	109	10	v	v	NOUN
ejpam-5125	109	11	,	,	PUNCT
ejpam-5125	109	12	p1	p1	NOUN
ejpam-5125	109	13	∨	∨	NOUN
ejpam-5125	109	14	p2	p2	PROPN
ejpam-5125	109	15	=	=	SYM
ejpam-5125	109	16	p1	p1	NOUN
ejpam-5125	109	17	if	if	SCONJ
ejpam-5125	109	18	and	and	CCONJ
ejpam-5125	109	19	only	only	ADV
ejpam-5125	109	20	if	if	SCONJ
ejpam-5125	109	21	p1	p1	PROPN
ejpam-5125	109	22	∧	∧	NOUN
ejpam-5125	109	23	p2	p2	NOUN
ejpam-5125	109	24	=	=	NOUN
ejpam-5125	109	25	p2	p2	NOUN
ejpam-5125	109	26	.	.	PUNCT
ejpam-5125	110	1	proof	proof	NOUN
ejpam-5125	110	2	.	.	PUNCT
ejpam-5125	111	1	let	let	VERB
ejpam-5125	111	2	p1∨p2	p1∨p2	NOUN
ejpam-5125	111	3	=	=	SYM
ejpam-5125	111	4	p1	p1	PROPN
ejpam-5125	111	5	.	.	PUNCT
ejpam-5125	112	1	then	then	ADV
ejpam-5125	112	2	p1∧p2	p1∧p2	PROPN
ejpam-5125	112	3	=	=	SYM
ejpam-5125	112	4	(	(	PUNCT
ejpam-5125	112	5	p1∨p2)∧p2	p1∨p2)∧p2	X
ejpam-5125	112	6	=	=	SYM
ejpam-5125	112	7	p2	p2	NOUN
ejpam-5125	112	8	.	.	PUNCT
ejpam-5125	113	1	similarly	similarly	ADV
ejpam-5125	113	2	,	,	PUNCT
ejpam-5125	113	3	for	for	ADP
ejpam-5125	113	4	p1∧p2	p1∧p2	PROPN
ejpam-5125	113	5	=	=	PUNCT
ejpam-5125	113	6	p2	p2	PROPN
ejpam-5125	113	7	,	,	PUNCT
ejpam-5125	113	8	we	we	PRON
ejpam-5125	113	9	have	have	VERB
ejpam-5125	113	10	,	,	PUNCT
ejpam-5125	113	11	p1	p1	NOUN
ejpam-5125	113	12	∨	∨	NOUN
ejpam-5125	113	13	p2	p2	PROPN
ejpam-5125	113	14	=	=	SYM
ejpam-5125	113	15	p1	p1	PROPN
ejpam-5125	113	16	∨	∨	NOUN
ejpam-5125	113	17	(	(	PUNCT
ejpam-5125	113	18	p1	p1	NOUN
ejpam-5125	113	19	∧	∧	NOUN
ejpam-5125	113	20	p2	p2	NOUN
ejpam-5125	113	21	)	)	PUNCT
ejpam-5125	113	22	=	=	SYM
ejpam-5125	113	23	p1	p1	PROPN
ejpam-5125	113	24	.	.	PUNCT
ejpam-5125	114	1	lemma	lemma	PROPN
ejpam-5125	114	2	6	6	NUM
ejpam-5125	114	3	.	.	PUNCT
ejpam-5125	115	1	for	for	ADP
ejpam-5125	115	2	any	any	DET
ejpam-5125	115	3	p1	p1	NOUN
ejpam-5125	115	4	,	,	PUNCT
ejpam-5125	115	5	p2	p2	PROPN
ejpam-5125	115	6	∈	∈	PROPN
ejpam-5125	115	7	v	v	NOUN
ejpam-5125	115	8	,	,	PUNCT
ejpam-5125	115	9	the	the	DET
ejpam-5125	115	10	following	follow	VERB
ejpam-5125	115	11	holds	hold	VERB
ejpam-5125	115	12	:	:	PUNCT
ejpam-5125	115	13	(	(	PUNCT
ejpam-5125	115	14	1	1	X
ejpam-5125	115	15	)	)	PUNCT
ejpam-5125	115	16	(	(	PUNCT
ejpam-5125	115	17	p1	p1	PROPN
ejpam-5125	115	18	∨	∨	NUM
ejpam-5125	115	19	p2	p2	PROPN
ejpam-5125	115	20	)	)	PUNCT
ejpam-5125	115	21	∨	∨	NUM
ejpam-5125	115	22	p2	p2	X
ejpam-5125	115	23	=	=	SYM
ejpam-5125	115	24	p1	p1	PROPN
ejpam-5125	115	25	∨	∨	NUM
ejpam-5125	115	26	p2	p2	NOUN
ejpam-5125	115	27	.	.	PUNCT
ejpam-5125	116	1	(	(	PUNCT
ejpam-5125	116	2	2	2	NUM
ejpam-5125	116	3	)	)	PUNCT
ejpam-5125	116	4	(	(	PUNCT
ejpam-5125	116	5	p1	p1	PROPN
ejpam-5125	116	6	∨	∨	NUM
ejpam-5125	116	7	p2	p2	PROPN
ejpam-5125	116	8	)	)	PUNCT
ejpam-5125	116	9	∨	∨	NUM
ejpam-5125	116	10	p1	p1	NOUN
ejpam-5125	116	11	=	=	PROPN
ejpam-5125	116	12	p1	p1	PROPN
ejpam-5125	116	13	∨	∨	NUM
ejpam-5125	116	14	p2	p2	NOUN
ejpam-5125	116	15	.	.	PUNCT
ejpam-5125	117	1	(	(	PUNCT
ejpam-5125	117	2	3	3	X
ejpam-5125	117	3	)	)	PUNCT
ejpam-5125	117	4	p1	p1	NOUN
ejpam-5125	117	5	∨	∨	NUM
ejpam-5125	117	6	(	(	PUNCT
ejpam-5125	117	7	p1	p1	PROPN
ejpam-5125	117	8	∨	∨	NUM
ejpam-5125	117	9	p2	p2	NOUN
ejpam-5125	117	10	)	)	PUNCT
ejpam-5125	117	11	=	=	SYM
ejpam-5125	117	12	p1	p1	PROPN
ejpam-5125	117	13	∨	∨	NUM
ejpam-5125	117	14	p2	p2	NOUN
ejpam-5125	117	15	.	.	PUNCT
ejpam-5125	118	1	(	(	PUNCT
ejpam-5125	118	2	4	4	X
ejpam-5125	118	3	)	)	PUNCT
ejpam-5125	118	4	p1	p1	NOUN
ejpam-5125	118	5	∧	∧	PROPN
ejpam-5125	118	6	(	(	PUNCT
ejpam-5125	118	7	p1	p1	NOUN
ejpam-5125	118	8	∧	∧	NOUN
ejpam-5125	118	9	p2	p2	NOUN
ejpam-5125	118	10	)	)	PUNCT
ejpam-5125	118	11	=	=	SYM
ejpam-5125	118	12	p1	p1	NOUN
ejpam-5125	118	13	∧	∧	PROPN
ejpam-5125	118	14	p2	p2	NOUN
ejpam-5125	118	15	.	.	PUNCT
ejpam-5125	119	1	(	(	PUNCT
ejpam-5125	119	2	5	5	NUM
ejpam-5125	119	3	)	)	PUNCT
ejpam-5125	119	4	(	(	PUNCT
ejpam-5125	119	5	p1	p1	NOUN
ejpam-5125	119	6	∧	∧	NOUN
ejpam-5125	119	7	p2	p2	NOUN
ejpam-5125	119	8	)	)	PUNCT
ejpam-5125	119	9	∧	∧	NOUN
ejpam-5125	119	10	p2	p2	NOUN
ejpam-5125	119	11	=	=	SYM
ejpam-5125	119	12	p1	p1	NOUN
ejpam-5125	119	13	∧	∧	PROPN
ejpam-5125	119	14	p2	p2	NOUN
ejpam-5125	119	15	.	.	PUNCT
ejpam-5125	120	1	(	(	PUNCT
ejpam-5125	120	2	6	6	NUM
ejpam-5125	120	3	)	)	PUNCT
ejpam-5125	120	4	p2	p2	PROPN
ejpam-5125	120	5	∧	∧	PROPN
ejpam-5125	120	6	(	(	PUNCT
ejpam-5125	120	7	p1	p1	NOUN
ejpam-5125	120	8	∧	∧	NOUN
ejpam-5125	120	9	p2	p2	NOUN
ejpam-5125	120	10	)	)	PUNCT
ejpam-5125	120	11	=	=	SYM
ejpam-5125	120	12	p1	p1	NOUN
ejpam-5125	120	13	∧	∧	NOUN
ejpam-5125	120	14	p2	p2	NOUN
ejpam-5125	120	15	.	.	PUNCT
ejpam-5125	121	1	proof	proof	NOUN
ejpam-5125	121	2	.	.	PUNCT
ejpam-5125	122	1	(	(	PUNCT
ejpam-5125	122	2	1	1	NUM
ejpam-5125	122	3	)	)	PUNCT
ejpam-5125	122	4	.	.	PUNCT
ejpam-5125	123	1	by	by	ADP
ejpam-5125	123	2	definition	definition	NOUN
ejpam-5125	123	3	1	1	NUM
ejpam-5125	123	4	,	,	PUNCT
ejpam-5125	123	5	we	we	PRON
ejpam-5125	123	6	have	have	AUX
ejpam-5125	123	7	(	(	PUNCT
ejpam-5125	123	8	p1	p1	PROPN
ejpam-5125	123	9	∨	∨	NUM
ejpam-5125	123	10	p2	p2	NOUN
ejpam-5125	123	11	)	)	PUNCT
ejpam-5125	123	12	∧	∧	NOUN
ejpam-5125	123	13	p2	p2	NOUN
ejpam-5125	123	14	=	=	NOUN
ejpam-5125	123	15	p2	p2	NOUN
ejpam-5125	123	16	.	.	PUNCT
ejpam-5125	124	1	now	now	ADV
ejpam-5125	124	2	,	,	PUNCT
ejpam-5125	124	3	by	by	ADP
ejpam-5125	124	4	lemma	lemma	PROPN
ejpam-5125	124	5	5	5	NUM
ejpam-5125	124	6	,	,	PUNCT
ejpam-5125	124	7	we	we	PRON
ejpam-5125	124	8	have	have	AUX
ejpam-5125	124	9	(	(	PUNCT
ejpam-5125	124	10	p1	p1	PROPN
ejpam-5125	124	11	∨	∨	NUM
ejpam-5125	124	12	p2	p2	PROPN
ejpam-5125	124	13	)	)	PUNCT
ejpam-5125	124	14	∨	∨	NUM
ejpam-5125	124	15	p2	p2	X
ejpam-5125	124	16	=	=	SYM
ejpam-5125	124	17	p1	p1	PROPN
ejpam-5125	124	18	∨	∨	NUM
ejpam-5125	124	19	p2	p2	NOUN
ejpam-5125	124	20	.	.	PUNCT
ejpam-5125	125	1	(	(	PUNCT
ejpam-5125	125	2	2	2	NUM
ejpam-5125	125	3	)	)	PUNCT
ejpam-5125	125	4	.	.	PUNCT
ejpam-5125	126	1	by	by	ADP
ejpam-5125	126	2	definition	definition	NOUN
ejpam-5125	126	3	1	1	NUM
ejpam-5125	126	4	,	,	PUNCT
ejpam-5125	126	5	we	we	PRON
ejpam-5125	126	6	have	have	VERB
ejpam-5125	126	7	(	(	PUNCT
ejpam-5125	126	8	p1∨p2)∧p1	p1∨p2)∧p1	NUM
ejpam-5125	126	9	=	=	NOUN
ejpam-5125	126	10	p1	p1	NOUN
ejpam-5125	126	11	.	.	PUNCT
ejpam-5125	127	1	now	now	ADV
ejpam-5125	127	2	,	,	PUNCT
ejpam-5125	127	3	by	by	ADP
ejpam-5125	127	4	lemma	lemma	PROPN
ejpam-5125	127	5	5	5	NUM
ejpam-5125	127	6	,	,	PUNCT
ejpam-5125	127	7	we	we	PRON
ejpam-5125	127	8	have	have	VERB
ejpam-5125	127	9	(	(	PUNCT
ejpam-5125	127	10	p1∨p2)∨p1	p1∨p2)∨p1	NOUN
ejpam-5125	127	11	=	=	SYM
ejpam-5125	127	12	p1	p1	PROPN
ejpam-5125	127	13	∨	∨	NUM
ejpam-5125	127	14	p2	p2	PROPN
ejpam-5125	127	15	.	.	PUNCT
ejpam-5125	128	1	r.	r.	PROPN
ejpam-5125	128	2	bandaru	bandaru	PROPN
ejpam-5125	128	3	,	,	PUNCT
ejpam-5125	128	4	s.	s.	PROPN
ejpam-5125	128	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	128	6	/	/	PUNCT
ejpam-5125	128	7	eur	eur	PROPN
ejpam-5125	128	8	.	.	PUNCT
ejpam-5125	129	1	j.	j.	PROPN
ejpam-5125	129	2	pure	pure	PROPN
ejpam-5125	129	3	appl	appl	PROPN
ejpam-5125	129	4	.	.	PROPN
ejpam-5125	129	5	math	math	PROPN
ejpam-5125	129	6	,	,	PUNCT
ejpam-5125	129	7	17	17	NUM
ejpam-5125	129	8	(	(	PUNCT
ejpam-5125	129	9	2	2	NUM
ejpam-5125	129	10	)	)	PUNCT
ejpam-5125	129	11	(	(	PUNCT
ejpam-5125	129	12	2024	2024	NUM
ejpam-5125	129	13	)	)	PUNCT
ejpam-5125	129	14	,	,	PUNCT
ejpam-5125	129	15	819	819	NUM
ejpam-5125	129	16	-	-	SYM
ejpam-5125	129	17	834	834	NUM
ejpam-5125	129	18	823	823	NUM
ejpam-5125	129	19	lemma	lemma	PROPN
ejpam-5125	129	20	7	7	NUM
ejpam-5125	129	21	.	.	X
ejpam-5125	130	1	for	for	ADP
ejpam-5125	130	2	any	any	DET
ejpam-5125	130	3	p1	p1	PROPN
ejpam-5125	130	4	∈	∈	PROPN
ejpam-5125	130	5	v	v	NOUN
ejpam-5125	130	6	,	,	PUNCT
ejpam-5125	130	7	we	we	PRON
ejpam-5125	130	8	have	have	VERB
ejpam-5125	130	9	(	(	PUNCT
ejpam-5125	130	10	i3	i3	NOUN
ejpam-5125	130	11	)	)	PUNCT
ejpam-5125	130	12	p1	p1	NOUN
ejpam-5125	130	13	∧	∧	PROPN
ejpam-5125	130	14	1	1	NUM
ejpam-5125	130	15	=	=	SYM
ejpam-5125	130	16	p1	p1	PROPN
ejpam-5125	130	17	.	.	PUNCT
ejpam-5125	131	1	(	(	PUNCT
ejpam-5125	131	2	i4	i4	PROPN
ejpam-5125	131	3	)	)	PUNCT
ejpam-5125	131	4	1	1	NUM
ejpam-5125	131	5	∧	∧	PROPN
ejpam-5125	131	6	p1	p1	NOUN
ejpam-5125	131	7	=	=	PROPN
ejpam-5125	131	8	p1	p1	PROPN
ejpam-5125	131	9	.	.	PUNCT
ejpam-5125	132	1	(	(	PUNCT
ejpam-5125	132	2	i5	i5	NOUN
ejpam-5125	132	3	)	)	PUNCT
ejpam-5125	132	4	1	1	NUM
ejpam-5125	132	5	∨	∨	NOUN
ejpam-5125	132	6	p1	p1	NOUN
ejpam-5125	132	7	=	=	NOUN
ejpam-5125	132	8	1	1	X
ejpam-5125	132	9	.	.	PUNCT
ejpam-5125	133	1	proof	proof	NOUN
ejpam-5125	133	2	.	.	PUNCT
ejpam-5125	134	1	let	let	VERB
ejpam-5125	134	2	p1	p1	PROPN
ejpam-5125	134	3	∈	∈	PROPN
ejpam-5125	134	4	v	v	NOUN
ejpam-5125	134	5	.	.	PUNCT
ejpam-5125	135	1	then	then	ADV
ejpam-5125	135	2	p1	p1	PROPN
ejpam-5125	135	3	∧	∧	PROPN
ejpam-5125	135	4	1	1	NUM
ejpam-5125	135	5	=	=	SYM
ejpam-5125	135	6	p1	p1	NOUN
ejpam-5125	135	7	∧	∧	PROPN
ejpam-5125	135	8	(	(	PUNCT
ejpam-5125	135	9	p1	p1	PROPN
ejpam-5125	135	10	∨	∨	NUM
ejpam-5125	135	11	1	1	NUM
ejpam-5125	135	12	)	)	PUNCT
ejpam-5125	135	13	=	=	SYM
ejpam-5125	135	14	p1	p1	NOUN
ejpam-5125	135	15	and	and	CCONJ
ejpam-5125	135	16	1	1	NUM
ejpam-5125	135	17	∧	∧	PROPN
ejpam-5125	135	18	p1	p1	NOUN
ejpam-5125	135	19	=	=	SYM
ejpam-5125	135	20	(	(	PUNCT
ejpam-5125	135	21	p1	p1	PROPN
ejpam-5125	135	22	∨	∨	NUM
ejpam-5125	135	23	1	1	NUM
ejpam-5125	135	24	)	)	PUNCT
ejpam-5125	135	25	∧	∧	PROPN
ejpam-5125	135	26	p1	p1	NOUN
ejpam-5125	135	27	=	=	SYM
ejpam-5125	135	28	p1	p1	PROPN
ejpam-5125	135	29	.	.	PUNCT
ejpam-5125	136	1	similarly	similarly	ADV
ejpam-5125	136	2	,	,	PUNCT
ejpam-5125	136	3	1	1	NUM
ejpam-5125	136	4	∨	∨	NOUN
ejpam-5125	136	5	p1	p1	NOUN
ejpam-5125	136	6	=	=	SYM
ejpam-5125	136	7	1	1	NUM
ejpam-5125	136	8	∨	∨	NUM
ejpam-5125	136	9	(	(	PUNCT
ejpam-5125	136	10	1	1	NUM
ejpam-5125	136	11	∧	∧	PROPN
ejpam-5125	136	12	p1	p1	NOUN
ejpam-5125	136	13	)	)	PUNCT
ejpam-5125	136	14	=	=	SYM
ejpam-5125	136	15	1	1	NUM
ejpam-5125	136	16	which	which	PRON
ejpam-5125	136	17	proves	prove	VERB
ejpam-5125	136	18	(	(	PUNCT
ejpam-5125	136	19	i5	i5	NOUN
ejpam-5125	136	20	)	)	PUNCT
ejpam-5125	136	21	.	.	PUNCT
ejpam-5125	137	1	theorem	theorem	NOUN
ejpam-5125	137	2	1	1	NUM
ejpam-5125	137	3	.	.	X
ejpam-5125	137	4	for	for	ADP
ejpam-5125	137	5	any	any	DET
ejpam-5125	137	6	p1	p1	NOUN
ejpam-5125	137	7	,	,	PUNCT
ejpam-5125	137	8	p2	p2	NOUN
ejpam-5125	137	9	,	,	PUNCT
ejpam-5125	137	10	p3	p3	PROPN
ejpam-5125	137	11	∈	∈	PROPN
ejpam-5125	137	12	v	v	NOUN
ejpam-5125	137	13	,	,	PUNCT
ejpam-5125	137	14	(	(	PUNCT
ejpam-5125	137	15	rd∧	rd∧	PROPN
ejpam-5125	137	16	)	)	PUNCT
ejpam-5125	137	17	(	(	PUNCT
ejpam-5125	137	18	p1	p1	PROPN
ejpam-5125	137	19	∨	∨	NUM
ejpam-5125	137	20	p2	p2	NOUN
ejpam-5125	137	21	)	)	PUNCT
ejpam-5125	137	22	∧	∧	PROPN
ejpam-5125	137	23	p3	p3	NOUN
ejpam-5125	137	24	=	=	SYM
ejpam-5125	137	25	(	(	PUNCT
ejpam-5125	137	26	p1	p1	PROPN
ejpam-5125	137	27	∧	∧	PROPN
ejpam-5125	137	28	p3	p3	PROPN
ejpam-5125	137	29	)	)	PUNCT
ejpam-5125	137	30	∨	∨	NUM
ejpam-5125	137	31	(	(	PUNCT
ejpam-5125	137	32	p2	p2	PROPN
ejpam-5125	137	33	∧	∧	PROPN
ejpam-5125	137	34	p3	p3	PROPN
ejpam-5125	137	35	)	)	PUNCT
ejpam-5125	137	36	.	.	PUNCT
ejpam-5125	138	1	proof	proof	NOUN
ejpam-5125	138	2	.	.	PUNCT
ejpam-5125	139	1	let	let	VERB
ejpam-5125	139	2	p1	p1	NOUN
ejpam-5125	139	3	,	,	PUNCT
ejpam-5125	139	4	p2	p2	NOUN
ejpam-5125	139	5	,	,	PUNCT
ejpam-5125	139	6	p3	p3	PROPN
ejpam-5125	139	7	∈	∈	PROPN
ejpam-5125	139	8	v	v	NOUN
ejpam-5125	139	9	.	.	PUNCT
ejpam-5125	140	1	write	write	VERB
ejpam-5125	140	2	x	x	PUNCT
ejpam-5125	141	1	=	=	SYM
ejpam-5125	141	2	(	(	PUNCT
ejpam-5125	141	3	p1	p1	PROPN
ejpam-5125	141	4	∨	∨	NUM
ejpam-5125	141	5	p2	p2	NOUN
ejpam-5125	141	6	)	)	PUNCT
ejpam-5125	141	7	∧	∧	PROPN
ejpam-5125	141	8	p3	p3	NOUN
ejpam-5125	141	9	and	and	CCONJ
ejpam-5125	141	10	y	y	NOUN
ejpam-5125	141	11	=	=	PUNCT
ejpam-5125	141	12	(	(	PUNCT
ejpam-5125	141	13	p1	p1	PROPN
ejpam-5125	141	14	∧	∧	PROPN
ejpam-5125	141	15	p3	p3	PROPN
ejpam-5125	141	16	)	)	PUNCT
ejpam-5125	141	17	∨	∨	NUM
ejpam-5125	141	18	(	(	PUNCT
ejpam-5125	141	19	p2	p2	PROPN
ejpam-5125	141	20	∧	∧	PROPN
ejpam-5125	141	21	p3	p3	PROPN
ejpam-5125	141	22	)	)	PUNCT
ejpam-5125	141	23	=	=	PUNCT
ejpam-5125	142	1	[	[	X
ejpam-5125	142	2	(	(	PUNCT
ejpam-5125	142	3	p1	p1	PROPN
ejpam-5125	142	4	∧	∧	PROPN
ejpam-5125	142	5	p3	p3	PROPN
ejpam-5125	142	6	)	)	PUNCT
ejpam-5125	142	7	∨	∨	NUM
ejpam-5125	142	8	p2	p2	X
ejpam-5125	142	9	]	]	PUNCT
ejpam-5125	142	10	∧	∧	PROPN
ejpam-5125	142	11	p3	p3	PROPN
ejpam-5125	142	12	.	.	PUNCT
ejpam-5125	143	1	then	then	ADV
ejpam-5125	143	2	,	,	PUNCT
ejpam-5125	143	3	by	by	ADP
ejpam-5125	143	4	(	(	PUNCT
ejpam-5125	143	5	rd∨	rd∨	NOUN
ejpam-5125	143	6	)	)	PUNCT
ejpam-5125	143	7	,	,	PUNCT
ejpam-5125	143	8	(	(	PUNCT
ejpam-5125	143	9	ld∨	ld∨	NOUN
ejpam-5125	143	10	)	)	PUNCT
ejpam-5125	143	11	,	,	PUNCT
ejpam-5125	143	12	(	(	PUNCT
ejpam-5125	143	13	l3	l3	NOUN
ejpam-5125	143	14	)	)	PUNCT
ejpam-5125	143	15	,	,	PUNCT
ejpam-5125	143	16	(	(	PUNCT
ejpam-5125	143	17	l5	l5	PROPN
ejpam-5125	143	18	)	)	PUNCT
ejpam-5125	143	19	and	and	CCONJ
ejpam-5125	143	20	(	(	PUNCT
ejpam-5125	143	21	l6	l6	PROPN
ejpam-5125	143	22	)	)	PUNCT
ejpam-5125	143	23	,	,	PUNCT
ejpam-5125	143	24	we	we	PRON
ejpam-5125	143	25	have	have	VERB
ejpam-5125	143	26	x	x	NOUN
ejpam-5125	143	27	∨	∨	NUM
ejpam-5125	143	28	y	y	NOUN
ejpam-5125	143	29	=	=	PUNCT
ejpam-5125	144	1	[	[	X
ejpam-5125	144	2	(	(	PUNCT
ejpam-5125	144	3	p1	p1	PROPN
ejpam-5125	144	4	∨	∨	NUM
ejpam-5125	144	5	p2	p2	NOUN
ejpam-5125	144	6	)	)	PUNCT
ejpam-5125	144	7	∧	∧	PROPN
ejpam-5125	144	8	p3	p3	PROPN
ejpam-5125	144	9	]	]	PUNCT
ejpam-5125	144	10	∨	∨	X
ejpam-5125	144	11	[	[	X
ejpam-5125	144	12	(	(	PUNCT
ejpam-5125	144	13	p1	p1	PROPN
ejpam-5125	144	14	∧	∧	PROPN
ejpam-5125	144	15	p3	p3	PROPN
ejpam-5125	144	16	)	)	PUNCT
ejpam-5125	144	17	∨	∨	NUM
ejpam-5125	144	18	(	(	PUNCT
ejpam-5125	144	19	p2	p2	PROPN
ejpam-5125	144	20	∧	∧	PROPN
ejpam-5125	144	21	p3	p3	PROPN
ejpam-5125	144	22	)	)	PUNCT
ejpam-5125	144	23	]	]	PUNCT
ejpam-5125	145	1	=	=	PUNCT
ejpam-5125	146	1	[	[	X
ejpam-5125	146	2	(	(	PUNCT
ejpam-5125	146	3	p1	p1	PROPN
ejpam-5125	146	4	∨	∨	NUM
ejpam-5125	146	5	p2	p2	NOUN
ejpam-5125	146	6	)	)	PUNCT
ejpam-5125	146	7	∨	∨	NOUN
ejpam-5125	147	1	[	[	X
ejpam-5125	147	2	(	(	PUNCT
ejpam-5125	147	3	p1	p1	PROPN
ejpam-5125	147	4	∧	∧	PROPN
ejpam-5125	147	5	p3	p3	PROPN
ejpam-5125	147	6	)	)	PUNCT
ejpam-5125	147	7	∨	∨	NUM
ejpam-5125	147	8	(	(	PUNCT
ejpam-5125	147	9	p2	p2	PROPN
ejpam-5125	147	10	∧	∧	PROPN
ejpam-5125	147	11	p3	p3	PROPN
ejpam-5125	147	12	)	)	PUNCT
ejpam-5125	147	13	]	]	PUNCT
ejpam-5125	147	14	]	]	X
ejpam-5125	147	15	∧	∧	PROPN
ejpam-5125	147	16	p3	p3	PROPN
ejpam-5125	147	17	=	=	PUNCT
ejpam-5125	148	1	[	[	X
ejpam-5125	148	2	(	(	PUNCT
ejpam-5125	148	3	p1	p1	PROPN
ejpam-5125	148	4	∨	∨	NUM
ejpam-5125	148	5	p2	p2	NOUN
ejpam-5125	148	6	)	)	PUNCT
ejpam-5125	148	7	∨	∨	NOUN
ejpam-5125	149	1	[	[	X
ejpam-5125	149	2	(	(	PUNCT
ejpam-5125	149	3	(	(	PUNCT
ejpam-5125	149	4	p1	p1	PROPN
ejpam-5125	149	5	∧	∧	PROPN
ejpam-5125	149	6	p3	p3	PROPN
ejpam-5125	149	7	)	)	PUNCT
ejpam-5125	149	8	∨	∨	NUM
ejpam-5125	149	9	p2	p2	NOUN
ejpam-5125	149	10	)	)	PUNCT
ejpam-5125	149	11	∧	∧	PROPN
ejpam-5125	149	12	p3	p3	PROPN
ejpam-5125	149	13	]	]	X
ejpam-5125	149	14	]	]	X
ejpam-5125	149	15	∧	∧	PROPN
ejpam-5125	149	16	p3	p3	PROPN
ejpam-5125	149	17	=	=	PUNCT
ejpam-5125	150	1	[	[	X
ejpam-5125	150	2	(	(	PUNCT
ejpam-5125	150	3	p1	p1	PROPN
ejpam-5125	150	4	∨	∨	NUM
ejpam-5125	150	5	p2	p2	NOUN
ejpam-5125	150	6	)	)	PUNCT
ejpam-5125	150	7	∨	∨	NOUN
ejpam-5125	151	1	[	[	X
ejpam-5125	151	2	(	(	PUNCT
ejpam-5125	151	3	(	(	PUNCT
ejpam-5125	151	4	p1	p1	PROPN
ejpam-5125	151	5	∨	∨	NUM
ejpam-5125	151	6	p2	p2	NOUN
ejpam-5125	151	7	)	)	PUNCT
ejpam-5125	151	8	∧	∧	PROPN
ejpam-5125	151	9	(	(	PUNCT
ejpam-5125	151	10	p3	p3	PROPN
ejpam-5125	151	11	∨	∨	NUM
ejpam-5125	151	12	p2	p2	NOUN
ejpam-5125	151	13	)	)	PUNCT
ejpam-5125	151	14	)	)	PUNCT
ejpam-5125	151	15	∧	∧	PROPN
ejpam-5125	151	16	p3	p3	PROPN
ejpam-5125	151	17	]	]	X
ejpam-5125	151	18	]	]	X
ejpam-5125	151	19	∧	∧	PROPN
ejpam-5125	151	20	p3	p3	PROPN
ejpam-5125	151	21	=	=	PUNCT
ejpam-5125	152	1	[	[	X
ejpam-5125	152	2	(	(	PUNCT
ejpam-5125	152	3	p1	p1	PROPN
ejpam-5125	152	4	∨	∨	NUM
ejpam-5125	152	5	p2	p2	NOUN
ejpam-5125	152	6	)	)	PUNCT
ejpam-5125	152	7	∧	∧	PROPN
ejpam-5125	152	8	[	[	X
ejpam-5125	152	9	(	(	PUNCT
ejpam-5125	152	10	p1	p1	PROPN
ejpam-5125	152	11	∨	∨	NUM
ejpam-5125	152	12	p2	p2	PROPN
ejpam-5125	152	13	)	)	PUNCT
ejpam-5125	152	14	∨	∨	NUM
ejpam-5125	152	15	p3	p3	PROPN
ejpam-5125	152	16	]	]	PUNCT
ejpam-5125	152	17	]	]	X
ejpam-5125	152	18	∧	∧	PROPN
ejpam-5125	152	19	p3	p3	PROPN
ejpam-5125	152	20	=	=	PUNCT
ejpam-5125	153	1	[	[	X
ejpam-5125	153	2	(	(	PUNCT
ejpam-5125	153	3	p1	p1	PROPN
ejpam-5125	153	4	∨	∨	NUM
ejpam-5125	153	5	p2	p2	NOUN
ejpam-5125	153	6	)	)	PUNCT
ejpam-5125	153	7	∧	∧	PROPN
ejpam-5125	153	8	p3	p3	NOUN
ejpam-5125	153	9	]	]	PUNCT
ejpam-5125	154	1	=	=	PUNCT
ejpam-5125	154	2	x	x	PUNCT
ejpam-5125	154	3	also	also	ADV
ejpam-5125	154	4	,	,	PUNCT
ejpam-5125	154	5	x	x	PROPN
ejpam-5125	154	6	∨	∨	NUM
ejpam-5125	154	7	y	y	NOUN
ejpam-5125	154	8	=	=	PUNCT
ejpam-5125	155	1	[	[	X
ejpam-5125	155	2	(	(	PUNCT
ejpam-5125	155	3	p1	p1	PROPN
ejpam-5125	155	4	∨	∨	NUM
ejpam-5125	155	5	p2	p2	NOUN
ejpam-5125	155	6	)	)	PUNCT
ejpam-5125	155	7	∧	∧	PROPN
ejpam-5125	155	8	p3	p3	PROPN
ejpam-5125	155	9	]	]	PUNCT
ejpam-5125	155	10	∨	∨	X
ejpam-5125	155	11	[	[	X
ejpam-5125	155	12	(	(	PUNCT
ejpam-5125	155	13	(	(	PUNCT
ejpam-5125	155	14	p1	p1	PROPN
ejpam-5125	155	15	∧	∧	PROPN
ejpam-5125	155	16	p3	p3	PROPN
ejpam-5125	155	17	)	)	PUNCT
ejpam-5125	155	18	∨	∨	NUM
ejpam-5125	155	19	p2	p2	NOUN
ejpam-5125	155	20	)	)	PUNCT
ejpam-5125	155	21	∧	∧	PROPN
ejpam-5125	155	22	p3	p3	NOUN
ejpam-5125	155	23	]	]	PUNCT
ejpam-5125	155	24	=	=	PUNCT
ejpam-5125	156	1	[	[	X
ejpam-5125	156	2	(	(	PUNCT
ejpam-5125	156	3	(	(	PUNCT
ejpam-5125	156	4	p1	p1	PROPN
ejpam-5125	156	5	∨	∨	NUM
ejpam-5125	156	6	p2	p2	PROPN
ejpam-5125	156	7	)	)	PUNCT
ejpam-5125	156	8	∨	∨	NUM
ejpam-5125	156	9	(	(	PUNCT
ejpam-5125	156	10	(	(	PUNCT
ejpam-5125	156	11	p1	p1	PROPN
ejpam-5125	156	12	∧	∧	PROPN
ejpam-5125	156	13	p3	p3	PROPN
ejpam-5125	156	14	)	)	PUNCT
ejpam-5125	156	15	∨	∨	NUM
ejpam-5125	156	16	p2	p2	NOUN
ejpam-5125	156	17	)	)	PUNCT
ejpam-5125	156	18	)	)	PUNCT
ejpam-5125	156	19	∧	∧	PROPN
ejpam-5125	156	20	(	(	PUNCT
ejpam-5125	156	21	p3	p3	PROPN
ejpam-5125	156	22	∨	∨	PROPN
ejpam-5125	156	23	(	(	PUNCT
ejpam-5125	156	24	(	(	PUNCT
ejpam-5125	156	25	p1	p1	PROPN
ejpam-5125	156	26	∧	∧	PROPN
ejpam-5125	156	27	p3	p3	PROPN
ejpam-5125	156	28	)	)	PUNCT
ejpam-5125	156	29	∨	∨	NUM
ejpam-5125	156	30	p2	p2	NOUN
ejpam-5125	156	31	)	)	PUNCT
ejpam-5125	156	32	)	)	PUNCT
ejpam-5125	156	33	]	]	PUNCT
ejpam-5125	157	1	∧	∧	PROPN
ejpam-5125	157	2	p3	p3	PROPN
ejpam-5125	157	3	=	=	PUNCT
ejpam-5125	158	1	[	[	X
ejpam-5125	158	2	(	(	PUNCT
ejpam-5125	158	3	(	(	PUNCT
ejpam-5125	158	4	p1	p1	PROPN
ejpam-5125	158	5	∨	∨	NUM
ejpam-5125	158	6	p2	p2	PROPN
ejpam-5125	158	7	)	)	PUNCT
ejpam-5125	158	8	∨	∨	NUM
ejpam-5125	158	9	(	(	PUNCT
ejpam-5125	158	10	(	(	PUNCT
ejpam-5125	158	11	p1	p1	PROPN
ejpam-5125	158	12	∨	∨	NUM
ejpam-5125	158	13	p2	p2	NOUN
ejpam-5125	158	14	)	)	PUNCT
ejpam-5125	158	15	∧	∧	PROPN
ejpam-5125	158	16	(	(	PUNCT
ejpam-5125	158	17	p3	p3	PROPN
ejpam-5125	158	18	∨	∨	NUM
ejpam-5125	158	19	p2	p2	NOUN
ejpam-5125	158	20	)	)	PUNCT
ejpam-5125	158	21	)	)	PUNCT
ejpam-5125	158	22	)	)	PUNCT
ejpam-5125	159	1	∧	∧	PROPN
ejpam-5125	159	2	(	(	PUNCT
ejpam-5125	159	3	p3	p3	PROPN
ejpam-5125	159	4	∨	∨	PROPN
ejpam-5125	159	5	(	(	PUNCT
ejpam-5125	159	6	(	(	PUNCT
ejpam-5125	159	7	p1	p1	PROPN
ejpam-5125	159	8	∨	∨	NUM
ejpam-5125	159	9	p2	p2	NOUN
ejpam-5125	159	10	)	)	PUNCT
ejpam-5125	159	11	∧	∧	PROPN
ejpam-5125	159	12	(	(	PUNCT
ejpam-5125	159	13	p3	p3	PROPN
ejpam-5125	159	14	∨	∨	NUM
ejpam-5125	159	15	p2	p2	NOUN
ejpam-5125	159	16	)	)	PUNCT
ejpam-5125	159	17	)	)	PUNCT
ejpam-5125	159	18	)	)	PUNCT
ejpam-5125	159	19	]	]	PUNCT
ejpam-5125	160	1	∧	∧	PROPN
ejpam-5125	160	2	p3	p3	PROPN
ejpam-5125	160	3	=	=	PUNCT
ejpam-5125	161	1	[	[	X
ejpam-5125	161	2	(	(	PUNCT
ejpam-5125	161	3	p1	p1	PROPN
ejpam-5125	161	4	∨	∨	NUM
ejpam-5125	161	5	p2	p2	NOUN
ejpam-5125	161	6	)	)	PUNCT
ejpam-5125	161	7	∧	∧	NOUN
ejpam-5125	161	8	(	(	PUNCT
ejpam-5125	161	9	(	(	PUNCT
ejpam-5125	161	10	p3	p3	PROPN
ejpam-5125	161	11	∨	∨	PROPN
ejpam-5125	161	12	(	(	PUNCT
ejpam-5125	161	13	p1	p1	PROPN
ejpam-5125	161	14	∨	∨	NUM
ejpam-5125	161	15	p2	p2	NOUN
ejpam-5125	161	16	)	)	PUNCT
ejpam-5125	161	17	)	)	PUNCT
ejpam-5125	162	1	∧	∧	PROPN
ejpam-5125	162	2	(	(	PUNCT
ejpam-5125	162	3	p3	p3	PROPN
ejpam-5125	162	4	∨	∨	NUM
ejpam-5125	162	5	p2	p2	NOUN
ejpam-5125	162	6	)	)	PUNCT
ejpam-5125	162	7	)	)	PUNCT
ejpam-5125	162	8	]	]	PUNCT
ejpam-5125	163	1	∧	∧	PROPN
ejpam-5125	163	2	p3	p3	PROPN
ejpam-5125	163	3	=	=	PUNCT
ejpam-5125	164	1	[	[	X
ejpam-5125	164	2	(	(	PUNCT
ejpam-5125	164	3	p1	p1	PROPN
ejpam-5125	164	4	∨	∨	NUM
ejpam-5125	164	5	p2	p2	NOUN
ejpam-5125	164	6	)	)	PUNCT
ejpam-5125	164	7	∧	∧	PROPN
ejpam-5125	164	8	(	(	PUNCT
ejpam-5125	164	9	p3	p3	PROPN
ejpam-5125	164	10	∨	∨	PROPN
ejpam-5125	164	11	(	(	PUNCT
ejpam-5125	164	12	(	(	PUNCT
ejpam-5125	164	13	p1	p1	PROPN
ejpam-5125	164	14	∨	∨	NUM
ejpam-5125	164	15	p2	p2	NOUN
ejpam-5125	164	16	)	)	PUNCT
ejpam-5125	164	17	∧	∧	NOUN
ejpam-5125	164	18	p2	p2	NOUN
ejpam-5125	164	19	)	)	PUNCT
ejpam-5125	164	20	)	)	PUNCT
ejpam-5125	164	21	]	]	PUNCT
ejpam-5125	165	1	∧	∧	PROPN
ejpam-5125	165	2	p3	p3	PROPN
ejpam-5125	165	3	=	=	PUNCT
ejpam-5125	166	1	[	[	X
ejpam-5125	166	2	(	(	PUNCT
ejpam-5125	166	3	p1	p1	PROPN
ejpam-5125	166	4	∨	∨	NUM
ejpam-5125	166	5	p2	p2	NOUN
ejpam-5125	166	6	)	)	PUNCT
ejpam-5125	166	7	∧	∧	PROPN
ejpam-5125	166	8	(	(	PUNCT
ejpam-5125	166	9	p3	p3	PROPN
ejpam-5125	166	10	∨	∨	NUM
ejpam-5125	166	11	p2	p2	NOUN
ejpam-5125	166	12	)	)	PUNCT
ejpam-5125	166	13	]	]	PUNCT
ejpam-5125	166	14	∧	∧	PROPN
ejpam-5125	166	15	p3	p3	PROPN
ejpam-5125	166	16	=	=	PUNCT
ejpam-5125	167	1	[	[	X
ejpam-5125	167	2	(	(	PUNCT
ejpam-5125	167	3	p1	p1	PROPN
ejpam-5125	167	4	∧	∧	PROPN
ejpam-5125	167	5	p3	p3	PROPN
ejpam-5125	167	6	)	)	PUNCT
ejpam-5125	167	7	∨	∨	NUM
ejpam-5125	167	8	p2	p2	X
ejpam-5125	167	9	]	]	PUNCT
ejpam-5125	167	10	∧	∧	PROPN
ejpam-5125	167	11	p3	p3	PROPN
ejpam-5125	167	12	.	.	PUNCT
ejpam-5125	168	1	=	=	SYM
ejpam-5125	168	2	y	y	PROPN
ejpam-5125	168	3	therefore	therefore	ADV
ejpam-5125	168	4	,	,	PUNCT
ejpam-5125	168	5	(	(	PUNCT
ejpam-5125	168	6	p1	p1	PROPN
ejpam-5125	168	7	∨	∨	NUM
ejpam-5125	168	8	p2	p2	NOUN
ejpam-5125	168	9	)	)	PUNCT
ejpam-5125	168	10	∧	∧	PROPN
ejpam-5125	168	11	p3	p3	NOUN
ejpam-5125	168	12	=	=	SYM
ejpam-5125	168	13	(	(	PUNCT
ejpam-5125	168	14	p1	p1	PROPN
ejpam-5125	168	15	∧	∧	PROPN
ejpam-5125	168	16	p3	p3	PROPN
ejpam-5125	168	17	)	)	PUNCT
ejpam-5125	168	18	∨	∨	NUM
ejpam-5125	168	19	(	(	PUNCT
ejpam-5125	168	20	p2	p2	PROPN
ejpam-5125	168	21	∧	∧	PROPN
ejpam-5125	168	22	p3	p3	PROPN
ejpam-5125	168	23	)	)	PUNCT
ejpam-5125	168	24	.	.	PUNCT
ejpam-5125	169	1	theorem	theorem	NOUN
ejpam-5125	169	2	2	2	NUM
ejpam-5125	169	3	.	.	X
ejpam-5125	169	4	for	for	ADP
ejpam-5125	169	5	any	any	DET
ejpam-5125	169	6	p1	p1	NOUN
ejpam-5125	169	7	,	,	PUNCT
ejpam-5125	169	8	p2	p2	PROPN
ejpam-5125	169	9	∈	∈	PROPN
ejpam-5125	169	10	v	v	NOUN
ejpam-5125	169	11	,	,	PUNCT
ejpam-5125	169	12	the	the	DET
ejpam-5125	169	13	following	follow	VERB
ejpam-5125	169	14	are	be	AUX
ejpam-5125	169	15	equivalent	equivalent	ADJ
ejpam-5125	169	16	:	:	PUNCT
ejpam-5125	169	17	(	(	PUNCT
ejpam-5125	169	18	1	1	X
ejpam-5125	169	19	)	)	PUNCT
ejpam-5125	169	20	(	(	PUNCT
ejpam-5125	169	21	p1	p1	NOUN
ejpam-5125	169	22	∧	∧	NOUN
ejpam-5125	169	23	p2	p2	NOUN
ejpam-5125	169	24	)	)	PUNCT
ejpam-5125	169	25	∨	∨	NUM
ejpam-5125	169	26	p1	p1	NOUN
ejpam-5125	169	27	=	=	PROPN
ejpam-5125	169	28	p1	p1	PROPN
ejpam-5125	169	29	.	.	PUNCT
ejpam-5125	170	1	(	(	PUNCT
ejpam-5125	170	2	2	2	X
ejpam-5125	170	3	)	)	PUNCT
ejpam-5125	170	4	p1	p1	NOUN
ejpam-5125	170	5	∧	∧	PROPN
ejpam-5125	170	6	(	(	PUNCT
ejpam-5125	170	7	p2	p2	PROPN
ejpam-5125	170	8	∨	∨	NUM
ejpam-5125	170	9	p1	p1	NOUN
ejpam-5125	170	10	)	)	PUNCT
ejpam-5125	170	11	=	=	SYM
ejpam-5125	170	12	p1	p1	PROPN
ejpam-5125	170	13	.	.	PUNCT
ejpam-5125	171	1	(	(	PUNCT
ejpam-5125	171	2	3	3	NUM
ejpam-5125	171	3	)	)	PUNCT
ejpam-5125	171	4	(	(	PUNCT
ejpam-5125	171	5	p2	p2	PROPN
ejpam-5125	171	6	∧	∧	PROPN
ejpam-5125	171	7	p1	p1	PROPN
ejpam-5125	171	8	)	)	PUNCT
ejpam-5125	171	9	∨	∨	NOUN
ejpam-5125	171	10	p2	p2	NOUN
ejpam-5125	171	11	=	=	SYM
ejpam-5125	171	12	p2	p2	X
ejpam-5125	171	13	.	.	PUNCT
ejpam-5125	172	1	(	(	PUNCT
ejpam-5125	172	2	4	4	X
ejpam-5125	172	3	)	)	PUNCT
ejpam-5125	172	4	p2	p2	PROPN
ejpam-5125	172	5	∧	∧	PROPN
ejpam-5125	172	6	(	(	PUNCT
ejpam-5125	172	7	p1	p1	PROPN
ejpam-5125	172	8	∨	∨	NUM
ejpam-5125	172	9	p2	p2	NOUN
ejpam-5125	172	10	)	)	PUNCT
ejpam-5125	172	11	=	=	SYM
ejpam-5125	172	12	p2	p2	X
ejpam-5125	172	13	.	.	PUNCT
ejpam-5125	173	1	(	(	PUNCT
ejpam-5125	173	2	5	5	X
ejpam-5125	173	3	)	)	PUNCT
ejpam-5125	173	4	p1	p1	NOUN
ejpam-5125	173	5	∧	∧	NOUN
ejpam-5125	173	6	p2	p2	NOUN
ejpam-5125	173	7	=	=	SYM
ejpam-5125	173	8	p2	p2	PROPN
ejpam-5125	173	9	∧	∧	PROPN
ejpam-5125	173	10	p1	p1	PROPN
ejpam-5125	173	11	.	.	PUNCT
ejpam-5125	174	1	(	(	PUNCT
ejpam-5125	174	2	6	6	NUM
ejpam-5125	174	3	)	)	PUNCT
ejpam-5125	174	4	p1	p1	NOUN
ejpam-5125	174	5	∨	∨	NUM
ejpam-5125	174	6	p2	p2	X
ejpam-5125	174	7	=	=	SYM
ejpam-5125	174	8	p2	p2	PROPN
ejpam-5125	174	9	∨	∨	NUM
ejpam-5125	174	10	p1	p1	NOUN
ejpam-5125	174	11	.	.	PUNCT
ejpam-5125	175	1	(	(	PUNCT
ejpam-5125	175	2	7	7	X
ejpam-5125	175	3	)	)	PUNCT
ejpam-5125	175	4	p1	p1	NOUN
ejpam-5125	175	5	∧	∧	PROPN
ejpam-5125	175	6	p2	p2	PROPN
ejpam-5125	175	7	≤	≤	PROPN
ejpam-5125	175	8	p1	p1	NOUN
ejpam-5125	175	9	.	.	PUNCT
ejpam-5125	176	1	(	(	PUNCT
ejpam-5125	176	2	8)	8)	NUM
ejpam-5125	176	3	there	there	ADV
ejpam-5125	176	4	exists	exist	VERB
ejpam-5125	176	5	a	a	DET
ejpam-5125	176	6	∈	∈	NOUN
ejpam-5125	176	7	v	v	ADP
ejpam-5125	176	8	such	such	DET
ejpam-5125	176	9	that	that	SCONJ
ejpam-5125	176	10	a	a	DET
ejpam-5125	176	11	≤	≤	ADJ
ejpam-5125	176	12	p1	p1	NOUN
ejpam-5125	176	13	and	and	CCONJ
ejpam-5125	176	14	a	a	DET
ejpam-5125	176	15	≤	≤	ADJ
ejpam-5125	176	16	p2	p2	NOUN
ejpam-5125	176	17	.	.	PUNCT
ejpam-5125	177	1	(	(	PUNCT
ejpam-5125	177	2	9	9	X
ejpam-5125	177	3	)	)	PUNCT
ejpam-5125	177	4	the	the	DET
ejpam-5125	177	5	g.l.b	g.l.b	NOUN
ejpam-5125	177	6	of	of	ADP
ejpam-5125	177	7	p1	p1	NOUN
ejpam-5125	177	8	and	and	CCONJ
ejpam-5125	177	9	p2	p2	PROPN
ejpam-5125	177	10	exists	exist	VERB
ejpam-5125	177	11	in	in	ADP
ejpam-5125	177	12	v	v	PRON
ejpam-5125	177	13	and	and	CCONJ
ejpam-5125	177	14	equals	equal	VERB
ejpam-5125	177	15	p1	p1	PROPN
ejpam-5125	177	16	∧	∧	PROPN
ejpam-5125	177	17	p2	p2	NOUN
ejpam-5125	177	18	.	.	PUNCT
ejpam-5125	178	1	r.	r.	PROPN
ejpam-5125	178	2	bandaru	bandaru	PROPN
ejpam-5125	178	3	,	,	PUNCT
ejpam-5125	178	4	s.	s.	PROPN
ejpam-5125	178	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	178	6	/	/	PUNCT
ejpam-5125	178	7	eur	eur	PROPN
ejpam-5125	178	8	.	.	PUNCT
ejpam-5125	179	1	j.	j.	PROPN
ejpam-5125	179	2	pure	pure	PROPN
ejpam-5125	179	3	appl	appl	PROPN
ejpam-5125	179	4	.	.	PROPN
ejpam-5125	179	5	math	math	PROPN
ejpam-5125	179	6	,	,	PUNCT
ejpam-5125	179	7	17	17	NUM
ejpam-5125	179	8	(	(	PUNCT
ejpam-5125	179	9	2	2	NUM
ejpam-5125	179	10	)	)	PUNCT
ejpam-5125	179	11	(	(	PUNCT
ejpam-5125	179	12	2024	2024	NUM
ejpam-5125	179	13	)	)	PUNCT
ejpam-5125	179	14	,	,	PUNCT
ejpam-5125	179	15	819	819	NUM
ejpam-5125	179	16	-	-	SYM
ejpam-5125	179	17	834	834	NUM
ejpam-5125	179	18	824	824	NUM
ejpam-5125	179	19	proof	proof	NOUN
ejpam-5125	179	20	.	.	PUNCT
ejpam-5125	180	1	let	let	VERB
ejpam-5125	180	2	p1	p1	NOUN
ejpam-5125	180	3	,	,	PUNCT
ejpam-5125	180	4	p2	p2	PROPN
ejpam-5125	180	5	∈	∈	PROPN
ejpam-5125	180	6	v	v	NOUN
ejpam-5125	180	7	.	.	PUNCT
ejpam-5125	181	1	(	(	PUNCT
ejpam-5125	181	2	1	1	X
ejpam-5125	181	3	)	)	PUNCT
ejpam-5125	181	4	⇒	⇒	NOUN
ejpam-5125	181	5	(	(	PUNCT
ejpam-5125	181	6	2	2	NUM
ejpam-5125	181	7	)	)	PUNCT
ejpam-5125	181	8	:	:	PUNCT
ejpam-5125	181	9	assume	assume	VERB
ejpam-5125	181	10	(	(	PUNCT
ejpam-5125	181	11	1	1	NUM
ejpam-5125	181	12	)	)	PUNCT
ejpam-5125	181	13	.	.	PUNCT
ejpam-5125	182	1	then	then	ADV
ejpam-5125	182	2	p1	p1	PROPN
ejpam-5125	182	3	∧	∧	PROPN
ejpam-5125	182	4	(	(	PUNCT
ejpam-5125	182	5	p2	p2	PROPN
ejpam-5125	182	6	∨	∨	NUM
ejpam-5125	182	7	p1	p1	NOUN
ejpam-5125	182	8	)	)	PUNCT
ejpam-5125	182	9	=	=	PUNCT
ejpam-5125	182	10	(	(	PUNCT
ejpam-5125	182	11	p1	p1	PROPN
ejpam-5125	182	12	∨	∨	NUM
ejpam-5125	182	13	p1	p1	PROPN
ejpam-5125	182	14	)	)	PUNCT
ejpam-5125	182	15	∧	∧	PROPN
ejpam-5125	182	16	(	(	PUNCT
ejpam-5125	182	17	p2	p2	PROPN
ejpam-5125	182	18	∨	∨	NUM
ejpam-5125	182	19	p1	p1	NOUN
ejpam-5125	182	20	)	)	PUNCT
ejpam-5125	182	21	=	=	PUNCT
ejpam-5125	182	22	(	(	PUNCT
ejpam-5125	182	23	p1	p1	NOUN
ejpam-5125	182	24	∧	∧	NOUN
ejpam-5125	182	25	p2	p2	NOUN
ejpam-5125	182	26	)	)	PUNCT
ejpam-5125	182	27	∨	∨	NUM
ejpam-5125	182	28	p1	p1	NOUN
ejpam-5125	182	29	=	=	PROPN
ejpam-5125	182	30	p1	p1	PROPN
ejpam-5125	182	31	.	.	PUNCT
ejpam-5125	183	1	(	(	PUNCT
ejpam-5125	183	2	2	2	X
ejpam-5125	183	3	)	)	PUNCT
ejpam-5125	183	4	⇒	⇒	NOUN
ejpam-5125	183	5	(	(	PUNCT
ejpam-5125	183	6	1	1	NUM
ejpam-5125	183	7	)	)	PUNCT
ejpam-5125	183	8	:	:	PUNCT
ejpam-5125	183	9	assume	assume	VERB
ejpam-5125	183	10	(	(	PUNCT
ejpam-5125	183	11	2	2	NUM
ejpam-5125	183	12	)	)	PUNCT
ejpam-5125	183	13	.	.	PUNCT
ejpam-5125	184	1	then	then	ADV
ejpam-5125	184	2	(	(	PUNCT
ejpam-5125	184	3	p1	p1	PROPN
ejpam-5125	184	4	∧	∧	NOUN
ejpam-5125	184	5	p2	p2	NOUN
ejpam-5125	184	6	)	)	PUNCT
ejpam-5125	184	7	∨	∨	NUM
ejpam-5125	184	8	p1	p1	NOUN
ejpam-5125	184	9	=	=	SYM
ejpam-5125	184	10	(	(	PUNCT
ejpam-5125	184	11	p1	p1	PROPN
ejpam-5125	184	12	∨	∨	NUM
ejpam-5125	184	13	p1	p1	PROPN
ejpam-5125	184	14	)	)	PUNCT
ejpam-5125	184	15	∧	∧	PROPN
ejpam-5125	184	16	(	(	PUNCT
ejpam-5125	184	17	p2	p2	PROPN
ejpam-5125	184	18	∨	∨	NUM
ejpam-5125	184	19	p1	p1	NOUN
ejpam-5125	184	20	)	)	PUNCT
ejpam-5125	184	21	=	=	SYM
ejpam-5125	184	22	p1	p1	NOUN
ejpam-5125	184	23	∧	∧	PROPN
ejpam-5125	184	24	(	(	PUNCT
ejpam-5125	184	25	p2	p2	PROPN
ejpam-5125	184	26	∨	∨	NUM
ejpam-5125	184	27	p1	p1	NOUN
ejpam-5125	184	28	)	)	PUNCT
ejpam-5125	184	29	=	=	SYM
ejpam-5125	184	30	p1	p1	PROPN
ejpam-5125	184	31	.	.	PUNCT
ejpam-5125	185	1	(	(	PUNCT
ejpam-5125	185	2	3	3	X
ejpam-5125	185	3	)	)	PUNCT
ejpam-5125	185	4	⇒	⇒	NOUN
ejpam-5125	185	5	(	(	PUNCT
ejpam-5125	185	6	4	4	NUM
ejpam-5125	185	7	)	)	PUNCT
ejpam-5125	186	1	:	:	PUNCT
ejpam-5125	186	2	assume	assume	VERB
ejpam-5125	186	3	(	(	PUNCT
ejpam-5125	186	4	3	3	NUM
ejpam-5125	186	5	)	)	PUNCT
ejpam-5125	186	6	.	.	PUNCT
ejpam-5125	187	1	then	then	ADV
ejpam-5125	187	2	p2	p2	PROPN
ejpam-5125	187	3	∧	∧	PROPN
ejpam-5125	187	4	(	(	PUNCT
ejpam-5125	187	5	p1	p1	PROPN
ejpam-5125	187	6	∨	∨	NUM
ejpam-5125	187	7	p2	p2	NOUN
ejpam-5125	187	8	)	)	PUNCT
ejpam-5125	187	9	=	=	SYM
ejpam-5125	187	10	(	(	PUNCT
ejpam-5125	187	11	p2	p2	PROPN
ejpam-5125	187	12	∨	∨	NUM
ejpam-5125	187	13	p2	p2	NOUN
ejpam-5125	187	14	)	)	PUNCT
ejpam-5125	187	15	∧	∧	PROPN
ejpam-5125	187	16	(	(	PUNCT
ejpam-5125	187	17	p1	p1	PROPN
ejpam-5125	187	18	∨	∨	NUM
ejpam-5125	187	19	p2	p2	NOUN
ejpam-5125	187	20	)	)	PUNCT
ejpam-5125	187	21	=	=	SYM
ejpam-5125	187	22	(	(	PUNCT
ejpam-5125	187	23	p2	p2	PROPN
ejpam-5125	187	24	∧	∧	PROPN
ejpam-5125	187	25	p1	p1	PROPN
ejpam-5125	187	26	)	)	PUNCT
ejpam-5125	187	27	∨	∨	NOUN
ejpam-5125	187	28	p2	p2	NOUN
ejpam-5125	187	29	=	=	SYM
ejpam-5125	187	30	p2	p2	X
ejpam-5125	187	31	(	(	PUNCT
ejpam-5125	187	32	4	4	NUM
ejpam-5125	187	33	)	)	PUNCT
ejpam-5125	187	34	⇒	⇒	NOUN
ejpam-5125	187	35	(	(	PUNCT
ejpam-5125	187	36	3	3	NUM
ejpam-5125	187	37	)	)	PUNCT
ejpam-5125	187	38	:	:	PUNCT
ejpam-5125	187	39	assume	assume	VERB
ejpam-5125	187	40	(	(	PUNCT
ejpam-5125	187	41	4	4	NUM
ejpam-5125	187	42	)	)	PUNCT
ejpam-5125	187	43	.	.	PUNCT
ejpam-5125	188	1	then	then	ADV
ejpam-5125	188	2	(	(	PUNCT
ejpam-5125	188	3	p2	p2	PROPN
ejpam-5125	188	4	∧	∧	PROPN
ejpam-5125	188	5	p1	p1	PROPN
ejpam-5125	188	6	)	)	PUNCT
ejpam-5125	188	7	∨	∨	NOUN
ejpam-5125	188	8	p2	p2	X
ejpam-5125	188	9	=	=	SYM
ejpam-5125	188	10	(	(	PUNCT
ejpam-5125	188	11	p2	p2	PROPN
ejpam-5125	188	12	∨	∨	NUM
ejpam-5125	188	13	p2	p2	NOUN
ejpam-5125	188	14	)	)	PUNCT
ejpam-5125	188	15	∧	∧	PROPN
ejpam-5125	188	16	(	(	PUNCT
ejpam-5125	188	17	p1	p1	PROPN
ejpam-5125	188	18	∨	∨	NUM
ejpam-5125	188	19	p2	p2	NOUN
ejpam-5125	188	20	)	)	PUNCT
ejpam-5125	189	1	=	=	SYM
ejpam-5125	189	2	p2	p2	PROPN
ejpam-5125	189	3	∧	∧	PROPN
ejpam-5125	189	4	(	(	PUNCT
ejpam-5125	189	5	p1	p1	PROPN
ejpam-5125	189	6	∨	∨	NUM
ejpam-5125	189	7	p2	p2	NOUN
ejpam-5125	189	8	)	)	PUNCT
ejpam-5125	189	9	=	=	SYM
ejpam-5125	189	10	p2	p2	X
ejpam-5125	189	11	.	.	PUNCT
ejpam-5125	190	1	(	(	PUNCT
ejpam-5125	190	2	1	1	X
ejpam-5125	190	3	)	)	PUNCT
ejpam-5125	190	4	⇒	⇒	NOUN
ejpam-5125	190	5	(	(	PUNCT
ejpam-5125	190	6	5	5	NUM
ejpam-5125	190	7	)	)	PUNCT
ejpam-5125	190	8	:	:	PUNCT
ejpam-5125	190	9	assume	assume	VERB
ejpam-5125	190	10	(	(	PUNCT
ejpam-5125	190	11	1	1	NUM
ejpam-5125	190	12	)	)	PUNCT
ejpam-5125	190	13	.	.	PUNCT
ejpam-5125	191	1	then	then	ADV
ejpam-5125	191	2	p2	p2	PROPN
ejpam-5125	191	3	∧	∧	PROPN
ejpam-5125	191	4	p1	p1	NOUN
ejpam-5125	191	5	=	=	PUNCT
ejpam-5125	192	1	[	[	X
ejpam-5125	192	2	(	(	PUNCT
ejpam-5125	192	3	p1	p1	NOUN
ejpam-5125	192	4	∧	∧	NOUN
ejpam-5125	192	5	p2	p2	NOUN
ejpam-5125	192	6	)	)	PUNCT
ejpam-5125	192	7	∨	∨	NUM
ejpam-5125	192	8	p2	p2	X
ejpam-5125	192	9	]	]	X
ejpam-5125	192	10	∧	∧	PROPN
ejpam-5125	192	11	[	[	X
ejpam-5125	192	12	(	(	PUNCT
ejpam-5125	192	13	p1	p1	NOUN
ejpam-5125	192	14	∧	∧	NOUN
ejpam-5125	192	15	p2	p2	NOUN
ejpam-5125	192	16	)	)	PUNCT
ejpam-5125	192	17	∨	∨	NUM
ejpam-5125	192	18	p1	p1	NOUN
ejpam-5125	192	19	]	]	X
ejpam-5125	192	20	=	=	SYM
ejpam-5125	192	21	(	(	PUNCT
ejpam-5125	192	22	p1	p1	NOUN
ejpam-5125	192	23	∧	∧	NOUN
ejpam-5125	192	24	p2	p2	NOUN
ejpam-5125	192	25	)	)	PUNCT
ejpam-5125	192	26	∨	∨	NUM
ejpam-5125	192	27	(	(	PUNCT
ejpam-5125	192	28	p2	p2	PROPN
ejpam-5125	192	29	∧	∧	PROPN
ejpam-5125	192	30	p1	p1	NOUN
ejpam-5125	192	31	)	)	PUNCT
ejpam-5125	192	32	=	=	PUNCT
ejpam-5125	193	1	[	[	X
ejpam-5125	193	2	(	(	PUNCT
ejpam-5125	193	3	p1	p1	PROPN
ejpam-5125	193	4	∨	∨	NUM
ejpam-5125	193	5	(	(	PUNCT
ejpam-5125	193	6	p2	p2	PROPN
ejpam-5125	193	7	∧	∧	PROPN
ejpam-5125	193	8	p1	p1	PROPN
ejpam-5125	193	9	)	)	PUNCT
ejpam-5125	193	10	]	]	PUNCT
ejpam-5125	194	1	∧	∧	PROPN
ejpam-5125	194	2	[	[	X
ejpam-5125	194	3	(	(	PUNCT
ejpam-5125	194	4	p2	p2	PROPN
ejpam-5125	194	5	∨	∨	PROPN
ejpam-5125	194	6	(	(	PUNCT
ejpam-5125	194	7	p2	p2	PROPN
ejpam-5125	194	8	∧	∧	PROPN
ejpam-5125	194	9	p1	p1	PROPN
ejpam-5125	194	10	)	)	PUNCT
ejpam-5125	194	11	]	]	PUNCT
ejpam-5125	195	1	=	=	PUNCT
ejpam-5125	195	2	p1	p1	NOUN
ejpam-5125	195	3	∧	∧	PROPN
ejpam-5125	195	4	p2	p2	NOUN
ejpam-5125	195	5	.	.	PUNCT
ejpam-5125	196	1	(	(	PUNCT
ejpam-5125	196	2	5	5	X
ejpam-5125	196	3	)	)	PUNCT
ejpam-5125	196	4	⇒	⇒	NOUN
ejpam-5125	196	5	(	(	PUNCT
ejpam-5125	196	6	1	1	NUM
ejpam-5125	196	7	)	)	PUNCT
ejpam-5125	196	8	:	:	PUNCT
ejpam-5125	196	9	assume	assume	VERB
ejpam-5125	196	10	(	(	PUNCT
ejpam-5125	196	11	5	5	NUM
ejpam-5125	196	12	)	)	PUNCT
ejpam-5125	196	13	.	.	PUNCT
ejpam-5125	197	1	then	then	ADV
ejpam-5125	197	2	(	(	PUNCT
ejpam-5125	197	3	p1	p1	PROPN
ejpam-5125	197	4	∧	∧	NOUN
ejpam-5125	197	5	p2	p2	NOUN
ejpam-5125	197	6	)	)	PUNCT
ejpam-5125	197	7	∨	∨	NOUN
ejpam-5125	197	8	p2	p2	X
ejpam-5125	197	9	=	=	SYM
ejpam-5125	197	10	(	(	PUNCT
ejpam-5125	197	11	p2	p2	PROPN
ejpam-5125	197	12	∧	∧	PROPN
ejpam-5125	197	13	p1	p1	PROPN
ejpam-5125	197	14	)	)	PUNCT
ejpam-5125	197	15	∨	∨	NUM
ejpam-5125	197	16	p2	p2	NOUN
ejpam-5125	197	17	=	=	SYM
ejpam-5125	197	18	p1	p1	PROPN
ejpam-5125	197	19	.	.	PUNCT
ejpam-5125	198	1	(	(	PUNCT
ejpam-5125	198	2	5	5	X
ejpam-5125	198	3	)	)	PUNCT
ejpam-5125	198	4	⇒	⇒	NOUN
ejpam-5125	198	5	(	(	PUNCT
ejpam-5125	198	6	3	3	NUM
ejpam-5125	198	7	)	)	PUNCT
ejpam-5125	198	8	:	:	PUNCT
ejpam-5125	198	9	assume	assume	VERB
ejpam-5125	198	10	(	(	PUNCT
ejpam-5125	198	11	5	5	NUM
ejpam-5125	198	12	)	)	PUNCT
ejpam-5125	198	13	.	.	PUNCT
ejpam-5125	199	1	then	then	ADV
ejpam-5125	199	2	(	(	PUNCT
ejpam-5125	199	3	p2	p2	PROPN
ejpam-5125	199	4	∧	∧	PROPN
ejpam-5125	199	5	p1	p1	PROPN
ejpam-5125	199	6	)	)	PUNCT
ejpam-5125	199	7	∨	∨	NOUN
ejpam-5125	199	8	p2	p2	X
ejpam-5125	199	9	=	=	PUNCT
ejpam-5125	199	10	(	(	PUNCT
ejpam-5125	199	11	p1	p1	NOUN
ejpam-5125	199	12	∧	∧	NOUN
ejpam-5125	199	13	p2	p2	NOUN
ejpam-5125	199	14	)	)	PUNCT
ejpam-5125	199	15	∨	∨	NOUN
ejpam-5125	199	16	p2	p2	NOUN
ejpam-5125	199	17	=	=	SYM
ejpam-5125	199	18	p2	p2	X
ejpam-5125	199	19	.	.	PUNCT
ejpam-5125	200	1	(	(	PUNCT
ejpam-5125	200	2	3	3	X
ejpam-5125	200	3	)	)	PUNCT
ejpam-5125	200	4	⇒	⇒	NOUN
ejpam-5125	200	5	(	(	PUNCT
ejpam-5125	200	6	5	5	NUM
ejpam-5125	200	7	)	)	PUNCT
ejpam-5125	200	8	:	:	PUNCT
ejpam-5125	200	9	assume	assume	VERB
ejpam-5125	200	10	(	(	PUNCT
ejpam-5125	200	11	3	3	NUM
ejpam-5125	200	12	)	)	PUNCT
ejpam-5125	200	13	.	.	PUNCT
ejpam-5125	201	1	then	then	ADV
ejpam-5125	201	2	p1	p1	PROPN
ejpam-5125	201	3	∧	∧	PROPN
ejpam-5125	201	4	p2	p2	PROPN
ejpam-5125	201	5	=	=	PUNCT
ejpam-5125	202	1	[	[	X
ejpam-5125	202	2	(	(	PUNCT
ejpam-5125	202	3	p2	p2	PROPN
ejpam-5125	202	4	∧	∧	PROPN
ejpam-5125	202	5	p1	p1	PROPN
ejpam-5125	202	6	)	)	PUNCT
ejpam-5125	202	7	∨	∨	NUM
ejpam-5125	202	8	p1	p1	NOUN
ejpam-5125	202	9	]	]	X
ejpam-5125	202	10	∧	∧	PROPN
ejpam-5125	202	11	[	[	X
ejpam-5125	202	12	(	(	PUNCT
ejpam-5125	202	13	p2	p2	PROPN
ejpam-5125	202	14	∧	∧	PROPN
ejpam-5125	202	15	p1	p1	PROPN
ejpam-5125	202	16	)	)	PUNCT
ejpam-5125	202	17	∨	∨	NUM
ejpam-5125	202	18	p2	p2	X
ejpam-5125	202	19	]	]	PUNCT
ejpam-5125	202	20	=	=	PUNCT
ejpam-5125	203	1	[	[	X
ejpam-5125	203	2	p2	p2	PROPN
ejpam-5125	203	3	∧	∧	PROPN
ejpam-5125	203	4	p1	p1	NOUN
ejpam-5125	203	5	)	)	PUNCT
ejpam-5125	203	6	∧	∧	PROPN
ejpam-5125	203	7	[	[	X
ejpam-5125	203	8	(	(	PUNCT
ejpam-5125	203	9	p2	p2	PROPN
ejpam-5125	203	10	∧	∧	PROPN
ejpam-5125	203	11	p1	p1	PROPN
ejpam-5125	203	12	)	)	PUNCT
ejpam-5125	203	13	∨	∨	NUM
ejpam-5125	203	14	p2	p2	NOUN
ejpam-5125	203	15	]	]	X
ejpam-5125	203	16	]	]	PUNCT
ejpam-5125	203	17	∨	∨	PUNCT
ejpam-5125	204	1	[	[	X
ejpam-5125	204	2	p1	p1	PROPN
ejpam-5125	204	3	∧	∧	PROPN
ejpam-5125	204	4	[	[	X
ejpam-5125	204	5	(	(	PUNCT
ejpam-5125	204	6	p2	p2	PROPN
ejpam-5125	204	7	∧	∧	PROPN
ejpam-5125	204	8	p1	p1	PROPN
ejpam-5125	204	9	)	)	PUNCT
ejpam-5125	204	10	∨	∨	NUM
ejpam-5125	204	11	p2	p2	NOUN
ejpam-5125	204	12	)	)	PUNCT
ejpam-5125	204	13	]	]	PUNCT
ejpam-5125	205	1	=	=	SYM
ejpam-5125	205	2	(	(	PUNCT
ejpam-5125	205	3	p2	p2	PROPN
ejpam-5125	205	4	∧	∧	PROPN
ejpam-5125	205	5	p1	p1	PROPN
ejpam-5125	205	6	)	)	PUNCT
ejpam-5125	205	7	∨	∨	PROPN
ejpam-5125	205	8	(	(	PUNCT
ejpam-5125	205	9	p1	p1	NOUN
ejpam-5125	205	10	∧	∧	NOUN
ejpam-5125	205	11	p2	p2	NOUN
ejpam-5125	205	12	)	)	PUNCT
ejpam-5125	205	13	=	=	NOUN
ejpam-5125	206	1	[	[	X
ejpam-5125	206	2	p2	p2	PROPN
ejpam-5125	206	3	∨	∨	NOUN
ejpam-5125	206	4	(	(	PUNCT
ejpam-5125	206	5	p1	p1	NOUN
ejpam-5125	206	6	∧	∧	NOUN
ejpam-5125	206	7	p2	p2	NOUN
ejpam-5125	206	8	)	)	PUNCT
ejpam-5125	206	9	]	]	PUNCT
ejpam-5125	207	1	∧	∧	PROPN
ejpam-5125	208	1	[	[	X
ejpam-5125	208	2	p1	p1	PROPN
ejpam-5125	208	3	∨	∨	NOUN
ejpam-5125	208	4	(	(	PUNCT
ejpam-5125	208	5	p1	p1	NOUN
ejpam-5125	208	6	∧	∧	NOUN
ejpam-5125	208	7	p2	p2	NOUN
ejpam-5125	208	8	)	)	PUNCT
ejpam-5125	208	9	]	]	PUNCT
ejpam-5125	209	1	=	=	SYM
ejpam-5125	209	2	p2	p2	PROPN
ejpam-5125	209	3	∧	∧	PROPN
ejpam-5125	209	4	p1	p1	PROPN
ejpam-5125	209	5	.	.	PUNCT
ejpam-5125	210	1	(	(	PUNCT
ejpam-5125	210	2	6	6	NUM
ejpam-5125	210	3	)	)	PUNCT
ejpam-5125	210	4	⇒	⇒	NOUN
ejpam-5125	210	5	(	(	PUNCT
ejpam-5125	210	6	3	3	NUM
ejpam-5125	210	7	)	)	PUNCT
ejpam-5125	210	8	:	:	PUNCT
ejpam-5125	210	9	assume	assume	VERB
ejpam-5125	210	10	(	(	PUNCT
ejpam-5125	210	11	6	6	NUM
ejpam-5125	210	12	)	)	PUNCT
ejpam-5125	210	13	.	.	PUNCT
ejpam-5125	211	1	then	then	ADV
ejpam-5125	211	2	p1	p1	PROPN
ejpam-5125	211	3	∧	∧	PROPN
ejpam-5125	211	4	(	(	PUNCT
ejpam-5125	211	5	p2	p2	PROPN
ejpam-5125	211	6	∨	∨	NUM
ejpam-5125	211	7	p1	p1	NOUN
ejpam-5125	211	8	)	)	PUNCT
ejpam-5125	211	9	=	=	SYM
ejpam-5125	211	10	p1	p1	NOUN
ejpam-5125	211	11	∧	∧	PROPN
ejpam-5125	211	12	(	(	PUNCT
ejpam-5125	211	13	p1	p1	PROPN
ejpam-5125	211	14	∨	∨	NUM
ejpam-5125	211	15	p2	p2	NOUN
ejpam-5125	211	16	)	)	PUNCT
ejpam-5125	211	17	=	=	SYM
ejpam-5125	211	18	p1	p1	NOUN
ejpam-5125	211	19	.	.	PUNCT
ejpam-5125	212	1	(	(	PUNCT
ejpam-5125	212	2	3	3	X
ejpam-5125	212	3	)	)	PUNCT
ejpam-5125	212	4	⇒	⇒	NOUN
ejpam-5125	212	5	(	(	PUNCT
ejpam-5125	212	6	6	6	NUM
ejpam-5125	212	7	)	)	PUNCT
ejpam-5125	212	8	:	:	PUNCT
ejpam-5125	212	9	assume	assume	VERB
ejpam-5125	212	10	(	(	PUNCT
ejpam-5125	212	11	3	3	NUM
ejpam-5125	212	12	)	)	PUNCT
ejpam-5125	212	13	.	.	PUNCT
ejpam-5125	213	1	then	then	ADV
ejpam-5125	213	2	p2	p2	PROPN
ejpam-5125	213	3	∨	∨	NUM
ejpam-5125	213	4	p1	p1	NOUN
ejpam-5125	213	5	=	=	PUNCT
ejpam-5125	213	6	p2	p2	PROPN
ejpam-5125	213	7	∨	∨	NUM
ejpam-5125	213	8	(	(	PUNCT
ejpam-5125	213	9	(	(	PUNCT
ejpam-5125	213	10	p1	p1	PROPN
ejpam-5125	213	11	∨	∨	NUM
ejpam-5125	213	12	p2	p2	NOUN
ejpam-5125	213	13	)	)	PUNCT
ejpam-5125	213	14	∧	∧	PROPN
ejpam-5125	213	15	p1	p1	NOUN
ejpam-5125	213	16	)	)	PUNCT
ejpam-5125	213	17	=	=	PUNCT
ejpam-5125	213	18	(	(	PUNCT
ejpam-5125	213	19	p2	p2	PROPN
ejpam-5125	213	20	∨	∨	PROPN
ejpam-5125	213	21	(	(	PUNCT
ejpam-5125	213	22	p1	p1	PROPN
ejpam-5125	213	23	∨	∨	NUM
ejpam-5125	213	24	p2	p2	NOUN
ejpam-5125	213	25	)	)	PUNCT
ejpam-5125	213	26	)	)	PUNCT
ejpam-5125	214	1	∧	∧	NOUN
ejpam-5125	214	2	(	(	PUNCT
ejpam-5125	214	3	p2	p2	PROPN
ejpam-5125	214	4	∨	∨	NUM
ejpam-5125	214	5	p1	p1	NOUN
ejpam-5125	214	6	)	)	PUNCT
ejpam-5125	214	7	=	=	PUNCT
ejpam-5125	214	8	(	(	PUNCT
ejpam-5125	214	9	p1	p1	PROPN
ejpam-5125	214	10	∨	∨	NUM
ejpam-5125	214	11	p2	p2	NOUN
ejpam-5125	214	12	)	)	PUNCT
ejpam-5125	214	13	∧	∧	NOUN
ejpam-5125	214	14	(	(	PUNCT
ejpam-5125	214	15	p2	p2	PROPN
ejpam-5125	214	16	∨	∨	NUM
ejpam-5125	214	17	p1	p1	NOUN
ejpam-5125	214	18	)	)	PUNCT
ejpam-5125	214	19	=	=	PUNCT
ejpam-5125	214	20	(	(	PUNCT
ejpam-5125	214	21	p1	p1	PROPN
ejpam-5125	214	22	∨	∨	NUM
ejpam-5125	214	23	p2	p2	NOUN
ejpam-5125	214	24	)	)	PUNCT
ejpam-5125	214	25	∧	∧	PROPN
ejpam-5125	214	26	(	(	PUNCT
ejpam-5125	214	27	p1	p1	PROPN
ejpam-5125	214	28	∨	∨	NUM
ejpam-5125	214	29	(	(	PUNCT
ejpam-5125	214	30	p2	p2	PROPN
ejpam-5125	214	31	∨	∨	NUM
ejpam-5125	214	32	p1	p1	NOUN
ejpam-5125	214	33	)	)	PUNCT
ejpam-5125	214	34	)	)	PUNCT
ejpam-5125	215	1	=	=	SYM
ejpam-5125	215	2	p1	p1	PROPN
ejpam-5125	215	3	∨	∨	NOUN
ejpam-5125	215	4	(	(	PUNCT
ejpam-5125	215	5	p2	p2	PROPN
ejpam-5125	215	6	∧	∧	PROPN
ejpam-5125	215	7	(	(	PUNCT
ejpam-5125	215	8	p2	p2	PROPN
ejpam-5125	215	9	∨	∨	NUM
ejpam-5125	215	10	p1	p1	NOUN
ejpam-5125	215	11	)	)	PUNCT
ejpam-5125	215	12	)	)	PUNCT
ejpam-5125	216	1	=	=	SYM
ejpam-5125	216	2	p1	p1	PROPN
ejpam-5125	216	3	∨	∨	NUM
ejpam-5125	216	4	p2	p2	NOUN
ejpam-5125	216	5	.	.	PUNCT
ejpam-5125	217	1	now	now	ADV
ejpam-5125	217	2	we	we	PRON
ejpam-5125	217	3	prove	prove	VERB
ejpam-5125	217	4	the	the	DET
ejpam-5125	217	5	equivalence	equivalence	NOUN
ejpam-5125	217	6	of	of	ADP
ejpam-5125	217	7	(	(	PUNCT
ejpam-5125	217	8	5	5	NUM
ejpam-5125	217	9	)	)	PUNCT
ejpam-5125	217	10	,	,	PUNCT
ejpam-5125	217	11	(	(	PUNCT
ejpam-5125	217	12	7	7	NUM
ejpam-5125	217	13	)	)	PUNCT
ejpam-5125	217	14	,	,	PUNCT
ejpam-5125	217	15	(	(	PUNCT
ejpam-5125	217	16	8)	8)	NUM
ejpam-5125	217	17	and	and	CCONJ
ejpam-5125	217	18	(	(	PUNCT
ejpam-5125	217	19	9	9	NUM
ejpam-5125	217	20	)	)	PUNCT
ejpam-5125	217	21	.	.	PUNCT
ejpam-5125	218	1	(	(	PUNCT
ejpam-5125	218	2	5	5	X
ejpam-5125	218	3	)	)	PUNCT
ejpam-5125	218	4	⇒	⇒	NOUN
ejpam-5125	218	5	(	(	PUNCT
ejpam-5125	218	6	7	7	NUM
ejpam-5125	218	7	)	)	PUNCT
ejpam-5125	218	8	:	:	PUNCT
ejpam-5125	218	9	assume	assume	VERB
ejpam-5125	218	10	(	(	PUNCT
ejpam-5125	218	11	5	5	NUM
ejpam-5125	218	12	)	)	PUNCT
ejpam-5125	218	13	.	.	PUNCT
ejpam-5125	219	1	then	then	ADV
ejpam-5125	219	2	p1	p1	PROPN
ejpam-5125	219	3	∧	∧	PROPN
ejpam-5125	219	4	p2	p2	NOUN
ejpam-5125	219	5	=	=	SYM
ejpam-5125	219	6	p2	p2	NOUN
ejpam-5125	219	7	∧	∧	PROPN
ejpam-5125	219	8	p1	p1	PROPN
ejpam-5125	219	9	≤	≤	NOUN
ejpam-5125	219	10	p1	p1	NOUN
ejpam-5125	219	11	.	.	PUNCT
ejpam-5125	220	1	hence	hence	ADV
ejpam-5125	220	2	(	(	PUNCT
ejpam-5125	220	3	7	7	X
ejpam-5125	220	4	)	)	PUNCT
ejpam-5125	220	5	follows	follow	VERB
ejpam-5125	220	6	.	.	PUNCT
ejpam-5125	221	1	(	(	PUNCT
ejpam-5125	221	2	7	7	X
ejpam-5125	221	3	)	)	PUNCT
ejpam-5125	221	4	⇒	⇒	NOUN
ejpam-5125	221	5	(	(	PUNCT
ejpam-5125	221	6	8)	8)	NUM
ejpam-5125	221	7	:	:	PUNCT
ejpam-5125	221	8	assume	assume	VERB
ejpam-5125	221	9	(	(	PUNCT
ejpam-5125	221	10	7	7	NUM
ejpam-5125	221	11	)	)	PUNCT
ejpam-5125	221	12	.	.	PUNCT
ejpam-5125	222	1	then	then	ADV
ejpam-5125	222	2	p1	p1	PROPN
ejpam-5125	222	3	∧	∧	PROPN
ejpam-5125	222	4	p2	p2	NOUN
ejpam-5125	222	5	≤	≤	NOUN
ejpam-5125	222	6	p1	p1	NOUN
ejpam-5125	222	7	and	and	CCONJ
ejpam-5125	222	8	let	let	VERB
ejpam-5125	222	9	a	a	DET
ejpam-5125	222	10	=	=	SYM
ejpam-5125	222	11	p1	p1	NOUN
ejpam-5125	222	12	∧	∧	NOUN
ejpam-5125	222	13	p2	p2	NOUN
ejpam-5125	222	14	.	.	PUNCT
ejpam-5125	223	1	therefore	therefore	ADV
ejpam-5125	223	2	a	a	DET
ejpam-5125	223	3	≤	≤	ADJ
ejpam-5125	223	4	p1	p1	NOUN
ejpam-5125	223	5	and	and	CCONJ
ejpam-5125	223	6	a	a	DET
ejpam-5125	223	7	≤	≤	ADJ
ejpam-5125	223	8	p2	p2	NOUN
ejpam-5125	223	9	.	.	PUNCT
ejpam-5125	224	1	(	(	PUNCT
ejpam-5125	224	2	8)	8)	NUM
ejpam-5125	224	3	⇒	⇒	NOUN
ejpam-5125	224	4	(	(	PUNCT
ejpam-5125	224	5	9	9	NUM
ejpam-5125	224	6	)	)	PUNCT
ejpam-5125	224	7	:	:	PUNCT
ejpam-5125	224	8	assume	assume	VERB
ejpam-5125	224	9	(	(	PUNCT
ejpam-5125	224	10	8)	8)	NUM
ejpam-5125	224	11	.	.	PUNCT
ejpam-5125	225	1	then	then	ADV
ejpam-5125	225	2	there	there	PRON
ejpam-5125	225	3	exists	exist	VERB
ejpam-5125	225	4	a	a	DET
ejpam-5125	225	5	∈	∈	NOUN
ejpam-5125	225	6	v	v	ADP
ejpam-5125	225	7	such	such	DET
ejpam-5125	225	8	that	that	SCONJ
ejpam-5125	225	9	a	a	DET
ejpam-5125	225	10	≤	≤	ADJ
ejpam-5125	225	11	p1	p1	NOUN
ejpam-5125	225	12	and	and	CCONJ
ejpam-5125	225	13	a	a	DET
ejpam-5125	225	14	≤	≤	ADJ
ejpam-5125	225	15	p2	p2	NOUN
ejpam-5125	225	16	.	.	PUNCT
ejpam-5125	226	1	now	now	ADV
ejpam-5125	226	2	consider	consider	VERB
ejpam-5125	226	3	(	(	PUNCT
ejpam-5125	226	4	p1	p1	NOUN
ejpam-5125	226	5	∧	∧	NOUN
ejpam-5125	226	6	p2	p2	NOUN
ejpam-5125	226	7	)	)	PUNCT
ejpam-5125	226	8	∨	∨	NUM
ejpam-5125	226	9	p1	p1	NOUN
ejpam-5125	226	10	=	=	SYM
ejpam-5125	226	11	(	(	PUNCT
ejpam-5125	226	12	p1	p1	PROPN
ejpam-5125	226	13	∨	∨	NUM
ejpam-5125	226	14	p1	p1	PROPN
ejpam-5125	226	15	)	)	PUNCT
ejpam-5125	226	16	∧	∧	PROPN
ejpam-5125	226	17	(	(	PUNCT
ejpam-5125	226	18	p2	p2	PROPN
ejpam-5125	226	19	∨	∨	NUM
ejpam-5125	226	20	p1	p1	NOUN
ejpam-5125	226	21	)	)	PUNCT
ejpam-5125	226	22	=	=	SYM
ejpam-5125	226	23	p1	p1	NOUN
ejpam-5125	226	24	∧	∧	PROPN
ejpam-5125	226	25	(	(	PUNCT
ejpam-5125	226	26	p2	p2	PROPN
ejpam-5125	226	27	∨	∨	NUM
ejpam-5125	226	28	p1	p1	NOUN
ejpam-5125	226	29	)	)	PUNCT
ejpam-5125	226	30	=	=	PUNCT
ejpam-5125	226	31	(	(	PUNCT
ejpam-5125	226	32	a	a	DET
ejpam-5125	226	33	∨	∨	NUM
ejpam-5125	226	34	p1	p1	NOUN
ejpam-5125	226	35	)	)	PUNCT
ejpam-5125	226	36	∧	∧	PROPN
ejpam-5125	226	37	(	(	PUNCT
ejpam-5125	226	38	p2	p2	PROPN
ejpam-5125	226	39	∨	∨	NUM
ejpam-5125	226	40	p1	p1	NOUN
ejpam-5125	226	41	)	)	PUNCT
ejpam-5125	226	42	=	=	PUNCT
ejpam-5125	226	43	(	(	PUNCT
ejpam-5125	226	44	a	a	DET
ejpam-5125	226	45	∧	∧	NOUN
ejpam-5125	226	46	p2	p2	NOUN
ejpam-5125	226	47	)	)	PUNCT
ejpam-5125	226	48	∨	∨	NUM
ejpam-5125	226	49	p1	p1	NOUN
ejpam-5125	226	50	=	=	PUNCT
ejpam-5125	226	51	a	a	DET
ejpam-5125	226	52	∨	∨	NUM
ejpam-5125	226	53	p1	p1	NOUN
ejpam-5125	226	54	=	=	PROPN
ejpam-5125	226	55	p1	p1	PROPN
ejpam-5125	226	56	therefore	therefore	ADV
ejpam-5125	226	57	p1	p1	PROPN
ejpam-5125	226	58	∧	∧	PROPN
ejpam-5125	226	59	p2	p2	PROPN
ejpam-5125	226	60	is	be	AUX
ejpam-5125	226	61	the	the	DET
ejpam-5125	226	62	lower	low	ADJ
ejpam-5125	226	63	bound	bind	VERB
ejpam-5125	226	64	of	of	ADP
ejpam-5125	226	65	p1	p1	PROPN
ejpam-5125	226	66	and	and	CCONJ
ejpam-5125	226	67	p2	p2	NOUN
ejpam-5125	226	68	in	in	ADP
ejpam-5125	226	69	v	v	NOUN
ejpam-5125	226	70	.	.	PUNCT
ejpam-5125	227	1	now	now	ADV
ejpam-5125	227	2	,	,	PUNCT
ejpam-5125	227	3	for	for	ADP
ejpam-5125	227	4	c	c	PROPN
ejpam-5125	227	5	∈	∈	PROPN
ejpam-5125	227	6	v	v	ADP
ejpam-5125	227	7	such	such	ADJ
ejpam-5125	227	8	that	that	SCONJ
ejpam-5125	227	9	c	c	PROPN
ejpam-5125	227	10	≤	≤	PROPN
ejpam-5125	227	11	p1	p1	NOUN
ejpam-5125	227	12	and	and	CCONJ
ejpam-5125	227	13	c	c	NOUN
ejpam-5125	227	14	≤	≤	NUM
ejpam-5125	227	15	p2	p2	NOUN
ejpam-5125	227	16	,	,	PUNCT
ejpam-5125	227	17	we	we	PRON
ejpam-5125	227	18	have	have	VERB
ejpam-5125	227	19	c	c	NOUN
ejpam-5125	227	20	∨	∨	NOUN
ejpam-5125	227	21	(	(	PUNCT
ejpam-5125	227	22	p1	p1	NOUN
ejpam-5125	227	23	∧	∧	NOUN
ejpam-5125	227	24	p2	p2	NOUN
ejpam-5125	227	25	)	)	PUNCT
ejpam-5125	227	26	=	=	SYM
ejpam-5125	227	27	(	(	PUNCT
ejpam-5125	227	28	c	c	PROPN
ejpam-5125	227	29	∨	∨	NUM
ejpam-5125	227	30	p1	p1	PROPN
ejpam-5125	227	31	)	)	PUNCT
ejpam-5125	227	32	∧	∧	NOUN
ejpam-5125	227	33	(	(	PUNCT
ejpam-5125	227	34	c	c	NOUN
ejpam-5125	227	35	∨	∨	NUM
ejpam-5125	227	36	p2	p2	X
ejpam-5125	227	37	)	)	PUNCT
ejpam-5125	227	38	=	=	SYM
ejpam-5125	227	39	p1	p1	NOUN
ejpam-5125	227	40	∧	∧	NOUN
ejpam-5125	227	41	p2	p2	NOUN
ejpam-5125	227	42	,	,	PUNCT
ejpam-5125	227	43	implies	imply	VERB
ejpam-5125	227	44	c	c	NOUN
ejpam-5125	227	45	≤	≤	PROPN
ejpam-5125	227	46	p1	p1	NOUN
ejpam-5125	227	47	∧	∧	PROPN
ejpam-5125	227	48	p2	p2	NOUN
ejpam-5125	227	49	.	.	PUNCT
ejpam-5125	228	1	thus	thus	ADV
ejpam-5125	228	2	r.	r.	PROPN
ejpam-5125	228	3	bandaru	bandaru	PROPN
ejpam-5125	228	4	,	,	PUNCT
ejpam-5125	228	5	s.	s.	PROPN
ejpam-5125	228	6	ajjarapu	ajjarapu	PROPN
ejpam-5125	228	7	/	/	PUNCT
ejpam-5125	228	8	eur	eur	PROPN
ejpam-5125	228	9	.	.	PUNCT
ejpam-5125	229	1	j.	j.	PROPN
ejpam-5125	229	2	pure	pure	PROPN
ejpam-5125	229	3	appl	appl	PROPN
ejpam-5125	229	4	.	.	PROPN
ejpam-5125	229	5	math	math	PROPN
ejpam-5125	229	6	,	,	PUNCT
ejpam-5125	229	7	17	17	NUM
ejpam-5125	229	8	(	(	PUNCT
ejpam-5125	229	9	2	2	NUM
ejpam-5125	229	10	)	)	PUNCT
ejpam-5125	229	11	(	(	PUNCT
ejpam-5125	229	12	2024	2024	NUM
ejpam-5125	229	13	)	)	PUNCT
ejpam-5125	229	14	,	,	PUNCT
ejpam-5125	229	15	819	819	NUM
ejpam-5125	229	16	-	-	SYM
ejpam-5125	229	17	834	834	NUM
ejpam-5125	229	18	825	825	NUM
ejpam-5125	229	19	p1	p1	NOUN
ejpam-5125	229	20	∧	∧	PROPN
ejpam-5125	229	21	p2	p2	PROPN
ejpam-5125	229	22	is	be	AUX
ejpam-5125	229	23	the	the	DET
ejpam-5125	229	24	g.l.b	g.l.b	NOUN
ejpam-5125	229	25	of	of	ADP
ejpam-5125	229	26	p1	p1	PROPN
ejpam-5125	229	27	and	and	CCONJ
ejpam-5125	229	28	p2	p2	NOUN
ejpam-5125	229	29	in	in	ADP
ejpam-5125	229	30	v	v	NUM
ejpam-5125	229	31	.	.	PUNCT
ejpam-5125	230	1	(	(	PUNCT
ejpam-5125	230	2	9	9	X
ejpam-5125	230	3	)	)	PUNCT
ejpam-5125	230	4	⇒	⇒	NOUN
ejpam-5125	230	5	(	(	PUNCT
ejpam-5125	230	6	5	5	NUM
ejpam-5125	230	7	):	):	PUNCT
ejpam-5125	230	8	assume	assume	VERB
ejpam-5125	230	9	(	(	PUNCT
ejpam-5125	230	10	9	9	NUM
ejpam-5125	230	11	)	)	PUNCT
ejpam-5125	230	12	.	.	PUNCT
ejpam-5125	231	1	then	then	ADV
ejpam-5125	231	2	(	(	PUNCT
ejpam-5125	231	3	p1	p1	PROPN
ejpam-5125	231	4	∧	∧	NOUN
ejpam-5125	231	5	p2	p2	NOUN
ejpam-5125	231	6	)	)	PUNCT
ejpam-5125	231	7	∨	∨	NUM
ejpam-5125	231	8	p1	p1	NOUN
ejpam-5125	231	9	=	=	SYM
ejpam-5125	231	10	p1	p1	PROPN
ejpam-5125	231	11	,	,	PUNCT
ejpam-5125	231	12	since	since	SCONJ
ejpam-5125	231	13	p1	p1	PROPN
ejpam-5125	231	14	∧	∧	PROPN
ejpam-5125	231	15	p2	p2	NOUN
ejpam-5125	231	16	≤	≤	NOUN
ejpam-5125	231	17	p1	p1	NOUN
ejpam-5125	231	18	and	and	CCONJ
ejpam-5125	231	19	hence	hence	ADV
ejpam-5125	231	20	(	(	PUNCT
ejpam-5125	231	21	5	5	X
ejpam-5125	231	22	)	)	PUNCT
ejpam-5125	231	23	follows	follow	VERB
ejpam-5125	231	24	by	by	ADP
ejpam-5125	231	25	the	the	DET
ejpam-5125	231	26	equivalence	equivalence	NOUN
ejpam-5125	231	27	of	of	ADP
ejpam-5125	231	28	(	(	PUNCT
ejpam-5125	231	29	1	1	NUM
ejpam-5125	231	30	)	)	PUNCT
ejpam-5125	231	31	and	and	CCONJ
ejpam-5125	231	32	(	(	PUNCT
ejpam-5125	231	33	5	5	NUM
ejpam-5125	231	34	)	)	PUNCT
ejpam-5125	231	35	.	.	PUNCT
ejpam-5125	232	1	thus	thus	ADV
ejpam-5125	232	2	the	the	DET
ejpam-5125	232	3	conditions	condition	NOUN
ejpam-5125	232	4	(	(	PUNCT
ejpam-5125	232	5	5	5	NUM
ejpam-5125	232	6	)	)	PUNCT
ejpam-5125	232	7	,	,	PUNCT
ejpam-5125	232	8	(	(	PUNCT
ejpam-5125	232	9	7	7	NUM
ejpam-5125	232	10	)	)	PUNCT
ejpam-5125	232	11	,	,	PUNCT
ejpam-5125	232	12	(	(	PUNCT
ejpam-5125	232	13	8)	8)	NUM
ejpam-5125	232	14	and	and	CCONJ
ejpam-5125	232	15	(	(	PUNCT
ejpam-5125	232	16	9	9	NUM
ejpam-5125	232	17	)	)	PUNCT
ejpam-5125	232	18	are	be	AUX
ejpam-5125	232	19	equivalent	equivalent	ADJ
ejpam-5125	232	20	.	.	PUNCT
ejpam-5125	233	1	theorem	theorem	NOUN
ejpam-5125	233	2	3	3	NUM
ejpam-5125	233	3	.	.	X
ejpam-5125	233	4	for	for	ADP
ejpam-5125	233	5	any	any	DET
ejpam-5125	233	6	p1	p1	NOUN
ejpam-5125	233	7	,	,	PUNCT
ejpam-5125	233	8	p2	p2	NOUN
ejpam-5125	233	9	,	,	PUNCT
ejpam-5125	233	10	p3	p3	PROPN
ejpam-5125	233	11	∈	∈	PROPN
ejpam-5125	233	12	v	v	NOUN
ejpam-5125	233	13	,	,	PUNCT
ejpam-5125	233	14	p1	p1	PROPN
ejpam-5125	233	15	∨	∨	NUM
ejpam-5125	233	16	(	(	PUNCT
ejpam-5125	233	17	p2	p2	PROPN
ejpam-5125	233	18	∧	∧	PROPN
ejpam-5125	233	19	p3	p3	PROPN
ejpam-5125	233	20	)	)	PUNCT
ejpam-5125	234	1	=	=	PUNCT
ejpam-5125	234	2	(	(	PUNCT
ejpam-5125	234	3	p1	p1	PROPN
ejpam-5125	234	4	∨	∨	NUM
ejpam-5125	234	5	p2	p2	NOUN
ejpam-5125	234	6	)	)	PUNCT
ejpam-5125	234	7	∧	∧	PROPN
ejpam-5125	234	8	(	(	PUNCT
ejpam-5125	234	9	p3	p3	PROPN
ejpam-5125	234	10	∨	∨	NUM
ejpam-5125	234	11	p1	p1	PROPN
ejpam-5125	234	12	)	)	PUNCT
ejpam-5125	235	1	if	if	SCONJ
ejpam-5125	235	2	and	and	CCONJ
ejpam-5125	235	3	only	only	ADV
ejpam-5125	235	4	if	if	SCONJ
ejpam-5125	235	5	p1	p1	PROPN
ejpam-5125	235	6	∧	∧	PROPN
ejpam-5125	235	7	p2	p2	NOUN
ejpam-5125	235	8	=	=	SYM
ejpam-5125	235	9	p2	p2	PROPN
ejpam-5125	235	10	∧	∧	PROPN
ejpam-5125	235	11	p1	p1	NOUN
ejpam-5125	235	12	.	.	PUNCT
ejpam-5125	236	1	proof	proof	NOUN
ejpam-5125	236	2	.	.	PUNCT
ejpam-5125	237	1	let	let	VERB
ejpam-5125	237	2	p1	p1	NOUN
ejpam-5125	237	3	,	,	PUNCT
ejpam-5125	237	4	p2	p2	NOUN
ejpam-5125	237	5	,	,	PUNCT
ejpam-5125	237	6	p3	p3	PROPN
ejpam-5125	237	7	∈	∈	PROPN
ejpam-5125	237	8	v	v	AUX
ejpam-5125	237	9	be	be	AUX
ejpam-5125	237	10	such	such	ADJ
ejpam-5125	237	11	that	that	SCONJ
ejpam-5125	237	12	p1	p1	PROPN
ejpam-5125	237	13	∨	∨	NUM
ejpam-5125	237	14	(	(	PUNCT
ejpam-5125	237	15	p2	p2	PROPN
ejpam-5125	237	16	∧	∧	PROPN
ejpam-5125	237	17	p3	p3	PROPN
ejpam-5125	237	18	)	)	PUNCT
ejpam-5125	238	1	=	=	PUNCT
ejpam-5125	239	1	(	(	PUNCT
ejpam-5125	239	2	p1	p1	PROPN
ejpam-5125	239	3	∨	∨	NUM
ejpam-5125	239	4	p2	p2	NOUN
ejpam-5125	239	5	)	)	PUNCT
ejpam-5125	239	6	∧	∧	PROPN
ejpam-5125	239	7	(	(	PUNCT
ejpam-5125	239	8	p3	p3	PROPN
ejpam-5125	239	9	∨	∨	NUM
ejpam-5125	239	10	p1	p1	PROPN
ejpam-5125	239	11	)	)	PUNCT
ejpam-5125	239	12	.	.	PUNCT
ejpam-5125	240	1	then	then	ADV
ejpam-5125	240	2	p2	p2	PROPN
ejpam-5125	240	3	∧	∧	PROPN
ejpam-5125	240	4	p1	p1	NOUN
ejpam-5125	240	5	=	=	SYM
ejpam-5125	240	6	(	(	PUNCT
ejpam-5125	240	7	(	(	PUNCT
ejpam-5125	240	8	p1	p1	NOUN
ejpam-5125	240	9	∧	∧	NOUN
ejpam-5125	240	10	p2	p2	NOUN
ejpam-5125	240	11	)	)	PUNCT
ejpam-5125	240	12	∨	∨	NUM
ejpam-5125	240	13	p2	p2	NOUN
ejpam-5125	240	14	)	)	PUNCT
ejpam-5125	240	15	∧	∧	NOUN
ejpam-5125	240	16	(	(	PUNCT
ejpam-5125	240	17	(	(	PUNCT
ejpam-5125	240	18	p1	p1	PROPN
ejpam-5125	240	19	∨	∨	PROPN
ejpam-5125	240	20	(	(	PUNCT
ejpam-5125	240	21	p1	p1	NOUN
ejpam-5125	240	22	∧	∧	NOUN
ejpam-5125	240	23	p2	p2	NOUN
ejpam-5125	240	24	)	)	PUNCT
ejpam-5125	240	25	)	)	PUNCT
ejpam-5125	241	1	=	=	SYM
ejpam-5125	241	2	(	(	PUNCT
ejpam-5125	241	3	p1	p1	NOUN
ejpam-5125	241	4	∧	∧	NOUN
ejpam-5125	241	5	p2	p2	NOUN
ejpam-5125	241	6	)	)	PUNCT
ejpam-5125	241	7	∨	∨	NUM
ejpam-5125	241	8	(	(	PUNCT
ejpam-5125	241	9	p2	p2	PROPN
ejpam-5125	241	10	∧	∧	PROPN
ejpam-5125	241	11	p1	p1	PROPN
ejpam-5125	241	12	)	)	PUNCT
ejpam-5125	241	13	=	=	PUNCT
ejpam-5125	241	14	(	(	PUNCT
ejpam-5125	241	15	p1	p1	PROPN
ejpam-5125	241	16	∨	∨	PROPN
ejpam-5125	241	17	(	(	PUNCT
ejpam-5125	241	18	p2	p2	PROPN
ejpam-5125	241	19	∧	∧	PROPN
ejpam-5125	241	20	p1	p1	PROPN
ejpam-5125	241	21	)	)	PUNCT
ejpam-5125	241	22	)	)	PUNCT
ejpam-5125	241	23	∧	∧	NOUN
ejpam-5125	241	24	(	(	PUNCT
ejpam-5125	241	25	p2	p2	PROPN
ejpam-5125	241	26	∨	∨	PROPN
ejpam-5125	241	27	(	(	PUNCT
ejpam-5125	241	28	p2	p2	PROPN
ejpam-5125	241	29	∧	∧	PROPN
ejpam-5125	241	30	p1	p1	NOUN
ejpam-5125	241	31	)	)	PUNCT
ejpam-5125	241	32	)	)	PUNCT
ejpam-5125	242	1	=	=	SYM
ejpam-5125	242	2	p1	p1	NOUN
ejpam-5125	242	3	∧	∧	PROPN
ejpam-5125	242	4	p2	p2	NOUN
ejpam-5125	242	5	.	.	PUNCT
ejpam-5125	243	1	converse	converse	NOUN
ejpam-5125	243	2	follows	follow	VERB
ejpam-5125	243	3	from	from	ADP
ejpam-5125	243	4	theorem	theorem	ADJ
ejpam-5125	243	5	2	2	NUM
ejpam-5125	243	6	and	and	CCONJ
ejpam-5125	243	7	(	(	PUNCT
ejpam-5125	243	8	ld∨	ld∨	NOUN
ejpam-5125	243	9	)	)	PUNCT
ejpam-5125	243	10	.	.	PUNCT
ejpam-5125	244	1	lemma	lemma	PROPN
ejpam-5125	244	2	8	8	NUM
ejpam-5125	244	3	.	.	PUNCT
ejpam-5125	245	1	for	for	ADP
ejpam-5125	245	2	any	any	DET
ejpam-5125	245	3	p1	p1	NOUN
ejpam-5125	245	4	,	,	PUNCT
ejpam-5125	245	5	p2	p2	NOUN
ejpam-5125	245	6	,	,	PUNCT
ejpam-5125	245	7	p3	p3	PROPN
ejpam-5125	245	8	∈	∈	PROPN
ejpam-5125	245	9	v	v	NOUN
ejpam-5125	245	10	,	,	PUNCT
ejpam-5125	245	11	p1	p1	PROPN
ejpam-5125	245	12	∨	∨	NUM
ejpam-5125	245	13	(	(	PUNCT
ejpam-5125	245	14	p2	p2	PROPN
ejpam-5125	245	15	∧	∧	PROPN
ejpam-5125	245	16	p3	p3	PROPN
ejpam-5125	245	17	)	)	PUNCT
ejpam-5125	245	18	=	=	SYM
ejpam-5125	245	19	p1	p1	PROPN
ejpam-5125	245	20	∨	∨	NOUN
ejpam-5125	245	21	(	(	PUNCT
ejpam-5125	245	22	p3	p3	PROPN
ejpam-5125	245	23	∧	∧	NOUN
ejpam-5125	245	24	p2	p2	NOUN
ejpam-5125	245	25	)	)	PUNCT
ejpam-5125	245	26	.	.	PUNCT
ejpam-5125	246	1	proof	proof	NOUN
ejpam-5125	246	2	.	.	PUNCT
ejpam-5125	247	1	since	since	SCONJ
ejpam-5125	247	2	p1	p1	PROPN
ejpam-5125	247	3	≤	≤	PROPN
ejpam-5125	247	4	p1	p1	PROPN
ejpam-5125	247	5	∨	∨	NOUN
ejpam-5125	247	6	p2	p2	PROPN
ejpam-5125	247	7	and	and	CCONJ
ejpam-5125	247	8	p1	p1	PROPN
ejpam-5125	247	9	≤	≤	PROPN
ejpam-5125	247	10	p1	p1	PROPN
ejpam-5125	247	11	∨	∨	NUM
ejpam-5125	247	12	p3	p3	PROPN
ejpam-5125	247	13	,	,	PUNCT
ejpam-5125	247	14	we	we	PRON
ejpam-5125	247	15	have	have	VERB
ejpam-5125	247	16	(	(	PUNCT
ejpam-5125	247	17	p1	p1	PROPN
ejpam-5125	247	18	∨	∨	NUM
ejpam-5125	247	19	p2	p2	NOUN
ejpam-5125	247	20	)	)	PUNCT
ejpam-5125	247	21	∧	∧	PROPN
ejpam-5125	247	22	(	(	PUNCT
ejpam-5125	247	23	p1	p1	PROPN
ejpam-5125	247	24	∨	∨	NUM
ejpam-5125	247	25	p3	p3	PROPN
ejpam-5125	247	26	)	)	PUNCT
ejpam-5125	247	27	=	=	PUNCT
ejpam-5125	248	1	(	(	PUNCT
ejpam-5125	248	2	p1	p1	PROPN
ejpam-5125	248	3	∨	∨	NUM
ejpam-5125	248	4	p3	p3	PROPN
ejpam-5125	248	5	)	)	PUNCT
ejpam-5125	248	6	∧	∧	PROPN
ejpam-5125	248	7	(	(	PUNCT
ejpam-5125	248	8	p1	p1	PROPN
ejpam-5125	248	9	∨	∨	NUM
ejpam-5125	248	10	p2	p2	NOUN
ejpam-5125	248	11	)	)	PUNCT
ejpam-5125	248	12	.	.	PUNCT
ejpam-5125	249	1	therefore	therefore	ADV
ejpam-5125	249	2	p1	p1	PROPN
ejpam-5125	249	3	∨	∨	NUM
ejpam-5125	249	4	(	(	PUNCT
ejpam-5125	249	5	p2	p2	PROPN
ejpam-5125	249	6	∧	∧	PROPN
ejpam-5125	249	7	p3	p3	PROPN
ejpam-5125	249	8	)	)	PUNCT
ejpam-5125	249	9	=	=	PUNCT
ejpam-5125	249	10	(	(	PUNCT
ejpam-5125	249	11	p1	p1	PROPN
ejpam-5125	249	12	∨	∨	NUM
ejpam-5125	249	13	p2	p2	NOUN
ejpam-5125	249	14	)	)	PUNCT
ejpam-5125	249	15	∧	∧	PROPN
ejpam-5125	249	16	(	(	PUNCT
ejpam-5125	249	17	p1	p1	PROPN
ejpam-5125	249	18	∨	∨	NUM
ejpam-5125	249	19	p3	p3	PROPN
ejpam-5125	249	20	)	)	PUNCT
ejpam-5125	249	21	=	=	PUNCT
ejpam-5125	249	22	(	(	PUNCT
ejpam-5125	249	23	p1	p1	PROPN
ejpam-5125	249	24	∨	∨	NUM
ejpam-5125	249	25	p3	p3	PROPN
ejpam-5125	249	26	)	)	PUNCT
ejpam-5125	249	27	∧	∧	PROPN
ejpam-5125	249	28	(	(	PUNCT
ejpam-5125	249	29	p1	p1	PROPN
ejpam-5125	249	30	∨	∨	NUM
ejpam-5125	249	31	p2	p2	NOUN
ejpam-5125	249	32	)	)	PUNCT
ejpam-5125	249	33	=	=	SYM
ejpam-5125	249	34	p1	p1	PROPN
ejpam-5125	249	35	∨	∨	NOUN
ejpam-5125	249	36	(	(	PUNCT
ejpam-5125	249	37	p3	p3	PROPN
ejpam-5125	249	38	∧	∧	PROPN
ejpam-5125	249	39	p2	p2	NOUN
ejpam-5125	249	40	)	)	PUNCT
ejpam-5125	249	41	.	.	PUNCT
ejpam-5125	250	1	theorem	theorem	NOUN
ejpam-5125	250	2	4	4	NUM
ejpam-5125	250	3	.	.	PUNCT
ejpam-5125	251	1	the	the	DET
ejpam-5125	251	2	operation	operation	NOUN
ejpam-5125	251	3	∨	∨	NOUN
ejpam-5125	251	4	is	be	AUX
ejpam-5125	251	5	associative	associative	ADJ
ejpam-5125	251	6	in	in	ADP
ejpam-5125	251	7	a	a	DET
ejpam-5125	251	8	pdl	pdl	NOUN
ejpam-5125	251	9	v	v	NOUN
ejpam-5125	251	10	.	.	PUNCT
ejpam-5125	252	1	proof	proof	NOUN
ejpam-5125	252	2	.	.	PUNCT
ejpam-5125	253	1	let	let	VERB
ejpam-5125	253	2	p1	p1	NOUN
ejpam-5125	253	3	,	,	PUNCT
ejpam-5125	253	4	p2	p2	NOUN
ejpam-5125	253	5	,	,	PUNCT
ejpam-5125	253	6	p3	p3	PROPN
ejpam-5125	253	7	∈	∈	PROPN
ejpam-5125	253	8	v	v	NOUN
ejpam-5125	253	9	.	.	PUNCT
ejpam-5125	254	1	then	then	ADV
ejpam-5125	254	2	p1	p1	PROPN
ejpam-5125	254	3	∨	∨	NUM
ejpam-5125	254	4	(	(	PUNCT
ejpam-5125	254	5	p2	p2	PROPN
ejpam-5125	254	6	∨	∨	NUM
ejpam-5125	254	7	p3	p3	PROPN
ejpam-5125	254	8	)	)	PUNCT
ejpam-5125	254	9	=	=	PUNCT
ejpam-5125	255	1	[	[	X
ejpam-5125	255	2	p1	p1	PROPN
ejpam-5125	255	3	∨	∨	NOUN
ejpam-5125	255	4	(	(	PUNCT
ejpam-5125	255	5	p3	p3	PROPN
ejpam-5125	255	6	∧	∧	PROPN
ejpam-5125	255	7	p1	p1	PROPN
ejpam-5125	255	8	)	)	PUNCT
ejpam-5125	255	9	]	]	PUNCT
ejpam-5125	256	1	∨	∨	X
ejpam-5125	256	2	(	(	PUNCT
ejpam-5125	256	3	p2	p2	PROPN
ejpam-5125	256	4	∨	∨	NUM
ejpam-5125	256	5	p3	p3	PROPN
ejpam-5125	256	6	)	)	PUNCT
ejpam-5125	256	7	=	=	PUNCT
ejpam-5125	257	1	[	[	X
ejpam-5125	257	2	(	(	PUNCT
ejpam-5125	257	3	(	(	PUNCT
ejpam-5125	257	4	p1	p1	PROPN
ejpam-5125	257	5	∨	∨	NUM
ejpam-5125	257	6	p2	p2	NOUN
ejpam-5125	257	7	)	)	PUNCT
ejpam-5125	257	8	∧	∧	PROPN
ejpam-5125	257	9	p1	p1	NOUN
ejpam-5125	257	10	)	)	PUNCT
ejpam-5125	257	11	∨	∨	PROPN
ejpam-5125	257	12	(	(	PUNCT
ejpam-5125	257	13	p3	p3	PROPN
ejpam-5125	257	14	∧	∧	PROPN
ejpam-5125	257	15	p1	p1	PROPN
ejpam-5125	257	16	)	)	PUNCT
ejpam-5125	257	17	]	]	PUNCT
ejpam-5125	257	18	∨	∨	X
ejpam-5125	257	19	(	(	PUNCT
ejpam-5125	257	20	p2	p2	PROPN
ejpam-5125	257	21	∨	∨	NUM
ejpam-5125	257	22	p3	p3	PROPN
ejpam-5125	257	23	)	)	PUNCT
ejpam-5125	257	24	=	=	PUNCT
ejpam-5125	258	1	[	[	X
ejpam-5125	258	2	[	[	X
ejpam-5125	258	3	(	(	PUNCT
ejpam-5125	258	4	p1	p1	PROPN
ejpam-5125	258	5	∨	∨	NUM
ejpam-5125	258	6	p2	p2	PROPN
ejpam-5125	258	7	)	)	PUNCT
ejpam-5125	258	8	∨	∨	NUM
ejpam-5125	258	9	p3	p3	PROPN
ejpam-5125	258	10	]	]	PUNCT
ejpam-5125	258	11	∧	∧	PROPN
ejpam-5125	258	12	p1	p1	PROPN
ejpam-5125	258	13	]	]	PUNCT
ejpam-5125	258	14	∨	∨	X
ejpam-5125	258	15	(	(	PUNCT
ejpam-5125	258	16	p2	p2	PROPN
ejpam-5125	258	17	∨	∨	NUM
ejpam-5125	258	18	p3	p3	PROPN
ejpam-5125	258	19	)	)	PUNCT
ejpam-5125	259	1	=	=	PUNCT
ejpam-5125	260	1	[	[	X
ejpam-5125	260	2	[	[	X
ejpam-5125	260	3	(	(	PUNCT
ejpam-5125	260	4	p1	p1	PROPN
ejpam-5125	260	5	∨	∨	NUM
ejpam-5125	260	6	p2	p2	PROPN
ejpam-5125	260	7	)	)	PUNCT
ejpam-5125	260	8	∨	∨	NUM
ejpam-5125	260	9	p3	p3	PROPN
ejpam-5125	260	10	]	]	PUNCT
ejpam-5125	260	11	∨	∨	NUM
ejpam-5125	260	12	(	(	PUNCT
ejpam-5125	260	13	p2	p2	PROPN
ejpam-5125	260	14	∨	∨	NUM
ejpam-5125	260	15	p3	p3	PROPN
ejpam-5125	260	16	)	)	PUNCT
ejpam-5125	260	17	]	]	PUNCT
ejpam-5125	261	1	∧	∧	PROPN
ejpam-5125	262	1	[	[	X
ejpam-5125	262	2	p1	p1	PROPN
ejpam-5125	262	3	∨	∨	NUM
ejpam-5125	262	4	(	(	PUNCT
ejpam-5125	262	5	p2	p2	PROPN
ejpam-5125	262	6	∨	∨	NUM
ejpam-5125	262	7	p3	p3	PROPN
ejpam-5125	262	8	)	)	PUNCT
ejpam-5125	262	9	]	]	PUNCT
ejpam-5125	263	1	=	=	PUNCT
ejpam-5125	263	2	(	(	PUNCT
ejpam-5125	263	3	(	(	PUNCT
ejpam-5125	263	4	p1	p1	PROPN
ejpam-5125	263	5	∨	∨	NUM
ejpam-5125	263	6	p2	p2	PROPN
ejpam-5125	263	7	)	)	PUNCT
ejpam-5125	263	8	∨	∨	NUM
ejpam-5125	263	9	p3	p3	PROPN
ejpam-5125	263	10	)	)	PUNCT
ejpam-5125	263	11	∧	∧	PROPN
ejpam-5125	263	12	(	(	PUNCT
ejpam-5125	263	13	p1	p1	PROPN
ejpam-5125	263	14	∨	∨	NUM
ejpam-5125	263	15	(	(	PUNCT
ejpam-5125	263	16	p2	p2	PROPN
ejpam-5125	263	17	∨	∨	NUM
ejpam-5125	263	18	p3	p3	PROPN
ejpam-5125	263	19	)	)	PUNCT
ejpam-5125	263	20	)	)	PUNCT
ejpam-5125	264	1	=	=	SYM
ejpam-5125	264	2	(	(	PUNCT
ejpam-5125	264	3	(	(	PUNCT
ejpam-5125	264	4	p1	p1	PROPN
ejpam-5125	264	5	∨	∨	NUM
ejpam-5125	264	6	p2	p2	NOUN
ejpam-5125	264	7	)	)	PUNCT
ejpam-5125	264	8	∧	∧	PROPN
ejpam-5125	264	9	(	(	PUNCT
ejpam-5125	264	10	p1	p1	PROPN
ejpam-5125	264	11	∨	∨	NUM
ejpam-5125	264	12	(	(	PUNCT
ejpam-5125	264	13	p2	p2	PROPN
ejpam-5125	264	14	∨	∨	NUM
ejpam-5125	264	15	p3	p3	PROPN
ejpam-5125	264	16	)	)	PUNCT
ejpam-5125	264	17	)	)	PUNCT
ejpam-5125	264	18	)	)	PUNCT
ejpam-5125	264	19	∨	∨	NUM
ejpam-5125	264	20	(	(	PUNCT
ejpam-5125	264	21	p3	p3	PROPN
ejpam-5125	264	22	∧	∧	PROPN
ejpam-5125	264	23	(	(	PUNCT
ejpam-5125	264	24	p1	p1	PROPN
ejpam-5125	264	25	∨	∨	NUM
ejpam-5125	264	26	(	(	PUNCT
ejpam-5125	264	27	p2	p2	PROPN
ejpam-5125	264	28	∨	∨	NUM
ejpam-5125	264	29	p3	p3	PROPN
ejpam-5125	264	30	)	)	PUNCT
ejpam-5125	264	31	)	)	PUNCT
ejpam-5125	264	32	)	)	PUNCT
ejpam-5125	265	1	=	=	PUNCT
ejpam-5125	265	2	(	(	PUNCT
ejpam-5125	265	3	p1	p1	PROPN
ejpam-5125	265	4	∨	∨	NUM
ejpam-5125	265	5	p2	p2	PROPN
ejpam-5125	265	6	)	)	PUNCT
ejpam-5125	265	7	∨	∨	NUM
ejpam-5125	265	8	(	(	PUNCT
ejpam-5125	265	9	(	(	PUNCT
ejpam-5125	265	10	p1	p1	PROPN
ejpam-5125	265	11	∨	∨	PROPN
ejpam-5125	265	12	(	(	PUNCT
ejpam-5125	265	13	p2	p2	PROPN
ejpam-5125	265	14	∨	∨	NUM
ejpam-5125	265	15	p3	p3	PROPN
ejpam-5125	265	16	)	)	PUNCT
ejpam-5125	265	17	)	)	PUNCT
ejpam-5125	265	18	∧	∧	PROPN
ejpam-5125	265	19	p3	p3	PROPN
ejpam-5125	265	20	)	)	PUNCT
ejpam-5125	265	21	=	=	PUNCT
ejpam-5125	265	22	(	(	PUNCT
ejpam-5125	265	23	p1	p1	PROPN
ejpam-5125	265	24	∨	∨	NUM
ejpam-5125	265	25	p2	p2	NOUN
ejpam-5125	265	26	)	)	PUNCT
ejpam-5125	265	27	∨	∨	NOUN
ejpam-5125	266	1	[	[	X
ejpam-5125	266	2	(	(	PUNCT
ejpam-5125	266	3	p1	p1	PROPN
ejpam-5125	266	4	∧	∧	PROPN
ejpam-5125	266	5	p3	p3	PROPN
ejpam-5125	266	6	)	)	PUNCT
ejpam-5125	266	7	∨	∨	NUM
ejpam-5125	266	8	(	(	PUNCT
ejpam-5125	266	9	(	(	PUNCT
ejpam-5125	266	10	p2	p2	PROPN
ejpam-5125	266	11	∨	∨	NUM
ejpam-5125	266	12	p3	p3	PROPN
ejpam-5125	266	13	)	)	PUNCT
ejpam-5125	266	14	∧	∧	PROPN
ejpam-5125	266	15	p3	p3	PROPN
ejpam-5125	266	16	)	)	PUNCT
ejpam-5125	266	17	]	]	PUNCT
ejpam-5125	267	1	=	=	PUNCT
ejpam-5125	267	2	(	(	PUNCT
ejpam-5125	267	3	p1	p1	PROPN
ejpam-5125	267	4	∨	∨	NUM
ejpam-5125	267	5	p2	p2	NOUN
ejpam-5125	267	6	)	)	PUNCT
ejpam-5125	267	7	∨	∨	NOUN
ejpam-5125	268	1	[	[	X
ejpam-5125	268	2	(	(	PUNCT
ejpam-5125	268	3	p1	p1	PROPN
ejpam-5125	268	4	∧	∧	PROPN
ejpam-5125	268	5	p3	p3	PROPN
ejpam-5125	268	6	)	)	PUNCT
ejpam-5125	268	7	∨	∨	NUM
ejpam-5125	268	8	p3	p3	PROPN
ejpam-5125	268	9	]	]	PUNCT
ejpam-5125	269	1	=	=	SYM
ejpam-5125	269	2	(	(	PUNCT
ejpam-5125	269	3	p1	p1	PROPN
ejpam-5125	269	4	∨	∨	NUM
ejpam-5125	269	5	p2	p2	PROPN
ejpam-5125	269	6	)	)	PUNCT
ejpam-5125	269	7	∨	∨	NUM
ejpam-5125	269	8	p3	p3	PROPN
ejpam-5125	269	9	therefore	therefore	ADV
ejpam-5125	269	10	∨	∨	PROPN
ejpam-5125	269	11	is	be	AUX
ejpam-5125	269	12	associative	associative	ADJ
ejpam-5125	269	13	in	in	ADP
ejpam-5125	269	14	v	v	NOUN
ejpam-5125	269	15	.	.	PUNCT
ejpam-5125	270	1	“	"	PUNCT
ejpam-5125	270	2	note	note	VERB
ejpam-5125	270	3	that	that	SCONJ
ejpam-5125	270	4	,	,	PUNCT
ejpam-5125	270	5	since	since	SCONJ
ejpam-5125	270	6	∨	∨	NOUN
ejpam-5125	270	7	is	be	AUX
ejpam-5125	270	8	associative	associative	ADJ
ejpam-5125	270	9	in	in	ADP
ejpam-5125	270	10	a	a	DET
ejpam-5125	270	11	pdl	pdl	NOUN
ejpam-5125	270	12	,	,	PUNCT
ejpam-5125	270	13	we	we	PRON
ejpam-5125	270	14	can	can	AUX
ejpam-5125	270	15	write	write	VERB
ejpam-5125	270	16	(	(	PUNCT
ejpam-5125	270	17	p1	p1	PROPN
ejpam-5125	270	18	∨	∨	NUM
ejpam-5125	270	19	p2	p2	PROPN
ejpam-5125	270	20	)	)	PUNCT
ejpam-5125	270	21	∨	∨	NUM
ejpam-5125	270	22	p3	p3	PROPN
ejpam-5125	270	23	,	,	PUNCT
ejpam-5125	270	24	simply	simply	ADV
ejpam-5125	270	25	,	,	PUNCT
ejpam-5125	270	26	as	as	ADP
ejpam-5125	270	27	p1	p1	PROPN
ejpam-5125	270	28	∨	∨	NUM
ejpam-5125	270	29	p2	p2	PROPN
ejpam-5125	270	30	∨	∨	NUM
ejpam-5125	270	31	p3	p3	PROPN
ejpam-5125	270	32	.	.	PUNCT
ejpam-5125	271	1	hence	hence	ADV
ejpam-5125	271	2	the	the	DET
ejpam-5125	271	3	notation	notation	NOUN
ejpam-5125	271	4	(	(	PUNCT
ejpam-5125	271	5	n	n	CCONJ
ejpam-5125	271	6	∨	∨	NUM
ejpam-5125	271	7	i=1	i=1	PROPN
ejpam-5125	271	8	αi	αi	NOUN
ejpam-5125	271	9	)	)	PUNCT
ejpam-5125	271	10	is	be	AUX
ejpam-5125	271	11	meaningful	meaningful	ADJ
ejpam-5125	271	12	,	,	PUNCT
ejpam-5125	271	13	as	as	ADV
ejpam-5125	271	14	well	well	ADV
ejpam-5125	271	15	”	"	PUNCT
ejpam-5125	271	16	.	.	PUNCT
ejpam-5125	272	1	the	the	DET
ejpam-5125	272	2	operation	operation	NOUN
ejpam-5125	272	3	∧	∧	PROPN
ejpam-5125	272	4	is	be	AUX
ejpam-5125	272	5	not	not	PART
ejpam-5125	272	6	associative	associative	ADJ
ejpam-5125	272	7	,	,	PUNCT
ejpam-5125	272	8	as	as	SCONJ
ejpam-5125	272	9	shown	show	VERB
ejpam-5125	272	10	by	by	ADP
ejpam-5125	272	11	the	the	DET
ejpam-5125	272	12	following	follow	VERB
ejpam-5125	272	13	example	example	NOUN
ejpam-5125	272	14	:	:	PUNCT
ejpam-5125	272	15	r.	r.	PROPN
ejpam-5125	272	16	bandaru	bandaru	PROPN
ejpam-5125	272	17	,	,	PUNCT
ejpam-5125	272	18	s.	s.	PROPN
ejpam-5125	272	19	ajjarapu	ajjarapu	PROPN
ejpam-5125	272	20	/	/	PUNCT
ejpam-5125	272	21	eur	eur	PROPN
ejpam-5125	272	22	.	.	PUNCT
ejpam-5125	273	1	j.	j.	PROPN
ejpam-5125	273	2	pure	pure	PROPN
ejpam-5125	273	3	appl	appl	PROPN
ejpam-5125	273	4	.	.	PROPN
ejpam-5125	273	5	math	math	PROPN
ejpam-5125	273	6	,	,	PUNCT
ejpam-5125	273	7	17	17	NUM
ejpam-5125	273	8	(	(	PUNCT
ejpam-5125	273	9	2	2	NUM
ejpam-5125	273	10	)	)	PUNCT
ejpam-5125	273	11	(	(	PUNCT
ejpam-5125	273	12	2024	2024	NUM
ejpam-5125	273	13	)	)	PUNCT
ejpam-5125	273	14	,	,	PUNCT
ejpam-5125	273	15	819	819	NUM
ejpam-5125	273	16	-	-	SYM
ejpam-5125	273	17	834	834	NUM
ejpam-5125	273	18	826	826	NUM
ejpam-5125	273	19	example	example	NOUN
ejpam-5125	273	20	4	4	NUM
ejpam-5125	273	21	.	.	PUNCT
ejpam-5125	274	1	let	let	VERB
ejpam-5125	274	2	v	v	VERB
ejpam-5125	274	3	=	=	SYM
ejpam-5125	274	4	{	{	PUNCT
ejpam-5125	274	5	0	0	NUM
ejpam-5125	274	6	,	,	PUNCT
ejpam-5125	274	7	1	1	NUM
ejpam-5125	274	8	,	,	PUNCT
ejpam-5125	274	9	2	2	NUM
ejpam-5125	274	10	,	,	PUNCT
ejpam-5125	274	11	3	3	NUM
ejpam-5125	274	12	,	,	PUNCT
ejpam-5125	274	13	4	4	NUM
ejpam-5125	274	14	,	,	PUNCT
ejpam-5125	274	15	5	5	NUM
ejpam-5125	274	16	}	}	PUNCT
ejpam-5125	274	17	be	be	AUX
ejpam-5125	274	18	a	a	DET
ejpam-5125	274	19	set	set	NOUN
ejpam-5125	274	20	with	with	ADP
ejpam-5125	274	21	binary	binary	ADJ
ejpam-5125	274	22	operations	operation	NOUN
ejpam-5125	274	23	∨	∨	NOUN
ejpam-5125	274	24	and	and	CCONJ
ejpam-5125	274	25	∧	∧	NOUN
ejpam-5125	274	26	given	give	VERB
ejpam-5125	274	27	in	in	ADP
ejpam-5125	274	28	the	the	DET
ejpam-5125	274	29	following	following	ADJ
ejpam-5125	274	30	tables	table	NOUN
ejpam-5125	274	31	:	:	PUNCT
ejpam-5125	274	32	∨	∨	NOUN
ejpam-5125	274	33	0	0	NUM
ejpam-5125	274	34	1	1	NUM
ejpam-5125	274	35	2	2	NUM
ejpam-5125	274	36	3	3	NUM
ejpam-5125	274	37	4	4	NUM
ejpam-5125	274	38	5	5	NUM
ejpam-5125	274	39	0	0	NUM
ejpam-5125	274	40	0	0	NUM
ejpam-5125	274	41	1	1	NUM
ejpam-5125	274	42	4	4	NUM
ejpam-5125	274	43	0	0	NUM
ejpam-5125	274	44	4	4	NUM
ejpam-5125	274	45	0	0	NUM
ejpam-5125	274	46	1	1	NUM
ejpam-5125	274	47	1	1	NUM
ejpam-5125	274	48	1	1	NUM
ejpam-5125	274	49	1	1	NUM
ejpam-5125	274	50	1	1	NUM
ejpam-5125	274	51	1	1	NUM
ejpam-5125	274	52	1	1	NUM
ejpam-5125	274	53	2	2	NUM
ejpam-5125	274	54	2	2	NUM
ejpam-5125	274	55	1	1	NUM
ejpam-5125	274	56	2	2	NUM
ejpam-5125	274	57	2	2	NUM
ejpam-5125	274	58	2	2	NUM
ejpam-5125	274	59	2	2	NUM
ejpam-5125	274	60	3	3	NUM
ejpam-5125	274	61	3	3	NUM
ejpam-5125	274	62	1	1	NUM
ejpam-5125	274	63	2	2	NUM
ejpam-5125	274	64	3	3	NUM
ejpam-5125	274	65	2	2	NUM
ejpam-5125	274	66	3	3	NUM
ejpam-5125	274	67	4	4	NUM
ejpam-5125	274	68	4	4	NUM
ejpam-5125	274	69	1	1	NUM
ejpam-5125	274	70	4	4	NUM
ejpam-5125	274	71	4	4	NUM
ejpam-5125	274	72	4	4	NUM
ejpam-5125	274	73	4	4	NUM
ejpam-5125	274	74	5	5	NUM
ejpam-5125	274	75	5	5	NUM
ejpam-5125	274	76	1	1	NUM
ejpam-5125	274	77	4	4	NUM
ejpam-5125	274	78	5	5	NUM
ejpam-5125	274	79	4	4	NUM
ejpam-5125	274	80	5	5	NUM
ejpam-5125	274	81	∧	∧	NOUN
ejpam-5125	274	82	0	0	NUM
ejpam-5125	274	83	1	1	NUM
ejpam-5125	274	84	2	2	NUM
ejpam-5125	274	85	3	3	NUM
ejpam-5125	274	86	4	4	NUM
ejpam-5125	274	87	5	5	NUM
ejpam-5125	274	88	0	0	NUM
ejpam-5125	274	89	0	0	NUM
ejpam-5125	274	90	0	0	NUM
ejpam-5125	274	91	3	3	NUM
ejpam-5125	274	92	3	3	NUM
ejpam-5125	274	93	0	0	NUM
ejpam-5125	274	94	5	5	NUM
ejpam-5125	274	95	1	1	NUM
ejpam-5125	274	96	0	0	NUM
ejpam-5125	274	97	1	1	NUM
ejpam-5125	274	98	2	2	NUM
ejpam-5125	274	99	3	3	NUM
ejpam-5125	274	100	4	4	NUM
ejpam-5125	274	101	5	5	NUM
ejpam-5125	274	102	2	2	NUM
ejpam-5125	274	103	0	0	NUM
ejpam-5125	274	104	2	2	NUM
ejpam-5125	274	105	2	2	NUM
ejpam-5125	274	106	3	3	NUM
ejpam-5125	274	107	4	4	NUM
ejpam-5125	274	108	5	5	NUM
ejpam-5125	274	109	3	3	NUM
ejpam-5125	274	110	0	0	NUM
ejpam-5125	274	111	3	3	NUM
ejpam-5125	274	112	3	3	NUM
ejpam-5125	274	113	3	3	NUM
ejpam-5125	274	114	5	5	NUM
ejpam-5125	274	115	5	5	NUM
ejpam-5125	274	116	4	4	NUM
ejpam-5125	274	117	0	0	NUM
ejpam-5125	274	118	4	4	NUM
ejpam-5125	274	119	2	2	NUM
ejpam-5125	274	120	3	3	NUM
ejpam-5125	274	121	4	4	NUM
ejpam-5125	274	122	5	5	NUM
ejpam-5125	274	123	5	5	NUM
ejpam-5125	274	124	0	0	NUM
ejpam-5125	274	125	5	5	NUM
ejpam-5125	274	126	3	3	NUM
ejpam-5125	274	127	3	3	NUM
ejpam-5125	274	128	5	5	NUM
ejpam-5125	274	129	5	5	NUM
ejpam-5125	274	130	then	then	ADV
ejpam-5125	274	131	(	(	PUNCT
ejpam-5125	274	132	v	v	NOUN
ejpam-5125	274	133	;	;	PUNCT
ejpam-5125	274	134	∨	∨	NUM
ejpam-5125	274	135	∧	∧	PROPN
ejpam-5125	274	136	1	1	NUM
ejpam-5125	274	137	)	)	PUNCT
ejpam-5125	274	138	is	be	AUX
ejpam-5125	274	139	a	a	DET
ejpam-5125	274	140	paradistributive	paradistributive	ADJ
ejpam-5125	274	141	latticoid	latticoid	NOUN
ejpam-5125	274	142	.	.	PUNCT
ejpam-5125	275	1	but	but	CCONJ
ejpam-5125	275	2	the	the	DET
ejpam-5125	275	3	operation	operation	NOUN
ejpam-5125	275	4	∧	∧	PROPN
ejpam-5125	275	5	is	be	AUX
ejpam-5125	275	6	not	not	PART
ejpam-5125	275	7	associative	associative	ADJ
ejpam-5125	275	8	,	,	PUNCT
ejpam-5125	275	9	since	since	SCONJ
ejpam-5125	275	10	0	0	NUM
ejpam-5125	275	11	∧	∧	PROPN
ejpam-5125	275	12	(	(	PUNCT
ejpam-5125	275	13	2	2	NUM
ejpam-5125	275	14	∧	∧	PROPN
ejpam-5125	275	15	4	4	NUM
ejpam-5125	275	16	)	)	PUNCT
ejpam-5125	275	17	=	=	SYM
ejpam-5125	275	18	0	0	NUM
ejpam-5125	276	1	∧	∧	NOUN
ejpam-5125	276	2	4	4	NUM
ejpam-5125	276	3	=	=	SYM
ejpam-5125	276	4	0	0	NUM
ejpam-5125	277	1	̸=	̸=	NOUN
ejpam-5125	277	2	5	5	NUM
ejpam-5125	277	3	=	=	SYM
ejpam-5125	277	4	3	3	NUM
ejpam-5125	277	5	∧	∧	PROPN
ejpam-5125	277	6	4	4	NUM
ejpam-5125	277	7	=	=	SYM
ejpam-5125	277	8	(	(	PUNCT
ejpam-5125	277	9	0	0	NUM
ejpam-5125	277	10	∧	∧	NOUN
ejpam-5125	277	11	2	2	NUM
ejpam-5125	277	12	)	)	PUNCT
ejpam-5125	277	13	∧	∧	NOUN
ejpam-5125	277	14	4	4	NUM
ejpam-5125	277	15	.	.	PUNCT
ejpam-5125	277	16	definition	definition	NOUN
ejpam-5125	277	17	2	2	NUM
ejpam-5125	277	18	.	.	PUNCT
ejpam-5125	278	1	a	a	DET
ejpam-5125	278	2	paradistributive	paradistributive	ADJ
ejpam-5125	278	3	latticoid	latticoid	NOUN
ejpam-5125	278	4	(	(	PUNCT
ejpam-5125	278	5	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	278	6	,	,	PUNCT
ejpam-5125	278	7	1	1	NUM
ejpam-5125	278	8	)	)	PUNCT
ejpam-5125	278	9	is	be	AUX
ejpam-5125	278	10	said	say	VERB
ejpam-5125	278	11	to	to	PART
ejpam-5125	278	12	be	be	AUX
ejpam-5125	278	13	associative	associative	ADJ
ejpam-5125	278	14	if	if	SCONJ
ejpam-5125	278	15	it	it	PRON
ejpam-5125	278	16	satisfies	satisfy	VERB
ejpam-5125	278	17	the	the	DET
ejpam-5125	278	18	following	follow	VERB
ejpam-5125	278	19	condition	condition	NOUN
ejpam-5125	278	20	p1	p1	NOUN
ejpam-5125	278	21	∧	∧	PROPN
ejpam-5125	278	22	(	(	PUNCT
ejpam-5125	278	23	p2	p2	PROPN
ejpam-5125	278	24	∧	∧	PROPN
ejpam-5125	278	25	p3	p3	PROPN
ejpam-5125	278	26	)	)	PUNCT
ejpam-5125	278	27	=	=	SYM
ejpam-5125	279	1	(	(	PUNCT
ejpam-5125	279	2	p1	p1	NOUN
ejpam-5125	279	3	∧	∧	NOUN
ejpam-5125	279	4	p2	p2	NOUN
ejpam-5125	279	5	)	)	PUNCT
ejpam-5125	279	6	∧	∧	NOUN
ejpam-5125	279	7	p3	p3	NOUN
ejpam-5125	279	8	for	for	ADP
ejpam-5125	279	9	all	all	DET
ejpam-5125	279	10	p1	p1	NOUN
ejpam-5125	279	11	,	,	PUNCT
ejpam-5125	279	12	p2	p2	NOUN
ejpam-5125	279	13	,	,	PUNCT
ejpam-5125	279	14	p3	p3	PROPN
ejpam-5125	279	15	∈	∈	PROPN
ejpam-5125	280	1	v.	v.	ADP
ejpam-5125	280	2	example	example	NOUN
ejpam-5125	280	3	5	5	X
ejpam-5125	280	4	.	.	PUNCT
ejpam-5125	281	1	let	let	VERB
ejpam-5125	281	2	v	v	VERB
ejpam-5125	281	3	=	=	SYM
ejpam-5125	281	4	{	{	PUNCT
ejpam-5125	281	5	0	0	NUM
ejpam-5125	281	6	,	,	PUNCT
ejpam-5125	281	7	1	1	NUM
ejpam-5125	281	8	,	,	PUNCT
ejpam-5125	281	9	2	2	NUM
ejpam-5125	281	10	,	,	PUNCT
ejpam-5125	281	11	3	3	NUM
ejpam-5125	281	12	,	,	PUNCT
ejpam-5125	281	13	4	4	NUM
ejpam-5125	281	14	}	}	PUNCT
ejpam-5125	281	15	be	be	AUX
ejpam-5125	281	16	a	a	DET
ejpam-5125	281	17	set	set	NOUN
ejpam-5125	281	18	with	with	ADP
ejpam-5125	281	19	binary	binary	ADJ
ejpam-5125	281	20	operations	operation	NOUN
ejpam-5125	281	21	∨	∨	NOUN
ejpam-5125	281	22	and	and	CCONJ
ejpam-5125	281	23	∧	∧	NOUN
ejpam-5125	281	24	given	give	VERB
ejpam-5125	281	25	in	in	ADP
ejpam-5125	281	26	the	the	DET
ejpam-5125	281	27	following	following	ADJ
ejpam-5125	281	28	tables	table	NOUN
ejpam-5125	281	29	:	:	PUNCT
ejpam-5125	281	30	∨	∨	NOUN
ejpam-5125	281	31	0	0	NUM
ejpam-5125	281	32	1	1	NUM
ejpam-5125	281	33	2	2	NUM
ejpam-5125	281	34	3	3	NUM
ejpam-5125	281	35	4	4	NUM
ejpam-5125	281	36	0	0	NUM
ejpam-5125	281	37	0	0	NUM
ejpam-5125	281	38	1	1	NUM
ejpam-5125	281	39	0	0	NUM
ejpam-5125	281	40	3	3	NUM
ejpam-5125	281	41	3	3	NUM
ejpam-5125	281	42	1	1	NUM
ejpam-5125	281	43	1	1	NUM
ejpam-5125	281	44	1	1	NUM
ejpam-5125	281	45	1	1	NUM
ejpam-5125	281	46	1	1	NUM
ejpam-5125	281	47	1	1	NUM
ejpam-5125	281	48	2	2	NUM
ejpam-5125	281	49	2	2	NUM
ejpam-5125	281	50	1	1	NUM
ejpam-5125	281	51	2	2	NUM
ejpam-5125	281	52	4	4	NUM
ejpam-5125	281	53	4	4	NUM
ejpam-5125	281	54	3	3	NUM
ejpam-5125	281	55	3	3	NUM
ejpam-5125	281	56	1	1	NUM
ejpam-5125	281	57	3	3	NUM
ejpam-5125	281	58	3	3	NUM
ejpam-5125	281	59	3	3	NUM
ejpam-5125	281	60	4	4	NUM
ejpam-5125	281	61	4	4	NUM
ejpam-5125	281	62	1	1	NUM
ejpam-5125	281	63	4	4	NUM
ejpam-5125	281	64	4	4	NUM
ejpam-5125	281	65	4	4	NUM
ejpam-5125	281	66	∧	∧	NOUN
ejpam-5125	281	67	0	0	NUM
ejpam-5125	281	68	1	1	NUM
ejpam-5125	281	69	2	2	NUM
ejpam-5125	281	70	3	3	NUM
ejpam-5125	281	71	4	4	NUM
ejpam-5125	281	72	0	0	NUM
ejpam-5125	281	73	0	0	NUM
ejpam-5125	281	74	0	0	NUM
ejpam-5125	281	75	2	2	NUM
ejpam-5125	281	76	0	0	NUM
ejpam-5125	281	77	2	2	NUM
ejpam-5125	281	78	1	1	NUM
ejpam-5125	281	79	0	0	NUM
ejpam-5125	281	80	1	1	NUM
ejpam-5125	281	81	2	2	NUM
ejpam-5125	281	82	3	3	NUM
ejpam-5125	281	83	4	4	NUM
ejpam-5125	281	84	2	2	NUM
ejpam-5125	281	85	0	0	NUM
ejpam-5125	281	86	2	2	NUM
ejpam-5125	281	87	2	2	NUM
ejpam-5125	281	88	0	0	NUM
ejpam-5125	281	89	2	2	NUM
ejpam-5125	281	90	3	3	NUM
ejpam-5125	281	91	0	0	NUM
ejpam-5125	281	92	3	3	NUM
ejpam-5125	281	93	2	2	NUM
ejpam-5125	281	94	3	3	NUM
ejpam-5125	281	95	4	4	NUM
ejpam-5125	281	96	4	4	NUM
ejpam-5125	281	97	0	0	NUM
ejpam-5125	281	98	4	4	NUM
ejpam-5125	281	99	2	2	NUM
ejpam-5125	281	100	3	3	NUM
ejpam-5125	281	101	4	4	NUM
ejpam-5125	281	102	then	then	ADV
ejpam-5125	281	103	(	(	PUNCT
ejpam-5125	281	104	v	v	NOUN
ejpam-5125	281	105	;	;	PUNCT
ejpam-5125	281	106	∨	∨	NUM
ejpam-5125	281	107	∧	∧	PROPN
ejpam-5125	281	108	1	1	NUM
ejpam-5125	281	109	)	)	PUNCT
ejpam-5125	281	110	is	be	AUX
ejpam-5125	281	111	an	an	DET
ejpam-5125	281	112	associative	associative	ADJ
ejpam-5125	281	113	paradistributive	paradistributive	ADJ
ejpam-5125	281	114	latticoid	latticoid	NOUN
ejpam-5125	281	115	.	.	PUNCT
ejpam-5125	282	1	lemma	lemma	PROPN
ejpam-5125	282	2	9	9	NUM
ejpam-5125	282	3	.	.	PUNCT
ejpam-5125	283	1	for	for	ADP
ejpam-5125	283	2	any	any	DET
ejpam-5125	283	3	p1	p1	NOUN
ejpam-5125	283	4	,	,	PUNCT
ejpam-5125	283	5	p2	p2	NOUN
ejpam-5125	283	6	,	,	PUNCT
ejpam-5125	283	7	p3	p3	PROPN
ejpam-5125	283	8	∈	∈	PROPN
ejpam-5125	283	9	v	v	NOUN
ejpam-5125	283	10	,	,	PUNCT
ejpam-5125	283	11	p1	p1	PROPN
ejpam-5125	283	12	∨	∨	NOUN
ejpam-5125	283	13	p2	p2	PROPN
ejpam-5125	283	14	∨	∨	NUM
ejpam-5125	283	15	p3	p3	PROPN
ejpam-5125	283	16	=	=	PROPN
ejpam-5125	283	17	p1	p1	PROPN
ejpam-5125	283	18	∨	∨	PROPN
ejpam-5125	283	19	p3	p3	PROPN
ejpam-5125	283	20	∨	∨	NUM
ejpam-5125	283	21	p2	p2	NOUN
ejpam-5125	283	22	.	.	PUNCT
ejpam-5125	284	1	proof	proof	NOUN
ejpam-5125	284	2	.	.	PUNCT
ejpam-5125	285	1	since	since	SCONJ
ejpam-5125	285	2	p1	p1	PROPN
ejpam-5125	285	3	≤	≤	PROPN
ejpam-5125	285	4	p1	p1	PROPN
ejpam-5125	285	5	∨	∨	NOUN
ejpam-5125	285	6	p2	p2	PROPN
ejpam-5125	285	7	and	and	CCONJ
ejpam-5125	285	8	p1	p1	PROPN
ejpam-5125	285	9	≤	≤	PROPN
ejpam-5125	285	10	p1	p1	PROPN
ejpam-5125	285	11	∨	∨	NUM
ejpam-5125	285	12	p3	p3	PROPN
ejpam-5125	285	13	,	,	PUNCT
ejpam-5125	285	14	we	we	PRON
ejpam-5125	285	15	have	have	AUX
ejpam-5125	285	16	(	(	PUNCT
ejpam-5125	285	17	p1	p1	PROPN
ejpam-5125	285	18	∨	∨	NUM
ejpam-5125	285	19	p2	p2	NOUN
ejpam-5125	285	20	)	)	PUNCT
ejpam-5125	285	21	∨	∨	NOUN
ejpam-5125	285	22	(	(	PUNCT
ejpam-5125	285	23	p1	p1	PROPN
ejpam-5125	285	24	∨	∨	NUM
ejpam-5125	285	25	p3	p3	PROPN
ejpam-5125	285	26	)	)	PUNCT
ejpam-5125	285	27	=	=	PUNCT
ejpam-5125	286	1	(	(	PUNCT
ejpam-5125	286	2	p1	p1	PROPN
ejpam-5125	286	3	∨	∨	NUM
ejpam-5125	286	4	p3	p3	PROPN
ejpam-5125	286	5	)	)	PUNCT
ejpam-5125	286	6	∨	∨	PROPN
ejpam-5125	286	7	(	(	PUNCT
ejpam-5125	286	8	p1	p1	PROPN
ejpam-5125	286	9	∨	∨	NUM
ejpam-5125	286	10	p2	p2	NOUN
ejpam-5125	286	11	)	)	PUNCT
ejpam-5125	286	12	.	.	PUNCT
ejpam-5125	287	1	therefore	therefore	ADV
ejpam-5125	287	2	(	(	PUNCT
ejpam-5125	287	3	p1	p1	PROPN
ejpam-5125	287	4	∨	∨	NUM
ejpam-5125	287	5	p2	p2	PROPN
ejpam-5125	287	6	∨	∨	NUM
ejpam-5125	287	7	p1	p1	NOUN
ejpam-5125	287	8	)	)	PUNCT
ejpam-5125	287	9	∨	∨	NUM
ejpam-5125	287	10	p3	p3	PROPN
ejpam-5125	287	11	=	=	SYM
ejpam-5125	287	12	(	(	PUNCT
ejpam-5125	287	13	p1	p1	PROPN
ejpam-5125	287	14	∨	∨	NUM
ejpam-5125	287	15	p3	p3	PROPN
ejpam-5125	287	16	∨	∨	NUM
ejpam-5125	287	17	p1	p1	PROPN
ejpam-5125	287	18	)	)	PUNCT
ejpam-5125	287	19	∨	∨	NUM
ejpam-5125	287	20	p2	p2	NOUN
ejpam-5125	287	21	.	.	PUNCT
ejpam-5125	288	1	hence	hence	ADV
ejpam-5125	288	2	p1	p1	PROPN
ejpam-5125	288	3	∨	∨	NUM
ejpam-5125	288	4	p2	p2	PROPN
ejpam-5125	288	5	∨	∨	NUM
ejpam-5125	288	6	p3	p3	PROPN
ejpam-5125	288	7	=	=	PROPN
ejpam-5125	288	8	p1	p1	PROPN
ejpam-5125	288	9	∨	∨	PROPN
ejpam-5125	288	10	p3	p3	PROPN
ejpam-5125	288	11	∨	∨	NUM
ejpam-5125	288	12	p2	p2	NOUN
ejpam-5125	288	13	.	.	PUNCT
ejpam-5125	289	1	lemma	lemma	PROPN
ejpam-5125	289	2	10	10	NUM
ejpam-5125	289	3	.	.	PUNCT
ejpam-5125	290	1	for	for	ADP
ejpam-5125	290	2	any	any	DET
ejpam-5125	290	3	p1	p1	NOUN
ejpam-5125	290	4	,	,	PUNCT
ejpam-5125	290	5	p2	p2	PROPN
ejpam-5125	290	6	∈	∈	PROPN
ejpam-5125	290	7	v	v	NOUN
ejpam-5125	290	8	,	,	PUNCT
ejpam-5125	290	9	p1	p1	NOUN
ejpam-5125	290	10	∨	∨	NOUN
ejpam-5125	290	11	p2	p2	X
ejpam-5125	290	12	=	=	SYM
ejpam-5125	290	13	1	1	NUM
ejpam-5125	290	14	if	if	SCONJ
ejpam-5125	290	15	and	and	CCONJ
ejpam-5125	290	16	only	only	ADV
ejpam-5125	290	17	if	if	SCONJ
ejpam-5125	290	18	p2	p2	PROPN
ejpam-5125	290	19	∨	∨	NUM
ejpam-5125	290	20	p1	p1	NOUN
ejpam-5125	290	21	=	=	SYM
ejpam-5125	290	22	1	1	X
ejpam-5125	290	23	.	.	PUNCT
ejpam-5125	290	24	theorem	theorem	NOUN
ejpam-5125	290	25	5	5	NUM
ejpam-5125	290	26	.	.	PUNCT
ejpam-5125	291	1	let	let	AUX
ejpam-5125	291	2	(	(	PUNCT
ejpam-5125	291	3	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	291	4	,	,	PUNCT
ejpam-5125	291	5	1	1	NUM
ejpam-5125	291	6	)	)	PUNCT
ejpam-5125	291	7	be	be	AUX
ejpam-5125	291	8	a	a	DET
ejpam-5125	291	9	pdl	pdl	NOUN
ejpam-5125	291	10	.	.	PUNCT
ejpam-5125	292	1	then	then	ADV
ejpam-5125	292	2	the	the	DET
ejpam-5125	292	3	following	following	NOUN
ejpam-5125	292	4	are	be	AUX
ejpam-5125	292	5	equivalent	equivalent	ADJ
ejpam-5125	292	6	:	:	PUNCT
ejpam-5125	292	7	(	(	PUNCT
ejpam-5125	292	8	1	1	X
ejpam-5125	292	9	)	)	PUNCT
ejpam-5125	292	10	(	(	PUNCT
ejpam-5125	292	11	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	292	12	,	,	PUNCT
ejpam-5125	292	13	1	1	NUM
ejpam-5125	292	14	)	)	PUNCT
ejpam-5125	292	15	is	be	AUX
ejpam-5125	292	16	a	a	DET
ejpam-5125	292	17	distributive	distributive	ADJ
ejpam-5125	292	18	lattice	lattice	NOUN
ejpam-5125	292	19	.	.	PUNCT
ejpam-5125	293	1	(	(	PUNCT
ejpam-5125	293	2	2	2	X
ejpam-5125	293	3	)	)	PUNCT
ejpam-5125	293	4	the	the	DET
ejpam-5125	293	5	poset	poset	NOUN
ejpam-5125	293	6	(	(	PUNCT
ejpam-5125	293	7	v,≤	v,≤	X
ejpam-5125	293	8	)	)	PUNCT
ejpam-5125	293	9	is	be	AUX
ejpam-5125	293	10	directed	direct	VERB
ejpam-5125	293	11	below	below	ADV
ejpam-5125	293	12	.	.	PUNCT
ejpam-5125	294	1	(	(	PUNCT
ejpam-5125	294	2	3	3	X
ejpam-5125	294	3	)	)	PUNCT
ejpam-5125	294	4	(	(	PUNCT
ejpam-5125	294	5	p1	p1	NOUN
ejpam-5125	294	6	∧	∧	NOUN
ejpam-5125	294	7	p2	p2	NOUN
ejpam-5125	294	8	)	)	PUNCT
ejpam-5125	294	9	∨	∨	NUM
ejpam-5125	294	10	p1	p1	NOUN
ejpam-5125	294	11	=	=	PROPN
ejpam-5125	294	12	p1	p1	PROPN
ejpam-5125	294	13	.	.	PUNCT
ejpam-5125	295	1	(	(	PUNCT
ejpam-5125	295	2	4	4	X
ejpam-5125	295	3	)	)	PUNCT
ejpam-5125	295	4	the	the	DET
ejpam-5125	295	5	operation	operation	NOUN
ejpam-5125	295	6	∧	∧	PROPN
ejpam-5125	295	7	is	be	AUX
ejpam-5125	295	8	commutative	commutative	ADJ
ejpam-5125	295	9	.	.	PUNCT
ejpam-5125	296	1	(	(	PUNCT
ejpam-5125	296	2	5	5	X
ejpam-5125	296	3	)	)	PUNCT
ejpam-5125	296	4	the	the	DET
ejpam-5125	296	5	operation	operation	NOUN
ejpam-5125	296	6	∨	∨	NOUN
ejpam-5125	296	7	is	be	AUX
ejpam-5125	296	8	commutative	commutative	ADJ
ejpam-5125	296	9	.	.	PUNCT
ejpam-5125	297	1	(	(	PUNCT
ejpam-5125	297	2	6	6	NUM
ejpam-5125	297	3	)	)	PUNCT
ejpam-5125	297	4	the	the	DET
ejpam-5125	297	5	relation	relation	NOUN
ejpam-5125	297	6	χ	χ	X
ejpam-5125	298	1	=	=	PUNCT
ejpam-5125	298	2	{	{	PUNCT
ejpam-5125	298	3	(	(	PUNCT
ejpam-5125	298	4	p1	p1	NOUN
ejpam-5125	298	5	,	,	PUNCT
ejpam-5125	298	6	p2	p2	X
ejpam-5125	298	7	)	)	PUNCT
ejpam-5125	298	8	∈	∈	NOUN
ejpam-5125	298	9	v	v	ADP
ejpam-5125	298	10	×	×	NOUN
ejpam-5125	298	11	v	v	NOUN
ejpam-5125	298	12	|	|	ADV
ejpam-5125	298	13	p2	p2	PROPN
ejpam-5125	298	14	∨	∨	NUM
ejpam-5125	298	15	p1	p1	NOUN
ejpam-5125	298	16	=	=	PUNCT
ejpam-5125	298	17	p2	p2	PROPN
ejpam-5125	298	18	}	}	PUNCT
ejpam-5125	298	19	is	be	AUX
ejpam-5125	298	20	antisymmetric	antisymmetric	ADJ
ejpam-5125	298	21	.	.	PUNCT
ejpam-5125	299	1	r.	r.	PROPN
ejpam-5125	299	2	bandaru	bandaru	PROPN
ejpam-5125	299	3	,	,	PUNCT
ejpam-5125	299	4	s.	s.	PROPN
ejpam-5125	299	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	299	6	/	/	PUNCT
ejpam-5125	299	7	eur	eur	PROPN
ejpam-5125	299	8	.	.	PUNCT
ejpam-5125	300	1	j.	j.	PROPN
ejpam-5125	300	2	pure	pure	PROPN
ejpam-5125	300	3	appl	appl	PROPN
ejpam-5125	300	4	.	.	PROPN
ejpam-5125	300	5	math	math	PROPN
ejpam-5125	300	6	,	,	PUNCT
ejpam-5125	300	7	17	17	NUM
ejpam-5125	300	8	(	(	PUNCT
ejpam-5125	300	9	2	2	NUM
ejpam-5125	300	10	)	)	PUNCT
ejpam-5125	300	11	(	(	PUNCT
ejpam-5125	300	12	2024	2024	NUM
ejpam-5125	300	13	)	)	PUNCT
ejpam-5125	300	14	,	,	PUNCT
ejpam-5125	300	15	819	819	NUM
ejpam-5125	300	16	-	-	SYM
ejpam-5125	300	17	834	834	NUM
ejpam-5125	300	18	827	827	NUM
ejpam-5125	300	19	proof	proof	NOUN
ejpam-5125	300	20	.	.	PUNCT
ejpam-5125	301	1	(	(	PUNCT
ejpam-5125	301	2	1	1	X
ejpam-5125	301	3	)	)	PUNCT
ejpam-5125	301	4	⇒	⇒	NOUN
ejpam-5125	301	5	(	(	PUNCT
ejpam-5125	301	6	2	2	NUM
ejpam-5125	301	7	):	):	PUNCT
ejpam-5125	301	8	let	let	VERB
ejpam-5125	301	9	(	(	PUNCT
ejpam-5125	301	10	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	301	11	,	,	PUNCT
ejpam-5125	301	12	1	1	NUM
ejpam-5125	301	13	)	)	PUNCT
ejpam-5125	301	14	be	be	AUX
ejpam-5125	301	15	a	a	DET
ejpam-5125	301	16	distributive	distributive	ADJ
ejpam-5125	301	17	lattice	lattice	NOUN
ejpam-5125	301	18	.	.	PUNCT
ejpam-5125	302	1	then	then	ADV
ejpam-5125	302	2	,	,	PUNCT
ejpam-5125	302	3	by	by	ADP
ejpam-5125	302	4	theorem	theorem	NOUN
ejpam-5125	302	5	2	2	NUM
ejpam-5125	302	6	,	,	PUNCT
ejpam-5125	302	7	v	v	NOUN
ejpam-5125	302	8	is	be	AUX
ejpam-5125	302	9	directed	direct	VERB
ejpam-5125	302	10	below	below	ADV
ejpam-5125	302	11	.	.	PUNCT
ejpam-5125	303	1	the	the	DET
ejpam-5125	303	2	equivalence	equivalence	NOUN
ejpam-5125	303	3	of	of	ADP
ejpam-5125	303	4	(	(	PUNCT
ejpam-5125	303	5	2	2	NUM
ejpam-5125	303	6	)	)	PUNCT
ejpam-5125	303	7	,	,	PUNCT
ejpam-5125	303	8	(	(	PUNCT
ejpam-5125	303	9	3	3	NUM
ejpam-5125	303	10	)	)	PUNCT
ejpam-5125	303	11	,	,	PUNCT
ejpam-5125	303	12	(	(	PUNCT
ejpam-5125	303	13	4	4	NUM
ejpam-5125	303	14	)	)	PUNCT
ejpam-5125	303	15	,	,	PUNCT
ejpam-5125	303	16	(	(	PUNCT
ejpam-5125	303	17	5	5	X
ejpam-5125	303	18	)	)	PUNCT
ejpam-5125	303	19	also	also	ADV
ejpam-5125	303	20	follows	follow	VERB
ejpam-5125	303	21	from	from	ADP
ejpam-5125	303	22	theorem	theorem	ADJ
ejpam-5125	303	23	2	2	NUM
ejpam-5125	303	24	.	.	PUNCT
ejpam-5125	303	25	(	(	PUNCT
ejpam-5125	303	26	5	5	X
ejpam-5125	303	27	)	)	PUNCT
ejpam-5125	303	28	⇒	⇒	NOUN
ejpam-5125	303	29	(	(	PUNCT
ejpam-5125	303	30	6	6	NUM
ejpam-5125	303	31	):	):	PUNCT
ejpam-5125	303	32	given	give	VERB
ejpam-5125	303	33	χ	χ	ADP
ejpam-5125	303	34	=	=	PRON
ejpam-5125	303	35	{	{	PUNCT
ejpam-5125	303	36	(	(	PUNCT
ejpam-5125	303	37	p1	p1	NOUN
ejpam-5125	303	38	,	,	PUNCT
ejpam-5125	303	39	p2	p2	X
ejpam-5125	303	40	)	)	PUNCT
ejpam-5125	303	41	∈	∈	NOUN
ejpam-5125	303	42	v	v	ADP
ejpam-5125	303	43	×	×	NOUN
ejpam-5125	303	44	v	v	NOUN
ejpam-5125	303	45	|	|	ADV
ejpam-5125	303	46	p2	p2	PROPN
ejpam-5125	303	47	∨	∨	NUM
ejpam-5125	303	48	p1	p1	NOUN
ejpam-5125	303	49	=	=	PUNCT
ejpam-5125	303	50	p2	p2	PROPN
ejpam-5125	303	51	}	}	PUNCT
ejpam-5125	303	52	.	.	PUNCT
ejpam-5125	304	1	if	if	SCONJ
ejpam-5125	304	2	(	(	PUNCT
ejpam-5125	304	3	p1	p1	NOUN
ejpam-5125	304	4	,	,	PUNCT
ejpam-5125	304	5	p2	p2	X
ejpam-5125	304	6	)	)	PUNCT
ejpam-5125	304	7	∈	∈	PROPN
ejpam-5125	304	8	χ	χ	NOUN
ejpam-5125	304	9	,	,	PUNCT
ejpam-5125	304	10	then	then	ADV
ejpam-5125	304	11	p2	p2	PROPN
ejpam-5125	304	12	∨	∨	NUM
ejpam-5125	304	13	p1	p1	PROPN
ejpam-5125	304	14	=	=	PROPN
ejpam-5125	304	15	p1	p1	PROPN
ejpam-5125	304	16	and	and	CCONJ
ejpam-5125	304	17	(	(	PUNCT
ejpam-5125	304	18	p2	p2	PROPN
ejpam-5125	304	19	,	,	PUNCT
ejpam-5125	304	20	p1	p1	NOUN
ejpam-5125	304	21	)	)	PUNCT
ejpam-5125	304	22	∈	∈	PROPN
ejpam-5125	304	23	χ	χ	NOUN
ejpam-5125	304	24	and	and	CCONJ
ejpam-5125	304	25	hence	hence	ADV
ejpam-5125	304	26	p1	p1	NOUN
ejpam-5125	304	27	=	=	SYM
ejpam-5125	304	28	p2	p2	PROPN
ejpam-5125	304	29	.	.	PUNCT
ejpam-5125	305	1	(	(	PUNCT
ejpam-5125	305	2	6	6	NUM
ejpam-5125	305	3	)	)	PUNCT
ejpam-5125	305	4	⇒	⇒	NOUN
ejpam-5125	305	5	(	(	PUNCT
ejpam-5125	305	6	1	1	NUM
ejpam-5125	305	7	):	):	PUNCT
ejpam-5125	305	8	let	let	VERB
ejpam-5125	305	9	p1	p1	NOUN
ejpam-5125	305	10	,	,	PUNCT
ejpam-5125	305	11	p2	p2	PROPN
ejpam-5125	305	12	∈	∈	PROPN
ejpam-5125	305	13	v	v	NOUN
ejpam-5125	305	14	.	.	PUNCT
ejpam-5125	306	1	then	then	ADV
ejpam-5125	306	2	(	(	PUNCT
ejpam-5125	306	3	p1∨p2)∨(p2∨p1	p1∨p2)∨(p2∨p1	ADJ
ejpam-5125	306	4	)	)	PUNCT
ejpam-5125	306	5	=	=	SYM
ejpam-5125	306	6	p1∨p2∨p1	p1∨p2∨p1	PUNCT
ejpam-5125	306	7	=	=	PUNCT
ejpam-5125	306	8	p1∨p1∨p2	p1∨p1∨p2	PROPN
ejpam-5125	306	9	=	=	SYM
ejpam-5125	306	10	p1∨p2	p1∨p2	X
ejpam-5125	306	11	and	and	CCONJ
ejpam-5125	306	12	hence	hence	ADV
ejpam-5125	306	13	(	(	PUNCT
ejpam-5125	306	14	p1∨p2	p1∨p2	ADP
ejpam-5125	306	15	,	,	PUNCT
ejpam-5125	306	16	p2∨p1	p2∨p1	NOUN
ejpam-5125	306	17	)	)	PUNCT
ejpam-5125	306	18	∈	∈	PROPN
ejpam-5125	306	19	χ	χ	NOUN
ejpam-5125	306	20	.	.	PUNCT
ejpam-5125	307	1	also	also	ADV
ejpam-5125	307	2	,	,	PUNCT
ejpam-5125	307	3	(	(	PUNCT
ejpam-5125	307	4	p2∨p1)∨	p2∨p1)∨	PROPN
ejpam-5125	307	5	(	(	PUNCT
ejpam-5125	307	6	p1∨p2	p1∨p2	NOUN
ejpam-5125	307	7	)	)	PUNCT
ejpam-5125	307	8	=	=	PUNCT
ejpam-5125	307	9	p2∨p1∨p2	p2∨p1∨p2	X
ejpam-5125	307	10	=	=	PUNCT
ejpam-5125	307	11	p2∨p2∨p1	p2∨p2∨p1	PROPN
ejpam-5125	307	12	=	=	SYM
ejpam-5125	307	13	p2∨p1	p2∨p1	NOUN
ejpam-5125	307	14	which	which	PRON
ejpam-5125	307	15	shows	show	VERB
ejpam-5125	307	16	(	(	PUNCT
ejpam-5125	307	17	p2	p2	PROPN
ejpam-5125	307	18	∨	∨	NUM
ejpam-5125	307	19	p1	p1	NOUN
ejpam-5125	307	20	,	,	PUNCT
ejpam-5125	307	21	p1	p1	NOUN
ejpam-5125	307	22	∨	∨	NUM
ejpam-5125	307	23	p2	p2	X
ejpam-5125	307	24	)	)	PUNCT
ejpam-5125	307	25	∈	∈	PROPN
ejpam-5125	307	26	χ	χ	NOUN
ejpam-5125	307	27	.	.	PUNCT
ejpam-5125	308	1	since	since	SCONJ
ejpam-5125	308	2	χ	χ	PROPN
ejpam-5125	308	3	is	be	AUX
ejpam-5125	308	4	antisymmetric	antisymmetric	ADJ
ejpam-5125	308	5	,	,	PUNCT
ejpam-5125	308	6	we	we	PRON
ejpam-5125	308	7	have	have	AUX
ejpam-5125	308	8	p1	p1	NOUN
ejpam-5125	308	9	∨	∨	NUM
ejpam-5125	308	10	p2	p2	X
ejpam-5125	308	11	=	=	SYM
ejpam-5125	308	12	p2	p2	PROPN
ejpam-5125	308	13	∨	∨	NUM
ejpam-5125	308	14	p1	p1	NOUN
ejpam-5125	308	15	.	.	PUNCT
ejpam-5125	309	1	so	so	ADV
ejpam-5125	309	2	that	that	PRON
ejpam-5125	309	3	v	v	NOUN
ejpam-5125	309	4	is	be	AUX
ejpam-5125	309	5	a	a	DET
ejpam-5125	309	6	lattice	lattice	NOUN
ejpam-5125	309	7	and	and	CCONJ
ejpam-5125	309	8	hence	hence	ADV
ejpam-5125	309	9	distributive	distributive	ADJ
ejpam-5125	309	10	.	.	PUNCT
ejpam-5125	310	1	theorem	theorem	VERB
ejpam-5125	310	2	6	6	NUM
ejpam-5125	310	3	.	.	PUNCT
ejpam-5125	311	1	an	an	DET
ejpam-5125	311	2	algebra	algebra	NOUN
ejpam-5125	311	3	(	(	PUNCT
ejpam-5125	311	4	v,∨,∧	v,∨,∧	NOUN
ejpam-5125	311	5	,	,	PUNCT
ejpam-5125	311	6	1	1	NUM
ejpam-5125	311	7	)	)	PUNCT
ejpam-5125	311	8	of	of	ADP
ejpam-5125	311	9	type	type	NOUN
ejpam-5125	311	10	(	(	PUNCT
ejpam-5125	311	11	2	2	NUM
ejpam-5125	311	12	,	,	PUNCT
ejpam-5125	311	13	2	2	NUM
ejpam-5125	311	14	,	,	PUNCT
ejpam-5125	311	15	0	0	NUM
ejpam-5125	311	16	)	)	PUNCT
ejpam-5125	311	17	is	be	AUX
ejpam-5125	311	18	a	a	DET
ejpam-5125	311	19	pdl	pdl	NOUN
ejpam-5125	311	20	if	if	SCONJ
ejpam-5125	312	1	and	and	CCONJ
ejpam-5125	312	2	only	only	ADV
ejpam-5125	312	3	if	if	SCONJ
ejpam-5125	312	4	it	it	PRON
ejpam-5125	312	5	satisfies	satisfy	VERB
ejpam-5125	312	6	the	the	DET
ejpam-5125	312	7	following	following	NOUN
ejpam-5125	312	8	:	:	PUNCT
ejpam-5125	312	9	(	(	PUNCT
ejpam-5125	312	10	ld∨	ld∨	NOUN
ejpam-5125	312	11	)	)	PUNCT
ejpam-5125	312	12	p1	p1	NOUN
ejpam-5125	312	13	∨	∨	NUM
ejpam-5125	312	14	(	(	PUNCT
ejpam-5125	312	15	p2	p2	PROPN
ejpam-5125	312	16	∧	∧	PROPN
ejpam-5125	312	17	p3	p3	PROPN
ejpam-5125	312	18	)	)	PUNCT
ejpam-5125	313	1	=	=	PUNCT
ejpam-5125	313	2	(	(	PUNCT
ejpam-5125	313	3	p1	p1	PROPN
ejpam-5125	313	4	∨	∨	NUM
ejpam-5125	313	5	p2	p2	NOUN
ejpam-5125	313	6	)	)	PUNCT
ejpam-5125	313	7	∧	∧	PROPN
ejpam-5125	313	8	(	(	PUNCT
ejpam-5125	313	9	p1	p1	PROPN
ejpam-5125	313	10	∨	∨	NUM
ejpam-5125	313	11	p3	p3	PROPN
ejpam-5125	313	12	)	)	PUNCT
ejpam-5125	313	13	(	(	PUNCT
ejpam-5125	313	14	rd∨	rd∨	X
ejpam-5125	313	15	)	)	PUNCT
ejpam-5125	313	16	(	(	PUNCT
ejpam-5125	313	17	p1	p1	NOUN
ejpam-5125	313	18	∧	∧	NOUN
ejpam-5125	313	19	p2	p2	NOUN
ejpam-5125	313	20	)	)	PUNCT
ejpam-5125	313	21	∨	∨	NUM
ejpam-5125	313	22	p3	p3	NOUN
ejpam-5125	313	23	=	=	SYM
ejpam-5125	313	24	(	(	PUNCT
ejpam-5125	313	25	p1	p1	PROPN
ejpam-5125	313	26	∨	∨	NUM
ejpam-5125	313	27	p3	p3	PROPN
ejpam-5125	313	28	)	)	PUNCT
ejpam-5125	314	1	∧	∧	PROPN
ejpam-5125	314	2	(	(	PUNCT
ejpam-5125	314	3	p2	p2	PROPN
ejpam-5125	314	4	∨	∨	NUM
ejpam-5125	314	5	p3	p3	PROPN
ejpam-5125	314	6	)	)	PUNCT
ejpam-5125	314	7	(	(	PUNCT
ejpam-5125	314	8	rd∧	rd∧	PROPN
ejpam-5125	314	9	)	)	PUNCT
ejpam-5125	314	10	(	(	PUNCT
ejpam-5125	314	11	p1	p1	PROPN
ejpam-5125	314	12	∨	∨	NUM
ejpam-5125	314	13	p2	p2	NOUN
ejpam-5125	314	14	)	)	PUNCT
ejpam-5125	314	15	∧	∧	PROPN
ejpam-5125	314	16	p3	p3	NOUN
ejpam-5125	314	17	=	=	SYM
ejpam-5125	314	18	(	(	PUNCT
ejpam-5125	314	19	p1	p1	PROPN
ejpam-5125	314	20	∧	∧	PROPN
ejpam-5125	314	21	p3	p3	PROPN
ejpam-5125	314	22	)	)	PUNCT
ejpam-5125	314	23	∨	∨	NUM
ejpam-5125	314	24	(	(	PUNCT
ejpam-5125	314	25	p2	p2	PROPN
ejpam-5125	314	26	∧	∧	PROPN
ejpam-5125	314	27	p3	p3	PROPN
ejpam-5125	314	28	)	)	PUNCT
ejpam-5125	314	29	(	(	PUNCT
ejpam-5125	314	30	l1	l1	PROPN
ejpam-5125	314	31	)	)	PUNCT
ejpam-5125	314	32	(	(	PUNCT
ejpam-5125	314	33	p1	p1	PROPN
ejpam-5125	314	34	∨	∨	NUM
ejpam-5125	314	35	p2	p2	NOUN
ejpam-5125	314	36	)	)	PUNCT
ejpam-5125	314	37	∧	∧	NOUN
ejpam-5125	314	38	p2	p2	NOUN
ejpam-5125	314	39	=	=	SYM
ejpam-5125	314	40	p2	p2	PROPN
ejpam-5125	314	41	(	(	PUNCT
ejpam-5125	314	42	l3	l3	NOUN
ejpam-5125	314	43	)	)	PUNCT
ejpam-5125	314	44	p1	p1	PROPN
ejpam-5125	314	45	∨	∨	NUM
ejpam-5125	314	46	(	(	PUNCT
ejpam-5125	314	47	p1	p1	NOUN
ejpam-5125	314	48	∧	∧	NOUN
ejpam-5125	314	49	p2	p2	NOUN
ejpam-5125	314	50	)	)	PUNCT
ejpam-5125	314	51	=	=	SYM
ejpam-5125	314	52	p1	p1	NOUN
ejpam-5125	314	53	(	(	PUNCT
ejpam-5125	314	54	i1	i1	PROPN
ejpam-5125	314	55	)	)	PUNCT
ejpam-5125	314	56	p1	p1	PROPN
ejpam-5125	314	57	∨	∨	NUM
ejpam-5125	314	58	1	1	NUM
ejpam-5125	314	59	=	=	SYM
ejpam-5125	314	60	1	1	NUM
ejpam-5125	314	61	(	(	PUNCT
ejpam-5125	314	62	i2	i2	PROPN
ejpam-5125	314	63	)	)	PUNCT
ejpam-5125	314	64	1	1	NUM
ejpam-5125	314	65	∧	∧	PROPN
ejpam-5125	314	66	p1	p1	NOUN
ejpam-5125	314	67	=	=	SYM
ejpam-5125	314	68	p1	p1	PROPN
ejpam-5125	314	69	.	.	PUNCT
ejpam-5125	315	1	for	for	ADP
ejpam-5125	315	2	all	all	DET
ejpam-5125	315	3	p1	p1	NOUN
ejpam-5125	315	4	,	,	PUNCT
ejpam-5125	315	5	p2	p2	NOUN
ejpam-5125	315	6	,	,	PUNCT
ejpam-5125	315	7	p3	p3	PROPN
ejpam-5125	315	8	∈	∈	PROPN
ejpam-5125	315	9	v	v	NOUN
ejpam-5125	315	10	.	.	PUNCT
ejpam-5125	316	1	corollary	corollary	ADJ
ejpam-5125	316	2	3	3	X
ejpam-5125	316	3	.	.	PUNCT
ejpam-5125	317	1	for	for	ADP
ejpam-5125	317	2	all	all	DET
ejpam-5125	317	3	a	a	DET
ejpam-5125	317	4	∈	∈	PROPN
ejpam-5125	317	5	v	v	NOUN
ejpam-5125	317	6	,	,	PUNCT
ejpam-5125	317	7	v	v	NOUN
ejpam-5125	317	8	contains	contain	VERB
ejpam-5125	317	9	an	an	DET
ejpam-5125	317	10	element	element	NOUN
ejpam-5125	317	11	0	0	NUM
ejpam-5125	317	12	such	such	ADJ
ejpam-5125	317	13	that	that	DET
ejpam-5125	317	14	0	0	NUM
ejpam-5125	317	15	∨	∨	NUM
ejpam-5125	317	16	a	a	DET
ejpam-5125	317	17	=	=	NOUN
ejpam-5125	317	18	a	a	PRON
ejpam-5125	317	19	if	if	NOUN
ejpam-5125	318	1	and	and	CCONJ
ejpam-5125	318	2	only	only	ADV
ejpam-5125	318	3	if	if	SCONJ
ejpam-5125	318	4	v	v	NOUN
ejpam-5125	318	5	is	be	AUX
ejpam-5125	318	6	a	a	DET
ejpam-5125	318	7	bounded	bounded	ADJ
ejpam-5125	318	8	distributive	distributive	ADJ
ejpam-5125	318	9	lattice	lattice	NOUN
ejpam-5125	318	10	,	,	PUNCT
ejpam-5125	318	11	and	and	CCONJ
ejpam-5125	318	12	hence	hence	ADV
ejpam-5125	318	13	for	for	ADP
ejpam-5125	318	14	any	any	DET
ejpam-5125	318	15	a	a	DET
ejpam-5125	318	16	∈	∈	PROPN
ejpam-5125	318	17	v	v	NOUN
ejpam-5125	318	18	,	,	PUNCT
ejpam-5125	318	19	the	the	DET
ejpam-5125	318	20	set	set	NOUN
ejpam-5125	318	21	va	va	NOUN
ejpam-5125	318	22	=	=	PUNCT
ejpam-5125	318	23	{	{	PUNCT
ejpam-5125	318	24	a	a	DET
ejpam-5125	318	25	≤	≤	PROPN
ejpam-5125	318	26	p1	p1	NOUN
ejpam-5125	318	27	|	|	NOUN
ejpam-5125	318	28	p1	p1	PROPN
ejpam-5125	318	29	∈	∈	PROPN
ejpam-5125	318	30	v	v	NOUN
ejpam-5125	318	31	}	}	PUNCT
ejpam-5125	318	32	is	be	AUX
ejpam-5125	318	33	a	a	DET
ejpam-5125	318	34	bounded	bounded	ADJ
ejpam-5125	318	35	distributive	distributive	ADJ
ejpam-5125	318	36	lattice	lattice	NOUN
ejpam-5125	318	37	under	under	ADP
ejpam-5125	318	38	the	the	DET
ejpam-5125	318	39	induced	induced	ADJ
ejpam-5125	318	40	operations	operation	NOUN
ejpam-5125	318	41	∨	∨	NOUN
ejpam-5125	318	42	and	and	CCONJ
ejpam-5125	318	43	∧	∧	NOUN
ejpam-5125	318	44	with	with	ADP
ejpam-5125	318	45	a	a	PRON
ejpam-5125	318	46	as	as	ADP
ejpam-5125	318	47	its	its	PRON
ejpam-5125	318	48	least	least	ADJ
ejpam-5125	318	49	element	element	NOUN
ejpam-5125	318	50	.	.	PUNCT
ejpam-5125	319	1	3	3	X
ejpam-5125	319	2	.	.	X
ejpam-5125	319	3	ideals	ideal	NOUN
ejpam-5125	319	4	and	and	CCONJ
ejpam-5125	319	5	filters	filter	NOUN
ejpam-5125	319	6	in	in	ADP
ejpam-5125	319	7	this	this	DET
ejpam-5125	319	8	section	section	NOUN
ejpam-5125	319	9	,	,	PUNCT
ejpam-5125	319	10	we	we	PRON
ejpam-5125	319	11	introduce	introduce	VERB
ejpam-5125	319	12	the	the	DET
ejpam-5125	319	13	notion	notion	NOUN
ejpam-5125	319	14	of	of	ADP
ejpam-5125	319	15	an	an	DET
ejpam-5125	319	16	ideal	ideal	NOUN
ejpam-5125	319	17	and	and	CCONJ
ejpam-5125	319	18	a	a	DET
ejpam-5125	319	19	filter	filter	NOUN
ejpam-5125	319	20	in	in	ADP
ejpam-5125	319	21	a	a	DET
ejpam-5125	319	22	paradistributive	paradistributive	ADJ
ejpam-5125	319	23	latticoid	latticoid	NOUN
ejpam-5125	319	24	and	and	CCONJ
ejpam-5125	319	25	investigate	investigate	VERB
ejpam-5125	319	26	its	its	PRON
ejpam-5125	319	27	important	important	ADJ
ejpam-5125	319	28	properties	property	NOUN
ejpam-5125	319	29	.	.	PUNCT
ejpam-5125	320	1	definition	definition	NOUN
ejpam-5125	320	2	4	4	NUM
ejpam-5125	320	3	.	.	PUNCT
ejpam-5125	321	1	a	a	DET
ejpam-5125	321	2	non	non	ADJ
ejpam-5125	321	3	-	-	ADJ
ejpam-5125	321	4	empty	empty	ADJ
ejpam-5125	321	5	subset	subset	ADJ
ejpam-5125	321	6	u	u	NOUN
ejpam-5125	321	7	of	of	ADP
ejpam-5125	321	8	v	v	NOUN
ejpam-5125	321	9	is	be	AUX
ejpam-5125	321	10	said	say	VERB
ejpam-5125	321	11	to	to	PART
ejpam-5125	321	12	be	be	AUX
ejpam-5125	321	13	an	an	DET
ejpam-5125	321	14	ideal	ideal	NOUN
ejpam-5125	321	15	if	if	SCONJ
ejpam-5125	321	16	it	it	PRON
ejpam-5125	321	17	satisfies	satisfy	VERB
ejpam-5125	321	18	the	the	DET
ejpam-5125	321	19	following	following	NOUN
ejpam-5125	321	20	:	:	PUNCT
ejpam-5125	321	21	p1	p1	NOUN
ejpam-5125	321	22	,	,	PUNCT
ejpam-5125	321	23	p2	p2	PROPN
ejpam-5125	321	24	∈	∈	PROPN
ejpam-5125	321	25	u	u	PROPN
ejpam-5125	321	26	⇒	⇒	PROPN
ejpam-5125	321	27	p1	p1	PROPN
ejpam-5125	321	28	∨	∨	NUM
ejpam-5125	321	29	p2	p2	PROPN
ejpam-5125	321	30	∈	∈	PROPN
ejpam-5125	321	31	u.	u.	PROPN
ejpam-5125	321	32	p1	p1	PROPN
ejpam-5125	321	33	∈	∈	PROPN
ejpam-5125	321	34	u	u	PROPN
ejpam-5125	321	35	,	,	PUNCT
ejpam-5125	321	36	a	a	DET
ejpam-5125	321	37	∈	∈	NOUN
ejpam-5125	321	38	v	v	ADP
ejpam-5125	321	39	⇒	⇒	NOUN
ejpam-5125	321	40	p1	p1	PROPN
ejpam-5125	321	41	∧	∧	PROPN
ejpam-5125	321	42	a	a	DET
ejpam-5125	321	43	∈	∈	PROPN
ejpam-5125	321	44	u.	u.	NOUN
ejpam-5125	322	1	it	it	PRON
ejpam-5125	322	2	should	should	AUX
ejpam-5125	322	3	be	be	AUX
ejpam-5125	322	4	noted	note	VERB
ejpam-5125	322	5	that	that	SCONJ
ejpam-5125	322	6	every	every	DET
ejpam-5125	322	7	ideal	ideal	NOUN
ejpam-5125	322	8	of	of	ADP
ejpam-5125	322	9	v	v	NOUN
ejpam-5125	322	10	is	be	AUX
ejpam-5125	322	11	a	a	DET
ejpam-5125	322	12	pdl	pdl	NOUN
ejpam-5125	322	13	.	.	PUNCT
ejpam-5125	323	1	next	next	ADJ
ejpam-5125	323	2	theorem	theorem	NOUN
ejpam-5125	323	3	describes	describe	VERB
ejpam-5125	323	4	the	the	DET
ejpam-5125	323	5	ideal	ideal	NOUN
ejpam-5125	323	6	generated	generate	VERB
ejpam-5125	323	7	by	by	ADP
ejpam-5125	323	8	a	a	DET
ejpam-5125	323	9	non	non	ADJ
ejpam-5125	323	10	-	-	ADJ
ejpam-5125	323	11	empty	empty	ADJ
ejpam-5125	323	12	subset	subset	NOUN
ejpam-5125	323	13	s	s	NOUN
ejpam-5125	323	14	of	of	ADP
ejpam-5125	323	15	v	v	NUM
ejpam-5125	323	16	.	.	PUNCT
ejpam-5125	324	1	theorem	theorem	ADJ
ejpam-5125	324	2	7	7	NUM
ejpam-5125	324	3	.	.	PUNCT
ejpam-5125	325	1	let	let	VERB
ejpam-5125	325	2	s	s	PRON
ejpam-5125	325	3	be	be	AUX
ejpam-5125	325	4	a	a	DET
ejpam-5125	325	5	non	non	ADJ
ejpam-5125	325	6	-	-	ADJ
ejpam-5125	325	7	empty	empty	ADJ
ejpam-5125	325	8	subset	subset	NOUN
ejpam-5125	325	9	of	of	ADP
ejpam-5125	325	10	v	v	NOUN
ejpam-5125	325	11	.	.	PUNCT
ejpam-5125	326	1	then	then	ADV
ejpam-5125	326	2	(	(	PUNCT
ejpam-5125	326	3	s	s	X
ejpam-5125	326	4	]	]	X
ejpam-5125	326	5	=	=	X
ejpam-5125	326	6	{	{	PUNCT
ejpam-5125	326	7	(	(	PUNCT
ejpam-5125	326	8	n	n	NUM
ejpam-5125	326	9	∨	∨	NUM
ejpam-5125	326	10	i=1	i=1	PROPN
ejpam-5125	326	11	αi	αi	PROPN
ejpam-5125	326	12	)	)	PUNCT
ejpam-5125	327	1	∧	∧	PROPN
ejpam-5125	327	2	s	s	PART
ejpam-5125	327	3	|	|	ADV
ejpam-5125	327	4	αi	αi	VERB
ejpam-5125	327	5	∈	∈	PROPN
ejpam-5125	327	6	s	s	X
ejpam-5125	327	7	,	,	PUNCT
ejpam-5125	327	8	s	s	NOUN
ejpam-5125	327	9	∈	∈	PROPN
ejpam-5125	327	10	v	v	NOUN
ejpam-5125	327	11	and	and	CCONJ
ejpam-5125	327	12	n	n	PRON
ejpam-5125	327	13	is	be	AUX
ejpam-5125	327	14	a	a	DET
ejpam-5125	327	15	positive	positive	ADJ
ejpam-5125	327	16	integer	integer	NOUN
ejpam-5125	327	17	}	}	PUNCT
ejpam-5125	327	18	is	be	AUX
ejpam-5125	327	19	the	the	DET
ejpam-5125	327	20	smallest	small	ADJ
ejpam-5125	327	21	ideal	ideal	NOUN
ejpam-5125	327	22	of	of	ADP
ejpam-5125	327	23	v	v	NOUN
ejpam-5125	327	24	containing	contain	VERB
ejpam-5125	327	25	s.	s.	PROPN
ejpam-5125	327	26	r.	r.	PROPN
ejpam-5125	327	27	bandaru	bandaru	PROPN
ejpam-5125	327	28	,	,	PUNCT
ejpam-5125	327	29	s.	s.	PROPN
ejpam-5125	327	30	ajjarapu	ajjarapu	PROPN
ejpam-5125	327	31	/	/	PUNCT
ejpam-5125	327	32	eur	eur	PROPN
ejpam-5125	327	33	.	.	PUNCT
ejpam-5125	328	1	j.	j.	PROPN
ejpam-5125	328	2	pure	pure	PROPN
ejpam-5125	328	3	appl	appl	PROPN
ejpam-5125	328	4	.	.	PROPN
ejpam-5125	328	5	math	math	PROPN
ejpam-5125	328	6	,	,	PUNCT
ejpam-5125	328	7	17	17	NUM
ejpam-5125	328	8	(	(	PUNCT
ejpam-5125	328	9	2	2	NUM
ejpam-5125	328	10	)	)	PUNCT
ejpam-5125	328	11	(	(	PUNCT
ejpam-5125	328	12	2024	2024	NUM
ejpam-5125	328	13	)	)	PUNCT
ejpam-5125	328	14	,	,	PUNCT
ejpam-5125	328	15	819	819	NUM
ejpam-5125	328	16	-	-	SYM
ejpam-5125	328	17	834	834	NUM
ejpam-5125	328	18	828	828	NUM
ejpam-5125	328	19	proof	proof	NOUN
ejpam-5125	328	20	.	.	PUNCT
ejpam-5125	329	1	let	let	VERB
ejpam-5125	329	2	s	s	PRON
ejpam-5125	329	3	be	be	AUX
ejpam-5125	329	4	a	a	DET
ejpam-5125	329	5	non	non	ADJ
ejpam-5125	329	6	-	-	ADJ
ejpam-5125	329	7	empty	empty	ADJ
ejpam-5125	329	8	subset	subset	NOUN
ejpam-5125	329	9	of	of	ADP
ejpam-5125	329	10	v	v	NOUN
ejpam-5125	329	11	.	.	PUNCT
ejpam-5125	330	1	choose	choose	VERB
ejpam-5125	330	2	t	t	PROPN
ejpam-5125	330	3	=	=	PUNCT
ejpam-5125	330	4	{	{	PUNCT
ejpam-5125	330	5	n	n	PRON
ejpam-5125	330	6	∨	∨	NUM
ejpam-5125	331	1	i=1	i=1	PROPN
ejpam-5125	331	2	αi	αi	VERB
ejpam-5125	331	3	|	|	ADV
ejpam-5125	331	4	αi	αi	VERB
ejpam-5125	331	5	∈	∈	PROPN
ejpam-5125	332	1	s	s	X
ejpam-5125	332	2	for	for	ADP
ejpam-5125	332	3	1	1	NUM
ejpam-5125	332	4	≤	≤	NUM
ejpam-5125	332	5	i	i	PRON
ejpam-5125	332	6	≤	≤	ADJ
ejpam-5125	333	1	n	n	CCONJ
ejpam-5125	333	2	and	and	CCONJ
ejpam-5125	333	3	n	n	PROPN
ejpam-5125	333	4	is	be	AUX
ejpam-5125	333	5	a	a	DET
ejpam-5125	333	6	positive	positive	ADJ
ejpam-5125	333	7	integer	integer	NOUN
ejpam-5125	333	8	}	}	PUNCT
ejpam-5125	333	9	clearly	clearly	ADV
ejpam-5125	333	10	s	s	VERB
ejpam-5125	333	11	⊆	⊆	NUM
ejpam-5125	333	12	t	t	NOUN
ejpam-5125	333	13	⊆	⊆	NUM
ejpam-5125	333	14	(	(	PUNCT
ejpam-5125	333	15	s	s	X
ejpam-5125	333	16	]	]	X
ejpam-5125	333	17	.	.	PUNCT
ejpam-5125	334	1	first	first	ADV
ejpam-5125	334	2	we	we	PRON
ejpam-5125	334	3	prove	prove	VERB
ejpam-5125	334	4	that	that	SCONJ
ejpam-5125	334	5	(	(	PUNCT
ejpam-5125	334	6	s	s	X
ejpam-5125	334	7	]	]	X
ejpam-5125	334	8	=	=	SYM
ejpam-5125	334	9	{	{	PUNCT
ejpam-5125	334	10	p1	p1	PROPN
ejpam-5125	334	11	∈	∈	PROPN
ejpam-5125	334	12	v	v	ADP
ejpam-5125	334	13	|	|	NOUN
ejpam-5125	334	14	t	t	PROPN
ejpam-5125	334	15	∨	∨	NOUN
ejpam-5125	334	16	p1	p1	PROPN
ejpam-5125	334	17	=	=	SYM
ejpam-5125	334	18	t	t	PROPN
ejpam-5125	334	19	for	for	ADP
ejpam-5125	334	20	some	some	DET
ejpam-5125	334	21	t	t	NOUN
ejpam-5125	334	22	∈	∈	PROPN
ejpam-5125	334	23	t	t	PROPN
ejpam-5125	334	24	}	}	PUNCT
ejpam-5125	334	25	=	=	NOUN
ejpam-5125	334	26	m.	m.	NOUN
ejpam-5125	334	27	let	let	VERB
ejpam-5125	334	28	x	x	X
ejpam-5125	334	29	∈	∈	PROPN
ejpam-5125	334	30	(	(	PUNCT
ejpam-5125	334	31	s	s	NOUN
ejpam-5125	334	32	]	]	X
ejpam-5125	334	33	.	.	PUNCT
ejpam-5125	335	1	then	then	ADV
ejpam-5125	335	2	x	x	X
ejpam-5125	335	3	=	=	PRON
ejpam-5125	335	4	(	(	PUNCT
ejpam-5125	335	5	n	n	CCONJ
ejpam-5125	335	6	∨	∨	NUM
ejpam-5125	335	7	i=1	i=1	PROPN
ejpam-5125	335	8	αi	αi	PROPN
ejpam-5125	335	9	)	)	PUNCT
ejpam-5125	335	10	∧	∧	PROPN
ejpam-5125	335	11	s	s	VERB
ejpam-5125	335	12	where	where	SCONJ
ejpam-5125	335	13	αi	αi	PRON
ejpam-5125	335	14	∈	∈	PROPN
ejpam-5125	335	15	s	s	PROPN
ejpam-5125	335	16	,	,	PUNCT
ejpam-5125	335	17	s	s	NOUN
ejpam-5125	335	18	∈	∈	PROPN
ejpam-5125	335	19	v	v	NOUN
ejpam-5125	335	20	.	.	PUNCT
ejpam-5125	336	1	now	now	ADV
ejpam-5125	336	2	(	(	PUNCT
ejpam-5125	336	3	n	n	NUM
ejpam-5125	336	4	∨	∨	NUM
ejpam-5125	336	5	i=1	i=1	PROPN
ejpam-5125	336	6	αi	αi	PROPN
ejpam-5125	336	7	)	)	PUNCT
ejpam-5125	336	8	∨	∨	NOUN
ejpam-5125	336	9	x	x	X
ejpam-5125	336	10	=	=	SYM
ejpam-5125	336	11	(	(	PUNCT
ejpam-5125	336	12	n	n	CCONJ
ejpam-5125	336	13	∨	∨	NUM
ejpam-5125	336	14	i=1	i=1	PROPN
ejpam-5125	336	15	αi	αi	PROPN
ejpam-5125	336	16	)	)	PUNCT
ejpam-5125	336	17	∨	∨	PROPN
ejpam-5125	336	18	(	(	PUNCT
ejpam-5125	336	19	(	(	PUNCT
ejpam-5125	336	20	n	n	CCONJ
ejpam-5125	336	21	∨	∨	NUM
ejpam-5125	336	22	i=1	i=1	PROPN
ejpam-5125	336	23	αi	αi	PROPN
ejpam-5125	336	24	)	)	PUNCT
ejpam-5125	337	1	∧	∧	PROPN
ejpam-5125	337	2	s	s	PART
ejpam-5125	337	3	)	)	PUNCT
ejpam-5125	337	4	.	.	PUNCT
ejpam-5125	338	1	=	=	PUNCT
ejpam-5125	338	2	n	n	NUM
ejpam-5125	338	3	∨	∨	NOUN
ejpam-5125	338	4	i=1	i=1	PROPN
ejpam-5125	338	5	αi	αi	X
ejpam-5125	338	6	which	which	PRON
ejpam-5125	338	7	implies	imply	VERB
ejpam-5125	338	8	x	x	X
ejpam-5125	338	9	∈	∈	PROPN
ejpam-5125	338	10	m	m	VERB
ejpam-5125	338	11	.	.	PUNCT
ejpam-5125	339	1	therefore	therefore	ADV
ejpam-5125	339	2	(	(	PUNCT
ejpam-5125	339	3	s	s	X
ejpam-5125	339	4	]	]	X
ejpam-5125	339	5	⊆	⊆	NUM
ejpam-5125	339	6	m	m	NOUN
ejpam-5125	339	7	.	.	PUNCT
ejpam-5125	340	1	conversely	conversely	ADV
ejpam-5125	340	2	,	,	PUNCT
ejpam-5125	340	3	let	let	VERB
ejpam-5125	340	4	s	s	PRON
ejpam-5125	340	5	∈	∈	VERB
ejpam-5125	340	6	m	m	NOUN
ejpam-5125	340	7	.	.	PUNCT
ejpam-5125	341	1	then	then	ADV
ejpam-5125	341	2	t	t	PROPN
ejpam-5125	341	3	∨	∨	PROPN
ejpam-5125	341	4	s	s	PART
ejpam-5125	341	5	=	=	X
ejpam-5125	341	6	t	t	PROPN
ejpam-5125	341	7	for	for	ADP
ejpam-5125	341	8	some	some	DET
ejpam-5125	341	9	t	t	NOUN
ejpam-5125	341	10	=	=	PUNCT
ejpam-5125	341	11	n	n	NUM
ejpam-5125	341	12	∨	∨	NOUN
ejpam-5125	341	13	i=1	i=1	PROPN
ejpam-5125	341	14	αi	αi	PROPN
ejpam-5125	341	15	,	,	PUNCT
ejpam-5125	341	16	αi	αi	PROPN
ejpam-5125	341	17	∈	∈	PROPN
ejpam-5125	341	18	s.	s.	PROPN
ejpam-5125	341	19	now	now	ADV
ejpam-5125	341	20	s	s	VERB
ejpam-5125	341	21	=	=	PUNCT
ejpam-5125	341	22	t	t	PROPN
ejpam-5125	341	23	∧	∧	PROPN
ejpam-5125	341	24	s	s	PART
ejpam-5125	341	25	=	=	PUNCT
ejpam-5125	341	26	(	(	PUNCT
ejpam-5125	341	27	n	n	CCONJ
ejpam-5125	341	28	∨	∨	NUM
ejpam-5125	341	29	i=1	i=1	PROPN
ejpam-5125	341	30	αi	αi	PROPN
ejpam-5125	341	31	)	)	PUNCT
ejpam-5125	341	32	∧	∧	PROPN
ejpam-5125	341	33	s	s	PART
ejpam-5125	341	34	∈	∈	PROPN
ejpam-5125	341	35	(	(	PUNCT
ejpam-5125	341	36	s	s	NOUN
ejpam-5125	341	37	]	]	X
ejpam-5125	341	38	.	.	PUNCT
ejpam-5125	342	1	therefore	therefore	ADV
ejpam-5125	342	2	m	m	VERB
ejpam-5125	342	3	⊆	⊆	NUM
ejpam-5125	342	4	(	(	PUNCT
ejpam-5125	342	5	s	s	X
ejpam-5125	342	6	]	]	X
ejpam-5125	342	7	.	.	PUNCT
ejpam-5125	343	1	hence	hence	ADV
ejpam-5125	343	2	(	(	PUNCT
ejpam-5125	343	3	s	s	X
ejpam-5125	343	4	]	]	X
ejpam-5125	343	5	=	=	PUNCT
ejpam-5125	343	6	{	{	PUNCT
ejpam-5125	343	7	s	s	NOUN
ejpam-5125	343	8	∈	∈	NOUN
ejpam-5125	343	9	v	v	ADP
ejpam-5125	343	10	|	|	ADV
ejpam-5125	343	11	t	t	PROPN
ejpam-5125	343	12	∨	∨	NUM
ejpam-5125	343	13	s	s	PART
ejpam-5125	343	14	=	=	X
ejpam-5125	343	15	t	t	PROPN
ejpam-5125	343	16	for	for	ADP
ejpam-5125	343	17	some	some	DET
ejpam-5125	343	18	t	t	NOUN
ejpam-5125	343	19	∈	∈	PROPN
ejpam-5125	343	20	t	t	PROPN
ejpam-5125	343	21	}	}	PUNCT
ejpam-5125	343	22	=	=	NOUN
ejpam-5125	343	23	m.	m.	NOUN
ejpam-5125	343	24	let	let	VERB
ejpam-5125	343	25	s	s	NOUN
ejpam-5125	343	26	,	,	PUNCT
ejpam-5125	343	27	l	l	PROPN
ejpam-5125	343	28	∈	∈	PROPN
ejpam-5125	343	29	(	(	PUNCT
ejpam-5125	343	30	s	s	X
ejpam-5125	343	31	]	]	X
ejpam-5125	343	32	,	,	PUNCT
ejpam-5125	343	33	then	then	ADV
ejpam-5125	343	34	there	there	PRON
ejpam-5125	343	35	exists	exist	VERB
ejpam-5125	343	36	t1	t1	NOUN
ejpam-5125	343	37	,	,	PUNCT
ejpam-5125	343	38	t2	t2	PROPN
ejpam-5125	343	39	∈	∈	PROPN
ejpam-5125	343	40	t	t	NOUN
ejpam-5125	343	41	such	such	ADJ
ejpam-5125	343	42	that	that	SCONJ
ejpam-5125	343	43	t1	t1	PROPN
ejpam-5125	343	44	∨	∨	NUM
ejpam-5125	343	45	s	s	PART
ejpam-5125	343	46	=	=	X
ejpam-5125	343	47	t1	t1	NOUN
ejpam-5125	343	48	and	and	CCONJ
ejpam-5125	343	49	t2	t2	NOUN
ejpam-5125	343	50	∨	∨	NUM
ejpam-5125	343	51	l	l	NOUN
ejpam-5125	343	52	=	=	SYM
ejpam-5125	343	53	t2	t2	NOUN
ejpam-5125	343	54	.	.	PUNCT
ejpam-5125	344	1	then	then	ADV
ejpam-5125	344	2	(	(	PUNCT
ejpam-5125	344	3	s	s	X
ejpam-5125	344	4	]	]	X
ejpam-5125	344	5	is	be	AUX
ejpam-5125	344	6	an	an	DET
ejpam-5125	344	7	ideal	ideal	NOUN
ejpam-5125	344	8	as	as	ADP
ejpam-5125	344	9	(	(	PUNCT
ejpam-5125	344	10	t1	t1	PROPN
ejpam-5125	344	11	∨	∨	NUM
ejpam-5125	344	12	t2	t2	PROPN
ejpam-5125	344	13	)	)	PUNCT
ejpam-5125	344	14	∨	∨	NUM
ejpam-5125	344	15	(	(	PUNCT
ejpam-5125	344	16	s	s	NOUN
ejpam-5125	344	17	∨	∨	NUM
ejpam-5125	344	18	l	l	NOUN
ejpam-5125	344	19	)	)	PUNCT
ejpam-5125	344	20	=	=	SYM
ejpam-5125	344	21	t1	t1	PROPN
ejpam-5125	344	22	∨	∨	NOUN
ejpam-5125	344	23	(	(	PUNCT
ejpam-5125	344	24	t2	t2	PROPN
ejpam-5125	344	25	∨	∨	NUM
ejpam-5125	344	26	(	(	PUNCT
ejpam-5125	344	27	s	s	NOUN
ejpam-5125	344	28	∨	∨	NUM
ejpam-5125	344	29	l	l	NOUN
ejpam-5125	344	30	)	)	PUNCT
ejpam-5125	344	31	)	)	PUNCT
ejpam-5125	345	1	=	=	SYM
ejpam-5125	345	2	t1	t1	NOUN
ejpam-5125	345	3	∨	∨	NOUN
ejpam-5125	345	4	(	(	PUNCT
ejpam-5125	345	5	t2	t2	PROPN
ejpam-5125	345	6	∨	∨	PROPN
ejpam-5125	345	7	l	l	PROPN
ejpam-5125	345	8	∨	∨	NUM
ejpam-5125	345	9	s	s	PART
ejpam-5125	345	10	)	)	PUNCT
ejpam-5125	345	11	=	=	PUNCT
ejpam-5125	346	1	t1	t1	PROPN
ejpam-5125	346	2	∨	∨	NUM
ejpam-5125	346	3	t2	t2	PROPN
ejpam-5125	346	4	∨	∨	NUM
ejpam-5125	346	5	s	s	PART
ejpam-5125	346	6	=	=	X
ejpam-5125	346	7	t1	t1	PROPN
ejpam-5125	346	8	∨	∨	PROPN
ejpam-5125	346	9	s	s	PART
ejpam-5125	346	10	∨	∨	NOUN
ejpam-5125	346	11	t2	t2	NOUN
ejpam-5125	346	12	=	=	PROPN
ejpam-5125	346	13	t1	t1	PROPN
ejpam-5125	346	14	∨	∨	NUM
ejpam-5125	346	15	t2	t2	PROPN
ejpam-5125	346	16	therefore	therefore	ADV
ejpam-5125	346	17	s∨	s∨	VERB
ejpam-5125	346	18	l	l	PROPN
ejpam-5125	346	19	∈	∈	PROPN
ejpam-5125	346	20	(	(	PUNCT
ejpam-5125	346	21	s	s	NOUN
ejpam-5125	346	22	]	]	X
ejpam-5125	346	23	.	.	PUNCT
ejpam-5125	347	1	also	also	ADV
ejpam-5125	347	2	,	,	PUNCT
ejpam-5125	347	3	for	for	ADP
ejpam-5125	347	4	s	s	PROPN
ejpam-5125	347	5	∈	∈	PROPN
ejpam-5125	347	6	(	(	PUNCT
ejpam-5125	347	7	s	s	X
ejpam-5125	347	8	]	]	X
ejpam-5125	347	9	and	and	CCONJ
ejpam-5125	347	10	u	u	PROPN
ejpam-5125	347	11	∈	∈	PROPN
ejpam-5125	347	12	v	v	NOUN
ejpam-5125	347	13	,	,	PUNCT
ejpam-5125	347	14	we	we	PRON
ejpam-5125	347	15	have	have	VERB
ejpam-5125	347	16	t	t	PROPN
ejpam-5125	347	17	∈	∈	PROPN
ejpam-5125	347	18	t	t	NOUN
ejpam-5125	347	19	such	such	ADJ
ejpam-5125	347	20	that	that	SCONJ
ejpam-5125	347	21	t∨	t∨	PROPN
ejpam-5125	347	22	s	s	X
ejpam-5125	347	23	=	=	PUNCT
ejpam-5125	347	24	t.	t.	NOUN
ejpam-5125	347	25	now	now	ADV
ejpam-5125	347	26	t	t	PROPN
ejpam-5125	347	27	∨	∨	NUM
ejpam-5125	347	28	(	(	PUNCT
ejpam-5125	347	29	s	s	VERB
ejpam-5125	347	30	∧	∧	PROPN
ejpam-5125	347	31	u	u	NOUN
ejpam-5125	347	32	)	)	PUNCT
ejpam-5125	347	33	=	=	SYM
ejpam-5125	348	1	(	(	PUNCT
ejpam-5125	348	2	t	t	PROPN
ejpam-5125	348	3	∨	∨	NUM
ejpam-5125	348	4	s	s	PROPN
ejpam-5125	348	5	)	)	PUNCT
ejpam-5125	348	6	∧	∧	PROPN
ejpam-5125	348	7	(	(	PUNCT
ejpam-5125	348	8	t	t	PROPN
ejpam-5125	348	9	∨	∨	NUM
ejpam-5125	348	10	u	u	NOUN
ejpam-5125	348	11	)	)	PUNCT
ejpam-5125	348	12	=	=	SYM
ejpam-5125	348	13	t	t	PROPN
ejpam-5125	348	14	∧	∧	PROPN
ejpam-5125	348	15	(	(	PUNCT
ejpam-5125	348	16	t	t	PROPN
ejpam-5125	348	17	∨	∨	NUM
ejpam-5125	348	18	u	u	NOUN
ejpam-5125	348	19	)	)	PUNCT
ejpam-5125	348	20	=	=	PUNCT
ejpam-5125	349	1	t.	t.	NOUN
ejpam-5125	349	2	hence	hence	ADV
ejpam-5125	349	3	s	s	PART
ejpam-5125	349	4	∧	∧	PROPN
ejpam-5125	349	5	u	u	X
ejpam-5125	349	6	∈	∈	PROPN
ejpam-5125	349	7	(	(	PUNCT
ejpam-5125	349	8	s	s	NOUN
ejpam-5125	349	9	]	]	X
ejpam-5125	349	10	.	.	PUNCT
ejpam-5125	350	1	thus	thus	ADV
ejpam-5125	350	2	(	(	PUNCT
ejpam-5125	350	3	s	s	X
ejpam-5125	350	4	]	]	X
ejpam-5125	350	5	is	be	AUX
ejpam-5125	350	6	an	an	DET
ejpam-5125	350	7	ideal	ideal	NOUN
ejpam-5125	350	8	of	of	ADP
ejpam-5125	350	9	v	v	NOUN
ejpam-5125	350	10	containing	contain	VERB
ejpam-5125	350	11	s.	s.	PROPN
ejpam-5125	350	12	now	now	ADV
ejpam-5125	350	13	,	,	PUNCT
ejpam-5125	350	14	let	let	VERB
ejpam-5125	350	15	u	u	PRON
ejpam-5125	350	16	be	be	AUX
ejpam-5125	350	17	any	any	DET
ejpam-5125	350	18	ideal	ideal	NOUN
ejpam-5125	350	19	of	of	ADP
ejpam-5125	350	20	v	v	NOUN
ejpam-5125	350	21	such	such	ADJ
ejpam-5125	350	22	that	that	PRON
ejpam-5125	350	23	s	s	VERB
ejpam-5125	350	24	⊆	⊆	NUM
ejpam-5125	350	25	u	u	NOUN
ejpam-5125	350	26	.	.	PUNCT
ejpam-5125	351	1	let	let	VERB
ejpam-5125	351	2	s	s	PRON
ejpam-5125	351	3	∈	∈	PROPN
ejpam-5125	351	4	(	(	PUNCT
ejpam-5125	351	5	s	s	NOUN
ejpam-5125	351	6	]	]	X
ejpam-5125	351	7	.	.	PUNCT
ejpam-5125	352	1	then	then	ADV
ejpam-5125	352	2	s	s	VERB
ejpam-5125	352	3	=	=	PUNCT
ejpam-5125	352	4	(	(	PUNCT
ejpam-5125	352	5	n	n	CCONJ
ejpam-5125	352	6	∨	∨	NUM
ejpam-5125	352	7	i=1	i=1	PROPN
ejpam-5125	352	8	αi	αi	PROPN
ejpam-5125	352	9	)	)	PUNCT
ejpam-5125	352	10	∧	∧	NOUN
ejpam-5125	352	11	p2	p2	NOUN
ejpam-5125	352	12	where	where	SCONJ
ejpam-5125	352	13	αi	αi	PRON
ejpam-5125	352	14	∈	∈	PROPN
ejpam-5125	352	15	s	s	PART
ejpam-5125	352	16	⊆	⊆	NUM
ejpam-5125	352	17	u	u	NOUN
ejpam-5125	352	18	for	for	ADP
ejpam-5125	352	19	1	1	NUM
ejpam-5125	352	20	≤	≤	NUM
ejpam-5125	352	21	i	i	PRON
ejpam-5125	352	22	≤	≤	ADJ
ejpam-5125	352	23	n	n	CCONJ
ejpam-5125	352	24	and	and	CCONJ
ejpam-5125	352	25	p2	p2	PROPN
ejpam-5125	352	26	∈	∈	PROPN
ejpam-5125	352	27	v	v	NOUN
ejpam-5125	352	28	.	.	PUNCT
ejpam-5125	353	1	since	since	SCONJ
ejpam-5125	353	2	u	u	NOUN
ejpam-5125	353	3	is	be	AUX
ejpam-5125	353	4	an	an	DET
ejpam-5125	353	5	ideal	ideal	NOUN
ejpam-5125	353	6	of	of	ADP
ejpam-5125	353	7	v	v	NUM
ejpam-5125	353	8	,	,	PUNCT
ejpam-5125	353	9	we	we	PRON
ejpam-5125	353	10	have	have	VERB
ejpam-5125	353	11	s	s	NOUN
ejpam-5125	353	12	=	=	PUNCT
ejpam-5125	353	13	(	(	PUNCT
ejpam-5125	353	14	n	n	ADV
ejpam-5125	353	15	∨	∨	NUM
ejpam-5125	353	16	i=1	i=1	PROPN
ejpam-5125	353	17	αi	αi	PROPN
ejpam-5125	353	18	)	)	PUNCT
ejpam-5125	354	1	∧	∧	NOUN
ejpam-5125	354	2	p2	p2	PROPN
ejpam-5125	354	3	∈	∈	PROPN
ejpam-5125	354	4	u.	u.	NOUN
ejpam-5125	354	5	hence	hence	ADV
ejpam-5125	354	6	(	(	PUNCT
ejpam-5125	354	7	s	s	X
ejpam-5125	354	8	]	]	X
ejpam-5125	354	9	⊆	⊆	NUM
ejpam-5125	354	10	u	u	NOUN
ejpam-5125	354	11	.	.	PUNCT
ejpam-5125	355	1	therefore	therefore	ADV
ejpam-5125	355	2	(	(	PUNCT
ejpam-5125	355	3	s	s	X
ejpam-5125	355	4	]	]	X
ejpam-5125	355	5	is	be	AUX
ejpam-5125	355	6	the	the	DET
ejpam-5125	355	7	smallest	small	ADJ
ejpam-5125	355	8	ideal	ideal	NOUN
ejpam-5125	355	9	of	of	ADP
ejpam-5125	355	10	v	v	NOUN
ejpam-5125	355	11	containing	contain	VERB
ejpam-5125	355	12	s.	s.	PROPN
ejpam-5125	355	13	note	note	VERB
ejpam-5125	355	14	that	that	SCONJ
ejpam-5125	355	15	if	if	SCONJ
ejpam-5125	355	16	s	s	VERB
ejpam-5125	355	17	=	=	X
ejpam-5125	355	18	{	{	PUNCT
ejpam-5125	355	19	a	a	NOUN
ejpam-5125	355	20	}	}	PUNCT
ejpam-5125	355	21	,	,	PUNCT
ejpam-5125	355	22	then	then	ADV
ejpam-5125	355	23	we	we	PRON
ejpam-5125	355	24	write	write	VERB
ejpam-5125	355	25	(	(	PUNCT
ejpam-5125	355	26	s	s	X
ejpam-5125	355	27	]	]	X
ejpam-5125	355	28	=	=	X
ejpam-5125	355	29	(	(	PUNCT
ejpam-5125	355	30	a	a	X
ejpam-5125	355	31	]	]	X
ejpam-5125	355	32	,	,	PUNCT
ejpam-5125	355	33	the	the	DET
ejpam-5125	355	34	principal	principal	ADJ
ejpam-5125	355	35	ideal	ideal	NOUN
ejpam-5125	355	36	of	of	ADP
ejpam-5125	355	37	v	v	NUM
ejpam-5125	355	38	generated	generate	VERB
ejpam-5125	355	39	by	by	ADP
ejpam-5125	355	40	‘	'	PUNCT
ejpam-5125	355	41	a	a	PRON
ejpam-5125	355	42	’	'	PUNCT
ejpam-5125	355	43	.	.	PUNCT
ejpam-5125	356	1	r.	r.	PROPN
ejpam-5125	356	2	bandaru	bandaru	PROPN
ejpam-5125	356	3	,	,	PUNCT
ejpam-5125	356	4	s.	s.	PROPN
ejpam-5125	356	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	356	6	/	/	PUNCT
ejpam-5125	356	7	eur	eur	PROPN
ejpam-5125	356	8	.	.	PUNCT
ejpam-5125	357	1	j.	j.	PROPN
ejpam-5125	357	2	pure	pure	PROPN
ejpam-5125	357	3	appl	appl	PROPN
ejpam-5125	357	4	.	.	PROPN
ejpam-5125	357	5	math	math	PROPN
ejpam-5125	357	6	,	,	PUNCT
ejpam-5125	357	7	17	17	NUM
ejpam-5125	357	8	(	(	PUNCT
ejpam-5125	357	9	2	2	NUM
ejpam-5125	357	10	)	)	PUNCT
ejpam-5125	357	11	(	(	PUNCT
ejpam-5125	357	12	2024	2024	NUM
ejpam-5125	357	13	)	)	PUNCT
ejpam-5125	357	14	,	,	PUNCT
ejpam-5125	357	15	819	819	NUM
ejpam-5125	357	16	-	-	SYM
ejpam-5125	357	17	834	834	NUM
ejpam-5125	357	18	829	829	NUM
ejpam-5125	357	19	corollary	corollary	ADJ
ejpam-5125	357	20	5	5	NUM
ejpam-5125	357	21	.	.	PUNCT
ejpam-5125	358	1	p1	p1	PROPN
ejpam-5125	358	2	∈	∈	PROPN
ejpam-5125	358	3	(	(	PUNCT
ejpam-5125	358	4	p2	p2	X
ejpam-5125	358	5	]	]	PUNCT
ejpam-5125	358	6	if	if	SCONJ
ejpam-5125	358	7	and	and	CCONJ
ejpam-5125	358	8	only	only	ADV
ejpam-5125	358	9	if	if	SCONJ
ejpam-5125	358	10	p2	p2	PROPN
ejpam-5125	358	11	∧	∧	PROPN
ejpam-5125	358	12	p1	p1	PROPN
ejpam-5125	358	13	=	=	SYM
ejpam-5125	358	14	p1	p1	PROPN
ejpam-5125	358	15	,	,	PUNCT
ejpam-5125	358	16	where	where	SCONJ
ejpam-5125	358	17	p1	p1	NOUN
ejpam-5125	358	18	,	,	PUNCT
ejpam-5125	358	19	p2	p2	PROPN
ejpam-5125	358	20	∈	∈	PROPN
ejpam-5125	358	21	v	v	NOUN
ejpam-5125	358	22	.	.	PUNCT
ejpam-5125	359	1	lemma	lemma	PROPN
ejpam-5125	359	2	11	11	NUM
ejpam-5125	359	3	.	.	PUNCT
ejpam-5125	360	1	let	let	VERB
ejpam-5125	360	2	u	u	PRON
ejpam-5125	360	3	be	be	AUX
ejpam-5125	360	4	an	an	DET
ejpam-5125	360	5	ideal	ideal	NOUN
ejpam-5125	360	6	of	of	ADP
ejpam-5125	360	7	v	v	NOUN
ejpam-5125	360	8	.	.	PUNCT
ejpam-5125	361	1	then	then	ADV
ejpam-5125	361	2	,	,	PUNCT
ejpam-5125	361	3	for	for	ADP
ejpam-5125	361	4	any	any	DET
ejpam-5125	361	5	p1	p1	NOUN
ejpam-5125	361	6	,	,	PUNCT
ejpam-5125	361	7	p2	p2	PROPN
ejpam-5125	361	8	∈	∈	PROPN
ejpam-5125	361	9	v	v	NOUN
ejpam-5125	361	10	,	,	PUNCT
ejpam-5125	361	11	p1	p1	NOUN
ejpam-5125	361	12	∧	∧	PROPN
ejpam-5125	361	13	p2	p2	NOUN
ejpam-5125	361	14	∈	∈	PROPN
ejpam-5125	361	15	u	u	NOUN
ejpam-5125	361	16	if	if	SCONJ
ejpam-5125	361	17	and	and	CCONJ
ejpam-5125	361	18	only	only	ADV
ejpam-5125	361	19	if	if	SCONJ
ejpam-5125	361	20	p2	p2	PROPN
ejpam-5125	361	21	∧	∧	PROPN
ejpam-5125	361	22	p1	p1	PROPN
ejpam-5125	361	23	∈	∈	PROPN
ejpam-5125	361	24	u	u	PROPN
ejpam-5125	361	25	.	.	PUNCT
ejpam-5125	362	1	corollary	corollary	ADJ
ejpam-5125	362	2	6	6	NUM
ejpam-5125	362	3	.	.	PUNCT
ejpam-5125	363	1	for	for	ADP
ejpam-5125	363	2	any	any	DET
ejpam-5125	363	3	p1	p1	NOUN
ejpam-5125	363	4	,	,	PUNCT
ejpam-5125	363	5	p2	p2	PROPN
ejpam-5125	363	6	∈	∈	PROPN
ejpam-5125	363	7	v	v	NOUN
ejpam-5125	363	8	(	(	PUNCT
ejpam-5125	363	9	p1	p1	PROPN
ejpam-5125	363	10	]	]	PUNCT
ejpam-5125	363	11	∨	∨	X
ejpam-5125	363	12	(	(	PUNCT
ejpam-5125	363	13	p2	p2	X
ejpam-5125	363	14	]	]	X
ejpam-5125	363	15	=	=	SYM
ejpam-5125	363	16	(	(	PUNCT
ejpam-5125	363	17	p1	p1	PROPN
ejpam-5125	363	18	∨	∨	NUM
ejpam-5125	363	19	p2	p2	PROPN
ejpam-5125	363	20	]	]	PUNCT
ejpam-5125	363	21	=	=	SYM
ejpam-5125	363	22	(	(	PUNCT
ejpam-5125	363	23	p2	p2	PROPN
ejpam-5125	363	24	∨	∨	NUM
ejpam-5125	363	25	p1	p1	PROPN
ejpam-5125	363	26	]	]	PUNCT
ejpam-5125	363	27	(	(	PUNCT
ejpam-5125	363	28	p1	p1	PROPN
ejpam-5125	363	29	]	]	X
ejpam-5125	363	30	∧	∧	PROPN
ejpam-5125	363	31	(	(	PUNCT
ejpam-5125	363	32	p2	p2	X
ejpam-5125	363	33	]	]	X
ejpam-5125	363	34	=	=	SYM
ejpam-5125	363	35	(	(	PUNCT
ejpam-5125	363	36	p1	p1	NOUN
ejpam-5125	363	37	∧	∧	NOUN
ejpam-5125	363	38	p2	p2	NOUN
ejpam-5125	363	39	]	]	X
ejpam-5125	363	40	=	=	SYM
ejpam-5125	363	41	(	(	PUNCT
ejpam-5125	363	42	p2	p2	PROPN
ejpam-5125	363	43	∧	∧	PROPN
ejpam-5125	363	44	p1	p1	PROPN
ejpam-5125	363	45	]	]	PUNCT
ejpam-5125	363	46	theorem	theorem	VERB
ejpam-5125	363	47	8	8	NUM
ejpam-5125	363	48	.	.	PUNCT
ejpam-5125	364	1	the	the	DET
ejpam-5125	364	2	set	set	NOUN
ejpam-5125	364	3	p	p	PROPN
ejpam-5125	364	4	(	(	PUNCT
ejpam-5125	364	5	v	v	NOUN
ejpam-5125	364	6	)	)	PUNCT
ejpam-5125	364	7	of	of	ADP
ejpam-5125	364	8	all	all	DET
ejpam-5125	364	9	ideals	ideal	NOUN
ejpam-5125	364	10	of	of	ADP
ejpam-5125	364	11	v	v	NOUN
ejpam-5125	364	12	forms	form	VERB
ejpam-5125	364	13	a	a	DET
ejpam-5125	364	14	distributive	distributive	ADJ
ejpam-5125	364	15	lattice	lattice	NOUN
ejpam-5125	364	16	with	with	ADP
ejpam-5125	364	17	greatest	great	ADJ
ejpam-5125	364	18	element	element	NOUN
ejpam-5125	364	19	under	under	ADP
ejpam-5125	364	20	the	the	DET
ejpam-5125	364	21	set	set	VERB
ejpam-5125	364	22	inclusion	inclusion	NOUN
ejpam-5125	364	23	in	in	ADP
ejpam-5125	364	24	which	which	PRON
ejpam-5125	364	25	the	the	DET
ejpam-5125	364	26	g.l.b	g.l.b	NOUN
ejpam-5125	364	27	and	and	CCONJ
ejpam-5125	364	28	l.u.b	l.u.b	NOUN
ejpam-5125	364	29	for	for	ADP
ejpam-5125	364	30	any	any	DET
ejpam-5125	364	31	p	p	NOUN
ejpam-5125	364	32	and	and	CCONJ
ejpam-5125	364	33	q	q	NOUN
ejpam-5125	364	34	are	be	AUX
ejpam-5125	364	35	respectively	respectively	ADV
ejpam-5125	364	36	,	,	PUNCT
ejpam-5125	364	37	p	p	NOUN
ejpam-5125	364	38	∧q	∧q	PROPN
ejpam-5125	365	1	=	=	PRON
ejpam-5125	365	2	p	p	X
ejpam-5125	365	3	∩q	∩q	PROPN
ejpam-5125	365	4	and	and	CCONJ
ejpam-5125	365	5	p	p	NOUN
ejpam-5125	365	6	∨q	∨q	NOUN
ejpam-5125	365	7	=	=	SYM
ejpam-5125	365	8	{	{	PUNCT
ejpam-5125	365	9	p1	p1	PROPN
ejpam-5125	365	10	∨	∨	NUM
ejpam-5125	365	11	p2	p2	PROPN
ejpam-5125	365	12	|	|	ADV
ejpam-5125	365	13	p1	p1	PROPN
ejpam-5125	365	14	∈	∈	PROPN
ejpam-5125	365	15	p	p	NOUN
ejpam-5125	365	16	,	,	PUNCT
ejpam-5125	365	17	p2	p2	PROPN
ejpam-5125	365	18	∈	∈	PROPN
ejpam-5125	365	19	q	q	NOUN
ejpam-5125	365	20	}	}	PUNCT
ejpam-5125	365	21	.	.	PUNCT
ejpam-5125	366	1	definition	definition	NOUN
ejpam-5125	366	2	7	7	NUM
ejpam-5125	366	3	.	.	PUNCT
ejpam-5125	367	1	a	a	DET
ejpam-5125	367	2	non	non	ADJ
ejpam-5125	367	3	-	-	ADJ
ejpam-5125	367	4	empty	empty	ADJ
ejpam-5125	367	5	subset	subset	NOUN
ejpam-5125	367	6	f	f	PROPN
ejpam-5125	367	7	of	of	ADP
ejpam-5125	367	8	v	v	PROPN
ejpam-5125	367	9	is	be	AUX
ejpam-5125	367	10	said	say	VERB
ejpam-5125	367	11	to	to	PART
ejpam-5125	367	12	be	be	AUX
ejpam-5125	367	13	a	a	DET
ejpam-5125	367	14	filter	filter	NOUN
ejpam-5125	367	15	if	if	SCONJ
ejpam-5125	367	16	it	it	PRON
ejpam-5125	367	17	satisfies	satisfy	VERB
ejpam-5125	367	18	the	the	DET
ejpam-5125	367	19	following	following	NOUN
ejpam-5125	367	20	:	:	PUNCT
ejpam-5125	367	21	p1	p1	NOUN
ejpam-5125	367	22	,	,	PUNCT
ejpam-5125	367	23	p2	p2	PROPN
ejpam-5125	367	24	∈	∈	PROPN
ejpam-5125	367	25	f	f	PROPN
ejpam-5125	367	26	⇒	⇒	PROPN
ejpam-5125	367	27	p1	p1	PROPN
ejpam-5125	368	1	∧	∧	PROPN
ejpam-5125	368	2	p2	p2	PROPN
ejpam-5125	368	3	∈	∈	PROPN
ejpam-5125	368	4	f.	f.	PROPN
ejpam-5125	368	5	p1	p1	PROPN
ejpam-5125	368	6	∈	∈	PROPN
ejpam-5125	369	1	f	f	PROPN
ejpam-5125	369	2	,	,	PUNCT
ejpam-5125	369	3	a	a	DET
ejpam-5125	369	4	∈	∈	NOUN
ejpam-5125	369	5	v	v	ADP
ejpam-5125	369	6	⇒	⇒	PROPN
ejpam-5125	369	7	a	a	DET
ejpam-5125	369	8	∨	∨	PROPN
ejpam-5125	369	9	p1	p1	PROPN
ejpam-5125	369	10	∈	∈	PROPN
ejpam-5125	369	11	f.	f.	PROPN
ejpam-5125	369	12	theorem	theorem	VERB
ejpam-5125	369	13	9	9	X
ejpam-5125	369	14	.	.	PUNCT
ejpam-5125	370	1	let	let	VERB
ejpam-5125	370	2	s	s	PRON
ejpam-5125	370	3	be	be	AUX
ejpam-5125	370	4	a	a	DET
ejpam-5125	370	5	non	non	ADJ
ejpam-5125	370	6	-	-	ADJ
ejpam-5125	370	7	empty	empty	ADJ
ejpam-5125	370	8	subset	subset	NOUN
ejpam-5125	370	9	of	of	ADP
ejpam-5125	370	10	v	v	NOUN
ejpam-5125	370	11	.	.	PUNCT
ejpam-5125	371	1	then	then	ADV
ejpam-5125	371	2	[	[	X
ejpam-5125	371	3	s	s	X
ejpam-5125	371	4	)	)	PUNCT
ejpam-5125	371	5	=	=	SYM
ejpam-5125	371	6	{	{	PUNCT
ejpam-5125	371	7	p1	p1	PROPN
ejpam-5125	371	8	∨	∨	NUM
ejpam-5125	371	9	(	(	PUNCT
ejpam-5125	371	10	n	n	CCONJ
ejpam-5125	371	11	∧	∧	PROPN
ejpam-5125	371	12	i=1	i=1	PROPN
ejpam-5125	371	13	si	si	NOUN
ejpam-5125	371	14	)	)	PUNCT
ejpam-5125	372	1	|	|	ADV
ejpam-5125	372	2	si	si	PROPN
ejpam-5125	372	3	∈	∈	PROPN
ejpam-5125	372	4	s	s	PROPN
ejpam-5125	372	5	,	,	PUNCT
ejpam-5125	372	6	p1	p1	PROPN
ejpam-5125	372	7	∈	∈	PROPN
ejpam-5125	372	8	v	v	NOUN
ejpam-5125	372	9	,	,	PUNCT
ejpam-5125	372	10	1	1	NUM
ejpam-5125	372	11	≤	≤	NUM
ejpam-5125	372	12	i	i	PRON
ejpam-5125	372	13	≤	≤	ADJ
ejpam-5125	372	14	n	n	CCONJ
ejpam-5125	372	15	and	and	CCONJ
ejpam-5125	372	16	n	n	PROPN
ejpam-5125	372	17	is	be	AUX
ejpam-5125	372	18	a	a	DET
ejpam-5125	372	19	positive	positive	ADJ
ejpam-5125	372	20	integer	integer	NOUN
ejpam-5125	372	21	}	}	PUNCT
ejpam-5125	372	22	is	be	AUX
ejpam-5125	372	23	the	the	DET
ejpam-5125	372	24	smallest	small	ADJ
ejpam-5125	372	25	filter	filter	NOUN
ejpam-5125	372	26	of	of	ADP
ejpam-5125	372	27	v	v	NOUN
ejpam-5125	372	28	containing	contain	VERB
ejpam-5125	372	29	s.	s.	PROPN
ejpam-5125	372	30	proof	proof	NOUN
ejpam-5125	372	31	.	.	PUNCT
ejpam-5125	373	1	let	let	VERB
ejpam-5125	373	2	a	a	DET
ejpam-5125	373	3	,	,	PUNCT
ejpam-5125	373	4	b	b	X
ejpam-5125	373	5	∈	∈	PROPN
ejpam-5125	373	6	[	[	X
ejpam-5125	373	7	s	s	NOUN
ejpam-5125	373	8	)	)	PUNCT
ejpam-5125	373	9	.	.	PUNCT
ejpam-5125	374	1	then	then	ADV
ejpam-5125	374	2	a	a	DET
ejpam-5125	374	3	=	=	X
ejpam-5125	374	4	p1	p1	PROPN
ejpam-5125	374	5	∨	∨	NUM
ejpam-5125	374	6	(	(	PUNCT
ejpam-5125	374	7	n	n	CCONJ
ejpam-5125	374	8	∧	∧	PROPN
ejpam-5125	374	9	i=1	i=1	PROPN
ejpam-5125	374	10	si	si	PROPN
ejpam-5125	374	11	)	)	PUNCT
ejpam-5125	374	12	,	,	PUNCT
ejpam-5125	374	13	b	b	X
ejpam-5125	374	14	=	=	NOUN
ejpam-5125	374	15	p2	p2	PROPN
ejpam-5125	374	16	∨	∨	NOUN
ejpam-5125	374	17	(	(	PUNCT
ejpam-5125	374	18	m	m	PROPN
ejpam-5125	374	19	∧	∧	PROPN
ejpam-5125	374	20	i=1	i=1	PROPN
ejpam-5125	374	21	tj	tj	PROPN
ejpam-5125	374	22	)	)	PUNCT
ejpam-5125	374	23	a	a	DET
ejpam-5125	374	24	∧	∧	PROPN
ejpam-5125	374	25	b	b	NOUN
ejpam-5125	375	1	=	=	SYM
ejpam-5125	376	1	[	[	X
ejpam-5125	376	2	p1	p1	PROPN
ejpam-5125	376	3	∨	∨	NOUN
ejpam-5125	376	4	(	(	PUNCT
ejpam-5125	376	5	n	n	CCONJ
ejpam-5125	376	6	∧	∧	PROPN
ejpam-5125	376	7	i=1	i=1	PROPN
ejpam-5125	376	8	si	si	PROPN
ejpam-5125	376	9	)	)	PUNCT
ejpam-5125	376	10	]	]	PUNCT
ejpam-5125	377	1	∧	∧	PROPN
ejpam-5125	377	2	[	[	X
ejpam-5125	377	3	p2	p2	PROPN
ejpam-5125	377	4	∨	∨	NOUN
ejpam-5125	377	5	(	(	PUNCT
ejpam-5125	377	6	m	m	PROPN
ejpam-5125	377	7	∧	∧	PROPN
ejpam-5125	377	8	j=1	j=1	ADJ
ejpam-5125	377	9	tj	tj	PROPN
ejpam-5125	377	10	)	)	PUNCT
ejpam-5125	377	11	]	]	PUNCT
ejpam-5125	378	1	=	=	PUNCT
ejpam-5125	379	1	[	[	X
ejpam-5125	379	2	p1	p1	PROPN
ejpam-5125	379	3	∧	∧	PROPN
ejpam-5125	379	4	(	(	PUNCT
ejpam-5125	379	5	p2	p2	PROPN
ejpam-5125	379	6	∨	∨	NUM
ejpam-5125	379	7	(	(	PUNCT
ejpam-5125	379	8	m	m	PROPN
ejpam-5125	379	9	∧	∧	PROPN
ejpam-5125	379	10	j=1	j=1	ADJ
ejpam-5125	379	11	tj	tj	PROPN
ejpam-5125	379	12	)	)	PUNCT
ejpam-5125	379	13	)	)	PUNCT
ejpam-5125	379	14	]	]	PUNCT
ejpam-5125	380	1	∨	∨	NUM
ejpam-5125	380	2	[	[	X
ejpam-5125	380	3	(	(	PUNCT
ejpam-5125	380	4	n	n	X
ejpam-5125	380	5	∧	∧	PROPN
ejpam-5125	380	6	i=1	i=1	PROPN
ejpam-5125	380	7	si	si	PROPN
ejpam-5125	380	8	)	)	PUNCT
ejpam-5125	380	9	∧	∧	PROPN
ejpam-5125	380	10	(	(	PUNCT
ejpam-5125	380	11	p2	p2	PROPN
ejpam-5125	380	12	∨	∨	NUM
ejpam-5125	380	13	(	(	PUNCT
ejpam-5125	380	14	m	m	PROPN
ejpam-5125	380	15	∧	∧	PROPN
ejpam-5125	380	16	j=1	j=1	ADJ
ejpam-5125	380	17	tj	tj	PROPN
ejpam-5125	380	18	)	)	PUNCT
ejpam-5125	380	19	)	)	PUNCT
ejpam-5125	380	20	]	]	PUNCT
ejpam-5125	380	21	=	=	PUNCT
ejpam-5125	381	1	[	[	X
ejpam-5125	381	2	p1	p1	PROPN
ejpam-5125	381	3	∧	∧	PROPN
ejpam-5125	381	4	(	(	PUNCT
ejpam-5125	381	5	p2	p2	PROPN
ejpam-5125	381	6	∨	∨	NUM
ejpam-5125	381	7	(	(	PUNCT
ejpam-5125	381	8	m	m	PROPN
ejpam-5125	381	9	∧	∧	PROPN
ejpam-5125	381	10	j=1	j=1	ADJ
ejpam-5125	381	11	tj	tj	PROPN
ejpam-5125	381	12	)	)	PUNCT
ejpam-5125	381	13	)	)	PUNCT
ejpam-5125	381	14	]	]	PUNCT
ejpam-5125	382	1	∨	∨	NUM
ejpam-5125	382	2	[	[	X
ejpam-5125	382	3	(	(	PUNCT
ejpam-5125	382	4	p2	p2	PROPN
ejpam-5125	382	5	∨	∨	NOUN
ejpam-5125	382	6	(	(	PUNCT
ejpam-5125	382	7	m	m	PROPN
ejpam-5125	382	8	∧	∧	PROPN
ejpam-5125	382	9	j=1	j=1	ADJ
ejpam-5125	382	10	tj	tj	PROPN
ejpam-5125	382	11	)	)	PUNCT
ejpam-5125	382	12	)	)	PUNCT
ejpam-5125	382	13	∧	∧	NOUN
ejpam-5125	382	14	(	(	PUNCT
ejpam-5125	382	15	n	n	CCONJ
ejpam-5125	382	16	∧	∧	PROPN
ejpam-5125	382	17	i=1	i=1	PROPN
ejpam-5125	382	18	si	si	PROPN
ejpam-5125	382	19	)	)	PUNCT
ejpam-5125	382	20	]	]	PUNCT
ejpam-5125	382	21	=	=	SYM
ejpam-5125	382	22	p3	p3	PROPN
ejpam-5125	382	23	∨	∨	PROPN
ejpam-5125	382	24	(	(	PUNCT
ejpam-5125	382	25	m	m	PROPN
ejpam-5125	382	26	∧	∧	NOUN
ejpam-5125	382	27	j=1	j=1	ADJ
ejpam-5125	382	28	tj	tj	PROPN
ejpam-5125	382	29	∧	∧	PROPN
ejpam-5125	382	30	n	n	PROPN
ejpam-5125	382	31	∧	∧	PROPN
ejpam-5125	382	32	i=1	i=1	PROPN
ejpam-5125	382	33	si	si	PROPN
ejpam-5125	382	34	)	)	PUNCT
ejpam-5125	382	35	(	(	PUNCT
ejpam-5125	382	36	where	where	SCONJ
ejpam-5125	382	37	p3	p3	PROPN
ejpam-5125	382	38	=	=	SYM
ejpam-5125	382	39	(	(	PUNCT
ejpam-5125	382	40	p1	p1	PROPN
ejpam-5125	382	41	∧	∧	PROPN
ejpam-5125	382	42	(	(	PUNCT
ejpam-5125	382	43	p2	p2	PROPN
ejpam-5125	382	44	∨	∨	NUM
ejpam-5125	382	45	(	(	PUNCT
ejpam-5125	382	46	m	m	PROPN
ejpam-5125	382	47	∧	∧	PROPN
ejpam-5125	382	48	j=1	j=1	ADJ
ejpam-5125	382	49	tj	tj	PROPN
ejpam-5125	382	50	)	)	PUNCT
ejpam-5125	382	51	)	)	PUNCT
ejpam-5125	382	52	)	)	PUNCT
ejpam-5125	383	1	∨	∨	NUM
ejpam-5125	383	2	(	(	PUNCT
ejpam-5125	383	3	p2	p2	PROPN
ejpam-5125	383	4	∧	∧	PROPN
ejpam-5125	383	5	(	(	PUNCT
ejpam-5125	383	6	n	n	CCONJ
ejpam-5125	383	7	∧	∧	PROPN
ejpam-5125	383	8	i=1	i=1	PROPN
ejpam-5125	383	9	si	si	NOUN
ejpam-5125	383	10	)	)	PUNCT
ejpam-5125	383	11	)	)	PUNCT
ejpam-5125	383	12	and	and	CCONJ
ejpam-5125	383	13	hence	hence	ADV
ejpam-5125	383	14	a	a	DET
ejpam-5125	383	15	∧	∧	PROPN
ejpam-5125	383	16	b	b	PROPN
ejpam-5125	383	17	∈	∈	PROPN
ejpam-5125	383	18	[	[	X
ejpam-5125	383	19	s	s	NOUN
ejpam-5125	383	20	)	)	PUNCT
ejpam-5125	383	21	.	.	PUNCT
ejpam-5125	384	1	now	now	ADV
ejpam-5125	384	2	,	,	PUNCT
ejpam-5125	384	3	we	we	PRON
ejpam-5125	384	4	prove	prove	VERB
ejpam-5125	384	5	u	u	NOUN
ejpam-5125	384	6	∨	∨	NUM
ejpam-5125	384	7	a	a	DET
ejpam-5125	384	8	∈	∈	PROPN
ejpam-5125	384	9	[	[	X
ejpam-5125	384	10	s	s	X
ejpam-5125	384	11	)	)	PUNCT
ejpam-5125	384	12	for	for	ADP
ejpam-5125	384	13	u	u	PROPN
ejpam-5125	384	14	∈	∈	PROPN
ejpam-5125	384	15	v	v	NOUN
ejpam-5125	384	16	.	.	PUNCT
ejpam-5125	385	1	consider	consider	VERB
ejpam-5125	385	2	u	u	PRON
ejpam-5125	385	3	∨	∨	NUM
ejpam-5125	385	4	a	a	DET
ejpam-5125	385	5	=	=	X
ejpam-5125	385	6	u	u	NOUN
ejpam-5125	385	7	∨	∨	NOUN
ejpam-5125	385	8	(	(	PUNCT
ejpam-5125	385	9	p1	p1	PROPN
ejpam-5125	385	10	∨	∨	PROPN
ejpam-5125	385	11	(	(	PUNCT
ejpam-5125	385	12	n	n	CCONJ
ejpam-5125	385	13	∧	∧	PROPN
ejpam-5125	385	14	i=1	i=1	PROPN
ejpam-5125	385	15	si	si	NOUN
ejpam-5125	385	16	)	)	PUNCT
ejpam-5125	385	17	)	)	PUNCT
ejpam-5125	386	1	=	=	PRON
ejpam-5125	386	2	(	(	PUNCT
ejpam-5125	386	3	u	u	PROPN
ejpam-5125	386	4	∨	∨	PROPN
ejpam-5125	386	5	p1	p1	PROPN
ejpam-5125	386	6	)	)	PUNCT
ejpam-5125	386	7	∨	∨	NUM
ejpam-5125	386	8	(	(	PUNCT
ejpam-5125	386	9	n	n	CCONJ
ejpam-5125	386	10	∧	∧	PROPN
ejpam-5125	386	11	i=1	i=1	PROPN
ejpam-5125	386	12	si	si	PROPN
ejpam-5125	386	13	)	)	PUNCT
ejpam-5125	386	14	∈	∈	PROPN
ejpam-5125	387	1	[	[	X
ejpam-5125	387	2	s	s	X
ejpam-5125	387	3	)	)	PUNCT
ejpam-5125	387	4	therefore	therefore	ADV
ejpam-5125	388	1	[	[	X
ejpam-5125	388	2	s	s	X
ejpam-5125	388	3	)	)	PUNCT
ejpam-5125	388	4	is	be	AUX
ejpam-5125	388	5	a	a	DET
ejpam-5125	388	6	filter	filter	NOUN
ejpam-5125	388	7	of	of	ADP
ejpam-5125	388	8	v	v	NOUN
ejpam-5125	388	9	,	,	PUNCT
ejpam-5125	388	10	and	and	CCONJ
ejpam-5125	388	11	clearly	clearly	ADV
ejpam-5125	388	12	it	it	PRON
ejpam-5125	388	13	is	be	AUX
ejpam-5125	388	14	the	the	DET
ejpam-5125	388	15	smallest	small	ADJ
ejpam-5125	388	16	filter	filter	NOUN
ejpam-5125	388	17	of	of	ADP
ejpam-5125	388	18	v	v	NOUN
ejpam-5125	388	19	containing	contain	VERB
ejpam-5125	388	20	s.	s.	PROPN
ejpam-5125	388	21	note	note	VERB
ejpam-5125	388	22	that	that	SCONJ
ejpam-5125	388	23	if	if	SCONJ
ejpam-5125	388	24	s	s	VERB
ejpam-5125	388	25	=	=	X
ejpam-5125	388	26	{	{	PUNCT
ejpam-5125	388	27	a	a	NOUN
ejpam-5125	388	28	}	}	PUNCT
ejpam-5125	388	29	,	,	PUNCT
ejpam-5125	388	30	then	then	ADV
ejpam-5125	388	31	we	we	PRON
ejpam-5125	388	32	write	write	VERB
ejpam-5125	388	33	[	[	X
ejpam-5125	388	34	s	s	X
ejpam-5125	388	35	)	)	PUNCT
ejpam-5125	388	36	=	=	PUNCT
ejpam-5125	389	1	[	[	X
ejpam-5125	389	2	a	a	X
ejpam-5125	389	3	)	)	PUNCT
ejpam-5125	389	4	,	,	PUNCT
ejpam-5125	389	5	the	the	DET
ejpam-5125	389	6	principal	principal	ADJ
ejpam-5125	389	7	filter	filter	NOUN
ejpam-5125	389	8	of	of	ADP
ejpam-5125	389	9	v	v	NUM
ejpam-5125	389	10	generated	generate	VERB
ejpam-5125	389	11	by	by	ADP
ejpam-5125	389	12	‘	'	PUNCT
ejpam-5125	389	13	a	a	PRON
ejpam-5125	389	14	’	'	PUNCT
ejpam-5125	389	15	.	.	PUNCT
ejpam-5125	390	1	corollary	corollary	ADJ
ejpam-5125	390	2	8	8	NUM
ejpam-5125	390	3	.	.	PUNCT
ejpam-5125	391	1	p1	p1	PROPN
ejpam-5125	391	2	∈	∈	PROPN
ejpam-5125	392	1	[	[	X
ejpam-5125	392	2	p2	p2	X
ejpam-5125	392	3	)	)	PUNCT
ejpam-5125	393	1	if	if	SCONJ
ejpam-5125	393	2	and	and	CCONJ
ejpam-5125	393	3	only	only	ADV
ejpam-5125	393	4	if	if	SCONJ
ejpam-5125	393	5	p1	p1	PROPN
ejpam-5125	393	6	=	=	SYM
ejpam-5125	393	7	p1	p1	PROPN
ejpam-5125	393	8	∨	∨	NUM
ejpam-5125	393	9	p2	p2	PROPN
ejpam-5125	393	10	for	for	ADP
ejpam-5125	393	11	all	all	DET
ejpam-5125	393	12	p1	p1	NOUN
ejpam-5125	393	13	,	,	PUNCT
ejpam-5125	393	14	p2	p2	PROPN
ejpam-5125	393	15	∈	∈	PROPN
ejpam-5125	393	16	v	v	NOUN
ejpam-5125	393	17	.	.	PUNCT
ejpam-5125	394	1	r.	r.	PROPN
ejpam-5125	394	2	bandaru	bandaru	PROPN
ejpam-5125	394	3	,	,	PUNCT
ejpam-5125	394	4	s.	s.	PROPN
ejpam-5125	394	5	ajjarapu	ajjarapu	PROPN
ejpam-5125	394	6	/	/	PUNCT
ejpam-5125	394	7	eur	eur	PROPN
ejpam-5125	394	8	.	.	PUNCT
ejpam-5125	395	1	j.	j.	PROPN
ejpam-5125	395	2	pure	pure	PROPN
ejpam-5125	395	3	appl	appl	PROPN
ejpam-5125	395	4	.	.	PROPN
ejpam-5125	395	5	math	math	PROPN
ejpam-5125	395	6	,	,	PUNCT
ejpam-5125	395	7	17	17	NUM
ejpam-5125	395	8	(	(	PUNCT
ejpam-5125	395	9	2	2	NUM
ejpam-5125	395	10	)	)	PUNCT
ejpam-5125	395	11	(	(	PUNCT
ejpam-5125	395	12	2024	2024	NUM
ejpam-5125	395	13	)	)	PUNCT
ejpam-5125	395	14	,	,	PUNCT
ejpam-5125	395	15	819	819	NUM
ejpam-5125	395	16	-	-	SYM
ejpam-5125	395	17	834	834	NUM
ejpam-5125	395	18	830	830	NUM
ejpam-5125	395	19	lemma	lemma	PROPN
ejpam-5125	395	20	12	12	NUM
ejpam-5125	395	21	.	.	PUNCT
ejpam-5125	396	1	let	let	VERB
ejpam-5125	396	2	f	f	PRON
ejpam-5125	396	3	be	be	AUX
ejpam-5125	396	4	a	a	DET
ejpam-5125	396	5	filter	filter	NOUN
ejpam-5125	396	6	of	of	ADP
ejpam-5125	396	7	v	v	NOUN
ejpam-5125	396	8	and	and	CCONJ
ejpam-5125	396	9	p1	p1	NOUN
ejpam-5125	396	10	,	,	PUNCT
ejpam-5125	396	11	p2	p2	PROPN
ejpam-5125	396	12	∈	∈	PROPN
ejpam-5125	396	13	v	v	NOUN
ejpam-5125	396	14	.	.	PUNCT
ejpam-5125	397	1	then	then	ADV
ejpam-5125	397	2	(	(	PUNCT
ejpam-5125	397	3	1	1	X
ejpam-5125	397	4	)	)	PUNCT
ejpam-5125	397	5	p1	p1	NOUN
ejpam-5125	397	6	∨	∨	NUM
ejpam-5125	397	7	p2	p2	PROPN
ejpam-5125	397	8	∈	∈	PROPN
ejpam-5125	398	1	f	f	NOUN
ejpam-5125	398	2	if	if	SCONJ
ejpam-5125	398	3	and	and	CCONJ
ejpam-5125	398	4	only	only	ADV
ejpam-5125	398	5	if	if	SCONJ
ejpam-5125	398	6	p2	p2	PROPN
ejpam-5125	398	7	∨	∨	NUM
ejpam-5125	398	8	p1	p1	PROPN
ejpam-5125	398	9	∈	∈	PROPN
ejpam-5125	398	10	f	f	X
ejpam-5125	398	11	.	.	PUNCT
ejpam-5125	399	1	(	(	PUNCT
ejpam-5125	399	2	2	2	X
ejpam-5125	399	3	)	)	PUNCT
ejpam-5125	399	4	for	for	ADP
ejpam-5125	399	5	any	any	DET
ejpam-5125	399	6	p1	p1	NOUN
ejpam-5125	399	7	,	,	PUNCT
ejpam-5125	399	8	p2	p2	PROPN
ejpam-5125	399	9	∈	∈	PROPN
ejpam-5125	399	10	v	v	NOUN
ejpam-5125	399	11	,	,	PUNCT
ejpam-5125	399	12	[	[	X
ejpam-5125	399	13	p1	p1	PROPN
ejpam-5125	399	14	∨	∨	NUM
ejpam-5125	399	15	p2	p2	NOUN
ejpam-5125	399	16	)	)	PUNCT
ejpam-5125	399	17	=	=	PUNCT
ejpam-5125	400	1	[	[	X
ejpam-5125	400	2	p2	p2	PROPN
ejpam-5125	400	3	∨	∨	NUM
ejpam-5125	400	4	p1	p1	NOUN
ejpam-5125	400	5	)	)	PUNCT
ejpam-5125	400	6	.	.	PUNCT
ejpam-5125	401	1	(	(	PUNCT
ejpam-5125	401	2	3	3	X
ejpam-5125	401	3	)	)	PUNCT
ejpam-5125	401	4	for	for	ADP
ejpam-5125	401	5	any	any	DET
ejpam-5125	401	6	p1	p1	NOUN
ejpam-5125	401	7	,	,	PUNCT
ejpam-5125	401	8	p2	p2	PROPN
ejpam-5125	401	9	∈	∈	PROPN
ejpam-5125	401	10	v	v	NOUN
ejpam-5125	401	11	,	,	PUNCT
ejpam-5125	401	12	[	[	X
ejpam-5125	401	13	p1	p1	NOUN
ejpam-5125	401	14	∧	∧	NOUN
ejpam-5125	401	15	p2	p2	NOUN
ejpam-5125	401	16	)	)	PUNCT
ejpam-5125	401	17	=	=	NOUN
ejpam-5125	402	1	[	[	X
ejpam-5125	402	2	p2	p2	X
ejpam-5125	402	3	∧	∧	PROPN
ejpam-5125	402	4	p1	p1	NOUN
ejpam-5125	402	5	)	)	PUNCT
ejpam-5125	402	6	=	=	PUNCT
ejpam-5125	403	1	[	[	X
ejpam-5125	403	2	p1	p1	NOUN
ejpam-5125	403	3	)	)	PUNCT
ejpam-5125	403	4	∨	∨	NOUN
ejpam-5125	403	5	[	[	X
ejpam-5125	403	6	p2	p2	X
ejpam-5125	403	7	)	)	PUNCT
ejpam-5125	403	8	.	.	PUNCT
ejpam-5125	404	1	theorem	theorem	ADJ
ejpam-5125	404	2	10	10	NUM
ejpam-5125	404	3	.	.	PUNCT
ejpam-5125	405	1	let	let	VERB
ejpam-5125	405	2	p1	p1	PROPN
ejpam-5125	405	3	,	,	PUNCT
ejpam-5125	405	4	p2	p2	PROPN
ejpam-5125	405	5	∈	∈	PROPN
ejpam-5125	405	6	v	v	NOUN
ejpam-5125	405	7	.	.	PUNCT
ejpam-5125	406	1	then	then	ADV
ejpam-5125	406	2	the	the	DET
ejpam-5125	406	3	following	following	NOUN
ejpam-5125	406	4	are	be	AUX
ejpam-5125	406	5	equivalent	equivalent	ADJ
ejpam-5125	406	6	.	.	PUNCT
ejpam-5125	407	1	(	(	PUNCT
ejpam-5125	407	2	1	1	X
ejpam-5125	407	3	)	)	PUNCT
ejpam-5125	407	4	(	(	PUNCT
ejpam-5125	407	5	p1	p1	PROPN
ejpam-5125	407	6	]	]	X
ejpam-5125	407	7	⊆	⊆	NUM
ejpam-5125	407	8	(	(	PUNCT
ejpam-5125	407	9	p2	p2	PROPN
ejpam-5125	407	10	]	]	PUNCT
ejpam-5125	407	11	.	.	PUNCT
ejpam-5125	408	1	(	(	PUNCT
ejpam-5125	408	2	2	2	X
ejpam-5125	408	3	)	)	PUNCT
ejpam-5125	408	4	p2	p2	PROPN
ejpam-5125	408	5	∧	∧	PROPN
ejpam-5125	408	6	p1	p1	PROPN
ejpam-5125	408	7	=	=	PROPN
ejpam-5125	408	8	p1	p1	PROPN
ejpam-5125	408	9	.	.	PUNCT
ejpam-5125	409	1	(	(	PUNCT
ejpam-5125	409	2	3	3	X
ejpam-5125	409	3	)	)	PUNCT
ejpam-5125	409	4	p2	p2	PROPN
ejpam-5125	409	5	∨	∨	NUM
ejpam-5125	409	6	p1	p1	NOUN
ejpam-5125	409	7	=	=	PUNCT
ejpam-5125	409	8	p2	p2	PROPN
ejpam-5125	409	9	.	.	PUNCT
ejpam-5125	410	1	(	(	PUNCT
ejpam-5125	410	2	4	4	X
ejpam-5125	410	3	)	)	PUNCT
ejpam-5125	411	1	[	[	X
ejpam-5125	411	2	p2	p2	NOUN
ejpam-5125	411	3	)	)	PUNCT
ejpam-5125	411	4	⊆	⊆	NUM
ejpam-5125	411	5	[	[	X
ejpam-5125	411	6	p1	p1	NOUN
ejpam-5125	411	7	)	)	PUNCT
ejpam-5125	411	8	.	.	PUNCT
ejpam-5125	412	1	theorem	theorem	VERB
ejpam-5125	412	2	11	11	NUM
ejpam-5125	412	3	.	.	PUNCT
ejpam-5125	413	1	the	the	DET
ejpam-5125	413	2	collection	collection	NOUN
ejpam-5125	413	3	f	f	X
ejpam-5125	413	4	(	(	PUNCT
ejpam-5125	413	5	v	v	NOUN
ejpam-5125	413	6	)	)	PUNCT
ejpam-5125	413	7	of	of	ADP
ejpam-5125	413	8	all	all	DET
ejpam-5125	413	9	filters	filter	NOUN
ejpam-5125	413	10	of	of	ADP
ejpam-5125	413	11	a	a	DET
ejpam-5125	413	12	pdl	pdl	NOUN
ejpam-5125	413	13	v	v	NOUN
ejpam-5125	413	14	forms	form	NOUN
ejpam-5125	413	15	a	a	DET
ejpam-5125	413	16	distributive	distributive	ADJ
ejpam-5125	413	17	lattice	lattice	NOUN
ejpam-5125	413	18	under	under	ADP
ejpam-5125	413	19	set	set	ADJ
ejpam-5125	413	20	inclusion	inclusion	NOUN
ejpam-5125	413	21	,	,	PUNCT
ejpam-5125	413	22	in	in	ADP
ejpam-5125	413	23	which	which	PRON
ejpam-5125	413	24	,	,	PUNCT
ejpam-5125	413	25	the	the	DET
ejpam-5125	413	26	glb	glb	NOUN
ejpam-5125	413	27	and	and	CCONJ
ejpam-5125	413	28	lub	lub	NOUN
ejpam-5125	413	29	of	of	ADP
ejpam-5125	413	30	any	any	DET
ejpam-5125	413	31	f	f	PROPN
ejpam-5125	413	32	and	and	CCONJ
ejpam-5125	413	33	g	g	PROPN
ejpam-5125	413	34	are	be	AUX
ejpam-5125	413	35	given	give	VERB
ejpam-5125	413	36	respectively	respectively	ADV
ejpam-5125	413	37	by	by	ADP
ejpam-5125	413	38	f	f	PROPN
ejpam-5125	413	39	∧g	∧g	PROPN
ejpam-5125	413	40	=	=	SYM
ejpam-5125	413	41	f	f	PROPN
ejpam-5125	413	42	∩g	∩g	NOUN
ejpam-5125	413	43	and	and	CCONJ
ejpam-5125	413	44	f	f	PROPN
ejpam-5125	413	45	∨g	∨g	PROPN
ejpam-5125	413	46	=	=	SYM
ejpam-5125	413	47	{	{	PUNCT
ejpam-5125	413	48	p1	p1	NOUN
ejpam-5125	413	49	∧	∧	PROPN
ejpam-5125	413	50	p2	p2	NOUN
ejpam-5125	413	51	|	|	ADV
ejpam-5125	413	52	p1	p1	PROPN
ejpam-5125	413	53	∈	∈	PROPN
ejpam-5125	413	54	f	f	PROPN
ejpam-5125	413	55	and	and	CCONJ
ejpam-5125	413	56	p2	p2	PROPN
ejpam-5125	413	57	∈	∈	PROPN
ejpam-5125	413	58	g	g	PROPN
ejpam-5125	413	59	}	}	PUNCT
ejpam-5125	413	60	.	.	PUNCT
ejpam-5125	414	1	theorem	theorem	NOUN
ejpam-5125	414	2	12	12	NUM
ejpam-5125	414	3	.	.	PUNCT
ejpam-5125	415	1	the	the	DET
ejpam-5125	415	2	class	class	NOUN
ejpam-5125	415	3	pu(v	pu(v	NOUN
ejpam-5125	415	4	)	)	PUNCT
ejpam-5125	415	5	(	(	PUNCT
ejpam-5125	415	6	pf	pf	X
ejpam-5125	415	7	(	(	PUNCT
ejpam-5125	415	8	v	v	NOUN
ejpam-5125	415	9	)	)	PUNCT
ejpam-5125	415	10	)	)	PUNCT
ejpam-5125	415	11	of	of	ADP
ejpam-5125	415	12	all	all	DET
ejpam-5125	415	13	principal	principal	ADJ
ejpam-5125	415	14	ideals(filters	ideals(filter	NOUN
ejpam-5125	415	15	)	)	PUNCT
ejpam-5125	415	16	of	of	ADP
ejpam-5125	415	17	v	v	NUM
ejpam-5125	415	18	is	be	AUX
ejpam-5125	415	19	a	a	DET
ejpam-5125	415	20	sublattice	sublattice	NOUN
ejpam-5125	415	21	of	of	ADP
ejpam-5125	415	22	the	the	DET
ejpam-5125	415	23	distributive	distributive	ADJ
ejpam-5125	415	24	lattice	lattice	NOUN
ejpam-5125	415	25	u(v	u(v	PROPN
ejpam-5125	415	26	)	)	PUNCT
ejpam-5125	416	1	(	(	PUNCT
ejpam-5125	416	2	f	f	X
ejpam-5125	416	3	(	(	PUNCT
ejpam-5125	416	4	v	v	NOUN
ejpam-5125	416	5	)	)	PUNCT
ejpam-5125	416	6	)	)	PUNCT
ejpam-5125	416	7	of	of	ADP
ejpam-5125	416	8	all	all	DET
ejpam-5125	416	9	the	the	DET
ejpam-5125	416	10	ideals(filters	ideals(filter	NOUN
ejpam-5125	416	11	)	)	PUNCT
ejpam-5125	416	12	of	of	ADP
ejpam-5125	416	13	v	v	NOUN
ejpam-5125	416	14	.	.	PUNCT
ejpam-5125	417	1	moreover	moreover	ADV
ejpam-5125	417	2	,	,	PUNCT
ejpam-5125	417	3	the	the	DET
ejpam-5125	417	4	lattice	lattice	PROPN
ejpam-5125	417	5	pu(v	pu(v	NOUN
ejpam-5125	417	6	)	)	PUNCT
ejpam-5125	417	7	is	be	AUX
ejpam-5125	417	8	“	"	PUNCT
ejpam-5125	417	9	dually	dually	ADV
ejpam-5125	417	10	isomorphic	isomorphic	ADJ
ejpam-5125	417	11	”	"	PUNCT
ejpam-5125	417	12	on	on	ADP
ejpam-5125	417	13	to	to	ADP
ejpam-5125	417	14	the	the	DET
ejpam-5125	417	15	lattice	lattice	NOUN
ejpam-5125	417	16	pf	pf	PROPN
ejpam-5125	417	17	(	(	PUNCT
ejpam-5125	417	18	v	v	NOUN
ejpam-5125	417	19	)	)	PUNCT
ejpam-5125	417	20	.	.	PUNCT
ejpam-5125	418	1	proof	proof	NOUN
ejpam-5125	418	2	.	.	PUNCT
ejpam-5125	419	1	let	let	VERB
ejpam-5125	419	2	pu(v	pu(v	PRON
ejpam-5125	419	3	)	)	PUNCT
ejpam-5125	419	4	be	be	AUX
ejpam-5125	419	5	the	the	DET
ejpam-5125	419	6	set	set	NOUN
ejpam-5125	419	7	of	of	ADP
ejpam-5125	419	8	all	all	DET
ejpam-5125	419	9	principal	principal	ADJ
ejpam-5125	419	10	ideals	ideal	NOUN
ejpam-5125	419	11	of	of	ADP
ejpam-5125	419	12	the	the	DET
ejpam-5125	419	13	pdl	pdl	PROPN
ejpam-5125	419	14	v	v	NOUN
ejpam-5125	419	15	and	and	CCONJ
ejpam-5125	419	16	pu(v	pu(v	NOUN
ejpam-5125	419	17	)	)	PUNCT
ejpam-5125	420	1	=	=	SYM
ejpam-5125	420	2	{	{	PUNCT
ejpam-5125	420	3	(	(	PUNCT
ejpam-5125	420	4	a	a	X
ejpam-5125	420	5	]	]	X
ejpam-5125	420	6	|	|	ADP
ejpam-5125	420	7	a	a	DET
ejpam-5125	420	8	∈	∈	NOUN
ejpam-5125	420	9	v	v	ADP
ejpam-5125	420	10	}	}	PUNCT
ejpam-5125	420	11	.	.	PUNCT
ejpam-5125	421	1	first	first	ADV
ejpam-5125	421	2	,	,	PUNCT
ejpam-5125	421	3	we	we	PRON
ejpam-5125	421	4	prove	prove	VERB
ejpam-5125	421	5	that	that	SCONJ
ejpam-5125	421	6	pu(v	pu(v	NOUN
ejpam-5125	421	7	)	)	PUNCT
ejpam-5125	421	8	is	be	AUX
ejpam-5125	421	9	a	a	DET
ejpam-5125	421	10	sublattice	sublattice	NOUN
ejpam-5125	421	11	of	of	ADP
ejpam-5125	421	12	u(v	u(v	PROPN
ejpam-5125	421	13	)	)	PUNCT
ejpam-5125	421	14	.	.	PUNCT
ejpam-5125	422	1	let	let	VERB
ejpam-5125	422	2	(	(	PUNCT
ejpam-5125	422	3	a	a	DET
ejpam-5125	422	4	]	]	X
ejpam-5125	422	5	,	,	PUNCT
ejpam-5125	422	6	(	(	PUNCT
ejpam-5125	422	7	b	b	X
ejpam-5125	422	8	]	]	X
ejpam-5125	422	9	∈	∈	NOUN
ejpam-5125	422	10	pu(v	pu(v	NOUN
ejpam-5125	422	11	)	)	PUNCT
ejpam-5125	422	12	.	.	PUNCT
ejpam-5125	423	1	then	then	ADV
ejpam-5125	423	2	(	(	PUNCT
ejpam-5125	423	3	a]∨(b	a]∨(b	PROPN
ejpam-5125	423	4	]	]	X
ejpam-5125	423	5	=	=	X
ejpam-5125	423	6	(	(	PUNCT
ejpam-5125	423	7	a	a	DET
ejpam-5125	423	8	∨	∨	NUM
ejpam-5125	423	9	b	b	NOUN
ejpam-5125	423	10	]	]	X
ejpam-5125	423	11	for	for	ADP
ejpam-5125	423	12	a	a	DET
ejpam-5125	423	13	,	,	PUNCT
ejpam-5125	423	14	b	b	PROPN
ejpam-5125	423	15	∈	∈	PROPN
ejpam-5125	423	16	v	v	NOUN
ejpam-5125	423	17	.	.	PUNCT
ejpam-5125	424	1	hence	hence	ADV
ejpam-5125	424	2	(	(	PUNCT
ejpam-5125	424	3	a	a	X
ejpam-5125	424	4	]	]	X
ejpam-5125	424	5	∨	∨	X
ejpam-5125	424	6	(	(	PUNCT
ejpam-5125	424	7	b	b	X
ejpam-5125	424	8	]	]	X
ejpam-5125	424	9	∈	∈	NOUN
ejpam-5125	424	10	pu(v	pu(v	NOUN
ejpam-5125	424	11	)	)	PUNCT
ejpam-5125	424	12	.	.	PUNCT
ejpam-5125	425	1	similarly	similarly	ADV
ejpam-5125	425	2	,	,	PUNCT
ejpam-5125	425	3	(	(	PUNCT
ejpam-5125	425	4	a	a	PRON
ejpam-5125	425	5	]	]	X
ejpam-5125	425	6	∧	∧	NOUN
ejpam-5125	425	7	(	(	PUNCT
ejpam-5125	425	8	b	b	NOUN
ejpam-5125	425	9	]	]	X
ejpam-5125	425	10	=	=	X
ejpam-5125	425	11	(	(	PUNCT
ejpam-5125	425	12	a	a	DET
ejpam-5125	425	13	∧	∧	PROPN
ejpam-5125	425	14	b	b	PROPN
ejpam-5125	425	15	]	]	X
ejpam-5125	425	16	∈	∈	NOUN
ejpam-5125	425	17	pu(v	pu(v	NOUN
ejpam-5125	425	18	)	)	PUNCT
ejpam-5125	425	19	.	.	PUNCT
ejpam-5125	426	1	therefore	therefore	ADV
ejpam-5125	426	2	pu(v	pu(v	VERB
ejpam-5125	426	3	)	)	PUNCT
ejpam-5125	426	4	is	be	AUX
ejpam-5125	426	5	a	a	DET
ejpam-5125	426	6	sublattice	sublattice	NOUN
ejpam-5125	426	7	of	of	ADP
ejpam-5125	426	8	u(v	u(v	PROPN
ejpam-5125	426	9	)	)	PUNCT
ejpam-5125	426	10	.	.	PUNCT
ejpam-5125	427	1	finally	finally	ADV
ejpam-5125	427	2	,	,	PUNCT
ejpam-5125	427	3	we	we	PRON
ejpam-5125	427	4	prove	prove	VERB
ejpam-5125	427	5	that	that	SCONJ
ejpam-5125	427	6	there	there	PRON
ejpam-5125	427	7	exists	exist	VERB
ejpam-5125	427	8	a	a	DET
ejpam-5125	427	9	dual	dual	ADJ
ejpam-5125	427	10	isomorphism	isomorphism	NOUN
ejpam-5125	427	11	from	from	ADP
ejpam-5125	427	12	pu(v	pu(v	NOUN
ejpam-5125	427	13	)	)	PUNCT
ejpam-5125	427	14	to	to	ADP
ejpam-5125	427	15	pf	pf	PROPN
ejpam-5125	427	16	(	(	PUNCT
ejpam-5125	427	17	v	v	NOUN
ejpam-5125	427	18	)	)	PUNCT
ejpam-5125	427	19	.	.	PUNCT
ejpam-5125	428	1	define	define	VERB
ejpam-5125	428	2	ζ	ζ	NOUN
ejpam-5125	428	3	:	:	PUNCT
ejpam-5125	428	4	pu(v	pu(v	NOUN
ejpam-5125	428	5	)	)	PUNCT
ejpam-5125	428	6	→	→	SYM
ejpam-5125	428	7	pf	pf	X
ejpam-5125	428	8	(	(	PUNCT
ejpam-5125	428	9	v	v	NOUN
ejpam-5125	428	10	)	)	PUNCT
ejpam-5125	428	11	by	by	ADP
ejpam-5125	428	12	ζ{(a	ζ{(a	ADV
ejpam-5125	428	13	]	]	PUNCT
ejpam-5125	428	14	}	}	PUNCT
ejpam-5125	428	15	=	=	PUNCT
ejpam-5125	429	1	[	[	X
ejpam-5125	429	2	a	a	X
ejpam-5125	429	3	)	)	PUNCT
ejpam-5125	429	4	,	,	PUNCT
ejpam-5125	429	5	a	a	DET
ejpam-5125	429	6	∈	∈	NOUN
ejpam-5125	429	7	v	v	NOUN
ejpam-5125	429	8	.	.	PUNCT
ejpam-5125	430	1	(	(	PUNCT
ejpam-5125	430	2	i	i	NOUN
ejpam-5125	430	3	)	)	PUNCT
ejpam-5125	430	4	ζ	ζ	PROPN
ejpam-5125	430	5	is	be	AUX
ejpam-5125	430	6	a	a	DET
ejpam-5125	430	7	homomorphism	homomorphism	NOUN
ejpam-5125	430	8	:	:	PUNCT
ejpam-5125	430	9	ζ{(a	ζ{(a	X
ejpam-5125	430	10	]	]	PUNCT
ejpam-5125	430	11	∨	∨	X
ejpam-5125	430	12	(	(	PUNCT
ejpam-5125	430	13	b	b	NOUN
ejpam-5125	430	14	]	]	X
ejpam-5125	430	15	}	}	PUNCT
ejpam-5125	430	16	=	=	PUNCT
ejpam-5125	430	17	ζ{(a	ζ{(a	ADP
ejpam-5125	430	18	∨	∨	PROPN
ejpam-5125	430	19	b	b	NOUN
ejpam-5125	430	20	]	]	X
ejpam-5125	430	21	}	}	PUNCT
ejpam-5125	430	22	=	=	PUNCT
ejpam-5125	431	1	[	[	PUNCT
ejpam-5125	431	2	a	a	DET
ejpam-5125	431	3	∨	∨	NUM
ejpam-5125	431	4	b	b	NOUN
ejpam-5125	431	5	)	)	PUNCT
ejpam-5125	431	6	=	=	PUNCT
ejpam-5125	432	1	[	[	X
ejpam-5125	432	2	a	a	X
ejpam-5125	432	3	)	)	PUNCT
ejpam-5125	432	4	∧	∧	NOUN
ejpam-5125	432	5	[	[	X
ejpam-5125	432	6	b	b	NOUN
ejpam-5125	432	7	)	)	PUNCT
ejpam-5125	432	8	=	=	PUNCT
ejpam-5125	433	1	ζ{(a	ζ{(a	ADJ
ejpam-5125	433	2	]	]	PUNCT
ejpam-5125	433	3	}	}	PUNCT
ejpam-5125	433	4	∧	∧	PROPN
ejpam-5125	433	5	ζ{(b	ζ{(b	PROPN
ejpam-5125	433	6	]	]	PUNCT
ejpam-5125	433	7	}	}	PUNCT
ejpam-5125	433	8	also	also	ADV
ejpam-5125	433	9	,	,	PUNCT
ejpam-5125	433	10	ζ{(a	ζ{(a	ADV
ejpam-5125	433	11	]	]	X
ejpam-5125	433	12	∧	∧	PROPN
ejpam-5125	433	13	(	(	PUNCT
ejpam-5125	433	14	b	b	NOUN
ejpam-5125	433	15	]	]	X
ejpam-5125	433	16	}	}	PUNCT
ejpam-5125	433	17	=	=	PUNCT
ejpam-5125	433	18	ζ{(a	ζ{(a	ADV
ejpam-5125	433	19	∧	∧	PROPN
ejpam-5125	433	20	b	b	PROPN
ejpam-5125	433	21	]	]	X
ejpam-5125	433	22	}	}	PUNCT
ejpam-5125	433	23	=	=	PUNCT
ejpam-5125	434	1	[	[	X
ejpam-5125	434	2	a	a	DET
ejpam-5125	434	3	∧	∧	PROPN
ejpam-5125	434	4	b	b	NOUN
ejpam-5125	434	5	)	)	PUNCT
ejpam-5125	434	6	=	=	PUNCT
ejpam-5125	435	1	[	[	X
ejpam-5125	435	2	a	a	X
ejpam-5125	435	3	)	)	PUNCT
ejpam-5125	435	4	∨	∨	NOUN
ejpam-5125	436	1	[	[	X
ejpam-5125	436	2	b	b	X
ejpam-5125	436	3	)	)	PUNCT
ejpam-5125	436	4	=	=	PUNCT
ejpam-5125	437	1	ζ{(a	ζ{(a	PROPN
ejpam-5125	437	2	]	]	PUNCT
ejpam-5125	437	3	}	}	PUNCT
ejpam-5125	437	4	∨	∨	X
ejpam-5125	437	5	ζ{(b	ζ{(b	PROPN
ejpam-5125	437	6	]	]	PUNCT
ejpam-5125	437	7	}	}	PUNCT
ejpam-5125	437	8	(	(	PUNCT
ejpam-5125	437	9	ii	ii	NOUN
ejpam-5125	437	10	)	)	PUNCT
ejpam-5125	437	11	ζ	ζ	NOUN
ejpam-5125	437	12	is	be	AUX
ejpam-5125	437	13	one	one	NUM
ejpam-5125	437	14	-	-	PUNCT
ejpam-5125	437	15	one	one	NUM
ejpam-5125	437	16	:	:	PUNCT
ejpam-5125	437	17	let	let	VERB
ejpam-5125	437	18	a	a	DET
ejpam-5125	437	19	,	,	PUNCT
ejpam-5125	437	20	b	b	PROPN
ejpam-5125	437	21	∈	∈	PROPN
ejpam-5125	437	22	v	v	NOUN
ejpam-5125	437	23	.	.	PUNCT
ejpam-5125	438	1	then	then	ADV
ejpam-5125	438	2	ζ{(a	ζ{(a	ADV
ejpam-5125	438	3	]	]	PUNCT
ejpam-5125	438	4	}	}	PUNCT
ejpam-5125	438	5	=	=	SYM
ejpam-5125	438	6	ζ{(b	ζ{(b	X
ejpam-5125	438	7	]	]	PUNCT
ejpam-5125	438	8	}	}	PUNCT
ejpam-5125	438	9	⇒	⇒	VERB
ejpam-5125	438	10	[	[	X
ejpam-5125	438	11	a	a	X
ejpam-5125	438	12	)	)	PUNCT
ejpam-5125	438	13	=	=	PUNCT
ejpam-5125	439	1	[	[	X
ejpam-5125	439	2	b	b	X
ejpam-5125	439	3	)	)	PUNCT
ejpam-5125	439	4	⇒	⇒	NOUN
ejpam-5125	439	5	(	(	PUNCT
ejpam-5125	439	6	a	a	X
ejpam-5125	439	7	]	]	X
ejpam-5125	439	8	=	=	SYM
ejpam-5125	439	9	(	(	PUNCT
ejpam-5125	439	10	b	b	X
ejpam-5125	439	11	]	]	X
ejpam-5125	439	12	r.	r.	PROPN
ejpam-5125	439	13	bandaru	bandaru	PROPN
ejpam-5125	439	14	,	,	PUNCT
ejpam-5125	439	15	s.	s.	PROPN
ejpam-5125	439	16	ajjarapu	ajjarapu	PROPN
ejpam-5125	439	17	/	/	PUNCT
ejpam-5125	439	18	eur	eur	PROPN
ejpam-5125	439	19	.	.	PUNCT
ejpam-5125	440	1	j.	j.	PROPN
ejpam-5125	440	2	pure	pure	PROPN
ejpam-5125	440	3	appl	appl	PROPN
ejpam-5125	440	4	.	.	PROPN
ejpam-5125	440	5	math	math	PROPN
ejpam-5125	440	6	,	,	PUNCT
ejpam-5125	440	7	17	17	NUM
ejpam-5125	440	8	(	(	PUNCT
ejpam-5125	440	9	2	2	NUM
ejpam-5125	440	10	)	)	PUNCT
ejpam-5125	440	11	(	(	PUNCT
ejpam-5125	440	12	2024	2024	NUM
ejpam-5125	440	13	)	)	PUNCT
ejpam-5125	440	14	,	,	PUNCT
ejpam-5125	440	15	819	819	NUM
ejpam-5125	440	16	-	-	SYM
ejpam-5125	440	17	834	834	NUM
ejpam-5125	440	18	831	831	NUM
ejpam-5125	440	19	(	(	PUNCT
ejpam-5125	440	20	iii	iii	NOUN
ejpam-5125	440	21	)	)	PUNCT
ejpam-5125	440	22	ζ	ζ	NOUN
ejpam-5125	440	23	is	be	AUX
ejpam-5125	440	24	onto	onto	ADP
ejpam-5125	440	25	:	:	PUNCT
ejpam-5125	440	26	let	let	VERB
ejpam-5125	440	27	b	b	X
ejpam-5125	440	28	∈	∈	PROPN
ejpam-5125	440	29	pf	pf	X
ejpam-5125	440	30	(	(	PUNCT
ejpam-5125	440	31	v	v	NOUN
ejpam-5125	440	32	)	)	PUNCT
ejpam-5125	440	33	.	.	PUNCT
ejpam-5125	441	1	then	then	ADV
ejpam-5125	441	2	b	b	X
ejpam-5125	441	3	=	=	PUNCT
ejpam-5125	442	1	[	[	X
ejpam-5125	442	2	a	a	X
ejpam-5125	442	3	)	)	PUNCT
ejpam-5125	442	4	for	for	ADP
ejpam-5125	442	5	some	some	PRON
ejpam-5125	442	6	a	a	DET
ejpam-5125	442	7	∈	∈	PROPN
ejpam-5125	442	8	v	v	NOUN
ejpam-5125	442	9	.	.	PUNCT
ejpam-5125	443	1	hence	hence	ADV
ejpam-5125	443	2	(	(	PUNCT
ejpam-5125	443	3	a	a	DET
ejpam-5125	443	4	]	]	X
ejpam-5125	443	5	∈	∈	NOUN
ejpam-5125	443	6	pu(v	pu(v	NOUN
ejpam-5125	443	7	)	)	PUNCT
ejpam-5125	443	8	.	.	PUNCT
ejpam-5125	444	1	therefore	therefore	ADV
ejpam-5125	444	2	ζ((a	ζ((a	PROPN
ejpam-5125	444	3	]	]	PUNCT
ejpam-5125	444	4	)	)	PUNCT
ejpam-5125	444	5	=	=	PUNCT
ejpam-5125	445	1	[	[	X
ejpam-5125	445	2	a	a	X
ejpam-5125	445	3	)	)	PUNCT
ejpam-5125	445	4	=	=	SYM
ejpam-5125	445	5	b.	b.	PROPN
ejpam-5125	445	6	therefore	therefore	ADV
ejpam-5125	445	7	there	there	PRON
ejpam-5125	445	8	exists	exist	VERB
ejpam-5125	445	9	a	a	DET
ejpam-5125	445	10	dual	dual	ADJ
ejpam-5125	445	11	isomorphism	isomorphism	NOUN
ejpam-5125	445	12	from	from	ADP
ejpam-5125	445	13	pu(v	pu(v	NOUN
ejpam-5125	445	14	)	)	PUNCT
ejpam-5125	445	15	onto	onto	ADP
ejpam-5125	445	16	pf	pf	PROPN
ejpam-5125	445	17	(	(	PUNCT
ejpam-5125	445	18	v	v	NOUN
ejpam-5125	445	19	)	)	PUNCT
ejpam-5125	445	20	.	.	PUNCT
ejpam-5125	446	1	4	4	X
ejpam-5125	446	2	.	.	X
ejpam-5125	446	3	subdirectly	subdirectly	ADV
ejpam-5125	446	4	irreducible	irreducible	ADJ
ejpam-5125	446	5	pdls	pdl	NOUN
ejpam-5125	446	6	for	for	ADP
ejpam-5125	446	7	any	any	DET
ejpam-5125	446	8	algebra	algebra	NOUN
ejpam-5125	446	9	a	a	X
ejpam-5125	446	10	,	,	PUNCT
ejpam-5125	446	11	we	we	PRON
ejpam-5125	446	12	denote	denote	VERB
ejpam-5125	446	13	the	the	DET
ejpam-5125	446	14	structure	structure	NOUN
ejpam-5125	446	15	lattice	lattice	NOUN
ejpam-5125	446	16	of	of	ADP
ejpam-5125	446	17	a	a	PRON
ejpam-5125	446	18	,	,	PUNCT
ejpam-5125	446	19	that	that	PRON
ejpam-5125	446	20	is	be	AUX
ejpam-5125	446	21	the	the	DET
ejpam-5125	446	22	lattice	lattice	NOUN
ejpam-5125	446	23	of	of	ADP
ejpam-5125	446	24	all	all	DET
ejpam-5125	446	25	congruence	congruence	NOUN
ejpam-5125	446	26	relations	relation	NOUN
ejpam-5125	446	27	on	on	ADP
ejpam-5125	446	28	a	a	PRON
ejpam-5125	446	29	,	,	PUNCT
ejpam-5125	446	30	by	by	ADP
ejpam-5125	446	31	b	b	PROPN
ejpam-5125	446	32	in	in	ADP
ejpam-5125	446	33	which	which	PRON
ejpam-5125	446	34	the	the	DET
ejpam-5125	446	35	least	least	ADJ
ejpam-5125	446	36	element	element	NOUN
ejpam-5125	446	37	is	be	AUX
ejpam-5125	446	38	∆a	∆a	VERB
ejpam-5125	446	39	where	where	SCONJ
ejpam-5125	446	40	∆a	∆a	VERB
ejpam-5125	446	41	=	=	SYM
ejpam-5125	446	42	{	{	PUNCT
ejpam-5125	446	43	(	(	PUNCT
ejpam-5125	446	44	x	x	NOUN
ejpam-5125	446	45	,	,	PUNCT
ejpam-5125	446	46	y	y	PROPN
ejpam-5125	446	47	)	)	PUNCT
ejpam-5125	446	48	∈	∈	PROPN
ejpam-5125	446	49	a	a	DET
ejpam-5125	446	50	×	×	NOUN
ejpam-5125	446	51	a	a	DET
ejpam-5125	446	52	|	|	NOUN
ejpam-5125	446	53	x	x	ADP
ejpam-5125	446	54	=	=	SYM
ejpam-5125	446	55	y	y	PROPN
ejpam-5125	446	56	}	}	PUNCT
ejpam-5125	446	57	and	and	CCONJ
ejpam-5125	446	58	greatest	great	ADJ
ejpam-5125	446	59	element	element	NOUN
ejpam-5125	446	60	is	be	AUX
ejpam-5125	446	61	a	a	DET
ejpam-5125	446	62	×	×	PROPN
ejpam-5125	446	63	a.	a.	NOUN
ejpam-5125	446	64	recall	recall	NOUN
ejpam-5125	446	65	that	that	SCONJ
ejpam-5125	446	66	a	a	DET
ejpam-5125	446	67	non	non	ADJ
ejpam-5125	446	68	-	-	ADJ
ejpam-5125	446	69	trivial	trivial	ADJ
ejpam-5125	446	70	algebra	algebra	NOUN
ejpam-5125	446	71	a	a	PRON
ejpam-5125	446	72	is	be	AUX
ejpam-5125	446	73	said	say	VERB
ejpam-5125	446	74	to	to	PART
ejpam-5125	446	75	be	be	AUX
ejpam-5125	446	76	subdirectly	subdirectly	ADV
ejpam-5125	446	77	irreducible	irreducible	ADJ
ejpam-5125	446	78	if	if	SCONJ
ejpam-5125	446	79	intersection	intersection	NOUN
ejpam-5125	446	80	of	of	ADP
ejpam-5125	446	81	any	any	DET
ejpam-5125	446	82	family	family	NOUN
ejpam-5125	446	83	of	of	ADP
ejpam-5125	446	84	nonzero	nonzero	PROPN
ejpam-5125	446	85	congruences	congruence	NOUN
ejpam-5125	446	86	is	be	AUX
ejpam-5125	446	87	again	again	ADV
ejpam-5125	446	88	non	non	ADJ
ejpam-5125	446	89	-	-	ADJ
ejpam-5125	446	90	zero	zero	NUM
ejpam-5125	446	91	;	;	PUNCT
ejpam-5125	446	92	or	or	CCONJ
ejpam-5125	446	93	equivalently	equivalently	ADV
ejpam-5125	446	94	,	,	PUNCT
ejpam-5125	446	95	b	b	PROPN
ejpam-5125	446	96	has	have	VERB
ejpam-5125	446	97	smallest	small	ADJ
ejpam-5125	446	98	non	non	ADJ
ejpam-5125	446	99	-	-	ADJ
ejpam-5125	446	100	zero	zero	NUM
ejpam-5125	446	101	congruence	congruence	NOUN
ejpam-5125	446	102	.	.	PUNCT
ejpam-5125	447	1	we	we	PRON
ejpam-5125	447	2	characterize	characterize	VERB
ejpam-5125	447	3	subdirectly	subdirectly	ADV
ejpam-5125	447	4	irreducible	irreducible	ADJ
ejpam-5125	447	5	associative	associative	ADJ
ejpam-5125	447	6	pdls	pdl	NOUN
ejpam-5125	447	7	in	in	ADP
ejpam-5125	447	8	this	this	DET
ejpam-5125	447	9	section	section	NOUN
ejpam-5125	447	10	,	,	PUNCT
ejpam-5125	447	11	and	and	CCONJ
ejpam-5125	447	12	then	then	ADV
ejpam-5125	447	13	use	use	VERB
ejpam-5125	447	14	birkhoff	birkhoff	NOUN
ejpam-5125	447	15	’s	’s	PART
ejpam-5125	447	16	subdirect	subdirect	PROPN
ejpam-5125	447	17	representation	representation	NOUN
ejpam-5125	447	18	theorem	theorem	VERB
ejpam-5125	447	19	to	to	PART
ejpam-5125	447	20	obtain	obtain	VERB
ejpam-5125	447	21	a	a	DET
ejpam-5125	447	22	subdirect	subdirect	NOUN
ejpam-5125	447	23	representation	representation	NOUN
ejpam-5125	447	24	for	for	ADP
ejpam-5125	447	25	an	an	DET
ejpam-5125	447	26	associative	associative	ADJ
ejpam-5125	447	27	pdl	pdl	NOUN
ejpam-5125	447	28	.	.	PUNCT
ejpam-5125	448	1	this	this	DET
ejpam-5125	448	2	subdirect	subdirect	NOUN
ejpam-5125	448	3	representation	representation	NOUN
ejpam-5125	448	4	of	of	ADP
ejpam-5125	448	5	a	a	DET
ejpam-5125	448	6	pdl	pdl	NOUN
ejpam-5125	448	7	v	v	NOUN
ejpam-5125	448	8	is	be	AUX
ejpam-5125	448	9	crucial	crucial	ADJ
ejpam-5125	448	10	in	in	ADP
ejpam-5125	448	11	the	the	DET
ejpam-5125	448	12	theory	theory	NOUN
ejpam-5125	448	13	of	of	ADP
ejpam-5125	448	14	pdls	pdl	NOUN
ejpam-5125	448	15	since	since	SCONJ
ejpam-5125	448	16	it	it	PRON
ejpam-5125	448	17	simplifies	simplify	VERB
ejpam-5125	448	18	numerous	numerous	ADJ
ejpam-5125	448	19	lattice	lattice	ADJ
ejpam-5125	448	20	theoretic	theoretic	ADJ
ejpam-5125	448	21	computations	computation	NOUN
ejpam-5125	448	22	.	.	PUNCT
ejpam-5125	449	1	definition	definition	NOUN
ejpam-5125	449	2	9	9	NUM
ejpam-5125	449	3	.	.	PUNCT
ejpam-5125	450	1	let	let	VERB
ejpam-5125	450	2	v	v	PART
ejpam-5125	450	3	be	be	AUX
ejpam-5125	450	4	a	a	DET
ejpam-5125	450	5	pdl	pdl	NOUN
ejpam-5125	450	6	,	,	PUNCT
ejpam-5125	450	7	an	an	DET
ejpam-5125	450	8	element	element	NOUN
ejpam-5125	450	9	a	a	DET
ejpam-5125	450	10	∈	∈	PROPN
ejpam-5125	450	11	v	v	NOUN
ejpam-5125	450	12	is	be	AUX
ejpam-5125	450	13	said	say	VERB
ejpam-5125	450	14	to	to	PART
ejpam-5125	450	15	be	be	AUX
ejpam-5125	450	16	minimal	minimal	ADJ
ejpam-5125	450	17	if	if	SCONJ
ejpam-5125	450	18	for	for	ADP
ejpam-5125	450	19	any	any	DET
ejpam-5125	450	20	u	u	PROPN
ejpam-5125	450	21	∈	∈	PROPN
ejpam-5125	450	22	v	v	NOUN
ejpam-5125	450	23	,	,	PUNCT
ejpam-5125	450	24	u	u	NOUN
ejpam-5125	450	25	≤	≤	X
ejpam-5125	450	26	a⇒	a⇒	PRON
ejpam-5125	450	27	u	u	NOUN
ejpam-5125	450	28	=	=	NOUN
ejpam-5125	450	29	a.	a.	PROPN
ejpam-5125	450	30	lemma	lemma	PROPN
ejpam-5125	450	31	13	13	NUM
ejpam-5125	450	32	.	.	PUNCT
ejpam-5125	451	1	let	let	VERB
ejpam-5125	451	2	v	v	PART
ejpam-5125	451	3	be	be	AUX
ejpam-5125	451	4	a	a	DET
ejpam-5125	451	5	pdl	pdl	NOUN
ejpam-5125	451	6	.	.	PUNCT
ejpam-5125	452	1	then	then	ADV
ejpam-5125	452	2	for	for	ADP
ejpam-5125	452	3	any	any	DET
ejpam-5125	452	4	a	a	DET
ejpam-5125	452	5	∈	∈	PROPN
ejpam-5125	452	6	v	v	NOUN
ejpam-5125	452	7	,	,	PUNCT
ejpam-5125	452	8	the	the	DET
ejpam-5125	452	9	following	follow	VERB
ejpam-5125	452	10	are	be	AUX
ejpam-5125	452	11	equivalent	equivalent	ADJ
ejpam-5125	452	12	:	:	PUNCT
ejpam-5125	452	13	(	(	PUNCT
ejpam-5125	452	14	1	1	NUM
ejpam-5125	452	15	)	)	PUNCT
ejpam-5125	452	16	.	.	PUNCT
ejpam-5125	453	1	a	a	PRON
ejpam-5125	453	2	is	be	AUX
ejpam-5125	453	3	minimal	minimal	ADJ
ejpam-5125	453	4	(	(	PUNCT
ejpam-5125	453	5	2	2	NUM
ejpam-5125	453	6	)	)	PUNCT
ejpam-5125	453	7	.	.	PUNCT
ejpam-5125	454	1	p1	p1	PROPN
ejpam-5125	454	2	∧	∧	PROPN
ejpam-5125	454	3	a	a	DET
ejpam-5125	454	4	=	=	X
ejpam-5125	454	5	a	a	NOUN
ejpam-5125	454	6	for	for	ADP
ejpam-5125	454	7	all	all	DET
ejpam-5125	454	8	p1	p1	PROPN
ejpam-5125	454	9	∈	∈	PROPN
ejpam-5125	454	10	v	v	NOUN
ejpam-5125	454	11	(	(	PUNCT
ejpam-5125	454	12	3	3	NUM
ejpam-5125	454	13	)	)	PUNCT
ejpam-5125	454	14	.	.	PUNCT
ejpam-5125	455	1	p1	p1	PROPN
ejpam-5125	455	2	∨	∨	NUM
ejpam-5125	455	3	a	a	DET
ejpam-5125	455	4	=	=	X
ejpam-5125	455	5	p1	p1	NOUN
ejpam-5125	455	6	for	for	ADP
ejpam-5125	455	7	all	all	DET
ejpam-5125	455	8	p1	p1	PROPN
ejpam-5125	455	9	∈	∈	PROPN
ejpam-5125	455	10	v	v	NOUN
ejpam-5125	455	11	.	.	PUNCT
ejpam-5125	456	1	definition	definition	NOUN
ejpam-5125	456	2	10	10	NUM
ejpam-5125	456	3	.	.	PUNCT
ejpam-5125	457	1	an	an	DET
ejpam-5125	457	2	equivalence	equivalence	NOUN
ejpam-5125	457	3	relation	relation	NOUN
ejpam-5125	457	4	θ	θ	PROPN
ejpam-5125	457	5	on	on	ADP
ejpam-5125	457	6	a	a	DET
ejpam-5125	457	7	pdl	pdl	NOUN
ejpam-5125	457	8	v	v	NOUN
ejpam-5125	457	9	,	,	PUNCT
ejpam-5125	457	10	is	be	AUX
ejpam-5125	457	11	called	call	VERB
ejpam-5125	457	12	a	a	DET
ejpam-5125	457	13	congruence	congruence	NOUN
ejpam-5125	457	14	relation	relation	NOUN
ejpam-5125	457	15	on	on	ADP
ejpam-5125	457	16	v	v	NOUN
ejpam-5125	457	17	if	if	SCONJ
ejpam-5125	457	18	(	(	PUNCT
ejpam-5125	457	19	a	a	DET
ejpam-5125	457	20	∧	∧	PROPN
ejpam-5125	457	21	c	c	PROPN
ejpam-5125	457	22	,	,	PUNCT
ejpam-5125	457	23	b	b	PROPN
ejpam-5125	457	24	∧	∧	PROPN
ejpam-5125	457	25	d	d	PROPN
ejpam-5125	457	26	)	)	PUNCT
ejpam-5125	457	27	,	,	PUNCT
ejpam-5125	457	28	(	(	PUNCT
ejpam-5125	457	29	a	a	DET
ejpam-5125	457	30	∨	∨	NUM
ejpam-5125	457	31	c	c	NOUN
ejpam-5125	457	32	,	,	PUNCT
ejpam-5125	457	33	b	b	PROPN
ejpam-5125	457	34	∨	∨	NUM
ejpam-5125	457	35	d	d	NOUN
ejpam-5125	457	36	)	)	PUNCT
ejpam-5125	457	37	∈	∈	PROPN
ejpam-5125	457	38	θ	θ	PROPN
ejpam-5125	457	39	,	,	PUNCT
ejpam-5125	457	40	for	for	ADP
ejpam-5125	457	41	all	all	PRON
ejpam-5125	457	42	(	(	PUNCT
ejpam-5125	457	43	a	a	DET
ejpam-5125	457	44	,	,	PUNCT
ejpam-5125	457	45	b	b	NOUN
ejpam-5125	457	46	)	)	PUNCT
ejpam-5125	457	47	,	,	PUNCT
ejpam-5125	457	48	(	(	PUNCT
ejpam-5125	457	49	c	c	X
ejpam-5125	457	50	,	,	PUNCT
ejpam-5125	457	51	d	d	NOUN
ejpam-5125	457	52	)	)	PUNCT
ejpam-5125	457	53	∈	∈	PROPN
ejpam-5125	457	54	θ	θ	PROPN
ejpam-5125	457	55	.	.	PUNCT
ejpam-5125	457	56	lemma	lemma	PROPN
ejpam-5125	457	57	14	14	NUM
ejpam-5125	457	58	.	.	PUNCT
ejpam-5125	458	1	for	for	ADP
ejpam-5125	458	2	any	any	DET
ejpam-5125	458	3	a	a	DET
ejpam-5125	458	4	∈	∈	PROPN
ejpam-5125	458	5	v	v	NOUN
ejpam-5125	458	6	,	,	PUNCT
ejpam-5125	458	7	φa	φa	ADP
ejpam-5125	458	8	=	=	SYM
ejpam-5125	458	9	{	{	PUNCT
ejpam-5125	458	10	(	(	PUNCT
ejpam-5125	458	11	p1	p1	NOUN
ejpam-5125	458	12	,	,	PUNCT
ejpam-5125	458	13	p2	p2	X
ejpam-5125	458	14	)	)	PUNCT
ejpam-5125	458	15	∈	∈	NOUN
ejpam-5125	459	1	v	v	AUX
ejpam-5125	459	2	×	×	NOUN
ejpam-5125	459	3	v	v	ADP
ejpam-5125	459	4	|	|	ADV
ejpam-5125	459	5	p1	p1	PROPN
ejpam-5125	459	6	∨	∨	NUM
ejpam-5125	459	7	a	a	DET
ejpam-5125	459	8	=	=	NOUN
ejpam-5125	459	9	p2	p2	PROPN
ejpam-5125	459	10	∨	∨	NOUN
ejpam-5125	459	11	a	a	PRON
ejpam-5125	459	12	}	}	PUNCT
ejpam-5125	459	13	is	be	AUX
ejpam-5125	459	14	a	a	DET
ejpam-5125	459	15	congruence	congruence	NOUN
ejpam-5125	459	16	relation	relation	NOUN
ejpam-5125	459	17	on	on	ADP
ejpam-5125	459	18	v	v	NUM
ejpam-5125	459	19	.	.	PUNCT
ejpam-5125	460	1	further	far	ADV
ejpam-5125	460	2	,	,	PUNCT
ejpam-5125	460	3	φa	φa	ADP
ejpam-5125	460	4	=	=	SYM
ejpam-5125	460	5	∆v	∆v	PROPN
ejpam-5125	460	6	if	if	SCONJ
ejpam-5125	460	7	and	and	CCONJ
ejpam-5125	460	8	only	only	ADV
ejpam-5125	460	9	if	if	SCONJ
ejpam-5125	460	10	a	a	PRON
ejpam-5125	460	11	is	be	AUX
ejpam-5125	460	12	minimal	minimal	ADJ
ejpam-5125	460	13	element	element	NOUN
ejpam-5125	460	14	of	of	ADP
ejpam-5125	460	15	v	v	NOUN
ejpam-5125	460	16	.	.	PUNCT
ejpam-5125	461	1	proof	proof	NOUN
ejpam-5125	461	2	.	.	PUNCT
ejpam-5125	462	1	given	give	VERB
ejpam-5125	462	2	φa	φa	ADP
ejpam-5125	462	3	=	=	SYM
ejpam-5125	462	4	{	{	PUNCT
ejpam-5125	462	5	(	(	PUNCT
ejpam-5125	462	6	p1	p1	NOUN
ejpam-5125	462	7	,	,	PUNCT
ejpam-5125	462	8	p2	p2	X
ejpam-5125	462	9	)	)	PUNCT
ejpam-5125	462	10	∈	∈	NOUN
ejpam-5125	462	11	v	v	ADP
ejpam-5125	462	12	×	×	NOUN
ejpam-5125	462	13	v	v	ADP
ejpam-5125	462	14	|	|	ADV
ejpam-5125	462	15	p1	p1	PROPN
ejpam-5125	462	16	∨	∨	NUM
ejpam-5125	462	17	a	a	DET
ejpam-5125	462	18	=	=	NOUN
ejpam-5125	462	19	p2	p2	PROPN
ejpam-5125	462	20	∨	∨	NUM
ejpam-5125	462	21	a	a	PRON
ejpam-5125	462	22	}	}	PUNCT
ejpam-5125	462	23	.	.	PUNCT
ejpam-5125	463	1	clearly	clearly	ADV
ejpam-5125	463	2	φa	φa	INTJ
ejpam-5125	463	3	is	be	AUX
ejpam-5125	463	4	an	an	DET
ejpam-5125	463	5	equivalence	equivalence	NOUN
ejpam-5125	463	6	relation	relation	NOUN
ejpam-5125	463	7	on	on	ADP
ejpam-5125	463	8	v	v	NUM
ejpam-5125	463	9	.	.	PUNCT
ejpam-5125	464	1	let	let	VERB
ejpam-5125	464	2	(	(	PUNCT
ejpam-5125	464	3	p1	p1	NOUN
ejpam-5125	464	4	,	,	PUNCT
ejpam-5125	464	5	p2	p2	PROPN
ejpam-5125	464	6	)	)	PUNCT
ejpam-5125	464	7	,	,	PUNCT
ejpam-5125	464	8	(	(	PUNCT
ejpam-5125	464	9	p3	p3	PROPN
ejpam-5125	464	10	,	,	PUNCT
ejpam-5125	464	11	s	s	X
ejpam-5125	464	12	)	)	PUNCT
ejpam-5125	464	13	∈	∈	PROPN
ejpam-5125	464	14	φa	φa	PROPN
ejpam-5125	464	15	.	.	PUNCT
ejpam-5125	465	1	then	then	ADV
ejpam-5125	465	2	p1	p1	PROPN
ejpam-5125	465	3	∨	∨	NUM
ejpam-5125	465	4	a	a	DET
ejpam-5125	465	5	=	=	NOUN
ejpam-5125	465	6	p2	p2	PROPN
ejpam-5125	465	7	∨	∨	NUM
ejpam-5125	465	8	a	a	PRON
ejpam-5125	465	9	and	and	CCONJ
ejpam-5125	465	10	p3	p3	PROPN
ejpam-5125	465	11	∨	∨	NUM
ejpam-5125	465	12	a	a	DET
ejpam-5125	465	13	=	=	SYM
ejpam-5125	465	14	s	s	NOUN
ejpam-5125	465	15	∨	∨	NOUN
ejpam-5125	465	16	a.	a.	NOUN
ejpam-5125	465	17	hence	hence	ADV
ejpam-5125	465	18	(	(	PUNCT
ejpam-5125	465	19	p1	p1	PROPN
ejpam-5125	465	20	∨	∨	NUM
ejpam-5125	465	21	p3	p3	PROPN
ejpam-5125	465	22	)	)	PUNCT
ejpam-5125	465	23	∨	∨	NUM
ejpam-5125	465	24	a	a	DET
ejpam-5125	465	25	=	=	PROPN
ejpam-5125	465	26	p1	p1	PROPN
ejpam-5125	465	27	∨	∨	PROPN
ejpam-5125	465	28	p3	p3	PROPN
ejpam-5125	465	29	∨	∨	NUM
ejpam-5125	465	30	a	a	DET
ejpam-5125	465	31	=	=	PROPN
ejpam-5125	465	32	p1	p1	PROPN
ejpam-5125	465	33	∨	∨	NUM
ejpam-5125	465	34	s	s	PROPN
ejpam-5125	465	35	∨	∨	NOUN
ejpam-5125	465	36	a	a	DET
ejpam-5125	465	37	=	=	PROPN
ejpam-5125	465	38	p1	p1	PROPN
ejpam-5125	465	39	∨	∨	NUM
ejpam-5125	465	40	a	a	DET
ejpam-5125	465	41	∨	∨	NUM
ejpam-5125	465	42	s	s	PART
ejpam-5125	465	43	=	=	NOUN
ejpam-5125	465	44	p2	p2	PROPN
ejpam-5125	465	45	∨	∨	NUM
ejpam-5125	465	46	a	a	DET
ejpam-5125	465	47	∨	∨	NUM
ejpam-5125	465	48	s	s	PART
ejpam-5125	465	49	=	=	NOUN
ejpam-5125	465	50	p2	p2	AUX
ejpam-5125	465	51	∨	∨	NUM
ejpam-5125	465	52	s	s	PART
ejpam-5125	465	53	∨	∨	NOUN
ejpam-5125	465	54	a	a	DET
ejpam-5125	465	55	and	and	CCONJ
ejpam-5125	465	56	(	(	PUNCT
ejpam-5125	465	57	p1	p1	PROPN
ejpam-5125	465	58	∧	∧	PROPN
ejpam-5125	465	59	p3	p3	PROPN
ejpam-5125	465	60	)	)	PUNCT
ejpam-5125	465	61	∨	∨	ADP
ejpam-5125	466	1	a	a	DET
ejpam-5125	466	2	=	=	PUNCT
ejpam-5125	466	3	(	(	PUNCT
ejpam-5125	466	4	p1	p1	PROPN
ejpam-5125	466	5	∨	∨	NUM
ejpam-5125	466	6	a	a	PRON
ejpam-5125	466	7	)	)	PUNCT
ejpam-5125	466	8	∧	∧	PROPN
ejpam-5125	466	9	(	(	PUNCT
ejpam-5125	466	10	p3	p3	PROPN
ejpam-5125	466	11	∨	∨	NUM
ejpam-5125	466	12	a	a	PRON
ejpam-5125	466	13	)	)	PUNCT
ejpam-5125	466	14	=	=	SYM
ejpam-5125	466	15	(	(	PUNCT
ejpam-5125	466	16	p2	p2	PROPN
ejpam-5125	466	17	∨	∨	NUM
ejpam-5125	466	18	a	a	PRON
ejpam-5125	466	19	)	)	PUNCT
ejpam-5125	466	20	∧	∧	NOUN
ejpam-5125	466	21	(	(	PUNCT
ejpam-5125	466	22	s	s	PROPN
ejpam-5125	466	23	∨	∨	NOUN
ejpam-5125	466	24	a	a	PRON
ejpam-5125	466	25	)	)	PUNCT
ejpam-5125	466	26	=	=	SYM
ejpam-5125	466	27	(	(	PUNCT
ejpam-5125	467	1	p2	p2	PROPN
ejpam-5125	467	2	∧	∧	PROPN
ejpam-5125	467	3	s	s	PART
ejpam-5125	467	4	)	)	PUNCT
ejpam-5125	467	5	∨	∨	PROPN
ejpam-5125	467	6	a.	a.	PROPN
ejpam-5125	467	7	r.	r.	PROPN
ejpam-5125	467	8	bandaru	bandaru	PROPN
ejpam-5125	467	9	,	,	PUNCT
ejpam-5125	467	10	s.	s.	PROPN
ejpam-5125	467	11	ajjarapu	ajjarapu	PROPN
ejpam-5125	467	12	/	/	PUNCT
ejpam-5125	467	13	eur	eur	PROPN
ejpam-5125	467	14	.	.	PUNCT
ejpam-5125	468	1	j.	j.	PROPN
ejpam-5125	468	2	pure	pure	PROPN
ejpam-5125	468	3	appl	appl	PROPN
ejpam-5125	468	4	.	.	PROPN
ejpam-5125	468	5	math	math	PROPN
ejpam-5125	468	6	,	,	PUNCT
ejpam-5125	468	7	17	17	NUM
ejpam-5125	468	8	(	(	PUNCT
ejpam-5125	468	9	2	2	NUM
ejpam-5125	468	10	)	)	PUNCT
ejpam-5125	468	11	(	(	PUNCT
ejpam-5125	468	12	2024	2024	NUM
ejpam-5125	468	13	)	)	PUNCT
ejpam-5125	468	14	,	,	PUNCT
ejpam-5125	468	15	819	819	NUM
ejpam-5125	468	16	-	-	SYM
ejpam-5125	468	17	834	834	NUM
ejpam-5125	468	18	832	832	NUM
ejpam-5125	468	19	therefore	therefore	ADV
ejpam-5125	468	20	(	(	PUNCT
ejpam-5125	468	21	p1	p1	PROPN
ejpam-5125	468	22	∨	∨	NUM
ejpam-5125	468	23	p3	p3	PROPN
ejpam-5125	468	24	,	,	PUNCT
ejpam-5125	468	25	p2	p2	PROPN
ejpam-5125	468	26	∨	∨	NUM
ejpam-5125	468	27	s),∈	s),∈	PROPN
ejpam-5125	468	28	φa	φa	PROPN
ejpam-5125	469	1	and	and	CCONJ
ejpam-5125	469	2	(	(	PUNCT
ejpam-5125	469	3	p1	p1	PROPN
ejpam-5125	469	4	∧	∧	PROPN
ejpam-5125	469	5	p3	p3	PROPN
ejpam-5125	469	6	,	,	PUNCT
ejpam-5125	469	7	p2	p2	PROPN
ejpam-5125	469	8	∧	∧	PROPN
ejpam-5125	469	9	s	s	PART
ejpam-5125	469	10	)	)	PUNCT
ejpam-5125	469	11	∈	∈	PROPN
ejpam-5125	469	12	φa	φa	PROPN
ejpam-5125	469	13	.	.	PUNCT
ejpam-5125	470	1	therefore	therefore	ADV
ejpam-5125	470	2	φa	φa	INTJ
ejpam-5125	470	3	is	be	AUX
ejpam-5125	470	4	a	a	DET
ejpam-5125	470	5	congruence	congruence	NOUN
ejpam-5125	470	6	relation	relation	NOUN
ejpam-5125	470	7	on	on	ADP
ejpam-5125	470	8	v	v	NUM
ejpam-5125	470	9	.	.	PUNCT
ejpam-5125	471	1	assume	assume	VERB
ejpam-5125	471	2	φa	φa	ADP
ejpam-5125	471	3	=	=	PROPN
ejpam-5125	471	4	∆v	∆v	PROPN
ejpam-5125	471	5	.	.	PUNCT
ejpam-5125	472	1	let	let	VERB
ejpam-5125	472	2	p1	p1	PROPN
ejpam-5125	472	3	∈	∈	PROPN
ejpam-5125	472	4	v	v	AUX
ejpam-5125	472	5	be	be	AUX
ejpam-5125	472	6	such	such	ADJ
ejpam-5125	472	7	that	that	SCONJ
ejpam-5125	472	8	p1	p1	PROPN
ejpam-5125	472	9	≤	≤	NUM
ejpam-5125	472	10	a.	a.	NOUN
ejpam-5125	472	11	then	then	ADV
ejpam-5125	472	12	a	a	DET
ejpam-5125	472	13	∨	∨	NOUN
ejpam-5125	472	14	a	a	DET
ejpam-5125	472	15	=	=	NOUN
ejpam-5125	472	16	a	a	DET
ejpam-5125	472	17	=	=	PROPN
ejpam-5125	472	18	p1	p1	PROPN
ejpam-5125	472	19	∨	∨	NUM
ejpam-5125	472	20	a	a	PRON
ejpam-5125	472	21	which	which	PRON
ejpam-5125	472	22	implies	imply	VERB
ejpam-5125	472	23	(	(	PUNCT
ejpam-5125	472	24	a	a	PRON
ejpam-5125	472	25	,	,	PUNCT
ejpam-5125	472	26	p1	p1	NOUN
ejpam-5125	472	27	)	)	PUNCT
ejpam-5125	472	28	∈	∈	NOUN
ejpam-5125	472	29	φa	φa	ADP
ejpam-5125	472	30	=	=	SYM
ejpam-5125	472	31	∆v	∆v	PROPN
ejpam-5125	472	32	.	.	PUNCT
ejpam-5125	473	1	therefore	therefore	ADV
ejpam-5125	473	2	a	a	DET
ejpam-5125	473	3	=	=	SYM
ejpam-5125	473	4	p1	p1	NOUN
ejpam-5125	473	5	.	.	PUNCT
ejpam-5125	474	1	hence	hence	ADV
ejpam-5125	474	2	a	a	PRON
ejpam-5125	474	3	is	be	AUX
ejpam-5125	474	4	minimal	minimal	ADJ
ejpam-5125	474	5	element	element	NOUN
ejpam-5125	474	6	.	.	PUNCT
ejpam-5125	475	1	let	let	VERB
ejpam-5125	475	2	(	(	PUNCT
ejpam-5125	475	3	p1	p1	NOUN
ejpam-5125	475	4	,	,	PUNCT
ejpam-5125	475	5	p2	p2	X
ejpam-5125	475	6	)	)	PUNCT
ejpam-5125	475	7	∈	∈	PROPN
ejpam-5125	475	8	φa	φa	PROPN
ejpam-5125	475	9	.	.	PUNCT
ejpam-5125	476	1	then	then	ADV
ejpam-5125	476	2	p1	p1	PROPN
ejpam-5125	476	3	∨	∨	NUM
ejpam-5125	476	4	a	a	DET
ejpam-5125	476	5	=	=	NOUN
ejpam-5125	476	6	p2	p2	PROPN
ejpam-5125	476	7	∨	∨	NUM
ejpam-5125	476	8	a.	a.	NOUN
ejpam-5125	476	9	since	since	SCONJ
ejpam-5125	476	10	a	a	PRON
ejpam-5125	476	11	is	be	AUX
ejpam-5125	476	12	minimal	minimal	ADJ
ejpam-5125	476	13	element	element	NOUN
ejpam-5125	476	14	,	,	PUNCT
ejpam-5125	476	15	we	we	PRON
ejpam-5125	476	16	have	have	VERB
ejpam-5125	476	17	p1	p1	NOUN
ejpam-5125	476	18	=	=	NOUN
ejpam-5125	476	19	p2	p2	NOUN
ejpam-5125	476	20	.	.	PUNCT
ejpam-5125	477	1	hence	hence	ADV
ejpam-5125	477	2	φ	φ	PROPN
ejpam-5125	477	3	a	a	DET
ejpam-5125	477	4	=	=	X
ejpam-5125	477	5	∆v	∆v	PROPN
ejpam-5125	477	6	.	.	PUNCT
ejpam-5125	478	1	lemma	lemma	PROPN
ejpam-5125	478	2	15	15	NUM
ejpam-5125	478	3	.	.	PUNCT
ejpam-5125	479	1	if	if	SCONJ
ejpam-5125	479	2	v	v	NOUN
ejpam-5125	479	3	is	be	AUX
ejpam-5125	479	4	an	an	DET
ejpam-5125	479	5	associative	associative	ADJ
ejpam-5125	479	6	pdl	pdl	NOUN
ejpam-5125	479	7	,	,	PUNCT
ejpam-5125	479	8	then	then	ADV
ejpam-5125	479	9	for	for	ADP
ejpam-5125	479	10	any	any	DET
ejpam-5125	479	11	a	a	DET
ejpam-5125	479	12	∈	∈	PROPN
ejpam-5125	479	13	v	v	NOUN
ejpam-5125	479	14	,	,	PUNCT
ejpam-5125	479	15	θa	θa	X
ejpam-5125	479	16	=	=	SYM
ejpam-5125	479	17	{	{	PUNCT
ejpam-5125	479	18	(	(	PUNCT
ejpam-5125	479	19	p1	p1	NOUN
ejpam-5125	479	20	,	,	PUNCT
ejpam-5125	479	21	p2	p2	X
ejpam-5125	479	22	)	)	PUNCT
ejpam-5125	479	23	∈	∈	NOUN
ejpam-5125	479	24	v	v	NOUN
ejpam-5125	479	25	×v	×v	NOUN
ejpam-5125	479	26	|	|	ADV
ejpam-5125	479	27	p1∧a	p1∧a	PROPN
ejpam-5125	479	28	=	=	SYM
ejpam-5125	479	29	p2∧a	p2∧a	PROPN
ejpam-5125	479	30	}	}	PUNCT
ejpam-5125	479	31	is	be	AUX
ejpam-5125	479	32	a	a	DET
ejpam-5125	479	33	congruence	congruence	NOUN
ejpam-5125	479	34	relation	relation	NOUN
ejpam-5125	479	35	on	on	ADP
ejpam-5125	479	36	v	v	NUM
ejpam-5125	479	37	.	.	PUNCT
ejpam-5125	480	1	further	far	ADV
ejpam-5125	480	2	,	,	PUNCT
ejpam-5125	480	3	θa	θa	X
ejpam-5125	480	4	=	=	SYM
ejpam-5125	480	5	∆v	∆v	PROPN
ejpam-5125	480	6	if	if	SCONJ
ejpam-5125	480	7	and	and	CCONJ
ejpam-5125	480	8	only	only	ADV
ejpam-5125	480	9	if	if	SCONJ
ejpam-5125	480	10	a	a	PRON
ejpam-5125	480	11	is	be	AUX
ejpam-5125	480	12	unity	unity	NOUN
ejpam-5125	480	13	(	(	PUNCT
ejpam-5125	480	14	greatest	great	ADJ
ejpam-5125	480	15	)	)	PUNCT
ejpam-5125	480	16	element	element	NOUN
ejpam-5125	480	17	of	of	ADP
ejpam-5125	480	18	v	v	NOUN
ejpam-5125	480	19	.	.	PUNCT
ejpam-5125	481	1	proof	proof	NOUN
ejpam-5125	481	2	.	.	PUNCT
ejpam-5125	482	1	let	let	VERB
ejpam-5125	482	2	θa	θa	NOUN
ejpam-5125	482	3	=	=	VERB
ejpam-5125	482	4	{	{	PUNCT
ejpam-5125	482	5	(	(	PUNCT
ejpam-5125	482	6	p1	p1	NOUN
ejpam-5125	482	7	,	,	PUNCT
ejpam-5125	482	8	p2	p2	X
ejpam-5125	482	9	)	)	PUNCT
ejpam-5125	482	10	∈	∈	NOUN
ejpam-5125	482	11	v	v	ADP
ejpam-5125	482	12	×	×	NOUN
ejpam-5125	482	13	v	v	ADP
ejpam-5125	482	14	|	|	NOUN
ejpam-5125	482	15	p1	p1	NOUN
ejpam-5125	482	16	∧	∧	PROPN
ejpam-5125	483	1	a	a	DET
ejpam-5125	483	2	=	=	NOUN
ejpam-5125	483	3	p2	p2	X
ejpam-5125	483	4	∧	∧	PROPN
ejpam-5125	483	5	a	a	PRON
ejpam-5125	483	6	}	}	PUNCT
ejpam-5125	483	7	.	.	PUNCT
ejpam-5125	484	1	clearly	clearly	ADV
ejpam-5125	484	2	θa	θa	PRON
ejpam-5125	484	3	is	be	AUX
ejpam-5125	484	4	an	an	DET
ejpam-5125	484	5	equivalence	equivalence	NOUN
ejpam-5125	484	6	relation	relation	NOUN
ejpam-5125	484	7	on	on	ADP
ejpam-5125	484	8	v	v	NUM
ejpam-5125	484	9	.	.	PUNCT
ejpam-5125	485	1	let	let	VERB
ejpam-5125	485	2	(	(	PUNCT
ejpam-5125	485	3	p1	p1	NOUN
ejpam-5125	485	4	,	,	PUNCT
ejpam-5125	485	5	p2	p2	X
ejpam-5125	485	6	)	)	PUNCT
ejpam-5125	485	7	∈	∈	NOUN
ejpam-5125	485	8	θa	θa	NUM
ejpam-5125	485	9	and	and	CCONJ
ejpam-5125	485	10	(	(	PUNCT
ejpam-5125	485	11	p3	p3	PROPN
ejpam-5125	485	12	,	,	PUNCT
ejpam-5125	485	13	t	t	PROPN
ejpam-5125	485	14	)	)	PUNCT
ejpam-5125	485	15	∈	∈	PROPN
ejpam-5125	485	16	θa	θa	NUM
ejpam-5125	485	17	.	.	PUNCT
ejpam-5125	486	1	then	then	ADV
ejpam-5125	486	2	p1	p1	PROPN
ejpam-5125	486	3	∧	∧	PROPN
ejpam-5125	486	4	a	a	DET
ejpam-5125	486	5	=	=	NOUN
ejpam-5125	486	6	p2	p2	X
ejpam-5125	486	7	∧	∧	PROPN
ejpam-5125	486	8	a	a	PROPN
ejpam-5125	486	9	and	and	CCONJ
ejpam-5125	486	10	p3	p3	PROPN
ejpam-5125	486	11	∧	∧	PROPN
ejpam-5125	486	12	a	a	DET
ejpam-5125	486	13	=	=	SYM
ejpam-5125	486	14	t	t	NOUN
ejpam-5125	486	15	∧	∧	PROPN
ejpam-5125	486	16	a.	a.	NOUN
ejpam-5125	486	17	hence	hence	ADV
ejpam-5125	486	18	(	(	PUNCT
ejpam-5125	486	19	p1	p1	PROPN
ejpam-5125	486	20	∧	∧	PROPN
ejpam-5125	486	21	p3	p3	PROPN
ejpam-5125	486	22	)	)	PUNCT
ejpam-5125	487	1	∧	∧	NOUN
ejpam-5125	487	2	a	a	DET
ejpam-5125	487	3	=	=	SYM
ejpam-5125	487	4	p1	p1	NOUN
ejpam-5125	487	5	∧	∧	PROPN
ejpam-5125	487	6	p3	p3	PROPN
ejpam-5125	487	7	∧	∧	PROPN
ejpam-5125	487	8	a	a	DET
ejpam-5125	487	9	=	=	X
ejpam-5125	487	10	p1	p1	NOUN
ejpam-5125	487	11	∧	∧	PROPN
ejpam-5125	487	12	t	t	PROPN
ejpam-5125	487	13	∧	∧	PROPN
ejpam-5125	487	14	a	a	DET
ejpam-5125	487	15	=	=	X
ejpam-5125	487	16	p1	p1	PROPN
ejpam-5125	487	17	∧	∧	PROPN
ejpam-5125	487	18	a	a	DET
ejpam-5125	487	19	∧	∧	PROPN
ejpam-5125	487	20	t	t	PROPN
ejpam-5125	487	21	∧	∧	PROPN
ejpam-5125	487	22	a	a	DET
ejpam-5125	487	23	=	=	NOUN
ejpam-5125	487	24	p2	p2	X
ejpam-5125	487	25	∧	∧	PROPN
ejpam-5125	487	26	a	a	DET
ejpam-5125	487	27	∧	∧	PROPN
ejpam-5125	487	28	t	t	PROPN
ejpam-5125	487	29	∧	∧	PROPN
ejpam-5125	487	30	a	a	DET
ejpam-5125	487	31	=	=	NOUN
ejpam-5125	487	32	p2	p2	NOUN
ejpam-5125	487	33	∧	∧	PROPN
ejpam-5125	487	34	t	t	PROPN
ejpam-5125	487	35	∧	∧	PROPN
ejpam-5125	487	36	a	a	PRON
ejpam-5125	487	37	=	=	X
ejpam-5125	487	38	(	(	PUNCT
ejpam-5125	487	39	p2	p2	PROPN
ejpam-5125	487	40	∧	∧	PROPN
ejpam-5125	487	41	t	t	PROPN
ejpam-5125	487	42	)	)	PUNCT
ejpam-5125	487	43	∧	∧	PROPN
ejpam-5125	487	44	a	a	PRON
ejpam-5125	487	45	and	and	CCONJ
ejpam-5125	487	46	(	(	PUNCT
ejpam-5125	487	47	p1	p1	PROPN
ejpam-5125	487	48	∨	∨	NUM
ejpam-5125	487	49	p3	p3	PROPN
ejpam-5125	487	50	)	)	PUNCT
ejpam-5125	487	51	∧	∧	NOUN
ejpam-5125	487	52	a	a	NOUN
ejpam-5125	487	53	=	=	X
ejpam-5125	487	54	(	(	PUNCT
ejpam-5125	487	55	p1	p1	PROPN
ejpam-5125	487	56	∧	∧	PROPN
ejpam-5125	487	57	a	a	PRON
ejpam-5125	487	58	)	)	PUNCT
ejpam-5125	487	59	∨	∨	PROPN
ejpam-5125	487	60	(	(	PUNCT
ejpam-5125	487	61	p3	p3	PROPN
ejpam-5125	487	62	∧	∧	PROPN
ejpam-5125	487	63	a	a	PRON
ejpam-5125	487	64	)	)	PUNCT
ejpam-5125	487	65	=	=	SYM
ejpam-5125	487	66	(	(	PUNCT
ejpam-5125	487	67	p2	p2	PROPN
ejpam-5125	487	68	∧	∧	PROPN
ejpam-5125	487	69	a	a	PRON
ejpam-5125	487	70	)	)	PUNCT
ejpam-5125	487	71	∨	∨	PROPN
ejpam-5125	487	72	(	(	PUNCT
ejpam-5125	487	73	t	t	PROPN
ejpam-5125	487	74	∧	∧	PROPN
ejpam-5125	487	75	a	a	NOUN
ejpam-5125	487	76	)	)	PUNCT
ejpam-5125	487	77	=	=	SYM
ejpam-5125	487	78	(	(	PUNCT
ejpam-5125	487	79	p2	p2	PROPN
ejpam-5125	487	80	∨	∨	NUM
ejpam-5125	487	81	t	t	PROPN
ejpam-5125	487	82	)	)	PUNCT
ejpam-5125	487	83	∧	∧	PROPN
ejpam-5125	487	84	a.	a.	NOUN
ejpam-5125	487	85	therefore	therefore	ADV
ejpam-5125	487	86	(	(	PUNCT
ejpam-5125	487	87	p1	p1	PROPN
ejpam-5125	487	88	∧	∧	PROPN
ejpam-5125	487	89	p3	p3	PROPN
ejpam-5125	487	90	,	,	PUNCT
ejpam-5125	487	91	p2	p2	PROPN
ejpam-5125	487	92	∧	∧	PROPN
ejpam-5125	487	93	t	t	PROPN
ejpam-5125	487	94	)	)	PUNCT
ejpam-5125	487	95	,	,	PUNCT
ejpam-5125	487	96	(	(	PUNCT
ejpam-5125	487	97	p1	p1	PROPN
ejpam-5125	487	98	∨	∨	NUM
ejpam-5125	487	99	p3	p3	PROPN
ejpam-5125	487	100	,	,	PUNCT
ejpam-5125	487	101	p2	p2	PROPN
ejpam-5125	487	102	∨	∨	NUM
ejpam-5125	487	103	t	t	PROPN
ejpam-5125	487	104	)	)	PUNCT
ejpam-5125	487	105	∈	∈	PROPN
ejpam-5125	487	106	θa	θa	NUM
ejpam-5125	487	107	.	.	PUNCT
ejpam-5125	488	1	hence	hence	ADV
ejpam-5125	488	2	θa	θa	PRON
ejpam-5125	488	3	is	be	AUX
ejpam-5125	488	4	a	a	DET
ejpam-5125	488	5	congruence	congruence	NOUN
ejpam-5125	488	6	relation	relation	NOUN
ejpam-5125	488	7	on	on	ADP
ejpam-5125	488	8	v	v	NUM
ejpam-5125	488	9	.	.	PUNCT
ejpam-5125	489	1	now	now	ADV
ejpam-5125	489	2	assume	assume	VERB
ejpam-5125	489	3	θa	θa	NUM
ejpam-5125	489	4	=	=	PROPN
ejpam-5125	489	5	∆v	∆v	PROPN
ejpam-5125	489	6	.	.	PUNCT
ejpam-5125	490	1	then	then	ADV
ejpam-5125	490	2	,	,	PUNCT
ejpam-5125	490	3	for	for	ADP
ejpam-5125	490	4	any	any	DET
ejpam-5125	490	5	p1	p1	PROPN
ejpam-5125	490	6	∈	∈	PROPN
ejpam-5125	490	7	v	v	NOUN
ejpam-5125	490	8	,	,	PUNCT
ejpam-5125	490	9	a	a	DET
ejpam-5125	490	10	∧	∧	PROPN
ejpam-5125	490	11	a	a	PRON
ejpam-5125	490	12	=	=	PUNCT
ejpam-5125	490	13	a	a	NOUN
ejpam-5125	490	14	=	=	PUNCT
ejpam-5125	490	15	(	(	PUNCT
ejpam-5125	490	16	p1	p1	PROPN
ejpam-5125	490	17	∨	∨	NUM
ejpam-5125	490	18	a	a	PRON
ejpam-5125	490	19	)	)	PUNCT
ejpam-5125	490	20	∧	∧	PROPN
ejpam-5125	490	21	a	a	DET
ejpam-5125	490	22	implies	implie	NOUN
ejpam-5125	490	23	(	(	PUNCT
ejpam-5125	490	24	a	a	PRON
ejpam-5125	490	25	,	,	PUNCT
ejpam-5125	490	26	p1	p1	PROPN
ejpam-5125	490	27	∨	∨	NUM
ejpam-5125	490	28	a	a	DET
ejpam-5125	490	29	)	)	PUNCT
ejpam-5125	490	30	∈	∈	NOUN
ejpam-5125	490	31	θa	θa	NUM
ejpam-5125	490	32	=	=	SYM
ejpam-5125	490	33	∆v	∆v	PROPN
ejpam-5125	490	34	which	which	PRON
ejpam-5125	490	35	shows	show	VERB
ejpam-5125	490	36	that	that	SCONJ
ejpam-5125	490	37	a	a	PRON
ejpam-5125	490	38	is	be	AUX
ejpam-5125	490	39	unity	unity	NOUN
ejpam-5125	490	40	element	element	NOUN
ejpam-5125	490	41	of	of	ADP
ejpam-5125	490	42	v	v	NOUN
ejpam-5125	490	43	.	.	PUNCT
ejpam-5125	491	1	conversely	conversely	ADV
ejpam-5125	491	2	,	,	PUNCT
ejpam-5125	491	3	suppose	suppose	VERB
ejpam-5125	491	4	that	that	SCONJ
ejpam-5125	491	5	a	a	PRON
ejpam-5125	491	6	is	be	AUX
ejpam-5125	491	7	unity	unity	NOUN
ejpam-5125	491	8	element	element	NOUN
ejpam-5125	491	9	of	of	ADP
ejpam-5125	491	10	v	v	NOUN
ejpam-5125	491	11	.	.	PUNCT
ejpam-5125	492	1	then	then	ADV
ejpam-5125	492	2	,	,	PUNCT
ejpam-5125	492	3	for	for	ADP
ejpam-5125	492	4	(	(	PUNCT
ejpam-5125	492	5	p1	p1	NOUN
ejpam-5125	492	6	,	,	PUNCT
ejpam-5125	492	7	p2	p2	NOUN
ejpam-5125	492	8	)	)	PUNCT
ejpam-5125	492	9	∈	∈	NOUN
ejpam-5125	492	10	θa	θa	NUM
ejpam-5125	492	11	,	,	PUNCT
ejpam-5125	492	12	we	we	PRON
ejpam-5125	492	13	have	have	VERB
ejpam-5125	492	14	p1∧a	p1∧a	PROPN
ejpam-5125	492	15	=	=	SYM
ejpam-5125	492	16	p2	p2	PROPN
ejpam-5125	492	17	∧	∧	PROPN
ejpam-5125	492	18	a	a	PRON
ejpam-5125	492	19	which	which	PRON
ejpam-5125	492	20	implies	imply	VERB
ejpam-5125	492	21	p1	p1	PROPN
ejpam-5125	492	22	=	=	SYM
ejpam-5125	492	23	p2	p2	NOUN
ejpam-5125	492	24	.	.	PUNCT
ejpam-5125	493	1	therefore	therefore	ADV
ejpam-5125	493	2	θa	θa	X
ejpam-5125	493	3	=	=	SYM
ejpam-5125	493	4	∆v	∆v	PROPN
ejpam-5125	493	5	,	,	PUNCT
ejpam-5125	493	6	which	which	PRON
ejpam-5125	493	7	concludes	conclude	VERB
ejpam-5125	493	8	the	the	DET
ejpam-5125	493	9	lemma	lemma	PROPN
ejpam-5125	493	10	.	.	PUNCT
ejpam-5125	493	11	theorem	theorem	VERB
ejpam-5125	493	12	13	13	NUM
ejpam-5125	493	13	.	.	PUNCT
ejpam-5125	494	1	let	let	VERB
ejpam-5125	494	2	v	v	PART
ejpam-5125	494	3	be	be	AUX
ejpam-5125	494	4	a	a	DET
ejpam-5125	494	5	subdirectly	subdirectly	ADV
ejpam-5125	494	6	irreducible	irreducible	ADJ
ejpam-5125	494	7	associative	associative	ADJ
ejpam-5125	494	8	pdl	pdl	PROPN
ejpam-5125	494	9	.	.	PUNCT
ejpam-5125	495	1	then	then	ADV
ejpam-5125	495	2	every	every	DET
ejpam-5125	495	3	non	non	ADJ
ejpam-5125	495	4	-	-	NOUN
ejpam-5125	495	5	unity	unity	NOUN
ejpam-5125	495	6	of	of	ADP
ejpam-5125	495	7	v	v	NOUN
ejpam-5125	495	8	is	be	AUX
ejpam-5125	495	9	minimal	minimal	ADJ
ejpam-5125	495	10	and	and	CCONJ
ejpam-5125	495	11	v	v	NOUN
ejpam-5125	495	12	contains	contain	VERB
ejpam-5125	495	13	atmost	atmost	PROPN
ejpam-5125	495	14	two	two	NUM
ejpam-5125	495	15	non	non	ADJ
ejpam-5125	495	16	-	-	ADJ
ejpam-5125	495	17	unity	unity	ADJ
ejpam-5125	495	18	elements	element	NOUN
ejpam-5125	495	19	.	.	PUNCT
ejpam-5125	496	1	proof	proof	NOUN
ejpam-5125	496	2	.	.	PUNCT
ejpam-5125	497	1	let	let	VERB
ejpam-5125	497	2	v	v	PART
ejpam-5125	497	3	be	be	AUX
ejpam-5125	497	4	an	an	DET
ejpam-5125	497	5	associative	associative	ADJ
ejpam-5125	497	6	pdl	pdl	NOUN
ejpam-5125	497	7	.	.	PUNCT
ejpam-5125	497	8	suppose	suppose	VERB
ejpam-5125	497	9	v	v	NOUN
ejpam-5125	497	10	is	be	AUX
ejpam-5125	497	11	subdirectly	subdirectly	ADV
ejpam-5125	497	12	irreducible	irreducible	ADJ
ejpam-5125	497	13	.	.	PUNCT
ejpam-5125	498	1	let	let	VERB
ejpam-5125	498	2	θ1	θ1	NOUN
ejpam-5125	498	3	be	be	AUX
ejpam-5125	498	4	the	the	DET
ejpam-5125	498	5	smallest	small	ADJ
ejpam-5125	498	6	non	non	ADJ
ejpam-5125	498	7	-	-	ADJ
ejpam-5125	498	8	zero	zero	NUM
ejpam-5125	498	9	congruence	congruence	NOUN
ejpam-5125	498	10	on	on	ADP
ejpam-5125	498	11	v	v	PROPN
ejpam-5125	498	12	.	.	PUNCT
ejpam-5125	499	1	select	select	ADJ
ejpam-5125	499	2	p1	p1	PROPN
ejpam-5125	499	3	,	,	PUNCT
ejpam-5125	499	4	p2	p2	PROPN
ejpam-5125	499	5	∈	∈	PROPN
ejpam-5125	499	6	v	v	NOUN
ejpam-5125	499	7	with	with	ADP
ejpam-5125	499	8	p1	p1	PROPN
ejpam-5125	499	9	̸=	̸=	PROPN
ejpam-5125	499	10	p2	p2	NOUN
ejpam-5125	499	11	such	such	ADJ
ejpam-5125	499	12	that	that	SCONJ
ejpam-5125	499	13	(	(	PUNCT
ejpam-5125	499	14	p1	p1	NOUN
ejpam-5125	499	15	,	,	PUNCT
ejpam-5125	499	16	p2	p2	NOUN
ejpam-5125	499	17	)	)	PUNCT
ejpam-5125	499	18	∈	∈	PROPN
ejpam-5125	499	19	θ1	θ1	NOUN
ejpam-5125	499	20	.	.	PUNCT
ejpam-5125	500	1	then	then	ADV
ejpam-5125	500	2	atleast	atleast	VERB
ejpam-5125	500	3	one	one	NUM
ejpam-5125	500	4	of	of	ADP
ejpam-5125	500	5	p1	p1	PROPN
ejpam-5125	500	6	and	and	CCONJ
ejpam-5125	500	7	p2	p2	PROPN
ejpam-5125	500	8	is	be	AUX
ejpam-5125	500	9	minimal	minimal	ADJ
ejpam-5125	500	10	.	.	PUNCT
ejpam-5125	501	1	for	for	ADP
ejpam-5125	501	2	if	if	SCONJ
ejpam-5125	501	3	,	,	PUNCT
ejpam-5125	501	4	assume	assume	VERB
ejpam-5125	501	5	that	that	SCONJ
ejpam-5125	501	6	p1	p1	NOUN
ejpam-5125	501	7	and	and	CCONJ
ejpam-5125	501	8	p2	p2	NOUN
ejpam-5125	501	9	both	both	PRON
ejpam-5125	501	10	are	be	AUX
ejpam-5125	501	11	not	not	PART
ejpam-5125	501	12	minimal	minimal	ADJ
ejpam-5125	501	13	elements	element	NOUN
ejpam-5125	501	14	of	of	ADP
ejpam-5125	501	15	v	v	NOUN
ejpam-5125	501	16	,	,	PUNCT
ejpam-5125	501	17	then	then	ADV
ejpam-5125	501	18	φp1	φp1	NOUN
ejpam-5125	501	19	̸=	̸=	PROPN
ejpam-5125	501	20	∆v	∆v	PROPN
ejpam-5125	501	21	̸=	̸=	PROPN
ejpam-5125	501	22	φp2	φp2	NOUN
ejpam-5125	501	23	,	,	PUNCT
ejpam-5125	501	24	implies	imply	VERB
ejpam-5125	501	25	(	(	PUNCT
ejpam-5125	501	26	p1	p1	NOUN
ejpam-5125	501	27	,	,	PUNCT
ejpam-5125	501	28	p2	p2	NOUN
ejpam-5125	501	29	)	)	PUNCT
ejpam-5125	501	30	∈	∈	NOUN
ejpam-5125	501	31	φp1	φp1	NOUN
ejpam-5125	501	32	∩	∩	NOUN
ejpam-5125	501	33	φp2	φp2	PROPN
ejpam-5125	501	34	.	.	PUNCT
ejpam-5125	502	1	since	since	SCONJ
ejpam-5125	502	2	p1	p1	PROPN
ejpam-5125	502	3	=	=	SYM
ejpam-5125	502	4	p1	p1	PROPN
ejpam-5125	502	5	∨	∨	NOUN
ejpam-5125	502	6	p1	p1	NOUN
ejpam-5125	502	7	=	=	SYM
ejpam-5125	502	8	p2	p2	PROPN
ejpam-5125	502	9	∨	∨	NUM
ejpam-5125	502	10	p1	p1	NOUN
ejpam-5125	502	11	and	and	CCONJ
ejpam-5125	502	12	p2	p2	PROPN
ejpam-5125	502	13	=	=	SYM
ejpam-5125	502	14	p1	p1	PROPN
ejpam-5125	502	15	∨	∨	NUM
ejpam-5125	502	16	p2	p2	NOUN
ejpam-5125	502	17	.	.	PUNCT
ejpam-5125	503	1	thus	thus	ADV
ejpam-5125	503	2	p1	p1	NOUN
ejpam-5125	503	3	=	=	SYM
ejpam-5125	503	4	p2	p2	NOUN
ejpam-5125	503	5	,	,	PUNCT
ejpam-5125	503	6	which	which	PRON
ejpam-5125	503	7	is	be	AUX
ejpam-5125	503	8	a	a	DET
ejpam-5125	503	9	contradiction	contradiction	NOUN
ejpam-5125	503	10	.	.	PUNCT
ejpam-5125	504	1	thus	thus	ADV
ejpam-5125	504	2	,	,	PUNCT
ejpam-5125	504	3	atleast	atleast	ADJ
ejpam-5125	504	4	one	one	NUM
ejpam-5125	504	5	of	of	ADP
ejpam-5125	504	6	p1	p1	NOUN
ejpam-5125	504	7	,	,	PUNCT
ejpam-5125	504	8	p2	p2	PROPN
ejpam-5125	504	9	is	be	AUX
ejpam-5125	504	10	minimal	minimal	ADJ
ejpam-5125	504	11	.	.	PUNCT
ejpam-5125	505	1	without	without	ADP
ejpam-5125	505	2	loss	loss	NOUN
ejpam-5125	505	3	of	of	ADP
ejpam-5125	505	4	generality	generality	NOUN
ejpam-5125	505	5	,	,	PUNCT
ejpam-5125	505	6	let	let	VERB
ejpam-5125	505	7	us	we	PRON
ejpam-5125	505	8	assume	assume	VERB
ejpam-5125	505	9	that	that	SCONJ
ejpam-5125	505	10	p1	p1	NOUN
ejpam-5125	505	11	is	be	AUX
ejpam-5125	505	12	minimal	minimal	ADJ
ejpam-5125	505	13	.	.	PUNCT
ejpam-5125	506	1	let	let	VERB
ejpam-5125	506	2	a	a	PRON
ejpam-5125	506	3	be	be	AUX
ejpam-5125	506	4	a	a	DET
ejpam-5125	506	5	non	non	ADJ
ejpam-5125	506	6	-	-	ADJ
ejpam-5125	506	7	unity	unity	ADJ
ejpam-5125	506	8	element	element	NOUN
ejpam-5125	506	9	of	of	ADP
ejpam-5125	506	10	v	v	PROPN
ejpam-5125	506	11	.	.	PUNCT
ejpam-5125	507	1	suppose	suppose	VERB
ejpam-5125	507	2	,	,	PUNCT
ejpam-5125	507	3	a	a	PRON
ejpam-5125	507	4	is	be	AUX
ejpam-5125	507	5	not	not	PART
ejpam-5125	507	6	minimal	minimal	ADJ
ejpam-5125	507	7	,	,	PUNCT
ejpam-5125	507	8	then	then	ADV
ejpam-5125	507	9	p1	p1	PROPN
ejpam-5125	507	10	being	be	AUX
ejpam-5125	507	11	the	the	DET
ejpam-5125	507	12	minimal	minimal	ADJ
ejpam-5125	507	13	element	element	NOUN
ejpam-5125	507	14	we	we	PRON
ejpam-5125	507	15	have	have	VERB
ejpam-5125	507	16	a∧	a∧	NOUN
ejpam-5125	507	17	p1	p1	PROPN
ejpam-5125	507	18	=	=	SYM
ejpam-5125	507	19	p1	p1	PROPN
ejpam-5125	507	20	.	.	PUNCT
ejpam-5125	508	1	hence	hence	ADV
ejpam-5125	508	2	a∨	a∨	PROPN
ejpam-5125	508	3	p1	p1	PROPN
ejpam-5125	508	4	=	=	PROPN
ejpam-5125	508	5	a∨	a∨	PROPN
ejpam-5125	508	6	(	(	PUNCT
ejpam-5125	508	7	a∧	a∧	NOUN
ejpam-5125	508	8	p1	p1	PROPN
ejpam-5125	508	9	)	)	PUNCT
ejpam-5125	508	10	=	=	SYM
ejpam-5125	508	11	a	a	PRON
ejpam-5125	508	12	,	,	PUNCT
ejpam-5125	508	13	so	so	SCONJ
ejpam-5125	508	14	that	that	SCONJ
ejpam-5125	508	15	p1	p1	PROPN
ejpam-5125	508	16	∨	∨	NOUN
ejpam-5125	508	17	a	a	PRON
ejpam-5125	508	18	is	be	AUX
ejpam-5125	508	19	a	a	DET
ejpam-5125	508	20	non	non	ADJ
ejpam-5125	508	21	-	-	ADJ
ejpam-5125	508	22	unity	unity	ADJ
ejpam-5125	508	23	element	element	NOUN
ejpam-5125	508	24	of	of	ADP
ejpam-5125	508	25	v	v	NOUN
ejpam-5125	508	26	.	.	PUNCT
ejpam-5125	509	1	therefore	therefore	ADV
ejpam-5125	509	2	θp1∨a	θp1∨a	VERB
ejpam-5125	509	3	̸=	̸=	PROPN
ejpam-5125	509	4	∆v	∆v	PROPN
ejpam-5125	509	5	.	.	PUNCT
ejpam-5125	510	1	hence	hence	ADV
ejpam-5125	510	2	(	(	PUNCT
ejpam-5125	510	3	p1	p1	NOUN
ejpam-5125	510	4	,	,	PUNCT
ejpam-5125	510	5	p2	p2	NOUN
ejpam-5125	510	6	)	)	PUNCT
ejpam-5125	510	7	∈	∈	PROPN
ejpam-5125	510	8	θp1∨a	θp1∨a	PROPN
ejpam-5125	510	9	.	.	PUNCT
ejpam-5125	511	1	also	also	ADV
ejpam-5125	511	2	,	,	PUNCT
ejpam-5125	511	3	since	since	SCONJ
ejpam-5125	511	4	a	a	PRON
ejpam-5125	511	5	is	be	AUX
ejpam-5125	511	6	not	not	PART
ejpam-5125	511	7	minimal	minimal	ADJ
ejpam-5125	511	8	,	,	PUNCT
ejpam-5125	511	9	we	we	PRON
ejpam-5125	511	10	obtain	obtain	VERB
ejpam-5125	511	11	(	(	PUNCT
ejpam-5125	511	12	p1	p1	NOUN
ejpam-5125	511	13	,	,	PUNCT
ejpam-5125	511	14	p2	p2	X
ejpam-5125	511	15	)	)	PUNCT
ejpam-5125	511	16	∈	∈	PROPN
ejpam-5125	511	17	φa	φa	PROPN
ejpam-5125	511	18	.	.	PUNCT
ejpam-5125	512	1	now	now	ADV
ejpam-5125	512	2	p1	p1	PROPN
ejpam-5125	512	3	=	=	SYM
ejpam-5125	512	4	p1	p1	PROPN
ejpam-5125	512	5	∧	∧	PROPN
ejpam-5125	512	6	(	(	PUNCT
ejpam-5125	512	7	p1	p1	PROPN
ejpam-5125	512	8	∨	∨	NUM
ejpam-5125	512	9	a	a	PRON
ejpam-5125	512	10	)	)	PUNCT
ejpam-5125	512	11	=	=	SYM
ejpam-5125	512	12	p2	p2	PROPN
ejpam-5125	512	13	∧	∧	PROPN
ejpam-5125	512	14	(	(	PUNCT
ejpam-5125	512	15	p1	p1	PROPN
ejpam-5125	512	16	∨	∨	NUM
ejpam-5125	512	17	a	a	PRON
ejpam-5125	512	18	)	)	PUNCT
ejpam-5125	512	19	=	=	SYM
ejpam-5125	512	20	p2	p2	PROPN
ejpam-5125	512	21	∧	∧	PROPN
ejpam-5125	512	22	(	(	PUNCT
ejpam-5125	512	23	p2	p2	PROPN
ejpam-5125	512	24	∨	∨	NUM
ejpam-5125	512	25	a	a	PRON
ejpam-5125	512	26	)	)	PUNCT
ejpam-5125	512	27	=	=	SYM
ejpam-5125	512	28	p2	p2	NOUN
ejpam-5125	512	29	which	which	PRON
ejpam-5125	512	30	r.	r.	PROPN
ejpam-5125	512	31	bandaru	bandaru	PROPN
ejpam-5125	512	32	,	,	PUNCT
ejpam-5125	512	33	s.	s.	PROPN
ejpam-5125	512	34	ajjarapu	ajjarapu	PROPN
ejpam-5125	512	35	/	/	PUNCT
ejpam-5125	512	36	eur	eur	PROPN
ejpam-5125	512	37	.	.	PUNCT
ejpam-5125	513	1	j.	j.	PROPN
ejpam-5125	513	2	pure	pure	PROPN
ejpam-5125	513	3	appl	appl	PROPN
ejpam-5125	513	4	.	.	PROPN
ejpam-5125	513	5	math	math	PROPN
ejpam-5125	513	6	,	,	PUNCT
ejpam-5125	513	7	17	17	NUM
ejpam-5125	513	8	(	(	PUNCT
ejpam-5125	513	9	2	2	NUM
ejpam-5125	513	10	)	)	PUNCT
ejpam-5125	513	11	(	(	PUNCT
ejpam-5125	513	12	2024	2024	NUM
ejpam-5125	513	13	)	)	PUNCT
ejpam-5125	513	14	,	,	PUNCT
ejpam-5125	513	15	819	819	NUM
ejpam-5125	513	16	-	-	SYM
ejpam-5125	513	17	834	834	NUM
ejpam-5125	513	18	833	833	NUM
ejpam-5125	513	19	is	be	AUX
ejpam-5125	513	20	a	a	DET
ejpam-5125	513	21	contradiction	contradiction	NOUN
ejpam-5125	513	22	.	.	PUNCT
ejpam-5125	514	1	thus	thus	ADV
ejpam-5125	514	2	a	a	DET
ejpam-5125	514	3	is	be	AUX
ejpam-5125	514	4	minimal	minimal	ADJ
ejpam-5125	514	5	.	.	PUNCT
ejpam-5125	515	1	suppose	suppose	VERB
ejpam-5125	515	2	a	a	DET
ejpam-5125	515	3	,	,	PUNCT
ejpam-5125	515	4	b	b	NOUN
ejpam-5125	515	5	,	,	PUNCT
ejpam-5125	515	6	c	c	PROPN
ejpam-5125	515	7	∈	∈	PROPN
ejpam-5125	515	8	v	v	NOUN
ejpam-5125	515	9	be	be	AUX
ejpam-5125	515	10	three	three	NUM
ejpam-5125	515	11	distinct	distinct	ADJ
ejpam-5125	515	12	non	non	ADJ
ejpam-5125	515	13	-	-	ADJ
ejpam-5125	515	14	unity	unity	ADJ
ejpam-5125	515	15	elements	element	NOUN
ejpam-5125	515	16	of	of	ADP
ejpam-5125	515	17	v	v	NOUN
ejpam-5125	515	18	.	.	PUNCT
ejpam-5125	516	1	then	then	ADV
ejpam-5125	516	2	since	since	SCONJ
ejpam-5125	516	3	a	a	DET
ejpam-5125	516	4	,	,	PUNCT
ejpam-5125	516	5	b	b	NOUN
ejpam-5125	516	6	,	,	PUNCT
ejpam-5125	516	7	c	c	PROPN
ejpam-5125	516	8	are	be	AUX
ejpam-5125	516	9	minimal	minimal	ADJ
ejpam-5125	516	10	elements	element	NOUN
ejpam-5125	516	11	,	,	PUNCT
ejpam-5125	516	12	we	we	PRON
ejpam-5125	516	13	have	have	VERB
ejpam-5125	516	14	φ	φ	NOUN
ejpam-5125	516	15	=	=	SYM
ejpam-5125	516	16	∆v	∆v	PROPN
ejpam-5125	516	17	∪	∪	X
ejpam-5125	516	18	{	{	PUNCT
ejpam-5125	516	19	(	(	PUNCT
ejpam-5125	516	20	a	a	DET
ejpam-5125	516	21	,	,	PUNCT
ejpam-5125	516	22	b	b	NOUN
ejpam-5125	516	23	)	)	PUNCT
ejpam-5125	516	24	,	,	PUNCT
ejpam-5125	516	25	(	(	PUNCT
ejpam-5125	516	26	b	b	X
ejpam-5125	516	27	,	,	PUNCT
ejpam-5125	516	28	a	a	NOUN
ejpam-5125	516	29	)	)	PUNCT
ejpam-5125	516	30	}	}	PUNCT
ejpam-5125	516	31	and	and	CCONJ
ejpam-5125	516	32	ψ	ψ	X
ejpam-5125	516	33	=	=	X
ejpam-5125	516	34	∆v	∆v	PROPN
ejpam-5125	516	35	∪	∪	X
ejpam-5125	516	36	{	{	PUNCT
ejpam-5125	516	37	(	(	PUNCT
ejpam-5125	516	38	b	b	NOUN
ejpam-5125	516	39	,	,	PUNCT
ejpam-5125	516	40	c	c	NOUN
ejpam-5125	516	41	)	)	PUNCT
ejpam-5125	516	42	,	,	PUNCT
ejpam-5125	516	43	(	(	PUNCT
ejpam-5125	516	44	c	c	X
ejpam-5125	516	45	,	,	PUNCT
ejpam-5125	516	46	b	b	NOUN
ejpam-5125	516	47	)	)	PUNCT
ejpam-5125	516	48	}	}	PUNCT
ejpam-5125	516	49	are	be	AUX
ejpam-5125	516	50	two	two	NUM
ejpam-5125	516	51	non	non	ADJ
ejpam-5125	516	52	-	-	ADJ
ejpam-5125	516	53	zero	zero	ADJ
ejpam-5125	516	54	congruences	congruence	NOUN
ejpam-5125	516	55	on	on	ADP
ejpam-5125	516	56	v	v	ADP
ejpam-5125	516	57	such	such	ADJ
ejpam-5125	516	58	that	that	SCONJ
ejpam-5125	516	59	φ	φ	PROPN
ejpam-5125	516	60	∩	∩	PROPN
ejpam-5125	516	61	ψ	ψ	PROPN
ejpam-5125	516	62	=	=	SYM
ejpam-5125	516	63	∆v	∆v	PROPN
ejpam-5125	516	64	which	which	PRON
ejpam-5125	516	65	contradicts	contradict	VERB
ejpam-5125	516	66	the	the	DET
ejpam-5125	516	67	fact	fact	NOUN
ejpam-5125	516	68	that	that	SCONJ
ejpam-5125	516	69	v	v	NOUN
ejpam-5125	516	70	is	be	AUX
ejpam-5125	516	71	subdirectly	subdirectly	ADV
ejpam-5125	516	72	irreducible	irreducible	ADJ
ejpam-5125	516	73	.	.	PUNCT
ejpam-5125	517	1	hence	hence	ADV
ejpam-5125	517	2	v	v	NOUN
ejpam-5125	517	3	has	have	AUX
ejpam-5125	517	4	atmost	atmost	PROPN
ejpam-5125	517	5	two	two	NUM
ejpam-5125	517	6	non	non	ADJ
ejpam-5125	517	7	-	-	ADJ
ejpam-5125	517	8	unity	unity	ADJ
ejpam-5125	517	9	elements	element	NOUN
ejpam-5125	517	10	.	.	PUNCT
ejpam-5125	518	1	corollary	corollary	ADJ
ejpam-5125	518	2	11	11	NUM
ejpam-5125	518	3	.	.	PUNCT
ejpam-5125	519	1	a	a	DET
ejpam-5125	519	2	subdirectly	subdirectly	ADV
ejpam-5125	519	3	irreducible	irreducible	ADJ
ejpam-5125	519	4	distributive	distributive	ADJ
ejpam-5125	519	5	lattice	lattice	NOUN
ejpam-5125	519	6	v	v	NOUN
ejpam-5125	519	7	is	be	AUX
ejpam-5125	519	8	a	a	DET
ejpam-5125	519	9	two	two	NUM
ejpam-5125	519	10	element	element	NOUN
ejpam-5125	519	11	chain	chain	NOUN
ejpam-5125	519	12	.	.	PUNCT
ejpam-5125	520	1	remark	remark	NOUN
ejpam-5125	520	2	:	:	PUNCT
ejpam-5125	520	3	it	it	PRON
ejpam-5125	520	4	is	be	AUX
ejpam-5125	520	5	evident	evident	ADJ
ejpam-5125	520	6	that	that	SCONJ
ejpam-5125	520	7	the	the	DET
ejpam-5125	520	8	converse	converse	NOUN
ejpam-5125	520	9	of	of	ADP
ejpam-5125	520	10	theorem	theorem	NOUN
ejpam-5125	520	11	13	13	NUM
ejpam-5125	520	12	is	be	AUX
ejpam-5125	520	13	also	also	ADV
ejpam-5125	520	14	true	true	ADJ
ejpam-5125	520	15	.	.	PUNCT
ejpam-5125	521	1	i.e.	i.e.	X
ejpam-5125	521	2	,	,	PUNCT
ejpam-5125	521	3	if	if	SCONJ
ejpam-5125	521	4	v	v	NOUN
ejpam-5125	521	5	is	be	AUX
ejpam-5125	521	6	a	a	DET
ejpam-5125	521	7	pdl	pdl	NOUN
ejpam-5125	521	8	containing	contain	VERB
ejpam-5125	521	9	atmost	atmost	PROPN
ejpam-5125	521	10	two	two	NUM
ejpam-5125	521	11	non	non	ADJ
ejpam-5125	521	12	-	-	ADJ
ejpam-5125	521	13	unity	unity	ADJ
ejpam-5125	521	14	elements	element	NOUN
ejpam-5125	521	15	and	and	CCONJ
ejpam-5125	521	16	every	every	DET
ejpam-5125	521	17	non	non	ADJ
ejpam-5125	521	18	-	-	ADJ
ejpam-5125	521	19	unity	unity	ADJ
ejpam-5125	521	20	element	element	NOUN
ejpam-5125	521	21	of	of	ADP
ejpam-5125	521	22	v	v	NOUN
ejpam-5125	521	23	is	be	AUX
ejpam-5125	521	24	minimal	minimal	ADJ
ejpam-5125	521	25	.	.	PUNCT
ejpam-5125	522	1	theorem	theorem	NOUN
ejpam-5125	522	2	14	14	NUM
ejpam-5125	522	3	.	.	PUNCT
ejpam-5125	523	1	let	let	VERB
ejpam-5125	523	2	v	v	PART
ejpam-5125	523	3	be	be	AUX
ejpam-5125	523	4	a	a	DET
ejpam-5125	523	5	pdl	pdl	NOUN
ejpam-5125	523	6	.	.	PUNCT
ejpam-5125	524	1	then	then	ADV
ejpam-5125	524	2	the	the	DET
ejpam-5125	524	3	following	following	NOUN
ejpam-5125	524	4	are	be	AUX
ejpam-5125	524	5	equivalent	equivalent	ADJ
ejpam-5125	524	6	:	:	PUNCT
ejpam-5125	524	7	(	(	PUNCT
ejpam-5125	524	8	1	1	X
ejpam-5125	524	9	)	)	PUNCT
ejpam-5125	524	10	v	v	NOUN
ejpam-5125	524	11	is	be	AUX
ejpam-5125	524	12	associative	associative	ADJ
ejpam-5125	524	13	.	.	PUNCT
ejpam-5125	525	1	(	(	PUNCT
ejpam-5125	525	2	2	2	X
ejpam-5125	525	3	)	)	PUNCT
ejpam-5125	525	4	θa	θa	PRON
ejpam-5125	525	5	is	be	AUX
ejpam-5125	525	6	a	a	DET
ejpam-5125	525	7	congruence	congruence	NOUN
ejpam-5125	525	8	relation	relation	NOUN
ejpam-5125	525	9	on	on	ADP
ejpam-5125	525	10	v	v	NOUN
ejpam-5125	525	11	for	for	ADP
ejpam-5125	525	12	all	all	DET
ejpam-5125	525	13	a	a	DET
ejpam-5125	525	14	∈	∈	PROPN
ejpam-5125	525	15	v	v	NOUN
ejpam-5125	525	16	.	.	PUNCT
ejpam-5125	526	1	(	(	PUNCT
ejpam-5125	526	2	3	3	X
ejpam-5125	526	3	)	)	PUNCT
ejpam-5125	526	4	v	v	NOUN
ejpam-5125	526	5	is	be	AUX
ejpam-5125	526	6	a	a	DET
ejpam-5125	526	7	subdirect	subdirect	NOUN
ejpam-5125	526	8	product	product	NOUN
ejpam-5125	526	9	of	of	ADP
ejpam-5125	526	10	pdls	pdl	NOUN
ejpam-5125	526	11	in	in	ADP
ejpam-5125	526	12	each	each	PRON
ejpam-5125	526	13	of	of	ADP
ejpam-5125	526	14	which	which	PRON
ejpam-5125	526	15	there	there	PRON
ejpam-5125	526	16	are	be	VERB
ejpam-5125	526	17	atmost	atmost	PROPN
ejpam-5125	526	18	two	two	NUM
ejpam-5125	526	19	non	non	ADJ
ejpam-5125	526	20	-	-	ADJ
ejpam-5125	526	21	unity	unity	ADJ
ejpam-5125	526	22	elements	element	NOUN
ejpam-5125	526	23	and	and	CCONJ
ejpam-5125	526	24	every	every	DET
ejpam-5125	526	25	non	non	ADJ
ejpam-5125	526	26	-	-	ADJ
ejpam-5125	526	27	unity	unity	ADJ
ejpam-5125	526	28	element	element	NOUN
ejpam-5125	526	29	is	be	AUX
ejpam-5125	526	30	minimal	minimal	ADJ
ejpam-5125	526	31	.	.	PUNCT
ejpam-5125	527	1	proof	proof	NOUN
ejpam-5125	527	2	.	.	PUNCT
ejpam-5125	528	1	let	let	VERB
ejpam-5125	528	2	v	v	PART
ejpam-5125	528	3	be	be	AUX
ejpam-5125	528	4	a	a	DET
ejpam-5125	528	5	pdl	pdl	NOUN
ejpam-5125	528	6	.	.	PUNCT
ejpam-5125	529	1	we	we	PRON
ejpam-5125	529	2	have	have	VERB
ejpam-5125	529	3	(	(	PUNCT
ejpam-5125	529	4	1	1	X
ejpam-5125	529	5	)	)	PUNCT
ejpam-5125	529	6	⇒	⇒	NOUN
ejpam-5125	529	7	(	(	PUNCT
ejpam-5125	529	8	2	2	NUM
ejpam-5125	529	9	)	)	PUNCT
ejpam-5125	529	10	by	by	ADP
ejpam-5125	529	11	lemma	lemma	PROPN
ejpam-5125	529	12	15	15	NUM
ejpam-5125	529	13	.	.	PUNCT
ejpam-5125	530	1	assume	assume	VERB
ejpam-5125	530	2	(	(	PUNCT
ejpam-5125	530	3	2	2	NUM
ejpam-5125	530	4	)	)	PUNCT
ejpam-5125	530	5	.	.	PUNCT
ejpam-5125	531	1	then	then	ADV
ejpam-5125	531	2	,	,	PUNCT
ejpam-5125	531	3	for	for	ADP
ejpam-5125	531	4	any	any	DET
ejpam-5125	531	5	congruence	congruence	PROPN
ejpam-5125	531	6	η	η	PROPN
ejpam-5125	531	7	on	on	ADP
ejpam-5125	531	8	v	v	NUM
ejpam-5125	531	9	,	,	PUNCT
ejpam-5125	531	10	p1	p1	NOUN
ejpam-5125	531	11	,	,	PUNCT
ejpam-5125	531	12	p2	p2	PROPN
ejpam-5125	531	13	and	and	CCONJ
ejpam-5125	531	14	p3	p3	PROPN
ejpam-5125	531	15	∈	∈	PROPN
ejpam-5125	531	16	v	v	ADP
ejpam-5125	531	17	we	we	PRON
ejpam-5125	531	18	have	have	VERB
ejpam-5125	531	19	(	(	PUNCT
ejpam-5125	531	20	p1	p1	PROPN
ejpam-5125	531	21	∧	∧	PROPN
ejpam-5125	531	22	p3	p3	PROPN
ejpam-5125	531	23	,	,	PUNCT
ejpam-5125	531	24	p2	p2	PROPN
ejpam-5125	531	25	∧	∧	PROPN
ejpam-5125	531	26	p3	p3	PROPN
ejpam-5125	531	27	)	)	PUNCT
ejpam-5125	531	28	∈	∈	PROPN
ejpam-5125	531	29	η	η	PROPN
ejpam-5125	531	30	if	if	SCONJ
ejpam-5125	531	31	and	and	CCONJ
ejpam-5125	531	32	only	only	ADV
ejpam-5125	531	33	if	if	SCONJ
ejpam-5125	531	34	(	(	PUNCT
ejpam-5125	531	35	p1	p1	NOUN
ejpam-5125	531	36	,	,	PUNCT
ejpam-5125	531	37	p2	p2	X
ejpam-5125	531	38	)	)	PUNCT
ejpam-5125	531	39	∈	∈	PROPN
ejpam-5125	531	40	θp3	θp3	NOUN
ejpam-5125	531	41	∧	∧	PROPN
ejpam-5125	531	42	η	η	PROPN
ejpam-5125	531	43	.	.	PROPN
ejpam-5125	531	44	hence	hence	ADV
ejpam-5125	531	45	for	for	ADP
ejpam-5125	531	46	any	any	DET
ejpam-5125	531	47	congruence	congruence	PROPN
ejpam-5125	531	48	relation	relation	PROPN
ejpam-5125	531	49	η	η	PROPN
ejpam-5125	531	50	on	on	ADP
ejpam-5125	531	51	v	v	NUM
ejpam-5125	531	52	,	,	PUNCT
ejpam-5125	531	53	the	the	DET
ejpam-5125	531	54	quotient	quotient	NOUN
ejpam-5125	531	55	pdl	pdl	PROPN
ejpam-5125	531	56	v	v	INTJ
ejpam-5125	531	57	|η	|η	NOUN
ejpam-5125	531	58	also	also	ADV
ejpam-5125	531	59	satisfies	satisfy	VERB
ejpam-5125	531	60	the	the	DET
ejpam-5125	531	61	assumption	assumption	NOUN
ejpam-5125	531	62	(	(	PUNCT
ejpam-5125	531	63	2	2	NUM
ejpam-5125	531	64	)	)	PUNCT
ejpam-5125	531	65	.	.	PUNCT
ejpam-5125	532	1	by	by	ADP
ejpam-5125	532	2	birkhoff	birkhoff	NOUN
ejpam-5125	532	3	’s	’s	PART
ejpam-5125	532	4	theorem	theorem	NOUN
ejpam-5125	532	5	and	and	CCONJ
ejpam-5125	532	6	theorem	theorem	VERB
ejpam-5125	532	7	13	13	NUM
ejpam-5125	532	8	,	,	PUNCT
ejpam-5125	532	9	we	we	PRON
ejpam-5125	532	10	have	have	VERB
ejpam-5125	532	11	(	(	PUNCT
ejpam-5125	532	12	3	3	NUM
ejpam-5125	532	13	)	)	PUNCT
ejpam-5125	532	14	.	.	PUNCT
ejpam-5125	533	1	(	(	PUNCT
ejpam-5125	533	2	3	3	X
ejpam-5125	533	3	)	)	PUNCT
ejpam-5125	533	4	⇒	⇒	NOUN
ejpam-5125	533	5	(	(	PUNCT
ejpam-5125	533	6	1	1	X
ejpam-5125	533	7	)	)	PUNCT
ejpam-5125	533	8	is	be	AUX
ejpam-5125	533	9	clear	clear	ADJ
ejpam-5125	533	10	.	.	PUNCT
ejpam-5125	534	1	5	5	X
ejpam-5125	534	2	.	.	X
ejpam-5125	534	3	conclusions	conclusion	NOUN
ejpam-5125	534	4	in	in	ADP
ejpam-5125	534	5	this	this	DET
ejpam-5125	534	6	paper	paper	NOUN
ejpam-5125	534	7	,	,	PUNCT
ejpam-5125	534	8	we	we	PRON
ejpam-5125	534	9	have	have	AUX
ejpam-5125	534	10	introduced	introduce	VERB
ejpam-5125	534	11	the	the	DET
ejpam-5125	534	12	concept	concept	NOUN
ejpam-5125	534	13	of	of	ADP
ejpam-5125	534	14	paradistributive	paradistributive	ADJ
ejpam-5125	534	15	latticoid	latticoid	NOUN
ejpam-5125	534	16	and	and	CCONJ
ejpam-5125	534	17	studied	study	VERB
ejpam-5125	534	18	certain	certain	ADJ
ejpam-5125	534	19	properties	property	NOUN
ejpam-5125	534	20	related	relate	VERB
ejpam-5125	534	21	to	to	ADP
ejpam-5125	534	22	the	the	DET
ejpam-5125	534	23	structure	structure	NOUN
ejpam-5125	534	24	.	.	PUNCT
ejpam-5125	535	1	further	far	ADV
ejpam-5125	535	2	,	,	PUNCT
ejpam-5125	535	3	provided	provide	VERB
ejpam-5125	535	4	a	a	DET
ejpam-5125	535	5	set	set	NOUN
ejpam-5125	535	6	of	of	ADP
ejpam-5125	535	7	equivalence	equivalence	NOUN
ejpam-5125	535	8	conditions	condition	NOUN
ejpam-5125	535	9	for	for	SCONJ
ejpam-5125	535	10	the	the	DET
ejpam-5125	535	11	pdl	pdl	NOUN
ejpam-5125	535	12	to	to	PART
ejpam-5125	535	13	become	become	VERB
ejpam-5125	535	14	a	a	DET
ejpam-5125	535	15	distributive	distributive	ADJ
ejpam-5125	535	16	lattice	lattice	NOUN
ejpam-5125	535	17	.	.	PUNCT
ejpam-5125	536	1	we	we	PRON
ejpam-5125	536	2	have	have	AUX
ejpam-5125	536	3	introduced	introduce	VERB
ejpam-5125	536	4	the	the	DET
ejpam-5125	536	5	notions	notion	NOUN
ejpam-5125	536	6	of	of	ADP
ejpam-5125	536	7	an	an	DET
ejpam-5125	536	8	ideal	ideal	NOUN
ejpam-5125	536	9	and	and	CCONJ
ejpam-5125	536	10	a	a	DET
ejpam-5125	536	11	filter	filter	NOUN
ejpam-5125	536	12	in	in	ADP
ejpam-5125	536	13	a	a	DET
ejpam-5125	536	14	pdl	pdl	NOUN
ejpam-5125	536	15	and	and	CCONJ
ejpam-5125	536	16	studied	study	VERB
ejpam-5125	536	17	their	their	PRON
ejpam-5125	536	18	properties	property	NOUN
ejpam-5125	536	19	.	.	PUNCT
ejpam-5125	537	1	we	we	PRON
ejpam-5125	537	2	have	have	AUX
ejpam-5125	537	3	obtained	obtain	VERB
ejpam-5125	537	4	subdirect	subdirect	NOUN
ejpam-5125	537	5	representation	representation	NOUN
ejpam-5125	537	6	of	of	ADP
ejpam-5125	537	7	a	a	DET
ejpam-5125	537	8	pdl	pdl	NOUN
ejpam-5125	537	9	.	.	PUNCT
ejpam-5125	538	1	in	in	ADP
ejpam-5125	538	2	future	future	NOUN
ejpam-5125	538	3	,	,	PUNCT
ejpam-5125	538	4	our	our	PRON
ejpam-5125	538	5	work	work	NOUN
ejpam-5125	538	6	will	will	AUX
ejpam-5125	538	7	focus	focus	VERB
ejpam-5125	538	8	on	on	ADP
ejpam-5125	538	9	parapseudo	parapseudo	NOUN
ejpam-5125	538	10	-	-	NOUN
ejpam-5125	538	11	complementation	complementation	NOUN
ejpam-5125	538	12	on	on	ADP
ejpam-5125	538	13	a	a	DET
ejpam-5125	538	14	pdl	pdl	NOUN
ejpam-5125	538	15	,	,	PUNCT
ejpam-5125	538	16	stone	stone	NOUN
ejpam-5125	538	17	pdl	pdl	PROPN
ejpam-5125	538	18	,	,	PUNCT
ejpam-5125	538	19	normal	normal	ADJ
ejpam-5125	538	20	pdl	pdl	NOUN
ejpam-5125	538	21	and	and	CCONJ
ejpam-5125	538	22	study	study	VERB
ejpam-5125	538	23	their	their	PRON
ejpam-5125	538	24	topological	topological	ADJ
ejpam-5125	538	25	properties	property	NOUN
ejpam-5125	538	26	.	.	PUNCT
ejpam-5125	539	1	conflicts	conflict	NOUN
ejpam-5125	539	2	of	of	ADP
ejpam-5125	539	3	interest	interest	NOUN
ejpam-5125	539	4	or	or	CCONJ
ejpam-5125	539	5	competing	compete	VERB
ejpam-5125	539	6	interests	interest	NOUN
ejpam-5125	539	7	the	the	DET
ejpam-5125	539	8	authors	author	NOUN
ejpam-5125	539	9	declare	declare	VERB
ejpam-5125	539	10	that	that	SCONJ
ejpam-5125	539	11	they	they	PRON
ejpam-5125	539	12	have	have	VERB
ejpam-5125	539	13	no	no	DET
ejpam-5125	539	14	conflicts	conflict	NOUN
ejpam-5125	539	15	of	of	ADP
ejpam-5125	539	16	interest	interest	NOUN
ejpam-5125	539	17	.	.	PUNCT
ejpam-5125	540	1	data	datum	NOUN
ejpam-5125	540	2	and	and	CCONJ
ejpam-5125	540	3	code	code	NOUN
ejpam-5125	540	4	availability	availability	NOUN
ejpam-5125	540	5	no	no	DET
ejpam-5125	540	6	data	datum	NOUN
ejpam-5125	540	7	were	be	AUX
ejpam-5125	540	8	used	use	VERB
ejpam-5125	540	9	to	to	PART
ejpam-5125	540	10	support	support	VERB
ejpam-5125	540	11	this	this	DET
ejpam-5125	540	12	study	study	NOUN
ejpam-5125	540	13	supplementary	supplementary	ADJ
ejpam-5125	540	14	information	information	NOUN
ejpam-5125	540	15	not	not	PART
ejpam-5125	540	16	applicable	applicable	ADJ
ejpam-5125	540	17	references	reference	NOUN
ejpam-5125	540	18	834	834	NUM
ejpam-5125	540	19	ethical	ethical	ADJ
ejpam-5125	540	20	approval	approval	NOUN
ejpam-5125	540	21	this	this	DET
ejpam-5125	540	22	article	article	NOUN
ejpam-5125	540	23	does	do	AUX
ejpam-5125	540	24	not	not	PART
ejpam-5125	540	25	contain	contain	VERB
ejpam-5125	540	26	any	any	DET
ejpam-5125	540	27	studies	study	NOUN
ejpam-5125	540	28	with	with	ADP
ejpam-5125	540	29	human	human	ADJ
ejpam-5125	540	30	participants	participant	NOUN
ejpam-5125	540	31	or	or	CCONJ
ejpam-5125	540	32	animals	animal	NOUN
ejpam-5125	540	33	performed	perform	VERB
ejpam-5125	540	34	by	by	ADP
ejpam-5125	540	35	any	any	PRON
ejpam-5125	540	36	of	of	ADP
ejpam-5125	540	37	the	the	DET
ejpam-5125	540	38	authors	author	NOUN
ejpam-5125	540	39	informed	inform	VERB
ejpam-5125	540	40	consent	consent	VERB
ejpam-5125	540	41	the	the	DET
ejpam-5125	540	42	authors	author	NOUN
ejpam-5125	540	43	are	be	AUX
ejpam-5125	540	44	fully	fully	ADV
ejpam-5125	540	45	aware	aware	ADJ
ejpam-5125	540	46	and	and	CCONJ
ejpam-5125	540	47	satisfied	satisfied	ADJ
ejpam-5125	540	48	with	with	ADP
ejpam-5125	540	49	the	the	DET
ejpam-5125	540	50	contents	content	NOUN
ejpam-5125	540	51	of	of	ADP
ejpam-5125	540	52	the	the	DET
ejpam-5125	540	53	article	article	NOUN
ejpam-5125	540	54	.	.	PUNCT
ejpam-5125	541	1	acknowledgements	acknowledgement	NOUN
ejpam-5125	541	2	the	the	DET
ejpam-5125	541	3	authors	author	NOUN
ejpam-5125	541	4	wish	wish	VERB
ejpam-5125	541	5	to	to	PART
ejpam-5125	541	6	thank	thank	VERB
ejpam-5125	541	7	the	the	DET
ejpam-5125	541	8	anonymous	anonymous	ADJ
ejpam-5125	541	9	reviewers	reviewer	NOUN
ejpam-5125	541	10	for	for	ADP
ejpam-5125	541	11	their	their	PRON
ejpam-5125	541	12	valuable	valuable	ADJ
ejpam-5125	541	13	suggestions	suggestion	NOUN
ejpam-5125	541	14	.	.	PUNCT
ejpam-5125	542	1	references	reference	NOUN
ejpam-5125	542	2	[	[	X
ejpam-5125	542	3	1	1	NUM
ejpam-5125	542	4	]	]	PUNCT
ejpam-5125	542	5	g	g	NOUN
ejpam-5125	542	6	birkhoff	birkhoff	NOUN
ejpam-5125	542	7	.	.	PUNCT
ejpam-5125	543	1	lattice	lattice	PROPN
ejpam-5125	543	2	theory	theory	NOUN
ejpam-5125	543	3	.	.	PUNCT
ejpam-5125	544	1	colloquium	colloquium	NOUN
ejpam-5125	544	2	publications	publication	NOUN
ejpam-5125	544	3	,	,	PUNCT
ejpam-5125	544	4	american	american	PROPN
ejpam-5125	544	5	mathematical	mathematical	PROPN
ejpam-5125	544	6	society	society	NOUN
ejpam-5125	544	7	,	,	PUNCT
ejpam-5125	544	8	new	new	PROPN
ejpam-5125	544	9	york	york	PROPN
ejpam-5125	544	10	,	,	PUNCT
ejpam-5125	544	11	1940	1940	NUM
ejpam-5125	544	12	.	.	PUNCT
ejpam-5125	545	1	[	[	X
ejpam-5125	545	2	2	2	NUM
ejpam-5125	545	3	]	]	X
ejpam-5125	545	4	g	g	PROPN
ejpam-5125	545	5	boole	boole	PROPN
ejpam-5125	545	6	.	.	PUNCT
ejpam-5125	546	1	an	an	DET
ejpam-5125	546	2	investigation	investigation	NOUN
ejpam-5125	546	3	of	of	ADP
ejpam-5125	546	4	the	the	DET
ejpam-5125	546	5	laws	law	NOUN
ejpam-5125	546	6	of	of	ADP
ejpam-5125	546	7	thought	thought	NOUN
ejpam-5125	546	8	on	on	ADP
ejpam-5125	546	9	which	which	PRON
ejpam-5125	546	10	are	be	AUX
ejpam-5125	546	11	founded	found	VERB
ejpam-5125	546	12	the	the	DET
ejpam-5125	546	13	mathematical	mathematical	ADJ
ejpam-5125	546	14	theories	theory	NOUN
ejpam-5125	546	15	of	of	ADP
ejpam-5125	546	16	logic	logic	NOUN
ejpam-5125	546	17	and	and	CCONJ
ejpam-5125	546	18	probabilities	probability	NOUN
ejpam-5125	546	19	.	.	PUNCT
ejpam-5125	547	1	dover	dover	PROPN
ejpam-5125	547	2	publications	publication	NOUN
ejpam-5125	547	3	,	,	PUNCT
ejpam-5125	547	4	new	new	PROPN
ejpam-5125	547	5	york	york	PROPN
ejpam-5125	547	6	,	,	PUNCT
ejpam-5125	547	7	1958	1958	NUM
ejpam-5125	547	8	.	.	PUNCT
ejpam-5125	548	1	[	[	X
ejpam-5125	548	2	3	3	X
ejpam-5125	548	3	]	]	X
ejpam-5125	548	4	j	j	PROPN
ejpam-5125	548	5	m	m	PROPN
ejpam-5125	548	6	cornejo	cornejo	PROPN
ejpam-5125	548	7	and	and	CCONJ
ejpam-5125	548	8	h	h	NOUN
ejpam-5125	548	9	p	p	PROPN
ejpam-5125	548	10	sankappanavar	sankappanavar	NOUN
ejpam-5125	548	11	.	.	PUNCT
ejpam-5125	549	1	implication	implication	NOUN
ejpam-5125	549	2	zroupoids	zroupoid	NOUN
ejpam-5125	549	3	and	and	CCONJ
ejpam-5125	549	4	birkhoff	birkhoff	NOUN
ejpam-5125	549	5	systems	system	NOUN
ejpam-5125	549	6	.	.	PUNCT
ejpam-5125	550	1	journal	journal	NOUN
ejpam-5125	550	2	of	of	ADP
ejpam-5125	550	3	algebraic	algebraic	PROPN
ejpam-5125	550	4	hyperstructures	hyperstructure	NOUN
ejpam-5125	550	5	and	and	CCONJ
ejpam-5125	550	6	logical	logical	ADJ
ejpam-5125	550	7	algebras	algebra	NOUN
ejpam-5125	550	8	,	,	PUNCT
ejpam-5125	550	9	2(4):1–12	2(4):1–12	NUM
ejpam-5125	550	10	,	,	PUNCT
ejpam-5125	550	11	2021	2021	NUM
ejpam-5125	550	12	.	.	PUNCT
ejpam-5125	551	1	[	[	X
ejpam-5125	551	2	4	4	NUM
ejpam-5125	551	3	]	]	X
ejpam-5125	551	4	j	j	PROPN
ejpam-5125	551	5	harding	harding	PROPN
ejpam-5125	551	6	and	and	CCONJ
ejpam-5125	551	7	a	a	DET
ejpam-5125	551	8	b	b	NOUN
ejpam-5125	551	9	romanowska	romanowska	NOUN
ejpam-5125	551	10	.	.	PUNCT
ejpam-5125	552	1	varieties	variety	NOUN
ejpam-5125	552	2	of	of	ADP
ejpam-5125	552	3	birkhoff	birkhoff	NOUN
ejpam-5125	552	4	systems	system	NOUN
ejpam-5125	552	5	.	.	PUNCT
ejpam-5125	553	1	part	part	NOUN
ejpam-5125	553	2	i.	i.	PROPN
ejpam-5125	553	3	order	order	PROPN
ejpam-5125	553	4	,	,	PUNCT
ejpam-5125	553	5	34:45–68	34:45–68	NUM
ejpam-5125	553	6	,	,	PUNCT
ejpam-5125	553	7	2017	2017	NUM
ejpam-5125	553	8	.	.	PUNCT
ejpam-5125	554	1	[	[	X
ejpam-5125	554	2	5	5	NUM
ejpam-5125	554	3	]	]	PUNCT
ejpam-5125	554	4	j	j	PROPN
ejpam-5125	554	5	a	a	DET
ejpam-5125	554	6	kalman	kalman	PROPN
ejpam-5125	554	7	.	.	PUNCT
ejpam-5125	555	1	subdirect	subdirect	PROPN
ejpam-5125	555	2	decomposition	decomposition	NOUN
ejpam-5125	555	3	of	of	ADP
ejpam-5125	555	4	distributive	distributive	ADJ
ejpam-5125	555	5	quasilattices	quasilattice	NOUN
ejpam-5125	555	6	.	.	PUNCT
ejpam-5125	556	1	fundamenta	fundamenta	PROPN
ejpam-5125	556	2	mathematicae	mathematicae	PROPN
ejpam-5125	556	3	,	,	PUNCT
ejpam-5125	556	4	71:161–163	71:161–163	PROPN
ejpam-5125	556	5	,	,	PUNCT
ejpam-5125	556	6	1971	1971	NUM
ejpam-5125	556	7	.	.	PUNCT
ejpam-5125	557	1	[	[	X
ejpam-5125	557	2	6	6	NUM
ejpam-5125	557	3	]	]	PUNCT
ejpam-5125	557	4	n	n	PRON
ejpam-5125	557	5	v	v	X
ejpam-5125	557	6	subrahmanyam	subrahmanyam	NOUN
ejpam-5125	557	7	.	.	PUNCT
ejpam-5125	558	1	lattice	lattice	PROPN
ejpam-5125	558	2	theory	theory	NOUN
ejpam-5125	558	3	for	for	ADP
ejpam-5125	558	4	certain	certain	ADJ
ejpam-5125	558	5	classes	class	NOUN
ejpam-5125	558	6	of	of	ADP
ejpam-5125	558	7	rings	ring	NOUN
ejpam-5125	558	8	.	.	PUNCT
ejpam-5125	559	1	mathematiche	mathematiche	PROPN
ejpam-5125	559	2	annalen	annalen	PROPN
ejpam-5125	559	3	,	,	PUNCT
ejpam-5125	559	4	139:275–286	139:275–286	NUM
ejpam-5125	559	5	,	,	PUNCT
ejpam-5125	559	6	1960	1960	NUM
ejpam-5125	559	7	.	.	PUNCT
ejpam-5125	560	1	[	[	X
ejpam-5125	560	2	7	7	X
ejpam-5125	560	3	]	]	SYM
ejpam-5125	560	4	n	n	PRON
ejpam-5125	560	5	v	v	X
ejpam-5125	560	6	subrahmanyam	subrahmanyam	NOUN
ejpam-5125	560	7	.	.	PUNCT
ejpam-5125	561	1	an	an	DET
ejpam-5125	561	2	extension	extension	NOUN
ejpam-5125	561	3	of	of	ADP
ejpam-5125	561	4	boolean	boolean	ADJ
ejpam-5125	561	5	lattice	lattice	NOUN
ejpam-5125	561	6	theory	theory	NOUN
ejpam-5125	561	7	.	.	PUNCT
ejpam-5125	562	1	mathematiche	mathematiche	PROPN
ejpam-5125	562	2	annalen	annalen	PROPN
ejpam-5125	562	3	,	,	PUNCT
ejpam-5125	562	4	151:332–345	151:332–345	NUM
ejpam-5125	562	5	,	,	PUNCT
ejpam-5125	562	6	1963	1963	NUM
ejpam-5125	562	7	.	.	PUNCT
ejpam-5125	563	1	[	[	X
ejpam-5125	563	2	8	8	NUM
ejpam-5125	563	3	]	]	X
ejpam-5125	563	4	u	u	NOUN
ejpam-5125	563	5	m	m	NOUN
ejpam-5125	563	6	swamy	swamy	NOUN
ejpam-5125	563	7	and	and	CCONJ
ejpam-5125	563	8	g	g	PROPN
ejpam-5125	563	9	c	c	PROPN
ejpam-5125	563	10	rao	rao	PROPN
ejpam-5125	563	11	.	.	PUNCT
ejpam-5125	564	1	almost	almost	ADV
ejpam-5125	564	2	distributive	distributive	ADJ
ejpam-5125	564	3	lattices	lattice	NOUN
ejpam-5125	564	4	.	.	PUNCT
ejpam-5125	565	1	journal	journal	NOUN
ejpam-5125	565	2	of	of	ADP
ejpam-5125	565	3	the	the	DET
ejpam-5125	565	4	australian	australian	ADJ
ejpam-5125	565	5	mathematical	mathematical	ADJ
ejpam-5125	565	6	society	society	NOUN
ejpam-5125	565	7	.	.	PUNCT
ejpam-5125	566	1	series	series	PROPN
ejpam-5125	566	2	a.	a.	PROPN
ejpam-5125	566	3	,	,	PUNCT
ejpam-5125	566	4	31:77–91	31:77–91	NUM
ejpam-5125	566	5	,	,	PUNCT
ejpam-5125	566	6	1981	1981	NUM
ejpam-5125	566	7	.	.	PUNCT