id	sid	tid	token	lemma	pos
ejpam-3036	1	1	european	european	PROPN
ejpam-3036	1	2	journal	journal	PROPN
ejpam-3036	1	3	of	of	ADP
ejpam-3036	1	4	pure	pure	ADJ
ejpam-3036	1	5	and	and	CCONJ
ejpam-3036	1	6	applied	apply	VERB
ejpam-3036	1	7	mathematics	mathematic	NOUN
ejpam-3036	1	8	vol	vol	NOUN
ejpam-3036	1	9	.	.	PROPN
ejpam-3036	2	1	10	10	NUM
ejpam-3036	2	2	,	,	PUNCT
ejpam-3036	2	3	no	no	INTJ
ejpam-3036	2	4	.	.	NOUN
ejpam-3036	2	5	4	4	NUM
ejpam-3036	2	6	,	,	PUNCT
ejpam-3036	2	7	2017	2017	NUM
ejpam-3036	2	8	,	,	PUNCT
ejpam-3036	2	9	717	717	NUM
ejpam-3036	2	10	-	-	SYM
ejpam-3036	2	11	729	729	NUM
ejpam-3036	2	12	issn	issn	PROPN
ejpam-3036	2	13	1307	1307	NUM
ejpam-3036	2	14	-	-	SYM
ejpam-3036	2	15	5543	5543	NUM
ejpam-3036	2	16	–	–	PUNCT
ejpam-3036	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3036	2	18	published	publish	VERB
ejpam-3036	2	19	by	by	ADP
ejpam-3036	2	20	new	new	PROPN
ejpam-3036	2	21	york	york	PROPN
ejpam-3036	2	22	business	business	PROPN
ejpam-3036	2	23	global	global	ADJ
ejpam-3036	2	24	centre	centre	PROPN
ejpam-3036	2	25	of	of	ADP
ejpam-3036	2	26	core	core	NOUN
ejpam-3036	2	27	regular	regular	ADJ
ejpam-3036	2	28	double	double	ADJ
ejpam-3036	2	29	stone	stone	NOUN
ejpam-3036	2	30	algebra	algebra	NOUN
ejpam-3036	2	31	a	a	DET
ejpam-3036	2	32	r	r	NOUN
ejpam-3036	2	33	j	j	PROPN
ejpam-3036	2	34	srikanth1,2,∗	srikanth1,2,∗	NOUN
ejpam-3036	2	35	,	,	PUNCT
ejpam-3036	2	36	r	r	NOUN
ejpam-3036	2	37	v	v	NUM
ejpam-3036	2	38	g	g	NOUN
ejpam-3036	2	39	ravi	ravi	NOUN
ejpam-3036	2	40	kumar3	kumar3	NOUN
ejpam-3036	2	41	1	1	NUM
ejpam-3036	2	42	g	g	NOUN
ejpam-3036	2	43	v	v	ADP
ejpam-3036	2	44	p	p	PRON
ejpam-3036	2	45	college	college	NOUN
ejpam-3036	2	46	of	of	ADP
ejpam-3036	2	47	engineering	engineering	PROPN
ejpam-3036	2	48	,	,	PUNCT
ejpam-3036	2	49	andhra	andhra	PROPN
ejpam-3036	2	50	pradesh	pradesh	PROPN
ejpam-3036	2	51	,	,	PUNCT
ejpam-3036	2	52	india	india	PROPN
ejpam-3036	2	53	2	2	PROPN
ejpam-3036	2	54	jawaharlal	jawaharlal	PROPN
ejpam-3036	2	55	nehru	nehru	PROPN
ejpam-3036	2	56	technological	technological	PROPN
ejpam-3036	2	57	university	university	PROPN
ejpam-3036	2	58	kakinada	kakinada	PROPN
ejpam-3036	2	59	,	,	PUNCT
ejpam-3036	2	60	andhra	andhra	PROPN
ejpam-3036	2	61	pradesh	pradesh	PROPN
ejpam-3036	2	62	,	,	PUNCT
ejpam-3036	2	63	india	india	PROPN
ejpam-3036	2	64	3	3	NUM
ejpam-3036	2	65	g	g	NOUN
ejpam-3036	2	66	v	v	ADP
ejpam-3036	2	67	p	p	PRON
ejpam-3036	2	68	college	college	NOUN
ejpam-3036	2	69	of	of	ADP
ejpam-3036	2	70	engineering	engineering	PROPN
ejpam-3036	2	71	,	,	PUNCT
ejpam-3036	2	72	andhra	andhra	PROPN
ejpam-3036	2	73	pradesh	pradesh	PROPN
ejpam-3036	2	74	,	,	PUNCT
ejpam-3036	2	75	india	india	PROPN
ejpam-3036	2	76	abstract	abstract	NOUN
ejpam-3036	2	77	.	.	PUNCT
ejpam-3036	3	1	in	in	ADP
ejpam-3036	3	2	literature	literature	NOUN
ejpam-3036	3	3	there	there	PRON
ejpam-3036	3	4	is	be	VERB
ejpam-3036	3	5	an	an	DET
ejpam-3036	3	6	elegant	elegant	ADJ
ejpam-3036	3	7	characterization	characterization	NOUN
ejpam-3036	3	8	of	of	ADP
ejpam-3036	3	9	factor	factor	NOUN
ejpam-3036	3	10	congruences	congruence	VERB
ejpam-3036	3	11	on	on	ADP
ejpam-3036	3	12	a	a	DET
ejpam-3036	3	13	distributive	distributive	ADJ
ejpam-3036	3	14	lattice	lattice	NOUN
ejpam-3036	3	15	.	.	PUNCT
ejpam-3036	4	1	in	in	ADP
ejpam-3036	4	2	this	this	DET
ejpam-3036	4	3	paper	paper	NOUN
ejpam-3036	4	4	,	,	PUNCT
ejpam-3036	4	5	we	we	PRON
ejpam-3036	4	6	make	make	VERB
ejpam-3036	4	7	an	an	DET
ejpam-3036	4	8	attempt	attempt	NOUN
ejpam-3036	4	9	such	such	ADJ
ejpam-3036	4	10	type	type	NOUN
ejpam-3036	4	11	of	of	ADP
ejpam-3036	4	12	characterization	characterization	NOUN
ejpam-3036	4	13	of	of	ADP
ejpam-3036	4	14	factor	factor	NOUN
ejpam-3036	4	15	congruences	congruence	VERB
ejpam-3036	4	16	on	on	ADP
ejpam-3036	4	17	a	a	DET
ejpam-3036	4	18	core	core	NOUN
ejpam-3036	4	19	regular	regular	ADJ
ejpam-3036	4	20	double	double	ADJ
ejpam-3036	4	21	stone	stone	NOUN
ejpam-3036	4	22	algebra	algebra	NOUN
ejpam-3036	4	23	(	(	PUNCT
ejpam-3036	4	24	crdsa	crdsa	PROPN
ejpam-3036	4	25	)	)	PUNCT
ejpam-3036	4	26	and	and	CCONJ
ejpam-3036	4	27	we	we	PRON
ejpam-3036	4	28	identify	identify	VERB
ejpam-3036	4	29	that	that	SCONJ
ejpam-3036	4	30	the	the	DET
ejpam-3036	4	31	factor	factor	NOUN
ejpam-3036	4	32	congruences	congruence	VERB
ejpam-3036	4	33	on	on	ADP
ejpam-3036	4	34	a	a	DET
ejpam-3036	4	35	crdsa	crdsa	NOUN
ejpam-3036	4	36	a	a	PRON
ejpam-3036	4	37	with	with	ADP
ejpam-3036	4	38	certain	certain	ADJ
ejpam-3036	4	39	elements	element	NOUN
ejpam-3036	4	40	of	of	ADP
ejpam-3036	4	41	a	a	PRON
ejpam-3036	4	42	and	and	CCONJ
ejpam-3036	4	43	proved	prove	VERB
ejpam-3036	4	44	that	that	SCONJ
ejpam-3036	4	45	set	set	NOUN
ejpam-3036	4	46	of	of	ADP
ejpam-3036	4	47	all	all	DET
ejpam-3036	4	48	factor	factor	NOUN
ejpam-3036	4	49	congruences	congruence	VERB
ejpam-3036	4	50	forms	form	VERB
ejpam-3036	4	51	a	a	DET
ejpam-3036	4	52	boolean	boolean	ADJ
ejpam-3036	4	53	centre	centre	NOUN
ejpam-3036	4	54	for	for	ADP
ejpam-3036	4	55	a.	a.	NOUN
ejpam-3036	4	56	further	further	ADJ
ejpam-3036	4	57	birkhoff	birkhoff	NOUN
ejpam-3036	4	58	centre	centre	PROPN
ejpam-3036	4	59	is	be	AUX
ejpam-3036	4	60	defined	define	VERB
ejpam-3036	4	61	for	for	ADP
ejpam-3036	4	62	crdsa	crdsa	ADV
ejpam-3036	4	63	and	and	CCONJ
ejpam-3036	4	64	finally	finally	ADV
ejpam-3036	4	65	it	it	PRON
ejpam-3036	4	66	is	be	AUX
ejpam-3036	4	67	shown	show	VERB
ejpam-3036	4	68	that	that	SCONJ
ejpam-3036	4	69	birkhoff	birkhoff	NOUN
ejpam-3036	4	70	centre	centre	NOUN
ejpam-3036	4	71	of	of	ADP
ejpam-3036	4	72	crdsa	crdsa	PROPN
ejpam-3036	4	73	is	be	AUX
ejpam-3036	4	74	isomorphic	isomorphic	ADJ
ejpam-3036	4	75	to	to	ADP
ejpam-3036	4	76	its	its	PRON
ejpam-3036	4	77	boolean	boolean	ADJ
ejpam-3036	4	78	centre	centre	NOUN
ejpam-3036	4	79	2010	2010	NUM
ejpam-3036	4	80	mathematics	mathematic	NOUN
ejpam-3036	4	81	subject	subject	NOUN
ejpam-3036	4	82	classifications	classification	NOUN
ejpam-3036	4	83	:	:	PUNCT
ejpam-3036	4	84	06d99	06d99	NUM
ejpam-3036	4	85	,	,	PUNCT
ejpam-3036	4	86	03g10,06d15	03g10,06d15	NUM
ejpam-3036	4	87	key	key	ADJ
ejpam-3036	4	88	words	word	NOUN
ejpam-3036	4	89	and	and	CCONJ
ejpam-3036	4	90	phrases	phrase	NOUN
ejpam-3036	4	91	:	:	PUNCT
ejpam-3036	4	92	core	core	NOUN
ejpam-3036	4	93	regular	regular	ADJ
ejpam-3036	4	94	double	double	ADJ
ejpam-3036	4	95	stone	stone	NOUN
ejpam-3036	4	96	algebra	algebra	NOUN
ejpam-3036	4	97	,	,	PUNCT
ejpam-3036	4	98	boolean	boolean	ADJ
ejpam-3036	4	99	centre	centre	NOUN
ejpam-3036	4	100	,	,	PUNCT
ejpam-3036	4	101	birkhoff	birkhoff	NOUN
ejpam-3036	4	102	centre	centre	NOUN
ejpam-3036	4	103	1	1	NUM
ejpam-3036	4	104	.	.	PUNCT
ejpam-3036	5	1	introduction	introduction	NOUN
ejpam-3036	5	2	the	the	DET
ejpam-3036	5	3	concept	concept	NOUN
ejpam-3036	5	4	of	of	ADP
ejpam-3036	5	5	a	a	DET
ejpam-3036	5	6	core	core	NOUN
ejpam-3036	5	7	regular	regular	ADJ
ejpam-3036	5	8	double	double	ADJ
ejpam-3036	5	9	stone	stone	NOUN
ejpam-3036	5	10	algebra	algebra	NOUN
ejpam-3036	5	11	was	be	AUX
ejpam-3036	5	12	introduced	introduce	VERB
ejpam-3036	5	13	by	by	ADP
ejpam-3036	5	14	ravi	ravi	PROPN
ejpam-3036	5	15	kumar	kumar	PROPN
ejpam-3036	5	16	etal	etal	PROPN
ejpam-3036	5	17	and	and	CCONJ
ejpam-3036	5	18	obtained	obtain	VERB
ejpam-3036	5	19	a	a	DET
ejpam-3036	5	20	decomposition	decomposition	NOUN
ejpam-3036	5	21	theorem	theorem	NOUN
ejpam-3036	5	22	for	for	ADP
ejpam-3036	5	23	a	a	DET
ejpam-3036	5	24	complete	complete	ADJ
ejpam-3036	5	25	atomic	atomic	ADJ
ejpam-3036	5	26	core	core	NOUN
ejpam-3036	5	27	regular	regular	ADJ
ejpam-3036	5	28	double	double	ADJ
ejpam-3036	5	29	stone	stone	NOUN
ejpam-3036	5	30	algebra	algebra	NOUN
ejpam-3036	6	1	[	[	X
ejpam-3036	6	2	5	5	NUM
ejpam-3036	6	3	]	]	PUNCT
ejpam-3036	6	4	.	.	PUNCT
ejpam-3036	7	1	in	in	ADP
ejpam-3036	7	2	[	[	X
ejpam-3036	7	3	6	6	NUM
ejpam-3036	7	4	]	]	PUNCT
ejpam-3036	7	5	,	,	PUNCT
ejpam-3036	7	6	u.m	u.m	PROPN
ejpam-3036	7	7	.	.	PROPN
ejpam-3036	7	8	swamy	swamy	PROPN
ejpam-3036	7	9	and	and	CCONJ
ejpam-3036	7	10	g.s	g.s	PROPN
ejpam-3036	7	11	.	.	PROPN
ejpam-3036	7	12	murti	murti	PROPN
ejpam-3036	7	13	introduced	introduce	VERB
ejpam-3036	7	14	the	the	DET
ejpam-3036	7	15	concept	concept	NOUN
ejpam-3036	7	16	of	of	ADP
ejpam-3036	7	17	the	the	DET
ejpam-3036	7	18	boolean	boolean	ADJ
ejpam-3036	7	19	center	center	NOUN
ejpam-3036	7	20	of	of	ADP
ejpam-3036	7	21	an	an	DET
ejpam-3036	7	22	universal	universal	ADJ
ejpam-3036	7	23	algebra	algebra	NOUN
ejpam-3036	7	24	.	.	PUNCT
ejpam-3036	8	1	in	in	ADP
ejpam-3036	8	2	this	this	DET
ejpam-3036	8	3	paper	paper	NOUN
ejpam-3036	8	4	we	we	PRON
ejpam-3036	8	5	make	make	VERB
ejpam-3036	8	6	an	an	DET
ejpam-3036	8	7	attempt	attempt	NOUN
ejpam-3036	8	8	to	to	PART
ejpam-3036	8	9	characterize	characterize	VERB
ejpam-3036	8	10	the	the	DET
ejpam-3036	8	11	boolean	boolean	ADJ
ejpam-3036	8	12	centre	centre	NOUN
ejpam-3036	8	13	of	of	ADP
ejpam-3036	8	14	a	a	DET
ejpam-3036	8	15	crdsa	crdsa	NOUN
ejpam-3036	8	16	a	a	PRON
ejpam-3036	8	17	and	and	CCONJ
ejpam-3036	8	18	the	the	DET
ejpam-3036	8	19	concept	concept	NOUN
ejpam-3036	8	20	of	of	ADP
ejpam-3036	8	21	birkhoff	birkhoff	NOUN
ejpam-3036	8	22	’s	’s	PART
ejpam-3036	8	23	‘	'	PUNCT
ejpam-3036	8	24	centre	centre	NOUN
ejpam-3036	8	25	’	'	PUNCT
ejpam-3036	8	26	of	of	ADP
ejpam-3036	8	27	a	a	DET
ejpam-3036	8	28	bounded	bounded	ADJ
ejpam-3036	8	29	poset	poset	NOUN
ejpam-3036	8	30	is	be	AUX
ejpam-3036	8	31	extended	extend	VERB
ejpam-3036	8	32	to	to	ADP
ejpam-3036	8	33	crdsa	crdsa	PROPN
ejpam-3036	8	34	a	a	PRON
ejpam-3036	8	35	and	and	CCONJ
ejpam-3036	8	36	referred	refer	VERB
ejpam-3036	8	37	to	to	ADP
ejpam-3036	8	38	this	this	PRON
ejpam-3036	8	39	,	,	PUNCT
ejpam-3036	8	40	as	as	ADP
ejpam-3036	8	41	‘	'	PUNCT
ejpam-3036	8	42	birkhoff	birkhoff	NOUN
ejpam-3036	8	43	centre	centre	NOUN
ejpam-3036	8	44	’	'	PUNCT
ejpam-3036	8	45	of	of	ADP
ejpam-3036	8	46	a.	a.	NOUN
ejpam-3036	8	47	it	it	PRON
ejpam-3036	8	48	is	be	AUX
ejpam-3036	8	49	also	also	ADV
ejpam-3036	8	50	proved	prove	VERB
ejpam-3036	8	51	that	that	SCONJ
ejpam-3036	8	52	birkhoff	birkhoff	NOUN
ejpam-3036	8	53	centre	centre	PROPN
ejpam-3036	8	54	bc(a	bc(a	NUM
ejpam-3036	8	55	)	)	PUNCT
ejpam-3036	8	56	is	be	AUX
ejpam-3036	8	57	isomorphic	isomorphic	ADJ
ejpam-3036	8	58	to	to	ADP
ejpam-3036	8	59	boolean	boolean	ADJ
ejpam-3036	8	60	centre	centre	NOUN
ejpam-3036	8	61	of	of	ADP
ejpam-3036	8	62	a.	a.	NOUN
ejpam-3036	8	63	2	2	NUM
ejpam-3036	9	1	.	.	PUNCT
ejpam-3036	9	2	preliminaries	preliminary	NOUN
ejpam-3036	9	3	in	in	ADP
ejpam-3036	9	4	this	this	DET
ejpam-3036	9	5	section	section	NOUN
ejpam-3036	9	6	the	the	DET
ejpam-3036	9	7	concept	concept	NOUN
ejpam-3036	9	8	of	of	ADP
ejpam-3036	9	9	the	the	DET
ejpam-3036	9	10	isomorphism	isomorphism	NOUN
ejpam-3036	9	11	of	of	ADP
ejpam-3036	9	12	rdsa	rdsa	NOUN
ejpam-3036	9	13	is	be	AUX
ejpam-3036	9	14	extended	extend	VERB
ejpam-3036	9	15	to	to	ADP
ejpam-3036	9	16	crdsa	crdsa	PROPN
ejpam-3036	9	17	and	and	CCONJ
ejpam-3036	9	18	a	a	DET
ejpam-3036	9	19	new	new	ADJ
ejpam-3036	9	20	characterization	characterization	NOUN
ejpam-3036	9	21	for	for	ADP
ejpam-3036	9	22	centre	centre	NOUN
ejpam-3036	9	23	of	of	ADP
ejpam-3036	9	24	a	a	DET
ejpam-3036	9	25	crdsa	crdsa	NOUN
ejpam-3036	9	26	based	base	VERB
ejpam-3036	9	27	on	on	ADP
ejpam-3036	9	28	core	core	NOUN
ejpam-3036	9	29	element	element	NOUN
ejpam-3036	9	30	is	be	AUX
ejpam-3036	9	31	done	do	VERB
ejpam-3036	9	32	.	.	PUNCT
ejpam-3036	10	1	we	we	PRON
ejpam-3036	10	2	start	start	VERB
ejpam-3036	10	3	with	with	ADP
ejpam-3036	10	4	certain	certain	ADJ
ejpam-3036	10	5	basic	basic	ADJ
ejpam-3036	10	6	definitions	definition	NOUN
ejpam-3036	10	7	and	and	CCONJ
ejpam-3036	10	8	properties	property	NOUN
ejpam-3036	10	9	of	of	ADP
ejpam-3036	10	10	rdsa	rdsa	NOUN
ejpam-3036	10	11	.	.	PUNCT
ejpam-3036	11	1	∗corresponding	∗corresponde	VERB
ejpam-3036	11	2	author	author	NOUN
ejpam-3036	11	3	.	.	PUNCT
ejpam-3036	12	1	email	email	NOUN
ejpam-3036	12	2	addresses	address	NOUN
ejpam-3036	12	3	:	:	PUNCT
ejpam-3036	13	1	rvgravikumar@yahoo.com	rvgravikumar@yahoo.com	X
ejpam-3036	13	2	(	(	PUNCT
ejpam-3036	13	3	r	r	NOUN
ejpam-3036	13	4	v	v	NUM
ejpam-3036	13	5	g	g	PROPN
ejpam-3036	13	6	ravi	ravi	PROPN
ejpam-3036	13	7	kumar	kumar	PROPN
ejpam-3036	13	8	)	)	PUNCT
ejpam-3036	13	9	,	,	PUNCT
ejpam-3036	13	10	seeku.ammu@gmail.com	seeku.ammu@gmail.com	X
ejpam-3036	13	11	(	(	PUNCT
ejpam-3036	13	12	a	a	DET
ejpam-3036	13	13	r	r	NOUN
ejpam-3036	13	14	j	j	PROPN
ejpam-3036	13	15	srikanth	srikanth	PROPN
ejpam-3036	13	16	)	)	PUNCT
ejpam-3036	13	17	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3036	14	1	717	717	NUM
ejpam-3036	14	2	c	c	NOUN
ejpam-3036	14	3	©	©	PROPN
ejpam-3036	14	4	2017	2017	NUM
ejpam-3036	14	5	ejpam	ejpam	VERB
ejpam-3036	14	6	all	all	DET
ejpam-3036	14	7	rights	right	NOUN
ejpam-3036	14	8	reserved	reserve	VERB
ejpam-3036	14	9	.	.	PUNCT
ejpam-3036	15	1	a	a	DET
ejpam-3036	15	2	r	r	NOUN
ejpam-3036	15	3	j	j	PROPN
ejpam-3036	15	4	srikanth	srikanth	PROPN
ejpam-3036	15	5	,	,	PUNCT
ejpam-3036	15	6	r	r	NOUN
ejpam-3036	15	7	v	v	NUM
ejpam-3036	15	8	g	g	NOUN
ejpam-3036	15	9	ravi	ravi	PROPN
ejpam-3036	15	10	kumar	kumar	PROPN
ejpam-3036	15	11	/	/	SYM
ejpam-3036	15	12	eur	eur	PROPN
ejpam-3036	15	13	.	.	PUNCT
ejpam-3036	16	1	j.	j.	PROPN
ejpam-3036	16	2	pure	pure	PROPN
ejpam-3036	16	3	appl	appl	PROPN
ejpam-3036	16	4	.	.	PROPN
ejpam-3036	16	5	math	math	PROPN
ejpam-3036	16	6	,	,	PUNCT
ejpam-3036	16	7	10	10	NUM
ejpam-3036	16	8	(	(	PUNCT
ejpam-3036	16	9	4	4	NUM
ejpam-3036	16	10	)	)	PUNCT
ejpam-3036	16	11	(	(	PUNCT
ejpam-3036	16	12	2017	2017	NUM
ejpam-3036	16	13	)	)	PUNCT
ejpam-3036	16	14	,	,	PUNCT
ejpam-3036	16	15	717	717	NUM
ejpam-3036	16	16	-	-	SYM
ejpam-3036	16	17	729	729	NUM
ejpam-3036	16	18	718	718	NUM
ejpam-3036	16	19	definition	definition	NOUN
ejpam-3036	16	20	1	1	NUM
ejpam-3036	16	21	.	.	PUNCT
ejpam-3036	17	1	a	a	DET
ejpam-3036	17	2	regular	regular	ADJ
ejpam-3036	17	3	double	double	ADJ
ejpam-3036	17	4	stone	stone	NOUN
ejpam-3036	17	5	algebra	algebra	NOUN
ejpam-3036	17	6	(	(	PUNCT
ejpam-3036	17	7	rdsa	rdsa	PROPN
ejpam-3036	17	8	)	)	PUNCT
ejpam-3036	17	9	<	<	X
ejpam-3036	17	10	a,∧,∨	a,∧,∨	X
ejpam-3036	17	11	,	,	PUNCT
ejpam-3036	17	12	∗,+	∗,+	NOUN
ejpam-3036	17	13	,	,	PUNCT
ejpam-3036	17	14	0	0	NUM
ejpam-3036	17	15	,	,	PUNCT
ejpam-3036	17	16	1	1	NUM
ejpam-3036	17	17	>	>	X
ejpam-3036	17	18	is	be	AUX
ejpam-3036	17	19	an	an	DET
ejpam-3036	17	20	algebra	algebra	NOUN
ejpam-3036	17	21	of	of	ADP
ejpam-3036	17	22	type	type	NOUN
ejpam-3036	17	23	<	<	X
ejpam-3036	17	24	2	2	NUM
ejpam-3036	17	25	,	,	PUNCT
ejpam-3036	17	26	2	2	NUM
ejpam-3036	17	27	,	,	PUNCT
ejpam-3036	17	28	1	1	NUM
ejpam-3036	17	29	,	,	PUNCT
ejpam-3036	17	30	1	1	NUM
ejpam-3036	17	31	,	,	PUNCT
ejpam-3036	17	32	0	0	NUM
ejpam-3036	17	33	,	,	PUNCT
ejpam-3036	17	34	0	0	NUM
ejpam-3036	17	35	>	>	X
ejpam-3036	17	36	such	such	ADJ
ejpam-3036	17	37	that	that	SCONJ
ejpam-3036	17	38	(	(	PUNCT
ejpam-3036	17	39	i	i	NOUN
ejpam-3036	17	40	)	)	PUNCT
ejpam-3036	17	41	<	<	X
ejpam-3036	17	42	a,∧,∨	a,∧,∨	PROPN
ejpam-3036	17	43	,	,	PUNCT
ejpam-3036	17	44	0	0	NUM
ejpam-3036	17	45	,	,	PUNCT
ejpam-3036	17	46	1	1	NUM
ejpam-3036	17	47	>	>	X
ejpam-3036	17	48	is	be	AUX
ejpam-3036	17	49	a	a	DET
ejpam-3036	17	50	bounded	bounded	ADJ
ejpam-3036	17	51	distributive	distributive	ADJ
ejpam-3036	17	52	lattice	lattice	NOUN
ejpam-3036	17	53	.	.	PUNCT
ejpam-3036	18	1	(	(	PUNCT
ejpam-3036	18	2	ii	ii	NOUN
ejpam-3036	18	3	)	)	PUNCT
ejpam-3036	18	4	∗	∗	NOUN
ejpam-3036	18	5	is	be	AUX
ejpam-3036	18	6	a	a	DET
ejpam-3036	18	7	pseudo	pseudo	NOUN
ejpam-3036	18	8	complementation	complementation	NOUN
ejpam-3036	18	9	satisfying	satisfy	VERB
ejpam-3036	18	10	the	the	DET
ejpam-3036	18	11	stone	stone	NOUN
ejpam-3036	18	12	identity	identity	NOUN
ejpam-3036	18	13	x∗	x∗	PROPN
ejpam-3036	18	14	∨	∨	NUM
ejpam-3036	18	15	x∗∗	x∗∗	PROPN
ejpam-3036	19	1	=	=	SYM
ejpam-3036	19	2	1	1	NUM
ejpam-3036	19	3	(	(	PUNCT
ejpam-3036	19	4	iii	iii	NOUN
ejpam-3036	19	5	)	)	PUNCT
ejpam-3036	19	6	+	+	CCONJ
ejpam-3036	19	7	is	be	AUX
ejpam-3036	19	8	a	a	DET
ejpam-3036	19	9	dual	dual	ADJ
ejpam-3036	19	10	pseudo	pseudo	NOUN
ejpam-3036	19	11	complementation	complementation	NOUN
ejpam-3036	19	12	satisfying	satisfy	VERB
ejpam-3036	19	13	the	the	DET
ejpam-3036	19	14	dual	dual	ADJ
ejpam-3036	19	15	stone	stone	NOUN
ejpam-3036	19	16	x+	x+	PUNCT
ejpam-3036	19	17	∧	∧	NOUN
ejpam-3036	19	18	x++	x++	PUNCT
ejpam-3036	19	19	=	=	SYM
ejpam-3036	19	20	0	0	PUNCT
ejpam-3036	19	21	(	(	PUNCT
ejpam-3036	19	22	iv	iv	X
ejpam-3036	19	23	)	)	PUNCT
ejpam-3036	19	24	for	for	ADP
ejpam-3036	19	25	any	any	DET
ejpam-3036	19	26	x	x	NOUN
ejpam-3036	19	27	,	,	PUNCT
ejpam-3036	19	28	y	y	PROPN
ejpam-3036	19	29	∈	∈	PROPN
ejpam-3036	19	30	a	a	PRON
ejpam-3036	19	31	,	,	PUNCT
ejpam-3036	19	32	x∗	x∗	PROPN
ejpam-3036	19	33	=	=	X
ejpam-3036	20	1	y∗	y∗	PROPN
ejpam-3036	20	2	and	and	CCONJ
ejpam-3036	20	3	x+	x+	NUM
ejpam-3036	20	4	=	=	NOUN
ejpam-3036	20	5	y+	y+	NOUN
ejpam-3036	20	6	then	then	ADV
ejpam-3036	20	7	x	x	X
ejpam-3036	20	8	=	=	PUNCT
ejpam-3036	20	9	y	y	PROPN
ejpam-3036	20	10	example	example	NOUN
ejpam-3036	20	11	1	1	X
ejpam-3036	20	12	.	.	X
ejpam-3036	21	1	consider	consider	VERB
ejpam-3036	21	2	the	the	DET
ejpam-3036	21	3	hasse	hasse	NOUN
ejpam-3036	21	4	diagrams	diagram	NOUN
ejpam-3036	21	5	of	of	ADP
ejpam-3036	21	6	lattices	lattice	NOUN
ejpam-3036	21	7	l1	l1	PROPN
ejpam-3036	21	8	and	and	CCONJ
ejpam-3036	21	9	l2	l2	NOUN
ejpam-3036	21	10	.	.	PUNCT
ejpam-3036	22	1	1	1	NUM
ejpam-3036	22	2	c	c	X
ejpam-3036	22	3	d	d	NOUN
ejpam-3036	22	4	a	a	DET
ejpam-3036	22	5	b	b	PROPN
ejpam-3036	22	6	0	0	NUM
ejpam-3036	22	7	figure	figure	NOUN
ejpam-3036	22	8	1	1	NUM
ejpam-3036	22	9	:	:	PUNCT
ejpam-3036	22	10	l1	l1	PROPN
ejpam-3036	22	11	1	1	NUM
ejpam-3036	22	12	d	d	PROPN
ejpam-3036	22	13	b	b	PROPN
ejpam-3036	22	14	c	c	NOUN
ejpam-3036	22	15	a	a	DET
ejpam-3036	22	16	0	0	NUM
ejpam-3036	22	17	figure	figure	NOUN
ejpam-3036	22	18	2	2	NUM
ejpam-3036	22	19	:	:	PUNCT
ejpam-3036	22	20	l2	l2	VERB
ejpam-3036	22	21	clearly	clearly	ADV
ejpam-3036	22	22	l1	l1	PROPN
ejpam-3036	22	23	,	,	PUNCT
ejpam-3036	22	24	l2	l2	NOUN
ejpam-3036	22	25	are	be	AUX
ejpam-3036	22	26	bounded	bound	VERB
ejpam-3036	22	27	distributive	distributive	ADJ
ejpam-3036	22	28	lattices	lattice	NOUN
ejpam-3036	22	29	,	,	PUNCT
ejpam-3036	22	30	pseudo	pseudo	NOUN
ejpam-3036	22	31	complemented	complemented	ADJ
ejpam-3036	22	32	and	and	CCONJ
ejpam-3036	22	33	dual	dual	ADJ
ejpam-3036	22	34	pseudo	pseudo	NOUN
ejpam-3036	22	35	complemented	complement	VERB
ejpam-3036	22	36	,	,	PUNCT
ejpam-3036	22	37	and	and	CCONJ
ejpam-3036	22	38	in	in	ADP
ejpam-3036	22	39	l1	l1	PROPN
ejpam-3036	22	40	,	,	PUNCT
ejpam-3036	22	41	0	0	NUM
ejpam-3036	22	42	∗	∗	NOUN
ejpam-3036	22	43	=	=	SYM
ejpam-3036	22	44	1	1	NUM
ejpam-3036	22	45	,	,	PUNCT
ejpam-3036	22	46	a∗	a∗	PROPN
ejpam-3036	22	47	=	=	SYM
ejpam-3036	22	48	b	b	PROPN
ejpam-3036	22	49	,	,	PUNCT
ejpam-3036	22	50	b∗	b∗	ADJ
ejpam-3036	22	51	=	=	SYM
ejpam-3036	22	52	c	c	NOUN
ejpam-3036	22	53	,	,	PUNCT
ejpam-3036	22	54	c∗	c∗	PROPN
ejpam-3036	22	55	=	=	SYM
ejpam-3036	22	56	b	b	PROPN
ejpam-3036	22	57	,	,	PUNCT
ejpam-3036	22	58	d∗	d∗	NOUN
ejpam-3036	22	59	=	=	SYM
ejpam-3036	22	60	0	0	NUM
ejpam-3036	22	61	,	,	PUNCT
ejpam-3036	22	62	1∗	1∗	NUM
ejpam-3036	22	63	=	=	SYM
ejpam-3036	22	64	0	0	NUM
ejpam-3036	22	65	and	and	CCONJ
ejpam-3036	22	66	0	0	NUM
ejpam-3036	22	67	+	+	NUM
ejpam-3036	22	68	=	=	SYM
ejpam-3036	22	69	1	1	NUM
ejpam-3036	22	70	,	,	PUNCT
ejpam-3036	22	71	a+	a+	PUNCT
ejpam-3036	22	72	=	=	NOUN
ejpam-3036	22	73	1	1	NUM
ejpam-3036	22	74	,	,	PUNCT
ejpam-3036	22	75	b+	b+	X
ejpam-3036	22	76	=	=	SYM
ejpam-3036	22	77	c	c	X
ejpam-3036	22	78	,	,	PUNCT
ejpam-3036	22	79	c+	c+	VERB
ejpam-3036	22	80	=	=	SYM
ejpam-3036	22	81	b	b	NUM
ejpam-3036	22	82	,	,	PUNCT
ejpam-3036	22	83	d+	d+	PUNCT
ejpam-3036	22	84	=	=	SYM
ejpam-3036	22	85	c	c	X
ejpam-3036	22	86	,	,	PUNCT
ejpam-3036	22	87	1	1	NUM
ejpam-3036	22	88	+	+	NUM
ejpam-3036	22	89	=	=	SYM
ejpam-3036	22	90	0	0	X
ejpam-3036	22	91	.	.	PUNCT
ejpam-3036	23	1	clearly	clearly	ADV
ejpam-3036	23	2	l1	l1	PROPN
ejpam-3036	23	3	is	be	AUX
ejpam-3036	23	4	a	a	DET
ejpam-3036	23	5	regular	regular	ADJ
ejpam-3036	23	6	double	double	ADJ
ejpam-3036	23	7	stone	stone	NOUN
ejpam-3036	23	8	algebra	algebra	NOUN
ejpam-3036	23	9	where	where	SCONJ
ejpam-3036	23	10	as	as	ADP
ejpam-3036	23	11	in	in	ADP
ejpam-3036	23	12	l2	l2	NOUN
ejpam-3036	23	13	,	,	PUNCT
ejpam-3036	23	14	a	a	DET
ejpam-3036	23	15	∗	∗	NOUN
ejpam-3036	23	16	=	=	SYM
ejpam-3036	23	17	b∗	b∗	ADJ
ejpam-3036	23	18	=	=	SYM
ejpam-3036	23	19	c∗	c∗	NOUN
ejpam-3036	23	20	=	=	SYM
ejpam-3036	23	21	d∗	d∗	PROPN
ejpam-3036	23	22	=	=	SYM
ejpam-3036	23	23	1∗	1∗	NUM
ejpam-3036	23	24	=	=	SYM
ejpam-3036	23	25	0	0	NUM
ejpam-3036	23	26	,	,	PUNCT
ejpam-3036	23	27	0∗	0∗	X
ejpam-3036	24	1	=	=	SYM
ejpam-3036	24	2	1	1	NUM
ejpam-3036	24	3	and	and	CCONJ
ejpam-3036	24	4	a+	a+	PUNCT
ejpam-3036	24	5	=	=	NOUN
ejpam-3036	24	6	b+	b+	X
ejpam-3036	24	7	=	=	X
ejpam-3036	24	8	c+	c+	X
ejpam-3036	24	9	=	=	PUNCT
ejpam-3036	24	10	d+	d+	PUNCT
ejpam-3036	24	11	=	=	SYM
ejpam-3036	24	12	0	0	NUM
ejpam-3036	24	13	+	+	NUM
ejpam-3036	24	14	=	=	SYM
ejpam-3036	24	15	1	1	NUM
ejpam-3036	24	16	,	,	PUNCT
ejpam-3036	24	17	1	1	NUM
ejpam-3036	24	18	+	+	NUM
ejpam-3036	24	19	=	=	SYM
ejpam-3036	24	20	0	0	X
ejpam-3036	24	21	.	.	PUNCT
ejpam-3036	25	1	here	here	ADV
ejpam-3036	25	2	a∗=b∗	a∗=b∗	PROPN
ejpam-3036	25	3	and	and	CCONJ
ejpam-3036	25	4	a+=b+but	a+=b+but	PROPN
ejpam-3036	25	5	a	a	DET
ejpam-3036	25	6	6=	6=	NOUN
ejpam-3036	25	7	b	b	X
ejpam-3036	25	8	therefore	therefore	ADV
ejpam-3036	25	9	l2	l2	NOUN
ejpam-3036	25	10	is	be	AUX
ejpam-3036	25	11	not	not	PART
ejpam-3036	25	12	a	a	DET
ejpam-3036	25	13	regular	regular	ADJ
ejpam-3036	25	14	double	double	ADJ
ejpam-3036	25	15	stone	stone	NOUN
ejpam-3036	25	16	algebra	algebra	NOUN
ejpam-3036	25	17	.	.	PUNCT
ejpam-3036	26	1	definition	definition	NOUN
ejpam-3036	26	2	2	2	NUM
ejpam-3036	26	3	.	.	PUNCT
ejpam-3036	26	4	let	let	VERB
ejpam-3036	26	5	a	a	PRON
ejpam-3036	26	6	be	be	AUX
ejpam-3036	26	7	a	a	DET
ejpam-3036	26	8	regular	regular	ADJ
ejpam-3036	26	9	double	double	ADJ
ejpam-3036	26	10	stone	stone	NOUN
ejpam-3036	26	11	algebra	algebra	NOUN
ejpam-3036	26	12	.	.	PUNCT
ejpam-3036	27	1	an	an	DET
ejpam-3036	27	2	element	element	NOUN
ejpam-3036	27	3	a	a	PRON
ejpam-3036	27	4	of	of	ADP
ejpam-3036	27	5	a	a	PRON
ejpam-3036	27	6	is	be	AUX
ejpam-3036	27	7	called	call	VERB
ejpam-3036	27	8	a	a	DET
ejpam-3036	27	9	central	central	ADJ
ejpam-3036	27	10	element	element	NOUN
ejpam-3036	27	11	of	of	ADP
ejpam-3036	27	12	a	a	DET
ejpam-3036	27	13	if	if	SCONJ
ejpam-3036	27	14	a∗	a∗	NOUN
ejpam-3036	27	15	=	=	SYM
ejpam-3036	27	16	a+	a+	X
ejpam-3036	27	17	.	.	PUNCT
ejpam-3036	28	1	the	the	DET
ejpam-3036	28	2	set	set	NOUN
ejpam-3036	28	3	of	of	ADP
ejpam-3036	28	4	all	all	DET
ejpam-3036	28	5	central	central	ADJ
ejpam-3036	28	6	elements	element	NOUN
ejpam-3036	28	7	of	of	ADP
ejpam-3036	28	8	a	a	PRON
ejpam-3036	28	9	is	be	AUX
ejpam-3036	28	10	called	call	VERB
ejpam-3036	28	11	the	the	DET
ejpam-3036	28	12	centre	centre	NOUN
ejpam-3036	28	13	of	of	ADP
ejpam-3036	28	14	a	a	PRON
ejpam-3036	28	15	and	and	CCONJ
ejpam-3036	28	16	is	be	AUX
ejpam-3036	28	17	denoted	denote	VERB
ejpam-3036	28	18	by	by	ADP
ejpam-3036	28	19	c(a	c(a	NOUN
ejpam-3036	28	20	)	)	PUNCT
ejpam-3036	28	21	;	;	PUNCT
ejpam-3036	28	22	that	that	PRON
ejpam-3036	28	23	is	be	AUX
ejpam-3036	28	24	,	,	PUNCT
ejpam-3036	28	25	c(a	c(a	ADV
ejpam-3036	28	26	)	)	PUNCT
ejpam-3036	28	27	=	=	SYM
ejpam-3036	28	28	{	{	PUNCT
ejpam-3036	28	29	a	a	DET
ejpam-3036	28	30	∈	∈	NOUN
ejpam-3036	28	31	a|a∗	a|a∗	NOUN
ejpam-3036	28	32	=	=	PUNCT
ejpam-3036	28	33	a+	a+	PUNCT
ejpam-3036	28	34	}	}	PUNCT
ejpam-3036	28	35	note	note	NOUN
ejpam-3036	28	36	that	that	SCONJ
ejpam-3036	28	37	c(a	c(a	ADV
ejpam-3036	28	38	)	)	PUNCT
ejpam-3036	28	39	can	can	AUX
ejpam-3036	28	40	be	be	AUX
ejpam-3036	28	41	described	describe	VERB
ejpam-3036	28	42	in	in	ADP
ejpam-3036	28	43	various	various	ADJ
ejpam-3036	28	44	ways	way	NOUN
ejpam-3036	28	45	as	as	SCONJ
ejpam-3036	28	46	follows	follow	VERB
ejpam-3036	28	47	;	;	PUNCT
ejpam-3036	28	48	c(a	c(a	ADV
ejpam-3036	28	49	)	)	PUNCT
ejpam-3036	28	50	=	=	PRON
ejpam-3036	28	51	{	{	PUNCT
ejpam-3036	28	52	a	a	DET
ejpam-3036	28	53	∈	∈	NOUN
ejpam-3036	28	54	a|a	a|a	PUNCT
ejpam-3036	29	1	=	=	PUNCT
ejpam-3036	29	2	a∗∗	a∗∗	X
ejpam-3036	29	3	}	}	PUNCT
ejpam-3036	29	4	=	=	PUNCT
ejpam-3036	29	5	{	{	PUNCT
ejpam-3036	29	6	a∗|a	a∗|a	PROPN
ejpam-3036	29	7	∈	∈	PROPN
ejpam-3036	29	8	a	a	DET
ejpam-3036	29	9	}	}	PUNCT
ejpam-3036	29	10	=	=	SYM
ejpam-3036	29	11	{	{	PUNCT
ejpam-3036	29	12	a	a	DET
ejpam-3036	29	13	∈	∈	NOUN
ejpam-3036	29	14	a|a	a|a	PUNCT
ejpam-3036	29	15	=	=	PUNCT
ejpam-3036	29	16	a++	a++	NOUN
ejpam-3036	29	17	}	}	PUNCT
ejpam-3036	29	18	a	a	DET
ejpam-3036	29	19	r	r	NOUN
ejpam-3036	29	20	j	j	PROPN
ejpam-3036	29	21	srikanth	srikanth	NOUN
ejpam-3036	29	22	,	,	PUNCT
ejpam-3036	29	23	r	r	NOUN
ejpam-3036	29	24	v	v	NUM
ejpam-3036	29	25	g	g	NOUN
ejpam-3036	29	26	ravi	ravi	PROPN
ejpam-3036	29	27	kumar	kumar	PROPN
ejpam-3036	29	28	/	/	SYM
ejpam-3036	29	29	eur	eur	PROPN
ejpam-3036	29	30	.	.	PUNCT
ejpam-3036	30	1	j.	j.	PROPN
ejpam-3036	30	2	pure	pure	PROPN
ejpam-3036	30	3	appl	appl	PROPN
ejpam-3036	30	4	.	.	PROPN
ejpam-3036	30	5	math	math	PROPN
ejpam-3036	30	6	,	,	PUNCT
ejpam-3036	30	7	10	10	NUM
ejpam-3036	30	8	(	(	PUNCT
ejpam-3036	30	9	4	4	NUM
ejpam-3036	30	10	)	)	PUNCT
ejpam-3036	30	11	(	(	PUNCT
ejpam-3036	30	12	2017	2017	NUM
ejpam-3036	30	13	)	)	PUNCT
ejpam-3036	30	14	,	,	PUNCT
ejpam-3036	30	15	717	717	NUM
ejpam-3036	30	16	-	-	SYM
ejpam-3036	30	17	729	729	NUM
ejpam-3036	30	18	719	719	NUM
ejpam-3036	30	19	=	=	SYM
ejpam-3036	30	20	{	{	PUNCT
ejpam-3036	30	21	a+|a	a+|a	X
ejpam-3036	30	22	∈	∈	PROPN
ejpam-3036	30	23	a	a	PRON
ejpam-3036	30	24	}	}	PUNCT
ejpam-3036	30	25	=	=	SYM
ejpam-3036	30	26	{	{	PUNCT
ejpam-3036	30	27	a	a	DET
ejpam-3036	30	28	∈	∈	PROPN
ejpam-3036	30	29	a|a	a|a	NOUN
ejpam-3036	30	30	∨	∨	NUM
ejpam-3036	30	31	a∗	a∗	NOUN
ejpam-3036	30	32	=	=	SYM
ejpam-3036	30	33	1	1	X
ejpam-3036	30	34	}	}	PUNCT
ejpam-3036	30	35	=	=	PRON
ejpam-3036	30	36	{	{	PUNCT
ejpam-3036	30	37	a	a	DET
ejpam-3036	30	38	∈	∈	PROPN
ejpam-3036	30	39	a	a	DET
ejpam-3036	30	40	|	|	NOUN
ejpam-3036	30	41	a	a	DET
ejpam-3036	30	42	∧	∧	NOUN
ejpam-3036	30	43	a+	a+	PUNCT
ejpam-3036	30	44	=	=	NOUN
ejpam-3036	30	45	0	0	NUM
ejpam-3036	30	46	}	}	PUNCT
ejpam-3036	30	47	=	=	SYM
ejpam-3036	30	48	{	{	PUNCT
ejpam-3036	30	49	a	a	DET
ejpam-3036	30	50	∈	∈	PROPN
ejpam-3036	30	51	a	a	DET
ejpam-3036	30	52	|	|	NOUN
ejpam-3036	30	53	a	a	DET
ejpam-3036	30	54	∧	∧	PROPN
ejpam-3036	30	55	b	b	NOUN
ejpam-3036	30	56	=	=	SYM
ejpam-3036	30	57	0	0	PROPN
ejpam-3036	30	58	and	and	CCONJ
ejpam-3036	30	59	a	a	DET
ejpam-3036	30	60	∨	∨	NUM
ejpam-3036	30	61	b	b	NOUN
ejpam-3036	30	62	=	=	SYM
ejpam-3036	30	63	1	1	NUM
ejpam-3036	30	64	for	for	ADP
ejpam-3036	30	65	some	some	DET
ejpam-3036	30	66	b	b	PROPN
ejpam-3036	30	67	∈	∈	PROPN
ejpam-3036	30	68	a	a	PRON
ejpam-3036	30	69	}	}	PUNCT
ejpam-3036	30	70	theorem	theorem	NOUN
ejpam-3036	30	71	2.1	2.1	NUM
ejpam-3036	30	72	.	.	PUNCT
ejpam-3036	31	1	let	let	VERB
ejpam-3036	31	2	a	a	DET
ejpam-3036	31	3	be	be	AUX
ejpam-3036	31	4	a	a	DET
ejpam-3036	31	5	regular	regular	ADJ
ejpam-3036	31	6	double	double	ADJ
ejpam-3036	31	7	stone	stone	NOUN
ejpam-3036	31	8	algebra.then	algebra.then	NOUN
ejpam-3036	31	9	c(a	c(a	PROPN
ejpam-3036	31	10	)	)	PUNCT
ejpam-3036	31	11	is	be	AUX
ejpam-3036	31	12	a	a	DET
ejpam-3036	31	13	boolean	boolean	ADJ
ejpam-3036	31	14	sub	sub	NOUN
ejpam-3036	31	15	algebra	algebra	NOUN
ejpam-3036	31	16	of	of	ADP
ejpam-3036	31	17	a	a	PRON
ejpam-3036	31	18	with	with	ADP
ejpam-3036	31	19	respect	respect	NOUN
ejpam-3036	31	20	to	to	ADP
ejpam-3036	31	21	the	the	DET
ejpam-3036	31	22	induced	induced	ADJ
ejpam-3036	31	23	operations	operation	NOUN
ejpam-3036	31	24	∧,∨	∧,∨	ADV
ejpam-3036	31	25	and	and	CCONJ
ejpam-3036	31	26	∗.	∗.	PROPN
ejpam-3036	31	27	definition	definition	NOUN
ejpam-3036	31	28	3	3	NUM
ejpam-3036	31	29	.	.	PUNCT
ejpam-3036	32	1	let	let	VERB
ejpam-3036	32	2	a	a	PRON
ejpam-3036	32	3	be	be	AUX
ejpam-3036	32	4	a	a	DET
ejpam-3036	32	5	regular	regular	ADJ
ejpam-3036	32	6	double	double	ADJ
ejpam-3036	32	7	stone	stone	NOUN
ejpam-3036	32	8	algebra	algebra	NOUN
ejpam-3036	32	9	.	.	PUNCT
ejpam-3036	33	1	the	the	DET
ejpam-3036	33	2	set	set	NOUN
ejpam-3036	33	3	d(a	d(a	PROPN
ejpam-3036	33	4	)	)	PUNCT
ejpam-3036	33	5	:	:	PUNCT
ejpam-3036	33	6	=	=	X
ejpam-3036	33	7	{	{	PUNCT
ejpam-3036	33	8	a	a	DET
ejpam-3036	33	9	∈	∈	PROPN
ejpam-3036	33	10	a	a	DET
ejpam-3036	33	11	|	|	ADV
ejpam-3036	33	12	a∗	a∗	NOUN
ejpam-3036	33	13	=	=	SYM
ejpam-3036	33	14	0	0	NUM
ejpam-3036	33	15	}	}	PUNCT
ejpam-3036	33	16	is	be	AUX
ejpam-3036	33	17	called	call	VERB
ejpam-3036	33	18	the	the	DET
ejpam-3036	33	19	dense	dense	ADJ
ejpam-3036	33	20	set	set	NOUN
ejpam-3036	33	21	of	of	ADP
ejpam-3036	33	22	a	a	PRON
ejpam-3036	33	23	and	and	CCONJ
ejpam-3036	33	24	the	the	DET
ejpam-3036	33	25	elements	element	NOUN
ejpam-3036	33	26	of	of	ADP
ejpam-3036	33	27	d(a	d(a	PROPN
ejpam-3036	33	28	)	)	PUNCT
ejpam-3036	33	29	are	be	AUX
ejpam-3036	33	30	called	call	VERB
ejpam-3036	33	31	dense	dense	ADJ
ejpam-3036	33	32	elements	element	NOUN
ejpam-3036	33	33	of	of	ADP
ejpam-3036	33	34	a.	a.	NOUN
ejpam-3036	33	35	the	the	DET
ejpam-3036	33	36	dual	dual	ADJ
ejpam-3036	33	37	of	of	ADP
ejpam-3036	33	38	d(a	d(a	PROPN
ejpam-3036	33	39	)	)	PUNCT
ejpam-3036	33	40	:	:	PUNCT
ejpam-3036	34	1	=	=	X
ejpam-3036	34	2	{	{	PUNCT
ejpam-3036	34	3	a	a	DET
ejpam-3036	34	4	∈	∈	PROPN
ejpam-3036	34	5	a	a	DET
ejpam-3036	34	6	|	|	NOUN
ejpam-3036	34	7	a+	a+	PUNCT
ejpam-3036	34	8	=	=	NOUN
ejpam-3036	34	9	1	1	X
ejpam-3036	34	10	}	}	PUNCT
ejpam-3036	34	11	is	be	AUX
ejpam-3036	34	12	called	call	VERB
ejpam-3036	34	13	dual	dual	ADV
ejpam-3036	34	14	dense	dense	ADJ
ejpam-3036	34	15	set	set	NOUN
ejpam-3036	34	16	of	of	ADP
ejpam-3036	34	17	a	a	PRON
ejpam-3036	34	18	and	and	CCONJ
ejpam-3036	34	19	denoted	denote	VERB
ejpam-3036	34	20	by	by	ADP
ejpam-3036	34	21	d(a	d(a	PROPN
ejpam-3036	34	22	)	)	PUNCT
ejpam-3036	34	23	.	.	PUNCT
ejpam-3036	35	1	the	the	DET
ejpam-3036	35	2	elements	element	NOUN
ejpam-3036	35	3	of	of	ADP
ejpam-3036	35	4	d(a	d(a	PROPN
ejpam-3036	35	5	)	)	PUNCT
ejpam-3036	35	6	are	be	AUX
ejpam-3036	35	7	called	call	VERB
ejpam-3036	35	8	dual	dual	ADJ
ejpam-3036	35	9	dense	dense	ADJ
ejpam-3036	35	10	elements	element	NOUN
ejpam-3036	35	11	of	of	ADP
ejpam-3036	35	12	a.	a.	NOUN
ejpam-3036	35	13	note	note	NOUN
ejpam-3036	35	14	that	that	SCONJ
ejpam-3036	35	15	d(a	d(a	PROPN
ejpam-3036	35	16	)	)	PUNCT
ejpam-3036	35	17	=	=	PRON
ejpam-3036	35	18	{	{	PUNCT
ejpam-3036	35	19	a	a	DET
ejpam-3036	35	20	∨	∨	NUM
ejpam-3036	35	21	a∗	a∗	NOUN
ejpam-3036	35	22	|	|	ADP
ejpam-3036	35	23	a	a	DET
ejpam-3036	35	24	∈	∈	PROPN
ejpam-3036	36	1	a	a	PRON
ejpam-3036	36	2	}	}	PUNCT
ejpam-3036	36	3	and	and	CCONJ
ejpam-3036	36	4	d(a	d(a	PROPN
ejpam-3036	36	5	)	)	PUNCT
ejpam-3036	36	6	=	=	PRON
ejpam-3036	36	7	{	{	PUNCT
ejpam-3036	36	8	a	a	DET
ejpam-3036	36	9	∧	∧	PROPN
ejpam-3036	36	10	a+	a+	PUNCT
ejpam-3036	36	11	|	|	ADV
ejpam-3036	36	12	a	a	DET
ejpam-3036	36	13	∈	∈	PROPN
ejpam-3036	36	14	a	a	PRON
ejpam-3036	36	15	}	}	PUNCT
ejpam-3036	36	16	.	.	PUNCT
ejpam-3036	37	1	theorem	theorem	NOUN
ejpam-3036	37	2	2.2	2.2	NUM
ejpam-3036	37	3	.	.	PUNCT
ejpam-3036	38	1	let	let	VERB
ejpam-3036	38	2	a	a	DET
ejpam-3036	38	3	be	be	AUX
ejpam-3036	38	4	a	a	DET
ejpam-3036	38	5	regular	regular	ADJ
ejpam-3036	38	6	double	double	ADJ
ejpam-3036	38	7	stone	stone	NOUN
ejpam-3036	38	8	algebra	algebra	NOUN
ejpam-3036	38	9	.	.	PUNCT
ejpam-3036	39	1	then	then	ADV
ejpam-3036	39	2	d(a	d(a	PROPN
ejpam-3036	39	3	)	)	PUNCT
ejpam-3036	39	4	is	be	AUX
ejpam-3036	39	5	a	a	DET
ejpam-3036	39	6	filter	filter	NOUN
ejpam-3036	39	7	of	of	ADP
ejpam-3036	39	8	a	a	PRON
ejpam-3036	39	9	and	and	CCONJ
ejpam-3036	39	10	d(a	d(a	PROPN
ejpam-3036	39	11	)	)	PUNCT
ejpam-3036	39	12	is	be	AUX
ejpam-3036	39	13	an	an	DET
ejpam-3036	39	14	ideal	ideal	NOUN
ejpam-3036	39	15	of	of	ADP
ejpam-3036	39	16	a.	a.	NOUN
ejpam-3036	39	17	definition	definition	NOUN
ejpam-3036	39	18	4	4	NUM
ejpam-3036	39	19	.	.	PUNCT
ejpam-3036	40	1	the	the	DET
ejpam-3036	40	2	core	core	NOUN
ejpam-3036	40	3	of	of	ADP
ejpam-3036	40	4	a	a	DET
ejpam-3036	40	5	double	double	ADJ
ejpam-3036	40	6	stone	stone	NOUN
ejpam-3036	40	7	algebra	algebra	NOUN
ejpam-3036	40	8	a	a	PRON
ejpam-3036	40	9	is	be	AUX
ejpam-3036	40	10	defined	define	VERB
ejpam-3036	40	11	to	to	PART
ejpam-3036	40	12	be	be	AUX
ejpam-3036	40	13	k(a	k(a	NOUN
ejpam-3036	40	14	)	)	PUNCT
ejpam-3036	41	1	=	=	SYM
ejpam-3036	41	2	d(a)∩d(a	d(a)∩d(a	NOUN
ejpam-3036	41	3	)	)	PUNCT
ejpam-3036	41	4	k(a	k(a	PROPN
ejpam-3036	41	5	)	)	PUNCT
ejpam-3036	41	6	is	be	AUX
ejpam-3036	41	7	non	non	X
ejpam-3036	41	8	empty	empty	ADJ
ejpam-3036	41	9	if	if	SCONJ
ejpam-3036	41	10	and	and	CCONJ
ejpam-3036	41	11	only	only	ADV
ejpam-3036	41	12	if	if	SCONJ
ejpam-3036	41	13	a	a	PRON
ejpam-3036	41	14	does	do	AUX
ejpam-3036	41	15	not	not	PART
ejpam-3036	41	16	have	have	VERB
ejpam-3036	41	17	2	2	NUM
ejpam-3036	41	18	=	=	NOUN
ejpam-3036	41	19	{	{	PUNCT
ejpam-3036	41	20	0	0	NUM
ejpam-3036	41	21	,	,	PUNCT
ejpam-3036	41	22	1	1	NUM
ejpam-3036	41	23	}	}	PUNCT
ejpam-3036	41	24	as	as	ADP
ejpam-3036	41	25	a	a	DET
ejpam-3036	41	26	factor	factor	NOUN
ejpam-3036	41	27	.	.	PUNCT
ejpam-3036	42	1	when	when	SCONJ
ejpam-3036	42	2	it	it	PRON
ejpam-3036	42	3	is	be	AUX
ejpam-3036	42	4	non	non	X
ejpam-3036	42	5	empty	empty	ADJ
ejpam-3036	42	6	the	the	DET
ejpam-3036	42	7	behavior	behavior	NOUN
ejpam-3036	42	8	of	of	ADP
ejpam-3036	42	9	k(a	k(a	NOUN
ejpam-3036	42	10	)	)	PUNCT
ejpam-3036	42	11	in	in	ADP
ejpam-3036	42	12	certain	certain	ADJ
ejpam-3036	42	13	respects	respect	NOUN
ejpam-3036	42	14	governs	govern	VERB
ejpam-3036	42	15	the	the	DET
ejpam-3036	42	16	behaviour	behaviour	NOUN
ejpam-3036	42	17	of	of	ADP
ejpam-3036	42	18	a.	a.	NOUN
ejpam-3036	42	19	it	it	PRON
ejpam-3036	42	20	is	be	AUX
ejpam-3036	42	21	easy	easy	ADJ
ejpam-3036	42	22	to	to	PART
ejpam-3036	42	23	prove	prove	VERB
ejpam-3036	42	24	that	that	SCONJ
ejpam-3036	42	25	in	in	ADP
ejpam-3036	42	26	any	any	DET
ejpam-3036	42	27	rdsa	rdsa	NOUN
ejpam-3036	42	28	there	there	PRON
ejpam-3036	42	29	exists	exist	VERB
ejpam-3036	42	30	at	at	ADP
ejpam-3036	42	31	most	most	ADV
ejpam-3036	42	32	one	one	NUM
ejpam-3036	42	33	core	core	NOUN
ejpam-3036	42	34	element	element	NOUN
ejpam-3036	42	35	.	.	PUNCT
ejpam-3036	43	1	we	we	PRON
ejpam-3036	43	2	call	call	VERB
ejpam-3036	43	3	a	a	DET
ejpam-3036	43	4	regular	regular	ADJ
ejpam-3036	43	5	double	double	ADJ
ejpam-3036	43	6	stone	stone	NOUN
ejpam-3036	43	7	algebra	algebra	NOUN
ejpam-3036	43	8	with	with	ADP
ejpam-3036	43	9	non	non	ADJ
ejpam-3036	43	10	empty	empty	ADJ
ejpam-3036	43	11	core	core	NOUN
ejpam-3036	43	12	as	as	ADP
ejpam-3036	43	13	core	core	NOUN
ejpam-3036	43	14	regular	regular	ADJ
ejpam-3036	43	15	double	double	ADJ
ejpam-3036	43	16	stone	stone	NOUN
ejpam-3036	43	17	algebra(crdsa	algebra(crdsa	PROPN
ejpam-3036	43	18	)	)	PUNCT
ejpam-3036	43	19	.	.	PUNCT
ejpam-3036	44	1	note	note	NOUN
ejpam-3036	44	2	:	:	PUNCT
ejpam-3036	44	3	in	in	ADP
ejpam-3036	44	4	any	any	DET
ejpam-3036	44	5	crdsa	crdsa	NOUN
ejpam-3036	44	6	a	a	PRON
ejpam-3036	44	7	,	,	PUNCT
ejpam-3036	44	8	|k(a)|	|k(a)|	PROPN
ejpam-3036	44	9	=	=	SYM
ejpam-3036	44	10	1	1	X
ejpam-3036	44	11	.	.	NOUN
ejpam-3036	44	12	example	example	NOUN
ejpam-3036	45	1	2	2	NUM
ejpam-3036	45	2	.	.	PUNCT
ejpam-3036	46	1	every	every	DET
ejpam-3036	46	2	three	three	NUM
ejpam-3036	46	3	element	element	NOUN
ejpam-3036	46	4	chain	chain	NOUN
ejpam-3036	46	5	is	be	AUX
ejpam-3036	46	6	crdsa	crdsa	ADJ
ejpam-3036	46	7	.	.	PUNCT
ejpam-3036	47	1	we	we	PRON
ejpam-3036	47	2	call	call	VERB
ejpam-3036	47	3	it	it	PRON
ejpam-3036	47	4	as	as	ADP
ejpam-3036	47	5	a	a	DET
ejpam-3036	47	6	discrete	discrete	NOUN
ejpam-3036	47	7	crdsa	crdsa	INTJ
ejpam-3036	47	8	example	example	NOUN
ejpam-3036	47	9	3	3	X
ejpam-3036	47	10	.	.	X
ejpam-3036	48	1	consider	consider	VERB
ejpam-3036	48	2	the	the	DET
ejpam-3036	48	3	hasse	hasse	NOUN
ejpam-3036	48	4	diagrams	diagram	NOUN
ejpam-3036	48	5	of	of	ADP
ejpam-3036	48	6	rdsas	rdsas	PROPN
ejpam-3036	48	7	l1	l1	PROPN
ejpam-3036	48	8	=	=	SYM
ejpam-3036	48	9	(	(	PUNCT
ejpam-3036	48	10	l1,∧,∨	l1,∧,∨	PROPN
ejpam-3036	48	11	,	,	PUNCT
ejpam-3036	48	12	∗,+	∗,+	NOUN
ejpam-3036	48	13	,	,	PUNCT
ejpam-3036	48	14	0	0	NUM
ejpam-3036	48	15	,	,	PUNCT
ejpam-3036	48	16	1	1	NUM
ejpam-3036	48	17	)	)	PUNCT
ejpam-3036	48	18	and	and	CCONJ
ejpam-3036	48	19	l3	l3	PROPN
ejpam-3036	48	20	=	=	SYM
ejpam-3036	48	21	(	(	PUNCT
ejpam-3036	48	22	l3,∧,∨	l3,∧,∨	PROPN
ejpam-3036	48	23	,	,	PUNCT
ejpam-3036	48	24	∗,+	∗,+	NOUN
ejpam-3036	48	25	,	,	PUNCT
ejpam-3036	48	26	0	0	NUM
ejpam-3036	48	27	,	,	PUNCT
ejpam-3036	48	28	1	1	NUM
ejpam-3036	48	29	)	)	PUNCT
ejpam-3036	48	30	.	.	PUNCT
ejpam-3036	49	1	1	1	NUM
ejpam-3036	49	2	c	c	X
ejpam-3036	49	3	d	d	NOUN
ejpam-3036	49	4	a	a	DET
ejpam-3036	49	5	b	b	PROPN
ejpam-3036	49	6	0	0	NUM
ejpam-3036	49	7	figure	figure	NOUN
ejpam-3036	49	8	3	3	NUM
ejpam-3036	49	9	:	:	PUNCT
ejpam-3036	49	10	l1	l1	PROPN
ejpam-3036	49	11	1	1	NUM
ejpam-3036	49	12	e	e	NOUN
ejpam-3036	49	13	f	f	PROPN
ejpam-3036	49	14	c	c	PROPN
ejpam-3036	49	15	k	k	PROPN
ejpam-3036	50	1	d	d	X
ejpam-3036	50	2	a	a	DET
ejpam-3036	50	3	b	b	PROPN
ejpam-3036	50	4	0	0	NUM
ejpam-3036	50	5	figure	figure	NOUN
ejpam-3036	50	6	4	4	NUM
ejpam-3036	50	7	:	:	PUNCT
ejpam-3036	50	8	l3	l3	NOUN
ejpam-3036	50	9	a	a	DET
ejpam-3036	50	10	r	r	NOUN
ejpam-3036	50	11	j	j	PROPN
ejpam-3036	50	12	srikanth	srikanth	NOUN
ejpam-3036	50	13	,	,	PUNCT
ejpam-3036	50	14	r	r	NOUN
ejpam-3036	50	15	v	v	NUM
ejpam-3036	50	16	g	g	NOUN
ejpam-3036	50	17	ravi	ravi	PROPN
ejpam-3036	50	18	kumar	kumar	PROPN
ejpam-3036	50	19	/	/	SYM
ejpam-3036	50	20	eur	eur	PROPN
ejpam-3036	50	21	.	.	PUNCT
ejpam-3036	51	1	j.	j.	PROPN
ejpam-3036	51	2	pure	pure	PROPN
ejpam-3036	51	3	appl	appl	PROPN
ejpam-3036	51	4	.	.	PROPN
ejpam-3036	51	5	math	math	PROPN
ejpam-3036	51	6	,	,	PUNCT
ejpam-3036	51	7	10	10	NUM
ejpam-3036	51	8	(	(	PUNCT
ejpam-3036	51	9	4	4	NUM
ejpam-3036	51	10	)	)	PUNCT
ejpam-3036	51	11	(	(	PUNCT
ejpam-3036	51	12	2017	2017	NUM
ejpam-3036	51	13	)	)	PUNCT
ejpam-3036	51	14	,	,	PUNCT
ejpam-3036	51	15	717	717	NUM
ejpam-3036	51	16	-	-	SYM
ejpam-3036	51	17	729	729	NUM
ejpam-3036	51	18	720	720	NUM
ejpam-3036	51	19	clearly	clearly	ADV
ejpam-3036	51	20	l1	l1	PROPN
ejpam-3036	51	21	,	,	PUNCT
ejpam-3036	51	22	l3	l3	PROPN
ejpam-3036	51	23	are	be	AUX
ejpam-3036	51	24	rdsas	rdsa	NOUN
ejpam-3036	51	25	,	,	PUNCT
ejpam-3036	51	26	and	and	CCONJ
ejpam-3036	51	27	it	it	PRON
ejpam-3036	51	28	is	be	AUX
ejpam-3036	51	29	seen	see	VERB
ejpam-3036	51	30	that	that	SCONJ
ejpam-3036	51	31	core	core	NOUN
ejpam-3036	51	32	of	of	ADP
ejpam-3036	51	33	l1	l1	PROPN
ejpam-3036	51	34	is	be	AUX
ejpam-3036	51	35	empty	empty	ADJ
ejpam-3036	51	36	where	where	SCONJ
ejpam-3036	51	37	as	as	SCONJ
ejpam-3036	51	38	l3	l3	PROPN
ejpam-3036	51	39	has	have	VERB
ejpam-3036	51	40	the	the	DET
ejpam-3036	51	41	core	core	NOUN
ejpam-3036	51	42	element	element	NOUN
ejpam-3036	52	1	k	k	PROPN
ejpam-3036	52	2	hence	hence	ADV
ejpam-3036	52	3	it	it	PRON
ejpam-3036	52	4	a	a	DET
ejpam-3036	52	5	crdsa	crdsa	NOUN
ejpam-3036	52	6	theorem	theorem	NOUN
ejpam-3036	52	7	2.3	2.3	NUM
ejpam-3036	52	8	.	.	PUNCT
ejpam-3036	53	1	if	if	SCONJ
ejpam-3036	53	2	a	a	PRON
ejpam-3036	53	3	is	be	AUX
ejpam-3036	53	4	crdsa	crdsa	ADJ
ejpam-3036	53	5	with	with	ADP
ejpam-3036	53	6	core	core	NOUN
ejpam-3036	53	7	element	element	NOUN
ejpam-3036	53	8	k	k	PROPN
ejpam-3036	53	9	,	,	PUNCT
ejpam-3036	53	10	then	then	ADV
ejpam-3036	53	11	every	every	DET
ejpam-3036	53	12	element	element	NOUN
ejpam-3036	53	13	x	x	X
ejpam-3036	53	14	of	of	ADP
ejpam-3036	53	15	a	a	PRON
ejpam-3036	53	16	can	can	AUX
ejpam-3036	53	17	be	be	AUX
ejpam-3036	53	18	written	write	VERB
ejpam-3036	53	19	as	as	ADP
ejpam-3036	53	20	x	x	X
ejpam-3036	53	21	=	=	SYM
ejpam-3036	53	22	x∗∗	x∗∗	X
ejpam-3036	53	23	∧	∧	PROPN
ejpam-3036	53	24	(	(	PUNCT
ejpam-3036	53	25	x++	x++	PROPN
ejpam-3036	53	26	∨	∨	PROPN
ejpam-3036	53	27	k	k	NOUN
ejpam-3036	53	28	)	)	PUNCT
ejpam-3036	53	29	and	and	CCONJ
ejpam-3036	53	30	x	x	X
ejpam-3036	53	31	=	=	SYM
ejpam-3036	53	32	x++	x++	X
ejpam-3036	53	33	∨	∨	X
ejpam-3036	53	34	(	(	PUNCT
ejpam-3036	53	35	x∗∗	x∗∗	PROPN
ejpam-3036	53	36	∧	∧	PROPN
ejpam-3036	53	37	k	k	NOUN
ejpam-3036	53	38	)	)	PUNCT
ejpam-3036	53	39	proof	proof	NOUN
ejpam-3036	53	40	.	.	PUNCT
ejpam-3036	54	1	let	let	VERB
ejpam-3036	54	2	y	y	NOUN
ejpam-3036	54	3	=	=	PUNCT
ejpam-3036	54	4	x∗∗	x∗∗	PROPN
ejpam-3036	54	5	∧	∧	PROPN
ejpam-3036	54	6	(	(	PUNCT
ejpam-3036	54	7	x++	x++	PROPN
ejpam-3036	54	8	∨	∨	NUM
ejpam-3036	54	9	k	k	NOUN
ejpam-3036	54	10	)	)	PUNCT
ejpam-3036	54	11	.	.	PUNCT
ejpam-3036	55	1	then	then	ADV
ejpam-3036	55	2	y∗∗	y∗∗	ADV
ejpam-3036	55	3	=	=	PUNCT
ejpam-3036	55	4	(	(	PUNCT
ejpam-3036	55	5	x∗∗	x∗∗	PROPN
ejpam-3036	55	6	∧	∧	PROPN
ejpam-3036	55	7	(	(	PUNCT
ejpam-3036	55	8	x++	x++	X
ejpam-3036	55	9	∨	∨	NUM
ejpam-3036	55	10	k))∗∗	k))∗∗	X
ejpam-3036	55	11	=	=	PUNCT
ejpam-3036	55	12	x∗∗	x∗∗	PROPN
ejpam-3036	55	13	and	and	CCONJ
ejpam-3036	55	14	y++	y++	NOUN
ejpam-3036	55	15	=	=	SYM
ejpam-3036	55	16	(	(	PUNCT
ejpam-3036	55	17	x++	x++	X
ejpam-3036	55	18	∧	∧	PROPN
ejpam-3036	55	19	(	(	PUNCT
ejpam-3036	55	20	x++	x++	PROPN
ejpam-3036	55	21	∨	∨	PROPN
ejpam-3036	55	22	k))++	k))++	PROPN
ejpam-3036	55	23	=	=	PUNCT
ejpam-3036	55	24	x++	x++	PROPN
ejpam-3036	55	25	.	.	PUNCT
ejpam-3036	56	1	thus	thus	ADV
ejpam-3036	56	2	by	by	ADP
ejpam-3036	56	3	regularity	regularity	NOUN
ejpam-3036	56	4	x	x	X
ejpam-3036	56	5	=	=	SYM
ejpam-3036	56	6	y.	y.	NOUN
ejpam-3036	56	7	other	other	ADJ
ejpam-3036	56	8	one	one	NUM
ejpam-3036	56	9	follows	follow	VERB
ejpam-3036	56	10	from	from	ADP
ejpam-3036	56	11	duality	duality	NOUN
ejpam-3036	56	12	.	.	PUNCT
ejpam-3036	57	1	definition	definition	NOUN
ejpam-3036	57	2	5	5	NUM
ejpam-3036	57	3	.	.	PUNCT
ejpam-3036	57	4	suppose	suppose	VERB
ejpam-3036	57	5	that	that	SCONJ
ejpam-3036	57	6	a	a	PRON
ejpam-3036	57	7	and	and	CCONJ
ejpam-3036	57	8	b	b	NOUN
ejpam-3036	57	9	are	be	AUX
ejpam-3036	57	10	two	two	NUM
ejpam-3036	57	11	crdsas	crdsa	NOUN
ejpam-3036	57	12	with	with	ADP
ejpam-3036	57	13	core	core	NOUN
ejpam-3036	57	14	elements	element	NOUN
ejpam-3036	57	15	k1	k1	NOUN
ejpam-3036	57	16	,	,	PUNCT
ejpam-3036	57	17	k2	k2	NOUN
ejpam-3036	57	18	respectively	respectively	ADV
ejpam-3036	57	19	.	.	PUNCT
ejpam-3036	58	1	a	a	DET
ejpam-3036	58	2	mapping	mapping	NOUN
ejpam-3036	58	3	f	f	NOUN
ejpam-3036	58	4	:	:	PUNCT
ejpam-3036	58	5	a	a	DET
ejpam-3036	58	6	−→	−→	NOUN
ejpam-3036	58	7	b	b	PROPN
ejpam-3036	58	8	is	be	AUX
ejpam-3036	58	9	called	call	VERB
ejpam-3036	58	10	a	a	DET
ejpam-3036	58	11	homomorphism	homomorphism	NOUN
ejpam-3036	58	12	from	from	ADP
ejpam-3036	58	13	a	a	PRON
ejpam-3036	58	14	to	to	PART
ejpam-3036	58	15	b	b	NOUN
ejpam-3036	58	16	if	if	SCONJ
ejpam-3036	58	17	(	(	PUNCT
ejpam-3036	58	18	i	i	NOUN
ejpam-3036	58	19	)	)	PUNCT
ejpam-3036	58	20	f	f	PROPN
ejpam-3036	58	21	is	be	AUX
ejpam-3036	58	22	lattice	lattice	ADJ
ejpam-3036	58	23	homomorphism	homomorphism	NOUN
ejpam-3036	58	24	from	from	ADP
ejpam-3036	58	25	a	a	DET
ejpam-3036	58	26	to	to	PART
ejpam-3036	58	27	b	b	PROPN
ejpam-3036	58	28	(	(	PUNCT
ejpam-3036	58	29	ii	ii	NOUN
ejpam-3036	58	30	)	)	PUNCT
ejpam-3036	58	31	for	for	ADP
ejpam-3036	58	32	a	a	DET
ejpam-3036	58	33	∈	∈	PROPN
ejpam-3036	58	34	a	a	PRON
ejpam-3036	58	35	,	,	PUNCT
ejpam-3036	58	36	f(a∗	f(a∗	PRON
ejpam-3036	58	37	)	)	PUNCT
ejpam-3036	58	38	=	=	SYM
ejpam-3036	58	39	f(a)∗	f(a)∗	PROPN
ejpam-3036	58	40	and	and	CCONJ
ejpam-3036	58	41	f(a+	f(a+	NOUN
ejpam-3036	58	42	)	)	PUNCT
ejpam-3036	59	1	=	=	SYM
ejpam-3036	59	2	f(a)+	f(a)+	NOUN
ejpam-3036	59	3	(	(	PUNCT
ejpam-3036	59	4	iii	iii	NOUN
ejpam-3036	59	5	)	)	PUNCT
ejpam-3036	59	6	f(k1	f(k1	NOUN
ejpam-3036	59	7	)	)	PUNCT
ejpam-3036	59	8	=	=	SYM
ejpam-3036	59	9	k2	k2	X
ejpam-3036	59	10	a	a	DET
ejpam-3036	59	11	necessary	necessary	ADJ
ejpam-3036	59	12	and	and	CCONJ
ejpam-3036	59	13	sufficient	sufficient	ADJ
ejpam-3036	59	14	condition	condition	NOUN
ejpam-3036	59	15	for	for	ADP
ejpam-3036	59	16	two	two	NUM
ejpam-3036	59	17	crdsas	crdsa	NOUN
ejpam-3036	59	18	is	be	AUX
ejpam-3036	59	19	isomorphic	isomorphic	ADJ
ejpam-3036	59	20	is	be	AUX
ejpam-3036	59	21	discussed	discuss	VERB
ejpam-3036	59	22	in	in	ADP
ejpam-3036	59	23	the	the	DET
ejpam-3036	59	24	following	follow	VERB
ejpam-3036	59	25	theorem	theorem	NOUN
ejpam-3036	59	26	.	.	PUNCT
ejpam-3036	60	1	theorem	theorem	VERB
ejpam-3036	60	2	2.4	2.4	NUM
ejpam-3036	60	3	.	.	PUNCT
ejpam-3036	61	1	two	two	NUM
ejpam-3036	61	2	crdsas	crdsa	NOUN
ejpam-3036	61	3	are	be	AUX
ejpam-3036	61	4	isomorphic	isomorphic	ADJ
ejpam-3036	61	5	if	if	SCONJ
ejpam-3036	61	6	and	and	CCONJ
ejpam-3036	61	7	only	only	ADV
ejpam-3036	61	8	if	if	SCONJ
ejpam-3036	61	9	their	their	PRON
ejpam-3036	61	10	centers	center	NOUN
ejpam-3036	61	11	are	be	AUX
ejpam-3036	61	12	isomorphic	isomorphic	ADJ
ejpam-3036	61	13	proof	proof	NOUN
ejpam-3036	61	14	.	.	PUNCT
ejpam-3036	62	1	let	let	VERB
ejpam-3036	62	2	a1	a1	NOUN
ejpam-3036	62	3	,	,	PUNCT
ejpam-3036	62	4	a2	a2	PROPN
ejpam-3036	62	5	be	be	VERB
ejpam-3036	62	6	crdsas	crdsa	NOUN
ejpam-3036	62	7	with	with	ADP
ejpam-3036	62	8	core	core	NOUN
ejpam-3036	62	9	elements	element	NOUN
ejpam-3036	62	10	k1	k1	NOUN
ejpam-3036	62	11	,	,	PUNCT
ejpam-3036	62	12	k2	k2	PROPN
ejpam-3036	62	13	respectively	respectively	ADV
ejpam-3036	62	14	.	.	PUNCT
ejpam-3036	63	1	first	first	ADV
ejpam-3036	63	2	suppose	suppose	VERB
ejpam-3036	63	3	that	that	SCONJ
ejpam-3036	63	4	f	f	PROPN
ejpam-3036	63	5	:	:	PUNCT
ejpam-3036	63	6	c(a1	c(a1	X
ejpam-3036	63	7	)	)	PUNCT
ejpam-3036	63	8	−→	−→	NOUN
ejpam-3036	63	9	c(a2	c(a2	NOUN
ejpam-3036	63	10	)	)	PUNCT
ejpam-3036	63	11	is	be	AUX
ejpam-3036	63	12	an	an	DET
ejpam-3036	63	13	isomorphism	isomorphism	NOUN
ejpam-3036	63	14	.	.	PUNCT
ejpam-3036	64	1	define	define	VERB
ejpam-3036	64	2	the	the	DET
ejpam-3036	64	3	map	map	NOUN
ejpam-3036	64	4	φ	φ	PROPN
ejpam-3036	64	5	on	on	ADP
ejpam-3036	64	6	a1	a1	NOUN
ejpam-3036	64	7	to	to	ADP
ejpam-3036	64	8	a2	a2	PROPN
ejpam-3036	64	9	by	by	ADP
ejpam-3036	64	10	φ(x	φ(x	NOUN
ejpam-3036	64	11	)	)	PUNCT
ejpam-3036	64	12	=	=	SYM
ejpam-3036	64	13	f(x∗∗	f(x∗∗	PROPN
ejpam-3036	64	14	)	)	PUNCT
ejpam-3036	64	15	∧	∧	PROPN
ejpam-3036	64	16	(	(	PUNCT
ejpam-3036	64	17	f(x++	f(x++	PROPN
ejpam-3036	64	18	)	)	PUNCT
ejpam-3036	64	19	∨	∨	NOUN
ejpam-3036	64	20	k2	k2	NOUN
ejpam-3036	64	21	)	)	PUNCT
ejpam-3036	64	22	.	.	PUNCT
ejpam-3036	65	1	by	by	ADP
ejpam-3036	65	2	using	use	VERB
ejpam-3036	65	3	distributive	distributive	ADJ
ejpam-3036	65	4	property	property	NOUN
ejpam-3036	65	5	and	and	CCONJ
ejpam-3036	65	6	the	the	DET
ejpam-3036	65	7	fact	fact	NOUN
ejpam-3036	65	8	that	that	SCONJ
ejpam-3036	65	9	f	f	PROPN
ejpam-3036	65	10	is	be	AUX
ejpam-3036	65	11	a	a	DET
ejpam-3036	65	12	homomorphism	homomorphism	NOUN
ejpam-3036	65	13	it	it	PRON
ejpam-3036	65	14	can	can	AUX
ejpam-3036	65	15	be	be	AUX
ejpam-3036	65	16	easily	easily	ADV
ejpam-3036	65	17	verify	verify	VERB
ejpam-3036	65	18	that	that	SCONJ
ejpam-3036	65	19	φ(x	φ(x	NOUN
ejpam-3036	65	20	)	)	PUNCT
ejpam-3036	65	21	=	=	SYM
ejpam-3036	65	22	f(x++	f(x++	ADJ
ejpam-3036	65	23	)	)	PUNCT
ejpam-3036	65	24	∨	∨	PROPN
ejpam-3036	65	25	(	(	PUNCT
ejpam-3036	65	26	f(x∗∗	f(x∗∗	PROPN
ejpam-3036	65	27	)	)	PUNCT
ejpam-3036	65	28	∧	∧	PROPN
ejpam-3036	65	29	k2	k2	NOUN
ejpam-3036	65	30	)	)	PUNCT
ejpam-3036	65	31	.	.	PUNCT
ejpam-3036	66	1	and	and	CCONJ
ejpam-3036	66	2	also	also	ADV
ejpam-3036	66	3	observe	observe	VERB
ejpam-3036	66	4	that	that	SCONJ
ejpam-3036	66	5	,	,	PUNCT
ejpam-3036	66	6	for	for	ADP
ejpam-3036	66	7	x	x	PROPN
ejpam-3036	66	8	∈	∈	PROPN
ejpam-3036	66	9	c(a1	c(a1	NOUN
ejpam-3036	66	10	)	)	PUNCT
ejpam-3036	66	11	,	,	PUNCT
ejpam-3036	66	12	φ(x	φ(x	NOUN
ejpam-3036	66	13	)	)	PUNCT
ejpam-3036	66	14	=	=	SYM
ejpam-3036	67	1	f(x∗∗)∧	f(x∗∗)∧	NOUN
ejpam-3036	67	2	(	(	PUNCT
ejpam-3036	67	3	f(x++	f(x++	PROPN
ejpam-3036	67	4	)	)	PUNCT
ejpam-3036	67	5	∨	∨	NOUN
ejpam-3036	67	6	k2	k2	NOUN
ejpam-3036	67	7	)	)	PUNCT
ejpam-3036	67	8	=	=	NUM
ejpam-3036	67	9	f(x)∨	f(x)∨	X
ejpam-3036	67	10	(	(	PUNCT
ejpam-3036	67	11	f(x	f(x	PROPN
ejpam-3036	67	12	)	)	PUNCT
ejpam-3036	67	13	∧	∧	PROPN
ejpam-3036	67	14	k2	k2	NOUN
ejpam-3036	67	15	)	)	PUNCT
ejpam-3036	67	16	=	=	SYM
ejpam-3036	67	17	f(x	f(x	PROPN
ejpam-3036	67	18	)	)	PUNCT
ejpam-3036	67	19	,	,	PUNCT
ejpam-3036	67	20	i.e.	i.e.	X
ejpam-3036	67	21	φ	φ	PROPN
ejpam-3036	67	22	coincides	coincide	VERB
ejpam-3036	67	23	with	with	ADP
ejpam-3036	67	24	f	f	PROPN
ejpam-3036	67	25	on	on	ADP
ejpam-3036	67	26	c(a1	c(a1	NOUN
ejpam-3036	67	27	)	)	PUNCT
ejpam-3036	67	28	to	to	PART
ejpam-3036	67	29	show	show	VERB
ejpam-3036	67	30	that	that	SCONJ
ejpam-3036	67	31	φ	φ	PROPN
ejpam-3036	67	32	is	be	AUX
ejpam-3036	67	33	one	one	NUM
ejpam-3036	67	34	-	-	PUNCT
ejpam-3036	67	35	one	one	NUM
ejpam-3036	67	36	suppose	suppose	VERB
ejpam-3036	67	37	that	that	SCONJ
ejpam-3036	67	38	φ(x	φ(x	NOUN
ejpam-3036	67	39	)	)	PUNCT
ejpam-3036	67	40	=	=	SYM
ejpam-3036	67	41	φ(y	φ(y	NOUN
ejpam-3036	67	42	)	)	PUNCT
ejpam-3036	67	43	for	for	ADP
ejpam-3036	67	44	x	x	SYM
ejpam-3036	67	45	,	,	PUNCT
ejpam-3036	67	46	y	y	PROPN
ejpam-3036	67	47	in	in	ADP
ejpam-3036	67	48	a1	a1	PROPN
ejpam-3036	67	49	.	.	PUNCT
ejpam-3036	68	1	then	then	ADV
ejpam-3036	68	2	(	(	PUNCT
ejpam-3036	68	3	φ(x))∗	φ(x))∗	X
ejpam-3036	68	4	=	=	SYM
ejpam-3036	68	5	(	(	PUNCT
ejpam-3036	68	6	φ(y))∗	φ(y))∗	NOUN
ejpam-3036	68	7	and	and	CCONJ
ejpam-3036	68	8	(	(	PUNCT
ejpam-3036	68	9	φ(x))+	φ(x))+	NOUN
ejpam-3036	68	10	=	=	SYM
ejpam-3036	68	11	(	(	PUNCT
ejpam-3036	68	12	φ(y))+	φ(y))+	PROPN
ejpam-3036	68	13	,	,	PUNCT
ejpam-3036	68	14	by	by	ADP
ejpam-3036	68	15	using	use	VERB
ejpam-3036	68	16	the	the	DET
ejpam-3036	68	17	definition	definition	NOUN
ejpam-3036	68	18	of	of	ADP
ejpam-3036	68	19	φ	φ	PROPN
ejpam-3036	68	20	and	and	CCONJ
ejpam-3036	68	21	the	the	DET
ejpam-3036	68	22	fact	fact	NOUN
ejpam-3036	68	23	that	that	SCONJ
ejpam-3036	68	24	f	f	PROPN
ejpam-3036	68	25	is	be	AUX
ejpam-3036	68	26	one	one	NUM
ejpam-3036	68	27	-	-	PUNCT
ejpam-3036	68	28	toone	toone	NOUN
ejpam-3036	68	29	,	,	PUNCT
ejpam-3036	68	30	it	it	PRON
ejpam-3036	68	31	gives	give	VERB
ejpam-3036	68	32	x∗	x∗	PROPN
ejpam-3036	68	33	=	=	PUNCT
ejpam-3036	69	1	y∗	y∗	PROPN
ejpam-3036	69	2	and	and	CCONJ
ejpam-3036	69	3	x+	x+	NUM
ejpam-3036	69	4	=	=	NOUN
ejpam-3036	69	5	y+	y+	PROPN
ejpam-3036	69	6	and	and	CCONJ
ejpam-3036	69	7	by	by	ADP
ejpam-3036	69	8	regularity	regularity	NOUN
ejpam-3036	69	9	x	x	X
ejpam-3036	69	10	=	=	SYM
ejpam-3036	69	11	y.	y.	PROPN
ejpam-3036	69	12	hence	hence	PROPN
ejpam-3036	69	13	φ	φ	PROPN
ejpam-3036	69	14	is	be	AUX
ejpam-3036	69	15	one	one	NUM
ejpam-3036	69	16	-	-	PUNCT
ejpam-3036	69	17	one	one	NUM
ejpam-3036	69	18	.	.	PUNCT
ejpam-3036	70	1	to	to	PART
ejpam-3036	70	2	show	show	VERB
ejpam-3036	70	3	that	that	SCONJ
ejpam-3036	70	4	φ	φ	PROPN
ejpam-3036	70	5	is	be	AUX
ejpam-3036	70	6	onto	onto	ADP
ejpam-3036	70	7	,	,	PUNCT
ejpam-3036	70	8	y	y	PROPN
ejpam-3036	70	9	∈	∈	PROPN
ejpam-3036	70	10	a2	a2	PROPN
ejpam-3036	70	11	and	and	CCONJ
ejpam-3036	70	12	consider	consider	VERB
ejpam-3036	70	13	the	the	DET
ejpam-3036	70	14	following	follow	VERB
ejpam-3036	70	15	cases	case	NOUN
ejpam-3036	70	16	a	a	DET
ejpam-3036	70	17	r	r	NOUN
ejpam-3036	70	18	j	j	PROPN
ejpam-3036	70	19	srikanth	srikanth	NOUN
ejpam-3036	70	20	,	,	PUNCT
ejpam-3036	70	21	r	r	NOUN
ejpam-3036	70	22	v	v	NUM
ejpam-3036	70	23	g	g	NOUN
ejpam-3036	70	24	ravi	ravi	PROPN
ejpam-3036	70	25	kumar	kumar	PROPN
ejpam-3036	70	26	/	/	SYM
ejpam-3036	70	27	eur	eur	PROPN
ejpam-3036	70	28	.	.	PUNCT
ejpam-3036	71	1	j.	j.	PROPN
ejpam-3036	71	2	pure	pure	PROPN
ejpam-3036	71	3	appl	appl	PROPN
ejpam-3036	71	4	.	.	PROPN
ejpam-3036	71	5	math	math	PROPN
ejpam-3036	71	6	,	,	PUNCT
ejpam-3036	71	7	10	10	NUM
ejpam-3036	71	8	(	(	PUNCT
ejpam-3036	71	9	4	4	NUM
ejpam-3036	71	10	)	)	PUNCT
ejpam-3036	71	11	(	(	PUNCT
ejpam-3036	71	12	2017	2017	NUM
ejpam-3036	71	13	)	)	PUNCT
ejpam-3036	71	14	,	,	PUNCT
ejpam-3036	71	15	717	717	NUM
ejpam-3036	71	16	-	-	SYM
ejpam-3036	71	17	729	729	NUM
ejpam-3036	71	18	721	721	NUM
ejpam-3036	71	19	case	case	NOUN
ejpam-3036	71	20	(	(	PUNCT
ejpam-3036	71	21	i	i	NOUN
ejpam-3036	71	22	)	)	PUNCT
ejpam-3036	71	23	:	:	PUNCT
ejpam-3036	71	24	y	y	PROPN
ejpam-3036	71	25	∈	∈	PROPN
ejpam-3036	71	26	c(a2	c(a2	NOUN
ejpam-3036	71	27	)	)	PUNCT
ejpam-3036	71	28	.	.	PUNCT
ejpam-3036	72	1	since	since	SCONJ
ejpam-3036	72	2	f	f	PROPN
ejpam-3036	72	3	is	be	AUX
ejpam-3036	72	4	onto	onto	ADP
ejpam-3036	72	5	from	from	ADP
ejpam-3036	72	6	c(a1	c(a1	NOUN
ejpam-3036	72	7	)	)	PUNCT
ejpam-3036	72	8	to	to	PART
ejpam-3036	72	9	c(a2	c(a2	VERB
ejpam-3036	72	10	)	)	PUNCT
ejpam-3036	72	11	,	,	PUNCT
ejpam-3036	72	12	there	there	PRON
ejpam-3036	72	13	exists	exist	VERB
ejpam-3036	72	14	an	an	DET
ejpam-3036	72	15	element	element	NOUN
ejpam-3036	72	16	x	x	SYM
ejpam-3036	72	17	∈	∈	PROPN
ejpam-3036	72	18	c(a1	c(a1	NOUN
ejpam-3036	72	19	)	)	PUNCT
ejpam-3036	72	20	such	such	ADJ
ejpam-3036	72	21	that	that	SCONJ
ejpam-3036	72	22	f(x	f(x	NOUN
ejpam-3036	72	23	)	)	PUNCT
ejpam-3036	73	1	=	=	SYM
ejpam-3036	73	2	y	y	PROPN
ejpam-3036	73	3	and	and	CCONJ
ejpam-3036	73	4	φ(x	φ(x	NOUN
ejpam-3036	73	5	)	)	PUNCT
ejpam-3036	73	6	=	=	SYM
ejpam-3036	73	7	f(x∗∗	f(x∗∗	PROPN
ejpam-3036	73	8	)	)	PUNCT
ejpam-3036	73	9	∧	∧	PROPN
ejpam-3036	73	10	(	(	PUNCT
ejpam-3036	73	11	f(x++	f(x++	PROPN
ejpam-3036	73	12	)	)	PUNCT
ejpam-3036	73	13	∨	∨	PROPN
ejpam-3036	73	14	k2	k2	NOUN
ejpam-3036	73	15	)	)	PUNCT
ejpam-3036	73	16	=	=	SYM
ejpam-3036	73	17	f(x	f(x	PROPN
ejpam-3036	73	18	)	)	PUNCT
ejpam-3036	73	19	∧	∧	PROPN
ejpam-3036	73	20	(	(	PUNCT
ejpam-3036	73	21	f(x	f(x	PROPN
ejpam-3036	73	22	)	)	PUNCT
ejpam-3036	73	23	∨	∨	NUM
ejpam-3036	73	24	k2	k2	NOUN
ejpam-3036	73	25	)	)	PUNCT
ejpam-3036	73	26	=	=	SYM
ejpam-3036	73	27	f(x	f(x	PROPN
ejpam-3036	73	28	)	)	PUNCT
ejpam-3036	74	1	=	=	SYM
ejpam-3036	74	2	y	y	PROPN
ejpam-3036	74	3	case	case	NOUN
ejpam-3036	74	4	(	(	PUNCT
ejpam-3036	74	5	ii	ii	NOUN
ejpam-3036	74	6	)	)	PUNCT
ejpam-3036	74	7	:	:	PUNCT
ejpam-3036	75	1	y	y	PROPN
ejpam-3036	75	2	=	=	SYM
ejpam-3036	75	3	k2	k2	PROPN
ejpam-3036	75	4	then	then	ADV
ejpam-3036	75	5	φ(k1	φ(k1	ADJ
ejpam-3036	75	6	)	)	PUNCT
ejpam-3036	75	7	=	=	SYM
ejpam-3036	75	8	f(k∗∗1	f(k∗∗1	ADJ
ejpam-3036	75	9	)	)	PUNCT
ejpam-3036	75	10	∧	∧	PROPN
ejpam-3036	75	11	(	(	PUNCT
ejpam-3036	75	12	f(k++	f(k++	ADJ
ejpam-3036	75	13	1	1	NUM
ejpam-3036	75	14	)	)	PUNCT
ejpam-3036	75	15	∨	∨	PROPN
ejpam-3036	75	16	k2	k2	NOUN
ejpam-3036	75	17	)	)	PUNCT
ejpam-3036	75	18	=	=	SYM
ejpam-3036	75	19	1	1	NUM
ejpam-3036	75	20	∧	∧	PROPN
ejpam-3036	75	21	k2	k2	PROPN
ejpam-3036	75	22	=	=	PROPN
ejpam-3036	75	23	k2	k2	PROPN
ejpam-3036	75	24	case	case	NOUN
ejpam-3036	75	25	(	(	PUNCT
ejpam-3036	75	26	iii	iii	NOUN
ejpam-3036	75	27	)	)	PUNCT
ejpam-3036	75	28	:	:	PUNCT
ejpam-3036	75	29	k2	k2	PROPN
ejpam-3036	75	30	6=	6=	PROPN
ejpam-3036	75	31	y	y	PROPN
ejpam-3036	75	32	and	and	CCONJ
ejpam-3036	75	33	y	y	PROPN
ejpam-3036	75	34	/∈	/∈	PUNCT
ejpam-3036	75	35	c(a2	c(a2	NOUN
ejpam-3036	75	36	)	)	PUNCT
ejpam-3036	75	37	then	then	ADV
ejpam-3036	75	38	y∗∗	y∗∗	X
ejpam-3036	75	39	,	,	PUNCT
ejpam-3036	75	40	y++	y++	NOUN
ejpam-3036	75	41	∈	∈	PROPN
ejpam-3036	75	42	c(a2	c(a2	NOUN
ejpam-3036	75	43	)	)	PUNCT
ejpam-3036	75	44	and	and	CCONJ
ejpam-3036	75	45	from	from	ADP
ejpam-3036	75	46	the	the	DET
ejpam-3036	75	47	fact	fact	NOUN
ejpam-3036	75	48	that	that	SCONJ
ejpam-3036	75	49	f	f	PROPN
ejpam-3036	75	50	is	be	AUX
ejpam-3036	75	51	onto	onto	ADP
ejpam-3036	75	52	there	there	ADV
ejpam-3036	75	53	exists	exist	VERB
ejpam-3036	75	54	x1	x1	PROPN
ejpam-3036	75	55	,	,	PUNCT
ejpam-3036	75	56	x2	x2	PROPN
ejpam-3036	75	57	∈	∈	PROPN
ejpam-3036	75	58	c(a1	c(a1	NOUN
ejpam-3036	75	59	)	)	PUNCT
ejpam-3036	76	1	such	such	ADJ
ejpam-3036	76	2	that	that	SCONJ
ejpam-3036	76	3	φ	φ	PROPN
ejpam-3036	76	4	(	(	PUNCT
ejpam-3036	76	5	x1	x1	PROPN
ejpam-3036	76	6	)	)	PUNCT
ejpam-3036	76	7	=	=	SYM
ejpam-3036	76	8	f	f	PROPN
ejpam-3036	76	9	(	(	PUNCT
ejpam-3036	76	10	x1	x1	PROPN
ejpam-3036	76	11	)	)	PUNCT
ejpam-3036	76	12	=	=	SYM
ejpam-3036	76	13	y	y	PROPN
ejpam-3036	76	14	∗∗	∗∗	PROPN
ejpam-3036	76	15	and	and	CCONJ
ejpam-3036	76	16	φ	φ	PROPN
ejpam-3036	76	17	(	(	PUNCT
ejpam-3036	76	18	x2	x2	PROPN
ejpam-3036	76	19	)	)	PUNCT
ejpam-3036	76	20	=	=	SYM
ejpam-3036	77	1	f	f	X
ejpam-3036	77	2	(	(	PUNCT
ejpam-3036	77	3	x2	x2	PROPN
ejpam-3036	77	4	)	)	PUNCT
ejpam-3036	77	5	=	=	PUNCT
ejpam-3036	78	1	y	y	PROPN
ejpam-3036	79	1	+	+	PROPN
ejpam-3036	79	2	+	+	PROPN
ejpam-3036	79	3	,	,	PUNCT
ejpam-3036	79	4	now	now	ADV
ejpam-3036	79	5	φ	φ	X
ejpam-3036	79	6	(	(	PUNCT
ejpam-3036	79	7	x1	x1	PROPN
ejpam-3036	79	8	∧	∧	PROPN
ejpam-3036	79	9	(	(	PUNCT
ejpam-3036	79	10	x2	x2	PROPN
ejpam-3036	79	11	∨	∨	NUM
ejpam-3036	79	12	k1	k1	NOUN
ejpam-3036	79	13	)	)	PUNCT
ejpam-3036	79	14	)	)	PUNCT
ejpam-3036	79	15	=	=	SYM
ejpam-3036	79	16	f(x1	f(x1	X
ejpam-3036	79	17	)	)	PUNCT
ejpam-3036	79	18	∧	∧	PROPN
ejpam-3036	79	19	(	(	PUNCT
ejpam-3036	79	20	f	f	PROPN
ejpam-3036	79	21	(	(	PUNCT
ejpam-3036	79	22	x1	x1	PROPN
ejpam-3036	79	23	∧	∧	PROPN
ejpam-3036	79	24	f	f	PROPN
ejpam-3036	79	25	(	(	PUNCT
ejpam-3036	79	26	x2	x2	PROPN
ejpam-3036	79	27	)	)	PUNCT
ejpam-3036	79	28	)	)	PUNCT
ejpam-3036	79	29	∨	∨	NUM
ejpam-3036	79	30	k2	k2	NOUN
ejpam-3036	79	31	)	)	PUNCT
ejpam-3036	79	32	=	=	SYM
ejpam-3036	80	1	f	f	PROPN
ejpam-3036	80	2	(	(	PUNCT
ejpam-3036	80	3	x1	x1	ADJ
ejpam-3036	80	4	)	)	PUNCT
ejpam-3036	80	5	∧	∧	PROPN
ejpam-3036	80	6	(	(	PUNCT
ejpam-3036	80	7	f	f	PROPN
ejpam-3036	80	8	(	(	PUNCT
ejpam-3036	80	9	x2	x2	PROPN
ejpam-3036	80	10	)	)	PUNCT
ejpam-3036	80	11	∨	∨	NOUN
ejpam-3036	80	12	k2	k2	NOUN
ejpam-3036	80	13	)	)	PUNCT
ejpam-3036	80	14	=	=	SYM
ejpam-3036	80	15	y∗∗	y∗∗	ADP
ejpam-3036	80	16	∧	∧	PROPN
ejpam-3036	80	17	(	(	PUNCT
ejpam-3036	80	18	y++	y++	PROPN
ejpam-3036	80	19	∨	∨	NUM
ejpam-3036	80	20	k2	k2	PROPN
ejpam-3036	80	21	)	)	PUNCT
ejpam-3036	80	22	=	=	SYM
ejpam-3036	81	1	y	y	PROPN
ejpam-3036	81	2	hence	hence	ADV
ejpam-3036	81	3	φ	φ	PROPN
ejpam-3036	81	4	is	be	AUX
ejpam-3036	81	5	onto	onto	ADP
ejpam-3036	81	6	.	.	PUNCT
ejpam-3036	82	1	the	the	DET
ejpam-3036	82	2	remaining	remain	VERB
ejpam-3036	82	3	conditions	condition	NOUN
ejpam-3036	82	4	which	which	PRON
ejpam-3036	82	5	verifies	verifie	NOUN
ejpam-3036	82	6	that	that	SCONJ
ejpam-3036	82	7	φ	φ	PROPN
ejpam-3036	82	8	is	be	AUX
ejpam-3036	82	9	homomorphism	homomorphism	NOUN
ejpam-3036	82	10	is	be	AUX
ejpam-3036	82	11	straightforward	straightforward	ADJ
ejpam-3036	82	12	.	.	PUNCT
ejpam-3036	83	1	conversely	conversely	ADV
ejpam-3036	83	2	suppose	suppose	VERB
ejpam-3036	83	3	that	that	SCONJ
ejpam-3036	83	4	φ	φ	PROPN
ejpam-3036	83	5	:	:	PUNCT
ejpam-3036	83	6	a1	a1	NOUN
ejpam-3036	83	7	−→	−→	NOUN
ejpam-3036	83	8	a2	a2	PROPN
ejpam-3036	83	9	is	be	AUX
ejpam-3036	83	10	an	an	DET
ejpam-3036	83	11	isomorphism	isomorphism	NOUN
ejpam-3036	83	12	.	.	PUNCT
ejpam-3036	84	1	let	let	VERB
ejpam-3036	84	2	x	x	PUNCT
ejpam-3036	84	3	∈	∈	PROPN
ejpam-3036	84	4	c(a1	c(a1	NOUN
ejpam-3036	84	5	)	)	PUNCT
ejpam-3036	84	6	be	be	AUX
ejpam-3036	84	7	any	any	DET
ejpam-3036	84	8	element	element	NOUN
ejpam-3036	84	9	,	,	PUNCT
ejpam-3036	84	10	then	then	ADV
ejpam-3036	84	11	(	(	PUNCT
ejpam-3036	84	12	φ(x))∗∗	φ(x))∗∗	NOUN
ejpam-3036	84	13	=	=	SYM
ejpam-3036	84	14	φ(x∗∗	φ(x∗∗	PROPN
ejpam-3036	84	15	)	)	PUNCT
ejpam-3036	84	16	=	=	SYM
ejpam-3036	84	17	φ(x	φ(x	NOUN
ejpam-3036	84	18	)	)	PUNCT
ejpam-3036	84	19	.	.	PUNCT
ejpam-3036	85	1	therefore	therefore	ADV
ejpam-3036	85	2	φ(x	φ(x	NOUN
ejpam-3036	85	3	)	)	PUNCT
ejpam-3036	85	4	∈	∈	PROPN
ejpam-3036	85	5	c(a2	c(a2	NOUN
ejpam-3036	85	6	)	)	PUNCT
ejpam-3036	85	7	,	,	PUNCT
ejpam-3036	85	8	and	and	CCONJ
ejpam-3036	85	9	hence	hence	ADV
ejpam-3036	85	10	φ(c(a1	φ(c(a1	NOUN
ejpam-3036	85	11	)	)	PUNCT
ejpam-3036	85	12	)	)	PUNCT
ejpam-3036	86	1	⊆	⊆	NUM
ejpam-3036	86	2	c(a2	c(a2	NOUN
ejpam-3036	86	3	)	)	PUNCT
ejpam-3036	86	4	.	.	PUNCT
ejpam-3036	87	1	on	on	ADP
ejpam-3036	87	2	the	the	DET
ejpam-3036	87	3	other	other	ADJ
ejpam-3036	87	4	hand	hand	NOUN
ejpam-3036	87	5	,	,	PUNCT
ejpam-3036	87	6	if	if	SCONJ
ejpam-3036	87	7	y	y	PROPN
ejpam-3036	87	8	∈	∈	PROPN
ejpam-3036	87	9	c(a2	c(a2	NOUN
ejpam-3036	87	10	)	)	PUNCT
ejpam-3036	87	11	then	then	ADV
ejpam-3036	87	12	there	there	PRON
ejpam-3036	87	13	exist	exist	VERB
ejpam-3036	87	14	x	x	SYM
ejpam-3036	87	15	∈	∈	NOUN
ejpam-3036	87	16	a1	a1	NOUN
ejpam-3036	87	17	and	and	CCONJ
ejpam-3036	87	18	φ(x	φ(x	NOUN
ejpam-3036	87	19	)	)	PUNCT
ejpam-3036	87	20	=	=	VERB
ejpam-3036	88	1	y.	y.	NOUN
ejpam-3036	88	2	now	now	ADV
ejpam-3036	88	3	φ(x∗∗	φ(x∗∗	NOUN
ejpam-3036	88	4	)	)	PUNCT
ejpam-3036	88	5	=	=	PUNCT
ejpam-3036	88	6	(	(	PUNCT
ejpam-3036	88	7	φ(x))∗∗	φ(x))∗∗	X
ejpam-3036	88	8	=	=	SYM
ejpam-3036	88	9	y∗∗	y∗∗	NOUN
ejpam-3036	88	10	=	=	SYM
ejpam-3036	88	11	y	y	PROPN
ejpam-3036	88	12	=	=	PUNCT
ejpam-3036	88	13	φ(x	φ(x	PROPN
ejpam-3036	88	14	)	)	PUNCT
ejpam-3036	88	15	as	as	SCONJ
ejpam-3036	88	16	φ	φ	PROPN
ejpam-3036	88	17	is	be	AUX
ejpam-3036	88	18	one	one	NUM
ejpam-3036	88	19	-	-	PUNCT
ejpam-3036	88	20	one	one	NUM
ejpam-3036	88	21	,	,	PUNCT
ejpam-3036	88	22	we	we	PRON
ejpam-3036	88	23	get	get	VERB
ejpam-3036	88	24	x∗∗	x∗∗	NOUN
ejpam-3036	89	1	=	=	PUNCT
ejpam-3036	89	2	x	x	X
ejpam-3036	89	3	and	and	CCONJ
ejpam-3036	89	4	hence	hence	ADV
ejpam-3036	89	5	x	x	X
ejpam-3036	89	6	∈	∈	PROPN
ejpam-3036	89	7	c(a1	c(a1	NOUN
ejpam-3036	89	8	)	)	PUNCT
ejpam-3036	89	9	.	.	PUNCT
ejpam-3036	90	1	therefore	therefore	ADV
ejpam-3036	90	2	φ(c(a1	φ(c(a1	NOUN
ejpam-3036	90	3	)	)	PUNCT
ejpam-3036	90	4	)	)	PUNCT
ejpam-3036	91	1	=	=	SYM
ejpam-3036	91	2	c(a2	c(a2	NOUN
ejpam-3036	91	3	)	)	PUNCT
ejpam-3036	91	4	and	and	CCONJ
ejpam-3036	91	5	hence	hence	ADV
ejpam-3036	91	6	they	they	PRON
ejpam-3036	91	7	are	be	AUX
ejpam-3036	91	8	isomorphic	isomorphic	ADJ
ejpam-3036	91	9	.	.	PUNCT
ejpam-3036	92	1	hence	hence	ADV
ejpam-3036	92	2	boolean	boolean	ADJ
ejpam-3036	92	3	isomorphism	isomorphism	NOUN
ejpam-3036	92	4	between	between	ADP
ejpam-3036	92	5	centre	centre	NOUN
ejpam-3036	92	6	of	of	ADP
ejpam-3036	92	7	a	a	DET
ejpam-3036	92	8	crdsa	crdsa	NOUN
ejpam-3036	92	9	can	can	AUX
ejpam-3036	92	10	be	be	AUX
ejpam-3036	92	11	extended	extend	VERB
ejpam-3036	92	12	to	to	ADP
ejpam-3036	92	13	whole	whole	ADJ
ejpam-3036	92	14	algebra	algebra	NOUN
ejpam-3036	92	15	so	so	SCONJ
ejpam-3036	92	16	that	that	SCONJ
ejpam-3036	92	17	core	core	NOUN
ejpam-3036	92	18	elements	element	NOUN
ejpam-3036	92	19	are	be	AUX
ejpam-3036	92	20	mapped	map	VERB
ejpam-3036	92	21	each	each	DET
ejpam-3036	92	22	other	other	ADJ
ejpam-3036	92	23	.	.	PUNCT
ejpam-3036	93	1	leta	leta	PROPN
ejpam-3036	93	2	be	be	AUX
ejpam-3036	93	3	a	a	DET
ejpam-3036	93	4	regular	regular	ADJ
ejpam-3036	93	5	double	double	ADJ
ejpam-3036	93	6	stone	stone	NOUN
ejpam-3036	93	7	algebra	algebra	NOUN
ejpam-3036	93	8	.	.	PUNCT
ejpam-3036	94	1	for	for	ADP
ejpam-3036	94	2	a	a	DET
ejpam-3036	94	3	∈	∈	PROPN
ejpam-3036	94	4	a	a	DET
ejpam-3036	94	5	the	the	DET
ejpam-3036	94	6	∗−	∗−	ADJ
ejpam-3036	94	7	centralizer	centralizer	NOUN
ejpam-3036	94	8	of	of	ADP
ejpam-3036	94	9	a	a	PRON
ejpam-3036	94	10	is	be	AUX
ejpam-3036	94	11	denoted	denote	VERB
ejpam-3036	94	12	by	by	ADP
ejpam-3036	94	13	a∗a	a∗a	PUNCT
ejpam-3036	94	14	and	and	CCONJ
ejpam-3036	94	15	defined	define	VERB
ejpam-3036	94	16	as	as	ADP
ejpam-3036	94	17	a∗a	a∗a	X
ejpam-3036	94	18	=	=	SYM
ejpam-3036	94	19	{	{	PUNCT
ejpam-3036	94	20	x∗∗	x∗∗	PROPN
ejpam-3036	95	1	|	|	ADV
ejpam-3036	95	2	x	x	SYM
ejpam-3036	95	3	≤	≤	PROPN
ejpam-3036	95	4	a	a	PRON
ejpam-3036	95	5	}	}	PUNCT
ejpam-3036	95	6	=	=	SYM
ejpam-3036	95	7	{	{	PUNCT
ejpam-3036	95	8	x∗∗	x∗∗	PROPN
ejpam-3036	95	9	∧	∧	PROPN
ejpam-3036	95	10	a∗∗	a∗∗	NOUN
ejpam-3036	95	11	|	|	ADV
ejpam-3036	95	12	x	x	X
ejpam-3036	95	13	∈	∈	PROPN
ejpam-3036	95	14	a	a	PRON
ejpam-3036	95	15	}	}	PUNCT
ejpam-3036	95	16	.	.	PUNCT
ejpam-3036	96	1	a	a	DET
ejpam-3036	96	2	r	r	NOUN
ejpam-3036	96	3	j	j	PROPN
ejpam-3036	96	4	srikanth	srikanth	PROPN
ejpam-3036	96	5	,	,	PUNCT
ejpam-3036	96	6	r	r	NOUN
ejpam-3036	96	7	v	v	NUM
ejpam-3036	96	8	g	g	NOUN
ejpam-3036	96	9	ravi	ravi	PROPN
ejpam-3036	96	10	kumar	kumar	PROPN
ejpam-3036	96	11	/	/	SYM
ejpam-3036	96	12	eur	eur	PROPN
ejpam-3036	96	13	.	.	PUNCT
ejpam-3036	97	1	j.	j.	PROPN
ejpam-3036	97	2	pure	pure	PROPN
ejpam-3036	97	3	appl	appl	PROPN
ejpam-3036	97	4	.	.	PROPN
ejpam-3036	97	5	math	math	PROPN
ejpam-3036	97	6	,	,	PUNCT
ejpam-3036	97	7	10	10	NUM
ejpam-3036	97	8	(	(	PUNCT
ejpam-3036	97	9	4	4	NUM
ejpam-3036	97	10	)	)	PUNCT
ejpam-3036	97	11	(	(	PUNCT
ejpam-3036	97	12	2017	2017	NUM
ejpam-3036	97	13	)	)	PUNCT
ejpam-3036	97	14	,	,	PUNCT
ejpam-3036	97	15	717	717	NUM
ejpam-3036	97	16	-	-	SYM
ejpam-3036	97	17	729	729	NUM
ejpam-3036	97	18	722	722	NUM
ejpam-3036	97	19	definition	definition	NOUN
ejpam-3036	97	20	7	7	NUM
ejpam-3036	97	21	.	.	PUNCT
ejpam-3036	98	1	leta	leta	PROPN
ejpam-3036	98	2	be	be	AUX
ejpam-3036	98	3	a	a	DET
ejpam-3036	98	4	regular	regular	ADJ
ejpam-3036	98	5	double	double	ADJ
ejpam-3036	98	6	stone	stone	NOUN
ejpam-3036	98	7	algebra	algebra	NOUN
ejpam-3036	98	8	.	.	PUNCT
ejpam-3036	99	1	for	for	ADP
ejpam-3036	99	2	a	a	DET
ejpam-3036	99	3	∈	∈	PROPN
ejpam-3036	99	4	a	a	DET
ejpam-3036	99	5	the	the	DET
ejpam-3036	99	6	+	+	ADJ
ejpam-3036	99	7	−	−	NOUN
ejpam-3036	99	8	centralizer	centralizer	NOUN
ejpam-3036	99	9	of	of	ADP
ejpam-3036	99	10	a	a	PRON
ejpam-3036	99	11	is	be	AUX
ejpam-3036	99	12	denoted	denote	VERB
ejpam-3036	99	13	by	by	ADP
ejpam-3036	99	14	a+	a+	PUNCT
ejpam-3036	99	15	a	a	PROPN
ejpam-3036	99	16	and	and	CCONJ
ejpam-3036	99	17	defined	define	VERB
ejpam-3036	99	18	as	as	ADP
ejpam-3036	99	19	a+	a+	PRON
ejpam-3036	99	20	a	a	X
ejpam-3036	99	21	=	=	X
ejpam-3036	99	22	{	{	PUNCT
ejpam-3036	99	23	x++	x++	X
ejpam-3036	99	24	|	|	ADV
ejpam-3036	99	25	x	x	X
ejpam-3036	99	26	≥	≥	X
ejpam-3036	99	27	a	a	PRON
ejpam-3036	99	28	}	}	PUNCT
ejpam-3036	99	29	=	=	NOUN
ejpam-3036	99	30	=	=	SYM
ejpam-3036	99	31	{	{	PUNCT
ejpam-3036	99	32	x++	x++	ADJ
ejpam-3036	99	33	∨	∨	NUM
ejpam-3036	99	34	a++	a++	NOUN
ejpam-3036	100	1	|	|	ADV
ejpam-3036	100	2	x	x	SYM
ejpam-3036	100	3	∈	∈	PROPN
ejpam-3036	100	4	a	a	PRON
ejpam-3036	100	5	}	}	PUNCT
ejpam-3036	100	6	.	.	PUNCT
ejpam-3036	101	1	theorem	theorem	VERB
ejpam-3036	101	2	2.5	2.5	NUM
ejpam-3036	101	3	.	.	PUNCT
ejpam-3036	102	1	let	let	VERB
ejpam-3036	102	2	a	a	PRON
ejpam-3036	102	3	be	be	AUX
ejpam-3036	102	4	a	a	DET
ejpam-3036	102	5	core	core	NOUN
ejpam-3036	102	6	regular	regular	ADJ
ejpam-3036	102	7	double	double	ADJ
ejpam-3036	102	8	stone	stone	NOUN
ejpam-3036	102	9	algebra	algebra	NOUN
ejpam-3036	102	10	.	.	PUNCT
ejpam-3036	103	1	the	the	DET
ejpam-3036	103	2	relativized	relativized	ADJ
ejpam-3036	103	3	algebra	algebra	NOUN
ejpam-3036	103	4	a∗a	a∗a	X
ejpam-3036	103	5	=	=	X
ejpam-3036	103	6	<	<	X
ejpam-3036	103	7	a∗a,∧,∨,′	a∗a,∧,∨,′	NUM
ejpam-3036	103	8	0	0	NUM
ejpam-3036	103	9	,	,	PUNCT
ejpam-3036	103	10	a∗∗	a∗∗	PROPN
ejpam-3036	103	11	>	>	X
ejpam-3036	103	12	is	be	AUX
ejpam-3036	103	13	a	a	DET
ejpam-3036	103	14	boolean	boolean	ADJ
ejpam-3036	103	15	algebra	algebra	NOUN
ejpam-3036	103	16	proof	proof	NOUN
ejpam-3036	103	17	.	.	PUNCT
ejpam-3036	104	1	let	let	VERB
ejpam-3036	104	2	x	x	PRON
ejpam-3036	104	3	,	,	PUNCT
ejpam-3036	104	4	y	y	PROPN
ejpam-3036	104	5	∈	∈	PROPN
ejpam-3036	104	6	a∗a	a∗a	PROPN
ejpam-3036	104	7	.	.	PUNCT
ejpam-3036	105	1	then	then	ADV
ejpam-3036	105	2	x	x	X
ejpam-3036	105	3	=	=	SYM
ejpam-3036	105	4	p∗∗	p∗∗	PROPN
ejpam-3036	105	5	,	,	PUNCT
ejpam-3036	105	6	y	y	NOUN
ejpam-3036	105	7	=	=	PUNCT
ejpam-3036	105	8	q∗∗	q∗∗	NOUN
ejpam-3036	105	9	for	for	ADP
ejpam-3036	105	10	some	some	DET
ejpam-3036	105	11	p	p	NOUN
ejpam-3036	105	12	,	,	PUNCT
ejpam-3036	105	13	q	q	PROPN
ejpam-3036	105	14	∈	∈	PROPN
ejpam-3036	105	15	a	a	PRON
ejpam-3036	105	16	and	and	CCONJ
ejpam-3036	105	17	p	p	NOUN
ejpam-3036	105	18	,	,	PUNCT
ejpam-3036	105	19	q	q	PROPN
ejpam-3036	105	20	≤	≤	NUM
ejpam-3036	105	21	a.	a.	NOUN
ejpam-3036	105	22	which	which	PRON
ejpam-3036	105	23	gives	give	VERB
ejpam-3036	105	24	p∨	p∨	PROPN
ejpam-3036	105	25	q	q	PROPN
ejpam-3036	105	26	≤	≤	PROPN
ejpam-3036	105	27	a	a	PRON
ejpam-3036	105	28	,	,	PUNCT
ejpam-3036	105	29	p∧	p∧	NOUN
ejpam-3036	105	30	q	q	PROPN
ejpam-3036	105	31	≤	≤	NUM
ejpam-3036	105	32	a.	a.	NOUN
ejpam-3036	105	33	hence	hence	ADV
ejpam-3036	105	34	(	(	PUNCT
ejpam-3036	105	35	p∨	p∨	PROPN
ejpam-3036	105	36	q)∗∗	q)∗∗	NOUN
ejpam-3036	105	37	=	=	SYM
ejpam-3036	105	38	p∗∗	p∗∗	PROPN
ejpam-3036	105	39	∨	∨	NOUN
ejpam-3036	105	40	q∗∗	q∗∗	NOUN
ejpam-3036	105	41	≤	≤	NUM
ejpam-3036	105	42	a∗∗	a∗∗	NOUN
ejpam-3036	105	43	and	and	CCONJ
ejpam-3036	105	44	(	(	PUNCT
ejpam-3036	105	45	p∧	p∧	NOUN
ejpam-3036	105	46	q)∗∗	q)∗∗	X
ejpam-3036	105	47	=	=	SYM
ejpam-3036	105	48	p∗∗	p∗∗	PROPN
ejpam-3036	105	49	∧	∧	NOUN
ejpam-3036	105	50	q∗∗	q∗∗	NOUN
ejpam-3036	105	51	≤	≤	NUM
ejpam-3036	105	52	a∗∗	a∗∗	NOUN
ejpam-3036	105	53	.	.	PUNCT
ejpam-3036	106	1	therefore	therefore	ADV
ejpam-3036	106	2	x	x	PROPN
ejpam-3036	106	3	∨	∨	NUM
ejpam-3036	106	4	y	y	PROPN
ejpam-3036	106	5	,	,	PUNCT
ejpam-3036	106	6	x	x	PUNCT
ejpam-3036	106	7	∧	∧	NOUN
ejpam-3036	106	8	y	y	PROPN
ejpam-3036	106	9	∈	∈	PROPN
ejpam-3036	106	10	a∗a	a∗a	PROPN
ejpam-3036	106	11	.	.	PUNCT
ejpam-3036	107	1	therefore	therefore	ADV
ejpam-3036	107	2	a∗a	a∗a	NUM
ejpam-3036	107	3	is	be	AUX
ejpam-3036	107	4	closed	close	VERB
ejpam-3036	107	5	with	with	ADP
ejpam-3036	107	6	respect	respect	NOUN
ejpam-3036	107	7	to	to	ADP
ejpam-3036	107	8	∨	∨	NUM
ejpam-3036	107	9	and	and	CCONJ
ejpam-3036	107	10	∧.	∧.	NOUN
ejpam-3036	107	11	it	it	PRON
ejpam-3036	107	12	is	be	AUX
ejpam-3036	107	13	a	a	DET
ejpam-3036	107	14	routine	routine	ADJ
ejpam-3036	107	15	verification	verification	NOUN
ejpam-3036	107	16	that	that	PRON
ejpam-3036	107	17	<	<	AUX
ejpam-3036	107	18	a∗a,∧,∨	a∗a,∧,∨	PROPN
ejpam-3036	107	19	>	>	PUNCT
ejpam-3036	107	20	is	be	AUX
ejpam-3036	107	21	distributive	distributive	ADJ
ejpam-3036	107	22	lattice	lattice	NOUN
ejpam-3036	107	23	.	.	PUNCT
ejpam-3036	108	1	clearly	clearly	ADV
ejpam-3036	108	2	0∗∗	0∗∗	X
ejpam-3036	108	3	=	=	SYM
ejpam-3036	108	4	0	0	NUM
ejpam-3036	108	5	≤	≤	NOUN
ejpam-3036	108	6	a	a	DET
ejpam-3036	108	7	,	,	PUNCT
ejpam-3036	108	8	so	so	ADV
ejpam-3036	108	9	0	0	NUM
ejpam-3036	108	10	∈	∈	PROPN
ejpam-3036	108	11	a∗a	a∗a	PROPN
ejpam-3036	108	12	.	.	PUNCT
ejpam-3036	109	1	since	since	SCONJ
ejpam-3036	109	2	a	a	DET
ejpam-3036	109	3	≤	≤	NOUN
ejpam-3036	109	4	a	a	PRON
ejpam-3036	109	5	we	we	PRON
ejpam-3036	109	6	get	get	VERB
ejpam-3036	109	7	a∗∗	a∗∗	PROPN
ejpam-3036	109	8	∈	∈	PROPN
ejpam-3036	109	9	a∗a	a∗a	PROPN
ejpam-3036	109	10	.	.	PUNCT
ejpam-3036	109	11	let	let	VERB
ejpam-3036	109	12	x	x	SYM
ejpam-3036	109	13	∈	∈	PROPN
ejpam-3036	109	14	a∗a	a∗a	PUNCT
ejpam-3036	109	15	be	be	AUX
ejpam-3036	109	16	any	any	DET
ejpam-3036	109	17	element	element	NOUN
ejpam-3036	109	18	then	then	ADV
ejpam-3036	109	19	x	x	PUNCT
ejpam-3036	109	20	=	=	SYM
ejpam-3036	109	21	p∗∗	p∗∗	NOUN
ejpam-3036	109	22	for	for	ADP
ejpam-3036	109	23	some	some	DET
ejpam-3036	109	24	p	p	NOUN
ejpam-3036	109	25	∈	∈	PROPN
ejpam-3036	109	26	a	a	PRON
ejpam-3036	109	27	and	and	CCONJ
ejpam-3036	109	28	p	p	NOUN
ejpam-3036	109	29	≤	≤	NOUN
ejpam-3036	109	30	a	a	PRON
ejpam-3036	109	31	which	which	PRON
ejpam-3036	109	32	gives	give	VERB
ejpam-3036	109	33	p∗∗	p∗∗	NOUN
ejpam-3036	109	34	=	=	SYM
ejpam-3036	109	35	x	x	SYM
ejpam-3036	109	36	≤	≤	X
ejpam-3036	109	37	a∗∗.	a∗∗.	X
ejpam-3036	109	38	therefore	therefore	ADV
ejpam-3036	109	39	a∗∗	a∗∗	PROPN
ejpam-3036	109	40	is	be	AUX
ejpam-3036	109	41	the	the	DET
ejpam-3036	109	42	greatest	great	ADJ
ejpam-3036	109	43	element	element	NOUN
ejpam-3036	109	44	of	of	ADP
ejpam-3036	109	45	a∗a	a∗a	PROPN
ejpam-3036	109	46	.	.	PUNCT
ejpam-3036	110	1	hence	hence	ADV
ejpam-3036	110	2	<	<	X
ejpam-3036	110	3	a∗a,∧,∨	a∗a,∧,∨	PROPN
ejpam-3036	110	4	,	,	PUNCT
ejpam-3036	110	5	0	0	NUM
ejpam-3036	110	6	,	,	PUNCT
ejpam-3036	110	7	a∗∗	a∗∗	AUX
ejpam-3036	110	8	>	>	X
ejpam-3036	110	9	is	be	AUX
ejpam-3036	110	10	a	a	DET
ejpam-3036	110	11	bounded	bounded	ADJ
ejpam-3036	110	12	distributive	distributive	ADJ
ejpam-3036	110	13	lattice	lattice	NOUN
ejpam-3036	110	14	.	.	PUNCT
ejpam-3036	111	1	finally	finally	ADV
ejpam-3036	111	2	for	for	ADP
ejpam-3036	111	3	x	x	SYM
ejpam-3036	111	4	=	=	SYM
ejpam-3036	111	5	p∗∗	p∗∗	PROPN
ejpam-3036	111	6	∈	∈	NOUN
ejpam-3036	111	7	a∗a	a∗a	PUNCT
ejpam-3036	111	8	we	we	PRON
ejpam-3036	111	9	have	have	VERB
ejpam-3036	111	10	p	p	NOUN
ejpam-3036	111	11	≤	≤	PROPN
ejpam-3036	111	12	a	a	PRON
ejpam-3036	111	13	and	and	CCONJ
ejpam-3036	111	14	x∗	x∗	PROPN
ejpam-3036	111	15	∧	∧	PROPN
ejpam-3036	111	16	a	a	DET
ejpam-3036	111	17	=	=	X
ejpam-3036	111	18	p∗	p∗	ADJ
ejpam-3036	111	19	∧	∧	PROPN
ejpam-3036	111	20	a	a	DET
ejpam-3036	111	21	≤	≤	NOUN
ejpam-3036	111	22	a	a	PRON
ejpam-3036	111	23	which	which	PRON
ejpam-3036	111	24	gives	give	VERB
ejpam-3036	111	25	(	(	PUNCT
ejpam-3036	111	26	x∗	x∗	PROPN
ejpam-3036	111	27	∧	∧	PROPN
ejpam-3036	111	28	a)∗∗	a)∗∗	X
ejpam-3036	111	29	=	=	PUNCT
ejpam-3036	112	1	x∗	x∗	PROPN
ejpam-3036	112	2	∧	∧	NOUN
ejpam-3036	112	3	a∗∗	a∗∗	PROPN
ejpam-3036	112	4	∈	∈	PROPN
ejpam-3036	112	5	a∗a	a∗a	PUNCT
ejpam-3036	112	6	and	and	CCONJ
ejpam-3036	112	7	x	x	PART
ejpam-3036	112	8	∧	∧	PROPN
ejpam-3036	112	9	(	(	PUNCT
ejpam-3036	112	10	x∗	x∗	PROPN
ejpam-3036	112	11	∧	∧	PROPN
ejpam-3036	112	12	a∗∗	a∗∗	PROPN
ejpam-3036	112	13	)	)	PUNCT
ejpam-3036	112	14	=	=	SYM
ejpam-3036	112	15	0	0	NUM
ejpam-3036	112	16	and	and	CCONJ
ejpam-3036	112	17	x	x	SYM
ejpam-3036	112	18	∨	∨	NOUN
ejpam-3036	112	19	(	(	PUNCT
ejpam-3036	112	20	x∗	x∗	PROPN
ejpam-3036	112	21	∧	∧	PROPN
ejpam-3036	112	22	a∗∗	a∗∗	PROPN
ejpam-3036	112	23	)	)	PUNCT
ejpam-3036	113	1	=	=	PUNCT
ejpam-3036	113	2	a∗∗.	a∗∗.	X
ejpam-3036	113	3	therefore	therefore	ADV
ejpam-3036	113	4	x∗	x∗	PROPN
ejpam-3036	113	5	∧	∧	PROPN
ejpam-3036	113	6	a∗∗	a∗∗	VERB
ejpam-3036	113	7	is	be	AUX
ejpam-3036	113	8	the	the	DET
ejpam-3036	113	9	compliment	compliment	NOUN
ejpam-3036	113	10	of	of	ADP
ejpam-3036	113	11	x	x	PUNCT
ejpam-3036	113	12	in	in	ADP
ejpam-3036	113	13	a∗a	a∗a	X
ejpam-3036	113	14	i.e.	i.e.	X
ejpam-3036	113	15	x′	x′	X
ejpam-3036	114	1	=	=	SYM
ejpam-3036	114	2	x∗	x∗	PROPN
ejpam-3036	114	3	∧	∧	NOUN
ejpam-3036	114	4	a∗∗.	a∗∗.	X
ejpam-3036	114	5	hence	hence	ADV
ejpam-3036	114	6	a∗a	a∗a	PUNCT
ejpam-3036	114	7	=	=	SYM
ejpam-3036	114	8	<	<	X
ejpam-3036	114	9	a∗a,∧,∨,′	a∗a,∧,∨,′	NUM
ejpam-3036	114	10	0	0	NUM
ejpam-3036	114	11	,	,	PUNCT
ejpam-3036	114	12	a∗∗	a∗∗	PROPN
ejpam-3036	114	13	>	>	X
ejpam-3036	114	14	is	be	AUX
ejpam-3036	114	15	a	a	DET
ejpam-3036	114	16	boolean	boolean	ADJ
ejpam-3036	114	17	algebra	algebra	NOUN
ejpam-3036	114	18	theorem	theorem	VERB
ejpam-3036	114	19	2.6	2.6	NUM
ejpam-3036	114	20	.	.	PUNCT
ejpam-3036	115	1	let	let	VERB
ejpam-3036	115	2	a	a	PRON
ejpam-3036	115	3	be	be	AUX
ejpam-3036	115	4	a	a	DET
ejpam-3036	115	5	core	core	NOUN
ejpam-3036	115	6	regular	regular	ADJ
ejpam-3036	115	7	double	double	ADJ
ejpam-3036	115	8	stone	stone	NOUN
ejpam-3036	115	9	algebra	algebra	NOUN
ejpam-3036	115	10	and	and	CCONJ
ejpam-3036	115	11	k	k	PROPN
ejpam-3036	115	12	is	be	AUX
ejpam-3036	115	13	the	the	DET
ejpam-3036	115	14	core	core	ADJ
ejpam-3036	115	15	element	element	NOUN
ejpam-3036	115	16	of	of	ADP
ejpam-3036	115	17	a.	a.	NOUN
ejpam-3036	115	18	then	then	ADV
ejpam-3036	115	19	a∗k	a∗k	PROPN
ejpam-3036	115	20	=	=	SYM
ejpam-3036	115	21	a+	a+	PUNCT
ejpam-3036	115	22	k	k	PROPN
ejpam-3036	115	23	proof	proof	NOUN
ejpam-3036	115	24	.	.	PUNCT
ejpam-3036	116	1	let	let	VERB
ejpam-3036	116	2	x∗∗	x∗∗	PROPN
ejpam-3036	116	3	∈	∈	PROPN
ejpam-3036	116	4	a∗k	a∗k	PROPN
ejpam-3036	116	5	and	and	CCONJ
ejpam-3036	116	6	y	y	PROPN
ejpam-3036	116	7	=	=	SYM
ejpam-3036	116	8	k	k	PROPN
ejpam-3036	116	9	∨	∨	NUM
ejpam-3036	116	10	x∗∗.	x∗∗.	X
ejpam-3036	117	1	then	then	ADV
ejpam-3036	117	2	y	y	PROPN
ejpam-3036	117	3	≥	≥	NUM
ejpam-3036	117	4	k	k	PROPN
ejpam-3036	117	5	and	and	CCONJ
ejpam-3036	117	6	y++	y++	NOUN
ejpam-3036	117	7	=	=	PUNCT
ejpam-3036	117	8	(	(	PUNCT
ejpam-3036	117	9	k	k	PROPN
ejpam-3036	117	10	∨	∨	NUM
ejpam-3036	117	11	x∗∗)++	x∗∗)++	PROPN
ejpam-3036	118	1	=	=	PUNCT
ejpam-3036	119	1	x∗∗.therefore	x∗∗.therefore	NOUN
ejpam-3036	119	2	x∗∗	x∗∗	PROPN
ejpam-3036	119	3	∈	∈	PROPN
ejpam-3036	119	4	a+	a+	PUNCT
ejpam-3036	119	5	and	and	CCONJ
ejpam-3036	119	6	hence	hence	ADV
ejpam-3036	119	7	a∗k	a∗k	NUM
ejpam-3036	119	8	⊆	⊆	NUM
ejpam-3036	119	9	a+	a+	X
ejpam-3036	119	10	k	k	NOUN
ejpam-3036	119	11	.	.	PUNCT
ejpam-3036	120	1	now	now	ADV
ejpam-3036	120	2	take	take	VERB
ejpam-3036	120	3	x++	x++	X
ejpam-3036	120	4	∈	∈	PROPN
ejpam-3036	120	5	a+	a+	PUNCT
ejpam-3036	121	1	k	k	PROPN
ejpam-3036	121	2	put	put	VERB
ejpam-3036	121	3	y	y	PROPN
ejpam-3036	121	4	=	=	PUNCT
ejpam-3036	122	1	k	k	PROPN
ejpam-3036	122	2	∧	∧	PROPN
ejpam-3036	122	3	x++	x++	SCONJ
ejpam-3036	122	4	then	then	ADV
ejpam-3036	122	5	y	y	PROPN
ejpam-3036	122	6	≤	≤	PROPN
ejpam-3036	122	7	k	k	PROPN
ejpam-3036	122	8	and	and	CCONJ
ejpam-3036	122	9	hence	hence	ADV
ejpam-3036	122	10	y∗∗	y∗∗	NOUN
ejpam-3036	122	11	=	=	SYM
ejpam-3036	122	12	(	(	PUNCT
ejpam-3036	122	13	k	k	PROPN
ejpam-3036	122	14	∧	∧	PROPN
ejpam-3036	122	15	x++)∗∗	x++)∗∗	X
ejpam-3036	122	16	=	=	SYM
ejpam-3036	122	17	x++	x++	X
ejpam-3036	122	18	∈	∈	PROPN
ejpam-3036	122	19	a∗k	a∗k	PROPN
ejpam-3036	122	20	.	.	PUNCT
ejpam-3036	123	1	so	so	ADV
ejpam-3036	123	2	a+	a+	PUNCT
ejpam-3036	124	1	k	k	PROPN
ejpam-3036	124	2	⊆	⊆	PROPN
ejpam-3036	124	3	a	a	DET
ejpam-3036	124	4	∗	∗	X
ejpam-3036	124	5	k	k	NOUN
ejpam-3036	124	6	and	and	CCONJ
ejpam-3036	124	7	hence	hence	ADV
ejpam-3036	124	8	a∗k	a∗k	PROPN
ejpam-3036	124	9	=	=	SYM
ejpam-3036	124	10	a+	a+	PUNCT
ejpam-3036	124	11	k	k	PROPN
ejpam-3036	124	12	.	.	PUNCT
ejpam-3036	125	1	in	in	ADP
ejpam-3036	125	2	fact	fact	NOUN
ejpam-3036	125	3	we	we	PRON
ejpam-3036	125	4	have	have	VERB
ejpam-3036	125	5	the	the	DET
ejpam-3036	125	6	stronger	strong	ADJ
ejpam-3036	125	7	result	result	NOUN
ejpam-3036	125	8	in	in	ADP
ejpam-3036	125	9	the	the	DET
ejpam-3036	125	10	following	follow	VERB
ejpam-3036	125	11	theorem	theorem	PROPN
ejpam-3036	125	12	.	.	PUNCT
ejpam-3036	125	13	theorem	theorem	PROPN
ejpam-3036	125	14	2.7	2.7	NUM
ejpam-3036	125	15	.	.	PUNCT
ejpam-3036	126	1	let	let	VERB
ejpam-3036	126	2	a	a	PRON
ejpam-3036	126	3	be	be	AUX
ejpam-3036	126	4	a	a	DET
ejpam-3036	126	5	core	core	NOUN
ejpam-3036	126	6	regular	regular	ADJ
ejpam-3036	126	7	double	double	ADJ
ejpam-3036	126	8	stone	stone	NOUN
ejpam-3036	126	9	algebra	algebra	NOUN
ejpam-3036	126	10	and	and	CCONJ
ejpam-3036	126	11	k	k	PROPN
ejpam-3036	126	12	is	be	AUX
ejpam-3036	126	13	the	the	DET
ejpam-3036	126	14	core	core	ADJ
ejpam-3036	126	15	element	element	NOUN
ejpam-3036	126	16	of	of	ADP
ejpam-3036	126	17	a	a	DET
ejpam-3036	126	18	then	then	ADV
ejpam-3036	126	19	a∗a	a∗a	ADP
ejpam-3036	126	20	=	=	SYM
ejpam-3036	126	21	a+	a+	PUNCT
ejpam-3036	126	22	a	a	DET
ejpam-3036	126	23	if	if	NOUN
ejpam-3036	127	1	and	and	CCONJ
ejpam-3036	127	2	only	only	ADV
ejpam-3036	127	3	if	if	SCONJ
ejpam-3036	127	4	a	a	DET
ejpam-3036	127	5	=	=	X
ejpam-3036	127	6	k	k	NOUN
ejpam-3036	127	7	proof	proof	NOUN
ejpam-3036	127	8	.	.	PUNCT
ejpam-3036	128	1	first	first	ADV
ejpam-3036	128	2	suppose	suppose	VERB
ejpam-3036	128	3	that	that	SCONJ
ejpam-3036	128	4	for	for	ADP
ejpam-3036	128	5	some	some	DET
ejpam-3036	128	6	a	a	DET
ejpam-3036	128	7	∈	∈	PROPN
ejpam-3036	128	8	a	a	PRON
ejpam-3036	128	9	,	,	PUNCT
ejpam-3036	128	10	a∗a	a∗a	X
ejpam-3036	128	11	=	=	SYM
ejpam-3036	128	12	a+	a+	PUNCT
ejpam-3036	128	13	a	a	X
ejpam-3036	128	14	.	.	PUNCT
ejpam-3036	129	1	since	since	SCONJ
ejpam-3036	129	2	0	0	NUM
ejpam-3036	129	3	∈	∈	PROPN
ejpam-3036	129	4	a∗a	a∗a	X
ejpam-3036	129	5	=	=	SYM
ejpam-3036	129	6	a+	a+	PUNCT
ejpam-3036	129	7	a	a	PRON
ejpam-3036	129	8	there	there	PRON
ejpam-3036	129	9	exists	exist	VERB
ejpam-3036	129	10	b	b	PROPN
ejpam-3036	129	11	∈	∈	PROPN
ejpam-3036	129	12	a	a	DET
ejpam-3036	129	13	such	such	ADJ
ejpam-3036	129	14	that	that	SCONJ
ejpam-3036	129	15	a	a	DET
ejpam-3036	129	16	≤	≤	PROPN
ejpam-3036	129	17	b	b	NOUN
ejpam-3036	129	18	and	and	CCONJ
ejpam-3036	129	19	0	0	NUM
ejpam-3036	129	20	=	=	SYM
ejpam-3036	129	21	b++	b++	X
ejpam-3036	129	22	.	.	PUNCT
ejpam-3036	130	1	so	so	ADV
ejpam-3036	130	2	b+	b+	VERB
ejpam-3036	130	3	=	=	SYM
ejpam-3036	130	4	1	1	NUM
ejpam-3036	130	5	and	and	CCONJ
ejpam-3036	130	6	b+	b+	VERB
ejpam-3036	130	7	≤	≤	NUM
ejpam-3036	130	8	a+	a+	PUNCT
ejpam-3036	130	9	.	.	PROPN
ejpam-3036	131	1	hence	hence	ADV
ejpam-3036	131	2	a+	a+	PUNCT
ejpam-3036	131	3	=	=	NOUN
ejpam-3036	131	4	1	1	X
ejpam-3036	131	5	.	.	PUNCT
ejpam-3036	132	1	as	as	ADP
ejpam-3036	132	2	1	1	NUM
ejpam-3036	132	3	∈	∈	NOUN
ejpam-3036	132	4	a+	a+	PUNCT
ejpam-3036	132	5	a	a	DET
ejpam-3036	132	6	=	=	SYM
ejpam-3036	132	7	a∗a	a∗a	PUNCT
ejpam-3036	132	8	there	there	PRON
ejpam-3036	132	9	exists	exist	VERB
ejpam-3036	132	10	c	c	PROPN
ejpam-3036	132	11	∈	∈	PROPN
ejpam-3036	132	12	a	a	DET
ejpam-3036	132	13	such	such	ADJ
ejpam-3036	132	14	that	that	SCONJ
ejpam-3036	132	15	c	c	PROPN
ejpam-3036	132	16	≤	≤	NOUN
ejpam-3036	132	17	a	a	DET
ejpam-3036	132	18	and	and	CCONJ
ejpam-3036	132	19	1	1	NUM
ejpam-3036	132	20	=	=	NOUN
ejpam-3036	132	21	c∗∗.	c∗∗.	PROPN
ejpam-3036	132	22	so	so	ADV
ejpam-3036	132	23	c∗	c∗	PROPN
ejpam-3036	132	24	=	=	SYM
ejpam-3036	132	25	0	0	NUM
ejpam-3036	132	26	and	and	CCONJ
ejpam-3036	132	27	a∗	a∗	PROPN
ejpam-3036	132	28	≤	≤	NUM
ejpam-3036	132	29	c∗.	c∗.	NOUN
ejpam-3036	132	30	hence	hence	ADV
ejpam-3036	132	31	a∗	a∗	NOUN
ejpam-3036	132	32	=	=	SYM
ejpam-3036	132	33	0	0	X
ejpam-3036	132	34	.	.	PUNCT
ejpam-3036	133	1	therefore	therefore	ADV
ejpam-3036	133	2	a	a	DET
ejpam-3036	133	3	∈	∈	PROPN
ejpam-3036	133	4	k(a	k(a	PROPN
ejpam-3036	133	5	)	)	PUNCT
ejpam-3036	133	6	=	=	VERB
ejpam-3036	134	1	k.	k.	PROPN
ejpam-3036	134	2	other	other	ADJ
ejpam-3036	134	3	part	part	NOUN
ejpam-3036	134	4	is	be	AUX
ejpam-3036	134	5	clear	clear	ADJ
ejpam-3036	134	6	from	from	ADP
ejpam-3036	134	7	theorem	theorem	ADJ
ejpam-3036	134	8	2.6	2.6	NUM
ejpam-3036	134	9	.	.	PUNCT
ejpam-3036	135	1	throughout	throughout	ADP
ejpam-3036	135	2	this	this	DET
ejpam-3036	135	3	paper	paper	NOUN
ejpam-3036	135	4	we	we	PRON
ejpam-3036	135	5	denote	denote	VERB
ejpam-3036	135	6	a∗k	a∗k	PROPN
ejpam-3036	135	7	=	=	SYM
ejpam-3036	135	8	a+	a+	PRON
ejpam-3036	135	9	k	k	NOUN
ejpam-3036	135	10	by	by	ADP
ejpam-3036	135	11	k(a	k(a	NOUN
ejpam-3036	135	12	)	)	PUNCT
ejpam-3036	135	13	.	.	PUNCT
ejpam-3036	136	1	the	the	DET
ejpam-3036	136	2	following	follow	VERB
ejpam-3036	136	3	theorem	theorem	NOUN
ejpam-3036	136	4	discuss	discuss	VERB
ejpam-3036	136	5	the	the	DET
ejpam-3036	136	6	relation	relation	NOUN
ejpam-3036	136	7	between	between	ADP
ejpam-3036	136	8	centralizer	centralizer	NOUN
ejpam-3036	136	9	of	of	ADP
ejpam-3036	136	10	core	core	NOUN
ejpam-3036	136	11	and	and	CCONJ
ejpam-3036	136	12	centre	centre	NOUN
ejpam-3036	136	13	of	of	ADP
ejpam-3036	136	14	crdsa	crdsa	PROPN
ejpam-3036	136	15	.	.	PUNCT
ejpam-3036	137	1	theorem	theorem	VERB
ejpam-3036	137	2	2.8	2.8	NUM
ejpam-3036	137	3	.	.	PUNCT
ejpam-3036	138	1	let	let	VERB
ejpam-3036	138	2	a	a	PRON
ejpam-3036	138	3	be	be	AUX
ejpam-3036	138	4	a	a	DET
ejpam-3036	138	5	core	core	NOUN
ejpam-3036	138	6	regular	regular	ADJ
ejpam-3036	138	7	double	double	ADJ
ejpam-3036	138	8	stone	stone	NOUN
ejpam-3036	138	9	algebra	algebra	NOUN
ejpam-3036	138	10	and	and	CCONJ
ejpam-3036	138	11	k	k	PROPN
ejpam-3036	138	12	is	be	AUX
ejpam-3036	138	13	the	the	DET
ejpam-3036	138	14	core	core	ADJ
ejpam-3036	138	15	element	element	NOUN
ejpam-3036	138	16	of	of	ADP
ejpam-3036	138	17	a	a	DET
ejpam-3036	138	18	then	then	ADV
ejpam-3036	138	19	k(a	k(a	NOUN
ejpam-3036	138	20	)	)	PUNCT
ejpam-3036	138	21	=	=	SYM
ejpam-3036	138	22	c(a	c(a	PROPN
ejpam-3036	138	23	)	)	PUNCT
ejpam-3036	138	24	.	.	PUNCT
ejpam-3036	139	1	proof	proof	NOUN
ejpam-3036	139	2	.	.	PUNCT
ejpam-3036	140	1	clearly	clearly	ADV
ejpam-3036	140	2	k(a	k(a	NOUN
ejpam-3036	140	3	)	)	PUNCT
ejpam-3036	140	4	⊆	⊆	NUM
ejpam-3036	140	5	c(a	c(a	NOUN
ejpam-3036	140	6	)	)	PUNCT
ejpam-3036	140	7	.	.	PUNCT
ejpam-3036	141	1	let	let	VERB
ejpam-3036	141	2	a	a	DET
ejpam-3036	141	3	be	be	AUX
ejpam-3036	141	4	any	any	DET
ejpam-3036	141	5	element	element	NOUN
ejpam-3036	141	6	of	of	ADP
ejpam-3036	141	7	c(a	c(a	PROPN
ejpam-3036	141	8	)	)	PUNCT
ejpam-3036	141	9	,	,	PUNCT
ejpam-3036	141	10	then	then	ADV
ejpam-3036	141	11	(	(	PUNCT
ejpam-3036	141	12	a	a	DET
ejpam-3036	141	13	∧	∧	PROPN
ejpam-3036	141	14	k)∗∗	k)∗∗	X
ejpam-3036	141	15	=	=	PUNCT
ejpam-3036	142	1	a∗∗	a∗∗	PROPN
ejpam-3036	142	2	=	=	PUNCT
ejpam-3036	142	3	a	a	DET
ejpam-3036	142	4	∈	∈	PROPN
ejpam-3036	142	5	k(a	k(a	NOUN
ejpam-3036	142	6	)	)	PUNCT
ejpam-3036	142	7	.	.	PUNCT
ejpam-3036	143	1	hence	hence	ADV
ejpam-3036	143	2	k(a	k(a	PROPN
ejpam-3036	143	3	)	)	PUNCT
ejpam-3036	143	4	=	=	SYM
ejpam-3036	143	5	c(a	c(a	PROPN
ejpam-3036	143	6	)	)	PUNCT
ejpam-3036	143	7	.	.	PUNCT
ejpam-3036	144	1	theorem	theorem	ADJ
ejpam-3036	144	2	2.8	2.8	NUM
ejpam-3036	144	3	gives	give	VERB
ejpam-3036	144	4	another	another	DET
ejpam-3036	144	5	characterization	characterization	NOUN
ejpam-3036	144	6	for	for	ADP
ejpam-3036	144	7	centre	centre	NOUN
ejpam-3036	144	8	of	of	ADP
ejpam-3036	144	9	a	a	DET
ejpam-3036	144	10	crdsa	crdsa	NOUN
ejpam-3036	144	11	based	base	VERB
ejpam-3036	144	12	on	on	ADP
ejpam-3036	144	13	the	the	DET
ejpam-3036	144	14	core	core	NOUN
ejpam-3036	144	15	element	element	NOUN
ejpam-3036	144	16	.	.	PUNCT
ejpam-3036	145	1	a	a	DET
ejpam-3036	145	2	r	r	NOUN
ejpam-3036	145	3	j	j	PROPN
ejpam-3036	145	4	srikanth	srikanth	PROPN
ejpam-3036	145	5	,	,	PUNCT
ejpam-3036	145	6	r	r	NOUN
ejpam-3036	145	7	v	v	NUM
ejpam-3036	145	8	g	g	NOUN
ejpam-3036	145	9	ravi	ravi	PROPN
ejpam-3036	145	10	kumar	kumar	PROPN
ejpam-3036	145	11	/	/	SYM
ejpam-3036	145	12	eur	eur	PROPN
ejpam-3036	145	13	.	.	PUNCT
ejpam-3036	146	1	j.	j.	PROPN
ejpam-3036	146	2	pure	pure	PROPN
ejpam-3036	146	3	appl	appl	PROPN
ejpam-3036	146	4	.	.	PROPN
ejpam-3036	146	5	math	math	PROPN
ejpam-3036	146	6	,	,	PUNCT
ejpam-3036	146	7	10	10	NUM
ejpam-3036	146	8	(	(	PUNCT
ejpam-3036	146	9	4	4	NUM
ejpam-3036	146	10	)	)	PUNCT
ejpam-3036	146	11	(	(	PUNCT
ejpam-3036	146	12	2017	2017	NUM
ejpam-3036	146	13	)	)	PUNCT
ejpam-3036	146	14	,	,	PUNCT
ejpam-3036	146	15	717	717	NUM
ejpam-3036	146	16	-	-	SYM
ejpam-3036	146	17	729	729	NUM
ejpam-3036	146	18	723	723	NUM
ejpam-3036	146	19	3	3	NUM
ejpam-3036	146	20	.	.	PUNCT
ejpam-3036	146	21	boolean	boolean	ADJ
ejpam-3036	146	22	centre	centre	NOUN
ejpam-3036	146	23	in	in	ADP
ejpam-3036	146	24	[	[	X
ejpam-3036	146	25	6	6	NUM
ejpam-3036	146	26	]	]	PUNCT
ejpam-3036	146	27	,	,	PUNCT
ejpam-3036	146	28	swamy	swamy	PROPN
ejpam-3036	146	29	and	and	CCONJ
ejpam-3036	146	30	murthy	murthy	PROPN
ejpam-3036	146	31	introduced	introduce	VERB
ejpam-3036	146	32	the	the	DET
ejpam-3036	146	33	concept	concept	NOUN
ejpam-3036	146	34	of	of	ADP
ejpam-3036	146	35	‘	'	PUNCT
ejpam-3036	146	36	balanced	balanced	ADJ
ejpam-3036	146	37	congruence	congruence	NOUN
ejpam-3036	146	38	’	'	PUNCT
ejpam-3036	146	39	on	on	ADP
ejpam-3036	146	40	any	any	DET
ejpam-3036	146	41	algebraa	algebraa	NOUN
ejpam-3036	146	42	and	and	CCONJ
ejpam-3036	146	43	showed	show	VERB
ejpam-3036	146	44	the	the	DET
ejpam-3036	146	45	set	set	NOUN
ejpam-3036	146	46	b(a	b(a	NOUN
ejpam-3036	146	47	)	)	PUNCT
ejpam-3036	146	48	of	of	ADP
ejpam-3036	146	49	all	all	DET
ejpam-3036	146	50	balanced	balanced	ADJ
ejpam-3036	146	51	(	(	PUNCT
ejpam-3036	146	52	direct	direct	ADJ
ejpam-3036	146	53	)	)	PUNCT
ejpam-3036	146	54	factor	factor	NOUN
ejpam-3036	146	55	congruences	congruence	NOUN
ejpam-3036	146	56	which	which	PRON
ejpam-3036	146	57	admit	admit	VERB
ejpam-3036	146	58	a	a	DET
ejpam-3036	146	59	balanced	balanced	ADJ
ejpam-3036	146	60	complement	complement	NOUN
ejpam-3036	146	61	as	as	ADP
ejpam-3036	146	62	a	a	DET
ejpam-3036	146	63	permutable	permutable	ADJ
ejpam-3036	146	64	boolean	boolean	ADJ
ejpam-3036	146	65	sublattice	sublattice	NOUN
ejpam-3036	146	66	of	of	ADP
ejpam-3036	146	67	the	the	DET
ejpam-3036	146	68	lattice	lattice	PROPN
ejpam-3036	146	69	c(a	c(a	PROPN
ejpam-3036	146	70	)	)	PUNCT
ejpam-3036	146	71	of	of	ADP
ejpam-3036	146	72	all	all	DET
ejpam-3036	146	73	congruences	congruence	NOUN
ejpam-3036	146	74	on	on	ADP
ejpam-3036	146	75	a.	a.	NOUN
ejpam-3036	146	76	they	they	PRON
ejpam-3036	146	77	referred	refer	VERB
ejpam-3036	146	78	to	to	ADP
ejpam-3036	146	79	b(a	b(a	NOUN
ejpam-3036	146	80	)	)	PUNCT
ejpam-3036	146	81	as	as	ADP
ejpam-3036	146	82	the	the	DET
ejpam-3036	146	83	‘	'	PUNCT
ejpam-3036	146	84	boolean	boolean	ADJ
ejpam-3036	146	85	centre	centre	NOUN
ejpam-3036	146	86	’	'	PUNCT
ejpam-3036	146	87	of	of	ADP
ejpam-3036	146	88	a.	a.	NOUN
ejpam-3036	146	89	the	the	DET
ejpam-3036	146	90	main	main	ADJ
ejpam-3036	146	91	goal	goal	NOUN
ejpam-3036	146	92	of	of	ADP
ejpam-3036	146	93	this	this	DET
ejpam-3036	146	94	section	section	NOUN
ejpam-3036	146	95	is	be	AUX
ejpam-3036	146	96	to	to	PART
ejpam-3036	146	97	characterize	characterize	VERB
ejpam-3036	146	98	the	the	DET
ejpam-3036	146	99	boolean	boolean	ADJ
ejpam-3036	146	100	centre	centre	NOUN
ejpam-3036	146	101	of	of	ADP
ejpam-3036	146	102	a	a	DET
ejpam-3036	146	103	crdsa	crdsa	NOUN
ejpam-3036	146	104	a	a	PRON
ejpam-3036	146	105	in	in	ADP
ejpam-3036	146	106	terms	term	NOUN
ejpam-3036	146	107	of	of	ADP
ejpam-3036	146	108	central	central	ADJ
ejpam-3036	146	109	elements	element	NOUN
ejpam-3036	146	110	.	.	PUNCT
ejpam-3036	147	1	let	let	VERB
ejpam-3036	147	2	a	a	PRON
ejpam-3036	147	3	be	be	AUX
ejpam-3036	147	4	a	a	DET
ejpam-3036	147	5	core	core	NOUN
ejpam-3036	147	6	regular	regular	ADJ
ejpam-3036	147	7	double	double	ADJ
ejpam-3036	147	8	stone	stone	NOUN
ejpam-3036	147	9	algebra	algebra	NOUN
ejpam-3036	147	10	.	.	PUNCT
ejpam-3036	148	1	let	let	VERB
ejpam-3036	148	2	θx	θx	PRON
ejpam-3036	148	3	denote	denote	VERB
ejpam-3036	148	4	the	the	DET
ejpam-3036	148	5	equivalence	equivalence	NOUN
ejpam-3036	148	6	relation	relation	NOUN
ejpam-3036	148	7	associated	associate	VERB
ejpam-3036	148	8	to	to	ADP
ejpam-3036	148	9	the	the	DET
ejpam-3036	148	10	function	function	NOUN
ejpam-3036	148	11	x→	x→	PROPN
ejpam-3036	149	1	x∧p	x∧p	PROPN
ejpam-3036	150	1	from	from	ADP
ejpam-3036	150	2	a	a	PRON
ejpam-3036	150	3	to	to	ADP
ejpam-3036	150	4	itself	itself	PRON
ejpam-3036	150	5	:	:	PUNCT
ejpam-3036	150	6	θx	θx	X
ejpam-3036	150	7	=	=	SYM
ejpam-3036	150	8	{	{	PUNCT
ejpam-3036	150	9	(	(	PUNCT
ejpam-3036	150	10	p	p	X
ejpam-3036	150	11	,	,	PUNCT
ejpam-3036	150	12	q	q	NOUN
ejpam-3036	150	13	)	)	PUNCT
ejpam-3036	150	14	∈	∈	PROPN
ejpam-3036	150	15	a×a	a×a	PROPN
ejpam-3036	150	16	|x	|x	NOUN
ejpam-3036	150	17	∧	∧	NOUN
ejpam-3036	150	18	p	p	NOUN
ejpam-3036	150	19	=	=	NOUN
ejpam-3036	150	20	x	x	SYM
ejpam-3036	150	21	∧	∧	PROPN
ejpam-3036	150	22	q	q	NOUN
ejpam-3036	150	23	}	}	PUNCT
ejpam-3036	150	24	.	.	PUNCT
ejpam-3036	151	1	we	we	PRON
ejpam-3036	151	2	will	will	AUX
ejpam-3036	151	3	write	write	VERB
ejpam-3036	151	4	pθxq	pθxq	PROPN
ejpam-3036	151	5	to	to	PART
ejpam-3036	151	6	indicate	indicate	VERB
ejpam-3036	151	7	(	(	PUNCT
ejpam-3036	151	8	p	p	X
ejpam-3036	151	9	,	,	PUNCT
ejpam-3036	151	10	q	q	NOUN
ejpam-3036	151	11	)	)	PUNCT
ejpam-3036	151	12	∈	∈	PROPN
ejpam-3036	151	13	θx	θx	PROPN
ejpam-3036	151	14	.	.	PROPN
ejpam-3036	151	15	theorem	theorem	PROPN
ejpam-3036	151	16	3.1	3.1	NUM
ejpam-3036	151	17	.	.	PUNCT
ejpam-3036	152	1	let	let	VERB
ejpam-3036	152	2	a	a	PRON
ejpam-3036	152	3	be	be	AUX
ejpam-3036	152	4	a	a	DET
ejpam-3036	152	5	core	core	NOUN
ejpam-3036	152	6	regular	regular	ADJ
ejpam-3036	152	7	double	double	ADJ
ejpam-3036	152	8	stone	stone	NOUN
ejpam-3036	152	9	algebra	algebra	NOUN
ejpam-3036	152	10	and	and	CCONJ
ejpam-3036	152	11	x	x	NOUN
ejpam-3036	152	12	,	,	PUNCT
ejpam-3036	152	13	y	y	PROPN
ejpam-3036	152	14	∈	∈	PROPN
ejpam-3036	152	15	a	a	DET
ejpam-3036	152	16	then	then	ADV
ejpam-3036	152	17	(	(	PUNCT
ejpam-3036	152	18	i	i	NOUN
ejpam-3036	152	19	)	)	PUNCT
ejpam-3036	152	20	θy	θy	VERB
ejpam-3036	152	21	⊆	⊆	NUM
ejpam-3036	152	22	θx	θx	NOUN
ejpam-3036	152	23	if	if	SCONJ
ejpam-3036	152	24	and	and	CCONJ
ejpam-3036	152	25	only	only	ADV
ejpam-3036	152	26	if	if	SCONJ
ejpam-3036	152	27	x	x	NOUN
ejpam-3036	152	28	=	=	SYM
ejpam-3036	152	29	x	x	SYM
ejpam-3036	152	30	∧	∧	PROPN
ejpam-3036	152	31	y.	y.	PROPN
ejpam-3036	152	32	(	(	PUNCT
ejpam-3036	152	33	ii	ii	NOUN
ejpam-3036	152	34	)	)	PUNCT
ejpam-3036	152	35	θy	θy	PROPN
ejpam-3036	152	36	=	=	PUNCT
ejpam-3036	153	1	θx	θx	PRON
ejpam-3036	153	2	if	if	SCONJ
ejpam-3036	153	3	and	and	CCONJ
ejpam-3036	153	4	only	only	ADV
ejpam-3036	153	5	if	if	SCONJ
ejpam-3036	153	6	x	x	X
ejpam-3036	153	7	=	=	SYM
ejpam-3036	153	8	y.	y.	NOUN
ejpam-3036	153	9	(	(	PUNCT
ejpam-3036	153	10	iii	iii	NOUN
ejpam-3036	153	11	)	)	PUNCT
ejpam-3036	153	12	θx	θx	NOUN
ejpam-3036	153	13	is	be	AUX
ejpam-3036	153	14	compatible	compatible	ADJ
ejpam-3036	153	15	with	with	ADP
ejpam-3036	153	16	∧,∨	∧,∨	ADV
ejpam-3036	153	17	,	,	PUNCT
ejpam-3036	153	18	∗	∗	NOUN
ejpam-3036	153	19	(	(	PUNCT
ejpam-3036	153	20	iv	iv	X
ejpam-3036	153	21	)	)	PUNCT
ejpam-3036	153	22	θx	θx	NOUN
ejpam-3036	153	23	is	be	AUX
ejpam-3036	153	24	compatible	compatible	ADJ
ejpam-3036	153	25	with	with	ADP
ejpam-3036	153	26	+	+	CCONJ
ejpam-3036	153	27	if	if	SCONJ
ejpam-3036	153	28	and	and	CCONJ
ejpam-3036	153	29	only	only	ADV
ejpam-3036	153	30	if	if	SCONJ
ejpam-3036	153	31	x	x	PROPN
ejpam-3036	153	32	∈	∈	PROPN
ejpam-3036	153	33	k(a	k(a	PROPN
ejpam-3036	153	34	)	)	PUNCT
ejpam-3036	153	35	(	(	PUNCT
ejpam-3036	153	36	v	v	NOUN
ejpam-3036	153	37	)	)	PUNCT
ejpam-3036	153	38	θx	θx	ADP
ejpam-3036	153	39	congruence	congruence	NOUN
ejpam-3036	153	40	on	on	ADP
ejpam-3036	153	41	a	a	DET
ejpam-3036	153	42	if	if	NOUN
ejpam-3036	153	43	and	and	CCONJ
ejpam-3036	153	44	only	only	ADV
ejpam-3036	153	45	if	if	SCONJ
ejpam-3036	153	46	x	x	PROPN
ejpam-3036	153	47	∈	∈	PROPN
ejpam-3036	153	48	k(a	k(a	PROPN
ejpam-3036	153	49	)	)	PUNCT
ejpam-3036	153	50	.	.	PUNCT
ejpam-3036	154	1	(	(	PUNCT
ejpam-3036	154	2	vi	vi	NOUN
ejpam-3036	154	3	)	)	PUNCT
ejpam-3036	154	4	θ0	θ0	NOUN
ejpam-3036	154	5	=	=	SYM
ejpam-3036	154	6	a×a	a×a	PROPN
ejpam-3036	154	7	(	(	PUNCT
ejpam-3036	154	8	vii	vii	PROPN
ejpam-3036	154	9	)	)	PUNCT
ejpam-3036	154	10	θ1	θ1	NOUN
ejpam-3036	154	11	=	=	SYM
ejpam-3036	154	12	∆a	∆a	PROPN
ejpam-3036	154	13	(	(	PUNCT
ejpam-3036	154	14	viii	viii	PROPN
ejpam-3036	154	15	)	)	PUNCT
ejpam-3036	154	16	θx	θx	X
ejpam-3036	154	17	∩	∩	NOUN
ejpam-3036	154	18	θy	θy	X
ejpam-3036	154	19	=	=	SYM
ejpam-3036	154	20	θx∨y	θx∨y	X
ejpam-3036	154	21	(	(	PUNCT
ejpam-3036	154	22	ix	ix	PROPN
ejpam-3036	154	23	)	)	PUNCT
ejpam-3036	154	24	θx	θx	ADP
ejpam-3036	154	25	◦	◦	NOUN
ejpam-3036	154	26	θy	θy	ADP
ejpam-3036	154	27	=	=	NOUN
ejpam-3036	154	28	θy	θy	PART
ejpam-3036	154	29	◦	◦	VERB
ejpam-3036	154	30	θx	θx	NUM
ejpam-3036	154	31	(	(	PUNCT
ejpam-3036	154	32	x	x	X
ejpam-3036	154	33	)	)	PUNCT
ejpam-3036	154	34	θx	θx	ADP
ejpam-3036	154	35	◦	◦	NOUN
ejpam-3036	154	36	θy	θy	X
ejpam-3036	154	37	=	=	NOUN
ejpam-3036	154	38	θx∧y	θx∧y	NOUN
ejpam-3036	154	39	(	(	PUNCT
ejpam-3036	154	40	xi	xi	NOUN
ejpam-3036	154	41	)	)	PUNCT
ejpam-3036	154	42	θx	θx	ADP
ejpam-3036	154	43	◦	◦	NOUN
ejpam-3036	154	44	θx∗	θx∗	NOUN
ejpam-3036	154	45	=	=	NUM
ejpam-3036	154	46	θx∗	θx∗	NOUN
ejpam-3036	154	47	◦	◦	VERB
ejpam-3036	154	48	θx	θx	NOUN
ejpam-3036	154	49	=	=	SYM
ejpam-3036	154	50	a×a	a×a	PROPN
ejpam-3036	154	51	(	(	PUNCT
ejpam-3036	154	52	xii	xii	NOUN
ejpam-3036	154	53	)	)	PUNCT
ejpam-3036	154	54	θx∨x∗	θx∨x∗	NOUN
ejpam-3036	155	1	=	=	SYM
ejpam-3036	156	1	∆a	∆a	VERB
ejpam-3036	156	2	if	if	SCONJ
ejpam-3036	157	1	and	and	CCONJ
ejpam-3036	157	2	only	only	ADV
ejpam-3036	157	3	if	if	SCONJ
ejpam-3036	157	4	θx	θx	INTJ
ejpam-3036	157	5	is	be	AUX
ejpam-3036	157	6	a	a	DET
ejpam-3036	157	7	congruence	congruence	NOUN
ejpam-3036	157	8	relation	relation	NOUN
ejpam-3036	157	9	(	(	PUNCT
ejpam-3036	157	10	xiii	xiii	PROPN
ejpam-3036	157	11	)	)	PUNCT
ejpam-3036	157	12	for	for	ADP
ejpam-3036	157	13	x	x	PROPN
ejpam-3036	157	14	∈	∈	PROPN
ejpam-3036	157	15	k(a	k(a	PROPN
ejpam-3036	157	16	)	)	PUNCT
ejpam-3036	157	17	,	,	PUNCT
ejpam-3036	157	18	θx	θx	X
ejpam-3036	157	19	is	be	AUX
ejpam-3036	157	20	the	the	DET
ejpam-3036	157	21	smallest	small	ADJ
ejpam-3036	157	22	congruence	congruence	NOUN
ejpam-3036	157	23	containing	contain	VERB
ejpam-3036	157	24	(	(	PUNCT
ejpam-3036	157	25	1	1	NUM
ejpam-3036	157	26	,	,	PUNCT
ejpam-3036	157	27	x	x	NOUN
ejpam-3036	157	28	)	)	PUNCT
ejpam-3036	157	29	proof	proof	NOUN
ejpam-3036	157	30	.	.	PUNCT
ejpam-3036	158	1	(	(	PUNCT
ejpam-3036	158	2	i	i	NOUN
ejpam-3036	158	3	)	)	PUNCT
ejpam-3036	158	4	let	let	VERB
ejpam-3036	158	5	x	x	PRON
ejpam-3036	158	6	,	,	PUNCT
ejpam-3036	158	7	y	y	PROPN
ejpam-3036	158	8	∈	∈	PROPN
ejpam-3036	158	9	a	a	PRON
ejpam-3036	158	10	and	and	CCONJ
ejpam-3036	158	11	suppose	suppose	VERB
ejpam-3036	158	12	that	that	SCONJ
ejpam-3036	158	13	θy	θy	PRON
ejpam-3036	158	14	⊆	⊆	NUM
ejpam-3036	158	15	θx	θx	NOUN
ejpam-3036	158	16	.	.	PUNCT
ejpam-3036	159	1	since	since	SCONJ
ejpam-3036	159	2	y	y	PROPN
ejpam-3036	159	3	∧	∧	PROPN
ejpam-3036	159	4	(	(	PUNCT
ejpam-3036	159	5	x	x	PROPN
ejpam-3036	159	6	∨	∨	NUM
ejpam-3036	159	7	y	y	PROPN
ejpam-3036	159	8	)	)	PUNCT
ejpam-3036	159	9	=	=	SYM
ejpam-3036	159	10	y	y	PROPN
ejpam-3036	159	11	=	=	SYM
ejpam-3036	159	12	y	y	PROPN
ejpam-3036	159	13	∧	∧	PROPN
ejpam-3036	159	14	y	y	PROPN
ejpam-3036	159	15	we	we	PRON
ejpam-3036	159	16	have	have	VERB
ejpam-3036	159	17	(	(	PUNCT
ejpam-3036	159	18	y	y	NOUN
ejpam-3036	159	19	,	,	PUNCT
ejpam-3036	159	20	x∨	x∨	PROPN
ejpam-3036	159	21	y	y	PROPN
ejpam-3036	159	22	)	)	PUNCT
ejpam-3036	159	23	∈	∈	PROPN
ejpam-3036	159	24	θy	θy	ADP
ejpam-3036	159	25	,	,	PUNCT
ejpam-3036	159	26	by	by	ADP
ejpam-3036	159	27	our	our	PRON
ejpam-3036	159	28	supposition	supposition	NOUN
ejpam-3036	159	29	(	(	PUNCT
ejpam-3036	159	30	y	y	PROPN
ejpam-3036	159	31	,	,	PUNCT
ejpam-3036	159	32	x∨	x∨	PROPN
ejpam-3036	159	33	y	y	PROPN
ejpam-3036	159	34	)	)	PUNCT
ejpam-3036	159	35	∈	∈	PROPN
ejpam-3036	159	36	θx	θx	X
ejpam-3036	159	37	;	;	PUNCT
ejpam-3036	159	38	that	that	PRON
ejpam-3036	159	39	is	be	AUX
ejpam-3036	159	40	x∧	x∧	PROPN
ejpam-3036	159	41	y	y	PROPN
ejpam-3036	159	42	=	=	SYM
ejpam-3036	159	43	x∧	x∧	PROPN
ejpam-3036	159	44	(	(	PUNCT
ejpam-3036	159	45	x∨	x∨	PROPN
ejpam-3036	159	46	y	y	PROPN
ejpam-3036	159	47	)	)	PUNCT
ejpam-3036	159	48	or	or	CCONJ
ejpam-3036	159	49	x∧	x∧	PROPN
ejpam-3036	159	50	y	y	PROPN
ejpam-3036	159	51	=	=	PUNCT
ejpam-3036	159	52	x.	x.	NOUN
ejpam-3036	159	53	conversely	conversely	ADV
ejpam-3036	159	54	suppose	suppose	VERB
ejpam-3036	159	55	that	that	SCONJ
ejpam-3036	159	56	x	x	PUNCT
ejpam-3036	159	57	∧	∧	NOUN
ejpam-3036	159	58	y	y	NOUN
ejpam-3036	159	59	=	=	PUNCT
ejpam-3036	159	60	x.	x.	NOUN
ejpam-3036	159	61	let	let	VERB
ejpam-3036	159	62	(	(	PUNCT
ejpam-3036	159	63	p	p	X
ejpam-3036	159	64	,	,	PUNCT
ejpam-3036	159	65	q	q	NOUN
ejpam-3036	159	66	)	)	PUNCT
ejpam-3036	159	67	∈	∈	PROPN
ejpam-3036	159	68	θy	θy	NOUN
ejpam-3036	159	69	.	.	PUNCT
ejpam-3036	160	1	then	then	ADV
ejpam-3036	160	2	y	y	PROPN
ejpam-3036	160	3	∧	∧	PROPN
ejpam-3036	160	4	p	p	NOUN
ejpam-3036	160	5	=	=	PUNCT
ejpam-3036	160	6	y	y	PROPN
ejpam-3036	160	7	∧	∧	PROPN
ejpam-3036	160	8	q.	q.	VERB
ejpam-3036	160	9	now	now	ADV
ejpam-3036	160	10	,	,	PUNCT
ejpam-3036	160	11	x	x	PUNCT
ejpam-3036	160	12	∧	∧	NOUN
ejpam-3036	160	13	p	p	NOUN
ejpam-3036	160	14	=	=	X
ejpam-3036	160	15	(	(	PUNCT
ejpam-3036	160	16	x	x	PROPN
ejpam-3036	160	17	∧	∧	PROPN
ejpam-3036	160	18	y	y	NOUN
ejpam-3036	160	19	)	)	PUNCT
ejpam-3036	160	20	∧	∧	NOUN
ejpam-3036	160	21	p	p	NOUN
ejpam-3036	160	22	=	=	NOUN
ejpam-3036	160	23	x	x	SYM
ejpam-3036	160	24	∧	∧	PROPN
ejpam-3036	160	25	(	(	PUNCT
ejpam-3036	160	26	y	y	PROPN
ejpam-3036	160	27	∧	∧	PROPN
ejpam-3036	160	28	p	p	PROPN
ejpam-3036	160	29	)	)	PUNCT
ejpam-3036	160	30	a	a	DET
ejpam-3036	160	31	r	r	NOUN
ejpam-3036	160	32	j	j	PROPN
ejpam-3036	160	33	srikanth	srikanth	NOUN
ejpam-3036	160	34	,	,	PUNCT
ejpam-3036	160	35	r	r	NOUN
ejpam-3036	160	36	v	v	NUM
ejpam-3036	160	37	g	g	NOUN
ejpam-3036	160	38	ravi	ravi	PROPN
ejpam-3036	160	39	kumar	kumar	PROPN
ejpam-3036	160	40	/	/	SYM
ejpam-3036	160	41	eur	eur	PROPN
ejpam-3036	160	42	.	.	PUNCT
ejpam-3036	161	1	j.	j.	PROPN
ejpam-3036	161	2	pure	pure	PROPN
ejpam-3036	161	3	appl	appl	PROPN
ejpam-3036	161	4	.	.	PROPN
ejpam-3036	161	5	math	math	PROPN
ejpam-3036	161	6	,	,	PUNCT
ejpam-3036	161	7	10	10	NUM
ejpam-3036	161	8	(	(	PUNCT
ejpam-3036	161	9	4	4	NUM
ejpam-3036	161	10	)	)	PUNCT
ejpam-3036	161	11	(	(	PUNCT
ejpam-3036	161	12	2017	2017	NUM
ejpam-3036	161	13	)	)	PUNCT
ejpam-3036	161	14	,	,	PUNCT
ejpam-3036	161	15	717	717	NUM
ejpam-3036	161	16	-	-	SYM
ejpam-3036	161	17	729	729	NUM
ejpam-3036	161	18	724	724	NUM
ejpam-3036	161	19	=	=	NOUN
ejpam-3036	161	20	x	x	SYM
ejpam-3036	161	21	∧	∧	PROPN
ejpam-3036	161	22	(	(	PUNCT
ejpam-3036	161	23	y	y	PROPN
ejpam-3036	161	24	∧	∧	PROPN
ejpam-3036	161	25	q	q	PROPN
ejpam-3036	161	26	)	)	PUNCT
ejpam-3036	161	27	=	=	SYM
ejpam-3036	161	28	(	(	PUNCT
ejpam-3036	161	29	x	x	PUNCT
ejpam-3036	161	30	∧	∧	PROPN
ejpam-3036	161	31	y	y	NOUN
ejpam-3036	161	32	)	)	PUNCT
ejpam-3036	161	33	∧	∧	NOUN
ejpam-3036	161	34	q	q	NOUN
ejpam-3036	162	1	=	=	PUNCT
ejpam-3036	162	2	x	x	SYM
ejpam-3036	162	3	∧	∧	PROPN
ejpam-3036	162	4	q	q	X
ejpam-3036	162	5	therefore	therefore	ADV
ejpam-3036	162	6	,	,	PUNCT
ejpam-3036	162	7	(	(	PUNCT
ejpam-3036	162	8	p	p	X
ejpam-3036	162	9	,	,	PUNCT
ejpam-3036	162	10	q	q	NOUN
ejpam-3036	162	11	)	)	PUNCT
ejpam-3036	162	12	∈	∈	NOUN
ejpam-3036	162	13	θx	θx	X
ejpam-3036	163	1	and	and	CCONJ
ejpam-3036	163	2	hence	hence	ADV
ejpam-3036	163	3	θy	θy	VERB
ejpam-3036	163	4	⊆	⊆	NUM
ejpam-3036	163	5	θx	θx	NOUN
ejpam-3036	163	6	.	.	PROPN
ejpam-3036	163	7	(	(	PUNCT
ejpam-3036	163	8	ii	ii	PROPN
ejpam-3036	163	9	)	)	PUNCT
ejpam-3036	163	10	clear	clear	ADV
ejpam-3036	163	11	from	from	ADP
ejpam-3036	163	12	(	(	PUNCT
ejpam-3036	163	13	i	i	NOUN
ejpam-3036	163	14	)	)	PUNCT
ejpam-3036	163	15	.	.	PUNCT
ejpam-3036	164	1	(	(	PUNCT
ejpam-3036	164	2	iii	iii	X
ejpam-3036	164	3	)	)	PUNCT
ejpam-3036	164	4	if	if	SCONJ
ejpam-3036	164	5	p	p	X
ejpam-3036	164	6	,	,	PUNCT
ejpam-3036	164	7	q	q	ADJ
ejpam-3036	164	8	,	,	PUNCT
ejpam-3036	164	9	r	r	NOUN
ejpam-3036	164	10	,	,	PUNCT
ejpam-3036	164	11	s	s	NOUN
ejpam-3036	164	12	∈	∈	PROPN
ejpam-3036	164	13	a	a	DET
ejpam-3036	164	14	satisfy	satisfy	NOUN
ejpam-3036	164	15	(	(	PUNCT
ejpam-3036	164	16	p	p	X
ejpam-3036	164	17	,	,	PUNCT
ejpam-3036	164	18	q	q	NOUN
ejpam-3036	164	19	)	)	PUNCT
ejpam-3036	164	20	∈	∈	NOUN
ejpam-3036	164	21	θx	θx	X
ejpam-3036	164	22	and	and	CCONJ
ejpam-3036	164	23	(	(	PUNCT
ejpam-3036	164	24	r	r	NOUN
ejpam-3036	164	25	,	,	PUNCT
ejpam-3036	164	26	s	s	PART
ejpam-3036	164	27	)	)	PUNCT
ejpam-3036	164	28	∈	∈	PROPN
ejpam-3036	164	29	θx	θx	NOUN
ejpam-3036	164	30	.	.	PROPN
ejpam-3036	164	31	from	from	ADP
ejpam-3036	164	32	associativity	associativity	NOUN
ejpam-3036	164	33	and	and	CCONJ
ejpam-3036	164	34	distributivity	distributivity	NOUN
ejpam-3036	164	35	in	in	ADP
ejpam-3036	164	36	a	a	PRON
ejpam-3036	164	37	it	it	PRON
ejpam-3036	164	38	follows	follow	VERB
ejpam-3036	164	39	that	that	SCONJ
ejpam-3036	164	40	(	(	PUNCT
ejpam-3036	164	41	(	(	PUNCT
ejpam-3036	164	42	p	p	X
ejpam-3036	164	43	∧	∧	PROPN
ejpam-3036	164	44	r	r	NOUN
ejpam-3036	164	45	)	)	PUNCT
ejpam-3036	164	46	,	,	PUNCT
ejpam-3036	164	47	(	(	PUNCT
ejpam-3036	164	48	q	q	PROPN
ejpam-3036	164	49	∧	∧	PROPN
ejpam-3036	164	50	s	s	PART
ejpam-3036	164	51	)	)	PUNCT
ejpam-3036	164	52	)	)	PUNCT
ejpam-3036	165	1	∈	∈	NOUN
ejpam-3036	165	2	θx	θx	X
ejpam-3036	166	1	and	and	CCONJ
ejpam-3036	166	2	(	(	PUNCT
ejpam-3036	166	3	(	(	PUNCT
ejpam-3036	166	4	p	p	X
ejpam-3036	166	5	∨	∨	PROPN
ejpam-3036	166	6	r	r	NOUN
ejpam-3036	166	7	)	)	PUNCT
ejpam-3036	166	8	,	,	PUNCT
ejpam-3036	166	9	(	(	PUNCT
ejpam-3036	166	10	q	q	PROPN
ejpam-3036	166	11	∨	∨	NUM
ejpam-3036	166	12	s	s	NOUN
ejpam-3036	166	13	)	)	PUNCT
ejpam-3036	166	14	)	)	PUNCT
ejpam-3036	167	1	∈	∈	PROPN
ejpam-3036	167	2	θx	θx	PRON
ejpam-3036	167	3	.	.	PROPN
ejpam-3036	168	1	also	also	ADV
ejpam-3036	168	2	if	if	SCONJ
ejpam-3036	168	3	p	p	X
ejpam-3036	168	4	,	,	PUNCT
ejpam-3036	168	5	q	q	PROPN
ejpam-3036	168	6	∈	∈	PROPN
ejpam-3036	168	7	a	a	PRON
ejpam-3036	169	1	and	and	CCONJ
ejpam-3036	169	2	(	(	PUNCT
ejpam-3036	169	3	p	p	X
ejpam-3036	169	4	,	,	PUNCT
ejpam-3036	169	5	q	q	NOUN
ejpam-3036	169	6	)	)	PUNCT
ejpam-3036	169	7	∈	∈	PROPN
ejpam-3036	169	8	θx	θx	NOUN
ejpam-3036	169	9	,	,	PUNCT
ejpam-3036	169	10	it	it	PRON
ejpam-3036	169	11	follows	follow	VERB
ejpam-3036	169	12	that	that	SCONJ
ejpam-3036	169	13	(	(	PUNCT
ejpam-3036	169	14	x∗	x∗	PROPN
ejpam-3036	169	15	∨	∨	NOUN
ejpam-3036	169	16	p∗	p∗	PROPN
ejpam-3036	169	17	)	)	PUNCT
ejpam-3036	170	1	=	=	PUNCT
ejpam-3036	170	2	(	(	PUNCT
ejpam-3036	170	3	x∗	x∗	PROPN
ejpam-3036	170	4	∨	∨	NUM
ejpam-3036	170	5	q∗	q∗	PROPN
ejpam-3036	170	6	)	)	PUNCT
ejpam-3036	170	7	,	,	PUNCT
ejpam-3036	171	1	so	so	SCONJ
ejpam-3036	171	2	that	that	SCONJ
ejpam-3036	171	3	x∧	x∧	PROPN
ejpam-3036	171	4	(	(	PUNCT
ejpam-3036	171	5	x∗	x∗	PROPN
ejpam-3036	171	6	∨	∨	PROPN
ejpam-3036	171	7	p∗	p∗	PROPN
ejpam-3036	171	8	)	)	PUNCT
ejpam-3036	171	9	=	=	SYM
ejpam-3036	171	10	x∧	x∧	PROPN
ejpam-3036	171	11	(	(	PUNCT
ejpam-3036	171	12	x∗	x∗	PROPN
ejpam-3036	171	13	∨	∨	NUM
ejpam-3036	171	14	q∗	q∗	PROPN
ejpam-3036	171	15	)	)	PUNCT
ejpam-3036	171	16	using	use	VERB
ejpam-3036	171	17	distributivity	distributivity	NOUN
ejpam-3036	171	18	we	we	PRON
ejpam-3036	171	19	conclude	conclude	VERB
ejpam-3036	171	20	that	that	PRON
ejpam-3036	171	21	(	(	PUNCT
ejpam-3036	171	22	p∗	p∗	ADJ
ejpam-3036	171	23	,	,	PUNCT
ejpam-3036	171	24	q∗	q∗	NOUN
ejpam-3036	171	25	)	)	PUNCT
ejpam-3036	171	26	∈	∈	PROPN
ejpam-3036	171	27	θx	θx	NOUN
ejpam-3036	171	28	.	.	PROPN
ejpam-3036	171	29	(	(	PUNCT
ejpam-3036	171	30	iv	iv	X
ejpam-3036	171	31	)	)	PUNCT
ejpam-3036	171	32	suppose	suppose	VERB
ejpam-3036	171	33	that	that	SCONJ
ejpam-3036	171	34	θx	θx	PROPN
ejpam-3036	171	35	is	be	AUX
ejpam-3036	171	36	compatible	compatible	ADJ
ejpam-3036	171	37	with	with	ADP
ejpam-3036	171	38	+	+	PROPN
ejpam-3036	171	39	.	.	PUNCT
ejpam-3036	172	1	put	put	VERB
ejpam-3036	172	2	y	y	NOUN
ejpam-3036	172	3	=	=	PUNCT
ejpam-3036	172	4	x	x	PROPN
ejpam-3036	173	1	∨	∨	PROPN
ejpam-3036	173	2	k	k	X
ejpam-3036	173	3	then	then	ADV
ejpam-3036	173	4	y++	y++	VERB
ejpam-3036	173	5	=	=	PUNCT
ejpam-3036	173	6	x++	x++	PROPN
ejpam-3036	173	7	.	.	PUNCT
ejpam-3036	174	1	as	as	ADP
ejpam-3036	174	2	(	(	PUNCT
ejpam-3036	174	3	1	1	NUM
ejpam-3036	174	4	,	,	PUNCT
ejpam-3036	174	5	x	x	NOUN
ejpam-3036	174	6	)	)	PUNCT
ejpam-3036	174	7	and	and	CCONJ
ejpam-3036	174	8	(	(	PUNCT
ejpam-3036	174	9	k	k	X
ejpam-3036	174	10	,	,	PUNCT
ejpam-3036	174	11	k	k	NOUN
ejpam-3036	174	12	)	)	PUNCT
ejpam-3036	174	13	∈	∈	NOUN
ejpam-3036	174	14	θx	θx	X
ejpam-3036	174	15	which	which	PRON
ejpam-3036	174	16	gives	give	VERB
ejpam-3036	174	17	(	(	PUNCT
ejpam-3036	174	18	1	1	NUM
ejpam-3036	174	19	,	,	PUNCT
ejpam-3036	174	20	x	x	PROPN
ejpam-3036	174	21	∨	∨	NUM
ejpam-3036	174	22	k	k	NOUN
ejpam-3036	174	23	)	)	PUNCT
ejpam-3036	174	24	=	=	PUNCT
ejpam-3036	174	25	(	(	PUNCT
ejpam-3036	174	26	1	1	NUM
ejpam-3036	174	27	,	,	PUNCT
ejpam-3036	174	28	y	y	NOUN
ejpam-3036	174	29	)	)	PUNCT
ejpam-3036	174	30	∈	∈	PROPN
ejpam-3036	174	31	θx	θx	NOUN
ejpam-3036	174	32	.	.	PUNCT
ejpam-3036	175	1	since	since	SCONJ
ejpam-3036	175	2	θx	θx	PRON
ejpam-3036	175	3	is	be	AUX
ejpam-3036	175	4	compatible	compatible	ADJ
ejpam-3036	175	5	with	with	ADP
ejpam-3036	175	6	+	+	PROPN
ejpam-3036	175	7	,	,	PUNCT
ejpam-3036	175	8	we	we	PRON
ejpam-3036	175	9	get(1	get(1	VERB
ejpam-3036	175	10	,	,	PUNCT
ejpam-3036	175	11	y++	y++	NOUN
ejpam-3036	175	12	)	)	PUNCT
ejpam-3036	176	1	∈	∈	PROPN
ejpam-3036	176	2	θx	θx	NOUN
ejpam-3036	176	3	.	.	PUNCT
ejpam-3036	176	4	hence	hence	ADV
ejpam-3036	176	5	x	x	X
ejpam-3036	177	1	=	=	PUNCT
ejpam-3036	177	2	x	x	SYM
ejpam-3036	177	3	∧	∧	NOUN
ejpam-3036	177	4	y++	y++	NOUN
ejpam-3036	177	5	=	=	PUNCT
ejpam-3036	177	6	y++	y++	NOUN
ejpam-3036	177	7	.	.	PUNCT
ejpam-3036	178	1	therefor	therefor	ADP
ejpam-3036	178	2	x	x	PROPN
ejpam-3036	178	3	∈	∈	PROPN
ejpam-3036	178	4	k(a	k(a	PROPN
ejpam-3036	178	5	)	)	PUNCT
ejpam-3036	178	6	.	.	PUNCT
ejpam-3036	179	1	conversely	conversely	ADV
ejpam-3036	179	2	suppose	suppose	VERB
ejpam-3036	179	3	that	that	SCONJ
ejpam-3036	179	4	x	x	PROPN
ejpam-3036	179	5	∈	∈	PROPN
ejpam-3036	179	6	k(a	k(a	PROPN
ejpam-3036	179	7	)	)	PUNCT
ejpam-3036	179	8	then	then	ADV
ejpam-3036	179	9	x	x	X
ejpam-3036	179	10	=	=	X
ejpam-3036	179	11	y++	y++	NOUN
ejpam-3036	179	12	for	for	ADP
ejpam-3036	179	13	some	some	DET
ejpam-3036	179	14	y	y	PROPN
ejpam-3036	179	15	≥	≥	NUM
ejpam-3036	179	16	k	k	NOUN
ejpam-3036	179	17	,	,	PUNCT
ejpam-3036	179	18	if	if	SCONJ
ejpam-3036	179	19	p	p	X
ejpam-3036	179	20	,	,	PUNCT
ejpam-3036	179	21	q	q	PROPN
ejpam-3036	179	22	∈	∈	PROPN
ejpam-3036	179	23	a	a	DET
ejpam-3036	179	24	satisfy	satisfy	NOUN
ejpam-3036	179	25	(	(	PUNCT
ejpam-3036	179	26	p	p	X
ejpam-3036	179	27	,	,	PUNCT
ejpam-3036	179	28	q	q	NOUN
ejpam-3036	179	29	)	)	PUNCT
ejpam-3036	179	30	∈	∈	NOUN
ejpam-3036	179	31	θx	θx	NOUN
ejpam-3036	179	32	=	=	PUNCT
ejpam-3036	179	33	θy++	θy++	PROPN
ejpam-3036	179	34	then	then	ADV
ejpam-3036	179	35	y++	y++	VERB
ejpam-3036	179	36	∧	∧	PROPN
ejpam-3036	179	37	p	p	NOUN
ejpam-3036	179	38	=	=	PROPN
ejpam-3036	179	39	y++	y++	NOUN
ejpam-3036	179	40	∧	∧	PROPN
ejpam-3036	179	41	q	q	NOUN
ejpam-3036	179	42	and	and	CCONJ
ejpam-3036	179	43	hence	hence	ADV
ejpam-3036	179	44	y+	y+	NUM
ejpam-3036	179	45	∨	∨	PROPN
ejpam-3036	179	46	p+	p+	NOUN
ejpam-3036	179	47	=	=	SYM
ejpam-3036	179	48	y+	y+	NUM
ejpam-3036	179	49	∨	∨	NUM
ejpam-3036	179	50	q+	q+	ADV
ejpam-3036	179	51	,	,	PUNCT
ejpam-3036	179	52	it	it	PRON
ejpam-3036	179	53	follows	follow	VERB
ejpam-3036	179	54	that	that	SCONJ
ejpam-3036	179	55	y++	y++	NOUN
ejpam-3036	179	56	∧	∧	PROPN
ejpam-3036	179	57	(	(	PUNCT
ejpam-3036	179	58	y+	y+	PROPN
ejpam-3036	179	59	∨	∨	NUM
ejpam-3036	179	60	p+	p+	PROPN
ejpam-3036	179	61	)	)	PUNCT
ejpam-3036	179	62	=	=	PUNCT
ejpam-3036	179	63	y++	y++	NOUN
ejpam-3036	179	64	∧	∧	PROPN
ejpam-3036	179	65	(	(	PUNCT
ejpam-3036	179	66	y+	y+	NUM
ejpam-3036	179	67	∨	∨	NUM
ejpam-3036	179	68	q+	q+	ADP
ejpam-3036	179	69	)	)	PUNCT
ejpam-3036	179	70	which	which	PRON
ejpam-3036	179	71	gives	give	VERB
ejpam-3036	179	72	(	(	PUNCT
ejpam-3036	179	73	y++	y++	NOUN
ejpam-3036	179	74	∧	∧	PROPN
ejpam-3036	179	75	p+	p+	NOUN
ejpam-3036	179	76	)	)	PUNCT
ejpam-3036	179	77	=	=	SYM
ejpam-3036	179	78	(	(	PUNCT
ejpam-3036	179	79	y++	y++	NOUN
ejpam-3036	179	80	∧	∧	PROPN
ejpam-3036	179	81	q+	q+	ADP
ejpam-3036	179	82	)	)	PUNCT
ejpam-3036	179	83	hence	hence	ADV
ejpam-3036	179	84	(	(	PUNCT
ejpam-3036	179	85	p+	p+	NOUN
ejpam-3036	179	86	,	,	PUNCT
ejpam-3036	179	87	q+	q+	ADJ
ejpam-3036	179	88	)	)	PUNCT
ejpam-3036	179	89	∈	∈	PROPN
ejpam-3036	179	90	θy++	θy++	PUNCT
ejpam-3036	179	91	=	=	SYM
ejpam-3036	180	1	θx	θx	PROPN
ejpam-3036	180	2	.	.	PUNCT
ejpam-3036	180	3	(	(	PUNCT
ejpam-3036	180	4	v	v	NOUN
ejpam-3036	180	5	)	)	PUNCT
ejpam-3036	180	6	is	be	AUX
ejpam-3036	180	7	clear	clear	ADJ
ejpam-3036	180	8	from	from	ADP
ejpam-3036	180	9	(	(	PUNCT
ejpam-3036	180	10	iii	iii	NOUN
ejpam-3036	180	11	)	)	PUNCT
ejpam-3036	180	12	and	and	CCONJ
ejpam-3036	180	13	(	(	PUNCT
ejpam-3036	180	14	iv	iv	X
ejpam-3036	180	15	)	)	PUNCT
ejpam-3036	180	16	.	.	PUNCT
ejpam-3036	181	1	(	(	PUNCT
ejpam-3036	181	2	vi	vi	NOUN
ejpam-3036	181	3	)	)	PUNCT
ejpam-3036	181	4	and	and	CCONJ
ejpam-3036	181	5	(	(	PUNCT
ejpam-3036	181	6	vii	vii	PROPN
ejpam-3036	181	7	)	)	PUNCT
ejpam-3036	181	8	are	be	AUX
ejpam-3036	181	9	clear	clear	ADJ
ejpam-3036	181	10	from	from	ADP
ejpam-3036	181	11	the	the	DET
ejpam-3036	181	12	definition	definition	NOUN
ejpam-3036	181	13	of	of	ADP
ejpam-3036	181	14	θx	θx	PROPN
ejpam-3036	181	15	.	.	PROPN
ejpam-3036	182	1	(	(	PUNCT
ejpam-3036	182	2	viii	viii	NOUN
ejpam-3036	182	3	)	)	PUNCT
ejpam-3036	182	4	let	let	VERB
ejpam-3036	182	5	(	(	PUNCT
ejpam-3036	182	6	p	p	X
ejpam-3036	182	7	,	,	PUNCT
ejpam-3036	182	8	q	q	NOUN
ejpam-3036	182	9	)	)	PUNCT
ejpam-3036	182	10	∈	∈	NOUN
ejpam-3036	182	11	θx	θx	ADP
ejpam-3036	182	12	∩	∩	NOUN
ejpam-3036	182	13	θy	θy	PART
ejpam-3036	182	14	.	.	PUNCT
ejpam-3036	183	1	then	then	ADV
ejpam-3036	183	2	x	x	X
ejpam-3036	183	3	∧	∧	NOUN
ejpam-3036	183	4	p	p	NOUN
ejpam-3036	183	5	=	=	NOUN
ejpam-3036	183	6	x	x	SYM
ejpam-3036	183	7	∧	∧	PROPN
ejpam-3036	183	8	q	q	NOUN
ejpam-3036	183	9	and	and	CCONJ
ejpam-3036	183	10	y	y	PROPN
ejpam-3036	183	11	∧	∧	PROPN
ejpam-3036	183	12	p	p	NOUN
ejpam-3036	183	13	=	=	PUNCT
ejpam-3036	183	14	y	y	PROPN
ejpam-3036	183	15	∧	∧	PROPN
ejpam-3036	183	16	q.	q.	PROPN
ejpam-3036	183	17	now	now	ADV
ejpam-3036	183	18	,	,	PUNCT
ejpam-3036	183	19	(	(	PUNCT
ejpam-3036	183	20	x	x	PROPN
ejpam-3036	183	21	∨	∨	NUM
ejpam-3036	183	22	y	y	NOUN
ejpam-3036	183	23	)	)	PUNCT
ejpam-3036	183	24	∧	∧	NOUN
ejpam-3036	183	25	p	p	NOUN
ejpam-3036	183	26	=	=	X
ejpam-3036	183	27	(	(	PUNCT
ejpam-3036	183	28	x	x	PUNCT
ejpam-3036	183	29	∧	∧	NOUN
ejpam-3036	183	30	p	p	NOUN
ejpam-3036	183	31	)	)	PUNCT
ejpam-3036	183	32	∨	∨	PROPN
ejpam-3036	183	33	(	(	PUNCT
ejpam-3036	183	34	y	y	PROPN
ejpam-3036	183	35	∧	∧	PROPN
ejpam-3036	183	36	p	p	NOUN
ejpam-3036	183	37	)	)	PUNCT
ejpam-3036	183	38	=	=	SYM
ejpam-3036	184	1	(	(	PUNCT
ejpam-3036	184	2	x	x	PART
ejpam-3036	184	3	∧	∧	NOUN
ejpam-3036	184	4	q	q	NOUN
ejpam-3036	184	5	)	)	PUNCT
ejpam-3036	184	6	∨	∨	PROPN
ejpam-3036	184	7	(	(	PUNCT
ejpam-3036	184	8	y	y	PROPN
ejpam-3036	184	9	∧	∧	PROPN
ejpam-3036	184	10	q	q	PROPN
ejpam-3036	184	11	)	)	PUNCT
ejpam-3036	184	12	=	=	SYM
ejpam-3036	184	13	(	(	PUNCT
ejpam-3036	184	14	x	x	PROPN
ejpam-3036	184	15	∨	∨	NUM
ejpam-3036	184	16	y	y	NOUN
ejpam-3036	184	17	)	)	PUNCT
ejpam-3036	184	18	∧	∧	PROPN
ejpam-3036	184	19	q	q	X
ejpam-3036	185	1	therefore	therefore	ADV
ejpam-3036	185	2	(	(	PUNCT
ejpam-3036	185	3	p	p	X
ejpam-3036	185	4	,	,	PUNCT
ejpam-3036	185	5	q	q	NOUN
ejpam-3036	185	6	)	)	PUNCT
ejpam-3036	185	7	∈	∈	PROPN
ejpam-3036	185	8	θx∨y	θx∨y	NOUN
ejpam-3036	185	9	.	.	PROPN
ejpam-3036	185	10	hence	hence	ADV
ejpam-3036	185	11	θx	θx	ADP
ejpam-3036	185	12	∩	∩	NOUN
ejpam-3036	185	13	θy	θy	ADP
ejpam-3036	185	14	⊆	⊆	NUM
ejpam-3036	185	15	θx∨y	θx∨y	NOUN
ejpam-3036	185	16	.	.	PUNCT
ejpam-3036	185	17	conversely	conversely	ADV
ejpam-3036	185	18	suppose	suppose	VERB
ejpam-3036	185	19	that	that	SCONJ
ejpam-3036	185	20	(	(	PUNCT
ejpam-3036	185	21	p	p	X
ejpam-3036	185	22	,	,	PUNCT
ejpam-3036	185	23	q	q	NOUN
ejpam-3036	185	24	)	)	PUNCT
ejpam-3036	185	25	∈	∈	NOUN
ejpam-3036	186	1	θx∨y	θx∨y	ADP
ejpam-3036	186	2	then	then	ADV
ejpam-3036	186	3	(	(	PUNCT
ejpam-3036	186	4	x	x	PROPN
ejpam-3036	186	5	∨	∨	NUM
ejpam-3036	186	6	y	y	NOUN
ejpam-3036	186	7	)	)	PUNCT
ejpam-3036	186	8	∧	∧	NOUN
ejpam-3036	186	9	p	p	NOUN
ejpam-3036	186	10	=	=	X
ejpam-3036	186	11	(	(	PUNCT
ejpam-3036	186	12	x	x	PROPN
ejpam-3036	186	13	∨	∨	NUM
ejpam-3036	186	14	y	y	NOUN
ejpam-3036	186	15	)	)	PUNCT
ejpam-3036	186	16	∧	∧	PROPN
ejpam-3036	186	17	q.	q.	NOUN
ejpam-3036	186	18	now	now	ADV
ejpam-3036	186	19	,	,	PUNCT
ejpam-3036	186	20	x	x	PART
ejpam-3036	186	21	∧	∧	NOUN
ejpam-3036	186	22	(	(	PUNCT
ejpam-3036	186	23	(	(	PUNCT
ejpam-3036	186	24	x	x	PROPN
ejpam-3036	186	25	∨	∨	NUM
ejpam-3036	186	26	y	y	NOUN
ejpam-3036	186	27	)	)	PUNCT
ejpam-3036	186	28	∧	∧	PROPN
ejpam-3036	186	29	p	p	NOUN
ejpam-3036	186	30	)	)	PUNCT
ejpam-3036	186	31	=	=	PUNCT
ejpam-3036	186	32	x	x	SYM
ejpam-3036	186	33	∧	∧	PROPN
ejpam-3036	186	34	(	(	PUNCT
ejpam-3036	186	35	(	(	PUNCT
ejpam-3036	186	36	x	x	PROPN
ejpam-3036	186	37	∨	∨	NUM
ejpam-3036	186	38	y	y	NOUN
ejpam-3036	186	39	)	)	PUNCT
ejpam-3036	186	40	∧	∧	PROPN
ejpam-3036	186	41	q	q	NOUN
ejpam-3036	186	42	)	)	PUNCT
ejpam-3036	186	43	(	(	PUNCT
ejpam-3036	186	44	x	x	SYM
ejpam-3036	186	45	∧	∧	NOUN
ejpam-3036	186	46	(	(	PUNCT
ejpam-3036	186	47	x	x	PROPN
ejpam-3036	186	48	∨	∨	NUM
ejpam-3036	186	49	y	y	PROPN
ejpam-3036	186	50	)	)	PUNCT
ejpam-3036	186	51	)	)	PUNCT
ejpam-3036	187	1	∧	∧	NOUN
ejpam-3036	187	2	p	p	NOUN
ejpam-3036	187	3	=	=	X
ejpam-3036	187	4	(	(	PUNCT
ejpam-3036	187	5	x	x	PART
ejpam-3036	187	6	∧	∧	NOUN
ejpam-3036	187	7	(	(	PUNCT
ejpam-3036	187	8	x	x	PROPN
ejpam-3036	187	9	∨	∨	NUM
ejpam-3036	187	10	y	y	PROPN
ejpam-3036	187	11	)	)	PUNCT
ejpam-3036	187	12	)	)	PUNCT
ejpam-3036	188	1	∧	∧	PROPN
ejpam-3036	188	2	q	q	NOUN
ejpam-3036	188	3	)	)	PUNCT
ejpam-3036	188	4	x	x	SYM
ejpam-3036	189	1	∧	∧	NOUN
ejpam-3036	189	2	p	p	NOUN
ejpam-3036	189	3	=	=	NOUN
ejpam-3036	189	4	x	x	PUNCT
ejpam-3036	189	5	∧	∧	PROPN
ejpam-3036	189	6	q	q	NOUN
ejpam-3036	189	7	a	a	DET
ejpam-3036	189	8	r	r	NOUN
ejpam-3036	189	9	j	j	PROPN
ejpam-3036	189	10	srikanth	srikanth	NOUN
ejpam-3036	189	11	,	,	PUNCT
ejpam-3036	189	12	r	r	NOUN
ejpam-3036	189	13	v	v	NUM
ejpam-3036	189	14	g	g	NOUN
ejpam-3036	189	15	ravi	ravi	PROPN
ejpam-3036	189	16	kumar	kumar	PROPN
ejpam-3036	189	17	/	/	SYM
ejpam-3036	189	18	eur	eur	PROPN
ejpam-3036	189	19	.	.	PUNCT
ejpam-3036	190	1	j.	j.	PROPN
ejpam-3036	190	2	pure	pure	PROPN
ejpam-3036	190	3	appl	appl	PROPN
ejpam-3036	190	4	.	.	PROPN
ejpam-3036	190	5	math	math	PROPN
ejpam-3036	190	6	,	,	PUNCT
ejpam-3036	190	7	10	10	NUM
ejpam-3036	190	8	(	(	PUNCT
ejpam-3036	190	9	4	4	NUM
ejpam-3036	190	10	)	)	PUNCT
ejpam-3036	190	11	(	(	PUNCT
ejpam-3036	190	12	2017	2017	NUM
ejpam-3036	190	13	)	)	PUNCT
ejpam-3036	190	14	,	,	PUNCT
ejpam-3036	190	15	717	717	NUM
ejpam-3036	190	16	-	-	SYM
ejpam-3036	190	17	729	729	NUM
ejpam-3036	190	18	725	725	NUM
ejpam-3036	190	19	therefore	therefore	ADV
ejpam-3036	190	20	(	(	PUNCT
ejpam-3036	190	21	p	p	X
ejpam-3036	190	22	,	,	PUNCT
ejpam-3036	190	23	q	q	NOUN
ejpam-3036	190	24	)	)	PUNCT
ejpam-3036	190	25	∈	∈	PROPN
ejpam-3036	190	26	θx	θx	NOUN
ejpam-3036	190	27	,	,	PUNCT
ejpam-3036	190	28	similarly	similarly	ADV
ejpam-3036	190	29	it	it	PRON
ejpam-3036	190	30	can	can	AUX
ejpam-3036	190	31	be	be	AUX
ejpam-3036	190	32	shown	show	VERB
ejpam-3036	190	33	that	that	SCONJ
ejpam-3036	190	34	(	(	PUNCT
ejpam-3036	190	35	p	p	X
ejpam-3036	190	36	,	,	PUNCT
ejpam-3036	190	37	q	q	NOUN
ejpam-3036	190	38	)	)	PUNCT
ejpam-3036	190	39	∈	∈	PROPN
ejpam-3036	190	40	θy	θy	NOUN
ejpam-3036	190	41	.	.	PUNCT
ejpam-3036	191	1	so	so	ADV
ejpam-3036	191	2	(	(	PUNCT
ejpam-3036	191	3	p	p	X
ejpam-3036	191	4	,	,	PUNCT
ejpam-3036	191	5	q	q	NOUN
ejpam-3036	191	6	)	)	PUNCT
ejpam-3036	191	7	∈	∈	NOUN
ejpam-3036	191	8	θx	θx	X
ejpam-3036	191	9	∩	∩	NOUN
ejpam-3036	191	10	θy	θy	X
ejpam-3036	191	11	and	and	CCONJ
ejpam-3036	191	12	hence	hence	ADV
ejpam-3036	191	13	θx∨y	θx∨y	X
ejpam-3036	191	14	⊆	⊆	NUM
ejpam-3036	191	15	θx	θx	ADP
ejpam-3036	191	16	∩	∩	NOUN
ejpam-3036	191	17	θy	θy	PART
ejpam-3036	191	18	.	.	PUNCT
ejpam-3036	192	1	therefore	therefore	ADV
ejpam-3036	192	2	θx	θx	X
ejpam-3036	192	3	∩	∩	NOUN
ejpam-3036	192	4	θy	θy	X
ejpam-3036	192	5	=	=	SYM
ejpam-3036	192	6	θx∨y	θx∨y	X
ejpam-3036	192	7	.	.	PROPN
ejpam-3036	192	8	(	(	PUNCT
ejpam-3036	192	9	ix	ix	ADV
ejpam-3036	192	10	)	)	PUNCT
ejpam-3036	192	11	let	let	VERB
ejpam-3036	192	12	(	(	PUNCT
ejpam-3036	192	13	p	p	X
ejpam-3036	192	14	,	,	PUNCT
ejpam-3036	192	15	r	r	NOUN
ejpam-3036	192	16	)	)	PUNCT
ejpam-3036	192	17	∈	∈	NOUN
ejpam-3036	192	18	θx	θx	ADP
ejpam-3036	192	19	◦	◦	NOUN
ejpam-3036	192	20	θy	θy	PRON
ejpam-3036	192	21	.	.	PUNCT
ejpam-3036	193	1	then	then	ADV
ejpam-3036	193	2	there	there	PRON
ejpam-3036	193	3	exists	exist	VERB
ejpam-3036	193	4	q	q	PROPN
ejpam-3036	193	5	∈	∈	PROPN
ejpam-3036	193	6	a	a	DET
ejpam-3036	193	7	such	such	ADJ
ejpam-3036	193	8	that	that	SCONJ
ejpam-3036	193	9	(	(	PUNCT
ejpam-3036	193	10	p	p	X
ejpam-3036	193	11	,	,	PUNCT
ejpam-3036	193	12	q	q	NOUN
ejpam-3036	193	13	)	)	PUNCT
ejpam-3036	193	14	∈	∈	NOUN
ejpam-3036	193	15	θx	θx	X
ejpam-3036	194	1	and	and	CCONJ
ejpam-3036	194	2	(	(	PUNCT
ejpam-3036	194	3	q	q	ADJ
ejpam-3036	194	4	,	,	PUNCT
ejpam-3036	194	5	r	r	NOUN
ejpam-3036	194	6	)	)	PUNCT
ejpam-3036	194	7	∈	∈	PROPN
ejpam-3036	194	8	θy	θy	NOUN
ejpam-3036	194	9	.	.	PUNCT
ejpam-3036	195	1	so	so	ADV
ejpam-3036	195	2	x	x	PUNCT
ejpam-3036	195	3	∧	∧	NOUN
ejpam-3036	195	4	p	p	NOUN
ejpam-3036	196	1	=	=	NOUN
ejpam-3036	196	2	x	x	SYM
ejpam-3036	196	3	∧	∧	PROPN
ejpam-3036	196	4	q	q	NOUN
ejpam-3036	196	5	and	and	CCONJ
ejpam-3036	196	6	y	y	PROPN
ejpam-3036	196	7	∧	∧	PROPN
ejpam-3036	196	8	q	q	PROPN
ejpam-3036	196	9	=	=	SYM
ejpam-3036	196	10	y	y	PROPN
ejpam-3036	196	11	∧	∧	PROPN
ejpam-3036	196	12	r.	r.	PROPN
ejpam-3036	196	13	put	put	VERB
ejpam-3036	196	14	t	t	PROPN
ejpam-3036	196	15	=	=	SYM
ejpam-3036	196	16	(	(	PUNCT
ejpam-3036	196	17	x	x	PUNCT
ejpam-3036	196	18	∧	∧	NOUN
ejpam-3036	196	19	r	r	NOUN
ejpam-3036	196	20	)	)	PUNCT
ejpam-3036	196	21	∨	∨	NOUN
ejpam-3036	196	22	(	(	PUNCT
ejpam-3036	196	23	y	y	PROPN
ejpam-3036	196	24	∧	∧	PROPN
ejpam-3036	196	25	p	p	PROPN
ejpam-3036	196	26	)	)	PUNCT
ejpam-3036	196	27	.	.	PUNCT
ejpam-3036	197	1	then	then	ADV
ejpam-3036	197	2	,	,	PUNCT
ejpam-3036	197	3	x	x	PUNCT
ejpam-3036	197	4	∧	∧	NOUN
ejpam-3036	197	5	t	t	NOUN
ejpam-3036	197	6	=	=	SYM
ejpam-3036	197	7	x	x	SYM
ejpam-3036	197	8	∧	∧	PROPN
ejpam-3036	197	9	(	(	PUNCT
ejpam-3036	197	10	(	(	PUNCT
ejpam-3036	197	11	x	x	SYM
ejpam-3036	197	12	∧	∧	NOUN
ejpam-3036	197	13	r	r	NOUN
ejpam-3036	197	14	)	)	PUNCT
ejpam-3036	197	15	∨	∨	NOUN
ejpam-3036	197	16	(	(	PUNCT
ejpam-3036	197	17	y	y	PROPN
ejpam-3036	197	18	∧	∧	PROPN
ejpam-3036	197	19	p	p	NOUN
ejpam-3036	197	20	)	)	PUNCT
ejpam-3036	197	21	)	)	PUNCT
ejpam-3036	197	22	=	=	SYM
ejpam-3036	198	1	(	(	PUNCT
ejpam-3036	198	2	x	x	PUNCT
ejpam-3036	198	3	∧	∧	NOUN
ejpam-3036	198	4	r	r	NOUN
ejpam-3036	198	5	)	)	PUNCT
ejpam-3036	198	6	∨	∨	NOUN
ejpam-3036	198	7	(	(	PUNCT
ejpam-3036	198	8	x	x	SYM
ejpam-3036	198	9	∧	∧	PROPN
ejpam-3036	198	10	(	(	PUNCT
ejpam-3036	198	11	y	y	PROPN
ejpam-3036	198	12	∧	∧	PROPN
ejpam-3036	198	13	p	p	NOUN
ejpam-3036	198	14	)	)	PUNCT
ejpam-3036	198	15	)	)	PUNCT
ejpam-3036	198	16	=	=	SYM
ejpam-3036	199	1	(	(	PUNCT
ejpam-3036	199	2	x	x	PUNCT
ejpam-3036	199	3	∧	∧	NOUN
ejpam-3036	199	4	r	r	NOUN
ejpam-3036	199	5	)	)	PUNCT
ejpam-3036	199	6	∨	∨	NOUN
ejpam-3036	199	7	x	x	SYM
ejpam-3036	199	8	∧	∧	PROPN
ejpam-3036	199	9	(	(	PUNCT
ejpam-3036	199	10	y	y	PROPN
ejpam-3036	199	11	∧	∧	PROPN
ejpam-3036	199	12	p	p	NOUN
ejpam-3036	199	13	)	)	PUNCT
ejpam-3036	199	14	=	=	SYM
ejpam-3036	199	15	(	(	PUNCT
ejpam-3036	199	16	x	x	PUNCT
ejpam-3036	199	17	∧	∧	NOUN
ejpam-3036	199	18	r	r	NOUN
ejpam-3036	199	19	)	)	PUNCT
ejpam-3036	199	20	∨	∨	NOUN
ejpam-3036	199	21	(	(	PUNCT
ejpam-3036	199	22	x	x	X
ejpam-3036	199	23	∧	∧	NOUN
ejpam-3036	199	24	p	p	NOUN
ejpam-3036	199	25	∧	∧	PROPN
ejpam-3036	199	26	p	p	NOUN
ejpam-3036	199	27	)	)	PUNCT
ejpam-3036	199	28	=	=	SYM
ejpam-3036	199	29	(	(	PUNCT
ejpam-3036	199	30	x	x	PUNCT
ejpam-3036	199	31	∧	∧	NOUN
ejpam-3036	199	32	r	r	NOUN
ejpam-3036	199	33	)	)	PUNCT
ejpam-3036	199	34	∨	∨	NOUN
ejpam-3036	199	35	(	(	PUNCT
ejpam-3036	199	36	x	x	SYM
ejpam-3036	199	37	∧	∧	NOUN
ejpam-3036	199	38	q	q	PROPN
ejpam-3036	199	39	∧	∧	PROPN
ejpam-3036	199	40	y	y	PROPN
ejpam-3036	199	41	)	)	PUNCT
ejpam-3036	199	42	,	,	PUNCT
ejpam-3036	199	43	since	since	SCONJ
ejpam-3036	199	44	x	x	X
ejpam-3036	199	45	∧	∧	NOUN
ejpam-3036	199	46	p	p	NOUN
ejpam-3036	199	47	=	=	NOUN
ejpam-3036	199	48	x	x	SYM
ejpam-3036	199	49	∧	∧	NOUN
ejpam-3036	199	50	q	q	NOUN
ejpam-3036	199	51	=	=	PUNCT
ejpam-3036	199	52	(	(	PUNCT
ejpam-3036	199	53	x	x	PUNCT
ejpam-3036	199	54	∧	∧	NOUN
ejpam-3036	199	55	r	r	NOUN
ejpam-3036	199	56	)	)	PUNCT
ejpam-3036	199	57	∨	∨	NOUN
ejpam-3036	199	58	(	(	PUNCT
ejpam-3036	199	59	x	x	SYM
ejpam-3036	199	60	∧	∧	NOUN
ejpam-3036	199	61	y	y	PROPN
ejpam-3036	199	62	∧	∧	PROPN
ejpam-3036	199	63	r	r	PROPN
ejpam-3036	199	64	)	)	PUNCT
ejpam-3036	199	65	,	,	PUNCT
ejpam-3036	199	66	since	since	SCONJ
ejpam-3036	199	67	q	q	PROPN
ejpam-3036	199	68	∧	∧	PROPN
ejpam-3036	199	69	y	y	PROPN
ejpam-3036	199	70	=	=	SYM
ejpam-3036	199	71	y	y	PROPN
ejpam-3036	199	72	∧	∧	PROPN
ejpam-3036	199	73	q	q	PROPN
ejpam-3036	199	74	=	=	PUNCT
ejpam-3036	199	75	y	y	NOUN
ejpam-3036	199	76	∧	∧	PROPN
ejpam-3036	199	77	r	r	NOUN
ejpam-3036	199	78	=	=	NOUN
ejpam-3036	199	79	x	x	SYM
ejpam-3036	199	80	∧	∧	NOUN
ejpam-3036	199	81	r	r	NOUN
ejpam-3036	199	82	hence	hence	ADV
ejpam-3036	199	83	(	(	PUNCT
ejpam-3036	199	84	t	t	PROPN
ejpam-3036	199	85	,	,	PUNCT
ejpam-3036	199	86	r	r	NOUN
ejpam-3036	199	87	)	)	PUNCT
ejpam-3036	199	88	∈	∈	PROPN
ejpam-3036	199	89	θx	θx	NOUN
ejpam-3036	199	90	.	.	PROPN
ejpam-3036	199	91	similarly	similarly	ADV
ejpam-3036	199	92	it	it	PRON
ejpam-3036	199	93	can	can	AUX
ejpam-3036	199	94	be	be	AUX
ejpam-3036	199	95	shown	show	VERB
ejpam-3036	199	96	that	that	SCONJ
ejpam-3036	199	97	y	y	PROPN
ejpam-3036	199	98	∧	∧	PROPN
ejpam-3036	199	99	t	t	PROPN
ejpam-3036	199	100	=	=	SYM
ejpam-3036	199	101	y	y	PROPN
ejpam-3036	199	102	∧	∧	PROPN
ejpam-3036	199	103	p	p	PROPN
ejpam-3036	199	104	which	which	PRON
ejpam-3036	199	105	gives	give	VERB
ejpam-3036	199	106	(	(	PUNCT
ejpam-3036	199	107	p	p	X
ejpam-3036	199	108	,	,	PUNCT
ejpam-3036	199	109	t	t	PROPN
ejpam-3036	199	110	)	)	PUNCT
ejpam-3036	199	111	∈	∈	PROPN
ejpam-3036	199	112	θy	θy	PROPN
ejpam-3036	199	113	.	.	PUNCT
ejpam-3036	200	1	therefore	therefore	ADV
ejpam-3036	200	2	(	(	PUNCT
ejpam-3036	200	3	p	p	X
ejpam-3036	200	4	,	,	PUNCT
ejpam-3036	200	5	r	r	NOUN
ejpam-3036	200	6	)	)	PUNCT
ejpam-3036	200	7	∈	∈	NOUN
ejpam-3036	200	8	θy	θy	AUX
ejpam-3036	200	9	◦	◦	NOUN
ejpam-3036	200	10	θx	θx	PRON
ejpam-3036	200	11	i.e.	i.e.	X
ejpam-3036	200	12	θx	θx	ADP
ejpam-3036	200	13	◦	◦	VERB
ejpam-3036	200	14	θy	θy	PRON
ejpam-3036	200	15	⊆	⊆	NUM
ejpam-3036	200	16	θy	θy	NOUN
ejpam-3036	200	17	◦	◦	VERB
ejpam-3036	200	18	θx	θx	NOUN
ejpam-3036	200	19	.	.	PUNCT
ejpam-3036	201	1	conversely	conversely	ADV
ejpam-3036	201	2	suppose	suppose	VERB
ejpam-3036	201	3	that	that	SCONJ
ejpam-3036	201	4	(	(	PUNCT
ejpam-3036	201	5	p	p	X
ejpam-3036	201	6	,	,	PUNCT
ejpam-3036	201	7	r	r	NOUN
ejpam-3036	201	8	)	)	PUNCT
ejpam-3036	201	9	∈	∈	NOUN
ejpam-3036	201	10	θy	θy	AUX
ejpam-3036	201	11	◦	◦	NOUN
ejpam-3036	201	12	θx	θx	PRON
ejpam-3036	201	13	.	.	PUNCT
ejpam-3036	201	14	now	now	ADV
ejpam-3036	201	15	by	by	ADP
ejpam-3036	201	16	setting	set	VERB
ejpam-3036	201	17	t	t	NOUN
ejpam-3036	201	18	=	=	SYM
ejpam-3036	201	19	(	(	PUNCT
ejpam-3036	201	20	y	y	PROPN
ejpam-3036	201	21	∧	∧	PROPN
ejpam-3036	201	22	r	r	NOUN
ejpam-3036	201	23	)	)	PUNCT
ejpam-3036	201	24	∨	∨	NOUN
ejpam-3036	201	25	(	(	PUNCT
ejpam-3036	201	26	x	x	PROPN
ejpam-3036	201	27	∧	∧	PROPN
ejpam-3036	201	28	p	p	NOUN
ejpam-3036	201	29	)	)	PUNCT
ejpam-3036	201	30	and	and	CCONJ
ejpam-3036	201	31	proceeding	proceed	VERB
ejpam-3036	201	32	as	as	ADP
ejpam-3036	201	33	above	above	ADV
ejpam-3036	201	34	it	it	PRON
ejpam-3036	201	35	can	can	AUX
ejpam-3036	201	36	be	be	AUX
ejpam-3036	201	37	shown	show	VERB
ejpam-3036	201	38	that	that	SCONJ
ejpam-3036	201	39	θy	θy	PART
ejpam-3036	201	40	◦	◦	VERB
ejpam-3036	201	41	θx	θx	PRON
ejpam-3036	201	42	⊆	⊆	NUM
ejpam-3036	201	43	θx	θx	ADP
ejpam-3036	201	44	◦	◦	NOUN
ejpam-3036	201	45	θy	θy	PRON
ejpam-3036	201	46	.	.	PUNCT
ejpam-3036	202	1	finally	finally	ADV
ejpam-3036	202	2	it	it	PRON
ejpam-3036	202	3	gives	give	VERB
ejpam-3036	202	4	θx	θx	PRON
ejpam-3036	202	5	◦	◦	VERB
ejpam-3036	202	6	θy	θy	ADJ
ejpam-3036	203	1	=	=	NOUN
ejpam-3036	203	2	θy	θy	AUX
ejpam-3036	203	3	◦	◦	VERB
ejpam-3036	203	4	θx	θx	PRON
ejpam-3036	203	5	.	.	PUNCT
ejpam-3036	204	1	(	(	PUNCT
ejpam-3036	204	2	x	x	X
ejpam-3036	204	3	)	)	PUNCT
ejpam-3036	204	4	let	let	VERB
ejpam-3036	204	5	(	(	PUNCT
ejpam-3036	204	6	p	p	X
ejpam-3036	204	7	,	,	PUNCT
ejpam-3036	204	8	r	r	NOUN
ejpam-3036	204	9	)	)	PUNCT
ejpam-3036	204	10	∈	∈	NOUN
ejpam-3036	204	11	θx	θx	ADP
ejpam-3036	204	12	◦	◦	NOUN
ejpam-3036	204	13	θy	θy	PRON
ejpam-3036	204	14	.	.	PUNCT
ejpam-3036	205	1	then	then	ADV
ejpam-3036	205	2	there	there	PRON
ejpam-3036	205	3	exists	exist	VERB
ejpam-3036	205	4	q	q	PROPN
ejpam-3036	205	5	∈	∈	PROPN
ejpam-3036	205	6	a	a	DET
ejpam-3036	205	7	such	such	ADJ
ejpam-3036	205	8	that	that	SCONJ
ejpam-3036	205	9	(	(	PUNCT
ejpam-3036	205	10	p	p	X
ejpam-3036	205	11	,	,	PUNCT
ejpam-3036	205	12	q	q	NOUN
ejpam-3036	205	13	)	)	PUNCT
ejpam-3036	205	14	∈	∈	NOUN
ejpam-3036	205	15	θx	θx	X
ejpam-3036	206	1	and	and	CCONJ
ejpam-3036	206	2	(	(	PUNCT
ejpam-3036	206	3	q	q	ADJ
ejpam-3036	206	4	,	,	PUNCT
ejpam-3036	206	5	r	r	NOUN
ejpam-3036	206	6	)	)	PUNCT
ejpam-3036	206	7	∈	∈	PROPN
ejpam-3036	206	8	θy	θy	NOUN
ejpam-3036	206	9	.	.	PUNCT
ejpam-3036	207	1	so	so	ADV
ejpam-3036	207	2	x	x	PUNCT
ejpam-3036	207	3	∧	∧	NOUN
ejpam-3036	207	4	p	p	NOUN
ejpam-3036	208	1	=	=	NOUN
ejpam-3036	208	2	x	x	SYM
ejpam-3036	208	3	∧	∧	PROPN
ejpam-3036	208	4	q	q	NOUN
ejpam-3036	208	5	and	and	CCONJ
ejpam-3036	208	6	y	y	PROPN
ejpam-3036	208	7	∧	∧	PROPN
ejpam-3036	208	8	q	q	PROPN
ejpam-3036	208	9	=	=	PUNCT
ejpam-3036	208	10	y	y	PROPN
ejpam-3036	208	11	∧	∧	PROPN
ejpam-3036	208	12	r	r	NOUN
ejpam-3036	208	13	now	now	ADV
ejpam-3036	208	14	,	,	PUNCT
ejpam-3036	208	15	(	(	PUNCT
ejpam-3036	208	16	x	x	PUNCT
ejpam-3036	208	17	∧	∧	PROPN
ejpam-3036	208	18	y	y	NOUN
ejpam-3036	208	19	)	)	PUNCT
ejpam-3036	208	20	∧	∧	NOUN
ejpam-3036	208	21	p	p	NOUN
ejpam-3036	208	22	=	=	X
ejpam-3036	208	23	(	(	PUNCT
ejpam-3036	208	24	x	x	PUNCT
ejpam-3036	208	25	∧	∧	NOUN
ejpam-3036	208	26	p	p	NOUN
ejpam-3036	208	27	)	)	PUNCT
ejpam-3036	208	28	∧	∧	NOUN
ejpam-3036	208	29	y	y	NOUN
ejpam-3036	208	30	=	=	SYM
ejpam-3036	208	31	(	(	PUNCT
ejpam-3036	208	32	x	x	PUNCT
ejpam-3036	208	33	∧	∧	PROPN
ejpam-3036	208	34	q	q	NOUN
ejpam-3036	208	35	)	)	PUNCT
ejpam-3036	208	36	∧	∧	PROPN
ejpam-3036	208	37	y	y	PROPN
ejpam-3036	208	38	,	,	PUNCT
ejpam-3036	208	39	since	since	SCONJ
ejpam-3036	208	40	x	x	X
ejpam-3036	208	41	∧	∧	NOUN
ejpam-3036	208	42	p	p	NOUN
ejpam-3036	208	43	=	=	NOUN
ejpam-3036	208	44	x	x	SYM
ejpam-3036	208	45	∧	∧	NOUN
ejpam-3036	208	46	q	q	NOUN
ejpam-3036	208	47	=	=	PUNCT
ejpam-3036	208	48	x	x	SYM
ejpam-3036	208	49	∧	∧	PROPN
ejpam-3036	208	50	(	(	PUNCT
ejpam-3036	208	51	y	y	PROPN
ejpam-3036	208	52	∧	∧	PROPN
ejpam-3036	208	53	q	q	PROPN
ejpam-3036	208	54	)	)	PUNCT
ejpam-3036	208	55	=	=	SYM
ejpam-3036	209	1	x	x	SYM
ejpam-3036	209	2	∧	∧	PROPN
ejpam-3036	209	3	(	(	PUNCT
ejpam-3036	209	4	y	y	PROPN
ejpam-3036	209	5	∧	∧	PROPN
ejpam-3036	209	6	r	r	PROPN
ejpam-3036	209	7	)	)	PUNCT
ejpam-3036	209	8	,	,	PUNCT
ejpam-3036	209	9	since	since	SCONJ
ejpam-3036	209	10	y	y	PROPN
ejpam-3036	209	11	∧	∧	PROPN
ejpam-3036	209	12	q	q	PROPN
ejpam-3036	209	13	=	=	PUNCT
ejpam-3036	209	14	y	y	NOUN
ejpam-3036	209	15	∧	∧	PROPN
ejpam-3036	209	16	r	r	NOUN
ejpam-3036	209	17	=	=	SYM
ejpam-3036	209	18	(	(	PUNCT
ejpam-3036	209	19	x	x	PUNCT
ejpam-3036	209	20	∧	∧	PROPN
ejpam-3036	209	21	y	y	NOUN
ejpam-3036	209	22	)	)	PUNCT
ejpam-3036	209	23	∧	∧	PROPN
ejpam-3036	209	24	r.	r.	PROPN
ejpam-3036	209	25	therefore	therefore	ADV
ejpam-3036	209	26	(	(	PUNCT
ejpam-3036	209	27	p	p	X
ejpam-3036	209	28	,	,	PUNCT
ejpam-3036	209	29	r	r	NOUN
ejpam-3036	209	30	)	)	PUNCT
ejpam-3036	209	31	∈	∈	NOUN
ejpam-3036	209	32	θx∧y	θx∧y	NOUN
ejpam-3036	209	33	and	and	CCONJ
ejpam-3036	209	34	hence	hence	ADV
ejpam-3036	209	35	θx	θx	ADP
ejpam-3036	209	36	◦	◦	NOUN
ejpam-3036	209	37	θy	θy	NUM
ejpam-3036	209	38	⊆	⊆	NUM
ejpam-3036	209	39	θx∧y	θx∧y	NOUN
ejpam-3036	209	40	.	.	PUNCT
ejpam-3036	210	1	conversely	conversely	ADV
ejpam-3036	210	2	suppose	suppose	VERB
ejpam-3036	210	3	that	that	SCONJ
ejpam-3036	210	4	(	(	PUNCT
ejpam-3036	210	5	p	p	X
ejpam-3036	210	6	,	,	PUNCT
ejpam-3036	210	7	r	r	NOUN
ejpam-3036	210	8	)	)	PUNCT
ejpam-3036	210	9	∈	∈	NOUN
ejpam-3036	210	10	θx∧y	θx∧y	NOUN
ejpam-3036	210	11	then	then	ADV
ejpam-3036	210	12	(	(	PUNCT
ejpam-3036	210	13	x∧y)∧p	x∧y)∧p	X
ejpam-3036	210	14	=	=	SYM
ejpam-3036	210	15	(	(	PUNCT
ejpam-3036	210	16	x∧y)∧r	x∧y)∧r	PROPN
ejpam-3036	210	17	.	.	PUNCT
ejpam-3036	211	1	put	put	VERB
ejpam-3036	211	2	q	q	NOUN
ejpam-3036	211	3	=	=	SYM
ejpam-3036	211	4	(	(	PUNCT
ejpam-3036	211	5	x∧p)∨(y∧r	x∧p)∨(y∧r	PROPN
ejpam-3036	211	6	)	)	PUNCT
ejpam-3036	211	7	.	.	PUNCT
ejpam-3036	212	1	then	then	ADV
ejpam-3036	212	2	,	,	PUNCT
ejpam-3036	212	3	x	x	PUNCT
ejpam-3036	212	4	∧	∧	NOUN
ejpam-3036	212	5	q	q	NOUN
ejpam-3036	212	6	=	=	PUNCT
ejpam-3036	212	7	x	x	SYM
ejpam-3036	212	8	∧	∧	PROPN
ejpam-3036	212	9	(	(	PUNCT
ejpam-3036	212	10	x	x	PROPN
ejpam-3036	212	11	∧	∧	PROPN
ejpam-3036	212	12	p	p	NOUN
ejpam-3036	212	13	)	)	PUNCT
ejpam-3036	212	14	∨	∨	PROPN
ejpam-3036	212	15	(	(	PUNCT
ejpam-3036	212	16	y	y	PROPN
ejpam-3036	212	17	∧	∧	PROPN
ejpam-3036	212	18	r	r	NOUN
ejpam-3036	212	19	)	)	PUNCT
ejpam-3036	212	20	=	=	SYM
ejpam-3036	212	21	(	(	PUNCT
ejpam-3036	212	22	x	x	PUNCT
ejpam-3036	212	23	∧	∧	NOUN
ejpam-3036	212	24	p	p	NOUN
ejpam-3036	212	25	)	)	PUNCT
ejpam-3036	212	26	∨	∨	NOUN
ejpam-3036	212	27	(	(	PUNCT
ejpam-3036	212	28	x	x	SYM
ejpam-3036	212	29	∧	∧	NOUN
ejpam-3036	212	30	y	y	PROPN
ejpam-3036	212	31	∧	∧	PROPN
ejpam-3036	212	32	r	r	NOUN
ejpam-3036	212	33	)	)	PUNCT
ejpam-3036	212	34	=	=	SYM
ejpam-3036	212	35	(	(	PUNCT
ejpam-3036	212	36	x	x	PUNCT
ejpam-3036	212	37	∧	∧	NOUN
ejpam-3036	212	38	p	p	NOUN
ejpam-3036	212	39	)	)	PUNCT
ejpam-3036	212	40	∨	∨	NOUN
ejpam-3036	212	41	(	(	PUNCT
ejpam-3036	212	42	x	x	SYM
ejpam-3036	212	43	∧	∧	NOUN
ejpam-3036	212	44	y	y	PROPN
ejpam-3036	212	45	∧	∧	PROPN
ejpam-3036	212	46	p	p	PROPN
ejpam-3036	212	47	)	)	PUNCT
ejpam-3036	212	48	,	,	PUNCT
ejpam-3036	212	49	since	since	SCONJ
ejpam-3036	212	50	(	(	PUNCT
ejpam-3036	212	51	x	x	PROPN
ejpam-3036	212	52	∧	∧	PROPN
ejpam-3036	212	53	y	y	NOUN
ejpam-3036	212	54	)	)	PUNCT
ejpam-3036	212	55	∧	∧	NOUN
ejpam-3036	212	56	p	p	NOUN
ejpam-3036	212	57	=	=	X
ejpam-3036	212	58	(	(	PUNCT
ejpam-3036	212	59	x	x	PROPN
ejpam-3036	212	60	∧	∧	PROPN
ejpam-3036	212	61	y	y	NOUN
ejpam-3036	212	62	)	)	PUNCT
ejpam-3036	212	63	∧	∧	NOUN
ejpam-3036	212	64	r	r	NOUN
ejpam-3036	212	65	=	=	SYM
ejpam-3036	212	66	(	(	PUNCT
ejpam-3036	212	67	x	x	PUNCT
ejpam-3036	212	68	∧	∧	PROPN
ejpam-3036	212	69	p	p	NOUN
ejpam-3036	212	70	)	)	PUNCT
ejpam-3036	212	71	a	a	DET
ejpam-3036	212	72	r	r	NOUN
ejpam-3036	212	73	j	j	PROPN
ejpam-3036	212	74	srikanth	srikanth	NOUN
ejpam-3036	212	75	,	,	PUNCT
ejpam-3036	212	76	r	r	NOUN
ejpam-3036	212	77	v	v	NUM
ejpam-3036	212	78	g	g	NOUN
ejpam-3036	212	79	ravi	ravi	PROPN
ejpam-3036	212	80	kumar	kumar	PROPN
ejpam-3036	212	81	/	/	SYM
ejpam-3036	212	82	eur	eur	PROPN
ejpam-3036	212	83	.	.	PUNCT
ejpam-3036	213	1	j.	j.	PROPN
ejpam-3036	213	2	pure	pure	PROPN
ejpam-3036	213	3	appl	appl	PROPN
ejpam-3036	213	4	.	.	PROPN
ejpam-3036	213	5	math	math	PROPN
ejpam-3036	213	6	,	,	PUNCT
ejpam-3036	213	7	10	10	NUM
ejpam-3036	213	8	(	(	PUNCT
ejpam-3036	213	9	4	4	NUM
ejpam-3036	213	10	)	)	PUNCT
ejpam-3036	213	11	(	(	PUNCT
ejpam-3036	213	12	2017	2017	NUM
ejpam-3036	213	13	)	)	PUNCT
ejpam-3036	213	14	,	,	PUNCT
ejpam-3036	213	15	717	717	NUM
ejpam-3036	213	16	-	-	SYM
ejpam-3036	213	17	729	729	NUM
ejpam-3036	213	18	726	726	NUM
ejpam-3036	213	19	hence	hence	ADV
ejpam-3036	213	20	(	(	PUNCT
ejpam-3036	213	21	p	p	X
ejpam-3036	213	22	,	,	PUNCT
ejpam-3036	213	23	q	q	NOUN
ejpam-3036	213	24	)	)	PUNCT
ejpam-3036	213	25	∈	∈	PROPN
ejpam-3036	213	26	θx	θx	NOUN
ejpam-3036	213	27	.	.	PUNCT
ejpam-3036	213	28	by	by	ADP
ejpam-3036	213	29	considering	consider	VERB
ejpam-3036	213	30	y	y	PROPN
ejpam-3036	213	31	∧	∧	PROPN
ejpam-3036	213	32	q	q	PROPN
ejpam-3036	213	33	and	and	CCONJ
ejpam-3036	213	34	proceeding	proceeding	NOUN
ejpam-3036	213	35	as	as	ADP
ejpam-3036	213	36	above	above	ADV
ejpam-3036	213	37	it	it	PRON
ejpam-3036	213	38	can	can	AUX
ejpam-3036	213	39	shown	show	VERB
ejpam-3036	213	40	that	that	SCONJ
ejpam-3036	213	41	y	y	PROPN
ejpam-3036	213	42	∧	∧	PROPN
ejpam-3036	213	43	q	q	PROPN
ejpam-3036	213	44	=	=	PUNCT
ejpam-3036	213	45	y	y	PROPN
ejpam-3036	213	46	∧	∧	PROPN
ejpam-3036	213	47	r	r	NOUN
ejpam-3036	213	48	,	,	PUNCT
ejpam-3036	213	49	so	so	CCONJ
ejpam-3036	213	50	(	(	PUNCT
ejpam-3036	213	51	q	q	X
ejpam-3036	213	52	,	,	PUNCT
ejpam-3036	213	53	r	r	NOUN
ejpam-3036	213	54	)	)	PUNCT
ejpam-3036	213	55	∈	∈	NOUN
ejpam-3036	213	56	θy	θy	X
ejpam-3036	214	1	and	and	CCONJ
ejpam-3036	214	2	hence	hence	ADV
ejpam-3036	214	3	(	(	PUNCT
ejpam-3036	214	4	p	p	X
ejpam-3036	214	5	,	,	PUNCT
ejpam-3036	214	6	r	r	NOUN
ejpam-3036	214	7	)	)	PUNCT
ejpam-3036	214	8	∈	∈	NOUN
ejpam-3036	214	9	θx	θx	ADP
ejpam-3036	214	10	◦	◦	NOUN
ejpam-3036	214	11	θy	θy	PROPN
ejpam-3036	214	12	.	.	PUNCT
ejpam-3036	215	1	therefore	therefore	ADV
ejpam-3036	215	2	θx∧y	θx∧y	VERB
ejpam-3036	215	3	⊆	⊆	NUM
ejpam-3036	215	4	θx	θx	ADP
ejpam-3036	215	5	◦	◦	NOUN
ejpam-3036	215	6	θy	θy	NUM
ejpam-3036	215	7	,	,	PUNCT
ejpam-3036	215	8	which	which	PRON
ejpam-3036	215	9	completes	complete	VERB
ejpam-3036	215	10	the	the	DET
ejpam-3036	215	11	proof	proof	NOUN
ejpam-3036	215	12	.	.	PUNCT
ejpam-3036	216	1	(	(	PUNCT
ejpam-3036	216	2	xi	xi	X
ejpam-3036	216	3	)	)	PUNCT
ejpam-3036	216	4	is	be	AUX
ejpam-3036	216	5	clear	clear	ADJ
ejpam-3036	216	6	from	from	ADP
ejpam-3036	216	7	(	(	PUNCT
ejpam-3036	216	8	x	x	NOUN
ejpam-3036	216	9	)	)	PUNCT
ejpam-3036	216	10	,	,	PUNCT
ejpam-3036	216	11	(	(	PUNCT
ejpam-3036	216	12	vi	vi	NOUN
ejpam-3036	216	13	)	)	PUNCT
ejpam-3036	216	14	and	and	CCONJ
ejpam-3036	216	15	the	the	DET
ejpam-3036	216	16	fact	fact	NOUN
ejpam-3036	216	17	that	that	SCONJ
ejpam-3036	216	18	x	x	PUNCT
ejpam-3036	216	19	∧	∧	NOUN
ejpam-3036	216	20	x∗	x∗	X
ejpam-3036	216	21	=	=	SYM
ejpam-3036	216	22	0	0	X
ejpam-3036	216	23	.	.	PUNCT
ejpam-3036	217	1	(	(	PUNCT
ejpam-3036	217	2	xii	xii	NOUN
ejpam-3036	217	3	)	)	PUNCT
ejpam-3036	217	4	follows	follow	VERB
ejpam-3036	217	5	from	from	ADP
ejpam-3036	217	6	(	(	PUNCT
ejpam-3036	217	7	v	v	NOUN
ejpam-3036	217	8	)	)	PUNCT
ejpam-3036	217	9	,	,	PUNCT
ejpam-3036	217	10	(	(	PUNCT
ejpam-3036	217	11	ii	ii	NOUN
ejpam-3036	217	12	)	)	PUNCT
ejpam-3036	217	13	and	and	CCONJ
ejpam-3036	217	14	(	(	PUNCT
ejpam-3036	217	15	vii	vii	PROPN
ejpam-3036	217	16	)	)	PUNCT
ejpam-3036	217	17	.	.	PUNCT
ejpam-3036	218	1	(	(	PUNCT
ejpam-3036	218	2	xiii	xiii	X
ejpam-3036	218	3	)	)	PUNCT
ejpam-3036	218	4	let	let	VERB
ejpam-3036	218	5	x	x	PUNCT
ejpam-3036	218	6	∈	∈	PROPN
ejpam-3036	218	7	k(a	k(a	PROPN
ejpam-3036	218	8	)	)	PUNCT
ejpam-3036	218	9	.	.	PUNCT
ejpam-3036	219	1	then	then	ADV
ejpam-3036	219	2	θx	θx	INTJ
ejpam-3036	219	3	is	be	AUX
ejpam-3036	219	4	congruence	congruence	ADJ
ejpam-3036	219	5	and	and	CCONJ
ejpam-3036	219	6	(	(	PUNCT
ejpam-3036	219	7	1	1	NUM
ejpam-3036	219	8	,	,	PUNCT
ejpam-3036	219	9	x	x	X
ejpam-3036	219	10	)	)	PUNCT
ejpam-3036	219	11	∈	∈	PROPN
ejpam-3036	219	12	θx	θx	PRON
ejpam-3036	219	13	.	.	PUNCT
ejpam-3036	219	14	suppose	suppose	VERB
ejpam-3036	219	15	that	that	SCONJ
ejpam-3036	219	16	θ	θ	PROPN
ejpam-3036	219	17	be	be	VERB
ejpam-3036	219	18	any	any	DET
ejpam-3036	219	19	congruence	congruence	NOUN
ejpam-3036	219	20	containing	contain	VERB
ejpam-3036	219	21	(	(	PUNCT
ejpam-3036	219	22	1	1	NUM
ejpam-3036	219	23	,	,	PUNCT
ejpam-3036	219	24	x	x	NOUN
ejpam-3036	219	25	)	)	PUNCT
ejpam-3036	219	26	and	and	CCONJ
ejpam-3036	219	27	(	(	PUNCT
ejpam-3036	219	28	a	a	PRON
ejpam-3036	219	29	,	,	PUNCT
ejpam-3036	219	30	b	b	NOUN
ejpam-3036	219	31	)	)	PUNCT
ejpam-3036	219	32	∈	∈	NOUN
ejpam-3036	219	33	θx	θx	X
ejpam-3036	219	34	i.e.	i.e.	X
ejpam-3036	219	35	a	a	DET
ejpam-3036	219	36	∧	∧	NOUN
ejpam-3036	219	37	x	x	X
ejpam-3036	219	38	=	=	SYM
ejpam-3036	219	39	b	b	SYM
ejpam-3036	219	40	∧	∧	PROPN
ejpam-3036	219	41	x.	x.	NOUN
ejpam-3036	219	42	since	since	SCONJ
ejpam-3036	219	43	θ	θ	PROPN
ejpam-3036	219	44	is	be	AUX
ejpam-3036	219	45	reflexive	reflexive	ADJ
ejpam-3036	219	46	and	and	CCONJ
ejpam-3036	219	47	(	(	PUNCT
ejpam-3036	219	48	1	1	NUM
ejpam-3036	219	49	,	,	PUNCT
ejpam-3036	219	50	x	x	X
ejpam-3036	219	51	)	)	PUNCT
ejpam-3036	219	52	∈	∈	PROPN
ejpam-3036	219	53	θ	θ	NOUN
ejpam-3036	219	54	we	we	PRON
ejpam-3036	219	55	get	get	VERB
ejpam-3036	219	56	(	(	PUNCT
ejpam-3036	219	57	a	a	PRON
ejpam-3036	219	58	,	,	PUNCT
ejpam-3036	219	59	a	a	PRON
ejpam-3036	219	60	)	)	PUNCT
ejpam-3036	219	61	∧	∧	NOUN
ejpam-3036	219	62	(	(	PUNCT
ejpam-3036	219	63	1	1	NUM
ejpam-3036	219	64	,	,	PUNCT
ejpam-3036	219	65	x	x	NOUN
ejpam-3036	219	66	)	)	PUNCT
ejpam-3036	219	67	=	=	SYM
ejpam-3036	219	68	(	(	PUNCT
ejpam-3036	219	69	a	a	PRON
ejpam-3036	219	70	,	,	PUNCT
ejpam-3036	219	71	a	a	DET
ejpam-3036	219	72	∧	∧	PROPN
ejpam-3036	219	73	x	x	NOUN
ejpam-3036	219	74	)	)	PUNCT
ejpam-3036	219	75	∈	∈	PROPN
ejpam-3036	219	76	θ	θ	PROPN
ejpam-3036	219	77	and	and	CCONJ
ejpam-3036	219	78	(	(	PUNCT
ejpam-3036	219	79	b	b	PROPN
ejpam-3036	219	80	,	,	PUNCT
ejpam-3036	219	81	b	b	NOUN
ejpam-3036	219	82	)	)	PUNCT
ejpam-3036	219	83	∧	∧	NOUN
ejpam-3036	219	84	(	(	PUNCT
ejpam-3036	219	85	1	1	NUM
ejpam-3036	219	86	,	,	PUNCT
ejpam-3036	219	87	x	x	NOUN
ejpam-3036	219	88	)	)	PUNCT
ejpam-3036	219	89	=	=	SYM
ejpam-3036	219	90	(	(	PUNCT
ejpam-3036	219	91	b	b	NOUN
ejpam-3036	219	92	,	,	PUNCT
ejpam-3036	219	93	b	b	PROPN
ejpam-3036	219	94	∧	∧	PROPN
ejpam-3036	219	95	x	x	NOUN
ejpam-3036	219	96	)	)	PUNCT
ejpam-3036	219	97	∈	∈	PROPN
ejpam-3036	219	98	θ	θ	NOUN
ejpam-3036	219	99	which	which	PRON
ejpam-3036	219	100	in	in	ADP
ejpam-3036	219	101	turn	turn	NOUN
ejpam-3036	219	102	gives	give	VERB
ejpam-3036	219	103	(	(	PUNCT
ejpam-3036	219	104	a	a	PRON
ejpam-3036	219	105	,	,	PUNCT
ejpam-3036	219	106	b	b	PROPN
ejpam-3036	219	107	∧	∧	PROPN
ejpam-3036	219	108	x	x	NOUN
ejpam-3036	219	109	)	)	PUNCT
ejpam-3036	219	110	∈	∈	PROPN
ejpam-3036	219	111	θ	θ	PROPN
ejpam-3036	219	112	and	and	CCONJ
ejpam-3036	219	113	(	(	PUNCT
ejpam-3036	219	114	b	b	NOUN
ejpam-3036	219	115	,	,	PUNCT
ejpam-3036	219	116	b	b	PROPN
ejpam-3036	219	117	∧	∧	PROPN
ejpam-3036	219	118	x	x	NOUN
ejpam-3036	219	119	)	)	PUNCT
ejpam-3036	219	120	∈	∈	PROPN
ejpam-3036	219	121	θ	θ	PROPN
ejpam-3036	219	122	.	.	PUNCT
ejpam-3036	220	1	hence	hence	ADV
ejpam-3036	220	2	(	(	PUNCT
ejpam-3036	220	3	a	a	PRON
ejpam-3036	220	4	,	,	PUNCT
ejpam-3036	220	5	b	b	NOUN
ejpam-3036	220	6	)	)	PUNCT
ejpam-3036	220	7	∈	∈	PROPN
ejpam-3036	220	8	θ	θ	PROPN
ejpam-3036	220	9	and	and	CCONJ
ejpam-3036	220	10	θx	θx	ADP
ejpam-3036	220	11	⊆	⊆	NUM
ejpam-3036	220	12	θ	θ	NOUN
ejpam-3036	220	13	.	.	PUNCT
ejpam-3036	221	1	therefore	therefore	ADV
ejpam-3036	221	2	θx	θx	PROPN
ejpam-3036	221	3	is	be	AUX
ejpam-3036	221	4	the	the	DET
ejpam-3036	221	5	smallest	small	ADJ
ejpam-3036	221	6	congruence	congruence	NOUN
ejpam-3036	221	7	containing	contain	VERB
ejpam-3036	221	8	(	(	PUNCT
ejpam-3036	221	9	1	1	NUM
ejpam-3036	221	10	,	,	PUNCT
ejpam-3036	221	11	x	x	NOUN
ejpam-3036	221	12	)	)	PUNCT
ejpam-3036	221	13	for	for	ADP
ejpam-3036	221	14	x	x	PROPN
ejpam-3036	221	15	∈	∈	PROPN
ejpam-3036	221	16	k(a	k(a	PROPN
ejpam-3036	221	17	)	)	PUNCT
ejpam-3036	221	18	.	.	PUNCT
ejpam-3036	222	1	recall	recall	VERB
ejpam-3036	222	2	that	that	SCONJ
ejpam-3036	222	3	a	a	DET
ejpam-3036	222	4	congruence	congruence	ADJ
ejpam-3036	222	5	θ	θ	NOUN
ejpam-3036	222	6	on	on	ADP
ejpam-3036	222	7	an	an	DET
ejpam-3036	222	8	algebra	algebra	NOUN
ejpam-3036	222	9	a	a	PRON
ejpam-3036	222	10	is	be	AUX
ejpam-3036	222	11	said	say	VERB
ejpam-3036	222	12	to	to	PART
ejpam-3036	222	13	be	be	AUX
ejpam-3036	222	14	factor	factor	NOUN
ejpam-3036	222	15	congruence	congruence	NOUN
ejpam-3036	222	16	if	if	SCONJ
ejpam-3036	222	17	there	there	PRON
ejpam-3036	222	18	is	be	VERB
ejpam-3036	222	19	a	a	DET
ejpam-3036	222	20	congruence	congruence	NOUN
ejpam-3036	222	21	ψ	ψ	NOUN
ejpam-3036	222	22	on	on	ADP
ejpam-3036	222	23	a	a	DET
ejpam-3036	222	24	such	such	ADJ
ejpam-3036	222	25	that	that	SCONJ
ejpam-3036	222	26	θ	θ	PROPN
ejpam-3036	222	27	∧	∧	PROPN
ejpam-3036	222	28	ψ	ψ	NOUN
ejpam-3036	222	29	=	=	NOUN
ejpam-3036	222	30	∆	∆	PROPN
ejpam-3036	222	31	θ	θ	NOUN
ejpam-3036	222	32	∨	∨	NUM
ejpam-3036	222	33	ψ	ψ	X
ejpam-3036	222	34	=	=	PUNCT
ejpam-3036	222	35	a×a	a×a	PROPN
ejpam-3036	222	36	and	and	CCONJ
ejpam-3036	222	37	θ	θ	PROPN
ejpam-3036	222	38	permutes	permute	NOUN
ejpam-3036	222	39	with	with	ADP
ejpam-3036	222	40	ψ	ψ	PRON
ejpam-3036	222	41	in	in	ADP
ejpam-3036	222	42	the	the	DET
ejpam-3036	222	43	following	following	NOUN
ejpam-3036	222	44	theorem	theorem	VERB
ejpam-3036	222	45	the	the	DET
ejpam-3036	222	46	factor	factor	NOUN
ejpam-3036	222	47	congruences	congruence	NOUN
ejpam-3036	222	48	of	of	ADP
ejpam-3036	222	49	core	core	NOUN
ejpam-3036	222	50	regular	regular	ADJ
ejpam-3036	222	51	double	double	ADJ
ejpam-3036	222	52	stone	stone	NOUN
ejpam-3036	222	53	algebras	algebra	NOUN
ejpam-3036	222	54	were	be	AUX
ejpam-3036	222	55	characterized	characterize	VERB
ejpam-3036	222	56	.	.	PUNCT
ejpam-3036	223	1	theorem	theorem	ADJ
ejpam-3036	223	2	3.2	3.2	NUM
ejpam-3036	223	3	.	.	PUNCT
ejpam-3036	224	1	let	let	VERB
ejpam-3036	224	2	a	a	PRON
ejpam-3036	224	3	be	be	AUX
ejpam-3036	224	4	a	a	DET
ejpam-3036	224	5	core	core	NOUN
ejpam-3036	224	6	regular	regular	ADJ
ejpam-3036	224	7	double	double	ADJ
ejpam-3036	224	8	stone	stone	NOUN
ejpam-3036	224	9	algebra	algebra	NOUN
ejpam-3036	224	10	and	and	CCONJ
ejpam-3036	224	11	θ	θ	PROPN
ejpam-3036	224	12	be	be	AUX
ejpam-3036	224	13	congruence	congruence	ADJ
ejpam-3036	224	14	on	on	ADP
ejpam-3036	224	15	a.	a.	NOUN
ejpam-3036	224	16	then	then	ADV
ejpam-3036	224	17	θ	θ	PROPN
ejpam-3036	224	18	is	be	AUX
ejpam-3036	224	19	factor	factor	NOUN
ejpam-3036	224	20	congruence	congruence	NOUN
ejpam-3036	224	21	on	on	ADP
ejpam-3036	224	22	a	a	DET
ejpam-3036	224	23	if	if	NOUN
ejpam-3036	224	24	and	and	CCONJ
ejpam-3036	224	25	only	only	ADV
ejpam-3036	224	26	if	if	SCONJ
ejpam-3036	224	27	θ	θ	X
ejpam-3036	224	28	=	=	PUNCT
ejpam-3036	224	29	θx	θx	NOUN
ejpam-3036	224	30	for	for	ADP
ejpam-3036	224	31	some	some	DET
ejpam-3036	224	32	x	x	SYM
ejpam-3036	224	33	∈	∈	PROPN
ejpam-3036	224	34	k(a	k(a	PROPN
ejpam-3036	224	35	)	)	PUNCT
ejpam-3036	224	36	.	.	PUNCT
ejpam-3036	225	1	proof	proof	NOUN
ejpam-3036	225	2	.	.	PUNCT
ejpam-3036	226	1	suppose	suppose	VERB
ejpam-3036	226	2	that	that	SCONJ
ejpam-3036	226	3	θ	θ	PROPN
ejpam-3036	226	4	=	=	PUNCT
ejpam-3036	226	5	θx	θx	X
ejpam-3036	226	6	for	for	ADP
ejpam-3036	226	7	some	some	DET
ejpam-3036	226	8	x	x	SYM
ejpam-3036	226	9	∈	∈	PROPN
ejpam-3036	226	10	k(a	k(a	PROPN
ejpam-3036	226	11	)	)	PUNCT
ejpam-3036	226	12	,	,	PUNCT
ejpam-3036	226	13	then	then	ADV
ejpam-3036	226	14	from	from	ADP
ejpam-3036	226	15	(	(	PUNCT
ejpam-3036	226	16	vi	vi	NOUN
ejpam-3036	226	17	)	)	PUNCT
ejpam-3036	226	18	,	,	PUNCT
ejpam-3036	226	19	(	(	PUNCT
ejpam-3036	226	20	vii	vii	PROPN
ejpam-3036	226	21	)	)	PUNCT
ejpam-3036	226	22	,	,	PUNCT
ejpam-3036	226	23	(	(	PUNCT
ejpam-3036	226	24	viii	viii	NOUN
ejpam-3036	226	25	)	)	PUNCT
ejpam-3036	226	26	and	and	CCONJ
ejpam-3036	226	27	(	(	PUNCT
ejpam-3036	226	28	x	x	X
ejpam-3036	226	29	)	)	PUNCT
ejpam-3036	226	30	of	of	ADP
ejpam-3036	226	31	theorem	theorem	ADJ
ejpam-3036	226	32	3.1	3.1	NUM
ejpam-3036	226	33	and	and	CCONJ
ejpam-3036	226	34	theorem	theorem	VERB
ejpam-3036	226	35	2.8	2.8	NUM
ejpam-3036	226	36	,	,	PUNCT
ejpam-3036	226	37	we	we	PRON
ejpam-3036	226	38	have	have	VERB
ejpam-3036	226	39	θx	θx	PRON
ejpam-3036	226	40	∧	∧	NOUN
ejpam-3036	226	41	θx∗	θx∗	NOUN
ejpam-3036	226	42	=	=	SYM
ejpam-3036	226	43	∆	∆	PROPN
ejpam-3036	226	44	and	and	CCONJ
ejpam-3036	226	45	θx	θx	ADP
ejpam-3036	226	46	∨	∨	NOUN
ejpam-3036	226	47	θx∗	θx∗	NOUN
ejpam-3036	226	48	=	=	PUNCT
ejpam-3036	226	49	a	a	DET
ejpam-3036	226	50	×	×	NOUN
ejpam-3036	226	51	a	a	NOUN
ejpam-3036	226	52	and	and	CCONJ
ejpam-3036	226	53	hence	hence	ADV
ejpam-3036	226	54	θ	θ	NOUN
ejpam-3036	226	55	=	=	PUNCT
ejpam-3036	226	56	θx	θx	X
ejpam-3036	226	57	is	be	AUX
ejpam-3036	226	58	factor	factor	NOUN
ejpam-3036	226	59	congruence	congruence	NOUN
ejpam-3036	226	60	on	on	ADP
ejpam-3036	226	61	a	a	DET
ejpam-3036	226	62	conversely	conversely	ADV
ejpam-3036	226	63	suppose	suppose	VERB
ejpam-3036	226	64	that	that	SCONJ
ejpam-3036	226	65	θ	θ	PROPN
ejpam-3036	226	66	is	be	AUX
ejpam-3036	226	67	factor	factor	NOUN
ejpam-3036	226	68	congruence	congruence	NOUN
ejpam-3036	226	69	on	on	ADP
ejpam-3036	226	70	a.	a.	NOUN
ejpam-3036	226	71	then	then	ADV
ejpam-3036	226	72	there	there	PRON
ejpam-3036	226	73	exists	exist	VERB
ejpam-3036	226	74	a	a	DET
ejpam-3036	226	75	congruence	congruence	NOUN
ejpam-3036	226	76	ψ	ψ	NOUN
ejpam-3036	226	77	on	on	ADP
ejpam-3036	226	78	a	a	DET
ejpam-3036	226	79	such	such	ADJ
ejpam-3036	226	80	that	that	PRON
ejpam-3036	227	1	θ∧ψ	θ∧ψ	NUM
ejpam-3036	227	2	=	=	NOUN
ejpam-3036	227	3	∆	∆	PROPN
ejpam-3036	227	4	and	and	CCONJ
ejpam-3036	227	5	θ∨ψ	θ∨ψ	NOUN
ejpam-3036	227	6	=	=	SYM
ejpam-3036	227	7	a×a	a×a	PROPN
ejpam-3036	227	8	.	.	PUNCT
ejpam-3036	228	1	since	since	SCONJ
ejpam-3036	228	2	(	(	PUNCT
ejpam-3036	228	3	1	1	NUM
ejpam-3036	228	4	,	,	PUNCT
ejpam-3036	228	5	0	0	NUM
ejpam-3036	228	6	)	)	PUNCT
ejpam-3036	228	7	∈	∈	NOUN
ejpam-3036	228	8	a×a	a×a	PROPN
ejpam-3036	228	9	=	=	SYM
ejpam-3036	228	10	θ∨ψ	θ∨ψ	NOUN
ejpam-3036	228	11	,	,	PUNCT
ejpam-3036	228	12	there	there	PRON
ejpam-3036	228	13	exists	exist	VERB
ejpam-3036	228	14	x	x	X
ejpam-3036	228	15	∈	∈	PROPN
ejpam-3036	228	16	a	a	DET
ejpam-3036	228	17	such	such	ADJ
ejpam-3036	228	18	that	that	SCONJ
ejpam-3036	228	19	(	(	PUNCT
ejpam-3036	228	20	1	1	NUM
ejpam-3036	228	21	,	,	PUNCT
ejpam-3036	228	22	x	x	X
ejpam-3036	228	23	)	)	PUNCT
ejpam-3036	228	24	∈	∈	PROPN
ejpam-3036	228	25	θ	θ	PROPN
ejpam-3036	228	26	and	and	CCONJ
ejpam-3036	228	27	(	(	PUNCT
ejpam-3036	228	28	x	x	NOUN
ejpam-3036	228	29	,	,	PUNCT
ejpam-3036	228	30	0	0	NUM
ejpam-3036	228	31	)	)	PUNCT
ejpam-3036	228	32	∈	∈	PROPN
ejpam-3036	229	1	ψ	ψ	X
ejpam-3036	229	2	.	.	PUNCT
ejpam-3036	229	3	now	now	ADV
ejpam-3036	229	4	put	put	VERB
ejpam-3036	229	5	y	y	NOUN
ejpam-3036	229	6	=	=	PUNCT
ejpam-3036	229	7	x	x	PUNCT
ejpam-3036	230	1	∧	∧	PROPN
ejpam-3036	230	2	k	k	PROPN
ejpam-3036	230	3	then	then	ADV
ejpam-3036	230	4	y∗∗	y∗∗	PROPN
ejpam-3036	230	5	∈	∈	PROPN
ejpam-3036	230	6	k(a	k(a	PROPN
ejpam-3036	230	7	)	)	PUNCT
ejpam-3036	230	8	and	and	CCONJ
ejpam-3036	230	9	from	from	ADP
ejpam-3036	230	10	the	the	DET
ejpam-3036	230	11	fact	fact	NOUN
ejpam-3036	230	12	that	that	SCONJ
ejpam-3036	230	13	(	(	PUNCT
ejpam-3036	230	14	1	1	NUM
ejpam-3036	230	15	,	,	PUNCT
ejpam-3036	230	16	x	x	NOUN
ejpam-3036	230	17	)	)	PUNCT
ejpam-3036	230	18	,	,	PUNCT
ejpam-3036	230	19	(	(	PUNCT
ejpam-3036	230	20	k	k	X
ejpam-3036	230	21	,	,	PUNCT
ejpam-3036	230	22	k	k	NOUN
ejpam-3036	230	23	)	)	PUNCT
ejpam-3036	230	24	∈	∈	PROPN
ejpam-3036	230	25	θ	θ	NOUN
ejpam-3036	230	26	it	it	PRON
ejpam-3036	230	27	follows	follow	VERB
ejpam-3036	230	28	that	that	SCONJ
ejpam-3036	230	29	(	(	PUNCT
ejpam-3036	230	30	1	1	NUM
ejpam-3036	230	31	,	,	PUNCT
ejpam-3036	230	32	y	y	NOUN
ejpam-3036	230	33	)	)	PUNCT
ejpam-3036	230	34	∈	∈	PROPN
ejpam-3036	230	35	θ	θ	PROPN
ejpam-3036	230	36	and	and	CCONJ
ejpam-3036	230	37	hence	hence	ADV
ejpam-3036	230	38	(	(	PUNCT
ejpam-3036	230	39	1	1	NUM
ejpam-3036	230	40	,	,	PUNCT
ejpam-3036	230	41	y∗∗	y∗∗	NOUN
ejpam-3036	230	42	)	)	PUNCT
ejpam-3036	230	43	∈	∈	PROPN
ejpam-3036	230	44	θ	θ	PROPN
ejpam-3036	230	45	.	.	PUNCT
ejpam-3036	230	46	also	also	ADV
ejpam-3036	230	47	observe	observe	VERB
ejpam-3036	230	48	that	that	SCONJ
ejpam-3036	230	49	(	(	PUNCT
ejpam-3036	230	50	x	x	X
ejpam-3036	230	51	,	,	PUNCT
ejpam-3036	230	52	0	0	NUM
ejpam-3036	230	53	)	)	PUNCT
ejpam-3036	230	54	,	,	PUNCT
ejpam-3036	230	55	(	(	PUNCT
ejpam-3036	230	56	k	k	X
ejpam-3036	230	57	,	,	PUNCT
ejpam-3036	230	58	k	k	NOUN
ejpam-3036	230	59	)	)	PUNCT
ejpam-3036	230	60	∈	∈	PROPN
ejpam-3036	230	61	ψ	ψ	NOUN
ejpam-3036	230	62	gives	give	VERB
ejpam-3036	230	63	(	(	PUNCT
ejpam-3036	230	64	y	y	PROPN
ejpam-3036	230	65	,	,	PUNCT
ejpam-3036	230	66	0	0	NUM
ejpam-3036	230	67	)	)	PUNCT
ejpam-3036	230	68	∈	∈	NOUN
ejpam-3036	230	69	ψ	ψ	NOUN
ejpam-3036	230	70	and	and	CCONJ
ejpam-3036	230	71	hence	hence	ADV
ejpam-3036	230	72	(	(	PUNCT
ejpam-3036	230	73	y∗∗	y∗∗	PROPN
ejpam-3036	230	74	,	,	PUNCT
ejpam-3036	230	75	0	0	NUM
ejpam-3036	230	76	)	)	PUNCT
ejpam-3036	230	77	∈	∈	PROPN
ejpam-3036	230	78	ψ	ψ	X
ejpam-3036	230	79	.	.	PUNCT
ejpam-3036	231	1	now	now	ADV
ejpam-3036	231	2	we	we	PRON
ejpam-3036	231	3	show	show	VERB
ejpam-3036	231	4	that	that	SCONJ
ejpam-3036	231	5	θ	θ	PROPN
ejpam-3036	231	6	=	=	PUNCT
ejpam-3036	231	7	θy∗∗.	θy∗∗.	PUNCT
ejpam-3036	231	8	since	since	SCONJ
ejpam-3036	231	9	(	(	PUNCT
ejpam-3036	231	10	1	1	NUM
ejpam-3036	231	11	,	,	PUNCT
ejpam-3036	231	12	y∗∗	y∗∗	NOUN
ejpam-3036	231	13	)	)	PUNCT
ejpam-3036	231	14	∈	∈	PROPN
ejpam-3036	231	15	θ	θ	PROPN
ejpam-3036	231	16	,	,	PUNCT
ejpam-3036	231	17	by	by	ADP
ejpam-3036	231	18	(	(	PUNCT
ejpam-3036	231	19	xiii	xiii	PROPN
ejpam-3036	231	20	)	)	PUNCT
ejpam-3036	231	21	of	of	ADP
ejpam-3036	231	22	theorem	theorem	ADJ
ejpam-3036	231	23	3.1	3.1	NUM
ejpam-3036	231	24	,	,	PUNCT
ejpam-3036	231	25	we	we	PRON
ejpam-3036	231	26	have	have	VERB
ejpam-3036	231	27	θy∗∗	θy∗∗	NOUN
ejpam-3036	231	28	⊆	⊆	NUM
ejpam-3036	231	29	θ	θ	PROPN
ejpam-3036	231	30	.	.	PUNCT
ejpam-3036	232	1	next	next	ADV
ejpam-3036	232	2	suppose	suppose	VERB
ejpam-3036	233	1	that	that	SCONJ
ejpam-3036	233	2	(	(	PUNCT
ejpam-3036	233	3	p	p	X
ejpam-3036	233	4	,	,	PUNCT
ejpam-3036	233	5	q	q	NOUN
ejpam-3036	233	6	)	)	PUNCT
ejpam-3036	233	7	∈	∈	PROPN
ejpam-3036	233	8	θ	θ	PROPN
ejpam-3036	233	9	then	then	ADV
ejpam-3036	233	10	(	(	PUNCT
ejpam-3036	233	11	y∗∗	y∗∗	PROPN
ejpam-3036	233	12	∧	∧	PROPN
ejpam-3036	233	13	p	p	PROPN
ejpam-3036	233	14	,	,	PUNCT
ejpam-3036	233	15	y∗∗	y∗∗	PROPN
ejpam-3036	233	16	∧	∧	PROPN
ejpam-3036	233	17	q	q	NOUN
ejpam-3036	233	18	)	)	PUNCT
ejpam-3036	233	19	∈	∈	PROPN
ejpam-3036	233	20	θ	θ	PROPN
ejpam-3036	233	21	.	.	PUNCT
ejpam-3036	233	22	since	since	SCONJ
ejpam-3036	233	23	(	(	PUNCT
ejpam-3036	233	24	y∗∗	y∗∗	PROPN
ejpam-3036	233	25	,	,	PUNCT
ejpam-3036	233	26	0	0	NUM
ejpam-3036	233	27	)	)	PUNCT
ejpam-3036	233	28	,	,	PUNCT
ejpam-3036	233	29	(	(	PUNCT
ejpam-3036	233	30	p	p	X
ejpam-3036	233	31	,	,	PUNCT
ejpam-3036	233	32	p	p	NOUN
ejpam-3036	233	33	)	)	PUNCT
ejpam-3036	233	34	and	and	CCONJ
ejpam-3036	233	35	(	(	PUNCT
ejpam-3036	233	36	q	q	ADJ
ejpam-3036	233	37	,	,	PUNCT
ejpam-3036	233	38	q	q	NOUN
ejpam-3036	233	39	)	)	PUNCT
ejpam-3036	233	40	∈	∈	PROPN
ejpam-3036	233	41	ψ	ψ	NOUN
ejpam-3036	233	42	,	,	PUNCT
ejpam-3036	233	43	we	we	PRON
ejpam-3036	233	44	have	have	VERB
ejpam-3036	233	45	(	(	PUNCT
ejpam-3036	233	46	y∗∗	y∗∗	X
ejpam-3036	233	47	∧	∧	PROPN
ejpam-3036	233	48	p	p	PROPN
ejpam-3036	233	49	,	,	PUNCT
ejpam-3036	233	50	0	0	NUM
ejpam-3036	233	51	∧	∧	PROPN
ejpam-3036	233	52	p	p	NOUN
ejpam-3036	233	53	)	)	PUNCT
ejpam-3036	233	54	and	and	CCONJ
ejpam-3036	233	55	(	(	PUNCT
ejpam-3036	233	56	y∗∗	y∗∗	X
ejpam-3036	233	57	∧	∧	PROPN
ejpam-3036	233	58	q	q	PROPN
ejpam-3036	233	59	,	,	PUNCT
ejpam-3036	233	60	0	0	NUM
ejpam-3036	233	61	∧	∧	PROPN
ejpam-3036	233	62	q	q	NOUN
ejpam-3036	233	63	)	)	PUNCT
ejpam-3036	233	64	∈	∈	PROPN
ejpam-3036	233	65	ψ	ψ	NOUN
ejpam-3036	233	66	;	;	PUNCT
ejpam-3036	233	67	that	that	ADV
ejpam-3036	233	68	is	is	ADV
ejpam-3036	233	69	(	(	PUNCT
ejpam-3036	233	70	y∗∗	y∗∗	X
ejpam-3036	233	71	∧	∧	PROPN
ejpam-3036	233	72	p	p	PROPN
ejpam-3036	233	73	,	,	PUNCT
ejpam-3036	233	74	0	0	NUM
ejpam-3036	233	75	)	)	PUNCT
ejpam-3036	233	76	and	and	CCONJ
ejpam-3036	233	77	(	(	PUNCT
ejpam-3036	233	78	0	0	NUM
ejpam-3036	233	79	,	,	PUNCT
ejpam-3036	233	80	y∗∗∧q	y∗∗∧q	NOUN
ejpam-3036	233	81	)	)	PUNCT
ejpam-3036	233	82	∈	∈	PROPN
ejpam-3036	233	83	ψ	ψ	X
ejpam-3036	233	84	which	which	PRON
ejpam-3036	233	85	imply	imply	VERB
ejpam-3036	233	86	that	that	PRON
ejpam-3036	233	87	(	(	PUNCT
ejpam-3036	233	88	y∗∗∧p	y∗∗∧p	NOUN
ejpam-3036	233	89	,	,	PUNCT
ejpam-3036	233	90	y∗∗∧q	y∗∗∧q	NOUN
ejpam-3036	233	91	)	)	PUNCT
ejpam-3036	233	92	∈	∈	PROPN
ejpam-3036	234	1	ψ	ψ	PROPN
ejpam-3036	234	2	.	.	PUNCT
ejpam-3036	234	3	therefore,(y∗∗∧p	therefore,(y∗∗∧p	PROPN
ejpam-3036	234	4	,	,	PUNCT
ejpam-3036	234	5	y∗∗∧q	y∗∗∧q	NOUN
ejpam-3036	234	6	)	)	PUNCT
ejpam-3036	234	7	∈	∈	PROPN
ejpam-3036	234	8	θ∩ψ	θ∩ψ	PROPN
ejpam-3036	234	9	=	=	PUNCT
ejpam-3036	234	10	∆	∆	PROPN
ejpam-3036	234	11	and	and	CCONJ
ejpam-3036	234	12	hence	hence	ADV
ejpam-3036	234	13	y∗∗	y∗∗	PROPN
ejpam-3036	234	14	∧	∧	PROPN
ejpam-3036	234	15	p	p	NOUN
ejpam-3036	234	16	=	=	SYM
ejpam-3036	234	17	y∗∗	y∗∗	PROPN
ejpam-3036	234	18	∧	∧	PROPN
ejpam-3036	234	19	q.	q.	PROPN
ejpam-3036	234	20	therefore	therefore	ADV
ejpam-3036	234	21	(	(	PUNCT
ejpam-3036	234	22	p	p	X
ejpam-3036	234	23	,	,	PUNCT
ejpam-3036	234	24	q	q	NOUN
ejpam-3036	234	25	)	)	PUNCT
ejpam-3036	234	26	∈	∈	PROPN
ejpam-3036	234	27	θy∗∗	θy∗∗	NOUN
ejpam-3036	234	28	,	,	PUNCT
ejpam-3036	234	29	hence	hence	ADV
ejpam-3036	234	30	θ	θ	NOUN
ejpam-3036	234	31	⊆	⊆	NUM
ejpam-3036	234	32	θy∗∗.	θy∗∗.	PUNCT
ejpam-3036	234	33	thus	thus	ADV
ejpam-3036	234	34	θ	θ	X
ejpam-3036	234	35	=	=	SYM
ejpam-3036	234	36	θy∗∗.	θy∗∗.	PUNCT
ejpam-3036	234	37	recall	recall	VERB
ejpam-3036	234	38	that	that	SCONJ
ejpam-3036	234	39	a	a	DET
ejpam-3036	234	40	congruence	congruence	ADJ
ejpam-3036	234	41	θ	θ	NOUN
ejpam-3036	234	42	on	on	ADP
ejpam-3036	234	43	any	any	DET
ejpam-3036	234	44	universal	universal	ADJ
ejpam-3036	234	45	algebra	algebra	NOUN
ejpam-3036	234	46	a	a	PRON
ejpam-3036	234	47	of	of	ADP
ejpam-3036	234	48	any	any	DET
ejpam-3036	234	49	type	type	NOUN
ejpam-3036	234	50	,	,	PUNCT
ejpam-3036	234	51	is	be	AUX
ejpam-3036	234	52	called	call	VERB
ejpam-3036	234	53	balanced	balanced	ADJ
ejpam-3036	234	54	if	if	SCONJ
ejpam-3036	234	55	(	(	PUNCT
ejpam-3036	234	56	θ∨ψ)∩	θ∨ψ)∩	NOUN
ejpam-3036	234	57	(	(	PUNCT
ejpam-3036	234	58	θ∨ψ′	θ∨ψ′	PROPN
ejpam-3036	234	59	)	)	PUNCT
ejpam-3036	234	60	=	=	SYM
ejpam-3036	234	61	θ	θ	PROPN
ejpam-3036	234	62	for	for	ADP
ejpam-3036	234	63	all	all	DET
ejpam-3036	234	64	factor	factor	NOUN
ejpam-3036	234	65	congruence	congruence	NOUN
ejpam-3036	234	66	ψ	ψ	PROPN
ejpam-3036	234	67	and	and	CCONJ
ejpam-3036	234	68	its	its	PRON
ejpam-3036	234	69	complements	complement	NOUN
ejpam-3036	234	70	ψ′	ψ′	PUNCT
ejpam-3036	234	71	and	and	CCONJ
ejpam-3036	234	72	the	the	DET
ejpam-3036	234	73	set	set	NOUN
ejpam-3036	234	74	b(a	b(a	NOUN
ejpam-3036	234	75	)	)	PUNCT
ejpam-3036	234	76	a	a	DET
ejpam-3036	234	77	r	r	NOUN
ejpam-3036	234	78	j	j	PROPN
ejpam-3036	234	79	srikanth	srikanth	NOUN
ejpam-3036	234	80	,	,	PUNCT
ejpam-3036	234	81	r	r	NOUN
ejpam-3036	234	82	v	v	NUM
ejpam-3036	234	83	g	g	NOUN
ejpam-3036	234	84	ravi	ravi	PROPN
ejpam-3036	234	85	kumar	kumar	PROPN
ejpam-3036	234	86	/	/	SYM
ejpam-3036	234	87	eur	eur	PROPN
ejpam-3036	234	88	.	.	PUNCT
ejpam-3036	235	1	j.	j.	PROPN
ejpam-3036	235	2	pure	pure	PROPN
ejpam-3036	235	3	appl	appl	PROPN
ejpam-3036	235	4	.	.	PROPN
ejpam-3036	235	5	math	math	PROPN
ejpam-3036	235	6	,	,	PUNCT
ejpam-3036	235	7	10	10	NUM
ejpam-3036	235	8	(	(	PUNCT
ejpam-3036	235	9	4	4	NUM
ejpam-3036	235	10	)	)	PUNCT
ejpam-3036	235	11	(	(	PUNCT
ejpam-3036	235	12	2017	2017	NUM
ejpam-3036	235	13	)	)	PUNCT
ejpam-3036	235	14	,	,	PUNCT
ejpam-3036	235	15	717	717	NUM
ejpam-3036	235	16	-	-	SYM
ejpam-3036	235	17	729	729	NUM
ejpam-3036	235	18	727	727	NUM
ejpam-3036	235	19	of	of	ADP
ejpam-3036	235	20	all	all	DET
ejpam-3036	235	21	balanced	balanced	ADJ
ejpam-3036	235	22	factor	factor	NOUN
ejpam-3036	235	23	congruences	congruence	NOUN
ejpam-3036	235	24	which	which	PRON
ejpam-3036	235	25	admit	admit	VERB
ejpam-3036	235	26	a	a	DET
ejpam-3036	235	27	balanced	balanced	ADJ
ejpam-3036	235	28	complement	complement	NOUN
ejpam-3036	235	29	is	be	AUX
ejpam-3036	235	30	called	call	VERB
ejpam-3036	235	31	the	the	DET
ejpam-3036	235	32	boolean	boolean	ADJ
ejpam-3036	235	33	centre	centre	NOUN
ejpam-3036	235	34	of	of	ADP
ejpam-3036	235	35	a.	a.	NOUN
ejpam-3036	235	36	now	now	ADV
ejpam-3036	235	37	we	we	PRON
ejpam-3036	235	38	conclude	conclude	VERB
ejpam-3036	235	39	this	this	DET
ejpam-3036	235	40	section	section	NOUN
ejpam-3036	235	41	by	by	ADP
ejpam-3036	235	42	proving	prove	VERB
ejpam-3036	235	43	that	that	SCONJ
ejpam-3036	235	44	,	,	PUNCT
ejpam-3036	235	45	if	if	SCONJ
ejpam-3036	235	46	a	a	PRON
ejpam-3036	235	47	is	be	AUX
ejpam-3036	235	48	a	a	DET
ejpam-3036	235	49	core	core	NOUN
ejpam-3036	235	50	regular	regular	ADJ
ejpam-3036	235	51	double	double	ADJ
ejpam-3036	235	52	stone	stone	NOUN
ejpam-3036	235	53	algebra	algebra	NOUN
ejpam-3036	235	54	,	,	PUNCT
ejpam-3036	235	55	then	then	ADV
ejpam-3036	235	56	the	the	DET
ejpam-3036	235	57	boolean	boolean	ADJ
ejpam-3036	235	58	centre	centre	NOUN
ejpam-3036	235	59	b(a	b(a	PROPN
ejpam-3036	235	60	)	)	PUNCT
ejpam-3036	235	61	is	be	AUX
ejpam-3036	235	62	precisely	precisely	ADV
ejpam-3036	235	63	the	the	DET
ejpam-3036	235	64	set	set	NOUN
ejpam-3036	235	65	d	d	NOUN
ejpam-3036	235	66	=	=	PUNCT
ejpam-3036	235	67	{	{	PUNCT
ejpam-3036	235	68	θx	θx	INTJ
ejpam-3036	235	69	|	|	ADV
ejpam-3036	235	70	x	x	SYM
ejpam-3036	235	71	∈	∈	PROPN
ejpam-3036	235	72	k(a	k(a	PROPN
ejpam-3036	235	73	)	)	PUNCT
ejpam-3036	235	74	}	}	PUNCT
ejpam-3036	235	75	and	and	CCONJ
ejpam-3036	235	76	that	that	SCONJ
ejpam-3036	235	77	the	the	DET
ejpam-3036	235	78	map	map	NOUN
ejpam-3036	235	79	x	x	NOUN
ejpam-3036	235	80	7−→	7−→	NOUN
ejpam-3036	235	81	θx	θx	NOUN
ejpam-3036	235	82	is	be	AUX
ejpam-3036	235	83	an	an	DET
ejpam-3036	235	84	isomorphism	isomorphism	NOUN
ejpam-3036	235	85	of	of	ADP
ejpam-3036	235	86	k(a	k(a	NOUN
ejpam-3036	235	87	)	)	PUNCT
ejpam-3036	235	88	onto	onto	ADP
ejpam-3036	235	89	b(a	b(a	NOUN
ejpam-3036	235	90	)	)	PUNCT
ejpam-3036	235	91	.	.	PUNCT
ejpam-3036	236	1	first	first	ADV
ejpam-3036	236	2	we	we	PRON
ejpam-3036	236	3	prove	prove	VERB
ejpam-3036	236	4	the	the	DET
ejpam-3036	236	5	following	following	NOUN
ejpam-3036	236	6	.	.	PUNCT
ejpam-3036	237	1	lemma	lemma	PROPN
ejpam-3036	237	2	1	1	X
ejpam-3036	237	3	.	.	PUNCT
ejpam-3036	238	1	let	let	VERB
ejpam-3036	238	2	a	a	PRON
ejpam-3036	238	3	be	be	AUX
ejpam-3036	238	4	a	a	DET
ejpam-3036	238	5	crdsa	crdsa	NOUN
ejpam-3036	238	6	and	and	CCONJ
ejpam-3036	238	7	x	x	PART
ejpam-3036	238	8	∈	∈	PROPN
ejpam-3036	238	9	k(a	k(a	PROPN
ejpam-3036	238	10	)	)	PUNCT
ejpam-3036	238	11	.	.	PUNCT
ejpam-3036	239	1	then	then	ADV
ejpam-3036	239	2	θx	θx	INTJ
ejpam-3036	239	3	is	be	AUX
ejpam-3036	239	4	balanced	balanced	ADJ
ejpam-3036	239	5	.	.	PUNCT
ejpam-3036	240	1	proof	proof	NOUN
ejpam-3036	240	2	.	.	PUNCT
ejpam-3036	241	1	let	let	VERB
ejpam-3036	241	2	ψ	ψ	PART
ejpam-3036	241	3	be	be	AUX
ejpam-3036	241	4	a	a	DET
ejpam-3036	241	5	factor	factor	NOUN
ejpam-3036	241	6	congruence	congruence	NOUN
ejpam-3036	241	7	on	on	ADP
ejpam-3036	241	8	a	a	PRON
ejpam-3036	241	9	and	and	CCONJ
ejpam-3036	241	10	ψ′	ψ′	PUNCT
ejpam-3036	241	11	be	be	AUX
ejpam-3036	241	12	its	its	PRON
ejpam-3036	241	13	complement	complement	NOUN
ejpam-3036	241	14	.	.	PUNCT
ejpam-3036	242	1	then	then	ADV
ejpam-3036	242	2	there	there	PRON
ejpam-3036	242	3	exist	exist	VERB
ejpam-3036	242	4	y	y	PROPN
ejpam-3036	242	5	,	,	PUNCT
ejpam-3036	242	6	z	z	PROPN
ejpam-3036	242	7	∈	∈	PROPN
ejpam-3036	242	8	k(a	k(a	PROPN
ejpam-3036	242	9	)	)	PUNCT
ejpam-3036	242	10	such	such	ADJ
ejpam-3036	242	11	that	that	SCONJ
ejpam-3036	242	12	ψ	ψ	X
ejpam-3036	242	13	=	=	X
ejpam-3036	242	14	θy	θy	X
ejpam-3036	242	15	and	and	CCONJ
ejpam-3036	242	16	ψ′	ψ′	PROPN
ejpam-3036	242	17	=	=	SYM
ejpam-3036	242	18	θz	θz	PROPN
ejpam-3036	242	19	.	.	PUNCT
ejpam-3036	243	1	now	now	ADV
ejpam-3036	243	2	,	,	PUNCT
ejpam-3036	243	3	(	(	PUNCT
ejpam-3036	243	4	θx	θx	X
ejpam-3036	243	5	∨	∨	NUM
ejpam-3036	243	6	ψ	ψ	NOUN
ejpam-3036	243	7	)	)	PUNCT
ejpam-3036	243	8	∩	∩	NOUN
ejpam-3036	243	9	(	(	PUNCT
ejpam-3036	243	10	θx	θx	PROPN
ejpam-3036	243	11	∨	∨	NUM
ejpam-3036	243	12	ψ′	ψ′	NUM
ejpam-3036	243	13	)	)	PUNCT
ejpam-3036	243	14	=	=	PUNCT
ejpam-3036	243	15	(	(	PUNCT
ejpam-3036	243	16	θx	θx	PROPN
ejpam-3036	243	17	∨	∨	NUM
ejpam-3036	243	18	θy	θy	PART
ejpam-3036	243	19	)	)	PUNCT
ejpam-3036	243	20	∩	∩	NOUN
ejpam-3036	243	21	(	(	PUNCT
ejpam-3036	243	22	θx	θx	PROPN
ejpam-3036	243	23	∨	∨	NUM
ejpam-3036	243	24	θz	θz	NOUN
ejpam-3036	243	25	)	)	PUNCT
ejpam-3036	243	26	=	=	VERB
ejpam-3036	243	27	θx∧y	θx∧y	NOUN
ejpam-3036	243	28	∩	∩	ADJ
ejpam-3036	243	29	θx∧z	θx∧z	NOUN
ejpam-3036	243	30	=	=	SYM
ejpam-3036	243	31	θ(x∧y)∨(x∧z	θ(x∧y)∨(x∧z	NUM
ejpam-3036	243	32	)	)	PUNCT
ejpam-3036	243	33	=	=	SYM
ejpam-3036	243	34	θx∧(y∨z	θx∧(y∨z	NOUN
ejpam-3036	243	35	)	)	PUNCT
ejpam-3036	243	36	=	=	PUNCT
ejpam-3036	243	37	θx	θx	ADP
ejpam-3036	243	38	∨	∨	NUM
ejpam-3036	243	39	θy∨z	θy∨z	ADP
ejpam-3036	243	40	=	=	PUNCT
ejpam-3036	243	41	θx	θx	NUM
ejpam-3036	243	42	∨	∨	NOUN
ejpam-3036	243	43	(	(	PUNCT
ejpam-3036	243	44	θy	θy	PROPN
ejpam-3036	243	45	∩	∩	NOUN
ejpam-3036	243	46	θz	θz	NOUN
ejpam-3036	243	47	)	)	PUNCT
ejpam-3036	243	48	=	=	PUNCT
ejpam-3036	243	49	θx	θx	PROPN
ejpam-3036	243	50	∨	∨	NOUN
ejpam-3036	243	51	(	(	PUNCT
ejpam-3036	243	52	ψ	ψ	X
ejpam-3036	243	53	∩	∩	ADJ
ejpam-3036	243	54	ψ′	ψ′	NOUN
ejpam-3036	243	55	)	)	PUNCT
ejpam-3036	243	56	=	=	PUNCT
ejpam-3036	243	57	θx	θx	NOUN
ejpam-3036	243	58	∨∆a	∨∆a	ADV
ejpam-3036	243	59	=	=	PUNCT
ejpam-3036	243	60	θx	θx	X
ejpam-3036	243	61	therefore	therefore	ADV
ejpam-3036	243	62	,	,	PUNCT
ejpam-3036	243	63	θx	θx	X
ejpam-3036	243	64	is	be	AUX
ejpam-3036	243	65	balanced	balanced	ADJ
ejpam-3036	243	66	.	.	PUNCT
ejpam-3036	244	1	thus	thus	ADV
ejpam-3036	244	2	we	we	PRON
ejpam-3036	244	3	have	have	AUX
ejpam-3036	244	4	proved	prove	VERB
ejpam-3036	244	5	the	the	DET
ejpam-3036	244	6	following	following	NOUN
ejpam-3036	244	7	.	.	PUNCT
ejpam-3036	245	1	theorem	theorem	VERB
ejpam-3036	245	2	3.3	3.3	NUM
ejpam-3036	245	3	.	.	PUNCT
ejpam-3036	246	1	let	let	VERB
ejpam-3036	246	2	a	a	PRON
ejpam-3036	246	3	be	be	AUX
ejpam-3036	246	4	a	a	DET
ejpam-3036	246	5	crdsa	crdsa	NOUN
ejpam-3036	246	6	.	.	PUNCT
ejpam-3036	247	1	then	then	ADV
ejpam-3036	247	2	the	the	DET
ejpam-3036	247	3	boolean	boolean	ADJ
ejpam-3036	247	4	centre	centre	NOUN
ejpam-3036	247	5	b(a	b(a	PROPN
ejpam-3036	247	6	)	)	PUNCT
ejpam-3036	247	7	of	of	ADP
ejpam-3036	247	8	a	a	PRON
ejpam-3036	247	9	is	be	AUX
ejpam-3036	247	10	precisely	precisely	ADV
ejpam-3036	247	11	the	the	DET
ejpam-3036	247	12	set	set	NOUN
ejpam-3036	247	13	{	{	PUNCT
ejpam-3036	247	14	θx	θx	INTJ
ejpam-3036	247	15	|	|	ADV
ejpam-3036	247	16	x	x	SYM
ejpam-3036	247	17	∈	∈	PROPN
ejpam-3036	247	18	k(a	k(a	PROPN
ejpam-3036	247	19	)	)	PUNCT
ejpam-3036	247	20	}	}	PUNCT
ejpam-3036	247	21	.	.	PUNCT
ejpam-3036	248	1	the	the	DET
ejpam-3036	248	2	following	follow	VERB
ejpam-3036	248	3	theorem	theorem	NOUN
ejpam-3036	248	4	is	be	AUX
ejpam-3036	248	5	a	a	DET
ejpam-3036	248	6	consequence	consequence	NOUN
ejpam-3036	248	7	of	of	ADP
ejpam-3036	248	8	lemma	lemma	PROPN
ejpam-3036	248	9	1	1	NUM
ejpam-3036	248	10	and	and	CCONJ
ejpam-3036	248	11	above	above	ADV
ejpam-3036	248	12	theorem	theorem	VERB
ejpam-3036	248	13	3.3	3.3	NUM
ejpam-3036	248	14	theorem	theorem	NOUN
ejpam-3036	248	15	3.4	3.4	NUM
ejpam-3036	248	16	.	.	PUNCT
ejpam-3036	249	1	let	let	VERB
ejpam-3036	249	2	a	a	PRON
ejpam-3036	249	3	be	be	AUX
ejpam-3036	249	4	a	a	DET
ejpam-3036	249	5	crdsa	crdsa	NOUN
ejpam-3036	249	6	.	.	PUNCT
ejpam-3036	250	1	then	then	ADV
ejpam-3036	250	2	the	the	DET
ejpam-3036	250	3	boolean	boolean	ADJ
ejpam-3036	250	4	centre	centre	NOUN
ejpam-3036	250	5	b(a	b(a	PROPN
ejpam-3036	250	6	)	)	PUNCT
ejpam-3036	250	7	=	=	PUNCT
ejpam-3036	250	8	{	{	PUNCT
ejpam-3036	250	9	θx	θx	INTJ
ejpam-3036	250	10	|	|	ADV
ejpam-3036	250	11	x	x	SYM
ejpam-3036	250	12	∈	∈	PROPN
ejpam-3036	250	13	k(a	k(a	PROPN
ejpam-3036	250	14	)	)	PUNCT
ejpam-3036	250	15	}	}	PUNCT
ejpam-3036	250	16	of	of	ADP
ejpam-3036	250	17	a	a	PRON
ejpam-3036	250	18	,	,	PUNCT
ejpam-3036	250	19	is	be	AUX
ejpam-3036	250	20	a	a	DET
ejpam-3036	250	21	boolean	boolean	ADJ
ejpam-3036	250	22	algebra	algebra	NOUN
ejpam-3036	250	23	and	and	CCONJ
ejpam-3036	250	24	the	the	DET
ejpam-3036	250	25	map	map	NOUN
ejpam-3036	250	26	x	x	INTJ
ejpam-3036	250	27	7−→	7−→	NOUN
ejpam-3036	250	28	θx	θx	NOUN
ejpam-3036	250	29	is	be	AUX
ejpam-3036	250	30	an	an	DET
ejpam-3036	250	31	isomorphism	isomorphism	NOUN
ejpam-3036	250	32	of	of	ADP
ejpam-3036	250	33	k(a	k(a	NOUN
ejpam-3036	250	34	)	)	PUNCT
ejpam-3036	250	35	onto	onto	ADP
ejpam-3036	250	36	b(a	b(a	NOUN
ejpam-3036	250	37	)	)	PUNCT
ejpam-3036	250	38	.	.	PUNCT
ejpam-3036	251	1	4	4	X
ejpam-3036	251	2	.	.	X
ejpam-3036	251	3	birkhoff	birkhoff	NOUN
ejpam-3036	251	4	centre	centre	VERB
ejpam-3036	251	5	an	an	DET
ejpam-3036	251	6	element	element	NOUN
ejpam-3036	251	7	a	a	PRON
ejpam-3036	251	8	of	of	ADP
ejpam-3036	251	9	a	a	DET
ejpam-3036	251	10	bounded	bounded	ADJ
ejpam-3036	251	11	poset	poset	NOUN
ejpam-3036	251	12	p	p	NOUN
ejpam-3036	251	13	is	be	AUX
ejpam-3036	251	14	called	call	VERB
ejpam-3036	251	15	a	a	DET
ejpam-3036	251	16	‘	'	PUNCT
ejpam-3036	251	17	central	central	ADJ
ejpam-3036	251	18	element	element	NOUN
ejpam-3036	251	19	’	'	PUNCT
ejpam-3036	251	20	of	of	ADP
ejpam-3036	251	21	p	p	PRON
ejpam-3036	251	22	if	if	SCONJ
ejpam-3036	251	23	there	there	PRON
ejpam-3036	251	24	exist	exist	VERB
ejpam-3036	251	25	bounded	bounded	ADJ
ejpam-3036	251	26	posets	poset	NOUN
ejpam-3036	251	27	p1	p1	NOUN
ejpam-3036	251	28	and	and	CCONJ
ejpam-3036	251	29	p2	p2	PROPN
ejpam-3036	251	30	and	and	CCONJ
ejpam-3036	251	31	an	an	DET
ejpam-3036	251	32	order	order	NOUN
ejpam-3036	251	33	isomorphism	isomorphism	NOUN
ejpam-3036	251	34	of	of	ADP
ejpam-3036	251	35	p	p	NOUN
ejpam-3036	251	36	onto	onto	ADP
ejpam-3036	251	37	p1	p1	PROPN
ejpam-3036	251	38	×	×	NOUN
ejpam-3036	251	39	p2	p2	NOUN
ejpam-3036	251	40	such	such	ADJ
ejpam-3036	251	41	that	that	SCONJ
ejpam-3036	251	42	a	a	PRON
ejpam-3036	251	43	is	be	AUX
ejpam-3036	251	44	mapped	map	VERB
ejpam-3036	251	45	onto	onto	ADP
ejpam-3036	251	46	(	(	PUNCT
ejpam-3036	251	47	1	1	NUM
ejpam-3036	251	48	,	,	PUNCT
ejpam-3036	251	49	0	0	NUM
ejpam-3036	251	50	)	)	PUNCT
ejpam-3036	251	51	.	.	PUNCT
ejpam-3036	252	1	the	the	DET
ejpam-3036	252	2	set	set	NOUN
ejpam-3036	252	3	of	of	ADP
ejpam-3036	252	4	all	all	DET
ejpam-3036	252	5	central	central	ADJ
ejpam-3036	252	6	elements	element	NOUN
ejpam-3036	252	7	of	of	ADP
ejpam-3036	252	8	p	p	NOUN
ejpam-3036	252	9	are	be	AUX
ejpam-3036	252	10	called	call	VERB
ejpam-3036	252	11	the	the	DET
ejpam-3036	252	12	’	'	PUNCT
ejpam-3036	252	13	birkhoff	birkhoff	NOUN
ejpam-3036	252	14	centre	centre	NOUN
ejpam-3036	252	15	’	'	PUNCT
ejpam-3036	252	16	of	of	ADP
ejpam-3036	252	17	p	p	NOUN
ejpam-3036	252	18	and	and	CCONJ
ejpam-3036	252	19	is	be	AUX
ejpam-3036	252	20	denoted	denote	VERB
ejpam-3036	252	21	by	by	ADP
ejpam-3036	252	22	bc(p	bc(p	NOUN
ejpam-3036	252	23	)	)	PUNCT
ejpam-3036	252	24	.	.	PUNCT
ejpam-3036	253	1	it	it	PRON
ejpam-3036	253	2	is	be	AUX
ejpam-3036	253	3	known	know	VERB
ejpam-3036	253	4	that	that	SCONJ
ejpam-3036	253	5	bc(p	bc(p	NOUN
ejpam-3036	253	6	)	)	PUNCT
ejpam-3036	253	7	is	be	AUX
ejpam-3036	253	8	a	a	DET
ejpam-3036	253	9	boolean	boolean	ADJ
ejpam-3036	253	10	algebra	algebra	NOUN
ejpam-3036	253	11	in	in	ADP
ejpam-3036	253	12	which	which	PRON
ejpam-3036	253	13	the	the	DET
ejpam-3036	253	14	operations	operation	NOUN
ejpam-3036	253	15	are	be	AUX
ejpam-3036	253	16	g.l.b	g.l.b	ADJ
ejpam-3036	253	17	and	and	CCONJ
ejpam-3036	253	18	l.u.b	l.u.b	NOUN
ejpam-3036	253	19	with	with	ADP
ejpam-3036	253	20	respect	respect	NOUN
ejpam-3036	253	21	to	to	ADP
ejpam-3036	253	22	the	the	DET
ejpam-3036	253	23	partial	partial	ADJ
ejpam-3036	253	24	order	order	NOUN
ejpam-3036	253	25	in	in	ADP
ejpam-3036	253	26	p	p	NOUN
ejpam-3036	253	27	.	.	PUNCT
ejpam-3036	254	1	in	in	ADP
ejpam-3036	254	2	this	this	DET
ejpam-3036	254	3	section	section	NOUN
ejpam-3036	254	4	we	we	PRON
ejpam-3036	254	5	extend	extend	VERB
ejpam-3036	254	6	the	the	DET
ejpam-3036	254	7	concept	concept	NOUN
ejpam-3036	254	8	of	of	ADP
ejpam-3036	254	9	birkhoff	birkhoff	NOUN
ejpam-3036	254	10	centre	centre	NOUN
ejpam-3036	254	11	to	to	ADP
ejpam-3036	254	12	core	core	ADJ
ejpam-3036	254	13	regualr	regualr	PROPN
ejpam-3036	254	14	double	double	ADJ
ejpam-3036	254	15	stone	stone	NOUN
ejpam-3036	254	16	algebra	algebra	NOUN
ejpam-3036	254	17	.	.	PUNCT
ejpam-3036	255	1	a	a	DET
ejpam-3036	255	2	r	r	NOUN
ejpam-3036	255	3	j	j	PROPN
ejpam-3036	255	4	srikanth	srikanth	PROPN
ejpam-3036	255	5	,	,	PUNCT
ejpam-3036	255	6	r	r	NOUN
ejpam-3036	255	7	v	v	NUM
ejpam-3036	255	8	g	g	NOUN
ejpam-3036	255	9	ravi	ravi	PROPN
ejpam-3036	255	10	kumar	kumar	PROPN
ejpam-3036	255	11	/	/	SYM
ejpam-3036	255	12	eur	eur	PROPN
ejpam-3036	255	13	.	.	PUNCT
ejpam-3036	256	1	j.	j.	PROPN
ejpam-3036	256	2	pure	pure	PROPN
ejpam-3036	256	3	appl	appl	PROPN
ejpam-3036	256	4	.	.	PROPN
ejpam-3036	256	5	math	math	PROPN
ejpam-3036	256	6	,	,	PUNCT
ejpam-3036	256	7	10	10	NUM
ejpam-3036	256	8	(	(	PUNCT
ejpam-3036	256	9	4	4	NUM
ejpam-3036	256	10	)	)	PUNCT
ejpam-3036	256	11	(	(	PUNCT
ejpam-3036	256	12	2017	2017	NUM
ejpam-3036	256	13	)	)	PUNCT
ejpam-3036	256	14	,	,	PUNCT
ejpam-3036	256	15	717	717	NUM
ejpam-3036	256	16	-	-	SYM
ejpam-3036	256	17	729	729	NUM
ejpam-3036	256	18	728	728	NUM
ejpam-3036	256	19	definition	definition	NOUN
ejpam-3036	256	20	8	8	NUM
ejpam-3036	256	21	.	.	PUNCT
ejpam-3036	257	1	an	an	DET
ejpam-3036	257	2	element	element	NOUN
ejpam-3036	257	3	a	a	PRON
ejpam-3036	257	4	of	of	ADP
ejpam-3036	257	5	an	an	DET
ejpam-3036	257	6	rdsa	rdsa	NOUN
ejpam-3036	257	7	a	a	PRON
ejpam-3036	257	8	is	be	AUX
ejpam-3036	257	9	called	call	VERB
ejpam-3036	257	10	a	a	DET
ejpam-3036	257	11	birkhoff	birkhoff	NOUN
ejpam-3036	257	12	central	central	ADJ
ejpam-3036	257	13	element	element	NOUN
ejpam-3036	257	14	if	if	SCONJ
ejpam-3036	257	15	there	there	PRON
ejpam-3036	257	16	exist	exist	VERB
ejpam-3036	257	17	rdsas	rdsa	NOUN
ejpam-3036	257	18	a1	a1	NOUN
ejpam-3036	257	19	and	and	CCONJ
ejpam-3036	257	20	a2	a2	PROPN
ejpam-3036	257	21	and	and	CCONJ
ejpam-3036	257	22	an	an	DET
ejpam-3036	257	23	isomorphism	isomorphism	NOUN
ejpam-3036	257	24	a	a	PRON
ejpam-3036	257	25	onto	onto	ADP
ejpam-3036	257	26	a1	a1	NOUN
ejpam-3036	257	27	×a2	×a2	PROPN
ejpam-3036	257	28	such	such	ADJ
ejpam-3036	257	29	that	that	SCONJ
ejpam-3036	257	30	a	a	PRON
ejpam-3036	257	31	is	be	AUX
ejpam-3036	257	32	mapped	map	VERB
ejpam-3036	257	33	onto	onto	ADP
ejpam-3036	257	34	(	(	PUNCT
ejpam-3036	257	35	1	1	NUM
ejpam-3036	257	36	,	,	PUNCT
ejpam-3036	257	37	0	0	NUM
ejpam-3036	257	38	)	)	PUNCT
ejpam-3036	257	39	.	.	PUNCT
ejpam-3036	258	1	the	the	DET
ejpam-3036	258	2	set	set	NOUN
ejpam-3036	258	3	bc(a	bc(a	NUM
ejpam-3036	258	4	)	)	PUNCT
ejpam-3036	258	5	of	of	ADP
ejpam-3036	258	6	all	all	DET
ejpam-3036	258	7	central	central	ADJ
ejpam-3036	258	8	elements	element	NOUN
ejpam-3036	258	9	of	of	ADP
ejpam-3036	258	10	p	p	NOUN
ejpam-3036	258	11	is	be	AUX
ejpam-3036	258	12	called	call	VERB
ejpam-3036	258	13	the	the	DET
ejpam-3036	258	14	birkhoff	birkhoff	NOUN
ejpam-3036	258	15	centre	centre	NOUN
ejpam-3036	258	16	.	.	PUNCT
ejpam-3036	259	1	recall	recall	VERB
ejpam-3036	259	2	that	that	SCONJ
ejpam-3036	259	3	the	the	DET
ejpam-3036	259	4	ideal	ideal	NOUN
ejpam-3036	259	5	generated	generate	VERB
ejpam-3036	259	6	by	by	ADP
ejpam-3036	259	7	an	an	DET
ejpam-3036	259	8	element	element	NOUN
ejpam-3036	259	9	x	x	PUNCT
ejpam-3036	259	10	of	of	ADP
ejpam-3036	259	11	a	a	PRON
ejpam-3036	259	12	in	in	ADP
ejpam-3036	259	13	a	a	DET
ejpam-3036	259	14	rdsa	rdsa	NOUN
ejpam-3036	259	15	is	be	AUX
ejpam-3036	259	16	called	call	VERB
ejpam-3036	259	17	a	a	DET
ejpam-3036	259	18	relativized	relativized	ADJ
ejpam-3036	259	19	algebra	algebra	NOUN
ejpam-3036	259	20	and	and	CCONJ
ejpam-3036	259	21	is	be	AUX
ejpam-3036	259	22	denoted	denote	VERB
ejpam-3036	259	23	by	by	ADP
ejpam-3036	259	24	(	(	PUNCT
ejpam-3036	259	25	x]a	x]a	ADJ
ejpam-3036	259	26	.	.	PUNCT
ejpam-3036	260	1	in	in	ADP
ejpam-3036	260	2	[	[	X
ejpam-3036	260	3	9	9	X
ejpam-3036	260	4	]	]	PUNCT
ejpam-3036	260	5	it	it	PRON
ejpam-3036	260	6	is	be	AUX
ejpam-3036	260	7	proved	prove	VERB
ejpam-3036	260	8	that	that	SCONJ
ejpam-3036	260	9	if	if	SCONJ
ejpam-3036	260	10	a	a	PRON
ejpam-3036	260	11	in	in	ADP
ejpam-3036	260	12	a	a	DET
ejpam-3036	260	13	crdsa	crdsa	NOUN
ejpam-3036	260	14	with	with	ADP
ejpam-3036	260	15	core	core	NOUN
ejpam-3036	260	16	element	element	NOUN
ejpam-3036	260	17	k	k	PROPN
ejpam-3036	260	18	then	then	ADV
ejpam-3036	260	19	for	for	ADP
ejpam-3036	260	20	x	x	PROPN
ejpam-3036	260	21	∈	∈	PROPN
ejpam-3036	260	22	at(c(a)),the	at(c(a)),the	DET
ejpam-3036	260	23	relativized	relativized	ADJ
ejpam-3036	260	24	algebra	algebra	NOUN
ejpam-3036	260	25	(	(	PUNCT
ejpam-3036	260	26	x]a	x]a	ADJ
ejpam-3036	260	27	is	be	AUX
ejpam-3036	260	28	a	a	DET
ejpam-3036	260	29	three	three	NUM
ejpam-3036	260	30	element	element	NOUN
ejpam-3036	260	31	chain	chain	NOUN
ejpam-3036	260	32	i.e.	i.e.	X
ejpam-3036	260	33	a	a	DET
ejpam-3036	260	34	discrete	discrete	NOUN
ejpam-3036	260	35	crdsa	crdsa	ADV
ejpam-3036	260	36	.	.	PUNCT
ejpam-3036	261	1	in	in	ADP
ejpam-3036	261	2	fact	fact	NOUN
ejpam-3036	261	3	we	we	PRON
ejpam-3036	261	4	have	have	VERB
ejpam-3036	261	5	the	the	DET
ejpam-3036	261	6	following	follow	VERB
ejpam-3036	261	7	theorem	theorem	VERB
ejpam-3036	261	8	.	.	PUNCT
ejpam-3036	261	9	theorem	theorem	VERB
ejpam-3036	261	10	4.1	4.1	NUM
ejpam-3036	261	11	.	.	PUNCT
ejpam-3036	262	1	let	let	VERB
ejpam-3036	262	2	a	a	PRON
ejpam-3036	262	3	be	be	AUX
ejpam-3036	262	4	a	a	DET
ejpam-3036	262	5	core	core	NOUN
ejpam-3036	262	6	regular	regular	ADJ
ejpam-3036	262	7	double	double	ADJ
ejpam-3036	262	8	stone	stone	NOUN
ejpam-3036	262	9	algebra	algebra	NOUN
ejpam-3036	262	10	.	.	PUNCT
ejpam-3036	263	1	the	the	DET
ejpam-3036	263	2	relativized	relativized	ADJ
ejpam-3036	263	3	algebra	algebra	NOUN
ejpam-3036	263	4	(	(	PUNCT
ejpam-3036	263	5	a]a	a]a	NOUN
ejpam-3036	263	6	is	be	AUX
ejpam-3036	263	7	a	a	DET
ejpam-3036	263	8	crdsa	crdsa	NOUN
ejpam-3036	263	9	if	if	SCONJ
ejpam-3036	264	1	and	and	CCONJ
ejpam-3036	264	2	only	only	ADV
ejpam-3036	264	3	if	if	SCONJ
ejpam-3036	264	4	a	a	DET
ejpam-3036	264	5	∈	∈	PROPN
ejpam-3036	264	6	k(a	k(a	NOUN
ejpam-3036	264	7	)	)	PUNCT
ejpam-3036	264	8	.	.	PUNCT
ejpam-3036	265	1	proof	proof	NOUN
ejpam-3036	265	2	.	.	PUNCT
ejpam-3036	266	1	assume	assume	VERB
ejpam-3036	266	2	that	that	SCONJ
ejpam-3036	266	3	a	a	DET
ejpam-3036	266	4	∈	∈	PROPN
ejpam-3036	266	5	k(a	k(a	NOUN
ejpam-3036	266	6	)	)	PUNCT
ejpam-3036	266	7	.	.	PUNCT
ejpam-3036	267	1	then	then	ADV
ejpam-3036	267	2	a∗	a∗	PROPN
ejpam-3036	267	3	∨	∨	NUM
ejpam-3036	267	4	a	a	DET
ejpam-3036	267	5	=	=	SYM
ejpam-3036	267	6	1	1	NUM
ejpam-3036	267	7	and	and	CCONJ
ejpam-3036	267	8	a+	a+	PUNCT
ejpam-3036	267	9	∧	∧	PROPN
ejpam-3036	267	10	a	a	PRON
ejpam-3036	267	11	=	=	NOUN
ejpam-3036	267	12	0	0	NUM
ejpam-3036	267	13	.	.	PUNCT
ejpam-3036	268	1	it	it	PRON
ejpam-3036	268	2	is	be	AUX
ejpam-3036	268	3	a	a	DET
ejpam-3036	268	4	routine	routine	ADJ
ejpam-3036	268	5	verification	verification	NOUN
ejpam-3036	268	6	that	that	PRON
ejpam-3036	268	7	(	(	PUNCT
ejpam-3036	268	8	a]a	a]a	NOUN
ejpam-3036	268	9	=	=	SYM
ejpam-3036	268	10	(	(	PUNCT
ejpam-3036	268	11	(	(	PUNCT
ejpam-3036	268	12	a],∧,∨	a],∧,∨	ADJ
ejpam-3036	268	13	,	,	PUNCT
ejpam-3036	268	14	∗a,+a	∗a,+a	NOUN
ejpam-3036	268	15	,	,	PUNCT
ejpam-3036	268	16	0	0	NUM
ejpam-3036	268	17	,	,	PUNCT
ejpam-3036	268	18	a	a	PRON
ejpam-3036	268	19	)	)	PUNCT
ejpam-3036	268	20	is	be	AUX
ejpam-3036	268	21	a	a	DET
ejpam-3036	268	22	double	double	ADJ
ejpam-3036	268	23	stone	stone	NOUN
ejpam-3036	268	24	algebra	algebra	NOUN
ejpam-3036	268	25	where	where	SCONJ
ejpam-3036	268	26	a	a	PRON
ejpam-3036	268	27	is	be	AUX
ejpam-3036	268	28	the	the	DET
ejpam-3036	268	29	greatest	great	ADJ
ejpam-3036	268	30	element	element	NOUN
ejpam-3036	268	31	and	and	CCONJ
ejpam-3036	268	32	for	for	ADP
ejpam-3036	268	33	x	x	PROPN
ejpam-3036	268	34	∈	∈	PROPN
ejpam-3036	268	35	(	(	PUNCT
ejpam-3036	268	36	a	a	X
ejpam-3036	268	37	]	]	X
ejpam-3036	268	38	,	,	PUNCT
ejpam-3036	268	39	x∗a	x∗a	PUNCT
ejpam-3036	268	40	=	=	SYM
ejpam-3036	268	41	x∗	x∗	PROPN
ejpam-3036	268	42	∧	∧	PROPN
ejpam-3036	268	43	a	a	PRON
ejpam-3036	268	44	and	and	CCONJ
ejpam-3036	268	45	x+a	x+a	NUM
ejpam-3036	268	46	=	=	SYM
ejpam-3036	268	47	x+	x+	PUNCT
ejpam-3036	268	48	∧	∧	NOUN
ejpam-3036	268	49	a.	a.	NOUN
ejpam-3036	268	50	to	to	PART
ejpam-3036	268	51	prove	prove	VERB
ejpam-3036	268	52	that	that	SCONJ
ejpam-3036	268	53	(	(	PUNCT
ejpam-3036	268	54	a]a	a]a	NOUN
ejpam-3036	268	55	is	be	AUX
ejpam-3036	268	56	regular	regular	ADJ
ejpam-3036	268	57	consider	consider	VERB
ejpam-3036	268	58	x	x	PRON
ejpam-3036	268	59	,	,	PUNCT
ejpam-3036	268	60	y	y	PROPN
ejpam-3036	268	61	∈	∈	PROPN
ejpam-3036	268	62	(	(	PUNCT
ejpam-3036	268	63	a]a	a]a	NOUN
ejpam-3036	268	64	such	such	ADJ
ejpam-3036	268	65	that	that	SCONJ
ejpam-3036	268	66	x∗a	x∗a	PUNCT
ejpam-3036	269	1	=	=	SYM
ejpam-3036	269	2	y∗a	y∗a	PROPN
ejpam-3036	269	3	and	and	CCONJ
ejpam-3036	269	4	x+a	x+a	NUM
ejpam-3036	269	5	=	=	SYM
ejpam-3036	269	6	y+a	y+a	NUM
ejpam-3036	269	7	,	,	PUNCT
ejpam-3036	269	8	that	that	ADV
ejpam-3036	269	9	is	is	ADV
ejpam-3036	269	10	,	,	PUNCT
ejpam-3036	269	11	x∗	x∗	PROPN
ejpam-3036	269	12	∧	∧	PROPN
ejpam-3036	269	13	a	a	DET
ejpam-3036	269	14	=	=	X
ejpam-3036	269	15	y∗	y∗	NOUN
ejpam-3036	269	16	∧	∧	PROPN
ejpam-3036	269	17	a	a	PRON
ejpam-3036	269	18	and	and	CCONJ
ejpam-3036	269	19	x+	x+	ADJ
ejpam-3036	269	20	∧	∧	PROPN
ejpam-3036	269	21	a	a	DET
ejpam-3036	269	22	=	=	SYM
ejpam-3036	269	23	y+	y+	NUM
ejpam-3036	269	24	∧	∧	PROPN
ejpam-3036	269	25	a.	a.	NOUN
ejpam-3036	269	26	then	then	ADV
ejpam-3036	269	27	(	(	PUNCT
ejpam-3036	269	28	x∗	x∗	PROPN
ejpam-3036	269	29	∧	∧	PROPN
ejpam-3036	269	30	a	a	PRON
ejpam-3036	269	31	)	)	PUNCT
ejpam-3036	269	32	∨	∨	NOUN
ejpam-3036	269	33	a∗	a∗	NOUN
ejpam-3036	269	34	=	=	SYM
ejpam-3036	269	35	(	(	PUNCT
ejpam-3036	269	36	y∗	y∗	ADV
ejpam-3036	269	37	∧	∧	PROPN
ejpam-3036	269	38	a	a	PRON
ejpam-3036	269	39	)	)	PUNCT
ejpam-3036	269	40	∨	∨	NUM
ejpam-3036	269	41	a∗	a∗	NOUN
ejpam-3036	269	42	and	and	CCONJ
ejpam-3036	269	43	(	(	PUNCT
ejpam-3036	269	44	x+	x+	X
ejpam-3036	269	45	∧	∧	PROPN
ejpam-3036	269	46	a	a	NOUN
ejpam-3036	269	47	)	)	PUNCT
ejpam-3036	269	48	∨	∨	NUM
ejpam-3036	269	49	a+	a+	PUNCT
ejpam-3036	269	50	=	=	SYM
ejpam-3036	269	51	y+	y+	NUM
ejpam-3036	269	52	∧	∧	PROPN
ejpam-3036	269	53	a	a	DET
ejpam-3036	269	54	⇒	⇒	NOUN
ejpam-3036	269	55	(	(	PUNCT
ejpam-3036	269	56	x∗	x∗	PROPN
ejpam-3036	269	57	∨	∨	NUM
ejpam-3036	269	58	a∗	a∗	PROPN
ejpam-3036	269	59	)	)	PUNCT
ejpam-3036	269	60	∧	∧	PROPN
ejpam-3036	269	61	(	(	PUNCT
ejpam-3036	269	62	a	a	DET
ejpam-3036	269	63	∨	∨	NOUN
ejpam-3036	269	64	a∗	a∗	NOUN
ejpam-3036	269	65	)	)	PUNCT
ejpam-3036	269	66	=	=	PUNCT
ejpam-3036	269	67	(	(	PUNCT
ejpam-3036	269	68	y∗	y∗	PROPN
ejpam-3036	269	69	∨	∨	NUM
ejpam-3036	269	70	a∗	a∗	ADJ
ejpam-3036	269	71	)	)	PUNCT
ejpam-3036	269	72	∧	∧	PROPN
ejpam-3036	269	73	(	(	PUNCT
ejpam-3036	269	74	a	a	DET
ejpam-3036	269	75	∨	∨	NOUN
ejpam-3036	269	76	a∗	a∗	NOUN
ejpam-3036	269	77	)	)	PUNCT
ejpam-3036	269	78	and	and	CCONJ
ejpam-3036	269	79	(	(	PUNCT
ejpam-3036	269	80	x+	x+	ADJ
ejpam-3036	269	81	∨	∨	NUM
ejpam-3036	269	82	a+	a+	NOUN
ejpam-3036	269	83	)	)	PUNCT
ejpam-3036	269	84	∧	∧	NOUN
ejpam-3036	269	85	(	(	PUNCT
ejpam-3036	269	86	a	a	DET
ejpam-3036	269	87	∨	∨	NUM
ejpam-3036	269	88	a+	a+	PUNCT
ejpam-3036	269	89	)	)	PUNCT
ejpam-3036	269	90	=	=	PUNCT
ejpam-3036	269	91	(	(	PUNCT
ejpam-3036	269	92	y+	y+	NUM
ejpam-3036	269	93	∨	∨	NUM
ejpam-3036	269	94	a+	a+	NOUN
ejpam-3036	269	95	)	)	PUNCT
ejpam-3036	269	96	∧	∧	NOUN
ejpam-3036	269	97	(	(	PUNCT
ejpam-3036	269	98	a	a	DET
ejpam-3036	269	99	∨	∨	NUM
ejpam-3036	269	100	a+)−	a+)−	NOUN
ejpam-3036	269	101	(	(	PUNCT
ejpam-3036	269	102	∗	∗	NOUN
ejpam-3036	269	103	)	)	PUNCT
ejpam-3036	269	104	since	since	SCONJ
ejpam-3036	269	105	x	x	X
ejpam-3036	269	106	,	,	PUNCT
ejpam-3036	269	107	y	y	PROPN
ejpam-3036	269	108	∈	∈	PROPN
ejpam-3036	269	109	(	(	PUNCT
ejpam-3036	269	110	a	a	X
ejpam-3036	269	111	]	]	X
ejpam-3036	269	112	we	we	PRON
ejpam-3036	269	113	have	have	AUX
ejpam-3036	269	114	x	x	X
ejpam-3036	269	115	,	,	PUNCT
ejpam-3036	269	116	y	y	PROPN
ejpam-3036	269	117	≤	≤	PROPN
ejpam-3036	269	118	a⇒	a⇒	PUNCT
ejpam-3036	269	119	a∗	a∗	PROPN
ejpam-3036	269	120	≤	≤	NUM
ejpam-3036	269	121	x∗	x∗	NOUN
ejpam-3036	269	122	,	,	PUNCT
ejpam-3036	269	123	y∗	y∗	ADV
ejpam-3036	269	124	and	and	CCONJ
ejpam-3036	269	125	a+	a+	PUNCT
ejpam-3036	269	126	≤	≤	NUM
ejpam-3036	269	127	x+	x+	NUM
ejpam-3036	269	128	,	,	PUNCT
ejpam-3036	269	129	y+	y+	X
ejpam-3036	269	130	.	.	PROPN
ejpam-3036	269	131	also	also	ADV
ejpam-3036	269	132	since	since	SCONJ
ejpam-3036	269	133	a∗∨a	a∗∨a	PROPN
ejpam-3036	269	134	=	=	NOUN
ejpam-3036	269	135	1	1	NUM
ejpam-3036	269	136	therefore	therefore	ADV
ejpam-3036	269	137	(	(	PUNCT
ejpam-3036	269	138	*	*	PUNCT
ejpam-3036	269	139	)	)	PUNCT
ejpam-3036	269	140	gives	give	VERB
ejpam-3036	269	141	x∗	x∗	NOUN
ejpam-3036	269	142	=	=	PUNCT
ejpam-3036	270	1	y∗	y∗	PROPN
ejpam-3036	270	2	and	and	CCONJ
ejpam-3036	270	3	x+	x+	NUM
ejpam-3036	270	4	=	=	NOUN
ejpam-3036	270	5	y+	y+	PROPN
ejpam-3036	270	6	and	and	CCONJ
ejpam-3036	270	7	by	by	ADP
ejpam-3036	270	8	regularity	regularity	NOUN
ejpam-3036	270	9	in	in	ADP
ejpam-3036	270	10	a	a	PRON
ejpam-3036	270	11	,	,	PUNCT
ejpam-3036	270	12	x	x	X
ejpam-3036	270	13	=	=	PUNCT
ejpam-3036	270	14	y.	y.	PROPN
ejpam-3036	270	15	hence	hence	ADV
ejpam-3036	270	16	(	(	PUNCT
ejpam-3036	270	17	a]a	a]a	NOUN
ejpam-3036	270	18	is	be	AUX
ejpam-3036	270	19	a	a	DET
ejpam-3036	270	20	regular	regular	ADJ
ejpam-3036	270	21	double	double	ADJ
ejpam-3036	270	22	stone	stone	NOUN
ejpam-3036	270	23	algebra	algebra	NOUN
ejpam-3036	270	24	.	.	PUNCT
ejpam-3036	271	1	moreover	moreover	ADV
ejpam-3036	271	2	a∧	a∧	PROPN
ejpam-3036	271	3	k	k	PROPN
ejpam-3036	271	4	∈	∈	PROPN
ejpam-3036	271	5	(	(	PUNCT
ejpam-3036	271	6	a	a	X
ejpam-3036	271	7	]	]	X
ejpam-3036	271	8	and	and	CCONJ
ejpam-3036	271	9	(	(	PUNCT
ejpam-3036	271	10	a∧	a∧	NOUN
ejpam-3036	271	11	k)∗a	k)∗a	NOUN
ejpam-3036	271	12	=	=	SYM
ejpam-3036	271	13	a∧	a∧	NOUN
ejpam-3036	271	14	k∗	k∗	NOUN
ejpam-3036	271	15	=	=	SYM
ejpam-3036	271	16	0	0	NUM
ejpam-3036	271	17	,	,	PUNCT
ejpam-3036	271	18	(	(	PUNCT
ejpam-3036	271	19	a∧	a∧	NOUN
ejpam-3036	271	20	k)+a	k)+a	PROPN
ejpam-3036	271	21	=	=	PUNCT
ejpam-3036	271	22	a	a	DET
ejpam-3036	271	23	∧	∧	PROPN
ejpam-3036	271	24	k+	k+	NOUN
ejpam-3036	271	25	=	=	NOUN
ejpam-3036	271	26	a.	a.	NOUN
ejpam-3036	271	27	theretofore	theretofore	VERB
ejpam-3036	271	28	a	a	DET
ejpam-3036	271	29	∧	∧	PROPN
ejpam-3036	271	30	k	k	PROPN
ejpam-3036	271	31	is	be	AUX
ejpam-3036	271	32	the	the	DET
ejpam-3036	271	33	core	core	NOUN
ejpam-3036	271	34	element	element	NOUN
ejpam-3036	271	35	of	of	ADP
ejpam-3036	271	36	(	(	PUNCT
ejpam-3036	271	37	a]a	a]a	NOUN
ejpam-3036	271	38	.	.	PUNCT
ejpam-3036	272	1	so	so	ADV
ejpam-3036	272	2	(	(	PUNCT
ejpam-3036	272	3	a]a	a]a	NOUN
ejpam-3036	272	4	is	be	AUX
ejpam-3036	272	5	a	a	DET
ejpam-3036	272	6	crdsa	crdsa	NOUN
ejpam-3036	272	7	.	.	PUNCT
ejpam-3036	273	1	conversely	conversely	ADV
ejpam-3036	273	2	suppose	suppose	VERB
ejpam-3036	273	3	that	that	SCONJ
ejpam-3036	273	4	for	for	ADP
ejpam-3036	273	5	a	a	DET
ejpam-3036	273	6	∈	∈	PROPN
ejpam-3036	273	7	a	a	PRON
ejpam-3036	273	8	,	,	PUNCT
ejpam-3036	273	9	(	(	PUNCT
ejpam-3036	273	10	a]a	a]a	NOUN
ejpam-3036	273	11	=	=	SYM
ejpam-3036	273	12	(	(	PUNCT
ejpam-3036	273	13	(	(	PUNCT
ejpam-3036	273	14	a],∧,∨	a],∧,∨	ADJ
ejpam-3036	273	15	,	,	PUNCT
ejpam-3036	273	16	∗a,+a	∗a,+a	NOUN
ejpam-3036	273	17	,	,	PUNCT
ejpam-3036	273	18	0	0	NUM
ejpam-3036	273	19	,	,	PUNCT
ejpam-3036	273	20	a	a	PRON
ejpam-3036	273	21	)	)	PUNCT
ejpam-3036	273	22	is	be	AUX
ejpam-3036	273	23	a	a	DET
ejpam-3036	273	24	crdsa	crdsa	NOUN
ejpam-3036	273	25	with	with	ADP
ejpam-3036	273	26	the	the	DET
ejpam-3036	273	27	above	above	ADV
ejpam-3036	273	28	defined	define	VERB
ejpam-3036	273	29	operations	operation	NOUN
ejpam-3036	273	30	.	.	PUNCT
ejpam-3036	274	1	since	since	SCONJ
ejpam-3036	274	2	a	a	PRON
ejpam-3036	274	3	is	be	AUX
ejpam-3036	274	4	the	the	DET
ejpam-3036	274	5	greatest	great	ADJ
ejpam-3036	274	6	element	element	NOUN
ejpam-3036	274	7	of	of	ADP
ejpam-3036	274	8	(	(	PUNCT
ejpam-3036	274	9	a]a	a]a	PROPN
ejpam-3036	274	10	,	,	PUNCT
ejpam-3036	274	11	we	we	PRON
ejpam-3036	274	12	have	have	VERB
ejpam-3036	274	13	a∗a	a∗a	X
ejpam-3036	274	14	=	=	SYM
ejpam-3036	274	15	a+a	a+a	X
ejpam-3036	274	16	and	and	CCONJ
ejpam-3036	274	17	therefore	therefore	ADV
ejpam-3036	274	18	a+	a+	PUNCT
ejpam-3036	274	19	∧	∧	PROPN
ejpam-3036	274	20	a	a	DET
ejpam-3036	274	21	=	=	NOUN
ejpam-3036	274	22	0	0	NUM
ejpam-3036	274	23	.	.	PUNCT
ejpam-3036	275	1	hence	hence	ADV
ejpam-3036	275	2	a	a	PRON
ejpam-3036	275	3	is	be	AUX
ejpam-3036	275	4	complimented	complimented	ADJ
ejpam-3036	275	5	element	element	NOUN
ejpam-3036	275	6	.	.	PUNCT
ejpam-3036	276	1	so	so	ADV
ejpam-3036	276	2	a	a	DET
ejpam-3036	276	3	∈	∈	PROPN
ejpam-3036	276	4	c(a	c(a	NOUN
ejpam-3036	276	5	)	)	PUNCT
ejpam-3036	276	6	=	=	SYM
ejpam-3036	276	7	k(a	k(a	NOUN
ejpam-3036	276	8	)	)	PUNCT
ejpam-3036	276	9	by	by	ADP
ejpam-3036	276	10	the	the	DET
ejpam-3036	276	11	principle	principle	NOUN
ejpam-3036	276	12	of	of	ADP
ejpam-3036	276	13	duality	duality	NOUN
ejpam-3036	276	14	and	and	CCONJ
ejpam-3036	276	15	theorem	theorem	VERB
ejpam-3036	276	16	2.6	2.6	NUM
ejpam-3036	276	17	we	we	PRON
ejpam-3036	276	18	have	have	VERB
ejpam-3036	276	19	the	the	DET
ejpam-3036	276	20	following	follow	VERB
ejpam-3036	276	21	theorem	theorem	VERB
ejpam-3036	276	22	.	.	PUNCT
ejpam-3036	277	1	theorem	theorem	VERB
ejpam-3036	277	2	4.2	4.2	NUM
ejpam-3036	277	3	.	.	PUNCT
ejpam-3036	278	1	let	let	VERB
ejpam-3036	278	2	a	a	PRON
ejpam-3036	278	3	be	be	AUX
ejpam-3036	278	4	a	a	DET
ejpam-3036	278	5	core	core	NOUN
ejpam-3036	278	6	regular	regular	ADJ
ejpam-3036	278	7	double	double	ADJ
ejpam-3036	278	8	stone	stone	NOUN
ejpam-3036	278	9	algebra	algebra	NOUN
ejpam-3036	278	10	.	.	PUNCT
ejpam-3036	279	1	then	then	ADV
ejpam-3036	279	2	relativized	relativized	ADJ
ejpam-3036	279	3	algebra	algebra	NOUN
ejpam-3036	279	4	[	[	X
ejpam-3036	279	5	a)a	a)a	X
ejpam-3036	279	6	=	=	SYM
ejpam-3036	279	7	(	(	PUNCT
ejpam-3036	279	8	[	[	X
ejpam-3036	279	9	a),∧,∨	a),∧,∨	ADJ
ejpam-3036	279	10	,	,	PUNCT
ejpam-3036	279	11	∗a,+a	∗a,+a	NOUN
ejpam-3036	279	12	,	,	PUNCT
ejpam-3036	279	13	a	a	DET
ejpam-3036	279	14	,	,	PUNCT
ejpam-3036	279	15	1	1	NUM
ejpam-3036	279	16	)	)	PUNCT
ejpam-3036	279	17	is	be	AUX
ejpam-3036	279	18	a	a	DET
ejpam-3036	279	19	crdsa	crdsa	ADJ
ejpam-3036	279	20	ifand	ifand	NOUN
ejpam-3036	279	21	only	only	ADV
ejpam-3036	279	22	if	if	SCONJ
ejpam-3036	279	23	a	a	DET
ejpam-3036	279	24	∈	∈	PROPN
ejpam-3036	279	25	k(a	k(a	NOUN
ejpam-3036	279	26	)	)	PUNCT
ejpam-3036	279	27	.	.	PUNCT
ejpam-3036	280	1	theorem	theorem	VERB
ejpam-3036	280	2	4.3	4.3	NUM
ejpam-3036	280	3	.	.	PUNCT
ejpam-3036	281	1	let	let	VERB
ejpam-3036	281	2	a	a	PRON
ejpam-3036	281	3	be	be	AUX
ejpam-3036	281	4	a	a	DET
ejpam-3036	281	5	core	core	NOUN
ejpam-3036	281	6	regular	regular	ADJ
ejpam-3036	281	7	double	double	ADJ
ejpam-3036	281	8	stone	stone	NOUN
ejpam-3036	281	9	algebra	algebra	NOUN
ejpam-3036	281	10	.	.	PUNCT
ejpam-3036	282	1	a	a	DET
ejpam-3036	282	2	∈	∈	PROPN
ejpam-3036	282	3	bc(a	bc(a	NUM
ejpam-3036	282	4	)	)	PUNCT
ejpam-3036	282	5	if	if	SCONJ
ejpam-3036	282	6	and	and	CCONJ
ejpam-3036	282	7	only	only	ADV
ejpam-3036	282	8	if	if	SCONJ
ejpam-3036	282	9	a	a	DET
ejpam-3036	282	10	∈	∈	PROPN
ejpam-3036	282	11	k(a	k(a	NOUN
ejpam-3036	282	12	)	)	PUNCT
ejpam-3036	282	13	.	.	PUNCT
ejpam-3036	283	1	proof	proof	NOUN
ejpam-3036	283	2	.	.	PUNCT
ejpam-3036	284	1	let	let	VERB
ejpam-3036	284	2	a	a	DET
ejpam-3036	284	3	∈	∈	NOUN
ejpam-3036	284	4	bc(a	bc(a	NUM
ejpam-3036	284	5	)	)	PUNCT
ejpam-3036	284	6	.	.	PUNCT
ejpam-3036	285	1	then	then	ADV
ejpam-3036	285	2	there	there	PRON
ejpam-3036	285	3	exist	exist	VERB
ejpam-3036	285	4	crdsas	crdsa	NOUN
ejpam-3036	285	5	a1	a1	NOUN
ejpam-3036	285	6	and	and	CCONJ
ejpam-3036	285	7	a2	a2	PROPN
ejpam-3036	285	8	and	and	CCONJ
ejpam-3036	285	9	an	an	DET
ejpam-3036	285	10	isomorphism	isomorphism	NOUN
ejpam-3036	285	11	f	f	PROPN
ejpam-3036	285	12	from	from	ADP
ejpam-3036	285	13	a	a	PRON
ejpam-3036	285	14	onto	onto	ADP
ejpam-3036	285	15	a1	a1	NOUN
ejpam-3036	285	16	×a2	×a2	PROPN
ejpam-3036	285	17	such	such	ADJ
ejpam-3036	285	18	that	that	SCONJ
ejpam-3036	285	19	a	a	PRON
ejpam-3036	285	20	is	be	AUX
ejpam-3036	285	21	mapped	map	VERB
ejpam-3036	285	22	onto	onto	ADP
ejpam-3036	285	23	(	(	PUNCT
ejpam-3036	285	24	1	1	NUM
ejpam-3036	285	25	,	,	PUNCT
ejpam-3036	285	26	0	0	NUM
ejpam-3036	285	27	)	)	PUNCT
ejpam-3036	285	28	.	.	PUNCT
ejpam-3036	286	1	by	by	ADP
ejpam-3036	286	2	theorem	theorem	ADJ
ejpam-3036	286	3	2.4	2.4	NUM
ejpam-3036	286	4	,	,	PUNCT
ejpam-3036	286	5	c(a	c(a	ADV
ejpam-3036	286	6	)	)	PUNCT
ejpam-3036	286	7	is	be	AUX
ejpam-3036	286	8	isomorphic	isomorphic	ADJ
ejpam-3036	286	9	to	to	PART
ejpam-3036	286	10	c(a1	c(a1	NOUN
ejpam-3036	286	11	)	)	PUNCT
ejpam-3036	286	12	×	×	NOUN
ejpam-3036	286	13	c(a2	c(a2	NOUN
ejpam-3036	286	14	)	)	PUNCT
ejpam-3036	286	15	and	and	CCONJ
ejpam-3036	286	16	(	(	PUNCT
ejpam-3036	286	17	1	1	NUM
ejpam-3036	286	18	,	,	PUNCT
ejpam-3036	286	19	0	0	NUM
ejpam-3036	286	20	)	)	PUNCT
ejpam-3036	286	21	∈	∈	PROPN
ejpam-3036	286	22	c(a1	c(a1	NOUN
ejpam-3036	286	23	)	)	PUNCT
ejpam-3036	286	24	×	×	NOUN
ejpam-3036	286	25	c(a2	c(a2	NOUN
ejpam-3036	286	26	)	)	PUNCT
ejpam-3036	286	27	which	which	PRON
ejpam-3036	286	28	in	in	ADP
ejpam-3036	286	29	turn	turn	NOUN
ejpam-3036	286	30	gives	give	VERB
ejpam-3036	286	31	a	a	DET
ejpam-3036	286	32	∈	∈	PROPN
ejpam-3036	286	33	c(a	c(a	NOUN
ejpam-3036	286	34	)	)	PUNCT
ejpam-3036	286	35	and	and	CCONJ
ejpam-3036	286	36	hence	hence	ADV
ejpam-3036	286	37	by	by	ADP
ejpam-3036	286	38	theorem	theorem	ADJ
ejpam-3036	286	39	2.8	2.8	NUM
ejpam-3036	286	40	a	a	DET
ejpam-3036	286	41	∈	∈	PROPN
ejpam-3036	286	42	k(a	k(a	NOUN
ejpam-3036	286	43	)	)	PUNCT
ejpam-3036	286	44	.	.	PUNCT
ejpam-3036	287	1	references	reference	NOUN
ejpam-3036	287	2	729	729	PRON
ejpam-3036	287	3	conversely	conversely	ADV
ejpam-3036	287	4	suppose	suppose	VERB
ejpam-3036	287	5	that	that	SCONJ
ejpam-3036	287	6	a	a	DET
ejpam-3036	287	7	∈	∈	PROPN
ejpam-3036	287	8	k(a	k(a	NOUN
ejpam-3036	287	9	)	)	PUNCT
ejpam-3036	287	10	.	.	PUNCT
ejpam-3036	288	1	by	by	ADP
ejpam-3036	288	2	theorems	theorem	NOUN
ejpam-3036	288	3	4.1	4.1	NUM
ejpam-3036	288	4	and	and	CCONJ
ejpam-3036	288	5	4.2	4.2	NUM
ejpam-3036	288	6	(	(	PUNCT
ejpam-3036	288	7	a]a	a]a	NOUN
ejpam-3036	288	8	and	and	CCONJ
ejpam-3036	288	9	[	[	X
ejpam-3036	288	10	a)a	a)a	X
ejpam-3036	288	11	are	be	AUX
ejpam-3036	288	12	crdsas	crdsa	NOUN
ejpam-3036	288	13	.	.	PUNCT
ejpam-3036	289	1	now	now	ADV
ejpam-3036	289	2	define	define	VERB
ejpam-3036	289	3	a	a	DET
ejpam-3036	289	4	map	map	NOUN
ejpam-3036	289	5	f	f	X
ejpam-3036	289	6	:	:	PUNCT
ejpam-3036	289	7	a→	a→	X
ejpam-3036	289	8	(	(	PUNCT
ejpam-3036	289	9	a]a	a]a	NOUN
ejpam-3036	289	10	×	×	NOUN
ejpam-3036	290	1	[	[	X
ejpam-3036	290	2	a)a	a)a	NOUN
ejpam-3036	290	3	by	by	ADP
ejpam-3036	290	4	f(x	f(x	PROPN
ejpam-3036	290	5	)	)	PUNCT
ejpam-3036	290	6	=	=	PUNCT
ejpam-3036	291	1	(	(	PUNCT
ejpam-3036	291	2	a∧	a∧	NOUN
ejpam-3036	291	3	x	x	PROPN
ejpam-3036	291	4	,	,	PUNCT
ejpam-3036	291	5	a∨	a∨	PROPN
ejpam-3036	291	6	x	x	X
ejpam-3036	291	7	)	)	PUNCT
ejpam-3036	291	8	.	.	PUNCT
ejpam-3036	292	1	then	then	ADV
ejpam-3036	292	2	f	f	PROPN
ejpam-3036	292	3	is	be	AUX
ejpam-3036	292	4	a	a	DET
ejpam-3036	292	5	isomorphism	isomorphism	NOUN
ejpam-3036	292	6	from	from	ADP
ejpam-3036	292	7	a	a	DET
ejpam-3036	292	8	onto	onto	NOUN
ejpam-3036	292	9	(	(	PUNCT
ejpam-3036	292	10	a]×	a]×	PROPN
ejpam-3036	292	11	[	[	X
ejpam-3036	292	12	a)a	a)a	X
ejpam-3036	292	13	,	,	PUNCT
ejpam-3036	292	14	such	such	ADJ
ejpam-3036	292	15	that	that	DET
ejpam-3036	292	16	f(a	f(a	NOUN
ejpam-3036	292	17	)	)	PUNCT
ejpam-3036	292	18	=	=	PRON
ejpam-3036	293	1	(	(	PUNCT
ejpam-3036	293	2	a	a	PRON
ejpam-3036	293	3	,	,	PUNCT
ejpam-3036	293	4	a	a	NOUN
ejpam-3036	293	5	)	)	PUNCT
ejpam-3036	293	6	=	=	SYM
ejpam-3036	293	7	(	(	PUNCT
ejpam-3036	293	8	1	1	NUM
ejpam-3036	293	9	,	,	PUNCT
ejpam-3036	293	10	0	0	NUM
ejpam-3036	293	11	)	)	PUNCT
ejpam-3036	293	12	.	.	PUNCT
ejpam-3036	294	1	hence	hence	ADV
ejpam-3036	294	2	a	a	DET
ejpam-3036	294	3	∈	∈	PROPN
ejpam-3036	294	4	bc(a	bc(a	NUM
ejpam-3036	294	5	)	)	PUNCT
ejpam-3036	294	6	.	.	PUNCT
ejpam-3036	295	1	thus	thus	ADV
ejpam-3036	295	2	we	we	PRON
ejpam-3036	295	3	have	have	AUX
ejpam-3036	295	4	proved	prove	VERB
ejpam-3036	295	5	the	the	DET
ejpam-3036	295	6	following	following	NOUN
ejpam-3036	295	7	.	.	PUNCT
ejpam-3036	296	1	theorem	theorem	VERB
ejpam-3036	296	2	4.4	4.4	NUM
ejpam-3036	296	3	.	.	PUNCT
ejpam-3036	297	1	let	let	VERB
ejpam-3036	297	2	a	a	PRON
ejpam-3036	297	3	be	be	AUX
ejpam-3036	297	4	a	a	DET
ejpam-3036	297	5	crdsa	crdsa	NOUN
ejpam-3036	297	6	.	.	PUNCT
ejpam-3036	298	1	then	then	ADV
ejpam-3036	298	2	the	the	DET
ejpam-3036	298	3	birkhoff	birkhoff	NOUN
ejpam-3036	298	4	centre	centre	PROPN
ejpam-3036	298	5	bc(a	bc(a	NUM
ejpam-3036	298	6	)	)	PUNCT
ejpam-3036	298	7	of	of	ADP
ejpam-3036	298	8	a	a	PRON
ejpam-3036	298	9	is	be	AUX
ejpam-3036	298	10	precisely	precisely	ADV
ejpam-3036	298	11	the	the	DET
ejpam-3036	298	12	set	set	NOUN
ejpam-3036	298	13	{	{	PUNCT
ejpam-3036	298	14	a	a	DET
ejpam-3036	298	15	|	|	NOUN
ejpam-3036	298	16	a	a	DET
ejpam-3036	298	17	∈	∈	PROPN
ejpam-3036	298	18	k(a	k(a	NOUN
ejpam-3036	298	19	)	)	PUNCT
ejpam-3036	298	20	}	}	PUNCT
ejpam-3036	298	21	=	=	SYM
ejpam-3036	298	22	{	{	PUNCT
ejpam-3036	298	23	a	a	DET
ejpam-3036	298	24	|	|	NOUN
ejpam-3036	298	25	a	a	DET
ejpam-3036	298	26	∈	∈	PROPN
ejpam-3036	298	27	c(a	c(a	NOUN
ejpam-3036	298	28	)	)	PUNCT
ejpam-3036	298	29	}	}	PUNCT
ejpam-3036	298	30	.	.	PUNCT
ejpam-3036	299	1	the	the	DET
ejpam-3036	299	2	following	follow	VERB
ejpam-3036	299	3	theorem	theorem	NOUN
ejpam-3036	299	4	is	be	AUX
ejpam-3036	299	5	a	a	DET
ejpam-3036	299	6	consequence	consequence	NOUN
ejpam-3036	299	7	of	of	ADP
ejpam-3036	299	8	theorem	theorem	ADJ
ejpam-3036	299	9	3.4	3.4	NUM
ejpam-3036	299	10	and	and	CCONJ
ejpam-3036	299	11	above	above	ADV
ejpam-3036	299	12	theorem	theorem	VERB
ejpam-3036	299	13	4.4	4.4	NUM
ejpam-3036	299	14	.	.	PUNCT
ejpam-3036	300	1	theorem	theorem	VERB
ejpam-3036	300	2	4.5	4.5	NUM
ejpam-3036	300	3	.	.	PUNCT
ejpam-3036	301	1	let	let	VERB
ejpam-3036	301	2	a	a	PRON
ejpam-3036	301	3	be	be	AUX
ejpam-3036	301	4	a	a	DET
ejpam-3036	301	5	crdsa	crdsa	NOUN
ejpam-3036	301	6	.	.	PUNCT
ejpam-3036	302	1	then	then	ADV
ejpam-3036	302	2	the	the	DET
ejpam-3036	302	3	boolean	boolean	ADJ
ejpam-3036	302	4	centre	centre	NOUN
ejpam-3036	302	5	b(a	b(a	PROPN
ejpam-3036	302	6	)	)	PUNCT
ejpam-3036	302	7	of	of	ADP
ejpam-3036	302	8	a	a	PRON
ejpam-3036	302	9	,	,	PUNCT
ejpam-3036	302	10	is	be	AUX
ejpam-3036	302	11	isomorphic	isomorphic	ADJ
ejpam-3036	302	12	to	to	ADP
ejpam-3036	302	13	birkhoff	birkhoff	PROPN
ejpam-3036	302	14	centre	centre	PROPN
ejpam-3036	302	15	bc(a	bc(a	NUM
ejpam-3036	302	16	)	)	PUNCT
ejpam-3036	302	17	of	of	ADP
ejpam-3036	302	18	a	a	PRON
ejpam-3036	302	19	.	.	PUNCT
ejpam-3036	303	1	references	reference	NOUN
ejpam-3036	303	2	[	[	X
ejpam-3036	303	3	1	1	NUM
ejpam-3036	303	4	]	]	X
ejpam-3036	303	5	balbes	balbe	NOUN
ejpam-3036	303	6	r.	r.	NOUN
ejpam-3036	303	7	and	and	CCONJ
ejpam-3036	303	8	dwinger	dwinger	PROPN
ejpam-3036	303	9	ph	ph	PROPN
ejpam-3036	303	10	.	.	PROPN
ejpam-3036	303	11	,	,	PUNCT
ejpam-3036	303	12	distributive	distributive	ADJ
ejpam-3036	303	13	lattices	lattice	NOUN
ejpam-3036	303	14	,	,	PUNCT
ejpam-3036	303	15	u.missouri	u.missouri	NOUN
ejpam-3036	303	16	press	press	NOUN
ejpam-3036	303	17	(	(	PUNCT
ejpam-3036	303	18	1974	1974	NUM
ejpam-3036	303	19	)	)	PUNCT
ejpam-3036	303	20	.	.	PUNCT
ejpam-3036	304	1	[	[	X
ejpam-3036	304	2	2	2	NUM
ejpam-3036	304	3	]	]	X
ejpam-3036	304	4	grätzer.george	grätzer.george	NOUN
ejpam-3036	304	5	and	and	CCONJ
ejpam-3036	304	6	schmidt	schmidt	PROPN
ejpam-3036	304	7	,	,	PUNCT
ejpam-3036	304	8	e.	e.	PROPN
ejpam-3036	304	9	t.	t.	PROPN
ejpam-3036	304	10	on	on	ADP
ejpam-3036	304	11	a	a	DET
ejpam-3036	304	12	problem	problem	NOUN
ejpam-3036	304	13	of	of	ADP
ejpam-3036	304	14	m.	m.	NOUN
ejpam-3036	304	15	h.	h.	PROPN
ejpam-3036	304	16	stone	stone	PROPN
ejpam-3036	304	17	,	,	PUNCT
ejpam-3036	304	18	acta	acta	PROPN
ejpam-3036	304	19	mathematica	mathematica	PROPN
ejpam-3036	304	20	academiae	academiae	PROPN
ejpam-3036	304	21	scientiarum	scientiarum	PROPN
ejpam-3036	304	22	hungaricae	hungaricae	PROPN
ejpam-3036	304	23	,	,	PUNCT
ejpam-3036	304	24	8	8	NUM
ejpam-3036	304	25	(	(	PUNCT
ejpam-3036	304	26	1957	1957	NUM
ejpam-3036	304	27	)	)	PUNCT
ejpam-3036	304	28	,	,	PUNCT
ejpam-3036	304	29	455–460	455–460	NUM
ejpam-3036	304	30	.	.	PUNCT
ejpam-3036	305	1	[	[	X
ejpam-3036	305	2	3	3	NUM
ejpam-3036	305	3	]	]	X
ejpam-3036	305	4	grätzer.george	grätzer.george	NOUN
ejpam-3036	305	5	,	,	PUNCT
ejpam-3036	305	6	lattice	lattice	PROPN
ejpam-3036	305	7	theory	theory	NOUN
ejpam-3036	305	8	.	.	PUNCT
ejpam-3036	306	1	first	first	ADJ
ejpam-3036	306	2	concepts	concept	NOUN
ejpam-3036	306	3	and	and	CCONJ
ejpam-3036	306	4	distributive	distributive	ADJ
ejpam-3036	306	5	lattices	lattice	NOUN
ejpam-3036	306	6	,	,	PUNCT
ejpam-3036	306	7	w.	w.	PROPN
ejpam-3036	306	8	h.	h.	PROPN
ejpam-3036	306	9	freeman	freeman	PROPN
ejpam-3036	306	10	and	and	CCONJ
ejpam-3036	306	11	co.	co.	PROPN
ejpam-3036	306	12	san	san	PROPN
ejpam-3036	306	13	francisco	francisco	PROPN
ejpam-3036	306	14	,	,	PUNCT
ejpam-3036	306	15	1971	1971	NUM
ejpam-3036	306	16	.	.	PUNCT
ejpam-3036	307	1	[	[	X
ejpam-3036	307	2	4	4	X
ejpam-3036	307	3	]	]	X
ejpam-3036	307	4	g.	g.	PROPN
ejpam-3036	307	5	c.	c.	PROPN
ejpam-3036	307	6	moisll	moisll	PROPN
ejpam-3036	307	7	,	,	PUNCT
ejpam-3036	307	8	sur	sur	PROPN
ejpam-3036	307	9	les	les	X
ejpam-3036	307	10	logiques	logique	NOUN
ejpam-3036	307	11	de	de	PROPN
ejpam-3036	307	12	,	,	PUNCT
ejpam-3036	307	13	lukasiewicz	lukasiewicz	VERB
ejpam-3036	307	14	a	a	DET
ejpam-3036	307	15	un	un	PROPN
ejpam-3036	307	16	nombre	nombre	PROPN
ejpam-3036	307	17	fini	fini	PROPN
ejpam-3036	307	18	de	de	X
ejpam-3036	307	19	valeurs	valeurs	X
ejpam-3036	307	20	,	,	PUNCT
ejpam-3036	307	21	rev	rev	PROPN
ejpam-3036	307	22	.	.	PROPN
ejpam-3036	307	23	roumaine	roumaine	PROPN
ejpam-3036	307	24	math	math	NOUN
ejpam-3036	307	25	.	.	PUNCT
ejpam-3036	308	1	pures	pure	NOUN
ejpam-3036	308	2	appl	appl	PROPN
ejpam-3036	308	3	,	,	PUNCT
ejpam-3036	308	4	9	9	NUM
ejpam-3036	308	5	(	(	PUNCT
ejpam-3036	308	6	1964	1964	NUM
ejpam-3036	308	7	)	)	PUNCT
ejpam-3036	308	8	,	,	PUNCT
ejpam-3036	308	9	905–920	905–920	NUM
ejpam-3036	308	10	.	.	PUNCT
ejpam-3036	309	1	[	[	X
ejpam-3036	309	2	5	5	X
ejpam-3036	309	3	]	]	PUNCT
ejpam-3036	309	4	ravi	ravi	PROPN
ejpam-3036	309	5	kumar	kumar	PROPN
ejpam-3036	309	6	.	.	PUNCT
ejpam-3036	310	1	r.v.g	r.v.g	NOUN
ejpam-3036	310	2	,	,	PUNCT
ejpam-3036	310	3	m.p.k.kishore	m.p.k.kishore	NOUN
ejpam-3036	310	4	and	and	CCONJ
ejpam-3036	310	5	a.r.j.srikanth	a.r.j.srikanth	PROPN
ejpam-3036	310	6	,	,	PUNCT
ejpam-3036	310	7	core	core	NOUN
ejpam-3036	310	8	regular	regular	ADJ
ejpam-3036	310	9	double	double	ADJ
ejpam-3036	310	10	stone	stone	NOUN
ejpam-3036	310	11	algebra	algebra	NOUN
ejpam-3036	310	12	,	,	PUNCT
ejpam-3036	310	13	journal	journal	NOUN
ejpam-3036	310	14	of	of	ADP
ejpam-3036	310	15	calcutta	calcutta	PROPN
ejpam-3036	310	16	mathematical	mathematical	ADJ
ejpam-3036	310	17	society	society	NOUN
ejpam-3036	310	18	,	,	PUNCT
ejpam-3036	310	19	11	11	NUM
ejpam-3036	310	20	(	(	PUNCT
ejpam-3036	310	21	2015	2015	NUM
ejpam-3036	310	22	)	)	PUNCT
ejpam-3036	310	23	1–10	1–10	NOUN
ejpam-3036	310	24	.	.	PUNCT
ejpam-3036	311	1	[	[	X
ejpam-3036	311	2	6	6	NUM
ejpam-3036	311	3	]	]	SYM
ejpam-3036	311	4	swamy.u.m	swamy.u.m	NOUN
ejpam-3036	311	5	.	.	PUNCT
ejpam-3036	312	1	and	and	CCONJ
ejpam-3036	312	2	murthy.g.s	murthy.g.s	PROPN
ejpam-3036	312	3	.	.	PROPN
ejpam-3036	312	4	,	,	PUNCT
ejpam-3036	312	5	boolean	boolean	ADJ
ejpam-3036	312	6	centre	centre	NOUN
ejpam-3036	312	7	of	of	ADP
ejpam-3036	312	8	a	a	DET
ejpam-3036	312	9	universal	universal	ADJ
ejpam-3036	312	10	algebra	algebra	NOUN
ejpam-3036	312	11	algebra	algebra	NOUN
ejpam-3036	312	12	universalis	universali	VERB
ejpam-3036	312	13	,	,	PUNCT
ejpam-3036	312	14	13	13	NUM
ejpam-3036	312	15	(	(	PUNCT
ejpam-3036	312	16	1981	1981	NUM
ejpam-3036	312	17	)	)	PUNCT
ejpam-3036	312	18	202–205	202–205	NUM
ejpam-3036	312	19	.	.	PUNCT
ejpam-3036	313	1	[	[	X
ejpam-3036	313	2	7	7	NUM
ejpam-3036	313	3	]	]	X
ejpam-3036	313	4	varlet	varlet	NOUN
ejpam-3036	313	5	,	,	PUNCT
ejpam-3036	313	6	j.	j.	PROPN
ejpam-3036	313	7	,	,	PUNCT
ejpam-3036	313	8	a	a	DET
ejpam-3036	313	9	regular	regular	ADJ
ejpam-3036	313	10	variety	variety	NOUN
ejpam-3036	313	11	of	of	ADP
ejpam-3036	313	12	type	type	NOUN
ejpam-3036	313	13	<	<	PROPN
ejpam-3036	313	14	2	2	NUM
ejpam-3036	313	15	,	,	PUNCT
ejpam-3036	313	16	2	2	NUM
ejpam-3036	313	17	,	,	PUNCT
ejpam-3036	313	18	1	1	NUM
ejpam-3036	313	19	,	,	PUNCT
ejpam-3036	313	20	1	1	NUM
ejpam-3036	313	21	,	,	PUNCT
ejpam-3036	313	22	0	0	NUM
ejpam-3036	313	23	,	,	PUNCT
ejpam-3036	313	24	0	0	NUM
ejpam-3036	313	25	>	>	PUNCT
ejpam-3036	313	26	,	,	PUNCT
ejpam-3036	313	27	algebra	algebra	PROPN
ejpam-3036	313	28	universalis	universali	VERB
ejpam-3036	313	29	,	,	PUNCT
ejpam-3036	313	30	2	2	NUM
ejpam-3036	313	31	(	(	PUNCT
ejpam-3036	313	32	1972	1972	NUM
ejpam-3036	313	33	)	)	PUNCT
ejpam-3036	313	34	218	218	NUM
ejpam-3036	313	35	-	-	SYM
ejpam-3036	313	36	223	223	NUM
ejpam-3036	313	37	.	.	PUNCT