id	sid	tid	token	lemma	pos
ejpam-5930	1	1	european	european	PROPN
ejpam-5930	1	2	journal	journal	PROPN
ejpam-5930	1	3	of	of	ADP
ejpam-5930	1	4	pure	pure	ADJ
ejpam-5930	1	5	and	and	CCONJ
ejpam-5930	1	6	applied	applied	ADJ
ejpam-5930	1	7	mathematics	mathematic	NOUN
ejpam-5930	1	8	2025	2025	NUM
ejpam-5930	1	9	,	,	PUNCT
ejpam-5930	1	10	vol	vol	NOUN
ejpam-5930	1	11	.	.	PROPN
ejpam-5930	1	12	18	18	NUM
ejpam-5930	1	13	,	,	PUNCT
ejpam-5930	1	14	issue	issue	NOUN
ejpam-5930	1	15	2	2	NUM
ejpam-5930	1	16	,	,	PUNCT
ejpam-5930	1	17	article	article	NOUN
ejpam-5930	1	18	number	number	NOUN
ejpam-5930	1	19	5930	5930	NUM
ejpam-5930	1	20	issn	issn	PROPN
ejpam-5930	1	21	1307	1307	NUM
ejpam-5930	1	22	-	-	SYM
ejpam-5930	1	23	5543	5543	NUM
ejpam-5930	1	24	–	–	PUNCT
ejpam-5930	1	25	ejpam.com	ejpam.com	X
ejpam-5930	1	26	published	publish	VERB
ejpam-5930	1	27	by	by	ADP
ejpam-5930	1	28	new	new	PROPN
ejpam-5930	1	29	york	york	PROPN
ejpam-5930	1	30	business	business	PROPN
ejpam-5930	1	31	global	global	PROPN
ejpam-5930	1	32	a	a	DET
ejpam-5930	1	33	theoretical	theoretical	ADJ
ejpam-5930	1	34	exploration	exploration	NOUN
ejpam-5930	1	35	of	of	ADP
ejpam-5930	1	36	rough	rough	ADJ
ejpam-5930	1	37	approximations	approximation	NOUN
ejpam-5930	1	38	in	in	ADP
ejpam-5930	1	39	hilbert	hilbert	PROPN
ejpam-5930	1	40	algebras	algebras	PROPN
ejpam-5930	1	41	aiyared	aiyare	VERB
ejpam-5930	1	42	iampan1,∗	iampan1,∗	PROPN
ejpam-5930	1	43	,	,	PUNCT
ejpam-5930	1	44	r.	r.	PROPN
ejpam-5930	1	45	vennila2	vennila2	PROPN
ejpam-5930	1	46	,	,	PUNCT
ejpam-5930	1	47	neelamegarajan	neelamegarajan	PROPN
ejpam-5930	1	48	rajesh3	rajesh3	PROPN
ejpam-5930	1	49	,	,	PUNCT
ejpam-5930	1	50	ramasamy	ramasamy	NOUN
ejpam-5930	1	51	subasini4	subasini4	PROPN
ejpam-5930	1	52	1	1	NUM
ejpam-5930	1	53	department	department	NOUN
ejpam-5930	1	54	of	of	ADP
ejpam-5930	1	55	mathematics	mathematic	NOUN
ejpam-5930	1	56	,	,	PUNCT
ejpam-5930	1	57	school	school	NOUN
ejpam-5930	1	58	of	of	ADP
ejpam-5930	1	59	science	science	NOUN
ejpam-5930	1	60	,	,	PUNCT
ejpam-5930	1	61	university	university	NOUN
ejpam-5930	1	62	of	of	ADP
ejpam-5930	1	63	phayao	phayao	NOUN
ejpam-5930	1	64	,	,	PUNCT
ejpam-5930	1	65	mae	mae	PROPN
ejpam-5930	1	66	ka	ka	PROPN
ejpam-5930	1	67	,	,	PUNCT
ejpam-5930	1	68	mueang	mueang	PROPN
ejpam-5930	1	69	,	,	PUNCT
ejpam-5930	1	70	phayao	phayao	NOUN
ejpam-5930	1	71	56000	56000	NUM
ejpam-5930	1	72	,	,	PUNCT
ejpam-5930	1	73	thailand	thailand	PROPN
ejpam-5930	1	74	2	2	NUM
ejpam-5930	1	75	7405	7405	NUM
ejpam-5930	1	76	goreway	goreway	NOUN
ejpam-5930	1	77	drive	drive	NOUN
ejpam-5930	1	78	,	,	PUNCT
ejpam-5930	1	79	mississauga	mississauga	PROPN
ejpam-5930	1	80	l4t0a3	l4t0a3	PROPN
ejpam-5930	1	81	,	,	PUNCT
ejpam-5930	1	82	canada	canada	PROPN
ejpam-5930	1	83	3	3	NUM
ejpam-5930	1	84	department	department	PROPN
ejpam-5930	1	85	of	of	ADP
ejpam-5930	1	86	mathematics	mathematics	PROPN
ejpam-5930	1	87	,	,	PUNCT
ejpam-5930	1	88	rajah	rajah	NOUN
ejpam-5930	1	89	serfoji	serfoji	ADJ
ejpam-5930	1	90	government	government	NOUN
ejpam-5930	1	91	college	college	NOUN
ejpam-5930	1	92	,	,	PUNCT
ejpam-5930	1	93	thanjavur-613005	thanjavur-613005	NOUN
ejpam-5930	1	94	,	,	PUNCT
ejpam-5930	1	95	tamil	tamil	PROPN
ejpam-5930	1	96	nadu	nadu	NOUN
ejpam-5930	1	97	,	,	PUNCT
ejpam-5930	1	98	india	india	PROPN
ejpam-5930	1	99	4	4	NUM
ejpam-5930	1	100	department	department	NOUN
ejpam-5930	1	101	of	of	ADP
ejpam-5930	1	102	mathematics	mathematics	PROPN
ejpam-5930	1	103	,	,	PUNCT
ejpam-5930	1	104	pollachi	pollachi	PROPN
ejpam-5930	1	105	institute	institute	PROPN
ejpam-5930	1	106	of	of	ADP
ejpam-5930	1	107	engineering	engineering	NOUN
ejpam-5930	1	108	and	and	CCONJ
ejpam-5930	1	109	technology	technology	NOUN
ejpam-5930	1	110	,	,	PUNCT
ejpam-5930	1	111	pollachi-642205	pollachi-642205	ADV
ejpam-5930	1	112	,	,	PUNCT
ejpam-5930	1	113	tamil	tamil	PROPN
ejpam-5930	1	114	nadu	nadu	NOUN
ejpam-5930	1	115	,	,	PUNCT
ejpam-5930	1	116	india	india	PROPN
ejpam-5930	1	117	abstract	abstract	NOUN
ejpam-5930	1	118	.	.	PUNCT
ejpam-5930	2	1	in	in	ADP
ejpam-5930	2	2	this	this	DET
ejpam-5930	2	3	paper	paper	NOUN
ejpam-5930	2	4	,	,	PUNCT
ejpam-5930	2	5	we	we	PRON
ejpam-5930	2	6	introduce	introduce	VERB
ejpam-5930	2	7	the	the	DET
ejpam-5930	2	8	concept	concept	NOUN
ejpam-5930	2	9	of	of	ADP
ejpam-5930	2	10	roughness	roughness	NOUN
ejpam-5930	2	11	in	in	ADP
ejpam-5930	2	12	the	the	DET
ejpam-5930	2	13	context	context	NOUN
ejpam-5930	2	14	of	of	ADP
ejpam-5930	2	15	hilbert	hilbert	PROPN
ejpam-5930	2	16	algebras	algebras	PROPN
ejpam-5930	2	17	,	,	PUNCT
ejpam-5930	2	18	a	a	DET
ejpam-5930	2	19	class	class	NOUN
ejpam-5930	2	20	of	of	ADP
ejpam-5930	2	21	algebraic	algebraic	ADJ
ejpam-5930	2	22	structures	structure	NOUN
ejpam-5930	2	23	fundamental	fundamental	ADJ
ejpam-5930	2	24	to	to	ADP
ejpam-5930	2	25	studying	study	VERB
ejpam-5930	2	26	non	non	ADJ
ejpam-5930	2	27	-	-	ADJ
ejpam-5930	2	28	classical	classical	ADJ
ejpam-5930	2	29	logic	logic	NOUN
ejpam-5930	2	30	.	.	PUNCT
ejpam-5930	3	1	by	by	ADP
ejpam-5930	3	2	integrating	integrate	VERB
ejpam-5930	3	3	rough	rough	ADJ
ejpam-5930	3	4	set	set	NOUN
ejpam-5930	3	5	theory	theory	NOUN
ejpam-5930	3	6	with	with	ADP
ejpam-5930	3	7	hilbert	hilbert	PROPN
ejpam-5930	3	8	algebras	algebras	PROPN
ejpam-5930	3	9	,	,	PUNCT
ejpam-5930	3	10	we	we	PRON
ejpam-5930	3	11	investigate	investigate	VERB
ejpam-5930	3	12	the	the	DET
ejpam-5930	3	13	lower	low	ADJ
ejpam-5930	3	14	and	and	CCONJ
ejpam-5930	3	15	upper	upper	ADJ
ejpam-5930	3	16	approximations	approximation	NOUN
ejpam-5930	3	17	of	of	ADP
ejpam-5930	3	18	subalgebras	subalgebra	NOUN
ejpam-5930	3	19	and	and	CCONJ
ejpam-5930	3	20	ideals	ideal	NOUN
ejpam-5930	3	21	.	.	PUNCT
ejpam-5930	4	1	we	we	PRON
ejpam-5930	4	2	show	show	VERB
ejpam-5930	4	3	that	that	SCONJ
ejpam-5930	4	4	the	the	DET
ejpam-5930	4	5	lower	low	ADJ
ejpam-5930	4	6	and	and	CCONJ
ejpam-5930	4	7	upper	upper	ADJ
ejpam-5930	4	8	approximations	approximation	NOUN
ejpam-5930	4	9	of	of	ADP
ejpam-5930	4	10	a	a	DET
ejpam-5930	4	11	subalgebra	subalgebra	NOUN
ejpam-5930	4	12	(	(	PUNCT
ejpam-5930	4	13	or	or	CCONJ
ejpam-5930	4	14	ideal	ideal	ADJ
ejpam-5930	4	15	)	)	PUNCT
ejpam-5930	4	16	in	in	ADP
ejpam-5930	4	17	a	a	DET
ejpam-5930	4	18	hilbert	hilbert	NOUN
ejpam-5930	4	19	algebra	algebra	NOUN
ejpam-5930	4	20	also	also	ADV
ejpam-5930	4	21	make	make	VERB
ejpam-5930	4	22	up	up	ADP
ejpam-5930	4	23	a	a	DET
ejpam-5930	4	24	subalgebra	subalgebra	NOUN
ejpam-5930	4	25	(	(	PUNCT
ejpam-5930	4	26	or	or	CCONJ
ejpam-5930	4	27	ideal	ideal	ADJ
ejpam-5930	4	28	)	)	PUNCT
ejpam-5930	4	29	.	.	PUNCT
ejpam-5930	5	1	this	this	PRON
ejpam-5930	5	2	implies	imply	VERB
ejpam-5930	5	3	that	that	SCONJ
ejpam-5930	5	4	algebraic	algebraic	ADJ
ejpam-5930	5	5	systems	system	NOUN
ejpam-5930	5	6	can	can	AUX
ejpam-5930	5	7	employ	employ	VERB
ejpam-5930	5	8	rough	rough	ADJ
ejpam-5930	5	9	set	set	NOUN
ejpam-5930	5	10	concepts	concept	NOUN
ejpam-5930	5	11	.	.	PUNCT
ejpam-5930	6	1	our	our	PRON
ejpam-5930	6	2	results	result	NOUN
ejpam-5930	6	3	demonstrate	demonstrate	VERB
ejpam-5930	6	4	that	that	SCONJ
ejpam-5930	6	5	the	the	DET
ejpam-5930	6	6	approximation	approximation	NOUN
ejpam-5930	6	7	spaces	space	NOUN
ejpam-5930	6	8	induced	induce	VERB
ejpam-5930	6	9	by	by	ADP
ejpam-5930	6	10	ideals	ideal	NOUN
ejpam-5930	6	11	in	in	ADP
ejpam-5930	6	12	hilbert	hilbert	PROPN
ejpam-5930	6	13	algebras	algebra	NOUN
ejpam-5930	6	14	provide	provide	VERB
ejpam-5930	6	15	a	a	DET
ejpam-5930	6	16	robust	robust	ADJ
ejpam-5930	6	17	framework	framework	NOUN
ejpam-5930	6	18	for	for	ADP
ejpam-5930	6	19	analyzing	analyze	VERB
ejpam-5930	6	20	algebraic	algebraic	ADJ
ejpam-5930	6	21	structures	structure	NOUN
ejpam-5930	6	22	under	under	ADP
ejpam-5930	6	23	incomplete	incomplete	ADJ
ejpam-5930	6	24	or	or	CCONJ
ejpam-5930	6	25	uncertain	uncertain	ADJ
ejpam-5930	6	26	information	information	NOUN
ejpam-5930	6	27	.	.	PUNCT
ejpam-5930	7	1	furthermore	furthermore	ADV
ejpam-5930	7	2	,	,	PUNCT
ejpam-5930	7	3	we	we	PRON
ejpam-5930	7	4	present	present	VERB
ejpam-5930	7	5	illustrative	illustrative	ADJ
ejpam-5930	7	6	examples	example	NOUN
ejpam-5930	7	7	to	to	PART
ejpam-5930	7	8	validate	validate	VERB
ejpam-5930	7	9	our	our	PRON
ejpam-5930	7	10	theoretical	theoretical	ADJ
ejpam-5930	7	11	findings	finding	NOUN
ejpam-5930	7	12	and	and	CCONJ
ejpam-5930	7	13	highlight	highlight	VERB
ejpam-5930	7	14	the	the	DET
ejpam-5930	7	15	practical	practical	ADJ
ejpam-5930	7	16	implications	implication	NOUN
ejpam-5930	7	17	of	of	ADP
ejpam-5930	7	18	this	this	DET
ejpam-5930	7	19	approach	approach	NOUN
ejpam-5930	7	20	.	.	PUNCT
ejpam-5930	8	1	this	this	DET
ejpam-5930	8	2	study	study	NOUN
ejpam-5930	8	3	not	not	PART
ejpam-5930	8	4	only	only	ADV
ejpam-5930	8	5	enriches	enrich	VERB
ejpam-5930	8	6	the	the	DET
ejpam-5930	8	7	theoretical	theoretical	ADJ
ejpam-5930	8	8	foundations	foundation	NOUN
ejpam-5930	8	9	of	of	ADP
ejpam-5930	8	10	rough	rough	ADJ
ejpam-5930	8	11	set	set	NOUN
ejpam-5930	8	12	theory	theory	NOUN
ejpam-5930	8	13	but	but	CCONJ
ejpam-5930	8	14	also	also	ADV
ejpam-5930	8	15	opens	open	VERB
ejpam-5930	8	16	new	new	ADJ
ejpam-5930	8	17	avenues	avenue	NOUN
ejpam-5930	8	18	for	for	ADP
ejpam-5930	8	19	its	its	PRON
ejpam-5930	8	20	application	application	NOUN
ejpam-5930	8	21	in	in	ADP
ejpam-5930	8	22	algebraic	algebraic	ADJ
ejpam-5930	8	23	logic	logic	NOUN
ejpam-5930	8	24	and	and	CCONJ
ejpam-5930	8	25	related	related	ADJ
ejpam-5930	8	26	fields	field	NOUN
ejpam-5930	8	27	.	.	PUNCT
ejpam-5930	9	1	2020	2020	NUM
ejpam-5930	9	2	mathematics	mathematic	NOUN
ejpam-5930	9	3	subject	subject	NOUN
ejpam-5930	9	4	classifications	classification	NOUN
ejpam-5930	9	5	:	:	PUNCT
ejpam-5930	9	6	03g25	03g25	NUM
ejpam-5930	9	7	,	,	PUNCT
ejpam-5930	9	8	03e72	03e72	AUX
ejpam-5930	9	9	key	key	ADJ
ejpam-5930	9	10	words	word	NOUN
ejpam-5930	9	11	and	and	CCONJ
ejpam-5930	9	12	phrases	phrase	NOUN
ejpam-5930	9	13	:	:	PUNCT
ejpam-5930	9	14	hilbert	hilbert	PROPN
ejpam-5930	9	15	algebra	algebra	PROPN
ejpam-5930	9	16	,	,	PUNCT
ejpam-5930	9	17	subalgebra	subalgebra	NOUN
ejpam-5930	9	18	,	,	PUNCT
ejpam-5930	9	19	ideal	ideal	ADJ
ejpam-5930	9	20	,	,	PUNCT
ejpam-5930	9	21	congruence	congruence	NOUN
ejpam-5930	9	22	,	,	PUNCT
ejpam-5930	9	23	rough	rough	ADJ
ejpam-5930	9	24	set	set	NOUN
ejpam-5930	9	25	,	,	PUNCT
ejpam-5930	9	26	lower	low	ADJ
ejpam-5930	9	27	and	and	CCONJ
ejpam-5930	9	28	upper	upper	ADJ
ejpam-5930	9	29	approximations	approximation	NOUN
ejpam-5930	9	30	1	1	NUM
ejpam-5930	9	31	.	.	PUNCT
ejpam-5930	10	1	introduction	introduction	NOUN
ejpam-5930	10	2	the	the	DET
ejpam-5930	10	3	concept	concept	NOUN
ejpam-5930	10	4	of	of	ADP
ejpam-5930	10	5	a	a	DET
ejpam-5930	10	6	rough	rough	ADJ
ejpam-5930	10	7	set	set	NOUN
ejpam-5930	10	8	was	be	AUX
ejpam-5930	10	9	originally	originally	ADV
ejpam-5930	10	10	proposed	propose	VERB
ejpam-5930	10	11	by	by	ADP
ejpam-5930	10	12	pawlak	pawlak	ADJ
ejpam-5930	10	13	[	[	X
ejpam-5930	10	14	1	1	NUM
ejpam-5930	10	15	,	,	PUNCT
ejpam-5930	10	16	2	2	NUM
ejpam-5930	10	17	]	]	PUNCT
ejpam-5930	10	18	as	as	ADP
ejpam-5930	10	19	a	a	DET
ejpam-5930	10	20	formal	formal	ADJ
ejpam-5930	10	21	tool	tool	NOUN
ejpam-5930	10	22	for	for	ADP
ejpam-5930	10	23	modeling	modeling	NOUN
ejpam-5930	10	24	and	and	CCONJ
ejpam-5930	10	25	processing	process	VERB
ejpam-5930	10	26	complete	complete	ADJ
ejpam-5930	10	27	information	information	NOUN
ejpam-5930	10	28	in	in	ADP
ejpam-5930	10	29	information	information	NOUN
ejpam-5930	10	30	systems	system	NOUN
ejpam-5930	10	31	.	.	PUNCT
ejpam-5930	11	1	it	it	PRON
ejpam-5930	11	2	seems	seem	VERB
ejpam-5930	11	3	that	that	SCONJ
ejpam-5930	11	4	the	the	DET
ejpam-5930	11	5	rough	rough	ADJ
ejpam-5930	11	6	set	set	NOUN
ejpam-5930	11	7	approach	approach	NOUN
ejpam-5930	11	8	is	be	AUX
ejpam-5930	11	9	fundamentally	fundamentally	ADV
ejpam-5930	11	10	important	important	ADJ
ejpam-5930	11	11	in	in	ADP
ejpam-5930	11	12	artificial	artificial	ADJ
ejpam-5930	11	13	intelligence	intelligence	NOUN
ejpam-5930	11	14	and	and	CCONJ
ejpam-5930	11	15	cognitive	cognitive	ADJ
ejpam-5930	11	16	sciences	science	NOUN
ejpam-5930	11	17	,	,	PUNCT
ejpam-5930	11	18	especially	especially	ADV
ejpam-5930	11	19	in	in	ADP
ejpam-5930	11	20	research	research	NOUN
ejpam-5930	11	21	areas	area	NOUN
ejpam-5930	11	22	such	such	ADJ
ejpam-5930	11	23	as	as	ADP
ejpam-5930	11	24	machine	machine	NOUN
ejpam-5930	11	25	learning	learning	NOUN
ejpam-5930	11	26	,	,	PUNCT
ejpam-5930	11	27	intelligent	intelligent	ADJ
ejpam-5930	11	28	systems	system	NOUN
ejpam-5930	11	29	,	,	PUNCT
ejpam-5930	11	30	inductive	inductive	ADJ
ejpam-5930	11	31	reasoning	reasoning	NOUN
ejpam-5930	11	32	,	,	PUNCT
ejpam-5930	11	33	pattern	pattern	NOUN
ejpam-5930	11	34	recognition	recognition	NOUN
ejpam-5930	11	35	,	,	PUNCT
ejpam-5930	11	36	knowledge	knowledge	NOUN
ejpam-5930	11	37	discovery	discovery	NOUN
ejpam-5930	11	38	,	,	PUNCT
ejpam-5930	11	39	decision	decision	NOUN
ejpam-5930	11	40	analysis	analysis	NOUN
ejpam-5930	11	41	,	,	PUNCT
ejpam-5930	11	42	and	and	CCONJ
ejpam-5930	11	43	expert	expert	NOUN
ejpam-5930	11	44	∗corresponding	∗corresponde	VERB
ejpam-5930	11	45	author	author	NOUN
ejpam-5930	11	46	.	.	PUNCT
ejpam-5930	12	1	doi	doi	NOUN
ejpam-5930	12	2	:	:	PUNCT
ejpam-5930	12	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5930	https://doi.org/10.29020/nybg.ejpam.v18i2.5930	PROPN
ejpam-5930	12	4	email	email	NOUN
ejpam-5930	12	5	addresses	address	VERB
ejpam-5930	12	6	:	:	PUNCT
ejpam-5930	13	1	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5930	13	2	(	(	PUNCT
ejpam-5930	13	3	a.	a.	NOUN
ejpam-5930	13	4	iampan	iampan	PROPN
ejpam-5930	13	5	)	)	PUNCT
ejpam-5930	13	6	,	,	PUNCT
ejpam-5930	13	7	vennilamaths@gmail.com	vennilamaths@gmail.com	X
ejpam-5930	13	8	(	(	PUNCT
ejpam-5930	13	9	r.	r.	PROPN
ejpam-5930	13	10	vennila	vennila	PROPN
ejpam-5930	13	11	)	)	PUNCT
ejpam-5930	13	12	,	,	PUNCT
ejpam-5930	13	13	nrajesh	nrajesh	PROPN
ejpam-5930	13	14	topology@yahoo.co.in	topology@yahoo.co.in	PROPN
ejpam-5930	13	15	(	(	PUNCT
ejpam-5930	13	16	n.	n.	PROPN
ejpam-5930	13	17	rajesh	rajesh	PROPN
ejpam-5930	13	18	)	)	PUNCT
ejpam-5930	13	19	,	,	PUNCT
ejpam-5930	13	20	subasinimaths@gmail.com	subasinimaths@gmail.com	PROPN
ejpam-5930	13	21	(	(	PUNCT
ejpam-5930	13	22	r.	r.	PROPN
ejpam-5930	13	23	subasini	subasini	PROPN
ejpam-5930	13	24	)	)	PUNCT
ejpam-5930	13	25	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5930	13	26	1	1	NUM
ejpam-5930	13	27	copyright	copyright	NOUN
ejpam-5930	13	28	:	:	PUNCT
ejpam-5930	13	29	©	©	PROPN
ejpam-5930	13	30	2025	2025	NUM
ejpam-5930	13	31	the	the	DET
ejpam-5930	13	32	author(s	author(s	NOUN
ejpam-5930	13	33	)	)	PUNCT
ejpam-5930	13	34	.	.	PUNCT
ejpam-5930	14	1	(	(	PUNCT
ejpam-5930	14	2	cc	cc	NOUN
ejpam-5930	14	3	by	by	ADP
ejpam-5930	14	4	-	-	PUNCT
ejpam-5930	14	5	nc	nc	PROPN
ejpam-5930	14	6	4.0	4.0	NUM
ejpam-5930	14	7	)	)	PUNCT
ejpam-5930	14	8	a.	a.	NOUN
ejpam-5930	14	9	iampan	iampan	NOUN
ejpam-5930	14	10	et	et	PROPN
ejpam-5930	14	11	al	al	PROPN
ejpam-5930	14	12	.	.	PUNCT
ejpam-5930	14	13	/	/	SYM
ejpam-5930	14	14	eur	eur	PROPN
ejpam-5930	14	15	.	.	PUNCT
ejpam-5930	15	1	j.	j.	PROPN
ejpam-5930	15	2	pure	pure	PROPN
ejpam-5930	15	3	appl	appl	PROPN
ejpam-5930	15	4	.	.	PROPN
ejpam-5930	15	5	math	math	PROPN
ejpam-5930	15	6	,	,	PUNCT
ejpam-5930	15	7	18	18	NUM
ejpam-5930	15	8	(	(	PUNCT
ejpam-5930	15	9	2	2	NUM
ejpam-5930	15	10	)	)	PUNCT
ejpam-5930	15	11	(	(	PUNCT
ejpam-5930	15	12	2025	2025	NUM
ejpam-5930	15	13	)	)	PUNCT
ejpam-5930	15	14	,	,	PUNCT
ejpam-5930	15	15	5930	5930	NUM
ejpam-5930	15	16	2	2	NUM
ejpam-5930	15	17	of	of	ADP
ejpam-5930	15	18	11	11	NUM
ejpam-5930	15	19	systems	system	NOUN
ejpam-5930	15	20	.	.	PUNCT
ejpam-5930	16	1	rough	rough	ADJ
ejpam-5930	16	2	set	set	NOUN
ejpam-5930	16	3	theory	theory	NOUN
ejpam-5930	16	4	(	(	PUNCT
ejpam-5930	16	5	rst	rst	PROPN
ejpam-5930	16	6	)	)	PUNCT
ejpam-5930	16	7	,	,	PUNCT
ejpam-5930	16	8	a	a	DET
ejpam-5930	16	9	new	new	ADJ
ejpam-5930	16	10	mathematical	mathematical	ADJ
ejpam-5930	16	11	approach	approach	NOUN
ejpam-5930	16	12	to	to	ADP
ejpam-5930	16	13	dealing	deal	VERB
ejpam-5930	16	14	with	with	ADP
ejpam-5930	16	15	inexact	inexact	ADJ
ejpam-5930	16	16	,	,	PUNCT
ejpam-5930	16	17	uncertain	uncertain	ADJ
ejpam-5930	16	18	,	,	PUNCT
ejpam-5930	16	19	or	or	CCONJ
ejpam-5930	16	20	vague	vague	ADJ
ejpam-5930	16	21	knowledge	knowledge	NOUN
ejpam-5930	16	22	,	,	PUNCT
ejpam-5930	16	23	has	have	AUX
ejpam-5930	16	24	recently	recently	ADV
ejpam-5930	16	25	received	receive	VERB
ejpam-5930	16	26	wide	wide	ADJ
ejpam-5930	16	27	attention	attention	NOUN
ejpam-5930	16	28	in	in	ADP
ejpam-5930	16	29	the	the	DET
ejpam-5930	16	30	research	research	NOUN
ejpam-5930	16	31	areas	area	NOUN
ejpam-5930	16	32	in	in	ADP
ejpam-5930	16	33	both	both	CCONJ
ejpam-5930	16	34	real	real	ADJ
ejpam-5930	16	35	-	-	PUNCT
ejpam-5930	16	36	life	life	NOUN
ejpam-5930	16	37	applications	application	NOUN
ejpam-5930	16	38	and	and	CCONJ
ejpam-5930	16	39	the	the	DET
ejpam-5930	16	40	theory	theory	NOUN
ejpam-5930	16	41	itself	itself	PRON
ejpam-5930	16	42	.	.	PUNCT
ejpam-5930	17	1	rst	rst	PROPN
ejpam-5930	17	2	is	be	AUX
ejpam-5930	17	3	an	an	DET
ejpam-5930	17	4	extension	extension	NOUN
ejpam-5930	17	5	of	of	ADP
ejpam-5930	17	6	set	set	NOUN
ejpam-5930	17	7	theory	theory	NOUN
ejpam-5930	17	8	in	in	ADP
ejpam-5930	17	9	which	which	PRON
ejpam-5930	17	10	a	a	DET
ejpam-5930	17	11	subset	subset	NOUN
ejpam-5930	17	12	of	of	ADP
ejpam-5930	17	13	a	a	DET
ejpam-5930	17	14	universe	universe	NOUN
ejpam-5930	17	15	is	be	AUX
ejpam-5930	17	16	described	describe	VERB
ejpam-5930	17	17	by	by	ADP
ejpam-5930	17	18	a	a	DET
ejpam-5930	17	19	pair	pair	NOUN
ejpam-5930	17	20	of	of	ADP
ejpam-5930	17	21	ordinary	ordinary	ADJ
ejpam-5930	17	22	sets	set	NOUN
ejpam-5930	17	23	called	call	VERB
ejpam-5930	17	24	the	the	DET
ejpam-5930	17	25	lower	low	ADJ
ejpam-5930	17	26	and	and	CCONJ
ejpam-5930	17	27	upper	upper	ADJ
ejpam-5930	17	28	approximations	approximation	NOUN
ejpam-5930	17	29	.	.	PUNCT
ejpam-5930	18	1	there	there	PRON
ejpam-5930	18	2	are	be	VERB
ejpam-5930	18	3	at	at	ADV
ejpam-5930	18	4	least	least	ADJ
ejpam-5930	18	5	two	two	NUM
ejpam-5930	18	6	methods	method	NOUN
ejpam-5930	18	7	for	for	ADP
ejpam-5930	18	8	developing	develop	VERB
ejpam-5930	18	9	this	this	DET
ejpam-5930	18	10	theory	theory	NOUN
ejpam-5930	18	11	,	,	PUNCT
ejpam-5930	18	12	the	the	DET
ejpam-5930	18	13	constructive	constructive	ADJ
ejpam-5930	18	14	and	and	CCONJ
ejpam-5930	18	15	axiomatic	axiomatic	ADJ
ejpam-5930	18	16	approaches	approach	NOUN
ejpam-5930	18	17	.	.	PUNCT
ejpam-5930	19	1	in	in	ADP
ejpam-5930	19	2	constructive	constructive	ADJ
ejpam-5930	19	3	methods	method	NOUN
ejpam-5930	19	4	,	,	PUNCT
ejpam-5930	19	5	lower	low	ADJ
ejpam-5930	19	6	and	and	CCONJ
ejpam-5930	19	7	upper	upper	ADJ
ejpam-5930	19	8	approximations	approximation	NOUN
ejpam-5930	19	9	are	be	AUX
ejpam-5930	19	10	constructed	construct	VERB
ejpam-5930	19	11	from	from	ADP
ejpam-5930	19	12	primitive	primitive	ADJ
ejpam-5930	19	13	notions	notion	NOUN
ejpam-5930	19	14	such	such	ADJ
ejpam-5930	19	15	as	as	ADP
ejpam-5930	19	16	equivalence	equivalence	NOUN
ejpam-5930	19	17	relations	relation	NOUN
ejpam-5930	19	18	on	on	ADP
ejpam-5930	19	19	a	a	DET
ejpam-5930	19	20	universe	universe	NOUN
ejpam-5930	19	21	[	[	X
ejpam-5930	19	22	2	2	NUM
ejpam-5930	19	23	,	,	PUNCT
ejpam-5930	19	24	3	3	NUM
ejpam-5930	19	25	]	]	PUNCT
ejpam-5930	19	26	and	and	CCONJ
ejpam-5930	19	27	neighborhood	neighborhood	NOUN
ejpam-5930	19	28	systems	system	NOUN
ejpam-5930	19	29	[	[	X
ejpam-5930	19	30	4	4	NUM
ejpam-5930	19	31	,	,	PUNCT
ejpam-5930	19	32	5	5	NUM
ejpam-5930	19	33	]	]	PUNCT
ejpam-5930	19	34	.	.	PUNCT
ejpam-5930	20	1	in	in	ADP
ejpam-5930	20	2	pawlak	pawlak	ADJ
ejpam-5930	20	3	’s	’s	PART
ejpam-5930	20	4	rough	rough	ADJ
ejpam-5930	20	5	set	set	NOUN
ejpam-5930	20	6	[	[	X
ejpam-5930	20	7	1	1	NUM
ejpam-5930	20	8	]	]	PUNCT
ejpam-5930	20	9	,	,	PUNCT
ejpam-5930	20	10	the	the	DET
ejpam-5930	20	11	equivalence	equivalence	NOUN
ejpam-5930	20	12	classes	class	NOUN
ejpam-5930	20	13	are	be	AUX
ejpam-5930	20	14	the	the	DET
ejpam-5930	20	15	building	building	NOUN
ejpam-5930	20	16	blocks	block	NOUN
ejpam-5930	20	17	for	for	ADP
ejpam-5930	20	18	constructing	construct	VERB
ejpam-5930	20	19	the	the	DET
ejpam-5930	20	20	lower	low	ADJ
ejpam-5930	20	21	and	and	CCONJ
ejpam-5930	20	22	upper	upper	ADJ
ejpam-5930	20	23	approximations	approximation	NOUN
ejpam-5930	20	24	.	.	PUNCT
ejpam-5930	21	1	comer	comer	NOUN
ejpam-5930	21	2	[	[	X
ejpam-5930	21	3	6	6	NUM
ejpam-5930	21	4	]	]	PUNCT
ejpam-5930	21	5	presented	present	VERB
ejpam-5930	21	6	an	an	DET
ejpam-5930	21	7	interesting	interesting	ADJ
ejpam-5930	21	8	discussion	discussion	NOUN
ejpam-5930	21	9	of	of	ADP
ejpam-5930	21	10	rough	rough	ADJ
ejpam-5930	21	11	sets	set	NOUN
ejpam-5930	21	12	and	and	CCONJ
ejpam-5930	21	13	various	various	ADJ
ejpam-5930	21	14	algebras	algebra	NOUN
ejpam-5930	21	15	related	relate	VERB
ejpam-5930	21	16	to	to	ADP
ejpam-5930	21	17	the	the	DET
ejpam-5930	21	18	study	study	NOUN
ejpam-5930	21	19	of	of	ADP
ejpam-5930	21	20	algebraic	algebraic	ADJ
ejpam-5930	21	21	logic	logic	NOUN
ejpam-5930	21	22	,	,	PUNCT
ejpam-5930	21	23	such	such	ADJ
ejpam-5930	21	24	as	as	ADP
ejpam-5930	21	25	stone	stone	NOUN
ejpam-5930	21	26	algebras	algebra	NOUN
ejpam-5930	21	27	and	and	CCONJ
ejpam-5930	21	28	relation	relation	NOUN
ejpam-5930	21	29	algebras	algebra	VERB
ejpam-5930	21	30	.	.	PUNCT
ejpam-5930	22	1	it	it	PRON
ejpam-5930	22	2	is	be	AUX
ejpam-5930	22	3	a	a	DET
ejpam-5930	22	4	natural	natural	ADJ
ejpam-5930	22	5	question	question	NOUN
ejpam-5930	22	6	to	to	PART
ejpam-5930	22	7	ask	ask	VERB
ejpam-5930	22	8	:	:	PUNCT
ejpam-5930	22	9	what	what	PRON
ejpam-5930	22	10	happens	happen	VERB
ejpam-5930	22	11	if	if	SCONJ
ejpam-5930	22	12	we	we	PRON
ejpam-5930	22	13	substitute	substitute	VERB
ejpam-5930	22	14	an	an	DET
ejpam-5930	22	15	algebraic	algebraic	ADJ
ejpam-5930	22	16	system	system	NOUN
ejpam-5930	22	17	instead	instead	ADV
ejpam-5930	22	18	of	of	ADP
ejpam-5930	22	19	the	the	DET
ejpam-5930	22	20	universe	universe	NOUN
ejpam-5930	22	21	set	set	NOUN
ejpam-5930	22	22	?	?	PUNCT
ejpam-5930	23	1	the	the	DET
ejpam-5930	23	2	concept	concept	NOUN
ejpam-5930	23	3	of	of	ADP
ejpam-5930	23	4	hilbert	hilbert	PROPN
ejpam-5930	23	5	algebra	algebra	PROPN
ejpam-5930	23	6	was	be	AUX
ejpam-5930	23	7	introduced	introduce	VERB
ejpam-5930	23	8	in	in	ADP
ejpam-5930	23	9	the	the	DET
ejpam-5930	23	10	early	early	ADJ
ejpam-5930	23	11	50s	50	NOUN
ejpam-5930	23	12	by	by	ADP
ejpam-5930	23	13	henkin	henkin	PROPN
ejpam-5930	23	14	for	for	ADP
ejpam-5930	23	15	some	some	DET
ejpam-5930	23	16	investigations	investigation	NOUN
ejpam-5930	23	17	of	of	ADP
ejpam-5930	23	18	implication	implication	NOUN
ejpam-5930	23	19	in	in	ADP
ejpam-5930	23	20	intuitionistic	intuitionistic	ADJ
ejpam-5930	23	21	and	and	CCONJ
ejpam-5930	23	22	other	other	ADJ
ejpam-5930	23	23	non	non	ADJ
ejpam-5930	23	24	-	-	ADJ
ejpam-5930	23	25	classical	classical	ADJ
ejpam-5930	23	26	logic	logic	NOUN
ejpam-5930	23	27	[	[	X
ejpam-5930	23	28	7	7	NUM
ejpam-5930	23	29	]	]	PUNCT
ejpam-5930	23	30	.	.	PUNCT
ejpam-5930	24	1	in	in	ADP
ejpam-5930	24	2	the	the	DET
ejpam-5930	24	3	60s	60	NOUN
ejpam-5930	24	4	,	,	PUNCT
ejpam-5930	24	5	these	these	DET
ejpam-5930	24	6	algebras	algebra	NOUN
ejpam-5930	24	7	were	be	AUX
ejpam-5930	24	8	studied	study	VERB
ejpam-5930	24	9	,	,	PUNCT
ejpam-5930	24	10	especially	especially	ADV
ejpam-5930	24	11	by	by	ADP
ejpam-5930	24	12	diego	diego	PROPN
ejpam-5930	24	13	,	,	PUNCT
ejpam-5930	24	14	from	from	ADP
ejpam-5930	24	15	an	an	DET
ejpam-5930	24	16	algebraic	algebraic	ADJ
ejpam-5930	24	17	point	point	NOUN
ejpam-5930	24	18	of	of	ADP
ejpam-5930	24	19	view	view	NOUN
ejpam-5930	24	20	.	.	PUNCT
ejpam-5930	25	1	diego	diego	PROPN
ejpam-5930	25	2	proved	prove	VERB
ejpam-5930	25	3	[	[	X
ejpam-5930	25	4	8	8	X
ejpam-5930	25	5	]	]	PUNCT
ejpam-5930	25	6	that	that	SCONJ
ejpam-5930	25	7	hilbert	hilbert	PROPN
ejpam-5930	25	8	algebras	algebras	PROPN
ejpam-5930	25	9	form	form	VERB
ejpam-5930	25	10	a	a	DET
ejpam-5930	25	11	locally	locally	ADV
ejpam-5930	25	12	finite	finite	ADJ
ejpam-5930	25	13	variety	variety	NOUN
ejpam-5930	25	14	.	.	PUNCT
ejpam-5930	26	1	hilbert	hilbert	PROPN
ejpam-5930	26	2	algebras	algebras	PROPN
ejpam-5930	26	3	were	be	AUX
ejpam-5930	26	4	treated	treat	VERB
ejpam-5930	26	5	by	by	ADP
ejpam-5930	26	6	busneag	busneag	NOUN
ejpam-5930	26	7	[	[	X
ejpam-5930	26	8	9	9	NUM
ejpam-5930	26	9	,	,	PUNCT
ejpam-5930	26	10	10	10	NUM
ejpam-5930	26	11	]	]	PUNCT
ejpam-5930	26	12	and	and	CCONJ
ejpam-5930	26	13	jun	jun	PROPN
ejpam-5930	27	1	[	[	X
ejpam-5930	27	2	11	11	NUM
ejpam-5930	27	3	]	]	PUNCT
ejpam-5930	27	4	,	,	PUNCT
ejpam-5930	27	5	and	and	CCONJ
ejpam-5930	27	6	some	some	PRON
ejpam-5930	27	7	of	of	ADP
ejpam-5930	27	8	their	their	PRON
ejpam-5930	27	9	filters	filter	NOUN
ejpam-5930	27	10	forming	form	VERB
ejpam-5930	27	11	deductive	deductive	ADJ
ejpam-5930	27	12	systems	system	NOUN
ejpam-5930	27	13	were	be	AUX
ejpam-5930	27	14	recognized	recognize	VERB
ejpam-5930	27	15	.	.	PUNCT
ejpam-5930	28	1	dudek	dudek	PROPN
ejpam-5930	29	1	[	[	X
ejpam-5930	29	2	12	12	NUM
ejpam-5930	29	3	]	]	PUNCT
ejpam-5930	29	4	considered	consider	VERB
ejpam-5930	29	5	the	the	DET
ejpam-5930	29	6	fuzzification	fuzzification	NOUN
ejpam-5930	29	7	of	of	ADP
ejpam-5930	29	8	subalgebras	subalgebras	PROPN
ejpam-5930	29	9	and	and	CCONJ
ejpam-5930	29	10	deductive	deductive	ADJ
ejpam-5930	29	11	systems	system	NOUN
ejpam-5930	29	12	in	in	ADP
ejpam-5930	29	13	hilbert	hilbert	PROPN
ejpam-5930	29	14	algebras	algebras	PROPN
ejpam-5930	29	15	.	.	PUNCT
ejpam-5930	30	1	the	the	DET
ejpam-5930	30	2	integration	integration	NOUN
ejpam-5930	30	3	of	of	ADP
ejpam-5930	30	4	rst	rst	PROPN
ejpam-5930	30	5	with	with	ADP
ejpam-5930	30	6	hilbert	hilbert	PROPN
ejpam-5930	30	7	algebras	algebras	PROPN
ejpam-5930	30	8	is	be	AUX
ejpam-5930	30	9	motivated	motivate	VERB
ejpam-5930	30	10	by	by	ADP
ejpam-5930	30	11	the	the	DET
ejpam-5930	30	12	need	need	NOUN
ejpam-5930	30	13	to	to	PART
ejpam-5930	30	14	analyze	analyze	VERB
ejpam-5930	30	15	algebraic	algebraic	ADJ
ejpam-5930	30	16	structures	structure	NOUN
ejpam-5930	30	17	under	under	ADP
ejpam-5930	30	18	conditions	condition	NOUN
ejpam-5930	30	19	of	of	ADP
ejpam-5930	30	20	uncertainty	uncertainty	NOUN
ejpam-5930	30	21	and	and	CCONJ
ejpam-5930	30	22	incomplete	incomplete	ADJ
ejpam-5930	30	23	information	information	NOUN
ejpam-5930	30	24	.	.	PUNCT
ejpam-5930	31	1	hilbert	hilbert	PROPN
ejpam-5930	31	2	algebras	algebras	PROPN
ejpam-5930	31	3	play	play	VERB
ejpam-5930	31	4	a	a	DET
ejpam-5930	31	5	fundamental	fundamental	ADJ
ejpam-5930	31	6	role	role	NOUN
ejpam-5930	31	7	in	in	ADP
ejpam-5930	31	8	non	non	ADJ
ejpam-5930	31	9	-	-	ADJ
ejpam-5930	31	10	classical	classical	ADJ
ejpam-5930	31	11	logic	logic	NOUN
ejpam-5930	31	12	,	,	PUNCT
ejpam-5930	31	13	particularly	particularly	ADV
ejpam-5930	31	14	in	in	ADP
ejpam-5930	31	15	the	the	DET
ejpam-5930	31	16	study	study	NOUN
ejpam-5930	31	17	of	of	ADP
ejpam-5930	31	18	implication	implication	NOUN
ejpam-5930	31	19	structures	structure	NOUN
ejpam-5930	31	20	,	,	PUNCT
ejpam-5930	31	21	making	make	VERB
ejpam-5930	31	22	them	they	PRON
ejpam-5930	31	23	a	a	DET
ejpam-5930	31	24	powerful	powerful	ADJ
ejpam-5930	31	25	tool	tool	NOUN
ejpam-5930	31	26	for	for	ADP
ejpam-5930	31	27	reasoning	reasoning	NOUN
ejpam-5930	31	28	in	in	ADP
ejpam-5930	31	29	mathematical	mathematical	ADJ
ejpam-5930	31	30	logic	logic	NOUN
ejpam-5930	31	31	and	and	CCONJ
ejpam-5930	31	32	artificial	artificial	ADJ
ejpam-5930	31	33	intelligence	intelligence	NOUN
ejpam-5930	31	34	.	.	PUNCT
ejpam-5930	32	1	however	however	ADV
ejpam-5930	32	2	,	,	PUNCT
ejpam-5930	32	3	real	real	ADJ
ejpam-5930	32	4	-	-	PUNCT
ejpam-5930	32	5	world	world	NOUN
ejpam-5930	32	6	applications	application	NOUN
ejpam-5930	32	7	often	often	ADV
ejpam-5930	32	8	involve	involve	VERB
ejpam-5930	32	9	imprecise	imprecise	ADV
ejpam-5930	32	10	or	or	CCONJ
ejpam-5930	32	11	vague	vague	ADJ
ejpam-5930	32	12	information	information	NOUN
ejpam-5930	32	13	,	,	PUNCT
ejpam-5930	32	14	where	where	SCONJ
ejpam-5930	32	15	classical	classical	ADJ
ejpam-5930	32	16	algebraic	algebraic	ADJ
ejpam-5930	32	17	methods	method	NOUN
ejpam-5930	32	18	may	may	AUX
ejpam-5930	32	19	not	not	PART
ejpam-5930	32	20	be	be	AUX
ejpam-5930	32	21	sufficient	sufficient	ADJ
ejpam-5930	32	22	.	.	PUNCT
ejpam-5930	33	1	rst	rst	PROPN
ejpam-5930	33	2	,	,	PUNCT
ejpam-5930	33	3	introduced	introduce	VERB
ejpam-5930	33	4	by	by	ADP
ejpam-5930	33	5	pawlak	pawlak	ADJ
ejpam-5930	33	6	[	[	X
ejpam-5930	33	7	1	1	NUM
ejpam-5930	33	8	,	,	PUNCT
ejpam-5930	33	9	2	2	NUM
ejpam-5930	33	10	]	]	PUNCT
ejpam-5930	33	11	,	,	PUNCT
ejpam-5930	33	12	provides	provide	VERB
ejpam-5930	33	13	a	a	DET
ejpam-5930	33	14	robust	robust	ADJ
ejpam-5930	33	15	framework	framework	NOUN
ejpam-5930	33	16	for	for	ADP
ejpam-5930	33	17	dealing	deal	VERB
ejpam-5930	33	18	with	with	ADP
ejpam-5930	33	19	such	such	ADJ
ejpam-5930	33	20	uncertainties	uncertainty	NOUN
ejpam-5930	33	21	by	by	ADP
ejpam-5930	33	22	defining	define	VERB
ejpam-5930	33	23	lower	low	ADJ
ejpam-5930	33	24	and	and	CCONJ
ejpam-5930	33	25	upper	upper	ADJ
ejpam-5930	33	26	approximations	approximation	NOUN
ejpam-5930	33	27	of	of	ADP
ejpam-5930	33	28	sets	set	NOUN
ejpam-5930	33	29	based	base	VERB
ejpam-5930	33	30	on	on	ADP
ejpam-5930	33	31	equivalence	equivalence	NOUN
ejpam-5930	33	32	relations	relation	NOUN
ejpam-5930	33	33	.	.	PUNCT
ejpam-5930	34	1	in	in	ADP
ejpam-5930	34	2	this	this	DET
ejpam-5930	34	3	context	context	NOUN
ejpam-5930	34	4	,	,	PUNCT
ejpam-5930	34	5	the	the	DET
ejpam-5930	34	6	work	work	NOUN
ejpam-5930	34	7	by	by	ADP
ejpam-5930	34	8	borumand	borumand	ADJ
ejpam-5930	34	9	saeid	saeid	PROPN
ejpam-5930	34	10	and	and	CCONJ
ejpam-5930	34	11	haveshki	haveshki	PROPN
ejpam-5930	35	1	[	[	X
ejpam-5930	35	2	13	13	NUM
ejpam-5930	35	3	]	]	PUNCT
ejpam-5930	35	4	represents	represent	VERB
ejpam-5930	35	5	an	an	DET
ejpam-5930	35	6	important	important	ADJ
ejpam-5930	35	7	step	step	NOUN
ejpam-5930	35	8	toward	toward	ADP
ejpam-5930	35	9	bridging	bridge	VERB
ejpam-5930	35	10	rough	rough	ADJ
ejpam-5930	35	11	set	set	NOUN
ejpam-5930	35	12	theory	theory	NOUN
ejpam-5930	35	13	and	and	CCONJ
ejpam-5930	35	14	hilbert	hilbert	PROPN
ejpam-5930	35	15	algebras	algebras	PROPN
ejpam-5930	35	16	.	.	PUNCT
ejpam-5930	36	1	their	their	PRON
ejpam-5930	36	2	study	study	NOUN
ejpam-5930	36	3	focused	focus	VERB
ejpam-5930	36	4	on	on	ADP
ejpam-5930	36	5	approximation	approximation	NOUN
ejpam-5930	36	6	techniques	technique	NOUN
ejpam-5930	36	7	within	within	ADP
ejpam-5930	36	8	hilbert	hilbert	PROPN
ejpam-5930	36	9	algebras	algebras	PROPN
ejpam-5930	36	10	and	and	CCONJ
ejpam-5930	36	11	highlighted	highlight	VERB
ejpam-5930	36	12	how	how	SCONJ
ejpam-5930	36	13	algebraic	algebraic	ADJ
ejpam-5930	36	14	operations	operation	NOUN
ejpam-5930	36	15	interact	interact	VERB
ejpam-5930	36	16	with	with	ADP
ejpam-5930	36	17	equivalence	equivalence	NOUN
ejpam-5930	36	18	-	-	PUNCT
ejpam-5930	36	19	based	base	VERB
ejpam-5930	36	20	approximations	approximation	NOUN
ejpam-5930	36	21	.	.	PUNCT
ejpam-5930	37	1	they	they	PRON
ejpam-5930	37	2	explored	explore	VERB
ejpam-5930	37	3	properties	property	NOUN
ejpam-5930	37	4	of	of	ADP
ejpam-5930	37	5	definable	definable	ADJ
ejpam-5930	37	6	sets	set	NOUN
ejpam-5930	37	7	and	and	CCONJ
ejpam-5930	37	8	congruence	congruence	NOUN
ejpam-5930	37	9	relations	relation	NOUN
ejpam-5930	37	10	,	,	PUNCT
ejpam-5930	37	11	providing	provide	VERB
ejpam-5930	37	12	foundational	foundational	ADJ
ejpam-5930	37	13	insights	insight	NOUN
ejpam-5930	37	14	that	that	PRON
ejpam-5930	37	15	guide	guide	VERB
ejpam-5930	37	16	subsequent	subsequent	ADJ
ejpam-5930	37	17	developments	development	NOUN
ejpam-5930	37	18	in	in	ADP
ejpam-5930	37	19	this	this	DET
ejpam-5930	37	20	area	area	NOUN
ejpam-5930	37	21	.	.	PUNCT
ejpam-5930	38	1	inspired	inspire	VERB
ejpam-5930	38	2	by	by	ADP
ejpam-5930	38	3	their	their	PRON
ejpam-5930	38	4	contribution	contribution	NOUN
ejpam-5930	38	5	,	,	PUNCT
ejpam-5930	38	6	our	our	PRON
ejpam-5930	38	7	study	study	NOUN
ejpam-5930	38	8	advances	advance	VERB
ejpam-5930	38	9	this	this	DET
ejpam-5930	38	10	line	line	NOUN
ejpam-5930	38	11	of	of	ADP
ejpam-5930	38	12	research	research	NOUN
ejpam-5930	38	13	by	by	ADP
ejpam-5930	38	14	formalizing	formalize	VERB
ejpam-5930	38	15	the	the	DET
ejpam-5930	38	16	structure	structure	NOUN
ejpam-5930	38	17	of	of	ADP
ejpam-5930	38	18	rough	rough	ADJ
ejpam-5930	38	19	subalgebras	subalgebra	NOUN
ejpam-5930	38	20	and	and	CCONJ
ejpam-5930	38	21	rough	rough	ADJ
ejpam-5930	38	22	ideals	ideal	NOUN
ejpam-5930	38	23	and	and	CCONJ
ejpam-5930	38	24	establishing	establish	VERB
ejpam-5930	38	25	algebraic	algebraic	ADJ
ejpam-5930	38	26	closure	closure	NOUN
ejpam-5930	38	27	under	under	ADP
ejpam-5930	38	28	approximation	approximation	NOUN
ejpam-5930	38	29	operations	operation	NOUN
ejpam-5930	38	30	.	.	PUNCT
ejpam-5930	39	1	by	by	ADP
ejpam-5930	39	2	applying	apply	VERB
ejpam-5930	39	3	rst	rst	PROPN
ejpam-5930	39	4	to	to	ADP
ejpam-5930	39	5	hilbert	hilbert	PROPN
ejpam-5930	39	6	algebras	algebras	PROPN
ejpam-5930	39	7	,	,	PUNCT
ejpam-5930	39	8	we	we	PRON
ejpam-5930	39	9	establish	establish	VERB
ejpam-5930	39	10	a	a	DET
ejpam-5930	39	11	novel	novel	ADJ
ejpam-5930	39	12	approach	approach	NOUN
ejpam-5930	39	13	to	to	ADP
ejpam-5930	39	14	approximating	approximate	VERB
ejpam-5930	39	15	subalgebras	subalgebra	NOUN
ejpam-5930	39	16	and	and	CCONJ
ejpam-5930	39	17	ideals	ideal	NOUN
ejpam-5930	39	18	,	,	PUNCT
ejpam-5930	39	19	ensuring	ensure	VERB
ejpam-5930	39	20	that	that	SCONJ
ejpam-5930	39	21	key	key	ADJ
ejpam-5930	39	22	algebraic	algebraic	ADJ
ejpam-5930	39	23	properties	property	NOUN
ejpam-5930	39	24	are	be	AUX
ejpam-5930	39	25	preserved	preserve	VERB
ejpam-5930	39	26	even	even	ADV
ejpam-5930	39	27	under	under	ADP
ejpam-5930	39	28	uncertainty	uncertainty	NOUN
ejpam-5930	39	29	.	.	PUNCT
ejpam-5930	40	1	this	this	DET
ejpam-5930	40	2	combination	combination	NOUN
ejpam-5930	40	3	not	not	PART
ejpam-5930	40	4	only	only	ADV
ejpam-5930	40	5	enriches	enrich	VERB
ejpam-5930	40	6	the	the	DET
ejpam-5930	40	7	theoretical	theoretical	ADJ
ejpam-5930	40	8	landscape	landscape	NOUN
ejpam-5930	40	9	of	of	ADP
ejpam-5930	40	10	algebraic	algebraic	ADJ
ejpam-5930	40	11	logic	logic	NOUN
ejpam-5930	40	12	but	but	CCONJ
ejpam-5930	40	13	also	also	ADV
ejpam-5930	40	14	opens	open	VERB
ejpam-5930	40	15	new	new	ADJ
ejpam-5930	40	16	avenues	avenue	NOUN
ejpam-5930	40	17	for	for	ADP
ejpam-5930	40	18	applications	application	NOUN
ejpam-5930	40	19	in	in	ADP
ejpam-5930	40	20	fuzzy	fuzzy	ADJ
ejpam-5930	40	21	logic	logic	NOUN
ejpam-5930	40	22	,	,	PUNCT
ejpam-5930	40	23	decision	decision	NOUN
ejpam-5930	40	24	-	-	PUNCT
ejpam-5930	40	25	making	make	VERB
ejpam-5930	40	26	systems	system	NOUN
ejpam-5930	40	27	,	,	PUNCT
ejpam-5930	40	28	and	and	CCONJ
ejpam-5930	40	29	knowledge	knowledge	NOUN
ejpam-5930	40	30	representation	representation	NOUN
ejpam-5930	40	31	.	.	PUNCT
ejpam-5930	41	1	the	the	DET
ejpam-5930	41	2	ability	ability	NOUN
ejpam-5930	41	3	to	to	PART
ejpam-5930	41	4	approximate	approximate	VERB
ejpam-5930	41	5	algebraic	algebraic	ADJ
ejpam-5930	41	6	structures	structure	NOUN
ejpam-5930	41	7	within	within	ADP
ejpam-5930	41	8	hilbert	hilbert	PROPN
ejpam-5930	41	9	algebras	algebras	PROPN
ejpam-5930	41	10	through	through	ADP
ejpam-5930	41	11	rough	rough	ADJ
ejpam-5930	41	12	set	set	NOUN
ejpam-5930	41	13	approximations	approximation	NOUN
ejpam-5930	41	14	provides	provide	VERB
ejpam-5930	41	15	a	a	DET
ejpam-5930	41	16	systematic	systematic	ADJ
ejpam-5930	41	17	way	way	NOUN
ejpam-5930	41	18	to	to	PART
ejpam-5930	41	19	manage	manage	VERB
ejpam-5930	41	20	and	and	CCONJ
ejpam-5930	41	21	process	process	VERB
ejpam-5930	41	22	incomplete	incomplete	ADJ
ejpam-5930	41	23	or	or	CCONJ
ejpam-5930	41	24	ambiguous	ambiguous	ADJ
ejpam-5930	41	25	data	datum	NOUN
ejpam-5930	41	26	,	,	PUNCT
ejpam-5930	41	27	making	make	VERB
ejpam-5930	41	28	this	this	DET
ejpam-5930	41	29	integration	integration	NOUN
ejpam-5930	41	30	both	both	PRON
ejpam-5930	41	31	mathematically	mathematically	ADV
ejpam-5930	41	32	significant	significant	ADJ
ejpam-5930	41	33	and	and	CCONJ
ejpam-5930	41	34	practically	practically	ADV
ejpam-5930	41	35	relevant	relevant	ADJ
ejpam-5930	41	36	.	.	PUNCT
ejpam-5930	42	1	this	this	DET
ejpam-5930	42	2	paper	paper	NOUN
ejpam-5930	42	3	explores	explore	VERB
ejpam-5930	42	4	the	the	DET
ejpam-5930	42	5	integration	integration	NOUN
ejpam-5930	42	6	of	of	ADP
ejpam-5930	42	7	rst	rst	PROPN
ejpam-5930	42	8	with	with	ADP
ejpam-5930	42	9	hilbert	hilbert	PROPN
ejpam-5930	42	10	algebras	algebras	PROPN
ejpam-5930	42	11	,	,	PUNCT
ejpam-5930	42	12	a	a	DET
ejpam-5930	42	13	class	class	NOUN
ejpam-5930	42	14	of	of	ADP
ejpam-5930	42	15	algebraic	algebraic	ADJ
ejpam-5930	42	16	a.	a.	NOUN
ejpam-5930	42	17	iampan	iampan	PROPN
ejpam-5930	42	18	et	et	PROPN
ejpam-5930	42	19	al	al	PROPN
ejpam-5930	42	20	.	.	PUNCT
ejpam-5930	42	21	/	/	SYM
ejpam-5930	42	22	eur	eur	PROPN
ejpam-5930	42	23	.	.	PUNCT
ejpam-5930	43	1	j.	j.	PROPN
ejpam-5930	43	2	pure	pure	PROPN
ejpam-5930	43	3	appl	appl	PROPN
ejpam-5930	43	4	.	.	PROPN
ejpam-5930	43	5	math	math	PROPN
ejpam-5930	43	6	,	,	PUNCT
ejpam-5930	43	7	18	18	NUM
ejpam-5930	43	8	(	(	PUNCT
ejpam-5930	43	9	2	2	NUM
ejpam-5930	43	10	)	)	PUNCT
ejpam-5930	43	11	(	(	PUNCT
ejpam-5930	43	12	2025	2025	NUM
ejpam-5930	43	13	)	)	PUNCT
ejpam-5930	43	14	,	,	PUNCT
ejpam-5930	43	15	5930	5930	NUM
ejpam-5930	43	16	3	3	NUM
ejpam-5930	43	17	of	of	ADP
ejpam-5930	43	18	11	11	NUM
ejpam-5930	43	19	structures	structure	NOUN
ejpam-5930	43	20	central	central	ADJ
ejpam-5930	43	21	to	to	ADP
ejpam-5930	43	22	non	non	ADJ
ejpam-5930	43	23	-	-	ADJ
ejpam-5930	43	24	classical	classical	ADJ
ejpam-5930	43	25	logic	logic	NOUN
ejpam-5930	43	26	.	.	PUNCT
ejpam-5930	44	1	we	we	PRON
ejpam-5930	44	2	introduce	introduce	VERB
ejpam-5930	44	3	the	the	DET
ejpam-5930	44	4	concept	concept	NOUN
ejpam-5930	44	5	of	of	ADP
ejpam-5930	44	6	roughness	roughness	NOUN
ejpam-5930	44	7	in	in	ADP
ejpam-5930	44	8	hilbert	hilbert	PROPN
ejpam-5930	44	9	algebras	algebras	PROPN
ejpam-5930	44	10	and	and	CCONJ
ejpam-5930	44	11	investigate	investigate	VERB
ejpam-5930	44	12	the	the	DET
ejpam-5930	44	13	lower	low	ADJ
ejpam-5930	44	14	and	and	CCONJ
ejpam-5930	44	15	upper	upper	ADJ
ejpam-5930	44	16	approximations	approximation	NOUN
ejpam-5930	44	17	of	of	ADP
ejpam-5930	44	18	subalgebras	subalgebra	NOUN
ejpam-5930	44	19	and	and	CCONJ
ejpam-5930	44	20	ideals	ideal	NOUN
ejpam-5930	44	21	.	.	PUNCT
ejpam-5930	45	1	our	our	PRON
ejpam-5930	45	2	main	main	ADJ
ejpam-5930	45	3	results	result	NOUN
ejpam-5930	45	4	show	show	VERB
ejpam-5930	45	5	that	that	SCONJ
ejpam-5930	45	6	these	these	DET
ejpam-5930	45	7	approximations	approximation	NOUN
ejpam-5930	45	8	preserve	preserve	VERB
ejpam-5930	45	9	the	the	DET
ejpam-5930	45	10	structure	structure	NOUN
ejpam-5930	45	11	of	of	ADP
ejpam-5930	45	12	subalgebras	subalgebra	NOUN
ejpam-5930	45	13	and	and	CCONJ
ejpam-5930	45	14	ideals	ideal	NOUN
ejpam-5930	45	15	,	,	PUNCT
ejpam-5930	45	16	extending	extend	VERB
ejpam-5930	45	17	the	the	DET
ejpam-5930	45	18	applicability	applicability	NOUN
ejpam-5930	45	19	of	of	ADP
ejpam-5930	45	20	rough	rough	ADJ
ejpam-5930	45	21	set	set	VERB
ejpam-5930	45	22	concepts	concept	NOUN
ejpam-5930	45	23	to	to	ADP
ejpam-5930	45	24	algebraic	algebraic	ADJ
ejpam-5930	45	25	systems	system	NOUN
ejpam-5930	45	26	.	.	PUNCT
ejpam-5930	46	1	the	the	DET
ejpam-5930	46	2	approximation	approximation	NOUN
ejpam-5930	46	3	spaces	space	NOUN
ejpam-5930	46	4	induced	induce	VERB
ejpam-5930	46	5	by	by	ADP
ejpam-5930	46	6	ideals	ideal	NOUN
ejpam-5930	46	7	in	in	ADP
ejpam-5930	46	8	hilbert	hilbert	PROPN
ejpam-5930	46	9	algebras	algebras	PROPN
ejpam-5930	46	10	offer	offer	VERB
ejpam-5930	46	11	a	a	DET
ejpam-5930	46	12	powerful	powerful	ADJ
ejpam-5930	46	13	framework	framework	NOUN
ejpam-5930	46	14	for	for	ADP
ejpam-5930	46	15	analyzing	analyze	VERB
ejpam-5930	46	16	algebraic	algebraic	ADJ
ejpam-5930	46	17	structures	structure	NOUN
ejpam-5930	46	18	under	under	ADP
ejpam-5930	46	19	conditions	condition	NOUN
ejpam-5930	46	20	of	of	ADP
ejpam-5930	46	21	uncertainty	uncertainty	NOUN
ejpam-5930	46	22	or	or	CCONJ
ejpam-5930	46	23	incomplete	incomplete	ADJ
ejpam-5930	46	24	information	information	NOUN
ejpam-5930	46	25	.	.	PUNCT
ejpam-5930	47	1	through	through	ADP
ejpam-5930	47	2	illustrative	illustrative	ADJ
ejpam-5930	47	3	examples	example	NOUN
ejpam-5930	47	4	,	,	PUNCT
ejpam-5930	47	5	we	we	PRON
ejpam-5930	47	6	validate	validate	VERB
ejpam-5930	47	7	our	our	PRON
ejpam-5930	47	8	theoretical	theoretical	ADJ
ejpam-5930	47	9	contributions	contribution	NOUN
ejpam-5930	47	10	and	and	CCONJ
ejpam-5930	47	11	underscore	underscore	VERB
ejpam-5930	47	12	the	the	DET
ejpam-5930	47	13	practical	practical	ADJ
ejpam-5930	47	14	relevance	relevance	NOUN
ejpam-5930	47	15	of	of	ADP
ejpam-5930	47	16	this	this	DET
ejpam-5930	47	17	approach	approach	NOUN
ejpam-5930	47	18	.	.	PUNCT
ejpam-5930	48	1	this	this	DET
ejpam-5930	48	2	work	work	NOUN
ejpam-5930	48	3	not	not	PART
ejpam-5930	48	4	only	only	ADV
ejpam-5930	48	5	advances	advance	VERB
ejpam-5930	48	6	the	the	DET
ejpam-5930	48	7	theoretical	theoretical	ADJ
ejpam-5930	48	8	foundations	foundation	NOUN
ejpam-5930	48	9	of	of	ADP
ejpam-5930	48	10	rst	rst	PROPN
ejpam-5930	48	11	but	but	CCONJ
ejpam-5930	48	12	also	also	ADV
ejpam-5930	48	13	paves	pave	VERB
ejpam-5930	48	14	the	the	DET
ejpam-5930	48	15	way	way	NOUN
ejpam-5930	48	16	for	for	ADP
ejpam-5930	48	17	novel	novel	ADJ
ejpam-5930	48	18	applications	application	NOUN
ejpam-5930	48	19	in	in	ADP
ejpam-5930	48	20	algebraic	algebraic	ADJ
ejpam-5930	48	21	logic	logic	NOUN
ejpam-5930	48	22	and	and	CCONJ
ejpam-5930	48	23	beyond	beyond	ADP
ejpam-5930	48	24	.	.	PUNCT
ejpam-5930	49	1	the	the	DET
ejpam-5930	49	2	rest	rest	NOUN
ejpam-5930	49	3	of	of	ADP
ejpam-5930	49	4	this	this	DET
ejpam-5930	49	5	paper	paper	NOUN
ejpam-5930	49	6	is	be	AUX
ejpam-5930	49	7	organized	organize	VERB
ejpam-5930	49	8	as	as	SCONJ
ejpam-5930	49	9	follows	follow	VERB
ejpam-5930	49	10	:	:	PUNCT
ejpam-5930	49	11	section	section	NOUN
ejpam-5930	49	12	2	2	NUM
ejpam-5930	49	13	presents	present	VERB
ejpam-5930	49	14	the	the	DET
ejpam-5930	49	15	necessary	necessary	ADJ
ejpam-5930	49	16	preliminaries	preliminary	NOUN
ejpam-5930	49	17	on	on	ADP
ejpam-5930	49	18	rough	rough	ADJ
ejpam-5930	49	19	sets	set	NOUN
ejpam-5930	49	20	and	and	CCONJ
ejpam-5930	49	21	hilbert	hilbert	PROPN
ejpam-5930	49	22	algebras	algebras	PROPN
ejpam-5930	49	23	.	.	PUNCT
ejpam-5930	50	1	section	section	NOUN
ejpam-5930	50	2	3	3	NUM
ejpam-5930	50	3	introduces	introduce	NOUN
ejpam-5930	50	4	and	and	CCONJ
ejpam-5930	50	5	investigates	investigate	VERB
ejpam-5930	50	6	the	the	DET
ejpam-5930	50	7	rough	rough	ADJ
ejpam-5930	50	8	approximations	approximation	NOUN
ejpam-5930	50	9	in	in	ADP
ejpam-5930	50	10	hilbert	hilbert	PROPN
ejpam-5930	50	11	algebras	algebras	PROPN
ejpam-5930	50	12	,	,	PUNCT
ejpam-5930	50	13	providing	provide	VERB
ejpam-5930	50	14	key	key	ADJ
ejpam-5930	50	15	results	result	NOUN
ejpam-5930	50	16	and	and	CCONJ
ejpam-5930	50	17	illustrative	illustrative	ADJ
ejpam-5930	50	18	examples	example	NOUN
ejpam-5930	50	19	.	.	PUNCT
ejpam-5930	51	1	finally	finally	ADV
ejpam-5930	51	2	,	,	PUNCT
ejpam-5930	51	3	section	section	NOUN
ejpam-5930	51	4	4	4	NUM
ejpam-5930	51	5	concludes	conclude	VERB
ejpam-5930	51	6	the	the	DET
ejpam-5930	51	7	paper	paper	NOUN
ejpam-5930	51	8	with	with	ADP
ejpam-5930	51	9	a	a	DET
ejpam-5930	51	10	summary	summary	NOUN
ejpam-5930	51	11	and	and	CCONJ
ejpam-5930	51	12	future	future	ADJ
ejpam-5930	51	13	research	research	NOUN
ejpam-5930	51	14	directions	direction	NOUN
ejpam-5930	51	15	.	.	PUNCT
ejpam-5930	52	1	2	2	X
ejpam-5930	52	2	.	.	X
ejpam-5930	52	3	preliminaries	preliminary	NOUN
ejpam-5930	52	4	let	let	VERB
ejpam-5930	52	5	u	u	PRON
ejpam-5930	52	6	be	be	AUX
ejpam-5930	52	7	a	a	DET
ejpam-5930	52	8	universal	universal	ADJ
ejpam-5930	52	9	set	set	NOUN
ejpam-5930	52	10	.	.	PUNCT
ejpam-5930	53	1	for	for	ADP
ejpam-5930	53	2	an	an	DET
ejpam-5930	53	3	equivalence	equivalence	NOUN
ejpam-5930	53	4	relation	relation	NOUN
ejpam-5930	53	5	θ	θ	PROPN
ejpam-5930	53	6	on	on	ADP
ejpam-5930	53	7	u	u	PROPN
ejpam-5930	53	8	,	,	PUNCT
ejpam-5930	53	9	the	the	DET
ejpam-5930	53	10	set	set	NOUN
ejpam-5930	53	11	of	of	ADP
ejpam-5930	53	12	elements	element	NOUN
ejpam-5930	53	13	of	of	ADP
ejpam-5930	53	14	u	u	PRON
ejpam-5930	53	15	that	that	PRON
ejpam-5930	53	16	are	be	AUX
ejpam-5930	53	17	related	relate	VERB
ejpam-5930	53	18	to	to	ADP
ejpam-5930	53	19	x	x	PART
ejpam-5930	53	20	∈	∈	PROPN
ejpam-5930	53	21	u	u	NOUN
ejpam-5930	53	22	is	be	AUX
ejpam-5930	53	23	called	call	VERB
ejpam-5930	53	24	the	the	DET
ejpam-5930	53	25	equivalence	equivalence	NOUN
ejpam-5930	53	26	class	class	NOUN
ejpam-5930	53	27	of	of	ADP
ejpam-5930	53	28	x	x	PUNCT
ejpam-5930	53	29	and	and	CCONJ
ejpam-5930	53	30	is	be	AUX
ejpam-5930	53	31	denoted	denote	VERB
ejpam-5930	53	32	by	by	ADP
ejpam-5930	53	33	[	[	X
ejpam-5930	53	34	x]θ	x]θ	PROPN
ejpam-5930	53	35	.	.	PUNCT
ejpam-5930	54	1	moreover	moreover	ADV
ejpam-5930	54	2	,	,	PUNCT
ejpam-5930	54	3	let	let	VERB
ejpam-5930	54	4	u	u	PRON
ejpam-5930	54	5	/	/	SYM
ejpam-5930	54	6	θ	θ	PROPN
ejpam-5930	54	7	denote	denote	VERB
ejpam-5930	54	8	the	the	DET
ejpam-5930	54	9	family	family	NOUN
ejpam-5930	54	10	of	of	ADP
ejpam-5930	54	11	all	all	DET
ejpam-5930	54	12	equivalence	equivalence	NOUN
ejpam-5930	54	13	classes	class	NOUN
ejpam-5930	54	14	induced	induce	VERB
ejpam-5930	54	15	on	on	ADP
ejpam-5930	54	16	u	u	NOUN
ejpam-5930	54	17	by	by	ADP
ejpam-5930	54	18	θ	θ	PROPN
ejpam-5930	54	19	.	.	PROPN
ejpam-5930	55	1	for	for	ADP
ejpam-5930	55	2	any	any	DET
ejpam-5930	55	3	x	x	SYM
ejpam-5930	55	4	⊆	⊆	NUM
ejpam-5930	55	5	u	u	NOUN
ejpam-5930	55	6	,	,	PUNCT
ejpam-5930	55	7	we	we	PRON
ejpam-5930	55	8	write	write	VERB
ejpam-5930	55	9	xc	xc	PROPN
ejpam-5930	55	10	to	to	PART
ejpam-5930	55	11	denote	denote	VERB
ejpam-5930	55	12	the	the	DET
ejpam-5930	55	13	complement	complement	NOUN
ejpam-5930	55	14	of	of	ADP
ejpam-5930	55	15	x	x	PUNCT
ejpam-5930	55	16	in	in	ADP
ejpam-5930	55	17	u	u	NOUN
ejpam-5930	55	18	,	,	PUNCT
ejpam-5930	55	19	the	the	DET
ejpam-5930	55	20	set	set	NOUN
ejpam-5930	55	21	u\x	u\x	NOUN
ejpam-5930	55	22	.	.	PUNCT
ejpam-5930	56	1	a	a	DET
ejpam-5930	56	2	pair	pair	NOUN
ejpam-5930	56	3	(	(	PUNCT
ejpam-5930	56	4	u	u	NOUN
ejpam-5930	56	5	,	,	PUNCT
ejpam-5930	56	6	θ	θ	PROPN
ejpam-5930	56	7	)	)	PUNCT
ejpam-5930	56	8	,	,	PUNCT
ejpam-5930	56	9	where	where	SCONJ
ejpam-5930	56	10	u	u	NOUN
ejpam-5930	56	11	̸=	̸=	PROPN
ejpam-5930	56	12	∅	∅	NOUN
ejpam-5930	56	13	and	and	CCONJ
ejpam-5930	56	14	θ	θ	PROPN
ejpam-5930	56	15	is	be	AUX
ejpam-5930	56	16	an	an	DET
ejpam-5930	56	17	equivalence	equivalence	NOUN
ejpam-5930	56	18	relation	relation	NOUN
ejpam-5930	56	19	on	on	ADP
ejpam-5930	56	20	u	u	NOUN
ejpam-5930	56	21	is	be	AUX
ejpam-5930	56	22	called	call	VERB
ejpam-5930	56	23	an	an	DET
ejpam-5930	56	24	approximation	approximation	NOUN
ejpam-5930	56	25	space	space	NOUN
ejpam-5930	56	26	.	.	PUNCT
ejpam-5930	57	1	the	the	DET
ejpam-5930	57	2	interpretation	interpretation	NOUN
ejpam-5930	57	3	in	in	ADP
ejpam-5930	57	4	rst	rst	PROPN
ejpam-5930	57	5	is	be	AUX
ejpam-5930	57	6	that	that	SCONJ
ejpam-5930	57	7	our	our	PRON
ejpam-5930	57	8	knowledge	knowledge	NOUN
ejpam-5930	57	9	of	of	ADP
ejpam-5930	57	10	the	the	DET
ejpam-5930	57	11	objects	object	NOUN
ejpam-5930	57	12	in	in	ADP
ejpam-5930	57	13	u	u	NOUN
ejpam-5930	57	14	extends	extend	VERB
ejpam-5930	57	15	only	only	ADV
ejpam-5930	57	16	up	up	ADP
ejpam-5930	57	17	to	to	ADP
ejpam-5930	57	18	membership	membership	NOUN
ejpam-5930	57	19	in	in	ADP
ejpam-5930	57	20	the	the	DET
ejpam-5930	57	21	class	class	NOUN
ejpam-5930	57	22	of	of	ADP
ejpam-5930	57	23	θ	θ	PROPN
ejpam-5930	57	24	and	and	CCONJ
ejpam-5930	57	25	our	our	PRON
ejpam-5930	57	26	knowledge	knowledge	NOUN
ejpam-5930	57	27	about	about	ADP
ejpam-5930	57	28	a	a	DET
ejpam-5930	57	29	subset	subset	NOUN
ejpam-5930	57	30	x	x	PUNCT
ejpam-5930	57	31	of	of	ADP
ejpam-5930	57	32	u	u	NOUN
ejpam-5930	57	33	is	be	AUX
ejpam-5930	57	34	limited	limit	VERB
ejpam-5930	57	35	to	to	ADP
ejpam-5930	57	36	the	the	DET
ejpam-5930	57	37	class	class	NOUN
ejpam-5930	57	38	of	of	ADP
ejpam-5930	57	39	θ	θ	PROPN
ejpam-5930	57	40	and	and	CCONJ
ejpam-5930	57	41	their	their	PRON
ejpam-5930	57	42	unions	union	NOUN
ejpam-5930	57	43	.	.	PUNCT
ejpam-5930	58	1	this	this	PRON
ejpam-5930	58	2	leads	lead	VERB
ejpam-5930	58	3	to	to	ADP
ejpam-5930	58	4	the	the	DET
ejpam-5930	58	5	following	follow	VERB
ejpam-5930	58	6	definition	definition	NOUN
ejpam-5930	58	7	.	.	PUNCT
ejpam-5930	59	1	definition	definition	NOUN
ejpam-5930	59	2	1	1	NUM
ejpam-5930	59	3	.	.	PUNCT
ejpam-5930	60	1	[	[	X
ejpam-5930	60	2	1	1	X
ejpam-5930	60	3	]	]	PUNCT
ejpam-5930	60	4	let	let	VERB
ejpam-5930	60	5	p(u	p(u	X
ejpam-5930	60	6	)	)	PUNCT
ejpam-5930	60	7	denote	denote	VERB
ejpam-5930	60	8	the	the	DET
ejpam-5930	60	9	power	power	NOUN
ejpam-5930	60	10	set	set	NOUN
ejpam-5930	60	11	of	of	ADP
ejpam-5930	60	12	a	a	DET
ejpam-5930	60	13	universal	universal	ADJ
ejpam-5930	60	14	set	set	NOUN
ejpam-5930	60	15	u	u	NOUN
ejpam-5930	60	16	.	.	PUNCT
ejpam-5930	61	1	for	for	ADP
ejpam-5930	61	2	an	an	DET
ejpam-5930	61	3	approximation	approximation	NOUN
ejpam-5930	61	4	space	space	NOUN
ejpam-5930	61	5	(	(	PUNCT
ejpam-5930	61	6	u	u	NOUN
ejpam-5930	61	7	,	,	PUNCT
ejpam-5930	61	8	θ	θ	PROPN
ejpam-5930	61	9	)	)	PUNCT
ejpam-5930	61	10	,	,	PUNCT
ejpam-5930	61	11	by	by	ADP
ejpam-5930	61	12	a	a	DET
ejpam-5930	61	13	rough	rough	ADJ
ejpam-5930	61	14	approximation	approximation	NOUN
ejpam-5930	61	15	in	in	ADP
ejpam-5930	61	16	(	(	PUNCT
ejpam-5930	61	17	u	u	NOUN
ejpam-5930	61	18	,	,	PUNCT
ejpam-5930	61	19	θ	θ	NOUN
ejpam-5930	61	20	)	)	PUNCT
ejpam-5930	61	21	we	we	PRON
ejpam-5930	61	22	mean	mean	VERB
ejpam-5930	61	23	a	a	DET
ejpam-5930	61	24	mapping	mapping	NOUN
ejpam-5930	61	25	apr	apr	NOUN
ejpam-5930	61	26	:	:	PUNCT
ejpam-5930	61	27	p(u	p(u	X
ejpam-5930	61	28	)	)	PUNCT
ejpam-5930	61	29	→	→	SYM
ejpam-5930	61	30	p(u	p(u	ADJ
ejpam-5930	61	31	)	)	PUNCT
ejpam-5930	61	32	×	×	NOUN
ejpam-5930	61	33	p(u	p(u	NOUN
ejpam-5930	61	34	)	)	PUNCT
ejpam-5930	61	35	defined	define	VERB
ejpam-5930	61	36	for	for	ADP
ejpam-5930	61	37	every	every	DET
ejpam-5930	61	38	x	x	SYM
ejpam-5930	61	39	∈	∈	PROPN
ejpam-5930	61	40	p(u	p(u	NOUN
ejpam-5930	61	41	)	)	PUNCT
ejpam-5930	61	42	by	by	ADP
ejpam-5930	61	43	apr(x	apr(x	PROPN
ejpam-5930	61	44	)	)	PUNCT
ejpam-5930	61	45	=	=	SYM
ejpam-5930	61	46	(	(	PUNCT
ejpam-5930	61	47	apr(x	apr(x	PROPN
ejpam-5930	61	48	)	)	PUNCT
ejpam-5930	61	49	,	,	PUNCT
ejpam-5930	61	50	apr(x	apr(x	PROPN
ejpam-5930	61	51	)	)	PUNCT
ejpam-5930	61	52	)	)	PUNCT
ejpam-5930	61	53	,	,	PUNCT
ejpam-5930	62	1	where	where	SCONJ
ejpam-5930	62	2	apr(x	apr(x	NOUN
ejpam-5930	62	3	)	)	PUNCT
ejpam-5930	62	4	=	=	PRON
ejpam-5930	62	5	{	{	PUNCT
ejpam-5930	62	6	x	x	PUNCT
ejpam-5930	62	7	∈	∈	PROPN
ejpam-5930	62	8	u	u	NOUN
ejpam-5930	62	9	:	:	PUNCT
ejpam-5930	62	10	[	[	X
ejpam-5930	62	11	x]θ	x]θ	NOUN
ejpam-5930	62	12	⊆	⊆	NUM
ejpam-5930	62	13	x	x	SYM
ejpam-5930	62	14	}	}	PUNCT
ejpam-5930	62	15	and	and	CCONJ
ejpam-5930	62	16	apr(x	apr(x	PROPN
ejpam-5930	62	17	)	)	PUNCT
ejpam-5930	62	18	=	=	PRON
ejpam-5930	62	19	{	{	PUNCT
ejpam-5930	62	20	x	x	PUNCT
ejpam-5930	62	21	∈	∈	PROPN
ejpam-5930	62	22	u	u	NOUN
ejpam-5930	62	23	:	:	PUNCT
ejpam-5930	63	1	[	[	X
ejpam-5930	63	2	x]θ	x]θ	X
ejpam-5930	63	3	∩	∩	X
ejpam-5930	63	4	x	x	SYM
ejpam-5930	63	5	=	=	VERB
ejpam-5930	63	6	∅	∅	NOUN
ejpam-5930	63	7	}	}	PUNCT
ejpam-5930	63	8	.	.	PUNCT
ejpam-5930	64	1	also	also	ADV
ejpam-5930	64	2	,	,	PUNCT
ejpam-5930	64	3	apr(x	apr(x	PROPN
ejpam-5930	64	4	)	)	PUNCT
ejpam-5930	64	5	is	be	AUX
ejpam-5930	64	6	called	call	VERB
ejpam-5930	64	7	a	a	DET
ejpam-5930	64	8	lower	low	ADJ
ejpam-5930	64	9	rough	rough	ADJ
ejpam-5930	64	10	approximation	approximation	NOUN
ejpam-5930	64	11	of	of	ADP
ejpam-5930	64	12	x	x	PUNCT
ejpam-5930	64	13	in	in	ADP
ejpam-5930	64	14	(	(	PUNCT
ejpam-5930	64	15	u	u	NOUN
ejpam-5930	64	16	,	,	PUNCT
ejpam-5930	64	17	θ	θ	PROPN
ejpam-5930	64	18	)	)	PUNCT
ejpam-5930	64	19	,	,	PUNCT
ejpam-5930	64	20	whereas	whereas	SCONJ
ejpam-5930	64	21	apr(x	apr(x	PROPN
ejpam-5930	64	22	)	)	PUNCT
ejpam-5930	64	23	is	be	AUX
ejpam-5930	64	24	called	call	VERB
ejpam-5930	64	25	an	an	DET
ejpam-5930	64	26	upper	upper	ADJ
ejpam-5930	64	27	rough	rough	ADJ
ejpam-5930	64	28	approximation	approximation	NOUN
ejpam-5930	64	29	of	of	ADP
ejpam-5930	64	30	x	x	PUNCT
ejpam-5930	64	31	in	in	ADP
ejpam-5930	64	32	(	(	PUNCT
ejpam-5930	64	33	u	u	NOUN
ejpam-5930	64	34	,	,	PUNCT
ejpam-5930	64	35	θ	θ	NOUN
ejpam-5930	64	36	)	)	PUNCT
ejpam-5930	64	37	.	.	PUNCT
ejpam-5930	65	1	definition	definition	NOUN
ejpam-5930	65	2	2	2	NUM
ejpam-5930	65	3	.	.	PUNCT
ejpam-5930	66	1	[	[	X
ejpam-5930	66	2	1	1	NUM
ejpam-5930	66	3	]	]	PUNCT
ejpam-5930	66	4	given	give	VERB
ejpam-5930	66	5	an	an	DET
ejpam-5930	66	6	approximation	approximation	NOUN
ejpam-5930	66	7	space	space	NOUN
ejpam-5930	66	8	(	(	PUNCT
ejpam-5930	66	9	u	u	NOUN
ejpam-5930	66	10	,	,	PUNCT
ejpam-5930	66	11	θ	θ	PROPN
ejpam-5930	66	12	)	)	PUNCT
ejpam-5930	66	13	,	,	PUNCT
ejpam-5930	66	14	a	a	DET
ejpam-5930	66	15	pair	pair	NOUN
ejpam-5930	66	16	(	(	PUNCT
ejpam-5930	66	17	a	a	PRON
ejpam-5930	66	18	,	,	PUNCT
ejpam-5930	66	19	b	b	NOUN
ejpam-5930	66	20	)	)	PUNCT
ejpam-5930	66	21	∈	∈	PROPN
ejpam-5930	66	22	p(u)×p(u	p(u)×p(u	NOUN
ejpam-5930	66	23	)	)	PUNCT
ejpam-5930	66	24	is	be	AUX
ejpam-5930	66	25	called	call	VERB
ejpam-5930	66	26	a	a	DET
ejpam-5930	66	27	rough	rough	ADJ
ejpam-5930	66	28	set	set	NOUN
ejpam-5930	66	29	in	in	ADP
ejpam-5930	66	30	(	(	PUNCT
ejpam-5930	66	31	u	u	NOUN
ejpam-5930	66	32	,	,	PUNCT
ejpam-5930	66	33	θ	θ	NOUN
ejpam-5930	66	34	)	)	PUNCT
ejpam-5930	66	35	if	if	SCONJ
ejpam-5930	66	36	(	(	PUNCT
ejpam-5930	66	37	a	a	DET
ejpam-5930	66	38	,	,	PUNCT
ejpam-5930	66	39	b	b	NOUN
ejpam-5930	66	40	)	)	PUNCT
ejpam-5930	66	41	=	=	SYM
ejpam-5930	66	42	apr(x	apr(x	PROPN
ejpam-5930	66	43	)	)	PUNCT
ejpam-5930	66	44	for	for	ADP
ejpam-5930	66	45	some	some	DET
ejpam-5930	66	46	x	x	SYM
ejpam-5930	66	47	∈	∈	PROPN
ejpam-5930	66	48	p(u	p(u	NOUN
ejpam-5930	66	49	)	)	PUNCT
ejpam-5930	66	50	.	.	PUNCT
ejpam-5930	67	1	definition	definition	NOUN
ejpam-5930	67	2	3	3	NUM
ejpam-5930	67	3	.	.	PUNCT
ejpam-5930	68	1	[	[	X
ejpam-5930	68	2	8	8	NUM
ejpam-5930	68	3	]	]	PUNCT
ejpam-5930	68	4	a	a	DET
ejpam-5930	68	5	hilbert	hilbert	NOUN
ejpam-5930	68	6	algebra	algebra	NOUN
ejpam-5930	68	7	is	be	AUX
ejpam-5930	68	8	a	a	DET
ejpam-5930	68	9	triplet	triplet	NOUN
ejpam-5930	68	10	with	with	ADP
ejpam-5930	68	11	the	the	DET
ejpam-5930	68	12	formula	formula	NOUN
ejpam-5930	68	13	a	a	DET
ejpam-5930	68	14	=	=	X
ejpam-5930	68	15	(	(	PUNCT
ejpam-5930	68	16	a	a	PRON
ejpam-5930	68	17	,	,	PUNCT
ejpam-5930	68	18	·	·	PUNCT
ejpam-5930	68	19	,	,	PUNCT
ejpam-5930	68	20	1	1	NUM
ejpam-5930	68	21	)	)	PUNCT
ejpam-5930	68	22	,	,	PUNCT
ejpam-5930	68	23	where	where	SCONJ
ejpam-5930	68	24	a	a	PRON
ejpam-5930	68	25	is	be	AUX
ejpam-5930	68	26	a	a	DET
ejpam-5930	68	27	nonempty	nonempty	ADJ
ejpam-5930	68	28	set	set	VERB
ejpam-5930	68	29	,	,	PUNCT
ejpam-5930	68	30	·	·	PUNCT
ejpam-5930	68	31	is	be	AUX
ejpam-5930	68	32	a	a	DET
ejpam-5930	68	33	binary	binary	ADJ
ejpam-5930	68	34	operation	operation	NOUN
ejpam-5930	68	35	,	,	PUNCT
ejpam-5930	68	36	and	and	CCONJ
ejpam-5930	68	37	1	1	NUM
ejpam-5930	68	38	is	be	AUX
ejpam-5930	68	39	a	a	DET
ejpam-5930	68	40	fixed	fix	VERB
ejpam-5930	68	41	member	member	NOUN
ejpam-5930	68	42	of	of	ADP
ejpam-5930	68	43	a	a	PRON
ejpam-5930	68	44	that	that	PRON
ejpam-5930	68	45	is	be	AUX
ejpam-5930	68	46	true	true	ADJ
ejpam-5930	68	47	according	accord	VERB
ejpam-5930	68	48	to	to	ADP
ejpam-5930	68	49	the	the	DET
ejpam-5930	68	50	axioms	axiom	NOUN
ejpam-5930	68	51	stated	state	VERB
ejpam-5930	68	52	below	below	ADV
ejpam-5930	68	53	:	:	PUNCT
ejpam-5930	68	54	(	(	PUNCT
ejpam-5930	68	55	∀x	∀x	X
ejpam-5930	68	56	,	,	PUNCT
ejpam-5930	68	57	y	y	PROPN
ejpam-5930	68	58	∈	∈	PROPN
ejpam-5930	68	59	a)(x	a)(x	PROPN
ejpam-5930	68	60	·	·	PUNCT
ejpam-5930	68	61	(	(	PUNCT
ejpam-5930	68	62	y	y	NOUN
ejpam-5930	68	63	·	·	PUNCT
ejpam-5930	68	64	x	x	X
ejpam-5930	68	65	)	)	PUNCT
ejpam-5930	68	66	=	=	SYM
ejpam-5930	68	67	1	1	X
ejpam-5930	68	68	)	)	PUNCT
ejpam-5930	68	69	(	(	PUNCT
ejpam-5930	68	70	1	1	X
ejpam-5930	68	71	)	)	PUNCT
ejpam-5930	68	72	(	(	PUNCT
ejpam-5930	68	73	∀x	∀x	X
ejpam-5930	68	74	,	,	PUNCT
ejpam-5930	68	75	y	y	PROPN
ejpam-5930	68	76	,	,	PUNCT
ejpam-5930	68	77	z	z	PROPN
ejpam-5930	68	78	∈	∈	PROPN
ejpam-5930	68	79	a)((x	a)((x	NOUN
ejpam-5930	68	80	·	·	PUNCT
ejpam-5930	68	81	(	(	PUNCT
ejpam-5930	68	82	y	y	PROPN
ejpam-5930	68	83	·	·	PUNCT
ejpam-5930	68	84	z	z	NOUN
ejpam-5930	68	85	)	)	PUNCT
ejpam-5930	68	86	)	)	PUNCT
ejpam-5930	68	87	·	·	PUNCT
ejpam-5930	69	1	(	(	PUNCT
ejpam-5930	69	2	(	(	PUNCT
ejpam-5930	69	3	x	x	SYM
ejpam-5930	69	4	·	·	PUNCT
ejpam-5930	69	5	y	y	X
ejpam-5930	69	6	)	)	PUNCT
ejpam-5930	69	7	·	·	PUNCT
ejpam-5930	70	1	(	(	PUNCT
ejpam-5930	70	2	x	x	X
ejpam-5930	70	3	·	·	PUNCT
ejpam-5930	70	4	z	z	NOUN
ejpam-5930	70	5	)	)	PUNCT
ejpam-5930	70	6	)	)	PUNCT
ejpam-5930	71	1	=	=	SYM
ejpam-5930	71	2	1	1	X
ejpam-5930	71	3	)	)	PUNCT
ejpam-5930	71	4	(	(	PUNCT
ejpam-5930	71	5	2	2	NUM
ejpam-5930	71	6	)	)	PUNCT
ejpam-5930	71	7	(	(	PUNCT
ejpam-5930	71	8	∀x	∀x	X
ejpam-5930	71	9	,	,	PUNCT
ejpam-5930	71	10	y	y	PROPN
ejpam-5930	71	11	∈	∈	PROPN
ejpam-5930	71	12	a)(x	a)(x	X
ejpam-5930	71	13	·	·	PUNCT
ejpam-5930	71	14	y	y	X
ejpam-5930	71	15	=	=	SYM
ejpam-5930	71	16	1	1	NUM
ejpam-5930	71	17	,	,	PUNCT
ejpam-5930	71	18	y	y	PROPN
ejpam-5930	71	19	·	·	PUNCT
ejpam-5930	71	20	x	x	PUNCT
ejpam-5930	71	21	=	=	SYM
ejpam-5930	71	22	1	1	NUM
ejpam-5930	71	23	⇒	⇒	NOUN
ejpam-5930	71	24	x	x	PUNCT
ejpam-5930	71	25	=	=	SYM
ejpam-5930	71	26	y	y	NOUN
ejpam-5930	71	27	)	)	PUNCT
ejpam-5930	71	28	(	(	PUNCT
ejpam-5930	71	29	3	3	X
ejpam-5930	71	30	)	)	PUNCT
ejpam-5930	71	31	a.	a.	NOUN
ejpam-5930	71	32	iampan	iampan	NOUN
ejpam-5930	71	33	et	et	PROPN
ejpam-5930	71	34	al	al	PROPN
ejpam-5930	71	35	.	.	PUNCT
ejpam-5930	71	36	/	/	SYM
ejpam-5930	71	37	eur	eur	PROPN
ejpam-5930	71	38	.	.	PUNCT
ejpam-5930	72	1	j.	j.	PROPN
ejpam-5930	72	2	pure	pure	PROPN
ejpam-5930	72	3	appl	appl	PROPN
ejpam-5930	72	4	.	.	PROPN
ejpam-5930	72	5	math	math	PROPN
ejpam-5930	72	6	,	,	PUNCT
ejpam-5930	72	7	18	18	NUM
ejpam-5930	72	8	(	(	PUNCT
ejpam-5930	72	9	2	2	NUM
ejpam-5930	72	10	)	)	PUNCT
ejpam-5930	72	11	(	(	PUNCT
ejpam-5930	72	12	2025	2025	NUM
ejpam-5930	72	13	)	)	PUNCT
ejpam-5930	72	14	,	,	PUNCT
ejpam-5930	72	15	5930	5930	NUM
ejpam-5930	72	16	4	4	NUM
ejpam-5930	72	17	of	of	ADP
ejpam-5930	72	18	11	11	NUM
ejpam-5930	72	19	in	in	ADP
ejpam-5930	72	20	[	[	X
ejpam-5930	72	21	12	12	NUM
ejpam-5930	72	22	]	]	PUNCT
ejpam-5930	72	23	,	,	PUNCT
ejpam-5930	72	24	the	the	DET
ejpam-5930	72	25	following	follow	VERB
ejpam-5930	72	26	conclusion	conclusion	NOUN
ejpam-5930	72	27	was	be	AUX
ejpam-5930	72	28	established	establish	VERB
ejpam-5930	72	29	.	.	PUNCT
ejpam-5930	73	1	lemma	lemma	PROPN
ejpam-5930	73	2	1	1	X
ejpam-5930	73	3	.	.	PUNCT
ejpam-5930	74	1	let	let	VERB
ejpam-5930	74	2	a	a	DET
ejpam-5930	74	3	=	=	X
ejpam-5930	74	4	(	(	PUNCT
ejpam-5930	74	5	a	a	PRON
ejpam-5930	74	6	,	,	PUNCT
ejpam-5930	74	7	·	·	PUNCT
ejpam-5930	74	8	,	,	PUNCT
ejpam-5930	74	9	1	1	X
ejpam-5930	74	10	)	)	PUNCT
ejpam-5930	74	11	be	be	AUX
ejpam-5930	74	12	a	a	DET
ejpam-5930	74	13	hilbert	hilbert	NOUN
ejpam-5930	74	14	algebra	algebra	NOUN
ejpam-5930	74	15	.	.	PUNCT
ejpam-5930	75	1	then	then	ADV
ejpam-5930	75	2	(	(	PUNCT
ejpam-5930	75	3	1	1	X
ejpam-5930	75	4	)	)	PUNCT
ejpam-5930	75	5	(	(	PUNCT
ejpam-5930	75	6	∀x	∀x	X
ejpam-5930	75	7	∈	∈	NOUN
ejpam-5930	75	8	a)(x	a)(x	NOUN
ejpam-5930	75	9	·	·	PUNCT
ejpam-5930	75	10	x	x	SYM
ejpam-5930	75	11	=	=	SYM
ejpam-5930	75	12	1	1	NUM
ejpam-5930	75	13	)	)	PUNCT
ejpam-5930	75	14	,	,	PUNCT
ejpam-5930	75	15	(	(	PUNCT
ejpam-5930	75	16	2	2	X
ejpam-5930	75	17	)	)	PUNCT
ejpam-5930	75	18	(	(	PUNCT
ejpam-5930	75	19	∀x	∀x	X
ejpam-5930	75	20	∈	∈	NOUN
ejpam-5930	75	21	a)(1	a)(1	X
ejpam-5930	75	22	·	·	PUNCT
ejpam-5930	75	23	x	x	X
ejpam-5930	76	1	=	=	PUNCT
ejpam-5930	76	2	x	x	X
ejpam-5930	76	3	)	)	PUNCT
ejpam-5930	76	4	,	,	PUNCT
ejpam-5930	76	5	(	(	PUNCT
ejpam-5930	76	6	3	3	X
ejpam-5930	76	7	)	)	PUNCT
ejpam-5930	76	8	(	(	PUNCT
ejpam-5930	76	9	∀x	∀x	X
ejpam-5930	76	10	∈	∈	PROPN
ejpam-5930	76	11	a)(x	a)(x	NOUN
ejpam-5930	76	12	·	·	PUNCT
ejpam-5930	76	13	1	1	NUM
ejpam-5930	76	14	=	=	SYM
ejpam-5930	76	15	1	1	NUM
ejpam-5930	76	16	)	)	PUNCT
ejpam-5930	76	17	,	,	PUNCT
ejpam-5930	76	18	(	(	PUNCT
ejpam-5930	77	1	4	4	X
ejpam-5930	77	2	)	)	PUNCT
ejpam-5930	77	3	(	(	PUNCT
ejpam-5930	77	4	∀x	∀x	X
ejpam-5930	77	5	,	,	PUNCT
ejpam-5930	77	6	y	y	PROPN
ejpam-5930	77	7	,	,	PUNCT
ejpam-5930	77	8	z	z	NOUN
ejpam-5930	77	9	∈	∈	NOUN
ejpam-5930	77	10	a)(x	a)(x	PROPN
ejpam-5930	77	11	·	·	PUNCT
ejpam-5930	77	12	(	(	PUNCT
ejpam-5930	77	13	y	y	PROPN
ejpam-5930	77	14	·	·	PUNCT
ejpam-5930	77	15	z	z	X
ejpam-5930	77	16	)	)	PUNCT
ejpam-5930	77	17	=	=	SYM
ejpam-5930	78	1	y	y	PROPN
ejpam-5930	78	2	·	·	PUNCT
ejpam-5930	78	3	(	(	PUNCT
ejpam-5930	78	4	x	x	X
ejpam-5930	78	5	·	·	PUNCT
ejpam-5930	78	6	z	z	NOUN
ejpam-5930	78	7	)	)	PUNCT
ejpam-5930	78	8	)	)	PUNCT
ejpam-5930	78	9	,	,	PUNCT
ejpam-5930	78	10	(	(	PUNCT
ejpam-5930	78	11	5	5	X
ejpam-5930	78	12	)	)	PUNCT
ejpam-5930	78	13	(	(	PUNCT
ejpam-5930	78	14	∀x	∀x	X
ejpam-5930	78	15	,	,	PUNCT
ejpam-5930	78	16	y	y	PROPN
ejpam-5930	78	17	,	,	PUNCT
ejpam-5930	78	18	z	z	PROPN
ejpam-5930	78	19	∈	∈	PROPN
ejpam-5930	78	20	a)((x	a)((x	NOUN
ejpam-5930	78	21	·	·	PUNCT
ejpam-5930	78	22	z	z	X
ejpam-5930	78	23	)	)	PUNCT
ejpam-5930	78	24	·	·	PUNCT
ejpam-5930	78	25	(	(	PUNCT
ejpam-5930	78	26	(	(	PUNCT
ejpam-5930	78	27	z	z	NOUN
ejpam-5930	78	28	·	·	PUNCT
ejpam-5930	78	29	y	y	X
ejpam-5930	78	30	)	)	PUNCT
ejpam-5930	78	31	·	·	PUNCT
ejpam-5930	79	1	(	(	PUNCT
ejpam-5930	79	2	x	x	X
ejpam-5930	79	3	·	·	PUNCT
ejpam-5930	79	4	y	y	X
ejpam-5930	79	5	)	)	PUNCT
ejpam-5930	79	6	)	)	PUNCT
ejpam-5930	80	1	=	=	SYM
ejpam-5930	80	2	1	1	NUM
ejpam-5930	80	3	)	)	PUNCT
ejpam-5930	80	4	.	.	PUNCT
ejpam-5930	81	1	in	in	ADP
ejpam-5930	81	2	a	a	DET
ejpam-5930	81	3	hilbert	hilbert	NOUN
ejpam-5930	81	4	algebra	algebra	NOUN
ejpam-5930	81	5	a	a	DET
ejpam-5930	81	6	=	=	X
ejpam-5930	81	7	(	(	PUNCT
ejpam-5930	81	8	a	a	PRON
ejpam-5930	81	9	,	,	PUNCT
ejpam-5930	81	10	·	·	PUNCT
ejpam-5930	81	11	,	,	PUNCT
ejpam-5930	81	12	1	1	NUM
ejpam-5930	81	13	)	)	PUNCT
ejpam-5930	81	14	,	,	PUNCT
ejpam-5930	81	15	the	the	DET
ejpam-5930	81	16	binary	binary	PROPN
ejpam-5930	81	17	relation	relation	PROPN
ejpam-5930	81	18	≤	≤	NUM
ejpam-5930	81	19	is	be	AUX
ejpam-5930	81	20	defined	define	VERB
ejpam-5930	81	21	by	by	ADP
ejpam-5930	81	22	(	(	PUNCT
ejpam-5930	81	23	∀x	∀x	NUM
ejpam-5930	81	24	,	,	PUNCT
ejpam-5930	81	25	y	y	PROPN
ejpam-5930	81	26	∈	∈	PROPN
ejpam-5930	81	27	a)(x	a)(x	PROPN
ejpam-5930	81	28	≤	≤	PUNCT
ejpam-5930	81	29	y	y	PROPN
ejpam-5930	81	30	⇔	⇔	PROPN
ejpam-5930	81	31	x	x	PROPN
ejpam-5930	81	32	·	·	PUNCT
ejpam-5930	81	33	y	y	SYM
ejpam-5930	81	34	=	=	SYM
ejpam-5930	81	35	1	1	NUM
ejpam-5930	81	36	)	)	PUNCT
ejpam-5930	81	37	,	,	PUNCT
ejpam-5930	81	38	which	which	PRON
ejpam-5930	81	39	is	be	AUX
ejpam-5930	81	40	a	a	DET
ejpam-5930	81	41	partial	partial	ADJ
ejpam-5930	81	42	order	order	NOUN
ejpam-5930	81	43	on	on	ADP
ejpam-5930	81	44	a	a	DET
ejpam-5930	81	45	with	with	ADP
ejpam-5930	81	46	1	1	NUM
ejpam-5930	81	47	as	as	ADP
ejpam-5930	81	48	the	the	DET
ejpam-5930	81	49	largest	large	ADJ
ejpam-5930	81	50	element	element	NOUN
ejpam-5930	81	51	.	.	PUNCT
ejpam-5930	82	1	definition	definition	NOUN
ejpam-5930	82	2	4	4	NUM
ejpam-5930	82	3	.	.	PUNCT
ejpam-5930	83	1	[	[	X
ejpam-5930	83	2	14	14	NUM
ejpam-5930	83	3	]	]	PUNCT
ejpam-5930	83	4	a	a	DET
ejpam-5930	83	5	nonempty	nonempty	NOUN
ejpam-5930	83	6	subset	subset	VERB
ejpam-5930	83	7	d	d	NOUN
ejpam-5930	83	8	of	of	ADP
ejpam-5930	83	9	a	a	DET
ejpam-5930	83	10	hilbert	hilbert	NOUN
ejpam-5930	83	11	algebra	algebra	NOUN
ejpam-5930	83	12	a	a	DET
ejpam-5930	83	13	=	=	X
ejpam-5930	83	14	(	(	PUNCT
ejpam-5930	83	15	a	a	PRON
ejpam-5930	83	16	,	,	PUNCT
ejpam-5930	83	17	·	·	PUNCT
ejpam-5930	83	18	,	,	PUNCT
ejpam-5930	83	19	1	1	NUM
ejpam-5930	83	20	)	)	PUNCT
ejpam-5930	83	21	is	be	AUX
ejpam-5930	83	22	called	call	VERB
ejpam-5930	83	23	a	a	DET
ejpam-5930	83	24	subalgebra	subalgebra	NOUN
ejpam-5930	83	25	of	of	ADP
ejpam-5930	83	26	a	a	DET
ejpam-5930	83	27	if	if	NOUN
ejpam-5930	83	28	x	x	X
ejpam-5930	83	29	·	·	PUNCT
ejpam-5930	83	30	y	y	X
ejpam-5930	83	31	∈	∈	PROPN
ejpam-5930	83	32	d	d	NOUN
ejpam-5930	83	33	for	for	ADP
ejpam-5930	83	34	all	all	DET
ejpam-5930	83	35	x	x	NOUN
ejpam-5930	83	36	,	,	PUNCT
ejpam-5930	83	37	y	y	PROPN
ejpam-5930	83	38	∈	∈	PROPN
ejpam-5930	83	39	d.	d.	PROPN
ejpam-5930	83	40	definition	definition	NOUN
ejpam-5930	83	41	5	5	NUM
ejpam-5930	83	42	.	.	PUNCT
ejpam-5930	84	1	[	[	X
ejpam-5930	84	2	15	15	NUM
ejpam-5930	84	3	]	]	X
ejpam-5930	84	4	a	a	DET
ejpam-5930	84	5	nonempty	nonempty	NOUN
ejpam-5930	84	6	subset	subset	VERB
ejpam-5930	84	7	d	d	NOUN
ejpam-5930	84	8	of	of	ADP
ejpam-5930	84	9	a	a	DET
ejpam-5930	84	10	hilbert	hilbert	NOUN
ejpam-5930	84	11	algebra	algebra	NOUN
ejpam-5930	84	12	a	a	DET
ejpam-5930	84	13	=	=	X
ejpam-5930	84	14	(	(	PUNCT
ejpam-5930	84	15	a	a	PRON
ejpam-5930	84	16	,	,	PUNCT
ejpam-5930	84	17	·	·	PUNCT
ejpam-5930	84	18	,	,	PUNCT
ejpam-5930	84	19	1	1	NUM
ejpam-5930	84	20	)	)	PUNCT
ejpam-5930	84	21	is	be	AUX
ejpam-5930	84	22	called	call	VERB
ejpam-5930	84	23	an	an	DET
ejpam-5930	84	24	ideal	ideal	NOUN
ejpam-5930	84	25	of	of	ADP
ejpam-5930	84	26	a	a	PRON
ejpam-5930	84	27	(	(	PUNCT
ejpam-5930	84	28	determined	determine	VERB
ejpam-5930	84	29	by	by	ADP
ejpam-5930	84	30	d	d	PROPN
ejpam-5930	84	31	▷	▷	PROPN
ejpam-5930	84	32	a	a	X
ejpam-5930	84	33	)	)	PUNCT
ejpam-5930	84	34	if	if	SCONJ
ejpam-5930	84	35	the	the	DET
ejpam-5930	84	36	following	follow	VERB
ejpam-5930	84	37	conditions	condition	NOUN
ejpam-5930	84	38	hold	hold	VERB
ejpam-5930	84	39	:	:	PUNCT
ejpam-5930	84	40	(	(	PUNCT
ejpam-5930	84	41	1	1	X
ejpam-5930	84	42	)	)	SYM
ejpam-5930	84	43	1	1	NUM
ejpam-5930	84	44	∈	∈	NOUN
ejpam-5930	84	45	d	d	NOUN
ejpam-5930	84	46	,	,	PUNCT
ejpam-5930	84	47	(	(	PUNCT
ejpam-5930	84	48	2	2	NUM
ejpam-5930	84	49	)	)	PUNCT
ejpam-5930	84	50	(	(	PUNCT
ejpam-5930	84	51	∀x	∀x	X
ejpam-5930	84	52	,	,	PUNCT
ejpam-5930	85	1	y	y	PROPN
ejpam-5930	85	2	∈	∈	PROPN
ejpam-5930	85	3	a)(y	a)(y	PUNCT
ejpam-5930	85	4	∈	∈	PROPN
ejpam-5930	85	5	d	d	NOUN
ejpam-5930	85	6	⇒	⇒	NOUN
ejpam-5930	85	7	x	x	X
ejpam-5930	85	8	·	·	PUNCT
ejpam-5930	85	9	y	y	X
ejpam-5930	85	10	∈	∈	PROPN
ejpam-5930	85	11	d	d	PROPN
ejpam-5930	85	12	)	)	PUNCT
ejpam-5930	85	13	,	,	PUNCT
ejpam-5930	85	14	(	(	PUNCT
ejpam-5930	85	15	3	3	X
ejpam-5930	85	16	)	)	PUNCT
ejpam-5930	85	17	(	(	PUNCT
ejpam-5930	85	18	∀x	∀x	X
ejpam-5930	85	19	,	,	PUNCT
ejpam-5930	85	20	y1	y1	X
ejpam-5930	85	21	,	,	PUNCT
ejpam-5930	85	22	y2	y2	PROPN
ejpam-5930	85	23	∈	∈	PROPN
ejpam-5930	85	24	a)(y1	a)(y1	PROPN
ejpam-5930	85	25	,	,	PUNCT
ejpam-5930	85	26	y2	y2	NOUN
ejpam-5930	85	27	∈	∈	PROPN
ejpam-5930	86	1	d	d	X
ejpam-5930	86	2	⇒	⇒	NOUN
ejpam-5930	86	3	(	(	PUNCT
ejpam-5930	86	4	y1	y1	INTJ
ejpam-5930	86	5	·	·	PUNCT
ejpam-5930	86	6	(	(	PUNCT
ejpam-5930	86	7	y2	y2	INTJ
ejpam-5930	86	8	·	·	PUNCT
ejpam-5930	86	9	x	x	X
ejpam-5930	86	10	)	)	PUNCT
ejpam-5930	86	11	)	)	PUNCT
ejpam-5930	86	12	·	·	PUNCT
ejpam-5930	87	1	x	x	PUNCT
ejpam-5930	87	2	∈	∈	PROPN
ejpam-5930	87	3	d	d	NOUN
ejpam-5930	87	4	)	)	PUNCT
ejpam-5930	87	5	.	.	PUNCT
ejpam-5930	88	1	3	3	X
ejpam-5930	88	2	.	.	X
ejpam-5930	88	3	rough	rough	ADJ
ejpam-5930	88	4	approximations	approximation	NOUN
ejpam-5930	88	5	in	in	ADP
ejpam-5930	88	6	hilbert	hilbert	PROPN
ejpam-5930	88	7	algebras	algebras	PROPN
ejpam-5930	88	8	let	let	VERB
ejpam-5930	88	9	v	v	PART
ejpam-5930	88	10	be	be	AUX
ejpam-5930	88	11	a	a	DET
ejpam-5930	88	12	set	set	NOUN
ejpam-5930	88	13	and	and	CCONJ
ejpam-5930	88	14	e	e	NOUN
ejpam-5930	88	15	an	an	DET
ejpam-5930	88	16	equivalence	equivalence	NOUN
ejpam-5930	88	17	relation	relation	NOUN
ejpam-5930	88	18	on	on	ADP
ejpam-5930	88	19	v	v	NUM
ejpam-5930	88	20	.	.	PUNCT
ejpam-5930	89	1	for	for	ADP
ejpam-5930	89	2	all	all	DET
ejpam-5930	89	3	a	a	DET
ejpam-5930	89	4	∈	∈	PROPN
ejpam-5930	89	5	v	v	NOUN
ejpam-5930	89	6	,	,	PUNCT
ejpam-5930	89	7	let	let	VERB
ejpam-5930	89	8	[	[	X
ejpam-5930	89	9	a]e	a]e	X
ejpam-5930	89	10	denote	denote	VERB
ejpam-5930	89	11	the	the	DET
ejpam-5930	89	12	equivalence	equivalence	NOUN
ejpam-5930	89	13	class	class	NOUN
ejpam-5930	89	14	of	of	ADP
ejpam-5930	89	15	a	a	PRON
ejpam-5930	89	16	with	with	ADP
ejpam-5930	89	17	respect	respect	NOUN
ejpam-5930	89	18	to	to	ADP
ejpam-5930	89	19	e.	e.	PROPN
ejpam-5930	89	20	define	define	VERB
ejpam-5930	89	21	the	the	DET
ejpam-5930	89	22	functions	function	NOUN
ejpam-5930	89	23	e−	e−	PROPN
ejpam-5930	89	24	,	,	PUNCT
ejpam-5930	90	1	e	e	X
ejpam-5930	90	2	−	−	PROPN
ejpam-5930	90	3	:	:	PUNCT
ejpam-5930	90	4	p(v	p(v	NOUN
ejpam-5930	90	5	)	)	PUNCT
ejpam-5930	90	6	→	→	SYM
ejpam-5930	90	7	p(v	p(v	NOUN
ejpam-5930	90	8	)	)	PUNCT
ejpam-5930	90	9	as	as	SCONJ
ejpam-5930	90	10	follows	follow	VERB
ejpam-5930	90	11	:	:	PUNCT
ejpam-5930	90	12	∀	∀	X
ejpam-5930	90	13	s	s	PART
ejpam-5930	90	14	∈	∈	PROPN
ejpam-5930	90	15	p(v	p(v	NOUN
ejpam-5930	90	16	)	)	PUNCT
ejpam-5930	90	17	,	,	PUNCT
ejpam-5930	90	18	e−(s	e−(s	PROPN
ejpam-5930	90	19	)	)	PUNCT
ejpam-5930	90	20	=	=	PRON
ejpam-5930	90	21	{	{	PUNCT
ejpam-5930	90	22	a	a	DET
ejpam-5930	90	23	∈	∈	NOUN
ejpam-5930	90	24	v	v	NOUN
ejpam-5930	90	25	:	:	PUNCT
ejpam-5930	91	1	[	[	X
ejpam-5930	91	2	a]e	a]e	X
ejpam-5930	91	3	⊆	⊆	NUM
ejpam-5930	91	4	s	s	NOUN
ejpam-5930	91	5	}	}	PUNCT
ejpam-5930	91	6	and	and	CCONJ
ejpam-5930	91	7	e−(s	e−(s	ADJ
ejpam-5930	91	8	)	)	PUNCT
ejpam-5930	91	9	=	=	PRON
ejpam-5930	91	10	{	{	PUNCT
ejpam-5930	91	11	x	x	PUNCT
ejpam-5930	91	12	∈	∈	NOUN
ejpam-5930	91	13	v	v	NOUN
ejpam-5930	91	14	:	:	PUNCT
ejpam-5930	91	15	[	[	X
ejpam-5930	91	16	a]e	a]e	X
ejpam-5930	91	17	∩	∩	NOUN
ejpam-5930	91	18	s	s	PART
ejpam-5930	91	19	̸=	̸=	PROPN
ejpam-5930	91	20	∅	∅	NOUN
ejpam-5930	91	21	}	}	PUNCT
ejpam-5930	91	22	.	.	PUNCT
ejpam-5930	92	1	the	the	DET
ejpam-5930	92	2	pair	pair	NOUN
ejpam-5930	92	3	(	(	PUNCT
ejpam-5930	92	4	v	v	NOUN
ejpam-5930	92	5	,	,	PUNCT
ejpam-5930	92	6	e	e	NOUN
ejpam-5930	92	7	)	)	PUNCT
ejpam-5930	92	8	is	be	AUX
ejpam-5930	92	9	called	call	VERB
ejpam-5930	92	10	an	an	DET
ejpam-5930	92	11	approximation	approximation	NOUN
ejpam-5930	92	12	space	space	NOUN
ejpam-5930	92	13	.	.	PUNCT
ejpam-5930	93	1	let	let	VERB
ejpam-5930	93	2	s	s	PRON
ejpam-5930	93	3	be	be	AUX
ejpam-5930	93	4	a	a	DET
ejpam-5930	93	5	subset	subset	NOUN
ejpam-5930	93	6	of	of	ADP
ejpam-5930	93	7	v	v	NOUN
ejpam-5930	93	8	.	.	PUNCT
ejpam-5930	94	1	then	then	ADV
ejpam-5930	94	2	s	s	VERB
ejpam-5930	94	3	is	be	AUX
ejpam-5930	94	4	said	say	VERB
ejpam-5930	94	5	to	to	PART
ejpam-5930	94	6	be	be	AUX
ejpam-5930	94	7	definable	definable	ADJ
ejpam-5930	94	8	if	if	SCONJ
ejpam-5930	94	9	e−(s	e−(s	ADJ
ejpam-5930	94	10	)	)	PUNCT
ejpam-5930	94	11	=	=	SYM
ejpam-5930	94	12	e−(s	e−(	VERB
ejpam-5930	94	13	)	)	PUNCT
ejpam-5930	94	14	and	and	CCONJ
ejpam-5930	94	15	rough	rough	ADJ
ejpam-5930	94	16	otherwise	otherwise	ADV
ejpam-5930	94	17	.	.	PUNCT
ejpam-5930	95	1	e−(s	e−(	VERB
ejpam-5930	95	2	)	)	PUNCT
ejpam-5930	95	3	is	be	AUX
ejpam-5930	95	4	called	call	VERB
ejpam-5930	95	5	the	the	DET
ejpam-5930	95	6	lower	low	ADJ
ejpam-5930	95	7	approximation	approximation	NOUN
ejpam-5930	95	8	of	of	ADP
ejpam-5930	95	9	s	s	PRON
ejpam-5930	95	10	while	while	SCONJ
ejpam-5930	95	11	e−(s	e−(s	PROPN
ejpam-5930	95	12	)	)	PUNCT
ejpam-5930	95	13	is	be	AUX
ejpam-5930	95	14	called	call	VERB
ejpam-5930	95	15	the	the	DET
ejpam-5930	95	16	upper	upper	ADJ
ejpam-5930	95	17	approximation	approximation	NOUN
ejpam-5930	95	18	.	.	PUNCT
ejpam-5930	96	1	throughout	throughout	ADP
ejpam-5930	96	2	this	this	DET
ejpam-5930	96	3	paper	paper	NOUN
ejpam-5930	96	4	,	,	PUNCT
ejpam-5930	96	5	a	a	DET
ejpam-5930	96	6	=	=	X
ejpam-5930	96	7	(	(	PUNCT
ejpam-5930	96	8	a	a	PRON
ejpam-5930	96	9	,	,	PUNCT
ejpam-5930	96	10	·	·	PUNCT
ejpam-5930	96	11	,	,	PUNCT
ejpam-5930	96	12	1	1	NUM
ejpam-5930	96	13	)	)	PUNCT
ejpam-5930	96	14	will	will	AUX
ejpam-5930	96	15	represent	represent	VERB
ejpam-5930	96	16	a	a	DET
ejpam-5930	96	17	hilbert	hilbert	NOUN
ejpam-5930	96	18	algebra	algebra	NOUN
ejpam-5930	96	19	.	.	PUNCT
ejpam-5930	97	1	let	let	VERB
ejpam-5930	97	2	i	i	PRON
ejpam-5930	97	3	be	be	AUX
ejpam-5930	97	4	an	an	DET
ejpam-5930	97	5	ideal	ideal	NOUN
ejpam-5930	97	6	of	of	ADP
ejpam-5930	97	7	a.	a.	NOUN
ejpam-5930	97	8	define	define	VERB
ejpam-5930	97	9	a	a	DET
ejpam-5930	97	10	relation	relation	NOUN
ejpam-5930	97	11	θ	θ	PROPN
ejpam-5930	97	12	on	on	ADP
ejpam-5930	97	13	a	a	DET
ejpam-5930	97	14	by	by	X
ejpam-5930	97	15	(	(	PUNCT
ejpam-5930	97	16	a	a	PRON
ejpam-5930	97	17	,	,	PUNCT
ejpam-5930	97	18	b	b	NOUN
ejpam-5930	97	19	)	)	PUNCT
ejpam-5930	97	20	∈	∈	PROPN
ejpam-5930	97	21	θ	θ	PROPN
ejpam-5930	98	1	if	if	SCONJ
ejpam-5930	98	2	and	and	CCONJ
ejpam-5930	98	3	only	only	ADV
ejpam-5930	98	4	if	if	SCONJ
ejpam-5930	98	5	a	a	PRON
ejpam-5930	98	6	·	·	SYM
ejpam-5930	98	7	b	b	X
ejpam-5930	98	8	∈	∈	ADV
ejpam-5930	98	9	i	i	PRON
ejpam-5930	98	10	and	and	CCONJ
ejpam-5930	98	11	b	b	X
ejpam-5930	98	12	·	·	PUNCT
ejpam-5930	98	13	a	a	DET
ejpam-5930	98	14	∈	∈	PROPN
ejpam-5930	98	15	i.	i.	NOUN
ejpam-5930	98	16	then	then	ADV
ejpam-5930	98	17	θ	θ	PROPN
ejpam-5930	98	18	is	be	AUX
ejpam-5930	98	19	an	an	DET
ejpam-5930	98	20	equivalence	equivalence	NOUN
ejpam-5930	98	21	relation	relation	NOUN
ejpam-5930	98	22	on	on	ADP
ejpam-5930	98	23	a	a	DET
ejpam-5930	98	24	related	relate	VERB
ejpam-5930	98	25	to	to	ADP
ejpam-5930	98	26	an	an	DET
ejpam-5930	98	27	ideal	ideal	NOUN
ejpam-5930	98	28	i	i	PRON
ejpam-5930	98	29	of	of	ADP
ejpam-5930	98	30	a.	a.	NOUN
ejpam-5930	98	31	moreover	moreover	ADV
ejpam-5930	98	32	,	,	PUNCT
ejpam-5930	98	33	(	(	PUNCT
ejpam-5930	98	34	a	a	PRON
ejpam-5930	98	35	,	,	PUNCT
ejpam-5930	98	36	b	b	NOUN
ejpam-5930	98	37	)	)	PUNCT
ejpam-5930	98	38	∈	∈	PROPN
ejpam-5930	98	39	θ	θ	PROPN
ejpam-5930	98	40	and	and	CCONJ
ejpam-5930	98	41	(	(	PUNCT
ejpam-5930	98	42	u	u	NOUN
ejpam-5930	98	43	,	,	PUNCT
ejpam-5930	98	44	v	v	NOUN
ejpam-5930	98	45	)	)	PUNCT
ejpam-5930	98	46	∈	∈	NOUN
ejpam-5930	98	47	θ	θ	X
ejpam-5930	98	48	imply	imply	ADV
ejpam-5930	98	49	(	(	PUNCT
ejpam-5930	98	50	a	a	DET
ejpam-5930	98	51	·	·	PUNCT
ejpam-5930	98	52	u	u	NOUN
ejpam-5930	98	53	,	,	PUNCT
ejpam-5930	98	54	b	b	PROPN
ejpam-5930	98	55	·	·	SYM
ejpam-5930	98	56	v	v	X
ejpam-5930	98	57	)	)	PUNCT
ejpam-5930	98	58	∈	∈	PROPN
ejpam-5930	98	59	θ	θ	PROPN
ejpam-5930	98	60	.	.	PUNCT
ejpam-5930	99	1	hence	hence	ADV
ejpam-5930	99	2	,	,	PUNCT
ejpam-5930	99	3	θ	θ	PROPN
ejpam-5930	99	4	is	be	AUX
ejpam-5930	99	5	a	a	DET
ejpam-5930	99	6	congruence	congruence	NOUN
ejpam-5930	99	7	relation	relation	NOUN
ejpam-5930	99	8	on	on	ADP
ejpam-5930	99	9	a.	a.	NOUN
ejpam-5930	99	10	a.	a.	NOUN
ejpam-5930	99	11	iampan	iampan	PROPN
ejpam-5930	99	12	et	et	PROPN
ejpam-5930	99	13	al	al	PROPN
ejpam-5930	99	14	.	.	PUNCT
ejpam-5930	99	15	/	/	SYM
ejpam-5930	99	16	eur	eur	PROPN
ejpam-5930	99	17	.	.	PUNCT
ejpam-5930	100	1	j.	j.	PROPN
ejpam-5930	100	2	pure	pure	PROPN
ejpam-5930	100	3	appl	appl	PROPN
ejpam-5930	100	4	.	.	PROPN
ejpam-5930	100	5	math	math	PROPN
ejpam-5930	100	6	,	,	PUNCT
ejpam-5930	100	7	18	18	NUM
ejpam-5930	100	8	(	(	PUNCT
ejpam-5930	100	9	2	2	NUM
ejpam-5930	100	10	)	)	PUNCT
ejpam-5930	100	11	(	(	PUNCT
ejpam-5930	100	12	2025	2025	NUM
ejpam-5930	100	13	)	)	PUNCT
ejpam-5930	100	14	,	,	PUNCT
ejpam-5930	100	15	5930	5930	NUM
ejpam-5930	100	16	5	5	NUM
ejpam-5930	100	17	of	of	ADP
ejpam-5930	100	18	11	11	NUM
ejpam-5930	100	19	let	let	VERB
ejpam-5930	100	20	ia	ia	PROPN
ejpam-5930	100	21	denote	denote	VERB
ejpam-5930	100	22	the	the	DET
ejpam-5930	100	23	equivalence	equivalence	NOUN
ejpam-5930	100	24	class	class	NOUN
ejpam-5930	100	25	of	of	ADP
ejpam-5930	100	26	a	a	PRON
ejpam-5930	100	27	with	with	ADP
ejpam-5930	100	28	respect	respect	NOUN
ejpam-5930	100	29	to	to	ADP
ejpam-5930	100	30	the	the	DET
ejpam-5930	100	31	equivalence	equivalence	NOUN
ejpam-5930	100	32	relation	relation	NOUN
ejpam-5930	100	33	θ	θ	PROPN
ejpam-5930	100	34	related	relate	VERB
ejpam-5930	100	35	to	to	ADP
ejpam-5930	100	36	an	an	DET
ejpam-5930	100	37	ideal	ideal	ADJ
ejpam-5930	100	38	i	i	PRON
ejpam-5930	100	39	of	of	ADP
ejpam-5930	100	40	a	a	PRON
ejpam-5930	100	41	,	,	PUNCT
ejpam-5930	100	42	and	and	CCONJ
ejpam-5930	100	43	a	a	X
ejpam-5930	100	44	/	/	SYM
ejpam-5930	100	45	i	i	PRON
ejpam-5930	100	46	denote	denote	VERB
ejpam-5930	100	47	the	the	DET
ejpam-5930	100	48	collection	collection	NOUN
ejpam-5930	100	49	of	of	ADP
ejpam-5930	100	50	all	all	DET
ejpam-5930	100	51	equivalence	equivalence	NOUN
ejpam-5930	100	52	classes	class	NOUN
ejpam-5930	100	53	,	,	PUNCT
ejpam-5930	100	54	that	that	ADV
ejpam-5930	100	55	is	is	ADV
ejpam-5930	100	56	,	,	PUNCT
ejpam-5930	100	57	a	a	X
ejpam-5930	100	58	/	/	SYM
ejpam-5930	100	59	i	i	NOUN
ejpam-5930	100	60	=	=	SYM
ejpam-5930	100	61	{	{	PUNCT
ejpam-5930	100	62	ia	ia	PROPN
ejpam-5930	100	63	:	:	PUNCT
ejpam-5930	100	64	a	a	DET
ejpam-5930	100	65	∈	∈	PROPN
ejpam-5930	100	66	a	a	PRON
ejpam-5930	100	67	}	}	PUNCT
ejpam-5930	100	68	.	.	PUNCT
ejpam-5930	101	1	then	then	ADV
ejpam-5930	101	2	i1	i1	PROPN
ejpam-5930	101	3	=	=	PROPN
ejpam-5930	101	4	i.	i.	PROPN
ejpam-5930	101	5	if	if	SCONJ
ejpam-5930	101	6	ia	ia	PROPN
ejpam-5930	101	7	·	·	PUNCT
ejpam-5930	101	8	ib	ib	PROPN
ejpam-5930	101	9	is	be	AUX
ejpam-5930	101	10	defined	define	VERB
ejpam-5930	101	11	as	as	ADP
ejpam-5930	101	12	the	the	DET
ejpam-5930	101	13	class	class	NOUN
ejpam-5930	101	14	containing	contain	VERB
ejpam-5930	101	15	a	a	DET
ejpam-5930	101	16	·	·	PUNCT
ejpam-5930	101	17	b	b	X
ejpam-5930	101	18	,	,	PUNCT
ejpam-5930	101	19	that	that	ADV
ejpam-5930	101	20	is	is	ADV
ejpam-5930	101	21	,	,	PUNCT
ejpam-5930	101	22	ia	ia	PROPN
ejpam-5930	101	23	·	·	PUNCT
ejpam-5930	101	24	ib	ib	NOUN
ejpam-5930	101	25	=	=	PUNCT
ejpam-5930	101	26	ia·b	ia·b	PROPN
ejpam-5930	101	27	,	,	PUNCT
ejpam-5930	101	28	then	then	ADV
ejpam-5930	101	29	(	(	PUNCT
ejpam-5930	101	30	a	a	X
ejpam-5930	101	31	/	/	SYM
ejpam-5930	101	32	i	i	PROPN
ejpam-5930	101	33	,	,	PUNCT
ejpam-5930	101	34	·	·	PROPN
ejpam-5930	101	35	,	,	PUNCT
ejpam-5930	101	36	i1	i1	PROPN
ejpam-5930	101	37	)	)	PUNCT
ejpam-5930	101	38	is	be	AUX
ejpam-5930	101	39	a	a	DET
ejpam-5930	101	40	hilbert	hilbert	NOUN
ejpam-5930	101	41	algebra	algebra	NOUN
ejpam-5930	101	42	.	.	PUNCT
ejpam-5930	102	1	let	let	VERB
ejpam-5930	102	2	θ	θ	NOUN
ejpam-5930	102	3	be	be	AUX
ejpam-5930	102	4	an	an	DET
ejpam-5930	102	5	equivalence	equivalence	NOUN
ejpam-5930	102	6	relation	relation	NOUN
ejpam-5930	102	7	on	on	ADP
ejpam-5930	102	8	a	a	DET
ejpam-5930	102	9	related	relate	VERB
ejpam-5930	102	10	to	to	ADP
ejpam-5930	102	11	an	an	DET
ejpam-5930	102	12	ideal	ideal	NOUN
ejpam-5930	102	13	i	i	PRON
ejpam-5930	102	14	of	of	ADP
ejpam-5930	102	15	a.	a.	NOUN
ejpam-5930	102	16	for	for	ADP
ejpam-5930	102	17	any	any	DET
ejpam-5930	102	18	nonempty	nonempty	NOUN
ejpam-5930	102	19	subset	subset	NOUN
ejpam-5930	102	20	s	s	PROPN
ejpam-5930	102	21	ofa	ofa	PROPN
ejpam-5930	102	22	,	,	PUNCT
ejpam-5930	102	23	the	the	DET
ejpam-5930	102	24	lower	low	ADJ
ejpam-5930	102	25	and	and	CCONJ
ejpam-5930	102	26	upper	upper	ADJ
ejpam-5930	102	27	approximations	approximation	NOUN
ejpam-5930	102	28	of	of	ADP
ejpam-5930	102	29	s	s	NOUN
ejpam-5930	102	30	are	be	AUX
ejpam-5930	102	31	denoted	denote	VERB
ejpam-5930	102	32	by	by	ADP
ejpam-5930	102	33	θ(i	θ(i	PROPN
ejpam-5930	102	34	,	,	PUNCT
ejpam-5930	102	35	s	s	PART
ejpam-5930	102	36	)	)	PUNCT
ejpam-5930	102	37	and	and	CCONJ
ejpam-5930	102	38	θ(i	θ(i	PROPN
ejpam-5930	102	39	,	,	PUNCT
ejpam-5930	102	40	s	s	PART
ejpam-5930	102	41	)	)	PUNCT
ejpam-5930	102	42	,	,	PUNCT
ejpam-5930	102	43	respectively	respectively	ADV
ejpam-5930	102	44	,	,	PUNCT
ejpam-5930	102	45	that	that	ADV
ejpam-5930	102	46	is	is	ADV
ejpam-5930	102	47	,	,	PUNCT
ejpam-5930	102	48	θ(i	θ(i	PROPN
ejpam-5930	102	49	,	,	PUNCT
ejpam-5930	102	50	s	s	PART
ejpam-5930	102	51	)	)	PUNCT
ejpam-5930	102	52	=	=	SYM
ejpam-5930	102	53	{	{	PUNCT
ejpam-5930	102	54	a	a	DET
ejpam-5930	102	55	∈	∈	PROPN
ejpam-5930	102	56	a	a	DET
ejpam-5930	102	57	:	:	PUNCT
ejpam-5930	102	58	ia	ia	PROPN
ejpam-5930	102	59	⊆	⊆	NUM
ejpam-5930	102	60	s	s	NOUN
ejpam-5930	102	61	}	}	PUNCT
ejpam-5930	102	62	and	and	CCONJ
ejpam-5930	102	63	θ(i	θ(i	PROPN
ejpam-5930	102	64	,	,	PUNCT
ejpam-5930	102	65	s	s	NOUN
ejpam-5930	102	66	)	)	PUNCT
ejpam-5930	102	67	=	=	SYM
ejpam-5930	102	68	{	{	PUNCT
ejpam-5930	102	69	a	a	DET
ejpam-5930	102	70	∈	∈	PROPN
ejpam-5930	102	71	a	a	DET
ejpam-5930	102	72	:	:	PUNCT
ejpam-5930	102	73	ia	ia	PROPN
ejpam-5930	102	74	∩	∩	PROPN
ejpam-5930	102	75	s	s	PART
ejpam-5930	102	76	̸=	̸=	PROPN
ejpam-5930	102	77	∅	∅	NOUN
ejpam-5930	102	78	}	}	PUNCT
ejpam-5930	102	79	.	.	PUNCT
ejpam-5930	103	1	if	if	SCONJ
ejpam-5930	103	2	i	i	PRON
ejpam-5930	103	3	=	=	SYM
ejpam-5930	103	4	s	s	PROPN
ejpam-5930	103	5	,	,	PUNCT
ejpam-5930	103	6	then	then	ADV
ejpam-5930	103	7	θ(i	θ(i	PROPN
ejpam-5930	103	8	,	,	PUNCT
ejpam-5930	103	9	s	s	PART
ejpam-5930	103	10	)	)	PUNCT
ejpam-5930	103	11	and	and	CCONJ
ejpam-5930	103	12	θ(i	θ(i	PROPN
ejpam-5930	103	13	,	,	PUNCT
ejpam-5930	103	14	s	s	PART
ejpam-5930	103	15	)	)	PUNCT
ejpam-5930	103	16	are	be	AUX
ejpam-5930	103	17	denoted	denote	VERB
ejpam-5930	103	18	by	by	ADP
ejpam-5930	103	19	θ(i	θ(i	PROPN
ejpam-5930	103	20	)	)	PUNCT
ejpam-5930	103	21	and	and	CCONJ
ejpam-5930	103	22	θ(i	θ(i	PROPN
ejpam-5930	103	23	)	)	PUNCT
ejpam-5930	103	24	,	,	PUNCT
ejpam-5930	103	25	respectively	respectively	ADV
ejpam-5930	103	26	.	.	PUNCT
ejpam-5930	104	1	definition	definition	NOUN
ejpam-5930	104	2	6	6	NUM
ejpam-5930	104	3	.	.	PUNCT
ejpam-5930	105	1	[	[	X
ejpam-5930	105	2	2	2	NUM
ejpam-5930	105	3	]	]	PUNCT
ejpam-5930	105	4	given	give	VERB
ejpam-5930	105	5	an	an	DET
ejpam-5930	105	6	approximation	approximation	NOUN
ejpam-5930	105	7	space	space	NOUN
ejpam-5930	105	8	(	(	PUNCT
ejpam-5930	105	9	u	u	NOUN
ejpam-5930	105	10	,	,	PUNCT
ejpam-5930	105	11	θ	θ	PROPN
ejpam-5930	105	12	)	)	PUNCT
ejpam-5930	105	13	,	,	PUNCT
ejpam-5930	105	14	a	a	DET
ejpam-5930	105	15	pair	pair	NOUN
ejpam-5930	105	16	(	(	PUNCT
ejpam-5930	105	17	a	a	PRON
ejpam-5930	105	18	,	,	PUNCT
ejpam-5930	105	19	b	b	NOUN
ejpam-5930	105	20	)	)	PUNCT
ejpam-5930	105	21	∈	∈	PROPN
ejpam-5930	105	22	p(u)×p(u	p(u)×p(u	NOUN
ejpam-5930	105	23	)	)	PUNCT
ejpam-5930	105	24	is	be	AUX
ejpam-5930	105	25	called	call	VERB
ejpam-5930	105	26	a	a	DET
ejpam-5930	105	27	rough	rough	ADJ
ejpam-5930	105	28	set	set	NOUN
ejpam-5930	105	29	in	in	ADP
ejpam-5930	105	30	(	(	PUNCT
ejpam-5930	105	31	u	u	NOUN
ejpam-5930	105	32	,	,	PUNCT
ejpam-5930	105	33	θ	θ	NOUN
ejpam-5930	105	34	)	)	PUNCT
ejpam-5930	105	35	if	if	SCONJ
ejpam-5930	105	36	(	(	PUNCT
ejpam-5930	105	37	a	a	DET
ejpam-5930	105	38	,	,	PUNCT
ejpam-5930	105	39	b	b	NOUN
ejpam-5930	105	40	)	)	PUNCT
ejpam-5930	105	41	=	=	SYM
ejpam-5930	105	42	apr(x	apr(x	PROPN
ejpam-5930	105	43	)	)	PUNCT
ejpam-5930	105	44	for	for	ADP
ejpam-5930	105	45	some	some	DET
ejpam-5930	105	46	x	x	SYM
ejpam-5930	105	47	∈	∈	PROPN
ejpam-5930	105	48	p(u	p(u	NOUN
ejpam-5930	105	49	)	)	PUNCT
ejpam-5930	105	50	.	.	PUNCT
ejpam-5930	106	1	definition	definition	NOUN
ejpam-5930	106	2	7	7	NUM
ejpam-5930	106	3	.	.	PUNCT
ejpam-5930	107	1	[	[	X
ejpam-5930	107	2	2	2	NUM
ejpam-5930	107	3	]	]	X
ejpam-5930	107	4	let	let	VERB
ejpam-5930	107	5	(	(	PUNCT
ejpam-5930	107	6	u	u	NOUN
ejpam-5930	107	7	,	,	PUNCT
ejpam-5930	107	8	θ	θ	NOUN
ejpam-5930	107	9	)	)	PUNCT
ejpam-5930	107	10	be	be	VERB
ejpam-5930	107	11	an	an	DET
ejpam-5930	107	12	approximation	approximation	NOUN
ejpam-5930	107	13	space	space	NOUN
ejpam-5930	107	14	and	and	CCONJ
ejpam-5930	107	15	x	x	ADP
ejpam-5930	107	16	a	a	DET
ejpam-5930	107	17	nonempty	nonempty	NOUN
ejpam-5930	107	18	subset	subset	NOUN
ejpam-5930	107	19	of	of	ADP
ejpam-5930	107	20	u	u	PROPN
ejpam-5930	107	21	.	.	PUNCT
ejpam-5930	108	1	(	(	PUNCT
ejpam-5930	108	2	1	1	X
ejpam-5930	108	3	)	)	PUNCT
ejpam-5930	108	4	if	if	SCONJ
ejpam-5930	108	5	apr(x	apr(x	PROPN
ejpam-5930	108	6	)	)	PUNCT
ejpam-5930	108	7	=	=	SYM
ejpam-5930	108	8	apr(x	apr(x	PROPN
ejpam-5930	108	9	)	)	PUNCT
ejpam-5930	108	10	,	,	PUNCT
ejpam-5930	108	11	then	then	ADV
ejpam-5930	108	12	x	x	PUNCT
ejpam-5930	108	13	is	be	AUX
ejpam-5930	108	14	called	call	VERB
ejpam-5930	108	15	definable	definable	ADJ
ejpam-5930	108	16	.	.	PUNCT
ejpam-5930	109	1	(	(	PUNCT
ejpam-5930	109	2	2	2	X
ejpam-5930	109	3	)	)	PUNCT
ejpam-5930	109	4	if	if	SCONJ
ejpam-5930	109	5	apr(x	apr(x	PROPN
ejpam-5930	109	6	)	)	PUNCT
ejpam-5930	109	7	=	=	SYM
ejpam-5930	109	8	∅	∅	NOUN
ejpam-5930	109	9	,	,	PUNCT
ejpam-5930	109	10	then	then	ADV
ejpam-5930	109	11	x	x	PUNCT
ejpam-5930	109	12	is	be	AUX
ejpam-5930	109	13	called	call	VERB
ejpam-5930	109	14	empty	empty	ADJ
ejpam-5930	109	15	interior	interior	NOUN
ejpam-5930	109	16	.	.	PUNCT
ejpam-5930	110	1	(	(	PUNCT
ejpam-5930	110	2	3	3	X
ejpam-5930	110	3	)	)	PUNCT
ejpam-5930	110	4	if	if	SCONJ
ejpam-5930	110	5	apr(x	apr(x	PROPN
ejpam-5930	110	6	)	)	PUNCT
ejpam-5930	110	7	=	=	SYM
ejpam-5930	110	8	u	u	PROPN
ejpam-5930	110	9	,	,	PUNCT
ejpam-5930	110	10	then	then	ADV
ejpam-5930	110	11	x	x	PUNCT
ejpam-5930	110	12	is	be	AUX
ejpam-5930	110	13	called	call	VERB
ejpam-5930	110	14	empty	empty	ADJ
ejpam-5930	110	15	exterior	exterior	NOUN
ejpam-5930	110	16	.	.	PUNCT
ejpam-5930	111	1	example	example	NOUN
ejpam-5930	112	1	1	1	NUM
ejpam-5930	112	2	.	.	PUNCT
ejpam-5930	112	3	let	let	VERB
ejpam-5930	112	4	a	a	DET
ejpam-5930	112	5	=	=	PUNCT
ejpam-5930	112	6	{	{	PUNCT
ejpam-5930	112	7	1	1	NUM
ejpam-5930	112	8	,	,	PUNCT
ejpam-5930	112	9	x	x	NOUN
ejpam-5930	112	10	,	,	PUNCT
ejpam-5930	112	11	y	y	PROPN
ejpam-5930	112	12	,	,	PUNCT
ejpam-5930	112	13	z	z	PROPN
ejpam-5930	112	14	,	,	PUNCT
ejpam-5930	112	15	0	0	NUM
ejpam-5930	112	16	}	}	PUNCT
ejpam-5930	112	17	with	with	ADP
ejpam-5930	112	18	the	the	DET
ejpam-5930	112	19	following	follow	VERB
ejpam-5930	112	20	cayley	cayley	ADJ
ejpam-5930	112	21	table	table	NOUN
ejpam-5930	112	22	:	:	PUNCT
ejpam-5930	112	23	·	·	PUNCT
ejpam-5930	112	24	1	1	X
ejpam-5930	112	25	x	x	SYM
ejpam-5930	113	1	y	y	PROPN
ejpam-5930	113	2	z	z	NOUN
ejpam-5930	113	3	0	0	NUM
ejpam-5930	113	4	1	1	NUM
ejpam-5930	113	5	1	1	NUM
ejpam-5930	113	6	x	x	SYM
ejpam-5930	113	7	y	y	NOUN
ejpam-5930	113	8	z	z	NOUN
ejpam-5930	113	9	0	0	NUM
ejpam-5930	114	1	x	x	SYM
ejpam-5930	114	2	1	1	NUM
ejpam-5930	114	3	1	1	NUM
ejpam-5930	114	4	y	y	PROPN
ejpam-5930	114	5	z	z	PROPN
ejpam-5930	114	6	0	0	PUNCT
ejpam-5930	115	1	y	y	PROPN
ejpam-5930	115	2	1	1	NUM
ejpam-5930	115	3	x	x	SYM
ejpam-5930	115	4	1	1	NUM
ejpam-5930	115	5	z	z	NOUN
ejpam-5930	115	6	z	z	NOUN
ejpam-5930	115	7	z	z	NOUN
ejpam-5930	116	1	1	1	NUM
ejpam-5930	116	2	1	1	NUM
ejpam-5930	116	3	y	y	PROPN
ejpam-5930	116	4	1	1	NUM
ejpam-5930	116	5	y	y	NOUN
ejpam-5930	116	6	0	0	NUM
ejpam-5930	116	7	1	1	NUM
ejpam-5930	116	8	1	1	NUM
ejpam-5930	116	9	1	1	NUM
ejpam-5930	116	10	1	1	NUM
ejpam-5930	116	11	1	1	NUM
ejpam-5930	116	12	then	then	ADV
ejpam-5930	116	13	a	a	PRON
ejpam-5930	116	14	=	=	X
ejpam-5930	116	15	(	(	PUNCT
ejpam-5930	116	16	a	a	PRON
ejpam-5930	116	17	,	,	PUNCT
ejpam-5930	116	18	·	·	PUNCT
ejpam-5930	116	19	,	,	PUNCT
ejpam-5930	116	20	1	1	NUM
ejpam-5930	116	21	)	)	PUNCT
ejpam-5930	116	22	is	be	AUX
ejpam-5930	116	23	a	a	DET
ejpam-5930	116	24	hilbert	hilbert	NOUN
ejpam-5930	116	25	algebra	algebra	NOUN
ejpam-5930	116	26	.	.	PUNCT
ejpam-5930	117	1	also	also	ADV
ejpam-5930	117	2	i	i	PRON
ejpam-5930	117	3	=	=	PUNCT
ejpam-5930	117	4	{	{	PUNCT
ejpam-5930	117	5	1	1	NUM
ejpam-5930	117	6	,	,	PUNCT
ejpam-5930	117	7	x	x	PRON
ejpam-5930	117	8	}	}	PUNCT
ejpam-5930	117	9	is	be	AUX
ejpam-5930	117	10	an	an	DET
ejpam-5930	117	11	ideal	ideal	NOUN
ejpam-5930	117	12	of	of	ADP
ejpam-5930	117	13	a	a	PRON
ejpam-5930	117	14	and	and	CCONJ
ejpam-5930	117	15	let	let	VERB
ejpam-5930	117	16	θ	θ	NOUN
ejpam-5930	117	17	be	be	AUX
ejpam-5930	117	18	an	an	DET
ejpam-5930	117	19	equivalence	equivalence	NOUN
ejpam-5930	117	20	relation	relation	NOUN
ejpam-5930	117	21	on	on	ADP
ejpam-5930	117	22	a	a	DET
ejpam-5930	117	23	related	relate	VERB
ejpam-5930	117	24	to	to	ADP
ejpam-5930	117	25	i.	i.	PROPN
ejpam-5930	117	26	then	then	ADV
ejpam-5930	117	27	i1	i1	PROPN
ejpam-5930	118	1	=	=	PUNCT
ejpam-5930	118	2	ix	ix	PROPN
ejpam-5930	119	1	=	=	PUNCT
ejpam-5930	119	2	i	i	PROPN
ejpam-5930	119	3	,	,	PUNCT
ejpam-5930	119	4	iy	iy	PROPN
ejpam-5930	119	5	=	=	PUNCT
ejpam-5930	119	6	{	{	PUNCT
ejpam-5930	119	7	y	y	PROPN
ejpam-5930	119	8	}	}	PUNCT
ejpam-5930	119	9	,	,	PUNCT
ejpam-5930	119	10	iz	iz	ADP
ejpam-5930	119	11	=	=	PRON
ejpam-5930	119	12	{	{	PUNCT
ejpam-5930	119	13	z	z	NOUN
ejpam-5930	119	14	}	}	PUNCT
ejpam-5930	119	15	,	,	PUNCT
ejpam-5930	119	16	and	and	CCONJ
ejpam-5930	119	17	i0	i0	PROPN
ejpam-5930	119	18	=	=	PUNCT
ejpam-5930	119	19	{	{	PUNCT
ejpam-5930	119	20	0	0	NUM
ejpam-5930	119	21	}	}	PUNCT
ejpam-5930	119	22	.	.	PUNCT
ejpam-5930	120	1	hence	hence	ADV
ejpam-5930	120	2	,	,	PUNCT
ejpam-5930	120	3	θ(i	θ(i	PROPN
ejpam-5930	120	4	,	,	PUNCT
ejpam-5930	120	5	{	{	PUNCT
ejpam-5930	120	6	1	1	NUM
ejpam-5930	120	7	,	,	PUNCT
ejpam-5930	120	8	x	x	NOUN
ejpam-5930	120	9	}	}	PUNCT
ejpam-5930	120	10	)	)	PUNCT
ejpam-5930	120	11	=	=	SYM
ejpam-5930	120	12	{	{	PUNCT
ejpam-5930	120	13	1	1	NUM
ejpam-5930	120	14	,	,	PUNCT
ejpam-5930	120	15	x	x	NOUN
ejpam-5930	120	16	}	}	PUNCT
ejpam-5930	120	17	▷	▷	ADP
ejpam-5930	120	18	a	a	DET
ejpam-5930	120	19	θ(i	θ(i	NOUN
ejpam-5930	120	20	,	,	PUNCT
ejpam-5930	120	21	{	{	PUNCT
ejpam-5930	120	22	1	1	NUM
ejpam-5930	120	23	,	,	PUNCT
ejpam-5930	120	24	y	y	NOUN
ejpam-5930	120	25	}	}	PUNCT
ejpam-5930	120	26	)	)	PUNCT
ejpam-5930	121	1	=	=	PRON
ejpam-5930	121	2	{	{	PUNCT
ejpam-5930	121	3	y	y	NOUN
ejpam-5930	121	4	}	}	PUNCT
ejpam-5930	121	5	θ(i	θ(i	PROPN
ejpam-5930	121	6	,	,	PUNCT
ejpam-5930	121	7	{	{	PUNCT
ejpam-5930	121	8	1	1	NUM
ejpam-5930	121	9	,	,	PUNCT
ejpam-5930	121	10	z	z	NOUN
ejpam-5930	121	11	}	}	PUNCT
ejpam-5930	121	12	)	)	PUNCT
ejpam-5930	122	1	=	=	PRON
ejpam-5930	122	2	{	{	PUNCT
ejpam-5930	122	3	z	z	NOUN
ejpam-5930	122	4	}	}	PUNCT
ejpam-5930	122	5	θ(i	θ(i	VERB
ejpam-5930	122	6	,	,	PUNCT
ejpam-5930	122	7	{	{	PUNCT
ejpam-5930	122	8	1	1	NUM
ejpam-5930	122	9	,	,	PUNCT
ejpam-5930	122	10	x	x	NOUN
ejpam-5930	122	11	,	,	PUNCT
ejpam-5930	122	12	y	y	PROPN
ejpam-5930	122	13	,	,	PUNCT
ejpam-5930	122	14	z	z	NOUN
ejpam-5930	122	15	}	}	PUNCT
ejpam-5930	122	16	)	)	PUNCT
ejpam-5930	123	1	=	=	PRON
ejpam-5930	123	2	{	{	PUNCT
ejpam-5930	123	3	1	1	NUM
ejpam-5930	123	4	,	,	PUNCT
ejpam-5930	123	5	x	x	NOUN
ejpam-5930	123	6	,	,	PUNCT
ejpam-5930	123	7	y	y	PROPN
ejpam-5930	123	8	,	,	PUNCT
ejpam-5930	123	9	z	z	NOUN
ejpam-5930	123	10	}	}	PUNCT
ejpam-5930	123	11	▷	▷	ADV
ejpam-5930	123	12	a	a	PRON
ejpam-5930	123	13	and	and	CCONJ
ejpam-5930	123	14	θ(i	θ(i	PROPN
ejpam-5930	123	15	,	,	PUNCT
ejpam-5930	123	16	{	{	PUNCT
ejpam-5930	123	17	1	1	NUM
ejpam-5930	123	18	,	,	PUNCT
ejpam-5930	123	19	x	x	NOUN
ejpam-5930	123	20	}	}	PUNCT
ejpam-5930	123	21	)	)	PUNCT
ejpam-5930	124	1	=	=	SYM
ejpam-5930	124	2	{	{	PUNCT
ejpam-5930	124	3	1	1	NUM
ejpam-5930	124	4	,	,	PUNCT
ejpam-5930	124	5	x	x	NOUN
ejpam-5930	124	6	}	}	PUNCT
ejpam-5930	124	7	▷	▷	ADP
ejpam-5930	124	8	a	a	DET
ejpam-5930	124	9	θ(i	θ(i	NOUN
ejpam-5930	124	10	,	,	PUNCT
ejpam-5930	124	11	{	{	PUNCT
ejpam-5930	124	12	1	1	NUM
ejpam-5930	124	13	,	,	PUNCT
ejpam-5930	124	14	y	y	NOUN
ejpam-5930	124	15	}	}	PUNCT
ejpam-5930	124	16	)	)	PUNCT
ejpam-5930	125	1	=	=	PUNCT
ejpam-5930	125	2	{	{	PUNCT
ejpam-5930	125	3	1	1	NUM
ejpam-5930	125	4	,	,	PUNCT
ejpam-5930	125	5	x	x	NOUN
ejpam-5930	125	6	,	,	PUNCT
ejpam-5930	125	7	y	y	NOUN
ejpam-5930	125	8	}	}	PUNCT
ejpam-5930	125	9	▷	▷	ADP
ejpam-5930	125	10	a	a	DET
ejpam-5930	125	11	θ(i	θ(i	NOUN
ejpam-5930	125	12	,	,	PUNCT
ejpam-5930	125	13	{	{	PUNCT
ejpam-5930	125	14	y	y	NOUN
ejpam-5930	125	15	}	}	PUNCT
ejpam-5930	125	16	)	)	PUNCT
ejpam-5930	126	1	=	=	PRON
ejpam-5930	126	2	{	{	PUNCT
ejpam-5930	126	3	y	y	NOUN
ejpam-5930	126	4	}	}	PUNCT
ejpam-5930	126	5	θ(i	θ(i	PROPN
ejpam-5930	126	6	,	,	PUNCT
ejpam-5930	126	7	{	{	PUNCT
ejpam-5930	126	8	x	x	NOUN
ejpam-5930	126	9	,	,	PUNCT
ejpam-5930	126	10	y	y	PROPN
ejpam-5930	126	11	,	,	PUNCT
ejpam-5930	126	12	z	z	NOUN
ejpam-5930	126	13	}	}	PUNCT
ejpam-5930	126	14	)	)	PUNCT
ejpam-5930	126	15	=	=	PRON
ejpam-5930	126	16	{	{	PUNCT
ejpam-5930	126	17	1	1	NUM
ejpam-5930	126	18	,	,	PUNCT
ejpam-5930	126	19	x	x	NOUN
ejpam-5930	126	20	,	,	PUNCT
ejpam-5930	126	21	y	y	PROPN
ejpam-5930	126	22	,	,	PUNCT
ejpam-5930	126	23	z	z	NOUN
ejpam-5930	126	24	}	}	PUNCT
ejpam-5930	126	25	▷	▷	ADP
ejpam-5930	126	26	a	a	DET
ejpam-5930	126	27	θ(i	θ(i	NOUN
ejpam-5930	126	28	,	,	PUNCT
ejpam-5930	126	29	{	{	PUNCT
ejpam-5930	126	30	1	1	NUM
ejpam-5930	126	31	,	,	PUNCT
ejpam-5930	126	32	y	y	PROPN
ejpam-5930	126	33	,	,	PUNCT
ejpam-5930	126	34	z	z	NOUN
ejpam-5930	126	35	}	}	PUNCT
ejpam-5930	126	36	)	)	PUNCT
ejpam-5930	127	1	=	=	PRON
ejpam-5930	127	2	{	{	PUNCT
ejpam-5930	127	3	1	1	NUM
ejpam-5930	127	4	,	,	PUNCT
ejpam-5930	127	5	x	x	NOUN
ejpam-5930	127	6	,	,	PUNCT
ejpam-5930	127	7	y	y	PROPN
ejpam-5930	127	8	,	,	PUNCT
ejpam-5930	127	9	z	z	NOUN
ejpam-5930	127	10	}	}	PUNCT
ejpam-5930	127	11	▷	▷	ADV
ejpam-5930	127	12	a	a	DET
ejpam-5930	127	13	a.	a.	NOUN
ejpam-5930	127	14	iampan	iampan	NOUN
ejpam-5930	127	15	et	et	PROPN
ejpam-5930	127	16	al	al	PROPN
ejpam-5930	127	17	.	.	PUNCT
ejpam-5930	127	18	/	/	SYM
ejpam-5930	127	19	eur	eur	PROPN
ejpam-5930	127	20	.	.	PUNCT
ejpam-5930	128	1	j.	j.	PROPN
ejpam-5930	128	2	pure	pure	PROPN
ejpam-5930	128	3	appl	appl	PROPN
ejpam-5930	128	4	.	.	PROPN
ejpam-5930	128	5	math	math	PROPN
ejpam-5930	128	6	,	,	PUNCT
ejpam-5930	128	7	18	18	NUM
ejpam-5930	128	8	(	(	PUNCT
ejpam-5930	128	9	2	2	NUM
ejpam-5930	128	10	)	)	PUNCT
ejpam-5930	128	11	(	(	PUNCT
ejpam-5930	128	12	2025	2025	NUM
ejpam-5930	128	13	)	)	PUNCT
ejpam-5930	128	14	,	,	PUNCT
ejpam-5930	128	15	5930	5930	NUM
ejpam-5930	128	16	6	6	NUM
ejpam-5930	128	17	of	of	ADP
ejpam-5930	128	18	11	11	NUM
ejpam-5930	128	19	θ(i	θ(i	VERB
ejpam-5930	128	20	,	,	PUNCT
ejpam-5930	128	21	{	{	PUNCT
ejpam-5930	128	22	x	x	NOUN
ejpam-5930	128	23	,	,	PUNCT
ejpam-5930	128	24	y	y	PROPN
ejpam-5930	128	25	,	,	PUNCT
ejpam-5930	128	26	z	z	PROPN
ejpam-5930	128	27	,	,	PUNCT
ejpam-5930	128	28	0	0	NUM
ejpam-5930	128	29	}	}	PUNCT
ejpam-5930	128	30	)	)	PUNCT
ejpam-5930	129	1	=	=	PUNCT
ejpam-5930	129	2	{	{	PUNCT
ejpam-5930	129	3	1	1	NUM
ejpam-5930	129	4	,	,	PUNCT
ejpam-5930	129	5	x	x	NOUN
ejpam-5930	129	6	,	,	PUNCT
ejpam-5930	129	7	y	y	PROPN
ejpam-5930	129	8	,	,	PUNCT
ejpam-5930	129	9	z	z	PROPN
ejpam-5930	129	10	,	,	PUNCT
ejpam-5930	129	11	0	0	NUM
ejpam-5930	129	12	}	}	PUNCT
ejpam-5930	129	13	▷	▷	ADJ
ejpam-5930	129	14	a.	a.	NOUN
ejpam-5930	129	15	in	in	ADP
ejpam-5930	129	16	this	this	DET
ejpam-5930	129	17	example	example	NOUN
ejpam-5930	129	18	,	,	PUNCT
ejpam-5930	129	19	we	we	PRON
ejpam-5930	129	20	know	know	VERB
ejpam-5930	129	21	that	that	SCONJ
ejpam-5930	129	22	there	there	PRON
ejpam-5930	129	23	exists	exist	VERB
ejpam-5930	129	24	a	a	DET
ejpam-5930	129	25	non	non	ADJ
ejpam-5930	129	26	-	-	ADJ
ejpam-5930	129	27	ideal	ideal	ADJ
ejpam-5930	129	28	s	s	NOUN
ejpam-5930	129	29	of	of	ADP
ejpam-5930	129	30	a	a	DET
ejpam-5930	129	31	such	such	ADJ
ejpam-5930	129	32	that	that	SCONJ
ejpam-5930	129	33	their	their	PRON
ejpam-5930	129	34	lower	low	ADJ
ejpam-5930	129	35	and	and	CCONJ
ejpam-5930	129	36	upper	upper	ADJ
ejpam-5930	129	37	approximations	approximation	NOUN
ejpam-5930	129	38	are	be	AUX
ejpam-5930	129	39	ideals	ideal	NOUN
ejpam-5930	129	40	of	of	ADP
ejpam-5930	129	41	a.	a.	NOUN
ejpam-5930	129	42	also	also	ADV
ejpam-5930	129	43	,	,	PUNCT
ejpam-5930	129	44	we	we	PRON
ejpam-5930	129	45	choose	choose	VERB
ejpam-5930	129	46	some	some	DET
ejpam-5930	129	47	non	non	ADJ
ejpam-5930	129	48	-	-	ADJ
ejpam-5930	129	49	ideal	ideal	ADJ
ejpam-5930	129	50	s	s	NOUN
ejpam-5930	129	51	of	of	ADP
ejpam-5930	129	52	a	a	DET
ejpam-5930	129	53	such	such	ADJ
ejpam-5930	129	54	that	that	SCONJ
ejpam-5930	129	55	their	their	PRON
ejpam-5930	129	56	lower	low	ADJ
ejpam-5930	129	57	and	and	CCONJ
ejpam-5930	129	58	upper	upper	ADJ
ejpam-5930	129	59	approximations	approximation	NOUN
ejpam-5930	129	60	are	be	AUX
ejpam-5930	129	61	ideals	ideal	NOUN
ejpam-5930	129	62	of	of	ADP
ejpam-5930	129	63	a.	a.	NOUN
ejpam-5930	129	64	proposition	proposition	NOUN
ejpam-5930	129	65	1	1	NUM
ejpam-5930	129	66	.	.	PUNCT
ejpam-5930	130	1	let	let	VERB
ejpam-5930	130	2	θ	θ	PROPN
ejpam-5930	130	3	and	and	CCONJ
ejpam-5930	130	4	ψ	ψ	X
ejpam-5930	130	5	be	be	AUX
ejpam-5930	130	6	equivalence	equivalence	NOUN
ejpam-5930	130	7	relations	relation	NOUN
ejpam-5930	130	8	on	on	ADP
ejpam-5930	130	9	a	a	DET
ejpam-5930	130	10	related	relate	VERB
ejpam-5930	130	11	to	to	ADP
ejpam-5930	130	12	ideals	ideal	NOUN
ejpam-5930	131	1	i	i	PRON
ejpam-5930	131	2	and	and	CCONJ
ejpam-5930	131	3	j	j	PROPN
ejpam-5930	131	4	of	of	ADP
ejpam-5930	131	5	a	a	PRON
ejpam-5930	131	6	,	,	PUNCT
ejpam-5930	131	7	respectively	respectively	ADV
ejpam-5930	131	8	.	.	PUNCT
ejpam-5930	132	1	if	if	SCONJ
ejpam-5930	132	2	s	s	PRON
ejpam-5930	132	3	and	and	CCONJ
ejpam-5930	132	4	t	t	PROPN
ejpam-5930	132	5	are	be	AUX
ejpam-5930	132	6	nonempty	nonempty	ADJ
ejpam-5930	132	7	subsets	subset	NOUN
ejpam-5930	132	8	of	of	ADP
ejpam-5930	132	9	a	a	PRON
ejpam-5930	132	10	,	,	PUNCT
ejpam-5930	132	11	then	then	ADV
ejpam-5930	132	12	:	:	PUNCT
ejpam-5930	132	13	(	(	PUNCT
ejpam-5930	132	14	1	1	X
ejpam-5930	132	15	)	)	PUNCT
ejpam-5930	132	16	θ(i	θ(i	PROPN
ejpam-5930	132	17	,	,	PUNCT
ejpam-5930	132	18	s	s	X
ejpam-5930	132	19	)	)	PUNCT
ejpam-5930	133	1	⊆	⊆	NUM
ejpam-5930	133	2	s	s	ADP
ejpam-5930	133	3	⊆	⊆	NUM
ejpam-5930	133	4	θ(i	θ(i	PROPN
ejpam-5930	133	5	,	,	PUNCT
ejpam-5930	133	6	s	s	PART
ejpam-5930	133	7	)	)	PUNCT
ejpam-5930	133	8	,	,	PUNCT
ejpam-5930	133	9	(	(	PUNCT
ejpam-5930	133	10	2	2	X
ejpam-5930	133	11	)	)	PUNCT
ejpam-5930	133	12	θ(i	θ(i	VERB
ejpam-5930	133	13	,	,	PUNCT
ejpam-5930	133	14	∅	∅	NOUN
ejpam-5930	133	15	)	)	PUNCT
ejpam-5930	133	16	=	=	NOUN
ejpam-5930	133	17	∅	∅	NOUN
ejpam-5930	133	18	=	=	SYM
ejpam-5930	133	19	θ(i	θ(i	PROPN
ejpam-5930	133	20	,	,	PUNCT
ejpam-5930	133	21	∅	∅	NOUN
ejpam-5930	133	22	)	)	PUNCT
ejpam-5930	133	23	,	,	PUNCT
ejpam-5930	133	24	(	(	PUNCT
ejpam-5930	133	25	3	3	X
ejpam-5930	133	26	)	)	PUNCT
ejpam-5930	133	27	θ(i	θ(i	VERB
ejpam-5930	133	28	,	,	PUNCT
ejpam-5930	133	29	s	s	X
ejpam-5930	133	30	∪	∪	PROPN
ejpam-5930	133	31	t	t	NOUN
ejpam-5930	133	32	)	)	PUNCT
ejpam-5930	134	1	=	=	SYM
ejpam-5930	134	2	θ(i	θ(i	PROPN
ejpam-5930	134	3	,	,	PUNCT
ejpam-5930	134	4	s	s	NOUN
ejpam-5930	134	5	)	)	PUNCT
ejpam-5930	134	6	∪θ(i	∪θ(i	PROPN
ejpam-5930	134	7	,	,	PUNCT
ejpam-5930	134	8	t	t	PROPN
ejpam-5930	134	9	)	)	PUNCT
ejpam-5930	134	10	,	,	PUNCT
ejpam-5930	134	11	(	(	PUNCT
ejpam-5930	134	12	4	4	X
ejpam-5930	134	13	)	)	PUNCT
ejpam-5930	134	14	θ(i	θ(i	VERB
ejpam-5930	134	15	,	,	PUNCT
ejpam-5930	134	16	s	s	NOUN
ejpam-5930	134	17	∩	∩	ADJ
ejpam-5930	134	18	t	t	NOUN
ejpam-5930	134	19	)	)	PUNCT
ejpam-5930	135	1	=	=	SYM
ejpam-5930	135	2	θ(i	θ(i	PROPN
ejpam-5930	135	3	,	,	PUNCT
ejpam-5930	135	4	s	s	PART
ejpam-5930	135	5	)	)	PUNCT
ejpam-5930	135	6	∩θ(i	∩θ(i	PROPN
ejpam-5930	135	7	,	,	PUNCT
ejpam-5930	135	8	t	t	PROPN
ejpam-5930	135	9	)	)	PUNCT
ejpam-5930	135	10	,	,	PUNCT
ejpam-5930	135	11	(	(	PUNCT
ejpam-5930	135	12	5	5	X
ejpam-5930	135	13	)	)	PUNCT
ejpam-5930	135	14	if	if	SCONJ
ejpam-5930	135	15	s	s	VERB
ejpam-5930	135	16	⊆	⊆	NUM
ejpam-5930	135	17	t	t	NOUN
ejpam-5930	135	18	,	,	PUNCT
ejpam-5930	135	19	then	then	ADV
ejpam-5930	135	20	θ(i	θ(i	PROPN
ejpam-5930	135	21	,	,	PUNCT
ejpam-5930	135	22	s	s	NOUN
ejpam-5930	135	23	)	)	PUNCT
ejpam-5930	135	24	⊆	⊆	NUM
ejpam-5930	135	25	θ(i	θ(i	PROPN
ejpam-5930	135	26	,	,	PUNCT
ejpam-5930	135	27	t	t	PROPN
ejpam-5930	135	28	)	)	PUNCT
ejpam-5930	135	29	and	and	CCONJ
ejpam-5930	135	30	θ(i	θ(i	PROPN
ejpam-5930	135	31	,	,	PUNCT
ejpam-5930	135	32	s	s	NOUN
ejpam-5930	135	33	)	)	PUNCT
ejpam-5930	135	34	⊆	⊆	NUM
ejpam-5930	135	35	θ(i	θ(i	PROPN
ejpam-5930	135	36	,	,	PUNCT
ejpam-5930	135	37	t	t	PROPN
ejpam-5930	135	38	)	)	PUNCT
ejpam-5930	135	39	,	,	PUNCT
ejpam-5930	135	40	(	(	PUNCT
ejpam-5930	135	41	6	6	X
ejpam-5930	135	42	)	)	PUNCT
ejpam-5930	135	43	θ(i	θ(i	PROPN
ejpam-5930	135	44	,	,	PUNCT
ejpam-5930	135	45	s	s	X
ejpam-5930	135	46	)	)	PUNCT
ejpam-5930	135	47	∪θ(i	∪θ(i	PROPN
ejpam-5930	135	48	,	,	PUNCT
ejpam-5930	135	49	t	t	PROPN
ejpam-5930	135	50	)	)	PUNCT
ejpam-5930	135	51	⊆	⊆	X
ejpam-5930	135	52	θ(i	θ(i	PROPN
ejpam-5930	135	53	,	,	PUNCT
ejpam-5930	135	54	s	s	NOUN
ejpam-5930	135	55	∪	∪	PROPN
ejpam-5930	135	56	t	t	PROPN
ejpam-5930	135	57	)	)	PUNCT
ejpam-5930	135	58	,	,	PUNCT
ejpam-5930	135	59	(	(	PUNCT
ejpam-5930	135	60	7	7	X
ejpam-5930	135	61	)	)	PUNCT
ejpam-5930	135	62	θ(i	θ(i	VERB
ejpam-5930	135	63	,	,	PUNCT
ejpam-5930	135	64	s	s	NOUN
ejpam-5930	135	65	∩	∩	ADJ
ejpam-5930	135	66	t	t	NOUN
ejpam-5930	135	67	)	)	PUNCT
ejpam-5930	135	68	⊆	⊆	X
ejpam-5930	135	69	θ(i	θ(i	PROPN
ejpam-5930	135	70	,	,	PUNCT
ejpam-5930	135	71	s	s	NOUN
ejpam-5930	135	72	)	)	PUNCT
ejpam-5930	135	73	∩θ(i	∩θ(i	PROPN
ejpam-5930	135	74	,	,	PUNCT
ejpam-5930	135	75	t	t	PROPN
ejpam-5930	135	76	)	)	PUNCT
ejpam-5930	135	77	,	,	PUNCT
ejpam-5930	135	78	(	(	PUNCT
ejpam-5930	135	79	8)	8)	NUM
ejpam-5930	135	80	if	if	SCONJ
ejpam-5930	135	81	θ	θ	PROPN
ejpam-5930	135	82	⊆	⊆	NUM
ejpam-5930	135	83	ψ	ψ	NOUN
ejpam-5930	135	84	and	and	CCONJ
ejpam-5930	135	85	i	i	PRON
ejpam-5930	135	86	⊆	⊆	NUM
ejpam-5930	135	87	j	j	PROPN
ejpam-5930	135	88	,	,	PUNCT
ejpam-5930	135	89	then	then	ADV
ejpam-5930	135	90	ψ(j	ψ(j	PROPN
ejpam-5930	135	91	,	,	PUNCT
ejpam-5930	135	92	s	s	X
ejpam-5930	135	93	)	)	PUNCT
ejpam-5930	135	94	⊆	⊆	NUM
ejpam-5930	135	95	θ(i	θ(i	PROPN
ejpam-5930	135	96	,	,	PUNCT
ejpam-5930	135	97	s	s	PART
ejpam-5930	135	98	)	)	PUNCT
ejpam-5930	135	99	and	and	CCONJ
ejpam-5930	135	100	θ(i	θ(i	PROPN
ejpam-5930	135	101	,	,	PUNCT
ejpam-5930	135	102	s	s	NOUN
ejpam-5930	135	103	)	)	PUNCT
ejpam-5930	135	104	⊆	⊆	NUM
ejpam-5930	135	105	ψ(j	ψ(j	PROPN
ejpam-5930	135	106	,	,	PUNCT
ejpam-5930	135	107	s	s	NOUN
ejpam-5930	135	108	)	)	PUNCT
ejpam-5930	135	109	.	.	PUNCT
ejpam-5930	136	1	proof	proof	NOUN
ejpam-5930	136	2	.	.	PUNCT
ejpam-5930	137	1	(	(	PUNCT
ejpam-5930	137	2	1	1	X
ejpam-5930	137	3	)	)	PUNCT
ejpam-5930	137	4	if	if	SCONJ
ejpam-5930	137	5	x	x	PROPN
ejpam-5930	137	6	∈	∈	PROPN
ejpam-5930	137	7	θ(i	θ(i	PROPN
ejpam-5930	137	8	,	,	PUNCT
ejpam-5930	137	9	s	s	PART
ejpam-5930	137	10	)	)	PUNCT
ejpam-5930	137	11	,	,	PUNCT
ejpam-5930	137	12	then	then	ADV
ejpam-5930	137	13	x	x	X
ejpam-5930	137	14	∈	∈	PROPN
ejpam-5930	137	15	ix	ix	ADP
ejpam-5930	137	16	⊆	⊆	NUM
ejpam-5930	137	17	s.	s.	PROPN
ejpam-5930	137	18	hence	hence	ADV
ejpam-5930	137	19	,	,	PUNCT
ejpam-5930	137	20	θ(i	θ(i	PROPN
ejpam-5930	137	21	,	,	PUNCT
ejpam-5930	137	22	s	s	NOUN
ejpam-5930	137	23	)	)	PUNCT
ejpam-5930	137	24	⊆	⊆	NUM
ejpam-5930	137	25	s.	s.	PROPN
ejpam-5930	137	26	next	next	ADV
ejpam-5930	137	27	,	,	PUNCT
ejpam-5930	137	28	if	if	SCONJ
ejpam-5930	137	29	x	x	PUNCT
ejpam-5930	137	30	∈	∈	PROPN
ejpam-5930	137	31	s	s	PROPN
ejpam-5930	137	32	,	,	PUNCT
ejpam-5930	137	33	then	then	ADV
ejpam-5930	137	34	ix	ix	ADP
ejpam-5930	137	35	∩	∩	PROPN
ejpam-5930	137	36	s	s	PART
ejpam-5930	137	37	=	=	NOUN
ejpam-5930	137	38	∅	∅	NOUN
ejpam-5930	137	39	because	because	SCONJ
ejpam-5930	137	40	x	x	PROPN
ejpam-5930	137	41	∈	∈	PROPN
ejpam-5930	137	42	ix	ix	X
ejpam-5930	137	43	,	,	PUNCT
ejpam-5930	137	44	and	and	CCONJ
ejpam-5930	137	45	so	so	ADV
ejpam-5930	137	46	x	x	SYM
ejpam-5930	137	47	∈	∈	PROPN
ejpam-5930	137	48	θ(i	θ(i	PROPN
ejpam-5930	137	49	,	,	PUNCT
ejpam-5930	137	50	s	s	NOUN
ejpam-5930	137	51	)	)	PUNCT
ejpam-5930	137	52	.	.	PUNCT
ejpam-5930	138	1	thus	thus	ADV
ejpam-5930	138	2	,	,	PUNCT
ejpam-5930	138	3	s	s	VERB
ejpam-5930	138	4	⊆	⊆	NUM
ejpam-5930	138	5	θ(i	θ(i	PROPN
ejpam-5930	138	6	,	,	PUNCT
ejpam-5930	138	7	s	s	NOUN
ejpam-5930	138	8	)	)	PUNCT
ejpam-5930	138	9	.	.	PUNCT
ejpam-5930	139	1	(	(	PUNCT
ejpam-5930	139	2	2	2	X
ejpam-5930	139	3	)	)	PUNCT
ejpam-5930	139	4	is	be	AUX
ejpam-5930	139	5	straightforward	straightforward	ADJ
ejpam-5930	139	6	.	.	PUNCT
ejpam-5930	140	1	(	(	PUNCT
ejpam-5930	140	2	3	3	X
ejpam-5930	140	3	)	)	PUNCT
ejpam-5930	140	4	note	note	VERB
ejpam-5930	140	5	that	that	SCONJ
ejpam-5930	140	6	x	x	PUNCT
ejpam-5930	140	7	∈	∈	PROPN
ejpam-5930	140	8	θ(i	θ(i	PROPN
ejpam-5930	140	9	,	,	PUNCT
ejpam-5930	140	10	s	s	NOUN
ejpam-5930	140	11	∪	∪	PROPN
ejpam-5930	140	12	t	t	PROPN
ejpam-5930	140	13	)	)	PUNCT
ejpam-5930	140	14	⇔	⇔	PROPN
ejpam-5930	140	15	ix	ix	PROPN
ejpam-5930	140	16	∩	∩	PROPN
ejpam-5930	140	17	(	(	PUNCT
ejpam-5930	140	18	s	s	X
ejpam-5930	140	19	∪	∪	PROPN
ejpam-5930	140	20	t	t	NOUN
ejpam-5930	140	21	)	)	PUNCT
ejpam-5930	140	22	̸=	̸=	PROPN
ejpam-5930	140	23	∅	∅	NOUN
ejpam-5930	140	24	⇔	⇔	X
ejpam-5930	140	25	(	(	PUNCT
ejpam-5930	140	26	ix	ix	PROPN
ejpam-5930	140	27	∩	∩	NOUN
ejpam-5930	140	28	s	s	PART
ejpam-5930	140	29	)	)	PUNCT
ejpam-5930	140	30	∪	∪	X
ejpam-5930	140	31	(	(	PUNCT
ejpam-5930	140	32	ix	ix	PROPN
ejpam-5930	140	33	∩	∩	PROPN
ejpam-5930	140	34	t	t	NOUN
ejpam-5930	140	35	)	)	PUNCT
ejpam-5930	140	36	̸=	̸=	PROPN
ejpam-5930	140	37	∅	∅	NOUN
ejpam-5930	140	38	⇔	⇔	X
ejpam-5930	140	39	ix	ix	ADP
ejpam-5930	140	40	∩	∩	PROPN
ejpam-5930	140	41	s	s	PART
ejpam-5930	140	42	=	=	NOUN
ejpam-5930	140	43	∅	∅	NOUN
ejpam-5930	140	44	or	or	CCONJ
ejpam-5930	140	45	ix	ix	ADP
ejpam-5930	140	46	∩	∩	NOUN
ejpam-5930	140	47	t	t	PROPN
ejpam-5930	140	48	̸=	̸=	PROPN
ejpam-5930	140	49	∅	∅	VERB
ejpam-5930	140	50	⇔	⇔	X
ejpam-5930	140	51	x	x	SYM
ejpam-5930	140	52	∈	∈	PROPN
ejpam-5930	140	53	θ(i	θ(i	PROPN
ejpam-5930	140	54	,	,	PUNCT
ejpam-5930	140	55	s	s	NOUN
ejpam-5930	140	56	)	)	PUNCT
ejpam-5930	140	57	or	or	CCONJ
ejpam-5930	140	58	a	a	DET
ejpam-5930	140	59	∈	∈	PROPN
ejpam-5930	140	60	θ(i	θ(i	PROPN
ejpam-5930	140	61	,	,	PUNCT
ejpam-5930	140	62	t	t	PROPN
ejpam-5930	140	63	)	)	PUNCT
ejpam-5930	140	64	⇔	⇔	PROPN
ejpam-5930	140	65	x	x	SYM
ejpam-5930	140	66	∈	∈	PROPN
ejpam-5930	140	67	θ(i	θ(i	PROPN
ejpam-5930	140	68	,	,	PUNCT
ejpam-5930	140	69	s	s	NOUN
ejpam-5930	140	70	)	)	PUNCT
ejpam-5930	140	71	∪θ(i	∪θ(i	PROPN
ejpam-5930	140	72	,	,	PUNCT
ejpam-5930	140	73	t	t	PROPN
ejpam-5930	140	74	)	)	PUNCT
ejpam-5930	140	75	.	.	PUNCT
ejpam-5930	141	1	thus	thus	ADV
ejpam-5930	141	2	,	,	PUNCT
ejpam-5930	141	3	θ(i	θ(i	PROPN
ejpam-5930	141	4	,	,	PUNCT
ejpam-5930	141	5	s	s	NOUN
ejpam-5930	141	6	∪	∪	PROPN
ejpam-5930	141	7	t	t	NOUN
ejpam-5930	141	8	)	)	PUNCT
ejpam-5930	141	9	=	=	SYM
ejpam-5930	141	10	θ(i	θ(i	PROPN
ejpam-5930	141	11	,	,	PUNCT
ejpam-5930	141	12	s	s	NOUN
ejpam-5930	141	13	)	)	PUNCT
ejpam-5930	141	14	∪θ(i	∪θ(i	PROPN
ejpam-5930	141	15	,	,	PUNCT
ejpam-5930	141	16	t	t	PROPN
ejpam-5930	141	17	)	)	PUNCT
ejpam-5930	141	18	.	.	PUNCT
ejpam-5930	142	1	(	(	PUNCT
ejpam-5930	142	2	4	4	X
ejpam-5930	142	3	)	)	PUNCT
ejpam-5930	142	4	note	note	VERB
ejpam-5930	142	5	that	that	SCONJ
ejpam-5930	142	6	x	x	PUNCT
ejpam-5930	142	7	∈	∈	PROPN
ejpam-5930	142	8	θ(i	θ(i	PROPN
ejpam-5930	142	9	,	,	PUNCT
ejpam-5930	142	10	s	s	NOUN
ejpam-5930	142	11	∩	∩	ADJ
ejpam-5930	142	12	t	t	NOUN
ejpam-5930	142	13	)	)	PUNCT
ejpam-5930	142	14	⇔	⇔	PROPN
ejpam-5930	142	15	ix	ix	ADP
ejpam-5930	142	16	⊆	⊆	NUM
ejpam-5930	142	17	s	s	NOUN
ejpam-5930	142	18	∩	∩	PROPN
ejpam-5930	142	19	t	t	PROPN
ejpam-5930	142	20	⇔	⇔	PROPN
ejpam-5930	142	21	ix	ix	PROPN
ejpam-5930	142	22	⊆	⊆	NUM
ejpam-5930	142	23	s	s	NOUN
ejpam-5930	142	24	and	and	CCONJ
ejpam-5930	142	25	ix	ix	ADJ
ejpam-5930	142	26	⊆	⊆	NUM
ejpam-5930	142	27	t	t	NOUN
ejpam-5930	142	28	⇔	⇔	X
ejpam-5930	142	29	x	x	SYM
ejpam-5930	142	30	∈	∈	PROPN
ejpam-5930	142	31	θ(i	θ(i	PROPN
ejpam-5930	142	32	,	,	PUNCT
ejpam-5930	142	33	s	s	PART
ejpam-5930	142	34	)	)	PUNCT
ejpam-5930	142	35	and	and	CCONJ
ejpam-5930	142	36	x	x	PUNCT
ejpam-5930	142	37	∈	∈	PROPN
ejpam-5930	142	38	θ(i	θ(i	PROPN
ejpam-5930	142	39	,	,	PUNCT
ejpam-5930	142	40	t	t	PROPN
ejpam-5930	142	41	)	)	PUNCT
ejpam-5930	142	42	⇔	⇔	PROPN
ejpam-5930	142	43	x	x	SYM
ejpam-5930	142	44	∈	∈	PROPN
ejpam-5930	142	45	θ(i	θ(i	PROPN
ejpam-5930	142	46	,	,	PUNCT
ejpam-5930	142	47	s	s	NOUN
ejpam-5930	142	48	)	)	PUNCT
ejpam-5930	142	49	∩θ(i	∩θ(i	PROPN
ejpam-5930	142	50	,	,	PUNCT
ejpam-5930	142	51	t	t	NOUN
ejpam-5930	142	52	)	)	PUNCT
ejpam-5930	142	53	.	.	PUNCT
ejpam-5930	143	1	thus	thus	ADV
ejpam-5930	143	2	,	,	PUNCT
ejpam-5930	143	3	θ(i	θ(i	PROPN
ejpam-5930	143	4	,	,	PUNCT
ejpam-5930	143	5	s	s	NOUN
ejpam-5930	143	6	∩	∩	ADJ
ejpam-5930	143	7	t	t	NOUN
ejpam-5930	143	8	)	)	PUNCT
ejpam-5930	143	9	=	=	SYM
ejpam-5930	143	10	θ(i	θ(i	PROPN
ejpam-5930	143	11	,	,	PUNCT
ejpam-5930	143	12	s	s	PART
ejpam-5930	143	13	)	)	PUNCT
ejpam-5930	143	14	∩θ(i	∩θ(i	PROPN
ejpam-5930	143	15	,	,	PUNCT
ejpam-5930	143	16	t	t	NOUN
ejpam-5930	143	17	)	)	PUNCT
ejpam-5930	143	18	.	.	PUNCT
ejpam-5930	144	1	(	(	PUNCT
ejpam-5930	144	2	5	5	X
ejpam-5930	144	3	)	)	PUNCT
ejpam-5930	144	4	since	since	SCONJ
ejpam-5930	144	5	s	s	NOUN
ejpam-5930	144	6	⊆	⊆	NUM
ejpam-5930	144	7	t	t	NOUN
ejpam-5930	144	8	if	if	SCONJ
ejpam-5930	144	9	and	and	CCONJ
ejpam-5930	144	10	only	only	ADV
ejpam-5930	144	11	if	if	SCONJ
ejpam-5930	144	12	s	s	X
ejpam-5930	144	13	∩	∩	NOUN
ejpam-5930	144	14	t	t	NOUN
ejpam-5930	144	15	=	=	SYM
ejpam-5930	144	16	s	s	PROPN
ejpam-5930	144	17	,	,	PUNCT
ejpam-5930	144	18	it	it	PRON
ejpam-5930	144	19	follows	follow	VERB
ejpam-5930	144	20	from	from	ADP
ejpam-5930	144	21	(	(	PUNCT
ejpam-5930	144	22	3	3	NUM
ejpam-5930	144	23	)	)	PUNCT
ejpam-5930	144	24	that	that	PRON
ejpam-5930	144	25	θ(i	θ(i	PROPN
ejpam-5930	144	26	,	,	PUNCT
ejpam-5930	144	27	s	s	PART
ejpam-5930	144	28	)	)	PUNCT
ejpam-5930	144	29	=	=	SYM
ejpam-5930	144	30	θ(i	θ(i	PROPN
ejpam-5930	144	31	,	,	PUNCT
ejpam-5930	144	32	s	s	NOUN
ejpam-5930	144	33	∩	∩	ADJ
ejpam-5930	144	34	t	t	NOUN
ejpam-5930	144	35	)	)	PUNCT
ejpam-5930	144	36	=	=	SYM
ejpam-5930	144	37	θ(i	θ(i	PROPN
ejpam-5930	144	38	,	,	PUNCT
ejpam-5930	144	39	s	s	PART
ejpam-5930	144	40	)	)	PUNCT
ejpam-5930	144	41	∩θ(i	∩θ(i	PROPN
ejpam-5930	144	42	,	,	PUNCT
ejpam-5930	144	43	t	t	PROPN
ejpam-5930	144	44	)	)	PUNCT
ejpam-5930	144	45	.	.	PUNCT
ejpam-5930	145	1	a.	a.	PROPN
ejpam-5930	145	2	iampan	iampan	PROPN
ejpam-5930	145	3	et	et	PROPN
ejpam-5930	145	4	al	al	PROPN
ejpam-5930	145	5	.	.	PUNCT
ejpam-5930	145	6	/	/	SYM
ejpam-5930	145	7	eur	eur	PROPN
ejpam-5930	145	8	.	.	PUNCT
ejpam-5930	146	1	j.	j.	PROPN
ejpam-5930	146	2	pure	pure	PROPN
ejpam-5930	146	3	appl	appl	PROPN
ejpam-5930	146	4	.	.	PROPN
ejpam-5930	146	5	math	math	PROPN
ejpam-5930	146	6	,	,	PUNCT
ejpam-5930	146	7	18	18	NUM
ejpam-5930	146	8	(	(	PUNCT
ejpam-5930	146	9	2	2	NUM
ejpam-5930	146	10	)	)	PUNCT
ejpam-5930	146	11	(	(	PUNCT
ejpam-5930	146	12	2025	2025	NUM
ejpam-5930	146	13	)	)	PUNCT
ejpam-5930	146	14	,	,	PUNCT
ejpam-5930	146	15	5930	5930	NUM
ejpam-5930	146	16	7	7	NUM
ejpam-5930	146	17	of	of	ADP
ejpam-5930	146	18	11	11	NUM
ejpam-5930	146	19	this	this	PRON
ejpam-5930	146	20	implies	imply	VERB
ejpam-5930	146	21	that	that	SCONJ
ejpam-5930	146	22	θ(i	θ(i	PROPN
ejpam-5930	146	23	,	,	PUNCT
ejpam-5930	146	24	s	s	NOUN
ejpam-5930	146	25	)	)	PUNCT
ejpam-5930	146	26	⊆	⊆	NUM
ejpam-5930	146	27	θ(i	θ(i	PROPN
ejpam-5930	146	28	,	,	PUNCT
ejpam-5930	146	29	t	t	PROPN
ejpam-5930	146	30	)	)	PUNCT
ejpam-5930	146	31	.	.	PUNCT
ejpam-5930	147	1	note	note	VERB
ejpam-5930	147	2	also	also	ADV
ejpam-5930	147	3	that	that	PRON
ejpam-5930	147	4	s	s	VERB
ejpam-5930	147	5	⊆	⊆	NUM
ejpam-5930	147	6	t	t	NOUN
ejpam-5930	147	7	if	if	SCONJ
ejpam-5930	147	8	and	and	CCONJ
ejpam-5930	147	9	only	only	ADV
ejpam-5930	147	10	if	if	SCONJ
ejpam-5930	147	11	s	s	X
ejpam-5930	147	12	∪	∪	X
ejpam-5930	147	13	t	t	PROPN
ejpam-5930	147	14	=	=	SYM
ejpam-5930	147	15	t	t	PROPN
ejpam-5930	147	16	,	,	PUNCT
ejpam-5930	147	17	it	it	PRON
ejpam-5930	147	18	follows	follow	VERB
ejpam-5930	147	19	from	from	ADP
ejpam-5930	147	20	(	(	PUNCT
ejpam-5930	147	21	2	2	NUM
ejpam-5930	147	22	)	)	PUNCT
ejpam-5930	147	23	that	that	PRON
ejpam-5930	147	24	θ(i	θ(i	PROPN
ejpam-5930	147	25	,	,	PUNCT
ejpam-5930	147	26	t	t	NOUN
ejpam-5930	147	27	)	)	PUNCT
ejpam-5930	148	1	=	=	SYM
ejpam-5930	148	2	θ(i	θ(i	PROPN
ejpam-5930	148	3	,	,	PUNCT
ejpam-5930	148	4	s	s	NOUN
ejpam-5930	148	5	∪	∪	PROPN
ejpam-5930	148	6	t	t	NOUN
ejpam-5930	148	7	)	)	PUNCT
ejpam-5930	148	8	=	=	SYM
ejpam-5930	148	9	θ(i	θ(i	PROPN
ejpam-5930	148	10	,	,	PUNCT
ejpam-5930	148	11	s	s	NOUN
ejpam-5930	148	12	)	)	PUNCT
ejpam-5930	148	13	∪θ(i	∪θ(i	PROPN
ejpam-5930	148	14	,	,	PUNCT
ejpam-5930	148	15	t	t	PROPN
ejpam-5930	148	16	)	)	PUNCT
ejpam-5930	148	17	.	.	PUNCT
ejpam-5930	149	1	this	this	PRON
ejpam-5930	149	2	implies	imply	VERB
ejpam-5930	149	3	that	that	SCONJ
ejpam-5930	149	4	θ(i	θ(i	PROPN
ejpam-5930	149	5	,	,	PUNCT
ejpam-5930	149	6	s	s	NOUN
ejpam-5930	149	7	)	)	PUNCT
ejpam-5930	149	8	⊆	⊆	NUM
ejpam-5930	149	9	θ(i	θ(i	PROPN
ejpam-5930	149	10	,	,	PUNCT
ejpam-5930	149	11	t	t	PROPN
ejpam-5930	149	12	)	)	PUNCT
ejpam-5930	149	13	.	.	PUNCT
ejpam-5930	150	1	(	(	PUNCT
ejpam-5930	150	2	6	6	NUM
ejpam-5930	150	3	)	)	PUNCT
ejpam-5930	150	4	since	since	SCONJ
ejpam-5930	150	5	s	s	VERB
ejpam-5930	150	6	⊆	⊆	NUM
ejpam-5930	150	7	s	s	NOUN
ejpam-5930	150	8	∪t	∪t	NUM
ejpam-5930	150	9	and	and	CCONJ
ejpam-5930	150	10	t	t	VERB
ejpam-5930	150	11	⊆	⊆	NUM
ejpam-5930	150	12	s	s	NOUN
ejpam-5930	150	13	∪t	∪t	NUM
ejpam-5930	150	14	,	,	PUNCT
ejpam-5930	150	15	it	it	PRON
ejpam-5930	150	16	follows	follow	VERB
ejpam-5930	150	17	from	from	ADP
ejpam-5930	150	18	(	(	PUNCT
ejpam-5930	150	19	4	4	NUM
ejpam-5930	150	20	)	)	PUNCT
ejpam-5930	150	21	that	that	PRON
ejpam-5930	150	22	θ(i	θ(i	PROPN
ejpam-5930	150	23	,	,	PUNCT
ejpam-5930	150	24	s	s	NOUN
ejpam-5930	150	25	)	)	PUNCT
ejpam-5930	150	26	⊆	⊆	NUM
ejpam-5930	150	27	θ(i	θ(i	PROPN
ejpam-5930	150	28	,	,	PUNCT
ejpam-5930	150	29	s	s	NOUN
ejpam-5930	150	30	∪t	∪t	NUM
ejpam-5930	150	31	)	)	PUNCT
ejpam-5930	150	32	and	and	CCONJ
ejpam-5930	150	33	θ(i	θ(i	PROPN
ejpam-5930	150	34	,	,	PUNCT
ejpam-5930	150	35	t	t	PROPN
ejpam-5930	150	36	)	)	PUNCT
ejpam-5930	151	1	⊆	⊆	X
ejpam-5930	151	2	θ(i	θ(i	PROPN
ejpam-5930	151	3	,	,	PUNCT
ejpam-5930	151	4	s	s	NOUN
ejpam-5930	151	5	∪	∪	PROPN
ejpam-5930	151	6	t	t	PROPN
ejpam-5930	151	7	)	)	PUNCT
ejpam-5930	151	8	.	.	PUNCT
ejpam-5930	152	1	this	this	PRON
ejpam-5930	152	2	implies	imply	VERB
ejpam-5930	152	3	θ(i	θ(i	PROPN
ejpam-5930	152	4	,	,	PUNCT
ejpam-5930	152	5	s	s	PART
ejpam-5930	152	6	)	)	PUNCT
ejpam-5930	152	7	∪θ(i	∪θ(i	PROPN
ejpam-5930	152	8	,	,	PUNCT
ejpam-5930	152	9	t	t	PROPN
ejpam-5930	152	10	)	)	PUNCT
ejpam-5930	153	1	⊆	⊆	X
ejpam-5930	153	2	θ(i	θ(i	PROPN
ejpam-5930	153	3	,	,	PUNCT
ejpam-5930	153	4	s	s	NOUN
ejpam-5930	153	5	∪	∪	PROPN
ejpam-5930	153	6	t	t	PROPN
ejpam-5930	153	7	)	)	PUNCT
ejpam-5930	153	8	.	.	PUNCT
ejpam-5930	154	1	(	(	PUNCT
ejpam-5930	154	2	7	7	X
ejpam-5930	154	3	)	)	PUNCT
ejpam-5930	154	4	since	since	SCONJ
ejpam-5930	154	5	s	s	PART
ejpam-5930	154	6	∩t	∩t	NOUN
ejpam-5930	154	7	⊆	⊆	NUM
ejpam-5930	154	8	s	s	NOUN
ejpam-5930	154	9	and	and	CCONJ
ejpam-5930	154	10	s	s	VERB
ejpam-5930	154	11	∩t	∩t	NOUN
ejpam-5930	154	12	⊆	⊆	NUM
ejpam-5930	154	13	t	t	NOUN
ejpam-5930	154	14	,	,	PUNCT
ejpam-5930	154	15	it	it	PRON
ejpam-5930	154	16	follows	follow	VERB
ejpam-5930	154	17	from	from	ADP
ejpam-5930	154	18	(	(	PUNCT
ejpam-5930	154	19	4	4	NUM
ejpam-5930	154	20	)	)	PUNCT
ejpam-5930	155	1	that	that	PRON
ejpam-5930	155	2	θ(i	θ(i	VERB
ejpam-5930	155	3	,	,	PUNCT
ejpam-5930	155	4	s	s	PART
ejpam-5930	155	5	∩t	∩t	NOUN
ejpam-5930	155	6	)	)	PUNCT
ejpam-5930	156	1	⊆	⊆	X
ejpam-5930	156	2	θ(i	θ(i	PROPN
ejpam-5930	156	3	,	,	PUNCT
ejpam-5930	156	4	s	s	PART
ejpam-5930	156	5	)	)	PUNCT
ejpam-5930	156	6	and	and	CCONJ
ejpam-5930	156	7	θ(i	θ(i	PROPN
ejpam-5930	156	8	,	,	PUNCT
ejpam-5930	156	9	s	s	NOUN
ejpam-5930	156	10	∩	∩	ADJ
ejpam-5930	156	11	t	t	NOUN
ejpam-5930	156	12	)	)	PUNCT
ejpam-5930	156	13	⊆	⊆	X
ejpam-5930	156	14	θ(i	θ(i	PROPN
ejpam-5930	156	15	,	,	PUNCT
ejpam-5930	156	16	t	t	PROPN
ejpam-5930	156	17	)	)	PUNCT
ejpam-5930	156	18	.	.	PUNCT
ejpam-5930	157	1	this	this	PRON
ejpam-5930	157	2	implies	imply	VERB
ejpam-5930	157	3	θ(i	θ(i	PROPN
ejpam-5930	157	4	,	,	PUNCT
ejpam-5930	157	5	s	s	PART
ejpam-5930	157	6	∩	∩	ADJ
ejpam-5930	157	7	t	t	NOUN
ejpam-5930	157	8	)	)	PUNCT
ejpam-5930	157	9	⊆	⊆	X
ejpam-5930	157	10	θ(i	θ(i	PROPN
ejpam-5930	157	11	,	,	PUNCT
ejpam-5930	157	12	s	s	NOUN
ejpam-5930	157	13	)	)	PUNCT
ejpam-5930	157	14	∩θ(i	∩θ(i	PROPN
ejpam-5930	157	15	,	,	PUNCT
ejpam-5930	157	16	t	t	NOUN
ejpam-5930	157	17	)	)	PUNCT
ejpam-5930	157	18	.	.	PUNCT
ejpam-5930	158	1	(	(	PUNCT
ejpam-5930	158	2	8)	8)	NUM
ejpam-5930	158	3	since	since	SCONJ
ejpam-5930	158	4	θ	θ	PROPN
ejpam-5930	158	5	⊆	⊆	NUM
ejpam-5930	158	6	ψ	ψ	NOUN
ejpam-5930	158	7	,	,	PUNCT
ejpam-5930	158	8	if	if	SCONJ
ejpam-5930	158	9	x	x	PROPN
ejpam-5930	158	10	∈	∈	PROPN
ejpam-5930	158	11	ψ(j	ψ(j	PROPN
ejpam-5930	158	12	,	,	PUNCT
ejpam-5930	158	13	s	s	PART
ejpam-5930	158	14	)	)	PUNCT
ejpam-5930	158	15	,	,	PUNCT
ejpam-5930	158	16	then	then	ADV
ejpam-5930	158	17	jx	jx	PROPN
ejpam-5930	158	18	⊆	⊆	NUM
ejpam-5930	158	19	s.	s.	PROPN
ejpam-5930	158	20	but	but	CCONJ
ejpam-5930	158	21	θ	θ	PROPN
ejpam-5930	158	22	⊆	⊆	NUM
ejpam-5930	158	23	ψ	ψ	NOUN
ejpam-5930	158	24	,	,	PUNCT
ejpam-5930	158	25	then	then	ADV
ejpam-5930	158	26	ix	ix	PROPN
ejpam-5930	158	27	⊆	⊆	NUM
ejpam-5930	158	28	jx	jx	PROPN
ejpam-5930	158	29	⊆	⊆	NUM
ejpam-5930	158	30	s	s	NOUN
ejpam-5930	158	31	,	,	PUNCT
ejpam-5930	158	32	that	that	ADV
ejpam-5930	158	33	is	is	ADV
ejpam-5930	158	34	,	,	PUNCT
ejpam-5930	158	35	ix	ix	PROPN
ejpam-5930	158	36	⊆	⊆	NUM
ejpam-5930	158	37	s.	s.	PROPN
ejpam-5930	158	38	thus	thus	ADV
ejpam-5930	158	39	,	,	PUNCT
ejpam-5930	158	40	x	x	PROPN
ejpam-5930	158	41	∈	∈	PROPN
ejpam-5930	158	42	θ(i	θ(i	PROPN
ejpam-5930	158	43	,	,	PUNCT
ejpam-5930	158	44	s	s	NOUN
ejpam-5930	158	45	)	)	PUNCT
ejpam-5930	158	46	.	.	PUNCT
ejpam-5930	159	1	hence	hence	ADV
ejpam-5930	159	2	,	,	PUNCT
ejpam-5930	159	3	ψ(j	ψ(j	PROPN
ejpam-5930	159	4	,	,	PUNCT
ejpam-5930	159	5	s	s	X
ejpam-5930	159	6	)	)	PUNCT
ejpam-5930	159	7	⊆	⊆	NUM
ejpam-5930	159	8	θ(i	θ(i	PROPN
ejpam-5930	159	9	,	,	PUNCT
ejpam-5930	159	10	s	s	NOUN
ejpam-5930	159	11	)	)	PUNCT
ejpam-5930	159	12	.	.	PUNCT
ejpam-5930	160	1	now	now	ADV
ejpam-5930	160	2	let	let	VERB
ejpam-5930	160	3	x	x	PRON
ejpam-5930	160	4	be	be	AUX
ejpam-5930	160	5	any	any	DET
ejpam-5930	160	6	element	element	NOUN
ejpam-5930	160	7	of	of	ADP
ejpam-5930	160	8	θ(s	θ(s	PROPN
ejpam-5930	160	9	)	)	PUNCT
ejpam-5930	160	10	.	.	PUNCT
ejpam-5930	161	1	so	so	ADV
ejpam-5930	161	2	ix	ix	ADP
ejpam-5930	161	3	∩	∩	PROPN
ejpam-5930	161	4	s	s	PART
ejpam-5930	161	5	=	=	SYM
ejpam-5930	161	6	∅	∅	NOUN
ejpam-5930	161	7	,	,	PUNCT
ejpam-5930	161	8	there	there	PRON
ejpam-5930	161	9	exists	exist	VERB
ejpam-5930	161	10	y	y	PROPN
ejpam-5930	161	11	∈	∈	PROPN
ejpam-5930	161	12	iy	iy	PROPN
ejpam-5930	161	13	∩	∩	PROPN
ejpam-5930	161	14	s	s	VERB
ejpam-5930	161	15	such	such	ADJ
ejpam-5930	161	16	that	that	SCONJ
ejpam-5930	161	17	y	y	PROPN
ejpam-5930	161	18	∈	∈	PROPN
ejpam-5930	161	19	iy	iy	PROPN
ejpam-5930	161	20	and	and	CCONJ
ejpam-5930	161	21	y	y	PROPN
ejpam-5930	161	22	∈	∈	PROPN
ejpam-5930	161	23	s.	s.	PROPN
ejpam-5930	161	24	hence	hence	ADV
ejpam-5930	161	25	,	,	PUNCT
ejpam-5930	161	26	(	(	PUNCT
ejpam-5930	161	27	y	y	NOUN
ejpam-5930	161	28	,	,	PUNCT
ejpam-5930	161	29	x	x	NOUN
ejpam-5930	161	30	)	)	PUNCT
ejpam-5930	161	31	∈	∈	PROPN
ejpam-5930	161	32	θ	θ	PROPN
ejpam-5930	161	33	,	,	PUNCT
ejpam-5930	161	34	that	that	ADV
ejpam-5930	161	35	is	is	ADV
ejpam-5930	161	36	,	,	PUNCT
ejpam-5930	161	37	y	y	PROPN
ejpam-5930	161	38	·	·	PUNCT
ejpam-5930	161	39	x	x	SYM
ejpam-5930	161	40	∈	∈	PROPN
ejpam-5930	161	41	i.	i.	NOUN
ejpam-5930	161	42	since	since	SCONJ
ejpam-5930	161	43	i	i	PROPN
ejpam-5930	161	44	⊆	⊆	NUM
ejpam-5930	161	45	j	j	PROPN
ejpam-5930	161	46	,	,	PUNCT
ejpam-5930	161	47	it	it	PRON
ejpam-5930	161	48	follows	follow	VERB
ejpam-5930	161	49	that	that	SCONJ
ejpam-5930	161	50	y	y	PROPN
ejpam-5930	161	51	·	·	PUNCT
ejpam-5930	161	52	x	x	SYM
ejpam-5930	161	53	∈	∈	PROPN
ejpam-5930	161	54	j	j	PROPN
ejpam-5930	161	55	and	and	CCONJ
ejpam-5930	161	56	x	x	SYM
ejpam-5930	161	57	·	·	PUNCT
ejpam-5930	161	58	y	y	PROPN
ejpam-5930	161	59	∈	∈	PROPN
ejpam-5930	161	60	j	j	NOUN
ejpam-5930	161	61	so	so	SCONJ
ejpam-5930	161	62	that	that	SCONJ
ejpam-5930	161	63	(	(	PUNCT
ejpam-5930	161	64	y	y	NOUN
ejpam-5930	161	65	,	,	PUNCT
ejpam-5930	161	66	x	x	NOUN
ejpam-5930	161	67	)	)	PUNCT
ejpam-5930	161	68	∈	∈	PROPN
ejpam-5930	161	69	ψ	ψ	NOUN
ejpam-5930	161	70	,	,	PUNCT
ejpam-5930	161	71	that	that	ADV
ejpam-5930	161	72	is	is	ADV
ejpam-5930	161	73	,	,	PUNCT
ejpam-5930	161	74	y	y	PROPN
ejpam-5930	161	75	∈	∈	PROPN
ejpam-5930	161	76	jx	jx	PROPN
ejpam-5930	161	77	.	.	PUNCT
ejpam-5930	162	1	therefore	therefore	ADV
ejpam-5930	162	2	,	,	PUNCT
ejpam-5930	162	3	y	y	PROPN
ejpam-5930	162	4	∈	∈	PROPN
ejpam-5930	162	5	jx	jx	PROPN
ejpam-5930	162	6	∩	∩	PROPN
ejpam-5930	162	7	s	s	PART
ejpam-5930	162	8	,	,	PUNCT
ejpam-5930	162	9	which	which	PRON
ejpam-5930	162	10	means	mean	VERB
ejpam-5930	162	11	that	that	SCONJ
ejpam-5930	162	12	x	x	PROPN
ejpam-5930	162	13	∈	∈	PROPN
ejpam-5930	162	14	ψ(j	ψ(j	PROPN
ejpam-5930	162	15	,	,	PUNCT
ejpam-5930	162	16	s	s	NOUN
ejpam-5930	162	17	)	)	PUNCT
ejpam-5930	162	18	.	.	PUNCT
ejpam-5930	163	1	proposition	proposition	NOUN
ejpam-5930	163	2	2	2	X
ejpam-5930	163	3	.	.	PUNCT
ejpam-5930	164	1	let	let	VERB
ejpam-5930	164	2	θ	θ	NOUN
ejpam-5930	164	3	be	be	AUX
ejpam-5930	164	4	an	an	DET
ejpam-5930	164	5	equivalence	equivalence	NOUN
ejpam-5930	164	6	relation	relation	NOUN
ejpam-5930	164	7	on	on	ADP
ejpam-5930	164	8	a	a	DET
ejpam-5930	164	9	related	relate	VERB
ejpam-5930	164	10	to	to	ADP
ejpam-5930	164	11	an	an	DET
ejpam-5930	164	12	ideal	ideal	NOUN
ejpam-5930	164	13	i	i	PRON
ejpam-5930	164	14	of	of	ADP
ejpam-5930	164	15	a.	a.	NOUN
ejpam-5930	164	16	if	if	SCONJ
ejpam-5930	164	17	s	s	X
ejpam-5930	164	18	is	be	AUX
ejpam-5930	164	19	a	a	DET
ejpam-5930	164	20	nonempty	nonempty	ADJ
ejpam-5930	164	21	subset	subset	NOUN
ejpam-5930	164	22	of	of	ADP
ejpam-5930	164	23	a	a	PRON
ejpam-5930	164	24	,	,	PUNCT
ejpam-5930	164	25	then	then	ADV
ejpam-5930	164	26	:	:	PUNCT
ejpam-5930	164	27	(	(	PUNCT
ejpam-5930	164	28	1	1	X
ejpam-5930	164	29	)	)	PUNCT
ejpam-5930	164	30	θ(i	θ(i	PROPN
ejpam-5930	164	31	,	,	PUNCT
ejpam-5930	164	32	θ(i	θ(i	PROPN
ejpam-5930	164	33	,	,	PUNCT
ejpam-5930	164	34	s	s	NOUN
ejpam-5930	164	35	)	)	PUNCT
ejpam-5930	164	36	)	)	PUNCT
ejpam-5930	165	1	=	=	SYM
ejpam-5930	165	2	θ(i	θ(i	PROPN
ejpam-5930	165	3	,	,	PUNCT
ejpam-5930	165	4	s	s	PART
ejpam-5930	165	5	)	)	PUNCT
ejpam-5930	165	6	,	,	PUNCT
ejpam-5930	165	7	(	(	PUNCT
ejpam-5930	165	8	2	2	X
ejpam-5930	165	9	)	)	PUNCT
ejpam-5930	165	10	θ(i	θ(i	PROPN
ejpam-5930	165	11	,	,	PUNCT
ejpam-5930	165	12	θ(i	θ(i	PROPN
ejpam-5930	165	13	,	,	PUNCT
ejpam-5930	165	14	s	s	NOUN
ejpam-5930	165	15	)	)	PUNCT
ejpam-5930	165	16	)	)	PUNCT
ejpam-5930	166	1	=	=	SYM
ejpam-5930	166	2	θ(i	θ(i	PROPN
ejpam-5930	166	3	,	,	PUNCT
ejpam-5930	166	4	s	s	PART
ejpam-5930	166	5	)	)	PUNCT
ejpam-5930	166	6	,	,	PUNCT
ejpam-5930	166	7	(	(	PUNCT
ejpam-5930	166	8	3	3	X
ejpam-5930	166	9	)	)	PUNCT
ejpam-5930	166	10	θ(i	θ(i	PROPN
ejpam-5930	166	11	,	,	PUNCT
ejpam-5930	166	12	θ(i	θ(i	PROPN
ejpam-5930	166	13	,	,	PUNCT
ejpam-5930	166	14	s	s	NOUN
ejpam-5930	166	15	)	)	PUNCT
ejpam-5930	166	16	)	)	PUNCT
ejpam-5930	167	1	=	=	SYM
ejpam-5930	167	2	θ(i	θ(i	PROPN
ejpam-5930	167	3	,	,	PUNCT
ejpam-5930	167	4	s	s	PART
ejpam-5930	167	5	)	)	PUNCT
ejpam-5930	167	6	,	,	PUNCT
ejpam-5930	167	7	(	(	PUNCT
ejpam-5930	167	8	4	4	X
ejpam-5930	167	9	)	)	PUNCT
ejpam-5930	167	10	θ(i	θ(i	PROPN
ejpam-5930	167	11	,	,	PUNCT
ejpam-5930	167	12	θ(i	θ(i	PROPN
ejpam-5930	167	13	,	,	PUNCT
ejpam-5930	167	14	s	s	NOUN
ejpam-5930	167	15	)	)	PUNCT
ejpam-5930	167	16	)	)	PUNCT
ejpam-5930	168	1	=	=	SYM
ejpam-5930	168	2	θ(i	θ(i	PROPN
ejpam-5930	168	3	,	,	PUNCT
ejpam-5930	168	4	s	s	PART
ejpam-5930	168	5	)	)	PUNCT
ejpam-5930	168	6	,	,	PUNCT
ejpam-5930	168	7	(	(	PUNCT
ejpam-5930	168	8	5	5	X
ejpam-5930	168	9	)	)	PUNCT
ejpam-5930	168	10	θ(i	θ(i	PROPN
ejpam-5930	168	11	,	,	PUNCT
ejpam-5930	168	12	s	s	NOUN
ejpam-5930	168	13	)	)	PUNCT
ejpam-5930	168	14	=	=	SYM
ejpam-5930	168	15	(	(	PUNCT
ejpam-5930	168	16	θ(i	θ(i	PROPN
ejpam-5930	168	17	,	,	PUNCT
ejpam-5930	168	18	sc))c	sc))c	PROPN
ejpam-5930	168	19	,	,	PUNCT
ejpam-5930	168	20	(	(	PUNCT
ejpam-5930	168	21	6	6	NUM
ejpam-5930	168	22	)	)	PUNCT
ejpam-5930	168	23	θ(i	θ(i	PROPN
ejpam-5930	168	24	,	,	PUNCT
ejpam-5930	168	25	s	s	NOUN
ejpam-5930	168	26	)	)	PUNCT
ejpam-5930	168	27	=	=	SYM
ejpam-5930	168	28	(	(	PUNCT
ejpam-5930	168	29	θ(i	θ(i	PROPN
ejpam-5930	168	30	,	,	PUNCT
ejpam-5930	168	31	sc))c	sc))c	PROPN
ejpam-5930	168	32	,	,	PUNCT
ejpam-5930	168	33	(	(	PUNCT
ejpam-5930	168	34	7	7	NUM
ejpam-5930	168	35	)	)	PUNCT
ejpam-5930	168	36	θ(i	θ(i	PROPN
ejpam-5930	168	37	,	,	PUNCT
ejpam-5930	168	38	ix	ix	PROPN
ejpam-5930	168	39	)	)	PUNCT
ejpam-5930	168	40	=	=	SYM
ejpam-5930	168	41	a	a	DET
ejpam-5930	168	42	=	=	SYM
ejpam-5930	168	43	θ(i	θ(i	PROPN
ejpam-5930	168	44	,	,	PUNCT
ejpam-5930	168	45	ix	ix	PROPN
ejpam-5930	168	46	)	)	PUNCT
ejpam-5930	168	47	,	,	PUNCT
ejpam-5930	168	48	for	for	ADP
ejpam-5930	168	49	all	all	DET
ejpam-5930	168	50	x	x	SYM
ejpam-5930	168	51	∈	∈	NOUN
ejpam-5930	168	52	a.	a.	NOUN
ejpam-5930	168	53	proof	proof	NOUN
ejpam-5930	168	54	.	.	PUNCT
ejpam-5930	169	1	the	the	DET
ejpam-5930	169	2	proof	proof	NOUN
ejpam-5930	169	3	is	be	AUX
ejpam-5930	169	4	straightforward	straightforward	ADJ
ejpam-5930	169	5	.	.	PUNCT
ejpam-5930	170	1	proposition	proposition	NOUN
ejpam-5930	170	2	3	3	X
ejpam-5930	170	3	.	.	PUNCT
ejpam-5930	171	1	let	let	VERB
ejpam-5930	171	2	θ	θ	NOUN
ejpam-5930	171	3	be	be	AUX
ejpam-5930	171	4	an	an	DET
ejpam-5930	171	5	equivalence	equivalence	NOUN
ejpam-5930	171	6	relation	relation	NOUN
ejpam-5930	171	7	on	on	ADP
ejpam-5930	171	8	a	a	DET
ejpam-5930	171	9	related	relate	VERB
ejpam-5930	171	10	to	to	ADP
ejpam-5930	171	11	an	an	DET
ejpam-5930	171	12	ideal	ideal	NOUN
ejpam-5930	171	13	i	i	PRON
ejpam-5930	171	14	of	of	ADP
ejpam-5930	171	15	a.	a.	NOUN
ejpam-5930	171	16	if	if	SCONJ
ejpam-5930	171	17	s	s	X
ejpam-5930	171	18	is	be	AUX
ejpam-5930	171	19	a	a	DET
ejpam-5930	171	20	nonempty	nonempty	ADJ
ejpam-5930	171	21	subset	subset	NOUN
ejpam-5930	171	22	of	of	ADP
ejpam-5930	171	23	a	a	PRON
ejpam-5930	171	24	,	,	PUNCT
ejpam-5930	171	25	then	then	ADV
ejpam-5930	171	26	:	:	PUNCT
ejpam-5930	171	27	(	(	PUNCT
ejpam-5930	171	28	1	1	X
ejpam-5930	171	29	)	)	PUNCT
ejpam-5930	171	30	θ(i	θ(i	PROPN
ejpam-5930	171	31	,	,	PUNCT
ejpam-5930	171	32	s	s	NOUN
ejpam-5930	171	33	)	)	PUNCT
ejpam-5930	171	34	·	·	PUNCT
ejpam-5930	171	35	θ(i	θ(i	PROPN
ejpam-5930	171	36	,	,	PUNCT
ejpam-5930	171	37	t	t	PROPN
ejpam-5930	171	38	)	)	PUNCT
ejpam-5930	172	1	⊆	⊆	X
ejpam-5930	172	2	θ(i	θ(i	PROPN
ejpam-5930	172	3	,	,	PUNCT
ejpam-5930	172	4	s	s	PART
ejpam-5930	172	5	·	·	PUNCT
ejpam-5930	172	6	t	t	PROPN
ejpam-5930	172	7	)	)	PUNCT
ejpam-5930	172	8	,	,	PUNCT
ejpam-5930	172	9	(	(	PUNCT
ejpam-5930	172	10	2	2	X
ejpam-5930	172	11	)	)	PUNCT
ejpam-5930	172	12	if	if	SCONJ
ejpam-5930	172	13	θ	θ	PROPN
ejpam-5930	172	14	is	be	AUX
ejpam-5930	172	15	a	a	DET
ejpam-5930	172	16	congruence	congruence	NOUN
ejpam-5930	172	17	relation	relation	NOUN
ejpam-5930	172	18	on	on	ADP
ejpam-5930	172	19	a	a	DET
ejpam-5930	172	20	,	,	PUNCT
ejpam-5930	172	21	then	then	ADV
ejpam-5930	172	22	θ(i	θ(i	PROPN
ejpam-5930	172	23	,	,	PUNCT
ejpam-5930	172	24	s	s	NOUN
ejpam-5930	172	25	)	)	PUNCT
ejpam-5930	172	26	·	·	PUNCT
ejpam-5930	172	27	θ(i	θ(i	PROPN
ejpam-5930	172	28	,	,	PUNCT
ejpam-5930	172	29	t	t	PROPN
ejpam-5930	172	30	)	)	PUNCT
ejpam-5930	172	31	⊆	⊆	X
ejpam-5930	172	32	θ(i	θ(i	PROPN
ejpam-5930	172	33	,	,	PUNCT
ejpam-5930	172	34	s	s	PART
ejpam-5930	172	35	·	·	PUNCT
ejpam-5930	172	36	t	t	PROPN
ejpam-5930	172	37	)	)	PUNCT
ejpam-5930	172	38	.	.	PUNCT
ejpam-5930	173	1	proof	proof	NOUN
ejpam-5930	173	2	.	.	PUNCT
ejpam-5930	174	1	(	(	PUNCT
ejpam-5930	174	2	1	1	X
ejpam-5930	174	3	)	)	PUNCT
ejpam-5930	174	4	let	let	VERB
ejpam-5930	174	5	c	c	NOUN
ejpam-5930	174	6	be	be	AUX
ejpam-5930	174	7	any	any	DET
ejpam-5930	174	8	element	element	NOUN
ejpam-5930	174	9	of	of	ADP
ejpam-5930	174	10	θ(i	θ(i	PROPN
ejpam-5930	174	11	,	,	PUNCT
ejpam-5930	174	12	s	s	NOUN
ejpam-5930	174	13	)	)	PUNCT
ejpam-5930	174	14	·	·	PUNCT
ejpam-5930	175	1	θ(i	θ(i	VERB
ejpam-5930	175	2	,	,	PUNCT
ejpam-5930	175	3	t	t	PROPN
ejpam-5930	175	4	)	)	PUNCT
ejpam-5930	175	5	.	.	PUNCT
ejpam-5930	176	1	then	then	ADV
ejpam-5930	176	2	c	c	X
ejpam-5930	177	1	=	=	PUNCT
ejpam-5930	177	2	p	p	X
ejpam-5930	177	3	·	·	PUNCT
ejpam-5930	177	4	q	q	NOUN
ejpam-5930	177	5	with	with	ADP
ejpam-5930	177	6	p	p	PROPN
ejpam-5930	177	7	∈	∈	PROPN
ejpam-5930	177	8	θ(i	θ(i	PROPN
ejpam-5930	177	9	,	,	PUNCT
ejpam-5930	177	10	s	s	PART
ejpam-5930	177	11	)	)	PUNCT
ejpam-5930	177	12	and	and	CCONJ
ejpam-5930	177	13	q	q	PROPN
ejpam-5930	177	14	∈	∈	PROPN
ejpam-5930	177	15	θ(i	θ(i	PROPN
ejpam-5930	177	16	,	,	PUNCT
ejpam-5930	177	17	t	t	PROPN
ejpam-5930	177	18	)	)	PUNCT
ejpam-5930	177	19	.	.	PUNCT
ejpam-5930	178	1	so	so	ADV
ejpam-5930	178	2	there	there	PRON
ejpam-5930	178	3	exist	exist	VERB
ejpam-5930	178	4	elements	element	NOUN
ejpam-5930	178	5	x	x	X
ejpam-5930	178	6	,	,	PUNCT
ejpam-5930	178	7	y	y	PROPN
ejpam-5930	178	8	∈	∈	PROPN
ejpam-5930	178	9	s	s	VERB
ejpam-5930	178	10	such	such	ADJ
ejpam-5930	178	11	that	that	SCONJ
ejpam-5930	178	12	x	x	SYM
ejpam-5930	178	13	∈	∈	NOUN
ejpam-5930	178	14	ip	ip	NOUN
ejpam-5930	178	15	∩	∩	NOUN
ejpam-5930	178	16	s	s	PART
ejpam-5930	178	17	and	and	CCONJ
ejpam-5930	178	18	y	y	PROPN
ejpam-5930	178	19	∈	∈	PROPN
ejpam-5930	178	20	iq	iq	NOUN
ejpam-5930	178	21	∩	∩	PROPN
ejpam-5930	178	22	t	t	PROPN
ejpam-5930	178	23	.	.	PUNCT
ejpam-5930	179	1	thus	thus	ADV
ejpam-5930	179	2	,	,	PUNCT
ejpam-5930	179	3	x	x	SYM
ejpam-5930	179	4	∈	∈	PROPN
ejpam-5930	179	5	ip	ip	NOUN
ejpam-5930	179	6	,	,	PUNCT
ejpam-5930	179	7	y	y	PROPN
ejpam-5930	179	8	∈	∈	PROPN
ejpam-5930	179	9	iq	iq	NOUN
ejpam-5930	179	10	,	,	PUNCT
ejpam-5930	179	11	x	x	PUNCT
ejpam-5930	179	12	∈	∈	PROPN
ejpam-5930	179	13	s	s	NOUN
ejpam-5930	179	14	,	,	PUNCT
ejpam-5930	179	15	and	and	CCONJ
ejpam-5930	179	16	y	y	PROPN
ejpam-5930	179	17	∈	∈	PROPN
ejpam-5930	179	18	t	t	PROPN
ejpam-5930	179	19	.	.	PUNCT
ejpam-5930	180	1	so	so	ADV
ejpam-5930	180	2	x	x	X
ejpam-5930	180	3	·	·	PUNCT
ejpam-5930	180	4	y	y	X
ejpam-5930	180	5	∈	∈	PROPN
ejpam-5930	180	6	ip	ip	NOUN
ejpam-5930	180	7	·	·	PUNCT
ejpam-5930	180	8	iq	iq	VERB
ejpam-5930	180	9	⊆	⊆	NUM
ejpam-5930	180	10	ip·q	ip·q	NUM
ejpam-5930	180	11	.	.	PUNCT
ejpam-5930	181	1	on	on	ADP
ejpam-5930	181	2	the	the	DET
ejpam-5930	181	3	other	other	ADJ
ejpam-5930	181	4	hand	hand	NOUN
ejpam-5930	181	5	,	,	PUNCT
ejpam-5930	181	6	since	since	SCONJ
ejpam-5930	181	7	x	x	X
ejpam-5930	181	8	·	·	PUNCT
ejpam-5930	181	9	y	y	PROPN
ejpam-5930	181	10	∈	∈	PROPN
ejpam-5930	181	11	s	s	PART
ejpam-5930	181	12	·	·	PUNCT
ejpam-5930	181	13	t	t	NOUN
ejpam-5930	181	14	,	,	PUNCT
ejpam-5930	181	15	we	we	PRON
ejpam-5930	181	16	have	have	VERB
ejpam-5930	181	17	x	x	X
ejpam-5930	181	18	·	·	PUNCT
ejpam-5930	181	19	y	y	PROPN
ejpam-5930	181	20	∈	∈	PROPN
ejpam-5930	181	21	ip·q	ip·q	PROPN
ejpam-5930	181	22	∩	∩	NOUN
ejpam-5930	181	23	(	(	PUNCT
ejpam-5930	181	24	s	s	X
ejpam-5930	181	25	·	·	PUNCT
ejpam-5930	181	26	t	t	NOUN
ejpam-5930	181	27	)	)	PUNCT
ejpam-5930	181	28	,	,	PUNCT
ejpam-5930	181	29	and	and	CCONJ
ejpam-5930	181	30	so	so	ADV
ejpam-5930	181	31	c	c	NOUN
ejpam-5930	182	1	=	=	SYM
ejpam-5930	182	2	p	p	X
ejpam-5930	182	3	·	·	PUNCT
ejpam-5930	182	4	q	q	PROPN
ejpam-5930	182	5	∈	∈	PROPN
ejpam-5930	182	6	θ(i	θ(i	PROPN
ejpam-5930	182	7	,	,	PUNCT
ejpam-5930	182	8	s	s	PART
ejpam-5930	182	9	·	·	PUNCT
ejpam-5930	182	10	t	t	PROPN
ejpam-5930	182	11	)	)	PUNCT
ejpam-5930	182	12	.	.	PUNCT
ejpam-5930	183	1	hence	hence	ADV
ejpam-5930	183	2	,	,	PUNCT
ejpam-5930	183	3	θ(i	θ(i	PROPN
ejpam-5930	183	4	,	,	PUNCT
ejpam-5930	183	5	s	s	NOUN
ejpam-5930	183	6	)	)	PUNCT
ejpam-5930	183	7	·	·	PUNCT
ejpam-5930	183	8	θ(i	θ(i	PROPN
ejpam-5930	183	9	,	,	PUNCT
ejpam-5930	183	10	t	t	PROPN
ejpam-5930	183	11	)	)	PUNCT
ejpam-5930	184	1	⊆	⊆	X
ejpam-5930	184	2	θ(i	θ(i	PROPN
ejpam-5930	184	3	,	,	PUNCT
ejpam-5930	184	4	s	s	PART
ejpam-5930	184	5	·	·	PUNCT
ejpam-5930	184	6	t	t	PROPN
ejpam-5930	184	7	)	)	PUNCT
ejpam-5930	184	8	.	.	PUNCT
ejpam-5930	185	1	a.	a.	PROPN
ejpam-5930	185	2	iampan	iampan	PROPN
ejpam-5930	185	3	et	et	PROPN
ejpam-5930	185	4	al	al	PROPN
ejpam-5930	185	5	.	.	PUNCT
ejpam-5930	185	6	/	/	SYM
ejpam-5930	185	7	eur	eur	PROPN
ejpam-5930	185	8	.	.	PUNCT
ejpam-5930	186	1	j.	j.	PROPN
ejpam-5930	186	2	pure	pure	PROPN
ejpam-5930	186	3	appl	appl	PROPN
ejpam-5930	186	4	.	.	PROPN
ejpam-5930	186	5	math	math	PROPN
ejpam-5930	186	6	,	,	PUNCT
ejpam-5930	186	7	18	18	NUM
ejpam-5930	186	8	(	(	PUNCT
ejpam-5930	186	9	2	2	NUM
ejpam-5930	186	10	)	)	PUNCT
ejpam-5930	186	11	(	(	PUNCT
ejpam-5930	186	12	2025	2025	NUM
ejpam-5930	186	13	)	)	PUNCT
ejpam-5930	186	14	,	,	PUNCT
ejpam-5930	186	15	5930	5930	NUM
ejpam-5930	186	16	8	8	NUM
ejpam-5930	186	17	of	of	ADP
ejpam-5930	186	18	11	11	NUM
ejpam-5930	186	19	(	(	PUNCT
ejpam-5930	186	20	2	2	NUM
ejpam-5930	186	21	)	)	PUNCT
ejpam-5930	186	22	assume	assume	VERB
ejpam-5930	186	23	that	that	SCONJ
ejpam-5930	186	24	θ	θ	PROPN
ejpam-5930	186	25	is	be	AUX
ejpam-5930	186	26	a	a	DET
ejpam-5930	186	27	congruence	congruence	NOUN
ejpam-5930	186	28	relation	relation	NOUN
ejpam-5930	186	29	on	on	ADP
ejpam-5930	186	30	a	a	PRON
ejpam-5930	186	31	and	and	CCONJ
ejpam-5930	186	32	let	let	VERB
ejpam-5930	186	33	c	c	PRON
ejpam-5930	186	34	be	be	AUX
ejpam-5930	186	35	any	any	DET
ejpam-5930	186	36	element	element	NOUN
ejpam-5930	186	37	of	of	ADP
ejpam-5930	186	38	θ(i	θ(i	PROPN
ejpam-5930	186	39	,	,	PUNCT
ejpam-5930	186	40	s	s	NOUN
ejpam-5930	186	41	)	)	PUNCT
ejpam-5930	186	42	·	·	PUNCT
ejpam-5930	187	1	θ(i	θ(i	VERB
ejpam-5930	187	2	,	,	PUNCT
ejpam-5930	187	3	t	t	PROPN
ejpam-5930	187	4	)	)	PUNCT
ejpam-5930	187	5	.	.	PUNCT
ejpam-5930	188	1	then	then	ADV
ejpam-5930	188	2	c	c	X
ejpam-5930	189	1	=	=	PUNCT
ejpam-5930	189	2	p	p	X
ejpam-5930	189	3	·	·	PUNCT
ejpam-5930	189	4	q	q	NOUN
ejpam-5930	189	5	with	with	ADP
ejpam-5930	189	6	p	p	PROPN
ejpam-5930	189	7	∈	∈	PROPN
ejpam-5930	189	8	θ(i	θ(i	PROPN
ejpam-5930	189	9	,	,	PUNCT
ejpam-5930	189	10	s	s	PART
ejpam-5930	189	11	)	)	PUNCT
ejpam-5930	189	12	and	and	CCONJ
ejpam-5930	189	13	q	q	PROPN
ejpam-5930	189	14	∈	∈	PROPN
ejpam-5930	189	15	θ(i	θ(i	PROPN
ejpam-5930	189	16	,	,	PUNCT
ejpam-5930	189	17	t	t	PROPN
ejpam-5930	189	18	)	)	PUNCT
ejpam-5930	189	19	.	.	PUNCT
ejpam-5930	190	1	it	it	PRON
ejpam-5930	190	2	follows	follow	VERB
ejpam-5930	190	3	that	that	SCONJ
ejpam-5930	190	4	ip	ip	VERB
ejpam-5930	190	5	⊆	⊆	NUM
ejpam-5930	190	6	s	s	NOUN
ejpam-5930	190	7	and	and	CCONJ
ejpam-5930	190	8	iq	iq	VERB
ejpam-5930	190	9	⊆	⊆	NUM
ejpam-5930	190	10	t	t	NOUN
ejpam-5930	190	11	.	.	PUNCT
ejpam-5930	191	1	since	since	SCONJ
ejpam-5930	191	2	θ	θ	PROPN
ejpam-5930	191	3	is	be	AUX
ejpam-5930	191	4	a	a	DET
ejpam-5930	191	5	congruence	congruence	NOUN
ejpam-5930	191	6	relation	relation	NOUN
ejpam-5930	191	7	on	on	ADP
ejpam-5930	191	8	a	a	PRON
ejpam-5930	191	9	,	,	PUNCT
ejpam-5930	191	10	we	we	PRON
ejpam-5930	191	11	have	have	AUX
ejpam-5930	191	12	ip·q	ip·q	PROPN
ejpam-5930	191	13	=	=	NOUN
ejpam-5930	191	14	ip	ip	NOUN
ejpam-5930	191	15	·	·	PUNCT
ejpam-5930	191	16	iq	iq	VERB
ejpam-5930	191	17	⊆	⊆	NUM
ejpam-5930	191	18	s	s	PART
ejpam-5930	191	19	·	·	PUNCT
ejpam-5930	191	20	t	t	PROPN
ejpam-5930	191	21	.	.	PUNCT
ejpam-5930	192	1	so	so	ADV
ejpam-5930	192	2	c	c	NOUN
ejpam-5930	193	1	=	=	PUNCT
ejpam-5930	193	2	p	p	X
ejpam-5930	193	3	·	·	PUNCT
ejpam-5930	193	4	q	q	PROPN
ejpam-5930	193	5	∈	∈	PROPN
ejpam-5930	193	6	θ(i	θ(i	PROPN
ejpam-5930	193	7	,	,	PUNCT
ejpam-5930	193	8	s	s	PART
ejpam-5930	193	9	·	·	PUNCT
ejpam-5930	193	10	t	t	PROPN
ejpam-5930	193	11	)	)	PUNCT
ejpam-5930	193	12	.	.	PUNCT
ejpam-5930	194	1	thus	thus	ADV
ejpam-5930	194	2	θ(i	θ(i	VERB
ejpam-5930	194	3	,	,	PUNCT
ejpam-5930	194	4	s	s	NOUN
ejpam-5930	194	5	)	)	PUNCT
ejpam-5930	194	6	·	·	PUNCT
ejpam-5930	194	7	θ(i	θ(i	PROPN
ejpam-5930	194	8	,	,	PUNCT
ejpam-5930	194	9	t	t	PROPN
ejpam-5930	194	10	)	)	PUNCT
ejpam-5930	194	11	⊆	⊆	X
ejpam-5930	194	12	θ(i	θ(i	PROPN
ejpam-5930	194	13	,	,	PUNCT
ejpam-5930	194	14	s	s	PART
ejpam-5930	194	15	·	·	PUNCT
ejpam-5930	194	16	t	t	PROPN
ejpam-5930	194	17	)	)	PUNCT
ejpam-5930	194	18	.	.	PUNCT
ejpam-5930	195	1	proposition	proposition	NOUN
ejpam-5930	195	2	4	4	NUM
ejpam-5930	195	3	.	.	PUNCT
ejpam-5930	196	1	let	let	VERB
ejpam-5930	196	2	θ	θ	PROPN
ejpam-5930	196	3	and	and	CCONJ
ejpam-5930	196	4	ψ	ψ	X
ejpam-5930	196	5	be	be	AUX
ejpam-5930	196	6	equivalence	equivalence	NOUN
ejpam-5930	196	7	relations	relation	NOUN
ejpam-5930	196	8	on	on	ADP
ejpam-5930	196	9	a	a	DET
ejpam-5930	196	10	related	relate	VERB
ejpam-5930	196	11	to	to	ADP
ejpam-5930	196	12	ideals	ideal	NOUN
ejpam-5930	197	1	i	i	PRON
ejpam-5930	197	2	and	and	CCONJ
ejpam-5930	197	3	j	j	PROPN
ejpam-5930	197	4	of	of	ADP
ejpam-5930	197	5	a	a	PRON
ejpam-5930	197	6	,	,	PUNCT
ejpam-5930	197	7	respectively	respectively	ADV
ejpam-5930	197	8	.	.	PUNCT
ejpam-5930	198	1	if	if	SCONJ
ejpam-5930	198	2	s	s	PRON
ejpam-5930	198	3	and	and	CCONJ
ejpam-5930	198	4	t	t	PROPN
ejpam-5930	198	5	are	be	AUX
ejpam-5930	198	6	nonempty	nonempty	ADJ
ejpam-5930	198	7	subsets	subset	NOUN
ejpam-5930	198	8	of	of	ADP
ejpam-5930	198	9	a	a	PRON
ejpam-5930	198	10	,	,	PUNCT
ejpam-5930	198	11	then	then	ADV
ejpam-5930	198	12	:	:	PUNCT
ejpam-5930	198	13	(	(	PUNCT
ejpam-5930	198	14	1	1	X
ejpam-5930	198	15	)	)	PUNCT
ejpam-5930	198	16	θ	θ	NOUN
ejpam-5930	198	17	∩ψ(i	∩ψ(i	ADJ
ejpam-5930	198	18	∩	∩	ADJ
ejpam-5930	198	19	j	j	PROPN
ejpam-5930	198	20	,	,	PUNCT
ejpam-5930	198	21	s	s	PART
ejpam-5930	198	22	)	)	PUNCT
ejpam-5930	198	23	⊆	⊆	NUM
ejpam-5930	198	24	θ(i	θ(i	PROPN
ejpam-5930	198	25	,	,	PUNCT
ejpam-5930	198	26	s	s	X
ejpam-5930	198	27	)	)	PUNCT
ejpam-5930	199	1	∩ψ(j	∩ψ(j	PROPN
ejpam-5930	199	2	,	,	PUNCT
ejpam-5930	199	3	s	s	NOUN
ejpam-5930	199	4	)	)	PUNCT
ejpam-5930	199	5	,	,	PUNCT
ejpam-5930	199	6	(	(	PUNCT
ejpam-5930	199	7	2	2	X
ejpam-5930	199	8	)	)	PUNCT
ejpam-5930	199	9	θ	θ	NOUN
ejpam-5930	199	10	∩ψ(i	∩ψ(i	ADJ
ejpam-5930	199	11	∩	∩	ADJ
ejpam-5930	199	12	j	j	PROPN
ejpam-5930	199	13	,	,	PUNCT
ejpam-5930	199	14	s	s	PART
ejpam-5930	199	15	)	)	PUNCT
ejpam-5930	199	16	⊇	⊇	PROPN
ejpam-5930	199	17	θ(i	θ(i	PROPN
ejpam-5930	199	18	,	,	PUNCT
ejpam-5930	199	19	s	s	PART
ejpam-5930	199	20	)	)	PUNCT
ejpam-5930	199	21	∩ψ(j	∩ψ(j	PROPN
ejpam-5930	199	22	,	,	PUNCT
ejpam-5930	199	23	s	s	NOUN
ejpam-5930	199	24	)	)	PUNCT
ejpam-5930	199	25	.	.	PUNCT
ejpam-5930	200	1	proof	proof	NOUN
ejpam-5930	200	2	.	.	PUNCT
ejpam-5930	201	1	(	(	PUNCT
ejpam-5930	201	2	1	1	X
ejpam-5930	201	3	)	)	PUNCT
ejpam-5930	201	4	note	note	NOUN
ejpam-5930	201	5	that	that	SCONJ
ejpam-5930	201	6	θ∩ψ	θ∩ψ	PROPN
ejpam-5930	201	7	is	be	AUX
ejpam-5930	201	8	also	also	ADV
ejpam-5930	201	9	a	a	DET
ejpam-5930	201	10	congruence	congruence	NOUN
ejpam-5930	201	11	relation	relation	NOUN
ejpam-5930	201	12	on	on	ADP
ejpam-5930	201	13	s.	s.	PROPN
ejpam-5930	201	14	let	let	VERB
ejpam-5930	201	15	c	c	NOUN
ejpam-5930	201	16	∈	∈	NOUN
ejpam-5930	201	17	θ	θ	X
ejpam-5930	201	18	∩ψ(i	∩ψ(i	PROPN
ejpam-5930	201	19	∩j	∩j	PROPN
ejpam-5930	201	20	,	,	PUNCT
ejpam-5930	201	21	s	s	NOUN
ejpam-5930	201	22	)	)	PUNCT
ejpam-5930	201	23	.	.	PUNCT
ejpam-5930	202	1	then	then	ADV
ejpam-5930	202	2	(	(	PUNCT
ejpam-5930	202	3	i	i	PRON
ejpam-5930	202	4	∩j)c∩s	∩j)c∩s	VERB
ejpam-5930	202	5	=	=	PUNCT
ejpam-5930	202	6	∅.	∅.	NOUN
ejpam-5930	202	7	then	then	ADV
ejpam-5930	202	8	there	there	PRON
ejpam-5930	202	9	exists	exist	VERB
ejpam-5930	202	10	an	an	DET
ejpam-5930	202	11	element	element	NOUN
ejpam-5930	202	12	x	x	X
ejpam-5930	202	13	∈	∈	PROPN
ejpam-5930	202	14	(	(	PUNCT
ejpam-5930	202	15	i	i	PRON
ejpam-5930	202	16	∩j)c∩s	∩j)c∩s	AUX
ejpam-5930	202	17	.	.	PUNCT
ejpam-5930	203	1	since	since	SCONJ
ejpam-5930	203	2	(	(	PUNCT
ejpam-5930	203	3	x	x	NOUN
ejpam-5930	203	4	,	,	PUNCT
ejpam-5930	203	5	c	c	NOUN
ejpam-5930	203	6	)	)	PUNCT
ejpam-5930	203	7	∈	∈	PROPN
ejpam-5930	203	8	θ∩ψ	θ∩ψ	PROPN
ejpam-5930	203	9	,	,	PUNCT
ejpam-5930	203	10	we	we	PRON
ejpam-5930	203	11	have	have	VERB
ejpam-5930	203	12	(	(	PUNCT
ejpam-5930	203	13	x	x	NOUN
ejpam-5930	203	14	,	,	PUNCT
ejpam-5930	203	15	c	c	NOUN
ejpam-5930	203	16	)	)	PUNCT
ejpam-5930	203	17	∈	∈	PROPN
ejpam-5930	203	18	θ	θ	PROPN
ejpam-5930	203	19	and	and	CCONJ
ejpam-5930	203	20	(	(	PUNCT
ejpam-5930	203	21	x	x	NOUN
ejpam-5930	203	22	,	,	PUNCT
ejpam-5930	203	23	c	c	NOUN
ejpam-5930	203	24	)	)	PUNCT
ejpam-5930	203	25	∈	∈	PROPN
ejpam-5930	203	26	ψ	ψ	NOUN
ejpam-5930	203	27	.	.	PUNCT
ejpam-5930	204	1	thus	thus	ADV
ejpam-5930	204	2	,	,	PUNCT
ejpam-5930	204	3	x	x	SYM
ejpam-5930	204	4	∈	∈	PROPN
ejpam-5930	204	5	ic	ic	X
ejpam-5930	204	6	and	and	CCONJ
ejpam-5930	204	7	x	x	PROPN
ejpam-5930	204	8	∈	∈	PROPN
ejpam-5930	204	9	jc	jc	PROPN
ejpam-5930	204	10	.	.	PUNCT
ejpam-5930	205	1	since	since	SCONJ
ejpam-5930	205	2	x	x	PROPN
ejpam-5930	205	3	∈	∈	PROPN
ejpam-5930	205	4	s	s	PART
ejpam-5930	205	5	,	,	PUNCT
ejpam-5930	205	6	we	we	PRON
ejpam-5930	205	7	have	have	VERB
ejpam-5930	205	8	x	x	PROPN
ejpam-5930	205	9	∈	∈	PROPN
ejpam-5930	205	10	ic	ic	PROPN
ejpam-5930	205	11	,	,	PUNCT
ejpam-5930	205	12	x	x	PROPN
ejpam-5930	205	13	∈	∈	NOUN
ejpam-5930	205	14	s	s	PART
ejpam-5930	205	15	and	and	CCONJ
ejpam-5930	205	16	x	x	PROPN
ejpam-5930	205	17	∈	∈	PROPN
ejpam-5930	205	18	jc	jc	PROPN
ejpam-5930	205	19	,	,	PUNCT
ejpam-5930	205	20	x	x	PROPN
ejpam-5930	205	21	∈	∈	PROPN
ejpam-5930	205	22	s.	s.	PROPN
ejpam-5930	205	23	this	this	PRON
ejpam-5930	205	24	implies	imply	VERB
ejpam-5930	205	25	that	that	SCONJ
ejpam-5930	205	26	x	x	SYM
ejpam-5930	205	27	∈	∈	PROPN
ejpam-5930	205	28	ic	ic	PROPN
ejpam-5930	205	29	∩s	∩s	PROPN
ejpam-5930	205	30	and	and	CCONJ
ejpam-5930	205	31	x	x	PROPN
ejpam-5930	205	32	∈	∈	PROPN
ejpam-5930	205	33	jc	jc	PROPN
ejpam-5930	205	34	∩s	∩s	PROPN
ejpam-5930	205	35	,	,	PUNCT
ejpam-5930	205	36	so	so	ADV
ejpam-5930	206	1	ic	ic	PROPN
ejpam-5930	206	2	∩s	∩s	PROPN
ejpam-5930	206	3	=	=	PROPN
ejpam-5930	206	4	∅	∅	NOUN
ejpam-5930	206	5	and	and	CCONJ
ejpam-5930	206	6	jc	jc	PROPN
ejpam-5930	206	7	∩	∩	PROPN
ejpam-5930	206	8	s	s	PART
ejpam-5930	206	9	=	=	X
ejpam-5930	206	10	∅.	∅.	VERB
ejpam-5930	206	11	so	so	SCONJ
ejpam-5930	206	12	c	c	PROPN
ejpam-5930	206	13	∈	∈	PROPN
ejpam-5930	206	14	θ(i	θ(i	PROPN
ejpam-5930	206	15	,	,	PUNCT
ejpam-5930	206	16	s	s	PART
ejpam-5930	206	17	)	)	PUNCT
ejpam-5930	206	18	and	and	CCONJ
ejpam-5930	206	19	c	c	PROPN
ejpam-5930	206	20	∈	∈	PROPN
ejpam-5930	206	21	ψ(j	ψ(j	PROPN
ejpam-5930	206	22	,	,	PUNCT
ejpam-5930	206	23	s	s	PART
ejpam-5930	206	24	)	)	PUNCT
ejpam-5930	206	25	,	,	PUNCT
ejpam-5930	206	26	hence	hence	ADV
ejpam-5930	206	27	c	c	PROPN
ejpam-5930	206	28	∈	∈	PROPN
ejpam-5930	206	29	θ(i	θ(i	PROPN
ejpam-5930	206	30	,	,	PUNCT
ejpam-5930	206	31	s	s	NOUN
ejpam-5930	206	32	)	)	PUNCT
ejpam-5930	206	33	∩	∩	PROPN
ejpam-5930	206	34	ψ(j	ψ(j	PROPN
ejpam-5930	206	35	,	,	PUNCT
ejpam-5930	206	36	s	s	NOUN
ejpam-5930	206	37	)	)	PUNCT
ejpam-5930	206	38	.	.	PUNCT
ejpam-5930	207	1	thus	thus	ADV
ejpam-5930	207	2	,	,	PUNCT
ejpam-5930	207	3	θ	θ	PROPN
ejpam-5930	207	4	∩ψ(i	∩ψ(i	PROPN
ejpam-5930	207	5	∩	∩	PROPN
ejpam-5930	207	6	j	j	PROPN
ejpam-5930	207	7	,	,	PUNCT
ejpam-5930	207	8	s	s	PART
ejpam-5930	207	9	)	)	PUNCT
ejpam-5930	207	10	⊆	⊆	NUM
ejpam-5930	207	11	θ(i	θ(i	PROPN
ejpam-5930	207	12	,	,	PUNCT
ejpam-5930	207	13	s	s	X
ejpam-5930	207	14	)	)	PUNCT
ejpam-5930	207	15	∩ψ(j	∩ψ(j	PROPN
ejpam-5930	207	16	,	,	PUNCT
ejpam-5930	207	17	s	s	NOUN
ejpam-5930	207	18	)	)	PUNCT
ejpam-5930	207	19	.	.	PUNCT
ejpam-5930	208	1	(	(	PUNCT
ejpam-5930	208	2	2	2	X
ejpam-5930	208	3	)	)	PUNCT
ejpam-5930	208	4	since	since	SCONJ
ejpam-5930	208	5	θ∩ψ	θ∩ψ	PROPN
ejpam-5930	208	6	⊆	⊆	NUM
ejpam-5930	208	7	θ	θ	NOUN
ejpam-5930	208	8	and	and	CCONJ
ejpam-5930	208	9	θ∩ψ	θ∩ψ	PROPN
ejpam-5930	208	10	⊆	⊆	NUM
ejpam-5930	208	11	ψ	ψ	NOUN
ejpam-5930	208	12	,	,	PUNCT
ejpam-5930	208	13	we	we	PRON
ejpam-5930	208	14	have	have	AUX
ejpam-5930	208	15	θ(i	θ(i	VERB
ejpam-5930	208	16	,	,	PUNCT
ejpam-5930	208	17	s	s	NOUN
ejpam-5930	208	18	)	)	PUNCT
ejpam-5930	208	19	⊆	⊆	NUM
ejpam-5930	208	20	θ	θ	NOUN
ejpam-5930	208	21	∩ψ(i	∩ψ(i	PROPN
ejpam-5930	208	22	∩j	∩j	PROPN
ejpam-5930	208	23	,	,	PUNCT
ejpam-5930	208	24	s	s	NOUN
ejpam-5930	208	25	)	)	PUNCT
ejpam-5930	208	26	and	and	CCONJ
ejpam-5930	208	27	ψ(j	ψ(j	PROPN
ejpam-5930	208	28	,	,	PUNCT
ejpam-5930	208	29	s	s	X
ejpam-5930	208	30	)	)	PUNCT
ejpam-5930	208	31	⊆	⊆	NUM
ejpam-5930	208	32	θ	θ	NOUN
ejpam-5930	208	33	∩ψ(i	∩ψ(i	PROPN
ejpam-5930	208	34	∩	∩	PROPN
ejpam-5930	208	35	j	j	PROPN
ejpam-5930	208	36	,	,	PUNCT
ejpam-5930	208	37	s	s	PART
ejpam-5930	208	38	)	)	PUNCT
ejpam-5930	208	39	.	.	PUNCT
ejpam-5930	209	1	hence	hence	ADV
ejpam-5930	209	2	,	,	PUNCT
ejpam-5930	209	3	θ(i	θ(i	PROPN
ejpam-5930	209	4	,	,	PUNCT
ejpam-5930	209	5	s	s	X
ejpam-5930	209	6	)	)	PUNCT
ejpam-5930	209	7	∩ψ(j	∩ψ(j	ADJ
ejpam-5930	209	8	,	,	PUNCT
ejpam-5930	209	9	s	s	NOUN
ejpam-5930	209	10	)	)	PUNCT
ejpam-5930	209	11	⊆	⊆	NUM
ejpam-5930	209	12	θ	θ	NOUN
ejpam-5930	209	13	∩ψ(i	∩ψ(i	PROPN
ejpam-5930	209	14	∩	∩	PROPN
ejpam-5930	209	15	j	j	PROPN
ejpam-5930	209	16	,	,	PUNCT
ejpam-5930	209	17	s	s	PART
ejpam-5930	209	18	)	)	PUNCT
ejpam-5930	209	19	.	.	PUNCT
ejpam-5930	210	1	theorem	theorem	NOUN
ejpam-5930	210	2	1	1	NUM
ejpam-5930	210	3	.	.	PUNCT
ejpam-5930	211	1	let	let	AUX
ejpam-5930	211	2	(	(	PUNCT
ejpam-5930	211	3	a	a	DET
ejpam-5930	211	4	,	,	PUNCT
ejpam-5930	211	5	θ	θ	NOUN
ejpam-5930	211	6	)	)	PUNCT
ejpam-5930	211	7	be	be	VERB
ejpam-5930	211	8	an	an	DET
ejpam-5930	211	9	approximation	approximation	NOUN
ejpam-5930	211	10	space	space	NOUN
ejpam-5930	211	11	.	.	PUNCT
ejpam-5930	212	1	then	then	ADV
ejpam-5930	212	2	(	(	PUNCT
ejpam-5930	212	3	1	1	X
ejpam-5930	212	4	)	)	PUNCT
ejpam-5930	212	5	for	for	ADP
ejpam-5930	212	6	every	every	PRON
ejpam-5930	212	7	s	s	PROPN
ejpam-5930	212	8	⊆	⊆	NUM
ejpam-5930	212	9	a	a	PRON
ejpam-5930	212	10	,	,	PUNCT
ejpam-5930	212	11	θ(i	θ(i	PROPN
ejpam-5930	212	12	,	,	PUNCT
ejpam-5930	212	13	s	s	PART
ejpam-5930	212	14	)	)	PUNCT
ejpam-5930	212	15	and	and	CCONJ
ejpam-5930	212	16	θ(i	θ(i	PROPN
ejpam-5930	212	17	,	,	PUNCT
ejpam-5930	212	18	s	s	PART
ejpam-5930	212	19	)	)	PUNCT
ejpam-5930	212	20	are	be	AUX
ejpam-5930	212	21	definable	definable	ADJ
ejpam-5930	212	22	sets	set	NOUN
ejpam-5930	212	23	,	,	PUNCT
ejpam-5930	212	24	(	(	PUNCT
ejpam-5930	212	25	2	2	X
ejpam-5930	212	26	)	)	PUNCT
ejpam-5930	212	27	for	for	ADP
ejpam-5930	212	28	every	every	DET
ejpam-5930	212	29	x	x	PROPN
ejpam-5930	212	30	∈	∈	PROPN
ejpam-5930	212	31	a	a	PRON
ejpam-5930	212	32	,	,	PUNCT
ejpam-5930	212	33	ix	ix	ADV
ejpam-5930	212	34	is	be	AUX
ejpam-5930	212	35	a	a	DET
ejpam-5930	212	36	definable	definable	ADJ
ejpam-5930	212	37	set	set	NOUN
ejpam-5930	212	38	.	.	PUNCT
ejpam-5930	213	1	proof	proof	NOUN
ejpam-5930	213	2	.	.	PUNCT
ejpam-5930	214	1	(	(	PUNCT
ejpam-5930	214	2	1	1	X
ejpam-5930	214	3	)	)	PUNCT
ejpam-5930	214	4	by	by	ADP
ejpam-5930	214	5	proposition	proposition	NOUN
ejpam-5930	214	6	1	1	NUM
ejpam-5930	214	7	(	(	PUNCT
ejpam-5930	214	8	1	1	NUM
ejpam-5930	214	9	)	)	PUNCT
ejpam-5930	214	10	and	and	CCONJ
ejpam-5930	214	11	(	(	PUNCT
ejpam-5930	214	12	3	3	NUM
ejpam-5930	214	13	)	)	PUNCT
ejpam-5930	214	14	,	,	PUNCT
ejpam-5930	214	15	we	we	PRON
ejpam-5930	214	16	have	have	VERB
ejpam-5930	214	17	θ(i	θ(i	VERB
ejpam-5930	214	18	,	,	PUNCT
ejpam-5930	214	19	θ(i	θ(i	PROPN
ejpam-5930	214	20	,	,	PUNCT
ejpam-5930	214	21	s	s	NOUN
ejpam-5930	214	22	)	)	PUNCT
ejpam-5930	214	23	)	)	PUNCT
ejpam-5930	215	1	=	=	SYM
ejpam-5930	215	2	θ(i	θ(i	PROPN
ejpam-5930	215	3	,	,	PUNCT
ejpam-5930	215	4	s	s	NOUN
ejpam-5930	215	5	)	)	PUNCT
ejpam-5930	215	6	=	=	SYM
ejpam-5930	215	7	θ(i	θ(i	PROPN
ejpam-5930	215	8	,	,	PUNCT
ejpam-5930	215	9	θ(i	θ(i	PROPN
ejpam-5930	215	10	,	,	PUNCT
ejpam-5930	215	11	s	s	NOUN
ejpam-5930	215	12	)	)	PUNCT
ejpam-5930	215	13	)	)	PUNCT
ejpam-5930	215	14	.	.	PUNCT
ejpam-5930	216	1	hence	hence	ADV
ejpam-5930	216	2	,	,	PUNCT
ejpam-5930	216	3	θ(i	θ(i	PROPN
ejpam-5930	216	4	,	,	PUNCT
ejpam-5930	216	5	s	s	PART
ejpam-5930	216	6	)	)	PUNCT
ejpam-5930	216	7	is	be	AUX
ejpam-5930	216	8	definable	definable	ADJ
ejpam-5930	216	9	.	.	PUNCT
ejpam-5930	217	1	on	on	ADP
ejpam-5930	217	2	the	the	DET
ejpam-5930	217	3	other	other	ADJ
ejpam-5930	217	4	hand	hand	NOUN
ejpam-5930	217	5	,	,	PUNCT
ejpam-5930	217	6	by	by	ADP
ejpam-5930	217	7	proposition	proposition	NOUN
ejpam-5930	217	8	1	1	NUM
ejpam-5930	217	9	(	(	PUNCT
ejpam-5930	217	10	2	2	NUM
ejpam-5930	217	11	)	)	PUNCT
ejpam-5930	217	12	and	and	CCONJ
ejpam-5930	217	13	(	(	PUNCT
ejpam-5930	217	14	4	4	NUM
ejpam-5930	217	15	)	)	PUNCT
ejpam-5930	217	16	,	,	PUNCT
ejpam-5930	217	17	we	we	PRON
ejpam-5930	217	18	have	have	VERB
ejpam-5930	217	19	θ(i	θ(i	VERB
ejpam-5930	217	20	,	,	PUNCT
ejpam-5930	217	21	θ(i	θ(i	PROPN
ejpam-5930	217	22	,	,	PUNCT
ejpam-5930	217	23	s	s	NOUN
ejpam-5930	217	24	)	)	PUNCT
ejpam-5930	217	25	)	)	PUNCT
ejpam-5930	218	1	=	=	SYM
ejpam-5930	218	2	θ(i	θ(i	PROPN
ejpam-5930	218	3	,	,	PUNCT
ejpam-5930	218	4	s	s	NOUN
ejpam-5930	218	5	)	)	PUNCT
ejpam-5930	218	6	=	=	SYM
ejpam-5930	218	7	θ(i	θ(i	PROPN
ejpam-5930	218	8	,	,	PUNCT
ejpam-5930	218	9	θ(i	θ(i	PROPN
ejpam-5930	218	10	,	,	PUNCT
ejpam-5930	218	11	s	s	NOUN
ejpam-5930	218	12	)	)	PUNCT
ejpam-5930	218	13	)	)	PUNCT
ejpam-5930	218	14	.	.	PUNCT
ejpam-5930	219	1	therefore	therefore	ADV
ejpam-5930	219	2	,	,	PUNCT
ejpam-5930	219	3	θ(i	θ(i	PROPN
ejpam-5930	219	4	,	,	PUNCT
ejpam-5930	219	5	s	s	PART
ejpam-5930	219	6	)	)	PUNCT
ejpam-5930	219	7	is	be	AUX
ejpam-5930	219	8	a	a	DET
ejpam-5930	219	9	definable	definable	ADJ
ejpam-5930	219	10	set	set	NOUN
ejpam-5930	219	11	.	.	PUNCT
ejpam-5930	220	1	(	(	PUNCT
ejpam-5930	220	2	2	2	X
ejpam-5930	220	3	)	)	PUNCT
ejpam-5930	220	4	by	by	ADP
ejpam-5930	220	5	proposition	proposition	NOUN
ejpam-5930	220	6	1	1	NUM
ejpam-5930	220	7	(	(	PUNCT
ejpam-5930	220	8	7	7	NUM
ejpam-5930	220	9	)	)	PUNCT
ejpam-5930	220	10	,	,	PUNCT
ejpam-5930	220	11	the	the	DET
ejpam-5930	220	12	proof	proof	NOUN
ejpam-5930	220	13	is	be	AUX
ejpam-5930	220	14	clear	clear	ADJ
ejpam-5930	220	15	.	.	PUNCT
ejpam-5930	221	1	definition	definition	NOUN
ejpam-5930	221	2	8	8	NUM
ejpam-5930	221	3	.	.	PUNCT
ejpam-5930	222	1	a	a	DET
ejpam-5930	222	2	nonempty	nonempty	ADJ
ejpam-5930	222	3	subset	subset	VERB
ejpam-5930	222	4	s	s	NOUN
ejpam-5930	222	5	of	of	ADP
ejpam-5930	222	6	a	a	PRON
ejpam-5930	222	7	is	be	AUX
ejpam-5930	222	8	called	call	VERB
ejpam-5930	222	9	an	an	DET
ejpam-5930	222	10	upper	upper	ADJ
ejpam-5930	222	11	(	(	PUNCT
ejpam-5930	222	12	resp	resp	NOUN
ejpam-5930	222	13	.	.	PROPN
ejpam-5930	223	1	,	,	PUNCT
ejpam-5930	223	2	a	a	DET
ejpam-5930	223	3	lower	low	ADJ
ejpam-5930	223	4	)	)	PUNCT
ejpam-5930	223	5	rough	rough	ADJ
ejpam-5930	223	6	subalgebra	subalgebra	NOUN
ejpam-5930	223	7	of	of	ADP
ejpam-5930	223	8	a	a	PRON
ejpam-5930	223	9	if	if	SCONJ
ejpam-5930	223	10	the	the	DET
ejpam-5930	223	11	upper	upper	ADJ
ejpam-5930	223	12	(	(	PUNCT
ejpam-5930	223	13	resp	resp	NOUN
ejpam-5930	223	14	.	.	PUNCT
ejpam-5930	223	15	,	,	PUNCT
ejpam-5930	223	16	nonempty	nonempty	VERB
ejpam-5930	223	17	lower	low	ADJ
ejpam-5930	223	18	)	)	PUNCT
ejpam-5930	223	19	approximation	approximation	NOUN
ejpam-5930	223	20	of	of	ADP
ejpam-5930	223	21	s	s	PROPN
ejpam-5930	223	22	is	be	AUX
ejpam-5930	223	23	a	a	DET
ejpam-5930	223	24	subalgebra	subalgebra	NOUN
ejpam-5930	223	25	of	of	ADP
ejpam-5930	223	26	a.	a.	NOUN
ejpam-5930	223	27	if	if	SCONJ
ejpam-5930	223	28	s	s	VERB
ejpam-5930	223	29	is	be	AUX
ejpam-5930	223	30	both	both	PRON
ejpam-5930	223	31	an	an	DET
ejpam-5930	223	32	upper	upper	NOUN
ejpam-5930	223	33	and	and	CCONJ
ejpam-5930	223	34	a	a	DET
ejpam-5930	223	35	lower	low	ADJ
ejpam-5930	223	36	rough	rough	ADJ
ejpam-5930	223	37	subalgebra	subalgebra	NOUN
ejpam-5930	223	38	of	of	ADP
ejpam-5930	223	39	a	a	PRON
ejpam-5930	223	40	,	,	PUNCT
ejpam-5930	223	41	we	we	PRON
ejpam-5930	223	42	say	say	VERB
ejpam-5930	223	43	that	that	SCONJ
ejpam-5930	223	44	s	s	VERB
ejpam-5930	223	45	is	be	AUX
ejpam-5930	223	46	a	a	DET
ejpam-5930	223	47	rough	rough	ADJ
ejpam-5930	223	48	subalgebra	subalgebra	NOUN
ejpam-5930	223	49	of	of	ADP
ejpam-5930	223	50	a.	a.	NOUN
ejpam-5930	223	51	theorem	theorem	NOUN
ejpam-5930	223	52	2	2	X
ejpam-5930	223	53	.	.	PUNCT
ejpam-5930	224	1	let	let	VERB
ejpam-5930	224	2	θ	θ	NOUN
ejpam-5930	224	3	be	be	AUX
ejpam-5930	224	4	a	a	DET
ejpam-5930	224	5	congruence	congruence	NOUN
ejpam-5930	224	6	relation	relation	NOUN
ejpam-5930	224	7	on	on	ADP
ejpam-5930	224	8	a	a	DET
ejpam-5930	224	9	related	relate	VERB
ejpam-5930	224	10	to	to	ADP
ejpam-5930	224	11	an	an	DET
ejpam-5930	224	12	ideal	ideal	NOUN
ejpam-5930	224	13	i	i	PRON
ejpam-5930	224	14	of	of	ADP
ejpam-5930	224	15	a.	a.	NOUN
ejpam-5930	224	16	if	if	SCONJ
ejpam-5930	224	17	s	s	X
ejpam-5930	224	18	is	be	AUX
ejpam-5930	224	19	a	a	DET
ejpam-5930	224	20	subalgebra	subalgebra	NOUN
ejpam-5930	224	21	of	of	ADP
ejpam-5930	224	22	i	i	PRON
ejpam-5930	224	23	,	,	PUNCT
ejpam-5930	224	24	then	then	ADV
ejpam-5930	224	25	(	(	PUNCT
ejpam-5930	224	26	1	1	X
ejpam-5930	224	27	)	)	PUNCT
ejpam-5930	224	28	θ(i	θ(i	PROPN
ejpam-5930	224	29	,	,	PUNCT
ejpam-5930	224	30	s	s	PART
ejpam-5930	224	31	)	)	PUNCT
ejpam-5930	224	32	is	be	AUX
ejpam-5930	224	33	a	a	DET
ejpam-5930	224	34	subalgebra	subalgebra	NOUN
ejpam-5930	224	35	of	of	ADP
ejpam-5930	224	36	a	a	DET
ejpam-5930	224	37	,	,	PUNCT
ejpam-5930	224	38	(	(	PUNCT
ejpam-5930	224	39	2	2	NUM
ejpam-5930	224	40	)	)	PUNCT
ejpam-5930	224	41	θ(i	θ(i	PROPN
ejpam-5930	224	42	,	,	PUNCT
ejpam-5930	224	43	s	s	PART
ejpam-5930	224	44	)	)	PUNCT
ejpam-5930	224	45	is	be	AUX
ejpam-5930	224	46	a	a	DET
ejpam-5930	224	47	subalgebra	subalgebra	NOUN
ejpam-5930	224	48	of	of	ADP
ejpam-5930	224	49	a.	a.	NOUN
ejpam-5930	224	50	a.	a.	PROPN
ejpam-5930	224	51	iampan	iampan	PROPN
ejpam-5930	224	52	et	et	PROPN
ejpam-5930	224	53	al	al	PROPN
ejpam-5930	224	54	.	.	PUNCT
ejpam-5930	224	55	/	/	SYM
ejpam-5930	224	56	eur	eur	PROPN
ejpam-5930	224	57	.	.	PUNCT
ejpam-5930	225	1	j.	j.	PROPN
ejpam-5930	225	2	pure	pure	PROPN
ejpam-5930	225	3	appl	appl	PROPN
ejpam-5930	225	4	.	.	PROPN
ejpam-5930	225	5	math	math	PROPN
ejpam-5930	225	6	,	,	PUNCT
ejpam-5930	225	7	18	18	NUM
ejpam-5930	225	8	(	(	PUNCT
ejpam-5930	225	9	2	2	NUM
ejpam-5930	225	10	)	)	PUNCT
ejpam-5930	225	11	(	(	PUNCT
ejpam-5930	225	12	2025	2025	NUM
ejpam-5930	225	13	)	)	PUNCT
ejpam-5930	225	14	,	,	PUNCT
ejpam-5930	225	15	5930	5930	NUM
ejpam-5930	225	16	9	9	NUM
ejpam-5930	225	17	of	of	ADP
ejpam-5930	225	18	11	11	NUM
ejpam-5930	225	19	proof	proof	NOUN
ejpam-5930	225	20	.	.	PUNCT
ejpam-5930	226	1	(	(	PUNCT
ejpam-5930	226	2	1	1	X
ejpam-5930	226	3	)	)	PUNCT
ejpam-5930	226	4	let	let	VERB
ejpam-5930	226	5	x	x	PRON
ejpam-5930	226	6	,	,	PUNCT
ejpam-5930	226	7	y	y	PROPN
ejpam-5930	226	8	∈	∈	PROPN
ejpam-5930	226	9	θ(i	θ(i	PROPN
ejpam-5930	226	10	,	,	PUNCT
ejpam-5930	226	11	s	s	NOUN
ejpam-5930	226	12	)	)	PUNCT
ejpam-5930	226	13	.	.	PUNCT
ejpam-5930	227	1	then	then	ADV
ejpam-5930	227	2	ix	ix	PROPN
ejpam-5930	227	3	∩	∩	PROPN
ejpam-5930	227	4	s	s	PART
ejpam-5930	227	5	̸=	̸=	PROPN
ejpam-5930	227	6	∅	∅	NOUN
ejpam-5930	227	7	and	and	CCONJ
ejpam-5930	227	8	iy	iy	PROPN
ejpam-5930	227	9	∩	∩	PROPN
ejpam-5930	227	10	s	s	PART
ejpam-5930	227	11	̸=	̸=	PROPN
ejpam-5930	227	12	∅	∅	NOUN
ejpam-5930	227	13	,	,	PUNCT
ejpam-5930	227	14	and	and	CCONJ
ejpam-5930	227	15	so	so	ADV
ejpam-5930	227	16	there	there	PRON
ejpam-5930	227	17	exist	exist	VERB
ejpam-5930	227	18	a	a	DET
ejpam-5930	227	19	,	,	PUNCT
ejpam-5930	227	20	b	b	X
ejpam-5930	227	21	∈	∈	NOUN
ejpam-5930	227	22	s	s	VERB
ejpam-5930	227	23	such	such	ADJ
ejpam-5930	227	24	that	that	SCONJ
ejpam-5930	227	25	a	a	DET
ejpam-5930	227	26	∈	∈	NOUN
ejpam-5930	227	27	ix	ix	ADV
ejpam-5930	227	28	and	and	CCONJ
ejpam-5930	227	29	b	b	X
ejpam-5930	227	30	∈	∈	PROPN
ejpam-5930	227	31	iy	iy	INTJ
ejpam-5930	227	32	.	.	PUNCT
ejpam-5930	228	1	it	it	PRON
ejpam-5930	228	2	follows	follow	VERB
ejpam-5930	228	3	that	that	SCONJ
ejpam-5930	228	4	(	(	PUNCT
ejpam-5930	228	5	a	a	PRON
ejpam-5930	228	6	,	,	PUNCT
ejpam-5930	228	7	x	x	NOUN
ejpam-5930	228	8	)	)	PUNCT
ejpam-5930	228	9	∈	∈	PROPN
ejpam-5930	228	10	θ	θ	PROPN
ejpam-5930	228	11	and	and	CCONJ
ejpam-5930	228	12	(	(	PUNCT
ejpam-5930	228	13	b	b	PROPN
ejpam-5930	228	14	,	,	PUNCT
ejpam-5930	228	15	y	y	NOUN
ejpam-5930	228	16	)	)	PUNCT
ejpam-5930	228	17	∈	∈	PROPN
ejpam-5930	228	18	θ	θ	PROPN
ejpam-5930	228	19	.	.	PUNCT
ejpam-5930	229	1	since	since	SCONJ
ejpam-5930	229	2	θ	θ	PROPN
ejpam-5930	229	3	is	be	AUX
ejpam-5930	229	4	a	a	DET
ejpam-5930	229	5	congruence	congruence	NOUN
ejpam-5930	229	6	relation	relation	NOUN
ejpam-5930	229	7	on	on	ADP
ejpam-5930	229	8	a	a	PRON
ejpam-5930	229	9	,	,	PUNCT
ejpam-5930	229	10	we	we	PRON
ejpam-5930	229	11	have	have	VERB
ejpam-5930	229	12	(	(	PUNCT
ejpam-5930	229	13	a	a	DET
ejpam-5930	229	14	·	·	SYM
ejpam-5930	229	15	b	b	NOUN
ejpam-5930	229	16	,	,	PUNCT
ejpam-5930	229	17	x	x	PROPN
ejpam-5930	229	18	·	·	PUNCT
ejpam-5930	229	19	y	y	X
ejpam-5930	229	20	)	)	PUNCT
ejpam-5930	229	21	∈	∈	PROPN
ejpam-5930	229	22	θ	θ	PROPN
ejpam-5930	229	23	.	.	PUNCT
ejpam-5930	230	1	hence	hence	ADV
ejpam-5930	230	2	,	,	PUNCT
ejpam-5930	230	3	a	a	DET
ejpam-5930	230	4	·	·	PUNCT
ejpam-5930	230	5	b	b	X
ejpam-5930	230	6	∈	∈	PROPN
ejpam-5930	230	7	ix·y	ix·y	PROPN
ejpam-5930	230	8	.	.	PUNCT
ejpam-5930	231	1	since	since	SCONJ
ejpam-5930	231	2	s	s	PROPN
ejpam-5930	231	3	is	be	AUX
ejpam-5930	231	4	a	a	DET
ejpam-5930	231	5	subalgebra	subalgebra	NOUN
ejpam-5930	231	6	of	of	ADP
ejpam-5930	231	7	a	a	PRON
ejpam-5930	231	8	,	,	PUNCT
ejpam-5930	231	9	we	we	PRON
ejpam-5930	231	10	get	get	VERB
ejpam-5930	231	11	a	a	DET
ejpam-5930	231	12	·	·	PUNCT
ejpam-5930	231	13	b	b	X
ejpam-5930	231	14	∈	∈	ADJ
ejpam-5930	231	15	s	s	NOUN
ejpam-5930	231	16	,	,	PUNCT
ejpam-5930	231	17	and	and	CCONJ
ejpam-5930	231	18	therefore	therefore	ADV
ejpam-5930	231	19	,	,	PUNCT
ejpam-5930	231	20	a	a	DET
ejpam-5930	231	21	·	·	SYM
ejpam-5930	231	22	b	b	X
ejpam-5930	231	23	∈	∈	PROPN
ejpam-5930	231	24	ix·y	ix·y	PROPN
ejpam-5930	231	25	∩	∩	NOUN
ejpam-5930	231	26	s	s	PART
ejpam-5930	231	27	,	,	PUNCT
ejpam-5930	231	28	that	that	ADV
ejpam-5930	231	29	is	is	ADV
ejpam-5930	231	30	,	,	PUNCT
ejpam-5930	231	31	ix·y	ix·y	PROPN
ejpam-5930	231	32	∩	∩	NOUN
ejpam-5930	231	33	s	s	PART
ejpam-5930	231	34	̸=	̸=	PROPN
ejpam-5930	231	35	∅.	∅.	ADP
ejpam-5930	231	36	this	this	PRON
ejpam-5930	231	37	shows	show	VERB
ejpam-5930	231	38	that	that	SCONJ
ejpam-5930	231	39	x	x	X
ejpam-5930	231	40	·	·	PUNCT
ejpam-5930	231	41	y	y	PROPN
ejpam-5930	231	42	∈	∈	PROPN
ejpam-5930	231	43	θ(i	θ(i	PROPN
ejpam-5930	231	44	,	,	PUNCT
ejpam-5930	231	45	s	s	NOUN
ejpam-5930	231	46	)	)	PUNCT
ejpam-5930	231	47	,	,	PUNCT
ejpam-5930	231	48	and	and	CCONJ
ejpam-5930	231	49	consequently	consequently	ADV
ejpam-5930	231	50	θ(i	θ(i	VERB
ejpam-5930	231	51	,	,	PUNCT
ejpam-5930	231	52	s	s	PART
ejpam-5930	231	53	)	)	PUNCT
ejpam-5930	231	54	is	be	AUX
ejpam-5930	231	55	a	a	DET
ejpam-5930	231	56	subalgebra	subalgebra	NOUN
ejpam-5930	231	57	of	of	ADP
ejpam-5930	231	58	a.	a.	NOUN
ejpam-5930	231	59	(	(	PUNCT
ejpam-5930	231	60	2	2	X
ejpam-5930	231	61	)	)	PUNCT
ejpam-5930	231	62	let	let	VERB
ejpam-5930	231	63	x	x	PRON
ejpam-5930	231	64	,	,	PUNCT
ejpam-5930	231	65	y	y	PROPN
ejpam-5930	231	66	∈	∈	PROPN
ejpam-5930	231	67	θ(i	θ(i	PROPN
ejpam-5930	231	68	,	,	PUNCT
ejpam-5930	231	69	s	s	NOUN
ejpam-5930	231	70	)	)	PUNCT
ejpam-5930	231	71	.	.	PUNCT
ejpam-5930	232	1	then	then	ADV
ejpam-5930	232	2	ix	ix	ADP
ejpam-5930	232	3	⊆	⊆	NUM
ejpam-5930	232	4	s	s	NOUN
ejpam-5930	232	5	and	and	CCONJ
ejpam-5930	232	6	iy	iy	PROPN
ejpam-5930	232	7	⊆	⊆	NUM
ejpam-5930	232	8	s.	s.	PROPN
ejpam-5930	232	9	since	since	SCONJ
ejpam-5930	232	10	s	s	PROPN
ejpam-5930	232	11	is	be	AUX
ejpam-5930	232	12	a	a	DET
ejpam-5930	232	13	subalgebra	subalgebra	NOUN
ejpam-5930	232	14	of	of	ADP
ejpam-5930	232	15	a	a	PRON
ejpam-5930	232	16	,	,	PUNCT
ejpam-5930	232	17	we	we	PRON
ejpam-5930	232	18	have	have	VERB
ejpam-5930	232	19	ix·y	ix·y	PROPN
ejpam-5930	232	20	=	=	SYM
ejpam-5930	232	21	ix	ix	PROPN
ejpam-5930	232	22	·	·	PUNCT
ejpam-5930	232	23	iy	iy	PROPN
ejpam-5930	233	1	⊆	⊆	NUM
ejpam-5930	233	2	s	s	VERB
ejpam-5930	233	3	so	so	SCONJ
ejpam-5930	233	4	that	that	SCONJ
ejpam-5930	233	5	x	x	X
ejpam-5930	233	6	·	·	PUNCT
ejpam-5930	233	7	y	y	PROPN
ejpam-5930	233	8	∈	∈	PROPN
ejpam-5930	233	9	θ(i	θ(i	PROPN
ejpam-5930	233	10	,	,	PUNCT
ejpam-5930	233	11	s	s	NOUN
ejpam-5930	233	12	)	)	PUNCT
ejpam-5930	233	13	.	.	PUNCT
ejpam-5930	234	1	hence	hence	ADV
ejpam-5930	234	2	,	,	PUNCT
ejpam-5930	234	3	θ(i	θ(i	PROPN
ejpam-5930	234	4	,	,	PUNCT
ejpam-5930	234	5	s	s	PART
ejpam-5930	234	6	)	)	PUNCT
ejpam-5930	234	7	is	be	AUX
ejpam-5930	234	8	a	a	DET
ejpam-5930	234	9	subalgebra	subalgebra	NOUN
ejpam-5930	234	10	of	of	ADP
ejpam-5930	234	11	a.	a.	NOUN
ejpam-5930	234	12	remark	remark	PROPN
ejpam-5930	234	13	1	1	NUM
ejpam-5930	234	14	.	.	PUNCT
ejpam-5930	235	1	the	the	DET
ejpam-5930	235	2	converse	converse	NOUN
ejpam-5930	235	3	of	of	ADP
ejpam-5930	235	4	theorem	theorem	ADJ
ejpam-5930	235	5	2	2	NUM
ejpam-5930	235	6	(	(	PUNCT
ejpam-5930	235	7	1	1	NUM
ejpam-5930	235	8	)	)	PUNCT
ejpam-5930	235	9	may	may	AUX
ejpam-5930	235	10	not	not	PART
ejpam-5930	235	11	be	be	AUX
ejpam-5930	235	12	true	true	ADJ
ejpam-5930	235	13	.	.	PUNCT
ejpam-5930	236	1	for	for	ADP
ejpam-5930	236	2	the	the	DET
ejpam-5930	236	3	hilbert	hilbert	PROPN
ejpam-5930	236	4	algebra	algebra	NOUN
ejpam-5930	236	5	a	a	DET
ejpam-5930	236	6	=	=	X
ejpam-5930	236	7	(	(	PUNCT
ejpam-5930	236	8	a	a	PRON
ejpam-5930	236	9	,	,	PUNCT
ejpam-5930	236	10	·	·	PUNCT
ejpam-5930	236	11	,	,	PUNCT
ejpam-5930	236	12	1	1	NUM
ejpam-5930	236	13	)	)	PUNCT
ejpam-5930	236	14	as	as	ADP
ejpam-5930	236	15	in	in	ADP
ejpam-5930	236	16	example	example	NOUN
ejpam-5930	236	17	1	1	NUM
ejpam-5930	236	18	,	,	PUNCT
ejpam-5930	236	19	the	the	DET
ejpam-5930	236	20	subset	subset	NOUN
ejpam-5930	236	21	{	{	PUNCT
ejpam-5930	236	22	x	x	NOUN
ejpam-5930	236	23	,	,	PUNCT
ejpam-5930	236	24	y	y	PROPN
ejpam-5930	236	25	,	,	PUNCT
ejpam-5930	236	26	z	z	PROPN
ejpam-5930	236	27	,	,	PUNCT
ejpam-5930	236	28	0	0	NUM
ejpam-5930	236	29	}	}	PUNCT
ejpam-5930	236	30	is	be	AUX
ejpam-5930	236	31	not	not	PART
ejpam-5930	236	32	a	a	DET
ejpam-5930	236	33	subalgebra	subalgebra	NOUN
ejpam-5930	236	34	of	of	ADP
ejpam-5930	236	35	a	a	PRON
ejpam-5930	236	36	,	,	PUNCT
ejpam-5930	236	37	but	but	CCONJ
ejpam-5930	236	38	θ(i	θ(i	PROPN
ejpam-5930	236	39	,	,	PUNCT
ejpam-5930	236	40	{	{	PUNCT
ejpam-5930	236	41	x	x	NOUN
ejpam-5930	236	42	,	,	PUNCT
ejpam-5930	236	43	y	y	PROPN
ejpam-5930	236	44	,	,	PUNCT
ejpam-5930	236	45	z	z	PROPN
ejpam-5930	236	46	,	,	PUNCT
ejpam-5930	236	47	0	0	NUM
ejpam-5930	236	48	}	}	PUNCT
ejpam-5930	236	49	)	)	PUNCT
ejpam-5930	237	1	=	=	PUNCT
ejpam-5930	237	2	{	{	PUNCT
ejpam-5930	237	3	1	1	NUM
ejpam-5930	237	4	,	,	PUNCT
ejpam-5930	237	5	x	x	NOUN
ejpam-5930	237	6	,	,	PUNCT
ejpam-5930	237	7	y	y	PROPN
ejpam-5930	237	8	,	,	PUNCT
ejpam-5930	237	9	z	z	PROPN
ejpam-5930	237	10	,	,	PUNCT
ejpam-5930	237	11	0	0	NUM
ejpam-5930	237	12	}	}	PUNCT
ejpam-5930	237	13	is	be	AUX
ejpam-5930	237	14	a	a	DET
ejpam-5930	237	15	subalgebra	subalgebra	NOUN
ejpam-5930	237	16	of	of	ADP
ejpam-5930	237	17	a.	a.	NOUN
ejpam-5930	237	18	definition	definition	NOUN
ejpam-5930	237	19	9	9	NUM
ejpam-5930	237	20	.	.	PUNCT
ejpam-5930	238	1	a	a	DET
ejpam-5930	238	2	nonempty	nonempty	NOUN
ejpam-5930	238	3	subset	subset	VERB
ejpam-5930	238	4	s	s	NOUN
ejpam-5930	238	5	of	of	ADP
ejpam-5930	238	6	a	a	PRON
ejpam-5930	238	7	is	be	AUX
ejpam-5930	238	8	called	call	VERB
ejpam-5930	238	9	an	an	DET
ejpam-5930	238	10	upper	upper	ADJ
ejpam-5930	238	11	(	(	PUNCT
ejpam-5930	238	12	resp	resp	NOUN
ejpam-5930	238	13	.	.	PROPN
ejpam-5930	239	1	,	,	PUNCT
ejpam-5930	239	2	a	a	DET
ejpam-5930	239	3	lower	low	ADJ
ejpam-5930	239	4	)	)	PUNCT
ejpam-5930	239	5	rough	rough	ADJ
ejpam-5930	239	6	ideal	ideal	NOUN
ejpam-5930	239	7	of	of	ADP
ejpam-5930	239	8	a	a	DET
ejpam-5930	239	9	if	if	SCONJ
ejpam-5930	239	10	the	the	DET
ejpam-5930	239	11	upper	upper	ADJ
ejpam-5930	239	12	(	(	PUNCT
ejpam-5930	239	13	resp	resp	NOUN
ejpam-5930	239	14	.	.	PUNCT
ejpam-5930	239	15	,	,	PUNCT
ejpam-5930	239	16	nonempty	nonempty	VERB
ejpam-5930	239	17	lower	low	ADJ
ejpam-5930	239	18	)	)	PUNCT
ejpam-5930	239	19	approximation	approximation	NOUN
ejpam-5930	239	20	of	of	ADP
ejpam-5930	239	21	s	s	PROPN
ejpam-5930	239	22	is	be	AUX
ejpam-5930	239	23	an	an	DET
ejpam-5930	239	24	ideal	ideal	NOUN
ejpam-5930	239	25	of	of	ADP
ejpam-5930	239	26	a.	a.	NOUN
ejpam-5930	239	27	if	if	SCONJ
ejpam-5930	239	28	s	s	VERB
ejpam-5930	239	29	is	be	AUX
ejpam-5930	239	30	both	both	PRON
ejpam-5930	239	31	an	an	DET
ejpam-5930	239	32	upper	upper	NOUN
ejpam-5930	239	33	and	and	CCONJ
ejpam-5930	239	34	a	a	DET
ejpam-5930	239	35	lower	low	ADJ
ejpam-5930	239	36	rough	rough	ADJ
ejpam-5930	239	37	ideal	ideal	NOUN
ejpam-5930	239	38	of	of	ADP
ejpam-5930	239	39	a	a	PRON
ejpam-5930	239	40	,	,	PUNCT
ejpam-5930	239	41	we	we	PRON
ejpam-5930	239	42	say	say	VERB
ejpam-5930	239	43	that	that	SCONJ
ejpam-5930	239	44	s	s	VERB
ejpam-5930	239	45	is	be	AUX
ejpam-5930	239	46	a	a	DET
ejpam-5930	239	47	rough	rough	ADJ
ejpam-5930	239	48	ideal	ideal	NOUN
ejpam-5930	239	49	of	of	ADP
ejpam-5930	239	50	a.	a.	NOUN
ejpam-5930	239	51	theorem	theorem	NOUN
ejpam-5930	239	52	3	3	X
ejpam-5930	239	53	.	.	PUNCT
ejpam-5930	240	1	let	let	VERB
ejpam-5930	240	2	θ	θ	NOUN
ejpam-5930	240	3	be	be	AUX
ejpam-5930	240	4	a	a	DET
ejpam-5930	240	5	congruence	congruence	NOUN
ejpam-5930	240	6	relation	relation	NOUN
ejpam-5930	240	7	on	on	ADP
ejpam-5930	240	8	a	a	DET
ejpam-5930	240	9	related	relate	VERB
ejpam-5930	240	10	to	to	ADP
ejpam-5930	240	11	an	an	DET
ejpam-5930	240	12	ideal	ideal	NOUN
ejpam-5930	240	13	i	i	PRON
ejpam-5930	240	14	of	of	ADP
ejpam-5930	240	15	a.	a.	NOUN
ejpam-5930	240	16	if	if	SCONJ
ejpam-5930	240	17	s	s	X
ejpam-5930	240	18	is	be	AUX
ejpam-5930	240	19	an	an	DET
ejpam-5930	240	20	ideal	ideal	NOUN
ejpam-5930	240	21	of	of	ADP
ejpam-5930	240	22	a	a	DET
ejpam-5930	240	23	containing	contain	VERB
ejpam-5930	240	24	i	i	PRON
ejpam-5930	240	25	,	,	PUNCT
ejpam-5930	240	26	then	then	ADV
ejpam-5930	240	27	(	(	PUNCT
ejpam-5930	240	28	1	1	X
ejpam-5930	240	29	)	)	PUNCT
ejpam-5930	240	30	θ(i	θ(i	PROPN
ejpam-5930	240	31	,	,	PUNCT
ejpam-5930	240	32	s	s	PART
ejpam-5930	240	33	)	)	PUNCT
ejpam-5930	240	34	is	be	AUX
ejpam-5930	240	35	an	an	DET
ejpam-5930	240	36	ideal	ideal	NOUN
ejpam-5930	240	37	of	of	ADP
ejpam-5930	240	38	a	a	DET
ejpam-5930	240	39	,	,	PUNCT
ejpam-5930	240	40	(	(	PUNCT
ejpam-5930	240	41	2	2	NUM
ejpam-5930	240	42	)	)	PUNCT
ejpam-5930	240	43	θ(i	θ(i	PROPN
ejpam-5930	240	44	,	,	PUNCT
ejpam-5930	240	45	s	s	PART
ejpam-5930	240	46	)	)	PUNCT
ejpam-5930	240	47	is	be	AUX
ejpam-5930	240	48	an	an	DET
ejpam-5930	240	49	ideal	ideal	NOUN
ejpam-5930	240	50	of	of	ADP
ejpam-5930	240	51	a.	a.	NOUN
ejpam-5930	240	52	proof	proof	NOUN
ejpam-5930	240	53	.	.	PUNCT
ejpam-5930	241	1	(	(	PUNCT
ejpam-5930	241	2	1	1	X
ejpam-5930	241	3	)	)	PUNCT
ejpam-5930	241	4	let	let	VERB
ejpam-5930	241	5	s	s	PRON
ejpam-5930	241	6	be	be	AUX
ejpam-5930	241	7	an	an	DET
ejpam-5930	241	8	ideal	ideal	NOUN
ejpam-5930	241	9	of	of	ADP
ejpam-5930	241	10	a	a	DET
ejpam-5930	241	11	containing	contain	VERB
ejpam-5930	241	12	i.	i.	NOUN
ejpam-5930	241	13	obviously	obviously	ADV
ejpam-5930	241	14	,	,	PUNCT
ejpam-5930	241	15	1	1	NUM
ejpam-5930	241	16	∈	∈	PROPN
ejpam-5930	241	17	θ(i	θ(i	PROPN
ejpam-5930	241	18	,	,	PUNCT
ejpam-5930	241	19	s	s	NOUN
ejpam-5930	241	20	)	)	PUNCT
ejpam-5930	241	21	.	.	PUNCT
ejpam-5930	242	1	let	let	VERB
ejpam-5930	242	2	x	x	PRON
ejpam-5930	242	3	,	,	PUNCT
ejpam-5930	242	4	y	y	PROPN
ejpam-5930	242	5	∈	∈	PROPN
ejpam-5930	242	6	a	a	DET
ejpam-5930	242	7	be	be	AUX
ejpam-5930	242	8	such	such	ADJ
ejpam-5930	242	9	that	that	SCONJ
ejpam-5930	242	10	y	y	PROPN
ejpam-5930	242	11	∈	∈	PROPN
ejpam-5930	242	12	θ(i	θ(i	PROPN
ejpam-5930	242	13	,	,	PUNCT
ejpam-5930	242	14	s	s	NOUN
ejpam-5930	242	15	)	)	PUNCT
ejpam-5930	242	16	.	.	PUNCT
ejpam-5930	243	1	then	then	ADV
ejpam-5930	243	2	iy	iy	PROPN
ejpam-5930	243	3	∩	∩	PROPN
ejpam-5930	243	4	s	s	PART
ejpam-5930	243	5	̸=	̸=	PROPN
ejpam-5930	243	6	∅	∅	NOUN
ejpam-5930	244	1	and	and	CCONJ
ejpam-5930	244	2	so	so	ADV
ejpam-5930	244	3	there	there	PRON
ejpam-5930	244	4	exists	exist	VERB
ejpam-5930	244	5	a	a	DET
ejpam-5930	244	6	∈	∈	NOUN
ejpam-5930	244	7	s	s	VERB
ejpam-5930	244	8	such	such	ADJ
ejpam-5930	244	9	that	that	SCONJ
ejpam-5930	244	10	a	a	DET
ejpam-5930	244	11	∈	∈	PROPN
ejpam-5930	244	12	iy	iy	PROPN
ejpam-5930	244	13	.	.	PUNCT
ejpam-5930	245	1	hence	hence	ADV
ejpam-5930	245	2	,	,	PUNCT
ejpam-5930	245	3	(	(	PUNCT
ejpam-5930	245	4	a	a	PRON
ejpam-5930	245	5	,	,	PUNCT
ejpam-5930	245	6	y	y	NOUN
ejpam-5930	245	7	)	)	PUNCT
ejpam-5930	245	8	∈	∈	PROPN
ejpam-5930	245	9	θ	θ	PROPN
ejpam-5930	245	10	,	,	PUNCT
ejpam-5930	245	11	which	which	PRON
ejpam-5930	245	12	implies	imply	VERB
ejpam-5930	245	13	y	y	PROPN
ejpam-5930	245	14	·	·	PUNCT
ejpam-5930	245	15	a	a	DET
ejpam-5930	245	16	∈	∈	PROPN
ejpam-5930	245	17	i	i	NOUN
ejpam-5930	245	18	⊆	⊆	NUM
ejpam-5930	245	19	s.	s.	PROPN
ejpam-5930	245	20	since	since	SCONJ
ejpam-5930	245	21	a	a	DET
ejpam-5930	245	22	∈	∈	PROPN
ejpam-5930	245	23	s	s	PART
ejpam-5930	245	24	and	and	CCONJ
ejpam-5930	245	25	s	s	VERB
ejpam-5930	245	26	is	be	AUX
ejpam-5930	245	27	an	an	DET
ejpam-5930	245	28	ideal	ideal	NOUN
ejpam-5930	245	29	of	of	ADP
ejpam-5930	245	30	a	a	PRON
ejpam-5930	245	31	,	,	PUNCT
ejpam-5930	245	32	we	we	PRON
ejpam-5930	245	33	get	get	VERB
ejpam-5930	245	34	y	y	PROPN
ejpam-5930	245	35	∈	∈	PROPN
ejpam-5930	245	36	s.	s.	PROPN
ejpam-5930	245	37	then	then	ADV
ejpam-5930	245	38	x	x	X
ejpam-5930	245	39	·	·	PUNCT
ejpam-5930	245	40	y	y	PROPN
ejpam-5930	245	41	∈	∈	PROPN
ejpam-5930	245	42	s.	s.	PROPN
ejpam-5930	245	43	note	note	VERB
ejpam-5930	245	44	that	that	SCONJ
ejpam-5930	245	45	x	x	X
ejpam-5930	245	46	·	·	PUNCT
ejpam-5930	245	47	y	y	PROPN
ejpam-5930	245	48	∈	∈	PROPN
ejpam-5930	245	49	ix·y	ix·y	PROPN
ejpam-5930	245	50	,	,	PUNCT
ejpam-5930	245	51	thus	thus	ADV
ejpam-5930	245	52	x	x	X
ejpam-5930	245	53	·	·	PUNCT
ejpam-5930	245	54	y	y	X
ejpam-5930	245	55	∈	∈	PROPN
ejpam-5930	245	56	ix·y	ix·y	PROPN
ejpam-5930	245	57	∩	∩	NOUN
ejpam-5930	245	58	s	s	PART
ejpam-5930	245	59	,	,	PUNCT
ejpam-5930	245	60	that	that	ADV
ejpam-5930	245	61	is	is	ADV
ejpam-5930	245	62	,	,	PUNCT
ejpam-5930	245	63	ix·y	ix·y	PROPN
ejpam-5930	245	64	∩	∩	NOUN
ejpam-5930	245	65	s	s	PART
ejpam-5930	245	66	̸=	̸=	PROPN
ejpam-5930	245	67	∅.	∅.	PRON
ejpam-5930	245	68	hence	hence	ADV
ejpam-5930	245	69	,	,	PUNCT
ejpam-5930	245	70	x	x	X
ejpam-5930	245	71	·	·	PUNCT
ejpam-5930	245	72	y	y	PROPN
ejpam-5930	245	73	∈	∈	PROPN
ejpam-5930	245	74	θ(i	θ(i	PROPN
ejpam-5930	245	75	,	,	PUNCT
ejpam-5930	245	76	s	s	NOUN
ejpam-5930	245	77	)	)	PUNCT
ejpam-5930	245	78	.	.	PUNCT
ejpam-5930	246	1	let	let	VERB
ejpam-5930	246	2	x	x	PRON
ejpam-5930	246	3	,	,	PUNCT
ejpam-5930	246	4	y1	y1	INTJ
ejpam-5930	246	5	,	,	PUNCT
ejpam-5930	246	6	y2	y2	PROPN
ejpam-5930	246	7	∈	∈	PROPN
ejpam-5930	246	8	a	a	PRON
ejpam-5930	246	9	be	be	AUX
ejpam-5930	246	10	such	such	ADJ
ejpam-5930	246	11	that	that	DET
ejpam-5930	246	12	y1	y1	NOUN
ejpam-5930	246	13	,	,	PUNCT
ejpam-5930	246	14	y2	y2	PROPN
ejpam-5930	246	15	∈	∈	PROPN
ejpam-5930	246	16	θ(i	θ(i	PROPN
ejpam-5930	246	17	,	,	PUNCT
ejpam-5930	246	18	s	s	NOUN
ejpam-5930	246	19	)	)	PUNCT
ejpam-5930	246	20	.	.	PUNCT
ejpam-5930	247	1	then	then	ADV
ejpam-5930	247	2	iy1	iy1	VERB
ejpam-5930	247	3	∩	∩	NOUN
ejpam-5930	247	4	s	s	PART
ejpam-5930	247	5	̸=	̸=	PROPN
ejpam-5930	247	6	∅	∅	NOUN
ejpam-5930	247	7	and	and	CCONJ
ejpam-5930	247	8	iy2	iy2	ADP
ejpam-5930	247	9	∩	∩	NOUN
ejpam-5930	247	10	s	s	PART
ejpam-5930	247	11	̸=	̸=	PROPN
ejpam-5930	247	12	∅	∅	NOUN
ejpam-5930	248	1	and	and	CCONJ
ejpam-5930	248	2	so	so	ADV
ejpam-5930	248	3	there	there	PRON
ejpam-5930	248	4	exist	exist	VERB
ejpam-5930	248	5	a	a	DET
ejpam-5930	248	6	,	,	PUNCT
ejpam-5930	248	7	b	b	X
ejpam-5930	248	8	∈	∈	NOUN
ejpam-5930	248	9	s	s	VERB
ejpam-5930	248	10	such	such	ADJ
ejpam-5930	248	11	that	that	SCONJ
ejpam-5930	248	12	a	a	DET
ejpam-5930	248	13	∈	∈	PROPN
ejpam-5930	248	14	iy1	iy1	NOUN
ejpam-5930	248	15	and	and	CCONJ
ejpam-5930	248	16	b	b	NOUN
ejpam-5930	248	17	∈	∈	PROPN
ejpam-5930	248	18	iy2	iy2	NOUN
ejpam-5930	248	19	.	.	PUNCT
ejpam-5930	249	1	hence	hence	ADV
ejpam-5930	249	2	,	,	PUNCT
ejpam-5930	249	3	(	(	PUNCT
ejpam-5930	249	4	a	a	PRON
ejpam-5930	249	5	,	,	PUNCT
ejpam-5930	249	6	y1	y1	ADJ
ejpam-5930	249	7	)	)	PUNCT
ejpam-5930	249	8	∈	∈	PROPN
ejpam-5930	249	9	θ	θ	PROPN
ejpam-5930	249	10	and	and	CCONJ
ejpam-5930	249	11	(	(	PUNCT
ejpam-5930	249	12	b	b	NOUN
ejpam-5930	249	13	,	,	PUNCT
ejpam-5930	249	14	y2	y2	NOUN
ejpam-5930	249	15	)	)	PUNCT
ejpam-5930	249	16	∈	∈	PROPN
ejpam-5930	249	17	θ	θ	PROPN
ejpam-5930	249	18	,	,	PUNCT
ejpam-5930	249	19	which	which	PRON
ejpam-5930	249	20	implies	imply	VERB
ejpam-5930	249	21	y1	y1	NOUN
ejpam-5930	249	22	·	·	PUNCT
ejpam-5930	249	23	a	a	DET
ejpam-5930	249	24	∈	∈	NOUN
ejpam-5930	249	25	i	i	NOUN
ejpam-5930	249	26	⊆	⊆	NUM
ejpam-5930	249	27	s	s	NOUN
ejpam-5930	249	28	and	and	CCONJ
ejpam-5930	249	29	y2	y2	PROPN
ejpam-5930	249	30	·	·	PUNCT
ejpam-5930	250	1	b	b	X
ejpam-5930	250	2	∈	∈	PROPN
ejpam-5930	250	3	i	i	NOUN
ejpam-5930	250	4	⊆	⊆	NUM
ejpam-5930	250	5	s.	s.	PROPN
ejpam-5930	250	6	since	since	SCONJ
ejpam-5930	250	7	a	a	DET
ejpam-5930	250	8	,	,	PUNCT
ejpam-5930	250	9	b	b	PROPN
ejpam-5930	250	10	∈	∈	PROPN
ejpam-5930	250	11	s	s	X
ejpam-5930	250	12	and	and	CCONJ
ejpam-5930	250	13	s	s	VERB
ejpam-5930	250	14	is	be	AUX
ejpam-5930	250	15	an	an	DET
ejpam-5930	250	16	ideal	ideal	NOUN
ejpam-5930	250	17	of	of	ADP
ejpam-5930	250	18	a	a	PRON
ejpam-5930	250	19	,	,	PUNCT
ejpam-5930	250	20	we	we	PRON
ejpam-5930	250	21	have	have	VERB
ejpam-5930	250	22	y1	y1	NOUN
ejpam-5930	250	23	∈	∈	PROPN
ejpam-5930	250	24	s	s	PART
ejpam-5930	250	25	and	and	CCONJ
ejpam-5930	250	26	y2	y2	PROPN
ejpam-5930	250	27	∈	∈	PROPN
ejpam-5930	250	28	s	s	PART
ejpam-5930	250	29	and	and	CCONJ
ejpam-5930	250	30	(	(	PUNCT
ejpam-5930	250	31	y1	y1	INTJ
ejpam-5930	250	32	·	·	PUNCT
ejpam-5930	250	33	(	(	PUNCT
ejpam-5930	250	34	y2	y2	INTJ
ejpam-5930	250	35	·	·	PUNCT
ejpam-5930	250	36	x	x	X
ejpam-5930	250	37	)	)	PUNCT
ejpam-5930	250	38	)	)	PUNCT
ejpam-5930	250	39	·	·	PUNCT
ejpam-5930	251	1	x	x	PUNCT
ejpam-5930	251	2	∈	∈	PROPN
ejpam-5930	251	3	s.	s.	PROPN
ejpam-5930	251	4	note	note	VERB
ejpam-5930	251	5	that	that	SCONJ
ejpam-5930	251	6	(	(	PUNCT
ejpam-5930	251	7	y1	y1	INTJ
ejpam-5930	251	8	·	·	PUNCT
ejpam-5930	251	9	(	(	PUNCT
ejpam-5930	251	10	y2	y2	INTJ
ejpam-5930	251	11	·	·	PUNCT
ejpam-5930	251	12	x	x	X
ejpam-5930	251	13	)	)	PUNCT
ejpam-5930	251	14	)	)	PUNCT
ejpam-5930	251	15	·	·	PUNCT
ejpam-5930	252	1	x	x	SYM
ejpam-5930	252	2	∈	∈	NOUN
ejpam-5930	252	3	i(y1·(y2·x))·x	i(y1·(y2·x))·x	NOUN
ejpam-5930	252	4	,	,	PUNCT
ejpam-5930	252	5	thus	thus	ADV
ejpam-5930	252	6	(	(	PUNCT
ejpam-5930	252	7	y1	y1	INTJ
ejpam-5930	252	8	·	·	PUNCT
ejpam-5930	252	9	(	(	PUNCT
ejpam-5930	252	10	y2	y2	INTJ
ejpam-5930	252	11	·	·	PUNCT
ejpam-5930	252	12	x	x	X
ejpam-5930	252	13	)	)	PUNCT
ejpam-5930	252	14	)	)	PUNCT
ejpam-5930	252	15	·	·	PUNCT
ejpam-5930	253	1	x	x	PUNCT
ejpam-5930	253	2	∈	∈	NOUN
ejpam-5930	253	3	i(y1·(y2·x))·x	i(y1·(y2·x))·x	NOUN
ejpam-5930	253	4	∩	∩	NOUN
ejpam-5930	253	5	s	s	PART
ejpam-5930	253	6	,	,	PUNCT
ejpam-5930	253	7	that	that	ADV
ejpam-5930	253	8	is	is	ADV
ejpam-5930	253	9	,	,	PUNCT
ejpam-5930	253	10	i(y1·(y2·x))·x	i(y1·(y2·x))·x	NOUN
ejpam-5930	253	11	∩	∩	NOUN
ejpam-5930	253	12	s	s	PART
ejpam-5930	253	13	̸=	̸=	PROPN
ejpam-5930	253	14	∅.	∅.	PRON
ejpam-5930	253	15	hence	hence	ADV
ejpam-5930	253	16	,	,	PUNCT
ejpam-5930	253	17	(	(	PUNCT
ejpam-5930	253	18	y1	y1	INTJ
ejpam-5930	253	19	·	·	PUNCT
ejpam-5930	253	20	(	(	PUNCT
ejpam-5930	253	21	y2	y2	INTJ
ejpam-5930	253	22	·	·	PUNCT
ejpam-5930	253	23	x	x	X
ejpam-5930	253	24	)	)	PUNCT
ejpam-5930	253	25	)	)	PUNCT
ejpam-5930	253	26	·	·	PUNCT
ejpam-5930	254	1	x	x	PUNCT
ejpam-5930	254	2	∈	∈	PROPN
ejpam-5930	254	3	θ(i	θ(i	PROPN
ejpam-5930	254	4	,	,	PUNCT
ejpam-5930	254	5	s	s	PART
ejpam-5930	254	6	)	)	PUNCT
ejpam-5930	254	7	and	and	CCONJ
ejpam-5930	254	8	therefore	therefore	ADV
ejpam-5930	254	9	,	,	PUNCT
ejpam-5930	254	10	θ(i	θ(i	PROPN
ejpam-5930	254	11	,	,	PUNCT
ejpam-5930	254	12	s	s	PART
ejpam-5930	254	13	)	)	PUNCT
ejpam-5930	254	14	is	be	AUX
ejpam-5930	254	15	an	an	DET
ejpam-5930	254	16	ideal	ideal	NOUN
ejpam-5930	254	17	of	of	ADP
ejpam-5930	254	18	a.	a.	NOUN
ejpam-5930	254	19	(	(	PUNCT
ejpam-5930	254	20	2	2	X
ejpam-5930	254	21	)	)	PUNCT
ejpam-5930	254	22	let	let	VERB
ejpam-5930	254	23	s	s	PRON
ejpam-5930	254	24	be	be	AUX
ejpam-5930	254	25	an	an	DET
ejpam-5930	254	26	ideal	ideal	NOUN
ejpam-5930	254	27	of	of	ADP
ejpam-5930	254	28	a	a	DET
ejpam-5930	254	29	containing	contain	VERB
ejpam-5930	254	30	i.	i.	NOUN
ejpam-5930	254	31	let	let	VERB
ejpam-5930	254	32	x	x	PROPN
ejpam-5930	254	33	∈	∈	PROPN
ejpam-5930	254	34	i1	i1	PROPN
ejpam-5930	254	35	.	.	PUNCT
ejpam-5930	255	1	then	then	ADV
ejpam-5930	255	2	x	x	SYM
ejpam-5930	255	3	∈	∈	PROPN
ejpam-5930	255	4	i	i	NOUN
ejpam-5930	255	5	⊆	⊆	NUM
ejpam-5930	255	6	s	s	NOUN
ejpam-5930	255	7	,	,	PUNCT
ejpam-5930	255	8	and	and	CCONJ
ejpam-5930	255	9	so	so	ADV
ejpam-5930	255	10	i1	i1	PROPN
ejpam-5930	255	11	⊆	⊆	NUM
ejpam-5930	255	12	s.	s.	PROPN
ejpam-5930	255	13	hence	hence	ADV
ejpam-5930	255	14	,	,	PUNCT
ejpam-5930	255	15	1	1	NUM
ejpam-5930	255	16	∈	∈	PROPN
ejpam-5930	255	17	θ(i	θ(i	PROPN
ejpam-5930	255	18	,	,	PUNCT
ejpam-5930	255	19	s	s	NOUN
ejpam-5930	255	20	)	)	PUNCT
ejpam-5930	255	21	.	.	PUNCT
ejpam-5930	256	1	let	let	VERB
ejpam-5930	256	2	x	x	PRON
ejpam-5930	256	3	,	,	PUNCT
ejpam-5930	256	4	y	y	PROPN
ejpam-5930	256	5	∈	∈	PROPN
ejpam-5930	256	6	x	x	AUX
ejpam-5930	256	7	be	be	AUX
ejpam-5930	256	8	such	such	ADJ
ejpam-5930	256	9	that	that	SCONJ
ejpam-5930	256	10	y	y	PROPN
ejpam-5930	256	11	∈	∈	PROPN
ejpam-5930	256	12	θ(i	θ(i	PROPN
ejpam-5930	256	13	,	,	PUNCT
ejpam-5930	256	14	s	s	NOUN
ejpam-5930	256	15	)	)	PUNCT
ejpam-5930	256	16	.	.	PUNCT
ejpam-5930	257	1	then	then	ADV
ejpam-5930	257	2	iy	iy	PROPN
ejpam-5930	257	3	⊆	⊆	NUM
ejpam-5930	257	4	s.	s.	PROPN
ejpam-5930	257	5	let	let	VERB
ejpam-5930	257	6	w	w	PROPN
ejpam-5930	257	7	∈	∈	PROPN
ejpam-5930	257	8	ix·y	ix·y	PROPN
ejpam-5930	257	9	=	=	SYM
ejpam-5930	257	10	ix	ix	PROPN
ejpam-5930	257	11	·	·	PUNCT
ejpam-5930	257	12	iy	iy	PROPN
ejpam-5930	257	13	.	.	PUNCT
ejpam-5930	258	1	then	then	ADV
ejpam-5930	258	2	w	w	PROPN
ejpam-5930	258	3	∈	∈	PROPN
ejpam-5930	258	4	ix	ix	ADP
ejpam-5930	258	5	·	·	PUNCT
ejpam-5930	258	6	iy	iy	PROPN
ejpam-5930	258	7	for	for	ADP
ejpam-5930	258	8	some	some	DET
ejpam-5930	258	9	a	a	DET
ejpam-5930	258	10	∈	∈	NOUN
ejpam-5930	258	11	ix	ix	ADV
ejpam-5930	258	12	and	and	CCONJ
ejpam-5930	258	13	c	c	PROPN
ejpam-5930	258	14	∈	∈	PROPN
ejpam-5930	258	15	iy	iy	PROPN
ejpam-5930	258	16	.	.	PUNCT
ejpam-5930	259	1	from	from	ADP
ejpam-5930	259	2	a	a	DET
ejpam-5930	259	3	∈	∈	NOUN
ejpam-5930	259	4	ix	ix	ADV
ejpam-5930	259	5	and	and	CCONJ
ejpam-5930	260	1	c	c	PROPN
ejpam-5930	260	2	∈	∈	PROPN
ejpam-5930	260	3	iy	iy	PROPN
ejpam-5930	260	4	,	,	PUNCT
ejpam-5930	260	5	we	we	PRON
ejpam-5930	260	6	have	have	AUX
ejpam-5930	260	7	(	(	PUNCT
ejpam-5930	260	8	a	a	PRON
ejpam-5930	260	9	,	,	PUNCT
ejpam-5930	260	10	x	x	NOUN
ejpam-5930	260	11	)	)	PUNCT
ejpam-5930	260	12	∈	∈	PROPN
ejpam-5930	260	13	θ	θ	PROPN
ejpam-5930	260	14	and	and	CCONJ
ejpam-5930	260	15	(	(	PUNCT
ejpam-5930	260	16	c	c	X
ejpam-5930	260	17	,	,	PUNCT
ejpam-5930	260	18	y	y	NOUN
ejpam-5930	260	19	)	)	PUNCT
ejpam-5930	260	20	∈	∈	PROPN
ejpam-5930	260	21	θ	θ	PROPN
ejpam-5930	260	22	.	.	PUNCT
ejpam-5930	261	1	taking	take	VERB
ejpam-5930	261	2	b	b	PROPN
ejpam-5930	261	3	∈	∈	PROPN
ejpam-5930	261	4	iy	iy	NOUN
ejpam-5930	261	5	,	,	PUNCT
ejpam-5930	261	6	we	we	PRON
ejpam-5930	261	7	get	get	VERB
ejpam-5930	261	8	(	(	PUNCT
ejpam-5930	261	9	b	b	NOUN
ejpam-5930	261	10	,	,	PUNCT
ejpam-5930	261	11	y	y	NOUN
ejpam-5930	261	12	)	)	PUNCT
ejpam-5930	261	13	∈	∈	PROPN
ejpam-5930	261	14	θ	θ	PROPN
ejpam-5930	261	15	.	.	PUNCT
ejpam-5930	262	1	since	since	SCONJ
ejpam-5930	262	2	θ	θ	PROPN
ejpam-5930	262	3	is	be	AUX
ejpam-5930	262	4	a	a	DET
ejpam-5930	262	5	congruence	congruence	NOUN
ejpam-5930	262	6	relation	relation	NOUN
ejpam-5930	262	7	on	on	ADP
ejpam-5930	262	8	s	s	PROPN
ejpam-5930	262	9	,	,	PUNCT
ejpam-5930	262	10	we	we	PRON
ejpam-5930	262	11	get	get	VERB
ejpam-5930	262	12	(	(	PUNCT
ejpam-5930	262	13	a	a	DET
ejpam-5930	262	14	·	·	SYM
ejpam-5930	262	15	b	b	NOUN
ejpam-5930	262	16	,	,	PUNCT
ejpam-5930	262	17	x	x	PROPN
ejpam-5930	262	18	·	·	PUNCT
ejpam-5930	262	19	y	y	X
ejpam-5930	262	20	)	)	PUNCT
ejpam-5930	262	21	∈	∈	PROPN
ejpam-5930	262	22	θ	θ	PROPN
ejpam-5930	263	1	and	and	CCONJ
ejpam-5930	263	2	so	so	ADV
ejpam-5930	263	3	a	a	DET
ejpam-5930	263	4	·	·	PUNCT
ejpam-5930	263	5	b	b	X
ejpam-5930	263	6	∈	∈	PROPN
ejpam-5930	263	7	ix·y	ix·y	PROPN
ejpam-5930	263	8	⊆	⊆	NUM
ejpam-5930	263	9	s.	s.	PROPN
ejpam-5930	263	10	since	since	SCONJ
ejpam-5930	263	11	s	s	PROPN
ejpam-5930	263	12	is	be	AUX
ejpam-5930	263	13	an	an	DET
ejpam-5930	263	14	ideal	ideal	NOUN
ejpam-5930	263	15	of	of	ADP
ejpam-5930	263	16	a	a	PRON
ejpam-5930	263	17	,	,	PUNCT
ejpam-5930	263	18	we	we	PRON
ejpam-5930	263	19	have	have	VERB
ejpam-5930	263	20	w	w	NOUN
ejpam-5930	263	21	=	=	PUNCT
ejpam-5930	263	22	a	a	PRON
ejpam-5930	263	23	·	·	PUNCT
ejpam-5930	263	24	c	c	X
ejpam-5930	263	25	∈	∈	PROPN
ejpam-5930	263	26	s	s	NOUN
ejpam-5930	263	27	,	,	PUNCT
ejpam-5930	263	28	so	so	SCONJ
ejpam-5930	263	29	that	that	SCONJ
ejpam-5930	263	30	ix·y	ix·y	PROPN
ejpam-5930	263	31	⊆	⊆	NUM
ejpam-5930	263	32	s.	s.	PROPN
ejpam-5930	263	33	hence	hence	ADV
ejpam-5930	263	34	,	,	PUNCT
ejpam-5930	263	35	x	x	X
ejpam-5930	263	36	·	·	PUNCT
ejpam-5930	263	37	y	y	PROPN
ejpam-5930	263	38	∈	∈	PROPN
ejpam-5930	263	39	θ(i	θ(i	PROPN
ejpam-5930	263	40	,	,	PUNCT
ejpam-5930	263	41	s	s	NOUN
ejpam-5930	263	42	)	)	PUNCT
ejpam-5930	263	43	.	.	PUNCT
ejpam-5930	264	1	let	let	VERB
ejpam-5930	264	2	x	x	PRON
ejpam-5930	264	3	,	,	PUNCT
ejpam-5930	264	4	y1	y1	INTJ
ejpam-5930	264	5	,	,	PUNCT
ejpam-5930	264	6	y2	y2	PROPN
ejpam-5930	264	7	∈	∈	PROPN
ejpam-5930	264	8	x	x	AUX
ejpam-5930	264	9	be	be	AUX
ejpam-5930	264	10	such	such	ADJ
ejpam-5930	264	11	that	that	SCONJ
ejpam-5930	264	12	y1	y1	PROPN
ejpam-5930	264	13	∈	∈	PROPN
ejpam-5930	264	14	θ(i	θ(i	PROPN
ejpam-5930	264	15	,	,	PUNCT
ejpam-5930	264	16	s	s	PART
ejpam-5930	264	17	)	)	PUNCT
ejpam-5930	264	18	and	and	CCONJ
ejpam-5930	264	19	y2	y2	PROPN
ejpam-5930	264	20	∈	∈	PROPN
ejpam-5930	264	21	θ(i	θ(i	PROPN
ejpam-5930	264	22	,	,	PUNCT
ejpam-5930	264	23	s	s	NOUN
ejpam-5930	264	24	)	)	PUNCT
ejpam-5930	264	25	.	.	PUNCT
ejpam-5930	265	1	then	then	ADV
ejpam-5930	265	2	iy1	iy1	VERB
ejpam-5930	265	3	⊆	⊆	NUM
ejpam-5930	265	4	s	s	NOUN
ejpam-5930	265	5	and	and	CCONJ
ejpam-5930	265	6	iy2	iy2	NOUN
ejpam-5930	265	7	⊆	⊆	NUM
ejpam-5930	265	8	s.	s.	PROPN
ejpam-5930	265	9	let	let	VERB
ejpam-5930	265	10	w	w	NOUN
ejpam-5930	265	11	∈	∈	NOUN
ejpam-5930	265	12	i(y1·(y2·x))·x	i(y1·(y2·x))·x	NOUN
ejpam-5930	265	13	=	=	PUNCT
ejpam-5930	265	14	(	(	PUNCT
ejpam-5930	265	15	iy1	iy1	NOUN
ejpam-5930	265	16	·	·	PUNCT
ejpam-5930	265	17	(	(	PUNCT
ejpam-5930	265	18	iy2	iy2	NOUN
ejpam-5930	265	19	·	·	SYM
ejpam-5930	265	20	ix	ix	PROPN
ejpam-5930	265	21	)	)	PUNCT
ejpam-5930	265	22	)	)	PUNCT
ejpam-5930	265	23	·	·	PUNCT
ejpam-5930	265	24	ix	ix	PROPN
ejpam-5930	265	25	.	.	PUNCT
ejpam-5930	266	1	then	then	ADV
ejpam-5930	266	2	w	w	PROPN
ejpam-5930	266	3	∈	∈	PROPN
ejpam-5930	266	4	(	(	PUNCT
ejpam-5930	266	5	iy1	iy1	NOUN
ejpam-5930	266	6	·	·	PUNCT
ejpam-5930	266	7	(	(	PUNCT
ejpam-5930	266	8	iy2	iy2	NOUN
ejpam-5930	266	9	·	·	SYM
ejpam-5930	266	10	ix	ix	PROPN
ejpam-5930	266	11	)	)	PUNCT
ejpam-5930	266	12	)	)	PUNCT
ejpam-5930	266	13	·	·	PUNCT
ejpam-5930	266	14	ix	ix	X
ejpam-5930	266	15	for	for	ADP
ejpam-5930	266	16	some	some	DET
ejpam-5930	266	17	a	a	DET
ejpam-5930	266	18	∈	∈	PROPN
ejpam-5930	266	19	iy1	iy1	NOUN
ejpam-5930	266	20	,	,	PUNCT
ejpam-5930	266	21	b	b	PROPN
ejpam-5930	266	22	∈	∈	PROPN
ejpam-5930	266	23	iy2	iy2	NOUN
ejpam-5930	266	24	,	,	PUNCT
ejpam-5930	266	25	and	and	CCONJ
ejpam-5930	266	26	c	c	X
ejpam-5930	266	27	∈	∈	PROPN
ejpam-5930	266	28	ix	ix	INTJ
ejpam-5930	266	29	.	.	PUNCT
ejpam-5930	267	1	from	from	ADP
ejpam-5930	267	2	a	a	DET
ejpam-5930	267	3	∈	∈	PROPN
ejpam-5930	267	4	iy1	iy1	NOUN
ejpam-5930	267	5	,	,	PUNCT
ejpam-5930	267	6	b	b	PROPN
ejpam-5930	267	7	∈	∈	PROPN
ejpam-5930	267	8	iy2	iy2	NOUN
ejpam-5930	267	9	,	,	PUNCT
ejpam-5930	267	10	and	and	CCONJ
ejpam-5930	267	11	c	c	X
ejpam-5930	267	12	∈	∈	PROPN
ejpam-5930	268	1	ix	ix	INTJ
ejpam-5930	268	2	,	,	PUNCT
ejpam-5930	268	3	we	we	PRON
ejpam-5930	268	4	have	have	VERB
ejpam-5930	268	5	(	(	PUNCT
ejpam-5930	268	6	a	a	PRON
ejpam-5930	268	7	,	,	PUNCT
ejpam-5930	268	8	y1	y1	ADJ
ejpam-5930	268	9	)	)	PUNCT
ejpam-5930	268	10	∈	∈	PROPN
ejpam-5930	268	11	θ	θ	PROPN
ejpam-5930	268	12	,	,	PUNCT
ejpam-5930	268	13	(	(	PUNCT
ejpam-5930	268	14	b	b	NOUN
ejpam-5930	268	15	,	,	PUNCT
ejpam-5930	268	16	y2	y2	NOUN
ejpam-5930	268	17	)	)	PUNCT
ejpam-5930	268	18	∈	∈	PROPN
ejpam-5930	268	19	θ	θ	PROPN
ejpam-5930	268	20	,	,	PUNCT
ejpam-5930	268	21	and	and	CCONJ
ejpam-5930	268	22	(	(	PUNCT
ejpam-5930	268	23	c	c	NOUN
ejpam-5930	268	24	,	,	PUNCT
ejpam-5930	268	25	x	x	X
ejpam-5930	268	26	)	)	PUNCT
ejpam-5930	268	27	∈	∈	PROPN
ejpam-5930	268	28	θ	θ	PROPN
ejpam-5930	268	29	.	.	PUNCT
ejpam-5930	269	1	since	since	SCONJ
ejpam-5930	269	2	θ	θ	PROPN
ejpam-5930	269	3	is	be	AUX
ejpam-5930	269	4	a	a	DET
ejpam-5930	269	5	congruence	congruence	NOUN
ejpam-5930	269	6	relation	relation	NOUN
ejpam-5930	269	7	on	on	ADP
ejpam-5930	269	8	s	s	PROPN
ejpam-5930	269	9	,	,	PUNCT
ejpam-5930	269	10	we	we	PRON
ejpam-5930	269	11	get	get	VERB
ejpam-5930	269	12	(	(	PUNCT
ejpam-5930	269	13	(	(	PUNCT
ejpam-5930	269	14	a	a	DET
ejpam-5930	269	15	·	·	PUNCT
ejpam-5930	269	16	(	(	PUNCT
ejpam-5930	269	17	b	b	X
ejpam-5930	269	18	·	·	SYM
ejpam-5930	269	19	c	c	NOUN
ejpam-5930	269	20	)	)	PUNCT
ejpam-5930	269	21	)	)	PUNCT
ejpam-5930	270	1	·	·	PUNCT
ejpam-5930	270	2	c	c	X
ejpam-5930	270	3	,	,	PUNCT
ejpam-5930	270	4	(	(	PUNCT
ejpam-5930	270	5	y1	y1	INTJ
ejpam-5930	270	6	·	·	PUNCT
ejpam-5930	270	7	(	(	PUNCT
ejpam-5930	270	8	y2	y2	PROPN
ejpam-5930	270	9	·	·	SYM
ejpam-5930	270	10	x	x	NOUN
ejpam-5930	270	11	)	)	PUNCT
ejpam-5930	270	12	)	)	PUNCT
ejpam-5930	270	13	·	·	PUNCT
ejpam-5930	270	14	x	x	X
ejpam-5930	270	15	)	)	PUNCT
ejpam-5930	270	16	∈	∈	PROPN
ejpam-5930	270	17	θ	θ	PROPN
ejpam-5930	270	18	and	and	CCONJ
ejpam-5930	270	19	so	so	ADV
ejpam-5930	270	20	(	(	PUNCT
ejpam-5930	270	21	a·(b·c))·c	a·(b·c))·c	NOUN
ejpam-5930	270	22	∈	∈	NOUN
ejpam-5930	270	23	i(y1·(y2·x))·x	i(y1·(y2·x))·x	VERB
ejpam-5930	270	24	⊆	⊆	NUM
ejpam-5930	270	25	s.	s.	PROPN
ejpam-5930	270	26	since	since	SCONJ
ejpam-5930	270	27	s	s	PROPN
ejpam-5930	270	28	is	be	AUX
ejpam-5930	270	29	an	an	DET
ejpam-5930	270	30	ideal	ideal	NOUN
ejpam-5930	270	31	of	of	ADP
ejpam-5930	270	32	a	a	PRON
ejpam-5930	270	33	,	,	PUNCT
ejpam-5930	270	34	we	we	PRON
ejpam-5930	270	35	have	have	VERB
ejpam-5930	270	36	w	w	NOUN
ejpam-5930	270	37	=	=	SYM
ejpam-5930	270	38	(	(	PUNCT
ejpam-5930	270	39	a·(b·c))·c	a·(b·c))·c	NOUN
ejpam-5930	270	40	∈	∈	PROPN
ejpam-5930	270	41	s	s	PROPN
ejpam-5930	270	42	,	,	PUNCT
ejpam-5930	270	43	a.	a.	NOUN
ejpam-5930	270	44	iampan	iampan	NOUN
ejpam-5930	271	1	et	et	PROPN
ejpam-5930	271	2	al	al	PROPN
ejpam-5930	271	3	.	.	PUNCT
ejpam-5930	271	4	/	/	SYM
ejpam-5930	271	5	eur	eur	PROPN
ejpam-5930	271	6	.	.	PUNCT
ejpam-5930	272	1	j.	j.	PROPN
ejpam-5930	272	2	pure	pure	PROPN
ejpam-5930	272	3	appl	appl	PROPN
ejpam-5930	272	4	.	.	PROPN
ejpam-5930	272	5	math	math	PROPN
ejpam-5930	272	6	,	,	PUNCT
ejpam-5930	272	7	18	18	NUM
ejpam-5930	272	8	(	(	PUNCT
ejpam-5930	272	9	2	2	NUM
ejpam-5930	272	10	)	)	PUNCT
ejpam-5930	272	11	(	(	PUNCT
ejpam-5930	272	12	2025	2025	NUM
ejpam-5930	272	13	)	)	PUNCT
ejpam-5930	272	14	,	,	PUNCT
ejpam-5930	272	15	5930	5930	NUM
ejpam-5930	272	16	10	10	NUM
ejpam-5930	272	17	of	of	ADP
ejpam-5930	272	18	11	11	NUM
ejpam-5930	272	19	so	so	SCONJ
ejpam-5930	272	20	that	that	SCONJ
ejpam-5930	272	21	i(y1·(y2·x))·x	i(y1·(y2·x))·x	PROPN
ejpam-5930	272	22	⊆	⊆	NUM
ejpam-5930	272	23	s.	s.	PROPN
ejpam-5930	272	24	hence	hence	ADV
ejpam-5930	272	25	,	,	PUNCT
ejpam-5930	272	26	(	(	PUNCT
ejpam-5930	272	27	y1	y1	INTJ
ejpam-5930	272	28	·	·	PUNCT
ejpam-5930	272	29	(	(	PUNCT
ejpam-5930	272	30	y2	y2	INTJ
ejpam-5930	272	31	·	·	PUNCT
ejpam-5930	272	32	x	x	X
ejpam-5930	272	33	)	)	PUNCT
ejpam-5930	272	34	)	)	PUNCT
ejpam-5930	272	35	·	·	PUNCT
ejpam-5930	273	1	x	x	PUNCT
ejpam-5930	273	2	∈	∈	PROPN
ejpam-5930	273	3	θ(i	θ(i	PROPN
ejpam-5930	273	4	,	,	PUNCT
ejpam-5930	273	5	s	s	PART
ejpam-5930	273	6	)	)	PUNCT
ejpam-5930	273	7	and	and	CCONJ
ejpam-5930	273	8	therefore	therefore	ADV
ejpam-5930	273	9	,	,	PUNCT
ejpam-5930	273	10	θ(i	θ(i	PROPN
ejpam-5930	273	11	,	,	PUNCT
ejpam-5930	273	12	s	s	PART
ejpam-5930	273	13	)	)	PUNCT
ejpam-5930	273	14	is	be	AUX
ejpam-5930	273	15	an	an	DET
ejpam-5930	273	16	ideal	ideal	NOUN
ejpam-5930	273	17	of	of	ADP
ejpam-5930	273	18	a.	a.	NOUN
ejpam-5930	273	19	remark	remark	NOUN
ejpam-5930	273	20	2	2	NUM
ejpam-5930	273	21	.	.	PUNCT
ejpam-5930	274	1	the	the	DET
ejpam-5930	274	2	converse	converse	NOUN
ejpam-5930	274	3	of	of	ADP
ejpam-5930	274	4	theorem	theorem	ADJ
ejpam-5930	274	5	3	3	NUM
ejpam-5930	274	6	(	(	PUNCT
ejpam-5930	274	7	1	1	NUM
ejpam-5930	274	8	)	)	PUNCT
ejpam-5930	274	9	may	may	AUX
ejpam-5930	274	10	not	not	PART
ejpam-5930	274	11	be	be	AUX
ejpam-5930	274	12	true	true	ADJ
ejpam-5930	274	13	.	.	PUNCT
ejpam-5930	275	1	for	for	ADP
ejpam-5930	275	2	the	the	DET
ejpam-5930	275	3	hilbert	hilbert	PROPN
ejpam-5930	275	4	algebra	algebra	NOUN
ejpam-5930	275	5	a	a	DET
ejpam-5930	275	6	=	=	X
ejpam-5930	275	7	(	(	PUNCT
ejpam-5930	275	8	a	a	PRON
ejpam-5930	275	9	,	,	PUNCT
ejpam-5930	275	10	·	·	PUNCT
ejpam-5930	275	11	,	,	PUNCT
ejpam-5930	275	12	1	1	NUM
ejpam-5930	275	13	)	)	PUNCT
ejpam-5930	275	14	as	as	ADP
ejpam-5930	275	15	in	in	ADP
ejpam-5930	275	16	example	example	NOUN
ejpam-5930	275	17	1	1	NUM
ejpam-5930	275	18	,	,	PUNCT
ejpam-5930	275	19	the	the	DET
ejpam-5930	275	20	subset	subset	NOUN
ejpam-5930	275	21	{	{	PUNCT
ejpam-5930	275	22	1	1	NUM
ejpam-5930	275	23	,	,	PUNCT
ejpam-5930	275	24	x	x	PRON
ejpam-5930	275	25	}	}	PUNCT
ejpam-5930	275	26	is	be	AUX
ejpam-5930	275	27	an	an	DET
ejpam-5930	275	28	ideal	ideal	NOUN
ejpam-5930	275	29	of	of	ADP
ejpam-5930	275	30	a	a	PRON
ejpam-5930	275	31	,	,	PUNCT
ejpam-5930	275	32	but	but	CCONJ
ejpam-5930	275	33	{	{	PUNCT
ejpam-5930	275	34	x	x	X
ejpam-5930	275	35	}	}	PUNCT
ejpam-5930	275	36	is	be	AUX
ejpam-5930	275	37	not	not	PART
ejpam-5930	275	38	an	an	DET
ejpam-5930	275	39	ideal	ideal	NOUN
ejpam-5930	275	40	of	of	ADP
ejpam-5930	275	41	a.	a.	NOUN
ejpam-5930	275	42	also	also	ADV
ejpam-5930	275	43	,	,	PUNCT
ejpam-5930	275	44	θ(i	θ(i	PROPN
ejpam-5930	275	45	,	,	PUNCT
ejpam-5930	275	46	{	{	PUNCT
ejpam-5930	275	47	x	x	NOUN
ejpam-5930	275	48	}	}	PUNCT
ejpam-5930	275	49	)	)	PUNCT
ejpam-5930	276	1	=	=	PUNCT
ejpam-5930	276	2	{	{	PUNCT
ejpam-5930	276	3	1	1	NUM
ejpam-5930	276	4	,	,	PUNCT
ejpam-5930	276	5	x	x	PRON
ejpam-5930	276	6	}	}	PUNCT
ejpam-5930	276	7	is	be	AUX
ejpam-5930	276	8	an	an	DET
ejpam-5930	276	9	ideal	ideal	NOUN
ejpam-5930	276	10	of	of	ADP
ejpam-5930	276	11	a.	a.	NOUN
ejpam-5930	276	12	4	4	NUM
ejpam-5930	276	13	.	.	PUNCT
ejpam-5930	276	14	conclusion	conclusion	NOUN
ejpam-5930	276	15	this	this	DET
ejpam-5930	276	16	study	study	NOUN
ejpam-5930	276	17	has	have	AUX
ejpam-5930	276	18	explored	explore	VERB
ejpam-5930	276	19	the	the	DET
ejpam-5930	276	20	integration	integration	NOUN
ejpam-5930	276	21	of	of	ADP
ejpam-5930	276	22	rst	rst	PROPN
ejpam-5930	276	23	with	with	ADP
ejpam-5930	276	24	hilbert	hilbert	PROPN
ejpam-5930	276	25	algebras	algebras	PROPN
ejpam-5930	276	26	,	,	PUNCT
ejpam-5930	276	27	establishing	establish	VERB
ejpam-5930	276	28	a	a	DET
ejpam-5930	276	29	novel	novel	ADJ
ejpam-5930	276	30	framework	framework	NOUN
ejpam-5930	276	31	for	for	ADP
ejpam-5930	276	32	analyzing	analyze	VERB
ejpam-5930	276	33	algebraic	algebraic	ADJ
ejpam-5930	276	34	structures	structure	NOUN
ejpam-5930	276	35	under	under	ADP
ejpam-5930	276	36	uncertainty	uncertainty	NOUN
ejpam-5930	276	37	.	.	PUNCT
ejpam-5930	277	1	by	by	ADP
ejpam-5930	277	2	defining	define	VERB
ejpam-5930	277	3	lower	low	ADJ
ejpam-5930	277	4	and	and	CCONJ
ejpam-5930	277	5	upper	upper	ADJ
ejpam-5930	277	6	approximations	approximation	NOUN
ejpam-5930	277	7	within	within	ADP
ejpam-5930	277	8	hilbert	hilbert	PROPN
ejpam-5930	277	9	algebras	algebras	PROPN
ejpam-5930	277	10	,	,	PUNCT
ejpam-5930	277	11	we	we	PRON
ejpam-5930	277	12	have	have	AUX
ejpam-5930	277	13	demonstrated	demonstrate	VERB
ejpam-5930	277	14	that	that	SCONJ
ejpam-5930	277	15	these	these	DET
ejpam-5930	277	16	approximations	approximation	NOUN
ejpam-5930	277	17	preserve	preserve	VERB
ejpam-5930	277	18	subalgebra	subalgebra	NOUN
ejpam-5930	277	19	and	and	CCONJ
ejpam-5930	277	20	ideal	ideal	ADJ
ejpam-5930	277	21	structures	structure	NOUN
ejpam-5930	277	22	,	,	PUNCT
ejpam-5930	277	23	thereby	thereby	ADV
ejpam-5930	277	24	extending	extend	VERB
ejpam-5930	277	25	the	the	DET
ejpam-5930	277	26	applicability	applicability	NOUN
ejpam-5930	277	27	of	of	ADP
ejpam-5930	277	28	rst	rst	PROPN
ejpam-5930	277	29	to	to	ADP
ejpam-5930	277	30	algebraic	algebraic	ADJ
ejpam-5930	277	31	logic	logic	NOUN
ejpam-5930	277	32	.	.	PUNCT
ejpam-5930	278	1	the	the	DET
ejpam-5930	278	2	introduction	introduction	NOUN
ejpam-5930	278	3	of	of	ADP
ejpam-5930	278	4	approximation	approximation	NOUN
ejpam-5930	278	5	spaces	space	NOUN
ejpam-5930	278	6	induced	induce	VERB
ejpam-5930	278	7	by	by	ADP
ejpam-5930	278	8	ideals	ideal	NOUN
ejpam-5930	278	9	provides	provide	VERB
ejpam-5930	278	10	a	a	DET
ejpam-5930	278	11	systematic	systematic	ADJ
ejpam-5930	278	12	approach	approach	NOUN
ejpam-5930	278	13	to	to	ADP
ejpam-5930	278	14	dealing	deal	VERB
ejpam-5930	278	15	with	with	ADP
ejpam-5930	278	16	incomplete	incomplete	ADJ
ejpam-5930	278	17	or	or	CCONJ
ejpam-5930	278	18	vague	vague	ADJ
ejpam-5930	278	19	information	information	NOUN
ejpam-5930	278	20	in	in	ADP
ejpam-5930	278	21	algebraic	algebraic	ADJ
ejpam-5930	278	22	systems	system	NOUN
ejpam-5930	278	23	.	.	PUNCT
ejpam-5930	279	1	moreover	moreover	ADV
ejpam-5930	279	2	,	,	PUNCT
ejpam-5930	279	3	we	we	PRON
ejpam-5930	279	4	have	have	AUX
ejpam-5930	279	5	validated	validate	VERB
ejpam-5930	279	6	our	our	PRON
ejpam-5930	279	7	theoretical	theoretical	ADJ
ejpam-5930	279	8	findings	finding	NOUN
ejpam-5930	279	9	through	through	ADP
ejpam-5930	279	10	illustrative	illustrative	ADJ
ejpam-5930	279	11	examples	example	NOUN
ejpam-5930	279	12	,	,	PUNCT
ejpam-5930	279	13	reinforcing	reinforce	VERB
ejpam-5930	279	14	the	the	DET
ejpam-5930	279	15	practical	practical	ADJ
ejpam-5930	279	16	significance	significance	NOUN
ejpam-5930	279	17	of	of	ADP
ejpam-5930	279	18	this	this	DET
ejpam-5930	279	19	approach	approach	NOUN
ejpam-5930	279	20	.	.	PUNCT
ejpam-5930	280	1	the	the	DET
ejpam-5930	280	2	results	result	NOUN
ejpam-5930	280	3	presented	present	VERB
ejpam-5930	280	4	in	in	ADP
ejpam-5930	280	5	this	this	DET
ejpam-5930	280	6	work	work	NOUN
ejpam-5930	280	7	not	not	PART
ejpam-5930	280	8	only	only	ADV
ejpam-5930	280	9	enhance	enhance	VERB
ejpam-5930	280	10	the	the	DET
ejpam-5930	280	11	theoretical	theoretical	ADJ
ejpam-5930	280	12	foundations	foundation	NOUN
ejpam-5930	280	13	of	of	ADP
ejpam-5930	280	14	rst	rst	PROPN
ejpam-5930	280	15	but	but	CCONJ
ejpam-5930	280	16	also	also	ADV
ejpam-5930	280	17	open	open	VERB
ejpam-5930	280	18	new	new	ADJ
ejpam-5930	280	19	pathways	pathway	NOUN
ejpam-5930	280	20	for	for	ADP
ejpam-5930	280	21	applications	application	NOUN
ejpam-5930	280	22	in	in	ADP
ejpam-5930	280	23	mathematical	mathematical	ADJ
ejpam-5930	280	24	logic	logic	NOUN
ejpam-5930	280	25	,	,	PUNCT
ejpam-5930	280	26	fuzzy	fuzzy	ADJ
ejpam-5930	280	27	systems	system	NOUN
ejpam-5930	280	28	,	,	PUNCT
ejpam-5930	280	29	and	and	CCONJ
ejpam-5930	280	30	artificial	artificial	ADJ
ejpam-5930	280	31	intelligence	intelligence	NOUN
ejpam-5930	280	32	.	.	PUNCT
ejpam-5930	281	1	future	future	ADJ
ejpam-5930	281	2	research	research	NOUN
ejpam-5930	281	3	may	may	AUX
ejpam-5930	281	4	focus	focus	VERB
ejpam-5930	281	5	on	on	ADP
ejpam-5930	281	6	further	further	ADJ
ejpam-5930	281	7	generalizations	generalization	NOUN
ejpam-5930	281	8	of	of	ADP
ejpam-5930	281	9	rough	rough	ADJ
ejpam-5930	281	10	approximations	approximation	NOUN
ejpam-5930	281	11	in	in	ADP
ejpam-5930	281	12	broader	broad	ADJ
ejpam-5930	281	13	algebraic	algebraic	ADJ
ejpam-5930	281	14	settings	setting	NOUN
ejpam-5930	281	15	or	or	CCONJ
ejpam-5930	281	16	their	their	PRON
ejpam-5930	281	17	potential	potential	ADJ
ejpam-5930	281	18	applications	application	NOUN
ejpam-5930	281	19	in	in	ADP
ejpam-5930	281	20	knowledge	knowledge	NOUN
ejpam-5930	281	21	representation	representation	NOUN
ejpam-5930	281	22	and	and	CCONJ
ejpam-5930	281	23	uncertainty	uncertainty	NOUN
ejpam-5930	281	24	reasoning	reasoning	NOUN
ejpam-5930	281	25	.	.	PUNCT
ejpam-5930	282	1	acknowledgements	acknowledgement	NOUN
ejpam-5930	282	2	this	this	DET
ejpam-5930	282	3	research	research	NOUN
ejpam-5930	282	4	was	be	AUX
ejpam-5930	282	5	supported	support	VERB
ejpam-5930	282	6	by	by	ADP
ejpam-5930	282	7	university	university	NOUN
ejpam-5930	282	8	of	of	ADP
ejpam-5930	282	9	phayao	phayao	NOUN
ejpam-5930	282	10	and	and	CCONJ
ejpam-5930	282	11	thailand	thailand	PROPN
ejpam-5930	282	12	science	science	PROPN
ejpam-5930	282	13	research	research	PROPN
ejpam-5930	282	14	and	and	CCONJ
ejpam-5930	282	15	innovation	innovation	NOUN
ejpam-5930	282	16	fund	fund	NOUN
ejpam-5930	282	17	(	(	PUNCT
ejpam-5930	282	18	fundamental	fundamental	ADJ
ejpam-5930	282	19	fund	fund	NOUN
ejpam-5930	282	20	2025	2025	NUM
ejpam-5930	282	21	,	,	PUNCT
ejpam-5930	282	22	grant	grant	VERB
ejpam-5930	282	23	no	no	NOUN
ejpam-5930	282	24	.	.	PROPN
ejpam-5930	283	1	5027/2567	5027/2567	NUM
ejpam-5930	283	2	)	)	PUNCT
ejpam-5930	283	3	.	.	PUNCT
ejpam-5930	284	1	references	reference	NOUN
ejpam-5930	284	2	[	[	X
ejpam-5930	284	3	1	1	NUM
ejpam-5930	284	4	]	]	PUNCT
ejpam-5930	284	5	z.	z.	PROPN
ejpam-5930	284	6	pawlak	pawlak	PROPN
ejpam-5930	284	7	.	.	PUNCT
ejpam-5930	285	1	rough	rough	ADJ
ejpam-5930	285	2	sets	set	NOUN
ejpam-5930	285	3	.	.	PUNCT
ejpam-5930	286	1	int	int	NOUN
ejpam-5930	286	2	.	.	PUNCT
ejpam-5930	287	1	j.	j.	PROPN
ejpam-5930	287	2	comput	comput	PROPN
ejpam-5930	287	3	.	.	PUNCT
ejpam-5930	288	1	inf	inf	PROPN
ejpam-5930	288	2	.	.	PUNCT
ejpam-5930	289	1	sci	sci	PROPN
ejpam-5930	289	2	.	.	PROPN
ejpam-5930	289	3	,	,	PUNCT
ejpam-5930	289	4	11(5):341–356	11(5):341–356	PROPN
ejpam-5930	289	5	,	,	PUNCT
ejpam-5930	289	6	1982	1982	NUM
ejpam-5930	289	7	.	.	PUNCT
ejpam-5930	290	1	[	[	X
ejpam-5930	290	2	2	2	NUM
ejpam-5930	290	3	]	]	PUNCT
ejpam-5930	290	4	z.	z.	PROPN
ejpam-5930	290	5	pawlak	pawlak	PROPN
ejpam-5930	290	6	.	.	PUNCT
ejpam-5930	291	1	rough	rough	ADJ
ejpam-5930	291	2	sets	set	NOUN
ejpam-5930	291	3	-	-	PUNCT
ejpam-5930	291	4	theoretical	theoretical	ADJ
ejpam-5930	291	5	aspects	aspect	NOUN
ejpam-5930	291	6	of	of	ADP
ejpam-5930	291	7	reasoning	reasoning	NOUN
ejpam-5930	291	8	about	about	ADP
ejpam-5930	291	9	data	datum	NOUN
ejpam-5930	291	10	.	.	PUNCT
ejpam-5930	292	1	kluwer	kluwer	NOUN
ejpam-5930	292	2	academic	academic	ADJ
ejpam-5930	292	3	publishing	publishing	NOUN
ejpam-5930	292	4	,	,	PUNCT
ejpam-5930	292	5	dordrecht	dordrecht	PROPN
ejpam-5930	292	6	,	,	PUNCT
ejpam-5930	292	7	1991	1991	NUM
ejpam-5930	292	8	.	.	PUNCT
ejpam-5930	293	1	[	[	X
ejpam-5930	293	2	3	3	X
ejpam-5930	293	3	]	]	X
ejpam-5930	293	4	m.	m.	NOUN
ejpam-5930	293	5	quafafou	quafafou	PROPN
ejpam-5930	293	6	.	.	PUNCT
ejpam-5930	294	1	α	α	X
ejpam-5930	294	2	-	-	PUNCT
ejpam-5930	294	3	rst	rst	ADJ
ejpam-5930	294	4	:	:	PUNCT
ejpam-5930	294	5	a	a	DET
ejpam-5930	294	6	generalization	generalization	NOUN
ejpam-5930	294	7	of	of	ADP
ejpam-5930	294	8	rough	rough	ADJ
ejpam-5930	294	9	set	set	NOUN
ejpam-5930	294	10	theory	theory	NOUN
ejpam-5930	294	11	.	.	PUNCT
ejpam-5930	295	1	inf	inf	PROPN
ejpam-5930	295	2	.	.	PUNCT
ejpam-5930	296	1	sci	sci	PROPN
ejpam-5930	296	2	.	.	PROPN
ejpam-5930	296	3	,	,	PUNCT
ejpam-5930	296	4	124(1	124(1	NUM
ejpam-5930	296	5	-	-	SYM
ejpam-5930	296	6	4):301–316	4):301–316	NUM
ejpam-5930	296	7	,	,	PUNCT
ejpam-5930	296	8	2000	2000	NUM
ejpam-5930	296	9	.	.	PUNCT
ejpam-5930	297	1	[	[	X
ejpam-5930	297	2	4	4	NUM
ejpam-5930	297	3	]	]	X
ejpam-5930	297	4	w.-z	w.-z	NOUN
ejpam-5930	297	5	.	.	PUNCT
ejpam-5930	298	1	wu	wu	PROPN
ejpam-5930	298	2	and	and	CCONJ
ejpam-5930	298	3	w.-x	w.-x	PROPN
ejpam-5930	298	4	.	.	PUNCT
ejpam-5930	299	1	zhang	zhang	PROPN
ejpam-5930	299	2	.	.	PUNCT
ejpam-5930	299	3	neighborhood	neighborhood	NOUN
ejpam-5930	299	4	operator	operator	NOUN
ejpam-5930	299	5	systems	system	NOUN
ejpam-5930	299	6	and	and	CCONJ
ejpam-5930	299	7	approximations	approximation	NOUN
ejpam-5930	299	8	.	.	PUNCT
ejpam-5930	300	1	inf	inf	PROPN
ejpam-5930	300	2	.	.	PUNCT
ejpam-5930	301	1	sci	sci	PROPN
ejpam-5930	301	2	.	.	PROPN
ejpam-5930	301	3	,	,	PUNCT
ejpam-5930	301	4	144(1	144(1	PROPN
ejpam-5930	301	5	-	-	SYM
ejpam-5930	301	6	4):201–217	4):201–217	NUM
ejpam-5930	301	7	,	,	PUNCT
ejpam-5930	301	8	2002	2002	NUM
ejpam-5930	301	9	.	.	PUNCT
ejpam-5930	302	1	[	[	X
ejpam-5930	302	2	5	5	X
ejpam-5930	302	3	]	]	PUNCT
ejpam-5930	302	4	y.	y.	PROPN
ejpam-5930	302	5	y.	y.	PROPN
ejpam-5930	302	6	yao	yao	PROPN
ejpam-5930	302	7	.	.	PUNCT
ejpam-5930	303	1	relational	relational	ADJ
ejpam-5930	303	2	interpretations	interpretation	NOUN
ejpam-5930	303	3	of	of	ADP
ejpam-5930	303	4	neighborhood	neighborhood	NOUN
ejpam-5930	303	5	operators	operator	NOUN
ejpam-5930	303	6	and	and	CCONJ
ejpam-5930	303	7	rough	rough	ADJ
ejpam-5930	303	8	set	set	NOUN
ejpam-5930	303	9	approximation	approximation	NOUN
ejpam-5930	303	10	operators	operator	NOUN
ejpam-5930	303	11	.	.	PUNCT
ejpam-5930	304	1	inf	inf	PROPN
ejpam-5930	304	2	.	.	PUNCT
ejpam-5930	305	1	sci	sci	PROPN
ejpam-5930	305	2	.	.	PROPN
ejpam-5930	305	3	,	,	PUNCT
ejpam-5930	305	4	111(1	111(1	NUM
ejpam-5930	305	5	-	-	SYM
ejpam-5930	305	6	4):239–259	4):239–259	NUM
ejpam-5930	305	7	,	,	PUNCT
ejpam-5930	305	8	1998	1998	NUM
ejpam-5930	305	9	.	.	PUNCT
ejpam-5930	306	1	[	[	X
ejpam-5930	306	2	6	6	NUM
ejpam-5930	306	3	]	]	PUNCT
ejpam-5930	306	4	s.	s.	PROPN
ejpam-5930	306	5	d.	d.	PROPN
ejpam-5930	306	6	comer	comer	PROPN
ejpam-5930	306	7	.	.	PUNCT
ejpam-5930	307	1	on	on	ADP
ejpam-5930	307	2	connections	connection	NOUN
ejpam-5930	307	3	between	between	ADP
ejpam-5930	307	4	information	information	NOUN
ejpam-5930	307	5	systems	system	NOUN
ejpam-5930	307	6	,	,	PUNCT
ejpam-5930	307	7	rough	rough	ADJ
ejpam-5930	307	8	sets	set	NOUN
ejpam-5930	307	9	and	and	CCONJ
ejpam-5930	307	10	algebraic	algebraic	ADJ
ejpam-5930	307	11	logic	logic	NOUN
ejpam-5930	307	12	.	.	PUNCT
ejpam-5930	308	1	algebr	algebr	NOUN
ejpam-5930	308	2	.	.	PUNCT
ejpam-5930	309	1	methods	method	NOUN
ejpam-5930	309	2	log	log	VERB
ejpam-5930	309	3	.	.	PUNCT
ejpam-5930	310	1	comput	comput	NOUN
ejpam-5930	310	2	.	.	PUNCT
ejpam-5930	311	1	sci	sci	PROPN
ejpam-5930	311	2	.	.	PROPN
ejpam-5930	311	3	,	,	PUNCT
ejpam-5930	311	4	28(1):117–124	28(1):117–124	NUM
ejpam-5930	311	5	,	,	PUNCT
ejpam-5930	311	6	1993	1993	NUM
ejpam-5930	311	7	.	.	PUNCT
ejpam-5930	312	1	[	[	X
ejpam-5930	312	2	7	7	X
ejpam-5930	312	3	]	]	X
ejpam-5930	312	4	l.	l.	PROPN
ejpam-5930	312	5	henkin	henkin	PROPN
ejpam-5930	312	6	.	.	PUNCT
ejpam-5930	313	1	an	an	DET
ejpam-5930	313	2	algebraic	algebraic	ADJ
ejpam-5930	313	3	characterization	characterization	NOUN
ejpam-5930	313	4	of	of	ADP
ejpam-5930	313	5	quantifiers	quantifier	NOUN
ejpam-5930	313	6	.	.	PUNCT
ejpam-5930	314	1	fund	fund	NOUN
ejpam-5930	314	2	.	.	PUNCT
ejpam-5930	315	1	math	math	NOUN
ejpam-5930	315	2	.	.	PUNCT
ejpam-5930	315	3	,	,	PUNCT
ejpam-5930	316	1	37:63–74	37:63–74	NUM
ejpam-5930	316	2	,	,	PUNCT
ejpam-5930	316	3	1950	1950	NUM
ejpam-5930	316	4	.	.	PUNCT
ejpam-5930	317	1	a.	a.	NOUN
ejpam-5930	317	2	iampan	iampan	PROPN
ejpam-5930	317	3	et	et	PROPN
ejpam-5930	317	4	al	al	PROPN
ejpam-5930	317	5	.	.	PUNCT
ejpam-5930	317	6	/	/	SYM
ejpam-5930	317	7	eur	eur	PROPN
ejpam-5930	317	8	.	.	PUNCT
ejpam-5930	318	1	j.	j.	PROPN
ejpam-5930	318	2	pure	pure	PROPN
ejpam-5930	318	3	appl	appl	PROPN
ejpam-5930	318	4	.	.	PROPN
ejpam-5930	318	5	math	math	PROPN
ejpam-5930	318	6	,	,	PUNCT
ejpam-5930	318	7	18	18	NUM
ejpam-5930	318	8	(	(	PUNCT
ejpam-5930	318	9	2	2	NUM
ejpam-5930	318	10	)	)	PUNCT
ejpam-5930	318	11	(	(	PUNCT
ejpam-5930	318	12	2025	2025	NUM
ejpam-5930	318	13	)	)	PUNCT
ejpam-5930	318	14	,	,	PUNCT
ejpam-5930	318	15	5930	5930	NUM
ejpam-5930	318	16	11	11	NUM
ejpam-5930	318	17	of	of	ADP
ejpam-5930	318	18	11	11	NUM
ejpam-5930	318	19	[	[	SYM
ejpam-5930	318	20	8	8	NUM
ejpam-5930	318	21	]	]	PUNCT
ejpam-5930	318	22	a.	a.	NOUN
ejpam-5930	318	23	diego	diego	PROPN
ejpam-5930	318	24	.	.	PUNCT
ejpam-5930	319	1	sur	sur	PROPN
ejpam-5930	319	2	les	les	PROPN
ejpam-5930	319	3	algébres	algébres	PROPN
ejpam-5930	319	4	de	de	X
ejpam-5930	319	5	hilbert	hilbert	PROPN
ejpam-5930	319	6	.	.	PUNCT
ejpam-5930	320	1	collection	collection	PROPN
ejpam-5930	320	2	de	de	X
ejpam-5930	320	3	logique	logique	X
ejpam-5930	320	4	math	math	PROPN
ejpam-5930	320	5	.	.	PUNCT
ejpam-5930	321	1	ser	ser	PROPN
ejpam-5930	321	2	.	.	PUNCT
ejpam-5930	322	1	a	a	DET
ejpam-5930	322	2	(	(	PUNCT
ejpam-5930	322	3	ed	ed	NOUN
ejpam-5930	322	4	.	.	PUNCT
ejpam-5930	322	5	hermann	hermann	PROPN
ejpam-5930	322	6	,	,	PUNCT
ejpam-5930	322	7	paris	paris	PROPN
ejpam-5930	322	8	)	)	PUNCT
ejpam-5930	322	9	,	,	PUNCT
ejpam-5930	322	10	21:1–52	21:1–52	NUM
ejpam-5930	322	11	,	,	PUNCT
ejpam-5930	322	12	1966	1966	NUM
ejpam-5930	322	13	.	.	PUNCT
ejpam-5930	323	1	[	[	X
ejpam-5930	323	2	9	9	NUM
ejpam-5930	323	3	]	]	X
ejpam-5930	323	4	d.	d.	PROPN
ejpam-5930	323	5	busneag	busneag	PROPN
ejpam-5930	323	6	.	.	PUNCT
ejpam-5930	324	1	a	a	DET
ejpam-5930	324	2	note	note	NOUN
ejpam-5930	324	3	on	on	ADP
ejpam-5930	324	4	deductive	deductive	ADJ
ejpam-5930	324	5	systems	system	NOUN
ejpam-5930	324	6	of	of	ADP
ejpam-5930	324	7	a	a	DET
ejpam-5930	324	8	hilbert	hilbert	NOUN
ejpam-5930	324	9	algebra	algebra	NOUN
ejpam-5930	324	10	.	.	PUNCT
ejpam-5930	325	1	kobe	kobe	PROPN
ejpam-5930	325	2	j.	j.	PROPN
ejpam-5930	325	3	math	math	PROPN
ejpam-5930	325	4	.	.	PROPN
ejpam-5930	325	5	,	,	PUNCT
ejpam-5930	325	6	2:29–35	2:29–35	NUM
ejpam-5930	325	7	,	,	PUNCT
ejpam-5930	325	8	1985	1985	NUM
ejpam-5930	325	9	.	.	PUNCT
ejpam-5930	326	1	[	[	X
ejpam-5930	326	2	10	10	NUM
ejpam-5930	326	3	]	]	X
ejpam-5930	326	4	d.	d.	PROPN
ejpam-5930	326	5	busneag	busneag	PROPN
ejpam-5930	326	6	.	.	PUNCT
ejpam-5930	327	1	hilbert	hilbert	PROPN
ejpam-5930	327	2	algebras	algebras	PROPN
ejpam-5930	327	3	of	of	ADP
ejpam-5930	327	4	fractions	fraction	NOUN
ejpam-5930	327	5	and	and	CCONJ
ejpam-5930	327	6	maximal	maximal	ADJ
ejpam-5930	327	7	hilbert	hilbert	NOUN
ejpam-5930	327	8	algebras	algebra	NOUN
ejpam-5930	327	9	of	of	ADP
ejpam-5930	327	10	quotients	quotient	NOUN
ejpam-5930	327	11	.	.	PUNCT
ejpam-5930	328	1	kobe	kobe	PROPN
ejpam-5930	328	2	j.	j.	PROPN
ejpam-5930	328	3	math	math	PROPN
ejpam-5930	328	4	.	.	PUNCT
ejpam-5930	328	5	,	,	PUNCT
ejpam-5930	328	6	5:161–172	5:161–172	NOUN
ejpam-5930	328	7	,	,	PUNCT
ejpam-5930	328	8	1988	1988	NUM
ejpam-5930	328	9	.	.	PUNCT
ejpam-5930	329	1	[	[	X
ejpam-5930	329	2	11	11	NUM
ejpam-5930	329	3	]	]	X
ejpam-5930	329	4	y.	y.	PROPN
ejpam-5930	329	5	b.	b.	PROPN
ejpam-5930	329	6	jun	jun	PROPN
ejpam-5930	329	7	.	.	PROPN
ejpam-5930	329	8	deductive	deductive	ADJ
ejpam-5930	329	9	systems	system	NOUN
ejpam-5930	329	10	of	of	ADP
ejpam-5930	329	11	hilbert	hilbert	PROPN
ejpam-5930	329	12	algebras	algebras	PROPN
ejpam-5930	329	13	.	.	PUNCT
ejpam-5930	330	1	math	math	PROPN
ejpam-5930	330	2	.	.	PUNCT
ejpam-5930	331	1	japon	japon	PROPN
ejpam-5930	331	2	.	.	PROPN
ejpam-5930	331	3	,	,	PUNCT
ejpam-5930	331	4	43:51–54	43:51–54	NUM
ejpam-5930	331	5	,	,	PUNCT
ejpam-5930	331	6	1996	1996	NUM
ejpam-5930	331	7	.	.	PUNCT
ejpam-5930	332	1	[	[	X
ejpam-5930	332	2	12	12	NUM
ejpam-5930	332	3	]	]	PUNCT
ejpam-5930	332	4	w.	w.	PROPN
ejpam-5930	332	5	a.	a.	PROPN
ejpam-5930	332	6	dudek	dudek	PROPN
ejpam-5930	332	7	.	.	PUNCT
ejpam-5930	333	1	on	on	ADP
ejpam-5930	333	2	fuzzification	fuzzification	NOUN
ejpam-5930	333	3	in	in	ADP
ejpam-5930	333	4	hilbert	hilbert	PROPN
ejpam-5930	333	5	algebras	algebras	PROPN
ejpam-5930	333	6	.	.	PUNCT
ejpam-5930	334	1	contrib	contrib	PROPN
ejpam-5930	334	2	.	.	PUNCT
ejpam-5930	334	3	gen	gen	PROPN
ejpam-5930	334	4	.	.	PROPN
ejpam-5930	334	5	algebra	algebra	PROPN
ejpam-5930	334	6	,	,	PUNCT
ejpam-5930	334	7	11:77–83	11:77–83	NUM
ejpam-5930	334	8	,	,	PUNCT
ejpam-5930	334	9	1999	1999	NUM
ejpam-5930	334	10	.	.	PUNCT
ejpam-5930	335	1	[	[	X
ejpam-5930	335	2	13	13	NUM
ejpam-5930	335	3	]	]	PUNCT
ejpam-5930	335	4	a.	a.	NOUN
ejpam-5930	335	5	borumand	borumand	PROPN
ejpam-5930	335	6	saeid	saeid	PROPN
ejpam-5930	335	7	and	and	CCONJ
ejpam-5930	335	8	m.	m.	PROPN
ejpam-5930	335	9	haveshki	haveshki	PROPN
ejpam-5930	335	10	.	.	PUNCT
ejpam-5930	336	1	approximation	approximation	NOUN
ejpam-5930	336	2	in	in	ADP
ejpam-5930	336	3	hilbert	hilbert	PROPN
ejpam-5930	336	4	algebras	algebras	PROPN
ejpam-5930	336	5	.	.	PUNCT
ejpam-5930	337	1	sci	sci	PROPN
ejpam-5930	337	2	.	.	PROPN
ejpam-5930	337	3	magna	magna	PROPN
ejpam-5930	337	4	,	,	PUNCT
ejpam-5930	337	5	4(4):77–85	4(4):77–85	NUM
ejpam-5930	337	6	,	,	PUNCT
ejpam-5930	337	7	2008	2008	NUM
ejpam-5930	337	8	.	.	PUNCT
ejpam-5930	338	1	[	[	X
ejpam-5930	338	2	14	14	NUM
ejpam-5930	338	3	]	]	X
ejpam-5930	338	4	y.	y.	PROPN
ejpam-5930	338	5	b.	b.	PROPN
ejpam-5930	338	6	jun	jun	PROPN
ejpam-5930	338	7	,	,	PUNCT
ejpam-5930	338	8	j.-w	j.-w	PROPN
ejpam-5930	338	9	.	.	PUNCT
ejpam-5930	339	1	nam	nam	PROPN
ejpam-5930	339	2	,	,	PUNCT
ejpam-5930	339	3	and	and	CCONJ
ejpam-5930	339	4	s.	s.	PROPN
ejpam-5930	339	5	m.	m.	PROPN
ejpam-5930	339	6	hong	hong	PROPN
ejpam-5930	339	7	.	.	PUNCT
ejpam-5930	340	1	a	a	DET
ejpam-5930	340	2	note	note	NOUN
ejpam-5930	340	3	on	on	ADP
ejpam-5930	340	4	hilbert	hilbert	PROPN
ejpam-5930	340	5	algebras	algebras	PROPN
ejpam-5930	340	6	.	.	PUNCT
ejpam-5930	341	1	east	east	PROPN
ejpam-5930	341	2	asian	asian	PROPN
ejpam-5930	341	3	math	math	PROPN
ejpam-5930	341	4	.	.	PUNCT
ejpam-5930	342	1	j.	j.	PROPN
ejpam-5930	342	2	,	,	PUNCT
ejpam-5930	342	3	10(2):279–285	10(2):279–285	PROPN
ejpam-5930	342	4	,	,	PUNCT
ejpam-5930	342	5	1994	1994	NUM
ejpam-5930	342	6	.	.	PUNCT
ejpam-5930	343	1	[	[	X
ejpam-5930	343	2	15	15	NUM
ejpam-5930	343	3	]	]	X
ejpam-5930	343	4	i.	i.	NOUN
ejpam-5930	343	5	chajda	chajda	PROPN
ejpam-5930	343	6	and	and	CCONJ
ejpam-5930	343	7	r.	r.	PROPN
ejpam-5930	343	8	halas	halas	PROPN
ejpam-5930	343	9	.	.	PUNCT
ejpam-5930	344	1	congruences	congruence	NOUN
ejpam-5930	344	2	and	and	CCONJ
ejpam-5930	344	3	ideals	ideal	NOUN
ejpam-5930	344	4	in	in	ADP
ejpam-5930	344	5	hilbert	hilbert	PROPN
ejpam-5930	344	6	algebras	algebras	PROPN
ejpam-5930	344	7	.	.	PUNCT
ejpam-5930	345	1	kyungpook	kyungpook	PROPN
ejpam-5930	345	2	math	math	PROPN
ejpam-5930	345	3	.	.	PUNCT
ejpam-5930	346	1	j.	j.	PROPN
ejpam-5930	346	2	,	,	PUNCT
ejpam-5930	346	3	39(2):429–432	39(2):429–432	PROPN
ejpam-5930	346	4	,	,	PUNCT
ejpam-5930	346	5	1999	1999	NUM
ejpam-5930	346	6	.	.	PUNCT