id	sid	tid	token	lemma	pos
ejpam-3272	1	1	european	european	PROPN
ejpam-3272	1	2	journal	journal	PROPN
ejpam-3272	1	3	of	of	ADP
ejpam-3272	1	4	pure	pure	ADJ
ejpam-3272	1	5	and	and	CCONJ
ejpam-3272	1	6	applied	apply	VERB
ejpam-3272	1	7	mathematics	mathematic	NOUN
ejpam-3272	1	8	vol	vol	NOUN
ejpam-3272	1	9	.	.	PUNCT
ejpam-3272	2	1	11	11	NUM
ejpam-3272	2	2	,	,	PUNCT
ejpam-3272	2	3	no	no	INTJ
ejpam-3272	2	4	.	.	NOUN
ejpam-3272	2	5	3	3	NUM
ejpam-3272	2	6	,	,	PUNCT
ejpam-3272	2	7	2018	2018	NUM
ejpam-3272	2	8	,	,	PUNCT
ejpam-3272	2	9	612	612	NUM
ejpam-3272	2	10	-	-	SYM
ejpam-3272	2	11	627	627	NUM
ejpam-3272	2	12	issn	issn	PROPN
ejpam-3272	2	13	1307	1307	NUM
ejpam-3272	2	14	-	-	SYM
ejpam-3272	2	15	5543	5543	NUM
ejpam-3272	2	16	–	–	PUNCT
ejpam-3272	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3272	2	18	published	publish	VERB
ejpam-3272	2	19	by	by	ADP
ejpam-3272	2	20	new	new	PROPN
ejpam-3272	2	21	york	york	PROPN
ejpam-3272	2	22	business	business	PROPN
ejpam-3272	2	23	global	global	ADJ
ejpam-3272	2	24	separation	separation	NOUN
ejpam-3272	2	25	axioms	axiom	VERB
ejpam-3272	2	26	in	in	ADP
ejpam-3272	2	27	diframes	diframe	NOUN
ejpam-3272	2	28	esra	esra	PROPN
ejpam-3272	2	29	korkmaz1	korkmaz1	PROPN
ejpam-3272	2	30	,	,	PUNCT
ejpam-3272	2	31	rıza	rıza	VERB
ejpam-3272	2	32	ertürk1,∗	ertürk1,∗	PROPN
ejpam-3272	2	33	1	1	NUM
ejpam-3272	2	34	department	department	NOUN
ejpam-3272	2	35	of	of	ADP
ejpam-3272	2	36	mathematics	mathematic	NOUN
ejpam-3272	2	37	,	,	PUNCT
ejpam-3272	2	38	hacettepe	hacettepe	PROPN
ejpam-3272	2	39	university	university	NOUN
ejpam-3272	2	40	,	,	PUNCT
ejpam-3272	2	41	ankara	ankara	PROPN
ejpam-3272	2	42	,	,	PUNCT
ejpam-3272	2	43	turkey	turkey	PROPN
ejpam-3272	2	44	abstract	abstract	NOUN
ejpam-3272	2	45	.	.	PUNCT
ejpam-3272	3	1	ditopological	ditopological	ADJ
ejpam-3272	3	2	texture	texture	ADJ
ejpam-3272	3	3	spaces	space	NOUN
ejpam-3272	3	4	are	be	AUX
ejpam-3272	3	5	simultaneously	simultaneously	ADV
ejpam-3272	3	6	generalizations	generalization	NOUN
ejpam-3272	3	7	of	of	ADP
ejpam-3272	3	8	topological	topological	ADJ
ejpam-3272	3	9	,	,	PUNCT
ejpam-3272	3	10	bitopological	bitopological	ADJ
ejpam-3272	3	11	and	and	CCONJ
ejpam-3272	3	12	fuzzy	fuzzy	ADJ
ejpam-3272	3	13	topological	topological	ADJ
ejpam-3272	3	14	spaces	space	NOUN
ejpam-3272	3	15	,	,	PUNCT
ejpam-3272	3	16	and	and	CCONJ
ejpam-3272	3	17	diframes	diframe	NOUN
ejpam-3272	3	18	are	be	AUX
ejpam-3272	3	19	generalizations	generalization	NOUN
ejpam-3272	3	20	of	of	ADP
ejpam-3272	3	21	ditopological	ditopological	ADJ
ejpam-3272	3	22	texture	texture	ADJ
ejpam-3272	3	23	spaces	space	NOUN
ejpam-3272	3	24	.	.	PUNCT
ejpam-3272	4	1	in	in	ADP
ejpam-3272	4	2	this	this	DET
ejpam-3272	4	3	paper	paper	NOUN
ejpam-3272	4	4	we	we	PRON
ejpam-3272	4	5	define	define	VERB
ejpam-3272	4	6	and	and	CCONJ
ejpam-3272	4	7	study	study	VERB
ejpam-3272	4	8	the	the	DET
ejpam-3272	4	9	separation	separation	NOUN
ejpam-3272	4	10	axioms	axiom	VERB
ejpam-3272	4	11	in	in	ADP
ejpam-3272	4	12	diframe	diframe	NOUN
ejpam-3272	4	13	setting	setting	NOUN
ejpam-3272	4	14	.	.	PUNCT
ejpam-3272	5	1	2010	2010	NUM
ejpam-3272	5	2	mathematics	mathematic	NOUN
ejpam-3272	5	3	subject	subject	NOUN
ejpam-3272	5	4	classifications	classification	NOUN
ejpam-3272	5	5	:	:	PUNCT
ejpam-3272	5	6	06d22	06d22	NOUN
ejpam-3272	5	7	,	,	PUNCT
ejpam-3272	5	8	54a05	54a05	NUM
ejpam-3272	5	9	key	key	ADJ
ejpam-3272	5	10	words	word	NOUN
ejpam-3272	5	11	and	and	CCONJ
ejpam-3272	5	12	phrases	phrase	NOUN
ejpam-3272	5	13	:	:	PUNCT
ejpam-3272	5	14	diframe	diframe	NOUN
ejpam-3272	5	15	,	,	PUNCT
ejpam-3272	5	16	fr	fr	NOUN
ejpam-3272	5	17	-	-	PUNCT
ejpam-3272	5	18	below	below	NOUN
ejpam-3272	5	19	,	,	PUNCT
ejpam-3272	5	20	cf	cf	NOUN
ejpam-3272	5	21	-	-	PUNCT
ejpam-3272	5	22	below	below	NOUN
ejpam-3272	5	23	,	,	PUNCT
ejpam-3272	5	24	urysohn	urysohn	PROPN
ejpam-3272	5	25	relation	relation	NOUN
ejpam-3272	5	26	1	1	NUM
ejpam-3272	5	27	.	.	PUNCT
ejpam-3272	6	1	introduction	introduction	NOUN
ejpam-3272	6	2	the	the	DET
ejpam-3272	6	3	concept	concept	NOUN
ejpam-3272	6	4	of	of	ADP
ejpam-3272	6	5	ditopological	ditopological	ADJ
ejpam-3272	6	6	texture	texture	ADJ
ejpam-3272	6	7	spaces	space	NOUN
ejpam-3272	6	8	grew	grow	VERB
ejpam-3272	6	9	out	out	ADP
ejpam-3272	6	10	of	of	ADP
ejpam-3272	6	11	the	the	DET
ejpam-3272	6	12	study	study	NOUN
ejpam-3272	6	13	of	of	ADP
ejpam-3272	6	14	the	the	DET
ejpam-3272	6	15	representation	representation	NOUN
ejpam-3272	6	16	of	of	ADP
ejpam-3272	6	17	lattice	lattice	NOUN
ejpam-3272	6	18	-	-	PUNCT
ejpam-3272	6	19	valued	value	VERB
ejpam-3272	6	20	topologies	topology	NOUN
ejpam-3272	6	21	by	by	ADP
ejpam-3272	6	22	bitopologies	bitopologie	NOUN
ejpam-3272	6	23	.	.	PUNCT
ejpam-3272	7	1	however	however	ADV
ejpam-3272	7	2	,	,	PUNCT
ejpam-3272	7	3	as	as	ADP
ejpam-3272	7	4	distinct	distinct	ADJ
ejpam-3272	7	5	from	from	ADP
ejpam-3272	7	6	the	the	DET
ejpam-3272	7	7	theory	theory	NOUN
ejpam-3272	7	8	of	of	ADP
ejpam-3272	7	9	bitopological	bitopological	ADJ
ejpam-3272	7	10	spaces	space	NOUN
ejpam-3272	7	11	based	base	VERB
ejpam-3272	7	12	on	on	ADP
ejpam-3272	7	13	the	the	DET
ejpam-3272	7	14	notion	notion	NOUN
ejpam-3272	7	15	of	of	ADP
ejpam-3272	7	16	open	open	ADJ
ejpam-3272	7	17	sets	set	NOUN
ejpam-3272	7	18	,	,	PUNCT
ejpam-3272	7	19	it	it	PRON
ejpam-3272	7	20	is	be	AUX
ejpam-3272	7	21	a	a	DET
ejpam-3272	7	22	structure	structure	NOUN
ejpam-3272	7	23	in	in	ADP
ejpam-3272	7	24	which	which	PRON
ejpam-3272	7	25	the	the	DET
ejpam-3272	7	26	open	open	ADJ
ejpam-3272	7	27	and	and	CCONJ
ejpam-3272	7	28	closed	closed	ADJ
ejpam-3272	7	29	sets	set	NOUN
ejpam-3272	7	30	play	play	VERB
ejpam-3272	7	31	an	an	DET
ejpam-3272	7	32	equal	equal	ADJ
ejpam-3272	7	33	role	role	NOUN
ejpam-3272	7	34	.	.	PUNCT
ejpam-3272	8	1	ditopologies	ditopologie	NOUN
ejpam-3272	8	2	are	be	AUX
ejpam-3272	8	3	defined	define	VERB
ejpam-3272	8	4	on	on	ADP
ejpam-3272	8	5	a	a	DET
ejpam-3272	8	6	suitable	suitable	ADJ
ejpam-3272	8	7	subfamily	subfamily	NOUN
ejpam-3272	8	8	s	s	VERB
ejpam-3272	8	9	⊆	⊆	NUM
ejpam-3272	8	10	p(s	p(s	NOUN
ejpam-3272	8	11	)	)	PUNCT
ejpam-3272	8	12	which	which	PRON
ejpam-3272	8	13	is	be	AUX
ejpam-3272	8	14	,	,	PUNCT
ejpam-3272	8	15	in	in	ADP
ejpam-3272	8	16	fact	fact	NOUN
ejpam-3272	8	17	,	,	PUNCT
ejpam-3272	8	18	a	a	DET
ejpam-3272	8	19	complete	complete	ADJ
ejpam-3272	8	20	,	,	PUNCT
ejpam-3272	8	21	completely	completely	ADV
ejpam-3272	8	22	distributive	distributive	ADJ
ejpam-3272	8	23	lattice	lattice	NOUN
ejpam-3272	8	24	with	with	ADP
ejpam-3272	8	25	the	the	DET
ejpam-3272	8	26	relation	relation	NOUN
ejpam-3272	8	27	of	of	ADP
ejpam-3272	8	28	inclusion	inclusion	NOUN
ejpam-3272	8	29	.	.	PUNCT
ejpam-3272	9	1	ever	ever	ADV
ejpam-3272	9	2	since	since	SCONJ
ejpam-3272	9	3	the	the	DET
ejpam-3272	9	4	theory	theory	NOUN
ejpam-3272	9	5	was	be	AUX
ejpam-3272	9	6	first	first	ADV
ejpam-3272	9	7	introduced	introduce	VERB
ejpam-3272	9	8	by	by	ADP
ejpam-3272	9	9	l.m	l.m	PROPN
ejpam-3272	9	10	.	.	PROPN
ejpam-3272	9	11	brown	brown	PROPN
ejpam-3272	10	1	[	[	X
ejpam-3272	10	2	5	5	NUM
ejpam-3272	10	3	]	]	PUNCT
ejpam-3272	10	4	,	,	PUNCT
ejpam-3272	10	5	topological	topological	ADJ
ejpam-3272	10	6	concepts	concept	NOUN
ejpam-3272	10	7	,	,	PUNCT
ejpam-3272	10	8	such	such	ADJ
ejpam-3272	10	9	as	as	ADP
ejpam-3272	10	10	separation	separation	NOUN
ejpam-3272	10	11	axioms	axiom	NOUN
ejpam-3272	10	12	,	,	PUNCT
ejpam-3272	10	13	compactness	compactness	NOUN
ejpam-3272	10	14	and	and	CCONJ
ejpam-3272	10	15	compactifications	compactification	NOUN
ejpam-3272	10	16	,	,	PUNCT
ejpam-3272	10	17	have	have	AUX
ejpam-3272	10	18	been	be	AUX
ejpam-3272	10	19	studied	study	VERB
ejpam-3272	10	20	in	in	ADP
ejpam-3272	10	21	a	a	DET
ejpam-3272	10	22	series	series	NOUN
ejpam-3272	10	23	of	of	ADP
ejpam-3272	10	24	papers	paper	NOUN
ejpam-3272	10	25	by	by	ADP
ejpam-3272	10	26	l.m	l.m	PROPN
ejpam-3272	10	27	.	.	PROPN
ejpam-3272	10	28	brown	brown	PROPN
ejpam-3272	10	29	and	and	CCONJ
ejpam-3272	10	30	co	co	NOUN
ejpam-3272	10	31	-	-	NOUN
ejpam-3272	10	32	authors	author	NOUN
ejpam-3272	10	33	[	[	X
ejpam-3272	10	34	2–4	2–4	NUM
ejpam-3272	10	35	]	]	PUNCT
ejpam-3272	10	36	.	.	PUNCT
ejpam-3272	11	1	this	this	DET
ejpam-3272	11	2	work	work	NOUN
ejpam-3272	11	3	is	be	AUX
ejpam-3272	11	4	a	a	DET
ejpam-3272	11	5	continuation	continuation	NOUN
ejpam-3272	11	6	of	of	ADP
ejpam-3272	11	7	our	our	PRON
ejpam-3272	11	8	previous	previous	ADJ
ejpam-3272	11	9	paper	paper	NOUN
ejpam-3272	11	10	[	[	X
ejpam-3272	11	11	9	9	NUM
ejpam-3272	11	12	]	]	PUNCT
ejpam-3272	11	13	.	.	PUNCT
ejpam-3272	12	1	in	in	ADP
ejpam-3272	12	2	that	that	DET
ejpam-3272	12	3	paper	paper	NOUN
ejpam-3272	12	4	,	,	PUNCT
ejpam-3272	12	5	we	we	PRON
ejpam-3272	12	6	defined	define	VERB
ejpam-3272	12	7	the	the	DET
ejpam-3272	12	8	notion	notion	NOUN
ejpam-3272	12	9	of	of	ADP
ejpam-3272	12	10	diframe	diframe	NOUN
ejpam-3272	12	11	by	by	ADP
ejpam-3272	12	12	replacing	replace	VERB
ejpam-3272	12	13	a	a	DET
ejpam-3272	12	14	texturing	texturing	NOUN
ejpam-3272	12	15	of	of	ADP
ejpam-3272	12	16	a	a	DET
ejpam-3272	12	17	set	set	NOUN
ejpam-3272	12	18	with	with	ADP
ejpam-3272	12	19	a	a	DET
ejpam-3272	12	20	lattice	lattice	NOUN
ejpam-3272	12	21	which	which	PRON
ejpam-3272	12	22	is	be	AUX
ejpam-3272	12	23	both	both	CCONJ
ejpam-3272	12	24	a	a	DET
ejpam-3272	12	25	frame	frame	NOUN
ejpam-3272	12	26	and	and	CCONJ
ejpam-3272	12	27	a	a	DET
ejpam-3272	12	28	coframe	coframe	NOUN
ejpam-3272	12	29	.	.	PUNCT
ejpam-3272	13	1	we	we	PRON
ejpam-3272	13	2	also	also	ADV
ejpam-3272	13	3	provided	provide	VERB
ejpam-3272	13	4	a	a	DET
ejpam-3272	13	5	link	link	NOUN
ejpam-3272	13	6	between	between	ADP
ejpam-3272	13	7	the	the	DET
ejpam-3272	13	8	morphisms	morphism	NOUN
ejpam-3272	13	9	of	of	ADP
ejpam-3272	13	10	the	the	DET
ejpam-3272	13	11	category	category	NOUN
ejpam-3272	13	12	of	of	ADP
ejpam-3272	13	13	texture	texture	ADJ
ejpam-3272	13	14	spaces	space	NOUN
ejpam-3272	13	15	(	(	PUNCT
ejpam-3272	13	16	drtex	drtex	PROPN
ejpam-3272	13	17	)	)	PUNCT
ejpam-3272	13	18	and	and	CCONJ
ejpam-3272	13	19	the	the	DET
ejpam-3272	13	20	category	category	NOUN
ejpam-3272	13	21	of	of	ADP
ejpam-3272	13	22	frames	frame	NOUN
ejpam-3272	13	23	(	(	PUNCT
ejpam-3272	13	24	frm	frm	PROPN
ejpam-3272	13	25	)	)	PUNCT
ejpam-3272	13	26	.	.	PUNCT
ejpam-3272	14	1	this	this	DET
ejpam-3272	14	2	connection	connection	NOUN
ejpam-3272	14	3	allows	allow	VERB
ejpam-3272	14	4	us	we	PRON
ejpam-3272	14	5	to	to	PART
ejpam-3272	14	6	construct	construct	VERB
ejpam-3272	14	7	the	the	DET
ejpam-3272	14	8	category	category	NOUN
ejpam-3272	14	9	difrm	difrm	NOUN
ejpam-3272	14	10	of	of	ADP
ejpam-3272	14	11	diframes	diframe	NOUN
ejpam-3272	14	12	and	and	CCONJ
ejpam-3272	14	13	diframe	diframe	VERB
ejpam-3272	14	14	homomorphisms	homomorphism	NOUN
ejpam-3272	14	15	.	.	PUNCT
ejpam-3272	15	1	there	there	PRON
ejpam-3272	15	2	are	be	VERB
ejpam-3272	15	3	at	at	ADV
ejpam-3272	15	4	least	least	ADJ
ejpam-3272	15	5	two	two	NUM
ejpam-3272	15	6	reasons	reason	NOUN
ejpam-3272	15	7	why	why	SCONJ
ejpam-3272	15	8	the	the	DET
ejpam-3272	15	9	theory	theory	NOUN
ejpam-3272	15	10	of	of	ADP
ejpam-3272	15	11	diframes	diframe	NOUN
ejpam-3272	15	12	is	be	AUX
ejpam-3272	15	13	important	important	ADJ
ejpam-3272	15	14	.	.	PUNCT
ejpam-3272	16	1	dropping	drop	VERB
ejpam-3272	16	2	the	the	DET
ejpam-3272	16	3	complete	complete	ADJ
ejpam-3272	16	4	distributivity	distributivity	NOUN
ejpam-3272	16	5	condition	condition	NOUN
ejpam-3272	16	6	,	,	PUNCT
ejpam-3272	16	7	which	which	PRON
ejpam-3272	16	8	makes	make	VERB
ejpam-3272	16	9	the	the	DET
ejpam-3272	16	10	texture	texture	NOUN
ejpam-3272	16	11	a	a	DET
ejpam-3272	16	12	spatial	spatial	ADJ
ejpam-3272	16	13	frame	frame	NOUN
ejpam-3272	16	14	,	,	PUNCT
ejpam-3272	16	15	(	(	PUNCT
ejpam-3272	16	16	that	that	ADV
ejpam-3272	16	17	is	is	ADV
ejpam-3272	16	18	,	,	PUNCT
ejpam-3272	16	19	a	a	DET
ejpam-3272	16	20	frame	frame	NOUN
ejpam-3272	16	21	isomorphic	isomorphic	ADJ
ejpam-3272	16	22	to	to	ADP
ejpam-3272	16	23	the	the	DET
ejpam-3272	16	24	lattice	lattice	NOUN
ejpam-3272	16	25	of	of	ADP
ejpam-3272	16	26	open	open	ADJ
ejpam-3272	16	27	sets	set	NOUN
ejpam-3272	16	28	,	,	PUNCT
ejpam-3272	16	29	ω(x	ω(x	NOUN
ejpam-3272	16	30	)	)	PUNCT
ejpam-3272	16	31	,	,	PUNCT
ejpam-3272	16	32	of	of	ADP
ejpam-3272	16	33	a	a	DET
ejpam-3272	16	34	set	set	NOUN
ejpam-3272	16	35	x	x	NOUN
ejpam-3272	16	36	)	)	PUNCT
ejpam-3272	16	37	,	,	PUNCT
ejpam-3272	16	38	we	we	PRON
ejpam-3272	16	39	obtain	obtain	VERB
ejpam-3272	16	40	a	a	DET
ejpam-3272	16	41	larger	large	ADJ
ejpam-3272	16	42	family	family	NOUN
ejpam-3272	16	43	of	of	ADP
ejpam-3272	16	44	lattices	lattice	NOUN
ejpam-3272	16	45	.	.	PUNCT
ejpam-3272	17	1	besides	besides	ADV
ejpam-3272	17	2	,	,	PUNCT
ejpam-3272	17	3	diframe	diframe	NOUN
ejpam-3272	17	4	theory	theory	NOUN
ejpam-3272	17	5	initiates	initiate	VERB
ejpam-3272	17	6	the	the	DET
ejpam-3272	17	7	frame	frame	NOUN
ejpam-3272	17	8	-	-	PUNCT
ejpam-3272	17	9	theoretical	theoretical	ADJ
ejpam-3272	17	10	perspective	perspective	NOUN
ejpam-3272	17	11	in	in	ADP
ejpam-3272	17	12	the	the	DET
ejpam-3272	17	13	theory	theory	NOUN
ejpam-3272	17	14	of	of	ADP
ejpam-3272	17	15	ditopological	ditopological	ADJ
ejpam-3272	17	16	spaces	space	NOUN
ejpam-3272	17	17	.	.	PUNCT
ejpam-3272	18	1	it	it	PRON
ejpam-3272	18	2	is	be	AUX
ejpam-3272	18	3	well	well	ADV
ejpam-3272	18	4	-	-	PUNCT
ejpam-3272	18	5	known	know	VERB
ejpam-3272	18	6	that	that	SCONJ
ejpam-3272	18	7	the	the	DET
ejpam-3272	18	8	frame	frame	NOUN
ejpam-3272	18	9	(	(	PUNCT
ejpam-3272	18	10	locale	locale	NOUN
ejpam-3272	18	11	)	)	PUNCT
ejpam-3272	18	12	theory	theory	NOUN
ejpam-3272	18	13	is	be	AUX
ejpam-3272	18	14	an	an	DET
ejpam-3272	18	15	important	important	ADJ
ejpam-3272	18	16	area	area	NOUN
ejpam-3272	18	17	of	of	ADP
ejpam-3272	18	18	research	research	NOUN
ejpam-3272	18	19	and	and	CCONJ
ejpam-3272	18	20	it	it	PRON
ejpam-3272	18	21	translates	translate	VERB
ejpam-3272	18	22	the	the	DET
ejpam-3272	18	23	(	(	PUNCT
ejpam-3272	18	24	bi)topological	bi)topological	ADJ
ejpam-3272	18	25	concepts	concept	NOUN
ejpam-3272	18	26	into	into	ADP
ejpam-3272	18	27	the	the	DET
ejpam-3272	18	28	point	point	NOUN
ejpam-3272	18	29	-	-	PUNCT
ejpam-3272	18	30	free	free	ADJ
ejpam-3272	18	31	language	language	NOUN
ejpam-3272	18	32	[	[	X
ejpam-3272	18	33	1	1	NUM
ejpam-3272	18	34	,	,	PUNCT
ejpam-3272	18	35	10	10	NUM
ejpam-3272	18	36	]	]	PUNCT
ejpam-3272	18	37	.	.	PUNCT
ejpam-3272	19	1	∗corresponding	∗corresponde	VERB
ejpam-3272	19	2	author	author	NOUN
ejpam-3272	19	3	.	.	PUNCT
ejpam-3272	20	1	doi	doi	NOUN
ejpam-3272	20	2	:	:	PUNCT
ejpam-3272	20	3	https://doi.org/10.29020/nybg.ejpam.v11i3.3272	https://doi.org/10.29020/nybg.ejpam.v11i3.3272	ADJ
ejpam-3272	20	4	email	email	NOUN
ejpam-3272	20	5	addresses	address	NOUN
ejpam-3272	20	6	:	:	PUNCT
ejpam-3272	20	7	esrakaratas@hacettepe.edu.tr	esrakaratas@hacettepe.edu.tr	PROPN
ejpam-3272	20	8	(	(	PUNCT
ejpam-3272	20	9	e.	e.	PROPN
ejpam-3272	20	10	korkmaz	korkmaz	PROPN
ejpam-3272	20	11	)	)	PUNCT
ejpam-3272	20	12	,	,	PUNCT
ejpam-3272	20	13	rerturk@hacettepe.edu.tr	rerturk@hacettepe.edu.tr	X
ejpam-3272	20	14	(	(	PUNCT
ejpam-3272	20	15	r.	r.	PROPN
ejpam-3272	20	16	ertürk	ertürk	NOUN
ejpam-3272	20	17	)	)	PUNCT
ejpam-3272	20	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3272	21	1	612	612	NUM
ejpam-3272	21	2	c	c	NOUN
ejpam-3272	21	3	©	©	PROPN
ejpam-3272	21	4	2018	2018	NUM
ejpam-3272	21	5	ejpam	ejpam	VERB
ejpam-3272	21	6	all	all	DET
ejpam-3272	21	7	rights	right	NOUN
ejpam-3272	21	8	reserved	reserve	VERB
ejpam-3272	21	9	.	.	PUNCT
ejpam-3272	22	1	e.	e.	PROPN
ejpam-3272	22	2	korkmaz	korkmaz	PROPN
ejpam-3272	22	3	,	,	PUNCT
ejpam-3272	22	4	r.	r.	PROPN
ejpam-3272	22	5	ertürk	ertürk	PROPN
ejpam-3272	22	6	/	/	SYM
ejpam-3272	22	7	eur	eur	PROPN
ejpam-3272	22	8	.	.	PUNCT
ejpam-3272	23	1	j.	j.	PROPN
ejpam-3272	23	2	pure	pure	PROPN
ejpam-3272	23	3	appl	appl	PROPN
ejpam-3272	23	4	.	.	PROPN
ejpam-3272	23	5	math	math	PROPN
ejpam-3272	23	6	,	,	PUNCT
ejpam-3272	23	7	11	11	NUM
ejpam-3272	23	8	(	(	PUNCT
ejpam-3272	23	9	3	3	NUM
ejpam-3272	23	10	)	)	PUNCT
ejpam-3272	23	11	(	(	PUNCT
ejpam-3272	23	12	2018	2018	NUM
ejpam-3272	23	13	)	)	PUNCT
ejpam-3272	23	14	,	,	PUNCT
ejpam-3272	23	15	612	612	NUM
ejpam-3272	23	16	-	-	SYM
ejpam-3272	23	17	627	627	NUM
ejpam-3272	23	18	613	613	NUM
ejpam-3272	23	19	the	the	DET
ejpam-3272	23	20	rest	rest	NOUN
ejpam-3272	23	21	of	of	ADP
ejpam-3272	23	22	this	this	DET
ejpam-3272	23	23	paper	paper	NOUN
ejpam-3272	23	24	is	be	AUX
ejpam-3272	23	25	structured	structure	VERB
ejpam-3272	23	26	as	as	SCONJ
ejpam-3272	23	27	follows	follow	VERB
ejpam-3272	23	28	.	.	PUNCT
ejpam-3272	24	1	in	in	ADP
ejpam-3272	24	2	the	the	DET
ejpam-3272	24	3	second	second	ADJ
ejpam-3272	24	4	section	section	NOUN
ejpam-3272	24	5	,	,	PUNCT
ejpam-3272	24	6	some	some	DET
ejpam-3272	24	7	basic	basic	ADJ
ejpam-3272	24	8	concepts	concept	NOUN
ejpam-3272	24	9	and	and	CCONJ
ejpam-3272	24	10	properties	property	NOUN
ejpam-3272	24	11	of	of	ADP
ejpam-3272	24	12	ditopological	ditopological	ADJ
ejpam-3272	24	13	texture	texture	ADJ
ejpam-3272	24	14	spaces	space	NOUN
ejpam-3272	24	15	and	and	CCONJ
ejpam-3272	24	16	frames	frame	NOUN
ejpam-3272	24	17	are	be	AUX
ejpam-3272	24	18	introduced	introduce	VERB
ejpam-3272	24	19	to	to	PART
ejpam-3272	24	20	make	make	VERB
ejpam-3272	24	21	the	the	DET
ejpam-3272	24	22	paper	paper	NOUN
ejpam-3272	24	23	self	self	NOUN
ejpam-3272	24	24	-	-	PUNCT
ejpam-3272	24	25	contained	contain	VERB
ejpam-3272	24	26	.	.	PUNCT
ejpam-3272	25	1	in	in	ADP
ejpam-3272	25	2	the	the	DET
ejpam-3272	25	3	third	third	ADJ
ejpam-3272	25	4	section	section	NOUN
ejpam-3272	25	5	,	,	PUNCT
ejpam-3272	25	6	we	we	PRON
ejpam-3272	25	7	define	define	VERB
ejpam-3272	25	8	the	the	DET
ejpam-3272	25	9	separation	separation	NOUN
ejpam-3272	25	10	axioms	axiom	NOUN
ejpam-3272	25	11	in	in	ADP
ejpam-3272	25	12	the	the	DET
ejpam-3272	25	13	setting	setting	NOUN
ejpam-3272	25	14	of	of	ADP
ejpam-3272	25	15	diframes	diframe	NOUN
ejpam-3272	25	16	and	and	CCONJ
ejpam-3272	25	17	we	we	PRON
ejpam-3272	25	18	obtain	obtain	VERB
ejpam-3272	25	19	equivalent	equivalent	ADJ
ejpam-3272	25	20	characterizations	characterization	NOUN
ejpam-3272	25	21	of	of	ADP
ejpam-3272	25	22	these	these	DET
ejpam-3272	25	23	axioms	axiom	NOUN
ejpam-3272	25	24	.	.	PUNCT
ejpam-3272	26	1	finally	finally	ADV
ejpam-3272	26	2	,	,	PUNCT
ejpam-3272	26	3	the	the	DET
ejpam-3272	26	4	conclusion	conclusion	NOUN
ejpam-3272	26	5	of	of	ADP
ejpam-3272	26	6	this	this	DET
ejpam-3272	26	7	paper	paper	NOUN
ejpam-3272	26	8	and	and	CCONJ
ejpam-3272	26	9	some	some	DET
ejpam-3272	26	10	future	future	ADJ
ejpam-3272	26	11	works	work	NOUN
ejpam-3272	26	12	are	be	AUX
ejpam-3272	26	13	discussed	discuss	VERB
ejpam-3272	26	14	in	in	ADP
ejpam-3272	26	15	section	section	NOUN
ejpam-3272	26	16	4	4	NUM
ejpam-3272	26	17	.	.	NOUN
ejpam-3272	26	18	2	2	NUM
ejpam-3272	26	19	.	.	NUM
ejpam-3272	26	20	preliminaries	preliminary	NOUN
ejpam-3272	26	21	in	in	ADP
ejpam-3272	26	22	this	this	DET
ejpam-3272	26	23	section	section	NOUN
ejpam-3272	26	24	,	,	PUNCT
ejpam-3272	26	25	we	we	PRON
ejpam-3272	26	26	recall	recall	VERB
ejpam-3272	26	27	some	some	DET
ejpam-3272	26	28	pertinent	pertinent	ADJ
ejpam-3272	26	29	concepts	concept	NOUN
ejpam-3272	26	30	of	of	ADP
ejpam-3272	26	31	ditopological	ditopological	ADJ
ejpam-3272	26	32	texture	texture	ADJ
ejpam-3272	26	33	spaces	space	NOUN
ejpam-3272	26	34	,	,	PUNCT
ejpam-3272	26	35	(	(	PUNCT
ejpam-3272	26	36	co)frames	co)frame	NOUN
ejpam-3272	26	37	and	and	CCONJ
ejpam-3272	26	38	diframes	diframe	NOUN
ejpam-3272	26	39	.	.	PUNCT
ejpam-3272	27	1	we	we	PRON
ejpam-3272	27	2	refer	refer	VERB
ejpam-3272	27	3	to	to	ADP
ejpam-3272	27	4	[	[	X
ejpam-3272	27	5	2	2	NUM
ejpam-3272	27	6	,	,	PUNCT
ejpam-3272	27	7	3	3	NUM
ejpam-3272	27	8	]	]	PUNCT
ejpam-3272	27	9	and	and	CCONJ
ejpam-3272	27	10	[	[	X
ejpam-3272	27	11	4	4	X
ejpam-3272	27	12	]	]	PUNCT
ejpam-3272	27	13	for	for	ADP
ejpam-3272	27	14	ditopological	ditopological	ADJ
ejpam-3272	27	15	texture	texture	ADJ
ejpam-3272	27	16	spaces	space	NOUN
ejpam-3272	27	17	,	,	PUNCT
ejpam-3272	27	18	and	and	CCONJ
ejpam-3272	27	19	to	to	ADP
ejpam-3272	27	20	[	[	X
ejpam-3272	27	21	6	6	NUM
ejpam-3272	27	22	]	]	PUNCT
ejpam-3272	27	23	and	and	CCONJ
ejpam-3272	27	24	[	[	X
ejpam-3272	27	25	10	10	NUM
ejpam-3272	27	26	]	]	PUNCT
ejpam-3272	27	27	for	for	ADP
ejpam-3272	27	28	lattice	lattice	NOUN
ejpam-3272	27	29	and	and	CCONJ
ejpam-3272	27	30	frame	frame	NOUN
ejpam-3272	27	31	theory	theory	NOUN
ejpam-3272	27	32	.	.	PUNCT
ejpam-3272	28	1	ditopological	ditopological	ADJ
ejpam-3272	28	2	texture	texture	NOUN
ejpam-3272	28	3	spaces	space	VERB
ejpam-3272	28	4	:	:	PUNCT
ejpam-3272	28	5	a	a	DET
ejpam-3272	28	6	texturing	texturing	NOUN
ejpam-3272	28	7	on	on	ADP
ejpam-3272	28	8	a	a	DET
ejpam-3272	28	9	set	set	NOUN
ejpam-3272	28	10	s	s	PART
ejpam-3272	28	11	is	be	AUX
ejpam-3272	28	12	a	a	DET
ejpam-3272	28	13	point	point	NOUN
ejpam-3272	28	14	separating	separate	VERB
ejpam-3272	28	15	,	,	PUNCT
ejpam-3272	28	16	complete	complete	ADJ
ejpam-3272	28	17	,	,	PUNCT
ejpam-3272	28	18	completely	completely	ADV
ejpam-3272	28	19	distributive	distributive	ADJ
ejpam-3272	28	20	lattice	lattice	NOUN
ejpam-3272	28	21	s	s	NOUN
ejpam-3272	28	22	of	of	ADP
ejpam-3272	28	23	subsets	subset	NOUN
ejpam-3272	28	24	of	of	ADP
ejpam-3272	28	25	s	s	PRON
ejpam-3272	28	26	with	with	ADP
ejpam-3272	28	27	inclusion	inclusion	NOUN
ejpam-3272	28	28	relation	relation	NOUN
ejpam-3272	28	29	,	,	PUNCT
ejpam-3272	28	30	which	which	PRON
ejpam-3272	28	31	contains	contain	VERB
ejpam-3272	28	32	s	s	PRON
ejpam-3272	28	33	and	and	CCONJ
ejpam-3272	28	34	∅	∅	NOUN
ejpam-3272	28	35	and	and	CCONJ
ejpam-3272	28	36	for	for	ADP
ejpam-3272	28	37	which	which	PRON
ejpam-3272	28	38	arbitrary	arbitrary	ADJ
ejpam-3272	28	39	meet	meet	VERB
ejpam-3272	28	40	coincides	coincide	VERB
ejpam-3272	28	41	with	with	ADP
ejpam-3272	28	42	intersection	intersection	NOUN
ejpam-3272	28	43	and	and	CCONJ
ejpam-3272	28	44	finite	finite	NOUN
ejpam-3272	28	45	joins	join	VERB
ejpam-3272	28	46	coincide	coincide	NOUN
ejpam-3272	28	47	with	with	ADP
ejpam-3272	28	48	the	the	DET
ejpam-3272	28	49	union	union	NOUN
ejpam-3272	28	50	.	.	PUNCT
ejpam-3272	29	1	the	the	DET
ejpam-3272	29	2	pair	pair	NOUN
ejpam-3272	29	3	(	(	PUNCT
ejpam-3272	29	4	s	s	X
ejpam-3272	29	5	,	,	PUNCT
ejpam-3272	29	6	s	s	PART
ejpam-3272	29	7	)	)	PUNCT
ejpam-3272	29	8	is	be	AUX
ejpam-3272	29	9	known	know	VERB
ejpam-3272	29	10	as	as	ADP
ejpam-3272	29	11	a	a	DET
ejpam-3272	29	12	texture	texture	ADJ
ejpam-3272	29	13	space	space	NOUN
ejpam-3272	29	14	,	,	PUNCT
ejpam-3272	29	15	or	or	CCONJ
ejpam-3272	29	16	shortly	shortly	ADV
ejpam-3272	29	17	a	a	DET
ejpam-3272	29	18	texture	texture	NOUN
ejpam-3272	29	19	.	.	PUNCT
ejpam-3272	30	1	a	a	DET
ejpam-3272	30	2	dichotomous	dichotomous	ADJ
ejpam-3272	30	3	topology	topology	NOUN
ejpam-3272	30	4	,	,	PUNCT
ejpam-3272	30	5	or	or	CCONJ
ejpam-3272	30	6	ditopology	ditopology	NOUN
ejpam-3272	30	7	for	for	ADP
ejpam-3272	30	8	short	short	ADJ
ejpam-3272	30	9	,	,	PUNCT
ejpam-3272	30	10	on	on	ADP
ejpam-3272	30	11	a	a	DET
ejpam-3272	30	12	texture	texture	NOUN
ejpam-3272	30	13	(	(	PUNCT
ejpam-3272	30	14	s	s	PROPN
ejpam-3272	30	15	,	,	PUNCT
ejpam-3272	30	16	s	s	PART
ejpam-3272	30	17	)	)	PUNCT
ejpam-3272	30	18	is	be	AUX
ejpam-3272	30	19	a	a	DET
ejpam-3272	30	20	pair	pair	NOUN
ejpam-3272	30	21	(	(	PUNCT
ejpam-3272	30	22	τ	τ	PROPN
ejpam-3272	30	23	,	,	PUNCT
ejpam-3272	30	24	κ	κ	NOUN
ejpam-3272	30	25	)	)	PUNCT
ejpam-3272	30	26	of	of	ADP
ejpam-3272	30	27	subsets	subset	NOUN
ejpam-3272	30	28	of	of	ADP
ejpam-3272	30	29	s	s	NOUN
ejpam-3272	30	30	,	,	PUNCT
ejpam-3272	30	31	where	where	SCONJ
ejpam-3272	30	32	the	the	DET
ejpam-3272	30	33	set	set	NOUN
ejpam-3272	30	34	of	of	ADP
ejpam-3272	30	35	open	open	ADJ
ejpam-3272	30	36	sets	set	NOUN
ejpam-3272	30	37	τ	τ	PROPN
ejpam-3272	30	38	satisfies	satisfie	NOUN
ejpam-3272	30	39	(	(	PUNCT
ejpam-3272	30	40	t1	t1	NOUN
ejpam-3272	30	41	)	)	PUNCT
ejpam-3272	30	42	s	s	PROPN
ejpam-3272	30	43	,	,	PUNCT
ejpam-3272	30	44	∅	∅	NOUN
ejpam-3272	30	45	∈	∈	PROPN
ejpam-3272	30	46	τ	τ	X
ejpam-3272	30	47	,	,	PUNCT
ejpam-3272	30	48	(	(	PUNCT
ejpam-3272	30	49	t2	t2	NOUN
ejpam-3272	30	50	)	)	PUNCT
ejpam-3272	30	51	g1	g1	NOUN
ejpam-3272	30	52	,	,	PUNCT
ejpam-3272	30	53	g2	g2	PROPN
ejpam-3272	30	54	∈	∈	PROPN
ejpam-3272	30	55	τ	τ	PROPN
ejpam-3272	30	56	⇒	⇒	PROPN
ejpam-3272	30	57	g1	g1	PROPN
ejpam-3272	30	58	∩g2	∩g2	PROPN
ejpam-3272	30	59	∈	∈	PROPN
ejpam-3272	30	60	τ	τ	X
ejpam-3272	30	61	,	,	PUNCT
ejpam-3272	30	62	(	(	PUNCT
ejpam-3272	30	63	t3	t3	PROPN
ejpam-3272	30	64	)	)	PUNCT
ejpam-3272	30	65	gi	gi	NOUN
ejpam-3272	31	1	∈	∈	PROPN
ejpam-3272	32	1	τ	τ	PROPN
ejpam-3272	32	2	,	,	PUNCT
ejpam-3272	32	3	i	i	PRON
ejpam-3272	32	4	∈	∈	VERB
ejpam-3272	32	5	i	i	PRON
ejpam-3272	32	6	⇒	⇒	VERB
ejpam-3272	32	7	∨	∨	NUM
ejpam-3272	32	8	igi	igi	PROPN
ejpam-3272	32	9	∈	∈	PROPN
ejpam-3272	32	10	τ	τ	X
ejpam-3272	32	11	,	,	PUNCT
ejpam-3272	32	12	and	and	CCONJ
ejpam-3272	32	13	the	the	DET
ejpam-3272	32	14	set	set	NOUN
ejpam-3272	32	15	of	of	ADP
ejpam-3272	32	16	closed	closed	ADJ
ejpam-3272	32	17	sets	set	NOUN
ejpam-3272	32	18	κ	κ	ADP
ejpam-3272	32	19	satisfies	satisfie	NOUN
ejpam-3272	32	20	(	(	PUNCT
ejpam-3272	32	21	ct1	ct1	PROPN
ejpam-3272	32	22	)	)	PUNCT
ejpam-3272	32	23	s	s	PROPN
ejpam-3272	32	24	,	,	PUNCT
ejpam-3272	32	25	∅	∅	NOUN
ejpam-3272	32	26	∈	∈	PROPN
ejpam-3272	32	27	κ	κ	NOUN
ejpam-3272	32	28	,	,	PUNCT
ejpam-3272	32	29	(	(	PUNCT
ejpam-3272	32	30	ct2	ct2	NOUN
ejpam-3272	32	31	)	)	PUNCT
ejpam-3272	33	1	k1,k2	k1,k2	PROPN
ejpam-3272	33	2	∈	∈	PROPN
ejpam-3272	33	3	κ⇒	κ⇒	VERB
ejpam-3272	33	4	k1	k1	X
ejpam-3272	33	5	∪k2	∪k2	X
ejpam-3272	34	1	∈	∈	PROPN
ejpam-3272	34	2	κ	κ	PROPN
ejpam-3272	34	3	,	,	PUNCT
ejpam-3272	34	4	(	(	PUNCT
ejpam-3272	34	5	ct3	ct3	PROPN
ejpam-3272	34	6	)	)	PUNCT
ejpam-3272	34	7	ki	ki	PROPN
ejpam-3272	35	1	∈	∈	PROPN
ejpam-3272	35	2	κ	κ	PROPN
ejpam-3272	35	3	,	,	PUNCT
ejpam-3272	35	4	i	i	PRON
ejpam-3272	35	5	∈	∈	VERB
ejpam-3272	35	6	i	i	PRON
ejpam-3272	35	7	⇒	⇒	VERB
ejpam-3272	35	8	⋂	⋂	PROPN
ejpam-3272	35	9	iki	iki	PROPN
ejpam-3272	35	10	∈	∈	PROPN
ejpam-3272	35	11	κ	κ	PROPN
ejpam-3272	35	12	.	.	PUNCT
ejpam-3272	36	1	a	a	DET
ejpam-3272	36	2	ditopology	ditopology	NOUN
ejpam-3272	36	3	can	can	AUX
ejpam-3272	36	4	be	be	AUX
ejpam-3272	36	5	considered	consider	VERB
ejpam-3272	36	6	as	as	ADP
ejpam-3272	36	7	a	a	DET
ejpam-3272	36	8	representation	representation	NOUN
ejpam-3272	36	9	of	of	ADP
ejpam-3272	36	10	lattice	lattice	NOUN
ejpam-3272	36	11	-	-	PUNCT
ejpam-3272	36	12	valued	value	VERB
ejpam-3272	36	13	topologies	topology	NOUN
ejpam-3272	36	14	by	by	ADP
ejpam-3272	36	15	bitopologies	bitopologie	NOUN
ejpam-3272	36	16	and	and	CCONJ
ejpam-3272	36	17	one	one	PRON
ejpam-3272	36	18	can	can	AUX
ejpam-3272	36	19	simply	simply	ADV
ejpam-3272	36	20	infer	infer	VERB
ejpam-3272	36	21	that	that	SCONJ
ejpam-3272	36	22	it	it	PRON
ejpam-3272	36	23	is	be	AUX
ejpam-3272	36	24	a	a	DET
ejpam-3272	36	25	structure	structure	NOUN
ejpam-3272	36	26	in	in	ADP
ejpam-3272	36	27	which	which	PRON
ejpam-3272	36	28	the	the	DET
ejpam-3272	36	29	open	open	ADJ
ejpam-3272	36	30	and	and	CCONJ
ejpam-3272	36	31	closed	closed	ADJ
ejpam-3272	36	32	sets	set	NOUN
ejpam-3272	36	33	play	play	VERB
ejpam-3272	36	34	an	an	DET
ejpam-3272	36	35	equal	equal	ADJ
ejpam-3272	36	36	role	role	NOUN
ejpam-3272	36	37	.	.	PUNCT
ejpam-3272	37	1	(	(	PUNCT
ejpam-3272	37	2	co)frames	co)frame	NOUN
ejpam-3272	37	3	and	and	CCONJ
ejpam-3272	37	4	(	(	PUNCT
ejpam-3272	37	5	co)locales	co)locale	NOUN
ejpam-3272	37	6	:	:	PUNCT
ejpam-3272	37	7	our	our	PRON
ejpam-3272	37	8	notation	notation	NOUN
ejpam-3272	37	9	for	for	ADP
ejpam-3272	37	10	the	the	DET
ejpam-3272	37	11	theory	theory	NOUN
ejpam-3272	37	12	of	of	ADP
ejpam-3272	37	13	(	(	PUNCT
ejpam-3272	37	14	co)frames	co)frame	NOUN
ejpam-3272	37	15	and	and	CCONJ
ejpam-3272	37	16	(	(	PUNCT
ejpam-3272	37	17	co)locales	co)locale	NOUN
ejpam-3272	37	18	is	be	AUX
ejpam-3272	37	19	that	that	PRON
ejpam-3272	37	20	of	of	ADP
ejpam-3272	37	21	[	[	X
ejpam-3272	37	22	10	10	NUM
ejpam-3272	37	23	]	]	PUNCT
ejpam-3272	37	24	and	and	CCONJ
ejpam-3272	37	25	[	[	X
ejpam-3272	37	26	9	9	NUM
ejpam-3272	37	27	]	]	PUNCT
ejpam-3272	37	28	.	.	PUNCT
ejpam-3272	38	1	first	first	ADV
ejpam-3272	38	2	we	we	PRON
ejpam-3272	38	3	recall	recall	VERB
ejpam-3272	38	4	the	the	DET
ejpam-3272	38	5	following	follow	VERB
ejpam-3272	38	6	definitions	definition	NOUN
ejpam-3272	38	7	for	for	ADP
ejpam-3272	38	8	a	a	DET
ejpam-3272	38	9	lattice	lattice	NOUN
ejpam-3272	38	10	l	l	NOUN
ejpam-3272	38	11	:	:	PUNCT
ejpam-3272	38	12	let	let	VERB
ejpam-3272	38	13	l	l	NOUN
ejpam-3272	38	14	and	and	CCONJ
ejpam-3272	38	15	m	m	VERB
ejpam-3272	38	16	be	be	VERB
ejpam-3272	38	17	posets	poset	NOUN
ejpam-3272	38	18	.	.	PUNCT
ejpam-3272	39	1	a	a	DET
ejpam-3272	39	2	pair	pair	NOUN
ejpam-3272	39	3	(	(	PUNCT
ejpam-3272	39	4	f	f	X
ejpam-3272	39	5	,	,	PUNCT
ejpam-3272	39	6	g	g	NOUN
ejpam-3272	39	7	)	)	PUNCT
ejpam-3272	39	8	of	of	ADP
ejpam-3272	39	9	monotone	monotone	ADJ
ejpam-3272	39	10	functions	function	NOUN
ejpam-3272	39	11	f	f	X
ejpam-3272	39	12	:	:	PUNCT
ejpam-3272	39	13	l→m	l→m	NUM
ejpam-3272	39	14	,	,	PUNCT
ejpam-3272	39	15	g	g	NOUN
ejpam-3272	39	16	:	:	PUNCT
ejpam-3272	39	17	m	m	VERB
ejpam-3272	39	18	→	→	SYM
ejpam-3272	39	19	l	l	NOUN
ejpam-3272	39	20	is	be	AUX
ejpam-3272	39	21	called	call	VERB
ejpam-3272	39	22	a	a	DET
ejpam-3272	39	23	galois	galois	PROPN
ejpam-3272	39	24	adjunction	adjunction	NOUN
ejpam-3272	39	25	if	if	SCONJ
ejpam-3272	39	26	,	,	PUNCT
ejpam-3272	39	27	for	for	ADP
ejpam-3272	39	28	all	all	PRON
ejpam-3272	39	29	x	x	SYM
ejpam-3272	39	30	∈	∈	PROPN
ejpam-3272	39	31	l	l	NOUN
ejpam-3272	39	32	and	and	CCONJ
ejpam-3272	39	33	y	y	PROPN
ejpam-3272	39	34	∈m	∈m	NOUN
ejpam-3272	39	35	,	,	PUNCT
ejpam-3272	39	36	f(x	f(x	PROPN
ejpam-3272	39	37	)	)	PUNCT
ejpam-3272	39	38	≤	≤	PUNCT
ejpam-3272	40	1	y	y	PROPN
ejpam-3272	40	2	iff	iff	PROPN
ejpam-3272	40	3	x	x	PROPN
ejpam-3272	40	4	≤	≤	PROPN
ejpam-3272	40	5	g(y	g(y	NOUN
ejpam-3272	40	6	)	)	PUNCT
ejpam-3272	40	7	.	.	PUNCT
ejpam-3272	41	1	in	in	ADP
ejpam-3272	41	2	this	this	DET
ejpam-3272	41	3	case	case	NOUN
ejpam-3272	41	4	,	,	PUNCT
ejpam-3272	41	5	f	f	PROPN
ejpam-3272	41	6	is	be	AUX
ejpam-3272	41	7	called	call	VERB
ejpam-3272	41	8	the	the	DET
ejpam-3272	41	9	left	left	ADJ
ejpam-3272	41	10	adjoint	adjoint	NOUN
ejpam-3272	41	11	of	of	ADP
ejpam-3272	41	12	g	g	PROPN
ejpam-3272	41	13	(	(	PUNCT
ejpam-3272	41	14	denoted	denote	VERB
ejpam-3272	41	15	by	by	ADP
ejpam-3272	41	16	f	f	PROPN
ejpam-3272	41	17	=	=	SYM
ejpam-3272	41	18	g∗	g∗	PROPN
ejpam-3272	41	19	)	)	PUNCT
ejpam-3272	41	20	,	,	PUNCT
ejpam-3272	41	21	and	and	CCONJ
ejpam-3272	41	22	g	g	PROPN
ejpam-3272	41	23	is	be	AUX
ejpam-3272	41	24	called	call	VERB
ejpam-3272	41	25	the	the	DET
ejpam-3272	41	26	right	right	ADJ
ejpam-3272	41	27	adjoint	adjoint	NOUN
ejpam-3272	41	28	of	of	ADP
ejpam-3272	41	29	f	f	PROPN
ejpam-3272	41	30	(	(	PUNCT
ejpam-3272	41	31	denoted	denote	VERB
ejpam-3272	41	32	by	by	ADP
ejpam-3272	41	33	g	g	PROPN
ejpam-3272	41	34	=	=	SYM
ejpam-3272	41	35	f∗	f∗	NOUN
ejpam-3272	41	36	)	)	PUNCT
ejpam-3272	41	37	.	.	PUNCT
ejpam-3272	42	1	proposition	proposition	NOUN
ejpam-3272	42	2	1	1	NUM
ejpam-3272	42	3	.	.	PUNCT
ejpam-3272	43	1	let	let	VERB
ejpam-3272	43	2	(	(	PUNCT
ejpam-3272	43	3	f	f	X
ejpam-3272	43	4	,	,	PUNCT
ejpam-3272	43	5	g	g	NOUN
ejpam-3272	43	6	)	)	PUNCT
ejpam-3272	43	7	be	be	VERB
ejpam-3272	43	8	a	a	DET
ejpam-3272	43	9	galois	galois	PROPN
ejpam-3272	43	10	adjunction	adjunction	NOUN
ejpam-3272	43	11	.	.	PUNCT
ejpam-3272	44	1	then	then	ADV
ejpam-3272	44	2	(	(	PUNCT
ejpam-3272	44	3	i	i	NOUN
ejpam-3272	44	4	)	)	PUNCT
ejpam-3272	44	5	f	f	PROPN
ejpam-3272	44	6	preserves	preserve	VERB
ejpam-3272	44	7	arbitrary	arbitrary	ADJ
ejpam-3272	44	8	join	join	NOUN
ejpam-3272	44	9	,	,	PUNCT
ejpam-3272	44	10	and	and	CCONJ
ejpam-3272	44	11	g	g	NOUN
ejpam-3272	44	12	preserves	preserve	VERB
ejpam-3272	44	13	arbitrary	arbitrary	ADJ
ejpam-3272	44	14	meet	meet	NOUN
ejpam-3272	44	15	.	.	PUNCT
ejpam-3272	45	1	(	(	PUNCT
ejpam-3272	45	2	ii	ii	NOUN
ejpam-3272	45	3	)	)	PUNCT
ejpam-3272	45	4	g	g	PROPN
ejpam-3272	45	5	is	be	AUX
ejpam-3272	45	6	one	one	NUM
ejpam-3272	45	7	-	-	PUNCT
ejpam-3272	45	8	one	one	NUM
ejpam-3272	45	9	iff	iff	PROPN
ejpam-3272	45	10	f	f	PROPN
ejpam-3272	45	11	is	be	AUX
ejpam-3272	45	12	onto	onto	ADP
ejpam-3272	45	13	.	.	PUNCT
ejpam-3272	46	1	e.	e.	PROPN
ejpam-3272	46	2	korkmaz	korkmaz	PROPN
ejpam-3272	46	3	,	,	PUNCT
ejpam-3272	46	4	r.	r.	PROPN
ejpam-3272	46	5	ertürk	ertürk	PROPN
ejpam-3272	46	6	/	/	SYM
ejpam-3272	46	7	eur	eur	PROPN
ejpam-3272	46	8	.	.	PUNCT
ejpam-3272	47	1	j.	j.	PROPN
ejpam-3272	47	2	pure	pure	PROPN
ejpam-3272	47	3	appl	appl	PROPN
ejpam-3272	47	4	.	.	PROPN
ejpam-3272	47	5	math	math	PROPN
ejpam-3272	47	6	,	,	PUNCT
ejpam-3272	47	7	11	11	NUM
ejpam-3272	47	8	(	(	PUNCT
ejpam-3272	47	9	3	3	NUM
ejpam-3272	47	10	)	)	PUNCT
ejpam-3272	47	11	(	(	PUNCT
ejpam-3272	47	12	2018	2018	NUM
ejpam-3272	47	13	)	)	PUNCT
ejpam-3272	47	14	,	,	PUNCT
ejpam-3272	47	15	612	612	NUM
ejpam-3272	47	16	-	-	SYM
ejpam-3272	47	17	627	627	NUM
ejpam-3272	47	18	614	614	NUM
ejpam-3272	47	19	(	(	PUNCT
ejpam-3272	47	20	iii	iii	NOUN
ejpam-3272	47	21	)	)	PUNCT
ejpam-3272	47	22	if	if	SCONJ
ejpam-3272	47	23	f	f	PROPN
ejpam-3272	47	24	is	be	AUX
ejpam-3272	47	25	one	one	NUM
ejpam-3272	47	26	-	-	PUNCT
ejpam-3272	47	27	one	one	NOUN
ejpam-3272	47	28	then	then	ADV
ejpam-3272	47	29	gf	gf	X
ejpam-3272	47	30	=	=	NOUN
ejpam-3272	47	31	id	id	NOUN
ejpam-3272	47	32	,	,	PUNCT
ejpam-3272	47	33	if	if	SCONJ
ejpam-3272	47	34	it	it	PRON
ejpam-3272	47	35	is	be	AUX
ejpam-3272	47	36	onto	onto	ADP
ejpam-3272	47	37	then	then	ADV
ejpam-3272	47	38	fg	fg	PROPN
ejpam-3272	47	39	=	=	NOUN
ejpam-3272	47	40	id	id	NOUN
ejpam-3272	47	41	.	.	PUNCT
ejpam-3272	48	1	now	now	ADV
ejpam-3272	48	2	let	let	VERB
ejpam-3272	48	3	us	we	PRON
ejpam-3272	48	4	recall	recall	VERB
ejpam-3272	48	5	the	the	DET
ejpam-3272	48	6	other	other	ADJ
ejpam-3272	48	7	required	require	VERB
ejpam-3272	48	8	notions	notion	NOUN
ejpam-3272	48	9	for	for	ADP
ejpam-3272	48	10	the	the	DET
ejpam-3272	48	11	present	present	ADJ
ejpam-3272	48	12	paper	paper	NOUN
ejpam-3272	48	13	:	:	PUNCT
ejpam-3272	48	14	l	l	NOUN
ejpam-3272	48	15	is	be	AUX
ejpam-3272	48	16	called	call	VERB
ejpam-3272	48	17	a	a	DET
ejpam-3272	48	18	frame	frame	NOUN
ejpam-3272	48	19	if	if	SCONJ
ejpam-3272	48	20	it	it	PRON
ejpam-3272	48	21	is	be	AUX
ejpam-3272	48	22	a	a	DET
ejpam-3272	48	23	complete	complete	ADJ
ejpam-3272	48	24	lattice	lattice	NOUN
ejpam-3272	48	25	with	with	ADP
ejpam-3272	48	26	the	the	DET
ejpam-3272	48	27	property	property	NOUN
ejpam-3272	48	28	b	b	PROPN
ejpam-3272	48	29	∧	∧	PROPN
ejpam-3272	48	30	(	(	PUNCT
ejpam-3272	48	31	∨	∨	NOUN
ejpam-3272	48	32	a	a	PRON
ejpam-3272	48	33	)	)	PUNCT
ejpam-3272	48	34	=	=	SYM
ejpam-3272	48	35	∨	∨	X
ejpam-3272	48	36	{	{	PUNCT
ejpam-3272	48	37	b	b	PROPN
ejpam-3272	48	38	∧	∧	PROPN
ejpam-3272	48	39	a	a	PRON
ejpam-3272	48	40	:	:	PUNCT
ejpam-3272	48	41	a	a	DET
ejpam-3272	48	42	∈	∈	PROPN
ejpam-3272	48	43	a	a	X
ejpam-3272	48	44	}	}	PUNCT
ejpam-3272	48	45	for	for	ADP
ejpam-3272	48	46	any	any	DET
ejpam-3272	48	47	b	b	PROPN
ejpam-3272	48	48	∈	∈	PROPN
ejpam-3272	48	49	l	l	NOUN
ejpam-3272	48	50	and	and	CCONJ
ejpam-3272	48	51	any	any	PRON
ejpam-3272	48	52	subset	subset	NOUN
ejpam-3272	48	53	a	a	DET
ejpam-3272	48	54	⊆	⊆	NUM
ejpam-3272	48	55	l.	l.	NOUN
ejpam-3272	48	56	dually	dually	PROPN
ejpam-3272	48	57	,	,	PUNCT
ejpam-3272	48	58	m	m	VERB
ejpam-3272	48	59	is	be	AUX
ejpam-3272	48	60	called	call	VERB
ejpam-3272	48	61	a	a	DET
ejpam-3272	48	62	coframe	coframe	NOUN
ejpam-3272	48	63	if	if	SCONJ
ejpam-3272	48	64	it	it	PRON
ejpam-3272	48	65	is	be	AUX
ejpam-3272	48	66	a	a	DET
ejpam-3272	48	67	complete	complete	ADJ
ejpam-3272	48	68	lattice	lattice	NOUN
ejpam-3272	48	69	with	with	ADP
ejpam-3272	48	70	the	the	DET
ejpam-3272	48	71	property	property	NOUN
ejpam-3272	48	72	b	b	PROPN
ejpam-3272	48	73	∨	∨	X
ejpam-3272	48	74	(	(	PUNCT
ejpam-3272	48	75	∧	∧	PROPN
ejpam-3272	48	76	a	a	NOUN
ejpam-3272	48	77	)	)	PUNCT
ejpam-3272	48	78	=	=	SYM
ejpam-3272	48	79	∧	∧	PROPN
ejpam-3272	48	80	{	{	PUNCT
ejpam-3272	48	81	b	b	PROPN
ejpam-3272	48	82	∨	∨	NUM
ejpam-3272	48	83	a	a	PRON
ejpam-3272	48	84	:	:	PUNCT
ejpam-3272	48	85	a	a	DET
ejpam-3272	48	86	∈	∈	PROPN
ejpam-3272	48	87	a	a	X
ejpam-3272	48	88	}	}	PUNCT
ejpam-3272	48	89	for	for	ADP
ejpam-3272	48	90	any	any	DET
ejpam-3272	48	91	b	b	PROPN
ejpam-3272	48	92	∈	∈	PROPN
ejpam-3272	48	93	l	l	NOUN
ejpam-3272	48	94	and	and	CCONJ
ejpam-3272	48	95	any	any	DET
ejpam-3272	48	96	subset	subset	NOUN
ejpam-3272	48	97	a	a	DET
ejpam-3272	48	98	⊆	⊆	NUM
ejpam-3272	48	99	l.	l.	NOUN
ejpam-3272	48	100	a	a	DET
ejpam-3272	48	101	frame	frame	NOUN
ejpam-3272	48	102	(	(	PUNCT
ejpam-3272	48	103	resp	resp	NOUN
ejpam-3272	48	104	.	.	PUNCT
ejpam-3272	49	1	coframe	coframe	PROPN
ejpam-3272	49	2	)	)	PUNCT
ejpam-3272	50	1	homomorphism	homomorphism	NOUN
ejpam-3272	50	2	is	be	AUX
ejpam-3272	50	3	a	a	DET
ejpam-3272	50	4	map	map	NOUN
ejpam-3272	50	5	between	between	ADP
ejpam-3272	50	6	frames	frame	NOUN
ejpam-3272	50	7	(	(	PUNCT
ejpam-3272	50	8	resp	resp	NOUN
ejpam-3272	50	9	.	.	PUNCT
ejpam-3272	51	1	coframes	coframe	NOUN
ejpam-3272	51	2	)	)	PUNCT
ejpam-3272	51	3	preserving	preserve	VERB
ejpam-3272	51	4	arbitrary	arbitrary	ADJ
ejpam-3272	51	5	joins	join	NOUN
ejpam-3272	51	6	(	(	PUNCT
ejpam-3272	51	7	resp	resp	PROPN
ejpam-3272	51	8	.	.	PUNCT
ejpam-3272	51	9	meets	meet	VERB
ejpam-3272	51	10	)	)	PUNCT
ejpam-3272	51	11	and	and	CCONJ
ejpam-3272	51	12	finite	finite	PROPN
ejpam-3272	51	13	meets	meet	VERB
ejpam-3272	51	14	(	(	PUNCT
ejpam-3272	51	15	resp	resp	NOUN
ejpam-3272	51	16	.	.	PUNCT
ejpam-3272	52	1	joins	joins	PROPN
ejpam-3272	52	2	)	)	PUNCT
ejpam-3272	52	3	.	.	PUNCT
ejpam-3272	53	1	the	the	DET
ejpam-3272	53	2	category	category	NOUN
ejpam-3272	53	3	of	of	ADP
ejpam-3272	53	4	frames	frame	NOUN
ejpam-3272	53	5	(	(	PUNCT
ejpam-3272	53	6	resp	resp	NOUN
ejpam-3272	53	7	.	.	PUNCT
ejpam-3272	54	1	co	co	NOUN
ejpam-3272	54	2	-	-	NOUN
ejpam-3272	54	3	frames	frame	NOUN
ejpam-3272	54	4	)	)	PUNCT
ejpam-3272	54	5	and	and	CCONJ
ejpam-3272	54	6	frame	frame	NOUN
ejpam-3272	54	7	(	(	PUNCT
ejpam-3272	54	8	resp	resp	NOUN
ejpam-3272	54	9	.	.	PUNCT
ejpam-3272	55	1	co	co	NOUN
ejpam-3272	55	2	-	-	NOUN
ejpam-3272	55	3	frame	frame	NOUN
ejpam-3272	55	4	)	)	PUNCT
ejpam-3272	55	5	homomorphisms	homomorphism	NOUN
ejpam-3272	55	6	is	be	AUX
ejpam-3272	55	7	denoted	denote	VERB
ejpam-3272	55	8	by	by	ADP
ejpam-3272	55	9	frm	frm	PROPN
ejpam-3272	55	10	(	(	PUNCT
ejpam-3272	55	11	resp	resp	PROPN
ejpam-3272	55	12	.	.	PUNCT
ejpam-3272	56	1	cofrm	cofrm	PROPN
ejpam-3272	56	2	)	)	PUNCT
ejpam-3272	56	3	,	,	PUNCT
ejpam-3272	56	4	and	and	CCONJ
ejpam-3272	56	5	the	the	DET
ejpam-3272	56	6	opposite	opposite	ADJ
ejpam-3272	56	7	category	category	NOUN
ejpam-3272	56	8	of	of	ADP
ejpam-3272	56	9	frm	frm	PROPN
ejpam-3272	56	10	(	(	PUNCT
ejpam-3272	56	11	resp	resp	PROPN
ejpam-3272	56	12	.	.	PUNCT
ejpam-3272	57	1	cofrm	cofrm	PROPN
ejpam-3272	57	2	)	)	PUNCT
ejpam-3272	57	3	is	be	AUX
ejpam-3272	57	4	denoted	denote	VERB
ejpam-3272	57	5	by	by	ADP
ejpam-3272	57	6	loc	loc	PROPN
ejpam-3272	57	7	(	(	PUNCT
ejpam-3272	57	8	resp	resp	NOUN
ejpam-3272	57	9	.	.	PUNCT
ejpam-3272	58	1	coloc	coloc	PROPN
ejpam-3272	58	2	)	)	PUNCT
ejpam-3272	58	3	.	.	PUNCT
ejpam-3272	59	1	a	a	DET
ejpam-3272	59	2	heyting	heyte	VERB
ejpam-3272	59	3	algebra	algebra	NOUN
ejpam-3272	59	4	is	be	AUX
ejpam-3272	59	5	a	a	DET
ejpam-3272	59	6	bounded	bounded	ADJ
ejpam-3272	59	7	lattice	lattice	PROPN
ejpam-3272	59	8	l	l	PROPN
ejpam-3272	59	9	equipped	equip	VERB
ejpam-3272	59	10	with	with	ADP
ejpam-3272	59	11	a	a	DET
ejpam-3272	59	12	binary	binary	ADJ
ejpam-3272	59	13	operation→	operation→	NOUN
ejpam-3272	59	14	:	:	PUNCT
ejpam-3272	59	15	l×l→	l×l→	NOUN
ejpam-3272	59	16	l	l	NOUN
ejpam-3272	59	17	satisfying	satisfy	VERB
ejpam-3272	59	18	c	c	PROPN
ejpam-3272	59	19	≤	≤	X
ejpam-3272	59	20	a→	a→	PUNCT
ejpam-3272	59	21	b⇔	b⇔	NOUN
ejpam-3272	59	22	c	c	NOUN
ejpam-3272	59	23	∧	∧	PROPN
ejpam-3272	59	24	a	a	DET
ejpam-3272	59	25	≤	≤	NUM
ejpam-3272	59	26	b	b	NOUN
ejpam-3272	59	27	for	for	ADP
ejpam-3272	59	28	all	all	DET
ejpam-3272	59	29	a	a	DET
ejpam-3272	59	30	,	,	PUNCT
ejpam-3272	59	31	b	b	NOUN
ejpam-3272	59	32	,	,	PUNCT
ejpam-3272	59	33	c	c	PROPN
ejpam-3272	59	34	∈	∈	PROPN
ejpam-3272	59	35	l.	l.	NOUN
ejpam-3272	59	36	a	a	DET
ejpam-3272	59	37	coheyting	coheyting	NOUN
ejpam-3272	59	38	algebra	algebra	NOUN
ejpam-3272	59	39	[	[	X
ejpam-3272	59	40	11	11	NUM
ejpam-3272	59	41	]	]	PUNCT
ejpam-3272	59	42	is	be	AUX
ejpam-3272	59	43	a	a	DET
ejpam-3272	59	44	bounded	bounded	ADJ
ejpam-3272	59	45	lattice	lattice	NOUN
ejpam-3272	59	46	m	m	AUX
ejpam-3272	59	47	equipped	equip	VERB
ejpam-3272	59	48	with	with	ADP
ejpam-3272	59	49	a	a	DET
ejpam-3272	59	50	binary	binary	ADJ
ejpam-3272	59	51	operation	operation	NOUN
ejpam-3272	59	52	←	←	PROPN
ejpam-3272	59	53	:	:	PUNCT
ejpam-3272	59	54	m	m	VERB
ejpam-3272	59	55	×m	×m	NOUN
ejpam-3272	59	56	→m	→m	X
ejpam-3272	59	57	satisfying	satisfy	VERB
ejpam-3272	59	58	a←	a←	PROPN
ejpam-3272	59	59	b	b	PROPN
ejpam-3272	59	60	≤	≤	PROPN
ejpam-3272	59	61	c⇔	c⇔	VERB
ejpam-3272	59	62	a	a	DET
ejpam-3272	59	63	≤	≤	NUM
ejpam-3272	59	64	b	b	PROPN
ejpam-3272	59	65	∨	∨	NUM
ejpam-3272	59	66	c	c	NOUN
ejpam-3272	59	67	for	for	ADP
ejpam-3272	59	68	all	all	DET
ejpam-3272	59	69	a	a	DET
ejpam-3272	59	70	,	,	PUNCT
ejpam-3272	59	71	b	b	NOUN
ejpam-3272	59	72	,	,	PUNCT
ejpam-3272	59	73	c	c	NOUN
ejpam-3272	59	74	∈m	∈m	NOUN
ejpam-3272	59	75	.	.	PUNCT
ejpam-3272	60	1	every	every	DET
ejpam-3272	60	2	complete	complete	ADJ
ejpam-3272	60	3	boolean	boolean	ADJ
ejpam-3272	60	4	algebra	algebra	NOUN
ejpam-3272	60	5	is	be	AUX
ejpam-3272	60	6	both	both	PRON
ejpam-3272	60	7	a	a	DET
ejpam-3272	60	8	heyting	heyting	NOUN
ejpam-3272	60	9	and	and	CCONJ
ejpam-3272	60	10	a	a	DET
ejpam-3272	60	11	coheyting	coheyting	NOUN
ejpam-3272	60	12	algebra	algebra	NOUN
ejpam-3272	60	13	.	.	PUNCT
ejpam-3272	61	1	the	the	DET
ejpam-3272	61	2	binary	binary	ADJ
ejpam-3272	61	3	operations	operation	NOUN
ejpam-3272	61	4	are	be	AUX
ejpam-3272	61	5	defined	define	VERB
ejpam-3272	61	6	by	by	ADP
ejpam-3272	61	7	x→	x→	X
ejpam-3272	61	8	y	y	PROPN
ejpam-3272	61	9	=	=	PUNCT
ejpam-3272	61	10	x∗	x∗	PROPN
ejpam-3272	61	11	∨	∨	NUM
ejpam-3272	61	12	y	y	PROPN
ejpam-3272	61	13	and	and	CCONJ
ejpam-3272	61	14	x←	x←	PROPN
ejpam-3272	61	15	y	y	PROPN
ejpam-3272	61	16	=	=	SYM
ejpam-3272	61	17	x∧	x∧	PROPN
ejpam-3272	61	18	y∗	y∗	PROPN
ejpam-3272	61	19	,	,	PUNCT
ejpam-3272	61	20	where	where	SCONJ
ejpam-3272	61	21	the	the	DET
ejpam-3272	61	22	exponent	exponent	NOUN
ejpam-3272	61	23	∗	∗	NOUN
ejpam-3272	61	24	denotes	denote	VERB
ejpam-3272	61	25	the	the	DET
ejpam-3272	61	26	complement	complement	NOUN
ejpam-3272	61	27	of	of	ADP
ejpam-3272	61	28	an	an	DET
ejpam-3272	61	29	element	element	NOUN
ejpam-3272	61	30	.	.	PUNCT
ejpam-3272	62	1	both	both	DET
ejpam-3272	62	2	x	x	X
ejpam-3272	62	3	→	→	SYM
ejpam-3272	62	4	0	0	NUM
ejpam-3272	62	5	and	and	CCONJ
ejpam-3272	62	6	1	1	NUM
ejpam-3272	62	7	←	←	PROPN
ejpam-3272	62	8	x	x	PUNCT
ejpam-3272	62	9	coincide	coincide	NOUN
ejpam-3272	62	10	with	with	ADP
ejpam-3272	62	11	the	the	DET
ejpam-3272	62	12	complement	complement	NOUN
ejpam-3272	62	13	present	present	NOUN
ejpam-3272	62	14	in	in	ADP
ejpam-3272	62	15	the	the	DET
ejpam-3272	62	16	boolean	boolean	ADJ
ejpam-3272	62	17	algebra	algebra	NOUN
ejpam-3272	62	18	.	.	PUNCT
ejpam-3272	63	1	any	any	PRON
ejpam-3272	63	2	(	(	PUNCT
ejpam-3272	63	3	co)frame	co)frame	NOUN
ejpam-3272	63	4	is	be	AUX
ejpam-3272	63	5	a	a	DET
ejpam-3272	63	6	complete	complete	ADJ
ejpam-3272	63	7	(	(	PUNCT
ejpam-3272	63	8	co)heyting	co)heyting	NOUN
ejpam-3272	63	9	algebra	algebra	NOUN
ejpam-3272	63	10	,	,	PUNCT
ejpam-3272	63	11	and	and	CCONJ
ejpam-3272	63	12	vice	vice	ADV
ejpam-3272	63	13	versa	versa	ADV
ejpam-3272	63	14	,	,	PUNCT
ejpam-3272	63	15	hence	hence	ADV
ejpam-3272	63	16	each	each	DET
ejpam-3272	63	17	frame	frame	NOUN
ejpam-3272	63	18	(	(	PUNCT
ejpam-3272	63	19	coframe	coframe	NOUN
ejpam-3272	63	20	)	)	PUNCT
ejpam-3272	63	21	carries	carry	VERB
ejpam-3272	63	22	a	a	DET
ejpam-3272	63	23	(	(	PUNCT
ejpam-3272	63	24	co)heyting	co)heyting	NOUN
ejpam-3272	63	25	operation	operation	NOUN
ejpam-3272	63	26	.	.	PUNCT
ejpam-3272	64	1	the	the	PRON
ejpam-3272	64	2	(	(	PUNCT
ejpam-3272	64	3	co)heyting	co)heyte	VERB
ejpam-3272	64	4	operation	operation	NOUN
ejpam-3272	64	5	plays	play	VERB
ejpam-3272	64	6	a	a	DET
ejpam-3272	64	7	crucial	crucial	ADJ
ejpam-3272	64	8	role	role	NOUN
ejpam-3272	64	9	in	in	ADP
ejpam-3272	64	10	defining	define	VERB
ejpam-3272	64	11	a	a	DET
ejpam-3272	64	12	sub(co)locale	sub(co)locale	NOUN
ejpam-3272	64	13	which	which	PRON
ejpam-3272	64	14	is	be	AUX
ejpam-3272	64	15	a	a	DET
ejpam-3272	64	16	subobject	subobject	NOUN
ejpam-3272	64	17	of	of	ADP
ejpam-3272	64	18	a	a	DET
ejpam-3272	64	19	(	(	PUNCT
ejpam-3272	64	20	co)locale	co)locale	NOUN
ejpam-3272	64	21	l	l	NOUN
ejpam-3272	64	22	in	in	ADP
ejpam-3272	64	23	the	the	DET
ejpam-3272	64	24	category	category	NOUN
ejpam-3272	64	25	of	of	ADP
ejpam-3272	64	26	(	(	PUNCT
ejpam-3272	64	27	co)loc	co)loc	NOUN
ejpam-3272	64	28	.	.	PUNCT
ejpam-3272	65	1	given	give	VERB
ejpam-3272	65	2	a	a	DET
ejpam-3272	65	3	frame	frame	NOUN
ejpam-3272	65	4	l	l	NOUN
ejpam-3272	65	5	,	,	PUNCT
ejpam-3272	65	6	a	a	DET
ejpam-3272	65	7	subframe	subframe	NOUN
ejpam-3272	65	8	is	be	AUX
ejpam-3272	65	9	a	a	DET
ejpam-3272	65	10	subset	subset	NOUN
ejpam-3272	65	11	l′	l′	NOUN
ejpam-3272	65	12	⊆	⊆	NUM
ejpam-3272	65	13	l	l	NOUN
ejpam-3272	65	14	that	that	PRON
ejpam-3272	65	15	is	be	AUX
ejpam-3272	65	16	closed	close	VERB
ejpam-3272	65	17	under	under	ADP
ejpam-3272	65	18	arbitrary	arbitrary	ADJ
ejpam-3272	65	19	join	join	NOUN
ejpam-3272	65	20	and	and	CCONJ
ejpam-3272	65	21	finite	finite	PROPN
ejpam-3272	65	22	meets	meet	NOUN
ejpam-3272	65	23	.	.	PUNCT
ejpam-3272	66	1	dually	dually	PROPN
ejpam-3272	66	2	,	,	PUNCT
ejpam-3272	66	3	a	a	DET
ejpam-3272	66	4	subcoframe	subcoframe	NOUN
ejpam-3272	66	5	is	be	AUX
ejpam-3272	66	6	a	a	DET
ejpam-3272	66	7	subset	subset	NOUN
ejpam-3272	66	8	m	m	VERB
ejpam-3272	66	9	′	′	NUM
ejpam-3272	66	10	⊆	⊆	NUM
ejpam-3272	66	11	m	m	NOUN
ejpam-3272	66	12	which	which	PRON
ejpam-3272	66	13	is	be	AUX
ejpam-3272	66	14	closed	close	VERB
ejpam-3272	66	15	under	under	ADP
ejpam-3272	66	16	arbitrary	arbitrary	ADJ
ejpam-3272	66	17	meet	meet	NOUN
ejpam-3272	66	18	and	and	CCONJ
ejpam-3272	66	19	finite	finite	NOUN
ejpam-3272	66	20	joins	join	NOUN
ejpam-3272	66	21	.	.	PUNCT
ejpam-3272	67	1	according	accord	VERB
ejpam-3272	67	2	to	to	ADP
ejpam-3272	67	3	[	[	X
ejpam-3272	67	4	10	10	NUM
ejpam-3272	67	5	]	]	PUNCT
ejpam-3272	67	6	,	,	PUNCT
ejpam-3272	67	7	a	a	DET
ejpam-3272	67	8	sublocale	sublocale	NOUN
ejpam-3272	67	9	is	be	AUX
ejpam-3272	67	10	a	a	DET
ejpam-3272	67	11	subset	subset	NOUN
ejpam-3272	67	12	s	s	VERB
ejpam-3272	67	13	⊆	⊆	NUM
ejpam-3272	67	14	l	l	NOUN
ejpam-3272	67	15	with	with	ADP
ejpam-3272	67	16	the	the	DET
ejpam-3272	67	17	following	follow	VERB
ejpam-3272	67	18	conditions	condition	NOUN
ejpam-3272	67	19	:	:	PUNCT
ejpam-3272	67	20	(	(	PUNCT
ejpam-3272	67	21	s1	s1	NOUN
ejpam-3272	67	22	)	)	PUNCT
ejpam-3272	67	23	for	for	ADP
ejpam-3272	67	24	all	all	DET
ejpam-3272	67	25	n	n	PRON
ejpam-3272	67	26	⊆	⊆	NUM
ejpam-3272	67	27	s	s	NOUN
ejpam-3272	67	28	,	,	PUNCT
ejpam-3272	67	29	∧	∧	PROPN
ejpam-3272	67	30	n	n	CCONJ
ejpam-3272	67	31	∈	∈	PROPN
ejpam-3272	67	32	s	s	PROPN
ejpam-3272	67	33	,	,	PUNCT
ejpam-3272	67	34	(	(	PUNCT
ejpam-3272	67	35	s2	s2	PROPN
ejpam-3272	67	36	)	)	PUNCT
ejpam-3272	67	37	x→	x→	X
ejpam-3272	68	1	s	s	PART
ejpam-3272	68	2	∈	∈	PROPN
ejpam-3272	68	3	s	s	NOUN
ejpam-3272	68	4	for	for	ADP
ejpam-3272	68	5	all	all	DET
ejpam-3272	68	6	s	s	PART
ejpam-3272	68	7	∈	∈	PROPN
ejpam-3272	68	8	s	s	PART
ejpam-3272	68	9	and	and	CCONJ
ejpam-3272	68	10	x	x	PROPN
ejpam-3272	68	11	∈	∈	PROPN
ejpam-3272	68	12	l.	l.	NOUN
ejpam-3272	68	13	similarly	similarly	ADV
ejpam-3272	68	14	,	,	PUNCT
ejpam-3272	68	15	we	we	PRON
ejpam-3272	68	16	define	define	VERB
ejpam-3272	68	17	a	a	DET
ejpam-3272	68	18	subcolocale	subcolocale	NOUN
ejpam-3272	68	19	of	of	ADP
ejpam-3272	68	20	a	a	DET
ejpam-3272	68	21	colocale	colocale	NOUN
ejpam-3272	68	22	m	m	NOUN
ejpam-3272	68	23	as	as	ADP
ejpam-3272	68	24	a	a	DET
ejpam-3272	68	25	subset	subset	NOUN
ejpam-3272	68	26	s	s	VERB
ejpam-3272	68	27	⊆	⊆	NUM
ejpam-3272	68	28	m	m	NOUN
ejpam-3272	68	29	satisfying	satisfy	VERB
ejpam-3272	68	30	the	the	DET
ejpam-3272	68	31	following	follow	VERB
ejpam-3272	68	32	conditions	condition	NOUN
ejpam-3272	68	33	:	:	PUNCT
ejpam-3272	68	34	e.	e.	PROPN
ejpam-3272	68	35	korkmaz	korkmaz	PROPN
ejpam-3272	68	36	,	,	PUNCT
ejpam-3272	68	37	r.	r.	PROPN
ejpam-3272	68	38	ertürk	ertürk	PROPN
ejpam-3272	68	39	/	/	SYM
ejpam-3272	68	40	eur	eur	PROPN
ejpam-3272	68	41	.	.	PUNCT
ejpam-3272	69	1	j.	j.	PROPN
ejpam-3272	69	2	pure	pure	PROPN
ejpam-3272	69	3	appl	appl	PROPN
ejpam-3272	69	4	.	.	PROPN
ejpam-3272	69	5	math	math	PROPN
ejpam-3272	69	6	,	,	PUNCT
ejpam-3272	69	7	11	11	NUM
ejpam-3272	69	8	(	(	PUNCT
ejpam-3272	69	9	3	3	NUM
ejpam-3272	69	10	)	)	PUNCT
ejpam-3272	69	11	(	(	PUNCT
ejpam-3272	69	12	2018	2018	NUM
ejpam-3272	69	13	)	)	PUNCT
ejpam-3272	69	14	,	,	PUNCT
ejpam-3272	69	15	612	612	NUM
ejpam-3272	69	16	-	-	SYM
ejpam-3272	69	17	627	627	NUM
ejpam-3272	69	18	615	615	NUM
ejpam-3272	69	19	(	(	PUNCT
ejpam-3272	69	20	cs1	cs1	NOUN
ejpam-3272	69	21	)	)	PUNCT
ejpam-3272	69	22	∨	∨	NUM
ejpam-3272	69	23	n	n	CCONJ
ejpam-3272	69	24	∈	∈	PROPN
ejpam-3272	69	25	s	s	NOUN
ejpam-3272	69	26	for	for	ADP
ejpam-3272	69	27	all	all	PRON
ejpam-3272	69	28	n	n	PRON
ejpam-3272	69	29	⊆	⊆	NUM
ejpam-3272	69	30	s	s	NOUN
ejpam-3272	69	31	,	,	PUNCT
ejpam-3272	69	32	(	(	PUNCT
ejpam-3272	69	33	cs2	cs2	NOUN
ejpam-3272	69	34	)	)	PUNCT
ejpam-3272	69	35	s←	s←	PUNCT
ejpam-3272	70	1	x	x	X
ejpam-3272	70	2	∈	∈	NOUN
ejpam-3272	70	3	s	s	X
ejpam-3272	70	4	for	for	ADP
ejpam-3272	70	5	all	all	DET
ejpam-3272	70	6	s	s	PART
ejpam-3272	70	7	∈	∈	PROPN
ejpam-3272	70	8	s	s	NOUN
ejpam-3272	70	9	and	and	CCONJ
ejpam-3272	70	10	x	x	SYM
ejpam-3272	70	11	∈m	∈m	NOUN
ejpam-3272	70	12	.	.	PUNCT
ejpam-3272	71	1	observe	observe	VERB
ejpam-3272	71	2	that	that	SCONJ
ejpam-3272	71	3	s	s	VERB
ejpam-3272	71	4	⊆	⊆	NUM
ejpam-3272	71	5	m	m	NOUN
ejpam-3272	71	6	is	be	AUX
ejpam-3272	71	7	a	a	DET
ejpam-3272	71	8	subcolocale	subcolocale	NOUN
ejpam-3272	71	9	if	if	SCONJ
ejpam-3272	71	10	and	and	CCONJ
ejpam-3272	71	11	only	only	ADV
ejpam-3272	71	12	if	if	SCONJ
ejpam-3272	71	13	s	s	NOUN
ejpam-3272	71	14	is	be	AUX
ejpam-3272	71	15	a	a	DET
ejpam-3272	71	16	colocale	colocale	NOUN
ejpam-3272	71	17	with	with	ADP
ejpam-3272	71	18	the	the	DET
ejpam-3272	71	19	induced	induced	ADJ
ejpam-3272	71	20	order	order	NOUN
ejpam-3272	71	21	and	and	CCONJ
ejpam-3272	71	22	the	the	DET
ejpam-3272	71	23	embedding	embed	VERB
ejpam-3272	71	24	ic	ic	PROPN
ejpam-3272	71	25	:	:	PUNCT
ejpam-3272	71	26	s	s	X
ejpam-3272	72	1	→m	→m	X
ejpam-3272	72	2	is	be	AUX
ejpam-3272	72	3	a	a	DET
ejpam-3272	72	4	morphism	morphism	NOUN
ejpam-3272	72	5	of	of	ADP
ejpam-3272	72	6	loc	loc	PROPN
ejpam-3272	72	7	.	.	PUNCT
ejpam-3272	73	1	the	the	DET
ejpam-3272	73	2	lattice	lattice	NOUN
ejpam-3272	73	3	of	of	ADP
ejpam-3272	73	4	all	all	DET
ejpam-3272	73	5	sublocales	sublocale	NOUN
ejpam-3272	73	6	of	of	ADP
ejpam-3272	73	7	locale	locale	PROPN
ejpam-3272	73	8	l	l	PROPN
ejpam-3272	73	9	and	and	CCONJ
ejpam-3272	73	10	the	the	DET
ejpam-3272	73	11	lattice	lattice	NOUN
ejpam-3272	73	12	of	of	ADP
ejpam-3272	73	13	all	all	DET
ejpam-3272	73	14	subcolocales	subcolocale	NOUN
ejpam-3272	73	15	of	of	ADP
ejpam-3272	73	16	colocale	colocale	NOUN
ejpam-3272	73	17	m	m	VERB
ejpam-3272	73	18	are	be	AUX
ejpam-3272	73	19	denoted	denote	VERB
ejpam-3272	73	20	by	by	ADP
ejpam-3272	73	21	sl(l	sl(l	PROPN
ejpam-3272	73	22	)	)	PUNCT
ejpam-3272	73	23	and	and	CCONJ
ejpam-3272	73	24	scl(m	scl(m	PROPN
ejpam-3272	73	25	)	)	PUNCT
ejpam-3272	73	26	,	,	PUNCT
ejpam-3272	73	27	respectively	respectively	ADV
ejpam-3272	73	28	.	.	PUNCT
ejpam-3272	74	1	note	note	VERB
ejpam-3272	74	2	that	that	SCONJ
ejpam-3272	74	3	these	these	DET
ejpam-3272	74	4	two	two	NUM
ejpam-3272	74	5	lattices	lattice	NOUN
ejpam-3272	74	6	are	be	AUX
ejpam-3272	74	7	both	both	DET
ejpam-3272	74	8	coframes	coframe	NOUN
ejpam-3272	74	9	and	and	CCONJ
ejpam-3272	74	10	hence	hence	ADV
ejpam-3272	74	11	they	they	PRON
ejpam-3272	74	12	satisfy	satisfy	VERB
ejpam-3272	74	13	de	de	PROPN
ejpam-3272	74	14	morgan	morgan	PROPN
ejpam-3272	74	15	’s	’s	PART
ejpam-3272	74	16	second	second	ADJ
ejpam-3272	74	17	law	law	NOUN
ejpam-3272	74	18	stating	state	VERB
ejpam-3272	74	19	that	that	SCONJ
ejpam-3272	74	20	(	(	PUNCT
ejpam-3272	74	21	∧	∧	PROPN
ejpam-3272	74	22	i∈i	i∈i	ADJ
ejpam-3272	74	23	ai	ai	VERB
ejpam-3272	74	24	)	)	PUNCT
ejpam-3272	74	25	∗	∗	NOUN
ejpam-3272	74	26	=	=	PUNCT
ejpam-3272	75	1	∨	∨	NOUN
ejpam-3272	75	2	i∈i	i∈i	ADJ
ejpam-3272	75	3	ai	ai	VERB
ejpam-3272	75	4	∗	∗	NOUN
ejpam-3272	75	5	whenever	whenever	SCONJ
ejpam-3272	75	6	∧	∧	PROPN
ejpam-3272	75	7	i∈i	i∈i	ADJ
ejpam-3272	75	8	ai	ai	VERB
ejpam-3272	75	9	exists	exist	NOUN
ejpam-3272	75	10	.	.	PUNCT
ejpam-3272	76	1	(	(	PUNCT
ejpam-3272	76	2	here	here	ADV
ejpam-3272	76	3	ai	ai	VERB
ejpam-3272	76	4	∗	∗	NOUN
ejpam-3272	76	5	denotes	denote	NOUN
ejpam-3272	76	6	the	the	DET
ejpam-3272	76	7	pseudocomplement	pseudocomplement	NOUN
ejpam-3272	76	8	of	of	ADP
ejpam-3272	76	9	ai	ai	NOUN
ejpam-3272	76	10	)	)	PUNCT
ejpam-3272	76	11	.	.	PUNCT
ejpam-3272	77	1	all	all	PRON
ejpam-3272	77	2	joins	join	VERB
ejpam-3272	77	3	and	and	CCONJ
ejpam-3272	77	4	meets	meet	NOUN
ejpam-3272	77	5	of	of	ADP
ejpam-3272	77	6	sublocales	sublocale	NOUN
ejpam-3272	77	7	(	(	PUNCT
ejpam-3272	77	8	resp	resp	NOUN
ejpam-3272	77	9	.	.	PUNCT
ejpam-3272	78	1	subcolocales	subcolocale	NOUN
ejpam-3272	78	2	)	)	PUNCT
ejpam-3272	78	3	are	be	AUX
ejpam-3272	78	4	taken	take	VERB
ejpam-3272	78	5	in	in	ADP
ejpam-3272	78	6	the	the	DET
ejpam-3272	78	7	lattice	lattice	NOUN
ejpam-3272	78	8	sl(l	sl(l	NOUN
ejpam-3272	78	9	)	)	PUNCT
ejpam-3272	78	10	(	(	PUNCT
ejpam-3272	78	11	resp	resp	NOUN
ejpam-3272	78	12	.	.	PUNCT
ejpam-3272	79	1	scl(m	scl(m	PROPN
ejpam-3272	79	2	)	)	PUNCT
ejpam-3272	79	3	)	)	PUNCT
ejpam-3272	79	4	.	.	PUNCT
ejpam-3272	80	1	let	let	VERB
ejpam-3272	80	2	l	l	NOUN
ejpam-3272	80	3	be	be	AUX
ejpam-3272	80	4	a	a	DET
ejpam-3272	80	5	locale	locale	NOUN
ejpam-3272	80	6	.	.	PUNCT
ejpam-3272	81	1	then	then	ADV
ejpam-3272	81	2	the	the	DET
ejpam-3272	81	3	elements	element	NOUN
ejpam-3272	81	4	o(a	o(a	NOUN
ejpam-3272	81	5	)	)	PUNCT
ejpam-3272	81	6	=	=	PRON
ejpam-3272	81	7	{	{	PUNCT
ejpam-3272	81	8	a→	a→	PROPN
ejpam-3272	81	9	x	x	SYM
ejpam-3272	81	10	:	:	PUNCT
ejpam-3272	81	11	x	x	SYM
ejpam-3272	81	12	∈	∈	PROPN
ejpam-3272	81	13	l	l	NOUN
ejpam-3272	81	14	}	}	PUNCT
ejpam-3272	81	15	and	and	CCONJ
ejpam-3272	81	16	c(a	c(a	ADV
ejpam-3272	81	17	)	)	PUNCT
ejpam-3272	82	1	=	=	SYM
ejpam-3272	82	2	↑	↑	NOUN
ejpam-3272	82	3	a	a	PRON
ejpam-3272	82	4	of	of	ADP
ejpam-3272	82	5	sl(l	sl(l	NUM
ejpam-3272	82	6	)	)	PUNCT
ejpam-3272	82	7	are	be	AUX
ejpam-3272	82	8	referred	refer	VERB
ejpam-3272	82	9	to	to	ADP
ejpam-3272	82	10	as	as	ADV
ejpam-3272	82	11	open	open	ADJ
ejpam-3272	82	12	and	and	CCONJ
ejpam-3272	82	13	closed	closed	ADJ
ejpam-3272	82	14	sublocales	sublocale	NOUN
ejpam-3272	82	15	corresponding	correspond	VERB
ejpam-3272	82	16	a	a	DET
ejpam-3272	82	17	∈	∈	PROPN
ejpam-3272	82	18	l	l	NOUN
ejpam-3272	82	19	,	,	PUNCT
ejpam-3272	82	20	respectively	respectively	ADV
ejpam-3272	82	21	.	.	PUNCT
ejpam-3272	83	1	dually	dually	ADV
ejpam-3272	83	2	,	,	PUNCT
ejpam-3272	83	3	given	give	VERB
ejpam-3272	83	4	a	a	DET
ejpam-3272	83	5	coframe	coframe	NOUN
ejpam-3272	83	6	m	m	VERB
ejpam-3272	83	7	,	,	PUNCT
ejpam-3272	83	8	we	we	PRON
ejpam-3272	83	9	define	define	VERB
ejpam-3272	83	10	the	the	DET
ejpam-3272	83	11	subcolocales	subcolocale	NOUN
ejpam-3272	83	12	oc(k	oc(k	NUM
ejpam-3272	83	13	)	)	PUNCT
ejpam-3272	83	14	=	=	PRON
ejpam-3272	84	1	{	{	PUNCT
ejpam-3272	84	2	x←	x←	PROPN
ejpam-3272	84	3	k	k	PROPN
ejpam-3272	84	4	:	:	PUNCT
ejpam-3272	85	1	k	k	PROPN
ejpam-3272	85	2	∈m	∈m	NOUN
ejpam-3272	85	3	}	}	PUNCT
ejpam-3272	85	4	=	=	SYM
ejpam-3272	85	5	{	{	PUNCT
ejpam-3272	85	6	x	x	X
ejpam-3272	85	7	∈m	∈m	NOUN
ejpam-3272	85	8	:	:	PUNCT
ejpam-3272	85	9	x←	x←	X
ejpam-3272	86	1	k	k	X
ejpam-3272	86	2	=	=	PUNCT
ejpam-3272	86	3	x	x	NOUN
ejpam-3272	86	4	}	}	PUNCT
ejpam-3272	86	5	and	and	CCONJ
ejpam-3272	86	6	cc(k	cc(k	NOUN
ejpam-3272	86	7	)	)	PUNCT
ejpam-3272	86	8	=	=	NUM
ejpam-3272	86	9	↓	↓	PROPN
ejpam-3272	86	10	k.	k.	PROPN
ejpam-3272	87	1	the	the	DET
ejpam-3272	87	2	former	former	ADJ
ejpam-3272	87	3	is	be	AUX
ejpam-3272	87	4	referred	refer	VERB
ejpam-3272	87	5	to	to	ADP
ejpam-3272	87	6	as	as	ADV
ejpam-3272	87	7	open	open	ADJ
ejpam-3272	87	8	subcolocale	subcolocale	NOUN
ejpam-3272	87	9	and	and	CCONJ
ejpam-3272	87	10	the	the	DET
ejpam-3272	87	11	latter	latter	ADJ
ejpam-3272	87	12	is	be	AUX
ejpam-3272	87	13	referred	refer	VERB
ejpam-3272	87	14	to	to	ADP
ejpam-3272	87	15	as	as	ADP
ejpam-3272	87	16	closed	close	VERB
ejpam-3272	87	17	subcolocale	subcolocale	NOUN
ejpam-3272	87	18	corresponding	correspond	VERB
ejpam-3272	87	19	k	k	PROPN
ejpam-3272	87	20	∈	∈	PROPN
ejpam-3272	87	21	m	m	VERB
ejpam-3272	87	22	.	.	PUNCT
ejpam-3272	88	1	unlike	unlike	ADP
ejpam-3272	88	2	subspaces	subspace	NOUN
ejpam-3272	88	3	,	,	PUNCT
ejpam-3272	88	4	not	not	PART
ejpam-3272	88	5	every	every	DET
ejpam-3272	88	6	sublocale	sublocale	NOUN
ejpam-3272	88	7	is	be	AUX
ejpam-3272	88	8	complemented	complement	VERB
ejpam-3272	88	9	in	in	ADP
ejpam-3272	88	10	the	the	DET
ejpam-3272	88	11	lattice	lattice	NOUN
ejpam-3272	88	12	of	of	ADP
ejpam-3272	88	13	sublocales	sublocale	NOUN
ejpam-3272	88	14	,	,	PUNCT
ejpam-3272	88	15	however	however	ADV
ejpam-3272	88	16	,	,	PUNCT
ejpam-3272	88	17	o(a	o(a	NUM
ejpam-3272	88	18	)	)	PUNCT
ejpam-3272	88	19	and	and	CCONJ
ejpam-3272	88	20	c(a	c(a	ADV
ejpam-3272	88	21	)	)	PUNCT
ejpam-3272	88	22	are	be	AUX
ejpam-3272	88	23	complementary	complementary	ADJ
ejpam-3272	88	24	pairs	pair	NOUN
ejpam-3272	88	25	in	in	ADP
ejpam-3272	88	26	sl(l	sl(l	NOUN
ejpam-3272	88	27	)	)	PUNCT
ejpam-3272	88	28	.	.	PUNCT
ejpam-3272	89	1	similarly	similarly	ADV
ejpam-3272	89	2	,	,	PUNCT
ejpam-3272	89	3	oc(k	oc(k	NUM
ejpam-3272	89	4	)	)	PUNCT
ejpam-3272	89	5	is	be	AUX
ejpam-3272	89	6	a	a	DET
ejpam-3272	89	7	complement	complement	NOUN
ejpam-3272	89	8	of	of	ADP
ejpam-3272	89	9	cc(k	cc(k	NOUN
ejpam-3272	89	10	)	)	PUNCT
ejpam-3272	89	11	in	in	ADP
ejpam-3272	89	12	scl(m	scl(m	PROPN
ejpam-3272	89	13	)	)	PUNCT
ejpam-3272	89	14	.	.	PUNCT
ejpam-3272	90	1	there	there	PRON
ejpam-3272	90	2	is	be	VERB
ejpam-3272	90	3	another	another	DET
ejpam-3272	90	4	way	way	NOUN
ejpam-3272	90	5	of	of	ADP
ejpam-3272	90	6	defining	define	VERB
ejpam-3272	90	7	sublocales	sublocale	NOUN
ejpam-3272	90	8	(	(	PUNCT
ejpam-3272	90	9	resp	resp	NOUN
ejpam-3272	90	10	.	.	PUNCT
ejpam-3272	91	1	subcolocales	subcolocale	NOUN
ejpam-3272	91	2	)	)	PUNCT
ejpam-3272	91	3	by	by	ADP
ejpam-3272	91	4	using	use	VERB
ejpam-3272	91	5	the	the	DET
ejpam-3272	91	6	notion	notion	NOUN
ejpam-3272	91	7	of	of	ADP
ejpam-3272	91	8	nuclei	nucleus	NOUN
ejpam-3272	91	9	(	(	PUNCT
ejpam-3272	91	10	resp	resp	NOUN
ejpam-3272	91	11	.	.	PUNCT
ejpam-3272	91	12	conuclei	conuclei	PROPN
ejpam-3272	91	13	)	)	PUNCT
ejpam-3272	91	14	.	.	PUNCT
ejpam-3272	92	1	a	a	DET
ejpam-3272	92	2	nucleus	nucleus	NOUN
ejpam-3272	92	3	on	on	ADP
ejpam-3272	92	4	a	a	DET
ejpam-3272	92	5	frame	frame	NOUN
ejpam-3272	92	6	l	l	NOUN
ejpam-3272	92	7	is	be	AUX
ejpam-3272	92	8	a	a	DET
ejpam-3272	92	9	closure	closure	NOUN
ejpam-3272	92	10	operator	operator	NOUN
ejpam-3272	92	11	v	v	NOUN
ejpam-3272	92	12	:	:	PUNCT
ejpam-3272	92	13	l→	l→	X
ejpam-3272	92	14	l	l	NOUN
ejpam-3272	92	15	preserving	preserve	VERB
ejpam-3272	92	16	finite	finite	NOUN
ejpam-3272	92	17	meets	meet	VERB
ejpam-3272	92	18	.	.	PUNCT
ejpam-3272	93	1	for	for	ADP
ejpam-3272	93	2	a	a	DET
ejpam-3272	93	3	sublocale	sublocale	NOUN
ejpam-3272	93	4	s	s	VERB
ejpam-3272	93	5	⊆	⊆	NUM
ejpam-3272	93	6	l	l	NOUN
ejpam-3272	93	7	,	,	PUNCT
ejpam-3272	93	8	vs(a	vs(a	NOUN
ejpam-3272	93	9	)	)	PUNCT
ejpam-3272	93	10	=	=	PUNCT
ejpam-3272	93	11	∧	∧	NOUN
ejpam-3272	93	12	{	{	PUNCT
ejpam-3272	93	13	s	s	NOUN
ejpam-3272	93	14	∈	∈	NOUN
ejpam-3272	93	15	s	s	PART
ejpam-3272	93	16	:	:	PUNCT
ejpam-3272	93	17	a	a	DET
ejpam-3272	93	18	≤	≤	NUM
ejpam-3272	93	19	s	s	VERB
ejpam-3272	93	20	}	}	PUNCT
ejpam-3272	93	21	is	be	AUX
ejpam-3272	93	22	a	a	DET
ejpam-3272	93	23	nucleus	nucleus	NOUN
ejpam-3272	93	24	,	,	PUNCT
ejpam-3272	93	25	and	and	CCONJ
ejpam-3272	93	26	given	give	VERB
ejpam-3272	93	27	a	a	DET
ejpam-3272	93	28	nucleus	nucleus	NOUN
ejpam-3272	93	29	v	v	NOUN
ejpam-3272	93	30	on	on	ADP
ejpam-3272	93	31	l	l	PROPN
ejpam-3272	93	32	,	,	PUNCT
ejpam-3272	93	33	sv	sv	PROPN
ejpam-3272	93	34	=	=	SYM
ejpam-3272	93	35	v(l	v(l	PROPN
ejpam-3272	93	36	)	)	PUNCT
ejpam-3272	93	37	is	be	AUX
ejpam-3272	93	38	a	a	DET
ejpam-3272	93	39	sublocale	sublocale	NOUN
ejpam-3272	93	40	.	.	PUNCT
ejpam-3272	94	1	further	far	ADV
ejpam-3272	94	2	we	we	PRON
ejpam-3272	94	3	have	have	VERB
ejpam-3272	94	4	vsv	vsv	PROPN
ejpam-3272	94	5	=	=	PROPN
ejpam-3272	94	6	v	v	PROPN
ejpam-3272	94	7	and	and	CCONJ
ejpam-3272	94	8	svs	sv	VERB
ejpam-3272	94	9	=	=	PUNCT
ejpam-3272	94	10	s.	s.	PROPN
ejpam-3272	94	11	a	a	DET
ejpam-3272	94	12	conucleus	conucleus	PROPN
ejpam-3272	94	13	on	on	ADP
ejpam-3272	94	14	a	a	DET
ejpam-3272	94	15	coframe	coframe	NOUN
ejpam-3272	94	16	m	m	VERB
ejpam-3272	94	17	is	be	AUX
ejpam-3272	94	18	a	a	DET
ejpam-3272	94	19	kernel	kernel	NOUN
ejpam-3272	94	20	operator	operator	NOUN
ejpam-3272	94	21	t	t	NOUN
ejpam-3272	94	22	:	:	PUNCT
ejpam-3272	94	23	m	m	AUX
ejpam-3272	94	24	→	→	AUX
ejpam-3272	94	25	m	m	AUX
ejpam-3272	94	26	preserving	preserve	VERB
ejpam-3272	94	27	finite	finite	NOUN
ejpam-3272	94	28	joins	join	NOUN
ejpam-3272	94	29	.	.	PUNCT
ejpam-3272	95	1	the	the	DET
ejpam-3272	95	2	subcolocale	subcolocale	NOUN
ejpam-3272	95	3	generated	generate	VERB
ejpam-3272	95	4	by	by	ADP
ejpam-3272	95	5	the	the	DET
ejpam-3272	95	6	conucleus	conucleus	PROPN
ejpam-3272	95	7	t	t	PROPN
ejpam-3272	95	8	:	:	PUNCT
ejpam-3272	95	9	m	m	VERB
ejpam-3272	95	10	→	→	NOUN
ejpam-3272	95	11	m	m	PROPN
ejpam-3272	95	12	is	be	AUX
ejpam-3272	95	13	st	st	PROPN
ejpam-3272	95	14	=	=	PUNCT
ejpam-3272	95	15	t(m	t(m	PROPN
ejpam-3272	95	16	)	)	PUNCT
ejpam-3272	95	17	.	.	PUNCT
ejpam-3272	96	1	on	on	ADP
ejpam-3272	96	2	the	the	DET
ejpam-3272	96	3	other	other	ADJ
ejpam-3272	96	4	hand	hand	NOUN
ejpam-3272	96	5	,	,	PUNCT
ejpam-3272	96	6	for	for	ADP
ejpam-3272	96	7	a	a	DET
ejpam-3272	96	8	subcolocale	subcolocale	NOUN
ejpam-3272	96	9	s	s	PROPN
ejpam-3272	96	10	⊆	⊆	NUM
ejpam-3272	96	11	m	m	NOUN
ejpam-3272	96	12	,	,	PUNCT
ejpam-3272	96	13	the	the	DET
ejpam-3272	96	14	corresponding	correspond	VERB
ejpam-3272	96	15	conuclei	conuclei	NOUN
ejpam-3272	96	16	ts	ts	ADP
ejpam-3272	96	17	:	:	PUNCT
ejpam-3272	96	18	m	m	VERB
ejpam-3272	96	19	→	→	NOUN
ejpam-3272	96	20	m	m	VERB
ejpam-3272	96	21	is	be	AUX
ejpam-3272	96	22	defined	define	VERB
ejpam-3272	96	23	by	by	ADP
ejpam-3272	96	24	ts(a	ts(a	NOUN
ejpam-3272	96	25	)	)	PUNCT
ejpam-3272	97	1	=	=	VERB
ejpam-3272	97	2	ic∗(a	ic∗(a	ADJ
ejpam-3272	97	3	)	)	PUNCT
ejpam-3272	97	4	=	=	SYM
ejpam-3272	97	5	∨	∨	X
ejpam-3272	97	6	{	{	PUNCT
ejpam-3272	97	7	s	s	NOUN
ejpam-3272	97	8	∈	∈	NOUN
ejpam-3272	97	9	s	s	PART
ejpam-3272	97	10	:	:	PUNCT
ejpam-3272	97	11	s	s	VERB
ejpam-3272	97	12	≤	≤	NUM
ejpam-3272	97	13	a	a	PRON
ejpam-3272	97	14	}	}	PUNCT
ejpam-3272	97	15	.	.	PUNCT
ejpam-3272	98	1	moreover	moreover	ADV
ejpam-3272	98	2	,	,	PUNCT
ejpam-3272	98	3	there	there	PRON
ejpam-3272	98	4	is	be	VERB
ejpam-3272	98	5	a	a	DET
ejpam-3272	98	6	one	one	NUM
ejpam-3272	98	7	-	-	PUNCT
ejpam-3272	98	8	one	one	NUM
ejpam-3272	98	9	correspondence	correspondence	NOUN
ejpam-3272	98	10	between	between	ADP
ejpam-3272	98	11	the	the	DET
ejpam-3272	98	12	subcolocales	subcolocale	NOUN
ejpam-3272	98	13	of	of	ADP
ejpam-3272	98	14	m	m	PRON
ejpam-3272	98	15	and	and	CCONJ
ejpam-3272	98	16	the	the	DET
ejpam-3272	98	17	conuclei	conuclei	NOUN
ejpam-3272	98	18	defined	define	VERB
ejpam-3272	98	19	on	on	ADP
ejpam-3272	98	20	m	m	PROPN
ejpam-3272	98	21	.	.	PUNCT
ejpam-3272	99	1	proposition	proposition	NOUN
ejpam-3272	99	2	2	2	NUM
ejpam-3272	99	3	.	.	PUNCT
ejpam-3272	100	1	let	let	VERB
ejpam-3272	100	2	m	m	PRON
ejpam-3272	100	3	be	be	AUX
ejpam-3272	100	4	a	a	DET
ejpam-3272	100	5	coframe	coframe	NOUN
ejpam-3272	100	6	.	.	PUNCT
ejpam-3272	101	1	then	then	ADV
ejpam-3272	101	2	(	(	PUNCT
ejpam-3272	101	3	i	i	NOUN
ejpam-3272	101	4	)	)	PUNCT
ejpam-3272	101	5	a	a	DET
ejpam-3272	101	6	≤	≤	PROPN
ejpam-3272	101	7	b	b	X
ejpam-3272	101	8	iff	iff	PROPN
ejpam-3272	101	9	cc(a	cc(a	NUM
ejpam-3272	101	10	)	)	PUNCT
ejpam-3272	101	11	⊆	⊆	NUM
ejpam-3272	101	12	cc(b	cc(b	NOUN
ejpam-3272	101	13	)	)	PUNCT
ejpam-3272	101	14	iff	iff	NOUN
ejpam-3272	101	15	oc(b	oc(b	NOUN
ejpam-3272	101	16	)	)	PUNCT
ejpam-3272	101	17	⊆	⊆	NUM
ejpam-3272	101	18	oc(a	oc(a	NUM
ejpam-3272	101	19	)	)	PUNCT
ejpam-3272	101	20	.	.	PUNCT
ejpam-3272	102	1	(	(	PUNCT
ejpam-3272	102	2	ii	ii	X
ejpam-3272	102	3	)	)	PUNCT
ejpam-3272	102	4	⋂	⋂	PROPN
ejpam-3272	102	5	i∈i	i∈i	ADJ
ejpam-3272	102	6	cc(ai	cc(ai	PROPN
ejpam-3272	102	7	)	)	PUNCT
ejpam-3272	102	8	=	=	SYM
ejpam-3272	102	9	cc	cc	PROPN
ejpam-3272	102	10	(	(	PUNCT
ejpam-3272	102	11	∧	∧	PROPN
ejpam-3272	102	12	i∈i	i∈i	ADJ
ejpam-3272	102	13	ai	ai	PROPN
ejpam-3272	102	14	)	)	PUNCT
ejpam-3272	102	15	.	.	PUNCT
ejpam-3272	103	1	(	(	PUNCT
ejpam-3272	103	2	iii	iii	NOUN
ejpam-3272	103	3	)	)	PUNCT
ejpam-3272	103	4	cc(a	cc(a	PUNCT
ejpam-3272	103	5	)	)	PUNCT
ejpam-3272	103	6	∨	∨	NUM
ejpam-3272	103	7	cc(b	cc(b	NOUN
ejpam-3272	103	8	)	)	PUNCT
ejpam-3272	103	9	=	=	PUNCT
ejpam-3272	103	10	cc(a	cc(a	PUNCT
ejpam-3272	103	11	∨	∨	PROPN
ejpam-3272	103	12	b	b	NOUN
ejpam-3272	103	13	)	)	PUNCT
ejpam-3272	103	14	.	.	PUNCT
ejpam-3272	104	1	(	(	PUNCT
ejpam-3272	104	2	iv	iv	X
ejpam-3272	104	3	)	)	PUNCT
ejpam-3272	104	4	∨	∨	NUM
ejpam-3272	104	5	i∈i	i∈i	ADJ
ejpam-3272	104	6	oc(ai	oc(ai	PROPN
ejpam-3272	104	7	)	)	PUNCT
ejpam-3272	105	1	=	=	SYM
ejpam-3272	105	2	oc	oc	PROPN
ejpam-3272	105	3	(	(	PUNCT
ejpam-3272	105	4	∧	∧	PROPN
ejpam-3272	105	5	i∈i	i∈i	ADJ
ejpam-3272	105	6	ai	ai	PROPN
ejpam-3272	105	7	)	)	PUNCT
ejpam-3272	105	8	.	.	PUNCT
ejpam-3272	106	1	(	(	PUNCT
ejpam-3272	106	2	v	v	NOUN
ejpam-3272	106	3	)	)	PUNCT
ejpam-3272	106	4	oc(a	oc(a	NUM
ejpam-3272	106	5	)	)	PUNCT
ejpam-3272	106	6	∩	∩	NOUN
ejpam-3272	106	7	oc(b	oc(b	X
ejpam-3272	106	8	)	)	PUNCT
ejpam-3272	106	9	=	=	SYM
ejpam-3272	107	1	oc(a	oc(a	NUM
ejpam-3272	107	2	∨	∨	NUM
ejpam-3272	107	3	b.	b.	PROPN
ejpam-3272	107	4	)	)	PUNCT
ejpam-3272	107	5	see	see	VERB
ejpam-3272	107	6	[	[	X
ejpam-3272	107	7	10	10	NUM
ejpam-3272	107	8	,	,	PUNCT
ejpam-3272	107	9	iii	iii	NUM
ejpam-3272	107	10	6.1.5	6.1.5	NOUN
ejpam-3272	107	11	]	]	X
ejpam-3272	107	12	for	for	ADP
ejpam-3272	107	13	the	the	DET
ejpam-3272	107	14	frame	frame	NOUN
ejpam-3272	107	15	version	version	NOUN
ejpam-3272	107	16	of	of	ADP
ejpam-3272	107	17	the	the	DET
ejpam-3272	107	18	proposition	proposition	NOUN
ejpam-3272	107	19	above	above	ADV
ejpam-3272	107	20	.	.	PUNCT
ejpam-3272	108	1	recall	recall	VERB
ejpam-3272	108	2	that	that	SCONJ
ejpam-3272	108	3	a	a	DET
ejpam-3272	108	4	diframe	diframe	NOUN
ejpam-3272	108	5	is	be	AUX
ejpam-3272	108	6	a	a	DET
ejpam-3272	108	7	triple	triple	ADJ
ejpam-3272	108	8	l	l	NOUN
ejpam-3272	108	9	=	=	SYM
ejpam-3272	108	10	(	(	PUNCT
ejpam-3272	108	11	le	le	X
ejpam-3272	108	12	,	,	PUNCT
ejpam-3272	108	13	lfr	lfr	PROPN
ejpam-3272	108	14	,	,	PUNCT
ejpam-3272	108	15	lcf	lcf	PROPN
ejpam-3272	108	16	)	)	PUNCT
ejpam-3272	108	17	in	in	ADP
ejpam-3272	108	18	which	which	PRON
ejpam-3272	108	19	le	le	X
ejpam-3272	108	20	is	be	AUX
ejpam-3272	108	21	both	both	PRON
ejpam-3272	108	22	a	a	DET
ejpam-3272	108	23	frame	frame	NOUN
ejpam-3272	108	24	and	and	CCONJ
ejpam-3272	108	25	a	a	DET
ejpam-3272	108	26	coframe	coframe	NOUN
ejpam-3272	108	27	,	,	PUNCT
ejpam-3272	108	28	lfr	lfr	PROPN
ejpam-3272	108	29	is	be	AUX
ejpam-3272	108	30	a	a	DET
ejpam-3272	108	31	subframe	subframe	NOUN
ejpam-3272	108	32	and	and	CCONJ
ejpam-3272	108	33	lcf	lcf	PROPN
ejpam-3272	108	34	is	be	AUX
ejpam-3272	108	35	a	a	DET
ejpam-3272	108	36	subcoframe	subcoframe	NOUN
ejpam-3272	108	37	of	of	ADP
ejpam-3272	108	38	le	le	X
ejpam-3272	108	39	.	.	PUNCT
ejpam-3272	109	1	a	a	DET
ejpam-3272	109	2	diframe	diframe	NOUN
ejpam-3272	109	3	homomorphism	homomorphism	NOUN
ejpam-3272	109	4	is	be	AUX
ejpam-3272	109	5	a	a	DET
ejpam-3272	109	6	triple	triple	ADJ
ejpam-3272	109	7	(	(	PUNCT
ejpam-3272	109	8	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	109	9	)	)	PUNCT
ejpam-3272	109	10	with	with	ADP
ejpam-3272	109	11	the	the	DET
ejpam-3272	109	12	following	follow	VERB
ejpam-3272	109	13	properties	property	NOUN
ejpam-3272	109	14	:	:	PUNCT
ejpam-3272	109	15	(	(	PUNCT
ejpam-3272	109	16	i	i	NOUN
ejpam-3272	109	17	)	)	PUNCT
ejpam-3272	109	18	ϕ	ϕ	NOUN
ejpam-3272	109	19	:	:	PUNCT
ejpam-3272	109	20	le	le	X
ejpam-3272	109	21	→me	→me	X
ejpam-3272	109	22	is	be	AUX
ejpam-3272	109	23	a	a	DET
ejpam-3272	109	24	frame	frame	NOUN
ejpam-3272	109	25	homomorphism	homomorphism	NOUN
ejpam-3272	109	26	and	and	CCONJ
ejpam-3272	109	27	ϕ[lfr	ϕ[lfr	PROPN
ejpam-3272	109	28	]	]	PUNCT
ejpam-3272	109	29	⊆mfr	⊆mfr	PROPN
ejpam-3272	109	30	,	,	PUNCT
ejpam-3272	109	31	e.	e.	PROPN
ejpam-3272	109	32	korkmaz	korkmaz	PROPN
ejpam-3272	109	33	,	,	PUNCT
ejpam-3272	109	34	r.	r.	PROPN
ejpam-3272	109	35	ertürk	ertürk	PROPN
ejpam-3272	109	36	/	/	SYM
ejpam-3272	109	37	eur	eur	PROPN
ejpam-3272	109	38	.	.	PUNCT
ejpam-3272	110	1	j.	j.	PROPN
ejpam-3272	110	2	pure	pure	PROPN
ejpam-3272	110	3	appl	appl	PROPN
ejpam-3272	110	4	.	.	PROPN
ejpam-3272	110	5	math	math	PROPN
ejpam-3272	110	6	,	,	PUNCT
ejpam-3272	110	7	11	11	NUM
ejpam-3272	110	8	(	(	PUNCT
ejpam-3272	110	9	3	3	NUM
ejpam-3272	110	10	)	)	PUNCT
ejpam-3272	110	11	(	(	PUNCT
ejpam-3272	110	12	2018	2018	NUM
ejpam-3272	110	13	)	)	PUNCT
ejpam-3272	110	14	,	,	PUNCT
ejpam-3272	110	15	612	612	NUM
ejpam-3272	110	16	-	-	SYM
ejpam-3272	110	17	627	627	NUM
ejpam-3272	110	18	616	616	NUM
ejpam-3272	110	19	(	(	PUNCT
ejpam-3272	110	20	ii	ii	NOUN
ejpam-3272	110	21	)	)	PUNCT
ejpam-3272	110	22	ψ	ψ	X
ejpam-3272	110	23	:	:	PUNCT
ejpam-3272	110	24	le	le	X
ejpam-3272	110	25	→me	→me	PUNCT
ejpam-3272	110	26	is	be	AUX
ejpam-3272	110	27	a	a	DET
ejpam-3272	110	28	coframe	coframe	NOUN
ejpam-3272	110	29	homomorphism	homomorphism	NOUN
ejpam-3272	110	30	and	and	CCONJ
ejpam-3272	110	31	ψ[lcf	ψ[lcf	NOUN
ejpam-3272	110	32	]	]	PUNCT
ejpam-3272	110	33	⊆mcf	⊆mcf	NOUN
ejpam-3272	110	34	.	.	PUNCT
ejpam-3272	111	1	the	the	DET
ejpam-3272	111	2	category	category	NOUN
ejpam-3272	111	3	of	of	ADP
ejpam-3272	111	4	diframes	diframe	NOUN
ejpam-3272	111	5	and	and	CCONJ
ejpam-3272	111	6	diframe	diframe	VERB
ejpam-3272	111	7	homomorphisms	homomorphism	NOUN
ejpam-3272	111	8	is	be	AUX
ejpam-3272	111	9	denoted	denote	VERB
ejpam-3272	111	10	by	by	ADP
ejpam-3272	111	11	difrm	difrm	NOUN
ejpam-3272	111	12	.	.	PUNCT
ejpam-3272	112	1	the	the	DET
ejpam-3272	112	2	opposite	opposite	ADJ
ejpam-3272	112	3	category	category	NOUN
ejpam-3272	112	4	of	of	ADP
ejpam-3272	112	5	difrm	difrm	NOUN
ejpam-3272	112	6	is	be	AUX
ejpam-3272	112	7	called	call	VERB
ejpam-3272	112	8	the	the	DET
ejpam-3272	112	9	category	category	NOUN
ejpam-3272	112	10	of	of	ADP
ejpam-3272	112	11	dilocales	dilocale	NOUN
ejpam-3272	112	12	and	and	CCONJ
ejpam-3272	112	13	denoted	denote	VERB
ejpam-3272	112	14	by	by	ADP
ejpam-3272	112	15	diloc	diloc	NOUN
ejpam-3272	112	16	.	.	PUNCT
ejpam-3272	113	1	the	the	DET
ejpam-3272	113	2	following	follow	VERB
ejpam-3272	113	3	examples	example	NOUN
ejpam-3272	113	4	will	will	AUX
ejpam-3272	113	5	be	be	AUX
ejpam-3272	113	6	useful	useful	ADJ
ejpam-3272	113	7	in	in	ADP
ejpam-3272	113	8	the	the	DET
ejpam-3272	113	9	sequel	sequel	NOUN
ejpam-3272	113	10	.	.	PUNCT
ejpam-3272	114	1	example	example	NOUN
ejpam-3272	115	1	1	1	NUM
ejpam-3272	115	2	.	.	PUNCT
ejpam-3272	116	1	(	(	PUNCT
ejpam-3272	116	2	i	i	NOUN
ejpam-3272	116	3	)	)	PUNCT
ejpam-3272	116	4	let	let	VERB
ejpam-3272	116	5	us	we	PRON
ejpam-3272	116	6	see	see	VERB
ejpam-3272	116	7	the	the	DET
ejpam-3272	116	8	motivating	motivating	NOUN
ejpam-3272	116	9	example	example	NOUN
ejpam-3272	116	10	:	:	PUNCT
ejpam-3272	116	11	given	give	VERB
ejpam-3272	116	12	a	a	DET
ejpam-3272	116	13	topological	topological	ADJ
ejpam-3272	116	14	space	space	NOUN
ejpam-3272	116	15	x	x	NOUN
ejpam-3272	116	16	,	,	PUNCT
ejpam-3272	116	17	denote	denote	VERB
ejpam-3272	116	18	by	by	ADP
ejpam-3272	116	19	ω(x	ω(x	NOUN
ejpam-3272	116	20	)	)	PUNCT
ejpam-3272	116	21	(	(	PUNCT
ejpam-3272	116	22	resp	resp	NOUN
ejpam-3272	116	23	.	.	PUNCT
ejpam-3272	117	1	c(x	c(x	NOUN
ejpam-3272	117	2	)	)	PUNCT
ejpam-3272	117	3	)	)	PUNCT
ejpam-3272	118	1	the	the	DET
ejpam-3272	118	2	lattice	lattice	NOUN
ejpam-3272	118	3	of	of	ADP
ejpam-3272	118	4	open	open	ADJ
ejpam-3272	118	5	(	(	PUNCT
ejpam-3272	118	6	resp	resp	NOUN
ejpam-3272	118	7	.	.	PUNCT
ejpam-3272	118	8	closed	closed	ADJ
ejpam-3272	118	9	)	)	PUNCT
ejpam-3272	118	10	sets	set	NOUN
ejpam-3272	118	11	of	of	ADP
ejpam-3272	118	12	x.	x.	NOUN
ejpam-3272	118	13	then	then	ADV
ejpam-3272	118	14	(	(	PUNCT
ejpam-3272	118	15	p(x),ω(x),c(x	p(x),ω(x),c(x	NOUN
ejpam-3272	118	16	)	)	PUNCT
ejpam-3272	118	17	)	)	PUNCT
ejpam-3272	118	18	is	be	AUX
ejpam-3272	118	19	a	a	DET
ejpam-3272	118	20	diframe	diframe	NOUN
ejpam-3272	118	21	.	.	PUNCT
ejpam-3272	119	1	for	for	ADP
ejpam-3272	119	2	a	a	DET
ejpam-3272	119	3	continuous	continuous	ADJ
ejpam-3272	119	4	map	map	NOUN
ejpam-3272	119	5	f	f	NOUN
ejpam-3272	119	6	:	:	PUNCT
ejpam-3272	119	7	x	x	X
ejpam-3272	119	8	→	→	SYM
ejpam-3272	119	9	y	y	PROPN
ejpam-3272	119	10	,	,	PUNCT
ejpam-3272	119	11	the	the	DET
ejpam-3272	119	12	pair	pair	NOUN
ejpam-3272	119	13	(	(	PUNCT
ejpam-3272	119	14	f−1	f−1	PROPN
ejpam-3272	119	15	,	,	PUNCT
ejpam-3272	119	16	f−1	f−1	PROPN
ejpam-3272	119	17	)	)	PUNCT
ejpam-3272	119	18	:	:	PUNCT
ejpam-3272	119	19	(	(	PUNCT
ejpam-3272	119	20	p(y	p(y	PROPN
ejpam-3272	119	21	)	)	PUNCT
ejpam-3272	119	22	,	,	PUNCT
ejpam-3272	119	23	ω(y	ω(y	PROPN
ejpam-3272	119	24	)	)	PUNCT
ejpam-3272	119	25	,	,	PUNCT
ejpam-3272	119	26	c(y	c(y	PROPN
ejpam-3272	119	27	)	)	PUNCT
ejpam-3272	119	28	)	)	PUNCT
ejpam-3272	120	1	→	→	SYM
ejpam-3272	120	2	(	(	PUNCT
ejpam-3272	120	3	p(x),ω(x),c(x	p(x),ω(x),c(x	NOUN
ejpam-3272	120	4	)	)	PUNCT
ejpam-3272	120	5	)	)	PUNCT
ejpam-3272	120	6	is	be	AUX
ejpam-3272	120	7	trivially	trivially	ADV
ejpam-3272	120	8	a	a	DET
ejpam-3272	120	9	diframe	diframe	NOUN
ejpam-3272	120	10	homomorphism	homomorphism	NOUN
ejpam-3272	120	11	.	.	PUNCT
ejpam-3272	121	1	(	(	PUNCT
ejpam-3272	121	2	ii	ii	NOUN
ejpam-3272	121	3	)	)	PUNCT
ejpam-3272	121	4	let	let	AUX
ejpam-3272	121	5	ωreg(r	ωreg(r	NUM
ejpam-3272	121	6	)	)	PUNCT
ejpam-3272	121	7	be	be	AUX
ejpam-3272	121	8	the	the	DET
ejpam-3272	121	9	complete	complete	ADJ
ejpam-3272	121	10	boolean	boolean	ADJ
ejpam-3272	121	11	algebra	algebra	NOUN
ejpam-3272	121	12	of	of	ADP
ejpam-3272	121	13	regular	regular	ADJ
ejpam-3272	121	14	open	open	ADJ
ejpam-3272	121	15	sets	set	NOUN
ejpam-3272	121	16	of	of	ADP
ejpam-3272	121	17	r	r	NOUN
ejpam-3272	121	18	(	(	PUNCT
ejpam-3272	121	19	with	with	ADP
ejpam-3272	121	20	usual	usual	ADJ
ejpam-3272	121	21	topology	topology	NOUN
ejpam-3272	121	22	)	)	PUNCT
ejpam-3272	121	23	.	.	PUNCT
ejpam-3272	122	1	let	let	VERB
ejpam-3272	122	2	le	le	X
ejpam-3272	122	3	=	=	SYM
ejpam-3272	122	4	lcf	lcf	PROPN
ejpam-3272	122	5	=	=	PROPN
ejpam-3272	122	6	ωreg(r	ωreg(r	PROPN
ejpam-3272	122	7	)	)	PUNCT
ejpam-3272	122	8	and	and	CCONJ
ejpam-3272	122	9	lfr	lfr	X
ejpam-3272	122	10	=	=	SYM
ejpam-3272	122	11	{	{	PUNCT
ejpam-3272	122	12	(	(	PUNCT
ejpam-3272	122	13	−∞	−∞	NOUN
ejpam-3272	122	14	,	,	PUNCT
ejpam-3272	122	15	a	a	NOUN
ejpam-3272	122	16	)	)	PUNCT
ejpam-3272	122	17	:	:	PUNCT
ejpam-3272	122	18	a	a	DET
ejpam-3272	122	19	∈	∈	PROPN
ejpam-3272	122	20	r	r	NOUN
ejpam-3272	122	21	}	}	PUNCT
ejpam-3272	122	22	∪	∪	X
ejpam-3272	122	23	{	{	PUNCT
ejpam-3272	122	24	∅,r	∅,r	ADV
ejpam-3272	122	25	}	}	PUNCT
ejpam-3272	122	26	.	.	PUNCT
ejpam-3272	123	1	then	then	ADV
ejpam-3272	123	2	the	the	DET
ejpam-3272	123	3	triple	triple	ADJ
ejpam-3272	123	4	l	l	NOUN
ejpam-3272	123	5	=	=	SYM
ejpam-3272	123	6	(	(	PUNCT
ejpam-3272	123	7	le	le	X
ejpam-3272	123	8	,	,	PUNCT
ejpam-3272	123	9	lfr	lfr	PROPN
ejpam-3272	123	10	,	,	PUNCT
ejpam-3272	123	11	lcf	lcf	PROPN
ejpam-3272	123	12	)	)	PUNCT
ejpam-3272	123	13	is	be	AUX
ejpam-3272	123	14	a	a	DET
ejpam-3272	123	15	diframe	diframe	NOUN
ejpam-3272	123	16	.	.	PUNCT
ejpam-3272	124	1	(	(	PUNCT
ejpam-3272	124	2	iii	iii	X
ejpam-3272	124	3	)	)	PUNCT
ejpam-3272	124	4	let	let	VERB
ejpam-3272	124	5	le	le	X
ejpam-3272	124	6	=	=	SYM
ejpam-3272	124	7	ωreg(r	ωreg(r	PROPN
ejpam-3272	124	8	)	)	PUNCT
ejpam-3272	124	9	,	,	PUNCT
ejpam-3272	124	10	lfr	lfr	X
ejpam-3272	124	11	=	=	SYM
ejpam-3272	124	12	{	{	PUNCT
ejpam-3272	124	13	(	(	PUNCT
ejpam-3272	124	14	−∞	−∞	NOUN
ejpam-3272	124	15	,	,	PUNCT
ejpam-3272	124	16	a	a	NOUN
ejpam-3272	124	17	)	)	PUNCT
ejpam-3272	124	18	:	:	PUNCT
ejpam-3272	124	19	a	a	DET
ejpam-3272	124	20	∈	∈	PROPN
ejpam-3272	124	21	r	r	NOUN
ejpam-3272	124	22	}	}	PUNCT
ejpam-3272	124	23	∪	∪	X
ejpam-3272	124	24	{	{	PUNCT
ejpam-3272	124	25	∅,r	∅,r	PRON
ejpam-3272	124	26	}	}	PUNCT
ejpam-3272	124	27	and	and	CCONJ
ejpam-3272	124	28	lcf	lcf	PROPN
ejpam-3272	124	29	=	=	SYM
ejpam-3272	124	30	{	{	PUNCT
ejpam-3272	124	31	(	(	PUNCT
ejpam-3272	124	32	a,∞	a,∞	PROPN
ejpam-3272	124	33	)	)	PUNCT
ejpam-3272	124	34	:	:	PUNCT
ejpam-3272	124	35	a	a	X
ejpam-3272	124	36	,	,	PUNCT
ejpam-3272	124	37	b	b	X
ejpam-3272	124	38	∈	∈	ADJ
ejpam-3272	124	39	r	r	NOUN
ejpam-3272	124	40	}	}	PUNCT
ejpam-3272	124	41	∪	∪	X
ejpam-3272	124	42	{	{	PUNCT
ejpam-3272	124	43	∅,r	∅,r	ADV
ejpam-3272	124	44	}	}	PUNCT
ejpam-3272	124	45	.	.	PUNCT
ejpam-3272	125	1	then	then	ADV
ejpam-3272	125	2	l	l	NOUN
ejpam-3272	125	3	=	=	SYM
ejpam-3272	125	4	(	(	PUNCT
ejpam-3272	125	5	le	le	X
ejpam-3272	125	6	,	,	PUNCT
ejpam-3272	125	7	lfr	lfr	PROPN
ejpam-3272	125	8	,	,	PUNCT
ejpam-3272	125	9	lcf	lcf	PROPN
ejpam-3272	125	10	)	)	PUNCT
ejpam-3272	125	11	is	be	AUX
ejpam-3272	125	12	a	a	DET
ejpam-3272	125	13	diframe	diframe	NOUN
ejpam-3272	125	14	.	.	PUNCT
ejpam-3272	126	1	(	(	PUNCT
ejpam-3272	126	2	iv	iv	X
ejpam-3272	126	3	)	)	PUNCT
ejpam-3272	126	4	if	if	SCONJ
ejpam-3272	126	5	(	(	PUNCT
ejpam-3272	126	6	s	s	X
ejpam-3272	126	7	,	,	PUNCT
ejpam-3272	126	8	s	s	PROPN
ejpam-3272	126	9	,	,	PUNCT
ejpam-3272	126	10	τ	τ	PROPN
ejpam-3272	126	11	,	,	PUNCT
ejpam-3272	126	12	κ	κ	NOUN
ejpam-3272	126	13	)	)	PUNCT
ejpam-3272	126	14	is	be	AUX
ejpam-3272	126	15	a	a	DET
ejpam-3272	126	16	ditopological	ditopological	ADJ
ejpam-3272	126	17	space	space	NOUN
ejpam-3272	126	18	then	then	ADV
ejpam-3272	126	19	(	(	PUNCT
ejpam-3272	126	20	s	s	X
ejpam-3272	126	21	,	,	PUNCT
ejpam-3272	126	22	s	s	PROPN
ejpam-3272	126	23	,	,	PUNCT
ejpam-3272	126	24	τ	τ	PROPN
ejpam-3272	126	25	,	,	PUNCT
ejpam-3272	126	26	κ	κ	NOUN
ejpam-3272	126	27	)	)	PUNCT
ejpam-3272	126	28	is	be	AUX
ejpam-3272	126	29	a	a	DET
ejpam-3272	126	30	diframe	diframe	NOUN
ejpam-3272	126	31	.	.	PUNCT
ejpam-3272	127	1	now	now	ADV
ejpam-3272	127	2	recall	recall	VERB
ejpam-3272	127	3	the	the	DET
ejpam-3272	127	4	category	category	NOUN
ejpam-3272	127	5	dfditop	dfditop	NOUN
ejpam-3272	127	6	of	of	ADP
ejpam-3272	127	7	ditopological	ditopological	ADJ
ejpam-3272	127	8	texture	texture	ADJ
ejpam-3272	127	9	spaces	space	NOUN
ejpam-3272	127	10	and	and	CCONJ
ejpam-3272	127	11	bicontinuous	bicontinuous	ADJ
ejpam-3272	127	12	difunctions	difunction	NOUN
ejpam-3272	127	13	[	[	X
ejpam-3272	127	14	3	3	NUM
ejpam-3272	127	15	]	]	PUNCT
ejpam-3272	127	16	.	.	PUNCT
ejpam-3272	128	1	we	we	PRON
ejpam-3272	128	2	have	have	VERB
ejpam-3272	128	3	the	the	DET
ejpam-3272	128	4	following	follow	VERB
ejpam-3272	128	5	functor	functor	PROPN
ejpam-3272	128	6	e	e	PROPN
ejpam-3272	128	7	:	:	PUNCT
ejpam-3272	128	8	dfditop→	dfditop→	PROPN
ejpam-3272	128	9	diloc	diloc	PROPN
ejpam-3272	128	10	e((s1	e((s1	PROPN
ejpam-3272	128	11	,	,	PUNCT
ejpam-3272	128	12	s1	s1	NOUN
ejpam-3272	128	13	,	,	PUNCT
ejpam-3272	128	14	τ1	τ1	NOUN
ejpam-3272	128	15	,	,	PUNCT
ejpam-3272	128	16	κ1	κ1	NOUN
ejpam-3272	128	17	)	)	PUNCT
ejpam-3272	128	18	(	(	PUNCT
ejpam-3272	128	19	f	f	X
ejpam-3272	128	20	,	,	PUNCT
ejpam-3272	128	21	f	f	PROPN
ejpam-3272	128	22	)	)	PUNCT
ejpam-3272	128	23	−−−→	−−−→	PROPN
ejpam-3272	128	24	(	(	PUNCT
ejpam-3272	128	25	s2	s2	PROPN
ejpam-3272	128	26	,	,	PUNCT
ejpam-3272	128	27	s2	s2	NOUN
ejpam-3272	128	28	,	,	PUNCT
ejpam-3272	128	29	τ2	τ2	PROPN
ejpam-3272	128	30	,	,	PUNCT
ejpam-3272	128	31	κ2	κ2	NOUN
ejpam-3272	128	32	)	)	PUNCT
ejpam-3272	128	33	=	=	SYM
ejpam-3272	128	34	(	(	PUNCT
ejpam-3272	128	35	s1	s1	NOUN
ejpam-3272	128	36	,	,	PUNCT
ejpam-3272	128	37	τ1	τ1	NOUN
ejpam-3272	128	38	,	,	PUNCT
ejpam-3272	128	39	κ1	κ1	NOUN
ejpam-3272	128	40	)	)	PUNCT
ejpam-3272	128	41	(	(	PUNCT
ejpam-3272	128	42	ϕf	ϕf	INTJ
ejpam-3272	128	43	,	,	PUNCT
ejpam-3272	128	44	ψf	ψf	NOUN
ejpam-3272	128	45	)	)	PUNCT
ejpam-3272	128	46	−−−−−→	−−−−−→	PUNCT
ejpam-3272	128	47	(	(	PUNCT
ejpam-3272	128	48	s2	s2	PROPN
ejpam-3272	128	49	,	,	PUNCT
ejpam-3272	128	50	τ2	τ2	PROPN
ejpam-3272	128	51	,	,	PUNCT
ejpam-3272	128	52	κ2	κ2	NOUN
ejpam-3272	128	53	)	)	PUNCT
ejpam-3272	128	54	,	,	PUNCT
ejpam-3272	128	55	where	where	SCONJ
ejpam-3272	128	56	the	the	DET
ejpam-3272	128	57	arrow	arrow	NOUN
ejpam-3272	128	58	on	on	ADP
ejpam-3272	128	59	the	the	DET
ejpam-3272	128	60	right	right	NOUN
ejpam-3272	128	61	represents	represent	VERB
ejpam-3272	128	62	the	the	DET
ejpam-3272	128	63	diloc	diloc	NOUN
ejpam-3272	128	64	morphism	morphism	NOUN
ejpam-3272	128	65	corresponding	correspond	VERB
ejpam-3272	128	66	to	to	ADP
ejpam-3272	128	67	the	the	DET
ejpam-3272	128	68	difrm	difrm	NOUN
ejpam-3272	128	69	morphism	morphism	NOUN
ejpam-3272	128	70	(	(	PUNCT
ejpam-3272	128	71	s2	s2	PROPN
ejpam-3272	128	72	,	,	PUNCT
ejpam-3272	128	73	τ2	τ2	PROPN
ejpam-3272	128	74	,	,	PUNCT
ejpam-3272	128	75	κ2	κ2	NOUN
ejpam-3272	128	76	)	)	PUNCT
ejpam-3272	128	77	(	(	PUNCT
ejpam-3272	128	78	ϕf←	ϕf←	INTJ
ejpam-3272	128	79	,	,	PUNCT
ejpam-3272	128	80	ψf←	ψf←	ADJ
ejpam-3272	128	81	)	)	PUNCT
ejpam-3272	128	82	=(	=(	NOUN
ejpam-3272	128	83	(	(	PUNCT
ejpam-3272	128	84	ψf	ψf	NOUN
ejpam-3272	128	85	)	)	PUNCT
ejpam-3272	128	86	∗,(ϕf	∗,(ϕf	NOUN
ejpam-3272	128	87	)	)	PUNCT
ejpam-3272	128	88	∗)−−−−−−−−−−−−−−−−−→	∗)−−−−−−−−−−−−−−−−−→	NOUN
ejpam-3272	128	89	(	(	PUNCT
ejpam-3272	128	90	s1	s1	NOUN
ejpam-3272	128	91	,	,	PUNCT
ejpam-3272	128	92	τ1	τ1	NOUN
ejpam-3272	128	93	,	,	PUNCT
ejpam-3272	128	94	κ1	κ1	NOUN
ejpam-3272	128	95	)	)	PUNCT
ejpam-3272	128	96	.	.	PUNCT
ejpam-3272	129	1	a	a	DET
ejpam-3272	129	2	hutton	hutton	PROPN
ejpam-3272	129	3	dispace	dispace	NOUN
ejpam-3272	129	4	is	be	AUX
ejpam-3272	129	5	a	a	DET
ejpam-3272	129	6	triple	triple	ADJ
ejpam-3272	129	7	(	(	PUNCT
ejpam-3272	129	8	l	l	NOUN
ejpam-3272	129	9	,	,	PUNCT
ejpam-3272	129	10	τ	τ	PROPN
ejpam-3272	129	11	,	,	PUNCT
ejpam-3272	129	12	κ	κ	NOUN
ejpam-3272	129	13	)	)	PUNCT
ejpam-3272	129	14	where	where	SCONJ
ejpam-3272	129	15	l	l	NOUN
ejpam-3272	129	16	is	be	AUX
ejpam-3272	129	17	a	a	DET
ejpam-3272	129	18	complete	complete	ADJ
ejpam-3272	129	19	,	,	PUNCT
ejpam-3272	129	20	completely	completely	ADV
ejpam-3272	129	21	distributive	distributive	ADJ
ejpam-3272	129	22	lattice	lattice	NOUN
ejpam-3272	129	23	and	and	CCONJ
ejpam-3272	129	24	(	(	PUNCT
ejpam-3272	129	25	τ	τ	PROPN
ejpam-3272	129	26	,	,	PUNCT
ejpam-3272	129	27	κ	κ	NOUN
ejpam-3272	129	28	)	)	PUNCT
ejpam-3272	129	29	is	be	AUX
ejpam-3272	129	30	a	a	DET
ejpam-3272	129	31	ditopology	ditopology	NOUN
ejpam-3272	129	32	.	.	PUNCT
ejpam-3272	130	1	consider	consider	VERB
ejpam-3272	130	2	the	the	DET
ejpam-3272	130	3	mappings	mapping	NOUN
ejpam-3272	130	4	ϕ	ϕ	NOUN
ejpam-3272	130	5	:	:	PUNCT
ejpam-3272	130	6	(	(	PUNCT
ejpam-3272	130	7	l1	l1	PROPN
ejpam-3272	130	8	,	,	PUNCT
ejpam-3272	130	9	τ1	τ1	NOUN
ejpam-3272	130	10	,	,	PUNCT
ejpam-3272	130	11	κ1	κ1	NOUN
ejpam-3272	130	12	)	)	PUNCT
ejpam-3272	130	13	→	→	SYM
ejpam-3272	130	14	(	(	PUNCT
ejpam-3272	130	15	l2	l2	NOUN
ejpam-3272	130	16	,	,	PUNCT
ejpam-3272	130	17	τ2	τ2	PROPN
ejpam-3272	130	18	,	,	PUNCT
ejpam-3272	130	19	κ2	κ2	NOUN
ejpam-3272	130	20	)	)	PUNCT
ejpam-3272	130	21	preserving	preserve	VERB
ejpam-3272	130	22	arbitrary	arbitrary	ADJ
ejpam-3272	130	23	meets	meet	NOUN
ejpam-3272	130	24	and	and	CCONJ
ejpam-3272	130	25	joins	join	VERB
ejpam-3272	130	26	and	and	CCONJ
ejpam-3272	130	27	satisfying	satisfy	VERB
ejpam-3272	130	28	ϕ[τ1	ϕ[τ1	NOUN
ejpam-3272	130	29	]	]	X
ejpam-3272	130	30	⊆	⊆	NUM
ejpam-3272	130	31	τ2	τ2	NOUN
ejpam-3272	130	32	,	,	PUNCT
ejpam-3272	130	33	ϕ[κ1	ϕ[κ1	NOUN
ejpam-3272	130	34	]	]	PUNCT
ejpam-3272	130	35	⊆	⊆	NUM
ejpam-3272	130	36	κ2	κ2	NOUN
ejpam-3272	130	37	.	.	PUNCT
ejpam-3272	131	1	the	the	DET
ejpam-3272	131	2	resulting	result	VERB
ejpam-3272	131	3	category	category	NOUN
ejpam-3272	131	4	is	be	AUX
ejpam-3272	131	5	denoted	denote	VERB
ejpam-3272	131	6	by	by	ADP
ejpam-3272	131	7	dih	dih	PROPN
ejpam-3272	131	8	.	.	PROPN
ejpam-3272	131	9	by	by	ADP
ejpam-3272	131	10	hdifrm	hdifrm	PROPN
ejpam-3272	131	11	,	,	PUNCT
ejpam-3272	131	12	we	we	PRON
ejpam-3272	131	13	shall	shall	AUX
ejpam-3272	131	14	denote	denote	VERB
ejpam-3272	131	15	the	the	DET
ejpam-3272	131	16	category	category	NOUN
ejpam-3272	131	17	of	of	ADP
ejpam-3272	131	18	diframes	diframe	NOUN
ejpam-3272	131	19	and	and	CCONJ
ejpam-3272	131	20	diframe	diframe	VERB
ejpam-3272	131	21	homomorphism	homomorphism	NOUN
ejpam-3272	131	22	with	with	ADP
ejpam-3272	131	23	ϕ	ϕ	NOUN
ejpam-3272	131	24	=	=	SYM
ejpam-3272	131	25	ψ	ψ	X
ejpam-3272	131	26	.	.	PUNCT
ejpam-3272	132	1	obviously	obviously	ADV
ejpam-3272	132	2	,	,	PUNCT
ejpam-3272	132	3	dih	dih	PROPN
ejpam-3272	132	4	is	be	AUX
ejpam-3272	132	5	a	a	DET
ejpam-3272	132	6	full	full	ADJ
ejpam-3272	132	7	subcategory	subcategory	NOUN
ejpam-3272	132	8	of	of	ADP
ejpam-3272	132	9	hdifrm	hdifrm	NOUN
ejpam-3272	132	10	,	,	PUNCT
ejpam-3272	132	11	and	and	CCONJ
ejpam-3272	132	12	hdifrm	hdifrm	PROPN
ejpam-3272	132	13	is	be	AUX
ejpam-3272	132	14	a	a	DET
ejpam-3272	132	15	non	non	ADJ
ejpam-3272	132	16	-	-	ADJ
ejpam-3272	132	17	full	full	ADJ
ejpam-3272	132	18	subcategory	subcategory	NOUN
ejpam-3272	132	19	of	of	ADP
ejpam-3272	132	20	difrm	difrm	PROPN
ejpam-3272	132	21	.	.	PUNCT
ejpam-3272	133	1	note	note	VERB
ejpam-3272	133	2	that	that	SCONJ
ejpam-3272	133	3	,	,	PUNCT
ejpam-3272	133	4	due	due	ADP
ejpam-3272	133	5	to	to	ADP
ejpam-3272	133	6	the	the	DET
ejpam-3272	133	7	lack	lack	NOUN
ejpam-3272	133	8	of	of	ADP
ejpam-3272	133	9	space	space	NOUN
ejpam-3272	133	10	,	,	PUNCT
ejpam-3272	133	11	the	the	DET
ejpam-3272	133	12	separation	separation	NOUN
ejpam-3272	133	13	axioms	axiom	VERB
ejpam-3272	133	14	for	for	ADP
ejpam-3272	133	15	ditopological	ditopological	ADJ
ejpam-3272	133	16	texture	texture	ADJ
ejpam-3272	133	17	spaces	space	NOUN
ejpam-3272	133	18	is	be	AUX
ejpam-3272	133	19	not	not	PART
ejpam-3272	133	20	repeated	repeat	VERB
ejpam-3272	133	21	here	here	ADV
ejpam-3272	133	22	.	.	PUNCT
ejpam-3272	134	1	the	the	DET
ejpam-3272	134	2	reader	reader	NOUN
ejpam-3272	134	3	is	be	AUX
ejpam-3272	134	4	referred	refer	VERB
ejpam-3272	134	5	to	to	ADP
ejpam-3272	134	6	[	[	X
ejpam-3272	134	7	4	4	X
ejpam-3272	134	8	]	]	PUNCT
ejpam-3272	134	9	for	for	ADP
ejpam-3272	134	10	a	a	DET
ejpam-3272	134	11	detailed	detailed	ADJ
ejpam-3272	134	12	discussion	discussion	NOUN
ejpam-3272	134	13	on	on	ADP
ejpam-3272	134	14	this	this	DET
ejpam-3272	134	15	subject	subject	NOUN
ejpam-3272	134	16	.	.	PUNCT
ejpam-3272	135	1	3	3	X
ejpam-3272	135	2	.	.	X
ejpam-3272	135	3	separation	separation	NOUN
ejpam-3272	135	4	axioms	axiom	NOUN
ejpam-3272	135	5	in	in	ADP
ejpam-3272	135	6	this	this	DET
ejpam-3272	135	7	section	section	NOUN
ejpam-3272	135	8	,	,	PUNCT
ejpam-3272	135	9	we	we	PRON
ejpam-3272	135	10	define	define	VERB
ejpam-3272	135	11	the	the	DET
ejpam-3272	135	12	separation	separation	NOUN
ejpam-3272	135	13	axioms	axiom	NOUN
ejpam-3272	135	14	on	on	ADP
ejpam-3272	135	15	diframes	diframe	NOUN
ejpam-3272	135	16	.	.	PUNCT
ejpam-3272	136	1	we	we	PRON
ejpam-3272	136	2	also	also	ADV
ejpam-3272	136	3	give	give	VERB
ejpam-3272	136	4	several	several	ADJ
ejpam-3272	136	5	characterizations	characterization	NOUN
ejpam-3272	136	6	of	of	ADP
ejpam-3272	136	7	these	these	DET
ejpam-3272	136	8	axioms	axiom	NOUN
ejpam-3272	136	9	and	and	CCONJ
ejpam-3272	136	10	discuss	discuss	VERB
ejpam-3272	136	11	the	the	DET
ejpam-3272	136	12	relationship	relationship	NOUN
ejpam-3272	136	13	between	between	ADP
ejpam-3272	136	14	them	they	PRON
ejpam-3272	136	15	.	.	PUNCT
ejpam-3272	137	1	e.	e.	PROPN
ejpam-3272	137	2	korkmaz	korkmaz	PROPN
ejpam-3272	137	3	,	,	PUNCT
ejpam-3272	137	4	r.	r.	PROPN
ejpam-3272	137	5	ertürk	ertürk	PROPN
ejpam-3272	137	6	/	/	SYM
ejpam-3272	137	7	eur	eur	PROPN
ejpam-3272	137	8	.	.	PUNCT
ejpam-3272	138	1	j.	j.	PROPN
ejpam-3272	138	2	pure	pure	PROPN
ejpam-3272	138	3	appl	appl	PROPN
ejpam-3272	138	4	.	.	PROPN
ejpam-3272	138	5	math	math	PROPN
ejpam-3272	138	6	,	,	PUNCT
ejpam-3272	138	7	11	11	NUM
ejpam-3272	138	8	(	(	PUNCT
ejpam-3272	138	9	3	3	NUM
ejpam-3272	138	10	)	)	PUNCT
ejpam-3272	138	11	(	(	PUNCT
ejpam-3272	138	12	2018	2018	NUM
ejpam-3272	138	13	)	)	PUNCT
ejpam-3272	138	14	,	,	PUNCT
ejpam-3272	138	15	612	612	NUM
ejpam-3272	138	16	-	-	SYM
ejpam-3272	138	17	627	627	NUM
ejpam-3272	138	18	617	617	NUM
ejpam-3272	138	19	definition	definition	NOUN
ejpam-3272	138	20	1	1	NUM
ejpam-3272	138	21	.	.	PUNCT
ejpam-3272	139	1	a	a	DET
ejpam-3272	139	2	diframe	diframe	NOUN
ejpam-3272	139	3	l	l	NOUN
ejpam-3272	139	4	=	=	SYM
ejpam-3272	139	5	(	(	PUNCT
ejpam-3272	139	6	le	le	X
ejpam-3272	139	7	,	,	PUNCT
ejpam-3272	139	8	lfr	lfr	PROPN
ejpam-3272	139	9	,	,	PUNCT
ejpam-3272	139	10	lcf	lcf	PROPN
ejpam-3272	139	11	)	)	PUNCT
ejpam-3272	139	12	is	be	AUX
ejpam-3272	139	13	said	say	VERB
ejpam-3272	139	14	to	to	PART
ejpam-3272	139	15	be	be	AUX
ejpam-3272	139	16	(	(	PUNCT
ejpam-3272	139	17	i	i	NOUN
ejpam-3272	139	18	)	)	PUNCT
ejpam-3272	139	19	t0	t0	PROPN
ejpam-3272	139	20	if	if	SCONJ
ejpam-3272	139	21	,	,	PUNCT
ejpam-3272	139	22	for	for	ADP
ejpam-3272	139	23	all	all	DET
ejpam-3272	139	24	a	a	DET
ejpam-3272	139	25	∈	∈	PROPN
ejpam-3272	139	26	le	le	X
ejpam-3272	139	27	,	,	PUNCT
ejpam-3272	139	28	there	there	PRON
ejpam-3272	139	29	exists	exist	VERB
ejpam-3272	139	30	cji	cji	NOUN
ejpam-3272	139	31	∈	∈	PROPN
ejpam-3272	139	32	lfr	lfr	PROPN
ejpam-3272	139	33	∪	∪	PROPN
ejpam-3272	139	34	lcf	lcf	PROPN
ejpam-3272	139	35	,	,	PUNCT
ejpam-3272	140	1	i	i	PROPN
ejpam-3272	140	2	∈	∈	PROPN
ejpam-3272	141	1	i	i	PRON
ejpam-3272	141	2	,	,	PUNCT
ejpam-3272	141	3	j	j	PROPN
ejpam-3272	141	4	∈	∈	PROPN
ejpam-3272	141	5	j	j	PROPN
ejpam-3272	141	6	such	such	ADJ
ejpam-3272	141	7	that	that	SCONJ
ejpam-3272	141	8	a	a	DET
ejpam-3272	141	9	=	=	NUM
ejpam-3272	141	10	∨	∨	NOUN
ejpam-3272	141	11	j∈j	j∈j	NOUN
ejpam-3272	141	12	∧	∧	PROPN
ejpam-3272	141	13	i∈i	i∈i	NOUN
ejpam-3272	141	14	c	c	PROPN
ejpam-3272	141	15	j	j	PROPN
ejpam-3272	141	16	i	i	PRON
ejpam-3272	141	17	.	.	PUNCT
ejpam-3272	142	1	(	(	PUNCT
ejpam-3272	142	2	ii	ii	NOUN
ejpam-3272	142	3	)	)	PUNCT
ejpam-3272	142	4	co	co	NOUN
ejpam-3272	142	5	-	-	NOUN
ejpam-3272	142	6	t0	t0	PRON
ejpam-3272	142	7	if	if	SCONJ
ejpam-3272	142	8	,	,	PUNCT
ejpam-3272	142	9	for	for	ADP
ejpam-3272	142	10	all	all	DET
ejpam-3272	142	11	a	a	DET
ejpam-3272	142	12	∈	∈	PROPN
ejpam-3272	142	13	le	le	X
ejpam-3272	142	14	,	,	PUNCT
ejpam-3272	142	15	there	there	PRON
ejpam-3272	142	16	exists	exist	VERB
ejpam-3272	142	17	cji	cji	NOUN
ejpam-3272	142	18	∈	∈	PROPN
ejpam-3272	142	19	lfr	lfr	PROPN
ejpam-3272	142	20	∪	∪	PROPN
ejpam-3272	142	21	lcf	lcf	PROPN
ejpam-3272	142	22	,	,	PUNCT
ejpam-3272	142	23	i	i	PROPN
ejpam-3272	142	24	∈	∈	PROPN
ejpam-3272	143	1	i	i	PRON
ejpam-3272	143	2	,	,	PUNCT
ejpam-3272	143	3	j	j	PROPN
ejpam-3272	143	4	∈	∈	PROPN
ejpam-3272	143	5	j	j	PROPN
ejpam-3272	144	1	such	such	ADJ
ejpam-3272	144	2	that	that	SCONJ
ejpam-3272	144	3	a	a	DET
ejpam-3272	144	4	=	=	ADJ
ejpam-3272	144	5	∧	∧	PROPN
ejpam-3272	144	6	j∈j	j∈j	NOUN
ejpam-3272	144	7	∨	∨	NUM
ejpam-3272	144	8	i∈i	i∈i	PROPN
ejpam-3272	145	1	c	c	PROPN
ejpam-3272	145	2	j	j	PROPN
ejpam-3272	146	1	i	i	INTJ
ejpam-3272	146	2	.	.	PUNCT
ejpam-3272	147	1	note	note	VERB
ejpam-3272	147	2	that	that	SCONJ
ejpam-3272	147	3	the	the	DET
ejpam-3272	147	4	axiom	axiom	NOUN
ejpam-3272	147	5	t0	t0	PROPN
ejpam-3272	147	6	is	be	AUX
ejpam-3272	147	7	not	not	PART
ejpam-3272	147	8	self	self	NOUN
ejpam-3272	147	9	-	-	PUNCT
ejpam-3272	147	10	dual	dual	ADJ
ejpam-3272	147	11	,	,	PUNCT
ejpam-3272	147	12	and	and	CCONJ
ejpam-3272	147	13	that	that	SCONJ
ejpam-3272	147	14	t0	t0	PROPN
ejpam-3272	147	15	and	and	CCONJ
ejpam-3272	147	16	co	co	NOUN
ejpam-3272	147	17	-	-	NOUN
ejpam-3272	147	18	t0	t0	NOUN
ejpam-3272	147	19	are	be	AUX
ejpam-3272	147	20	equivalent	equivalent	ADJ
ejpam-3272	147	21	if	if	SCONJ
ejpam-3272	147	22	le	le	X
ejpam-3272	147	23	is	be	AUX
ejpam-3272	147	24	completely	completely	ADV
ejpam-3272	147	25	distributive	distributive	ADJ
ejpam-3272	147	26	.	.	PUNCT
ejpam-3272	148	1	remark	remark	NOUN
ejpam-3272	148	2	1	1	NUM
ejpam-3272	148	3	.	.	PUNCT
ejpam-3272	149	1	(	(	PUNCT
ejpam-3272	149	2	i	i	NOUN
ejpam-3272	149	3	)	)	PUNCT
ejpam-3272	149	4	we	we	PRON
ejpam-3272	149	5	say	say	VERB
ejpam-3272	149	6	u	u	PRON
ejpam-3272	149	7	⊆	⊆	NUM
ejpam-3272	149	8	l	l	NOUN
ejpam-3272	149	9	generates	generate	VERB
ejpam-3272	149	10	v	v	ADP
ejpam-3272	149	11	⊆	⊆	NUM
ejpam-3272	149	12	l	l	NOUN
ejpam-3272	149	13	if	if	SCONJ
ejpam-3272	149	14	v	v	NOUN
ejpam-3272	149	15	is	be	AUX
ejpam-3272	149	16	the	the	DET
ejpam-3272	149	17	smallest	small	ADJ
ejpam-3272	149	18	subset	subset	NOUN
ejpam-3272	149	19	of	of	ADP
ejpam-3272	149	20	l	l	NOUN
ejpam-3272	149	21	containing	contain	VERB
ejpam-3272	149	22	u	u	NOUN
ejpam-3272	149	23	and	and	CCONJ
ejpam-3272	149	24	closed	close	VERB
ejpam-3272	149	25	under	under	ADP
ejpam-3272	149	26	arbitrary	arbitrary	ADJ
ejpam-3272	149	27	meet	meet	NOUN
ejpam-3272	149	28	and	and	CCONJ
ejpam-3272	149	29	join	join	VERB
ejpam-3272	149	30	.	.	PUNCT
ejpam-3272	150	1	(	(	PUNCT
ejpam-3272	150	2	ii	ii	NOUN
ejpam-3272	150	3	)	)	PUNCT
ejpam-3272	150	4	in	in	ADP
ejpam-3272	150	5	a	a	DET
ejpam-3272	150	6	diframe	diframe	NOUN
ejpam-3272	150	7	l	l	NOUN
ejpam-3272	150	8	=	=	SYM
ejpam-3272	150	9	(	(	PUNCT
ejpam-3272	150	10	le	le	X
ejpam-3272	150	11	,	,	PUNCT
ejpam-3272	150	12	lfr	lfr	PROPN
ejpam-3272	150	13	,	,	PUNCT
ejpam-3272	150	14	lcf	lcf	PROPN
ejpam-3272	150	15	)	)	PUNCT
ejpam-3272	150	16	,	,	PUNCT
ejpam-3272	150	17	le	le	X
ejpam-3272	150	18	need	need	VERB
ejpam-3272	150	19	not	not	PART
ejpam-3272	150	20	to	to	PART
ejpam-3272	150	21	be	be	AUX
ejpam-3272	150	22	generated	generate	VERB
ejpam-3272	150	23	by	by	ADP
ejpam-3272	150	24	lfr	lfr	PROPN
ejpam-3272	150	25	∪	∪	X
ejpam-3272	150	26	lcf	lcf	PROPN
ejpam-3272	150	27	.	.	PUNCT
ejpam-3272	151	1	if	if	SCONJ
ejpam-3272	151	2	le	le	X
ejpam-3272	151	3	=	=	SYM
ejpam-3272	151	4	p(x	p(x	PROPN
ejpam-3272	151	5	)	)	PUNCT
ejpam-3272	151	6	,	,	PUNCT
ejpam-3272	151	7	lfr	lfr	PROPN
ejpam-3272	151	8	=	=	SYM
ejpam-3272	151	9	lcf	lcf	PROPN
ejpam-3272	151	10	=	=	SYM
ejpam-3272	151	11	{	{	PUNCT
ejpam-3272	151	12	∅	∅	NOUN
ejpam-3272	151	13	,	,	PUNCT
ejpam-3272	151	14	x	x	NOUN
ejpam-3272	151	15	}	}	PUNCT
ejpam-3272	151	16	,	,	PUNCT
ejpam-3272	151	17	le	le	X
ejpam-3272	151	18	is	be	AUX
ejpam-3272	151	19	not	not	PART
ejpam-3272	151	20	generated	generate	VERB
ejpam-3272	151	21	by	by	ADP
ejpam-3272	151	22	lfr	lfr	PROPN
ejpam-3272	151	23	∪	∪	PROPN
ejpam-3272	151	24	lcf	lcf	PROPN
ejpam-3272	151	25	.	.	PUNCT
ejpam-3272	152	1	however	however	ADV
ejpam-3272	152	2	,	,	PUNCT
ejpam-3272	152	3	this	this	DET
ejpam-3272	152	4	property	property	NOUN
ejpam-3272	152	5	holds	hold	VERB
ejpam-3272	152	6	for	for	ADP
ejpam-3272	152	7	t0	t0	PROPN
ejpam-3272	152	8	or	or	CCONJ
ejpam-3272	152	9	co	co	NOUN
ejpam-3272	152	10	-	-	NOUN
ejpam-3272	152	11	t0	t0	NOUN
ejpam-3272	152	12	diframes	diframe	NOUN
ejpam-3272	152	13	.	.	PUNCT
ejpam-3272	153	1	indeed	indeed	ADV
ejpam-3272	153	2	,	,	PUNCT
ejpam-3272	153	3	if	if	SCONJ
ejpam-3272	153	4	l	l	NOUN
ejpam-3272	153	5	is	be	AUX
ejpam-3272	153	6	t0	t0	NOUN
ejpam-3272	153	7	,	,	PUNCT
ejpam-3272	153	8	for	for	ADP
ejpam-3272	153	9	all	all	DET
ejpam-3272	153	10	a	a	DET
ejpam-3272	153	11	∈	∈	PROPN
ejpam-3272	153	12	le	le	X
ejpam-3272	153	13	,	,	PUNCT
ejpam-3272	153	14	a	a	DET
ejpam-3272	153	15	=	=	ADJ
ejpam-3272	153	16	∨	∨	NUM
ejpam-3272	153	17	j∈j	j∈j	NOUN
ejpam-3272	153	18	∧	∧	PROPN
ejpam-3272	153	19	i∈i	i∈i	NOUN
ejpam-3272	154	1	c	c	PROPN
ejpam-3272	154	2	j	j	PROPN
ejpam-3272	155	1	i	i	PRON
ejpam-3272	155	2	where	where	SCONJ
ejpam-3272	155	3	cji	cji	NOUN
ejpam-3272	155	4	∈	∈	PROPN
ejpam-3272	155	5	lfr	lfr	PROPN
ejpam-3272	155	6	∪	∪	PROPN
ejpam-3272	155	7	lcf	lcf	PROPN
ejpam-3272	155	8	.	.	PUNCT
ejpam-3272	156	1	this	this	PRON
ejpam-3272	156	2	means	mean	VERB
ejpam-3272	156	3	that	that	SCONJ
ejpam-3272	156	4	a	a	PRON
ejpam-3272	156	5	is	be	AUX
ejpam-3272	156	6	an	an	DET
ejpam-3272	156	7	element	element	NOUN
ejpam-3272	156	8	of	of	ADP
ejpam-3272	156	9	the	the	DET
ejpam-3272	156	10	set	set	NOUN
ejpam-3272	156	11	generated	generate	VERB
ejpam-3272	156	12	by	by	ADP
ejpam-3272	156	13	lfr	lfr	PROPN
ejpam-3272	156	14	∪lcf	∪lcf	PROPN
ejpam-3272	156	15	.	.	PUNCT
ejpam-3272	157	1	the	the	DET
ejpam-3272	157	2	other	other	ADJ
ejpam-3272	157	3	inclusion	inclusion	NOUN
ejpam-3272	157	4	is	be	AUX
ejpam-3272	157	5	an	an	DET
ejpam-3272	157	6	immediate	immediate	ADJ
ejpam-3272	157	7	consequence	consequence	NOUN
ejpam-3272	157	8	of	of	ADP
ejpam-3272	157	9	the	the	DET
ejpam-3272	157	10	fact	fact	NOUN
ejpam-3272	157	11	that	that	SCONJ
ejpam-3272	157	12	le	le	PROPN
ejpam-3272	157	13	is	be	AUX
ejpam-3272	157	14	closed	close	VERB
ejpam-3272	157	15	under	under	ADP
ejpam-3272	157	16	arbitrary	arbitrary	ADJ
ejpam-3272	157	17	meets	meet	NOUN
ejpam-3272	157	18	and	and	CCONJ
ejpam-3272	157	19	joins	join	VERB
ejpam-3272	157	20	.	.	PUNCT
ejpam-3272	158	1	(	(	PUNCT
ejpam-3272	158	2	iii	iii	X
ejpam-3272	158	3	)	)	PUNCT
ejpam-3272	158	4	if	if	SCONJ
ejpam-3272	158	5	(	(	PUNCT
ejpam-3272	158	6	s	s	X
ejpam-3272	158	7	,	,	PUNCT
ejpam-3272	158	8	s	s	PROPN
ejpam-3272	158	9	,	,	PUNCT
ejpam-3272	158	10	τ	τ	PROPN
ejpam-3272	158	11	,	,	PUNCT
ejpam-3272	158	12	κ	κ	NOUN
ejpam-3272	158	13	)	)	PUNCT
ejpam-3272	158	14	is	be	AUX
ejpam-3272	158	15	t0	t0	NOUN
ejpam-3272	158	16	as	as	ADP
ejpam-3272	158	17	a	a	DET
ejpam-3272	158	18	diframe	diframe	NOUN
ejpam-3272	158	19	,	,	PUNCT
ejpam-3272	158	20	it	it	PRON
ejpam-3272	158	21	is	be	AUX
ejpam-3272	158	22	not	not	PART
ejpam-3272	158	23	necessarily	necessarily	ADV
ejpam-3272	158	24	t0	t0	NOUN
ejpam-3272	158	25	as	as	ADP
ejpam-3272	158	26	a	a	DET
ejpam-3272	158	27	ditopological	ditopological	ADJ
ejpam-3272	158	28	space	space	NOUN
ejpam-3272	158	29	.	.	PUNCT
ejpam-3272	159	1	definition	definition	NOUN
ejpam-3272	159	2	2	2	NUM
ejpam-3272	159	3	.	.	PUNCT
ejpam-3272	159	4	a	a	DET
ejpam-3272	159	5	diframe	diframe	NOUN
ejpam-3272	159	6	l	l	NOUN
ejpam-3272	159	7	=	=	SYM
ejpam-3272	159	8	(	(	PUNCT
ejpam-3272	159	9	le	le	X
ejpam-3272	159	10	,	,	PUNCT
ejpam-3272	159	11	lfr	lfr	PROPN
ejpam-3272	159	12	,	,	PUNCT
ejpam-3272	159	13	lcf	lcf	PROPN
ejpam-3272	159	14	)	)	PUNCT
ejpam-3272	159	15	is	be	AUX
ejpam-3272	159	16	said	say	VERB
ejpam-3272	159	17	to	to	PART
ejpam-3272	159	18	be	be	AUX
ejpam-3272	159	19	(	(	PUNCT
ejpam-3272	159	20	i	i	NOUN
ejpam-3272	159	21	)	)	PUNCT
ejpam-3272	159	22	r0	r0	NOUN
ejpam-3272	159	23	if	if	SCONJ
ejpam-3272	159	24	every	every	DET
ejpam-3272	159	25	element	element	NOUN
ejpam-3272	159	26	of	of	ADP
ejpam-3272	159	27	lfr	lfr	PROPN
ejpam-3272	159	28	can	can	AUX
ejpam-3272	159	29	be	be	AUX
ejpam-3272	159	30	written	write	VERB
ejpam-3272	159	31	as	as	ADP
ejpam-3272	159	32	a	a	DET
ejpam-3272	159	33	join	join	NOUN
ejpam-3272	159	34	of	of	ADP
ejpam-3272	159	35	elements	element	NOUN
ejpam-3272	159	36	from	from	ADP
ejpam-3272	159	37	lcf	lcf	PROPN
ejpam-3272	159	38	.	.	PUNCT
ejpam-3272	160	1	(	(	PUNCT
ejpam-3272	160	2	ii	ii	NOUN
ejpam-3272	160	3	)	)	PUNCT
ejpam-3272	160	4	co	co	NOUN
ejpam-3272	160	5	-	-	NOUN
ejpam-3272	160	6	r0	r0	NOUN
ejpam-3272	160	7	if	if	SCONJ
ejpam-3272	160	8	every	every	DET
ejpam-3272	160	9	element	element	NOUN
ejpam-3272	160	10	of	of	ADP
ejpam-3272	160	11	lcf	lcf	PROPN
ejpam-3272	160	12	can	can	AUX
ejpam-3272	160	13	be	be	AUX
ejpam-3272	160	14	written	write	VERB
ejpam-3272	160	15	as	as	ADP
ejpam-3272	160	16	a	a	DET
ejpam-3272	160	17	meet	meet	NOUN
ejpam-3272	160	18	of	of	ADP
ejpam-3272	160	19	elements	element	NOUN
ejpam-3272	160	20	from	from	ADP
ejpam-3272	160	21	lfr	lfr	PROPN
ejpam-3272	160	22	.	.	PUNCT
ejpam-3272	161	1	(	(	PUNCT
ejpam-3272	161	2	iii	iii	X
ejpam-3272	161	3	)	)	PUNCT
ejpam-3272	161	4	t1	t1	NOUN
ejpam-3272	162	1	if	if	SCONJ
ejpam-3272	162	2	t0	t0	PROPN
ejpam-3272	162	3	and	and	CCONJ
ejpam-3272	162	4	r0	r0	NOUN
ejpam-3272	162	5	.	.	PUNCT
ejpam-3272	163	1	(	(	PUNCT
ejpam-3272	163	2	iv	iv	X
ejpam-3272	163	3	)	)	PUNCT
ejpam-3272	163	4	co	co	NOUN
ejpam-3272	163	5	-	-	NOUN
ejpam-3272	163	6	t1	t1	NOUN
ejpam-3272	163	7	if	if	SCONJ
ejpam-3272	163	8	co	co	NOUN
ejpam-3272	163	9	-	-	NOUN
ejpam-3272	163	10	t0	t0	NOUN
ejpam-3272	163	11	and	and	CCONJ
ejpam-3272	163	12	co	co	NOUN
ejpam-3272	163	13	-	-	NOUN
ejpam-3272	163	14	r0	r0	NOUN
ejpam-3272	163	15	.	.	PUNCT
ejpam-3272	164	1	for	for	ADP
ejpam-3272	164	2	each	each	DET
ejpam-3272	164	3	property	property	NOUN
ejpam-3272	164	4	p	p	NOUN
ejpam-3272	164	5	,	,	PUNCT
ejpam-3272	164	6	the	the	DET
ejpam-3272	164	7	diframe	diframe	NOUN
ejpam-3272	164	8	l	l	NOUN
ejpam-3272	164	9	=	=	SYM
ejpam-3272	164	10	(	(	PUNCT
ejpam-3272	164	11	le	le	X
ejpam-3272	164	12	,	,	PUNCT
ejpam-3272	164	13	lfr	lfr	PROPN
ejpam-3272	164	14	,	,	PUNCT
ejpam-3272	164	15	lcf	lcf	PROPN
ejpam-3272	164	16	)	)	PUNCT
ejpam-3272	164	17	is	be	AUX
ejpam-3272	164	18	said	say	VERB
ejpam-3272	164	19	to	to	PART
ejpam-3272	164	20	be	be	AUX
ejpam-3272	164	21	bi	bi	NOUN
ejpam-3272	164	22	-	-	NOUN
ejpam-3272	164	23	p	p	NOUN
ejpam-3272	164	24	if	if	SCONJ
ejpam-3272	164	25	it	it	PRON
ejpam-3272	164	26	is	be	AUX
ejpam-3272	164	27	p	p	NOUN
ejpam-3272	164	28	and	and	CCONJ
ejpam-3272	164	29	co	co	ADJ
ejpam-3272	164	30	-	-	PROPN
ejpam-3272	164	31	p.	p.	NOUN
ejpam-3272	164	32	note	note	NOUN
ejpam-3272	164	33	that	that	SCONJ
ejpam-3272	164	34	,	,	PUNCT
ejpam-3272	164	35	kopperman	kopperman	NOUN
ejpam-3272	164	36	was	be	AUX
ejpam-3272	164	37	studied	study	VERB
ejpam-3272	164	38	r0	r0	NOUN
ejpam-3272	164	39	in	in	ADP
ejpam-3272	164	40	[	[	X
ejpam-3272	164	41	8	8	NUM
ejpam-3272	164	42	]	]	PUNCT
ejpam-3272	164	43	,	,	PUNCT
ejpam-3272	164	44	under	under	ADP
ejpam-3272	164	45	the	the	DET
ejpam-3272	164	46	name	name	NOUN
ejpam-3272	164	47	of	of	ADP
ejpam-3272	164	48	“	"	PUNCT
ejpam-3272	164	49	weak	weak	ADJ
ejpam-3272	164	50	symmetry	symmetry	NOUN
ejpam-3272	164	51	”	"	PUNCT
ejpam-3272	164	52	.	.	PUNCT
ejpam-3272	165	1	example	example	NOUN
ejpam-3272	166	1	2	2	NUM
ejpam-3272	166	2	.	.	X
ejpam-3272	166	3	consider	consider	VERB
ejpam-3272	166	4	the	the	DET
ejpam-3272	166	5	diframe	diframe	NOUN
ejpam-3272	166	6	l	l	NOUN
ejpam-3272	166	7	=	=	SYM
ejpam-3272	166	8	(	(	PUNCT
ejpam-3272	166	9	le	le	X
ejpam-3272	166	10	,	,	PUNCT
ejpam-3272	166	11	lfr	lfr	PROPN
ejpam-3272	166	12	,	,	PUNCT
ejpam-3272	166	13	lcf	lcf	PROPN
ejpam-3272	166	14	)	)	PUNCT
ejpam-3272	166	15	of	of	ADP
ejpam-3272	166	16	example	example	NOUN
ejpam-3272	166	17	1	1	NUM
ejpam-3272	166	18	(	(	PUNCT
ejpam-3272	166	19	ii	ii	NOUN
ejpam-3272	166	20	)	)	PUNCT
ejpam-3272	166	21	.	.	PUNCT
ejpam-3272	167	1	l	l	NOUN
ejpam-3272	167	2	is	be	AUX
ejpam-3272	167	3	r0	r0	NOUN
ejpam-3272	167	4	since	since	SCONJ
ejpam-3272	167	5	(	(	PUNCT
ejpam-3272	167	6	−∞	−∞	NOUN
ejpam-3272	167	7	,	,	PUNCT
ejpam-3272	167	8	a	a	PRON
ejpam-3272	167	9	)	)	PUNCT
ejpam-3272	167	10	=	=	SYM
ejpam-3272	167	11	∨	∨	PROPN
ejpam-3272	167	12	n∈n(a	n∈n(a	PROPN
ejpam-3272	167	13	−	−	PROPN
ejpam-3272	167	14	n	n	CCONJ
ejpam-3272	167	15	,	,	PUNCT
ejpam-3272	167	16	a	a	X
ejpam-3272	167	17	)	)	PUNCT
ejpam-3272	167	18	for	for	ADP
ejpam-3272	167	19	all	all	DET
ejpam-3272	167	20	a	a	DET
ejpam-3272	167	21	∈	∈	PROPN
ejpam-3272	167	22	r.	r.	NOUN
ejpam-3272	167	23	however	however	ADV
ejpam-3272	167	24	,	,	PUNCT
ejpam-3272	167	25	l	l	NOUN
ejpam-3272	167	26	is	be	AUX
ejpam-3272	167	27	not	not	PART
ejpam-3272	167	28	co	co	NOUN
ejpam-3272	167	29	-	-	NOUN
ejpam-3272	167	30	r0	r0	ADJ
ejpam-3272	167	31	because	because	SCONJ
ejpam-3272	167	32	the	the	DET
ejpam-3272	167	33	bounded	bounded	ADJ
ejpam-3272	167	34	intervals	interval	NOUN
ejpam-3272	167	35	(	(	PUNCT
ejpam-3272	167	36	a	a	DET
ejpam-3272	167	37	,	,	PUNCT
ejpam-3272	167	38	b	b	NOUN
ejpam-3272	167	39	)	)	PUNCT
ejpam-3272	167	40	∈	∈	PROPN
ejpam-3272	167	41	lcf	lcf	NOUN
ejpam-3272	167	42	can	can	AUX
ejpam-3272	167	43	not	not	PART
ejpam-3272	167	44	be	be	AUX
ejpam-3272	167	45	expressed	express	VERB
ejpam-3272	167	46	as	as	ADP
ejpam-3272	167	47	a	a	DET
ejpam-3272	167	48	meet	meet	NOUN
ejpam-3272	167	49	of	of	ADP
ejpam-3272	167	50	elements	element	NOUN
ejpam-3272	167	51	from	from	ADP
ejpam-3272	167	52	lfr	lfr	PROPN
ejpam-3272	167	53	.	.	PUNCT
ejpam-3272	168	1	here	here	ADV
ejpam-3272	168	2	are	be	AUX
ejpam-3272	168	3	some	some	DET
ejpam-3272	168	4	statements	statement	NOUN
ejpam-3272	168	5	equivalent	equivalent	ADJ
ejpam-3272	168	6	to	to	PART
ejpam-3272	168	7	r0	r0	VERB
ejpam-3272	168	8	and	and	CCONJ
ejpam-3272	168	9	co	co	NOUN
ejpam-3272	168	10	-	-	NOUN
ejpam-3272	168	11	r0	r0	NOUN
ejpam-3272	168	12	.	.	PUNCT
ejpam-3272	169	1	proposition	proposition	NOUN
ejpam-3272	169	2	3	3	X
ejpam-3272	169	3	.	.	PUNCT
ejpam-3272	170	1	let	let	AUX
ejpam-3272	170	2	l	l	NOUN
ejpam-3272	170	3	=	=	SYM
ejpam-3272	170	4	(	(	PUNCT
ejpam-3272	170	5	le	le	X
ejpam-3272	170	6	,	,	PUNCT
ejpam-3272	170	7	lfr	lfr	PROPN
ejpam-3272	170	8	,	,	PUNCT
ejpam-3272	170	9	lcf	lcf	PROPN
ejpam-3272	170	10	)	)	PUNCT
ejpam-3272	170	11	be	be	AUX
ejpam-3272	170	12	a	a	DET
ejpam-3272	170	13	diframe	diframe	NOUN
ejpam-3272	170	14	.	.	PUNCT
ejpam-3272	171	1	(	(	PUNCT
ejpam-3272	171	2	i	i	NOUN
ejpam-3272	171	3	)	)	PUNCT
ejpam-3272	171	4	the	the	DET
ejpam-3272	171	5	following	follow	VERB
ejpam-3272	171	6	are	be	AUX
ejpam-3272	171	7	equivalent	equivalent	ADJ
ejpam-3272	171	8	:	:	PUNCT
ejpam-3272	171	9	(	(	PUNCT
ejpam-3272	171	10	a	a	X
ejpam-3272	171	11	)	)	PUNCT
ejpam-3272	171	12	l	l	NOUN
ejpam-3272	171	13	is	be	AUX
ejpam-3272	171	14	r0	r0	NOUN
ejpam-3272	171	15	.	.	PUNCT
ejpam-3272	172	1	e.	e.	PROPN
ejpam-3272	172	2	korkmaz	korkmaz	PROPN
ejpam-3272	172	3	,	,	PUNCT
ejpam-3272	172	4	r.	r.	PROPN
ejpam-3272	172	5	ertürk	ertürk	PROPN
ejpam-3272	172	6	/	/	SYM
ejpam-3272	172	7	eur	eur	PROPN
ejpam-3272	172	8	.	.	PUNCT
ejpam-3272	173	1	j.	j.	PROPN
ejpam-3272	173	2	pure	pure	PROPN
ejpam-3272	173	3	appl	appl	PROPN
ejpam-3272	173	4	.	.	PROPN
ejpam-3272	173	5	math	math	PROPN
ejpam-3272	173	6	,	,	PUNCT
ejpam-3272	173	7	11	11	NUM
ejpam-3272	173	8	(	(	PUNCT
ejpam-3272	173	9	3	3	NUM
ejpam-3272	173	10	)	)	PUNCT
ejpam-3272	173	11	(	(	PUNCT
ejpam-3272	173	12	2018	2018	NUM
ejpam-3272	173	13	)	)	PUNCT
ejpam-3272	173	14	,	,	PUNCT
ejpam-3272	173	15	612	612	NUM
ejpam-3272	173	16	-	-	SYM
ejpam-3272	173	17	627	627	NUM
ejpam-3272	173	18	618	618	NUM
ejpam-3272	173	19	(	(	PUNCT
ejpam-3272	173	20	b	b	NOUN
ejpam-3272	173	21	)	)	PUNCT
ejpam-3272	173	22	every	every	DET
ejpam-3272	173	23	open	open	ADJ
ejpam-3272	173	24	sublocale	sublocale	NOUN
ejpam-3272	173	25	associated	associate	VERB
ejpam-3272	173	26	with	with	ADP
ejpam-3272	173	27	the	the	DET
ejpam-3272	173	28	elements	element	NOUN
ejpam-3272	173	29	of	of	ADP
ejpam-3272	173	30	lfr	lfr	PROPN
ejpam-3272	173	31	can	can	AUX
ejpam-3272	173	32	be	be	AUX
ejpam-3272	173	33	written	write	VERB
ejpam-3272	173	34	as	as	ADP
ejpam-3272	173	35	a	a	DET
ejpam-3272	173	36	join	join	NOUN
ejpam-3272	173	37	of	of	ADP
ejpam-3272	173	38	the	the	DET
ejpam-3272	173	39	open	open	ADJ
ejpam-3272	173	40	sublocales	sublocale	NOUN
ejpam-3272	173	41	associated	associate	VERB
ejpam-3272	173	42	with	with	ADP
ejpam-3272	173	43	the	the	DET
ejpam-3272	173	44	elements	element	NOUN
ejpam-3272	173	45	of	of	ADP
ejpam-3272	173	46	lcf	lcf	PROPN
ejpam-3272	173	47	,	,	PUNCT
ejpam-3272	173	48	that	that	ADV
ejpam-3272	173	49	is	is	ADV
ejpam-3272	173	50	,	,	PUNCT
ejpam-3272	173	51	o(a	o(a	NOUN
ejpam-3272	173	52	)	)	PUNCT
ejpam-3272	173	53	=	=	SYM
ejpam-3272	173	54	∨	∨	X
ejpam-3272	173	55	{	{	PUNCT
ejpam-3272	173	56	o(k	o(k	PROPN
ejpam-3272	173	57	)	)	PUNCT
ejpam-3272	173	58	:	:	PUNCT
ejpam-3272	174	1	k	k	PROPN
ejpam-3272	174	2	∈	∈	PROPN
ejpam-3272	174	3	lcf	lcf	PROPN
ejpam-3272	174	4	and	and	CCONJ
ejpam-3272	174	5	k	k	PROPN
ejpam-3272	174	6	≤	≤	PROPN
ejpam-3272	174	7	a	a	PRON
ejpam-3272	174	8	}	}	PUNCT
ejpam-3272	174	9	for	for	ADP
ejpam-3272	174	10	all	all	DET
ejpam-3272	174	11	a	a	DET
ejpam-3272	174	12	∈	∈	PROPN
ejpam-3272	174	13	lfr	lfr	NOUN
ejpam-3272	174	14	.	.	PUNCT
ejpam-3272	175	1	(	(	PUNCT
ejpam-3272	175	2	c	c	X
ejpam-3272	175	3	)	)	PUNCT
ejpam-3272	175	4	every	every	DET
ejpam-3272	175	5	closed	closed	ADJ
ejpam-3272	175	6	sublocale	sublocale	NOUN
ejpam-3272	175	7	associated	associate	VERB
ejpam-3272	175	8	with	with	ADP
ejpam-3272	175	9	the	the	DET
ejpam-3272	175	10	elements	element	NOUN
ejpam-3272	175	11	of	of	ADP
ejpam-3272	175	12	lfr	lfr	PROPN
ejpam-3272	175	13	can	can	AUX
ejpam-3272	175	14	be	be	AUX
ejpam-3272	175	15	written	write	VERB
ejpam-3272	175	16	as	as	ADP
ejpam-3272	175	17	an	an	DET
ejpam-3272	175	18	intersection	intersection	NOUN
ejpam-3272	175	19	of	of	ADP
ejpam-3272	175	20	the	the	DET
ejpam-3272	175	21	closed	closed	ADJ
ejpam-3272	175	22	sublocales	sublocale	NOUN
ejpam-3272	175	23	associated	associate	VERB
ejpam-3272	175	24	with	with	ADP
ejpam-3272	175	25	the	the	DET
ejpam-3272	175	26	elements	element	NOUN
ejpam-3272	175	27	of	of	ADP
ejpam-3272	175	28	lcf	lcf	PROPN
ejpam-3272	175	29	,	,	PUNCT
ejpam-3272	175	30	that	that	ADV
ejpam-3272	175	31	is	is	ADV
ejpam-3272	175	32	,	,	PUNCT
ejpam-3272	175	33	c(a	c(a	ADV
ejpam-3272	175	34	)	)	PUNCT
ejpam-3272	175	35	=	=	SYM
ejpam-3272	175	36	⋂	⋂	PROPN
ejpam-3272	175	37	{	{	PUNCT
ejpam-3272	175	38	c(k	c(k	PROPN
ejpam-3272	175	39	)	)	PUNCT
ejpam-3272	175	40	:	:	PUNCT
ejpam-3272	176	1	k	k	PROPN
ejpam-3272	176	2	∈	∈	PROPN
ejpam-3272	176	3	lcf	lcf	PROPN
ejpam-3272	176	4	and	and	CCONJ
ejpam-3272	176	5	k	k	PROPN
ejpam-3272	176	6	≤	≤	PROPN
ejpam-3272	176	7	a	a	PRON
ejpam-3272	176	8	}	}	PUNCT
ejpam-3272	176	9	for	for	ADP
ejpam-3272	176	10	all	all	DET
ejpam-3272	176	11	a	a	DET
ejpam-3272	176	12	∈	∈	PROPN
ejpam-3272	176	13	lfr	lfr	NOUN
ejpam-3272	176	14	.	.	PUNCT
ejpam-3272	177	1	(	(	PUNCT
ejpam-3272	177	2	d	d	X
ejpam-3272	177	3	)	)	PUNCT
ejpam-3272	177	4	∀a	∀a	NOUN
ejpam-3272	177	5	∈	∈	PROPN
ejpam-3272	177	6	lfr	lfr	PROPN
ejpam-3272	177	7	,	,	PUNCT
ejpam-3272	177	8	∀x	∀x	NUM
ejpam-3272	177	9	,	,	PUNCT
ejpam-3272	177	10	y	y	PROPN
ejpam-3272	177	11	∈	∈	PROPN
ejpam-3272	177	12	le	le	PROPN
ejpam-3272	177	13	,	,	PUNCT
ejpam-3272	177	14	a	a	DET
ejpam-3272	177	15	�	�	PROPN
ejpam-3272	177	16	y	y	PROPN
ejpam-3272	177	17	→	→	PUNCT
ejpam-3272	177	18	x⇒	x⇒	PROPN
ejpam-3272	178	1	k	k	PROPN
ejpam-3272	178	2	∈	∈	PROPN
ejpam-3272	178	3	lcf	lcf	PROPN
ejpam-3272	178	4	;	;	PUNCT
ejpam-3272	178	5	k	k	PROPN
ejpam-3272	178	6	≤	≤	PROPN
ejpam-3272	178	7	a	a	X
ejpam-3272	178	8	,	,	PUNCT
ejpam-3272	178	9	y	y	PROPN
ejpam-3272	178	10	�	�	PROPN
ejpam-3272	178	11	k	k	PROPN
ejpam-3272	178	12	→	→	PROPN
ejpam-3272	178	13	x.	x.	PROPN
ejpam-3272	178	14	(	(	PUNCT
ejpam-3272	178	15	ii	ii	PROPN
ejpam-3272	178	16	)	)	PUNCT
ejpam-3272	178	17	the	the	DET
ejpam-3272	178	18	following	follow	VERB
ejpam-3272	178	19	are	be	AUX
ejpam-3272	178	20	equivalent	equivalent	ADJ
ejpam-3272	178	21	:	:	PUNCT
ejpam-3272	178	22	(	(	PUNCT
ejpam-3272	178	23	a	a	X
ejpam-3272	178	24	)	)	PUNCT
ejpam-3272	178	25	l	l	NOUN
ejpam-3272	178	26	is	be	AUX
ejpam-3272	178	27	co	co	ADJ
ejpam-3272	178	28	-	-	NOUN
ejpam-3272	178	29	r0	r0	NOUN
ejpam-3272	178	30	.	.	PUNCT
ejpam-3272	179	1	(	(	PUNCT
ejpam-3272	179	2	b	b	X
ejpam-3272	179	3	)	)	PUNCT
ejpam-3272	179	4	every	every	DET
ejpam-3272	179	5	open	open	ADJ
ejpam-3272	179	6	subcolocale	subcolocale	NOUN
ejpam-3272	179	7	associated	associate	VERB
ejpam-3272	179	8	with	with	ADP
ejpam-3272	179	9	the	the	DET
ejpam-3272	179	10	elements	element	NOUN
ejpam-3272	179	11	of	of	ADP
ejpam-3272	179	12	lcf	lcf	PROPN
ejpam-3272	179	13	can	can	AUX
ejpam-3272	179	14	be	be	AUX
ejpam-3272	179	15	written	write	VERB
ejpam-3272	179	16	as	as	ADP
ejpam-3272	179	17	a	a	DET
ejpam-3272	179	18	join	join	NOUN
ejpam-3272	179	19	of	of	ADP
ejpam-3272	179	20	the	the	DET
ejpam-3272	179	21	open	open	ADJ
ejpam-3272	179	22	subcolocales	subcolocale	NOUN
ejpam-3272	179	23	associated	associate	VERB
ejpam-3272	179	24	with	with	ADP
ejpam-3272	179	25	the	the	DET
ejpam-3272	179	26	elements	element	NOUN
ejpam-3272	179	27	of	of	ADP
ejpam-3272	179	28	lfr	lfr	PROPN
ejpam-3272	179	29	,	,	PUNCT
ejpam-3272	179	30	that	that	ADV
ejpam-3272	179	31	is	is	ADV
ejpam-3272	179	32	,	,	PUNCT
ejpam-3272	179	33	oc(k	oc(k	X
ejpam-3272	179	34	)	)	PUNCT
ejpam-3272	179	35	=	=	SYM
ejpam-3272	179	36	∨	∨	X
ejpam-3272	179	37	{	{	PUNCT
ejpam-3272	179	38	oc(a	oc(a	NUM
ejpam-3272	179	39	)	)	PUNCT
ejpam-3272	179	40	:	:	PUNCT
ejpam-3272	179	41	a	a	DET
ejpam-3272	179	42	∈	∈	PROPN
ejpam-3272	179	43	lfr	lfr	NOUN
ejpam-3272	179	44	and	and	CCONJ
ejpam-3272	179	45	k	k	PROPN
ejpam-3272	179	46	≤	≤	PROPN
ejpam-3272	179	47	a	a	PRON
ejpam-3272	179	48	}	}	PUNCT
ejpam-3272	179	49	for	for	ADP
ejpam-3272	179	50	all	all	DET
ejpam-3272	179	51	k	k	PROPN
ejpam-3272	179	52	∈	∈	PROPN
ejpam-3272	179	53	lcf	lcf	PROPN
ejpam-3272	179	54	.	.	PUNCT
ejpam-3272	180	1	(	(	PUNCT
ejpam-3272	180	2	c	c	X
ejpam-3272	180	3	)	)	PUNCT
ejpam-3272	180	4	every	every	DET
ejpam-3272	180	5	closed	closed	ADJ
ejpam-3272	180	6	subcolocale	subcolocale	NOUN
ejpam-3272	180	7	associated	associate	VERB
ejpam-3272	180	8	with	with	ADP
ejpam-3272	180	9	the	the	DET
ejpam-3272	180	10	elements	element	NOUN
ejpam-3272	180	11	of	of	ADP
ejpam-3272	180	12	lcf	lcf	PROPN
ejpam-3272	180	13	can	can	AUX
ejpam-3272	180	14	be	be	AUX
ejpam-3272	180	15	written	write	VERB
ejpam-3272	180	16	as	as	ADP
ejpam-3272	180	17	an	an	DET
ejpam-3272	180	18	intersection	intersection	NOUN
ejpam-3272	180	19	of	of	ADP
ejpam-3272	180	20	the	the	DET
ejpam-3272	180	21	closed	closed	ADJ
ejpam-3272	180	22	subcolocales	subcolocale	NOUN
ejpam-3272	180	23	associated	associate	VERB
ejpam-3272	180	24	with	with	ADP
ejpam-3272	180	25	the	the	DET
ejpam-3272	180	26	elements	element	NOUN
ejpam-3272	180	27	of	of	ADP
ejpam-3272	180	28	lfr	lfr	PROPN
ejpam-3272	180	29	,	,	PUNCT
ejpam-3272	180	30	that	that	ADV
ejpam-3272	180	31	is	be	AUX
ejpam-3272	180	32	,	,	PUNCT
ejpam-3272	180	33	cc(k	cc(k	NOUN
ejpam-3272	180	34	)	)	PUNCT
ejpam-3272	180	35	=	=	SYM
ejpam-3272	180	36	⋂	⋂	PROPN
ejpam-3272	180	37	{	{	PUNCT
ejpam-3272	180	38	cc(a	cc(a	PROPN
ejpam-3272	180	39	)	)	PUNCT
ejpam-3272	180	40	:	:	PUNCT
ejpam-3272	180	41	a	a	DET
ejpam-3272	180	42	∈	∈	PROPN
ejpam-3272	180	43	lfr	lfr	NOUN
ejpam-3272	180	44	and	and	CCONJ
ejpam-3272	180	45	k	k	PROPN
ejpam-3272	180	46	≤	≤	PROPN
ejpam-3272	180	47	a	a	PRON
ejpam-3272	180	48	}	}	PUNCT
ejpam-3272	180	49	for	for	ADP
ejpam-3272	180	50	all	all	DET
ejpam-3272	180	51	k	k	PROPN
ejpam-3272	180	52	∈	∈	PROPN
ejpam-3272	180	53	lcf	lcf	PROPN
ejpam-3272	180	54	.	.	PUNCT
ejpam-3272	181	1	(	(	PUNCT
ejpam-3272	181	2	d	d	X
ejpam-3272	181	3	)	)	PUNCT
ejpam-3272	181	4	∀k	∀k	NOUN
ejpam-3272	181	5	∈	∈	PROPN
ejpam-3272	181	6	lcf	lcf	PROPN
ejpam-3272	181	7	,	,	PUNCT
ejpam-3272	181	8	∀x	∀x	NUM
ejpam-3272	181	9	,	,	PUNCT
ejpam-3272	181	10	y	y	PROPN
ejpam-3272	181	11	∈	∈	PROPN
ejpam-3272	181	12	le	le	X
ejpam-3272	181	13	,	,	PUNCT
ejpam-3272	181	14	x←	x←	PROPN
ejpam-3272	182	1	y	y	PROPN
ejpam-3272	182	2	�	�	PROPN
ejpam-3272	182	3	k	k	PROPN
ejpam-3272	182	4	⇒	⇒	VERB
ejpam-3272	182	5	a	a	DET
ejpam-3272	182	6	∈	∈	PROPN
ejpam-3272	182	7	lfr	lfr	X
ejpam-3272	182	8	;	;	PUNCT
ejpam-3272	182	9	k	k	PROPN
ejpam-3272	182	10	≤	≤	PROPN
ejpam-3272	182	11	a	a	PRON
ejpam-3272	182	12	,	,	PUNCT
ejpam-3272	182	13	x←	x←	PROPN
ejpam-3272	183	1	a	a	DET
ejpam-3272	183	2	�	�	PROPN
ejpam-3272	183	3	y.	y.	PROPN
ejpam-3272	183	4	proof	proof	NOUN
ejpam-3272	183	5	.	.	PUNCT
ejpam-3272	184	1	(	(	PUNCT
ejpam-3272	184	2	ii	ii	NUM
ejpam-3272	184	3	):	):	PUNCT
ejpam-3272	184	4	(	(	PUNCT
ejpam-3272	184	5	a	a	X
ejpam-3272	184	6	)	)	PUNCT
ejpam-3272	184	7	and	and	CCONJ
ejpam-3272	184	8	(	(	PUNCT
ejpam-3272	184	9	b	b	X
ejpam-3272	184	10	)	)	PUNCT
ejpam-3272	184	11	are	be	AUX
ejpam-3272	184	12	equivalent	equivalent	ADJ
ejpam-3272	184	13	since	since	SCONJ
ejpam-3272	184	14	the	the	DET
ejpam-3272	184	15	equality	equality	NOUN
ejpam-3272	184	16	∨	∨	NUM
ejpam-3272	184	17	i∈i	i∈i	ADJ
ejpam-3272	184	18	oc(a	oc(a	NOUN
ejpam-3272	184	19	)	)	PUNCT
ejpam-3272	184	20	=	=	SYM
ejpam-3272	184	21	oc	oc	PROPN
ejpam-3272	184	22	(	(	PUNCT
ejpam-3272	184	23	∧	∧	PROPN
ejpam-3272	184	24	i∈i	i∈i	ADJ
ejpam-3272	184	25	ai	ai	NOUN
ejpam-3272	184	26	)	)	PUNCT
ejpam-3272	184	27	holds	hold	VERB
ejpam-3272	184	28	.	.	PUNCT
ejpam-3272	185	1	similarly	similarly	ADV
ejpam-3272	185	2	,	,	PUNCT
ejpam-3272	185	3	(	(	PUNCT
ejpam-3272	185	4	a	a	X
ejpam-3272	185	5	)	)	PUNCT
ejpam-3272	185	6	and	and	CCONJ
ejpam-3272	185	7	(	(	PUNCT
ejpam-3272	185	8	c	c	X
ejpam-3272	185	9	)	)	PUNCT
ejpam-3272	185	10	are	be	AUX
ejpam-3272	185	11	equivalent	equivalent	ADJ
ejpam-3272	185	12	by	by	ADP
ejpam-3272	185	13	the	the	DET
ejpam-3272	185	14	property	property	NOUN
ejpam-3272	185	15	⋂	⋂	PROPN
ejpam-3272	185	16	i∈i	i∈i	ADJ
ejpam-3272	185	17	cc(ai	cc(ai	PROPN
ejpam-3272	185	18	)	)	PUNCT
ejpam-3272	186	1	=	=	SYM
ejpam-3272	186	2	cc	cc	PROPN
ejpam-3272	186	3	(	(	PUNCT
ejpam-3272	186	4	∧	∧	PROPN
ejpam-3272	186	5	i∈i	i∈i	ADJ
ejpam-3272	186	6	ai	ai	PROPN
ejpam-3272	186	7	)	)	PUNCT
ejpam-3272	186	8	.	.	PUNCT
ejpam-3272	187	1	for	for	ADP
ejpam-3272	187	2	(	(	PUNCT
ejpam-3272	187	3	b	b	NOUN
ejpam-3272	187	4	)	)	PUNCT
ejpam-3272	187	5	implies	imply	VERB
ejpam-3272	187	6	(	(	PUNCT
ejpam-3272	187	7	d	d	NOUN
ejpam-3272	187	8	)	)	PUNCT
ejpam-3272	187	9	,	,	PUNCT
ejpam-3272	187	10	let	let	VERB
ejpam-3272	187	11	x←	x←	PROPN
ejpam-3272	187	12	y	y	PROPN
ejpam-3272	187	13	�	�	PROPN
ejpam-3272	187	14	k	k	PROPN
ejpam-3272	187	15	for	for	ADP
ejpam-3272	187	16	k	k	PROPN
ejpam-3272	187	17	∈	∈	PROPN
ejpam-3272	187	18	lcf	lcf	PROPN
ejpam-3272	187	19	and	and	CCONJ
ejpam-3272	187	20	x	x	NOUN
ejpam-3272	187	21	,	,	PUNCT
ejpam-3272	187	22	y	y	PROPN
ejpam-3272	187	23	∈	∈	PROPN
ejpam-3272	187	24	le	le	PROPN
ejpam-3272	187	25	.	.	PUNCT
ejpam-3272	188	1	then	then	ADV
ejpam-3272	188	2	,	,	PUNCT
ejpam-3272	188	3	oc(k	oc(k	X
ejpam-3272	188	4	)	)	PUNCT
ejpam-3272	188	5	=	=	SYM
ejpam-3272	188	6	∨	∨	X
ejpam-3272	188	7	{	{	PUNCT
ejpam-3272	188	8	oc(a	oc(a	NUM
ejpam-3272	188	9	)	)	PUNCT
ejpam-3272	188	10	:	:	PUNCT
ejpam-3272	188	11	a	a	DET
ejpam-3272	188	12	∈	∈	PROPN
ejpam-3272	188	13	lfr	lfr	NOUN
ejpam-3272	188	14	and	and	CCONJ
ejpam-3272	188	15	k	k	PROPN
ejpam-3272	188	16	≤	≤	PROPN
ejpam-3272	188	17	a	a	DET
ejpam-3272	188	18	}	}	PUNCT
ejpam-3272	188	19	6⊆	6⊆	PROPN
ejpam-3272	188	20	oc(x←	oc(x←	NUM
ejpam-3272	188	21	y	y	NOUN
ejpam-3272	188	22	)	)	PUNCT
ejpam-3272	188	23	and	and	CCONJ
ejpam-3272	188	24	hence	hence	ADV
ejpam-3272	188	25	there	there	PRON
ejpam-3272	188	26	exists	exist	VERB
ejpam-3272	188	27	an	an	DET
ejpam-3272	188	28	a	a	DET
ejpam-3272	188	29	∈	∈	PROPN
ejpam-3272	188	30	lfr	lfr	NOUN
ejpam-3272	189	1	such	such	ADJ
ejpam-3272	189	2	that	that	SCONJ
ejpam-3272	189	3	k	k	PROPN
ejpam-3272	189	4	≤	≤	PROPN
ejpam-3272	189	5	a	a	PRON
ejpam-3272	189	6	and	and	CCONJ
ejpam-3272	189	7	oc(a	oc(a	NUM
ejpam-3272	189	8	)	)	PUNCT
ejpam-3272	189	9	6⊆	6⊆	PROPN
ejpam-3272	189	10	oc(x←	oc(x←	NUM
ejpam-3272	189	11	y	y	NOUN
ejpam-3272	189	12	)	)	PUNCT
ejpam-3272	189	13	,	,	PUNCT
ejpam-3272	189	14	which	which	PRON
ejpam-3272	189	15	implies	imply	VERB
ejpam-3272	189	16	the	the	DET
ejpam-3272	189	17	existence	existence	NOUN
ejpam-3272	189	18	of	of	ADP
ejpam-3272	189	19	an	an	DET
ejpam-3272	189	20	a	a	DET
ejpam-3272	189	21	∈	∈	PROPN
ejpam-3272	189	22	lfr	lfr	NOUN
ejpam-3272	189	23	such	such	ADJ
ejpam-3272	189	24	that	that	SCONJ
ejpam-3272	189	25	k	k	PROPN
ejpam-3272	189	26	≤	≤	PROPN
ejpam-3272	189	27	a	a	PRON
ejpam-3272	189	28	and	and	CCONJ
ejpam-3272	189	29	x←	x←	X
ejpam-3272	189	30	a	a	DET
ejpam-3272	189	31	�	�	PROPN
ejpam-3272	189	32	y.	y.	PROPN
ejpam-3272	189	33	for	for	ADP
ejpam-3272	189	34	the	the	DET
ejpam-3272	189	35	converse	converse	NOUN
ejpam-3272	189	36	,	,	PUNCT
ejpam-3272	189	37	assume	assume	VERB
ejpam-3272	189	38	contrary	contrary	ADJ
ejpam-3272	189	39	that	that	SCONJ
ejpam-3272	189	40	l	l	NOUN
ejpam-3272	189	41	=	=	SYM
ejpam-3272	189	42	(	(	PUNCT
ejpam-3272	189	43	le	le	X
ejpam-3272	189	44	,	,	PUNCT
ejpam-3272	189	45	lfr	lfr	PROPN
ejpam-3272	189	46	,	,	PUNCT
ejpam-3272	189	47	lcf	lcf	PROPN
ejpam-3272	189	48	)	)	PUNCT
ejpam-3272	189	49	does	do	AUX
ejpam-3272	189	50	not	not	PART
ejpam-3272	189	51	satisfy	satisfy	VERB
ejpam-3272	189	52	(	(	PUNCT
ejpam-3272	189	53	b	b	NOUN
ejpam-3272	189	54	)	)	PUNCT
ejpam-3272	189	55	.	.	PUNCT
ejpam-3272	190	1	then	then	ADV
ejpam-3272	190	2	there	there	PRON
ejpam-3272	190	3	is	be	VERB
ejpam-3272	190	4	a	a	DET
ejpam-3272	190	5	k	k	PROPN
ejpam-3272	190	6	∈	∈	PROPN
ejpam-3272	190	7	lcf	lcf	NOUN
ejpam-3272	190	8	such	such	ADJ
ejpam-3272	190	9	that	that	PRON
ejpam-3272	190	10	oc(k	oc(k	X
ejpam-3272	190	11	)	)	PUNCT
ejpam-3272	190	12	6⊆	6⊆	PROPN
ejpam-3272	190	13	∨	∨	NUM
ejpam-3272	190	14	{	{	PUNCT
ejpam-3272	190	15	oc(a	oc(a	NUM
ejpam-3272	190	16	)	)	PUNCT
ejpam-3272	190	17	:	:	PUNCT
ejpam-3272	190	18	a	a	DET
ejpam-3272	190	19	∈	∈	PROPN
ejpam-3272	190	20	lfr	lfr	NOUN
ejpam-3272	190	21	and	and	CCONJ
ejpam-3272	190	22	k	k	PROPN
ejpam-3272	190	23	≤	≤	PROPN
ejpam-3272	190	24	a	a	PRON
ejpam-3272	190	25	}	}	PUNCT
ejpam-3272	190	26	.	.	PUNCT
ejpam-3272	191	1	thus	thus	ADV
ejpam-3272	191	2	,	,	PUNCT
ejpam-3272	191	3	there	there	PRON
ejpam-3272	191	4	exists	exist	VERB
ejpam-3272	191	5	an	an	DET
ejpam-3272	191	6	x	x	SYM
ejpam-3272	191	7	∈	∈	NOUN
ejpam-3272	191	8	le	le	ADP
ejpam-3272	191	9	such	such	ADJ
ejpam-3272	191	10	that	that	SCONJ
ejpam-3272	191	11	x	x	SYM
ejpam-3272	191	12	∈	∈	NOUN
ejpam-3272	191	13	oc(k	oc(k	NOUN
ejpam-3272	191	14	)	)	PUNCT
ejpam-3272	191	15	and	and	CCONJ
ejpam-3272	191	16	x	x	X
ejpam-3272	191	17	/∈	/∈	PUNCT
ejpam-3272	191	18	oc(a	oc(a	NUM
ejpam-3272	191	19	)	)	PUNCT
ejpam-3272	191	20	for	for	ADP
ejpam-3272	191	21	all	all	DET
ejpam-3272	191	22	a	a	DET
ejpam-3272	191	23	∈	∈	PROPN
ejpam-3272	191	24	lfr	lfr	NOUN
ejpam-3272	191	25	satisfying	satisfy	VERB
ejpam-3272	191	26	k	k	PROPN
ejpam-3272	191	27	≤	≤	NUM
ejpam-3272	191	28	a.	a.	NOUN
ejpam-3272	191	29	now	now	ADV
ejpam-3272	191	30	we	we	PRON
ejpam-3272	191	31	obtain	obtain	VERB
ejpam-3272	191	32	x	x	ADJ
ejpam-3272	191	33	←	←	PROPN
ejpam-3272	191	34	k	k	NOUN
ejpam-3272	191	35	=	=	PUNCT
ejpam-3272	191	36	x	x	PROPN
ejpam-3272	192	1	6=	6=	NUM
ejpam-3272	192	2	x	x	X
ejpam-3272	192	3	←	←	PROPN
ejpam-3272	192	4	a	a	X
ejpam-3272	192	5	,	,	PUNCT
ejpam-3272	192	6	and	and	CCONJ
ejpam-3272	192	7	hence	hence	ADV
ejpam-3272	192	8	x	x	PROPN
ejpam-3272	192	9	←	←	PROPN
ejpam-3272	192	10	k	k	PROPN
ejpam-3272	192	11	�	�	PROPN
ejpam-3272	192	12	x	x	PROPN
ejpam-3272	192	13	←	←	PROPN
ejpam-3272	192	14	a	a	PRON
ejpam-3272	192	15	since	since	SCONJ
ejpam-3272	192	16	the	the	DET
ejpam-3272	192	17	converse	converse	NOUN
ejpam-3272	192	18	inequality	inequality	NOUN
ejpam-3272	192	19	is	be	AUX
ejpam-3272	192	20	always	always	ADV
ejpam-3272	192	21	valid	valid	ADJ
ejpam-3272	192	22	.	.	PUNCT
ejpam-3272	193	1	thereby	thereby	ADV
ejpam-3272	193	2	,	,	PUNCT
ejpam-3272	193	3	there	there	PRON
ejpam-3272	193	4	exists	exist	VERB
ejpam-3272	193	5	a	a	DET
ejpam-3272	193	6	y	y	PROPN
ejpam-3272	193	7	∈	∈	PROPN
ejpam-3272	193	8	le	le	ADP
ejpam-3272	193	9	such	such	ADJ
ejpam-3272	193	10	that	that	SCONJ
ejpam-3272	193	11	x	x	PRON
ejpam-3272	193	12	←	←	PROPN
ejpam-3272	193	13	a	a	DET
ejpam-3272	193	14	≤	≤	NUM
ejpam-3272	193	15	y	y	PROPN
ejpam-3272	193	16	and	and	CCONJ
ejpam-3272	193	17	x←	x←	PROPN
ejpam-3272	193	18	k	k	PROPN
ejpam-3272	193	19	�	�	PROPN
ejpam-3272	193	20	y.	y.	PROPN
ejpam-3272	193	21	we	we	PRON
ejpam-3272	193	22	now	now	ADV
ejpam-3272	193	23	obtain	obtain	VERB
ejpam-3272	193	24	x←	x←	X
ejpam-3272	194	1	y	y	PROPN
ejpam-3272	194	2	≤	≤	PROPN
ejpam-3272	194	3	a	a	PRON
ejpam-3272	194	4	and	and	CCONJ
ejpam-3272	194	5	x←	x←	PROPN
ejpam-3272	194	6	y	y	PROPN
ejpam-3272	194	7	�	�	PROPN
ejpam-3272	194	8	k	k	PROPN
ejpam-3272	194	9	for	for	ADP
ejpam-3272	194	10	all	all	DET
ejpam-3272	194	11	a	a	DET
ejpam-3272	194	12	∈	∈	PROPN
ejpam-3272	194	13	lfr	lfr	NOUN
ejpam-3272	194	14	satisfying	satisfy	VERB
ejpam-3272	194	15	k	k	PROPN
ejpam-3272	194	16	≤	≤	PROPN
ejpam-3272	194	17	a	a	PRON
ejpam-3272	194	18	,	,	PUNCT
ejpam-3272	194	19	which	which	PRON
ejpam-3272	194	20	contradicts	contradict	VERB
ejpam-3272	194	21	with	with	ADP
ejpam-3272	194	22	the	the	DET
ejpam-3272	194	23	assumption	assumption	NOUN
ejpam-3272	194	24	.	.	PUNCT
ejpam-3272	195	1	the	the	DET
ejpam-3272	195	2	proof	proof	NOUN
ejpam-3272	195	3	of	of	ADP
ejpam-3272	195	4	(	(	PUNCT
ejpam-3272	195	5	i	i	NOUN
ejpam-3272	195	6	)	)	PUNCT
ejpam-3272	195	7	is	be	AUX
ejpam-3272	195	8	omitted	omit	VERB
ejpam-3272	195	9	since	since	SCONJ
ejpam-3272	195	10	it	it	PRON
ejpam-3272	195	11	can	can	AUX
ejpam-3272	195	12	be	be	AUX
ejpam-3272	195	13	proved	prove	VERB
ejpam-3272	195	14	in	in	ADP
ejpam-3272	195	15	a	a	DET
ejpam-3272	195	16	similar	similar	ADJ
ejpam-3272	195	17	way	way	NOUN
ejpam-3272	195	18	as	as	ADP
ejpam-3272	195	19	above	above	ADV
ejpam-3272	195	20	.	.	PUNCT
ejpam-3272	196	1	e.	e.	PROPN
ejpam-3272	196	2	korkmaz	korkmaz	PROPN
ejpam-3272	196	3	,	,	PUNCT
ejpam-3272	196	4	r.	r.	PROPN
ejpam-3272	196	5	ertürk	ertürk	PROPN
ejpam-3272	196	6	/	/	SYM
ejpam-3272	196	7	eur	eur	PROPN
ejpam-3272	196	8	.	.	PUNCT
ejpam-3272	197	1	j.	j.	PROPN
ejpam-3272	197	2	pure	pure	PROPN
ejpam-3272	197	3	appl	appl	PROPN
ejpam-3272	197	4	.	.	PROPN
ejpam-3272	197	5	math	math	PROPN
ejpam-3272	197	6	,	,	PUNCT
ejpam-3272	197	7	11	11	NUM
ejpam-3272	197	8	(	(	PUNCT
ejpam-3272	197	9	3	3	NUM
ejpam-3272	197	10	)	)	PUNCT
ejpam-3272	197	11	(	(	PUNCT
ejpam-3272	197	12	2018	2018	NUM
ejpam-3272	197	13	)	)	PUNCT
ejpam-3272	197	14	,	,	PUNCT
ejpam-3272	197	15	612	612	NUM
ejpam-3272	197	16	-	-	SYM
ejpam-3272	197	17	627	627	NUM
ejpam-3272	197	18	619	619	NUM
ejpam-3272	197	19	remark	remark	NOUN
ejpam-3272	197	20	2	2	NUM
ejpam-3272	197	21	.	.	PUNCT
ejpam-3272	198	1	the	the	DET
ejpam-3272	198	2	closure	closure	NOUN
ejpam-3272	198	3	of	of	ADP
ejpam-3272	198	4	an	an	DET
ejpam-3272	198	5	element	element	NOUN
ejpam-3272	198	6	a	a	DET
ejpam-3272	198	7	∈	∈	NOUN
ejpam-3272	198	8	le	le	X
ejpam-3272	198	9	is	be	AUX
ejpam-3272	198	10	given	give	VERB
ejpam-3272	198	11	by	by	ADP
ejpam-3272	198	12	[	[	X
ejpam-3272	198	13	a	a	X
ejpam-3272	198	14	]	]	X
ejpam-3272	198	15	=	=	SYM
ejpam-3272	198	16	∧	∧	NOUN
ejpam-3272	198	17	{	{	PUNCT
ejpam-3272	198	18	c	c	PROPN
ejpam-3272	198	19	∈	∈	PROPN
ejpam-3272	198	20	lcf	lcf	NOUN
ejpam-3272	198	21	:	:	PUNCT
ejpam-3272	198	22	a	a	DET
ejpam-3272	198	23	≤	≤	NUM
ejpam-3272	198	24	c	c	NOUN
ejpam-3272	198	25	}	}	PUNCT
ejpam-3272	198	26	,	,	PUNCT
ejpam-3272	198	27	and	and	CCONJ
ejpam-3272	198	28	the	the	DET
ejpam-3272	198	29	interior	interior	NOUN
ejpam-3272	198	30	by	by	ADP
ejpam-3272	198	31	]	]	X
ejpam-3272	198	32	a[=	a[=	PROPN
ejpam-3272	198	33	∨	∨	NUM
ejpam-3272	198	34	{	{	PUNCT
ejpam-3272	198	35	b	b	PROPN
ejpam-3272	198	36	∈	∈	PROPN
ejpam-3272	198	37	lfr	lfr	NOUN
ejpam-3272	198	38	:	:	PUNCT
ejpam-3272	198	39	b	b	X
ejpam-3272	198	40	≤	≤	ADV
ejpam-3272	198	41	a	a	PRON
ejpam-3272	198	42	}	}	PUNCT
ejpam-3272	198	43	.	.	PUNCT
ejpam-3272	199	1	definition	definition	NOUN
ejpam-3272	199	2	3	3	NUM
ejpam-3272	199	3	.	.	PUNCT
ejpam-3272	200	1	a	a	DET
ejpam-3272	200	2	diframe	diframe	NOUN
ejpam-3272	200	3	is	be	AUX
ejpam-3272	200	4	said	say	VERB
ejpam-3272	200	5	to	to	PART
ejpam-3272	200	6	be	be	AUX
ejpam-3272	200	7	(	(	PUNCT
ejpam-3272	200	8	i	i	NOUN
ejpam-3272	200	9	)	)	PUNCT
ejpam-3272	200	10	r1	r1	PROPN
ejpam-3272	200	11	if	if	SCONJ
ejpam-3272	200	12	,	,	PUNCT
ejpam-3272	200	13	for	for	ADP
ejpam-3272	200	14	all	all	DET
ejpam-3272	200	15	a	a	DET
ejpam-3272	200	16	∈	∈	PROPN
ejpam-3272	200	17	lfr	lfr	NOUN
ejpam-3272	200	18	,	,	PUNCT
ejpam-3272	200	19	a	a	DET
ejpam-3272	200	20	=	=	PUNCT
ejpam-3272	200	21	∨	∨	NUM
ejpam-3272	200	22	j∈j	j∈j	NOUN
ejpam-3272	200	23	∧	∧	PROPN
ejpam-3272	200	24	i∈i	i∈i	ADJ
ejpam-3272	200	25	cji	cji	NOUN
ejpam-3272	200	26	=	=	PUNCT
ejpam-3272	201	1	∨	∨	NUM
ejpam-3272	201	2	j∈j	j∈j	NOUN
ejpam-3272	201	3	∧	∧	PROPN
ejpam-3272	201	4	i∈i	i∈i	ADJ
ejpam-3272	201	5	[	[	X
ejpam-3272	201	6	cji	cji	NOUN
ejpam-3272	201	7	]	]	PUNCT
ejpam-3272	201	8	where	where	SCONJ
ejpam-3272	201	9	cji	cji	NOUN
ejpam-3272	201	10	∈	∈	PROPN
ejpam-3272	201	11	lfr	lfr	PROPN
ejpam-3272	201	12	.	.	PUNCT
ejpam-3272	202	1	(	(	PUNCT
ejpam-3272	202	2	ii	ii	NOUN
ejpam-3272	202	3	)	)	PUNCT
ejpam-3272	202	4	co	co	NOUN
ejpam-3272	202	5	-	-	NOUN
ejpam-3272	202	6	r1	r1	ADJ
ejpam-3272	202	7	if	if	SCONJ
ejpam-3272	202	8	,	,	PUNCT
ejpam-3272	202	9	for	for	ADP
ejpam-3272	202	10	all	all	DET
ejpam-3272	202	11	k	k	PROPN
ejpam-3272	202	12	∈	∈	PROPN
ejpam-3272	202	13	lcf	lcf	PROPN
ejpam-3272	202	14	,	,	PUNCT
ejpam-3272	202	15	k	k	PROPN
ejpam-3272	203	1	=	=	SYM
ejpam-3272	203	2	∧	∧	PROPN
ejpam-3272	203	3	j∈j	j∈j	NOUN
ejpam-3272	203	4	∨	∨	NUM
ejpam-3272	203	5	i∈i	i∈i	PROPN
ejpam-3272	203	6	f	f	PROPN
ejpam-3272	203	7	ji	ji	PROPN
ejpam-3272	204	1	=	=	PROPN
ejpam-3272	204	2	∧	∧	PROPN
ejpam-3272	204	3	j∈j	j∈j	NOUN
ejpam-3272	204	4	∨	∨	PROPN
ejpam-3272	204	5	i∈i	i∈i	ADJ
ejpam-3272	204	6	]	]	X
ejpam-3272	205	1	f	f	X
ejpam-3272	205	2	ji	ji	PROPN
ejpam-3272	205	3	[	[	PUNCT
ejpam-3272	205	4	where	where	SCONJ
ejpam-3272	205	5	f	f	PROPN
ejpam-3272	205	6	ji	ji	PROPN
ejpam-3272	205	7	∈	∈	PROPN
ejpam-3272	205	8	lcf	lcf	PROPN
ejpam-3272	205	9	.	.	PUNCT
ejpam-3272	206	1	(	(	PUNCT
ejpam-3272	206	2	iii	iii	X
ejpam-3272	206	3	)	)	PUNCT
ejpam-3272	206	4	t2	t2	NOUN
ejpam-3272	206	5	if	if	SCONJ
ejpam-3272	206	6	r1	r1	PROPN
ejpam-3272	206	7	and	and	CCONJ
ejpam-3272	206	8	t0	t0	PROPN
ejpam-3272	206	9	(	(	PUNCT
ejpam-3272	206	10	iv	iv	X
ejpam-3272	206	11	)	)	PUNCT
ejpam-3272	206	12	co	co	NOUN
ejpam-3272	206	13	-	-	NOUN
ejpam-3272	206	14	t2	t2	NOUN
ejpam-3272	206	15	if	if	SCONJ
ejpam-3272	206	16	co	co	NOUN
ejpam-3272	206	17	-	-	NOUN
ejpam-3272	206	18	r1	r1	ADJ
ejpam-3272	206	19	and	and	CCONJ
ejpam-3272	206	20	co	co	NOUN
ejpam-3272	206	21	-	-	NOUN
ejpam-3272	206	22	t0	t0	NOUN
ejpam-3272	206	23	.	.	PUNCT
ejpam-3272	207	1	note	note	VERB
ejpam-3272	207	2	that	that	SCONJ
ejpam-3272	207	3	,	,	PUNCT
ejpam-3272	207	4	r1	r1	PROPN
ejpam-3272	207	5	was	be	AUX
ejpam-3272	207	6	also	also	ADV
ejpam-3272	207	7	studied	study	VERB
ejpam-3272	207	8	in	in	ADP
ejpam-3272	207	9	[	[	X
ejpam-3272	207	10	8	8	NUM
ejpam-3272	207	11	]	]	PUNCT
ejpam-3272	207	12	,	,	PUNCT
ejpam-3272	207	13	under	under	ADP
ejpam-3272	207	14	the	the	DET
ejpam-3272	207	15	name	name	NOUN
ejpam-3272	207	16	“	"	PUNCT
ejpam-3272	207	17	pseudo	pseudo	NOUN
ejpam-3272	207	18	hausdorff	hausdorff	NOUN
ejpam-3272	207	19	”	"	PUNCT
ejpam-3272	207	20	.	.	PUNCT
ejpam-3272	208	1	proposition	proposition	NOUN
ejpam-3272	208	2	4	4	NUM
ejpam-3272	208	3	.	.	PUNCT
ejpam-3272	209	1	every	every	DET
ejpam-3272	209	2	r1	r1	NOUN
ejpam-3272	209	3	diframe	diframe	NOUN
ejpam-3272	209	4	is	be	AUX
ejpam-3272	209	5	r0	r0	NOUN
ejpam-3272	209	6	.	.	PUNCT
ejpam-3272	210	1	dually	dually	PROPN
ejpam-3272	210	2	,	,	PUNCT
ejpam-3272	210	3	every	every	DET
ejpam-3272	210	4	co	co	NOUN
ejpam-3272	210	5	-	-	ADJ
ejpam-3272	210	6	r1	r1	ADJ
ejpam-3272	210	7	diframe	diframe	NOUN
ejpam-3272	210	8	is	be	AUX
ejpam-3272	210	9	co	co	NOUN
ejpam-3272	210	10	-	-	NOUN
ejpam-3272	210	11	r0	r0	NOUN
ejpam-3272	210	12	.	.	PUNCT
ejpam-3272	211	1	proof	proof	NOUN
ejpam-3272	211	2	.	.	PUNCT
ejpam-3272	212	1	straightforward	straightforward	ADJ
ejpam-3272	212	2	by	by	ADP
ejpam-3272	212	3	definitions	definition	NOUN
ejpam-3272	212	4	.	.	PUNCT
ejpam-3272	213	1	remark	remark	NOUN
ejpam-3272	213	2	3	3	NUM
ejpam-3272	213	3	.	.	PUNCT
ejpam-3272	214	1	as	as	SCONJ
ejpam-3272	214	2	is	be	AUX
ejpam-3272	214	3	well	well	ADV
ejpam-3272	214	4	known	know	VERB
ejpam-3272	214	5	,	,	PUNCT
ejpam-3272	214	6	a	a	DET
ejpam-3272	214	7	bitopological	bitopological	ADJ
ejpam-3272	214	8	space	space	NOUN
ejpam-3272	214	9	(	(	PUNCT
ejpam-3272	214	10	x	x	X
ejpam-3272	214	11	,	,	PUNCT
ejpam-3272	214	12	t	t	PROPN
ejpam-3272	214	13	,	,	PUNCT
ejpam-3272	214	14	t∗	t∗	PROPN
ejpam-3272	214	15	)	)	PUNCT
ejpam-3272	214	16	is	be	AUX
ejpam-3272	214	17	regular	regular	ADJ
ejpam-3272	214	18	if	if	SCONJ
ejpam-3272	214	19	for	for	ADP
ejpam-3272	214	20	all	all	DET
ejpam-3272	214	21	g	g	PROPN
ejpam-3272	214	22	∈	∈	PROPN
ejpam-3272	214	23	t	t	NOUN
ejpam-3272	214	24	and	and	CCONJ
ejpam-3272	214	25	x	x	PROPN
ejpam-3272	214	26	∈	∈	PROPN
ejpam-3272	214	27	g	g	NOUN
ejpam-3272	214	28	,	,	PUNCT
ejpam-3272	214	29	there	there	PRON
ejpam-3272	214	30	exist	exist	VERB
ejpam-3272	214	31	a	a	DET
ejpam-3272	214	32	t	t	NOUN
ejpam-3272	214	33	-	-	PUNCT
ejpam-3272	214	34	open	open	ADJ
ejpam-3272	214	35	set	set	VERB
ejpam-3272	214	36	h	h	NOUN
ejpam-3272	214	37	and	and	CCONJ
ejpam-3272	214	38	a	a	DET
ejpam-3272	214	39	t∗-closed	t∗-close	VERB
ejpam-3272	214	40	set	set	NOUN
ejpam-3272	214	41	f	f	PROPN
ejpam-3272	214	42	such	such	ADJ
ejpam-3272	214	43	that	that	SCONJ
ejpam-3272	214	44	x	x	SYM
ejpam-3272	214	45	∈	∈	NOUN
ejpam-3272	214	46	h	h	NOUN
ejpam-3272	214	47	⊆	⊆	NUM
ejpam-3272	214	48	f	f	PROPN
ejpam-3272	214	49	⊆	⊆	NUM
ejpam-3272	214	50	g	g	NOUN
ejpam-3272	214	51	,	,	PUNCT
ejpam-3272	214	52	or	or	CCONJ
ejpam-3272	214	53	equivalently	equivalently	ADV
ejpam-3272	214	54	,	,	PUNCT
ejpam-3272	214	55	each	each	DET
ejpam-3272	214	56	g	g	PROPN
ejpam-3272	214	57	∈	∈	PROPN
ejpam-3272	214	58	t	t	NOUN
ejpam-3272	214	59	can	can	AUX
ejpam-3272	214	60	be	be	AUX
ejpam-3272	214	61	expressed	express	VERB
ejpam-3272	214	62	as	as	SCONJ
ejpam-3272	214	63	follows	follow	VERB
ejpam-3272	214	64	:	:	PUNCT
ejpam-3272	214	65	g	g	PROPN
ejpam-3272	214	66	=	=	SYM
ejpam-3272	214	67	⋃	⋃	NOUN
ejpam-3272	214	68	{	{	PUNCT
ejpam-3272	214	69	h	h	NOUN
ejpam-3272	214	70	∈	∈	PROPN
ejpam-3272	214	71	t	t	PROPN
ejpam-3272	214	72	:	:	PUNCT
ejpam-3272	214	73	∃f	∃f	PROPN
ejpam-3272	214	74	t∗-closed	t∗-close	VERB
ejpam-3272	214	75	;	;	PUNCT
ejpam-3272	214	76	h	h	PROPN
ejpam-3272	214	77	⊆	⊆	NUM
ejpam-3272	214	78	f	f	NOUN
ejpam-3272	214	79	⊆	⊆	NUM
ejpam-3272	214	80	g	g	NOUN
ejpam-3272	214	81	}	}	PUNCT
ejpam-3272	214	82	similarly	similarly	ADV
ejpam-3272	214	83	,	,	PUNCT
ejpam-3272	214	84	the	the	DET
ejpam-3272	214	85	dual	dual	ADJ
ejpam-3272	214	86	space	space	NOUN
ejpam-3272	214	87	(	(	PUNCT
ejpam-3272	214	88	x	x	NOUN
ejpam-3272	214	89	,	,	PUNCT
ejpam-3272	214	90	t∗,t	t∗,t	NOUN
ejpam-3272	214	91	)	)	PUNCT
ejpam-3272	214	92	is	be	AUX
ejpam-3272	214	93	regular	regular	ADJ
ejpam-3272	214	94	if	if	SCONJ
ejpam-3272	214	95	,	,	PUNCT
ejpam-3272	214	96	for	for	ADP
ejpam-3272	214	97	all	all	PRON
ejpam-3272	214	98	t∗-closed	t∗-close	VERB
ejpam-3272	214	99	set	set	VERB
ejpam-3272	214	100	f	f	PROPN
ejpam-3272	214	101	,	,	PUNCT
ejpam-3272	214	102	f	f	PROPN
ejpam-3272	214	103	=	=	SYM
ejpam-3272	214	104	⋂	⋂	PROPN
ejpam-3272	214	105	{	{	PUNCT
ejpam-3272	214	106	k	k	PROPN
ejpam-3272	214	107	t∗-closed	t∗-close	VERB
ejpam-3272	214	108	:	:	PUNCT
ejpam-3272	214	109	∃g	∃g	PROPN
ejpam-3272	214	110	∈	∈	PROPN
ejpam-3272	214	111	t	t	NOUN
ejpam-3272	214	112	;	;	PUNCT
ejpam-3272	214	113	f	f	PROPN
ejpam-3272	214	114	⊆	⊆	NUM
ejpam-3272	214	115	g	g	PROPN
ejpam-3272	214	116	⊆	⊆	NUM
ejpam-3272	214	117	k	k	NOUN
ejpam-3272	214	118	}	}	PUNCT
ejpam-3272	214	119	.	.	PUNCT
ejpam-3272	215	1	now	now	ADV
ejpam-3272	215	2	define	define	VERB
ejpam-3272	215	3	the	the	DET
ejpam-3272	215	4	relations	relation	NOUN
ejpam-3272	215	5	≺fr	≺fr	X
ejpam-3272	215	6	and	and	CCONJ
ejpam-3272	215	7	≺cf	≺cf	AUX
ejpam-3272	215	8	on	on	ADP
ejpam-3272	215	9	p(x	p(x	PROPN
ejpam-3272	215	10	)	)	PUNCT
ejpam-3272	215	11	by	by	ADP
ejpam-3272	215	12	declaring	declare	VERB
ejpam-3272	215	13	that	that	SCONJ
ejpam-3272	215	14	h	h	NOUN
ejpam-3272	215	15	≺fr	≺fr	X
ejpam-3272	215	16	g	g	PROPN
ejpam-3272	215	17	iff	iff	PROPN
ejpam-3272	215	18	there	there	PRON
ejpam-3272	215	19	exists	exist	VERB
ejpam-3272	215	20	an	an	DET
ejpam-3272	215	21	f	f	PROPN
ejpam-3272	215	22	∈	∈	PROPN
ejpam-3272	215	23	c(x	c(x	NOUN
ejpam-3272	215	24	)	)	PUNCT
ejpam-3272	215	25	such	such	ADJ
ejpam-3272	215	26	that	that	SCONJ
ejpam-3272	215	27	h	h	NOUN
ejpam-3272	215	28	⊆	⊆	NUM
ejpam-3272	215	29	f	f	NOUN
ejpam-3272	215	30	⊆	⊆	NUM
ejpam-3272	215	31	g	g	NOUN
ejpam-3272	215	32	and	and	CCONJ
ejpam-3272	215	33	f	f	PROPN
ejpam-3272	215	34	≺cf	≺cf	PROPN
ejpam-3272	215	35	k	k	PROPN
ejpam-3272	215	36	iff	iff	PROPN
ejpam-3272	215	37	there	there	PRON
ejpam-3272	215	38	exists	exist	VERB
ejpam-3272	215	39	a	a	DET
ejpam-3272	215	40	g	g	PROPN
ejpam-3272	215	41	∈	∈	PROPN
ejpam-3272	215	42	ω(x	ω(x	NOUN
ejpam-3272	215	43	)	)	PUNCT
ejpam-3272	216	1	such	such	ADJ
ejpam-3272	216	2	that	that	SCONJ
ejpam-3272	216	3	f	f	PROPN
ejpam-3272	216	4	⊆	⊆	NUM
ejpam-3272	216	5	g	g	PROPN
ejpam-3272	216	6	⊆	⊆	NUM
ejpam-3272	216	7	k.	k.	NOUN
ejpam-3272	216	8	on	on	ADP
ejpam-3272	216	9	the	the	DET
ejpam-3272	216	10	basis	basis	NOUN
ejpam-3272	216	11	of	of	ADP
ejpam-3272	216	12	the	the	DET
ejpam-3272	216	13	previous	previous	ADJ
ejpam-3272	216	14	discussion	discussion	NOUN
ejpam-3272	216	15	,	,	PUNCT
ejpam-3272	216	16	we	we	PRON
ejpam-3272	216	17	introduce	introduce	VERB
ejpam-3272	216	18	the	the	DET
ejpam-3272	216	19	following	follow	VERB
ejpam-3272	216	20	relations	relation	NOUN
ejpam-3272	216	21	on	on	ADP
ejpam-3272	216	22	le	le	X
ejpam-3272	216	23	:	:	PUNCT
ejpam-3272	216	24	we	we	PRON
ejpam-3272	216	25	say	say	VERB
ejpam-3272	216	26	that	that	SCONJ
ejpam-3272	216	27	a	a	PRON
ejpam-3272	216	28	is	be	AUX
ejpam-3272	216	29	fr	fr	NOUN
ejpam-3272	216	30	-	-	PUNCT
ejpam-3272	216	31	below	below	NOUN
ejpam-3272	216	32	b	b	NOUN
ejpam-3272	216	33	,	,	PUNCT
ejpam-3272	216	34	in	in	ADP
ejpam-3272	216	35	symbols	symbol	NOUN
ejpam-3272	216	36	a	a	DET
ejpam-3272	216	37	≺fr	≺fr	PROPN
ejpam-3272	216	38	b	b	PROPN
ejpam-3272	216	39	,	,	PUNCT
ejpam-3272	216	40	iff	iff	PROPN
ejpam-3272	216	41	a	a	PRON
ejpam-3272	216	42	,	,	PUNCT
ejpam-3272	216	43	b	b	PROPN
ejpam-3272	216	44	∈	∈	PROPN
ejpam-3272	216	45	lfr	lfr	NOUN
ejpam-3272	217	1	and	and	CCONJ
ejpam-3272	217	2	there	there	PRON
ejpam-3272	217	3	exists	exist	VERB
ejpam-3272	217	4	a	a	DET
ejpam-3272	217	5	c	c	PROPN
ejpam-3272	217	6	∈	∈	PROPN
ejpam-3272	217	7	lcf	lcf	NOUN
ejpam-3272	217	8	such	such	ADJ
ejpam-3272	217	9	that	that	SCONJ
ejpam-3272	217	10	a	a	DET
ejpam-3272	217	11	≤	≤	PROPN
ejpam-3272	217	12	c	c	PROPN
ejpam-3272	217	13	≤	≤	PROPN
ejpam-3272	217	14	b.	b.	PROPN
ejpam-3272	217	15	dually	dually	PROPN
ejpam-3272	217	16	,	,	PUNCT
ejpam-3272	217	17	we	we	PRON
ejpam-3272	217	18	say	say	VERB
ejpam-3272	217	19	that	that	SCONJ
ejpam-3272	217	20	f	f	PROPN
ejpam-3272	217	21	is	be	AUX
ejpam-3272	217	22	cf	cf	NOUN
ejpam-3272	217	23	-	-	PUNCT
ejpam-3272	217	24	below	below	NOUN
ejpam-3272	217	25	k	k	NOUN
ejpam-3272	217	26	,	,	PUNCT
ejpam-3272	217	27	in	in	ADP
ejpam-3272	217	28	symbols	symbol	NOUN
ejpam-3272	218	1	f	f	PROPN
ejpam-3272	218	2	≺cf	≺cf	PROPN
ejpam-3272	218	3	k	k	PROPN
ejpam-3272	218	4	,	,	PUNCT
ejpam-3272	218	5	iff	iff	PROPN
ejpam-3272	218	6	f	f	PROPN
ejpam-3272	218	7	,	,	PUNCT
ejpam-3272	218	8	k	k	PROPN
ejpam-3272	218	9	∈	∈	PROPN
ejpam-3272	218	10	lcf	lcf	PROPN
ejpam-3272	218	11	and	and	CCONJ
ejpam-3272	218	12	there	there	PRON
ejpam-3272	218	13	exists	exist	VERB
ejpam-3272	218	14	an	an	DET
ejpam-3272	218	15	a	a	DET
ejpam-3272	218	16	∈	∈	PROPN
ejpam-3272	218	17	lfr	lfr	NOUN
ejpam-3272	218	18	such	such	ADJ
ejpam-3272	218	19	that	that	SCONJ
ejpam-3272	218	20	f	f	PROPN
ejpam-3272	218	21	≤	≤	ADV
ejpam-3272	218	22	a	a	DET
ejpam-3272	218	23	≤	≤	PROPN
ejpam-3272	218	24	k.	k.	PROPN
ejpam-3272	218	25	e.	e.	PROPN
ejpam-3272	218	26	korkmaz	korkmaz	PROPN
ejpam-3272	218	27	,	,	PUNCT
ejpam-3272	218	28	r.	r.	PROPN
ejpam-3272	218	29	ertürk	ertürk	PROPN
ejpam-3272	218	30	/	/	SYM
ejpam-3272	218	31	eur	eur	PROPN
ejpam-3272	218	32	.	.	PUNCT
ejpam-3272	219	1	j.	j.	PROPN
ejpam-3272	219	2	pure	pure	PROPN
ejpam-3272	219	3	appl	appl	PROPN
ejpam-3272	219	4	.	.	PROPN
ejpam-3272	219	5	math	math	PROPN
ejpam-3272	219	6	,	,	PUNCT
ejpam-3272	219	7	11	11	NUM
ejpam-3272	219	8	(	(	PUNCT
ejpam-3272	219	9	3	3	NUM
ejpam-3272	219	10	)	)	PUNCT
ejpam-3272	219	11	(	(	PUNCT
ejpam-3272	219	12	2018	2018	NUM
ejpam-3272	219	13	)	)	PUNCT
ejpam-3272	219	14	,	,	PUNCT
ejpam-3272	219	15	612	612	NUM
ejpam-3272	219	16	-	-	SYM
ejpam-3272	219	17	627	627	NUM
ejpam-3272	219	18	620	620	NUM
ejpam-3272	219	19	proposition	proposition	NOUN
ejpam-3272	219	20	5	5	NUM
ejpam-3272	219	21	.	.	PUNCT
ejpam-3272	220	1	in	in	ADP
ejpam-3272	220	2	a	a	DET
ejpam-3272	220	3	diframe	diframe	NOUN
ejpam-3272	220	4	l	l	NOUN
ejpam-3272	220	5	,	,	PUNCT
ejpam-3272	220	6	the	the	DET
ejpam-3272	220	7	relations	relation	NOUN
ejpam-3272	220	8	≺fr	≺fr	X
ejpam-3272	220	9	and	and	CCONJ
ejpam-3272	220	10	≺cf	≺cf	PROPN
ejpam-3272	220	11	satisfy	satisfy	VERB
ejpam-3272	220	12	the	the	DET
ejpam-3272	220	13	following	follow	VERB
ejpam-3272	220	14	conditions	condition	NOUN
ejpam-3272	220	15	:	:	PUNCT
ejpam-3272	220	16	(	(	PUNCT
ejpam-3272	220	17	i	i	NOUN
ejpam-3272	220	18	)	)	PUNCT
ejpam-3272	220	19	0	0	PUNCT
ejpam-3272	221	1	≺fr	≺fr	X
ejpam-3272	221	2	a	a	DET
ejpam-3272	221	3	≺fr	≺fr	NOUN
ejpam-3272	221	4	1	1	NUM
ejpam-3272	221	5	for	for	ADP
ejpam-3272	221	6	all	all	DET
ejpam-3272	221	7	a	a	DET
ejpam-3272	221	8	∈	∈	PROPN
ejpam-3272	221	9	lfr	lfr	NOUN
ejpam-3272	221	10	,	,	PUNCT
ejpam-3272	221	11	and	and	CCONJ
ejpam-3272	221	12	0	0	NUM
ejpam-3272	221	13	≺cf	≺cf	VERB
ejpam-3272	221	14	k	k	PROPN
ejpam-3272	221	15	≺cf	≺cf	PROPN
ejpam-3272	221	16	1	1	NUM
ejpam-3272	221	17	for	for	ADP
ejpam-3272	221	18	all	all	DET
ejpam-3272	221	19	k	k	PROPN
ejpam-3272	221	20	∈	∈	PROPN
ejpam-3272	221	21	lcf	lcf	PROPN
ejpam-3272	221	22	.	.	PUNCT
ejpam-3272	222	1	(	(	PUNCT
ejpam-3272	222	2	ii	ii	NOUN
ejpam-3272	222	3	)	)	PUNCT
ejpam-3272	222	4	a	a	DET
ejpam-3272	222	5	≺fr	≺fr	PROPN
ejpam-3272	222	6	b	b	PROPN
ejpam-3272	222	7	implies	imply	VERB
ejpam-3272	222	8	a	a	DET
ejpam-3272	222	9	≤	≤	NUM
ejpam-3272	222	10	b	b	NUM
ejpam-3272	222	11	,	,	PUNCT
ejpam-3272	222	12	and	and	CCONJ
ejpam-3272	222	13	f	f	PROPN
ejpam-3272	222	14	≺cf	≺cf	PROPN
ejpam-3272	222	15	k	k	PROPN
ejpam-3272	222	16	implies	imply	VERB
ejpam-3272	222	17	f	f	PROPN
ejpam-3272	222	18	≤	≤	PROPN
ejpam-3272	222	19	k.	k.	PROPN
ejpam-3272	222	20	(	(	PUNCT
ejpam-3272	222	21	iii	iii	X
ejpam-3272	222	22	)	)	PUNCT
ejpam-3272	222	23	if	if	SCONJ
ejpam-3272	222	24	a	a	DET
ejpam-3272	222	25	≤	≤	NUM
ejpam-3272	222	26	b	b	NOUN
ejpam-3272	223	1	≺fr	≺fr	X
ejpam-3272	223	2	c	c	NOUN
ejpam-3272	223	3	≤	≤	PROPN
ejpam-3272	223	4	d	d	NOUN
ejpam-3272	223	5	then	then	ADV
ejpam-3272	223	6	a	a	DET
ejpam-3272	223	7	≺fr	≺fr	X
ejpam-3272	223	8	d.	d.	NOUN
ejpam-3272	223	9	if	if	SCONJ
ejpam-3272	223	10	f	f	PROPN
ejpam-3272	223	11	≤	≤	X
ejpam-3272	224	1	c	c	PROPN
ejpam-3272	224	2	≺cf	≺cf	ADP
ejpam-3272	224	3	d	d	NOUN
ejpam-3272	224	4	≤	≤	PROPN
ejpam-3272	225	1	k	k	NOUN
ejpam-3272	225	2	then	then	ADV
ejpam-3272	225	3	f	f	PROPN
ejpam-3272	225	4	≺cf	≺cf	PROPN
ejpam-3272	225	5	k.	k.	PROPN
ejpam-3272	225	6	(	(	PUNCT
ejpam-3272	225	7	iv	iv	X
ejpam-3272	225	8	)	)	PUNCT
ejpam-3272	225	9	for	for	ADP
ejpam-3272	225	10	i	i	PROPN
ejpam-3272	225	11	=	=	SYM
ejpam-3272	225	12	1	1	NUM
ejpam-3272	225	13	,	,	PUNCT
ejpam-3272	225	14	2	2	NUM
ejpam-3272	225	15	if	if	SCONJ
ejpam-3272	225	16	ai	ai	VERB
ejpam-3272	225	17	≺fr	≺fr	NOUN
ejpam-3272	225	18	bi	bi	NOUN
ejpam-3272	225	19	then	then	ADV
ejpam-3272	225	20	a1	a1	PROPN
ejpam-3272	225	21	∨	∨	PROPN
ejpam-3272	225	22	a2	a2	PROPN
ejpam-3272	225	23	≺fr	≺fr	PUNCT
ejpam-3272	225	24	b1	b1	PROPN
ejpam-3272	225	25	∨	∨	NUM
ejpam-3272	225	26	b2	b2	PROPN
ejpam-3272	225	27	and	and	CCONJ
ejpam-3272	225	28	a1	a1	NOUN
ejpam-3272	225	29	∧	∧	PROPN
ejpam-3272	225	30	a2	a2	PROPN
ejpam-3272	225	31	≺fr	≺fr	NOUN
ejpam-3272	225	32	b1	b1	VERB
ejpam-3272	225	33	∧	∧	PROPN
ejpam-3272	225	34	b2	b2	NOUN
ejpam-3272	225	35	.	.	PUNCT
ejpam-3272	226	1	moreover	moreover	ADV
ejpam-3272	226	2	,	,	PUNCT
ejpam-3272	226	3	if	if	SCONJ
ejpam-3272	226	4	fi	fi	NOUN
ejpam-3272	226	5	≺cf	≺cf	VERB
ejpam-3272	226	6	ki	ki	PROPN
ejpam-3272	226	7	then	then	ADV
ejpam-3272	226	8	f1	f1	PROPN
ejpam-3272	226	9	∨	∨	PROPN
ejpam-3272	226	10	f2	f2	PROPN
ejpam-3272	226	11	≺cf	≺cf	ADP
ejpam-3272	226	12	k1	k1	PROPN
ejpam-3272	226	13	∨	∨	NUM
ejpam-3272	226	14	k2	k2	PROPN
ejpam-3272	226	15	and	and	CCONJ
ejpam-3272	226	16	f1	f1	PROPN
ejpam-3272	226	17	∧	∧	PROPN
ejpam-3272	226	18	f2	f2	PRON
ejpam-3272	226	19	≺cf	≺cf	ADP
ejpam-3272	226	20	k1	k1	PROPN
ejpam-3272	226	21	∧	∧	PROPN
ejpam-3272	226	22	k2	k2	PROPN
ejpam-3272	226	23	.	.	PUNCT
ejpam-3272	227	1	clearly	clearly	ADV
ejpam-3272	227	2	,	,	PUNCT
ejpam-3272	227	3	≺fr	≺fr	PROPN
ejpam-3272	227	4	and	and	CCONJ
ejpam-3272	227	5	≺cf	≺cf	PROPN
ejpam-3272	227	6	are	be	AUX
ejpam-3272	227	7	auxiliary	auxiliary	ADJ
ejpam-3272	227	8	relations	relation	NOUN
ejpam-3272	227	9	in	in	ADP
ejpam-3272	227	10	the	the	DET
ejpam-3272	227	11	sense	sense	NOUN
ejpam-3272	227	12	of	of	ADP
ejpam-3272	227	13	definition	definition	NOUN
ejpam-3272	227	14	i.1.9	i.1.9	VERB
ejpam-3272	227	15	in	in	ADP
ejpam-3272	227	16	[	[	X
ejpam-3272	227	17	6	6	NUM
ejpam-3272	227	18	]	]	PUNCT
ejpam-3272	227	19	.	.	PUNCT
ejpam-3272	228	1	definition	definition	NOUN
ejpam-3272	228	2	4	4	NUM
ejpam-3272	228	3	.	.	PUNCT
ejpam-3272	229	1	a	a	DET
ejpam-3272	229	2	diframe	diframe	NOUN
ejpam-3272	229	3	is	be	AUX
ejpam-3272	229	4	said	say	VERB
ejpam-3272	229	5	to	to	PART
ejpam-3272	229	6	be	be	AUX
ejpam-3272	229	7	(	(	PUNCT
ejpam-3272	229	8	i	i	NOUN
ejpam-3272	229	9	)	)	PUNCT
ejpam-3272	229	10	regular	regular	ADV
ejpam-3272	229	11	if	if	SCONJ
ejpam-3272	229	12	a	a	DET
ejpam-3272	229	13	=	=	SYM
ejpam-3272	229	14	∨	∨	X
ejpam-3272	229	15	{	{	PUNCT
ejpam-3272	229	16	x	x	SYM
ejpam-3272	229	17	∈	∈	PROPN
ejpam-3272	229	18	lfr	lfr	NOUN
ejpam-3272	229	19	:	:	PUNCT
ejpam-3272	229	20	x	x	PUNCT
ejpam-3272	230	1	≺fr	≺fr	NOUN
ejpam-3272	230	2	a	a	X
ejpam-3272	230	3	}	}	PUNCT
ejpam-3272	230	4	for	for	ADP
ejpam-3272	230	5	all	all	DET
ejpam-3272	230	6	a	a	DET
ejpam-3272	230	7	∈	∈	PROPN
ejpam-3272	230	8	lfr	lfr	X
ejpam-3272	230	9	.	.	PUNCT
ejpam-3272	230	10	(	(	PUNCT
ejpam-3272	230	11	ii	ii	NOUN
ejpam-3272	230	12	)	)	PUNCT
ejpam-3272	230	13	co	co	NOUN
ejpam-3272	230	14	-	-	NOUN
ejpam-3272	230	15	regular	regular	ADJ
ejpam-3272	230	16	if	if	SCONJ
ejpam-3272	230	17	c	c	NOUN
ejpam-3272	230	18	=	=	SYM
ejpam-3272	230	19	∧	∧	PROPN
ejpam-3272	230	20	{	{	PUNCT
ejpam-3272	230	21	x	x	PROPN
ejpam-3272	230	22	∈	∈	PROPN
ejpam-3272	230	23	lcf	lcf	NOUN
ejpam-3272	230	24	:	:	PUNCT
ejpam-3272	230	25	c	c	X
ejpam-3272	230	26	≺cf	≺cf	VERB
ejpam-3272	230	27	x	x	X
ejpam-3272	230	28	}	}	PUNCT
ejpam-3272	230	29	for	for	ADP
ejpam-3272	230	30	all	all	DET
ejpam-3272	230	31	c	c	PROPN
ejpam-3272	230	32	∈	∈	PROPN
ejpam-3272	230	33	lcf	lcf	PROPN
ejpam-3272	230	34	.	.	PUNCT
ejpam-3272	231	1	(	(	PUNCT
ejpam-3272	231	2	iii	iii	X
ejpam-3272	231	3	)	)	PUNCT
ejpam-3272	231	4	t3	t3	NOUN
ejpam-3272	231	5	if	if	SCONJ
ejpam-3272	231	6	regular	regular	ADJ
ejpam-3272	231	7	and	and	CCONJ
ejpam-3272	231	8	t0	t0	NOUN
ejpam-3272	231	9	.	.	PUNCT
ejpam-3272	232	1	(	(	PUNCT
ejpam-3272	232	2	iv	iv	X
ejpam-3272	232	3	)	)	PUNCT
ejpam-3272	232	4	co	co	NOUN
ejpam-3272	232	5	-	-	NOUN
ejpam-3272	232	6	t3	t3	NOUN
ejpam-3272	232	7	if	if	SCONJ
ejpam-3272	232	8	co	co	NOUN
ejpam-3272	232	9	-	-	NOUN
ejpam-3272	232	10	regular	regular	ADJ
ejpam-3272	232	11	and	and	CCONJ
ejpam-3272	232	12	co	co	NOUN
ejpam-3272	232	13	-	-	NOUN
ejpam-3272	232	14	t0	t0	NOUN
ejpam-3272	232	15	.	.	PUNCT
ejpam-3272	233	1	the	the	DET
ejpam-3272	233	2	following	follow	VERB
ejpam-3272	233	3	proposition	proposition	NOUN
ejpam-3272	233	4	is	be	AUX
ejpam-3272	233	5	immediate	immediate	ADJ
ejpam-3272	233	6	by	by	ADP
ejpam-3272	233	7	definitions	definition	NOUN
ejpam-3272	233	8	:	:	PUNCT
ejpam-3272	233	9	proposition	proposition	NOUN
ejpam-3272	233	10	6	6	NUM
ejpam-3272	233	11	.	.	PUNCT
ejpam-3272	234	1	(	(	PUNCT
ejpam-3272	234	2	i	i	NOUN
ejpam-3272	234	3	)	)	PUNCT
ejpam-3272	234	4	a	a	DET
ejpam-3272	234	5	diframe	diframe	NOUN
ejpam-3272	234	6	l	l	NOUN
ejpam-3272	234	7	is	be	AUX
ejpam-3272	234	8	regular	regular	ADJ
ejpam-3272	234	9	iff	iff	PROPN
ejpam-3272	234	10	a	a	DET
ejpam-3272	234	11	=	=	SYM
ejpam-3272	234	12	∨	∨	X
ejpam-3272	234	13	{	{	PUNCT
ejpam-3272	234	14	x	x	SYM
ejpam-3272	234	15	∈	∈	PROPN
ejpam-3272	234	16	lfr	lfr	NOUN
ejpam-3272	234	17	:	:	PUNCT
ejpam-3272	235	1	[	[	X
ejpam-3272	235	2	x	x	X
ejpam-3272	235	3	]	]	X
ejpam-3272	235	4	≤	≤	NUM
ejpam-3272	235	5	a	a	X
ejpam-3272	235	6	}	}	PUNCT
ejpam-3272	235	7	for	for	ADP
ejpam-3272	235	8	all	all	DET
ejpam-3272	235	9	a	a	DET
ejpam-3272	235	10	∈	∈	PROPN
ejpam-3272	235	11	lfr	lfr	X
ejpam-3272	235	12	.	.	PUNCT
ejpam-3272	235	13	(	(	PUNCT
ejpam-3272	235	14	ii	ii	NOUN
ejpam-3272	235	15	)	)	PUNCT
ejpam-3272	235	16	a	a	DET
ejpam-3272	235	17	diframe	diframe	NOUN
ejpam-3272	235	18	l	l	NOUN
ejpam-3272	235	19	is	be	AUX
ejpam-3272	235	20	co	co	ADJ
ejpam-3272	235	21	-	-	ADJ
ejpam-3272	235	22	regular	regular	ADJ
ejpam-3272	235	23	iff	iff	PROPN
ejpam-3272	235	24	c	c	PROPN
ejpam-3272	235	25	=	=	SYM
ejpam-3272	235	26	∧	∧	PROPN
ejpam-3272	235	27	{	{	PUNCT
ejpam-3272	235	28	x	x	PROPN
ejpam-3272	235	29	∈	∈	PROPN
ejpam-3272	235	30	lcf	lcf	NOUN
ejpam-3272	235	31	:	:	PUNCT
ejpam-3272	236	1	c	c	X
ejpam-3272	236	2	≤]x	≤]x	PROPN
ejpam-3272	236	3	[	[	X
ejpam-3272	236	4	}	}	PUNCT
ejpam-3272	236	5	for	for	ADP
ejpam-3272	236	6	all	all	DET
ejpam-3272	236	7	c	c	PROPN
ejpam-3272	236	8	∈	∈	PROPN
ejpam-3272	236	9	lcf	lcf	PROPN
ejpam-3272	236	10	.	.	PUNCT
ejpam-3272	236	11	example	example	NOUN
ejpam-3272	237	1	3	3	X
ejpam-3272	237	2	.	.	PUNCT
ejpam-3272	237	3	let	let	VERB
ejpam-3272	237	4	i	i	PRON
ejpam-3272	237	5	=	=	PUNCT
ejpam-3272	238	1	[	[	X
ejpam-3272	238	2	0	0	NUM
ejpam-3272	238	3	,	,	PUNCT
ejpam-3272	238	4	1	1	NUM
ejpam-3272	238	5	]	]	PUNCT
ejpam-3272	238	6	be	be	AUX
ejpam-3272	238	7	the	the	DET
ejpam-3272	238	8	unit	unit	NOUN
ejpam-3272	238	9	interval	interval	NOUN
ejpam-3272	238	10	,	,	PUNCT
ejpam-3272	238	11	le	le	X
ejpam-3272	238	12	=	=	PUNCT
ejpam-3272	238	13	{	{	PUNCT
ejpam-3272	239	1	[	[	X
ejpam-3272	239	2	0	0	NUM
ejpam-3272	239	3	,	,	PUNCT
ejpam-3272	239	4	r	r	NOUN
ejpam-3272	239	5	]	]	X
ejpam-3272	239	6	,	,	PUNCT
ejpam-3272	239	7	[	[	X
ejpam-3272	239	8	0	0	NUM
ejpam-3272	239	9	,	,	PUNCT
ejpam-3272	239	10	r	r	NOUN
ejpam-3272	239	11	)	)	PUNCT
ejpam-3272	239	12	:	:	PUNCT
ejpam-3272	240	1	0	0	NUM
ejpam-3272	240	2	≤	≤	NUM
ejpam-3272	240	3	r	r	NOUN
ejpam-3272	240	4	≤	≤	NUM
ejpam-3272	240	5	1	1	NUM
ejpam-3272	240	6	}	}	PUNCT
ejpam-3272	240	7	,	,	PUNCT
ejpam-3272	240	8	lfr	lfr	X
ejpam-3272	240	9	=	=	SYM
ejpam-3272	240	10	{	{	PUNCT
ejpam-3272	240	11	[	[	X
ejpam-3272	240	12	0	0	NUM
ejpam-3272	240	13	,	,	PUNCT
ejpam-3272	240	14	r	r	NOUN
ejpam-3272	240	15	)	)	PUNCT
ejpam-3272	240	16	:	:	PUNCT
ejpam-3272	240	17	0	0	NUM
ejpam-3272	240	18	≤	≤	NUM
ejpam-3272	240	19	r	r	NOUN
ejpam-3272	240	20	≤	≤	NUM
ejpam-3272	240	21	1	1	NUM
ejpam-3272	240	22	}	}	PUNCT
ejpam-3272	240	23	∪	∪	X
ejpam-3272	240	24	{	{	PUNCT
ejpam-3272	240	25	i	i	NOUN
ejpam-3272	240	26	}	}	PUNCT
ejpam-3272	240	27	and	and	CCONJ
ejpam-3272	240	28	lcf	lcf	PROPN
ejpam-3272	240	29	=	=	SYM
ejpam-3272	240	30	{	{	PUNCT
ejpam-3272	240	31	[	[	X
ejpam-3272	240	32	0	0	NUM
ejpam-3272	240	33	,	,	PUNCT
ejpam-3272	240	34	r	r	NOUN
ejpam-3272	240	35	]	]	X
ejpam-3272	240	36	:	:	PUNCT
ejpam-3272	240	37	0	0	NUM
ejpam-3272	240	38	≤	≤	NUM
ejpam-3272	240	39	r	r	NOUN
ejpam-3272	240	40	≤	≤	NUM
ejpam-3272	240	41	1	1	NUM
ejpam-3272	240	42	}	}	PUNCT
ejpam-3272	240	43	∪	∪	ADJ
ejpam-3272	240	44	{	{	PUNCT
ejpam-3272	240	45	∅	∅	NOUN
ejpam-3272	240	46	}	}	PUNCT
ejpam-3272	240	47	.	.	PUNCT
ejpam-3272	241	1	trivially	trivially	ADV
ejpam-3272	241	2	,	,	PUNCT
ejpam-3272	241	3	for	for	ADP
ejpam-3272	241	4	[	[	X
ejpam-3272	241	5	0	0	NUM
ejpam-3272	241	6	,	,	PUNCT
ejpam-3272	241	7	r	r	NOUN
ejpam-3272	241	8	)	)	PUNCT
ejpam-3272	241	9	,	,	PUNCT
ejpam-3272	242	1	[	[	X
ejpam-3272	242	2	0	0	NUM
ejpam-3272	242	3	,	,	PUNCT
ejpam-3272	242	4	s	s	NOUN
ejpam-3272	242	5	)	)	PUNCT
ejpam-3272	242	6	∈	∈	PROPN
ejpam-3272	242	7	lfr	lfr	PROPN
ejpam-3272	242	8	,	,	PUNCT
ejpam-3272	242	9	[	[	X
ejpam-3272	242	10	0	0	NUM
ejpam-3272	242	11	,	,	PUNCT
ejpam-3272	242	12	r	r	NOUN
ejpam-3272	242	13	)	)	PUNCT
ejpam-3272	242	14	≺fr	≺fr	X
ejpam-3272	243	1	[	[	X
ejpam-3272	243	2	0	0	NUM
ejpam-3272	243	3	,	,	PUNCT
ejpam-3272	243	4	s	s	PART
ejpam-3272	243	5	)	)	PUNCT
ejpam-3272	243	6	iff	iff	PROPN
ejpam-3272	243	7	r	r	NOUN
ejpam-3272	243	8	<	<	X
ejpam-3272	243	9	s.	s.	PROPN
ejpam-3272	243	10	for	for	ADP
ejpam-3272	243	11	each	each	DET
ejpam-3272	243	12	u	u	NOUN
ejpam-3272	243	13	=	=	PUNCT
ejpam-3272	244	1	[	[	X
ejpam-3272	244	2	0	0	NUM
ejpam-3272	244	3	,	,	PUNCT
ejpam-3272	244	4	r	r	NOUN
ejpam-3272	244	5	)	)	PUNCT
ejpam-3272	244	6	∈	∈	PROPN
ejpam-3272	244	7	lfr	lfr	PROPN
ejpam-3272	244	8	,	,	PUNCT
ejpam-3272	244	9	u	u	NOUN
ejpam-3272	244	10	=	=	PROPN
ejpam-3272	244	11	∨	∨	X
ejpam-3272	244	12	{	{	PUNCT
ejpam-3272	244	13	[	[	X
ejpam-3272	244	14	0	0	NUM
ejpam-3272	244	15	,	,	PUNCT
ejpam-3272	244	16	r	r	NOUN
ejpam-3272	244	17	−	−	PROPN
ejpam-3272	244	18	1	1	NUM
ejpam-3272	244	19	n	n	CCONJ
ejpam-3272	244	20	)	)	PUNCT
ejpam-3272	244	21	:	:	PUNCT
ejpam-3272	245	1	[	[	X
ejpam-3272	245	2	0	0	NUM
ejpam-3272	245	3	,	,	PUNCT
ejpam-3272	245	4	r	r	NOUN
ejpam-3272	245	5	−	−	PROPN
ejpam-3272	245	6	1	1	NUM
ejpam-3272	245	7	n	n	CCONJ
ejpam-3272	245	8	)	)	PUNCT
ejpam-3272	245	9	≺fr	≺fr	X
ejpam-3272	245	10	[	[	X
ejpam-3272	245	11	0	0	NUM
ejpam-3272	245	12	,	,	PUNCT
ejpam-3272	245	13	r	r	NOUN
ejpam-3272	245	14	)	)	PUNCT
ejpam-3272	245	15	}	}	PUNCT
ejpam-3272	245	16	.	.	PUNCT
ejpam-3272	246	1	thus	thus	ADV
ejpam-3272	246	2	,	,	PUNCT
ejpam-3272	246	3	l	l	NOUN
ejpam-3272	246	4	=	=	SYM
ejpam-3272	246	5	(	(	PUNCT
ejpam-3272	246	6	le	le	X
ejpam-3272	246	7	,	,	PUNCT
ejpam-3272	246	8	lfr	lfr	PROPN
ejpam-3272	246	9	,	,	PUNCT
ejpam-3272	246	10	lcf	lcf	PROPN
ejpam-3272	246	11	)	)	PUNCT
ejpam-3272	246	12	is	be	AUX
ejpam-3272	246	13	regular	regular	ADJ
ejpam-3272	246	14	.	.	PUNCT
ejpam-3272	247	1	similarly	similarly	ADV
ejpam-3272	247	2	,	,	PUNCT
ejpam-3272	247	3	we	we	PRON
ejpam-3272	247	4	can	can	AUX
ejpam-3272	247	5	show	show	VERB
ejpam-3272	247	6	the	the	DET
ejpam-3272	247	7	co	co	NOUN
ejpam-3272	247	8	-	-	NOUN
ejpam-3272	247	9	regularity	regularity	NOUN
ejpam-3272	247	10	of	of	ADP
ejpam-3272	247	11	l.	l.	NOUN
ejpam-3272	247	12	the	the	DET
ejpam-3272	247	13	proof	proof	NOUN
ejpam-3272	247	14	of	of	ADP
ejpam-3272	247	15	the	the	DET
ejpam-3272	247	16	next	next	ADJ
ejpam-3272	247	17	proposition	proposition	NOUN
ejpam-3272	247	18	is	be	AUX
ejpam-3272	247	19	quite	quite	ADV
ejpam-3272	247	20	standard	standard	ADJ
ejpam-3272	247	21	and	and	CCONJ
ejpam-3272	247	22	will	will	AUX
ejpam-3272	247	23	therefore	therefore	ADV
ejpam-3272	247	24	be	be	AUX
ejpam-3272	247	25	omitted	omit	VERB
ejpam-3272	247	26	.	.	PUNCT
ejpam-3272	248	1	proposition	proposition	NOUN
ejpam-3272	248	2	7	7	NUM
ejpam-3272	248	3	.	.	PUNCT
ejpam-3272	249	1	if	if	SCONJ
ejpam-3272	249	2	l	l	NOUN
ejpam-3272	249	3	=	=	SYM
ejpam-3272	249	4	(	(	PUNCT
ejpam-3272	249	5	le	le	X
ejpam-3272	249	6	,	,	PUNCT
ejpam-3272	249	7	lfr	lfr	PROPN
ejpam-3272	249	8	,	,	PUNCT
ejpam-3272	249	9	lcf	lcf	PROPN
ejpam-3272	249	10	)	)	PUNCT
ejpam-3272	249	11	is	be	AUX
ejpam-3272	249	12	r0	r0	NOUN
ejpam-3272	249	13	(	(	PUNCT
ejpam-3272	249	14	r1	r1	PROPN
ejpam-3272	249	15	,	,	PUNCT
ejpam-3272	249	16	regular	regular	ADJ
ejpam-3272	249	17	)	)	PUNCT
ejpam-3272	249	18	and	and	CCONJ
ejpam-3272	249	19	l′cf	l′cf	PROPN
ejpam-3272	249	20	is	be	AUX
ejpam-3272	249	21	a	a	DET
ejpam-3272	249	22	coframe	coframe	NOUN
ejpam-3272	249	23	with	with	ADP
ejpam-3272	249	24	lcf	lcf	PROPN
ejpam-3272	249	25	⊆	⊆	NUM
ejpam-3272	249	26	l′cf	l′cf	NOUN
ejpam-3272	249	27	then	then	ADV
ejpam-3272	249	28	l′	l′	PUNCT
ejpam-3272	249	29	=	=	SYM
ejpam-3272	249	30	(	(	PUNCT
ejpam-3272	249	31	le	le	X
ejpam-3272	249	32	,	,	PUNCT
ejpam-3272	249	33	lfr	lfr	PROPN
ejpam-3272	249	34	,	,	PUNCT
ejpam-3272	249	35	l	l	NOUN
ejpam-3272	249	36	′	′	NUM
ejpam-3272	249	37	cf	cf	NOUN
ejpam-3272	249	38	)	)	PUNCT
ejpam-3272	249	39	is	be	AUX
ejpam-3272	249	40	r0	r0	NOUN
ejpam-3272	249	41	(	(	PUNCT
ejpam-3272	249	42	r1	r1	NOUN
ejpam-3272	249	43	,	,	PUNCT
ejpam-3272	249	44	regular	regular	ADJ
ejpam-3272	249	45	)	)	PUNCT
ejpam-3272	249	46	.	.	PUNCT
ejpam-3272	250	1	dually	dually	PROPN
ejpam-3272	250	2	,	,	PUNCT
ejpam-3272	250	3	if	if	SCONJ
ejpam-3272	250	4	l	l	NOUN
ejpam-3272	250	5	=	=	SYM
ejpam-3272	250	6	(	(	PUNCT
ejpam-3272	250	7	le	le	X
ejpam-3272	250	8	,	,	PUNCT
ejpam-3272	250	9	lfr	lfr	PROPN
ejpam-3272	250	10	,	,	PUNCT
ejpam-3272	250	11	lcf	lcf	PROPN
ejpam-3272	250	12	)	)	PUNCT
ejpam-3272	250	13	is	be	AUX
ejpam-3272	250	14	co	co	ADJ
ejpam-3272	250	15	-	-	NOUN
ejpam-3272	250	16	r0	r0	ADJ
ejpam-3272	250	17	(	(	PUNCT
ejpam-3272	250	18	co	co	NOUN
ejpam-3272	250	19	-	-	NOUN
ejpam-3272	250	20	r1	r1	ADJ
ejpam-3272	250	21	,	,	PUNCT
ejpam-3272	250	22	co	co	NOUN
ejpam-3272	250	23	-	-	NOUN
ejpam-3272	250	24	regular	regular	ADJ
ejpam-3272	250	25	)	)	PUNCT
ejpam-3272	250	26	and	and	CCONJ
ejpam-3272	250	27	l′fr	l′fr	NOUN
ejpam-3272	250	28	is	be	AUX
ejpam-3272	250	29	a	a	DET
ejpam-3272	250	30	frame	frame	NOUN
ejpam-3272	250	31	with	with	ADP
ejpam-3272	250	32	lfr	lfr	PROPN
ejpam-3272	250	33	⊆	⊆	NUM
ejpam-3272	250	34	l′fr	l′fr	NOUN
ejpam-3272	250	35	then	then	ADV
ejpam-3272	250	36	l′	l′	PUNCT
ejpam-3272	250	37	=	=	SYM
ejpam-3272	250	38	(	(	PUNCT
ejpam-3272	250	39	le	le	X
ejpam-3272	250	40	,	,	PUNCT
ejpam-3272	250	41	l	l	NOUN
ejpam-3272	250	42	′	′	NUM
ejpam-3272	250	43	fr	fr	PROPN
ejpam-3272	250	44	,	,	PUNCT
ejpam-3272	250	45	lcf	lcf	PROPN
ejpam-3272	250	46	)	)	PUNCT
ejpam-3272	250	47	is	be	AUX
ejpam-3272	250	48	co	co	ADJ
ejpam-3272	250	49	-	-	NOUN
ejpam-3272	250	50	r0	r0	ADJ
ejpam-3272	250	51	(	(	PUNCT
ejpam-3272	250	52	co	co	NOUN
ejpam-3272	250	53	-	-	NOUN
ejpam-3272	250	54	r1	r1	ADJ
ejpam-3272	250	55	,	,	PUNCT
ejpam-3272	250	56	co	co	NOUN
ejpam-3272	250	57	-	-	NOUN
ejpam-3272	250	58	regular	regular	ADJ
ejpam-3272	250	59	)	)	PUNCT
ejpam-3272	250	60	.	.	PUNCT
ejpam-3272	251	1	proposition	proposition	NOUN
ejpam-3272	251	2	8	8	NUM
ejpam-3272	251	3	.	.	PUNCT
ejpam-3272	252	1	(	(	PUNCT
ejpam-3272	252	2	i	i	NOUN
ejpam-3272	252	3	)	)	PUNCT
ejpam-3272	252	4	a	a	DET
ejpam-3272	252	5	regular	regular	ADJ
ejpam-3272	252	6	diframe	diframe	NOUN
ejpam-3272	252	7	is	be	AUX
ejpam-3272	252	8	r1	r1	VERB
ejpam-3272	252	9	.	.	PUNCT
ejpam-3272	253	1	(	(	PUNCT
ejpam-3272	253	2	ii	ii	NOUN
ejpam-3272	253	3	)	)	PUNCT
ejpam-3272	253	4	a	a	DET
ejpam-3272	253	5	co	co	ADJ
ejpam-3272	253	6	-	-	ADJ
ejpam-3272	253	7	regular	regular	ADJ
ejpam-3272	253	8	diframe	diframe	NOUN
ejpam-3272	253	9	is	be	AUX
ejpam-3272	253	10	co	co	NOUN
ejpam-3272	253	11	-	-	NOUN
ejpam-3272	253	12	r1	r1	ADJ
ejpam-3272	253	13	.	.	PUNCT
ejpam-3272	254	1	e.	e.	PROPN
ejpam-3272	254	2	korkmaz	korkmaz	PROPN
ejpam-3272	254	3	,	,	PUNCT
ejpam-3272	254	4	r.	r.	PROPN
ejpam-3272	254	5	ertürk	ertürk	PROPN
ejpam-3272	254	6	/	/	SYM
ejpam-3272	254	7	eur	eur	PROPN
ejpam-3272	254	8	.	.	PUNCT
ejpam-3272	255	1	j.	j.	PROPN
ejpam-3272	255	2	pure	pure	PROPN
ejpam-3272	255	3	appl	appl	PROPN
ejpam-3272	255	4	.	.	PROPN
ejpam-3272	255	5	math	math	PROPN
ejpam-3272	255	6	,	,	PUNCT
ejpam-3272	255	7	11	11	NUM
ejpam-3272	255	8	(	(	PUNCT
ejpam-3272	255	9	3	3	NUM
ejpam-3272	255	10	)	)	PUNCT
ejpam-3272	255	11	(	(	PUNCT
ejpam-3272	255	12	2018	2018	NUM
ejpam-3272	255	13	)	)	PUNCT
ejpam-3272	255	14	,	,	PUNCT
ejpam-3272	255	15	612	612	NUM
ejpam-3272	255	16	-	-	SYM
ejpam-3272	255	17	627	627	NUM
ejpam-3272	255	18	621	621	NUM
ejpam-3272	255	19	proof	proof	NOUN
ejpam-3272	255	20	.	.	PUNCT
ejpam-3272	256	1	(	(	PUNCT
ejpam-3272	256	2	i	i	NOUN
ejpam-3272	256	3	)	)	PUNCT
ejpam-3272	256	4	given	give	VERB
ejpam-3272	256	5	a	a	DET
ejpam-3272	256	6	∈	∈	NOUN
ejpam-3272	256	7	lfr	lfr	NOUN
ejpam-3272	256	8	we	we	PRON
ejpam-3272	256	9	have	have	VERB
ejpam-3272	256	10	a	a	DET
ejpam-3272	256	11	=	=	PUNCT
ejpam-3272	256	12	∨	∨	NUM
ejpam-3272	256	13	i∈i{ci	i∈i{ci	NOUN
ejpam-3272	256	14	∈	∈	PROPN
ejpam-3272	256	15	lfr	lfr	NOUN
ejpam-3272	256	16	:	:	PUNCT
ejpam-3272	256	17	ci	ci	PROPN
ejpam-3272	257	1	≺fr	≺fr	X
ejpam-3272	257	2	a	a	X
ejpam-3272	257	3	}	}	PUNCT
ejpam-3272	257	4	by	by	ADP
ejpam-3272	257	5	regularity	regularity	NOUN
ejpam-3272	257	6	of	of	ADP
ejpam-3272	257	7	l.	l.	PROPN
ejpam-3272	257	8	further	far	ADV
ejpam-3272	257	9	,	,	PUNCT
ejpam-3272	257	10	if	if	SCONJ
ejpam-3272	257	11	ci	ci	PROPN
ejpam-3272	257	12	≺fr	≺fr	PROPN
ejpam-3272	257	13	a	a	PRON
ejpam-3272	257	14	then	then	ADV
ejpam-3272	257	15	there	there	PRON
ejpam-3272	257	16	exists	exist	VERB
ejpam-3272	257	17	ki	ki	PROPN
ejpam-3272	257	18	∈	∈	PROPN
ejpam-3272	257	19	lcf	lcf	PROPN
ejpam-3272	257	20	such	such	ADJ
ejpam-3272	257	21	that	that	DET
ejpam-3272	257	22	ci	ci	PROPN
ejpam-3272	257	23	≤	≤	X
ejpam-3272	257	24	ki	ki	AUX
ejpam-3272	257	25	≤	≤	PROPN
ejpam-3272	257	26	a.	a.	NOUN
ejpam-3272	257	27	setting	set	VERB
ejpam-3272	257	28	j	j	PROPN
ejpam-3272	257	29	=	=	PRON
ejpam-3272	257	30	{	{	PUNCT
ejpam-3272	257	31	j	j	NOUN
ejpam-3272	257	32	}	}	PUNCT
ejpam-3272	257	33	and	and	CCONJ
ejpam-3272	257	34	cji	cji	NOUN
ejpam-3272	257	35	=	=	SYM
ejpam-3272	257	36	ci	ci	PROPN
ejpam-3272	257	37	,	,	PUNCT
ejpam-3272	257	38	for	for	ADP
ejpam-3272	257	39	all	all	PRON
ejpam-3272	257	40	i	i	PRON
ejpam-3272	257	41	∈	∈	PROPN
ejpam-3272	258	1	i	i	PRON
ejpam-3272	258	2	,	,	PUNCT
ejpam-3272	258	3	we	we	PRON
ejpam-3272	258	4	obtain	obtain	VERB
ejpam-3272	258	5	a	a	DET
ejpam-3272	258	6	=	=	SYM
ejpam-3272	258	7	∨	∨	NOUN
ejpam-3272	258	8	i∈i	i∈i	ADJ
ejpam-3272	258	9	∧	∧	PROPN
ejpam-3272	258	10	j∈j	j∈j	NOUN
ejpam-3272	258	11	cji	cji	NOUN
ejpam-3272	258	12	≤	≤	NOUN
ejpam-3272	258	13	∨	∨	NUM
ejpam-3272	258	14	i∈i	i∈i	ADJ
ejpam-3272	258	15	∧	∧	PROPN
ejpam-3272	258	16	j∈j	j∈j	NOUN
ejpam-3272	259	1	[	[	X
ejpam-3272	259	2	cji	cji	NOUN
ejpam-3272	259	3	]	]	PUNCT
ejpam-3272	259	4	≤	≤	NUM
ejpam-3272	259	5	∨	∨	NUM
ejpam-3272	259	6	i∈i	i∈i	ADJ
ejpam-3272	259	7	∧	∧	PROPN
ejpam-3272	259	8	j∈j	j∈j	NOUN
ejpam-3272	259	9	kji	kji	ADJ
ejpam-3272	259	10	≤	≤	NOUN
ejpam-3272	259	11	a	a	DET
ejpam-3272	259	12	thus	thus	ADV
ejpam-3272	259	13	a	a	DET
ejpam-3272	259	14	=	=	SYM
ejpam-3272	259	15	∨	∨	NOUN
ejpam-3272	259	16	i∈i	i∈i	ADJ
ejpam-3272	259	17	∧	∧	PROPN
ejpam-3272	259	18	j∈j	j∈j	NOUN
ejpam-3272	260	1	c	c	PROPN
ejpam-3272	260	2	j	j	PROPN
ejpam-3272	261	1	i	i	PRON
ejpam-3272	261	2	=	=	SYM
ejpam-3272	261	3	∨	∨	NUM
ejpam-3272	261	4	i∈i	i∈i	ADJ
ejpam-3272	261	5	∧	∧	PROPN
ejpam-3272	261	6	j∈j	j∈j	NOUN
ejpam-3272	261	7	[	[	X
ejpam-3272	261	8	cji	cji	NOUN
ejpam-3272	261	9	]	]	PUNCT
ejpam-3272	261	10	,	,	PUNCT
ejpam-3272	261	11	showing	show	VERB
ejpam-3272	261	12	that	that	SCONJ
ejpam-3272	261	13	l	l	NOUN
ejpam-3272	261	14	is	be	AUX
ejpam-3272	261	15	r1	r1	VERB
ejpam-3272	261	16	.	.	PUNCT
ejpam-3272	262	1	proposition	proposition	NOUN
ejpam-3272	262	2	9	9	NUM
ejpam-3272	262	3	.	.	PUNCT
ejpam-3272	263	1	(	(	PUNCT
ejpam-3272	263	2	i	i	NOUN
ejpam-3272	263	3	)	)	PUNCT
ejpam-3272	263	4	every	every	DET
ejpam-3272	263	5	regular	regular	ADJ
ejpam-3272	263	6	co	co	NOUN
ejpam-3272	263	7	-	-	ADJ
ejpam-3272	263	8	r0	r0	ADJ
ejpam-3272	263	9	diframe	diframe	NOUN
ejpam-3272	263	10	is	be	AUX
ejpam-3272	263	11	co	co	NOUN
ejpam-3272	263	12	-	-	NOUN
ejpam-3272	263	13	r1	r1	ADJ
ejpam-3272	263	14	.	.	PUNCT
ejpam-3272	264	1	(	(	PUNCT
ejpam-3272	264	2	ii	ii	NOUN
ejpam-3272	264	3	)	)	PUNCT
ejpam-3272	264	4	every	every	DET
ejpam-3272	264	5	co	co	NOUN
ejpam-3272	264	6	-regular	-regular	ADJ
ejpam-3272	264	7	r0	r0	NOUN
ejpam-3272	264	8	diframe	diframe	NOUN
ejpam-3272	264	9	is	be	AUX
ejpam-3272	264	10	r1	r1	VERB
ejpam-3272	264	11	.	.	PUNCT
ejpam-3272	265	1	proof	proof	NOUN
ejpam-3272	265	2	.	.	PUNCT
ejpam-3272	266	1	(	(	PUNCT
ejpam-3272	266	2	i	i	NOUN
ejpam-3272	266	3	)	)	PUNCT
ejpam-3272	266	4	let	let	VERB
ejpam-3272	266	5	l	l	NOUN
ejpam-3272	266	6	be	be	AUX
ejpam-3272	266	7	regular	regular	ADJ
ejpam-3272	266	8	,	,	PUNCT
ejpam-3272	266	9	co	co	NOUN
ejpam-3272	266	10	-	-	NOUN
ejpam-3272	266	11	r0	r0	VERB
ejpam-3272	266	12	and	and	CCONJ
ejpam-3272	266	13	let	let	VERB
ejpam-3272	266	14	k	k	PROPN
ejpam-3272	266	15	∈	∈	PROPN
ejpam-3272	266	16	lcf	lcf	PROPN
ejpam-3272	266	17	.	.	PUNCT
ejpam-3272	267	1	first	first	ADV
ejpam-3272	267	2	we	we	PRON
ejpam-3272	267	3	have	have	AUX
ejpam-3272	267	4	ai	ai	PROPN
ejpam-3272	267	5	∈	∈	PROPN
ejpam-3272	267	6	lfr	lfr	NOUN
ejpam-3272	267	7	such	such	ADJ
ejpam-3272	267	8	that	that	SCONJ
ejpam-3272	267	9	k	k	PROPN
ejpam-3272	267	10	=	=	SYM
ejpam-3272	267	11	∧	∧	PROPN
ejpam-3272	267	12	i∈i	i∈i	ADJ
ejpam-3272	267	13	ai	ai	VERB
ejpam-3272	267	14	.	.	PUNCT
ejpam-3272	268	1	now	now	ADV
ejpam-3272	268	2	,	,	PUNCT
ejpam-3272	268	3	by	by	ADP
ejpam-3272	268	4	regularity	regularity	NOUN
ejpam-3272	268	5	of	of	ADP
ejpam-3272	268	6	l	l	NOUN
ejpam-3272	268	7	,	,	PUNCT
ejpam-3272	268	8	ai	ai	VERB
ejpam-3272	268	9	=	=	PUNCT
ejpam-3272	268	10	∨	∨	PROPN
ejpam-3272	268	11	j∈j	j∈j	NOUN
ejpam-3272	268	12	{	{	PUNCT
ejpam-3272	268	13	bij	bij	PROPN
ejpam-3272	268	14	∈	∈	PROPN
ejpam-3272	268	15	lfr	lfr	NOUN
ejpam-3272	268	16	:	:	PUNCT
ejpam-3272	268	17	∃fij	∃fij	PROPN
ejpam-3272	268	18	∈	∈	PROPN
ejpam-3272	268	19	lcf	lcf	PROPN
ejpam-3272	268	20	;	;	PUNCT
ejpam-3272	268	21	bij	bij	VERB
ejpam-3272	268	22	≤	≤	PROPN
ejpam-3272	268	23	fij	fij	PROPN
ejpam-3272	268	24	≤	≤	PROPN
ejpam-3272	268	25	ai	ai	VERB
ejpam-3272	268	26	}	}	PUNCT
ejpam-3272	268	27	for	for	ADP
ejpam-3272	268	28	all	all	DET
ejpam-3272	268	29	i	i	PRON
ejpam-3272	268	30	∈	∈	PROPN
ejpam-3272	268	31	i.	i.	NOUN
ejpam-3272	269	1	but	but	CCONJ
ejpam-3272	269	2	then	then	ADV
ejpam-3272	269	3	,	,	PUNCT
ejpam-3272	269	4	k	k	PROPN
ejpam-3272	269	5	≤	≤	PROPN
ejpam-3272	269	6	∧	∧	PROPN
ejpam-3272	269	7	i∈i	i∈i	ADJ
ejpam-3272	269	8	∨	∨	NOUN
ejpam-3272	269	9	j∈j	j∈j	NOUN
ejpam-3272	269	10	bij	bij	NOUN
ejpam-3272	269	11	≤	≤	X
ejpam-3272	269	12	∧	∧	PROPN
ejpam-3272	269	13	i∈i	i∈i	ADJ
ejpam-3272	269	14	∨	∨	PROPN
ejpam-3272	269	15	j∈j	j∈j	NOUN
ejpam-3272	269	16	]	]	X
ejpam-3272	269	17	fij	fij	PROPN
ejpam-3272	270	1	[	[	X
ejpam-3272	270	2	≤	≤	PROPN
ejpam-3272	270	3	∧	∧	PROPN
ejpam-3272	270	4	i∈i	i∈i	ADJ
ejpam-3272	270	5	∨	∨	PROPN
ejpam-3272	270	6	j∈j	j∈j	PROPN
ejpam-3272	270	7	fij	fij	PROPN
ejpam-3272	270	8	≤	≤	PROPN
ejpam-3272	270	9	∧	∧	PROPN
ejpam-3272	270	10	i∈i	i∈i	ADJ
ejpam-3272	270	11	ai	ai	VERB
ejpam-3272	270	12	≤	≤	PROPN
ejpam-3272	270	13	k	k	PROPN
ejpam-3272	271	1	and	and	CCONJ
ejpam-3272	271	2	hence	hence	ADV
ejpam-3272	271	3	k	k	PROPN
ejpam-3272	271	4	=	=	SYM
ejpam-3272	271	5	∧	∧	PROPN
ejpam-3272	271	6	i∈i	i∈i	ADJ
ejpam-3272	271	7	∨	∨	PROPN
ejpam-3272	271	8	j∈j	j∈j	PROPN
ejpam-3272	271	9	fij	fij	PROPN
ejpam-3272	271	10	=	=	PROPN
ejpam-3272	271	11	∧	∧	PROPN
ejpam-3272	271	12	i∈i	i∈i	ADJ
ejpam-3272	271	13	∨	∨	PROPN
ejpam-3272	271	14	j∈j	j∈j	NOUN
ejpam-3272	271	15	]	]	X
ejpam-3272	271	16	fij	fij	PROPN
ejpam-3272	272	1	[	[	X
ejpam-3272	272	2	.	.	PUNCT
ejpam-3272	273	1	therefore	therefore	ADV
ejpam-3272	273	2	,	,	PUNCT
ejpam-3272	273	3	l	l	NOUN
ejpam-3272	273	4	is	be	AUX
ejpam-3272	273	5	co	co	NOUN
ejpam-3272	273	6	-	-	NOUN
ejpam-3272	273	7	r1	r1	ADJ
ejpam-3272	273	8	.	.	PUNCT
ejpam-3272	274	1	(	(	PUNCT
ejpam-3272	274	2	ii	ii	NOUN
ejpam-3272	274	3	)	)	PUNCT
ejpam-3272	274	4	dual	dual	ADV
ejpam-3272	274	5	to	to	ADP
ejpam-3272	274	6	(	(	PUNCT
ejpam-3272	274	7	i	i	NOUN
ejpam-3272	274	8	)	)	PUNCT
ejpam-3272	274	9	,	,	PUNCT
ejpam-3272	274	10	so	so	ADV
ejpam-3272	274	11	we	we	PRON
ejpam-3272	274	12	omit	omit	VERB
ejpam-3272	274	13	the	the	DET
ejpam-3272	274	14	details	detail	NOUN
ejpam-3272	274	15	.	.	PUNCT
ejpam-3272	275	1	note	note	VERB
ejpam-3272	275	2	that	that	SCONJ
ejpam-3272	275	3	complete	complete	ADJ
ejpam-3272	275	4	regularity	regularity	NOUN
ejpam-3272	275	5	also	also	ADV
ejpam-3272	275	6	has	have	VERB
ejpam-3272	275	7	a	a	DET
ejpam-3272	275	8	counterpart	counterpart	NOUN
ejpam-3272	275	9	in	in	ADP
ejpam-3272	275	10	the	the	DET
ejpam-3272	275	11	theory	theory	NOUN
ejpam-3272	275	12	of	of	ADP
ejpam-3272	275	13	diframes	diframe	NOUN
ejpam-3272	275	14	.	.	PUNCT
ejpam-3272	276	1	but	but	CCONJ
ejpam-3272	276	2	first	first	ADV
ejpam-3272	276	3	we	we	PRON
ejpam-3272	276	4	need	need	VERB
ejpam-3272	276	5	the	the	DET
ejpam-3272	276	6	following	follow	VERB
ejpam-3272	276	7	binary	binary	ADJ
ejpam-3272	276	8	relations	relation	NOUN
ejpam-3272	276	9	on	on	ADP
ejpam-3272	276	10	le	le	PROPN
ejpam-3272	276	11	.	.	PUNCT
ejpam-3272	276	12	remark	remark	PROPN
ejpam-3272	276	13	4	4	NUM
ejpam-3272	276	14	.	.	PUNCT
ejpam-3272	277	1	let	let	VERB
ejpam-3272	277	2	d	d	NOUN
ejpam-3272	277	3	=	=	PUNCT
ejpam-3272	277	4	{	{	PUNCT
ejpam-3272	277	5	k/2n	k/2n	X
ejpam-3272	277	6	:	:	PUNCT
ejpam-3272	278	1	k	k	X
ejpam-3272	278	2	,	,	PUNCT
ejpam-3272	278	3	n	n	PROPN
ejpam-3272	278	4	∈	∈	PROPN
ejpam-3272	278	5	n	n	CCONJ
ejpam-3272	278	6	,	,	PUNCT
ejpam-3272	278	7	k	k	PROPN
ejpam-3272	278	8	=	=	PUNCT
ejpam-3272	278	9	0	0	PROPN
ejpam-3272	278	10	,	,	PUNCT
ejpam-3272	278	11	.	.	PUNCT
ejpam-3272	278	12	.	.	PUNCT
ejpam-3272	278	13	.	.	PUNCT
ejpam-3272	279	1	2n	2n	NUM
ejpam-3272	279	2	}	}	PUNCT
ejpam-3272	279	3	be	be	VERB
ejpam-3272	279	4	the	the	DET
ejpam-3272	279	5	set	set	NOUN
ejpam-3272	279	6	of	of	ADP
ejpam-3272	279	7	dyadic	dyadic	ADJ
ejpam-3272	279	8	rationals	rational	NOUN
ejpam-3272	279	9	.	.	PUNCT
ejpam-3272	280	1	we	we	PRON
ejpam-3272	280	2	can	can	AUX
ejpam-3272	280	3	define	define	VERB
ejpam-3272	280	4	a	a	DET
ejpam-3272	280	5	binary	binary	ADJ
ejpam-3272	280	6	relation	relation	NOUN
ejpam-3272	280	7	on	on	ADP
ejpam-3272	280	8	le	le	X
ejpam-3272	280	9	by	by	ADP
ejpam-3272	280	10	setting	set	VERB
ejpam-3272	280	11	a	a	DET
ejpam-3272	280	12	≺≺fr	≺≺fr	PROPN
ejpam-3272	280	13	b	b	PROPN
ejpam-3272	280	14	iff	iff	PROPN
ejpam-3272	280	15	a	a	PRON
ejpam-3272	280	16	,	,	PUNCT
ejpam-3272	280	17	b	b	PROPN
ejpam-3272	280	18	∈	∈	PROPN
ejpam-3272	280	19	lfr	lfr	NOUN
ejpam-3272	280	20	and	and	CCONJ
ejpam-3272	280	21	there	there	PRON
ejpam-3272	280	22	exists	exist	VERB
ejpam-3272	280	23	aq	aq	PROPN
ejpam-3272	280	24	∈	∈	PROPN
ejpam-3272	280	25	lfr	lfr	X
ejpam-3272	280	26	(	(	PUNCT
ejpam-3272	280	27	q	q	NOUN
ejpam-3272	280	28	∈	∈	PROPN
ejpam-3272	280	29	d	d	NOUN
ejpam-3272	280	30	)	)	PUNCT
ejpam-3272	280	31	satisfying	satisfy	VERB
ejpam-3272	280	32	a0	a0	NOUN
ejpam-3272	280	33	=	=	PUNCT
ejpam-3272	280	34	a	a	PROPN
ejpam-3272	280	35	,	,	PUNCT
ejpam-3272	280	36	a1	a1	NOUN
ejpam-3272	280	37	=	=	SYM
ejpam-3272	280	38	b	b	PROPN
ejpam-3272	280	39	,	,	PUNCT
ejpam-3272	280	40	and	and	CCONJ
ejpam-3272	280	41	aq	aq	PROPN
ejpam-3272	280	42	≺fr	≺fr	PUNCT
ejpam-3272	280	43	ar	ar	NOUN
ejpam-3272	280	44	for	for	ADP
ejpam-3272	280	45	q	q	X
ejpam-3272	280	46	<	<	X
ejpam-3272	280	47	r.	r.	PROPN
ejpam-3272	280	48	if	if	SCONJ
ejpam-3272	280	49	a	a	DET
ejpam-3272	280	50	≺≺fr	≺≺fr	PROPN
ejpam-3272	280	51	b	b	PROPN
ejpam-3272	280	52	then	then	ADV
ejpam-3272	280	53	we	we	PRON
ejpam-3272	280	54	say	say	VERB
ejpam-3272	280	55	that	that	SCONJ
ejpam-3272	280	56	a	a	PRON
ejpam-3272	280	57	is	be	AUX
ejpam-3272	280	58	completely	completely	ADV
ejpam-3272	280	59	fr	fr	ADJ
ejpam-3272	280	60	-	-	PUNCT
ejpam-3272	280	61	below	below	NOUN
ejpam-3272	280	62	b.	b.	PROPN
ejpam-3272	280	63	similarly	similarly	ADV
ejpam-3272	280	64	,	,	PUNCT
ejpam-3272	280	65	the	the	DET
ejpam-3272	280	66	dual	dual	ADJ
ejpam-3272	280	67	relation	relation	NOUN
ejpam-3272	280	68	can	can	AUX
ejpam-3272	280	69	be	be	AUX
ejpam-3272	280	70	defined	define	VERB
ejpam-3272	280	71	by	by	ADP
ejpam-3272	280	72	setting	set	VERB
ejpam-3272	280	73	k	k	PROPN
ejpam-3272	280	74	≺≺cf	≺≺cf	PROPN
ejpam-3272	280	75	f	f	PROPN
ejpam-3272	280	76	iff	iff	PROPN
ejpam-3272	280	77	k	k	PROPN
ejpam-3272	280	78	,	,	PUNCT
ejpam-3272	280	79	f	f	PROPN
ejpam-3272	280	80	∈	∈	PROPN
ejpam-3272	280	81	lcf	lcf	PROPN
ejpam-3272	280	82	and	and	CCONJ
ejpam-3272	280	83	there	there	ADV
ejpam-3272	280	84	exists	exist	VERB
ejpam-3272	280	85	kq	kq	PROPN
ejpam-3272	280	86	∈	∈	PROPN
ejpam-3272	280	87	lcf	lcf	PROPN
ejpam-3272	280	88	(	(	PUNCT
ejpam-3272	280	89	q	q	NOUN
ejpam-3272	280	90	∈	∈	PROPN
ejpam-3272	280	91	d	d	NOUN
ejpam-3272	280	92	)	)	PUNCT
ejpam-3272	280	93	satisfying	satisfy	VERB
ejpam-3272	280	94	k0	k0	PROPN
ejpam-3272	280	95	=	=	SYM
ejpam-3272	280	96	k	k	PROPN
ejpam-3272	280	97	,	,	PUNCT
ejpam-3272	280	98	k1	k1	NOUN
ejpam-3272	280	99	=	=	SYM
ejpam-3272	280	100	f	f	PROPN
ejpam-3272	280	101	,	,	PUNCT
ejpam-3272	280	102	and	and	CCONJ
ejpam-3272	280	103	kq	kq	PROPN
ejpam-3272	280	104	≺cf	≺cf	VERB
ejpam-3272	280	105	kr	kr	PROPN
ejpam-3272	280	106	for	for	ADP
ejpam-3272	280	107	q	q	PROPN
ejpam-3272	280	108	<	<	X
ejpam-3272	280	109	r.	r.	PROPN
ejpam-3272	280	110	if	if	SCONJ
ejpam-3272	280	111	k	k	PROPN
ejpam-3272	280	112	≺≺cf	≺≺cf	PROPN
ejpam-3272	280	113	f	f	PROPN
ejpam-3272	280	114	then	then	ADV
ejpam-3272	280	115	we	we	PRON
ejpam-3272	280	116	say	say	VERB
ejpam-3272	280	117	that	that	SCONJ
ejpam-3272	280	118	k	k	PROPN
ejpam-3272	280	119	is	be	AUX
ejpam-3272	280	120	completely	completely	ADV
ejpam-3272	280	121	cf	cf	NOUN
ejpam-3272	280	122	-	-	PUNCT
ejpam-3272	280	123	below	below	NOUN
ejpam-3272	280	124	f.	f.	PROPN
ejpam-3272	281	1	the	the	DET
ejpam-3272	281	2	relations	relation	NOUN
ejpam-3272	281	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	281	4	and	and	CCONJ
ejpam-3272	281	5	≺≺cf	≺≺cf	PROPN
ejpam-3272	281	6	have	have	VERB
ejpam-3272	281	7	similar	similar	ADJ
ejpam-3272	281	8	properties	property	NOUN
ejpam-3272	281	9	like	like	ADP
ejpam-3272	281	10	those	those	PRON
ejpam-3272	281	11	in	in	ADP
ejpam-3272	281	12	proposition	proposition	NOUN
ejpam-3272	281	13	5	5	NUM
ejpam-3272	281	14	.	.	PUNCT
ejpam-3272	281	15	proposition	proposition	NOUN
ejpam-3272	281	16	10	10	NUM
ejpam-3272	281	17	.	.	PUNCT
ejpam-3272	282	1	the	the	DET
ejpam-3272	282	2	relations	relation	NOUN
ejpam-3272	282	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	282	4	and	and	CCONJ
ejpam-3272	282	5	≺≺cf	≺≺cf	PROPN
ejpam-3272	282	6	on	on	ADP
ejpam-3272	282	7	le	le	PART
ejpam-3272	282	8	satisfy	satisfy	VERB
ejpam-3272	282	9	the	the	DET
ejpam-3272	282	10	following	follow	VERB
ejpam-3272	282	11	properties	property	NOUN
ejpam-3272	282	12	:	:	PUNCT
ejpam-3272	282	13	(	(	PUNCT
ejpam-3272	282	14	i	i	NOUN
ejpam-3272	282	15	)	)	PUNCT
ejpam-3272	282	16	0	0	PUNCT
ejpam-3272	283	1	≺≺fr	≺≺fr	PROPN
ejpam-3272	283	2	a	a	DET
ejpam-3272	283	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	283	4	1	1	NUM
ejpam-3272	283	5	for	for	ADP
ejpam-3272	283	6	all	all	DET
ejpam-3272	283	7	a	a	DET
ejpam-3272	283	8	∈	∈	PROPN
ejpam-3272	283	9	lfr	lfr	NOUN
ejpam-3272	283	10	,	,	PUNCT
ejpam-3272	283	11	and	and	CCONJ
ejpam-3272	283	12	0	0	NUM
ejpam-3272	283	13	≺≺cf	≺≺cf	PROPN
ejpam-3272	283	14	k	k	PROPN
ejpam-3272	283	15	≺≺fr	≺≺fr	PROPN
ejpam-3272	283	16	1	1	NUM
ejpam-3272	283	17	for	for	ADP
ejpam-3272	283	18	all	all	DET
ejpam-3272	283	19	k	k	PROPN
ejpam-3272	283	20	∈	∈	PROPN
ejpam-3272	283	21	lcf	lcf	PROPN
ejpam-3272	283	22	.	.	PUNCT
ejpam-3272	284	1	e.	e.	PROPN
ejpam-3272	284	2	korkmaz	korkmaz	PROPN
ejpam-3272	284	3	,	,	PUNCT
ejpam-3272	284	4	r.	r.	PROPN
ejpam-3272	284	5	ertürk	ertürk	PROPN
ejpam-3272	284	6	/	/	SYM
ejpam-3272	284	7	eur	eur	PROPN
ejpam-3272	284	8	.	.	PUNCT
ejpam-3272	285	1	j.	j.	PROPN
ejpam-3272	285	2	pure	pure	PROPN
ejpam-3272	285	3	appl	appl	PROPN
ejpam-3272	285	4	.	.	PROPN
ejpam-3272	285	5	math	math	PROPN
ejpam-3272	285	6	,	,	PUNCT
ejpam-3272	285	7	11	11	NUM
ejpam-3272	285	8	(	(	PUNCT
ejpam-3272	285	9	3	3	NUM
ejpam-3272	285	10	)	)	PUNCT
ejpam-3272	285	11	(	(	PUNCT
ejpam-3272	285	12	2018	2018	NUM
ejpam-3272	285	13	)	)	PUNCT
ejpam-3272	285	14	,	,	PUNCT
ejpam-3272	285	15	612	612	NUM
ejpam-3272	285	16	-	-	SYM
ejpam-3272	285	17	627	627	NUM
ejpam-3272	285	18	622	622	NUM
ejpam-3272	285	19	(	(	PUNCT
ejpam-3272	285	20	ii	ii	NOUN
ejpam-3272	285	21	)	)	PUNCT
ejpam-3272	285	22	a	a	DET
ejpam-3272	285	23	≺≺fr	≺≺fr	PROPN
ejpam-3272	285	24	b	b	PROPN
ejpam-3272	285	25	implies	imply	VERB
ejpam-3272	285	26	a	a	DET
ejpam-3272	285	27	≤	≤	PROPN
ejpam-3272	285	28	b.	b.	NOUN
ejpam-3272	286	1	moreover	moreover	ADV
ejpam-3272	286	2	,	,	PUNCT
ejpam-3272	286	3	f	f	PROPN
ejpam-3272	286	4	≺≺cf	≺≺cf	PROPN
ejpam-3272	286	5	k	k	PROPN
ejpam-3272	286	6	implies	imply	VERB
ejpam-3272	286	7	f	f	PROPN
ejpam-3272	286	8	≤	≤	PROPN
ejpam-3272	286	9	k.	k.	PROPN
ejpam-3272	287	1	(	(	PUNCT
ejpam-3272	287	2	iii	iii	X
ejpam-3272	287	3	)	)	PUNCT
ejpam-3272	287	4	if	if	SCONJ
ejpam-3272	287	5	a	a	DET
ejpam-3272	287	6	≤	≤	PROPN
ejpam-3272	287	7	b	b	PROPN
ejpam-3272	287	8	≺≺fr	≺≺fr	PROPN
ejpam-3272	287	9	c	c	PROPN
ejpam-3272	287	10	≤	≤	PROPN
ejpam-3272	288	1	d	d	NOUN
ejpam-3272	288	2	then	then	ADV
ejpam-3272	288	3	a	a	DET
ejpam-3272	288	4	≺≺fr	≺≺fr	PROPN
ejpam-3272	288	5	d	d	PROPN
ejpam-3272	288	6	,	,	PUNCT
ejpam-3272	288	7	and	and	CCONJ
ejpam-3272	288	8	if	if	SCONJ
ejpam-3272	288	9	f	f	PROPN
ejpam-3272	288	10	≤	≤	X
ejpam-3272	288	11	c	c	PROPN
ejpam-3272	288	12	≺≺cf	≺≺cf	NOUN
ejpam-3272	288	13	d	d	X
ejpam-3272	288	14	≤	≤	NOUN
ejpam-3272	289	1	k	k	NOUN
ejpam-3272	289	2	then	then	ADV
ejpam-3272	289	3	f	f	PROPN
ejpam-3272	289	4	≺≺cf	≺≺cf	PROPN
ejpam-3272	289	5	k.	k.	PROPN
ejpam-3272	289	6	(	(	PUNCT
ejpam-3272	289	7	iv	iv	X
ejpam-3272	289	8	)	)	PUNCT
ejpam-3272	289	9	if	if	SCONJ
ejpam-3272	289	10	ai	ai	ADP
ejpam-3272	289	11	≺≺fr	≺≺fr	PROPN
ejpam-3272	289	12	bi	bi	NOUN
ejpam-3272	289	13	for	for	ADP
ejpam-3272	289	14	i	i	PROPN
ejpam-3272	289	15	=	=	NOUN
ejpam-3272	289	16	1	1	NUM
ejpam-3272	289	17	,	,	PUNCT
ejpam-3272	289	18	2	2	NUM
ejpam-3272	289	19	then	then	ADV
ejpam-3272	289	20	a1∨a2	a1∨a2	PROPN
ejpam-3272	289	21	≺≺fr	≺≺fr	PROPN
ejpam-3272	289	22	b1∨	b1∨	VERB
ejpam-3272	289	23	b2	b2	PROPN
ejpam-3272	289	24	and	and	CCONJ
ejpam-3272	289	25	a1∧a2	a1∧a2	PROPN
ejpam-3272	289	26	≺≺fr	≺≺fr	PROPN
ejpam-3272	289	27	b1∧	b1∧	NOUN
ejpam-3272	289	28	b2	b2	NOUN
ejpam-3272	289	29	.	.	PUNCT
ejpam-3272	290	1	similarly	similarly	ADV
ejpam-3272	290	2	,	,	PUNCT
ejpam-3272	290	3	if	if	SCONJ
ejpam-3272	290	4	fi	fi	ADJ
ejpam-3272	290	5	≺≺cf	≺≺cf	PROPN
ejpam-3272	290	6	ki	ki	PROPN
ejpam-3272	290	7	for	for	ADP
ejpam-3272	290	8	i	i	PROPN
ejpam-3272	290	9	=	=	NOUN
ejpam-3272	290	10	1	1	NUM
ejpam-3272	290	11	,	,	PUNCT
ejpam-3272	290	12	2	2	NUM
ejpam-3272	290	13	then	then	ADV
ejpam-3272	290	14	f1	f1	PROPN
ejpam-3272	290	15	∨	∨	NUM
ejpam-3272	290	16	f2	f2	PROPN
ejpam-3272	290	17	≺≺cf	≺≺cf	PROPN
ejpam-3272	290	18	k1	k1	PROPN
ejpam-3272	290	19	∨	∨	NUM
ejpam-3272	290	20	k2	k2	PROPN
ejpam-3272	290	21	and	and	CCONJ
ejpam-3272	290	22	f1	f1	PROPN
ejpam-3272	290	23	∧	∧	PROPN
ejpam-3272	290	24	f2	f2	PROPN
ejpam-3272	290	25	≺≺cf	≺≺cf	PROPN
ejpam-3272	290	26	k1	k1	NOUN
ejpam-3272	290	27	∧	∧	PROPN
ejpam-3272	290	28	k2	k2	PROPN
ejpam-3272	290	29	.	.	PUNCT
ejpam-3272	291	1	(	(	PUNCT
ejpam-3272	291	2	v	v	NOUN
ejpam-3272	291	3	)	)	PUNCT
ejpam-3272	292	1	if	if	SCONJ
ejpam-3272	292	2	a	a	DET
ejpam-3272	292	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	292	4	b	b	NOUN
ejpam-3272	292	5	then	then	ADV
ejpam-3272	292	6	there	there	PRON
ejpam-3272	292	7	exists	exist	VERB
ejpam-3272	292	8	a	a	DET
ejpam-3272	292	9	c	c	PROPN
ejpam-3272	292	10	∈	∈	PROPN
ejpam-3272	292	11	lfr	lfr	PROPN
ejpam-3272	292	12	with	with	ADP
ejpam-3272	292	13	a	a	DET
ejpam-3272	292	14	≺≺fr	≺≺fr	PROPN
ejpam-3272	292	15	c	c	PROPN
ejpam-3272	292	16	≺≺fr	≺≺fr	PROPN
ejpam-3272	292	17	b	b	X
ejpam-3272	292	18	,	,	PUNCT
ejpam-3272	292	19	that	that	ADV
ejpam-3272	292	20	is	is	ADV
ejpam-3272	292	21	,	,	PUNCT
ejpam-3272	292	22	the	the	DET
ejpam-3272	292	23	relation	relation	NOUN
ejpam-3272	292	24	≺≺fr	≺≺fr	PROPN
ejpam-3272	292	25	is	be	AUX
ejpam-3272	292	26	interpolative	interpolative	ADJ
ejpam-3272	292	27	.	.	PUNCT
ejpam-3272	293	1	moreover	moreover	ADV
ejpam-3272	293	2	,	,	PUNCT
ejpam-3272	293	3	it	it	PRON
ejpam-3272	293	4	is	be	AUX
ejpam-3272	293	5	the	the	DET
ejpam-3272	293	6	largest	large	ADJ
ejpam-3272	293	7	interpolative	interpolative	ADJ
ejpam-3272	293	8	relation	relation	NOUN
ejpam-3272	293	9	contained	contain	VERB
ejpam-3272	293	10	in	in	ADP
ejpam-3272	293	11	≺fr	≺fr	PROPN
ejpam-3272	293	12	.	.	PUNCT
ejpam-3272	294	1	similarly	similarly	ADV
ejpam-3272	294	2	,	,	PUNCT
ejpam-3272	294	3	the	the	DET
ejpam-3272	294	4	relation	relation	NOUN
ejpam-3272	294	5	≺≺cf	≺≺cf	PROPN
ejpam-3272	294	6	is	be	AUX
ejpam-3272	294	7	interpolative	interpolative	ADJ
ejpam-3272	294	8	and	and	CCONJ
ejpam-3272	294	9	it	it	PRON
ejpam-3272	294	10	is	be	AUX
ejpam-3272	294	11	the	the	DET
ejpam-3272	294	12	largest	large	ADJ
ejpam-3272	294	13	interpolative	interpolative	ADJ
ejpam-3272	294	14	relation	relation	NOUN
ejpam-3272	294	15	contained	contain	VERB
ejpam-3272	294	16	in	in	ADP
ejpam-3272	294	17	≺cf	≺cf	PROPN
ejpam-3272	294	18	.	.	PUNCT
ejpam-3272	295	1	proof	proof	NOUN
ejpam-3272	295	2	.	.	PUNCT
ejpam-3272	296	1	the	the	DET
ejpam-3272	296	2	facts	fact	NOUN
ejpam-3272	296	3	(	(	PUNCT
ejpam-3272	296	4	i)−	i)−	PROPN
ejpam-3272	296	5	(	(	PUNCT
ejpam-3272	296	6	iv	iv	X
ejpam-3272	296	7	)	)	PUNCT
ejpam-3272	296	8	are	be	AUX
ejpam-3272	296	9	immediate	immediate	ADJ
ejpam-3272	296	10	consequences	consequence	NOUN
ejpam-3272	296	11	of	of	ADP
ejpam-3272	296	12	the	the	DET
ejpam-3272	296	13	definitions	definition	NOUN
ejpam-3272	296	14	.	.	PUNCT
ejpam-3272	297	1	(	(	PUNCT
ejpam-3272	297	2	v	v	NOUN
ejpam-3272	297	3	)	)	PUNCT
ejpam-3272	297	4	if	if	SCONJ
ejpam-3272	297	5	a	a	DET
ejpam-3272	297	6	≺≺fr	≺≺fr	PROPN
ejpam-3272	297	7	b	b	PROPN
ejpam-3272	297	8	then	then	ADV
ejpam-3272	297	9	we	we	PRON
ejpam-3272	297	10	have	have	VERB
ejpam-3272	297	11	aq	aq	VERB
ejpam-3272	297	12	∈	∈	PROPN
ejpam-3272	297	13	lfr	lfr	X
ejpam-3272	297	14	(	(	PUNCT
ejpam-3272	297	15	q	q	NOUN
ejpam-3272	297	16	∈	∈	PROPN
ejpam-3272	297	17	d	d	NOUN
ejpam-3272	297	18	)	)	PUNCT
ejpam-3272	297	19	with	with	ADP
ejpam-3272	297	20	a0	a0	PROPN
ejpam-3272	297	21	=	=	PUNCT
ejpam-3272	297	22	a	a	PROPN
ejpam-3272	297	23	,	,	PUNCT
ejpam-3272	297	24	a1	a1	NOUN
ejpam-3272	297	25	=	=	SYM
ejpam-3272	297	26	b	b	PROPN
ejpam-3272	297	27	and	and	CCONJ
ejpam-3272	297	28	aq	aq	PROPN
ejpam-3272	297	29	≺fr	≺fr	PROPN
ejpam-3272	297	30	ar	ar	PROPN
ejpam-3272	297	31	for	for	ADP
ejpam-3272	297	32	q	q	PROPN
ejpam-3272	297	33	<	<	X
ejpam-3272	297	34	r.	r.	PROPN
ejpam-3272	297	35	setting	set	VERB
ejpam-3272	297	36	c	c	NOUN
ejpam-3272	298	1	=	=	SYM
ejpam-3272	298	2	a1/2	a1/2	PROPN
ejpam-3272	298	3	we	we	PRON
ejpam-3272	298	4	obtain	obtain	VERB
ejpam-3272	298	5	a	a	DET
ejpam-3272	298	6	sequence	sequence	NOUN
ejpam-3272	298	7	of	of	ADP
ejpam-3272	298	8	elements	element	NOUN
ejpam-3272	299	1	such	such	ADJ
ejpam-3272	299	2	that	that	SCONJ
ejpam-3272	299	3	x0	x0	PROPN
ejpam-3272	299	4	=	=	PUNCT
ejpam-3272	299	5	a	a	X
ejpam-3272	299	6	,	,	PUNCT
ejpam-3272	299	7	x1	x1	PROPN
ejpam-3272	299	8	=	=	SYM
ejpam-3272	299	9	c	c	PROPN
ejpam-3272	299	10	and	and	CCONJ
ejpam-3272	299	11	xk/2n	xk/2n	NOUN
ejpam-3272	299	12	=	=	PROPN
ejpam-3272	299	13	ak/2n+1	ak/2n+1	PROPN
ejpam-3272	299	14	.	.	PUNCT
ejpam-3272	300	1	clearly	clearly	ADV
ejpam-3272	300	2	,	,	PUNCT
ejpam-3272	300	3	xq	xq	PROPN
ejpam-3272	300	4	≺fr	≺fr	PROPN
ejpam-3272	300	5	xr	xr	PROPN
ejpam-3272	300	6	for	for	ADP
ejpam-3272	300	7	q	q	ADJ
ejpam-3272	300	8	<	<	X
ejpam-3272	300	9	r	r	NOUN
ejpam-3272	300	10	,	,	PUNCT
ejpam-3272	300	11	and	and	CCONJ
ejpam-3272	300	12	consequently	consequently	ADV
ejpam-3272	300	13	a	a	DET
ejpam-3272	300	14	≺≺fr	≺≺fr	PROPN
ejpam-3272	300	15	c.	c.	PROPN
ejpam-3272	300	16	similarly	similarly	ADV
ejpam-3272	300	17	,	,	PUNCT
ejpam-3272	300	18	we	we	PRON
ejpam-3272	300	19	can	can	AUX
ejpam-3272	300	20	find	find	VERB
ejpam-3272	300	21	a	a	DET
ejpam-3272	300	22	sequence	sequence	NOUN
ejpam-3272	300	23	of	of	ADP
ejpam-3272	300	24	elements	element	NOUN
ejpam-3272	300	25	such	such	ADJ
ejpam-3272	300	26	that	that	SCONJ
ejpam-3272	300	27	y0	y0	NOUN
ejpam-3272	300	28	=	=	SYM
ejpam-3272	300	29	c	c	X
ejpam-3272	300	30	,	,	PUNCT
ejpam-3272	300	31	y1	y1	NOUN
ejpam-3272	300	32	=	=	SYM
ejpam-3272	300	33	b	b	PROPN
ejpam-3272	300	34	and	and	CCONJ
ejpam-3272	300	35	yq	yq	PROPN
ejpam-3272	301	1	≺fr	≺fr	X
ejpam-3272	301	2	yr	yr	VERB
ejpam-3272	301	3	for	for	ADP
ejpam-3272	301	4	q	q	ADJ
ejpam-3272	301	5	<	<	X
ejpam-3272	301	6	r.	r.	X
ejpam-3272	301	7	thus	thus	ADV
ejpam-3272	301	8	c	c	PROPN
ejpam-3272	301	9	≺≺fr	≺≺fr	PROPN
ejpam-3272	301	10	b	b	PROPN
ejpam-3272	301	11	and	and	CCONJ
ejpam-3272	301	12	hence	hence	ADV
ejpam-3272	301	13	the	the	DET
ejpam-3272	301	14	relation	relation	NOUN
ejpam-3272	301	15	≺≺fr	≺≺fr	PROPN
ejpam-3272	301	16	is	be	AUX
ejpam-3272	301	17	interpolative	interpolative	ADJ
ejpam-3272	301	18	.	.	PUNCT
ejpam-3272	302	1	further	far	ADV
ejpam-3272	302	2	,	,	PUNCT
ejpam-3272	302	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	302	4	is	be	AUX
ejpam-3272	302	5	obviously	obviously	ADV
ejpam-3272	302	6	contained	contain	VERB
ejpam-3272	302	7	in	in	ADP
ejpam-3272	302	8	≺fr	≺fr	PROPN
ejpam-3272	302	9	.	.	PUNCT
ejpam-3272	303	1	for	for	ADP
ejpam-3272	303	2	the	the	DET
ejpam-3272	303	3	remaining	remain	VERB
ejpam-3272	303	4	assertion	assertion	NOUN
ejpam-3272	303	5	,	,	PUNCT
ejpam-3272	303	6	let	let	VERB
ejpam-3272	303	7	≺	≺	NOUN
ejpam-3272	303	8	be	be	AUX
ejpam-3272	303	9	any	any	DET
ejpam-3272	303	10	interpolative	interpolative	ADJ
ejpam-3272	303	11	relation	relation	NOUN
ejpam-3272	303	12	contained	contain	VERB
ejpam-3272	303	13	in	in	ADP
ejpam-3272	303	14	≺fr	≺fr	PROPN
ejpam-3272	303	15	.	.	PUNCT
ejpam-3272	304	1	if	if	SCONJ
ejpam-3272	304	2	a	a	DET
ejpam-3272	304	3	≺	≺	NOUN
ejpam-3272	304	4	b	b	NOUN
ejpam-3272	304	5	for	for	ADP
ejpam-3272	304	6	a	a	DET
ejpam-3272	304	7	,	,	PUNCT
ejpam-3272	304	8	b	b	X
ejpam-3272	304	9	∈	∈	NOUN
ejpam-3272	304	10	le	le	X
ejpam-3272	304	11	then	then	ADV
ejpam-3272	304	12	,	,	PUNCT
ejpam-3272	304	13	by	by	ADP
ejpam-3272	304	14	induction	induction	NOUN
ejpam-3272	304	15	,	,	PUNCT
ejpam-3272	304	16	we	we	PRON
ejpam-3272	304	17	obtain	obtain	VERB
ejpam-3272	304	18	a	a	DET
ejpam-3272	304	19	sequence	sequence	NOUN
ejpam-3272	304	20	of	of	ADP
ejpam-3272	304	21	elements	element	NOUN
ejpam-3272	304	22	with	with	ADP
ejpam-3272	304	23	a0	a0	PROPN
ejpam-3272	304	24	=	=	PUNCT
ejpam-3272	304	25	a	a	PROPN
ejpam-3272	304	26	,	,	PUNCT
ejpam-3272	304	27	a1	a1	NOUN
ejpam-3272	304	28	=	=	SYM
ejpam-3272	304	29	b	b	PROPN
ejpam-3272	304	30	and	and	CCONJ
ejpam-3272	304	31	aq	aq	PROPN
ejpam-3272	304	32	≺	≺	NOUN
ejpam-3272	304	33	ar	ar	NOUN
ejpam-3272	304	34	for	for	ADP
ejpam-3272	304	35	q	q	NOUN
ejpam-3272	304	36	<	<	X
ejpam-3272	304	37	r.	r.	X
ejpam-3272	304	38	we	we	PRON
ejpam-3272	304	39	also	also	ADV
ejpam-3272	304	40	have	have	VERB
ejpam-3272	304	41	“	"	PUNCT
ejpam-3272	304	42	aq	aq	X
ejpam-3272	304	43	≺	≺	NOUN
ejpam-3272	304	44	ar	ar	NOUN
ejpam-3272	304	45	⇒	⇒	NOUN
ejpam-3272	304	46	aq	aq	PROPN
ejpam-3272	304	47	≺fr	≺fr	PROPN
ejpam-3272	304	48	ar	ar	NOUN
ejpam-3272	304	49	by	by	ADP
ejpam-3272	304	50	assumption	assumption	NOUN
ejpam-3272	304	51	.	.	PUNCT
ejpam-3272	305	1	thus	thus	ADV
ejpam-3272	305	2	,	,	PUNCT
ejpam-3272	305	3	a	a	DET
ejpam-3272	305	4	≺≺fr	≺≺fr	PROPN
ejpam-3272	305	5	b.	b.	PROPN
ejpam-3272	305	6	definition	definition	NOUN
ejpam-3272	305	7	5	5	NUM
ejpam-3272	305	8	.	.	PUNCT
ejpam-3272	306	1	a	a	DET
ejpam-3272	306	2	diframe	diframe	NOUN
ejpam-3272	306	3	is	be	AUX
ejpam-3272	306	4	said	say	VERB
ejpam-3272	306	5	to	to	PART
ejpam-3272	306	6	be	be	AUX
ejpam-3272	306	7	(	(	PUNCT
ejpam-3272	306	8	i	i	NOUN
ejpam-3272	306	9	)	)	PUNCT
ejpam-3272	306	10	completely	completely	ADV
ejpam-3272	306	11	regular	regular	ADJ
ejpam-3272	306	12	if	if	SCONJ
ejpam-3272	306	13	a	a	DET
ejpam-3272	306	14	=	=	SYM
ejpam-3272	306	15	∨	∨	X
ejpam-3272	306	16	{	{	PUNCT
ejpam-3272	306	17	x	x	SYM
ejpam-3272	306	18	∈	∈	PROPN
ejpam-3272	306	19	lfr	lfr	NOUN
ejpam-3272	306	20	:	:	PUNCT
ejpam-3272	306	21	x	x	X
ejpam-3272	306	22	≺≺fr	≺≺fr	PROPN
ejpam-3272	306	23	a	a	X
ejpam-3272	306	24	}	}	PUNCT
ejpam-3272	306	25	for	for	ADP
ejpam-3272	306	26	all	all	DET
ejpam-3272	306	27	a	a	DET
ejpam-3272	306	28	∈	∈	PROPN
ejpam-3272	306	29	lfr	lfr	X
ejpam-3272	306	30	.	.	PUNCT
ejpam-3272	306	31	(	(	PUNCT
ejpam-3272	306	32	ii	ii	NOUN
ejpam-3272	306	33	)	)	PUNCT
ejpam-3272	306	34	completely	completely	ADV
ejpam-3272	306	35	co	co	VERB
ejpam-3272	306	36	-	-	NOUN
ejpam-3272	306	37	regular	regular	ADJ
ejpam-3272	306	38	if	if	SCONJ
ejpam-3272	306	39	c	c	NOUN
ejpam-3272	306	40	=	=	SYM
ejpam-3272	306	41	∧	∧	PROPN
ejpam-3272	306	42	{	{	PUNCT
ejpam-3272	306	43	x	x	PROPN
ejpam-3272	306	44	∈	∈	PROPN
ejpam-3272	306	45	lcf	lcf	NOUN
ejpam-3272	306	46	:	:	PUNCT
ejpam-3272	306	47	c	c	X
ejpam-3272	306	48	≺≺cf	≺≺cf	PROPN
ejpam-3272	306	49	x	x	X
ejpam-3272	306	50	}	}	PUNCT
ejpam-3272	306	51	for	for	ADP
ejpam-3272	306	52	all	all	DET
ejpam-3272	306	53	c	c	PROPN
ejpam-3272	306	54	∈	∈	PROPN
ejpam-3272	306	55	lcf	lcf	PROPN
ejpam-3272	306	56	.	.	PUNCT
ejpam-3272	307	1	(	(	PUNCT
ejpam-3272	307	2	iii	iii	X
ejpam-3272	307	3	)	)	PUNCT
ejpam-3272	307	4	t3	t3	NOUN
ejpam-3272	307	5	1	1	NUM
ejpam-3272	307	6	2	2	NUM
ejpam-3272	307	7	if	if	SCONJ
ejpam-3272	307	8	completely	completely	ADV
ejpam-3272	307	9	regular	regular	ADJ
ejpam-3272	307	10	and	and	CCONJ
ejpam-3272	307	11	t0	t0	NOUN
ejpam-3272	307	12	.	.	PUNCT
ejpam-3272	308	1	(	(	PUNCT
ejpam-3272	308	2	iv	iv	X
ejpam-3272	308	3	)	)	PUNCT
ejpam-3272	308	4	co	co	NOUN
ejpam-3272	308	5	-	-	NOUN
ejpam-3272	308	6	t3	t3	ADJ
ejpam-3272	308	7	1	1	NUM
ejpam-3272	308	8	2	2	NUM
ejpam-3272	308	9	if	if	SCONJ
ejpam-3272	308	10	completely	completely	ADV
ejpam-3272	308	11	co	co	ADJ
ejpam-3272	308	12	-	-	ADJ
ejpam-3272	308	13	regular	regular	ADJ
ejpam-3272	308	14	and	and	CCONJ
ejpam-3272	308	15	co	co	NOUN
ejpam-3272	308	16	-	-	NOUN
ejpam-3272	308	17	t0	t0	NOUN
ejpam-3272	308	18	.	.	PUNCT
ejpam-3272	309	1	as	as	SCONJ
ejpam-3272	309	2	mentioned	mention	VERB
ejpam-3272	309	3	before	before	ADV
ejpam-3272	309	4	,	,	PUNCT
ejpam-3272	309	5	complete	complete	ADJ
ejpam-3272	309	6	(	(	PUNCT
ejpam-3272	309	7	co-	co-	NOUN
ejpam-3272	309	8	)	)	PUNCT
ejpam-3272	309	9	regularity	regularity	NOUN
ejpam-3272	309	10	is	be	AUX
ejpam-3272	309	11	defined	define	VERB
ejpam-3272	309	12	using	use	VERB
ejpam-3272	309	13	bicontinuous	bicontinuous	ADJ
ejpam-3272	309	14	difunctions	difunction	NOUN
ejpam-3272	309	15	in	in	ADP
ejpam-3272	309	16	ditopological	ditopological	ADJ
ejpam-3272	309	17	spaces	space	NOUN
ejpam-3272	309	18	.	.	PUNCT
ejpam-3272	310	1	here	here	ADV
ejpam-3272	310	2	,	,	PUNCT
ejpam-3272	310	3	we	we	PRON
ejpam-3272	310	4	leave	leave	VERB
ejpam-3272	310	5	the	the	DET
ejpam-3272	310	6	following	follow	VERB
ejpam-3272	310	7	questions	question	NOUN
ejpam-3272	310	8	as	as	ADP
ejpam-3272	310	9	open	open	ADJ
ejpam-3272	310	10	problems	problem	NOUN
ejpam-3272	310	11	:	:	PUNCT
ejpam-3272	310	12	(	(	PUNCT
ejpam-3272	310	13	1	1	X
ejpam-3272	310	14	)	)	PUNCT
ejpam-3272	310	15	can	can	AUX
ejpam-3272	310	16	we	we	PRON
ejpam-3272	310	17	construct	construct	VERB
ejpam-3272	310	18	a	a	DET
ejpam-3272	310	19	diframe	diframe	NOUN
ejpam-3272	310	20	corresponding	correspond	VERB
ejpam-3272	310	21	to	to	ADP
ejpam-3272	310	22	the	the	DET
ejpam-3272	310	23	ditopological	ditopological	ADJ
ejpam-3272	310	24	unit	unit	NOUN
ejpam-3272	310	25	interval	interval	NOUN
ejpam-3272	310	26	texture	texture	NOUN
ejpam-3272	310	27	space	space	NOUN
ejpam-3272	310	28	(	(	PUNCT
ejpam-3272	310	29	i	i	PROPN
ejpam-3272	310	30	,	,	PUNCT
ejpam-3272	310	31	j	j	PROPN
ejpam-3272	310	32	,	,	PUNCT
ejpam-3272	310	33	τi	τi	ADP
ejpam-3272	310	34	,	,	PUNCT
ejpam-3272	310	35	κi	κi	NOUN
ejpam-3272	310	36	)	)	PUNCT
ejpam-3272	310	37	?	?	PUNCT
ejpam-3272	311	1	(	(	PUNCT
ejpam-3272	311	2	2	2	X
ejpam-3272	311	3	)	)	PUNCT
ejpam-3272	311	4	how	how	SCONJ
ejpam-3272	311	5	do	do	AUX
ejpam-3272	311	6	we	we	PRON
ejpam-3272	311	7	characterize	characterize	VERB
ejpam-3272	311	8	complete	complete	ADJ
ejpam-3272	311	9	regularity	regularity	NOUN
ejpam-3272	311	10	by	by	ADP
ejpam-3272	311	11	using	use	VERB
ejpam-3272	311	12	diframe	diframe	NOUN
ejpam-3272	311	13	homomorphisms	homomorphism	NOUN
ejpam-3272	311	14	?	?	PUNCT
ejpam-3272	312	1	(	(	PUNCT
ejpam-3272	312	2	3	3	X
ejpam-3272	312	3	)	)	PUNCT
ejpam-3272	312	4	what	what	PRON
ejpam-3272	312	5	is	be	AUX
ejpam-3272	312	6	the	the	DET
ejpam-3272	312	7	relation	relation	NOUN
ejpam-3272	312	8	between	between	ADP
ejpam-3272	312	9	these	these	DET
ejpam-3272	312	10	two	two	NUM
ejpam-3272	312	11	characterizations	characterization	NOUN
ejpam-3272	312	12	of	of	ADP
ejpam-3272	312	13	completely	completely	ADV
ejpam-3272	312	14	regularity	regularity	NOUN
ejpam-3272	312	15	?	?	PUNCT
ejpam-3272	313	1	proposition	proposition	NOUN
ejpam-3272	313	2	11	11	NUM
ejpam-3272	313	3	.	.	PUNCT
ejpam-3272	314	1	(	(	PUNCT
ejpam-3272	314	2	i	i	NOUN
ejpam-3272	314	3	)	)	PUNCT
ejpam-3272	314	4	a	a	DET
ejpam-3272	314	5	completely	completely	ADV
ejpam-3272	314	6	regular	regular	ADJ
ejpam-3272	314	7	diframe	diframe	NOUN
ejpam-3272	314	8	is	be	AUX
ejpam-3272	314	9	regular	regular	ADJ
ejpam-3272	314	10	.	.	PUNCT
ejpam-3272	315	1	(	(	PUNCT
ejpam-3272	315	2	ii	ii	NOUN
ejpam-3272	315	3	)	)	PUNCT
ejpam-3272	315	4	a	a	DET
ejpam-3272	315	5	completely	completely	ADV
ejpam-3272	315	6	co	co	ADJ
ejpam-3272	315	7	-	-	ADJ
ejpam-3272	315	8	regular	regular	ADJ
ejpam-3272	315	9	diframe	diframe	NOUN
ejpam-3272	315	10	is	be	AUX
ejpam-3272	315	11	co	co	ADJ
ejpam-3272	315	12	-	-	NOUN
ejpam-3272	315	13	regular	regular	ADJ
ejpam-3272	315	14	.	.	PUNCT
ejpam-3272	316	1	e.	e.	PROPN
ejpam-3272	316	2	korkmaz	korkmaz	PROPN
ejpam-3272	316	3	,	,	PUNCT
ejpam-3272	316	4	r.	r.	PROPN
ejpam-3272	316	5	ertürk	ertürk	PROPN
ejpam-3272	316	6	/	/	SYM
ejpam-3272	316	7	eur	eur	PROPN
ejpam-3272	316	8	.	.	PUNCT
ejpam-3272	317	1	j.	j.	PROPN
ejpam-3272	317	2	pure	pure	PROPN
ejpam-3272	317	3	appl	appl	PROPN
ejpam-3272	317	4	.	.	PROPN
ejpam-3272	317	5	math	math	PROPN
ejpam-3272	317	6	,	,	PUNCT
ejpam-3272	317	7	11	11	NUM
ejpam-3272	317	8	(	(	PUNCT
ejpam-3272	317	9	3	3	NUM
ejpam-3272	317	10	)	)	PUNCT
ejpam-3272	317	11	(	(	PUNCT
ejpam-3272	317	12	2018	2018	NUM
ejpam-3272	317	13	)	)	PUNCT
ejpam-3272	317	14	,	,	PUNCT
ejpam-3272	317	15	612	612	NUM
ejpam-3272	317	16	-	-	SYM
ejpam-3272	317	17	627	627	NUM
ejpam-3272	317	18	623	623	NUM
ejpam-3272	317	19	proof	proof	NOUN
ejpam-3272	317	20	.	.	PUNCT
ejpam-3272	318	1	this	this	PRON
ejpam-3272	318	2	is	be	AUX
ejpam-3272	318	3	an	an	DET
ejpam-3272	318	4	immediate	immediate	ADJ
ejpam-3272	318	5	consequence	consequence	NOUN
ejpam-3272	318	6	of	of	ADP
ejpam-3272	318	7	the	the	DET
ejpam-3272	318	8	following	follow	VERB
ejpam-3272	318	9	facts	fact	NOUN
ejpam-3272	318	10	:	:	PUNCT
ejpam-3272	318	11	a	a	DET
ejpam-3272	318	12	≺≺fr	≺≺fr	PROPN
ejpam-3272	318	13	b	b	PROPN
ejpam-3272	318	14	implies	imply	VERB
ejpam-3272	318	15	a	a	DET
ejpam-3272	318	16	≺fr	≺fr	PROPN
ejpam-3272	318	17	b	b	NOUN
ejpam-3272	318	18	,	,	PUNCT
ejpam-3272	318	19	and	and	CCONJ
ejpam-3272	318	20	a	a	DET
ejpam-3272	318	21	≺≺cf	≺≺cf	PROPN
ejpam-3272	318	22	b	b	PROPN
ejpam-3272	318	23	implies	imply	VERB
ejpam-3272	318	24	a	a	DET
ejpam-3272	318	25	≺cf	≺cf	PROPN
ejpam-3272	318	26	b.	b.	PROPN
ejpam-3272	318	27	there	there	PRON
ejpam-3272	318	28	is	be	VERB
ejpam-3272	318	29	another	another	DET
ejpam-3272	318	30	way	way	NOUN
ejpam-3272	318	31	of	of	ADP
ejpam-3272	318	32	characterizing	characterize	VERB
ejpam-3272	318	33	complete	complete	ADJ
ejpam-3272	318	34	regularity	regularity	NOUN
ejpam-3272	318	35	of	of	ADP
ejpam-3272	318	36	a	a	DET
ejpam-3272	318	37	bitopological	bitopological	ADJ
ejpam-3272	318	38	space	space	NOUN
ejpam-3272	318	39	in	in	ADP
ejpam-3272	318	40	terms	term	NOUN
ejpam-3272	318	41	of	of	ADP
ejpam-3272	318	42	a	a	DET
ejpam-3272	318	43	urysohn	urysohn	PROPN
ejpam-3272	318	44	relation	relation	NOUN
ejpam-3272	318	45	due	due	ADP
ejpam-3272	318	46	to	to	ADP
ejpam-3272	318	47	kopperman	kopperman	NOUN
ejpam-3272	318	48	[	[	X
ejpam-3272	318	49	8	8	NUM
ejpam-3272	318	50	]	]	PUNCT
ejpam-3272	318	51	.	.	PUNCT
ejpam-3272	319	1	now	now	ADV
ejpam-3272	319	2	we	we	PRON
ejpam-3272	319	3	will	will	AUX
ejpam-3272	319	4	generalize	generalize	VERB
ejpam-3272	319	5	this	this	DET
ejpam-3272	319	6	idea	idea	NOUN
ejpam-3272	319	7	to	to	ADP
ejpam-3272	319	8	diframes	diframe	NOUN
ejpam-3272	319	9	.	.	PUNCT
ejpam-3272	320	1	we	we	PRON
ejpam-3272	320	2	start	start	VERB
ejpam-3272	320	3	by	by	ADP
ejpam-3272	320	4	recalling	recall	VERB
ejpam-3272	320	5	the	the	DET
ejpam-3272	320	6	definition	definition	NOUN
ejpam-3272	320	7	of	of	ADP
ejpam-3272	320	8	a	a	DET
ejpam-3272	320	9	urysohn	urysohn	PROPN
ejpam-3272	320	10	relation	relation	NOUN
ejpam-3272	320	11	[	[	X
ejpam-3272	320	12	7	7	NUM
ejpam-3272	320	13	]	]	PUNCT
ejpam-3272	320	14	.	.	PUNCT
ejpam-3272	321	1	a	a	DET
ejpam-3272	321	2	binary	binary	PROPN
ejpam-3272	321	3	relation	relation	NOUN
ejpam-3272	321	4	c	c	PROPN
ejpam-3272	321	5	on	on	ADP
ejpam-3272	321	6	a	a	DET
ejpam-3272	321	7	partially	partially	ADV
ejpam-3272	321	8	ordered	order	VERB
ejpam-3272	321	9	set	set	NOUN
ejpam-3272	321	10	(	(	PUNCT
ejpam-3272	321	11	l,≤	l,≤	PROPN
ejpam-3272	321	12	)	)	PUNCT
ejpam-3272	321	13	is	be	AUX
ejpam-3272	321	14	called	call	VERB
ejpam-3272	321	15	a	a	DET
ejpam-3272	321	16	urysohn	urysohn	PROPN
ejpam-3272	321	17	relation	relation	NOUN
ejpam-3272	321	18	if	if	SCONJ
ejpam-3272	321	19	it	it	PRON
ejpam-3272	321	20	satisfies	satisfy	VERB
ejpam-3272	321	21	the	the	DET
ejpam-3272	321	22	following	follow	VERB
ejpam-3272	321	23	conditions	condition	NOUN
ejpam-3272	321	24	:	:	PUNCT
ejpam-3272	321	25	(	(	PUNCT
ejpam-3272	321	26	u1	u1	PROPN
ejpam-3272	321	27	)	)	PUNCT
ejpam-3272	322	1	ac	ac	PROPN
ejpam-3272	323	1	b	b	PROPN
ejpam-3272	323	2	implies	imply	VERB
ejpam-3272	323	3	a	a	DET
ejpam-3272	323	4	≤	≤	NUM
ejpam-3272	323	5	b	b	NOUN
ejpam-3272	323	6	for	for	ADP
ejpam-3272	323	7	all	all	DET
ejpam-3272	323	8	a	a	PRON
ejpam-3272	323	9	,	,	PUNCT
ejpam-3272	323	10	b	b	PROPN
ejpam-3272	323	11	∈	∈	PROPN
ejpam-3272	323	12	l	l	NOUN
ejpam-3272	323	13	,	,	PUNCT
ejpam-3272	323	14	(	(	PUNCT
ejpam-3272	323	15	u2	u2	PROPN
ejpam-3272	323	16	)	)	PUNCT
ejpam-3272	323	17	a	a	DET
ejpam-3272	323	18	≤	≤	NUM
ejpam-3272	323	19	bc	bc	PROPN
ejpam-3272	323	20	c	c	PROPN
ejpam-3272	323	21	≤	≤	PROPN
ejpam-3272	323	22	d	d	PROPN
ejpam-3272	323	23	implies	imply	VERB
ejpam-3272	323	24	ac	ac	PROPN
ejpam-3272	323	25	d	d	NOUN
ejpam-3272	323	26	for	for	ADP
ejpam-3272	323	27	all	all	DET
ejpam-3272	323	28	a	a	DET
ejpam-3272	323	29	,	,	PUNCT
ejpam-3272	323	30	b	b	NOUN
ejpam-3272	323	31	,	,	PUNCT
ejpam-3272	323	32	c	c	NOUN
ejpam-3272	323	33	,	,	PUNCT
ejpam-3272	323	34	d	d	PROPN
ejpam-3272	323	35	∈	∈	PROPN
ejpam-3272	323	36	l	l	NOUN
ejpam-3272	323	37	,	,	PUNCT
ejpam-3272	323	38	(	(	PUNCT
ejpam-3272	323	39	u3	u3	PROPN
ejpam-3272	323	40	)	)	PUNCT
ejpam-3272	323	41	ac	ac	PROPN
ejpam-3272	323	42	b	b	PROPN
ejpam-3272	323	43	implies	imply	VERB
ejpam-3272	323	44	the	the	DET
ejpam-3272	323	45	existence	existence	NOUN
ejpam-3272	323	46	of	of	ADP
ejpam-3272	323	47	c	c	PROPN
ejpam-3272	323	48	∈	∈	PROPN
ejpam-3272	323	49	l	l	NOUN
ejpam-3272	323	50	such	such	ADJ
ejpam-3272	323	51	that	that	DET
ejpam-3272	323	52	ac	ac	PROPN
ejpam-3272	323	53	cc	cc	PROPN
ejpam-3272	323	54	b	b	PROPN
ejpam-3272	323	55	for	for	ADP
ejpam-3272	323	56	all	all	DET
ejpam-3272	323	57	a	a	PRON
ejpam-3272	323	58	,	,	PUNCT
ejpam-3272	323	59	b	b	X
ejpam-3272	323	60	∈	∈	ADJ
ejpam-3272	323	61	l	l	NOUN
ejpam-3272	323	62	(	(	PUNCT
ejpam-3272	323	63	that	that	PRON
ejpam-3272	323	64	is	is	ADV
ejpam-3272	323	65	,	,	PUNCT
ejpam-3272	323	66	c	c	PROPN
ejpam-3272	323	67	is	be	AUX
ejpam-3272	323	68	an	an	DET
ejpam-3272	323	69	interpolative	interpolative	ADJ
ejpam-3272	323	70	relation	relation	NOUN
ejpam-3272	323	71	)	)	PUNCT
ejpam-3272	323	72	.	.	PUNCT
ejpam-3272	324	1	if	if	SCONJ
ejpam-3272	324	2	l	l	NOUN
ejpam-3272	324	3	is	be	AUX
ejpam-3272	324	4	a	a	DET
ejpam-3272	324	5	lattice	lattice	NOUN
ejpam-3272	324	6	and	and	CCONJ
ejpam-3272	324	7	c	c	NOUN
ejpam-3272	324	8	is	be	AUX
ejpam-3272	324	9	a	a	DET
ejpam-3272	324	10	urysohn	urysohn	PROPN
ejpam-3272	324	11	relation	relation	NOUN
ejpam-3272	324	12	on	on	ADP
ejpam-3272	324	13	l	l	NOUN
ejpam-3272	324	14	,	,	PUNCT
ejpam-3272	324	15	we	we	PRON
ejpam-3272	324	16	call	call	VERB
ejpam-3272	324	17	the	the	DET
ejpam-3272	324	18	pair	pair	NOUN
ejpam-3272	324	19	(	(	PUNCT
ejpam-3272	324	20	l	l	NOUN
ejpam-3272	324	21	,	,	PUNCT
ejpam-3272	324	22	c	c	NOUN
ejpam-3272	324	23	)	)	PUNCT
ejpam-3272	324	24	a	a	DET
ejpam-3272	324	25	urysohn	urysohn	PROPN
ejpam-3272	324	26	lattice	lattice	NOUN
ejpam-3272	324	27	.	.	PUNCT
ejpam-3272	325	1	the	the	DET
ejpam-3272	325	2	following	follow	VERB
ejpam-3272	325	3	are	be	AUX
ejpam-3272	325	4	some	some	DET
ejpam-3272	325	5	basic	basic	ADJ
ejpam-3272	325	6	examples	example	NOUN
ejpam-3272	325	7	of	of	ADP
ejpam-3272	325	8	urysohn	urysohn	PROPN
ejpam-3272	325	9	relations	relation	NOUN
ejpam-3272	325	10	.	.	PUNCT
ejpam-3272	325	11	example	example	NOUN
ejpam-3272	326	1	4	4	NUM
ejpam-3272	326	2	.	.	PUNCT
ejpam-3272	327	1	(	(	PUNCT
ejpam-3272	327	2	i	i	NOUN
ejpam-3272	327	3	)	)	PUNCT
ejpam-3272	327	4	let	let	VERB
ejpam-3272	327	5	x	x	PRON
ejpam-3272	327	6	be	be	AUX
ejpam-3272	327	7	a	a	DET
ejpam-3272	327	8	normal	normal	ADJ
ejpam-3272	327	9	space	space	NOUN
ejpam-3272	327	10	.	.	PUNCT
ejpam-3272	328	1	for	for	ADP
ejpam-3272	328	2	u	u	NOUN
ejpam-3272	328	3	,	,	PUNCT
ejpam-3272	328	4	v	v	NOUN
ejpam-3272	328	5	∈	∈	PROPN
ejpam-3272	328	6	ω(x	ω(x	NOUN
ejpam-3272	328	7	)	)	PUNCT
ejpam-3272	328	8	,	,	PUNCT
ejpam-3272	328	9	define	define	VERB
ejpam-3272	328	10	a	a	DET
ejpam-3272	328	11	relation	relation	NOUN
ejpam-3272	328	12	c	c	NOUN
ejpam-3272	328	13	by	by	ADP
ejpam-3272	328	14	setting	set	VERB
ejpam-3272	328	15	u	u	PROPN
ejpam-3272	328	16	c	c	PROPN
ejpam-3272	328	17	v	v	ADP
ejpam-3272	328	18	iff	iff	PROPN
ejpam-3272	328	19	u	u	PROPN
ejpam-3272	328	20	⊆	⊆	NUM
ejpam-3272	328	21	v	v	NOUN
ejpam-3272	328	22	.	.	PUNCT
ejpam-3272	329	1	then	then	ADV
ejpam-3272	329	2	c	c	PROPN
ejpam-3272	329	3	is	be	AUX
ejpam-3272	329	4	a	a	DET
ejpam-3272	329	5	urysohn	urysohn	PROPN
ejpam-3272	329	6	relation	relation	NOUN
ejpam-3272	329	7	.	.	PUNCT
ejpam-3272	330	1	(	(	PUNCT
ejpam-3272	330	2	ii	ii	X
ejpam-3272	330	3	)	)	PUNCT
ejpam-3272	330	4	the	the	DET
ejpam-3272	330	5	relations	relation	NOUN
ejpam-3272	330	6	≺fr	≺fr	PROPN
ejpam-3272	330	7	and	and	CCONJ
ejpam-3272	330	8	≺cf	≺cf	PROPN
ejpam-3272	330	9	are	be	AUX
ejpam-3272	330	10	not	not	PART
ejpam-3272	330	11	urysohn	urysohn	ADJ
ejpam-3272	330	12	since	since	SCONJ
ejpam-3272	330	13	the	the	DET
ejpam-3272	330	14	interpolation	interpolation	NOUN
ejpam-3272	330	15	property	property	NOUN
ejpam-3272	330	16	does	do	AUX
ejpam-3272	330	17	not	not	PART
ejpam-3272	330	18	hold	hold	VERB
ejpam-3272	330	19	.	.	PUNCT
ejpam-3272	331	1	however	however	ADV
ejpam-3272	331	2	,	,	PUNCT
ejpam-3272	331	3	≺≺fr	≺≺fr	PROPN
ejpam-3272	331	4	and	and	CCONJ
ejpam-3272	331	5	≺≺cf	≺≺cf	PROPN
ejpam-3272	331	6	are	be	AUX
ejpam-3272	331	7	obviously	obviously	ADV
ejpam-3272	331	8	urysohn	urysohn	PROPN
ejpam-3272	331	9	relations	relation	NOUN
ejpam-3272	331	10	by	by	ADP
ejpam-3272	331	11	proposition	proposition	NOUN
ejpam-3272	331	12	10	10	NUM
ejpam-3272	331	13	.	.	PUNCT
ejpam-3272	332	1	proposition	proposition	NOUN
ejpam-3272	332	2	12	12	NUM
ejpam-3272	332	3	.	.	PUNCT
ejpam-3272	333	1	let	let	VERB
ejpam-3272	333	2	l	l	NOUN
ejpam-3272	333	3	=	=	SYM
ejpam-3272	333	4	(	(	PUNCT
ejpam-3272	333	5	le	le	X
ejpam-3272	333	6	,	,	PUNCT
ejpam-3272	333	7	lfr	lfr	PROPN
ejpam-3272	333	8	,	,	PUNCT
ejpam-3272	333	9	lcf	lcf	PROPN
ejpam-3272	333	10	)	)	PUNCT
ejpam-3272	333	11	be	be	AUX
ejpam-3272	333	12	a	a	DET
ejpam-3272	333	13	diframe	diframe	NOUN
ejpam-3272	333	14	.	.	PUNCT
ejpam-3272	334	1	(	(	PUNCT
ejpam-3272	334	2	i	i	NOUN
ejpam-3272	334	3	)	)	PUNCT
ejpam-3272	334	4	l	l	NOUN
ejpam-3272	334	5	is	be	AUX
ejpam-3272	334	6	completely	completely	ADV
ejpam-3272	334	7	regular	regular	ADJ
ejpam-3272	334	8	if	if	SCONJ
ejpam-3272	334	9	and	and	CCONJ
ejpam-3272	334	10	only	only	ADV
ejpam-3272	334	11	if	if	SCONJ
ejpam-3272	334	12	there	there	PRON
ejpam-3272	334	13	exists	exist	VERB
ejpam-3272	334	14	a	a	DET
ejpam-3272	334	15	urysohn	urysohn	PROPN
ejpam-3272	334	16	relation	relation	NOUN
ejpam-3272	334	17	c	c	PROPN
ejpam-3272	334	18	on	on	X
ejpam-3272	334	19	le	le	X
ejpam-3272	334	20	satisfying	satisfy	VERB
ejpam-3272	334	21	the	the	DET
ejpam-3272	334	22	following	follow	VERB
ejpam-3272	334	23	conditions	condition	NOUN
ejpam-3272	334	24	:	:	PUNCT
ejpam-3272	334	25	(	(	PUNCT
ejpam-3272	334	26	a	a	X
ejpam-3272	334	27	)	)	PUNCT
ejpam-3272	334	28	ac	ac	PROPN
ejpam-3272	334	29	b	b	PROPN
ejpam-3272	334	30	implies	imply	VERB
ejpam-3272	334	31	[	[	X
ejpam-3272	334	32	a	a	X
ejpam-3272	334	33	]	]	X
ejpam-3272	334	34	≤]b	≤]b	ADP
ejpam-3272	334	35	[	[	X
ejpam-3272	334	36	,	,	PUNCT
ejpam-3272	334	37	(	(	PUNCT
ejpam-3272	334	38	b	b	NOUN
ejpam-3272	334	39	)	)	PUNCT
ejpam-3272	334	40	for	for	ADP
ejpam-3272	334	41	every	every	DET
ejpam-3272	334	42	a	a	DET
ejpam-3272	334	43	∈	∈	PROPN
ejpam-3272	334	44	lfr	lfr	NOUN
ejpam-3272	334	45	,	,	PUNCT
ejpam-3272	334	46	a	a	DET
ejpam-3272	334	47	=	=	SYM
ejpam-3272	334	48	∨	∨	X
ejpam-3272	334	49	{	{	PUNCT
ejpam-3272	334	50	x	x	SYM
ejpam-3272	334	51	∈	∈	PROPN
ejpam-3272	334	52	lfr	lfr	NOUN
ejpam-3272	334	53	:	:	PUNCT
ejpam-3272	334	54	xc	xc	PROPN
ejpam-3272	334	55	a	a	PRON
ejpam-3272	334	56	}	}	PUNCT
ejpam-3272	334	57	.	.	PUNCT
ejpam-3272	335	1	(	(	PUNCT
ejpam-3272	335	2	ii	ii	NOUN
ejpam-3272	335	3	)	)	PUNCT
ejpam-3272	335	4	l	l	NOUN
ejpam-3272	335	5	is	be	AUX
ejpam-3272	335	6	completely	completely	ADV
ejpam-3272	335	7	co	co	ADJ
ejpam-3272	335	8	-	-	NOUN
ejpam-3272	335	9	regular	regular	ADJ
ejpam-3272	335	10	if	if	SCONJ
ejpam-3272	336	1	and	and	CCONJ
ejpam-3272	336	2	only	only	ADV
ejpam-3272	336	3	if	if	SCONJ
ejpam-3272	336	4	there	there	PRON
ejpam-3272	336	5	exists	exist	VERB
ejpam-3272	336	6	a	a	DET
ejpam-3272	336	7	urysohn	urysohn	PROPN
ejpam-3272	336	8	relation	relation	NOUN
ejpam-3272	336	9	c	c	PROPN
ejpam-3272	336	10	on	on	X
ejpam-3272	336	11	le	le	X
ejpam-3272	336	12	satisfying	satisfy	VERB
ejpam-3272	336	13	the	the	DET
ejpam-3272	336	14	following	follow	VERB
ejpam-3272	336	15	conditions	condition	NOUN
ejpam-3272	336	16	:	:	PUNCT
ejpam-3272	336	17	(	(	PUNCT
ejpam-3272	336	18	a	a	X
ejpam-3272	336	19	)	)	PUNCT
ejpam-3272	336	20	ac	ac	PROPN
ejpam-3272	336	21	b	b	PROPN
ejpam-3272	336	22	implies	imply	VERB
ejpam-3272	336	23	[	[	X
ejpam-3272	336	24	a	a	X
ejpam-3272	336	25	]	]	X
ejpam-3272	336	26	≤]b	≤]b	ADP
ejpam-3272	336	27	[	[	X
ejpam-3272	336	28	,	,	PUNCT
ejpam-3272	336	29	(	(	PUNCT
ejpam-3272	336	30	b	b	NOUN
ejpam-3272	336	31	)	)	PUNCT
ejpam-3272	336	32	for	for	ADP
ejpam-3272	336	33	every	every	DET
ejpam-3272	336	34	c	c	PROPN
ejpam-3272	336	35	∈	∈	PROPN
ejpam-3272	336	36	lcf	lcf	PROPN
ejpam-3272	336	37	,	,	PUNCT
ejpam-3272	336	38	c	c	NOUN
ejpam-3272	336	39	=	=	SYM
ejpam-3272	336	40	∧	∧	PROPN
ejpam-3272	336	41	{	{	PUNCT
ejpam-3272	336	42	x	x	PROPN
ejpam-3272	336	43	∈	∈	PROPN
ejpam-3272	336	44	lcf	lcf	NOUN
ejpam-3272	336	45	:	:	PUNCT
ejpam-3272	336	46	cc	cc	NOUN
ejpam-3272	336	47	x	x	X
ejpam-3272	336	48	}	}	PUNCT
ejpam-3272	336	49	.	.	PUNCT
ejpam-3272	337	1	proof	proof	NOUN
ejpam-3272	337	2	.	.	PUNCT
ejpam-3272	338	1	here	here	ADV
ejpam-3272	338	2	,	,	PUNCT
ejpam-3272	338	3	we	we	PRON
ejpam-3272	338	4	just	just	ADV
ejpam-3272	338	5	prove	prove	VERB
ejpam-3272	338	6	(	(	PUNCT
ejpam-3272	338	7	i	i	NOUN
ejpam-3272	338	8	)	)	PUNCT
ejpam-3272	338	9	,	,	PUNCT
ejpam-3272	338	10	since	since	SCONJ
ejpam-3272	338	11	(	(	PUNCT
ejpam-3272	338	12	ii	ii	NOUN
ejpam-3272	338	13	)	)	PUNCT
ejpam-3272	338	14	can	can	AUX
ejpam-3272	338	15	be	be	AUX
ejpam-3272	338	16	proven	prove	VERB
ejpam-3272	338	17	similarly	similarly	ADV
ejpam-3272	338	18	.	.	PUNCT
ejpam-3272	339	1	if	if	SCONJ
ejpam-3272	339	2	l	l	NOUN
ejpam-3272	339	3	is	be	AUX
ejpam-3272	339	4	a	a	DET
ejpam-3272	339	5	completely	completely	ADV
ejpam-3272	339	6	regular	regular	ADJ
ejpam-3272	339	7	diframe	diframe	NOUN
ejpam-3272	339	8	then	then	ADV
ejpam-3272	339	9	≺≺fr	≺≺fr	PROPN
ejpam-3272	339	10	is	be	AUX
ejpam-3272	339	11	the	the	DET
ejpam-3272	339	12	desired	desire	VERB
ejpam-3272	339	13	relation	relation	NOUN
ejpam-3272	339	14	.	.	PUNCT
ejpam-3272	340	1	indeed	indeed	ADV
ejpam-3272	340	2	,	,	PUNCT
ejpam-3272	340	3	as	as	SCONJ
ejpam-3272	340	4	can	can	AUX
ejpam-3272	340	5	be	be	AUX
ejpam-3272	340	6	easily	easily	ADV
ejpam-3272	340	7	checked	check	VERB
ejpam-3272	340	8	,	,	PUNCT
ejpam-3272	340	9	it	it	PRON
ejpam-3272	340	10	is	be	AUX
ejpam-3272	340	11	a	a	DET
ejpam-3272	340	12	urysohn	urysohn	PROPN
ejpam-3272	340	13	relation	relation	NOUN
ejpam-3272	340	14	.	.	PUNCT
ejpam-3272	341	1	further	far	ADV
ejpam-3272	341	2	,	,	PUNCT
ejpam-3272	341	3	the	the	DET
ejpam-3272	341	4	condition	condition	NOUN
ejpam-3272	341	5	(	(	PUNCT
ejpam-3272	341	6	b	b	NOUN
ejpam-3272	341	7	)	)	PUNCT
ejpam-3272	341	8	is	be	AUX
ejpam-3272	341	9	a	a	DET
ejpam-3272	341	10	direct	direct	ADJ
ejpam-3272	341	11	result	result	NOUN
ejpam-3272	341	12	of	of	ADP
ejpam-3272	341	13	the	the	DET
ejpam-3272	341	14	definition	definition	NOUN
ejpam-3272	341	15	.	.	PUNCT
ejpam-3272	342	1	now	now	ADV
ejpam-3272	342	2	let	let	VERB
ejpam-3272	342	3	a	a	DET
ejpam-3272	342	4	≺≺fr	≺≺fr	PROPN
ejpam-3272	342	5	b.	b.	PROPN
ejpam-3272	342	6	then	then	ADV
ejpam-3272	342	7	applying	apply	VERB
ejpam-3272	342	8	the	the	DET
ejpam-3272	342	9	definitions	definition	NOUN
ejpam-3272	342	10	of	of	ADP
ejpam-3272	342	11	≺≺fr	≺≺fr	PROPN
ejpam-3272	342	12	and	and	CCONJ
ejpam-3272	342	13	≺fr	≺fr	PROPN
ejpam-3272	342	14	,	,	PUNCT
ejpam-3272	342	15	respectively	respectively	ADV
ejpam-3272	342	16	,	,	PUNCT
ejpam-3272	342	17	we	we	PRON
ejpam-3272	342	18	obtain	obtain	VERB
ejpam-3272	342	19	aq	aq	PRON
ejpam-3272	342	20	∈	∈	PROPN
ejpam-3272	342	21	lfr	lfr	NOUN
ejpam-3272	342	22	and	and	CCONJ
ejpam-3272	342	23	cq	cq	NOUN
ejpam-3272	342	24	∈	∈	PROPN
ejpam-3272	342	25	lcf	lcf	PROPN
ejpam-3272	342	26	(	(	PUNCT
ejpam-3272	342	27	q	q	NOUN
ejpam-3272	342	28	∈	∈	PROPN
ejpam-3272	342	29	d	d	NOUN
ejpam-3272	342	30	)	)	PUNCT
ejpam-3272	342	31	such	such	ADJ
ejpam-3272	342	32	that	that	SCONJ
ejpam-3272	342	33	a	a	DET
ejpam-3272	342	34	≤	≤	NOUN
ejpam-3272	342	35	.	.	PUNCT
ejpam-3272	342	36	.	.	PUNCT
ejpam-3272	342	37	.	.	PUNCT
ejpam-3272	343	1	aq	aq	VERB
ejpam-3272	343	2	≤	≤	NUM
ejpam-3272	343	3	cq	cq	NOUN
ejpam-3272	343	4	≤	≤	PROPN
ejpam-3272	343	5	ar	ar	NOUN
ejpam-3272	343	6	≤	≤	NOUN
ejpam-3272	343	7	.	.	PUNCT
ejpam-3272	343	8	.	.	PUNCT
ejpam-3272	343	9	.	.	PUNCT
ejpam-3272	344	1	≤	≤	NUM
ejpam-3272	344	2	b	b	X
ejpam-3272	344	3	e.	e.	PROPN
ejpam-3272	344	4	korkmaz	korkmaz	PROPN
ejpam-3272	344	5	,	,	PUNCT
ejpam-3272	344	6	r.	r.	PROPN
ejpam-3272	344	7	ertürk	ertürk	PROPN
ejpam-3272	344	8	/	/	SYM
ejpam-3272	344	9	eur	eur	PROPN
ejpam-3272	344	10	.	.	PUNCT
ejpam-3272	345	1	j.	j.	PROPN
ejpam-3272	345	2	pure	pure	PROPN
ejpam-3272	345	3	appl	appl	PROPN
ejpam-3272	345	4	.	.	PROPN
ejpam-3272	345	5	math	math	PROPN
ejpam-3272	345	6	,	,	PUNCT
ejpam-3272	345	7	11	11	NUM
ejpam-3272	345	8	(	(	PUNCT
ejpam-3272	345	9	3	3	NUM
ejpam-3272	345	10	)	)	PUNCT
ejpam-3272	345	11	(	(	PUNCT
ejpam-3272	345	12	2018	2018	NUM
ejpam-3272	345	13	)	)	PUNCT
ejpam-3272	345	14	,	,	PUNCT
ejpam-3272	345	15	612	612	NUM
ejpam-3272	345	16	-	-	SYM
ejpam-3272	345	17	627	627	NUM
ejpam-3272	345	18	624	624	NUM
ejpam-3272	345	19	and	and	CCONJ
ejpam-3272	345	20	hence	hence	ADV
ejpam-3272	345	21	[	[	X
ejpam-3272	345	22	a	a	X
ejpam-3272	345	23	]	]	X
ejpam-3272	345	24	≤	≤	NOUN
ejpam-3272	345	25	.	.	PUNCT
ejpam-3272	345	26	.	.	PUNCT
ejpam-3272	345	27	.	.	PUNCT
ejpam-3272	346	1	≤	≤	PUNCT
ejpam-3272	347	1	[	[	X
ejpam-3272	347	2	aq	aq	X
ejpam-3272	347	3	]	]	X
ejpam-3272	347	4	≤	≤	NOUN
ejpam-3272	348	1	[	[	X
ejpam-3272	348	2	cq	cq	X
ejpam-3272	348	3	]	]	X
ejpam-3272	348	4	=	=	PUNCT
ejpam-3272	348	5	cq	cq	PROPN
ejpam-3272	348	6	≤	≤	PROPN
ejpam-3272	348	7	ar	ar	NOUN
ejpam-3272	348	8	≤	≤	NOUN
ejpam-3272	348	9	.	.	PUNCT
ejpam-3272	348	10	.	.	PUNCT
ejpam-3272	348	11	.	.	PUNCT
ejpam-3272	349	1	≤	≤	NUM
ejpam-3272	349	2	b	b	X
ejpam-3272	349	3	=]	=]	NOUN
ejpam-3272	349	4	b	b	PROPN
ejpam-3272	350	1	[	[	X
ejpam-3272	350	2	.	.	PUNCT
ejpam-3272	351	1	where	where	SCONJ
ejpam-3272	351	2	q	q	X
ejpam-3272	351	3	<	<	X
ejpam-3272	351	4	r.	r.	PROPN
ejpam-3272	351	5	thus	thus	ADV
ejpam-3272	351	6	,	,	PUNCT
ejpam-3272	351	7	the	the	DET
ejpam-3272	351	8	relation	relation	NOUN
ejpam-3272	351	9	≺≺fr	≺≺fr	PROPN
ejpam-3272	351	10	satisfies	satisfy	VERB
ejpam-3272	351	11	(	(	PUNCT
ejpam-3272	351	12	a	a	X
ejpam-3272	351	13	)	)	PUNCT
ejpam-3272	351	14	.	.	PUNCT
ejpam-3272	352	1	conversely	conversely	ADV
ejpam-3272	352	2	,	,	PUNCT
ejpam-3272	352	3	suppose	suppose	VERB
ejpam-3272	352	4	that	that	SCONJ
ejpam-3272	352	5	we	we	PRON
ejpam-3272	352	6	have	have	VERB
ejpam-3272	352	7	a	a	DET
ejpam-3272	352	8	urysohn	urysohn	PROPN
ejpam-3272	352	9	relation	relation	NOUN
ejpam-3272	352	10	c	c	PROPN
ejpam-3272	352	11	on	on	X
ejpam-3272	352	12	le	le	X
ejpam-3272	352	13	satisfying	satisfy	VERB
ejpam-3272	352	14	the	the	DET
ejpam-3272	352	15	conditions	condition	NOUN
ejpam-3272	352	16	(	(	PUNCT
ejpam-3272	352	17	a	a	X
ejpam-3272	352	18	)	)	PUNCT
ejpam-3272	352	19	and	and	CCONJ
ejpam-3272	352	20	(	(	PUNCT
ejpam-3272	352	21	b	b	NOUN
ejpam-3272	352	22	)	)	PUNCT
ejpam-3272	352	23	.	.	PUNCT
ejpam-3272	353	1	let	let	VERB
ejpam-3272	353	2	xc	xc	PROPN
ejpam-3272	353	3	a	a	PRON
ejpam-3272	353	4	for	for	ADP
ejpam-3272	353	5	x	x	X
ejpam-3272	353	6	,	,	PUNCT
ejpam-3272	353	7	a	a	DET
ejpam-3272	353	8	∈	∈	PROPN
ejpam-3272	353	9	lfr	lfr	NOUN
ejpam-3272	353	10	.	.	PUNCT
ejpam-3272	354	1	by	by	ADP
ejpam-3272	354	2	(	(	PUNCT
ejpam-3272	354	3	u3	u3	PROPN
ejpam-3272	354	4	)	)	PUNCT
ejpam-3272	354	5	,	,	PUNCT
ejpam-3272	354	6	there	there	PRON
ejpam-3272	354	7	exists	exist	VERB
ejpam-3272	354	8	yq	yq	PROPN
ejpam-3272	354	9	∈	∈	PROPN
ejpam-3272	354	10	le	le	X
ejpam-3272	354	11	(	(	PUNCT
ejpam-3272	354	12	q	q	PROPN
ejpam-3272	354	13	∈	∈	PROPN
ejpam-3272	354	14	d	d	NOUN
ejpam-3272	354	15	)	)	PUNCT
ejpam-3272	354	16	such	such	ADJ
ejpam-3272	354	17	that	that	SCONJ
ejpam-3272	354	18	xc	xc	PROPN
ejpam-3272	354	19	.	.	PUNCT
ejpam-3272	354	20	.	.	PUNCT
ejpam-3272	354	21	.	.	PUNCT
ejpam-3272	355	1	yq	yq	PROPN
ejpam-3272	356	1	c	c	PROPN
ejpam-3272	356	2	yr	yr	INTJ
ejpam-3272	356	3	.	.	PUNCT
ejpam-3272	356	4	.	.	PUNCT
ejpam-3272	357	1	.c	.c	VERB
ejpam-3272	358	1	a	a	DET
ejpam-3272	358	2	where	where	SCONJ
ejpam-3272	358	3	q	q	X
ejpam-3272	358	4	<	<	X
ejpam-3272	358	5	r.	r.	PROPN
ejpam-3272	358	6	since	since	SCONJ
ejpam-3272	358	7	]	]	PUNCT
ejpam-3272	358	8	yq[≤	yq[≤	PROPN
ejpam-3272	358	9	[	[	X
ejpam-3272	358	10	yq	yq	X
ejpam-3272	358	11	]	]	X
ejpam-3272	358	12	≤]yr	≤]yr	PROPN
ejpam-3272	358	13	[	[	PUNCT
ejpam-3272	358	14	by	by	ADP
ejpam-3272	358	15	(	(	PUNCT
ejpam-3272	358	16	a	a	X
ejpam-3272	358	17	)	)	PUNCT
ejpam-3272	358	18	,	,	PUNCT
ejpam-3272	358	19	we	we	PRON
ejpam-3272	358	20	have	have	VERB
ejpam-3272	358	21	x	x	X
ejpam-3272	358	22	≺fr	≺fr	PROPN
ejpam-3272	358	23	.	.	PUNCT
ejpam-3272	358	24	.	.	PUNCT
ejpam-3272	358	25	.	.	PUNCT
ejpam-3272	359	1	≺fr]yq[≺fr]yr[≺fr	≺fr]yq[≺fr]yr[≺fr	PUNCT
ejpam-3272	359	2	.	.	PUNCT
ejpam-3272	359	3	.	.	PUNCT
ejpam-3272	360	1	.	.	PUNCT
ejpam-3272	361	1	≺fr	≺fr	NOUN
ejpam-3272	361	2	a.	a.	NOUN
ejpam-3272	362	1	we	we	PRON
ejpam-3272	362	2	now	now	ADV
ejpam-3272	362	3	obtain	obtain	VERB
ejpam-3272	362	4	xc	xc	PROPN
ejpam-3272	362	5	a	a	DET
ejpam-3272	362	6	implies	implie	NOUN
ejpam-3272	362	7	x	x	X
ejpam-3272	362	8	≺≺fr	≺≺fr	PROPN
ejpam-3272	362	9	a.	a.	PROPN
ejpam-3272	362	10	therefore	therefore	ADV
ejpam-3272	362	11	,	,	PUNCT
ejpam-3272	362	12	for	for	ADP
ejpam-3272	362	13	all	all	DET
ejpam-3272	362	14	a	a	DET
ejpam-3272	362	15	∈	∈	PROPN
ejpam-3272	362	16	lfr	lfr	NOUN
ejpam-3272	362	17	,	,	PUNCT
ejpam-3272	362	18	a	a	DET
ejpam-3272	362	19	=	=	SYM
ejpam-3272	362	20	∨	∨	X
ejpam-3272	362	21	{	{	PUNCT
ejpam-3272	362	22	x	x	SYM
ejpam-3272	362	23	∈	∈	PROPN
ejpam-3272	362	24	lfr	lfr	NOUN
ejpam-3272	362	25	:	:	PUNCT
ejpam-3272	362	26	xc	xc	PROPN
ejpam-3272	362	27	a	a	DET
ejpam-3272	362	28	}	}	PUNCT
ejpam-3272	362	29	≤	≤	NUM
ejpam-3272	362	30	∨	∨	NUM
ejpam-3272	362	31	{	{	PUNCT
ejpam-3272	362	32	x	x	SYM
ejpam-3272	362	33	∈	∈	PROPN
ejpam-3272	362	34	lfr	lfr	NOUN
ejpam-3272	362	35	:	:	PUNCT
ejpam-3272	362	36	x	x	X
ejpam-3272	362	37	≺≺fr	≺≺fr	PROPN
ejpam-3272	362	38	a	a	DET
ejpam-3272	362	39	}	}	PUNCT
ejpam-3272	362	40	≤	≤	NOUN
ejpam-3272	362	41	a	a	PRON
ejpam-3272	362	42	and	and	CCONJ
ejpam-3272	362	43	hence	hence	ADV
ejpam-3272	362	44	l	l	NOUN
ejpam-3272	362	45	=	=	SYM
ejpam-3272	362	46	(	(	PUNCT
ejpam-3272	362	47	le	le	X
ejpam-3272	362	48	,	,	PUNCT
ejpam-3272	362	49	lfr	lfr	PROPN
ejpam-3272	362	50	,	,	PUNCT
ejpam-3272	362	51	lcf	lcf	PROPN
ejpam-3272	362	52	)	)	PUNCT
ejpam-3272	362	53	is	be	AUX
ejpam-3272	362	54	completely	completely	ADV
ejpam-3272	362	55	regular	regular	ADJ
ejpam-3272	362	56	.	.	PUNCT
ejpam-3272	363	1	as	as	SCONJ
ejpam-3272	363	2	is	be	AUX
ejpam-3272	363	3	well	well	ADV
ejpam-3272	363	4	known	know	VERB
ejpam-3272	363	5	,	,	PUNCT
ejpam-3272	363	6	normality	normality	NOUN
ejpam-3272	363	7	is	be	AUX
ejpam-3272	363	8	a	a	DET
ejpam-3272	363	9	separation	separation	NOUN
ejpam-3272	363	10	axiom	axiom	NOUN
ejpam-3272	363	11	that	that	PRON
ejpam-3272	363	12	can	can	AUX
ejpam-3272	363	13	be	be	AUX
ejpam-3272	363	14	defined	define	VERB
ejpam-3272	363	15	purely	purely	ADV
ejpam-3272	363	16	in	in	ADP
ejpam-3272	363	17	terms	term	NOUN
ejpam-3272	363	18	of	of	ADP
ejpam-3272	363	19	the	the	DET
ejpam-3272	363	20	open	open	ADJ
ejpam-3272	363	21	and	and	CCONJ
ejpam-3272	363	22	closed	closed	ADJ
ejpam-3272	363	23	sets	set	NOUN
ejpam-3272	363	24	.	.	PUNCT
ejpam-3272	364	1	in	in	ADP
ejpam-3272	364	2	other	other	ADJ
ejpam-3272	364	3	words	word	NOUN
ejpam-3272	364	4	,	,	PUNCT
ejpam-3272	364	5	its	its	PRON
ejpam-3272	364	6	definition	definition	NOUN
ejpam-3272	364	7	is	be	AUX
ejpam-3272	364	8	not	not	PART
ejpam-3272	364	9	based	base	VERB
ejpam-3272	364	10	on	on	ADP
ejpam-3272	364	11	points	point	NOUN
ejpam-3272	364	12	,	,	PUNCT
ejpam-3272	364	13	which	which	PRON
ejpam-3272	364	14	makes	make	VERB
ejpam-3272	364	15	it	it	PRON
ejpam-3272	364	16	easier	easy	ADJ
ejpam-3272	364	17	to	to	PART
ejpam-3272	364	18	discuss	discuss	VERB
ejpam-3272	364	19	them	they	PRON
ejpam-3272	364	20	in	in	ADP
ejpam-3272	364	21	the	the	DET
ejpam-3272	364	22	point	point	NOUN
ejpam-3272	364	23	-	-	PUNCT
ejpam-3272	364	24	free	free	ADJ
ejpam-3272	364	25	context	context	NOUN
ejpam-3272	364	26	.	.	PUNCT
ejpam-3272	365	1	definition	definition	NOUN
ejpam-3272	365	2	6	6	NUM
ejpam-3272	365	3	.	.	PUNCT
ejpam-3272	366	1	a	a	DET
ejpam-3272	366	2	diframe	diframe	NOUN
ejpam-3272	366	3	is	be	AUX
ejpam-3272	366	4	said	say	VERB
ejpam-3272	366	5	to	to	PART
ejpam-3272	366	6	be	be	AUX
ejpam-3272	366	7	(	(	PUNCT
ejpam-3272	366	8	i	i	NOUN
ejpam-3272	366	9	)	)	PUNCT
ejpam-3272	366	10	normal	normal	ADJ
ejpam-3272	366	11	if	if	SCONJ
ejpam-3272	366	12	,	,	PUNCT
ejpam-3272	366	13	for	for	ADP
ejpam-3272	366	14	any	any	DET
ejpam-3272	366	15	c	c	PROPN
ejpam-3272	366	16	∈	∈	PROPN
ejpam-3272	366	17	lcf	lcf	NOUN
ejpam-3272	366	18	and	and	CCONJ
ejpam-3272	366	19	a	a	DET
ejpam-3272	366	20	∈	∈	PROPN
ejpam-3272	366	21	lfr	lfr	NOUN
ejpam-3272	366	22	such	such	ADJ
ejpam-3272	366	23	that	that	SCONJ
ejpam-3272	366	24	c	c	PROPN
ejpam-3272	366	25	≤	≤	PROPN
ejpam-3272	367	1	a	a	PRON
ejpam-3272	368	1	,	,	PUNCT
ejpam-3272	368	2	there	there	PRON
ejpam-3272	368	3	exists	exist	VERB
ejpam-3272	368	4	a	a	DET
ejpam-3272	368	5	b	b	PROPN
ejpam-3272	368	6	∈	∈	PROPN
ejpam-3272	368	7	lfr	lfr	NOUN
ejpam-3272	368	8	such	such	ADJ
ejpam-3272	369	1	that	that	SCONJ
ejpam-3272	369	2	c	c	PROPN
ejpam-3272	369	3	≤	≤	NUM
ejpam-3272	369	4	b	b	X
ejpam-3272	369	5	≤	≤	NOUN
ejpam-3272	369	6	[	[	X
ejpam-3272	369	7	b	b	X
ejpam-3272	369	8	]	]	PUNCT
ejpam-3272	369	9	≤	≤	NUM
ejpam-3272	369	10	a.	a.	NOUN
ejpam-3272	369	11	(	(	PUNCT
ejpam-3272	369	12	ii	ii	PROPN
ejpam-3272	369	13	)	)	PUNCT
ejpam-3272	369	14	t4	t4	PROPN
ejpam-3272	369	15	if	if	SCONJ
ejpam-3272	369	16	normal	normal	ADJ
ejpam-3272	369	17	and	and	CCONJ
ejpam-3272	369	18	t1	t1	NOUN
ejpam-3272	369	19	.	.	PUNCT
ejpam-3272	370	1	(	(	PUNCT
ejpam-3272	370	2	iii	iii	NOUN
ejpam-3272	370	3	)	)	PUNCT
ejpam-3272	370	4	co	co	NOUN
ejpam-3272	370	5	-	-	NOUN
ejpam-3272	370	6	t4	t4	PROPN
ejpam-3272	370	7	if	if	SCONJ
ejpam-3272	370	8	normal	normal	ADJ
ejpam-3272	370	9	and	and	CCONJ
ejpam-3272	370	10	co	co	NOUN
ejpam-3272	370	11	-	-	NOUN
ejpam-3272	370	12	t1	t1	NOUN
ejpam-3272	370	13	.	.	PUNCT
ejpam-3272	371	1	remark	remark	NOUN
ejpam-3272	371	2	5	5	NUM
ejpam-3272	371	3	.	.	PUNCT
ejpam-3272	372	1	normality	normality	NOUN
ejpam-3272	372	2	is	be	AUX
ejpam-3272	372	3	self	self	NOUN
ejpam-3272	372	4	-	-	PUNCT
ejpam-3272	372	5	dual	dual	ADJ
ejpam-3272	372	6	.	.	PUNCT
ejpam-3272	373	1	hence	hence	ADV
ejpam-3272	373	2	we	we	PRON
ejpam-3272	373	3	can	can	AUX
ejpam-3272	373	4	use	use	VERB
ejpam-3272	373	5	the	the	DET
ejpam-3272	373	6	equivalent	equivalent	ADJ
ejpam-3272	373	7	definition	definition	NOUN
ejpam-3272	373	8	:	:	PUNCT
ejpam-3272	373	9	“	"	PUNCT
ejpam-3272	373	10	for	for	ADP
ejpam-3272	373	11	any	any	DET
ejpam-3272	373	12	c	c	PROPN
ejpam-3272	373	13	∈	∈	PROPN
ejpam-3272	373	14	lcf	lcf	NOUN
ejpam-3272	373	15	and	and	CCONJ
ejpam-3272	373	16	a	a	DET
ejpam-3272	373	17	∈	∈	PROPN
ejpam-3272	373	18	lfr	lfr	NOUN
ejpam-3272	374	1	such	such	ADJ
ejpam-3272	374	2	that	that	SCONJ
ejpam-3272	374	3	c	c	PROPN
ejpam-3272	374	4	≤	≤	PROPN
ejpam-3272	375	1	a	a	PRON
ejpam-3272	376	1	there	there	PRON
ejpam-3272	376	2	exists	exist	VERB
ejpam-3272	376	3	a	a	DET
ejpam-3272	376	4	k	k	PROPN
ejpam-3272	376	5	∈	∈	PROPN
ejpam-3272	376	6	lcf	lcf	NOUN
ejpam-3272	376	7	such	such	ADJ
ejpam-3272	376	8	that	that	SCONJ
ejpam-3272	376	9	c	c	PROPN
ejpam-3272	376	10	≤]k[≤	≤]k[≤	NOUN
ejpam-3272	376	11	k	k	PROPN
ejpam-3272	376	12	≤	≤	NUM
ejpam-3272	376	13	a.	a.	NOUN
ejpam-3272	376	14	”	"	PUNCT
ejpam-3272	376	15	this	this	PRON
ejpam-3272	376	16	is	be	AUX
ejpam-3272	376	17	easily	easily	ADV
ejpam-3272	376	18	obtained	obtain	VERB
ejpam-3272	376	19	by	by	ADP
ejpam-3272	376	20	setting	set	VERB
ejpam-3272	376	21	k	k	X
ejpam-3272	377	1	=	=	PUNCT
ejpam-3272	378	1	[	[	X
ejpam-3272	378	2	b	b	X
ejpam-3272	378	3	]	]	X
ejpam-3272	378	4	in	in	ADP
ejpam-3272	378	5	the	the	DET
ejpam-3272	378	6	definition	definition	NOUN
ejpam-3272	378	7	of	of	ADP
ejpam-3272	378	8	normality	normality	NOUN
ejpam-3272	378	9	.	.	PUNCT
ejpam-3272	379	1	proposition	proposition	NOUN
ejpam-3272	379	2	13	13	NUM
ejpam-3272	379	3	.	.	PUNCT
ejpam-3272	380	1	let	let	VERB
ejpam-3272	380	2	c	c	PRON
ejpam-3272	380	3	be	be	AUX
ejpam-3272	380	4	a	a	DET
ejpam-3272	380	5	binary	binary	ADJ
ejpam-3272	380	6	relation	relation	NOUN
ejpam-3272	380	7	on	on	ADP
ejpam-3272	380	8	le	le	ADP
ejpam-3272	381	1	such	such	ADJ
ejpam-3272	381	2	that	that	SCONJ
ejpam-3272	381	3	“	"	PUNCT
ejpam-3272	381	4	a	a	DET
ejpam-3272	381	5	c	c	PROPN
ejpam-3272	381	6	b	b	PROPN
ejpam-3272	381	7	iff	iff	PROPN
ejpam-3272	382	1	[	[	X
ejpam-3272	382	2	a	a	X
ejpam-3272	382	3	]	]	X
ejpam-3272	382	4	≤]b	≤]b	ADP
ejpam-3272	382	5	[	[	X
ejpam-3272	382	6	”	"	PUNCT
ejpam-3272	382	7	.	.	PUNCT
ejpam-3272	383	1	then	then	ADV
ejpam-3272	383	2	l	l	NOUN
ejpam-3272	383	3	=	=	SYM
ejpam-3272	383	4	(	(	PUNCT
ejpam-3272	383	5	le	le	X
ejpam-3272	383	6	,	,	PUNCT
ejpam-3272	383	7	lfr	lfr	PROPN
ejpam-3272	383	8	,	,	PUNCT
ejpam-3272	383	9	lcf	lcf	PROPN
ejpam-3272	383	10	)	)	PUNCT
ejpam-3272	383	11	is	be	AUX
ejpam-3272	383	12	normal	normal	ADJ
ejpam-3272	383	13	if	if	SCONJ
ejpam-3272	383	14	and	and	CCONJ
ejpam-3272	383	15	only	only	ADV
ejpam-3272	383	16	if	if	SCONJ
ejpam-3272	383	17	c	c	PROPN
ejpam-3272	383	18	is	be	AUX
ejpam-3272	383	19	a	a	DET
ejpam-3272	383	20	urysohn	urysohn	PROPN
ejpam-3272	383	21	relation	relation	NOUN
ejpam-3272	383	22	on	on	ADP
ejpam-3272	383	23	le	le	X
ejpam-3272	383	24	.	.	PUNCT
ejpam-3272	384	1	proof	proof	NOUN
ejpam-3272	384	2	.	.	PUNCT
ejpam-3272	385	1	suppose	suppose	VERB
ejpam-3272	385	2	l	l	NOUN
ejpam-3272	385	3	is	be	AUX
ejpam-3272	385	4	a	a	DET
ejpam-3272	385	5	normal	normal	ADJ
ejpam-3272	385	6	diframe	diframe	NOUN
ejpam-3272	385	7	.	.	PUNCT
ejpam-3272	386	1	then	then	ADV
ejpam-3272	386	2	we	we	PRON
ejpam-3272	386	3	claim	claim	VERB
ejpam-3272	386	4	that	that	SCONJ
ejpam-3272	386	5	the	the	DET
ejpam-3272	386	6	relation	relation	NOUN
ejpam-3272	386	7	c	c	NOUN
ejpam-3272	386	8	given	give	VERB
ejpam-3272	386	9	in	in	ADP
ejpam-3272	386	10	the	the	DET
ejpam-3272	386	11	proposition	proposition	NOUN
ejpam-3272	386	12	satisfies	satisfy	VERB
ejpam-3272	386	13	the	the	DET
ejpam-3272	386	14	properties	property	NOUN
ejpam-3272	386	15	(	(	PUNCT
ejpam-3272	386	16	u1)−	u1)−	PROPN
ejpam-3272	386	17	(	(	PUNCT
ejpam-3272	386	18	u3	u3	PROPN
ejpam-3272	386	19	)	)	PUNCT
ejpam-3272	386	20	.	.	PUNCT
ejpam-3272	387	1	we	we	PRON
ejpam-3272	387	2	only	only	ADV
ejpam-3272	387	3	prove	prove	VERB
ejpam-3272	387	4	(	(	PUNCT
ejpam-3272	387	5	u3	u3	NOUN
ejpam-3272	387	6	)	)	PUNCT
ejpam-3272	387	7	since	since	SCONJ
ejpam-3272	387	8	(	(	PUNCT
ejpam-3272	387	9	u1	u1	NOUN
ejpam-3272	387	10	)	)	PUNCT
ejpam-3272	387	11	and	and	CCONJ
ejpam-3272	387	12	(	(	PUNCT
ejpam-3272	387	13	u2	u2	NOUN
ejpam-3272	387	14	)	)	PUNCT
ejpam-3272	387	15	are	be	AUX
ejpam-3272	387	16	straightforward	straightforward	ADJ
ejpam-3272	387	17	.	.	PUNCT
ejpam-3272	388	1	let	let	VERB
ejpam-3272	388	2	a	a	DET
ejpam-3272	388	3	c	c	PROPN
ejpam-3272	388	4	b.	b.	PROPN
ejpam-3272	389	1	then	then	ADV
ejpam-3272	389	2	[	[	X
ejpam-3272	389	3	a	a	X
ejpam-3272	389	4	]	]	X
ejpam-3272	389	5	≤]b	≤]b	ADP
ejpam-3272	389	6	[	[	PUNCT
ejpam-3272	389	7	and	and	CCONJ
ejpam-3272	389	8	hence	hence	ADV
ejpam-3272	389	9	,	,	PUNCT
ejpam-3272	389	10	by	by	ADP
ejpam-3272	389	11	normality	normality	NOUN
ejpam-3272	389	12	,	,	PUNCT
ejpam-3272	389	13	there	there	PRON
ejpam-3272	389	14	exists	exist	VERB
ejpam-3272	389	15	a	a	DET
ejpam-3272	389	16	c	c	PROPN
ejpam-3272	389	17	∈	∈	PROPN
ejpam-3272	389	18	lfr	lfr	NOUN
ejpam-3272	389	19	such	such	ADJ
ejpam-3272	389	20	that	that	SCONJ
ejpam-3272	389	21	[	[	X
ejpam-3272	389	22	a	a	X
ejpam-3272	389	23	]	]	X
ejpam-3272	389	24	≤]c[=	≤]c[=	PUNCT
ejpam-3272	389	25	c	c	NOUN
ejpam-3272	389	26	≤	≤	X
ejpam-3272	390	1	[	[	X
ejpam-3272	390	2	c	c	X
ejpam-3272	390	3	]	]	X
ejpam-3272	390	4	≤]b	≤]b	ADP
ejpam-3272	390	5	[	[	NOUN
ejpam-3272	390	6	.	.	PUNCT
ejpam-3272	391	1	thus	thus	ADV
ejpam-3272	391	2	we	we	PRON
ejpam-3272	391	3	have	have	VERB
ejpam-3272	391	4	ac	ac	PROPN
ejpam-3272	391	5	cc	cc	PROPN
ejpam-3272	391	6	b.	b.	PROPN
ejpam-3272	391	7	for	for	ADP
ejpam-3272	391	8	the	the	DET
ejpam-3272	391	9	converse	converse	NOUN
ejpam-3272	391	10	,	,	PUNCT
ejpam-3272	391	11	let	let	VERB
ejpam-3272	391	12	c	c	NOUN
ejpam-3272	391	13	≤	≤	VERB
ejpam-3272	391	14	a	a	PRON
ejpam-3272	391	15	for	for	ADP
ejpam-3272	391	16	any	any	DET
ejpam-3272	391	17	c	c	PROPN
ejpam-3272	391	18	∈	∈	PROPN
ejpam-3272	391	19	lcf	lcf	NOUN
ejpam-3272	391	20	and	and	CCONJ
ejpam-3272	391	21	a	a	DET
ejpam-3272	391	22	∈	∈	PROPN
ejpam-3272	391	23	lfr	lfr	NOUN
ejpam-3272	391	24	.	.	PUNCT
ejpam-3272	392	1	then	then	ADV
ejpam-3272	392	2	c	c	PROPN
ejpam-3272	392	3	c	c	PROPN
ejpam-3272	392	4	a	a	NOUN
ejpam-3272	392	5	and	and	CCONJ
ejpam-3272	392	6	hence	hence	ADV
ejpam-3272	392	7	,	,	PUNCT
ejpam-3272	392	8	by	by	ADP
ejpam-3272	392	9	(	(	PUNCT
ejpam-3272	392	10	u3	u3	PROPN
ejpam-3272	392	11	)	)	PUNCT
ejpam-3272	392	12	,	,	PUNCT
ejpam-3272	392	13	there	there	PRON
ejpam-3272	392	14	exists	exist	VERB
ejpam-3272	392	15	a	a	DET
ejpam-3272	392	16	b	b	NOUN
ejpam-3272	392	17	∈	∈	NOUN
ejpam-3272	392	18	le	le	ADP
ejpam-3272	392	19	such	such	ADJ
ejpam-3272	392	20	that	that	SCONJ
ejpam-3272	392	21	cc	cc	PROPN
ejpam-3272	392	22	bc	bc	PROPN
ejpam-3272	392	23	a.	a.	PROPN
ejpam-3272	392	24	now	now	ADV
ejpam-3272	392	25	we	we	PRON
ejpam-3272	392	26	have	have	AUX
ejpam-3272	392	27	c	c	NOUN
ejpam-3272	392	28	≤	≤	X
ejpam-3272	393	1	[	[	X
ejpam-3272	393	2	c	c	X
ejpam-3272	393	3	]	]	X
ejpam-3272	393	4	≤]b[≤	≤]b[≤	NOUN
ejpam-3272	394	1	b	b	X
ejpam-3272	394	2	≤	≤	NOUN
ejpam-3272	394	3	[	[	X
ejpam-3272	394	4	b	b	X
ejpam-3272	394	5	]	]	X
ejpam-3272	394	6	≤]a[≤	≤]a[≤	NOUN
ejpam-3272	394	7	a.	a.	NOUN
ejpam-3272	394	8	setting	set	VERB
ejpam-3272	394	9	d	d	PROPN
ejpam-3272	394	10	=]	=]	PROPN
ejpam-3272	394	11	b	b	PROPN
ejpam-3272	394	12	[	[	PUNCT
ejpam-3272	394	13	we	we	PRON
ejpam-3272	394	14	obtain	obtain	VERB
ejpam-3272	394	15	c	c	NOUN
ejpam-3272	394	16	≤	≤	NUM
ejpam-3272	394	17	d	d	NOUN
ejpam-3272	394	18	≤	≤	NOUN
ejpam-3272	395	1	[	[	X
ejpam-3272	395	2	d	d	X
ejpam-3272	395	3	]	]	X
ejpam-3272	395	4	≤	≤	NUM
ejpam-3272	395	5	a.	a.	NOUN
ejpam-3272	395	6	thus	thus	ADV
ejpam-3272	395	7	l	l	NOUN
ejpam-3272	395	8	is	be	AUX
ejpam-3272	395	9	a	a	DET
ejpam-3272	395	10	normal	normal	ADJ
ejpam-3272	395	11	diframe	diframe	NOUN
ejpam-3272	395	12	.	.	PUNCT
ejpam-3272	396	1	e.	e.	PROPN
ejpam-3272	396	2	korkmaz	korkmaz	PROPN
ejpam-3272	396	3	,	,	PUNCT
ejpam-3272	396	4	r.	r.	PROPN
ejpam-3272	396	5	ertürk	ertürk	PROPN
ejpam-3272	396	6	/	/	SYM
ejpam-3272	396	7	eur	eur	PROPN
ejpam-3272	396	8	.	.	PUNCT
ejpam-3272	397	1	j.	j.	PROPN
ejpam-3272	397	2	pure	pure	PROPN
ejpam-3272	397	3	appl	appl	PROPN
ejpam-3272	397	4	.	.	PROPN
ejpam-3272	397	5	math	math	PROPN
ejpam-3272	397	6	,	,	PUNCT
ejpam-3272	397	7	11	11	NUM
ejpam-3272	397	8	(	(	PUNCT
ejpam-3272	397	9	3	3	NUM
ejpam-3272	397	10	)	)	PUNCT
ejpam-3272	397	11	(	(	PUNCT
ejpam-3272	397	12	2018	2018	NUM
ejpam-3272	397	13	)	)	PUNCT
ejpam-3272	397	14	,	,	PUNCT
ejpam-3272	397	15	612	612	NUM
ejpam-3272	397	16	-	-	SYM
ejpam-3272	397	17	627	627	NUM
ejpam-3272	397	18	625	625	NUM
ejpam-3272	397	19	example	example	NOUN
ejpam-3272	397	20	5	5	NUM
ejpam-3272	397	21	.	.	PUNCT
ejpam-3272	397	22	normality	normality	NOUN
ejpam-3272	397	23	does	do	AUX
ejpam-3272	397	24	not	not	PART
ejpam-3272	397	25	imply	imply	VERB
ejpam-3272	397	26	regularity	regularity	NOUN
ejpam-3272	397	27	.	.	PUNCT
ejpam-3272	398	1	consider	consider	VERB
ejpam-3272	398	2	the	the	DET
ejpam-3272	398	3	diframe	diframe	NOUN
ejpam-3272	398	4	l	l	NOUN
ejpam-3272	398	5	of	of	ADP
ejpam-3272	398	6	example	example	NOUN
ejpam-3272	398	7	1	1	NUM
ejpam-3272	398	8	(	(	PUNCT
ejpam-3272	398	9	iii	iii	NOUN
ejpam-3272	398	10	)	)	PUNCT
ejpam-3272	398	11	.	.	PUNCT
ejpam-3272	399	1	l	l	NOUN
ejpam-3272	399	2	is	be	AUX
ejpam-3272	399	3	normal	normal	ADJ
ejpam-3272	399	4	:	:	PUNCT
ejpam-3272	399	5	let	let	VERB
ejpam-3272	399	6	c	c	PROPN
ejpam-3272	399	7	∈	∈	PROPN
ejpam-3272	399	8	lcf	lcf	PROPN
ejpam-3272	399	9	,	,	PUNCT
ejpam-3272	399	10	a	a	DET
ejpam-3272	399	11	∈	∈	PROPN
ejpam-3272	399	12	lfr	lfr	NOUN
ejpam-3272	399	13	with	with	ADP
ejpam-3272	399	14	c	c	PROPN
ejpam-3272	399	15	⊆	⊆	NUM
ejpam-3272	399	16	a.	a.	NOUN
ejpam-3272	399	17	then	then	ADV
ejpam-3272	399	18	there	there	PRON
ejpam-3272	399	19	are	be	VERB
ejpam-3272	399	20	three	three	NUM
ejpam-3272	399	21	cases	case	NOUN
ejpam-3272	399	22	to	to	PART
ejpam-3272	399	23	consider	consider	VERB
ejpam-3272	399	24	:	:	PUNCT
ejpam-3272	399	25	(	(	PUNCT
ejpam-3272	399	26	i	i	NOUN
ejpam-3272	399	27	)	)	PUNCT
ejpam-3272	399	28	c	c	PROPN
ejpam-3272	400	1	=	=	PUNCT
ejpam-3272	400	2	a	a	DET
ejpam-3272	400	3	=	=	NOUN
ejpam-3272	400	4	∅	∅	NOUN
ejpam-3272	400	5	,	,	PUNCT
ejpam-3272	400	6	(	(	PUNCT
ejpam-3272	400	7	ii	ii	NOUN
ejpam-3272	400	8	)	)	PUNCT
ejpam-3272	400	9	c	c	NOUN
ejpam-3272	400	10	=	=	PUNCT
ejpam-3272	400	11	a	a	PRON
ejpam-3272	400	12	=	=	SYM
ejpam-3272	400	13	r	r	NOUN
ejpam-3272	400	14	,	,	PUNCT
ejpam-3272	400	15	(	(	PUNCT
ejpam-3272	400	16	iii	iii	NOUN
ejpam-3272	400	17	)	)	PUNCT
ejpam-3272	400	18	c	c	NOUN
ejpam-3272	400	19	6=	6=	SYM
ejpam-3272	400	20	r	r	NOUN
ejpam-3272	400	21	,	,	PUNCT
ejpam-3272	400	22	a	a	DET
ejpam-3272	400	23	=	=	X
ejpam-3272	400	24	r.	r.	NOUN
ejpam-3272	400	25	we	we	PRON
ejpam-3272	400	26	may	may	AUX
ejpam-3272	400	27	take	take	VERB
ejpam-3272	400	28	b	b	NOUN
ejpam-3272	400	29	=	=	NOUN
ejpam-3272	400	30	∅	∅	NOUN
ejpam-3272	400	31	in	in	ADP
ejpam-3272	400	32	case	case	NOUN
ejpam-3272	400	33	(	(	PUNCT
ejpam-3272	400	34	i	i	NOUN
ejpam-3272	400	35	)	)	PUNCT
ejpam-3272	400	36	,	,	PUNCT
ejpam-3272	400	37	and	and	CCONJ
ejpam-3272	400	38	b	b	X
ejpam-3272	400	39	=	=	SYM
ejpam-3272	400	40	r	r	NOUN
ejpam-3272	400	41	in	in	ADP
ejpam-3272	400	42	cases	case	NOUN
ejpam-3272	400	43	(	(	PUNCT
ejpam-3272	400	44	ii	ii	NOUN
ejpam-3272	400	45	)	)	PUNCT
ejpam-3272	400	46	and	and	CCONJ
ejpam-3272	400	47	(	(	PUNCT
ejpam-3272	400	48	iii	iii	NOUN
ejpam-3272	400	49	)	)	PUNCT
ejpam-3272	400	50	,	,	PUNCT
ejpam-3272	400	51	showing	show	VERB
ejpam-3272	400	52	l	l	NOUN
ejpam-3272	400	53	is	be	AUX
ejpam-3272	400	54	regular	regular	ADJ
ejpam-3272	400	55	.	.	PUNCT
ejpam-3272	401	1	however	however	ADV
ejpam-3272	401	2	,	,	PUNCT
ejpam-3272	401	3	l	l	NOUN
ejpam-3272	401	4	is	be	AUX
ejpam-3272	401	5	obviously	obviously	ADV
ejpam-3272	401	6	not	not	PART
ejpam-3272	401	7	normal	normal	ADJ
ejpam-3272	401	8	.	.	PUNCT
ejpam-3272	402	1	proposition	proposition	NOUN
ejpam-3272	402	2	14	14	NUM
ejpam-3272	402	3	.	.	PUNCT
ejpam-3272	403	1	(	(	PUNCT
ejpam-3272	403	2	i	i	NOUN
ejpam-3272	403	3	)	)	PUNCT
ejpam-3272	403	4	every	every	DET
ejpam-3272	403	5	normal	normal	ADJ
ejpam-3272	403	6	r0	r0	NOUN
ejpam-3272	403	7	diframe	diframe	NOUN
ejpam-3272	403	8	is	be	AUX
ejpam-3272	403	9	regular	regular	ADJ
ejpam-3272	403	10	.	.	PUNCT
ejpam-3272	404	1	(	(	PUNCT
ejpam-3272	404	2	ii	ii	NOUN
ejpam-3272	404	3	)	)	PUNCT
ejpam-3272	404	4	every	every	DET
ejpam-3272	404	5	normal	normal	ADJ
ejpam-3272	404	6	co	co	ADJ
ejpam-3272	404	7	-	-	ADJ
ejpam-3272	404	8	r0	r0	ADJ
ejpam-3272	404	9	diframe	diframe	NOUN
ejpam-3272	404	10	is	be	AUX
ejpam-3272	404	11	co	co	ADJ
ejpam-3272	404	12	-	-	ADJ
ejpam-3272	404	13	regular	regular	ADJ
ejpam-3272	404	14	.	.	PUNCT
ejpam-3272	405	1	proof	proof	NOUN
ejpam-3272	405	2	.	.	PUNCT
ejpam-3272	406	1	(	(	PUNCT
ejpam-3272	406	2	i	i	NOUN
ejpam-3272	406	3	)	)	PUNCT
ejpam-3272	406	4	let	let	VERB
ejpam-3272	406	5	a	a	DET
ejpam-3272	406	6	∈	∈	PROPN
ejpam-3272	406	7	lfr	lfr	NOUN
ejpam-3272	406	8	and	and	CCONJ
ejpam-3272	406	9	set	set	VERB
ejpam-3272	406	10	c	c	PROPN
ejpam-3272	406	11	=	=	SYM
ejpam-3272	406	12	∨	∨	X
ejpam-3272	406	13	{	{	PUNCT
ejpam-3272	406	14	b	b	PROPN
ejpam-3272	406	15	∈	∈	PROPN
ejpam-3272	406	16	lfr	lfr	NOUN
ejpam-3272	406	17	:	:	PUNCT
ejpam-3272	406	18	b	b	X
ejpam-3272	407	1	≺fr	≺fr	X
ejpam-3272	407	2	a	a	X
ejpam-3272	407	3	}	}	PUNCT
ejpam-3272	407	4	.	.	PUNCT
ejpam-3272	408	1	clearly	clearly	ADV
ejpam-3272	408	2	,	,	PUNCT
ejpam-3272	408	3	c	c	PROPN
ejpam-3272	408	4	≤	≤	NUM
ejpam-3272	408	5	a.	a.	NOUN
ejpam-3272	408	6	on	on	ADP
ejpam-3272	408	7	the	the	DET
ejpam-3272	408	8	other	other	ADJ
ejpam-3272	408	9	hand	hand	NOUN
ejpam-3272	408	10	,	,	PUNCT
ejpam-3272	408	11	o(a	o(a	NUM
ejpam-3272	408	12	)	)	PUNCT
ejpam-3272	408	13	=	=	SYM
ejpam-3272	408	14	∨	∨	X
ejpam-3272	408	15	{	{	PUNCT
ejpam-3272	408	16	o(k	o(k	PROPN
ejpam-3272	408	17	)	)	PUNCT
ejpam-3272	408	18	:	:	PUNCT
ejpam-3272	408	19	k	k	PROPN
ejpam-3272	408	20	∈	∈	PROPN
ejpam-3272	408	21	lcf	lcf	PROPN
ejpam-3272	408	22	and	and	CCONJ
ejpam-3272	408	23	k	k	PROPN
ejpam-3272	408	24	≤	≤	PROPN
ejpam-3272	408	25	a	a	X
ejpam-3272	408	26	}	}	PUNCT
ejpam-3272	408	27	since	since	SCONJ
ejpam-3272	408	28	l	l	NOUN
ejpam-3272	408	29	is	be	AUX
ejpam-3272	408	30	r0	r0	NOUN
ejpam-3272	408	31	.	.	PUNCT
ejpam-3272	409	1	hence	hence	ADV
ejpam-3272	409	2	,	,	PUNCT
ejpam-3272	409	3	to	to	PART
ejpam-3272	409	4	prove	prove	VERB
ejpam-3272	409	5	a	a	DET
ejpam-3272	409	6	≤	≤	NUM
ejpam-3272	409	7	c	c	NOUN
ejpam-3272	409	8	,	,	PUNCT
ejpam-3272	409	9	it	it	PRON
ejpam-3272	409	10	is	be	AUX
ejpam-3272	409	11	enough	enough	ADJ
ejpam-3272	409	12	to	to	PART
ejpam-3272	409	13	show	show	VERB
ejpam-3272	409	14	that	that	SCONJ
ejpam-3272	409	15	o(a	o(a	NOUN
ejpam-3272	409	16	)	)	PUNCT
ejpam-3272	409	17	⊆	⊆	NUM
ejpam-3272	409	18	o(c	o(c	NUM
ejpam-3272	409	19	)	)	PUNCT
ejpam-3272	409	20	,	,	PUNCT
ejpam-3272	409	21	that	that	ADV
ejpam-3272	409	22	is	be	AUX
ejpam-3272	409	23	,	,	PUNCT
ejpam-3272	409	24	o(k	o(k	PROPN
ejpam-3272	409	25	)	)	PUNCT
ejpam-3272	409	26	⊆	⊆	NUM
ejpam-3272	409	27	o(c	o(c	ADP
ejpam-3272	409	28	)	)	PUNCT
ejpam-3272	409	29	for	for	ADP
ejpam-3272	409	30	all	all	DET
ejpam-3272	409	31	k	k	PROPN
ejpam-3272	409	32	∈	∈	PROPN
ejpam-3272	409	33	lcf	lcf	NOUN
ejpam-3272	409	34	with	with	ADP
ejpam-3272	409	35	k	k	PROPN
ejpam-3272	409	36	≤	≤	PROPN
ejpam-3272	409	37	a.	a.	NOUN
ejpam-3272	409	38	so	so	ADV
ejpam-3272	409	39	take	take	VERB
ejpam-3272	409	40	an	an	DET
ejpam-3272	409	41	element	element	NOUN
ejpam-3272	409	42	k	k	PROPN
ejpam-3272	409	43	∈	∈	PROPN
ejpam-3272	409	44	lcf	lcf	NOUN
ejpam-3272	409	45	such	such	ADJ
ejpam-3272	409	46	that	that	SCONJ
ejpam-3272	409	47	k	k	PROPN
ejpam-3272	409	48	≤	≤	PROPN
ejpam-3272	409	49	a.	a.	NOUN
ejpam-3272	409	50	then	then	ADV
ejpam-3272	409	51	,	,	PUNCT
ejpam-3272	409	52	by	by	ADP
ejpam-3272	409	53	normality	normality	NOUN
ejpam-3272	409	54	,	,	PUNCT
ejpam-3272	409	55	there	there	PRON
ejpam-3272	409	56	exists	exist	VERB
ejpam-3272	409	57	a	a	DET
ejpam-3272	409	58	b	b	PROPN
ejpam-3272	409	59	∈	∈	PROPN
ejpam-3272	409	60	lfr	lfr	NOUN
ejpam-3272	410	1	such	such	ADJ
ejpam-3272	410	2	that	that	SCONJ
ejpam-3272	410	3	k	k	PROPN
ejpam-3272	410	4	≤	≤	PROPN
ejpam-3272	410	5	b	b	X
ejpam-3272	410	6	≤	≤	NOUN
ejpam-3272	411	1	[	[	X
ejpam-3272	411	2	b	b	X
ejpam-3272	411	3	]	]	PUNCT
ejpam-3272	411	4	≤	≤	NUM
ejpam-3272	411	5	a	a	PRON
ejpam-3272	411	6	,	,	PUNCT
ejpam-3272	411	7	yielding	yield	VERB
ejpam-3272	411	8	b	b	PROPN
ejpam-3272	411	9	≺fr	≺fr	ADJ
ejpam-3272	411	10	a	a	PROPN
ejpam-3272	411	11	and	and	CCONJ
ejpam-3272	411	12	k	k	PROPN
ejpam-3272	411	13	≤	≤	PROPN
ejpam-3272	411	14	b.	b.	PROPN
ejpam-3272	412	1	thus	thus	ADV
ejpam-3272	412	2	k	k	X
ejpam-3272	412	3	≤	≤	PROPN
ejpam-3272	412	4	b	b	X
ejpam-3272	412	5	≤	≤	NUM
ejpam-3272	412	6	c	c	NOUN
ejpam-3272	412	7	,	,	PUNCT
ejpam-3272	412	8	and	and	CCONJ
ejpam-3272	412	9	hence	hence	ADV
ejpam-3272	412	10	o(k	o(k	NUM
ejpam-3272	412	11	)	)	PUNCT
ejpam-3272	412	12	⊆	⊆	NUM
ejpam-3272	412	13	o(c	o(c	NUM
ejpam-3272	412	14	)	)	PUNCT
ejpam-3272	412	15	,	,	PUNCT
ejpam-3272	412	16	as	as	SCONJ
ejpam-3272	412	17	required	require	VERB
ejpam-3272	412	18	.	.	PUNCT
ejpam-3272	413	1	proposition	proposition	NOUN
ejpam-3272	413	2	15	15	NUM
ejpam-3272	413	3	.	.	PUNCT
ejpam-3272	414	1	(	(	PUNCT
ejpam-3272	414	2	i	i	NOUN
ejpam-3272	414	3	)	)	PUNCT
ejpam-3272	414	4	a	a	DET
ejpam-3272	414	5	normal	normal	ADJ
ejpam-3272	414	6	r0	r0	NOUN
ejpam-3272	414	7	diframe	diframe	NOUN
ejpam-3272	414	8	is	be	AUX
ejpam-3272	414	9	completely	completely	ADV
ejpam-3272	414	10	regular	regular	ADJ
ejpam-3272	414	11	.	.	PUNCT
ejpam-3272	415	1	(	(	PUNCT
ejpam-3272	415	2	ii	ii	NOUN
ejpam-3272	415	3	)	)	PUNCT
ejpam-3272	415	4	a	a	DET
ejpam-3272	415	5	normal	normal	ADJ
ejpam-3272	415	6	co	co	NOUN
ejpam-3272	415	7	-	-	ADJ
ejpam-3272	415	8	r0	r0	ADJ
ejpam-3272	415	9	diframe	diframe	NOUN
ejpam-3272	415	10	is	be	AUX
ejpam-3272	415	11	completely	completely	ADV
ejpam-3272	415	12	co	co	ADJ
ejpam-3272	415	13	-	-	ADJ
ejpam-3272	415	14	regular	regular	ADJ
ejpam-3272	415	15	.	.	PUNCT
ejpam-3272	416	1	proof	proof	NOUN
ejpam-3272	416	2	.	.	PUNCT
ejpam-3272	417	1	we	we	PRON
ejpam-3272	417	2	will	will	AUX
ejpam-3272	417	3	just	just	ADV
ejpam-3272	417	4	prove	prove	VERB
ejpam-3272	417	5	the	the	DET
ejpam-3272	417	6	first	first	ADJ
ejpam-3272	417	7	statement	statement	NOUN
ejpam-3272	417	8	and	and	CCONJ
ejpam-3272	417	9	leave	leave	VERB
ejpam-3272	417	10	the	the	DET
ejpam-3272	417	11	other	other	ADJ
ejpam-3272	417	12	statement	statement	NOUN
ejpam-3272	417	13	to	to	ADP
ejpam-3272	417	14	the	the	DET
ejpam-3272	417	15	reader	reader	NOUN
ejpam-3272	417	16	.	.	PUNCT
ejpam-3272	418	1	since	since	SCONJ
ejpam-3272	418	2	each	each	DET
ejpam-3272	418	3	normal	normal	ADJ
ejpam-3272	418	4	r0	r0	NOUN
ejpam-3272	418	5	diframe	diframe	NOUN
ejpam-3272	418	6	is	be	AUX
ejpam-3272	418	7	regular	regular	ADJ
ejpam-3272	418	8	it	it	PRON
ejpam-3272	418	9	is	be	AUX
ejpam-3272	418	10	enough	enough	ADJ
ejpam-3272	418	11	to	to	PART
ejpam-3272	418	12	show	show	VERB
ejpam-3272	418	13	that	that	SCONJ
ejpam-3272	418	14	the	the	DET
ejpam-3272	418	15	relations	relation	NOUN
ejpam-3272	418	16	≺fr	≺fr	PROPN
ejpam-3272	418	17	and	and	CCONJ
ejpam-3272	418	18	≺≺fr	≺≺fr	PROPN
ejpam-3272	418	19	coincide	coincide	NOUN
ejpam-3272	418	20	in	in	ADP
ejpam-3272	418	21	a	a	DET
ejpam-3272	418	22	normal	normal	ADJ
ejpam-3272	418	23	diframe	diframe	NOUN
ejpam-3272	418	24	.	.	PUNCT
ejpam-3272	419	1	for	for	ADP
ejpam-3272	419	2	this	this	PRON
ejpam-3272	419	3	,	,	PUNCT
ejpam-3272	419	4	we	we	PRON
ejpam-3272	419	5	have	have	VERB
ejpam-3272	419	6	to	to	PART
ejpam-3272	419	7	prove	prove	VERB
ejpam-3272	419	8	that	that	SCONJ
ejpam-3272	419	9	≺fr	≺fr	PROPN
ejpam-3272	419	10	is	be	AUX
ejpam-3272	419	11	interpolative	interpolative	ADJ
ejpam-3272	419	12	.	.	PUNCT
ejpam-3272	420	1	if	if	SCONJ
ejpam-3272	420	2	a	a	DET
ejpam-3272	420	3	≺fr	≺fr	X
ejpam-3272	420	4	b	b	NOUN
ejpam-3272	420	5	then	then	ADV
ejpam-3272	420	6	there	there	PRON
ejpam-3272	420	7	exists	exist	VERB
ejpam-3272	420	8	a	a	DET
ejpam-3272	420	9	k	k	PROPN
ejpam-3272	420	10	∈	∈	PROPN
ejpam-3272	420	11	lcf	lcf	NOUN
ejpam-3272	420	12	such	such	ADJ
ejpam-3272	420	13	that	that	SCONJ
ejpam-3272	420	14	a	a	DET
ejpam-3272	420	15	≤	≤	PROPN
ejpam-3272	420	16	k	k	PROPN
ejpam-3272	420	17	≤	≤	PROPN
ejpam-3272	420	18	b.	b.	PROPN
ejpam-3272	420	19	moreover	moreover	ADV
ejpam-3272	420	20	,	,	PUNCT
ejpam-3272	420	21	by	by	ADP
ejpam-3272	420	22	normality	normality	NOUN
ejpam-3272	420	23	,	,	PUNCT
ejpam-3272	420	24	there	there	PRON
ejpam-3272	420	25	is	be	VERB
ejpam-3272	420	26	a	a	DET
ejpam-3272	420	27	d	d	PROPN
ejpam-3272	420	28	∈	∈	PROPN
ejpam-3272	420	29	lfr	lfr	NOUN
ejpam-3272	420	30	such	such	ADJ
ejpam-3272	420	31	that	that	SCONJ
ejpam-3272	420	32	a	a	DET
ejpam-3272	420	33	≤	≤	X
ejpam-3272	420	34	k	k	NOUN
ejpam-3272	420	35	≤	≤	NUM
ejpam-3272	421	1	d	d	NOUN
ejpam-3272	421	2	≤	≤	NOUN
ejpam-3272	422	1	[	[	X
ejpam-3272	422	2	d	d	X
ejpam-3272	422	3	]	]	X
ejpam-3272	422	4	≤	≤	NUM
ejpam-3272	422	5	b	b	NOUN
ejpam-3272	422	6	.	.	PUNCT
ejpam-3272	423	1	thus	thus	ADV
ejpam-3272	423	2	,	,	PUNCT
ejpam-3272	423	3	a	a	DET
ejpam-3272	423	4	≺fr	≺fr	X
ejpam-3272	423	5	d	d	X
ejpam-3272	423	6	≺fr	≺fr	X
ejpam-3272	423	7	b	b	PROPN
ejpam-3272	423	8	,	,	PUNCT
ejpam-3272	423	9	which	which	PRON
ejpam-3272	423	10	means	mean	VERB
ejpam-3272	423	11	that	that	SCONJ
ejpam-3272	423	12	≺fr	≺fr	PROPN
ejpam-3272	423	13	is	be	AUX
ejpam-3272	423	14	interpolative	interpolative	ADJ
ejpam-3272	423	15	.	.	PUNCT
ejpam-3272	424	1	thus	thus	ADV
ejpam-3272	424	2	we	we	PRON
ejpam-3272	424	3	have	have	AUX
ejpam-3272	424	4	,	,	PUNCT
ejpam-3272	424	5	by	by	ADP
ejpam-3272	424	6	proposition	proposition	NOUN
ejpam-3272	424	7	10	10	NUM
ejpam-3272	424	8	(	(	PUNCT
ejpam-3272	424	9	v	v	NOUN
ejpam-3272	424	10	)	)	PUNCT
ejpam-3272	424	11	,	,	PUNCT
ejpam-3272	424	12	≺fr=≺≺fr	≺fr=≺≺fr	PUNCT
ejpam-3272	424	13	.	.	PUNCT
ejpam-3272	425	1	corollary	corollary	ADJ
ejpam-3272	425	2	1	1	NUM
ejpam-3272	425	3	.	.	PUNCT
ejpam-3272	426	1	we	we	PRON
ejpam-3272	426	2	have	have	VERB
ejpam-3272	426	3	the	the	DET
ejpam-3272	426	4	following	follow	VERB
ejpam-3272	426	5	implications	implication	NOUN
ejpam-3272	426	6	in	in	ADP
ejpam-3272	426	7	a	a	DET
ejpam-3272	426	8	diframe	diframe	NOUN
ejpam-3272	426	9	:	:	PUNCT
ejpam-3272	426	10	normal	normal	ADJ
ejpam-3272	426	11	and	and	CCONJ
ejpam-3272	426	12	r0	r0	VERB
ejpam-3272	426	13	⇒	⇒	NOUN
ejpam-3272	426	14	completely	completely	ADV
ejpam-3272	426	15	regular⇒	regular⇒	ADJ
ejpam-3272	426	16	regular⇒	regular⇒	ADJ
ejpam-3272	426	17	r0	r0	NOUN
ejpam-3272	426	18	.	.	PUNCT
ejpam-3272	427	1	normal	normal	ADJ
ejpam-3272	427	2	and	and	CCONJ
ejpam-3272	427	3	co	co	ADJ
ejpam-3272	427	4	-	-	ADJ
ejpam-3272	427	5	r0	r0	ADJ
ejpam-3272	427	6	⇒	⇒	NOUN
ejpam-3272	427	7	completely	completely	ADV
ejpam-3272	427	8	co	co	ADJ
ejpam-3272	427	9	-	-	ADJ
ejpam-3272	427	10	regular⇒	regular⇒	ADJ
ejpam-3272	427	11	co	co	ADJ
ejpam-3272	427	12	-	-	ADJ
ejpam-3272	427	13	regular⇒	regular⇒	ADJ
ejpam-3272	427	14	co	co	NOUN
ejpam-3272	427	15	-	-	NOUN
ejpam-3272	427	16	r0	r0	NOUN
ejpam-3272	427	17	.	.	PUNCT
ejpam-3272	428	1	(	(	PUNCT
ejpam-3272	428	2	co-)t4	co-)t4	ADJ
ejpam-3272	428	3	⇒	⇒	NOUN
ejpam-3272	428	4	(	(	PUNCT
ejpam-3272	428	5	co-)t3	co-)t3	NOUN
ejpam-3272	428	6	1	1	NUM
ejpam-3272	428	7	2	2	NUM
ejpam-3272	428	8	⇒	⇒	NOUN
ejpam-3272	428	9	(	(	PUNCT
ejpam-3272	428	10	co-)t3	co-)t3	NOUN
ejpam-3272	428	11	⇒	⇒	NOUN
ejpam-3272	428	12	(	(	PUNCT
ejpam-3272	428	13	co-)t2	co-)t2	NOUN
ejpam-3272	428	14	⇒	⇒	NOUN
ejpam-3272	428	15	(	(	PUNCT
ejpam-3272	428	16	co-)t1	co-)t1	PROPN
ejpam-3272	428	17	⇒	⇒	NOUN
ejpam-3272	428	18	(	(	PUNCT
ejpam-3272	428	19	co-)t0	co-)t0	ADJ
ejpam-3272	428	20	.	.	PUNCT
ejpam-3272	429	1	we	we	PRON
ejpam-3272	429	2	end	end	VERB
ejpam-3272	429	3	this	this	DET
ejpam-3272	429	4	section	section	NOUN
ejpam-3272	429	5	by	by	ADP
ejpam-3272	429	6	investigating	investigate	VERB
ejpam-3272	429	7	the	the	DET
ejpam-3272	429	8	image	image	NOUN
ejpam-3272	429	9	of	of	ADP
ejpam-3272	429	10	a	a	DET
ejpam-3272	429	11	diframe	diframe	NOUN
ejpam-3272	429	12	with	with	ADP
ejpam-3272	429	13	a	a	DET
ejpam-3272	429	14	property	property	NOUN
ejpam-3272	429	15	p	p	NOUN
ejpam-3272	429	16	under	under	ADP
ejpam-3272	429	17	a	a	DET
ejpam-3272	429	18	special	special	ADJ
ejpam-3272	429	19	kind	kind	NOUN
ejpam-3272	429	20	of	of	ADP
ejpam-3272	429	21	homomorphism	homomorphism	NOUN
ejpam-3272	429	22	.	.	PUNCT
ejpam-3272	430	1	definition	definition	NOUN
ejpam-3272	430	2	7	7	NUM
ejpam-3272	430	3	.	.	PUNCT
ejpam-3272	431	1	a	a	DET
ejpam-3272	431	2	diframe	diframe	NOUN
ejpam-3272	431	3	homomorphism	homomorphism	NOUN
ejpam-3272	431	4	(	(	PUNCT
ejpam-3272	431	5	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	431	6	)	)	PUNCT
ejpam-3272	431	7	:	:	PUNCT
ejpam-3272	431	8	l→m	l→m	NUM
ejpam-3272	431	9	is	be	AUX
ejpam-3272	431	10	called	call	VERB
ejpam-3272	431	11	(	(	PUNCT
ejpam-3272	431	12	i	i	NOUN
ejpam-3272	431	13	)	)	PUNCT
ejpam-3272	431	14	open	open	ADJ
ejpam-3272	431	15	(	(	PUNCT
ejpam-3272	431	16	respectively	respectively	ADV
ejpam-3272	431	17	,	,	PUNCT
ejpam-3272	431	18	co	co	ADJ
ejpam-3272	431	19	-	-	ADJ
ejpam-3272	431	20	open	open	ADJ
ejpam-3272	431	21	)	)	PUNCT
ejpam-3272	431	22	if	if	SCONJ
ejpam-3272	431	23	ψ∗(a	ψ∗(a	PROPN
ejpam-3272	431	24	)	)	PUNCT
ejpam-3272	431	25	∈	∈	PROPN
ejpam-3272	431	26	lfr	lfr	PROPN
ejpam-3272	431	27	(	(	PUNCT
ejpam-3272	431	28	resp	resp	NOUN
ejpam-3272	431	29	.	.	PUNCT
ejpam-3272	432	1	ϕ∗(a	ϕ∗(a	PROPN
ejpam-3272	432	2	)	)	PUNCT
ejpam-3272	432	3	∈	∈	PROPN
ejpam-3272	432	4	lfr	lfr	PROPN
ejpam-3272	432	5	)	)	PUNCT
ejpam-3272	432	6	for	for	ADP
ejpam-3272	432	7	all	all	DET
ejpam-3272	432	8	a	a	DET
ejpam-3272	432	9	∈mfr	∈mfr	PROPN
ejpam-3272	432	10	.	.	PUNCT
ejpam-3272	432	11	(	(	PUNCT
ejpam-3272	432	12	ii	ii	NOUN
ejpam-3272	432	13	)	)	PUNCT
ejpam-3272	432	14	closed	close	VERB
ejpam-3272	432	15	(	(	PUNCT
ejpam-3272	432	16	respectively	respectively	ADV
ejpam-3272	432	17	,	,	PUNCT
ejpam-3272	432	18	co	co	ADJ
ejpam-3272	432	19	-	-	VERB
ejpam-3272	432	20	closed	closed	ADJ
ejpam-3272	432	21	)	)	PUNCT
ejpam-3272	432	22	if	if	SCONJ
ejpam-3272	432	23	ψ∗(k	ψ∗(k	NUM
ejpam-3272	432	24	)	)	PUNCT
ejpam-3272	432	25	∈	∈	PROPN
ejpam-3272	432	26	lcf	lcf	PROPN
ejpam-3272	432	27	(	(	PUNCT
ejpam-3272	432	28	resp	resp	NOUN
ejpam-3272	432	29	.	.	PUNCT
ejpam-3272	433	1	ϕ∗(k	ϕ∗(k	NUM
ejpam-3272	433	2	)	)	PUNCT
ejpam-3272	433	3	∈	∈	PROPN
ejpam-3272	433	4	lcf	lcf	PROPN
ejpam-3272	433	5	)	)	PUNCT
ejpam-3272	433	6	for	for	ADP
ejpam-3272	433	7	all	all	PRON
ejpam-3272	433	8	k	k	NOUN
ejpam-3272	433	9	∈mcf	∈mcf	NOUN
ejpam-3272	433	10	.	.	PUNCT
ejpam-3272	434	1	proposition	proposition	NOUN
ejpam-3272	434	2	16	16	NUM
ejpam-3272	434	3	.	.	PUNCT
ejpam-3272	435	1	let	let	VERB
ejpam-3272	435	2	l	l	NOUN
ejpam-3272	435	3	and	and	CCONJ
ejpam-3272	435	4	m	m	AUX
ejpam-3272	435	5	be	be	AUX
ejpam-3272	435	6	diframes	diframe	NOUN
ejpam-3272	435	7	and	and	CCONJ
ejpam-3272	435	8	let	let	VERB
ejpam-3272	435	9	(	(	PUNCT
ejpam-3272	435	10	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	435	11	)	)	PUNCT
ejpam-3272	435	12	:	:	PUNCT
ejpam-3272	436	1	l	l	X
ejpam-3272	436	2	→	→	PUNCT
ejpam-3272	436	3	m	m	AUX
ejpam-3272	436	4	be	be	AUX
ejpam-3272	436	5	a	a	DET
ejpam-3272	436	6	one	one	NUM
ejpam-3272	436	7	-	-	PUNCT
ejpam-3272	436	8	one	one	NOUN
ejpam-3272	436	9	onto	onto	ADP
ejpam-3272	436	10	diframe	diframe	NOUN
ejpam-3272	436	11	homomorphism	homomorphism	NOUN
ejpam-3272	436	12	.	.	PUNCT
ejpam-3272	437	1	(	(	PUNCT
ejpam-3272	437	2	i	i	NOUN
ejpam-3272	437	3	)	)	PUNCT
ejpam-3272	437	4	if	if	SCONJ
ejpam-3272	437	5	(	(	PUNCT
ejpam-3272	437	6	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	437	7	)	)	PUNCT
ejpam-3272	437	8	is	be	AUX
ejpam-3272	437	9	open	open	ADJ
ejpam-3272	437	10	(	(	PUNCT
ejpam-3272	437	11	resp	resp	NOUN
ejpam-3272	437	12	.	.	PUNCT
ejpam-3272	438	1	co	co	VERB
ejpam-3272	438	2	-	-	ADJ
ejpam-3272	438	3	open	open	ADJ
ejpam-3272	438	4	)	)	PUNCT
ejpam-3272	438	5	then	then	ADV
ejpam-3272	438	6	,	,	PUNCT
ejpam-3272	438	7	for	for	ADP
ejpam-3272	438	8	all	all	DET
ejpam-3272	438	9	b	b	NOUN
ejpam-3272	438	10	∈mfr	∈mfr	NUM
ejpam-3272	438	11	,	,	PUNCT
ejpam-3272	438	12	there	there	PRON
ejpam-3272	438	13	exists	exist	VERB
ejpam-3272	438	14	an	an	DET
ejpam-3272	438	15	a	a	DET
ejpam-3272	438	16	∈	∈	PROPN
ejpam-3272	438	17	lfr	lfr	NOUN
ejpam-3272	438	18	such	such	ADJ
ejpam-3272	438	19	that	that	PRON
ejpam-3272	438	20	ψ(a	ψ(a	PROPN
ejpam-3272	438	21	)	)	PUNCT
ejpam-3272	439	1	=	=	SYM
ejpam-3272	439	2	b	b	PROPN
ejpam-3272	439	3	(	(	PUNCT
ejpam-3272	439	4	resp	resp	NOUN
ejpam-3272	439	5	.	.	PUNCT
ejpam-3272	440	1	ϕ(a	ϕ(a	NOUN
ejpam-3272	440	2	)	)	PUNCT
ejpam-3272	440	3	=	=	SYM
ejpam-3272	440	4	b	b	X
ejpam-3272	440	5	)	)	PUNCT
ejpam-3272	440	6	.	.	PUNCT
ejpam-3272	441	1	e.	e.	PROPN
ejpam-3272	441	2	korkmaz	korkmaz	PROPN
ejpam-3272	441	3	,	,	PUNCT
ejpam-3272	441	4	r.	r.	PROPN
ejpam-3272	441	5	ertürk	ertürk	PROPN
ejpam-3272	441	6	/	/	SYM
ejpam-3272	441	7	eur	eur	PROPN
ejpam-3272	441	8	.	.	PUNCT
ejpam-3272	442	1	j.	j.	PROPN
ejpam-3272	442	2	pure	pure	PROPN
ejpam-3272	442	3	appl	appl	PROPN
ejpam-3272	442	4	.	.	PROPN
ejpam-3272	442	5	math	math	PROPN
ejpam-3272	442	6	,	,	PUNCT
ejpam-3272	442	7	11	11	NUM
ejpam-3272	442	8	(	(	PUNCT
ejpam-3272	442	9	3	3	NUM
ejpam-3272	442	10	)	)	PUNCT
ejpam-3272	442	11	(	(	PUNCT
ejpam-3272	442	12	2018	2018	NUM
ejpam-3272	442	13	)	)	PUNCT
ejpam-3272	442	14	,	,	PUNCT
ejpam-3272	442	15	612	612	NUM
ejpam-3272	442	16	-	-	SYM
ejpam-3272	442	17	627	627	NUM
ejpam-3272	442	18	626	626	NUM
ejpam-3272	442	19	(	(	PUNCT
ejpam-3272	442	20	ii	ii	NOUN
ejpam-3272	442	21	)	)	PUNCT
ejpam-3272	442	22	if	if	SCONJ
ejpam-3272	442	23	(	(	PUNCT
ejpam-3272	442	24	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	442	25	)	)	PUNCT
ejpam-3272	442	26	is	be	AUX
ejpam-3272	442	27	closed	close	VERB
ejpam-3272	442	28	(	(	PUNCT
ejpam-3272	442	29	resp	resp	NOUN
ejpam-3272	442	30	.	.	PUNCT
ejpam-3272	443	1	co	co	VERB
ejpam-3272	443	2	-	-	VERB
ejpam-3272	443	3	closed	closed	ADJ
ejpam-3272	443	4	)	)	PUNCT
ejpam-3272	443	5	then	then	ADV
ejpam-3272	443	6	,	,	PUNCT
ejpam-3272	443	7	for	for	ADP
ejpam-3272	443	8	all	all	DET
ejpam-3272	443	9	k	k	PROPN
ejpam-3272	443	10	∈	∈	PROPN
ejpam-3272	443	11	mcf	mcf	NOUN
ejpam-3272	443	12	,	,	PUNCT
ejpam-3272	443	13	there	there	PRON
ejpam-3272	443	14	exists	exist	VERB
ejpam-3272	443	15	an	an	DET
ejpam-3272	443	16	f	f	PROPN
ejpam-3272	443	17	∈	∈	PROPN
ejpam-3272	443	18	lcf	lcf	NOUN
ejpam-3272	443	19	such	such	ADJ
ejpam-3272	443	20	that	that	DET
ejpam-3272	443	21	ψ(f	ψ(f	NOUN
ejpam-3272	443	22	)	)	PUNCT
ejpam-3272	444	1	=	=	SYM
ejpam-3272	444	2	k	k	PROPN
ejpam-3272	444	3	(	(	PUNCT
ejpam-3272	444	4	resp	resp	NOUN
ejpam-3272	444	5	.	.	PUNCT
ejpam-3272	445	1	ϕ(f	ϕ(f	NOUN
ejpam-3272	445	2	)	)	PUNCT
ejpam-3272	446	1	=	=	SYM
ejpam-3272	446	2	k	k	X
ejpam-3272	446	3	)	)	PUNCT
ejpam-3272	446	4	.	.	PUNCT
ejpam-3272	447	1	proof	proof	NOUN
ejpam-3272	447	2	.	.	PUNCT
ejpam-3272	448	1	suppose	suppose	VERB
ejpam-3272	448	2	(	(	PUNCT
ejpam-3272	448	3	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	448	4	)	)	PUNCT
ejpam-3272	448	5	:	:	PUNCT
ejpam-3272	448	6	le	le	X
ejpam-3272	448	7	→	→	PUNCT
ejpam-3272	448	8	me	i	PRON
ejpam-3272	448	9	is	be	AUX
ejpam-3272	448	10	open	open	ADJ
ejpam-3272	448	11	and	and	CCONJ
ejpam-3272	448	12	b	b	X
ejpam-3272	448	13	∈	∈	PROPN
ejpam-3272	448	14	mfr	mfr	PROPN
ejpam-3272	448	15	.	.	PUNCT
ejpam-3272	449	1	since	since	SCONJ
ejpam-3272	449	2	ψ	ψ	NOUN
ejpam-3272	449	3	is	be	AUX
ejpam-3272	449	4	onto	onto	ADP
ejpam-3272	449	5	,	,	PUNCT
ejpam-3272	449	6	there	there	PRON
ejpam-3272	449	7	is	be	VERB
ejpam-3272	449	8	an	an	DET
ejpam-3272	449	9	a	a	DET
ejpam-3272	449	10	∈	∈	NOUN
ejpam-3272	449	11	le	le	X
ejpam-3272	449	12	with	with	ADP
ejpam-3272	449	13	ψ(a	ψ(a	PROPN
ejpam-3272	449	14	)	)	PUNCT
ejpam-3272	449	15	=	=	SYM
ejpam-3272	449	16	b.	b.	PROPN
ejpam-3272	449	17	now	now	ADV
ejpam-3272	449	18	,	,	PUNCT
ejpam-3272	449	19	by	by	ADP
ejpam-3272	449	20	proposition	proposition	NOUN
ejpam-3272	449	21	1	1	NUM
ejpam-3272	449	22	,	,	PUNCT
ejpam-3272	449	23	we	we	PRON
ejpam-3272	449	24	have	have	AUX
ejpam-3272	449	25	ψ∗ψ(a	ψ∗ψ(a	PROPN
ejpam-3272	449	26	)	)	PUNCT
ejpam-3272	449	27	=	=	SYM
ejpam-3272	450	1	a	a	PRON
ejpam-3272	450	2	=	=	SYM
ejpam-3272	450	3	ψ∗(b	ψ∗(b	PROPN
ejpam-3272	450	4	)	)	PUNCT
ejpam-3272	450	5	,	,	PUNCT
ejpam-3272	450	6	and	and	CCONJ
ejpam-3272	450	7	hence	hence	ADV
ejpam-3272	450	8	a	a	DET
ejpam-3272	450	9	=	=	SYM
ejpam-3272	450	10	ψ∗(b	ψ∗(b	PROPN
ejpam-3272	450	11	)	)	PUNCT
ejpam-3272	450	12	∈	∈	PROPN
ejpam-3272	450	13	lfr	lfr	NOUN
ejpam-3272	450	14	by	by	ADP
ejpam-3272	450	15	openness	openness	NOUN
ejpam-3272	450	16	of	of	ADP
ejpam-3272	450	17	(	(	PUNCT
ejpam-3272	450	18	ϕ,ψ	ϕ,ψ	NOUN
ejpam-3272	450	19	)	)	PUNCT
ejpam-3272	450	20	.	.	PUNCT
ejpam-3272	451	1	the	the	DET
ejpam-3272	451	2	other	other	ADJ
ejpam-3272	451	3	cases	case	NOUN
ejpam-3272	451	4	can	can	AUX
ejpam-3272	451	5	be	be	AUX
ejpam-3272	451	6	proved	prove	VERB
ejpam-3272	451	7	similarly	similarly	ADV
ejpam-3272	451	8	.	.	PUNCT
ejpam-3272	452	1	remark	remark	VERB
ejpam-3272	452	2	6	6	NUM
ejpam-3272	452	3	.	.	PUNCT
ejpam-3272	453	1	if	if	SCONJ
ejpam-3272	453	2	ϕ	ϕ	NOUN
ejpam-3272	453	3	is	be	AUX
ejpam-3272	453	4	one	one	NUM
ejpam-3272	453	5	-	-	PUNCT
ejpam-3272	453	6	one	one	NUM
ejpam-3272	453	7	and	and	CCONJ
ejpam-3272	453	8	onto	onto	ADP
ejpam-3272	453	9	then	then	ADV
ejpam-3272	453	10	,	,	PUNCT
ejpam-3272	453	11	by	by	ADP
ejpam-3272	453	12	proposition	proposition	NOUN
ejpam-3272	453	13	1	1	NUM
ejpam-3272	453	14	(	(	PUNCT
ejpam-3272	453	15	iii	iii	NOUN
ejpam-3272	453	16	)	)	PUNCT
ejpam-3272	453	17	,	,	PUNCT
ejpam-3272	453	18	ϕ∗ϕ	ϕ∗ϕ	PUNCT
ejpam-3272	453	19	=	=	SYM
ejpam-3272	454	1	1le	1le	ADJ
ejpam-3272	454	2	and	and	CCONJ
ejpam-3272	454	3	ϕϕ∗	ϕϕ∗	NOUN
ejpam-3272	454	4	=	=	NOUN
ejpam-3272	454	5	1me	1me	NOUN
ejpam-3272	454	6	,	,	PUNCT
ejpam-3272	454	7	that	that	ADV
ejpam-3272	454	8	is	is	ADV
ejpam-3272	454	9	,	,	PUNCT
ejpam-3272	454	10	ϕ−1	ϕ−1	PROPN
ejpam-3272	454	11	=	=	SYM
ejpam-3272	454	12	ϕ∗.	ϕ∗.	CCONJ
ejpam-3272	454	13	similarly	similarly	ADV
ejpam-3272	454	14	,	,	PUNCT
ejpam-3272	454	15	if	if	SCONJ
ejpam-3272	454	16	ψ	ψ	NOUN
ejpam-3272	454	17	is	be	AUX
ejpam-3272	454	18	one	one	NUM
ejpam-3272	454	19	-	-	PUNCT
ejpam-3272	454	20	one	one	NUM
ejpam-3272	454	21	and	and	CCONJ
ejpam-3272	454	22	onto	onto	ADP
ejpam-3272	454	23	then	then	ADV
ejpam-3272	454	24	ψ−1	ψ−1	PROPN
ejpam-3272	454	25	=	=	SYM
ejpam-3272	454	26	ψ∗.	ψ∗.	PROPN
ejpam-3272	454	27	thus	thus	ADV
ejpam-3272	454	28	,	,	PUNCT
ejpam-3272	454	29	if	if	SCONJ
ejpam-3272	454	30	(	(	PUNCT
ejpam-3272	454	31	ϕ,ϕ	ϕ,ϕ	NOUN
ejpam-3272	454	32	)	)	PUNCT
ejpam-3272	454	33	=	=	SYM
ejpam-3272	454	34	ϕ	ϕ	NOUN
ejpam-3272	454	35	:	:	PUNCT
ejpam-3272	454	36	l	l	X
ejpam-3272	454	37	→	→	PUNCT
ejpam-3272	454	38	m	m	VERB
ejpam-3272	454	39	is	be	AUX
ejpam-3272	454	40	a	a	DET
ejpam-3272	454	41	one	one	NUM
ejpam-3272	454	42	-	-	PUNCT
ejpam-3272	454	43	one	one	NOUN
ejpam-3272	454	44	onto	onto	ADP
ejpam-3272	454	45	hdifrm	hdifrm	NOUN
ejpam-3272	454	46	homomorphism	homomorphism	PROPN
ejpam-3272	454	47	then	then	ADV
ejpam-3272	454	48	ϕ∗	ϕ∗	PROPN
ejpam-3272	454	49	=	=	SYM
ejpam-3272	454	50	ϕ∗	ϕ∗	PROPN
ejpam-3272	454	51	,	,	PUNCT
ejpam-3272	454	52	and	and	CCONJ
ejpam-3272	454	53	hence	hence	ADV
ejpam-3272	454	54	the	the	DET
ejpam-3272	454	55	concept	concept	NOUN
ejpam-3272	454	56	of	of	ADP
ejpam-3272	454	57	openness	openness	NOUN
ejpam-3272	454	58	(	(	PUNCT
ejpam-3272	454	59	resp	resp	NOUN
ejpam-3272	454	60	.	.	PUNCT
ejpam-3272	454	61	,	,	PUNCT
ejpam-3272	454	62	closedness	closedness	ADJ
ejpam-3272	454	63	)	)	PUNCT
ejpam-3272	454	64	coincides	coincide	VERB
ejpam-3272	454	65	with	with	ADP
ejpam-3272	454	66	co	co	NOUN
ejpam-3272	454	67	-	-	NOUN
ejpam-3272	454	68	openness	openness	ADJ
ejpam-3272	454	69	(	(	PUNCT
ejpam-3272	454	70	resp	resp	NOUN
ejpam-3272	454	71	.	.	PUNCT
ejpam-3272	455	1	coclosedness	coclosedness	NOUN
ejpam-3272	455	2	)	)	PUNCT
ejpam-3272	455	3	.	.	PUNCT
ejpam-3272	456	1	definition	definition	NOUN
ejpam-3272	456	2	8	8	NUM
ejpam-3272	456	3	.	.	PUNCT
ejpam-3272	457	1	if	if	SCONJ
ejpam-3272	457	2	l	l	PROPN
ejpam-3272	457	3	and	and	CCONJ
ejpam-3272	457	4	m	m	PROPN
ejpam-3272	457	5	are	be	AUX
ejpam-3272	457	6	diframes	diframe	NOUN
ejpam-3272	457	7	,	,	PUNCT
ejpam-3272	457	8	a	a	DET
ejpam-3272	457	9	hdifrm	hdifrm	NOUN
ejpam-3272	457	10	homomorphism	homomorphism	NOUN
ejpam-3272	457	11	(	(	PUNCT
ejpam-3272	457	12	ϕ,ϕ	ϕ,ϕ	NOUN
ejpam-3272	457	13	)	)	PUNCT
ejpam-3272	457	14	=	=	SYM
ejpam-3272	457	15	ϕ	ϕ	NOUN
ejpam-3272	457	16	:	:	PUNCT
ejpam-3272	457	17	l→	l→	NOUN
ejpam-3272	457	18	m	m	VERB
ejpam-3272	457	19	is	be	AUX
ejpam-3272	457	20	called	call	VERB
ejpam-3272	457	21	an	an	DET
ejpam-3272	457	22	isomorphism	isomorphism	NOUN
ejpam-3272	457	23	if	if	SCONJ
ejpam-3272	457	24	it	it	PRON
ejpam-3272	457	25	is	be	AUX
ejpam-3272	457	26	one	one	NUM
ejpam-3272	457	27	-	-	PUNCT
ejpam-3272	457	28	one	one	NUM
ejpam-3272	457	29	,	,	PUNCT
ejpam-3272	457	30	onto	onto	ADP
ejpam-3272	457	31	,	,	PUNCT
ejpam-3272	457	32	open	open	ADJ
ejpam-3272	457	33	and	and	CCONJ
ejpam-3272	457	34	closed	closed	ADJ
ejpam-3272	457	35	,	,	PUNCT
ejpam-3272	457	36	.	.	PUNCT
ejpam-3272	458	1	proposition	proposition	NOUN
ejpam-3272	458	2	17	17	NUM
ejpam-3272	458	3	.	.	PUNCT
ejpam-3272	459	1	let	let	VERB
ejpam-3272	459	2	l	l	NOUN
ejpam-3272	459	3	,	,	PUNCT
ejpam-3272	459	4	m	m	VERB
ejpam-3272	459	5	be	be	VERB
ejpam-3272	459	6	diframes	diframe	NOUN
ejpam-3272	459	7	and	and	CCONJ
ejpam-3272	459	8	ϕ	ϕ	NOUN
ejpam-3272	459	9	:	:	PUNCT
ejpam-3272	459	10	l	l	X
ejpam-3272	459	11	→	→	PUNCT
ejpam-3272	459	12	m	m	AUX
ejpam-3272	459	13	be	be	AUX
ejpam-3272	459	14	a	a	DET
ejpam-3272	459	15	hdifrm	hdifrm	NOUN
ejpam-3272	459	16	isomorphism	isomorphism	NOUN
ejpam-3272	459	17	.	.	PUNCT
ejpam-3272	460	1	then	then	ADV
ejpam-3272	460	2	,	,	PUNCT
ejpam-3272	460	3	l	l	NOUN
ejpam-3272	460	4	is	be	AUX
ejpam-3272	460	5	bi	bi	ADJ
ejpam-3272	460	6	-	-	ADJ
ejpam-3272	460	7	r0	r0	ADJ
ejpam-3272	460	8	(	(	PUNCT
ejpam-3272	460	9	respectively	respectively	ADV
ejpam-3272	460	10	,	,	PUNCT
ejpam-3272	460	11	bi	bi	NOUN
ejpam-3272	460	12	-	-	ADJ
ejpam-3272	460	13	r1	r1	ADJ
ejpam-3272	460	14	,	,	PUNCT
ejpam-3272	460	15	bi	bi	NOUN
ejpam-3272	460	16	-	-	ADJ
ejpam-3272	460	17	regular	regular	ADJ
ejpam-3272	460	18	,	,	PUNCT
ejpam-3272	460	19	completely	completely	ADV
ejpam-3272	460	20	bi	bi	ADJ
ejpam-3272	460	21	-	-	ADJ
ejpam-3272	460	22	regular	regular	ADJ
ejpam-3272	460	23	,	,	PUNCT
ejpam-3272	460	24	normal	normal	ADJ
ejpam-3272	460	25	)	)	PUNCT
ejpam-3272	460	26	if	if	SCONJ
ejpam-3272	461	1	and	and	CCONJ
ejpam-3272	461	2	only	only	ADV
ejpam-3272	461	3	if	if	SCONJ
ejpam-3272	461	4	m	m	NOUN
ejpam-3272	461	5	is	be	AUX
ejpam-3272	461	6	bi	bi	ADJ
ejpam-3272	461	7	-	-	NOUN
ejpam-3272	461	8	r0	r0	ADJ
ejpam-3272	461	9	(	(	PUNCT
ejpam-3272	461	10	respectively	respectively	ADV
ejpam-3272	461	11	,	,	PUNCT
ejpam-3272	461	12	bi	bi	NOUN
ejpam-3272	461	13	-	-	ADJ
ejpam-3272	461	14	r1	r1	ADJ
ejpam-3272	461	15	,	,	PUNCT
ejpam-3272	461	16	bi	bi	NOUN
ejpam-3272	461	17	-	-	ADJ
ejpam-3272	461	18	regular	regular	ADJ
ejpam-3272	461	19	,	,	PUNCT
ejpam-3272	461	20	completely	completely	ADV
ejpam-3272	461	21	bi	bi	ADJ
ejpam-3272	461	22	-	-	ADJ
ejpam-3272	461	23	regular	regular	ADJ
ejpam-3272	461	24	,	,	PUNCT
ejpam-3272	461	25	normal	normal	ADJ
ejpam-3272	461	26	)	)	PUNCT
ejpam-3272	461	27	.	.	PUNCT
ejpam-3272	462	1	proof	proof	NOUN
ejpam-3272	462	2	.	.	PUNCT
ejpam-3272	463	1	we	we	PRON
ejpam-3272	463	2	will	will	AUX
ejpam-3272	463	3	just	just	ADV
ejpam-3272	463	4	prove	prove	VERB
ejpam-3272	463	5	the	the	DET
ejpam-3272	463	6	regularity	regularity	NOUN
ejpam-3272	463	7	and	and	CCONJ
ejpam-3272	463	8	the	the	DET
ejpam-3272	463	9	other	other	ADJ
ejpam-3272	463	10	axioms	axiom	NOUN
ejpam-3272	463	11	are	be	AUX
ejpam-3272	463	12	left	leave	VERB
ejpam-3272	463	13	to	to	ADP
ejpam-3272	463	14	the	the	DET
ejpam-3272	463	15	interested	interested	ADJ
ejpam-3272	463	16	reader	reader	NOUN
ejpam-3272	463	17	.	.	PUNCT
ejpam-3272	464	1	let	let	VERB
ejpam-3272	464	2	l	l	NOUN
ejpam-3272	464	3	be	be	AUX
ejpam-3272	464	4	regular	regular	ADJ
ejpam-3272	464	5	and	and	CCONJ
ejpam-3272	464	6	b	b	NOUN
ejpam-3272	464	7	∈mfr	∈mfr	PROPN
ejpam-3272	464	8	.	.	PUNCT
ejpam-3272	465	1	then	then	ADV
ejpam-3272	465	2	,	,	PUNCT
ejpam-3272	465	3	by	by	ADP
ejpam-3272	465	4	proposition	proposition	NOUN
ejpam-3272	465	5	16	16	NUM
ejpam-3272	465	6	,	,	PUNCT
ejpam-3272	465	7	there	there	PRON
ejpam-3272	465	8	is	be	VERB
ejpam-3272	465	9	an	an	DET
ejpam-3272	465	10	a	a	DET
ejpam-3272	465	11	∈	∈	PROPN
ejpam-3272	465	12	lfr	lfr	NOUN
ejpam-3272	465	13	such	such	ADJ
ejpam-3272	465	14	that	that	SCONJ
ejpam-3272	465	15	ϕ(a	ϕ(a	NOUN
ejpam-3272	465	16	)	)	PUNCT
ejpam-3272	466	1	=	=	SYM
ejpam-3272	466	2	b	b	PROPN
ejpam-3272	466	3	and	and	CCONJ
ejpam-3272	466	4	,	,	PUNCT
ejpam-3272	466	5	by	by	ADP
ejpam-3272	466	6	regularity	regularity	NOUN
ejpam-3272	466	7	of	of	ADP
ejpam-3272	466	8	l	l	NOUN
ejpam-3272	466	9	,	,	PUNCT
ejpam-3272	466	10	a	a	DET
ejpam-3272	466	11	=	=	SYM
ejpam-3272	466	12	∨	∨	X
ejpam-3272	466	13	{	{	PUNCT
ejpam-3272	466	14	x	x	SYM
ejpam-3272	466	15	∈	∈	PROPN
ejpam-3272	466	16	lfr	lfr	NOUN
ejpam-3272	466	17	:	:	PUNCT
ejpam-3272	467	1	x	x	PUNCT
ejpam-3272	467	2	≺fr	≺fr	NOUN
ejpam-3272	467	3	a	a	X
ejpam-3272	467	4	}	}	PUNCT
ejpam-3272	467	5	.	.	PUNCT
ejpam-3272	468	1	moreover	moreover	ADV
ejpam-3272	468	2	,	,	PUNCT
ejpam-3272	468	3	x	x	PROPN
ejpam-3272	468	4	≺fr	≺fr	NOUN
ejpam-3272	468	5	a	a	DET
ejpam-3272	468	6	implies	imply	VERB
ejpam-3272	468	7	ϕ(x	ϕ(x	X
ejpam-3272	468	8	)	)	PUNCT
ejpam-3272	469	1	≺fr	≺fr	X
ejpam-3272	469	2	b	b	X
ejpam-3272	469	3	by	by	ADP
ejpam-3272	469	4	definition	definition	NOUN
ejpam-3272	469	5	of	of	ADP
ejpam-3272	469	6	≺fr	≺fr	PROPN
ejpam-3272	469	7	.	.	PUNCT
ejpam-3272	470	1	now	now	ADV
ejpam-3272	470	2	we	we	PRON
ejpam-3272	470	3	have	have	VERB
ejpam-3272	470	4	b	b	NOUN
ejpam-3272	470	5	=	=	SYM
ejpam-3272	470	6	ϕ(a	ϕ(a	PROPN
ejpam-3272	470	7	)	)	PUNCT
ejpam-3272	471	1	=	=	SYM
ejpam-3272	471	2	ϕ	ϕ	NOUN
ejpam-3272	471	3	(	(	PUNCT
ejpam-3272	471	4	∨	∨	X
ejpam-3272	471	5	{	{	PUNCT
ejpam-3272	471	6	x	x	SYM
ejpam-3272	471	7	∈	∈	PROPN
ejpam-3272	471	8	lfr	lfr	NOUN
ejpam-3272	471	9	:	:	PUNCT
ejpam-3272	471	10	x	x	PUNCT
ejpam-3272	471	11	≺fr	≺fr	X
ejpam-3272	471	12	a	a	X
ejpam-3272	471	13	}	}	PUNCT
ejpam-3272	471	14	)	)	PUNCT
ejpam-3272	471	15	≤	≤	NUM
ejpam-3272	471	16	∨	∨	NUM
ejpam-3272	471	17	{	{	PUNCT
ejpam-3272	471	18	ϕ(x	ϕ(x	X
ejpam-3272	471	19	)	)	PUNCT
ejpam-3272	471	20	∈mfr	∈mfr	ADP
ejpam-3272	471	21	:	:	PUNCT
ejpam-3272	471	22	ϕ(x	ϕ(x	X
ejpam-3272	471	23	)	)	PUNCT
ejpam-3272	472	1	≺fr	≺fr	X
ejpam-3272	472	2	b	b	X
ejpam-3272	472	3	)	)	PUNCT
ejpam-3272	472	4	}	}	PUNCT
ejpam-3272	472	5	≤	≤	NUM
ejpam-3272	472	6	b	b	NOUN
ejpam-3272	472	7	and	and	CCONJ
ejpam-3272	472	8	hence	hence	ADV
ejpam-3272	472	9	m	m	VERB
ejpam-3272	472	10	is	be	AUX
ejpam-3272	472	11	regular	regular	ADJ
ejpam-3272	472	12	.	.	PUNCT
ejpam-3272	473	1	conversely	conversely	ADV
ejpam-3272	473	2	,	,	PUNCT
ejpam-3272	473	3	suppose	suppose	VERB
ejpam-3272	473	4	that	that	SCONJ
ejpam-3272	473	5	m	m	PROPN
ejpam-3272	473	6	is	be	AUX
ejpam-3272	473	7	regular	regular	ADJ
ejpam-3272	473	8	and	and	CCONJ
ejpam-3272	473	9	a	a	DET
ejpam-3272	473	10	∈	∈	PROPN
ejpam-3272	473	11	lfr	lfr	NOUN
ejpam-3272	473	12	.	.	PUNCT
ejpam-3272	474	1	then	then	ADV
ejpam-3272	474	2	ϕ(a	ϕ(a	NOUN
ejpam-3272	474	3	)	)	PUNCT
ejpam-3272	474	4	∈	∈	PROPN
ejpam-3272	474	5	mfr	mfr	NOUN
ejpam-3272	474	6	and	and	CCONJ
ejpam-3272	474	7	hence	hence	ADV
ejpam-3272	474	8	,	,	PUNCT
ejpam-3272	474	9	by	by	ADP
ejpam-3272	474	10	regularity	regularity	NOUN
ejpam-3272	474	11	,	,	PUNCT
ejpam-3272	474	12	ϕ(a	ϕ(a	NOUN
ejpam-3272	474	13	)	)	PUNCT
ejpam-3272	475	1	=	=	PUNCT
ejpam-3272	475	2	∨	∨	X
ejpam-3272	475	3	{	{	PUNCT
ejpam-3272	475	4	x	x	SYM
ejpam-3272	475	5	∈	∈	PROPN
ejpam-3272	475	6	mfr	mfr	NOUN
ejpam-3272	475	7	:	:	PUNCT
ejpam-3272	475	8	x	x	PUNCT
ejpam-3272	475	9	≺fr	≺fr	PUNCT
ejpam-3272	475	10	ϕ(a	ϕ(a	NOUN
ejpam-3272	475	11	)	)	PUNCT
ejpam-3272	475	12	}	}	PUNCT
ejpam-3272	475	13	.	.	PUNCT
ejpam-3272	476	1	now	now	ADV
ejpam-3272	476	2	if	if	SCONJ
ejpam-3272	476	3	x	x	X
ejpam-3272	476	4	≺fr	≺fr	PUNCT
ejpam-3272	476	5	ϕ(a	ϕ(a	NOUN
ejpam-3272	476	6	)	)	PUNCT
ejpam-3272	476	7	then	then	ADV
ejpam-3272	476	8	,	,	PUNCT
ejpam-3272	476	9	by	by	ADP
ejpam-3272	476	10	proposition	proposition	NOUN
ejpam-3272	476	11	1	1	NUM
ejpam-3272	476	12	together	together	ADV
ejpam-3272	476	13	with	with	ADP
ejpam-3272	476	14	the	the	DET
ejpam-3272	476	15	closedness	closedness	NOUN
ejpam-3272	476	16	of	of	ADP
ejpam-3272	476	17	ϕ	ϕ	NOUN
ejpam-3272	476	18	,	,	PUNCT
ejpam-3272	476	19	we	we	PRON
ejpam-3272	476	20	have	have	VERB
ejpam-3272	476	21	ϕ∗(x	ϕ∗(x	PRON
ejpam-3272	476	22	)	)	PUNCT
ejpam-3272	477	1	≺fr	≺fr	NOUN
ejpam-3272	477	2	a.	a.	NOUN
ejpam-3272	478	1	but	but	CCONJ
ejpam-3272	478	2	then	then	ADV
ejpam-3272	478	3	a	a	DET
ejpam-3272	478	4	=	=	PUNCT
ejpam-3272	478	5	ϕ∗ϕ(a	ϕ∗ϕ(a	PROPN
ejpam-3272	478	6	)	)	PUNCT
ejpam-3272	478	7	=	=	SYM
ejpam-3272	478	8	ϕ∗	ϕ∗	PROPN
ejpam-3272	478	9	(	(	PUNCT
ejpam-3272	478	10	∨	∨	X
ejpam-3272	478	11	{	{	PUNCT
ejpam-3272	478	12	x	x	X
ejpam-3272	478	13	∈mfr	∈mfr	PROPN
ejpam-3272	478	14	:	:	PUNCT
ejpam-3272	478	15	x	x	PUNCT
ejpam-3272	478	16	≺fr	≺fr	NOUN
ejpam-3272	478	17	ϕ(a	ϕ(a	NOUN
ejpam-3272	478	18	)	)	PUNCT
ejpam-3272	478	19	}	}	PUNCT
ejpam-3272	478	20	)	)	PUNCT
ejpam-3272	478	21	≤	≤	ADV
ejpam-3272	478	22	∨	∨	NUM
ejpam-3272	478	23	{	{	PUNCT
ejpam-3272	478	24	ϕ∗(x	ϕ∗(x	PROPN
ejpam-3272	478	25	)	)	PUNCT
ejpam-3272	478	26	∈	∈	PROPN
ejpam-3272	478	27	lfr	lfr	NOUN
ejpam-3272	478	28	:	:	PUNCT
ejpam-3272	478	29	ϕ∗(x	ϕ∗(x	X
ejpam-3272	478	30	)	)	PUNCT
ejpam-3272	478	31	≺fr	≺fr	X
ejpam-3272	478	32	a	a	DET
ejpam-3272	478	33	}	}	PUNCT
ejpam-3272	478	34	≤	≤	NOUN
ejpam-3272	478	35	a	a	PRON
ejpam-3272	478	36	and	and	CCONJ
ejpam-3272	478	37	hence	hence	ADV
ejpam-3272	478	38	l	l	NOUN
ejpam-3272	478	39	is	be	AUX
ejpam-3272	478	40	regular	regular	ADJ
ejpam-3272	478	41	.	.	PUNCT
ejpam-3272	479	1	4	4	X
ejpam-3272	479	2	.	.	X
ejpam-3272	479	3	conclusion	conclusion	NOUN
ejpam-3272	479	4	in	in	ADP
ejpam-3272	479	5	this	this	DET
ejpam-3272	479	6	paper	paper	NOUN
ejpam-3272	479	7	we	we	PRON
ejpam-3272	479	8	have	have	AUX
ejpam-3272	479	9	studied	study	VERB
ejpam-3272	479	10	the	the	DET
ejpam-3272	479	11	separation	separation	NOUN
ejpam-3272	479	12	axioms	axiom	NOUN
ejpam-3272	479	13	in	in	ADP
ejpam-3272	479	14	diframes	diframe	NOUN
ejpam-3272	479	15	and	and	CCONJ
ejpam-3272	479	16	examined	examine	VERB
ejpam-3272	479	17	the	the	DET
ejpam-3272	479	18	relations	relation	NOUN
ejpam-3272	479	19	between	between	ADP
ejpam-3272	479	20	them	they	PRON
ejpam-3272	479	21	.	.	PUNCT
ejpam-3272	480	1	we	we	PRON
ejpam-3272	480	2	have	have	AUX
ejpam-3272	480	3	defined	define	VERB
ejpam-3272	480	4	new	new	ADJ
ejpam-3272	480	5	binary	binary	ADJ
ejpam-3272	480	6	relations	relation	NOUN
ejpam-3272	480	7	on	on	ADP
ejpam-3272	480	8	a	a	DET
ejpam-3272	480	9	diframe	diframe	NOUN
ejpam-3272	480	10	and	and	CCONJ
ejpam-3272	480	11	obtained	obtain	VERB
ejpam-3272	480	12	a	a	DET
ejpam-3272	480	13	characterization	characterization	NOUN
ejpam-3272	480	14	of	of	ADP
ejpam-3272	480	15	regularity	regularity	NOUN
ejpam-3272	480	16	and	and	CCONJ
ejpam-3272	480	17	complete	complete	ADJ
ejpam-3272	480	18	regularity	regularity	NOUN
ejpam-3272	480	19	by	by	ADP
ejpam-3272	480	20	using	use	VERB
ejpam-3272	480	21	these	these	DET
ejpam-3272	480	22	relations	relation	NOUN
ejpam-3272	480	23	.	.	PUNCT
ejpam-3272	481	1	as	as	ADP
ejpam-3272	481	2	a	a	DET
ejpam-3272	481	3	future	future	ADJ
ejpam-3272	481	4	work	work	NOUN
ejpam-3272	481	5	,	,	PUNCT
ejpam-3272	481	6	other	other	ADJ
ejpam-3272	481	7	topological	topological	ADJ
ejpam-3272	481	8	and	and	CCONJ
ejpam-3272	481	9	bitopological	bitopological	ADJ
ejpam-3272	481	10	structures	structure	NOUN
ejpam-3272	481	11	such	such	ADJ
ejpam-3272	481	12	as	as	ADP
ejpam-3272	481	13	compactness	compactness	NOUN
ejpam-3272	481	14	,	,	PUNCT
ejpam-3272	481	15	stability	stability	NOUN
ejpam-3272	481	16	,	,	PUNCT
ejpam-3272	481	17	join	join	VERB
ejpam-3272	481	18	compactness	compactness	NOUN
ejpam-3272	481	19	and	and	CCONJ
ejpam-3272	481	20	connectedness	connectedness	NOUN
ejpam-3272	481	21	,	,	PUNCT
ejpam-3272	481	22	etc	etc	X
ejpam-3272	481	23	.	.	X
ejpam-3272	481	24	can	can	AUX
ejpam-3272	481	25	be	be	AUX
ejpam-3272	481	26	constructed	construct	VERB
ejpam-3272	481	27	on	on	ADP
ejpam-3272	481	28	diframes	diframe	NOUN
ejpam-3272	481	29	.	.	PUNCT
ejpam-3272	482	1	references	reference	NOUN
ejpam-3272	482	2	627	627	NUM
ejpam-3272	482	3	acknowledgements	acknowledgement	NOUN
ejpam-3272	482	4	the	the	DET
ejpam-3272	482	5	authors	author	NOUN
ejpam-3272	482	6	thank	thank	VERB
ejpam-3272	482	7	the	the	DET
ejpam-3272	482	8	referees	referee	NOUN
ejpam-3272	482	9	for	for	ADP
ejpam-3272	482	10	valuable	valuable	ADJ
ejpam-3272	482	11	comments	comment	NOUN
ejpam-3272	482	12	and	and	CCONJ
ejpam-3272	482	13	suggestions	suggestion	NOUN
ejpam-3272	482	14	that	that	PRON
ejpam-3272	482	15	improved	improve	VERB
ejpam-3272	482	16	the	the	DET
ejpam-3272	482	17	quality	quality	NOUN
ejpam-3272	482	18	of	of	ADP
ejpam-3272	482	19	this	this	DET
ejpam-3272	482	20	manuscript	manuscript	NOUN
ejpam-3272	482	21	.	.	PUNCT
ejpam-3272	483	1	references	reference	NOUN
ejpam-3272	483	2	[	[	X
ejpam-3272	483	3	1	1	NUM
ejpam-3272	483	4	]	]	PUNCT
ejpam-3272	483	5	b.	b.	PROPN
ejpam-3272	483	6	banaschewski	banaschewski	PROPN
ejpam-3272	483	7	,	,	PUNCT
ejpam-3272	483	8	g.c.l	g.c.l	PROPN
ejpam-3272	483	9	.	.	PUNCT
ejpam-3272	484	1	brümmer	brümmer	PROPN
ejpam-3272	484	2	and	and	CCONJ
ejpam-3272	484	3	k.a	k.a	PROPN
ejpam-3272	484	4	.	.	PROPN
ejpam-3272	484	5	hardie	hardie	PROPN
ejpam-3272	484	6	.	.	PUNCT
ejpam-3272	485	1	biframes	biframe	NOUN
ejpam-3272	485	2	and	and	CCONJ
ejpam-3272	485	3	bispaces	bispace	NOUN
ejpam-3272	485	4	.	.	PUNCT
ejpam-3272	486	1	quaest	qua	ADJ
ejpam-3272	486	2	.	.	PUNCT
ejpam-3272	487	1	math	math	NOUN
ejpam-3272	487	2	.	.	PUNCT
ejpam-3272	487	3	,	,	PUNCT
ejpam-3272	487	4	6(1	6(1	NUM
ejpam-3272	487	5	-	-	SYM
ejpam-3272	487	6	3):13–25	3):13–25	NUM
ejpam-3272	487	7	,	,	PUNCT
ejpam-3272	487	8	1983	1983	NUM
ejpam-3272	487	9	.	.	PUNCT
ejpam-3272	488	1	[	[	X
ejpam-3272	488	2	2	2	NUM
ejpam-3272	488	3	]	]	PUNCT
ejpam-3272	488	4	l.	l.	PROPN
ejpam-3272	488	5	m.	m.	PROPN
ejpam-3272	488	6	brown	brown	PROPN
ejpam-3272	488	7	,	,	PUNCT
ejpam-3272	488	8	r.	r.	PROPN
ejpam-3272	488	9	ertürk	ertürk	PROPN
ejpam-3272	488	10	and	and	CCONJ
ejpam-3272	488	11	ş.	ş.	ADV
ejpam-3272	488	12	dost	dost	ADJ
ejpam-3272	488	13	.	.	PUNCT
ejpam-3272	489	1	ditopological	ditopological	ADJ
ejpam-3272	489	2	texture	texture	ADJ
ejpam-3272	489	3	spaces	space	NOUN
ejpam-3272	489	4	and	and	CCONJ
ejpam-3272	489	5	fuzzy	fuzzy	ADJ
ejpam-3272	489	6	topology	topology	NOUN
ejpam-3272	490	1	i	i	PRON
ejpam-3272	490	2	:	:	PUNCT
ejpam-3272	490	3	basic	basic	ADJ
ejpam-3272	490	4	concepts	concept	NOUN
ejpam-3272	490	5	.	.	PUNCT
ejpam-3272	491	1	fuzzy	fuzzy	ADJ
ejpam-3272	491	2	sets	set	NOUN
ejpam-3272	491	3	and	and	CCONJ
ejpam-3272	491	4	systems	system	NOUN
ejpam-3272	491	5	,	,	PUNCT
ejpam-3272	491	6	147(2):171–199	147(2):171–199	NUM
ejpam-3272	491	7	,	,	PUNCT
ejpam-3272	491	8	2004	2004	NUM
ejpam-3272	491	9	.	.	PUNCT
ejpam-3272	492	1	[	[	X
ejpam-3272	492	2	3	3	X
ejpam-3272	492	3	]	]	X
ejpam-3272	492	4	l.	l.	PROPN
ejpam-3272	492	5	m.	m.	PROPN
ejpam-3272	492	6	brown	brown	PROPN
ejpam-3272	492	7	,	,	PUNCT
ejpam-3272	492	8	r.	r.	PROPN
ejpam-3272	492	9	ertürk	ertürk	PROPN
ejpam-3272	492	10	and	and	CCONJ
ejpam-3272	492	11	ş.	ş.	ADV
ejpam-3272	492	12	dost	dost	ADJ
ejpam-3272	492	13	.	.	PUNCT
ejpam-3272	493	1	ditopological	ditopological	ADJ
ejpam-3272	493	2	texture	texture	ADJ
ejpam-3272	493	3	spaces	space	NOUN
ejpam-3272	493	4	and	and	CCONJ
ejpam-3272	493	5	fuzzy	fuzzy	ADJ
ejpam-3272	493	6	topology	topology	NOUN
ejpam-3272	493	7	ii	ii	PROPN
ejpam-3272	493	8	:	:	PUNCT
ejpam-3272	493	9	topological	topological	ADJ
ejpam-3272	493	10	consideration	consideration	NOUN
ejpam-3272	493	11	.	.	PUNCT
ejpam-3272	494	1	fuzzy	fuzzy	ADJ
ejpam-3272	494	2	sets	set	NOUN
ejpam-3272	494	3	and	and	CCONJ
ejpam-3272	494	4	systems	system	NOUN
ejpam-3272	494	5	,	,	PUNCT
ejpam-3272	494	6	147(2):201–231	147(2):201–231	NUM
ejpam-3272	494	7	,	,	PUNCT
ejpam-3272	494	8	2004	2004	NUM
ejpam-3272	494	9	.	.	PUNCT
ejpam-3272	495	1	[	[	X
ejpam-3272	495	2	4	4	NUM
ejpam-3272	495	3	]	]	X
ejpam-3272	495	4	l.	l.	PROPN
ejpam-3272	495	5	m.	m.	PROPN
ejpam-3272	495	6	brown	brown	PROPN
ejpam-3272	495	7	,	,	PUNCT
ejpam-3272	495	8	r.	r.	PROPN
ejpam-3272	495	9	ertürk	ertürk	PROPN
ejpam-3272	495	10	and	and	CCONJ
ejpam-3272	495	11	ş.	ş.	ADV
ejpam-3272	495	12	dost	dost	ADJ
ejpam-3272	495	13	.	.	PUNCT
ejpam-3272	496	1	ditopological	ditopological	ADJ
ejpam-3272	496	2	texture	texture	ADJ
ejpam-3272	496	3	spaces	space	NOUN
ejpam-3272	496	4	and	and	CCONJ
ejpam-3272	496	5	fuzzy	fuzzy	ADJ
ejpam-3272	496	6	topology	topology	NOUN
ejpam-3272	496	7	iii	iii	PROPN
ejpam-3272	496	8	:	:	PUNCT
ejpam-3272	496	9	separation	separation	NOUN
ejpam-3272	496	10	axioms	axiom	VERB
ejpam-3272	496	11	.	.	PUNCT
ejpam-3272	497	1	fuzzy	fuzzy	ADJ
ejpam-3272	497	2	sets	set	NOUN
ejpam-3272	497	3	and	and	CCONJ
ejpam-3272	497	4	systems	system	NOUN
ejpam-3272	497	5	,	,	PUNCT
ejpam-3272	497	6	157(14):1886–1912	157(14):1886–1912	NUM
ejpam-3272	497	7	,	,	PUNCT
ejpam-3272	497	8	2006	2006	NUM
ejpam-3272	497	9	.	.	PUNCT
ejpam-3272	498	1	[	[	X
ejpam-3272	498	2	5	5	X
ejpam-3272	498	3	]	]	PUNCT
ejpam-3272	498	4	l.	l.	PROPN
ejpam-3272	498	5	m.	m.	PROPN
ejpam-3272	498	6	brown	brown	PROPN
ejpam-3272	498	7	and	and	CCONJ
ejpam-3272	498	8	m.	m.	NOUN
ejpam-3272	498	9	diker	diker	NOUN
ejpam-3272	498	10	.	.	PUNCT
ejpam-3272	499	1	ditopological	ditopological	ADJ
ejpam-3272	499	2	texture	texture	ADJ
ejpam-3272	499	3	spaces	space	NOUN
ejpam-3272	499	4	and	and	CCONJ
ejpam-3272	499	5	intuitionistic	intuitionistic	ADJ
ejpam-3272	499	6	sets	set	NOUN
ejpam-3272	499	7	.	.	PUNCT
ejpam-3272	500	1	fuzzy	fuzzy	ADJ
ejpam-3272	500	2	sets	set	NOUN
ejpam-3272	500	3	and	and	CCONJ
ejpam-3272	500	4	systems	system	NOUN
ejpam-3272	500	5	,	,	PUNCT
ejpam-3272	500	6	98:217–224	98:217–224	NUM
ejpam-3272	500	7	,	,	PUNCT
ejpam-3272	500	8	1998	1998	NUM
ejpam-3272	500	9	.	.	PUNCT
ejpam-3272	501	1	[	[	X
ejpam-3272	501	2	6	6	NUM
ejpam-3272	501	3	]	]	PUNCT
ejpam-3272	501	4	g.	g.	NOUN
ejpam-3272	501	5	gierz	gierz	PROPN
ejpam-3272	501	6	,	,	PUNCT
ejpam-3272	501	7	k.h	k.h	PROPN
ejpam-3272	501	8	.	.	PROPN
ejpam-3272	501	9	hofmann	hofmann	PROPN
ejpam-3272	501	10	,	,	PUNCT
ejpam-3272	501	11	k.	k.	PROPN
ejpam-3272	501	12	keimel	keimel	PROPN
ejpam-3272	501	13	,	,	PUNCT
ejpam-3272	501	14	j.d	j.d	PROPN
ejpam-3272	501	15	.	.	PROPN
ejpam-3272	501	16	lawson	lawson	PROPN
ejpam-3272	501	17	,	,	PUNCT
ejpam-3272	501	18	m.w	m.w	PROPN
ejpam-3272	501	19	.	.	PROPN
ejpam-3272	501	20	mislove	mislove	PROPN
ejpam-3272	501	21	and	and	CCONJ
ejpam-3272	501	22	d.s	d.s	PROPN
ejpam-3272	501	23	.	.	PROPN
ejpam-3272	501	24	scott	scott	PROPN
ejpam-3272	501	25	.	.	PUNCT
ejpam-3272	502	1	a	a	DET
ejpam-3272	502	2	compendium	compendium	NOUN
ejpam-3272	502	3	of	of	ADP
ejpam-3272	502	4	continuous	continuous	ADJ
ejpam-3272	502	5	lattices	lattice	NOUN
ejpam-3272	502	6	.	.	PUNCT
ejpam-3272	503	1	springer	springer	NOUN
ejpam-3272	503	2	-	-	PUNCT
ejpam-3272	503	3	verlag	verlag	PROPN
ejpam-3272	503	4	berlin	berlin	PROPN
ejpam-3272	503	5	,	,	PUNCT
ejpam-3272	503	6	heidelberg	heidelberg	PROPN
ejpam-3272	503	7	,	,	PUNCT
ejpam-3272	503	8	1980	1980	NUM
ejpam-3272	503	9	.	.	PUNCT
ejpam-3272	504	1	[	[	X
ejpam-3272	504	2	7	7	X
ejpam-3272	504	3	]	]	X
ejpam-3272	504	4	c.	c.	NOUN
ejpam-3272	504	5	good	good	PROPN
ejpam-3272	504	6	,	,	PUNCT
ejpam-3272	504	7	r.	r.	PROPN
ejpam-3272	504	8	kopperman	kopperman	PROPN
ejpam-3272	504	9	and	and	CCONJ
ejpam-3272	504	10	f.	f.	PROPN
ejpam-3272	504	11	yıldız	yıldız	PROPN
ejpam-3272	504	12	.	.	PUNCT
ejpam-3272	505	1	interpolating	interpolate	VERB
ejpam-3272	505	2	functions	function	NOUN
ejpam-3272	505	3	.	.	PUNCT
ejpam-3272	506	1	topology	topology	NOUN
ejpam-3272	506	2	and	and	CCONJ
ejpam-3272	506	3	its	its	PRON
ejpam-3272	506	4	applications	application	NOUN
ejpam-3272	506	5	,	,	PUNCT
ejpam-3272	506	6	158(4):582–593	158(4):582–593	NUM
ejpam-3272	506	7	,	,	PUNCT
ejpam-3272	506	8	2011	2011	NUM
ejpam-3272	506	9	.	.	PUNCT
ejpam-3272	507	1	[	[	X
ejpam-3272	507	2	8	8	NUM
ejpam-3272	507	3	]	]	X
ejpam-3272	507	4	r.	r.	PROPN
ejpam-3272	507	5	kopperman	kopperman	PROPN
ejpam-3272	507	6	.	.	PUNCT
ejpam-3272	508	1	asymmetry	asymmetry	NOUN
ejpam-3272	508	2	and	and	CCONJ
ejpam-3272	508	3	duality	duality	NOUN
ejpam-3272	508	4	in	in	ADP
ejpam-3272	508	5	topology	topology	NOUN
ejpam-3272	508	6	.	.	PUNCT
ejpam-3272	509	1	topology	topology	NOUN
ejpam-3272	509	2	and	and	CCONJ
ejpam-3272	509	3	its	its	PRON
ejpam-3272	509	4	applications	application	NOUN
ejpam-3272	509	5	,	,	PUNCT
ejpam-3272	509	6	66(1):1–39	66(1):1–39	NUM
ejpam-3272	509	7	,	,	PUNCT
ejpam-3272	509	8	1995	1995	NUM
ejpam-3272	509	9	.	.	PUNCT
ejpam-3272	510	1	[	[	X
ejpam-3272	510	2	9	9	NUM
ejpam-3272	510	3	]	]	X
ejpam-3272	510	4	e.	e.	PROPN
ejpam-3272	510	5	korkmaz	korkmaz	PROPN
ejpam-3272	510	6	and	and	CCONJ
ejpam-3272	510	7	r.	r.	PROPN
ejpam-3272	510	8	ertürk	ertürk	PROPN
ejpam-3272	510	9	.	.	PUNCT
ejpam-3272	511	1	on	on	ADP
ejpam-3272	511	2	a	a	DET
ejpam-3272	511	3	new	new	ADJ
ejpam-3272	511	4	generalization	generalization	NOUN
ejpam-3272	511	5	of	of	ADP
ejpam-3272	511	6	ditopological	ditopological	ADJ
ejpam-3272	511	7	texture	texture	ADJ
ejpam-3272	511	8	spaces	space	NOUN
ejpam-3272	511	9	.	.	PUNCT
ejpam-3272	512	1	submitted	submit	VERB
ejpam-3272	512	2	.	.	PUNCT
ejpam-3272	513	1	[	[	X
ejpam-3272	513	2	10	10	NUM
ejpam-3272	513	3	]	]	X
ejpam-3272	513	4	j.	j.	PROPN
ejpam-3272	513	5	picado	picado	PROPN
ejpam-3272	513	6	and	and	CCONJ
ejpam-3272	513	7	a.	a.	NOUN
ejpam-3272	513	8	pultr	pultr	NOUN
ejpam-3272	513	9	.	.	PUNCT
ejpam-3272	514	1	frames	frame	NOUN
ejpam-3272	514	2	and	and	CCONJ
ejpam-3272	514	3	locales	locale	NOUN
ejpam-3272	514	4	.	.	PUNCT
ejpam-3272	515	1	topology	topology	NOUN
ejpam-3272	515	2	without	without	ADP
ejpam-3272	515	3	points	point	NOUN
ejpam-3272	515	4	.	.	PUNCT
ejpam-3272	516	1	springer	springer	NOUN
ejpam-3272	516	2	basel	basel	PROPN
ejpam-3272	516	3	ag	ag	PROPN
ejpam-3272	516	4	,	,	PUNCT
ejpam-3272	516	5	2012	2012	NUM
ejpam-3272	516	6	.	.	PUNCT
ejpam-3272	517	1	[	[	X
ejpam-3272	517	2	11	11	NUM
ejpam-3272	517	3	]	]	X
ejpam-3272	517	4	h.	h.	PROPN
ejpam-3272	517	5	rasiowa	rasiowa	PROPN
ejpam-3272	517	6	and	and	CCONJ
ejpam-3272	517	7	r.	r.	PROPN
ejpam-3272	517	8	sikorski	sikorski	PROPN
ejpam-3272	517	9	.	.	PUNCT
ejpam-3272	518	1	the	the	DET
ejpam-3272	518	2	mathematics	mathematic	NOUN
ejpam-3272	518	3	of	of	ADP
ejpam-3272	518	4	metamathematics	metamathematic	NOUN
ejpam-3272	518	5	.	.	PUNCT
ejpam-3272	519	1	panstwowe	panstwowe	PROPN
ejpam-3272	519	2	wydawnictwo	wydawnictwo	PROPN
ejpam-3272	519	3	,	,	PUNCT
ejpam-3272	519	4	naukowe	naukowe	NOUN
ejpam-3272	519	5	,	,	PUNCT
ejpam-3272	519	6	1963	1963	NUM
ejpam-3272	519	7	.	.	PUNCT