id	sid	tid	token	lemma	pos
ejpam-3418	1	1	european	european	PROPN
ejpam-3418	1	2	journal	journal	PROPN
ejpam-3418	1	3	of	of	ADP
ejpam-3418	1	4	pure	pure	ADJ
ejpam-3418	1	5	and	and	CCONJ
ejpam-3418	1	6	applied	apply	VERB
ejpam-3418	1	7	mathematics	mathematic	NOUN
ejpam-3418	1	8	vol	vol	NOUN
ejpam-3418	1	9	.	.	PROPN
ejpam-3418	2	1	12	12	NUM
ejpam-3418	2	2	,	,	PUNCT
ejpam-3418	2	3	no	no	INTJ
ejpam-3418	2	4	.	.	NOUN
ejpam-3418	2	5	2	2	NUM
ejpam-3418	2	6	,	,	PUNCT
ejpam-3418	2	7	2019	2019	NUM
ejpam-3418	2	8	,	,	PUNCT
ejpam-3418	2	9	553	553	NUM
ejpam-3418	2	10	-	-	SYM
ejpam-3418	2	11	570	570	NUM
ejpam-3418	2	12	issn	issn	PROPN
ejpam-3418	2	13	1307	1307	NUM
ejpam-3418	2	14	-	-	SYM
ejpam-3418	2	15	5543	5543	NUM
ejpam-3418	2	16	–	–	PUNCT
ejpam-3418	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3418	2	18	published	publish	VERB
ejpam-3418	2	19	by	by	ADP
ejpam-3418	2	20	new	new	PROPN
ejpam-3418	2	21	york	york	PROPN
ejpam-3418	2	22	business	business	PROPN
ejpam-3418	2	23	global	global	ADJ
ejpam-3418	2	24	on	on	ADP
ejpam-3418	2	25	interval	interval	NOUN
ejpam-3418	2	26	-	-	PUNCT
ejpam-3418	2	27	valued	value	VERB
ejpam-3418	2	28	fuzzy	fuzzy	NOUN
ejpam-3418	2	29	on	on	ADP
ejpam-3418	2	30	ideal	ideal	ADJ
ejpam-3418	2	31	sets	set	NOUN
ejpam-3418	2	32	mary	mary	PROPN
ejpam-3418	2	33	joy	joy	PROPN
ejpam-3418	2	34	s.	s.	PROPN
ejpam-3418	2	35	togonon1	togonon1	PROPN
ejpam-3418	2	36	,	,	PUNCT
ejpam-3418	2	37	randy	randy	PROPN
ejpam-3418	2	38	l.	l.	PROPN
ejpam-3418	2	39	caga	caga	PROPN
ejpam-3418	2	40	-	-	PUNCT
ejpam-3418	2	41	anan2,∗	anan2,∗	ADJ
ejpam-3418	2	42	1	1	NUM
ejpam-3418	2	43	bukidnon	bukidnon	NOUN
ejpam-3418	2	44	state	state	PROPN
ejpam-3418	2	45	university	university	PROPN
ejpam-3418	2	46	-	-	PUNCT
ejpam-3418	2	47	baungon	baungon	PROPN
ejpam-3418	2	48	satellite	satellite	PROPN
ejpam-3418	2	49	campus	campus	PROPN
ejpam-3418	2	50	,	,	PUNCT
ejpam-3418	2	51	bukidnon	bukidnon	NOUN
ejpam-3418	2	52	,	,	PUNCT
ejpam-3418	2	53	philippines	philippines	PROPN
ejpam-3418	2	54	2	2	NUM
ejpam-3418	2	55	department	department	NOUN
ejpam-3418	2	56	of	of	ADP
ejpam-3418	2	57	mathematics	mathematic	NOUN
ejpam-3418	2	58	and	and	CCONJ
ejpam-3418	2	59	statistics	statistic	NOUN
ejpam-3418	2	60	,	,	PUNCT
ejpam-3418	2	61	college	college	NOUN
ejpam-3418	2	62	of	of	ADP
ejpam-3418	2	63	science	science	NOUN
ejpam-3418	2	64	and	and	CCONJ
ejpam-3418	2	65	mathematics	mathematic	NOUN
ejpam-3418	2	66	,	,	PUNCT
ejpam-3418	2	67	and	and	CCONJ
ejpam-3418	2	68	premier	premier	PROPN
ejpam-3418	2	69	research	research	PROPN
ejpam-3418	2	70	institute	institute	PROPN
ejpam-3418	2	71	of	of	ADP
ejpam-3418	2	72	science	science	NOUN
ejpam-3418	2	73	and	and	CCONJ
ejpam-3418	2	74	mathematics	mathematic	NOUN
ejpam-3418	2	75	,	,	PUNCT
ejpam-3418	2	76	msu	msu	PROPN
ejpam-3418	2	77	-	-	PUNCT
ejpam-3418	2	78	iligan	iligan	PROPN
ejpam-3418	2	79	institute	institute	PROPN
ejpam-3418	2	80	of	of	ADP
ejpam-3418	2	81	technology	technology	PROPN
ejpam-3418	2	82	,	,	PUNCT
ejpam-3418	2	83	iligan	iligan	PROPN
ejpam-3418	2	84	city	city	PROPN
ejpam-3418	2	85	,	,	PUNCT
ejpam-3418	2	86	philippines	philippine	NOUN
ejpam-3418	2	87	abstract	abstract	ADJ
ejpam-3418	2	88	.	.	PUNCT
ejpam-3418	3	1	fuzzy	fuzzy	ADJ
ejpam-3418	3	2	sets	set	NOUN
ejpam-3418	3	3	,	,	PUNCT
ejpam-3418	3	4	formalized	formalize	VERB
ejpam-3418	3	5	by	by	ADP
ejpam-3418	3	6	zadeh	zadeh	PROPN
ejpam-3418	3	7	in	in	ADP
ejpam-3418	3	8	1965	1965	NUM
ejpam-3418	3	9	,	,	PUNCT
ejpam-3418	3	10	generalizes	generalize	VERB
ejpam-3418	3	11	the	the	DET
ejpam-3418	3	12	classical	classical	ADJ
ejpam-3418	3	13	idea	idea	NOUN
ejpam-3418	3	14	of	of	ADP
ejpam-3418	3	15	sets	set	NOUN
ejpam-3418	3	16	.	.	PUNCT
ejpam-3418	4	1	the	the	DET
ejpam-3418	4	2	idea	idea	NOUN
ejpam-3418	4	3	itself	itself	PRON
ejpam-3418	4	4	was	be	AUX
ejpam-3418	4	5	generalized	generalize	VERB
ejpam-3418	4	6	in	in	ADP
ejpam-3418	4	7	1975	1975	NUM
ejpam-3418	4	8	when	when	SCONJ
ejpam-3418	4	9	zadeh	zadeh	PROPN
ejpam-3418	4	10	introduced	introduce	VERB
ejpam-3418	4	11	the	the	DET
ejpam-3418	4	12	interval	interval	NOUN
ejpam-3418	4	13	-	-	PUNCT
ejpam-3418	4	14	valued	value	VERB
ejpam-3418	4	15	fuzzy	fuzzy	ADJ
ejpam-3418	4	16	sets	set	NOUN
ejpam-3418	4	17	.	.	PUNCT
ejpam-3418	5	1	in	in	ADP
ejpam-3418	5	2	this	this	DET
ejpam-3418	5	3	paper	paper	NOUN
ejpam-3418	5	4	,	,	PUNCT
ejpam-3418	5	5	we	we	PRON
ejpam-3418	5	6	generalize	generalize	VERB
ejpam-3418	5	7	further	far	ADV
ejpam-3418	5	8	the	the	DET
ejpam-3418	5	9	above	above	ADJ
ejpam-3418	5	10	concepts	concept	NOUN
ejpam-3418	5	11	by	by	ADP
ejpam-3418	5	12	introducing	introduce	VERB
ejpam-3418	5	13	interval	interval	NOUN
ejpam-3418	5	14	-	-	PUNCT
ejpam-3418	5	15	valued	value	VERB
ejpam-3418	5	16	fuzzy	fuzzy	NOUN
ejpam-3418	5	17	on	on	ADP
ejpam-3418	5	18	ideal	ideal	ADJ
ejpam-3418	5	19	sets	set	NOUN
ejpam-3418	5	20	,	,	PUNCT
ejpam-3418	5	21	where	where	SCONJ
ejpam-3418	5	22	an	an	DET
ejpam-3418	5	23	ideal	ideal	NOUN
ejpam-3418	5	24	is	be	AUX
ejpam-3418	5	25	a	a	DET
ejpam-3418	5	26	nonempty	nonempty	ADJ
ejpam-3418	5	27	collection	collection	NOUN
ejpam-3418	5	28	of	of	ADP
ejpam-3418	5	29	sets	set	NOUN
ejpam-3418	5	30	with	with	ADP
ejpam-3418	5	31	a	a	DET
ejpam-3418	5	32	property	property	NOUN
ejpam-3418	5	33	describing	describe	VERB
ejpam-3418	5	34	the	the	DET
ejpam-3418	5	35	notion	notion	NOUN
ejpam-3418	5	36	of	of	ADP
ejpam-3418	5	37	smallness	smallness	NOUN
ejpam-3418	5	38	.	.	PUNCT
ejpam-3418	6	1	we	we	PRON
ejpam-3418	6	2	develop	develop	VERB
ejpam-3418	6	3	its	its	PRON
ejpam-3418	6	4	basic	basic	ADJ
ejpam-3418	6	5	concepts	concept	NOUN
ejpam-3418	6	6	and	and	CCONJ
ejpam-3418	6	7	properties	property	NOUN
ejpam-3418	6	8	and	and	CCONJ
ejpam-3418	6	9	consider	consider	VERB
ejpam-3418	6	10	how	how	SCONJ
ejpam-3418	6	11	one	one	PRON
ejpam-3418	6	12	can	can	AUX
ejpam-3418	6	13	create	create	VERB
ejpam-3418	6	14	mappings	mapping	NOUN
ejpam-3418	6	15	of	of	ADP
ejpam-3418	6	16	intervalvalued	intervalvalue	VERB
ejpam-3418	6	17	fuzzy	fuzzy	ADJ
ejpam-3418	6	18	on	on	ADP
ejpam-3418	6	19	ideal	ideal	ADJ
ejpam-3418	6	20	sets	set	NOUN
ejpam-3418	6	21	from	from	ADP
ejpam-3418	6	22	mappings	mapping	NOUN
ejpam-3418	6	23	of	of	ADP
ejpam-3418	6	24	ordinary	ordinary	ADJ
ejpam-3418	6	25	sets	set	NOUN
ejpam-3418	6	26	.	.	PUNCT
ejpam-3418	7	1	we	we	PRON
ejpam-3418	7	2	then	then	ADV
ejpam-3418	7	3	consider	consider	VERB
ejpam-3418	7	4	topology	topology	NOUN
ejpam-3418	7	5	and	and	CCONJ
ejpam-3418	7	6	continuity	continuity	NOUN
ejpam-3418	7	7	with	with	ADP
ejpam-3418	7	8	respect	respect	NOUN
ejpam-3418	7	9	to	to	ADP
ejpam-3418	7	10	these	these	DET
ejpam-3418	7	11	sets	set	NOUN
ejpam-3418	7	12	.	.	PUNCT
ejpam-3418	8	1	2010	2010	NUM
ejpam-3418	8	2	mathematics	mathematic	NOUN
ejpam-3418	8	3	subject	subject	NOUN
ejpam-3418	8	4	classifications	classification	NOUN
ejpam-3418	8	5	:	:	PUNCT
ejpam-3418	8	6	03e72	03e72	NUM
ejpam-3418	8	7	,	,	PUNCT
ejpam-3418	8	8	62b86	62b86	NUM
ejpam-3418	8	9	,	,	PUNCT
ejpam-3418	8	10	94d05	94d05	NUM
ejpam-3418	8	11	key	key	ADJ
ejpam-3418	8	12	words	word	NOUN
ejpam-3418	8	13	and	and	CCONJ
ejpam-3418	8	14	phrases	phrase	NOUN
ejpam-3418	8	15	:	:	PUNCT
ejpam-3418	8	16	fuzzy	fuzzy	ADJ
ejpam-3418	8	17	sets	set	NOUN
ejpam-3418	8	18	,	,	PUNCT
ejpam-3418	8	19	interval	interval	NOUN
ejpam-3418	8	20	-	-	PUNCT
ejpam-3418	8	21	valued	value	VERB
ejpam-3418	8	22	,	,	PUNCT
ejpam-3418	8	23	ideal	ideal	ADJ
ejpam-3418	8	24	1	1	NUM
ejpam-3418	8	25	.	.	PUNCT
ejpam-3418	9	1	introduction	introduction	NOUN
ejpam-3418	9	2	in	in	ADP
ejpam-3418	9	3	classical	classical	ADJ
ejpam-3418	9	4	set	set	NOUN
ejpam-3418	9	5	theory	theory	NOUN
ejpam-3418	9	6	,	,	PUNCT
ejpam-3418	9	7	an	an	DET
ejpam-3418	9	8	element	element	NOUN
ejpam-3418	9	9	either	either	CCONJ
ejpam-3418	9	10	belongs	belong	VERB
ejpam-3418	9	11	or	or	CCONJ
ejpam-3418	9	12	does	do	AUX
ejpam-3418	9	13	not	not	PART
ejpam-3418	9	14	belong	belong	VERB
ejpam-3418	9	15	to	to	ADP
ejpam-3418	9	16	a	a	DET
ejpam-3418	9	17	given	give	VERB
ejpam-3418	9	18	set	set	NOUN
ejpam-3418	9	19	.	.	PUNCT
ejpam-3418	10	1	that	that	PRON
ejpam-3418	10	2	is	be	AUX
ejpam-3418	10	3	,	,	PUNCT
ejpam-3418	10	4	the	the	DET
ejpam-3418	10	5	membership	membership	NOUN
ejpam-3418	10	6	of	of	ADP
ejpam-3418	10	7	elements	element	NOUN
ejpam-3418	10	8	to	to	ADP
ejpam-3418	10	9	a	a	DET
ejpam-3418	10	10	given	give	VERB
ejpam-3418	10	11	set	set	NOUN
ejpam-3418	10	12	is	be	AUX
ejpam-3418	10	13	assessed	assess	VERB
ejpam-3418	10	14	in	in	ADP
ejpam-3418	10	15	binary	binary	ADJ
ejpam-3418	10	16	terms	term	NOUN
ejpam-3418	10	17	.	.	PUNCT
ejpam-3418	11	1	thus	thus	ADV
ejpam-3418	11	2	,	,	PUNCT
ejpam-3418	11	3	one	one	PRON
ejpam-3418	11	4	may	may	AUX
ejpam-3418	11	5	associate	associate	VERB
ejpam-3418	11	6	a	a	DET
ejpam-3418	11	7	set	set	NOUN
ejpam-3418	11	8	a	a	PRON
ejpam-3418	11	9	on	on	ADP
ejpam-3418	11	10	a	a	DET
ejpam-3418	11	11	universal	universal	ADJ
ejpam-3418	11	12	set	set	VERB
ejpam-3418	11	13	u	u	NOUN
ejpam-3418	11	14	to	to	ADP
ejpam-3418	11	15	the	the	DET
ejpam-3418	11	16	characteristic	characteristic	ADJ
ejpam-3418	11	17	function	function	NOUN
ejpam-3418	11	18	of	of	ADP
ejpam-3418	11	19	a	a	PRON
ejpam-3418	11	20	with	with	ADP
ejpam-3418	11	21	values	value	NOUN
ejpam-3418	11	22	0	0	NUM
ejpam-3418	11	23	or	or	CCONJ
ejpam-3418	11	24	1	1	NUM
ejpam-3418	11	25	.	.	PUNCT
ejpam-3418	12	1	however	however	ADV
ejpam-3418	12	2	,	,	PUNCT
ejpam-3418	12	3	there	there	PRON
ejpam-3418	12	4	are	be	VERB
ejpam-3418	12	5	informations	information	NOUN
ejpam-3418	12	6	that	that	PRON
ejpam-3418	12	7	can	can	AUX
ejpam-3418	12	8	not	not	PART
ejpam-3418	12	9	be	be	AUX
ejpam-3418	12	10	precisely	precisely	ADV
ejpam-3418	12	11	assessed	assess	VERB
ejpam-3418	12	12	as	as	ADP
ejpam-3418	12	13	belonging	belong	VERB
ejpam-3418	12	14	to	to	ADP
ejpam-3418	12	15	or	or	CCONJ
ejpam-3418	12	16	not	not	PART
ejpam-3418	12	17	to	to	ADP
ejpam-3418	12	18	a	a	DET
ejpam-3418	12	19	given	give	VERB
ejpam-3418	12	20	set	set	NOUN
ejpam-3418	12	21	,	,	PUNCT
ejpam-3418	12	22	like	like	ADP
ejpam-3418	12	23	the	the	DET
ejpam-3418	12	24	set	set	NOUN
ejpam-3418	12	25	of	of	ADP
ejpam-3418	12	26	young	young	ADJ
ejpam-3418	12	27	people	people	NOUN
ejpam-3418	12	28	in	in	ADP
ejpam-3418	12	29	a	a	DET
ejpam-3418	12	30	group	group	NOUN
ejpam-3418	12	31	.	.	PUNCT
ejpam-3418	13	1	to	to	PART
ejpam-3418	13	2	address	address	VERB
ejpam-3418	13	3	this	this	DET
ejpam-3418	13	4	problem	problem	NOUN
ejpam-3418	13	5	,	,	PUNCT
ejpam-3418	13	6	in	in	ADP
ejpam-3418	13	7	1965	1965	NUM
ejpam-3418	13	8	,	,	PUNCT
ejpam-3418	13	9	zadeh	zadeh	PROPN
ejpam-3418	14	1	[	[	X
ejpam-3418	14	2	9	9	NUM
ejpam-3418	14	3	]	]	PUNCT
ejpam-3418	14	4	and	and	CCONJ
ejpam-3418	14	5	klaua	klaua	NOUN
ejpam-3418	14	6	[	[	X
ejpam-3418	14	7	4	4	X
ejpam-3418	14	8	]	]	PUNCT
ejpam-3418	14	9	introduced	introduce	VERB
ejpam-3418	14	10	fuzzy	fuzzy	ADJ
ejpam-3418	14	11	sets	set	NOUN
ejpam-3418	14	12	,	,	PUNCT
ejpam-3418	14	13	where	where	SCONJ
ejpam-3418	14	14	elements	element	NOUN
ejpam-3418	14	15	have	have	VERB
ejpam-3418	14	16	degrees	degree	NOUN
ejpam-3418	14	17	of	of	ADP
ejpam-3418	14	18	membership	membership	NOUN
ejpam-3418	14	19	,	,	PUNCT
ejpam-3418	14	20	not	not	PART
ejpam-3418	14	21	just	just	ADV
ejpam-3418	14	22	0	0	NUM
ejpam-3418	14	23	or	or	CCONJ
ejpam-3418	14	24	1	1	NUM
ejpam-3418	14	25	.	.	PUNCT
ejpam-3418	14	26	formally	formally	ADV
ejpam-3418	14	27	defined	define	VERB
ejpam-3418	14	28	,	,	PUNCT
ejpam-3418	14	29	a	a	DET
ejpam-3418	14	30	fuzzy	fuzzy	ADJ
ejpam-3418	14	31	set	set	NOUN
ejpam-3418	14	32	is	be	AUX
ejpam-3418	14	33	a	a	DET
ejpam-3418	14	34	mapping	mapping	NOUN
ejpam-3418	14	35	from	from	ADP
ejpam-3418	14	36	u	u	NOUN
ejpam-3418	14	37	into	into	ADP
ejpam-3418	14	38	the	the	DET
ejpam-3418	14	39	unit	unit	NOUN
ejpam-3418	14	40	interval	interval	NOUN
ejpam-3418	14	41	[	[	X
ejpam-3418	14	42	0	0	NUM
ejpam-3418	14	43	,	,	PUNCT
ejpam-3418	14	44	1	1	NUM
ejpam-3418	14	45	]	]	PUNCT
ejpam-3418	14	46	.	.	PUNCT
ejpam-3418	15	1	in	in	ADP
ejpam-3418	15	2	our	our	PRON
ejpam-3418	15	3	example	example	NOUN
ejpam-3418	15	4	,	,	PUNCT
ejpam-3418	15	5	for	for	ADP
ejpam-3418	15	6	a	a	DET
ejpam-3418	15	7	not	not	PART
ejpam-3418	15	8	so	so	ADV
ejpam-3418	15	9	young	young	ADJ
ejpam-3418	15	10	member	member	NOUN
ejpam-3418	15	11	of	of	ADP
ejpam-3418	15	12	the	the	DET
ejpam-3418	15	13	group	group	NOUN
ejpam-3418	15	14	,	,	PUNCT
ejpam-3418	15	15	a	a	DET
ejpam-3418	15	16	degree	degree	NOUN
ejpam-3418	15	17	of	of	ADP
ejpam-3418	15	18	membership	membership	NOUN
ejpam-3418	15	19	equal	equal	ADJ
ejpam-3418	15	20	to	to	ADP
ejpam-3418	15	21	0.2	0.2	NUM
ejpam-3418	15	22	can	can	AUX
ejpam-3418	15	23	be	be	AUX
ejpam-3418	15	24	assigned	assign	VERB
ejpam-3418	15	25	.	.	PUNCT
ejpam-3418	16	1	in	in	ADP
ejpam-3418	16	2	1978	1978	NUM
ejpam-3418	16	3	,	,	PUNCT
ejpam-3418	16	4	zadeh	zadeh	PROPN
ejpam-3418	16	5	used	use	VERB
ejpam-3418	16	6	his	his	PRON
ejpam-3418	16	7	theory	theory	NOUN
ejpam-3418	16	8	of	of	ADP
ejpam-3418	16	9	fuzzy	fuzzy	ADJ
ejpam-3418	16	10	sets	set	NOUN
ejpam-3418	16	11	and	and	CCONJ
ejpam-3418	16	12	fuzzy	fuzzy	ADJ
ejpam-3418	16	13	logic	logic	NOUN
ejpam-3418	16	14	to	to	PART
ejpam-3418	16	15	introduce	introduce	VERB
ejpam-3418	16	16	possibility	possibility	NOUN
ejpam-3418	16	17	theory	theory	NOUN
ejpam-3418	16	18	[	[	X
ejpam-3418	16	19	11	11	NUM
ejpam-3418	16	20	]	]	PUNCT
ejpam-3418	16	21	.	.	PUNCT
ejpam-3418	17	1	the	the	DET
ejpam-3418	17	2	theory	theory	NOUN
ejpam-3418	17	3	uses	use	VERB
ejpam-3418	17	4	a	a	DET
ejpam-3418	17	5	possibility	possibility	NOUN
ejpam-3418	17	6	distribution	distribution	NOUN
ejpam-3418	17	7	which	which	PRON
ejpam-3418	17	8	should	should	AUX
ejpam-3418	17	9	not	not	PART
ejpam-3418	17	10	be	be	AUX
ejpam-3418	17	11	confused	confuse	VERB
ejpam-3418	17	12	with	with	ADP
ejpam-3418	17	13	a	a	DET
ejpam-3418	17	14	probability	probability	NOUN
ejpam-3418	17	15	distribution	distribution	NOUN
ejpam-3418	17	16	.	.	PUNCT
ejpam-3418	18	1	both	both	PRON
ejpam-3418	18	2	are	be	AUX
ejpam-3418	18	3	fuzzy	fuzzy	ADJ
ejpam-3418	18	4	sets	set	NOUN
ejpam-3418	18	5	but	but	CCONJ
ejpam-3418	18	6	the	the	DET
ejpam-3418	18	7	sum	sum	NOUN
ejpam-3418	18	8	of	of	ADP
ejpam-3418	18	9	the	the	DET
ejpam-3418	18	10	values	value	NOUN
ejpam-3418	18	11	of	of	ADP
ejpam-3418	18	12	a	a	DET
ejpam-3418	18	13	possibility	possibility	NOUN
ejpam-3418	18	14	distribution	distribution	NOUN
ejpam-3418	18	15	need	need	AUX
ejpam-3418	18	16	not	not	PART
ejpam-3418	18	17	be	be	AUX
ejpam-3418	18	18	1	1	NUM
ejpam-3418	18	19	while	while	SCONJ
ejpam-3418	18	20	it	it	PRON
ejpam-3418	18	21	should	should	AUX
ejpam-3418	18	22	be	be	AUX
ejpam-3418	18	23	1	1	NUM
ejpam-3418	18	24	in	in	ADP
ejpam-3418	18	25	a	a	DET
ejpam-3418	18	26	probability	probability	NOUN
ejpam-3418	18	27	distribution	distribution	NOUN
ejpam-3418	18	28	.	.	PUNCT
ejpam-3418	19	1	for	for	ADP
ejpam-3418	19	2	instance	instance	NOUN
ejpam-3418	19	3	,	,	PUNCT
ejpam-3418	19	4	∗corresponding	∗corresponde	VERB
ejpam-3418	19	5	author	author	NOUN
ejpam-3418	19	6	.	.	PUNCT
ejpam-3418	20	1	doi	doi	PROPN
ejpam-3418	20	2	:	:	PUNCT
ejpam-3418	20	3	https://doi.org/10.29020/nybg.ejpam.v12i2.3418	https://doi.org/10.29020/nybg.ejpam.v12i2.3418	PRON
ejpam-3418	20	4	email	email	NOUN
ejpam-3418	20	5	addresses	address	NOUN
ejpam-3418	20	6	:	:	PUNCT
ejpam-3418	20	7	maryjoy.togonon@g.msuiit.edu.ph	maryjoy.togonon@g.msuiit.edu.ph	PROPN
ejpam-3418	20	8	(	(	PUNCT
ejpam-3418	20	9	mj	mj	PROPN
ejpam-3418	20	10	togonon	togonon	PROPN
ejpam-3418	20	11	)	)	PUNCT
ejpam-3418	20	12	,	,	PUNCT
ejpam-3418	20	13	randy.caga-anan@g.msuiit.edu.ph	randy.caga-anan@g.msuiit.edu.ph	PUNCT
ejpam-3418	20	14	(	(	PUNCT
ejpam-3418	20	15	r.	r.	PROPN
ejpam-3418	20	16	caga	caga	PROPN
ejpam-3418	20	17	-	-	PUNCT
ejpam-3418	20	18	anan	anan	PROPN
ejpam-3418	20	19	)	)	PUNCT
ejpam-3418	20	20	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3418	21	1	553	553	NUM
ejpam-3418	21	2	c	c	X
ejpam-3418	21	3	©	©	PROPN
ejpam-3418	21	4	2019	2019	NUM
ejpam-3418	21	5	ejpam	ejpam	NOUN
ejpam-3418	21	6	all	all	DET
ejpam-3418	21	7	rights	right	NOUN
ejpam-3418	21	8	reserved	reserve	VERB
ejpam-3418	21	9	.	.	PUNCT
ejpam-3418	22	1	mj	mj	PROPN
ejpam-3418	22	2	togonon	togonon	PROPN
ejpam-3418	22	3	,	,	PUNCT
ejpam-3418	22	4	r	r	NOUN
ejpam-3418	22	5	caga	caga	NOUN
ejpam-3418	22	6	-	-	PUNCT
ejpam-3418	22	7	anan	anan	PROPN
ejpam-3418	22	8	/	/	SYM
ejpam-3418	22	9	eur	eur	PROPN
ejpam-3418	22	10	.	.	PUNCT
ejpam-3418	23	1	j.	j.	PROPN
ejpam-3418	23	2	pure	pure	PROPN
ejpam-3418	23	3	appl	appl	PROPN
ejpam-3418	23	4	.	.	PROPN
ejpam-3418	23	5	math	math	PROPN
ejpam-3418	23	6	,	,	PUNCT
ejpam-3418	23	7	12	12	NUM
ejpam-3418	23	8	(	(	PUNCT
ejpam-3418	23	9	2	2	NUM
ejpam-3418	23	10	)	)	PUNCT
ejpam-3418	23	11	(	(	PUNCT
ejpam-3418	23	12	2019	2019	NUM
ejpam-3418	23	13	)	)	PUNCT
ejpam-3418	23	14	,	,	PUNCT
ejpam-3418	23	15	553	553	NUM
ejpam-3418	23	16	-	-	SYM
ejpam-3418	23	17	570	570	NUM
ejpam-3418	23	18	554	554	NUM
ejpam-3418	23	19	we	we	PRON
ejpam-3418	23	20	may	may	AUX
ejpam-3418	23	21	assign	assign	VERB
ejpam-3418	23	22	a	a	DET
ejpam-3418	23	23	value	value	NOUN
ejpam-3418	23	24	of	of	ADP
ejpam-3418	23	25	0.4	0.4	NUM
ejpam-3418	23	26	for	for	ADP
ejpam-3418	23	27	the	the	DET
ejpam-3418	23	28	possibility	possibility	NOUN
ejpam-3418	23	29	that	that	SCONJ
ejpam-3418	23	30	tomorrow	tomorrow	NOUN
ejpam-3418	23	31	there	there	PRON
ejpam-3418	23	32	will	will	AUX
ejpam-3418	23	33	be	be	AUX
ejpam-3418	23	34	rain	rain	NOUN
ejpam-3418	23	35	and	and	CCONJ
ejpam-3418	23	36	a	a	DET
ejpam-3418	23	37	value	value	NOUN
ejpam-3418	23	38	of	of	ADP
ejpam-3418	23	39	0.7	0.7	NUM
ejpam-3418	23	40	for	for	ADP
ejpam-3418	23	41	the	the	DET
ejpam-3418	23	42	possibility	possibility	NOUN
ejpam-3418	23	43	that	that	SCONJ
ejpam-3418	23	44	tomorrow	tomorrow	NOUN
ejpam-3418	23	45	will	will	AUX
ejpam-3418	23	46	be	be	AUX
ejpam-3418	23	47	sunny	sunny	ADJ
ejpam-3418	23	48	.	.	PUNCT
ejpam-3418	24	1	the	the	DET
ejpam-3418	24	2	sum	sum	NOUN
ejpam-3418	24	3	of	of	ADP
ejpam-3418	24	4	these	these	DET
ejpam-3418	24	5	two	two	NUM
ejpam-3418	24	6	possibilities	possibility	NOUN
ejpam-3418	24	7	is	be	AUX
ejpam-3418	24	8	already	already	ADV
ejpam-3418	24	9	greater	great	ADJ
ejpam-3418	24	10	than	than	ADP
ejpam-3418	24	11	1	1	NUM
ejpam-3418	24	12	,	,	PUNCT
ejpam-3418	24	13	but	but	CCONJ
ejpam-3418	24	14	our	our	PRON
ejpam-3418	24	15	assignment	assignment	NOUN
ejpam-3418	24	16	may	may	AUX
ejpam-3418	24	17	represent	represent	VERB
ejpam-3418	24	18	best	good	ADJ
ejpam-3418	24	19	the	the	DET
ejpam-3418	24	20	information	information	NOUN
ejpam-3418	24	21	that	that	PRON
ejpam-3418	24	22	we	we	PRON
ejpam-3418	24	23	know	know	VERB
ejpam-3418	24	24	about	about	ADP
ejpam-3418	24	25	what	what	PRON
ejpam-3418	24	26	will	will	AUX
ejpam-3418	24	27	be	be	AUX
ejpam-3418	24	28	the	the	DET
ejpam-3418	24	29	weather	weather	NOUN
ejpam-3418	24	30	for	for	ADP
ejpam-3418	24	31	tomorrow	tomorrow	NOUN
ejpam-3418	24	32	.	.	PUNCT
ejpam-3418	25	1	informations	information	NOUN
ejpam-3418	25	2	like	like	ADP
ejpam-3418	25	3	these	these	PRON
ejpam-3418	25	4	,	,	PUNCT
ejpam-3418	25	5	with	with	ADP
ejpam-3418	25	6	a	a	DET
ejpam-3418	25	7	lot	lot	NOUN
ejpam-3418	25	8	of	of	ADP
ejpam-3418	25	9	uncertainties	uncertainty	NOUN
ejpam-3418	25	10	,	,	PUNCT
ejpam-3418	25	11	are	be	AUX
ejpam-3418	25	12	not	not	PART
ejpam-3418	25	13	suited	suit	VERB
ejpam-3418	25	14	to	to	PART
ejpam-3418	25	15	be	be	AUX
ejpam-3418	25	16	expressed	express	VERB
ejpam-3418	25	17	using	use	VERB
ejpam-3418	25	18	a	a	DET
ejpam-3418	25	19	probability	probability	NOUN
ejpam-3418	25	20	distribution	distribution	NOUN
ejpam-3418	25	21	.	.	PUNCT
ejpam-3418	26	1	now	now	ADV
ejpam-3418	26	2	,	,	PUNCT
ejpam-3418	26	3	consider	consider	VERB
ejpam-3418	26	4	the	the	DET
ejpam-3418	26	5	possibility	possibility	NOUN
ejpam-3418	26	6	that	that	SCONJ
ejpam-3418	26	7	tomorrow	tomorrow	NOUN
ejpam-3418	26	8	there	there	PRON
ejpam-3418	26	9	will	will	AUX
ejpam-3418	26	10	be	be	AUX
ejpam-3418	26	11	rain	rain	NOUN
ejpam-3418	26	12	and	and	CCONJ
ejpam-3418	26	13	at	at	ADP
ejpam-3418	26	14	the	the	DET
ejpam-3418	26	15	same	same	ADJ
ejpam-3418	26	16	time	time	NOUN
ejpam-3418	26	17	it	it	PRON
ejpam-3418	26	18	will	will	AUX
ejpam-3418	26	19	be	be	AUX
ejpam-3418	26	20	sunny	sunny	ADJ
ejpam-3418	26	21	.	.	PUNCT
ejpam-3418	27	1	this	this	DET
ejpam-3418	27	2	case	case	NOUN
ejpam-3418	27	3	is	be	AUX
ejpam-3418	27	4	not	not	PART
ejpam-3418	27	5	impossible	impossible	ADJ
ejpam-3418	27	6	as	as	SCONJ
ejpam-3418	27	7	it	it	PRON
ejpam-3418	27	8	happens	happen	VERB
ejpam-3418	27	9	rarely	rarely	ADV
ejpam-3418	27	10	in	in	ADP
ejpam-3418	27	11	the	the	DET
ejpam-3418	27	12	philippines	philippine	NOUN
ejpam-3418	27	13	.	.	PUNCT
ejpam-3418	28	1	but	but	CCONJ
ejpam-3418	28	2	its	its	PRON
ejpam-3418	28	3	possibility	possibility	NOUN
ejpam-3418	28	4	should	should	AUX
ejpam-3418	28	5	be	be	AUX
ejpam-3418	28	6	far	far	ADV
ejpam-3418	28	7	less	less	ADJ
ejpam-3418	28	8	than	than	ADP
ejpam-3418	28	9	any	any	PRON
ejpam-3418	28	10	of	of	ADP
ejpam-3418	28	11	the	the	DET
ejpam-3418	28	12	two	two	NUM
ejpam-3418	28	13	separate	separate	ADJ
ejpam-3418	28	14	possibilities	possibility	NOUN
ejpam-3418	28	15	.	.	PUNCT
ejpam-3418	29	1	we	we	PRON
ejpam-3418	29	2	could	could	AUX
ejpam-3418	29	3	not	not	PART
ejpam-3418	29	4	just	just	ADV
ejpam-3418	29	5	give	give	VERB
ejpam-3418	29	6	it	it	PRON
ejpam-3418	29	7	a	a	DET
ejpam-3418	29	8	value	value	NOUN
ejpam-3418	29	9	equal	equal	ADJ
ejpam-3418	29	10	to	to	ADP
ejpam-3418	29	11	the	the	DET
ejpam-3418	29	12	minimum	minimum	NOUN
ejpam-3418	29	13	of	of	ADP
ejpam-3418	29	14	the	the	DET
ejpam-3418	29	15	two	two	NUM
ejpam-3418	29	16	separate	separate	ADJ
ejpam-3418	29	17	possibilities	possibility	NOUN
ejpam-3418	29	18	.	.	PUNCT
ejpam-3418	30	1	expressing	express	VERB
ejpam-3418	30	2	information	information	NOUN
ejpam-3418	30	3	like	like	ADP
ejpam-3418	30	4	this	this	PRON
ejpam-3418	30	5	motivated	motivate	VERB
ejpam-3418	30	6	the	the	DET
ejpam-3418	30	7	introduction	introduction	NOUN
ejpam-3418	30	8	of	of	ADP
ejpam-3418	30	9	fuzzy	fuzzy	ADJ
ejpam-3418	30	10	on	on	ADP
ejpam-3418	30	11	ideal	ideal	ADJ
ejpam-3418	30	12	sets	set	NOUN
ejpam-3418	30	13	in	in	ADP
ejpam-3418	30	14	[	[	X
ejpam-3418	30	15	6	6	NUM
ejpam-3418	30	16	]	]	PUNCT
ejpam-3418	30	17	by	by	ADP
ejpam-3418	30	18	mernilo	mernilo	ADJ
ejpam-3418	30	19	and	and	CCONJ
ejpam-3418	30	20	caga	caga	PROPN
ejpam-3418	30	21	-	-	PUNCT
ejpam-3418	30	22	anan	anan	PROPN
ejpam-3418	30	23	.	.	PUNCT
ejpam-3418	31	1	an	an	DET
ejpam-3418	31	2	ideal	ideal	NOUN
ejpam-3418	31	3	here	here	ADV
ejpam-3418	31	4	is	be	AUX
ejpam-3418	31	5	a	a	DET
ejpam-3418	31	6	nonempty	nonempty	ADJ
ejpam-3418	31	7	collection	collection	NOUN
ejpam-3418	31	8	of	of	ADP
ejpam-3418	31	9	subsets	subset	NOUN
ejpam-3418	31	10	of	of	ADP
ejpam-3418	31	11	a	a	DET
ejpam-3418	31	12	set	set	NOUN
ejpam-3418	31	13	x	x	NOUN
ejpam-3418	31	14	,	,	PUNCT
ejpam-3418	31	15	denoted	denote	VERB
ejpam-3418	31	16	by	by	ADP
ejpam-3418	31	17	i(x	i(x	PROPN
ejpam-3418	31	18	)	)	PUNCT
ejpam-3418	31	19	,	,	PUNCT
ejpam-3418	31	20	that	that	PRON
ejpam-3418	31	21	satisfies	satisfy	VERB
ejpam-3418	31	22	:	:	PUNCT
ejpam-3418	31	23	i.	i.	NOUN
ejpam-3418	31	24	a	a	DET
ejpam-3418	31	25	∈	∈	PROPN
ejpam-3418	31	26	i(x	i(x	NOUN
ejpam-3418	31	27	)	)	PUNCT
ejpam-3418	31	28	and	and	CCONJ
ejpam-3418	31	29	b	b	X
ejpam-3418	31	30	⊆	⊆	NUM
ejpam-3418	31	31	a	a	PRON
ejpam-3418	31	32	implies	imply	VERB
ejpam-3418	31	33	b	b	PROPN
ejpam-3418	31	34	∈	∈	PROPN
ejpam-3418	31	35	i(x	i(x	PROPN
ejpam-3418	31	36	)	)	PUNCT
ejpam-3418	31	37	;	;	PUNCT
ejpam-3418	31	38	and	and	CCONJ
ejpam-3418	31	39	ii	ii	X
ejpam-3418	31	40	.	.	PUNCT
ejpam-3418	32	1	a	a	DET
ejpam-3418	32	2	∈	∈	PROPN
ejpam-3418	32	3	i(x	i(x	NOUN
ejpam-3418	32	4	)	)	PUNCT
ejpam-3418	32	5	and	and	CCONJ
ejpam-3418	32	6	b	b	PROPN
ejpam-3418	32	7	∈	∈	PROPN
ejpam-3418	32	8	i(x	i(x	NOUN
ejpam-3418	32	9	)	)	PUNCT
ejpam-3418	32	10	implies	imply	VERB
ejpam-3418	32	11	a	a	DET
ejpam-3418	32	12	∪b	∪b	PUNCT
ejpam-3418	32	13	∈	∈	PROPN
ejpam-3418	32	14	i(x	i(x	NOUN
ejpam-3418	32	15	)	)	PUNCT
ejpam-3418	32	16	.	.	PUNCT
ejpam-3418	33	1	the	the	DET
ejpam-3418	33	2	first	first	ADJ
ejpam-3418	33	3	property	property	NOUN
ejpam-3418	33	4	is	be	AUX
ejpam-3418	33	5	the	the	DET
ejpam-3418	33	6	reason	reason	NOUN
ejpam-3418	33	7	why	why	SCONJ
ejpam-3418	33	8	an	an	DET
ejpam-3418	33	9	ideal	ideal	NOUN
ejpam-3418	33	10	is	be	AUX
ejpam-3418	33	11	said	say	VERB
ejpam-3418	33	12	to	to	PART
ejpam-3418	33	13	be	be	AUX
ejpam-3418	33	14	a	a	DET
ejpam-3418	33	15	collection	collection	NOUN
ejpam-3418	33	16	of	of	ADP
ejpam-3418	33	17	sets	set	NOUN
ejpam-3418	33	18	that	that	PRON
ejpam-3418	33	19	are	be	AUX
ejpam-3418	33	20	considered	consider	VERB
ejpam-3418	33	21	small	small	ADJ
ejpam-3418	33	22	.	.	PUNCT
ejpam-3418	34	1	ideal	ideal	ADJ
ejpam-3418	34	2	spaces	space	NOUN
ejpam-3418	34	3	were	be	AUX
ejpam-3418	34	4	first	first	ADV
ejpam-3418	34	5	studied	study	VERB
ejpam-3418	34	6	by	by	ADP
ejpam-3418	34	7	kuratowski	kuratowski	ADJ
ejpam-3418	34	8	[	[	X
ejpam-3418	34	9	5	5	NUM
ejpam-3418	34	10	]	]	PUNCT
ejpam-3418	34	11	and	and	CCONJ
ejpam-3418	34	12	vaidyanathaswamy	vaidyanathaswamy	VERB
ejpam-3418	34	13	[	[	X
ejpam-3418	34	14	8	8	NUM
ejpam-3418	34	15	]	]	PUNCT
ejpam-3418	34	16	.	.	PUNCT
ejpam-3418	35	1	formally	formally	ADV
ejpam-3418	35	2	,	,	PUNCT
ejpam-3418	35	3	given	give	VERB
ejpam-3418	35	4	a	a	DET
ejpam-3418	35	5	nonempty	nonempty	ADJ
ejpam-3418	35	6	set	set	VERB
ejpam-3418	35	7	x	x	PUNCT
ejpam-3418	35	8	and	and	CCONJ
ejpam-3418	35	9	an	an	DET
ejpam-3418	35	10	ideal	ideal	ADJ
ejpam-3418	35	11	i(x	i(x	NOUN
ejpam-3418	35	12	)	)	PUNCT
ejpam-3418	35	13	on	on	ADP
ejpam-3418	35	14	x	x	SYM
ejpam-3418	35	15	,	,	PUNCT
ejpam-3418	35	16	a	a	DET
ejpam-3418	35	17	fuzzy	fuzzy	ADJ
ejpam-3418	35	18	on	on	ADP
ejpam-3418	35	19	ideal	ideal	ADJ
ejpam-3418	35	20	set	set	NOUN
ejpam-3418	35	21	is	be	AUX
ejpam-3418	35	22	a	a	DET
ejpam-3418	35	23	mapping	mapping	NOUN
ejpam-3418	35	24	µ	µ	NOUN
ejpam-3418	35	25	:	:	PUNCT
ejpam-3418	35	26	i(x)→	i(x)→	PROPN
ejpam-3418	35	27	[	[	X
ejpam-3418	35	28	0	0	NUM
ejpam-3418	35	29	,	,	PUNCT
ejpam-3418	35	30	1	1	NUM
ejpam-3418	35	31	]	]	PUNCT
ejpam-3418	35	32	such	such	ADJ
ejpam-3418	35	33	that	that	SCONJ
ejpam-3418	35	34	:	:	PUNCT
ejpam-3418	35	35	i.	i.	NOUN
ejpam-3418	35	36	µ(∅	µ(∅	PROPN
ejpam-3418	35	37	)	)	PUNCT
ejpam-3418	35	38	=	=	SYM
ejpam-3418	35	39	0	0	NUM
ejpam-3418	35	40	;	;	PUNCT
ejpam-3418	35	41	and	and	CCONJ
ejpam-3418	35	42	ii	ii	X
ejpam-3418	35	43	.	.	PUNCT
ejpam-3418	36	1	for	for	ADP
ejpam-3418	36	2	nonempty	nonempty	NOUN
ejpam-3418	36	3	sets	set	VERB
ejpam-3418	36	4	a	a	DET
ejpam-3418	36	5	,	,	PUNCT
ejpam-3418	36	6	b	b	PROPN
ejpam-3418	36	7	∈	∈	PROPN
ejpam-3418	36	8	i(x	i(x	PROPN
ejpam-3418	36	9	)	)	PUNCT
ejpam-3418	36	10	,	,	PUNCT
ejpam-3418	36	11	with	with	ADP
ejpam-3418	36	12	a	a	DET
ejpam-3418	36	13	⊆	⊆	NUM
ejpam-3418	36	14	b	b	NOUN
ejpam-3418	36	15	,	,	PUNCT
ejpam-3418	36	16	we	we	PRON
ejpam-3418	36	17	have	have	VERB
ejpam-3418	36	18	µ(b	µ(b	NOUN
ejpam-3418	36	19	)	)	PUNCT
ejpam-3418	36	20	≤	≤	NOUN
ejpam-3418	36	21	µ(a	µ(a	PROPN
ejpam-3418	36	22	)	)	PUNCT
ejpam-3418	36	23	.	.	PUNCT
ejpam-3418	37	1	the	the	DET
ejpam-3418	37	2	set	set	NOUN
ejpam-3418	37	3	of	of	ADP
ejpam-3418	37	4	all	all	PRON
ejpam-3418	37	5	such	such	ADJ
ejpam-3418	37	6	µ	µ	NOUN
ejpam-3418	37	7	is	be	AUX
ejpam-3418	37	8	denoted	denote	VERB
ejpam-3418	37	9	by	by	ADP
ejpam-3418	37	10	ii(x	ii(x	NOUN
ejpam-3418	37	11	)	)	PUNCT
ejpam-3418	37	12	.	.	PUNCT
ejpam-3418	38	1	observe	observe	VERB
ejpam-3418	38	2	that	that	SCONJ
ejpam-3418	38	3	the	the	DET
ejpam-3418	38	4	reverse	reverse	ADJ
ejpam-3418	38	5	inequality	inequality	NOUN
ejpam-3418	38	6	µ(b	µ(b	NOUN
ejpam-3418	38	7	)	)	PUNCT
ejpam-3418	38	8	≤	≤	NOUN
ejpam-3418	38	9	µ(a	µ(a	NOUN
ejpam-3418	38	10	)	)	PUNCT
ejpam-3418	38	11	encapsulates	encapsulate	VERB
ejpam-3418	38	12	the	the	DET
ejpam-3418	38	13	preceding	precede	VERB
ejpam-3418	38	14	idea	idea	NOUN
ejpam-3418	38	15	that	that	SCONJ
ejpam-3418	38	16	the	the	DET
ejpam-3418	38	17	possibility	possibility	NOUN
ejpam-3418	38	18	that	that	SCONJ
ejpam-3418	38	19	tomorrow	tomorrow	NOUN
ejpam-3418	38	20	there	there	PRON
ejpam-3418	38	21	will	will	AUX
ejpam-3418	38	22	be	be	AUX
ejpam-3418	38	23	rain	rain	NOUN
ejpam-3418	38	24	and	and	CCONJ
ejpam-3418	38	25	at	at	ADP
ejpam-3418	38	26	the	the	DET
ejpam-3418	38	27	same	same	ADJ
ejpam-3418	38	28	time	time	NOUN
ejpam-3418	38	29	it	it	PRON
ejpam-3418	38	30	will	will	AUX
ejpam-3418	38	31	be	be	AUX
ejpam-3418	38	32	sunny	sunny	ADJ
ejpam-3418	38	33	should	should	AUX
ejpam-3418	38	34	not	not	PART
ejpam-3418	38	35	just	just	ADV
ejpam-3418	38	36	be	be	AUX
ejpam-3418	38	37	equal	equal	ADJ
ejpam-3418	38	38	to	to	ADP
ejpam-3418	38	39	the	the	DET
ejpam-3418	38	40	minimum	minimum	NOUN
ejpam-3418	38	41	of	of	ADP
ejpam-3418	38	42	the	the	DET
ejpam-3418	38	43	separate	separate	ADJ
ejpam-3418	38	44	possibilies	possibilie	NOUN
ejpam-3418	38	45	as	as	SCONJ
ejpam-3418	38	46	it	it	PRON
ejpam-3418	38	47	could	could	AUX
ejpam-3418	38	48	be	be	AUX
ejpam-3418	38	49	far	far	ADV
ejpam-3418	38	50	less	less	ADJ
ejpam-3418	38	51	.	.	PUNCT
ejpam-3418	39	1	it	it	PRON
ejpam-3418	39	2	is	be	AUX
ejpam-3418	39	3	also	also	ADV
ejpam-3418	39	4	important	important	ADJ
ejpam-3418	39	5	to	to	PART
ejpam-3418	39	6	note	note	VERB
ejpam-3418	39	7	that	that	SCONJ
ejpam-3418	39	8	the	the	DET
ejpam-3418	39	9	preceding	precede	VERB
ejpam-3418	39	10	definition	definition	NOUN
ejpam-3418	39	11	does	do	AUX
ejpam-3418	39	12	not	not	PART
ejpam-3418	39	13	define	define	VERB
ejpam-3418	39	14	a	a	DET
ejpam-3418	39	15	measure	measure	NOUN
ejpam-3418	39	16	.	.	PUNCT
ejpam-3418	40	1	for	for	ADP
ejpam-3418	40	2	a	a	DET
ejpam-3418	40	3	⊆	⊆	NUM
ejpam-3418	40	4	b	b	NOUN
ejpam-3418	40	5	,	,	PUNCT
ejpam-3418	40	6	a	a	DET
ejpam-3418	40	7	measure	measure	NOUN
ejpam-3418	40	8	m	m	NOUN
ejpam-3418	40	9	should	should	AUX
ejpam-3418	40	10	have	have	VERB
ejpam-3418	40	11	m(a	m(a	NOUN
ejpam-3418	40	12	)	)	PUNCT
ejpam-3418	40	13	≤	≤	NUM
ejpam-3418	40	14	m(b	m(b	NOUN
ejpam-3418	40	15	)	)	PUNCT
ejpam-3418	40	16	,	,	PUNCT
ejpam-3418	40	17	not	not	PART
ejpam-3418	40	18	the	the	DET
ejpam-3418	40	19	reverse	reverse	ADJ
ejpam-3418	40	20	inequality	inequality	NOUN
ejpam-3418	40	21	,	,	PUNCT
ejpam-3418	40	22	as	as	ADP
ejpam-3418	40	23	in	in	ADP
ejpam-3418	40	24	our	our	PRON
ejpam-3418	40	25	definition	definition	NOUN
ejpam-3418	40	26	.	.	PUNCT
ejpam-3418	41	1	moreover	moreover	ADV
ejpam-3418	41	2	,	,	PUNCT
ejpam-3418	41	3	any	any	DET
ejpam-3418	41	4	fuzzy	fuzzy	ADJ
ejpam-3418	41	5	set	set	VERB
ejpam-3418	41	6	α	α	PRON
ejpam-3418	41	7	defined	define	VERB
ejpam-3418	41	8	on	on	ADP
ejpam-3418	41	9	a	a	DET
ejpam-3418	41	10	set	set	NOUN
ejpam-3418	41	11	x	x	PUNCT
ejpam-3418	41	12	can	can	AUX
ejpam-3418	41	13	be	be	AUX
ejpam-3418	41	14	embedded	embed	VERB
ejpam-3418	41	15	as	as	ADP
ejpam-3418	41	16	an	an	DET
ejpam-3418	41	17	element	element	NOUN
ejpam-3418	41	18	of	of	ADP
ejpam-3418	41	19	ip(x	ip(x	PRON
ejpam-3418	41	20	)	)	PUNCT
ejpam-3418	41	21	by	by	ADP
ejpam-3418	41	22	associating	associate	VERB
ejpam-3418	41	23	it	it	PRON
ejpam-3418	41	24	with	with	ADP
ejpam-3418	41	25	the	the	DET
ejpam-3418	41	26	fuzzy	fuzzy	ADJ
ejpam-3418	41	27	on	on	ADP
ejpam-3418	41	28	ideal	ideal	ADJ
ejpam-3418	41	29	set	set	VERB
ejpam-3418	41	30	µα	µα	ADP
ejpam-3418	41	31	defined	define	VERB
ejpam-3418	41	32	by	by	ADP
ejpam-3418	41	33	µα(a	µα(a	NOUN
ejpam-3418	41	34	)	)	PUNCT
ejpam-3418	42	1	=	=	SYM
ejpam-3418	42	2	{	{	PUNCT
ejpam-3418	42	3	α(x	α(x	PROPN
ejpam-3418	42	4	)	)	PUNCT
ejpam-3418	42	5	,	,	PUNCT
ejpam-3418	42	6	if	if	SCONJ
ejpam-3418	42	7	a	a	PRON
ejpam-3418	42	8	=	=	X
ejpam-3418	42	9	{	{	PUNCT
ejpam-3418	42	10	x	x	NOUN
ejpam-3418	42	11	}	}	PUNCT
ejpam-3418	42	12	,	,	PUNCT
ejpam-3418	42	13	x	x	PUNCT
ejpam-3418	42	14	∈	∈	NOUN
ejpam-3418	42	15	x	x	X
ejpam-3418	42	16	0	0	NUM
ejpam-3418	42	17	,	,	PUNCT
ejpam-3418	42	18	otherwise	otherwise	ADV
ejpam-3418	42	19	,	,	PUNCT
ejpam-3418	42	20	where	where	SCONJ
ejpam-3418	42	21	p(x	p(x	NOUN
ejpam-3418	42	22	)	)	PUNCT
ejpam-3418	42	23	is	be	AUX
ejpam-3418	42	24	the	the	DET
ejpam-3418	42	25	powerset	powerset	NOUN
ejpam-3418	42	26	of	of	ADP
ejpam-3418	42	27	x−the	x−the	DET
ejpam-3418	42	28	largest	large	ADJ
ejpam-3418	42	29	ideal	ideal	NOUN
ejpam-3418	42	30	of	of	ADP
ejpam-3418	42	31	x.	x.	NOUN
ejpam-3418	42	32	thus	thus	ADV
ejpam-3418	42	33	,	,	PUNCT
ejpam-3418	42	34	fuzzy	fuzzy	ADJ
ejpam-3418	42	35	on	on	ADP
ejpam-3418	42	36	ideal	ideal	ADJ
ejpam-3418	42	37	sets	set	NOUN
ejpam-3418	42	38	generalize	generalize	VERB
ejpam-3418	42	39	fuzzy	fuzzy	ADJ
ejpam-3418	42	40	sets	set	NOUN
ejpam-3418	42	41	.	.	PUNCT
ejpam-3418	43	1	with	with	ADP
ejpam-3418	43	2	regard	regard	NOUN
ejpam-3418	43	3	to	to	ADP
ejpam-3418	43	4	uncertainty	uncertainty	NOUN
ejpam-3418	43	5	,	,	PUNCT
ejpam-3418	43	6	there	there	PRON
ejpam-3418	43	7	are	be	VERB
ejpam-3418	43	8	cases	case	NOUN
ejpam-3418	43	9	that	that	SCONJ
ejpam-3418	43	10	even	even	ADV
ejpam-3418	43	11	the	the	DET
ejpam-3418	43	12	assigning	assigning	NOUN
ejpam-3418	43	13	of	of	ADP
ejpam-3418	43	14	degrees	degree	NOUN
ejpam-3418	43	15	of	of	ADP
ejpam-3418	43	16	membership	membership	NOUN
ejpam-3418	43	17	on	on	ADP
ejpam-3418	43	18	a	a	DET
ejpam-3418	43	19	fuzzy	fuzzy	ADJ
ejpam-3418	43	20	set	set	NOUN
ejpam-3418	43	21	or	or	CCONJ
ejpam-3418	43	22	fuzzy	fuzzy	ADJ
ejpam-3418	43	23	on	on	ADP
ejpam-3418	43	24	ideal	ideal	ADJ
ejpam-3418	43	25	set	set	NOUN
ejpam-3418	43	26	has	have	VERB
ejpam-3418	43	27	its	its	PRON
ejpam-3418	43	28	own	own	ADJ
ejpam-3418	43	29	uncertainties	uncertainty	NOUN
ejpam-3418	43	30	.	.	PUNCT
ejpam-3418	44	1	in	in	ADP
ejpam-3418	44	2	these	these	DET
ejpam-3418	44	3	cases	case	NOUN
ejpam-3418	44	4	,	,	PUNCT
ejpam-3418	44	5	it	it	PRON
ejpam-3418	44	6	is	be	AUX
ejpam-3418	44	7	better	well	ADJ
ejpam-3418	44	8	to	to	PART
ejpam-3418	44	9	give	give	VERB
ejpam-3418	44	10	the	the	DET
ejpam-3418	44	11	degree	degree	NOUN
ejpam-3418	44	12	of	of	ADP
ejpam-3418	44	13	membership	membership	NOUN
ejpam-3418	44	14	as	as	ADP
ejpam-3418	44	15	an	an	DET
ejpam-3418	44	16	interval	interval	NOUN
ejpam-3418	44	17	rather	rather	ADV
ejpam-3418	44	18	than	than	ADP
ejpam-3418	44	19	as	as	ADP
ejpam-3418	44	20	a	a	DET
ejpam-3418	44	21	single	single	ADJ
ejpam-3418	44	22	number	number	NOUN
ejpam-3418	44	23	.	.	PUNCT
ejpam-3418	45	1	for	for	ADP
ejpam-3418	45	2	instance	instance	NOUN
ejpam-3418	45	3	,	,	PUNCT
ejpam-3418	45	4	when	when	SCONJ
ejpam-3418	45	5	one	one	PRON
ejpam-3418	45	6	is	be	AUX
ejpam-3418	45	7	estimating	estimate	VERB
ejpam-3418	45	8	the	the	DET
ejpam-3418	45	9	age	age	NOUN
ejpam-3418	45	10	of	of	ADP
ejpam-3418	45	11	a	a	DET
ejpam-3418	45	12	person	person	NOUN
ejpam-3418	45	13	,	,	PUNCT
ejpam-3418	45	14	one	one	PRON
ejpam-3418	45	15	has	have	VERB
ejpam-3418	45	16	a	a	DET
ejpam-3418	45	17	better	well	ADJ
ejpam-3418	45	18	chance	chance	NOUN
ejpam-3418	45	19	of	of	ADP
ejpam-3418	45	20	capturing	capture	VERB
ejpam-3418	45	21	the	the	DET
ejpam-3418	45	22	real	real	ADJ
ejpam-3418	45	23	age	age	NOUN
ejpam-3418	45	24	by	by	ADP
ejpam-3418	45	25	giving	give	VERB
ejpam-3418	45	26	a	a	DET
ejpam-3418	45	27	possible	possible	ADJ
ejpam-3418	45	28	range	range	NOUN
ejpam-3418	45	29	of	of	ADP
ejpam-3418	45	30	the	the	DET
ejpam-3418	45	31	age	age	NOUN
ejpam-3418	45	32	rather	rather	ADV
ejpam-3418	45	33	than	than	ADP
ejpam-3418	45	34	estimating	estimate	VERB
ejpam-3418	45	35	it	it	PRON
ejpam-3418	45	36	with	with	ADP
ejpam-3418	45	37	a	a	DET
ejpam-3418	45	38	single	single	ADJ
ejpam-3418	45	39	number	number	NOUN
ejpam-3418	45	40	.	.	PUNCT
ejpam-3418	46	1	this	this	DET
ejpam-3418	46	2	way	way	NOUN
ejpam-3418	46	3	,	,	PUNCT
ejpam-3418	46	4	one	one	PRON
ejpam-3418	46	5	captures	capture	VERB
ejpam-3418	46	6	the	the	DET
ejpam-3418	46	7	imprecision	imprecision	NOUN
ejpam-3418	46	8	better	well	ADV
ejpam-3418	46	9	.	.	PUNCT
ejpam-3418	47	1	this	this	DET
ejpam-3418	47	2	lead	lead	NOUN
ejpam-3418	47	3	to	to	ADP
ejpam-3418	47	4	the	the	DET
ejpam-3418	47	5	introduction	introduction	NOUN
ejpam-3418	47	6	of	of	ADP
ejpam-3418	47	7	mj	mj	PROPN
ejpam-3418	47	8	togonon	togonon	PROPN
ejpam-3418	47	9	,	,	PUNCT
ejpam-3418	47	10	r	r	NOUN
ejpam-3418	47	11	caga	caga	NOUN
ejpam-3418	47	12	-	-	PUNCT
ejpam-3418	47	13	anan	anan	PROPN
ejpam-3418	47	14	/	/	SYM
ejpam-3418	47	15	eur	eur	PROPN
ejpam-3418	47	16	.	.	PUNCT
ejpam-3418	48	1	j.	j.	PROPN
ejpam-3418	48	2	pure	pure	PROPN
ejpam-3418	48	3	appl	appl	PROPN
ejpam-3418	48	4	.	.	PROPN
ejpam-3418	48	5	math	math	PROPN
ejpam-3418	48	6	,	,	PUNCT
ejpam-3418	48	7	12	12	NUM
ejpam-3418	48	8	(	(	PUNCT
ejpam-3418	48	9	2	2	NUM
ejpam-3418	48	10	)	)	PUNCT
ejpam-3418	48	11	(	(	PUNCT
ejpam-3418	48	12	2019	2019	NUM
ejpam-3418	48	13	)	)	PUNCT
ejpam-3418	48	14	,	,	PUNCT
ejpam-3418	48	15	553	553	NUM
ejpam-3418	48	16	-	-	SYM
ejpam-3418	48	17	570	570	NUM
ejpam-3418	48	18	555	555	NUM
ejpam-3418	48	19	interval	interval	NOUN
ejpam-3418	48	20	-	-	PUNCT
ejpam-3418	48	21	valued	value	VERB
ejpam-3418	48	22	fuzzy	fuzzy	ADJ
ejpam-3418	48	23	sets	set	NOUN
ejpam-3418	48	24	in	in	ADP
ejpam-3418	48	25	1975	1975	NUM
ejpam-3418	48	26	by	by	ADP
ejpam-3418	48	27	zadeh	zadeh	PROPN
ejpam-3418	49	1	[	[	X
ejpam-3418	49	2	10	10	NUM
ejpam-3418	49	3	]	]	PUNCT
ejpam-3418	49	4	.	.	PUNCT
ejpam-3418	50	1	in	in	ADP
ejpam-3418	50	2	that	that	DET
ejpam-3418	50	3	same	same	ADJ
ejpam-3418	50	4	year	year	NOUN
ejpam-3418	50	5	,	,	PUNCT
ejpam-3418	50	6	it	it	PRON
ejpam-3418	50	7	was	be	AUX
ejpam-3418	50	8	also	also	ADV
ejpam-3418	50	9	considered	consider	VERB
ejpam-3418	50	10	by	by	ADP
ejpam-3418	50	11	grattan	grattan	PROPN
ejpam-3418	50	12	-	-	PUNCT
ejpam-3418	50	13	guiness	guiness	PROPN
ejpam-3418	51	1	[	[	X
ejpam-3418	51	2	2	2	NUM
ejpam-3418	51	3	]	]	PUNCT
ejpam-3418	51	4	,	,	PUNCT
ejpam-3418	51	5	jahn	jahn	NOUN
ejpam-3418	52	1	[	[	X
ejpam-3418	52	2	3	3	X
ejpam-3418	52	3	]	]	PUNCT
ejpam-3418	52	4	and	and	CCONJ
ejpam-3418	52	5	sambuc	sambuc	ADV
ejpam-3418	53	1	[	[	X
ejpam-3418	53	2	7	7	NUM
ejpam-3418	53	3	]	]	PUNCT
ejpam-3418	53	4	.	.	PUNCT
ejpam-3418	54	1	in	in	ADP
ejpam-3418	54	2	this	this	DET
ejpam-3418	54	3	study	study	NOUN
ejpam-3418	54	4	,	,	PUNCT
ejpam-3418	54	5	we	we	PRON
ejpam-3418	54	6	introduce	introduce	VERB
ejpam-3418	54	7	and	and	CCONJ
ejpam-3418	54	8	develop	develop	VERB
ejpam-3418	54	9	the	the	DET
ejpam-3418	54	10	interval	interval	NOUN
ejpam-3418	54	11	-	-	PUNCT
ejpam-3418	54	12	valued	value	VERB
ejpam-3418	54	13	fuzzy	fuzzy	NOUN
ejpam-3418	54	14	on	on	ADP
ejpam-3418	54	15	ideal	ideal	ADJ
ejpam-3418	54	16	sets	set	NOUN
ejpam-3418	54	17	.	.	PUNCT
ejpam-3418	55	1	this	this	DET
ejpam-3418	55	2	concept	concept	NOUN
ejpam-3418	55	3	generalizes	generalize	VERB
ejpam-3418	55	4	the	the	DET
ejpam-3418	55	5	above	above	ADJ
ejpam-3418	55	6	discussed	discuss	VERB
ejpam-3418	55	7	fuzzy	fuzzy	ADJ
ejpam-3418	55	8	sets	set	NOUN
ejpam-3418	55	9	,	,	PUNCT
ejpam-3418	55	10	fuzzy	fuzzy	ADJ
ejpam-3418	55	11	on	on	ADP
ejpam-3418	55	12	ideal	ideal	ADJ
ejpam-3418	55	13	sets	set	NOUN
ejpam-3418	55	14	,	,	PUNCT
ejpam-3418	55	15	and	and	CCONJ
ejpam-3418	55	16	interval	interval	NOUN
ejpam-3418	55	17	-	-	PUNCT
ejpam-3418	55	18	valued	value	VERB
ejpam-3418	55	19	fuzzy	fuzzy	ADJ
ejpam-3418	55	20	sets	set	NOUN
ejpam-3418	55	21	.	.	PUNCT
ejpam-3418	56	1	we	we	PRON
ejpam-3418	56	2	formally	formally	ADV
ejpam-3418	56	3	define	define	VERB
ejpam-3418	56	4	it	it	PRON
ejpam-3418	56	5	in	in	ADP
ejpam-3418	56	6	the	the	DET
ejpam-3418	56	7	next	next	ADJ
ejpam-3418	56	8	section	section	NOUN
ejpam-3418	56	9	.	.	PUNCT
ejpam-3418	57	1	2	2	X
ejpam-3418	57	2	.	.	NUM
ejpam-3418	57	3	basic	basic	ADJ
ejpam-3418	57	4	concepts	concept	NOUN
ejpam-3418	57	5	and	and	CCONJ
ejpam-3418	57	6	properties	property	NOUN
ejpam-3418	57	7	let	let	VERB
ejpam-3418	57	8	us	we	PRON
ejpam-3418	57	9	first	first	ADV
ejpam-3418	57	10	introduce	introduce	VERB
ejpam-3418	57	11	some	some	DET
ejpam-3418	57	12	useful	useful	ADJ
ejpam-3418	57	13	notations	notation	NOUN
ejpam-3418	57	14	.	.	PUNCT
ejpam-3418	58	1	we	we	PRON
ejpam-3418	58	2	denote	denote	VERB
ejpam-3418	58	3	by	by	ADP
ejpam-3418	58	4	i	i	PRON
ejpam-3418	58	5	the	the	DET
ejpam-3418	58	6	set	set	NOUN
ejpam-3418	58	7	of	of	ADP
ejpam-3418	58	8	all	all	DET
ejpam-3418	58	9	closed	closed	ADJ
ejpam-3418	58	10	subintervals	subinterval	NOUN
ejpam-3418	58	11	of	of	ADP
ejpam-3418	58	12	[	[	X
ejpam-3418	58	13	0,1	0,1	NUM
ejpam-3418	58	14	]	]	PUNCT
ejpam-3418	58	15	.	.	PUNCT
ejpam-3418	59	1	for	for	ADP
ejpam-3418	59	2	α	α	PRON
ejpam-3418	59	3	∈	∈	PROPN
ejpam-3418	59	4	i	i	PRON
ejpam-3418	59	5	,	,	PUNCT
ejpam-3418	59	6	let	let	VERB
ejpam-3418	59	7	α−	α−	PART
ejpam-3418	59	8	be	be	AUX
ejpam-3418	59	9	the	the	DET
ejpam-3418	59	10	left	left	ADJ
ejpam-3418	59	11	endpoint	endpoint	NOUN
ejpam-3418	59	12	of	of	ADP
ejpam-3418	59	13	α	α	PROPN
ejpam-3418	59	14	and	and	CCONJ
ejpam-3418	59	15	α+	α+	NOUN
ejpam-3418	59	16	be	be	AUX
ejpam-3418	59	17	the	the	DET
ejpam-3418	59	18	right	right	ADJ
ejpam-3418	59	19	endpoint	endpoint	NOUN
ejpam-3418	59	20	of	of	ADP
ejpam-3418	59	21	α	α	PRON
ejpam-3418	59	22	,	,	PUNCT
ejpam-3418	59	23	so	so	SCONJ
ejpam-3418	59	24	that	that	SCONJ
ejpam-3418	59	25	α	α	PRON
ejpam-3418	60	1	=	=	X
ejpam-3418	61	1	[	[	X
ejpam-3418	61	2	α−	α−	ADP
ejpam-3418	61	3	,	,	PUNCT
ejpam-3418	61	4	α+	α+	PRON
ejpam-3418	61	5	]	]	PUNCT
ejpam-3418	61	6	.	.	PUNCT
ejpam-3418	62	1	let	let	VERB
ejpam-3418	62	2	α1	α1	PROPN
ejpam-3418	62	3	=	=	PUNCT
ejpam-3418	63	1	[	[	X
ejpam-3418	63	2	α−1	α−1	PROPN
ejpam-3418	63	3	,	,	PUNCT
ejpam-3418	63	4	α	α	PROPN
ejpam-3418	63	5	+	+	ADJ
ejpam-3418	63	6	1	1	NUM
ejpam-3418	63	7	]	]	PUNCT
ejpam-3418	63	8	and	and	CCONJ
ejpam-3418	63	9	α2	α2	ADJ
ejpam-3418	63	10	=	=	PUNCT
ejpam-3418	64	1	[	[	X
ejpam-3418	64	2	α−2	α−2	PROPN
ejpam-3418	64	3	,	,	PUNCT
ejpam-3418	64	4	α	α	PROPN
ejpam-3418	64	5	+	+	NOUN
ejpam-3418	64	6	2	2	NUM
ejpam-3418	64	7	]	]	PUNCT
ejpam-3418	64	8	be	be	AUX
ejpam-3418	64	9	closed	close	VERB
ejpam-3418	64	10	subintervals	subinterval	NOUN
ejpam-3418	64	11	of	of	ADP
ejpam-3418	64	12	i	i	PRON
ejpam-3418	64	13	.	.	PUNCT
ejpam-3418	65	1	we	we	PRON
ejpam-3418	65	2	use	use	VERB
ejpam-3418	65	3	the	the	DET
ejpam-3418	65	4	inequality	inequality	NOUN
ejpam-3418	65	5	notation	notation	NOUN
ejpam-3418	65	6	“	"	PUNCT
ejpam-3418	65	7	≤i	≤i	PROPN
ejpam-3418	65	8	”	"	PUNCT
ejpam-3418	65	9	,	,	PUNCT
ejpam-3418	65	10	say	say	VERB
ejpam-3418	65	11	α1	α1	PROPN
ejpam-3418	65	12	≤i	≤i	PROPN
ejpam-3418	65	13	α2	α2	PROPN
ejpam-3418	65	14	,	,	PUNCT
ejpam-3418	65	15	to	to	PART
ejpam-3418	65	16	mean	mean	VERB
ejpam-3418	65	17	α−1	α−1	PROPN
ejpam-3418	65	18	≤	≤	NOUN
ejpam-3418	66	1	α	α	DET
ejpam-3418	66	2	−	−	PROPN
ejpam-3418	66	3	2	2	NUM
ejpam-3418	66	4	and	and	CCONJ
ejpam-3418	66	5	α+	α+	PUNCT
ejpam-3418	66	6	1	1	NUM
ejpam-3418	66	7	≤	≤	NUM
ejpam-3418	66	8	α	α	NOUN
ejpam-3418	66	9	+	+	ADP
ejpam-3418	66	10	2	2	NUM
ejpam-3418	66	11	.	.	PUNCT
ejpam-3418	67	1	let	let	VERB
ejpam-3418	67	2	a	a	DET
ejpam-3418	67	3	be	be	AUX
ejpam-3418	67	4	an	an	DET
ejpam-3418	67	5	index	index	NOUN
ejpam-3418	67	6	set	set	NOUN
ejpam-3418	67	7	and	and	CCONJ
ejpam-3418	67	8	αi	αi	PRON
ejpam-3418	67	9	∈	∈	PROPN
ejpam-3418	68	1	i	i	PRON
ejpam-3418	68	2	,	,	PUNCT
ejpam-3418	68	3	for	for	ADP
ejpam-3418	68	4	each	each	PRON
ejpam-3418	68	5	i	i	PRON
ejpam-3418	68	6	∈	∈	NOUN
ejpam-3418	68	7	a.	a.	NOUN
ejpam-3418	68	8	we	we	PRON
ejpam-3418	68	9	define	define	VERB
ejpam-3418	68	10	the	the	DET
ejpam-3418	68	11	supremum	supremum	NOUN
ejpam-3418	68	12	of	of	ADP
ejpam-3418	68	13	αi	αi	NOUN
ejpam-3418	68	14	by	by	ADP
ejpam-3418	68	15	sup	sup	PROPN
ejpam-3418	68	16	i∈a	i∈a	ADJ
ejpam-3418	68	17	αi	αi	VERB
ejpam-3418	68	18	=	=	PUNCT
ejpam-3418	69	1	[	[	X
ejpam-3418	69	2	sup	sup	NOUN
ejpam-3418	69	3	i∈a	i∈a	ADJ
ejpam-3418	69	4	α−i	α−i	NOUN
ejpam-3418	69	5	,	,	PUNCT
ejpam-3418	69	6	sup	sup	NOUN
ejpam-3418	69	7	i∈a	i∈a	ADJ
ejpam-3418	69	8	α+	α+	PUNCT
ejpam-3418	69	9	i	i	X
ejpam-3418	69	10	]	]	PUNCT
ejpam-3418	69	11	and	and	CCONJ
ejpam-3418	69	12	the	the	DET
ejpam-3418	69	13	infimum	infimum	NOUN
ejpam-3418	69	14	of	of	ADP
ejpam-3418	69	15	αi	αi	NOUN
ejpam-3418	69	16	by	by	ADP
ejpam-3418	69	17	inf	inf	PROPN
ejpam-3418	69	18	i∈a	i∈a	PROPN
ejpam-3418	69	19	αi	αi	PART
ejpam-3418	69	20	=	=	PUNCT
ejpam-3418	70	1	[	[	X
ejpam-3418	70	2	inf	inf	NOUN
ejpam-3418	70	3	i∈a	i∈a	ADJ
ejpam-3418	70	4	α−i	α−i	PROPN
ejpam-3418	70	5	,	,	PUNCT
ejpam-3418	70	6	inf	inf	PROPN
ejpam-3418	70	7	i∈a	i∈a	ADJ
ejpam-3418	70	8	α+	α+	PUNCT
ejpam-3418	70	9	i	i	X
ejpam-3418	70	10	]	]	PUNCT
ejpam-3418	70	11	.	.	PUNCT
ejpam-3418	71	1	we	we	PRON
ejpam-3418	71	2	define	define	VERB
ejpam-3418	71	3	formally	formally	ADV
ejpam-3418	71	4	an	an	DET
ejpam-3418	71	5	interval	interval	NOUN
ejpam-3418	71	6	-	-	PUNCT
ejpam-3418	71	7	valued	value	VERB
ejpam-3418	71	8	fuzzy	fuzzy	NOUN
ejpam-3418	71	9	on	on	ADP
ejpam-3418	71	10	ideal	ideal	ADJ
ejpam-3418	71	11	set	set	NOUN
ejpam-3418	71	12	as	as	SCONJ
ejpam-3418	71	13	follows	follow	VERB
ejpam-3418	71	14	.	.	PUNCT
ejpam-3418	72	1	definition	definition	NOUN
ejpam-3418	72	2	1	1	NUM
ejpam-3418	72	3	.	.	PUNCT
ejpam-3418	73	1	let	let	VERB
ejpam-3418	73	2	x	x	PRON
ejpam-3418	73	3	be	be	AUX
ejpam-3418	73	4	a	a	DET
ejpam-3418	73	5	nonempty	nonempty	ADV
ejpam-3418	73	6	set	set	VERB
ejpam-3418	73	7	and	and	CCONJ
ejpam-3418	73	8	i(x	i(x	NOUN
ejpam-3418	73	9	)	)	PUNCT
ejpam-3418	73	10	be	be	AUX
ejpam-3418	73	11	an	an	DET
ejpam-3418	73	12	ideal	ideal	NOUN
ejpam-3418	73	13	on	on	ADP
ejpam-3418	73	14	x.	x.	NOUN
ejpam-3418	73	15	an	an	DET
ejpam-3418	73	16	interval	interval	NOUN
ejpam-3418	73	17	-	-	PUNCT
ejpam-3418	73	18	valued	value	VERB
ejpam-3418	73	19	fuzzy	fuzzy	NOUN
ejpam-3418	73	20	on	on	ADP
ejpam-3418	73	21	ideal	ideal	ADJ
ejpam-3418	73	22	set	set	NOUN
ejpam-3418	73	23	(	(	PUNCT
ejpam-3418	73	24	briefly	briefly	NOUN
ejpam-3418	73	25	an	an	DET
ejpam-3418	73	26	ivfi	ivfi	NOUN
ejpam-3418	73	27	set	set	NOUN
ejpam-3418	73	28	)	)	PUNCT
ejpam-3418	73	29	is	be	AUX
ejpam-3418	73	30	a	a	DET
ejpam-3418	73	31	mapping	mapping	NOUN
ejpam-3418	73	32	ι̂	ι̂	NOUN
ejpam-3418	73	33	:	:	PUNCT
ejpam-3418	73	34	i(x	i(x	NOUN
ejpam-3418	73	35	)	)	PUNCT
ejpam-3418	73	36	→	→	PUNCT
ejpam-3418	73	37	i	i	PRON
ejpam-3418	73	38	that	that	PRON
ejpam-3418	73	39	satisfies	satisfy	VERB
ejpam-3418	73	40	the	the	DET
ejpam-3418	73	41	following	follow	VERB
ejpam-3418	73	42	:	:	PUNCT
ejpam-3418	73	43	i.	i.	PROPN
ejpam-3418	73	44	ι̂(∅	ι̂(∅	PROPN
ejpam-3418	73	45	)	)	PUNCT
ejpam-3418	74	1	=	=	PUNCT
ejpam-3418	75	1	[	[	X
ejpam-3418	75	2	0	0	NUM
ejpam-3418	75	3	,	,	PUNCT
ejpam-3418	75	4	0	0	NUM
ejpam-3418	75	5	]	]	PUNCT
ejpam-3418	75	6	;	;	PUNCT
ejpam-3418	75	7	and	and	CCONJ
ejpam-3418	75	8	ii	ii	X
ejpam-3418	75	9	.	.	PUNCT
ejpam-3418	76	1	for	for	ADP
ejpam-3418	76	2	nonempty	nonempty	NOUN
ejpam-3418	76	3	sets	set	VERB
ejpam-3418	76	4	a	a	DET
ejpam-3418	76	5	,	,	PUNCT
ejpam-3418	76	6	b	b	PROPN
ejpam-3418	76	7	∈	∈	PROPN
ejpam-3418	76	8	i(x	i(x	NOUN
ejpam-3418	76	9	)	)	PUNCT
ejpam-3418	76	10	with	with	ADP
ejpam-3418	76	11	a	a	DET
ejpam-3418	76	12	⊆	⊆	NUM
ejpam-3418	76	13	b	b	NOUN
ejpam-3418	76	14	,	,	PUNCT
ejpam-3418	76	15	ι̂(b	ι̂(b	PUNCT
ejpam-3418	76	16	)	)	PUNCT
ejpam-3418	76	17	≤i	≤i	PROPN
ejpam-3418	76	18	ι̂(a	ι̂(a	PUNCT
ejpam-3418	76	19	)	)	PUNCT
ejpam-3418	76	20	.	.	PUNCT
ejpam-3418	77	1	we	we	PRON
ejpam-3418	77	2	denote	denote	VERB
ejpam-3418	77	3	the	the	DET
ejpam-3418	77	4	set	set	NOUN
ejpam-3418	77	5	of	of	ADP
ejpam-3418	77	6	all	all	DET
ejpam-3418	77	7	such	such	ADJ
ejpam-3418	77	8	ι̂	ι̂	NUM
ejpam-3418	77	9	by	by	ADP
ejpam-3418	77	10	i	i	PRON
ejpam-3418	77	11	i(x	i(x	PROPN
ejpam-3418	77	12	)	)	PUNCT
ejpam-3418	77	13	.	.	PUNCT
ejpam-3418	78	1	remark	remark	PROPN
ejpam-3418	78	2	1	1	NUM
ejpam-3418	78	3	.	.	PUNCT
ejpam-3418	79	1	in	in	ADP
ejpam-3418	79	2	a	a	DET
ejpam-3418	79	3	similar	similar	ADJ
ejpam-3418	79	4	way	way	NOUN
ejpam-3418	79	5	that	that	PRON
ejpam-3418	79	6	fuzzy	fuzzy	ADJ
ejpam-3418	79	7	on	on	ADP
ejpam-3418	79	8	ideal	ideal	ADJ
ejpam-3418	79	9	sets	set	NOUN
ejpam-3418	79	10	generalize	generalize	VERB
ejpam-3418	79	11	fuzzy	fuzzy	ADJ
ejpam-3418	79	12	sets	set	NOUN
ejpam-3418	79	13	,	,	PUNCT
ejpam-3418	79	14	as	as	SCONJ
ejpam-3418	79	15	discussed	discuss	VERB
ejpam-3418	79	16	above	above	ADV
ejpam-3418	79	17	,	,	PUNCT
ejpam-3418	79	18	ivfi	ivfi	NOUN
ejpam-3418	79	19	sets	set	VERB
ejpam-3418	79	20	generalize	generalize	VERB
ejpam-3418	79	21	interval	interval	NOUN
ejpam-3418	79	22	-	-	PUNCT
ejpam-3418	79	23	valued	value	VERB
ejpam-3418	79	24	fuzzy	fuzzy	ADJ
ejpam-3418	79	25	sets	set	NOUN
ejpam-3418	79	26	.	.	PUNCT
ejpam-3418	80	1	example	example	NOUN
ejpam-3418	81	1	1	1	NUM
ejpam-3418	81	2	.	.	PUNCT
ejpam-3418	82	1	let	let	VERB
ejpam-3418	82	2	x	x	PRON
ejpam-3418	82	3	be	be	AUX
ejpam-3418	82	4	a	a	DET
ejpam-3418	82	5	nonempty	nonempty	ADV
ejpam-3418	82	6	set	set	VERB
ejpam-3418	82	7	and	and	CCONJ
ejpam-3418	82	8	π	π	NOUN
ejpam-3418	82	9	:	:	PUNCT
ejpam-3418	82	10	x	x	X
ejpam-3418	82	11	→	→	SYM
ejpam-3418	82	12	i	i	PRON
ejpam-3418	82	13	be	be	VERB
ejpam-3418	82	14	an	an	DET
ejpam-3418	82	15	interval	interval	NOUN
ejpam-3418	82	16	-	-	PUNCT
ejpam-3418	82	17	valued	value	VERB
ejpam-3418	82	18	fuzzy	fuzzy	ADJ
ejpam-3418	82	19	set	set	NOUN
ejpam-3418	82	20	.	.	PUNCT
ejpam-3418	83	1	we	we	PRON
ejpam-3418	83	2	can	can	AUX
ejpam-3418	83	3	define	define	VERB
ejpam-3418	83	4	an	an	DET
ejpam-3418	83	5	ivfi	ivfi	NOUN
ejpam-3418	83	6	set	set	VERB
ejpam-3418	83	7	π̂	π̂	NUM
ejpam-3418	83	8	:	:	PUNCT
ejpam-3418	83	9	p(x)→	p(x)→	NOUN
ejpam-3418	83	10	i	i	PRON
ejpam-3418	83	11	by	by	ADP
ejpam-3418	83	12	π̂(a	π̂(a	NOUN
ejpam-3418	83	13	)	)	PUNCT
ejpam-3418	83	14	=	=	PRON
ejpam-3418	83	15	{	{	PUNCT
ejpam-3418	84	1	[	[	X
ejpam-3418	84	2	0	0	NUM
ejpam-3418	84	3	,	,	PUNCT
ejpam-3418	84	4	0	0	NUM
ejpam-3418	84	5	]	]	PUNCT
ejpam-3418	84	6	,	,	PUNCT
ejpam-3418	84	7	if	if	SCONJ
ejpam-3418	84	8	a	a	DET
ejpam-3418	84	9	=	=	NOUN
ejpam-3418	84	10	∅	∅	NOUN
ejpam-3418	84	11	;	;	PUNCT
ejpam-3418	84	12	infx∈a	infx∈a	ADJ
ejpam-3418	84	13	π(x	π(x	ADP
ejpam-3418	84	14	)	)	PUNCT
ejpam-3418	84	15	,	,	PUNCT
ejpam-3418	84	16	if	if	SCONJ
ejpam-3418	84	17	a	a	DET
ejpam-3418	84	18	6=	6=	NUM
ejpam-3418	84	19	∅	∅	NOUN
ejpam-3418	84	20	,	,	PUNCT
ejpam-3418	84	21	a	a	DET
ejpam-3418	84	22	∈	∈	PROPN
ejpam-3418	84	23	p(x	p(x	NOUN
ejpam-3418	84	24	)	)	PUNCT
ejpam-3418	84	25	.	.	PUNCT
ejpam-3418	85	1	this	this	PRON
ejpam-3418	85	2	is	be	AUX
ejpam-3418	85	3	similar	similar	ADJ
ejpam-3418	85	4	to	to	ADP
ejpam-3418	85	5	the	the	DET
ejpam-3418	85	6	guaranteed	guarantee	VERB
ejpam-3418	85	7	possibility	possibility	NOUN
ejpam-3418	85	8	given	give	VERB
ejpam-3418	85	9	in	in	ADP
ejpam-3418	85	10	[	[	X
ejpam-3418	85	11	1	1	NUM
ejpam-3418	85	12	]	]	PUNCT
ejpam-3418	85	13	.	.	PUNCT
ejpam-3418	86	1	we	we	PRON
ejpam-3418	86	2	call	call	VERB
ejpam-3418	86	3	an	an	DET
ejpam-3418	86	4	ivfi	ivfi	NOUN
ejpam-3418	86	5	set	set	VERB
ejpam-3418	86	6	ι̂	ι̂	NOUN
ejpam-3418	86	7	:	:	PUNCT
ejpam-3418	86	8	i(x	i(x	NOUN
ejpam-3418	86	9	)	)	PUNCT
ejpam-3418	86	10	→	→	PUNCT
ejpam-3418	86	11	i	i	PRON
ejpam-3418	86	12	with	with	ADP
ejpam-3418	86	13	the	the	DET
ejpam-3418	86	14	property	property	NOUN
ejpam-3418	86	15	that	that	PRON
ejpam-3418	86	16	for	for	ADP
ejpam-3418	86	17	all	all	DET
ejpam-3418	86	18	nonempty	nonempty	ADV
ejpam-3418	86	19	set	set	VERB
ejpam-3418	86	20	a	a	DET
ejpam-3418	86	21	∈	∈	PROPN
ejpam-3418	86	22	i(x	i(x	NOUN
ejpam-3418	86	23	)	)	PUNCT
ejpam-3418	86	24	,	,	PUNCT
ejpam-3418	86	25	ι̂(a	ι̂(a	PUNCT
ejpam-3418	86	26	)	)	PUNCT
ejpam-3418	86	27	=	=	SYM
ejpam-3418	86	28	inf	inf	NOUN
ejpam-3418	86	29	x∈a	x∈a	VERB
ejpam-3418	86	30	ι̂({x	ι̂({x	PROPN
ejpam-3418	86	31	}	}	PUNCT
ejpam-3418	86	32	)	)	PUNCT
ejpam-3418	86	33	,	,	PUNCT
ejpam-3418	86	34	a	a	DET
ejpam-3418	86	35	guaranteed	guarantee	VERB
ejpam-3418	86	36	possibility	possibility	NOUN
ejpam-3418	86	37	ivfi	ivfi	NOUN
ejpam-3418	86	38	set	set	VERB
ejpam-3418	86	39	.	.	PUNCT
ejpam-3418	87	1	remark	remark	PROPN
ejpam-3418	87	2	2	2	NUM
ejpam-3418	87	3	.	.	PUNCT
ejpam-3418	88	1	let	let	VERB
ejpam-3418	88	2	x	x	PRON
ejpam-3418	88	3	be	be	AUX
ejpam-3418	88	4	a	a	DET
ejpam-3418	88	5	nonempty	nonempty	ADV
ejpam-3418	88	6	set	set	VERB
ejpam-3418	88	7	and	and	CCONJ
ejpam-3418	88	8	i(x	i(x	NOUN
ejpam-3418	88	9	)	)	PUNCT
ejpam-3418	88	10	be	be	AUX
ejpam-3418	88	11	an	an	DET
ejpam-3418	88	12	ideal	ideal	NOUN
ejpam-3418	88	13	on	on	ADP
ejpam-3418	88	14	x.	x.	NOUN
ejpam-3418	88	15	we	we	PRON
ejpam-3418	88	16	denote	denote	VERB
ejpam-3418	88	17	by	by	ADP
ejpam-3418	88	18	0̃i(x	0̃i(x	NUM
ejpam-3418	88	19	)	)	PUNCT
ejpam-3418	88	20	the	the	DET
ejpam-3418	88	21	ivfi	ivfi	NOUN
ejpam-3418	88	22	set	set	VERB
ejpam-3418	88	23	0̃i(x	0̃i(x	NUM
ejpam-3418	88	24	)	)	PUNCT
ejpam-3418	89	1	:	:	PUNCT
ejpam-3418	89	2	i(x)→	i(x)→	PUNCT
ejpam-3418	89	3	{	{	PUNCT
ejpam-3418	89	4	[	[	X
ejpam-3418	89	5	0	0	NUM
ejpam-3418	89	6	,	,	PUNCT
ejpam-3418	89	7	0	0	NUM
ejpam-3418	89	8	]	]	PUNCT
ejpam-3418	89	9	}	}	PUNCT
ejpam-3418	89	10	and	and	CCONJ
ejpam-3418	89	11	by	by	ADP
ejpam-3418	89	12	1̃i(x	1̃i(x	NUM
ejpam-3418	89	13	)	)	PUNCT
ejpam-3418	89	14	the	the	DET
ejpam-3418	89	15	ivfi	ivfi	NOUN
ejpam-3418	89	16	set	set	VERB
ejpam-3418	89	17	1̃i(x)(a	1̃i(x)(a	NUM
ejpam-3418	89	18	)	)	PUNCT
ejpam-3418	90	1	=	=	PRON
ejpam-3418	90	2	{	{	PUNCT
ejpam-3418	91	1	[	[	X
ejpam-3418	91	2	0	0	NUM
ejpam-3418	91	3	,	,	PUNCT
ejpam-3418	91	4	0	0	NUM
ejpam-3418	91	5	]	]	PUNCT
ejpam-3418	91	6	,	,	PUNCT
ejpam-3418	91	7	if	if	SCONJ
ejpam-3418	91	8	a	a	DET
ejpam-3418	91	9	=	=	NOUN
ejpam-3418	91	10	∅	∅	NOUN
ejpam-3418	91	11	;	;	PUNCT
ejpam-3418	92	1	[	[	X
ejpam-3418	92	2	1	1	NUM
ejpam-3418	92	3	,	,	PUNCT
ejpam-3418	92	4	1	1	NUM
ejpam-3418	92	5	]	]	PUNCT
ejpam-3418	92	6	,	,	PUNCT
ejpam-3418	92	7	if	if	SCONJ
ejpam-3418	92	8	a	a	DET
ejpam-3418	92	9	6=	6=	NUM
ejpam-3418	92	10	∅	∅	NOUN
ejpam-3418	92	11	,	,	PUNCT
ejpam-3418	92	12	a	a	DET
ejpam-3418	92	13	∈	∈	PROPN
ejpam-3418	92	14	i(x	i(x	NOUN
ejpam-3418	92	15	)	)	PUNCT
ejpam-3418	92	16	.	.	PUNCT
ejpam-3418	93	1	mj	mj	PROPN
ejpam-3418	93	2	togonon	togonon	PROPN
ejpam-3418	93	3	,	,	PUNCT
ejpam-3418	93	4	r	r	NOUN
ejpam-3418	93	5	caga	caga	NOUN
ejpam-3418	93	6	-	-	PUNCT
ejpam-3418	93	7	anan	anan	PROPN
ejpam-3418	93	8	/	/	SYM
ejpam-3418	93	9	eur	eur	PROPN
ejpam-3418	93	10	.	.	PUNCT
ejpam-3418	94	1	j.	j.	PROPN
ejpam-3418	94	2	pure	pure	PROPN
ejpam-3418	94	3	appl	appl	PROPN
ejpam-3418	94	4	.	.	PROPN
ejpam-3418	94	5	math	math	PROPN
ejpam-3418	94	6	,	,	PUNCT
ejpam-3418	94	7	12	12	NUM
ejpam-3418	94	8	(	(	PUNCT
ejpam-3418	94	9	2	2	NUM
ejpam-3418	94	10	)	)	PUNCT
ejpam-3418	94	11	(	(	PUNCT
ejpam-3418	94	12	2019	2019	NUM
ejpam-3418	94	13	)	)	PUNCT
ejpam-3418	94	14	,	,	PUNCT
ejpam-3418	94	15	553	553	NUM
ejpam-3418	94	16	-	-	SYM
ejpam-3418	94	17	570	570	NUM
ejpam-3418	94	18	556	556	NUM
ejpam-3418	94	19	next	next	ADJ
ejpam-3418	94	20	,	,	PUNCT
ejpam-3418	94	21	we	we	PRON
ejpam-3418	94	22	define	define	VERB
ejpam-3418	94	23	some	some	DET
ejpam-3418	94	24	relational	relational	ADJ
ejpam-3418	94	25	operators	operator	NOUN
ejpam-3418	94	26	between	between	ADP
ejpam-3418	94	27	ivfi	ivfi	NOUN
ejpam-3418	94	28	sets	set	NOUN
ejpam-3418	94	29	.	.	PUNCT
ejpam-3418	95	1	definition	definition	NOUN
ejpam-3418	95	2	2	2	NUM
ejpam-3418	95	3	.	.	PUNCT
ejpam-3418	96	1	let	let	VERB
ejpam-3418	96	2	x	x	PRON
ejpam-3418	96	3	be	be	AUX
ejpam-3418	96	4	a	a	DET
ejpam-3418	96	5	nonempty	nonempty	ADV
ejpam-3418	96	6	set	set	VERB
ejpam-3418	96	7	and	and	CCONJ
ejpam-3418	96	8	i(x	i(x	NOUN
ejpam-3418	96	9	)	)	PUNCT
ejpam-3418	96	10	be	be	AUX
ejpam-3418	96	11	an	an	DET
ejpam-3418	96	12	ideal	ideal	NOUN
ejpam-3418	96	13	on	on	ADP
ejpam-3418	96	14	x.	x.	NOUN
ejpam-3418	96	15	let	let	VERB
ejpam-3418	96	16	ι̂	ι̂	NOUN
ejpam-3418	96	17	,	,	PUNCT
ejpam-3418	96	18	τ̂	τ̂	PUNCT
ejpam-3418	96	19	∈	∈	PROPN
ejpam-3418	96	20	i	i	PRON
ejpam-3418	96	21	i(x	i(x	PROPN
ejpam-3418	96	22	)	)	PUNCT
ejpam-3418	96	23	.	.	PUNCT
ejpam-3418	97	1	we	we	PRON
ejpam-3418	97	2	say	say	VERB
ejpam-3418	97	3	(	(	PUNCT
ejpam-3418	97	4	i	i	NOUN
ejpam-3418	97	5	)	)	PUNCT
ejpam-3418	97	6	ι̂	ι̂	X
ejpam-3418	97	7	is	be	AUX
ejpam-3418	97	8	a	a	DET
ejpam-3418	97	9	subset	subset	NOUN
ejpam-3418	97	10	of	of	ADP
ejpam-3418	97	11	τ̂	τ̂	NUM
ejpam-3418	97	12	,	,	PUNCT
ejpam-3418	97	13	denoted	denote	VERB
ejpam-3418	97	14	by	by	ADP
ejpam-3418	97	15	ι̂	ι̂	X
ejpam-3418	97	16	6	6	NUM
ejpam-3418	97	17	τ̂	τ̂	PUNCT
ejpam-3418	97	18	,	,	PUNCT
ejpam-3418	97	19	if	if	SCONJ
ejpam-3418	97	20	ι̂(a	ι̂(a	NOUN
ejpam-3418	97	21	)	)	PUNCT
ejpam-3418	97	22	≤i	≤i	NOUN
ejpam-3418	97	23	τ̂(a	τ̂(a	NUM
ejpam-3418	97	24	)	)	PUNCT
ejpam-3418	97	25	,	,	PUNCT
ejpam-3418	97	26	for	for	ADP
ejpam-3418	97	27	all	all	DET
ejpam-3418	97	28	a	a	DET
ejpam-3418	97	29	∈	∈	PROPN
ejpam-3418	97	30	i(x	i(x	NOUN
ejpam-3418	97	31	)	)	PUNCT
ejpam-3418	97	32	;	;	PUNCT
ejpam-3418	97	33	and	and	CCONJ
ejpam-3418	97	34	(	(	PUNCT
ejpam-3418	97	35	ii	ii	NOUN
ejpam-3418	97	36	)	)	PUNCT
ejpam-3418	97	37	ι̂	ι̂	X
ejpam-3418	97	38	is	be	AUX
ejpam-3418	97	39	equal	equal	ADJ
ejpam-3418	97	40	to	to	ADP
ejpam-3418	97	41	τ̂	τ̂	NUM
ejpam-3418	97	42	,	,	PUNCT
ejpam-3418	97	43	denoted	denote	VERB
ejpam-3418	97	44	by	by	ADP
ejpam-3418	97	45	ι̂	ι̂	NOUN
ejpam-3418	97	46	=	=	PUNCT
ejpam-3418	97	47	τ̂	τ̂	PUNCT
ejpam-3418	97	48	,	,	PUNCT
ejpam-3418	97	49	if	if	SCONJ
ejpam-3418	97	50	ι̂(a	ι̂(a	NOUN
ejpam-3418	97	51	)	)	PUNCT
ejpam-3418	97	52	=	=	PUNCT
ejpam-3418	97	53	τ̂(a	τ̂(a	NUM
ejpam-3418	97	54	)	)	PUNCT
ejpam-3418	97	55	,	,	PUNCT
ejpam-3418	97	56	for	for	ADP
ejpam-3418	97	57	all	all	DET
ejpam-3418	97	58	a	a	DET
ejpam-3418	97	59	∈	∈	PROPN
ejpam-3418	97	60	i(x	i(x	NOUN
ejpam-3418	97	61	)	)	PUNCT
ejpam-3418	97	62	.	.	PUNCT
ejpam-3418	98	1	remark	remark	PROPN
ejpam-3418	98	2	3	3	NUM
ejpam-3418	98	3	.	.	PUNCT
ejpam-3418	98	4	to	to	PART
ejpam-3418	98	5	avoid	avoid	VERB
ejpam-3418	98	6	confusion	confusion	NOUN
ejpam-3418	98	7	,	,	PUNCT
ejpam-3418	98	8	we	we	PRON
ejpam-3418	98	9	summarize	summarize	VERB
ejpam-3418	98	10	first	first	ADV
ejpam-3418	98	11	our	our	PRON
ejpam-3418	98	12	“	"	PUNCT
ejpam-3418	98	13	inequality	inequality	NOUN
ejpam-3418	98	14	”	"	PUNCT
ejpam-3418	98	15	notations	notation	NOUN
ejpam-3418	98	16	.	.	PUNCT
ejpam-3418	99	1	i.	i.	PROPN
ejpam-3418	99	2	the	the	DET
ejpam-3418	99	3	symbol	symbol	NOUN
ejpam-3418	99	4	“	"	PUNCT
ejpam-3418	99	5	≤	≤	NUM
ejpam-3418	99	6	”	"	PUNCT
ejpam-3418	99	7	for	for	ADP
ejpam-3418	99	8	the	the	DET
ejpam-3418	99	9	usual	usual	ADJ
ejpam-3418	99	10	inequality	inequality	NOUN
ejpam-3418	99	11	with	with	ADP
ejpam-3418	99	12	the	the	DET
ejpam-3418	99	13	real	real	ADJ
ejpam-3418	99	14	numbers	number	NOUN
ejpam-3418	99	15	.	.	PUNCT
ejpam-3418	100	1	ii	ii	PROPN
ejpam-3418	100	2	.	.	PUNCT
ejpam-3418	101	1	the	the	DET
ejpam-3418	101	2	symbol	symbol	NOUN
ejpam-3418	101	3	“	"	PUNCT
ejpam-3418	101	4	≤i	≤i	NOUN
ejpam-3418	101	5	”	"	PUNCT
ejpam-3418	101	6	for	for	ADP
ejpam-3418	101	7	the	the	DET
ejpam-3418	101	8	inequality	inequality	NOUN
ejpam-3418	101	9	with	with	ADP
ejpam-3418	101	10	intervals	interval	NOUN
ejpam-3418	101	11	.	.	PUNCT
ejpam-3418	102	1	iii	iii	X
ejpam-3418	102	2	.	.	PUNCT
ejpam-3418	103	1	the	the	DET
ejpam-3418	103	2	symbol	symbol	NOUN
ejpam-3418	103	3	“	"	PUNCT
ejpam-3418	103	4	6	6	NUM
ejpam-3418	103	5	”	"	PUNCT
ejpam-3418	103	6	to	to	PART
ejpam-3418	103	7	denote	denote	VERB
ejpam-3418	103	8	the	the	DET
ejpam-3418	103	9	subset	subset	NOUN
ejpam-3418	103	10	relation	relation	NOUN
ejpam-3418	103	11	with	with	ADP
ejpam-3418	103	12	ivfi	ivfi	NOUN
ejpam-3418	103	13	sets	set	NOUN
ejpam-3418	103	14	.	.	PUNCT
ejpam-3418	104	1	next	next	ADV
ejpam-3418	104	2	,	,	PUNCT
ejpam-3418	104	3	we	we	PRON
ejpam-3418	104	4	define	define	VERB
ejpam-3418	104	5	complement	complement	NOUN
ejpam-3418	104	6	,	,	PUNCT
ejpam-3418	104	7	union	union	NOUN
ejpam-3418	104	8	,	,	PUNCT
ejpam-3418	104	9	and	and	CCONJ
ejpam-3418	104	10	intersection	intersection	NOUN
ejpam-3418	104	11	of	of	ADP
ejpam-3418	104	12	ivfi	ivfi	NOUN
ejpam-3418	104	13	sets	set	NOUN
ejpam-3418	104	14	.	.	PUNCT
ejpam-3418	105	1	we	we	PRON
ejpam-3418	105	2	then	then	ADV
ejpam-3418	105	3	prove	prove	VERB
ejpam-3418	105	4	that	that	SCONJ
ejpam-3418	105	5	the	the	DET
ejpam-3418	105	6	resulting	result	VERB
ejpam-3418	105	7	mappings	mapping	NOUN
ejpam-3418	105	8	are	be	AUX
ejpam-3418	105	9	also	also	ADV
ejpam-3418	105	10	ivfi	ivfi	NOUN
ejpam-3418	105	11	sets	set	NOUN
ejpam-3418	105	12	,	,	PUNCT
ejpam-3418	105	13	showing	show	VERB
ejpam-3418	105	14	that	that	SCONJ
ejpam-3418	105	15	our	our	PRON
ejpam-3418	105	16	definitions	definition	NOUN
ejpam-3418	105	17	are	be	AUX
ejpam-3418	105	18	well	well	ADV
ejpam-3418	105	19	-	-	PUNCT
ejpam-3418	105	20	defined	define	VERB
ejpam-3418	105	21	.	.	PUNCT
ejpam-3418	106	1	definition	definition	NOUN
ejpam-3418	106	2	3	3	X
ejpam-3418	106	3	.	.	PUNCT
ejpam-3418	107	1	let	let	VERB
ejpam-3418	107	2	x	x	PRON
ejpam-3418	107	3	be	be	AUX
ejpam-3418	107	4	a	a	DET
ejpam-3418	107	5	nonempty	nonempty	ADV
ejpam-3418	107	6	set	set	VERB
ejpam-3418	107	7	and	and	CCONJ
ejpam-3418	107	8	i(x	i(x	NOUN
ejpam-3418	107	9	)	)	PUNCT
ejpam-3418	107	10	be	be	AUX
ejpam-3418	107	11	an	an	DET
ejpam-3418	107	12	ideal	ideal	NOUN
ejpam-3418	107	13	on	on	ADP
ejpam-3418	107	14	x.	x.	NOUN
ejpam-3418	107	15	let	let	VERB
ejpam-3418	107	16	ι̂	ι̂	PUNCT
ejpam-3418	107	17	∈	∈	VERB
ejpam-3418	107	18	i	i	PRON
ejpam-3418	107	19	i(x	i(x	PROPN
ejpam-3418	107	20	)	)	PUNCT
ejpam-3418	107	21	.	.	PUNCT
ejpam-3418	108	1	the	the	DET
ejpam-3418	108	2	complement	complement	NOUN
ejpam-3418	108	3	of	of	ADP
ejpam-3418	108	4	ι̂	ι̂	NOUN
ejpam-3418	108	5	,	,	PUNCT
ejpam-3418	108	6	denoted	denote	VERB
ejpam-3418	108	7	by	by	ADP
ejpam-3418	108	8	ι̂c	ι̂c	PROPN
ejpam-3418	108	9	,	,	PUNCT
ejpam-3418	108	10	is	be	AUX
ejpam-3418	108	11	defined	define	VERB
ejpam-3418	108	12	by	by	ADP
ejpam-3418	108	13	,	,	PUNCT
ejpam-3418	108	14	ι̂c(∅	ι̂c(∅	NOUN
ejpam-3418	108	15	)	)	PUNCT
ejpam-3418	108	16	=	=	PUNCT
ejpam-3418	109	1	[	[	X
ejpam-3418	109	2	0	0	NUM
ejpam-3418	109	3	,	,	PUNCT
ejpam-3418	109	4	0	0	NUM
ejpam-3418	109	5	]	]	PUNCT
ejpam-3418	109	6	and	and	CCONJ
ejpam-3418	109	7	for	for	ADP
ejpam-3418	109	8	every	every	DET
ejpam-3418	109	9	nonempty	nonempty	NOUN
ejpam-3418	109	10	set	set	VERB
ejpam-3418	109	11	a	a	DET
ejpam-3418	109	12	∈	∈	PROPN
ejpam-3418	109	13	i(x	i(x	NOUN
ejpam-3418	109	14	)	)	PUNCT
ejpam-3418	109	15	,	,	PUNCT
ejpam-3418	109	16	ι̂c(a	ι̂c(a	NUM
ejpam-3418	109	17	)	)	PUNCT
ejpam-3418	109	18	=	=	PUNCT
ejpam-3418	110	1	[	[	PUNCT
ejpam-3418	110	2	inf	inf	ADJ
ejpam-3418	110	3	x∈a	x∈a	NOUN
ejpam-3418	110	4	{	{	PUNCT
ejpam-3418	110	5	1−	1−	NUM
ejpam-3418	110	6	[	[	X
ejpam-3418	110	7	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	110	8	}	}	PUNCT
ejpam-3418	110	9	,	,	PUNCT
ejpam-3418	110	10	inf	inf	PROPN
ejpam-3418	110	11	x∈a	x∈a	NOUN
ejpam-3418	110	12	{	{	PUNCT
ejpam-3418	110	13	1−	1−	NUM
ejpam-3418	111	1	[	[	X
ejpam-3418	111	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	111	3	}	}	PUNCT
ejpam-3418	111	4	]	]	PUNCT
ejpam-3418	111	5	.	.	PUNCT
ejpam-3418	112	1	proposition	proposition	NOUN
ejpam-3418	112	2	1	1	X
ejpam-3418	112	3	.	.	PUNCT
ejpam-3418	113	1	let	let	VERB
ejpam-3418	113	2	x	x	PRON
ejpam-3418	113	3	be	be	AUX
ejpam-3418	113	4	a	a	DET
ejpam-3418	113	5	nonempty	nonempty	ADV
ejpam-3418	113	6	set	set	VERB
ejpam-3418	113	7	and	and	CCONJ
ejpam-3418	113	8	i(x	i(x	NOUN
ejpam-3418	113	9	)	)	PUNCT
ejpam-3418	113	10	be	be	AUX
ejpam-3418	113	11	an	an	DET
ejpam-3418	113	12	ideal	ideal	NOUN
ejpam-3418	113	13	on	on	ADP
ejpam-3418	113	14	x.	x.	NOUN
ejpam-3418	113	15	if	if	SCONJ
ejpam-3418	113	16	ι̂	ι̂	NUM
ejpam-3418	113	17	∈	∈	PROPN
ejpam-3418	113	18	i	i	PRON
ejpam-3418	113	19	i(x	i(x	PROPN
ejpam-3418	113	20	)	)	PUNCT
ejpam-3418	113	21	,	,	PUNCT
ejpam-3418	113	22	then	then	ADV
ejpam-3418	113	23	the	the	DET
ejpam-3418	113	24	complement	complement	NOUN
ejpam-3418	113	25	of	of	ADP
ejpam-3418	113	26	ι̂	ι̂	NUM
ejpam-3418	113	27	is	be	AUX
ejpam-3418	113	28	an	an	DET
ejpam-3418	113	29	ivfi	ivfi	NOUN
ejpam-3418	113	30	set	set	VERB
ejpam-3418	113	31	.	.	PUNCT
ejpam-3418	114	1	proof	proof	NOUN
ejpam-3418	114	2	.	.	PUNCT
ejpam-3418	115	1	let	let	VERB
ejpam-3418	115	2	ι̂	ι̂	PRON
ejpam-3418	115	3	∈	∈	VERB
ejpam-3418	115	4	i	i	PRON
ejpam-3418	115	5	i(x	i(x	PROPN
ejpam-3418	115	6	)	)	PUNCT
ejpam-3418	115	7	.	.	PUNCT
ejpam-3418	116	1	let	let	VERB
ejpam-3418	116	2	a	a	DET
ejpam-3418	116	3	∈	∈	PROPN
ejpam-3418	116	4	i(x	i(x	NOUN
ejpam-3418	116	5	)	)	PUNCT
ejpam-3418	116	6	,	,	PUNCT
ejpam-3418	116	7	x	x	PUNCT
ejpam-3418	116	8	∈	∈	PROPN
ejpam-3418	116	9	a	a	PRON
ejpam-3418	116	10	and	and	CCONJ
ejpam-3418	116	11	ι̂({x	ι̂({x	NOUN
ejpam-3418	116	12	}	}	PUNCT
ejpam-3418	116	13	)	)	PUNCT
ejpam-3418	117	1	=	=	PUNCT
ejpam-3418	118	1	[	[	X
ejpam-3418	118	2	[	[	X
ejpam-3418	118	3	̃ι({x})]−	̃ι({x})]−	X
ejpam-3418	118	4	,	,	PUNCT
ejpam-3418	118	5	[	[	X
ejpam-3418	118	6	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	118	7	]	]	X
ejpam-3418	118	8	.	.	PUNCT
ejpam-3418	119	1	since	since	SCONJ
ejpam-3418	119	2	[	[	X
ejpam-3418	119	3	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	119	4	≤	≤	PROPN
ejpam-3418	119	5	[	[	X
ejpam-3418	119	6	̂ι({x})]+	̂ι({x})]+	INTJ
ejpam-3418	119	7	,	,	PUNCT
ejpam-3418	119	8	we	we	PRON
ejpam-3418	119	9	have	have	VERB
ejpam-3418	119	10	1−	1−	NUM
ejpam-3418	120	1	[	[	X
ejpam-3418	120	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	120	3	≤	≤	ADJ
ejpam-3418	120	4	1−	1−	NUM
ejpam-3418	121	1	[	[	X
ejpam-3418	121	2	̂ι({x})]−.	̂ι({x})]−.	X
ejpam-3418	121	3	hence	hence	ADV
ejpam-3418	121	4	,	,	PUNCT
ejpam-3418	121	5	inf	inf	PROPN
ejpam-3418	121	6	x∈a	x∈a	NOUN
ejpam-3418	121	7	{	{	PUNCT
ejpam-3418	121	8	1−	1−	NUM
ejpam-3418	121	9	[	[	X
ejpam-3418	121	10	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	121	11	}	}	PUNCT
ejpam-3418	121	12	≤	≤	NUM
ejpam-3418	121	13	inf	inf	NOUN
ejpam-3418	121	14	x∈a	x∈a	NOUN
ejpam-3418	121	15	{	{	PUNCT
ejpam-3418	121	16	1−	1−	NUM
ejpam-3418	122	1	[	[	X
ejpam-3418	122	2	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	122	3	}	}	PUNCT
ejpam-3418	122	4	,	,	PUNCT
ejpam-3418	122	5	and	and	CCONJ
ejpam-3418	122	6	indeed	indeed	ADV
ejpam-3418	122	7	we	we	PRON
ejpam-3418	122	8	have	have	VERB
ejpam-3418	122	9	a	a	DET
ejpam-3418	122	10	closed	closed	ADJ
ejpam-3418	122	11	interval	interval	NOUN
ejpam-3418	122	12	.	.	PUNCT
ejpam-3418	123	1	we	we	PRON
ejpam-3418	123	2	need	need	VERB
ejpam-3418	123	3	to	to	PART
ejpam-3418	123	4	show	show	VERB
ejpam-3418	123	5	that	that	SCONJ
ejpam-3418	123	6	the	the	DET
ejpam-3418	123	7	reverse	reverse	ADJ
ejpam-3418	123	8	inequality	inequality	NOUN
ejpam-3418	123	9	of	of	ADP
ejpam-3418	123	10	an	an	DET
ejpam-3418	123	11	ivfi	ivfi	NOUN
ejpam-3418	123	12	set	set	VERB
ejpam-3418	123	13	holds	hold	VERB
ejpam-3418	123	14	.	.	PUNCT
ejpam-3418	124	1	let	let	VERB
ejpam-3418	124	2	∅	∅	NOUN
ejpam-3418	124	3	6=	6=	ADP
ejpam-3418	124	4	a	a	DET
ejpam-3418	124	5	,	,	PUNCT
ejpam-3418	124	6	b	b	PROPN
ejpam-3418	124	7	∈	∈	PROPN
ejpam-3418	124	8	i(x	i(x	NOUN
ejpam-3418	124	9	)	)	PUNCT
ejpam-3418	124	10	such	such	ADJ
ejpam-3418	124	11	that	that	SCONJ
ejpam-3418	124	12	a	a	DET
ejpam-3418	124	13	⊆	⊆	NUM
ejpam-3418	124	14	b.	b.	NOUN
ejpam-3418	124	15	then	then	ADV
ejpam-3418	124	16	,	,	PUNCT
ejpam-3418	124	17	{	{	PUNCT
ejpam-3418	124	18	1−	1−	NUM
ejpam-3418	124	19	[	[	X
ejpam-3418	124	20	̂ι({x})]−	̂ι({x})]−	NOUN
ejpam-3418	124	21	:	:	PUNCT
ejpam-3418	124	22	x	x	SYM
ejpam-3418	124	23	∈	∈	PROPN
ejpam-3418	124	24	a	a	DET
ejpam-3418	124	25	}	}	PUNCT
ejpam-3418	124	26	⊆	⊆	NUM
ejpam-3418	124	27	{	{	PUNCT
ejpam-3418	124	28	1−	1−	NUM
ejpam-3418	125	1	[	[	X
ejpam-3418	125	2	̂ι({x})]−	̂ι({x})]−	NOUN
ejpam-3418	125	3	:	:	PUNCT
ejpam-3418	125	4	x	x	SYM
ejpam-3418	125	5	∈	∈	PROPN
ejpam-3418	125	6	b	b	NOUN
ejpam-3418	125	7	}	}	PUNCT
ejpam-3418	125	8	and	and	CCONJ
ejpam-3418	125	9	{	{	PUNCT
ejpam-3418	125	10	1−	1−	NUM
ejpam-3418	126	1	[	[	X
ejpam-3418	126	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	126	3	:	:	PUNCT
ejpam-3418	126	4	x	x	SYM
ejpam-3418	126	5	∈	∈	PROPN
ejpam-3418	126	6	a	a	PRON
ejpam-3418	126	7	}	}	PUNCT
ejpam-3418	126	8	⊆	⊆	NUM
ejpam-3418	126	9	{	{	PUNCT
ejpam-3418	126	10	1−	1−	NUM
ejpam-3418	127	1	[	[	X
ejpam-3418	127	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	127	3	:	:	PUNCT
ejpam-3418	127	4	x	x	SYM
ejpam-3418	127	5	∈	∈	PROPN
ejpam-3418	127	6	b	b	NOUN
ejpam-3418	127	7	}	}	PUNCT
ejpam-3418	127	8	.	.	PUNCT
ejpam-3418	128	1	hence	hence	ADV
ejpam-3418	128	2	,	,	PUNCT
ejpam-3418	128	3	ι̂c(b	ι̂c(b	PROPN
ejpam-3418	128	4	)	)	PUNCT
ejpam-3418	129	1	=	=	PUNCT
ejpam-3418	129	2	[	[	PUNCT
ejpam-3418	129	3	inf	inf	NOUN
ejpam-3418	129	4	x∈b	x∈b	NOUN
ejpam-3418	129	5	{	{	PUNCT
ejpam-3418	129	6	1−	1−	NUM
ejpam-3418	129	7	[	[	X
ejpam-3418	129	8	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	129	9	}	}	PUNCT
ejpam-3418	129	10	,	,	PUNCT
ejpam-3418	129	11	inf	inf	PROPN
ejpam-3418	129	12	x∈b	x∈b	NOUN
ejpam-3418	129	13	{	{	PUNCT
ejpam-3418	129	14	1−	1−	NUM
ejpam-3418	130	1	[	[	X
ejpam-3418	130	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	130	3	}	}	PUNCT
ejpam-3418	130	4	]	]	X
ejpam-3418	130	5	≤i	≤i	PROPN
ejpam-3418	130	6	[	[	PUNCT
ejpam-3418	130	7	inf	inf	ADJ
ejpam-3418	130	8	x∈a	x∈a	NOUN
ejpam-3418	130	9	{	{	PUNCT
ejpam-3418	130	10	1−	1−	NUM
ejpam-3418	130	11	[	[	X
ejpam-3418	130	12	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	130	13	}	}	PUNCT
ejpam-3418	130	14	,	,	PUNCT
ejpam-3418	130	15	inf	inf	PROPN
ejpam-3418	130	16	x∈a	x∈a	NOUN
ejpam-3418	130	17	{	{	PUNCT
ejpam-3418	130	18	1−	1−	NUM
ejpam-3418	131	1	[	[	X
ejpam-3418	131	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	131	3	}	}	PUNCT
ejpam-3418	131	4	]	]	PUNCT
ejpam-3418	131	5	=	=	PUNCT
ejpam-3418	131	6	ι̂c(a	ι̂c(a	NUM
ejpam-3418	131	7	)	)	PUNCT
ejpam-3418	131	8	.	.	PUNCT
ejpam-3418	132	1	therefore	therefore	ADV
ejpam-3418	132	2	,	,	PUNCT
ejpam-3418	132	3	ι̂c	ι̂c	PROPN
ejpam-3418	132	4	is	be	AUX
ejpam-3418	132	5	an	an	DET
ejpam-3418	132	6	ivfi	ivfi	NOUN
ejpam-3418	132	7	set	set	VERB
ejpam-3418	132	8	.	.	PUNCT
ejpam-3418	133	1	remark	remark	PROPN
ejpam-3418	133	2	4	4	NUM
ejpam-3418	133	3	.	.	PUNCT
ejpam-3418	134	1	for	for	ADP
ejpam-3418	134	2	a	a	DET
ejpam-3418	134	3	singleton	singleton	NOUN
ejpam-3418	134	4	set	set	VERB
ejpam-3418	134	5	a	a	DET
ejpam-3418	134	6	=	=	SYM
ejpam-3418	134	7	{	{	PUNCT
ejpam-3418	134	8	x	x	NOUN
ejpam-3418	134	9	}	}	PUNCT
ejpam-3418	134	10	∈	∈	PROPN
ejpam-3418	134	11	i(x	i(x	NOUN
ejpam-3418	134	12	)	)	PUNCT
ejpam-3418	134	13	,	,	PUNCT
ejpam-3418	134	14	the	the	DET
ejpam-3418	134	15	preceding	precede	VERB
ejpam-3418	134	16	definition	definition	NOUN
ejpam-3418	134	17	coincides	coincide	VERB
ejpam-3418	134	18	with	with	ADP
ejpam-3418	134	19	the	the	DET
ejpam-3418	134	20	definition	definition	NOUN
ejpam-3418	134	21	of	of	ADP
ejpam-3418	134	22	the	the	DET
ejpam-3418	134	23	complement	complement	NOUN
ejpam-3418	134	24	of	of	ADP
ejpam-3418	134	25	an	an	DET
ejpam-3418	134	26	interval	interval	NOUN
ejpam-3418	134	27	-	-	PUNCT
ejpam-3418	134	28	valued	value	VERB
ejpam-3418	134	29	fuzzy	fuzzy	ADJ
ejpam-3418	134	30	set	set	NOUN
ejpam-3418	134	31	.	.	PUNCT
ejpam-3418	135	1	mj	mj	PROPN
ejpam-3418	135	2	togonon	togonon	PROPN
ejpam-3418	135	3	,	,	PUNCT
ejpam-3418	135	4	r	r	NOUN
ejpam-3418	135	5	caga	caga	NOUN
ejpam-3418	135	6	-	-	PUNCT
ejpam-3418	135	7	anan	anan	PROPN
ejpam-3418	135	8	/	/	SYM
ejpam-3418	135	9	eur	eur	PROPN
ejpam-3418	135	10	.	.	PUNCT
ejpam-3418	136	1	j.	j.	PROPN
ejpam-3418	136	2	pure	pure	PROPN
ejpam-3418	136	3	appl	appl	PROPN
ejpam-3418	136	4	.	.	PROPN
ejpam-3418	136	5	math	math	PROPN
ejpam-3418	136	6	,	,	PUNCT
ejpam-3418	136	7	12	12	NUM
ejpam-3418	136	8	(	(	PUNCT
ejpam-3418	136	9	2	2	NUM
ejpam-3418	136	10	)	)	PUNCT
ejpam-3418	136	11	(	(	PUNCT
ejpam-3418	136	12	2019	2019	NUM
ejpam-3418	136	13	)	)	PUNCT
ejpam-3418	136	14	,	,	PUNCT
ejpam-3418	136	15	553	553	NUM
ejpam-3418	136	16	-	-	SYM
ejpam-3418	136	17	570	570	NUM
ejpam-3418	136	18	557	557	NUM
ejpam-3418	136	19	definition	definition	NOUN
ejpam-3418	136	20	4	4	NUM
ejpam-3418	136	21	.	.	PUNCT
ejpam-3418	137	1	let	let	VERB
ejpam-3418	137	2	x	x	PRON
ejpam-3418	137	3	be	be	AUX
ejpam-3418	137	4	a	a	DET
ejpam-3418	137	5	nonempty	nonempty	ADV
ejpam-3418	137	6	set	set	VERB
ejpam-3418	137	7	and	and	CCONJ
ejpam-3418	137	8	i(x	i(x	NOUN
ejpam-3418	137	9	)	)	PUNCT
ejpam-3418	137	10	be	be	AUX
ejpam-3418	137	11	an	an	DET
ejpam-3418	137	12	ideal	ideal	NOUN
ejpam-3418	137	13	on	on	ADP
ejpam-3418	137	14	x.	x.	NOUN
ejpam-3418	137	15	let	let	VERB
ejpam-3418	137	16	ι̂	ι̂	NOUN
ejpam-3418	137	17	,	,	PUNCT
ejpam-3418	137	18	τ̂	τ̂	PUNCT
ejpam-3418	137	19	∈	∈	PROPN
ejpam-3418	137	20	i	i	PRON
ejpam-3418	137	21	i(x	i(x	PROPN
ejpam-3418	137	22	)	)	PUNCT
ejpam-3418	137	23	.	.	PUNCT
ejpam-3418	138	1	the	the	DET
ejpam-3418	138	2	union	union	NOUN
ejpam-3418	138	3	and	and	CCONJ
ejpam-3418	138	4	intersection	intersection	NOUN
ejpam-3418	138	5	of	of	ADP
ejpam-3418	138	6	ι̂	ι̂	PUNCT
ejpam-3418	138	7	and	and	CCONJ
ejpam-3418	138	8	τ̂	τ̂	PUNCT
ejpam-3418	138	9	,	,	PUNCT
ejpam-3418	138	10	denoted	denote	VERB
ejpam-3418	138	11	by	by	ADP
ejpam-3418	138	12	ι̂∨	ι̂∨	PROPN
ejpam-3418	138	13	τ̂	τ̂	PUNCT
ejpam-3418	138	14	and	and	CCONJ
ejpam-3418	138	15	ι̂∧	ι̂∧	PROPN
ejpam-3418	138	16	τ̂	τ̂	PUNCT
ejpam-3418	138	17	,	,	PUNCT
ejpam-3418	138	18	respectively	respectively	ADV
ejpam-3418	138	19	,	,	PUNCT
ejpam-3418	138	20	are	be	AUX
ejpam-3418	138	21	given	give	VERB
ejpam-3418	138	22	by	by	ADP
ejpam-3418	138	23	(	(	PUNCT
ejpam-3418	138	24	ι̂	ι̂	NOUN
ejpam-3418	138	25	∨	∨	ADJ
ejpam-3418	138	26	τ̂)(a	τ̂)(a	NUM
ejpam-3418	138	27	)	)	PUNCT
ejpam-3418	138	28	=	=	PUNCT
ejpam-3418	139	1	[	[	X
ejpam-3418	139	2	max{[̂ι(a)]−	max{[̂ι(a)]−	X
ejpam-3418	139	3	,	,	PUNCT
ejpam-3418	139	4	[	[	X
ejpam-3418	139	5	τ̂(a)]−},max{[̂ι(a)]+	τ̂(a)]−},max{[̂ι(a)]+	X
ejpam-3418	139	6	,	,	PUNCT
ejpam-3418	139	7	[	[	X
ejpam-3418	139	8	τ̂(a)]+	τ̂(a)]+	X
ejpam-3418	139	9	}	}	PUNCT
ejpam-3418	139	10	]	]	PUNCT
ejpam-3418	139	11	and	and	CCONJ
ejpam-3418	139	12	(	(	PUNCT
ejpam-3418	139	13	ι̂	ι̂	NUM
ejpam-3418	139	14	∧	∧	PROPN
ejpam-3418	139	15	τ̂)(a	τ̂)(a	PROPN
ejpam-3418	139	16	)	)	PUNCT
ejpam-3418	139	17	=	=	PUNCT
ejpam-3418	140	1	[	[	X
ejpam-3418	140	2	min{[̂ι(a)]−	min{[̂ι(a)]−	X
ejpam-3418	140	3	,	,	PUNCT
ejpam-3418	140	4	[	[	X
ejpam-3418	140	5	τ̂(a)]−},min{[̂ι(a)]+	τ̂(a)]−},min{[̂ι(a)]+	X
ejpam-3418	140	6	,	,	PUNCT
ejpam-3418	140	7	[	[	X
ejpam-3418	140	8	τ̂(a)]+	τ̂(a)]+	X
ejpam-3418	140	9	}	}	PUNCT
ejpam-3418	140	10	]	]	PUNCT
ejpam-3418	140	11	,	,	PUNCT
ejpam-3418	140	12	for	for	ADP
ejpam-3418	140	13	all	all	DET
ejpam-3418	140	14	a	a	DET
ejpam-3418	140	15	∈	∈	PROPN
ejpam-3418	140	16	i(x	i(x	NOUN
ejpam-3418	140	17	)	)	PUNCT
ejpam-3418	140	18	,	,	PUNCT
ejpam-3418	140	19	respectively	respectively	ADV
ejpam-3418	140	20	.	.	PUNCT
ejpam-3418	141	1	in	in	ADP
ejpam-3418	141	2	general	general	ADJ
ejpam-3418	141	3	,	,	PUNCT
ejpam-3418	141	4	the	the	DET
ejpam-3418	141	5	union	union	NOUN
ejpam-3418	141	6	and	and	CCONJ
ejpam-3418	141	7	intersection	intersection	NOUN
ejpam-3418	141	8	of	of	ADP
ejpam-3418	141	9	a	a	DET
ejpam-3418	141	10	collection	collection	NOUN
ejpam-3418	141	11	of	of	ADP
ejpam-3418	141	12	ivfi	ivfi	NOUN
ejpam-3418	141	13	sets	set	VERB
ejpam-3418	141	14	{	{	PUNCT
ejpam-3418	141	15	ι̂j	ι̂j	NOUN
ejpam-3418	141	16	:	:	PUNCT
ejpam-3418	141	17	j	j	PROPN
ejpam-3418	141	18	∈	∈	PROPN
ejpam-3418	141	19	j	j	PROPN
ejpam-3418	141	20	}	}	PUNCT
ejpam-3418	141	21	,	,	PUNCT
ejpam-3418	141	22	denoted	denote	VERB
ejpam-3418	141	23	by	by	ADP
ejpam-3418	141	24	∨	∨	NUM
ejpam-3418	141	25	j∈j	j∈j	NOUN
ejpam-3418	141	26	ι̂j	ι̂j	PROPN
ejpam-3418	141	27	and	and	CCONJ
ejpam-3418	141	28	∧	∧	PROPN
ejpam-3418	141	29	j∈j	j∈j	NOUN
ejpam-3418	141	30	ι̂j	ι̂j	PROPN
ejpam-3418	141	31	,	,	PUNCT
ejpam-3418	141	32	are	be	AUX
ejpam-3418	141	33	given	give	VERB
ejpam-3418	141	34	by∨	by∨	PROPN
ejpam-3418	141	35	j∈j	j∈j	NOUN
ejpam-3418	141	36	ι̂j	ι̂j	X
ejpam-3418	141	37			PROPN
ejpam-3418	141	38	(	(	PUNCT
ejpam-3418	141	39	a	a	X
ejpam-3418	141	40	)	)	PUNCT
ejpam-3418	141	41	=	=	NOUN
ejpam-3418	142	1	[	[	X
ejpam-3418	142	2	sup{[̂ιj(a)]−	sup{[̂ιj(a)]−	ADV
ejpam-3418	142	3	:	:	PUNCT
ejpam-3418	142	4	j	j	PROPN
ejpam-3418	142	5	∈	∈	PROPN
ejpam-3418	142	6	j	j	PROPN
ejpam-3418	142	7	}	}	PUNCT
ejpam-3418	142	8	,	,	PUNCT
ejpam-3418	142	9	sup{[̂ιj(a)]+	sup{[̂ιj(a)]+	NOUN
ejpam-3418	142	10	:	:	PUNCT
ejpam-3418	142	11	j	j	PROPN
ejpam-3418	142	12	∈	∈	PROPN
ejpam-3418	142	13	j	j	PROPN
ejpam-3418	142	14	}	}	PUNCT
ejpam-3418	142	15	]	]	PUNCT
ejpam-3418	142	16	and	and	CCONJ
ejpam-3418	142	17	∧	∧	PROPN
ejpam-3418	142	18	j∈j	j∈j	NOUN
ejpam-3418	142	19	ι̂j	ι̂j	PROPN
ejpam-3418	142	20			PROPN
ejpam-3418	142	21	(	(	PUNCT
ejpam-3418	142	22	a	a	X
ejpam-3418	142	23	)	)	PUNCT
ejpam-3418	142	24	=	=	NOUN
ejpam-3418	143	1	[	[	X
ejpam-3418	143	2	inf{[̂ιj(a)]−	inf{[̂ιj(a)]−	ADV
ejpam-3418	143	3	:	:	PUNCT
ejpam-3418	143	4	j	j	PROPN
ejpam-3418	143	5	∈	∈	PROPN
ejpam-3418	143	6	j	j	PROPN
ejpam-3418	143	7	}	}	PUNCT
ejpam-3418	143	8	,	,	PUNCT
ejpam-3418	143	9	inf{[̂ιj(a)]+	inf{[̂ιj(a)]+	NOUN
ejpam-3418	143	10	:	:	PUNCT
ejpam-3418	143	11	j	j	PROPN
ejpam-3418	143	12	∈	∈	PROPN
ejpam-3418	143	13	j	j	PROPN
ejpam-3418	143	14	}	}	PUNCT
ejpam-3418	143	15	]	]	PUNCT
ejpam-3418	143	16	,	,	PUNCT
ejpam-3418	143	17	for	for	ADP
ejpam-3418	143	18	all	all	DET
ejpam-3418	143	19	a	a	DET
ejpam-3418	143	20	∈	∈	PROPN
ejpam-3418	143	21	i(x	i(x	NOUN
ejpam-3418	143	22	)	)	PUNCT
ejpam-3418	143	23	,	,	PUNCT
ejpam-3418	143	24	respectively	respectively	ADV
ejpam-3418	143	25	.	.	PUNCT
ejpam-3418	144	1	one	one	PRON
ejpam-3418	144	2	can	can	AUX
ejpam-3418	144	3	easily	easily	ADV
ejpam-3418	144	4	check	check	VERB
ejpam-3418	144	5	that	that	SCONJ
ejpam-3418	144	6	the	the	DET
ejpam-3418	144	7	arbitrary	arbitrary	ADJ
ejpam-3418	144	8	union	union	NOUN
ejpam-3418	144	9	or	or	CCONJ
ejpam-3418	144	10	intersection	intersection	NOUN
ejpam-3418	144	11	of	of	ADP
ejpam-3418	144	12	ivfi	ivfi	NOUN
ejpam-3418	144	13	sets	set	NOUN
ejpam-3418	144	14	is	be	AUX
ejpam-3418	144	15	an	an	DET
ejpam-3418	144	16	ivfi	ivfi	NOUN
ejpam-3418	144	17	set	set	VERB
ejpam-3418	144	18	.	.	PUNCT
ejpam-3418	145	1	the	the	DET
ejpam-3418	145	2	following	follow	VERB
ejpam-3418	145	3	are	be	AUX
ejpam-3418	145	4	some	some	DET
ejpam-3418	145	5	properties	property	NOUN
ejpam-3418	145	6	of	of	ADP
ejpam-3418	145	7	the	the	DET
ejpam-3418	145	8	operations	operation	NOUN
ejpam-3418	145	9	on	on	ADP
ejpam-3418	145	10	ivfi	ivfi	NOUN
ejpam-3418	145	11	sets	set	NOUN
ejpam-3418	145	12	.	.	PUNCT
ejpam-3418	146	1	theorem	theorem	NOUN
ejpam-3418	146	2	1	1	NUM
ejpam-3418	146	3	.	.	PUNCT
ejpam-3418	147	1	let	let	VERB
ejpam-3418	147	2	x	x	PRON
ejpam-3418	147	3	be	be	AUX
ejpam-3418	147	4	a	a	DET
ejpam-3418	147	5	nonempty	nonempty	ADV
ejpam-3418	147	6	set	set	VERB
ejpam-3418	147	7	and	and	CCONJ
ejpam-3418	147	8	i(x	i(x	NOUN
ejpam-3418	147	9	)	)	PUNCT
ejpam-3418	147	10	be	be	AUX
ejpam-3418	147	11	an	an	DET
ejpam-3418	147	12	ideal	ideal	NOUN
ejpam-3418	147	13	on	on	ADP
ejpam-3418	147	14	x.	x.	NOUN
ejpam-3418	147	15	let	let	VERB
ejpam-3418	147	16	ι̂	ι̂	NOUN
ejpam-3418	147	17	,	,	PUNCT
ejpam-3418	147	18	τ̂	τ̂	PUNCT
ejpam-3418	147	19	,	,	PUNCT
ejpam-3418	147	20	η̂	η̂	NUM
ejpam-3418	147	21	∈	∈	PROPN
ejpam-3418	147	22	i	i	PRON
ejpam-3418	147	23	i(x	i(x	PROPN
ejpam-3418	147	24	)	)	PUNCT
ejpam-3418	147	25	.	.	PUNCT
ejpam-3418	148	1	then	then	ADV
ejpam-3418	148	2	,	,	PUNCT
ejpam-3418	148	3	i.	i.	PROPN
ejpam-3418	148	4	(	(	PUNCT
ejpam-3418	148	5	commutativity	commutativity	NOUN
ejpam-3418	148	6	):	):	PUNCT
ejpam-3418	148	7	ι̂	ι̂	X
ejpam-3418	148	8	∨	∨	X
ejpam-3418	148	9	τ̂	τ̂	X
ejpam-3418	148	10	=	=	SYM
ejpam-3418	148	11	τ̂	τ̂	X
ejpam-3418	148	12	∨	∨	NUM
ejpam-3418	148	13	ι̂	ι̂	NUM
ejpam-3418	148	14	and	and	CCONJ
ejpam-3418	148	15	ι̂	ι̂	ADJ
ejpam-3418	148	16	∧	∧	NOUN
ejpam-3418	148	17	τ̂	τ̂	PUNCT
ejpam-3418	148	18	=	=	SYM
ejpam-3418	148	19	τ̂	τ̂	PUNCT
ejpam-3418	148	20	∧	∧	PROPN
ejpam-3418	148	21	ι̂.	ι̂.	PROPN
ejpam-3418	148	22	ii	ii	PROPN
ejpam-3418	148	23	.	.	PUNCT
ejpam-3418	149	1	(	(	PUNCT
ejpam-3418	149	2	associativity	associativity	NOUN
ejpam-3418	149	3	):	):	PUNCT
ejpam-3418	149	4	(	(	PUNCT
ejpam-3418	149	5	ι̂	ι̂	X
ejpam-3418	149	6	∨	∨	NUM
ejpam-3418	149	7	τ̂	τ̂	NUM
ejpam-3418	149	8	)	)	PUNCT
ejpam-3418	149	9	∨	∨	NUM
ejpam-3418	149	10	η̂	η̂	PROPN
ejpam-3418	149	11	=	=	SYM
ejpam-3418	149	12	ι̂	ι̂	PUNCT
ejpam-3418	149	13	∨	∨	NUM
ejpam-3418	149	14	(	(	PUNCT
ejpam-3418	149	15	τ̂	τ̂	X
ejpam-3418	149	16	∨	∨	NUM
ejpam-3418	149	17	η̂	η̂	NUM
ejpam-3418	149	18	)	)	PUNCT
ejpam-3418	149	19	and	and	CCONJ
ejpam-3418	149	20	(	(	PUNCT
ejpam-3418	149	21	ι̂	ι̂	NUM
ejpam-3418	149	22	∧	∧	NOUN
ejpam-3418	149	23	τ̂	τ̂	NUM
ejpam-3418	149	24	)	)	PUNCT
ejpam-3418	149	25	∧	∧	NOUN
ejpam-3418	149	26	η̂	η̂	NUM
ejpam-3418	149	27	=	=	SYM
ejpam-3418	149	28	ι̂	ι̂	PUNCT
ejpam-3418	149	29	∧	∧	PROPN
ejpam-3418	149	30	(	(	PUNCT
ejpam-3418	149	31	τ̂	τ̂	PUNCT
ejpam-3418	149	32	∧	∧	NOUN
ejpam-3418	149	33	η̂	η̂	NUM
ejpam-3418	149	34	)	)	PUNCT
ejpam-3418	149	35	.	.	PUNCT
ejpam-3418	150	1	iii	iii	X
ejpam-3418	150	2	.	.	PUNCT
ejpam-3418	151	1	(	(	PUNCT
ejpam-3418	151	2	transitivity	transitivity	NOUN
ejpam-3418	151	3	):	):	PUNCT
ejpam-3418	151	4	if	if	SCONJ
ejpam-3418	151	5	ι̂	ι̂	NUM
ejpam-3418	151	6	6	6	NUM
ejpam-3418	151	7	τ̂	τ̂	PUNCT
ejpam-3418	151	8	and	and	CCONJ
ejpam-3418	151	9	τ̂	τ̂	ADP
ejpam-3418	151	10	6	6	NUM
ejpam-3418	151	11	η̂	η̂	NOUN
ejpam-3418	151	12	,	,	PUNCT
ejpam-3418	151	13	then	then	ADV
ejpam-3418	151	14	ι̂	ι̂	PUNCT
ejpam-3418	151	15	6	6	NUM
ejpam-3418	151	16	η̂.	η̂.	NOUN
ejpam-3418	151	17	iv	iv	NUM
ejpam-3418	151	18	.	.	PUNCT
ejpam-3418	152	1	(	(	PUNCT
ejpam-3418	152	2	distributivity	distributivity	NOUN
ejpam-3418	152	3	):	):	PUNCT
ejpam-3418	152	4	ι̂	ι̂	X
ejpam-3418	152	5	∨	∨	NOUN
ejpam-3418	152	6	(	(	PUNCT
ejpam-3418	152	7	τ̂	τ̂	PUNCT
ejpam-3418	152	8	∧	∧	NOUN
ejpam-3418	152	9	η̂	η̂	NUM
ejpam-3418	152	10	)	)	PUNCT
ejpam-3418	152	11	=	=	SYM
ejpam-3418	152	12	(	(	PUNCT
ejpam-3418	152	13	ι̂	ι̂	X
ejpam-3418	152	14	∨	∨	NUM
ejpam-3418	152	15	τ̂	τ̂	NUM
ejpam-3418	152	16	)	)	PUNCT
ejpam-3418	152	17	∧	∧	NOUN
ejpam-3418	152	18	(	(	PUNCT
ejpam-3418	152	19	ι̂	ι̂	X
ejpam-3418	152	20	∨	∨	NUM
ejpam-3418	152	21	η̂	η̂	NUM
ejpam-3418	152	22	)	)	PUNCT
ejpam-3418	152	23	and	and	CCONJ
ejpam-3418	152	24	ι̂	ι̂	NUM
ejpam-3418	152	25	∧	∧	PROPN
ejpam-3418	152	26	(	(	PUNCT
ejpam-3418	152	27	τ̂	τ̂	X
ejpam-3418	152	28	∨	∨	NUM
ejpam-3418	152	29	η̂	η̂	NUM
ejpam-3418	152	30	)	)	PUNCT
ejpam-3418	152	31	=	=	SYM
ejpam-3418	152	32	(	(	PUNCT
ejpam-3418	152	33	ι̂	ι̂	NUM
ejpam-3418	152	34	∧	∧	NOUN
ejpam-3418	152	35	τ̂	τ̂	NUM
ejpam-3418	152	36	)	)	PUNCT
ejpam-3418	152	37	∨	∨	NUM
ejpam-3418	152	38	(	(	PUNCT
ejpam-3418	152	39	ι̂	ι̂	X
ejpam-3418	152	40	∧	∧	NOUN
ejpam-3418	152	41	η̂	η̂	NUM
ejpam-3418	152	42	)	)	PUNCT
ejpam-3418	152	43	v.	v.	PROPN
ejpam-3418	152	44	(	(	PUNCT
ejpam-3418	152	45	de	de	PROPN
ejpam-3418	152	46	morgan	morgan	PROPN
ejpam-3418	152	47	’s	’s	PART
ejpam-3418	152	48	law	law	NOUN
ejpam-3418	152	49	):	):	PUNCT
ejpam-3418	152	50	(	(	PUNCT
ejpam-3418	152	51	ι̂	ι̂	X
ejpam-3418	152	52	∨	∨	X
ejpam-3418	152	53	τ̂)c	τ̂)c	X
ejpam-3418	152	54	=	=	SYM
ejpam-3418	152	55	ι̂c	ι̂c	X
ejpam-3418	152	56	∧	∧	NOUN
ejpam-3418	152	57	τ̂	τ̂	PUNCT
ejpam-3418	152	58	c	c	NOUN
ejpam-3418	152	59	and	and	CCONJ
ejpam-3418	152	60	(	(	PUNCT
ejpam-3418	152	61	ι̂	ι̂	NOUN
ejpam-3418	152	62	∧	∧	NOUN
ejpam-3418	152	63	τ̂)c	τ̂)c	NOUN
ejpam-3418	152	64	=	=	SYM
ejpam-3418	152	65	ι̂c	ι̂c	X
ejpam-3418	152	66	∨	∨	X
ejpam-3418	152	67	τ̂	τ̂	PUNCT
ejpam-3418	152	68	c	c	NOUN
ejpam-3418	152	69	proof	proof	NOUN
ejpam-3418	152	70	.	.	PUNCT
ejpam-3418	153	1	let	let	VERB
ejpam-3418	153	2	ι̂	ι̂	NOUN
ejpam-3418	153	3	,	,	PUNCT
ejpam-3418	153	4	τ̂	τ̂	PUNCT
ejpam-3418	153	5	,	,	PUNCT
ejpam-3418	153	6	η̂	η̂	NUM
ejpam-3418	153	7	∈	∈	PROPN
ejpam-3418	153	8	i	i	PRON
ejpam-3418	153	9	i(x	i(x	NOUN
ejpam-3418	153	10	)	)	PUNCT
ejpam-3418	153	11	.	.	PUNCT
ejpam-3418	154	1	properties	property	NOUN
ejpam-3418	154	2	(	(	PUNCT
ejpam-3418	154	3	i	i	NOUN
ejpam-3418	154	4	)	)	PUNCT
ejpam-3418	154	5	and	and	CCONJ
ejpam-3418	154	6	(	(	PUNCT
ejpam-3418	154	7	ii	ii	NOUN
ejpam-3418	154	8	)	)	PUNCT
ejpam-3418	154	9	follows	follow	VERB
ejpam-3418	154	10	from	from	ADP
ejpam-3418	154	11	the	the	DET
ejpam-3418	154	12	commutativity	commutativity	NOUN
ejpam-3418	154	13	and	and	CCONJ
ejpam-3418	154	14	associativity	associativity	NOUN
ejpam-3418	154	15	of	of	ADP
ejpam-3418	154	16	the	the	DET
ejpam-3418	154	17	maximum	maximum	ADJ
ejpam-3418	154	18	and	and	CCONJ
ejpam-3418	154	19	minimum	minimum	ADJ
ejpam-3418	154	20	operations	operation	NOUN
ejpam-3418	154	21	.	.	PUNCT
ejpam-3418	155	1	suppose	suppose	VERB
ejpam-3418	155	2	that	that	SCONJ
ejpam-3418	155	3	ι̂	ι̂	PRON
ejpam-3418	155	4	6	6	NUM
ejpam-3418	155	5	τ̂	τ̂	PUNCT
ejpam-3418	155	6	and	and	CCONJ
ejpam-3418	155	7	τ̂	τ̂	ADP
ejpam-3418	155	8	6	6	NUM
ejpam-3418	155	9	η̂.	η̂.	NOUN
ejpam-3418	155	10	then	then	ADV
ejpam-3418	155	11	for	for	ADP
ejpam-3418	155	12	all	all	DET
ejpam-3418	155	13	a	a	DET
ejpam-3418	155	14	∈	∈	PROPN
ejpam-3418	155	15	i(x	i(x	NOUN
ejpam-3418	155	16	)	)	PUNCT
ejpam-3418	155	17	,	,	PUNCT
ejpam-3418	155	18	ι̂(a	ι̂(a	PUNCT
ejpam-3418	155	19	)	)	PUNCT
ejpam-3418	155	20	≤i	≤i	NOUN
ejpam-3418	155	21	τ̂(a	τ̂(a	NUM
ejpam-3418	155	22	)	)	PUNCT
ejpam-3418	155	23	and	and	CCONJ
ejpam-3418	155	24	τ̂(a	τ̂(a	NUM
ejpam-3418	155	25	)	)	PUNCT
ejpam-3418	155	26	≤i	≤i	NOUN
ejpam-3418	155	27	η̂(a	η̂(a	NOUN
ejpam-3418	155	28	)	)	PUNCT
ejpam-3418	155	29	,	,	PUNCT
ejpam-3418	155	30	which	which	PRON
ejpam-3418	155	31	implies	imply	VERB
ejpam-3418	155	32	that	that	SCONJ
ejpam-3418	155	33	ι̂(a	ι̂(a	PUNCT
ejpam-3418	155	34	)	)	PUNCT
ejpam-3418	155	35	≤i	≤i	PROPN
ejpam-3418	155	36	η̂(a	η̂(a	NOUN
ejpam-3418	155	37	)	)	PUNCT
ejpam-3418	155	38	,	,	PUNCT
ejpam-3418	155	39	for	for	ADP
ejpam-3418	155	40	all	all	DET
ejpam-3418	155	41	a	a	DET
ejpam-3418	155	42	∈	∈	PROPN
ejpam-3418	155	43	i(x	i(x	NOUN
ejpam-3418	155	44	)	)	PUNCT
ejpam-3418	155	45	.	.	PUNCT
ejpam-3418	156	1	that	that	PRON
ejpam-3418	156	2	is	be	AUX
ejpam-3418	156	3	,	,	PUNCT
ejpam-3418	156	4	ι̂	ι̂	X
ejpam-3418	156	5	6	6	NUM
ejpam-3418	156	6	η̂	η̂	ADJ
ejpam-3418	156	7	,	,	PUNCT
ejpam-3418	156	8	easily	easily	ADV
ejpam-3418	156	9	proving	prove	VERB
ejpam-3418	156	10	(	(	PUNCT
ejpam-3418	156	11	iii	iii	NOUN
ejpam-3418	156	12	)	)	PUNCT
ejpam-3418	156	13	.	.	PUNCT
ejpam-3418	157	1	to	to	PART
ejpam-3418	157	2	prove	prove	VERB
ejpam-3418	157	3	(	(	PUNCT
ejpam-3418	157	4	iv	iv	NUM
ejpam-3418	157	5	)	)	PUNCT
ejpam-3418	157	6	,	,	PUNCT
ejpam-3418	157	7	let	let	VERB
ejpam-3418	157	8	a	a	DET
ejpam-3418	157	9	∈	∈	PROPN
ejpam-3418	157	10	i(x	i(x	NOUN
ejpam-3418	157	11	)	)	PUNCT
ejpam-3418	157	12	and	and	CCONJ
ejpam-3418	157	13	consider	consider	VERB
ejpam-3418	157	14	that	that	PRON
ejpam-3418	157	15	(	(	PUNCT
ejpam-3418	157	16	ι̂	ι̂	X
ejpam-3418	157	17	∨	∨	NUM
ejpam-3418	157	18	(	(	PUNCT
ejpam-3418	157	19	τ̂	τ̂	PUNCT
ejpam-3418	157	20	∧	∧	PROPN
ejpam-3418	157	21	η̂))(a	η̂))(a	NOUN
ejpam-3418	157	22	)	)	PUNCT
ejpam-3418	158	1	=	=	PUNCT
ejpam-3418	159	1	[	[	X
ejpam-3418	159	2	max{[̂ι(a)]−	max{[̂ι(a)]−	X
ejpam-3418	159	3	,	,	PUNCT
ejpam-3418	159	4	[	[	X
ejpam-3418	159	5	(	(	PUNCT
ejpam-3418	159	6	τ̂	τ̂	PUNCT
ejpam-3418	159	7	∧	∧	PROPN
ejpam-3418	159	8	η̂)(a)]−},max{[̂ι(a)]+	η̂)(a)]−},max{[̂ι(a)]+	PROPN
ejpam-3418	159	9	,	,	PUNCT
ejpam-3418	159	10	[	[	X
ejpam-3418	159	11	(	(	PUNCT
ejpam-3418	159	12	τ̂	τ̂	PUNCT
ejpam-3418	159	13	∧	∧	PROPN
ejpam-3418	159	14	η̂)(a)]+	η̂)(a)]+	NOUN
ejpam-3418	159	15	}	}	PUNCT
ejpam-3418	159	16	]	]	PUNCT
ejpam-3418	159	17	=	=	PUNCT
ejpam-3418	160	1	[	[	X
ejpam-3418	160	2	max{[̂ι(a)]−,min{[τ̂(a)]−	max{[̂ι(a)]−,min{[τ̂(a)]−	X
ejpam-3418	160	3	,	,	PUNCT
ejpam-3418	160	4	[	[	X
ejpam-3418	160	5	η̂(a)]−}},max{[̂ι(a)]+,min{[τ̂(a)]+	η̂(a)]−}},max{[̂ι(a)]+,min{[τ̂(a)]+	PROPN
ejpam-3418	160	6	,	,	PUNCT
ejpam-3418	160	7	[	[	X
ejpam-3418	160	8	η̂(a)]+	η̂(a)]+	X
ejpam-3418	160	9	}	}	PUNCT
ejpam-3418	160	10	}	}	PUNCT
ejpam-3418	160	11	]	]	PUNCT
ejpam-3418	160	12	.	.	PUNCT
ejpam-3418	161	1	mj	mj	PROPN
ejpam-3418	161	2	togonon	togonon	PROPN
ejpam-3418	161	3	,	,	PUNCT
ejpam-3418	161	4	r	r	NOUN
ejpam-3418	161	5	caga	caga	NOUN
ejpam-3418	161	6	-	-	PUNCT
ejpam-3418	161	7	anan	anan	PROPN
ejpam-3418	161	8	/	/	SYM
ejpam-3418	161	9	eur	eur	PROPN
ejpam-3418	161	10	.	.	PUNCT
ejpam-3418	162	1	j.	j.	PROPN
ejpam-3418	162	2	pure	pure	PROPN
ejpam-3418	162	3	appl	appl	PROPN
ejpam-3418	162	4	.	.	PROPN
ejpam-3418	162	5	math	math	PROPN
ejpam-3418	162	6	,	,	PUNCT
ejpam-3418	162	7	12	12	NUM
ejpam-3418	162	8	(	(	PUNCT
ejpam-3418	162	9	2	2	NUM
ejpam-3418	162	10	)	)	PUNCT
ejpam-3418	162	11	(	(	PUNCT
ejpam-3418	162	12	2019	2019	NUM
ejpam-3418	162	13	)	)	PUNCT
ejpam-3418	162	14	,	,	PUNCT
ejpam-3418	162	15	553	553	NUM
ejpam-3418	162	16	-	-	SYM
ejpam-3418	162	17	570	570	NUM
ejpam-3418	162	18	558	558	NUM
ejpam-3418	162	19	we	we	PRON
ejpam-3418	162	20	consider	consider	VERB
ejpam-3418	162	21	two	two	NUM
ejpam-3418	162	22	cases	case	NOUN
ejpam-3418	162	23	.	.	PUNCT
ejpam-3418	163	1	first	first	ADV
ejpam-3418	163	2	,	,	PUNCT
ejpam-3418	163	3	if	if	SCONJ
ejpam-3418	163	4	τ̂(a	τ̂(a	NOUN
ejpam-3418	163	5	)	)	PUNCT
ejpam-3418	163	6	≤i	≤i	NOUN
ejpam-3418	163	7	η̂(a	η̂(a	NOUN
ejpam-3418	163	8	)	)	PUNCT
ejpam-3418	163	9	,	,	PUNCT
ejpam-3418	163	10	then	then	ADV
ejpam-3418	163	11	(	(	PUNCT
ejpam-3418	163	12	ι̂	ι̂	X
ejpam-3418	163	13	∨	∨	X
ejpam-3418	163	14	(	(	PUNCT
ejpam-3418	163	15	τ̂	τ̂	PUNCT
ejpam-3418	163	16	∧	∧	PROPN
ejpam-3418	163	17	η̂))(a	η̂))(a	NOUN
ejpam-3418	163	18	)	)	PUNCT
ejpam-3418	163	19	=	=	PUNCT
ejpam-3418	164	1	[	[	X
ejpam-3418	164	2	max{[̂ι(a)]−	max{[̂ι(a)]−	X
ejpam-3418	164	3	,	,	PUNCT
ejpam-3418	164	4	[	[	X
ejpam-3418	164	5	τ̂(a)]−},max{[̂ι(a)]+	τ̂(a)]−},max{[̂ι(a)]+	X
ejpam-3418	164	6	,	,	PUNCT
ejpam-3418	164	7	[	[	X
ejpam-3418	164	8	τ̂(a)]+	τ̂(a)]+	X
ejpam-3418	164	9	}	}	PUNCT
ejpam-3418	164	10	]	]	PUNCT
ejpam-3418	164	11	.	.	PUNCT
ejpam-3418	165	1	second	second	ADJ
ejpam-3418	165	2	,	,	PUNCT
ejpam-3418	165	3	if	if	SCONJ
ejpam-3418	165	4	η̂(a	η̂(a	NOUN
ejpam-3418	165	5	)	)	PUNCT
ejpam-3418	165	6	≤i	≤i	NOUN
ejpam-3418	165	7	τ̂(a	τ̂(a	NUM
ejpam-3418	165	8	)	)	PUNCT
ejpam-3418	165	9	,	,	PUNCT
ejpam-3418	165	10	then	then	ADV
ejpam-3418	165	11	(	(	PUNCT
ejpam-3418	165	12	ι̂	ι̂	X
ejpam-3418	165	13	∨	∨	X
ejpam-3418	165	14	(	(	PUNCT
ejpam-3418	165	15	τ̂	τ̂	PUNCT
ejpam-3418	165	16	∧	∧	PROPN
ejpam-3418	165	17	η̂))(a	η̂))(a	NOUN
ejpam-3418	165	18	)	)	PUNCT
ejpam-3418	165	19	=	=	PUNCT
ejpam-3418	166	1	[	[	X
ejpam-3418	166	2	max{[̂ι(a)]−	max{[̂ι(a)]−	X
ejpam-3418	166	3	,	,	PUNCT
ejpam-3418	166	4	[	[	X
ejpam-3418	166	5	η̂(a)]−},max{[̂ι(a)]+	η̂(a)]−},max{[̂ι(a)]+	PROPN
ejpam-3418	166	6	,	,	PUNCT
ejpam-3418	166	7	[	[	X
ejpam-3418	166	8	η̂(a)]+	η̂(a)]+	X
ejpam-3418	166	9	}	}	PUNCT
ejpam-3418	166	10	]	]	PUNCT
ejpam-3418	166	11	.	.	PUNCT
ejpam-3418	167	1	summarizing	summarize	VERB
ejpam-3418	167	2	the	the	DET
ejpam-3418	167	3	two	two	NUM
ejpam-3418	167	4	cases	case	NOUN
ejpam-3418	167	5	,	,	PUNCT
ejpam-3418	167	6	we	we	PRON
ejpam-3418	167	7	have	have	VERB
ejpam-3418	167	8	(	(	PUNCT
ejpam-3418	167	9	ι̂	ι̂	X
ejpam-3418	167	10	∨	∨	X
ejpam-3418	167	11	(	(	PUNCT
ejpam-3418	167	12	τ̂	τ̂	PUNCT
ejpam-3418	167	13	∧	∧	PROPN
ejpam-3418	167	14	η̂))(a	η̂))(a	NOUN
ejpam-3418	167	15	)	)	PUNCT
ejpam-3418	167	16	=	=	PUNCT
ejpam-3418	168	1	[	[	X
ejpam-3418	168	2	min{max{[̂ι(a)]−	min{max{[̂ι(a)]−	X
ejpam-3418	168	3	,	,	PUNCT
ejpam-3418	168	4	[	[	X
ejpam-3418	168	5	τ̂(a)]−},max{[̂ι(a)]−	τ̂(a)]−},max{[̂ι(a)]−	ADJ
ejpam-3418	168	6	,	,	PUNCT
ejpam-3418	168	7	[	[	X
ejpam-3418	168	8	η̂(a)]−	η̂(a)]−	X
ejpam-3418	168	9	}	}	PUNCT
ejpam-3418	168	10	}	}	PUNCT
ejpam-3418	168	11	,	,	PUNCT
ejpam-3418	168	12	min{max{[̂ι(a)]+	min{max{[̂ι(a)]+	PROPN
ejpam-3418	168	13	,	,	PUNCT
ejpam-3418	168	14	[	[	X
ejpam-3418	168	15	τ̂(a)]+},max{[̂ι(a)]+	τ̂(a)]+},max{[̂ι(a)]+	X
ejpam-3418	168	16	,	,	PUNCT
ejpam-3418	168	17	[	[	X
ejpam-3418	168	18	η̂(a)]+	η̂(a)]+	X
ejpam-3418	168	19	}	}	PUNCT
ejpam-3418	168	20	}	}	PUNCT
ejpam-3418	168	21	]	]	PUNCT
ejpam-3418	169	1	=	=	SYM
ejpam-3418	169	2	(	(	PUNCT
ejpam-3418	169	3	(	(	PUNCT
ejpam-3418	169	4	ι̂	ι̂	X
ejpam-3418	169	5	∨	∨	NUM
ejpam-3418	169	6	τ̂	τ̂	NUM
ejpam-3418	169	7	)	)	PUNCT
ejpam-3418	169	8	∧	∧	NOUN
ejpam-3418	169	9	(	(	PUNCT
ejpam-3418	169	10	ι̂	ι̂	NOUN
ejpam-3418	169	11	∨	∨	NUM
ejpam-3418	169	12	η̂))(a	η̂))(a	NOUN
ejpam-3418	169	13	)	)	PUNCT
ejpam-3418	169	14	.	.	PUNCT
ejpam-3418	170	1	thus	thus	ADV
ejpam-3418	170	2	,	,	PUNCT
ejpam-3418	170	3	ι̂∨	ι̂∨	INTJ
ejpam-3418	170	4	(	(	PUNCT
ejpam-3418	170	5	τ̂	τ̂	ADP
ejpam-3418	170	6	∧	∧	NOUN
ejpam-3418	170	7	η̂	η̂	NUM
ejpam-3418	170	8	)	)	PUNCT
ejpam-3418	170	9	=	=	SYM
ejpam-3418	170	10	(	(	PUNCT
ejpam-3418	170	11	ι̂∨	ι̂∨	PROPN
ejpam-3418	170	12	τ̂)∧	τ̂)∧	PROPN
ejpam-3418	170	13	(	(	PUNCT
ejpam-3418	170	14	ι̂∨	ι̂∨	ADJ
ejpam-3418	170	15	η̂	η̂	NUM
ejpam-3418	170	16	)	)	PUNCT
ejpam-3418	170	17	.	.	PUNCT
ejpam-3418	171	1	similarly	similarly	ADV
ejpam-3418	171	2	,	,	PUNCT
ejpam-3418	171	3	we	we	PRON
ejpam-3418	171	4	can	can	AUX
ejpam-3418	171	5	show	show	VERB
ejpam-3418	171	6	that	that	SCONJ
ejpam-3418	171	7	ι̂∧	ι̂∧	PROPN
ejpam-3418	171	8	(	(	PUNCT
ejpam-3418	171	9	τ̂	τ̂	X
ejpam-3418	171	10	∨	∨	NUM
ejpam-3418	171	11	η̂	η̂	NUM
ejpam-3418	171	12	)	)	PUNCT
ejpam-3418	171	13	=	=	SYM
ejpam-3418	171	14	(	(	PUNCT
ejpam-3418	171	15	ι̂∧	ι̂∧	PROPN
ejpam-3418	171	16	τ̂)∨	τ̂)∨	NOUN
ejpam-3418	171	17	(	(	PUNCT
ejpam-3418	171	18	ι̂∧	ι̂∧	PROPN
ejpam-3418	171	19	η̂	η̂	NUM
ejpam-3418	171	20	)	)	PUNCT
ejpam-3418	171	21	.	.	PUNCT
ejpam-3418	172	1	to	to	PART
ejpam-3418	172	2	prove	prove	VERB
ejpam-3418	172	3	(	(	PUNCT
ejpam-3418	172	4	v	v	NOUN
ejpam-3418	172	5	)	)	PUNCT
ejpam-3418	172	6	,	,	PUNCT
ejpam-3418	172	7	let	let	VERB
ejpam-3418	172	8	∅	∅	NOUN
ejpam-3418	172	9	6=	6=	ADP
ejpam-3418	172	10	a	a	DET
ejpam-3418	172	11	∈	∈	PROPN
ejpam-3418	172	12	i(x	i(x	NOUN
ejpam-3418	172	13	)	)	PUNCT
ejpam-3418	172	14	and	and	CCONJ
ejpam-3418	172	15	note	note	VERB
ejpam-3418	172	16	that	that	SCONJ
ejpam-3418	172	17	(	(	PUNCT
ejpam-3418	172	18	ι̂	ι̂	X
ejpam-3418	172	19	∨	∨	NOUN
ejpam-3418	172	20	τ̂)c(a	τ̂)c(a	NUM
ejpam-3418	172	21	)	)	PUNCT
ejpam-3418	172	22	=	=	PUNCT
ejpam-3418	173	1	[	[	PUNCT
ejpam-3418	173	2	inf	inf	ADJ
ejpam-3418	173	3	x∈a	x∈a	NOUN
ejpam-3418	173	4	{	{	PUNCT
ejpam-3418	173	5	1−	1−	NUM
ejpam-3418	174	1	[	[	X
ejpam-3418	174	2	(	(	PUNCT
ejpam-3418	174	3	ι̂	ι̂	X
ejpam-3418	174	4	∨	∨	NOUN
ejpam-3418	174	5	τ̂)({x})]+	τ̂)({x})]+	NUM
ejpam-3418	174	6	}	}	PUNCT
ejpam-3418	174	7	,	,	PUNCT
ejpam-3418	174	8	inf	inf	PROPN
ejpam-3418	174	9	x∈a	x∈a	NOUN
ejpam-3418	174	10	{	{	PUNCT
ejpam-3418	174	11	1−	1−	NUM
ejpam-3418	174	12	[	[	X
ejpam-3418	174	13	(	(	PUNCT
ejpam-3418	174	14	ι̂	ι̂	X
ejpam-3418	174	15	∨	∨	NOUN
ejpam-3418	174	16	τ̂)({x})]−	τ̂)({x})]−	PROPN
ejpam-3418	174	17	}	}	PUNCT
ejpam-3418	174	18	]	]	PUNCT
ejpam-3418	174	19	=	=	PUNCT
ejpam-3418	174	20	[	[	PUNCT
ejpam-3418	174	21	inf	inf	ADJ
ejpam-3418	174	22	x∈a	x∈a	PROPN
ejpam-3418	174	23	{	{	PUNCT
ejpam-3418	174	24	1−max{[̂ι({x})]+	1−max{[̂ι({x})]+	NUM
ejpam-3418	174	25	,	,	PUNCT
ejpam-3418	174	26	[	[	X
ejpam-3418	174	27	τ̂({x})]+	τ̂({x})]+	X
ejpam-3418	174	28	}	}	PUNCT
ejpam-3418	174	29	}	}	PUNCT
ejpam-3418	174	30	,	,	PUNCT
ejpam-3418	174	31	inf	inf	PROPN
ejpam-3418	174	32	x∈a	x∈a	PROPN
ejpam-3418	174	33	{	{	PUNCT
ejpam-3418	174	34	1−max{[̂ι({x})]−	1−max{[̂ι({x})]−	NUM
ejpam-3418	174	35	,	,	PUNCT
ejpam-3418	174	36	[	[	X
ejpam-3418	174	37	τ̂({x})]−	τ̂({x})]−	PUNCT
ejpam-3418	174	38	}	}	PUNCT
ejpam-3418	174	39	}	}	PUNCT
ejpam-3418	174	40	]	]	PUNCT
ejpam-3418	174	41	.	.	PUNCT
ejpam-3418	175	1	we	we	PRON
ejpam-3418	175	2	also	also	ADV
ejpam-3418	175	3	consider	consider	VERB
ejpam-3418	175	4	two	two	NUM
ejpam-3418	175	5	cases	case	NOUN
ejpam-3418	175	6	.	.	PUNCT
ejpam-3418	176	1	first	first	ADV
ejpam-3418	176	2	,	,	PUNCT
ejpam-3418	176	3	if	if	SCONJ
ejpam-3418	176	4	τ̂({x	τ̂({x	PRON
ejpam-3418	176	5	}	}	PUNCT
ejpam-3418	176	6	)	)	PUNCT
ejpam-3418	176	7	≤i	≤i	PROPN
ejpam-3418	176	8	ι̂({x	ι̂({x	PROPN
ejpam-3418	176	9	}	}	PUNCT
ejpam-3418	176	10	)	)	PUNCT
ejpam-3418	176	11	,	,	PUNCT
ejpam-3418	176	12	then	then	ADV
ejpam-3418	176	13	(	(	PUNCT
ejpam-3418	176	14	ι̂	ι̂	NOUN
ejpam-3418	176	15	∨	∨	NOUN
ejpam-3418	176	16	τ̂)c(a	τ̂)c(a	NUM
ejpam-3418	176	17	)	)	PUNCT
ejpam-3418	176	18	=	=	PUNCT
ejpam-3418	176	19	[	[	PUNCT
ejpam-3418	176	20	inf	inf	ADJ
ejpam-3418	176	21	x∈a	x∈a	NOUN
ejpam-3418	176	22	{	{	PUNCT
ejpam-3418	176	23	1−	1−	NUM
ejpam-3418	177	1	[	[	X
ejpam-3418	177	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	177	3	}	}	PUNCT
ejpam-3418	177	4	,	,	PUNCT
ejpam-3418	177	5	inf	inf	PROPN
ejpam-3418	177	6	x∈a	x∈a	NOUN
ejpam-3418	177	7	{	{	PUNCT
ejpam-3418	177	8	1−	1−	NUM
ejpam-3418	178	1	[	[	X
ejpam-3418	178	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	178	3	}	}	PUNCT
ejpam-3418	178	4	]	]	PUNCT
ejpam-3418	178	5	.	.	PUNCT
ejpam-3418	179	1	second	second	ADJ
ejpam-3418	179	2	,	,	PUNCT
ejpam-3418	179	3	if	if	SCONJ
ejpam-3418	179	4	ι̂({x	ι̂({x	PRON
ejpam-3418	179	5	}	}	PUNCT
ejpam-3418	179	6	)	)	PUNCT
ejpam-3418	179	7	≤i	≤i	PROPN
ejpam-3418	179	8	τ̂({x	τ̂({x	NOUN
ejpam-3418	179	9	}	}	PUNCT
ejpam-3418	179	10	)	)	PUNCT
ejpam-3418	179	11	,	,	PUNCT
ejpam-3418	179	12	then	then	ADV
ejpam-3418	179	13	(	(	PUNCT
ejpam-3418	179	14	ι̂	ι̂	NOUN
ejpam-3418	179	15	∨	∨	NOUN
ejpam-3418	179	16	τ̂)c(a	τ̂)c(a	NUM
ejpam-3418	179	17	)	)	PUNCT
ejpam-3418	179	18	=	=	PUNCT
ejpam-3418	180	1	[	[	PUNCT
ejpam-3418	180	2	inf	inf	ADJ
ejpam-3418	180	3	x∈a	x∈a	NOUN
ejpam-3418	180	4	{	{	PUNCT
ejpam-3418	180	5	1−	1−	NUM
ejpam-3418	180	6	[	[	X
ejpam-3418	180	7	τ̂({x})]+	τ̂({x})]+	PROPN
ejpam-3418	180	8	}	}	PUNCT
ejpam-3418	180	9	,	,	PUNCT
ejpam-3418	180	10	inf	inf	PROPN
ejpam-3418	180	11	x∈a	x∈a	NOUN
ejpam-3418	180	12	{	{	PUNCT
ejpam-3418	180	13	1−	1−	NUM
ejpam-3418	180	14	[	[	X
ejpam-3418	180	15	τ̂({x})]−	τ̂({x})]−	PUNCT
ejpam-3418	180	16	}	}	PUNCT
ejpam-3418	180	17	]	]	PUNCT
ejpam-3418	180	18	.	.	PUNCT
ejpam-3418	181	1	combining	combine	VERB
ejpam-3418	181	2	the	the	DET
ejpam-3418	181	3	two	two	NUM
ejpam-3418	181	4	cases	case	NOUN
ejpam-3418	181	5	,	,	PUNCT
ejpam-3418	181	6	we	we	PRON
ejpam-3418	181	7	have	have	VERB
ejpam-3418	181	8	(	(	PUNCT
ejpam-3418	181	9	ι̂	ι̂	X
ejpam-3418	181	10	∨	∨	NOUN
ejpam-3418	181	11	τ̂)c(a	τ̂)c(a	NUM
ejpam-3418	181	12	)	)	PUNCT
ejpam-3418	181	13	=	=	PRON
ejpam-3418	181	14	[	[	PUNCT
ejpam-3418	181	15	min	min	NOUN
ejpam-3418	181	16	{	{	PUNCT
ejpam-3418	181	17	inf	inf	PROPN
ejpam-3418	181	18	x∈a	x∈a	PROPN
ejpam-3418	181	19	{	{	PUNCT
ejpam-3418	181	20	1−	1−	NUM
ejpam-3418	181	21	[	[	X
ejpam-3418	181	22	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	181	23	}	}	PUNCT
ejpam-3418	181	24	,	,	PUNCT
ejpam-3418	181	25	inf	inf	PROPN
ejpam-3418	181	26	x∈a	x∈a	NOUN
ejpam-3418	181	27	{	{	PUNCT
ejpam-3418	181	28	1−	1−	NUM
ejpam-3418	182	1	[	[	X
ejpam-3418	182	2	τ̂({x})]+	τ̂({x})]+	PROPN
ejpam-3418	182	3	}	}	PUNCT
ejpam-3418	182	4	}	}	PUNCT
ejpam-3418	182	5	,	,	PUNCT
ejpam-3418	182	6	min	min	PROPN
ejpam-3418	182	7	{	{	PUNCT
ejpam-3418	182	8	inf	inf	PROPN
ejpam-3418	182	9	x∈a	x∈a	PROPN
ejpam-3418	182	10	{	{	PUNCT
ejpam-3418	182	11	1−	1−	NUM
ejpam-3418	183	1	[	[	X
ejpam-3418	183	2	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	183	3	}	}	PUNCT
ejpam-3418	183	4	,	,	PUNCT
ejpam-3418	183	5	inf	inf	PROPN
ejpam-3418	183	6	x∈a	x∈a	NOUN
ejpam-3418	183	7	{	{	PUNCT
ejpam-3418	183	8	1−	1−	NUM
ejpam-3418	183	9	[	[	X
ejpam-3418	183	10	τ̂({x})]−	τ̂({x})]−	PUNCT
ejpam-3418	183	11	}	}	PUNCT
ejpam-3418	183	12	}	}	PUNCT
ejpam-3418	183	13	]	]	PUNCT
ejpam-3418	183	14	=	=	SYM
ejpam-3418	183	15	(	(	PUNCT
ejpam-3418	183	16	ι̂c	ι̂c	X
ejpam-3418	183	17	∧	∧	NOUN
ejpam-3418	183	18	τ̂	τ̂	PUNCT
ejpam-3418	183	19	c)(a	c)(a	NOUN
ejpam-3418	183	20	)	)	PUNCT
ejpam-3418	183	21	.	.	PUNCT
ejpam-3418	184	1	hence	hence	ADV
ejpam-3418	184	2	,	,	PUNCT
ejpam-3418	184	3	(	(	PUNCT
ejpam-3418	184	4	ι̂∨	ι̂∨	INTJ
ejpam-3418	184	5	τ̂)c	τ̂)c	PROPN
ejpam-3418	184	6	=	=	SYM
ejpam-3418	184	7	ι̂c∧	ι̂c∧	X
ejpam-3418	184	8	τ̂	τ̂	PUNCT
ejpam-3418	184	9	c.	c.	NOUN
ejpam-3418	184	10	using	use	VERB
ejpam-3418	184	11	the	the	DET
ejpam-3418	184	12	same	same	ADJ
ejpam-3418	184	13	argument	argument	NOUN
ejpam-3418	184	14	,	,	PUNCT
ejpam-3418	184	15	we	we	PRON
ejpam-3418	184	16	can	can	AUX
ejpam-3418	184	17	also	also	ADV
ejpam-3418	184	18	show	show	VERB
ejpam-3418	184	19	that	that	SCONJ
ejpam-3418	184	20	(	(	PUNCT
ejpam-3418	184	21	ι̂∧	ι̂∧	PROPN
ejpam-3418	184	22	τ̂)c	τ̂)c	PROPN
ejpam-3418	184	23	=	=	SYM
ejpam-3418	184	24	ι̂c∨	ι̂c∨	NOUN
ejpam-3418	184	25	τ̂	τ̂	PUNCT
ejpam-3418	185	1	c	c	X
ejpam-3418	185	2	,	,	PUNCT
ejpam-3418	185	3	and	and	CCONJ
ejpam-3418	185	4	our	our	PRON
ejpam-3418	185	5	proof	proof	NOUN
ejpam-3418	185	6	is	be	AUX
ejpam-3418	185	7	complete	complete	ADJ
ejpam-3418	185	8	.	.	PUNCT
ejpam-3418	186	1	the	the	DET
ejpam-3418	186	2	next	next	ADJ
ejpam-3418	186	3	theorem	theorem	NOUN
ejpam-3418	186	4	states	state	VERB
ejpam-3418	186	5	some	some	DET
ejpam-3418	186	6	interesting	interesting	ADJ
ejpam-3418	186	7	properties	property	NOUN
ejpam-3418	186	8	of	of	ADP
ejpam-3418	186	9	the	the	DET
ejpam-3418	186	10	complement	complement	NOUN
ejpam-3418	186	11	of	of	ADP
ejpam-3418	186	12	ivfi	ivfi	NOUN
ejpam-3418	186	13	sets	set	NOUN
ejpam-3418	186	14	.	.	PUNCT
ejpam-3418	187	1	theorem	theorem	NOUN
ejpam-3418	187	2	2	2	NUM
ejpam-3418	187	3	.	.	PUNCT
ejpam-3418	188	1	let	let	VERB
ejpam-3418	188	2	x	x	PRON
ejpam-3418	188	3	be	be	AUX
ejpam-3418	188	4	a	a	DET
ejpam-3418	188	5	nonempty	nonempty	ADV
ejpam-3418	188	6	set	set	VERB
ejpam-3418	188	7	and	and	CCONJ
ejpam-3418	188	8	i(x	i(x	NOUN
ejpam-3418	188	9	)	)	PUNCT
ejpam-3418	188	10	be	be	AUX
ejpam-3418	188	11	an	an	DET
ejpam-3418	188	12	ideal	ideal	NOUN
ejpam-3418	188	13	on	on	ADP
ejpam-3418	188	14	x.	x.	NOUN
ejpam-3418	188	15	let	let	VERB
ejpam-3418	188	16	ι̂	ι̂	NOUN
ejpam-3418	188	17	,	,	PUNCT
ejpam-3418	188	18	τ̂	τ̂	PUNCT
ejpam-3418	188	19	∈	∈	PROPN
ejpam-3418	188	20	i	i	PRON
ejpam-3418	188	21	i(x	i(x	PROPN
ejpam-3418	188	22	)	)	PUNCT
ejpam-3418	188	23	.	.	PUNCT
ejpam-3418	189	1	then	then	ADV
ejpam-3418	189	2	,	,	PUNCT
ejpam-3418	189	3	i.	i.	PROPN
ejpam-3418	189	4	ι̂	ι̂	PUNCT
ejpam-3418	189	5	6	6	NUM
ejpam-3418	189	6	(	(	PUNCT
ejpam-3418	189	7	ι̂c)c	ι̂c)c	X
ejpam-3418	189	8	;	;	PUNCT
ejpam-3418	189	9	mj	mj	PROPN
ejpam-3418	189	10	togonon	togonon	PROPN
ejpam-3418	189	11	,	,	PUNCT
ejpam-3418	189	12	r	r	NOUN
ejpam-3418	189	13	caga	caga	NOUN
ejpam-3418	189	14	-	-	PUNCT
ejpam-3418	189	15	anan	anan	PROPN
ejpam-3418	189	16	/	/	SYM
ejpam-3418	189	17	eur	eur	PROPN
ejpam-3418	189	18	.	.	PUNCT
ejpam-3418	190	1	j.	j.	PROPN
ejpam-3418	190	2	pure	pure	PROPN
ejpam-3418	190	3	appl	appl	PROPN
ejpam-3418	190	4	.	.	PROPN
ejpam-3418	190	5	math	math	PROPN
ejpam-3418	190	6	,	,	PUNCT
ejpam-3418	190	7	12	12	NUM
ejpam-3418	190	8	(	(	PUNCT
ejpam-3418	190	9	2	2	NUM
ejpam-3418	190	10	)	)	PUNCT
ejpam-3418	190	11	(	(	PUNCT
ejpam-3418	190	12	2019	2019	NUM
ejpam-3418	190	13	)	)	PUNCT
ejpam-3418	190	14	,	,	PUNCT
ejpam-3418	190	15	553	553	NUM
ejpam-3418	190	16	-	-	SYM
ejpam-3418	190	17	570	570	NUM
ejpam-3418	190	18	559	559	NUM
ejpam-3418	190	19	ii	ii	NOUN
ejpam-3418	190	20	.	.	PUNCT
ejpam-3418	190	21	ι̂	ι̂	PUNCT
ejpam-3418	191	1	=	=	PUNCT
ejpam-3418	191	2	(	(	PUNCT
ejpam-3418	191	3	ι̂c)c	ι̂c)c	ADJ
ejpam-3418	191	4	if	if	SCONJ
ejpam-3418	191	5	and	and	CCONJ
ejpam-3418	191	6	only	only	ADV
ejpam-3418	191	7	if	if	SCONJ
ejpam-3418	191	8	ι̂	ι̂	PRON
ejpam-3418	191	9	is	be	AUX
ejpam-3418	191	10	a	a	DET
ejpam-3418	191	11	guaranteed	guarantee	VERB
ejpam-3418	191	12	possibility	possibility	NOUN
ejpam-3418	191	13	ivfi	ivfi	NOUN
ejpam-3418	191	14	set	set	VERB
ejpam-3418	191	15	;	;	PUNCT
ejpam-3418	191	16	and	and	CCONJ
ejpam-3418	191	17	iii	iii	X
ejpam-3418	191	18	.	.	PUNCT
ejpam-3418	192	1	if	if	SCONJ
ejpam-3418	192	2	ι̂	ι̂	NUM
ejpam-3418	192	3	6	6	NUM
ejpam-3418	192	4	τ̂	τ̂	NUM
ejpam-3418	192	5	,	,	PUNCT
ejpam-3418	192	6	then	then	ADV
ejpam-3418	192	7	τ̂	τ̂	PUNCT
ejpam-3418	192	8	c	c	PROPN
ejpam-3418	192	9	6	6	NUM
ejpam-3418	192	10	ι̂c	ι̂c	NOUN
ejpam-3418	192	11	.	.	PUNCT
ejpam-3418	193	1	proof	proof	NOUN
ejpam-3418	193	2	.	.	PUNCT
ejpam-3418	194	1	let	let	VERB
ejpam-3418	194	2	ι̂	ι̂	PRON
ejpam-3418	194	3	∈	∈	VERB
ejpam-3418	194	4	i	i	PRON
ejpam-3418	194	5	i(x	i(x	PROPN
ejpam-3418	194	6	)	)	PUNCT
ejpam-3418	194	7	and	and	CCONJ
ejpam-3418	194	8	∅	∅	NOUN
ejpam-3418	194	9	6=	6=	ADP
ejpam-3418	194	10	a	a	DET
ejpam-3418	194	11	∈	∈	PROPN
ejpam-3418	194	12	i(x	i(x	NOUN
ejpam-3418	194	13	)	)	PUNCT
ejpam-3418	194	14	.	.	PUNCT
ejpam-3418	195	1	note	note	VERB
ejpam-3418	195	2	first	first	ADV
ejpam-3418	195	3	that	that	SCONJ
ejpam-3418	195	4	for	for	ADP
ejpam-3418	195	5	singleton	singleton	NOUN
ejpam-3418	195	6	sets	set	NOUN
ejpam-3418	195	7	{	{	PUNCT
ejpam-3418	195	8	x	x	NOUN
ejpam-3418	195	9	}	}	PUNCT
ejpam-3418	195	10	∈	∈	PROPN
ejpam-3418	195	11	i(x	i(x	NOUN
ejpam-3418	195	12	)	)	PUNCT
ejpam-3418	195	13	,	,	PUNCT
ejpam-3418	195	14	we	we	PRON
ejpam-3418	195	15	have	have	VERB
ejpam-3418	195	16	ι̂c({x	ι̂c({x	NOUN
ejpam-3418	195	17	}	}	PUNCT
ejpam-3418	195	18	)	)	PUNCT
ejpam-3418	195	19	=	=	PUNCT
ejpam-3418	196	1	[	[	X
ejpam-3418	196	2	1−	1−	NUM
ejpam-3418	196	3	[	[	X
ejpam-3418	196	4	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	196	5	,	,	PUNCT
ejpam-3418	196	6	1−	1−	NUM
ejpam-3418	197	1	[	[	X
ejpam-3418	197	2	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	197	3	]	]	PUNCT
ejpam-3418	197	4	.	.	PUNCT
ejpam-3418	198	1	then	then	ADV
ejpam-3418	198	2	,	,	PUNCT
ejpam-3418	198	3	consider	consider	VERB
ejpam-3418	198	4	that	that	PRON
ejpam-3418	198	5	(	(	PUNCT
ejpam-3418	198	6	ι̂c)c(a	ι̂c)c(a	PROPN
ejpam-3418	198	7	)	)	PUNCT
ejpam-3418	198	8	=	=	PUNCT
ejpam-3418	199	1	[	[	PUNCT
ejpam-3418	199	2	inf	inf	ADJ
ejpam-3418	199	3	x∈a	x∈a	NOUN
ejpam-3418	199	4	{	{	PUNCT
ejpam-3418	199	5	1−	1−	NUM
ejpam-3418	199	6	[	[	X
ejpam-3418	199	7	̂ιc({x})]+	̂ιc({x})]+	NUM
ejpam-3418	199	8	}	}	PUNCT
ejpam-3418	199	9	,	,	PUNCT
ejpam-3418	199	10	inf	inf	PROPN
ejpam-3418	199	11	x∈a	x∈a	NOUN
ejpam-3418	199	12	{	{	PUNCT
ejpam-3418	199	13	1−	1−	NUM
ejpam-3418	200	1	[	[	X
ejpam-3418	200	2	̂ιc({x})]−	̂ιc({x})]−	X
ejpam-3418	200	3	}	}	PUNCT
ejpam-3418	200	4	]	]	PUNCT
ejpam-3418	200	5	=	=	PUNCT
ejpam-3418	200	6	[	[	PUNCT
ejpam-3418	200	7	inf	inf	ADJ
ejpam-3418	200	8	x∈a	x∈a	NOUN
ejpam-3418	200	9	{	{	PUNCT
ejpam-3418	200	10	1−	1−	NUM
ejpam-3418	200	11	(	(	PUNCT
ejpam-3418	200	12	1−	1−	NUM
ejpam-3418	200	13	[	[	X
ejpam-3418	200	14	̂ι({x})]−	̂ι({x})]−	NUM
ejpam-3418	200	15	)	)	PUNCT
ejpam-3418	200	16	}	}	PUNCT
ejpam-3418	200	17	,	,	PUNCT
ejpam-3418	200	18	inf	inf	PROPN
ejpam-3418	200	19	x∈a	x∈a	NOUN
ejpam-3418	200	20	{	{	PUNCT
ejpam-3418	200	21	1−	1−	NUM
ejpam-3418	200	22	(	(	PUNCT
ejpam-3418	200	23	1−	1−	NUM
ejpam-3418	200	24	ι̂[({x})]+	ι̂[({x})]+	NUM
ejpam-3418	200	25	)	)	PUNCT
ejpam-3418	200	26	}	}	PUNCT
ejpam-3418	200	27	]	]	PUNCT
ejpam-3418	201	1	=	=	PUNCT
ejpam-3418	201	2	[	[	PUNCT
ejpam-3418	201	3	inf	inf	ADJ
ejpam-3418	201	4	x∈a	x∈a	NOUN
ejpam-3418	201	5	[	[	X
ejpam-3418	201	6	̂ι({x})]−	̂ι({x})]−	NUM
ejpam-3418	201	7	,	,	PUNCT
ejpam-3418	201	8	inf	inf	NOUN
ejpam-3418	201	9	x∈a	x∈a	NOUN
ejpam-3418	202	1	[	[	X
ejpam-3418	202	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	202	3	]	]	PUNCT
ejpam-3418	202	4	.	.	PUNCT
ejpam-3418	203	1	(	(	PUNCT
ejpam-3418	203	2	1	1	X
ejpam-3418	203	3	)	)	PUNCT
ejpam-3418	203	4	since	since	SCONJ
ejpam-3418	203	5	ι̂	ι̂	NUM
ejpam-3418	203	6	is	be	AUX
ejpam-3418	203	7	an	an	DET
ejpam-3418	203	8	ivfi	ivfi	NOUN
ejpam-3418	203	9	set	set	VERB
ejpam-3418	203	10	,	,	PUNCT
ejpam-3418	203	11	the	the	DET
ejpam-3418	203	12	reverse	reverse	ADJ
ejpam-3418	203	13	inequality	inequality	NOUN
ejpam-3418	203	14	property	property	NOUN
ejpam-3418	203	15	implies	imply	VERB
ejpam-3418	203	16	that	that	PRON
ejpam-3418	203	17	ι̂(a	ι̂(a	PUNCT
ejpam-3418	203	18	)	)	PUNCT
ejpam-3418	203	19	=	=	PUNCT
ejpam-3418	204	1	[	[	X
ejpam-3418	204	2	[	[	X
ejpam-3418	204	3	̂ι(a)]−	̂ι(a)]−	X
ejpam-3418	204	4	,	,	PUNCT
ejpam-3418	204	5	[	[	X
ejpam-3418	204	6	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	204	7	]	]	X
ejpam-3418	204	8	≤i	≤i	PROPN
ejpam-3418	204	9	[	[	PUNCT
ejpam-3418	204	10	inf	inf	NOUN
ejpam-3418	204	11	x∈a	x∈a	NOUN
ejpam-3418	204	12	[	[	X
ejpam-3418	204	13	̂ι({x})]−	̂ι({x})]−	NUM
ejpam-3418	204	14	,	,	PUNCT
ejpam-3418	204	15	inf	inf	NOUN
ejpam-3418	204	16	x∈a	x∈a	NOUN
ejpam-3418	205	1	[	[	X
ejpam-3418	205	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	205	3	]	]	PUNCT
ejpam-3418	205	4	.	.	PUNCT
ejpam-3418	206	1	thus	thus	ADV
ejpam-3418	206	2	,	,	PUNCT
ejpam-3418	206	3	ι̂	ι̂	X
ejpam-3418	206	4	6	6	NUM
ejpam-3418	206	5	(	(	PUNCT
ejpam-3418	206	6	ι̂c)c	ι̂c)c	X
ejpam-3418	206	7	,	,	PUNCT
ejpam-3418	206	8	proving	prove	VERB
ejpam-3418	206	9	(	(	PUNCT
ejpam-3418	206	10	i	i	NOUN
ejpam-3418	206	11	)	)	PUNCT
ejpam-3418	206	12	.	.	PUNCT
ejpam-3418	207	1	to	to	PART
ejpam-3418	207	2	prove	prove	VERB
ejpam-3418	207	3	(	(	PUNCT
ejpam-3418	207	4	ii	ii	NOUN
ejpam-3418	207	5	)	)	PUNCT
ejpam-3418	207	6	,	,	PUNCT
ejpam-3418	207	7	recall	recall	VERB
ejpam-3418	207	8	first	first	ADV
ejpam-3418	207	9	the	the	DET
ejpam-3418	207	10	definition	definition	NOUN
ejpam-3418	207	11	of	of	ADP
ejpam-3418	207	12	a	a	DET
ejpam-3418	207	13	guaranteed	guarantee	VERB
ejpam-3418	207	14	possibility	possibility	NOUN
ejpam-3418	207	15	ivfi	ivfi	NOUN
ejpam-3418	207	16	set	set	VERB
ejpam-3418	207	17	after	after	ADP
ejpam-3418	207	18	example	example	NOUN
ejpam-3418	207	19	1	1	NUM
ejpam-3418	207	20	.	.	PUNCT
ejpam-3418	208	1	now	now	ADV
ejpam-3418	208	2	,	,	PUNCT
ejpam-3418	208	3	suppose	suppose	VERB
ejpam-3418	208	4	that	that	SCONJ
ejpam-3418	208	5	ι̂	ι̂	PRON
ejpam-3418	208	6	=	=	PUNCT
ejpam-3418	208	7	(	(	PUNCT
ejpam-3418	208	8	ι̂c)c	ι̂c)c	X
ejpam-3418	208	9	.	.	PUNCT
ejpam-3418	209	1	then	then	ADV
ejpam-3418	209	2	,	,	PUNCT
ejpam-3418	209	3	ι̂(a	ι̂(a	PUNCT
ejpam-3418	209	4	)	)	PUNCT
ejpam-3418	209	5	=	=	SYM
ejpam-3418	209	6	(	(	PUNCT
ejpam-3418	209	7	ι̂c)c(a	ι̂c)c(a	PROPN
ejpam-3418	209	8	)	)	PUNCT
ejpam-3418	209	9	,	,	PUNCT
ejpam-3418	209	10	for	for	ADP
ejpam-3418	209	11	all	all	DET
ejpam-3418	209	12	a	a	DET
ejpam-3418	209	13	∈	∈	PROPN
ejpam-3418	209	14	i(x	i(x	NOUN
ejpam-3418	209	15	)	)	PUNCT
ejpam-3418	209	16	.	.	PUNCT
ejpam-3418	210	1	note	note	VERB
ejpam-3418	210	2	from	from	ADP
ejpam-3418	210	3	(	(	PUNCT
ejpam-3418	210	4	1	1	NUM
ejpam-3418	210	5	)	)	PUNCT
ejpam-3418	210	6	that	that	SCONJ
ejpam-3418	210	7	for	for	ADP
ejpam-3418	210	8	∅	∅	NOUN
ejpam-3418	210	9	6=	6=	ADP
ejpam-3418	210	10	a	a	DET
ejpam-3418	210	11	∈	∈	PROPN
ejpam-3418	210	12	i(x	i(x	NOUN
ejpam-3418	210	13	)	)	PUNCT
ejpam-3418	210	14	,	,	PUNCT
ejpam-3418	210	15	(	(	PUNCT
ejpam-3418	210	16	ι̂c)c(a	ι̂c)c(a	PROPN
ejpam-3418	210	17	)	)	PUNCT
ejpam-3418	210	18	=	=	PUNCT
ejpam-3418	211	1	[	[	PUNCT
ejpam-3418	211	2	inf	inf	ADJ
ejpam-3418	211	3	x∈a	x∈a	NOUN
ejpam-3418	211	4	[	[	X
ejpam-3418	211	5	̂ι({x})]−	̂ι({x})]−	NUM
ejpam-3418	211	6	,	,	PUNCT
ejpam-3418	211	7	inf	inf	NOUN
ejpam-3418	211	8	x∈a	x∈a	NOUN
ejpam-3418	212	1	[	[	X
ejpam-3418	212	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	212	3	]	]	PUNCT
ejpam-3418	212	4	.	.	PUNCT
ejpam-3418	213	1	thus	thus	ADV
ejpam-3418	213	2	,	,	PUNCT
ejpam-3418	213	3	ι̂(a	ι̂(a	PUNCT
ejpam-3418	213	4	)	)	PUNCT
ejpam-3418	213	5	=	=	SYM
ejpam-3418	213	6	[	[	PUNCT
ejpam-3418	213	7	infx∈a	infx∈a	PROPN
ejpam-3418	213	8	[	[	X
ejpam-3418	213	9	̂ι({x})]−	̂ι({x})]−	NUM
ejpam-3418	213	10	,	,	PUNCT
ejpam-3418	213	11	infx∈a	infx∈a	ADJ
ejpam-3418	213	12	[	[	X
ejpam-3418	213	13	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	213	14	]	]	X
ejpam-3418	213	15	=	=	SYM
ejpam-3418	213	16	infx∈a	infx∈a	VERB
ejpam-3418	213	17	ι̂({x	ι̂({x	NOUN
ejpam-3418	213	18	}	}	PUNCT
ejpam-3418	213	19	)	)	PUNCT
ejpam-3418	213	20	.	.	PUNCT
ejpam-3418	214	1	that	that	PRON
ejpam-3418	214	2	is	be	AUX
ejpam-3418	214	3	,	,	PUNCT
ejpam-3418	214	4	ι̂	ι̂	X
ejpam-3418	214	5	is	be	AUX
ejpam-3418	214	6	a	a	DET
ejpam-3418	214	7	guaranteed	guarantee	VERB
ejpam-3418	214	8	possibility	possibility	NOUN
ejpam-3418	214	9	ivfi	ivfi	NOUN
ejpam-3418	214	10	set	set	VERB
ejpam-3418	214	11	.	.	PUNCT
ejpam-3418	215	1	conversely	conversely	ADV
ejpam-3418	215	2	,	,	PUNCT
ejpam-3418	215	3	suppose	suppose	VERB
ejpam-3418	215	4	that	that	SCONJ
ejpam-3418	215	5	ι̂(a	ι̂(a	PUNCT
ejpam-3418	215	6	)	)	PUNCT
ejpam-3418	215	7	=	=	SYM
ejpam-3418	215	8	[	[	PUNCT
ejpam-3418	215	9	inf	inf	ADJ
ejpam-3418	215	10	x∈a	x∈a	PROPN
ejpam-3418	215	11	ι̂({x})−	ι̂({x})−	NOUN
ejpam-3418	215	12	,	,	PUNCT
ejpam-3418	215	13	inf	inf	ADJ
ejpam-3418	215	14	x∈a	x∈a	NOUN
ejpam-3418	215	15	ι̂({x})+	ι̂({x})+	VERB
ejpam-3418	215	16	]	]	PUNCT
ejpam-3418	215	17	,	,	PUNCT
ejpam-3418	215	18	for	for	ADP
ejpam-3418	215	19	∅	∅	NOUN
ejpam-3418	215	20	6=	6=	ADP
ejpam-3418	215	21	a	a	DET
ejpam-3418	215	22	∈	∈	PROPN
ejpam-3418	215	23	i(x	i(x	NOUN
ejpam-3418	215	24	)	)	PUNCT
ejpam-3418	215	25	.	.	PUNCT
ejpam-3418	216	1	then	then	ADV
ejpam-3418	216	2	by	by	ADP
ejpam-3418	216	3	(	(	PUNCT
ejpam-3418	216	4	1	1	NUM
ejpam-3418	216	5	)	)	PUNCT
ejpam-3418	216	6	,	,	PUNCT
ejpam-3418	216	7	ι̂(a	ι̂(a	PUNCT
ejpam-3418	216	8	)	)	PUNCT
ejpam-3418	216	9	=	=	SYM
ejpam-3418	216	10	(	(	PUNCT
ejpam-3418	216	11	ι̂c)c(a	ι̂c)c(a	PROPN
ejpam-3418	216	12	)	)	PUNCT
ejpam-3418	216	13	.	.	PUNCT
ejpam-3418	217	1	thus	thus	ADV
ejpam-3418	217	2	,	,	PUNCT
ejpam-3418	217	3	ι̂	ι̂	PUNCT
ejpam-3418	217	4	=	=	PUNCT
ejpam-3418	217	5	(	(	PUNCT
ejpam-3418	217	6	ι̂c)c	ι̂c)c	X
ejpam-3418	217	7	.	.	PUNCT
ejpam-3418	218	1	to	to	PART
ejpam-3418	218	2	prove	prove	VERB
ejpam-3418	218	3	(	(	PUNCT
ejpam-3418	218	4	iii	iii	NOUN
ejpam-3418	218	5	)	)	PUNCT
ejpam-3418	218	6	,	,	PUNCT
ejpam-3418	218	7	suppose	suppose	VERB
ejpam-3418	218	8	that	that	SCONJ
ejpam-3418	218	9	ι̂	ι̂	PRON
ejpam-3418	218	10	6	6	NUM
ejpam-3418	218	11	τ̂	τ̂	PUNCT
ejpam-3418	218	12	.	.	PUNCT
ejpam-3418	219	1	let	let	VERB
ejpam-3418	219	2	∅	∅	NOUN
ejpam-3418	219	3	6=	6=	ADP
ejpam-3418	219	4	a	a	DET
ejpam-3418	219	5	∈	∈	PROPN
ejpam-3418	219	6	i(x	i(x	NOUN
ejpam-3418	219	7	)	)	PUNCT
ejpam-3418	219	8	.	.	PUNCT
ejpam-3418	220	1	then	then	ADV
ejpam-3418	220	2	,	,	PUNCT
ejpam-3418	220	3	ι̂({x	ι̂({x	PROPN
ejpam-3418	220	4	}	}	PUNCT
ejpam-3418	220	5	)	)	PUNCT
ejpam-3418	220	6	≤i	≤i	NOUN
ejpam-3418	220	7	τ̂({x	τ̂({x	NOUN
ejpam-3418	220	8	}	}	PUNCT
ejpam-3418	220	9	)	)	PUNCT
ejpam-3418	220	10	,	,	PUNCT
ejpam-3418	220	11	for	for	ADP
ejpam-3418	220	12	all	all	PRON
ejpam-3418	220	13	x	x	SYM
ejpam-3418	220	14	∈	∈	NOUN
ejpam-3418	220	15	a.	a.	NOUN
ejpam-3418	220	16	thus	thus	ADV
ejpam-3418	220	17	,	,	PUNCT
ejpam-3418	220	18	for	for	ADP
ejpam-3418	220	19	every	every	DET
ejpam-3418	220	20	x	x	PROPN
ejpam-3418	220	21	∈	∈	PROPN
ejpam-3418	220	22	a	a	PRON
ejpam-3418	220	23	,	,	PUNCT
ejpam-3418	220	24	[	[	PUNCT
ejpam-3418	220	25	1−	1−	NUM
ejpam-3418	220	26	[	[	X
ejpam-3418	220	27	τ̂({x})]+	τ̂({x})]+	PROPN
ejpam-3418	220	28	,	,	PUNCT
ejpam-3418	220	29	1−	1−	NUM
ejpam-3418	221	1	[	[	X
ejpam-3418	221	2	τ̂({x})]−	τ̂({x})]−	PUNCT
ejpam-3418	221	3	]	]	X
ejpam-3418	221	4	≤i	≤i	PROPN
ejpam-3418	221	5	[	[	PUNCT
ejpam-3418	221	6	1−	1−	NUM
ejpam-3418	222	1	[	[	X
ejpam-3418	222	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	222	3	,	,	PUNCT
ejpam-3418	222	4	1−	1−	NUM
ejpam-3418	223	1	[	[	X
ejpam-3418	223	2	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	223	3	]	]	PUNCT
ejpam-3418	223	4	.	.	PUNCT
ejpam-3418	224	1	hence	hence	ADV
ejpam-3418	224	2	,	,	PUNCT
ejpam-3418	224	3	[	[	PUNCT
ejpam-3418	224	4	inf	inf	ADJ
ejpam-3418	224	5	x∈a	x∈a	PROPN
ejpam-3418	224	6	{	{	PUNCT
ejpam-3418	224	7	1−[τ̂({x})]+	1−[τ̂({x})]+	PROPN
ejpam-3418	224	8	}	}	PUNCT
ejpam-3418	224	9	,	,	PUNCT
ejpam-3418	224	10	inf	inf	PROPN
ejpam-3418	224	11	x∈a	x∈a	PROPN
ejpam-3418	224	12	{	{	PUNCT
ejpam-3418	224	13	1−[τ̂({x})]−	1−[τ̂({x})]−	NUM
ejpam-3418	224	14	}	}	PUNCT
ejpam-3418	224	15	]	]	X
ejpam-3418	225	1	≤i	≤i	PROPN
ejpam-3418	225	2	[	[	PUNCT
ejpam-3418	225	3	inf	inf	ADJ
ejpam-3418	225	4	x∈a	x∈a	NOUN
ejpam-3418	225	5	{	{	PUNCT
ejpam-3418	225	6	1−	1−	NUM
ejpam-3418	226	1	[	[	X
ejpam-3418	226	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	226	3	}	}	PUNCT
ejpam-3418	226	4	,	,	PUNCT
ejpam-3418	226	5	inf	inf	PROPN
ejpam-3418	226	6	x∈a	x∈a	NOUN
ejpam-3418	226	7	{	{	PUNCT
ejpam-3418	226	8	1−	1−	NUM
ejpam-3418	227	1	[	[	X
ejpam-3418	227	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	227	3	}	}	PUNCT
ejpam-3418	227	4	]	]	PUNCT
ejpam-3418	227	5	.	.	PUNCT
ejpam-3418	228	1	thus	thus	ADV
ejpam-3418	228	2	,	,	PUNCT
ejpam-3418	228	3	τ̂	τ̂	PUNCT
ejpam-3418	228	4	c(a	c(a	PROPN
ejpam-3418	228	5	)	)	PUNCT
ejpam-3418	228	6	≤i	≤i	PROPN
ejpam-3418	228	7	ι̂c(a	ι̂c(a	NUM
ejpam-3418	228	8	)	)	PUNCT
ejpam-3418	228	9	,	,	PUNCT
ejpam-3418	228	10	for	for	ADP
ejpam-3418	228	11	all	all	DET
ejpam-3418	228	12	a	a	DET
ejpam-3418	228	13	∈	∈	PROPN
ejpam-3418	228	14	i(x	i(x	NOUN
ejpam-3418	228	15	)	)	PUNCT
ejpam-3418	228	16	.	.	PUNCT
ejpam-3418	229	1	therefore	therefore	ADV
ejpam-3418	229	2	,	,	PUNCT
ejpam-3418	229	3	τ̂	τ̂	PUNCT
ejpam-3418	229	4	c	c	PROPN
ejpam-3418	229	5	6	6	NUM
ejpam-3418	229	6	ι̂c	ι̂c	NOUN
ejpam-3418	229	7	.	.	PUNCT
ejpam-3418	230	1	3	3	X
ejpam-3418	230	2	.	.	X
ejpam-3418	230	3	mappings	mapping	NOUN
ejpam-3418	230	4	let	let	VERB
ejpam-3418	230	5	x	x	PRON
ejpam-3418	230	6	and	and	CCONJ
ejpam-3418	230	7	y	y	PROPN
ejpam-3418	230	8	be	be	AUX
ejpam-3418	230	9	nonempty	nonempty	X
ejpam-3418	230	10	sets	set	NOUN
ejpam-3418	230	11	and	and	CCONJ
ejpam-3418	230	12	f	f	NOUN
ejpam-3418	230	13	:	:	PUNCT
ejpam-3418	230	14	x	x	X
ejpam-3418	230	15	→	→	SYM
ejpam-3418	230	16	y	y	X
ejpam-3418	230	17	be	be	AUX
ejpam-3418	230	18	a	a	DET
ejpam-3418	230	19	mapping	mapping	NOUN
ejpam-3418	230	20	.	.	PUNCT
ejpam-3418	231	1	moreover	moreover	ADV
ejpam-3418	231	2	,	,	PUNCT
ejpam-3418	231	3	let	let	VERB
ejpam-3418	231	4	i(x	i(x	PROPN
ejpam-3418	231	5	)	)	PUNCT
ejpam-3418	231	6	and	and	CCONJ
ejpam-3418	231	7	i(y	i(y	NOUN
ejpam-3418	231	8	)	)	PUNCT
ejpam-3418	231	9	be	be	AUX
ejpam-3418	231	10	ideals	ideal	NOUN
ejpam-3418	231	11	on	on	ADP
ejpam-3418	231	12	x	x	PUNCT
ejpam-3418	231	13	and	and	CCONJ
ejpam-3418	231	14	y	y	PROPN
ejpam-3418	231	15	,	,	PUNCT
ejpam-3418	231	16	respectively	respectively	ADV
ejpam-3418	231	17	.	.	PUNCT
ejpam-3418	232	1	we	we	PRON
ejpam-3418	232	2	define	define	VERB
ejpam-3418	232	3	the	the	DET
ejpam-3418	232	4	image	image	NOUN
ejpam-3418	232	5	and	and	CCONJ
ejpam-3418	232	6	pre	pre	NOUN
ejpam-3418	232	7	-	-	NOUN
ejpam-3418	232	8	image	image	NOUN
ejpam-3418	232	9	of	of	ADP
ejpam-3418	232	10	the	the	DET
ejpam-3418	232	11	ideals	ideal	NOUN
ejpam-3418	232	12	under	under	ADP
ejpam-3418	232	13	f	f	PROPN
ejpam-3418	232	14	by	by	ADP
ejpam-3418	232	15	f(i(x	f(i(x	NOUN
ejpam-3418	232	16	)	)	PUNCT
ejpam-3418	232	17	)	)	PUNCT
ejpam-3418	233	1	=	=	PRON
ejpam-3418	233	2	{	{	PUNCT
ejpam-3418	233	3	f(a	f(a	NOUN
ejpam-3418	233	4	)	)	PUNCT
ejpam-3418	233	5	:	:	PUNCT
ejpam-3418	233	6	a	a	DET
ejpam-3418	233	7	∈	∈	PROPN
ejpam-3418	233	8	i(x	i(x	NOUN
ejpam-3418	233	9	)	)	PUNCT
ejpam-3418	233	10	}	}	PUNCT
ejpam-3418	233	11	and	and	CCONJ
ejpam-3418	233	12	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	233	13	)	)	PUNCT
ejpam-3418	233	14	)	)	PUNCT
ejpam-3418	234	1	=	=	PRON
ejpam-3418	234	2	{	{	PUNCT
ejpam-3418	234	3	a	a	X
ejpam-3418	234	4	:	:	PUNCT
ejpam-3418	234	5	a	a	DET
ejpam-3418	234	6	⊆	⊆	NUM
ejpam-3418	234	7	f−1(b	f−1(b	PROPN
ejpam-3418	234	8	)	)	PUNCT
ejpam-3418	234	9	,	,	PUNCT
ejpam-3418	234	10	b	b	X
ejpam-3418	234	11	∈	∈	PROPN
ejpam-3418	234	12	i(y	i(y	NOUN
ejpam-3418	234	13	)	)	PUNCT
ejpam-3418	234	14	}	}	PUNCT
ejpam-3418	234	15	,	,	PUNCT
ejpam-3418	234	16	where	where	SCONJ
ejpam-3418	234	17	f(a	f(a	NOUN
ejpam-3418	234	18	)	)	PUNCT
ejpam-3418	234	19	and	and	CCONJ
ejpam-3418	234	20	f−1(b	f−1(b	PROPN
ejpam-3418	234	21	)	)	PUNCT
ejpam-3418	234	22	is	be	AUX
ejpam-3418	234	23	the	the	DET
ejpam-3418	234	24	usual	usual	ADJ
ejpam-3418	234	25	image	image	NOUN
ejpam-3418	234	26	and	and	CCONJ
ejpam-3418	234	27	preimage	preimage	NOUN
ejpam-3418	234	28	of	of	ADP
ejpam-3418	234	29	a	a	DET
ejpam-3418	234	30	⊆	⊆	NUM
ejpam-3418	234	31	x	x	NOUN
ejpam-3418	234	32	and	and	CCONJ
ejpam-3418	234	33	b	b	NOUN
ejpam-3418	234	34	⊆	⊆	NUM
ejpam-3418	234	35	y	y	PROPN
ejpam-3418	234	36	,	,	PUNCT
ejpam-3418	234	37	respectively	respectively	ADV
ejpam-3418	234	38	.	.	PUNCT
ejpam-3418	235	1	the	the	DET
ejpam-3418	235	2	next	next	ADJ
ejpam-3418	235	3	theorem	theorem	NOUN
ejpam-3418	235	4	is	be	AUX
ejpam-3418	235	5	important	important	ADJ
ejpam-3418	235	6	because	because	SCONJ
ejpam-3418	235	7	it	it	PRON
ejpam-3418	235	8	shows	show	VERB
ejpam-3418	235	9	that	that	SCONJ
ejpam-3418	235	10	these	these	DET
ejpam-3418	235	11	image	image	NOUN
ejpam-3418	235	12	and	and	CCONJ
ejpam-3418	235	13	preimage	preimage	NOUN
ejpam-3418	235	14	of	of	ADP
ejpam-3418	235	15	ideals	ideal	NOUN
ejpam-3418	235	16	are	be	AUX
ejpam-3418	235	17	also	also	ADV
ejpam-3418	235	18	ideals	ideal	NOUN
ejpam-3418	235	19	.	.	PUNCT
ejpam-3418	236	1	the	the	DET
ejpam-3418	236	2	proof	proof	NOUN
ejpam-3418	236	3	can	can	AUX
ejpam-3418	236	4	be	be	AUX
ejpam-3418	236	5	found	find	VERB
ejpam-3418	236	6	in	in	ADP
ejpam-3418	236	7	[	[	X
ejpam-3418	236	8	6	6	NUM
ejpam-3418	236	9	]	]	PUNCT
ejpam-3418	236	10	.	.	PUNCT
ejpam-3418	237	1	mj	mj	PROPN
ejpam-3418	237	2	togonon	togonon	PROPN
ejpam-3418	237	3	,	,	PUNCT
ejpam-3418	237	4	r	r	NOUN
ejpam-3418	237	5	caga	caga	NOUN
ejpam-3418	237	6	-	-	PUNCT
ejpam-3418	237	7	anan	anan	PROPN
ejpam-3418	237	8	/	/	SYM
ejpam-3418	237	9	eur	eur	PROPN
ejpam-3418	237	10	.	.	PUNCT
ejpam-3418	238	1	j.	j.	PROPN
ejpam-3418	238	2	pure	pure	PROPN
ejpam-3418	238	3	appl	appl	PROPN
ejpam-3418	238	4	.	.	PROPN
ejpam-3418	238	5	math	math	PROPN
ejpam-3418	238	6	,	,	PUNCT
ejpam-3418	238	7	12	12	NUM
ejpam-3418	238	8	(	(	PUNCT
ejpam-3418	238	9	2	2	NUM
ejpam-3418	238	10	)	)	PUNCT
ejpam-3418	238	11	(	(	PUNCT
ejpam-3418	238	12	2019	2019	NUM
ejpam-3418	238	13	)	)	PUNCT
ejpam-3418	238	14	,	,	PUNCT
ejpam-3418	238	15	553	553	NUM
ejpam-3418	238	16	-	-	SYM
ejpam-3418	238	17	570	570	NUM
ejpam-3418	238	18	560	560	NUM
ejpam-3418	238	19	theorem	theorem	NOUN
ejpam-3418	238	20	3	3	NUM
ejpam-3418	238	21	(	(	PUNCT
ejpam-3418	238	22	[	[	X
ejpam-3418	238	23	6	6	NUM
ejpam-3418	238	24	]	]	PUNCT
ejpam-3418	238	25	)	)	PUNCT
ejpam-3418	238	26	.	.	PUNCT
ejpam-3418	239	1	let	let	VERB
ejpam-3418	239	2	x	x	PRON
ejpam-3418	239	3	and	and	CCONJ
ejpam-3418	239	4	y	y	PROPN
ejpam-3418	239	5	be	be	AUX
ejpam-3418	239	6	nonempty	nonempty	ADJ
ejpam-3418	239	7	sets	set	NOUN
ejpam-3418	239	8	and	and	CCONJ
ejpam-3418	239	9	let	let	VERB
ejpam-3418	239	10	f	f	NOUN
ejpam-3418	239	11	:	:	PUNCT
ejpam-3418	239	12	x	x	X
ejpam-3418	239	13	→	→	SYM
ejpam-3418	239	14	y	y	X
ejpam-3418	239	15	be	be	AUX
ejpam-3418	239	16	a	a	DET
ejpam-3418	239	17	mapping	mapping	NOUN
ejpam-3418	239	18	.	.	PUNCT
ejpam-3418	240	1	if	if	SCONJ
ejpam-3418	240	2	i(x	i(x	PROPN
ejpam-3418	240	3	)	)	PUNCT
ejpam-3418	240	4	and	and	CCONJ
ejpam-3418	240	5	i(y	i(y	NOUN
ejpam-3418	240	6	)	)	PUNCT
ejpam-3418	240	7	are	be	AUX
ejpam-3418	240	8	ideals	ideal	NOUN
ejpam-3418	240	9	on	on	ADP
ejpam-3418	240	10	x	x	PUNCT
ejpam-3418	240	11	and	and	CCONJ
ejpam-3418	240	12	y	y	PROPN
ejpam-3418	240	13	,	,	PUNCT
ejpam-3418	240	14	respectively	respectively	ADV
ejpam-3418	240	15	,	,	PUNCT
ejpam-3418	240	16	then	then	ADV
ejpam-3418	240	17	f(i(x	f(i(x	NOUN
ejpam-3418	240	18	)	)	PUNCT
ejpam-3418	240	19	)	)	PUNCT
ejpam-3418	240	20	and	and	CCONJ
ejpam-3418	240	21	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	240	22	)	)	PUNCT
ejpam-3418	240	23	)	)	PUNCT
ejpam-3418	241	1	are	be	AUX
ejpam-3418	241	2	ideals	ideal	NOUN
ejpam-3418	241	3	on	on	ADP
ejpam-3418	241	4	y	y	PROPN
ejpam-3418	241	5	and	and	CCONJ
ejpam-3418	241	6	x	x	NOUN
ejpam-3418	241	7	,	,	PUNCT
ejpam-3418	241	8	respectively	respectively	ADV
ejpam-3418	241	9	.	.	PUNCT
ejpam-3418	242	1	given	give	VERB
ejpam-3418	242	2	a	a	DET
ejpam-3418	242	3	mapping	mapping	NOUN
ejpam-3418	242	4	of	of	ADP
ejpam-3418	242	5	two	two	NUM
ejpam-3418	242	6	ordinary	ordinary	ADJ
ejpam-3418	242	7	sets	set	NOUN
ejpam-3418	242	8	,	,	PUNCT
ejpam-3418	242	9	we	we	PRON
ejpam-3418	242	10	define	define	VERB
ejpam-3418	242	11	the	the	DET
ejpam-3418	242	12	image	image	NOUN
ejpam-3418	242	13	and	and	CCONJ
ejpam-3418	242	14	pre	pre	NOUN
ejpam-3418	242	15	-	-	NOUN
ejpam-3418	242	16	image	image	NOUN
ejpam-3418	242	17	of	of	ADP
ejpam-3418	242	18	ivfi	ivfi	NOUN
ejpam-3418	242	19	sets	set	NOUN
ejpam-3418	242	20	.	.	PUNCT
ejpam-3418	243	1	we	we	PRON
ejpam-3418	243	2	then	then	ADV
ejpam-3418	243	3	prove	prove	VERB
ejpam-3418	243	4	that	that	SCONJ
ejpam-3418	243	5	these	these	DET
ejpam-3418	243	6	image	image	NOUN
ejpam-3418	243	7	and	and	CCONJ
ejpam-3418	243	8	pre	pre	ADJ
ejpam-3418	243	9	-	-	NOUN
ejpam-3418	243	10	image	image	NOUN
ejpam-3418	243	11	are	be	AUX
ejpam-3418	243	12	also	also	ADV
ejpam-3418	243	13	ivfi	ivfi	NOUN
ejpam-3418	243	14	sets	set	NOUN
ejpam-3418	243	15	,	,	PUNCT
ejpam-3418	243	16	showing	show	VERB
ejpam-3418	243	17	that	that	SCONJ
ejpam-3418	243	18	they	they	PRON
ejpam-3418	243	19	are	be	AUX
ejpam-3418	243	20	well	well	ADV
ejpam-3418	243	21	-	-	PUNCT
ejpam-3418	243	22	defined	define	VERB
ejpam-3418	243	23	.	.	PUNCT
ejpam-3418	244	1	definition	definition	NOUN
ejpam-3418	244	2	5	5	NUM
ejpam-3418	244	3	.	.	PUNCT
ejpam-3418	245	1	let	let	VERB
ejpam-3418	245	2	x	x	PRON
ejpam-3418	245	3	and	and	CCONJ
ejpam-3418	245	4	y	y	PROPN
ejpam-3418	245	5	be	be	AUX
ejpam-3418	245	6	nonempty	nonempty	X
ejpam-3418	245	7	sets	set	NOUN
ejpam-3418	245	8	and	and	CCONJ
ejpam-3418	245	9	f	f	NOUN
ejpam-3418	245	10	:	:	PUNCT
ejpam-3418	245	11	x	x	X
ejpam-3418	245	12	→	→	SYM
ejpam-3418	245	13	y	y	X
ejpam-3418	245	14	be	be	AUX
ejpam-3418	245	15	a	a	DET
ejpam-3418	245	16	mapping	mapping	NOUN
ejpam-3418	245	17	.	.	PUNCT
ejpam-3418	246	1	moreover	moreover	ADV
ejpam-3418	246	2	,	,	PUNCT
ejpam-3418	246	3	let	let	VERB
ejpam-3418	246	4	i(x	i(x	PROPN
ejpam-3418	246	5	)	)	PUNCT
ejpam-3418	246	6	and	and	CCONJ
ejpam-3418	246	7	i(y	i(y	NOUN
ejpam-3418	246	8	)	)	PUNCT
ejpam-3418	246	9	be	be	AUX
ejpam-3418	246	10	ideals	ideal	NOUN
ejpam-3418	246	11	on	on	ADP
ejpam-3418	246	12	x	x	PUNCT
ejpam-3418	246	13	and	and	CCONJ
ejpam-3418	246	14	y	y	PROPN
ejpam-3418	246	15	,	,	PUNCT
ejpam-3418	246	16	respectively	respectively	ADV
ejpam-3418	246	17	.	.	PUNCT
ejpam-3418	247	1	i.	i.	PROPN
ejpam-3418	248	1	if	if	SCONJ
ejpam-3418	248	2	ι̂	ι̂	NUM
ejpam-3418	248	3	∈	∈	PROPN
ejpam-3418	249	1	i	i	PRON
ejpam-3418	249	2	i(x	i(x	PROPN
ejpam-3418	249	3	)	)	PUNCT
ejpam-3418	249	4	,	,	PUNCT
ejpam-3418	249	5	then	then	ADV
ejpam-3418	249	6	the	the	DET
ejpam-3418	249	7	image	image	NOUN
ejpam-3418	249	8	of	of	ADP
ejpam-3418	249	9	ι̂	ι̂	NOUN
ejpam-3418	249	10	under	under	ADP
ejpam-3418	249	11	f	f	PROPN
ejpam-3418	249	12	,	,	PUNCT
ejpam-3418	249	13	denoted	denote	VERB
ejpam-3418	249	14	by	by	ADP
ejpam-3418	249	15	f	f	PROPN
ejpam-3418	250	1	[	[	X
ejpam-3418	250	2	̂ι	̂ι	PROPN
ejpam-3418	250	3	]	]	PUNCT
ejpam-3418	250	4	,	,	PUNCT
ejpam-3418	250	5	is	be	AUX
ejpam-3418	250	6	the	the	DET
ejpam-3418	250	7	mapping	mapping	NOUN
ejpam-3418	250	8	f	f	X
ejpam-3418	251	1	[	[	X
ejpam-3418	251	2	̂ι	̂ι	X
ejpam-3418	251	3	]	]	X
ejpam-3418	251	4	:	:	PUNCT
ejpam-3418	252	1	f(i(x))→	f(i(x))→	CCONJ
ejpam-3418	252	2	i	i	PRON
ejpam-3418	252	3	given	give	VERB
ejpam-3418	252	4	by	by	ADP
ejpam-3418	252	5	(	(	PUNCT
ejpam-3418	252	6	f	f	X
ejpam-3418	253	1	[	[	X
ejpam-3418	253	2	̂ι])(b	̂ι])(b	X
ejpam-3418	253	3	)	)	PUNCT
ejpam-3418	253	4	=	=	PUNCT
ejpam-3418	254	1	[	[	PUNCT
ejpam-3418	254	2	sup	sup	NUM
ejpam-3418	254	3	a∈s	a∈s	PROPN
ejpam-3418	254	4	[	[	X
ejpam-3418	254	5	̂ι(a)]−	̂ι(a)]−	PROPN
ejpam-3418	254	6	,	,	PUNCT
ejpam-3418	254	7	sup	sup	NOUN
ejpam-3418	254	8	a∈s	a∈s	ADJ
ejpam-3418	254	9	[	[	X
ejpam-3418	254	10	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	254	11	]	]	PUNCT
ejpam-3418	254	12	,	,	PUNCT
ejpam-3418	254	13	where	where	SCONJ
ejpam-3418	254	14	s	s	VERB
ejpam-3418	254	15	=	=	X
ejpam-3418	254	16	{	{	PUNCT
ejpam-3418	254	17	a	a	DET
ejpam-3418	254	18	∈	∈	PROPN
ejpam-3418	254	19	i(x	i(x	NOUN
ejpam-3418	254	20	)	)	PUNCT
ejpam-3418	254	21	:	:	PUNCT
ejpam-3418	254	22	f(a	f(a	X
ejpam-3418	254	23	)	)	PUNCT
ejpam-3418	254	24	=	=	SYM
ejpam-3418	255	1	b	b	X
ejpam-3418	255	2	}	}	PUNCT
ejpam-3418	255	3	.	.	PUNCT
ejpam-3418	256	1	ii	ii	PROPN
ejpam-3418	256	2	.	.	PUNCT
ejpam-3418	257	1	if	if	SCONJ
ejpam-3418	257	2	τ̂	τ̂	NUM
ejpam-3418	257	3	∈	∈	PROPN
ejpam-3418	257	4	i	i	PRON
ejpam-3418	257	5	i(y	i(y	NOUN
ejpam-3418	257	6	)	)	PUNCT
ejpam-3418	257	7	,	,	PUNCT
ejpam-3418	257	8	then	then	ADV
ejpam-3418	257	9	the	the	DET
ejpam-3418	257	10	pre	pre	NOUN
ejpam-3418	257	11	-	-	NOUN
ejpam-3418	257	12	image	image	NOUN
ejpam-3418	257	13	of	of	ADP
ejpam-3418	257	14	τ̂	τ̂	PUNCT
ejpam-3418	257	15	under	under	ADP
ejpam-3418	257	16	f	f	PROPN
ejpam-3418	257	17	,	,	PUNCT
ejpam-3418	257	18	denoted	denote	VERB
ejpam-3418	257	19	by	by	ADP
ejpam-3418	257	20	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	257	21	]	]	PUNCT
ejpam-3418	257	22	,	,	PUNCT
ejpam-3418	257	23	is	be	AUX
ejpam-3418	257	24	the	the	DET
ejpam-3418	257	25	mapping	mapping	NOUN
ejpam-3418	257	26	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	257	27	]	]	PUNCT
ejpam-3418	257	28	:	:	PUNCT
ejpam-3418	257	29	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	257	30	)	)	PUNCT
ejpam-3418	257	31	)	)	PUNCT
ejpam-3418	258	1	→	→	SYM
ejpam-3418	258	2	i	i	PRON
ejpam-3418	258	3	given	give	VERB
ejpam-3418	258	4	by	by	ADP
ejpam-3418	258	5	(	(	PUNCT
ejpam-3418	258	6	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	258	7	]	]	PUNCT
ejpam-3418	258	8	)	)	PUNCT
ejpam-3418	258	9	(	(	PUNCT
ejpam-3418	258	10	a	a	X
ejpam-3418	258	11	)	)	PUNCT
ejpam-3418	258	12	=	=	SYM
ejpam-3418	258	13	(	(	PUNCT
ejpam-3418	258	14	τ̂	τ̂	PUNCT
ejpam-3418	258	15	◦	◦	NOUN
ejpam-3418	258	16	f)(a	f)(a	NUM
ejpam-3418	258	17	)	)	PUNCT
ejpam-3418	258	18	,	,	PUNCT
ejpam-3418	258	19	where	where	SCONJ
ejpam-3418	258	20	(	(	PUNCT
ejpam-3418	258	21	τ̂	τ̂	NUM
ejpam-3418	258	22	◦	◦	NOUN
ejpam-3418	258	23	f)(a	f)(a	NOUN
ejpam-3418	258	24	)	)	PUNCT
ejpam-3418	258	25	is	be	AUX
ejpam-3418	258	26	the	the	DET
ejpam-3418	258	27	composition	composition	NOUN
ejpam-3418	258	28	τ̂(f(a	τ̂(f(a	NOUN
ejpam-3418	258	29	)	)	PUNCT
ejpam-3418	258	30	)	)	PUNCT
ejpam-3418	258	31	.	.	PUNCT
ejpam-3418	259	1	let	let	VERB
ejpam-3418	259	2	b	b	X
ejpam-3418	259	3	∈	∈	PROPN
ejpam-3418	259	4	f(i(x	f(i(x	NOUN
ejpam-3418	259	5	)	)	PUNCT
ejpam-3418	259	6	)	)	PUNCT
ejpam-3418	259	7	and	and	CCONJ
ejpam-3418	259	8	s	s	X
ejpam-3418	259	9	=	=	X
ejpam-3418	259	10	{	{	PUNCT
ejpam-3418	259	11	a	a	DET
ejpam-3418	259	12	∈	∈	PROPN
ejpam-3418	259	13	i(x	i(x	NOUN
ejpam-3418	259	14	)	)	PUNCT
ejpam-3418	259	15	:	:	PUNCT
ejpam-3418	260	1	f(a	f(a	X
ejpam-3418	260	2	)	)	PUNCT
ejpam-3418	260	3	=	=	SYM
ejpam-3418	261	1	b	b	X
ejpam-3418	261	2	}	}	PUNCT
ejpam-3418	261	3	.	.	PUNCT
ejpam-3418	262	1	if	if	SCONJ
ejpam-3418	262	2	b	b	NOUN
ejpam-3418	262	3	=	=	SYM
ejpam-3418	262	4	∅	∅	NOUN
ejpam-3418	262	5	,	,	PUNCT
ejpam-3418	262	6	then	then	ADV
ejpam-3418	262	7	s	s	VERB
ejpam-3418	262	8	=	=	NOUN
ejpam-3418	262	9	{	{	PUNCT
ejpam-3418	262	10	∅	∅	NOUN
ejpam-3418	262	11	}	}	PUNCT
ejpam-3418	262	12	,	,	PUNCT
ejpam-3418	262	13	and	and	CCONJ
ejpam-3418	262	14	so	so	ADV
ejpam-3418	262	15	sup	sup	PROPN
ejpam-3418	262	16	a∈s	a∈s	PROPN
ejpam-3418	262	17	ι̂(a	ι̂(a	PUNCT
ejpam-3418	262	18	)	)	PUNCT
ejpam-3418	262	19	=	=	PUNCT
ejpam-3418	263	1	[	[	X
ejpam-3418	263	2	0	0	NUM
ejpam-3418	263	3	,	,	PUNCT
ejpam-3418	263	4	0	0	NUM
ejpam-3418	263	5	]	]	PUNCT
ejpam-3418	263	6	.	.	PUNCT
ejpam-3418	264	1	also	also	ADV
ejpam-3418	264	2	,	,	PUNCT
ejpam-3418	264	3	if	if	SCONJ
ejpam-3418	264	4	∅	∅	NOUN
ejpam-3418	264	5	=	=	PUNCT
ejpam-3418	264	6	a	a	DET
ejpam-3418	264	7	∈	∈	NOUN
ejpam-3418	264	8	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	264	9	)	)	PUNCT
ejpam-3418	264	10	)	)	PUNCT
ejpam-3418	264	11	,	,	PUNCT
ejpam-3418	264	12	then	then	ADV
ejpam-3418	264	13	f(a	f(a	PUNCT
ejpam-3418	264	14	)	)	PUNCT
ejpam-3418	265	1	=	=	NOUN
ejpam-3418	265	2	∅	∅	NOUN
ejpam-3418	265	3	and	and	CCONJ
ejpam-3418	265	4	so	so	ADV
ejpam-3418	265	5	τ̂(f(a	τ̂(f(a	NOUN
ejpam-3418	265	6	)	)	PUNCT
ejpam-3418	265	7	)	)	PUNCT
ejpam-3418	266	1	=	=	PUNCT
ejpam-3418	267	1	[	[	X
ejpam-3418	267	2	0	0	NUM
ejpam-3418	267	3	,	,	PUNCT
ejpam-3418	267	4	0	0	NUM
ejpam-3418	267	5	]	]	PUNCT
ejpam-3418	267	6	.	.	PUNCT
ejpam-3418	268	1	hence	hence	ADV
ejpam-3418	268	2	,	,	PUNCT
ejpam-3418	268	3	we	we	PRON
ejpam-3418	268	4	have	have	VERB
ejpam-3418	268	5	the	the	DET
ejpam-3418	268	6	following	follow	VERB
ejpam-3418	268	7	remark	remark	NOUN
ejpam-3418	268	8	.	.	PUNCT
ejpam-3418	269	1	remark	remark	PROPN
ejpam-3418	269	2	5	5	NUM
ejpam-3418	269	3	.	.	PUNCT
ejpam-3418	270	1	let	let	VERB
ejpam-3418	270	2	x	x	PRON
ejpam-3418	270	3	and	and	CCONJ
ejpam-3418	270	4	y	y	PROPN
ejpam-3418	270	5	be	be	AUX
ejpam-3418	270	6	nonempty	nonempty	X
ejpam-3418	270	7	sets	set	NOUN
ejpam-3418	270	8	and	and	CCONJ
ejpam-3418	270	9	f	f	NOUN
ejpam-3418	270	10	:	:	PUNCT
ejpam-3418	270	11	x	x	X
ejpam-3418	270	12	→	→	SYM
ejpam-3418	270	13	y	y	X
ejpam-3418	270	14	be	be	AUX
ejpam-3418	270	15	a	a	DET
ejpam-3418	270	16	mapping	mapping	NOUN
ejpam-3418	270	17	.	.	PUNCT
ejpam-3418	271	1	let	let	VERB
ejpam-3418	271	2	ι̂	ι̂	PRON
ejpam-3418	271	3	and	and	CCONJ
ejpam-3418	271	4	τ̂	τ̂	AUX
ejpam-3418	271	5	be	be	AUX
ejpam-3418	271	6	ivfi	ivfi	NOUN
ejpam-3418	271	7	sets	set	NOUN
ejpam-3418	271	8	in	in	ADP
ejpam-3418	271	9	i	i	PRON
ejpam-3418	271	10	i(x	i(x	PROPN
ejpam-3418	271	11	)	)	PUNCT
ejpam-3418	271	12	and	and	CCONJ
ejpam-3418	271	13	i	i	PRON
ejpam-3418	271	14	i(y	i(y	NOUN
ejpam-3418	271	15	)	)	PUNCT
ejpam-3418	271	16	,	,	PUNCT
ejpam-3418	271	17	respectively	respectively	ADV
ejpam-3418	271	18	.	.	PUNCT
ejpam-3418	272	1	then	then	ADV
ejpam-3418	272	2	,	,	PUNCT
ejpam-3418	272	3	we	we	PRON
ejpam-3418	272	4	have	have	VERB
ejpam-3418	272	5	f	f	X
ejpam-3418	272	6	[	[	X
ejpam-3418	272	7	̂ι](∅	̂ι](∅	NOUN
ejpam-3418	272	8	)	)	PUNCT
ejpam-3418	272	9	=	=	PUNCT
ejpam-3418	273	1	[	[	X
ejpam-3418	273	2	0	0	NUM
ejpam-3418	273	3	,	,	PUNCT
ejpam-3418	273	4	0	0	NUM
ejpam-3418	273	5	]	]	PUNCT
ejpam-3418	273	6	and	and	CCONJ
ejpam-3418	273	7	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	273	8	]	]	PUNCT
ejpam-3418	273	9	(	(	PUNCT
ejpam-3418	273	10	∅	∅	NOUN
ejpam-3418	273	11	)	)	PUNCT
ejpam-3418	273	12	=	=	PUNCT
ejpam-3418	274	1	[	[	X
ejpam-3418	274	2	0	0	NUM
ejpam-3418	274	3	,	,	PUNCT
ejpam-3418	274	4	0	0	NUM
ejpam-3418	274	5	]	]	PUNCT
ejpam-3418	274	6	.	.	PUNCT
ejpam-3418	275	1	theorem	theorem	ADJ
ejpam-3418	275	2	4	4	NUM
ejpam-3418	275	3	.	.	PUNCT
ejpam-3418	276	1	let	let	AUX
ejpam-3418	276	2	x	x	PRON
ejpam-3418	276	3	and	and	CCONJ
ejpam-3418	276	4	y	y	PROPN
ejpam-3418	276	5	be	be	AUX
ejpam-3418	276	6	nonempty	nonempty	X
ejpam-3418	276	7	sets	set	NOUN
ejpam-3418	276	8	and	and	CCONJ
ejpam-3418	276	9	f	f	NOUN
ejpam-3418	276	10	:	:	PUNCT
ejpam-3418	276	11	x	x	X
ejpam-3418	276	12	→	→	SYM
ejpam-3418	276	13	y	y	X
ejpam-3418	276	14	be	be	AUX
ejpam-3418	276	15	a	a	DET
ejpam-3418	276	16	mapping	mapping	NOUN
ejpam-3418	276	17	.	.	PUNCT
ejpam-3418	277	1	let	let	VERB
ejpam-3418	277	2	ι̂	ι̂	PRON
ejpam-3418	277	3	and	and	CCONJ
ejpam-3418	277	4	τ̂	τ̂	AUX
ejpam-3418	277	5	be	be	AUX
ejpam-3418	277	6	ivfi	ivfi	NOUN
ejpam-3418	277	7	sets	set	NOUN
ejpam-3418	277	8	defined	define	VERB
ejpam-3418	277	9	on	on	ADP
ejpam-3418	277	10	the	the	DET
ejpam-3418	277	11	ideals	ideal	NOUN
ejpam-3418	277	12	i(x	i(x	PROPN
ejpam-3418	277	13	)	)	PUNCT
ejpam-3418	277	14	and	and	CCONJ
ejpam-3418	277	15	i(y	i(y	NOUN
ejpam-3418	277	16	)	)	PUNCT
ejpam-3418	277	17	,	,	PUNCT
ejpam-3418	277	18	respectively	respectively	ADV
ejpam-3418	277	19	.	.	PUNCT
ejpam-3418	278	1	then	then	ADV
ejpam-3418	278	2	,	,	PUNCT
ejpam-3418	278	3	f	f	PROPN
ejpam-3418	279	1	[	[	X
ejpam-3418	279	2	̂ι	̂ι	X
ejpam-3418	279	3	]	]	PUNCT
ejpam-3418	279	4	and	and	CCONJ
ejpam-3418	279	5	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	279	6	]	]	PUNCT
ejpam-3418	279	7	are	be	AUX
ejpam-3418	279	8	ivfi	ivfi	NOUN
ejpam-3418	279	9	sets	set	NOUN
ejpam-3418	279	10	defined	define	VERB
ejpam-3418	279	11	on	on	ADP
ejpam-3418	279	12	the	the	DET
ejpam-3418	279	13	ideals	ideal	NOUN
ejpam-3418	279	14	f(i(x	f(i(x	NUM
ejpam-3418	279	15	)	)	PUNCT
ejpam-3418	279	16	)	)	PUNCT
ejpam-3418	279	17	and	and	CCONJ
ejpam-3418	279	18	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	279	19	)	)	PUNCT
ejpam-3418	279	20	)	)	PUNCT
ejpam-3418	279	21	,	,	PUNCT
ejpam-3418	279	22	respectively	respectively	ADV
ejpam-3418	279	23	.	.	PUNCT
ejpam-3418	280	1	proof	proof	NOUN
ejpam-3418	280	2	.	.	PUNCT
ejpam-3418	281	1	we	we	PRON
ejpam-3418	281	2	first	first	ADV
ejpam-3418	281	3	show	show	VERB
ejpam-3418	281	4	that	that	SCONJ
ejpam-3418	282	1	f	f	PROPN
ejpam-3418	283	1	[	[	X
ejpam-3418	283	2	̂ι	̂ι	X
ejpam-3418	283	3	]	]	PUNCT
ejpam-3418	283	4	is	be	AUX
ejpam-3418	283	5	an	an	DET
ejpam-3418	283	6	ivfi	ivfi	NOUN
ejpam-3418	283	7	set	set	VERB
ejpam-3418	283	8	defined	define	VERB
ejpam-3418	283	9	on	on	ADP
ejpam-3418	283	10	the	the	DET
ejpam-3418	283	11	ideal	ideal	ADJ
ejpam-3418	283	12	f(i(x	f(i(x	NOUN
ejpam-3418	283	13	)	)	PUNCT
ejpam-3418	283	14	)	)	PUNCT
ejpam-3418	283	15	.	.	PUNCT
ejpam-3418	284	1	let	let	VERB
ejpam-3418	284	2	∅	∅	NOUN
ejpam-3418	284	3	6=	6=	ADP
ejpam-3418	284	4	b1	b1	NOUN
ejpam-3418	284	5	,	,	PUNCT
ejpam-3418	284	6	b2	b2	NOUN
ejpam-3418	284	7	∈	∈	PROPN
ejpam-3418	284	8	f(i(x	f(i(x	NOUN
ejpam-3418	284	9	)	)	PUNCT
ejpam-3418	284	10	)	)	PUNCT
ejpam-3418	285	1	such	such	ADJ
ejpam-3418	285	2	that	that	PRON
ejpam-3418	285	3	b1	b1	VERB
ejpam-3418	285	4	⊆	⊆	NUM
ejpam-3418	285	5	b2	b2	NOUN
ejpam-3418	285	6	.	.	PUNCT
ejpam-3418	286	1	let	let	VERB
ejpam-3418	286	2	s1	s1	PROPN
ejpam-3418	286	3	=	=	PUNCT
ejpam-3418	286	4	{	{	PUNCT
ejpam-3418	286	5	a	a	DET
ejpam-3418	286	6	∈	∈	PROPN
ejpam-3418	286	7	i(x	i(x	NOUN
ejpam-3418	286	8	)	)	PUNCT
ejpam-3418	286	9	:	:	PUNCT
ejpam-3418	286	10	f(a	f(a	X
ejpam-3418	286	11	)	)	PUNCT
ejpam-3418	286	12	=	=	SYM
ejpam-3418	286	13	b1	b1	NOUN
ejpam-3418	286	14	}	}	PUNCT
ejpam-3418	286	15	and	and	CCONJ
ejpam-3418	286	16	s2	s2	PROPN
ejpam-3418	286	17	=	=	SYM
ejpam-3418	286	18	{	{	PUNCT
ejpam-3418	286	19	a	a	DET
ejpam-3418	286	20	∈	∈	PROPN
ejpam-3418	286	21	i(x	i(x	NOUN
ejpam-3418	286	22	)	)	PUNCT
ejpam-3418	286	23	:	:	PUNCT
ejpam-3418	286	24	f(a	f(a	X
ejpam-3418	286	25	)	)	PUNCT
ejpam-3418	286	26	=	=	SYM
ejpam-3418	286	27	b2	b2	NOUN
ejpam-3418	286	28	}	}	PUNCT
ejpam-3418	286	29	.	.	PUNCT
ejpam-3418	287	1	since	since	SCONJ
ejpam-3418	287	2	i(x	i(x	PROPN
ejpam-3418	287	3	)	)	PUNCT
ejpam-3418	287	4	is	be	AUX
ejpam-3418	287	5	an	an	DET
ejpam-3418	287	6	ideal	ideal	NOUN
ejpam-3418	287	7	,	,	PUNCT
ejpam-3418	287	8	for	for	ADP
ejpam-3418	287	9	every	every	DET
ejpam-3418	287	10	a	a	DET
ejpam-3418	287	11	∈	∈	PROPN
ejpam-3418	287	12	i(x	i(x	NOUN
ejpam-3418	287	13	)	)	PUNCT
ejpam-3418	287	14	such	such	ADJ
ejpam-3418	287	15	that	that	DET
ejpam-3418	287	16	f(a	f(a	NOUN
ejpam-3418	287	17	)	)	PUNCT
ejpam-3418	287	18	=	=	SYM
ejpam-3418	287	19	b2	b2	NOUN
ejpam-3418	287	20	,	,	PUNCT
ejpam-3418	287	21	there	there	PRON
ejpam-3418	287	22	exists	exist	VERB
ejpam-3418	287	23	a1	a1	PROPN
ejpam-3418	287	24	∈	∈	PROPN
ejpam-3418	287	25	i(x	i(x	NOUN
ejpam-3418	287	26	)	)	PUNCT
ejpam-3418	287	27	such	such	ADJ
ejpam-3418	287	28	that	that	DET
ejpam-3418	287	29	a1	a1	NOUN
ejpam-3418	287	30	⊆	⊆	NUM
ejpam-3418	287	31	a	a	PRON
ejpam-3418	287	32	and	and	CCONJ
ejpam-3418	287	33	f(a1	f(a1	NOUN
ejpam-3418	287	34	)	)	PUNCT
ejpam-3418	287	35	=	=	SYM
ejpam-3418	287	36	b1	b1	NOUN
ejpam-3418	287	37	.	.	PUNCT
ejpam-3418	288	1	since	since	SCONJ
ejpam-3418	288	2	ι̂	ι̂	NUM
ejpam-3418	288	3	is	be	AUX
ejpam-3418	288	4	an	an	DET
ejpam-3418	288	5	ivfi	ivfi	NOUN
ejpam-3418	288	6	set	set	VERB
ejpam-3418	288	7	and	and	CCONJ
ejpam-3418	288	8	a1	a1	VERB
ejpam-3418	288	9	⊆	⊆	NUM
ejpam-3418	288	10	a	a	PRON
ejpam-3418	288	11	,	,	PUNCT
ejpam-3418	288	12	we	we	PRON
ejpam-3418	288	13	have	have	VERB
ejpam-3418	288	14	ι̂(a	ι̂(a	NOUN
ejpam-3418	288	15	)	)	PUNCT
ejpam-3418	288	16	≤i	≤i	NOUN
ejpam-3418	288	17	ι̂(a1	ι̂(a1	PROPN
ejpam-3418	288	18	)	)	PUNCT
ejpam-3418	288	19	.	.	PUNCT
ejpam-3418	289	1	hence	hence	ADV
ejpam-3418	289	2	,	,	PUNCT
ejpam-3418	289	3	supa∈s2	supa∈s2	PROPN
ejpam-3418	289	4	ι̂(a	ι̂(a	PUNCT
ejpam-3418	289	5	)	)	PUNCT
ejpam-3418	289	6	≤i	≤i	PROPN
ejpam-3418	289	7	supa1∈s1	supa1∈s1	PROPN
ejpam-3418	289	8	ι̂(a1	ι̂(a1	PROPN
ejpam-3418	289	9	)	)	PUNCT
ejpam-3418	289	10	.	.	PUNCT
ejpam-3418	290	1	thus	thus	ADV
ejpam-3418	290	2	,	,	PUNCT
ejpam-3418	290	3	f	f	PROPN
ejpam-3418	290	4	[	[	X
ejpam-3418	290	5	̂ι](b2	̂ι](b2	X
ejpam-3418	290	6	)	)	PUNCT
ejpam-3418	290	7	≤i	≤i	PROPN
ejpam-3418	290	8	f	f	PROPN
ejpam-3418	291	1	[	[	X
ejpam-3418	291	2	̂ι](b1	̂ι](b1	NOUN
ejpam-3418	291	3	)	)	PUNCT
ejpam-3418	291	4	.	.	PUNCT
ejpam-3418	292	1	therefore	therefore	ADV
ejpam-3418	292	2	,	,	PUNCT
ejpam-3418	292	3	with	with	ADP
ejpam-3418	292	4	remark	remark	NOUN
ejpam-3418	292	5	5	5	NUM
ejpam-3418	292	6	and	and	CCONJ
ejpam-3418	292	7	theorem	theorem	VERB
ejpam-3418	292	8	3	3	NUM
ejpam-3418	292	9	,	,	PUNCT
ejpam-3418	292	10	f	f	PROPN
ejpam-3418	293	1	[	[	X
ejpam-3418	293	2	̂ι	̂ι	X
ejpam-3418	293	3	]	]	PUNCT
ejpam-3418	293	4	is	be	AUX
ejpam-3418	293	5	an	an	DET
ejpam-3418	293	6	ivfi	ivfi	NOUN
ejpam-3418	293	7	set	set	VERB
ejpam-3418	293	8	defined	define	VERB
ejpam-3418	293	9	on	on	ADP
ejpam-3418	293	10	the	the	DET
ejpam-3418	293	11	ideal	ideal	ADJ
ejpam-3418	293	12	f(i(x	f(i(x	NOUN
ejpam-3418	293	13	)	)	PUNCT
ejpam-3418	293	14	)	)	PUNCT
ejpam-3418	293	15	.	.	PUNCT
ejpam-3418	294	1	next	next	ADV
ejpam-3418	294	2	,	,	PUNCT
ejpam-3418	294	3	to	to	PART
ejpam-3418	294	4	show	show	VERB
ejpam-3418	294	5	that	that	SCONJ
ejpam-3418	294	6	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	294	7	]	]	PUNCT
ejpam-3418	294	8	is	be	AUX
ejpam-3418	294	9	an	an	DET
ejpam-3418	294	10	ivfi	ivfi	NOUN
ejpam-3418	294	11	set	set	VERB
ejpam-3418	294	12	defined	define	VERB
ejpam-3418	294	13	on	on	ADP
ejpam-3418	294	14	the	the	DET
ejpam-3418	294	15	ideal	ideal	ADJ
ejpam-3418	294	16	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	294	17	)	)	PUNCT
ejpam-3418	294	18	)	)	PUNCT
ejpam-3418	294	19	,	,	PUNCT
ejpam-3418	294	20	let	let	VERB
ejpam-3418	294	21	a	a	DET
ejpam-3418	294	22	,	,	PUNCT
ejpam-3418	294	23	a1	a1	NOUN
ejpam-3418	294	24	∈	∈	NOUN
ejpam-3418	294	25	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	294	26	)	)	PUNCT
ejpam-3418	294	27	)	)	PUNCT
ejpam-3418	294	28	such	such	ADJ
ejpam-3418	294	29	that	that	DET
ejpam-3418	294	30	a1	a1	NOUN
ejpam-3418	294	31	⊆	⊆	NUM
ejpam-3418	294	32	a.	a.	NOUN
ejpam-3418	294	33	then	then	ADV
ejpam-3418	294	34	,	,	PUNCT
ejpam-3418	294	35	f(a1	f(a1	NOUN
ejpam-3418	294	36	)	)	PUNCT
ejpam-3418	294	37	⊆	⊆	NUM
ejpam-3418	294	38	f(a	f(a	NOUN
ejpam-3418	294	39	)	)	PUNCT
ejpam-3418	294	40	and	and	CCONJ
ejpam-3418	294	41	f(a1	f(a1	NOUN
ejpam-3418	294	42	)	)	PUNCT
ejpam-3418	294	43	,	,	PUNCT
ejpam-3418	294	44	f(a	f(a	NOUN
ejpam-3418	294	45	)	)	PUNCT
ejpam-3418	294	46	∈	∈	PROPN
ejpam-3418	294	47	i(y	i(y	NOUN
ejpam-3418	294	48	)	)	PUNCT
ejpam-3418	294	49	.	.	PUNCT
ejpam-3418	295	1	since	since	SCONJ
ejpam-3418	295	2	τ̂	τ̂	NUM
ejpam-3418	295	3	is	be	AUX
ejpam-3418	295	4	an	an	DET
ejpam-3418	295	5	ivfi	ivfi	NOUN
ejpam-3418	295	6	set	set	VERB
ejpam-3418	295	7	on	on	ADP
ejpam-3418	295	8	i(y	i(y	NOUN
ejpam-3418	295	9	)	)	PUNCT
ejpam-3418	295	10	,	,	PUNCT
ejpam-3418	295	11	τ̂(f(a	τ̂(f(a	NOUN
ejpam-3418	295	12	)	)	PUNCT
ejpam-3418	295	13	)	)	PUNCT
ejpam-3418	295	14	≤i	≤i	PROPN
ejpam-3418	295	15	τ̂(f(a1	τ̂(f(a1	PROPN
ejpam-3418	295	16	)	)	PUNCT
ejpam-3418	295	17	)	)	PUNCT
ejpam-3418	295	18	.	.	PUNCT
ejpam-3418	296	1	thus	thus	ADV
ejpam-3418	296	2	,	,	PUNCT
ejpam-3418	296	3	mj	mj	PROPN
ejpam-3418	296	4	togonon	togonon	PROPN
ejpam-3418	296	5	,	,	PUNCT
ejpam-3418	296	6	r	r	NOUN
ejpam-3418	296	7	caga	caga	NOUN
ejpam-3418	296	8	-	-	PUNCT
ejpam-3418	296	9	anan	anan	PROPN
ejpam-3418	296	10	/	/	SYM
ejpam-3418	296	11	eur	eur	PROPN
ejpam-3418	296	12	.	.	PUNCT
ejpam-3418	297	1	j.	j.	PROPN
ejpam-3418	297	2	pure	pure	PROPN
ejpam-3418	297	3	appl	appl	PROPN
ejpam-3418	297	4	.	.	PROPN
ejpam-3418	297	5	math	math	PROPN
ejpam-3418	297	6	,	,	PUNCT
ejpam-3418	297	7	12	12	NUM
ejpam-3418	297	8	(	(	PUNCT
ejpam-3418	297	9	2	2	NUM
ejpam-3418	297	10	)	)	PUNCT
ejpam-3418	297	11	(	(	PUNCT
ejpam-3418	297	12	2019	2019	NUM
ejpam-3418	297	13	)	)	PUNCT
ejpam-3418	297	14	,	,	PUNCT
ejpam-3418	297	15	553	553	NUM
ejpam-3418	297	16	-	-	SYM
ejpam-3418	297	17	570	570	NUM
ejpam-3418	297	18	561	561	NUM
ejpam-3418	297	19	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	297	20	]	]	PUNCT
ejpam-3418	297	21	(	(	PUNCT
ejpam-3418	297	22	a	a	X
ejpam-3418	297	23	)	)	PUNCT
ejpam-3418	297	24	≤i	≤i	NOUN
ejpam-3418	297	25	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	297	26	]	]	PUNCT
ejpam-3418	297	27	(	(	PUNCT
ejpam-3418	297	28	a1	a1	NOUN
ejpam-3418	297	29	)	)	PUNCT
ejpam-3418	297	30	.	.	PUNCT
ejpam-3418	298	1	therefore	therefore	ADV
ejpam-3418	298	2	,	,	PUNCT
ejpam-3418	298	3	with	with	ADP
ejpam-3418	298	4	remark	remark	NOUN
ejpam-3418	298	5	5	5	NUM
ejpam-3418	298	6	and	and	CCONJ
ejpam-3418	298	7	theorem	theorem	VERB
ejpam-3418	298	8	3	3	NUM
ejpam-3418	298	9	,	,	PUNCT
ejpam-3418	298	10	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	298	11	]	]	PUNCT
ejpam-3418	298	12	is	be	AUX
ejpam-3418	298	13	an	an	DET
ejpam-3418	298	14	ivfi	ivfi	NOUN
ejpam-3418	298	15	set	set	VERB
ejpam-3418	298	16	defined	define	VERB
ejpam-3418	298	17	on	on	ADP
ejpam-3418	298	18	the	the	DET
ejpam-3418	298	19	ideal	ideal	ADJ
ejpam-3418	298	20	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	298	21	)	)	PUNCT
ejpam-3418	298	22	)	)	PUNCT
ejpam-3418	298	23	.	.	PUNCT
ejpam-3418	299	1	we	we	PRON
ejpam-3418	299	2	can	can	AUX
ejpam-3418	299	3	extend	extend	VERB
ejpam-3418	299	4	our	our	PRON
ejpam-3418	299	5	result	result	NOUN
ejpam-3418	299	6	to	to	ADP
ejpam-3418	299	7	composition	composition	NOUN
ejpam-3418	299	8	of	of	ADP
ejpam-3418	299	9	mappings	mapping	NOUN
ejpam-3418	299	10	.	.	PUNCT
ejpam-3418	300	1	the	the	DET
ejpam-3418	300	2	following	follow	VERB
ejpam-3418	300	3	corollaries	corollary	NOUN
ejpam-3418	300	4	are	be	AUX
ejpam-3418	300	5	immediate	immediate	ADJ
ejpam-3418	300	6	consequences	consequence	NOUN
ejpam-3418	300	7	of	of	ADP
ejpam-3418	300	8	theorem	theorem	ADJ
ejpam-3418	300	9	3	3	NUM
ejpam-3418	300	10	and	and	CCONJ
ejpam-3418	300	11	theorem	theorem	VERB
ejpam-3418	300	12	5	5	NUM
ejpam-3418	300	13	.	.	PUNCT
ejpam-3418	300	14	corollary	corollary	ADJ
ejpam-3418	300	15	1	1	NUM
ejpam-3418	300	16	.	.	PUNCT
ejpam-3418	301	1	let	let	VERB
ejpam-3418	301	2	x	x	PRON
ejpam-3418	301	3	,	,	PUNCT
ejpam-3418	301	4	y	y	PROPN
ejpam-3418	301	5	,	,	PUNCT
ejpam-3418	301	6	and	and	CCONJ
ejpam-3418	301	7	z	z	NOUN
ejpam-3418	301	8	be	be	VERB
ejpam-3418	301	9	nonempty	nonempty	ADJ
ejpam-3418	301	10	sets	set	NOUN
ejpam-3418	301	11	and	and	CCONJ
ejpam-3418	301	12	f	f	NOUN
ejpam-3418	301	13	:	:	PUNCT
ejpam-3418	301	14	x	x	X
ejpam-3418	301	15	→	→	SYM
ejpam-3418	301	16	y	y	PROPN
ejpam-3418	301	17	and	and	CCONJ
ejpam-3418	301	18	g	g	PROPN
ejpam-3418	301	19	:	:	PUNCT
ejpam-3418	301	20	y	y	PROPN
ejpam-3418	301	21	→	→	SYM
ejpam-3418	301	22	z	z	AUX
ejpam-3418	301	23	be	be	AUX
ejpam-3418	301	24	mappings	mapping	NOUN
ejpam-3418	301	25	.	.	PUNCT
ejpam-3418	302	1	let	let	VERB
ejpam-3418	302	2	g	g	NOUN
ejpam-3418	302	3	◦	◦	VERB
ejpam-3418	302	4	f	f	NOUN
ejpam-3418	302	5	:	:	PUNCT
ejpam-3418	302	6	x	x	X
ejpam-3418	302	7	→	→	SYM
ejpam-3418	302	8	z	z	NOUN
ejpam-3418	302	9	be	be	AUX
ejpam-3418	302	10	a	a	DET
ejpam-3418	302	11	composition	composition	NOUN
ejpam-3418	302	12	map	map	NOUN
ejpam-3418	302	13	.	.	PUNCT
ejpam-3418	303	1	if	if	SCONJ
ejpam-3418	303	2	i(x	i(x	PROPN
ejpam-3418	303	3	)	)	PUNCT
ejpam-3418	303	4	and	and	CCONJ
ejpam-3418	303	5	i(z	i(z	NOUN
ejpam-3418	303	6	)	)	PUNCT
ejpam-3418	303	7	are	be	AUX
ejpam-3418	303	8	ideals	ideal	NOUN
ejpam-3418	303	9	on	on	ADP
ejpam-3418	303	10	x	x	X
ejpam-3418	303	11	and	and	CCONJ
ejpam-3418	303	12	z	z	PROPN
ejpam-3418	303	13	,	,	PUNCT
ejpam-3418	303	14	respectively	respectively	ADV
ejpam-3418	303	15	,	,	PUNCT
ejpam-3418	303	16	then	then	ADV
ejpam-3418	303	17	,	,	PUNCT
ejpam-3418	303	18	(	(	PUNCT
ejpam-3418	303	19	g	g	ADP
ejpam-3418	303	20	◦	◦	NOUN
ejpam-3418	303	21	f)(i(x	f)(i(x	NUM
ejpam-3418	303	22	)	)	PUNCT
ejpam-3418	303	23	)	)	PUNCT
ejpam-3418	304	1	=	=	PUNCT
ejpam-3418	304	2	g(f(i(x	g(f(i(x	PROPN
ejpam-3418	304	3	)	)	PUNCT
ejpam-3418	304	4	)	)	PUNCT
ejpam-3418	304	5	)	)	PUNCT
ejpam-3418	305	1	and	and	CCONJ
ejpam-3418	305	2	(	(	PUNCT
ejpam-3418	305	3	g	g	PROPN
ejpam-3418	305	4	◦	◦	NOUN
ejpam-3418	305	5	f)−1(i(z	f)−1(i(z	NOUN
ejpam-3418	305	6	)	)	PUNCT
ejpam-3418	305	7	)	)	PUNCT
ejpam-3418	306	1	=	=	SYM
ejpam-3418	306	2	f−1(g−1(i(z	f−1(g−1(i(z	NOUN
ejpam-3418	306	3	)	)	PUNCT
ejpam-3418	306	4	)	)	PUNCT
ejpam-3418	306	5	)	)	PUNCT
ejpam-3418	306	6	are	be	AUX
ejpam-3418	306	7	ideals	ideal	NOUN
ejpam-3418	306	8	on	on	ADP
ejpam-3418	306	9	z	z	PROPN
ejpam-3418	306	10	and	and	CCONJ
ejpam-3418	306	11	x	x	NOUN
ejpam-3418	306	12	,	,	PUNCT
ejpam-3418	306	13	respectively	respectively	ADV
ejpam-3418	306	14	.	.	PUNCT
ejpam-3418	307	1	corollary	corollary	ADJ
ejpam-3418	307	2	2	2	NUM
ejpam-3418	307	3	.	.	PUNCT
ejpam-3418	308	1	let	let	VERB
ejpam-3418	308	2	x	x	PRON
ejpam-3418	308	3	,	,	PUNCT
ejpam-3418	308	4	y	y	PROPN
ejpam-3418	308	5	,	,	PUNCT
ejpam-3418	308	6	and	and	CCONJ
ejpam-3418	308	7	z	z	NOUN
ejpam-3418	308	8	be	be	VERB
ejpam-3418	308	9	nonempty	nonempty	ADJ
ejpam-3418	308	10	sets	set	NOUN
ejpam-3418	308	11	and	and	CCONJ
ejpam-3418	308	12	f	f	NOUN
ejpam-3418	308	13	:	:	PUNCT
ejpam-3418	308	14	x	x	X
ejpam-3418	308	15	→	→	SYM
ejpam-3418	308	16	y	y	PROPN
ejpam-3418	308	17	and	and	CCONJ
ejpam-3418	308	18	g	g	PROPN
ejpam-3418	308	19	:	:	PUNCT
ejpam-3418	308	20	y	y	PROPN
ejpam-3418	308	21	→	→	SYM
ejpam-3418	308	22	z	z	AUX
ejpam-3418	308	23	be	be	AUX
ejpam-3418	308	24	mappings	mapping	NOUN
ejpam-3418	308	25	.	.	PUNCT
ejpam-3418	309	1	let	let	VERB
ejpam-3418	309	2	g	g	NOUN
ejpam-3418	309	3	◦	◦	VERB
ejpam-3418	310	1	f	f	X
ejpam-3418	310	2	:	:	PUNCT
ejpam-3418	310	3	x	x	X
ejpam-3418	310	4	→	→	SYM
ejpam-3418	310	5	z	z	NOUN
ejpam-3418	310	6	be	be	AUX
ejpam-3418	310	7	a	a	DET
ejpam-3418	310	8	composition	composition	NOUN
ejpam-3418	310	9	map	map	NOUN
ejpam-3418	310	10	and	and	CCONJ
ejpam-3418	310	11	,	,	PUNCT
ejpam-3418	310	12	i(x	i(x	PROPN
ejpam-3418	310	13	)	)	PUNCT
ejpam-3418	310	14	and	and	CCONJ
ejpam-3418	310	15	i(z	i(z	NOUN
ejpam-3418	310	16	)	)	PUNCT
ejpam-3418	310	17	are	be	AUX
ejpam-3418	310	18	ideals	ideal	NOUN
ejpam-3418	310	19	on	on	ADP
ejpam-3418	310	20	x	x	X
ejpam-3418	310	21	and	and	CCONJ
ejpam-3418	310	22	z	z	PROPN
ejpam-3418	310	23	,	,	PUNCT
ejpam-3418	310	24	respectively	respectively	ADV
ejpam-3418	310	25	.	.	PUNCT
ejpam-3418	311	1	if	if	SCONJ
ejpam-3418	311	2	ι̂	ι̂	NUM
ejpam-3418	311	3	∈	∈	PROPN
ejpam-3418	312	1	i	i	PRON
ejpam-3418	312	2	i(x	i(x	PROPN
ejpam-3418	312	3	)	)	PUNCT
ejpam-3418	312	4	and	and	CCONJ
ejpam-3418	312	5	η̂	η̂	PRON
ejpam-3418	312	6	∈	∈	NOUN
ejpam-3418	312	7	i	i	PRON
ejpam-3418	312	8	i(z	i(z	VERB
ejpam-3418	312	9	)	)	PUNCT
ejpam-3418	312	10	,	,	PUNCT
ejpam-3418	312	11	then	then	ADV
ejpam-3418	312	12	(	(	PUNCT
ejpam-3418	312	13	g	g	NOUN
ejpam-3418	312	14	◦	◦	NOUN
ejpam-3418	312	15	f)[̂ι	f)[̂ι	NOUN
ejpam-3418	312	16	]	]	PUNCT
ejpam-3418	312	17	and	and	CCONJ
ejpam-3418	312	18	(	(	PUNCT
ejpam-3418	312	19	g	g	NOUN
ejpam-3418	312	20	◦	◦	NOUN
ejpam-3418	312	21	f)−1(η̂	f)−1(η̂	ADJ
ejpam-3418	312	22	)	)	PUNCT
ejpam-3418	312	23	are	be	AUX
ejpam-3418	312	24	ivfi	ivfi	NOUN
ejpam-3418	312	25	sets	set	NOUN
ejpam-3418	312	26	defined	define	VERB
ejpam-3418	312	27	on	on	ADP
ejpam-3418	312	28	(	(	PUNCT
ejpam-3418	312	29	g	g	ADP
ejpam-3418	312	30	◦	◦	NOUN
ejpam-3418	312	31	f)(i(x	f)(i(x	NUM
ejpam-3418	312	32	)	)	PUNCT
ejpam-3418	312	33	)	)	PUNCT
ejpam-3418	313	1	and	and	CCONJ
ejpam-3418	313	2	(	(	PUNCT
ejpam-3418	313	3	g	g	PROPN
ejpam-3418	313	4	◦	◦	NOUN
ejpam-3418	313	5	f)−1(i(z	f)−1(i(z	NOUN
ejpam-3418	313	6	)	)	PUNCT
ejpam-3418	313	7	)	)	PUNCT
ejpam-3418	313	8	,	,	PUNCT
ejpam-3418	313	9	respectively	respectively	ADV
ejpam-3418	313	10	.	.	PUNCT
ejpam-3418	314	1	the	the	DET
ejpam-3418	314	2	next	next	ADJ
ejpam-3418	314	3	theorem	theorem	ADJ
ejpam-3418	314	4	state	state	NOUN
ejpam-3418	314	5	some	some	DET
ejpam-3418	314	6	properties	property	NOUN
ejpam-3418	314	7	of	of	ADP
ejpam-3418	314	8	the	the	DET
ejpam-3418	314	9	defined	define	VERB
ejpam-3418	314	10	mappings	mapping	NOUN
ejpam-3418	314	11	of	of	ADP
ejpam-3418	314	12	ivfi	ivfi	NOUN
ejpam-3418	314	13	sets	set	NOUN
ejpam-3418	314	14	.	.	PUNCT
ejpam-3418	315	1	we	we	PRON
ejpam-3418	315	2	start	start	VERB
ejpam-3418	315	3	with	with	ADP
ejpam-3418	315	4	the	the	DET
ejpam-3418	315	5	following	follow	VERB
ejpam-3418	315	6	needed	need	VERB
ejpam-3418	315	7	proposition	proposition	NOUN
ejpam-3418	315	8	.	.	PUNCT
ejpam-3418	316	1	proposition	proposition	NOUN
ejpam-3418	316	2	2	2	NUM
ejpam-3418	316	3	.	.	PUNCT
ejpam-3418	317	1	let	let	AUX
ejpam-3418	317	2	x	x	PRON
ejpam-3418	317	3	and	and	CCONJ
ejpam-3418	317	4	y	y	PROPN
ejpam-3418	317	5	be	be	AUX
ejpam-3418	317	6	nonempty	nonempty	X
ejpam-3418	317	7	sets	set	NOUN
ejpam-3418	317	8	and	and	CCONJ
ejpam-3418	317	9	,	,	PUNCT
ejpam-3418	317	10	i(x	i(x	PROPN
ejpam-3418	317	11	)	)	PUNCT
ejpam-3418	317	12	and	and	CCONJ
ejpam-3418	317	13	i(y	i(y	NOUN
ejpam-3418	317	14	)	)	PUNCT
ejpam-3418	317	15	be	be	AUX
ejpam-3418	317	16	ideals	ideal	NOUN
ejpam-3418	317	17	in	in	ADP
ejpam-3418	317	18	x	x	PUNCT
ejpam-3418	317	19	and	and	CCONJ
ejpam-3418	317	20	y	y	PROPN
ejpam-3418	317	21	,	,	PUNCT
ejpam-3418	317	22	respectively	respectively	ADV
ejpam-3418	317	23	.	.	PUNCT
ejpam-3418	318	1	let	let	VERB
ejpam-3418	318	2	f	f	NOUN
ejpam-3418	318	3	:	:	PUNCT
ejpam-3418	318	4	x	x	X
ejpam-3418	318	5	→	→	SYM
ejpam-3418	318	6	y	y	X
ejpam-3418	318	7	be	be	AUX
ejpam-3418	318	8	a	a	DET
ejpam-3418	318	9	mapping	mapping	NOUN
ejpam-3418	318	10	.	.	PUNCT
ejpam-3418	319	1	then	then	ADV
ejpam-3418	319	2	,	,	PUNCT
ejpam-3418	319	3	i.	i.	PROPN
ejpam-3418	319	4	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	319	5	)	)	PUNCT
ejpam-3418	319	6	)	)	PUNCT
ejpam-3418	319	7	)	)	PUNCT
ejpam-3418	320	1	=	=	PUNCT
ejpam-3418	320	2	i(y	i(y	NOUN
ejpam-3418	320	3	)	)	PUNCT
ejpam-3418	320	4	,	,	PUNCT
ejpam-3418	320	5	if	if	SCONJ
ejpam-3418	320	6	f	f	PROPN
ejpam-3418	320	7	is	be	AUX
ejpam-3418	320	8	onto	onto	ADP
ejpam-3418	320	9	;	;	PUNCT
ejpam-3418	320	10	and	and	CCONJ
ejpam-3418	320	11	ii	ii	PROPN
ejpam-3418	320	12	.	.	PUNCT
ejpam-3418	320	13	f−1(f(i(x	f−1(f(i(x	PROPN
ejpam-3418	320	14	)	)	PUNCT
ejpam-3418	320	15	)	)	PUNCT
ejpam-3418	320	16	)	)	PUNCT
ejpam-3418	321	1	=	=	SYM
ejpam-3418	321	2	i(x	i(x	NOUN
ejpam-3418	321	3	)	)	PUNCT
ejpam-3418	321	4	,	,	PUNCT
ejpam-3418	321	5	if	if	SCONJ
ejpam-3418	321	6	f	f	PROPN
ejpam-3418	321	7	is	be	AUX
ejpam-3418	321	8	one	one	NUM
ejpam-3418	321	9	-	-	PUNCT
ejpam-3418	321	10	to	to	ADP
ejpam-3418	321	11	-	-	PUNCT
ejpam-3418	321	12	one	one	NUM
ejpam-3418	321	13	.	.	PUNCT
ejpam-3418	322	1	proof	proof	NOUN
ejpam-3418	322	2	.	.	PUNCT
ejpam-3418	323	1	suppose	suppose	VERB
ejpam-3418	323	2	that	that	SCONJ
ejpam-3418	323	3	f	f	PROPN
ejpam-3418	323	4	is	be	AUX
ejpam-3418	323	5	onto	onto	ADP
ejpam-3418	323	6	.	.	PUNCT
ejpam-3418	324	1	let	let	VERB
ejpam-3418	324	2	b	b	NOUN
ejpam-3418	324	3	∈	∈	PROPN
ejpam-3418	324	4	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	324	5	)	)	PUNCT
ejpam-3418	324	6	)	)	PUNCT
ejpam-3418	324	7	)	)	PUNCT
ejpam-3418	324	8	.	.	PUNCT
ejpam-3418	325	1	then	then	ADV
ejpam-3418	325	2	there	there	PRON
ejpam-3418	325	3	exists	exist	VERB
ejpam-3418	325	4	a	a	DET
ejpam-3418	325	5	∈	∈	NOUN
ejpam-3418	325	6	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	325	7	)	)	PUNCT
ejpam-3418	325	8	)	)	PUNCT
ejpam-3418	325	9	such	such	ADJ
ejpam-3418	325	10	that	that	DET
ejpam-3418	325	11	f(a	f(a	NOUN
ejpam-3418	325	12	)	)	PUNCT
ejpam-3418	326	1	=	=	SYM
ejpam-3418	326	2	b.	b.	PROPN
ejpam-3418	326	3	since	since	SCONJ
ejpam-3418	326	4	a	a	DET
ejpam-3418	326	5	∈	∈	NOUN
ejpam-3418	326	6	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	326	7	)	)	PUNCT
ejpam-3418	326	8	)	)	PUNCT
ejpam-3418	326	9	,	,	PUNCT
ejpam-3418	326	10	a	a	DET
ejpam-3418	326	11	⊆	⊆	NUM
ejpam-3418	326	12	f−1(b1	f−1(b1	NOUN
ejpam-3418	326	13	)	)	PUNCT
ejpam-3418	326	14	for	for	ADP
ejpam-3418	326	15	some	some	DET
ejpam-3418	326	16	b1	b1	NOUN
ejpam-3418	326	17	∈	∈	PROPN
ejpam-3418	326	18	i(y	i(y	NOUN
ejpam-3418	326	19	)	)	PUNCT
ejpam-3418	326	20	.	.	PUNCT
ejpam-3418	327	1	note	note	VERB
ejpam-3418	327	2	that	that	SCONJ
ejpam-3418	327	3	f(a	f(a	NOUN
ejpam-3418	327	4	)	)	PUNCT
ejpam-3418	327	5	⊆	⊆	NUM
ejpam-3418	327	6	f(f−1(b1	f(f−1(b1	NUM
ejpam-3418	327	7	)	)	PUNCT
ejpam-3418	327	8	)	)	PUNCT
ejpam-3418	328	1	=	=	SYM
ejpam-3418	328	2	b1	b1	NOUN
ejpam-3418	328	3	,	,	PUNCT
ejpam-3418	328	4	since	since	SCONJ
ejpam-3418	328	5	f	f	PROPN
ejpam-3418	328	6	is	be	AUX
ejpam-3418	328	7	onto	onto	ADP
ejpam-3418	328	8	.	.	PUNCT
ejpam-3418	329	1	thus	thus	ADV
ejpam-3418	329	2	,	,	PUNCT
ejpam-3418	329	3	b	b	PROPN
ejpam-3418	329	4	⊆	⊆	NUM
ejpam-3418	329	5	b1	b1	NOUN
ejpam-3418	329	6	.	.	PUNCT
ejpam-3418	330	1	by	by	ADP
ejpam-3418	330	2	the	the	DET
ejpam-3418	330	3	definition	definition	NOUN
ejpam-3418	330	4	of	of	ADP
ejpam-3418	330	5	an	an	DET
ejpam-3418	330	6	ideal	ideal	NOUN
ejpam-3418	330	7	,	,	PUNCT
ejpam-3418	330	8	b	b	PROPN
ejpam-3418	330	9	∈	∈	PROPN
ejpam-3418	330	10	i(y	i(y	NOUN
ejpam-3418	330	11	)	)	PUNCT
ejpam-3418	330	12	.	.	PUNCT
ejpam-3418	331	1	hence	hence	ADV
ejpam-3418	331	2	,	,	PUNCT
ejpam-3418	331	3	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	331	4	)	)	PUNCT
ejpam-3418	331	5	)	)	PUNCT
ejpam-3418	331	6	)	)	PUNCT
ejpam-3418	332	1	⊆	⊆	NUM
ejpam-3418	332	2	i(y	i(y	NOUN
ejpam-3418	332	3	)	)	PUNCT
ejpam-3418	332	4	.	.	PUNCT
ejpam-3418	333	1	conversely	conversely	ADV
ejpam-3418	333	2	,	,	PUNCT
ejpam-3418	333	3	let	let	VERB
ejpam-3418	333	4	b	b	X
ejpam-3418	333	5	∈	∈	PROPN
ejpam-3418	333	6	i(y	i(y	PROPN
ejpam-3418	333	7	)	)	PUNCT
ejpam-3418	333	8	and	and	CCONJ
ejpam-3418	333	9	c	c	NOUN
ejpam-3418	333	10	=	=	SYM
ejpam-3418	333	11	f−1(b	f−1(b	PROPN
ejpam-3418	333	12	)	)	PUNCT
ejpam-3418	333	13	.	.	PUNCT
ejpam-3418	334	1	then	then	ADV
ejpam-3418	334	2	c	c	PROPN
ejpam-3418	334	3	∈	∈	PROPN
ejpam-3418	334	4	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	334	5	)	)	PUNCT
ejpam-3418	334	6	)	)	PUNCT
ejpam-3418	334	7	.	.	PUNCT
ejpam-3418	335	1	we	we	PRON
ejpam-3418	335	2	thus	thus	ADV
ejpam-3418	335	3	have	have	VERB
ejpam-3418	335	4	f(c	f(c	PROPN
ejpam-3418	335	5	)	)	PUNCT
ejpam-3418	335	6	∈	∈	PROPN
ejpam-3418	335	7	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	335	8	)	)	PUNCT
ejpam-3418	335	9	)	)	PUNCT
ejpam-3418	335	10	)	)	PUNCT
ejpam-3418	335	11	.	.	PUNCT
ejpam-3418	336	1	since	since	SCONJ
ejpam-3418	336	2	f	f	PROPN
ejpam-3418	336	3	is	be	AUX
ejpam-3418	336	4	onto	onto	ADP
ejpam-3418	336	5	,	,	PUNCT
ejpam-3418	336	6	b	b	NOUN
ejpam-3418	336	7	=	=	SYM
ejpam-3418	336	8	f(f−1(b	f(f−1(b	PROPN
ejpam-3418	336	9	)	)	PUNCT
ejpam-3418	336	10	)	)	PUNCT
ejpam-3418	337	1	=	=	SYM
ejpam-3418	337	2	f(c	f(c	PROPN
ejpam-3418	337	3	)	)	PUNCT
ejpam-3418	337	4	∈	∈	PROPN
ejpam-3418	337	5	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	337	6	)	)	PUNCT
ejpam-3418	337	7	)	)	PUNCT
ejpam-3418	337	8	)	)	PUNCT
ejpam-3418	337	9	.	.	PUNCT
ejpam-3418	338	1	thus	thus	ADV
ejpam-3418	338	2	,	,	PUNCT
ejpam-3418	338	3	i(y	i(y	NOUN
ejpam-3418	338	4	)	)	PUNCT
ejpam-3418	338	5	⊆	⊆	NUM
ejpam-3418	338	6	f(f−1(i(y	f(f−1(i(y	ADJ
ejpam-3418	338	7	)	)	PUNCT
ejpam-3418	338	8	)	)	PUNCT
ejpam-3418	338	9	)	)	PUNCT
ejpam-3418	338	10	.	.	PUNCT
ejpam-3418	339	1	therefore	therefore	ADV
ejpam-3418	339	2	,	,	PUNCT
ejpam-3418	339	3	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	339	4	)	)	PUNCT
ejpam-3418	339	5	)	)	PUNCT
ejpam-3418	339	6	)	)	PUNCT
ejpam-3418	340	1	=	=	PUNCT
ejpam-3418	340	2	i(y	i(y	NOUN
ejpam-3418	340	3	)	)	PUNCT
ejpam-3418	340	4	.	.	PUNCT
ejpam-3418	340	5	suppose	suppose	VERB
ejpam-3418	340	6	that	that	SCONJ
ejpam-3418	340	7	f	f	PROPN
ejpam-3418	340	8	is	be	AUX
ejpam-3418	340	9	one	one	NUM
ejpam-3418	340	10	-	-	PUNCT
ejpam-3418	340	11	to	to	ADP
ejpam-3418	340	12	-	-	PUNCT
ejpam-3418	340	13	one	one	NUM
ejpam-3418	340	14	.	.	PUNCT
ejpam-3418	341	1	let	let	VERB
ejpam-3418	341	2	a	a	DET
ejpam-3418	341	3	∈	∈	PROPN
ejpam-3418	341	4	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	341	5	)	)	PUNCT
ejpam-3418	341	6	)	)	PUNCT
ejpam-3418	341	7	)	)	PUNCT
ejpam-3418	341	8	.	.	PUNCT
ejpam-3418	342	1	then	then	ADV
ejpam-3418	342	2	there	there	PRON
ejpam-3418	342	3	exists	exist	VERB
ejpam-3418	342	4	b	b	PROPN
ejpam-3418	342	5	∈	∈	PROPN
ejpam-3418	342	6	f(i(x	f(i(x	NOUN
ejpam-3418	342	7	)	)	PUNCT
ejpam-3418	342	8	)	)	PUNCT
ejpam-3418	342	9	such	such	ADJ
ejpam-3418	342	10	that	that	DET
ejpam-3418	342	11	f−1(b	f−1(b	PROPN
ejpam-3418	342	12	)	)	PUNCT
ejpam-3418	342	13	=	=	PUNCT
ejpam-3418	343	1	a.	a.	NOUN
ejpam-3418	343	2	since	since	SCONJ
ejpam-3418	343	3	b	b	PROPN
ejpam-3418	343	4	∈	∈	PROPN
ejpam-3418	343	5	f(i(x	f(i(x	NOUN
ejpam-3418	343	6	)	)	PUNCT
ejpam-3418	343	7	)	)	PUNCT
ejpam-3418	343	8	,	,	PUNCT
ejpam-3418	343	9	b	b	X
ejpam-3418	343	10	=	=	PUNCT
ejpam-3418	343	11	f(a1	f(a1	NOUN
ejpam-3418	343	12	)	)	PUNCT
ejpam-3418	343	13	for	for	ADP
ejpam-3418	343	14	some	some	DET
ejpam-3418	343	15	a1	a1	NOUN
ejpam-3418	343	16	∈	∈	PROPN
ejpam-3418	343	17	i(x	i(x	PROPN
ejpam-3418	343	18	)	)	PUNCT
ejpam-3418	343	19	.	.	PUNCT
ejpam-3418	344	1	since	since	SCONJ
ejpam-3418	344	2	f	f	PROPN
ejpam-3418	344	3	is	be	AUX
ejpam-3418	344	4	one	one	NUM
ejpam-3418	344	5	-	-	PUNCT
ejpam-3418	344	6	to	to	ADP
ejpam-3418	344	7	-	-	PUNCT
ejpam-3418	344	8	one	one	NUM
ejpam-3418	344	9	,	,	PUNCT
ejpam-3418	344	10	we	we	PRON
ejpam-3418	344	11	have	have	VERB
ejpam-3418	344	12	a	a	DET
ejpam-3418	344	13	=	=	SYM
ejpam-3418	344	14	f−1(b	f−1(b	PROPN
ejpam-3418	344	15	)	)	PUNCT
ejpam-3418	345	1	=	=	SYM
ejpam-3418	345	2	f−1(f(a1	f−1(f(a1	NOUN
ejpam-3418	345	3	)	)	PUNCT
ejpam-3418	345	4	)	)	PUNCT
ejpam-3418	346	1	=	=	SYM
ejpam-3418	346	2	a1	a1	NOUN
ejpam-3418	346	3	.	.	PUNCT
ejpam-3418	347	1	thus	thus	ADV
ejpam-3418	347	2	,	,	PUNCT
ejpam-3418	347	3	a	a	DET
ejpam-3418	347	4	∈	∈	PROPN
ejpam-3418	347	5	i(x	i(x	NOUN
ejpam-3418	347	6	)	)	PUNCT
ejpam-3418	347	7	.	.	PUNCT
ejpam-3418	348	1	hence	hence	ADV
ejpam-3418	348	2	,	,	PUNCT
ejpam-3418	348	3	f−1(f(i(x	f−1(f(i(x	PROPN
ejpam-3418	348	4	)	)	PUNCT
ejpam-3418	348	5	)	)	PUNCT
ejpam-3418	348	6	)	)	PUNCT
ejpam-3418	349	1	⊆	⊆	NUM
ejpam-3418	349	2	i(x	i(x	NOUN
ejpam-3418	349	3	)	)	PUNCT
ejpam-3418	349	4	.	.	PUNCT
ejpam-3418	350	1	conversely	conversely	ADV
ejpam-3418	350	2	,	,	PUNCT
ejpam-3418	350	3	let	let	VERB
ejpam-3418	350	4	a	a	DET
ejpam-3418	350	5	∈	∈	PROPN
ejpam-3418	350	6	i(x	i(x	NOUN
ejpam-3418	350	7	)	)	PUNCT
ejpam-3418	350	8	and	and	CCONJ
ejpam-3418	350	9	d	d	NOUN
ejpam-3418	350	10	=	=	SYM
ejpam-3418	350	11	f(a	f(a	PROPN
ejpam-3418	350	12	)	)	PUNCT
ejpam-3418	350	13	.	.	PUNCT
ejpam-3418	351	1	then	then	ADV
ejpam-3418	351	2	,	,	PUNCT
ejpam-3418	351	3	d	d	PROPN
ejpam-3418	351	4	∈	∈	PROPN
ejpam-3418	351	5	f(i(x	f(i(x	NOUN
ejpam-3418	351	6	)	)	PUNCT
ejpam-3418	351	7	)	)	PUNCT
ejpam-3418	351	8	.	.	PUNCT
ejpam-3418	352	1	we	we	PRON
ejpam-3418	352	2	thus	thus	ADV
ejpam-3418	352	3	have	have	VERB
ejpam-3418	352	4	f−1(d	f−1(d	NOUN
ejpam-3418	352	5	)	)	PUNCT
ejpam-3418	352	6	∈	∈	PROPN
ejpam-3418	352	7	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	352	8	)	)	PUNCT
ejpam-3418	352	9	)	)	PUNCT
ejpam-3418	352	10	)	)	PUNCT
ejpam-3418	352	11	.	.	PUNCT
ejpam-3418	353	1	since	since	SCONJ
ejpam-3418	353	2	f	f	PROPN
ejpam-3418	353	3	is	be	AUX
ejpam-3418	353	4	one	one	NUM
ejpam-3418	353	5	-	-	PUNCT
ejpam-3418	353	6	to	to	ADP
ejpam-3418	353	7	-	-	PUNCT
ejpam-3418	353	8	one	one	NOUN
ejpam-3418	353	9	,	,	PUNCT
ejpam-3418	353	10	a	a	DET
ejpam-3418	353	11	=	=	SYM
ejpam-3418	353	12	f−1(f(a	f−1(f(a	NOUN
ejpam-3418	353	13	)	)	PUNCT
ejpam-3418	353	14	)	)	PUNCT
ejpam-3418	354	1	=	=	SYM
ejpam-3418	354	2	f−1(d	f−1(d	PROPN
ejpam-3418	354	3	)	)	PUNCT
ejpam-3418	354	4	∈	∈	PROPN
ejpam-3418	354	5	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	354	6	)	)	PUNCT
ejpam-3418	354	7	)	)	PUNCT
ejpam-3418	354	8	)	)	PUNCT
ejpam-3418	354	9	.	.	PUNCT
ejpam-3418	355	1	thus	thus	ADV
ejpam-3418	355	2	,	,	PUNCT
ejpam-3418	355	3	i(x	i(x	PROPN
ejpam-3418	355	4	)	)	PUNCT
ejpam-3418	355	5	⊆	⊆	NUM
ejpam-3418	355	6	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	355	7	)	)	PUNCT
ejpam-3418	355	8	)	)	PUNCT
ejpam-3418	355	9	)	)	PUNCT
ejpam-3418	355	10	.	.	PUNCT
ejpam-3418	356	1	therefore	therefore	ADV
ejpam-3418	356	2	,	,	PUNCT
ejpam-3418	356	3	f−1(f(i(x	f−1(f(i(x	PROPN
ejpam-3418	356	4	)	)	PUNCT
ejpam-3418	356	5	)	)	PUNCT
ejpam-3418	356	6	)	)	PUNCT
ejpam-3418	357	1	=	=	SYM
ejpam-3418	357	2	i(x	i(x	NOUN
ejpam-3418	357	3	)	)	PUNCT
ejpam-3418	357	4	.	.	PUNCT
ejpam-3418	358	1	theorem	theorem	NOUN
ejpam-3418	358	2	5	5	NUM
ejpam-3418	358	3	.	.	PUNCT
ejpam-3418	359	1	let	let	AUX
ejpam-3418	359	2	x	x	PRON
ejpam-3418	359	3	and	and	CCONJ
ejpam-3418	359	4	y	y	PROPN
ejpam-3418	359	5	be	be	AUX
ejpam-3418	359	6	nonempty	nonempty	X
ejpam-3418	359	7	sets	set	NOUN
ejpam-3418	359	8	and	and	CCONJ
ejpam-3418	359	9	,	,	PUNCT
ejpam-3418	359	10	i(x	i(x	PROPN
ejpam-3418	359	11	)	)	PUNCT
ejpam-3418	359	12	and	and	CCONJ
ejpam-3418	359	13	i(y	i(y	NOUN
ejpam-3418	359	14	)	)	PUNCT
ejpam-3418	359	15	be	be	AUX
ejpam-3418	359	16	ideals	ideal	NOUN
ejpam-3418	359	17	in	in	ADP
ejpam-3418	359	18	x	x	PUNCT
ejpam-3418	359	19	and	and	CCONJ
ejpam-3418	359	20	y	y	PROPN
ejpam-3418	359	21	,	,	PUNCT
ejpam-3418	359	22	respectively	respectively	ADV
ejpam-3418	359	23	.	.	PUNCT
ejpam-3418	360	1	let	let	VERB
ejpam-3418	360	2	f	f	NOUN
ejpam-3418	360	3	:	:	PUNCT
ejpam-3418	360	4	x	x	X
ejpam-3418	360	5	→	→	SYM
ejpam-3418	360	6	y	y	X
ejpam-3418	360	7	be	be	AUX
ejpam-3418	360	8	a	a	DET
ejpam-3418	360	9	mapping	mapping	NOUN
ejpam-3418	360	10	.	.	PUNCT
ejpam-3418	361	1	if	if	SCONJ
ejpam-3418	361	2	ι̂	ι̂	NOUN
ejpam-3418	361	3	,	,	PUNCT
ejpam-3418	361	4	τ̂	τ̂	PUNCT
ejpam-3418	361	5	∈	∈	PROPN
ejpam-3418	361	6	i	i	PRON
ejpam-3418	361	7	i(x	i(x	PROPN
ejpam-3418	361	8	)	)	PUNCT
ejpam-3418	361	9	and	and	CCONJ
ejpam-3418	361	10	ω̂	ω̂	NUM
ejpam-3418	361	11	,	,	PUNCT
ejpam-3418	361	12	η̂	η̂	NUM
ejpam-3418	361	13	∈	∈	PROPN
ejpam-3418	361	14	i	i	PRON
ejpam-3418	361	15	i(y	i(y	NOUN
ejpam-3418	361	16	)	)	PUNCT
ejpam-3418	361	17	,	,	PUNCT
ejpam-3418	361	18	then	then	ADV
ejpam-3418	361	19	i.	i.	PROPN
ejpam-3418	361	20	f−1	f−1	PROPN
ejpam-3418	362	1	[	[	X
ejpam-3418	362	2	η̂c	η̂c	X
ejpam-3418	362	3	]	]	X
ejpam-3418	362	4	=	=	SYM
ejpam-3418	362	5	(	(	PUNCT
ejpam-3418	362	6	f−1	f−1	PROPN
ejpam-3418	362	7	[	[	X
ejpam-3418	362	8	η̂])c	η̂])c	PROPN
ejpam-3418	362	9	;	;	PUNCT
ejpam-3418	362	10	ii	ii	X
ejpam-3418	362	11	.	.	PUNCT
ejpam-3418	363	1	(	(	PUNCT
ejpam-3418	363	2	f	f	X
ejpam-3418	364	1	[	[	X
ejpam-3418	364	2	̂ι])c	̂ι])c	PROPN
ejpam-3418	364	3	6	6	NUM
ejpam-3418	364	4	f	f	NOUN
ejpam-3418	364	5	[	[	X
ejpam-3418	364	6	̂ιc	̂ιc	NOUN
ejpam-3418	364	7	]	]	X
ejpam-3418	364	8	;	;	PUNCT
ejpam-3418	364	9	mj	mj	PROPN
ejpam-3418	364	10	togonon	togonon	PROPN
ejpam-3418	364	11	,	,	PUNCT
ejpam-3418	364	12	r	r	NOUN
ejpam-3418	364	13	caga	caga	NOUN
ejpam-3418	364	14	-	-	PUNCT
ejpam-3418	364	15	anan	anan	PROPN
ejpam-3418	364	16	/	/	SYM
ejpam-3418	364	17	eur	eur	PROPN
ejpam-3418	364	18	.	.	PUNCT
ejpam-3418	365	1	j.	j.	PROPN
ejpam-3418	365	2	pure	pure	PROPN
ejpam-3418	365	3	appl	appl	PROPN
ejpam-3418	365	4	.	.	PROPN
ejpam-3418	365	5	math	math	PROPN
ejpam-3418	365	6	,	,	PUNCT
ejpam-3418	365	7	12	12	NUM
ejpam-3418	365	8	(	(	PUNCT
ejpam-3418	365	9	2	2	NUM
ejpam-3418	365	10	)	)	PUNCT
ejpam-3418	365	11	(	(	PUNCT
ejpam-3418	365	12	2019	2019	NUM
ejpam-3418	365	13	)	)	PUNCT
ejpam-3418	365	14	,	,	PUNCT
ejpam-3418	365	15	553	553	NUM
ejpam-3418	365	16	-	-	SYM
ejpam-3418	365	17	570	570	NUM
ejpam-3418	365	18	562	562	NUM
ejpam-3418	365	19	iii	iii	NOUN
ejpam-3418	365	20	.	.	PUNCT
ejpam-3418	366	1	if	if	SCONJ
ejpam-3418	366	2	ω̂	ω̂	PROPN
ejpam-3418	366	3	6	6	NUM
ejpam-3418	366	4	η̂	η̂	NOUN
ejpam-3418	366	5	,	,	PUNCT
ejpam-3418	366	6	then	then	ADV
ejpam-3418	366	7	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	366	8	]	]	X
ejpam-3418	366	9	6	6	NUM
ejpam-3418	366	10	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	366	11	]	]	PUNCT
ejpam-3418	366	12	;	;	PUNCT
ejpam-3418	366	13	iv	iv	X
ejpam-3418	366	14	.	.	PUNCT
ejpam-3418	367	1	if	if	SCONJ
ejpam-3418	367	2	ι̂	ι̂	NUM
ejpam-3418	367	3	6	6	NUM
ejpam-3418	367	4	τ̂	τ̂	NUM
ejpam-3418	367	5	,	,	PUNCT
ejpam-3418	367	6	then	then	ADV
ejpam-3418	367	7	f	f	PROPN
ejpam-3418	368	1	[	[	X
ejpam-3418	368	2	̂ι	̂ι	X
ejpam-3418	368	3	]	]	X
ejpam-3418	368	4	6	6	NUM
ejpam-3418	368	5	f	f	X
ejpam-3418	368	6	[	[	X
ejpam-3418	368	7	τ̂	τ̂	X
ejpam-3418	368	8	]	]	PUNCT
ejpam-3418	368	9	;	;	PUNCT
ejpam-3418	368	10	v.	v.	CCONJ
ejpam-3418	368	11	if	if	SCONJ
ejpam-3418	368	12	f	f	PROPN
ejpam-3418	368	13	is	be	AUX
ejpam-3418	368	14	onto	onto	ADP
ejpam-3418	368	15	,	,	PUNCT
ejpam-3418	368	16	then	then	ADV
ejpam-3418	368	17	f	f	X
ejpam-3418	369	1	[	[	X
ejpam-3418	369	2	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	369	3	]	]	X
ejpam-3418	369	4	]	]	X
ejpam-3418	369	5	=	=	PUNCT
ejpam-3418	369	6	η̂	η̂	X
ejpam-3418	369	7	;	;	PUNCT
ejpam-3418	369	8	and	and	CCONJ
ejpam-3418	369	9	vi	vi	NOUN
ejpam-3418	369	10	.	.	NOUN
ejpam-3418	370	1	if	if	SCONJ
ejpam-3418	370	2	f	f	PROPN
ejpam-3418	370	3	is	be	AUX
ejpam-3418	370	4	one	one	NUM
ejpam-3418	370	5	-	-	PUNCT
ejpam-3418	370	6	to	to	ADP
ejpam-3418	370	7	-	-	PUNCT
ejpam-3418	370	8	one	one	NUM
ejpam-3418	370	9	,	,	PUNCT
ejpam-3418	370	10	then	then	ADV
ejpam-3418	370	11	ι̂	ι̂	PUNCT
ejpam-3418	370	12	6	6	NUM
ejpam-3418	370	13	f−1[f	f−1[f	NOUN
ejpam-3418	370	14	[	[	X
ejpam-3418	370	15	̂ι	̂ι	X
ejpam-3418	370	16	]	]	X
ejpam-3418	370	17	]	]	PUNCT
ejpam-3418	370	18	.	.	PUNCT
ejpam-3418	371	1	proof	proof	NOUN
ejpam-3418	371	2	.	.	PUNCT
ejpam-3418	372	1	i.	i.	PROPN
ejpam-3418	372	2	let	let	VERB
ejpam-3418	372	3	∅	∅	NOUN
ejpam-3418	372	4	6=	6=	ADP
ejpam-3418	372	5	a	a	DET
ejpam-3418	372	6	∈	∈	NOUN
ejpam-3418	372	7	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	372	8	)	)	PUNCT
ejpam-3418	372	9	)	)	PUNCT
ejpam-3418	372	10	.	.	PUNCT
ejpam-3418	373	1	then	then	ADV
ejpam-3418	373	2	(	(	PUNCT
ejpam-3418	373	3	f−1[η̂])c(a	f−1[η̂])c(a	PROPN
ejpam-3418	373	4	)	)	PUNCT
ejpam-3418	373	5	=	=	SYM
ejpam-3418	373	6	[	[	PUNCT
ejpam-3418	373	7	inf	inf	ADJ
ejpam-3418	373	8	x∈a	x∈a	NOUN
ejpam-3418	373	9	{	{	PUNCT
ejpam-3418	373	10	1−	1−	NUM
ejpam-3418	373	11	[	[	X
ejpam-3418	373	12	(	(	PUNCT
ejpam-3418	373	13	f−1[η̂])({x})]+	f−1[η̂])({x})]+	PROPN
ejpam-3418	373	14	}	}	PUNCT
ejpam-3418	373	15	,	,	PUNCT
ejpam-3418	373	16	inf	inf	PROPN
ejpam-3418	373	17	x∈a	x∈a	NOUN
ejpam-3418	373	18	{	{	PUNCT
ejpam-3418	373	19	1−	1−	NUM
ejpam-3418	374	1	[	[	X
ejpam-3418	374	2	(	(	PUNCT
ejpam-3418	374	3	f−1[η̂])({x})]−	f−1[η̂])({x})]−	NOUN
ejpam-3418	374	4	}	}	PUNCT
ejpam-3418	374	5	]	]	PUNCT
ejpam-3418	375	1	=	=	PUNCT
ejpam-3418	375	2	[	[	PUNCT
ejpam-3418	375	3	inf	inf	ADJ
ejpam-3418	375	4	x∈a	x∈a	NOUN
ejpam-3418	375	5	{	{	PUNCT
ejpam-3418	375	6	1−	1−	NUM
ejpam-3418	375	7	[	[	X
ejpam-3418	375	8	(	(	PUNCT
ejpam-3418	375	9	η̂	η̂	ADJ
ejpam-3418	375	10	◦	◦	NOUN
ejpam-3418	375	11	f)({x})]+	f)({x})]+	NOUN
ejpam-3418	375	12	}	}	PUNCT
ejpam-3418	375	13	,	,	PUNCT
ejpam-3418	375	14	inf	inf	PROPN
ejpam-3418	375	15	x∈a	x∈a	NOUN
ejpam-3418	375	16	{	{	PUNCT
ejpam-3418	375	17	1−	1−	NUM
ejpam-3418	375	18	[	[	X
ejpam-3418	375	19	(	(	PUNCT
ejpam-3418	375	20	η̂	η̂	X
ejpam-3418	375	21	◦	◦	NOUN
ejpam-3418	375	22	f)({x})]−	f)({x})]−	PUNCT
ejpam-3418	375	23	}	}	PUNCT
ejpam-3418	375	24	]	]	PUNCT
ejpam-3418	376	1	=	=	PUNCT
ejpam-3418	376	2	[	[	PUNCT
ejpam-3418	376	3	inf	inf	ADJ
ejpam-3418	376	4	x∈a	x∈a	NOUN
ejpam-3418	376	5	{	{	PUNCT
ejpam-3418	376	6	1−	1−	NUM
ejpam-3418	377	1	[	[	X
ejpam-3418	377	2	η̂(f({x}))]+	η̂(f({x}))]+	NOUN
ejpam-3418	377	3	}	}	PUNCT
ejpam-3418	377	4	,	,	PUNCT
ejpam-3418	377	5	inf	inf	PROPN
ejpam-3418	377	6	x∈a	x∈a	NOUN
ejpam-3418	377	7	{	{	PUNCT
ejpam-3418	377	8	1−	1−	NUM
ejpam-3418	378	1	[	[	X
ejpam-3418	378	2	η̂(f({x}))]−	η̂(f({x}))]−	PROPN
ejpam-3418	378	3	}	}	PUNCT
ejpam-3418	378	4	]	]	PUNCT
ejpam-3418	379	1	=	=	PUNCT
ejpam-3418	379	2	[	[	PUNCT
ejpam-3418	379	3	inf	inf	NOUN
ejpam-3418	379	4	f(x)∈f(a	f(x)∈f(a	NOUN
ejpam-3418	379	5	)	)	PUNCT
ejpam-3418	379	6	{	{	PUNCT
ejpam-3418	379	7	1−	1−	NUM
ejpam-3418	379	8	[	[	X
ejpam-3418	379	9	η̂(f({x}))]+	η̂(f({x}))]+	NOUN
ejpam-3418	379	10	}	}	PUNCT
ejpam-3418	379	11	,	,	PUNCT
ejpam-3418	379	12	inf	inf	PROPN
ejpam-3418	379	13	f(x)∈f(a	f(x)∈f(a	NOUN
ejpam-3418	379	14	)	)	PUNCT
ejpam-3418	379	15	{	{	PUNCT
ejpam-3418	379	16	1−	1−	NUM
ejpam-3418	380	1	[	[	X
ejpam-3418	380	2	η̂(f({x}))]−	η̂(f({x}))]−	PROPN
ejpam-3418	380	3	}	}	PUNCT
ejpam-3418	380	4	]	]	PUNCT
ejpam-3418	381	1	=	=	PUNCT
ejpam-3418	382	1	[	[	X
ejpam-3418	382	2	[	[	X
ejpam-3418	382	3	η̂c(f(a))]−	η̂c(f(a))]−	PROPN
ejpam-3418	382	4	,	,	PUNCT
ejpam-3418	382	5	[	[	X
ejpam-3418	382	6	η̂c(f(a))]+	η̂c(f(a))]+	NOUN
ejpam-3418	382	7	]	]	X
ejpam-3418	382	8	=	=	PUNCT
ejpam-3418	383	1	[	[	X
ejpam-3418	383	2	[	[	X
ejpam-3418	383	3	(	(	PUNCT
ejpam-3418	383	4	η̂c	η̂c	ADP
ejpam-3418	383	5	◦	◦	NOUN
ejpam-3418	383	6	f)(a)]−	f)(a)]−	PROPN
ejpam-3418	383	7	,	,	PUNCT
ejpam-3418	383	8	[	[	X
ejpam-3418	383	9	(	(	PUNCT
ejpam-3418	383	10	η̂c	η̂c	PART
ejpam-3418	383	11	◦	◦	NOUN
ejpam-3418	383	12	f)(a)]+	f)(a)]+	NUM
ejpam-3418	383	13	]	]	X
ejpam-3418	383	14	=	=	PUNCT
ejpam-3418	384	1	[	[	X
ejpam-3418	384	2	[	[	X
ejpam-3418	384	3	(	(	PUNCT
ejpam-3418	384	4	f−1[η̂c])(a)]−	f−1[η̂c])(a)]−	ADJ
ejpam-3418	384	5	,	,	PUNCT
ejpam-3418	384	6	[	[	X
ejpam-3418	384	7	(	(	PUNCT
ejpam-3418	384	8	f−1[η̂c])(a)]+	f−1[η̂c])(a)]+	PROPN
ejpam-3418	384	9	]	]	X
ejpam-3418	384	10	=	=	SYM
ejpam-3418	384	11	f−1[η̂c](a	f−1[η̂c](a	NOUN
ejpam-3418	384	12	)	)	PUNCT
ejpam-3418	384	13	.	.	PUNCT
ejpam-3418	385	1	hence	hence	ADV
ejpam-3418	385	2	,	,	PUNCT
ejpam-3418	385	3	(	(	PUNCT
ejpam-3418	385	4	f−1[η̂])c	f−1[η̂])c	NOUN
ejpam-3418	385	5	=	=	SYM
ejpam-3418	385	6	f−1[η̂c	f−1[η̂c	PROPN
ejpam-3418	385	7	]	]	PUNCT
ejpam-3418	385	8	.	.	PUNCT
ejpam-3418	386	1	ii	ii	PROPN
ejpam-3418	386	2	.	.	PUNCT
ejpam-3418	387	1	let	let	VERB
ejpam-3418	387	2	∅	∅	NOUN
ejpam-3418	387	3	6=	6=	ADP
ejpam-3418	387	4	b	b	X
ejpam-3418	387	5	∈	∈	PROPN
ejpam-3418	387	6	f(i(x	f(i(x	NOUN
ejpam-3418	387	7	)	)	PUNCT
ejpam-3418	387	8	)	)	PUNCT
ejpam-3418	387	9	and	and	CCONJ
ejpam-3418	387	10	s	s	X
ejpam-3418	387	11	=	=	X
ejpam-3418	387	12	{	{	PUNCT
ejpam-3418	387	13	a	a	DET
ejpam-3418	387	14	∈	∈	PROPN
ejpam-3418	387	15	i(x	i(x	NOUN
ejpam-3418	387	16	)	)	PUNCT
ejpam-3418	387	17	:	:	PUNCT
ejpam-3418	387	18	f(a	f(a	X
ejpam-3418	387	19	)	)	PUNCT
ejpam-3418	388	1	=	=	SYM
ejpam-3418	388	2	b	b	X
ejpam-3418	388	3	}	}	PUNCT
ejpam-3418	388	4	.	.	PUNCT
ejpam-3418	389	1	then	then	ADV
ejpam-3418	389	2	(	(	PUNCT
ejpam-3418	389	3	f	f	X
ejpam-3418	390	1	[	[	X
ejpam-3418	390	2	̂ι])c(b	̂ι])c(b	X
ejpam-3418	390	3	)	)	PUNCT
ejpam-3418	390	4	=	=	PUNCT
ejpam-3418	391	1	[	[	PUNCT
ejpam-3418	391	2	inf	inf	ADJ
ejpam-3418	391	3	y∈b	y∈b	NOUN
ejpam-3418	391	4	{	{	PUNCT
ejpam-3418	391	5	1−	1−	NUM
ejpam-3418	392	1	[	[	X
ejpam-3418	392	2	f	f	X
ejpam-3418	392	3	[	[	X
ejpam-3418	392	4	̂ι]({y})]−	̂ι]({y})]−	NOUN
ejpam-3418	392	5	}	}	PUNCT
ejpam-3418	392	6	,	,	PUNCT
ejpam-3418	392	7	inf	inf	PROPN
ejpam-3418	392	8	y∈b	y∈b	NOUN
ejpam-3418	392	9	{	{	PUNCT
ejpam-3418	392	10	1−	1−	NUM
ejpam-3418	393	1	[	[	X
ejpam-3418	393	2	f	f	X
ejpam-3418	394	1	[	[	X
ejpam-3418	394	2	̂ι]({y})]+	̂ι]({y})]+	NUM
ejpam-3418	394	3	}	}	PUNCT
ejpam-3418	394	4	]	]	PUNCT
ejpam-3418	394	5	=	=	PUNCT
ejpam-3418	394	6	[	[	PUNCT
ejpam-3418	394	7	inf	inf	ADJ
ejpam-3418	394	8	y∈b	y∈b	NOUN
ejpam-3418	394	9	{	{	PUNCT
ejpam-3418	394	10	1−	1−	NUM
ejpam-3418	394	11	sup	sup	NOUN
ejpam-3418	394	12	a∈s′	a∈s′	X
ejpam-3418	394	13	[	[	X
ejpam-3418	394	14	̂ι(a)]−	̂ι(a)]−	PUNCT
ejpam-3418	394	15	}	}	PUNCT
ejpam-3418	394	16	,	,	PUNCT
ejpam-3418	394	17	inf	inf	PROPN
ejpam-3418	394	18	y∈b	y∈b	NOUN
ejpam-3418	394	19	{	{	PUNCT
ejpam-3418	394	20	1−	1−	NUM
ejpam-3418	394	21	sup	sup	NOUN
ejpam-3418	394	22	a∈s′	a∈s′	NOUN
ejpam-3418	394	23	[	[	X
ejpam-3418	394	24	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	394	25	}	}	PUNCT
ejpam-3418	394	26	]	]	PUNCT
ejpam-3418	394	27	;	;	PUNCT
ejpam-3418	394	28	where	where	SCONJ
ejpam-3418	394	29	s′	s′	ADJ
ejpam-3418	394	30	=	=	PUNCT
ejpam-3418	394	31	{	{	PUNCT
ejpam-3418	394	32	a	a	DET
ejpam-3418	394	33	∈	∈	PROPN
ejpam-3418	394	34	i(x	i(x	NOUN
ejpam-3418	394	35	)	)	PUNCT
ejpam-3418	394	36	:	:	PUNCT
ejpam-3418	394	37	f(a	f(a	X
ejpam-3418	394	38	)	)	PUNCT
ejpam-3418	394	39	=	=	PRON
ejpam-3418	394	40	{	{	PUNCT
ejpam-3418	394	41	y	y	NOUN
ejpam-3418	394	42	}	}	PUNCT
ejpam-3418	394	43	}	}	PUNCT
ejpam-3418	394	44	.	.	PUNCT
ejpam-3418	395	1	since	since	SCONJ
ejpam-3418	395	2	1−	1−	NUM
ejpam-3418	395	3	sup	sup	NOUN
ejpam-3418	395	4	a∈s′	a∈s′	X
ejpam-3418	395	5	ι̂(a	ι̂(a	PUNCT
ejpam-3418	395	6	)	)	PUNCT
ejpam-3418	395	7	=	=	SYM
ejpam-3418	395	8	inf	inf	NOUN
ejpam-3418	395	9	a∈s′	a∈s′	X
ejpam-3418	395	10	{	{	PUNCT
ejpam-3418	395	11	1−	1−	NUM
ejpam-3418	395	12	ι̂(a	ι̂(a	PUNCT
ejpam-3418	395	13	)	)	PUNCT
ejpam-3418	395	14	}	}	PUNCT
ejpam-3418	395	15	,	,	PUNCT
ejpam-3418	395	16	we	we	PRON
ejpam-3418	395	17	have	have	VERB
ejpam-3418	395	18	(	(	PUNCT
ejpam-3418	395	19	f	f	X
ejpam-3418	396	1	[	[	X
ejpam-3418	396	2	̂ι])c(b	̂ι])c(b	X
ejpam-3418	396	3	)	)	PUNCT
ejpam-3418	396	4	=	=	PUNCT
ejpam-3418	396	5	[	[	PUNCT
ejpam-3418	396	6	inf	inf	ADJ
ejpam-3418	396	7	y∈b	y∈b	NOUN
ejpam-3418	396	8	{	{	PUNCT
ejpam-3418	396	9	inf	inf	PROPN
ejpam-3418	396	10	a∈s′	a∈s′	X
ejpam-3418	396	11	{	{	PUNCT
ejpam-3418	396	12	1−	1−	NUM
ejpam-3418	396	13	[	[	X
ejpam-3418	396	14	̂ι(a)]−	̂ι(a)]−	X
ejpam-3418	396	15	}	}	PUNCT
ejpam-3418	396	16	}	}	PUNCT
ejpam-3418	396	17	,	,	PUNCT
ejpam-3418	396	18	inf	inf	PROPN
ejpam-3418	396	19	y∈b	y∈b	PROPN
ejpam-3418	396	20	{	{	PUNCT
ejpam-3418	396	21	inf	inf	PROPN
ejpam-3418	396	22	a∈s′	a∈s′	X
ejpam-3418	396	23	{	{	PUNCT
ejpam-3418	396	24	1−	1−	NUM
ejpam-3418	396	25	[	[	X
ejpam-3418	396	26	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	396	27	}	}	PUNCT
ejpam-3418	396	28	}	}	PUNCT
ejpam-3418	396	29	]	]	PUNCT
ejpam-3418	396	30	.	.	PUNCT
ejpam-3418	397	1	let	let	VERB
ejpam-3418	397	2	s′y	s′y	VERB
ejpam-3418	397	3	=	=	SYM
ejpam-3418	397	4	{	{	PUNCT
ejpam-3418	397	5	{	{	PUNCT
ejpam-3418	397	6	x	x	NOUN
ejpam-3418	397	7	}	}	PUNCT
ejpam-3418	397	8	∈	∈	PROPN
ejpam-3418	397	9	i(x	i(x	NOUN
ejpam-3418	397	10	)	)	PUNCT
ejpam-3418	397	11	:	:	PUNCT
ejpam-3418	397	12	f({x	f({x	NOUN
ejpam-3418	397	13	}	}	PUNCT
ejpam-3418	397	14	)	)	PUNCT
ejpam-3418	398	1	=	=	PRON
ejpam-3418	398	2	{	{	PUNCT
ejpam-3418	398	3	y	y	NOUN
ejpam-3418	398	4	}	}	PUNCT
ejpam-3418	398	5	}	}	PUNCT
ejpam-3418	398	6	.	.	PUNCT
ejpam-3418	399	1	observe	observe	VERB
ejpam-3418	399	2	that	that	SCONJ
ejpam-3418	399	3	since	since	SCONJ
ejpam-3418	399	4	ι̂	ι̂	NUM
ejpam-3418	399	5	is	be	AUX
ejpam-3418	399	6	an	an	DET
ejpam-3418	399	7	ivfi	ivfi	NOUN
ejpam-3418	399	8	set	set	VERB
ejpam-3418	399	9	,	,	PUNCT
ejpam-3418	399	10	if	if	SCONJ
ejpam-3418	399	11	{	{	PUNCT
ejpam-3418	399	12	x	x	NOUN
ejpam-3418	399	13	}	}	PUNCT
ejpam-3418	399	14	⊆	⊆	NUM
ejpam-3418	399	15	a	a	PRON
ejpam-3418	399	16	,	,	PUNCT
ejpam-3418	399	17	then	then	ADV
ejpam-3418	399	18	ι̂(a	ι̂(a	PUNCT
ejpam-3418	399	19	)	)	PUNCT
ejpam-3418	399	20	≤i	≤i	PROPN
ejpam-3418	399	21	ι̂({x	ι̂({x	PROPN
ejpam-3418	399	22	}	}	PUNCT
ejpam-3418	399	23	)	)	PUNCT
ejpam-3418	399	24	and	and	CCONJ
ejpam-3418	399	25	1−	1−	NUM
ejpam-3418	399	26	ι̂({x	ι̂({x	PROPN
ejpam-3418	399	27	}	}	PUNCT
ejpam-3418	399	28	)	)	PUNCT
ejpam-3418	399	29	≤i	≤i	PROPN
ejpam-3418	399	30	1−	1−	NUM
ejpam-3418	399	31	ι̂(a	ι̂(a	PUNCT
ejpam-3418	399	32	)	)	PUNCT
ejpam-3418	399	33	.	.	PUNCT
ejpam-3418	400	1	noting	note	VERB
ejpam-3418	400	2	that	that	SCONJ
ejpam-3418	400	3	s′y	s′y	VERB
ejpam-3418	400	4	⊆	⊆	NUM
ejpam-3418	400	5	s′	s′	NOUN
ejpam-3418	400	6	,	,	PUNCT
ejpam-3418	400	7	we	we	PRON
ejpam-3418	400	8	thus	thus	ADV
ejpam-3418	400	9	have	have	VERB
ejpam-3418	400	10	[	[	PUNCT
ejpam-3418	400	11	inf	inf	NOUN
ejpam-3418	400	12	a∈s′	a∈s′	X
ejpam-3418	400	13	{	{	PUNCT
ejpam-3418	400	14	1−	1−	NUM
ejpam-3418	400	15	[	[	X
ejpam-3418	400	16	̂ι(a)]−	̂ι(a)]−	X
ejpam-3418	400	17	}	}	PUNCT
ejpam-3418	400	18	,	,	PUNCT
ejpam-3418	400	19	inf	inf	NOUN
ejpam-3418	400	20	a∈s′	a∈s′	X
ejpam-3418	400	21	{	{	PUNCT
ejpam-3418	400	22	1−	1−	NUM
ejpam-3418	401	1	[	[	X
ejpam-3418	401	2	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	401	3	}	}	PUNCT
ejpam-3418	401	4	]	]	PUNCT
ejpam-3418	402	1	=	=	PUNCT
ejpam-3418	402	2	[	[	PUNCT
ejpam-3418	402	3	inf	inf	ADJ
ejpam-3418	402	4	a∈s′y	a∈s′y	PROPN
ejpam-3418	402	5	{	{	PUNCT
ejpam-3418	402	6	1−	1−	NUM
ejpam-3418	402	7	[	[	X
ejpam-3418	402	8	̂ι(a)]−	̂ι(a)]−	PUNCT
ejpam-3418	402	9	}	}	PUNCT
ejpam-3418	402	10	,	,	PUNCT
ejpam-3418	402	11	inf	inf	PROPN
ejpam-3418	402	12	a∈s′y	a∈s′y	PROPN
ejpam-3418	402	13	{	{	PUNCT
ejpam-3418	402	14	1−	1−	NUM
ejpam-3418	402	15	[	[	X
ejpam-3418	402	16	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	402	17	}	}	PUNCT
ejpam-3418	402	18	]	]	PUNCT
ejpam-3418	402	19	.	.	PUNCT
ejpam-3418	403	1	hence	hence	ADV
ejpam-3418	403	2	,	,	PUNCT
ejpam-3418	403	3	(	(	PUNCT
ejpam-3418	403	4	f	f	X
ejpam-3418	404	1	[	[	X
ejpam-3418	404	2	̂ι])c(b	̂ι])c(b	X
ejpam-3418	404	3	)	)	PUNCT
ejpam-3418	404	4	=	=	PUNCT
ejpam-3418	405	1	[	[	PUNCT
ejpam-3418	405	2	inf	inf	ADJ
ejpam-3418	405	3	y∈b	y∈b	NOUN
ejpam-3418	405	4	{	{	PUNCT
ejpam-3418	405	5	inf	inf	PROPN
ejpam-3418	405	6	{	{	PUNCT
ejpam-3418	405	7	x}∈s′y	x}∈s′y	PROPN
ejpam-3418	405	8	{	{	PUNCT
ejpam-3418	405	9	1−	1−	NUM
ejpam-3418	406	1	[	[	X
ejpam-3418	406	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	406	3	}	}	PUNCT
ejpam-3418	406	4	}	}	PUNCT
ejpam-3418	406	5	,	,	PUNCT
ejpam-3418	406	6	inf	inf	PROPN
ejpam-3418	406	7	y∈b	y∈b	PROPN
ejpam-3418	406	8	{	{	PUNCT
ejpam-3418	406	9	inf	inf	PROPN
ejpam-3418	406	10	{	{	PUNCT
ejpam-3418	406	11	x}∈s′y	x}∈s′y	PROPN
ejpam-3418	406	12	{	{	PUNCT
ejpam-3418	406	13	1−	1−	NUM
ejpam-3418	407	1	[	[	X
ejpam-3418	407	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	407	3	}	}	PUNCT
ejpam-3418	407	4	}	}	PUNCT
ejpam-3418	407	5	]	]	PUNCT
ejpam-3418	407	6	.	.	PUNCT
ejpam-3418	408	1	mj	mj	PROPN
ejpam-3418	408	2	togonon	togonon	PROPN
ejpam-3418	408	3	,	,	PUNCT
ejpam-3418	408	4	r	r	NOUN
ejpam-3418	408	5	caga	caga	NOUN
ejpam-3418	408	6	-	-	PUNCT
ejpam-3418	408	7	anan	anan	PROPN
ejpam-3418	408	8	/	/	SYM
ejpam-3418	408	9	eur	eur	PROPN
ejpam-3418	408	10	.	.	PUNCT
ejpam-3418	409	1	j.	j.	PROPN
ejpam-3418	409	2	pure	pure	PROPN
ejpam-3418	409	3	appl	appl	PROPN
ejpam-3418	409	4	.	.	PROPN
ejpam-3418	409	5	math	math	PROPN
ejpam-3418	409	6	,	,	PUNCT
ejpam-3418	409	7	12	12	NUM
ejpam-3418	409	8	(	(	PUNCT
ejpam-3418	409	9	2	2	NUM
ejpam-3418	409	10	)	)	PUNCT
ejpam-3418	409	11	(	(	PUNCT
ejpam-3418	409	12	2019	2019	NUM
ejpam-3418	409	13	)	)	PUNCT
ejpam-3418	409	14	,	,	PUNCT
ejpam-3418	409	15	553	553	NUM
ejpam-3418	409	16	-	-	SYM
ejpam-3418	409	17	570	570	NUM
ejpam-3418	409	18	563	563	NUM
ejpam-3418	409	19	consider	consider	VERB
ejpam-3418	409	20	that	that	PRON
ejpam-3418	409	21	[	[	PUNCT
ejpam-3418	409	22	inf	inf	ADJ
ejpam-3418	409	23	y∈b	y∈b	NOUN
ejpam-3418	409	24	{	{	PUNCT
ejpam-3418	409	25	inf	inf	PROPN
ejpam-3418	409	26	{	{	PUNCT
ejpam-3418	409	27	x}∈s′y	x}∈s′y	PROPN
ejpam-3418	409	28	{	{	PUNCT
ejpam-3418	409	29	1−	1−	NUM
ejpam-3418	410	1	[	[	X
ejpam-3418	410	2	̂ι({x})]−	̂ι({x})]−	PROPN
ejpam-3418	410	3	}	}	PUNCT
ejpam-3418	410	4	}	}	PUNCT
ejpam-3418	410	5	,	,	PUNCT
ejpam-3418	410	6	inf	inf	PROPN
ejpam-3418	410	7	y∈b	y∈b	PROPN
ejpam-3418	410	8	{	{	PUNCT
ejpam-3418	410	9	inf	inf	PROPN
ejpam-3418	410	10	{	{	PUNCT
ejpam-3418	410	11	x}∈s′y	x}∈s′y	PROPN
ejpam-3418	410	12	{	{	PUNCT
ejpam-3418	410	13	1−	1−	NUM
ejpam-3418	411	1	[	[	X
ejpam-3418	411	2	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	411	3	}	}	PUNCT
ejpam-3418	411	4	}	}	PUNCT
ejpam-3418	411	5	]	]	PUNCT
ejpam-3418	412	1	=	=	PUNCT
ejpam-3418	412	2	[	[	PUNCT
ejpam-3418	412	3	inf	inf	NOUN
ejpam-3418	412	4	x∈ab	x∈ab	PROPN
ejpam-3418	412	5	{	{	PUNCT
ejpam-3418	412	6	1−	1−	NUM
ejpam-3418	412	7	[	[	X
ejpam-3418	412	8	̂ι({x})]−	̂ι({x})]−	X
ejpam-3418	412	9	}	}	PUNCT
ejpam-3418	412	10	,	,	PUNCT
ejpam-3418	412	11	inf	inf	NOUN
ejpam-3418	412	12	x∈ab	x∈ab	PROPN
ejpam-3418	412	13	{	{	PUNCT
ejpam-3418	412	14	1−	1−	NUM
ejpam-3418	412	15	[	[	X
ejpam-3418	412	16	̂ι({x})]+	̂ι({x})]+	X
ejpam-3418	412	17	}	}	PUNCT
ejpam-3418	412	18	]	]	PUNCT
ejpam-3418	412	19	=	=	PUNCT
ejpam-3418	413	1	[	[	X
ejpam-3418	413	2	[	[	X
ejpam-3418	413	3	̂ιc(ab)]−	̂ιc(ab)]−	NUM
ejpam-3418	413	4	,	,	PUNCT
ejpam-3418	413	5	[	[	X
ejpam-3418	413	6	̂ιc(ab)]+	̂ιc(ab)]+	X
ejpam-3418	413	7	]	]	X
ejpam-3418	413	8	,	,	PUNCT
ejpam-3418	413	9	where	where	SCONJ
ejpam-3418	413	10	ab	ab	PROPN
ejpam-3418	413	11	=	=	PUNCT
ejpam-3418	413	12	{	{	PUNCT
ejpam-3418	413	13	x	x	X
ejpam-3418	413	14	:	:	PUNCT
ejpam-3418	413	15	{	{	PUNCT
ejpam-3418	413	16	x	x	NOUN
ejpam-3418	413	17	}	}	PUNCT
ejpam-3418	413	18	∈	∈	NOUN
ejpam-3418	413	19	s′y	s′y	NOUN
ejpam-3418	413	20	,	,	PUNCT
ejpam-3418	413	21	y	y	PROPN
ejpam-3418	413	22	∈	∈	PROPN
ejpam-3418	413	23	b	b	PROPN
ejpam-3418	413	24	}	}	PUNCT
ejpam-3418	413	25	.	.	PUNCT
ejpam-3418	414	1	thus	thus	ADV
ejpam-3418	414	2	,	,	PUNCT
ejpam-3418	414	3	(	(	PUNCT
ejpam-3418	414	4	f	f	X
ejpam-3418	415	1	[	[	X
ejpam-3418	415	2	̂ι])c(b	̂ι])c(b	X
ejpam-3418	415	3	)	)	PUNCT
ejpam-3418	415	4	=	=	PUNCT
ejpam-3418	416	1	[	[	X
ejpam-3418	416	2	[	[	X
ejpam-3418	416	3	̂ιc(ab)]−	̂ιc(ab)]−	NUM
ejpam-3418	416	4	,	,	PUNCT
ejpam-3418	416	5	[	[	X
ejpam-3418	416	6	̂ιc(ab)]+	̂ιc(ab)]+	X
ejpam-3418	416	7	]	]	X
ejpam-3418	416	8	.	.	PUNCT
ejpam-3418	417	1	note	note	VERB
ejpam-3418	417	2	that	that	SCONJ
ejpam-3418	417	3	ab	ab	PROPN
ejpam-3418	417	4	∈	∈	PROPN
ejpam-3418	417	5	s.	s.	PROPN
ejpam-3418	417	6	then	then	ADV
ejpam-3418	417	7	,	,	PUNCT
ejpam-3418	417	8	(	(	PUNCT
ejpam-3418	417	9	f	f	X
ejpam-3418	418	1	[	[	X
ejpam-3418	418	2	̂ι])c(b	̂ι])c(b	X
ejpam-3418	418	3	)	)	PUNCT
ejpam-3418	418	4	=	=	PUNCT
ejpam-3418	419	1	[	[	X
ejpam-3418	419	2	[	[	X
ejpam-3418	419	3	̂ιc(ab)]−	̂ιc(ab)]−	NUM
ejpam-3418	419	4	,	,	PUNCT
ejpam-3418	419	5	[	[	X
ejpam-3418	419	6	̂ιc(ab)]+	̂ιc(ab)]+	X
ejpam-3418	419	7	]	]	X
ejpam-3418	419	8	≤i	≤i	NOUN
ejpam-3418	419	9	[	[	PUNCT
ejpam-3418	419	10	sup	sup	NOUN
ejpam-3418	419	11	a∈s	a∈s	PROPN
ejpam-3418	419	12	[	[	X
ejpam-3418	419	13	̂ιc(a)]−	̂ιc(a)]−	ADJ
ejpam-3418	419	14	,	,	PUNCT
ejpam-3418	419	15	sup	sup	NOUN
ejpam-3418	419	16	a∈s	a∈s	PROPN
ejpam-3418	419	17	[	[	X
ejpam-3418	419	18	̂ιc(a)]+	̂ιc(a)]+	X
ejpam-3418	419	19	]	]	X
ejpam-3418	419	20	=	=	PUNCT
ejpam-3418	420	1	[	[	X
ejpam-3418	420	2	[	[	X
ejpam-3418	420	3	f	f	X
ejpam-3418	420	4	[	[	X
ejpam-3418	420	5	̂ιc](b)]−	̂ιc](b)]−	PROPN
ejpam-3418	420	6	,	,	PUNCT
ejpam-3418	420	7	[	[	X
ejpam-3418	420	8	f	f	X
ejpam-3418	420	9	[	[	X
ejpam-3418	420	10	̂ιc](b)]+	̂ιc](b)]+	X
ejpam-3418	420	11	]	]	X
ejpam-3418	420	12	=	=	SYM
ejpam-3418	420	13	f	f	X
ejpam-3418	421	1	[	[	X
ejpam-3418	421	2	̂ιc](b	̂ιc](b	NOUN
ejpam-3418	421	3	)	)	PUNCT
ejpam-3418	421	4	.	.	PUNCT
ejpam-3418	422	1	hence	hence	ADV
ejpam-3418	422	2	,	,	PUNCT
ejpam-3418	422	3	(	(	PUNCT
ejpam-3418	422	4	f	f	X
ejpam-3418	423	1	[	[	X
ejpam-3418	423	2	̂ι])c	̂ι])c	PROPN
ejpam-3418	423	3	6	6	NUM
ejpam-3418	423	4	f	f	NOUN
ejpam-3418	424	1	[	[	X
ejpam-3418	424	2	̂ιc	̂ιc	NOUN
ejpam-3418	424	3	]	]	PUNCT
ejpam-3418	424	4	.	.	PUNCT
ejpam-3418	425	1	iii	iii	X
ejpam-3418	425	2	.	.	PUNCT
ejpam-3418	425	3	let	let	VERB
ejpam-3418	425	4	a	a	DET
ejpam-3418	425	5	∈	∈	NOUN
ejpam-3418	425	6	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	425	7	)	)	PUNCT
ejpam-3418	425	8	)	)	PUNCT
ejpam-3418	425	9	.	.	PUNCT
ejpam-3418	426	1	then	then	ADV
ejpam-3418	426	2	,	,	PUNCT
ejpam-3418	426	3	a	a	DET
ejpam-3418	426	4	⊆	⊆	NUM
ejpam-3418	426	5	f−1(b	f−1(b	NOUN
ejpam-3418	426	6	)	)	PUNCT
ejpam-3418	426	7	,	,	PUNCT
ejpam-3418	426	8	for	for	ADP
ejpam-3418	426	9	some	some	DET
ejpam-3418	426	10	b	b	PROPN
ejpam-3418	426	11	∈	∈	PROPN
ejpam-3418	426	12	i(y	i(y	NOUN
ejpam-3418	426	13	)	)	PUNCT
ejpam-3418	426	14	.	.	PUNCT
ejpam-3418	427	1	note	note	VERB
ejpam-3418	427	2	that	that	SCONJ
ejpam-3418	427	3	f(a	f(a	NOUN
ejpam-3418	427	4	)	)	PUNCT
ejpam-3418	427	5	⊆	⊆	NUM
ejpam-3418	427	6	f(f−1(b	f(f−1(b	PROPN
ejpam-3418	427	7	)	)	PUNCT
ejpam-3418	427	8	)	)	PUNCT
ejpam-3418	428	1	⊆	⊆	NUM
ejpam-3418	428	2	b	b	X
ejpam-3418	428	3	∈	∈	PROPN
ejpam-3418	428	4	i(y	i(y	NOUN
ejpam-3418	428	5	)	)	PUNCT
ejpam-3418	428	6	.	.	PUNCT
ejpam-3418	429	1	thus	thus	ADV
ejpam-3418	429	2	,	,	PUNCT
ejpam-3418	429	3	f(a	f(a	PROPN
ejpam-3418	429	4	)	)	PUNCT
ejpam-3418	429	5	∈	∈	PROPN
ejpam-3418	429	6	i(y	i(y	NOUN
ejpam-3418	429	7	)	)	PUNCT
ejpam-3418	429	8	.	.	PUNCT
ejpam-3418	430	1	if	if	SCONJ
ejpam-3418	430	2	ω̂	ω̂	PROPN
ejpam-3418	430	3	6	6	NUM
ejpam-3418	430	4	η̂	η̂	NUM
ejpam-3418	430	5	,	,	PUNCT
ejpam-3418	430	6	then	then	ADV
ejpam-3418	430	7	ω̂(f(a	ω̂(f(a	NOUN
ejpam-3418	430	8	)	)	PUNCT
ejpam-3418	430	9	)	)	PUNCT
ejpam-3418	430	10	≤i	≤i	PROPN
ejpam-3418	430	11	η̂(f(a	η̂(f(a	NOUN
ejpam-3418	430	12	)	)	PUNCT
ejpam-3418	430	13	)	)	PUNCT
ejpam-3418	430	14	.	.	PUNCT
ejpam-3418	431	1	consider	consider	VERB
ejpam-3418	431	2	that	that	PRON
ejpam-3418	431	3	(	(	PUNCT
ejpam-3418	431	4	f−1[ω̂])(a	f−1[ω̂])(a	NOUN
ejpam-3418	431	5	)	)	PUNCT
ejpam-3418	431	6	=	=	SYM
ejpam-3418	431	7	(	(	PUNCT
ejpam-3418	431	8	ω̂	ω̂	PUNCT
ejpam-3418	431	9	◦	◦	NOUN
ejpam-3418	431	10	f)(a	f)(a	NUM
ejpam-3418	431	11	)	)	PUNCT
ejpam-3418	431	12	=	=	SYM
ejpam-3418	431	13	ω̂(f(a	ω̂(f(a	NOUN
ejpam-3418	431	14	)	)	PUNCT
ejpam-3418	431	15	)	)	PUNCT
ejpam-3418	432	1	≤i	≤i	PROPN
ejpam-3418	432	2	η̂(f(a	η̂(f(a	NOUN
ejpam-3418	432	3	)	)	PUNCT
ejpam-3418	432	4	)	)	PUNCT
ejpam-3418	433	1	=	=	SYM
ejpam-3418	433	2	(	(	PUNCT
ejpam-3418	433	3	η̂	η̂	X
ejpam-3418	433	4	◦	◦	NOUN
ejpam-3418	433	5	f)(a	f)(a	NUM
ejpam-3418	433	6	)	)	PUNCT
ejpam-3418	433	7	=	=	PRON
ejpam-3418	433	8	(	(	PUNCT
ejpam-3418	433	9	f−1[η̂])(a	f−1[η̂])(a	NOUN
ejpam-3418	433	10	)	)	PUNCT
ejpam-3418	433	11	.	.	PUNCT
ejpam-3418	434	1	therefore	therefore	ADV
ejpam-3418	434	2	,	,	PUNCT
ejpam-3418	434	3	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	434	4	]	]	X
ejpam-3418	434	5	6	6	NUM
ejpam-3418	434	6	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	434	7	]	]	PUNCT
ejpam-3418	434	8	.	.	PUNCT
ejpam-3418	435	1	iv	iv	X
ejpam-3418	435	2	.	.	PUNCT
ejpam-3418	436	1	let	let	VERB
ejpam-3418	436	2	b	b	X
ejpam-3418	436	3	∈	∈	PROPN
ejpam-3418	436	4	f(i(x	f(i(x	NOUN
ejpam-3418	436	5	)	)	PUNCT
ejpam-3418	436	6	)	)	PUNCT
ejpam-3418	437	1	and	and	CCONJ
ejpam-3418	437	2	s	s	X
ejpam-3418	437	3	=	=	X
ejpam-3418	437	4	{	{	PUNCT
ejpam-3418	437	5	a	a	DET
ejpam-3418	437	6	∈	∈	PROPN
ejpam-3418	437	7	i(x	i(x	NOUN
ejpam-3418	437	8	)	)	PUNCT
ejpam-3418	437	9	:	:	PUNCT
ejpam-3418	437	10	f(a	f(a	X
ejpam-3418	437	11	)	)	PUNCT
ejpam-3418	438	1	=	=	SYM
ejpam-3418	438	2	b	b	X
ejpam-3418	438	3	}	}	PUNCT
ejpam-3418	438	4	.	.	PUNCT
ejpam-3418	439	1	if	if	SCONJ
ejpam-3418	439	2	ι̂	ι̂	PRON
ejpam-3418	439	3	6	6	NUM
ejpam-3418	439	4	τ̂	τ̂	NUM
ejpam-3418	439	5	,	,	PUNCT
ejpam-3418	439	6	then	then	ADV
ejpam-3418	439	7	ι̂(a	ι̂(a	PUNCT
ejpam-3418	439	8	)	)	PUNCT
ejpam-3418	439	9	≤i	≤i	NOUN
ejpam-3418	439	10	τ̂(a	τ̂(a	NUM
ejpam-3418	439	11	)	)	PUNCT
ejpam-3418	439	12	,	,	PUNCT
ejpam-3418	439	13	for	for	ADP
ejpam-3418	439	14	all	all	DET
ejpam-3418	439	15	a	a	DET
ejpam-3418	439	16	∈	∈	PROPN
ejpam-3418	439	17	i(x	i(x	NOUN
ejpam-3418	439	18	)	)	PUNCT
ejpam-3418	439	19	.	.	PUNCT
ejpam-3418	440	1	hence	hence	ADV
ejpam-3418	440	2	,	,	PUNCT
ejpam-3418	440	3	(	(	PUNCT
ejpam-3418	440	4	f	f	X
ejpam-3418	441	1	[	[	X
ejpam-3418	441	2	̂ι])(b	̂ι])(b	X
ejpam-3418	441	3	)	)	PUNCT
ejpam-3418	441	4	=	=	PUNCT
ejpam-3418	442	1	[	[	PUNCT
ejpam-3418	442	2	sup	sup	NUM
ejpam-3418	442	3	a∈s	a∈s	PROPN
ejpam-3418	442	4	[	[	X
ejpam-3418	442	5	̂ι(a)]−	̂ι(a)]−	PROPN
ejpam-3418	442	6	,	,	PUNCT
ejpam-3418	442	7	sup	sup	NOUN
ejpam-3418	442	8	a∈s	a∈s	ADJ
ejpam-3418	442	9	[	[	X
ejpam-3418	442	10	̂ι(a)]+	̂ι(a)]+	X
ejpam-3418	442	11	]	]	X
ejpam-3418	442	12	≤i	≤i	PROPN
ejpam-3418	442	13	[	[	PUNCT
ejpam-3418	442	14	sup	sup	NOUN
ejpam-3418	442	15	a∈s	a∈s	ADJ
ejpam-3418	442	16	[	[	X
ejpam-3418	442	17	τ̂(a)]−	τ̂(a)]−	PROPN
ejpam-3418	442	18	,	,	PUNCT
ejpam-3418	442	19	sup	sup	NOUN
ejpam-3418	442	20	a∈s	a∈s	PROPN
ejpam-3418	442	21	[	[	X
ejpam-3418	442	22	τ̂(a)]+	τ̂(a)]+	X
ejpam-3418	442	23	]	]	X
ejpam-3418	442	24	=	=	PUNCT
ejpam-3418	442	25	(	(	PUNCT
ejpam-3418	442	26	f	f	X
ejpam-3418	443	1	[	[	X
ejpam-3418	443	2	τ̂	τ̂	X
ejpam-3418	443	3	]	]	PUNCT
ejpam-3418	443	4	)	)	PUNCT
ejpam-3418	443	5	(	(	PUNCT
ejpam-3418	443	6	b	b	NOUN
ejpam-3418	443	7	)	)	PUNCT
ejpam-3418	443	8	.	.	PUNCT
ejpam-3418	444	1	therefore	therefore	ADV
ejpam-3418	444	2	,	,	PUNCT
ejpam-3418	444	3	f	f	PROPN
ejpam-3418	445	1	[	[	X
ejpam-3418	445	2	̂ι	̂ι	X
ejpam-3418	445	3	]	]	X
ejpam-3418	445	4	6	6	NUM
ejpam-3418	445	5	f	f	X
ejpam-3418	445	6	[	[	X
ejpam-3418	445	7	τ̂	τ̂	X
ejpam-3418	445	8	]	]	PUNCT
ejpam-3418	445	9	.	.	PUNCT
ejpam-3418	446	1	v.	v.	CCONJ
ejpam-3418	446	2	suppose	suppose	VERB
ejpam-3418	446	3	that	that	SCONJ
ejpam-3418	446	4	f	f	PROPN
ejpam-3418	446	5	is	be	AUX
ejpam-3418	446	6	onto	onto	ADP
ejpam-3418	446	7	.	.	PUNCT
ejpam-3418	447	1	let	let	VERB
ejpam-3418	447	2	b	b	NOUN
ejpam-3418	447	3	∈	∈	PROPN
ejpam-3418	447	4	f(f−1(i(y	f(f−1(i(y	PROPN
ejpam-3418	447	5	)	)	PUNCT
ejpam-3418	447	6	)	)	PUNCT
ejpam-3418	447	7	)	)	PUNCT
ejpam-3418	448	1	=	=	PUNCT
ejpam-3418	448	2	i(y	i(y	NOUN
ejpam-3418	448	3	)	)	PUNCT
ejpam-3418	448	4	and	and	CCONJ
ejpam-3418	448	5	s	s	X
ejpam-3418	448	6	=	=	X
ejpam-3418	448	7	{	{	PUNCT
ejpam-3418	448	8	a	a	DET
ejpam-3418	448	9	∈	∈	NOUN
ejpam-3418	448	10	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	448	11	)	)	PUNCT
ejpam-3418	448	12	)	)	PUNCT
ejpam-3418	448	13	:	:	PUNCT
ejpam-3418	449	1	f(a	f(a	X
ejpam-3418	449	2	)	)	PUNCT
ejpam-3418	449	3	=	=	SYM
ejpam-3418	450	1	b	b	X
ejpam-3418	450	2	}	}	PUNCT
ejpam-3418	450	3	.	.	PUNCT
ejpam-3418	451	1	then	then	ADV
ejpam-3418	451	2	f	f	X
ejpam-3418	452	1	[	[	X
ejpam-3418	452	2	f−1[η̂]](b	f−1[η̂]](b	NOUN
ejpam-3418	452	3	)	)	PUNCT
ejpam-3418	452	4	=	=	PUNCT
ejpam-3418	453	1	[	[	PUNCT
ejpam-3418	453	2	sup	sup	NUM
ejpam-3418	453	3	a∈s	a∈s	ADJ
ejpam-3418	453	4	[	[	X
ejpam-3418	453	5	(	(	PUNCT
ejpam-3418	453	6	f−1[η̂])(a)]−	f−1[η̂])(a)]−	ADJ
ejpam-3418	453	7	,	,	PUNCT
ejpam-3418	453	8	sup	sup	NOUN
ejpam-3418	453	9	a∈s	a∈s	PROPN
ejpam-3418	453	10	[	[	X
ejpam-3418	453	11	(	(	PUNCT
ejpam-3418	453	12	f−1[η̂])(a)]+	f−1[η̂])(a)]+	NOUN
ejpam-3418	453	13	]	]	PUNCT
ejpam-3418	453	14	=	=	PUNCT
ejpam-3418	453	15	[	[	PUNCT
ejpam-3418	453	16	sup	sup	NUM
ejpam-3418	453	17	a∈s	a∈s	ADJ
ejpam-3418	453	18	[	[	X
ejpam-3418	453	19	(	(	PUNCT
ejpam-3418	453	20	η̂	η̂	NUM
ejpam-3418	453	21	◦	◦	NOUN
ejpam-3418	453	22	f)(a)]−	f)(a)]−	PROPN
ejpam-3418	453	23	,	,	PUNCT
ejpam-3418	453	24	sup	sup	NOUN
ejpam-3418	453	25	a∈s	a∈s	ADJ
ejpam-3418	453	26	[	[	X
ejpam-3418	453	27	(	(	PUNCT
ejpam-3418	453	28	η̂	η̂	ADJ
ejpam-3418	453	29	◦	◦	NOUN
ejpam-3418	453	30	f)(a)]+	f)(a)]+	NOUN
ejpam-3418	453	31	]	]	PUNCT
ejpam-3418	454	1	=	=	PUNCT
ejpam-3418	455	1	[	[	PUNCT
ejpam-3418	455	2	sup	sup	NUM
ejpam-3418	455	3	a∈s	a∈s	PROPN
ejpam-3418	455	4	[	[	X
ejpam-3418	455	5	η̂(f(a))]−	η̂(f(a))]−	ADJ
ejpam-3418	455	6	,	,	PUNCT
ejpam-3418	455	7	sup	sup	NOUN
ejpam-3418	455	8	a∈s	a∈s	PROPN
ejpam-3418	455	9	[	[	X
ejpam-3418	455	10	η̂(f(a))]+	η̂(f(a))]+	X
ejpam-3418	455	11	]	]	X
ejpam-3418	455	12	mj	mj	PROPN
ejpam-3418	455	13	togonon	togonon	PROPN
ejpam-3418	455	14	,	,	PUNCT
ejpam-3418	455	15	r	r	NOUN
ejpam-3418	455	16	caga	caga	NOUN
ejpam-3418	455	17	-	-	PUNCT
ejpam-3418	455	18	anan	anan	PROPN
ejpam-3418	455	19	/	/	SYM
ejpam-3418	455	20	eur	eur	PROPN
ejpam-3418	455	21	.	.	PUNCT
ejpam-3418	456	1	j.	j.	PROPN
ejpam-3418	456	2	pure	pure	PROPN
ejpam-3418	456	3	appl	appl	PROPN
ejpam-3418	456	4	.	.	PROPN
ejpam-3418	456	5	math	math	PROPN
ejpam-3418	456	6	,	,	PUNCT
ejpam-3418	456	7	12	12	NUM
ejpam-3418	456	8	(	(	PUNCT
ejpam-3418	456	9	2	2	NUM
ejpam-3418	456	10	)	)	PUNCT
ejpam-3418	456	11	(	(	PUNCT
ejpam-3418	456	12	2019	2019	NUM
ejpam-3418	456	13	)	)	PUNCT
ejpam-3418	456	14	,	,	PUNCT
ejpam-3418	456	15	553	553	NUM
ejpam-3418	456	16	-	-	SYM
ejpam-3418	456	17	570	570	NUM
ejpam-3418	456	18	564	564	NUM
ejpam-3418	456	19	=	=	PUNCT
ejpam-3418	457	1	[	[	X
ejpam-3418	457	2	[	[	X
ejpam-3418	457	3	η̂(b)]−	η̂(b)]−	X
ejpam-3418	457	4	,	,	PUNCT
ejpam-3418	457	5	[	[	X
ejpam-3418	457	6	η̂(b)]+	η̂(b)]+	X
ejpam-3418	457	7	]	]	X
ejpam-3418	457	8	=	=	SYM
ejpam-3418	457	9	η̂(b	η̂(b	NOUN
ejpam-3418	457	10	)	)	PUNCT
ejpam-3418	457	11	.	.	PUNCT
ejpam-3418	458	1	thus	thus	ADV
ejpam-3418	458	2	,	,	PUNCT
ejpam-3418	458	3	f	f	PROPN
ejpam-3418	458	4	[	[	X
ejpam-3418	458	5	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	458	6	]	]	X
ejpam-3418	458	7	]	]	X
ejpam-3418	458	8	=	=	SYM
ejpam-3418	458	9	η̂.	η̂.	PROPN
ejpam-3418	458	10	vi	vi	PROPN
ejpam-3418	458	11	.	.	PROPN
ejpam-3418	458	12	suppose	suppose	VERB
ejpam-3418	458	13	that	that	SCONJ
ejpam-3418	458	14	f	f	PROPN
ejpam-3418	458	15	is	be	AUX
ejpam-3418	458	16	one	one	NUM
ejpam-3418	458	17	-	-	PUNCT
ejpam-3418	458	18	to	to	ADP
ejpam-3418	458	19	-	-	PUNCT
ejpam-3418	458	20	one	one	NUM
ejpam-3418	458	21	.	.	PUNCT
ejpam-3418	459	1	let	let	VERB
ejpam-3418	459	2	a	a	DET
ejpam-3418	459	3	∈	∈	PROPN
ejpam-3418	459	4	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	459	5	)	)	PUNCT
ejpam-3418	459	6	)	)	PUNCT
ejpam-3418	459	7	)	)	PUNCT
ejpam-3418	460	1	=	=	SYM
ejpam-3418	460	2	i(x	i(x	NOUN
ejpam-3418	460	3	)	)	PUNCT
ejpam-3418	460	4	.	.	PUNCT
ejpam-3418	461	1	then	then	ADV
ejpam-3418	461	2	,	,	PUNCT
ejpam-3418	461	3	a	a	DET
ejpam-3418	461	4	⊆	⊆	NUM
ejpam-3418	461	5	f−1(b	f−1(b	NOUN
ejpam-3418	461	6	)	)	PUNCT
ejpam-3418	461	7	,	,	PUNCT
ejpam-3418	461	8	for	for	ADP
ejpam-3418	461	9	some	some	DET
ejpam-3418	461	10	b	b	PROPN
ejpam-3418	461	11	∈	∈	PROPN
ejpam-3418	461	12	f(i(x	f(i(x	NOUN
ejpam-3418	461	13	)	)	PUNCT
ejpam-3418	461	14	)	)	PUNCT
ejpam-3418	461	15	.	.	PUNCT
ejpam-3418	462	1	hence	hence	ADV
ejpam-3418	462	2	,	,	PUNCT
ejpam-3418	462	3	f(a	f(a	PROPN
ejpam-3418	462	4	)	)	PUNCT
ejpam-3418	462	5	⊆	⊆	NUM
ejpam-3418	462	6	b.	b.	NOUN
ejpam-3418	462	7	since	since	SCONJ
ejpam-3418	462	8	f(i(x	f(i(x	NOUN
ejpam-3418	462	9	)	)	PUNCT
ejpam-3418	462	10	)	)	PUNCT
ejpam-3418	462	11	is	be	AUX
ejpam-3418	462	12	an	an	DET
ejpam-3418	462	13	ideal	ideal	ADJ
ejpam-3418	462	14	,	,	PUNCT
ejpam-3418	462	15	f(a	f(a	NOUN
ejpam-3418	462	16	)	)	PUNCT
ejpam-3418	462	17	∈	∈	PROPN
ejpam-3418	462	18	f(i(x	f(i(x	NOUN
ejpam-3418	462	19	)	)	PUNCT
ejpam-3418	462	20	)	)	PUNCT
ejpam-3418	462	21	.	.	PUNCT
ejpam-3418	463	1	then	then	ADV
ejpam-3418	463	2	,	,	PUNCT
ejpam-3418	463	3	f−1[f	f−1[f	VERB
ejpam-3418	463	4	[	[	NOUN
ejpam-3418	463	5	̂ι]](a	̂ι]](a	NOUN
ejpam-3418	463	6	)	)	PUNCT
ejpam-3418	463	7	=	=	PUNCT
ejpam-3418	464	1	(	(	PUNCT
ejpam-3418	464	2	f	f	X
ejpam-3418	465	1	[	[	X
ejpam-3418	465	2	̂ι	̂ι	X
ejpam-3418	465	3	]	]	X
ejpam-3418	465	4	◦	◦	NOUN
ejpam-3418	465	5	f)(a	f)(a	NUM
ejpam-3418	465	6	)	)	PUNCT
ejpam-3418	466	1	=	=	SYM
ejpam-3418	466	2	f	f	X
ejpam-3418	467	1	[	[	X
ejpam-3418	467	2	̂ι](f(a	̂ι](f(a	NOUN
ejpam-3418	467	3	)	)	PUNCT
ejpam-3418	467	4	)	)	PUNCT
ejpam-3418	468	1	=	=	PUNCT
ejpam-3418	468	2	[	[	PUNCT
ejpam-3418	468	3	sup	sup	NOUN
ejpam-3418	468	4	c∈s	c∈	NOUN
ejpam-3418	468	5	[	[	X
ejpam-3418	468	6	̂ι(c)]−	̂ι(c)]−	NOUN
ejpam-3418	468	7	,	,	PUNCT
ejpam-3418	468	8	sup	sup	NOUN
ejpam-3418	468	9	c∈s	c∈	NOUN
ejpam-3418	468	10	[	[	X
ejpam-3418	468	11	̂ι(c)]+	̂ι(c)]+	X
ejpam-3418	468	12	]	]	PUNCT
ejpam-3418	468	13	,	,	PUNCT
ejpam-3418	468	14	where	where	SCONJ
ejpam-3418	468	15	s	s	VERB
ejpam-3418	468	16	=	=	PUNCT
ejpam-3418	468	17	{	{	PUNCT
ejpam-3418	468	18	c	c	NOUN
ejpam-3418	468	19	∈	∈	PROPN
ejpam-3418	468	20	i(x	i(x	PROPN
ejpam-3418	468	21	)	)	PUNCT
ejpam-3418	468	22	:	:	PUNCT
ejpam-3418	469	1	f(c	f(c	NOUN
ejpam-3418	469	2	)	)	PUNCT
ejpam-3418	469	3	=	=	SYM
ejpam-3418	469	4	f(a	f(a	NOUN
ejpam-3418	469	5	)	)	PUNCT
ejpam-3418	469	6	}	}	PUNCT
ejpam-3418	469	7	.	.	PUNCT
ejpam-3418	470	1	since	since	SCONJ
ejpam-3418	470	2	a	a	DET
ejpam-3418	470	3	∈	∈	PROPN
ejpam-3418	470	4	s	s	NOUN
ejpam-3418	470	5	,	,	PUNCT
ejpam-3418	470	6	we	we	PRON
ejpam-3418	470	7	have	have	VERB
ejpam-3418	470	8	ι̂(a	ι̂(a	NOUN
ejpam-3418	470	9	)	)	PUNCT
ejpam-3418	470	10	≤i	≤i	NOUN
ejpam-3418	470	11	[	[	PUNCT
ejpam-3418	470	12	sup	sup	NOUN
ejpam-3418	470	13	c∈s	c∈	NOUN
ejpam-3418	470	14	[	[	X
ejpam-3418	470	15	̂ι(c)]−	̂ι(c)]−	NOUN
ejpam-3418	470	16	,	,	PUNCT
ejpam-3418	470	17	sup	sup	NOUN
ejpam-3418	470	18	c∈s	c∈	NOUN
ejpam-3418	470	19	[	[	X
ejpam-3418	470	20	̂ι(c)]+	̂ι(c)]+	X
ejpam-3418	470	21	]	]	PUNCT
ejpam-3418	470	22	.	.	PUNCT
ejpam-3418	471	1	thus	thus	ADV
ejpam-3418	471	2	,	,	PUNCT
ejpam-3418	471	3	ι̂	ι̂	X
ejpam-3418	471	4	6	6	NUM
ejpam-3418	471	5	f−1[f	f−1[f	NOUN
ejpam-3418	471	6	[	[	X
ejpam-3418	471	7	̂ι	̂ι	X
ejpam-3418	471	8	]	]	X
ejpam-3418	471	9	]	]	PUNCT
ejpam-3418	471	10	.	.	PUNCT
ejpam-3418	472	1	the	the	DET
ejpam-3418	472	2	pre	pre	NOUN
ejpam-3418	472	3	-	-	NOUN
ejpam-3418	472	4	image	image	NOUN
ejpam-3418	472	5	of	of	ADP
ejpam-3418	472	6	the	the	DET
ejpam-3418	472	7	arbitrary	arbitrary	ADJ
ejpam-3418	472	8	union	union	NOUN
ejpam-3418	472	9	and	and	CCONJ
ejpam-3418	472	10	intersection	intersection	NOUN
ejpam-3418	472	11	of	of	ADP
ejpam-3418	472	12	ivfi	ivfi	NOUN
ejpam-3418	472	13	sets	set	NOUN
ejpam-3418	472	14	is	be	AUX
ejpam-3418	472	15	just	just	ADV
ejpam-3418	472	16	the	the	DET
ejpam-3418	472	17	union	union	NOUN
ejpam-3418	472	18	and	and	CCONJ
ejpam-3418	472	19	intersection	intersection	NOUN
ejpam-3418	472	20	of	of	ADP
ejpam-3418	472	21	the	the	DET
ejpam-3418	472	22	pre	pre	NOUN
ejpam-3418	472	23	-	-	NOUN
ejpam-3418	472	24	images	image	NOUN
ejpam-3418	472	25	as	as	SCONJ
ejpam-3418	472	26	proved	prove	VERB
ejpam-3418	472	27	below	below	ADV
ejpam-3418	472	28	.	.	PUNCT
ejpam-3418	473	1	theorem	theorem	VERB
ejpam-3418	473	2	6	6	NUM
ejpam-3418	473	3	.	.	PUNCT
ejpam-3418	474	1	let	let	AUX
ejpam-3418	474	2	x	x	PRON
ejpam-3418	474	3	and	and	CCONJ
ejpam-3418	474	4	y	y	PROPN
ejpam-3418	474	5	be	be	AUX
ejpam-3418	474	6	nonempty	nonempty	X
ejpam-3418	474	7	sets	set	NOUN
ejpam-3418	474	8	and	and	CCONJ
ejpam-3418	474	9	i(y	i(y	NOUN
ejpam-3418	474	10	)	)	PUNCT
ejpam-3418	474	11	be	be	AUX
ejpam-3418	474	12	an	an	DET
ejpam-3418	474	13	ideal	ideal	NOUN
ejpam-3418	474	14	on	on	ADP
ejpam-3418	474	15	y	y	PROPN
ejpam-3418	474	16	.	.	PUNCT
ejpam-3418	475	1	moreover	moreover	ADV
ejpam-3418	475	2	,	,	PUNCT
ejpam-3418	475	3	let	let	VERB
ejpam-3418	475	4	f	f	PRON
ejpam-3418	475	5	:	:	PUNCT
ejpam-3418	475	6	x	x	X
ejpam-3418	475	7	→	→	SYM
ejpam-3418	475	8	y	y	X
ejpam-3418	475	9	be	be	AUX
ejpam-3418	475	10	a	a	DET
ejpam-3418	475	11	mapping	mapping	NOUN
ejpam-3418	475	12	and	and	CCONJ
ejpam-3418	475	13	{	{	PUNCT
ejpam-3418	475	14	ι̂j	ι̂j	PROPN
ejpam-3418	475	15	:	:	PUNCT
ejpam-3418	475	16	j	j	PROPN
ejpam-3418	475	17	∈	∈	PROPN
ejpam-3418	475	18	j	j	PROPN
ejpam-3418	475	19	}	}	PUNCT
ejpam-3418	475	20	be	be	AUX
ejpam-3418	475	21	a	a	DET
ejpam-3418	475	22	collection	collection	NOUN
ejpam-3418	475	23	of	of	ADP
ejpam-3418	475	24	ivfi	ivfi	NOUN
ejpam-3418	475	25	sets	set	NOUN
ejpam-3418	475	26	in	in	ADP
ejpam-3418	475	27	i	i	PRON
ejpam-3418	475	28	i(y	i(y	NOUN
ejpam-3418	475	29	)	)	PUNCT
ejpam-3418	475	30	.	.	PUNCT
ejpam-3418	476	1	then	then	ADV
ejpam-3418	476	2	i.	i.	PROPN
ejpam-3418	476	3	f−1	f−1	PROPN
ejpam-3418	476	4	∨	∨	PROPN
ejpam-3418	476	5	j∈j	j∈j	NOUN
ejpam-3418	476	6	ι̂j	ι̂j	NOUN
ejpam-3418	476	7			NOUN
ejpam-3418	476	8	=	=	SYM
ejpam-3418	476	9	∨	∨	NUM
ejpam-3418	476	10	j∈j	j∈j	NOUN
ejpam-3418	476	11	f−1	f−1	PROPN
ejpam-3418	477	1	[	[	X
ejpam-3418	477	2	̂ιj	̂ιj	X
ejpam-3418	477	3	]	]	X
ejpam-3418	477	4	;	;	PUNCT
ejpam-3418	477	5	and	and	CCONJ
ejpam-3418	477	6	ii	ii	X
ejpam-3418	477	7	.	.	PUNCT
ejpam-3418	478	1	f−1	f−1	PROPN
ejpam-3418	478	2	∧	∧	PROPN
ejpam-3418	478	3	j∈j	j∈j	NOUN
ejpam-3418	478	4	ι̂j	ι̂j	NOUN
ejpam-3418	478	5			NOUN
ejpam-3418	478	6	=	=	PUNCT
ejpam-3418	478	7	∧	∧	PROPN
ejpam-3418	478	8	j∈j	j∈j	NOUN
ejpam-3418	478	9	f−1	f−1	PROPN
ejpam-3418	479	1	[	[	X
ejpam-3418	479	2	̂ιj	̂ιj	X
ejpam-3418	479	3	]	]	PUNCT
ejpam-3418	479	4	.	.	PUNCT
ejpam-3418	480	1	proof	proof	NOUN
ejpam-3418	480	2	.	.	PUNCT
ejpam-3418	481	1	let	let	VERB
ejpam-3418	481	2	a	a	DET
ejpam-3418	481	3	∈	∈	NOUN
ejpam-3418	481	4	f−1(i(y	f−1(i(y	NOUN
ejpam-3418	481	5	)	)	PUNCT
ejpam-3418	481	6	)	)	PUNCT
ejpam-3418	481	7	.	.	PUNCT
ejpam-3418	482	1	consider	consider	VERB
ejpam-3418	482	2	thatf−1	thatf−1	NOUN
ejpam-3418	482	3	∨	∨	PROPN
ejpam-3418	482	4	j∈j	j∈j	NOUN
ejpam-3418	482	5	ι̂j	ι̂j	PROPN
ejpam-3418	483	1			NOUN
ejpam-3418	483	2	(	(	PUNCT
ejpam-3418	483	3	a	a	NOUN
ejpam-3418	483	4	)	)	PUNCT
ejpam-3418	483	5	=	=	SYM
ejpam-3418	483	6	∨	∨	PROPN
ejpam-3418	483	7	j∈j	j∈j	NOUN
ejpam-3418	483	8	ι̂j	ι̂j	PROPN
ejpam-3418	483	9	◦	◦	NOUN
ejpam-3418	483	10	f	f	X
ejpam-3418	484	1			PROPN
ejpam-3418	484	2	(	(	PUNCT
ejpam-3418	484	3	a	a	X
ejpam-3418	484	4	)	)	PUNCT
ejpam-3418	484	5	=	=	SYM
ejpam-3418	484	6	∨	∨	PROPN
ejpam-3418	484	7	j∈j	j∈j	NOUN
ejpam-3418	484	8	ι̂j	ι̂j	PROPN
ejpam-3418	485	1			PROPN
ejpam-3418	485	2	(	(	PUNCT
ejpam-3418	485	3	f(a	f(a	NOUN
ejpam-3418	485	4	)	)	PUNCT
ejpam-3418	485	5	)	)	PUNCT
ejpam-3418	486	1	=	=	PUNCT
ejpam-3418	486	2	[	[	PUNCT
ejpam-3418	486	3	sup	sup	NOUN
ejpam-3418	486	4	j∈j	j∈j	NOUN
ejpam-3418	486	5	{	{	PUNCT
ejpam-3418	486	6	[	[	X
ejpam-3418	486	7	̂ιj(f(a))]−	̂ιj(f(a))]−	X
ejpam-3418	486	8	}	}	PUNCT
ejpam-3418	486	9	,	,	PUNCT
ejpam-3418	486	10	sup	sup	NOUN
ejpam-3418	486	11	j∈j	j∈j	NOUN
ejpam-3418	486	12	{	{	PUNCT
ejpam-3418	486	13	[	[	X
ejpam-3418	486	14	̂ιj(f(a))]+	̂ιj(f(a))]+	NOUN
ejpam-3418	486	15	}	}	PUNCT
ejpam-3418	486	16	]	]	PUNCT
ejpam-3418	487	1	=	=	PUNCT
ejpam-3418	487	2	[	[	PUNCT
ejpam-3418	487	3	sup	sup	NOUN
ejpam-3418	487	4	j∈j	j∈j	NOUN
ejpam-3418	487	5	{	{	PUNCT
ejpam-3418	487	6	[	[	X
ejpam-3418	487	7	(	(	PUNCT
ejpam-3418	487	8	ι̂j	ι̂j	PROPN
ejpam-3418	487	9	◦	◦	NOUN
ejpam-3418	487	10	f)(a)]−	f)(a)]−	PROPN
ejpam-3418	487	11	}	}	PUNCT
ejpam-3418	487	12	,	,	PUNCT
ejpam-3418	487	13	sup	sup	NOUN
ejpam-3418	487	14	j∈j	j∈j	NOUN
ejpam-3418	487	15	{	{	PUNCT
ejpam-3418	487	16	[	[	X
ejpam-3418	487	17	(	(	PUNCT
ejpam-3418	487	18	ι̂j	ι̂j	NOUN
ejpam-3418	487	19	◦	◦	NOUN
ejpam-3418	487	20	f)(a)]+	f)(a)]+	NOUN
ejpam-3418	487	21	}	}	PUNCT
ejpam-3418	487	22	]	]	PUNCT
ejpam-3418	487	23	mj	mj	PROPN
ejpam-3418	487	24	togonon	togonon	PROPN
ejpam-3418	487	25	,	,	PUNCT
ejpam-3418	487	26	r	r	NOUN
ejpam-3418	487	27	caga	caga	NOUN
ejpam-3418	487	28	-	-	PUNCT
ejpam-3418	487	29	anan	anan	PROPN
ejpam-3418	487	30	/	/	SYM
ejpam-3418	487	31	eur	eur	PROPN
ejpam-3418	487	32	.	.	PUNCT
ejpam-3418	488	1	j.	j.	PROPN
ejpam-3418	488	2	pure	pure	PROPN
ejpam-3418	488	3	appl	appl	PROPN
ejpam-3418	488	4	.	.	PROPN
ejpam-3418	488	5	math	math	PROPN
ejpam-3418	488	6	,	,	PUNCT
ejpam-3418	488	7	12	12	NUM
ejpam-3418	488	8	(	(	PUNCT
ejpam-3418	488	9	2	2	NUM
ejpam-3418	488	10	)	)	PUNCT
ejpam-3418	488	11	(	(	PUNCT
ejpam-3418	488	12	2019	2019	NUM
ejpam-3418	488	13	)	)	PUNCT
ejpam-3418	488	14	,	,	PUNCT
ejpam-3418	488	15	553	553	NUM
ejpam-3418	488	16	-	-	SYM
ejpam-3418	488	17	570	570	NUM
ejpam-3418	488	18	565	565	NUM
ejpam-3418	488	19	=	=	SYM
ejpam-3418	488	20	∨	∨	NUM
ejpam-3418	488	21	j∈j	j∈j	NOUN
ejpam-3418	488	22	(	(	PUNCT
ejpam-3418	488	23	ι̂j	ι̂j	NOUN
ejpam-3418	488	24	◦	◦	NOUN
ejpam-3418	488	25	f)(a	f)(a	NOUN
ejpam-3418	488	26	)	)	PUNCT
ejpam-3418	488	27	=	=	SYM
ejpam-3418	489	1	∨	∨	PROPN
ejpam-3418	489	2	j∈j	j∈j	NOUN
ejpam-3418	489	3	f−1	f−1	PROPN
ejpam-3418	489	4	[	[	X
ejpam-3418	489	5	̂ιj	̂ιj	X
ejpam-3418	489	6	]	]	X
ejpam-3418	490	1			PROPN
ejpam-3418	490	2	(	(	PUNCT
ejpam-3418	490	3	a	a	NOUN
ejpam-3418	490	4	)	)	PUNCT
ejpam-3418	490	5	.	.	PUNCT
ejpam-3418	491	1	thus	thus	ADV
ejpam-3418	491	2	,	,	PUNCT
ejpam-3418	491	3	f−1	f−1	PROPN
ejpam-3418	491	4	∨	∨	PROPN
ejpam-3418	491	5	j∈j	j∈j	NOUN
ejpam-3418	491	6	ι̂j	ι̂j	NOUN
ejpam-3418	491	7			NOUN
ejpam-3418	491	8	=	=	SYM
ejpam-3418	491	9	∨	∨	NUM
ejpam-3418	491	10	j∈j	j∈j	NOUN
ejpam-3418	491	11	f−1	f−1	PROPN
ejpam-3418	492	1	[	[	X
ejpam-3418	492	2	̂ιj	̂ιj	X
ejpam-3418	492	3	]	]	PUNCT
ejpam-3418	492	4	,	,	PUNCT
ejpam-3418	492	5	proving	prove	VERB
ejpam-3418	492	6	(	(	PUNCT
ejpam-3418	492	7	i	i	NOUN
ejpam-3418	492	8	)	)	PUNCT
ejpam-3418	492	9	.	.	PUNCT
ejpam-3418	493	1	result	result	NOUN
ejpam-3418	493	2	(	(	PUNCT
ejpam-3418	493	3	ii	ii	NOUN
ejpam-3418	493	4	)	)	PUNCT
ejpam-3418	493	5	can	can	AUX
ejpam-3418	493	6	be	be	AUX
ejpam-3418	493	7	proved	prove	VERB
ejpam-3418	493	8	similarly	similarly	ADV
ejpam-3418	493	9	.	.	PUNCT
ejpam-3418	494	1	4	4	X
ejpam-3418	494	2	.	.	X
ejpam-3418	494	3	topology	topology	NOUN
ejpam-3418	494	4	and	and	CCONJ
ejpam-3418	494	5	continuity	continuity	NOUN
ejpam-3418	494	6	with	with	ADP
ejpam-3418	494	7	the	the	DET
ejpam-3418	494	8	operations	operation	NOUN
ejpam-3418	494	9	on	on	ADP
ejpam-3418	494	10	ivfi	ivfi	NOUN
ejpam-3418	494	11	sets	set	NOUN
ejpam-3418	494	12	,	,	PUNCT
ejpam-3418	494	13	we	we	PRON
ejpam-3418	494	14	can	can	AUX
ejpam-3418	494	15	have	have	VERB
ejpam-3418	494	16	an	an	DET
ejpam-3418	494	17	analogue	analogue	NOUN
ejpam-3418	494	18	of	of	ADP
ejpam-3418	494	19	the	the	DET
ejpam-3418	494	20	classical	classical	ADJ
ejpam-3418	494	21	topology	topology	NOUN
ejpam-3418	494	22	.	.	PUNCT
ejpam-3418	495	1	definition	definition	NOUN
ejpam-3418	495	2	6	6	NUM
ejpam-3418	495	3	.	.	PUNCT
ejpam-3418	496	1	let	let	VERB
ejpam-3418	496	2	x	x	PRON
ejpam-3418	496	3	be	be	AUX
ejpam-3418	496	4	a	a	DET
ejpam-3418	496	5	nonempty	nonempty	ADV
ejpam-3418	496	6	set	set	VERB
ejpam-3418	496	7	and	and	CCONJ
ejpam-3418	496	8	i(x	i(x	NOUN
ejpam-3418	496	9	)	)	PUNCT
ejpam-3418	496	10	be	be	AUX
ejpam-3418	496	11	an	an	DET
ejpam-3418	496	12	ideal	ideal	NOUN
ejpam-3418	496	13	on	on	ADP
ejpam-3418	496	14	x.	x.	NOUN
ejpam-3418	496	15	an	an	DET
ejpam-3418	496	16	ivfi	ivfi	NOUN
ejpam-3418	496	17	topology	topology	NOUN
ejpam-3418	496	18	is	be	AUX
ejpam-3418	496	19	a	a	DET
ejpam-3418	496	20	family	family	NOUN
ejpam-3418	496	21	t	t	NOUN
ejpam-3418	496	22	′	′	NUM
ejpam-3418	496	23	of	of	ADP
ejpam-3418	496	24	ivfi	ivfi	NOUN
ejpam-3418	496	25	sets	set	VERB
ejpam-3418	496	26	such	such	ADJ
ejpam-3418	496	27	that	that	PRON
ejpam-3418	496	28	:	:	PUNCT
ejpam-3418	496	29	(	(	PUNCT
ejpam-3418	496	30	i	i	NOUN
ejpam-3418	496	31	)	)	PUNCT
ejpam-3418	496	32	0̃i(x	0̃i(x	NUM
ejpam-3418	496	33	)	)	PUNCT
ejpam-3418	496	34	,	,	PUNCT
ejpam-3418	496	35	1̃i(x	1̃i(x	NUM
ejpam-3418	496	36	)	)	PUNCT
ejpam-3418	496	37	∈	∈	PROPN
ejpam-3418	496	38	t	t	PROPN
ejpam-3418	496	39	′	′	NOUN
ejpam-3418	496	40	;	;	PUNCT
ejpam-3418	496	41	(	(	PUNCT
ejpam-3418	496	42	ii	ii	NOUN
ejpam-3418	496	43	)	)	PUNCT
ejpam-3418	496	44	if	if	SCONJ
ejpam-3418	496	45	ι̂	ι̂	NOUN
ejpam-3418	496	46	,	,	PUNCT
ejpam-3418	496	47	τ̂	τ̂	PUNCT
ejpam-3418	496	48	∈	∈	PROPN
ejpam-3418	496	49	t	t	PROPN
ejpam-3418	496	50	′	′	NOUN
ejpam-3418	496	51	,	,	PUNCT
ejpam-3418	496	52	then	then	ADV
ejpam-3418	496	53	ι̂	ι̂	VERB
ejpam-3418	496	54	∧	∧	NOUN
ejpam-3418	496	55	τ̂	τ̂	PUNCT
ejpam-3418	496	56	∈	∈	PROPN
ejpam-3418	496	57	t	t	PROPN
ejpam-3418	496	58	′	′	NUM
ejpam-3418	496	59	;	;	PUNCT
ejpam-3418	496	60	and	and	CCONJ
ejpam-3418	496	61	(	(	PUNCT
ejpam-3418	496	62	iii	iii	X
ejpam-3418	496	63	)	)	PUNCT
ejpam-3418	496	64	if	if	SCONJ
ejpam-3418	496	65	{	{	PUNCT
ejpam-3418	496	66	ι̂j	ι̂j	NOUN
ejpam-3418	496	67	:	:	PUNCT
ejpam-3418	496	68	j	j	PROPN
ejpam-3418	496	69	∈	∈	PROPN
ejpam-3418	496	70	j	j	PROPN
ejpam-3418	496	71	}	}	PUNCT
ejpam-3418	496	72	⊆	⊆	NUM
ejpam-3418	496	73	t	t	NOUN
ejpam-3418	496	74	′	′	NUM
ejpam-3418	496	75	,	,	PUNCT
ejpam-3418	496	76	then	then	ADV
ejpam-3418	496	77	∨	∨	NUM
ejpam-3418	496	78	j∈j	j∈j	NOUN
ejpam-3418	496	79	ι̂j	ι̂j	PROPN
ejpam-3418	496	80	∈	∈	PROPN
ejpam-3418	496	81	t	t	NOUN
ejpam-3418	496	82	′.	′.	NOUN
ejpam-3418	496	83	we	we	PRON
ejpam-3418	496	84	call	call	VERB
ejpam-3418	496	85	the	the	DET
ejpam-3418	496	86	ordered	order	VERB
ejpam-3418	496	87	pair	pair	NOUN
ejpam-3418	496	88	(	(	PUNCT
ejpam-3418	496	89	i(x),t	i(x),t	NOUN
ejpam-3418	496	90	′	′	NUM
ejpam-3418	496	91	)	)	PUNCT
ejpam-3418	496	92	an	an	DET
ejpam-3418	496	93	ivfi	ivfi	NOUN
ejpam-3418	496	94	space	space	NOUN
ejpam-3418	496	95	and	and	CCONJ
ejpam-3418	496	96	an	an	DET
ejpam-3418	496	97	element	element	NOUN
ejpam-3418	496	98	of	of	ADP
ejpam-3418	496	99	t	t	PROPN
ejpam-3418	496	100	′	′	NUM
ejpam-3418	496	101	an	an	DET
ejpam-3418	496	102	ivfi	ivfi	NOUN
ejpam-3418	496	103	open	open	ADJ
ejpam-3418	496	104	set	set	NOUN
ejpam-3418	496	105	.	.	PUNCT
ejpam-3418	497	1	an	an	DET
ejpam-3418	497	2	ivfi	ivfi	NOUN
ejpam-3418	497	3	set	set	VERB
ejpam-3418	497	4	ι̂	ι̂	PRON
ejpam-3418	497	5	will	will	AUX
ejpam-3418	497	6	be	be	AUX
ejpam-3418	497	7	called	call	VERB
ejpam-3418	497	8	ivfi	ivfi	NOUN
ejpam-3418	497	9	closed	close	VERB
ejpam-3418	497	10	if	if	SCONJ
ejpam-3418	497	11	its	its	PRON
ejpam-3418	497	12	complement	complement	NOUN
ejpam-3418	497	13	is	be	AUX
ejpam-3418	497	14	ivfi	ivfi	NOUN
ejpam-3418	497	15	open	open	ADJ
ejpam-3418	497	16	.	.	PUNCT
ejpam-3418	498	1	it	it	PRON
ejpam-3418	498	2	can	can	AUX
ejpam-3418	498	3	be	be	AUX
ejpam-3418	498	4	easily	easily	ADV
ejpam-3418	498	5	seen	see	VERB
ejpam-3418	498	6	that	that	SCONJ
ejpam-3418	498	7	if	if	SCONJ
ejpam-3418	498	8	{	{	PUNCT
ejpam-3418	498	9	t	t	NOUN
ejpam-3418	498	10	′α	′α	NOUN
ejpam-3418	498	11	:	:	PUNCT
ejpam-3418	498	12	α	α	PROPN
ejpam-3418	498	13	∈	∈	PROPN
ejpam-3418	498	14	a	a	PRON
ejpam-3418	498	15	}	}	PUNCT
ejpam-3418	498	16	is	be	AUX
ejpam-3418	498	17	a	a	DET
ejpam-3418	498	18	family	family	NOUN
ejpam-3418	498	19	of	of	ADP
ejpam-3418	498	20	ivfi	ivfi	NOUN
ejpam-3418	498	21	topologies	topology	NOUN
ejpam-3418	498	22	on	on	ADP
ejpam-3418	498	23	i(x	i(x	PROPN
ejpam-3418	498	24	)	)	PUNCT
ejpam-3418	498	25	,	,	PUNCT
ejpam-3418	498	26	then⋂	then⋂	PROPN
ejpam-3418	498	27	α∈a	α∈a	PROPN
ejpam-3418	498	28	t	t	PROPN
ejpam-3418	498	29	′α	′α	PROPN
ejpam-3418	498	30	is	be	AUX
ejpam-3418	498	31	also	also	ADV
ejpam-3418	498	32	an	an	DET
ejpam-3418	498	33	ivfi	ivfi	NOUN
ejpam-3418	498	34	topology	topology	NOUN
ejpam-3418	498	35	.	.	PUNCT
ejpam-3418	499	1	however	however	ADV
ejpam-3418	499	2	,	,	PUNCT
ejpam-3418	499	3	⋃	⋃	PROPN
ejpam-3418	499	4	α∈a	α∈a	PROPN
ejpam-3418	499	5	t	t	PROPN
ejpam-3418	499	6	′α	′α	NOUN
ejpam-3418	499	7	need	need	AUX
ejpam-3418	499	8	not	not	PART
ejpam-3418	499	9	be	be	AUX
ejpam-3418	499	10	.	.	PUNCT
ejpam-3418	500	1	let	let	AUX
ejpam-3418	500	2	(	(	PUNCT
ejpam-3418	500	3	i(x),t	i(x),t	NOUN
ejpam-3418	500	4	′	′	NUM
ejpam-3418	500	5	)	)	PUNCT
ejpam-3418	500	6	be	be	VERB
ejpam-3418	500	7	an	an	DET
ejpam-3418	500	8	ivfi	ivfi	NOUN
ejpam-3418	500	9	space	space	NOUN
ejpam-3418	500	10	.	.	PUNCT
ejpam-3418	501	1	we	we	PRON
ejpam-3418	501	2	call	call	VERB
ejpam-3418	501	3	the	the	DET
ejpam-3418	501	4	subcollection	subcollection	NOUN
ejpam-3418	501	5	b	b	PROPN
ejpam-3418	501	6	of	of	ADP
ejpam-3418	501	7	t	t	PROPN
ejpam-3418	501	8	′	′	NUM
ejpam-3418	501	9	an	an	DET
ejpam-3418	501	10	ivfi	ivfi	NOUN
ejpam-3418	501	11	base	base	NOUN
ejpam-3418	501	12	for	for	ADP
ejpam-3418	501	13	t	t	NOUN
ejpam-3418	501	14	′	′	NUM
ejpam-3418	501	15	if	if	SCONJ
ejpam-3418	501	16	every	every	DET
ejpam-3418	501	17	member	member	NOUN
ejpam-3418	501	18	of	of	ADP
ejpam-3418	501	19	t	t	PROPN
ejpam-3418	501	20	′	′	NUM
ejpam-3418	501	21	can	can	AUX
ejpam-3418	501	22	be	be	AUX
ejpam-3418	501	23	expressed	express	VERB
ejpam-3418	501	24	as	as	ADP
ejpam-3418	501	25	a	a	DET
ejpam-3418	501	26	union	union	NOUN
ejpam-3418	501	27	of	of	ADP
ejpam-3418	501	28	members	member	NOUN
ejpam-3418	501	29	of	of	ADP
ejpam-3418	501	30	b.	b.	PROPN
ejpam-3418	501	31	let	let	VERB
ejpam-3418	501	32	[	[	X
ejpam-3418	501	33	0	0	NUM
ejpam-3418	501	34	,	,	PUNCT
ejpam-3418	501	35	0	0	NUM
ejpam-3418	501	36	]	]	PUNCT
ejpam-3418	501	37	6=	6=	ADP
ejpam-3418	501	38	α	α	X
ejpam-3418	501	39	∈	∈	X
ejpam-3418	502	1	i	i	PRON
ejpam-3418	502	2	and	and	CCONJ
ejpam-3418	502	3	∅	∅	NOUN
ejpam-3418	502	4	6=	6=	PUNCT
ejpam-3418	502	5	b	b	PROPN
ejpam-3418	502	6	∈	∈	PROPN
ejpam-3418	502	7	i(x	i(x	NOUN
ejpam-3418	502	8	)	)	PUNCT
ejpam-3418	502	9	.	.	PUNCT
ejpam-3418	503	1	we	we	PRON
ejpam-3418	503	2	call	call	VERB
ejpam-3418	503	3	the	the	DET
ejpam-3418	503	4	ivfi	ivfi	NOUN
ejpam-3418	503	5	set	set	VERB
ejpam-3418	503	6	given	give	VERB
ejpam-3418	503	7	by	by	ADP
ejpam-3418	503	8	pα	pα	PROPN
ejpam-3418	503	9	,	,	PUNCT
ejpam-3418	503	10	b(a	b(a	ADJ
ejpam-3418	503	11	)	)	PUNCT
ejpam-3418	504	1	=	=	SYM
ejpam-3418	504	2	{	{	PUNCT
ejpam-3418	504	3	α	α	NOUN
ejpam-3418	504	4	,	,	PUNCT
ejpam-3418	504	5	if	if	SCONJ
ejpam-3418	504	6	a	a	DET
ejpam-3418	504	7	⊆	⊆	NUM
ejpam-3418	504	8	b	b	NOUN
ejpam-3418	504	9	,	,	PUNCT
ejpam-3418	504	10	a	a	DET
ejpam-3418	504	11	6=	6=	NUM
ejpam-3418	504	12	∅	∅	NOUN
ejpam-3418	504	13	;	;	PUNCT
ejpam-3418	504	14	[	[	X
ejpam-3418	504	15	0	0	NUM
ejpam-3418	504	16	,	,	PUNCT
ejpam-3418	504	17	0	0	NUM
ejpam-3418	504	18	]	]	PUNCT
ejpam-3418	504	19	,	,	PUNCT
ejpam-3418	504	20	otherwise	otherwise	ADV
ejpam-3418	504	21	.	.	PUNCT
ejpam-3418	505	1	an	an	DET
ejpam-3418	505	2	ivfi	ivfi	NOUN
ejpam-3418	505	3	point	point	NOUN
ejpam-3418	505	4	.	.	PUNCT
ejpam-3418	506	1	one	one	PRON
ejpam-3418	506	2	can	can	AUX
ejpam-3418	506	3	calculate	calculate	VERB
ejpam-3418	506	4	that	that	SCONJ
ejpam-3418	506	5	the	the	DET
ejpam-3418	506	6	explicit	explicit	ADJ
ejpam-3418	506	7	form	form	NOUN
ejpam-3418	506	8	of	of	ADP
ejpam-3418	506	9	the	the	DET
ejpam-3418	506	10	complement	complement	NOUN
ejpam-3418	506	11	of	of	ADP
ejpam-3418	506	12	pα	pα	PROPN
ejpam-3418	506	13	,	,	PUNCT
ejpam-3418	506	14	b	b	PROPN
ejpam-3418	506	15	is	be	AUX
ejpam-3418	506	16	given	give	VERB
ejpam-3418	506	17	by	by	ADP
ejpam-3418	506	18	p	p	PROPN
ejpam-3418	506	19	cα	cα	PROPN
ejpam-3418	506	20	,	,	PUNCT
ejpam-3418	506	21	b(a	b(a	X
ejpam-3418	506	22	)	)	PUNCT
ejpam-3418	507	1	=	=	SYM
ejpam-3418	507	2			PRON
ejpam-3418	508	1	[	[	X
ejpam-3418	508	2	0	0	NUM
ejpam-3418	508	3	,	,	PUNCT
ejpam-3418	508	4	0	0	NUM
ejpam-3418	508	5	]	]	PUNCT
ejpam-3418	508	6	,	,	PUNCT
ejpam-3418	508	7	if	if	SCONJ
ejpam-3418	508	8	a	a	DET
ejpam-3418	508	9	=	=	NOUN
ejpam-3418	508	10	∅	∅	NOUN
ejpam-3418	508	11	;	;	PUNCT
ejpam-3418	508	12	[	[	X
ejpam-3418	508	13	1−	1−	NUM
ejpam-3418	508	14	α+	α+	NOUN
ejpam-3418	508	15	,	,	PUNCT
ejpam-3418	508	16	1−	1−	NUM
ejpam-3418	508	17	α−	α−	ADP
ejpam-3418	508	18	]	]	PUNCT
ejpam-3418	508	19	,	,	PUNCT
ejpam-3418	508	20	if	if	SCONJ
ejpam-3418	508	21	a	a	DET
ejpam-3418	508	22	∩b	∩b	NOUN
ejpam-3418	508	23	6=	6=	NOUN
ejpam-3418	508	24	∅	∅	NOUN
ejpam-3418	508	25	;	;	PUNCT
ejpam-3418	509	1	[	[	X
ejpam-3418	509	2	1	1	NUM
ejpam-3418	509	3	,	,	PUNCT
ejpam-3418	509	4	1	1	NUM
ejpam-3418	509	5	]	]	PUNCT
ejpam-3418	509	6	,	,	PUNCT
ejpam-3418	509	7	if	if	SCONJ
ejpam-3418	509	8	a	a	DET
ejpam-3418	509	9	6=	6=	NOUN
ejpam-3418	509	10	∅	∅	NOUN
ejpam-3418	509	11	and	and	CCONJ
ejpam-3418	509	12	a	a	DET
ejpam-3418	509	13	∩b	∩b	NOUN
ejpam-3418	509	14	=	=	PUNCT
ejpam-3418	509	15	∅.	∅.	NOUN
ejpam-3418	509	16	we	we	PRON
ejpam-3418	509	17	say	say	VERB
ejpam-3418	509	18	that	that	PRON
ejpam-3418	509	19	pα	pα	NOUN
ejpam-3418	509	20	,	,	PUNCT
ejpam-3418	509	21	b	b	PROPN
ejpam-3418	509	22	is	be	AUX
ejpam-3418	509	23	contained	contain	VERB
ejpam-3418	509	24	in	in	ADP
ejpam-3418	509	25	an	an	DET
ejpam-3418	509	26	ivfi	ivfi	NOUN
ejpam-3418	509	27	set	set	VERB
ejpam-3418	509	28	ι̂	ι̂	NOUN
ejpam-3418	509	29	,	,	PUNCT
ejpam-3418	509	30	denoted	denote	VERB
ejpam-3418	509	31	by	by	ADP
ejpam-3418	509	32	pα	pα	PROPN
ejpam-3418	509	33	,	,	PUNCT
ejpam-3418	509	34	b	b	PROPN
ejpam-3418	509	35	∈	∈	PROPN
ejpam-3418	509	36	ι̂	ι̂	NOUN
ejpam-3418	509	37	,	,	PUNCT
ejpam-3418	509	38	if	if	SCONJ
ejpam-3418	509	39	and	and	CCONJ
ejpam-3418	509	40	only	only	ADV
ejpam-3418	509	41	if	if	SCONJ
ejpam-3418	509	42	α	α	PRON
ejpam-3418	509	43	≤i	≤i	PROPN
ejpam-3418	509	44	ι̂(b	ι̂(b	NOUN
ejpam-3418	509	45	)	)	PUNCT
ejpam-3418	509	46	.	.	PUNCT
ejpam-3418	510	1	with	with	ADP
ejpam-3418	510	2	this	this	PRON
ejpam-3418	510	3	,	,	PUNCT
ejpam-3418	510	4	we	we	PRON
ejpam-3418	510	5	can	can	AUX
ejpam-3418	510	6	characterize	characterize	VERB
ejpam-3418	510	7	an	an	DET
ejpam-3418	510	8	ivfi	ivfi	NOUN
ejpam-3418	510	9	base	base	NOUN
ejpam-3418	510	10	and	and	CCONJ
ejpam-3418	510	11	ivfi	ivfi	VERB
ejpam-3418	510	12	open	open	ADJ
ejpam-3418	510	13	sets	set	NOUN
ejpam-3418	510	14	.	.	PUNCT
ejpam-3418	511	1	remark	remark	NOUN
ejpam-3418	511	2	6	6	NUM
ejpam-3418	511	3	.	.	PUNCT
ejpam-3418	512	1	every	every	DET
ejpam-3418	512	2	ivfi	ivfi	NOUN
ejpam-3418	512	3	set	set	VERB
ejpam-3418	512	4	ι̂	ι̂	PRON
ejpam-3418	512	5	can	can	AUX
ejpam-3418	512	6	be	be	AUX
ejpam-3418	512	7	expressed	express	VERB
ejpam-3418	512	8	as	as	ADP
ejpam-3418	512	9	the	the	DET
ejpam-3418	512	10	union	union	NOUN
ejpam-3418	512	11	of	of	ADP
ejpam-3418	512	12	all	all	DET
ejpam-3418	512	13	ivfi	ivfi	NOUN
ejpam-3418	512	14	points	point	NOUN
ejpam-3418	512	15	which	which	PRON
ejpam-3418	512	16	is	be	AUX
ejpam-3418	512	17	contained	contain	VERB
ejpam-3418	512	18	in	in	ADP
ejpam-3418	512	19	ι̂.	ι̂.	NOUN
ejpam-3418	512	20	that	that	PRON
ejpam-3418	512	21	is	be	AUX
ejpam-3418	512	22	,	,	PUNCT
ejpam-3418	512	23	if	if	SCONJ
ejpam-3418	512	24	ι̂(b	ι̂(b	NOUN
ejpam-3418	512	25	)	)	PUNCT
ejpam-3418	512	26	is	be	AUX
ejpam-3418	512	27	not	not	PART
ejpam-3418	512	28	zero	zero	NUM
ejpam-3418	512	29	for	for	ADP
ejpam-3418	512	30	b	b	PROPN
ejpam-3418	512	31	∈	∈	PROPN
ejpam-3418	512	32	i(x	i(x	PROPN
ejpam-3418	512	33	)	)	PUNCT
ejpam-3418	512	34	,	,	PUNCT
ejpam-3418	512	35	then	then	ADV
ejpam-3418	512	36	ι̂(b	ι̂(b	NOUN
ejpam-3418	512	37	)	)	PUNCT
ejpam-3418	512	38	=	=	SYM
ejpam-3418	512	39	sup{α	sup{α	NOUN
ejpam-3418	512	40	:	:	PUNCT
ejpam-3418	512	41	pα	pα	VERB
ejpam-3418	512	42	,	,	PUNCT
ejpam-3418	512	43	b	b	PROPN
ejpam-3418	512	44	is	be	AUX
ejpam-3418	512	45	an	an	DET
ejpam-3418	512	46	ivfi	ivfi	NOUN
ejpam-3418	512	47	point	point	NOUN
ejpam-3418	512	48	and	and	CCONJ
ejpam-3418	512	49	[	[	X
ejpam-3418	512	50	0	0	NUM
ejpam-3418	512	51	,	,	PUNCT
ejpam-3418	512	52	0	0	NUM
ejpam-3418	512	53	]	]	PUNCT
ejpam-3418	513	1	<	<	X
ejpam-3418	513	2	i	i	X
ejpam-3418	513	3	α	α	PROPN
ejpam-3418	513	4	≤i	≤i	PROPN
ejpam-3418	513	5	ι̂(b	ι̂(b	NOUN
ejpam-3418	513	6	)	)	PUNCT
ejpam-3418	513	7	}	}	PUNCT
ejpam-3418	513	8	.	.	PUNCT
ejpam-3418	514	1	mj	mj	PROPN
ejpam-3418	514	2	togonon	togonon	PROPN
ejpam-3418	514	3	,	,	PUNCT
ejpam-3418	514	4	r	r	NOUN
ejpam-3418	514	5	caga	caga	NOUN
ejpam-3418	514	6	-	-	PUNCT
ejpam-3418	514	7	anan	anan	PROPN
ejpam-3418	514	8	/	/	SYM
ejpam-3418	514	9	eur	eur	PROPN
ejpam-3418	514	10	.	.	PUNCT
ejpam-3418	515	1	j.	j.	PROPN
ejpam-3418	515	2	pure	pure	PROPN
ejpam-3418	515	3	appl	appl	PROPN
ejpam-3418	515	4	.	.	PROPN
ejpam-3418	515	5	math	math	PROPN
ejpam-3418	515	6	,	,	PUNCT
ejpam-3418	515	7	12	12	NUM
ejpam-3418	515	8	(	(	PUNCT
ejpam-3418	515	9	2	2	NUM
ejpam-3418	515	10	)	)	PUNCT
ejpam-3418	515	11	(	(	PUNCT
ejpam-3418	515	12	2019	2019	NUM
ejpam-3418	515	13	)	)	PUNCT
ejpam-3418	515	14	,	,	PUNCT
ejpam-3418	515	15	553	553	NUM
ejpam-3418	515	16	-	-	SYM
ejpam-3418	515	17	570	570	NUM
ejpam-3418	515	18	566	566	NUM
ejpam-3418	515	19	theorem	theorem	NOUN
ejpam-3418	515	20	7	7	NUM
ejpam-3418	515	21	.	.	PUNCT
ejpam-3418	516	1	let	let	AUX
ejpam-3418	516	2	(	(	PUNCT
ejpam-3418	516	3	i(x),t	i(x),t	NOUN
ejpam-3418	516	4	′	′	NUM
ejpam-3418	516	5	)	)	PUNCT
ejpam-3418	516	6	be	be	VERB
ejpam-3418	516	7	an	an	DET
ejpam-3418	516	8	ivfi	ivfi	NOUN
ejpam-3418	516	9	space	space	NOUN
ejpam-3418	516	10	and	and	CCONJ
ejpam-3418	516	11	b	b	NOUN
ejpam-3418	516	12	⊆	⊆	NUM
ejpam-3418	516	13	t	t	NOUN
ejpam-3418	516	14	′.	′.	NOUN
ejpam-3418	516	15	then	then	ADV
ejpam-3418	516	16	b	b	PROPN
ejpam-3418	516	17	is	be	AUX
ejpam-3418	516	18	an	an	DET
ejpam-3418	516	19	ivfi	ivfi	NOUN
ejpam-3418	516	20	base	base	NOUN
ejpam-3418	516	21	for	for	ADP
ejpam-3418	516	22	(	(	PUNCT
ejpam-3418	516	23	i(x),t	i(x),t	NOUN
ejpam-3418	516	24	′	′	NUM
ejpam-3418	516	25	)	)	PUNCT
ejpam-3418	516	26	if	if	SCONJ
ejpam-3418	516	27	and	and	CCONJ
ejpam-3418	516	28	only	only	ADV
ejpam-3418	516	29	if	if	SCONJ
ejpam-3418	516	30	for	for	ADP
ejpam-3418	516	31	each	each	DET
ejpam-3418	516	32	ι̂	ι̂	NUM
ejpam-3418	516	33	∈	∈	PROPN
ejpam-3418	516	34	t	t	NOUN
ejpam-3418	516	35	′	′	NOUN
ejpam-3418	516	36	and	and	CCONJ
ejpam-3418	516	37	for	for	ADP
ejpam-3418	516	38	each	each	DET
ejpam-3418	516	39	ivfi	ivfi	NOUN
ejpam-3418	516	40	point	point	NOUN
ejpam-3418	516	41	pα	pα	NOUN
ejpam-3418	516	42	,	,	PUNCT
ejpam-3418	516	43	b	b	PROPN
ejpam-3418	516	44	∈	∈	PROPN
ejpam-3418	516	45	ι̂	ι̂	PUNCT
ejpam-3418	516	46	where	where	SCONJ
ejpam-3418	516	47	[	[	X
ejpam-3418	516	48	0	0	NUM
ejpam-3418	516	49	,	,	PUNCT
ejpam-3418	516	50	0	0	NUM
ejpam-3418	516	51	]	]	PUNCT
ejpam-3418	516	52	6=	6=	ADP
ejpam-3418	516	53	α	α	X
ejpam-3418	516	54	∈	∈	X
ejpam-3418	517	1	i	i	PRON
ejpam-3418	517	2	and	and	CCONJ
ejpam-3418	517	3	∅	∅	NOUN
ejpam-3418	517	4	6=	6=	PUNCT
ejpam-3418	517	5	b	b	PROPN
ejpam-3418	517	6	∈	∈	PROPN
ejpam-3418	517	7	i(x	i(x	PROPN
ejpam-3418	517	8	)	)	PUNCT
ejpam-3418	517	9	,	,	PUNCT
ejpam-3418	517	10	there	there	PRON
ejpam-3418	517	11	exists	exist	VERB
ejpam-3418	517	12	ρ̂	ρ̂	NUM
ejpam-3418	517	13	∈	∈	PROPN
ejpam-3418	517	14	b	b	PROPN
ejpam-3418	517	15	such	such	ADJ
ejpam-3418	517	16	that	that	DET
ejpam-3418	517	17	pα	pα	NOUN
ejpam-3418	517	18	,	,	PUNCT
ejpam-3418	517	19	b	b	PROPN
ejpam-3418	517	20	∈	∈	PROPN
ejpam-3418	517	21	ρ̂	ρ̂	NUM
ejpam-3418	517	22	6	6	NUM
ejpam-3418	517	23	ι̂.	ι̂.	NOUN
ejpam-3418	517	24	proof	proof	NOUN
ejpam-3418	517	25	.	.	PUNCT
ejpam-3418	518	1	suppose	suppose	VERB
ejpam-3418	518	2	that	that	SCONJ
ejpam-3418	518	3	b	b	PROPN
ejpam-3418	518	4	is	be	AUX
ejpam-3418	518	5	an	an	DET
ejpam-3418	518	6	ivfi	ivfi	NOUN
ejpam-3418	518	7	base	base	NOUN
ejpam-3418	518	8	for	for	ADP
ejpam-3418	518	9	(	(	PUNCT
ejpam-3418	518	10	i(x),t	i(x),t	NOUN
ejpam-3418	518	11	′	′	NUM
ejpam-3418	518	12	)	)	PUNCT
ejpam-3418	518	13	.	.	PUNCT
ejpam-3418	519	1	let	let	VERB
ejpam-3418	519	2	ι̂	ι̂	PRON
ejpam-3418	519	3	∈	∈	NOUN
ejpam-3418	519	4	t	t	NOUN
ejpam-3418	519	5	′	′	NOUN
ejpam-3418	519	6	and	and	CCONJ
ejpam-3418	519	7	pα	pα	VERB
ejpam-3418	519	8	,	,	PUNCT
ejpam-3418	519	9	b	b	PROPN
ejpam-3418	519	10	∈	∈	PROPN
ejpam-3418	519	11	ι̂	ι̂	NOUN
ejpam-3418	519	12	,	,	PUNCT
ejpam-3418	519	13	where	where	SCONJ
ejpam-3418	519	14	[	[	X
ejpam-3418	519	15	0	0	NUM
ejpam-3418	519	16	,	,	PUNCT
ejpam-3418	519	17	0	0	NUM
ejpam-3418	519	18	]	]	PUNCT
ejpam-3418	519	19	6=	6=	ADP
ejpam-3418	519	20	α	α	X
ejpam-3418	519	21	∈	∈	X
ejpam-3418	520	1	i	i	PRON
ejpam-3418	520	2	and	and	CCONJ
ejpam-3418	520	3	∅	∅	NOUN
ejpam-3418	520	4	6=	6=	PUNCT
ejpam-3418	520	5	b	b	PROPN
ejpam-3418	520	6	∈	∈	PROPN
ejpam-3418	520	7	i(x	i(x	PROPN
ejpam-3418	520	8	)	)	PUNCT
ejpam-3418	520	9	.	.	PUNCT
ejpam-3418	521	1	then	then	ADV
ejpam-3418	521	2	by	by	ADP
ejpam-3418	521	3	the	the	DET
ejpam-3418	521	4	definition	definition	NOUN
ejpam-3418	521	5	of	of	ADP
ejpam-3418	521	6	an	an	DET
ejpam-3418	521	7	ivfi	ivfi	NOUN
ejpam-3418	521	8	base	base	NOUN
ejpam-3418	521	9	,	,	PUNCT
ejpam-3418	521	10	there	there	PRON
ejpam-3418	521	11	exists	exist	VERB
ejpam-3418	521	12	c	c	NOUN
ejpam-3418	521	13	⊆	⊆	NUM
ejpam-3418	521	14	b	b	NUM
ejpam-3418	521	15	such	such	ADJ
ejpam-3418	521	16	that	that	DET
ejpam-3418	521	17	ι̂	ι̂	NUM
ejpam-3418	521	18	=	=	SYM
ejpam-3418	521	19	∨	∨	NUM
ejpam-3418	521	20	ρ̂∈c	ρ̂∈c	PROPN
ejpam-3418	521	21	ρ̂.	ρ̂.	NOUN
ejpam-3418	521	22	since	since	SCONJ
ejpam-3418	521	23	pα	pα	PROPN
ejpam-3418	521	24	,	,	PUNCT
ejpam-3418	521	25	b	b	PROPN
ejpam-3418	521	26	∈	∈	PROPN
ejpam-3418	521	27	ι̂	ι̂	NOUN
ejpam-3418	521	28	,	,	PUNCT
ejpam-3418	521	29	pα	pα	INTJ
ejpam-3418	521	30	,	,	PUNCT
ejpam-3418	521	31	b	b	PROPN
ejpam-3418	521	32	∈	∈	PROPN
ejpam-3418	521	33	ρ̂	ρ̂	NUM
ejpam-3418	521	34	for	for	ADP
ejpam-3418	521	35	some	some	DET
ejpam-3418	521	36	ρ̂	ρ̂	NUM
ejpam-3418	521	37	∈	∈	PROPN
ejpam-3418	521	38	c.	c.	NOUN
ejpam-3418	521	39	consequently	consequently	ADV
ejpam-3418	521	40	,	,	PUNCT
ejpam-3418	521	41	pα	pα	INTJ
ejpam-3418	521	42	,	,	PUNCT
ejpam-3418	521	43	b	b	PROPN
ejpam-3418	521	44	∈	∈	PROPN
ejpam-3418	521	45	ρ̂	ρ̂	NUM
ejpam-3418	521	46	6	6	NUM
ejpam-3418	521	47	ι̂.	ι̂.	NOUN
ejpam-3418	521	48	conversely	conversely	ADV
ejpam-3418	521	49	,	,	PUNCT
ejpam-3418	521	50	suppose	suppose	VERB
ejpam-3418	521	51	that	that	SCONJ
ejpam-3418	521	52	for	for	ADP
ejpam-3418	521	53	each	each	DET
ejpam-3418	521	54	ι̂	ι̂	NUM
ejpam-3418	521	55	∈	∈	PROPN
ejpam-3418	521	56	t	t	NOUN
ejpam-3418	521	57	′	′	NOUN
ejpam-3418	521	58	and	and	CCONJ
ejpam-3418	521	59	for	for	ADP
ejpam-3418	521	60	each	each	DET
ejpam-3418	521	61	ivfi	ivfi	NOUN
ejpam-3418	521	62	point	point	NOUN
ejpam-3418	521	63	pα	pα	NOUN
ejpam-3418	521	64	,	,	PUNCT
ejpam-3418	521	65	b	b	PROPN
ejpam-3418	521	66	∈	∈	PROPN
ejpam-3418	521	67	ι̂	ι̂	PUNCT
ejpam-3418	521	68	where	where	SCONJ
ejpam-3418	521	69	[	[	X
ejpam-3418	521	70	0	0	NUM
ejpam-3418	521	71	,	,	PUNCT
ejpam-3418	521	72	0	0	NUM
ejpam-3418	521	73	]	]	PUNCT
ejpam-3418	521	74	6=	6=	ADP
ejpam-3418	521	75	α	α	X
ejpam-3418	521	76	∈	∈	X
ejpam-3418	522	1	i	i	PRON
ejpam-3418	522	2	and	and	CCONJ
ejpam-3418	522	3	∅	∅	NOUN
ejpam-3418	522	4	6=	6=	PUNCT
ejpam-3418	522	5	b	b	PROPN
ejpam-3418	522	6	∈	∈	PROPN
ejpam-3418	522	7	i(x	i(x	PROPN
ejpam-3418	522	8	)	)	PUNCT
ejpam-3418	522	9	,	,	PUNCT
ejpam-3418	522	10	there	there	PRON
ejpam-3418	522	11	exists	exist	VERB
ejpam-3418	522	12	ρ̂α	ρ̂α	NUM
ejpam-3418	522	13	,	,	PUNCT
ejpam-3418	522	14	b	b	X
ejpam-3418	522	15	∈	∈	PROPN
ejpam-3418	522	16	b	b	NOUN
ejpam-3418	522	17	such	such	ADJ
ejpam-3418	522	18	that	that	PRON
ejpam-3418	522	19	pα	pα	NOUN
ejpam-3418	522	20	,	,	PUNCT
ejpam-3418	522	21	b	b	PROPN
ejpam-3418	522	22	∈	∈	PROPN
ejpam-3418	522	23	ρ̂α	ρ̂α	NUM
ejpam-3418	522	24	,	,	PUNCT
ejpam-3418	522	25	b	b	PROPN
ejpam-3418	522	26	6	6	NUM
ejpam-3418	522	27	ι̂.	ι̂.	NOUN
ejpam-3418	522	28	let	let	VERB
ejpam-3418	522	29	ι̂	ι̂	PUNCT
ejpam-3418	522	30	∈	∈	NOUN
ejpam-3418	522	31	t	t	NOUN
ejpam-3418	522	32	′	′	NOUN
ejpam-3418	522	33	and	and	CCONJ
ejpam-3418	522	34	consider	consider	VERB
ejpam-3418	522	35	any	any	DET
ejpam-3418	522	36	arbitrary	arbitrary	ADJ
ejpam-3418	522	37	pα	pα	NOUN
ejpam-3418	522	38	,	,	PUNCT
ejpam-3418	522	39	b	b	PROPN
ejpam-3418	522	40	∈	∈	PROPN
ejpam-3418	522	41	ι̂.	ι̂.	NOUN
ejpam-3418	522	42	by	by	ADP
ejpam-3418	522	43	remark	remark	NOUN
ejpam-3418	522	44	6	6	NUM
ejpam-3418	522	45	,	,	PUNCT
ejpam-3418	522	46	we	we	PRON
ejpam-3418	522	47	have	have	AUX
ejpam-3418	522	48	ι̂	ι̂	NOUN
ejpam-3418	522	49	=	=	SYM
ejpam-3418	522	50	∨	∨	NUM
ejpam-3418	522	51	pα	pα	PROPN
ejpam-3418	522	52	,	,	PUNCT
ejpam-3418	522	53	b∈ι̂	b∈ι̂	VERB
ejpam-3418	522	54	pα	pα	PROPN
ejpam-3418	522	55	,	,	PUNCT
ejpam-3418	522	56	b.	b.	PROPN
ejpam-3418	522	57	since	since	SCONJ
ejpam-3418	522	58	pα	pα	PROPN
ejpam-3418	522	59	,	,	PUNCT
ejpam-3418	522	60	b	b	PROPN
ejpam-3418	522	61	∈	∈	PROPN
ejpam-3418	522	62	ρ̂α	ρ̂α	NUM
ejpam-3418	522	63	,	,	PUNCT
ejpam-3418	522	64	b	b	AUX
ejpam-3418	522	65	,	,	PUNCT
ejpam-3418	522	66	it	it	PRON
ejpam-3418	522	67	follows	follow	VERB
ejpam-3418	522	68	that	that	SCONJ
ejpam-3418	522	69	∨	∨	PROPN
ejpam-3418	522	70	pα	pα	PROPN
ejpam-3418	522	71	,	,	PUNCT
ejpam-3418	522	72	b∈ι̂	b∈ι̂	VERB
ejpam-3418	522	73	pα	pα	PROPN
ejpam-3418	522	74	,	,	PUNCT
ejpam-3418	522	75	b	b	PROPN
ejpam-3418	522	76	6	6	NUM
ejpam-3418	522	77	∨	∨	NUM
ejpam-3418	522	78	pα	pα	NOUN
ejpam-3418	522	79	,	,	PUNCT
ejpam-3418	522	80	b∈ι̂	b∈ι̂	X
ejpam-3418	522	81	ρ̂α	ρ̂α	NUM
ejpam-3418	522	82	,	,	PUNCT
ejpam-3418	522	83	b	b	NOUN
ejpam-3418	522	84	,	,	PUNCT
ejpam-3418	522	85	and	and	CCONJ
ejpam-3418	522	86	so	so	ADV
ejpam-3418	522	87	ι̂	ι̂	PUNCT
ejpam-3418	522	88	6	6	NUM
ejpam-3418	522	89	∨	∨	NUM
ejpam-3418	522	90	pα	pα	NOUN
ejpam-3418	522	91	,	,	PUNCT
ejpam-3418	522	92	b∈ι̂	b∈ι̂	X
ejpam-3418	522	93	ρ̂α	ρ̂α	NUM
ejpam-3418	522	94	,	,	PUNCT
ejpam-3418	522	95	b.	b.	NOUN
ejpam-3418	522	96	but	but	CCONJ
ejpam-3418	522	97	note	note	VERB
ejpam-3418	522	98	that	that	SCONJ
ejpam-3418	522	99	ρ̂α	ρ̂α	NUM
ejpam-3418	522	100	,	,	PUNCT
ejpam-3418	522	101	b	b	PROPN
ejpam-3418	522	102	6	6	NUM
ejpam-3418	522	103	ι̂.	ι̂.	NOUN
ejpam-3418	522	104	thus	thus	ADV
ejpam-3418	522	105	,	,	PUNCT
ejpam-3418	522	106	we	we	PRON
ejpam-3418	522	107	have	have	VERB
ejpam-3418	522	108	∨	∨	NUM
ejpam-3418	522	109	pα	pα	PROPN
ejpam-3418	522	110	,	,	PUNCT
ejpam-3418	522	111	b∈ι̂	b∈ι̂	X
ejpam-3418	522	112	ρ̂α	ρ̂α	NUM
ejpam-3418	522	113	,	,	PUNCT
ejpam-3418	522	114	b	b	PROPN
ejpam-3418	522	115	6	6	NUM
ejpam-3418	522	116	∨	∨	NUM
ejpam-3418	522	117	pα	pα	NOUN
ejpam-3418	522	118	,	,	PUNCT
ejpam-3418	522	119	b∈ι̂	b∈ι̂	X
ejpam-3418	522	120	ι̂	ι̂	X
ejpam-3418	523	1	=	=	SYM
ejpam-3418	523	2	ι̂.	ι̂.	NOUN
ejpam-3418	523	3	hence	hence	ADV
ejpam-3418	523	4	,	,	PUNCT
ejpam-3418	523	5	ι̂	ι̂	PUNCT
ejpam-3418	523	6	=	=	SYM
ejpam-3418	523	7	∨	∨	NUM
ejpam-3418	523	8	pα	pα	PROPN
ejpam-3418	523	9	,	,	PUNCT
ejpam-3418	523	10	b∈ι̂	b∈ι̂	X
ejpam-3418	523	11	ρ̂α	ρ̂α	NUM
ejpam-3418	523	12	,	,	PUNCT
ejpam-3418	523	13	b.	b.	PROPN
ejpam-3418	523	14	therefore	therefore	ADV
ejpam-3418	523	15	,	,	PUNCT
ejpam-3418	523	16	b	b	PROPN
ejpam-3418	523	17	is	be	AUX
ejpam-3418	523	18	an	an	DET
ejpam-3418	523	19	ivfi	ivfi	NOUN
ejpam-3418	523	20	base	base	NOUN
ejpam-3418	523	21	for	for	ADP
ejpam-3418	523	22	t	t	PROPN
ejpam-3418	523	23	′.	′.	NOUN
ejpam-3418	523	24	corollary	corollary	ADJ
ejpam-3418	523	25	3	3	X
ejpam-3418	523	26	.	.	PUNCT
ejpam-3418	524	1	let	let	AUX
ejpam-3418	524	2	(	(	PUNCT
ejpam-3418	524	3	i(x),t	i(x),t	NOUN
ejpam-3418	524	4	′	′	NUM
ejpam-3418	524	5	)	)	PUNCT
ejpam-3418	524	6	be	be	VERB
ejpam-3418	524	7	an	an	DET
ejpam-3418	524	8	ivfi	ivfi	NOUN
ejpam-3418	524	9	space	space	NOUN
ejpam-3418	524	10	and	and	CCONJ
ejpam-3418	524	11	b	b	NOUN
ejpam-3418	524	12	an	an	DET
ejpam-3418	524	13	ivfi	ivfi	NOUN
ejpam-3418	524	14	base	base	NOUN
ejpam-3418	524	15	.	.	PUNCT
ejpam-3418	525	1	then	then	ADV
ejpam-3418	525	2	,	,	PUNCT
ejpam-3418	525	3	ι̂	ι̂	PUNCT
ejpam-3418	525	4	∈	∈	PROPN
ejpam-3418	525	5	t	t	NOUN
ejpam-3418	525	6	′	′	NOUN
ejpam-3418	525	7	if	if	SCONJ
ejpam-3418	525	8	and	and	CCONJ
ejpam-3418	525	9	only	only	ADV
ejpam-3418	525	10	if	if	SCONJ
ejpam-3418	525	11	for	for	ADP
ejpam-3418	525	12	each	each	DET
ejpam-3418	525	13	pα	pα	NOUN
ejpam-3418	525	14	,	,	PUNCT
ejpam-3418	525	15	b	b	PROPN
ejpam-3418	525	16	∈	∈	PROPN
ejpam-3418	525	17	ι̂	ι̂	PUNCT
ejpam-3418	525	18	where	where	SCONJ
ejpam-3418	525	19	[	[	X
ejpam-3418	525	20	0	0	NUM
ejpam-3418	525	21	,	,	PUNCT
ejpam-3418	525	22	0	0	NUM
ejpam-3418	525	23	]	]	PUNCT
ejpam-3418	525	24	6=	6=	ADP
ejpam-3418	525	25	α	α	X
ejpam-3418	525	26	∈	∈	X
ejpam-3418	526	1	i	i	PRON
ejpam-3418	526	2	and	and	CCONJ
ejpam-3418	526	3	∅	∅	NOUN
ejpam-3418	526	4	6=	6=	PUNCT
ejpam-3418	526	5	b	b	PROPN
ejpam-3418	526	6	∈	∈	PROPN
ejpam-3418	526	7	i(x	i(x	PROPN
ejpam-3418	526	8	)	)	PUNCT
ejpam-3418	526	9	,	,	PUNCT
ejpam-3418	526	10	there	there	PRON
ejpam-3418	526	11	exists	exist	VERB
ejpam-3418	526	12	ω̂	ω̂	PUNCT
ejpam-3418	526	13	∈	∈	PROPN
ejpam-3418	526	14	b	b	PROPN
ejpam-3418	526	15	such	such	ADJ
ejpam-3418	526	16	that	that	DET
ejpam-3418	526	17	pα	pα	NOUN
ejpam-3418	526	18	,	,	PUNCT
ejpam-3418	526	19	b	b	PROPN
ejpam-3418	526	20	∈	∈	PROPN
ejpam-3418	526	21	ω̂	ω̂	NUM
ejpam-3418	526	22	6	6	NUM
ejpam-3418	526	23	ι̂.	ι̂.	NOUN
ejpam-3418	526	24	proof	proof	NOUN
ejpam-3418	526	25	.	.	PUNCT
ejpam-3418	527	1	let	let	VERB
ejpam-3418	527	2	ι̂	ι̂	PRON
ejpam-3418	527	3	∈	∈	NOUN
ejpam-3418	527	4	t	t	NOUN
ejpam-3418	527	5	′	′	NOUN
ejpam-3418	527	6	and	and	CCONJ
ejpam-3418	527	7	pα	pα	VERB
ejpam-3418	527	8	,	,	PUNCT
ejpam-3418	527	9	b	b	PROPN
ejpam-3418	527	10	∈	∈	PROPN
ejpam-3418	527	11	ι̂	ι̂	NOUN
ejpam-3418	527	12	,	,	PUNCT
ejpam-3418	527	13	where	where	SCONJ
ejpam-3418	527	14	[	[	X
ejpam-3418	527	15	0	0	NUM
ejpam-3418	527	16	,	,	PUNCT
ejpam-3418	527	17	0	0	NUM
ejpam-3418	527	18	]	]	PUNCT
ejpam-3418	527	19	6=	6=	ADP
ejpam-3418	527	20	α	α	X
ejpam-3418	527	21	∈	∈	X
ejpam-3418	528	1	i	i	PRON
ejpam-3418	528	2	and	and	CCONJ
ejpam-3418	528	3	∅	∅	NOUN
ejpam-3418	528	4	6=	6=	PUNCT
ejpam-3418	528	5	b	b	PROPN
ejpam-3418	528	6	∈	∈	PROPN
ejpam-3418	528	7	i(x	i(x	PROPN
ejpam-3418	528	8	)	)	PUNCT
ejpam-3418	528	9	.	.	PUNCT
ejpam-3418	529	1	then	then	ADV
ejpam-3418	529	2	by	by	ADP
ejpam-3418	529	3	theorem	theorem	NOUN
ejpam-3418	529	4	7	7	NUM
ejpam-3418	529	5	,	,	PUNCT
ejpam-3418	529	6	there	there	PRON
ejpam-3418	529	7	exists	exist	VERB
ejpam-3418	529	8	ω̂	ω̂	PUNCT
ejpam-3418	529	9	∈	∈	PROPN
ejpam-3418	529	10	b	b	PROPN
ejpam-3418	529	11	such	such	ADJ
ejpam-3418	529	12	that	that	DET
ejpam-3418	529	13	pα	pα	NOUN
ejpam-3418	529	14	,	,	PUNCT
ejpam-3418	529	15	b	b	PROPN
ejpam-3418	529	16	∈	∈	PROPN
ejpam-3418	529	17	ω̂	ω̂	NUM
ejpam-3418	529	18	6	6	NUM
ejpam-3418	529	19	ι̂.	ι̂.	NOUN
ejpam-3418	529	20	conversely	conversely	ADV
ejpam-3418	529	21	,	,	PUNCT
ejpam-3418	529	22	suppose	suppose	VERB
ejpam-3418	529	23	that	that	SCONJ
ejpam-3418	529	24	for	for	ADP
ejpam-3418	529	25	each	each	DET
ejpam-3418	529	26	pα	pα	NOUN
ejpam-3418	529	27	,	,	PUNCT
ejpam-3418	529	28	b	b	PROPN
ejpam-3418	529	29	∈	∈	PROPN
ejpam-3418	529	30	ι̂	ι̂	PUNCT
ejpam-3418	529	31	where	where	SCONJ
ejpam-3418	529	32	[	[	X
ejpam-3418	529	33	0	0	NUM
ejpam-3418	529	34	,	,	PUNCT
ejpam-3418	529	35	0	0	NUM
ejpam-3418	529	36	]	]	PUNCT
ejpam-3418	529	37	6=	6=	ADP
ejpam-3418	529	38	α	α	X
ejpam-3418	529	39	∈	∈	X
ejpam-3418	530	1	i	i	PRON
ejpam-3418	530	2	and	and	CCONJ
ejpam-3418	530	3	∅	∅	NOUN
ejpam-3418	530	4	6=	6=	PUNCT
ejpam-3418	530	5	b	b	PROPN
ejpam-3418	530	6	∈	∈	PROPN
ejpam-3418	530	7	i(x	i(x	PROPN
ejpam-3418	530	8	)	)	PUNCT
ejpam-3418	530	9	,	,	PUNCT
ejpam-3418	530	10	there	there	PRON
ejpam-3418	530	11	exists	exist	VERB
ejpam-3418	530	12	ω̂α	ω̂α	PRON
ejpam-3418	530	13	,	,	PUNCT
ejpam-3418	530	14	b	b	PROPN
ejpam-3418	530	15	∈	∈	PROPN
ejpam-3418	530	16	b	b	NOUN
ejpam-3418	530	17	such	such	ADJ
ejpam-3418	530	18	that	that	DET
ejpam-3418	530	19	pα	pα	NOUN
ejpam-3418	530	20	,	,	PUNCT
ejpam-3418	530	21	b	b	PROPN
ejpam-3418	530	22	∈	∈	PROPN
ejpam-3418	530	23	ω̂α	ω̂α	PRON
ejpam-3418	530	24	,	,	PUNCT
ejpam-3418	530	25	b	b	PROPN
ejpam-3418	530	26	6	6	NUM
ejpam-3418	530	27	ι̂.	ι̂.	NOUN
ejpam-3418	530	28	then	then	ADV
ejpam-3418	530	29	by	by	ADP
ejpam-3418	530	30	theorem	theorem	NOUN
ejpam-3418	530	31	7	7	NUM
ejpam-3418	530	32	,	,	PUNCT
ejpam-3418	530	33	ι̂	ι̂	PUNCT
ejpam-3418	530	34	=	=	SYM
ejpam-3418	530	35	∨	∨	NUM
ejpam-3418	530	36	pα	pα	PROPN
ejpam-3418	530	37	,	,	PUNCT
ejpam-3418	530	38	b∈ι̂	b∈ι̂	X
ejpam-3418	530	39	ω̂α	ω̂α	X
ejpam-3418	530	40	,	,	PUNCT
ejpam-3418	530	41	b	b	PROPN
ejpam-3418	530	42	where	where	SCONJ
ejpam-3418	530	43	ω̂α	ω̂α	X
ejpam-3418	530	44	,	,	PUNCT
ejpam-3418	530	45	b	b	PROPN
ejpam-3418	530	46	∈	∈	PROPN
ejpam-3418	530	47	b.	b.	PROPN
ejpam-3418	530	48	since	since	SCONJ
ejpam-3418	530	49	b	b	PROPN
ejpam-3418	530	50	is	be	AUX
ejpam-3418	530	51	an	an	DET
ejpam-3418	530	52	ivfi	ivfi	NOUN
ejpam-3418	530	53	base	base	NOUN
ejpam-3418	530	54	,	,	PUNCT
ejpam-3418	530	55	we	we	PRON
ejpam-3418	530	56	should	should	AUX
ejpam-3418	530	57	have	have	VERB
ejpam-3418	530	58	ι̂	ι̂	NOUN
ejpam-3418	530	59	=	=	SYM
ejpam-3418	530	60	∨	∨	NUM
ejpam-3418	530	61	pα	pα	PROPN
ejpam-3418	530	62	,	,	PUNCT
ejpam-3418	530	63	b∈ι̂	b∈ι̂	X
ejpam-3418	530	64	ω̂α	ω̂α	X
ejpam-3418	530	65	,	,	PUNCT
ejpam-3418	530	66	b	b	PROPN
ejpam-3418	530	67	∈	∈	PROPN
ejpam-3418	530	68	t	t	NOUN
ejpam-3418	530	69	′.	′.	NOUN
ejpam-3418	530	70	let	let	VERB
ejpam-3418	530	71	ι̂	ι̂	PRON
ejpam-3418	530	72	be	be	AUX
ejpam-3418	530	73	an	an	DET
ejpam-3418	530	74	ivfi	ivfi	NOUN
ejpam-3418	530	75	set	set	VERB
ejpam-3418	530	76	in	in	ADP
ejpam-3418	530	77	an	an	DET
ejpam-3418	530	78	ivfi	ivfi	NOUN
ejpam-3418	530	79	space	space	NOUN
ejpam-3418	530	80	(	(	PUNCT
ejpam-3418	530	81	i(x),t	i(x),t	NOUN
ejpam-3418	530	82	′	′	NUM
ejpam-3418	530	83	)	)	PUNCT
ejpam-3418	530	84	.	.	PUNCT
ejpam-3418	531	1	we	we	PRON
ejpam-3418	531	2	define	define	VERB
ejpam-3418	531	3	the	the	DET
ejpam-3418	531	4	ivfi	ivfi	NOUN
ejpam-3418	531	5	interior	interior	NOUN
ejpam-3418	531	6	of	of	ADP
ejpam-3418	531	7	ι̂	ι̂	NOUN
ejpam-3418	531	8	,	,	PUNCT
ejpam-3418	531	9	denoted	denote	VERB
ejpam-3418	531	10	by	by	ADP
ejpam-3418	531	11	int	int	NOUN
ejpam-3418	531	12	ι̂	ι̂	NOUN
ejpam-3418	531	13	,	,	PUNCT
ejpam-3418	531	14	by	by	ADP
ejpam-3418	531	15	int	int	NOUN
ejpam-3418	531	16	ι̂	ι̂	NOUN
ejpam-3418	531	17	=	=	SYM
ejpam-3418	531	18	∨	∨	X
ejpam-3418	531	19	{	{	PUNCT
ejpam-3418	531	20	τ̂	τ̂	NUM
ejpam-3418	531	21	:	:	PUNCT
ejpam-3418	531	22	τ̂	τ̂	NUM
ejpam-3418	531	23	6	6	NUM
ejpam-3418	531	24	ι̂	ι̂	NOUN
ejpam-3418	531	25	,	,	PUNCT
ejpam-3418	531	26	τ̂	τ̂	PUNCT
ejpam-3418	531	27	∈	∈	PROPN
ejpam-3418	531	28	t	t	PROPN
ejpam-3418	531	29	′	′	NUM
ejpam-3418	531	30	}	}	PUNCT
ejpam-3418	531	31	,	,	PUNCT
ejpam-3418	531	32	and	and	CCONJ
ejpam-3418	531	33	the	the	DET
ejpam-3418	531	34	ivfi	ivfi	NOUN
ejpam-3418	531	35	closure	closure	NOUN
ejpam-3418	531	36	of	of	ADP
ejpam-3418	531	37	ι̂	ι̂	NOUN
ejpam-3418	531	38	,	,	PUNCT
ejpam-3418	531	39	denoted	denote	VERB
ejpam-3418	531	40	by	by	ADP
ejpam-3418	531	41	cl	cl	NOUN
ejpam-3418	531	42	ι̂	ι̂	NOUN
ejpam-3418	531	43	,	,	PUNCT
ejpam-3418	531	44	by	by	ADP
ejpam-3418	531	45	cl	cl	NOUN
ejpam-3418	531	46	ι̂	ι̂	NOUN
ejpam-3418	531	47	=	=	PUNCT
ejpam-3418	531	48	∧	∧	NOUN
ejpam-3418	531	49	{	{	PUNCT
ejpam-3418	531	50	ω̂	ω̂	NUM
ejpam-3418	531	51	:	:	PUNCT
ejpam-3418	531	52	ι̂	ι̂	NUM
ejpam-3418	531	53	6	6	NUM
ejpam-3418	531	54	ω̂	ω̂	NUM
ejpam-3418	531	55	,	,	PUNCT
ejpam-3418	531	56	ω̂c	ω̂c	PROPN
ejpam-3418	531	57	∈	∈	PROPN
ejpam-3418	531	58	t	t	PROPN
ejpam-3418	531	59	′	′	NUM
ejpam-3418	531	60	}	}	PUNCT
ejpam-3418	531	61	.	.	PUNCT
ejpam-3418	532	1	one	one	PRON
ejpam-3418	532	2	can	can	AUX
ejpam-3418	532	3	show	show	VERB
ejpam-3418	532	4	,	,	PUNCT
ejpam-3418	532	5	that	that	SCONJ
ejpam-3418	532	6	indeed	indeed	ADV
ejpam-3418	532	7	just	just	ADV
ejpam-3418	532	8	like	like	ADP
ejpam-3418	532	9	the	the	DET
ejpam-3418	532	10	usual	usual	ADJ
ejpam-3418	532	11	topology	topology	NOUN
ejpam-3418	532	12	,	,	PUNCT
ejpam-3418	532	13	the	the	DET
ejpam-3418	532	14	ivfi	ivfi	NOUN
ejpam-3418	532	15	interior	interior	NOUN
ejpam-3418	532	16	is	be	AUX
ejpam-3418	532	17	the	the	DET
ejpam-3418	532	18	largest	large	ADJ
ejpam-3418	532	19	ivfi	ivfi	NOUN
ejpam-3418	532	20	open	open	ADJ
ejpam-3418	532	21	set	set	NOUN
ejpam-3418	532	22	contained	contain	VERB
ejpam-3418	532	23	in	in	ADP
ejpam-3418	532	24	ι̂	ι̂	NOUN
ejpam-3418	532	25	and	and	CCONJ
ejpam-3418	532	26	the	the	DET
ejpam-3418	532	27	ivfi	ivfi	NOUN
ejpam-3418	532	28	closure	closure	NOUN
ejpam-3418	532	29	is	be	AUX
ejpam-3418	532	30	the	the	DET
ejpam-3418	532	31	smallest	small	ADJ
ejpam-3418	532	32	ivfi	ivfi	NOUN
ejpam-3418	532	33	closed	close	VERB
ejpam-3418	532	34	set	set	ADJ
ejpam-3418	532	35	containing	contain	VERB
ejpam-3418	532	36	ι̂.	ι̂.	NOUN
ejpam-3418	532	37	moreover	moreover	ADV
ejpam-3418	532	38	,	,	PUNCT
ejpam-3418	532	39	ι̂	ι̂	PRON
ejpam-3418	532	40	is	be	AUX
ejpam-3418	532	41	ivfi	ivfi	ADV
ejpam-3418	532	42	open	open	ADJ
ejpam-3418	532	43	if	if	SCONJ
ejpam-3418	532	44	and	and	CCONJ
ejpam-3418	532	45	only	only	ADV
ejpam-3418	532	46	if	if	SCONJ
ejpam-3418	532	47	ι̂	ι̂	NUM
ejpam-3418	532	48	=	=	SYM
ejpam-3418	532	49	int	int	NOUN
ejpam-3418	532	50	ι̂	ι̂	NOUN
ejpam-3418	532	51	and	and	CCONJ
ejpam-3418	532	52	ι̂	ι̂	PRON
ejpam-3418	532	53	is	be	AUX
ejpam-3418	532	54	ivfi	ivfi	NOUN
ejpam-3418	532	55	closed	close	VERB
ejpam-3418	532	56	if	if	SCONJ
ejpam-3418	532	57	and	and	CCONJ
ejpam-3418	532	58	only	only	ADV
ejpam-3418	532	59	if	if	SCONJ
ejpam-3418	532	60	ι̂	ι̂	NOUN
ejpam-3418	532	61	=	=	SYM
ejpam-3418	533	1	cl	cl	NOUN
ejpam-3418	533	2	ι̂.	ι̂.	NOUN
ejpam-3418	533	3	the	the	DET
ejpam-3418	533	4	importance	importance	NOUN
ejpam-3418	533	5	of	of	ADP
ejpam-3418	533	6	the	the	DET
ejpam-3418	533	7	following	follow	VERB
ejpam-3418	533	8	concept	concept	NOUN
ejpam-3418	533	9	will	will	AUX
ejpam-3418	533	10	be	be	AUX
ejpam-3418	533	11	seen	see	VERB
ejpam-3418	533	12	when	when	SCONJ
ejpam-3418	533	13	dealing	deal	VERB
ejpam-3418	533	14	with	with	ADP
ejpam-3418	533	15	continuity	continuity	NOUN
ejpam-3418	533	16	with	with	ADP
ejpam-3418	533	17	respect	respect	NOUN
ejpam-3418	533	18	to	to	ADP
ejpam-3418	533	19	ivfi	ivfi	NOUN
ejpam-3418	533	20	sets	set	NOUN
ejpam-3418	533	21	.	.	PUNCT
ejpam-3418	534	1	definition	definition	NOUN
ejpam-3418	534	2	7	7	NUM
ejpam-3418	534	3	.	.	PUNCT
ejpam-3418	535	1	let	let	AUX
ejpam-3418	535	2	(	(	PUNCT
ejpam-3418	535	3	i(x),t	i(x),t	NOUN
ejpam-3418	535	4	′	′	NUM
ejpam-3418	535	5	)	)	PUNCT
ejpam-3418	535	6	be	be	VERB
ejpam-3418	535	7	an	an	DET
ejpam-3418	535	8	ivfi	ivfi	NOUN
ejpam-3418	535	9	space	space	NOUN
ejpam-3418	535	10	.	.	PUNCT
ejpam-3418	536	1	an	an	DET
ejpam-3418	536	2	ivfi	ivfi	NOUN
ejpam-3418	536	3	set	set	VERB
ejpam-3418	536	4	η̂	η̂	PUNCT
ejpam-3418	536	5	is	be	AUX
ejpam-3418	536	6	said	say	VERB
ejpam-3418	536	7	to	to	PART
ejpam-3418	536	8	be	be	AUX
ejpam-3418	536	9	an	an	DET
ejpam-3418	536	10	ivfi	ivfi	NOUN
ejpam-3418	536	11	neighborhood	neighborhood	NOUN
ejpam-3418	536	12	of	of	ADP
ejpam-3418	536	13	an	an	DET
ejpam-3418	536	14	ivfi	ivfi	NOUN
ejpam-3418	536	15	point	point	NOUN
ejpam-3418	536	16	pα	pα	NOUN
ejpam-3418	536	17	,	,	PUNCT
ejpam-3418	536	18	b	b	NOUN
ejpam-3418	536	19	,	,	PUNCT
ejpam-3418	536	20	where	where	SCONJ
ejpam-3418	536	21	[	[	X
ejpam-3418	536	22	0	0	NUM
ejpam-3418	536	23	,	,	PUNCT
ejpam-3418	536	24	0	0	NUM
ejpam-3418	536	25	]	]	PUNCT
ejpam-3418	536	26	6=	6=	ADP
ejpam-3418	536	27	α	α	X
ejpam-3418	536	28	∈	∈	X
ejpam-3418	537	1	i	i	PRON
ejpam-3418	537	2	and	and	CCONJ
ejpam-3418	537	3	∅	∅	NOUN
ejpam-3418	537	4	6=	6=	PUNCT
ejpam-3418	537	5	b	b	PROPN
ejpam-3418	537	6	∈	∈	PROPN
ejpam-3418	537	7	i(x	i(x	PROPN
ejpam-3418	537	8	)	)	PUNCT
ejpam-3418	537	9	,	,	PUNCT
ejpam-3418	537	10	if	if	SCONJ
ejpam-3418	537	11	there	there	PRON
ejpam-3418	537	12	exists	exist	VERB
ejpam-3418	537	13	ω̂	ω̂	PUNCT
ejpam-3418	537	14	∈	∈	PROPN
ejpam-3418	537	15	t	t	NOUN
ejpam-3418	537	16	′	′	NUM
ejpam-3418	537	17	such	such	ADJ
ejpam-3418	537	18	that	that	DET
ejpam-3418	537	19	pα	pα	NOUN
ejpam-3418	537	20	,	,	PUNCT
ejpam-3418	537	21	b	b	PROPN
ejpam-3418	537	22	∈	∈	PROPN
ejpam-3418	537	23	ω̂	ω̂	NUM
ejpam-3418	537	24	6	6	NUM
ejpam-3418	537	25	η̂.	η̂.	NOUN
ejpam-3418	537	26	an	an	DET
ejpam-3418	537	27	ivfi	ivfi	NOUN
ejpam-3418	537	28	neighborhood	neighborhood	NOUN
ejpam-3418	537	29	η̂	η̂	NUM
ejpam-3418	537	30	of	of	ADP
ejpam-3418	537	31	an	an	DET
ejpam-3418	537	32	ivfi	ivfi	NOUN
ejpam-3418	537	33	point	point	NOUN
ejpam-3418	537	34	pα	pα	NOUN
ejpam-3418	537	35	,	,	PUNCT
ejpam-3418	537	36	b	b	PROPN
ejpam-3418	537	37	is	be	AUX
ejpam-3418	537	38	said	say	VERB
ejpam-3418	537	39	to	to	PART
ejpam-3418	537	40	be	be	AUX
ejpam-3418	537	41	an	an	DET
ejpam-3418	537	42	ivfi	ivfi	NOUN
ejpam-3418	537	43	open	open	ADJ
ejpam-3418	537	44	neighborhood	neighborhood	NOUN
ejpam-3418	538	1	if	if	SCONJ
ejpam-3418	538	2	η̂	η̂	NUM
ejpam-3418	538	3	∈	∈	PROPN
ejpam-3418	538	4	t	t	NOUN
ejpam-3418	538	5	′.	′.	PROPN
ejpam-3418	538	6	mj	mj	PROPN
ejpam-3418	538	7	togonon	togonon	PROPN
ejpam-3418	538	8	,	,	PUNCT
ejpam-3418	538	9	r	r	NOUN
ejpam-3418	538	10	caga	caga	NOUN
ejpam-3418	538	11	-	-	PUNCT
ejpam-3418	538	12	anan	anan	PROPN
ejpam-3418	538	13	/	/	SYM
ejpam-3418	538	14	eur	eur	PROPN
ejpam-3418	538	15	.	.	PUNCT
ejpam-3418	539	1	j.	j.	PROPN
ejpam-3418	539	2	pure	pure	PROPN
ejpam-3418	539	3	appl	appl	PROPN
ejpam-3418	539	4	.	.	PROPN
ejpam-3418	539	5	math	math	PROPN
ejpam-3418	539	6	,	,	PUNCT
ejpam-3418	539	7	12	12	NUM
ejpam-3418	539	8	(	(	PUNCT
ejpam-3418	539	9	2	2	NUM
ejpam-3418	539	10	)	)	PUNCT
ejpam-3418	539	11	(	(	PUNCT
ejpam-3418	539	12	2019	2019	NUM
ejpam-3418	539	13	)	)	PUNCT
ejpam-3418	539	14	,	,	PUNCT
ejpam-3418	539	15	553	553	NUM
ejpam-3418	539	16	-	-	SYM
ejpam-3418	539	17	570	570	NUM
ejpam-3418	539	18	567	567	NUM
ejpam-3418	539	19	theorem	theorem	NOUN
ejpam-3418	539	20	8	8	NUM
ejpam-3418	539	21	.	.	PUNCT
ejpam-3418	540	1	let	let	AUX
ejpam-3418	540	2	(	(	PUNCT
ejpam-3418	540	3	i(x),t	i(x),t	NOUN
ejpam-3418	540	4	′	′	NUM
ejpam-3418	540	5	)	)	PUNCT
ejpam-3418	540	6	be	be	VERB
ejpam-3418	540	7	an	an	DET
ejpam-3418	540	8	ivfi	ivfi	NOUN
ejpam-3418	540	9	space	space	NOUN
ejpam-3418	540	10	.	.	PUNCT
ejpam-3418	541	1	then	then	ADV
ejpam-3418	541	2	η̂	η̂	PROPN
ejpam-3418	541	3	∈	∈	PROPN
ejpam-3418	541	4	t	t	NOUN
ejpam-3418	541	5	′	′	NOUN
ejpam-3418	541	6	if	if	SCONJ
ejpam-3418	541	7	and	and	CCONJ
ejpam-3418	541	8	only	only	ADV
ejpam-3418	541	9	if	if	SCONJ
ejpam-3418	541	10	for	for	ADP
ejpam-3418	541	11	every	every	DET
ejpam-3418	541	12	ivfi	ivfi	NOUN
ejpam-3418	541	13	point	point	NOUN
ejpam-3418	541	14	pα	pα	NOUN
ejpam-3418	541	15	,	,	PUNCT
ejpam-3418	541	16	b	b	PROPN
ejpam-3418	541	17	∈	∈	PROPN
ejpam-3418	541	18	η̂	η̂	NUM
ejpam-3418	541	19	where	where	SCONJ
ejpam-3418	541	20	[	[	X
ejpam-3418	541	21	0	0	NUM
ejpam-3418	541	22	,	,	PUNCT
ejpam-3418	541	23	0	0	NUM
ejpam-3418	541	24	]	]	PUNCT
ejpam-3418	541	25	6=	6=	ADP
ejpam-3418	541	26	α	α	X
ejpam-3418	541	27	∈	∈	X
ejpam-3418	542	1	i	i	PRON
ejpam-3418	542	2	and	and	CCONJ
ejpam-3418	542	3	∅	∅	NOUN
ejpam-3418	542	4	6=	6=	PUNCT
ejpam-3418	542	5	b	b	PROPN
ejpam-3418	542	6	∈	∈	PROPN
ejpam-3418	542	7	i(x	i(x	NOUN
ejpam-3418	542	8	)	)	PUNCT
ejpam-3418	542	9	,	,	PUNCT
ejpam-3418	542	10	η̂	η̂	PROPN
ejpam-3418	542	11	is	be	AUX
ejpam-3418	542	12	an	an	DET
ejpam-3418	542	13	ivfi	ivfi	NOUN
ejpam-3418	542	14	neighborhood	neighborhood	NOUN
ejpam-3418	542	15	of	of	ADP
ejpam-3418	542	16	pα	pα	PROPN
ejpam-3418	542	17	,	,	PUNCT
ejpam-3418	542	18	b.	b.	PROPN
ejpam-3418	542	19	proof	proof	NOUN
ejpam-3418	542	20	.	.	PUNCT
ejpam-3418	543	1	let	let	VERB
ejpam-3418	543	2	η̂	η̂	NUM
ejpam-3418	543	3	∈	∈	PROPN
ejpam-3418	543	4	t	t	NOUN
ejpam-3418	543	5	′	′	NOUN
ejpam-3418	543	6	and	and	CCONJ
ejpam-3418	543	7	pα	pα	VERB
ejpam-3418	543	8	,	,	PUNCT
ejpam-3418	543	9	b	b	PROPN
ejpam-3418	543	10	∈	∈	PROPN
ejpam-3418	543	11	η̂	η̂	NUM
ejpam-3418	543	12	be	be	AUX
ejpam-3418	543	13	an	an	DET
ejpam-3418	543	14	ivfi	ivfi	NOUN
ejpam-3418	543	15	point	point	NOUN
ejpam-3418	544	1	where	where	SCONJ
ejpam-3418	544	2	[	[	X
ejpam-3418	544	3	0	0	NUM
ejpam-3418	544	4	,	,	PUNCT
ejpam-3418	544	5	0	0	NUM
ejpam-3418	544	6	]	]	PUNCT
ejpam-3418	544	7	6=	6=	ADP
ejpam-3418	544	8	α	α	X
ejpam-3418	544	9	∈	∈	X
ejpam-3418	545	1	i	i	PRON
ejpam-3418	545	2	and	and	CCONJ
ejpam-3418	545	3	∅	∅	NOUN
ejpam-3418	545	4	6=	6=	PUNCT
ejpam-3418	545	5	b	b	PROPN
ejpam-3418	545	6	∈	∈	PROPN
ejpam-3418	545	7	i(x	i(x	PROPN
ejpam-3418	545	8	)	)	PUNCT
ejpam-3418	545	9	.	.	PUNCT
ejpam-3418	546	1	then	then	ADV
ejpam-3418	546	2	by	by	ADP
ejpam-3418	546	3	theorem	theorem	NOUN
ejpam-3418	546	4	3	3	NUM
ejpam-3418	546	5	,	,	PUNCT
ejpam-3418	546	6	there	there	PRON
ejpam-3418	546	7	exists	exist	VERB
ejpam-3418	546	8	ω̂	ω̂	PROPN
ejpam-3418	546	9	∈	∈	PROPN
ejpam-3418	546	10	b	b	PROPN
ejpam-3418	546	11	where	where	SCONJ
ejpam-3418	546	12	b	b	NOUN
ejpam-3418	546	13	is	be	AUX
ejpam-3418	546	14	an	an	DET
ejpam-3418	546	15	ivfi	ivfi	NOUN
ejpam-3418	546	16	base	base	NOUN
ejpam-3418	546	17	for	for	ADP
ejpam-3418	546	18	t	t	NOUN
ejpam-3418	546	19	′	′	NUM
ejpam-3418	546	20	such	such	ADJ
ejpam-3418	546	21	that	that	DET
ejpam-3418	546	22	pα	pα	NOUN
ejpam-3418	546	23	,	,	PUNCT
ejpam-3418	546	24	b	b	PROPN
ejpam-3418	546	25	∈	∈	PROPN
ejpam-3418	546	26	ω̂	ω̂	NUM
ejpam-3418	546	27	6	6	NUM
ejpam-3418	546	28	η̂.	η̂.	NOUN
ejpam-3418	546	29	but	but	CCONJ
ejpam-3418	546	30	note	note	VERB
ejpam-3418	546	31	that	that	SCONJ
ejpam-3418	546	32	every	every	DET
ejpam-3418	546	33	members	member	NOUN
ejpam-3418	546	34	of	of	ADP
ejpam-3418	546	35	b	b	NOUN
ejpam-3418	546	36	are	be	AUX
ejpam-3418	546	37	basic	basic	ADJ
ejpam-3418	546	38	ivfi	ivfi	NOUN
ejpam-3418	546	39	open	open	ADJ
ejpam-3418	546	40	sets	set	NOUN
ejpam-3418	546	41	of	of	ADP
ejpam-3418	546	42	t	t	NOUN
ejpam-3418	546	43	′	′	NUM
ejpam-3418	546	44	,	,	PUNCT
ejpam-3418	546	45	and	and	CCONJ
ejpam-3418	546	46	so	so	ADV
ejpam-3418	546	47	ω̂	ω̂	PROPN
ejpam-3418	546	48	∈	∈	PROPN
ejpam-3418	546	49	t	t	PROPN
ejpam-3418	546	50	′.	′.	NOUN
ejpam-3418	546	51	thus	thus	ADV
ejpam-3418	546	52	,	,	PUNCT
ejpam-3418	546	53	there	there	PRON
ejpam-3418	546	54	exists	exist	VERB
ejpam-3418	546	55	ω̂	ω̂	PUNCT
ejpam-3418	546	56	∈	∈	PROPN
ejpam-3418	546	57	t	t	NOUN
ejpam-3418	546	58	′	′	NUM
ejpam-3418	546	59	such	such	ADJ
ejpam-3418	546	60	that	that	DET
ejpam-3418	546	61	pα	pα	NOUN
ejpam-3418	546	62	,	,	PUNCT
ejpam-3418	546	63	b	b	PROPN
ejpam-3418	546	64	∈	∈	PROPN
ejpam-3418	546	65	ω̂	ω̂	NUM
ejpam-3418	546	66	6	6	NUM
ejpam-3418	546	67	η̂.	η̂.	NOUN
ejpam-3418	546	68	hence	hence	ADV
ejpam-3418	546	69	,	,	PUNCT
ejpam-3418	546	70	η̂	η̂	PROPN
ejpam-3418	546	71	is	be	AUX
ejpam-3418	546	72	an	an	DET
ejpam-3418	546	73	ivfi	ivfi	NOUN
ejpam-3418	546	74	neighborhood	neighborhood	NOUN
ejpam-3418	546	75	of	of	ADP
ejpam-3418	546	76	pα	pα	PROPN
ejpam-3418	546	77	,	,	PUNCT
ejpam-3418	546	78	b.	b.	PROPN
ejpam-3418	546	79	conversely	conversely	ADV
ejpam-3418	546	80	,	,	PUNCT
ejpam-3418	546	81	suppose	suppose	VERB
ejpam-3418	546	82	that	that	SCONJ
ejpam-3418	546	83	for	for	ADP
ejpam-3418	546	84	every	every	DET
ejpam-3418	546	85	ivfi	ivfi	NOUN
ejpam-3418	546	86	point	point	NOUN
ejpam-3418	546	87	pα	pα	NOUN
ejpam-3418	546	88	,	,	PUNCT
ejpam-3418	546	89	b	b	PROPN
ejpam-3418	546	90	∈	∈	PROPN
ejpam-3418	546	91	η̂	η̂	NUM
ejpam-3418	546	92	where	where	SCONJ
ejpam-3418	546	93	[	[	X
ejpam-3418	546	94	0	0	NUM
ejpam-3418	546	95	,	,	PUNCT
ejpam-3418	546	96	0	0	NUM
ejpam-3418	546	97	]	]	PUNCT
ejpam-3418	546	98	6=	6=	ADP
ejpam-3418	546	99	α	α	X
ejpam-3418	546	100	∈	∈	X
ejpam-3418	547	1	i	i	PRON
ejpam-3418	547	2	and	and	CCONJ
ejpam-3418	547	3	∅	∅	NOUN
ejpam-3418	547	4	6=	6=	PUNCT
ejpam-3418	547	5	b	b	PROPN
ejpam-3418	547	6	∈	∈	PROPN
ejpam-3418	547	7	i(x	i(x	NOUN
ejpam-3418	547	8	)	)	PUNCT
ejpam-3418	547	9	,	,	PUNCT
ejpam-3418	547	10	η̂	η̂	PROPN
ejpam-3418	547	11	is	be	AUX
ejpam-3418	547	12	an	an	DET
ejpam-3418	547	13	ivfi	ivfi	NOUN
ejpam-3418	547	14	neighborhood	neighborhood	NOUN
ejpam-3418	547	15	of	of	ADP
ejpam-3418	547	16	pα	pα	PROPN
ejpam-3418	547	17	,	,	PUNCT
ejpam-3418	547	18	b.	b.	PROPN
ejpam-3418	547	19	then	then	ADV
ejpam-3418	547	20	there	there	PRON
ejpam-3418	547	21	exists	exist	VERB
ejpam-3418	547	22	ω̂α	ω̂α	PRON
ejpam-3418	547	23	,	,	PUNCT
ejpam-3418	547	24	b	b	PROPN
ejpam-3418	547	25	∈	∈	PROPN
ejpam-3418	547	26	t	t	NOUN
ejpam-3418	547	27	′	′	NUM
ejpam-3418	547	28	such	such	ADJ
ejpam-3418	547	29	that	that	DET
ejpam-3418	547	30	pα	pα	NOUN
ejpam-3418	547	31	,	,	PUNCT
ejpam-3418	547	32	b	b	PROPN
ejpam-3418	547	33	∈	∈	PROPN
ejpam-3418	547	34	ω̂pα	ω̂pα	PROPN
ejpam-3418	547	35	,	,	PUNCT
ejpam-3418	547	36	b	b	PROPN
ejpam-3418	547	37	6	6	NUM
ejpam-3418	547	38	η̂.	η̂.	NOUN
ejpam-3418	547	39	by	by	ADP
ejpam-3418	547	40	theorem	theorem	NOUN
ejpam-3418	547	41	7	7	NUM
ejpam-3418	547	42	,	,	PUNCT
ejpam-3418	547	43	η̂	η̂	NUM
ejpam-3418	547	44	=	=	SYM
ejpam-3418	547	45	∨	∨	NUM
ejpam-3418	547	46	pα	pα	NOUN
ejpam-3418	547	47	,	,	PUNCT
ejpam-3418	547	48	b∈η̂	b∈η̂	PROPN
ejpam-3418	547	49	ω̂α	ω̂α	PROPN
ejpam-3418	547	50	,	,	PUNCT
ejpam-3418	547	51	b.	b.	PROPN
ejpam-3418	547	52	since	since	SCONJ
ejpam-3418	547	53	each	each	DET
ejpam-3418	547	54	ω̂α	ω̂α	NOUN
ejpam-3418	547	55	,	,	PUNCT
ejpam-3418	547	56	b	b	PROPN
ejpam-3418	547	57	∈	∈	PROPN
ejpam-3418	547	58	t	t	PROPN
ejpam-3418	547	59	′	′	NOUN
ejpam-3418	547	60	,	,	PUNCT
ejpam-3418	547	61	we	we	PRON
ejpam-3418	547	62	have∨	have∨	VERB
ejpam-3418	547	63	pα	pα	VERB
ejpam-3418	547	64	,	,	PUNCT
ejpam-3418	547	65	b∈η̂	b∈η̂	PROPN
ejpam-3418	547	66	ω̂pα	ω̂pα	PROPN
ejpam-3418	547	67	,	,	PUNCT
ejpam-3418	547	68	b	b	PROPN
ejpam-3418	547	69	∈	∈	PROPN
ejpam-3418	547	70	t	t	NOUN
ejpam-3418	547	71	′.	′.	NOUN
ejpam-3418	547	72	hence	hence	ADV
ejpam-3418	547	73	,	,	PUNCT
ejpam-3418	547	74	η̂	η̂	PROPN
ejpam-3418	547	75	∈	∈	PROPN
ejpam-3418	547	76	t	t	NOUN
ejpam-3418	547	77	′.	′.	NOUN
ejpam-3418	547	78	due	due	ADP
ejpam-3418	547	79	to	to	ADP
ejpam-3418	547	80	theorem	theorem	NOUN
ejpam-3418	547	81	2	2	NUM
ejpam-3418	547	82	,	,	PUNCT
ejpam-3418	547	83	we	we	PRON
ejpam-3418	547	84	have	have	VERB
ejpam-3418	547	85	the	the	DET
ejpam-3418	547	86	following	following	ADJ
ejpam-3418	547	87	result	result	NOUN
ejpam-3418	547	88	which	which	PRON
ejpam-3418	547	89	is	be	AUX
ejpam-3418	547	90	an	an	DET
ejpam-3418	547	91	instance	instance	NOUN
ejpam-3418	547	92	of	of	ADP
ejpam-3418	547	93	the	the	DET
ejpam-3418	547	94	difference	difference	NOUN
ejpam-3418	547	95	of	of	ADP
ejpam-3418	547	96	the	the	DET
ejpam-3418	547	97	ivfi	ivfi	NOUN
ejpam-3418	547	98	topology	topology	NOUN
ejpam-3418	547	99	from	from	ADP
ejpam-3418	547	100	the	the	DET
ejpam-3418	547	101	usual	usual	ADJ
ejpam-3418	547	102	topology	topology	NOUN
ejpam-3418	547	103	.	.	PUNCT
ejpam-3418	548	1	theorem	theorem	VERB
ejpam-3418	548	2	9	9	NUM
ejpam-3418	548	3	.	.	PUNCT
ejpam-3418	549	1	let	let	VERB
ejpam-3418	549	2	ι̂	ι̂	PRON
ejpam-3418	549	3	be	be	AUX
ejpam-3418	549	4	an	an	DET
ejpam-3418	549	5	ivfi	ivfi	NOUN
ejpam-3418	549	6	set	set	VERB
ejpam-3418	549	7	in	in	ADP
ejpam-3418	549	8	an	an	DET
ejpam-3418	549	9	ivfi	ivfi	NOUN
ejpam-3418	549	10	space	space	NOUN
ejpam-3418	549	11	(	(	PUNCT
ejpam-3418	549	12	i(x),t	i(x),t	NOUN
ejpam-3418	549	13	′	′	NUM
ejpam-3418	549	14	)	)	PUNCT
ejpam-3418	549	15	.	.	PUNCT
ejpam-3418	550	1	then	then	ADV
ejpam-3418	550	2	,	,	PUNCT
ejpam-3418	550	3	int	int	NOUN
ejpam-3418	550	4	ι̂	ι̂	NUM
ejpam-3418	550	5	6	6	NUM
ejpam-3418	550	6	(	(	PUNCT
ejpam-3418	550	7	cl	cl	NOUN
ejpam-3418	550	8	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	550	9	and	and	CCONJ
ejpam-3418	550	10	cl	cl	NOUN
ejpam-3418	550	11	ι̂	ι̂	PUNCT
ejpam-3418	550	12	6	6	NUM
ejpam-3418	550	13	(	(	PUNCT
ejpam-3418	550	14	int	int	NOUN
ejpam-3418	550	15	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	550	16	.	.	PUNCT
ejpam-3418	551	1	proof	proof	NOUN
ejpam-3418	551	2	.	.	PUNCT
ejpam-3418	552	1	let	let	VERB
ejpam-3418	552	2	ι̂	ι̂	PRON
ejpam-3418	552	3	be	be	AUX
ejpam-3418	552	4	an	an	DET
ejpam-3418	552	5	ivfi	ivfi	NOUN
ejpam-3418	552	6	set	set	VERB
ejpam-3418	552	7	in	in	ADP
ejpam-3418	552	8	an	an	DET
ejpam-3418	552	9	ivfi	ivfi	NOUN
ejpam-3418	552	10	space	space	NOUN
ejpam-3418	552	11	(	(	PUNCT
ejpam-3418	552	12	i(x),t	i(x),t	NOUN
ejpam-3418	552	13	′	′	NUM
ejpam-3418	552	14	)	)	PUNCT
ejpam-3418	552	15	.	.	PUNCT
ejpam-3418	553	1	then	then	ADV
ejpam-3418	553	2	int	int	VERB
ejpam-3418	553	3	ι̂	ι̂	PUNCT
ejpam-3418	553	4	=	=	SYM
ejpam-3418	553	5	∨	∨	X
ejpam-3418	553	6	{	{	PUNCT
ejpam-3418	553	7	τ̂	τ̂	NUM
ejpam-3418	553	8	:	:	PUNCT
ejpam-3418	553	9	τ̂	τ̂	NUM
ejpam-3418	553	10	6	6	NUM
ejpam-3418	553	11	ι̂	ι̂	NOUN
ejpam-3418	553	12	,	,	PUNCT
ejpam-3418	553	13	τ̂	τ̂	PUNCT
ejpam-3418	553	14	∈	∈	PROPN
ejpam-3418	553	15	t	t	PROPN
ejpam-3418	553	16	′	′	NOUN
ejpam-3418	553	17	}	}	PUNCT
ejpam-3418	553	18	6	6	NUM
ejpam-3418	553	19	∨	∨	NUM
ejpam-3418	553	20	{	{	PUNCT
ejpam-3418	553	21	(	(	PUNCT
ejpam-3418	553	22	τ̂	τ̂	X
ejpam-3418	553	23	c)c	c)c	NOUN
ejpam-3418	553	24	:	:	PUNCT
ejpam-3418	553	25	(	(	PUNCT
ejpam-3418	553	26	τ̂	τ̂	X
ejpam-3418	553	27	c)c	c)c	X
ejpam-3418	553	28	6	6	NUM
ejpam-3418	553	29	(	(	PUNCT
ejpam-3418	553	30	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	553	31	,	,	PUNCT
ejpam-3418	553	32	τ̂	τ̂	PUNCT
ejpam-3418	553	33	∈	∈	PROPN
ejpam-3418	553	34	t	t	NOUN
ejpam-3418	553	35	′	′	NOUN
ejpam-3418	553	36	}	}	PUNCT
ejpam-3418	553	37	=	=	SYM
ejpam-3418	553	38	(	(	PUNCT
ejpam-3418	553	39	∧	∧	PROPN
ejpam-3418	553	40	{	{	PUNCT
ejpam-3418	553	41	τ̂	τ̂	X
ejpam-3418	553	42	c	c	NOUN
ejpam-3418	553	43	:	:	PUNCT
ejpam-3418	553	44	ι̂c	ι̂c	X
ejpam-3418	553	45	6	6	NUM
ejpam-3418	553	46	τ̂	τ̂	SYM
ejpam-3418	553	47	c	c	NOUN
ejpam-3418	553	48	,	,	PUNCT
ejpam-3418	553	49	τ̂	τ̂	PUNCT
ejpam-3418	553	50	∈	∈	PROPN
ejpam-3418	553	51	t	t	PROPN
ejpam-3418	553	52	′	′	NOUN
ejpam-3418	553	53	}	}	PUNCT
ejpam-3418	553	54	)	)	PUNCT
ejpam-3418	554	1	c	c	X
ejpam-3418	554	2	=	=	SYM
ejpam-3418	554	3	(	(	PUNCT
ejpam-3418	554	4	cl	cl	NOUN
ejpam-3418	554	5	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	554	6	and	and	CCONJ
ejpam-3418	554	7	cl	cl	NOUN
ejpam-3418	554	8	ι̂	ι̂	NOUN
ejpam-3418	555	1	=	=	PUNCT
ejpam-3418	555	2	∧	∧	NOUN
ejpam-3418	555	3	{	{	PUNCT
ejpam-3418	555	4	ω̂	ω̂	NUM
ejpam-3418	555	5	:	:	PUNCT
ejpam-3418	555	6	ι̂	ι̂	NUM
ejpam-3418	555	7	6	6	NUM
ejpam-3418	555	8	ω̂	ω̂	NUM
ejpam-3418	555	9	,	,	PUNCT
ejpam-3418	555	10	ω̂c	ω̂c	PROPN
ejpam-3418	555	11	∈	∈	PROPN
ejpam-3418	555	12	t	t	PROPN
ejpam-3418	555	13	′	′	NOUN
ejpam-3418	555	14	}	}	PUNCT
ejpam-3418	555	15	6	6	NUM
ejpam-3418	555	16	∧	∧	PROPN
ejpam-3418	555	17	{	{	PUNCT
ejpam-3418	555	18	(	(	PUNCT
ejpam-3418	555	19	ω̂c)c	ω̂c)c	NOUN
ejpam-3418	555	20	:	:	PUNCT
ejpam-3418	555	21	(	(	PUNCT
ejpam-3418	555	22	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	555	23	6	6	NUM
ejpam-3418	555	24	(	(	PUNCT
ejpam-3418	555	25	ω̂c)c	ω̂c)c	NOUN
ejpam-3418	555	26	,	,	PUNCT
ejpam-3418	555	27	ω̂c	ω̂c	PROPN
ejpam-3418	555	28	∈	∈	PROPN
ejpam-3418	555	29	t	t	NOUN
ejpam-3418	555	30	′	′	NOUN
ejpam-3418	555	31	}	}	PUNCT
ejpam-3418	555	32	=	=	SYM
ejpam-3418	555	33	(	(	PUNCT
ejpam-3418	555	34	∨	∨	X
ejpam-3418	555	35	{	{	PUNCT
ejpam-3418	555	36	ω̂c	ω̂c	PROPN
ejpam-3418	555	37	:	:	PUNCT
ejpam-3418	555	38	ω̂c	ω̂c	PROPN
ejpam-3418	555	39	6	6	NUM
ejpam-3418	555	40	ι̂c	ι̂c	NOUN
ejpam-3418	555	41	,	,	PUNCT
ejpam-3418	555	42	ω̂c	ω̂c	PROPN
ejpam-3418	555	43	∈	∈	PROPN
ejpam-3418	555	44	t	t	PROPN
ejpam-3418	555	45	′	′	NOUN
ejpam-3418	555	46	}	}	PUNCT
ejpam-3418	555	47	)	)	PUNCT
ejpam-3418	556	1	c	c	X
ejpam-3418	556	2	=	=	SYM
ejpam-3418	556	3	(	(	PUNCT
ejpam-3418	556	4	int	int	NOUN
ejpam-3418	556	5	ι̂c)c	ι̂c)c	NOUN
ejpam-3418	556	6	.	.	PUNCT
ejpam-3418	557	1	we	we	PRON
ejpam-3418	557	2	next	next	ADV
ejpam-3418	557	3	make	make	VERB
ejpam-3418	557	4	precise	precise	ADJ
ejpam-3418	557	5	what	what	PRON
ejpam-3418	557	6	we	we	PRON
ejpam-3418	557	7	mean	mean	VERB
ejpam-3418	557	8	by	by	ADP
ejpam-3418	557	9	continuity	continuity	NOUN
ejpam-3418	557	10	with	with	ADP
ejpam-3418	557	11	respect	respect	NOUN
ejpam-3418	557	12	to	to	ADP
ejpam-3418	557	13	ivfi	ivfi	NOUN
ejpam-3418	557	14	sets	set	NOUN
ejpam-3418	557	15	.	.	PUNCT
ejpam-3418	558	1	let	let	VERB
ejpam-3418	558	2	f	f	NOUN
ejpam-3418	558	3	:	:	PUNCT
ejpam-3418	558	4	x	x	X
ejpam-3418	558	5	→	→	SYM
ejpam-3418	558	6	y	y	PROPN
ejpam-3418	558	7	be	be	AUX
ejpam-3418	558	8	a	a	DET
ejpam-3418	558	9	one	one	NUM
ejpam-3418	558	10	-	-	PUNCT
ejpam-3418	558	11	to	to	ADP
ejpam-3418	558	12	-	-	PUNCT
ejpam-3418	558	13	one	one	NUM
ejpam-3418	558	14	map	map	NOUN
ejpam-3418	558	15	and	and	CCONJ
ejpam-3418	558	16	i(x	i(x	NOUN
ejpam-3418	558	17	)	)	PUNCT
ejpam-3418	558	18	be	be	AUX
ejpam-3418	558	19	an	an	DET
ejpam-3418	558	20	ideal	ideal	NOUN
ejpam-3418	558	21	on	on	ADP
ejpam-3418	558	22	x.	x.	NOUN
ejpam-3418	558	23	then	then	ADV
ejpam-3418	558	24	,	,	PUNCT
ejpam-3418	558	25	f(i(x	f(i(x	NOUN
ejpam-3418	558	26	)	)	PUNCT
ejpam-3418	558	27	)	)	PUNCT
ejpam-3418	558	28	is	be	AUX
ejpam-3418	558	29	an	an	DET
ejpam-3418	558	30	ideal	ideal	NOUN
ejpam-3418	558	31	on	on	ADP
ejpam-3418	558	32	y	y	PROPN
ejpam-3418	558	33	,	,	PUNCT
ejpam-3418	558	34	by	by	ADP
ejpam-3418	558	35	theorem	theorem	NOUN
ejpam-3418	558	36	3	3	NUM
ejpam-3418	558	37	,	,	PUNCT
ejpam-3418	558	38	and	and	CCONJ
ejpam-3418	558	39	by	by	ADP
ejpam-3418	558	40	proposition	proposition	NOUN
ejpam-3418	558	41	2	2	NUM
ejpam-3418	558	42	,	,	PUNCT
ejpam-3418	558	43	f−1(f(i(x	f−1(f(i(x	NOUN
ejpam-3418	558	44	)	)	PUNCT
ejpam-3418	558	45	)	)	PUNCT
ejpam-3418	558	46	)	)	PUNCT
ejpam-3418	559	1	=	=	SYM
ejpam-3418	559	2	i(x	i(x	NOUN
ejpam-3418	559	3	)	)	PUNCT
ejpam-3418	559	4	.	.	PUNCT
ejpam-3418	560	1	definition	definition	NOUN
ejpam-3418	560	2	8	8	NUM
ejpam-3418	560	3	.	.	PUNCT
ejpam-3418	561	1	let	let	VERB
ejpam-3418	561	2	f	f	NOUN
ejpam-3418	561	3	:	:	PUNCT
ejpam-3418	561	4	x	x	X
ejpam-3418	561	5	→	→	SYM
ejpam-3418	561	6	y	y	PROPN
ejpam-3418	561	7	be	be	AUX
ejpam-3418	561	8	a	a	DET
ejpam-3418	561	9	one	one	NUM
ejpam-3418	561	10	-	-	PUNCT
ejpam-3418	561	11	to	to	ADP
ejpam-3418	561	12	-	-	PUNCT
ejpam-3418	561	13	one	one	NUM
ejpam-3418	561	14	map	map	NOUN
ejpam-3418	561	15	and	and	CCONJ
ejpam-3418	561	16	i(x	i(x	NOUN
ejpam-3418	561	17	)	)	PUNCT
ejpam-3418	561	18	be	be	AUX
ejpam-3418	561	19	an	an	DET
ejpam-3418	561	20	ideal	ideal	NOUN
ejpam-3418	561	21	on	on	ADP
ejpam-3418	561	22	x.	x.	NOUN
ejpam-3418	561	23	let	let	VERB
ejpam-3418	561	24	t	t	PROPN
ejpam-3418	561	25	′1	′1	NOUN
ejpam-3418	561	26	and	and	CCONJ
ejpam-3418	561	27	t	t	PROPN
ejpam-3418	561	28	′2	′2	PROPN
ejpam-3418	561	29	be	be	AUX
ejpam-3418	561	30	ivfi	ivfi	NOUN
ejpam-3418	561	31	topologies	topology	NOUN
ejpam-3418	561	32	on	on	ADP
ejpam-3418	561	33	i(x	i(x	PROPN
ejpam-3418	561	34	)	)	PUNCT
ejpam-3418	561	35	and	and	CCONJ
ejpam-3418	561	36	f(i(x	f(i(x	NOUN
ejpam-3418	561	37	)	)	PUNCT
ejpam-3418	561	38	)	)	PUNCT
ejpam-3418	561	39	,	,	PUNCT
ejpam-3418	561	40	respectively	respectively	ADV
ejpam-3418	561	41	.	.	PUNCT
ejpam-3418	562	1	the	the	DET
ejpam-3418	562	2	map	map	NOUN
ejpam-3418	562	3	f	f	NOUN
ejpam-3418	562	4	is	be	AUX
ejpam-3418	562	5	said	say	VERB
ejpam-3418	562	6	to	to	PART
ejpam-3418	562	7	be	be	AUX
ejpam-3418	562	8	ivfi	ivfi	NOUN
ejpam-3418	562	9	continuous	continuous	ADJ
ejpam-3418	562	10	if	if	SCONJ
ejpam-3418	562	11	f−1	f−1	PROPN
ejpam-3418	562	12	[	[	X
ejpam-3418	562	13	̂ι	̂ι	X
ejpam-3418	562	14	]	]	X
ejpam-3418	562	15	∈	∈	PROPN
ejpam-3418	562	16	t	t	PROPN
ejpam-3418	562	17	′1	′1	X
ejpam-3418	562	18	,	,	PUNCT
ejpam-3418	562	19	for	for	ADP
ejpam-3418	562	20	all	all	DET
ejpam-3418	562	21	ι̂	ι̂	NUM
ejpam-3418	562	22	∈	∈	PROPN
ejpam-3418	562	23	t	t	NOUN
ejpam-3418	562	24	′2	′2	X
ejpam-3418	562	25	.	.	PUNCT
ejpam-3418	563	1	mj	mj	PROPN
ejpam-3418	563	2	togonon	togonon	PROPN
ejpam-3418	563	3	,	,	PUNCT
ejpam-3418	563	4	r	r	NOUN
ejpam-3418	563	5	caga	caga	NOUN
ejpam-3418	563	6	-	-	PUNCT
ejpam-3418	563	7	anan	anan	PROPN
ejpam-3418	563	8	/	/	SYM
ejpam-3418	563	9	eur	eur	PROPN
ejpam-3418	563	10	.	.	PUNCT
ejpam-3418	564	1	j.	j.	PROPN
ejpam-3418	564	2	pure	pure	PROPN
ejpam-3418	564	3	appl	appl	PROPN
ejpam-3418	564	4	.	.	PROPN
ejpam-3418	564	5	math	math	PROPN
ejpam-3418	564	6	,	,	PUNCT
ejpam-3418	564	7	12	12	NUM
ejpam-3418	564	8	(	(	PUNCT
ejpam-3418	564	9	2	2	NUM
ejpam-3418	564	10	)	)	PUNCT
ejpam-3418	564	11	(	(	PUNCT
ejpam-3418	564	12	2019	2019	NUM
ejpam-3418	564	13	)	)	PUNCT
ejpam-3418	564	14	,	,	PUNCT
ejpam-3418	564	15	553	553	NUM
ejpam-3418	564	16	-	-	SYM
ejpam-3418	564	17	570	570	NUM
ejpam-3418	564	18	568	568	NUM
ejpam-3418	564	19	the	the	DET
ejpam-3418	564	20	next	next	ADJ
ejpam-3418	564	21	theorem	theorem	NOUN
ejpam-3418	564	22	characterizes	characterize	VERB
ejpam-3418	564	23	the	the	DET
ejpam-3418	564	24	ivfi	ivfi	NOUN
ejpam-3418	564	25	continuous	continuous	ADJ
ejpam-3418	564	26	maps	map	NOUN
ejpam-3418	564	27	.	.	PUNCT
ejpam-3418	565	1	theorem	theorem	NOUN
ejpam-3418	565	2	10	10	NUM
ejpam-3418	565	3	.	.	PUNCT
ejpam-3418	566	1	let	let	VERB
ejpam-3418	566	2	f	f	NOUN
ejpam-3418	566	3	:	:	PUNCT
ejpam-3418	566	4	x	x	X
ejpam-3418	566	5	→	→	SYM
ejpam-3418	566	6	y	y	PROPN
ejpam-3418	566	7	be	be	AUX
ejpam-3418	566	8	a	a	DET
ejpam-3418	566	9	one	one	NUM
ejpam-3418	566	10	-	-	PUNCT
ejpam-3418	566	11	to	to	ADP
ejpam-3418	566	12	-	-	PUNCT
ejpam-3418	566	13	one	one	NUM
ejpam-3418	566	14	map	map	NOUN
ejpam-3418	566	15	and	and	CCONJ
ejpam-3418	566	16	i(x	i(x	NOUN
ejpam-3418	566	17	)	)	PUNCT
ejpam-3418	566	18	be	be	AUX
ejpam-3418	566	19	an	an	DET
ejpam-3418	566	20	ideal	ideal	NOUN
ejpam-3418	566	21	on	on	ADP
ejpam-3418	566	22	x.	x.	NOUN
ejpam-3418	566	23	let	let	VERB
ejpam-3418	566	24	i(y	i(y	NOUN
ejpam-3418	566	25	)	)	PUNCT
ejpam-3418	567	1	=	=	SYM
ejpam-3418	567	2	f(i(x	f(i(x	NOUN
ejpam-3418	567	3	)	)	PUNCT
ejpam-3418	567	4	)	)	PUNCT
ejpam-3418	567	5	.	.	PUNCT
ejpam-3418	568	1	moreover	moreover	ADV
ejpam-3418	568	2	,	,	PUNCT
ejpam-3418	568	3	let	let	VERB
ejpam-3418	568	4	(	(	PUNCT
ejpam-3418	568	5	i(x),t	i(x),t	VERB
ejpam-3418	568	6	′1	′1	NOUN
ejpam-3418	568	7	)	)	PUNCT
ejpam-3418	568	8	and	and	CCONJ
ejpam-3418	568	9	(	(	PUNCT
ejpam-3418	568	10	i(y	i(y	NOUN
ejpam-3418	568	11	)	)	PUNCT
ejpam-3418	568	12	,	,	PUNCT
ejpam-3418	568	13	t	t	PROPN
ejpam-3418	568	14	′2	′2	X
ejpam-3418	568	15	)	)	PUNCT
ejpam-3418	568	16	be	be	AUX
ejpam-3418	568	17	ivfi	ivfi	NOUN
ejpam-3418	568	18	spaces	space	NOUN
ejpam-3418	568	19	of	of	ADP
ejpam-3418	568	20	i(x	i(x	PROPN
ejpam-3418	568	21	)	)	PUNCT
ejpam-3418	568	22	and	and	CCONJ
ejpam-3418	568	23	i(y	i(y	NOUN
ejpam-3418	568	24	)	)	PUNCT
ejpam-3418	568	25	,	,	PUNCT
ejpam-3418	568	26	respectively	respectively	ADV
ejpam-3418	568	27	.	.	PUNCT
ejpam-3418	569	1	then	then	ADV
ejpam-3418	569	2	the	the	DET
ejpam-3418	569	3	following	follow	VERB
ejpam-3418	569	4	statements	statement	NOUN
ejpam-3418	569	5	are	be	AUX
ejpam-3418	569	6	equivalent	equivalent	ADJ
ejpam-3418	569	7	:	:	PUNCT
ejpam-3418	569	8	(	(	PUNCT
ejpam-3418	569	9	i	i	NOUN
ejpam-3418	569	10	)	)	PUNCT
ejpam-3418	569	11	the	the	DET
ejpam-3418	569	12	function	function	NOUN
ejpam-3418	569	13	f	f	PROPN
ejpam-3418	569	14	is	be	AUX
ejpam-3418	569	15	ivfi	ivfi	NOUN
ejpam-3418	569	16	continuous	continuous	ADJ
ejpam-3418	569	17	;	;	PUNCT
ejpam-3418	569	18	(	(	PUNCT
ejpam-3418	569	19	ii	ii	NOUN
ejpam-3418	569	20	)	)	PUNCT
ejpam-3418	569	21	the	the	DET
ejpam-3418	569	22	inverse	inverse	ADJ
ejpam-3418	569	23	image	image	NOUN
ejpam-3418	569	24	of	of	ADP
ejpam-3418	569	25	every	every	DET
ejpam-3418	569	26	ivfi	ivfi	NOUN
ejpam-3418	569	27	closed	close	VERB
ejpam-3418	569	28	set	set	VERB
ejpam-3418	569	29	is	be	AUX
ejpam-3418	569	30	ivfi	ivfi	NOUN
ejpam-3418	569	31	closed	close	VERB
ejpam-3418	569	32	;	;	PUNCT
ejpam-3418	569	33	(	(	PUNCT
ejpam-3418	569	34	iii	iii	NOUN
ejpam-3418	569	35	)	)	PUNCT
ejpam-3418	569	36	for	for	ADP
ejpam-3418	569	37	each	each	DET
ejpam-3418	569	38	ivfi	ivfi	NOUN
ejpam-3418	569	39	point	point	NOUN
ejpam-3418	569	40	pα	pα	NOUN
ejpam-3418	569	41	,	,	PUNCT
ejpam-3418	569	42	b	b	NOUN
ejpam-3418	569	43	,	,	PUNCT
ejpam-3418	569	44	the	the	DET
ejpam-3418	569	45	inverse	inverse	NOUN
ejpam-3418	569	46	of	of	ADP
ejpam-3418	569	47	every	every	DET
ejpam-3418	569	48	neighborhood	neighborhood	NOUN
ejpam-3418	569	49	of	of	ADP
ejpam-3418	569	50	f	f	PROPN
ejpam-3418	570	1	[	[	X
ejpam-3418	570	2	pα	pα	NOUN
ejpam-3418	570	3	,	,	PUNCT
ejpam-3418	570	4	b	b	NOUN
ejpam-3418	570	5	]	]	X
ejpam-3418	570	6	under	under	ADP
ejpam-3418	570	7	f	f	PROPN
ejpam-3418	570	8	is	be	AUX
ejpam-3418	570	9	neighborhood	neighborhood	NOUN
ejpam-3418	570	10	of	of	ADP
ejpam-3418	570	11	pα	pα	PROPN
ejpam-3418	570	12	,	,	PUNCT
ejpam-3418	570	13	b	b	NOUN
ejpam-3418	570	14	;	;	PUNCT
ejpam-3418	570	15	(	(	PUNCT
ejpam-3418	570	16	iv	iv	X
ejpam-3418	570	17	)	)	PUNCT
ejpam-3418	570	18	for	for	ADP
ejpam-3418	570	19	each	each	DET
ejpam-3418	570	20	ivfi	ivfi	NOUN
ejpam-3418	570	21	point	point	NOUN
ejpam-3418	570	22	pα	pα	NOUN
ejpam-3418	570	23	,	,	PUNCT
ejpam-3418	570	24	b	b	NOUN
ejpam-3418	570	25	and	and	CCONJ
ejpam-3418	570	26	each	each	DET
ejpam-3418	570	27	neighborhood	neighborhood	NOUN
ejpam-3418	570	28	η̂	η̂	NUM
ejpam-3418	570	29	of	of	ADP
ejpam-3418	570	30	f	f	PROPN
ejpam-3418	571	1	[	[	X
ejpam-3418	571	2	pα	pα	NOUN
ejpam-3418	571	3	,	,	PUNCT
ejpam-3418	571	4	b	b	NOUN
ejpam-3418	571	5	]	]	X
ejpam-3418	571	6	,	,	PUNCT
ejpam-3418	571	7	there	there	PRON
ejpam-3418	571	8	is	be	VERB
ejpam-3418	571	9	a	a	DET
ejpam-3418	571	10	neighborhood	neighborhood	NOUN
ejpam-3418	571	11	η̂′	η̂′	PROPN
ejpam-3418	571	12	of	of	ADP
ejpam-3418	571	13	pα	pα	PROPN
ejpam-3418	571	14	,	,	PUNCT
ejpam-3418	571	15	b	b	NOUN
ejpam-3418	571	16	such	such	ADJ
ejpam-3418	572	1	that	that	SCONJ
ejpam-3418	572	2	f	f	PROPN
ejpam-3418	573	1	[	[	X
ejpam-3418	573	2	η̂′	η̂′	X
ejpam-3418	573	3	]	]	X
ejpam-3418	573	4	=	=	PUNCT
ejpam-3418	573	5	η̂	η̂	X
ejpam-3418	573	6	whenever	whenever	SCONJ
ejpam-3418	573	7	f	f	PROPN
ejpam-3418	573	8	is	be	AUX
ejpam-3418	573	9	onto	onto	ADP
ejpam-3418	573	10	;	;	PUNCT
ejpam-3418	573	11	(	(	PUNCT
ejpam-3418	573	12	v	v	NOUN
ejpam-3418	573	13	)	)	PUNCT
ejpam-3418	573	14	f	f	NOUN
ejpam-3418	574	1	[	[	X
ejpam-3418	574	2	cl	cl	NOUN
ejpam-3418	574	3	ι̂	ι̂	X
ejpam-3418	574	4	]	]	X
ejpam-3418	574	5	6	6	NUM
ejpam-3418	574	6	cl	cl	NOUN
ejpam-3418	574	7	f	f	PROPN
ejpam-3418	575	1	[	[	X
ejpam-3418	575	2	̂ι	̂ι	X
ejpam-3418	575	3	]	]	PUNCT
ejpam-3418	575	4	for	for	ADP
ejpam-3418	575	5	all	all	DET
ejpam-3418	575	6	ι̂	ι̂	NUM
ejpam-3418	575	7	∈	∈	NOUN
ejpam-3418	575	8	i	i	PRON
ejpam-3418	575	9	i(x	i(x	PROPN
ejpam-3418	575	10	)	)	PUNCT
ejpam-3418	575	11	;	;	PUNCT
ejpam-3418	575	12	and	and	CCONJ
ejpam-3418	575	13	(	(	PUNCT
ejpam-3418	575	14	vi	vi	NOUN
ejpam-3418	575	15	)	)	PUNCT
ejpam-3418	575	16	cl	cl	NOUN
ejpam-3418	575	17	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	575	18	]	]	X
ejpam-3418	575	19	6	6	NUM
ejpam-3418	575	20	f−1[cl	f−1[cl	NOUN
ejpam-3418	575	21	ω̂	ω̂	NUM
ejpam-3418	575	22	]	]	PUNCT
ejpam-3418	575	23	for	for	ADP
ejpam-3418	575	24	all	all	DET
ejpam-3418	575	25	ω̂	ω̂	NUM
ejpam-3418	575	26	∈	∈	PROPN
ejpam-3418	575	27	i	i	PRON
ejpam-3418	575	28	i(y	i(y	NOUN
ejpam-3418	575	29	)	)	PUNCT
ejpam-3418	575	30	,	,	PUNCT
ejpam-3418	575	31	whenever	whenever	SCONJ
ejpam-3418	575	32	f	f	PROPN
ejpam-3418	575	33	is	be	AUX
ejpam-3418	575	34	onto	onto	ADP
ejpam-3418	575	35	.	.	PUNCT
ejpam-3418	576	1	proof	proof	NOUN
ejpam-3418	576	2	.	.	PUNCT
ejpam-3418	577	1	(	(	PUNCT
ejpam-3418	577	2	i	i	NOUN
ejpam-3418	577	3	)	)	PUNCT
ejpam-3418	577	4	⇐	⇐	ADJ
ejpam-3418	577	5	⇒	⇒	PROPN
ejpam-3418	577	6	(	(	PUNCT
ejpam-3418	577	7	ii	ii	NOUN
ejpam-3418	577	8	)	)	PUNCT
ejpam-3418	577	9	suppose	suppose	VERB
ejpam-3418	577	10	that	that	SCONJ
ejpam-3418	577	11	f	f	PROPN
ejpam-3418	577	12	is	be	AUX
ejpam-3418	577	13	ivfi	ivfi	NOUN
ejpam-3418	577	14	continuous	continuous	ADJ
ejpam-3418	577	15	.	.	PUNCT
ejpam-3418	578	1	let	let	VERB
ejpam-3418	578	2	ι̂	ι̂	PRON
ejpam-3418	578	3	be	be	AUX
ejpam-3418	578	4	an	an	DET
ejpam-3418	578	5	ivfi	ivfi	NOUN
ejpam-3418	578	6	closed	close	VERB
ejpam-3418	578	7	set	set	VERB
ejpam-3418	578	8	in	in	ADP
ejpam-3418	578	9	i	i	PROPN
ejpam-3418	578	10	i(y	i(y	NOUN
ejpam-3418	578	11	)	)	PUNCT
ejpam-3418	578	12	.	.	PUNCT
ejpam-3418	579	1	then	then	ADV
ejpam-3418	579	2	ι̂c	ι̂c	PUNCT
ejpam-3418	579	3	∈	∈	PROPN
ejpam-3418	579	4	t	t	NOUN
ejpam-3418	579	5	′2	′2	X
ejpam-3418	579	6	.	.	PUNCT
ejpam-3418	580	1	since	since	SCONJ
ejpam-3418	580	2	f	f	PROPN
ejpam-3418	580	3	is	be	AUX
ejpam-3418	580	4	ivfi	ivfi	NOUN
ejpam-3418	580	5	continuous	continuous	ADJ
ejpam-3418	580	6	,	,	PUNCT
ejpam-3418	580	7	f−1	f−1	PROPN
ejpam-3418	580	8	[	[	X
ejpam-3418	580	9	̂ιc	̂ιc	X
ejpam-3418	580	10	]	]	X
ejpam-3418	580	11	∈	∈	PROPN
ejpam-3418	580	12	t	t	PROPN
ejpam-3418	580	13	′1	′1	PUNCT
ejpam-3418	580	14	.	.	PUNCT
ejpam-3418	581	1	note	note	VERB
ejpam-3418	581	2	that	that	SCONJ
ejpam-3418	581	3	by	by	ADP
ejpam-3418	581	4	theorem	theorem	NOUN
ejpam-3418	581	5	5	5	NUM
ejpam-3418	581	6	(	(	PUNCT
ejpam-3418	581	7	i	i	NOUN
ejpam-3418	581	8	)	)	PUNCT
ejpam-3418	581	9	,	,	PUNCT
ejpam-3418	581	10	f−1	f−1	PROPN
ejpam-3418	582	1	[	[	X
ejpam-3418	582	2	̂ιc	̂ιc	X
ejpam-3418	582	3	]	]	X
ejpam-3418	582	4	=	=	SYM
ejpam-3418	582	5	(	(	PUNCT
ejpam-3418	582	6	f−1	f−1	PROPN
ejpam-3418	583	1	[	[	X
ejpam-3418	583	2	̂ι])c	̂ι])c	PROPN
ejpam-3418	583	3	.	.	PUNCT
ejpam-3418	584	1	thus	thus	ADV
ejpam-3418	584	2	,	,	PUNCT
ejpam-3418	584	3	f−1	f−1	PROPN
ejpam-3418	584	4	[	[	X
ejpam-3418	584	5	̂ι	̂ι	X
ejpam-3418	584	6	]	]	PUNCT
ejpam-3418	584	7	is	be	AUX
ejpam-3418	584	8	an	an	DET
ejpam-3418	584	9	ivfi	ivfi	NOUN
ejpam-3418	584	10	closed	close	VERB
ejpam-3418	584	11	set	set	VERB
ejpam-3418	584	12	in	in	ADP
ejpam-3418	584	13	i	i	PRON
ejpam-3418	584	14	i(x	i(x	PROPN
ejpam-3418	584	15	)	)	PUNCT
ejpam-3418	584	16	.	.	PUNCT
ejpam-3418	585	1	conversely	conversely	ADV
ejpam-3418	585	2	,	,	PUNCT
ejpam-3418	585	3	suppose	suppose	VERB
ejpam-3418	585	4	that	that	SCONJ
ejpam-3418	585	5	the	the	DET
ejpam-3418	585	6	inverse	inverse	ADJ
ejpam-3418	585	7	image	image	NOUN
ejpam-3418	585	8	of	of	ADP
ejpam-3418	585	9	every	every	DET
ejpam-3418	585	10	ivfi	ivfi	NOUN
ejpam-3418	585	11	closed	close	VERB
ejpam-3418	585	12	set	set	VERB
ejpam-3418	585	13	is	be	AUX
ejpam-3418	585	14	ivfi	ivfi	NOUN
ejpam-3418	585	15	closed	closed	ADJ
ejpam-3418	585	16	.	.	PUNCT
ejpam-3418	586	1	let	let	VERB
ejpam-3418	586	2	τ̂	τ̂	VERB
ejpam-3418	586	3	∈	∈	PROPN
ejpam-3418	586	4	t	t	NOUN
ejpam-3418	586	5	′2	′2	X
ejpam-3418	586	6	.	.	PUNCT
ejpam-3418	587	1	then	then	ADV
ejpam-3418	587	2	τ̂	τ̂	PUNCT
ejpam-3418	587	3	c	c	NOUN
ejpam-3418	587	4	is	be	AUX
ejpam-3418	587	5	an	an	DET
ejpam-3418	587	6	ivfi	ivfi	NOUN
ejpam-3418	587	7	closed	close	VERB
ejpam-3418	587	8	set	set	VERB
ejpam-3418	587	9	in	in	ADP
ejpam-3418	587	10	i	i	PROPN
ejpam-3418	587	11	i(y	i(y	NOUN
ejpam-3418	587	12	)	)	PUNCT
ejpam-3418	587	13	.	.	PUNCT
ejpam-3418	588	1	by	by	ADP
ejpam-3418	588	2	assumption	assumption	NOUN
ejpam-3418	588	3	,	,	PUNCT
ejpam-3418	588	4	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	588	5	c	c	X
ejpam-3418	588	6	]	]	X
ejpam-3418	588	7	is	be	AUX
ejpam-3418	588	8	an	an	DET
ejpam-3418	588	9	ivfi	ivfi	NOUN
ejpam-3418	588	10	closed	close	VERB
ejpam-3418	588	11	set	set	VERB
ejpam-3418	588	12	in	in	ADP
ejpam-3418	588	13	i	i	PRON
ejpam-3418	588	14	i(x	i(x	PROPN
ejpam-3418	588	15	)	)	PUNCT
ejpam-3418	588	16	.	.	PUNCT
ejpam-3418	589	1	note	note	VERB
ejpam-3418	589	2	that	that	SCONJ
ejpam-3418	589	3	by	by	ADP
ejpam-3418	589	4	theorem	theorem	NOUN
ejpam-3418	589	5	5	5	NUM
ejpam-3418	589	6	(	(	PUNCT
ejpam-3418	589	7	i	i	NOUN
ejpam-3418	589	8	)	)	PUNCT
ejpam-3418	589	9	,	,	PUNCT
ejpam-3418	589	10	(	(	PUNCT
ejpam-3418	589	11	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	589	12	]	]	PUNCT
ejpam-3418	589	13	)	)	PUNCT
ejpam-3418	589	14	c	c	NOUN
ejpam-3418	589	15	=	=	PUNCT
ejpam-3418	589	16	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	589	17	c	c	X
ejpam-3418	589	18	]	]	PUNCT
ejpam-3418	589	19	.	.	PUNCT
ejpam-3418	590	1	thus	thus	ADV
ejpam-3418	590	2	,	,	PUNCT
ejpam-3418	590	3	f−1[τ̂	f−1[τ̂	NOUN
ejpam-3418	590	4	]	]	X
ejpam-3418	590	5	∈	∈	PROPN
ejpam-3418	590	6	t	t	PROPN
ejpam-3418	590	7	′1	′1	PROPN
ejpam-3418	590	8	.	.	PUNCT
ejpam-3418	591	1	hence	hence	ADV
ejpam-3418	591	2	,	,	PUNCT
ejpam-3418	591	3	f	f	PROPN
ejpam-3418	591	4	is	be	AUX
ejpam-3418	591	5	ivfi	ivfi	NOUN
ejpam-3418	591	6	continuous	continuous	ADJ
ejpam-3418	591	7	.	.	PUNCT
ejpam-3418	592	1	(	(	PUNCT
ejpam-3418	592	2	ii	ii	NOUN
ejpam-3418	592	3	)	)	PUNCT
ejpam-3418	592	4	⇐	⇐	ADJ
ejpam-3418	592	5	⇒	⇒	NOUN
ejpam-3418	592	6	(	(	PUNCT
ejpam-3418	592	7	vi	vi	X
ejpam-3418	592	8	)	)	PUNCT
ejpam-3418	592	9	suppose	suppose	VERB
ejpam-3418	592	10	that	that	SCONJ
ejpam-3418	592	11	the	the	DET
ejpam-3418	592	12	inverse	inverse	ADJ
ejpam-3418	592	13	image	image	NOUN
ejpam-3418	592	14	of	of	ADP
ejpam-3418	592	15	every	every	DET
ejpam-3418	592	16	ivfi	ivfi	NOUN
ejpam-3418	592	17	closed	close	VERB
ejpam-3418	592	18	set	set	VERB
ejpam-3418	592	19	is	be	AUX
ejpam-3418	592	20	ivfi	ivfi	NOUN
ejpam-3418	592	21	closed	closed	ADJ
ejpam-3418	592	22	.	.	PUNCT
ejpam-3418	593	1	let	let	VERB
ejpam-3418	593	2	ω̂	ω̂	PUNCT
ejpam-3418	593	3	be	be	AUX
ejpam-3418	593	4	an	an	DET
ejpam-3418	593	5	ivfi	ivfi	NOUN
ejpam-3418	593	6	closed	close	VERB
ejpam-3418	593	7	set	set	VERB
ejpam-3418	593	8	in	in	ADP
ejpam-3418	593	9	i	i	PROPN
ejpam-3418	593	10	i(y	i(y	NOUN
ejpam-3418	593	11	)	)	PUNCT
ejpam-3418	593	12	.	.	PUNCT
ejpam-3418	594	1	then	then	ADV
ejpam-3418	594	2	by	by	ADP
ejpam-3418	594	3	assumption	assumption	NOUN
ejpam-3418	594	4	,	,	PUNCT
ejpam-3418	594	5	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	594	6	]	]	PUNCT
ejpam-3418	594	7	is	be	AUX
ejpam-3418	594	8	an	an	DET
ejpam-3418	594	9	ivfi	ivfi	NOUN
ejpam-3418	594	10	closed	close	VERB
ejpam-3418	594	11	set	set	VERB
ejpam-3418	594	12	in	in	ADP
ejpam-3418	594	13	i	i	PRON
ejpam-3418	594	14	i(x	i(x	PROPN
ejpam-3418	594	15	)	)	PUNCT
ejpam-3418	594	16	.	.	PUNCT
ejpam-3418	595	1	thus	thus	ADV
ejpam-3418	595	2	,	,	PUNCT
ejpam-3418	595	3	cl	cl	INTJ
ejpam-3418	595	4	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	595	5	]	]	X
ejpam-3418	595	6	=	=	SYM
ejpam-3418	595	7	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	595	8	]	]	X
ejpam-3418	595	9	.	.	PUNCT
ejpam-3418	596	1	but	but	CCONJ
ejpam-3418	596	2	ω̂	ω̂	NUM
ejpam-3418	596	3	6	6	NUM
ejpam-3418	596	4	cl	cl	NOUN
ejpam-3418	596	5	ω̂	ω̂	PUNCT
ejpam-3418	596	6	for	for	ADP
ejpam-3418	596	7	all	all	DET
ejpam-3418	596	8	ω̂	ω̂	NUM
ejpam-3418	596	9	∈	∈	PROPN
ejpam-3418	596	10	i	i	PRON
ejpam-3418	596	11	i(y	i(y	NOUN
ejpam-3418	596	12	)	)	PUNCT
ejpam-3418	596	13	,	,	PUNCT
ejpam-3418	596	14	and	and	CCONJ
ejpam-3418	596	15	so	so	ADV
ejpam-3418	596	16	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	596	17	]	]	X
ejpam-3418	596	18	6	6	NUM
ejpam-3418	596	19	f−1[cl	f−1[cl	NOUN
ejpam-3418	596	20	ω̂	ω̂	NUM
ejpam-3418	596	21	]	]	X
ejpam-3418	596	22	.	.	PUNCT
ejpam-3418	597	1	hence	hence	ADV
ejpam-3418	597	2	,	,	PUNCT
ejpam-3418	597	3	cl	cl	INTJ
ejpam-3418	597	4	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	597	5	]	]	X
ejpam-3418	597	6	6	6	NUM
ejpam-3418	597	7	f−1[cl	f−1[cl	NOUN
ejpam-3418	597	8	ω̂	ω̂	NUM
ejpam-3418	597	9	]	]	PUNCT
ejpam-3418	597	10	for	for	ADP
ejpam-3418	597	11	all	all	DET
ejpam-3418	597	12	ω̂	ω̂	NUM
ejpam-3418	597	13	∈	∈	PROPN
ejpam-3418	597	14	i	i	PRON
ejpam-3418	597	15	i(y	i(y	NOUN
ejpam-3418	597	16	)	)	PUNCT
ejpam-3418	597	17	.	.	PUNCT
ejpam-3418	598	1	conversely	conversely	ADV
ejpam-3418	598	2	,	,	PUNCT
ejpam-3418	598	3	suppose	suppose	VERB
ejpam-3418	598	4	that	that	SCONJ
ejpam-3418	598	5	cl	cl	NOUN
ejpam-3418	598	6	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	598	7	]	]	X
ejpam-3418	598	8	6	6	NUM
ejpam-3418	598	9	f−1[cl	f−1[cl	NOUN
ejpam-3418	598	10	ω̂	ω̂	NUM
ejpam-3418	598	11	]	]	PUNCT
ejpam-3418	598	12	for	for	ADP
ejpam-3418	598	13	all	all	DET
ejpam-3418	598	14	ω̂	ω̂	NUM
ejpam-3418	598	15	∈	∈	PROPN
ejpam-3418	598	16	i	i	PRON
ejpam-3418	598	17	i(y	i(y	VERB
ejpam-3418	598	18	)	)	PUNCT
ejpam-3418	598	19	whenever	whenever	SCONJ
ejpam-3418	598	20	f	f	PROPN
ejpam-3418	598	21	is	be	AUX
ejpam-3418	598	22	onto	onto	ADP
ejpam-3418	598	23	.	.	PUNCT
ejpam-3418	599	1	let	let	VERB
ejpam-3418	599	2	ω̂	ω̂	PUNCT
ejpam-3418	599	3	be	be	AUX
ejpam-3418	599	4	an	an	DET
ejpam-3418	599	5	ivfi	ivfi	NOUN
ejpam-3418	599	6	closed	close	VERB
ejpam-3418	599	7	set	set	VERB
ejpam-3418	599	8	in	in	ADP
ejpam-3418	599	9	i	i	PROPN
ejpam-3418	599	10	i(y	i(y	NOUN
ejpam-3418	599	11	)	)	PUNCT
ejpam-3418	599	12	.	.	PUNCT
ejpam-3418	600	1	then	then	ADV
ejpam-3418	600	2	by	by	ADP
ejpam-3418	600	3	assumption	assumption	NOUN
ejpam-3418	600	4	,	,	PUNCT
ejpam-3418	600	5	cl	cl	NOUN
ejpam-3418	600	6	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	600	7	]	]	X
ejpam-3418	600	8	6	6	NUM
ejpam-3418	600	9	f−1[cl	f−1[cl	NOUN
ejpam-3418	600	10	ω̂	ω̂	NUM
ejpam-3418	600	11	]	]	PUNCT
ejpam-3418	600	12	.	.	PUNCT
ejpam-3418	601	1	since	since	SCONJ
ejpam-3418	601	2	ω̂	ω̂	NUM
ejpam-3418	601	3	be	be	AUX
ejpam-3418	601	4	an	an	DET
ejpam-3418	601	5	ivfi	ivfi	NOUN
ejpam-3418	601	6	closed	close	VERB
ejpam-3418	601	7	set	set	VERB
ejpam-3418	601	8	in	in	ADP
ejpam-3418	601	9	i	i	PROPN
ejpam-3418	601	10	i(y	i(y	NOUN
ejpam-3418	601	11	)	)	PUNCT
ejpam-3418	601	12	,	,	PUNCT
ejpam-3418	601	13	cl	cl	INTJ
ejpam-3418	601	14	ω̂	ω̂	PUNCT
ejpam-3418	601	15	=	=	SYM
ejpam-3418	601	16	ω̂	ω̂	NUM
ejpam-3418	601	17	,	,	PUNCT
ejpam-3418	601	18	and	and	CCONJ
ejpam-3418	601	19	so	so	ADV
ejpam-3418	601	20	f−1[cl	f−1[cl	PROPN
ejpam-3418	601	21	ω̂	ω̂	PROPN
ejpam-3418	601	22	]	]	X
ejpam-3418	601	23	=	=	SYM
ejpam-3418	601	24	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	601	25	]	]	X
ejpam-3418	601	26	.	.	PUNCT
ejpam-3418	602	1	but	but	CCONJ
ejpam-3418	602	2	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	602	3	]	]	X
ejpam-3418	602	4	6	6	NUM
ejpam-3418	602	5	cl	cl	NOUN
ejpam-3418	602	6	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	602	7	]	]	X
ejpam-3418	602	8	,	,	PUNCT
ejpam-3418	602	9	and	and	CCONJ
ejpam-3418	602	10	so	so	ADV
ejpam-3418	602	11	cl	cl	VERB
ejpam-3418	602	12	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	602	13	]	]	X
ejpam-3418	602	14	6	6	NUM
ejpam-3418	602	15	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	602	16	]	]	PUNCT
ejpam-3418	602	17	6	6	NUM
ejpam-3418	602	18	cl	cl	NOUN
ejpam-3418	602	19	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	602	20	]	]	X
ejpam-3418	602	21	.	.	PUNCT
ejpam-3418	603	1	thus	thus	ADV
ejpam-3418	603	2	,	,	PUNCT
ejpam-3418	603	3	cl	cl	INTJ
ejpam-3418	603	4	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	603	5	]	]	X
ejpam-3418	603	6	=	=	SYM
ejpam-3418	603	7	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	603	8	]	]	X
ejpam-3418	603	9	.	.	PUNCT
ejpam-3418	604	1	hence	hence	ADV
ejpam-3418	604	2	,	,	PUNCT
ejpam-3418	604	3	f−1[ω̂	f−1[ω̂	PROPN
ejpam-3418	604	4	]	]	PUNCT
ejpam-3418	604	5	is	be	AUX
ejpam-3418	604	6	an	an	DET
ejpam-3418	604	7	ivfi	ivfi	NOUN
ejpam-3418	604	8	closed	close	VERB
ejpam-3418	604	9	set	set	VERB
ejpam-3418	604	10	in	in	ADP
ejpam-3418	604	11	i	i	PRON
ejpam-3418	604	12	i(x	i(x	PROPN
ejpam-3418	604	13	)	)	PUNCT
ejpam-3418	604	14	.	.	PUNCT
ejpam-3418	605	1	(	(	PUNCT
ejpam-3418	605	2	vi	vi	NOUN
ejpam-3418	605	3	)	)	PUNCT
ejpam-3418	605	4	⇐	⇐	ADJ
ejpam-3418	605	5	⇒	⇒	NOUN
ejpam-3418	605	6	(	(	PUNCT
ejpam-3418	605	7	v	v	NOUN
ejpam-3418	605	8	)	)	PUNCT
ejpam-3418	605	9	suppose	suppose	VERB
ejpam-3418	605	10	that	that	SCONJ
ejpam-3418	605	11	cl	cl	NOUN
ejpam-3418	605	12	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	605	13	]	]	X
ejpam-3418	605	14	6	6	NUM
ejpam-3418	605	15	f−1[cl	f−1[cl	NOUN
ejpam-3418	605	16	ω̂	ω̂	NUM
ejpam-3418	605	17	]	]	PUNCT
ejpam-3418	605	18	for	for	ADP
ejpam-3418	605	19	all	all	DET
ejpam-3418	605	20	ω̂	ω̂	NUM
ejpam-3418	605	21	∈	∈	PROPN
ejpam-3418	605	22	i	i	PRON
ejpam-3418	605	23	i(y	i(y	VERB
ejpam-3418	605	24	)	)	PUNCT
ejpam-3418	605	25	whenever	whenever	SCONJ
ejpam-3418	605	26	f	f	PROPN
ejpam-3418	605	27	is	be	AUX
ejpam-3418	605	28	onto	onto	ADP
ejpam-3418	605	29	.	.	PUNCT
ejpam-3418	606	1	let	let	VERB
ejpam-3418	606	2	ι̂	ι̂	PRON
ejpam-3418	606	3	∈	∈	VERB
ejpam-3418	606	4	i	i	PRON
ejpam-3418	606	5	i(x	i(x	PROPN
ejpam-3418	606	6	)	)	PUNCT
ejpam-3418	606	7	and	and	CCONJ
ejpam-3418	606	8	put	put	VERB
ejpam-3418	606	9	ω̂	ω̂	PUNCT
ejpam-3418	606	10	=	=	SYM
ejpam-3418	606	11	f	f	PROPN
ejpam-3418	607	1	[	[	X
ejpam-3418	607	2	̂ι	̂ι	X
ejpam-3418	607	3	]	]	X
ejpam-3418	607	4	∈	∈	PROPN
ejpam-3418	607	5	i	i	PRON
ejpam-3418	607	6	i(y	i(y	NOUN
ejpam-3418	607	7	)	)	PUNCT
ejpam-3418	607	8	.	.	PUNCT
ejpam-3418	608	1	then	then	ADV
ejpam-3418	608	2	by	by	ADP
ejpam-3418	608	3	assumption	assumption	NOUN
ejpam-3418	608	4	,	,	PUNCT
ejpam-3418	608	5	cl	cl	NOUN
ejpam-3418	608	6	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	608	7	]	]	X
ejpam-3418	608	8	6	6	NUM
ejpam-3418	608	9	f−1[cl	f−1[cl	NOUN
ejpam-3418	608	10	ω̂	ω̂	PROPN
ejpam-3418	608	11	]	]	PUNCT
ejpam-3418	608	12	,	,	PUNCT
ejpam-3418	608	13	that	that	PRON
ejpam-3418	608	14	is	be	AUX
ejpam-3418	608	15	cl	cl	NOUN
ejpam-3418	608	16	f−1[f	f−1[f	VERB
ejpam-3418	609	1	[	[	X
ejpam-3418	609	2	̂ι	̂ι	X
ejpam-3418	609	3	]	]	X
ejpam-3418	609	4	]	]	X
ejpam-3418	609	5	6	6	NUM
ejpam-3418	609	6	f−1[cl	f−1[cl	NOUN
ejpam-3418	609	7	f	f	PROPN
ejpam-3418	610	1	[	[	X
ejpam-3418	610	2	̂ι	̂ι	X
ejpam-3418	610	3	]	]	X
ejpam-3418	610	4	]	]	PUNCT
ejpam-3418	610	5	.	.	PUNCT
ejpam-3418	611	1	note	note	VERB
ejpam-3418	611	2	that	that	SCONJ
ejpam-3418	611	3	by	by	ADP
ejpam-3418	611	4	theorem	theorem	NOUN
ejpam-3418	611	5	5	5	NUM
ejpam-3418	611	6	(	(	PUNCT
ejpam-3418	611	7	vi	vi	NOUN
ejpam-3418	611	8	)	)	PUNCT
ejpam-3418	611	9	,	,	PUNCT
ejpam-3418	611	10	ι̂	ι̂	NUM
ejpam-3418	611	11	6	6	NUM
ejpam-3418	611	12	f−1[f	f−1[f	VERB
ejpam-3418	611	13	[	[	X
ejpam-3418	611	14	̂ι	̂ι	X
ejpam-3418	611	15	]	]	X
ejpam-3418	611	16	]	]	X
ejpam-3418	611	17	,	,	PUNCT
ejpam-3418	611	18	and	and	CCONJ
ejpam-3418	611	19	so	so	ADV
ejpam-3418	611	20	cl	cl	NOUN
ejpam-3418	611	21	ι̂	ι̂	NUM
ejpam-3418	611	22	6	6	NUM
ejpam-3418	611	23	cl	cl	NOUN
ejpam-3418	611	24	f−1[f	f−1[f	NOUN
ejpam-3418	611	25	[	[	X
ejpam-3418	611	26	̂ι	̂ι	X
ejpam-3418	611	27	]	]	X
ejpam-3418	611	28	]	]	PUNCT
ejpam-3418	611	29	.	.	PUNCT
ejpam-3418	612	1	thus	thus	ADV
ejpam-3418	612	2	,	,	PUNCT
ejpam-3418	612	3	cl	cl	INTJ
ejpam-3418	612	4	ι̂	ι̂	NUM
ejpam-3418	612	5	6	6	NUM
ejpam-3418	612	6	f−1[cl	f−1[cl	NOUN
ejpam-3418	612	7	f	f	PROPN
ejpam-3418	613	1	[	[	X
ejpam-3418	613	2	̂ι	̂ι	X
ejpam-3418	613	3	]	]	X
ejpam-3418	613	4	]	]	PUNCT
ejpam-3418	613	5	.	.	PUNCT
ejpam-3418	614	1	taking	take	VERB
ejpam-3418	614	2	the	the	DET
ejpam-3418	614	3	images	image	NOUN
ejpam-3418	614	4	,	,	PUNCT
ejpam-3418	614	5	we	we	PRON
ejpam-3418	614	6	have	have	VERB
ejpam-3418	614	7	f	f	PRON
ejpam-3418	615	1	[	[	X
ejpam-3418	615	2	cl	cl	NOUN
ejpam-3418	615	3	ι̂	ι̂	X
ejpam-3418	615	4	]	]	X
ejpam-3418	615	5	6	6	NUM
ejpam-3418	615	6	f	f	NOUN
ejpam-3418	615	7	[	[	X
ejpam-3418	615	8	f−1[cl	f−1[cl	X
ejpam-3418	615	9	f	f	X
ejpam-3418	616	1	[	[	X
ejpam-3418	616	2	̂ι	̂ι	X
ejpam-3418	616	3	]	]	X
ejpam-3418	616	4	]	]	X
ejpam-3418	616	5	]	]	PUNCT
ejpam-3418	616	6	.	.	PUNCT
ejpam-3418	617	1	since	since	SCONJ
ejpam-3418	617	2	f	f	PROPN
ejpam-3418	617	3	is	be	AUX
ejpam-3418	617	4	onto	onto	ADP
ejpam-3418	617	5	,	,	PUNCT
ejpam-3418	617	6	by	by	ADP
ejpam-3418	617	7	theorem	theorem	NOUN
ejpam-3418	617	8	5	5	NUM
ejpam-3418	617	9	(	(	PUNCT
ejpam-3418	617	10	v	v	NOUN
ejpam-3418	617	11	)	)	PUNCT
ejpam-3418	617	12	,	,	PUNCT
ejpam-3418	617	13	f	f	PROPN
ejpam-3418	618	1	[	[	X
ejpam-3418	618	2	f−1[cl	f−1[cl	X
ejpam-3418	618	3	f	f	X
ejpam-3418	619	1	[	[	X
ejpam-3418	619	2	̂ι	̂ι	X
ejpam-3418	619	3	]	]	X
ejpam-3418	619	4	]	]	X
ejpam-3418	619	5	]	]	X
ejpam-3418	619	6	=	=	PUNCT
ejpam-3418	619	7	cl	cl	NOUN
ejpam-3418	619	8	f	f	PROPN
ejpam-3418	620	1	[	[	X
ejpam-3418	620	2	̂ι	̂ι	X
ejpam-3418	620	3	]	]	X
ejpam-3418	620	4	.	.	PUNCT
ejpam-3418	621	1	hence	hence	ADV
ejpam-3418	621	2	,	,	PUNCT
ejpam-3418	621	3	f	f	PROPN
ejpam-3418	622	1	[	[	X
ejpam-3418	622	2	cl	cl	X
ejpam-3418	622	3	ι̂	ι̂	X
ejpam-3418	622	4	]	]	X
ejpam-3418	622	5	6	6	NUM
ejpam-3418	622	6	cl	cl	NOUN
ejpam-3418	622	7	f	f	PROPN
ejpam-3418	623	1	[	[	X
ejpam-3418	623	2	̂ι	̂ι	X
ejpam-3418	623	3	]	]	X
ejpam-3418	623	4	.	.	PUNCT
ejpam-3418	624	1	conversely	conversely	ADV
ejpam-3418	624	2	,	,	PUNCT
ejpam-3418	624	3	suppose	suppose	VERB
ejpam-3418	624	4	that	that	SCONJ
ejpam-3418	624	5	f	f	PROPN
ejpam-3418	625	1	[	[	X
ejpam-3418	625	2	cl	cl	X
ejpam-3418	625	3	ι̂	ι̂	X
ejpam-3418	625	4	]	]	X
ejpam-3418	625	5	6	6	NUM
ejpam-3418	625	6	cl	cl	NOUN
ejpam-3418	625	7	f	f	PROPN
ejpam-3418	626	1	[	[	X
ejpam-3418	626	2	̂ι	̂ι	X
ejpam-3418	626	3	]	]	PUNCT
ejpam-3418	626	4	for	for	ADP
ejpam-3418	626	5	all	all	DET
ejpam-3418	626	6	ι̂	ι̂	NUM
ejpam-3418	626	7	∈	∈	NOUN
ejpam-3418	626	8	i	i	PRON
ejpam-3418	626	9	i(x	i(x	PROPN
ejpam-3418	626	10	)	)	PUNCT
ejpam-3418	626	11	.	.	PUNCT
ejpam-3418	627	1	let	let	VERB
ejpam-3418	627	2	ι̂	ι̂	PRON
ejpam-3418	627	3	∈	∈	VERB
ejpam-3418	627	4	i	i	PRON
ejpam-3418	627	5	i(y	i(y	VERB
ejpam-3418	627	6	)	)	PUNCT
ejpam-3418	627	7	and	and	CCONJ
ejpam-3418	627	8	put	put	VERB
ejpam-3418	627	9	ι̂	ι̂	NOUN
ejpam-3418	627	10	=	=	PUNCT
ejpam-3418	627	11	f−1[ω̂	f−1[ω̂	X
ejpam-3418	627	12	]	]	X
ejpam-3418	627	13	∈	∈	PROPN
ejpam-3418	628	1	i	i	PRON
ejpam-3418	628	2	i(x	i(x	PROPN
ejpam-3418	628	3	)	)	PUNCT
ejpam-3418	628	4	.	.	PUNCT
ejpam-3418	629	1	then	then	ADV
ejpam-3418	629	2	by	by	ADP
ejpam-3418	629	3	assumption	assumption	NOUN
ejpam-3418	629	4	,	,	PUNCT
ejpam-3418	629	5	f	f	PROPN
ejpam-3418	630	1	[	[	X
ejpam-3418	630	2	cl	cl	X
ejpam-3418	630	3	ι̂	ι̂	X
ejpam-3418	630	4	]	]	X
ejpam-3418	630	5	6	6	NUM
ejpam-3418	630	6	cl	cl	NOUN
ejpam-3418	630	7	f	f	PROPN
ejpam-3418	631	1	[	[	X
ejpam-3418	631	2	̂ι	̂ι	X
ejpam-3418	631	3	]	]	X
ejpam-3418	631	4	,	,	PUNCT
ejpam-3418	631	5	that	that	PRON
ejpam-3418	631	6	is	be	AUX
ejpam-3418	631	7	f	f	PROPN
ejpam-3418	632	1	[	[	X
ejpam-3418	632	2	cl	cl	INTJ
ejpam-3418	632	3	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	632	4	]	]	X
ejpam-3418	632	5	]	]	X
ejpam-3418	632	6	6	6	NUM
ejpam-3418	632	7	references	reference	NOUN
ejpam-3418	632	8	569	569	NUM
ejpam-3418	632	9	cl	cl	NOUN
ejpam-3418	632	10	f	f	X
ejpam-3418	633	1	[	[	X
ejpam-3418	633	2	f−1[ω̂	f−1[ω̂	X
ejpam-3418	633	3	]	]	X
ejpam-3418	633	4	]	]	PUNCT
ejpam-3418	633	5	.	.	PUNCT
ejpam-3418	634	1	let	let	VERB
ejpam-3418	634	2	f	f	PRON
ejpam-3418	634	3	be	be	AUX
ejpam-3418	634	4	an	an	PRON
ejpam-3418	634	5	onto	onto	ADP
ejpam-3418	634	6	function	function	NOUN
ejpam-3418	634	7	.	.	PUNCT
ejpam-3418	635	1	note	note	VERB
ejpam-3418	635	2	that	that	SCONJ
ejpam-3418	635	3	by	by	ADP
ejpam-3418	635	4	theorem	theorem	NOUN
ejpam-3418	635	5	5	5	NUM
ejpam-3418	635	6	(	(	PUNCT
ejpam-3418	635	7	v	v	NOUN
ejpam-3418	635	8	)	)	PUNCT
ejpam-3418	635	9	,	,	PUNCT
ejpam-3418	635	10	f	f	X
ejpam-3418	636	1	[	[	X
ejpam-3418	636	2	f−1[ω̂	f−1[ω̂	X
ejpam-3418	636	3	]	]	X
ejpam-3418	636	4	]	]	X
ejpam-3418	636	5	=	=	SYM
ejpam-3418	636	6	ω̂	ω̂	PROPN
ejpam-3418	636	7	,	,	PUNCT
ejpam-3418	636	8	and	and	CCONJ
ejpam-3418	636	9	so	so	ADV
ejpam-3418	636	10	cl	cl	NOUN
ejpam-3418	636	11	f	f	X
ejpam-3418	637	1	[	[	X
ejpam-3418	637	2	f−1[ω̂	f−1[ω̂	ADJ
ejpam-3418	637	3	]	]	X
ejpam-3418	637	4	]	]	X
ejpam-3418	637	5	=	=	SYM
ejpam-3418	637	6	cl	cl	NOUN
ejpam-3418	637	7	ω̂.	ω̂.	NOUN
ejpam-3418	637	8	thus	thus	ADV
ejpam-3418	637	9	,	,	PUNCT
ejpam-3418	637	10	f	f	PROPN
ejpam-3418	638	1	[	[	X
ejpam-3418	638	2	cl	cl	INTJ
ejpam-3418	638	3	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	638	4	]	]	X
ejpam-3418	638	5	]	]	X
ejpam-3418	638	6	6	6	NUM
ejpam-3418	638	7	cl	cl	NOUN
ejpam-3418	638	8	ω̂.	ω̂.	NOUN
ejpam-3418	638	9	taking	take	VERB
ejpam-3418	638	10	the	the	DET
ejpam-3418	638	11	inverse	inverse	NOUN
ejpam-3418	638	12	images	image	NOUN
ejpam-3418	638	13	,	,	PUNCT
ejpam-3418	638	14	we	we	PRON
ejpam-3418	638	15	have	have	AUX
ejpam-3418	638	16	f−1[f	f−1[f	VERB
ejpam-3418	638	17	[	[	X
ejpam-3418	638	18	cl	cl	NOUN
ejpam-3418	638	19	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	638	20	]	]	X
ejpam-3418	638	21	]	]	X
ejpam-3418	638	22	]	]	X
ejpam-3418	638	23	6	6	NUM
ejpam-3418	638	24	f−1[cl	f−1[cl	NOUN
ejpam-3418	638	25	ω̂	ω̂	NUM
ejpam-3418	638	26	]	]	X
ejpam-3418	638	27	.	.	PUNCT
ejpam-3418	639	1	by	by	ADP
ejpam-3418	639	2	theorem	theorem	NOUN
ejpam-3418	639	3	5	5	NUM
ejpam-3418	639	4	(	(	PUNCT
ejpam-3418	639	5	vi	vi	NOUN
ejpam-3418	639	6	)	)	PUNCT
ejpam-3418	639	7	,	,	PUNCT
ejpam-3418	639	8	we	we	PRON
ejpam-3418	639	9	have	have	VERB
ejpam-3418	639	10	cl	cl	NOUN
ejpam-3418	639	11	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	639	12	]	]	X
ejpam-3418	639	13	6	6	NUM
ejpam-3418	639	14	f−1[f	f−1[f	NOUN
ejpam-3418	639	15	[	[	X
ejpam-3418	639	16	cl	cl	NOUN
ejpam-3418	639	17	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	639	18	]	]	X
ejpam-3418	639	19	]	]	X
ejpam-3418	639	20	]	]	PUNCT
ejpam-3418	639	21	.	.	PUNCT
ejpam-3418	640	1	thus	thus	ADV
ejpam-3418	640	2	,	,	PUNCT
ejpam-3418	640	3	cl	cl	INTJ
ejpam-3418	640	4	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	640	5	]	]	X
ejpam-3418	640	6	6	6	NUM
ejpam-3418	640	7	f−1[cl	f−1[cl	NOUN
ejpam-3418	640	8	ω̂	ω̂	PROPN
ejpam-3418	640	9	]	]	X
ejpam-3418	640	10	.	.	PUNCT
ejpam-3418	641	1	(	(	PUNCT
ejpam-3418	641	2	iii	iii	NOUN
ejpam-3418	641	3	)	)	PUNCT
ejpam-3418	641	4	⇐	⇐	ADJ
ejpam-3418	641	5	⇒	⇒	PROPN
ejpam-3418	641	6	(	(	PUNCT
ejpam-3418	641	7	iv	iv	X
ejpam-3418	641	8	)	)	PUNCT
ejpam-3418	641	9	suppose	suppose	VERB
ejpam-3418	641	10	that	that	SCONJ
ejpam-3418	641	11	for	for	ADP
ejpam-3418	641	12	each	each	DET
ejpam-3418	641	13	ivfi	ivfi	NOUN
ejpam-3418	641	14	point	point	NOUN
ejpam-3418	641	15	pα	pα	NOUN
ejpam-3418	641	16	,	,	PUNCT
ejpam-3418	641	17	b	b	NOUN
ejpam-3418	641	18	,	,	PUNCT
ejpam-3418	641	19	the	the	DET
ejpam-3418	641	20	inverse	inverse	NOUN
ejpam-3418	641	21	of	of	ADP
ejpam-3418	641	22	every	every	DET
ejpam-3418	641	23	neighborhood	neighborhood	NOUN
ejpam-3418	641	24	of	of	ADP
ejpam-3418	641	25	f	f	PROPN
ejpam-3418	642	1	[	[	X
ejpam-3418	642	2	pα	pα	NOUN
ejpam-3418	642	3	,	,	PUNCT
ejpam-3418	642	4	b	b	NOUN
ejpam-3418	642	5	]	]	X
ejpam-3418	642	6	under	under	ADP
ejpam-3418	642	7	f	f	PROPN
ejpam-3418	642	8	is	be	AUX
ejpam-3418	642	9	a	a	DET
ejpam-3418	642	10	neighborhood	neighborhood	NOUN
ejpam-3418	642	11	of	of	ADP
ejpam-3418	642	12	pα	pα	PROPN
ejpam-3418	642	13	,	,	PUNCT
ejpam-3418	642	14	b.	b.	PROPN
ejpam-3418	642	15	let	let	AUX
ejpam-3418	642	16	pα	pα	VERB
ejpam-3418	642	17	,	,	PUNCT
ejpam-3418	642	18	b	b	PROPN
ejpam-3418	642	19	be	be	AUX
ejpam-3418	642	20	an	an	DET
ejpam-3418	642	21	ivfi	ivfi	NOUN
ejpam-3418	642	22	point	point	NOUN
ejpam-3418	642	23	in	in	ADP
ejpam-3418	642	24	i	i	PRON
ejpam-3418	642	25	i(x	i(x	NOUN
ejpam-3418	642	26	)	)	PUNCT
ejpam-3418	642	27	and	and	CCONJ
ejpam-3418	642	28	η̂	η̂	PROPN
ejpam-3418	642	29	be	be	AUX
ejpam-3418	642	30	a	a	DET
ejpam-3418	642	31	neighborhood	neighborhood	NOUN
ejpam-3418	642	32	of	of	ADP
ejpam-3418	642	33	f	f	PROPN
ejpam-3418	643	1	[	[	X
ejpam-3418	643	2	pα	pα	NOUN
ejpam-3418	643	3	,	,	PUNCT
ejpam-3418	643	4	b	b	NOUN
ejpam-3418	643	5	]	]	X
ejpam-3418	643	6	.	.	PUNCT
ejpam-3418	644	1	then	then	ADV
ejpam-3418	644	2	by	by	ADP
ejpam-3418	644	3	assumption	assumption	NOUN
ejpam-3418	644	4	,	,	PUNCT
ejpam-3418	644	5	there	there	PRON
ejpam-3418	644	6	is	be	VERB
ejpam-3418	644	7	a	a	DET
ejpam-3418	644	8	neighborhood	neighborhood	NOUN
ejpam-3418	644	9	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	644	10	]	]	PUNCT
ejpam-3418	644	11	of	of	ADP
ejpam-3418	644	12	pα	pα	PROPN
ejpam-3418	644	13	,	,	PUNCT
ejpam-3418	644	14	b.	b.	PROPN
ejpam-3418	644	15	take	take	VERB
ejpam-3418	644	16	η̂′	η̂′	PROPN
ejpam-3418	644	17	=	=	PUNCT
ejpam-3418	644	18	f−1[η̂	f−1[η̂	PROPN
ejpam-3418	644	19	]	]	PUNCT
ejpam-3418	644	20	.	.	PUNCT
ejpam-3418	645	1	thus	thus	ADV
ejpam-3418	645	2	,	,	PUNCT
ejpam-3418	645	3	f	f	PROPN
ejpam-3418	646	1	[	[	X
ejpam-3418	646	2	η̂′	η̂′	X
ejpam-3418	646	3	]	]	X
ejpam-3418	646	4	=	=	SYM
ejpam-3418	646	5	f	f	X
ejpam-3418	647	1	[	[	X
ejpam-3418	647	2	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	647	3	]	]	X
ejpam-3418	647	4	]	]	X
ejpam-3418	647	5	=	=	PUNCT
ejpam-3418	647	6	η̂	η̂	X
ejpam-3418	647	7	whenever	whenever	SCONJ
ejpam-3418	647	8	f	f	PROPN
ejpam-3418	647	9	is	be	AUX
ejpam-3418	647	10	onto	onto	ADP
ejpam-3418	647	11	.	.	PUNCT
ejpam-3418	648	1	conversely	conversely	ADV
ejpam-3418	648	2	,	,	PUNCT
ejpam-3418	648	3	suppose	suppose	VERB
ejpam-3418	648	4	that	that	SCONJ
ejpam-3418	648	5	for	for	SCONJ
ejpam-3418	648	6	each	each	DET
ejpam-3418	648	7	ivfi	ivfi	NOUN
ejpam-3418	648	8	point	point	NOUN
ejpam-3418	648	9	pα	pα	NOUN
ejpam-3418	648	10	,	,	PUNCT
ejpam-3418	648	11	b	b	NOUN
ejpam-3418	648	12	and	and	CCONJ
ejpam-3418	648	13	each	each	DET
ejpam-3418	648	14	neighborhood	neighborhood	NOUN
ejpam-3418	648	15	η̂	η̂	NUM
ejpam-3418	648	16	of	of	ADP
ejpam-3418	648	17	f	f	PROPN
ejpam-3418	649	1	[	[	X
ejpam-3418	649	2	pα	pα	NOUN
ejpam-3418	649	3	,	,	PUNCT
ejpam-3418	649	4	b	b	NOUN
ejpam-3418	649	5	]	]	X
ejpam-3418	649	6	,	,	PUNCT
ejpam-3418	649	7	there	there	PRON
ejpam-3418	649	8	is	be	VERB
ejpam-3418	649	9	a	a	DET
ejpam-3418	649	10	neighborhood	neighborhood	NOUN
ejpam-3418	649	11	η̂′	η̂′	PROPN
ejpam-3418	649	12	of	of	ADP
ejpam-3418	649	13	pα	pα	PROPN
ejpam-3418	649	14	,	,	PUNCT
ejpam-3418	649	15	b	b	NOUN
ejpam-3418	649	16	such	such	ADJ
ejpam-3418	650	1	that	that	SCONJ
ejpam-3418	650	2	f	f	PROPN
ejpam-3418	651	1	[	[	X
ejpam-3418	651	2	η̂′	η̂′	X
ejpam-3418	651	3	]	]	X
ejpam-3418	651	4	=	=	PUNCT
ejpam-3418	651	5	η̂	η̂	X
ejpam-3418	651	6	whenever	whenever	SCONJ
ejpam-3418	651	7	f	f	PROPN
ejpam-3418	651	8	is	be	AUX
ejpam-3418	651	9	onto	onto	ADP
ejpam-3418	651	10	.	.	PUNCT
ejpam-3418	652	1	let	let	AUX
ejpam-3418	652	2	pα	pα	VERB
ejpam-3418	652	3	,	,	PUNCT
ejpam-3418	652	4	b	b	NOUN
ejpam-3418	652	5	be	be	AUX
ejpam-3418	652	6	an	an	DET
ejpam-3418	652	7	ivfi	ivfi	NOUN
ejpam-3418	652	8	point	point	NOUN
ejpam-3418	652	9	in	in	ADP
ejpam-3418	652	10	i	i	PRON
ejpam-3418	652	11	i(x	i(x	NOUN
ejpam-3418	652	12	)	)	PUNCT
ejpam-3418	652	13	and	and	CCONJ
ejpam-3418	652	14	η̂	η̂	PROPN
ejpam-3418	652	15	be	be	AUX
ejpam-3418	652	16	a	a	DET
ejpam-3418	652	17	neighborhood	neighborhood	NOUN
ejpam-3418	652	18	of	of	ADP
ejpam-3418	652	19	f	f	PROPN
ejpam-3418	653	1	[	[	X
ejpam-3418	653	2	pα	pα	NOUN
ejpam-3418	653	3	,	,	PUNCT
ejpam-3418	653	4	b	b	NOUN
ejpam-3418	653	5	]	]	X
ejpam-3418	653	6	.	.	PUNCT
ejpam-3418	654	1	then	then	ADV
ejpam-3418	654	2	by	by	ADP
ejpam-3418	654	3	assumption	assumption	NOUN
ejpam-3418	654	4	,	,	PUNCT
ejpam-3418	654	5	there	there	PRON
ejpam-3418	654	6	is	be	VERB
ejpam-3418	654	7	a	a	DET
ejpam-3418	654	8	neighborhood	neighborhood	NOUN
ejpam-3418	654	9	η̂′	η̂′	PROPN
ejpam-3418	654	10	of	of	ADP
ejpam-3418	654	11	pα	pα	PROPN
ejpam-3418	654	12	,	,	PUNCT
ejpam-3418	654	13	b	b	NOUN
ejpam-3418	654	14	such	such	ADJ
ejpam-3418	655	1	that	that	SCONJ
ejpam-3418	655	2	f	f	PROPN
ejpam-3418	656	1	[	[	X
ejpam-3418	656	2	η̂′	η̂′	X
ejpam-3418	656	3	]	]	X
ejpam-3418	656	4	=	=	SYM
ejpam-3418	656	5	η̂.	η̂.	NOUN
ejpam-3418	656	6	but	but	CCONJ
ejpam-3418	656	7	note	note	VERB
ejpam-3418	656	8	that	that	SCONJ
ejpam-3418	656	9	by	by	ADP
ejpam-3418	656	10	theorem	theorem	NOUN
ejpam-3418	656	11	5	5	NUM
ejpam-3418	656	12	(	(	PUNCT
ejpam-3418	656	13	v	v	NOUN
ejpam-3418	656	14	)	)	PUNCT
ejpam-3418	656	15	,	,	PUNCT
ejpam-3418	656	16	f	f	X
ejpam-3418	657	1	[	[	X
ejpam-3418	657	2	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	657	3	]	]	X
ejpam-3418	657	4	]	]	X
ejpam-3418	657	5	=	=	PUNCT
ejpam-3418	657	6	η̂	η̂	X
ejpam-3418	657	7	whenever	whenever	SCONJ
ejpam-3418	657	8	f	f	PROPN
ejpam-3418	657	9	is	be	AUX
ejpam-3418	657	10	onto	onto	ADP
ejpam-3418	657	11	.	.	PUNCT
ejpam-3418	658	1	take	take	VERB
ejpam-3418	658	2	η̂′	η̂′	NOUN
ejpam-3418	658	3	=	=	SYM
ejpam-3418	658	4	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	658	5	]	]	PUNCT
ejpam-3418	658	6	so	so	SCONJ
ejpam-3418	658	7	that	that	SCONJ
ejpam-3418	658	8	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	658	9	]	]	PUNCT
ejpam-3418	658	10	is	be	AUX
ejpam-3418	658	11	a	a	DET
ejpam-3418	658	12	neighborhood	neighborhood	NOUN
ejpam-3418	658	13	of	of	ADP
ejpam-3418	658	14	pα	pα	PROPN
ejpam-3418	658	15	,	,	PUNCT
ejpam-3418	658	16	b.	b.	PROPN
ejpam-3418	658	17	(	(	PUNCT
ejpam-3418	658	18	iv	iv	X
ejpam-3418	658	19	)	)	PUNCT
ejpam-3418	658	20	⇐	⇐	ADJ
ejpam-3418	658	21	⇒	⇒	NOUN
ejpam-3418	658	22	(	(	PUNCT
ejpam-3418	658	23	i	i	NOUN
ejpam-3418	658	24	)	)	PUNCT
ejpam-3418	659	1	suppose	suppose	VERB
ejpam-3418	659	2	that	that	SCONJ
ejpam-3418	659	3	for	for	ADP
ejpam-3418	659	4	each	each	DET
ejpam-3418	659	5	ivfi	ivfi	NOUN
ejpam-3418	659	6	point	point	NOUN
ejpam-3418	659	7	pα	pα	NOUN
ejpam-3418	659	8	,	,	PUNCT
ejpam-3418	659	9	b	b	NOUN
ejpam-3418	659	10	and	and	CCONJ
ejpam-3418	659	11	each	each	DET
ejpam-3418	659	12	neighborhood	neighborhood	NOUN
ejpam-3418	659	13	η̂	η̂	NUM
ejpam-3418	659	14	of	of	ADP
ejpam-3418	659	15	f	f	PROPN
ejpam-3418	660	1	[	[	X
ejpam-3418	660	2	pα	pα	NOUN
ejpam-3418	660	3	,	,	PUNCT
ejpam-3418	660	4	b	b	NOUN
ejpam-3418	660	5	]	]	X
ejpam-3418	660	6	,	,	PUNCT
ejpam-3418	660	7	there	there	PRON
ejpam-3418	660	8	is	be	VERB
ejpam-3418	660	9	a	a	DET
ejpam-3418	660	10	neighborhood	neighborhood	NOUN
ejpam-3418	660	11	η̂′	η̂′	PROPN
ejpam-3418	660	12	of	of	ADP
ejpam-3418	660	13	pα	pα	PROPN
ejpam-3418	660	14	,	,	PUNCT
ejpam-3418	660	15	b	b	NOUN
ejpam-3418	660	16	such	such	ADJ
ejpam-3418	661	1	that	that	SCONJ
ejpam-3418	661	2	f	f	PROPN
ejpam-3418	662	1	[	[	X
ejpam-3418	662	2	η̂′	η̂′	X
ejpam-3418	662	3	]	]	X
ejpam-3418	662	4	=	=	PUNCT
ejpam-3418	662	5	η̂	η̂	X
ejpam-3418	662	6	whenever	whenever	SCONJ
ejpam-3418	662	7	f	f	PROPN
ejpam-3418	662	8	is	be	AUX
ejpam-3418	662	9	onto	onto	ADP
ejpam-3418	662	10	.	.	PUNCT
ejpam-3418	663	1	let	let	VERB
ejpam-3418	663	2	η̂	η̂	NUM
ejpam-3418	663	3	∈	∈	PROPN
ejpam-3418	663	4	t	t	NOUN
ejpam-3418	663	5	′2	′2	X
ejpam-3418	663	6	and	and	CCONJ
ejpam-3418	663	7	pα	pα	VERB
ejpam-3418	663	8	,	,	PUNCT
ejpam-3418	663	9	b	b	PROPN
ejpam-3418	663	10	be	be	AUX
ejpam-3418	663	11	an	an	DET
ejpam-3418	663	12	ivfi	ivfi	NOUN
ejpam-3418	663	13	point	point	NOUN
ejpam-3418	663	14	in	in	ADP
ejpam-3418	663	15	i	i	PRON
ejpam-3418	663	16	i(x	i(x	NOUN
ejpam-3418	663	17	)	)	PUNCT
ejpam-3418	663	18	.	.	PUNCT
ejpam-3418	664	1	by	by	ADP
ejpam-3418	664	2	assumption	assumption	NOUN
ejpam-3418	664	3	,	,	PUNCT
ejpam-3418	664	4	η̂	η̂	NUM
ejpam-3418	664	5	is	be	AUX
ejpam-3418	664	6	a	a	DET
ejpam-3418	664	7	neighborhood	neighborhood	NOUN
ejpam-3418	664	8	of	of	ADP
ejpam-3418	664	9	f	f	PROPN
ejpam-3418	665	1	[	[	X
ejpam-3418	665	2	pα	pα	NOUN
ejpam-3418	665	3	,	,	PUNCT
ejpam-3418	665	4	b	b	NOUN
ejpam-3418	665	5	]	]	X
ejpam-3418	665	6	.	.	PUNCT
ejpam-3418	666	1	thus	thus	ADV
ejpam-3418	666	2	,	,	PUNCT
ejpam-3418	666	3	there	there	PRON
ejpam-3418	666	4	exists	exist	VERB
ejpam-3418	666	5	a	a	DET
ejpam-3418	666	6	neighborhood	neighborhood	NOUN
ejpam-3418	666	7	η̂′	η̂′	PROPN
ejpam-3418	666	8	of	of	ADP
ejpam-3418	666	9	pα	pα	PROPN
ejpam-3418	666	10	,	,	PUNCT
ejpam-3418	666	11	b	b	NOUN
ejpam-3418	666	12	such	such	ADJ
ejpam-3418	667	1	that	that	SCONJ
ejpam-3418	667	2	f	f	PROPN
ejpam-3418	668	1	[	[	X
ejpam-3418	668	2	η̂′	η̂′	X
ejpam-3418	668	3	]	]	X
ejpam-3418	668	4	=	=	SYM
ejpam-3418	668	5	η̂.	η̂.	NOUN
ejpam-3418	668	6	taking	take	VERB
ejpam-3418	668	7	the	the	DET
ejpam-3418	668	8	inverse	inverse	NOUN
ejpam-3418	668	9	images	image	NOUN
ejpam-3418	668	10	,	,	PUNCT
ejpam-3418	668	11	f−1[f	f−1[f	VERB
ejpam-3418	668	12	[	[	X
ejpam-3418	668	13	η̂′	η̂′	PROPN
ejpam-3418	668	14	]	]	X
ejpam-3418	668	15	]	]	X
ejpam-3418	668	16	=	=	PUNCT
ejpam-3418	668	17	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	668	18	]	]	PUNCT
ejpam-3418	668	19	.	.	PUNCT
ejpam-3418	669	1	note	note	VERB
ejpam-3418	669	2	that	that	SCONJ
ejpam-3418	669	3	by	by	ADP
ejpam-3418	669	4	theorem	theorem	NOUN
ejpam-3418	669	5	5	5	NUM
ejpam-3418	669	6	(	(	PUNCT
ejpam-3418	669	7	vi	vi	NOUN
ejpam-3418	669	8	)	)	PUNCT
ejpam-3418	669	9	,	,	PUNCT
ejpam-3418	669	10	η̂′	η̂′	PROPN
ejpam-3418	669	11	6	6	NUM
ejpam-3418	669	12	f−1[f	f−1[f	NOUN
ejpam-3418	669	13	[	[	X
ejpam-3418	669	14	η̂′	η̂′	PROPN
ejpam-3418	669	15	]	]	X
ejpam-3418	669	16	]	]	PUNCT
ejpam-3418	669	17	,	,	PUNCT
ejpam-3418	669	18	and	and	CCONJ
ejpam-3418	669	19	so	so	ADV
ejpam-3418	669	20	η̂′	η̂′	PROPN
ejpam-3418	669	21	=	=	PUNCT
ejpam-3418	669	22	f−1[η̂	f−1[η̂	PROPN
ejpam-3418	669	23	]	]	PUNCT
ejpam-3418	669	24	.	.	PUNCT
ejpam-3418	670	1	thus	thus	ADV
ejpam-3418	670	2	,	,	PUNCT
ejpam-3418	670	3	by	by	ADP
ejpam-3418	670	4	theorem	theorem	NOUN
ejpam-3418	670	5	8	8	NUM
ejpam-3418	670	6	,	,	PUNCT
ejpam-3418	670	7	f−1[η̂	f−1[η̂	PROPN
ejpam-3418	670	8	]	]	X
ejpam-3418	670	9	∈	∈	PROPN
ejpam-3418	670	10	t	t	PROPN
ejpam-3418	670	11	′1	′1	PROPN
ejpam-3418	670	12	.	.	PUNCT
ejpam-3418	671	1	hence	hence	ADV
ejpam-3418	671	2	,	,	PUNCT
ejpam-3418	671	3	f	f	PROPN
ejpam-3418	671	4	is	be	AUX
ejpam-3418	671	5	ivfi	ivfi	NOUN
ejpam-3418	671	6	continuous	continuous	ADJ
ejpam-3418	671	7	.	.	PUNCT
ejpam-3418	672	1	conversely	conversely	ADV
ejpam-3418	672	2	,	,	PUNCT
ejpam-3418	672	3	suppose	suppose	VERB
ejpam-3418	672	4	that	that	SCONJ
ejpam-3418	672	5	f	f	PROPN
ejpam-3418	672	6	is	be	AUX
ejpam-3418	672	7	ivfi	ivfi	NOUN
ejpam-3418	672	8	continuous	continuous	ADJ
ejpam-3418	672	9	.	.	PUNCT
ejpam-3418	673	1	let	let	AUX
ejpam-3418	673	2	pα	pα	VERB
ejpam-3418	673	3	,	,	PUNCT
ejpam-3418	673	4	b	b	NOUN
ejpam-3418	673	5	be	be	AUX
ejpam-3418	673	6	an	an	DET
ejpam-3418	673	7	ivfi	ivfi	NOUN
ejpam-3418	673	8	point	point	NOUN
ejpam-3418	673	9	in	in	ADP
ejpam-3418	673	10	i	i	PRON
ejpam-3418	673	11	i(x	i(x	NOUN
ejpam-3418	673	12	)	)	PUNCT
ejpam-3418	673	13	and	and	CCONJ
ejpam-3418	673	14	η̂	η̂	PROPN
ejpam-3418	673	15	be	be	AUX
ejpam-3418	673	16	a	a	DET
ejpam-3418	673	17	neighborhood	neighborhood	NOUN
ejpam-3418	673	18	of	of	ADP
ejpam-3418	673	19	f	f	PROPN
ejpam-3418	674	1	[	[	X
ejpam-3418	674	2	pα	pα	NOUN
ejpam-3418	674	3	,	,	PUNCT
ejpam-3418	674	4	b	b	NOUN
ejpam-3418	674	5	]	]	PUNCT
ejpam-3418	674	6	.	.	PUNCT
ejpam-3418	675	1	then	then	ADV
ejpam-3418	675	2	there	there	PRON
ejpam-3418	675	3	exists	exist	VERB
ejpam-3418	675	4	ω̂	ω̂	PROPN
ejpam-3418	675	5	∈	∈	PROPN
ejpam-3418	675	6	t	t	NOUN
ejpam-3418	675	7	′2	′2	X
ejpam-3418	675	8	such	such	ADJ
ejpam-3418	675	9	that	that	SCONJ
ejpam-3418	675	10	f	f	PROPN
ejpam-3418	676	1	[	[	X
ejpam-3418	676	2	pα	pα	NOUN
ejpam-3418	676	3	,	,	PUNCT
ejpam-3418	676	4	a	a	DET
ejpam-3418	676	5	]	]	X
ejpam-3418	676	6	∈	∈	NOUN
ejpam-3418	676	7	ω̂	ω̂	NUM
ejpam-3418	676	8	6	6	NUM
ejpam-3418	676	9	η̂.	η̂.	NOUN
ejpam-3418	676	10	taking	take	VERB
ejpam-3418	676	11	the	the	DET
ejpam-3418	676	12	inverse	inverse	NOUN
ejpam-3418	676	13	images	image	NOUN
ejpam-3418	676	14	,	,	PUNCT
ejpam-3418	676	15	we	we	PRON
ejpam-3418	676	16	have	have	AUX
ejpam-3418	676	17	f−1[f	f−1[f	VERB
ejpam-3418	676	18	[	[	X
ejpam-3418	676	19	pα	pα	NOUN
ejpam-3418	676	20	,	,	PUNCT
ejpam-3418	676	21	b	b	NOUN
ejpam-3418	676	22	]	]	X
ejpam-3418	676	23	]	]	X
ejpam-3418	676	24	∈	∈	PROPN
ejpam-3418	676	25	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	676	26	]	]	X
ejpam-3418	676	27	6	6	NUM
ejpam-3418	676	28	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	676	29	]	]	PUNCT
ejpam-3418	676	30	.	.	PUNCT
ejpam-3418	677	1	note	note	VERB
ejpam-3418	677	2	that	that	SCONJ
ejpam-3418	677	3	by	by	ADP
ejpam-3418	677	4	theorem	theorem	NOUN
ejpam-3418	677	5	5	5	NUM
ejpam-3418	677	6	(	(	PUNCT
ejpam-3418	677	7	vi	vi	NOUN
ejpam-3418	677	8	)	)	PUNCT
ejpam-3418	677	9	,	,	PUNCT
ejpam-3418	677	10	pα	pα	INTJ
ejpam-3418	677	11	,	,	PUNCT
ejpam-3418	677	12	b	b	PROPN
ejpam-3418	677	13	6	6	NUM
ejpam-3418	677	14	f−1[f	f−1[f	NOUN
ejpam-3418	677	15	[	[	X
ejpam-3418	677	16	pα	pα	NOUN
ejpam-3418	677	17	,	,	PUNCT
ejpam-3418	677	18	b	b	NOUN
ejpam-3418	677	19	]	]	X
ejpam-3418	677	20	]	]	X
ejpam-3418	677	21	,	,	PUNCT
ejpam-3418	677	22	and	and	CCONJ
ejpam-3418	677	23	so	so	ADV
ejpam-3418	677	24	pα	pα	VERB
ejpam-3418	677	25	,	,	PUNCT
ejpam-3418	677	26	b	b	PROPN
ejpam-3418	677	27	∈	∈	PROPN
ejpam-3418	677	28	f−1[ω̂	f−1[ω̂	NOUN
ejpam-3418	677	29	]	]	X
ejpam-3418	677	30	6	6	NUM
ejpam-3418	677	31	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	677	32	]	]	PUNCT
ejpam-3418	677	33	.	.	PUNCT
ejpam-3418	678	1	since	since	SCONJ
ejpam-3418	678	2	f	f	PROPN
ejpam-3418	678	3	is	be	AUX
ejpam-3418	678	4	continuous	continuous	ADJ
ejpam-3418	678	5	,	,	PUNCT
ejpam-3418	678	6	f−1[ω̂	f−1[ω̂	ADJ
ejpam-3418	678	7	]	]	X
ejpam-3418	678	8	∈	∈	PROPN
ejpam-3418	678	9	t	t	PROPN
ejpam-3418	678	10	′1	′1	PROPN
ejpam-3418	678	11	.	.	PUNCT
ejpam-3418	679	1	hence	hence	ADV
ejpam-3418	679	2	,	,	PUNCT
ejpam-3418	679	3	f−1[η̂	f−1[η̂	PROPN
ejpam-3418	679	4	]	]	PUNCT
ejpam-3418	679	5	is	be	AUX
ejpam-3418	679	6	a	a	DET
ejpam-3418	679	7	neighborhood	neighborhood	NOUN
ejpam-3418	679	8	of	of	ADP
ejpam-3418	679	9	pα	pα	PROPN
ejpam-3418	679	10	,	,	PUNCT
ejpam-3418	679	11	b.	b.	PROPN
ejpam-3418	679	12	take	take	VERB
ejpam-3418	679	13	η̂′	η̂′	PROPN
ejpam-3418	679	14	=	=	PUNCT
ejpam-3418	679	15	f−1[η̂	f−1[η̂	PROPN
ejpam-3418	679	16	]	]	PUNCT
ejpam-3418	679	17	.	.	PUNCT
ejpam-3418	680	1	taking	take	VERB
ejpam-3418	680	2	the	the	DET
ejpam-3418	680	3	images	image	NOUN
ejpam-3418	680	4	,	,	PUNCT
ejpam-3418	680	5	we	we	PRON
ejpam-3418	680	6	have	have	VERB
ejpam-3418	680	7	f	f	PROPN
ejpam-3418	681	1	[	[	X
ejpam-3418	681	2	η̂′	η̂′	X
ejpam-3418	681	3	]	]	X
ejpam-3418	681	4	=	=	SYM
ejpam-3418	681	5	f	f	X
ejpam-3418	682	1	[	[	X
ejpam-3418	682	2	f−1[η̂	f−1[η̂	NOUN
ejpam-3418	682	3	]	]	X
ejpam-3418	682	4	]	]	X
ejpam-3418	682	5	=	=	PUNCT
ejpam-3418	682	6	η̂	η̂	X
ejpam-3418	682	7	,	,	PUNCT
ejpam-3418	682	8	whenever	whenever	SCONJ
ejpam-3418	682	9	f	f	PROPN
ejpam-3418	682	10	is	be	AUX
ejpam-3418	682	11	onto	onto	ADP
ejpam-3418	682	12	.	.	PUNCT
ejpam-3418	683	1	hence	hence	ADV
ejpam-3418	683	2	,	,	PUNCT
ejpam-3418	683	3	f	f	PROPN
ejpam-3418	684	1	[	[	X
ejpam-3418	684	2	η̂′	η̂′	X
ejpam-3418	684	3	]	]	X
ejpam-3418	684	4	=	=	SYM
ejpam-3418	684	5	η̂.	η̂.	NOUN
ejpam-3418	684	6	acknowledgements	acknowledgement	NOUN
ejpam-3418	684	7	the	the	DET
ejpam-3418	684	8	first	first	ADJ
ejpam-3418	684	9	author	author	NOUN
ejpam-3418	684	10	was	be	AUX
ejpam-3418	684	11	supported	support	VERB
ejpam-3418	684	12	by	by	ADP
ejpam-3418	684	13	the	the	DET
ejpam-3418	684	14	department	department	PROPN
ejpam-3418	684	15	of	of	ADP
ejpam-3418	684	16	science	science	NOUN
ejpam-3418	684	17	and	and	CCONJ
ejpam-3418	684	18	technology	technology	NOUN
ejpam-3418	684	19	(	(	PUNCT
ejpam-3418	684	20	dost	dost	NOUN
ejpam-3418	684	21	)	)	PUNCT
ejpam-3418	684	22	of	of	ADP
ejpam-3418	684	23	the	the	DET
ejpam-3418	684	24	philippines	philippine	NOUN
ejpam-3418	684	25	and	and	CCONJ
ejpam-3418	684	26	the	the	DET
ejpam-3418	684	27	second	second	ADJ
ejpam-3418	684	28	author	author	NOUN
ejpam-3418	684	29	by	by	ADP
ejpam-3418	684	30	the	the	DET
ejpam-3418	684	31	premier	premier	PROPN
ejpam-3418	684	32	research	research	PROPN
ejpam-3418	684	33	institute	institute	PROPN
ejpam-3418	684	34	of	of	ADP
ejpam-3418	684	35	science	science	NOUN
ejpam-3418	684	36	and	and	CCONJ
ejpam-3418	684	37	mathematics	mathematics	PROPN
ejpam-3418	684	38	(	(	PUNCT
ejpam-3418	684	39	prism	prism	NOUN
ejpam-3418	684	40	)	)	PUNCT
ejpam-3418	684	41	of	of	ADP
ejpam-3418	684	42	the	the	DET
ejpam-3418	684	43	mindanao	mindanao	PROPN
ejpam-3418	684	44	state	state	PROPN
ejpam-3418	684	45	university	university	PROPN
ejpam-3418	684	46	–	–	PUNCT
ejpam-3418	684	47	iligan	iligan	PROPN
ejpam-3418	684	48	institute	institute	PROPN
ejpam-3418	684	49	of	of	ADP
ejpam-3418	684	50	technology	technology	PROPN
ejpam-3418	684	51	(	(	PUNCT
ejpam-3418	684	52	msu	msu	PROPN
ejpam-3418	684	53	–	–	PUNCT
ejpam-3418	684	54	iit	iit	NOUN
ejpam-3418	684	55	)	)	PUNCT
ejpam-3418	684	56	.	.	PUNCT
ejpam-3418	685	1	references	reference	NOUN
ejpam-3418	685	2	[	[	X
ejpam-3418	685	3	1	1	NUM
ejpam-3418	685	4	]	]	X
ejpam-3418	685	5	d	d	X
ejpam-3418	685	6	dubois	dubois	PROPN
ejpam-3418	685	7	and	and	CCONJ
ejpam-3418	685	8	h	h	PROPN
ejpam-3418	685	9	prade	prade	NOUN
ejpam-3418	685	10	(	(	PUNCT
ejpam-3418	685	11	eds	ed	NOUN
ejpam-3418	685	12	.	.	PUNCT
ejpam-3418	685	13	)	)	PUNCT
ejpam-3418	685	14	.	.	PUNCT
ejpam-3418	686	1	fundamentals	fundamental	NOUN
ejpam-3418	686	2	of	of	ADP
ejpam-3418	686	3	fuzzy	fuzzy	ADJ
ejpam-3418	686	4	sets	set	NOUN
ejpam-3418	686	5	.	.	PUNCT
ejpam-3418	687	1	springer	springer	NOUN
ejpam-3418	687	2	science	science	NOUN
ejpam-3418	687	3	and	and	CCONJ
ejpam-3418	687	4	bussiness	bussiness	NOUN
ejpam-3418	687	5	media	medium	NOUN
ejpam-3418	687	6	,	,	PUNCT
ejpam-3418	687	7	2012	2012	NUM
ejpam-3418	687	8	.	.	PUNCT
ejpam-3418	688	1	[	[	X
ejpam-3418	688	2	2	2	X
ejpam-3418	688	3	]	]	PUNCT
ejpam-3418	688	4	i	i	PRON
ejpam-3418	688	5	grattan	grattan	PROPN
ejpam-3418	688	6	-	-	PUNCT
ejpam-3418	688	7	guiness	guiness	PROPN
ejpam-3418	688	8	.	.	PUNCT
ejpam-3418	689	1	fuzzy	fuzzy	ADJ
ejpam-3418	689	2	membership	membership	NOUN
ejpam-3418	689	3	mapped	map	VERB
ejpam-3418	689	4	onto	onto	ADP
ejpam-3418	689	5	interval	interval	NOUN
ejpam-3418	689	6	and	and	CCONJ
ejpam-3418	689	7	many	many	ADV
ejpam-3418	689	8	-	-	PUNCT
ejpam-3418	689	9	valued	value	VERB
ejpam-3418	689	10	quantities	quantity	NOUN
ejpam-3418	689	11	.	.	PUNCT
ejpam-3418	690	1	z.	z.	PROPN
ejpam-3418	690	2	math	math	PROPN
ejpam-3418	690	3	.	.	PUNCT
ejpam-3418	691	1	logik	logik	PROPN
ejpam-3418	691	2	.	.	PUNCT
ejpam-3418	692	1	grundladen	grundladen	PROPN
ejpam-3418	692	2	math	math	NOUN
ejpam-3418	692	3	.	.	PUNCT
ejpam-3418	692	4	,	,	PUNCT
ejpam-3418	692	5	22:149–160	22:149–160	PROPN
ejpam-3418	692	6	,	,	PUNCT
ejpam-3418	692	7	1975	1975	NUM
ejpam-3418	692	8	.	.	PUNCT
ejpam-3418	693	1	references	reference	NOUN
ejpam-3418	693	2	570	570	NUM
ejpam-3418	693	3	[	[	X
ejpam-3418	693	4	3	3	NUM
ejpam-3418	693	5	]	]	X
ejpam-3418	693	6	ku	ku	PROPN
ejpam-3418	693	7	jahn	jahn	PROPN
ejpam-3418	693	8	.	.	PUNCT
ejpam-3418	694	1	intervall	intervall	VERB
ejpam-3418	694	2	-	-	PUNCT
ejpam-3418	694	3	wertige	wertige	ADJ
ejpam-3418	694	4	mengen	mengen	PROPN
ejpam-3418	694	5	.	.	PUNCT
ejpam-3418	695	1	math.nach	math.nach	PROPN
ejpam-3418	695	2	.	.	PROPN
ejpam-3418	695	3	southern	southern	ADJ
ejpam-3418	695	4	economic	economic	ADJ
ejpam-3418	695	5	journal	journal	PROPN
ejpam-3418	695	6	,	,	PUNCT
ejpam-3418	695	7	68:115	68:115	NUM
ejpam-3418	695	8	–	–	PUNCT
ejpam-3418	695	9	132	132	NUM
ejpam-3418	695	10	,	,	PUNCT
ejpam-3418	695	11	1975	1975	NUM
ejpam-3418	695	12	.	.	PUNCT
ejpam-3418	696	1	[	[	X
ejpam-3418	696	2	4	4	NUM
ejpam-3418	696	3	]	]	X
ejpam-3418	696	4	d	d	PROPN
ejpam-3418	696	5	klaua	klaua	PROPN
ejpam-3418	696	6	.	.	PUNCT
ejpam-3418	697	1	ber	ber	PROPN
ejpam-3418	697	2	einen	einen	PROPN
ejpam-3418	697	3	ansatz	ansatz	ADJ
ejpam-3418	697	4	zur	zur	NOUN
ejpam-3418	697	5	mehrwertigen	mehrwertigen	NOUN
ejpam-3418	697	6	mengenlehre	mengenlehre	PROPN
ejpam-3418	697	7	.	.	PUNCT
ejpam-3418	698	1	monatsberichte	monatsberichte	PROPN
ejpam-3418	698	2	der	der	PROPN
ejpam-3418	698	3	kniglichen	kniglichen	PROPN
ejpam-3418	698	4	preussische	preussische	PROPN
ejpam-3418	698	5	akademie	akademie	PROPN
ejpam-3418	698	6	des	des	PROPN
ejpam-3418	698	7	wissenschaften	wissenschaften	PROPN
ejpam-3418	698	8	zu	zu	PROPN
ejpam-3418	698	9	berlin	berlin	PROPN
ejpam-3418	698	10	.	.	PROPN
ejpam-3418	698	11	,	,	PUNCT
ejpam-3418	698	12	7:859–876	7:859–876	NUM
ejpam-3418	698	13	,	,	PUNCT
ejpam-3418	698	14	1965	1965	NUM
ejpam-3418	698	15	.	.	PUNCT
ejpam-3418	699	1	[	[	X
ejpam-3418	699	2	5	5	NUM
ejpam-3418	699	3	]	]	X
ejpam-3418	699	4	k	k	PROPN
ejpam-3418	699	5	kuratowski	kuratowski	PROPN
ejpam-3418	699	6	.	.	PUNCT
ejpam-3418	700	1	topology	topology	PROPN
ejpam-3418	700	2	.	.	PUNCT
ejpam-3418	701	1	new	new	PROPN
ejpam-3418	701	2	york	york	PROPN
ejpam-3418	701	3	:	:	PUNCT
ejpam-3418	701	4	academic	academic	ADJ
ejpam-3418	701	5	press	press	NOUN
ejpam-3418	701	6	,	,	PUNCT
ejpam-3418	701	7	1966	1966	NUM
ejpam-3418	701	8	.	.	PUNCT
ejpam-3418	702	1	[	[	X
ejpam-3418	702	2	6	6	NUM
ejpam-3418	702	3	]	]	PUNCT
ejpam-3418	702	4	l	l	NOUN
ejpam-3418	702	5	mernilo	mernilo	ADJ
ejpam-3418	702	6	-	-	PUNCT
ejpam-3418	702	7	tutanes	tutane	NOUN
ejpam-3418	702	8	and	and	CCONJ
ejpam-3418	702	9	r	r	NOUN
ejpam-3418	702	10	caga	caga	NOUN
ejpam-3418	702	11	-	-	PUNCT
ejpam-3418	702	12	anan	anan	PROPN
ejpam-3418	702	13	.	.	PUNCT
ejpam-3418	703	1	fuzzy	fuzzy	ADJ
ejpam-3418	703	2	on	on	ADP
ejpam-3418	703	3	ideal	ideal	ADJ
ejpam-3418	703	4	sets	set	NOUN
ejpam-3418	703	5	and	and	CCONJ
ejpam-3418	703	6	a	a	DET
ejpam-3418	703	7	fuzzy	fuzzy	ADJ
ejpam-3418	703	8	on	on	ADP
ejpam-3418	703	9	ideal	ideal	ADJ
ejpam-3418	703	10	hahnbanach	hahnbanach	PROPN
ejpam-3418	703	11	theorem	theorem	PROPN
ejpam-3418	703	12	.	.	PROPN
ejpam-3418	703	13	science	science	PROPN
ejpam-3418	703	14	diliman	diliman	PROPN
ejpam-3418	703	15	.	.	PUNCT
ejpam-3418	703	16	,	,	PUNCT
ejpam-3418	703	17	30(2):70–86	30(2):70–86	NUM
ejpam-3418	703	18	,	,	PUNCT
ejpam-3418	703	19	2018	2018	NUM
ejpam-3418	703	20	.	.	PUNCT
ejpam-3418	704	1	[	[	X
ejpam-3418	704	2	7	7	NUM
ejpam-3418	704	3	]	]	X
ejpam-3418	704	4	r	r	NOUN
ejpam-3418	704	5	sambuc	sambuc	NOUN
ejpam-3418	704	6	.	.	PUNCT
ejpam-3418	705	1	functions	function	NOUN
ejpam-3418	705	2	φ	φ	NOUN
ejpam-3418	705	3	-	-	NOUN
ejpam-3418	705	4	floues	floue	NOUN
ejpam-3418	705	5	.	.	PUNCT
ejpam-3418	706	1	application	application	NOUN
ejpam-3418	706	2	laide	laide	NOUN
ejpam-3418	706	3	au	au	PROPN
ejpam-3418	706	4	diagnostic	diagnostic	PROPN
ejpam-3418	706	5	en	en	PROPN
ejpam-3418	706	6	pathologie	pathologie	PROPN
ejpam-3418	706	7	thyroidienne	thyroidienne	PROPN
ejpam-3418	706	8	.	.	PUNCT
ejpam-3418	707	1	phd	phd	NOUN
ejpam-3418	707	2	thesis	thesis	PROPN
ejpam-3418	707	3	,	,	PUNCT
ejpam-3418	707	4	phd	phd	NOUN
ejpam-3418	707	5	,	,	PUNCT
ejpam-3418	707	6	université	université	ADJ
ejpam-3418	707	7	de	de	X
ejpam-3418	707	8	marseille	marseille	PROPN
ejpam-3418	707	9	,	,	PUNCT
ejpam-3418	707	10	france	france	PROPN
ejpam-3418	707	11	,	,	PUNCT
ejpam-3418	707	12	1975	1975	NUM
ejpam-3418	707	13	.	.	PUNCT
ejpam-3418	708	1	[	[	X
ejpam-3418	708	2	8	8	NUM
ejpam-3418	708	3	]	]	X
ejpam-3418	708	4	r	r	NOUN
ejpam-3418	708	5	vaidyanathaswamy	vaidyanathaswamy	NOUN
ejpam-3418	708	6	.	.	PUNCT
ejpam-3418	709	1	the	the	DET
ejpam-3418	709	2	localisation	localisation	NOUN
ejpam-3418	709	3	theory	theory	NOUN
ejpam-3418	709	4	in	in	ADP
ejpam-3418	709	5	set	set	NOUN
ejpam-3418	709	6	-	-	PUNCT
ejpam-3418	709	7	topology	topology	NOUN
ejpam-3418	709	8	.	.	PUNCT
ejpam-3418	710	1	in	in	ADP
ejpam-3418	710	2	proceedings	proceeding	NOUN
ejpam-3418	710	3	of	of	ADP
ejpam-3418	710	4	the	the	DET
ejpam-3418	710	5	indian	indian	PROPN
ejpam-3418	710	6	academy	academy	PROPN
ejpam-3418	710	7	of	of	ADP
ejpam-3418	710	8	sciences	science	NOUN
ejpam-3418	710	9	-	-	PUNCT
ejpam-3418	710	10	section	section	NOUN
ejpam-3418	710	11	a	a	PRON
ejpam-3418	710	12	;	;	PUNCT
ejpam-3418	710	13	springer	springer	NOUN
ejpam-3418	710	14	india	india	PROPN
ejpam-3418	710	15	.	.	PROPN
ejpam-3418	710	16	,	,	PUNCT
ejpam-3418	710	17	volume	volume	NOUN
ejpam-3418	710	18	20(1	20(1	NUM
ejpam-3418	710	19	)	)	PUNCT
ejpam-3418	710	20	,	,	PUNCT
ejpam-3418	710	21	pages	page	NOUN
ejpam-3418	710	22	51–61	51–61	NUM
ejpam-3418	710	23	,	,	PUNCT
ejpam-3418	710	24	1944	1944	NUM
ejpam-3418	710	25	.	.	PUNCT
ejpam-3418	711	1	[	[	X
ejpam-3418	711	2	9	9	NUM
ejpam-3418	711	3	]	]	X
ejpam-3418	711	4	la	la	PROPN
ejpam-3418	711	5	zadeh	zadeh	PROPN
ejpam-3418	711	6	.	.	PUNCT
ejpam-3418	711	7	fuzzy	fuzzy	ADJ
ejpam-3418	711	8	sets	set	NOUN
ejpam-3418	711	9	.	.	PUNCT
ejpam-3418	712	1	information	information	NOUN
ejpam-3418	712	2	control	control	PROPN
ejpam-3418	712	3	.	.	PUNCT
ejpam-3418	712	4	,	,	PUNCT
ejpam-3418	712	5	8:338–353	8:338–353	NUM
ejpam-3418	712	6	,	,	PUNCT
ejpam-3418	712	7	1965	1965	NUM
ejpam-3418	712	8	.	.	PUNCT
ejpam-3418	713	1	[	[	X
ejpam-3418	713	2	10	10	NUM
ejpam-3418	713	3	]	]	X
ejpam-3418	713	4	la	la	PROPN
ejpam-3418	713	5	zadeh	zadeh	PROPN
ejpam-3418	713	6	.	.	PUNCT
ejpam-3418	714	1	the	the	DET
ejpam-3418	714	2	concept	concept	NOUN
ejpam-3418	714	3	of	of	ADP
ejpam-3418	714	4	a	a	DET
ejpam-3418	714	5	linguistic	linguistic	ADJ
ejpam-3418	714	6	variable	variable	NOUN
ejpam-3418	714	7	and	and	CCONJ
ejpam-3418	714	8	its	its	PRON
ejpam-3418	714	9	application	application	NOUN
ejpam-3418	714	10	to	to	PART
ejpam-3418	714	11	approximate	approximate	ADJ
ejpam-3418	714	12	reasoning	reasoning	NOUN
ejpam-3418	714	13	i.	i.	PROPN
ejpam-3418	714	14	information	information	PROPN
ejpam-3418	714	15	sciences	sciences	PROPN
ejpam-3418	714	16	.	.	PUNCT
ejpam-3418	714	17	,	,	PUNCT
ejpam-3418	714	18	8:199–249	8:199–249	NUM
ejpam-3418	714	19	,	,	PUNCT
ejpam-3418	714	20	1975	1975	NUM
ejpam-3418	714	21	.	.	PUNCT
ejpam-3418	715	1	[	[	X
ejpam-3418	715	2	11	11	NUM
ejpam-3418	715	3	]	]	X
ejpam-3418	715	4	la	la	PROPN
ejpam-3418	715	5	zadeh	zadeh	PROPN
ejpam-3418	715	6	.	.	PUNCT
ejpam-3418	715	7	fuzzy	fuzzy	ADJ
ejpam-3418	715	8	sets	set	NOUN
ejpam-3418	715	9	as	as	ADP
ejpam-3418	715	10	a	a	DET
ejpam-3418	715	11	basis	basis	NOUN
ejpam-3418	715	12	for	for	ADP
ejpam-3418	715	13	a	a	DET
ejpam-3418	715	14	theory	theory	NOUN
ejpam-3418	715	15	of	of	ADP
ejpam-3418	715	16	possibility	possibility	NOUN
ejpam-3418	715	17	.	.	PUNCT
ejpam-3418	716	1	fuzzy	fuzzy	ADJ
ejpam-3418	716	2	sets	set	NOUN
ejpam-3418	716	3	and	and	CCONJ
ejpam-3418	716	4	systems	system	NOUN
ejpam-3418	716	5	.	.	PUNCT
ejpam-3418	716	6	,	,	PUNCT
ejpam-3418	716	7	1:3–28	1:3–28	NUM
ejpam-3418	716	8	,	,	PUNCT
ejpam-3418	716	9	1978	1978	NUM
ejpam-3418	716	10	.	.	PUNCT