id	sid	tid	token	lemma	pos
ejpam-5367	1	1	european	european	PROPN
ejpam-5367	1	2	journal	journal	PROPN
ejpam-5367	1	3	of	of	ADP
ejpam-5367	1	4	pure	pure	ADJ
ejpam-5367	1	5	and	and	CCONJ
ejpam-5367	1	6	applied	apply	VERB
ejpam-5367	1	7	mathematics	mathematic	NOUN
ejpam-5367	1	8	vol	vol	NOUN
ejpam-5367	1	9	.	.	PROPN
ejpam-5367	2	1	17	17	NUM
ejpam-5367	2	2	,	,	PUNCT
ejpam-5367	2	3	no	no	INTJ
ejpam-5367	2	4	.	.	NOUN
ejpam-5367	2	5	4	4	NUM
ejpam-5367	2	6	,	,	PUNCT
ejpam-5367	2	7	2024	2024	NUM
ejpam-5367	2	8	,	,	PUNCT
ejpam-5367	2	9	3022	3022	NUM
ejpam-5367	2	10	-	-	SYM
ejpam-5367	2	11	3042	3042	NUM
ejpam-5367	2	12	issn	issn	PROPN
ejpam-5367	2	13	1307	1307	NUM
ejpam-5367	2	14	-	-	SYM
ejpam-5367	2	15	5543	5543	NUM
ejpam-5367	2	16	–	–	PUNCT
ejpam-5367	3	1	ejpam.com	ejpam.com	X
ejpam-5367	3	2	published	publish	VERB
ejpam-5367	3	3	by	by	ADP
ejpam-5367	3	4	new	new	PROPN
ejpam-5367	3	5	york	york	PROPN
ejpam-5367	3	6	business	business	PROPN
ejpam-5367	3	7	global	global	ADJ
ejpam-5367	3	8	fermatean	fermatean	PROPN
ejpam-5367	3	9	fuzzy	fuzzy	ADJ
ejpam-5367	3	10	set	set	VERB
ejpam-5367	3	11	theory	theory	NOUN
ejpam-5367	3	12	applied	apply	VERB
ejpam-5367	3	13	to	to	ADP
ejpam-5367	3	14	iup	iup	VERB
ejpam-5367	3	15	-	-	PUNCT
ejpam-5367	3	16	algebras	algebras	PROPN
ejpam-5367	3	17	kannirun	kannirun	PROPN
ejpam-5367	3	18	suayngam1	suayngam1	PROPN
ejpam-5367	3	19	,	,	PUNCT
ejpam-5367	3	20	rukchart	rukchart	NOUN
ejpam-5367	3	21	prasertpong2	prasertpong2	NOUN
ejpam-5367	3	22	,	,	PUNCT
ejpam-5367	3	23	nareupanat	nareupanat	ADJ
ejpam-5367	3	24	lekkoksung3	lekkoksung3	PROPN
ejpam-5367	3	25	,	,	PUNCT
ejpam-5367	3	26	pongpun	pongpun	ADJ
ejpam-5367	3	27	julatha4	julatha4	PROPN
ejpam-5367	3	28	,	,	PUNCT
ejpam-5367	3	29	aiyared	aiyare	VERB
ejpam-5367	3	30	iampan1,∗	iampan1,∗	NOUN
ejpam-5367	3	31	1	1	NUM
ejpam-5367	3	32	department	department	NOUN
ejpam-5367	3	33	of	of	ADP
ejpam-5367	3	34	mathematics	mathematic	NOUN
ejpam-5367	3	35	,	,	PUNCT
ejpam-5367	3	36	school	school	NOUN
ejpam-5367	3	37	of	of	ADP
ejpam-5367	3	38	science	science	NOUN
ejpam-5367	3	39	,	,	PUNCT
ejpam-5367	3	40	university	university	NOUN
ejpam-5367	3	41	of	of	ADP
ejpam-5367	3	42	phayao	phayao	NOUN
ejpam-5367	3	43	,	,	PUNCT
ejpam-5367	3	44	mae	mae	PROPN
ejpam-5367	3	45	ka	ka	PROPN
ejpam-5367	3	46	,	,	PUNCT
ejpam-5367	3	47	mueang	mueang	PROPN
ejpam-5367	3	48	,	,	PUNCT
ejpam-5367	3	49	phayao	phayao	NOUN
ejpam-5367	3	50	56000	56000	NUM
ejpam-5367	3	51	,	,	PUNCT
ejpam-5367	3	52	thailand	thailand	PROPN
ejpam-5367	3	53	2	2	NUM
ejpam-5367	3	54	division	division	NOUN
ejpam-5367	3	55	of	of	ADP
ejpam-5367	3	56	mathematics	mathematic	NOUN
ejpam-5367	3	57	and	and	CCONJ
ejpam-5367	3	58	statistics	statistic	NOUN
ejpam-5367	3	59	,	,	PUNCT
ejpam-5367	3	60	faculty	faculty	NOUN
ejpam-5367	3	61	of	of	ADP
ejpam-5367	3	62	science	science	NOUN
ejpam-5367	3	63	and	and	CCONJ
ejpam-5367	3	64	technology	technology	NOUN
ejpam-5367	3	65	,	,	PUNCT
ejpam-5367	3	66	nakhon	nakhon	PROPN
ejpam-5367	3	67	sawan	sawan	PROPN
ejpam-5367	3	68	rajabhat	rajabhat	PROPN
ejpam-5367	3	69	university	university	PROPN
ejpam-5367	3	70	,	,	PUNCT
ejpam-5367	3	71	nakhon	nakhon	PROPN
ejpam-5367	3	72	sawan	sawan	PROPN
ejpam-5367	3	73	60000	60000	NUM
ejpam-5367	3	74	,	,	PUNCT
ejpam-5367	3	75	thailand	thailand	PROPN
ejpam-5367	3	76	3	3	NUM
ejpam-5367	3	77	division	division	NOUN
ejpam-5367	3	78	of	of	ADP
ejpam-5367	3	79	mathematics	mathematic	NOUN
ejpam-5367	3	80	,	,	PUNCT
ejpam-5367	3	81	faculty	faculty	NOUN
ejpam-5367	3	82	of	of	ADP
ejpam-5367	3	83	engineering	engineering	NOUN
ejpam-5367	3	84	,	,	PUNCT
ejpam-5367	3	85	rajamangala	rajamangala	PROPN
ejpam-5367	3	86	university	university	PROPN
ejpam-5367	3	87	of	of	ADP
ejpam-5367	3	88	technology	technology	PROPN
ejpam-5367	3	89	isan	isan	PROPN
ejpam-5367	3	90	,	,	PUNCT
ejpam-5367	3	91	khon	khon	PROPN
ejpam-5367	3	92	kaen	kaen	PROPN
ejpam-5367	3	93	campus	campus	PROPN
ejpam-5367	3	94	,	,	PUNCT
ejpam-5367	3	95	khon	khon	PROPN
ejpam-5367	3	96	kaen	kaen	PROPN
ejpam-5367	3	97	40000	40000	NUM
ejpam-5367	3	98	,	,	PUNCT
ejpam-5367	3	99	thailand	thailand	PROPN
ejpam-5367	3	100	4	4	NUM
ejpam-5367	3	101	department	department	NOUN
ejpam-5367	3	102	of	of	ADP
ejpam-5367	3	103	mathematics	mathematic	NOUN
ejpam-5367	3	104	,	,	PUNCT
ejpam-5367	3	105	faculty	faculty	NOUN
ejpam-5367	3	106	of	of	ADP
ejpam-5367	3	107	science	science	NOUN
ejpam-5367	3	108	and	and	CCONJ
ejpam-5367	3	109	technology	technology	NOUN
ejpam-5367	3	110	,	,	PUNCT
ejpam-5367	3	111	pibulsongkram	pibulsongkram	PROPN
ejpam-5367	3	112	rajabhat	rajabhat	PROPN
ejpam-5367	3	113	university	university	NOUN
ejpam-5367	3	114	,	,	PUNCT
ejpam-5367	3	115	phitsanulok	phitsanulok	NOUN
ejpam-5367	3	116	65000	65000	NUM
ejpam-5367	3	117	,	,	PUNCT
ejpam-5367	3	118	thailand	thailand	PROPN
ejpam-5367	3	119	abstract	abstract	NOUN
ejpam-5367	3	120	.	.	PUNCT
ejpam-5367	4	1	in	in	ADP
ejpam-5367	4	2	1965	1965	NUM
ejpam-5367	4	3	,	,	PUNCT
ejpam-5367	4	4	zadeh	zadeh	PROPN
ejpam-5367	4	5	introduced	introduce	VERB
ejpam-5367	4	6	the	the	DET
ejpam-5367	4	7	foundational	foundational	ADJ
ejpam-5367	4	8	concept	concept	NOUN
ejpam-5367	4	9	of	of	ADP
ejpam-5367	4	10	fuzzy	fuzzy	ADJ
ejpam-5367	4	11	sets	set	NOUN
ejpam-5367	4	12	,	,	PUNCT
ejpam-5367	4	13	followed	follow	VERB
ejpam-5367	4	14	by	by	ADP
ejpam-5367	4	15	atanassov	atanassov	PROPN
ejpam-5367	4	16	’s	’s	PART
ejpam-5367	4	17	introduction	introduction	NOUN
ejpam-5367	4	18	of	of	ADP
ejpam-5367	4	19	intuitionistic	intuitionistic	ADJ
ejpam-5367	4	20	fuzzy	fuzzy	ADJ
ejpam-5367	4	21	sets	set	NOUN
ejpam-5367	4	22	in	in	ADP
ejpam-5367	4	23	1986	1986	NUM
ejpam-5367	4	24	.	.	PUNCT
ejpam-5367	5	1	yager	yager	NOUN
ejpam-5367	5	2	expanded	expand	VERB
ejpam-5367	5	3	this	this	DET
ejpam-5367	5	4	field	field	NOUN
ejpam-5367	5	5	with	with	ADP
ejpam-5367	5	6	pythagorean	pythagorean	PROPN
ejpam-5367	5	7	fuzzy	fuzzy	ADJ
ejpam-5367	5	8	sets	set	NOUN
ejpam-5367	5	9	in	in	ADP
ejpam-5367	5	10	2013	2013	NUM
ejpam-5367	5	11	,	,	PUNCT
ejpam-5367	5	12	and	and	CCONJ
ejpam-5367	5	13	in	in	ADP
ejpam-5367	5	14	2020	2020	NUM
ejpam-5367	5	15	,	,	PUNCT
ejpam-5367	5	16	senapati	senapati	PROPN
ejpam-5367	5	17	and	and	CCONJ
ejpam-5367	5	18	yager	yager	NOUN
ejpam-5367	5	19	further	far	ADV
ejpam-5367	5	20	advanced	advance	VERB
ejpam-5367	5	21	the	the	DET
ejpam-5367	5	22	theory	theory	NOUN
ejpam-5367	5	23	by	by	ADP
ejpam-5367	5	24	proposing	propose	VERB
ejpam-5367	5	25	fermatean	fermatean	ADJ
ejpam-5367	5	26	fuzzy	fuzzy	ADJ
ejpam-5367	5	27	sets	set	NOUN
ejpam-5367	5	28	.	.	PUNCT
ejpam-5367	6	1	this	this	DET
ejpam-5367	6	2	study	study	NOUN
ejpam-5367	6	3	applies	apply	VERB
ejpam-5367	6	4	fermatean	fermatean	ADJ
ejpam-5367	6	5	fuzzy	fuzzy	ADJ
ejpam-5367	6	6	sets	set	NOUN
ejpam-5367	6	7	to	to	PART
ejpam-5367	6	8	iup	iup	VERB
ejpam-5367	6	9	-	-	PUNCT
ejpam-5367	6	10	algebras	algebras	X
ejpam-5367	6	11	,	,	PUNCT
ejpam-5367	6	12	focusing	focus	VERB
ejpam-5367	6	13	on	on	ADP
ejpam-5367	6	14	fermatean	fermatean	ADJ
ejpam-5367	6	15	fuzzy	fuzzy	ADJ
ejpam-5367	6	16	iup	iup	PROPN
ejpam-5367	6	17	-	-	PUNCT
ejpam-5367	6	18	subalgebras	subalgebras	PROPN
ejpam-5367	6	19	,	,	PUNCT
ejpam-5367	6	20	iup	iup	NOUN
ejpam-5367	6	21	-	-	PUNCT
ejpam-5367	6	22	ideals	ideal	NOUN
ejpam-5367	6	23	,	,	PUNCT
ejpam-5367	6	24	iup	iup	NOUN
ejpam-5367	6	25	-	-	PUNCT
ejpam-5367	6	26	filters	filter	NOUN
ejpam-5367	6	27	,	,	PUNCT
ejpam-5367	6	28	and	and	CCONJ
ejpam-5367	6	29	strong	strong	ADJ
ejpam-5367	6	30	iup	iup	NOUN
ejpam-5367	6	31	-	-	PUNCT
ejpam-5367	6	32	ideals	ideal	NOUN
ejpam-5367	6	33	.	.	PUNCT
ejpam-5367	7	1	we	we	PRON
ejpam-5367	7	2	examine	examine	VERB
ejpam-5367	7	3	their	their	PRON
ejpam-5367	7	4	properties	property	NOUN
ejpam-5367	7	5	,	,	PUNCT
ejpam-5367	7	6	including	include	VERB
ejpam-5367	7	7	characteristic	characteristic	ADJ
ejpam-5367	7	8	fermatean	fermatean	NOUN
ejpam-5367	7	9	fuzzy	fuzzy	ADJ
ejpam-5367	7	10	sets	set	NOUN
ejpam-5367	7	11	and	and	CCONJ
ejpam-5367	7	12	upper	upper	ADJ
ejpam-5367	7	13	and	and	CCONJ
ejpam-5367	7	14	lower	low	ADJ
ejpam-5367	7	15	t-(strong	t-(strong	NOUN
ejpam-5367	7	16	)	)	PUNCT
ejpam-5367	7	17	level	level	NOUN
ejpam-5367	7	18	subsets	subset	NOUN
ejpam-5367	7	19	,	,	PUNCT
ejpam-5367	7	20	offering	offer	VERB
ejpam-5367	7	21	deeper	deep	ADJ
ejpam-5367	7	22	insights	insight	NOUN
ejpam-5367	7	23	into	into	ADP
ejpam-5367	7	24	their	their	PRON
ejpam-5367	7	25	structural	structural	ADJ
ejpam-5367	7	26	relationships	relationship	NOUN
ejpam-5367	7	27	.	.	PUNCT
ejpam-5367	8	1	2020	2020	NUM
ejpam-5367	8	2	mathematics	mathematic	NOUN
ejpam-5367	8	3	subject	subject	NOUN
ejpam-5367	8	4	classifications	classification	NOUN
ejpam-5367	8	5	:	:	PUNCT
ejpam-5367	8	6	03g25	03g25	NUM
ejpam-5367	8	7	,	,	PUNCT
ejpam-5367	8	8	03e72	03e72	NUM
ejpam-5367	8	9	,	,	PUNCT
ejpam-5367	8	10	08a72	08a72	NOUN
ejpam-5367	8	11	key	key	ADJ
ejpam-5367	8	12	words	word	NOUN
ejpam-5367	8	13	and	and	CCONJ
ejpam-5367	8	14	phrases	phrase	NOUN
ejpam-5367	8	15	:	:	PUNCT
ejpam-5367	8	16	iup	iup	NOUN
ejpam-5367	8	17	-	-	PUNCT
ejpam-5367	8	18	algebra	algebra	NOUN
ejpam-5367	8	19	,	,	PUNCT
ejpam-5367	8	20	fermatean	fermatean	ADJ
ejpam-5367	8	21	fuzzy	fuzzy	ADJ
ejpam-5367	8	22	set	set	NOUN
ejpam-5367	8	23	,	,	PUNCT
ejpam-5367	8	24	fermatean	fermatean	ADJ
ejpam-5367	8	25	fuzzy	fuzzy	ADJ
ejpam-5367	8	26	iup	iup	NOUN
ejpam-5367	8	27	-	-	PUNCT
ejpam-5367	8	28	subalgebra	subalgebra	NOUN
ejpam-5367	8	29	,	,	PUNCT
ejpam-5367	8	30	fermatean	fermatean	ADJ
ejpam-5367	8	31	fuzzy	fuzzy	ADJ
ejpam-5367	8	32	iup	iup	NOUN
ejpam-5367	8	33	-	-	PUNCT
ejpam-5367	8	34	ideal	ideal	ADJ
ejpam-5367	8	35	,	,	PUNCT
ejpam-5367	8	36	fermatean	fermatean	ADJ
ejpam-5367	8	37	fuzzy	fuzzy	ADJ
ejpam-5367	8	38	iup	iup	NOUN
ejpam-5367	8	39	-	-	PUNCT
ejpam-5367	8	40	filter	filter	NOUN
ejpam-5367	8	41	,	,	PUNCT
ejpam-5367	8	42	fermatean	fermatean	ADJ
ejpam-5367	8	43	fuzzy	fuzzy	ADJ
ejpam-5367	8	44	strong	strong	ADJ
ejpam-5367	8	45	iup	iup	NOUN
ejpam-5367	8	46	-	-	PUNCT
ejpam-5367	8	47	ideal	ideal	NOUN
ejpam-5367	8	48	,	,	PUNCT
ejpam-5367	8	49	upper	upper	ADJ
ejpam-5367	8	50	t-(strong	t-(strong	NUM
ejpam-5367	8	51	)	)	PUNCT
ejpam-5367	8	52	level	level	NOUN
ejpam-5367	8	53	subset	subset	NOUN
ejpam-5367	8	54	,	,	PUNCT
ejpam-5367	8	55	lower	low	ADJ
ejpam-5367	8	56	t-(strong	t-(strong	NOUN
ejpam-5367	8	57	)	)	PUNCT
ejpam-5367	8	58	level	level	NOUN
ejpam-5367	8	59	subset	subset	NOUN
ejpam-5367	8	60	1	1	X
ejpam-5367	8	61	.	.	X
ejpam-5367	8	62	introduction	introduction	NOUN
ejpam-5367	8	63	the	the	DET
ejpam-5367	8	64	concept	concept	NOUN
ejpam-5367	8	65	of	of	ADP
ejpam-5367	8	66	fuzzy	fuzzy	ADJ
ejpam-5367	8	67	sets	set	NOUN
ejpam-5367	8	68	(	(	PUNCT
ejpam-5367	8	69	fss	fss	NOUN
ejpam-5367	8	70	)	)	PUNCT
ejpam-5367	8	71	,	,	PUNCT
ejpam-5367	8	72	introduced	introduce	VERB
ejpam-5367	8	73	by	by	ADP
ejpam-5367	8	74	zadeh	zadeh	PROPN
ejpam-5367	9	1	[	[	X
ejpam-5367	9	2	15	15	NUM
ejpam-5367	9	3	]	]	PUNCT
ejpam-5367	9	4	,	,	PUNCT
ejpam-5367	9	5	revolutionized	revolutionize	VERB
ejpam-5367	9	6	the	the	DET
ejpam-5367	9	7	handling	handling	NOUN
ejpam-5367	9	8	of	of	ADP
ejpam-5367	9	9	uncertainty	uncertainty	NOUN
ejpam-5367	9	10	by	by	ADP
ejpam-5367	9	11	allowing	allow	VERB
ejpam-5367	9	12	elements	element	NOUN
ejpam-5367	9	13	to	to	PART
ejpam-5367	9	14	have	have	VERB
ejpam-5367	9	15	varying	vary	VERB
ejpam-5367	9	16	degrees	degree	NOUN
ejpam-5367	9	17	of	of	ADP
ejpam-5367	9	18	membership	membership	NOUN
ejpam-5367	9	19	.	.	PUNCT
ejpam-5367	10	1	this	this	DET
ejpam-5367	10	2	foundational	foundational	ADJ
ejpam-5367	10	3	idea	idea	NOUN
ejpam-5367	10	4	was	be	AUX
ejpam-5367	10	5	extended	extend	VERB
ejpam-5367	10	6	by	by	ADP
ejpam-5367	10	7	atanassov	atanassov	NOUN
ejpam-5367	10	8	[	[	X
ejpam-5367	10	9	2	2	NUM
ejpam-5367	10	10	]	]	PUNCT
ejpam-5367	10	11	with	with	ADP
ejpam-5367	10	12	intuitionistic	intuitionistic	ADJ
ejpam-5367	10	13	fuzzy	fuzzy	ADJ
ejpam-5367	10	14	sets	set	NOUN
ejpam-5367	10	15	(	(	PUNCT
ejpam-5367	10	16	ifss	ifss	NOUN
ejpam-5367	10	17	)	)	PUNCT
ejpam-5367	10	18	,	,	PUNCT
ejpam-5367	10	19	which	which	PRON
ejpam-5367	10	20	added	add	VERB
ejpam-5367	10	21	a	a	DET
ejpam-5367	10	22	degree	degree	NOUN
ejpam-5367	10	23	of	of	ADP
ejpam-5367	10	24	non	non	ADJ
ejpam-5367	10	25	-	-	NOUN
ejpam-5367	10	26	membership	membership	NOUN
ejpam-5367	10	27	.	.	PUNCT
ejpam-5367	11	1	yager	yager	NOUN
ejpam-5367	12	1	[	[	X
ejpam-5367	12	2	14	14	NUM
ejpam-5367	12	3	]	]	X
ejpam-5367	12	4	further	far	ADV
ejpam-5367	12	5	advanced	advance	VERB
ejpam-5367	12	6	this	this	DET
ejpam-5367	12	7	field	field	NOUN
ejpam-5367	12	8	with	with	ADP
ejpam-5367	12	9	pythagorean	pythagorean	PROPN
ejpam-5367	12	10	fuzzy	fuzzy	ADJ
ejpam-5367	12	11	sets	set	NOUN
ejpam-5367	12	12	(	(	PUNCT
ejpam-5367	12	13	pfss	pfss	NOUN
ejpam-5367	12	14	)	)	PUNCT
ejpam-5367	12	15	,	,	PUNCT
ejpam-5367	12	16	where	where	SCONJ
ejpam-5367	12	17	the	the	DET
ejpam-5367	12	18	square	square	ADJ
ejpam-5367	12	19	sum	sum	NOUN
ejpam-5367	12	20	of	of	ADP
ejpam-5367	12	21	membership	membership	NOUN
ejpam-5367	12	22	and	and	CCONJ
ejpam-5367	12	23	non	non	ADJ
ejpam-5367	12	24	-	-	ADJ
ejpam-5367	12	25	membership	membership	ADJ
ejpam-5367	12	26	degrees	degree	NOUN
ejpam-5367	12	27	is	be	AUX
ejpam-5367	12	28	≤	≤	NUM
ejpam-5367	12	29	1	1	NUM
ejpam-5367	12	30	.	.	PUNCT
ejpam-5367	13	1	the	the	DET
ejpam-5367	13	2	most	most	ADV
ejpam-5367	13	3	recent	recent	ADJ
ejpam-5367	13	4	development	development	NOUN
ejpam-5367	13	5	,	,	PUNCT
ejpam-5367	13	6	fermatean	fermatean	ADJ
ejpam-5367	13	7	fuzzy	fuzzy	ADJ
ejpam-5367	13	8	sets	set	NOUN
ejpam-5367	13	9	(	(	PUNCT
ejpam-5367	13	10	ffss	ffss	NOUN
ejpam-5367	13	11	)	)	PUNCT
ejpam-5367	13	12	,	,	PUNCT
ejpam-5367	13	13	was	be	AUX
ejpam-5367	13	14	introduced	introduce	VERB
ejpam-5367	13	15	by	by	ADP
ejpam-5367	13	16	senapati	senapati	PROPN
ejpam-5367	13	17	∗corresponding	∗corresponding	NOUN
ejpam-5367	13	18	author	author	NOUN
ejpam-5367	13	19	.	.	PUNCT
ejpam-5367	14	1	doi	doi	NOUN
ejpam-5367	14	2	:	:	PUNCT
ejpam-5367	14	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5367	https://doi.org/10.29020/nybg.ejpam.v17i4.5367	PROPN
ejpam-5367	14	4	email	email	NOUN
ejpam-5367	14	5	addresses	address	NOUN
ejpam-5367	14	6	:	:	PUNCT
ejpam-5367	15	1	kannirun.s@gmail.com	kannirun.s@gmail.com	X
ejpam-5367	15	2	(	(	PUNCT
ejpam-5367	15	3	k.	k.	NOUN
ejpam-5367	15	4	suayngam	suayngam	PROPN
ejpam-5367	15	5	)	)	PUNCT
ejpam-5367	15	6	,	,	PUNCT
ejpam-5367	15	7	rukchart.p@nsru.ac.th	rukchart.p@nsru.ac.th	PROPN
ejpam-5367	15	8	(	(	PUNCT
ejpam-5367	15	9	r.	r.	PROPN
ejpam-5367	15	10	prasertpong	prasertpong	PROPN
ejpam-5367	15	11	)	)	PUNCT
ejpam-5367	15	12	,	,	PUNCT
ejpam-5367	15	13	nareupanat.le@rmuti.ac.th	nareupanat.le@rmuti.ac.th	PROPN
ejpam-5367	15	14	(	(	PUNCT
ejpam-5367	15	15	n.	n.	PROPN
ejpam-5367	15	16	lekkoksung	lekkoksung	PROPN
ejpam-5367	15	17	)	)	PUNCT
ejpam-5367	15	18	,	,	PUNCT
ejpam-5367	15	19	pongpun.j@psru.ac.th	pongpun.j@psru.ac.th	PROPN
ejpam-5367	15	20	(	(	PUNCT
ejpam-5367	15	21	p.	p.	NOUN
ejpam-5367	15	22	julatha	julatha	NOUN
ejpam-5367	15	23	)	)	PUNCT
ejpam-5367	15	24	,	,	PUNCT
ejpam-5367	15	25	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5367	15	26	(	(	PUNCT
ejpam-5367	15	27	a.	a.	NOUN
ejpam-5367	15	28	iampan	iampan	PROPN
ejpam-5367	15	29	)	)	PUNCT
ejpam-5367	15	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5367	15	31	3022	3022	NUM
ejpam-5367	16	1	copyright	copyright	NOUN
ejpam-5367	16	2	:	:	PUNCT
ejpam-5367	16	3	©	©	PROPN
ejpam-5367	16	4	2024	2024	NUM
ejpam-5367	16	5	the	the	DET
ejpam-5367	16	6	author(s	author(s	NOUN
ejpam-5367	16	7	)	)	PUNCT
ejpam-5367	16	8	.	.	PUNCT
ejpam-5367	17	1	(	(	PUNCT
ejpam-5367	17	2	cc	cc	NOUN
ejpam-5367	17	3	by	by	ADP
ejpam-5367	17	4	-	-	PUNCT
ejpam-5367	17	5	nc	nc	PROPN
ejpam-5367	17	6	4.0	4.0	NUM
ejpam-5367	17	7	)	)	PUNCT
ejpam-5367	17	8	a.	a.	NOUN
ejpam-5367	17	9	iampan	iampan	NOUN
ejpam-5367	17	10	et	et	PROPN
ejpam-5367	17	11	al	al	PROPN
ejpam-5367	17	12	.	.	PUNCT
ejpam-5367	17	13	/	/	SYM
ejpam-5367	17	14	eur	eur	PROPN
ejpam-5367	17	15	.	.	PUNCT
ejpam-5367	18	1	j.	j.	PROPN
ejpam-5367	18	2	pure	pure	PROPN
ejpam-5367	18	3	appl	appl	PROPN
ejpam-5367	18	4	.	.	PROPN
ejpam-5367	18	5	math	math	PROPN
ejpam-5367	18	6	,	,	PUNCT
ejpam-5367	18	7	17	17	NUM
ejpam-5367	18	8	(	(	PUNCT
ejpam-5367	18	9	4	4	NUM
ejpam-5367	18	10	)	)	PUNCT
ejpam-5367	18	11	(	(	PUNCT
ejpam-5367	18	12	2024	2024	NUM
ejpam-5367	18	13	)	)	PUNCT
ejpam-5367	18	14	,	,	PUNCT
ejpam-5367	18	15	3022	3022	NUM
ejpam-5367	18	16	-	-	SYM
ejpam-5367	18	17	3042	3042	NUM
ejpam-5367	18	18	3023	3023	NUM
ejpam-5367	18	19	and	and	CCONJ
ejpam-5367	18	20	yager	yager	NOUN
ejpam-5367	19	1	[	[	X
ejpam-5367	19	2	11	11	NUM
ejpam-5367	19	3	]	]	PUNCT
ejpam-5367	19	4	.	.	PUNCT
ejpam-5367	20	1	fermatean	fermatean	PROPN
ejpam-5367	20	2	fuzzy	fuzzy	ADJ
ejpam-5367	20	3	sets	set	NOUN
ejpam-5367	20	4	allow	allow	VERB
ejpam-5367	20	5	the	the	DET
ejpam-5367	20	6	sum	sum	NOUN
ejpam-5367	20	7	of	of	ADP
ejpam-5367	20	8	the	the	DET
ejpam-5367	20	9	cubes	cube	NOUN
ejpam-5367	20	10	of	of	ADP
ejpam-5367	20	11	membership	membership	NOUN
ejpam-5367	20	12	and	and	CCONJ
ejpam-5367	20	13	non	non	ADJ
ejpam-5367	20	14	-	-	ADJ
ejpam-5367	20	15	membership	membership	ADJ
ejpam-5367	20	16	degrees	degree	NOUN
ejpam-5367	20	17	to	to	PART
ejpam-5367	20	18	be	be	AUX
ejpam-5367	20	19	≤	≤	NUM
ejpam-5367	20	20	1	1	NUM
ejpam-5367	20	21	,	,	PUNCT
ejpam-5367	20	22	providing	provide	VERB
ejpam-5367	20	23	even	even	ADV
ejpam-5367	20	24	greater	great	ADJ
ejpam-5367	20	25	flexibility	flexibility	NOUN
ejpam-5367	20	26	and	and	CCONJ
ejpam-5367	20	27	precision	precision	NOUN
ejpam-5367	20	28	.	.	PUNCT
ejpam-5367	21	1	these	these	DET
ejpam-5367	21	2	progressive	progressive	ADJ
ejpam-5367	21	3	enhancements	enhancement	NOUN
ejpam-5367	21	4	have	have	VERB
ejpam-5367	21	5	significantly	significantly	ADV
ejpam-5367	21	6	enriched	enrich	VERB
ejpam-5367	21	7	decision	decision	NOUN
ejpam-5367	21	8	-	-	PUNCT
ejpam-5367	21	9	making	making	NOUN
ejpam-5367	21	10	,	,	PUNCT
ejpam-5367	21	11	medical	medical	ADJ
ejpam-5367	21	12	diagnosis	diagnosis	NOUN
ejpam-5367	21	13	,	,	PUNCT
ejpam-5367	21	14	and	and	CCONJ
ejpam-5367	21	15	risk	risk	NOUN
ejpam-5367	21	16	assessment	assessment	NOUN
ejpam-5367	21	17	,	,	PUNCT
ejpam-5367	21	18	showcasing	showcase	VERB
ejpam-5367	21	19	fuzzy	fuzzy	ADJ
ejpam-5367	21	20	logic	logic	NOUN
ejpam-5367	21	21	’s	’s	PART
ejpam-5367	21	22	dynamic	dynamic	ADJ
ejpam-5367	21	23	evolution	evolution	NOUN
ejpam-5367	21	24	and	and	CCONJ
ejpam-5367	21	25	growing	grow	VERB
ejpam-5367	21	26	sophistication	sophistication	NOUN
ejpam-5367	21	27	in	in	ADP
ejpam-5367	21	28	modelling	model	VERB
ejpam-5367	21	29	complex	complex	ADJ
ejpam-5367	21	30	uncertainties	uncertainty	NOUN
ejpam-5367	21	31	.	.	PUNCT
ejpam-5367	22	1	after	after	ADP
ejpam-5367	22	2	that	that	PRON
ejpam-5367	22	3	,	,	PUNCT
ejpam-5367	22	4	the	the	DET
ejpam-5367	22	5	concept	concept	NOUN
ejpam-5367	22	6	of	of	ADP
ejpam-5367	22	7	fermatean	fermatean	ADJ
ejpam-5367	22	8	fuzzy	fuzzy	ADJ
ejpam-5367	22	9	sets	set	NOUN
ejpam-5367	22	10	has	have	AUX
ejpam-5367	22	11	been	be	AUX
ejpam-5367	22	12	studied	study	VERB
ejpam-5367	22	13	extensively	extensively	ADV
ejpam-5367	22	14	and	and	CCONJ
ejpam-5367	22	15	continuously	continuously	ADV
ejpam-5367	22	16	in	in	ADP
ejpam-5367	22	17	many	many	ADJ
ejpam-5367	22	18	spaces	space	NOUN
ejpam-5367	22	19	,	,	PUNCT
ejpam-5367	22	20	such	such	ADJ
ejpam-5367	22	21	as	as	ADP
ejpam-5367	22	22	lalitha	lalitha	NOUN
ejpam-5367	22	23	and	and	CCONJ
ejpam-5367	22	24	buvaneswari	buvaneswari	VERB
ejpam-5367	23	1	[	[	X
ejpam-5367	23	2	10	10	NUM
ejpam-5367	23	3	]	]	PUNCT
ejpam-5367	23	4	identified	identify	VERB
ejpam-5367	23	5	and	and	CCONJ
ejpam-5367	23	6	proved	prove	VERB
ejpam-5367	23	7	various	various	ADJ
ejpam-5367	23	8	properties	property	NOUN
ejpam-5367	23	9	,	,	PUNCT
ejpam-5367	23	10	especially	especially	ADV
ejpam-5367	23	11	those	those	PRON
ejpam-5367	23	12	involving	involve	VERB
ejpam-5367	23	13	the	the	DET
ejpam-5367	23	14	operation	operation	NOUN
ejpam-5367	23	15	a	a	DET
ejpam-5367	23	16	⇁	⇁	PROPN
ejpam-5367	23	17	b	b	PROPN
ejpam-5367	23	18	defined	define	VERB
ejpam-5367	23	19	as	as	ADP
ejpam-5367	23	20	fermatean	fermatean	NOUN
ejpam-5367	23	21	fuzzy	fuzzy	ADJ
ejpam-5367	23	22	implication	implication	NOUN
ejpam-5367	23	23	with	with	ADP
ejpam-5367	23	24	other	other	ADJ
ejpam-5367	23	25	operations	operation	NOUN
ejpam-5367	23	26	.	.	PUNCT
ejpam-5367	24	1	muhammad	muhammad	PROPN
ejpam-5367	24	2	et	et	PROPN
ejpam-5367	24	3	al	al	PROPN
ejpam-5367	24	4	.	.	PUNCT
ejpam-5367	25	1	[	[	X
ejpam-5367	25	2	8	8	NUM
ejpam-5367	25	3	]	]	PUNCT
ejpam-5367	25	4	proposed	propose	VERB
ejpam-5367	25	5	a	a	DET
ejpam-5367	25	6	new	new	ADJ
ejpam-5367	25	7	type	type	NOUN
ejpam-5367	25	8	of	of	ADP
ejpam-5367	25	9	fuzzy	fuzzy	ADJ
ejpam-5367	25	10	system	system	NOUN
ejpam-5367	25	11	known	know	VERB
ejpam-5367	25	12	as	as	ADP
ejpam-5367	25	13	the	the	DET
ejpam-5367	25	14	fermatean	fermatean	ADJ
ejpam-5367	25	15	fuzzy	fuzzy	ADJ
ejpam-5367	25	16	system	system	NOUN
ejpam-5367	25	17	.	.	PUNCT
ejpam-5367	26	1	more	more	ADV
ejpam-5367	26	2	precisely	precisely	ADV
ejpam-5367	26	3	,	,	PUNCT
ejpam-5367	26	4	they	they	PRON
ejpam-5367	26	5	presented	present	VERB
ejpam-5367	26	6	the	the	DET
ejpam-5367	26	7	notion	notion	NOUN
ejpam-5367	26	8	of	of	ADP
ejpam-5367	26	9	fermatean	fermatean	ADJ
ejpam-5367	26	10	fuzzy	fuzzy	ADJ
ejpam-5367	26	11	ideal	ideal	PROPN
ejpam-5367	26	12	theory	theory	NOUN
ejpam-5367	26	13	and	and	CCONJ
ejpam-5367	26	14	rough	rough	ADJ
ejpam-5367	26	15	fermatean	fermatean	NOUN
ejpam-5367	26	16	fuzzy	fuzzy	ADJ
ejpam-5367	26	17	sets	set	NOUN
ejpam-5367	26	18	in	in	ADP
ejpam-5367	26	19	semigroups	semigroup	NOUN
ejpam-5367	26	20	and	and	CCONJ
ejpam-5367	26	21	initiated	initiate	VERB
ejpam-5367	26	22	the	the	DET
ejpam-5367	26	23	idea	idea	NOUN
ejpam-5367	26	24	of	of	ADP
ejpam-5367	26	25	lower	low	ADJ
ejpam-5367	26	26	and	and	CCONJ
ejpam-5367	26	27	upper	upper	ADJ
ejpam-5367	26	28	approximations	approximation	NOUN
ejpam-5367	26	29	in	in	ADP
ejpam-5367	26	30	fermatean	fermatean	ADJ
ejpam-5367	26	31	fuzzy	fuzzy	ADJ
ejpam-5367	26	32	sets	set	NOUN
ejpam-5367	26	33	.	.	PUNCT
ejpam-5367	27	1	they	they	PRON
ejpam-5367	27	2	extended	extend	VERB
ejpam-5367	27	3	the	the	DET
ejpam-5367	27	4	study	study	NOUN
ejpam-5367	27	5	to	to	ADP
ejpam-5367	27	6	rough	rough	ADJ
ejpam-5367	27	7	fermatean	fermatean	NOUN
ejpam-5367	27	8	fuzzy	fuzzy	ADJ
ejpam-5367	27	9	left	left	ADJ
ejpam-5367	27	10	(	(	PUNCT
ejpam-5367	27	11	resp	resp	NOUN
ejpam-5367	27	12	.	.	PUNCT
ejpam-5367	27	13	,	,	PUNCT
ejpam-5367	27	14	right	right	ADJ
ejpam-5367	27	15	,	,	PUNCT
ejpam-5367	27	16	interior	interior	ADJ
ejpam-5367	27	17	)	)	PUNCT
ejpam-5367	27	18	ideals	ideal	NOUN
ejpam-5367	27	19	in	in	ADP
ejpam-5367	27	20	semigroups	semigroup	NOUN
ejpam-5367	27	21	.	.	PUNCT
ejpam-5367	28	1	balamurugan	balamurugan	NOUN
ejpam-5367	28	2	and	and	CCONJ
ejpam-5367	28	3	nagarajan	nagarajan	NOUN
ejpam-5367	28	4	[	[	X
ejpam-5367	28	5	3	3	NUM
ejpam-5367	28	6	]	]	PUNCT
ejpam-5367	28	7	came	come	VERB
ejpam-5367	28	8	up	up	ADP
ejpam-5367	28	9	with	with	ADP
ejpam-5367	28	10	the	the	DET
ejpam-5367	28	11	idea	idea	NOUN
ejpam-5367	28	12	of	of	ADP
ejpam-5367	28	13	a	a	DET
ejpam-5367	28	14	fermatean	fermatean	ADJ
ejpam-5367	28	15	fuzzy	fuzzy	ADJ
ejpam-5367	28	16	soft	soft	ADV
ejpam-5367	28	17	-	-	PUNCT
ejpam-5367	28	18	covered	cover	VERB
ejpam-5367	28	19	generalized	generalized	ADJ
ejpam-5367	28	20	bi	bi	NOUN
ejpam-5367	28	21	-	-	NOUN
ejpam-5367	28	22	ideal	ideal	NOUN
ejpam-5367	28	23	on	on	ADP
ejpam-5367	28	24	a	a	DET
ejpam-5367	28	25	semigroup	semigroup	NOUN
ejpam-5367	28	26	.	.	PUNCT
ejpam-5367	29	1	this	this	PRON
ejpam-5367	29	2	extends	extend	VERB
ejpam-5367	29	3	the	the	DET
ejpam-5367	29	4	idea	idea	NOUN
ejpam-5367	29	5	of	of	ADP
ejpam-5367	29	6	a	a	DET
ejpam-5367	29	7	fermatean	fermatean	ADJ
ejpam-5367	29	8	fuzzy	fuzzy	ADJ
ejpam-5367	29	9	soft	soft	ADJ
ejpam-5367	29	10	bi	bi	NOUN
ejpam-5367	29	11	-	-	NOUN
ejpam-5367	29	12	ideal	ideal	NOUN
ejpam-5367	29	13	and	and	CCONJ
ejpam-5367	29	14	describes	describe	VERB
ejpam-5367	29	15	regular	regular	ADJ
ejpam-5367	29	16	semigroups	semigroup	NOUN
ejpam-5367	29	17	in	in	ADP
ejpam-5367	29	18	terms	term	NOUN
ejpam-5367	29	19	of	of	ADP
ejpam-5367	29	20	fermatean	fermatean	ADJ
ejpam-5367	29	21	fuzzy	fuzzy	ADJ
ejpam-5367	29	22	soft	soft	ADJ
ejpam-5367	29	23	generalized	generalized	ADJ
ejpam-5367	29	24	bi	bi	NOUN
ejpam-5367	29	25	-	-	NOUN
ejpam-5367	29	26	ideals	ideal	NOUN
ejpam-5367	29	27	.	.	PUNCT
ejpam-5367	30	1	they	they	PRON
ejpam-5367	30	2	framed	frame	VERB
ejpam-5367	30	3	the	the	DET
ejpam-5367	30	4	combining	combining	NOUN
ejpam-5367	30	5	of	of	ADP
ejpam-5367	30	6	fuzzy	fuzzy	ADJ
ejpam-5367	30	7	relations	relation	NOUN
ejpam-5367	30	8	,	,	PUNCT
ejpam-5367	30	9	composition	composition	NOUN
ejpam-5367	30	10	relations	relation	NOUN
ejpam-5367	30	11	,	,	PUNCT
ejpam-5367	30	12	and	and	CCONJ
ejpam-5367	30	13	compatible	compatible	ADJ
ejpam-5367	30	14	relations	relation	NOUN
ejpam-5367	30	15	with	with	ADP
ejpam-5367	30	16	fermatean	fermatean	ADJ
ejpam-5367	30	17	fuzzy	fuzzy	ADJ
ejpam-5367	30	18	sets	set	NOUN
ejpam-5367	30	19	.	.	PUNCT
ejpam-5367	31	1	they	they	PRON
ejpam-5367	31	2	also	also	ADV
ejpam-5367	31	3	introduced	introduce	VERB
ejpam-5367	31	4	the	the	DET
ejpam-5367	31	5	notions	notion	NOUN
ejpam-5367	31	6	of	of	ADP
ejpam-5367	31	7	a	a	DET
ejpam-5367	31	8	fermatean	fermatean	ADJ
ejpam-5367	31	9	fuzzy	fuzzy	ADJ
ejpam-5367	31	10	soft	soft	ADJ
ejpam-5367	31	11	equivalence	equivalence	NOUN
ejpam-5367	31	12	relation	relation	NOUN
ejpam-5367	31	13	and	and	CCONJ
ejpam-5367	31	14	a	a	DET
ejpam-5367	31	15	fermatean	fermatean	ADJ
ejpam-5367	31	16	fuzzy	fuzzy	ADJ
ejpam-5367	31	17	soft	soft	ADJ
ejpam-5367	31	18	compatible	compatible	ADJ
ejpam-5367	31	19	relation	relation	NOUN
ejpam-5367	31	20	on	on	ADP
ejpam-5367	31	21	a	a	DET
ejpam-5367	31	22	semigroup	semigroup	NOUN
ejpam-5367	31	23	.	.	PUNCT
ejpam-5367	32	1	finally	finally	ADV
ejpam-5367	32	2	,	,	PUNCT
ejpam-5367	32	3	they	they	PRON
ejpam-5367	32	4	provided	provide	VERB
ejpam-5367	32	5	a	a	DET
ejpam-5367	32	6	fermatean	fermatean	ADJ
ejpam-5367	32	7	fuzzy	fuzzy	ADJ
ejpam-5367	32	8	soft	soft	ADJ
ejpam-5367	32	9	inverse	inverse	NOUN
ejpam-5367	32	10	relation	relation	NOUN
ejpam-5367	32	11	and	and	CCONJ
ejpam-5367	32	12	a	a	DET
ejpam-5367	32	13	fermatean	fermatean	ADJ
ejpam-5367	32	14	fuzzy	fuzzy	ADJ
ejpam-5367	32	15	soft	soft	ADJ
ejpam-5367	32	16	congruence	congruence	NOUN
ejpam-5367	32	17	on	on	ADP
ejpam-5367	32	18	a	a	DET
ejpam-5367	32	19	semigroup	semigroup	NOUN
ejpam-5367	32	20	.	.	PUNCT
ejpam-5367	32	21	balamurugan	balamurugan	PROPN
ejpam-5367	32	22	and	and	CCONJ
ejpam-5367	32	23	nagarajan	nagarajan	VERB
ejpam-5367	32	24	[	[	X
ejpam-5367	32	25	4	4	NUM
ejpam-5367	32	26	]	]	PUNCT
ejpam-5367	32	27	first	first	ADV
ejpam-5367	32	28	discussed	discuss	VERB
ejpam-5367	32	29	bipolar	bipolar	ADJ
ejpam-5367	32	30	fermatean	fermatean	ADJ
ejpam-5367	32	31	uncertainty	uncertainty	NOUN
ejpam-5367	32	32	subalgebras	subalgebras	PROPN
ejpam-5367	32	33	regarding	regard	VERB
ejpam-5367	32	34	r	r	NOUN
ejpam-5367	32	35	-	-	PUNCT
ejpam-5367	32	36	ideals	ideal	NOUN
ejpam-5367	32	37	.	.	PUNCT
ejpam-5367	33	1	they	they	PRON
ejpam-5367	33	2	also	also	ADV
ejpam-5367	33	3	discussed	discuss	VERB
ejpam-5367	33	4	some	some	DET
ejpam-5367	33	5	exciting	exciting	ADJ
ejpam-5367	33	6	ideas	idea	NOUN
ejpam-5367	33	7	and	and	CCONJ
ejpam-5367	33	8	examined	examine	VERB
ejpam-5367	33	9	how	how	SCONJ
ejpam-5367	33	10	bipolar	bipolar	ADJ
ejpam-5367	33	11	fermatean	fermatean	ADJ
ejpam-5367	33	12	uncertainty	uncertainty	NOUN
ejpam-5367	33	13	soft	soft	ADJ
ejpam-5367	33	14	ideals	ideal	NOUN
ejpam-5367	33	15	and	and	CCONJ
ejpam-5367	33	16	bipolar	bipolar	ADJ
ejpam-5367	33	17	fermatean	fermatean	ADJ
ejpam-5367	33	18	uncertainty	uncertainty	NOUN
ejpam-5367	33	19	soft	soft	ADJ
ejpam-5367	33	20	r	r	NOUN
ejpam-5367	33	21	-	-	PUNCT
ejpam-5367	33	22	ideals	ideal	NOUN
ejpam-5367	33	23	are	be	AUX
ejpam-5367	33	24	related	relate	VERB
ejpam-5367	33	25	.	.	PUNCT
ejpam-5367	34	1	adak	adak	NOUN
ejpam-5367	34	2	et	et	PROPN
ejpam-5367	34	3	al	al	PROPN
ejpam-5367	34	4	.	.	PUNCT
ejpam-5367	35	1	[	[	X
ejpam-5367	35	2	1	1	X
ejpam-5367	35	3	]	]	PUNCT
ejpam-5367	35	4	introduced	introduce	VERB
ejpam-5367	35	5	the	the	DET
ejpam-5367	35	6	concept	concept	NOUN
ejpam-5367	35	7	of	of	ADP
ejpam-5367	35	8	fermatean	fermatean	ADJ
ejpam-5367	35	9	fuzzy	fuzzy	ADJ
ejpam-5367	35	10	semi	semi	ADJ
ejpam-5367	35	11	-	-	ADJ
ejpam-5367	35	12	prime	prime	ADJ
ejpam-5367	35	13	ideals	ideal	NOUN
ejpam-5367	35	14	and	and	CCONJ
ejpam-5367	35	15	fermatean	fermatean	ADJ
ejpam-5367	35	16	fuzzy	fuzzy	ADJ
ejpam-5367	35	17	prime	prime	ADJ
ejpam-5367	35	18	ideals	ideal	NOUN
ejpam-5367	35	19	of	of	ADP
ejpam-5367	35	20	ordered	order	VERB
ejpam-5367	35	21	semigroups	semigroup	NOUN
ejpam-5367	35	22	.	.	PUNCT
ejpam-5367	36	1	they	they	PRON
ejpam-5367	36	2	illustrated	illustrate	VERB
ejpam-5367	36	3	some	some	DET
ejpam-5367	36	4	novel	novel	ADJ
ejpam-5367	36	5	concepts	concept	NOUN
ejpam-5367	36	6	to	to	PART
ejpam-5367	36	7	construct	construct	VERB
ejpam-5367	36	8	fermatean	fermatean	PROPN
ejpam-5367	36	9	fuzzy	fuzzy	ADJ
ejpam-5367	36	10	intra	intra	ADJ
ejpam-5367	36	11	-	-	ADJ
ejpam-5367	36	12	regular	regular	ADJ
ejpam-5367	36	13	and	and	CCONJ
ejpam-5367	36	14	regular	regular	ADJ
ejpam-5367	36	15	ideals	ideal	NOUN
ejpam-5367	36	16	and	and	CCONJ
ejpam-5367	36	17	gave	give	VERB
ejpam-5367	36	18	several	several	ADJ
ejpam-5367	36	19	relations	relation	NOUN
ejpam-5367	36	20	for	for	ADP
ejpam-5367	36	21	the	the	DET
ejpam-5367	36	22	family	family	NOUN
ejpam-5367	36	23	of	of	ADP
ejpam-5367	36	24	fermatean	fermatean	PROPN
ejpam-5367	36	25	fuzzy	fuzzy	ADJ
ejpam-5367	36	26	regular	regular	ADJ
ejpam-5367	36	27	ideals	ideal	NOUN
ejpam-5367	36	28	of	of	ADP
ejpam-5367	36	29	ordered	order	VERB
ejpam-5367	36	30	semigroups	semigroup	NOUN
ejpam-5367	36	31	.	.	PUNCT
ejpam-5367	37	1	iampan	iampan	NOUN
ejpam-5367	37	2	et	et	PROPN
ejpam-5367	37	3	al	al	PROPN
ejpam-5367	37	4	.	.	PUNCT
ejpam-5367	38	1	[	[	X
ejpam-5367	38	2	7	7	X
ejpam-5367	38	3	]	]	PUNCT
ejpam-5367	38	4	introduced	introduce	VERB
ejpam-5367	38	5	the	the	DET
ejpam-5367	38	6	groundbreaking	groundbreake	VERB
ejpam-5367	38	7	concept	concept	NOUN
ejpam-5367	38	8	of	of	ADP
ejpam-5367	38	9	iup	iup	NOUN
ejpam-5367	38	10	-	-	PUNCT
ejpam-5367	38	11	algebras	algebras	PROPN
ejpam-5367	38	12	.	.	PUNCT
ejpam-5367	39	1	this	this	DET
ejpam-5367	39	2	innovative	innovative	ADJ
ejpam-5367	39	3	theory	theory	NOUN
ejpam-5367	39	4	defines	define	VERB
ejpam-5367	39	5	four	four	NUM
ejpam-5367	39	6	key	key	ADJ
ejpam-5367	39	7	subsets	subset	NOUN
ejpam-5367	39	8	:	:	PUNCT
ejpam-5367	39	9	iup	iup	PROPN
ejpam-5367	39	10	-	-	PUNCT
ejpam-5367	39	11	subalgebras	subalgebras	PROPN
ejpam-5367	39	12	,	,	PUNCT
ejpam-5367	39	13	iup	iup	NOUN
ejpam-5367	39	14	-	-	PUNCT
ejpam-5367	39	15	filters	filter	NOUN
ejpam-5367	39	16	,	,	PUNCT
ejpam-5367	39	17	iup	iup	NOUN
ejpam-5367	39	18	-	-	PUNCT
ejpam-5367	39	19	ideals	ideal	NOUN
ejpam-5367	39	20	,	,	PUNCT
ejpam-5367	39	21	and	and	CCONJ
ejpam-5367	39	22	strong	strong	ADJ
ejpam-5367	39	23	iup	iup	NOUN
ejpam-5367	39	24	-	-	PUNCT
ejpam-5367	39	25	ideals	ideal	NOUN
ejpam-5367	39	26	.	.	PUNCT
ejpam-5367	40	1	each	each	DET
ejpam-5367	40	2	subset	subset	NOUN
ejpam-5367	40	3	’s	’s	PART
ejpam-5367	40	4	fundamental	fundamental	ADJ
ejpam-5367	40	5	properties	property	NOUN
ejpam-5367	40	6	were	be	AUX
ejpam-5367	40	7	meticulously	meticulously	ADV
ejpam-5367	40	8	examined	examine	VERB
ejpam-5367	40	9	,	,	PUNCT
ejpam-5367	40	10	unveiling	unveil	VERB
ejpam-5367	40	11	new	new	ADJ
ejpam-5367	40	12	research	research	NOUN
ejpam-5367	40	13	avenues	avenue	NOUN
ejpam-5367	40	14	and	and	CCONJ
ejpam-5367	40	15	applications	application	NOUN
ejpam-5367	40	16	in	in	ADP
ejpam-5367	40	17	the	the	DET
ejpam-5367	40	18	mathematical	mathematical	ADJ
ejpam-5367	40	19	world	world	NOUN
ejpam-5367	40	20	.	.	PUNCT
ejpam-5367	41	1	since	since	SCONJ
ejpam-5367	41	2	its	its	PRON
ejpam-5367	41	3	introduction	introduction	NOUN
ejpam-5367	41	4	,	,	PUNCT
ejpam-5367	41	5	the	the	DET
ejpam-5367	41	6	mathematical	mathematical	ADJ
ejpam-5367	41	7	structure	structure	NOUN
ejpam-5367	41	8	of	of	ADP
ejpam-5367	41	9	iup	iup	NOUN
ejpam-5367	41	10	-	-	PUNCT
ejpam-5367	41	11	algebras	algebras	PROPN
ejpam-5367	41	12	has	have	AUX
ejpam-5367	41	13	captivated	captivate	VERB
ejpam-5367	41	14	numerous	numerous	ADJ
ejpam-5367	41	15	researchers	researcher	NOUN
ejpam-5367	41	16	,	,	PUNCT
ejpam-5367	41	17	sparking	spark	VERB
ejpam-5367	41	18	extensive	extensive	ADJ
ejpam-5367	41	19	studies	study	NOUN
ejpam-5367	41	20	that	that	PRON
ejpam-5367	41	21	continue	continue	VERB
ejpam-5367	41	22	to	to	ADP
ejpam-5367	41	23	this	this	DET
ejpam-5367	41	24	day	day	NOUN
ejpam-5367	41	25	.	.	PUNCT
ejpam-5367	42	1	enthusiastic	enthusiastic	ADJ
ejpam-5367	42	2	scholars	scholar	NOUN
ejpam-5367	42	3	have	have	AUX
ejpam-5367	42	4	delved	delve	VERB
ejpam-5367	42	5	deep	deep	ADV
ejpam-5367	42	6	into	into	ADP
ejpam-5367	42	7	the	the	DET
ejpam-5367	42	8	intricacies	intricacy	NOUN
ejpam-5367	42	9	of	of	ADP
ejpam-5367	42	10	iup	iup	NOUN
ejpam-5367	42	11	-	-	PUNCT
ejpam-5367	42	12	algebras	algebras	PROPN
ejpam-5367	42	13	and	and	CCONJ
ejpam-5367	42	14	applied	apply	VERB
ejpam-5367	42	15	its	its	PRON
ejpam-5367	42	16	principles	principle	NOUN
ejpam-5367	42	17	to	to	ADP
ejpam-5367	42	18	various	various	ADJ
ejpam-5367	42	19	other	other	ADJ
ejpam-5367	42	20	concepts	concept	NOUN
ejpam-5367	42	21	.	.	PUNCT
ejpam-5367	43	1	this	this	PRON
ejpam-5367	43	2	has	have	AUX
ejpam-5367	43	3	led	lead	VERB
ejpam-5367	43	4	to	to	ADP
ejpam-5367	43	5	the	the	DET
ejpam-5367	43	6	creation	creation	NOUN
ejpam-5367	43	7	of	of	ADP
ejpam-5367	43	8	many	many	ADJ
ejpam-5367	43	9	new	new	ADJ
ejpam-5367	43	10	definitions	definition	NOUN
ejpam-5367	43	11	and	and	CCONJ
ejpam-5367	43	12	theories	theory	NOUN
ejpam-5367	43	13	,	,	PUNCT
ejpam-5367	43	14	significantly	significantly	ADV
ejpam-5367	43	15	expanding	expand	VERB
ejpam-5367	43	16	the	the	DET
ejpam-5367	43	17	field	field	NOUN
ejpam-5367	43	18	and	and	CCONJ
ejpam-5367	43	19	demonstrating	demonstrate	VERB
ejpam-5367	43	20	the	the	DET
ejpam-5367	43	21	far	far	ADV
ejpam-5367	43	22	-	-	PUNCT
ejpam-5367	43	23	reaching	reach	VERB
ejpam-5367	43	24	impact	impact	NOUN
ejpam-5367	43	25	of	of	ADP
ejpam-5367	43	26	iup	iup	NOUN
ejpam-5367	43	27	-	-	PUNCT
ejpam-5367	43	28	algebras	algebras	PROPN
ejpam-5367	43	29	on	on	ADP
ejpam-5367	43	30	modern	modern	ADJ
ejpam-5367	43	31	mathematics	mathematic	NOUN
ejpam-5367	43	32	.	.	PUNCT
ejpam-5367	44	1	chanmanee	chanmanee	NOUN
ejpam-5367	44	2	et	et	PROPN
ejpam-5367	44	3	al	al	PROPN
ejpam-5367	44	4	.	.	PUNCT
ejpam-5367	45	1	[	[	X
ejpam-5367	45	2	6	6	NUM
ejpam-5367	45	3	]	]	PUNCT
ejpam-5367	45	4	introduced	introduce	VERB
ejpam-5367	45	5	the	the	DET
ejpam-5367	45	6	concept	concept	NOUN
ejpam-5367	45	7	of	of	ADP
ejpam-5367	45	8	the	the	DET
ejpam-5367	45	9	direct	direct	ADJ
ejpam-5367	45	10	product	product	NOUN
ejpam-5367	45	11	of	of	ADP
ejpam-5367	45	12	an	an	DET
ejpam-5367	45	13	infinite	infinite	ADJ
ejpam-5367	45	14	family	family	NOUN
ejpam-5367	45	15	of	of	ADP
ejpam-5367	45	16	iup	iup	NOUN
ejpam-5367	45	17	-	-	PUNCT
ejpam-5367	45	18	algebras	algebras	PROPN
ejpam-5367	45	19	.	.	PUNCT
ejpam-5367	46	1	they	they	PRON
ejpam-5367	46	2	explored	explore	VERB
ejpam-5367	46	3	the	the	DET
ejpam-5367	46	4	external	external	ADJ
ejpam-5367	46	5	direct	direct	ADJ
ejpam-5367	46	6	product	product	NOUN
ejpam-5367	46	7	of	of	ADP
ejpam-5367	46	8	specific	specific	ADJ
ejpam-5367	46	9	subsets	subset	NOUN
ejpam-5367	46	10	and	and	CCONJ
ejpam-5367	46	11	introduced	introduce	VERB
ejpam-5367	46	12	the	the	DET
ejpam-5367	46	13	weak	weak	ADJ
ejpam-5367	46	14	direct	direct	ADJ
ejpam-5367	46	15	product	product	NOUN
ejpam-5367	46	16	.	.	PUNCT
ejpam-5367	47	1	additionally	additionally	ADV
ejpam-5367	47	2	,	,	PUNCT
ejpam-5367	47	3	they	they	PRON
ejpam-5367	47	4	presented	present	VERB
ejpam-5367	47	5	fundamental	fundamental	ADJ
ejpam-5367	47	6	theorems	theorem	NOUN
ejpam-5367	47	7	on	on	ADP
ejpam-5367	47	8	(	(	PUNCT
ejpam-5367	47	9	anti-)iup	anti-)iup	ADJ
ejpam-5367	47	10	-	-	PUNCT
ejpam-5367	47	11	homomorphisms	homomorphism	NOUN
ejpam-5367	47	12	within	within	ADP
ejpam-5367	47	13	this	this	DET
ejpam-5367	47	14	context	context	NOUN
ejpam-5367	47	15	.	.	PUNCT
ejpam-5367	48	1	their	their	PRON
ejpam-5367	48	2	work	work	NOUN
ejpam-5367	48	3	significantly	significantly	ADV
ejpam-5367	48	4	advances	advance	VERB
ejpam-5367	48	5	both	both	CCONJ
ejpam-5367	48	6	the	the	DET
ejpam-5367	48	7	theoretical	theoretical	ADJ
ejpam-5367	48	8	framework	framework	NOUN
ejpam-5367	48	9	and	and	CCONJ
ejpam-5367	48	10	practical	practical	ADJ
ejpam-5367	48	11	understanding	understanding	NOUN
ejpam-5367	48	12	of	of	ADP
ejpam-5367	48	13	iup	iup	NOUN
ejpam-5367	48	14	-	-	PUNCT
ejpam-5367	48	15	algebras	algebras	PROPN
ejpam-5367	48	16	.	.	PUNCT
ejpam-5367	49	1	chanmanee	chanmanee	PROPN
ejpam-5367	49	2	et	et	PROPN
ejpam-5367	49	3	al	al	PROPN
ejpam-5367	49	4	.	.	PUNCT
ejpam-5367	50	1	[	[	X
ejpam-5367	50	2	5	5	NUM
ejpam-5367	50	3	]	]	PUNCT
ejpam-5367	50	4	pioneered	pioneer	VERB
ejpam-5367	50	5	the	the	DET
ejpam-5367	50	6	concept	concept	NOUN
ejpam-5367	50	7	of	of	ADP
ejpam-5367	50	8	the	the	DET
ejpam-5367	50	9	direct	direct	ADJ
ejpam-5367	50	10	product	product	NOUN
ejpam-5367	50	11	for	for	ADP
ejpam-5367	50	12	an	an	DET
ejpam-5367	50	13	infinite	infinite	ADJ
ejpam-5367	50	14	family	family	NOUN
ejpam-5367	50	15	of	of	ADP
ejpam-5367	50	16	iup	iup	PROPN
ejpam-5367	50	17	-	-	PUNCT
ejpam-5367	50	18	algebras	algebras	PROPN
ejpam-5367	50	19	,	,	PUNCT
ejpam-5367	50	20	a.	a.	NOUN
ejpam-5367	50	21	iampan	iampan	NOUN
ejpam-5367	50	22	et	et	PROPN
ejpam-5367	50	23	al	al	PROPN
ejpam-5367	50	24	.	.	PUNCT
ejpam-5367	50	25	/	/	SYM
ejpam-5367	50	26	eur	eur	PROPN
ejpam-5367	50	27	.	.	PUNCT
ejpam-5367	51	1	j.	j.	PROPN
ejpam-5367	51	2	pure	pure	PROPN
ejpam-5367	51	3	appl	appl	PROPN
ejpam-5367	51	4	.	.	PROPN
ejpam-5367	51	5	math	math	PROPN
ejpam-5367	51	6	,	,	PUNCT
ejpam-5367	51	7	17	17	NUM
ejpam-5367	51	8	(	(	PUNCT
ejpam-5367	51	9	4	4	NUM
ejpam-5367	51	10	)	)	PUNCT
ejpam-5367	51	11	(	(	PUNCT
ejpam-5367	51	12	2024	2024	NUM
ejpam-5367	51	13	)	)	PUNCT
ejpam-5367	51	14	,	,	PUNCT
ejpam-5367	51	15	3022	3022	NUM
ejpam-5367	51	16	-	-	SYM
ejpam-5367	51	17	3042	3042	NUM
ejpam-5367	51	18	3024	3024	NUM
ejpam-5367	51	19	demonstrating	demonstrating	NOUN
ejpam-5367	51	20	that	that	SCONJ
ejpam-5367	51	21	it	it	PRON
ejpam-5367	51	22	forms	form	VERB
ejpam-5367	51	23	a	a	DET
ejpam-5367	51	24	diup	diup	NOUN
ejpam-5367	51	25	-	-	PUNCT
ejpam-5367	51	26	algebra	algebra	NOUN
ejpam-5367	51	27	.	.	PUNCT
ejpam-5367	52	1	additionally	additionally	ADV
ejpam-5367	52	2	,	,	PUNCT
ejpam-5367	52	3	they	they	PRON
ejpam-5367	52	4	introduced	introduce	VERB
ejpam-5367	52	5	the	the	DET
ejpam-5367	52	6	innovative	innovative	ADJ
ejpam-5367	52	7	idea	idea	NOUN
ejpam-5367	52	8	of	of	ADP
ejpam-5367	52	9	weak	weak	ADJ
ejpam-5367	52	10	direct	direct	ADJ
ejpam-5367	52	11	product	product	NOUN
ejpam-5367	52	12	diup	diup	NOUN
ejpam-5367	52	13	-	-	PUNCT
ejpam-5367	52	14	algebras	algebras	PROPN
ejpam-5367	52	15	,	,	PUNCT
ejpam-5367	52	16	further	far	ADV
ejpam-5367	52	17	expanding	expand	VERB
ejpam-5367	52	18	the	the	DET
ejpam-5367	52	19	theoretical	theoretical	ADJ
ejpam-5367	52	20	framework	framework	NOUN
ejpam-5367	52	21	of	of	ADP
ejpam-5367	52	22	iup	iup	NOUN
ejpam-5367	52	23	-	-	PUNCT
ejpam-5367	52	24	algebras	algebras	PROPN
ejpam-5367	52	25	.	.	PUNCT
ejpam-5367	53	1	kuntama	kuntama	PROPN
ejpam-5367	53	2	et	et	PROPN
ejpam-5367	53	3	al	al	PROPN
ejpam-5367	53	4	.	.	PUNCT
ejpam-5367	54	1	[	[	X
ejpam-5367	54	2	9	9	NUM
ejpam-5367	54	3	]	]	PUNCT
ejpam-5367	54	4	has	have	AUX
ejpam-5367	54	5	revolutionized	revolutionize	VERB
ejpam-5367	54	6	the	the	DET
ejpam-5367	54	7	application	application	NOUN
ejpam-5367	54	8	of	of	ADP
ejpam-5367	54	9	fuzzy	fuzzy	ADJ
ejpam-5367	54	10	set	set	NOUN
ejpam-5367	54	11	theory	theory	NOUN
ejpam-5367	54	12	to	to	PART
ejpam-5367	54	13	iup	iup	VERB
ejpam-5367	54	14	-	-	PUNCT
ejpam-5367	54	15	algebras	algebras	NOUN
ejpam-5367	54	16	by	by	ADP
ejpam-5367	54	17	introducing	introduce	VERB
ejpam-5367	54	18	four	four	NUM
ejpam-5367	54	19	groundbreaking	groundbreaking	ADJ
ejpam-5367	54	20	concepts	concept	NOUN
ejpam-5367	54	21	:	:	PUNCT
ejpam-5367	54	22	fuzzy	fuzzy	ADJ
ejpam-5367	54	23	iup	iup	PROPN
ejpam-5367	54	24	-	-	PUNCT
ejpam-5367	54	25	subalgebras	subalgebras	PROPN
ejpam-5367	54	26	,	,	PUNCT
ejpam-5367	54	27	fuzzy	fuzzy	ADJ
ejpam-5367	54	28	iup	iup	NOUN
ejpam-5367	54	29	-	-	PUNCT
ejpam-5367	54	30	ideals	ideal	NOUN
ejpam-5367	54	31	,	,	PUNCT
ejpam-5367	54	32	fuzzy	fuzzy	ADJ
ejpam-5367	54	33	iup	iup	NOUN
ejpam-5367	54	34	-	-	PUNCT
ejpam-5367	54	35	filters	filter	NOUN
ejpam-5367	54	36	,	,	PUNCT
ejpam-5367	54	37	and	and	CCONJ
ejpam-5367	54	38	fuzzy	fuzzy	ADJ
ejpam-5367	54	39	strong	strong	ADJ
ejpam-5367	54	40	iup	iup	NOUN
ejpam-5367	54	41	-	-	PUNCT
ejpam-5367	54	42	ideals	ideal	NOUN
ejpam-5367	54	43	.	.	PUNCT
ejpam-5367	55	1	their	their	PRON
ejpam-5367	55	2	study	study	NOUN
ejpam-5367	55	3	delves	delve	VERB
ejpam-5367	55	4	deep	deep	ADV
ejpam-5367	55	5	into	into	ADP
ejpam-5367	55	6	these	these	DET
ejpam-5367	55	7	innovative	innovative	ADJ
ejpam-5367	55	8	ideas	idea	NOUN
ejpam-5367	55	9	,	,	PUNCT
ejpam-5367	55	10	meticulously	meticulously	ADV
ejpam-5367	55	11	exploring	explore	VERB
ejpam-5367	55	12	their	their	PRON
ejpam-5367	55	13	unique	unique	ADJ
ejpam-5367	55	14	properties	property	NOUN
ejpam-5367	55	15	and	and	CCONJ
ejpam-5367	55	16	intricate	intricate	ADJ
ejpam-5367	55	17	interrelationships	interrelationship	NOUN
ejpam-5367	55	18	.	.	PUNCT
ejpam-5367	56	1	this	this	DET
ejpam-5367	56	2	work	work	NOUN
ejpam-5367	56	3	marks	mark	VERB
ejpam-5367	56	4	a	a	DET
ejpam-5367	56	5	significant	significant	ADJ
ejpam-5367	56	6	advancement	advancement	NOUN
ejpam-5367	56	7	in	in	ADP
ejpam-5367	56	8	the	the	DET
ejpam-5367	56	9	field	field	NOUN
ejpam-5367	56	10	,	,	PUNCT
ejpam-5367	56	11	opening	open	VERB
ejpam-5367	56	12	new	new	ADJ
ejpam-5367	56	13	avenues	avenue	NOUN
ejpam-5367	56	14	for	for	ADP
ejpam-5367	56	15	research	research	NOUN
ejpam-5367	56	16	and	and	CCONJ
ejpam-5367	56	17	application	application	NOUN
ejpam-5367	56	18	.	.	PUNCT
ejpam-5367	57	1	suayngam	suayngam	INTJ
ejpam-5367	57	2	et	et	PROPN
ejpam-5367	57	3	al	al	PROPN
ejpam-5367	57	4	.	.	PUNCT
ejpam-5367	58	1	[	[	X
ejpam-5367	58	2	13	13	NUM
ejpam-5367	58	3	]	]	PUNCT
ejpam-5367	58	4	made	make	VERB
ejpam-5367	58	5	significant	significant	ADJ
ejpam-5367	58	6	steps	step	NOUN
ejpam-5367	58	7	forward	forward	ADV
ejpam-5367	58	8	in	in	ADP
ejpam-5367	58	9	the	the	DET
ejpam-5367	58	10	study	study	NOUN
ejpam-5367	58	11	of	of	ADP
ejpam-5367	58	12	iup	iup	NOUN
ejpam-5367	58	13	-	-	PUNCT
ejpam-5367	58	14	algebras	algebras	PROPN
ejpam-5367	58	15	in	in	ADP
ejpam-5367	58	16	2024	2024	NUM
ejpam-5367	58	17	by	by	ADP
ejpam-5367	58	18	coming	come	VERB
ejpam-5367	58	19	up	up	ADP
ejpam-5367	58	20	with	with	ADP
ejpam-5367	58	21	the	the	DET
ejpam-5367	58	22	ideas	idea	NOUN
ejpam-5367	58	23	of	of	ADP
ejpam-5367	58	24	intuitionistic	intuitionistic	ADJ
ejpam-5367	58	25	fuzzy	fuzzy	ADJ
ejpam-5367	58	26	iup	iup	NOUN
ejpam-5367	58	27	-	-	PUNCT
ejpam-5367	58	28	subalgebras	subalgebras	PROPN
ejpam-5367	58	29	,	,	PUNCT
ejpam-5367	58	30	intuitionistic	intuitionistic	ADJ
ejpam-5367	58	31	fuzzy	fuzzy	ADJ
ejpam-5367	58	32	iup	iup	NOUN
ejpam-5367	58	33	-	-	PUNCT
ejpam-5367	58	34	ideals	ideal	NOUN
ejpam-5367	58	35	,	,	PUNCT
ejpam-5367	58	36	intuitionistic	intuitionistic	ADJ
ejpam-5367	58	37	fuzzy	fuzzy	ADJ
ejpam-5367	58	38	iup	iup	NOUN
ejpam-5367	58	39	-	-	PUNCT
ejpam-5367	58	40	filters	filter	NOUN
ejpam-5367	58	41	,	,	PUNCT
ejpam-5367	58	42	and	and	CCONJ
ejpam-5367	58	43	intuitionistic	intuitionistic	ADJ
ejpam-5367	58	44	fuzzy	fuzzy	ADJ
ejpam-5367	58	45	strong	strong	ADJ
ejpam-5367	58	46	iup	iup	NOUN
ejpam-5367	58	47	-	-	PUNCT
ejpam-5367	58	48	ideals	ideal	NOUN
ejpam-5367	58	49	.	.	PUNCT
ejpam-5367	59	1	this	this	DET
ejpam-5367	59	2	pioneering	pioneer	VERB
ejpam-5367	59	3	work	work	NOUN
ejpam-5367	59	4	expands	expand	VERB
ejpam-5367	59	5	the	the	DET
ejpam-5367	59	6	theoretical	theoretical	ADJ
ejpam-5367	59	7	landscape	landscape	NOUN
ejpam-5367	59	8	of	of	ADP
ejpam-5367	59	9	iup	iup	NOUN
ejpam-5367	59	10	-	-	PUNCT
ejpam-5367	59	11	algebras	algebras	PROPN
ejpam-5367	59	12	,	,	PUNCT
ejpam-5367	59	13	blending	blend	VERB
ejpam-5367	59	14	intuitionistic	intuitionistic	ADJ
ejpam-5367	59	15	fuzzy	fuzzy	ADJ
ejpam-5367	59	16	set	set	NOUN
ejpam-5367	59	17	theory	theory	NOUN
ejpam-5367	59	18	with	with	ADP
ejpam-5367	59	19	algebraic	algebraic	ADJ
ejpam-5367	59	20	structures	structure	NOUN
ejpam-5367	59	21	in	in	ADP
ejpam-5367	59	22	innovative	innovative	ADJ
ejpam-5367	59	23	ways	way	NOUN
ejpam-5367	59	24	.	.	PUNCT
ejpam-5367	60	1	building	build	VERB
ejpam-5367	60	2	on	on	ADP
ejpam-5367	60	3	extensive	extensive	ADJ
ejpam-5367	60	4	research	research	NOUN
ejpam-5367	60	5	into	into	ADP
ejpam-5367	60	6	fermatean	fermatean	ADJ
ejpam-5367	60	7	fuzzy	fuzzy	ADJ
ejpam-5367	60	8	sets	set	NOUN
ejpam-5367	60	9	,	,	PUNCT
ejpam-5367	60	10	this	this	DET
ejpam-5367	60	11	paper	paper	NOUN
ejpam-5367	60	12	aims	aim	VERB
ejpam-5367	60	13	to	to	PART
ejpam-5367	60	14	extend	extend	VERB
ejpam-5367	60	15	these	these	DET
ejpam-5367	60	16	concepts	concept	NOUN
ejpam-5367	60	17	to	to	PART
ejpam-5367	60	18	iup	iup	VERB
ejpam-5367	60	19	-	-	PUNCT
ejpam-5367	60	20	algebras	algebras	PROPN
ejpam-5367	60	21	.	.	PUNCT
ejpam-5367	61	1	we	we	PRON
ejpam-5367	61	2	introduce	introduce	VERB
ejpam-5367	61	3	and	and	CCONJ
ejpam-5367	61	4	explore	explore	VERB
ejpam-5367	61	5	fermatean	fermatean	ADJ
ejpam-5367	61	6	fuzzy	fuzzy	ADJ
ejpam-5367	61	7	iupsubalgebras	iupsubalgebra	NOUN
ejpam-5367	61	8	,	,	PUNCT
ejpam-5367	61	9	fermatean	fermatean	ADJ
ejpam-5367	61	10	fuzzy	fuzzy	ADJ
ejpam-5367	61	11	iup	iup	NOUN
ejpam-5367	61	12	-	-	PUNCT
ejpam-5367	61	13	ideals	ideal	NOUN
ejpam-5367	61	14	,	,	PUNCT
ejpam-5367	61	15	fermatean	fermatean	NOUN
ejpam-5367	61	16	fuzzy	fuzzy	ADJ
ejpam-5367	61	17	iup	iup	NOUN
ejpam-5367	61	18	-	-	PUNCT
ejpam-5367	61	19	filters	filter	NOUN
ejpam-5367	61	20	,	,	PUNCT
ejpam-5367	61	21	and	and	CCONJ
ejpam-5367	61	22	fermatean	fermatean	ADJ
ejpam-5367	61	23	fuzzy	fuzzy	ADJ
ejpam-5367	61	24	strong	strong	ADJ
ejpam-5367	61	25	iup	iup	NOUN
ejpam-5367	61	26	-	-	PUNCT
ejpam-5367	61	27	ideals	ideal	NOUN
ejpam-5367	61	28	.	.	PUNCT
ejpam-5367	62	1	our	our	PRON
ejpam-5367	62	2	study	study	NOUN
ejpam-5367	62	3	investigates	investigate	VERB
ejpam-5367	62	4	their	their	PRON
ejpam-5367	62	5	properties	property	NOUN
ejpam-5367	62	6	,	,	PUNCT
ejpam-5367	62	7	focusing	focus	VERB
ejpam-5367	62	8	on	on	ADP
ejpam-5367	62	9	characteristic	characteristic	ADJ
ejpam-5367	62	10	fermatean	fermatean	NOUN
ejpam-5367	62	11	fuzzy	fuzzy	ADJ
ejpam-5367	62	12	sets	set	NOUN
ejpam-5367	62	13	,	,	PUNCT
ejpam-5367	62	14	upper	upper	ADJ
ejpam-5367	62	15	t-(strong	t-(strong	NUM
ejpam-5367	62	16	)	)	PUNCT
ejpam-5367	62	17	level	level	NOUN
ejpam-5367	62	18	subsets	subset	NOUN
ejpam-5367	62	19	,	,	PUNCT
ejpam-5367	62	20	and	and	CCONJ
ejpam-5367	62	21	lower	low	ADJ
ejpam-5367	62	22	t-(strong	t-(strong	NUM
ejpam-5367	62	23	)	)	PUNCT
ejpam-5367	62	24	level	level	NOUN
ejpam-5367	62	25	subsets	subset	NOUN
ejpam-5367	62	26	.	.	PUNCT
ejpam-5367	63	1	2	2	X
ejpam-5367	63	2	.	.	X
ejpam-5367	63	3	preliminaries	preliminary	NOUN
ejpam-5367	63	4	before	before	ADP
ejpam-5367	63	5	delving	delve	VERB
ejpam-5367	63	6	into	into	ADP
ejpam-5367	63	7	our	our	PRON
ejpam-5367	63	8	study	study	NOUN
ejpam-5367	63	9	,	,	PUNCT
ejpam-5367	63	10	let	let	VERB
ejpam-5367	63	11	’s	’s	PRON
ejpam-5367	63	12	review	review	VERB
ejpam-5367	63	13	the	the	DET
ejpam-5367	63	14	foundational	foundational	ADJ
ejpam-5367	63	15	concepts	concept	NOUN
ejpam-5367	63	16	of	of	ADP
ejpam-5367	63	17	iup	iup	NOUN
ejpam-5367	63	18	-	-	PUNCT
ejpam-5367	63	19	algebras	algebra	NOUN
ejpam-5367	63	20	,	,	PUNCT
ejpam-5367	63	21	including	include	VERB
ejpam-5367	63	22	their	their	PRON
ejpam-5367	63	23	various	various	ADJ
ejpam-5367	63	24	properties	property	NOUN
ejpam-5367	63	25	and	and	CCONJ
ejpam-5367	63	26	pertinent	pertinent	ADJ
ejpam-5367	63	27	definitions	definition	NOUN
ejpam-5367	63	28	crucial	crucial	ADJ
ejpam-5367	63	29	to	to	ADP
ejpam-5367	63	30	this	this	DET
ejpam-5367	63	31	research	research	NOUN
ejpam-5367	63	32	.	.	PUNCT
ejpam-5367	64	1	definition	definition	NOUN
ejpam-5367	64	2	1	1	NUM
ejpam-5367	64	3	.	.	PUNCT
ejpam-5367	65	1	[	[	X
ejpam-5367	65	2	7	7	X
ejpam-5367	65	3	]	]	X
ejpam-5367	65	4	an	an	DET
ejpam-5367	65	5	algebra	algebra	NOUN
ejpam-5367	65	6	x	x	X
ejpam-5367	65	7	=	=	SYM
ejpam-5367	65	8	(	(	PUNCT
ejpam-5367	65	9	x	x	NOUN
ejpam-5367	65	10	;	;	PUNCT
ejpam-5367	65	11	·	·	PUNCT
ejpam-5367	65	12	,	,	PUNCT
ejpam-5367	65	13	0	0	NUM
ejpam-5367	65	14	)	)	PUNCT
ejpam-5367	65	15	of	of	ADP
ejpam-5367	65	16	type	type	NOUN
ejpam-5367	65	17	(	(	PUNCT
ejpam-5367	65	18	2	2	NUM
ejpam-5367	65	19	,	,	PUNCT
ejpam-5367	65	20	0	0	NUM
ejpam-5367	65	21	)	)	PUNCT
ejpam-5367	65	22	is	be	AUX
ejpam-5367	65	23	called	call	VERB
ejpam-5367	65	24	an	an	DET
ejpam-5367	65	25	iup	iup	NOUN
ejpam-5367	65	26	-	-	PUNCT
ejpam-5367	65	27	algebra	algebra	NOUN
ejpam-5367	65	28	,	,	PUNCT
ejpam-5367	65	29	where	where	SCONJ
ejpam-5367	65	30	x	x	PRON
ejpam-5367	65	31	is	be	AUX
ejpam-5367	65	32	a	a	DET
ejpam-5367	65	33	non	non	ADJ
ejpam-5367	65	34	-	-	ADJ
ejpam-5367	65	35	empty	empty	ADJ
ejpam-5367	65	36	set	set	NOUN
ejpam-5367	65	37	,	,	PUNCT
ejpam-5367	65	38	·	·	PUNCT
ejpam-5367	65	39	is	be	AUX
ejpam-5367	65	40	a	a	DET
ejpam-5367	65	41	binary	binary	ADJ
ejpam-5367	65	42	operation	operation	NOUN
ejpam-5367	65	43	on	on	ADP
ejpam-5367	65	44	x	x	NOUN
ejpam-5367	65	45	,	,	PUNCT
ejpam-5367	65	46	and	and	CCONJ
ejpam-5367	65	47	0	0	NUM
ejpam-5367	65	48	is	be	AUX
ejpam-5367	65	49	a	a	DET
ejpam-5367	65	50	fixed	fix	VERB
ejpam-5367	65	51	element	element	NOUN
ejpam-5367	65	52	of	of	ADP
ejpam-5367	65	53	x	x	PRON
ejpam-5367	65	54	if	if	SCONJ
ejpam-5367	65	55	it	it	PRON
ejpam-5367	65	56	satisfies	satisfy	VERB
ejpam-5367	65	57	the	the	DET
ejpam-5367	65	58	following	follow	VERB
ejpam-5367	65	59	axioms	axiom	NOUN
ejpam-5367	65	60	:	:	PUNCT
ejpam-5367	65	61	(	(	PUNCT
ejpam-5367	65	62	∀x	∀x	X
ejpam-5367	65	63	∈	∈	NOUN
ejpam-5367	65	64	x)(0	x)(0	X
ejpam-5367	65	65	·	·	PUNCT
ejpam-5367	66	1	x	x	X
ejpam-5367	66	2	=	=	SYM
ejpam-5367	66	3	x	x	X
ejpam-5367	66	4	)	)	PUNCT
ejpam-5367	66	5	(	(	PUNCT
ejpam-5367	66	6	iup-1	iup-1	X
ejpam-5367	66	7	)	)	PUNCT
ejpam-5367	66	8	(	(	PUNCT
ejpam-5367	66	9	∀x	∀x	X
ejpam-5367	66	10	∈	∈	PROPN
ejpam-5367	66	11	x)(x	x)(x	PROPN
ejpam-5367	66	12	·	·	PUNCT
ejpam-5367	66	13	x	x	PUNCT
ejpam-5367	67	1	=	=	PUNCT
ejpam-5367	67	2	0	0	NUM
ejpam-5367	67	3	)	)	PUNCT
ejpam-5367	67	4	(	(	PUNCT
ejpam-5367	67	5	iup-2	iup-2	NUM
ejpam-5367	67	6	)	)	PUNCT
ejpam-5367	67	7	(	(	PUNCT
ejpam-5367	67	8	∀x	∀x	X
ejpam-5367	67	9	,	,	PUNCT
ejpam-5367	67	10	y	y	PROPN
ejpam-5367	67	11	,	,	PUNCT
ejpam-5367	67	12	z	z	PROPN
ejpam-5367	67	13	∈	∈	PROPN
ejpam-5367	67	14	x)((x	x)((x	NOUN
ejpam-5367	67	15	·	·	PUNCT
ejpam-5367	67	16	y	y	X
ejpam-5367	67	17	)	)	PUNCT
ejpam-5367	67	18	·	·	PUNCT
ejpam-5367	67	19	(	(	PUNCT
ejpam-5367	67	20	x	x	X
ejpam-5367	67	21	·	·	PUNCT
ejpam-5367	67	22	z	z	X
ejpam-5367	67	23	)	)	PUNCT
ejpam-5367	67	24	=	=	SYM
ejpam-5367	68	1	y	y	PROPN
ejpam-5367	68	2	·	·	PUNCT
ejpam-5367	68	3	z	z	X
ejpam-5367	68	4	)	)	PUNCT
ejpam-5367	68	5	(	(	PUNCT
ejpam-5367	68	6	iup-3	iup-3	NOUN
ejpam-5367	68	7	)	)	PUNCT
ejpam-5367	68	8	example	example	NOUN
ejpam-5367	69	1	1	1	NUM
ejpam-5367	69	2	.	.	PUNCT
ejpam-5367	70	1	[	[	X
ejpam-5367	70	2	7	7	X
ejpam-5367	70	3	]	]	X
ejpam-5367	70	4	let	let	VERB
ejpam-5367	70	5	(	(	PUNCT
ejpam-5367	70	6	g	g	NOUN
ejpam-5367	70	7	,	,	PUNCT
ejpam-5367	70	8	•	•	NUM
ejpam-5367	70	9	,	,	PUNCT
ejpam-5367	70	10	e	e	NOUN
ejpam-5367	70	11	)	)	PUNCT
ejpam-5367	70	12	be	be	AUX
ejpam-5367	70	13	a	a	DET
ejpam-5367	70	14	group	group	NOUN
ejpam-5367	70	15	such	such	ADJ
ejpam-5367	70	16	that	that	SCONJ
ejpam-5367	70	17	all	all	DET
ejpam-5367	70	18	elements	element	NOUN
ejpam-5367	70	19	self	self	NOUN
ejpam-5367	70	20	-	-	PUNCT
ejpam-5367	70	21	inverse	inverse	NOUN
ejpam-5367	70	22	.	.	PUNCT
ejpam-5367	71	1	then	then	ADV
ejpam-5367	71	2	(	(	PUNCT
ejpam-5367	71	3	g	g	NOUN
ejpam-5367	71	4	,	,	PUNCT
ejpam-5367	71	5	•	•	NUM
ejpam-5367	71	6	,	,	PUNCT
ejpam-5367	71	7	e	e	NOUN
ejpam-5367	71	8	)	)	PUNCT
ejpam-5367	71	9	is	be	AUX
ejpam-5367	71	10	an	an	DET
ejpam-5367	71	11	iup	iup	NOUN
ejpam-5367	71	12	-	-	PUNCT
ejpam-5367	71	13	algebra	algebra	NOUN
ejpam-5367	71	14	.	.	PUNCT
ejpam-5367	71	15	example	example	NOUN
ejpam-5367	72	1	2	2	NUM
ejpam-5367	72	2	.	.	PUNCT
ejpam-5367	73	1	[	[	X
ejpam-5367	73	2	7	7	X
ejpam-5367	73	3	]	]	X
ejpam-5367	73	4	let	let	VERB
ejpam-5367	73	5	x	x	PRON
ejpam-5367	73	6	be	be	AUX
ejpam-5367	73	7	a	a	DET
ejpam-5367	73	8	set	set	NOUN
ejpam-5367	73	9	and	and	CCONJ
ejpam-5367	73	10	p(x	p(x	PROPN
ejpam-5367	73	11	)	)	PUNCT
ejpam-5367	73	12	means	mean	VERB
ejpam-5367	73	13	the	the	DET
ejpam-5367	73	14	power	power	NOUN
ejpam-5367	73	15	set	set	NOUN
ejpam-5367	73	16	of	of	ADP
ejpam-5367	73	17	x.	x.	NOUN
ejpam-5367	73	18	it	it	PRON
ejpam-5367	73	19	follows	follow	VERB
ejpam-5367	73	20	from	from	ADP
ejpam-5367	73	21	example	example	NOUN
ejpam-5367	73	22	1	1	NUM
ejpam-5367	73	23	that	that	PRON
ejpam-5367	73	24	(	(	PUNCT
ejpam-5367	73	25	p(x	p(x	PROPN
ejpam-5367	73	26	)	)	PUNCT
ejpam-5367	73	27	,	,	PUNCT
ejpam-5367	73	28	△	△	X
ejpam-5367	73	29	,	,	PUNCT
ejpam-5367	73	30	∅	∅	NOUN
ejpam-5367	73	31	)	)	PUNCT
ejpam-5367	73	32	is	be	AUX
ejpam-5367	73	33	an	an	DET
ejpam-5367	73	34	iup	iup	NOUN
ejpam-5367	73	35	-	-	PUNCT
ejpam-5367	73	36	algebra	algebra	NOUN
ejpam-5367	73	37	where	where	SCONJ
ejpam-5367	73	38	the	the	DET
ejpam-5367	73	39	binary	binary	PROPN
ejpam-5367	73	40	operation	operation	PROPN
ejpam-5367	73	41	△	△	PROPN
ejpam-5367	73	42	is	be	AUX
ejpam-5367	73	43	defined	define	VERB
ejpam-5367	73	44	as	as	ADP
ejpam-5367	73	45	the	the	DET
ejpam-5367	73	46	symmetric	symmetric	ADJ
ejpam-5367	73	47	difference	difference	NOUN
ejpam-5367	73	48	of	of	ADP
ejpam-5367	73	49	any	any	DET
ejpam-5367	73	50	two	two	NUM
ejpam-5367	73	51	sets	set	NOUN
ejpam-5367	73	52	.	.	PUNCT
ejpam-5367	74	1	example	example	NOUN
ejpam-5367	75	1	3	3	NUM
ejpam-5367	75	2	.	.	PUNCT
ejpam-5367	76	1	[	[	X
ejpam-5367	76	2	7	7	X
ejpam-5367	76	3	]	]	X
ejpam-5367	76	4	let	let	VERB
ejpam-5367	76	5	(	(	PUNCT
ejpam-5367	76	6	g	g	NOUN
ejpam-5367	76	7	,	,	PUNCT
ejpam-5367	76	8	•	•	NUM
ejpam-5367	76	9	,	,	PUNCT
ejpam-5367	76	10	e	e	NOUN
ejpam-5367	76	11	)	)	PUNCT
ejpam-5367	76	12	be	be	AUX
ejpam-5367	76	13	a	a	DET
ejpam-5367	76	14	group	group	NOUN
ejpam-5367	76	15	with	with	ADP
ejpam-5367	76	16	the	the	DET
ejpam-5367	76	17	identity	identity	NOUN
ejpam-5367	76	18	element	element	NOUN
ejpam-5367	76	19	e.	e.	PROPN
ejpam-5367	76	20	define	define	VERB
ejpam-5367	76	21	a	a	DET
ejpam-5367	76	22	binary	binary	ADJ
ejpam-5367	76	23	operation	operation	NOUN
ejpam-5367	76	24	•	•	NOUN
ejpam-5367	76	25	on	on	ADP
ejpam-5367	76	26	g	g	NOUN
ejpam-5367	76	27	by	by	ADP
ejpam-5367	76	28	:	:	PUNCT
ejpam-5367	76	29	(	(	PUNCT
ejpam-5367	76	30	∀x	∀x	X
ejpam-5367	76	31	,	,	PUNCT
ejpam-5367	76	32	y	y	PROPN
ejpam-5367	76	33	∈	∈	PROPN
ejpam-5367	76	34	g)(x	g)(x	PROPN
ejpam-5367	76	35	•	•	NOUN
ejpam-5367	76	36	y	y	PROPN
ejpam-5367	76	37	=	=	SYM
ejpam-5367	76	38	yx−1	yx−1	NOUN
ejpam-5367	76	39	)	)	PUNCT
ejpam-5367	76	40	(	(	PUNCT
ejpam-5367	76	41	2.1	2.1	NUM
ejpam-5367	76	42	)	)	PUNCT
ejpam-5367	76	43	then	then	ADV
ejpam-5367	76	44	(	(	PUNCT
ejpam-5367	76	45	g	g	NOUN
ejpam-5367	76	46	,	,	PUNCT
ejpam-5367	76	47	•	•	NUM
ejpam-5367	76	48	,	,	PUNCT
ejpam-5367	76	49	e	e	NOUN
ejpam-5367	76	50	)	)	PUNCT
ejpam-5367	76	51	is	be	AUX
ejpam-5367	76	52	an	an	DET
ejpam-5367	76	53	iup	iup	NOUN
ejpam-5367	76	54	-	-	PUNCT
ejpam-5367	76	55	algebra	algebra	NOUN
ejpam-5367	76	56	.	.	PUNCT
ejpam-5367	77	1	a.	a.	NOUN
ejpam-5367	77	2	iampan	iampan	PROPN
ejpam-5367	77	3	et	et	PROPN
ejpam-5367	77	4	al	al	PROPN
ejpam-5367	77	5	.	.	PUNCT
ejpam-5367	77	6	/	/	SYM
ejpam-5367	77	7	eur	eur	PROPN
ejpam-5367	77	8	.	.	PUNCT
ejpam-5367	78	1	j.	j.	PROPN
ejpam-5367	78	2	pure	pure	PROPN
ejpam-5367	78	3	appl	appl	PROPN
ejpam-5367	78	4	.	.	PROPN
ejpam-5367	78	5	math	math	PROPN
ejpam-5367	78	6	,	,	PUNCT
ejpam-5367	78	7	17	17	NUM
ejpam-5367	78	8	(	(	PUNCT
ejpam-5367	78	9	4	4	NUM
ejpam-5367	78	10	)	)	PUNCT
ejpam-5367	78	11	(	(	PUNCT
ejpam-5367	78	12	2024	2024	NUM
ejpam-5367	78	13	)	)	PUNCT
ejpam-5367	78	14	,	,	PUNCT
ejpam-5367	78	15	3022	3022	NUM
ejpam-5367	78	16	-	-	SYM
ejpam-5367	78	17	3042	3042	NUM
ejpam-5367	78	18	3025	3025	NUM
ejpam-5367	78	19	for	for	ADP
ejpam-5367	78	20	convenience	convenience	NOUN
ejpam-5367	78	21	,	,	PUNCT
ejpam-5367	78	22	we	we	PRON
ejpam-5367	78	23	refer	refer	VERB
ejpam-5367	78	24	to	to	ADP
ejpam-5367	78	25	x	x	PUNCT
ejpam-5367	78	26	as	as	ADP
ejpam-5367	78	27	an	an	DET
ejpam-5367	78	28	iup	iup	NOUN
ejpam-5367	78	29	-	-	PUNCT
ejpam-5367	78	30	algebra	algebra	NOUN
ejpam-5367	78	31	x	x	PUNCT
ejpam-5367	78	32	=	=	SYM
ejpam-5367	78	33	(	(	PUNCT
ejpam-5367	78	34	x	x	NOUN
ejpam-5367	78	35	;	;	PUNCT
ejpam-5367	78	36	·	·	PUNCT
ejpam-5367	78	37	,	,	PUNCT
ejpam-5367	78	38	0	0	NUM
ejpam-5367	78	39	)	)	PUNCT
ejpam-5367	78	40	until	until	SCONJ
ejpam-5367	78	41	otherwise	otherwise	ADV
ejpam-5367	78	42	specified	specify	VERB
ejpam-5367	78	43	.	.	PUNCT
ejpam-5367	79	1	proposition	proposition	NOUN
ejpam-5367	79	2	1	1	NUM
ejpam-5367	79	3	.	.	PUNCT
ejpam-5367	80	1	[	[	X
ejpam-5367	80	2	7	7	X
ejpam-5367	80	3	]	]	PUNCT
ejpam-5367	80	4	in	in	ADP
ejpam-5367	80	5	x	x	SYM
ejpam-5367	80	6	,	,	PUNCT
ejpam-5367	80	7	the	the	DET
ejpam-5367	80	8	following	follow	VERB
ejpam-5367	80	9	assertions	assertion	NOUN
ejpam-5367	80	10	are	be	AUX
ejpam-5367	80	11	valid	valid	ADJ
ejpam-5367	80	12	(	(	PUNCT
ejpam-5367	80	13	see	see	VERB
ejpam-5367	80	14	[	[	X
ejpam-5367	80	15	7	7	NUM
ejpam-5367	80	16	]	]	NUM
ejpam-5367	80	17	)	)	PUNCT
ejpam-5367	80	18	.	.	PUNCT
ejpam-5367	81	1	(	(	PUNCT
ejpam-5367	81	2	∀x	∀x	X
ejpam-5367	81	3	,	,	PUNCT
ejpam-5367	81	4	y	y	PROPN
ejpam-5367	81	5	∈	∈	PROPN
ejpam-5367	81	6	x)((x	x)((x	PROPN
ejpam-5367	81	7	·	·	PUNCT
ejpam-5367	81	8	0	0	NUM
ejpam-5367	81	9	)	)	PUNCT
ejpam-5367	81	10	·	·	PUNCT
ejpam-5367	81	11	(	(	PUNCT
ejpam-5367	81	12	x	x	X
ejpam-5367	81	13	·	·	PUNCT
ejpam-5367	81	14	y	y	X
ejpam-5367	81	15	)	)	PUNCT
ejpam-5367	81	16	=	=	SYM
ejpam-5367	81	17	y	y	NOUN
ejpam-5367	81	18	)	)	PUNCT
ejpam-5367	81	19	(	(	PUNCT
ejpam-5367	81	20	2.2	2.2	NUM
ejpam-5367	81	21	)	)	PUNCT
ejpam-5367	81	22	(	(	PUNCT
ejpam-5367	81	23	∀x	∀x	X
ejpam-5367	81	24	∈	∈	PROPN
ejpam-5367	81	25	x)((x	x)((x	NOUN
ejpam-5367	81	26	·	·	PUNCT
ejpam-5367	81	27	0	0	NUM
ejpam-5367	81	28	)	)	PUNCT
ejpam-5367	81	29	·	·	PUNCT
ejpam-5367	82	1	(	(	PUNCT
ejpam-5367	82	2	x	x	X
ejpam-5367	82	3	·	·	PUNCT
ejpam-5367	82	4	0	0	NUM
ejpam-5367	82	5	)	)	PUNCT
ejpam-5367	82	6	=	=	SYM
ejpam-5367	82	7	0	0	X
ejpam-5367	82	8	)	)	PUNCT
ejpam-5367	82	9	(	(	PUNCT
ejpam-5367	82	10	2.3	2.3	NUM
ejpam-5367	82	11	)	)	PUNCT
ejpam-5367	82	12	(	(	PUNCT
ejpam-5367	82	13	∀x	∀x	X
ejpam-5367	82	14	,	,	PUNCT
ejpam-5367	82	15	y	y	PROPN
ejpam-5367	82	16	∈	∈	PROPN
ejpam-5367	82	17	x)((x	x)((x	NOUN
ejpam-5367	82	18	·	·	PUNCT
ejpam-5367	82	19	y	y	X
ejpam-5367	82	20	)	)	PUNCT
ejpam-5367	82	21	·	·	PUNCT
ejpam-5367	82	22	0	0	PUNCT
ejpam-5367	83	1	=	=	SYM
ejpam-5367	83	2	y	y	PROPN
ejpam-5367	83	3	·	·	PUNCT
ejpam-5367	83	4	x	x	X
ejpam-5367	83	5	)	)	PUNCT
ejpam-5367	83	6	(	(	PUNCT
ejpam-5367	83	7	2.4	2.4	NUM
ejpam-5367	83	8	)	)	PUNCT
ejpam-5367	83	9	(	(	PUNCT
ejpam-5367	83	10	∀x	∀x	X
ejpam-5367	83	11	∈	∈	PROPN
ejpam-5367	83	12	x)((x	x)((x	NOUN
ejpam-5367	83	13	·	·	PUNCT
ejpam-5367	83	14	0	0	NUM
ejpam-5367	83	15	)	)	PUNCT
ejpam-5367	83	16	·	·	PUNCT
ejpam-5367	83	17	0	0	PUNCT
ejpam-5367	84	1	=	=	SYM
ejpam-5367	84	2	x	x	X
ejpam-5367	84	3	)	)	PUNCT
ejpam-5367	84	4	(	(	PUNCT
ejpam-5367	84	5	2.5	2.5	NUM
ejpam-5367	84	6	)	)	PUNCT
ejpam-5367	84	7	(	(	PUNCT
ejpam-5367	84	8	∀x	∀x	X
ejpam-5367	84	9	,	,	PUNCT
ejpam-5367	84	10	y	y	PROPN
ejpam-5367	84	11	∈	∈	PROPN
ejpam-5367	84	12	x)(x	x)(x	PROPN
ejpam-5367	84	13	·	·	PUNCT
ejpam-5367	84	14	(	(	PUNCT
ejpam-5367	84	15	(	(	PUNCT
ejpam-5367	84	16	x	x	X
ejpam-5367	84	17	·	·	PUNCT
ejpam-5367	84	18	0	0	NUM
ejpam-5367	84	19	)	)	PUNCT
ejpam-5367	84	20	·	·	PUNCT
ejpam-5367	85	1	y	y	X
ejpam-5367	85	2	)	)	PUNCT
ejpam-5367	85	3	=	=	SYM
ejpam-5367	85	4	y	y	NOUN
ejpam-5367	85	5	)	)	PUNCT
ejpam-5367	85	6	(	(	PUNCT
ejpam-5367	85	7	2.6	2.6	NUM
ejpam-5367	85	8	)	)	PUNCT
ejpam-5367	85	9	(	(	PUNCT
ejpam-5367	85	10	∀x	∀x	X
ejpam-5367	85	11	,	,	PUNCT
ejpam-5367	85	12	y	y	PROPN
ejpam-5367	85	13	∈	∈	PROPN
ejpam-5367	85	14	x)(((x	x)(((x	PUNCT
ejpam-5367	85	15	·	·	PUNCT
ejpam-5367	85	16	0	0	NUM
ejpam-5367	85	17	)	)	PUNCT
ejpam-5367	85	18	·	·	PUNCT
ejpam-5367	86	1	y	y	X
ejpam-5367	86	2	)	)	PUNCT
ejpam-5367	86	3	·	·	PUNCT
ejpam-5367	87	1	x	x	PUNCT
ejpam-5367	87	2	=	=	PUNCT
ejpam-5367	87	3	y	y	PROPN
ejpam-5367	87	4	·	·	PUNCT
ejpam-5367	87	5	0	0	NUM
ejpam-5367	87	6	)	)	PUNCT
ejpam-5367	87	7	(	(	PUNCT
ejpam-5367	87	8	2.7	2.7	NUM
ejpam-5367	87	9	)	)	PUNCT
ejpam-5367	87	10	(	(	PUNCT
ejpam-5367	87	11	∀x	∀x	X
ejpam-5367	87	12	,	,	PUNCT
ejpam-5367	87	13	y	y	PROPN
ejpam-5367	87	14	,	,	PUNCT
ejpam-5367	87	15	z	z	PROPN
ejpam-5367	87	16	∈	∈	PROPN
ejpam-5367	87	17	x)(x	x)(x	PROPN
ejpam-5367	87	18	·	·	PUNCT
ejpam-5367	87	19	y	y	X
ejpam-5367	87	20	=	=	PUNCT
ejpam-5367	87	21	x	x	PUNCT
ejpam-5367	87	22	·	·	PUNCT
ejpam-5367	87	23	z	z	PROPN
ejpam-5367	87	24	⇔	⇔	PROPN
ejpam-5367	87	25	y	y	PROPN
ejpam-5367	87	26	=	=	SYM
ejpam-5367	87	27	z	z	PROPN
ejpam-5367	87	28	)	)	PUNCT
ejpam-5367	87	29	(	(	PUNCT
ejpam-5367	87	30	2.8	2.8	NUM
ejpam-5367	87	31	)	)	PUNCT
ejpam-5367	87	32	(	(	PUNCT
ejpam-5367	87	33	∀x	∀x	X
ejpam-5367	87	34	,	,	PUNCT
ejpam-5367	87	35	y	y	PROPN
ejpam-5367	87	36	∈	∈	PROPN
ejpam-5367	87	37	x)(x	x)(x	PROPN
ejpam-5367	87	38	·	·	PUNCT
ejpam-5367	87	39	y	y	X
ejpam-5367	87	40	=	=	SYM
ejpam-5367	87	41	0	0	NUM
ejpam-5367	87	42	⇔	⇔	X
ejpam-5367	87	43	x	x	X
ejpam-5367	87	44	=	=	SYM
ejpam-5367	87	45	y	y	PROPN
ejpam-5367	87	46	)	)	PUNCT
ejpam-5367	87	47	(	(	PUNCT
ejpam-5367	87	48	2.9	2.9	NUM
ejpam-5367	87	49	)	)	PUNCT
ejpam-5367	87	50	(	(	PUNCT
ejpam-5367	87	51	∀x	∀x	X
ejpam-5367	87	52	∈	∈	PROPN
ejpam-5367	87	53	x)(x	x)(x	PROPN
ejpam-5367	87	54	·	·	PUNCT
ejpam-5367	87	55	0	0	PUNCT
ejpam-5367	88	1	=	=	SYM
ejpam-5367	88	2	0	0	NUM
ejpam-5367	88	3	⇔	⇔	X
ejpam-5367	88	4	x	x	PUNCT
ejpam-5367	88	5	=	=	SYM
ejpam-5367	88	6	0	0	NUM
ejpam-5367	88	7	)	)	PUNCT
ejpam-5367	88	8	(	(	PUNCT
ejpam-5367	88	9	2.10	2.10	NUM
ejpam-5367	88	10	)	)	PUNCT
ejpam-5367	88	11	(	(	PUNCT
ejpam-5367	88	12	∀x	∀x	X
ejpam-5367	88	13	,	,	PUNCT
ejpam-5367	88	14	y	y	PROPN
ejpam-5367	88	15	,	,	PUNCT
ejpam-5367	88	16	z	z	PROPN
ejpam-5367	88	17	∈	∈	PROPN
ejpam-5367	88	18	x)(y	x)(y	PUNCT
ejpam-5367	88	19	·	·	PUNCT
ejpam-5367	88	20	x	x	PUNCT
ejpam-5367	88	21	=	=	PUNCT
ejpam-5367	88	22	z	z	X
ejpam-5367	88	23	·	·	PUNCT
ejpam-5367	88	24	x	x	SYM
ejpam-5367	88	25	⇔	⇔	PROPN
ejpam-5367	88	26	y	y	PROPN
ejpam-5367	88	27	=	=	SYM
ejpam-5367	88	28	z	z	PROPN
ejpam-5367	88	29	)	)	PUNCT
ejpam-5367	88	30	(	(	PUNCT
ejpam-5367	88	31	2.11	2.11	NUM
ejpam-5367	88	32	)	)	PUNCT
ejpam-5367	88	33	(	(	PUNCT
ejpam-5367	88	34	∀x	∀x	X
ejpam-5367	88	35	,	,	PUNCT
ejpam-5367	88	36	y	y	PROPN
ejpam-5367	88	37	∈	∈	PROPN
ejpam-5367	88	38	x)(x	x)(x	PROPN
ejpam-5367	88	39	·	·	PUNCT
ejpam-5367	88	40	y	y	X
ejpam-5367	88	41	=	=	PUNCT
ejpam-5367	88	42	y	y	PROPN
ejpam-5367	88	43	⇒	⇒	VERB
ejpam-5367	88	44	x	x	PUNCT
ejpam-5367	89	1	=	=	NOUN
ejpam-5367	89	2	0	0	NUM
ejpam-5367	89	3	)	)	PUNCT
ejpam-5367	89	4	(	(	PUNCT
ejpam-5367	89	5	2.12	2.12	NUM
ejpam-5367	89	6	)	)	PUNCT
ejpam-5367	89	7	(	(	PUNCT
ejpam-5367	89	8	∀x	∀x	X
ejpam-5367	89	9	,	,	PUNCT
ejpam-5367	89	10	y	y	PROPN
ejpam-5367	89	11	,	,	PUNCT
ejpam-5367	89	12	z	z	PROPN
ejpam-5367	89	13	∈	∈	PROPN
ejpam-5367	89	14	x)((x	x)((x	NOUN
ejpam-5367	89	15	·	·	PUNCT
ejpam-5367	89	16	y	y	X
ejpam-5367	89	17	)	)	PUNCT
ejpam-5367	89	18	·	·	PUNCT
ejpam-5367	89	19	0	0	PUNCT
ejpam-5367	90	1	=	=	SYM
ejpam-5367	90	2	(	(	PUNCT
ejpam-5367	90	3	z	z	NOUN
ejpam-5367	90	4	·	·	PUNCT
ejpam-5367	90	5	y	y	X
ejpam-5367	90	6	)	)	PUNCT
ejpam-5367	90	7	·	·	PUNCT
ejpam-5367	91	1	(	(	PUNCT
ejpam-5367	91	2	z	z	NOUN
ejpam-5367	91	3	·	·	PUNCT
ejpam-5367	91	4	x	x	X
ejpam-5367	91	5	)	)	PUNCT
ejpam-5367	91	6	)	)	PUNCT
ejpam-5367	91	7	(	(	PUNCT
ejpam-5367	91	8	2.13	2.13	NUM
ejpam-5367	91	9	)	)	PUNCT
ejpam-5367	91	10	(	(	PUNCT
ejpam-5367	91	11	∀x	∀x	X
ejpam-5367	91	12	,	,	PUNCT
ejpam-5367	91	13	y	y	PROPN
ejpam-5367	91	14	,	,	PUNCT
ejpam-5367	91	15	z	z	PROPN
ejpam-5367	91	16	∈	∈	PROPN
ejpam-5367	91	17	x)(x	x)(x	PROPN
ejpam-5367	91	18	·	·	PUNCT
ejpam-5367	91	19	y	y	X
ejpam-5367	91	20	=	=	SYM
ejpam-5367	91	21	0	0	PROPN
ejpam-5367	91	22	⇔	⇔	X
ejpam-5367	91	23	(	(	PUNCT
ejpam-5367	91	24	z	z	NOUN
ejpam-5367	91	25	·	·	PUNCT
ejpam-5367	91	26	x	x	X
ejpam-5367	91	27	)	)	PUNCT
ejpam-5367	91	28	·	·	PUNCT
ejpam-5367	91	29	(	(	PUNCT
ejpam-5367	91	30	z	z	X
ejpam-5367	91	31	·	·	PUNCT
ejpam-5367	91	32	y	y	X
ejpam-5367	91	33	)	)	PUNCT
ejpam-5367	91	34	=	=	SYM
ejpam-5367	91	35	0	0	NUM
ejpam-5367	91	36	)	)	PUNCT
ejpam-5367	91	37	(	(	PUNCT
ejpam-5367	91	38	2.14	2.14	NUM
ejpam-5367	91	39	)	)	PUNCT
ejpam-5367	91	40	(	(	PUNCT
ejpam-5367	91	41	∀x	∀x	X
ejpam-5367	91	42	,	,	PUNCT
ejpam-5367	91	43	y	y	PROPN
ejpam-5367	91	44	,	,	PUNCT
ejpam-5367	91	45	z	z	PROPN
ejpam-5367	91	46	∈	∈	PROPN
ejpam-5367	91	47	x)(x	x)(x	PROPN
ejpam-5367	91	48	·	·	PUNCT
ejpam-5367	91	49	y	y	X
ejpam-5367	91	50	=	=	SYM
ejpam-5367	91	51	0	0	PROPN
ejpam-5367	91	52	⇔	⇔	X
ejpam-5367	91	53	(	(	PUNCT
ejpam-5367	91	54	x	x	PROPN
ejpam-5367	91	55	·	·	PUNCT
ejpam-5367	91	56	z	z	X
ejpam-5367	91	57	)	)	PUNCT
ejpam-5367	91	58	·	·	PUNCT
ejpam-5367	91	59	(	(	PUNCT
ejpam-5367	91	60	y	y	PROPN
ejpam-5367	91	61	·	·	PUNCT
ejpam-5367	91	62	z	z	X
ejpam-5367	91	63	)	)	PUNCT
ejpam-5367	91	64	=	=	SYM
ejpam-5367	91	65	0	0	NUM
ejpam-5367	91	66	)	)	PUNCT
ejpam-5367	91	67	(	(	PUNCT
ejpam-5367	91	68	2.15	2.15	NUM
ejpam-5367	91	69	)	)	PUNCT
ejpam-5367	91	70	the	the	DET
ejpam-5367	91	71	right	right	NOUN
ejpam-5367	91	72	and	and	CCONJ
ejpam-5367	91	73	the	the	DET
ejpam-5367	91	74	left	left	ADJ
ejpam-5367	91	75	cancellation	cancellation	NOUN
ejpam-5367	91	76	laws	law	NOUN
ejpam-5367	91	77	hold	hold	VERB
ejpam-5367	91	78	(	(	PUNCT
ejpam-5367	91	79	2.16	2.16	NUM
ejpam-5367	91	80	)	)	PUNCT
ejpam-5367	91	81	in	in	ADP
ejpam-5367	91	82	the	the	DET
ejpam-5367	91	83	realm	realm	NOUN
ejpam-5367	91	84	of	of	ADP
ejpam-5367	91	85	iup	iup	NOUN
ejpam-5367	91	86	-	-	PUNCT
ejpam-5367	91	87	algebras	algebras	PROPN
ejpam-5367	91	88	,	,	PUNCT
ejpam-5367	91	89	four	four	NUM
ejpam-5367	91	90	key	key	ADJ
ejpam-5367	91	91	subsets	subset	NOUN
ejpam-5367	91	92	are	be	AUX
ejpam-5367	91	93	crucial	crucial	ADJ
ejpam-5367	91	94	:	:	PUNCT
ejpam-5367	91	95	iup	iup	PROPN
ejpam-5367	91	96	-	-	PUNCT
ejpam-5367	91	97	subalgebras	subalgebras	PROPN
ejpam-5367	91	98	,	,	PUNCT
ejpam-5367	91	99	iupfilters	iupfilter	NOUN
ejpam-5367	91	100	,	,	PUNCT
ejpam-5367	91	101	iup	iup	NOUN
ejpam-5367	91	102	-	-	PUNCT
ejpam-5367	91	103	ideals	ideal	NOUN
ejpam-5367	91	104	,	,	PUNCT
ejpam-5367	91	105	and	and	CCONJ
ejpam-5367	91	106	strong	strong	ADJ
ejpam-5367	91	107	iup	iup	NOUN
ejpam-5367	91	108	-	-	PUNCT
ejpam-5367	91	109	ideals	ideal	NOUN
ejpam-5367	91	110	.	.	PUNCT
ejpam-5367	92	1	these	these	DET
ejpam-5367	92	2	subsets	subset	NOUN
ejpam-5367	92	3	provide	provide	VERB
ejpam-5367	92	4	a	a	DET
ejpam-5367	92	5	nuanced	nuanced	ADJ
ejpam-5367	92	6	framework	framework	NOUN
ejpam-5367	92	7	essential	essential	ADJ
ejpam-5367	92	8	for	for	ADP
ejpam-5367	92	9	understanding	understanding	NOUN
ejpam-5367	92	10	and	and	CCONJ
ejpam-5367	92	11	applying	apply	VERB
ejpam-5367	92	12	iup	iup	NOUN
ejpam-5367	92	13	-	-	PUNCT
ejpam-5367	92	14	algebras	algebras	PROPN
ejpam-5367	92	15	in	in	ADP
ejpam-5367	92	16	various	various	ADJ
ejpam-5367	92	17	mathematical	mathematical	ADJ
ejpam-5367	92	18	contexts	contexts	NOUN
ejpam-5367	92	19	.	.	PUNCT
ejpam-5367	93	1	definition	definition	NOUN
ejpam-5367	93	2	2	2	NUM
ejpam-5367	93	3	.	.	PUNCT
ejpam-5367	94	1	[	[	X
ejpam-5367	94	2	7	7	X
ejpam-5367	94	3	]	]	X
ejpam-5367	94	4	a	a	DET
ejpam-5367	94	5	non	non	ADJ
ejpam-5367	94	6	-	-	ADJ
ejpam-5367	94	7	empty	empty	ADJ
ejpam-5367	94	8	subset	subset	NOUN
ejpam-5367	94	9	s	s	NOUN
ejpam-5367	94	10	of	of	ADP
ejpam-5367	94	11	x	x	PRON
ejpam-5367	94	12	is	be	AUX
ejpam-5367	94	13	called	call	VERB
ejpam-5367	94	14	(	(	PUNCT
ejpam-5367	94	15	i	i	NOUN
ejpam-5367	94	16	)	)	PUNCT
ejpam-5367	94	17	an	an	DET
ejpam-5367	94	18	iup	iup	NOUN
ejpam-5367	94	19	-	-	PUNCT
ejpam-5367	94	20	subalgebra	subalgebra	NOUN
ejpam-5367	94	21	of	of	ADP
ejpam-5367	94	22	x	x	PRON
ejpam-5367	94	23	if	if	SCONJ
ejpam-5367	94	24	it	it	PRON
ejpam-5367	94	25	satisfies	satisfy	VERB
ejpam-5367	94	26	the	the	DET
ejpam-5367	94	27	following	follow	VERB
ejpam-5367	94	28	condition	condition	NOUN
ejpam-5367	94	29	:	:	PUNCT
ejpam-5367	94	30	(	(	PUNCT
ejpam-5367	94	31	∀x	∀x	X
ejpam-5367	94	32	,	,	PUNCT
ejpam-5367	94	33	y	y	PROPN
ejpam-5367	94	34	∈	∈	PROPN
ejpam-5367	94	35	s)(x	s)(x	PROPN
ejpam-5367	94	36	·	·	PUNCT
ejpam-5367	95	1	y	y	PROPN
ejpam-5367	95	2	∈	∈	PROPN
ejpam-5367	95	3	s	s	PART
ejpam-5367	95	4	)	)	PUNCT
ejpam-5367	95	5	(	(	PUNCT
ejpam-5367	95	6	2.17	2.17	NUM
ejpam-5367	95	7	)	)	PUNCT
ejpam-5367	95	8	(	(	PUNCT
ejpam-5367	95	9	ii	ii	NOUN
ejpam-5367	95	10	)	)	PUNCT
ejpam-5367	95	11	an	an	DET
ejpam-5367	95	12	iup	iup	NOUN
ejpam-5367	95	13	-	-	PUNCT
ejpam-5367	95	14	filter	filter	NOUN
ejpam-5367	95	15	of	of	ADP
ejpam-5367	95	16	x	x	PRON
ejpam-5367	95	17	if	if	SCONJ
ejpam-5367	95	18	it	it	PRON
ejpam-5367	95	19	satisfies	satisfy	VERB
ejpam-5367	95	20	the	the	DET
ejpam-5367	95	21	following	follow	VERB
ejpam-5367	95	22	conditions	condition	NOUN
ejpam-5367	95	23	:	:	PUNCT
ejpam-5367	95	24	the	the	DET
ejpam-5367	95	25	constant	constant	ADJ
ejpam-5367	95	26	0	0	NUM
ejpam-5367	95	27	of	of	ADP
ejpam-5367	95	28	x	x	PRON
ejpam-5367	95	29	is	be	AUX
ejpam-5367	95	30	in	in	ADP
ejpam-5367	95	31	s	s	PROPN
ejpam-5367	95	32	(	(	PUNCT
ejpam-5367	95	33	2.18	2.18	NUM
ejpam-5367	95	34	)	)	PUNCT
ejpam-5367	95	35	(	(	PUNCT
ejpam-5367	95	36	∀x	∀x	X
ejpam-5367	95	37	,	,	PUNCT
ejpam-5367	95	38	y	y	PROPN
ejpam-5367	95	39	∈	∈	PROPN
ejpam-5367	95	40	x)(x	x)(x	PROPN
ejpam-5367	95	41	·	·	PUNCT
ejpam-5367	96	1	y	y	PROPN
ejpam-5367	96	2	∈	∈	PROPN
ejpam-5367	96	3	s	s	PART
ejpam-5367	96	4	and	and	CCONJ
ejpam-5367	96	5	x	x	PUNCT
ejpam-5367	96	6	∈	∈	PROPN
ejpam-5367	96	7	s	s	PART
ejpam-5367	96	8	⇒	⇒	NOUN
ejpam-5367	96	9	y	y	PROPN
ejpam-5367	96	10	∈	∈	PROPN
ejpam-5367	96	11	s	s	PART
ejpam-5367	96	12	)	)	PUNCT
ejpam-5367	96	13	(	(	PUNCT
ejpam-5367	96	14	2.19	2.19	NUM
ejpam-5367	96	15	)	)	PUNCT
ejpam-5367	96	16	(	(	PUNCT
ejpam-5367	96	17	iii	iii	X
ejpam-5367	96	18	)	)	PUNCT
ejpam-5367	96	19	an	an	DET
ejpam-5367	96	20	iup	iup	NOUN
ejpam-5367	96	21	-	-	PUNCT
ejpam-5367	96	22	ideal	ideal	NOUN
ejpam-5367	96	23	of	of	ADP
ejpam-5367	96	24	x	x	PRON
ejpam-5367	96	25	if	if	SCONJ
ejpam-5367	96	26	it	it	PRON
ejpam-5367	96	27	satisfies	satisfy	VERB
ejpam-5367	96	28	the	the	DET
ejpam-5367	96	29	condition	condition	NOUN
ejpam-5367	96	30	(	(	PUNCT
ejpam-5367	96	31	2.18	2.18	NUM
ejpam-5367	96	32	)	)	PUNCT
ejpam-5367	96	33	and	and	CCONJ
ejpam-5367	96	34	the	the	DET
ejpam-5367	96	35	following	follow	VERB
ejpam-5367	96	36	condition	condition	NOUN
ejpam-5367	96	37	:	:	PUNCT
ejpam-5367	96	38	(	(	PUNCT
ejpam-5367	96	39	∀x	∀x	X
ejpam-5367	96	40	,	,	PUNCT
ejpam-5367	96	41	y	y	PROPN
ejpam-5367	96	42	,	,	PUNCT
ejpam-5367	96	43	z	z	PROPN
ejpam-5367	96	44	∈	∈	PROPN
ejpam-5367	96	45	x)(x	x)(x	PROPN
ejpam-5367	96	46	·	·	PUNCT
ejpam-5367	96	47	(	(	PUNCT
ejpam-5367	96	48	y	y	PROPN
ejpam-5367	96	49	·	·	PUNCT
ejpam-5367	96	50	z	z	X
ejpam-5367	96	51	)	)	PUNCT
ejpam-5367	96	52	∈	∈	PROPN
ejpam-5367	96	53	s	s	PART
ejpam-5367	96	54	and	and	CCONJ
ejpam-5367	96	55	y	y	PROPN
ejpam-5367	96	56	∈	∈	PROPN
ejpam-5367	96	57	s	s	PART
ejpam-5367	96	58	⇒	⇒	NOUN
ejpam-5367	96	59	x	x	PUNCT
ejpam-5367	96	60	·	·	PUNCT
ejpam-5367	96	61	z	z	PUNCT
ejpam-5367	96	62	∈	∈	PROPN
ejpam-5367	96	63	s	s	PART
ejpam-5367	96	64	)	)	PUNCT
ejpam-5367	96	65	(	(	PUNCT
ejpam-5367	96	66	2.20	2.20	NUM
ejpam-5367	96	67	)	)	PUNCT
ejpam-5367	96	68	(	(	PUNCT
ejpam-5367	96	69	iv	iv	X
ejpam-5367	96	70	)	)	PUNCT
ejpam-5367	96	71	a	a	DET
ejpam-5367	96	72	strong	strong	ADJ
ejpam-5367	96	73	iup	iup	NOUN
ejpam-5367	96	74	-	-	PUNCT
ejpam-5367	96	75	ideal	ideal	NOUN
ejpam-5367	96	76	of	of	ADP
ejpam-5367	96	77	x	x	PRON
ejpam-5367	96	78	if	if	SCONJ
ejpam-5367	96	79	it	it	PRON
ejpam-5367	96	80	satisfies	satisfy	VERB
ejpam-5367	96	81	the	the	DET
ejpam-5367	96	82	following	follow	VERB
ejpam-5367	96	83	condition	condition	NOUN
ejpam-5367	96	84	:	:	PUNCT
ejpam-5367	96	85	(	(	PUNCT
ejpam-5367	96	86	∀x	∀x	X
ejpam-5367	96	87	,	,	PUNCT
ejpam-5367	96	88	y	y	PROPN
ejpam-5367	96	89	∈	∈	PROPN
ejpam-5367	96	90	x)(y	x)(y	PUNCT
ejpam-5367	97	1	∈	∈	PROPN
ejpam-5367	97	2	s	s	PART
ejpam-5367	97	3	⇒	⇒	NOUN
ejpam-5367	97	4	x	x	X
ejpam-5367	97	5	·	·	PUNCT
ejpam-5367	97	6	y	y	X
ejpam-5367	97	7	∈	∈	PROPN
ejpam-5367	97	8	s	s	PART
ejpam-5367	97	9	)	)	PUNCT
ejpam-5367	97	10	(	(	PUNCT
ejpam-5367	97	11	2.21	2.21	NUM
ejpam-5367	97	12	)	)	PUNCT
ejpam-5367	97	13	a.	a.	NOUN
ejpam-5367	97	14	iampan	iampan	NOUN
ejpam-5367	97	15	et	et	PROPN
ejpam-5367	97	16	al	al	PROPN
ejpam-5367	97	17	.	.	PUNCT
ejpam-5367	97	18	/	/	SYM
ejpam-5367	97	19	eur	eur	PROPN
ejpam-5367	97	20	.	.	PUNCT
ejpam-5367	98	1	j.	j.	PROPN
ejpam-5367	98	2	pure	pure	PROPN
ejpam-5367	98	3	appl	appl	PROPN
ejpam-5367	98	4	.	.	PROPN
ejpam-5367	98	5	math	math	PROPN
ejpam-5367	98	6	,	,	PUNCT
ejpam-5367	98	7	17	17	NUM
ejpam-5367	98	8	(	(	PUNCT
ejpam-5367	98	9	4	4	NUM
ejpam-5367	98	10	)	)	PUNCT
ejpam-5367	98	11	(	(	PUNCT
ejpam-5367	98	12	2024	2024	NUM
ejpam-5367	98	13	)	)	PUNCT
ejpam-5367	98	14	,	,	PUNCT
ejpam-5367	98	15	3022	3022	NUM
ejpam-5367	98	16	-	-	SYM
ejpam-5367	98	17	3042	3042	NUM
ejpam-5367	98	18	3026	3026	NUM
ejpam-5367	98	19	according	accord	VERB
ejpam-5367	98	20	to	to	ADP
ejpam-5367	98	21	[	[	X
ejpam-5367	98	22	7	7	NUM
ejpam-5367	98	23	]	]	PUNCT
ejpam-5367	98	24	,	,	PUNCT
ejpam-5367	98	25	the	the	DET
ejpam-5367	98	26	concept	concept	NOUN
ejpam-5367	98	27	of	of	ADP
ejpam-5367	98	28	iup	iup	NOUN
ejpam-5367	98	29	-	-	PUNCT
ejpam-5367	98	30	filters	filter	NOUN
ejpam-5367	98	31	serves	serve	VERB
ejpam-5367	98	32	as	as	ADP
ejpam-5367	98	33	a	a	DET
ejpam-5367	98	34	generalization	generalization	NOUN
ejpam-5367	98	35	encompassing	encompass	VERB
ejpam-5367	98	36	iup	iup	NOUN
ejpam-5367	98	37	-	-	PUNCT
ejpam-5367	98	38	ideals	ideal	NOUN
ejpam-5367	98	39	and	and	CCONJ
ejpam-5367	98	40	iup	iup	NOUN
ejpam-5367	98	41	-	-	PUNCT
ejpam-5367	98	42	subalgebras	subalgebras	PROPN
ejpam-5367	98	43	.	.	PUNCT
ejpam-5367	99	1	both	both	DET
ejpam-5367	99	2	iup	iup	NOUN
ejpam-5367	99	3	-	-	PUNCT
ejpam-5367	99	4	ideals	ideal	NOUN
ejpam-5367	99	5	and	and	CCONJ
ejpam-5367	99	6	iup	iup	NOUN
ejpam-5367	99	7	-	-	PUNCT
ejpam-5367	99	8	subalgebras	subalgebras	PROPN
ejpam-5367	99	9	,	,	PUNCT
ejpam-5367	99	10	in	in	ADP
ejpam-5367	99	11	turn	turn	NOUN
ejpam-5367	99	12	,	,	PUNCT
ejpam-5367	99	13	generalize	generalize	VERB
ejpam-5367	99	14	strong	strong	ADJ
ejpam-5367	99	15	iup	iup	NOUN
ejpam-5367	99	16	-	-	PUNCT
ejpam-5367	99	17	ideals	ideal	NOUN
ejpam-5367	99	18	.	.	PUNCT
ejpam-5367	100	1	in	in	ADP
ejpam-5367	100	2	an	an	DET
ejpam-5367	100	3	iup	iup	NOUN
ejpam-5367	100	4	-	-	PUNCT
ejpam-5367	100	5	algebra	algebra	NOUN
ejpam-5367	100	6	x	x	NOUN
ejpam-5367	100	7	,	,	PUNCT
ejpam-5367	100	8	it	it	PRON
ejpam-5367	100	9	is	be	AUX
ejpam-5367	100	10	observed	observe	VERB
ejpam-5367	100	11	that	that	SCONJ
ejpam-5367	100	12	strong	strong	ADJ
ejpam-5367	100	13	iup	iup	NOUN
ejpam-5367	100	14	-	-	PUNCT
ejpam-5367	100	15	ideals	ideal	NOUN
ejpam-5367	100	16	coincide	coincide	VERB
ejpam-5367	100	17	with	with	ADP
ejpam-5367	100	18	x	x	X
ejpam-5367	100	19	itself	itself	PRON
ejpam-5367	100	20	.	.	PUNCT
ejpam-5367	101	1	this	this	DET
ejpam-5367	101	2	relationship	relationship	NOUN
ejpam-5367	101	3	is	be	AUX
ejpam-5367	101	4	illustrated	illustrate	VERB
ejpam-5367	101	5	in	in	ADP
ejpam-5367	101	6	the	the	DET
ejpam-5367	101	7	diagram	diagram	NOUN
ejpam-5367	101	8	of	of	ADP
ejpam-5367	101	9	special	special	ADJ
ejpam-5367	101	10	subsets	subset	NOUN
ejpam-5367	101	11	of	of	ADP
ejpam-5367	101	12	iup	iup	NOUN
ejpam-5367	101	13	-	-	PUNCT
ejpam-5367	101	14	algebras	algebras	PROPN
ejpam-5367	101	15	,	,	PUNCT
ejpam-5367	101	16	depicted	depict	VERB
ejpam-5367	101	17	in	in	ADP
ejpam-5367	101	18	figure	figure	NOUN
ejpam-5367	101	19	1	1	NUM
ejpam-5367	101	20	.	.	PUNCT
ejpam-5367	102	1	figure	figure	NOUN
ejpam-5367	102	2	1	1	NUM
ejpam-5367	102	3	:	:	PUNCT
ejpam-5367	102	4	special	special	ADJ
ejpam-5367	102	5	subsets	subset	NOUN
ejpam-5367	102	6	of	of	ADP
ejpam-5367	102	7	iup	iup	NOUN
ejpam-5367	102	8	-	-	PUNCT
ejpam-5367	102	9	algebras	algebras	PROPN
ejpam-5367	102	10	3	3	NUM
ejpam-5367	102	11	.	.	NOUN
ejpam-5367	102	12	main	main	ADJ
ejpam-5367	102	13	results	result	NOUN
ejpam-5367	102	14	before	before	ADP
ejpam-5367	102	15	diving	diving	NOUN
ejpam-5367	102	16	into	into	ADP
ejpam-5367	102	17	the	the	DET
ejpam-5367	102	18	definition	definition	NOUN
ejpam-5367	102	19	of	of	ADP
ejpam-5367	102	20	fermatean	fermatean	ADJ
ejpam-5367	102	21	fuzzy	fuzzy	ADJ
ejpam-5367	102	22	sets	set	NOUN
ejpam-5367	102	23	,	,	PUNCT
ejpam-5367	102	24	it	it	PRON
ejpam-5367	102	25	’s	’	VERB
ejpam-5367	102	26	essential	essential	ADJ
ejpam-5367	102	27	to	to	PART
ejpam-5367	102	28	revisit	revisit	VERB
ejpam-5367	102	29	and	and	CCONJ
ejpam-5367	102	30	understand	understand	VERB
ejpam-5367	102	31	the	the	DET
ejpam-5367	102	32	foundational	foundational	ADJ
ejpam-5367	102	33	concepts	concept	NOUN
ejpam-5367	102	34	that	that	PRON
ejpam-5367	102	35	underpin	underpin	VERB
ejpam-5367	102	36	them	they	PRON
ejpam-5367	102	37	.	.	PUNCT
ejpam-5367	103	1	this	this	DET
ejpam-5367	103	2	background	background	NOUN
ejpam-5367	103	3	will	will	AUX
ejpam-5367	103	4	provide	provide	VERB
ejpam-5367	103	5	the	the	DET
ejpam-5367	103	6	necessary	necessary	ADJ
ejpam-5367	103	7	context	context	NOUN
ejpam-5367	103	8	and	and	CCONJ
ejpam-5367	103	9	enhance	enhance	VERB
ejpam-5367	103	10	our	our	PRON
ejpam-5367	103	11	comprehension	comprehension	NOUN
ejpam-5367	103	12	of	of	ADP
ejpam-5367	103	13	fermatean	fermatean	ADJ
ejpam-5367	103	14	fuzzy	fuzzy	ADJ
ejpam-5367	103	15	sets	set	NOUN
ejpam-5367	103	16	.	.	PUNCT
ejpam-5367	104	1	from	from	ADP
ejpam-5367	104	2	now	now	ADV
ejpam-5367	104	3	on	on	ADV
ejpam-5367	104	4	,	,	PUNCT
ejpam-5367	104	5	we	we	PRON
ejpam-5367	104	6	will	will	AUX
ejpam-5367	104	7	use	use	VERB
ejpam-5367	104	8	abbreviations	abbreviation	NOUN
ejpam-5367	104	9	to	to	PART
ejpam-5367	104	10	represent	represent	VERB
ejpam-5367	104	11	the	the	DET
ejpam-5367	104	12	following	follow	VERB
ejpam-5367	104	13	technical	technical	ADJ
ejpam-5367	104	14	terms	term	NOUN
ejpam-5367	104	15	.	.	PUNCT
ejpam-5367	105	1	technical	technical	ADJ
ejpam-5367	105	2	terms	term	NOUN
ejpam-5367	105	3	abbreviations	abbreviation	NOUN
ejpam-5367	105	4	fuzzy	fuzzy	ADJ
ejpam-5367	105	5	set	set	VERB
ejpam-5367	105	6	fs	fs	ADP
ejpam-5367	105	7	fermatean	fermatean	ADJ
ejpam-5367	105	8	fuzzy	fuzzy	PROPN
ejpam-5367	105	9	set	set	VERB
ejpam-5367	105	10	ffs	ffs	NOUN
ejpam-5367	105	11	fermatean	fermatean	PROPN
ejpam-5367	105	12	fuzzy	fuzzy	ADJ
ejpam-5367	105	13	iup	iup	NOUN
ejpam-5367	105	14	-	-	PUNCT
ejpam-5367	105	15	subalgebra	subalgebra	NOUN
ejpam-5367	105	16	ffiup	ffiup	ADJ
ejpam-5367	105	17	-	-	PUNCT
ejpam-5367	105	18	subalgebra	subalgebra	NOUN
ejpam-5367	105	19	fermatean	fermatean	NOUN
ejpam-5367	105	20	fuzzy	fuzzy	ADJ
ejpam-5367	105	21	iup	iup	NOUN
ejpam-5367	105	22	-	-	PUNCT
ejpam-5367	105	23	ideal	ideal	NOUN
ejpam-5367	105	24	ffiup	ffiup	NOUN
ejpam-5367	105	25	-	-	PUNCT
ejpam-5367	105	26	ideal	ideal	NOUN
ejpam-5367	105	27	fermatean	fermatean	NOUN
ejpam-5367	105	28	fuzzy	fuzzy	ADJ
ejpam-5367	105	29	iup	iup	NOUN
ejpam-5367	105	30	-	-	PUNCT
ejpam-5367	105	31	filter	filter	NOUN
ejpam-5367	105	32	ffiup	ffiup	ADJ
ejpam-5367	105	33	-	-	PUNCT
ejpam-5367	105	34	filter	filter	NOUN
ejpam-5367	105	35	fermatean	fermatean	NOUN
ejpam-5367	105	36	fuzzy	fuzzy	ADJ
ejpam-5367	105	37	strong	strong	ADJ
ejpam-5367	105	38	iup	iup	NOUN
ejpam-5367	105	39	-	-	PUNCT
ejpam-5367	105	40	ideal	ideal	NOUN
ejpam-5367	105	41	ffsiup	ffsiup	NOUN
ejpam-5367	105	42	-	-	PUNCT
ejpam-5367	105	43	ideal	ideal	NOUN
ejpam-5367	105	44	definition	definition	NOUN
ejpam-5367	105	45	3	3	NUM
ejpam-5367	105	46	.	.	PUNCT
ejpam-5367	106	1	[	[	X
ejpam-5367	106	2	2	2	X
ejpam-5367	106	3	]	]	PUNCT
ejpam-5367	106	4	let	let	VERB
ejpam-5367	106	5	x	x	PRON
ejpam-5367	106	6	be	be	AUX
ejpam-5367	106	7	a	a	DET
ejpam-5367	106	8	universe	universe	NOUN
ejpam-5367	106	9	of	of	ADP
ejpam-5367	106	10	discourse	discourse	NOUN
ejpam-5367	106	11	.	.	PUNCT
ejpam-5367	107	1	a	a	DET
ejpam-5367	107	2	fermatean	fermatean	ADJ
ejpam-5367	107	3	fuzzy	fuzzy	NOUN
ejpam-5367	107	4	set	set	VERB
ejpam-5367	107	5	f	f	PROPN
ejpam-5367	107	6	(	(	PUNCT
ejpam-5367	107	7	ffs	ffs	PROPN
ejpam-5367	107	8	)	)	PUNCT
ejpam-5367	107	9	in	in	ADP
ejpam-5367	107	10	x	x	PRON
ejpam-5367	107	11	is	be	AUX
ejpam-5367	107	12	an	an	DET
ejpam-5367	107	13	object	object	NOUN
ejpam-5367	107	14	having	have	VERB
ejpam-5367	107	15	the	the	DET
ejpam-5367	107	16	form	form	NOUN
ejpam-5367	107	17	f	f	X
ejpam-5367	107	18	=	=	PRON
ejpam-5367	107	19	{	{	PUNCT
ejpam-5367	107	20	(	(	PUNCT
ejpam-5367	107	21	x	x	X
ejpam-5367	107	22	,	,	PUNCT
ejpam-5367	107	23	αf	αf	PROPN
ejpam-5367	107	24	(	(	PUNCT
ejpam-5367	107	25	x	x	NOUN
ejpam-5367	107	26	)	)	PUNCT
ejpam-5367	107	27	,	,	PUNCT
ejpam-5367	107	28	βf	βf	CCONJ
ejpam-5367	107	29	(	(	PUNCT
ejpam-5367	107	30	x	x	NOUN
ejpam-5367	107	31	)	)	PUNCT
ejpam-5367	107	32	)	)	PUNCT
ejpam-5367	107	33	:	:	PUNCT
ejpam-5367	108	1	x	x	X
ejpam-5367	108	2	∈	∈	NOUN
ejpam-5367	108	3	x	x	X
ejpam-5367	108	4	}	}	PUNCT
ejpam-5367	108	5	,	,	PUNCT
ejpam-5367	108	6	where	where	SCONJ
ejpam-5367	108	7	αf	αf	VERB
ejpam-5367	108	8	(	(	PUNCT
ejpam-5367	108	9	x	x	NOUN
ejpam-5367	108	10	)	)	PUNCT
ejpam-5367	108	11	:	:	PUNCT
ejpam-5367	108	12	x	x	X
ejpam-5367	108	13	→	→	PUNCT
ejpam-5367	108	14	[	[	X
ejpam-5367	108	15	0	0	NUM
ejpam-5367	108	16	,	,	PUNCT
ejpam-5367	108	17	1	1	NUM
ejpam-5367	108	18	]	]	PUNCT
ejpam-5367	108	19	and	and	CCONJ
ejpam-5367	108	20	βf	βf	INTJ
ejpam-5367	108	21	(	(	PUNCT
ejpam-5367	108	22	x	x	X
ejpam-5367	108	23	)	)	PUNCT
ejpam-5367	108	24	:	:	PUNCT
ejpam-5367	108	25	x	x	X
ejpam-5367	108	26	→	→	PUNCT
ejpam-5367	108	27	[	[	X
ejpam-5367	108	28	0	0	NUM
ejpam-5367	108	29	,	,	PUNCT
ejpam-5367	108	30	1	1	NUM
ejpam-5367	108	31	]	]	PUNCT
ejpam-5367	108	32	,	,	PUNCT
ejpam-5367	108	33	including	include	VERB
ejpam-5367	108	34	the	the	DET
ejpam-5367	108	35	following	follow	VERB
ejpam-5367	108	36	condition	condition	NOUN
ejpam-5367	108	37	:	:	PUNCT
ejpam-5367	108	38	(	(	PUNCT
ejpam-5367	108	39	∀x	∀x	X
ejpam-5367	108	40	∈	∈	NOUN
ejpam-5367	108	41	x)(0	x)(0	X
ejpam-5367	108	42	≤	≤	X
ejpam-5367	108	43	(	(	PUNCT
ejpam-5367	108	44	αf	αf	X
ejpam-5367	108	45	(	(	PUNCT
ejpam-5367	108	46	x	x	NOUN
ejpam-5367	108	47	)	)	PUNCT
ejpam-5367	108	48	)	)	PUNCT
ejpam-5367	108	49	3	3	NUM
ejpam-5367	109	1	+	+	CCONJ
ejpam-5367	109	2	(	(	PUNCT
ejpam-5367	109	3	βf	βf	INTJ
ejpam-5367	109	4	(	(	PUNCT
ejpam-5367	109	5	x	x	NOUN
ejpam-5367	109	6	)	)	PUNCT
ejpam-5367	109	7	)	)	PUNCT
ejpam-5367	109	8	3	3	NUM
ejpam-5367	109	9	≤	≤	NUM
ejpam-5367	109	10	1	1	NUM
ejpam-5367	109	11	)	)	PUNCT
ejpam-5367	109	12	(	(	PUNCT
ejpam-5367	109	13	3.1	3.1	NUM
ejpam-5367	109	14	)	)	PUNCT
ejpam-5367	109	15	the	the	DET
ejpam-5367	109	16	numbers	number	NOUN
ejpam-5367	109	17	αf	αf	VERB
ejpam-5367	109	18	(	(	PUNCT
ejpam-5367	109	19	x	x	NOUN
ejpam-5367	109	20	)	)	PUNCT
ejpam-5367	109	21	and	and	CCONJ
ejpam-5367	109	22	βf	βf	INTJ
ejpam-5367	109	23	(	(	PUNCT
ejpam-5367	109	24	x	x	X
ejpam-5367	109	25	)	)	PUNCT
ejpam-5367	109	26	denote	denote	NOUN
ejpam-5367	109	27	,	,	PUNCT
ejpam-5367	109	28	respectively	respectively	ADV
ejpam-5367	109	29	,	,	PUNCT
ejpam-5367	109	30	the	the	DET
ejpam-5367	109	31	degree	degree	NOUN
ejpam-5367	109	32	of	of	ADP
ejpam-5367	109	33	membership	membership	NOUN
ejpam-5367	109	34	and	and	CCONJ
ejpam-5367	109	35	the	the	DET
ejpam-5367	109	36	degree	degree	NOUN
ejpam-5367	109	37	of	of	ADP
ejpam-5367	109	38	non	non	ADJ
ejpam-5367	109	39	-	-	NOUN
ejpam-5367	109	40	membership	membership	NOUN
ejpam-5367	109	41	of	of	ADP
ejpam-5367	109	42	the	the	DET
ejpam-5367	109	43	element	element	NOUN
ejpam-5367	109	44	x	x	PUNCT
ejpam-5367	109	45	in	in	ADP
ejpam-5367	109	46	the	the	DET
ejpam-5367	109	47	set	set	NOUN
ejpam-5367	109	48	f	f	PROPN
ejpam-5367	109	49	.	.	PUNCT
ejpam-5367	110	1	for	for	ADP
ejpam-5367	110	2	any	any	DET
ejpam-5367	110	3	ffs	ffs	NOUN
ejpam-5367	110	4	f	f	PROPN
ejpam-5367	110	5	and	and	CCONJ
ejpam-5367	110	6	x	x	PUNCT
ejpam-5367	110	7	∈	∈	PROPN
ejpam-5367	110	8	x	x	X
ejpam-5367	110	9	,	,	PUNCT
ejpam-5367	110	10	πf	πf	INTJ
ejpam-5367	110	11	(	(	PUNCT
ejpam-5367	110	12	x	x	X
ejpam-5367	110	13	)	)	PUNCT
ejpam-5367	110	14	=	=	SYM
ejpam-5367	110	15	3	3	NUM
ejpam-5367	110	16	√	√	NUM
ejpam-5367	110	17	1−	1−	NUM
ejpam-5367	110	18	(	(	PUNCT
ejpam-5367	110	19	αf	αf	X
ejpam-5367	110	20	(	(	PUNCT
ejpam-5367	110	21	x))3	x))3	PROPN
ejpam-5367	110	22	−	−	PROPN
ejpam-5367	111	1	(	(	PUNCT
ejpam-5367	111	2	βf	βf	INTJ
ejpam-5367	111	3	(	(	PUNCT
ejpam-5367	111	4	x))3	x))3	PROPN
ejpam-5367	111	5	is	be	AUX
ejpam-5367	111	6	identified	identify	VERB
ejpam-5367	111	7	as	as	ADP
ejpam-5367	111	8	the	the	DET
ejpam-5367	111	9	degree	degree	NOUN
ejpam-5367	111	10	of	of	ADP
ejpam-5367	111	11	indeterminacy	indeterminacy	NOUN
ejpam-5367	111	12	of	of	ADP
ejpam-5367	111	13	x	x	PUNCT
ejpam-5367	111	14	to	to	ADP
ejpam-5367	111	15	f	f	PROPN
ejpam-5367	111	16	.	.	PUNCT
ejpam-5367	112	1	in	in	ADP
ejpam-5367	112	2	the	the	DET
ejpam-5367	112	3	interest	interest	NOUN
ejpam-5367	112	4	of	of	ADP
ejpam-5367	112	5	simplicity	simplicity	NOUN
ejpam-5367	112	6	,	,	PUNCT
ejpam-5367	112	7	we	we	PRON
ejpam-5367	112	8	shall	shall	AUX
ejpam-5367	112	9	mention	mention	VERB
ejpam-5367	112	10	the	the	DET
ejpam-5367	112	11	symbol	symbol	NOUN
ejpam-5367	112	12	f	f	NOUN
ejpam-5367	113	1	=	=	PUNCT
ejpam-5367	113	2	(	(	PUNCT
ejpam-5367	113	3	αf	αf	NOUN
ejpam-5367	113	4	,	,	PUNCT
ejpam-5367	113	5	βf	βf	CCONJ
ejpam-5367	113	6	)	)	PUNCT
ejpam-5367	113	7	for	for	ADP
ejpam-5367	113	8	the	the	DET
ejpam-5367	113	9	ffs	ffs	PROPN
ejpam-5367	113	10	f	f	PROPN
ejpam-5367	113	11	=	=	PRON
ejpam-5367	113	12	{	{	PUNCT
ejpam-5367	113	13	(	(	PUNCT
ejpam-5367	113	14	x	x	X
ejpam-5367	113	15	,	,	PUNCT
ejpam-5367	113	16	αf	αf	PROPN
ejpam-5367	113	17	(	(	PUNCT
ejpam-5367	113	18	x	x	NOUN
ejpam-5367	113	19	)	)	PUNCT
ejpam-5367	113	20	,	,	PUNCT
ejpam-5367	113	21	βf	βf	CCONJ
ejpam-5367	113	22	(	(	PUNCT
ejpam-5367	113	23	x	x	NOUN
ejpam-5367	113	24	)	)	PUNCT
ejpam-5367	113	25	)	)	PUNCT
ejpam-5367	113	26	:	:	PUNCT
ejpam-5367	114	1	x	x	X
ejpam-5367	114	2	∈	∈	NOUN
ejpam-5367	114	3	x	x	X
ejpam-5367	114	4	}	}	PUNCT
ejpam-5367	114	5	.	.	PUNCT
ejpam-5367	115	1	a.	a.	NOUN
ejpam-5367	115	2	iampan	iampan	PROPN
ejpam-5367	115	3	et	et	PROPN
ejpam-5367	115	4	al	al	PROPN
ejpam-5367	115	5	.	.	PUNCT
ejpam-5367	115	6	/	/	SYM
ejpam-5367	115	7	eur	eur	PROPN
ejpam-5367	115	8	.	.	PUNCT
ejpam-5367	116	1	j.	j.	PROPN
ejpam-5367	116	2	pure	pure	PROPN
ejpam-5367	116	3	appl	appl	PROPN
ejpam-5367	116	4	.	.	PROPN
ejpam-5367	116	5	math	math	PROPN
ejpam-5367	116	6	,	,	PUNCT
ejpam-5367	116	7	17	17	NUM
ejpam-5367	116	8	(	(	PUNCT
ejpam-5367	116	9	4	4	NUM
ejpam-5367	116	10	)	)	PUNCT
ejpam-5367	116	11	(	(	PUNCT
ejpam-5367	116	12	2024	2024	NUM
ejpam-5367	116	13	)	)	PUNCT
ejpam-5367	116	14	,	,	PUNCT
ejpam-5367	116	15	3022	3022	NUM
ejpam-5367	116	16	-	-	SYM
ejpam-5367	116	17	3042	3042	NUM
ejpam-5367	116	18	3027	3027	NUM
ejpam-5367	116	19	for	for	ADP
ejpam-5367	116	20	a	a	DET
ejpam-5367	116	21	subset	subset	NOUN
ejpam-5367	116	22	g	g	NOUN
ejpam-5367	116	23	of	of	ADP
ejpam-5367	116	24	a	a	DET
ejpam-5367	116	25	non	non	ADJ
ejpam-5367	116	26	-	-	ADJ
ejpam-5367	116	27	empty	empty	ADJ
ejpam-5367	116	28	set	set	NOUN
ejpam-5367	116	29	x	x	NOUN
ejpam-5367	116	30	,	,	PUNCT
ejpam-5367	116	31	the	the	DET
ejpam-5367	116	32	characteristic	characteristic	ADJ
ejpam-5367	116	33	functions	function	NOUN
ejpam-5367	116	34	αfg	αfg	NOUN
ejpam-5367	116	35	and	and	CCONJ
ejpam-5367	116	36	βfg	βfg	PROPN
ejpam-5367	116	37	are	be	AUX
ejpam-5367	116	38	functions	function	NOUN
ejpam-5367	116	39	of	of	ADP
ejpam-5367	116	40	x	x	PUNCT
ejpam-5367	116	41	into	into	ADP
ejpam-5367	116	42	{	{	PUNCT
ejpam-5367	116	43	0	0	NUM
ejpam-5367	116	44	,	,	PUNCT
ejpam-5367	116	45	1	1	NUM
ejpam-5367	116	46	}	}	PUNCT
ejpam-5367	116	47	defined	define	VERB
ejpam-5367	116	48	as	as	SCONJ
ejpam-5367	116	49	follows	follow	VERB
ejpam-5367	116	50	:	:	PUNCT
ejpam-5367	116	51	αfg	αfg	PROPN
ejpam-5367	116	52	(	(	PUNCT
ejpam-5367	116	53	x	x	X
ejpam-5367	116	54	)	)	PUNCT
ejpam-5367	116	55	=	=	SYM
ejpam-5367	116	56	{	{	PUNCT
ejpam-5367	116	57	1	1	NUM
ejpam-5367	116	58	if	if	SCONJ
ejpam-5367	116	59	x	x	PROPN
ejpam-5367	116	60	∈	∈	PROPN
ejpam-5367	116	61	g	g	NOUN
ejpam-5367	116	62	0	0	NUM
ejpam-5367	116	63	otherwise	otherwise	ADV
ejpam-5367	116	64	βfg	βfg	PROPN
ejpam-5367	116	65	(	(	PUNCT
ejpam-5367	116	66	x	x	X
ejpam-5367	116	67	)	)	PUNCT
ejpam-5367	116	68	=	=	PRON
ejpam-5367	116	69	{	{	PUNCT
ejpam-5367	116	70	0	0	NUM
ejpam-5367	116	71	if	if	SCONJ
ejpam-5367	116	72	x	x	PROPN
ejpam-5367	116	73	∈	∈	NOUN
ejpam-5367	116	74	g	g	NOUN
ejpam-5367	116	75	1	1	NUM
ejpam-5367	116	76	otherwise	otherwise	ADV
ejpam-5367	116	77	by	by	ADP
ejpam-5367	116	78	the	the	DET
ejpam-5367	116	79	definition	definition	NOUN
ejpam-5367	116	80	of	of	ADP
ejpam-5367	116	81	the	the	DET
ejpam-5367	116	82	characteristic	characteristic	ADJ
ejpam-5367	116	83	function	function	NOUN
ejpam-5367	116	84	,	,	PUNCT
ejpam-5367	116	85	αfg	αfg	PROPN
ejpam-5367	116	86	and	and	CCONJ
ejpam-5367	116	87	βfg	βfg	PROPN
ejpam-5367	116	88	are	be	AUX
ejpam-5367	116	89	functions	function	NOUN
ejpam-5367	116	90	of	of	ADP
ejpam-5367	116	91	x	x	PUNCT
ejpam-5367	116	92	into	into	ADP
ejpam-5367	116	93	{	{	PUNCT
ejpam-5367	116	94	0	0	NUM
ejpam-5367	116	95	,	,	PUNCT
ejpam-5367	116	96	1	1	NUM
ejpam-5367	116	97	}	}	PUNCT
ejpam-5367	116	98	⊂	⊂	PROPN
ejpam-5367	117	1	[	[	X
ejpam-5367	117	2	0	0	NUM
ejpam-5367	117	3	,	,	PUNCT
ejpam-5367	117	4	1	1	NUM
ejpam-5367	117	5	]	]	PUNCT
ejpam-5367	117	6	.	.	PUNCT
ejpam-5367	118	1	therefore	therefore	ADV
ejpam-5367	118	2	,	,	PUNCT
ejpam-5367	118	3	the	the	DET
ejpam-5367	118	4	ffs	ffs	NOUN
ejpam-5367	118	5	fg	fg	NOUN
ejpam-5367	118	6	=	=	X
ejpam-5367	118	7	(	(	PUNCT
ejpam-5367	118	8	αfg	αfg	NOUN
ejpam-5367	118	9	,	,	PUNCT
ejpam-5367	118	10	βfg	βfg	PROPN
ejpam-5367	118	11	)	)	PUNCT
ejpam-5367	118	12	is	be	AUX
ejpam-5367	118	13	defined	define	VERB
ejpam-5367	118	14	as	as	ADP
ejpam-5367	118	15	the	the	DET
ejpam-5367	118	16	characteristic	characteristic	ADJ
ejpam-5367	118	17	ffs	ffs	NOUN
ejpam-5367	118	18	of	of	ADP
ejpam-5367	118	19	g	g	PROPN
ejpam-5367	118	20	in	in	ADP
ejpam-5367	118	21	x.	x.	NOUN
ejpam-5367	118	22	definition	definition	NOUN
ejpam-5367	118	23	4	4	NUM
ejpam-5367	118	24	.	.	PUNCT
ejpam-5367	119	1	let	let	VERB
ejpam-5367	119	2	f	f	PRON
ejpam-5367	119	3	be	be	AUX
ejpam-5367	119	4	an	an	DET
ejpam-5367	119	5	fs	fs	NOUN
ejpam-5367	119	6	in	in	ADP
ejpam-5367	119	7	a	a	DET
ejpam-5367	119	8	non	non	ADJ
ejpam-5367	119	9	-	-	ADJ
ejpam-5367	119	10	empty	empty	ADJ
ejpam-5367	119	11	set	set	NOUN
ejpam-5367	119	12	x.	x.	NOUN
ejpam-5367	119	13	then	then	ADV
ejpam-5367	119	14	the	the	DET
ejpam-5367	119	15	fs	fs	PROPN
ejpam-5367	119	16	f	f	PROPN
ejpam-5367	119	17	defined	define	VERB
ejpam-5367	119	18	by	by	ADP
ejpam-5367	119	19	f(x	f(x	PROPN
ejpam-5367	119	20	)	)	PUNCT
ejpam-5367	119	21	=	=	SYM
ejpam-5367	120	1	1−	1−	NUM
ejpam-5367	120	2	f(x	f(x	PROPN
ejpam-5367	120	3	)	)	PUNCT
ejpam-5367	120	4	for	for	ADP
ejpam-5367	120	5	all	all	DET
ejpam-5367	120	6	x	x	SYM
ejpam-5367	120	7	∈	∈	NOUN
ejpam-5367	120	8	x	x	PUNCT
ejpam-5367	120	9	is	be	AUX
ejpam-5367	120	10	called	call	VERB
ejpam-5367	120	11	the	the	DET
ejpam-5367	120	12	complement	complement	NOUN
ejpam-5367	120	13	of	of	ADP
ejpam-5367	120	14	f	f	PROPN
ejpam-5367	120	15	in	in	ADP
ejpam-5367	120	16	x.	x.	NOUN
ejpam-5367	120	17	definition	definition	NOUN
ejpam-5367	120	18	5	5	NUM
ejpam-5367	120	19	.	.	PUNCT
ejpam-5367	121	1	let	let	VERB
ejpam-5367	121	2	f	f	PRON
ejpam-5367	121	3	be	be	AUX
ejpam-5367	121	4	an	an	DET
ejpam-5367	121	5	ffs	ffs	NOUN
ejpam-5367	121	6	in	in	ADP
ejpam-5367	121	7	a	a	DET
ejpam-5367	121	8	non	non	ADJ
ejpam-5367	121	9	-	-	ADJ
ejpam-5367	121	10	empty	empty	ADJ
ejpam-5367	121	11	set	set	NOUN
ejpam-5367	121	12	x.	x.	NOUN
ejpam-5367	121	13	then	then	ADV
ejpam-5367	121	14	the	the	DET
ejpam-5367	121	15	ffs	ffs	NOUN
ejpam-5367	121	16	f	f	PROPN
ejpam-5367	122	1	=	=	PRON
ejpam-5367	122	2	(	(	PUNCT
ejpam-5367	122	3	αf	αf	X
ejpam-5367	122	4	,	,	PUNCT
ejpam-5367	122	5	βf	βf	CCONJ
ejpam-5367	122	6	)	)	PUNCT
ejpam-5367	122	7	is	be	AUX
ejpam-5367	122	8	called	call	VERB
ejpam-5367	122	9	the	the	DET
ejpam-5367	122	10	complement	complement	NOUN
ejpam-5367	122	11	of	of	ADP
ejpam-5367	122	12	f	f	PROPN
ejpam-5367	122	13	in	in	ADP
ejpam-5367	122	14	x.	x.	NOUN
ejpam-5367	122	15	we	we	PRON
ejpam-5367	122	16	extend	extend	VERB
ejpam-5367	122	17	ffss	ffss	NOUN
ejpam-5367	122	18	to	to	PART
ejpam-5367	122	19	iup	iup	VERB
ejpam-5367	122	20	-	-	PUNCT
ejpam-5367	122	21	algebras	algebras	X
ejpam-5367	122	22	,	,	PUNCT
ejpam-5367	122	23	introducing	introduce	VERB
ejpam-5367	122	24	four	four	NUM
ejpam-5367	122	25	innovative	innovative	ADJ
ejpam-5367	122	26	types	type	NOUN
ejpam-5367	122	27	:	:	PUNCT
ejpam-5367	122	28	fermatean	fermatean	NOUN
ejpam-5367	122	29	fuzzy	fuzzy	NOUN
ejpam-5367	122	30	iup	iup	NOUN
ejpam-5367	122	31	-	-	PUNCT
ejpam-5367	122	32	subalgebras	subalgebras	PROPN
ejpam-5367	122	33	,	,	PUNCT
ejpam-5367	122	34	iup	iup	NOUN
ejpam-5367	122	35	-	-	PUNCT
ejpam-5367	122	36	ideals	ideal	NOUN
ejpam-5367	122	37	,	,	PUNCT
ejpam-5367	122	38	iup	iup	NOUN
ejpam-5367	122	39	-	-	PUNCT
ejpam-5367	122	40	filters	filter	NOUN
ejpam-5367	122	41	,	,	PUNCT
ejpam-5367	122	42	and	and	CCONJ
ejpam-5367	122	43	strong	strong	ADJ
ejpam-5367	122	44	iup	iup	NOUN
ejpam-5367	122	45	-	-	PUNCT
ejpam-5367	122	46	ideals	ideal	NOUN
ejpam-5367	122	47	.	.	PUNCT
ejpam-5367	123	1	this	this	DET
ejpam-5367	123	2	application	application	NOUN
ejpam-5367	123	3	opens	open	VERB
ejpam-5367	123	4	new	new	ADJ
ejpam-5367	123	5	dimensions	dimension	NOUN
ejpam-5367	123	6	in	in	ADP
ejpam-5367	123	7	the	the	DET
ejpam-5367	123	8	study	study	NOUN
ejpam-5367	123	9	of	of	ADP
ejpam-5367	123	10	iup	iup	NOUN
ejpam-5367	123	11	-	-	PUNCT
ejpam-5367	123	12	algebras	algebras	PROPN
ejpam-5367	123	13	,	,	PUNCT
ejpam-5367	123	14	enriching	enrich	VERB
ejpam-5367	123	15	both	both	CCONJ
ejpam-5367	123	16	their	their	PRON
ejpam-5367	123	17	theoretical	theoretical	ADJ
ejpam-5367	123	18	and	and	CCONJ
ejpam-5367	123	19	practical	practical	ADJ
ejpam-5367	123	20	frameworks	framework	NOUN
ejpam-5367	123	21	.	.	PUNCT
ejpam-5367	124	1	definition	definition	NOUN
ejpam-5367	124	2	6	6	NUM
ejpam-5367	124	3	.	.	PUNCT
ejpam-5367	125	1	an	an	DET
ejpam-5367	125	2	ffs	ffs	NOUN
ejpam-5367	125	3	f	f	PROPN
ejpam-5367	125	4	in	in	ADP
ejpam-5367	125	5	x	x	PROPN
ejpam-5367	125	6	is	be	AUX
ejpam-5367	125	7	called	call	VERB
ejpam-5367	125	8	a	a	DET
ejpam-5367	125	9	fermatean	fermatean	ADJ
ejpam-5367	125	10	fuzzy	fuzzy	ADJ
ejpam-5367	125	11	iup	iup	NOUN
ejpam-5367	125	12	-	-	PUNCT
ejpam-5367	125	13	subalgebra	subalgebra	NOUN
ejpam-5367	125	14	(	(	PUNCT
ejpam-5367	125	15	ffiupsubalgebra	ffiupsubalgebra	NOUN
ejpam-5367	125	16	)	)	PUNCT
ejpam-5367	125	17	of	of	ADP
ejpam-5367	125	18	x	x	PRON
ejpam-5367	125	19	if	if	SCONJ
ejpam-5367	125	20	it	it	PRON
ejpam-5367	125	21	satisfies	satisfy	VERB
ejpam-5367	125	22	the	the	DET
ejpam-5367	125	23	following	follow	VERB
ejpam-5367	125	24	properties	property	NOUN
ejpam-5367	125	25	:	:	PUNCT
ejpam-5367	125	26	(	(	PUNCT
ejpam-5367	125	27	∀x	∀x	X
ejpam-5367	125	28	,	,	PUNCT
ejpam-5367	125	29	y	y	PROPN
ejpam-5367	125	30	∈	∈	PROPN
ejpam-5367	125	31	x)(αf	x)(αf	X
ejpam-5367	126	1	(	(	PUNCT
ejpam-5367	126	2	x	x	X
ejpam-5367	126	3	·	·	PUNCT
ejpam-5367	126	4	y	y	X
ejpam-5367	126	5	)	)	PUNCT
ejpam-5367	126	6	≥	≥	NOUN
ejpam-5367	126	7	min{αf	min{αf	PUNCT
ejpam-5367	126	8	(	(	PUNCT
ejpam-5367	126	9	x	x	X
ejpam-5367	126	10	)	)	PUNCT
ejpam-5367	126	11	,	,	PUNCT
ejpam-5367	126	12	αf	αf	X
ejpam-5367	126	13	(	(	PUNCT
ejpam-5367	126	14	y	y	NOUN
ejpam-5367	126	15	)	)	PUNCT
ejpam-5367	126	16	}	}	PUNCT
ejpam-5367	126	17	)	)	PUNCT
ejpam-5367	126	18	(	(	PUNCT
ejpam-5367	126	19	3.2	3.2	NUM
ejpam-5367	126	20	)	)	PUNCT
ejpam-5367	126	21	(	(	PUNCT
ejpam-5367	126	22	∀x	∀x	X
ejpam-5367	126	23	,	,	PUNCT
ejpam-5367	126	24	y	y	PROPN
ejpam-5367	126	25	∈	∈	PROPN
ejpam-5367	126	26	x)(βf	x)(βf	PROPN
ejpam-5367	127	1	(	(	PUNCT
ejpam-5367	127	2	x	x	X
ejpam-5367	127	3	·	·	PUNCT
ejpam-5367	127	4	y	y	X
ejpam-5367	127	5	)	)	PUNCT
ejpam-5367	127	6	≤	≤	NUM
ejpam-5367	127	7	max{βf	max{βf	INTJ
ejpam-5367	128	1	(	(	PUNCT
ejpam-5367	128	2	x	x	NOUN
ejpam-5367	128	3	)	)	PUNCT
ejpam-5367	128	4	,	,	PUNCT
ejpam-5367	128	5	βf	βf	CCONJ
ejpam-5367	128	6	(	(	PUNCT
ejpam-5367	128	7	y	y	NOUN
ejpam-5367	128	8	)	)	PUNCT
ejpam-5367	128	9	}	}	PUNCT
ejpam-5367	128	10	)	)	PUNCT
ejpam-5367	128	11	(	(	PUNCT
ejpam-5367	128	12	3.3	3.3	NUM
ejpam-5367	128	13	)	)	PUNCT
ejpam-5367	128	14	definition	definition	NOUN
ejpam-5367	128	15	7	7	NUM
ejpam-5367	128	16	.	.	PUNCT
ejpam-5367	129	1	an	an	DET
ejpam-5367	129	2	ffs	ffs	NOUN
ejpam-5367	129	3	f	f	PROPN
ejpam-5367	129	4	in	in	ADP
ejpam-5367	129	5	x	x	PROPN
ejpam-5367	129	6	is	be	AUX
ejpam-5367	129	7	called	call	VERB
ejpam-5367	129	8	a	a	DET
ejpam-5367	129	9	fermatean	fermatean	ADJ
ejpam-5367	129	10	fuzzy	fuzzy	ADJ
ejpam-5367	129	11	iup	iup	NOUN
ejpam-5367	129	12	-	-	PUNCT
ejpam-5367	129	13	ideal	ideal	NOUN
ejpam-5367	129	14	(	(	PUNCT
ejpam-5367	129	15	ffiup	ffiup	NOUN
ejpam-5367	129	16	-	-	PUNCT
ejpam-5367	129	17	ideal	ideal	NOUN
ejpam-5367	129	18	)	)	PUNCT
ejpam-5367	129	19	of	of	ADP
ejpam-5367	129	20	x	x	PRON
ejpam-5367	129	21	if	if	SCONJ
ejpam-5367	129	22	it	it	PRON
ejpam-5367	129	23	satisfies	satisfy	VERB
ejpam-5367	129	24	the	the	DET
ejpam-5367	129	25	following	follow	VERB
ejpam-5367	129	26	properties	property	NOUN
ejpam-5367	129	27	:	:	PUNCT
ejpam-5367	129	28	(	(	PUNCT
ejpam-5367	129	29	∀x	∀x	X
ejpam-5367	129	30	∈	∈	PROPN
ejpam-5367	129	31	x)(αf	x)(αf	X
ejpam-5367	129	32	(	(	PUNCT
ejpam-5367	129	33	0	0	NUM
ejpam-5367	129	34	)	)	PUNCT
ejpam-5367	129	35	≥	≥	NOUN
ejpam-5367	129	36	αf	αf	X
ejpam-5367	129	37	(	(	PUNCT
ejpam-5367	129	38	x	x	NOUN
ejpam-5367	129	39	)	)	PUNCT
ejpam-5367	129	40	)	)	PUNCT
ejpam-5367	129	41	(	(	PUNCT
ejpam-5367	129	42	3.4	3.4	NUM
ejpam-5367	129	43	)	)	PUNCT
ejpam-5367	129	44	(	(	PUNCT
ejpam-5367	129	45	∀x	∀x	X
ejpam-5367	129	46	∈	∈	NOUN
ejpam-5367	129	47	x)(βf	x)(βf	NOUN
ejpam-5367	129	48	(	(	PUNCT
ejpam-5367	129	49	0	0	NUM
ejpam-5367	129	50	)	)	PUNCT
ejpam-5367	129	51	≤	≤	NOUN
ejpam-5367	130	1	βf	βf	CCONJ
ejpam-5367	130	2	(	(	PUNCT
ejpam-5367	130	3	x	x	NOUN
ejpam-5367	130	4	)	)	PUNCT
ejpam-5367	130	5	)	)	PUNCT
ejpam-5367	130	6	(	(	PUNCT
ejpam-5367	130	7	3.5	3.5	NUM
ejpam-5367	130	8	)	)	PUNCT
ejpam-5367	130	9	(	(	PUNCT
ejpam-5367	130	10	∀x	∀x	X
ejpam-5367	130	11	,	,	PUNCT
ejpam-5367	130	12	y	y	PROPN
ejpam-5367	130	13	,	,	PUNCT
ejpam-5367	130	14	z	z	PROPN
ejpam-5367	130	15	∈	∈	PROPN
ejpam-5367	130	16	x)(αf	x)(αf	X
ejpam-5367	131	1	(	(	PUNCT
ejpam-5367	131	2	x	x	X
ejpam-5367	131	3	·	·	PUNCT
ejpam-5367	131	4	z	z	X
ejpam-5367	131	5	)	)	PUNCT
ejpam-5367	131	6	≥	≥	NOUN
ejpam-5367	131	7	min{αf	min{αf	PUNCT
ejpam-5367	131	8	(	(	PUNCT
ejpam-5367	131	9	x	x	X
ejpam-5367	131	10	·	·	PUNCT
ejpam-5367	131	11	(	(	PUNCT
ejpam-5367	131	12	y	y	PROPN
ejpam-5367	131	13	·	·	PUNCT
ejpam-5367	131	14	z	z	NOUN
ejpam-5367	131	15	)	)	PUNCT
ejpam-5367	131	16	)	)	PUNCT
ejpam-5367	131	17	,	,	PUNCT
ejpam-5367	131	18	αf	αf	X
ejpam-5367	131	19	(	(	PUNCT
ejpam-5367	131	20	y	y	NOUN
ejpam-5367	131	21	)	)	PUNCT
ejpam-5367	131	22	}	}	PUNCT
ejpam-5367	131	23	)	)	PUNCT
ejpam-5367	131	24	(	(	PUNCT
ejpam-5367	131	25	3.6	3.6	NUM
ejpam-5367	131	26	)	)	PUNCT
ejpam-5367	131	27	(	(	PUNCT
ejpam-5367	131	28	∀x	∀x	X
ejpam-5367	131	29	,	,	PUNCT
ejpam-5367	131	30	y	y	PROPN
ejpam-5367	131	31	,	,	PUNCT
ejpam-5367	131	32	z	z	PROPN
ejpam-5367	131	33	∈	∈	PROPN
ejpam-5367	131	34	x)(βf	x)(βf	PUNCT
ejpam-5367	132	1	(	(	PUNCT
ejpam-5367	132	2	x	x	X
ejpam-5367	132	3	·	·	PUNCT
ejpam-5367	132	4	z	z	X
ejpam-5367	132	5	)	)	PUNCT
ejpam-5367	132	6	≤	≤	NUM
ejpam-5367	132	7	max{βf	max{βf	INTJ
ejpam-5367	133	1	(	(	PUNCT
ejpam-5367	133	2	x	x	X
ejpam-5367	133	3	·	·	PUNCT
ejpam-5367	133	4	(	(	PUNCT
ejpam-5367	133	5	y	y	PROPN
ejpam-5367	133	6	·	·	PUNCT
ejpam-5367	133	7	z	z	NOUN
ejpam-5367	133	8	)	)	PUNCT
ejpam-5367	133	9	)	)	PUNCT
ejpam-5367	133	10	,	,	PUNCT
ejpam-5367	133	11	βf	βf	CCONJ
ejpam-5367	133	12	(	(	PUNCT
ejpam-5367	133	13	y	y	NOUN
ejpam-5367	133	14	)	)	PUNCT
ejpam-5367	133	15	}	}	PUNCT
ejpam-5367	133	16	)	)	PUNCT
ejpam-5367	133	17	(	(	PUNCT
ejpam-5367	133	18	3.7	3.7	NUM
ejpam-5367	133	19	)	)	PUNCT
ejpam-5367	133	20	definition	definition	NOUN
ejpam-5367	133	21	8	8	NUM
ejpam-5367	133	22	.	.	PUNCT
ejpam-5367	134	1	an	an	DET
ejpam-5367	134	2	ffs	ffs	NOUN
ejpam-5367	134	3	f	f	PROPN
ejpam-5367	134	4	in	in	ADP
ejpam-5367	134	5	x	x	PROPN
ejpam-5367	134	6	is	be	AUX
ejpam-5367	134	7	called	call	VERB
ejpam-5367	134	8	a	a	DET
ejpam-5367	134	9	fermatean	fermatean	ADJ
ejpam-5367	134	10	fuzzy	fuzzy	ADJ
ejpam-5367	134	11	iup	iup	NOUN
ejpam-5367	134	12	-	-	PUNCT
ejpam-5367	134	13	filter	filter	NOUN
ejpam-5367	134	14	(	(	PUNCT
ejpam-5367	134	15	ffiup	ffiup	ADJ
ejpam-5367	134	16	-	-	PUNCT
ejpam-5367	134	17	filter	filter	NOUN
ejpam-5367	134	18	)	)	PUNCT
ejpam-5367	134	19	of	of	ADP
ejpam-5367	134	20	x	x	PRON
ejpam-5367	134	21	if	if	SCONJ
ejpam-5367	134	22	it	it	PRON
ejpam-5367	134	23	satisfies	satisfy	VERB
ejpam-5367	134	24	(	(	PUNCT
ejpam-5367	134	25	3.4	3.4	NUM
ejpam-5367	134	26	)	)	PUNCT
ejpam-5367	134	27	,	,	PUNCT
ejpam-5367	134	28	(	(	PUNCT
ejpam-5367	134	29	3.5	3.5	NUM
ejpam-5367	134	30	)	)	PUNCT
ejpam-5367	134	31	,	,	PUNCT
ejpam-5367	134	32	and	and	CCONJ
ejpam-5367	134	33	the	the	DET
ejpam-5367	134	34	following	follow	VERB
ejpam-5367	134	35	properties	property	NOUN
ejpam-5367	134	36	:	:	PUNCT
ejpam-5367	134	37	(	(	PUNCT
ejpam-5367	134	38	∀x	∀x	X
ejpam-5367	134	39	,	,	PUNCT
ejpam-5367	134	40	y	y	PROPN
ejpam-5367	134	41	∈	∈	PROPN
ejpam-5367	134	42	x)(αf	x)(αf	X
ejpam-5367	135	1	(	(	PUNCT
ejpam-5367	135	2	y	y	X
ejpam-5367	135	3	)	)	PUNCT
ejpam-5367	135	4	≥	≥	NOUN
ejpam-5367	135	5	min{αf	min{αf	PUNCT
ejpam-5367	135	6	(	(	PUNCT
ejpam-5367	135	7	x	x	X
ejpam-5367	135	8	·	·	PUNCT
ejpam-5367	135	9	y	y	X
ejpam-5367	135	10	)	)	PUNCT
ejpam-5367	135	11	,	,	PUNCT
ejpam-5367	135	12	αf	αf	X
ejpam-5367	135	13	(	(	PUNCT
ejpam-5367	135	14	x	x	NOUN
ejpam-5367	135	15	)	)	PUNCT
ejpam-5367	135	16	}	}	PUNCT
ejpam-5367	135	17	)	)	PUNCT
ejpam-5367	135	18	(	(	PUNCT
ejpam-5367	135	19	3.8	3.8	NUM
ejpam-5367	135	20	)	)	PUNCT
ejpam-5367	135	21	(	(	PUNCT
ejpam-5367	135	22	∀x	∀x	X
ejpam-5367	135	23	,	,	PUNCT
ejpam-5367	135	24	y	y	PROPN
ejpam-5367	135	25	∈	∈	PROPN
ejpam-5367	135	26	x)(βf	x)(βf	PROPN
ejpam-5367	136	1	(	(	PUNCT
ejpam-5367	136	2	y	y	NOUN
ejpam-5367	136	3	)	)	PUNCT
ejpam-5367	136	4	≤	≤	NOUN
ejpam-5367	136	5	max{βf	max{βf	INTJ
ejpam-5367	137	1	(	(	PUNCT
ejpam-5367	137	2	x	x	X
ejpam-5367	137	3	·	·	PUNCT
ejpam-5367	137	4	y	y	X
ejpam-5367	137	5	)	)	PUNCT
ejpam-5367	137	6	,	,	PUNCT
ejpam-5367	137	7	βf	βf	CCONJ
ejpam-5367	137	8	(	(	PUNCT
ejpam-5367	137	9	x	x	NOUN
ejpam-5367	137	10	)	)	PUNCT
ejpam-5367	137	11	}	}	PUNCT
ejpam-5367	137	12	)	)	PUNCT
ejpam-5367	137	13	(	(	PUNCT
ejpam-5367	137	14	3.9	3.9	NUM
ejpam-5367	137	15	)	)	PUNCT
ejpam-5367	137	16	a.	a.	NOUN
ejpam-5367	137	17	iampan	iampan	NOUN
ejpam-5367	137	18	et	et	PROPN
ejpam-5367	137	19	al	al	PROPN
ejpam-5367	137	20	.	.	PUNCT
ejpam-5367	137	21	/	/	SYM
ejpam-5367	137	22	eur	eur	PROPN
ejpam-5367	137	23	.	.	PUNCT
ejpam-5367	138	1	j.	j.	PROPN
ejpam-5367	138	2	pure	pure	PROPN
ejpam-5367	138	3	appl	appl	PROPN
ejpam-5367	138	4	.	.	PROPN
ejpam-5367	138	5	math	math	PROPN
ejpam-5367	138	6	,	,	PUNCT
ejpam-5367	138	7	17	17	NUM
ejpam-5367	138	8	(	(	PUNCT
ejpam-5367	138	9	4	4	NUM
ejpam-5367	138	10	)	)	PUNCT
ejpam-5367	138	11	(	(	PUNCT
ejpam-5367	138	12	2024	2024	NUM
ejpam-5367	138	13	)	)	PUNCT
ejpam-5367	138	14	,	,	PUNCT
ejpam-5367	138	15	3022	3022	NUM
ejpam-5367	138	16	-	-	SYM
ejpam-5367	138	17	3042	3042	NUM
ejpam-5367	138	18	3028	3028	NUM
ejpam-5367	138	19	definition	definition	NOUN
ejpam-5367	138	20	9	9	NUM
ejpam-5367	138	21	.	.	PUNCT
ejpam-5367	139	1	an	an	DET
ejpam-5367	139	2	ffs	ffs	NOUN
ejpam-5367	139	3	f	f	PROPN
ejpam-5367	139	4	in	in	ADP
ejpam-5367	139	5	x	x	PROPN
ejpam-5367	139	6	is	be	AUX
ejpam-5367	139	7	called	call	VERB
ejpam-5367	139	8	a	a	DET
ejpam-5367	139	9	fermatean	fermatean	ADJ
ejpam-5367	139	10	fuzzy	fuzzy	ADJ
ejpam-5367	139	11	strong	strong	ADJ
ejpam-5367	139	12	iup	iup	NOUN
ejpam-5367	139	13	-	-	PUNCT
ejpam-5367	139	14	ideal	ideal	NOUN
ejpam-5367	139	15	(	(	PUNCT
ejpam-5367	139	16	ffsiupideal	ffsiupideal	NOUN
ejpam-5367	139	17	)	)	PUNCT
ejpam-5367	139	18	of	of	ADP
ejpam-5367	139	19	x	x	PRON
ejpam-5367	139	20	if	if	SCONJ
ejpam-5367	139	21	it	it	PRON
ejpam-5367	139	22	satisfies	satisfy	VERB
ejpam-5367	139	23	the	the	DET
ejpam-5367	139	24	following	follow	VERB
ejpam-5367	139	25	properties	property	NOUN
ejpam-5367	139	26	:	:	PUNCT
ejpam-5367	139	27	(	(	PUNCT
ejpam-5367	139	28	∀x	∀x	X
ejpam-5367	139	29	,	,	PUNCT
ejpam-5367	139	30	y	y	PROPN
ejpam-5367	139	31	∈	∈	PROPN
ejpam-5367	139	32	x)(αf	x)(αf	X
ejpam-5367	140	1	(	(	PUNCT
ejpam-5367	140	2	x	x	X
ejpam-5367	140	3	·	·	PUNCT
ejpam-5367	140	4	y	y	X
ejpam-5367	140	5	)	)	PUNCT
ejpam-5367	140	6	≥	≥	NOUN
ejpam-5367	140	7	αf	αf	X
ejpam-5367	140	8	(	(	PUNCT
ejpam-5367	140	9	y	y	NOUN
ejpam-5367	140	10	)	)	PUNCT
ejpam-5367	140	11	)	)	PUNCT
ejpam-5367	140	12	(	(	PUNCT
ejpam-5367	140	13	3.10	3.10	NUM
ejpam-5367	140	14	)	)	PUNCT
ejpam-5367	140	15	(	(	PUNCT
ejpam-5367	140	16	∀x	∀x	X
ejpam-5367	140	17	,	,	PUNCT
ejpam-5367	140	18	y	y	PROPN
ejpam-5367	140	19	∈	∈	PROPN
ejpam-5367	140	20	x)(βf	x)(βf	PROPN
ejpam-5367	141	1	(	(	PUNCT
ejpam-5367	141	2	x	x	X
ejpam-5367	141	3	·	·	PUNCT
ejpam-5367	141	4	y	y	X
ejpam-5367	141	5	)	)	PUNCT
ejpam-5367	141	6	≤	≤	NOUN
ejpam-5367	142	1	βf	βf	CCONJ
ejpam-5367	142	2	(	(	PUNCT
ejpam-5367	142	3	y	y	NOUN
ejpam-5367	142	4	)	)	PUNCT
ejpam-5367	142	5	)	)	PUNCT
ejpam-5367	143	1	(	(	PUNCT
ejpam-5367	143	2	3.11	3.11	NUM
ejpam-5367	143	3	)	)	PUNCT
ejpam-5367	143	4	lemma	lemma	PROPN
ejpam-5367	143	5	1	1	NUM
ejpam-5367	143	6	.	.	PUNCT
ejpam-5367	144	1	every	every	DET
ejpam-5367	144	2	ffiup	ffiup	NOUN
ejpam-5367	144	3	-	-	PUNCT
ejpam-5367	144	4	subalgebra	subalgebra	NOUN
ejpam-5367	144	5	of	of	ADP
ejpam-5367	144	6	x	x	PUNCT
ejpam-5367	144	7	satisfies	satisfie	NOUN
ejpam-5367	144	8	(	(	PUNCT
ejpam-5367	144	9	3.4	3.4	NUM
ejpam-5367	144	10	)	)	PUNCT
ejpam-5367	144	11	and	and	CCONJ
ejpam-5367	144	12	(	(	PUNCT
ejpam-5367	144	13	3.5	3.5	NUM
ejpam-5367	144	14	)	)	PUNCT
ejpam-5367	144	15	.	.	PUNCT
ejpam-5367	145	1	proof	proof	NOUN
ejpam-5367	145	2	.	.	PUNCT
ejpam-5367	146	1	assume	assume	VERB
ejpam-5367	146	2	that	that	SCONJ
ejpam-5367	146	3	f	f	PROPN
ejpam-5367	146	4	is	be	AUX
ejpam-5367	146	5	an	an	DET
ejpam-5367	146	6	ffiup	ffiup	NOUN
ejpam-5367	146	7	-	-	PUNCT
ejpam-5367	146	8	subalgebra	subalgebra	NOUN
ejpam-5367	146	9	of	of	ADP
ejpam-5367	146	10	x.	x.	NOUN
ejpam-5367	146	11	let	let	VERB
ejpam-5367	146	12	x	x	SYM
ejpam-5367	146	13	∈	∈	PROPN
ejpam-5367	146	14	x.	x.	NOUN
ejpam-5367	146	15	then	then	ADV
ejpam-5367	146	16	αf	αf	VERB
ejpam-5367	146	17	(	(	PUNCT
ejpam-5367	146	18	0	0	NUM
ejpam-5367	146	19	)	)	PUNCT
ejpam-5367	146	20	=	=	SYM
ejpam-5367	146	21	αf	αf	X
ejpam-5367	146	22	(	(	PUNCT
ejpam-5367	146	23	x	x	X
ejpam-5367	146	24	·	·	PUNCT
ejpam-5367	146	25	x	x	X
ejpam-5367	146	26	)	)	PUNCT
ejpam-5367	146	27	(	(	PUNCT
ejpam-5367	146	28	by	by	ADP
ejpam-5367	146	29	(	(	PUNCT
ejpam-5367	146	30	iup-2	iup-2	NUM
ejpam-5367	146	31	)	)	PUNCT
ejpam-5367	146	32	)	)	PUNCT
ejpam-5367	146	33	≥	≥	NOUN
ejpam-5367	146	34	min{αf	min{αf	PUNCT
ejpam-5367	146	35	(	(	PUNCT
ejpam-5367	146	36	x	x	X
ejpam-5367	146	37	)	)	PUNCT
ejpam-5367	146	38	,	,	PUNCT
ejpam-5367	146	39	αf	αf	X
ejpam-5367	146	40	(	(	PUNCT
ejpam-5367	146	41	x	x	NOUN
ejpam-5367	146	42	)	)	PUNCT
ejpam-5367	146	43	}	}	PUNCT
ejpam-5367	146	44	(	(	PUNCT
ejpam-5367	146	45	by	by	ADP
ejpam-5367	146	46	(	(	PUNCT
ejpam-5367	146	47	3.2	3.2	NUM
ejpam-5367	146	48	)	)	PUNCT
ejpam-5367	146	49	)	)	PUNCT
ejpam-5367	147	1	=	=	SYM
ejpam-5367	147	2	αf	αf	X
ejpam-5367	147	3	(	(	PUNCT
ejpam-5367	147	4	x	x	NOUN
ejpam-5367	147	5	)	)	PUNCT
ejpam-5367	147	6	,	,	PUNCT
ejpam-5367	147	7	βf	βf	CCONJ
ejpam-5367	147	8	(	(	PUNCT
ejpam-5367	147	9	0	0	NUM
ejpam-5367	147	10	)	)	PUNCT
ejpam-5367	147	11	=	=	PUNCT
ejpam-5367	147	12	βf	βf	INTJ
ejpam-5367	147	13	(	(	PUNCT
ejpam-5367	147	14	x	x	X
ejpam-5367	147	15	·	·	PUNCT
ejpam-5367	147	16	x	x	X
ejpam-5367	147	17	)	)	PUNCT
ejpam-5367	147	18	(	(	PUNCT
ejpam-5367	147	19	by	by	ADP
ejpam-5367	147	20	(	(	PUNCT
ejpam-5367	147	21	iup-2	iup-2	NUM
ejpam-5367	147	22	)	)	PUNCT
ejpam-5367	147	23	)	)	PUNCT
ejpam-5367	147	24	≤	≤	NUM
ejpam-5367	147	25	max{βf	max{βf	INTJ
ejpam-5367	147	26	(	(	PUNCT
ejpam-5367	147	27	x	x	NOUN
ejpam-5367	147	28	)	)	PUNCT
ejpam-5367	147	29	,	,	PUNCT
ejpam-5367	147	30	βf	βf	CCONJ
ejpam-5367	147	31	(	(	PUNCT
ejpam-5367	147	32	x	x	NOUN
ejpam-5367	147	33	)	)	PUNCT
ejpam-5367	147	34	}	}	PUNCT
ejpam-5367	147	35	.	.	PUNCT
ejpam-5367	148	1	(	(	PUNCT
ejpam-5367	148	2	by	by	ADP
ejpam-5367	148	3	(	(	PUNCT
ejpam-5367	148	4	3.3	3.3	NUM
ejpam-5367	148	5	)	)	PUNCT
ejpam-5367	148	6	)	)	PUNCT
ejpam-5367	148	7	hence	hence	ADV
ejpam-5367	148	8	,	,	PUNCT
ejpam-5367	148	9	f	f	PROPN
ejpam-5367	148	10	satisfies	satisfie	NOUN
ejpam-5367	148	11	(	(	PUNCT
ejpam-5367	148	12	3.4	3.4	NUM
ejpam-5367	148	13	)	)	PUNCT
ejpam-5367	148	14	and	and	CCONJ
ejpam-5367	148	15	(	(	PUNCT
ejpam-5367	148	16	3.5	3.5	NUM
ejpam-5367	148	17	)	)	PUNCT
ejpam-5367	148	18	.	.	PUNCT
ejpam-5367	149	1	theorem	theorem	NOUN
ejpam-5367	149	2	1	1	NUM
ejpam-5367	149	3	.	.	PUNCT
ejpam-5367	150	1	every	every	DET
ejpam-5367	150	2	ffsiup	ffsiup	NOUN
ejpam-5367	150	3	-	-	PUNCT
ejpam-5367	150	4	ideal	ideal	NOUN
ejpam-5367	150	5	of	of	ADP
ejpam-5367	150	6	x	x	SYM
ejpam-5367	150	7	satisfies	satisfie	NOUN
ejpam-5367	150	8	(	(	PUNCT
ejpam-5367	150	9	3.4	3.4	NUM
ejpam-5367	150	10	)	)	PUNCT
ejpam-5367	150	11	and	and	CCONJ
ejpam-5367	150	12	(	(	PUNCT
ejpam-5367	150	13	3.5	3.5	NUM
ejpam-5367	150	14	)	)	PUNCT
ejpam-5367	150	15	.	.	PUNCT
ejpam-5367	151	1	proof	proof	NOUN
ejpam-5367	151	2	.	.	PUNCT
ejpam-5367	152	1	assume	assume	VERB
ejpam-5367	152	2	that	that	SCONJ
ejpam-5367	152	3	f	f	PROPN
ejpam-5367	152	4	is	be	AUX
ejpam-5367	152	5	an	an	DET
ejpam-5367	152	6	ffsiup	ffsiup	NOUN
ejpam-5367	152	7	-	-	PUNCT
ejpam-5367	152	8	ideal	ideal	NOUN
ejpam-5367	152	9	of	of	ADP
ejpam-5367	152	10	x.	x.	NOUN
ejpam-5367	152	11	let	let	VERB
ejpam-5367	152	12	x	x	SYM
ejpam-5367	152	13	∈	∈	PROPN
ejpam-5367	152	14	x.	x.	NOUN
ejpam-5367	152	15	then	then	ADV
ejpam-5367	152	16	αf	αf	VERB
ejpam-5367	152	17	(	(	PUNCT
ejpam-5367	152	18	0	0	NUM
ejpam-5367	152	19	)	)	PUNCT
ejpam-5367	152	20	=	=	SYM
ejpam-5367	152	21	αf	αf	X
ejpam-5367	152	22	(	(	PUNCT
ejpam-5367	152	23	x	x	X
ejpam-5367	152	24	·	·	PUNCT
ejpam-5367	152	25	x	x	X
ejpam-5367	152	26	)	)	PUNCT
ejpam-5367	152	27	(	(	PUNCT
ejpam-5367	152	28	by	by	ADP
ejpam-5367	152	29	(	(	PUNCT
ejpam-5367	152	30	iup-2	iup-2	NUM
ejpam-5367	152	31	)	)	PUNCT
ejpam-5367	152	32	)	)	PUNCT
ejpam-5367	152	33	≥	≥	NOUN
ejpam-5367	153	1	αf	αf	X
ejpam-5367	153	2	(	(	PUNCT
ejpam-5367	153	3	x	x	NOUN
ejpam-5367	153	4	)	)	PUNCT
ejpam-5367	153	5	,	,	PUNCT
ejpam-5367	153	6	(	(	PUNCT
ejpam-5367	153	7	by	by	ADP
ejpam-5367	153	8	(	(	PUNCT
ejpam-5367	153	9	3.10	3.10	NUM
ejpam-5367	153	10	)	)	PUNCT
ejpam-5367	153	11	)	)	PUNCT
ejpam-5367	153	12	βf	βf	CCONJ
ejpam-5367	153	13	(	(	PUNCT
ejpam-5367	153	14	0	0	NUM
ejpam-5367	153	15	)	)	PUNCT
ejpam-5367	153	16	=	=	PUNCT
ejpam-5367	153	17	βf	βf	INTJ
ejpam-5367	153	18	(	(	PUNCT
ejpam-5367	153	19	x	x	X
ejpam-5367	153	20	·	·	PUNCT
ejpam-5367	153	21	x	x	X
ejpam-5367	153	22	)	)	PUNCT
ejpam-5367	153	23	(	(	PUNCT
ejpam-5367	153	24	by	by	ADP
ejpam-5367	153	25	(	(	PUNCT
ejpam-5367	153	26	iup-2	iup-2	NUM
ejpam-5367	153	27	)	)	PUNCT
ejpam-5367	153	28	)	)	PUNCT
ejpam-5367	153	29	≤	≤	NUM
ejpam-5367	153	30	βf	βf	CCONJ
ejpam-5367	153	31	(	(	PUNCT
ejpam-5367	153	32	x	x	NOUN
ejpam-5367	153	33	)	)	PUNCT
ejpam-5367	153	34	.	.	PUNCT
ejpam-5367	154	1	(	(	PUNCT
ejpam-5367	154	2	by	by	ADP
ejpam-5367	154	3	(	(	PUNCT
ejpam-5367	154	4	3.11	3.11	NUM
ejpam-5367	154	5	)	)	PUNCT
ejpam-5367	154	6	)	)	PUNCT
ejpam-5367	154	7	hence	hence	ADV
ejpam-5367	154	8	,	,	PUNCT
ejpam-5367	154	9	f	f	PROPN
ejpam-5367	154	10	satisfies	satisfie	NOUN
ejpam-5367	154	11	(	(	PUNCT
ejpam-5367	154	12	3.4	3.4	NUM
ejpam-5367	154	13	)	)	PUNCT
ejpam-5367	154	14	and	and	CCONJ
ejpam-5367	154	15	(	(	PUNCT
ejpam-5367	154	16	3.5	3.5	NUM
ejpam-5367	154	17	)	)	PUNCT
ejpam-5367	154	18	.	.	PUNCT
ejpam-5367	155	1	theorem	theorem	NOUN
ejpam-5367	155	2	2	2	NUM
ejpam-5367	155	3	.	.	PUNCT
ejpam-5367	155	4	an	an	DET
ejpam-5367	155	5	ffsiup	ffsiup	NOUN
ejpam-5367	155	6	-	-	PUNCT
ejpam-5367	155	7	ideal	ideal	ADJ
ejpam-5367	155	8	and	and	CCONJ
ejpam-5367	155	9	constant	constant	ADJ
ejpam-5367	155	10	ffs	ffs	NOUN
ejpam-5367	155	11	coincide	coincide	NOUN
ejpam-5367	155	12	.	.	PUNCT
ejpam-5367	156	1	proof	proof	NOUN
ejpam-5367	156	2	.	.	PUNCT
ejpam-5367	157	1	assume	assume	VERB
ejpam-5367	157	2	that	that	SCONJ
ejpam-5367	157	3	f	f	PROPN
ejpam-5367	157	4	is	be	AUX
ejpam-5367	157	5	an	an	DET
ejpam-5367	157	6	ffsiup	ffsiup	NOUN
ejpam-5367	157	7	-	-	PUNCT
ejpam-5367	157	8	ideal	ideal	NOUN
ejpam-5367	157	9	of	of	ADP
ejpam-5367	157	10	x.	x.	NOUN
ejpam-5367	157	11	let	let	VERB
ejpam-5367	157	12	x	x	SYM
ejpam-5367	157	13	∈	∈	PROPN
ejpam-5367	157	14	x.	x.	NOUN
ejpam-5367	157	15	then	then	ADV
ejpam-5367	157	16	αf	αf	VERB
ejpam-5367	157	17	(	(	PUNCT
ejpam-5367	157	18	x	x	X
ejpam-5367	157	19	)	)	PUNCT
ejpam-5367	157	20	=	=	SYM
ejpam-5367	157	21	αf	αf	X
ejpam-5367	157	22	(	(	PUNCT
ejpam-5367	157	23	(	(	PUNCT
ejpam-5367	157	24	x	x	X
ejpam-5367	157	25	·	·	PUNCT
ejpam-5367	157	26	0	0	NUM
ejpam-5367	157	27	)	)	PUNCT
ejpam-5367	157	28	·	·	PUNCT
ejpam-5367	157	29	0	0	X
ejpam-5367	157	30	)	)	PUNCT
ejpam-5367	157	31	(	(	PUNCT
ejpam-5367	157	32	by	by	ADP
ejpam-5367	157	33	(	(	PUNCT
ejpam-5367	157	34	2.5	2.5	NUM
ejpam-5367	157	35	)	)	PUNCT
ejpam-5367	157	36	)	)	PUNCT
ejpam-5367	157	37	≥	≥	NOUN
ejpam-5367	157	38	αf	αf	X
ejpam-5367	157	39	(	(	PUNCT
ejpam-5367	157	40	0	0	NUM
ejpam-5367	157	41	)	)	PUNCT
ejpam-5367	157	42	,	,	PUNCT
ejpam-5367	157	43	(	(	PUNCT
ejpam-5367	157	44	by	by	ADP
ejpam-5367	157	45	(	(	PUNCT
ejpam-5367	157	46	3.10	3.10	NUM
ejpam-5367	157	47	)	)	PUNCT
ejpam-5367	157	48	)	)	PUNCT
ejpam-5367	158	1	βf	βf	X
ejpam-5367	159	1	(	(	PUNCT
ejpam-5367	159	2	x	x	X
ejpam-5367	159	3	)	)	PUNCT
ejpam-5367	159	4	=	=	SYM
ejpam-5367	159	5	βf	βf	PUNCT
ejpam-5367	159	6	(	(	PUNCT
ejpam-5367	159	7	(	(	PUNCT
ejpam-5367	159	8	x	x	X
ejpam-5367	159	9	·	·	PUNCT
ejpam-5367	159	10	0	0	NUM
ejpam-5367	159	11	)	)	PUNCT
ejpam-5367	159	12	·	·	PUNCT
ejpam-5367	159	13	0	0	X
ejpam-5367	159	14	)	)	PUNCT
ejpam-5367	159	15	(	(	PUNCT
ejpam-5367	159	16	by	by	ADP
ejpam-5367	159	17	(	(	PUNCT
ejpam-5367	159	18	2.5	2.5	NUM
ejpam-5367	159	19	)	)	PUNCT
ejpam-5367	159	20	)	)	PUNCT
ejpam-5367	159	21	≤	≤	NUM
ejpam-5367	159	22	βf	βf	CCONJ
ejpam-5367	160	1	(	(	PUNCT
ejpam-5367	160	2	0	0	NUM
ejpam-5367	160	3	)	)	PUNCT
ejpam-5367	160	4	.	.	PUNCT
ejpam-5367	161	1	(	(	PUNCT
ejpam-5367	161	2	by	by	ADP
ejpam-5367	161	3	(	(	PUNCT
ejpam-5367	161	4	3.11	3.11	NUM
ejpam-5367	161	5	)	)	PUNCT
ejpam-5367	161	6	)	)	PUNCT
ejpam-5367	162	1	it	it	PRON
ejpam-5367	162	2	follows	follow	VERB
ejpam-5367	162	3	from	from	ADP
ejpam-5367	162	4	theorem	theorem	ADJ
ejpam-5367	162	5	1	1	NUM
ejpam-5367	162	6	that	that	SCONJ
ejpam-5367	162	7	f	f	PROPN
ejpam-5367	162	8	is	be	AUX
ejpam-5367	162	9	a	a	DET
ejpam-5367	162	10	constant	constant	ADJ
ejpam-5367	162	11	ffs	ffs	NOUN
ejpam-5367	162	12	of	of	ADP
ejpam-5367	162	13	x.	x.	NOUN
ejpam-5367	162	14	conversely	conversely	ADV
ejpam-5367	162	15	,	,	PUNCT
ejpam-5367	162	16	it	it	PRON
ejpam-5367	162	17	is	be	AUX
ejpam-5367	162	18	obviously	obviously	ADV
ejpam-5367	162	19	true	true	ADJ
ejpam-5367	162	20	that	that	SCONJ
ejpam-5367	162	21	every	every	DET
ejpam-5367	162	22	constant	constant	ADJ
ejpam-5367	162	23	ffs	ffs	NOUN
ejpam-5367	162	24	is	be	AUX
ejpam-5367	162	25	an	an	DET
ejpam-5367	162	26	ffsiup	ffsiup	NOUN
ejpam-5367	162	27	-	-	PUNCT
ejpam-5367	162	28	ideal	ideal	NOUN
ejpam-5367	162	29	of	of	ADP
ejpam-5367	162	30	x.	x.	NOUN
ejpam-5367	162	31	the	the	DET
ejpam-5367	162	32	following	follow	VERB
ejpam-5367	162	33	theorem	theorem	NOUN
ejpam-5367	162	34	is	be	AUX
ejpam-5367	162	35	a	a	DET
ejpam-5367	162	36	direct	direct	ADJ
ejpam-5367	162	37	consequence	consequence	NOUN
ejpam-5367	162	38	of	of	ADP
ejpam-5367	162	39	theorem	theorem	ADJ
ejpam-5367	162	40	2	2	NUM
ejpam-5367	162	41	.	.	PUNCT
ejpam-5367	162	42	theorem	theorem	NOUN
ejpam-5367	162	43	3	3	NUM
ejpam-5367	162	44	.	.	PUNCT
ejpam-5367	163	1	every	every	DET
ejpam-5367	163	2	ffsiup	ffsiup	NOUN
ejpam-5367	163	3	-	-	PUNCT
ejpam-5367	163	4	ideal	ideal	NOUN
ejpam-5367	163	5	of	of	ADP
ejpam-5367	163	6	x	x	PUNCT
ejpam-5367	163	7	is	be	AUX
ejpam-5367	163	8	an	an	DET
ejpam-5367	163	9	ffiup	ffiup	NOUN
ejpam-5367	163	10	-	-	PUNCT
ejpam-5367	163	11	subalgebra	subalgebra	NOUN
ejpam-5367	163	12	of	of	ADP
ejpam-5367	163	13	x.	x.	PROPN
ejpam-5367	163	14	a.	a.	PROPN
ejpam-5367	163	15	iampan	iampan	PROPN
ejpam-5367	163	16	et	et	PROPN
ejpam-5367	163	17	al	al	PROPN
ejpam-5367	163	18	.	.	PUNCT
ejpam-5367	163	19	/	/	SYM
ejpam-5367	163	20	eur	eur	PROPN
ejpam-5367	163	21	.	.	PUNCT
ejpam-5367	164	1	j.	j.	PROPN
ejpam-5367	164	2	pure	pure	PROPN
ejpam-5367	164	3	appl	appl	PROPN
ejpam-5367	164	4	.	.	PROPN
ejpam-5367	164	5	math	math	PROPN
ejpam-5367	164	6	,	,	PUNCT
ejpam-5367	164	7	17	17	NUM
ejpam-5367	164	8	(	(	PUNCT
ejpam-5367	164	9	4	4	NUM
ejpam-5367	164	10	)	)	PUNCT
ejpam-5367	164	11	(	(	PUNCT
ejpam-5367	164	12	2024	2024	NUM
ejpam-5367	164	13	)	)	PUNCT
ejpam-5367	164	14	,	,	PUNCT
ejpam-5367	164	15	3022	3022	NUM
ejpam-5367	164	16	-	-	SYM
ejpam-5367	164	17	3042	3042	NUM
ejpam-5367	164	18	3029	3029	NUM
ejpam-5367	164	19	example	example	NOUN
ejpam-5367	164	20	4	4	NUM
ejpam-5367	164	21	.	.	PUNCT
ejpam-5367	165	1	let	let	VERB
ejpam-5367	165	2	x	x	PUNCT
ejpam-5367	165	3	=	=	PUNCT
ejpam-5367	165	4	{	{	PUNCT
ejpam-5367	165	5	0	0	NUM
ejpam-5367	165	6	,	,	PUNCT
ejpam-5367	165	7	1	1	NUM
ejpam-5367	165	8	,	,	PUNCT
ejpam-5367	165	9	2	2	NUM
ejpam-5367	165	10	,	,	PUNCT
ejpam-5367	165	11	3	3	NUM
ejpam-5367	165	12	,	,	PUNCT
ejpam-5367	165	13	4	4	NUM
ejpam-5367	165	14	,	,	PUNCT
ejpam-5367	165	15	5	5	NUM
ejpam-5367	165	16	}	}	PUNCT
ejpam-5367	165	17	with	with	ADP
ejpam-5367	165	18	the	the	DET
ejpam-5367	165	19	following	follow	VERB
ejpam-5367	165	20	cayley	cayley	ADJ
ejpam-5367	165	21	table	table	NOUN
ejpam-5367	165	22	:	:	PUNCT
ejpam-5367	165	23	·	·	PUNCT
ejpam-5367	165	24	0	0	NUM
ejpam-5367	166	1	1	1	NUM
ejpam-5367	166	2	2	2	NUM
ejpam-5367	166	3	3	3	NUM
ejpam-5367	166	4	4	4	NUM
ejpam-5367	166	5	5	5	NUM
ejpam-5367	166	6	0	0	NUM
ejpam-5367	166	7	0	0	NUM
ejpam-5367	166	8	1	1	NUM
ejpam-5367	166	9	2	2	NUM
ejpam-5367	166	10	3	3	NUM
ejpam-5367	166	11	4	4	NUM
ejpam-5367	166	12	5	5	NUM
ejpam-5367	166	13	1	1	NUM
ejpam-5367	166	14	1	1	NUM
ejpam-5367	166	15	0	0	NUM
ejpam-5367	166	16	4	4	NUM
ejpam-5367	166	17	5	5	NUM
ejpam-5367	166	18	2	2	NUM
ejpam-5367	166	19	3	3	NUM
ejpam-5367	166	20	2	2	NUM
ejpam-5367	166	21	3	3	NUM
ejpam-5367	166	22	5	5	NUM
ejpam-5367	166	23	0	0	NUM
ejpam-5367	166	24	4	4	NUM
ejpam-5367	166	25	1	1	NUM
ejpam-5367	166	26	2	2	NUM
ejpam-5367	166	27	3	3	NUM
ejpam-5367	166	28	2	2	NUM
ejpam-5367	166	29	4	4	NUM
ejpam-5367	166	30	5	5	NUM
ejpam-5367	166	31	0	0	NUM
ejpam-5367	166	32	3	3	NUM
ejpam-5367	166	33	1	1	NUM
ejpam-5367	166	34	4	4	NUM
ejpam-5367	166	35	5	5	NUM
ejpam-5367	166	36	3	3	NUM
ejpam-5367	166	37	1	1	NUM
ejpam-5367	166	38	2	2	NUM
ejpam-5367	166	39	0	0	NUM
ejpam-5367	166	40	4	4	NUM
ejpam-5367	166	41	5	5	NUM
ejpam-5367	166	42	4	4	NUM
ejpam-5367	166	43	2	2	NUM
ejpam-5367	166	44	3	3	NUM
ejpam-5367	166	45	1	1	NUM
ejpam-5367	166	46	5	5	NUM
ejpam-5367	166	47	0	0	NUM
ejpam-5367	166	48	then	then	ADV
ejpam-5367	166	49	x	x	PUNCT
ejpam-5367	166	50	is	be	AUX
ejpam-5367	166	51	an	an	DET
ejpam-5367	166	52	iup	iup	NOUN
ejpam-5367	166	53	-	-	PUNCT
ejpam-5367	166	54	algebra	algebra	NOUN
ejpam-5367	166	55	.	.	PUNCT
ejpam-5367	167	1	we	we	PRON
ejpam-5367	167	2	define	define	VERB
ejpam-5367	167	3	an	an	DET
ejpam-5367	167	4	ffs	ffs	NOUN
ejpam-5367	167	5	f	f	PROPN
ejpam-5367	167	6	in	in	ADP
ejpam-5367	167	7	x	x	PROPN
ejpam-5367	167	8	as	as	SCONJ
ejpam-5367	167	9	follows	follow	VERB
ejpam-5367	167	10	:	:	PUNCT
ejpam-5367	167	11	αf	αf	NOUN
ejpam-5367	167	12	=	=	PUNCT
ejpam-5367	167	13	(	(	PUNCT
ejpam-5367	167	14	0	0	NUM
ejpam-5367	167	15	0.9	0.9	NUM
ejpam-5367	167	16	1	1	NUM
ejpam-5367	167	17	0.1	0.1	NUM
ejpam-5367	167	18	2	2	NUM
ejpam-5367	167	19	0.1	0.1	NUM
ejpam-5367	167	20	3	3	NUM
ejpam-5367	167	21	0.1	0.1	NUM
ejpam-5367	167	22	4	4	NUM
ejpam-5367	167	23	0.5	0.5	NUM
ejpam-5367	167	24	5	5	NUM
ejpam-5367	167	25	0.5	0.5	NUM
ejpam-5367	167	26	)	)	PUNCT
ejpam-5367	167	27	βf	βf	PUNCT
ejpam-5367	168	1	=	=	PUNCT
ejpam-5367	168	2	(	(	PUNCT
ejpam-5367	168	3	0	0	NUM
ejpam-5367	168	4	0.2	0.2	NUM
ejpam-5367	168	5	1	1	NUM
ejpam-5367	168	6	0.8	0.8	NUM
ejpam-5367	168	7	2	2	NUM
ejpam-5367	168	8	0.8	0.8	NUM
ejpam-5367	168	9	3	3	NUM
ejpam-5367	168	10	0.8	0.8	NUM
ejpam-5367	168	11	4	4	NUM
ejpam-5367	168	12	0.6	0.6	NUM
ejpam-5367	168	13	5	5	NUM
ejpam-5367	168	14	0.6	0.6	NUM
ejpam-5367	168	15	)	)	PUNCT
ejpam-5367	169	1	then	then	ADV
ejpam-5367	169	2	f	f	PROPN
ejpam-5367	169	3	is	be	AUX
ejpam-5367	169	4	an	an	DET
ejpam-5367	169	5	ffiup	ffiup	NOUN
ejpam-5367	169	6	-	-	PUNCT
ejpam-5367	169	7	subalgebra	subalgebra	NOUN
ejpam-5367	169	8	of	of	ADP
ejpam-5367	169	9	x.	x.	NOUN
ejpam-5367	169	10	since	since	SCONJ
ejpam-5367	169	11	αf	αf	PROPN
ejpam-5367	169	12	(	(	PUNCT
ejpam-5367	169	13	1	1	NUM
ejpam-5367	169	14	·	·	SYM
ejpam-5367	169	15	4	4	NUM
ejpam-5367	169	16	)	)	PUNCT
ejpam-5367	169	17	=	=	SYM
ejpam-5367	169	18	αf	αf	X
ejpam-5367	169	19	(	(	PUNCT
ejpam-5367	169	20	2	2	NUM
ejpam-5367	169	21	)	)	PUNCT
ejpam-5367	169	22	=	=	SYM
ejpam-5367	169	23	0.1	0.1	NUM
ejpam-5367	169	24	≱	≱	PROPN
ejpam-5367	169	25	0.5	0.5	NUM
ejpam-5367	169	26	=	=	SYM
ejpam-5367	169	27	αf	αf	X
ejpam-5367	169	28	(	(	PUNCT
ejpam-5367	169	29	4	4	NUM
ejpam-5367	169	30	)	)	PUNCT
ejpam-5367	169	31	and	and	CCONJ
ejpam-5367	169	32	βf	βf	X
ejpam-5367	169	33	(	(	PUNCT
ejpam-5367	169	34	3	3	NUM
ejpam-5367	169	35	·	·	SYM
ejpam-5367	169	36	4	4	NUM
ejpam-5367	169	37	)	)	PUNCT
ejpam-5367	169	38	=	=	SYM
ejpam-5367	169	39	βf	βf	PUNCT
ejpam-5367	169	40	(	(	PUNCT
ejpam-5367	169	41	3	3	X
ejpam-5367	169	42	)	)	PUNCT
ejpam-5367	169	43	=	=	SYM
ejpam-5367	169	44	0.8	0.8	NUM
ejpam-5367	169	45	≰	≰	PROPN
ejpam-5367	169	46	0.6	0.6	NUM
ejpam-5367	169	47	=	=	SYM
ejpam-5367	169	48	βf	βf	PUNCT
ejpam-5367	169	49	(	(	PUNCT
ejpam-5367	169	50	4	4	NUM
ejpam-5367	169	51	)	)	PUNCT
ejpam-5367	169	52	.	.	PUNCT
ejpam-5367	170	1	hence	hence	ADV
ejpam-5367	170	2	,	,	PUNCT
ejpam-5367	170	3	f	f	PROPN
ejpam-5367	170	4	is	be	AUX
ejpam-5367	170	5	not	not	PART
ejpam-5367	170	6	an	an	DET
ejpam-5367	170	7	ffsiup	ffsiup	NOUN
ejpam-5367	170	8	-	-	PUNCT
ejpam-5367	170	9	ideal	ideal	NOUN
ejpam-5367	170	10	of	of	ADP
ejpam-5367	170	11	x.	x.	NOUN
ejpam-5367	170	12	the	the	DET
ejpam-5367	170	13	following	follow	VERB
ejpam-5367	170	14	theorem	theorem	NOUN
ejpam-5367	170	15	is	be	AUX
ejpam-5367	170	16	a	a	DET
ejpam-5367	170	17	direct	direct	ADJ
ejpam-5367	170	18	consequence	consequence	NOUN
ejpam-5367	170	19	of	of	ADP
ejpam-5367	170	20	theorem	theorem	ADJ
ejpam-5367	170	21	2	2	NUM
ejpam-5367	170	22	.	.	PUNCT
ejpam-5367	170	23	theorem	theorem	NOUN
ejpam-5367	170	24	4	4	NUM
ejpam-5367	170	25	.	.	PUNCT
ejpam-5367	171	1	every	every	DET
ejpam-5367	171	2	ffsiup	ffsiup	NOUN
ejpam-5367	171	3	-	-	PUNCT
ejpam-5367	171	4	ideal	ideal	NOUN
ejpam-5367	171	5	of	of	ADP
ejpam-5367	171	6	x	x	PUNCT
ejpam-5367	171	7	is	be	AUX
ejpam-5367	171	8	an	an	DET
ejpam-5367	171	9	ffiup	ffiup	NOUN
ejpam-5367	171	10	-	-	PUNCT
ejpam-5367	171	11	ideal	ideal	NOUN
ejpam-5367	171	12	of	of	ADP
ejpam-5367	171	13	x.	x.	PROPN
ejpam-5367	171	14	example	example	NOUN
ejpam-5367	171	15	5	5	NUM
ejpam-5367	171	16	.	.	PUNCT
ejpam-5367	172	1	let	let	VERB
ejpam-5367	172	2	x	x	PUNCT
ejpam-5367	172	3	=	=	PUNCT
ejpam-5367	172	4	{	{	PUNCT
ejpam-5367	172	5	0	0	NUM
ejpam-5367	172	6	,	,	PUNCT
ejpam-5367	172	7	1	1	NUM
ejpam-5367	172	8	,	,	PUNCT
ejpam-5367	172	9	2	2	NUM
ejpam-5367	172	10	,	,	PUNCT
ejpam-5367	172	11	3	3	NUM
ejpam-5367	172	12	,	,	PUNCT
ejpam-5367	172	13	4	4	NUM
ejpam-5367	172	14	,	,	PUNCT
ejpam-5367	172	15	5	5	NUM
ejpam-5367	172	16	}	}	PUNCT
ejpam-5367	172	17	with	with	ADP
ejpam-5367	172	18	the	the	DET
ejpam-5367	172	19	following	follow	VERB
ejpam-5367	172	20	cayley	cayley	ADJ
ejpam-5367	172	21	table	table	NOUN
ejpam-5367	172	22	:	:	PUNCT
ejpam-5367	172	23	·	·	PUNCT
ejpam-5367	172	24	0	0	NUM
ejpam-5367	172	25	1	1	NUM
ejpam-5367	172	26	2	2	NUM
ejpam-5367	172	27	3	3	NUM
ejpam-5367	172	28	4	4	NUM
ejpam-5367	172	29	5	5	NUM
ejpam-5367	172	30	0	0	NUM
ejpam-5367	172	31	0	0	NUM
ejpam-5367	172	32	1	1	NUM
ejpam-5367	172	33	2	2	NUM
ejpam-5367	172	34	3	3	NUM
ejpam-5367	172	35	4	4	NUM
ejpam-5367	172	36	5	5	NUM
ejpam-5367	172	37	1	1	NUM
ejpam-5367	172	38	4	4	NUM
ejpam-5367	172	39	0	0	NUM
ejpam-5367	172	40	3	3	NUM
ejpam-5367	172	41	1	1	NUM
ejpam-5367	172	42	5	5	NUM
ejpam-5367	172	43	2	2	NUM
ejpam-5367	172	44	2	2	NUM
ejpam-5367	172	45	2	2	NUM
ejpam-5367	172	46	5	5	NUM
ejpam-5367	172	47	0	0	NUM
ejpam-5367	172	48	4	4	NUM
ejpam-5367	172	49	3	3	NUM
ejpam-5367	172	50	1	1	NUM
ejpam-5367	172	51	3	3	NUM
ejpam-5367	172	52	5	5	NUM
ejpam-5367	172	53	4	4	NUM
ejpam-5367	172	54	1	1	NUM
ejpam-5367	172	55	0	0	NUM
ejpam-5367	172	56	2	2	NUM
ejpam-5367	172	57	3	3	NUM
ejpam-5367	172	58	4	4	NUM
ejpam-5367	172	59	1	1	NUM
ejpam-5367	172	60	3	3	NUM
ejpam-5367	172	61	5	5	NUM
ejpam-5367	172	62	2	2	NUM
ejpam-5367	172	63	0	0	NUM
ejpam-5367	172	64	4	4	NUM
ejpam-5367	172	65	5	5	NUM
ejpam-5367	172	66	3	3	NUM
ejpam-5367	172	67	2	2	NUM
ejpam-5367	172	68	4	4	NUM
ejpam-5367	172	69	5	5	NUM
ejpam-5367	172	70	1	1	NUM
ejpam-5367	172	71	0	0	NUM
ejpam-5367	172	72	then	then	ADV
ejpam-5367	172	73	x	x	PUNCT
ejpam-5367	172	74	is	be	AUX
ejpam-5367	172	75	an	an	DET
ejpam-5367	172	76	iup	iup	NOUN
ejpam-5367	172	77	-	-	PUNCT
ejpam-5367	172	78	algebra	algebra	NOUN
ejpam-5367	172	79	.	.	PUNCT
ejpam-5367	173	1	we	we	PRON
ejpam-5367	173	2	define	define	VERB
ejpam-5367	173	3	an	an	DET
ejpam-5367	173	4	ffs	ffs	NOUN
ejpam-5367	173	5	f	f	PROPN
ejpam-5367	173	6	in	in	ADP
ejpam-5367	173	7	x	x	PROPN
ejpam-5367	173	8	as	as	SCONJ
ejpam-5367	173	9	follows	follow	VERB
ejpam-5367	173	10	:	:	PUNCT
ejpam-5367	173	11	αf	αf	NOUN
ejpam-5367	173	12	=	=	PUNCT
ejpam-5367	173	13	(	(	PUNCT
ejpam-5367	173	14	0	0	NUM
ejpam-5367	173	15	0.5	0.5	NUM
ejpam-5367	173	16	1	1	NUM
ejpam-5367	173	17	0.1	0.1	NUM
ejpam-5367	173	18	2	2	NUM
ejpam-5367	173	19	0.1	0.1	NUM
ejpam-5367	173	20	3	3	NUM
ejpam-5367	173	21	0.3	0.3	NUM
ejpam-5367	173	22	4	4	NUM
ejpam-5367	173	23	0.1	0.1	NUM
ejpam-5367	173	24	5	5	NUM
ejpam-5367	173	25	0.3	0.3	NUM
ejpam-5367	173	26	)	)	PUNCT
ejpam-5367	173	27	βf	βf	PUNCT
ejpam-5367	174	1	=	=	PUNCT
ejpam-5367	174	2	(	(	PUNCT
ejpam-5367	174	3	0	0	NUM
ejpam-5367	174	4	0.6	0.6	NUM
ejpam-5367	174	5	1	1	NUM
ejpam-5367	174	6	0.9	0.9	NUM
ejpam-5367	174	7	2	2	NUM
ejpam-5367	174	8	0.9	0.9	NUM
ejpam-5367	174	9	3	3	NUM
ejpam-5367	174	10	0.7	0.7	NUM
ejpam-5367	174	11	4	4	NUM
ejpam-5367	174	12	0.9	0.9	NUM
ejpam-5367	174	13	5	5	NUM
ejpam-5367	174	14	0.7	0.7	NUM
ejpam-5367	174	15	)	)	PUNCT
ejpam-5367	174	16	then	then	ADV
ejpam-5367	174	17	f	f	PROPN
ejpam-5367	174	18	is	be	AUX
ejpam-5367	174	19	an	an	DET
ejpam-5367	174	20	ffiup	ffiup	NOUN
ejpam-5367	174	21	-	-	PUNCT
ejpam-5367	174	22	ideal	ideal	NOUN
ejpam-5367	174	23	of	of	ADP
ejpam-5367	174	24	x.	x.	NOUN
ejpam-5367	174	25	since	since	SCONJ
ejpam-5367	174	26	αf	αf	PROPN
ejpam-5367	174	27	(	(	PUNCT
ejpam-5367	174	28	5	5	NUM
ejpam-5367	174	29	·	·	SYM
ejpam-5367	174	30	0	0	NUM
ejpam-5367	174	31	)	)	PUNCT
ejpam-5367	175	1	=	=	SYM
ejpam-5367	175	2	αf	αf	X
ejpam-5367	175	3	(	(	PUNCT
ejpam-5367	175	4	3	3	NUM
ejpam-5367	175	5	)	)	PUNCT
ejpam-5367	175	6	=	=	PUNCT
ejpam-5367	175	7	0.3	0.3	NUM
ejpam-5367	175	8	≱	≱	PROPN
ejpam-5367	175	9	0.5	0.5	NUM
ejpam-5367	175	10	=	=	SYM
ejpam-5367	175	11	αf	αf	X
ejpam-5367	175	12	(	(	PUNCT
ejpam-5367	175	13	0	0	NUM
ejpam-5367	175	14	)	)	PUNCT
ejpam-5367	175	15	and	and	CCONJ
ejpam-5367	175	16	βf	βf	X
ejpam-5367	175	17	(	(	PUNCT
ejpam-5367	175	18	1	1	NUM
ejpam-5367	175	19	·	·	SYM
ejpam-5367	175	20	5	5	NUM
ejpam-5367	175	21	)	)	PUNCT
ejpam-5367	175	22	=	=	SYM
ejpam-5367	175	23	βf	βf	PUNCT
ejpam-5367	175	24	(	(	PUNCT
ejpam-5367	175	25	2	2	X
ejpam-5367	175	26	)	)	PUNCT
ejpam-5367	175	27	=	=	SYM
ejpam-5367	175	28	0.9	0.9	NUM
ejpam-5367	175	29	≰	≰	VERB
ejpam-5367	175	30	0.7	0.7	NUM
ejpam-5367	175	31	=	=	SYM
ejpam-5367	175	32	βf	βf	PRON
ejpam-5367	175	33	(	(	PUNCT
ejpam-5367	175	34	5	5	NUM
ejpam-5367	175	35	)	)	PUNCT
ejpam-5367	175	36	.	.	PUNCT
ejpam-5367	176	1	hence	hence	ADV
ejpam-5367	176	2	,	,	PUNCT
ejpam-5367	176	3	f	f	PROPN
ejpam-5367	176	4	is	be	AUX
ejpam-5367	176	5	not	not	PART
ejpam-5367	176	6	an	an	DET
ejpam-5367	176	7	ffsiup	ffsiup	NOUN
ejpam-5367	176	8	-	-	PUNCT
ejpam-5367	176	9	ideal	ideal	NOUN
ejpam-5367	176	10	of	of	ADP
ejpam-5367	176	11	x.	x.	PROPN
ejpam-5367	176	12	theorem	theorem	VERB
ejpam-5367	176	13	5	5	NUM
ejpam-5367	176	14	.	.	PUNCT
ejpam-5367	177	1	every	every	DET
ejpam-5367	177	2	ffiup	ffiup	NOUN
ejpam-5367	177	3	-	-	PUNCT
ejpam-5367	177	4	ideal	ideal	NOUN
ejpam-5367	177	5	of	of	ADP
ejpam-5367	177	6	x	x	PUNCT
ejpam-5367	177	7	is	be	AUX
ejpam-5367	177	8	an	an	DET
ejpam-5367	177	9	ffiup	ffiup	NOUN
ejpam-5367	177	10	-	-	PUNCT
ejpam-5367	177	11	filter	filter	NOUN
ejpam-5367	177	12	of	of	ADP
ejpam-5367	177	13	x.	x.	PROPN
ejpam-5367	177	14	a.	a.	PROPN
ejpam-5367	177	15	iampan	iampan	PROPN
ejpam-5367	177	16	et	et	PROPN
ejpam-5367	177	17	al	al	PROPN
ejpam-5367	177	18	.	.	PUNCT
ejpam-5367	177	19	/	/	SYM
ejpam-5367	177	20	eur	eur	PROPN
ejpam-5367	177	21	.	.	PUNCT
ejpam-5367	178	1	j.	j.	PROPN
ejpam-5367	178	2	pure	pure	PROPN
ejpam-5367	178	3	appl	appl	PROPN
ejpam-5367	178	4	.	.	PROPN
ejpam-5367	178	5	math	math	PROPN
ejpam-5367	178	6	,	,	PUNCT
ejpam-5367	178	7	17	17	NUM
ejpam-5367	178	8	(	(	PUNCT
ejpam-5367	178	9	4	4	NUM
ejpam-5367	178	10	)	)	PUNCT
ejpam-5367	178	11	(	(	PUNCT
ejpam-5367	178	12	2024	2024	NUM
ejpam-5367	178	13	)	)	PUNCT
ejpam-5367	178	14	,	,	PUNCT
ejpam-5367	178	15	3022	3022	NUM
ejpam-5367	178	16	-	-	SYM
ejpam-5367	178	17	3042	3042	NUM
ejpam-5367	178	18	3030	3030	NUM
ejpam-5367	178	19	proof	proof	NOUN
ejpam-5367	178	20	.	.	PUNCT
ejpam-5367	179	1	assume	assume	VERB
ejpam-5367	179	2	that	that	SCONJ
ejpam-5367	179	3	f	f	PROPN
ejpam-5367	179	4	is	be	AUX
ejpam-5367	179	5	an	an	DET
ejpam-5367	179	6	ffiup	ffiup	NOUN
ejpam-5367	179	7	-	-	PUNCT
ejpam-5367	179	8	ideal	ideal	NOUN
ejpam-5367	179	9	of	of	ADP
ejpam-5367	179	10	x.	x.	NOUN
ejpam-5367	179	11	by	by	ADP
ejpam-5367	179	12	the	the	DET
ejpam-5367	179	13	assumption	assumption	NOUN
ejpam-5367	179	14	,	,	PUNCT
ejpam-5367	179	15	it	it	PRON
ejpam-5367	179	16	satisfies	satisfy	VERB
ejpam-5367	179	17	(	(	PUNCT
ejpam-5367	179	18	3.4	3.4	NUM
ejpam-5367	179	19	)	)	PUNCT
ejpam-5367	179	20	and	and	CCONJ
ejpam-5367	179	21	(	(	PUNCT
ejpam-5367	179	22	3.5	3.5	NUM
ejpam-5367	179	23	)	)	PUNCT
ejpam-5367	179	24	.	.	PUNCT
ejpam-5367	180	1	let	let	VERB
ejpam-5367	180	2	x	x	PRON
ejpam-5367	180	3	,	,	PUNCT
ejpam-5367	180	4	y	y	PROPN
ejpam-5367	180	5	∈	∈	PROPN
ejpam-5367	180	6	x.	x.	NOUN
ejpam-5367	181	1	then	then	ADV
ejpam-5367	181	2	αf	αf	VERB
ejpam-5367	181	3	(	(	PUNCT
ejpam-5367	181	4	y	y	NOUN
ejpam-5367	181	5	)	)	PUNCT
ejpam-5367	181	6	=	=	SYM
ejpam-5367	181	7	αf	αf	X
ejpam-5367	181	8	(	(	PUNCT
ejpam-5367	181	9	0	0	NUM
ejpam-5367	181	10	·	·	PUNCT
ejpam-5367	181	11	y	y	X
ejpam-5367	181	12	)	)	PUNCT
ejpam-5367	181	13	(	(	PUNCT
ejpam-5367	181	14	by	by	ADP
ejpam-5367	181	15	(	(	PUNCT
ejpam-5367	181	16	iup-1	iup-1	NOUN
ejpam-5367	181	17	)	)	PUNCT
ejpam-5367	181	18	)	)	PUNCT
ejpam-5367	181	19	≥	≥	NOUN
ejpam-5367	181	20	min{αf	min{αf	PUNCT
ejpam-5367	181	21	(	(	PUNCT
ejpam-5367	181	22	0	0	NUM
ejpam-5367	181	23	·	·	PUNCT
ejpam-5367	181	24	(	(	PUNCT
ejpam-5367	181	25	x	x	X
ejpam-5367	181	26	·	·	PUNCT
ejpam-5367	181	27	y	y	NOUN
ejpam-5367	181	28	)	)	PUNCT
ejpam-5367	181	29	)	)	PUNCT
ejpam-5367	181	30	,	,	PUNCT
ejpam-5367	181	31	αf	αf	X
ejpam-5367	181	32	(	(	PUNCT
ejpam-5367	181	33	x	x	NOUN
ejpam-5367	181	34	)	)	PUNCT
ejpam-5367	181	35	}	}	PUNCT
ejpam-5367	181	36	(	(	PUNCT
ejpam-5367	181	37	by	by	ADP
ejpam-5367	181	38	(	(	PUNCT
ejpam-5367	181	39	3.6	3.6	NUM
ejpam-5367	181	40	)	)	PUNCT
ejpam-5367	181	41	)	)	PUNCT
ejpam-5367	182	1	=	=	PUNCT
ejpam-5367	182	2	min{αf	min{αf	PUNCT
ejpam-5367	182	3	(	(	PUNCT
ejpam-5367	182	4	x	x	X
ejpam-5367	182	5	·	·	PUNCT
ejpam-5367	182	6	y	y	X
ejpam-5367	182	7	)	)	PUNCT
ejpam-5367	182	8	,	,	PUNCT
ejpam-5367	182	9	αf	αf	X
ejpam-5367	182	10	(	(	PUNCT
ejpam-5367	182	11	x	x	NOUN
ejpam-5367	182	12	)	)	PUNCT
ejpam-5367	182	13	}	}	PUNCT
ejpam-5367	182	14	,	,	PUNCT
ejpam-5367	182	15	(	(	PUNCT
ejpam-5367	182	16	by	by	ADP
ejpam-5367	182	17	(	(	PUNCT
ejpam-5367	182	18	iup-1	iup-1	NOUN
ejpam-5367	182	19	)	)	PUNCT
ejpam-5367	182	20	)	)	PUNCT
ejpam-5367	182	21	βf	βf	X
ejpam-5367	183	1	(	(	PUNCT
ejpam-5367	183	2	y	y	NOUN
ejpam-5367	183	3	)	)	PUNCT
ejpam-5367	183	4	=	=	VERB
ejpam-5367	184	1	βf	βf	PUNCT
ejpam-5367	184	2	(	(	PUNCT
ejpam-5367	184	3	0	0	NUM
ejpam-5367	184	4	·	·	PUNCT
ejpam-5367	184	5	y	y	X
ejpam-5367	184	6	)	)	PUNCT
ejpam-5367	184	7	(	(	PUNCT
ejpam-5367	184	8	by	by	ADP
ejpam-5367	184	9	(	(	PUNCT
ejpam-5367	184	10	iup-1	iup-1	NOUN
ejpam-5367	184	11	)	)	PUNCT
ejpam-5367	184	12	)	)	PUNCT
ejpam-5367	184	13	≤	≤	NUM
ejpam-5367	185	1	max{βf	max{βf	INTJ
ejpam-5367	185	2	(	(	PUNCT
ejpam-5367	185	3	0	0	NUM
ejpam-5367	185	4	·	·	PUNCT
ejpam-5367	185	5	(	(	PUNCT
ejpam-5367	185	6	x	x	X
ejpam-5367	185	7	·	·	PUNCT
ejpam-5367	185	8	y	y	NOUN
ejpam-5367	185	9	)	)	PUNCT
ejpam-5367	185	10	)	)	PUNCT
ejpam-5367	185	11	,	,	PUNCT
ejpam-5367	185	12	βf	βf	CCONJ
ejpam-5367	185	13	(	(	PUNCT
ejpam-5367	185	14	x	x	X
ejpam-5367	185	15	)	)	PUNCT
ejpam-5367	185	16	}	}	PUNCT
ejpam-5367	185	17	(	(	PUNCT
ejpam-5367	185	18	by	by	ADP
ejpam-5367	185	19	(	(	PUNCT
ejpam-5367	185	20	3.7	3.7	NUM
ejpam-5367	185	21	)	)	PUNCT
ejpam-5367	185	22	)	)	PUNCT
ejpam-5367	186	1	=	=	PUNCT
ejpam-5367	186	2	max{βf	max{βf	INTJ
ejpam-5367	187	1	(	(	PUNCT
ejpam-5367	187	2	x	x	X
ejpam-5367	187	3	·	·	PUNCT
ejpam-5367	187	4	y	y	X
ejpam-5367	187	5	)	)	PUNCT
ejpam-5367	187	6	,	,	PUNCT
ejpam-5367	187	7	βf	βf	CCONJ
ejpam-5367	187	8	(	(	PUNCT
ejpam-5367	187	9	x	x	NOUN
ejpam-5367	187	10	)	)	PUNCT
ejpam-5367	187	11	}	}	PUNCT
ejpam-5367	187	12	.	.	PUNCT
ejpam-5367	188	1	(	(	PUNCT
ejpam-5367	188	2	by	by	ADP
ejpam-5367	188	3	(	(	PUNCT
ejpam-5367	188	4	iup-1	iup-1	NOUN
ejpam-5367	188	5	)	)	PUNCT
ejpam-5367	188	6	)	)	PUNCT
ejpam-5367	188	7	hence	hence	ADV
ejpam-5367	188	8	,	,	PUNCT
ejpam-5367	188	9	f	f	PROPN
ejpam-5367	188	10	is	be	AUX
ejpam-5367	188	11	an	an	DET
ejpam-5367	188	12	ffiup	ffiup	ADJ
ejpam-5367	188	13	-	-	PUNCT
ejpam-5367	188	14	filter	filter	NOUN
ejpam-5367	188	15	of	of	ADP
ejpam-5367	188	16	x.	x.	PROPN
ejpam-5367	188	17	example	example	NOUN
ejpam-5367	188	18	6	6	NUM
ejpam-5367	188	19	.	.	PUNCT
ejpam-5367	189	1	let	let	VERB
ejpam-5367	189	2	x	x	PUNCT
ejpam-5367	189	3	=	=	PUNCT
ejpam-5367	189	4	{	{	PUNCT
ejpam-5367	189	5	0	0	NUM
ejpam-5367	189	6	,	,	PUNCT
ejpam-5367	189	7	1	1	NUM
ejpam-5367	189	8	,	,	PUNCT
ejpam-5367	189	9	2	2	NUM
ejpam-5367	189	10	,	,	PUNCT
ejpam-5367	189	11	3	3	NUM
ejpam-5367	189	12	,	,	PUNCT
ejpam-5367	189	13	4	4	NUM
ejpam-5367	189	14	,	,	PUNCT
ejpam-5367	189	15	5	5	NUM
ejpam-5367	189	16	}	}	PUNCT
ejpam-5367	189	17	with	with	ADP
ejpam-5367	189	18	the	the	DET
ejpam-5367	189	19	following	follow	VERB
ejpam-5367	189	20	cayley	cayley	ADJ
ejpam-5367	189	21	table	table	NOUN
ejpam-5367	189	22	:	:	PUNCT
ejpam-5367	189	23	·	·	PUNCT
ejpam-5367	189	24	0	0	NUM
ejpam-5367	189	25	1	1	NUM
ejpam-5367	189	26	2	2	NUM
ejpam-5367	189	27	3	3	NUM
ejpam-5367	189	28	4	4	NUM
ejpam-5367	189	29	5	5	NUM
ejpam-5367	189	30	0	0	NUM
ejpam-5367	189	31	0	0	NUM
ejpam-5367	189	32	1	1	NUM
ejpam-5367	189	33	2	2	NUM
ejpam-5367	189	34	3	3	NUM
ejpam-5367	189	35	4	4	NUM
ejpam-5367	189	36	5	5	NUM
ejpam-5367	189	37	1	1	NUM
ejpam-5367	189	38	1	1	NUM
ejpam-5367	189	39	0	0	NUM
ejpam-5367	189	40	5	5	NUM
ejpam-5367	189	41	4	4	NUM
ejpam-5367	189	42	3	3	NUM
ejpam-5367	189	43	2	2	NUM
ejpam-5367	189	44	2	2	NUM
ejpam-5367	189	45	2	2	NUM
ejpam-5367	189	46	4	4	NUM
ejpam-5367	189	47	0	0	NUM
ejpam-5367	189	48	5	5	NUM
ejpam-5367	189	49	1	1	NUM
ejpam-5367	189	50	3	3	NUM
ejpam-5367	189	51	3	3	NUM
ejpam-5367	189	52	3	3	NUM
ejpam-5367	189	53	5	5	NUM
ejpam-5367	189	54	4	4	NUM
ejpam-5367	189	55	0	0	NUM
ejpam-5367	189	56	2	2	NUM
ejpam-5367	189	57	1	1	NUM
ejpam-5367	189	58	4	4	NUM
ejpam-5367	189	59	5	5	NUM
ejpam-5367	189	60	3	3	NUM
ejpam-5367	189	61	1	1	NUM
ejpam-5367	189	62	2	2	NUM
ejpam-5367	189	63	0	0	NUM
ejpam-5367	189	64	4	4	NUM
ejpam-5367	189	65	5	5	NUM
ejpam-5367	189	66	4	4	NUM
ejpam-5367	189	67	2	2	NUM
ejpam-5367	189	68	3	3	NUM
ejpam-5367	189	69	1	1	NUM
ejpam-5367	189	70	5	5	NUM
ejpam-5367	189	71	0	0	NUM
ejpam-5367	189	72	then	then	ADV
ejpam-5367	189	73	x	x	PUNCT
ejpam-5367	189	74	is	be	AUX
ejpam-5367	189	75	an	an	DET
ejpam-5367	189	76	iup	iup	NOUN
ejpam-5367	189	77	-	-	PUNCT
ejpam-5367	189	78	algebra	algebra	NOUN
ejpam-5367	189	79	.	.	PUNCT
ejpam-5367	190	1	we	we	PRON
ejpam-5367	190	2	define	define	VERB
ejpam-5367	190	3	an	an	DET
ejpam-5367	190	4	ffs	ffs	NOUN
ejpam-5367	190	5	f	f	PROPN
ejpam-5367	190	6	in	in	ADP
ejpam-5367	190	7	x	x	PROPN
ejpam-5367	190	8	as	as	SCONJ
ejpam-5367	190	9	follows	follow	VERB
ejpam-5367	190	10	:	:	PUNCT
ejpam-5367	190	11	αf	αf	NOUN
ejpam-5367	190	12	=	=	PUNCT
ejpam-5367	190	13	(	(	PUNCT
ejpam-5367	190	14	0	0	NUM
ejpam-5367	190	15	0.6	0.6	NUM
ejpam-5367	190	16	1	1	NUM
ejpam-5367	190	17	0.5	0.5	NUM
ejpam-5367	190	18	2	2	NUM
ejpam-5367	190	19	0.2	0.2	NUM
ejpam-5367	190	20	3	3	NUM
ejpam-5367	190	21	0.2	0.2	NUM
ejpam-5367	190	22	4	4	NUM
ejpam-5367	190	23	0.2	0.2	NUM
ejpam-5367	190	24	5	5	NUM
ejpam-5367	190	25	0.2	0.2	NUM
ejpam-5367	190	26	)	)	PUNCT
ejpam-5367	190	27	βf	βf	PUNCT
ejpam-5367	191	1	=	=	PUNCT
ejpam-5367	191	2	(	(	PUNCT
ejpam-5367	191	3	0	0	NUM
ejpam-5367	191	4	0.1	0.1	NUM
ejpam-5367	191	5	1	1	NUM
ejpam-5367	191	6	0.7	0.7	NUM
ejpam-5367	191	7	2	2	NUM
ejpam-5367	191	8	0.9	0.9	NUM
ejpam-5367	191	9	3	3	NUM
ejpam-5367	191	10	0.9	0.9	NUM
ejpam-5367	191	11	4	4	NUM
ejpam-5367	191	12	0.9	0.9	NUM
ejpam-5367	191	13	5	5	NUM
ejpam-5367	191	14	0.9	0.9	NUM
ejpam-5367	191	15	)	)	PUNCT
ejpam-5367	192	1	then	then	ADV
ejpam-5367	192	2	f	f	PROPN
ejpam-5367	192	3	is	be	AUX
ejpam-5367	192	4	an	an	DET
ejpam-5367	192	5	ffiup	ffiup	ADJ
ejpam-5367	192	6	-	-	PUNCT
ejpam-5367	192	7	filter	filter	NOUN
ejpam-5367	192	8	of	of	ADP
ejpam-5367	192	9	x.	x.	NOUN
ejpam-5367	192	10	since	since	SCONJ
ejpam-5367	192	11	αf	αf	PROPN
ejpam-5367	192	12	(	(	PUNCT
ejpam-5367	192	13	2	2	NUM
ejpam-5367	192	14	·	·	SYM
ejpam-5367	192	15	5	5	NUM
ejpam-5367	192	16	)	)	PUNCT
ejpam-5367	192	17	=	=	SYM
ejpam-5367	192	18	αf	αf	X
ejpam-5367	192	19	(	(	PUNCT
ejpam-5367	192	20	3	3	NUM
ejpam-5367	192	21	)	)	PUNCT
ejpam-5367	192	22	=	=	SYM
ejpam-5367	192	23	0.2	0.2	NUM
ejpam-5367	192	24	≱	≱	PROPN
ejpam-5367	192	25	0.5	0.5	NUM
ejpam-5367	192	26	=	=	SYM
ejpam-5367	192	27	min{0.6	min{0.6	PROPN
ejpam-5367	192	28	,	,	PUNCT
ejpam-5367	192	29	0.5	0.5	NUM
ejpam-5367	192	30	}	}	PUNCT
ejpam-5367	192	31	=	=	PUNCT
ejpam-5367	192	32	min{αf	min{αf	X
ejpam-5367	192	33	(	(	PUNCT
ejpam-5367	192	34	0	0	NUM
ejpam-5367	192	35	)	)	PUNCT
ejpam-5367	192	36	,	,	PUNCT
ejpam-5367	192	37	αf	αf	X
ejpam-5367	192	38	(	(	PUNCT
ejpam-5367	192	39	1	1	NUM
ejpam-5367	192	40	)	)	PUNCT
ejpam-5367	192	41	}	}	PUNCT
ejpam-5367	192	42	=	=	SYM
ejpam-5367	192	43	min{αf	min{αf	X
ejpam-5367	192	44	(	(	PUNCT
ejpam-5367	192	45	2	2	NUM
ejpam-5367	192	46	·	·	SYM
ejpam-5367	192	47	2	2	NUM
ejpam-5367	192	48	)	)	PUNCT
ejpam-5367	192	49	,	,	PUNCT
ejpam-5367	192	50	αf	αf	X
ejpam-5367	192	51	(	(	PUNCT
ejpam-5367	192	52	1	1	NUM
ejpam-5367	192	53	)	)	PUNCT
ejpam-5367	192	54	}	}	PUNCT
ejpam-5367	192	55	=	=	SYM
ejpam-5367	192	56	min{αf	min{αf	X
ejpam-5367	192	57	(	(	PUNCT
ejpam-5367	192	58	2	2	NUM
ejpam-5367	192	59	·	·	PUNCT
ejpam-5367	192	60	(	(	PUNCT
ejpam-5367	192	61	1	1	NUM
ejpam-5367	192	62	·	·	SYM
ejpam-5367	192	63	5	5	NUM
ejpam-5367	192	64	)	)	PUNCT
ejpam-5367	192	65	)	)	PUNCT
ejpam-5367	192	66	,	,	PUNCT
ejpam-5367	192	67	αf	αf	X
ejpam-5367	192	68	(	(	PUNCT
ejpam-5367	192	69	1	1	NUM
ejpam-5367	192	70	)	)	PUNCT
ejpam-5367	192	71	}	}	PUNCT
ejpam-5367	192	72	and	and	CCONJ
ejpam-5367	192	73	βf	βf	X
ejpam-5367	192	74	(	(	PUNCT
ejpam-5367	192	75	3	3	NUM
ejpam-5367	192	76	·	·	SYM
ejpam-5367	192	77	4	4	NUM
ejpam-5367	192	78	)	)	PUNCT
ejpam-5367	192	79	=	=	SYM
ejpam-5367	193	1	βf	βf	PUNCT
ejpam-5367	193	2	(	(	PUNCT
ejpam-5367	193	3	2	2	X
ejpam-5367	193	4	)	)	PUNCT
ejpam-5367	193	5	=	=	SYM
ejpam-5367	194	1	0.9	0.9	NUM
ejpam-5367	194	2	≰	≰	VERB
ejpam-5367	194	3	0.7	0.7	NUM
ejpam-5367	194	4	=	=	SYM
ejpam-5367	194	5	max{0.1	max{0.1	PROPN
ejpam-5367	194	6	,	,	PUNCT
ejpam-5367	194	7	0.7	0.7	NUM
ejpam-5367	194	8	}	}	PUNCT
ejpam-5367	194	9	=	=	SYM
ejpam-5367	194	10	max{βf	max{βf	PROPN
ejpam-5367	194	11	(	(	PUNCT
ejpam-5367	194	12	0	0	NUM
ejpam-5367	194	13	)	)	PUNCT
ejpam-5367	194	14	,	,	PUNCT
ejpam-5367	194	15	βf	βf	CCONJ
ejpam-5367	194	16	(	(	PUNCT
ejpam-5367	194	17	1	1	NUM
ejpam-5367	194	18	)	)	PUNCT
ejpam-5367	194	19	}	}	PUNCT
ejpam-5367	194	20	=	=	SYM
ejpam-5367	194	21	max{βf	max{βf	PROPN
ejpam-5367	194	22	(	(	PUNCT
ejpam-5367	194	23	3	3	NUM
ejpam-5367	194	24	·	·	SYM
ejpam-5367	194	25	3	3	NUM
ejpam-5367	194	26	)	)	PUNCT
ejpam-5367	194	27	,	,	PUNCT
ejpam-5367	194	28	βf	βf	CCONJ
ejpam-5367	194	29	(	(	PUNCT
ejpam-5367	194	30	1	1	NUM
ejpam-5367	194	31	)	)	PUNCT
ejpam-5367	194	32	}	}	PUNCT
ejpam-5367	194	33	=	=	SYM
ejpam-5367	194	34	max{βf	max{βf	PROPN
ejpam-5367	194	35	(	(	PUNCT
ejpam-5367	194	36	3	3	NUM
ejpam-5367	194	37	·	·	PUNCT
ejpam-5367	194	38	(	(	PUNCT
ejpam-5367	194	39	1	1	NUM
ejpam-5367	194	40	·	·	SYM
ejpam-5367	194	41	4	4	NUM
ejpam-5367	194	42	)	)	PUNCT
ejpam-5367	194	43	)	)	PUNCT
ejpam-5367	194	44	,	,	PUNCT
ejpam-5367	194	45	βf	βf	CCONJ
ejpam-5367	194	46	(	(	PUNCT
ejpam-5367	194	47	1	1	NUM
ejpam-5367	194	48	)	)	PUNCT
ejpam-5367	194	49	}	}	PUNCT
ejpam-5367	194	50	.	.	PUNCT
ejpam-5367	195	1	hence	hence	ADV
ejpam-5367	195	2	,	,	PUNCT
ejpam-5367	195	3	f	f	PROPN
ejpam-5367	195	4	is	be	AUX
ejpam-5367	195	5	not	not	PART
ejpam-5367	195	6	an	an	DET
ejpam-5367	195	7	ffiup	ffiup	NOUN
ejpam-5367	195	8	-	-	PUNCT
ejpam-5367	195	9	ideal	ideal	NOUN
ejpam-5367	195	10	of	of	ADP
ejpam-5367	195	11	x.	x.	PROPN
ejpam-5367	195	12	theorem	theorem	VERB
ejpam-5367	195	13	6	6	NUM
ejpam-5367	195	14	.	.	PUNCT
ejpam-5367	196	1	every	every	DET
ejpam-5367	196	2	ffiup	ffiup	NOUN
ejpam-5367	196	3	-	-	PUNCT
ejpam-5367	196	4	subalgebra	subalgebra	NOUN
ejpam-5367	196	5	of	of	ADP
ejpam-5367	196	6	x	x	PUNCT
ejpam-5367	196	7	is	be	AUX
ejpam-5367	196	8	an	an	DET
ejpam-5367	196	9	ffiup	ffiup	NOUN
ejpam-5367	196	10	-	-	PUNCT
ejpam-5367	196	11	filter	filter	NOUN
ejpam-5367	196	12	of	of	ADP
ejpam-5367	196	13	x.	x.	NOUN
ejpam-5367	196	14	proof	proof	PROPN
ejpam-5367	196	15	.	.	PUNCT
ejpam-5367	197	1	assume	assume	VERB
ejpam-5367	197	2	that	that	SCONJ
ejpam-5367	197	3	f	f	PROPN
ejpam-5367	197	4	is	be	AUX
ejpam-5367	197	5	an	an	DET
ejpam-5367	197	6	ffiup	ffiup	NOUN
ejpam-5367	197	7	-	-	PUNCT
ejpam-5367	197	8	subalgebra	subalgebra	NOUN
ejpam-5367	197	9	of	of	ADP
ejpam-5367	197	10	x.	x.	NOUN
ejpam-5367	197	11	by	by	ADP
ejpam-5367	197	12	lemma	lemma	PROPN
ejpam-5367	197	13	1	1	NUM
ejpam-5367	197	14	,	,	PUNCT
ejpam-5367	197	15	we	we	PRON
ejpam-5367	197	16	have	have	VERB
ejpam-5367	197	17	f	f	PROPN
ejpam-5367	197	18	satisfies	satisfie	NOUN
ejpam-5367	197	19	(	(	PUNCT
ejpam-5367	197	20	3.4	3.4	NUM
ejpam-5367	197	21	)	)	PUNCT
ejpam-5367	197	22	and	and	CCONJ
ejpam-5367	197	23	(	(	PUNCT
ejpam-5367	197	24	3.5	3.5	NUM
ejpam-5367	197	25	)	)	PUNCT
ejpam-5367	197	26	.	.	PUNCT
ejpam-5367	198	1	let	let	VERB
ejpam-5367	198	2	x	x	PRON
ejpam-5367	198	3	,	,	PUNCT
ejpam-5367	198	4	y	y	PROPN
ejpam-5367	198	5	∈	∈	PROPN
ejpam-5367	198	6	x.	x.	NOUN
ejpam-5367	199	1	then	then	ADV
ejpam-5367	199	2	αf	αf	VERB
ejpam-5367	199	3	(	(	PUNCT
ejpam-5367	199	4	y	y	NOUN
ejpam-5367	199	5	)	)	PUNCT
ejpam-5367	199	6	=	=	SYM
ejpam-5367	199	7	αf	αf	X
ejpam-5367	199	8	(	(	PUNCT
ejpam-5367	199	9	0	0	NUM
ejpam-5367	199	10	·	·	PUNCT
ejpam-5367	199	11	y	y	X
ejpam-5367	199	12	)	)	PUNCT
ejpam-5367	199	13	(	(	PUNCT
ejpam-5367	199	14	by	by	ADP
ejpam-5367	199	15	(	(	PUNCT
ejpam-5367	199	16	iup-1	iup-1	NOUN
ejpam-5367	199	17	)	)	PUNCT
ejpam-5367	199	18	)	)	PUNCT
ejpam-5367	200	1	=	=	SYM
ejpam-5367	200	2	αf	αf	X
ejpam-5367	200	3	(	(	PUNCT
ejpam-5367	200	4	(	(	PUNCT
ejpam-5367	200	5	x	x	X
ejpam-5367	200	6	·	·	PUNCT
ejpam-5367	200	7	0	0	NUM
ejpam-5367	200	8	)	)	PUNCT
ejpam-5367	200	9	·	·	PUNCT
ejpam-5367	200	10	(	(	PUNCT
ejpam-5367	200	11	x	x	X
ejpam-5367	200	12	·	·	PUNCT
ejpam-5367	200	13	y	y	NOUN
ejpam-5367	200	14	)	)	PUNCT
ejpam-5367	200	15	)	)	PUNCT
ejpam-5367	200	16	(	(	PUNCT
ejpam-5367	200	17	by	by	ADP
ejpam-5367	200	18	(	(	PUNCT
ejpam-5367	200	19	iup-3	iup-3	NUM
ejpam-5367	200	20	)	)	PUNCT
ejpam-5367	200	21	)	)	PUNCT
ejpam-5367	200	22	≥	≥	NOUN
ejpam-5367	200	23	min{αf	min{αf	PUNCT
ejpam-5367	200	24	(	(	PUNCT
ejpam-5367	200	25	x	x	X
ejpam-5367	200	26	·	·	PUNCT
ejpam-5367	200	27	0	0	NUM
ejpam-5367	200	28	)	)	PUNCT
ejpam-5367	200	29	,	,	PUNCT
ejpam-5367	200	30	αf	αf	X
ejpam-5367	200	31	(	(	PUNCT
ejpam-5367	200	32	x	x	X
ejpam-5367	200	33	·	·	PUNCT
ejpam-5367	200	34	y	y	X
ejpam-5367	200	35	)	)	PUNCT
ejpam-5367	200	36	}	}	PUNCT
ejpam-5367	200	37	(	(	PUNCT
ejpam-5367	200	38	by	by	ADP
ejpam-5367	200	39	(	(	PUNCT
ejpam-5367	200	40	3.2	3.2	NUM
ejpam-5367	200	41	)	)	PUNCT
ejpam-5367	200	42	)	)	PUNCT
ejpam-5367	200	43	≥	≥	NOUN
ejpam-5367	200	44	min{min{αf	min{min{αf	NOUN
ejpam-5367	200	45	(	(	PUNCT
ejpam-5367	200	46	x	x	X
ejpam-5367	200	47	)	)	PUNCT
ejpam-5367	200	48	,	,	PUNCT
ejpam-5367	200	49	αf	αf	ADP
ejpam-5367	200	50	(	(	PUNCT
ejpam-5367	200	51	0	0	NUM
ejpam-5367	200	52	)	)	PUNCT
ejpam-5367	200	53	}	}	PUNCT
ejpam-5367	200	54	,	,	PUNCT
ejpam-5367	200	55	αf	αf	ADP
ejpam-5367	200	56	(	(	PUNCT
ejpam-5367	200	57	x	x	X
ejpam-5367	200	58	·	·	PUNCT
ejpam-5367	200	59	y	y	X
ejpam-5367	200	60	)	)	PUNCT
ejpam-5367	200	61	}	}	PUNCT
ejpam-5367	200	62	(	(	PUNCT
ejpam-5367	200	63	by	by	ADP
ejpam-5367	200	64	(	(	PUNCT
ejpam-5367	200	65	3.2	3.2	NUM
ejpam-5367	200	66	)	)	PUNCT
ejpam-5367	200	67	)	)	PUNCT
ejpam-5367	200	68	a.	a.	NOUN
ejpam-5367	200	69	iampan	iampan	NOUN
ejpam-5367	200	70	et	et	PROPN
ejpam-5367	200	71	al	al	PROPN
ejpam-5367	200	72	.	.	PUNCT
ejpam-5367	200	73	/	/	SYM
ejpam-5367	200	74	eur	eur	PROPN
ejpam-5367	200	75	.	.	PUNCT
ejpam-5367	201	1	j.	j.	PROPN
ejpam-5367	201	2	pure	pure	PROPN
ejpam-5367	201	3	appl	appl	PROPN
ejpam-5367	201	4	.	.	PROPN
ejpam-5367	201	5	math	math	PROPN
ejpam-5367	201	6	,	,	PUNCT
ejpam-5367	201	7	17	17	NUM
ejpam-5367	201	8	(	(	PUNCT
ejpam-5367	201	9	4	4	NUM
ejpam-5367	201	10	)	)	PUNCT
ejpam-5367	201	11	(	(	PUNCT
ejpam-5367	201	12	2024	2024	NUM
ejpam-5367	201	13	)	)	PUNCT
ejpam-5367	201	14	,	,	PUNCT
ejpam-5367	201	15	3022	3022	NUM
ejpam-5367	201	16	-	-	SYM
ejpam-5367	201	17	3042	3042	NUM
ejpam-5367	201	18	3031	3031	NUM
ejpam-5367	201	19	=	=	SYM
ejpam-5367	201	20	min{αf	min{αf	PUNCT
ejpam-5367	201	21	(	(	PUNCT
ejpam-5367	201	22	x	x	X
ejpam-5367	201	23	)	)	PUNCT
ejpam-5367	201	24	,	,	PUNCT
ejpam-5367	201	25	αf	αf	X
ejpam-5367	201	26	(	(	PUNCT
ejpam-5367	201	27	x	x	X
ejpam-5367	201	28	·	·	PUNCT
ejpam-5367	201	29	y	y	NOUN
ejpam-5367	201	30	)	)	PUNCT
ejpam-5367	201	31	}	}	PUNCT
ejpam-5367	201	32	,	,	PUNCT
ejpam-5367	201	33	(	(	PUNCT
ejpam-5367	201	34	by	by	ADP
ejpam-5367	201	35	(	(	PUNCT
ejpam-5367	201	36	3.4	3.4	NUM
ejpam-5367	201	37	)	)	PUNCT
ejpam-5367	201	38	)	)	PUNCT
ejpam-5367	202	1	βf	βf	X
ejpam-5367	202	2	(	(	PUNCT
ejpam-5367	202	3	y	y	NOUN
ejpam-5367	202	4	)	)	PUNCT
ejpam-5367	202	5	=	=	VERB
ejpam-5367	203	1	βf	βf	PUNCT
ejpam-5367	203	2	(	(	PUNCT
ejpam-5367	203	3	0	0	NUM
ejpam-5367	203	4	·	·	PUNCT
ejpam-5367	203	5	y	y	X
ejpam-5367	203	6	)	)	PUNCT
ejpam-5367	203	7	(	(	PUNCT
ejpam-5367	203	8	by	by	ADP
ejpam-5367	203	9	(	(	PUNCT
ejpam-5367	203	10	iup-1	iup-1	NOUN
ejpam-5367	203	11	)	)	PUNCT
ejpam-5367	203	12	)	)	PUNCT
ejpam-5367	204	1	=	=	SYM
ejpam-5367	204	2	βf	βf	PUNCT
ejpam-5367	204	3	(	(	PUNCT
ejpam-5367	204	4	(	(	PUNCT
ejpam-5367	204	5	x	x	X
ejpam-5367	204	6	·	·	PUNCT
ejpam-5367	204	7	0	0	NUM
ejpam-5367	204	8	)	)	PUNCT
ejpam-5367	204	9	·	·	PUNCT
ejpam-5367	205	1	(	(	PUNCT
ejpam-5367	205	2	x	x	X
ejpam-5367	205	3	·	·	PUNCT
ejpam-5367	205	4	y	y	NOUN
ejpam-5367	205	5	)	)	PUNCT
ejpam-5367	205	6	)	)	PUNCT
ejpam-5367	205	7	(	(	PUNCT
ejpam-5367	205	8	by	by	ADP
ejpam-5367	205	9	(	(	PUNCT
ejpam-5367	205	10	iup-3	iup-3	NUM
ejpam-5367	205	11	)	)	PUNCT
ejpam-5367	205	12	)	)	PUNCT
ejpam-5367	205	13	≤	≤	NUM
ejpam-5367	206	1	max{βf	max{βf	INTJ
ejpam-5367	206	2	(	(	PUNCT
ejpam-5367	206	3	x	x	SYM
ejpam-5367	206	4	·	·	PUNCT
ejpam-5367	206	5	0	0	NUM
ejpam-5367	206	6	)	)	PUNCT
ejpam-5367	206	7	,	,	PUNCT
ejpam-5367	206	8	βf	βf	CCONJ
ejpam-5367	206	9	(	(	PUNCT
ejpam-5367	206	10	x	x	X
ejpam-5367	206	11	·	·	PUNCT
ejpam-5367	206	12	y	y	X
ejpam-5367	206	13	)	)	PUNCT
ejpam-5367	206	14	}	}	PUNCT
ejpam-5367	206	15	(	(	PUNCT
ejpam-5367	206	16	by	by	ADP
ejpam-5367	206	17	(	(	PUNCT
ejpam-5367	206	18	3.3	3.3	NUM
ejpam-5367	206	19	)	)	PUNCT
ejpam-5367	206	20	)	)	PUNCT
ejpam-5367	206	21	≤	≤	NUM
ejpam-5367	206	22	max{max{βf	max{max{βf	NOUN
ejpam-5367	206	23	(	(	PUNCT
ejpam-5367	206	24	x	x	NOUN
ejpam-5367	206	25	)	)	PUNCT
ejpam-5367	206	26	,	,	PUNCT
ejpam-5367	206	27	βf	βf	CCONJ
ejpam-5367	206	28	(	(	PUNCT
ejpam-5367	206	29	0	0	NUM
ejpam-5367	206	30	)	)	PUNCT
ejpam-5367	206	31	}	}	PUNCT
ejpam-5367	206	32	,	,	PUNCT
ejpam-5367	206	33	βf	βf	CCONJ
ejpam-5367	206	34	(	(	PUNCT
ejpam-5367	206	35	x	x	X
ejpam-5367	206	36	·	·	PUNCT
ejpam-5367	206	37	y	y	X
ejpam-5367	206	38	)	)	PUNCT
ejpam-5367	206	39	}	}	PUNCT
ejpam-5367	206	40	(	(	PUNCT
ejpam-5367	206	41	by	by	ADP
ejpam-5367	206	42	(	(	PUNCT
ejpam-5367	206	43	3.3	3.3	NUM
ejpam-5367	206	44	)	)	PUNCT
ejpam-5367	206	45	)	)	PUNCT
ejpam-5367	207	1	=	=	PUNCT
ejpam-5367	207	2	max{βf	max{βf	INTJ
ejpam-5367	208	1	(	(	PUNCT
ejpam-5367	208	2	x	x	NOUN
ejpam-5367	208	3	)	)	PUNCT
ejpam-5367	208	4	,	,	PUNCT
ejpam-5367	208	5	βf	βf	CCONJ
ejpam-5367	208	6	(	(	PUNCT
ejpam-5367	208	7	x	x	X
ejpam-5367	208	8	·	·	PUNCT
ejpam-5367	208	9	y	y	X
ejpam-5367	208	10	)	)	PUNCT
ejpam-5367	208	11	}	}	PUNCT
ejpam-5367	208	12	.	.	PUNCT
ejpam-5367	209	1	(	(	PUNCT
ejpam-5367	209	2	by	by	ADP
ejpam-5367	209	3	(	(	PUNCT
ejpam-5367	209	4	3.5	3.5	NUM
ejpam-5367	209	5	)	)	PUNCT
ejpam-5367	209	6	)	)	PUNCT
ejpam-5367	209	7	hence	hence	ADV
ejpam-5367	209	8	,	,	PUNCT
ejpam-5367	209	9	f	f	PROPN
ejpam-5367	209	10	is	be	AUX
ejpam-5367	209	11	an	an	DET
ejpam-5367	209	12	ffiup	ffiup	ADJ
ejpam-5367	209	13	-	-	PUNCT
ejpam-5367	209	14	filter	filter	NOUN
ejpam-5367	209	15	of	of	ADP
ejpam-5367	209	16	x.	x.	NOUN
ejpam-5367	209	17	example	example	NOUN
ejpam-5367	209	18	7	7	NUM
ejpam-5367	209	19	.	.	PUNCT
ejpam-5367	210	1	[	[	X
ejpam-5367	210	2	7	7	X
ejpam-5367	210	3	]	]	PUNCT
ejpam-5367	210	4	let	let	VERB
ejpam-5367	210	5	r∗	r∗	PROPN
ejpam-5367	210	6	be	be	AUX
ejpam-5367	210	7	the	the	DET
ejpam-5367	210	8	set	set	NOUN
ejpam-5367	210	9	of	of	ADP
ejpam-5367	210	10	all	all	DET
ejpam-5367	210	11	nonzero	nonzero	ADJ
ejpam-5367	210	12	real	real	ADJ
ejpam-5367	210	13	numbers	number	NOUN
ejpam-5367	210	14	.	.	PUNCT
ejpam-5367	211	1	define	define	VERB
ejpam-5367	211	2	a	a	DET
ejpam-5367	211	3	binary	binary	ADJ
ejpam-5367	211	4	operation	operation	NOUN
ejpam-5367	211	5	·	·	PUNCT
ejpam-5367	211	6	on	on	ADP
ejpam-5367	211	7	r∗	r∗	VERB
ejpam-5367	211	8	by	by	ADP
ejpam-5367	211	9	:	:	PUNCT
ejpam-5367	211	10	(	(	PUNCT
ejpam-5367	211	11	∀x	∀x	X
ejpam-5367	211	12	,	,	PUNCT
ejpam-5367	211	13	y	y	PROPN
ejpam-5367	211	14	∈	∈	PROPN
ejpam-5367	211	15	r∗)(x	r∗)(x	NOUN
ejpam-5367	211	16	·	·	PUNCT
ejpam-5367	211	17	y	y	X
ejpam-5367	211	18	=	=	SYM
ejpam-5367	211	19	y	y	PROPN
ejpam-5367	211	20	x	x	PROPN
ejpam-5367	211	21	)	)	PUNCT
ejpam-5367	211	22	.	.	PUNCT
ejpam-5367	212	1	thus	thus	ADV
ejpam-5367	212	2	,	,	PUNCT
ejpam-5367	212	3	(	(	PUNCT
ejpam-5367	212	4	r∗	r∗	PROPN
ejpam-5367	212	5	,	,	PUNCT
ejpam-5367	212	6	·	·	PUNCT
ejpam-5367	212	7	,	,	PUNCT
ejpam-5367	212	8	1	1	NUM
ejpam-5367	212	9	)	)	PUNCT
ejpam-5367	212	10	is	be	AUX
ejpam-5367	212	11	an	an	DET
ejpam-5367	212	12	iup	iup	NOUN
ejpam-5367	212	13	-	-	PUNCT
ejpam-5367	212	14	algebra	algebra	NOUN
ejpam-5367	212	15	.	.	PUNCT
ejpam-5367	212	16	example	example	NOUN
ejpam-5367	212	17	8	8	NUM
ejpam-5367	212	18	.	.	PUNCT
ejpam-5367	213	1	from	from	ADP
ejpam-5367	213	2	example	example	NOUN
ejpam-5367	213	3	7	7	NUM
ejpam-5367	213	4	,	,	PUNCT
ejpam-5367	213	5	let	let	VERB
ejpam-5367	213	6	s	s	AUX
ejpam-5367	213	7	=	=	PUNCT
ejpam-5367	213	8	{	{	PUNCT
ejpam-5367	213	9	x	x	PUNCT
ejpam-5367	213	10	∈	∈	PROPN
ejpam-5367	213	11	r∗	r∗	NOUN
ejpam-5367	213	12	|	|	ADV
ejpam-5367	213	13	x	x	NOUN
ejpam-5367	213	14	≥	≥	NOUN
ejpam-5367	213	15	1	1	NUM
ejpam-5367	213	16	}	}	PUNCT
ejpam-5367	213	17	.	.	PUNCT
ejpam-5367	214	1	then	then	ADV
ejpam-5367	214	2	1	1	NUM
ejpam-5367	214	3	∈	∈	PROPN
ejpam-5367	214	4	s.	s.	PROPN
ejpam-5367	214	5	next	next	ADV
ejpam-5367	214	6	,	,	PUNCT
ejpam-5367	214	7	let	let	VERB
ejpam-5367	214	8	x	x	PRON
ejpam-5367	214	9	,	,	PUNCT
ejpam-5367	214	10	y	y	PROPN
ejpam-5367	214	11	,	,	PUNCT
ejpam-5367	214	12	z	z	PROPN
ejpam-5367	214	13	∈	∈	PROPN
ejpam-5367	214	14	r∗	r∗	NOUN
ejpam-5367	214	15	be	be	VERB
ejpam-5367	214	16	such	such	ADJ
ejpam-5367	214	17	that	that	SCONJ
ejpam-5367	214	18	x	x	PART
ejpam-5367	214	19	·	·	PUNCT
ejpam-5367	214	20	(	(	PUNCT
ejpam-5367	214	21	y	y	PROPN
ejpam-5367	214	22	·	·	PUNCT
ejpam-5367	214	23	z	z	X
ejpam-5367	214	24	)	)	PUNCT
ejpam-5367	214	25	≥	≥	NOUN
ejpam-5367	214	26	1	1	NUM
ejpam-5367	214	27	and	and	CCONJ
ejpam-5367	214	28	y	y	PROPN
ejpam-5367	214	29	≥	≥	NUM
ejpam-5367	214	30	1	1	NUM
ejpam-5367	214	31	.	.	PUNCT
ejpam-5367	215	1	then	then	ADV
ejpam-5367	215	2	z	z	PROPN
ejpam-5367	215	3	yx	yx	PROPN
ejpam-5367	215	4	≥	≥	NUM
ejpam-5367	215	5	1	1	NUM
ejpam-5367	215	6	.	.	PUNCT
ejpam-5367	216	1	thus	thus	ADV
ejpam-5367	216	2	,	,	PUNCT
ejpam-5367	216	3	x	x	X
ejpam-5367	216	4	·	·	PUNCT
ejpam-5367	216	5	z	z	X
ejpam-5367	216	6	=	=	PUNCT
ejpam-5367	216	7	z	z	NOUN
ejpam-5367	216	8	x	x	SYM
ejpam-5367	216	9	=	=	PUNCT
ejpam-5367	216	10	(	(	PUNCT
ejpam-5367	216	11	z	z	NOUN
ejpam-5367	216	12	yx	yx	PROPN
ejpam-5367	216	13	)	)	PUNCT
ejpam-5367	216	14	y	y	PROPN
ejpam-5367	216	15	≥	≥	NUM
ejpam-5367	216	16	1	1	NUM
ejpam-5367	216	17	,	,	PUNCT
ejpam-5367	216	18	that	that	ADV
ejpam-5367	216	19	is	is	ADV
ejpam-5367	216	20	,	,	PUNCT
ejpam-5367	216	21	x	x	X
ejpam-5367	216	22	·	·	PUNCT
ejpam-5367	216	23	z	z	PROPN
ejpam-5367	216	24	∈	∈	PROPN
ejpam-5367	216	25	s.	s.	PROPN
ejpam-5367	216	26	hence	hence	ADV
ejpam-5367	216	27	,	,	PUNCT
ejpam-5367	216	28	s	s	VERB
ejpam-5367	216	29	is	be	AUX
ejpam-5367	216	30	an	an	DET
ejpam-5367	216	31	iup	iup	NOUN
ejpam-5367	216	32	-	-	PUNCT
ejpam-5367	216	33	ideal	ideal	NOUN
ejpam-5367	216	34	of	of	ADP
ejpam-5367	216	35	r∗.	r∗.	NOUN
ejpam-5367	216	36	then	then	ADV
ejpam-5367	216	37	s	s	AUX
ejpam-5367	216	38	is	be	AUX
ejpam-5367	216	39	an	an	DET
ejpam-5367	216	40	iup	iup	NOUN
ejpam-5367	216	41	-	-	PUNCT
ejpam-5367	216	42	filter	filter	NOUN
ejpam-5367	216	43	of	of	ADP
ejpam-5367	216	44	r∗.	r∗.	NOUN
ejpam-5367	216	45	by	by	ADP
ejpam-5367	216	46	theorems	theorem	NOUN
ejpam-5367	216	47	9	9	NUM
ejpam-5367	216	48	and	and	CCONJ
ejpam-5367	216	49	10	10	NUM
ejpam-5367	216	50	,	,	PUNCT
ejpam-5367	216	51	we	we	PRON
ejpam-5367	216	52	have	have	VERB
ejpam-5367	216	53	fs	fs	INTJ
ejpam-5367	216	54	is	be	AUX
ejpam-5367	216	55	an	an	DET
ejpam-5367	216	56	ffiup	ffiup	NOUN
ejpam-5367	216	57	-	-	PUNCT
ejpam-5367	216	58	ideal	ideal	NOUN
ejpam-5367	216	59	and	and	CCONJ
ejpam-5367	216	60	an	an	DET
ejpam-5367	216	61	ffiup	ffiup	NOUN
ejpam-5367	216	62	-	-	PUNCT
ejpam-5367	216	63	filter	filter	NOUN
ejpam-5367	216	64	of	of	ADP
ejpam-5367	216	65	r∗.	r∗.	NOUN
ejpam-5367	216	66	since	since	SCONJ
ejpam-5367	216	67	1	1	NUM
ejpam-5367	216	68	,	,	PUNCT
ejpam-5367	216	69	3	3	NUM
ejpam-5367	216	70	∈	∈	NOUN
ejpam-5367	216	71	s	s	X
ejpam-5367	216	72	but	but	CCONJ
ejpam-5367	216	73	3	3	NUM
ejpam-5367	216	74	·	·	SYM
ejpam-5367	216	75	1	1	NUM
ejpam-5367	216	76	=	=	SYM
ejpam-5367	216	77	1	1	NUM
ejpam-5367	216	78	3	3	NUM
ejpam-5367	216	79	∈	∈	NOUN
ejpam-5367	216	80	s	s	NOUN
ejpam-5367	216	81	,	,	PUNCT
ejpam-5367	216	82	we	we	PRON
ejpam-5367	216	83	have	have	AUX
ejpam-5367	216	84	s	s	PROPN
ejpam-5367	216	85	is	be	AUX
ejpam-5367	216	86	not	not	PART
ejpam-5367	216	87	an	an	DET
ejpam-5367	216	88	iup	iup	NOUN
ejpam-5367	216	89	-	-	PUNCT
ejpam-5367	216	90	subalgebra	subalgebra	NOUN
ejpam-5367	216	91	of	of	ADP
ejpam-5367	216	92	r∗.	r∗.	NOUN
ejpam-5367	216	93	by	by	ADP
ejpam-5367	216	94	theorem	theorem	NOUN
ejpam-5367	216	95	8	8	NUM
ejpam-5367	216	96	,	,	PUNCT
ejpam-5367	216	97	we	we	PRON
ejpam-5367	216	98	have	have	VERB
ejpam-5367	216	99	fs	fs	INTJ
ejpam-5367	216	100	is	be	AUX
ejpam-5367	216	101	not	not	PART
ejpam-5367	216	102	an	an	DET
ejpam-5367	216	103	ffiup	ffiup	NOUN
ejpam-5367	216	104	-	-	PUNCT
ejpam-5367	216	105	subalgebra	subalgebra	NOUN
ejpam-5367	216	106	of	of	ADP
ejpam-5367	216	107	r∗.	r∗.	NOUN
ejpam-5367	216	108	example	example	NOUN
ejpam-5367	216	109	9	9	NUM
ejpam-5367	216	110	.	.	PUNCT
ejpam-5367	217	1	let	let	VERB
ejpam-5367	217	2	x	x	PUNCT
ejpam-5367	217	3	=	=	PUNCT
ejpam-5367	217	4	{	{	PUNCT
ejpam-5367	217	5	0	0	NUM
ejpam-5367	217	6	,	,	PUNCT
ejpam-5367	217	7	1	1	NUM
ejpam-5367	217	8	,	,	PUNCT
ejpam-5367	217	9	2	2	NUM
ejpam-5367	217	10	,	,	PUNCT
ejpam-5367	217	11	3	3	NUM
ejpam-5367	217	12	,	,	PUNCT
ejpam-5367	217	13	4	4	NUM
ejpam-5367	217	14	,	,	PUNCT
ejpam-5367	217	15	5	5	NUM
ejpam-5367	217	16	}	}	PUNCT
ejpam-5367	217	17	with	with	ADP
ejpam-5367	217	18	the	the	DET
ejpam-5367	217	19	following	follow	VERB
ejpam-5367	217	20	cayley	cayley	ADJ
ejpam-5367	217	21	table	table	NOUN
ejpam-5367	217	22	:	:	PUNCT
ejpam-5367	217	23	·	·	PUNCT
ejpam-5367	217	24	0	0	NUM
ejpam-5367	217	25	1	1	NUM
ejpam-5367	217	26	2	2	NUM
ejpam-5367	217	27	3	3	NUM
ejpam-5367	217	28	4	4	NUM
ejpam-5367	217	29	5	5	NUM
ejpam-5367	217	30	0	0	NUM
ejpam-5367	217	31	0	0	NUM
ejpam-5367	217	32	1	1	NUM
ejpam-5367	217	33	2	2	NUM
ejpam-5367	217	34	3	3	NUM
ejpam-5367	217	35	4	4	NUM
ejpam-5367	217	36	5	5	NUM
ejpam-5367	217	37	1	1	NUM
ejpam-5367	217	38	2	2	NUM
ejpam-5367	217	39	0	0	NUM
ejpam-5367	217	40	1	1	NUM
ejpam-5367	217	41	4	4	NUM
ejpam-5367	217	42	5	5	NUM
ejpam-5367	217	43	3	3	NUM
ejpam-5367	217	44	2	2	NUM
ejpam-5367	217	45	1	1	NUM
ejpam-5367	217	46	2	2	NUM
ejpam-5367	217	47	0	0	NUM
ejpam-5367	217	48	5	5	NUM
ejpam-5367	217	49	3	3	NUM
ejpam-5367	217	50	4	4	NUM
ejpam-5367	217	51	3	3	NUM
ejpam-5367	217	52	3	3	NUM
ejpam-5367	217	53	4	4	NUM
ejpam-5367	217	54	5	5	NUM
ejpam-5367	217	55	0	0	NUM
ejpam-5367	217	56	1	1	NUM
ejpam-5367	217	57	2	2	NUM
ejpam-5367	217	58	4	4	NUM
ejpam-5367	217	59	4	4	NUM
ejpam-5367	217	60	5	5	NUM
ejpam-5367	217	61	3	3	NUM
ejpam-5367	217	62	2	2	NUM
ejpam-5367	217	63	0	0	NUM
ejpam-5367	217	64	1	1	NUM
ejpam-5367	217	65	5	5	NUM
ejpam-5367	217	66	5	5	NUM
ejpam-5367	217	67	3	3	NUM
ejpam-5367	217	68	4	4	NUM
ejpam-5367	217	69	1	1	NUM
ejpam-5367	217	70	2	2	NUM
ejpam-5367	217	71	0	0	NUM
ejpam-5367	217	72	then	then	ADV
ejpam-5367	217	73	x	x	PUNCT
ejpam-5367	217	74	is	be	AUX
ejpam-5367	217	75	an	an	DET
ejpam-5367	217	76	iup	iup	NOUN
ejpam-5367	217	77	-	-	PUNCT
ejpam-5367	217	78	algebra	algebra	NOUN
ejpam-5367	217	79	.	.	PUNCT
ejpam-5367	218	1	we	we	PRON
ejpam-5367	218	2	define	define	VERB
ejpam-5367	218	3	an	an	DET
ejpam-5367	218	4	ffs	ffs	NOUN
ejpam-5367	218	5	f	f	PROPN
ejpam-5367	218	6	in	in	ADP
ejpam-5367	218	7	x	x	PROPN
ejpam-5367	218	8	as	as	SCONJ
ejpam-5367	218	9	follows	follow	VERB
ejpam-5367	218	10	:	:	PUNCT
ejpam-5367	218	11	αf	αf	NOUN
ejpam-5367	218	12	=	=	PUNCT
ejpam-5367	218	13	(	(	PUNCT
ejpam-5367	218	14	0	0	NUM
ejpam-5367	218	15	0.8	0.8	NUM
ejpam-5367	218	16	1	1	NUM
ejpam-5367	218	17	0.2	0.2	NUM
ejpam-5367	218	18	2	2	NUM
ejpam-5367	218	19	0.2	0.2	NUM
ejpam-5367	218	20	3	3	NUM
ejpam-5367	218	21	0.7	0.7	NUM
ejpam-5367	218	22	4	4	NUM
ejpam-5367	218	23	0.2	0.2	NUM
ejpam-5367	218	24	5	5	NUM
ejpam-5367	218	25	0.2	0.2	NUM
ejpam-5367	218	26	)	)	PUNCT
ejpam-5367	218	27	βf	βf	PUNCT
ejpam-5367	219	1	=	=	PUNCT
ejpam-5367	219	2	(	(	PUNCT
ejpam-5367	219	3	0	0	NUM
ejpam-5367	219	4	0.1	0.1	NUM
ejpam-5367	219	5	1	1	NUM
ejpam-5367	219	6	0.9	0.9	NUM
ejpam-5367	219	7	2	2	NUM
ejpam-5367	219	8	0.9	0.9	NUM
ejpam-5367	219	9	3	3	NUM
ejpam-5367	219	10	0.5	0.5	NUM
ejpam-5367	219	11	4	4	NUM
ejpam-5367	219	12	0.9	0.9	NUM
ejpam-5367	219	13	5	5	NUM
ejpam-5367	219	14	0.9	0.9	NUM
ejpam-5367	219	15	)	)	PUNCT
ejpam-5367	220	1	then	then	ADV
ejpam-5367	220	2	f	f	PROPN
ejpam-5367	220	3	is	be	AUX
ejpam-5367	220	4	an	an	DET
ejpam-5367	220	5	ffiup	ffiup	NOUN
ejpam-5367	220	6	-	-	PUNCT
ejpam-5367	220	7	subalgebra	subalgebra	NOUN
ejpam-5367	220	8	of	of	ADP
ejpam-5367	220	9	x.	x.	NOUN
ejpam-5367	220	10	since	since	SCONJ
ejpam-5367	220	11	αf	αf	PROPN
ejpam-5367	220	12	(	(	PUNCT
ejpam-5367	220	13	1·4	1·4	NUM
ejpam-5367	220	14	)	)	PUNCT
ejpam-5367	221	1	=	=	SYM
ejpam-5367	221	2	αf	αf	X
ejpam-5367	221	3	(	(	PUNCT
ejpam-5367	221	4	5	5	NUM
ejpam-5367	221	5	)	)	PUNCT
ejpam-5367	221	6	=	=	SYM
ejpam-5367	221	7	0.2	0.2	NUM
ejpam-5367	221	8	≱	≱	PROPN
ejpam-5367	221	9	0.7	0.7	NUM
ejpam-5367	221	10	=	=	SYM
ejpam-5367	221	11	min{0.8	min{0.8	PROPN
ejpam-5367	221	12	,	,	PUNCT
ejpam-5367	221	13	0.7	0.7	NUM
ejpam-5367	221	14	}	}	PUNCT
ejpam-5367	221	15	=	=	PUNCT
ejpam-5367	221	16	min{αf	min{αf	X
ejpam-5367	221	17	(	(	PUNCT
ejpam-5367	221	18	0	0	NUM
ejpam-5367	221	19	)	)	PUNCT
ejpam-5367	221	20	,	,	PUNCT
ejpam-5367	221	21	αf	αf	X
ejpam-5367	221	22	(	(	PUNCT
ejpam-5367	221	23	3	3	NUM
ejpam-5367	221	24	)	)	PUNCT
ejpam-5367	221	25	}	}	PUNCT
ejpam-5367	221	26	=	=	SYM
ejpam-5367	221	27	min{αf	min{αf	PUNCT
ejpam-5367	221	28	(	(	PUNCT
ejpam-5367	221	29	1	1	NUM
ejpam-5367	221	30	·	·	PUNCT
ejpam-5367	221	31	(	(	PUNCT
ejpam-5367	221	32	3	3	NUM
ejpam-5367	221	33	·	·	SYM
ejpam-5367	221	34	4	4	NUM
ejpam-5367	221	35	)	)	PUNCT
ejpam-5367	221	36	)	)	PUNCT
ejpam-5367	221	37	,	,	PUNCT
ejpam-5367	221	38	αf	αf	X
ejpam-5367	221	39	(	(	PUNCT
ejpam-5367	221	40	3	3	NUM
ejpam-5367	221	41	)	)	PUNCT
ejpam-5367	221	42	}	}	PUNCT
ejpam-5367	221	43	and	and	CCONJ
ejpam-5367	221	44	βf	βf	X
ejpam-5367	221	45	(	(	PUNCT
ejpam-5367	221	46	1	1	NUM
ejpam-5367	221	47	·	·	SYM
ejpam-5367	221	48	2	2	X
ejpam-5367	221	49	)	)	PUNCT
ejpam-5367	221	50	=	=	SYM
ejpam-5367	221	51	βf	βf	PUNCT
ejpam-5367	221	52	(	(	PUNCT
ejpam-5367	221	53	1	1	X
ejpam-5367	221	54	)	)	PUNCT
ejpam-5367	221	55	=	=	SYM
ejpam-5367	221	56	0.9	0.9	NUM
ejpam-5367	221	57	≰	≰	PROPN
ejpam-5367	221	58	0.5	0.5	NUM
ejpam-5367	221	59	=	=	SYM
ejpam-5367	221	60	max{0.5	max{0.5	PROPN
ejpam-5367	221	61	,	,	PUNCT
ejpam-5367	221	62	0.5	0.5	NUM
ejpam-5367	221	63	}	}	PUNCT
ejpam-5367	221	64	=	=	SYM
ejpam-5367	221	65	max{βf	max{βf	PROPN
ejpam-5367	221	66	(	(	PUNCT
ejpam-5367	221	67	3	3	NUM
ejpam-5367	221	68	)	)	PUNCT
ejpam-5367	221	69	,	,	PUNCT
ejpam-5367	221	70	βf	βf	CCONJ
ejpam-5367	221	71	(	(	PUNCT
ejpam-5367	221	72	3	3	NUM
ejpam-5367	221	73	)	)	PUNCT
ejpam-5367	221	74	}	}	PUNCT
ejpam-5367	221	75	=	=	SYM
ejpam-5367	221	76	max{βf	max{βf	PROPN
ejpam-5367	221	77	(	(	PUNCT
ejpam-5367	221	78	1	1	NUM
ejpam-5367	221	79	·	·	PUNCT
ejpam-5367	221	80	(	(	PUNCT
ejpam-5367	221	81	3	3	NUM
ejpam-5367	221	82	·	·	SYM
ejpam-5367	221	83	2	2	NUM
ejpam-5367	221	84	)	)	PUNCT
ejpam-5367	221	85	)	)	PUNCT
ejpam-5367	221	86	,	,	PUNCT
ejpam-5367	221	87	βf	βf	CCONJ
ejpam-5367	221	88	(	(	PUNCT
ejpam-5367	221	89	3	3	NUM
ejpam-5367	221	90	)	)	PUNCT
ejpam-5367	221	91	}	}	PUNCT
ejpam-5367	221	92	.	.	PUNCT
ejpam-5367	222	1	hence	hence	ADV
ejpam-5367	222	2	,	,	PUNCT
ejpam-5367	222	3	f	f	PROPN
ejpam-5367	222	4	is	be	AUX
ejpam-5367	222	5	not	not	PART
ejpam-5367	222	6	an	an	DET
ejpam-5367	222	7	ffiup	ffiup	NOUN
ejpam-5367	222	8	-	-	PUNCT
ejpam-5367	222	9	ideal	ideal	NOUN
ejpam-5367	222	10	of	of	ADP
ejpam-5367	222	11	x.	x.	PROPN
ejpam-5367	222	12	a.	a.	PROPN
ejpam-5367	222	13	iampan	iampan	PROPN
ejpam-5367	222	14	et	et	PROPN
ejpam-5367	222	15	al	al	PROPN
ejpam-5367	222	16	.	.	PUNCT
ejpam-5367	222	17	/	/	SYM
ejpam-5367	222	18	eur	eur	PROPN
ejpam-5367	222	19	.	.	PUNCT
ejpam-5367	223	1	j.	j.	PROPN
ejpam-5367	223	2	pure	pure	PROPN
ejpam-5367	223	3	appl	appl	PROPN
ejpam-5367	223	4	.	.	PROPN
ejpam-5367	223	5	math	math	PROPN
ejpam-5367	223	6	,	,	PUNCT
ejpam-5367	223	7	17	17	NUM
ejpam-5367	223	8	(	(	PUNCT
ejpam-5367	223	9	4	4	NUM
ejpam-5367	223	10	)	)	PUNCT
ejpam-5367	223	11	(	(	PUNCT
ejpam-5367	223	12	2024	2024	NUM
ejpam-5367	223	13	)	)	PUNCT
ejpam-5367	223	14	,	,	PUNCT
ejpam-5367	223	15	3022	3022	NUM
ejpam-5367	223	16	-	-	SYM
ejpam-5367	223	17	3042	3042	NUM
ejpam-5367	223	18	3032	3032	NUM
ejpam-5367	223	19	the	the	DET
ejpam-5367	223	20	study	study	NOUN
ejpam-5367	223	21	revealed	reveal	VERB
ejpam-5367	223	22	a	a	DET
ejpam-5367	223	23	relationship	relationship	NOUN
ejpam-5367	223	24	between	between	ADP
ejpam-5367	223	25	the	the	DET
ejpam-5367	223	26	four	four	NUM
ejpam-5367	223	27	concepts	concept	NOUN
ejpam-5367	223	28	:	:	PUNCT
ejpam-5367	223	29	ffiup	ffiup	ADJ
ejpam-5367	223	30	-	-	PUNCT
ejpam-5367	223	31	ideals	ideal	NOUN
ejpam-5367	223	32	and	and	CCONJ
ejpam-5367	223	33	ffiupsubalgebras	ffiupsubalgebra	NOUN
ejpam-5367	223	34	are	be	AUX
ejpam-5367	223	35	generalizations	generalization	NOUN
ejpam-5367	223	36	of	of	ADP
ejpam-5367	223	37	ffsiup	ffsiup	NOUN
ejpam-5367	223	38	-	-	PUNCT
ejpam-5367	223	39	ideals	ideal	NOUN
ejpam-5367	223	40	of	of	ADP
ejpam-5367	223	41	iup	iup	NOUN
ejpam-5367	223	42	-	-	PUNCT
ejpam-5367	223	43	algebras	algebra	NOUN
ejpam-5367	223	44	,	,	PUNCT
ejpam-5367	223	45	where	where	SCONJ
ejpam-5367	223	46	ffsiup	ffsiup	NOUN
ejpam-5367	223	47	-	-	PUNCT
ejpam-5367	223	48	ideals	ideal	NOUN
ejpam-5367	223	49	of	of	ADP
ejpam-5367	223	50	iup	iup	NOUN
ejpam-5367	223	51	-	-	PUNCT
ejpam-5367	223	52	algebras	algebras	PROPN
ejpam-5367	223	53	can	can	AUX
ejpam-5367	223	54	only	only	ADV
ejpam-5367	223	55	be	be	AUX
ejpam-5367	223	56	a	a	DET
ejpam-5367	223	57	constant	constant	ADJ
ejpam-5367	223	58	ffs	ff	NOUN
ejpam-5367	223	59	.	.	PUNCT
ejpam-5367	223	60	ffiup	ffiup	ADJ
ejpam-5367	223	61	-	-	PUNCT
ejpam-5367	223	62	filters	filter	NOUN
ejpam-5367	223	63	are	be	AUX
ejpam-5367	223	64	a	a	DET
ejpam-5367	223	65	generalization	generalization	NOUN
ejpam-5367	223	66	of	of	ADP
ejpam-5367	223	67	ffiupideals	ffiupideal	NOUN
ejpam-5367	223	68	and	and	CCONJ
ejpam-5367	223	69	ffiup	ffiup	NOUN
ejpam-5367	223	70	-	-	PUNCT
ejpam-5367	223	71	subalgebras	subalgebras	PROPN
ejpam-5367	223	72	.	.	PUNCT
ejpam-5367	224	1	we	we	PRON
ejpam-5367	224	2	summarize	summarize	VERB
ejpam-5367	224	3	the	the	DET
ejpam-5367	224	4	relationship	relationship	NOUN
ejpam-5367	224	5	between	between	ADP
ejpam-5367	224	6	these	these	DET
ejpam-5367	224	7	four	four	NUM
ejpam-5367	224	8	concepts	concept	NOUN
ejpam-5367	224	9	,	,	PUNCT
ejpam-5367	224	10	shown	show	VERB
ejpam-5367	224	11	in	in	ADP
ejpam-5367	224	12	figure	figure	NOUN
ejpam-5367	224	13	2	2	NUM
ejpam-5367	224	14	.	.	PUNCT
ejpam-5367	224	15	figure	figure	NOUN
ejpam-5367	224	16	2	2	NUM
ejpam-5367	224	17	:	:	PUNCT
ejpam-5367	224	18	ffss	ffss	NOUN
ejpam-5367	224	19	in	in	ADP
ejpam-5367	224	20	iup	iup	NOUN
ejpam-5367	224	21	-	-	PUNCT
ejpam-5367	224	22	algebras	algebras	PROPN
ejpam-5367	224	23	theorem	theorem	VERB
ejpam-5367	224	24	7	7	NUM
ejpam-5367	224	25	.	.	PUNCT
ejpam-5367	225	1	if	if	SCONJ
ejpam-5367	225	2	f	f	PROPN
ejpam-5367	225	3	is	be	AUX
ejpam-5367	225	4	an	an	DET
ejpam-5367	225	5	ffiup	ffiup	ADJ
ejpam-5367	225	6	-	-	PUNCT
ejpam-5367	225	7	filter	filter	NOUN
ejpam-5367	225	8	of	of	ADP
ejpam-5367	225	9	x	x	PUNCT
ejpam-5367	225	10	satisfying	satisfy	VERB
ejpam-5367	225	11	the	the	DET
ejpam-5367	225	12	following	follow	VERB
ejpam-5367	225	13	condition	condition	NOUN
ejpam-5367	225	14	:	:	PUNCT
ejpam-5367	225	15	(	(	PUNCT
ejpam-5367	225	16	∀x	∀x	X
ejpam-5367	225	17	,	,	PUNCT
ejpam-5367	225	18	y	y	PROPN
ejpam-5367	225	19	,	,	PUNCT
ejpam-5367	225	20	z	z	NOUN
ejpam-5367	225	21	∈	∈	PROPN
ejpam-5367	225	22	x	x	X
ejpam-5367	225	23	)	)	PUNCT
ejpam-5367	225	24	(	(	PUNCT
ejpam-5367	225	25	αf	αf	X
ejpam-5367	225	26	(	(	PUNCT
ejpam-5367	225	27	y	y	NOUN
ejpam-5367	225	28	·	·	PUNCT
ejpam-5367	225	29	(	(	PUNCT
ejpam-5367	225	30	x	x	X
ejpam-5367	225	31	·	·	PUNCT
ejpam-5367	225	32	z	z	X
ejpam-5367	225	33	)	)	PUNCT
ejpam-5367	225	34	)	)	PUNCT
ejpam-5367	226	1	=	=	SYM
ejpam-5367	226	2	αf	αf	X
ejpam-5367	226	3	(	(	PUNCT
ejpam-5367	226	4	x	x	X
ejpam-5367	226	5	·	·	PUNCT
ejpam-5367	226	6	(	(	PUNCT
ejpam-5367	226	7	y	y	PROPN
ejpam-5367	226	8	·	·	PUNCT
ejpam-5367	226	9	z	z	X
ejpam-5367	226	10	)	)	PUNCT
ejpam-5367	226	11	)	)	PUNCT
ejpam-5367	227	1	βf	βf	X
ejpam-5367	227	2	(	(	PUNCT
ejpam-5367	227	3	y	y	PROPN
ejpam-5367	227	4	·	·	PUNCT
ejpam-5367	227	5	(	(	PUNCT
ejpam-5367	227	6	x	x	X
ejpam-5367	227	7	·	·	PUNCT
ejpam-5367	227	8	z	z	NOUN
ejpam-5367	227	9	)	)	PUNCT
ejpam-5367	227	10	)	)	PUNCT
ejpam-5367	228	1	=	=	PUNCT
ejpam-5367	229	1	βf	βf	INTJ
ejpam-5367	229	2	(	(	PUNCT
ejpam-5367	229	3	x	x	PART
ejpam-5367	229	4	·	·	PUNCT
ejpam-5367	229	5	(	(	PUNCT
ejpam-5367	229	6	y	y	PROPN
ejpam-5367	229	7	·	·	PUNCT
ejpam-5367	229	8	z	z	NOUN
ejpam-5367	229	9	)	)	PUNCT
ejpam-5367	229	10	)	)	PUNCT
ejpam-5367	229	11	)	)	PUNCT
ejpam-5367	230	1	(	(	PUNCT
ejpam-5367	230	2	3.12	3.12	NUM
ejpam-5367	230	3	)	)	PUNCT
ejpam-5367	230	4	then	then	ADV
ejpam-5367	230	5	f	f	PROPN
ejpam-5367	230	6	is	be	AUX
ejpam-5367	230	7	an	an	DET
ejpam-5367	230	8	ffiup	ffiup	NOUN
ejpam-5367	230	9	-	-	PUNCT
ejpam-5367	230	10	ideal	ideal	NOUN
ejpam-5367	230	11	of	of	ADP
ejpam-5367	230	12	x.	x.	NOUN
ejpam-5367	230	13	proof	proof	PROPN
ejpam-5367	230	14	.	.	PUNCT
ejpam-5367	231	1	assume	assume	VERB
ejpam-5367	231	2	that	that	SCONJ
ejpam-5367	231	3	f	f	PROPN
ejpam-5367	231	4	is	be	AUX
ejpam-5367	231	5	an	an	DET
ejpam-5367	231	6	ffiup	ffiup	ADJ
ejpam-5367	231	7	-	-	PUNCT
ejpam-5367	231	8	filter	filter	NOUN
ejpam-5367	231	9	of	of	ADP
ejpam-5367	231	10	x	x	PUNCT
ejpam-5367	231	11	satisfying	satisfy	VERB
ejpam-5367	231	12	the	the	DET
ejpam-5367	231	13	condition	condition	NOUN
ejpam-5367	231	14	(	(	PUNCT
ejpam-5367	231	15	3.12	3.12	NUM
ejpam-5367	231	16	)	)	PUNCT
ejpam-5367	231	17	.	.	PUNCT
ejpam-5367	232	1	by	by	ADP
ejpam-5367	232	2	the	the	DET
ejpam-5367	232	3	assumption	assumption	NOUN
ejpam-5367	232	4	,	,	PUNCT
ejpam-5367	232	5	it	it	PRON
ejpam-5367	232	6	satisfies	satisfy	VERB
ejpam-5367	232	7	(	(	PUNCT
ejpam-5367	232	8	3.4	3.4	NUM
ejpam-5367	232	9	)	)	PUNCT
ejpam-5367	232	10	and	and	CCONJ
ejpam-5367	232	11	(	(	PUNCT
ejpam-5367	232	12	3.5	3.5	NUM
ejpam-5367	232	13	)	)	PUNCT
ejpam-5367	232	14	.	.	PUNCT
ejpam-5367	233	1	let	let	VERB
ejpam-5367	233	2	x	x	PRON
ejpam-5367	233	3	,	,	PUNCT
ejpam-5367	233	4	y	y	PROPN
ejpam-5367	233	5	,	,	PUNCT
ejpam-5367	233	6	z	z	PROPN
ejpam-5367	233	7	∈	∈	PROPN
ejpam-5367	233	8	x.	x.	NOUN
ejpam-5367	234	1	then	then	ADV
ejpam-5367	234	2	αf	αf	VERB
ejpam-5367	234	3	(	(	PUNCT
ejpam-5367	234	4	x	x	X
ejpam-5367	234	5	·	·	PUNCT
ejpam-5367	234	6	z	z	X
ejpam-5367	234	7	)	)	PUNCT
ejpam-5367	234	8	≥	≥	NOUN
ejpam-5367	234	9	min{αf	min{αf	PUNCT
ejpam-5367	234	10	(	(	PUNCT
ejpam-5367	234	11	y	y	PROPN
ejpam-5367	234	12	·	·	PUNCT
ejpam-5367	234	13	(	(	PUNCT
ejpam-5367	234	14	x	x	X
ejpam-5367	234	15	·	·	PUNCT
ejpam-5367	234	16	z	z	NOUN
ejpam-5367	234	17	)	)	PUNCT
ejpam-5367	234	18	)	)	PUNCT
ejpam-5367	234	19	,	,	PUNCT
ejpam-5367	234	20	αf	αf	X
ejpam-5367	234	21	(	(	PUNCT
ejpam-5367	234	22	y	y	NOUN
ejpam-5367	234	23	)	)	PUNCT
ejpam-5367	234	24	}	}	PUNCT
ejpam-5367	234	25	(	(	PUNCT
ejpam-5367	234	26	by	by	ADP
ejpam-5367	234	27	(	(	PUNCT
ejpam-5367	234	28	3.8	3.8	NUM
ejpam-5367	234	29	)	)	PUNCT
ejpam-5367	234	30	)	)	PUNCT
ejpam-5367	235	1	=	=	PUNCT
ejpam-5367	235	2	min{αf	min{αf	PUNCT
ejpam-5367	235	3	(	(	PUNCT
ejpam-5367	235	4	x	x	X
ejpam-5367	235	5	·	·	PUNCT
ejpam-5367	235	6	(	(	PUNCT
ejpam-5367	235	7	y	y	PROPN
ejpam-5367	235	8	·	·	PUNCT
ejpam-5367	235	9	z	z	NOUN
ejpam-5367	235	10	)	)	PUNCT
ejpam-5367	235	11	)	)	PUNCT
ejpam-5367	235	12	,	,	PUNCT
ejpam-5367	235	13	αf	αf	X
ejpam-5367	235	14	(	(	PUNCT
ejpam-5367	235	15	y	y	NOUN
ejpam-5367	235	16	)	)	PUNCT
ejpam-5367	235	17	}	}	PUNCT
ejpam-5367	235	18	,	,	PUNCT
ejpam-5367	235	19	(	(	PUNCT
ejpam-5367	235	20	by	by	ADP
ejpam-5367	235	21	(	(	PUNCT
ejpam-5367	235	22	3.12	3.12	NUM
ejpam-5367	235	23	)	)	PUNCT
ejpam-5367	235	24	)	)	PUNCT
ejpam-5367	235	25	βf	βf	X
ejpam-5367	236	1	(	(	PUNCT
ejpam-5367	236	2	x	x	X
ejpam-5367	236	3	·	·	PUNCT
ejpam-5367	236	4	z	z	X
ejpam-5367	236	5	)	)	PUNCT
ejpam-5367	236	6	≤	≤	NUM
ejpam-5367	236	7	max{βf	max{βf	INTJ
ejpam-5367	237	1	(	(	PUNCT
ejpam-5367	237	2	y	y	PROPN
ejpam-5367	237	3	·	·	PUNCT
ejpam-5367	237	4	(	(	PUNCT
ejpam-5367	237	5	x	x	X
ejpam-5367	237	6	·	·	PUNCT
ejpam-5367	237	7	z	z	NOUN
ejpam-5367	237	8	)	)	PUNCT
ejpam-5367	237	9	)	)	PUNCT
ejpam-5367	237	10	,	,	PUNCT
ejpam-5367	237	11	βf	βf	CCONJ
ejpam-5367	237	12	(	(	PUNCT
ejpam-5367	237	13	y	y	NOUN
ejpam-5367	237	14	)	)	PUNCT
ejpam-5367	237	15	}	}	PUNCT
ejpam-5367	237	16	(	(	PUNCT
ejpam-5367	237	17	by	by	ADP
ejpam-5367	237	18	(	(	PUNCT
ejpam-5367	237	19	3.9	3.9	NUM
ejpam-5367	237	20	)	)	PUNCT
ejpam-5367	237	21	)	)	PUNCT
ejpam-5367	238	1	=	=	PUNCT
ejpam-5367	239	1	max{βf	max{βf	INTJ
ejpam-5367	240	1	(	(	PUNCT
ejpam-5367	240	2	x	x	X
ejpam-5367	240	3	·	·	PUNCT
ejpam-5367	240	4	(	(	PUNCT
ejpam-5367	240	5	y	y	PROPN
ejpam-5367	240	6	·	·	PUNCT
ejpam-5367	240	7	z	z	NOUN
ejpam-5367	240	8	)	)	PUNCT
ejpam-5367	240	9	)	)	PUNCT
ejpam-5367	240	10	,	,	PUNCT
ejpam-5367	240	11	βf	βf	CCONJ
ejpam-5367	240	12	(	(	PUNCT
ejpam-5367	240	13	y	y	NOUN
ejpam-5367	240	14	)	)	PUNCT
ejpam-5367	240	15	}	}	PUNCT
ejpam-5367	240	16	.	.	PUNCT
ejpam-5367	241	1	(	(	PUNCT
ejpam-5367	241	2	by	by	ADP
ejpam-5367	241	3	(	(	PUNCT
ejpam-5367	241	4	3.12	3.12	NUM
ejpam-5367	241	5	)	)	PUNCT
ejpam-5367	241	6	)	)	PUNCT
ejpam-5367	241	7	hence	hence	ADV
ejpam-5367	241	8	,	,	PUNCT
ejpam-5367	241	9	f	f	PROPN
ejpam-5367	241	10	is	be	AUX
ejpam-5367	241	11	an	an	DET
ejpam-5367	241	12	ffiup	ffiup	NOUN
ejpam-5367	241	13	-	-	PUNCT
ejpam-5367	241	14	ideal	ideal	NOUN
ejpam-5367	241	15	of	of	ADP
ejpam-5367	241	16	x.	x.	PROPN
ejpam-5367	241	17	lemma	lemma	PROPN
ejpam-5367	242	1	2	2	X
ejpam-5367	242	2	.	.	PUNCT
ejpam-5367	242	3	let	let	VERB
ejpam-5367	242	4	g	g	PRON
ejpam-5367	242	5	be	be	AUX
ejpam-5367	242	6	a	a	DET
ejpam-5367	242	7	non	non	ADJ
ejpam-5367	242	8	-	-	ADJ
ejpam-5367	242	9	empty	empty	ADJ
ejpam-5367	242	10	subset	subset	NOUN
ejpam-5367	242	11	of	of	ADP
ejpam-5367	242	12	x.	x.	NOUN
ejpam-5367	242	13	then	then	ADV
ejpam-5367	242	14	the	the	DET
ejpam-5367	242	15	constant	constant	ADJ
ejpam-5367	242	16	0	0	NUM
ejpam-5367	242	17	is	be	AUX
ejpam-5367	242	18	in	in	ADP
ejpam-5367	242	19	g	g	PROPN
ejpam-5367	242	20	if	if	SCONJ
ejpam-5367	243	1	and	and	CCONJ
ejpam-5367	243	2	only	only	ADV
ejpam-5367	243	3	if	if	SCONJ
ejpam-5367	243	4	the	the	DET
ejpam-5367	243	5	characteristic	characteristic	ADJ
ejpam-5367	243	6	ffs	ffs	NOUN
ejpam-5367	243	7	fg	fg	PROPN
ejpam-5367	243	8	satisfies	satisfie	NOUN
ejpam-5367	243	9	(	(	PUNCT
ejpam-5367	243	10	3.4	3.4	NUM
ejpam-5367	243	11	)	)	PUNCT
ejpam-5367	243	12	and	and	CCONJ
ejpam-5367	243	13	(	(	PUNCT
ejpam-5367	243	14	3.5	3.5	NUM
ejpam-5367	243	15	)	)	PUNCT
ejpam-5367	243	16	.	.	PUNCT
ejpam-5367	244	1	proof	proof	NOUN
ejpam-5367	244	2	.	.	PUNCT
ejpam-5367	245	1	assume	assume	VERB
ejpam-5367	245	2	that	that	SCONJ
ejpam-5367	245	3	the	the	DET
ejpam-5367	245	4	constant	constant	ADJ
ejpam-5367	245	5	0	0	NUM
ejpam-5367	245	6	is	be	AUX
ejpam-5367	245	7	in	in	ADP
ejpam-5367	245	8	g.	g.	PROPN
ejpam-5367	245	9	then	then	ADV
ejpam-5367	245	10	αfg	αfg	PROPN
ejpam-5367	245	11	(	(	PUNCT
ejpam-5367	245	12	0	0	NUM
ejpam-5367	245	13	)	)	PUNCT
ejpam-5367	246	1	=	=	SYM
ejpam-5367	246	2	1	1	NUM
ejpam-5367	246	3	and	and	CCONJ
ejpam-5367	246	4	βfg	βfg	PROPN
ejpam-5367	246	5	(	(	PUNCT
ejpam-5367	246	6	0	0	NUM
ejpam-5367	246	7	)	)	PUNCT
ejpam-5367	246	8	=	=	SYM
ejpam-5367	246	9	0	0	X
ejpam-5367	246	10	.	.	PUNCT
ejpam-5367	247	1	thus	thus	ADV
ejpam-5367	247	2	,	,	PUNCT
ejpam-5367	247	3	αfg	αfg	NOUN
ejpam-5367	247	4	(	(	PUNCT
ejpam-5367	247	5	0	0	NUM
ejpam-5367	247	6	)	)	PUNCT
ejpam-5367	247	7	=	=	SYM
ejpam-5367	247	8	1	1	NUM
ejpam-5367	247	9	≥	≥	NOUN
ejpam-5367	247	10	αfg	αfg	NOUN
ejpam-5367	247	11	(	(	PUNCT
ejpam-5367	247	12	x	x	NOUN
ejpam-5367	247	13	)	)	PUNCT
ejpam-5367	247	14	and	and	CCONJ
ejpam-5367	247	15	βfg	βfg	PROPN
ejpam-5367	247	16	(	(	PUNCT
ejpam-5367	247	17	0	0	NUM
ejpam-5367	247	18	)	)	PUNCT
ejpam-5367	247	19	=	=	SYM
ejpam-5367	247	20	0	0	X
ejpam-5367	247	21	≤	≤	NUM
ejpam-5367	247	22	βfg	βfg	NOUN
ejpam-5367	247	23	(	(	PUNCT
ejpam-5367	247	24	x	x	X
ejpam-5367	247	25	)	)	PUNCT
ejpam-5367	247	26	for	for	ADP
ejpam-5367	247	27	all	all	DET
ejpam-5367	247	28	x	x	SYM
ejpam-5367	247	29	∈	∈	PROPN
ejpam-5367	247	30	x	x	NOUN
ejpam-5367	247	31	,	,	PUNCT
ejpam-5367	247	32	that	that	ADV
ejpam-5367	247	33	is	is	ADV
ejpam-5367	247	34	,	,	PUNCT
ejpam-5367	247	35	fg	fg	PROPN
ejpam-5367	247	36	satisfies	satisfie	NOUN
ejpam-5367	247	37	(	(	PUNCT
ejpam-5367	247	38	3.4	3.4	NUM
ejpam-5367	247	39	)	)	PUNCT
ejpam-5367	247	40	and	and	CCONJ
ejpam-5367	247	41	(	(	PUNCT
ejpam-5367	247	42	3.5	3.5	NUM
ejpam-5367	247	43	)	)	PUNCT
ejpam-5367	247	44	.	.	PUNCT
ejpam-5367	248	1	conversely	conversely	ADV
ejpam-5367	248	2	,	,	PUNCT
ejpam-5367	248	3	assume	assume	VERB
ejpam-5367	248	4	that	that	SCONJ
ejpam-5367	248	5	the	the	DET
ejpam-5367	248	6	characteristic	characteristic	ADJ
ejpam-5367	248	7	ffs	ffs	PROPN
ejpam-5367	248	8	fg	fg	PROPN
ejpam-5367	248	9	satisfies	satisfie	NOUN
ejpam-5367	248	10	(	(	PUNCT
ejpam-5367	248	11	3.4	3.4	NUM
ejpam-5367	248	12	)	)	PUNCT
ejpam-5367	248	13	and	and	CCONJ
ejpam-5367	248	14	(	(	PUNCT
ejpam-5367	248	15	3.5	3.5	NUM
ejpam-5367	248	16	)	)	PUNCT
ejpam-5367	248	17	.	.	PUNCT
ejpam-5367	249	1	then	then	ADV
ejpam-5367	249	2	αfg	αfg	PROPN
ejpam-5367	249	3	(	(	PUNCT
ejpam-5367	249	4	0	0	NUM
ejpam-5367	249	5	)	)	PUNCT
ejpam-5367	249	6	≥	≥	NOUN
ejpam-5367	249	7	αfg	αfg	INTJ
ejpam-5367	249	8	(	(	PUNCT
ejpam-5367	249	9	x	x	X
ejpam-5367	249	10	)	)	PUNCT
ejpam-5367	249	11	for	for	ADP
ejpam-5367	249	12	all	all	PRON
ejpam-5367	249	13	x	x	SYM
ejpam-5367	249	14	∈	∈	PROPN
ejpam-5367	249	15	x.	x.	NOUN
ejpam-5367	249	16	since	since	SCONJ
ejpam-5367	249	17	g	g	PROPN
ejpam-5367	249	18	is	be	AUX
ejpam-5367	249	19	a	a	DET
ejpam-5367	249	20	non	non	ADJ
ejpam-5367	249	21	-	-	ADJ
ejpam-5367	249	22	empty	empty	ADJ
ejpam-5367	249	23	subset	subset	NOUN
ejpam-5367	249	24	of	of	ADP
ejpam-5367	249	25	x	x	PRON
ejpam-5367	249	26	,	,	PUNCT
ejpam-5367	249	27	we	we	PRON
ejpam-5367	249	28	let	let	VERB
ejpam-5367	249	29	a	a	DET
ejpam-5367	249	30	∈	∈	PROPN
ejpam-5367	249	31	g.	g.	NOUN
ejpam-5367	249	32	then	then	ADV
ejpam-5367	249	33	αfg	αfg	PROPN
ejpam-5367	249	34	(	(	PUNCT
ejpam-5367	249	35	0	0	NUM
ejpam-5367	249	36	)	)	PUNCT
ejpam-5367	249	37	≥	≥	NOUN
ejpam-5367	249	38	αfg	αfg	INTJ
ejpam-5367	249	39	(	(	PUNCT
ejpam-5367	249	40	a	a	NOUN
ejpam-5367	249	41	)	)	PUNCT
ejpam-5367	249	42	=	=	SYM
ejpam-5367	249	43	1	1	NUM
ejpam-5367	249	44	,	,	PUNCT
ejpam-5367	249	45	so	so	ADV
ejpam-5367	249	46	αfg	αfg	PROPN
ejpam-5367	249	47	(	(	PUNCT
ejpam-5367	249	48	0	0	NUM
ejpam-5367	249	49	)	)	PUNCT
ejpam-5367	249	50	=	=	SYM
ejpam-5367	250	1	1	1	X
ejpam-5367	250	2	.	.	PUNCT
ejpam-5367	251	1	hence	hence	ADV
ejpam-5367	251	2	,	,	PUNCT
ejpam-5367	251	3	the	the	DET
ejpam-5367	251	4	constant	constant	ADJ
ejpam-5367	251	5	0	0	NUM
ejpam-5367	251	6	is	be	AUX
ejpam-5367	251	7	in	in	ADP
ejpam-5367	251	8	g.	g.	PROPN
ejpam-5367	251	9	theorem	theorem	VERB
ejpam-5367	251	10	8	8	NUM
ejpam-5367	251	11	.	.	PUNCT
ejpam-5367	252	1	a	a	DET
ejpam-5367	252	2	non	non	ADJ
ejpam-5367	252	3	-	-	ADJ
ejpam-5367	252	4	empty	empty	ADJ
ejpam-5367	252	5	subset	subset	NOUN
ejpam-5367	252	6	g	g	NOUN
ejpam-5367	252	7	of	of	ADP
ejpam-5367	252	8	x	x	X
ejpam-5367	252	9	is	be	AUX
ejpam-5367	252	10	an	an	DET
ejpam-5367	252	11	iup	iup	NOUN
ejpam-5367	252	12	-	-	PUNCT
ejpam-5367	252	13	subalgebra	subalgebra	NOUN
ejpam-5367	252	14	of	of	ADP
ejpam-5367	252	15	x	x	PRON
ejpam-5367	252	16	if	if	SCONJ
ejpam-5367	252	17	and	and	CCONJ
ejpam-5367	253	1	only	only	ADV
ejpam-5367	253	2	if	if	SCONJ
ejpam-5367	253	3	the	the	DET
ejpam-5367	253	4	characteristic	characteristic	ADJ
ejpam-5367	253	5	ffs	ffs	PROPN
ejpam-5367	253	6	fg	fg	PROPN
ejpam-5367	253	7	is	be	AUX
ejpam-5367	253	8	an	an	DET
ejpam-5367	253	9	ffiup	ffiup	NOUN
ejpam-5367	253	10	-	-	PUNCT
ejpam-5367	253	11	subalgebra	subalgebra	NOUN
ejpam-5367	253	12	of	of	ADP
ejpam-5367	253	13	x.	x.	PROPN
ejpam-5367	253	14	a.	a.	PROPN
ejpam-5367	253	15	iampan	iampan	PROPN
ejpam-5367	253	16	et	et	PROPN
ejpam-5367	253	17	al	al	PROPN
ejpam-5367	253	18	.	.	PUNCT
ejpam-5367	253	19	/	/	SYM
ejpam-5367	253	20	eur	eur	PROPN
ejpam-5367	253	21	.	.	PUNCT
ejpam-5367	254	1	j.	j.	PROPN
ejpam-5367	254	2	pure	pure	PROPN
ejpam-5367	254	3	appl	appl	PROPN
ejpam-5367	254	4	.	.	PROPN
ejpam-5367	254	5	math	math	PROPN
ejpam-5367	254	6	,	,	PUNCT
ejpam-5367	254	7	17	17	NUM
ejpam-5367	254	8	(	(	PUNCT
ejpam-5367	254	9	4	4	NUM
ejpam-5367	254	10	)	)	PUNCT
ejpam-5367	254	11	(	(	PUNCT
ejpam-5367	254	12	2024	2024	NUM
ejpam-5367	254	13	)	)	PUNCT
ejpam-5367	254	14	,	,	PUNCT
ejpam-5367	254	15	3022	3022	NUM
ejpam-5367	254	16	-	-	SYM
ejpam-5367	254	17	3042	3042	NUM
ejpam-5367	254	18	3033	3033	NUM
ejpam-5367	254	19	proof	proof	NOUN
ejpam-5367	254	20	.	.	PUNCT
ejpam-5367	255	1	assume	assume	VERB
ejpam-5367	255	2	that	that	SCONJ
ejpam-5367	255	3	g	g	PROPN
ejpam-5367	255	4	is	be	AUX
ejpam-5367	255	5	an	an	DET
ejpam-5367	255	6	iup	iup	NOUN
ejpam-5367	255	7	-	-	PUNCT
ejpam-5367	255	8	subalgebra	subalgebra	NOUN
ejpam-5367	255	9	of	of	ADP
ejpam-5367	255	10	x.	x.	NOUN
ejpam-5367	255	11	let	let	VERB
ejpam-5367	255	12	x	x	PRON
ejpam-5367	255	13	,	,	PUNCT
ejpam-5367	255	14	y	y	PROPN
ejpam-5367	255	15	∈	∈	PROPN
ejpam-5367	255	16	x.	x.	NOUN
ejpam-5367	255	17	case	case	NOUN
ejpam-5367	255	18	1	1	NUM
ejpam-5367	255	19	:	:	PUNCT
ejpam-5367	255	20	suppose	suppose	VERB
ejpam-5367	255	21	x	x	PRON
ejpam-5367	255	22	,	,	PUNCT
ejpam-5367	255	23	y	y	PROPN
ejpam-5367	255	24	∈	∈	PROPN
ejpam-5367	255	25	g.	g.	NOUN
ejpam-5367	255	26	then	then	ADV
ejpam-5367	255	27	αfg	αfg	PROPN
ejpam-5367	255	28	(	(	PUNCT
ejpam-5367	255	29	x	x	NOUN
ejpam-5367	255	30	)	)	PUNCT
ejpam-5367	255	31	=	=	SYM
ejpam-5367	255	32	1	1	NUM
ejpam-5367	255	33	and	and	CCONJ
ejpam-5367	255	34	αfg	αfg	NOUN
ejpam-5367	255	35	(	(	PUNCT
ejpam-5367	255	36	y	y	NOUN
ejpam-5367	255	37	)	)	PUNCT
ejpam-5367	255	38	=	=	SYM
ejpam-5367	256	1	1	1	X
ejpam-5367	256	2	.	.	PUNCT
ejpam-5367	256	3	since	since	SCONJ
ejpam-5367	256	4	g	g	PROPN
ejpam-5367	256	5	is	be	AUX
ejpam-5367	256	6	an	an	DET
ejpam-5367	256	7	iupsubalgebra	iupsubalgebra	NOUN
ejpam-5367	256	8	ofx	ofx	NOUN
ejpam-5367	256	9	,	,	PUNCT
ejpam-5367	256	10	we	we	PRON
ejpam-5367	256	11	have	have	VERB
ejpam-5367	256	12	x·y	x·y	PROPN
ejpam-5367	256	13	∈	∈	PROPN
ejpam-5367	256	14	g.	g.	PROPN
ejpam-5367	256	15	thus	thus	ADV
ejpam-5367	256	16	,	,	PUNCT
ejpam-5367	256	17	αfg	αfg	PROPN
ejpam-5367	256	18	(	(	PUNCT
ejpam-5367	256	19	x·y	x·y	PROPN
ejpam-5367	256	20	)	)	PUNCT
ejpam-5367	256	21	=	=	SYM
ejpam-5367	256	22	1	1	NUM
ejpam-5367	256	23	≥	≥	NOUN
ejpam-5367	256	24	min{1	min{1	NOUN
ejpam-5367	256	25	,	,	PUNCT
ejpam-5367	256	26	1	1	NUM
ejpam-5367	256	27	}	}	PUNCT
ejpam-5367	256	28	=	=	SYM
ejpam-5367	256	29	min{αfg	min{αfg	NOUN
ejpam-5367	256	30	(	(	PUNCT
ejpam-5367	256	31	x	x	NOUN
ejpam-5367	256	32	)	)	PUNCT
ejpam-5367	256	33	,	,	PUNCT
ejpam-5367	256	34	αfg	αfg	NOUN
ejpam-5367	256	35	(	(	PUNCT
ejpam-5367	256	36	y	y	NOUN
ejpam-5367	256	37	)	)	PUNCT
ejpam-5367	256	38	}	}	PUNCT
ejpam-5367	256	39	.	.	PUNCT
ejpam-5367	257	1	case	case	NOUN
ejpam-5367	257	2	2	2	NUM
ejpam-5367	257	3	:	:	PUNCT
ejpam-5367	257	4	suppose	suppose	VERB
ejpam-5367	257	5	x	x	X
ejpam-5367	257	6	/∈	/∈	PUNCT
ejpam-5367	257	7	g	g	NOUN
ejpam-5367	257	8	or	or	CCONJ
ejpam-5367	257	9	y	y	PROPN
ejpam-5367	257	10	/∈	/∈	PUNCT
ejpam-5367	258	1	g.	g.	PROPN
ejpam-5367	258	2	then	then	ADV
ejpam-5367	258	3	αfg	αfg	PROPN
ejpam-5367	258	4	(	(	PUNCT
ejpam-5367	258	5	x	x	X
ejpam-5367	258	6	)	)	PUNCT
ejpam-5367	258	7	=	=	SYM
ejpam-5367	258	8	0	0	NUM
ejpam-5367	258	9	or	or	CCONJ
ejpam-5367	258	10	αfg	αfg	NOUN
ejpam-5367	258	11	(	(	PUNCT
ejpam-5367	258	12	y	y	NOUN
ejpam-5367	258	13	)	)	PUNCT
ejpam-5367	258	14	=	=	SYM
ejpam-5367	258	15	0	0	X
ejpam-5367	258	16	.	.	PUNCT
ejpam-5367	259	1	thus	thus	ADV
ejpam-5367	259	2	,	,	PUNCT
ejpam-5367	259	3	αfg	αfg	NOUN
ejpam-5367	259	4	(	(	PUNCT
ejpam-5367	259	5	x	x	NOUN
ejpam-5367	259	6	·	·	PUNCT
ejpam-5367	259	7	y	y	NOUN
ejpam-5367	259	8	)	)	PUNCT
ejpam-5367	259	9	≥	≥	NOUN
ejpam-5367	259	10	0	0	NUM
ejpam-5367	259	11	=	=	SYM
ejpam-5367	259	12	min{αfg	min{αfg	NOUN
ejpam-5367	259	13	(	(	PUNCT
ejpam-5367	259	14	x	x	NOUN
ejpam-5367	259	15	)	)	PUNCT
ejpam-5367	259	16	,	,	PUNCT
ejpam-5367	259	17	αfg	αfg	NOUN
ejpam-5367	259	18	(	(	PUNCT
ejpam-5367	259	19	y	y	NOUN
ejpam-5367	259	20	)	)	PUNCT
ejpam-5367	259	21	}	}	PUNCT
ejpam-5367	259	22	.	.	PUNCT
ejpam-5367	260	1	case	case	NOUN
ejpam-5367	260	2	1	1	NUM
ejpam-5367	260	3	’	'	PUNCT
ejpam-5367	260	4	:	:	PUNCT
ejpam-5367	260	5	suppose	suppose	VERB
ejpam-5367	260	6	x	x	PRON
ejpam-5367	260	7	,	,	PUNCT
ejpam-5367	260	8	y	y	PROPN
ejpam-5367	260	9	∈	∈	PROPN
ejpam-5367	260	10	g.	g.	NOUN
ejpam-5367	260	11	then	then	ADV
ejpam-5367	260	12	βfg	βfg	PROPN
ejpam-5367	260	13	(	(	PUNCT
ejpam-5367	260	14	x	x	X
ejpam-5367	260	15	)	)	PUNCT
ejpam-5367	260	16	=	=	SYM
ejpam-5367	260	17	0	0	NUM
ejpam-5367	260	18	and	and	CCONJ
ejpam-5367	260	19	βfg	βfg	PROPN
ejpam-5367	260	20	(	(	PUNCT
ejpam-5367	260	21	y	y	NOUN
ejpam-5367	260	22	)	)	PUNCT
ejpam-5367	260	23	=	=	NOUN
ejpam-5367	261	1	0	0	X
ejpam-5367	261	2	.	.	PUNCT
ejpam-5367	262	1	since	since	SCONJ
ejpam-5367	262	2	g	g	PROPN
ejpam-5367	262	3	is	be	AUX
ejpam-5367	262	4	an	an	DET
ejpam-5367	262	5	iupsubalgebra	iupsubalgebra	NOUN
ejpam-5367	262	6	of	of	ADP
ejpam-5367	262	7	x	x	PRON
ejpam-5367	262	8	,	,	PUNCT
ejpam-5367	262	9	we	we	PRON
ejpam-5367	262	10	have	have	VERB
ejpam-5367	262	11	x	x	X
ejpam-5367	262	12	·	·	PUNCT
ejpam-5367	262	13	y	y	PROPN
ejpam-5367	262	14	∈	∈	PROPN
ejpam-5367	262	15	g.	g.	PROPN
ejpam-5367	263	1	thus	thus	ADV
ejpam-5367	263	2	,	,	PUNCT
ejpam-5367	263	3	βfg	βfg	PROPN
ejpam-5367	263	4	(	(	PUNCT
ejpam-5367	263	5	x	x	X
ejpam-5367	263	6	·	·	PUNCT
ejpam-5367	263	7	y	y	X
ejpam-5367	263	8	)	)	PUNCT
ejpam-5367	263	9	=	=	SYM
ejpam-5367	263	10	0	0	NUM
ejpam-5367	263	11	≤	≤	NUM
ejpam-5367	263	12	0	0	NUM
ejpam-5367	263	13	=	=	SYM
ejpam-5367	263	14	max{βfg	max{βfg	NOUN
ejpam-5367	263	15	(	(	PUNCT
ejpam-5367	263	16	x	x	NOUN
ejpam-5367	263	17	)	)	PUNCT
ejpam-5367	263	18	,	,	PUNCT
ejpam-5367	263	19	βfg	βfg	PROPN
ejpam-5367	263	20	(	(	PUNCT
ejpam-5367	263	21	y	y	NOUN
ejpam-5367	263	22	)	)	PUNCT
ejpam-5367	263	23	}	}	PUNCT
ejpam-5367	263	24	.	.	PUNCT
ejpam-5367	264	1	case	case	NOUN
ejpam-5367	264	2	2	2	NUM
ejpam-5367	264	3	’	'	PUNCT
ejpam-5367	264	4	:	:	PUNCT
ejpam-5367	264	5	suppose	suppose	VERB
ejpam-5367	264	6	x	x	X
ejpam-5367	264	7	/∈	/∈	PUNCT
ejpam-5367	264	8	g	g	NOUN
ejpam-5367	264	9	or	or	CCONJ
ejpam-5367	264	10	y	y	PROPN
ejpam-5367	264	11	/∈	/∈	PUNCT
ejpam-5367	265	1	g.	g.	PROPN
ejpam-5367	266	1	then	then	ADV
ejpam-5367	266	2	βfg	βfg	PROPN
ejpam-5367	266	3	(	(	PUNCT
ejpam-5367	266	4	x	x	X
ejpam-5367	266	5	)	)	PUNCT
ejpam-5367	266	6	=	=	SYM
ejpam-5367	266	7	1	1	NUM
ejpam-5367	266	8	or	or	CCONJ
ejpam-5367	266	9	βfg	βfg	PROPN
ejpam-5367	266	10	(	(	PUNCT
ejpam-5367	266	11	y	y	NOUN
ejpam-5367	266	12	)	)	PUNCT
ejpam-5367	266	13	=	=	SYM
ejpam-5367	266	14	1	1	X
ejpam-5367	266	15	.	.	PUNCT
ejpam-5367	267	1	thus	thus	ADV
ejpam-5367	267	2	,	,	PUNCT
ejpam-5367	267	3	βfg	βfg	PROPN
ejpam-5367	267	4	(	(	PUNCT
ejpam-5367	267	5	x	x	X
ejpam-5367	267	6	·	·	PUNCT
ejpam-5367	267	7	y	y	X
ejpam-5367	267	8	)	)	PUNCT
ejpam-5367	267	9	≤	≤	NOUN
ejpam-5367	267	10	1	1	NUM
ejpam-5367	267	11	=	=	SYM
ejpam-5367	267	12	max{βfg	max{βfg	NOUN
ejpam-5367	267	13	(	(	PUNCT
ejpam-5367	267	14	x	x	NOUN
ejpam-5367	267	15	)	)	PUNCT
ejpam-5367	267	16	,	,	PUNCT
ejpam-5367	267	17	βfg	βfg	PROPN
ejpam-5367	267	18	(	(	PUNCT
ejpam-5367	267	19	y	y	NOUN
ejpam-5367	267	20	)	)	PUNCT
ejpam-5367	267	21	}	}	PUNCT
ejpam-5367	267	22	.	.	PUNCT
ejpam-5367	268	1	hence	hence	ADV
ejpam-5367	268	2	,	,	PUNCT
ejpam-5367	268	3	the	the	DET
ejpam-5367	268	4	characteristic	characteristic	ADJ
ejpam-5367	268	5	ffs	ffs	PROPN
ejpam-5367	268	6	fg	fg	PROPN
ejpam-5367	268	7	is	be	AUX
ejpam-5367	268	8	an	an	DET
ejpam-5367	268	9	ffiup	ffiup	NOUN
ejpam-5367	268	10	-	-	PUNCT
ejpam-5367	268	11	subalgebra	subalgebra	NOUN
ejpam-5367	268	12	of	of	ADP
ejpam-5367	268	13	x.	x.	NOUN
ejpam-5367	268	14	conversely	conversely	ADV
ejpam-5367	268	15	,	,	PUNCT
ejpam-5367	268	16	assume	assume	VERB
ejpam-5367	268	17	that	that	SCONJ
ejpam-5367	268	18	the	the	DET
ejpam-5367	268	19	characteristic	characteristic	ADJ
ejpam-5367	268	20	ffs	ffs	PROPN
ejpam-5367	269	1	fg	fg	PROPN
ejpam-5367	269	2	is	be	AUX
ejpam-5367	269	3	an	an	DET
ejpam-5367	269	4	ffiup	ffiup	NOUN
ejpam-5367	269	5	-	-	PUNCT
ejpam-5367	269	6	subalgebra	subalgebra	NOUN
ejpam-5367	269	7	of	of	ADP
ejpam-5367	269	8	x.	x.	NOUN
ejpam-5367	269	9	let	let	VERB
ejpam-5367	269	10	x	x	PRON
ejpam-5367	269	11	,	,	PUNCT
ejpam-5367	269	12	y	y	PROPN
ejpam-5367	269	13	∈	∈	PROPN
ejpam-5367	269	14	g.	g.	NOUN
ejpam-5367	269	15	then	then	ADV
ejpam-5367	269	16	αfg	αfg	PROPN
ejpam-5367	269	17	(	(	PUNCT
ejpam-5367	269	18	x	x	NOUN
ejpam-5367	269	19	)	)	PUNCT
ejpam-5367	269	20	=	=	SYM
ejpam-5367	269	21	1	1	NUM
ejpam-5367	269	22	and	and	CCONJ
ejpam-5367	269	23	αfg	αfg	NOUN
ejpam-5367	269	24	(	(	PUNCT
ejpam-5367	269	25	y	y	NOUN
ejpam-5367	269	26	)	)	PUNCT
ejpam-5367	269	27	=	=	SYM
ejpam-5367	270	1	1	1	X
ejpam-5367	270	2	.	.	PUNCT
ejpam-5367	270	3	by	by	ADP
ejpam-5367	270	4	(	(	PUNCT
ejpam-5367	270	5	3.2	3.2	NUM
ejpam-5367	270	6	)	)	PUNCT
ejpam-5367	270	7	,	,	PUNCT
ejpam-5367	270	8	we	we	PRON
ejpam-5367	270	9	have	have	VERB
ejpam-5367	270	10	αfg	αfg	NOUN
ejpam-5367	270	11	(	(	PUNCT
ejpam-5367	270	12	x	x	SYM
ejpam-5367	270	13	·	·	PUNCT
ejpam-5367	270	14	y	y	X
ejpam-5367	270	15	)	)	PUNCT
ejpam-5367	270	16	≥	≥	NOUN
ejpam-5367	270	17	min{αfg	min{αfg	PROPN
ejpam-5367	270	18	(	(	PUNCT
ejpam-5367	270	19	x	x	X
ejpam-5367	270	20	)	)	PUNCT
ejpam-5367	270	21	,	,	PUNCT
ejpam-5367	270	22	αfg	αfg	NOUN
ejpam-5367	270	23	(	(	PUNCT
ejpam-5367	270	24	y	y	NOUN
ejpam-5367	270	25	)	)	PUNCT
ejpam-5367	270	26	}	}	PUNCT
ejpam-5367	270	27	=	=	SYM
ejpam-5367	270	28	min{1	min{1	PROPN
ejpam-5367	270	29	,	,	PUNCT
ejpam-5367	270	30	1	1	NUM
ejpam-5367	270	31	}	}	PUNCT
ejpam-5367	270	32	=	=	SYM
ejpam-5367	270	33	1	1	NUM
ejpam-5367	270	34	.	.	PUNCT
ejpam-5367	271	1	thus	thus	ADV
ejpam-5367	271	2	,	,	PUNCT
ejpam-5367	271	3	αfg	αfg	NOUN
ejpam-5367	271	4	(	(	PUNCT
ejpam-5367	271	5	x	x	X
ejpam-5367	271	6	·	·	PUNCT
ejpam-5367	271	7	y	y	X
ejpam-5367	271	8	)	)	PUNCT
ejpam-5367	271	9	=	=	SYM
ejpam-5367	271	10	1	1	NUM
ejpam-5367	271	11	,	,	PUNCT
ejpam-5367	271	12	that	that	ADV
ejpam-5367	271	13	is	is	ADV
ejpam-5367	271	14	,	,	PUNCT
ejpam-5367	271	15	x	x	X
ejpam-5367	271	16	·	·	PUNCT
ejpam-5367	271	17	y	y	X
ejpam-5367	271	18	∈	∈	PROPN
ejpam-5367	271	19	g.	g.	NOUN
ejpam-5367	271	20	hence	hence	ADV
ejpam-5367	271	21	,	,	PUNCT
ejpam-5367	271	22	g	g	PROPN
ejpam-5367	271	23	is	be	AUX
ejpam-5367	271	24	an	an	DET
ejpam-5367	271	25	iup	iup	NOUN
ejpam-5367	271	26	-	-	PUNCT
ejpam-5367	271	27	subalgebra	subalgebra	NOUN
ejpam-5367	271	28	of	of	ADP
ejpam-5367	271	29	x.	x.	NOUN
ejpam-5367	271	30	theorem	theorem	VERB
ejpam-5367	271	31	9	9	NUM
ejpam-5367	271	32	.	.	PUNCT
ejpam-5367	272	1	a	a	DET
ejpam-5367	272	2	non	non	ADJ
ejpam-5367	272	3	-	-	ADJ
ejpam-5367	272	4	empty	empty	ADJ
ejpam-5367	272	5	subset	subset	NOUN
ejpam-5367	272	6	g	g	NOUN
ejpam-5367	272	7	of	of	ADP
ejpam-5367	272	8	x	x	X
ejpam-5367	272	9	is	be	AUX
ejpam-5367	272	10	an	an	DET
ejpam-5367	272	11	iup	iup	NOUN
ejpam-5367	272	12	-	-	PUNCT
ejpam-5367	272	13	ideal	ideal	NOUN
ejpam-5367	272	14	of	of	ADP
ejpam-5367	272	15	x	x	SYM
ejpam-5367	272	16	if	if	SCONJ
ejpam-5367	272	17	and	and	CCONJ
ejpam-5367	273	1	only	only	ADV
ejpam-5367	273	2	if	if	SCONJ
ejpam-5367	273	3	the	the	DET
ejpam-5367	273	4	characteristic	characteristic	ADJ
ejpam-5367	273	5	ffs	ffs	PROPN
ejpam-5367	273	6	fg	fg	PROPN
ejpam-5367	273	7	is	be	AUX
ejpam-5367	273	8	an	an	DET
ejpam-5367	273	9	ffiup	ffiup	NOUN
ejpam-5367	273	10	-	-	PUNCT
ejpam-5367	273	11	ideal	ideal	NOUN
ejpam-5367	273	12	of	of	ADP
ejpam-5367	273	13	x.	x.	NOUN
ejpam-5367	273	14	proof	proof	PROPN
ejpam-5367	273	15	.	.	PUNCT
ejpam-5367	274	1	assume	assume	VERB
ejpam-5367	274	2	that	that	SCONJ
ejpam-5367	274	3	g	g	PROPN
ejpam-5367	274	4	is	be	AUX
ejpam-5367	274	5	an	an	DET
ejpam-5367	274	6	iup	iup	NOUN
ejpam-5367	274	7	-	-	PUNCT
ejpam-5367	274	8	ideal	ideal	NOUN
ejpam-5367	274	9	of	of	ADP
ejpam-5367	274	10	x.	x.	NOUN
ejpam-5367	274	11	since	since	SCONJ
ejpam-5367	274	12	0	0	NUM
ejpam-5367	274	13	∈	∈	PROPN
ejpam-5367	274	14	g	g	NOUN
ejpam-5367	274	15	,	,	PUNCT
ejpam-5367	274	16	it	it	PRON
ejpam-5367	274	17	follows	follow	VERB
ejpam-5367	274	18	from	from	ADP
ejpam-5367	274	19	lemma	lemma	PROPN
ejpam-5367	274	20	2	2	NUM
ejpam-5367	274	21	that	that	SCONJ
ejpam-5367	274	22	αfg	αfg	NOUN
ejpam-5367	274	23	and	and	CCONJ
ejpam-5367	274	24	βfg	βfg	PROPN
ejpam-5367	274	25	satisfy	satisfy	NOUN
ejpam-5367	274	26	(	(	PUNCT
ejpam-5367	274	27	3.4	3.4	NUM
ejpam-5367	274	28	)	)	PUNCT
ejpam-5367	274	29	and	and	CCONJ
ejpam-5367	274	30	(	(	PUNCT
ejpam-5367	274	31	3.5	3.5	NUM
ejpam-5367	274	32	)	)	PUNCT
ejpam-5367	274	33	,	,	PUNCT
ejpam-5367	274	34	respectively	respectively	ADV
ejpam-5367	274	35	.	.	PUNCT
ejpam-5367	275	1	next	next	ADV
ejpam-5367	275	2	,	,	PUNCT
ejpam-5367	275	3	let	let	VERB
ejpam-5367	275	4	x	x	PRON
ejpam-5367	275	5	,	,	PUNCT
ejpam-5367	275	6	y	y	PROPN
ejpam-5367	275	7	,	,	PUNCT
ejpam-5367	275	8	z	z	PROPN
ejpam-5367	275	9	∈	∈	NOUN
ejpam-5367	275	10	x.	x.	NOUN
ejpam-5367	275	11	case	case	NOUN
ejpam-5367	275	12	1	1	NUM
ejpam-5367	275	13	:	:	PUNCT
ejpam-5367	275	14	suppose	suppose	VERB
ejpam-5367	275	15	x	x	X
ejpam-5367	275	16	·	·	PUNCT
ejpam-5367	275	17	(	(	PUNCT
ejpam-5367	275	18	y	y	PROPN
ejpam-5367	275	19	·	·	PUNCT
ejpam-5367	275	20	z	z	X
ejpam-5367	275	21	)	)	PUNCT
ejpam-5367	275	22	∈	∈	PROPN
ejpam-5367	275	23	g	g	PROPN
ejpam-5367	275	24	and	and	CCONJ
ejpam-5367	275	25	y	y	PROPN
ejpam-5367	275	26	∈	∈	PROPN
ejpam-5367	275	27	g.	g.	NOUN
ejpam-5367	275	28	since	since	SCONJ
ejpam-5367	275	29	g	g	PROPN
ejpam-5367	275	30	is	be	AUX
ejpam-5367	275	31	an	an	DET
ejpam-5367	275	32	iup	iup	NOUN
ejpam-5367	275	33	-	-	PUNCT
ejpam-5367	275	34	ideal	ideal	NOUN
ejpam-5367	275	35	of	of	ADP
ejpam-5367	275	36	x	x	SYM
ejpam-5367	275	37	,	,	PUNCT
ejpam-5367	275	38	we	we	PRON
ejpam-5367	275	39	have	have	VERB
ejpam-5367	275	40	x	x	X
ejpam-5367	275	41	·	·	PUNCT
ejpam-5367	275	42	z	z	SYM
ejpam-5367	275	43	∈	∈	PROPN
ejpam-5367	275	44	g.	g.	PROPN
ejpam-5367	275	45	thus	thus	ADV
ejpam-5367	275	46	,	,	PUNCT
ejpam-5367	275	47	αfg	αfg	PROPN
ejpam-5367	275	48	(	(	PUNCT
ejpam-5367	275	49	x	x	X
ejpam-5367	275	50	·	·	PUNCT
ejpam-5367	275	51	z	z	X
ejpam-5367	275	52	)	)	PUNCT
ejpam-5367	275	53	=	=	SYM
ejpam-5367	275	54	1	1	NUM
ejpam-5367	275	55	≥	≥	NOUN
ejpam-5367	275	56	1	1	NUM
ejpam-5367	275	57	=	=	SYM
ejpam-5367	275	58	min{1	min{1	PROPN
ejpam-5367	275	59	,	,	PUNCT
ejpam-5367	275	60	1	1	NUM
ejpam-5367	275	61	}	}	PUNCT
ejpam-5367	275	62	=	=	SYM
ejpam-5367	275	63	min{αfg	min{αfg	NOUN
ejpam-5367	275	64	(	(	PUNCT
ejpam-5367	275	65	x	x	X
ejpam-5367	275	66	·	·	PUNCT
ejpam-5367	275	67	(	(	PUNCT
ejpam-5367	275	68	y	y	PROPN
ejpam-5367	275	69	·	·	PUNCT
ejpam-5367	275	70	z	z	NOUN
ejpam-5367	275	71	)	)	PUNCT
ejpam-5367	275	72	)	)	PUNCT
ejpam-5367	275	73	,	,	PUNCT
ejpam-5367	275	74	αfg	αfg	NOUN
ejpam-5367	275	75	(	(	PUNCT
ejpam-5367	275	76	y	y	NOUN
ejpam-5367	275	77	)	)	PUNCT
ejpam-5367	275	78	}	}	PUNCT
ejpam-5367	275	79	.	.	PUNCT
ejpam-5367	276	1	case	case	NOUN
ejpam-5367	276	2	2	2	NUM
ejpam-5367	276	3	:	:	PUNCT
ejpam-5367	276	4	suppose	suppose	VERB
ejpam-5367	276	5	x	x	X
ejpam-5367	276	6	·	·	PUNCT
ejpam-5367	276	7	(	(	PUNCT
ejpam-5367	276	8	y	y	PROPN
ejpam-5367	276	9	·	·	PUNCT
ejpam-5367	276	10	z	z	X
ejpam-5367	276	11	)	)	PUNCT
ejpam-5367	276	12	/∈	/∈	PUNCT
ejpam-5367	277	1	g	g	NOUN
ejpam-5367	277	2	or	or	CCONJ
ejpam-5367	277	3	y	y	PROPN
ejpam-5367	277	4	/∈	/∈	PUNCT
ejpam-5367	278	1	g.	g.	PROPN
ejpam-5367	278	2	then	then	ADV
ejpam-5367	278	3	αfg	αfg	PROPN
ejpam-5367	278	4	(	(	PUNCT
ejpam-5367	278	5	x	x	X
ejpam-5367	278	6	·	·	PUNCT
ejpam-5367	278	7	(	(	PUNCT
ejpam-5367	278	8	y	y	PROPN
ejpam-5367	278	9	·	·	PUNCT
ejpam-5367	278	10	z	z	NOUN
ejpam-5367	278	11	)	)	PUNCT
ejpam-5367	278	12	)	)	PUNCT
ejpam-5367	279	1	=	=	SYM
ejpam-5367	279	2	0	0	NUM
ejpam-5367	279	3	or	or	CCONJ
ejpam-5367	279	4	αfg	αfg	NOUN
ejpam-5367	279	5	(	(	PUNCT
ejpam-5367	279	6	y	y	NOUN
ejpam-5367	279	7	)	)	PUNCT
ejpam-5367	279	8	=	=	SYM
ejpam-5367	279	9	0	0	X
ejpam-5367	279	10	.	.	PUNCT
ejpam-5367	280	1	thus	thus	ADV
ejpam-5367	280	2	,	,	PUNCT
ejpam-5367	280	3	αfg	αfg	NOUN
ejpam-5367	280	4	(	(	PUNCT
ejpam-5367	280	5	x	x	X
ejpam-5367	280	6	·	·	PUNCT
ejpam-5367	280	7	z	z	X
ejpam-5367	280	8	)	)	PUNCT
ejpam-5367	280	9	≥	≥	NOUN
ejpam-5367	280	10	0	0	NUM
ejpam-5367	280	11	=	=	SYM
ejpam-5367	280	12	min{αfg	min{αfg	NOUN
ejpam-5367	280	13	(	(	PUNCT
ejpam-5367	280	14	x	x	X
ejpam-5367	280	15	·	·	PUNCT
ejpam-5367	280	16	(	(	PUNCT
ejpam-5367	280	17	y	y	PROPN
ejpam-5367	280	18	·	·	PUNCT
ejpam-5367	280	19	z	z	NOUN
ejpam-5367	280	20	)	)	PUNCT
ejpam-5367	280	21	)	)	PUNCT
ejpam-5367	280	22	,	,	PUNCT
ejpam-5367	280	23	αfg	αfg	NOUN
ejpam-5367	280	24	(	(	PUNCT
ejpam-5367	280	25	y	y	NOUN
ejpam-5367	280	26	)	)	PUNCT
ejpam-5367	280	27	}	}	PUNCT
ejpam-5367	280	28	.	.	PUNCT
ejpam-5367	281	1	case	case	NOUN
ejpam-5367	281	2	1	1	NUM
ejpam-5367	281	3	’	'	PUNCT
ejpam-5367	281	4	:	:	PUNCT
ejpam-5367	281	5	suppose	suppose	VERB
ejpam-5367	281	6	x	x	X
ejpam-5367	281	7	·	·	PUNCT
ejpam-5367	281	8	(	(	PUNCT
ejpam-5367	281	9	y	y	PROPN
ejpam-5367	281	10	·	·	PUNCT
ejpam-5367	281	11	z	z	X
ejpam-5367	281	12	)	)	PUNCT
ejpam-5367	281	13	∈	∈	PROPN
ejpam-5367	281	14	g	g	PROPN
ejpam-5367	281	15	and	and	CCONJ
ejpam-5367	281	16	y	y	PROPN
ejpam-5367	281	17	∈	∈	PROPN
ejpam-5367	281	18	g.	g.	NOUN
ejpam-5367	281	19	since	since	SCONJ
ejpam-5367	281	20	g	g	PROPN
ejpam-5367	281	21	is	be	AUX
ejpam-5367	281	22	an	an	DET
ejpam-5367	281	23	iup	iup	NOUN
ejpam-5367	281	24	-	-	PUNCT
ejpam-5367	281	25	ideal	ideal	NOUN
ejpam-5367	281	26	of	of	ADP
ejpam-5367	281	27	x	x	SYM
ejpam-5367	281	28	,	,	PUNCT
ejpam-5367	281	29	we	we	PRON
ejpam-5367	281	30	have	have	VERB
ejpam-5367	281	31	x	x	X
ejpam-5367	281	32	·	·	PUNCT
ejpam-5367	281	33	z	z	SYM
ejpam-5367	281	34	∈	∈	PROPN
ejpam-5367	281	35	g.	g.	PROPN
ejpam-5367	281	36	thus	thus	ADV
ejpam-5367	281	37	,	,	PUNCT
ejpam-5367	281	38	βfg	βfg	PROPN
ejpam-5367	281	39	(	(	PUNCT
ejpam-5367	281	40	x	x	X
ejpam-5367	281	41	·	·	PUNCT
ejpam-5367	281	42	z	z	X
ejpam-5367	281	43	)	)	PUNCT
ejpam-5367	281	44	=	=	SYM
ejpam-5367	281	45	0	0	NUM
ejpam-5367	281	46	≤	≤	NUM
ejpam-5367	281	47	0	0	NUM
ejpam-5367	281	48	=	=	SYM
ejpam-5367	281	49	max{0	max{0	PROPN
ejpam-5367	281	50	,	,	PUNCT
ejpam-5367	281	51	0	0	NUM
ejpam-5367	281	52	}	}	PUNCT
ejpam-5367	281	53	=	=	SYM
ejpam-5367	281	54	max{βfg	max{βfg	NOUN
ejpam-5367	281	55	(	(	PUNCT
ejpam-5367	281	56	x	x	X
ejpam-5367	281	57	·	·	PUNCT
ejpam-5367	281	58	(	(	PUNCT
ejpam-5367	281	59	y	y	PROPN
ejpam-5367	281	60	·	·	PUNCT
ejpam-5367	281	61	z	z	NOUN
ejpam-5367	281	62	)	)	PUNCT
ejpam-5367	281	63	)	)	PUNCT
ejpam-5367	281	64	,	,	PUNCT
ejpam-5367	281	65	βfg	βfg	PROPN
ejpam-5367	281	66	(	(	PUNCT
ejpam-5367	281	67	y	y	NOUN
ejpam-5367	281	68	)	)	PUNCT
ejpam-5367	281	69	}	}	PUNCT
ejpam-5367	281	70	.	.	PUNCT
ejpam-5367	282	1	case	case	NOUN
ejpam-5367	282	2	2	2	NUM
ejpam-5367	282	3	’	'	PUNCT
ejpam-5367	282	4	:	:	PUNCT
ejpam-5367	282	5	suppose	suppose	VERB
ejpam-5367	282	6	x	x	X
ejpam-5367	282	7	·	·	PUNCT
ejpam-5367	282	8	(	(	PUNCT
ejpam-5367	282	9	y	y	PROPN
ejpam-5367	282	10	·	·	PUNCT
ejpam-5367	282	11	z	z	X
ejpam-5367	282	12	)	)	PUNCT
ejpam-5367	282	13	/∈	/∈	PUNCT
ejpam-5367	283	1	g	g	NOUN
ejpam-5367	283	2	or	or	CCONJ
ejpam-5367	283	3	y	y	PROPN
ejpam-5367	283	4	/∈	/∈	PUNCT
ejpam-5367	284	1	g.	g.	PROPN
ejpam-5367	285	1	then	then	ADV
ejpam-5367	285	2	βfg	βfg	PROPN
ejpam-5367	285	3	(	(	PUNCT
ejpam-5367	285	4	x	x	X
ejpam-5367	285	5	·	·	PUNCT
ejpam-5367	285	6	(	(	PUNCT
ejpam-5367	285	7	y	y	PROPN
ejpam-5367	285	8	·	·	PUNCT
ejpam-5367	285	9	z	z	NOUN
ejpam-5367	285	10	)	)	PUNCT
ejpam-5367	285	11	)	)	PUNCT
ejpam-5367	286	1	=	=	SYM
ejpam-5367	286	2	1	1	NUM
ejpam-5367	286	3	or	or	CCONJ
ejpam-5367	286	4	βfg	βfg	PROPN
ejpam-5367	286	5	(	(	PUNCT
ejpam-5367	286	6	y	y	NOUN
ejpam-5367	286	7	)	)	PUNCT
ejpam-5367	286	8	=	=	SYM
ejpam-5367	287	1	1	1	X
ejpam-5367	287	2	.	.	PUNCT
ejpam-5367	288	1	thus	thus	ADV
ejpam-5367	288	2	,	,	PUNCT
ejpam-5367	288	3	βfg	βfg	PROPN
ejpam-5367	288	4	(	(	PUNCT
ejpam-5367	288	5	x	x	X
ejpam-5367	288	6	·	·	PUNCT
ejpam-5367	288	7	z	z	X
ejpam-5367	288	8	)	)	PUNCT
ejpam-5367	288	9	≤	≤	NOUN
ejpam-5367	288	10	1	1	NUM
ejpam-5367	288	11	=	=	SYM
ejpam-5367	288	12	max{βfg	max{βfg	NOUN
ejpam-5367	288	13	(	(	PUNCT
ejpam-5367	288	14	x	x	X
ejpam-5367	288	15	·	·	PUNCT
ejpam-5367	288	16	(	(	PUNCT
ejpam-5367	288	17	y	y	PROPN
ejpam-5367	288	18	·	·	PUNCT
ejpam-5367	288	19	z	z	NOUN
ejpam-5367	288	20	)	)	PUNCT
ejpam-5367	288	21	)	)	PUNCT
ejpam-5367	288	22	,	,	PUNCT
ejpam-5367	288	23	βfg	βfg	PROPN
ejpam-5367	288	24	(	(	PUNCT
ejpam-5367	288	25	y	y	NOUN
ejpam-5367	288	26	)	)	PUNCT
ejpam-5367	288	27	}	}	PUNCT
ejpam-5367	288	28	.	.	PUNCT
ejpam-5367	289	1	hence	hence	ADV
ejpam-5367	289	2	,	,	PUNCT
ejpam-5367	289	3	fg	fg	PROPN
ejpam-5367	289	4	is	be	AUX
ejpam-5367	289	5	an	an	DET
ejpam-5367	289	6	ffiup	ffiup	NOUN
ejpam-5367	289	7	-	-	PUNCT
ejpam-5367	289	8	ideal	ideal	NOUN
ejpam-5367	289	9	of	of	ADP
ejpam-5367	289	10	x.	x.	NOUN
ejpam-5367	289	11	conversely	conversely	ADV
ejpam-5367	289	12	,	,	PUNCT
ejpam-5367	289	13	assume	assume	VERB
ejpam-5367	289	14	that	that	SCONJ
ejpam-5367	289	15	the	the	DET
ejpam-5367	289	16	characteristic	characteristic	ADJ
ejpam-5367	289	17	ffs	ffs	PROPN
ejpam-5367	290	1	fg	fg	PROPN
ejpam-5367	290	2	is	be	AUX
ejpam-5367	290	3	an	an	DET
ejpam-5367	290	4	ffiup	ffiup	NOUN
ejpam-5367	290	5	-	-	PUNCT
ejpam-5367	290	6	ideal	ideal	NOUN
ejpam-5367	290	7	of	of	ADP
ejpam-5367	290	8	x.	x.	NOUN
ejpam-5367	290	9	since	since	SCONJ
ejpam-5367	290	10	αfg	αfg	ADJ
ejpam-5367	290	11	satisfies	satisfie	NOUN
ejpam-5367	290	12	(	(	PUNCT
ejpam-5367	290	13	3.4	3.4	NUM
ejpam-5367	290	14	)	)	PUNCT
ejpam-5367	290	15	,	,	PUNCT
ejpam-5367	290	16	it	it	PRON
ejpam-5367	290	17	follows	follow	VERB
ejpam-5367	290	18	from	from	ADP
ejpam-5367	290	19	lemma	lemma	PROPN
ejpam-5367	290	20	2	2	NUM
ejpam-5367	290	21	that	that	SCONJ
ejpam-5367	290	22	0	0	NUM
ejpam-5367	290	23	∈	∈	NOUN
ejpam-5367	290	24	g.	g.	NOUN
ejpam-5367	290	25	next	next	ADV
ejpam-5367	290	26	,	,	PUNCT
ejpam-5367	290	27	let	let	VERB
ejpam-5367	290	28	x	x	PRON
ejpam-5367	290	29	,	,	PUNCT
ejpam-5367	290	30	y	y	PROPN
ejpam-5367	290	31	,	,	PUNCT
ejpam-5367	290	32	z	z	NOUN
ejpam-5367	290	33	∈	∈	PROPN
ejpam-5367	290	34	x	x	AUX
ejpam-5367	290	35	be	be	AUX
ejpam-5367	290	36	such	such	ADJ
ejpam-5367	290	37	that	that	SCONJ
ejpam-5367	290	38	x	x	PART
ejpam-5367	290	39	·	·	PUNCT
ejpam-5367	290	40	(	(	PUNCT
ejpam-5367	290	41	y	y	PROPN
ejpam-5367	290	42	·	·	PUNCT
ejpam-5367	290	43	z	z	X
ejpam-5367	290	44	)	)	PUNCT
ejpam-5367	290	45	∈	∈	PROPN
ejpam-5367	290	46	g	g	PROPN
ejpam-5367	290	47	and	and	CCONJ
ejpam-5367	290	48	y	y	PROPN
ejpam-5367	290	49	∈	∈	PROPN
ejpam-5367	291	1	g.	g.	NOUN
ejpam-5367	291	2	then	then	ADV
ejpam-5367	291	3	αfg	αfg	PROPN
ejpam-5367	291	4	(	(	PUNCT
ejpam-5367	291	5	x	x	X
ejpam-5367	291	6	·	·	PUNCT
ejpam-5367	291	7	(	(	PUNCT
ejpam-5367	291	8	y	y	PROPN
ejpam-5367	291	9	·	·	PUNCT
ejpam-5367	291	10	z	z	NOUN
ejpam-5367	291	11	)	)	PUNCT
ejpam-5367	291	12	)	)	PUNCT
ejpam-5367	292	1	=	=	SYM
ejpam-5367	292	2	1	1	NUM
ejpam-5367	292	3	and	and	CCONJ
ejpam-5367	292	4	αfg	αfg	NOUN
ejpam-5367	292	5	(	(	PUNCT
ejpam-5367	292	6	y	y	NOUN
ejpam-5367	292	7	)	)	PUNCT
ejpam-5367	292	8	=	=	SYM
ejpam-5367	293	1	1	1	X
ejpam-5367	293	2	.	.	PUNCT
ejpam-5367	293	3	thus	thus	ADV
ejpam-5367	293	4	,	,	PUNCT
ejpam-5367	293	5	min{αfg	min{αfg	PROPN
ejpam-5367	293	6	(	(	PUNCT
ejpam-5367	293	7	x	x	X
ejpam-5367	293	8	·	·	PUNCT
ejpam-5367	293	9	(	(	PUNCT
ejpam-5367	293	10	y	y	PROPN
ejpam-5367	293	11	·	·	PUNCT
ejpam-5367	293	12	z	z	NOUN
ejpam-5367	293	13	)	)	PUNCT
ejpam-5367	293	14	)	)	PUNCT
ejpam-5367	293	15	,	,	PUNCT
ejpam-5367	293	16	αfg	αfg	NOUN
ejpam-5367	293	17	(	(	PUNCT
ejpam-5367	293	18	y	y	NOUN
ejpam-5367	293	19	)	)	PUNCT
ejpam-5367	293	20	}	}	PUNCT
ejpam-5367	293	21	=	=	SYM
ejpam-5367	293	22	1	1	X
ejpam-5367	293	23	.	.	PUNCT
ejpam-5367	293	24	by	by	ADP
ejpam-5367	293	25	(	(	PUNCT
ejpam-5367	293	26	3.6	3.6	NUM
ejpam-5367	293	27	)	)	PUNCT
ejpam-5367	293	28	,	,	PUNCT
ejpam-5367	293	29	we	we	PRON
ejpam-5367	293	30	have	have	VERB
ejpam-5367	293	31	αfg	αfg	NOUN
ejpam-5367	293	32	(	(	PUNCT
ejpam-5367	293	33	x	x	X
ejpam-5367	293	34	·	·	PUNCT
ejpam-5367	293	35	z	z	X
ejpam-5367	293	36	)	)	PUNCT
ejpam-5367	293	37	≥	≥	NOUN
ejpam-5367	293	38	min{αfg	min{αfg	PROPN
ejpam-5367	293	39	(	(	PUNCT
ejpam-5367	293	40	x	x	X
ejpam-5367	293	41	·	·	PUNCT
ejpam-5367	293	42	(	(	PUNCT
ejpam-5367	293	43	y	y	PROPN
ejpam-5367	293	44	·	·	PUNCT
ejpam-5367	293	45	z	z	NOUN
ejpam-5367	293	46	)	)	PUNCT
ejpam-5367	293	47	)	)	PUNCT
ejpam-5367	293	48	,	,	PUNCT
ejpam-5367	293	49	αfg	αfg	NOUN
ejpam-5367	293	50	(	(	PUNCT
ejpam-5367	293	51	y	y	NOUN
ejpam-5367	293	52	)	)	PUNCT
ejpam-5367	293	53	}	}	PUNCT
ejpam-5367	293	54	=	=	SYM
ejpam-5367	293	55	1	1	NUM
ejpam-5367	293	56	,	,	PUNCT
ejpam-5367	293	57	that	that	ADV
ejpam-5367	293	58	is	is	ADV
ejpam-5367	293	59	,	,	PUNCT
ejpam-5367	293	60	αfg	αfg	PROPN
ejpam-5367	293	61	(	(	PUNCT
ejpam-5367	293	62	x	x	X
ejpam-5367	293	63	·	·	PUNCT
ejpam-5367	293	64	z	z	X
ejpam-5367	294	1	)	)	PUNCT
ejpam-5367	294	2	=	=	SYM
ejpam-5367	294	3	1	1	X
ejpam-5367	294	4	.	.	PUNCT
ejpam-5367	295	1	hence	hence	ADV
ejpam-5367	295	2	,	,	PUNCT
ejpam-5367	295	3	x	x	X
ejpam-5367	295	4	·	·	PUNCT
ejpam-5367	295	5	z	z	X
ejpam-5367	295	6	∈	∈	PROPN
ejpam-5367	295	7	g	g	NOUN
ejpam-5367	295	8	,	,	PUNCT
ejpam-5367	295	9	so	so	SCONJ
ejpam-5367	295	10	g	g	PROPN
ejpam-5367	295	11	is	be	AUX
ejpam-5367	295	12	an	an	DET
ejpam-5367	295	13	iup	iup	NOUN
ejpam-5367	295	14	-	-	PUNCT
ejpam-5367	295	15	ideal	ideal	NOUN
ejpam-5367	295	16	of	of	ADP
ejpam-5367	295	17	x.	x.	PROPN
ejpam-5367	295	18	theorem	theorem	VERB
ejpam-5367	295	19	10	10	NUM
ejpam-5367	295	20	.	.	PUNCT
ejpam-5367	296	1	a	a	DET
ejpam-5367	296	2	non	non	ADJ
ejpam-5367	296	3	-	-	ADJ
ejpam-5367	296	4	empty	empty	ADJ
ejpam-5367	296	5	subset	subset	NOUN
ejpam-5367	296	6	g	g	NOUN
ejpam-5367	296	7	of	of	ADP
ejpam-5367	296	8	x	x	X
ejpam-5367	296	9	is	be	AUX
ejpam-5367	296	10	an	an	DET
ejpam-5367	296	11	iup	iup	NOUN
ejpam-5367	296	12	-	-	PUNCT
ejpam-5367	296	13	filter	filter	NOUN
ejpam-5367	296	14	of	of	ADP
ejpam-5367	296	15	x	x	SYM
ejpam-5367	296	16	if	if	SCONJ
ejpam-5367	296	17	and	and	CCONJ
ejpam-5367	297	1	only	only	ADV
ejpam-5367	297	2	if	if	SCONJ
ejpam-5367	297	3	the	the	DET
ejpam-5367	297	4	characteristic	characteristic	ADJ
ejpam-5367	297	5	ffs	ffs	PROPN
ejpam-5367	297	6	fg	fg	PROPN
ejpam-5367	297	7	is	be	AUX
ejpam-5367	297	8	an	an	DET
ejpam-5367	297	9	ffiup	ffiup	ADJ
ejpam-5367	297	10	-	-	PUNCT
ejpam-5367	297	11	filter	filter	NOUN
ejpam-5367	297	12	of	of	ADP
ejpam-5367	297	13	x.	x.	NOUN
ejpam-5367	297	14	proof	proof	PROPN
ejpam-5367	297	15	.	.	PUNCT
ejpam-5367	298	1	assume	assume	VERB
ejpam-5367	298	2	that	that	SCONJ
ejpam-5367	298	3	g	g	PROPN
ejpam-5367	298	4	is	be	AUX
ejpam-5367	298	5	an	an	DET
ejpam-5367	298	6	iup	iup	NOUN
ejpam-5367	298	7	-	-	PUNCT
ejpam-5367	298	8	filter	filter	NOUN
ejpam-5367	298	9	of	of	ADP
ejpam-5367	298	10	x.	x.	NOUN
ejpam-5367	298	11	since	since	SCONJ
ejpam-5367	298	12	0	0	NUM
ejpam-5367	298	13	∈	∈	PROPN
ejpam-5367	298	14	g	g	NOUN
ejpam-5367	298	15	,	,	PUNCT
ejpam-5367	298	16	it	it	PRON
ejpam-5367	298	17	follows	follow	VERB
ejpam-5367	298	18	from	from	ADP
ejpam-5367	298	19	lemma	lemma	PROPN
ejpam-5367	298	20	2	2	NUM
ejpam-5367	298	21	that	that	SCONJ
ejpam-5367	298	22	αfg	αfg	NOUN
ejpam-5367	298	23	and	and	CCONJ
ejpam-5367	298	24	βfg	βfg	PROPN
ejpam-5367	298	25	satisfy	satisfy	NOUN
ejpam-5367	298	26	(	(	PUNCT
ejpam-5367	298	27	3.4	3.4	NUM
ejpam-5367	298	28	)	)	PUNCT
ejpam-5367	298	29	and	and	CCONJ
ejpam-5367	298	30	(	(	PUNCT
ejpam-5367	298	31	3.5	3.5	NUM
ejpam-5367	298	32	)	)	PUNCT
ejpam-5367	298	33	,	,	PUNCT
ejpam-5367	298	34	respectively	respectively	ADV
ejpam-5367	298	35	.	.	PUNCT
ejpam-5367	299	1	next	next	ADV
ejpam-5367	299	2	,	,	PUNCT
ejpam-5367	299	3	let	let	VERB
ejpam-5367	299	4	x	x	PRON
ejpam-5367	299	5	,	,	PUNCT
ejpam-5367	299	6	y	y	PROPN
ejpam-5367	299	7	∈	∈	PROPN
ejpam-5367	299	8	x.	x.	NOUN
ejpam-5367	299	9	case	case	NOUN
ejpam-5367	299	10	1	1	NUM
ejpam-5367	299	11	:	:	PUNCT
ejpam-5367	299	12	suppose	suppose	VERB
ejpam-5367	299	13	x	x	X
ejpam-5367	299	14	·	·	PUNCT
ejpam-5367	299	15	y	y	PROPN
ejpam-5367	299	16	∈	∈	PROPN
ejpam-5367	299	17	g	g	PROPN
ejpam-5367	299	18	and	and	CCONJ
ejpam-5367	299	19	x	x	PROPN
ejpam-5367	299	20	∈	∈	PROPN
ejpam-5367	299	21	g.	g.	NOUN
ejpam-5367	299	22	since	since	SCONJ
ejpam-5367	299	23	g	g	PROPN
ejpam-5367	299	24	is	be	AUX
ejpam-5367	299	25	an	an	DET
ejpam-5367	299	26	iup	iup	NOUN
ejpam-5367	299	27	-	-	PUNCT
ejpam-5367	299	28	filter	filter	NOUN
ejpam-5367	299	29	of	of	ADP
ejpam-5367	299	30	x	x	PRON
ejpam-5367	299	31	,	,	PUNCT
ejpam-5367	299	32	we	we	PRON
ejpam-5367	299	33	have	have	VERB
ejpam-5367	299	34	y	y	PROPN
ejpam-5367	299	35	∈	∈	PROPN
ejpam-5367	299	36	g.	g.	PROPN
ejpam-5367	299	37	thus	thus	ADV
ejpam-5367	299	38	,	,	PUNCT
ejpam-5367	299	39	αfg	αfg	PROPN
ejpam-5367	299	40	(	(	PUNCT
ejpam-5367	299	41	y	y	NOUN
ejpam-5367	299	42	)	)	PUNCT
ejpam-5367	299	43	=	=	SYM
ejpam-5367	299	44	1	1	NUM
ejpam-5367	299	45	≥	≥	NOUN
ejpam-5367	299	46	1	1	NUM
ejpam-5367	299	47	=	=	SYM
ejpam-5367	299	48	min{1	min{1	PROPN
ejpam-5367	299	49	,	,	PUNCT
ejpam-5367	299	50	1	1	NUM
ejpam-5367	299	51	}	}	PUNCT
ejpam-5367	299	52	=	=	SYM
ejpam-5367	299	53	min{αfg	min{αfg	NOUN
ejpam-5367	299	54	(	(	PUNCT
ejpam-5367	299	55	x	x	PROPN
ejpam-5367	299	56	·	·	PUNCT
ejpam-5367	299	57	y	y	X
ejpam-5367	299	58	)	)	PUNCT
ejpam-5367	299	59	,	,	PUNCT
ejpam-5367	299	60	αfg	αfg	NOUN
ejpam-5367	299	61	(	(	PUNCT
ejpam-5367	299	62	x	x	NOUN
ejpam-5367	299	63	)	)	PUNCT
ejpam-5367	299	64	}	}	PUNCT
ejpam-5367	299	65	.	.	PUNCT
ejpam-5367	300	1	case	case	NOUN
ejpam-5367	300	2	2	2	NUM
ejpam-5367	300	3	:	:	PUNCT
ejpam-5367	300	4	suppose	suppose	VERB
ejpam-5367	300	5	x	x	X
ejpam-5367	300	6	·	·	PUNCT
ejpam-5367	300	7	y	y	X
ejpam-5367	300	8	/∈	/∈	PUNCT
ejpam-5367	301	1	g	g	NOUN
ejpam-5367	301	2	or	or	CCONJ
ejpam-5367	301	3	x	x	PROPN
ejpam-5367	301	4	/∈	/∈	PROPN
ejpam-5367	302	1	g.	g.	PROPN
ejpam-5367	302	2	then	then	ADV
ejpam-5367	302	3	αfg	αfg	PROPN
ejpam-5367	302	4	(	(	PUNCT
ejpam-5367	302	5	x	x	SYM
ejpam-5367	302	6	·	·	PUNCT
ejpam-5367	302	7	y	y	X
ejpam-5367	302	8	)	)	PUNCT
ejpam-5367	302	9	=	=	SYM
ejpam-5367	302	10	0	0	NUM
ejpam-5367	302	11	or	or	CCONJ
ejpam-5367	302	12	αfg	αfg	NOUN
ejpam-5367	302	13	(	(	PUNCT
ejpam-5367	302	14	x	x	NOUN
ejpam-5367	302	15	)	)	PUNCT
ejpam-5367	302	16	=	=	SYM
ejpam-5367	302	17	0	0	X
ejpam-5367	302	18	.	.	PUNCT
ejpam-5367	303	1	thus	thus	ADV
ejpam-5367	303	2	,	,	PUNCT
ejpam-5367	303	3	αfg	αfg	PROPN
ejpam-5367	303	4	(	(	PUNCT
ejpam-5367	303	5	y	y	NOUN
ejpam-5367	303	6	)	)	PUNCT
ejpam-5367	303	7	≥	≥	NOUN
ejpam-5367	303	8	0	0	NUM
ejpam-5367	303	9	=	=	SYM
ejpam-5367	303	10	min{αfg	min{αfg	NOUN
ejpam-5367	303	11	(	(	PUNCT
ejpam-5367	303	12	x	x	PROPN
ejpam-5367	303	13	·	·	PUNCT
ejpam-5367	303	14	y	y	X
ejpam-5367	303	15	)	)	PUNCT
ejpam-5367	303	16	,	,	PUNCT
ejpam-5367	303	17	αfg	αfg	NOUN
ejpam-5367	303	18	(	(	PUNCT
ejpam-5367	303	19	x	x	NOUN
ejpam-5367	303	20	)	)	PUNCT
ejpam-5367	303	21	}	}	PUNCT
ejpam-5367	303	22	.	.	PUNCT
ejpam-5367	304	1	a.	a.	NOUN
ejpam-5367	304	2	iampan	iampan	PROPN
ejpam-5367	304	3	et	et	PROPN
ejpam-5367	304	4	al	al	PROPN
ejpam-5367	304	5	.	.	PUNCT
ejpam-5367	304	6	/	/	SYM
ejpam-5367	304	7	eur	eur	PROPN
ejpam-5367	304	8	.	.	PUNCT
ejpam-5367	305	1	j.	j.	PROPN
ejpam-5367	305	2	pure	pure	PROPN
ejpam-5367	305	3	appl	appl	PROPN
ejpam-5367	305	4	.	.	PROPN
ejpam-5367	305	5	math	math	PROPN
ejpam-5367	305	6	,	,	PUNCT
ejpam-5367	305	7	17	17	NUM
ejpam-5367	305	8	(	(	PUNCT
ejpam-5367	305	9	4	4	NUM
ejpam-5367	305	10	)	)	PUNCT
ejpam-5367	305	11	(	(	PUNCT
ejpam-5367	305	12	2024	2024	NUM
ejpam-5367	305	13	)	)	PUNCT
ejpam-5367	305	14	,	,	PUNCT
ejpam-5367	305	15	3022	3022	NUM
ejpam-5367	305	16	-	-	SYM
ejpam-5367	305	17	3042	3042	NUM
ejpam-5367	305	18	3034	3034	NUM
ejpam-5367	305	19	case	case	NOUN
ejpam-5367	305	20	1	1	NUM
ejpam-5367	305	21	’	'	PUNCT
ejpam-5367	305	22	:	:	PUNCT
ejpam-5367	305	23	suppose	suppose	VERB
ejpam-5367	305	24	x	x	X
ejpam-5367	305	25	·	·	PUNCT
ejpam-5367	305	26	y	y	PROPN
ejpam-5367	305	27	∈	∈	PROPN
ejpam-5367	305	28	g	g	PROPN
ejpam-5367	305	29	and	and	CCONJ
ejpam-5367	305	30	x	x	PROPN
ejpam-5367	305	31	∈	∈	PROPN
ejpam-5367	305	32	g.	g.	NOUN
ejpam-5367	305	33	since	since	SCONJ
ejpam-5367	305	34	g	g	PROPN
ejpam-5367	305	35	is	be	AUX
ejpam-5367	305	36	an	an	DET
ejpam-5367	305	37	iup	iup	NOUN
ejpam-5367	305	38	-	-	PUNCT
ejpam-5367	305	39	filter	filter	NOUN
ejpam-5367	305	40	of	of	ADP
ejpam-5367	305	41	x	x	PRON
ejpam-5367	305	42	,	,	PUNCT
ejpam-5367	305	43	we	we	PRON
ejpam-5367	305	44	have	have	VERB
ejpam-5367	305	45	y	y	PROPN
ejpam-5367	305	46	∈	∈	PROPN
ejpam-5367	305	47	g.	g.	PROPN
ejpam-5367	306	1	thus	thus	ADV
ejpam-5367	306	2	,	,	PUNCT
ejpam-5367	306	3	βfg	βfg	PROPN
ejpam-5367	306	4	(	(	PUNCT
ejpam-5367	306	5	y	y	X
ejpam-5367	306	6	)	)	PUNCT
ejpam-5367	306	7	=	=	SYM
ejpam-5367	306	8	0	0	NUM
ejpam-5367	306	9	≤	≤	NUM
ejpam-5367	306	10	0	0	NUM
ejpam-5367	306	11	=	=	SYM
ejpam-5367	306	12	max{0	max{0	PROPN
ejpam-5367	306	13	,	,	PUNCT
ejpam-5367	306	14	0	0	NUM
ejpam-5367	306	15	}	}	PUNCT
ejpam-5367	306	16	=	=	SYM
ejpam-5367	306	17	max{βfg	max{βfg	NOUN
ejpam-5367	306	18	(	(	PUNCT
ejpam-5367	306	19	x	x	PROPN
ejpam-5367	306	20	·	·	PUNCT
ejpam-5367	306	21	y	y	X
ejpam-5367	306	22	)	)	PUNCT
ejpam-5367	306	23	,	,	PUNCT
ejpam-5367	306	24	βfg	βfg	PROPN
ejpam-5367	306	25	(	(	PUNCT
ejpam-5367	306	26	x	x	X
ejpam-5367	306	27	)	)	PUNCT
ejpam-5367	306	28	}	}	PUNCT
ejpam-5367	306	29	.	.	PUNCT
ejpam-5367	307	1	case	case	NOUN
ejpam-5367	307	2	2	2	NUM
ejpam-5367	307	3	’	'	PUNCT
ejpam-5367	307	4	:	:	PUNCT
ejpam-5367	307	5	suppose	suppose	VERB
ejpam-5367	307	6	x	x	X
ejpam-5367	307	7	·	·	PUNCT
ejpam-5367	307	8	y	y	X
ejpam-5367	307	9	/∈	/∈	PUNCT
ejpam-5367	307	10	g	g	NOUN
ejpam-5367	307	11	or	or	CCONJ
ejpam-5367	307	12	x	x	PROPN
ejpam-5367	307	13	/∈	/∈	PROPN
ejpam-5367	308	1	g.	g.	PROPN
ejpam-5367	309	1	then	then	ADV
ejpam-5367	309	2	βfg	βfg	PROPN
ejpam-5367	309	3	(	(	PUNCT
ejpam-5367	309	4	x	x	X
ejpam-5367	309	5	·	·	PUNCT
ejpam-5367	309	6	y	y	X
ejpam-5367	309	7	)	)	PUNCT
ejpam-5367	309	8	=	=	SYM
ejpam-5367	309	9	1	1	NUM
ejpam-5367	309	10	or	or	CCONJ
ejpam-5367	309	11	βfg	βfg	PROPN
ejpam-5367	309	12	(	(	PUNCT
ejpam-5367	309	13	x	x	X
ejpam-5367	309	14	)	)	PUNCT
ejpam-5367	309	15	=	=	SYM
ejpam-5367	309	16	1	1	X
ejpam-5367	309	17	.	.	PUNCT
ejpam-5367	310	1	thus	thus	ADV
ejpam-5367	310	2	,	,	PUNCT
ejpam-5367	310	3	βfg	βfg	PROPN
ejpam-5367	310	4	(	(	PUNCT
ejpam-5367	310	5	y	y	NOUN
ejpam-5367	310	6	)	)	PUNCT
ejpam-5367	310	7	≤	≤	NOUN
ejpam-5367	310	8	1	1	NUM
ejpam-5367	310	9	=	=	SYM
ejpam-5367	310	10	max{βfg	max{βfg	NOUN
ejpam-5367	310	11	(	(	PUNCT
ejpam-5367	310	12	x	x	PROPN
ejpam-5367	310	13	·	·	PUNCT
ejpam-5367	310	14	y	y	X
ejpam-5367	310	15	)	)	PUNCT
ejpam-5367	310	16	,	,	PUNCT
ejpam-5367	310	17	βfg	βfg	PROPN
ejpam-5367	310	18	(	(	PUNCT
ejpam-5367	310	19	x	x	X
ejpam-5367	310	20	)	)	PUNCT
ejpam-5367	310	21	}	}	PUNCT
ejpam-5367	310	22	.	.	PUNCT
ejpam-5367	311	1	hence	hence	ADV
ejpam-5367	311	2	,	,	PUNCT
ejpam-5367	311	3	fg	fg	PROPN
ejpam-5367	311	4	is	be	AUX
ejpam-5367	311	5	an	an	DET
ejpam-5367	311	6	ffiup	ffiup	ADJ
ejpam-5367	311	7	-	-	PUNCT
ejpam-5367	311	8	filter	filter	NOUN
ejpam-5367	311	9	of	of	ADP
ejpam-5367	311	10	x.	x.	NOUN
ejpam-5367	311	11	conversely	conversely	ADV
ejpam-5367	311	12	,	,	PUNCT
ejpam-5367	311	13	assume	assume	VERB
ejpam-5367	311	14	that	that	SCONJ
ejpam-5367	311	15	the	the	DET
ejpam-5367	311	16	characteristic	characteristic	ADJ
ejpam-5367	311	17	ffs	ffs	PROPN
ejpam-5367	312	1	fg	fg	PROPN
ejpam-5367	312	2	is	be	AUX
ejpam-5367	312	3	an	an	DET
ejpam-5367	312	4	ffiup	ffiup	ADJ
ejpam-5367	312	5	-	-	PUNCT
ejpam-5367	312	6	filter	filter	NOUN
ejpam-5367	312	7	of	of	ADP
ejpam-5367	312	8	x.	x.	NOUN
ejpam-5367	312	9	since	since	SCONJ
ejpam-5367	312	10	αfg	αfg	ADJ
ejpam-5367	312	11	satisfies	satisfie	NOUN
ejpam-5367	312	12	(	(	PUNCT
ejpam-5367	312	13	3.4	3.4	NUM
ejpam-5367	312	14	)	)	PUNCT
ejpam-5367	312	15	,	,	PUNCT
ejpam-5367	312	16	it	it	PRON
ejpam-5367	312	17	follows	follow	VERB
ejpam-5367	312	18	from	from	ADP
ejpam-5367	312	19	lemma	lemma	PROPN
ejpam-5367	312	20	2	2	NUM
ejpam-5367	312	21	that	that	SCONJ
ejpam-5367	312	22	0	0	NUM
ejpam-5367	312	23	∈	∈	NOUN
ejpam-5367	312	24	g.	g.	NOUN
ejpam-5367	312	25	next	next	ADV
ejpam-5367	312	26	,	,	PUNCT
ejpam-5367	312	27	let	let	VERB
ejpam-5367	312	28	x	x	PRON
ejpam-5367	312	29	,	,	PUNCT
ejpam-5367	312	30	y	y	PROPN
ejpam-5367	312	31	∈	∈	PROPN
ejpam-5367	312	32	g	g	PROPN
ejpam-5367	312	33	be	be	VERB
ejpam-5367	312	34	such	such	ADJ
ejpam-5367	312	35	that	that	SCONJ
ejpam-5367	312	36	x·y	x·y	PROPN
ejpam-5367	312	37	∈	∈	PROPN
ejpam-5367	312	38	g	g	PROPN
ejpam-5367	312	39	and	and	CCONJ
ejpam-5367	312	40	x	x	PROPN
ejpam-5367	312	41	∈	∈	PROPN
ejpam-5367	312	42	g.	g.	NOUN
ejpam-5367	312	43	then	then	ADV
ejpam-5367	312	44	αfg	αfg	PROPN
ejpam-5367	312	45	(	(	PUNCT
ejpam-5367	312	46	x	x	SYM
ejpam-5367	312	47	·	·	PUNCT
ejpam-5367	312	48	y	y	X
ejpam-5367	312	49	)	)	PUNCT
ejpam-5367	312	50	=	=	SYM
ejpam-5367	312	51	1	1	NUM
ejpam-5367	312	52	and	and	CCONJ
ejpam-5367	312	53	αfg	αfg	NOUN
ejpam-5367	312	54	(	(	PUNCT
ejpam-5367	312	55	x	x	NOUN
ejpam-5367	312	56	)	)	PUNCT
ejpam-5367	312	57	=	=	SYM
ejpam-5367	312	58	1	1	X
ejpam-5367	312	59	.	.	PUNCT
ejpam-5367	313	1	thus	thus	ADV
ejpam-5367	313	2	,	,	PUNCT
ejpam-5367	313	3	min{αfg	min{αfg	PROPN
ejpam-5367	313	4	(	(	PUNCT
ejpam-5367	313	5	x	x	PROPN
ejpam-5367	313	6	·	·	PUNCT
ejpam-5367	313	7	y	y	X
ejpam-5367	313	8	)	)	PUNCT
ejpam-5367	313	9	,	,	PUNCT
ejpam-5367	313	10	αfg	αfg	NOUN
ejpam-5367	313	11	(	(	PUNCT
ejpam-5367	313	12	x	x	NOUN
ejpam-5367	313	13	)	)	PUNCT
ejpam-5367	313	14	}	}	PUNCT
ejpam-5367	313	15	=	=	SYM
ejpam-5367	313	16	1	1	X
ejpam-5367	313	17	.	.	PUNCT
ejpam-5367	314	1	by	by	ADP
ejpam-5367	314	2	(	(	PUNCT
ejpam-5367	314	3	3.8	3.8	NUM
ejpam-5367	314	4	)	)	PUNCT
ejpam-5367	314	5	,	,	PUNCT
ejpam-5367	314	6	we	we	PRON
ejpam-5367	314	7	have	have	VERB
ejpam-5367	314	8	αfg	αfg	NOUN
ejpam-5367	314	9	(	(	PUNCT
ejpam-5367	314	10	y	y	NOUN
ejpam-5367	314	11	)	)	PUNCT
ejpam-5367	314	12	=	=	SYM
ejpam-5367	314	13	min{αfg	min{αfg	NOUN
ejpam-5367	314	14	(	(	PUNCT
ejpam-5367	314	15	x	x	PROPN
ejpam-5367	314	16	·	·	PUNCT
ejpam-5367	314	17	y	y	X
ejpam-5367	314	18	)	)	PUNCT
ejpam-5367	314	19	,	,	PUNCT
ejpam-5367	314	20	αfg	αfg	NOUN
ejpam-5367	314	21	(	(	PUNCT
ejpam-5367	314	22	x	x	NOUN
ejpam-5367	314	23	)	)	PUNCT
ejpam-5367	314	24	}	}	PUNCT
ejpam-5367	314	25	=	=	SYM
ejpam-5367	314	26	1	1	NUM
ejpam-5367	314	27	,	,	PUNCT
ejpam-5367	314	28	that	that	ADV
ejpam-5367	314	29	is	is	ADV
ejpam-5367	314	30	,	,	PUNCT
ejpam-5367	314	31	αfg	αfg	PROPN
ejpam-5367	314	32	(	(	PUNCT
ejpam-5367	314	33	y	y	NOUN
ejpam-5367	314	34	)	)	PUNCT
ejpam-5367	314	35	=	=	SYM
ejpam-5367	315	1	1	1	X
ejpam-5367	315	2	.	.	PUNCT
ejpam-5367	316	1	hence	hence	ADV
ejpam-5367	316	2	,	,	PUNCT
ejpam-5367	316	3	y	y	PROPN
ejpam-5367	316	4	∈	∈	PROPN
ejpam-5367	316	5	g	g	PROPN
ejpam-5367	316	6	,	,	PUNCT
ejpam-5367	316	7	so	so	SCONJ
ejpam-5367	316	8	g	g	PROPN
ejpam-5367	316	9	is	be	AUX
ejpam-5367	316	10	an	an	DET
ejpam-5367	316	11	iup	iup	NOUN
ejpam-5367	316	12	-	-	PUNCT
ejpam-5367	316	13	filter	filter	NOUN
ejpam-5367	316	14	of	of	ADP
ejpam-5367	316	15	x.	x.	NOUN
ejpam-5367	316	16	the	the	DET
ejpam-5367	316	17	following	follow	VERB
ejpam-5367	316	18	theorem	theorem	NOUN
ejpam-5367	316	19	is	be	AUX
ejpam-5367	316	20	a	a	DET
ejpam-5367	316	21	direct	direct	ADJ
ejpam-5367	316	22	consequence	consequence	NOUN
ejpam-5367	316	23	of	of	ADP
ejpam-5367	316	24	theorem	theorem	ADJ
ejpam-5367	316	25	2	2	NUM
ejpam-5367	316	26	.	.	PUNCT
ejpam-5367	316	27	theorem	theorem	VERB
ejpam-5367	316	28	11	11	NUM
ejpam-5367	316	29	.	.	PUNCT
ejpam-5367	317	1	a	a	DET
ejpam-5367	317	2	non	non	ADJ
ejpam-5367	317	3	-	-	ADJ
ejpam-5367	317	4	empty	empty	ADJ
ejpam-5367	317	5	subset	subset	NOUN
ejpam-5367	317	6	g	g	NOUN
ejpam-5367	317	7	of	of	ADP
ejpam-5367	317	8	x	x	X
ejpam-5367	317	9	is	be	AUX
ejpam-5367	317	10	a	a	DET
ejpam-5367	317	11	strong	strong	ADJ
ejpam-5367	317	12	iup	iup	NOUN
ejpam-5367	317	13	-	-	PUNCT
ejpam-5367	317	14	ideal	ideal	NOUN
ejpam-5367	317	15	of	of	ADP
ejpam-5367	317	16	x	x	SYM
ejpam-5367	317	17	if	if	SCONJ
ejpam-5367	317	18	and	and	CCONJ
ejpam-5367	318	1	only	only	ADV
ejpam-5367	318	2	if	if	SCONJ
ejpam-5367	318	3	the	the	DET
ejpam-5367	318	4	characteristic	characteristic	ADJ
ejpam-5367	318	5	ffs	ffs	PROPN
ejpam-5367	318	6	fg	fg	PROPN
ejpam-5367	318	7	is	be	AUX
ejpam-5367	318	8	an	an	DET
ejpam-5367	318	9	ffsiup	ffsiup	NOUN
ejpam-5367	318	10	-	-	PUNCT
ejpam-5367	318	11	ideal	ideal	NOUN
ejpam-5367	318	12	of	of	ADP
ejpam-5367	318	13	x.	x.	PROPN
ejpam-5367	318	14	lemma	lemma	PROPN
ejpam-5367	319	1	3	3	X
ejpam-5367	319	2	.	.	PUNCT
ejpam-5367	320	1	[	[	X
ejpam-5367	320	2	13	13	NUM
ejpam-5367	320	3	]	]	PUNCT
ejpam-5367	320	4	let	let	VERB
ejpam-5367	320	5	f	f	PRON
ejpam-5367	320	6	be	be	AUX
ejpam-5367	320	7	an	an	DET
ejpam-5367	320	8	fs	fs	NOUN
ejpam-5367	320	9	in	in	ADP
ejpam-5367	320	10	a	a	DET
ejpam-5367	320	11	non	non	ADJ
ejpam-5367	320	12	-	-	ADJ
ejpam-5367	320	13	empty	empty	ADJ
ejpam-5367	320	14	set	set	NOUN
ejpam-5367	320	15	x.	x.	NOUN
ejpam-5367	320	16	then	then	ADV
ejpam-5367	320	17	the	the	DET
ejpam-5367	320	18	following	follow	VERB
ejpam-5367	320	19	statements	statement	NOUN
ejpam-5367	320	20	hold	hold	VERB
ejpam-5367	320	21	:	:	PUNCT
ejpam-5367	320	22	(	(	PUNCT
ejpam-5367	320	23	∀x	∀x	X
ejpam-5367	320	24	,	,	PUNCT
ejpam-5367	320	25	y	y	PROPN
ejpam-5367	320	26	∈	∈	PROPN
ejpam-5367	320	27	x)(1−max{f(x	x)(1−max{f(x	PROPN
ejpam-5367	320	28	)	)	PUNCT
ejpam-5367	320	29	,	,	PUNCT
ejpam-5367	320	30	f(y	f(y	NOUN
ejpam-5367	320	31	)	)	PUNCT
ejpam-5367	320	32	}	}	PUNCT
ejpam-5367	320	33	=	=	PUNCT
ejpam-5367	321	1	min{1−	min{1−	VERB
ejpam-5367	321	2	f(x	f(x	PROPN
ejpam-5367	321	3	)	)	PUNCT
ejpam-5367	321	4	,	,	PUNCT
ejpam-5367	321	5	1−	1−	NUM
ejpam-5367	321	6	f(y	f(y	NOUN
ejpam-5367	321	7	)	)	PUNCT
ejpam-5367	321	8	}	}	PUNCT
ejpam-5367	321	9	)	)	PUNCT
ejpam-5367	322	1	(	(	PUNCT
ejpam-5367	322	2	3.13	3.13	NUM
ejpam-5367	322	3	)	)	PUNCT
ejpam-5367	322	4	(	(	PUNCT
ejpam-5367	322	5	∀x	∀x	X
ejpam-5367	322	6	,	,	PUNCT
ejpam-5367	322	7	y	y	PROPN
ejpam-5367	322	8	∈	∈	PROPN
ejpam-5367	322	9	x)(1−min{f(x	x)(1−min{f(x	PROPN
ejpam-5367	322	10	)	)	PUNCT
ejpam-5367	322	11	,	,	PUNCT
ejpam-5367	322	12	f(y	f(y	NOUN
ejpam-5367	322	13	)	)	PUNCT
ejpam-5367	322	14	}	}	PUNCT
ejpam-5367	322	15	=	=	PUNCT
ejpam-5367	323	1	max{1−	max{1−	PROPN
ejpam-5367	323	2	f(x	f(x	PROPN
ejpam-5367	323	3	)	)	PUNCT
ejpam-5367	323	4	,	,	PUNCT
ejpam-5367	323	5	1−	1−	NUM
ejpam-5367	323	6	f(y	f(y	NOUN
ejpam-5367	323	7	)	)	PUNCT
ejpam-5367	323	8	}	}	PUNCT
ejpam-5367	323	9	)	)	PUNCT
ejpam-5367	323	10	(	(	PUNCT
ejpam-5367	323	11	3.14	3.14	X
ejpam-5367	323	12	)	)	PUNCT
ejpam-5367	323	13	lemma	lemma	PROPN
ejpam-5367	323	14	4	4	NUM
ejpam-5367	323	15	.	.	PUNCT
ejpam-5367	324	1	[	[	X
ejpam-5367	324	2	13	13	NUM
ejpam-5367	324	3	]	]	PUNCT
ejpam-5367	324	4	let	let	VERB
ejpam-5367	324	5	f	f	PRON
ejpam-5367	324	6	be	be	AUX
ejpam-5367	324	7	an	an	DET
ejpam-5367	324	8	fs	fs	NOUN
ejpam-5367	324	9	in	in	ADP
ejpam-5367	324	10	a	a	DET
ejpam-5367	324	11	non	non	ADJ
ejpam-5367	324	12	-	-	ADJ
ejpam-5367	324	13	empty	empty	ADJ
ejpam-5367	324	14	set	set	NOUN
ejpam-5367	324	15	x.	x.	NOUN
ejpam-5367	324	16	then	then	ADV
ejpam-5367	324	17	the	the	DET
ejpam-5367	324	18	following	follow	VERB
ejpam-5367	324	19	statements	statement	NOUN
ejpam-5367	324	20	hold	hold	VERB
ejpam-5367	324	21	:	:	PUNCT
ejpam-5367	324	22	(	(	PUNCT
ejpam-5367	324	23	∀x	∀x	X
ejpam-5367	324	24	,	,	PUNCT
ejpam-5367	324	25	y	y	PROPN
ejpam-5367	324	26	,	,	PUNCT
ejpam-5367	324	27	z	z	PROPN
ejpam-5367	324	28	∈	∈	PROPN
ejpam-5367	324	29	x)(f(z	x)(f(z	PROPN
ejpam-5367	324	30	)	)	PUNCT
ejpam-5367	324	31	≥	≥	NOUN
ejpam-5367	324	32	min{f(x	min{f(x	NOUN
ejpam-5367	324	33	)	)	PUNCT
ejpam-5367	324	34	,	,	PUNCT
ejpam-5367	324	35	f(y	f(y	NOUN
ejpam-5367	324	36	)	)	PUNCT
ejpam-5367	324	37	}	}	PUNCT
ejpam-5367	324	38	⇔	⇔	PROPN
ejpam-5367	324	39	f(z	f(z	PROPN
ejpam-5367	324	40	)	)	PUNCT
ejpam-5367	324	41	≤	≤	NUM
ejpam-5367	324	42	max{f(x	max{f(x	PROPN
ejpam-5367	324	43	)	)	PUNCT
ejpam-5367	324	44	,	,	PUNCT
ejpam-5367	324	45	f(y	f(y	NOUN
ejpam-5367	324	46	)	)	PUNCT
ejpam-5367	324	47	}	}	PUNCT
ejpam-5367	324	48	)	)	PUNCT
ejpam-5367	324	49	(	(	PUNCT
ejpam-5367	324	50	3.15	3.15	NUM
ejpam-5367	324	51	)	)	PUNCT
ejpam-5367	324	52	(	(	PUNCT
ejpam-5367	324	53	∀x	∀x	X
ejpam-5367	324	54	,	,	PUNCT
ejpam-5367	324	55	y	y	PROPN
ejpam-5367	324	56	,	,	PUNCT
ejpam-5367	324	57	z	z	PROPN
ejpam-5367	324	58	∈	∈	PROPN
ejpam-5367	324	59	x)(f(z	x)(f(z	PROPN
ejpam-5367	324	60	)	)	PUNCT
ejpam-5367	324	61	≤	≤	NUM
ejpam-5367	325	1	max{f(x	max{f(x	PROPN
ejpam-5367	325	2	)	)	PUNCT
ejpam-5367	325	3	,	,	PUNCT
ejpam-5367	325	4	f(y	f(y	NOUN
ejpam-5367	325	5	)	)	PUNCT
ejpam-5367	325	6	}	}	PUNCT
ejpam-5367	325	7	⇔	⇔	PROPN
ejpam-5367	325	8	f(z	f(z	PROPN
ejpam-5367	325	9	)	)	PUNCT
ejpam-5367	325	10	≥	≥	NOUN
ejpam-5367	325	11	min{f(x	min{f(x	NOUN
ejpam-5367	325	12	)	)	PUNCT
ejpam-5367	325	13	,	,	PUNCT
ejpam-5367	325	14	f(y	f(y	NOUN
ejpam-5367	325	15	)	)	PUNCT
ejpam-5367	325	16	}	}	PUNCT
ejpam-5367	325	17	)	)	PUNCT
ejpam-5367	325	18	(	(	PUNCT
ejpam-5367	325	19	3.16	3.16	NUM
ejpam-5367	325	20	)	)	PUNCT
ejpam-5367	325	21	before	before	ADP
ejpam-5367	325	22	presenting	present	VERB
ejpam-5367	325	23	theorems	theorem	NOUN
ejpam-5367	325	24	on	on	ADP
ejpam-5367	325	25	the	the	DET
ejpam-5367	325	26	relationship	relationship	NOUN
ejpam-5367	325	27	between	between	ADP
ejpam-5367	325	28	fsss	fsss	NOUN
ejpam-5367	325	29	and	and	CCONJ
ejpam-5367	325	30	their	their	PRON
ejpam-5367	325	31	complements	complement	NOUN
ejpam-5367	325	32	,	,	PUNCT
ejpam-5367	325	33	it	it	PRON
ejpam-5367	325	34	’s	’	VERB
ejpam-5367	325	35	crucial	crucial	ADJ
ejpam-5367	325	36	to	to	PART
ejpam-5367	325	37	grasp	grasp	VERB
ejpam-5367	325	38	their	their	PRON
ejpam-5367	325	39	basic	basic	ADJ
ejpam-5367	325	40	concept	concept	NOUN
ejpam-5367	325	41	.	.	PUNCT
ejpam-5367	326	1	fsss	fsss	NOUN
ejpam-5367	326	2	extend	extend	VERB
ejpam-5367	326	3	traditional	traditional	ADJ
ejpam-5367	326	4	fss	fss	NOUN
ejpam-5367	326	5	by	by	ADP
ejpam-5367	326	6	incorporating	incorporate	VERB
ejpam-5367	326	7	hesitation	hesitation	NOUN
ejpam-5367	326	8	degrees	degree	NOUN
ejpam-5367	326	9	.	.	PUNCT
ejpam-5367	327	1	the	the	DET
ejpam-5367	327	2	following	follow	VERB
ejpam-5367	327	3	theorem	theorem	ADJ
ejpam-5367	327	4	highlights	highlight	NOUN
ejpam-5367	327	5	the	the	DET
ejpam-5367	327	6	key	key	ADJ
ejpam-5367	327	7	relationship	relationship	NOUN
ejpam-5367	327	8	between	between	ADP
ejpam-5367	327	9	these	these	DET
ejpam-5367	327	10	sets	set	NOUN
ejpam-5367	327	11	and	and	CCONJ
ejpam-5367	327	12	their	their	PRON
ejpam-5367	327	13	complements	complement	NOUN
ejpam-5367	327	14	.	.	PUNCT
ejpam-5367	328	1	theorem	theorem	NOUN
ejpam-5367	328	2	12	12	NUM
ejpam-5367	328	3	.	.	PUNCT
ejpam-5367	329	1	an	an	DET
ejpam-5367	329	2	ffs	ffs	NOUN
ejpam-5367	329	3	f	f	PROPN
ejpam-5367	329	4	is	be	AUX
ejpam-5367	329	5	an	an	DET
ejpam-5367	329	6	ffiup	ffiup	NOUN
ejpam-5367	329	7	-	-	PUNCT
ejpam-5367	329	8	subalgebra	subalgebra	NOUN
ejpam-5367	329	9	of	of	ADP
ejpam-5367	329	10	x	x	PRON
ejpam-5367	329	11	if	if	SCONJ
ejpam-5367	329	12	and	and	CCONJ
ejpam-5367	329	13	only	only	ADV
ejpam-5367	329	14	if	if	SCONJ
ejpam-5367	329	15	the	the	DET
ejpam-5367	329	16	fss	fss	NOUN
ejpam-5367	329	17	αf	αf	VERB
ejpam-5367	329	18	and	and	CCONJ
ejpam-5367	329	19	βf	βf	ADV
ejpam-5367	329	20	satisfy	satisfy	VERB
ejpam-5367	329	21	(	(	PUNCT
ejpam-5367	329	22	3.2	3.2	NUM
ejpam-5367	329	23	)	)	PUNCT
ejpam-5367	329	24	,	,	PUNCT
ejpam-5367	329	25	and	and	CCONJ
ejpam-5367	329	26	the	the	DET
ejpam-5367	329	27	fss	fss	NOUN
ejpam-5367	329	28	αf	αf	VERB
ejpam-5367	329	29	and	and	CCONJ
ejpam-5367	329	30	βf	βf	ADV
ejpam-5367	329	31	satisfy	satisfy	VERB
ejpam-5367	329	32	(	(	PUNCT
ejpam-5367	329	33	3.3	3.3	NUM
ejpam-5367	329	34	)	)	PUNCT
ejpam-5367	329	35	.	.	PUNCT
ejpam-5367	330	1	proof	proof	NOUN
ejpam-5367	330	2	.	.	PUNCT
ejpam-5367	331	1	assume	assume	VERB
ejpam-5367	331	2	that	that	SCONJ
ejpam-5367	331	3	f	f	PROPN
ejpam-5367	331	4	is	be	AUX
ejpam-5367	331	5	an	an	DET
ejpam-5367	331	6	ffiup	ffiup	NOUN
ejpam-5367	331	7	-	-	PUNCT
ejpam-5367	331	8	subalgebra	subalgebra	NOUN
ejpam-5367	331	9	of	of	ADP
ejpam-5367	331	10	x.	x.	NOUN
ejpam-5367	331	11	then	then	ADV
ejpam-5367	331	12	αf	αf	VERB
ejpam-5367	331	13	(	(	PUNCT
ejpam-5367	331	14	x	x	PROPN
ejpam-5367	331	15	·	·	PUNCT
ejpam-5367	331	16	y	y	X
ejpam-5367	331	17	)	)	PUNCT
ejpam-5367	331	18	≥	≥	NOUN
ejpam-5367	331	19	min{αf	min{αf	PUNCT
ejpam-5367	331	20	(	(	PUNCT
ejpam-5367	331	21	x	x	X
ejpam-5367	331	22	)	)	PUNCT
ejpam-5367	331	23	,	,	PUNCT
ejpam-5367	331	24	αf	αf	X
ejpam-5367	331	25	(	(	PUNCT
ejpam-5367	331	26	y	y	NOUN
ejpam-5367	331	27	)	)	PUNCT
ejpam-5367	331	28	}	}	PUNCT
ejpam-5367	331	29	,	,	PUNCT
ejpam-5367	331	30	βf	βf	CCONJ
ejpam-5367	331	31	(	(	PUNCT
ejpam-5367	331	32	x	x	X
ejpam-5367	331	33	·	·	PUNCT
ejpam-5367	331	34	y	y	X
ejpam-5367	331	35	)	)	PUNCT
ejpam-5367	331	36	≤	≤	NUM
ejpam-5367	332	1	max{βf	max{βf	INTJ
ejpam-5367	332	2	(	(	PUNCT
ejpam-5367	332	3	x	x	NOUN
ejpam-5367	332	4	)	)	PUNCT
ejpam-5367	332	5	,	,	PUNCT
ejpam-5367	332	6	βf	βf	CCONJ
ejpam-5367	332	7	(	(	PUNCT
ejpam-5367	332	8	y	y	NOUN
ejpam-5367	332	9	)	)	PUNCT
ejpam-5367	332	10	}	}	PUNCT
ejpam-5367	332	11	.	.	PUNCT
ejpam-5367	333	1	thus	thus	ADV
ejpam-5367	333	2	,	,	PUNCT
ejpam-5367	333	3	αf	αf	ADP
ejpam-5367	333	4	(	(	PUNCT
ejpam-5367	333	5	x	x	PROPN
ejpam-5367	333	6	·	·	PUNCT
ejpam-5367	333	7	y	y	X
ejpam-5367	333	8	)	)	PUNCT
ejpam-5367	333	9	≤	≤	NOUN
ejpam-5367	333	10	max{αf	max{αf	NOUN
ejpam-5367	333	11	(	(	PUNCT
ejpam-5367	333	12	x	x	NOUN
ejpam-5367	333	13	)	)	PUNCT
ejpam-5367	333	14	,	,	PUNCT
ejpam-5367	333	15	αf	αf	X
ejpam-5367	333	16	(	(	PUNCT
ejpam-5367	333	17	y	y	NOUN
ejpam-5367	333	18	)	)	PUNCT
ejpam-5367	333	19	}	}	PUNCT
ejpam-5367	333	20	,	,	PUNCT
ejpam-5367	333	21	(	(	PUNCT
ejpam-5367	333	22	by	by	ADP
ejpam-5367	333	23	(	(	PUNCT
ejpam-5367	333	24	3.15	3.15	NUM
ejpam-5367	333	25	)	)	PUNCT
ejpam-5367	333	26	)	)	PUNCT
ejpam-5367	334	1	βf	βf	X
ejpam-5367	334	2	(	(	PUNCT
ejpam-5367	334	3	x	x	X
ejpam-5367	334	4	·	·	PUNCT
ejpam-5367	334	5	y	y	X
ejpam-5367	334	6	)	)	PUNCT
ejpam-5367	334	7	≥	≥	NOUN
ejpam-5367	334	8	min{βf	min{βf	INTJ
ejpam-5367	334	9	(	(	PUNCT
ejpam-5367	334	10	x	x	NOUN
ejpam-5367	334	11	)	)	PUNCT
ejpam-5367	334	12	,	,	PUNCT
ejpam-5367	334	13	βf	βf	CCONJ
ejpam-5367	334	14	(	(	PUNCT
ejpam-5367	334	15	y	y	NOUN
ejpam-5367	334	16	)	)	PUNCT
ejpam-5367	334	17	}	}	PUNCT
ejpam-5367	334	18	,	,	PUNCT
ejpam-5367	334	19	.	.	PUNCT
ejpam-5367	335	1	(	(	PUNCT
ejpam-5367	335	2	by	by	ADP
ejpam-5367	335	3	(	(	PUNCT
ejpam-5367	335	4	3.16	3.16	NUM
ejpam-5367	335	5	)	)	PUNCT
ejpam-5367	335	6	)	)	PUNCT
ejpam-5367	335	7	a.	a.	NOUN
ejpam-5367	335	8	iampan	iampan	NOUN
ejpam-5367	335	9	et	et	PROPN
ejpam-5367	335	10	al	al	PROPN
ejpam-5367	335	11	.	.	PUNCT
ejpam-5367	335	12	/	/	SYM
ejpam-5367	335	13	eur	eur	PROPN
ejpam-5367	335	14	.	.	PUNCT
ejpam-5367	336	1	j.	j.	PROPN
ejpam-5367	336	2	pure	pure	PROPN
ejpam-5367	336	3	appl	appl	PROPN
ejpam-5367	336	4	.	.	PROPN
ejpam-5367	336	5	math	math	PROPN
ejpam-5367	336	6	,	,	PUNCT
ejpam-5367	336	7	17	17	NUM
ejpam-5367	336	8	(	(	PUNCT
ejpam-5367	336	9	4	4	NUM
ejpam-5367	336	10	)	)	PUNCT
ejpam-5367	336	11	(	(	PUNCT
ejpam-5367	336	12	2024	2024	NUM
ejpam-5367	336	13	)	)	PUNCT
ejpam-5367	336	14	,	,	PUNCT
ejpam-5367	336	15	3022	3022	NUM
ejpam-5367	336	16	-	-	SYM
ejpam-5367	336	17	3042	3042	NUM
ejpam-5367	336	18	3035	3035	NUM
ejpam-5367	336	19	hence	hence	ADV
ejpam-5367	336	20	,	,	PUNCT
ejpam-5367	336	21	the	the	DET
ejpam-5367	336	22	fss	fss	NOUN
ejpam-5367	336	23	αf	αf	VERB
ejpam-5367	336	24	and	and	CCONJ
ejpam-5367	336	25	βf	βf	ADV
ejpam-5367	336	26	satisfy	satisfy	VERB
ejpam-5367	336	27	(	(	PUNCT
ejpam-5367	336	28	3.2	3.2	NUM
ejpam-5367	336	29	)	)	PUNCT
ejpam-5367	336	30	,	,	PUNCT
ejpam-5367	336	31	and	and	CCONJ
ejpam-5367	336	32	the	the	DET
ejpam-5367	336	33	fss	fss	NOUN
ejpam-5367	336	34	αf	αf	VERB
ejpam-5367	337	1	and	and	CCONJ
ejpam-5367	337	2	βf	βf	ADV
ejpam-5367	337	3	satisfy	satisfy	VERB
ejpam-5367	337	4	(	(	PUNCT
ejpam-5367	337	5	3.3	3.3	NUM
ejpam-5367	337	6	)	)	PUNCT
ejpam-5367	337	7	.	.	PUNCT
ejpam-5367	338	1	conversely	conversely	ADV
ejpam-5367	338	2	,	,	PUNCT
ejpam-5367	338	3	assume	assume	VERB
ejpam-5367	338	4	that	that	SCONJ
ejpam-5367	338	5	the	the	DET
ejpam-5367	338	6	fss	fss	NOUN
ejpam-5367	338	7	αf	αf	VERB
ejpam-5367	338	8	and	and	CCONJ
ejpam-5367	338	9	βf	βf	ADV
ejpam-5367	338	10	satisfy	satisfy	VERB
ejpam-5367	338	11	(	(	PUNCT
ejpam-5367	338	12	3.2	3.2	NUM
ejpam-5367	338	13	)	)	PUNCT
ejpam-5367	338	14	,	,	PUNCT
ejpam-5367	338	15	and	and	CCONJ
ejpam-5367	338	16	the	the	DET
ejpam-5367	338	17	fss	fss	NOUN
ejpam-5367	338	18	αf	αf	VERB
ejpam-5367	338	19	and	and	CCONJ
ejpam-5367	338	20	βf	βf	ADV
ejpam-5367	338	21	satisfy	satisfy	VERB
ejpam-5367	338	22	(	(	PUNCT
ejpam-5367	338	23	3.3	3.3	NUM
ejpam-5367	338	24	)	)	PUNCT
ejpam-5367	338	25	.	.	PUNCT
ejpam-5367	339	1	then	then	ADV
ejpam-5367	339	2	αf	αf	VERB
ejpam-5367	339	3	satisfies	satisfie	NOUN
ejpam-5367	339	4	(	(	PUNCT
ejpam-5367	339	5	3.2	3.2	NUM
ejpam-5367	339	6	)	)	PUNCT
ejpam-5367	339	7	,	,	PUNCT
ejpam-5367	339	8	and	and	CCONJ
ejpam-5367	339	9	βf	βf	ADV
ejpam-5367	339	10	satisfies	satisfie	NOUN
ejpam-5367	339	11	(	(	PUNCT
ejpam-5367	339	12	3.3	3.3	NUM
ejpam-5367	339	13	)	)	PUNCT
ejpam-5367	339	14	.	.	PUNCT
ejpam-5367	340	1	hence	hence	ADV
ejpam-5367	340	2	,	,	PUNCT
ejpam-5367	340	3	f	f	PROPN
ejpam-5367	340	4	is	be	AUX
ejpam-5367	340	5	an	an	DET
ejpam-5367	340	6	ffiup	ffiup	NOUN
ejpam-5367	340	7	-	-	PUNCT
ejpam-5367	340	8	subalgebra	subalgebra	NOUN
ejpam-5367	340	9	of	of	ADP
ejpam-5367	340	10	x.	x.	NOUN
ejpam-5367	340	11	theorem	theorem	VERB
ejpam-5367	340	12	13	13	NUM
ejpam-5367	340	13	.	.	PUNCT
ejpam-5367	341	1	an	an	DET
ejpam-5367	341	2	ffs	ffs	NOUN
ejpam-5367	341	3	f	f	PROPN
ejpam-5367	341	4	is	be	AUX
ejpam-5367	341	5	an	an	DET
ejpam-5367	341	6	ffiup	ffiup	NOUN
ejpam-5367	341	7	-	-	PUNCT
ejpam-5367	341	8	ideal	ideal	NOUN
ejpam-5367	341	9	of	of	ADP
ejpam-5367	341	10	x	x	SYM
ejpam-5367	341	11	if	if	SCONJ
ejpam-5367	341	12	and	and	CCONJ
ejpam-5367	341	13	only	only	ADV
ejpam-5367	341	14	if	if	SCONJ
ejpam-5367	341	15	the	the	DET
ejpam-5367	341	16	fss	fss	NOUN
ejpam-5367	341	17	αf	αf	VERB
ejpam-5367	341	18	and	and	CCONJ
ejpam-5367	341	19	βf	βf	ADV
ejpam-5367	341	20	satisfy	satisfy	VERB
ejpam-5367	341	21	(	(	PUNCT
ejpam-5367	341	22	3.4	3.4	NUM
ejpam-5367	341	23	)	)	PUNCT
ejpam-5367	341	24	and	and	CCONJ
ejpam-5367	341	25	(	(	PUNCT
ejpam-5367	341	26	3.6	3.6	NUM
ejpam-5367	341	27	)	)	PUNCT
ejpam-5367	341	28	,	,	PUNCT
ejpam-5367	341	29	and	and	CCONJ
ejpam-5367	341	30	the	the	DET
ejpam-5367	341	31	fss	fss	NOUN
ejpam-5367	341	32	αf	αf	VERB
ejpam-5367	341	33	and	and	CCONJ
ejpam-5367	341	34	βf	βf	ADV
ejpam-5367	341	35	satisfy	satisfy	VERB
ejpam-5367	341	36	(	(	PUNCT
ejpam-5367	341	37	3.5	3.5	NUM
ejpam-5367	341	38	)	)	PUNCT
ejpam-5367	341	39	and	and	CCONJ
ejpam-5367	341	40	(	(	PUNCT
ejpam-5367	341	41	3.7	3.7	NUM
ejpam-5367	341	42	)	)	PUNCT
ejpam-5367	341	43	.	.	PUNCT
ejpam-5367	342	1	proof	proof	NOUN
ejpam-5367	342	2	.	.	PUNCT
ejpam-5367	343	1	assume	assume	VERB
ejpam-5367	343	2	that	that	SCONJ
ejpam-5367	343	3	f	f	PROPN
ejpam-5367	343	4	is	be	AUX
ejpam-5367	343	5	an	an	DET
ejpam-5367	343	6	ffiup	ffiup	NOUN
ejpam-5367	343	7	-	-	PUNCT
ejpam-5367	343	8	ideal	ideal	NOUN
ejpam-5367	343	9	of	of	ADP
ejpam-5367	343	10	x.	x.	NOUN
ejpam-5367	343	11	then	then	ADV
ejpam-5367	343	12	αf	αf	VERB
ejpam-5367	343	13	(	(	PUNCT
ejpam-5367	343	14	0	0	NUM
ejpam-5367	343	15	)	)	PUNCT
ejpam-5367	343	16	≥	≥	NOUN
ejpam-5367	343	17	αf	αf	X
ejpam-5367	343	18	(	(	PUNCT
ejpam-5367	343	19	x	x	NOUN
ejpam-5367	343	20	)	)	PUNCT
ejpam-5367	343	21	,	,	PUNCT
ejpam-5367	343	22	βf	βf	CCONJ
ejpam-5367	343	23	(	(	PUNCT
ejpam-5367	343	24	0	0	X
ejpam-5367	343	25	)	)	PUNCT
ejpam-5367	343	26	≤	≤	NOUN
ejpam-5367	344	1	βf	βf	CCONJ
ejpam-5367	344	2	(	(	PUNCT
ejpam-5367	344	3	x	x	X
ejpam-5367	344	4	)	)	PUNCT
ejpam-5367	344	5	,	,	PUNCT
ejpam-5367	344	6	αf	αf	X
ejpam-5367	344	7	(	(	PUNCT
ejpam-5367	344	8	x	x	X
ejpam-5367	344	9	·	·	PUNCT
ejpam-5367	344	10	z	z	X
ejpam-5367	344	11	)	)	PUNCT
ejpam-5367	344	12	≥	≥	NOUN
ejpam-5367	344	13	min{αf	min{αf	PUNCT
ejpam-5367	344	14	(	(	PUNCT
ejpam-5367	344	15	x	x	X
ejpam-5367	344	16	·	·	PUNCT
ejpam-5367	344	17	(	(	PUNCT
ejpam-5367	344	18	y	y	PROPN
ejpam-5367	344	19	·	·	PUNCT
ejpam-5367	344	20	z	z	NOUN
ejpam-5367	344	21	)	)	PUNCT
ejpam-5367	344	22	)	)	PUNCT
ejpam-5367	344	23	,	,	PUNCT
ejpam-5367	344	24	αf	αf	X
ejpam-5367	344	25	(	(	PUNCT
ejpam-5367	344	26	y	y	NOUN
ejpam-5367	344	27	)	)	PUNCT
ejpam-5367	344	28	}	}	PUNCT
ejpam-5367	344	29	,	,	PUNCT
ejpam-5367	344	30	βf	βf	CCONJ
ejpam-5367	344	31	(	(	PUNCT
ejpam-5367	344	32	x	x	X
ejpam-5367	344	33	·	·	PUNCT
ejpam-5367	344	34	z	z	X
ejpam-5367	344	35	)	)	PUNCT
ejpam-5367	344	36	≤	≤	NUM
ejpam-5367	345	1	max{βf	max{βf	INTJ
ejpam-5367	345	2	(	(	PUNCT
ejpam-5367	345	3	x	x	X
ejpam-5367	345	4	·	·	PUNCT
ejpam-5367	345	5	(	(	PUNCT
ejpam-5367	345	6	y	y	PROPN
ejpam-5367	345	7	·	·	PUNCT
ejpam-5367	345	8	z	z	NOUN
ejpam-5367	345	9	)	)	PUNCT
ejpam-5367	345	10	)	)	PUNCT
ejpam-5367	345	11	,	,	PUNCT
ejpam-5367	345	12	βf	βf	CCONJ
ejpam-5367	345	13	(	(	PUNCT
ejpam-5367	345	14	y	y	NOUN
ejpam-5367	345	15	)	)	PUNCT
ejpam-5367	345	16	}	}	PUNCT
ejpam-5367	345	17	.	.	PUNCT
ejpam-5367	346	1	thus	thus	ADV
ejpam-5367	346	2	,	,	PUNCT
ejpam-5367	346	3	αf	αf	ADP
ejpam-5367	346	4	(	(	PUNCT
ejpam-5367	346	5	0	0	NUM
ejpam-5367	346	6	)	)	PUNCT
ejpam-5367	346	7	≤	≤	NOUN
ejpam-5367	346	8	αf	αf	ADP
ejpam-5367	346	9	(	(	PUNCT
ejpam-5367	346	10	x	x	NOUN
ejpam-5367	346	11	)	)	PUNCT
ejpam-5367	346	12	,	,	PUNCT
ejpam-5367	346	13	βf	βf	CCONJ
ejpam-5367	346	14	(	(	PUNCT
ejpam-5367	346	15	0	0	NUM
ejpam-5367	346	16	)	)	PUNCT
ejpam-5367	346	17	≥	≥	NOUN
ejpam-5367	346	18	βf	βf	INTJ
ejpam-5367	346	19	(	(	PUNCT
ejpam-5367	346	20	x	x	X
ejpam-5367	346	21	)	)	PUNCT
ejpam-5367	346	22	,	,	PUNCT
ejpam-5367	346	23	αf	αf	X
ejpam-5367	346	24	(	(	PUNCT
ejpam-5367	346	25	x	x	X
ejpam-5367	346	26	·	·	PUNCT
ejpam-5367	346	27	z	z	X
ejpam-5367	346	28	)	)	PUNCT
ejpam-5367	346	29	≤	≤	NOUN
ejpam-5367	346	30	max{αf	max{αf	NOUN
ejpam-5367	346	31	(	(	PUNCT
ejpam-5367	346	32	x	x	X
ejpam-5367	346	33	·	·	PUNCT
ejpam-5367	346	34	(	(	PUNCT
ejpam-5367	346	35	y	y	PROPN
ejpam-5367	346	36	·	·	PUNCT
ejpam-5367	346	37	z	z	NOUN
ejpam-5367	346	38	)	)	PUNCT
ejpam-5367	346	39	)	)	PUNCT
ejpam-5367	346	40	,	,	PUNCT
ejpam-5367	346	41	αf	αf	X
ejpam-5367	346	42	(	(	PUNCT
ejpam-5367	346	43	y	y	NOUN
ejpam-5367	346	44	)	)	PUNCT
ejpam-5367	346	45	}	}	PUNCT
ejpam-5367	346	46	,	,	PUNCT
ejpam-5367	346	47	βf	βf	CCONJ
ejpam-5367	346	48	(	(	PUNCT
ejpam-5367	346	49	x	x	X
ejpam-5367	346	50	·	·	PUNCT
ejpam-5367	346	51	z	z	X
ejpam-5367	346	52	)	)	PUNCT
ejpam-5367	346	53	≥	≥	NOUN
ejpam-5367	346	54	min{βf	min{βf	INTJ
ejpam-5367	346	55	(	(	PUNCT
ejpam-5367	346	56	x	x	X
ejpam-5367	346	57	·	·	PUNCT
ejpam-5367	346	58	(	(	PUNCT
ejpam-5367	346	59	y	y	PROPN
ejpam-5367	346	60	·	·	PUNCT
ejpam-5367	346	61	z	z	NOUN
ejpam-5367	346	62	)	)	PUNCT
ejpam-5367	346	63	)	)	PUNCT
ejpam-5367	346	64	,	,	PUNCT
ejpam-5367	346	65	βf	βf	CCONJ
ejpam-5367	346	66	(	(	PUNCT
ejpam-5367	346	67	y	y	NOUN
ejpam-5367	346	68	)	)	PUNCT
ejpam-5367	346	69	}	}	PUNCT
ejpam-5367	346	70	.	.	PUNCT
ejpam-5367	347	1	hence	hence	ADV
ejpam-5367	347	2	,	,	PUNCT
ejpam-5367	347	3	the	the	DET
ejpam-5367	347	4	fss	fss	NOUN
ejpam-5367	347	5	αf	αf	VERB
ejpam-5367	347	6	and	and	CCONJ
ejpam-5367	347	7	βf	βf	ADV
ejpam-5367	347	8	satisfy	satisfy	VERB
ejpam-5367	347	9	(	(	PUNCT
ejpam-5367	347	10	3.4	3.4	NUM
ejpam-5367	347	11	)	)	PUNCT
ejpam-5367	347	12	and	and	CCONJ
ejpam-5367	347	13	(	(	PUNCT
ejpam-5367	347	14	3.6	3.6	NUM
ejpam-5367	347	15	)	)	PUNCT
ejpam-5367	347	16	,	,	PUNCT
ejpam-5367	347	17	and	and	CCONJ
ejpam-5367	347	18	the	the	DET
ejpam-5367	347	19	fss	fss	NOUN
ejpam-5367	347	20	αf	αf	VERB
ejpam-5367	347	21	and	and	CCONJ
ejpam-5367	347	22	βf	βf	ADV
ejpam-5367	347	23	satisfy	satisfy	VERB
ejpam-5367	347	24	(	(	PUNCT
ejpam-5367	347	25	3.5	3.5	NUM
ejpam-5367	347	26	)	)	PUNCT
ejpam-5367	347	27	and	and	CCONJ
ejpam-5367	347	28	(	(	PUNCT
ejpam-5367	347	29	3.7	3.7	NUM
ejpam-5367	347	30	)	)	PUNCT
ejpam-5367	347	31	.	.	PUNCT
ejpam-5367	348	1	conversely	conversely	ADV
ejpam-5367	348	2	,	,	PUNCT
ejpam-5367	348	3	assume	assume	VERB
ejpam-5367	348	4	that	that	SCONJ
ejpam-5367	348	5	the	the	DET
ejpam-5367	348	6	fss	fss	NOUN
ejpam-5367	348	7	αf	αf	VERB
ejpam-5367	348	8	and	and	CCONJ
ejpam-5367	348	9	βf	βf	ADV
ejpam-5367	348	10	satisfy	satisfy	VERB
ejpam-5367	348	11	(	(	PUNCT
ejpam-5367	348	12	3.4	3.4	NUM
ejpam-5367	348	13	)	)	PUNCT
ejpam-5367	348	14	and	and	CCONJ
ejpam-5367	348	15	(	(	PUNCT
ejpam-5367	348	16	3.6	3.6	NUM
ejpam-5367	348	17	)	)	PUNCT
ejpam-5367	348	18	,	,	PUNCT
ejpam-5367	348	19	and	and	CCONJ
ejpam-5367	348	20	the	the	DET
ejpam-5367	348	21	fss	fss	NOUN
ejpam-5367	348	22	αf	αf	VERB
ejpam-5367	348	23	and	and	CCONJ
ejpam-5367	348	24	βf	βf	ADV
ejpam-5367	348	25	satisfy	satisfy	VERB
ejpam-5367	348	26	(	(	PUNCT
ejpam-5367	348	27	3.5	3.5	NUM
ejpam-5367	348	28	)	)	PUNCT
ejpam-5367	348	29	and	and	CCONJ
ejpam-5367	348	30	(	(	PUNCT
ejpam-5367	348	31	3.7	3.7	NUM
ejpam-5367	348	32	)	)	PUNCT
ejpam-5367	348	33	.	.	PUNCT
ejpam-5367	349	1	then	then	ADV
ejpam-5367	349	2	αf	αf	VERB
ejpam-5367	349	3	satisfies	satisfie	NOUN
ejpam-5367	349	4	(	(	PUNCT
ejpam-5367	349	5	3.4	3.4	NUM
ejpam-5367	349	6	)	)	PUNCT
ejpam-5367	349	7	and	and	CCONJ
ejpam-5367	349	8	(	(	PUNCT
ejpam-5367	349	9	3.6	3.6	NUM
ejpam-5367	349	10	)	)	PUNCT
ejpam-5367	349	11	,	,	PUNCT
ejpam-5367	349	12	and	and	CCONJ
ejpam-5367	349	13	βf	βf	ADV
ejpam-5367	349	14	satisfies	satisfie	NOUN
ejpam-5367	349	15	(	(	PUNCT
ejpam-5367	349	16	3.5	3.5	NUM
ejpam-5367	349	17	)	)	PUNCT
ejpam-5367	349	18	and	and	CCONJ
ejpam-5367	349	19	(	(	PUNCT
ejpam-5367	349	20	3.7	3.7	NUM
ejpam-5367	349	21	)	)	PUNCT
ejpam-5367	349	22	.	.	PUNCT
ejpam-5367	350	1	hence	hence	ADV
ejpam-5367	350	2	,	,	PUNCT
ejpam-5367	350	3	f	f	PROPN
ejpam-5367	350	4	is	be	AUX
ejpam-5367	350	5	an	an	DET
ejpam-5367	350	6	ffiup	ffiup	NOUN
ejpam-5367	350	7	-	-	PUNCT
ejpam-5367	350	8	ideal	ideal	NOUN
ejpam-5367	350	9	of	of	ADP
ejpam-5367	350	10	x.	x.	PROPN
ejpam-5367	350	11	theorem	theorem	VERB
ejpam-5367	350	12	14	14	NUM
ejpam-5367	350	13	.	.	PUNCT
ejpam-5367	351	1	an	an	DET
ejpam-5367	351	2	ffs	ffs	NOUN
ejpam-5367	351	3	f	f	PROPN
ejpam-5367	351	4	is	be	AUX
ejpam-5367	351	5	an	an	DET
ejpam-5367	351	6	ffiup	ffiup	ADJ
ejpam-5367	351	7	-	-	PUNCT
ejpam-5367	351	8	filter	filter	NOUN
ejpam-5367	351	9	of	of	ADP
ejpam-5367	351	10	x	x	SYM
ejpam-5367	351	11	if	if	SCONJ
ejpam-5367	351	12	and	and	CCONJ
ejpam-5367	351	13	only	only	ADV
ejpam-5367	351	14	if	if	SCONJ
ejpam-5367	351	15	the	the	DET
ejpam-5367	351	16	fss	fss	NOUN
ejpam-5367	351	17	αf	αf	VERB
ejpam-5367	351	18	and	and	CCONJ
ejpam-5367	351	19	βf	βf	ADV
ejpam-5367	351	20	satisfy	satisfy	VERB
ejpam-5367	351	21	(	(	PUNCT
ejpam-5367	351	22	3.4	3.4	NUM
ejpam-5367	351	23	)	)	PUNCT
ejpam-5367	351	24	and	and	CCONJ
ejpam-5367	351	25	(	(	PUNCT
ejpam-5367	351	26	3.8	3.8	NUM
ejpam-5367	351	27	)	)	PUNCT
ejpam-5367	351	28	,	,	PUNCT
ejpam-5367	351	29	and	and	CCONJ
ejpam-5367	351	30	the	the	DET
ejpam-5367	351	31	fss	fss	NOUN
ejpam-5367	351	32	αf	αf	VERB
ejpam-5367	351	33	and	and	CCONJ
ejpam-5367	351	34	βf	βf	ADV
ejpam-5367	351	35	satisfy	satisfy	VERB
ejpam-5367	351	36	(	(	PUNCT
ejpam-5367	351	37	3.5	3.5	NUM
ejpam-5367	351	38	)	)	PUNCT
ejpam-5367	351	39	and	and	CCONJ
ejpam-5367	351	40	(	(	PUNCT
ejpam-5367	351	41	3.9	3.9	NUM
ejpam-5367	351	42	)	)	PUNCT
ejpam-5367	351	43	.	.	PUNCT
ejpam-5367	352	1	proof	proof	NOUN
ejpam-5367	352	2	.	.	PUNCT
ejpam-5367	353	1	assume	assume	VERB
ejpam-5367	353	2	that	that	SCONJ
ejpam-5367	353	3	f	f	PROPN
ejpam-5367	353	4	is	be	AUX
ejpam-5367	353	5	an	an	DET
ejpam-5367	353	6	ffiup	ffiup	NOUN
ejpam-5367	353	7	-	-	PUNCT
ejpam-5367	353	8	ideal	ideal	NOUN
ejpam-5367	353	9	of	of	ADP
ejpam-5367	353	10	x.	x.	NOUN
ejpam-5367	353	11	then	then	ADV
ejpam-5367	353	12	αf	αf	VERB
ejpam-5367	353	13	(	(	PUNCT
ejpam-5367	353	14	0	0	NUM
ejpam-5367	353	15	)	)	PUNCT
ejpam-5367	353	16	≥	≥	NOUN
ejpam-5367	353	17	αf	αf	X
ejpam-5367	353	18	(	(	PUNCT
ejpam-5367	353	19	x	x	NOUN
ejpam-5367	353	20	)	)	PUNCT
ejpam-5367	353	21	,	,	PUNCT
ejpam-5367	353	22	βf	βf	CCONJ
ejpam-5367	353	23	(	(	PUNCT
ejpam-5367	353	24	0	0	X
ejpam-5367	353	25	)	)	PUNCT
ejpam-5367	353	26	≤	≤	NOUN
ejpam-5367	354	1	βf	βf	CCONJ
ejpam-5367	354	2	(	(	PUNCT
ejpam-5367	354	3	x	x	X
ejpam-5367	354	4	)	)	PUNCT
ejpam-5367	354	5	,	,	PUNCT
ejpam-5367	354	6	αf	αf	X
ejpam-5367	354	7	(	(	PUNCT
ejpam-5367	354	8	y	y	NOUN
ejpam-5367	354	9	)	)	PUNCT
ejpam-5367	354	10	≥	≥	NOUN
ejpam-5367	354	11	min{αf	min{αf	PUNCT
ejpam-5367	354	12	(	(	PUNCT
ejpam-5367	354	13	x	x	X
ejpam-5367	354	14	·	·	PUNCT
ejpam-5367	354	15	y	y	X
ejpam-5367	354	16	)	)	PUNCT
ejpam-5367	354	17	,	,	PUNCT
ejpam-5367	354	18	αf	αf	X
ejpam-5367	354	19	(	(	PUNCT
ejpam-5367	354	20	x	x	NOUN
ejpam-5367	354	21	)	)	PUNCT
ejpam-5367	354	22	}	}	PUNCT
ejpam-5367	354	23	,	,	PUNCT
ejpam-5367	354	24	βf	βf	CCONJ
ejpam-5367	354	25	(	(	PUNCT
ejpam-5367	354	26	y	y	NOUN
ejpam-5367	354	27	)	)	PUNCT
ejpam-5367	354	28	≤	≤	NOUN
ejpam-5367	355	1	max{βf	max{βf	INTJ
ejpam-5367	355	2	(	(	PUNCT
ejpam-5367	355	3	x	x	X
ejpam-5367	355	4	·	·	PUNCT
ejpam-5367	355	5	y	y	X
ejpam-5367	355	6	)	)	PUNCT
ejpam-5367	355	7	,	,	PUNCT
ejpam-5367	355	8	βf	βf	CCONJ
ejpam-5367	355	9	(	(	PUNCT
ejpam-5367	355	10	x	x	NOUN
ejpam-5367	355	11	)	)	PUNCT
ejpam-5367	355	12	}	}	PUNCT
ejpam-5367	355	13	.	.	PUNCT
ejpam-5367	356	1	thus	thus	ADV
ejpam-5367	356	2	,	,	PUNCT
ejpam-5367	356	3	αf	αf	ADP
ejpam-5367	356	4	(	(	PUNCT
ejpam-5367	356	5	0	0	NUM
ejpam-5367	356	6	)	)	PUNCT
ejpam-5367	356	7	≤	≤	NOUN
ejpam-5367	356	8	αf	αf	ADP
ejpam-5367	356	9	(	(	PUNCT
ejpam-5367	356	10	x	x	NOUN
ejpam-5367	356	11	)	)	PUNCT
ejpam-5367	356	12	,	,	PUNCT
ejpam-5367	356	13	βf	βf	CCONJ
ejpam-5367	356	14	(	(	PUNCT
ejpam-5367	356	15	0	0	NUM
ejpam-5367	356	16	)	)	PUNCT
ejpam-5367	356	17	≥	≥	NOUN
ejpam-5367	356	18	βf	βf	INTJ
ejpam-5367	356	19	(	(	PUNCT
ejpam-5367	356	20	x	x	X
ejpam-5367	356	21	)	)	PUNCT
ejpam-5367	356	22	,	,	PUNCT
ejpam-5367	356	23	αf	αf	X
ejpam-5367	356	24	(	(	PUNCT
ejpam-5367	356	25	y	y	NOUN
ejpam-5367	356	26	)	)	PUNCT
ejpam-5367	356	27	≤	≤	NOUN
ejpam-5367	356	28	max{αf	max{αf	NOUN
ejpam-5367	356	29	(	(	PUNCT
ejpam-5367	356	30	x	x	X
ejpam-5367	356	31	·	·	PUNCT
ejpam-5367	356	32	y	y	X
ejpam-5367	356	33	)	)	PUNCT
ejpam-5367	356	34	,	,	PUNCT
ejpam-5367	356	35	αf	αf	X
ejpam-5367	356	36	(	(	PUNCT
ejpam-5367	356	37	x	x	NOUN
ejpam-5367	356	38	)	)	PUNCT
ejpam-5367	356	39	}	}	PUNCT
ejpam-5367	356	40	,	,	PUNCT
ejpam-5367	356	41	a.	a.	NOUN
ejpam-5367	356	42	iampan	iampan	NOUN
ejpam-5367	356	43	et	et	PROPN
ejpam-5367	356	44	al	al	PROPN
ejpam-5367	356	45	.	.	PUNCT
ejpam-5367	356	46	/	/	SYM
ejpam-5367	356	47	eur	eur	PROPN
ejpam-5367	356	48	.	.	PUNCT
ejpam-5367	357	1	j.	j.	PROPN
ejpam-5367	357	2	pure	pure	PROPN
ejpam-5367	357	3	appl	appl	PROPN
ejpam-5367	357	4	.	.	PROPN
ejpam-5367	357	5	math	math	PROPN
ejpam-5367	357	6	,	,	PUNCT
ejpam-5367	357	7	17	17	NUM
ejpam-5367	357	8	(	(	PUNCT
ejpam-5367	357	9	4	4	NUM
ejpam-5367	357	10	)	)	PUNCT
ejpam-5367	357	11	(	(	PUNCT
ejpam-5367	357	12	2024	2024	NUM
ejpam-5367	357	13	)	)	PUNCT
ejpam-5367	357	14	,	,	PUNCT
ejpam-5367	357	15	3022	3022	NUM
ejpam-5367	357	16	-	-	SYM
ejpam-5367	357	17	3042	3042	NUM
ejpam-5367	357	18	3036	3036	NUM
ejpam-5367	357	19	βf	βf	SYM
ejpam-5367	357	20	(	(	PUNCT
ejpam-5367	357	21	y	y	NOUN
ejpam-5367	357	22	)	)	PUNCT
ejpam-5367	357	23	≥	≥	NOUN
ejpam-5367	357	24	min{βf	min{βf	INTJ
ejpam-5367	357	25	(	(	PUNCT
ejpam-5367	357	26	x	x	PROPN
ejpam-5367	357	27	·	·	PUNCT
ejpam-5367	357	28	y	y	X
ejpam-5367	357	29	)	)	PUNCT
ejpam-5367	357	30	,	,	PUNCT
ejpam-5367	357	31	βf	βf	CCONJ
ejpam-5367	357	32	(	(	PUNCT
ejpam-5367	357	33	x	x	NOUN
ejpam-5367	357	34	)	)	PUNCT
ejpam-5367	357	35	}	}	PUNCT
ejpam-5367	357	36	.	.	PUNCT
ejpam-5367	358	1	hence	hence	ADV
ejpam-5367	358	2	,	,	PUNCT
ejpam-5367	358	3	the	the	DET
ejpam-5367	358	4	fss	fss	NOUN
ejpam-5367	358	5	αf	αf	VERB
ejpam-5367	358	6	and	and	CCONJ
ejpam-5367	358	7	βf	βf	ADV
ejpam-5367	358	8	satisfy	satisfy	VERB
ejpam-5367	358	9	(	(	PUNCT
ejpam-5367	358	10	3.4	3.4	NUM
ejpam-5367	358	11	)	)	PUNCT
ejpam-5367	358	12	and	and	CCONJ
ejpam-5367	358	13	(	(	PUNCT
ejpam-5367	358	14	3.8	3.8	NUM
ejpam-5367	358	15	)	)	PUNCT
ejpam-5367	358	16	,	,	PUNCT
ejpam-5367	358	17	and	and	CCONJ
ejpam-5367	358	18	the	the	DET
ejpam-5367	358	19	fss	fss	NOUN
ejpam-5367	358	20	αf	αf	VERB
ejpam-5367	358	21	and	and	CCONJ
ejpam-5367	358	22	βf	βf	ADV
ejpam-5367	358	23	satisfy	satisfy	VERB
ejpam-5367	358	24	(	(	PUNCT
ejpam-5367	358	25	3.5	3.5	NUM
ejpam-5367	358	26	)	)	PUNCT
ejpam-5367	358	27	and	and	CCONJ
ejpam-5367	358	28	(	(	PUNCT
ejpam-5367	358	29	3.9	3.9	NUM
ejpam-5367	358	30	)	)	PUNCT
ejpam-5367	358	31	.	.	PUNCT
ejpam-5367	359	1	conversely	conversely	ADV
ejpam-5367	359	2	,	,	PUNCT
ejpam-5367	359	3	assume	assume	VERB
ejpam-5367	359	4	that	that	SCONJ
ejpam-5367	359	5	the	the	DET
ejpam-5367	359	6	fss	fss	NOUN
ejpam-5367	359	7	αf	αf	VERB
ejpam-5367	359	8	and	and	CCONJ
ejpam-5367	359	9	βf	βf	ADV
ejpam-5367	359	10	satisfy	satisfy	VERB
ejpam-5367	359	11	(	(	PUNCT
ejpam-5367	359	12	3.4	3.4	NUM
ejpam-5367	359	13	)	)	PUNCT
ejpam-5367	359	14	and	and	CCONJ
ejpam-5367	359	15	(	(	PUNCT
ejpam-5367	359	16	3.8	3.8	NUM
ejpam-5367	359	17	)	)	PUNCT
ejpam-5367	359	18	,	,	PUNCT
ejpam-5367	359	19	and	and	CCONJ
ejpam-5367	359	20	the	the	DET
ejpam-5367	359	21	fss	fss	NOUN
ejpam-5367	359	22	αf	αf	VERB
ejpam-5367	359	23	and	and	CCONJ
ejpam-5367	359	24	βf	βf	ADV
ejpam-5367	359	25	satisfy	satisfy	VERB
ejpam-5367	359	26	(	(	PUNCT
ejpam-5367	359	27	3.5	3.5	NUM
ejpam-5367	359	28	)	)	PUNCT
ejpam-5367	359	29	and	and	CCONJ
ejpam-5367	359	30	(	(	PUNCT
ejpam-5367	359	31	3.9	3.9	NUM
ejpam-5367	359	32	)	)	PUNCT
ejpam-5367	359	33	.	.	PUNCT
ejpam-5367	360	1	then	then	ADV
ejpam-5367	360	2	αf	αf	VERB
ejpam-5367	360	3	satisfies	satisfie	NOUN
ejpam-5367	360	4	(	(	PUNCT
ejpam-5367	360	5	3.4	3.4	NUM
ejpam-5367	360	6	)	)	PUNCT
ejpam-5367	360	7	and	and	CCONJ
ejpam-5367	360	8	(	(	PUNCT
ejpam-5367	360	9	3.8	3.8	NUM
ejpam-5367	360	10	)	)	PUNCT
ejpam-5367	360	11	,	,	PUNCT
ejpam-5367	360	12	and	and	CCONJ
ejpam-5367	360	13	βf	βf	ADV
ejpam-5367	360	14	satisfies	satisfie	NOUN
ejpam-5367	360	15	(	(	PUNCT
ejpam-5367	360	16	3.5	3.5	NUM
ejpam-5367	360	17	)	)	PUNCT
ejpam-5367	360	18	and	and	CCONJ
ejpam-5367	360	19	(	(	PUNCT
ejpam-5367	360	20	3.9	3.9	NUM
ejpam-5367	360	21	)	)	PUNCT
ejpam-5367	360	22	.	.	PUNCT
ejpam-5367	361	1	hence	hence	ADV
ejpam-5367	361	2	,	,	PUNCT
ejpam-5367	361	3	f	f	PROPN
ejpam-5367	361	4	is	be	AUX
ejpam-5367	361	5	an	an	DET
ejpam-5367	361	6	ffiup	ffiup	ADJ
ejpam-5367	361	7	-	-	PUNCT
ejpam-5367	361	8	filter	filter	NOUN
ejpam-5367	361	9	of	of	ADP
ejpam-5367	361	10	x.	x.	NOUN
ejpam-5367	361	11	the	the	DET
ejpam-5367	361	12	following	follow	VERB
ejpam-5367	361	13	theorem	theorem	NOUN
ejpam-5367	361	14	is	be	AUX
ejpam-5367	361	15	a	a	DET
ejpam-5367	361	16	direct	direct	ADJ
ejpam-5367	361	17	consequence	consequence	NOUN
ejpam-5367	361	18	of	of	ADP
ejpam-5367	361	19	theorem	theorem	ADJ
ejpam-5367	361	20	2	2	NUM
ejpam-5367	361	21	.	.	PUNCT
ejpam-5367	361	22	theorem	theorem	VERB
ejpam-5367	361	23	15	15	NUM
ejpam-5367	361	24	.	.	PUNCT
ejpam-5367	362	1	an	an	DET
ejpam-5367	362	2	ffs	ffs	NOUN
ejpam-5367	362	3	f	f	PROPN
ejpam-5367	362	4	is	be	AUX
ejpam-5367	362	5	an	an	DET
ejpam-5367	362	6	ffsiup	ffsiup	NOUN
ejpam-5367	362	7	-	-	PUNCT
ejpam-5367	362	8	ideal	ideal	NOUN
ejpam-5367	362	9	of	of	ADP
ejpam-5367	362	10	x	x	SYM
ejpam-5367	362	11	if	if	SCONJ
ejpam-5367	362	12	and	and	CCONJ
ejpam-5367	362	13	only	only	ADV
ejpam-5367	362	14	if	if	SCONJ
ejpam-5367	362	15	the	the	DET
ejpam-5367	362	16	fss	fss	NOUN
ejpam-5367	362	17	αf	αf	VERB
ejpam-5367	362	18	and	and	CCONJ
ejpam-5367	362	19	βf	βf	ADV
ejpam-5367	362	20	satisfy	satisfy	VERB
ejpam-5367	362	21	(	(	PUNCT
ejpam-5367	362	22	3.10	3.10	NUM
ejpam-5367	362	23	)	)	PUNCT
ejpam-5367	362	24	,	,	PUNCT
ejpam-5367	362	25	and	and	CCONJ
ejpam-5367	362	26	the	the	DET
ejpam-5367	362	27	fss	fss	NOUN
ejpam-5367	362	28	αf	αf	VERB
ejpam-5367	362	29	and	and	CCONJ
ejpam-5367	362	30	βf	βf	ADV
ejpam-5367	362	31	satisfy	satisfy	VERB
ejpam-5367	362	32	(	(	PUNCT
ejpam-5367	362	33	3.11	3.11	NUM
ejpam-5367	362	34	)	)	PUNCT
ejpam-5367	362	35	.	.	PUNCT
ejpam-5367	363	1	theorem	theorem	VERB
ejpam-5367	363	2	16	16	NUM
ejpam-5367	363	3	.	.	PUNCT
ejpam-5367	364	1	an	an	DET
ejpam-5367	364	2	ffs	ffs	NOUN
ejpam-5367	364	3	f	f	PROPN
ejpam-5367	364	4	is	be	AUX
ejpam-5367	364	5	an	an	DET
ejpam-5367	364	6	ffiup	ffiup	NOUN
ejpam-5367	364	7	-	-	PUNCT
ejpam-5367	364	8	subalgebra	subalgebra	NOUN
ejpam-5367	364	9	of	of	ADP
ejpam-5367	364	10	x	x	PRON
ejpam-5367	364	11	if	if	SCONJ
ejpam-5367	364	12	and	and	CCONJ
ejpam-5367	364	13	only	only	ADV
ejpam-5367	364	14	if	if	SCONJ
ejpam-5367	364	15	ffs	ff	NOUN
ejpam-5367	364	16	∗f	∗f	ADV
ejpam-5367	364	17	=	=	PUNCT
ejpam-5367	364	18	(	(	PUNCT
ejpam-5367	364	19	αf	αf	X
ejpam-5367	364	20	,	,	PUNCT
ejpam-5367	364	21	αf	αf	NOUN
ejpam-5367	364	22	)	)	PUNCT
ejpam-5367	364	23	and	and	CCONJ
ejpam-5367	364	24	△	△	NOUN
ejpam-5367	364	25	f	f	X
ejpam-5367	364	26	=	=	PUNCT
ejpam-5367	364	27	(	(	PUNCT
ejpam-5367	364	28	βf	βf	INTJ
ejpam-5367	364	29	,	,	PUNCT
ejpam-5367	364	30	βf	βf	CCONJ
ejpam-5367	364	31	)	)	PUNCT
ejpam-5367	364	32	are	be	AUX
ejpam-5367	364	33	ffiup	ffiup	ADJ
ejpam-5367	364	34	-	-	PUNCT
ejpam-5367	364	35	subalgebras	subalgebra	NOUN
ejpam-5367	364	36	of	of	ADP
ejpam-5367	364	37	x.	x.	NOUN
ejpam-5367	364	38	proof	proof	NOUN
ejpam-5367	364	39	.	.	PUNCT
ejpam-5367	365	1	it	it	PRON
ejpam-5367	365	2	is	be	AUX
ejpam-5367	365	3	straightforward	straightforward	ADJ
ejpam-5367	365	4	by	by	ADP
ejpam-5367	365	5	theorem	theorem	NOUN
ejpam-5367	365	6	12	12	NUM
ejpam-5367	365	7	.	.	PUNCT
ejpam-5367	366	1	theorem	theorem	VERB
ejpam-5367	366	2	17	17	NUM
ejpam-5367	366	3	.	.	PUNCT
ejpam-5367	367	1	an	an	DET
ejpam-5367	367	2	ffs	ffs	NOUN
ejpam-5367	367	3	f	f	PROPN
ejpam-5367	367	4	is	be	AUX
ejpam-5367	367	5	an	an	DET
ejpam-5367	367	6	ffiup	ffiup	NOUN
ejpam-5367	367	7	-	-	PUNCT
ejpam-5367	367	8	ideal	ideal	NOUN
ejpam-5367	367	9	of	of	ADP
ejpam-5367	367	10	x	x	SYM
ejpam-5367	367	11	if	if	SCONJ
ejpam-5367	367	12	and	and	CCONJ
ejpam-5367	367	13	only	only	ADV
ejpam-5367	367	14	if	if	SCONJ
ejpam-5367	367	15	ffs	ff	NOUN
ejpam-5367	367	16	∗f	∗f	ADV
ejpam-5367	367	17	=	=	PUNCT
ejpam-5367	367	18	(	(	PUNCT
ejpam-5367	367	19	αf	αf	X
ejpam-5367	367	20	,	,	PUNCT
ejpam-5367	367	21	αf	αf	NOUN
ejpam-5367	367	22	)	)	PUNCT
ejpam-5367	367	23	and	and	CCONJ
ejpam-5367	367	24	△	△	NOUN
ejpam-5367	367	25	f	f	X
ejpam-5367	367	26	=	=	SYM
ejpam-5367	367	27	(	(	PUNCT
ejpam-5367	367	28	βfβf	βfβf	PROPN
ejpam-5367	367	29	)	)	PUNCT
ejpam-5367	367	30	are	be	AUX
ejpam-5367	367	31	ffiup	ffiup	ADJ
ejpam-5367	367	32	-	-	PUNCT
ejpam-5367	367	33	ideals	ideal	NOUN
ejpam-5367	367	34	of	of	ADP
ejpam-5367	367	35	x.	x.	NOUN
ejpam-5367	367	36	proof	proof	NOUN
ejpam-5367	367	37	.	.	PUNCT
ejpam-5367	368	1	it	it	PRON
ejpam-5367	368	2	is	be	AUX
ejpam-5367	368	3	straightforward	straightforward	ADJ
ejpam-5367	368	4	by	by	ADP
ejpam-5367	368	5	theorem	theorem	ADJ
ejpam-5367	368	6	13	13	NUM
ejpam-5367	368	7	.	.	PUNCT
ejpam-5367	369	1	theorem	theorem	VERB
ejpam-5367	369	2	18	18	NUM
ejpam-5367	369	3	.	.	PUNCT
ejpam-5367	370	1	an	an	DET
ejpam-5367	370	2	ffs	ffs	NOUN
ejpam-5367	370	3	f	f	PROPN
ejpam-5367	370	4	is	be	AUX
ejpam-5367	370	5	an	an	DET
ejpam-5367	370	6	ffiup	ffiup	ADJ
ejpam-5367	370	7	-	-	PUNCT
ejpam-5367	370	8	filter	filter	NOUN
ejpam-5367	370	9	of	of	ADP
ejpam-5367	370	10	x	x	SYM
ejpam-5367	370	11	if	if	SCONJ
ejpam-5367	370	12	and	and	CCONJ
ejpam-5367	370	13	only	only	ADV
ejpam-5367	370	14	if	if	SCONJ
ejpam-5367	370	15	ffs	ff	NOUN
ejpam-5367	370	16	∗f	∗f	ADV
ejpam-5367	370	17	=	=	PUNCT
ejpam-5367	370	18	(	(	PUNCT
ejpam-5367	370	19	αf	αf	X
ejpam-5367	370	20	,	,	PUNCT
ejpam-5367	370	21	αf	αf	NOUN
ejpam-5367	370	22	)	)	PUNCT
ejpam-5367	370	23	and	and	CCONJ
ejpam-5367	370	24	△	△	NOUN
ejpam-5367	370	25	f	f	X
ejpam-5367	370	26	=	=	SYM
ejpam-5367	370	27	(	(	PUNCT
ejpam-5367	370	28	βfβf	βfβf	PROPN
ejpam-5367	370	29	)	)	PUNCT
ejpam-5367	370	30	are	be	AUX
ejpam-5367	370	31	ffiup	ffiup	ADJ
ejpam-5367	370	32	-	-	PUNCT
ejpam-5367	370	33	filters	filter	NOUN
ejpam-5367	370	34	of	of	ADP
ejpam-5367	370	35	x.	x.	NOUN
ejpam-5367	370	36	proof	proof	NOUN
ejpam-5367	370	37	.	.	PUNCT
ejpam-5367	371	1	it	it	PRON
ejpam-5367	371	2	is	be	AUX
ejpam-5367	371	3	straightforward	straightforward	ADJ
ejpam-5367	371	4	by	by	ADP
ejpam-5367	371	5	theorem	theorem	ADJ
ejpam-5367	371	6	14	14	NUM
ejpam-5367	371	7	.	.	PUNCT
ejpam-5367	372	1	theorem	theorem	VERB
ejpam-5367	372	2	19	19	NUM
ejpam-5367	372	3	.	.	PUNCT
ejpam-5367	373	1	an	an	DET
ejpam-5367	373	2	ffs	ffs	NOUN
ejpam-5367	373	3	f	f	PROPN
ejpam-5367	373	4	is	be	AUX
ejpam-5367	373	5	an	an	DET
ejpam-5367	373	6	ffsiup	ffsiup	NOUN
ejpam-5367	373	7	-	-	PUNCT
ejpam-5367	373	8	ideal	ideal	NOUN
ejpam-5367	373	9	of	of	ADP
ejpam-5367	373	10	x	x	SYM
ejpam-5367	373	11	if	if	SCONJ
ejpam-5367	373	12	and	and	CCONJ
ejpam-5367	373	13	only	only	ADV
ejpam-5367	373	14	if	if	SCONJ
ejpam-5367	373	15	ffs	ff	NOUN
ejpam-5367	373	16	∗f	∗f	ADV
ejpam-5367	373	17	=	=	PUNCT
ejpam-5367	373	18	(	(	PUNCT
ejpam-5367	373	19	αf	αf	X
ejpam-5367	373	20	,	,	PUNCT
ejpam-5367	373	21	αf	αf	NOUN
ejpam-5367	373	22	)	)	PUNCT
ejpam-5367	373	23	and	and	CCONJ
ejpam-5367	373	24	△	△	NOUN
ejpam-5367	373	25	f	f	X
ejpam-5367	373	26	=	=	SYM
ejpam-5367	373	27	(	(	PUNCT
ejpam-5367	373	28	βfβf	βfβf	PROPN
ejpam-5367	373	29	)	)	PUNCT
ejpam-5367	373	30	are	be	AUX
ejpam-5367	373	31	ffsiup	ffsiup	NOUN
ejpam-5367	373	32	-	-	PUNCT
ejpam-5367	373	33	ideals	ideal	NOUN
ejpam-5367	373	34	of	of	ADP
ejpam-5367	373	35	x.	x.	NOUN
ejpam-5367	373	36	proof	proof	NOUN
ejpam-5367	373	37	.	.	PUNCT
ejpam-5367	374	1	it	it	PRON
ejpam-5367	374	2	is	be	AUX
ejpam-5367	374	3	straightforward	straightforward	ADJ
ejpam-5367	374	4	by	by	ADP
ejpam-5367	374	5	theorem	theorem	NOUN
ejpam-5367	374	6	15	15	NUM
ejpam-5367	374	7	.	.	PUNCT
ejpam-5367	375	1	definition	definition	NOUN
ejpam-5367	375	2	10	10	NUM
ejpam-5367	375	3	.	.	PUNCT
ejpam-5367	376	1	[	[	X
ejpam-5367	376	2	12	12	NUM
ejpam-5367	376	3	]	]	PUNCT
ejpam-5367	376	4	let	let	VERB
ejpam-5367	376	5	f	f	PRON
ejpam-5367	376	6	be	be	AUX
ejpam-5367	376	7	an	an	DET
ejpam-5367	376	8	fs	fs	NOUN
ejpam-5367	376	9	in	in	ADP
ejpam-5367	376	10	a	a	DET
ejpam-5367	376	11	non	non	ADJ
ejpam-5367	376	12	-	-	ADJ
ejpam-5367	376	13	empty	empty	ADJ
ejpam-5367	376	14	set	set	VERB
ejpam-5367	376	15	x.	x.	NOUN
ejpam-5367	376	16	for	for	ADP
ejpam-5367	376	17	any	any	DET
ejpam-5367	376	18	t	t	NOUN
ejpam-5367	376	19	∈	∈	PROPN
ejpam-5367	377	1	[	[	X
ejpam-5367	377	2	0	0	NUM
ejpam-5367	377	3	,	,	PUNCT
ejpam-5367	377	4	1	1	NUM
ejpam-5367	377	5	]	]	PUNCT
ejpam-5367	377	6	,	,	PUNCT
ejpam-5367	377	7	the	the	DET
ejpam-5367	377	8	sets	set	VERB
ejpam-5367	377	9	u(f	u(f	PROPN
ejpam-5367	377	10	;	;	PUNCT
ejpam-5367	377	11	t	t	X
ejpam-5367	377	12	)	)	PUNCT
ejpam-5367	377	13	=	=	PRON
ejpam-5367	378	1	{	{	PUNCT
ejpam-5367	378	2	x	x	PUNCT
ejpam-5367	378	3	∈	∈	PROPN
ejpam-5367	378	4	x	x	X
ejpam-5367	378	5	|	|	ADV
ejpam-5367	378	6	f(x	f(x	PROPN
ejpam-5367	378	7	)	)	PUNCT
ejpam-5367	378	8	≥	≥	NOUN
ejpam-5367	378	9	t	t	PROPN
ejpam-5367	378	10	}	}	PUNCT
ejpam-5367	378	11	,	,	PUNCT
ejpam-5367	378	12	(	(	PUNCT
ejpam-5367	378	13	3.17	3.17	NUM
ejpam-5367	378	14	)	)	PUNCT
ejpam-5367	378	15	l(f	l(f	PROPN
ejpam-5367	378	16	;	;	PUNCT
ejpam-5367	378	17	t	t	X
ejpam-5367	378	18	)	)	PUNCT
ejpam-5367	378	19	=	=	PRON
ejpam-5367	378	20	{	{	PUNCT
ejpam-5367	378	21	x	x	PUNCT
ejpam-5367	378	22	∈	∈	PROPN
ejpam-5367	378	23	x	x	X
ejpam-5367	378	24	|	|	ADV
ejpam-5367	378	25	f(x	f(x	PROPN
ejpam-5367	378	26	)	)	PUNCT
ejpam-5367	378	27	≤	≤	NOUN
ejpam-5367	378	28	t	t	PROPN
ejpam-5367	378	29	}	}	PUNCT
ejpam-5367	378	30	(	(	PUNCT
ejpam-5367	378	31	3.18	3.18	NUM
ejpam-5367	378	32	)	)	PUNCT
ejpam-5367	378	33	are	be	AUX
ejpam-5367	378	34	called	call	VERB
ejpam-5367	378	35	an	an	DET
ejpam-5367	378	36	upper	upper	ADJ
ejpam-5367	378	37	t	t	NOUN
ejpam-5367	378	38	-	-	PUNCT
ejpam-5367	378	39	level	level	NOUN
ejpam-5367	378	40	subset	subset	NOUN
ejpam-5367	378	41	and	and	CCONJ
ejpam-5367	378	42	a	a	DET
ejpam-5367	378	43	lower	low	ADJ
ejpam-5367	378	44	t	t	NOUN
ejpam-5367	378	45	-	-	PUNCT
ejpam-5367	378	46	level	level	NOUN
ejpam-5367	378	47	subset	subset	NOUN
ejpam-5367	378	48	of	of	ADP
ejpam-5367	378	49	f	f	PROPN
ejpam-5367	378	50	,	,	PUNCT
ejpam-5367	378	51	respectively	respectively	ADV
ejpam-5367	378	52	.	.	PUNCT
ejpam-5367	379	1	the	the	DET
ejpam-5367	379	2	sets	set	VERB
ejpam-5367	379	3	u	u	NOUN
ejpam-5367	379	4	+	+	X
ejpam-5367	379	5	(	(	PUNCT
ejpam-5367	379	6	f	f	PROPN
ejpam-5367	379	7	;	;	PUNCT
ejpam-5367	379	8	t	t	PROPN
ejpam-5367	379	9	)	)	PUNCT
ejpam-5367	379	10	=	=	PRON
ejpam-5367	380	1	{	{	PUNCT
ejpam-5367	380	2	x	x	PUNCT
ejpam-5367	380	3	∈	∈	PROPN
ejpam-5367	380	4	x	x	X
ejpam-5367	380	5	|	|	ADV
ejpam-5367	380	6	f(x	f(x	PROPN
ejpam-5367	380	7	)	)	PUNCT
ejpam-5367	380	8	>	>	X
ejpam-5367	380	9	t	t	PROPN
ejpam-5367	380	10	}	}	PUNCT
ejpam-5367	380	11	,	,	PUNCT
ejpam-5367	380	12	(	(	PUNCT
ejpam-5367	380	13	3.19	3.19	NUM
ejpam-5367	380	14	)	)	PUNCT
ejpam-5367	380	15	l	l	NOUN
ejpam-5367	380	16	−	−	PROPN
ejpam-5367	380	17	(	(	PUNCT
ejpam-5367	380	18	f	f	PROPN
ejpam-5367	380	19	;	;	PUNCT
ejpam-5367	380	20	t	t	PROPN
ejpam-5367	380	21	)	)	PUNCT
ejpam-5367	380	22	=	=	PRON
ejpam-5367	380	23	{	{	PUNCT
ejpam-5367	380	24	x	x	PUNCT
ejpam-5367	380	25	∈	∈	PROPN
ejpam-5367	380	26	x	x	X
ejpam-5367	380	27	|	|	ADV
ejpam-5367	380	28	f(x	f(x	PROPN
ejpam-5367	380	29	)	)	PUNCT
ejpam-5367	380	30	<	<	X
ejpam-5367	380	31	t	t	PROPN
ejpam-5367	380	32	}	}	PUNCT
ejpam-5367	380	33	(	(	PUNCT
ejpam-5367	380	34	3.20	3.20	NUM
ejpam-5367	380	35	)	)	PUNCT
ejpam-5367	380	36	are	be	AUX
ejpam-5367	380	37	called	call	VERB
ejpam-5367	380	38	an	an	DET
ejpam-5367	380	39	upper	upper	ADJ
ejpam-5367	380	40	t	t	NOUN
ejpam-5367	380	41	-	-	PUNCT
ejpam-5367	380	42	strong	strong	ADJ
ejpam-5367	380	43	level	level	NOUN
ejpam-5367	380	44	subset	subset	NOUN
ejpam-5367	380	45	and	and	CCONJ
ejpam-5367	380	46	a	a	DET
ejpam-5367	380	47	lower	low	ADJ
ejpam-5367	380	48	t	t	NOUN
ejpam-5367	380	49	-	-	PUNCT
ejpam-5367	380	50	strong	strong	ADJ
ejpam-5367	380	51	level	level	NOUN
ejpam-5367	380	52	subset	subset	NOUN
ejpam-5367	380	53	of	of	ADP
ejpam-5367	380	54	f	f	PROPN
ejpam-5367	380	55	,	,	PUNCT
ejpam-5367	380	56	respectively	respectively	ADV
ejpam-5367	380	57	.	.	PUNCT
ejpam-5367	381	1	a.	a.	PROPN
ejpam-5367	381	2	iampan	iampan	PROPN
ejpam-5367	381	3	et	et	PROPN
ejpam-5367	381	4	al	al	PROPN
ejpam-5367	381	5	.	.	PUNCT
ejpam-5367	381	6	/	/	SYM
ejpam-5367	381	7	eur	eur	PROPN
ejpam-5367	381	8	.	.	PUNCT
ejpam-5367	382	1	j.	j.	PROPN
ejpam-5367	382	2	pure	pure	PROPN
ejpam-5367	382	3	appl	appl	PROPN
ejpam-5367	382	4	.	.	PROPN
ejpam-5367	382	5	math	math	PROPN
ejpam-5367	382	6	,	,	PUNCT
ejpam-5367	382	7	17	17	NUM
ejpam-5367	382	8	(	(	PUNCT
ejpam-5367	382	9	4	4	NUM
ejpam-5367	382	10	)	)	PUNCT
ejpam-5367	382	11	(	(	PUNCT
ejpam-5367	382	12	2024	2024	NUM
ejpam-5367	382	13	)	)	PUNCT
ejpam-5367	382	14	,	,	PUNCT
ejpam-5367	382	15	3022	3022	NUM
ejpam-5367	382	16	-	-	SYM
ejpam-5367	382	17	3042	3042	NUM
ejpam-5367	382	18	3037	3037	NUM
ejpam-5367	382	19	before	before	ADP
ejpam-5367	382	20	presenting	present	VERB
ejpam-5367	382	21	theorems	theorem	NOUN
ejpam-5367	382	22	on	on	ADP
ejpam-5367	382	23	the	the	DET
ejpam-5367	382	24	relationship	relationship	NOUN
ejpam-5367	382	25	between	between	ADP
ejpam-5367	382	26	level	level	NOUN
ejpam-5367	382	27	subsets	subset	NOUN
ejpam-5367	382	28	and	and	CCONJ
ejpam-5367	382	29	their	their	PRON
ejpam-5367	382	30	corresponding	correspond	VERB
ejpam-5367	382	31	ffss	ffss	NOUN
ejpam-5367	382	32	,	,	PUNCT
ejpam-5367	382	33	it	it	PRON
ejpam-5367	382	34	’s	’	VERB
ejpam-5367	382	35	essential	essential	ADJ
ejpam-5367	382	36	to	to	PART
ejpam-5367	382	37	grasp	grasp	VERB
ejpam-5367	382	38	the	the	DET
ejpam-5367	382	39	key	key	ADJ
ejpam-5367	382	40	concepts	concept	NOUN
ejpam-5367	382	41	.	.	PUNCT
ejpam-5367	383	1	level	level	NOUN
ejpam-5367	383	2	subsets	subset	NOUN
ejpam-5367	383	3	characterize	characterize	VERB
ejpam-5367	383	4	ffss	ffss	NOUN
ejpam-5367	383	5	by	by	ADP
ejpam-5367	383	6	detailing	detail	VERB
ejpam-5367	383	7	the	the	DET
ejpam-5367	383	8	distribution	distribution	NOUN
ejpam-5367	383	9	of	of	ADP
ejpam-5367	383	10	membership	membership	NOUN
ejpam-5367	383	11	degrees	degree	NOUN
ejpam-5367	383	12	.	.	PUNCT
ejpam-5367	384	1	the	the	DET
ejpam-5367	384	2	following	follow	VERB
ejpam-5367	384	3	theorem	theorem	NOUN
ejpam-5367	384	4	formalizes	formalize	VERB
ejpam-5367	384	5	this	this	DET
ejpam-5367	384	6	relationship	relationship	NOUN
ejpam-5367	384	7	,	,	PUNCT
ejpam-5367	384	8	providing	provide	VERB
ejpam-5367	384	9	insights	insight	NOUN
ejpam-5367	384	10	into	into	ADP
ejpam-5367	384	11	the	the	DET
ejpam-5367	384	12	structure	structure	NOUN
ejpam-5367	384	13	of	of	ADP
ejpam-5367	384	14	ffss	ffss	NOUN
ejpam-5367	384	15	.	.	PUNCT
ejpam-5367	385	1	theorem	theorem	VERB
ejpam-5367	385	2	20	20	NUM
ejpam-5367	385	3	.	.	PUNCT
ejpam-5367	386	1	an	an	DET
ejpam-5367	386	2	ffs	ffs	NOUN
ejpam-5367	386	3	f	f	PROPN
ejpam-5367	386	4	is	be	AUX
ejpam-5367	386	5	an	an	DET
ejpam-5367	386	6	ffiup	ffiup	NOUN
ejpam-5367	386	7	-	-	PUNCT
ejpam-5367	386	8	subalgebra	subalgebra	NOUN
ejpam-5367	386	9	of	of	ADP
ejpam-5367	386	10	x	x	PRON
ejpam-5367	386	11	if	if	SCONJ
ejpam-5367	386	12	and	and	CCONJ
ejpam-5367	386	13	only	only	ADV
ejpam-5367	386	14	if	if	SCONJ
ejpam-5367	386	15	for	for	ADP
ejpam-5367	386	16	all	all	DET
ejpam-5367	386	17	t	t	NOUN
ejpam-5367	386	18	,	,	PUNCT
ejpam-5367	386	19	s	s	PART
ejpam-5367	386	20	∈	∈	PROPN
ejpam-5367	387	1	[	[	X
ejpam-5367	387	2	0	0	NUM
ejpam-5367	387	3	,	,	PUNCT
ejpam-5367	387	4	1	1	NUM
ejpam-5367	387	5	]	]	PUNCT
ejpam-5367	387	6	,	,	PUNCT
ejpam-5367	387	7	the	the	DET
ejpam-5367	387	8	sets	set	NOUN
ejpam-5367	387	9	u(αf	u(αf	NOUN
ejpam-5367	387	10	;	;	PUNCT
ejpam-5367	387	11	t	t	X
ejpam-5367	387	12	)	)	PUNCT
ejpam-5367	387	13	and	and	CCONJ
ejpam-5367	387	14	l(βf	l(βf	PROPN
ejpam-5367	387	15	;	;	PUNCT
ejpam-5367	387	16	s	s	X
ejpam-5367	387	17	)	)	PUNCT
ejpam-5367	387	18	are	be	AUX
ejpam-5367	387	19	either	either	CCONJ
ejpam-5367	387	20	empty	empty	ADJ
ejpam-5367	387	21	or	or	CCONJ
ejpam-5367	387	22	iup	iup	NOUN
ejpam-5367	387	23	-	-	PUNCT
ejpam-5367	387	24	subalgebras	subalgebras	PROPN
ejpam-5367	387	25	of	of	ADP
ejpam-5367	387	26	x.	x.	PROPN
ejpam-5367	387	27	proof	proof	PROPN
ejpam-5367	387	28	.	.	PUNCT
ejpam-5367	388	1	assume	assume	VERB
ejpam-5367	388	2	that	that	SCONJ
ejpam-5367	388	3	f	f	PROPN
ejpam-5367	388	4	is	be	AUX
ejpam-5367	388	5	an	an	DET
ejpam-5367	388	6	ffiup	ffiup	NOUN
ejpam-5367	388	7	-	-	PUNCT
ejpam-5367	388	8	subalgebra	subalgebra	NOUN
ejpam-5367	388	9	of	of	ADP
ejpam-5367	388	10	x.	x.	NOUN
ejpam-5367	388	11	let	let	VERB
ejpam-5367	388	12	t	t	PROPN
ejpam-5367	388	13	∈	∈	PROPN
ejpam-5367	389	1	[	[	X
ejpam-5367	389	2	0	0	NUM
ejpam-5367	389	3	,	,	PUNCT
ejpam-5367	389	4	1	1	NUM
ejpam-5367	389	5	]	]	PUNCT
ejpam-5367	389	6	be	be	AUX
ejpam-5367	389	7	such	such	ADJ
ejpam-5367	389	8	that	that	SCONJ
ejpam-5367	389	9	u(αf	u(αf	NOUN
ejpam-5367	389	10	;	;	PUNCT
ejpam-5367	389	11	t	t	X
ejpam-5367	389	12	)	)	PUNCT
ejpam-5367	389	13	̸=	̸=	PROPN
ejpam-5367	389	14	∅.	∅.	ADV
ejpam-5367	389	15	let	let	VERB
ejpam-5367	389	16	x	x	PRON
ejpam-5367	389	17	,	,	PUNCT
ejpam-5367	389	18	y	y	PROPN
ejpam-5367	389	19	∈	∈	PROPN
ejpam-5367	389	20	u(αf	u(αf	VERB
ejpam-5367	389	21	;	;	PUNCT
ejpam-5367	389	22	t	t	PROPN
ejpam-5367	389	23	)	)	PUNCT
ejpam-5367	389	24	.	.	PUNCT
ejpam-5367	390	1	then	then	ADV
ejpam-5367	390	2	αf	αf	VERB
ejpam-5367	390	3	(	(	PUNCT
ejpam-5367	390	4	x	x	NOUN
ejpam-5367	390	5	)	)	PUNCT
ejpam-5367	390	6	≥	≥	PROPN
ejpam-5367	390	7	t	t	NOUN
ejpam-5367	390	8	and	and	CCONJ
ejpam-5367	390	9	αf	αf	PROPN
ejpam-5367	390	10	(	(	PUNCT
ejpam-5367	390	11	y	y	NOUN
ejpam-5367	390	12	)	)	PUNCT
ejpam-5367	390	13	≥	≥	NOUN
ejpam-5367	390	14	t.	t.	PROPN
ejpam-5367	390	15	thus	thus	ADV
ejpam-5367	390	16	,	,	PUNCT
ejpam-5367	390	17	min{αf	min{αf	PUNCT
ejpam-5367	390	18	(	(	PUNCT
ejpam-5367	390	19	x	x	X
ejpam-5367	390	20	)	)	PUNCT
ejpam-5367	390	21	,	,	PUNCT
ejpam-5367	390	22	αf	αf	X
ejpam-5367	390	23	(	(	PUNCT
ejpam-5367	390	24	y	y	NOUN
ejpam-5367	390	25	)	)	PUNCT
ejpam-5367	390	26	}	}	PUNCT
ejpam-5367	390	27	≥	≥	NOUN
ejpam-5367	390	28	t.	t.	NOUN
ejpam-5367	390	29	by	by	ADP
ejpam-5367	390	30	(	(	PUNCT
ejpam-5367	390	31	3.2	3.2	NUM
ejpam-5367	390	32	)	)	PUNCT
ejpam-5367	390	33	,	,	PUNCT
ejpam-5367	390	34	we	we	PRON
ejpam-5367	390	35	have	have	VERB
ejpam-5367	390	36	αf	αf	ADP
ejpam-5367	390	37	(	(	PUNCT
ejpam-5367	390	38	x	x	X
ejpam-5367	390	39	·	·	PUNCT
ejpam-5367	390	40	y	y	X
ejpam-5367	390	41	)	)	PUNCT
ejpam-5367	390	42	≥	≥	NOUN
ejpam-5367	390	43	min{αf	min{αf	PUNCT
ejpam-5367	390	44	(	(	PUNCT
ejpam-5367	390	45	x	x	X
ejpam-5367	390	46	)	)	PUNCT
ejpam-5367	390	47	,	,	PUNCT
ejpam-5367	390	48	αf	αf	X
ejpam-5367	390	49	(	(	PUNCT
ejpam-5367	390	50	y	y	NOUN
ejpam-5367	390	51	)	)	PUNCT
ejpam-5367	390	52	}	}	PUNCT
ejpam-5367	390	53	≥	≥	PROPN
ejpam-5367	390	54	t	t	PROPN
ejpam-5367	390	55	,	,	PUNCT
ejpam-5367	390	56	that	that	ADV
ejpam-5367	390	57	is	is	ADV
ejpam-5367	390	58	,	,	PUNCT
ejpam-5367	390	59	αf	αf	ADP
ejpam-5367	390	60	(	(	PUNCT
ejpam-5367	390	61	x	x	X
ejpam-5367	390	62	·	·	PUNCT
ejpam-5367	390	63	y	y	X
ejpam-5367	390	64	)	)	PUNCT
ejpam-5367	390	65	≥	≥	NOUN
ejpam-5367	390	66	t.	t.	PROPN
ejpam-5367	390	67	thus	thus	ADV
ejpam-5367	390	68	,	,	PUNCT
ejpam-5367	390	69	x	x	X
ejpam-5367	390	70	·	·	PUNCT
ejpam-5367	390	71	y	y	PROPN
ejpam-5367	390	72	∈	∈	PROPN
ejpam-5367	390	73	u(αf	u(αf	NOUN
ejpam-5367	390	74	;	;	PUNCT
ejpam-5367	390	75	t	t	PROPN
ejpam-5367	390	76	)	)	PUNCT
ejpam-5367	390	77	.	.	PUNCT
ejpam-5367	391	1	hence	hence	ADV
ejpam-5367	391	2	,	,	PUNCT
ejpam-5367	391	3	u(αf	u(αf	PROPN
ejpam-5367	391	4	;	;	PUNCT
ejpam-5367	391	5	t	t	X
ejpam-5367	391	6	)	)	PUNCT
ejpam-5367	391	7	is	be	AUX
ejpam-5367	391	8	an	an	DET
ejpam-5367	391	9	iup	iup	NOUN
ejpam-5367	391	10	-	-	PUNCT
ejpam-5367	391	11	subalgebra	subalgebra	NOUN
ejpam-5367	391	12	of	of	ADP
ejpam-5367	391	13	x.	x.	NOUN
ejpam-5367	391	14	let	let	VERB
ejpam-5367	392	1	s	s	PRON
ejpam-5367	392	2	∈	∈	NOUN
ejpam-5367	392	3	[	[	X
ejpam-5367	392	4	0	0	NUM
ejpam-5367	392	5	,	,	PUNCT
ejpam-5367	392	6	1	1	NUM
ejpam-5367	392	7	]	]	PUNCT
ejpam-5367	392	8	be	be	AUX
ejpam-5367	392	9	such	such	ADJ
ejpam-5367	392	10	that	that	SCONJ
ejpam-5367	392	11	l(βf	l(βf	PROPN
ejpam-5367	392	12	;	;	PUNCT
ejpam-5367	392	13	s	s	X
ejpam-5367	392	14	)	)	PUNCT
ejpam-5367	392	15	̸=	̸=	PROPN
ejpam-5367	392	16	∅.	∅.	ADV
ejpam-5367	392	17	let	let	VERB
ejpam-5367	392	18	x	x	PRON
ejpam-5367	392	19	,	,	PUNCT
ejpam-5367	392	20	y	y	PROPN
ejpam-5367	392	21	∈	∈	PROPN
ejpam-5367	392	22	l(βf	l(βf	PROPN
ejpam-5367	392	23	;	;	PUNCT
ejpam-5367	392	24	s	s	X
ejpam-5367	392	25	)	)	PUNCT
ejpam-5367	392	26	.	.	PUNCT
ejpam-5367	393	1	then	then	ADV
ejpam-5367	393	2	βf	βf	INTJ
ejpam-5367	393	3	(	(	PUNCT
ejpam-5367	393	4	x	x	X
ejpam-5367	393	5	)	)	PUNCT
ejpam-5367	393	6	≤	≤	NOUN
ejpam-5367	393	7	s	s	X
ejpam-5367	393	8	and	and	CCONJ
ejpam-5367	393	9	βf	βf	INTJ
ejpam-5367	393	10	(	(	PUNCT
ejpam-5367	393	11	y	y	NOUN
ejpam-5367	393	12	)	)	PUNCT
ejpam-5367	393	13	≤	≤	NOUN
ejpam-5367	394	1	s.	s.	PROPN
ejpam-5367	394	2	thus	thus	ADV
ejpam-5367	394	3	,	,	PUNCT
ejpam-5367	394	4	max{βf	max{βf	PROPN
ejpam-5367	394	5	(	(	PUNCT
ejpam-5367	394	6	x	x	NOUN
ejpam-5367	394	7	)	)	PUNCT
ejpam-5367	394	8	,	,	PUNCT
ejpam-5367	394	9	βf	βf	CCONJ
ejpam-5367	394	10	(	(	PUNCT
ejpam-5367	394	11	y	y	NOUN
ejpam-5367	394	12	)	)	PUNCT
ejpam-5367	394	13	}	}	PUNCT
ejpam-5367	394	14	≤	≤	NOUN
ejpam-5367	394	15	s.	s.	PROPN
ejpam-5367	394	16	by	by	ADP
ejpam-5367	394	17	(	(	PUNCT
ejpam-5367	394	18	3.3	3.3	NUM
ejpam-5367	394	19	)	)	PUNCT
ejpam-5367	394	20	,	,	PUNCT
ejpam-5367	394	21	we	we	PRON
ejpam-5367	394	22	have	have	VERB
ejpam-5367	394	23	βf	βf	INTJ
ejpam-5367	394	24	(	(	PUNCT
ejpam-5367	394	25	x	x	X
ejpam-5367	394	26	·	·	PUNCT
ejpam-5367	394	27	y	y	X
ejpam-5367	394	28	)	)	PUNCT
ejpam-5367	394	29	≤	≤	NUM
ejpam-5367	395	1	max{βf	max{βf	INTJ
ejpam-5367	395	2	(	(	PUNCT
ejpam-5367	395	3	x	x	NOUN
ejpam-5367	395	4	)	)	PUNCT
ejpam-5367	395	5	,	,	PUNCT
ejpam-5367	395	6	βf	βf	CCONJ
ejpam-5367	395	7	(	(	PUNCT
ejpam-5367	395	8	y	y	NOUN
ejpam-5367	395	9	)	)	PUNCT
ejpam-5367	395	10	}	}	PUNCT
ejpam-5367	395	11	≤	≤	NUM
ejpam-5367	395	12	s	s	X
ejpam-5367	395	13	,	,	PUNCT
ejpam-5367	395	14	that	that	ADV
ejpam-5367	395	15	is	is	ADV
ejpam-5367	395	16	,	,	PUNCT
ejpam-5367	395	17	βf	βf	INTJ
ejpam-5367	395	18	(	(	PUNCT
ejpam-5367	395	19	x	x	X
ejpam-5367	395	20	·	·	PUNCT
ejpam-5367	395	21	y	y	X
ejpam-5367	395	22	)	)	PUNCT
ejpam-5367	395	23	≤	≤	NOUN
ejpam-5367	396	1	s.	s.	PROPN
ejpam-5367	396	2	thus	thus	ADV
ejpam-5367	396	3	,	,	PUNCT
ejpam-5367	396	4	x	x	X
ejpam-5367	396	5	·	·	PUNCT
ejpam-5367	396	6	y	y	PROPN
ejpam-5367	396	7	∈	∈	PROPN
ejpam-5367	396	8	l(βf	l(βf	PROPN
ejpam-5367	396	9	;	;	PUNCT
ejpam-5367	396	10	s	s	X
ejpam-5367	396	11	)	)	PUNCT
ejpam-5367	396	12	.	.	PUNCT
ejpam-5367	397	1	hence	hence	ADV
ejpam-5367	397	2	,	,	PUNCT
ejpam-5367	397	3	l(βf	l(βf	PROPN
ejpam-5367	397	4	;	;	PUNCT
ejpam-5367	397	5	s	s	X
ejpam-5367	397	6	)	)	PUNCT
ejpam-5367	397	7	is	be	AUX
ejpam-5367	397	8	an	an	DET
ejpam-5367	397	9	iup	iup	NOUN
ejpam-5367	397	10	-	-	PUNCT
ejpam-5367	397	11	subalgebra	subalgebra	NOUN
ejpam-5367	397	12	of	of	ADP
ejpam-5367	397	13	x.	x.	NOUN
ejpam-5367	397	14	conversely	conversely	ADV
ejpam-5367	397	15	,	,	PUNCT
ejpam-5367	397	16	assume	assume	VERB
ejpam-5367	397	17	that	that	SCONJ
ejpam-5367	397	18	for	for	ADP
ejpam-5367	397	19	all	all	DET
ejpam-5367	397	20	t	t	PROPN
ejpam-5367	397	21	,	,	PUNCT
ejpam-5367	397	22	s	s	PART
ejpam-5367	397	23	∈	∈	PROPN
ejpam-5367	398	1	[	[	X
ejpam-5367	398	2	0	0	NUM
ejpam-5367	398	3	,	,	PUNCT
ejpam-5367	398	4	1	1	NUM
ejpam-5367	398	5	]	]	PUNCT
ejpam-5367	398	6	,	,	PUNCT
ejpam-5367	398	7	the	the	DET
ejpam-5367	398	8	sets	set	NOUN
ejpam-5367	398	9	u(αf	u(αf	NOUN
ejpam-5367	398	10	;	;	PUNCT
ejpam-5367	398	11	t	t	X
ejpam-5367	398	12	)	)	PUNCT
ejpam-5367	398	13	and	and	CCONJ
ejpam-5367	398	14	l(βf	l(βf	PROPN
ejpam-5367	398	15	;	;	PUNCT
ejpam-5367	398	16	s	s	X
ejpam-5367	398	17	)	)	PUNCT
ejpam-5367	398	18	are	be	AUX
ejpam-5367	398	19	either	either	CCONJ
ejpam-5367	398	20	empty	empty	ADJ
ejpam-5367	398	21	or	or	CCONJ
ejpam-5367	398	22	iup	iup	NOUN
ejpam-5367	398	23	-	-	PUNCT
ejpam-5367	398	24	subalgebras	subalgebras	PROPN
ejpam-5367	398	25	of	of	ADP
ejpam-5367	398	26	x.	x.	PROPN
ejpam-5367	398	27	let	let	VERB
ejpam-5367	398	28	x	x	PRON
ejpam-5367	398	29	,	,	PUNCT
ejpam-5367	398	30	y	y	PROPN
ejpam-5367	398	31	∈	∈	PROPN
ejpam-5367	398	32	x.	x.	NOUN
ejpam-5367	398	33	let	let	VERB
ejpam-5367	398	34	t	t	NOUN
ejpam-5367	398	35	=	=	PUNCT
ejpam-5367	398	36	min{αf	min{αf	PUNCT
ejpam-5367	398	37	(	(	PUNCT
ejpam-5367	398	38	x	x	X
ejpam-5367	398	39	)	)	PUNCT
ejpam-5367	398	40	,	,	PUNCT
ejpam-5367	398	41	αf	αf	X
ejpam-5367	398	42	(	(	PUNCT
ejpam-5367	398	43	y	y	NOUN
ejpam-5367	398	44	)	)	PUNCT
ejpam-5367	398	45	}	}	PUNCT
ejpam-5367	398	46	.	.	PUNCT
ejpam-5367	399	1	then	then	ADV
ejpam-5367	399	2	αf	αf	VERB
ejpam-5367	399	3	(	(	PUNCT
ejpam-5367	399	4	x	x	NOUN
ejpam-5367	399	5	)	)	PUNCT
ejpam-5367	399	6	≥	≥	PROPN
ejpam-5367	399	7	t	t	NOUN
ejpam-5367	399	8	and	and	CCONJ
ejpam-5367	399	9	αf	αf	PROPN
ejpam-5367	399	10	(	(	PUNCT
ejpam-5367	399	11	y	y	NOUN
ejpam-5367	399	12	)	)	PUNCT
ejpam-5367	399	13	≥	≥	NOUN
ejpam-5367	399	14	t.	t.	PROPN
ejpam-5367	399	15	thus	thus	ADV
ejpam-5367	399	16	,	,	PUNCT
ejpam-5367	399	17	x	x	PRON
ejpam-5367	399	18	,	,	PUNCT
ejpam-5367	399	19	y	y	PROPN
ejpam-5367	399	20	∈	∈	PROPN
ejpam-5367	399	21	u(αf	u(αf	VERB
ejpam-5367	399	22	;	;	PUNCT
ejpam-5367	399	23	t	t	X
ejpam-5367	399	24	)	)	PUNCT
ejpam-5367	399	25	̸=	̸=	PROPN
ejpam-5367	399	26	∅.	∅.	NOUN
ejpam-5367	399	27	by	by	ADP
ejpam-5367	399	28	the	the	DET
ejpam-5367	399	29	assumption	assumption	NOUN
ejpam-5367	399	30	,	,	PUNCT
ejpam-5367	399	31	we	we	PRON
ejpam-5367	399	32	have	have	VERB
ejpam-5367	399	33	u(αf	u(αf	NOUN
ejpam-5367	399	34	;	;	PUNCT
ejpam-5367	399	35	t	t	X
ejpam-5367	399	36	)	)	PUNCT
ejpam-5367	399	37	is	be	AUX
ejpam-5367	399	38	an	an	DET
ejpam-5367	399	39	iup	iup	NOUN
ejpam-5367	399	40	-	-	PUNCT
ejpam-5367	399	41	subalgebra	subalgebra	NOUN
ejpam-5367	399	42	of	of	ADP
ejpam-5367	399	43	x.	x.	NOUN
ejpam-5367	399	44	by	by	ADP
ejpam-5367	399	45	(	(	PUNCT
ejpam-5367	399	46	2.17	2.17	NUM
ejpam-5367	399	47	)	)	PUNCT
ejpam-5367	399	48	,	,	PUNCT
ejpam-5367	399	49	we	we	PRON
ejpam-5367	399	50	have	have	VERB
ejpam-5367	399	51	x	x	X
ejpam-5367	399	52	·	·	PUNCT
ejpam-5367	399	53	y	y	PROPN
ejpam-5367	399	54	∈	∈	PROPN
ejpam-5367	399	55	u(αf	u(αf	VERB
ejpam-5367	399	56	;	;	PUNCT
ejpam-5367	399	57	t	t	PROPN
ejpam-5367	399	58	)	)	PUNCT
ejpam-5367	399	59	.	.	PUNCT
ejpam-5367	400	1	thus	thus	ADV
ejpam-5367	400	2	,	,	PUNCT
ejpam-5367	400	3	αf	αf	ADP
ejpam-5367	400	4	(	(	PUNCT
ejpam-5367	400	5	x	x	NOUN
ejpam-5367	400	6	·	·	PUNCT
ejpam-5367	400	7	y	y	NOUN
ejpam-5367	400	8	)	)	PUNCT
ejpam-5367	400	9	≥	≥	NOUN
ejpam-5367	400	10	t	t	NOUN
ejpam-5367	400	11	=	=	PUNCT
ejpam-5367	400	12	min{αf	min{αf	X
ejpam-5367	400	13	(	(	PUNCT
ejpam-5367	400	14	x	x	X
ejpam-5367	400	15	)	)	PUNCT
ejpam-5367	400	16	,	,	PUNCT
ejpam-5367	400	17	αf	αf	X
ejpam-5367	400	18	(	(	PUNCT
ejpam-5367	400	19	y	y	NOUN
ejpam-5367	400	20	)	)	PUNCT
ejpam-5367	400	21	}	}	PUNCT
ejpam-5367	400	22	.	.	PUNCT
ejpam-5367	401	1	let	let	VERB
ejpam-5367	401	2	x	x	PRON
ejpam-5367	401	3	,	,	PUNCT
ejpam-5367	401	4	y	y	PROPN
ejpam-5367	401	5	∈	∈	PROPN
ejpam-5367	401	6	x.	x.	NOUN
ejpam-5367	401	7	let	let	VERB
ejpam-5367	401	8	s	s	AUX
ejpam-5367	401	9	=	=	VERB
ejpam-5367	401	10	max{βf	max{βf	INTJ
ejpam-5367	401	11	(	(	PUNCT
ejpam-5367	401	12	x	x	NOUN
ejpam-5367	401	13	)	)	PUNCT
ejpam-5367	401	14	,	,	PUNCT
ejpam-5367	401	15	βf	βf	CCONJ
ejpam-5367	401	16	(	(	PUNCT
ejpam-5367	401	17	y	y	NOUN
ejpam-5367	401	18	)	)	PUNCT
ejpam-5367	401	19	}	}	PUNCT
ejpam-5367	401	20	.	.	PUNCT
ejpam-5367	402	1	then	then	ADV
ejpam-5367	402	2	βf	βf	INTJ
ejpam-5367	402	3	(	(	PUNCT
ejpam-5367	402	4	x	x	X
ejpam-5367	402	5	)	)	PUNCT
ejpam-5367	402	6	≤	≤	NOUN
ejpam-5367	402	7	s	s	X
ejpam-5367	402	8	and	and	CCONJ
ejpam-5367	402	9	βf	βf	INTJ
ejpam-5367	402	10	(	(	PUNCT
ejpam-5367	402	11	y	y	NOUN
ejpam-5367	402	12	)	)	PUNCT
ejpam-5367	402	13	≤	≤	NOUN
ejpam-5367	403	1	s.	s.	PROPN
ejpam-5367	403	2	thus	thus	ADV
ejpam-5367	403	3	,	,	PUNCT
ejpam-5367	403	4	x	x	PRON
ejpam-5367	403	5	,	,	PUNCT
ejpam-5367	403	6	y	y	PROPN
ejpam-5367	403	7	∈	∈	PROPN
ejpam-5367	403	8	l(βf	l(βf	PROPN
ejpam-5367	403	9	;	;	PUNCT
ejpam-5367	403	10	s	s	X
ejpam-5367	403	11	)	)	PUNCT
ejpam-5367	403	12	̸=	̸=	PROPN
ejpam-5367	403	13	∅.	∅.	NOUN
ejpam-5367	403	14	by	by	ADP
ejpam-5367	403	15	the	the	DET
ejpam-5367	403	16	assumption	assumption	NOUN
ejpam-5367	403	17	,	,	PUNCT
ejpam-5367	403	18	we	we	PRON
ejpam-5367	403	19	have	have	VERB
ejpam-5367	403	20	l(βf	l(βf	NUM
ejpam-5367	403	21	;	;	PUNCT
ejpam-5367	403	22	s	s	X
ejpam-5367	403	23	)	)	PUNCT
ejpam-5367	403	24	is	be	AUX
ejpam-5367	403	25	an	an	DET
ejpam-5367	403	26	iup	iup	NOUN
ejpam-5367	403	27	-	-	PUNCT
ejpam-5367	403	28	subalgebra	subalgebra	NOUN
ejpam-5367	403	29	of	of	ADP
ejpam-5367	403	30	x.	x.	NOUN
ejpam-5367	403	31	by	by	ADP
ejpam-5367	403	32	(	(	PUNCT
ejpam-5367	403	33	2.17	2.17	NUM
ejpam-5367	403	34	)	)	PUNCT
ejpam-5367	403	35	,	,	PUNCT
ejpam-5367	403	36	we	we	PRON
ejpam-5367	403	37	have	have	VERB
ejpam-5367	403	38	x	x	X
ejpam-5367	403	39	·	·	PUNCT
ejpam-5367	403	40	y	y	PROPN
ejpam-5367	403	41	∈	∈	PROPN
ejpam-5367	403	42	l(βf	l(βf	PROPN
ejpam-5367	403	43	;	;	PUNCT
ejpam-5367	403	44	s	s	X
ejpam-5367	403	45	)	)	PUNCT
ejpam-5367	403	46	.	.	PUNCT
ejpam-5367	404	1	thus	thus	ADV
ejpam-5367	404	2	,	,	PUNCT
ejpam-5367	404	3	βf	βf	INTJ
ejpam-5367	404	4	(	(	PUNCT
ejpam-5367	404	5	x	x	X
ejpam-5367	404	6	·	·	PUNCT
ejpam-5367	404	7	y	y	X
ejpam-5367	404	8	)	)	PUNCT
ejpam-5367	404	9	≤	≤	NUM
ejpam-5367	404	10	s	s	PART
ejpam-5367	405	1	=	=	PUNCT
ejpam-5367	405	2	max{βf	max{βf	INTJ
ejpam-5367	405	3	(	(	PUNCT
ejpam-5367	405	4	x	x	NOUN
ejpam-5367	405	5	)	)	PUNCT
ejpam-5367	405	6	,	,	PUNCT
ejpam-5367	405	7	βf	βf	CCONJ
ejpam-5367	405	8	(	(	PUNCT
ejpam-5367	405	9	y	y	NOUN
ejpam-5367	405	10	)	)	PUNCT
ejpam-5367	405	11	}	}	PUNCT
ejpam-5367	405	12	.	.	PUNCT
ejpam-5367	406	1	hence	hence	ADV
ejpam-5367	406	2	,	,	PUNCT
ejpam-5367	406	3	f	f	PROPN
ejpam-5367	406	4	is	be	AUX
ejpam-5367	406	5	an	an	DET
ejpam-5367	406	6	ffiup	ffiup	NOUN
ejpam-5367	406	7	-	-	PUNCT
ejpam-5367	406	8	subalgebra	subalgebra	NOUN
ejpam-5367	406	9	of	of	ADP
ejpam-5367	406	10	x.	x.	PROPN
ejpam-5367	406	11	theorem	theorem	VERB
ejpam-5367	406	12	21	21	NUM
ejpam-5367	406	13	.	.	PUNCT
ejpam-5367	407	1	an	an	DET
ejpam-5367	407	2	ffs	ffs	NOUN
ejpam-5367	407	3	f	f	PROPN
ejpam-5367	407	4	in	in	ADP
ejpam-5367	407	5	x	x	PROPN
ejpam-5367	407	6	is	be	AUX
ejpam-5367	407	7	an	an	DET
ejpam-5367	407	8	ffiup	ffiup	NOUN
ejpam-5367	407	9	-	-	PUNCT
ejpam-5367	407	10	ideal	ideal	NOUN
ejpam-5367	407	11	of	of	ADP
ejpam-5367	407	12	x	x	SYM
ejpam-5367	407	13	if	if	SCONJ
ejpam-5367	407	14	and	and	CCONJ
ejpam-5367	407	15	only	only	ADV
ejpam-5367	407	16	if	if	SCONJ
ejpam-5367	407	17	for	for	ADP
ejpam-5367	407	18	all	all	DET
ejpam-5367	407	19	t	t	NOUN
ejpam-5367	407	20	,	,	PUNCT
ejpam-5367	407	21	s	s	PART
ejpam-5367	407	22	∈	∈	PROPN
ejpam-5367	408	1	[	[	X
ejpam-5367	408	2	0	0	NUM
ejpam-5367	408	3	,	,	PUNCT
ejpam-5367	408	4	1	1	NUM
ejpam-5367	408	5	]	]	PUNCT
ejpam-5367	408	6	,	,	PUNCT
ejpam-5367	408	7	the	the	DET
ejpam-5367	408	8	sets	set	NOUN
ejpam-5367	408	9	u(αf	u(αf	NOUN
ejpam-5367	408	10	;	;	PUNCT
ejpam-5367	408	11	t	t	X
ejpam-5367	408	12	)	)	PUNCT
ejpam-5367	408	13	and	and	CCONJ
ejpam-5367	408	14	l(βf	l(βf	PROPN
ejpam-5367	408	15	;	;	PUNCT
ejpam-5367	408	16	s	s	X
ejpam-5367	408	17	)	)	PUNCT
ejpam-5367	408	18	are	be	AUX
ejpam-5367	408	19	either	either	CCONJ
ejpam-5367	408	20	empty	empty	ADJ
ejpam-5367	408	21	or	or	CCONJ
ejpam-5367	408	22	iup	iup	NOUN
ejpam-5367	408	23	-	-	PUNCT
ejpam-5367	408	24	ideals	ideal	NOUN
ejpam-5367	408	25	of	of	ADP
ejpam-5367	408	26	x.	x.	NOUN
ejpam-5367	408	27	proof	proof	NOUN
ejpam-5367	408	28	.	.	PUNCT
ejpam-5367	409	1	assume	assume	VERB
ejpam-5367	409	2	that	that	SCONJ
ejpam-5367	409	3	f	f	PROPN
ejpam-5367	409	4	is	be	AUX
ejpam-5367	409	5	an	an	DET
ejpam-5367	409	6	ffiup	ffiup	NOUN
ejpam-5367	409	7	-	-	PUNCT
ejpam-5367	409	8	ideal	ideal	NOUN
ejpam-5367	409	9	of	of	ADP
ejpam-5367	409	10	x.	x.	NOUN
ejpam-5367	409	11	let	let	VERB
ejpam-5367	409	12	t	t	PROPN
ejpam-5367	409	13	∈	∈	PROPN
ejpam-5367	410	1	[	[	X
ejpam-5367	410	2	0	0	NUM
ejpam-5367	410	3	,	,	PUNCT
ejpam-5367	410	4	1	1	NUM
ejpam-5367	410	5	]	]	PUNCT
ejpam-5367	410	6	be	be	AUX
ejpam-5367	410	7	such	such	ADJ
ejpam-5367	410	8	that	that	SCONJ
ejpam-5367	410	9	u(αf	u(αf	NOUN
ejpam-5367	410	10	;	;	PUNCT
ejpam-5367	410	11	t	t	X
ejpam-5367	410	12	)	)	PUNCT
ejpam-5367	410	13	̸=	̸=	PROPN
ejpam-5367	410	14	∅.	∅.	ADV
ejpam-5367	410	15	let	let	VERB
ejpam-5367	410	16	r	r	NOUN
ejpam-5367	410	17	∈	∈	PROPN
ejpam-5367	410	18	u(αf	u(αf	NOUN
ejpam-5367	410	19	;	;	PUNCT
ejpam-5367	410	20	t	t	PROPN
ejpam-5367	410	21	)	)	PUNCT
ejpam-5367	410	22	.	.	PUNCT
ejpam-5367	411	1	then	then	ADV
ejpam-5367	411	2	αf	αf	VERB
ejpam-5367	411	3	(	(	PUNCT
ejpam-5367	411	4	r	r	NOUN
ejpam-5367	411	5	)	)	PUNCT
ejpam-5367	411	6	≥	≥	NOUN
ejpam-5367	411	7	t.	t.	NOUN
ejpam-5367	411	8	by	by	ADP
ejpam-5367	411	9	(	(	PUNCT
ejpam-5367	411	10	3.4	3.4	NUM
ejpam-5367	411	11	)	)	PUNCT
ejpam-5367	411	12	,	,	PUNCT
ejpam-5367	411	13	we	we	PRON
ejpam-5367	411	14	have	have	VERB
ejpam-5367	411	15	αf	αf	NUM
ejpam-5367	411	16	(	(	PUNCT
ejpam-5367	411	17	0	0	NUM
ejpam-5367	411	18	)	)	PUNCT
ejpam-5367	411	19	≥	≥	NOUN
ejpam-5367	411	20	αf	αf	X
ejpam-5367	411	21	(	(	PUNCT
ejpam-5367	411	22	r	r	NOUN
ejpam-5367	411	23	)	)	PUNCT
ejpam-5367	411	24	≥	≥	NOUN
ejpam-5367	411	25	t.	t.	PROPN
ejpam-5367	411	26	thus	thus	ADV
ejpam-5367	411	27	,	,	PUNCT
ejpam-5367	411	28	0	0	NUM
ejpam-5367	411	29	∈	∈	NOUN
ejpam-5367	411	30	u(αf	u(αf	NOUN
ejpam-5367	411	31	;	;	PUNCT
ejpam-5367	411	32	t	t	PROPN
ejpam-5367	411	33	)	)	PUNCT
ejpam-5367	411	34	.	.	PUNCT
ejpam-5367	412	1	let	let	VERB
ejpam-5367	412	2	x	x	PRON
ejpam-5367	412	3	,	,	PUNCT
ejpam-5367	412	4	y	y	PROPN
ejpam-5367	412	5	,	,	PUNCT
ejpam-5367	412	6	z	z	NOUN
ejpam-5367	412	7	∈	∈	PROPN
ejpam-5367	412	8	x	x	AUX
ejpam-5367	412	9	be	be	AUX
ejpam-5367	412	10	such	such	ADJ
ejpam-5367	412	11	that	that	SCONJ
ejpam-5367	412	12	x	x	PART
ejpam-5367	412	13	·	·	PUNCT
ejpam-5367	412	14	(	(	PUNCT
ejpam-5367	412	15	y	y	PROPN
ejpam-5367	412	16	·	·	PUNCT
ejpam-5367	412	17	z	z	X
ejpam-5367	412	18	)	)	PUNCT
ejpam-5367	412	19	∈	∈	PROPN
ejpam-5367	412	20	u(αf	u(αf	NOUN
ejpam-5367	412	21	;	;	PUNCT
ejpam-5367	412	22	t	t	X
ejpam-5367	412	23	)	)	PUNCT
ejpam-5367	412	24	and	and	CCONJ
ejpam-5367	412	25	y	y	PROPN
ejpam-5367	412	26	∈	∈	PROPN
ejpam-5367	412	27	u(αf	u(αf	VERB
ejpam-5367	412	28	;	;	PUNCT
ejpam-5367	412	29	t	t	PROPN
ejpam-5367	412	30	)	)	PUNCT
ejpam-5367	412	31	.	.	PUNCT
ejpam-5367	413	1	then	then	ADV
ejpam-5367	413	2	αf	αf	VERB
ejpam-5367	413	3	(	(	PUNCT
ejpam-5367	413	4	x	x	X
ejpam-5367	413	5	·	·	PUNCT
ejpam-5367	413	6	(	(	PUNCT
ejpam-5367	413	7	y	y	PROPN
ejpam-5367	413	8	·	·	PUNCT
ejpam-5367	413	9	z	z	NOUN
ejpam-5367	413	10	)	)	PUNCT
ejpam-5367	413	11	)	)	PUNCT
ejpam-5367	413	12	≥	≥	PROPN
ejpam-5367	413	13	t	t	NOUN
ejpam-5367	413	14	and	and	CCONJ
ejpam-5367	413	15	αf	αf	PROPN
ejpam-5367	413	16	(	(	PUNCT
ejpam-5367	413	17	y	y	NOUN
ejpam-5367	413	18	)	)	PUNCT
ejpam-5367	413	19	≥	≥	NOUN
ejpam-5367	413	20	t.	t.	PROPN
ejpam-5367	413	21	thus	thus	ADV
ejpam-5367	413	22	,	,	PUNCT
ejpam-5367	413	23	min{αf	min{αf	PUNCT
ejpam-5367	413	24	(	(	PUNCT
ejpam-5367	413	25	x	x	X
ejpam-5367	413	26	·	·	PUNCT
ejpam-5367	413	27	(	(	PUNCT
ejpam-5367	413	28	y	y	PROPN
ejpam-5367	413	29	·	·	PUNCT
ejpam-5367	413	30	z	z	NOUN
ejpam-5367	413	31	)	)	PUNCT
ejpam-5367	413	32	)	)	PUNCT
ejpam-5367	413	33	,	,	PUNCT
ejpam-5367	413	34	αf	αf	X
ejpam-5367	413	35	(	(	PUNCT
ejpam-5367	413	36	y	y	NOUN
ejpam-5367	413	37	)	)	PUNCT
ejpam-5367	413	38	}	}	PUNCT
ejpam-5367	413	39	≥	≥	NOUN
ejpam-5367	413	40	t.	t.	X
ejpam-5367	413	41	by	by	ADP
ejpam-5367	413	42	(	(	PUNCT
ejpam-5367	413	43	3.6	3.6	NUM
ejpam-5367	413	44	)	)	PUNCT
ejpam-5367	413	45	,	,	PUNCT
ejpam-5367	413	46	we	we	PRON
ejpam-5367	413	47	have	have	VERB
ejpam-5367	413	48	αf	αf	ADP
ejpam-5367	413	49	(	(	PUNCT
ejpam-5367	413	50	x	x	X
ejpam-5367	413	51	·	·	PUNCT
ejpam-5367	413	52	z	z	X
ejpam-5367	413	53	)	)	PUNCT
ejpam-5367	413	54	≥	≥	NOUN
ejpam-5367	413	55	min{αf	min{αf	PUNCT
ejpam-5367	413	56	(	(	PUNCT
ejpam-5367	413	57	x	x	X
ejpam-5367	413	58	·	·	PUNCT
ejpam-5367	413	59	(	(	PUNCT
ejpam-5367	413	60	y	y	PROPN
ejpam-5367	413	61	·	·	PUNCT
ejpam-5367	413	62	z	z	NOUN
ejpam-5367	413	63	)	)	PUNCT
ejpam-5367	413	64	)	)	PUNCT
ejpam-5367	413	65	,	,	PUNCT
ejpam-5367	413	66	αf	αf	X
ejpam-5367	413	67	(	(	PUNCT
ejpam-5367	413	68	y	y	NOUN
ejpam-5367	413	69	)	)	PUNCT
ejpam-5367	413	70	}	}	PUNCT
ejpam-5367	413	71	≥	≥	X
ejpam-5367	413	72	t.	t.	PROPN
ejpam-5367	413	73	thus	thus	ADV
ejpam-5367	413	74	,	,	PUNCT
ejpam-5367	413	75	x	x	X
ejpam-5367	413	76	·	·	PUNCT
ejpam-5367	413	77	z	z	X
ejpam-5367	413	78	∈	∈	PROPN
ejpam-5367	413	79	u(αf	u(αf	NOUN
ejpam-5367	413	80	;	;	PUNCT
ejpam-5367	413	81	t	t	PROPN
ejpam-5367	413	82	)	)	PUNCT
ejpam-5367	413	83	.	.	PUNCT
ejpam-5367	414	1	hence	hence	ADV
ejpam-5367	414	2	,	,	PUNCT
ejpam-5367	414	3	u(αf	u(αf	PROPN
ejpam-5367	414	4	;	;	PUNCT
ejpam-5367	414	5	t	t	X
ejpam-5367	414	6	)	)	PUNCT
ejpam-5367	414	7	is	be	AUX
ejpam-5367	414	8	an	an	DET
ejpam-5367	414	9	iup	iup	NOUN
ejpam-5367	414	10	-	-	PUNCT
ejpam-5367	414	11	ideal	ideal	NOUN
ejpam-5367	414	12	of	of	ADP
ejpam-5367	414	13	x.	x.	NOUN
ejpam-5367	414	14	let	let	VERB
ejpam-5367	415	1	s	s	PRON
ejpam-5367	415	2	∈	∈	NOUN
ejpam-5367	415	3	[	[	X
ejpam-5367	415	4	0	0	NUM
ejpam-5367	415	5	,	,	PUNCT
ejpam-5367	415	6	1	1	NUM
ejpam-5367	415	7	]	]	PUNCT
ejpam-5367	415	8	be	be	AUX
ejpam-5367	415	9	such	such	ADJ
ejpam-5367	415	10	that	that	SCONJ
ejpam-5367	415	11	l(βf	l(βf	PROPN
ejpam-5367	415	12	;	;	PUNCT
ejpam-5367	415	13	s	s	X
ejpam-5367	415	14	)	)	PUNCT
ejpam-5367	415	15	̸=	̸=	PROPN
ejpam-5367	415	16	∅.	∅.	ADV
ejpam-5367	415	17	let	let	VERB
ejpam-5367	415	18	β	β	NOUN
ejpam-5367	415	19	∈	∈	PROPN
ejpam-5367	415	20	l(βf	l(βf	PROPN
ejpam-5367	415	21	;	;	PUNCT
ejpam-5367	415	22	s	s	X
ejpam-5367	415	23	)	)	PUNCT
ejpam-5367	415	24	.	.	PUNCT
ejpam-5367	416	1	then	then	ADV
ejpam-5367	416	2	βf	βf	INTJ
ejpam-5367	416	3	(	(	PUNCT
ejpam-5367	416	4	l	l	NOUN
ejpam-5367	416	5	)	)	PUNCT
ejpam-5367	416	6	≤	≤	NOUN
ejpam-5367	416	7	s.	s.	PROPN
ejpam-5367	416	8	by	by	ADP
ejpam-5367	416	9	(	(	PUNCT
ejpam-5367	416	10	3.5	3.5	NUM
ejpam-5367	416	11	)	)	PUNCT
ejpam-5367	416	12	,	,	PUNCT
ejpam-5367	416	13	we	we	PRON
ejpam-5367	416	14	have	have	VERB
ejpam-5367	416	15	βf	βf	ADV
ejpam-5367	416	16	(	(	PUNCT
ejpam-5367	416	17	0	0	X
ejpam-5367	416	18	)	)	PUNCT
ejpam-5367	416	19	≤	≤	NOUN
ejpam-5367	417	1	βf	βf	CCONJ
ejpam-5367	417	2	(	(	PUNCT
ejpam-5367	417	3	l	l	NOUN
ejpam-5367	417	4	)	)	PUNCT
ejpam-5367	417	5	≤	≤	NOUN
ejpam-5367	418	1	s.	s.	PROPN
ejpam-5367	418	2	thus	thus	ADV
ejpam-5367	418	3	,	,	PUNCT
ejpam-5367	418	4	0	0	NUM
ejpam-5367	418	5	∈	∈	PROPN
ejpam-5367	418	6	l(βf	l(βf	PROPN
ejpam-5367	418	7	;	;	PUNCT
ejpam-5367	418	8	s	s	X
ejpam-5367	418	9	)	)	PUNCT
ejpam-5367	418	10	.	.	PUNCT
ejpam-5367	419	1	let	let	VERB
ejpam-5367	419	2	x	x	PRON
ejpam-5367	419	3	,	,	PUNCT
ejpam-5367	419	4	y	y	PROPN
ejpam-5367	419	5	,	,	PUNCT
ejpam-5367	419	6	z	z	NOUN
ejpam-5367	419	7	∈	∈	PROPN
ejpam-5367	419	8	x	x	AUX
ejpam-5367	419	9	be	be	AUX
ejpam-5367	419	10	such	such	ADJ
ejpam-5367	419	11	that	that	SCONJ
ejpam-5367	419	12	x	x	PART
ejpam-5367	419	13	·	·	PUNCT
ejpam-5367	419	14	(	(	PUNCT
ejpam-5367	419	15	y	y	PROPN
ejpam-5367	419	16	·	·	PUNCT
ejpam-5367	419	17	z	z	X
ejpam-5367	419	18	)	)	PUNCT
ejpam-5367	419	19	∈	∈	PROPN
ejpam-5367	419	20	l(βf	l(βf	PROPN
ejpam-5367	419	21	;	;	PUNCT
ejpam-5367	419	22	s	s	X
ejpam-5367	419	23	)	)	PUNCT
ejpam-5367	419	24	and	and	CCONJ
ejpam-5367	419	25	y	y	PROPN
ejpam-5367	419	26	∈	∈	PROPN
ejpam-5367	419	27	l(βf	l(βf	PROPN
ejpam-5367	419	28	;	;	PUNCT
ejpam-5367	419	29	s	s	X
ejpam-5367	419	30	)	)	PUNCT
ejpam-5367	419	31	.	.	PUNCT
ejpam-5367	420	1	then	then	ADV
ejpam-5367	420	2	βf	βf	INTJ
ejpam-5367	420	3	(	(	PUNCT
ejpam-5367	420	4	x	x	X
ejpam-5367	420	5	·	·	PUNCT
ejpam-5367	420	6	(	(	PUNCT
ejpam-5367	420	7	y	y	PROPN
ejpam-5367	420	8	·	·	PUNCT
ejpam-5367	420	9	z	z	NOUN
ejpam-5367	420	10	)	)	PUNCT
ejpam-5367	420	11	)	)	PUNCT
ejpam-5367	420	12	≤	≤	PROPN
ejpam-5367	420	13	s	s	X
ejpam-5367	420	14	and	and	CCONJ
ejpam-5367	420	15	βf	βf	INTJ
ejpam-5367	420	16	(	(	PUNCT
ejpam-5367	420	17	y	y	NOUN
ejpam-5367	420	18	)	)	PUNCT
ejpam-5367	420	19	≤	≤	NOUN
ejpam-5367	420	20	s.	s.	PROPN
ejpam-5367	420	21	thus	thus	ADV
ejpam-5367	420	22	,	,	PUNCT
ejpam-5367	420	23	max{βf	max{βf	PROPN
ejpam-5367	420	24	(	(	PUNCT
ejpam-5367	420	25	x·(y·z	x·(y·z	PROPN
ejpam-5367	420	26	)	)	PUNCT
ejpam-5367	420	27	)	)	PUNCT
ejpam-5367	420	28	,	,	PUNCT
ejpam-5367	420	29	βf	βf	CCONJ
ejpam-5367	420	30	(	(	PUNCT
ejpam-5367	420	31	y	y	NOUN
ejpam-5367	420	32	)	)	PUNCT
ejpam-5367	420	33	}	}	PUNCT
ejpam-5367	420	34	≤	≤	NOUN
ejpam-5367	420	35	s.	s.	PROPN
ejpam-5367	420	36	by	by	ADP
ejpam-5367	420	37	(	(	PUNCT
ejpam-5367	420	38	3.7	3.7	NUM
ejpam-5367	420	39	)	)	PUNCT
ejpam-5367	420	40	,	,	PUNCT
ejpam-5367	420	41	we	we	PRON
ejpam-5367	420	42	have	have	VERB
ejpam-5367	420	43	βf	βf	ADV
ejpam-5367	420	44	(	(	PUNCT
ejpam-5367	420	45	x·z	x·z	NOUN
ejpam-5367	420	46	)	)	PUNCT
ejpam-5367	420	47	≤	≤	NUM
ejpam-5367	421	1	max{βf	max{βf	INTJ
ejpam-5367	421	2	(	(	PUNCT
ejpam-5367	421	3	x·(y·z	x·(y·z	PROPN
ejpam-5367	421	4	)	)	PUNCT
ejpam-5367	421	5	)	)	PUNCT
ejpam-5367	421	6	,	,	PUNCT
ejpam-5367	421	7	βf	βf	CCONJ
ejpam-5367	421	8	(	(	PUNCT
ejpam-5367	421	9	y	y	NOUN
ejpam-5367	421	10	)	)	PUNCT
ejpam-5367	421	11	}	}	PUNCT
ejpam-5367	421	12	≤	≤	PROPN
ejpam-5367	421	13	s.	s.	PROPN
ejpam-5367	421	14	thus	thus	ADV
ejpam-5367	421	15	,	,	PUNCT
ejpam-5367	421	16	x	x	X
ejpam-5367	421	17	·	·	PUNCT
ejpam-5367	421	18	z	z	X
ejpam-5367	421	19	∈	∈	PROPN
ejpam-5367	421	20	l(βf	l(βf	PROPN
ejpam-5367	421	21	;	;	PUNCT
ejpam-5367	421	22	s	s	X
ejpam-5367	421	23	)	)	PUNCT
ejpam-5367	421	24	.	.	PUNCT
ejpam-5367	422	1	hence	hence	ADV
ejpam-5367	422	2	,	,	PUNCT
ejpam-5367	422	3	l(βf	l(βf	PROPN
ejpam-5367	422	4	;	;	PUNCT
ejpam-5367	422	5	s	s	X
ejpam-5367	422	6	)	)	PUNCT
ejpam-5367	422	7	is	be	AUX
ejpam-5367	422	8	an	an	DET
ejpam-5367	422	9	iup	iup	NOUN
ejpam-5367	422	10	-	-	PUNCT
ejpam-5367	422	11	ideal	ideal	NOUN
ejpam-5367	422	12	of	of	ADP
ejpam-5367	422	13	x.	x.	NOUN
ejpam-5367	422	14	conversely	conversely	ADV
ejpam-5367	422	15	,	,	PUNCT
ejpam-5367	422	16	assume	assume	VERB
ejpam-5367	422	17	that	that	SCONJ
ejpam-5367	422	18	for	for	ADP
ejpam-5367	422	19	all	all	DET
ejpam-5367	422	20	t	t	PROPN
ejpam-5367	422	21	,	,	PUNCT
ejpam-5367	422	22	s	s	PART
ejpam-5367	422	23	∈	∈	PROPN
ejpam-5367	423	1	[	[	X
ejpam-5367	423	2	0	0	NUM
ejpam-5367	423	3	,	,	PUNCT
ejpam-5367	423	4	1	1	NUM
ejpam-5367	423	5	]	]	PUNCT
ejpam-5367	423	6	,	,	PUNCT
ejpam-5367	423	7	the	the	DET
ejpam-5367	423	8	sets	set	NOUN
ejpam-5367	423	9	u(αf	u(αf	NOUN
ejpam-5367	423	10	;	;	PUNCT
ejpam-5367	423	11	t	t	X
ejpam-5367	423	12	)	)	PUNCT
ejpam-5367	423	13	and	and	CCONJ
ejpam-5367	423	14	l(βf	l(βf	PROPN
ejpam-5367	423	15	;	;	PUNCT
ejpam-5367	423	16	s	s	X
ejpam-5367	423	17	)	)	PUNCT
ejpam-5367	423	18	are	be	AUX
ejpam-5367	423	19	either	either	CCONJ
ejpam-5367	423	20	empty	empty	ADJ
ejpam-5367	423	21	or	or	CCONJ
ejpam-5367	423	22	iup	iup	NOUN
ejpam-5367	423	23	-	-	PUNCT
ejpam-5367	423	24	ideals	ideal	NOUN
ejpam-5367	423	25	of	of	ADP
ejpam-5367	423	26	x.	x.	NOUN
ejpam-5367	423	27	let	let	VERB
ejpam-5367	423	28	x	x	AUX
ejpam-5367	423	29	∈	∈	PROPN
ejpam-5367	423	30	x.	x.	NOUN
ejpam-5367	423	31	let	let	VERB
ejpam-5367	423	32	t	t	NOUN
ejpam-5367	423	33	=	=	PUNCT
ejpam-5367	423	34	αf	αf	X
ejpam-5367	423	35	(	(	PUNCT
ejpam-5367	423	36	x	x	NOUN
ejpam-5367	423	37	)	)	PUNCT
ejpam-5367	423	38	.	.	PUNCT
ejpam-5367	424	1	then	then	ADV
ejpam-5367	424	2	αf	αf	VERB
ejpam-5367	424	3	(	(	PUNCT
ejpam-5367	424	4	x	x	NOUN
ejpam-5367	424	5	)	)	PUNCT
ejpam-5367	424	6	≥	≥	NOUN
ejpam-5367	424	7	t.	t.	PROPN
ejpam-5367	424	8	thus	thus	ADV
ejpam-5367	424	9	,	,	PUNCT
ejpam-5367	424	10	x	x	SYM
ejpam-5367	424	11	∈	∈	NOUN
ejpam-5367	424	12	u(αf	u(αf	NOUN
ejpam-5367	424	13	;	;	PUNCT
ejpam-5367	424	14	t	t	X
ejpam-5367	424	15	)	)	PUNCT
ejpam-5367	424	16	̸=	̸=	PROPN
ejpam-5367	424	17	∅.	∅.	NOUN
ejpam-5367	424	18	by	by	ADP
ejpam-5367	424	19	the	the	DET
ejpam-5367	424	20	assumption	assumption	NOUN
ejpam-5367	424	21	,	,	PUNCT
ejpam-5367	424	22	we	we	PRON
ejpam-5367	424	23	have	have	VERB
ejpam-5367	424	24	u(αf	u(αf	NOUN
ejpam-5367	424	25	;	;	PUNCT
ejpam-5367	424	26	t	t	X
ejpam-5367	424	27	)	)	PUNCT
ejpam-5367	424	28	is	be	AUX
ejpam-5367	424	29	an	an	DET
ejpam-5367	424	30	iup	iup	NOUN
ejpam-5367	424	31	-	-	PUNCT
ejpam-5367	424	32	ideal	ideal	NOUN
ejpam-5367	424	33	of	of	ADP
ejpam-5367	424	34	x.	x.	NOUN
ejpam-5367	424	35	by	by	ADP
ejpam-5367	424	36	(	(	PUNCT
ejpam-5367	424	37	2.18	2.18	NUM
ejpam-5367	424	38	)	)	PUNCT
ejpam-5367	424	39	,	,	PUNCT
ejpam-5367	424	40	we	we	PRON
ejpam-5367	424	41	have	have	VERB
ejpam-5367	424	42	0	0	NUM
ejpam-5367	424	43	∈	∈	NOUN
ejpam-5367	424	44	u(αf	u(αf	NOUN
ejpam-5367	424	45	;	;	PUNCT
ejpam-5367	424	46	t	t	PROPN
ejpam-5367	424	47	)	)	PUNCT
ejpam-5367	424	48	.	.	PUNCT
ejpam-5367	425	1	then	then	ADV
ejpam-5367	425	2	αf	αf	VERB
ejpam-5367	425	3	(	(	PUNCT
ejpam-5367	425	4	0	0	NUM
ejpam-5367	425	5	)	)	PUNCT
ejpam-5367	425	6	≥	≥	NOUN
ejpam-5367	425	7	t	t	NOUN
ejpam-5367	425	8	=	=	SYM
ejpam-5367	425	9	αf	αf	X
ejpam-5367	425	10	(	(	PUNCT
ejpam-5367	425	11	x	x	NOUN
ejpam-5367	425	12	)	)	PUNCT
ejpam-5367	425	13	.	.	PUNCT
ejpam-5367	426	1	let	let	VERB
ejpam-5367	426	2	x	x	PRON
ejpam-5367	426	3	,	,	PUNCT
ejpam-5367	426	4	y	y	PROPN
ejpam-5367	426	5	,	,	PUNCT
ejpam-5367	426	6	z	z	PROPN
ejpam-5367	426	7	∈	∈	PROPN
ejpam-5367	426	8	x.	x.	NOUN
ejpam-5367	426	9	let	let	VERB
ejpam-5367	426	10	t	t	NOUN
ejpam-5367	426	11	=	=	PUNCT
ejpam-5367	426	12	min{αf	min{αf	PUNCT
ejpam-5367	426	13	(	(	PUNCT
ejpam-5367	426	14	x	x	X
ejpam-5367	426	15	·	·	PUNCT
ejpam-5367	426	16	(	(	PUNCT
ejpam-5367	426	17	y	y	X
ejpam-5367	426	18	·	·	PUNCT
ejpam-5367	426	19	a.	a.	NOUN
ejpam-5367	426	20	iampan	iampan	NOUN
ejpam-5367	426	21	et	et	PROPN
ejpam-5367	426	22	al	al	PROPN
ejpam-5367	426	23	.	.	PUNCT
ejpam-5367	426	24	/	/	SYM
ejpam-5367	426	25	eur	eur	PROPN
ejpam-5367	426	26	.	.	PUNCT
ejpam-5367	427	1	j.	j.	PROPN
ejpam-5367	427	2	pure	pure	PROPN
ejpam-5367	427	3	appl	appl	PROPN
ejpam-5367	427	4	.	.	PROPN
ejpam-5367	427	5	math	math	PROPN
ejpam-5367	427	6	,	,	PUNCT
ejpam-5367	427	7	17	17	NUM
ejpam-5367	427	8	(	(	PUNCT
ejpam-5367	427	9	4	4	NUM
ejpam-5367	427	10	)	)	PUNCT
ejpam-5367	427	11	(	(	PUNCT
ejpam-5367	427	12	2024	2024	NUM
ejpam-5367	427	13	)	)	PUNCT
ejpam-5367	427	14	,	,	PUNCT
ejpam-5367	427	15	3022	3022	NUM
ejpam-5367	427	16	-	-	SYM
ejpam-5367	427	17	3042	3042	NUM
ejpam-5367	427	18	3038	3038	NUM
ejpam-5367	427	19	z	z	NOUN
ejpam-5367	427	20	)	)	PUNCT
ejpam-5367	427	21	)	)	PUNCT
ejpam-5367	427	22	,	,	PUNCT
ejpam-5367	427	23	αf	αf	X
ejpam-5367	427	24	(	(	PUNCT
ejpam-5367	427	25	y	y	NOUN
ejpam-5367	427	26	)	)	PUNCT
ejpam-5367	427	27	}	}	PUNCT
ejpam-5367	427	28	.	.	PUNCT
ejpam-5367	428	1	then	then	ADV
ejpam-5367	428	2	αf	αf	VERB
ejpam-5367	428	3	(	(	PUNCT
ejpam-5367	428	4	x	x	X
ejpam-5367	428	5	·	·	PUNCT
ejpam-5367	428	6	(	(	PUNCT
ejpam-5367	428	7	y	y	PROPN
ejpam-5367	428	8	·	·	PUNCT
ejpam-5367	428	9	z	z	NOUN
ejpam-5367	428	10	)	)	PUNCT
ejpam-5367	428	11	)	)	PUNCT
ejpam-5367	428	12	≥	≥	PROPN
ejpam-5367	428	13	t	t	NOUN
ejpam-5367	428	14	and	and	CCONJ
ejpam-5367	428	15	αf	αf	PROPN
ejpam-5367	428	16	(	(	PUNCT
ejpam-5367	428	17	y	y	NOUN
ejpam-5367	428	18	)	)	PUNCT
ejpam-5367	428	19	≥	≥	NOUN
ejpam-5367	428	20	t.	t.	PROPN
ejpam-5367	428	21	thus	thus	ADV
ejpam-5367	428	22	,	,	PUNCT
ejpam-5367	428	23	x	x	X
ejpam-5367	428	24	·	·	PUNCT
ejpam-5367	428	25	(	(	PUNCT
ejpam-5367	428	26	y	y	PROPN
ejpam-5367	428	27	·	·	PUNCT
ejpam-5367	428	28	z	z	X
ejpam-5367	428	29	)	)	PUNCT
ejpam-5367	428	30	,	,	PUNCT
ejpam-5367	428	31	y	y	PROPN
ejpam-5367	428	32	∈	∈	PROPN
ejpam-5367	428	33	u(αf	u(αf	VERB
ejpam-5367	428	34	;	;	PUNCT
ejpam-5367	428	35	t	t	X
ejpam-5367	428	36	)	)	PUNCT
ejpam-5367	428	37	̸=	̸=	PROPN
ejpam-5367	428	38	∅.	∅.	NOUN
ejpam-5367	428	39	by	by	ADP
ejpam-5367	428	40	the	the	DET
ejpam-5367	428	41	assumption	assumption	NOUN
ejpam-5367	428	42	,	,	PUNCT
ejpam-5367	428	43	we	we	PRON
ejpam-5367	428	44	have	have	VERB
ejpam-5367	428	45	u(αf	u(αf	NOUN
ejpam-5367	428	46	;	;	PUNCT
ejpam-5367	428	47	t	t	X
ejpam-5367	428	48	)	)	PUNCT
ejpam-5367	428	49	is	be	AUX
ejpam-5367	428	50	an	an	DET
ejpam-5367	428	51	iup	iup	NOUN
ejpam-5367	428	52	-	-	PUNCT
ejpam-5367	428	53	ideal	ideal	NOUN
ejpam-5367	428	54	of	of	ADP
ejpam-5367	428	55	x.	x.	NOUN
ejpam-5367	428	56	by	by	ADP
ejpam-5367	428	57	(	(	PUNCT
ejpam-5367	428	58	2.20	2.20	NUM
ejpam-5367	428	59	)	)	PUNCT
ejpam-5367	428	60	,	,	PUNCT
ejpam-5367	428	61	we	we	PRON
ejpam-5367	428	62	have	have	VERB
ejpam-5367	428	63	x·z	x·z	PROPN
ejpam-5367	428	64	∈	∈	PROPN
ejpam-5367	428	65	u(αf	u(αf	NOUN
ejpam-5367	428	66	;	;	PUNCT
ejpam-5367	428	67	t	t	PROPN
ejpam-5367	428	68	)	)	PUNCT
ejpam-5367	428	69	.	.	PUNCT
ejpam-5367	429	1	thus	thus	ADV
ejpam-5367	429	2	,	,	PUNCT
ejpam-5367	429	3	αf	αf	ADP
ejpam-5367	429	4	(	(	PUNCT
ejpam-5367	429	5	x	x	X
ejpam-5367	429	6	·	·	PUNCT
ejpam-5367	429	7	z	z	X
ejpam-5367	429	8	)	)	PUNCT
ejpam-5367	429	9	≥	≥	NOUN
ejpam-5367	429	10	t	t	NOUN
ejpam-5367	429	11	=	=	PUNCT
ejpam-5367	429	12	min{αf	min{αf	X
ejpam-5367	429	13	(	(	PUNCT
ejpam-5367	429	14	x	x	X
ejpam-5367	429	15	·	·	PUNCT
ejpam-5367	429	16	(	(	PUNCT
ejpam-5367	429	17	y	y	PROPN
ejpam-5367	429	18	·	·	PUNCT
ejpam-5367	429	19	z	z	NOUN
ejpam-5367	429	20	)	)	PUNCT
ejpam-5367	429	21	)	)	PUNCT
ejpam-5367	429	22	,	,	PUNCT
ejpam-5367	429	23	αf	αf	X
ejpam-5367	429	24	(	(	PUNCT
ejpam-5367	429	25	y	y	NOUN
ejpam-5367	429	26	)	)	PUNCT
ejpam-5367	429	27	}	}	PUNCT
ejpam-5367	429	28	.	.	PUNCT
ejpam-5367	430	1	let	let	VERB
ejpam-5367	430	2	x	x	SYM
ejpam-5367	430	3	∈	∈	PROPN
ejpam-5367	430	4	x.	x.	NOUN
ejpam-5367	430	5	let	let	VERB
ejpam-5367	430	6	s	s	AUX
ejpam-5367	430	7	=	=	VERB
ejpam-5367	430	8	βf	βf	INTJ
ejpam-5367	430	9	(	(	PUNCT
ejpam-5367	430	10	x	x	NOUN
ejpam-5367	430	11	)	)	PUNCT
ejpam-5367	430	12	.	.	PUNCT
ejpam-5367	431	1	then	then	ADV
ejpam-5367	431	2	βf	βf	INTJ
ejpam-5367	431	3	(	(	PUNCT
ejpam-5367	431	4	x	x	X
ejpam-5367	431	5	)	)	PUNCT
ejpam-5367	431	6	≤	≤	NOUN
ejpam-5367	431	7	s.	s.	PROPN
ejpam-5367	431	8	thus	thus	ADV
ejpam-5367	431	9	,	,	PUNCT
ejpam-5367	431	10	x	x	PUNCT
ejpam-5367	431	11	∈	∈	PROPN
ejpam-5367	431	12	l(βf	l(βf	PROPN
ejpam-5367	431	13	;	;	PUNCT
ejpam-5367	431	14	s	s	X
ejpam-5367	431	15	)	)	PUNCT
ejpam-5367	431	16	̸=	̸=	PROPN
ejpam-5367	431	17	∅.	∅.	NOUN
ejpam-5367	431	18	by	by	ADP
ejpam-5367	431	19	the	the	DET
ejpam-5367	431	20	assumption	assumption	NOUN
ejpam-5367	431	21	,	,	PUNCT
ejpam-5367	431	22	we	we	PRON
ejpam-5367	431	23	have	have	VERB
ejpam-5367	431	24	l(βf	l(βf	NUM
ejpam-5367	431	25	;	;	PUNCT
ejpam-5367	431	26	s	s	X
ejpam-5367	431	27	)	)	PUNCT
ejpam-5367	431	28	is	be	AUX
ejpam-5367	431	29	an	an	DET
ejpam-5367	431	30	iup	iup	NOUN
ejpam-5367	431	31	-	-	PUNCT
ejpam-5367	431	32	ideal	ideal	NOUN
ejpam-5367	431	33	of	of	ADP
ejpam-5367	431	34	x.	x.	NOUN
ejpam-5367	431	35	by	by	ADP
ejpam-5367	431	36	(	(	PUNCT
ejpam-5367	431	37	2.18	2.18	NUM
ejpam-5367	431	38	)	)	PUNCT
ejpam-5367	431	39	,	,	PUNCT
ejpam-5367	431	40	we	we	PRON
ejpam-5367	431	41	have	have	VERB
ejpam-5367	431	42	0	0	NUM
ejpam-5367	431	43	∈	∈	PROPN
ejpam-5367	431	44	l(βf	l(βf	PROPN
ejpam-5367	431	45	;	;	PUNCT
ejpam-5367	431	46	s	s	X
ejpam-5367	431	47	)	)	PUNCT
ejpam-5367	431	48	.	.	PUNCT
ejpam-5367	432	1	then	then	ADV
ejpam-5367	432	2	βf	βf	INTJ
ejpam-5367	432	3	(	(	PUNCT
ejpam-5367	432	4	0	0	X
ejpam-5367	432	5	)	)	PUNCT
ejpam-5367	432	6	≤	≤	NOUN
ejpam-5367	432	7	s	s	PART
ejpam-5367	433	1	=	=	X
ejpam-5367	433	2	βf	βf	INTJ
ejpam-5367	433	3	(	(	PUNCT
ejpam-5367	433	4	x	x	NOUN
ejpam-5367	433	5	)	)	PUNCT
ejpam-5367	433	6	.	.	PUNCT
ejpam-5367	434	1	let	let	VERB
ejpam-5367	434	2	x	x	PRON
ejpam-5367	434	3	,	,	PUNCT
ejpam-5367	434	4	y	y	PROPN
ejpam-5367	434	5	,	,	PUNCT
ejpam-5367	434	6	z	z	PROPN
ejpam-5367	434	7	∈	∈	PROPN
ejpam-5367	434	8	x.	x.	NOUN
ejpam-5367	434	9	let	let	VERB
ejpam-5367	434	10	s	s	AUX
ejpam-5367	434	11	=	=	VERB
ejpam-5367	434	12	max{βf	max{βf	INTJ
ejpam-5367	434	13	(	(	PUNCT
ejpam-5367	434	14	x	x	X
ejpam-5367	434	15	·	·	PUNCT
ejpam-5367	434	16	(	(	PUNCT
ejpam-5367	434	17	y	y	PROPN
ejpam-5367	434	18	·	·	PUNCT
ejpam-5367	434	19	z	z	NOUN
ejpam-5367	434	20	)	)	PUNCT
ejpam-5367	434	21	)	)	PUNCT
ejpam-5367	434	22	,	,	PUNCT
ejpam-5367	434	23	βf	βf	CCONJ
ejpam-5367	434	24	(	(	PUNCT
ejpam-5367	434	25	y	y	NOUN
ejpam-5367	434	26	)	)	PUNCT
ejpam-5367	434	27	}	}	PUNCT
ejpam-5367	434	28	.	.	PUNCT
ejpam-5367	435	1	then	then	ADV
ejpam-5367	435	2	βf	βf	INTJ
ejpam-5367	435	3	(	(	PUNCT
ejpam-5367	435	4	x	x	X
ejpam-5367	435	5	·	·	PUNCT
ejpam-5367	435	6	(	(	PUNCT
ejpam-5367	435	7	y	y	PROPN
ejpam-5367	435	8	·	·	PUNCT
ejpam-5367	435	9	z	z	NOUN
ejpam-5367	435	10	)	)	PUNCT
ejpam-5367	435	11	)	)	PUNCT
ejpam-5367	436	1	≤	≤	PROPN
ejpam-5367	436	2	s	s	X
ejpam-5367	436	3	and	and	CCONJ
ejpam-5367	436	4	βf	βf	INTJ
ejpam-5367	436	5	(	(	PUNCT
ejpam-5367	436	6	y	y	NOUN
ejpam-5367	436	7	)	)	PUNCT
ejpam-5367	436	8	≤	≤	NOUN
ejpam-5367	436	9	s.	s.	PROPN
ejpam-5367	436	10	thus	thus	ADV
ejpam-5367	436	11	,	,	PUNCT
ejpam-5367	436	12	x	x	X
ejpam-5367	436	13	·	·	PUNCT
ejpam-5367	436	14	(	(	PUNCT
ejpam-5367	436	15	y	y	PROPN
ejpam-5367	436	16	·	·	PUNCT
ejpam-5367	436	17	z	z	X
ejpam-5367	436	18	)	)	PUNCT
ejpam-5367	436	19	,	,	PUNCT
ejpam-5367	436	20	y	y	PROPN
ejpam-5367	436	21	∈	∈	PROPN
ejpam-5367	436	22	l(βf	l(βf	PROPN
ejpam-5367	436	23	;	;	PUNCT
ejpam-5367	436	24	s	s	X
ejpam-5367	436	25	)	)	PUNCT
ejpam-5367	436	26	̸=	̸=	PROPN
ejpam-5367	436	27	∅.	∅.	NOUN
ejpam-5367	436	28	by	by	ADP
ejpam-5367	436	29	the	the	DET
ejpam-5367	436	30	assumption	assumption	NOUN
ejpam-5367	436	31	,	,	PUNCT
ejpam-5367	436	32	we	we	PRON
ejpam-5367	436	33	have	have	VERB
ejpam-5367	436	34	u(βf	u(βf	ADJ
ejpam-5367	436	35	;	;	PUNCT
ejpam-5367	436	36	s	s	X
ejpam-5367	436	37	)	)	PUNCT
ejpam-5367	436	38	is	be	AUX
ejpam-5367	436	39	an	an	DET
ejpam-5367	436	40	iup	iup	NOUN
ejpam-5367	436	41	-	-	PUNCT
ejpam-5367	436	42	ideal	ideal	NOUN
ejpam-5367	436	43	of	of	ADP
ejpam-5367	436	44	x.	x.	NOUN
ejpam-5367	436	45	by	by	ADP
ejpam-5367	436	46	(	(	PUNCT
ejpam-5367	436	47	2.20	2.20	NUM
ejpam-5367	436	48	)	)	PUNCT
ejpam-5367	436	49	,	,	PUNCT
ejpam-5367	436	50	we	we	PRON
ejpam-5367	436	51	have	have	VERB
ejpam-5367	436	52	x	x	X
ejpam-5367	436	53	·	·	PUNCT
ejpam-5367	436	54	z	z	SYM
ejpam-5367	436	55	∈	∈	PROPN
ejpam-5367	436	56	l(βf	l(βf	PROPN
ejpam-5367	436	57	;	;	PUNCT
ejpam-5367	436	58	s	s	X
ejpam-5367	436	59	)	)	PUNCT
ejpam-5367	436	60	.	.	PUNCT
ejpam-5367	437	1	thus	thus	ADV
ejpam-5367	437	2	,	,	PUNCT
ejpam-5367	437	3	βf	βf	INTJ
ejpam-5367	437	4	(	(	PUNCT
ejpam-5367	437	5	x	x	X
ejpam-5367	437	6	·	·	PUNCT
ejpam-5367	437	7	z	z	X
ejpam-5367	437	8	)	)	PUNCT
ejpam-5367	437	9	≤	≤	NUM
ejpam-5367	437	10	s	s	PART
ejpam-5367	437	11	=	=	PUNCT
ejpam-5367	437	12	max{βf	max{βf	INTJ
ejpam-5367	437	13	(	(	PUNCT
ejpam-5367	437	14	x	x	X
ejpam-5367	437	15	·	·	PUNCT
ejpam-5367	437	16	(	(	PUNCT
ejpam-5367	437	17	y	y	PROPN
ejpam-5367	437	18	·	·	PUNCT
ejpam-5367	437	19	z	z	NOUN
ejpam-5367	437	20	)	)	PUNCT
ejpam-5367	437	21	)	)	PUNCT
ejpam-5367	437	22	,	,	PUNCT
ejpam-5367	437	23	βf	βf	CCONJ
ejpam-5367	437	24	(	(	PUNCT
ejpam-5367	437	25	y	y	NOUN
ejpam-5367	437	26	)	)	PUNCT
ejpam-5367	437	27	}	}	PUNCT
ejpam-5367	437	28	.	.	PUNCT
ejpam-5367	438	1	hence	hence	ADV
ejpam-5367	438	2	,	,	PUNCT
ejpam-5367	438	3	f	f	PROPN
ejpam-5367	438	4	is	be	AUX
ejpam-5367	438	5	an	an	DET
ejpam-5367	438	6	ffiup	ffiup	NOUN
ejpam-5367	438	7	-	-	PUNCT
ejpam-5367	438	8	ideal	ideal	NOUN
ejpam-5367	438	9	of	of	ADP
ejpam-5367	438	10	x.	x.	PROPN
ejpam-5367	438	11	theorem	theorem	VERB
ejpam-5367	438	12	22	22	NUM
ejpam-5367	438	13	.	.	PUNCT
ejpam-5367	439	1	an	an	DET
ejpam-5367	439	2	ffs	ffs	NOUN
ejpam-5367	439	3	f	f	PROPN
ejpam-5367	439	4	in	in	ADP
ejpam-5367	439	5	x	x	PROPN
ejpam-5367	439	6	is	be	AUX
ejpam-5367	439	7	an	an	DET
ejpam-5367	439	8	ffiup	ffiup	NOUN
ejpam-5367	439	9	-	-	PUNCT
ejpam-5367	439	10	filter	filter	NOUN
ejpam-5367	439	11	of	of	ADP
ejpam-5367	439	12	x	x	SYM
ejpam-5367	439	13	if	if	SCONJ
ejpam-5367	439	14	and	and	CCONJ
ejpam-5367	439	15	only	only	ADV
ejpam-5367	439	16	if	if	SCONJ
ejpam-5367	439	17	for	for	ADP
ejpam-5367	439	18	all	all	DET
ejpam-5367	439	19	t	t	NOUN
ejpam-5367	439	20	,	,	PUNCT
ejpam-5367	439	21	s	s	PART
ejpam-5367	439	22	∈	∈	PROPN
ejpam-5367	440	1	[	[	X
ejpam-5367	440	2	0	0	NUM
ejpam-5367	440	3	,	,	PUNCT
ejpam-5367	440	4	1	1	NUM
ejpam-5367	440	5	]	]	PUNCT
ejpam-5367	440	6	,	,	PUNCT
ejpam-5367	440	7	the	the	DET
ejpam-5367	440	8	sets	set	NOUN
ejpam-5367	440	9	u(αf	u(αf	NOUN
ejpam-5367	440	10	;	;	PUNCT
ejpam-5367	440	11	t	t	X
ejpam-5367	440	12	)	)	PUNCT
ejpam-5367	440	13	and	and	CCONJ
ejpam-5367	440	14	l(βf	l(βf	PROPN
ejpam-5367	440	15	;	;	PUNCT
ejpam-5367	440	16	s	s	X
ejpam-5367	440	17	)	)	PUNCT
ejpam-5367	440	18	are	be	AUX
ejpam-5367	440	19	either	either	CCONJ
ejpam-5367	440	20	empty	empty	ADJ
ejpam-5367	440	21	or	or	CCONJ
ejpam-5367	440	22	iup	iup	NOUN
ejpam-5367	440	23	-	-	PUNCT
ejpam-5367	440	24	filters	filter	NOUN
ejpam-5367	440	25	of	of	ADP
ejpam-5367	440	26	x.	x.	NOUN
ejpam-5367	440	27	proof	proof	PROPN
ejpam-5367	440	28	.	.	PUNCT
ejpam-5367	441	1	assume	assume	VERB
ejpam-5367	441	2	that	that	SCONJ
ejpam-5367	441	3	f	f	PROPN
ejpam-5367	441	4	is	be	AUX
ejpam-5367	441	5	an	an	DET
ejpam-5367	441	6	ffiup	ffiup	ADJ
ejpam-5367	441	7	-	-	PUNCT
ejpam-5367	441	8	filter	filter	NOUN
ejpam-5367	441	9	of	of	ADP
ejpam-5367	441	10	x.	x.	NOUN
ejpam-5367	441	11	let	let	VERB
ejpam-5367	441	12	t	t	PROPN
ejpam-5367	441	13	∈	∈	PROPN
ejpam-5367	442	1	[	[	X
ejpam-5367	442	2	0	0	NUM
ejpam-5367	442	3	,	,	PUNCT
ejpam-5367	442	4	1	1	NUM
ejpam-5367	442	5	]	]	PUNCT
ejpam-5367	442	6	be	be	AUX
ejpam-5367	442	7	such	such	ADJ
ejpam-5367	442	8	that	that	SCONJ
ejpam-5367	442	9	u(αf	u(αf	NOUN
ejpam-5367	442	10	;	;	PUNCT
ejpam-5367	442	11	t	t	X
ejpam-5367	442	12	)	)	PUNCT
ejpam-5367	442	13	̸=	̸=	PROPN
ejpam-5367	442	14	∅.	∅.	ADV
ejpam-5367	442	15	let	let	VERB
ejpam-5367	442	16	r	r	NOUN
ejpam-5367	442	17	∈	∈	PROPN
ejpam-5367	442	18	u(αf	u(αf	NOUN
ejpam-5367	442	19	;	;	PUNCT
ejpam-5367	442	20	t	t	PROPN
ejpam-5367	442	21	)	)	PUNCT
ejpam-5367	442	22	.	.	PUNCT
ejpam-5367	443	1	then	then	ADV
ejpam-5367	443	2	αf	αf	VERB
ejpam-5367	443	3	(	(	PUNCT
ejpam-5367	443	4	r	r	NOUN
ejpam-5367	443	5	)	)	PUNCT
ejpam-5367	443	6	≥	≥	NOUN
ejpam-5367	443	7	t.	t.	NOUN
ejpam-5367	443	8	by	by	ADP
ejpam-5367	443	9	(	(	PUNCT
ejpam-5367	443	10	3.4	3.4	NUM
ejpam-5367	443	11	)	)	PUNCT
ejpam-5367	443	12	,	,	PUNCT
ejpam-5367	443	13	we	we	PRON
ejpam-5367	443	14	have	have	VERB
ejpam-5367	443	15	αf	αf	NUM
ejpam-5367	443	16	(	(	PUNCT
ejpam-5367	443	17	0	0	NUM
ejpam-5367	443	18	)	)	PUNCT
ejpam-5367	443	19	≥	≥	NOUN
ejpam-5367	443	20	αf	αf	X
ejpam-5367	443	21	(	(	PUNCT
ejpam-5367	443	22	r	r	NOUN
ejpam-5367	443	23	)	)	PUNCT
ejpam-5367	443	24	≥	≥	NOUN
ejpam-5367	443	25	t.	t.	PROPN
ejpam-5367	443	26	thus	thus	ADV
ejpam-5367	443	27	,	,	PUNCT
ejpam-5367	443	28	0	0	NUM
ejpam-5367	443	29	∈	∈	NOUN
ejpam-5367	443	30	u(αf	u(αf	NOUN
ejpam-5367	443	31	;	;	PUNCT
ejpam-5367	443	32	t	t	PROPN
ejpam-5367	443	33	)	)	PUNCT
ejpam-5367	443	34	.	.	PUNCT
ejpam-5367	444	1	let	let	VERB
ejpam-5367	444	2	x	x	PRON
ejpam-5367	444	3	,	,	PUNCT
ejpam-5367	444	4	y	y	PROPN
ejpam-5367	444	5	∈	∈	PROPN
ejpam-5367	444	6	x	x	AUX
ejpam-5367	444	7	be	be	AUX
ejpam-5367	444	8	such	such	ADJ
ejpam-5367	444	9	that	that	SCONJ
ejpam-5367	444	10	x	x	X
ejpam-5367	444	11	·	·	PUNCT
ejpam-5367	444	12	y	y	PROPN
ejpam-5367	444	13	∈	∈	PROPN
ejpam-5367	444	14	u(αf	u(αf	NOUN
ejpam-5367	444	15	;	;	PUNCT
ejpam-5367	444	16	t	t	X
ejpam-5367	444	17	)	)	PUNCT
ejpam-5367	444	18	and	and	CCONJ
ejpam-5367	444	19	x	x	PUNCT
ejpam-5367	444	20	∈	∈	NOUN
ejpam-5367	444	21	u(αf	u(αf	NOUN
ejpam-5367	444	22	;	;	PUNCT
ejpam-5367	444	23	t	t	PROPN
ejpam-5367	444	24	)	)	PUNCT
ejpam-5367	444	25	.	.	PUNCT
ejpam-5367	445	1	then	then	ADV
ejpam-5367	445	2	αf	αf	VERB
ejpam-5367	445	3	(	(	PUNCT
ejpam-5367	445	4	x	x	PROPN
ejpam-5367	445	5	·	·	PUNCT
ejpam-5367	445	6	y	y	X
ejpam-5367	445	7	)	)	PUNCT
ejpam-5367	445	8	≥	≥	NOUN
ejpam-5367	445	9	t	t	NOUN
ejpam-5367	445	10	and	and	CCONJ
ejpam-5367	445	11	αf	αf	PROPN
ejpam-5367	445	12	(	(	PUNCT
ejpam-5367	445	13	x	x	NOUN
ejpam-5367	445	14	)	)	PUNCT
ejpam-5367	445	15	≥	≥	NOUN
ejpam-5367	445	16	t.	t.	PROPN
ejpam-5367	445	17	thus	thus	ADV
ejpam-5367	445	18	,	,	PUNCT
ejpam-5367	445	19	min{αf	min{αf	PUNCT
ejpam-5367	445	20	(	(	PUNCT
ejpam-5367	445	21	x	x	X
ejpam-5367	445	22	·	·	PUNCT
ejpam-5367	445	23	y	y	X
ejpam-5367	445	24	)	)	PUNCT
ejpam-5367	445	25	,	,	PUNCT
ejpam-5367	445	26	αf	αf	X
ejpam-5367	445	27	(	(	PUNCT
ejpam-5367	445	28	x	x	NOUN
ejpam-5367	445	29	)	)	PUNCT
ejpam-5367	445	30	}	}	PUNCT
ejpam-5367	445	31	≥	≥	NOUN
ejpam-5367	445	32	t.	t.	X
ejpam-5367	445	33	by	by	ADP
ejpam-5367	445	34	(	(	PUNCT
ejpam-5367	445	35	3.8	3.8	NUM
ejpam-5367	445	36	)	)	PUNCT
ejpam-5367	445	37	,	,	PUNCT
ejpam-5367	445	38	we	we	PRON
ejpam-5367	445	39	have	have	VERB
ejpam-5367	445	40	αf	αf	NUM
ejpam-5367	445	41	(	(	PUNCT
ejpam-5367	445	42	y	y	NOUN
ejpam-5367	445	43	)	)	PUNCT
ejpam-5367	445	44	≥	≥	NOUN
ejpam-5367	445	45	min{αf	min{αf	PUNCT
ejpam-5367	445	46	(	(	PUNCT
ejpam-5367	445	47	x	x	X
ejpam-5367	445	48	·	·	PUNCT
ejpam-5367	445	49	y	y	X
ejpam-5367	445	50	)	)	PUNCT
ejpam-5367	445	51	,	,	PUNCT
ejpam-5367	445	52	αf	αf	X
ejpam-5367	445	53	(	(	PUNCT
ejpam-5367	445	54	x	x	NOUN
ejpam-5367	445	55	)	)	PUNCT
ejpam-5367	445	56	}	}	PUNCT
ejpam-5367	445	57	≥	≥	NOUN
ejpam-5367	445	58	t.	t.	PROPN
ejpam-5367	445	59	thus	thus	ADV
ejpam-5367	445	60	,	,	PUNCT
ejpam-5367	445	61	y	y	PROPN
ejpam-5367	445	62	∈	∈	PROPN
ejpam-5367	445	63	u(αf	u(αf	VERB
ejpam-5367	445	64	;	;	PUNCT
ejpam-5367	445	65	t	t	PROPN
ejpam-5367	445	66	)	)	PUNCT
ejpam-5367	445	67	.	.	PUNCT
ejpam-5367	446	1	hence	hence	ADV
ejpam-5367	446	2	,	,	PUNCT
ejpam-5367	446	3	u(αf	u(αf	PROPN
ejpam-5367	446	4	;	;	PUNCT
ejpam-5367	446	5	t	t	X
ejpam-5367	446	6	)	)	PUNCT
ejpam-5367	446	7	is	be	AUX
ejpam-5367	446	8	an	an	DET
ejpam-5367	446	9	iup	iup	NOUN
ejpam-5367	446	10	-	-	PUNCT
ejpam-5367	446	11	filter	filter	NOUN
ejpam-5367	446	12	of	of	ADP
ejpam-5367	446	13	x.	x.	NOUN
ejpam-5367	446	14	let	let	VERB
ejpam-5367	447	1	s	s	PRON
ejpam-5367	447	2	∈	∈	NOUN
ejpam-5367	447	3	[	[	X
ejpam-5367	447	4	0	0	NUM
ejpam-5367	447	5	,	,	PUNCT
ejpam-5367	447	6	1	1	NUM
ejpam-5367	447	7	]	]	PUNCT
ejpam-5367	447	8	be	be	AUX
ejpam-5367	447	9	such	such	ADJ
ejpam-5367	447	10	that	that	SCONJ
ejpam-5367	447	11	l(βf	l(βf	PROPN
ejpam-5367	447	12	;	;	PUNCT
ejpam-5367	447	13	s	s	X
ejpam-5367	447	14	)	)	PUNCT
ejpam-5367	447	15	̸=	̸=	PROPN
ejpam-5367	447	16	∅.	∅.	ADV
ejpam-5367	447	17	let	let	VERB
ejpam-5367	447	18	l	l	NOUN
ejpam-5367	447	19	∈	∈	PROPN
ejpam-5367	447	20	l(βf	l(βf	PROPN
ejpam-5367	447	21	;	;	PUNCT
ejpam-5367	447	22	s	s	X
ejpam-5367	447	23	)	)	PUNCT
ejpam-5367	447	24	.	.	PUNCT
ejpam-5367	448	1	then	then	ADV
ejpam-5367	448	2	βf	βf	INTJ
ejpam-5367	448	3	(	(	PUNCT
ejpam-5367	448	4	l	l	NOUN
ejpam-5367	448	5	)	)	PUNCT
ejpam-5367	448	6	≤	≤	NOUN
ejpam-5367	448	7	s.	s.	PROPN
ejpam-5367	448	8	by	by	ADP
ejpam-5367	448	9	(	(	PUNCT
ejpam-5367	448	10	3.5	3.5	NUM
ejpam-5367	448	11	)	)	PUNCT
ejpam-5367	448	12	,	,	PUNCT
ejpam-5367	448	13	we	we	PRON
ejpam-5367	448	14	have	have	VERB
ejpam-5367	448	15	βf	βf	ADV
ejpam-5367	448	16	(	(	PUNCT
ejpam-5367	448	17	0	0	X
ejpam-5367	448	18	)	)	PUNCT
ejpam-5367	448	19	≤	≤	NOUN
ejpam-5367	449	1	βf	βf	CCONJ
ejpam-5367	449	2	(	(	PUNCT
ejpam-5367	449	3	l	l	NOUN
ejpam-5367	449	4	)	)	PUNCT
ejpam-5367	449	5	≤	≤	NOUN
ejpam-5367	450	1	s.	s.	PROPN
ejpam-5367	450	2	thus	thus	ADV
ejpam-5367	450	3	,	,	PUNCT
ejpam-5367	450	4	0	0	NUM
ejpam-5367	450	5	∈	∈	PROPN
ejpam-5367	450	6	l(βf	l(βf	PROPN
ejpam-5367	450	7	;	;	PUNCT
ejpam-5367	450	8	s	s	X
ejpam-5367	450	9	)	)	PUNCT
ejpam-5367	450	10	.	.	PUNCT
ejpam-5367	451	1	let	let	VERB
ejpam-5367	451	2	x	x	PRON
ejpam-5367	451	3	,	,	PUNCT
ejpam-5367	451	4	y	y	PROPN
ejpam-5367	451	5	∈	∈	PROPN
ejpam-5367	451	6	x	x	AUX
ejpam-5367	451	7	be	be	AUX
ejpam-5367	451	8	such	such	ADJ
ejpam-5367	451	9	that	that	SCONJ
ejpam-5367	451	10	x	x	X
ejpam-5367	451	11	·	·	PUNCT
ejpam-5367	451	12	y	y	PROPN
ejpam-5367	451	13	∈	∈	PROPN
ejpam-5367	451	14	l(βf	l(βf	PROPN
ejpam-5367	451	15	;	;	PUNCT
ejpam-5367	451	16	s	s	X
ejpam-5367	451	17	)	)	PUNCT
ejpam-5367	451	18	and	and	CCONJ
ejpam-5367	451	19	x	x	PUNCT
ejpam-5367	451	20	∈	∈	PROPN
ejpam-5367	451	21	l(βf	l(βf	PROPN
ejpam-5367	451	22	;	;	PUNCT
ejpam-5367	451	23	s	s	X
ejpam-5367	451	24	)	)	PUNCT
ejpam-5367	451	25	.	.	PUNCT
ejpam-5367	452	1	then	then	ADV
ejpam-5367	452	2	βf	βf	INTJ
ejpam-5367	452	3	(	(	PUNCT
ejpam-5367	452	4	x·y	x·y	PROPN
ejpam-5367	452	5	)	)	PUNCT
ejpam-5367	452	6	≤	≤	PROPN
ejpam-5367	453	1	s	s	X
ejpam-5367	453	2	and	and	CCONJ
ejpam-5367	453	3	βf	βf	INTJ
ejpam-5367	453	4	(	(	PUNCT
ejpam-5367	453	5	x	x	X
ejpam-5367	453	6	)	)	PUNCT
ejpam-5367	453	7	≤	≤	NOUN
ejpam-5367	454	1	s.	s.	PROPN
ejpam-5367	454	2	thus	thus	ADV
ejpam-5367	454	3	,	,	PUNCT
ejpam-5367	454	4	max{βf	max{βf	PROPN
ejpam-5367	454	5	(	(	PUNCT
ejpam-5367	454	6	x·y	x·y	PROPN
ejpam-5367	454	7	)	)	PUNCT
ejpam-5367	454	8	,	,	PUNCT
ejpam-5367	454	9	βf	βf	CCONJ
ejpam-5367	454	10	(	(	PUNCT
ejpam-5367	454	11	x	x	X
ejpam-5367	454	12	)	)	PUNCT
ejpam-5367	454	13	}	}	PUNCT
ejpam-5367	454	14	≤	≤	NOUN
ejpam-5367	454	15	s.	s.	PROPN
ejpam-5367	454	16	by	by	ADP
ejpam-5367	454	17	(	(	PUNCT
ejpam-5367	454	18	3.9	3.9	NUM
ejpam-5367	454	19	)	)	PUNCT
ejpam-5367	454	20	,	,	PUNCT
ejpam-5367	454	21	we	we	PRON
ejpam-5367	454	22	have	have	VERB
ejpam-5367	454	23	βf	βf	NUM
ejpam-5367	454	24	(	(	PUNCT
ejpam-5367	454	25	y	y	NOUN
ejpam-5367	454	26	)	)	PUNCT
ejpam-5367	454	27	≤	≤	NOUN
ejpam-5367	455	1	max{βf	max{βf	INTJ
ejpam-5367	455	2	(	(	PUNCT
ejpam-5367	455	3	x	x	X
ejpam-5367	455	4	·	·	PUNCT
ejpam-5367	455	5	y	y	X
ejpam-5367	455	6	)	)	PUNCT
ejpam-5367	455	7	,	,	PUNCT
ejpam-5367	455	8	βf	βf	CCONJ
ejpam-5367	455	9	(	(	PUNCT
ejpam-5367	455	10	x	x	X
ejpam-5367	455	11	)	)	PUNCT
ejpam-5367	455	12	}	}	PUNCT
ejpam-5367	455	13	≤	≤	PROPN
ejpam-5367	455	14	s.	s.	PROPN
ejpam-5367	455	15	thus	thus	ADV
ejpam-5367	455	16	,	,	PUNCT
ejpam-5367	455	17	y	y	PROPN
ejpam-5367	455	18	∈	∈	PROPN
ejpam-5367	455	19	l(βf	l(βf	PROPN
ejpam-5367	455	20	;	;	PUNCT
ejpam-5367	455	21	s	s	X
ejpam-5367	455	22	)	)	PUNCT
ejpam-5367	455	23	.	.	PUNCT
ejpam-5367	456	1	hence	hence	ADV
ejpam-5367	456	2	,	,	PUNCT
ejpam-5367	456	3	l(βf	l(βf	PROPN
ejpam-5367	456	4	;	;	PUNCT
ejpam-5367	456	5	s	s	X
ejpam-5367	456	6	)	)	PUNCT
ejpam-5367	456	7	is	be	AUX
ejpam-5367	456	8	an	an	DET
ejpam-5367	456	9	iup	iup	NOUN
ejpam-5367	456	10	-	-	PUNCT
ejpam-5367	456	11	ideal	ideal	NOUN
ejpam-5367	456	12	of	of	ADP
ejpam-5367	456	13	x.	x.	NOUN
ejpam-5367	456	14	conversely	conversely	ADV
ejpam-5367	456	15	,	,	PUNCT
ejpam-5367	456	16	assume	assume	VERB
ejpam-5367	456	17	that	that	SCONJ
ejpam-5367	456	18	for	for	ADP
ejpam-5367	456	19	all	all	DET
ejpam-5367	456	20	t	t	PROPN
ejpam-5367	456	21	,	,	PUNCT
ejpam-5367	456	22	s	s	PART
ejpam-5367	456	23	∈	∈	PROPN
ejpam-5367	457	1	[	[	X
ejpam-5367	457	2	0	0	NUM
ejpam-5367	457	3	,	,	PUNCT
ejpam-5367	457	4	1	1	NUM
ejpam-5367	457	5	]	]	PUNCT
ejpam-5367	457	6	,	,	PUNCT
ejpam-5367	457	7	the	the	DET
ejpam-5367	457	8	sets	set	NOUN
ejpam-5367	457	9	u(αf	u(αf	NOUN
ejpam-5367	457	10	;	;	PUNCT
ejpam-5367	457	11	t	t	X
ejpam-5367	457	12	)	)	PUNCT
ejpam-5367	457	13	and	and	CCONJ
ejpam-5367	457	14	l(βf	l(βf	PROPN
ejpam-5367	457	15	;	;	PUNCT
ejpam-5367	457	16	s	s	X
ejpam-5367	457	17	)	)	PUNCT
ejpam-5367	457	18	are	be	AUX
ejpam-5367	457	19	either	either	CCONJ
ejpam-5367	457	20	empty	empty	ADJ
ejpam-5367	457	21	or	or	CCONJ
ejpam-5367	457	22	iup	iup	NOUN
ejpam-5367	457	23	-	-	PUNCT
ejpam-5367	457	24	filters	filter	NOUN
ejpam-5367	457	25	of	of	ADP
ejpam-5367	457	26	x.	x.	NOUN
ejpam-5367	457	27	let	let	VERB
ejpam-5367	457	28	x	x	AUX
ejpam-5367	457	29	∈	∈	PROPN
ejpam-5367	457	30	x.	x.	NOUN
ejpam-5367	457	31	let	let	VERB
ejpam-5367	457	32	t	t	NOUN
ejpam-5367	457	33	=	=	PUNCT
ejpam-5367	457	34	αf	αf	X
ejpam-5367	457	35	(	(	PUNCT
ejpam-5367	457	36	x	x	NOUN
ejpam-5367	457	37	)	)	PUNCT
ejpam-5367	457	38	.	.	PUNCT
ejpam-5367	458	1	then	then	ADV
ejpam-5367	458	2	αf	αf	VERB
ejpam-5367	458	3	(	(	PUNCT
ejpam-5367	458	4	x	x	NOUN
ejpam-5367	458	5	)	)	PUNCT
ejpam-5367	458	6	≥	≥	NOUN
ejpam-5367	458	7	t.	t.	PROPN
ejpam-5367	458	8	thus	thus	ADV
ejpam-5367	458	9	,	,	PUNCT
ejpam-5367	458	10	x	x	SYM
ejpam-5367	458	11	∈	∈	NOUN
ejpam-5367	458	12	u(αf	u(αf	NOUN
ejpam-5367	458	13	;	;	PUNCT
ejpam-5367	458	14	t	t	X
ejpam-5367	458	15	)	)	PUNCT
ejpam-5367	458	16	̸=	̸=	PROPN
ejpam-5367	458	17	∅.	∅.	NOUN
ejpam-5367	458	18	by	by	ADP
ejpam-5367	458	19	the	the	DET
ejpam-5367	458	20	assumption	assumption	NOUN
ejpam-5367	458	21	,	,	PUNCT
ejpam-5367	458	22	we	we	PRON
ejpam-5367	458	23	have	have	VERB
ejpam-5367	458	24	u(αf	u(αf	NOUN
ejpam-5367	458	25	;	;	PUNCT
ejpam-5367	458	26	t	t	X
ejpam-5367	458	27	)	)	PUNCT
ejpam-5367	458	28	is	be	AUX
ejpam-5367	458	29	an	an	DET
ejpam-5367	458	30	iup	iup	ADJ
ejpam-5367	458	31	-	-	PUNCT
ejpam-5367	458	32	filter	filter	NOUN
ejpam-5367	458	33	ofx	ofx	NOUN
ejpam-5367	458	34	.	.	PUNCT
ejpam-5367	459	1	by	by	ADP
ejpam-5367	459	2	(	(	PUNCT
ejpam-5367	459	3	2.18	2.18	NUM
ejpam-5367	459	4	)	)	PUNCT
ejpam-5367	459	5	,	,	PUNCT
ejpam-5367	459	6	we	we	PRON
ejpam-5367	459	7	have	have	VERB
ejpam-5367	459	8	0	0	NUM
ejpam-5367	459	9	∈	∈	NOUN
ejpam-5367	459	10	u(αf	u(αf	NOUN
ejpam-5367	459	11	;	;	PUNCT
ejpam-5367	459	12	t	t	PROPN
ejpam-5367	459	13	)	)	PUNCT
ejpam-5367	459	14	.	.	PUNCT
ejpam-5367	460	1	then	then	ADV
ejpam-5367	460	2	αf	αf	VERB
ejpam-5367	460	3	(	(	PUNCT
ejpam-5367	460	4	0	0	NUM
ejpam-5367	460	5	)	)	PUNCT
ejpam-5367	460	6	≥	≥	NOUN
ejpam-5367	460	7	t	t	NOUN
ejpam-5367	460	8	=	=	SYM
ejpam-5367	460	9	αf	αf	X
ejpam-5367	460	10	(	(	PUNCT
ejpam-5367	460	11	x	x	NOUN
ejpam-5367	460	12	)	)	PUNCT
ejpam-5367	460	13	.	.	PUNCT
ejpam-5367	461	1	let	let	VERB
ejpam-5367	461	2	x	x	PRON
ejpam-5367	461	3	,	,	PUNCT
ejpam-5367	461	4	y	y	PROPN
ejpam-5367	461	5	∈	∈	PROPN
ejpam-5367	461	6	x.	x.	NOUN
ejpam-5367	461	7	let	let	VERB
ejpam-5367	461	8	t	t	PROPN
ejpam-5367	461	9	=	=	PUNCT
ejpam-5367	461	10	min{αf	min{αf	X
ejpam-5367	461	11	(	(	PUNCT
ejpam-5367	461	12	x·y	x·y	PROPN
ejpam-5367	461	13	)	)	PUNCT
ejpam-5367	461	14	,	,	PUNCT
ejpam-5367	461	15	αf	αf	X
ejpam-5367	461	16	(	(	PUNCT
ejpam-5367	461	17	x	x	NOUN
ejpam-5367	461	18	)	)	PUNCT
ejpam-5367	461	19	}	}	PUNCT
ejpam-5367	461	20	.	.	PUNCT
ejpam-5367	462	1	then	then	ADV
ejpam-5367	462	2	αf	αf	VERB
ejpam-5367	462	3	(	(	PUNCT
ejpam-5367	462	4	x	x	PROPN
ejpam-5367	462	5	·	·	PUNCT
ejpam-5367	462	6	y	y	X
ejpam-5367	462	7	)	)	PUNCT
ejpam-5367	462	8	≥	≥	NOUN
ejpam-5367	462	9	t	t	NOUN
ejpam-5367	462	10	and	and	CCONJ
ejpam-5367	462	11	αf	αf	PROPN
ejpam-5367	462	12	(	(	PUNCT
ejpam-5367	462	13	x	x	NOUN
ejpam-5367	462	14	)	)	PUNCT
ejpam-5367	462	15	≥	≥	NOUN
ejpam-5367	462	16	t.	t.	PROPN
ejpam-5367	462	17	thus	thus	ADV
ejpam-5367	462	18	,	,	PUNCT
ejpam-5367	462	19	x	x	X
ejpam-5367	462	20	·	·	PUNCT
ejpam-5367	462	21	y	y	X
ejpam-5367	462	22	,	,	PUNCT
ejpam-5367	462	23	x	x	SYM
ejpam-5367	462	24	∈	∈	NOUN
ejpam-5367	462	25	u(αf	u(αf	NOUN
ejpam-5367	462	26	;	;	PUNCT
ejpam-5367	462	27	t	t	X
ejpam-5367	462	28	)	)	PUNCT
ejpam-5367	462	29	̸=	̸=	PROPN
ejpam-5367	462	30	∅.	∅.	NOUN
ejpam-5367	462	31	by	by	ADP
ejpam-5367	462	32	the	the	DET
ejpam-5367	462	33	assumption	assumption	NOUN
ejpam-5367	462	34	,	,	PUNCT
ejpam-5367	462	35	we	we	PRON
ejpam-5367	462	36	have	have	AUX
ejpam-5367	462	37	u(αf	u(αf	NOUN
ejpam-5367	462	38	;	;	PUNCT
ejpam-5367	462	39	t	t	X
ejpam-5367	462	40	)	)	PUNCT
ejpam-5367	462	41	is	be	AUX
ejpam-5367	462	42	an	an	DET
ejpam-5367	462	43	iup	iup	NOUN
ejpam-5367	462	44	-	-	PUNCT
ejpam-5367	462	45	filter	filter	NOUN
ejpam-5367	462	46	of	of	ADP
ejpam-5367	462	47	x.	x.	NOUN
ejpam-5367	462	48	by	by	ADP
ejpam-5367	462	49	(	(	PUNCT
ejpam-5367	462	50	2.19	2.19	NUM
ejpam-5367	462	51	)	)	PUNCT
ejpam-5367	462	52	,	,	PUNCT
ejpam-5367	462	53	we	we	PRON
ejpam-5367	462	54	have	have	VERB
ejpam-5367	462	55	y	y	PROPN
ejpam-5367	462	56	∈	∈	PROPN
ejpam-5367	462	57	u(αf	u(αf	NOUN
ejpam-5367	462	58	;	;	PUNCT
ejpam-5367	462	59	t	t	PROPN
ejpam-5367	462	60	)	)	PUNCT
ejpam-5367	462	61	.	.	PUNCT
ejpam-5367	463	1	thus	thus	ADV
ejpam-5367	463	2	,	,	PUNCT
ejpam-5367	463	3	αf	αf	ADP
ejpam-5367	463	4	(	(	PUNCT
ejpam-5367	463	5	y	y	NOUN
ejpam-5367	463	6	)	)	PUNCT
ejpam-5367	463	7	≥	≥	NOUN
ejpam-5367	463	8	t	t	NOUN
ejpam-5367	463	9	=	=	PUNCT
ejpam-5367	463	10	min{αf	min{αf	X
ejpam-5367	463	11	(	(	PUNCT
ejpam-5367	463	12	x	x	X
ejpam-5367	463	13	·	·	PUNCT
ejpam-5367	463	14	y	y	X
ejpam-5367	463	15	)	)	PUNCT
ejpam-5367	463	16	,	,	PUNCT
ejpam-5367	463	17	αf	αf	X
ejpam-5367	463	18	(	(	PUNCT
ejpam-5367	463	19	x	x	NOUN
ejpam-5367	463	20	)	)	PUNCT
ejpam-5367	463	21	}	}	PUNCT
ejpam-5367	463	22	.	.	PUNCT
ejpam-5367	464	1	let	let	VERB
ejpam-5367	464	2	x	x	SYM
ejpam-5367	464	3	∈	∈	PROPN
ejpam-5367	464	4	x.	x.	NOUN
ejpam-5367	464	5	let	let	VERB
ejpam-5367	464	6	s	s	AUX
ejpam-5367	464	7	=	=	VERB
ejpam-5367	464	8	βf	βf	INTJ
ejpam-5367	464	9	(	(	PUNCT
ejpam-5367	464	10	x	x	NOUN
ejpam-5367	464	11	)	)	PUNCT
ejpam-5367	464	12	.	.	PUNCT
ejpam-5367	465	1	then	then	ADV
ejpam-5367	465	2	βf	βf	INTJ
ejpam-5367	465	3	(	(	PUNCT
ejpam-5367	465	4	x	x	X
ejpam-5367	465	5	)	)	PUNCT
ejpam-5367	465	6	≤	≤	NOUN
ejpam-5367	465	7	s.	s.	PROPN
ejpam-5367	465	8	thus	thus	ADV
ejpam-5367	465	9	,	,	PUNCT
ejpam-5367	465	10	x	x	PUNCT
ejpam-5367	465	11	∈	∈	PROPN
ejpam-5367	465	12	l(βf	l(βf	PROPN
ejpam-5367	465	13	;	;	PUNCT
ejpam-5367	465	14	s	s	X
ejpam-5367	465	15	)	)	PUNCT
ejpam-5367	465	16	̸=	̸=	PROPN
ejpam-5367	465	17	∅.	∅.	NOUN
ejpam-5367	465	18	by	by	ADP
ejpam-5367	465	19	the	the	DET
ejpam-5367	465	20	assumption	assumption	NOUN
ejpam-5367	465	21	,	,	PUNCT
ejpam-5367	465	22	we	we	PRON
ejpam-5367	465	23	have	have	VERB
ejpam-5367	465	24	l(βf	l(βf	NUM
ejpam-5367	465	25	;	;	PUNCT
ejpam-5367	465	26	s	s	X
ejpam-5367	465	27	)	)	PUNCT
ejpam-5367	465	28	is	be	AUX
ejpam-5367	465	29	an	an	DET
ejpam-5367	465	30	iup	iup	NOUN
ejpam-5367	465	31	-	-	PUNCT
ejpam-5367	465	32	filter	filter	NOUN
ejpam-5367	465	33	of	of	ADP
ejpam-5367	465	34	x.	x.	NOUN
ejpam-5367	465	35	by	by	ADP
ejpam-5367	465	36	(	(	PUNCT
ejpam-5367	465	37	2.18	2.18	NUM
ejpam-5367	465	38	)	)	PUNCT
ejpam-5367	465	39	,	,	PUNCT
ejpam-5367	465	40	we	we	PRON
ejpam-5367	465	41	have	have	VERB
ejpam-5367	465	42	0	0	NUM
ejpam-5367	465	43	∈	∈	PROPN
ejpam-5367	465	44	l(βf	l(βf	PROPN
ejpam-5367	465	45	;	;	PUNCT
ejpam-5367	465	46	s	s	X
ejpam-5367	465	47	)	)	PUNCT
ejpam-5367	465	48	.	.	PUNCT
ejpam-5367	466	1	then	then	ADV
ejpam-5367	466	2	βf	βf	INTJ
ejpam-5367	466	3	(	(	PUNCT
ejpam-5367	466	4	0	0	X
ejpam-5367	466	5	)	)	PUNCT
ejpam-5367	466	6	≤	≤	NOUN
ejpam-5367	466	7	s	s	PART
ejpam-5367	467	1	=	=	X
ejpam-5367	467	2	βf	βf	INTJ
ejpam-5367	467	3	(	(	PUNCT
ejpam-5367	467	4	x	x	NOUN
ejpam-5367	467	5	)	)	PUNCT
ejpam-5367	467	6	.	.	PUNCT
ejpam-5367	468	1	let	let	VERB
ejpam-5367	468	2	x	x	PRON
ejpam-5367	468	3	,	,	PUNCT
ejpam-5367	468	4	y	y	PROPN
ejpam-5367	468	5	∈	∈	PROPN
ejpam-5367	468	6	x.	x.	NOUN
ejpam-5367	468	7	let	let	VERB
ejpam-5367	468	8	s	s	AUX
ejpam-5367	468	9	=	=	VERB
ejpam-5367	468	10	max{βf	max{βf	INTJ
ejpam-5367	468	11	(	(	PUNCT
ejpam-5367	468	12	x	x	X
ejpam-5367	468	13	·	·	PUNCT
ejpam-5367	468	14	y	y	X
ejpam-5367	468	15	)	)	PUNCT
ejpam-5367	468	16	,	,	PUNCT
ejpam-5367	468	17	βf	βf	CCONJ
ejpam-5367	468	18	(	(	PUNCT
ejpam-5367	468	19	x	x	NOUN
ejpam-5367	468	20	)	)	PUNCT
ejpam-5367	468	21	}	}	PUNCT
ejpam-5367	468	22	.	.	PUNCT
ejpam-5367	469	1	then	then	ADV
ejpam-5367	469	2	βf	βf	INTJ
ejpam-5367	469	3	(	(	PUNCT
ejpam-5367	469	4	x	x	X
ejpam-5367	469	5	·	·	PUNCT
ejpam-5367	469	6	y	y	X
ejpam-5367	469	7	)	)	PUNCT
ejpam-5367	469	8	≤	≤	PROPN
ejpam-5367	469	9	s	s	X
ejpam-5367	469	10	and	and	CCONJ
ejpam-5367	469	11	βf	βf	INTJ
ejpam-5367	469	12	(	(	PUNCT
ejpam-5367	469	13	x	x	X
ejpam-5367	469	14	)	)	PUNCT
ejpam-5367	469	15	≤	≤	NOUN
ejpam-5367	469	16	s.	s.	PROPN
ejpam-5367	469	17	thus	thus	ADV
ejpam-5367	469	18	,	,	PUNCT
ejpam-5367	469	19	x	x	X
ejpam-5367	469	20	·	·	PUNCT
ejpam-5367	469	21	y	y	X
ejpam-5367	469	22	,	,	PUNCT
ejpam-5367	469	23	x	x	X
ejpam-5367	469	24	∈	∈	SYM
ejpam-5367	469	25	l(βf	l(βf	PROPN
ejpam-5367	469	26	;	;	PUNCT
ejpam-5367	469	27	s	s	X
ejpam-5367	469	28	)	)	PUNCT
ejpam-5367	469	29	̸=	̸=	PROPN
ejpam-5367	469	30	∅.	∅.	NOUN
ejpam-5367	469	31	by	by	ADP
ejpam-5367	469	32	the	the	DET
ejpam-5367	469	33	assumption	assumption	NOUN
ejpam-5367	469	34	,	,	PUNCT
ejpam-5367	469	35	we	we	PRON
ejpam-5367	469	36	have	have	VERB
ejpam-5367	469	37	l(βf	l(βf	NUM
ejpam-5367	469	38	;	;	PUNCT
ejpam-5367	469	39	s	s	X
ejpam-5367	469	40	)	)	PUNCT
ejpam-5367	469	41	is	be	AUX
ejpam-5367	469	42	an	an	DET
ejpam-5367	469	43	iupfilter	iupfilter	NOUN
ejpam-5367	469	44	of	of	ADP
ejpam-5367	469	45	x.	x.	NOUN
ejpam-5367	469	46	by	by	ADP
ejpam-5367	469	47	(	(	PUNCT
ejpam-5367	469	48	2.19	2.19	NUM
ejpam-5367	469	49	)	)	PUNCT
ejpam-5367	469	50	,	,	PUNCT
ejpam-5367	469	51	we	we	PRON
ejpam-5367	469	52	have	have	VERB
ejpam-5367	469	53	y	y	PROPN
ejpam-5367	469	54	∈	∈	PROPN
ejpam-5367	469	55	l(βf	l(βf	PROPN
ejpam-5367	469	56	;	;	PUNCT
ejpam-5367	469	57	s	s	X
ejpam-5367	469	58	)	)	PUNCT
ejpam-5367	469	59	.	.	PUNCT
ejpam-5367	470	1	thus	thus	ADV
ejpam-5367	470	2	,	,	PUNCT
ejpam-5367	470	3	βf	βf	PRON
ejpam-5367	470	4	(	(	PUNCT
ejpam-5367	470	5	y	y	NOUN
ejpam-5367	470	6	)	)	PUNCT
ejpam-5367	470	7	≤	≤	NOUN
ejpam-5367	470	8	s	s	PART
ejpam-5367	470	9	=	=	PUNCT
ejpam-5367	470	10	max{βf	max{βf	INTJ
ejpam-5367	470	11	(	(	PUNCT
ejpam-5367	470	12	x	x	X
ejpam-5367	470	13	·	·	PUNCT
ejpam-5367	470	14	y	y	X
ejpam-5367	470	15	)	)	PUNCT
ejpam-5367	470	16	,	,	PUNCT
ejpam-5367	470	17	βf	βf	CCONJ
ejpam-5367	470	18	(	(	PUNCT
ejpam-5367	470	19	x	x	NOUN
ejpam-5367	470	20	)	)	PUNCT
ejpam-5367	470	21	}	}	PUNCT
ejpam-5367	470	22	.	.	PUNCT
ejpam-5367	471	1	hence	hence	ADV
ejpam-5367	471	2	,	,	PUNCT
ejpam-5367	471	3	f	f	PROPN
ejpam-5367	471	4	is	be	AUX
ejpam-5367	471	5	an	an	DET
ejpam-5367	471	6	ffiup	ffiup	ADJ
ejpam-5367	471	7	-	-	PUNCT
ejpam-5367	471	8	filter	filter	NOUN
ejpam-5367	471	9	of	of	ADP
ejpam-5367	471	10	x.	x.	NOUN
ejpam-5367	471	11	the	the	DET
ejpam-5367	471	12	following	follow	VERB
ejpam-5367	471	13	theorem	theorem	NOUN
ejpam-5367	471	14	is	be	AUX
ejpam-5367	471	15	a	a	DET
ejpam-5367	471	16	direct	direct	ADJ
ejpam-5367	471	17	consequence	consequence	NOUN
ejpam-5367	471	18	of	of	ADP
ejpam-5367	471	19	theorem	theorem	ADJ
ejpam-5367	471	20	2	2	NUM
ejpam-5367	471	21	.	.	PUNCT
ejpam-5367	471	22	theorem	theorem	VERB
ejpam-5367	471	23	23	23	NUM
ejpam-5367	471	24	.	.	PUNCT
ejpam-5367	472	1	an	an	DET
ejpam-5367	472	2	ffs	ffs	NOUN
ejpam-5367	472	3	f	f	PROPN
ejpam-5367	472	4	in	in	ADP
ejpam-5367	472	5	x	x	PROPN
ejpam-5367	472	6	is	be	AUX
ejpam-5367	472	7	an	an	DET
ejpam-5367	472	8	ffsiup	ffsiup	NOUN
ejpam-5367	472	9	-	-	PUNCT
ejpam-5367	472	10	ideal	ideal	NOUN
ejpam-5367	472	11	of	of	ADP
ejpam-5367	472	12	x	x	SYM
ejpam-5367	472	13	if	if	SCONJ
ejpam-5367	472	14	and	and	CCONJ
ejpam-5367	472	15	only	only	ADV
ejpam-5367	472	16	if	if	SCONJ
ejpam-5367	472	17	for	for	ADP
ejpam-5367	472	18	all	all	DET
ejpam-5367	472	19	t	t	NOUN
ejpam-5367	472	20	,	,	PUNCT
ejpam-5367	472	21	s	s	PART
ejpam-5367	472	22	∈	∈	PROPN
ejpam-5367	473	1	[	[	X
ejpam-5367	473	2	0	0	NUM
ejpam-5367	473	3	,	,	PUNCT
ejpam-5367	473	4	1	1	NUM
ejpam-5367	473	5	]	]	PUNCT
ejpam-5367	473	6	,	,	PUNCT
ejpam-5367	473	7	the	the	DET
ejpam-5367	473	8	sets	set	NOUN
ejpam-5367	473	9	u(αf	u(αf	NOUN
ejpam-5367	473	10	;	;	PUNCT
ejpam-5367	473	11	t	t	X
ejpam-5367	473	12	)	)	PUNCT
ejpam-5367	473	13	and	and	CCONJ
ejpam-5367	473	14	l(βf	l(βf	PROPN
ejpam-5367	473	15	;	;	PUNCT
ejpam-5367	473	16	s	s	X
ejpam-5367	473	17	)	)	PUNCT
ejpam-5367	473	18	are	be	AUX
ejpam-5367	473	19	either	either	CCONJ
ejpam-5367	473	20	empty	empty	ADJ
ejpam-5367	473	21	or	or	CCONJ
ejpam-5367	473	22	strong	strong	ADJ
ejpam-5367	473	23	iup	iup	NOUN
ejpam-5367	473	24	-	-	PUNCT
ejpam-5367	473	25	ideal	ideal	NOUN
ejpam-5367	473	26	of	of	ADP
ejpam-5367	473	27	x.	x.	PROPN
ejpam-5367	473	28	theorem	theorem	VERB
ejpam-5367	473	29	24	24	NUM
ejpam-5367	473	30	.	.	PUNCT
ejpam-5367	474	1	an	an	DET
ejpam-5367	474	2	ffs	ffs	NOUN
ejpam-5367	474	3	f	f	PROPN
ejpam-5367	474	4	in	in	ADP
ejpam-5367	474	5	x	x	PROPN
ejpam-5367	474	6	is	be	AUX
ejpam-5367	474	7	an	an	DET
ejpam-5367	474	8	ffiup	ffiup	NOUN
ejpam-5367	474	9	-	-	PUNCT
ejpam-5367	474	10	subalgebra	subalgebra	NOUN
ejpam-5367	474	11	of	of	ADP
ejpam-5367	474	12	x	x	PRON
ejpam-5367	474	13	if	if	SCONJ
ejpam-5367	474	14	and	and	CCONJ
ejpam-5367	474	15	only	only	ADV
ejpam-5367	474	16	if	if	SCONJ
ejpam-5367	474	17	for	for	ADP
ejpam-5367	474	18	all	all	DET
ejpam-5367	474	19	t	t	NOUN
ejpam-5367	474	20	,	,	PUNCT
ejpam-5367	474	21	s	s	PART
ejpam-5367	474	22	∈	∈	PROPN
ejpam-5367	475	1	[	[	X
ejpam-5367	475	2	0	0	NUM
ejpam-5367	475	3	,	,	PUNCT
ejpam-5367	475	4	1	1	NUM
ejpam-5367	475	5	]	]	PUNCT
ejpam-5367	475	6	,	,	PUNCT
ejpam-5367	475	7	the	the	DET
ejpam-5367	475	8	sets	set	NOUN
ejpam-5367	475	9	u	u	NOUN
ejpam-5367	475	10	+	+	X
ejpam-5367	475	11	(	(	PUNCT
ejpam-5367	475	12	αf	αf	X
ejpam-5367	475	13	;	;	PUNCT
ejpam-5367	475	14	t	t	PROPN
ejpam-5367	475	15	)	)	PUNCT
ejpam-5367	475	16	and	and	CCONJ
ejpam-5367	475	17	l	l	NOUN
ejpam-5367	475	18	−	−	PROPN
ejpam-5367	476	1	(	(	PUNCT
ejpam-5367	476	2	βf	βf	SYM
ejpam-5367	476	3	;	;	PUNCT
ejpam-5367	476	4	s	s	X
ejpam-5367	476	5	)	)	PUNCT
ejpam-5367	476	6	are	be	AUX
ejpam-5367	476	7	either	either	CCONJ
ejpam-5367	476	8	empty	empty	ADJ
ejpam-5367	476	9	or	or	CCONJ
ejpam-5367	476	10	iup	iup	NOUN
ejpam-5367	476	11	-	-	PUNCT
ejpam-5367	476	12	subalgebras	subalgebras	PROPN
ejpam-5367	476	13	of	of	ADP
ejpam-5367	476	14	x.	x.	PROPN
ejpam-5367	476	15	a.	a.	PROPN
ejpam-5367	476	16	iampan	iampan	PROPN
ejpam-5367	476	17	et	et	PROPN
ejpam-5367	476	18	al	al	PROPN
ejpam-5367	476	19	.	.	PUNCT
ejpam-5367	476	20	/	/	SYM
ejpam-5367	476	21	eur	eur	PROPN
ejpam-5367	476	22	.	.	PUNCT
ejpam-5367	477	1	j.	j.	PROPN
ejpam-5367	477	2	pure	pure	PROPN
ejpam-5367	477	3	appl	appl	PROPN
ejpam-5367	477	4	.	.	PROPN
ejpam-5367	477	5	math	math	PROPN
ejpam-5367	477	6	,	,	PUNCT
ejpam-5367	477	7	17	17	NUM
ejpam-5367	477	8	(	(	PUNCT
ejpam-5367	477	9	4	4	NUM
ejpam-5367	477	10	)	)	PUNCT
ejpam-5367	477	11	(	(	PUNCT
ejpam-5367	477	12	2024	2024	NUM
ejpam-5367	477	13	)	)	PUNCT
ejpam-5367	477	14	,	,	PUNCT
ejpam-5367	477	15	3022	3022	NUM
ejpam-5367	477	16	-	-	SYM
ejpam-5367	477	17	3042	3042	NUM
ejpam-5367	477	18	3039	3039	NUM
ejpam-5367	477	19	proof	proof	NOUN
ejpam-5367	477	20	.	.	PUNCT
ejpam-5367	478	1	assume	assume	VERB
ejpam-5367	478	2	that	that	SCONJ
ejpam-5367	478	3	f	f	PROPN
ejpam-5367	478	4	is	be	AUX
ejpam-5367	478	5	an	an	DET
ejpam-5367	478	6	ffiup	ffiup	NOUN
ejpam-5367	478	7	-	-	PUNCT
ejpam-5367	478	8	subalgebra	subalgebra	NOUN
ejpam-5367	478	9	of	of	ADP
ejpam-5367	478	10	x.	x.	NOUN
ejpam-5367	478	11	let	let	VERB
ejpam-5367	478	12	t	t	PROPN
ejpam-5367	478	13	∈	∈	PROPN
ejpam-5367	479	1	[	[	X
ejpam-5367	479	2	0	0	NUM
ejpam-5367	479	3	,	,	PUNCT
ejpam-5367	479	4	1	1	NUM
ejpam-5367	479	5	]	]	PUNCT
ejpam-5367	479	6	be	be	AUX
ejpam-5367	479	7	such	such	ADJ
ejpam-5367	479	8	that	that	SCONJ
ejpam-5367	479	9	u	u	NOUN
ejpam-5367	479	10	+	+	X
ejpam-5367	479	11	(	(	PUNCT
ejpam-5367	479	12	αf	αf	X
ejpam-5367	479	13	;	;	PUNCT
ejpam-5367	479	14	t	t	X
ejpam-5367	479	15	)	)	PUNCT
ejpam-5367	479	16	̸=	̸=	PROPN
ejpam-5367	479	17	∅.	∅.	ADV
ejpam-5367	479	18	let	let	VERB
ejpam-5367	479	19	x	x	PRON
ejpam-5367	479	20	,	,	PUNCT
ejpam-5367	479	21	y	y	PROPN
ejpam-5367	479	22	∈	∈	PROPN
ejpam-5367	479	23	u	u	NOUN
ejpam-5367	479	24	+	+	X
ejpam-5367	479	25	(	(	PUNCT
ejpam-5367	479	26	αf	αf	X
ejpam-5367	479	27	;	;	PUNCT
ejpam-5367	479	28	t	t	PROPN
ejpam-5367	479	29	)	)	PUNCT
ejpam-5367	479	30	.	.	PUNCT
ejpam-5367	480	1	then	then	ADV
ejpam-5367	480	2	αf	αf	VERB
ejpam-5367	480	3	(	(	PUNCT
ejpam-5367	480	4	x	x	X
ejpam-5367	480	5	)	)	PUNCT
ejpam-5367	480	6	>	>	X
ejpam-5367	480	7	t	t	PROPN
ejpam-5367	480	8	and	and	CCONJ
ejpam-5367	480	9	αf	αf	PROPN
ejpam-5367	480	10	(	(	PUNCT
ejpam-5367	480	11	y	y	NOUN
ejpam-5367	480	12	)	)	PUNCT
ejpam-5367	480	13	>	>	PUNCT
ejpam-5367	481	1	t.	t.	PROPN
ejpam-5367	481	2	thus	thus	ADV
ejpam-5367	481	3	,	,	PUNCT
ejpam-5367	481	4	min{αf	min{αf	PUNCT
ejpam-5367	481	5	(	(	PUNCT
ejpam-5367	481	6	x	x	X
ejpam-5367	481	7	)	)	PUNCT
ejpam-5367	481	8	,	,	PUNCT
ejpam-5367	481	9	αf	αf	X
ejpam-5367	481	10	(	(	PUNCT
ejpam-5367	481	11	y	y	NOUN
ejpam-5367	481	12	)	)	PUNCT
ejpam-5367	481	13	}	}	PUNCT
ejpam-5367	481	14	>	>	PUNCT
ejpam-5367	482	1	t.	t.	NOUN
ejpam-5367	482	2	by	by	ADP
ejpam-5367	482	3	(	(	PUNCT
ejpam-5367	482	4	3.2	3.2	NUM
ejpam-5367	482	5	)	)	PUNCT
ejpam-5367	482	6	,	,	PUNCT
ejpam-5367	482	7	we	we	PRON
ejpam-5367	482	8	have	have	VERB
ejpam-5367	482	9	αf	αf	ADP
ejpam-5367	482	10	(	(	PUNCT
ejpam-5367	482	11	x	x	X
ejpam-5367	482	12	·	·	PUNCT
ejpam-5367	482	13	y	y	X
ejpam-5367	482	14	)	)	PUNCT
ejpam-5367	482	15	≥	≥	NOUN
ejpam-5367	482	16	min{αf	min{αf	PUNCT
ejpam-5367	482	17	(	(	PUNCT
ejpam-5367	482	18	x	x	X
ejpam-5367	482	19	)	)	PUNCT
ejpam-5367	482	20	,	,	PUNCT
ejpam-5367	482	21	αf	αf	X
ejpam-5367	482	22	(	(	PUNCT
ejpam-5367	482	23	y	y	NOUN
ejpam-5367	482	24	)	)	PUNCT
ejpam-5367	482	25	}	}	PUNCT
ejpam-5367	482	26	>	>	PUNCT
ejpam-5367	482	27	t.	t.	PROPN
ejpam-5367	483	1	thus	thus	ADV
ejpam-5367	483	2	,	,	PUNCT
ejpam-5367	483	3	x	x	X
ejpam-5367	483	4	·	·	PUNCT
ejpam-5367	483	5	y	y	PROPN
ejpam-5367	483	6	∈	∈	PROPN
ejpam-5367	483	7	u	u	NOUN
ejpam-5367	483	8	+	+	X
ejpam-5367	483	9	(	(	PUNCT
ejpam-5367	483	10	αf	αf	X
ejpam-5367	483	11	;	;	PUNCT
ejpam-5367	483	12	t	t	PROPN
ejpam-5367	483	13	)	)	PUNCT
ejpam-5367	483	14	.	.	PUNCT
ejpam-5367	484	1	hence	hence	ADV
ejpam-5367	484	2	,	,	PUNCT
ejpam-5367	484	3	u	u	PROPN
ejpam-5367	484	4	+	+	X
ejpam-5367	484	5	(	(	PUNCT
ejpam-5367	484	6	αf	αf	X
ejpam-5367	484	7	;	;	PUNCT
ejpam-5367	484	8	t	t	X
ejpam-5367	484	9	)	)	PUNCT
ejpam-5367	484	10	is	be	AUX
ejpam-5367	484	11	an	an	DET
ejpam-5367	484	12	iup	iup	NOUN
ejpam-5367	484	13	-	-	PUNCT
ejpam-5367	484	14	subalgebra	subalgebra	NOUN
ejpam-5367	484	15	of	of	ADP
ejpam-5367	484	16	x.	x.	NOUN
ejpam-5367	484	17	let	let	VERB
ejpam-5367	485	1	s	s	PRON
ejpam-5367	485	2	∈	∈	NOUN
ejpam-5367	485	3	[	[	X
ejpam-5367	485	4	0	0	NUM
ejpam-5367	485	5	,	,	PUNCT
ejpam-5367	485	6	1	1	NUM
ejpam-5367	485	7	]	]	PUNCT
ejpam-5367	485	8	be	be	AUX
ejpam-5367	485	9	such	such	ADJ
ejpam-5367	485	10	that	that	SCONJ
ejpam-5367	485	11	l	l	NOUN
ejpam-5367	485	12	−	−	PROPN
ejpam-5367	486	1	(	(	PUNCT
ejpam-5367	486	2	βf	βf	SYM
ejpam-5367	486	3	;	;	PUNCT
ejpam-5367	486	4	s	s	X
ejpam-5367	486	5	)	)	PUNCT
ejpam-5367	486	6	̸=	̸=	PROPN
ejpam-5367	486	7	∅.	∅.	ADV
ejpam-5367	486	8	let	let	VERB
ejpam-5367	486	9	x	x	PRON
ejpam-5367	486	10	,	,	PUNCT
ejpam-5367	486	11	y	y	PROPN
ejpam-5367	486	12	∈	∈	PROPN
ejpam-5367	486	13	l	l	NOUN
ejpam-5367	486	14	−	−	PROPN
ejpam-5367	487	1	(	(	PUNCT
ejpam-5367	487	2	βf	βf	INTJ
ejpam-5367	487	3	;	;	PUNCT
ejpam-5367	487	4	s	s	X
ejpam-5367	487	5	)	)	PUNCT
ejpam-5367	487	6	.	.	PUNCT
ejpam-5367	488	1	then	then	ADV
ejpam-5367	488	2	βf	βf	INTJ
ejpam-5367	488	3	(	(	PUNCT
ejpam-5367	488	4	x	x	X
ejpam-5367	488	5	)	)	PUNCT
ejpam-5367	488	6	<	<	X
ejpam-5367	488	7	s	s	X
ejpam-5367	488	8	and	and	CCONJ
ejpam-5367	488	9	βf	βf	INTJ
ejpam-5367	488	10	(	(	PUNCT
ejpam-5367	488	11	y	y	NOUN
ejpam-5367	488	12	)	)	PUNCT
ejpam-5367	488	13	<	<	X
ejpam-5367	488	14	s.	s.	PROPN
ejpam-5367	489	1	thus	thus	ADV
ejpam-5367	489	2	,	,	PUNCT
ejpam-5367	489	3	max{βf	max{βf	PROPN
ejpam-5367	489	4	(	(	PUNCT
ejpam-5367	489	5	x	x	NOUN
ejpam-5367	489	6	)	)	PUNCT
ejpam-5367	489	7	,	,	PUNCT
ejpam-5367	489	8	βf	βf	CCONJ
ejpam-5367	489	9	(	(	PUNCT
ejpam-5367	489	10	y	y	NOUN
ejpam-5367	489	11	)	)	PUNCT
ejpam-5367	489	12	}	}	PUNCT
ejpam-5367	489	13	<	<	X
ejpam-5367	489	14	s.	s.	PROPN
ejpam-5367	489	15	by	by	ADP
ejpam-5367	489	16	(	(	PUNCT
ejpam-5367	489	17	3.3	3.3	NUM
ejpam-5367	489	18	)	)	PUNCT
ejpam-5367	489	19	,	,	PUNCT
ejpam-5367	489	20	we	we	PRON
ejpam-5367	489	21	have	have	VERB
ejpam-5367	489	22	βf	βf	INTJ
ejpam-5367	490	1	(	(	PUNCT
ejpam-5367	490	2	x	x	X
ejpam-5367	490	3	·	·	PUNCT
ejpam-5367	490	4	y	y	X
ejpam-5367	490	5	)	)	PUNCT
ejpam-5367	490	6	≤	≤	NUM
ejpam-5367	490	7	max{βf	max{βf	INTJ
ejpam-5367	491	1	(	(	PUNCT
ejpam-5367	491	2	x	x	NOUN
ejpam-5367	491	3	)	)	PUNCT
ejpam-5367	491	4	,	,	PUNCT
ejpam-5367	491	5	βf	βf	CCONJ
ejpam-5367	491	6	(	(	PUNCT
ejpam-5367	491	7	y	y	NOUN
ejpam-5367	491	8	)	)	PUNCT
ejpam-5367	491	9	}	}	PUNCT
ejpam-5367	491	10	<	<	X
ejpam-5367	491	11	s.	s.	PROPN
ejpam-5367	491	12	thus	thus	ADV
ejpam-5367	491	13	,	,	PUNCT
ejpam-5367	491	14	x	x	X
ejpam-5367	491	15	·	·	PUNCT
ejpam-5367	491	16	y	y	SYM
ejpam-5367	491	17	∈	∈	PROPN
ejpam-5367	491	18	l	l	NOUN
ejpam-5367	491	19	−	−	PROPN
ejpam-5367	491	20	(	(	PUNCT
ejpam-5367	491	21	βf	βf	INTJ
ejpam-5367	491	22	;	;	PUNCT
ejpam-5367	491	23	s	s	X
ejpam-5367	491	24	)	)	PUNCT
ejpam-5367	491	25	.	.	PUNCT
ejpam-5367	492	1	hence	hence	ADV
ejpam-5367	492	2	,	,	PUNCT
ejpam-5367	492	3	l	l	NOUN
ejpam-5367	492	4	−	−	PROPN
ejpam-5367	492	5	(	(	PUNCT
ejpam-5367	492	6	βf	βf	SYM
ejpam-5367	492	7	;	;	PUNCT
ejpam-5367	492	8	s	s	X
ejpam-5367	492	9	)	)	PUNCT
ejpam-5367	492	10	is	be	AUX
ejpam-5367	492	11	an	an	DET
ejpam-5367	492	12	iup	iup	NOUN
ejpam-5367	492	13	-	-	PUNCT
ejpam-5367	492	14	subalgebra	subalgebra	NOUN
ejpam-5367	492	15	of	of	ADP
ejpam-5367	492	16	x.	x.	NOUN
ejpam-5367	492	17	conversely	conversely	ADV
ejpam-5367	492	18	,	,	PUNCT
ejpam-5367	492	19	assume	assume	VERB
ejpam-5367	492	20	that	that	SCONJ
ejpam-5367	492	21	for	for	ADP
ejpam-5367	492	22	all	all	DET
ejpam-5367	492	23	t	t	PROPN
ejpam-5367	492	24	,	,	PUNCT
ejpam-5367	492	25	s	s	PART
ejpam-5367	492	26	∈	∈	PROPN
ejpam-5367	493	1	[	[	X
ejpam-5367	493	2	0	0	NUM
ejpam-5367	493	3	,	,	PUNCT
ejpam-5367	493	4	1	1	NUM
ejpam-5367	493	5	]	]	PUNCT
ejpam-5367	493	6	,	,	PUNCT
ejpam-5367	493	7	the	the	DET
ejpam-5367	493	8	sets	set	NOUN
ejpam-5367	493	9	u	u	NOUN
ejpam-5367	493	10	+	+	X
ejpam-5367	493	11	(	(	PUNCT
ejpam-5367	493	12	αf	αf	X
ejpam-5367	493	13	;	;	PUNCT
ejpam-5367	493	14	t	t	PROPN
ejpam-5367	493	15	)	)	PUNCT
ejpam-5367	493	16	and	and	CCONJ
ejpam-5367	493	17	l	l	NOUN
ejpam-5367	493	18	−	−	PROPN
ejpam-5367	494	1	(	(	PUNCT
ejpam-5367	494	2	βf	βf	SYM
ejpam-5367	494	3	;	;	PUNCT
ejpam-5367	494	4	s	s	X
ejpam-5367	494	5	)	)	PUNCT
ejpam-5367	494	6	are	be	AUX
ejpam-5367	494	7	either	either	CCONJ
ejpam-5367	494	8	empty	empty	ADJ
ejpam-5367	494	9	or	or	CCONJ
ejpam-5367	494	10	iup	iup	NOUN
ejpam-5367	494	11	-	-	PUNCT
ejpam-5367	494	12	subalgebras	subalgebras	PROPN
ejpam-5367	494	13	ofx	ofx	PROPN
ejpam-5367	494	14	.	.	PUNCT
ejpam-5367	495	1	let	let	VERB
ejpam-5367	495	2	x	x	PRON
ejpam-5367	495	3	,	,	PUNCT
ejpam-5367	495	4	y	y	PROPN
ejpam-5367	495	5	∈	∈	PROPN
ejpam-5367	495	6	x.	x.	NOUN
ejpam-5367	495	7	assume	assume	VERB
ejpam-5367	495	8	that	that	SCONJ
ejpam-5367	495	9	αf	αf	X
ejpam-5367	495	10	(	(	PUNCT
ejpam-5367	495	11	x·y	x·y	PROPN
ejpam-5367	495	12	)	)	PUNCT
ejpam-5367	495	13	<	<	X
ejpam-5367	495	14	min{αf	min{αf	PUNCT
ejpam-5367	495	15	(	(	PUNCT
ejpam-5367	495	16	x	x	X
ejpam-5367	495	17	)	)	PUNCT
ejpam-5367	495	18	,	,	PUNCT
ejpam-5367	495	19	αf	αf	X
ejpam-5367	495	20	(	(	PUNCT
ejpam-5367	495	21	y	y	NOUN
ejpam-5367	495	22	)	)	PUNCT
ejpam-5367	495	23	}	}	PUNCT
ejpam-5367	495	24	.	.	PUNCT
ejpam-5367	496	1	let	let	VERB
ejpam-5367	496	2	t	t	NOUN
ejpam-5367	496	3	=	=	SYM
ejpam-5367	496	4	αf	αf	X
ejpam-5367	496	5	(	(	PUNCT
ejpam-5367	496	6	x·y	x·y	PROPN
ejpam-5367	496	7	)	)	PUNCT
ejpam-5367	496	8	.	.	PUNCT
ejpam-5367	497	1	then	then	ADV
ejpam-5367	497	2	αf	αf	VERB
ejpam-5367	497	3	(	(	PUNCT
ejpam-5367	497	4	x	x	X
ejpam-5367	497	5	)	)	PUNCT
ejpam-5367	497	6	>	>	X
ejpam-5367	497	7	t	t	PROPN
ejpam-5367	497	8	and	and	CCONJ
ejpam-5367	497	9	αf	αf	PROPN
ejpam-5367	497	10	(	(	PUNCT
ejpam-5367	497	11	y	y	NOUN
ejpam-5367	497	12	)	)	PUNCT
ejpam-5367	497	13	>	>	PUNCT
ejpam-5367	498	1	t.	t.	PROPN
ejpam-5367	498	2	thus	thus	ADV
ejpam-5367	498	3	,	,	PUNCT
ejpam-5367	498	4	x	x	PRON
ejpam-5367	498	5	,	,	PUNCT
ejpam-5367	498	6	y	y	PROPN
ejpam-5367	498	7	∈	∈	PROPN
ejpam-5367	498	8	u	u	NOUN
ejpam-5367	498	9	+	+	X
ejpam-5367	498	10	(	(	PUNCT
ejpam-5367	498	11	αf	αf	X
ejpam-5367	498	12	;	;	PUNCT
ejpam-5367	498	13	t	t	PROPN
ejpam-5367	498	14	)	)	PUNCT
ejpam-5367	498	15	.	.	PUNCT
ejpam-5367	499	1	by	by	ADP
ejpam-5367	499	2	the	the	DET
ejpam-5367	499	3	assumption	assumption	NOUN
ejpam-5367	499	4	,	,	PUNCT
ejpam-5367	499	5	we	we	PRON
ejpam-5367	499	6	have	have	VERB
ejpam-5367	499	7	u	u	NOUN
ejpam-5367	499	8	+	+	CCONJ
ejpam-5367	499	9	(	(	PUNCT
ejpam-5367	499	10	αf	αf	X
ejpam-5367	499	11	;	;	PUNCT
ejpam-5367	499	12	t	t	X
ejpam-5367	499	13	)	)	PUNCT
ejpam-5367	499	14	is	be	AUX
ejpam-5367	499	15	an	an	DET
ejpam-5367	499	16	iup	iup	NOUN
ejpam-5367	499	17	-	-	PUNCT
ejpam-5367	499	18	subalgebra	subalgebra	NOUN
ejpam-5367	499	19	.	.	PUNCT
ejpam-5367	500	1	by	by	ADP
ejpam-5367	500	2	(	(	PUNCT
ejpam-5367	500	3	2.17	2.17	NUM
ejpam-5367	500	4	)	)	PUNCT
ejpam-5367	500	5	,	,	PUNCT
ejpam-5367	500	6	we	we	PRON
ejpam-5367	500	7	have	have	VERB
ejpam-5367	500	8	x	x	X
ejpam-5367	500	9	·	·	PUNCT
ejpam-5367	500	10	y	y	PROPN
ejpam-5367	500	11	∈	∈	PROPN
ejpam-5367	500	12	u	u	NOUN
ejpam-5367	500	13	+	+	X
ejpam-5367	500	14	(	(	PUNCT
ejpam-5367	500	15	αf	αf	X
ejpam-5367	500	16	;	;	PUNCT
ejpam-5367	500	17	t	t	PROPN
ejpam-5367	500	18	)	)	PUNCT
ejpam-5367	500	19	.	.	PUNCT
ejpam-5367	501	1	so	so	ADV
ejpam-5367	501	2	αf	αf	VERB
ejpam-5367	501	3	(	(	PUNCT
ejpam-5367	501	4	x	x	X
ejpam-5367	501	5	·	·	PUNCT
ejpam-5367	501	6	y	y	X
ejpam-5367	501	7	)	)	PUNCT
ejpam-5367	501	8	>	>	X
ejpam-5367	501	9	t	t	PROPN
ejpam-5367	502	1	=	=	PUNCT
ejpam-5367	502	2	αf	αf	X
ejpam-5367	502	3	(	(	PUNCT
ejpam-5367	502	4	x	x	PROPN
ejpam-5367	502	5	·	·	PUNCT
ejpam-5367	502	6	y	y	X
ejpam-5367	502	7	)	)	PUNCT
ejpam-5367	502	8	,	,	PUNCT
ejpam-5367	502	9	which	which	PRON
ejpam-5367	502	10	is	be	AUX
ejpam-5367	502	11	a	a	DET
ejpam-5367	502	12	contradiction	contradiction	NOUN
ejpam-5367	502	13	.	.	PUNCT
ejpam-5367	503	1	thus	thus	ADV
ejpam-5367	503	2	,	,	PUNCT
ejpam-5367	503	3	αf	αf	ADP
ejpam-5367	503	4	(	(	PUNCT
ejpam-5367	503	5	x	x	PROPN
ejpam-5367	503	6	·	·	PUNCT
ejpam-5367	503	7	y	y	X
ejpam-5367	503	8	)	)	PUNCT
ejpam-5367	503	9	≥	≥	NOUN
ejpam-5367	503	10	min{αf	min{αf	PUNCT
ejpam-5367	503	11	(	(	PUNCT
ejpam-5367	503	12	x	x	X
ejpam-5367	503	13	)	)	PUNCT
ejpam-5367	503	14	,	,	PUNCT
ejpam-5367	503	15	αf	αf	X
ejpam-5367	503	16	(	(	PUNCT
ejpam-5367	503	17	y	y	NOUN
ejpam-5367	503	18	)	)	PUNCT
ejpam-5367	503	19	}	}	PUNCT
ejpam-5367	503	20	.	.	PUNCT
ejpam-5367	504	1	let	let	VERB
ejpam-5367	504	2	x	x	PRON
ejpam-5367	504	3	,	,	PUNCT
ejpam-5367	504	4	y	y	PROPN
ejpam-5367	504	5	∈	∈	PROPN
ejpam-5367	504	6	x.	x.	NOUN
ejpam-5367	504	7	assume	assume	VERB
ejpam-5367	504	8	that	that	SCONJ
ejpam-5367	505	1	βf	βf	INTJ
ejpam-5367	505	2	(	(	PUNCT
ejpam-5367	505	3	x	x	X
ejpam-5367	505	4	·	·	PUNCT
ejpam-5367	505	5	y	y	X
ejpam-5367	505	6	)	)	PUNCT
ejpam-5367	505	7	>	>	X
ejpam-5367	506	1	max{βf	max{βf	PROPN
ejpam-5367	506	2	(	(	PUNCT
ejpam-5367	506	3	x	x	NOUN
ejpam-5367	506	4	)	)	PUNCT
ejpam-5367	506	5	,	,	PUNCT
ejpam-5367	506	6	βf	βf	CCONJ
ejpam-5367	506	7	(	(	PUNCT
ejpam-5367	506	8	y	y	NOUN
ejpam-5367	506	9	)	)	PUNCT
ejpam-5367	506	10	}	}	PUNCT
ejpam-5367	506	11	.	.	PUNCT
ejpam-5367	507	1	let	let	VERB
ejpam-5367	507	2	s	s	PRON
ejpam-5367	507	3	=	=	VERB
ejpam-5367	507	4	βf	βf	INTJ
ejpam-5367	507	5	(	(	PUNCT
ejpam-5367	507	6	x	x	X
ejpam-5367	507	7	·	·	PUNCT
ejpam-5367	507	8	y	y	X
ejpam-5367	507	9	)	)	PUNCT
ejpam-5367	507	10	.	.	PUNCT
ejpam-5367	508	1	then	then	ADV
ejpam-5367	508	2	βf	βf	INTJ
ejpam-5367	508	3	(	(	PUNCT
ejpam-5367	508	4	x	x	X
ejpam-5367	508	5	)	)	PUNCT
ejpam-5367	508	6	<	<	X
ejpam-5367	508	7	s	s	X
ejpam-5367	508	8	and	and	CCONJ
ejpam-5367	508	9	βf	βf	INTJ
ejpam-5367	508	10	(	(	PUNCT
ejpam-5367	508	11	y	y	NOUN
ejpam-5367	508	12	)	)	PUNCT
ejpam-5367	508	13	<	<	X
ejpam-5367	508	14	s.	s.	PROPN
ejpam-5367	508	15	thus	thus	ADV
ejpam-5367	508	16	,	,	PUNCT
ejpam-5367	508	17	x	x	PRON
ejpam-5367	508	18	,	,	PUNCT
ejpam-5367	508	19	y	y	PROPN
ejpam-5367	508	20	∈	∈	PROPN
ejpam-5367	508	21	l	l	NOUN
ejpam-5367	508	22	−	−	PROPN
ejpam-5367	508	23	(	(	PUNCT
ejpam-5367	508	24	βf	βf	INTJ
ejpam-5367	508	25	;	;	PUNCT
ejpam-5367	508	26	s	s	X
ejpam-5367	508	27	)	)	PUNCT
ejpam-5367	508	28	.	.	PUNCT
ejpam-5367	509	1	by	by	ADP
ejpam-5367	509	2	the	the	DET
ejpam-5367	509	3	assumption	assumption	NOUN
ejpam-5367	509	4	,	,	PUNCT
ejpam-5367	509	5	we	we	PRON
ejpam-5367	509	6	have	have	VERB
ejpam-5367	509	7	l	l	NOUN
ejpam-5367	509	8	−	−	PROPN
ejpam-5367	509	9	(	(	PUNCT
ejpam-5367	509	10	βf	βf	SYM
ejpam-5367	509	11	;	;	PUNCT
ejpam-5367	509	12	s	s	X
ejpam-5367	509	13	)	)	PUNCT
ejpam-5367	509	14	is	be	AUX
ejpam-5367	509	15	an	an	DET
ejpam-5367	509	16	iup	iup	NOUN
ejpam-5367	509	17	-	-	PUNCT
ejpam-5367	509	18	subalgebra	subalgebra	NOUN
ejpam-5367	509	19	.	.	PUNCT
ejpam-5367	510	1	by	by	ADP
ejpam-5367	510	2	(	(	PUNCT
ejpam-5367	510	3	2.17	2.17	NUM
ejpam-5367	510	4	)	)	PUNCT
ejpam-5367	510	5	,	,	PUNCT
ejpam-5367	510	6	we	we	PRON
ejpam-5367	510	7	have	have	VERB
ejpam-5367	510	8	x	x	X
ejpam-5367	510	9	·	·	PUNCT
ejpam-5367	510	10	y	y	SYM
ejpam-5367	510	11	∈	∈	PROPN
ejpam-5367	510	12	l	l	NOUN
ejpam-5367	510	13	−	−	PROPN
ejpam-5367	511	1	(	(	PUNCT
ejpam-5367	511	2	βf	βf	INTJ
ejpam-5367	511	3	;	;	PUNCT
ejpam-5367	511	4	s	s	X
ejpam-5367	511	5	)	)	PUNCT
ejpam-5367	511	6	.	.	PUNCT
ejpam-5367	512	1	so	so	ADV
ejpam-5367	512	2	βf	βf	INTJ
ejpam-5367	512	3	(	(	PUNCT
ejpam-5367	512	4	x	x	X
ejpam-5367	512	5	·	·	PUNCT
ejpam-5367	512	6	y	y	X
ejpam-5367	512	7	)	)	PUNCT
ejpam-5367	512	8	<	<	X
ejpam-5367	513	1	s	s	X
ejpam-5367	514	1	=	=	X
ejpam-5367	514	2	βf	βf	INTJ
ejpam-5367	514	3	(	(	PUNCT
ejpam-5367	514	4	x	x	X
ejpam-5367	514	5	·	·	PUNCT
ejpam-5367	514	6	y	y	X
ejpam-5367	514	7	)	)	PUNCT
ejpam-5367	514	8	,	,	PUNCT
ejpam-5367	514	9	which	which	PRON
ejpam-5367	514	10	is	be	AUX
ejpam-5367	514	11	a	a	DET
ejpam-5367	514	12	contradiction	contradiction	NOUN
ejpam-5367	514	13	.	.	PUNCT
ejpam-5367	515	1	thus	thus	ADV
ejpam-5367	515	2	,	,	PUNCT
ejpam-5367	515	3	βf	βf	INTJ
ejpam-5367	515	4	(	(	PUNCT
ejpam-5367	515	5	x	x	X
ejpam-5367	515	6	·	·	PUNCT
ejpam-5367	515	7	y	y	X
ejpam-5367	515	8	)	)	PUNCT
ejpam-5367	515	9	≤	≤	NUM
ejpam-5367	515	10	max{βf	max{βf	INTJ
ejpam-5367	515	11	(	(	PUNCT
ejpam-5367	515	12	x	x	NOUN
ejpam-5367	515	13	)	)	PUNCT
ejpam-5367	515	14	,	,	PUNCT
ejpam-5367	515	15	βf	βf	CCONJ
ejpam-5367	515	16	(	(	PUNCT
ejpam-5367	515	17	y	y	NOUN
ejpam-5367	515	18	)	)	PUNCT
ejpam-5367	515	19	}	}	PUNCT
ejpam-5367	515	20	.	.	PUNCT
ejpam-5367	516	1	hence	hence	ADV
ejpam-5367	516	2	,	,	PUNCT
ejpam-5367	516	3	f	f	PROPN
ejpam-5367	516	4	is	be	AUX
ejpam-5367	516	5	an	an	DET
ejpam-5367	516	6	ffiup	ffiup	NOUN
ejpam-5367	516	7	-	-	PUNCT
ejpam-5367	516	8	subalgebra	subalgebra	NOUN
ejpam-5367	516	9	of	of	ADP
ejpam-5367	516	10	x.	x.	PROPN
ejpam-5367	516	11	theorem	theorem	VERB
ejpam-5367	516	12	25	25	NUM
ejpam-5367	516	13	.	.	PUNCT
ejpam-5367	517	1	an	an	DET
ejpam-5367	517	2	ffs	ffs	NOUN
ejpam-5367	517	3	f	f	PROPN
ejpam-5367	517	4	in	in	ADP
ejpam-5367	517	5	x	x	PROPN
ejpam-5367	517	6	is	be	AUX
ejpam-5367	517	7	an	an	DET
ejpam-5367	517	8	ffiup	ffiup	NOUN
ejpam-5367	517	9	-	-	PUNCT
ejpam-5367	517	10	ideal	ideal	NOUN
ejpam-5367	517	11	of	of	ADP
ejpam-5367	517	12	x	x	SYM
ejpam-5367	517	13	if	if	SCONJ
ejpam-5367	517	14	and	and	CCONJ
ejpam-5367	517	15	only	only	ADV
ejpam-5367	517	16	if	if	SCONJ
ejpam-5367	517	17	for	for	ADP
ejpam-5367	517	18	all	all	DET
ejpam-5367	517	19	t	t	NOUN
ejpam-5367	517	20	,	,	PUNCT
ejpam-5367	517	21	s	s	PART
ejpam-5367	517	22	∈	∈	PROPN
ejpam-5367	518	1	[	[	X
ejpam-5367	518	2	0	0	NUM
ejpam-5367	518	3	,	,	PUNCT
ejpam-5367	518	4	1	1	NUM
ejpam-5367	518	5	]	]	PUNCT
ejpam-5367	518	6	,	,	PUNCT
ejpam-5367	518	7	the	the	DET
ejpam-5367	518	8	sets	set	NOUN
ejpam-5367	518	9	u	u	NOUN
ejpam-5367	518	10	+	+	X
ejpam-5367	518	11	(	(	PUNCT
ejpam-5367	518	12	αf	αf	X
ejpam-5367	518	13	;	;	PUNCT
ejpam-5367	518	14	t	t	PROPN
ejpam-5367	518	15	)	)	PUNCT
ejpam-5367	518	16	and	and	CCONJ
ejpam-5367	518	17	l	l	NOUN
ejpam-5367	518	18	−	−	PROPN
ejpam-5367	519	1	(	(	PUNCT
ejpam-5367	519	2	βf	βf	SYM
ejpam-5367	519	3	;	;	PUNCT
ejpam-5367	519	4	s	s	X
ejpam-5367	519	5	)	)	PUNCT
ejpam-5367	519	6	are	be	AUX
ejpam-5367	519	7	either	either	CCONJ
ejpam-5367	519	8	empty	empty	ADJ
ejpam-5367	519	9	or	or	CCONJ
ejpam-5367	519	10	iup	iup	NOUN
ejpam-5367	519	11	-	-	PUNCT
ejpam-5367	519	12	ideals	ideal	NOUN
ejpam-5367	519	13	of	of	ADP
ejpam-5367	519	14	x.	x.	NOUN
ejpam-5367	519	15	proof	proof	NOUN
ejpam-5367	519	16	.	.	PUNCT
ejpam-5367	520	1	assume	assume	VERB
ejpam-5367	520	2	that	that	SCONJ
ejpam-5367	520	3	f	f	PROPN
ejpam-5367	520	4	is	be	AUX
ejpam-5367	520	5	an	an	DET
ejpam-5367	520	6	ffiup	ffiup	NOUN
ejpam-5367	520	7	-	-	PUNCT
ejpam-5367	520	8	ideal	ideal	NOUN
ejpam-5367	520	9	of	of	ADP
ejpam-5367	520	10	x.	x.	NOUN
ejpam-5367	520	11	let	let	VERB
ejpam-5367	520	12	t	t	PROPN
ejpam-5367	520	13	∈	∈	PROPN
ejpam-5367	521	1	[	[	X
ejpam-5367	521	2	0	0	NUM
ejpam-5367	521	3	,	,	PUNCT
ejpam-5367	521	4	1	1	NUM
ejpam-5367	521	5	]	]	PUNCT
ejpam-5367	521	6	be	be	AUX
ejpam-5367	521	7	such	such	ADJ
ejpam-5367	521	8	that	that	SCONJ
ejpam-5367	521	9	u	u	NOUN
ejpam-5367	521	10	+	+	X
ejpam-5367	521	11	(	(	PUNCT
ejpam-5367	521	12	αf	αf	X
ejpam-5367	521	13	;	;	PUNCT
ejpam-5367	521	14	t	t	X
ejpam-5367	521	15	)	)	PUNCT
ejpam-5367	521	16	̸=	̸=	PROPN
ejpam-5367	521	17	∅.	∅.	ADV
ejpam-5367	521	18	let	let	VERB
ejpam-5367	521	19	a	a	DET
ejpam-5367	521	20	∈	∈	PROPN
ejpam-5367	521	21	u	u	NOUN
ejpam-5367	521	22	+	+	X
ejpam-5367	521	23	(	(	PUNCT
ejpam-5367	521	24	αf	αf	X
ejpam-5367	521	25	;	;	PUNCT
ejpam-5367	521	26	t	t	PROPN
ejpam-5367	521	27	)	)	PUNCT
ejpam-5367	521	28	.	.	PUNCT
ejpam-5367	522	1	then	then	ADV
ejpam-5367	522	2	αf	αf	VERB
ejpam-5367	522	3	(	(	PUNCT
ejpam-5367	522	4	a	a	NOUN
ejpam-5367	522	5	)	)	PUNCT
ejpam-5367	522	6	>	>	PUNCT
ejpam-5367	523	1	t.	t.	PROPN
ejpam-5367	523	2	by	by	ADP
ejpam-5367	523	3	(	(	PUNCT
ejpam-5367	523	4	3.4	3.4	NUM
ejpam-5367	523	5	)	)	PUNCT
ejpam-5367	523	6	,	,	PUNCT
ejpam-5367	523	7	we	we	PRON
ejpam-5367	523	8	have	have	VERB
ejpam-5367	523	9	αf	αf	NUM
ejpam-5367	523	10	(	(	PUNCT
ejpam-5367	523	11	0	0	NUM
ejpam-5367	523	12	)	)	PUNCT
ejpam-5367	523	13	≥	≥	NOUN
ejpam-5367	523	14	αf	αf	X
ejpam-5367	523	15	(	(	PUNCT
ejpam-5367	523	16	a	a	NOUN
ejpam-5367	523	17	)	)	PUNCT
ejpam-5367	523	18	>	>	PUNCT
ejpam-5367	524	1	t.	t.	PROPN
ejpam-5367	525	1	thus	thus	ADV
ejpam-5367	525	2	,	,	PUNCT
ejpam-5367	525	3	0	0	NUM
ejpam-5367	525	4	∈	∈	PROPN
ejpam-5367	525	5	u	u	NOUN
ejpam-5367	525	6	+	+	X
ejpam-5367	525	7	(	(	PUNCT
ejpam-5367	525	8	αf	αf	X
ejpam-5367	525	9	;	;	PUNCT
ejpam-5367	525	10	t	t	PROPN
ejpam-5367	525	11	)	)	PUNCT
ejpam-5367	525	12	.	.	PUNCT
ejpam-5367	526	1	let	let	VERB
ejpam-5367	526	2	x	x	PRON
ejpam-5367	526	3	,	,	PUNCT
ejpam-5367	526	4	y	y	PROPN
ejpam-5367	526	5	,	,	PUNCT
ejpam-5367	526	6	z	z	PROPN
ejpam-5367	526	7	∈	∈	PROPN
ejpam-5367	526	8	u	u	NOUN
ejpam-5367	526	9	+	+	X
ejpam-5367	526	10	(	(	PUNCT
ejpam-5367	526	11	αf	αf	X
ejpam-5367	526	12	;	;	PUNCT
ejpam-5367	526	13	t	t	X
ejpam-5367	526	14	)	)	PUNCT
ejpam-5367	526	15	be	be	AUX
ejpam-5367	526	16	such	such	ADJ
ejpam-5367	526	17	that	that	SCONJ
ejpam-5367	526	18	x	x	PART
ejpam-5367	526	19	·	·	PUNCT
ejpam-5367	526	20	(	(	PUNCT
ejpam-5367	526	21	y	y	PROPN
ejpam-5367	526	22	·	·	PUNCT
ejpam-5367	526	23	z	z	X
ejpam-5367	526	24	)	)	PUNCT
ejpam-5367	526	25	,	,	PUNCT
ejpam-5367	526	26	y	y	PROPN
ejpam-5367	526	27	∈	∈	PROPN
ejpam-5367	526	28	u	u	NOUN
ejpam-5367	526	29	+	+	X
ejpam-5367	526	30	(	(	PUNCT
ejpam-5367	526	31	αf	αf	X
ejpam-5367	526	32	;	;	PUNCT
ejpam-5367	526	33	t	t	PROPN
ejpam-5367	526	34	)	)	PUNCT
ejpam-5367	526	35	.	.	PUNCT
ejpam-5367	527	1	then	then	ADV
ejpam-5367	527	2	αf	αf	VERB
ejpam-5367	527	3	(	(	PUNCT
ejpam-5367	527	4	x	x	X
ejpam-5367	527	5	·	·	PUNCT
ejpam-5367	527	6	(	(	PUNCT
ejpam-5367	527	7	y	y	PROPN
ejpam-5367	527	8	·	·	PUNCT
ejpam-5367	527	9	z	z	X
ejpam-5367	527	10	)	)	PUNCT
ejpam-5367	527	11	)	)	PUNCT
ejpam-5367	527	12	>	>	X
ejpam-5367	527	13	t	t	PROPN
ejpam-5367	527	14	and	and	CCONJ
ejpam-5367	527	15	αf	αf	PROPN
ejpam-5367	527	16	(	(	PUNCT
ejpam-5367	527	17	y	y	NOUN
ejpam-5367	527	18	)	)	PUNCT
ejpam-5367	527	19	>	>	PUNCT
ejpam-5367	527	20	t.	t.	PROPN
ejpam-5367	528	1	thus	thus	ADV
ejpam-5367	528	2	,	,	PUNCT
ejpam-5367	528	3	min{αf	min{αf	PUNCT
ejpam-5367	528	4	(	(	PUNCT
ejpam-5367	528	5	x	x	X
ejpam-5367	528	6	·	·	PUNCT
ejpam-5367	528	7	(	(	PUNCT
ejpam-5367	528	8	y	y	PROPN
ejpam-5367	528	9	·	·	PUNCT
ejpam-5367	528	10	z	z	NOUN
ejpam-5367	528	11	)	)	PUNCT
ejpam-5367	528	12	)	)	PUNCT
ejpam-5367	528	13	,	,	PUNCT
ejpam-5367	528	14	αf	αf	X
ejpam-5367	528	15	(	(	PUNCT
ejpam-5367	528	16	y	y	NOUN
ejpam-5367	528	17	)	)	PUNCT
ejpam-5367	528	18	}	}	PUNCT
ejpam-5367	528	19	>	>	PUNCT
ejpam-5367	528	20	t.	t.	NOUN
ejpam-5367	528	21	by	by	ADP
ejpam-5367	528	22	(	(	PUNCT
ejpam-5367	528	23	3.6	3.6	NUM
ejpam-5367	528	24	)	)	PUNCT
ejpam-5367	528	25	.	.	PUNCT
ejpam-5367	529	1	we	we	PRON
ejpam-5367	529	2	have	have	VERB
ejpam-5367	529	3	αf	αf	ADP
ejpam-5367	529	4	(	(	PUNCT
ejpam-5367	529	5	x	x	X
ejpam-5367	529	6	·	·	PUNCT
ejpam-5367	529	7	z	z	X
ejpam-5367	529	8	)	)	PUNCT
ejpam-5367	529	9	≥	≥	NOUN
ejpam-5367	529	10	min{αf	min{αf	PUNCT
ejpam-5367	529	11	(	(	PUNCT
ejpam-5367	529	12	x	x	X
ejpam-5367	529	13	·	·	PUNCT
ejpam-5367	529	14	(	(	PUNCT
ejpam-5367	529	15	y	y	PROPN
ejpam-5367	529	16	·	·	PUNCT
ejpam-5367	529	17	z	z	NOUN
ejpam-5367	529	18	)	)	PUNCT
ejpam-5367	529	19	)	)	PUNCT
ejpam-5367	529	20	,	,	PUNCT
ejpam-5367	529	21	αf	αf	X
ejpam-5367	529	22	(	(	PUNCT
ejpam-5367	529	23	y	y	NOUN
ejpam-5367	529	24	)	)	PUNCT
ejpam-5367	529	25	}	}	PUNCT
ejpam-5367	529	26	>	>	PUNCT
ejpam-5367	529	27	t.	t.	PROPN
ejpam-5367	530	1	thus	thus	ADV
ejpam-5367	530	2	,	,	PUNCT
ejpam-5367	530	3	x	x	X
ejpam-5367	530	4	·	·	PUNCT
ejpam-5367	530	5	z	z	X
ejpam-5367	530	6	∈	∈	PROPN
ejpam-5367	530	7	u	u	NOUN
ejpam-5367	530	8	+	+	X
ejpam-5367	530	9	(	(	PUNCT
ejpam-5367	530	10	αf	αf	X
ejpam-5367	530	11	;	;	PUNCT
ejpam-5367	530	12	t	t	PROPN
ejpam-5367	530	13	)	)	PUNCT
ejpam-5367	530	14	.	.	PUNCT
ejpam-5367	531	1	hence	hence	ADV
ejpam-5367	531	2	,	,	PUNCT
ejpam-5367	531	3	u	u	PROPN
ejpam-5367	531	4	+	+	X
ejpam-5367	531	5	(	(	PUNCT
ejpam-5367	531	6	αf	αf	X
ejpam-5367	531	7	;	;	PUNCT
ejpam-5367	531	8	t	t	X
ejpam-5367	531	9	)	)	PUNCT
ejpam-5367	531	10	is	be	AUX
ejpam-5367	531	11	an	an	DET
ejpam-5367	531	12	iup	iup	NOUN
ejpam-5367	531	13	-	-	PUNCT
ejpam-5367	531	14	ideal	ideal	NOUN
ejpam-5367	531	15	of	of	ADP
ejpam-5367	531	16	x.	x.	NOUN
ejpam-5367	531	17	let	let	VERB
ejpam-5367	532	1	s	s	PRON
ejpam-5367	532	2	∈	∈	NOUN
ejpam-5367	532	3	[	[	X
ejpam-5367	532	4	0	0	NUM
ejpam-5367	532	5	,	,	PUNCT
ejpam-5367	532	6	1	1	NUM
ejpam-5367	532	7	]	]	PUNCT
ejpam-5367	532	8	be	be	AUX
ejpam-5367	532	9	such	such	ADJ
ejpam-5367	532	10	that	that	SCONJ
ejpam-5367	532	11	l	l	NOUN
ejpam-5367	532	12	−	−	PROPN
ejpam-5367	533	1	(	(	PUNCT
ejpam-5367	533	2	βf	βf	SYM
ejpam-5367	533	3	;	;	PUNCT
ejpam-5367	533	4	s	s	X
ejpam-5367	533	5	)	)	PUNCT
ejpam-5367	533	6	̸=	̸=	PROPN
ejpam-5367	533	7	∅.	∅.	ADV
ejpam-5367	533	8	let	let	VERB
ejpam-5367	533	9	β	β	X
ejpam-5367	533	10	∈	∈	NOUN
ejpam-5367	533	11	l	l	NOUN
ejpam-5367	533	12	−	−	PROPN
ejpam-5367	534	1	(	(	PUNCT
ejpam-5367	534	2	βf	βf	INTJ
ejpam-5367	534	3	;	;	PUNCT
ejpam-5367	534	4	s	s	X
ejpam-5367	534	5	)	)	PUNCT
ejpam-5367	534	6	.	.	PUNCT
ejpam-5367	535	1	then	then	ADV
ejpam-5367	535	2	βf	βf	INTJ
ejpam-5367	535	3	(	(	PUNCT
ejpam-5367	535	4	β	β	X
ejpam-5367	535	5	)	)	PUNCT
ejpam-5367	535	6	<	<	X
ejpam-5367	535	7	s.	s.	PROPN
ejpam-5367	535	8	by	by	ADP
ejpam-5367	535	9	(	(	PUNCT
ejpam-5367	535	10	3.5	3.5	NUM
ejpam-5367	535	11	)	)	PUNCT
ejpam-5367	535	12	,	,	PUNCT
ejpam-5367	535	13	we	we	PRON
ejpam-5367	535	14	have	have	VERB
ejpam-5367	535	15	βf	βf	ADV
ejpam-5367	535	16	(	(	PUNCT
ejpam-5367	535	17	0	0	X
ejpam-5367	535	18	)	)	PUNCT
ejpam-5367	535	19	≤	≤	NOUN
ejpam-5367	536	1	βf	βf	CCONJ
ejpam-5367	536	2	(	(	PUNCT
ejpam-5367	536	3	β	β	X
ejpam-5367	536	4	)	)	PUNCT
ejpam-5367	536	5	<	<	X
ejpam-5367	536	6	s.	s.	PROPN
ejpam-5367	536	7	thus	thus	ADV
ejpam-5367	536	8	,	,	PUNCT
ejpam-5367	536	9	0	0	NUM
ejpam-5367	536	10	∈	∈	PROPN
ejpam-5367	536	11	l	l	NOUN
ejpam-5367	536	12	−	−	PROPN
ejpam-5367	537	1	(	(	PUNCT
ejpam-5367	537	2	βf	βf	INTJ
ejpam-5367	537	3	;	;	PUNCT
ejpam-5367	537	4	s	s	X
ejpam-5367	537	5	)	)	PUNCT
ejpam-5367	537	6	.	.	PUNCT
ejpam-5367	538	1	let	let	VERB
ejpam-5367	538	2	x	x	PRON
ejpam-5367	538	3	,	,	PUNCT
ejpam-5367	538	4	y	y	PROPN
ejpam-5367	538	5	,	,	PUNCT
ejpam-5367	538	6	z	z	NOUN
ejpam-5367	538	7	∈	∈	PROPN
ejpam-5367	538	8	l	l	NOUN
ejpam-5367	538	9	−	−	PROPN
ejpam-5367	538	10	(	(	PUNCT
ejpam-5367	538	11	βf	βf	SYM
ejpam-5367	538	12	;	;	PUNCT
ejpam-5367	538	13	s	s	AUX
ejpam-5367	538	14	)	)	PUNCT
ejpam-5367	538	15	be	be	AUX
ejpam-5367	538	16	such	such	ADJ
ejpam-5367	538	17	that	that	SCONJ
ejpam-5367	539	1	x	x	PART
ejpam-5367	539	2	·	·	PUNCT
ejpam-5367	539	3	(	(	PUNCT
ejpam-5367	539	4	y	y	PROPN
ejpam-5367	539	5	·	·	PUNCT
ejpam-5367	539	6	z	z	X
ejpam-5367	539	7	)	)	PUNCT
ejpam-5367	539	8	,	,	PUNCT
ejpam-5367	539	9	y	y	PROPN
ejpam-5367	539	10	∈	∈	PROPN
ejpam-5367	539	11	l	l	NOUN
ejpam-5367	539	12	−	−	PROPN
ejpam-5367	540	1	(	(	PUNCT
ejpam-5367	540	2	βf	βf	INTJ
ejpam-5367	540	3	;	;	PUNCT
ejpam-5367	540	4	s	s	X
ejpam-5367	540	5	)	)	PUNCT
ejpam-5367	540	6	.	.	PUNCT
ejpam-5367	541	1	then	then	ADV
ejpam-5367	541	2	βf	βf	INTJ
ejpam-5367	541	3	(	(	PUNCT
ejpam-5367	541	4	x	x	X
ejpam-5367	541	5	·	·	PUNCT
ejpam-5367	541	6	(	(	PUNCT
ejpam-5367	541	7	y	y	PROPN
ejpam-5367	541	8	·	·	PUNCT
ejpam-5367	541	9	z	z	X
ejpam-5367	541	10	)	)	PUNCT
ejpam-5367	541	11	)	)	PUNCT
ejpam-5367	542	1	<	<	X
ejpam-5367	542	2	s	s	X
ejpam-5367	542	3	and	and	CCONJ
ejpam-5367	542	4	βf	βf	INTJ
ejpam-5367	542	5	(	(	PUNCT
ejpam-5367	542	6	y	y	NOUN
ejpam-5367	542	7	)	)	PUNCT
ejpam-5367	542	8	<	<	X
ejpam-5367	542	9	s.	s.	PROPN
ejpam-5367	543	1	thus	thus	ADV
ejpam-5367	543	2	,	,	PUNCT
ejpam-5367	543	3	max{βf	max{βf	PROPN
ejpam-5367	543	4	(	(	PUNCT
ejpam-5367	543	5	x·(y·z	x·(y·z	PROPN
ejpam-5367	543	6	)	)	PUNCT
ejpam-5367	543	7	)	)	PUNCT
ejpam-5367	543	8	,	,	PUNCT
ejpam-5367	543	9	βf	βf	CCONJ
ejpam-5367	543	10	(	(	PUNCT
ejpam-5367	543	11	y	y	NOUN
ejpam-5367	543	12	)	)	PUNCT
ejpam-5367	543	13	}	}	PUNCT
ejpam-5367	543	14	<	<	X
ejpam-5367	543	15	s.	s.	PROPN
ejpam-5367	543	16	by	by	ADP
ejpam-5367	543	17	(	(	PUNCT
ejpam-5367	543	18	3.7	3.7	NUM
ejpam-5367	543	19	)	)	PUNCT
ejpam-5367	543	20	.	.	PUNCT
ejpam-5367	544	1	we	we	PRON
ejpam-5367	544	2	have	have	VERB
ejpam-5367	544	3	βf	βf	ADV
ejpam-5367	544	4	(	(	PUNCT
ejpam-5367	544	5	x·z	x·z	NOUN
ejpam-5367	544	6	)	)	PUNCT
ejpam-5367	544	7	≤	≤	NUM
ejpam-5367	545	1	max{βf	max{βf	INTJ
ejpam-5367	545	2	(	(	PUNCT
ejpam-5367	545	3	x·(y·z	x·(y·z	PROPN
ejpam-5367	545	4	)	)	PUNCT
ejpam-5367	545	5	)	)	PUNCT
ejpam-5367	545	6	,	,	PUNCT
ejpam-5367	545	7	βf	βf	CCONJ
ejpam-5367	545	8	(	(	PUNCT
ejpam-5367	545	9	y	y	NOUN
ejpam-5367	545	10	)	)	PUNCT
ejpam-5367	545	11	}	}	PUNCT
ejpam-5367	545	12	>	>	PUNCT
ejpam-5367	546	1	s.	s.	PROPN
ejpam-5367	546	2	thus	thus	ADV
ejpam-5367	546	3	,	,	PUNCT
ejpam-5367	546	4	x	x	X
ejpam-5367	546	5	·	·	PUNCT
ejpam-5367	546	6	z	z	SYM
ejpam-5367	546	7	∈	∈	PROPN
ejpam-5367	546	8	l	l	NOUN
ejpam-5367	546	9	−	−	PROPN
ejpam-5367	547	1	(	(	PUNCT
ejpam-5367	547	2	βf	βf	INTJ
ejpam-5367	547	3	;	;	PUNCT
ejpam-5367	547	4	s	s	X
ejpam-5367	547	5	)	)	PUNCT
ejpam-5367	547	6	.	.	PUNCT
ejpam-5367	548	1	hence	hence	ADV
ejpam-5367	548	2	,	,	PUNCT
ejpam-5367	548	3	l	l	NOUN
ejpam-5367	548	4	−	−	PROPN
ejpam-5367	548	5	(	(	PUNCT
ejpam-5367	548	6	βf	βf	SYM
ejpam-5367	548	7	;	;	PUNCT
ejpam-5367	548	8	s	s	X
ejpam-5367	548	9	)	)	PUNCT
ejpam-5367	548	10	is	be	AUX
ejpam-5367	548	11	an	an	DET
ejpam-5367	548	12	iup	iup	NOUN
ejpam-5367	548	13	-	-	PUNCT
ejpam-5367	548	14	ideal	ideal	NOUN
ejpam-5367	548	15	of	of	ADP
ejpam-5367	548	16	x.	x.	NOUN
ejpam-5367	548	17	conversely	conversely	ADV
ejpam-5367	548	18	,	,	PUNCT
ejpam-5367	548	19	assume	assume	VERB
ejpam-5367	548	20	that	that	SCONJ
ejpam-5367	548	21	for	for	ADP
ejpam-5367	548	22	all	all	DET
ejpam-5367	548	23	t	t	PROPN
ejpam-5367	548	24	,	,	PUNCT
ejpam-5367	548	25	s	s	PART
ejpam-5367	548	26	∈	∈	PROPN
ejpam-5367	549	1	[	[	X
ejpam-5367	549	2	0	0	NUM
ejpam-5367	549	3	,	,	PUNCT
ejpam-5367	549	4	1	1	NUM
ejpam-5367	549	5	]	]	PUNCT
ejpam-5367	549	6	,	,	PUNCT
ejpam-5367	549	7	the	the	DET
ejpam-5367	549	8	sets	set	NOUN
ejpam-5367	549	9	u	u	NOUN
ejpam-5367	549	10	+	+	X
ejpam-5367	549	11	(	(	PUNCT
ejpam-5367	549	12	αf	αf	X
ejpam-5367	549	13	;	;	PUNCT
ejpam-5367	549	14	t	t	PROPN
ejpam-5367	549	15	)	)	PUNCT
ejpam-5367	549	16	and	and	CCONJ
ejpam-5367	549	17	l	l	NOUN
ejpam-5367	549	18	−	−	PROPN
ejpam-5367	550	1	(	(	PUNCT
ejpam-5367	550	2	βf	βf	SYM
ejpam-5367	550	3	;	;	PUNCT
ejpam-5367	550	4	s	s	X
ejpam-5367	550	5	)	)	PUNCT
ejpam-5367	550	6	are	be	AUX
ejpam-5367	550	7	either	either	CCONJ
ejpam-5367	550	8	empty	empty	ADJ
ejpam-5367	550	9	or	or	CCONJ
ejpam-5367	550	10	iup	iup	NOUN
ejpam-5367	550	11	-	-	PUNCT
ejpam-5367	550	12	ideals	ideal	NOUN
ejpam-5367	550	13	of	of	ADP
ejpam-5367	550	14	x.	x.	NOUN
ejpam-5367	550	15	let	let	VERB
ejpam-5367	550	16	x	x	SYM
ejpam-5367	550	17	∈	∈	PROPN
ejpam-5367	550	18	x.	x.	NOUN
ejpam-5367	550	19	assume	assume	VERB
ejpam-5367	550	20	that	that	SCONJ
ejpam-5367	550	21	αf	αf	VERB
ejpam-5367	550	22	(	(	PUNCT
ejpam-5367	550	23	0	0	NUM
ejpam-5367	550	24	)	)	PUNCT
ejpam-5367	550	25	<	<	X
ejpam-5367	550	26	αf	αf	X
ejpam-5367	550	27	(	(	PUNCT
ejpam-5367	550	28	x	x	NOUN
ejpam-5367	550	29	)	)	PUNCT
ejpam-5367	550	30	.	.	PUNCT
ejpam-5367	551	1	let	let	VERB
ejpam-5367	551	2	t	t	NOUN
ejpam-5367	551	3	=	=	PUNCT
ejpam-5367	551	4	αf	αf	X
ejpam-5367	551	5	(	(	PUNCT
ejpam-5367	551	6	0	0	NUM
ejpam-5367	551	7	)	)	PUNCT
ejpam-5367	551	8	.	.	PUNCT
ejpam-5367	552	1	then	then	ADV
ejpam-5367	552	2	x	x	SYM
ejpam-5367	552	3	∈	∈	PROPN
ejpam-5367	552	4	u	u	NOUN
ejpam-5367	552	5	+	+	X
ejpam-5367	552	6	(	(	PUNCT
ejpam-5367	552	7	αf	αf	X
ejpam-5367	552	8	;	;	PUNCT
ejpam-5367	552	9	t	t	X
ejpam-5367	552	10	)	)	PUNCT
ejpam-5367	552	11	̸=	̸=	PROPN
ejpam-5367	552	12	∅.	∅.	NOUN
ejpam-5367	552	13	by	by	ADP
ejpam-5367	552	14	the	the	DET
ejpam-5367	552	15	assumption	assumption	NOUN
ejpam-5367	552	16	,	,	PUNCT
ejpam-5367	552	17	we	we	PRON
ejpam-5367	552	18	have	have	VERB
ejpam-5367	552	19	u	u	NOUN
ejpam-5367	552	20	+	+	CCONJ
ejpam-5367	552	21	(	(	PUNCT
ejpam-5367	552	22	αf	αf	X
ejpam-5367	552	23	;	;	PUNCT
ejpam-5367	552	24	t	t	X
ejpam-5367	552	25	)	)	PUNCT
ejpam-5367	552	26	is	be	AUX
ejpam-5367	552	27	an	an	DET
ejpam-5367	552	28	iup	iup	NOUN
ejpam-5367	552	29	-	-	PUNCT
ejpam-5367	552	30	ideal	ideal	NOUN
ejpam-5367	552	31	of	of	ADP
ejpam-5367	552	32	x.	x.	NOUN
ejpam-5367	552	33	by	by	ADP
ejpam-5367	552	34	(	(	PUNCT
ejpam-5367	552	35	2.18	2.18	NUM
ejpam-5367	552	36	)	)	PUNCT
ejpam-5367	552	37	,	,	PUNCT
ejpam-5367	552	38	we	we	PRON
ejpam-5367	552	39	have	have	VERB
ejpam-5367	552	40	0	0	NUM
ejpam-5367	552	41	∈	∈	PROPN
ejpam-5367	552	42	u	u	NOUN
ejpam-5367	552	43	+	+	X
ejpam-5367	552	44	(	(	PUNCT
ejpam-5367	552	45	αf	αf	X
ejpam-5367	552	46	;	;	PUNCT
ejpam-5367	552	47	t	t	PROPN
ejpam-5367	552	48	)	)	PUNCT
ejpam-5367	552	49	.	.	PUNCT
ejpam-5367	553	1	so	so	ADV
ejpam-5367	553	2	αf	αf	VERB
ejpam-5367	553	3	(	(	PUNCT
ejpam-5367	553	4	0	0	NUM
ejpam-5367	553	5	)	)	PUNCT
ejpam-5367	553	6	>	>	X
ejpam-5367	553	7	t	t	PROPN
ejpam-5367	554	1	=	=	PUNCT
ejpam-5367	554	2	αf	αf	X
ejpam-5367	554	3	(	(	PUNCT
ejpam-5367	554	4	0	0	NUM
ejpam-5367	554	5	)	)	PUNCT
ejpam-5367	554	6	,	,	PUNCT
ejpam-5367	554	7	which	which	PRON
ejpam-5367	554	8	is	be	AUX
ejpam-5367	554	9	a	a	DET
ejpam-5367	554	10	contradiction	contradiction	NOUN
ejpam-5367	554	11	.	.	PUNCT
ejpam-5367	555	1	thus	thus	ADV
ejpam-5367	555	2	,	,	PUNCT
ejpam-5367	555	3	αf	αf	ADP
ejpam-5367	555	4	(	(	PUNCT
ejpam-5367	555	5	0	0	NUM
ejpam-5367	555	6	)	)	PUNCT
ejpam-5367	555	7	≥	≥	NOUN
ejpam-5367	555	8	αf	αf	X
ejpam-5367	555	9	(	(	PUNCT
ejpam-5367	555	10	x	x	NOUN
ejpam-5367	555	11	)	)	PUNCT
ejpam-5367	555	12	.	.	PUNCT
ejpam-5367	556	1	let	let	VERB
ejpam-5367	556	2	x	x	PRON
ejpam-5367	556	3	,	,	PUNCT
ejpam-5367	556	4	y	y	PROPN
ejpam-5367	556	5	,	,	PUNCT
ejpam-5367	556	6	z	z	PROPN
ejpam-5367	556	7	∈	∈	PROPN
ejpam-5367	556	8	x.	x.	NOUN
ejpam-5367	556	9	assume	assume	VERB
ejpam-5367	556	10	that	that	SCONJ
ejpam-5367	556	11	αf	αf	VERB
ejpam-5367	556	12	(	(	PUNCT
ejpam-5367	556	13	x	x	X
ejpam-5367	556	14	·	·	PUNCT
ejpam-5367	556	15	z	z	X
ejpam-5367	556	16	)	)	PUNCT
ejpam-5367	556	17	<	<	X
ejpam-5367	556	18	min{αf	min{αf	PUNCT
ejpam-5367	556	19	(	(	PUNCT
ejpam-5367	556	20	x	x	X
ejpam-5367	556	21	·	·	PUNCT
ejpam-5367	556	22	(	(	PUNCT
ejpam-5367	556	23	y	y	PROPN
ejpam-5367	556	24	·	·	PUNCT
ejpam-5367	556	25	z	z	NOUN
ejpam-5367	556	26	)	)	PUNCT
ejpam-5367	556	27	)	)	PUNCT
ejpam-5367	556	28	,	,	PUNCT
ejpam-5367	556	29	αf	αf	X
ejpam-5367	556	30	(	(	PUNCT
ejpam-5367	556	31	y	y	NOUN
ejpam-5367	556	32	)	)	PUNCT
ejpam-5367	556	33	}	}	PUNCT
ejpam-5367	556	34	.	.	PUNCT
ejpam-5367	557	1	let	let	VERB
ejpam-5367	557	2	t	t	NOUN
ejpam-5367	557	3	=	=	PUNCT
ejpam-5367	557	4	αf	αf	X
ejpam-5367	557	5	(	(	PUNCT
ejpam-5367	557	6	x	x	X
ejpam-5367	557	7	·	·	PUNCT
ejpam-5367	557	8	z	z	X
ejpam-5367	557	9	)	)	PUNCT
ejpam-5367	557	10	.	.	PUNCT
ejpam-5367	558	1	then	then	ADV
ejpam-5367	558	2	x	x	X
ejpam-5367	558	3	·	·	PUNCT
ejpam-5367	558	4	(	(	PUNCT
ejpam-5367	558	5	y	y	PROPN
ejpam-5367	558	6	·	·	PUNCT
ejpam-5367	558	7	z	z	X
ejpam-5367	558	8	)	)	PUNCT
ejpam-5367	558	9	,	,	PUNCT
ejpam-5367	558	10	y	y	PROPN
ejpam-5367	558	11	∈	∈	PROPN
ejpam-5367	558	12	u	u	NOUN
ejpam-5367	558	13	+	+	X
ejpam-5367	558	14	(	(	PUNCT
ejpam-5367	558	15	αf	αf	X
ejpam-5367	558	16	;	;	PUNCT
ejpam-5367	558	17	t	t	X
ejpam-5367	558	18	)	)	PUNCT
ejpam-5367	558	19	̸=	̸=	PROPN
ejpam-5367	558	20	∅.	∅.	NOUN
ejpam-5367	558	21	by	by	ADP
ejpam-5367	558	22	the	the	DET
ejpam-5367	558	23	assumption	assumption	NOUN
ejpam-5367	558	24	,	,	PUNCT
ejpam-5367	558	25	we	we	PRON
ejpam-5367	558	26	have	have	VERB
ejpam-5367	558	27	u	u	NOUN
ejpam-5367	558	28	+	+	CCONJ
ejpam-5367	558	29	(	(	PUNCT
ejpam-5367	558	30	αf	αf	X
ejpam-5367	558	31	;	;	PUNCT
ejpam-5367	558	32	t	t	X
ejpam-5367	558	33	)	)	PUNCT
ejpam-5367	558	34	is	be	AUX
ejpam-5367	558	35	an	an	DET
ejpam-5367	558	36	iup	iup	NOUN
ejpam-5367	558	37	-	-	PUNCT
ejpam-5367	558	38	ideal	ideal	NOUN
ejpam-5367	558	39	of	of	ADP
ejpam-5367	558	40	x.	x.	NOUN
ejpam-5367	558	41	by	by	ADP
ejpam-5367	558	42	(	(	PUNCT
ejpam-5367	558	43	2.20	2.20	NUM
ejpam-5367	558	44	)	)	PUNCT
ejpam-5367	558	45	,	,	PUNCT
ejpam-5367	558	46	we	we	PRON
ejpam-5367	558	47	have	have	VERB
ejpam-5367	558	48	x	x	X
ejpam-5367	558	49	·	·	PUNCT
ejpam-5367	558	50	z	z	PUNCT
ejpam-5367	558	51	∈	∈	PROPN
ejpam-5367	558	52	u	u	NOUN
ejpam-5367	558	53	+	+	X
ejpam-5367	558	54	(	(	PUNCT
ejpam-5367	558	55	αf	αf	X
ejpam-5367	558	56	;	;	PUNCT
ejpam-5367	558	57	t	t	PROPN
ejpam-5367	558	58	)	)	PUNCT
ejpam-5367	558	59	.	.	PUNCT
ejpam-5367	559	1	so	so	ADV
ejpam-5367	559	2	αf	αf	VERB
ejpam-5367	559	3	(	(	PUNCT
ejpam-5367	559	4	x	x	X
ejpam-5367	559	5	·	·	PUNCT
ejpam-5367	559	6	z	z	X
ejpam-5367	559	7	)	)	PUNCT
ejpam-5367	559	8	>	>	X
ejpam-5367	560	1	t	t	PROPN
ejpam-5367	561	1	=	=	PUNCT
ejpam-5367	561	2	αf	αf	X
ejpam-5367	561	3	(	(	PUNCT
ejpam-5367	561	4	x	x	X
ejpam-5367	561	5	·	·	PUNCT
ejpam-5367	561	6	z	z	X
ejpam-5367	561	7	)	)	PUNCT
ejpam-5367	561	8	,	,	PUNCT
ejpam-5367	561	9	which	which	PRON
ejpam-5367	561	10	is	be	AUX
ejpam-5367	561	11	a	a	DET
ejpam-5367	561	12	contradiction	contradiction	NOUN
ejpam-5367	561	13	.	.	PUNCT
ejpam-5367	562	1	thus	thus	ADV
ejpam-5367	562	2	,	,	PUNCT
ejpam-5367	562	3	αf	αf	ADP
ejpam-5367	562	4	(	(	PUNCT
ejpam-5367	562	5	x	x	X
ejpam-5367	562	6	·	·	PUNCT
ejpam-5367	562	7	z	z	X
ejpam-5367	562	8	)	)	PUNCT
ejpam-5367	562	9	≥	≥	NOUN
ejpam-5367	562	10	min{αf	min{αf	PUNCT
ejpam-5367	562	11	(	(	PUNCT
ejpam-5367	562	12	x	x	X
ejpam-5367	562	13	·	·	PUNCT
ejpam-5367	562	14	(	(	PUNCT
ejpam-5367	562	15	y	y	PROPN
ejpam-5367	562	16	·	·	PUNCT
ejpam-5367	562	17	z	z	NOUN
ejpam-5367	562	18	)	)	PUNCT
ejpam-5367	562	19	)	)	PUNCT
ejpam-5367	562	20	,	,	PUNCT
ejpam-5367	562	21	αf	αf	X
ejpam-5367	562	22	(	(	PUNCT
ejpam-5367	562	23	y	y	NOUN
ejpam-5367	562	24	)	)	PUNCT
ejpam-5367	562	25	}	}	PUNCT
ejpam-5367	562	26	.	.	PUNCT
ejpam-5367	563	1	let	let	VERB
ejpam-5367	563	2	x	x	SYM
ejpam-5367	563	3	∈	∈	PROPN
ejpam-5367	563	4	x.	x.	NOUN
ejpam-5367	563	5	assume	assume	VERB
ejpam-5367	563	6	that	that	SCONJ
ejpam-5367	563	7	βf	βf	INTJ
ejpam-5367	563	8	(	(	PUNCT
ejpam-5367	563	9	0	0	NUM
ejpam-5367	563	10	)	)	PUNCT
ejpam-5367	563	11	>	>	X
ejpam-5367	564	1	βf	βf	INTJ
ejpam-5367	564	2	(	(	PUNCT
ejpam-5367	564	3	x	x	NOUN
ejpam-5367	564	4	)	)	PUNCT
ejpam-5367	564	5	.	.	PUNCT
ejpam-5367	565	1	let	let	VERB
ejpam-5367	565	2	s	s	PRON
ejpam-5367	565	3	=	=	X
ejpam-5367	565	4	βf	βf	INTJ
ejpam-5367	565	5	(	(	PUNCT
ejpam-5367	565	6	0	0	NUM
ejpam-5367	565	7	)	)	PUNCT
ejpam-5367	565	8	.	.	PUNCT
ejpam-5367	566	1	then	then	ADV
ejpam-5367	566	2	x	x	SYM
ejpam-5367	566	3	∈	∈	PROPN
ejpam-5367	566	4	l	l	NOUN
ejpam-5367	566	5	−	−	PROPN
ejpam-5367	567	1	(	(	PUNCT
ejpam-5367	567	2	βf	βf	SYM
ejpam-5367	567	3	;	;	PUNCT
ejpam-5367	567	4	s	s	X
ejpam-5367	567	5	)	)	PUNCT
ejpam-5367	567	6	̸=	̸=	PROPN
ejpam-5367	567	7	∅.	∅.	NOUN
ejpam-5367	567	8	by	by	ADP
ejpam-5367	567	9	the	the	DET
ejpam-5367	567	10	assumption	assumption	NOUN
ejpam-5367	567	11	,	,	PUNCT
ejpam-5367	567	12	we	we	PRON
ejpam-5367	567	13	have	have	VERB
ejpam-5367	567	14	l	l	NOUN
ejpam-5367	567	15	−	−	PROPN
ejpam-5367	567	16	(	(	PUNCT
ejpam-5367	567	17	βf	βf	SYM
ejpam-5367	567	18	;	;	PUNCT
ejpam-5367	567	19	s	s	X
ejpam-5367	567	20	)	)	PUNCT
ejpam-5367	567	21	is	be	AUX
ejpam-5367	567	22	an	an	DET
ejpam-5367	567	23	iup	iup	NOUN
ejpam-5367	567	24	-	-	PUNCT
ejpam-5367	567	25	ideal	ideal	NOUN
ejpam-5367	567	26	of	of	ADP
ejpam-5367	567	27	x.	x.	NOUN
ejpam-5367	567	28	by	by	ADP
ejpam-5367	567	29	(	(	PUNCT
ejpam-5367	567	30	2.18	2.18	NUM
ejpam-5367	567	31	)	)	PUNCT
ejpam-5367	567	32	,	,	PUNCT
ejpam-5367	567	33	we	we	PRON
ejpam-5367	567	34	have	have	VERB
ejpam-5367	567	35	0	0	NUM
ejpam-5367	567	36	∈	∈	NOUN
ejpam-5367	567	37	a.	a.	NOUN
ejpam-5367	567	38	iampan	iampan	NOUN
ejpam-5367	567	39	et	et	PROPN
ejpam-5367	567	40	al	al	PROPN
ejpam-5367	567	41	.	.	PUNCT
ejpam-5367	567	42	/	/	SYM
ejpam-5367	567	43	eur	eur	PROPN
ejpam-5367	567	44	.	.	PUNCT
ejpam-5367	568	1	j.	j.	PROPN
ejpam-5367	568	2	pure	pure	PROPN
ejpam-5367	568	3	appl	appl	PROPN
ejpam-5367	568	4	.	.	PROPN
ejpam-5367	568	5	math	math	PROPN
ejpam-5367	568	6	,	,	PUNCT
ejpam-5367	568	7	17	17	NUM
ejpam-5367	568	8	(	(	PUNCT
ejpam-5367	568	9	4	4	NUM
ejpam-5367	568	10	)	)	PUNCT
ejpam-5367	568	11	(	(	PUNCT
ejpam-5367	568	12	2024	2024	NUM
ejpam-5367	568	13	)	)	PUNCT
ejpam-5367	568	14	,	,	PUNCT
ejpam-5367	568	15	3022	3022	NUM
ejpam-5367	568	16	-	-	SYM
ejpam-5367	568	17	3042	3042	NUM
ejpam-5367	568	18	3040	3040	NUM
ejpam-5367	568	19	l	l	NOUN
ejpam-5367	568	20	−	−	PROPN
ejpam-5367	568	21	(	(	PUNCT
ejpam-5367	568	22	βf	βf	INTJ
ejpam-5367	568	23	;	;	PUNCT
ejpam-5367	568	24	s	s	X
ejpam-5367	568	25	)	)	PUNCT
ejpam-5367	568	26	.	.	PUNCT
ejpam-5367	569	1	so	so	ADV
ejpam-5367	569	2	βf	βf	INTJ
ejpam-5367	569	3	(	(	PUNCT
ejpam-5367	569	4	0	0	NUM
ejpam-5367	569	5	)	)	PUNCT
ejpam-5367	569	6	<	<	X
ejpam-5367	569	7	s	s	X
ejpam-5367	569	8	=	=	X
ejpam-5367	569	9	βf	βf	INTJ
ejpam-5367	569	10	(	(	PUNCT
ejpam-5367	569	11	0	0	NUM
ejpam-5367	569	12	)	)	PUNCT
ejpam-5367	569	13	,	,	PUNCT
ejpam-5367	569	14	which	which	PRON
ejpam-5367	569	15	is	be	AUX
ejpam-5367	569	16	a	a	DET
ejpam-5367	569	17	contradiction	contradiction	NOUN
ejpam-5367	569	18	.	.	PUNCT
ejpam-5367	570	1	thus	thus	ADV
ejpam-5367	570	2	,	,	PUNCT
ejpam-5367	570	3	βf	βf	PRON
ejpam-5367	570	4	(	(	PUNCT
ejpam-5367	570	5	0	0	X
ejpam-5367	570	6	)	)	PUNCT
ejpam-5367	570	7	≤	≤	NOUN
ejpam-5367	570	8	βf	βf	CCONJ
ejpam-5367	570	9	(	(	PUNCT
ejpam-5367	570	10	x	x	NOUN
ejpam-5367	570	11	)	)	PUNCT
ejpam-5367	570	12	.	.	PUNCT
ejpam-5367	571	1	let	let	VERB
ejpam-5367	571	2	x	x	PRON
ejpam-5367	571	3	,	,	PUNCT
ejpam-5367	571	4	y	y	PROPN
ejpam-5367	571	5	,	,	PUNCT
ejpam-5367	571	6	z	z	PROPN
ejpam-5367	571	7	∈	∈	PROPN
ejpam-5367	571	8	x.	x.	NOUN
ejpam-5367	571	9	assume	assume	VERB
ejpam-5367	571	10	that	that	SCONJ
ejpam-5367	572	1	βf	βf	INTJ
ejpam-5367	572	2	(	(	PUNCT
ejpam-5367	572	3	x	x	X
ejpam-5367	572	4	·	·	PUNCT
ejpam-5367	573	1	z	z	X
ejpam-5367	573	2	)	)	PUNCT
ejpam-5367	573	3	>	>	X
ejpam-5367	574	1	max{βf	max{βf	PUNCT
ejpam-5367	574	2	(	(	PUNCT
ejpam-5367	574	3	x	x	X
ejpam-5367	574	4	·	·	PUNCT
ejpam-5367	574	5	(	(	PUNCT
ejpam-5367	574	6	y	y	PROPN
ejpam-5367	574	7	·	·	PUNCT
ejpam-5367	574	8	z	z	NOUN
ejpam-5367	574	9	)	)	PUNCT
ejpam-5367	574	10	)	)	PUNCT
ejpam-5367	574	11	,	,	PUNCT
ejpam-5367	574	12	βf	βf	CCONJ
ejpam-5367	574	13	(	(	PUNCT
ejpam-5367	574	14	y	y	NOUN
ejpam-5367	574	15	)	)	PUNCT
ejpam-5367	574	16	}	}	PUNCT
ejpam-5367	574	17	.	.	PUNCT
ejpam-5367	575	1	let	let	VERB
ejpam-5367	575	2	s	s	PRON
ejpam-5367	575	3	=	=	VERB
ejpam-5367	575	4	βf	βf	INTJ
ejpam-5367	575	5	(	(	PUNCT
ejpam-5367	575	6	x	x	X
ejpam-5367	575	7	·	·	PUNCT
ejpam-5367	575	8	z	z	X
ejpam-5367	575	9	)	)	PUNCT
ejpam-5367	575	10	.	.	PUNCT
ejpam-5367	576	1	then	then	ADV
ejpam-5367	576	2	x	x	X
ejpam-5367	576	3	·	·	PUNCT
ejpam-5367	576	4	(	(	PUNCT
ejpam-5367	576	5	y	y	PROPN
ejpam-5367	576	6	·	·	PUNCT
ejpam-5367	576	7	z	z	X
ejpam-5367	576	8	)	)	PUNCT
ejpam-5367	576	9	,	,	PUNCT
ejpam-5367	576	10	y	y	PROPN
ejpam-5367	576	11	∈	∈	PROPN
ejpam-5367	576	12	l	l	NOUN
ejpam-5367	576	13	−	−	PROPN
ejpam-5367	577	1	(	(	PUNCT
ejpam-5367	577	2	βf	βf	SYM
ejpam-5367	577	3	;	;	PUNCT
ejpam-5367	577	4	s	s	X
ejpam-5367	577	5	)	)	PUNCT
ejpam-5367	577	6	̸=	̸=	PROPN
ejpam-5367	577	7	∅.	∅.	NOUN
ejpam-5367	577	8	by	by	ADP
ejpam-5367	577	9	the	the	DET
ejpam-5367	577	10	assumption	assumption	NOUN
ejpam-5367	577	11	,	,	PUNCT
ejpam-5367	577	12	we	we	PRON
ejpam-5367	577	13	have	have	VERB
ejpam-5367	577	14	l	l	NOUN
ejpam-5367	577	15	−	−	PROPN
ejpam-5367	577	16	(	(	PUNCT
ejpam-5367	577	17	βf	βf	SYM
ejpam-5367	577	18	;	;	PUNCT
ejpam-5367	577	19	s	s	X
ejpam-5367	577	20	)	)	PUNCT
ejpam-5367	577	21	is	be	AUX
ejpam-5367	577	22	an	an	DET
ejpam-5367	577	23	iup	iup	NOUN
ejpam-5367	577	24	-	-	PUNCT
ejpam-5367	577	25	ideal	ideal	NOUN
ejpam-5367	577	26	of	of	ADP
ejpam-5367	577	27	x.	x.	NOUN
ejpam-5367	577	28	by	by	ADP
ejpam-5367	577	29	(	(	PUNCT
ejpam-5367	577	30	2.20	2.20	NUM
ejpam-5367	577	31	)	)	PUNCT
ejpam-5367	577	32	,	,	PUNCT
ejpam-5367	577	33	we	we	PRON
ejpam-5367	577	34	have	have	VERB
ejpam-5367	577	35	x	x	X
ejpam-5367	577	36	·	·	PUNCT
ejpam-5367	577	37	z	z	NOUN
ejpam-5367	577	38	∈	∈	PROPN
ejpam-5367	577	39	l	l	NOUN
ejpam-5367	577	40	−	−	PROPN
ejpam-5367	578	1	(	(	PUNCT
ejpam-5367	578	2	βf	βf	INTJ
ejpam-5367	578	3	;	;	PUNCT
ejpam-5367	578	4	s	s	X
ejpam-5367	578	5	)	)	PUNCT
ejpam-5367	578	6	.	.	PUNCT
ejpam-5367	579	1	so	so	ADV
ejpam-5367	579	2	βf	βf	INTJ
ejpam-5367	579	3	(	(	PUNCT
ejpam-5367	579	4	x	x	X
ejpam-5367	579	5	·	·	SYM
ejpam-5367	579	6	z	z	X
ejpam-5367	579	7	)	)	PUNCT
ejpam-5367	579	8	<	<	X
ejpam-5367	579	9	s	s	X
ejpam-5367	579	10	=	=	X
ejpam-5367	579	11	βf	βf	INTJ
ejpam-5367	579	12	(	(	PUNCT
ejpam-5367	579	13	x	x	X
ejpam-5367	579	14	·	·	SYM
ejpam-5367	579	15	z	z	NOUN
ejpam-5367	579	16	)	)	PUNCT
ejpam-5367	579	17	,	,	PUNCT
ejpam-5367	579	18	which	which	PRON
ejpam-5367	579	19	is	be	AUX
ejpam-5367	579	20	a	a	DET
ejpam-5367	579	21	contradiction	contradiction	NOUN
ejpam-5367	579	22	.	.	PUNCT
ejpam-5367	580	1	thus	thus	ADV
ejpam-5367	580	2	,	,	PUNCT
ejpam-5367	580	3	βf	βf	INTJ
ejpam-5367	580	4	(	(	PUNCT
ejpam-5367	580	5	x	x	X
ejpam-5367	580	6	·	·	PUNCT
ejpam-5367	580	7	z	z	X
ejpam-5367	580	8	)	)	PUNCT
ejpam-5367	580	9	≤	≤	NUM
ejpam-5367	580	10	max{βf	max{βf	INTJ
ejpam-5367	580	11	(	(	PUNCT
ejpam-5367	580	12	x	x	X
ejpam-5367	580	13	·	·	PUNCT
ejpam-5367	580	14	(	(	PUNCT
ejpam-5367	580	15	y	y	PROPN
ejpam-5367	580	16	·	·	PUNCT
ejpam-5367	580	17	z	z	NOUN
ejpam-5367	580	18	)	)	PUNCT
ejpam-5367	580	19	)	)	PUNCT
ejpam-5367	580	20	,	,	PUNCT
ejpam-5367	580	21	βf	βf	CCONJ
ejpam-5367	580	22	(	(	PUNCT
ejpam-5367	580	23	y	y	NOUN
ejpam-5367	580	24	)	)	PUNCT
ejpam-5367	580	25	}	}	PUNCT
ejpam-5367	580	26	.	.	PUNCT
ejpam-5367	581	1	hence	hence	ADV
ejpam-5367	581	2	,	,	PUNCT
ejpam-5367	581	3	f	f	PROPN
ejpam-5367	581	4	is	be	AUX
ejpam-5367	581	5	an	an	DET
ejpam-5367	581	6	ffiup	ffiup	NOUN
ejpam-5367	581	7	-	-	PUNCT
ejpam-5367	581	8	ideal	ideal	NOUN
ejpam-5367	581	9	of	of	ADP
ejpam-5367	581	10	x.	x.	PROPN
ejpam-5367	581	11	theorem	theorem	VERB
ejpam-5367	581	12	26	26	NUM
ejpam-5367	581	13	.	.	PUNCT
ejpam-5367	582	1	an	an	DET
ejpam-5367	582	2	ffs	ffs	NOUN
ejpam-5367	583	1	a	a	PRON
ejpam-5367	583	2	in	in	ADP
ejpam-5367	583	3	x	x	PROPN
ejpam-5367	583	4	is	be	AUX
ejpam-5367	583	5	an	an	DET
ejpam-5367	583	6	ffiup	ffiup	NOUN
ejpam-5367	583	7	-	-	PUNCT
ejpam-5367	583	8	filter	filter	NOUN
ejpam-5367	583	9	of	of	ADP
ejpam-5367	583	10	x	x	SYM
ejpam-5367	583	11	if	if	SCONJ
ejpam-5367	583	12	and	and	CCONJ
ejpam-5367	583	13	only	only	ADV
ejpam-5367	583	14	if	if	SCONJ
ejpam-5367	583	15	for	for	ADP
ejpam-5367	583	16	all	all	DET
ejpam-5367	583	17	t	t	NOUN
ejpam-5367	583	18	,	,	PUNCT
ejpam-5367	583	19	s	s	PART
ejpam-5367	583	20	∈	∈	PROPN
ejpam-5367	584	1	[	[	X
ejpam-5367	584	2	0	0	NUM
ejpam-5367	584	3	,	,	PUNCT
ejpam-5367	584	4	1	1	NUM
ejpam-5367	584	5	]	]	PUNCT
ejpam-5367	584	6	,	,	PUNCT
ejpam-5367	584	7	the	the	DET
ejpam-5367	584	8	sets	set	NOUN
ejpam-5367	584	9	u	u	NOUN
ejpam-5367	584	10	+	+	X
ejpam-5367	584	11	(	(	PUNCT
ejpam-5367	584	12	αf	αf	X
ejpam-5367	584	13	;	;	PUNCT
ejpam-5367	584	14	t	t	PROPN
ejpam-5367	584	15	)	)	PUNCT
ejpam-5367	584	16	and	and	CCONJ
ejpam-5367	584	17	l	l	NOUN
ejpam-5367	584	18	−	−	PROPN
ejpam-5367	585	1	(	(	PUNCT
ejpam-5367	585	2	βf	βf	SYM
ejpam-5367	585	3	;	;	PUNCT
ejpam-5367	585	4	s	s	X
ejpam-5367	585	5	)	)	PUNCT
ejpam-5367	585	6	are	be	AUX
ejpam-5367	585	7	either	either	CCONJ
ejpam-5367	585	8	empty	empty	ADJ
ejpam-5367	585	9	or	or	CCONJ
ejpam-5367	585	10	iup	iup	NOUN
ejpam-5367	585	11	-	-	PUNCT
ejpam-5367	585	12	filters	filter	NOUN
ejpam-5367	585	13	of	of	ADP
ejpam-5367	585	14	x.	x.	NOUN
ejpam-5367	585	15	proof	proof	PROPN
ejpam-5367	585	16	.	.	PUNCT
ejpam-5367	586	1	assume	assume	VERB
ejpam-5367	586	2	that	that	SCONJ
ejpam-5367	586	3	f	f	PROPN
ejpam-5367	586	4	is	be	AUX
ejpam-5367	586	5	an	an	DET
ejpam-5367	586	6	ffiup	ffiup	ADJ
ejpam-5367	586	7	-	-	PUNCT
ejpam-5367	586	8	filter	filter	NOUN
ejpam-5367	586	9	of	of	ADP
ejpam-5367	586	10	x.	x.	NOUN
ejpam-5367	586	11	let	let	VERB
ejpam-5367	586	12	t	t	PROPN
ejpam-5367	586	13	∈	∈	PROPN
ejpam-5367	587	1	[	[	X
ejpam-5367	587	2	0	0	NUM
ejpam-5367	587	3	,	,	PUNCT
ejpam-5367	587	4	1	1	NUM
ejpam-5367	587	5	]	]	PUNCT
ejpam-5367	587	6	be	be	AUX
ejpam-5367	587	7	such	such	ADJ
ejpam-5367	587	8	that	that	SCONJ
ejpam-5367	587	9	u	u	NOUN
ejpam-5367	587	10	+	+	X
ejpam-5367	587	11	(	(	PUNCT
ejpam-5367	587	12	αf	αf	X
ejpam-5367	587	13	;	;	PUNCT
ejpam-5367	587	14	t	t	X
ejpam-5367	587	15	)	)	PUNCT
ejpam-5367	587	16	̸=	̸=	PROPN
ejpam-5367	587	17	∅.	∅.	ADV
ejpam-5367	587	18	let	let	VERB
ejpam-5367	587	19	a	a	DET
ejpam-5367	587	20	∈	∈	PROPN
ejpam-5367	587	21	u	u	NOUN
ejpam-5367	587	22	+	+	X
ejpam-5367	587	23	(	(	PUNCT
ejpam-5367	587	24	αf	αf	X
ejpam-5367	587	25	;	;	PUNCT
ejpam-5367	587	26	t	t	PROPN
ejpam-5367	587	27	)	)	PUNCT
ejpam-5367	587	28	.	.	PUNCT
ejpam-5367	588	1	then	then	ADV
ejpam-5367	588	2	αf	αf	VERB
ejpam-5367	588	3	(	(	PUNCT
ejpam-5367	588	4	a	a	NOUN
ejpam-5367	588	5	)	)	PUNCT
ejpam-5367	588	6	>	>	PUNCT
ejpam-5367	589	1	t.	t.	PROPN
ejpam-5367	589	2	by	by	ADP
ejpam-5367	589	3	(	(	PUNCT
ejpam-5367	589	4	3.4	3.4	NUM
ejpam-5367	589	5	)	)	PUNCT
ejpam-5367	589	6	,	,	PUNCT
ejpam-5367	589	7	we	we	PRON
ejpam-5367	589	8	have	have	VERB
ejpam-5367	589	9	αf	αf	NUM
ejpam-5367	589	10	(	(	PUNCT
ejpam-5367	589	11	0	0	NUM
ejpam-5367	589	12	)	)	PUNCT
ejpam-5367	589	13	≥	≥	NOUN
ejpam-5367	589	14	αf	αf	X
ejpam-5367	589	15	(	(	PUNCT
ejpam-5367	589	16	a	a	NOUN
ejpam-5367	589	17	)	)	PUNCT
ejpam-5367	589	18	>	>	PUNCT
ejpam-5367	590	1	t.	t.	PROPN
ejpam-5367	591	1	thus	thus	ADV
ejpam-5367	591	2	,	,	PUNCT
ejpam-5367	591	3	0	0	NUM
ejpam-5367	591	4	∈	∈	PROPN
ejpam-5367	591	5	u	u	NOUN
ejpam-5367	591	6	+	+	X
ejpam-5367	591	7	(	(	PUNCT
ejpam-5367	591	8	αf	αf	X
ejpam-5367	591	9	;	;	PUNCT
ejpam-5367	591	10	t	t	PROPN
ejpam-5367	591	11	)	)	PUNCT
ejpam-5367	591	12	.	.	PUNCT
ejpam-5367	592	1	let	let	VERB
ejpam-5367	592	2	x	x	PRON
ejpam-5367	592	3	,	,	PUNCT
ejpam-5367	592	4	y	y	PROPN
ejpam-5367	592	5	∈	∈	PROPN
ejpam-5367	592	6	u	u	NOUN
ejpam-5367	592	7	+	+	X
ejpam-5367	592	8	(	(	PUNCT
ejpam-5367	592	9	αf	αf	X
ejpam-5367	592	10	;	;	PUNCT
ejpam-5367	592	11	t	t	X
ejpam-5367	592	12	)	)	PUNCT
ejpam-5367	592	13	be	be	AUX
ejpam-5367	592	14	such	such	ADJ
ejpam-5367	592	15	that	that	SCONJ
ejpam-5367	592	16	x	x	X
ejpam-5367	592	17	·	·	PUNCT
ejpam-5367	592	18	y	y	X
ejpam-5367	592	19	,	,	PUNCT
ejpam-5367	592	20	x	x	SYM
ejpam-5367	592	21	∈	∈	PROPN
ejpam-5367	592	22	u	u	NOUN
ejpam-5367	592	23	+	+	X
ejpam-5367	592	24	(	(	PUNCT
ejpam-5367	592	25	αf	αf	X
ejpam-5367	592	26	;	;	PUNCT
ejpam-5367	592	27	t	t	PROPN
ejpam-5367	592	28	)	)	PUNCT
ejpam-5367	592	29	.	.	PUNCT
ejpam-5367	593	1	then	then	ADV
ejpam-5367	593	2	αf	αf	VERB
ejpam-5367	593	3	(	(	PUNCT
ejpam-5367	593	4	x	x	PROPN
ejpam-5367	593	5	·	·	PUNCT
ejpam-5367	593	6	y	y	X
ejpam-5367	593	7	)	)	PUNCT
ejpam-5367	593	8	>	>	X
ejpam-5367	593	9	t	t	PROPN
ejpam-5367	593	10	and	and	CCONJ
ejpam-5367	593	11	αf	αf	PROPN
ejpam-5367	593	12	(	(	PUNCT
ejpam-5367	593	13	x	x	X
ejpam-5367	593	14	)	)	PUNCT
ejpam-5367	593	15	>	>	PUNCT
ejpam-5367	593	16	t.	t.	PROPN
ejpam-5367	594	1	thus	thus	ADV
ejpam-5367	594	2	,	,	PUNCT
ejpam-5367	594	3	min{αf	min{αf	PUNCT
ejpam-5367	594	4	(	(	PUNCT
ejpam-5367	594	5	x	x	X
ejpam-5367	594	6	·	·	PUNCT
ejpam-5367	594	7	y	y	X
ejpam-5367	594	8	)	)	PUNCT
ejpam-5367	594	9	,	,	PUNCT
ejpam-5367	594	10	αf	αf	X
ejpam-5367	594	11	(	(	PUNCT
ejpam-5367	594	12	x	x	NOUN
ejpam-5367	594	13	)	)	PUNCT
ejpam-5367	594	14	}	}	PUNCT
ejpam-5367	594	15	>	>	PUNCT
ejpam-5367	594	16	t.	t.	NOUN
ejpam-5367	594	17	by	by	ADP
ejpam-5367	594	18	(	(	PUNCT
ejpam-5367	594	19	3.8	3.8	NUM
ejpam-5367	594	20	)	)	PUNCT
ejpam-5367	594	21	.	.	PUNCT
ejpam-5367	595	1	we	we	PRON
ejpam-5367	595	2	have	have	VERB
ejpam-5367	595	3	αf	αf	ADP
ejpam-5367	595	4	(	(	PUNCT
ejpam-5367	595	5	y	y	NOUN
ejpam-5367	595	6	)	)	PUNCT
ejpam-5367	595	7	≥	≥	NOUN
ejpam-5367	595	8	min{αf	min{αf	PUNCT
ejpam-5367	595	9	(	(	PUNCT
ejpam-5367	595	10	x	x	X
ejpam-5367	595	11	·	·	PUNCT
ejpam-5367	595	12	y	y	X
ejpam-5367	595	13	)	)	PUNCT
ejpam-5367	595	14	,	,	PUNCT
ejpam-5367	595	15	αf	αf	X
ejpam-5367	595	16	(	(	PUNCT
ejpam-5367	595	17	x	x	NOUN
ejpam-5367	595	18	)	)	PUNCT
ejpam-5367	595	19	}	}	PUNCT
ejpam-5367	595	20	>	>	PUNCT
ejpam-5367	595	21	t.	t.	PROPN
ejpam-5367	596	1	thus	thus	ADV
ejpam-5367	596	2	,	,	PUNCT
ejpam-5367	596	3	y	y	PROPN
ejpam-5367	596	4	∈	∈	PROPN
ejpam-5367	596	5	u	u	NOUN
ejpam-5367	596	6	+	+	X
ejpam-5367	596	7	(	(	PUNCT
ejpam-5367	596	8	αf	αf	X
ejpam-5367	596	9	;	;	PUNCT
ejpam-5367	596	10	t	t	PROPN
ejpam-5367	596	11	)	)	PUNCT
ejpam-5367	596	12	.	.	PUNCT
ejpam-5367	597	1	hence	hence	ADV
ejpam-5367	597	2	,	,	PUNCT
ejpam-5367	597	3	u	u	PROPN
ejpam-5367	597	4	+	+	X
ejpam-5367	597	5	(	(	PUNCT
ejpam-5367	597	6	αf	αf	X
ejpam-5367	597	7	;	;	PUNCT
ejpam-5367	597	8	t	t	X
ejpam-5367	597	9	)	)	PUNCT
ejpam-5367	597	10	is	be	AUX
ejpam-5367	597	11	an	an	DET
ejpam-5367	597	12	iup	iup	NOUN
ejpam-5367	597	13	-	-	PUNCT
ejpam-5367	597	14	filter	filter	NOUN
ejpam-5367	597	15	of	of	ADP
ejpam-5367	597	16	x.	x.	NOUN
ejpam-5367	597	17	let	let	VERB
ejpam-5367	598	1	s	s	PRON
ejpam-5367	598	2	∈	∈	NOUN
ejpam-5367	598	3	[	[	X
ejpam-5367	598	4	0	0	NUM
ejpam-5367	598	5	,	,	PUNCT
ejpam-5367	598	6	1	1	NUM
ejpam-5367	598	7	]	]	PUNCT
ejpam-5367	598	8	be	be	AUX
ejpam-5367	598	9	such	such	ADJ
ejpam-5367	598	10	that	that	SCONJ
ejpam-5367	598	11	l	l	NOUN
ejpam-5367	598	12	−	−	PROPN
ejpam-5367	599	1	(	(	PUNCT
ejpam-5367	599	2	βf	βf	SYM
ejpam-5367	599	3	;	;	PUNCT
ejpam-5367	599	4	s	s	X
ejpam-5367	599	5	)	)	PUNCT
ejpam-5367	599	6	̸=	̸=	PROPN
ejpam-5367	599	7	∅.	∅.	ADV
ejpam-5367	599	8	let	let	VERB
ejpam-5367	599	9	β	β	X
ejpam-5367	599	10	∈	∈	NOUN
ejpam-5367	599	11	l	l	NOUN
ejpam-5367	599	12	−	−	PROPN
ejpam-5367	600	1	(	(	PUNCT
ejpam-5367	600	2	βf	βf	INTJ
ejpam-5367	600	3	;	;	PUNCT
ejpam-5367	600	4	s	s	X
ejpam-5367	600	5	)	)	PUNCT
ejpam-5367	600	6	.	.	PUNCT
ejpam-5367	601	1	then	then	ADV
ejpam-5367	601	2	βf	βf	INTJ
ejpam-5367	601	3	(	(	PUNCT
ejpam-5367	601	4	β	β	X
ejpam-5367	601	5	)	)	PUNCT
ejpam-5367	601	6	<	<	X
ejpam-5367	601	7	s.	s.	PROPN
ejpam-5367	601	8	by	by	ADP
ejpam-5367	601	9	(	(	PUNCT
ejpam-5367	601	10	3.5	3.5	NUM
ejpam-5367	601	11	)	)	PUNCT
ejpam-5367	601	12	,	,	PUNCT
ejpam-5367	601	13	we	we	PRON
ejpam-5367	601	14	have	have	VERB
ejpam-5367	601	15	βf	βf	ADV
ejpam-5367	601	16	(	(	PUNCT
ejpam-5367	601	17	0	0	X
ejpam-5367	601	18	)	)	PUNCT
ejpam-5367	601	19	≤	≤	NOUN
ejpam-5367	602	1	βf	βf	CCONJ
ejpam-5367	602	2	(	(	PUNCT
ejpam-5367	602	3	β	β	X
ejpam-5367	602	4	)	)	PUNCT
ejpam-5367	602	5	<	<	X
ejpam-5367	602	6	s.	s.	PROPN
ejpam-5367	602	7	thus	thus	ADV
ejpam-5367	602	8	,	,	PUNCT
ejpam-5367	602	9	0	0	NUM
ejpam-5367	602	10	∈	∈	PROPN
ejpam-5367	602	11	l	l	NOUN
ejpam-5367	602	12	−	−	PROPN
ejpam-5367	603	1	(	(	PUNCT
ejpam-5367	603	2	βf	βf	INTJ
ejpam-5367	603	3	;	;	PUNCT
ejpam-5367	603	4	s	s	X
ejpam-5367	603	5	)	)	PUNCT
ejpam-5367	603	6	.	.	PUNCT
ejpam-5367	604	1	let	let	VERB
ejpam-5367	604	2	x	x	PRON
ejpam-5367	604	3	,	,	PUNCT
ejpam-5367	604	4	y	y	PROPN
ejpam-5367	604	5	∈	∈	PROPN
ejpam-5367	604	6	l	l	NOUN
ejpam-5367	604	7	−	−	PROPN
ejpam-5367	605	1	(	(	PUNCT
ejpam-5367	605	2	βf	βf	SYM
ejpam-5367	605	3	;	;	PUNCT
ejpam-5367	605	4	s	s	AUX
ejpam-5367	605	5	)	)	PUNCT
ejpam-5367	605	6	be	be	AUX
ejpam-5367	605	7	such	such	ADJ
ejpam-5367	605	8	that	that	SCONJ
ejpam-5367	605	9	x	x	X
ejpam-5367	605	10	·	·	PUNCT
ejpam-5367	605	11	y	y	X
ejpam-5367	605	12	,	,	PUNCT
ejpam-5367	605	13	x	x	X
ejpam-5367	605	14	∈	∈	NOUN
ejpam-5367	605	15	l	l	NOUN
ejpam-5367	605	16	−	−	PROPN
ejpam-5367	606	1	(	(	PUNCT
ejpam-5367	606	2	βf	βf	INTJ
ejpam-5367	606	3	;	;	PUNCT
ejpam-5367	606	4	s	s	X
ejpam-5367	606	5	)	)	PUNCT
ejpam-5367	606	6	.	.	PUNCT
ejpam-5367	607	1	then	then	ADV
ejpam-5367	607	2	βf	βf	INTJ
ejpam-5367	607	3	(	(	PUNCT
ejpam-5367	607	4	x	x	X
ejpam-5367	607	5	·	·	PUNCT
ejpam-5367	607	6	y	y	X
ejpam-5367	607	7	)	)	PUNCT
ejpam-5367	607	8	<	<	X
ejpam-5367	607	9	s	s	X
ejpam-5367	607	10	and	and	CCONJ
ejpam-5367	607	11	βf	βf	INTJ
ejpam-5367	607	12	(	(	PUNCT
ejpam-5367	607	13	x	x	X
ejpam-5367	607	14	)	)	PUNCT
ejpam-5367	607	15	<	<	X
ejpam-5367	607	16	s.	s.	PROPN
ejpam-5367	608	1	thus	thus	ADV
ejpam-5367	608	2	,	,	PUNCT
ejpam-5367	608	3	max{βf	max{βf	PROPN
ejpam-5367	608	4	(	(	PUNCT
ejpam-5367	608	5	x	x	X
ejpam-5367	608	6	·	·	PUNCT
ejpam-5367	608	7	y	y	X
ejpam-5367	608	8	)	)	PUNCT
ejpam-5367	608	9	,	,	PUNCT
ejpam-5367	608	10	βf	βf	CCONJ
ejpam-5367	608	11	(	(	PUNCT
ejpam-5367	608	12	x	x	X
ejpam-5367	608	13	)	)	PUNCT
ejpam-5367	608	14	}	}	PUNCT
ejpam-5367	609	1	<	<	X
ejpam-5367	609	2	s.	s.	PROPN
ejpam-5367	609	3	by	by	ADP
ejpam-5367	609	4	(	(	PUNCT
ejpam-5367	609	5	3.9	3.9	NUM
ejpam-5367	609	6	)	)	PUNCT
ejpam-5367	609	7	.	.	PUNCT
ejpam-5367	610	1	we	we	PRON
ejpam-5367	610	2	have	have	VERB
ejpam-5367	610	3	βf	βf	NUM
ejpam-5367	610	4	(	(	PUNCT
ejpam-5367	610	5	y	y	NOUN
ejpam-5367	610	6	)	)	PUNCT
ejpam-5367	610	7	≤	≤	NOUN
ejpam-5367	611	1	max{βf	max{βf	INTJ
ejpam-5367	611	2	(	(	PUNCT
ejpam-5367	611	3	x	x	X
ejpam-5367	611	4	·	·	PUNCT
ejpam-5367	611	5	y	y	X
ejpam-5367	611	6	)	)	PUNCT
ejpam-5367	611	7	,	,	PUNCT
ejpam-5367	611	8	βf	βf	CCONJ
ejpam-5367	611	9	(	(	PUNCT
ejpam-5367	611	10	x	x	X
ejpam-5367	611	11	)	)	PUNCT
ejpam-5367	611	12	}	}	PUNCT
ejpam-5367	611	13	>	>	PUNCT
ejpam-5367	611	14	s.	s.	PROPN
ejpam-5367	612	1	thus	thus	ADV
ejpam-5367	612	2	,	,	PUNCT
ejpam-5367	612	3	y	y	PROPN
ejpam-5367	612	4	∈	∈	PROPN
ejpam-5367	612	5	l	l	NOUN
ejpam-5367	612	6	−	−	PROPN
ejpam-5367	612	7	(	(	PUNCT
ejpam-5367	612	8	βf	βf	INTJ
ejpam-5367	612	9	;	;	PUNCT
ejpam-5367	612	10	s	s	X
ejpam-5367	612	11	)	)	PUNCT
ejpam-5367	612	12	.	.	PUNCT
ejpam-5367	613	1	hence	hence	ADV
ejpam-5367	613	2	,	,	PUNCT
ejpam-5367	613	3	l	l	NOUN
ejpam-5367	613	4	−	−	PROPN
ejpam-5367	613	5	(	(	PUNCT
ejpam-5367	613	6	βf	βf	SYM
ejpam-5367	613	7	;	;	PUNCT
ejpam-5367	613	8	s	s	X
ejpam-5367	613	9	)	)	PUNCT
ejpam-5367	613	10	is	be	AUX
ejpam-5367	613	11	an	an	DET
ejpam-5367	613	12	iup	iup	NOUN
ejpam-5367	613	13	-	-	PUNCT
ejpam-5367	613	14	ideal	ideal	NOUN
ejpam-5367	613	15	of	of	ADP
ejpam-5367	613	16	x.	x.	NOUN
ejpam-5367	613	17	conversely	conversely	ADV
ejpam-5367	613	18	,	,	PUNCT
ejpam-5367	613	19	assume	assume	VERB
ejpam-5367	613	20	that	that	SCONJ
ejpam-5367	613	21	for	for	ADP
ejpam-5367	613	22	all	all	DET
ejpam-5367	613	23	t	t	PROPN
ejpam-5367	613	24	,	,	PUNCT
ejpam-5367	613	25	s	s	PART
ejpam-5367	613	26	∈	∈	PROPN
ejpam-5367	614	1	[	[	X
ejpam-5367	614	2	0	0	NUM
ejpam-5367	614	3	,	,	PUNCT
ejpam-5367	614	4	1	1	NUM
ejpam-5367	614	5	]	]	PUNCT
ejpam-5367	614	6	,	,	PUNCT
ejpam-5367	614	7	the	the	DET
ejpam-5367	614	8	sets	set	NOUN
ejpam-5367	614	9	u	u	NOUN
ejpam-5367	614	10	+	+	X
ejpam-5367	614	11	(	(	PUNCT
ejpam-5367	614	12	αf	αf	X
ejpam-5367	614	13	;	;	PUNCT
ejpam-5367	614	14	t	t	PROPN
ejpam-5367	614	15	)	)	PUNCT
ejpam-5367	614	16	and	and	CCONJ
ejpam-5367	614	17	l	l	NOUN
ejpam-5367	614	18	−	−	PROPN
ejpam-5367	615	1	(	(	PUNCT
ejpam-5367	615	2	βf	βf	SYM
ejpam-5367	615	3	;	;	PUNCT
ejpam-5367	615	4	s	s	X
ejpam-5367	615	5	)	)	PUNCT
ejpam-5367	615	6	are	be	AUX
ejpam-5367	615	7	either	either	CCONJ
ejpam-5367	615	8	empty	empty	ADJ
ejpam-5367	615	9	or	or	CCONJ
ejpam-5367	615	10	iup	iup	NOUN
ejpam-5367	615	11	-	-	PUNCT
ejpam-5367	615	12	filters	filter	NOUN
ejpam-5367	615	13	of	of	ADP
ejpam-5367	615	14	x.	x.	NOUN
ejpam-5367	615	15	let	let	VERB
ejpam-5367	615	16	x	x	SYM
ejpam-5367	615	17	∈	∈	PROPN
ejpam-5367	615	18	x.	x.	NOUN
ejpam-5367	615	19	assume	assume	VERB
ejpam-5367	615	20	that	that	SCONJ
ejpam-5367	615	21	αf	αf	VERB
ejpam-5367	615	22	(	(	PUNCT
ejpam-5367	615	23	0	0	NUM
ejpam-5367	615	24	)	)	PUNCT
ejpam-5367	615	25	<	<	X
ejpam-5367	615	26	αf	αf	X
ejpam-5367	615	27	(	(	PUNCT
ejpam-5367	615	28	x	x	NOUN
ejpam-5367	615	29	)	)	PUNCT
ejpam-5367	615	30	.	.	PUNCT
ejpam-5367	616	1	let	let	VERB
ejpam-5367	616	2	t	t	NOUN
ejpam-5367	616	3	=	=	PUNCT
ejpam-5367	616	4	αf	αf	X
ejpam-5367	616	5	(	(	PUNCT
ejpam-5367	616	6	0	0	NUM
ejpam-5367	616	7	)	)	PUNCT
ejpam-5367	616	8	.	.	PUNCT
ejpam-5367	617	1	then	then	ADV
ejpam-5367	617	2	x	x	SYM
ejpam-5367	617	3	∈	∈	PROPN
ejpam-5367	617	4	u	u	NOUN
ejpam-5367	617	5	+	+	X
ejpam-5367	617	6	(	(	PUNCT
ejpam-5367	617	7	αf	αf	X
ejpam-5367	617	8	;	;	PUNCT
ejpam-5367	617	9	t	t	X
ejpam-5367	617	10	)	)	PUNCT
ejpam-5367	617	11	̸=	̸=	PROPN
ejpam-5367	617	12	∅.	∅.	NOUN
ejpam-5367	617	13	by	by	ADP
ejpam-5367	617	14	the	the	DET
ejpam-5367	617	15	assumption	assumption	NOUN
ejpam-5367	617	16	,	,	PUNCT
ejpam-5367	617	17	we	we	PRON
ejpam-5367	617	18	have	have	VERB
ejpam-5367	617	19	u	u	NOUN
ejpam-5367	617	20	+	+	CCONJ
ejpam-5367	617	21	(	(	PUNCT
ejpam-5367	617	22	αf	αf	X
ejpam-5367	617	23	;	;	PUNCT
ejpam-5367	617	24	t	t	X
ejpam-5367	617	25	)	)	PUNCT
ejpam-5367	617	26	is	be	AUX
ejpam-5367	617	27	an	an	DET
ejpam-5367	617	28	iup	iup	NOUN
ejpam-5367	617	29	-	-	PUNCT
ejpam-5367	617	30	ideal	ideal	NOUN
ejpam-5367	617	31	of	of	ADP
ejpam-5367	617	32	x.	x.	NOUN
ejpam-5367	617	33	by	by	ADP
ejpam-5367	617	34	(	(	PUNCT
ejpam-5367	617	35	2.18	2.18	NUM
ejpam-5367	617	36	)	)	PUNCT
ejpam-5367	617	37	,	,	PUNCT
ejpam-5367	617	38	we	we	PRON
ejpam-5367	617	39	have	have	VERB
ejpam-5367	617	40	0	0	NUM
ejpam-5367	617	41	∈	∈	PROPN
ejpam-5367	617	42	u	u	NOUN
ejpam-5367	617	43	+	+	X
ejpam-5367	617	44	(	(	PUNCT
ejpam-5367	617	45	αf	αf	X
ejpam-5367	617	46	;	;	PUNCT
ejpam-5367	617	47	t	t	PROPN
ejpam-5367	617	48	)	)	PUNCT
ejpam-5367	617	49	.	.	PUNCT
ejpam-5367	618	1	so	so	ADV
ejpam-5367	618	2	αf	αf	VERB
ejpam-5367	618	3	(	(	PUNCT
ejpam-5367	618	4	0	0	NUM
ejpam-5367	618	5	)	)	PUNCT
ejpam-5367	618	6	>	>	X
ejpam-5367	618	7	t	t	PROPN
ejpam-5367	619	1	=	=	PUNCT
ejpam-5367	619	2	αf	αf	X
ejpam-5367	619	3	(	(	PUNCT
ejpam-5367	619	4	0	0	NUM
ejpam-5367	619	5	)	)	PUNCT
ejpam-5367	619	6	,	,	PUNCT
ejpam-5367	619	7	which	which	PRON
ejpam-5367	619	8	is	be	AUX
ejpam-5367	619	9	a	a	DET
ejpam-5367	619	10	contradiction	contradiction	NOUN
ejpam-5367	619	11	.	.	PUNCT
ejpam-5367	620	1	thus	thus	ADV
ejpam-5367	620	2	,	,	PUNCT
ejpam-5367	620	3	αf	αf	ADP
ejpam-5367	620	4	(	(	PUNCT
ejpam-5367	620	5	0	0	NUM
ejpam-5367	620	6	)	)	PUNCT
ejpam-5367	620	7	≥	≥	NOUN
ejpam-5367	620	8	αf	αf	X
ejpam-5367	620	9	(	(	PUNCT
ejpam-5367	620	10	x	x	NOUN
ejpam-5367	620	11	)	)	PUNCT
ejpam-5367	620	12	.	.	PUNCT
ejpam-5367	621	1	let	let	VERB
ejpam-5367	621	2	x	x	PRON
ejpam-5367	621	3	,	,	PUNCT
ejpam-5367	621	4	y	y	PROPN
ejpam-5367	621	5	∈	∈	PROPN
ejpam-5367	621	6	x.	x.	NOUN
ejpam-5367	621	7	assume	assume	VERB
ejpam-5367	621	8	that	that	SCONJ
ejpam-5367	621	9	αf	αf	VERB
ejpam-5367	621	10	(	(	PUNCT
ejpam-5367	621	11	y	y	NOUN
ejpam-5367	621	12	)	)	PUNCT
ejpam-5367	621	13	<	<	X
ejpam-5367	621	14	min{αf	min{αf	PUNCT
ejpam-5367	621	15	(	(	PUNCT
ejpam-5367	621	16	x	x	SYM
ejpam-5367	621	17	·	·	PUNCT
ejpam-5367	621	18	y	y	X
ejpam-5367	621	19	)	)	PUNCT
ejpam-5367	621	20	,	,	PUNCT
ejpam-5367	621	21	αf	αf	X
ejpam-5367	621	22	(	(	PUNCT
ejpam-5367	621	23	x	x	NOUN
ejpam-5367	621	24	)	)	PUNCT
ejpam-5367	621	25	}	}	PUNCT
ejpam-5367	621	26	.	.	PUNCT
ejpam-5367	622	1	let	let	VERB
ejpam-5367	622	2	t	t	NOUN
ejpam-5367	622	3	=	=	SYM
ejpam-5367	622	4	αf	αf	X
ejpam-5367	622	5	(	(	PUNCT
ejpam-5367	622	6	y	y	NOUN
ejpam-5367	622	7	)	)	PUNCT
ejpam-5367	622	8	.	.	PUNCT
ejpam-5367	623	1	then	then	ADV
ejpam-5367	623	2	x	x	X
ejpam-5367	623	3	·	·	PUNCT
ejpam-5367	623	4	y	y	X
ejpam-5367	623	5	,	,	PUNCT
ejpam-5367	623	6	x	x	SYM
ejpam-5367	623	7	∈	∈	PROPN
ejpam-5367	623	8	u	u	NOUN
ejpam-5367	623	9	+	+	X
ejpam-5367	623	10	(	(	PUNCT
ejpam-5367	623	11	αf	αf	X
ejpam-5367	623	12	;	;	PUNCT
ejpam-5367	623	13	t	t	X
ejpam-5367	623	14	)	)	PUNCT
ejpam-5367	623	15	̸=	̸=	PROPN
ejpam-5367	623	16	∅.	∅.	NOUN
ejpam-5367	623	17	by	by	ADP
ejpam-5367	623	18	the	the	DET
ejpam-5367	623	19	assumption	assumption	NOUN
ejpam-5367	623	20	,	,	PUNCT
ejpam-5367	623	21	we	we	PRON
ejpam-5367	623	22	have	have	VERB
ejpam-5367	623	23	u	u	NOUN
ejpam-5367	623	24	+	+	CCONJ
ejpam-5367	623	25	(	(	PUNCT
ejpam-5367	623	26	αf	αf	X
ejpam-5367	623	27	;	;	PUNCT
ejpam-5367	623	28	t	t	X
ejpam-5367	623	29	)	)	PUNCT
ejpam-5367	623	30	is	be	AUX
ejpam-5367	623	31	an	an	DET
ejpam-5367	623	32	iup	iup	NOUN
ejpam-5367	623	33	-	-	PUNCT
ejpam-5367	623	34	filter	filter	NOUN
ejpam-5367	623	35	of	of	ADP
ejpam-5367	623	36	x.	x.	NOUN
ejpam-5367	623	37	by	by	ADP
ejpam-5367	623	38	(	(	PUNCT
ejpam-5367	623	39	2.19	2.19	NUM
ejpam-5367	623	40	)	)	PUNCT
ejpam-5367	623	41	,	,	PUNCT
ejpam-5367	623	42	we	we	PRON
ejpam-5367	623	43	have	have	VERB
ejpam-5367	623	44	y	y	PROPN
ejpam-5367	623	45	∈	∈	PROPN
ejpam-5367	623	46	u	u	NOUN
ejpam-5367	623	47	+	+	X
ejpam-5367	623	48	(	(	PUNCT
ejpam-5367	623	49	αf	αf	X
ejpam-5367	623	50	;	;	PUNCT
ejpam-5367	623	51	t	t	PROPN
ejpam-5367	623	52	)	)	PUNCT
ejpam-5367	623	53	.	.	PUNCT
ejpam-5367	624	1	so	so	ADV
ejpam-5367	624	2	αf	αf	PROPN
ejpam-5367	624	3	(	(	PUNCT
ejpam-5367	624	4	y	y	NOUN
ejpam-5367	624	5	)	)	PUNCT
ejpam-5367	624	6	>	>	X
ejpam-5367	625	1	t	t	PROPN
ejpam-5367	626	1	=	=	PUNCT
ejpam-5367	626	2	αf	αf	X
ejpam-5367	626	3	(	(	PUNCT
ejpam-5367	626	4	y	y	NOUN
ejpam-5367	626	5	)	)	PUNCT
ejpam-5367	626	6	,	,	PUNCT
ejpam-5367	626	7	which	which	PRON
ejpam-5367	626	8	is	be	AUX
ejpam-5367	626	9	a	a	DET
ejpam-5367	626	10	contradiction	contradiction	NOUN
ejpam-5367	626	11	.	.	PUNCT
ejpam-5367	627	1	thus	thus	ADV
ejpam-5367	627	2	,	,	PUNCT
ejpam-5367	627	3	αf	αf	ADP
ejpam-5367	627	4	(	(	PUNCT
ejpam-5367	627	5	y	y	NOUN
ejpam-5367	627	6	)	)	PUNCT
ejpam-5367	627	7	≥	≥	NOUN
ejpam-5367	627	8	min{αf	min{αf	PUNCT
ejpam-5367	627	9	(	(	PUNCT
ejpam-5367	627	10	x	x	X
ejpam-5367	627	11	·	·	PUNCT
ejpam-5367	627	12	y	y	X
ejpam-5367	627	13	)	)	PUNCT
ejpam-5367	627	14	,	,	PUNCT
ejpam-5367	627	15	αf	αf	X
ejpam-5367	627	16	(	(	PUNCT
ejpam-5367	627	17	x	x	NOUN
ejpam-5367	627	18	)	)	PUNCT
ejpam-5367	627	19	}	}	PUNCT
ejpam-5367	627	20	.	.	PUNCT
ejpam-5367	628	1	let	let	VERB
ejpam-5367	628	2	x	x	SYM
ejpam-5367	628	3	∈	∈	PROPN
ejpam-5367	628	4	x.	x.	NOUN
ejpam-5367	628	5	assume	assume	VERB
ejpam-5367	628	6	that	that	SCONJ
ejpam-5367	628	7	βf	βf	INTJ
ejpam-5367	628	8	(	(	PUNCT
ejpam-5367	628	9	0	0	NUM
ejpam-5367	628	10	)	)	PUNCT
ejpam-5367	628	11	>	>	X
ejpam-5367	629	1	βf	βf	INTJ
ejpam-5367	629	2	(	(	PUNCT
ejpam-5367	629	3	x	x	NOUN
ejpam-5367	629	4	)	)	PUNCT
ejpam-5367	629	5	.	.	PUNCT
ejpam-5367	630	1	let	let	VERB
ejpam-5367	630	2	s	s	PRON
ejpam-5367	630	3	=	=	X
ejpam-5367	630	4	βf	βf	INTJ
ejpam-5367	630	5	(	(	PUNCT
ejpam-5367	630	6	0	0	NUM
ejpam-5367	630	7	)	)	PUNCT
ejpam-5367	630	8	.	.	PUNCT
ejpam-5367	631	1	then	then	ADV
ejpam-5367	631	2	x	x	SYM
ejpam-5367	631	3	∈	∈	PROPN
ejpam-5367	631	4	l	l	NOUN
ejpam-5367	631	5	−	−	PROPN
ejpam-5367	632	1	(	(	PUNCT
ejpam-5367	632	2	βf	βf	SYM
ejpam-5367	632	3	;	;	PUNCT
ejpam-5367	632	4	s	s	X
ejpam-5367	632	5	)	)	PUNCT
ejpam-5367	632	6	̸=	̸=	PROPN
ejpam-5367	632	7	∅.	∅.	NOUN
ejpam-5367	632	8	by	by	ADP
ejpam-5367	632	9	the	the	DET
ejpam-5367	632	10	assumption	assumption	NOUN
ejpam-5367	632	11	,	,	PUNCT
ejpam-5367	632	12	we	we	PRON
ejpam-5367	632	13	have	have	VERB
ejpam-5367	632	14	l	l	NOUN
ejpam-5367	632	15	−	−	PROPN
ejpam-5367	632	16	(	(	PUNCT
ejpam-5367	632	17	βf	βf	SYM
ejpam-5367	632	18	;	;	PUNCT
ejpam-5367	632	19	s	s	X
ejpam-5367	632	20	)	)	PUNCT
ejpam-5367	632	21	is	be	AUX
ejpam-5367	632	22	an	an	DET
ejpam-5367	632	23	iup	iup	NOUN
ejpam-5367	632	24	-	-	PUNCT
ejpam-5367	632	25	filter	filter	NOUN
ejpam-5367	632	26	of	of	ADP
ejpam-5367	632	27	x.	x.	NOUN
ejpam-5367	632	28	by	by	ADP
ejpam-5367	632	29	(	(	PUNCT
ejpam-5367	632	30	2.18	2.18	NUM
ejpam-5367	632	31	)	)	PUNCT
ejpam-5367	632	32	,	,	PUNCT
ejpam-5367	632	33	we	we	PRON
ejpam-5367	632	34	have	have	VERB
ejpam-5367	632	35	0	0	NUM
ejpam-5367	632	36	∈	∈	NOUN
ejpam-5367	632	37	l	l	NOUN
ejpam-5367	632	38	−	−	PROPN
ejpam-5367	633	1	(	(	PUNCT
ejpam-5367	633	2	βf	βf	INTJ
ejpam-5367	633	3	;	;	PUNCT
ejpam-5367	633	4	s	s	X
ejpam-5367	633	5	)	)	PUNCT
ejpam-5367	633	6	.	.	PUNCT
ejpam-5367	634	1	so	so	ADV
ejpam-5367	634	2	βf	βf	INTJ
ejpam-5367	634	3	(	(	PUNCT
ejpam-5367	634	4	0	0	NUM
ejpam-5367	634	5	)	)	PUNCT
ejpam-5367	634	6	<	<	X
ejpam-5367	634	7	s	s	X
ejpam-5367	634	8	=	=	X
ejpam-5367	634	9	βf	βf	INTJ
ejpam-5367	634	10	(	(	PUNCT
ejpam-5367	634	11	0	0	NUM
ejpam-5367	634	12	)	)	PUNCT
ejpam-5367	634	13	,	,	PUNCT
ejpam-5367	634	14	which	which	PRON
ejpam-5367	634	15	is	be	AUX
ejpam-5367	634	16	a	a	DET
ejpam-5367	634	17	contradiction	contradiction	NOUN
ejpam-5367	634	18	.	.	PUNCT
ejpam-5367	635	1	thus	thus	ADV
ejpam-5367	635	2	,	,	PUNCT
ejpam-5367	635	3	βf	βf	PRON
ejpam-5367	635	4	(	(	PUNCT
ejpam-5367	635	5	0	0	X
ejpam-5367	635	6	)	)	PUNCT
ejpam-5367	635	7	≤	≤	NOUN
ejpam-5367	635	8	βf	βf	CCONJ
ejpam-5367	635	9	(	(	PUNCT
ejpam-5367	635	10	x	x	NOUN
ejpam-5367	635	11	)	)	PUNCT
ejpam-5367	635	12	.	.	PUNCT
ejpam-5367	636	1	let	let	VERB
ejpam-5367	636	2	x	x	PRON
ejpam-5367	636	3	,	,	PUNCT
ejpam-5367	636	4	y	y	PROPN
ejpam-5367	636	5	∈	∈	PROPN
ejpam-5367	636	6	x.	x.	NOUN
ejpam-5367	636	7	assume	assume	VERB
ejpam-5367	636	8	that	that	SCONJ
ejpam-5367	636	9	βf	βf	INTJ
ejpam-5367	636	10	(	(	PUNCT
ejpam-5367	636	11	y	y	NOUN
ejpam-5367	636	12	)	)	PUNCT
ejpam-5367	636	13	>	>	X
ejpam-5367	637	1	max{βf	max{βf	PUNCT
ejpam-5367	637	2	(	(	PUNCT
ejpam-5367	637	3	x	x	X
ejpam-5367	637	4	·	·	PUNCT
ejpam-5367	637	5	y	y	X
ejpam-5367	637	6	)	)	PUNCT
ejpam-5367	637	7	,	,	PUNCT
ejpam-5367	637	8	βf	βf	CCONJ
ejpam-5367	637	9	(	(	PUNCT
ejpam-5367	637	10	x	x	NOUN
ejpam-5367	637	11	)	)	PUNCT
ejpam-5367	637	12	}	}	PUNCT
ejpam-5367	637	13	.	.	PUNCT
ejpam-5367	638	1	let	let	VERB
ejpam-5367	638	2	s	s	PRON
ejpam-5367	638	3	=	=	X
ejpam-5367	638	4	βf	βf	INTJ
ejpam-5367	638	5	(	(	PUNCT
ejpam-5367	638	6	y	y	NOUN
ejpam-5367	638	7	)	)	PUNCT
ejpam-5367	638	8	.	.	PUNCT
ejpam-5367	639	1	then	then	ADV
ejpam-5367	639	2	x	x	X
ejpam-5367	639	3	·	·	PUNCT
ejpam-5367	639	4	y	y	X
ejpam-5367	639	5	,	,	PUNCT
ejpam-5367	639	6	x	x	X
ejpam-5367	639	7	∈	∈	NOUN
ejpam-5367	639	8	l	l	NOUN
ejpam-5367	639	9	−	−	PROPN
ejpam-5367	639	10	(	(	PUNCT
ejpam-5367	639	11	βf	βf	SYM
ejpam-5367	639	12	;	;	PUNCT
ejpam-5367	639	13	s	s	X
ejpam-5367	639	14	)	)	PUNCT
ejpam-5367	639	15	̸=	̸=	PROPN
ejpam-5367	639	16	∅.	∅.	NOUN
ejpam-5367	639	17	by	by	ADP
ejpam-5367	639	18	the	the	DET
ejpam-5367	639	19	assumption	assumption	NOUN
ejpam-5367	639	20	,	,	PUNCT
ejpam-5367	639	21	we	we	PRON
ejpam-5367	639	22	have	have	VERB
ejpam-5367	639	23	l	l	NOUN
ejpam-5367	639	24	−	−	PROPN
ejpam-5367	639	25	(	(	PUNCT
ejpam-5367	639	26	βf	βf	SYM
ejpam-5367	639	27	;	;	PUNCT
ejpam-5367	639	28	s	s	X
ejpam-5367	639	29	)	)	PUNCT
ejpam-5367	639	30	is	be	AUX
ejpam-5367	639	31	an	an	DET
ejpam-5367	639	32	iup	iup	NOUN
ejpam-5367	639	33	-	-	PUNCT
ejpam-5367	639	34	filter	filter	NOUN
ejpam-5367	639	35	of	of	ADP
ejpam-5367	639	36	x.	x.	NOUN
ejpam-5367	639	37	by	by	ADP
ejpam-5367	639	38	(	(	PUNCT
ejpam-5367	639	39	2.19	2.19	NUM
ejpam-5367	639	40	)	)	PUNCT
ejpam-5367	639	41	,	,	PUNCT
ejpam-5367	639	42	we	we	PRON
ejpam-5367	639	43	have	have	VERB
ejpam-5367	639	44	y	y	PROPN
ejpam-5367	639	45	∈	∈	PROPN
ejpam-5367	639	46	l	l	NOUN
ejpam-5367	639	47	−	−	PROPN
ejpam-5367	640	1	(	(	PUNCT
ejpam-5367	640	2	βf	βf	INTJ
ejpam-5367	640	3	;	;	PUNCT
ejpam-5367	640	4	s	s	X
ejpam-5367	640	5	)	)	PUNCT
ejpam-5367	640	6	.	.	PUNCT
ejpam-5367	641	1	so	so	ADV
ejpam-5367	641	2	βf	βf	INTJ
ejpam-5367	641	3	(	(	PUNCT
ejpam-5367	641	4	y	y	NOUN
ejpam-5367	641	5	)	)	PUNCT
ejpam-5367	641	6	<	<	X
ejpam-5367	642	1	s	s	X
ejpam-5367	643	1	=	=	X
ejpam-5367	643	2	βf	βf	INTJ
ejpam-5367	643	3	(	(	PUNCT
ejpam-5367	643	4	y	y	NOUN
ejpam-5367	643	5	)	)	PUNCT
ejpam-5367	643	6	,	,	PUNCT
ejpam-5367	643	7	which	which	PRON
ejpam-5367	643	8	is	be	AUX
ejpam-5367	643	9	a	a	DET
ejpam-5367	643	10	contradiction	contradiction	NOUN
ejpam-5367	643	11	.	.	PUNCT
ejpam-5367	644	1	thus	thus	ADV
ejpam-5367	644	2	,	,	PUNCT
ejpam-5367	644	3	βf	βf	PRON
ejpam-5367	644	4	(	(	PUNCT
ejpam-5367	644	5	y	y	NOUN
ejpam-5367	644	6	)	)	PUNCT
ejpam-5367	644	7	≤	≤	NOUN
ejpam-5367	644	8	max{βf	max{βf	INTJ
ejpam-5367	644	9	(	(	PUNCT
ejpam-5367	644	10	x	x	X
ejpam-5367	644	11	·	·	PUNCT
ejpam-5367	644	12	y	y	X
ejpam-5367	644	13	)	)	PUNCT
ejpam-5367	644	14	,	,	PUNCT
ejpam-5367	644	15	βf	βf	CCONJ
ejpam-5367	644	16	(	(	PUNCT
ejpam-5367	644	17	x	x	NOUN
ejpam-5367	644	18	)	)	PUNCT
ejpam-5367	644	19	}	}	PUNCT
ejpam-5367	644	20	.	.	PUNCT
ejpam-5367	645	1	hence	hence	ADV
ejpam-5367	645	2	,	,	PUNCT
ejpam-5367	645	3	f	f	PROPN
ejpam-5367	645	4	is	be	AUX
ejpam-5367	645	5	an	an	DET
ejpam-5367	645	6	ffiup	ffiup	ADJ
ejpam-5367	645	7	-	-	PUNCT
ejpam-5367	645	8	filter	filter	NOUN
ejpam-5367	645	9	of	of	ADP
ejpam-5367	645	10	x.	x.	NOUN
ejpam-5367	645	11	the	the	DET
ejpam-5367	645	12	following	follow	VERB
ejpam-5367	645	13	theorem	theorem	NOUN
ejpam-5367	645	14	is	be	AUX
ejpam-5367	645	15	a	a	DET
ejpam-5367	645	16	direct	direct	ADJ
ejpam-5367	645	17	consequence	consequence	NOUN
ejpam-5367	645	18	of	of	ADP
ejpam-5367	645	19	theorem	theorem	ADJ
ejpam-5367	645	20	2	2	NUM
ejpam-5367	645	21	.	.	PUNCT
ejpam-5367	645	22	theorem	theorem	VERB
ejpam-5367	645	23	27	27	NUM
ejpam-5367	645	24	.	.	PUNCT
ejpam-5367	646	1	an	an	DET
ejpam-5367	646	2	ffs	ffs	NOUN
ejpam-5367	646	3	f	f	PROPN
ejpam-5367	646	4	in	in	ADP
ejpam-5367	646	5	x	x	PROPN
ejpam-5367	646	6	is	be	AUX
ejpam-5367	646	7	an	an	DET
ejpam-5367	646	8	ffsiup	ffsiup	NOUN
ejpam-5367	646	9	-	-	PUNCT
ejpam-5367	646	10	ideal	ideal	NOUN
ejpam-5367	646	11	of	of	ADP
ejpam-5367	646	12	x	x	SYM
ejpam-5367	646	13	if	if	SCONJ
ejpam-5367	646	14	and	and	CCONJ
ejpam-5367	646	15	only	only	ADV
ejpam-5367	646	16	if	if	SCONJ
ejpam-5367	646	17	for	for	ADP
ejpam-5367	646	18	all	all	DET
ejpam-5367	646	19	t	t	NOUN
ejpam-5367	646	20	,	,	PUNCT
ejpam-5367	646	21	s	s	PART
ejpam-5367	646	22	∈	∈	PROPN
ejpam-5367	647	1	[	[	X
ejpam-5367	647	2	0	0	NUM
ejpam-5367	647	3	,	,	PUNCT
ejpam-5367	647	4	1	1	NUM
ejpam-5367	647	5	]	]	PUNCT
ejpam-5367	647	6	,	,	PUNCT
ejpam-5367	647	7	the	the	DET
ejpam-5367	647	8	sets	set	NOUN
ejpam-5367	647	9	u	u	NOUN
ejpam-5367	647	10	+	+	X
ejpam-5367	647	11	(	(	PUNCT
ejpam-5367	647	12	αf	αf	X
ejpam-5367	647	13	;	;	PUNCT
ejpam-5367	647	14	t	t	PROPN
ejpam-5367	647	15	)	)	PUNCT
ejpam-5367	647	16	and	and	CCONJ
ejpam-5367	647	17	l	l	NOUN
ejpam-5367	647	18	−	−	PROPN
ejpam-5367	648	1	(	(	PUNCT
ejpam-5367	648	2	βf	βf	SYM
ejpam-5367	648	3	;	;	PUNCT
ejpam-5367	648	4	s	s	X
ejpam-5367	648	5	)	)	PUNCT
ejpam-5367	648	6	are	be	AUX
ejpam-5367	648	7	either	either	CCONJ
ejpam-5367	648	8	empty	empty	ADJ
ejpam-5367	648	9	or	or	CCONJ
ejpam-5367	648	10	strong	strong	ADJ
ejpam-5367	648	11	iup	iup	NOUN
ejpam-5367	648	12	-	-	PUNCT
ejpam-5367	648	13	ideals	ideal	NOUN
ejpam-5367	648	14	of	of	ADP
ejpam-5367	648	15	x.	x.	NOUN
ejpam-5367	648	16	references	reference	NOUN
ejpam-5367	648	17	3041	3041	NUM
ejpam-5367	648	18	4	4	NUM
ejpam-5367	648	19	.	.	PUNCT
ejpam-5367	649	1	conclusions	conclusion	NOUN
ejpam-5367	649	2	and	and	CCONJ
ejpam-5367	649	3	future	future	ADJ
ejpam-5367	649	4	work	work	NOUN
ejpam-5367	649	5	our	our	PRON
ejpam-5367	649	6	paper	paper	NOUN
ejpam-5367	649	7	introduces	introduce	NOUN
ejpam-5367	649	8	pioneering	pioneer	VERB
ejpam-5367	649	9	concepts	concept	NOUN
ejpam-5367	649	10	like	like	ADP
ejpam-5367	649	11	ffiup	ffiup	NOUN
ejpam-5367	649	12	-	-	PUNCT
ejpam-5367	649	13	subalgebras	subalgebra	NOUN
ejpam-5367	649	14	,	,	PUNCT
ejpam-5367	649	15	ffiup	ffiup	NOUN
ejpam-5367	649	16	-	-	PUNCT
ejpam-5367	649	17	ideals	ideal	NOUN
ejpam-5367	649	18	,	,	PUNCT
ejpam-5367	649	19	ffiupfilters	ffiupfilter	NOUN
ejpam-5367	649	20	,	,	PUNCT
ejpam-5367	649	21	and	and	CCONJ
ejpam-5367	649	22	ffsiup	ffsiup	NOUN
ejpam-5367	649	23	-	-	PUNCT
ejpam-5367	649	24	ideals	ideal	NOUN
ejpam-5367	649	25	.	.	PUNCT
ejpam-5367	650	1	we	we	PRON
ejpam-5367	650	2	explore	explore	VERB
ejpam-5367	650	3	their	their	PRON
ejpam-5367	650	4	crucial	crucial	ADJ
ejpam-5367	650	5	properties	property	NOUN
ejpam-5367	650	6	,	,	PUNCT
ejpam-5367	650	7	examining	examine	VERB
ejpam-5367	650	8	their	their	PRON
ejpam-5367	650	9	relationships	relationship	NOUN
ejpam-5367	650	10	to	to	ADP
ejpam-5367	650	11	complements	complement	NOUN
ejpam-5367	650	12	,	,	PUNCT
ejpam-5367	650	13	characteristic	characteristic	ADJ
ejpam-5367	650	14	functions	function	NOUN
ejpam-5367	650	15	,	,	PUNCT
ejpam-5367	650	16	and	and	CCONJ
ejpam-5367	650	17	level	level	NOUN
ejpam-5367	650	18	subsets	subset	NOUN
ejpam-5367	650	19	.	.	PUNCT
ejpam-5367	651	1	these	these	DET
ejpam-5367	651	2	insights	insight	NOUN
ejpam-5367	651	3	reveal	reveal	VERB
ejpam-5367	651	4	complex	complex	ADJ
ejpam-5367	651	5	structures	structure	NOUN
ejpam-5367	651	6	within	within	ADP
ejpam-5367	651	7	ffss	ffss	NOUN
ejpam-5367	651	8	and	and	CCONJ
ejpam-5367	651	9	iup	iup	NOUN
ejpam-5367	651	10	-	-	PUNCT
ejpam-5367	651	11	algebras	algebras	PROPN
ejpam-5367	651	12	,	,	PUNCT
ejpam-5367	651	13	offering	offer	VERB
ejpam-5367	651	14	new	new	ADJ
ejpam-5367	651	15	perspectives	perspective	NOUN
ejpam-5367	651	16	and	and	CCONJ
ejpam-5367	651	17	practical	practical	ADJ
ejpam-5367	651	18	applications	application	NOUN
ejpam-5367	651	19	in	in	ADP
ejpam-5367	651	20	mathematical	mathematical	ADJ
ejpam-5367	651	21	theory	theory	NOUN
ejpam-5367	651	22	.	.	PUNCT
ejpam-5367	652	1	the	the	DET
ejpam-5367	652	2	study	study	NOUN
ejpam-5367	652	3	of	of	ADP
ejpam-5367	652	4	ffss	ffss	NOUN
ejpam-5367	652	5	represents	represent	VERB
ejpam-5367	652	6	a	a	DET
ejpam-5367	652	7	dynamic	dynamic	ADJ
ejpam-5367	652	8	evolution	evolution	NOUN
ejpam-5367	652	9	in	in	ADP
ejpam-5367	652	10	fs	fs	ADP
ejpam-5367	652	11	theory	theory	NOUN
ejpam-5367	652	12	,	,	PUNCT
ejpam-5367	652	13	extending	extend	VERB
ejpam-5367	652	14	beyond	beyond	ADP
ejpam-5367	652	15	academic	academic	ADJ
ejpam-5367	652	16	curiosity	curiosity	NOUN
ejpam-5367	652	17	.	.	PUNCT
ejpam-5367	653	1	this	this	DET
ejpam-5367	653	2	expanding	expand	VERB
ejpam-5367	653	3	research	research	NOUN
ejpam-5367	653	4	inspires	inspire	VERB
ejpam-5367	653	5	scientists	scientist	NOUN
ejpam-5367	653	6	and	and	CCONJ
ejpam-5367	653	7	scholars	scholar	NOUN
ejpam-5367	653	8	to	to	PART
ejpam-5367	653	9	explore	explore	VERB
ejpam-5367	653	10	the	the	DET
ejpam-5367	653	11	synergy	synergy	NOUN
ejpam-5367	653	12	between	between	ADP
ejpam-5367	653	13	ffss	ffss	NOUN
ejpam-5367	653	14	and	and	CCONJ
ejpam-5367	653	15	iup	iup	NOUN
ejpam-5367	653	16	-	-	PUNCT
ejpam-5367	653	17	algebras	algebras	PROPN
ejpam-5367	653	18	.	.	PUNCT
ejpam-5367	654	1	this	this	DET
ejpam-5367	654	2	convergence	convergence	NOUN
ejpam-5367	654	3	is	be	AUX
ejpam-5367	654	4	set	set	VERB
ejpam-5367	654	5	to	to	PART
ejpam-5367	654	6	drive	drive	VERB
ejpam-5367	654	7	innovations	innovation	NOUN
ejpam-5367	654	8	,	,	PUNCT
ejpam-5367	654	9	shaping	shape	VERB
ejpam-5367	654	10	the	the	DET
ejpam-5367	654	11	future	future	NOUN
ejpam-5367	654	12	of	of	ADP
ejpam-5367	654	13	fuzzy	fuzzy	ADJ
ejpam-5367	654	14	mathematics	mathematic	NOUN
ejpam-5367	654	15	and	and	CCONJ
ejpam-5367	654	16	its	its	PRON
ejpam-5367	654	17	diverse	diverse	ADJ
ejpam-5367	654	18	applications	application	NOUN
ejpam-5367	654	19	.	.	PUNCT
ejpam-5367	655	1	acknowledgements	acknowledgement	NOUN
ejpam-5367	655	2	this	this	DET
ejpam-5367	655	3	research	research	NOUN
ejpam-5367	655	4	was	be	AUX
ejpam-5367	655	5	supported	support	VERB
ejpam-5367	655	6	by	by	ADP
ejpam-5367	655	7	university	university	NOUN
ejpam-5367	655	8	of	of	ADP
ejpam-5367	655	9	phayao	phayao	NOUN
ejpam-5367	655	10	and	and	CCONJ
ejpam-5367	655	11	thailand	thailand	PROPN
ejpam-5367	655	12	science	science	PROPN
ejpam-5367	655	13	research	research	PROPN
ejpam-5367	655	14	and	and	CCONJ
ejpam-5367	655	15	innovation	innovation	NOUN
ejpam-5367	655	16	fund	fund	NOUN
ejpam-5367	655	17	(	(	PUNCT
ejpam-5367	655	18	fundamental	fundamental	ADJ
ejpam-5367	655	19	fund	fund	NOUN
ejpam-5367	655	20	2025	2025	NUM
ejpam-5367	655	21	,	,	PUNCT
ejpam-5367	655	22	grant	grant	VERB
ejpam-5367	655	23	no	no	NOUN
ejpam-5367	655	24	.	.	PROPN
ejpam-5367	656	1	5027/2567	5027/2567	NUM
ejpam-5367	656	2	)	)	PUNCT
ejpam-5367	656	3	.	.	PUNCT
ejpam-5367	657	1	references	reference	NOUN
ejpam-5367	657	2	[	[	X
ejpam-5367	657	3	1	1	NUM
ejpam-5367	657	4	]	]	PUNCT
ejpam-5367	657	5	a.	a.	NOUN
ejpam-5367	657	6	k.	k.	PROPN
ejpam-5367	657	7	adak	adak	PROPN
ejpam-5367	657	8	,	,	PUNCT
ejpam-5367	657	9	nilkamal	nilkamal	NOUN
ejpam-5367	657	10	,	,	PUNCT
ejpam-5367	657	11	and	and	CCONJ
ejpam-5367	657	12	n.	n.	PROPN
ejpam-5367	657	13	barman	barman	NOUN
ejpam-5367	657	14	.	.	PUNCT
ejpam-5367	658	1	fermatean	fermatean	PROPN
ejpam-5367	658	2	fuzzy	fuzzy	ADJ
ejpam-5367	658	3	semi	semi	ADJ
ejpam-5367	658	4	-	-	ADJ
ejpam-5367	658	5	prime	prime	ADJ
ejpam-5367	658	6	ideals	ideal	NOUN
ejpam-5367	658	7	of	of	ADP
ejpam-5367	658	8	ordered	order	VERB
ejpam-5367	658	9	semigroups	semigroup	NOUN
ejpam-5367	658	10	.	.	PUNCT
ejpam-5367	659	1	topol	topol	NOUN
ejpam-5367	659	2	.	.	PUNCT
ejpam-5367	660	1	algebra	algebra	PROPN
ejpam-5367	660	2	appl	appl	PROPN
ejpam-5367	660	3	.	.	PROPN
ejpam-5367	660	4	,	,	PUNCT
ejpam-5367	660	5	11(1):20230102	11(1):20230102	NUM
ejpam-5367	660	6	,	,	PUNCT
ejpam-5367	660	7	2023	2023	NUM
ejpam-5367	660	8	.	.	PUNCT
ejpam-5367	661	1	[	[	X
ejpam-5367	661	2	2	2	NUM
ejpam-5367	661	3	]	]	PUNCT
ejpam-5367	661	4	k.	k.	PROPN
ejpam-5367	661	5	t.	t.	PROPN
ejpam-5367	661	6	atanassov	atanassov	PROPN
ejpam-5367	661	7	.	.	PUNCT
ejpam-5367	662	1	intuitionistic	intuitionistic	ADJ
ejpam-5367	662	2	fuzzy	fuzzy	ADJ
ejpam-5367	662	3	sets	set	NOUN
ejpam-5367	662	4	.	.	PUNCT
ejpam-5367	663	1	fuzzy	fuzzy	ADJ
ejpam-5367	663	2	sets	set	NOUN
ejpam-5367	663	3	syst	syst	PROPN
ejpam-5367	663	4	.	.	PUNCT
ejpam-5367	663	5	,	,	PUNCT
ejpam-5367	663	6	20(1):87–96	20(1):87–96	NUM
ejpam-5367	663	7	,	,	PUNCT
ejpam-5367	663	8	1986	1986	NUM
ejpam-5367	663	9	.	.	PUNCT
ejpam-5367	664	1	[	[	X
ejpam-5367	664	2	3	3	X
ejpam-5367	664	3	]	]	PUNCT
ejpam-5367	664	4	k.	k.	PROPN
ejpam-5367	664	5	balamurugan	balamurugan	PROPN
ejpam-5367	664	6	and	and	CCONJ
ejpam-5367	664	7	r.	r.	PROPN
ejpam-5367	664	8	nagarajan	nagarajan	PROPN
ejpam-5367	664	9	.	.	PUNCT
ejpam-5367	665	1	fermatean	fermatean	PROPN
ejpam-5367	665	2	fuzzy	fuzzy	ADJ
ejpam-5367	665	3	soft	soft	ADJ
ejpam-5367	665	4	covered	cover	VERB
ejpam-5367	665	5	congruence	congruence	NOUN
ejpam-5367	665	6	relations	relation	NOUN
ejpam-5367	665	7	acting	act	VERB
ejpam-5367	665	8	on	on	ADP
ejpam-5367	665	9	a	a	DET
ejpam-5367	665	10	semigroup	semigroup	NOUN
ejpam-5367	665	11	.	.	PUNCT
ejpam-5367	666	1	j.	j.	PROPN
ejpam-5367	666	2	basic	basic	PROPN
ejpam-5367	666	3	sci	sci	PROPN
ejpam-5367	666	4	.	.	PROPN
ejpam-5367	666	5	,	,	PUNCT
ejpam-5367	666	6	22(11):85–96	22(11):85–96	ADJ
ejpam-5367	666	7	,	,	PUNCT
ejpam-5367	666	8	2022	2022	NUM
ejpam-5367	666	9	.	.	PUNCT
ejpam-5367	667	1	[	[	X
ejpam-5367	667	2	4	4	NUM
ejpam-5367	667	3	]	]	PUNCT
ejpam-5367	667	4	k.	k.	PROPN
ejpam-5367	667	5	balamurugan	balamurugan	PROPN
ejpam-5367	667	6	and	and	CCONJ
ejpam-5367	667	7	r.	r.	PROPN
ejpam-5367	667	8	nagarajan	nagarajan	PROPN
ejpam-5367	667	9	.	.	PUNCT
ejpam-5367	668	1	fermatean	fermatean	ADJ
ejpam-5367	668	2	uncertainty	uncertainty	NOUN
ejpam-5367	668	3	soft	soft	ADJ
ejpam-5367	668	4	sub	sub	NOUN
ejpam-5367	668	5	algebra	algebra	NOUN
ejpam-5367	668	6	in	in	ADP
ejpam-5367	668	7	terms	term	NOUN
ejpam-5367	668	8	of	of	ADP
ejpam-5367	668	9	ideal	ideal	ADJ
ejpam-5367	668	10	structures	structure	NOUN
ejpam-5367	668	11	.	.	PUNCT
ejpam-5367	669	1	j.	j.	PROPN
ejpam-5367	669	2	propulsion	propulsion	PROPN
ejpam-5367	669	3	tech	tech	PROPN
ejpam-5367	669	4	.	.	PUNCT
ejpam-5367	669	5	,	,	PUNCT
ejpam-5367	669	6	44(4):1163–1174	44(4):1163–1174	PROPN
ejpam-5367	669	7	,	,	PUNCT
ejpam-5367	669	8	2023	2023	NUM
ejpam-5367	669	9	.	.	PUNCT
ejpam-5367	670	1	[	[	X
ejpam-5367	670	2	5	5	NUM
ejpam-5367	670	3	]	]	PUNCT
ejpam-5367	670	4	c.	c.	PROPN
ejpam-5367	670	5	chanmanee	chanmanee	PROPN
ejpam-5367	670	6	,	,	PUNCT
ejpam-5367	670	7	w.	w.	PROPN
ejpam-5367	670	8	nakkhasen	nakkhasen	PROPN
ejpam-5367	670	9	,	,	PUNCT
ejpam-5367	670	10	r.	r.	PROPN
ejpam-5367	670	11	prasertpong	prasertpong	PROPN
ejpam-5367	670	12	,	,	PUNCT
ejpam-5367	670	13	p.	p.	PROPN
ejpam-5367	670	14	julatha	julatha	PROPN
ejpam-5367	670	15	,	,	PUNCT
ejpam-5367	670	16	and	and	CCONJ
ejpam-5367	670	17	a.	a.	NOUN
ejpam-5367	670	18	iampan	iampan	PROPN
ejpam-5367	670	19	.	.	PUNCT
ejpam-5367	671	1	notes	note	NOUN
ejpam-5367	671	2	on	on	ADP
ejpam-5367	671	3	external	external	ADJ
ejpam-5367	671	4	direct	direct	ADJ
ejpam-5367	671	5	products	product	NOUN
ejpam-5367	671	6	of	of	ADP
ejpam-5367	671	7	dual	dual	ADJ
ejpam-5367	671	8	iup	iup	NOUN
ejpam-5367	671	9	-	-	PUNCT
ejpam-5367	671	10	algebras	algebras	PROPN
ejpam-5367	671	11	.	.	PUNCT
ejpam-5367	672	1	south	south	PROPN
ejpam-5367	672	2	east	east	PROPN
ejpam-5367	672	3	asian	asian	PROPN
ejpam-5367	672	4	j.	j.	PROPN
ejpam-5367	672	5	math	math	PROPN
ejpam-5367	672	6	.	.	PUNCT
ejpam-5367	673	1	math	math	NOUN
ejpam-5367	673	2	.	.	PUNCT
ejpam-5367	674	1	sci	sci	PROPN
ejpam-5367	674	2	.	.	PROPN
ejpam-5367	674	3	,	,	PUNCT
ejpam-5367	674	4	19(3):13–30	19(3):13–30	NUM
ejpam-5367	674	5	,	,	PUNCT
ejpam-5367	674	6	2023	2023	NUM
ejpam-5367	674	7	.	.	PUNCT
ejpam-5367	675	1	[	[	X
ejpam-5367	675	2	6	6	NUM
ejpam-5367	675	3	]	]	PUNCT
ejpam-5367	675	4	c.	c.	PROPN
ejpam-5367	675	5	chanmanee	chanmanee	PROPN
ejpam-5367	675	6	,	,	PUNCT
ejpam-5367	675	7	r.	r.	PROPN
ejpam-5367	675	8	prasertpong	prasertpong	PROPN
ejpam-5367	675	9	,	,	PUNCT
ejpam-5367	675	10	p.	p.	PROPN
ejpam-5367	675	11	julatha	julatha	PROPN
ejpam-5367	675	12	,	,	PUNCT
ejpam-5367	675	13	n.	n.	PROPN
ejpam-5367	675	14	lekkoksung	lekkoksung	PROPN
ejpam-5367	675	15	,	,	PUNCT
ejpam-5367	675	16	and	and	CCONJ
ejpam-5367	675	17	a.	a.	NOUN
ejpam-5367	675	18	iampan	iampan	PROPN
ejpam-5367	675	19	.	.	PUNCT
ejpam-5367	676	1	on	on	ADP
ejpam-5367	676	2	external	external	ADJ
ejpam-5367	676	3	direct	direct	ADJ
ejpam-5367	676	4	products	product	NOUN
ejpam-5367	676	5	of	of	ADP
ejpam-5367	676	6	iup	iup	NOUN
ejpam-5367	676	7	-	-	PUNCT
ejpam-5367	676	8	algebras	algebras	PROPN
ejpam-5367	676	9	.	.	PUNCT
ejpam-5367	677	1	int	int	NOUN
ejpam-5367	677	2	.	.	PUNCT
ejpam-5367	678	1	j.	j.	PROPN
ejpam-5367	678	2	innov	innov	PROPN
ejpam-5367	678	3	.	.	PUNCT
ejpam-5367	679	1	comput	comput	PROPN
ejpam-5367	679	2	.	.	PUNCT
ejpam-5367	680	1	inf	inf	PROPN
ejpam-5367	680	2	.	.	PUNCT
ejpam-5367	680	3	control	control	PROPN
ejpam-5367	680	4	,	,	PUNCT
ejpam-5367	680	5	19(3):775	19(3):775	NUM
ejpam-5367	680	6	–	–	PUNCT
ejpam-5367	680	7	787	787	NUM
ejpam-5367	680	8	,	,	PUNCT
ejpam-5367	680	9	2023	2023	NUM
ejpam-5367	680	10	.	.	PUNCT
ejpam-5367	681	1	[	[	X
ejpam-5367	681	2	7	7	NUM
ejpam-5367	681	3	]	]	PUNCT
ejpam-5367	681	4	a.	a.	NOUN
ejpam-5367	681	5	iampan	iampan	PROPN
ejpam-5367	681	6	,	,	PUNCT
ejpam-5367	681	7	p.	p.	PROPN
ejpam-5367	681	8	julatha	julatha	PROPN
ejpam-5367	681	9	,	,	PUNCT
ejpam-5367	681	10	p.	p.	NOUN
ejpam-5367	681	11	khamrot	khamrot	NOUN
ejpam-5367	681	12	,	,	PUNCT
ejpam-5367	681	13	and	and	CCONJ
ejpam-5367	681	14	d.	d.	PROPN
ejpam-5367	681	15	a.	a.	PROPN
ejpam-5367	681	16	romano	romano	PROPN
ejpam-5367	681	17	.	.	PUNCT
ejpam-5367	682	1	independent	independent	ADJ
ejpam-5367	682	2	up	up	ADP
ejpam-5367	682	3	-	-	PUNCT
ejpam-5367	682	4	algebras	algebras	X
ejpam-5367	682	5	.	.	PUNCT
ejpam-5367	683	1	j.	j.	PROPN
ejpam-5367	683	2	math	math	PROPN
ejpam-5367	683	3	.	.	PUNCT
ejpam-5367	684	1	comput	comput	NOUN
ejpam-5367	684	2	.	.	PUNCT
ejpam-5367	685	1	sci	sci	PROPN
ejpam-5367	685	2	.	.	PROPN
ejpam-5367	685	3	,	,	PUNCT
ejpam-5367	685	4	jmcs	jmcs	NOUN
ejpam-5367	685	5	,	,	PUNCT
ejpam-5367	685	6	27(1):65–76	27(1):65–76	NUM
ejpam-5367	685	7	,	,	PUNCT
ejpam-5367	685	8	2022	2022	NUM
ejpam-5367	685	9	.	.	PUNCT
ejpam-5367	686	1	[	[	X
ejpam-5367	686	2	8	8	NUM
ejpam-5367	686	3	]	]	X
ejpam-5367	686	4	f.	f.	PROPN
ejpam-5367	686	5	m.	m.	PROPN
ejpam-5367	686	6	khan	khan	PROPN
ejpam-5367	686	7	,	,	PUNCT
ejpam-5367	686	8	n.	n.	NOUN
ejpam-5367	686	9	bibi	bibi	PROPN
ejpam-5367	686	10	,	,	PUNCT
ejpam-5367	686	11	x.	x.	PROPN
ejpam-5367	686	12	l.	l.	PROPN
ejpam-5367	686	13	xin	xin	PROPN
ejpam-5367	686	14	,	,	PUNCT
ejpam-5367	686	15	muhsina	muhsina	NOUN
ejpam-5367	686	16	,	,	PUNCT
ejpam-5367	686	17	and	and	CCONJ
ejpam-5367	686	18	a.	a.	PROPN
ejpam-5367	686	19	alam	alam	PROPN
ejpam-5367	686	20	.	.	PUNCT
ejpam-5367	687	1	rough	rough	ADJ
ejpam-5367	687	2	fermatean	fermatean	ADJ
ejpam-5367	687	3	fuzzy	fuzzy	ADJ
ejpam-5367	687	4	ideals	ideal	NOUN
ejpam-5367	687	5	in	in	ADP
ejpam-5367	687	6	semigroups	semigroup	NOUN
ejpam-5367	687	7	.	.	PUNCT
ejpam-5367	688	1	j.	j.	PROPN
ejpam-5367	688	2	int	int	PROPN
ejpam-5367	688	3	.	.	PUNCT
ejpam-5367	689	1	fuzzy	fuzzy	ADJ
ejpam-5367	689	2	syst	syst	PROPN
ejpam-5367	689	3	.	.	PUNCT
ejpam-5367	689	4	,	,	PUNCT
ejpam-5367	689	5	42(6):5741–5752	42(6):5741–5752	NUM
ejpam-5367	689	6	,	,	PUNCT
ejpam-5367	689	7	2022	2022	NUM
ejpam-5367	689	8	.	.	PUNCT
ejpam-5367	690	1	[	[	X
ejpam-5367	690	2	9	9	NUM
ejpam-5367	690	3	]	]	PUNCT
ejpam-5367	690	4	k.	k.	PROPN
ejpam-5367	690	5	kuntama	kuntama	PROPN
ejpam-5367	690	6	,	,	PUNCT
ejpam-5367	690	7	p.	p.	NOUN
ejpam-5367	690	8	krongchai	krongchai	PROPN
ejpam-5367	690	9	,	,	PUNCT
ejpam-5367	690	10	r.	r.	PROPN
ejpam-5367	690	11	prasertpong	prasertpong	PROPN
ejpam-5367	690	12	,	,	PUNCT
ejpam-5367	690	13	p.	p.	PROPN
ejpam-5367	690	14	julatha	julatha	PROPN
ejpam-5367	690	15	,	,	PUNCT
ejpam-5367	690	16	and	and	CCONJ
ejpam-5367	690	17	a.	a.	NOUN
ejpam-5367	690	18	iampan	iampan	PROPN
ejpam-5367	690	19	.	.	PUNCT
ejpam-5367	691	1	fuzzy	fuzzy	ADJ
ejpam-5367	691	2	set	set	VERB
ejpam-5367	691	3	theory	theory	NOUN
ejpam-5367	691	4	applied	apply	VERB
ejpam-5367	691	5	to	to	ADP
ejpam-5367	691	6	iup	iup	VERB
ejpam-5367	691	7	-	-	PUNCT
ejpam-5367	691	8	algebras	algebras	PROPN
ejpam-5367	691	9	.	.	PUNCT
ejpam-5367	692	1	j.	j.	PROPN
ejpam-5367	692	2	math	math	PROPN
ejpam-5367	692	3	.	.	PUNCT
ejpam-5367	693	1	comput	comput	NOUN
ejpam-5367	693	2	.	.	PUNCT
ejpam-5367	694	1	sci	sci	PROPN
ejpam-5367	694	2	.	.	PROPN
ejpam-5367	694	3	,	,	PUNCT
ejpam-5367	694	4	jmcs	jmcs	NOUN
ejpam-5367	694	5	,	,	PUNCT
ejpam-5367	694	6	34(2):128–143	34(2):128–143	PROPN
ejpam-5367	694	7	,	,	PUNCT
ejpam-5367	694	8	2024	2024	NUM
ejpam-5367	694	9	.	.	PUNCT
ejpam-5367	695	1	references	reference	NOUN
ejpam-5367	695	2	3042	3042	NUM
ejpam-5367	695	3	[	[	X
ejpam-5367	695	4	10	10	NUM
ejpam-5367	695	5	]	]	PUNCT
ejpam-5367	695	6	k.	k.	PROPN
ejpam-5367	695	7	lalitha	lalitha	PROPN
ejpam-5367	695	8	and	and	CCONJ
ejpam-5367	695	9	n.	n.	PROPN
ejpam-5367	695	10	buvaneswari	buvaneswari	PROPN
ejpam-5367	695	11	.	.	PUNCT
ejpam-5367	696	1	some	some	DET
ejpam-5367	696	2	new	new	ADJ
ejpam-5367	696	3	results	result	NOUN
ejpam-5367	696	4	on	on	ADP
ejpam-5367	696	5	fermatean	fermatean	ADJ
ejpam-5367	696	6	fuzzy	fuzzy	ADJ
ejpam-5367	696	7	sets	set	NOUN
ejpam-5367	696	8	using	use	VERB
ejpam-5367	696	9	implication	implication	NOUN
ejpam-5367	696	10	.	.	PUNCT
ejpam-5367	697	1	adv	adv	PROPN
ejpam-5367	697	2	.	.	PUNCT
ejpam-5367	697	3	appl	appl	PROPN
ejpam-5367	697	4	.	.	PROPN
ejpam-5367	697	5	math	math	PROPN
ejpam-5367	697	6	.	.	PUNCT
ejpam-5367	698	1	sci	sci	PROPN
ejpam-5367	698	2	.	.	PROPN
ejpam-5367	698	3	,	,	PUNCT
ejpam-5367	698	4	21(5):2829–2841	21(5):2829–2841	NUM
ejpam-5367	698	5	,	,	PUNCT
ejpam-5367	698	6	2022	2022	NUM
ejpam-5367	698	7	.	.	PUNCT
ejpam-5367	699	1	[	[	X
ejpam-5367	699	2	11	11	NUM
ejpam-5367	699	3	]	]	PUNCT
ejpam-5367	699	4	t.	t.	NOUN
ejpam-5367	699	5	senapati	senapati	PROPN
ejpam-5367	699	6	and	and	CCONJ
ejpam-5367	699	7	r.	r.	PROPN
ejpam-5367	699	8	r.	r.	PROPN
ejpam-5367	699	9	yager	yager	PROPN
ejpam-5367	699	10	.	.	PUNCT
ejpam-5367	700	1	fermatean	fermatean	ADJ
ejpam-5367	700	2	fuzzy	fuzzy	ADJ
ejpam-5367	700	3	sets	set	NOUN
ejpam-5367	700	4	.	.	PUNCT
ejpam-5367	701	1	j.	j.	PROPN
ejpam-5367	701	2	ambient	ambient	PROPN
ejpam-5367	701	3	intell	intell	PROPN
ejpam-5367	701	4	.	.	PUNCT
ejpam-5367	702	1	humanized	humanize	VERB
ejpam-5367	702	2	comput	comput	NOUN
ejpam-5367	702	3	.	.	PUNCT
ejpam-5367	702	4	,	,	PUNCT
ejpam-5367	702	5	11:663–674	11:663–674	NUM
ejpam-5367	702	6	,	,	PUNCT
ejpam-5367	702	7	2020	2020	NUM
ejpam-5367	702	8	.	.	PUNCT
ejpam-5367	703	1	[	[	X
ejpam-5367	703	2	12	12	NUM
ejpam-5367	703	3	]	]	PUNCT
ejpam-5367	703	4	j.	j.	PROPN
ejpam-5367	703	5	somjanta	somjanta	PROPN
ejpam-5367	703	6	,	,	PUNCT
ejpam-5367	703	7	n.	n.	PROPN
ejpam-5367	703	8	thuekaew	thuekaew	PROPN
ejpam-5367	703	9	,	,	PUNCT
ejpam-5367	703	10	p.	p.	NOUN
ejpam-5367	703	11	kumpeangkeaw	kumpeangkeaw	PROPN
ejpam-5367	703	12	,	,	PUNCT
ejpam-5367	703	13	and	and	CCONJ
ejpam-5367	703	14	a.	a.	NOUN
ejpam-5367	703	15	iampan	iampan	PROPN
ejpam-5367	703	16	.	.	PUNCT
ejpam-5367	704	1	fuzzy	fuzzy	ADJ
ejpam-5367	704	2	sets	set	NOUN
ejpam-5367	704	3	in	in	ADP
ejpam-5367	704	4	upalgebras	upalgebra	NOUN
ejpam-5367	704	5	.	.	PUNCT
ejpam-5367	705	1	ann	ann	PROPN
ejpam-5367	705	2	.	.	PUNCT
ejpam-5367	705	3	fuzzy	fuzzy	ADJ
ejpam-5367	705	4	math	math	NOUN
ejpam-5367	705	5	.	.	PUNCT
ejpam-5367	706	1	inform	inform	NOUN
ejpam-5367	706	2	.	.	PUNCT
ejpam-5367	706	3	,	,	PUNCT
ejpam-5367	706	4	12(6):739–756	12(6):739–756	PROPN
ejpam-5367	706	5	,	,	PUNCT
ejpam-5367	706	6	2016	2016	NUM
ejpam-5367	706	7	.	.	PUNCT
ejpam-5367	707	1	[	[	X
ejpam-5367	707	2	13	13	NUM
ejpam-5367	707	3	]	]	PUNCT
ejpam-5367	707	4	k.	k.	NOUN
ejpam-5367	707	5	suayngam	suayngam	PROPN
ejpam-5367	707	6	,	,	PUNCT
ejpam-5367	707	7	t.	t.	PROPN
ejpam-5367	707	8	suwanklang	suwanklang	PROPN
ejpam-5367	707	9	,	,	PUNCT
ejpam-5367	707	10	p.	p.	PROPN
ejpam-5367	707	11	julatha	julatha	PROPN
ejpam-5367	707	12	,	,	PUNCT
ejpam-5367	707	13	r.	r.	PROPN
ejpam-5367	707	14	prasertpong	prasertpong	PROPN
ejpam-5367	707	15	,	,	PUNCT
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ejpam-5367	707	17	a.	a.	NOUN
ejpam-5367	707	18	iampan	iampan	PROPN
ejpam-5367	707	19	.	.	PUNCT
ejpam-5367	708	1	new	new	ADJ
ejpam-5367	708	2	results	result	NOUN
ejpam-5367	708	3	on	on	ADP
ejpam-5367	708	4	intuitionistic	intuitionistic	ADJ
ejpam-5367	708	5	fuzzy	fuzzy	ADJ
ejpam-5367	708	6	sets	set	NOUN
ejpam-5367	708	7	in	in	ADP
ejpam-5367	708	8	iup	iup	NOUN
ejpam-5367	708	9	-	-	PUNCT
ejpam-5367	708	10	algebras	algebras	PROPN
ejpam-5367	708	11	.	.	PUNCT
ejpam-5367	709	1	int	int	NOUN
ejpam-5367	709	2	.	.	PUNCT
ejpam-5367	710	1	j.	j.	PROPN
ejpam-5367	710	2	innov	innov	PROPN
ejpam-5367	710	3	.	.	PUNCT
ejpam-5367	711	1	comput	comput	PROPN
ejpam-5367	711	2	.	.	PUNCT
ejpam-5367	712	1	inf	inf	PROPN
ejpam-5367	712	2	.	.	PUNCT
ejpam-5367	712	3	control	control	PROPN
ejpam-5367	712	4	,	,	PUNCT
ejpam-5367	712	5	20(4):1125–1141	20(4):1125–1141	NUM
ejpam-5367	712	6	,	,	PUNCT
ejpam-5367	712	7	2024	2024	NUM
ejpam-5367	712	8	.	.	PUNCT
ejpam-5367	713	1	[	[	X
ejpam-5367	713	2	14	14	NUM
ejpam-5367	713	3	]	]	X
ejpam-5367	713	4	r.	r.	PROPN
ejpam-5367	713	5	r.	r.	PROPN
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ejpam-5367	713	7	.	.	PUNCT
ejpam-5367	714	1	pythagorean	pythagorean	PROPN
ejpam-5367	714	2	fuzzy	fuzzy	ADJ
ejpam-5367	714	3	subsets	subset	NOUN
ejpam-5367	714	4	.	.	PUNCT
ejpam-5367	715	1	in	in	ADP
ejpam-5367	715	2	2013	2013	NUM
ejpam-5367	715	3	joint	joint	ADJ
ejpam-5367	715	4	ifsa	ifsa	PROPN
ejpam-5367	715	5	world	world	PROPN
ejpam-5367	715	6	congress	congress	PROPN
ejpam-5367	715	7	and	and	CCONJ
ejpam-5367	715	8	nafips	nafip	NOUN
ejpam-5367	715	9	annual	annual	ADJ
ejpam-5367	715	10	meeting	meeting	NOUN
ejpam-5367	715	11	(	(	PUNCT
ejpam-5367	715	12	ifsa	ifsa	PROPN
ejpam-5367	715	13	/	/	SYM
ejpam-5367	715	14	nafips	nafip	NOUN
ejpam-5367	715	15	)	)	PUNCT
ejpam-5367	715	16	,	,	PUNCT
ejpam-5367	715	17	pages	page	NOUN
ejpam-5367	715	18	57–61	57–61	NUM
ejpam-5367	715	19	.	.	PUNCT
ejpam-5367	715	20	ieee	ieee	PROPN
ejpam-5367	715	21	,	,	PUNCT
ejpam-5367	715	22	2013	2013	NUM
ejpam-5367	715	23	.	.	PUNCT
ejpam-5367	716	1	[	[	X
ejpam-5367	716	2	15	15	NUM
ejpam-5367	716	3	]	]	X
ejpam-5367	716	4	l.	l.	PROPN
ejpam-5367	716	5	a.	a.	PROPN
ejpam-5367	716	6	zadeh	zadeh	PROPN
ejpam-5367	716	7	.	.	PUNCT
ejpam-5367	716	8	fuzzy	fuzzy	ADJ
ejpam-5367	716	9	sets	set	NOUN
ejpam-5367	716	10	.	.	PUNCT
ejpam-5367	717	1	inf	inf	PROPN
ejpam-5367	717	2	.	.	PUNCT
ejpam-5367	717	3	cont	cont	PROPN
ejpam-5367	717	4	.	.	PROPN
ejpam-5367	717	5	,	,	PUNCT
ejpam-5367	717	6	8(3):338–353	8(3):338–353	NUM
ejpam-5367	717	7	,	,	PUNCT
ejpam-5367	717	8	1965	1965	NUM
ejpam-5367	717	9	.	.	PUNCT