id	sid	tid	token	lemma	pos
ejpam-6171	1	1	european	european	PROPN
ejpam-6171	1	2	journal	journal	PROPN
ejpam-6171	1	3	of	of	ADP
ejpam-6171	1	4	pure	pure	ADJ
ejpam-6171	1	5	and	and	CCONJ
ejpam-6171	1	6	applied	applied	ADJ
ejpam-6171	1	7	mathematics	mathematic	NOUN
ejpam-6171	1	8	2025	2025	NUM
ejpam-6171	1	9	,	,	PUNCT
ejpam-6171	1	10	vol	vol	NOUN
ejpam-6171	1	11	.	.	PROPN
ejpam-6171	1	12	18	18	NUM
ejpam-6171	1	13	,	,	PUNCT
ejpam-6171	1	14	issue	issue	NOUN
ejpam-6171	1	15	3	3	NUM
ejpam-6171	1	16	,	,	PUNCT
ejpam-6171	1	17	article	article	NOUN
ejpam-6171	1	18	number	number	NOUN
ejpam-6171	1	19	6171	6171	NUM
ejpam-6171	1	20	issn	issn	PROPN
ejpam-6171	1	21	1307	1307	NUM
ejpam-6171	1	22	-	-	SYM
ejpam-6171	1	23	5543	5543	NUM
ejpam-6171	1	24	–	–	PUNCT
ejpam-6171	1	25	ejpam.com	ejpam.com	X
ejpam-6171	1	26	published	publish	VERB
ejpam-6171	1	27	by	by	ADP
ejpam-6171	1	28	new	new	PROPN
ejpam-6171	1	29	york	york	PROPN
ejpam-6171	1	30	business	business	PROPN
ejpam-6171	1	31	global	global	PROPN
ejpam-6171	1	32	pythagorean	pythagorean	PROPN
ejpam-6171	1	33	neutrosophic	neutrosophic	PROPN
ejpam-6171	1	34	iup	iup	PROPN
ejpam-6171	1	35	-	-	PUNCT
ejpam-6171	1	36	algebras	algebras	PROPN
ejpam-6171	1	37	:	:	PUNCT
ejpam-6171	1	38	theoretical	theoretical	ADJ
ejpam-6171	1	39	foundations	foundation	NOUN
ejpam-6171	1	40	and	and	CCONJ
ejpam-6171	1	41	extensions	extension	NOUN
ejpam-6171	1	42	kannirun	kannirun	VERB
ejpam-6171	1	43	suayngam1	suayngam1	PROPN
ejpam-6171	1	44	,	,	PUNCT
ejpam-6171	1	45	pongpun	pongpun	PROPN
ejpam-6171	1	46	julatha2	julatha2	PROPN
ejpam-6171	1	47	,	,	PUNCT
ejpam-6171	1	48	warud	warud	NOUN
ejpam-6171	1	49	nakkhasen3	nakkhasen3	PROPN
ejpam-6171	1	50	,	,	PUNCT
ejpam-6171	1	51	rukchart	rukchart	PROPN
ejpam-6171	1	52	prasertpong4	prasertpong4	PROPN
ejpam-6171	1	53	,	,	PUNCT
ejpam-6171	1	54	aiyared	aiyare	VERB
ejpam-6171	1	55	iampan1,∗	iampan1,∗	NOUN
ejpam-6171	1	56	1	1	NUM
ejpam-6171	1	57	department	department	NOUN
ejpam-6171	1	58	of	of	ADP
ejpam-6171	1	59	mathematics	mathematic	NOUN
ejpam-6171	1	60	,	,	PUNCT
ejpam-6171	1	61	school	school	NOUN
ejpam-6171	1	62	of	of	ADP
ejpam-6171	1	63	science	science	NOUN
ejpam-6171	1	64	,	,	PUNCT
ejpam-6171	1	65	university	university	NOUN
ejpam-6171	1	66	of	of	ADP
ejpam-6171	1	67	phayao	phayao	NOUN
ejpam-6171	1	68	,	,	PUNCT
ejpam-6171	1	69	mae	mae	PROPN
ejpam-6171	1	70	ka	ka	PROPN
ejpam-6171	1	71	,	,	PUNCT
ejpam-6171	1	72	mueang	mueang	PROPN
ejpam-6171	1	73	,	,	PUNCT
ejpam-6171	1	74	phayao	phayao	NOUN
ejpam-6171	1	75	56000	56000	NUM
ejpam-6171	1	76	,	,	PUNCT
ejpam-6171	1	77	thailand	thailand	PROPN
ejpam-6171	1	78	2	2	NUM
ejpam-6171	1	79	department	department	NOUN
ejpam-6171	1	80	of	of	ADP
ejpam-6171	1	81	mathematics	mathematic	NOUN
ejpam-6171	1	82	,	,	PUNCT
ejpam-6171	1	83	faculty	faculty	NOUN
ejpam-6171	1	84	of	of	ADP
ejpam-6171	1	85	science	science	NOUN
ejpam-6171	1	86	and	and	CCONJ
ejpam-6171	1	87	technology	technology	NOUN
ejpam-6171	1	88	,	,	PUNCT
ejpam-6171	1	89	pibulsongkram	pibulsongkram	PROPN
ejpam-6171	1	90	rajabhat	rajabhat	PROPN
ejpam-6171	1	91	university	university	PROPN
ejpam-6171	1	92	,	,	PUNCT
ejpam-6171	1	93	phlai	phlai	PROPN
ejpam-6171	1	94	chumphon	chumphon	PROPN
ejpam-6171	1	95	,	,	PUNCT
ejpam-6171	1	96	mueang	mueang	PROPN
ejpam-6171	1	97	,	,	PUNCT
ejpam-6171	1	98	phitsanulok	phitsanulok	NOUN
ejpam-6171	1	99	65000	65000	NUM
ejpam-6171	1	100	,	,	PUNCT
ejpam-6171	1	101	thailand	thailand	PROPN
ejpam-6171	1	102	3	3	NUM
ejpam-6171	1	103	department	department	NOUN
ejpam-6171	1	104	of	of	ADP
ejpam-6171	1	105	mathematics	mathematic	NOUN
ejpam-6171	1	106	,	,	PUNCT
ejpam-6171	1	107	faculty	faculty	NOUN
ejpam-6171	1	108	of	of	ADP
ejpam-6171	1	109	science	science	NOUN
ejpam-6171	1	110	,	,	PUNCT
ejpam-6171	1	111	mahasarakham	mahasarakham	PROPN
ejpam-6171	1	112	university	university	PROPN
ejpam-6171	1	113	,	,	PUNCT
ejpam-6171	1	114	khamriang	khamriang	PROPN
ejpam-6171	1	115	,	,	PUNCT
ejpam-6171	1	116	kantarawichai	kantarawichai	PROPN
ejpam-6171	1	117	,	,	PUNCT
ejpam-6171	1	118	maha	maha	PROPN
ejpam-6171	1	119	sarakham	sarakham	PROPN
ejpam-6171	1	120	44150	44150	NUM
ejpam-6171	1	121	,	,	PUNCT
ejpam-6171	1	122	thailand	thailand	PROPN
ejpam-6171	1	123	4	4	NUM
ejpam-6171	1	124	division	division	NOUN
ejpam-6171	1	125	of	of	ADP
ejpam-6171	1	126	mathematics	mathematic	NOUN
ejpam-6171	1	127	and	and	CCONJ
ejpam-6171	1	128	statistics	statistic	NOUN
ejpam-6171	1	129	,	,	PUNCT
ejpam-6171	1	130	faculty	faculty	NOUN
ejpam-6171	1	131	of	of	ADP
ejpam-6171	1	132	science	science	NOUN
ejpam-6171	1	133	and	and	CCONJ
ejpam-6171	1	134	technology	technology	NOUN
ejpam-6171	1	135	,	,	PUNCT
ejpam-6171	1	136	nakhon	nakhon	PROPN
ejpam-6171	1	137	sawan	sawan	PROPN
ejpam-6171	1	138	rajabhat	rajabhat	PROPN
ejpam-6171	1	139	university	university	PROPN
ejpam-6171	1	140	,	,	PUNCT
ejpam-6171	1	141	nakhon	nakhon	PROPN
ejpam-6171	1	142	sawan	sawan	PROPN
ejpam-6171	1	143	tok	tok	PROPN
ejpam-6171	1	144	,	,	PUNCT
ejpam-6171	1	145	mueang	mueang	NOUN
ejpam-6171	1	146	,	,	PUNCT
ejpam-6171	1	147	nakhon	nakhon	PROPN
ejpam-6171	1	148	sawan	sawan	PROPN
ejpam-6171	1	149	60000	60000	NUM
ejpam-6171	1	150	,	,	PUNCT
ejpam-6171	1	151	thailand	thailand	PROPN
ejpam-6171	1	152	abstract	abstract	NOUN
ejpam-6171	1	153	.	.	PUNCT
ejpam-6171	2	1	this	this	DET
ejpam-6171	2	2	paper	paper	NOUN
ejpam-6171	2	3	introduces	introduce	VERB
ejpam-6171	2	4	the	the	DET
ejpam-6171	2	5	concepts	concept	NOUN
ejpam-6171	2	6	of	of	ADP
ejpam-6171	2	7	pythagorean	pythagorean	PROPN
ejpam-6171	2	8	neutrosophic	neutrosophic	PROPN
ejpam-6171	2	9	iup	iup	PROPN
ejpam-6171	2	10	-	-	PUNCT
ejpam-6171	2	11	subalgebras	subalgebras	PROPN
ejpam-6171	2	12	,	,	PUNCT
ejpam-6171	2	13	pythagorean	pythagorean	PROPN
ejpam-6171	2	14	neutrosophic	neutrosophic	PROPN
ejpam-6171	2	15	iup	iup	PROPN
ejpam-6171	2	16	-	-	PUNCT
ejpam-6171	2	17	ideals	ideal	NOUN
ejpam-6171	2	18	,	,	PUNCT
ejpam-6171	2	19	pythagorean	pythagorean	PROPN
ejpam-6171	2	20	neutrosophic	neutrosophic	PROPN
ejpam-6171	2	21	iup	iup	NOUN
ejpam-6171	2	22	-	-	PUNCT
ejpam-6171	2	23	filters	filter	NOUN
ejpam-6171	2	24	,	,	PUNCT
ejpam-6171	2	25	and	and	CCONJ
ejpam-6171	2	26	pythagorean	pythagorean	PROPN
ejpam-6171	2	27	neutrosophic	neutrosophic	PROPN
ejpam-6171	2	28	strong	strong	ADJ
ejpam-6171	2	29	iup	iup	NOUN
ejpam-6171	2	30	-	-	PUNCT
ejpam-6171	2	31	ideals	ideal	NOUN
ejpam-6171	2	32	within	within	ADP
ejpam-6171	2	33	the	the	DET
ejpam-6171	2	34	framework	framework	NOUN
ejpam-6171	2	35	of	of	ADP
ejpam-6171	2	36	iup	iup	NOUN
ejpam-6171	2	37	-	-	PUNCT
ejpam-6171	2	38	algebras	algebras	PROPN
ejpam-6171	2	39	.	.	PUNCT
ejpam-6171	3	1	we	we	PRON
ejpam-6171	3	2	establish	establish	VERB
ejpam-6171	3	3	the	the	DET
ejpam-6171	3	4	fundamental	fundamental	ADJ
ejpam-6171	3	5	properties	property	NOUN
ejpam-6171	3	6	of	of	ADP
ejpam-6171	3	7	these	these	DET
ejpam-6171	3	8	structures	structure	NOUN
ejpam-6171	3	9	and	and	CCONJ
ejpam-6171	3	10	provide	provide	VERB
ejpam-6171	3	11	necessary	necessary	ADJ
ejpam-6171	3	12	and	and	CCONJ
ejpam-6171	3	13	sufficient	sufficient	ADJ
ejpam-6171	3	14	conditions	condition	NOUN
ejpam-6171	3	15	under	under	ADP
ejpam-6171	3	16	which	which	PRON
ejpam-6171	3	17	a	a	DET
ejpam-6171	3	18	pythagorean	pythagorean	PROPN
ejpam-6171	3	19	neutrosophic	neutrosophic	PROPN
ejpam-6171	3	20	set	set	NOUN
ejpam-6171	3	21	qualifies	qualifie	NOUN
ejpam-6171	3	22	as	as	ADP
ejpam-6171	3	23	one	one	NUM
ejpam-6171	3	24	of	of	ADP
ejpam-6171	3	25	these	these	DET
ejpam-6171	3	26	algebraic	algebraic	ADJ
ejpam-6171	3	27	subsets	subset	NOUN
ejpam-6171	3	28	.	.	PUNCT
ejpam-6171	4	1	additionally	additionally	ADV
ejpam-6171	4	2	,	,	PUNCT
ejpam-6171	4	3	we	we	PRON
ejpam-6171	4	4	explore	explore	VERB
ejpam-6171	4	5	the	the	DET
ejpam-6171	4	6	relationships	relationship	NOUN
ejpam-6171	4	7	between	between	ADP
ejpam-6171	4	8	these	these	DET
ejpam-6171	4	9	subsets	subset	NOUN
ejpam-6171	4	10	and	and	CCONJ
ejpam-6171	4	11	their	their	PRON
ejpam-6171	4	12	corresponding	corresponding	ADJ
ejpam-6171	4	13	level	level	NOUN
ejpam-6171	4	14	subsets	subset	NOUN
ejpam-6171	4	15	,	,	PUNCT
ejpam-6171	4	16	offering	offer	VERB
ejpam-6171	4	17	a	a	DET
ejpam-6171	4	18	deeper	deep	ADJ
ejpam-6171	4	19	understanding	understanding	NOUN
ejpam-6171	4	20	of	of	ADP
ejpam-6171	4	21	their	their	PRON
ejpam-6171	4	22	interconnections	interconnection	NOUN
ejpam-6171	4	23	.	.	PUNCT
ejpam-6171	5	1	by	by	ADP
ejpam-6171	5	2	extending	extend	VERB
ejpam-6171	5	3	the	the	DET
ejpam-6171	5	4	theoretical	theoretical	ADJ
ejpam-6171	5	5	foundation	foundation	NOUN
ejpam-6171	5	6	of	of	ADP
ejpam-6171	5	7	iup	iup	NOUN
ejpam-6171	5	8	-	-	PUNCT
ejpam-6171	5	9	algebras	algebras	NOUN
ejpam-6171	5	10	through	through	ADP
ejpam-6171	5	11	the	the	DET
ejpam-6171	5	12	integration	integration	NOUN
ejpam-6171	5	13	of	of	ADP
ejpam-6171	5	14	pythagorean	pythagorean	PROPN
ejpam-6171	5	15	neutrosophic	neutrosophic	ADJ
ejpam-6171	5	16	sets	set	NOUN
ejpam-6171	5	17	,	,	PUNCT
ejpam-6171	5	18	this	this	DET
ejpam-6171	5	19	study	study	NOUN
ejpam-6171	5	20	contributes	contribute	VERB
ejpam-6171	5	21	to	to	ADP
ejpam-6171	5	22	the	the	DET
ejpam-6171	5	23	broader	broad	ADJ
ejpam-6171	5	24	field	field	NOUN
ejpam-6171	5	25	of	of	ADP
ejpam-6171	5	26	algebraic	algebraic	ADJ
ejpam-6171	5	27	structures	structure	NOUN
ejpam-6171	5	28	and	and	CCONJ
ejpam-6171	5	29	uncertainty	uncertainty	NOUN
ejpam-6171	5	30	modeling	modeling	NOUN
ejpam-6171	5	31	.	.	PUNCT
ejpam-6171	6	1	beyond	beyond	ADP
ejpam-6171	6	2	its	its	PRON
ejpam-6171	6	3	theoretical	theoretical	ADJ
ejpam-6171	6	4	significance	significance	NOUN
ejpam-6171	6	5	,	,	PUNCT
ejpam-6171	6	6	this	this	DET
ejpam-6171	6	7	work	work	NOUN
ejpam-6171	6	8	promotes	promote	VERB
ejpam-6171	6	9	inclusive	inclusive	ADJ
ejpam-6171	6	10	and	and	CCONJ
ejpam-6171	6	11	equitable	equitable	ADJ
ejpam-6171	6	12	education	education	NOUN
ejpam-6171	6	13	by	by	ADP
ejpam-6171	6	14	making	make	VERB
ejpam-6171	6	15	abstract	abstract	ADJ
ejpam-6171	6	16	mathematical	mathematical	ADJ
ejpam-6171	6	17	concepts	concept	NOUN
ejpam-6171	6	18	more	more	ADV
ejpam-6171	6	19	accessible	accessible	ADJ
ejpam-6171	6	20	to	to	ADP
ejpam-6171	6	21	students	student	NOUN
ejpam-6171	6	22	,	,	PUNCT
ejpam-6171	6	23	educators	educator	NOUN
ejpam-6171	6	24	,	,	PUNCT
ejpam-6171	6	25	and	and	CCONJ
ejpam-6171	6	26	researchers	researcher	NOUN
ejpam-6171	6	27	at	at	ADP
ejpam-6171	6	28	both	both	CCONJ
ejpam-6171	6	29	the	the	DET
ejpam-6171	6	30	school	school	NOUN
ejpam-6171	6	31	and	and	CCONJ
ejpam-6171	6	32	university	university	NOUN
ejpam-6171	6	33	levels	level	NOUN
ejpam-6171	6	34	.	.	PUNCT
ejpam-6171	7	1	by	by	ADP
ejpam-6171	7	2	fostering	foster	VERB
ejpam-6171	7	3	mathematical	mathematical	ADJ
ejpam-6171	7	4	literacy	literacy	NOUN
ejpam-6171	7	5	and	and	CCONJ
ejpam-6171	7	6	critical	critical	ADJ
ejpam-6171	7	7	thinking	thinking	NOUN
ejpam-6171	7	8	,	,	PUNCT
ejpam-6171	7	9	this	this	DET
ejpam-6171	7	10	research	research	NOUN
ejpam-6171	7	11	equips	equip	VERB
ejpam-6171	7	12	learners	learner	NOUN
ejpam-6171	7	13	with	with	ADP
ejpam-6171	7	14	essential	essential	ADJ
ejpam-6171	7	15	problem	problem	NOUN
ejpam-6171	7	16	-	-	PUNCT
ejpam-6171	7	17	solving	solve	VERB
ejpam-6171	7	18	skills	skill	NOUN
ejpam-6171	7	19	applicable	applicable	ADJ
ejpam-6171	7	20	to	to	ADP
ejpam-6171	7	21	various	various	ADJ
ejpam-6171	7	22	fields	field	NOUN
ejpam-6171	7	23	,	,	PUNCT
ejpam-6171	7	24	including	include	VERB
ejpam-6171	7	25	mathematics	mathematic	NOUN
ejpam-6171	7	26	,	,	PUNCT
ejpam-6171	7	27	computer	computer	NOUN
ejpam-6171	7	28	science	science	NOUN
ejpam-6171	7	29	,	,	PUNCT
ejpam-6171	7	30	and	and	CCONJ
ejpam-6171	7	31	artificial	artificial	ADJ
ejpam-6171	7	32	intelligence	intelligence	NOUN
ejpam-6171	7	33	.	.	PUNCT
ejpam-6171	8	1	to	to	PART
ejpam-6171	8	2	enhance	enhance	VERB
ejpam-6171	8	3	accessibility	accessibility	NOUN
ejpam-6171	8	4	,	,	PUNCT
ejpam-6171	8	5	key	key	ADJ
ejpam-6171	8	6	concepts	concept	NOUN
ejpam-6171	8	7	are	be	AUX
ejpam-6171	8	8	presented	present	VERB
ejpam-6171	8	9	in	in	ADP
ejpam-6171	8	10	a	a	DET
ejpam-6171	8	11	structured	structured	ADJ
ejpam-6171	8	12	and	and	CCONJ
ejpam-6171	8	13	comprehensible	comprehensible	ADJ
ejpam-6171	8	14	manner	manner	NOUN
ejpam-6171	8	15	,	,	PUNCT
ejpam-6171	8	16	allowing	allow	VERB
ejpam-6171	8	17	students	student	NOUN
ejpam-6171	8	18	from	from	ADP
ejpam-6171	8	19	diverse	diverse	ADJ
ejpam-6171	8	20	backgrounds	background	NOUN
ejpam-6171	8	21	to	to	PART
ejpam-6171	8	22	engage	engage	VERB
ejpam-6171	8	23	with	with	ADP
ejpam-6171	8	24	complex	complex	ADJ
ejpam-6171	8	25	algebraic	algebraic	ADJ
ejpam-6171	8	26	theories	theory	NOUN
ejpam-6171	8	27	more	more	ADV
ejpam-6171	8	28	effectively	effectively	ADV
ejpam-6171	8	29	.	.	PUNCT
ejpam-6171	9	1	this	this	DET
ejpam-6171	9	2	study	study	NOUN
ejpam-6171	9	3	fosters	foster	VERB
ejpam-6171	9	4	lifelong	lifelong	ADJ
ejpam-6171	9	5	learning	learning	NOUN
ejpam-6171	9	6	and	and	CCONJ
ejpam-6171	9	7	innovation	innovation	NOUN
ejpam-6171	9	8	by	by	ADP
ejpam-6171	9	9	strengthening	strengthen	VERB
ejpam-6171	9	10	the	the	DET
ejpam-6171	9	11	integration	integration	NOUN
ejpam-6171	9	12	of	of	ADP
ejpam-6171	9	13	mathematical	mathematical	ADJ
ejpam-6171	9	14	education	education	NOUN
ejpam-6171	9	15	across	across	ADP
ejpam-6171	9	16	different	different	ADJ
ejpam-6171	9	17	learning	learning	NOUN
ejpam-6171	9	18	environments	environment	NOUN
ejpam-6171	9	19	.	.	PUNCT
ejpam-6171	10	1	2020	2020	NUM
ejpam-6171	10	2	mathematics	mathematic	NOUN
ejpam-6171	10	3	subject	subject	NOUN
ejpam-6171	10	4	classifications	classification	NOUN
ejpam-6171	10	5	:	:	PUNCT
ejpam-6171	10	6	03g25	03g25	NUM
ejpam-6171	10	7	,	,	PUNCT
ejpam-6171	10	8	03e72	03e72	NUM
ejpam-6171	10	9	,	,	PUNCT
ejpam-6171	10	10	08a72	08a72	NOUN
ejpam-6171	10	11	key	key	ADJ
ejpam-6171	10	12	words	word	NOUN
ejpam-6171	10	13	and	and	CCONJ
ejpam-6171	10	14	phrases	phrase	NOUN
ejpam-6171	10	15	:	:	PUNCT
ejpam-6171	10	16	iup	iup	NOUN
ejpam-6171	10	17	-	-	PUNCT
ejpam-6171	10	18	algebra	algebra	PROPN
ejpam-6171	10	19	,	,	PUNCT
ejpam-6171	10	20	pythagorean	pythagorean	PROPN
ejpam-6171	10	21	neutrosophic	neutrosophic	PROPN
ejpam-6171	10	22	iup	iup	PROPN
ejpam-6171	10	23	-	-	PUNCT
ejpam-6171	10	24	subalgebra	subalgebra	PROPN
ejpam-6171	10	25	,	,	PUNCT
ejpam-6171	10	26	pythagorean	pythagorean	PROPN
ejpam-6171	10	27	neutrosophic	neutrosophic	PROPN
ejpam-6171	10	28	iup	iup	PROPN
ejpam-6171	10	29	-	-	PUNCT
ejpam-6171	10	30	ideal	ideal	NOUN
ejpam-6171	10	31	,	,	PUNCT
ejpam-6171	10	32	pythagorean	pythagorean	PROPN
ejpam-6171	10	33	neutrosophic	neutrosophic	PROPN
ejpam-6171	10	34	iup	iup	NOUN
ejpam-6171	10	35	-	-	PUNCT
ejpam-6171	10	36	filter	filter	NOUN
ejpam-6171	10	37	,	,	PUNCT
ejpam-6171	10	38	and	and	CCONJ
ejpam-6171	10	39	pythagorean	pythagorean	PROPN
ejpam-6171	10	40	neutrosophic	neutrosophic	PROPN
ejpam-6171	10	41	strong	strong	ADJ
ejpam-6171	10	42	iup	iup	NOUN
ejpam-6171	10	43	-	-	PUNCT
ejpam-6171	10	44	ideal	ideal	NOUN
ejpam-6171	10	45	∗corresponding	∗corresponding	NOUN
ejpam-6171	10	46	author	author	NOUN
ejpam-6171	10	47	.	.	PUNCT
ejpam-6171	11	1	doi	doi	NOUN
ejpam-6171	11	2	:	:	PUNCT
ejpam-6171	11	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6171	https://doi.org/10.29020/nybg.ejpam.v18i3.6171	NOUN
ejpam-6171	11	4	email	email	NOUN
ejpam-6171	11	5	addresses	address	NOUN
ejpam-6171	11	6	:	:	PUNCT
ejpam-6171	12	1	kannirun.s@gmail.com	kannirun.s@gmail.com	X
ejpam-6171	12	2	(	(	PUNCT
ejpam-6171	12	3	k.	k.	NOUN
ejpam-6171	12	4	suayngam	suayngam	PROPN
ejpam-6171	12	5	)	)	PUNCT
ejpam-6171	12	6	,	,	PUNCT
ejpam-6171	12	7	pongpun.j@psru.ac.th	pongpun.j@psru.ac.th	PROPN
ejpam-6171	12	8	(	(	PUNCT
ejpam-6171	12	9	p.	p.	NOUN
ejpam-6171	12	10	julatha	julatha	NOUN
ejpam-6171	12	11	)	)	PUNCT
ejpam-6171	12	12	,	,	PUNCT
ejpam-6171	12	13	warud.n@msu.ac.th	warud.n@msu.ac.th	PRON
ejpam-6171	12	14	(	(	PUNCT
ejpam-6171	12	15	w.	w.	PROPN
ejpam-6171	12	16	nakkhasen	nakkhasen	PROPN
ejpam-6171	12	17	)	)	PUNCT
ejpam-6171	12	18	,	,	PUNCT
ejpam-6171	12	19	rukchart.p@nsru.ac.th	rukchart.p@nsru.ac.th	PROPN
ejpam-6171	12	20	(	(	PUNCT
ejpam-6171	12	21	r.	r.	PROPN
ejpam-6171	12	22	prasertpong	prasertpong	PROPN
ejpam-6171	12	23	)	)	PUNCT
ejpam-6171	12	24	,	,	PUNCT
ejpam-6171	12	25	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-6171	12	26	(	(	PUNCT
ejpam-6171	12	27	a.	a.	NOUN
ejpam-6171	12	28	iampan	iampan	PROPN
ejpam-6171	12	29	)	)	PUNCT
ejpam-6171	12	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6171	12	31	1	1	NUM
ejpam-6171	12	32	copyright	copyright	NOUN
ejpam-6171	12	33	:	:	PUNCT
ejpam-6171	12	34	©	©	PROPN
ejpam-6171	12	35	2025	2025	NUM
ejpam-6171	12	36	the	the	DET
ejpam-6171	12	37	author(s	author(s	NOUN
ejpam-6171	12	38	)	)	PUNCT
ejpam-6171	12	39	.	.	PUNCT
ejpam-6171	13	1	(	(	PUNCT
ejpam-6171	13	2	cc	cc	NOUN
ejpam-6171	13	3	by	by	ADP
ejpam-6171	13	4	-	-	PUNCT
ejpam-6171	13	5	nc	nc	PROPN
ejpam-6171	13	6	4.0	4.0	NUM
ejpam-6171	13	7	)	)	PUNCT
ejpam-6171	13	8	k.	k.	NOUN
ejpam-6171	14	1	suayngam	suayngam	INTJ
ejpam-6171	14	2	et	et	PROPN
ejpam-6171	14	3	al	al	PROPN
ejpam-6171	14	4	.	.	PUNCT
ejpam-6171	14	5	/	/	SYM
ejpam-6171	14	6	eur	eur	PROPN
ejpam-6171	14	7	.	.	PUNCT
ejpam-6171	15	1	j.	j.	PROPN
ejpam-6171	15	2	pure	pure	PROPN
ejpam-6171	15	3	appl	appl	PROPN
ejpam-6171	15	4	.	.	PROPN
ejpam-6171	15	5	math	math	PROPN
ejpam-6171	15	6	,	,	PUNCT
ejpam-6171	15	7	18	18	NUM
ejpam-6171	15	8	(	(	PUNCT
ejpam-6171	15	9	3	3	NUM
ejpam-6171	15	10	)	)	PUNCT
ejpam-6171	15	11	(	(	PUNCT
ejpam-6171	15	12	2025	2025	NUM
ejpam-6171	15	13	)	)	PUNCT
ejpam-6171	15	14	,	,	PUNCT
ejpam-6171	15	15	6171	6171	NUM
ejpam-6171	15	16	2	2	NUM
ejpam-6171	15	17	of	of	ADP
ejpam-6171	15	18	28	28	NUM
ejpam-6171	15	19	1	1	NUM
ejpam-6171	15	20	.	.	PUNCT
ejpam-6171	15	21	introduction	introduction	NOUN
ejpam-6171	15	22	in	in	ADP
ejpam-6171	15	23	1965	1965	NUM
ejpam-6171	15	24	,	,	PUNCT
ejpam-6171	15	25	zadeh	zadeh	PROPN
ejpam-6171	16	1	[	[	X
ejpam-6171	16	2	1	1	X
ejpam-6171	16	3	]	]	PUNCT
ejpam-6171	16	4	defined	define	VERB
ejpam-6171	16	5	the	the	DET
ejpam-6171	16	6	concept	concept	NOUN
ejpam-6171	16	7	of	of	ADP
ejpam-6171	16	8	fuzzy	fuzzy	ADJ
ejpam-6171	16	9	sets	set	NOUN
ejpam-6171	16	10	(	(	PUNCT
ejpam-6171	16	11	fss	fss	NOUN
ejpam-6171	16	12	)	)	PUNCT
ejpam-6171	16	13	,	,	PUNCT
ejpam-6171	16	14	an	an	DET
ejpam-6171	16	15	important	important	ADJ
ejpam-6171	16	16	idea	idea	NOUN
ejpam-6171	16	17	that	that	PRON
ejpam-6171	16	18	has	have	AUX
ejpam-6171	16	19	been	be	AUX
ejpam-6171	16	20	extensively	extensively	ADV
ejpam-6171	16	21	built	build	VERB
ejpam-6171	16	22	upon	upon	SCONJ
ejpam-6171	16	23	by	by	ADP
ejpam-6171	16	24	many	many	ADJ
ejpam-6171	16	25	.	.	PUNCT
ejpam-6171	17	1	later	later	ADV
ejpam-6171	17	2	,	,	PUNCT
ejpam-6171	17	3	in	in	ADP
ejpam-6171	17	4	1986	1986	NUM
ejpam-6171	17	5	,	,	PUNCT
ejpam-6171	17	6	atanasov	atanasov	NOUN
ejpam-6171	17	7	[	[	X
ejpam-6171	17	8	2	2	NUM
ejpam-6171	17	9	]	]	PUNCT
ejpam-6171	17	10	defined	define	VERB
ejpam-6171	17	11	the	the	DET
ejpam-6171	17	12	concept	concept	NOUN
ejpam-6171	17	13	of	of	ADP
ejpam-6171	17	14	intuitionistic	intuitionistic	ADJ
ejpam-6171	17	15	fuzzy	fuzzy	ADJ
ejpam-6171	17	16	sets	set	NOUN
ejpam-6171	17	17	(	(	PUNCT
ejpam-6171	17	18	ifss	ifss	NOUN
ejpam-6171	17	19	)	)	PUNCT
ejpam-6171	17	20	,	,	PUNCT
ejpam-6171	17	21	which	which	PRON
ejpam-6171	17	22	is	be	AUX
ejpam-6171	17	23	a	a	DET
ejpam-6171	17	24	generalization	generalization	NOUN
ejpam-6171	17	25	of	of	ADP
ejpam-6171	17	26	fuzzy	fuzzy	ADJ
ejpam-6171	17	27	sets	set	NOUN
ejpam-6171	17	28	(	(	PUNCT
ejpam-6171	17	29	fss	fss	NOUN
ejpam-6171	17	30	)	)	PUNCT
ejpam-6171	17	31	.	.	PUNCT
ejpam-6171	18	1	subsequently	subsequently	ADV
ejpam-6171	18	2	,	,	PUNCT
ejpam-6171	18	3	in	in	ADP
ejpam-6171	18	4	2004	2004	NUM
ejpam-6171	18	5	,	,	PUNCT
ejpam-6171	18	6	smarandache	smarandache	NOUN
ejpam-6171	18	7	defined	define	VERB
ejpam-6171	18	8	the	the	DET
ejpam-6171	18	9	concept	concept	NOUN
ejpam-6171	18	10	of	of	ADP
ejpam-6171	18	11	neutrosophic	neutrosophic	ADJ
ejpam-6171	18	12	sets	set	NOUN
ejpam-6171	18	13	,	,	PUNCT
ejpam-6171	18	14	a	a	DET
ejpam-6171	18	15	generalization	generalization	NOUN
ejpam-6171	18	16	of	of	ADP
ejpam-6171	18	17	intuitionistic	intuitionistic	ADJ
ejpam-6171	18	18	fuzzy	fuzzy	ADJ
ejpam-6171	18	19	sets	set	NOUN
ejpam-6171	18	20	(	(	PUNCT
ejpam-6171	18	21	ifss	ifss	NOUN
ejpam-6171	18	22	)	)	PUNCT
ejpam-6171	18	23	.	.	PUNCT
ejpam-6171	19	1	subsequently	subsequently	ADV
ejpam-6171	19	2	,	,	PUNCT
ejpam-6171	19	3	in	in	ADP
ejpam-6171	19	4	2004	2004	NUM
ejpam-6171	19	5	,	,	PUNCT
ejpam-6171	19	6	smarandache	smarandache	NOUN
ejpam-6171	20	1	[	[	X
ejpam-6171	20	2	3	3	NUM
ejpam-6171	20	3	]	]	PUNCT
ejpam-6171	20	4	introduced	introduce	VERB
ejpam-6171	20	5	the	the	DET
ejpam-6171	20	6	concept	concept	NOUN
ejpam-6171	20	7	of	of	ADP
ejpam-6171	20	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	20	9	sets	set	NOUN
ejpam-6171	20	10	(	(	PUNCT
ejpam-6171	20	11	nss	ns	NOUN
ejpam-6171	20	12	)	)	PUNCT
ejpam-6171	20	13	,	,	PUNCT
ejpam-6171	20	14	which	which	PRON
ejpam-6171	20	15	is	be	AUX
ejpam-6171	20	16	a	a	DET
ejpam-6171	20	17	generalization	generalization	NOUN
ejpam-6171	20	18	of	of	ADP
ejpam-6171	20	19	intuitionistic	intuitionistic	ADJ
ejpam-6171	20	20	fuzzy	fuzzy	ADJ
ejpam-6171	20	21	sets	set	NOUN
ejpam-6171	20	22	(	(	PUNCT
ejpam-6171	20	23	ifss	ifss	NOUN
ejpam-6171	20	24	)	)	PUNCT
ejpam-6171	20	25	.	.	PUNCT
ejpam-6171	21	1	afterwards	afterwards	ADV
ejpam-6171	21	2	,	,	PUNCT
ejpam-6171	21	3	in	in	ADP
ejpam-6171	21	4	2013	2013	NUM
ejpam-6171	21	5	,	,	PUNCT
ejpam-6171	21	6	yager	yager	NOUN
ejpam-6171	22	1	[	[	X
ejpam-6171	22	2	4	4	NUM
ejpam-6171	22	3	]	]	PUNCT
ejpam-6171	22	4	defined	define	VERB
ejpam-6171	22	5	the	the	DET
ejpam-6171	22	6	concept	concept	NOUN
ejpam-6171	22	7	of	of	ADP
ejpam-6171	22	8	pythagorean	pythagorean	PROPN
ejpam-6171	22	9	fuzzy	fuzzy	ADJ
ejpam-6171	22	10	subsets	subset	NOUN
ejpam-6171	22	11	(	(	PUNCT
ejpam-6171	22	12	pfss	pfss	NOUN
ejpam-6171	22	13	)	)	PUNCT
ejpam-6171	22	14	,	,	PUNCT
ejpam-6171	22	15	an	an	DET
ejpam-6171	22	16	extension	extension	NOUN
ejpam-6171	22	17	of	of	ADP
ejpam-6171	22	18	fuzzy	fuzzy	ADJ
ejpam-6171	22	19	sets	set	NOUN
ejpam-6171	22	20	.	.	PUNCT
ejpam-6171	23	1	later	later	ADV
ejpam-6171	23	2	,	,	PUNCT
ejpam-6171	23	3	in	in	ADP
ejpam-6171	23	4	2019	2019	NUM
ejpam-6171	23	5	,	,	PUNCT
ejpam-6171	23	6	jansi	jansi	PROPN
ejpam-6171	23	7	et	et	PROPN
ejpam-6171	23	8	al	al	PROPN
ejpam-6171	23	9	.	.	PUNCT
ejpam-6171	24	1	[	[	X
ejpam-6171	24	2	5	5	NUM
ejpam-6171	24	3	]	]	PUNCT
ejpam-6171	24	4	defined	define	VERB
ejpam-6171	24	5	a	a	DET
ejpam-6171	24	6	highly	highly	ADV
ejpam-6171	24	7	significant	significant	ADJ
ejpam-6171	24	8	and	and	CCONJ
ejpam-6171	24	9	interesting	interesting	ADJ
ejpam-6171	24	10	concept	concept	NOUN
ejpam-6171	24	11	known	know	VERB
ejpam-6171	24	12	as	as	ADP
ejpam-6171	24	13	the	the	DET
ejpam-6171	24	14	pythagorean	pythagorean	PROPN
ejpam-6171	24	15	neutrosophic	neutrosophic	ADJ
ejpam-6171	24	16	sets	set	NOUN
ejpam-6171	24	17	(	(	PUNCT
ejpam-6171	24	18	pnss	pns	NOUN
ejpam-6171	24	19	)	)	PUNCT
ejpam-6171	24	20	.	.	PUNCT
ejpam-6171	25	1	since	since	SCONJ
ejpam-6171	25	2	the	the	DET
ejpam-6171	25	3	idea	idea	NOUN
ejpam-6171	25	4	of	of	ADP
ejpam-6171	25	5	pythagorean	pythagorean	PROPN
ejpam-6171	25	6	neutrosophic	neutrosophic	ADJ
ejpam-6171	25	7	sets	set	NOUN
ejpam-6171	25	8	was	be	AUX
ejpam-6171	25	9	defined	define	VERB
ejpam-6171	25	10	,	,	PUNCT
ejpam-6171	25	11	many	many	ADJ
ejpam-6171	25	12	researchers	researcher	NOUN
ejpam-6171	25	13	have	have	AUX
ejpam-6171	25	14	continuously	continuously	ADV
ejpam-6171	25	15	studied	study	VERB
ejpam-6171	25	16	and	and	CCONJ
ejpam-6171	25	17	expanded	expand	VERB
ejpam-6171	25	18	upon	upon	SCONJ
ejpam-6171	25	19	this	this	DET
ejpam-6171	25	20	idea	idea	NOUN
ejpam-6171	25	21	.	.	PUNCT
ejpam-6171	26	1	in	in	ADP
ejpam-6171	26	2	2019	2019	NUM
ejpam-6171	26	3	,	,	PUNCT
ejpam-6171	26	4	jansi	jansi	PROPN
ejpam-6171	26	5	et	et	PROPN
ejpam-6171	26	6	al	al	PROPN
ejpam-6171	26	7	.	.	PUNCT
ejpam-6171	27	1	[	[	X
ejpam-6171	27	2	5	5	NUM
ejpam-6171	27	3	]	]	PUNCT
ejpam-6171	27	4	introduced	introduce	VERB
ejpam-6171	27	5	the	the	DET
ejpam-6171	27	6	notion	notion	NOUN
ejpam-6171	27	7	of	of	ADP
ejpam-6171	27	8	exploring	explore	VERB
ejpam-6171	27	9	the	the	DET
ejpam-6171	27	10	new	new	ADJ
ejpam-6171	27	11	concept	concept	NOUN
ejpam-6171	27	12	of	of	ADP
ejpam-6171	27	13	pythagorean	pythagorean	PROPN
ejpam-6171	27	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	27	15	sets	set	NOUN
ejpam-6171	27	16	(	(	PUNCT
ejpam-6171	27	17	pnss	pns	NOUN
ejpam-6171	27	18	)	)	PUNCT
ejpam-6171	27	19	with	with	ADP
ejpam-6171	27	20	t	t	PROPN
ejpam-6171	27	21	and	and	CCONJ
ejpam-6171	27	22	f	f	PROPN
ejpam-6171	27	23	as	as	ADP
ejpam-6171	27	24	dependent	dependent	ADJ
ejpam-6171	27	25	neutrosophic	neutrosophic	ADJ
ejpam-6171	27	26	components	component	NOUN
ejpam-6171	27	27	.	.	PUNCT
ejpam-6171	28	1	they	they	PRON
ejpam-6171	28	2	introduced	introduce	VERB
ejpam-6171	28	3	pnss	pns	NOUN
ejpam-6171	28	4	as	as	ADP
ejpam-6171	28	5	a	a	DET
ejpam-6171	28	6	generalization	generalization	NOUN
ejpam-6171	28	7	of	of	ADP
ejpam-6171	28	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	28	9	sets	set	NOUN
ejpam-6171	28	10	and	and	CCONJ
ejpam-6171	28	11	pythagorean	pythagorean	VERB
ejpam-6171	28	12	fuzzy	fuzzy	ADJ
ejpam-6171	28	13	sets	set	NOUN
ejpam-6171	28	14	.	.	PUNCT
ejpam-6171	29	1	they	they	PRON
ejpam-6171	29	2	investigated	investigate	VERB
ejpam-6171	29	3	the	the	DET
ejpam-6171	29	4	basic	basic	ADJ
ejpam-6171	29	5	operations	operation	NOUN
ejpam-6171	29	6	of	of	ADP
ejpam-6171	29	7	pnss	pns	NOUN
ejpam-6171	29	8	and	and	CCONJ
ejpam-6171	29	9	proposed	propose	VERB
ejpam-6171	29	10	a	a	DET
ejpam-6171	29	11	correlation	correlation	NOUN
ejpam-6171	29	12	measure	measure	NOUN
ejpam-6171	29	13	for	for	ADP
ejpam-6171	29	14	pnss	pns	NOUN
ejpam-6171	29	15	,	,	PUNCT
ejpam-6171	29	16	proving	prove	VERB
ejpam-6171	29	17	some	some	PRON
ejpam-6171	29	18	of	of	ADP
ejpam-6171	29	19	their	their	PRON
ejpam-6171	29	20	fundamental	fundamental	ADJ
ejpam-6171	29	21	properties	property	NOUN
ejpam-6171	29	22	.	.	PUNCT
ejpam-6171	30	1	they	they	PRON
ejpam-6171	30	2	extended	extend	VERB
ejpam-6171	30	3	the	the	DET
ejpam-6171	30	4	concept	concept	NOUN
ejpam-6171	30	5	of	of	ADP
ejpam-6171	30	6	correlation	correlation	NOUN
ejpam-6171	30	7	measures	measure	NOUN
ejpam-6171	30	8	from	from	ADP
ejpam-6171	30	9	pythagorean	pythagorean	PROPN
ejpam-6171	30	10	fuzzy	fuzzy	ADJ
ejpam-6171	30	11	sets	set	NOUN
ejpam-6171	30	12	and	and	CCONJ
ejpam-6171	30	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	30	14	sets	set	NOUN
ejpam-6171	30	15	.	.	PUNCT
ejpam-6171	31	1	finally	finally	ADV
ejpam-6171	31	2	,	,	PUNCT
ejpam-6171	31	3	they	they	PRON
ejpam-6171	31	4	applied	apply	VERB
ejpam-6171	31	5	the	the	DET
ejpam-6171	31	6	correlation	correlation	NOUN
ejpam-6171	31	7	measure	measure	NOUN
ejpam-6171	31	8	of	of	ADP
ejpam-6171	31	9	pnss	pns	NOUN
ejpam-6171	31	10	to	to	ADP
ejpam-6171	31	11	medical	medical	ADJ
ejpam-6171	31	12	diagnosis	diagnosis	NOUN
ejpam-6171	31	13	.	.	PUNCT
ejpam-6171	32	1	in	in	ADP
ejpam-6171	32	2	2021	2021	NUM
ejpam-6171	32	3	,	,	PUNCT
ejpam-6171	32	4	satirad	satirad	PROPN
ejpam-6171	32	5	et	et	PROPN
ejpam-6171	32	6	al	al	PROPN
ejpam-6171	32	7	.	.	PUNCT
ejpam-6171	33	1	[	[	X
ejpam-6171	33	2	6	6	NUM
ejpam-6171	33	3	]	]	PUNCT
ejpam-6171	33	4	extended	extend	VERB
ejpam-6171	33	5	the	the	DET
ejpam-6171	33	6	framework	framework	NOUN
ejpam-6171	33	7	of	of	ADP
ejpam-6171	33	8	up	up	ADV
ejpam-6171	33	9	-	-	PUNCT
ejpam-6171	33	10	algebras	algebra	VERB
ejpam-6171	33	11	by	by	ADP
ejpam-6171	33	12	incorporating	incorporate	VERB
ejpam-6171	33	13	pythagorean	pythagorean	PROPN
ejpam-6171	33	14	fuzzy	fuzzy	ADJ
ejpam-6171	33	15	sets	set	NOUN
ejpam-6171	33	16	to	to	PART
ejpam-6171	33	17	address	address	VERB
ejpam-6171	33	18	uncertainty	uncertainty	NOUN
ejpam-6171	33	19	in	in	ADP
ejpam-6171	33	20	algebraic	algebraic	ADJ
ejpam-6171	33	21	reasoning	reasoning	NOUN
ejpam-6171	33	22	.	.	PUNCT
ejpam-6171	34	1	they	they	PRON
ejpam-6171	34	2	define	define	VERB
ejpam-6171	34	3	lower	low	ADJ
ejpam-6171	34	4	and	and	CCONJ
ejpam-6171	34	5	upper	upper	ADJ
ejpam-6171	34	6	approximations	approximation	NOUN
ejpam-6171	34	7	of	of	ADP
ejpam-6171	34	8	pythagorean	pythagorean	PROPN
ejpam-6171	34	9	fuzzy	fuzzy	ADJ
ejpam-6171	34	10	sets	set	NOUN
ejpam-6171	34	11	within	within	ADP
ejpam-6171	34	12	up	up	ADV
ejpam-6171	34	13	-	-	PUNCT
ejpam-6171	34	14	algebras	algebra	NOUN
ejpam-6171	34	15	and	and	CCONJ
ejpam-6171	34	16	explore	explore	VERB
ejpam-6171	34	17	their	their	PRON
ejpam-6171	34	18	structural	structural	ADJ
ejpam-6171	34	19	properties	property	NOUN
ejpam-6171	34	20	.	.	PUNCT
ejpam-6171	35	1	the	the	DET
ejpam-6171	35	2	study	study	NOUN
ejpam-6171	35	3	enriches	enrich	VERB
ejpam-6171	35	4	fuzzy	fuzzy	ADJ
ejpam-6171	35	5	algebraic	algebraic	ADJ
ejpam-6171	35	6	theory	theory	NOUN
ejpam-6171	35	7	by	by	ADP
ejpam-6171	35	8	offering	offer	VERB
ejpam-6171	35	9	a	a	DET
ejpam-6171	35	10	more	more	ADV
ejpam-6171	35	11	flexible	flexible	ADJ
ejpam-6171	35	12	approximation	approximation	NOUN
ejpam-6171	35	13	model	model	NOUN
ejpam-6171	35	14	for	for	ADP
ejpam-6171	35	15	managing	manage	VERB
ejpam-6171	35	16	vague	vague	ADJ
ejpam-6171	35	17	or	or	CCONJ
ejpam-6171	35	18	partial	partial	ADJ
ejpam-6171	35	19	information	information	NOUN
ejpam-6171	35	20	.	.	PUNCT
ejpam-6171	36	1	in	in	ADP
ejpam-6171	36	2	2023	2023	NUM
ejpam-6171	36	3	,	,	PUNCT
ejpam-6171	36	4	ismail	ismail	NOUN
ejpam-6171	36	5	et	et	PROPN
ejpam-6171	36	6	al	al	PROPN
ejpam-6171	36	7	.	.	PUNCT
ejpam-6171	37	1	[	[	X
ejpam-6171	37	2	7	7	X
ejpam-6171	37	3	]	]	PUNCT
ejpam-6171	37	4	put	put	VERB
ejpam-6171	37	5	forth	forth	ADP
ejpam-6171	37	6	a	a	DET
ejpam-6171	37	7	set	set	NOUN
ejpam-6171	37	8	of	of	ADP
ejpam-6171	37	9	algebraic	algebraic	ADJ
ejpam-6171	37	10	operations	operation	NOUN
ejpam-6171	37	11	that	that	PRON
ejpam-6171	37	12	could	could	AUX
ejpam-6171	37	13	be	be	AUX
ejpam-6171	37	14	applied	apply	VERB
ejpam-6171	37	15	to	to	ADP
ejpam-6171	37	16	pnss	pns	NOUN
ejpam-6171	37	17	.	.	PUNCT
ejpam-6171	38	1	these	these	DET
ejpam-6171	38	2	operations	operation	NOUN
ejpam-6171	38	3	included	include	VERB
ejpam-6171	38	4	addition	addition	NOUN
ejpam-6171	38	5	,	,	PUNCT
ejpam-6171	38	6	multiplication	multiplication	NOUN
ejpam-6171	38	7	,	,	PUNCT
ejpam-6171	38	8	scalar	scalar	ADJ
ejpam-6171	38	9	multiplication	multiplication	NOUN
ejpam-6171	38	10	,	,	PUNCT
ejpam-6171	38	11	and	and	CCONJ
ejpam-6171	38	12	power	power	NOUN
ejpam-6171	38	13	.	.	PUNCT
ejpam-6171	39	1	these	these	DET
ejpam-6171	39	2	operations	operation	NOUN
ejpam-6171	39	3	facilitated	facilitate	VERB
ejpam-6171	39	4	the	the	DET
ejpam-6171	39	5	efficient	efficient	ADJ
ejpam-6171	39	6	manipulation	manipulation	NOUN
ejpam-6171	39	7	and	and	CCONJ
ejpam-6171	39	8	combination	combination	NOUN
ejpam-6171	39	9	of	of	ADP
ejpam-6171	39	10	pnss	pns	NOUN
ejpam-6171	39	11	,	,	PUNCT
ejpam-6171	39	12	thereby	thereby	ADV
ejpam-6171	39	13	enhancing	enhance	VERB
ejpam-6171	39	14	decision	decision	NOUN
ejpam-6171	39	15	-	-	PUNCT
ejpam-6171	39	16	making	making	NOUN
ejpam-6171	39	17	in	in	ADP
ejpam-6171	39	18	scenarios	scenario	NOUN
ejpam-6171	39	19	characterized	characterize	VERB
ejpam-6171	39	20	by	by	ADP
ejpam-6171	39	21	uncertainty	uncertainty	NOUN
ejpam-6171	39	22	and	and	CCONJ
ejpam-6171	39	23	vagueness	vagueness	NOUN
ejpam-6171	39	24	.	.	PUNCT
ejpam-6171	40	1	to	to	PART
ejpam-6171	40	2	demonstrate	demonstrate	VERB
ejpam-6171	40	3	the	the	DET
ejpam-6171	40	4	efficacy	efficacy	NOUN
ejpam-6171	40	5	of	of	ADP
ejpam-6171	40	6	these	these	DET
ejpam-6171	40	7	operations	operation	NOUN
ejpam-6171	40	8	,	,	PUNCT
ejpam-6171	40	9	they	they	PRON
ejpam-6171	40	10	presented	present	VERB
ejpam-6171	40	11	several	several	ADJ
ejpam-6171	40	12	illustrative	illustrative	ADJ
ejpam-6171	40	13	examples	example	NOUN
ejpam-6171	40	14	accompanied	accompany	VERB
ejpam-6171	40	15	by	by	ADP
ejpam-6171	40	16	corroborating	corroborate	VERB
ejpam-6171	40	17	proofs	proof	NOUN
ejpam-6171	40	18	.	.	PUNCT
ejpam-6171	41	1	the	the	DET
ejpam-6171	41	2	introduction	introduction	NOUN
ejpam-6171	41	3	of	of	ADP
ejpam-6171	41	4	algebraic	algebraic	ADJ
ejpam-6171	41	5	operations	operation	NOUN
ejpam-6171	41	6	enhanced	enhance	VERB
ejpam-6171	41	7	the	the	DET
ejpam-6171	41	8	capabilities	capability	NOUN
ejpam-6171	41	9	of	of	ADP
ejpam-6171	41	10	pnss	pns	NOUN
ejpam-6171	41	11	,	,	PUNCT
ejpam-6171	41	12	thereby	thereby	ADV
ejpam-6171	41	13	creating	create	VERB
ejpam-6171	41	14	opportunities	opportunity	NOUN
ejpam-6171	41	15	for	for	ADP
ejpam-6171	41	16	their	their	PRON
ejpam-6171	41	17	practical	practical	ADJ
ejpam-6171	41	18	application	application	NOUN
ejpam-6171	41	19	.	.	PUNCT
ejpam-6171	42	1	in	in	ADP
ejpam-6171	42	2	2024	2024	NUM
ejpam-6171	42	3	,	,	PUNCT
ejpam-6171	42	4	razak	razak	PROPN
ejpam-6171	42	5	et	et	PROPN
ejpam-6171	42	6	al	al	PROPN
ejpam-6171	42	7	.	.	PUNCT
ejpam-6171	43	1	[	[	X
ejpam-6171	43	2	8	8	NUM
ejpam-6171	43	3	]	]	X
ejpam-6171	43	4	propose	propose	VERB
ejpam-6171	43	5	the	the	DET
ejpam-6171	43	6	interval	interval	NOUN
ejpam-6171	43	7	valued	value	VERB
ejpam-6171	43	8	pythagorean	pythagorean	PROPN
ejpam-6171	43	9	neutrosophic	neutrosophic	PROPN
ejpam-6171	43	10	set	set	NOUN
ejpam-6171	43	11	(	(	PUNCT
ejpam-6171	43	12	ivpns	ivpns	NOUN
ejpam-6171	43	13	)	)	PUNCT
ejpam-6171	43	14	to	to	PART
ejpam-6171	43	15	better	well	ADV
ejpam-6171	43	16	capture	capture	VERB
ejpam-6171	43	17	uncertainty	uncertainty	NOUN
ejpam-6171	43	18	and	and	CCONJ
ejpam-6171	43	19	imprecision	imprecision	NOUN
ejpam-6171	43	20	in	in	ADP
ejpam-6171	43	21	real	real	ADJ
ejpam-6171	43	22	-	-	PUNCT
ejpam-6171	43	23	world	world	NOUN
ejpam-6171	43	24	data	datum	NOUN
ejpam-6171	43	25	.	.	PUNCT
ejpam-6171	44	1	they	they	PRON
ejpam-6171	44	2	define	define	VERB
ejpam-6171	44	3	key	key	ADJ
ejpam-6171	44	4	algebraic	algebraic	ADJ
ejpam-6171	44	5	operations	operation	NOUN
ejpam-6171	44	6	and	and	CCONJ
ejpam-6171	44	7	validate	validate	VERB
ejpam-6171	44	8	their	their	PRON
ejpam-6171	44	9	properties	property	NOUN
ejpam-6171	44	10	,	,	PUNCT
ejpam-6171	44	11	offering	offer	VERB
ejpam-6171	44	12	a	a	DET
ejpam-6171	44	13	comparative	comparative	ADJ
ejpam-6171	44	14	analysis	analysis	NOUN
ejpam-6171	44	15	with	with	ADP
ejpam-6171	44	16	related	related	ADJ
ejpam-6171	44	17	set	set	NOUN
ejpam-6171	44	18	models	model	NOUN
ejpam-6171	44	19	.	.	PUNCT
ejpam-6171	45	1	this	this	DET
ejpam-6171	45	2	framework	framework	NOUN
ejpam-6171	45	3	enhances	enhance	VERB
ejpam-6171	45	4	the	the	DET
ejpam-6171	45	5	robustness	robustness	NOUN
ejpam-6171	45	6	of	of	ADP
ejpam-6171	45	7	decision	decision	NOUN
ejpam-6171	45	8	-	-	PUNCT
ejpam-6171	45	9	making	making	NOUN
ejpam-6171	45	10	and	and	CCONJ
ejpam-6171	45	11	modeling	modeling	NOUN
ejpam-6171	45	12	under	under	ADP
ejpam-6171	45	13	complex	complex	ADJ
ejpam-6171	45	14	uncertainty	uncertainty	NOUN
ejpam-6171	45	15	.	.	PUNCT
ejpam-6171	46	1	in	in	ADP
ejpam-6171	46	2	2022	2022	NUM
ejpam-6171	46	3	,	,	PUNCT
ejpam-6171	46	4	iampan	iampan	NOUN
ejpam-6171	46	5	et	et	PROPN
ejpam-6171	46	6	al	al	PROPN
ejpam-6171	46	7	.	.	PUNCT
ejpam-6171	47	1	[	[	X
ejpam-6171	47	2	9	9	NUM
ejpam-6171	47	3	]	]	PUNCT
ejpam-6171	47	4	defined	define	VERB
ejpam-6171	47	5	a	a	DET
ejpam-6171	47	6	new	new	ADJ
ejpam-6171	47	7	concept	concept	NOUN
ejpam-6171	47	8	called	call	VERB
ejpam-6171	47	9	iup	iup	NOUN
ejpam-6171	47	10	-	-	PUNCT
ejpam-6171	47	11	algebras	algebra	NOUN
ejpam-6171	47	12	,	,	PUNCT
ejpam-6171	47	13	which	which	PRON
ejpam-6171	47	14	is	be	AUX
ejpam-6171	47	15	an	an	DET
ejpam-6171	47	16	algebraic	algebraic	ADJ
ejpam-6171	47	17	structure	structure	NOUN
ejpam-6171	47	18	with	with	ADP
ejpam-6171	47	19	four	four	NUM
ejpam-6171	47	20	special	special	ADJ
ejpam-6171	47	21	subsets	subset	NOUN
ejpam-6171	47	22	:	:	PUNCT
ejpam-6171	47	23	iup	iup	PROPN
ejpam-6171	47	24	-	-	PUNCT
ejpam-6171	47	25	subalgebras	subalgebras	PROPN
ejpam-6171	47	26	,	,	PUNCT
ejpam-6171	47	27	iup	iup	NOUN
ejpam-6171	47	28	-	-	PUNCT
ejpam-6171	47	29	filters	filter	NOUN
ejpam-6171	47	30	,	,	PUNCT
ejpam-6171	47	31	iup	iup	NOUN
ejpam-6171	47	32	-	-	PUNCT
ejpam-6171	47	33	ideals	ideal	NOUN
ejpam-6171	47	34	,	,	PUNCT
ejpam-6171	47	35	and	and	CCONJ
ejpam-6171	47	36	strong	strong	ADJ
ejpam-6171	47	37	iup	iup	NOUN
ejpam-6171	47	38	-	-	PUNCT
ejpam-6171	47	39	ideals	ideal	NOUN
ejpam-6171	47	40	.	.	PUNCT
ejpam-6171	48	1	they	they	PRON
ejpam-6171	48	2	also	also	ADV
ejpam-6171	48	3	discovered	discover	VERB
ejpam-6171	48	4	additional	additional	ADJ
ejpam-6171	48	5	properties	property	NOUN
ejpam-6171	48	6	,	,	PUNCT
ejpam-6171	48	7	which	which	PRON
ejpam-6171	48	8	have	have	AUX
ejpam-6171	48	9	inspired	inspire	VERB
ejpam-6171	48	10	many	many	ADJ
ejpam-6171	48	11	researchers	researcher	NOUN
ejpam-6171	48	12	to	to	PART
ejpam-6171	48	13	study	study	VERB
ejpam-6171	48	14	and	and	CCONJ
ejpam-6171	48	15	expand	expand	VERB
ejpam-6171	48	16	this	this	DET
ejpam-6171	48	17	knowledge	knowledge	NOUN
ejpam-6171	48	18	further	far	ADV
ejpam-6171	48	19	.	.	PUNCT
ejpam-6171	49	1	this	this	DET
ejpam-6171	49	2	concept	concept	NOUN
ejpam-6171	49	3	has	have	AUX
ejpam-6171	49	4	sparked	spark	VERB
ejpam-6171	49	5	inspiration	inspiration	NOUN
ejpam-6171	49	6	among	among	ADP
ejpam-6171	49	7	mathematicians	mathematician	NOUN
ejpam-6171	49	8	to	to	PART
ejpam-6171	49	9	explore	explore	VERB
ejpam-6171	49	10	this	this	DET
ejpam-6171	49	11	algebraic	algebraic	ADJ
ejpam-6171	49	12	structure	structure	NOUN
ejpam-6171	49	13	more	more	ADV
ejpam-6171	49	14	deeply	deeply	ADV
ejpam-6171	49	15	,	,	PUNCT
ejpam-6171	49	16	such	such	ADJ
ejpam-6171	49	17	as	as	ADP
ejpam-6171	49	18	in	in	ADP
ejpam-6171	49	19	2023	2023	NUM
ejpam-6171	49	20	,	,	PUNCT
ejpam-6171	49	21	when	when	SCONJ
ejpam-6171	49	22	chanmanee	chanmanee	PROPN
ejpam-6171	49	23	et	et	PROPN
ejpam-6171	49	24	al	al	PROPN
ejpam-6171	49	25	.	.	PUNCT
ejpam-6171	50	1	[	[	X
ejpam-6171	50	2	10	10	NUM
ejpam-6171	50	3	]	]	PUNCT
ejpam-6171	50	4	introduced	introduce	VERB
ejpam-6171	50	5	the	the	DET
ejpam-6171	50	6	concept	concept	NOUN
ejpam-6171	50	7	of	of	ADP
ejpam-6171	50	8	the	the	DET
ejpam-6171	50	9	direct	direct	ADJ
ejpam-6171	50	10	product	product	NOUN
ejpam-6171	50	11	of	of	ADP
ejpam-6171	50	12	infinite	infinite	ADJ
ejpam-6171	50	13	families	family	NOUN
ejpam-6171	50	14	of	of	ADP
ejpam-6171	50	15	iup	iup	NOUN
ejpam-6171	50	16	-	-	PUNCT
ejpam-6171	50	17	algebras	algebras	PROPN
ejpam-6171	50	18	.	.	PUNCT
ejpam-6171	51	1	their	their	PRON
ejpam-6171	51	2	research	research	NOUN
ejpam-6171	51	3	unveiled	unveil	VERB
ejpam-6171	51	4	the	the	DET
ejpam-6171	51	5	notion	notion	NOUN
ejpam-6171	51	6	of	of	ADP
ejpam-6171	51	7	weak	weak	ADJ
ejpam-6171	51	8	direct	direct	ADJ
ejpam-6171	51	9	k.	k.	PROPN
ejpam-6171	51	10	suayngam	suayngam	PROPN
ejpam-6171	51	11	et	et	PROPN
ejpam-6171	51	12	al	al	PROPN
ejpam-6171	51	13	.	.	PUNCT
ejpam-6171	51	14	/	/	SYM
ejpam-6171	51	15	eur	eur	PROPN
ejpam-6171	51	16	.	.	PUNCT
ejpam-6171	52	1	j.	j.	PROPN
ejpam-6171	52	2	pure	pure	PROPN
ejpam-6171	52	3	appl	appl	PROPN
ejpam-6171	52	4	.	.	PROPN
ejpam-6171	52	5	math	math	PROPN
ejpam-6171	52	6	,	,	PUNCT
ejpam-6171	52	7	18	18	NUM
ejpam-6171	52	8	(	(	PUNCT
ejpam-6171	52	9	3	3	NUM
ejpam-6171	52	10	)	)	PUNCT
ejpam-6171	52	11	(	(	PUNCT
ejpam-6171	52	12	2025	2025	NUM
ejpam-6171	52	13	)	)	PUNCT
ejpam-6171	52	14	,	,	PUNCT
ejpam-6171	52	15	6171	6171	NUM
ejpam-6171	52	16	3	3	NUM
ejpam-6171	52	17	of	of	ADP
ejpam-6171	52	18	28	28	NUM
ejpam-6171	52	19	products	product	NOUN
ejpam-6171	52	20	and	and	CCONJ
ejpam-6171	52	21	presented	present	VERB
ejpam-6171	52	22	pivotal	pivotal	ADJ
ejpam-6171	52	23	findings	finding	NOUN
ejpam-6171	52	24	regarding	regard	VERB
ejpam-6171	52	25	(	(	PUNCT
ejpam-6171	52	26	anti-)iup	anti-)iup	ADJ
ejpam-6171	52	27	-	-	PUNCT
ejpam-6171	52	28	homomorphisms	homomorphism	NOUN
ejpam-6171	52	29	within	within	ADP
ejpam-6171	52	30	this	this	DET
ejpam-6171	52	31	context	context	NOUN
ejpam-6171	52	32	.	.	PUNCT
ejpam-6171	53	1	these	these	DET
ejpam-6171	53	2	contributions	contribution	NOUN
ejpam-6171	53	3	significantly	significantly	ADV
ejpam-6171	53	4	deepened	deepen	VERB
ejpam-6171	53	5	the	the	DET
ejpam-6171	53	6	structural	structural	ADJ
ejpam-6171	53	7	comprehension	comprehension	NOUN
ejpam-6171	53	8	of	of	ADP
ejpam-6171	53	9	iupalgebras	iupalgebra	NOUN
ejpam-6171	53	10	and	and	CCONJ
ejpam-6171	53	11	established	establish	VERB
ejpam-6171	53	12	essential	essential	ADJ
ejpam-6171	53	13	tools	tool	NOUN
ejpam-6171	53	14	for	for	ADP
ejpam-6171	53	15	further	further	ADJ
ejpam-6171	53	16	exploration	exploration	NOUN
ejpam-6171	53	17	of	of	ADP
ejpam-6171	53	18	their	their	PRON
ejpam-6171	53	19	properties	property	NOUN
ejpam-6171	53	20	.	.	PUNCT
ejpam-6171	54	1	in	in	ADP
ejpam-6171	54	2	2024	2024	NUM
ejpam-6171	54	3	,	,	PUNCT
ejpam-6171	54	4	kuntama	kuntama	NOUN
ejpam-6171	54	5	et	et	NOUN
ejpam-6171	54	6	al	al	PROPN
ejpam-6171	54	7	.	.	PUNCT
ejpam-6171	55	1	[	[	X
ejpam-6171	55	2	11	11	NUM
ejpam-6171	55	3	]	]	PUNCT
ejpam-6171	55	4	made	make	VERB
ejpam-6171	55	5	substantial	substantial	ADJ
ejpam-6171	55	6	advancements	advancement	NOUN
ejpam-6171	55	7	by	by	ADP
ejpam-6171	55	8	integrating	integrate	VERB
ejpam-6171	55	9	fs	fs	ADP
ejpam-6171	55	10	theory	theory	NOUN
ejpam-6171	55	11	into	into	ADP
ejpam-6171	55	12	iupalgebras	iupalgebra	NOUN
ejpam-6171	55	13	.	.	PUNCT
ejpam-6171	56	1	they	they	PRON
ejpam-6171	56	2	introduced	introduce	VERB
ejpam-6171	56	3	fuzzy	fuzzy	ADJ
ejpam-6171	56	4	iup	iup	NOUN
ejpam-6171	56	5	-	-	PUNCT
ejpam-6171	56	6	subalgebras	subalgebras	PROPN
ejpam-6171	56	7	,	,	PUNCT
ejpam-6171	56	8	fuzzy	fuzzy	ADJ
ejpam-6171	56	9	iup	iup	NOUN
ejpam-6171	56	10	-	-	PUNCT
ejpam-6171	56	11	ideals	ideal	NOUN
ejpam-6171	56	12	,	,	PUNCT
ejpam-6171	56	13	fuzzy	fuzzy	ADJ
ejpam-6171	56	14	iup	iup	NOUN
ejpam-6171	56	15	-	-	PUNCT
ejpam-6171	56	16	filters	filter	NOUN
ejpam-6171	56	17	,	,	PUNCT
ejpam-6171	56	18	and	and	CCONJ
ejpam-6171	56	19	fuzzy	fuzzy	ADJ
ejpam-6171	56	20	strong	strong	ADJ
ejpam-6171	56	21	iup	iup	NOUN
ejpam-6171	56	22	-	-	PUNCT
ejpam-6171	56	23	ideals	ideal	NOUN
ejpam-6171	56	24	,	,	PUNCT
ejpam-6171	56	25	meticulously	meticulously	ADV
ejpam-6171	56	26	examining	examine	VERB
ejpam-6171	56	27	the	the	DET
ejpam-6171	56	28	properties	property	NOUN
ejpam-6171	56	29	and	and	CCONJ
ejpam-6171	56	30	interactions	interaction	NOUN
ejpam-6171	56	31	of	of	ADP
ejpam-6171	56	32	these	these	DET
ejpam-6171	56	33	subsets	subset	NOUN
ejpam-6171	56	34	.	.	PUNCT
ejpam-6171	57	1	suayngam	suayngam	INTJ
ejpam-6171	57	2	et	et	PROPN
ejpam-6171	57	3	al	al	PROPN
ejpam-6171	57	4	.	.	PUNCT
ejpam-6171	58	1	[	[	X
ejpam-6171	58	2	12	12	NUM
ejpam-6171	58	3	]	]	PUNCT
ejpam-6171	58	4	applied	apply	VERB
ejpam-6171	58	5	fermatean	fermatean	NOUN
ejpam-6171	58	6	fuzzy	fuzzy	ADJ
ejpam-6171	58	7	sets	set	NOUN
ejpam-6171	58	8	to	to	PART
ejpam-6171	58	9	iup	iup	VERB
ejpam-6171	58	10	-	-	PUNCT
ejpam-6171	58	11	algebras	algebras	X
ejpam-6171	58	12	,	,	PUNCT
ejpam-6171	58	13	focusing	focus	VERB
ejpam-6171	58	14	on	on	ADP
ejpam-6171	58	15	fermatean	fermatean	ADJ
ejpam-6171	58	16	fuzzy	fuzzy	ADJ
ejpam-6171	58	17	iup	iup	PROPN
ejpam-6171	58	18	-	-	PUNCT
ejpam-6171	58	19	subalgebras	subalgebras	PROPN
ejpam-6171	58	20	,	,	PUNCT
ejpam-6171	58	21	iup	iup	NOUN
ejpam-6171	58	22	-	-	PUNCT
ejpam-6171	58	23	ideals	ideal	NOUN
ejpam-6171	58	24	,	,	PUNCT
ejpam-6171	58	25	iup	iup	NOUN
ejpam-6171	58	26	-	-	PUNCT
ejpam-6171	58	27	filters	filter	NOUN
ejpam-6171	58	28	,	,	PUNCT
ejpam-6171	58	29	and	and	CCONJ
ejpam-6171	58	30	strong	strong	ADJ
ejpam-6171	58	31	iup	iup	NOUN
ejpam-6171	58	32	-	-	PUNCT
ejpam-6171	58	33	ideals	ideal	NOUN
ejpam-6171	58	34	.	.	PUNCT
ejpam-6171	59	1	they	they	PRON
ejpam-6171	59	2	examined	examine	VERB
ejpam-6171	59	3	their	their	PRON
ejpam-6171	59	4	properties	property	NOUN
ejpam-6171	59	5	,	,	PUNCT
ejpam-6171	59	6	including	include	VERB
ejpam-6171	59	7	characteristic	characteristic	ADJ
ejpam-6171	59	8	fermatean	fermatean	NOUN
ejpam-6171	59	9	fuzzy	fuzzy	ADJ
ejpam-6171	59	10	sets	set	NOUN
ejpam-6171	59	11	and	and	CCONJ
ejpam-6171	59	12	upper	upper	ADJ
ejpam-6171	59	13	and	and	CCONJ
ejpam-6171	59	14	lower	low	ADJ
ejpam-6171	59	15	t-(strong	t-(strong	NOUN
ejpam-6171	59	16	)	)	PUNCT
ejpam-6171	59	17	level	level	NOUN
ejpam-6171	59	18	subsets	subset	NOUN
ejpam-6171	59	19	,	,	PUNCT
ejpam-6171	59	20	offering	offer	VERB
ejpam-6171	59	21	deeper	deep	ADJ
ejpam-6171	59	22	insights	insight	NOUN
ejpam-6171	59	23	into	into	ADP
ejpam-6171	59	24	their	their	PRON
ejpam-6171	59	25	structural	structural	ADJ
ejpam-6171	59	26	relationships	relationship	NOUN
ejpam-6171	59	27	.	.	PUNCT
ejpam-6171	60	1	this	this	DET
ejpam-6171	60	2	research	research	NOUN
ejpam-6171	60	3	expanded	expand	VERB
ejpam-6171	60	4	the	the	DET
ejpam-6171	60	5	applicability	applicability	NOUN
ejpam-6171	60	6	of	of	ADP
ejpam-6171	60	7	iup	iup	NOUN
ejpam-6171	60	8	-	-	PUNCT
ejpam-6171	60	9	algebras	algebras	PROPN
ejpam-6171	60	10	,	,	PUNCT
ejpam-6171	60	11	offering	offer	VERB
ejpam-6171	60	12	fresh	fresh	ADJ
ejpam-6171	60	13	perspectives	perspective	NOUN
ejpam-6171	60	14	and	and	CCONJ
ejpam-6171	60	15	mathematical	mathematical	ADJ
ejpam-6171	60	16	tools	tool	NOUN
ejpam-6171	60	17	that	that	PRON
ejpam-6171	60	18	bridge	bridge	VERB
ejpam-6171	60	19	algebraic	algebraic	ADJ
ejpam-6171	60	20	structures	structure	NOUN
ejpam-6171	60	21	with	with	ADP
ejpam-6171	60	22	fuzzy	fuzzy	ADJ
ejpam-6171	60	23	logic	logic	NOUN
ejpam-6171	60	24	.	.	PUNCT
ejpam-6171	61	1	further	far	ADV
ejpam-6171	61	2	enriching	enrich	VERB
ejpam-6171	61	3	this	this	DET
ejpam-6171	61	4	theoretical	theoretical	ADJ
ejpam-6171	61	5	framework	framework	NOUN
ejpam-6171	61	6	,	,	PUNCT
ejpam-6171	61	7	suayngam	suayngam	NOUN
ejpam-6171	61	8	et	et	PROPN
ejpam-6171	61	9	al	al	PROPN
ejpam-6171	61	10	.	.	PUNCT
ejpam-6171	62	1	[	[	X
ejpam-6171	62	2	13	13	NUM
ejpam-6171	62	3	]	]	PUNCT
ejpam-6171	62	4	introduced	introduce	VERB
ejpam-6171	62	5	the	the	DET
ejpam-6171	62	6	concept	concept	NOUN
ejpam-6171	62	7	of	of	ADP
ejpam-6171	62	8	intuitionistic	intuitionistic	ADJ
ejpam-6171	62	9	fuzzy	fuzzy	ADJ
ejpam-6171	62	10	iup	iup	NOUN
ejpam-6171	62	11	-	-	PUNCT
ejpam-6171	62	12	algebras	algebras	PROPN
ejpam-6171	62	13	in	in	ADP
ejpam-6171	62	14	2024	2024	NUM
ejpam-6171	62	15	.	.	PUNCT
ejpam-6171	63	1	their	their	PRON
ejpam-6171	63	2	work	work	NOUN
ejpam-6171	63	3	amalgamated	amalgamate	VERB
ejpam-6171	63	4	ifs	ifs	PROPN
ejpam-6171	63	5	theory	theory	NOUN
ejpam-6171	63	6	with	with	ADP
ejpam-6171	63	7	iup	iup	NOUN
ejpam-6171	63	8	-	-	PUNCT
ejpam-6171	63	9	algebras	algebras	PROPN
ejpam-6171	63	10	,	,	PUNCT
ejpam-6171	63	11	leading	lead	VERB
ejpam-6171	63	12	to	to	ADP
ejpam-6171	63	13	the	the	DET
ejpam-6171	63	14	development	development	NOUN
ejpam-6171	63	15	of	of	ADP
ejpam-6171	63	16	intuitionistic	intuitionistic	ADJ
ejpam-6171	63	17	fuzzy	fuzzy	ADJ
ejpam-6171	63	18	iup	iup	NOUN
ejpam-6171	63	19	-	-	PUNCT
ejpam-6171	63	20	subalgebras	subalgebras	PROPN
ejpam-6171	63	21	,	,	PUNCT
ejpam-6171	63	22	ideals	ideal	NOUN
ejpam-6171	63	23	,	,	PUNCT
ejpam-6171	63	24	filters	filter	NOUN
ejpam-6171	63	25	,	,	PUNCT
ejpam-6171	63	26	and	and	CCONJ
ejpam-6171	63	27	strong	strong	ADJ
ejpam-6171	63	28	ideals	ideal	NOUN
ejpam-6171	63	29	.	.	PUNCT
ejpam-6171	64	1	in	in	ADP
ejpam-6171	64	2	2025	2025	NUM
ejpam-6171	64	3	,	,	PUNCT
ejpam-6171	64	4	suayngam	suayngam	PROPN
ejpam-6171	64	5	et	et	PROPN
ejpam-6171	64	6	al	al	PROPN
ejpam-6171	64	7	.	.	PUNCT
ejpam-6171	65	1	[	[	X
ejpam-6171	65	2	14	14	NUM
ejpam-6171	65	3	]	]	PUNCT
ejpam-6171	65	4	introduced	introduce	VERB
ejpam-6171	65	5	the	the	DET
ejpam-6171	65	6	notions	notion	NOUN
ejpam-6171	65	7	of	of	ADP
ejpam-6171	65	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	65	9	iup	iup	PROPN
ejpam-6171	65	10	-	-	PUNCT
ejpam-6171	65	11	subalgebras	subalgebras	PROPN
ejpam-6171	65	12	,	,	PUNCT
ejpam-6171	65	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	65	14	iup	iup	NOUN
ejpam-6171	65	15	-	-	PUNCT
ejpam-6171	65	16	ideals	ideal	NOUN
ejpam-6171	65	17	,	,	PUNCT
ejpam-6171	65	18	neutrosophic	neutrosophic	ADJ
ejpam-6171	65	19	iup	iup	NOUN
ejpam-6171	65	20	-	-	PUNCT
ejpam-6171	65	21	filters	filter	NOUN
ejpam-6171	65	22	,	,	PUNCT
ejpam-6171	65	23	and	and	CCONJ
ejpam-6171	65	24	neutrosophic	neutrosophic	ADJ
ejpam-6171	65	25	strong	strong	ADJ
ejpam-6171	65	26	iup	iup	NOUN
ejpam-6171	65	27	-	-	PUNCT
ejpam-6171	65	28	ideals	ideal	NOUN
ejpam-6171	65	29	of	of	ADP
ejpam-6171	65	30	iup	iup	NOUN
ejpam-6171	65	31	-	-	PUNCT
ejpam-6171	65	32	algebras	algebras	PROPN
ejpam-6171	65	33	and	and	CCONJ
ejpam-6171	65	34	investigated	investigate	VERB
ejpam-6171	65	35	their	their	PRON
ejpam-6171	65	36	basic	basic	ADJ
ejpam-6171	65	37	properties	property	NOUN
ejpam-6171	65	38	.	.	PUNCT
ejpam-6171	66	1	they	they	PRON
ejpam-6171	66	2	provided	provide	VERB
ejpam-6171	66	3	conditions	condition	NOUN
ejpam-6171	66	4	for	for	ADP
ejpam-6171	66	5	neutrosophic	neutrosophic	ADJ
ejpam-6171	66	6	sets	set	NOUN
ejpam-6171	66	7	to	to	PART
ejpam-6171	66	8	be	be	AUX
ejpam-6171	66	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	66	10	iup	iup	NOUN
ejpam-6171	66	11	-	-	PUNCT
ejpam-6171	66	12	subalgebras	subalgebras	PROPN
ejpam-6171	66	13	,	,	PUNCT
ejpam-6171	66	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	66	15	iup	iup	NOUN
ejpam-6171	66	16	-	-	PUNCT
ejpam-6171	66	17	ideals	ideal	NOUN
ejpam-6171	66	18	,	,	PUNCT
ejpam-6171	66	19	neutrosophic	neutrosophic	ADJ
ejpam-6171	66	20	iup	iup	NOUN
ejpam-6171	66	21	-	-	PUNCT
ejpam-6171	66	22	filters	filter	NOUN
ejpam-6171	66	23	,	,	PUNCT
ejpam-6171	66	24	and	and	CCONJ
ejpam-6171	66	25	neutrosophic	neutrosophic	ADJ
ejpam-6171	66	26	strong	strong	ADJ
ejpam-6171	66	27	iup	iup	NOUN
ejpam-6171	66	28	-	-	PUNCT
ejpam-6171	66	29	ideals	ideal	NOUN
ejpam-6171	66	30	of	of	ADP
ejpam-6171	66	31	iup	iup	NOUN
ejpam-6171	66	32	-	-	PUNCT
ejpam-6171	66	33	algebras	algebras	PROPN
ejpam-6171	66	34	.	.	PUNCT
ejpam-6171	67	1	they	they	PRON
ejpam-6171	67	2	considered	consider	VERB
ejpam-6171	67	3	relations	relation	NOUN
ejpam-6171	67	4	between	between	ADP
ejpam-6171	67	5	neutrosophic	neutrosophic	ADJ
ejpam-6171	67	6	iup	iup	PROPN
ejpam-6171	67	7	-	-	PUNCT
ejpam-6171	67	8	subalgebras	subalgebras	PROPN
ejpam-6171	67	9	(	(	PUNCT
ejpam-6171	67	10	resp	resp	NOUN
ejpam-6171	67	11	.	.	PUNCT
ejpam-6171	67	12	,	,	PUNCT
ejpam-6171	67	13	neutrosophic	neutrosophic	PROPN
ejpam-6171	67	14	iup	iup	NOUN
ejpam-6171	67	15	-	-	PUNCT
ejpam-6171	67	16	ideals	ideal	NOUN
ejpam-6171	67	17	,	,	PUNCT
ejpam-6171	67	18	neutrosophic	neutrosophic	ADJ
ejpam-6171	67	19	iup	iup	NOUN
ejpam-6171	67	20	-	-	PUNCT
ejpam-6171	67	21	filters	filter	NOUN
ejpam-6171	67	22	,	,	PUNCT
ejpam-6171	67	23	neutrosophic	neutrosophic	ADJ
ejpam-6171	67	24	strong	strong	ADJ
ejpam-6171	67	25	iup	iup	NOUN
ejpam-6171	67	26	-	-	PUNCT
ejpam-6171	67	27	ideals	ideal	NOUN
ejpam-6171	67	28	)	)	PUNCT
ejpam-6171	67	29	and	and	CCONJ
ejpam-6171	67	30	their	their	PRON
ejpam-6171	67	31	level	level	NOUN
ejpam-6171	67	32	subsets	subset	NOUN
ejpam-6171	67	33	.	.	PUNCT
ejpam-6171	68	1	suayngam	suayngam	INTJ
ejpam-6171	68	2	et	et	PROPN
ejpam-6171	68	3	al	al	PROPN
ejpam-6171	68	4	.	.	PUNCT
ejpam-6171	69	1	[	[	X
ejpam-6171	69	2	15	15	NUM
ejpam-6171	69	3	]	]	PUNCT
ejpam-6171	69	4	applied	apply	VERB
ejpam-6171	69	5	the	the	DET
ejpam-6171	69	6	concept	concept	NOUN
ejpam-6171	69	7	of	of	ADP
ejpam-6171	69	8	pythagorean	pythagorean	PROPN
ejpam-6171	69	9	fuzzy	fuzzy	ADJ
ejpam-6171	69	10	sets	set	NOUN
ejpam-6171	69	11	to	to	PART
ejpam-6171	69	12	iup	iup	VERB
ejpam-6171	69	13	-	-	PUNCT
ejpam-6171	69	14	algebras	algebras	PROPN
ejpam-6171	69	15	and	and	CCONJ
ejpam-6171	69	16	introduced	introduce	VERB
ejpam-6171	69	17	the	the	DET
ejpam-6171	69	18	notions	notion	NOUN
ejpam-6171	69	19	of	of	ADP
ejpam-6171	69	20	pythagorean	pythagorean	PROPN
ejpam-6171	69	21	fuzzy	fuzzy	PROPN
ejpam-6171	69	22	iup	iup	PROPN
ejpam-6171	69	23	-	-	PUNCT
ejpam-6171	69	24	subalgebras	subalgebras	PROPN
ejpam-6171	69	25	,	,	PUNCT
ejpam-6171	69	26	pythagorean	pythagorean	PROPN
ejpam-6171	69	27	fuzzy	fuzzy	ADJ
ejpam-6171	69	28	iup	iup	NOUN
ejpam-6171	69	29	-	-	PUNCT
ejpam-6171	69	30	ideals	ideal	NOUN
ejpam-6171	69	31	,	,	PUNCT
ejpam-6171	69	32	pythagorean	pythagorean	PROPN
ejpam-6171	69	33	fuzzy	fuzzy	ADJ
ejpam-6171	69	34	iup	iup	NOUN
ejpam-6171	69	35	-	-	PUNCT
ejpam-6171	69	36	filters	filter	NOUN
ejpam-6171	69	37	,	,	PUNCT
ejpam-6171	69	38	and	and	CCONJ
ejpam-6171	69	39	pythagorean	pythagorean	PROPN
ejpam-6171	69	40	fuzzy	fuzzy	ADV
ejpam-6171	69	41	strong	strong	ADJ
ejpam-6171	69	42	iup	iup	NOUN
ejpam-6171	69	43	-	-	PUNCT
ejpam-6171	69	44	ideals	ideal	NOUN
ejpam-6171	69	45	.	.	PUNCT
ejpam-6171	70	1	they	they	PRON
ejpam-6171	70	2	investigated	investigate	VERB
ejpam-6171	70	3	their	their	PRON
ejpam-6171	70	4	properties	property	NOUN
ejpam-6171	70	5	,	,	PUNCT
ejpam-6171	70	6	including	include	VERB
ejpam-6171	70	7	the	the	DET
ejpam-6171	70	8	characteristic	characteristic	ADJ
ejpam-6171	70	9	pythagorean	pythagorean	NOUN
ejpam-6171	70	10	fuzzy	fuzzy	ADJ
ejpam-6171	70	11	sets	set	NOUN
ejpam-6171	70	12	,	,	PUNCT
ejpam-6171	70	13	the	the	DET
ejpam-6171	70	14	upper	upper	ADJ
ejpam-6171	70	15	t-(strong	t-(strong	PROPN
ejpam-6171	70	16	)	)	PUNCT
ejpam-6171	70	17	level	level	NOUN
ejpam-6171	70	18	subsets	subset	NOUN
ejpam-6171	70	19	,	,	PUNCT
ejpam-6171	70	20	and	and	CCONJ
ejpam-6171	70	21	the	the	DET
ejpam-6171	70	22	lower	low	ADJ
ejpam-6171	70	23	t-(strong	t-(strong	NUM
ejpam-6171	70	24	)	)	PUNCT
ejpam-6171	70	25	level	level	NOUN
ejpam-6171	70	26	subsets	subset	NOUN
ejpam-6171	70	27	of	of	ADP
ejpam-6171	70	28	the	the	DET
ejpam-6171	70	29	pythagorean	pythagorean	PROPN
ejpam-6171	70	30	fuzzy	fuzzy	ADJ
ejpam-6171	70	31	set	set	NOUN
ejpam-6171	70	32	.	.	PUNCT
ejpam-6171	71	1	suayngam	suayngam	INTJ
ejpam-6171	71	2	et	et	PROPN
ejpam-6171	71	3	al	al	PROPN
ejpam-6171	71	4	.	.	PUNCT
ejpam-6171	72	1	[	[	X
ejpam-6171	72	2	16	16	NUM
ejpam-6171	72	3	]	]	X
ejpam-6171	72	4	advanced	advance	VERB
ejpam-6171	72	5	the	the	DET
ejpam-6171	72	6	study	study	NOUN
ejpam-6171	72	7	of	of	ADP
ejpam-6171	72	8	iup	iup	NOUN
ejpam-6171	72	9	-	-	PUNCT
ejpam-6171	72	10	algebras	algebras	PROPN
ejpam-6171	72	11	by	by	ADP
ejpam-6171	72	12	incorporating	incorporate	VERB
ejpam-6171	72	13	intuitionistic	intuitionistic	ADJ
ejpam-6171	72	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	72	15	sets	set	NOUN
ejpam-6171	72	16	to	to	PART
ejpam-6171	72	17	formalize	formalize	VERB
ejpam-6171	72	18	algebraic	algebraic	ADJ
ejpam-6171	72	19	reasoning	reasoning	NOUN
ejpam-6171	72	20	under	under	ADP
ejpam-6171	72	21	uncertainty	uncertainty	NOUN
ejpam-6171	72	22	.	.	PUNCT
ejpam-6171	73	1	they	they	PRON
ejpam-6171	73	2	introduced	introduce	VERB
ejpam-6171	73	3	and	and	CCONJ
ejpam-6171	73	4	characterized	characterize	VERB
ejpam-6171	73	5	key	key	ADJ
ejpam-6171	73	6	substructures	substructure	NOUN
ejpam-6171	73	7	—	—	PUNCT
ejpam-6171	73	8	such	such	ADJ
ejpam-6171	73	9	as	as	ADP
ejpam-6171	73	10	iup	iup	NOUN
ejpam-6171	73	11	-	-	PUNCT
ejpam-6171	73	12	subalgebras	subalgebras	PROPN
ejpam-6171	73	13	,	,	PUNCT
ejpam-6171	73	14	ideals	ideal	NOUN
ejpam-6171	73	15	,	,	PUNCT
ejpam-6171	73	16	filters	filter	NOUN
ejpam-6171	73	17	,	,	PUNCT
ejpam-6171	73	18	and	and	CCONJ
ejpam-6171	73	19	strong	strong	ADJ
ejpam-6171	73	20	ideals	ideal	NOUN
ejpam-6171	73	21	—	—	PUNCT
ejpam-6171	73	22	establishing	establish	VERB
ejpam-6171	73	23	necessary	necessary	ADJ
ejpam-6171	73	24	and	and	CCONJ
ejpam-6171	73	25	sufficient	sufficient	ADJ
ejpam-6171	73	26	conditions	condition	NOUN
ejpam-6171	73	27	within	within	ADP
ejpam-6171	73	28	the	the	DET
ejpam-6171	73	29	ins	in	NOUN
ejpam-6171	73	30	framework	framework	NOUN
ejpam-6171	73	31	.	.	PUNCT
ejpam-6171	74	1	this	this	DET
ejpam-6171	74	2	work	work	NOUN
ejpam-6171	74	3	laid	lay	VERB
ejpam-6171	74	4	important	important	ADJ
ejpam-6171	74	5	theoretical	theoretical	ADJ
ejpam-6171	74	6	foundations	foundation	NOUN
ejpam-6171	74	7	for	for	ADP
ejpam-6171	74	8	extending	extend	VERB
ejpam-6171	74	9	iup	iup	NOUN
ejpam-6171	74	10	-	-	PUNCT
ejpam-6171	74	11	algebras	algebras	PROPN
ejpam-6171	74	12	to	to	ADP
ejpam-6171	74	13	contexts	context	NOUN
ejpam-6171	74	14	involving	involve	VERB
ejpam-6171	74	15	indeterminate	indeterminate	ADJ
ejpam-6171	74	16	or	or	CCONJ
ejpam-6171	74	17	imprecise	imprecise	ADJ
ejpam-6171	74	18	data	datum	NOUN
ejpam-6171	74	19	.	.	PUNCT
ejpam-6171	75	1	this	this	DET
ejpam-6171	75	2	innovative	innovative	ADJ
ejpam-6171	75	3	approach	approach	NOUN
ejpam-6171	75	4	enhanced	enhance	VERB
ejpam-6171	75	5	the	the	DET
ejpam-6171	75	6	study	study	NOUN
ejpam-6171	75	7	of	of	ADP
ejpam-6171	75	8	iup	iup	NOUN
ejpam-6171	75	9	-	-	PUNCT
ejpam-6171	75	10	algebras	algebras	PROPN
ejpam-6171	75	11	,	,	PUNCT
ejpam-6171	75	12	presenting	present	VERB
ejpam-6171	75	13	new	new	ADJ
ejpam-6171	75	14	hybrid	hybrid	ADJ
ejpam-6171	75	15	structures	structure	NOUN
ejpam-6171	75	16	with	with	ADP
ejpam-6171	75	17	the	the	DET
ejpam-6171	75	18	potential	potential	NOUN
ejpam-6171	75	19	to	to	PART
ejpam-6171	75	20	inspire	inspire	VERB
ejpam-6171	75	21	a	a	DET
ejpam-6171	75	22	wide	wide	ADJ
ejpam-6171	75	23	array	array	NOUN
ejpam-6171	75	24	of	of	ADP
ejpam-6171	75	25	applications	application	NOUN
ejpam-6171	75	26	and	and	CCONJ
ejpam-6171	75	27	future	future	ADJ
ejpam-6171	75	28	research	research	NOUN
ejpam-6171	75	29	.	.	PUNCT
ejpam-6171	76	1	from	from	ADP
ejpam-6171	76	2	the	the	DET
ejpam-6171	76	3	literature	literature	NOUN
ejpam-6171	76	4	review	review	NOUN
ejpam-6171	76	5	,	,	PUNCT
ejpam-6171	76	6	it	it	PRON
ejpam-6171	76	7	has	have	AUX
ejpam-6171	76	8	been	be	AUX
ejpam-6171	76	9	found	find	VERB
ejpam-6171	76	10	that	that	SCONJ
ejpam-6171	76	11	since	since	SCONJ
ejpam-6171	76	12	the	the	DET
ejpam-6171	76	13	concept	concept	NOUN
ejpam-6171	76	14	of	of	ADP
ejpam-6171	76	15	pythagorean	pythagorean	PROPN
ejpam-6171	76	16	neutrosophic	neutrosophic	ADJ
ejpam-6171	76	17	sets	set	NOUN
ejpam-6171	76	18	(	(	PUNCT
ejpam-6171	76	19	pnss	pns	NOUN
ejpam-6171	76	20	)	)	PUNCT
ejpam-6171	76	21	was	be	AUX
ejpam-6171	76	22	defined	define	VERB
ejpam-6171	76	23	,	,	PUNCT
ejpam-6171	76	24	many	many	ADJ
ejpam-6171	76	25	researchers	researcher	NOUN
ejpam-6171	76	26	have	have	AUX
ejpam-6171	76	27	begun	begin	VERB
ejpam-6171	76	28	and	and	CCONJ
ejpam-6171	76	29	continued	continue	VERB
ejpam-6171	76	30	to	to	PART
ejpam-6171	76	31	study	study	VERB
ejpam-6171	76	32	this	this	DET
ejpam-6171	76	33	concept	concept	NOUN
ejpam-6171	76	34	.	.	PUNCT
ejpam-6171	77	1	many	many	ADJ
ejpam-6171	77	2	researchers	researcher	NOUN
ejpam-6171	77	3	have	have	AUX
ejpam-6171	77	4	studied	study	VERB
ejpam-6171	77	5	iup	iup	NOUN
ejpam-6171	77	6	-	-	PUNCT
ejpam-6171	77	7	algebras	algebra	NOUN
ejpam-6171	77	8	,	,	PUNCT
ejpam-6171	77	9	which	which	PRON
ejpam-6171	77	10	are	be	AUX
ejpam-6171	77	11	highly	highly	ADV
ejpam-6171	77	12	interesting	interesting	ADJ
ejpam-6171	77	13	algebraic	algebraic	ADJ
ejpam-6171	77	14	structures	structure	NOUN
ejpam-6171	77	15	.	.	PUNCT
ejpam-6171	78	1	our	our	PRON
ejpam-6171	78	2	researchers	researcher	NOUN
ejpam-6171	78	3	are	be	AUX
ejpam-6171	78	4	therefore	therefore	ADV
ejpam-6171	78	5	interested	interested	ADJ
ejpam-6171	78	6	in	in	ADP
ejpam-6171	78	7	applying	apply	VERB
ejpam-6171	78	8	the	the	DET
ejpam-6171	78	9	concept	concept	NOUN
ejpam-6171	78	10	of	of	ADP
ejpam-6171	78	11	pnss	pns	NOUN
ejpam-6171	78	12	to	to	PART
ejpam-6171	78	13	iup	iup	VERB
ejpam-6171	78	14	-	-	PUNCT
ejpam-6171	78	15	algebras	algebras	PROPN
ejpam-6171	78	16	.	.	PUNCT
ejpam-6171	79	1	thus	thus	ADV
ejpam-6171	79	2	,	,	PUNCT
ejpam-6171	79	3	we	we	PRON
ejpam-6171	79	4	will	will	AUX
ejpam-6171	79	5	study	study	VERB
ejpam-6171	79	6	the	the	DET
ejpam-6171	79	7	application	application	NOUN
ejpam-6171	79	8	of	of	ADP
ejpam-6171	79	9	pnss	pns	NOUN
ejpam-6171	79	10	to	to	ADP
ejpam-6171	79	11	the	the	DET
ejpam-6171	79	12	subsets	subset	NOUN
ejpam-6171	79	13	of	of	ADP
ejpam-6171	79	14	iupalgebras	iupalgebra	NOUN
ejpam-6171	79	15	,	,	PUNCT
ejpam-6171	79	16	including	include	VERB
ejpam-6171	79	17	iup	iup	NOUN
ejpam-6171	79	18	-	-	PUNCT
ejpam-6171	79	19	subalgebras	subalgebras	PROPN
ejpam-6171	79	20	,	,	PUNCT
ejpam-6171	79	21	iup	iup	NOUN
ejpam-6171	79	22	-	-	PUNCT
ejpam-6171	79	23	filters	filter	NOUN
ejpam-6171	79	24	,	,	PUNCT
ejpam-6171	79	25	iup	iup	NOUN
ejpam-6171	79	26	-	-	PUNCT
ejpam-6171	79	27	ideals	ideal	NOUN
ejpam-6171	79	28	,	,	PUNCT
ejpam-6171	79	29	and	and	CCONJ
ejpam-6171	79	30	strong	strong	ADJ
ejpam-6171	79	31	iup	iup	NOUN
ejpam-6171	79	32	-	-	PUNCT
ejpam-6171	79	33	ideals	ideal	NOUN
ejpam-6171	79	34	,	,	PUNCT
ejpam-6171	79	35	and	and	CCONJ
ejpam-6171	79	36	investigate	investigate	VERB
ejpam-6171	79	37	their	their	PRON
ejpam-6171	79	38	properties	property	NOUN
ejpam-6171	79	39	and	and	CCONJ
ejpam-6171	79	40	relationships	relationship	NOUN
ejpam-6171	79	41	.	.	PUNCT
ejpam-6171	80	1	we	we	PRON
ejpam-6171	80	2	will	will	AUX
ejpam-6171	80	3	examine	examine	VERB
ejpam-6171	80	4	the	the	DET
ejpam-6171	80	5	relationships	relationship	NOUN
ejpam-6171	80	6	between	between	ADP
ejpam-6171	80	7	pnss	pns	NOUN
ejpam-6171	80	8	and	and	CCONJ
ejpam-6171	80	9	these	these	DET
ejpam-6171	80	10	subsets	subset	NOUN
ejpam-6171	80	11	.	.	PUNCT
ejpam-6171	81	1	k.	k.	PROPN
ejpam-6171	81	2	suayngam	suayngam	PROPN
ejpam-6171	81	3	et	et	PROPN
ejpam-6171	81	4	al	al	PROPN
ejpam-6171	81	5	.	.	PUNCT
ejpam-6171	81	6	/	/	SYM
ejpam-6171	81	7	eur	eur	PROPN
ejpam-6171	81	8	.	.	PUNCT
ejpam-6171	82	1	j.	j.	PROPN
ejpam-6171	82	2	pure	pure	PROPN
ejpam-6171	82	3	appl	appl	PROPN
ejpam-6171	82	4	.	.	PROPN
ejpam-6171	82	5	math	math	PROPN
ejpam-6171	82	6	,	,	PUNCT
ejpam-6171	82	7	18	18	NUM
ejpam-6171	82	8	(	(	PUNCT
ejpam-6171	82	9	3	3	NUM
ejpam-6171	82	10	)	)	PUNCT
ejpam-6171	82	11	(	(	PUNCT
ejpam-6171	82	12	2025	2025	NUM
ejpam-6171	82	13	)	)	PUNCT
ejpam-6171	82	14	,	,	PUNCT
ejpam-6171	82	15	6171	6171	NUM
ejpam-6171	82	16	4	4	NUM
ejpam-6171	82	17	of	of	ADP
ejpam-6171	82	18	28	28	NUM
ejpam-6171	82	19	the	the	DET
ejpam-6171	82	20	content	content	NOUN
ejpam-6171	82	21	of	of	ADP
ejpam-6171	82	22	this	this	DET
ejpam-6171	82	23	paper	paper	NOUN
ejpam-6171	82	24	is	be	AUX
ejpam-6171	82	25	divided	divide	VERB
ejpam-6171	82	26	into	into	ADP
ejpam-6171	82	27	four	four	NUM
ejpam-6171	82	28	sections	section	NOUN
ejpam-6171	82	29	.	.	PUNCT
ejpam-6171	83	1	the	the	DET
ejpam-6171	83	2	first	first	ADJ
ejpam-6171	83	3	section	section	NOUN
ejpam-6171	83	4	explains	explain	VERB
ejpam-6171	83	5	the	the	DET
ejpam-6171	83	6	related	related	ADJ
ejpam-6171	83	7	research	research	NOUN
ejpam-6171	83	8	and	and	CCONJ
ejpam-6171	83	9	the	the	DET
ejpam-6171	83	10	inspiration	inspiration	NOUN
ejpam-6171	83	11	for	for	ADP
ejpam-6171	83	12	this	this	DET
ejpam-6171	83	13	paper	paper	NOUN
ejpam-6171	83	14	.	.	PUNCT
ejpam-6171	84	1	the	the	DET
ejpam-6171	84	2	second	second	ADJ
ejpam-6171	84	3	section	section	NOUN
ejpam-6171	84	4	introduces	introduce	VERB
ejpam-6171	84	5	the	the	DET
ejpam-6171	84	6	definitions	definition	NOUN
ejpam-6171	84	7	of	of	ADP
ejpam-6171	84	8	pnss	pns	NOUN
ejpam-6171	84	9	,	,	PUNCT
ejpam-6171	84	10	providing	provide	VERB
ejpam-6171	84	11	examples	example	NOUN
ejpam-6171	84	12	and	and	CCONJ
ejpam-6171	84	13	key	key	ADJ
ejpam-6171	84	14	properties	property	NOUN
ejpam-6171	84	15	.	.	PUNCT
ejpam-6171	85	1	additionally	additionally	ADV
ejpam-6171	85	2	,	,	PUNCT
ejpam-6171	85	3	we	we	PRON
ejpam-6171	85	4	will	will	AUX
ejpam-6171	85	5	review	review	VERB
ejpam-6171	85	6	the	the	DET
ejpam-6171	85	7	definitions	definition	NOUN
ejpam-6171	85	8	of	of	ADP
ejpam-6171	85	9	iup	iup	NOUN
ejpam-6171	85	10	-	-	PUNCT
ejpam-6171	85	11	subalgebras	subalgebras	PROPN
ejpam-6171	85	12	,	,	PUNCT
ejpam-6171	85	13	iup	iup	NOUN
ejpam-6171	85	14	-	-	PUNCT
ejpam-6171	85	15	filters	filter	NOUN
ejpam-6171	85	16	,	,	PUNCT
ejpam-6171	85	17	iup	iup	NOUN
ejpam-6171	85	18	-	-	PUNCT
ejpam-6171	85	19	ideals	ideal	NOUN
ejpam-6171	85	20	,	,	PUNCT
ejpam-6171	85	21	and	and	CCONJ
ejpam-6171	85	22	strong	strong	ADJ
ejpam-6171	85	23	iup	iup	NOUN
ejpam-6171	85	24	-	-	PUNCT
ejpam-6171	85	25	ideals	ideal	NOUN
ejpam-6171	85	26	,	,	PUNCT
ejpam-6171	85	27	showing	show	VERB
ejpam-6171	85	28	their	their	PRON
ejpam-6171	85	29	relationships	relationship	NOUN
ejpam-6171	85	30	.	.	PUNCT
ejpam-6171	86	1	the	the	DET
ejpam-6171	86	2	third	third	ADJ
ejpam-6171	86	3	section	section	NOUN
ejpam-6171	86	4	reviews	review	VERB
ejpam-6171	86	5	the	the	DET
ejpam-6171	86	6	definitions	definition	NOUN
ejpam-6171	86	7	of	of	ADP
ejpam-6171	86	8	pnss	pns	NOUN
ejpam-6171	86	9	,	,	PUNCT
ejpam-6171	86	10	including	include	VERB
ejpam-6171	86	11	the	the	DET
ejpam-6171	86	12	concepts	concept	NOUN
ejpam-6171	86	13	of	of	ADP
ejpam-6171	86	14	pythagorean	pythagorean	PROPN
ejpam-6171	86	15	neutrosophic	neutrosophic	PROPN
ejpam-6171	86	16	iup	iup	PROPN
ejpam-6171	86	17	-	-	PUNCT
ejpam-6171	86	18	subalgebras	subalgebras	PROPN
ejpam-6171	86	19	,	,	PUNCT
ejpam-6171	86	20	pythagorean	pythagorean	PROPN
ejpam-6171	86	21	neutrosophic	neutrosophic	PROPN
ejpam-6171	86	22	iup	iup	PROPN
ejpam-6171	86	23	-	-	PUNCT
ejpam-6171	86	24	filters	filter	NOUN
ejpam-6171	86	25	,	,	PUNCT
ejpam-6171	86	26	pythagorean	pythagorean	PROPN
ejpam-6171	86	27	neutrosophic	neutrosophic	PROPN
ejpam-6171	86	28	iup	iup	NOUN
ejpam-6171	86	29	-	-	PUNCT
ejpam-6171	86	30	ideals	ideal	NOUN
ejpam-6171	86	31	,	,	PUNCT
ejpam-6171	86	32	and	and	CCONJ
ejpam-6171	86	33	pythagorean	pythagorean	PROPN
ejpam-6171	86	34	neutrosophic	neutrosophic	PROPN
ejpam-6171	86	35	strong	strong	ADJ
ejpam-6171	86	36	iup	iup	NOUN
ejpam-6171	86	37	-	-	PUNCT
ejpam-6171	86	38	ideals	ideal	NOUN
ejpam-6171	86	39	,	,	PUNCT
ejpam-6171	86	40	with	with	ADP
ejpam-6171	86	41	examples	example	NOUN
ejpam-6171	86	42	and	and	CCONJ
ejpam-6171	86	43	explanations	explanation	NOUN
ejpam-6171	86	44	.	.	PUNCT
ejpam-6171	87	1	the	the	DET
ejpam-6171	87	2	fourth	fourth	ADJ
ejpam-6171	87	3	section	section	NOUN
ejpam-6171	87	4	examines	examine	VERB
ejpam-6171	87	5	the	the	DET
ejpam-6171	87	6	characteristic	characteristic	ADJ
ejpam-6171	87	7	functions	function	NOUN
ejpam-6171	87	8	for	for	ADP
ejpam-6171	87	9	these	these	DET
ejpam-6171	87	10	concepts	concept	NOUN
ejpam-6171	87	11	,	,	PUNCT
ejpam-6171	87	12	finds	find	VERB
ejpam-6171	87	13	general	general	ADJ
ejpam-6171	87	14	conclusions	conclusion	NOUN
ejpam-6171	87	15	,	,	PUNCT
ejpam-6171	87	16	and	and	CCONJ
ejpam-6171	87	17	shows	show	VERB
ejpam-6171	87	18	the	the	DET
ejpam-6171	87	19	relationships	relationship	NOUN
ejpam-6171	87	20	between	between	ADP
ejpam-6171	87	21	characteristic	characteristic	ADJ
ejpam-6171	87	22	functions	function	NOUN
ejpam-6171	87	23	,	,	PUNCT
ejpam-6171	87	24	level	level	NOUN
ejpam-6171	87	25	subsets	subset	NOUN
ejpam-6171	87	26	,	,	PUNCT
ejpam-6171	87	27	and	and	CCONJ
ejpam-6171	87	28	their	their	PRON
ejpam-6171	87	29	pnss	pns	NOUN
ejpam-6171	87	30	.	.	PUNCT
ejpam-6171	88	1	the	the	DET
ejpam-6171	88	2	final	final	ADJ
ejpam-6171	88	3	section	section	NOUN
ejpam-6171	88	4	summarizes	summarize	VERB
ejpam-6171	88	5	the	the	DET
ejpam-6171	88	6	research	research	NOUN
ejpam-6171	88	7	findings	finding	NOUN
ejpam-6171	88	8	and	and	CCONJ
ejpam-6171	88	9	suggests	suggest	VERB
ejpam-6171	88	10	further	further	ADJ
ejpam-6171	88	11	studies	study	NOUN
ejpam-6171	88	12	and	and	CCONJ
ejpam-6171	88	13	expansions	expansion	NOUN
ejpam-6171	88	14	of	of	ADP
ejpam-6171	88	15	this	this	DET
ejpam-6171	88	16	research	research	NOUN
ejpam-6171	88	17	.	.	PUNCT
ejpam-6171	89	1	2	2	X
ejpam-6171	89	2	.	.	X
ejpam-6171	89	3	preliminaries	preliminary	NOUN
ejpam-6171	89	4	the	the	DET
ejpam-6171	89	5	study	study	NOUN
ejpam-6171	89	6	of	of	ADP
ejpam-6171	89	7	algebraic	algebraic	ADJ
ejpam-6171	89	8	structures	structure	NOUN
ejpam-6171	89	9	has	have	AUX
ejpam-6171	89	10	evolved	evolve	VERB
ejpam-6171	89	11	significantly	significantly	ADV
ejpam-6171	89	12	to	to	PART
ejpam-6171	89	13	accommodate	accommodate	VERB
ejpam-6171	89	14	the	the	DET
ejpam-6171	89	15	complexities	complexity	NOUN
ejpam-6171	89	16	of	of	ADP
ejpam-6171	89	17	uncertainty	uncertainty	NOUN
ejpam-6171	89	18	and	and	CCONJ
ejpam-6171	89	19	imprecision	imprecision	NOUN
ejpam-6171	89	20	in	in	ADP
ejpam-6171	89	21	mathematical	mathematical	ADJ
ejpam-6171	89	22	modeling	modeling	NOUN
ejpam-6171	89	23	.	.	PUNCT
ejpam-6171	90	1	among	among	ADP
ejpam-6171	90	2	these	these	DET
ejpam-6171	90	3	structures	structure	NOUN
ejpam-6171	90	4	,	,	PUNCT
ejpam-6171	90	5	iup	iup	NOUN
ejpam-6171	90	6	-	-	PUNCT
ejpam-6171	90	7	algebras	algebra	NOUN
ejpam-6171	90	8	have	have	AUX
ejpam-6171	90	9	been	be	AUX
ejpam-6171	90	10	recognized	recognize	VERB
ejpam-6171	90	11	for	for	ADP
ejpam-6171	90	12	their	their	PRON
ejpam-6171	90	13	unique	unique	ADJ
ejpam-6171	90	14	properties	property	NOUN
ejpam-6171	90	15	,	,	PUNCT
ejpam-6171	90	16	making	make	VERB
ejpam-6171	90	17	them	they	PRON
ejpam-6171	90	18	a	a	DET
ejpam-6171	90	19	valuable	valuable	ADJ
ejpam-6171	90	20	tool	tool	NOUN
ejpam-6171	90	21	in	in	ADP
ejpam-6171	90	22	various	various	ADJ
ejpam-6171	90	23	mathematical	mathematical	ADJ
ejpam-6171	90	24	and	and	CCONJ
ejpam-6171	90	25	computational	computational	ADJ
ejpam-6171	90	26	applications	application	NOUN
ejpam-6171	90	27	.	.	PUNCT
ejpam-6171	91	1	simultaneously	simultaneously	ADV
ejpam-6171	91	2	,	,	PUNCT
ejpam-6171	91	3	the	the	DET
ejpam-6171	91	4	development	development	NOUN
ejpam-6171	91	5	of	of	ADP
ejpam-6171	91	6	neutrosophic	neutrosophic	ADJ
ejpam-6171	91	7	sets	set	NOUN
ejpam-6171	91	8	and	and	CCONJ
ejpam-6171	91	9	their	their	PRON
ejpam-6171	91	10	extensions	extension	NOUN
ejpam-6171	91	11	,	,	PUNCT
ejpam-6171	91	12	such	such	ADJ
ejpam-6171	91	13	as	as	ADP
ejpam-6171	91	14	pythagorean	pythagorean	PROPN
ejpam-6171	91	15	neutrosophic	neutrosophic	ADJ
ejpam-6171	91	16	sets	set	NOUN
ejpam-6171	91	17	(	(	PUNCT
ejpam-6171	91	18	pnss	pns	NOUN
ejpam-6171	91	19	)	)	PUNCT
ejpam-6171	91	20	,	,	PUNCT
ejpam-6171	91	21	has	have	AUX
ejpam-6171	91	22	provided	provide	VERB
ejpam-6171	91	23	a	a	DET
ejpam-6171	91	24	more	more	ADV
ejpam-6171	91	25	refined	refined	ADJ
ejpam-6171	91	26	approach	approach	NOUN
ejpam-6171	91	27	to	to	ADP
ejpam-6171	91	28	handling	handle	VERB
ejpam-6171	91	29	degrees	degree	NOUN
ejpam-6171	91	30	of	of	ADP
ejpam-6171	91	31	truth	truth	NOUN
ejpam-6171	91	32	,	,	PUNCT
ejpam-6171	91	33	indeterminacy	indeterminacy	NOUN
ejpam-6171	91	34	,	,	PUNCT
ejpam-6171	91	35	and	and	CCONJ
ejpam-6171	91	36	falsity	falsity	NOUN
ejpam-6171	91	37	.	.	PUNCT
ejpam-6171	92	1	the	the	DET
ejpam-6171	92	2	combination	combination	NOUN
ejpam-6171	92	3	of	of	ADP
ejpam-6171	92	4	these	these	DET
ejpam-6171	92	5	two	two	NUM
ejpam-6171	92	6	frameworks	framework	NOUN
ejpam-6171	92	7	—	—	PUNCT
ejpam-6171	92	8	iup	iup	NOUN
ejpam-6171	92	9	-	-	PUNCT
ejpam-6171	92	10	algebras	algebras	PROPN
ejpam-6171	92	11	and	and	CCONJ
ejpam-6171	92	12	pythagorean	pythagorean	PROPN
ejpam-6171	92	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	92	14	sets	set	NOUN
ejpam-6171	92	15	—	—	PUNCT
ejpam-6171	92	16	offers	offer	VERB
ejpam-6171	92	17	a	a	DET
ejpam-6171	92	18	novel	novel	ADJ
ejpam-6171	92	19	perspective	perspective	NOUN
ejpam-6171	92	20	on	on	ADP
ejpam-6171	92	21	algebraic	algebraic	ADJ
ejpam-6171	92	22	systems	system	NOUN
ejpam-6171	92	23	under	under	ADP
ejpam-6171	92	24	uncertainty	uncertainty	NOUN
ejpam-6171	92	25	.	.	PUNCT
ejpam-6171	93	1	before	before	ADP
ejpam-6171	93	2	delving	delve	VERB
ejpam-6171	93	3	into	into	ADP
ejpam-6171	93	4	the	the	DET
ejpam-6171	93	5	main	main	ADJ
ejpam-6171	93	6	results	result	NOUN
ejpam-6171	93	7	of	of	ADP
ejpam-6171	93	8	this	this	DET
ejpam-6171	93	9	study	study	NOUN
ejpam-6171	93	10	,	,	PUNCT
ejpam-6171	93	11	it	it	PRON
ejpam-6171	93	12	is	be	AUX
ejpam-6171	93	13	essential	essential	ADJ
ejpam-6171	93	14	to	to	PART
ejpam-6171	93	15	establish	establish	VERB
ejpam-6171	93	16	a	a	DET
ejpam-6171	93	17	strong	strong	ADJ
ejpam-6171	93	18	foundation	foundation	NOUN
ejpam-6171	93	19	by	by	ADP
ejpam-6171	93	20	reviewing	review	VERB
ejpam-6171	93	21	the	the	DET
ejpam-6171	93	22	fundamental	fundamental	ADJ
ejpam-6171	93	23	concepts	concept	NOUN
ejpam-6171	93	24	that	that	PRON
ejpam-6171	93	25	underpin	underpin	VERB
ejpam-6171	93	26	our	our	PRON
ejpam-6171	93	27	research	research	NOUN
ejpam-6171	93	28	.	.	PUNCT
ejpam-6171	94	1	this	this	DET
ejpam-6171	94	2	section	section	NOUN
ejpam-6171	94	3	introduces	introduce	VERB
ejpam-6171	94	4	key	key	ADJ
ejpam-6171	94	5	definitions	definition	NOUN
ejpam-6171	94	6	and	and	CCONJ
ejpam-6171	94	7	properties	property	NOUN
ejpam-6171	94	8	of	of	ADP
ejpam-6171	94	9	iup	iup	NOUN
ejpam-6171	94	10	-	-	PUNCT
ejpam-6171	94	11	algebras	algebras	PROPN
ejpam-6171	94	12	and	and	CCONJ
ejpam-6171	94	13	pythagorean	pythagorean	PROPN
ejpam-6171	94	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	94	15	sets	set	NOUN
ejpam-6171	94	16	,	,	PUNCT
ejpam-6171	94	17	ensuring	ensure	VERB
ejpam-6171	94	18	clarity	clarity	NOUN
ejpam-6171	94	19	in	in	ADP
ejpam-6171	94	20	their	their	PRON
ejpam-6171	94	21	mathematical	mathematical	ADJ
ejpam-6171	94	22	formulation	formulation	NOUN
ejpam-6171	94	23	.	.	PUNCT
ejpam-6171	95	1	by	by	ADP
ejpam-6171	95	2	revisiting	revisit	VERB
ejpam-6171	95	3	these	these	DET
ejpam-6171	95	4	preliminary	preliminary	ADJ
ejpam-6171	95	5	concepts	concept	NOUN
ejpam-6171	95	6	,	,	PUNCT
ejpam-6171	95	7	we	we	PRON
ejpam-6171	95	8	aim	aim	VERB
ejpam-6171	95	9	to	to	PART
ejpam-6171	95	10	provide	provide	VERB
ejpam-6171	95	11	a	a	DET
ejpam-6171	95	12	comprehensive	comprehensive	ADJ
ejpam-6171	95	13	background	background	NOUN
ejpam-6171	95	14	that	that	PRON
ejpam-6171	95	15	facilitates	facilitate	VERB
ejpam-6171	95	16	a	a	DET
ejpam-6171	95	17	deeper	deep	ADJ
ejpam-6171	95	18	understanding	understanding	NOUN
ejpam-6171	95	19	of	of	ADP
ejpam-6171	95	20	how	how	SCONJ
ejpam-6171	95	21	pythagorean	pythagorean	PROPN
ejpam-6171	95	22	neutrosophic	neutrosophic	ADJ
ejpam-6171	95	23	structures	structure	NOUN
ejpam-6171	95	24	can	can	AUX
ejpam-6171	95	25	be	be	AUX
ejpam-6171	95	26	integrated	integrate	VERB
ejpam-6171	95	27	into	into	ADP
ejpam-6171	95	28	iupalgebras	iupalgebra	NOUN
ejpam-6171	95	29	.	.	PUNCT
ejpam-6171	96	1	these	these	DET
ejpam-6171	96	2	insights	insight	NOUN
ejpam-6171	96	3	will	will	AUX
ejpam-6171	96	4	serve	serve	VERB
ejpam-6171	96	5	as	as	ADP
ejpam-6171	96	6	the	the	DET
ejpam-6171	96	7	basis	basis	NOUN
ejpam-6171	96	8	for	for	ADP
ejpam-6171	96	9	the	the	DET
ejpam-6171	96	10	subsequent	subsequent	ADJ
ejpam-6171	96	11	theoretical	theoretical	ADJ
ejpam-6171	96	12	developments	development	NOUN
ejpam-6171	96	13	presented	present	VERB
ejpam-6171	96	14	in	in	ADP
ejpam-6171	96	15	this	this	DET
ejpam-6171	96	16	study	study	NOUN
ejpam-6171	96	17	.	.	PUNCT
ejpam-6171	97	1	definition	definition	NOUN
ejpam-6171	97	2	1	1	NUM
ejpam-6171	97	3	.	.	PUNCT
ejpam-6171	98	1	[	[	X
ejpam-6171	98	2	9	9	NUM
ejpam-6171	98	3	]	]	PUNCT
ejpam-6171	98	4	an	an	DET
ejpam-6171	98	5	algebra	algebra	NOUN
ejpam-6171	98	6	x	x	X
ejpam-6171	98	7	=	=	SYM
ejpam-6171	98	8	(	(	PUNCT
ejpam-6171	98	9	x	x	X
ejpam-6171	98	10	,	,	PUNCT
ejpam-6171	98	11	?	?	PUNCT
ejpam-6171	98	12	,	,	PUNCT
ejpam-6171	98	13	0	0	NUM
ejpam-6171	98	14	)	)	PUNCT
ejpam-6171	98	15	of	of	ADP
ejpam-6171	98	16	type	type	NOUN
ejpam-6171	98	17	(	(	PUNCT
ejpam-6171	98	18	2	2	NUM
ejpam-6171	98	19	,	,	PUNCT
ejpam-6171	98	20	0	0	NUM
ejpam-6171	98	21	)	)	PUNCT
ejpam-6171	98	22	is	be	AUX
ejpam-6171	98	23	called	call	VERB
ejpam-6171	98	24	an	an	DET
ejpam-6171	98	25	iup	iup	NOUN
ejpam-6171	98	26	-	-	PUNCT
ejpam-6171	98	27	algebra	algebra	NOUN
ejpam-6171	98	28	,	,	PUNCT
ejpam-6171	98	29	where	where	SCONJ
ejpam-6171	98	30	x	x	PRON
ejpam-6171	98	31	is	be	AUX
ejpam-6171	98	32	a	a	DET
ejpam-6171	98	33	nonempty	nonempty	ADJ
ejpam-6171	98	34	set	set	VERB
ejpam-6171	98	35	,	,	PUNCT
ejpam-6171	98	36	?	?	PUNCT
ejpam-6171	99	1	is	be	AUX
ejpam-6171	99	2	a	a	DET
ejpam-6171	99	3	binary	binary	ADJ
ejpam-6171	99	4	operation	operation	NOUN
ejpam-6171	99	5	on	on	ADP
ejpam-6171	99	6	x	x	NOUN
ejpam-6171	99	7	,	,	PUNCT
ejpam-6171	99	8	and	and	CCONJ
ejpam-6171	99	9	0	0	NUM
ejpam-6171	99	10	is	be	AUX
ejpam-6171	99	11	a	a	DET
ejpam-6171	99	12	fixed	fix	VERB
ejpam-6171	99	13	element	element	NOUN
ejpam-6171	99	14	of	of	ADP
ejpam-6171	99	15	x	x	PRON
ejpam-6171	99	16	if	if	SCONJ
ejpam-6171	99	17	it	it	PRON
ejpam-6171	99	18	satisfies	satisfy	VERB
ejpam-6171	99	19	the	the	DET
ejpam-6171	99	20	following	follow	VERB
ejpam-6171	99	21	axioms	axiom	NOUN
ejpam-6171	99	22	:	:	PUNCT
ejpam-6171	99	23	(	(	PUNCT
ejpam-6171	99	24	∀x	∀x	X
ejpam-6171	99	25	∈	∈	NOUN
ejpam-6171	99	26	x)(0	x)(0	PUNCT
ejpam-6171	99	27	?	?	PUNCT
ejpam-6171	100	1	x	x	X
ejpam-6171	101	1	=	=	PUNCT
ejpam-6171	101	2	x	x	NOUN
ejpam-6171	101	3	)	)	PUNCT
ejpam-6171	101	4	(	(	PUNCT
ejpam-6171	101	5	iup-1	iup-1	X
ejpam-6171	101	6	)	)	PUNCT
ejpam-6171	101	7	(	(	PUNCT
ejpam-6171	101	8	∀x	∀x	X
ejpam-6171	101	9	∈	∈	PROPN
ejpam-6171	101	10	x)(x	x)(x	PROPN
ejpam-6171	101	11	?	?	PUNCT
ejpam-6171	102	1	x	x	X
ejpam-6171	103	1	=	=	NOUN
ejpam-6171	103	2	0	0	NUM
ejpam-6171	103	3	)	)	PUNCT
ejpam-6171	103	4	(	(	PUNCT
ejpam-6171	103	5	iup-2	iup-2	NUM
ejpam-6171	103	6	)	)	PUNCT
ejpam-6171	103	7	(	(	PUNCT
ejpam-6171	103	8	∀x	∀x	X
ejpam-6171	103	9	,	,	PUNCT
ejpam-6171	103	10	y	y	PROPN
ejpam-6171	103	11	,	,	PUNCT
ejpam-6171	103	12	z	z	PROPN
ejpam-6171	103	13	∈	∈	PROPN
ejpam-6171	103	14	x)((x	x)((x	NOUN
ejpam-6171	103	15	?	?	PUNCT
ejpam-6171	104	1	y	y	X
ejpam-6171	104	2	)	)	PUNCT
ejpam-6171	104	3	?	?	PUNCT
ejpam-6171	105	1	(	(	PUNCT
ejpam-6171	105	2	x	x	X
ejpam-6171	105	3	?	?	PUNCT
ejpam-6171	106	1	z	z	X
ejpam-6171	106	2	)	)	PUNCT
ejpam-6171	106	3	=	=	SYM
ejpam-6171	106	4	y	y	PROPN
ejpam-6171	106	5	?	?	PUNCT
ejpam-6171	107	1	z	z	X
ejpam-6171	107	2	)	)	PUNCT
ejpam-6171	107	3	(	(	PUNCT
ejpam-6171	107	4	iup-3	iup-3	NOUN
ejpam-6171	107	5	)	)	PUNCT
ejpam-6171	107	6	for	for	ADP
ejpam-6171	107	7	simplicity	simplicity	NOUN
ejpam-6171	107	8	,	,	PUNCT
ejpam-6171	107	9	we	we	PRON
ejpam-6171	107	10	will	will	AUX
ejpam-6171	107	11	refer	refer	VERB
ejpam-6171	107	12	to	to	ADP
ejpam-6171	107	13	x	x	PUNCT
ejpam-6171	107	14	as	as	ADP
ejpam-6171	107	15	the	the	DET
ejpam-6171	107	16	iup	iup	NOUN
ejpam-6171	107	17	-	-	PUNCT
ejpam-6171	107	18	algebra	algebra	NOUN
ejpam-6171	107	19	x	x	PUNCT
ejpam-6171	107	20	=	=	SYM
ejpam-6171	107	21	(	(	PUNCT
ejpam-6171	107	22	x	x	X
ejpam-6171	107	23	,	,	PUNCT
ejpam-6171	107	24	?	?	PUNCT
ejpam-6171	107	25	,	,	PUNCT
ejpam-6171	107	26	0	0	NUM
ejpam-6171	107	27	)	)	PUNCT
ejpam-6171	107	28	unless	unless	SCONJ
ejpam-6171	107	29	stated	state	VERB
ejpam-6171	107	30	otherwise	otherwise	ADV
ejpam-6171	107	31	.	.	PUNCT
ejpam-6171	108	1	k.	k.	PROPN
ejpam-6171	109	1	suayngam	suayngam	PROPN
ejpam-6171	109	2	et	et	PROPN
ejpam-6171	109	3	al	al	PROPN
ejpam-6171	109	4	.	.	PUNCT
ejpam-6171	109	5	/	/	SYM
ejpam-6171	109	6	eur	eur	PROPN
ejpam-6171	109	7	.	.	PUNCT
ejpam-6171	110	1	j.	j.	PROPN
ejpam-6171	110	2	pure	pure	PROPN
ejpam-6171	110	3	appl	appl	PROPN
ejpam-6171	110	4	.	.	PROPN
ejpam-6171	110	5	math	math	PROPN
ejpam-6171	110	6	,	,	PUNCT
ejpam-6171	110	7	18	18	NUM
ejpam-6171	110	8	(	(	PUNCT
ejpam-6171	110	9	3	3	NUM
ejpam-6171	110	10	)	)	PUNCT
ejpam-6171	110	11	(	(	PUNCT
ejpam-6171	110	12	2025	2025	NUM
ejpam-6171	110	13	)	)	PUNCT
ejpam-6171	110	14	,	,	PUNCT
ejpam-6171	110	15	6171	6171	NUM
ejpam-6171	110	16	5	5	NUM
ejpam-6171	110	17	of	of	ADP
ejpam-6171	110	18	28	28	NUM
ejpam-6171	110	19	example	example	NOUN
ejpam-6171	110	20	1	1	NUM
ejpam-6171	110	21	.	.	PUNCT
ejpam-6171	111	1	let	let	VERB
ejpam-6171	111	2	x	x	PUNCT
ejpam-6171	111	3	=	=	PUNCT
ejpam-6171	111	4	{	{	PUNCT
ejpam-6171	111	5	0	0	NUM
ejpam-6171	111	6	,	,	PUNCT
ejpam-6171	111	7	1	1	NUM
ejpam-6171	111	8	,	,	PUNCT
ejpam-6171	111	9	2	2	NUM
ejpam-6171	111	10	,	,	PUNCT
ejpam-6171	111	11	3	3	NUM
ejpam-6171	111	12	,	,	PUNCT
ejpam-6171	111	13	4	4	NUM
ejpam-6171	111	14	,	,	PUNCT
ejpam-6171	111	15	5	5	NUM
ejpam-6171	111	16	}	}	PUNCT
ejpam-6171	111	17	be	be	AUX
ejpam-6171	111	18	a	a	DET
ejpam-6171	111	19	set	set	NOUN
ejpam-6171	111	20	with	with	ADP
ejpam-6171	111	21	a	a	DET
ejpam-6171	111	22	binary	binary	ADJ
ejpam-6171	111	23	operation	operation	NOUN
ejpam-6171	111	24	?	?	PUNCT
ejpam-6171	112	1	defined	define	VERB
ejpam-6171	112	2	by	by	ADP
ejpam-6171	112	3	the	the	DET
ejpam-6171	112	4	following	following	ADJ
ejpam-6171	112	5	cayley	cayley	ADJ
ejpam-6171	112	6	table	table	NOUN
ejpam-6171	112	7	:	:	PUNCT
ejpam-6171	112	8	·	·	PUNCT
ejpam-6171	112	9	0	0	NUM
ejpam-6171	112	10	1	1	NUM
ejpam-6171	112	11	2	2	NUM
ejpam-6171	112	12	3	3	NUM
ejpam-6171	112	13	4	4	NUM
ejpam-6171	112	14	5	5	NUM
ejpam-6171	112	15	0	0	NUM
ejpam-6171	112	16	0	0	NUM
ejpam-6171	112	17	1	1	NUM
ejpam-6171	112	18	2	2	NUM
ejpam-6171	112	19	3	3	NUM
ejpam-6171	112	20	4	4	NUM
ejpam-6171	112	21	5	5	NUM
ejpam-6171	112	22	1	1	NUM
ejpam-6171	112	23	3	3	NUM
ejpam-6171	112	24	0	0	NUM
ejpam-6171	112	25	5	5	NUM
ejpam-6171	112	26	1	1	NUM
ejpam-6171	112	27	2	2	NUM
ejpam-6171	112	28	4	4	NUM
ejpam-6171	112	29	2	2	NUM
ejpam-6171	112	30	5	5	NUM
ejpam-6171	112	31	2	2	NUM
ejpam-6171	112	32	0	0	NUM
ejpam-6171	112	33	4	4	NUM
ejpam-6171	112	34	1	1	NUM
ejpam-6171	112	35	3	3	NUM
ejpam-6171	112	36	3	3	NUM
ejpam-6171	112	37	1	1	NUM
ejpam-6171	112	38	3	3	NUM
ejpam-6171	112	39	4	4	NUM
ejpam-6171	112	40	0	0	NUM
ejpam-6171	112	41	5	5	NUM
ejpam-6171	112	42	2	2	NUM
ejpam-6171	112	43	4	4	NUM
ejpam-6171	112	44	4	4	NUM
ejpam-6171	112	45	5	5	NUM
ejpam-6171	112	46	3	3	NUM
ejpam-6171	112	47	2	2	NUM
ejpam-6171	112	48	0	0	NUM
ejpam-6171	112	49	1	1	NUM
ejpam-6171	112	50	5	5	NUM
ejpam-6171	112	51	2	2	NUM
ejpam-6171	112	52	4	4	NUM
ejpam-6171	112	53	1	1	NUM
ejpam-6171	112	54	5	5	NUM
ejpam-6171	112	55	3	3	NUM
ejpam-6171	112	56	0	0	NUM
ejpam-6171	112	57	then	then	ADV
ejpam-6171	112	58	x	x	SYM
ejpam-6171	112	59	=	=	SYM
ejpam-6171	112	60	(	(	PUNCT
ejpam-6171	112	61	x	x	X
ejpam-6171	112	62	,	,	PUNCT
ejpam-6171	112	63	?	?	PUNCT
ejpam-6171	112	64	,	,	PUNCT
ejpam-6171	112	65	0	0	X
ejpam-6171	112	66	)	)	PUNCT
ejpam-6171	112	67	is	be	AUX
ejpam-6171	112	68	an	an	DET
ejpam-6171	112	69	iup	iup	NOUN
ejpam-6171	112	70	-	-	PUNCT
ejpam-6171	112	71	algebra	algebra	NOUN
ejpam-6171	112	72	.	.	PUNCT
ejpam-6171	113	1	example	example	NOUN
ejpam-6171	114	1	2	2	NUM
ejpam-6171	114	2	.	.	PUNCT
ejpam-6171	115	1	[	[	X
ejpam-6171	115	2	9	9	NUM
ejpam-6171	115	3	]	]	X
ejpam-6171	115	4	let	let	VERB
ejpam-6171	115	5	(	(	PUNCT
ejpam-6171	115	6	g	g	NOUN
ejpam-6171	115	7	,	,	PUNCT
ejpam-6171	115	8	•	•	NUM
ejpam-6171	115	9	,	,	PUNCT
ejpam-6171	115	10	e	e	NOUN
ejpam-6171	115	11	)	)	PUNCT
ejpam-6171	115	12	be	be	AUX
ejpam-6171	115	13	a	a	DET
ejpam-6171	115	14	group	group	NOUN
ejpam-6171	115	15	with	with	ADP
ejpam-6171	115	16	the	the	DET
ejpam-6171	115	17	identity	identity	NOUN
ejpam-6171	115	18	element	element	NOUN
ejpam-6171	115	19	e	e	NOUN
ejpam-6171	115	20	in	in	ADP
ejpam-6171	115	21	which	which	PRON
ejpam-6171	115	22	each	each	DET
ejpam-6171	115	23	element	element	NOUN
ejpam-6171	115	24	is	be	AUX
ejpam-6171	115	25	its	its	PRON
ejpam-6171	115	26	own	own	ADJ
ejpam-6171	115	27	inverse	inverse	NOUN
ejpam-6171	115	28	.	.	PUNCT
ejpam-6171	116	1	under	under	ADP
ejpam-6171	116	2	this	this	DET
ejpam-6171	116	3	condition	condition	NOUN
ejpam-6171	116	4	,	,	PUNCT
ejpam-6171	116	5	(	(	PUNCT
ejpam-6171	116	6	g	g	NOUN
ejpam-6171	116	7	,	,	PUNCT
ejpam-6171	116	8	•	•	NUM
ejpam-6171	116	9	,	,	PUNCT
ejpam-6171	116	10	e	e	NOUN
ejpam-6171	116	11	)	)	PUNCT
ejpam-6171	116	12	inherently	inherently	ADV
ejpam-6171	116	13	satisfies	satisfy	VERB
ejpam-6171	116	14	the	the	DET
ejpam-6171	116	15	axioms	axiom	NOUN
ejpam-6171	116	16	of	of	ADP
ejpam-6171	116	17	an	an	DET
ejpam-6171	116	18	iup	iup	NOUN
ejpam-6171	116	19	-	-	PUNCT
ejpam-6171	116	20	algebra	algebra	NOUN
ejpam-6171	116	21	.	.	PUNCT
ejpam-6171	116	22	example	example	NOUN
ejpam-6171	117	1	3	3	NUM
ejpam-6171	117	2	.	.	PUNCT
ejpam-6171	118	1	[	[	X
ejpam-6171	118	2	9	9	NUM
ejpam-6171	118	3	]	]	PUNCT
ejpam-6171	118	4	let	let	VERB
ejpam-6171	118	5	x	x	PRON
ejpam-6171	118	6	be	be	AUX
ejpam-6171	118	7	a	a	DET
ejpam-6171	118	8	set	set	NOUN
ejpam-6171	118	9	,	,	PUNCT
ejpam-6171	118	10	and	and	CCONJ
ejpam-6171	118	11	let	let	VERB
ejpam-6171	118	12	p(x	p(x	PROPN
ejpam-6171	118	13	)	)	PUNCT
ejpam-6171	118	14	denote	denote	VERB
ejpam-6171	118	15	its	its	PRON
ejpam-6171	118	16	power	power	NOUN
ejpam-6171	118	17	set	set	NOUN
ejpam-6171	118	18	.	.	PUNCT
ejpam-6171	119	1	as	as	SCONJ
ejpam-6171	119	2	shown	show	VERB
ejpam-6171	119	3	in	in	ADP
ejpam-6171	119	4	example	example	NOUN
ejpam-6171	119	5	2	2	NUM
ejpam-6171	119	6	,	,	PUNCT
ejpam-6171	119	7	(	(	PUNCT
ejpam-6171	119	8	p(x),4	p(x),4	NOUN
ejpam-6171	119	9	,	,	PUNCT
ejpam-6171	119	10	∅	∅	NOUN
ejpam-6171	119	11	)	)	PUNCT
ejpam-6171	119	12	forms	form	VERB
ejpam-6171	119	13	an	an	DET
ejpam-6171	119	14	iup	iup	NOUN
ejpam-6171	119	15	-	-	PUNCT
ejpam-6171	119	16	algebra	algebra	NOUN
ejpam-6171	119	17	,	,	PUNCT
ejpam-6171	119	18	where	where	SCONJ
ejpam-6171	119	19	4	4	NUM
ejpam-6171	119	20	represents	represent	VERB
ejpam-6171	119	21	the	the	DET
ejpam-6171	119	22	symmetric	symmetric	ADJ
ejpam-6171	119	23	difference	difference	NOUN
ejpam-6171	119	24	between	between	ADP
ejpam-6171	119	25	sets	set	NOUN
ejpam-6171	119	26	.	.	PUNCT
ejpam-6171	120	1	example	example	NOUN
ejpam-6171	120	2	4	4	NUM
ejpam-6171	120	3	.	.	PUNCT
ejpam-6171	121	1	[	[	X
ejpam-6171	121	2	9	9	NUM
ejpam-6171	121	3	]	]	X
ejpam-6171	121	4	let	let	VERB
ejpam-6171	121	5	(	(	PUNCT
ejpam-6171	121	6	g	g	NOUN
ejpam-6171	121	7	,	,	PUNCT
ejpam-6171	121	8	·	·	PUNCT
ejpam-6171	121	9	,	,	PUNCT
ejpam-6171	121	10	e	e	X
ejpam-6171	121	11	)	)	PUNCT
ejpam-6171	121	12	be	be	AUX
ejpam-6171	121	13	a	a	DET
ejpam-6171	121	14	group	group	NOUN
ejpam-6171	121	15	with	with	ADP
ejpam-6171	121	16	the	the	DET
ejpam-6171	121	17	identity	identity	NOUN
ejpam-6171	121	18	element	element	NOUN
ejpam-6171	121	19	e.	e.	PROPN
ejpam-6171	121	20	define	define	VERB
ejpam-6171	121	21	a	a	DET
ejpam-6171	121	22	binary	binary	ADJ
ejpam-6171	121	23	operation	operation	NOUN
ejpam-6171	121	24	•	•	NOUN
ejpam-6171	121	25	on	on	ADP
ejpam-6171	121	26	g	g	NOUN
ejpam-6171	121	27	by	by	ADP
ejpam-6171	121	28	:	:	PUNCT
ejpam-6171	121	29	(	(	PUNCT
ejpam-6171	121	30	∀x	∀x	X
ejpam-6171	121	31	,	,	PUNCT
ejpam-6171	121	32	y	y	PROPN
ejpam-6171	121	33	∈	∈	PROPN
ejpam-6171	121	34	g)(x	g)(x	PROPN
ejpam-6171	121	35	•	•	NOUN
ejpam-6171	121	36	y	y	PROPN
ejpam-6171	121	37	=	=	SYM
ejpam-6171	121	38	y	y	PROPN
ejpam-6171	121	39	·	·	PUNCT
ejpam-6171	121	40	x−1	x−1	NOUN
ejpam-6171	121	41	)	)	PUNCT
ejpam-6171	121	42	(	(	PUNCT
ejpam-6171	121	43	2.1	2.1	NUM
ejpam-6171	121	44	)	)	PUNCT
ejpam-6171	121	45	then	then	ADV
ejpam-6171	121	46	(	(	PUNCT
ejpam-6171	121	47	g	g	NOUN
ejpam-6171	121	48	,	,	PUNCT
ejpam-6171	121	49	•	•	NUM
ejpam-6171	121	50	,	,	PUNCT
ejpam-6171	121	51	e	e	NOUN
ejpam-6171	121	52	)	)	PUNCT
ejpam-6171	121	53	is	be	AUX
ejpam-6171	121	54	an	an	DET
ejpam-6171	121	55	iup	iup	NOUN
ejpam-6171	121	56	-	-	PUNCT
ejpam-6171	121	57	algebra	algebra	NOUN
ejpam-6171	121	58	.	.	PUNCT
ejpam-6171	122	1	proposition	proposition	NOUN
ejpam-6171	122	2	1	1	NUM
ejpam-6171	122	3	.	.	PUNCT
ejpam-6171	123	1	[	[	X
ejpam-6171	123	2	9	9	NUM
ejpam-6171	123	3	]	]	PUNCT
ejpam-6171	123	4	in	in	ADP
ejpam-6171	123	5	an	an	DET
ejpam-6171	123	6	iup	iup	NOUN
ejpam-6171	123	7	-	-	PUNCT
ejpam-6171	123	8	algebra	algebra	NOUN
ejpam-6171	123	9	x	x	PUNCT
ejpam-6171	123	10	=	=	SYM
ejpam-6171	123	11	(	(	PUNCT
ejpam-6171	123	12	x	x	X
ejpam-6171	123	13	,	,	PUNCT
ejpam-6171	123	14	?	?	PUNCT
ejpam-6171	123	15	,	,	PUNCT
ejpam-6171	123	16	0	0	NUM
ejpam-6171	123	17	)	)	PUNCT
ejpam-6171	123	18	,	,	PUNCT
ejpam-6171	123	19	the	the	DET
ejpam-6171	123	20	following	follow	VERB
ejpam-6171	123	21	assertions	assertion	NOUN
ejpam-6171	123	22	are	be	AUX
ejpam-6171	123	23	valid	valid	ADJ
ejpam-6171	123	24	:	:	PUNCT
ejpam-6171	123	25	(	(	PUNCT
ejpam-6171	123	26	∀x	∀x	X
ejpam-6171	123	27	,	,	PUNCT
ejpam-6171	123	28	y	y	PROPN
ejpam-6171	123	29	∈	∈	PROPN
ejpam-6171	123	30	x)((x	x)((x	NOUN
ejpam-6171	123	31	?	?	PUNCT
ejpam-6171	124	1	0	0	NUM
ejpam-6171	124	2	)	)	PUNCT
ejpam-6171	124	3	?	?	PUNCT
ejpam-6171	125	1	(	(	PUNCT
ejpam-6171	125	2	x	x	X
ejpam-6171	125	3	?	?	PUNCT
ejpam-6171	125	4	y	y	X
ejpam-6171	125	5	)	)	PUNCT
ejpam-6171	125	6	=	=	SYM
ejpam-6171	125	7	y	y	NOUN
ejpam-6171	125	8	)	)	PUNCT
ejpam-6171	125	9	(	(	PUNCT
ejpam-6171	125	10	2.2	2.2	NUM
ejpam-6171	125	11	)	)	PUNCT
ejpam-6171	125	12	(	(	PUNCT
ejpam-6171	125	13	∀x	∀x	X
ejpam-6171	125	14	∈	∈	PROPN
ejpam-6171	125	15	x)((x	x)((x	NOUN
ejpam-6171	125	16	?	?	PUNCT
ejpam-6171	125	17	0	0	NUM
ejpam-6171	125	18	)	)	PUNCT
ejpam-6171	125	19	?	?	PUNCT
ejpam-6171	126	1	(	(	PUNCT
ejpam-6171	126	2	x	x	X
ejpam-6171	126	3	?	?	PUNCT
ejpam-6171	126	4	0	0	X
ejpam-6171	126	5	)	)	PUNCT
ejpam-6171	126	6	=	=	SYM
ejpam-6171	126	7	0	0	X
ejpam-6171	126	8	)	)	PUNCT
ejpam-6171	126	9	(	(	PUNCT
ejpam-6171	126	10	2.3	2.3	NUM
ejpam-6171	126	11	)	)	PUNCT
ejpam-6171	126	12	(	(	PUNCT
ejpam-6171	126	13	∀x	∀x	X
ejpam-6171	126	14	,	,	PUNCT
ejpam-6171	126	15	y	y	PROPN
ejpam-6171	126	16	∈	∈	PROPN
ejpam-6171	126	17	x)((x	x)((x	NOUN
ejpam-6171	126	18	?	?	PUNCT
ejpam-6171	127	1	y	y	X
ejpam-6171	127	2	)	)	PUNCT
ejpam-6171	127	3	?	?	PUNCT
ejpam-6171	127	4	0	0	PUNCT
ejpam-6171	128	1	=	=	SYM
ejpam-6171	128	2	y	y	PROPN
ejpam-6171	128	3	?	?	PUNCT
ejpam-6171	129	1	x	x	X
ejpam-6171	129	2	)	)	PUNCT
ejpam-6171	129	3	(	(	PUNCT
ejpam-6171	129	4	2.4	2.4	NUM
ejpam-6171	129	5	)	)	PUNCT
ejpam-6171	129	6	(	(	PUNCT
ejpam-6171	129	7	∀x	∀x	X
ejpam-6171	129	8	∈	∈	PROPN
ejpam-6171	129	9	x)((x	x)((x	NOUN
ejpam-6171	129	10	?	?	PUNCT
ejpam-6171	129	11	0	0	NUM
ejpam-6171	129	12	)	)	PUNCT
ejpam-6171	129	13	?	?	PUNCT
ejpam-6171	129	14	0	0	PUNCT
ejpam-6171	130	1	=	=	SYM
ejpam-6171	130	2	x	x	X
ejpam-6171	130	3	)	)	PUNCT
ejpam-6171	130	4	(	(	PUNCT
ejpam-6171	130	5	2.5	2.5	NUM
ejpam-6171	130	6	)	)	PUNCT
ejpam-6171	130	7	(	(	PUNCT
ejpam-6171	130	8	∀x	∀x	X
ejpam-6171	130	9	,	,	PUNCT
ejpam-6171	130	10	y	y	PROPN
ejpam-6171	130	11	∈	∈	PROPN
ejpam-6171	130	12	x)(x	x)(x	PROPN
ejpam-6171	130	13	?	?	PUNCT
ejpam-6171	131	1	(	(	PUNCT
ejpam-6171	131	2	(	(	PUNCT
ejpam-6171	131	3	x	x	X
ejpam-6171	131	4	?	?	PUNCT
ejpam-6171	131	5	0	0	NUM
ejpam-6171	131	6	)	)	PUNCT
ejpam-6171	131	7	?	?	PUNCT
ejpam-6171	132	1	y	y	X
ejpam-6171	132	2	)	)	PUNCT
ejpam-6171	132	3	=	=	SYM
ejpam-6171	132	4	y	y	NOUN
ejpam-6171	132	5	)	)	PUNCT
ejpam-6171	132	6	(	(	PUNCT
ejpam-6171	132	7	2.6	2.6	NUM
ejpam-6171	132	8	)	)	PUNCT
ejpam-6171	132	9	(	(	PUNCT
ejpam-6171	132	10	∀x	∀x	X
ejpam-6171	132	11	,	,	PUNCT
ejpam-6171	132	12	y	y	PROPN
ejpam-6171	132	13	∈	∈	PROPN
ejpam-6171	132	14	x)(((x	x)(((x	X
ejpam-6171	132	15	?	?	NOUN
ejpam-6171	132	16	0	0	NUM
ejpam-6171	132	17	)	)	PUNCT
ejpam-6171	132	18	?	?	PUNCT
ejpam-6171	133	1	y	y	X
ejpam-6171	133	2	)	)	PUNCT
ejpam-6171	133	3	?	?	PUNCT
ejpam-6171	134	1	x	x	X
ejpam-6171	134	2	=	=	PUNCT
ejpam-6171	134	3	y	y	PROPN
ejpam-6171	134	4	?	?	PUNCT
ejpam-6171	134	5	0	0	X
ejpam-6171	134	6	)	)	PUNCT
ejpam-6171	134	7	(	(	PUNCT
ejpam-6171	134	8	2.7	2.7	NUM
ejpam-6171	134	9	)	)	PUNCT
ejpam-6171	134	10	(	(	PUNCT
ejpam-6171	134	11	∀x	∀x	X
ejpam-6171	134	12	,	,	PUNCT
ejpam-6171	134	13	y	y	PROPN
ejpam-6171	134	14	,	,	PUNCT
ejpam-6171	134	15	z	z	PROPN
ejpam-6171	134	16	∈	∈	PROPN
ejpam-6171	134	17	x)(x	x)(x	PROPN
ejpam-6171	134	18	?	?	PUNCT
ejpam-6171	135	1	y	y	NOUN
ejpam-6171	135	2	=	=	PUNCT
ejpam-6171	136	1	x	x	PROPN
ejpam-6171	136	2	?	?	PUNCT
ejpam-6171	137	1	z	z	PROPN
ejpam-6171	137	2	⇔	⇔	PROPN
ejpam-6171	137	3	y	y	PROPN
ejpam-6171	137	4	=	=	SYM
ejpam-6171	137	5	z	z	PROPN
ejpam-6171	137	6	)	)	PUNCT
ejpam-6171	137	7	(	(	PUNCT
ejpam-6171	137	8	2.8	2.8	NUM
ejpam-6171	137	9	)	)	PUNCT
ejpam-6171	137	10	(	(	PUNCT
ejpam-6171	137	11	∀x	∀x	X
ejpam-6171	137	12	,	,	PUNCT
ejpam-6171	138	1	y	y	PROPN
ejpam-6171	138	2	∈	∈	PROPN
ejpam-6171	138	3	x)(x	x)(x	PROPN
ejpam-6171	138	4	?	?	PUNCT
ejpam-6171	139	1	y	y	PROPN
ejpam-6171	139	2	=	=	SYM
ejpam-6171	139	3	0	0	NUM
ejpam-6171	139	4	⇔	⇔	X
ejpam-6171	139	5	x	x	X
ejpam-6171	139	6	=	=	SYM
ejpam-6171	139	7	y	y	PROPN
ejpam-6171	139	8	)	)	PUNCT
ejpam-6171	139	9	(	(	PUNCT
ejpam-6171	139	10	2.9	2.9	NUM
ejpam-6171	139	11	)	)	PUNCT
ejpam-6171	139	12	(	(	PUNCT
ejpam-6171	139	13	∀x	∀x	X
ejpam-6171	139	14	∈	∈	PROPN
ejpam-6171	139	15	x)(x	x)(x	PROPN
ejpam-6171	139	16	?	?	PUNCT
ejpam-6171	139	17	0	0	PUNCT
ejpam-6171	140	1	=	=	SYM
ejpam-6171	140	2	0	0	NUM
ejpam-6171	140	3	⇔	⇔	X
ejpam-6171	140	4	x	x	PUNCT
ejpam-6171	140	5	=	=	SYM
ejpam-6171	140	6	0	0	NUM
ejpam-6171	140	7	)	)	PUNCT
ejpam-6171	140	8	(	(	PUNCT
ejpam-6171	140	9	2.10	2.10	NUM
ejpam-6171	140	10	)	)	PUNCT
ejpam-6171	140	11	(	(	PUNCT
ejpam-6171	140	12	∀x	∀x	X
ejpam-6171	140	13	,	,	PUNCT
ejpam-6171	140	14	y	y	PROPN
ejpam-6171	140	15	,	,	PUNCT
ejpam-6171	140	16	z	z	PROPN
ejpam-6171	140	17	∈	∈	PROPN
ejpam-6171	140	18	x)(y	x)(y	PUNCT
ejpam-6171	140	19	?	?	PUNCT
ejpam-6171	141	1	x	x	PUNCT
ejpam-6171	142	1	=	=	PUNCT
ejpam-6171	142	2	z	z	NOUN
ejpam-6171	142	3	?	?	PUNCT
ejpam-6171	143	1	x	x	X
ejpam-6171	143	2	⇔	⇔	PROPN
ejpam-6171	143	3	y	y	PROPN
ejpam-6171	143	4	=	=	SYM
ejpam-6171	143	5	z	z	PROPN
ejpam-6171	143	6	)	)	PUNCT
ejpam-6171	143	7	(	(	PUNCT
ejpam-6171	143	8	2.11	2.11	NUM
ejpam-6171	143	9	)	)	PUNCT
ejpam-6171	143	10	(	(	PUNCT
ejpam-6171	143	11	∀x	∀x	X
ejpam-6171	143	12	,	,	PUNCT
ejpam-6171	143	13	y	y	PROPN
ejpam-6171	143	14	∈	∈	PROPN
ejpam-6171	143	15	x)(x	x)(x	PROPN
ejpam-6171	143	16	?	?	PUNCT
ejpam-6171	144	1	y	y	NOUN
ejpam-6171	144	2	=	=	SYM
ejpam-6171	144	3	y	y	PROPN
ejpam-6171	144	4	⇒	⇒	VERB
ejpam-6171	144	5	x	x	PUNCT
ejpam-6171	145	1	=	=	NOUN
ejpam-6171	145	2	0	0	NUM
ejpam-6171	145	3	)	)	PUNCT
ejpam-6171	145	4	(	(	PUNCT
ejpam-6171	145	5	2.12	2.12	NUM
ejpam-6171	145	6	)	)	PUNCT
ejpam-6171	145	7	(	(	PUNCT
ejpam-6171	145	8	∀x	∀x	X
ejpam-6171	145	9	,	,	PUNCT
ejpam-6171	145	10	y	y	PROPN
ejpam-6171	145	11	,	,	PUNCT
ejpam-6171	145	12	z	z	PROPN
ejpam-6171	145	13	∈	∈	PROPN
ejpam-6171	145	14	x)((x	x)((x	NOUN
ejpam-6171	145	15	?	?	PUNCT
ejpam-6171	146	1	y	y	X
ejpam-6171	146	2	)	)	PUNCT
ejpam-6171	146	3	?	?	PUNCT
ejpam-6171	146	4	0	0	PUNCT
ejpam-6171	147	1	=	=	SYM
ejpam-6171	147	2	(	(	PUNCT
ejpam-6171	147	3	z	z	NOUN
ejpam-6171	147	4	?	?	PUNCT
ejpam-6171	147	5	y	y	X
ejpam-6171	147	6	)	)	PUNCT
ejpam-6171	147	7	?	?	PUNCT
ejpam-6171	148	1	(	(	PUNCT
ejpam-6171	148	2	z	z	NOUN
ejpam-6171	148	3	?	?	PUNCT
ejpam-6171	148	4	x	x	X
ejpam-6171	148	5	)	)	PUNCT
ejpam-6171	148	6	)	)	PUNCT
ejpam-6171	148	7	(	(	PUNCT
ejpam-6171	148	8	2.13	2.13	NUM
ejpam-6171	148	9	)	)	PUNCT
ejpam-6171	148	10	(	(	PUNCT
ejpam-6171	148	11	∀x	∀x	X
ejpam-6171	148	12	,	,	PUNCT
ejpam-6171	148	13	y	y	PROPN
ejpam-6171	148	14	,	,	PUNCT
ejpam-6171	148	15	z	z	PROPN
ejpam-6171	148	16	∈	∈	PROPN
ejpam-6171	148	17	x)(x	x)(x	PROPN
ejpam-6171	148	18	?	?	PUNCT
ejpam-6171	149	1	y	y	PROPN
ejpam-6171	149	2	=	=	SYM
ejpam-6171	149	3	0	0	PROPN
ejpam-6171	149	4	⇔	⇔	X
ejpam-6171	149	5	(	(	PUNCT
ejpam-6171	149	6	z	z	NOUN
ejpam-6171	149	7	?	?	PUNCT
ejpam-6171	150	1	x	x	X
ejpam-6171	150	2	)	)	PUNCT
ejpam-6171	150	3	?	?	PUNCT
ejpam-6171	151	1	(	(	PUNCT
ejpam-6171	151	2	z	z	NOUN
ejpam-6171	151	3	?	?	PUNCT
ejpam-6171	152	1	y	y	X
ejpam-6171	152	2	)	)	PUNCT
ejpam-6171	152	3	=	=	SYM
ejpam-6171	152	4	0	0	NUM
ejpam-6171	152	5	)	)	PUNCT
ejpam-6171	152	6	(	(	PUNCT
ejpam-6171	152	7	2.14	2.14	NUM
ejpam-6171	152	8	)	)	PUNCT
ejpam-6171	152	9	(	(	PUNCT
ejpam-6171	152	10	∀x	∀x	X
ejpam-6171	152	11	,	,	PUNCT
ejpam-6171	152	12	y	y	PROPN
ejpam-6171	152	13	,	,	PUNCT
ejpam-6171	152	14	z	z	PROPN
ejpam-6171	152	15	∈	∈	PROPN
ejpam-6171	152	16	x)(x	x)(x	PROPN
ejpam-6171	152	17	?	?	PUNCT
ejpam-6171	153	1	y	y	PROPN
ejpam-6171	153	2	=	=	SYM
ejpam-6171	153	3	0	0	PROPN
ejpam-6171	153	4	⇔	⇔	X
ejpam-6171	153	5	(	(	PUNCT
ejpam-6171	153	6	x	x	PROPN
ejpam-6171	153	7	?	?	PUNCT
ejpam-6171	154	1	z	z	X
ejpam-6171	154	2	)	)	PUNCT
ejpam-6171	154	3	?	?	PUNCT
ejpam-6171	155	1	(	(	PUNCT
ejpam-6171	155	2	y	y	NOUN
ejpam-6171	155	3	?	?	PUNCT
ejpam-6171	156	1	z	z	X
ejpam-6171	156	2	)	)	PUNCT
ejpam-6171	156	3	=	=	SYM
ejpam-6171	156	4	0	0	X
ejpam-6171	156	5	)	)	PUNCT
ejpam-6171	156	6	(	(	PUNCT
ejpam-6171	156	7	2.15	2.15	NUM
ejpam-6171	156	8	)	)	PUNCT
ejpam-6171	156	9	the	the	DET
ejpam-6171	156	10	right	right	NOUN
ejpam-6171	156	11	and	and	CCONJ
ejpam-6171	156	12	the	the	DET
ejpam-6171	156	13	left	left	ADJ
ejpam-6171	156	14	cancellation	cancellation	NOUN
ejpam-6171	156	15	laws	law	NOUN
ejpam-6171	156	16	hold	hold	VERB
ejpam-6171	156	17	(	(	PUNCT
ejpam-6171	156	18	2.16	2.16	NUM
ejpam-6171	156	19	)	)	PUNCT
ejpam-6171	156	20	k.	k.	NOUN
ejpam-6171	157	1	suayngam	suayngam	INTJ
ejpam-6171	157	2	et	et	PROPN
ejpam-6171	157	3	al	al	PROPN
ejpam-6171	157	4	.	.	PUNCT
ejpam-6171	157	5	/	/	SYM
ejpam-6171	157	6	eur	eur	PROPN
ejpam-6171	157	7	.	.	PUNCT
ejpam-6171	158	1	j.	j.	PROPN
ejpam-6171	158	2	pure	pure	PROPN
ejpam-6171	158	3	appl	appl	PROPN
ejpam-6171	158	4	.	.	PROPN
ejpam-6171	158	5	math	math	PROPN
ejpam-6171	158	6	,	,	PUNCT
ejpam-6171	158	7	18	18	NUM
ejpam-6171	158	8	(	(	PUNCT
ejpam-6171	158	9	3	3	NUM
ejpam-6171	158	10	)	)	PUNCT
ejpam-6171	158	11	(	(	PUNCT
ejpam-6171	158	12	2025	2025	NUM
ejpam-6171	158	13	)	)	PUNCT
ejpam-6171	158	14	,	,	PUNCT
ejpam-6171	158	15	6171	6171	NUM
ejpam-6171	158	16	6	6	NUM
ejpam-6171	158	17	of	of	ADP
ejpam-6171	158	18	28	28	NUM
ejpam-6171	158	19	within	within	ADP
ejpam-6171	158	20	iup	iup	PROPN
ejpam-6171	158	21	-	-	PUNCT
ejpam-6171	158	22	algebras	algebras	PROPN
ejpam-6171	158	23	,	,	PUNCT
ejpam-6171	158	24	four	four	NUM
ejpam-6171	158	25	fundamental	fundamental	ADJ
ejpam-6171	158	26	subsets	subset	NOUN
ejpam-6171	158	27	stand	stand	VERB
ejpam-6171	158	28	out	out	ADP
ejpam-6171	158	29	:	:	PUNCT
ejpam-6171	158	30	iup	iup	PROPN
ejpam-6171	158	31	-	-	PUNCT
ejpam-6171	158	32	subalgebras	subalgebras	PROPN
ejpam-6171	158	33	,	,	PUNCT
ejpam-6171	158	34	iupfilters	iupfilter	NOUN
ejpam-6171	158	35	,	,	PUNCT
ejpam-6171	158	36	iup	iup	NOUN
ejpam-6171	158	37	-	-	PUNCT
ejpam-6171	158	38	ideals	ideal	NOUN
ejpam-6171	158	39	,	,	PUNCT
ejpam-6171	158	40	and	and	CCONJ
ejpam-6171	158	41	strong	strong	ADJ
ejpam-6171	158	42	iup	iup	NOUN
ejpam-6171	158	43	-	-	PUNCT
ejpam-6171	158	44	ideals	ideal	NOUN
ejpam-6171	158	45	.	.	PUNCT
ejpam-6171	159	1	these	these	DET
ejpam-6171	159	2	subsets	subset	NOUN
ejpam-6171	159	3	form	form	VERB
ejpam-6171	159	4	a	a	DET
ejpam-6171	159	5	critical	critical	ADJ
ejpam-6171	159	6	framework	framework	NOUN
ejpam-6171	159	7	that	that	PRON
ejpam-6171	159	8	deepens	deepen	VERB
ejpam-6171	159	9	our	our	PRON
ejpam-6171	159	10	understanding	understanding	NOUN
ejpam-6171	159	11	and	and	CCONJ
ejpam-6171	159	12	facilitates	facilitate	VERB
ejpam-6171	159	13	the	the	DET
ejpam-6171	159	14	application	application	NOUN
ejpam-6171	159	15	of	of	ADP
ejpam-6171	159	16	iup	iup	NOUN
ejpam-6171	159	17	-	-	PUNCT
ejpam-6171	159	18	algebras	algebras	PROPN
ejpam-6171	159	19	across	across	ADP
ejpam-6171	159	20	different	different	ADJ
ejpam-6171	159	21	mathematical	mathematical	ADJ
ejpam-6171	159	22	contexts	contexts	NOUN
ejpam-6171	159	23	.	.	PUNCT
ejpam-6171	160	1	definition	definition	NOUN
ejpam-6171	160	2	2	2	NUM
ejpam-6171	160	3	.	.	PUNCT
ejpam-6171	161	1	[	[	X
ejpam-6171	161	2	9	9	NUM
ejpam-6171	161	3	]	]	PUNCT
ejpam-6171	161	4	a	a	DET
ejpam-6171	161	5	nonempty	nonempty	NOUN
ejpam-6171	161	6	subset	subset	VERB
ejpam-6171	161	7	s	s	NOUN
ejpam-6171	161	8	of	of	ADP
ejpam-6171	161	9	x	x	PRON
ejpam-6171	161	10	is	be	AUX
ejpam-6171	161	11	called	call	VERB
ejpam-6171	161	12	(	(	PUNCT
ejpam-6171	161	13	i	i	NOUN
ejpam-6171	161	14	)	)	PUNCT
ejpam-6171	161	15	an	an	DET
ejpam-6171	161	16	iup	iup	NOUN
ejpam-6171	161	17	-	-	PUNCT
ejpam-6171	161	18	subalgebra	subalgebra	NOUN
ejpam-6171	161	19	of	of	ADP
ejpam-6171	161	20	x	x	PRON
ejpam-6171	161	21	if	if	SCONJ
ejpam-6171	161	22	it	it	PRON
ejpam-6171	161	23	satisfies	satisfy	VERB
ejpam-6171	161	24	the	the	DET
ejpam-6171	161	25	following	follow	VERB
ejpam-6171	161	26	condition	condition	NOUN
ejpam-6171	161	27	:	:	PUNCT
ejpam-6171	161	28	(	(	PUNCT
ejpam-6171	161	29	∀x	∀x	X
ejpam-6171	161	30	,	,	PUNCT
ejpam-6171	161	31	y	y	PROPN
ejpam-6171	161	32	∈	∈	PROPN
ejpam-6171	161	33	s)(x	s)(x	PROPN
ejpam-6171	161	34	?	?	PUNCT
ejpam-6171	162	1	y	y	PROPN
ejpam-6171	162	2	∈	∈	PROPN
ejpam-6171	162	3	s	s	PART
ejpam-6171	162	4	)	)	PUNCT
ejpam-6171	162	5	(	(	PUNCT
ejpam-6171	162	6	2.17	2.17	NUM
ejpam-6171	162	7	)	)	PUNCT
ejpam-6171	162	8	(	(	PUNCT
ejpam-6171	162	9	ii	ii	NOUN
ejpam-6171	162	10	)	)	PUNCT
ejpam-6171	162	11	an	an	DET
ejpam-6171	162	12	iup	iup	NOUN
ejpam-6171	162	13	-	-	PUNCT
ejpam-6171	162	14	filter	filter	NOUN
ejpam-6171	162	15	of	of	ADP
ejpam-6171	162	16	x	x	PRON
ejpam-6171	162	17	if	if	SCONJ
ejpam-6171	162	18	it	it	PRON
ejpam-6171	162	19	satisfies	satisfy	VERB
ejpam-6171	162	20	the	the	DET
ejpam-6171	162	21	following	follow	VERB
ejpam-6171	162	22	conditions	condition	NOUN
ejpam-6171	162	23	:	:	PUNCT
ejpam-6171	162	24	the	the	DET
ejpam-6171	162	25	constant	constant	ADJ
ejpam-6171	162	26	0	0	NUM
ejpam-6171	162	27	of	of	ADP
ejpam-6171	162	28	x	x	PRON
ejpam-6171	162	29	is	be	AUX
ejpam-6171	162	30	in	in	ADP
ejpam-6171	162	31	s	s	PROPN
ejpam-6171	162	32	(	(	PUNCT
ejpam-6171	162	33	2.18	2.18	NUM
ejpam-6171	162	34	)	)	PUNCT
ejpam-6171	162	35	(	(	PUNCT
ejpam-6171	162	36	∀x	∀x	X
ejpam-6171	162	37	,	,	PUNCT
ejpam-6171	162	38	y	y	PROPN
ejpam-6171	162	39	∈	∈	PROPN
ejpam-6171	162	40	x)(x	x)(x	PROPN
ejpam-6171	162	41	?	?	PUNCT
ejpam-6171	163	1	y	y	PROPN
ejpam-6171	163	2	∈	∈	PROPN
ejpam-6171	163	3	s	s	PROPN
ejpam-6171	163	4	,	,	PUNCT
ejpam-6171	163	5	x	x	SYM
ejpam-6171	163	6	∈	∈	PROPN
ejpam-6171	163	7	s	s	PART
ejpam-6171	163	8	⇒	⇒	NOUN
ejpam-6171	163	9	y	y	PROPN
ejpam-6171	163	10	∈	∈	PROPN
ejpam-6171	163	11	s	s	PART
ejpam-6171	163	12	)	)	PUNCT
ejpam-6171	163	13	(	(	PUNCT
ejpam-6171	163	14	2.19	2.19	NUM
ejpam-6171	163	15	)	)	PUNCT
ejpam-6171	163	16	(	(	PUNCT
ejpam-6171	163	17	iii	iii	X
ejpam-6171	163	18	)	)	PUNCT
ejpam-6171	163	19	an	an	DET
ejpam-6171	163	20	iup	iup	NOUN
ejpam-6171	163	21	-	-	PUNCT
ejpam-6171	163	22	ideal	ideal	NOUN
ejpam-6171	163	23	of	of	ADP
ejpam-6171	163	24	x	x	PRON
ejpam-6171	163	25	if	if	SCONJ
ejpam-6171	163	26	it	it	PRON
ejpam-6171	163	27	satisfies	satisfy	VERB
ejpam-6171	163	28	the	the	DET
ejpam-6171	163	29	condition	condition	NOUN
ejpam-6171	163	30	(	(	PUNCT
ejpam-6171	163	31	2.18	2.18	NUM
ejpam-6171	163	32	)	)	PUNCT
ejpam-6171	163	33	and	and	CCONJ
ejpam-6171	163	34	the	the	DET
ejpam-6171	163	35	following	follow	VERB
ejpam-6171	163	36	condition	condition	NOUN
ejpam-6171	163	37	:	:	PUNCT
ejpam-6171	163	38	(	(	PUNCT
ejpam-6171	163	39	∀x	∀x	X
ejpam-6171	163	40	,	,	PUNCT
ejpam-6171	163	41	y	y	PROPN
ejpam-6171	163	42	,	,	PUNCT
ejpam-6171	163	43	z	z	PROPN
ejpam-6171	163	44	∈	∈	PROPN
ejpam-6171	163	45	x)(x	x)(x	PROPN
ejpam-6171	163	46	?	?	PUNCT
ejpam-6171	164	1	(	(	PUNCT
ejpam-6171	164	2	y	y	NOUN
ejpam-6171	164	3	?	?	PUNCT
ejpam-6171	165	1	z	z	X
ejpam-6171	165	2	)	)	PUNCT
ejpam-6171	165	3	∈	∈	PROPN
ejpam-6171	165	4	s	s	PROPN
ejpam-6171	165	5	,	,	PUNCT
ejpam-6171	165	6	y	y	PROPN
ejpam-6171	165	7	∈	∈	PROPN
ejpam-6171	165	8	s	s	PART
ejpam-6171	165	9	⇒	⇒	NOUN
ejpam-6171	165	10	x	x	PUNCT
ejpam-6171	165	11	?	?	PUNCT
ejpam-6171	165	12	z	z	PUNCT
ejpam-6171	165	13	∈	∈	PROPN
ejpam-6171	165	14	s	s	PART
ejpam-6171	165	15	)	)	PUNCT
ejpam-6171	165	16	(	(	PUNCT
ejpam-6171	165	17	2.20	2.20	NUM
ejpam-6171	165	18	)	)	PUNCT
ejpam-6171	165	19	(	(	PUNCT
ejpam-6171	165	20	iv	iv	X
ejpam-6171	165	21	)	)	PUNCT
ejpam-6171	165	22	a	a	DET
ejpam-6171	165	23	strong	strong	ADJ
ejpam-6171	165	24	iup	iup	NOUN
ejpam-6171	165	25	-	-	PUNCT
ejpam-6171	165	26	ideal	ideal	NOUN
ejpam-6171	165	27	of	of	ADP
ejpam-6171	165	28	x	x	PRON
ejpam-6171	165	29	if	if	SCONJ
ejpam-6171	165	30	it	it	PRON
ejpam-6171	165	31	satisfies	satisfy	VERB
ejpam-6171	165	32	the	the	DET
ejpam-6171	165	33	following	follow	VERB
ejpam-6171	165	34	condition	condition	NOUN
ejpam-6171	165	35	:	:	PUNCT
ejpam-6171	165	36	(	(	PUNCT
ejpam-6171	165	37	∀x	∀x	X
ejpam-6171	165	38	,	,	PUNCT
ejpam-6171	165	39	y	y	PROPN
ejpam-6171	165	40	∈	∈	PROPN
ejpam-6171	165	41	x)(y	x)(y	PUNCT
ejpam-6171	166	1	∈	∈	PROPN
ejpam-6171	166	2	s	s	PART
ejpam-6171	166	3	⇒	⇒	NOUN
ejpam-6171	166	4	x	x	PUNCT
ejpam-6171	166	5	?	?	PUNCT
ejpam-6171	167	1	y	y	PROPN
ejpam-6171	167	2	∈	∈	PROPN
ejpam-6171	167	3	s	s	PART
ejpam-6171	167	4	)	)	PUNCT
ejpam-6171	167	5	(	(	PUNCT
ejpam-6171	167	6	2.21	2.21	NUM
ejpam-6171	167	7	)	)	PUNCT
ejpam-6171	167	8	according	accord	VERB
ejpam-6171	167	9	to	to	ADP
ejpam-6171	167	10	[	[	X
ejpam-6171	167	11	9	9	NUM
ejpam-6171	167	12	]	]	PUNCT
ejpam-6171	167	13	,	,	PUNCT
ejpam-6171	167	14	iup	iup	NOUN
ejpam-6171	167	15	-	-	PUNCT
ejpam-6171	167	16	filters	filter	NOUN
ejpam-6171	167	17	represent	represent	VERB
ejpam-6171	167	18	a	a	DET
ejpam-6171	167	19	unifying	unifying	ADJ
ejpam-6171	167	20	concept	concept	NOUN
ejpam-6171	167	21	encompassing	encompass	VERB
ejpam-6171	167	22	both	both	CCONJ
ejpam-6171	167	23	iupideals	iupideal	NOUN
ejpam-6171	167	24	and	and	CCONJ
ejpam-6171	167	25	iup	iup	NOUN
ejpam-6171	167	26	-	-	PUNCT
ejpam-6171	167	27	subalgebras	subalgebras	PROPN
ejpam-6171	167	28	.	.	PUNCT
ejpam-6171	168	1	these	these	DET
ejpam-6171	168	2	two	two	NUM
ejpam-6171	168	3	subsets	subset	NOUN
ejpam-6171	168	4	,	,	PUNCT
ejpam-6171	168	5	iup	iup	NOUN
ejpam-6171	168	6	-	-	PUNCT
ejpam-6171	168	7	ideals	ideal	NOUN
ejpam-6171	168	8	and	and	CCONJ
ejpam-6171	168	9	iup	iup	NOUN
ejpam-6171	168	10	-	-	PUNCT
ejpam-6171	168	11	subalgebras	subalgebras	PROPN
ejpam-6171	168	12	,	,	PUNCT
ejpam-6171	168	13	are	be	AUX
ejpam-6171	168	14	generalizations	generalization	NOUN
ejpam-6171	168	15	of	of	ADP
ejpam-6171	168	16	strong	strong	ADJ
ejpam-6171	168	17	iup	iup	NOUN
ejpam-6171	168	18	-	-	PUNCT
ejpam-6171	168	19	ideals	ideal	NOUN
ejpam-6171	168	20	.	.	PUNCT
ejpam-6171	169	1	particularly	particularly	ADV
ejpam-6171	169	2	,	,	PUNCT
ejpam-6171	169	3	in	in	ADP
ejpam-6171	169	4	an	an	DET
ejpam-6171	169	5	iup	iup	NOUN
ejpam-6171	169	6	-	-	PUNCT
ejpam-6171	169	7	algebra	algebra	NOUN
ejpam-6171	169	8	x	x	NOUN
ejpam-6171	169	9	,	,	PUNCT
ejpam-6171	169	10	strong	strong	ADJ
ejpam-6171	169	11	iup	iup	NOUN
ejpam-6171	169	12	-	-	PUNCT
ejpam-6171	169	13	ideals	ideal	NOUN
ejpam-6171	169	14	are	be	AUX
ejpam-6171	169	15	equivalent	equivalent	ADJ
ejpam-6171	169	16	to	to	ADP
ejpam-6171	169	17	the	the	DET
ejpam-6171	169	18	entire	entire	ADJ
ejpam-6171	169	19	algebra	algebra	NOUN
ejpam-6171	169	20	x	x	X
ejpam-6171	169	21	itself	itself	PRON
ejpam-6171	169	22	.	.	PUNCT
ejpam-6171	170	1	this	this	DET
ejpam-6171	170	2	hierarchical	hierarchical	ADJ
ejpam-6171	170	3	relationship	relationship	NOUN
ejpam-6171	170	4	among	among	ADP
ejpam-6171	170	5	these	these	DET
ejpam-6171	170	6	subsets	subset	NOUN
ejpam-6171	170	7	is	be	AUX
ejpam-6171	170	8	visually	visually	ADV
ejpam-6171	170	9	represented	represent	VERB
ejpam-6171	170	10	in	in	ADP
ejpam-6171	170	11	figure	figure	NOUN
ejpam-6171	170	12	1	1	NUM
ejpam-6171	170	13	,	,	PUNCT
ejpam-6171	170	14	illustrating	illustrate	VERB
ejpam-6171	170	15	the	the	DET
ejpam-6171	170	16	structure	structure	NOUN
ejpam-6171	170	17	of	of	ADP
ejpam-6171	170	18	special	special	ADJ
ejpam-6171	170	19	subsets	subset	NOUN
ejpam-6171	170	20	within	within	ADP
ejpam-6171	170	21	iup	iup	PROPN
ejpam-6171	170	22	-	-	PUNCT
ejpam-6171	170	23	algebras	algebras	PROPN
ejpam-6171	170	24	.	.	PUNCT
ejpam-6171	171	1	iup	iup	PROPN
ejpam-6171	171	2	-	-	PUNCT
ejpam-6171	171	3	filter	filter	NOUN
ejpam-6171	171	4	iup	iup	NOUN
ejpam-6171	171	5	-	-	PUNCT
ejpam-6171	171	6	ideal	ideal	NOUN
ejpam-6171	171	7	iup	iup	NOUN
ejpam-6171	171	8	-	-	PUNCT
ejpam-6171	171	9	subalgebra	subalgebra	NOUN
ejpam-6171	171	10	strong	strong	ADJ
ejpam-6171	171	11	iup	iup	NOUN
ejpam-6171	171	12	-	-	PUNCT
ejpam-6171	171	13	ideal	ideal	NOUN
ejpam-6171	171	14	an	an	DET
ejpam-6171	171	15	iup	iup	NOUN
ejpam-6171	171	16	-	-	PUNCT
ejpam-6171	171	17	algebra	algebra	NOUN
ejpam-6171	171	18	x	x	VERB
ejpam-6171	171	19	figure	figure	NOUN
ejpam-6171	171	20	1	1	NUM
ejpam-6171	171	21	:	:	PUNCT
ejpam-6171	171	22	special	special	ADJ
ejpam-6171	171	23	subsets	subset	NOUN
ejpam-6171	171	24	of	of	ADP
ejpam-6171	171	25	iup	iup	NOUN
ejpam-6171	171	26	-	-	PUNCT
ejpam-6171	171	27	algebras	algebras	PROPN
ejpam-6171	171	28	3	3	NUM
ejpam-6171	171	29	.	.	PUNCT
ejpam-6171	171	30	main	main	ADJ
ejpam-6171	171	31	results	result	NOUN
ejpam-6171	171	32	the	the	DET
ejpam-6171	171	33	study	study	NOUN
ejpam-6171	171	34	of	of	ADP
ejpam-6171	171	35	algebraic	algebraic	ADJ
ejpam-6171	171	36	structures	structure	NOUN
ejpam-6171	171	37	has	have	AUX
ejpam-6171	171	38	continually	continually	ADV
ejpam-6171	171	39	evolved	evolve	VERB
ejpam-6171	171	40	to	to	PART
ejpam-6171	171	41	incorporate	incorporate	VERB
ejpam-6171	171	42	concepts	concept	NOUN
ejpam-6171	171	43	that	that	PRON
ejpam-6171	171	44	better	well	ADV
ejpam-6171	171	45	represent	represent	VERB
ejpam-6171	171	46	uncertainty	uncertainty	NOUN
ejpam-6171	171	47	and	and	CCONJ
ejpam-6171	171	48	imprecision	imprecision	NOUN
ejpam-6171	171	49	in	in	ADP
ejpam-6171	171	50	mathematical	mathematical	ADJ
ejpam-6171	171	51	modeling	modeling	NOUN
ejpam-6171	171	52	.	.	PUNCT
ejpam-6171	172	1	among	among	ADP
ejpam-6171	172	2	these	these	DET
ejpam-6171	172	3	k.	k.	PROPN
ejpam-6171	172	4	suayngam	suayngam	PROPN
ejpam-6171	172	5	et	et	PROPN
ejpam-6171	172	6	al	al	PROPN
ejpam-6171	172	7	.	.	PUNCT
ejpam-6171	172	8	/	/	SYM
ejpam-6171	172	9	eur	eur	PROPN
ejpam-6171	172	10	.	.	PUNCT
ejpam-6171	173	1	j.	j.	PROPN
ejpam-6171	173	2	pure	pure	PROPN
ejpam-6171	173	3	appl	appl	PROPN
ejpam-6171	173	4	.	.	PROPN
ejpam-6171	173	5	math	math	PROPN
ejpam-6171	173	6	,	,	PUNCT
ejpam-6171	173	7	18	18	NUM
ejpam-6171	173	8	(	(	PUNCT
ejpam-6171	173	9	3	3	NUM
ejpam-6171	173	10	)	)	PUNCT
ejpam-6171	173	11	(	(	PUNCT
ejpam-6171	173	12	2025	2025	NUM
ejpam-6171	173	13	)	)	PUNCT
ejpam-6171	173	14	,	,	PUNCT
ejpam-6171	173	15	6171	6171	NUM
ejpam-6171	173	16	7	7	NUM
ejpam-6171	173	17	of	of	ADP
ejpam-6171	173	18	28	28	NUM
ejpam-6171	173	19	structures	structure	NOUN
ejpam-6171	173	20	,	,	PUNCT
ejpam-6171	173	21	iup	iup	NOUN
ejpam-6171	173	22	-	-	PUNCT
ejpam-6171	173	23	algebras	algebras	PROPN
ejpam-6171	173	24	have	have	AUX
ejpam-6171	173	25	gained	gain	VERB
ejpam-6171	173	26	attention	attention	NOUN
ejpam-6171	173	27	due	due	ADP
ejpam-6171	173	28	to	to	ADP
ejpam-6171	173	29	their	their	PRON
ejpam-6171	173	30	unique	unique	ADJ
ejpam-6171	173	31	operational	operational	ADJ
ejpam-6171	173	32	properties	property	NOUN
ejpam-6171	173	33	and	and	CCONJ
ejpam-6171	173	34	applicability	applicability	NOUN
ejpam-6171	173	35	in	in	ADP
ejpam-6171	173	36	logic	logic	NOUN
ejpam-6171	173	37	and	and	CCONJ
ejpam-6171	173	38	computational	computational	ADJ
ejpam-6171	173	39	mathematics	mathematic	NOUN
ejpam-6171	173	40	.	.	PUNCT
ejpam-6171	174	1	at	at	ADP
ejpam-6171	174	2	the	the	DET
ejpam-6171	174	3	same	same	ADJ
ejpam-6171	174	4	time	time	NOUN
ejpam-6171	174	5	,	,	PUNCT
ejpam-6171	174	6	pythagorean	pythagorean	PROPN
ejpam-6171	174	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	174	8	sets	set	NOUN
ejpam-6171	174	9	(	(	PUNCT
ejpam-6171	174	10	pnss	pns	NOUN
ejpam-6171	174	11	)	)	PUNCT
ejpam-6171	174	12	have	have	AUX
ejpam-6171	174	13	emerged	emerge	VERB
ejpam-6171	174	14	as	as	ADP
ejpam-6171	174	15	a	a	DET
ejpam-6171	174	16	powerful	powerful	ADJ
ejpam-6171	174	17	extension	extension	NOUN
ejpam-6171	174	18	of	of	ADP
ejpam-6171	174	19	fuzzy	fuzzy	ADJ
ejpam-6171	174	20	and	and	CCONJ
ejpam-6171	174	21	neutrosophic	neutrosophic	ADJ
ejpam-6171	174	22	sets	set	NOUN
ejpam-6171	174	23	,	,	PUNCT
ejpam-6171	174	24	offering	offer	VERB
ejpam-6171	174	25	enhanced	enhance	VERB
ejpam-6171	174	26	flexibility	flexibility	NOUN
ejpam-6171	174	27	in	in	ADP
ejpam-6171	174	28	handling	handle	VERB
ejpam-6171	174	29	degrees	degree	NOUN
ejpam-6171	174	30	of	of	ADP
ejpam-6171	174	31	truth	truth	NOUN
ejpam-6171	174	32	,	,	PUNCT
ejpam-6171	174	33	indeterminacy	indeterminacy	NOUN
ejpam-6171	174	34	,	,	PUNCT
ejpam-6171	174	35	and	and	CCONJ
ejpam-6171	174	36	falsity	falsity	NOUN
ejpam-6171	174	37	.	.	PUNCT
ejpam-6171	175	1	given	give	VERB
ejpam-6171	175	2	their	their	PRON
ejpam-6171	175	3	potential	potential	NOUN
ejpam-6171	175	4	,	,	PUNCT
ejpam-6171	175	5	integrating	integrate	VERB
ejpam-6171	175	6	pythagorean	pythagorean	PROPN
ejpam-6171	175	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	175	8	sets	set	NOUN
ejpam-6171	175	9	into	into	ADP
ejpam-6171	175	10	iup	iup	NOUN
ejpam-6171	175	11	-	-	PUNCT
ejpam-6171	175	12	algebras	algebras	PROPN
ejpam-6171	175	13	presents	present	VERB
ejpam-6171	175	14	an	an	DET
ejpam-6171	175	15	opportunity	opportunity	NOUN
ejpam-6171	175	16	to	to	PART
ejpam-6171	175	17	develop	develop	VERB
ejpam-6171	175	18	a	a	DET
ejpam-6171	175	19	more	more	ADV
ejpam-6171	175	20	robust	robust	ADJ
ejpam-6171	175	21	algebraic	algebraic	ADJ
ejpam-6171	175	22	framework	framework	NOUN
ejpam-6171	175	23	capable	capable	ADJ
ejpam-6171	175	24	of	of	ADP
ejpam-6171	175	25	capturing	capture	VERB
ejpam-6171	175	26	a	a	DET
ejpam-6171	175	27	wider	wide	ADJ
ejpam-6171	175	28	range	range	NOUN
ejpam-6171	175	29	of	of	ADP
ejpam-6171	175	30	uncertain	uncertain	ADJ
ejpam-6171	175	31	scenarios	scenario	NOUN
ejpam-6171	175	32	.	.	PUNCT
ejpam-6171	176	1	this	this	DET
ejpam-6171	176	2	section	section	NOUN
ejpam-6171	176	3	formalizes	formalize	VERB
ejpam-6171	176	4	the	the	DET
ejpam-6171	176	5	key	key	ADJ
ejpam-6171	176	6	notions	notion	NOUN
ejpam-6171	176	7	of	of	ADP
ejpam-6171	176	8	pythagorean	pythagorean	PROPN
ejpam-6171	176	9	neutrosophic	neutrosophic	PROPN
ejpam-6171	176	10	iup	iup	PROPN
ejpam-6171	176	11	-	-	PUNCT
ejpam-6171	176	12	subalgebras	subalgebras	PROPN
ejpam-6171	176	13	,	,	PUNCT
ejpam-6171	176	14	pythagorean	pythagorean	PROPN
ejpam-6171	176	15	neutrosophic	neutrosophic	PROPN
ejpam-6171	176	16	iup	iup	PROPN
ejpam-6171	176	17	-	-	PUNCT
ejpam-6171	176	18	ideals	ideal	NOUN
ejpam-6171	176	19	,	,	PUNCT
ejpam-6171	176	20	pythagorean	pythagorean	PROPN
ejpam-6171	176	21	neutrosophic	neutrosophic	PROPN
ejpam-6171	176	22	iup	iup	NOUN
ejpam-6171	176	23	-	-	PUNCT
ejpam-6171	176	24	filters	filter	NOUN
ejpam-6171	176	25	,	,	PUNCT
ejpam-6171	176	26	and	and	CCONJ
ejpam-6171	176	27	pythagorean	pythagorean	PROPN
ejpam-6171	176	28	neutrosophic	neutrosophic	PROPN
ejpam-6171	176	29	strong	strong	ADJ
ejpam-6171	176	30	iup	iup	NOUN
ejpam-6171	176	31	-	-	PUNCT
ejpam-6171	176	32	ideals	ideal	NOUN
ejpam-6171	176	33	and	and	CCONJ
ejpam-6171	176	34	explores	explore	VERB
ejpam-6171	176	35	their	their	PRON
ejpam-6171	176	36	fundamental	fundamental	ADJ
ejpam-6171	176	37	properties	property	NOUN
ejpam-6171	176	38	.	.	PUNCT
ejpam-6171	177	1	by	by	ADP
ejpam-6171	177	2	establishing	establish	VERB
ejpam-6171	177	3	essential	essential	ADJ
ejpam-6171	177	4	definitions	definition	NOUN
ejpam-6171	177	5	,	,	PUNCT
ejpam-6171	177	6	theorems	theorem	NOUN
ejpam-6171	177	7	,	,	PUNCT
ejpam-6171	177	8	and	and	CCONJ
ejpam-6171	177	9	proofs	proof	NOUN
ejpam-6171	177	10	,	,	PUNCT
ejpam-6171	177	11	we	we	PRON
ejpam-6171	177	12	aim	aim	VERB
ejpam-6171	177	13	to	to	PART
ejpam-6171	177	14	provide	provide	VERB
ejpam-6171	177	15	a	a	DET
ejpam-6171	177	16	structured	structured	ADJ
ejpam-6171	177	17	foundation	foundation	NOUN
ejpam-6171	177	18	for	for	ADP
ejpam-6171	177	19	understanding	understand	VERB
ejpam-6171	177	20	how	how	SCONJ
ejpam-6171	177	21	pythagorean	pythagorean	PROPN
ejpam-6171	177	22	neutrosophic	neutrosophic	ADJ
ejpam-6171	177	23	sets	set	NOUN
ejpam-6171	177	24	interact	interact	VERB
ejpam-6171	177	25	with	with	ADP
ejpam-6171	177	26	iup	iup	PROPN
ejpam-6171	177	27	-	-	PUNCT
ejpam-6171	177	28	algebraic	algebraic	ADJ
ejpam-6171	177	29	operations	operation	NOUN
ejpam-6171	177	30	.	.	PUNCT
ejpam-6171	178	1	furthermore	furthermore	ADV
ejpam-6171	178	2	,	,	PUNCT
ejpam-6171	178	3	we	we	PRON
ejpam-6171	178	4	analyze	analyze	VERB
ejpam-6171	178	5	the	the	DET
ejpam-6171	178	6	interrelations	interrelation	NOUN
ejpam-6171	178	7	between	between	ADP
ejpam-6171	178	8	these	these	DET
ejpam-6171	178	9	newly	newly	ADV
ejpam-6171	178	10	defined	define	VERB
ejpam-6171	178	11	subsets	subset	NOUN
ejpam-6171	178	12	and	and	CCONJ
ejpam-6171	178	13	their	their	PRON
ejpam-6171	178	14	respective	respective	ADJ
ejpam-6171	178	15	level	level	NOUN
ejpam-6171	178	16	subsets	subset	NOUN
ejpam-6171	178	17	,	,	PUNCT
ejpam-6171	178	18	revealing	reveal	VERB
ejpam-6171	178	19	deeper	deep	ADJ
ejpam-6171	178	20	insights	insight	NOUN
ejpam-6171	178	21	into	into	ADP
ejpam-6171	178	22	their	their	PRON
ejpam-6171	178	23	structural	structural	ADJ
ejpam-6171	178	24	coherence	coherence	NOUN
ejpam-6171	178	25	.	.	PUNCT
ejpam-6171	179	1	the	the	DET
ejpam-6171	179	2	results	result	NOUN
ejpam-6171	179	3	obtained	obtain	VERB
ejpam-6171	179	4	not	not	PART
ejpam-6171	179	5	only	only	ADV
ejpam-6171	179	6	extend	extend	VERB
ejpam-6171	179	7	the	the	DET
ejpam-6171	179	8	theoretical	theoretical	ADJ
ejpam-6171	179	9	framework	framework	NOUN
ejpam-6171	179	10	of	of	ADP
ejpam-6171	179	11	iup	iup	NOUN
ejpam-6171	179	12	-	-	PUNCT
ejpam-6171	179	13	algebras	algebras	PROPN
ejpam-6171	179	14	but	but	CCONJ
ejpam-6171	179	15	also	also	ADV
ejpam-6171	179	16	open	open	VERB
ejpam-6171	179	17	new	new	ADJ
ejpam-6171	179	18	avenues	avenue	NOUN
ejpam-6171	179	19	for	for	ADP
ejpam-6171	179	20	their	their	PRON
ejpam-6171	179	21	application	application	NOUN
ejpam-6171	179	22	in	in	ADP
ejpam-6171	179	23	areas	area	NOUN
ejpam-6171	179	24	such	such	ADJ
ejpam-6171	179	25	as	as	ADP
ejpam-6171	179	26	fuzzy	fuzzy	ADJ
ejpam-6171	179	27	logic	logic	NOUN
ejpam-6171	179	28	systems	system	NOUN
ejpam-6171	179	29	,	,	PUNCT
ejpam-6171	179	30	artificial	artificial	ADJ
ejpam-6171	179	31	intelligence	intelligence	NOUN
ejpam-6171	179	32	,	,	PUNCT
ejpam-6171	179	33	and	and	CCONJ
ejpam-6171	179	34	decision	decision	NOUN
ejpam-6171	179	35	analysis	analysis	NOUN
ejpam-6171	179	36	under	under	ADP
ejpam-6171	179	37	uncertainty	uncertainty	NOUN
ejpam-6171	179	38	.	.	PUNCT
ejpam-6171	180	1	definition	definition	NOUN
ejpam-6171	180	2	3	3	NUM
ejpam-6171	180	3	.	.	PUNCT
ejpam-6171	181	1	[	[	X
ejpam-6171	181	2	5	5	X
ejpam-6171	181	3	]	]	PUNCT
ejpam-6171	181	4	let	let	VERB
ejpam-6171	181	5	x	x	PRON
ejpam-6171	181	6	be	be	AUX
ejpam-6171	181	7	a	a	DET
ejpam-6171	181	8	nonempty	nonempty	ADJ
ejpam-6171	181	9	set	set	VERB
ejpam-6171	181	10	(	(	PUNCT
ejpam-6171	181	11	universe	universe	NOUN
ejpam-6171	181	12	)	)	PUNCT
ejpam-6171	181	13	.	.	PUNCT
ejpam-6171	182	1	a	a	DET
ejpam-6171	182	2	pythagorean	pythagorean	PROPN
ejpam-6171	182	3	neutrosophic	neutrosophic	ADJ
ejpam-6171	182	4	set	set	NOUN
ejpam-6171	182	5	(	(	PUNCT
ejpam-6171	182	6	pns	pns	PROPN
ejpam-6171	182	7	)	)	PUNCT
ejpam-6171	182	8	with	with	ADP
ejpam-6171	182	9	pt	pt	PROPN
ejpam-6171	182	10	and	and	CCONJ
ejpam-6171	182	11	pf	pf	PROPN
ejpam-6171	182	12	are	be	AUX
ejpam-6171	182	13	dependent	dependent	ADJ
ejpam-6171	182	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	182	15	components	component	NOUN
ejpam-6171	182	16	on	on	ADP
ejpam-6171	182	17	x	x	X
ejpam-6171	182	18	is	be	AUX
ejpam-6171	182	19	an	an	DET
ejpam-6171	182	20	object	object	NOUN
ejpam-6171	182	21	of	of	ADP
ejpam-6171	182	22	the	the	DET
ejpam-6171	182	23	form	form	NOUN
ejpam-6171	182	24	p	p	X
ejpam-6171	182	25	=	=	X
ejpam-6171	182	26	{	{	PUNCT
ejpam-6171	182	27	(	(	PUNCT
ejpam-6171	182	28	x	x	NOUN
ejpam-6171	182	29	,	,	PUNCT
ejpam-6171	182	30	pt	pt	X
ejpam-6171	182	31	(	(	PUNCT
ejpam-6171	182	32	x),pi(x),pf	x),pi(x),pf	PROPN
ejpam-6171	182	33	(	(	PUNCT
ejpam-6171	182	34	x	x	NOUN
ejpam-6171	182	35	)	)	PUNCT
ejpam-6171	182	36	)	)	PUNCT
ejpam-6171	183	1	|	|	ADV
ejpam-6171	183	2	x	x	SYM
ejpam-6171	183	3	∈	∈	NOUN
ejpam-6171	183	4	x	x	X
ejpam-6171	183	5	}	}	PUNCT
ejpam-6171	183	6	,	,	PUNCT
ejpam-6171	183	7	(	(	PUNCT
ejpam-6171	183	8	3.1	3.1	NUM
ejpam-6171	183	9	)	)	PUNCT
ejpam-6171	183	10	where	where	SCONJ
ejpam-6171	183	11	pt	pt	X
ejpam-6171	183	12	(	(	PUNCT
ejpam-6171	183	13	x),pi(x),pf	x),pi(x),pf	PROPN
ejpam-6171	183	14	(	(	PUNCT
ejpam-6171	183	15	x	x	X
ejpam-6171	183	16	)	)	PUNCT
ejpam-6171	183	17	∈	∈	PROPN
ejpam-6171	184	1	[	[	X
ejpam-6171	184	2	0	0	NUM
ejpam-6171	184	3	,	,	PUNCT
ejpam-6171	184	4	1	1	NUM
ejpam-6171	184	5	]	]	PUNCT
ejpam-6171	184	6	,	,	PUNCT
ejpam-6171	184	7	and	and	CCONJ
ejpam-6171	184	8	0	0	NUM
ejpam-6171	184	9	≤	≤	NOUN
ejpam-6171	184	10	(	(	PUNCT
ejpam-6171	184	11	pt	pt	X
ejpam-6171	184	12	(	(	PUNCT
ejpam-6171	184	13	x	x	NOUN
ejpam-6171	184	14	)	)	PUNCT
ejpam-6171	184	15	)	)	PUNCT
ejpam-6171	184	16	2	2	NUM
ejpam-6171	185	1	+	+	CCONJ
ejpam-6171	185	2	(	(	PUNCT
ejpam-6171	185	3	pi(x	pi(x	NUM
ejpam-6171	185	4	)	)	PUNCT
ejpam-6171	185	5	)	)	PUNCT
ejpam-6171	186	1	2	2	NUM
ejpam-6171	187	1	+	+	CCONJ
ejpam-6171	187	2	(	(	PUNCT
ejpam-6171	187	3	pf	pf	PROPN
ejpam-6171	187	4	(	(	PUNCT
ejpam-6171	187	5	x	x	NOUN
ejpam-6171	187	6	)	)	PUNCT
ejpam-6171	187	7	)	)	PUNCT
ejpam-6171	187	8	2	2	NUM
ejpam-6171	187	9	≤	≤	NUM
ejpam-6171	187	10	2	2	NUM
ejpam-6171	187	11	,	,	PUNCT
ejpam-6171	187	12	for	for	ADP
ejpam-6171	187	13	all	all	DET
ejpam-6171	187	14	x	x	NOUN
ejpam-6171	187	15	in	in	ADP
ejpam-6171	187	16	x.	x.	PROPN
ejpam-6171	187	17	pt	pt	PROPN
ejpam-6171	187	18	(	(	PUNCT
ejpam-6171	187	19	x	x	X
ejpam-6171	187	20	)	)	PUNCT
ejpam-6171	187	21	is	be	AUX
ejpam-6171	187	22	the	the	DET
ejpam-6171	187	23	degree	degree	NOUN
ejpam-6171	187	24	of	of	ADP
ejpam-6171	187	25	membership	membership	NOUN
ejpam-6171	187	26	,	,	PUNCT
ejpam-6171	187	27	pi(x	pi(x	NUM
ejpam-6171	187	28	)	)	PUNCT
ejpam-6171	187	29	is	be	AUX
ejpam-6171	187	30	the	the	DET
ejpam-6171	187	31	degree	degree	NOUN
ejpam-6171	187	32	of	of	ADP
ejpam-6171	187	33	inderminancy	inderminancy	NOUN
ejpam-6171	187	34	,	,	PUNCT
ejpam-6171	187	35	and	and	CCONJ
ejpam-6171	187	36	pf	pf	INTJ
ejpam-6171	187	37	(	(	PUNCT
ejpam-6171	187	38	x	x	X
ejpam-6171	187	39	)	)	PUNCT
ejpam-6171	187	40	is	be	AUX
ejpam-6171	187	41	the	the	DET
ejpam-6171	187	42	degree	degree	NOUN
ejpam-6171	187	43	of	of	ADP
ejpam-6171	187	44	non	non	ADJ
ejpam-6171	187	45	-	-	NOUN
ejpam-6171	187	46	membership	membership	NOUN
ejpam-6171	187	47	of	of	ADP
ejpam-6171	187	48	the	the	DET
ejpam-6171	187	49	element	element	NOUN
ejpam-6171	187	50	x	x	PUNCT
ejpam-6171	187	51	in	in	ADP
ejpam-6171	187	52	the	the	DET
ejpam-6171	187	53	set	set	NOUN
ejpam-6171	188	1	p.	p.	NOUN
ejpam-6171	188	2	here	here	ADV
ejpam-6171	189	1	pt	pt	X
ejpam-6171	189	2	(	(	PUNCT
ejpam-6171	189	3	x	x	NOUN
ejpam-6171	189	4	)	)	PUNCT
ejpam-6171	189	5	and	and	CCONJ
ejpam-6171	189	6	pf	pf	PROPN
ejpam-6171	189	7	(	(	PUNCT
ejpam-6171	189	8	x	x	X
ejpam-6171	189	9	)	)	PUNCT
ejpam-6171	189	10	are	be	AUX
ejpam-6171	189	11	dependent	dependent	ADJ
ejpam-6171	189	12	components	component	NOUN
ejpam-6171	189	13	and	and	CCONJ
ejpam-6171	189	14	pi(x	pi(x	NUM
ejpam-6171	189	15	)	)	PUNCT
ejpam-6171	189	16	is	be	AUX
ejpam-6171	189	17	an	an	DET
ejpam-6171	189	18	independent	independent	ADJ
ejpam-6171	189	19	component	component	NOUN
ejpam-6171	189	20	.	.	PUNCT
ejpam-6171	190	1	to	to	PART
ejpam-6171	190	2	streamline	streamline	VERB
ejpam-6171	190	3	notation	notation	NOUN
ejpam-6171	190	4	,	,	PUNCT
ejpam-6171	190	5	we	we	PRON
ejpam-6171	190	6	represent	represent	VERB
ejpam-6171	190	7	a	a	DET
ejpam-6171	190	8	pns	pns	NOUN
ejpam-6171	190	9	as	as	ADP
ejpam-6171	190	10	p	p	NOUN
ejpam-6171	190	11	=	=	X
ejpam-6171	190	12	(	(	PUNCT
ejpam-6171	190	13	x	x	PROPN
ejpam-6171	190	14	,	,	PUNCT
ejpam-6171	190	15	pt	pt	X
ejpam-6171	190	16	,	,	PUNCT
ejpam-6171	190	17	pi	pi	NOUN
ejpam-6171	190	18	,	,	PUNCT
ejpam-6171	190	19	pf	pf	PROPN
ejpam-6171	190	20	)	)	PUNCT
ejpam-6171	190	21	,	,	PUNCT
ejpam-6171	190	22	where	where	SCONJ
ejpam-6171	190	23	p	p	NOUN
ejpam-6171	190	24	is	be	AUX
ejpam-6171	190	25	defined	define	VERB
ejpam-6171	190	26	as	as	ADP
ejpam-6171	190	27	{	{	PUNCT
ejpam-6171	190	28	(	(	PUNCT
ejpam-6171	190	29	x	x	NOUN
ejpam-6171	190	30	,	,	PUNCT
ejpam-6171	190	31	pt	pt	X
ejpam-6171	190	32	(	(	PUNCT
ejpam-6171	190	33	x),pi(x),pf	x),pi(x),pf	PROPN
ejpam-6171	190	34	(	(	PUNCT
ejpam-6171	190	35	x	x	NOUN
ejpam-6171	190	36	)	)	PUNCT
ejpam-6171	190	37	)	)	PUNCT
ejpam-6171	191	1	|	|	ADV
ejpam-6171	191	2	x	x	SYM
ejpam-6171	191	3	∈	∈	NOUN
ejpam-6171	191	4	x	x	X
ejpam-6171	191	5	}	}	PUNCT
ejpam-6171	191	6	.	.	PUNCT
ejpam-6171	192	1	definition	definition	NOUN
ejpam-6171	192	2	4	4	NUM
ejpam-6171	192	3	.	.	PUNCT
ejpam-6171	193	1	let	let	VERB
ejpam-6171	193	2	f	f	PRON
ejpam-6171	193	3	be	be	AUX
ejpam-6171	193	4	an	an	DET
ejpam-6171	193	5	fs	fs	NOUN
ejpam-6171	193	6	in	in	ADP
ejpam-6171	193	7	x	x	PUNCT
ejpam-6171	193	8	and	and	CCONJ
ejpam-6171	193	9	let	let	VERB
ejpam-6171	193	10	n	n	PRON
ejpam-6171	193	11	be	be	AUX
ejpam-6171	193	12	a	a	DET
ejpam-6171	193	13	positive	positive	ADJ
ejpam-6171	193	14	integer	integer	NOUN
ejpam-6171	193	15	.	.	PUNCT
ejpam-6171	194	1	the	the	DET
ejpam-6171	194	2	fs	fs	PROPN
ejpam-6171	194	3	fn	fn	NOUN
ejpam-6171	194	4	defined	define	VERB
ejpam-6171	194	5	by	by	ADP
ejpam-6171	194	6	fn(x	fn(x	NOUN
ejpam-6171	194	7	)	)	PUNCT
ejpam-6171	194	8	=	=	SYM
ejpam-6171	194	9	f(x	f(x	PROPN
ejpam-6171	194	10	)	)	PUNCT
ejpam-6171	194	11	n	n	CCONJ
ejpam-6171	194	12	for	for	ADP
ejpam-6171	194	13	all	all	DET
ejpam-6171	194	14	x	x	SYM
ejpam-6171	194	15	∈	∈	NOUN
ejpam-6171	194	16	x	x	PUNCT
ejpam-6171	194	17	is	be	AUX
ejpam-6171	194	18	called	call	VERB
ejpam-6171	194	19	the	the	DET
ejpam-6171	194	20	n	n	NOUN
ejpam-6171	194	21	-	-	PUNCT
ejpam-6171	194	22	division	division	NOUN
ejpam-6171	194	23	of	of	ADP
ejpam-6171	194	24	f	f	PROPN
ejpam-6171	194	25	in	in	ADP
ejpam-6171	194	26	x.	x.	NOUN
ejpam-6171	194	27	definition	definition	NOUN
ejpam-6171	194	28	5	5	NUM
ejpam-6171	194	29	.	.	PUNCT
ejpam-6171	195	1	let	let	VERB
ejpam-6171	195	2	p	p	PRON
ejpam-6171	195	3	be	be	AUX
ejpam-6171	195	4	a	a	DET
ejpam-6171	195	5	pns	pns	NOUN
ejpam-6171	195	6	in	in	ADP
ejpam-6171	195	7	a	a	DET
ejpam-6171	195	8	nonempty	nonempty	ADV
ejpam-6171	195	9	set	set	VERB
ejpam-6171	195	10	x	x	PUNCT
ejpam-6171	195	11	and	and	CCONJ
ejpam-6171	195	12	let	let	VERB
ejpam-6171	195	13	n	n	PRON
ejpam-6171	195	14	be	be	AUX
ejpam-6171	195	15	a	a	DET
ejpam-6171	195	16	positive	positive	ADJ
ejpam-6171	195	17	integer	integer	NOUN
ejpam-6171	195	18	.	.	PUNCT
ejpam-6171	196	1	the	the	DET
ejpam-6171	196	2	pns	pns	PROPN
ejpam-6171	196	3	pn	pn	PROPN
ejpam-6171	196	4	=	=	SYM
ejpam-6171	196	5	(	(	PUNCT
ejpam-6171	196	6	x	x	PROPN
ejpam-6171	196	7	,	,	PUNCT
ejpam-6171	196	8	pt	pt	PROPN
ejpam-6171	196	9	n	n	CCONJ
ejpam-6171	196	10	,	,	PUNCT
ejpam-6171	196	11	pin	pin	NOUN
ejpam-6171	196	12	,	,	PUNCT
ejpam-6171	196	13	pf	pf	PROPN
ejpam-6171	196	14	n	n	CCONJ
ejpam-6171	196	15	)	)	PUNCT
ejpam-6171	196	16	is	be	AUX
ejpam-6171	196	17	called	call	VERB
ejpam-6171	196	18	the	the	DET
ejpam-6171	196	19	n	n	NOUN
ejpam-6171	196	20	-	-	PUNCT
ejpam-6171	196	21	division	division	NOUN
ejpam-6171	196	22	of	of	ADP
ejpam-6171	196	23	p	p	NOUN
ejpam-6171	196	24	in	in	ADP
ejpam-6171	196	25	x.	x.	NOUN
ejpam-6171	196	26	definition	definition	NOUN
ejpam-6171	196	27	6	6	NUM
ejpam-6171	196	28	.	.	PUNCT
ejpam-6171	197	1	a	a	DET
ejpam-6171	197	2	pns	pns	NOUN
ejpam-6171	197	3	p	p	NOUN
ejpam-6171	197	4	in	in	ADP
ejpam-6171	197	5	x	x	PROPN
ejpam-6171	197	6	is	be	AUX
ejpam-6171	197	7	called	call	VERB
ejpam-6171	197	8	a	a	DET
ejpam-6171	197	9	pythagorean	pythagorean	PROPN
ejpam-6171	197	10	neutrosophic	neutrosophic	ADJ
ejpam-6171	197	11	iup	iup	NOUN
ejpam-6171	197	12	-	-	PUNCT
ejpam-6171	197	13	subalgebra	subalgebra	NOUN
ejpam-6171	197	14	of	of	ADP
ejpam-6171	197	15	x	x	PRON
ejpam-6171	197	16	if	if	SCONJ
ejpam-6171	197	17	it	it	PRON
ejpam-6171	197	18	satisfies	satisfy	VERB
ejpam-6171	197	19	the	the	DET
ejpam-6171	197	20	following	follow	VERB
ejpam-6171	197	21	properties	property	NOUN
ejpam-6171	197	22	:	:	PUNCT
ejpam-6171	197	23	(	(	PUNCT
ejpam-6171	197	24	∀x	∀x	X
ejpam-6171	197	25	,	,	PUNCT
ejpam-6171	197	26	y	y	PROPN
ejpam-6171	197	27	∈	∈	PROPN
ejpam-6171	197	28	x)(pt	x)(pt	PUNCT
ejpam-6171	198	1	(	(	PUNCT
ejpam-6171	198	2	x	x	X
ejpam-6171	198	3	?	?	PUNCT
ejpam-6171	198	4	y	y	X
ejpam-6171	198	5	)	)	PUNCT
ejpam-6171	198	6	≥	≥	NOUN
ejpam-6171	198	7	min{pt	min{pt	X
ejpam-6171	199	1	(	(	PUNCT
ejpam-6171	199	2	x),pt	x),pt	PROPN
ejpam-6171	199	3	(	(	PUNCT
ejpam-6171	199	4	y	y	NOUN
ejpam-6171	199	5	)	)	PUNCT
ejpam-6171	199	6	}	}	PUNCT
ejpam-6171	199	7	)	)	PUNCT
ejpam-6171	199	8	(	(	PUNCT
ejpam-6171	199	9	3.2	3.2	NUM
ejpam-6171	199	10	)	)	PUNCT
ejpam-6171	199	11	(	(	PUNCT
ejpam-6171	199	12	∀x	∀x	X
ejpam-6171	199	13	,	,	PUNCT
ejpam-6171	199	14	y	y	PROPN
ejpam-6171	199	15	∈	∈	PROPN
ejpam-6171	199	16	x)(pi(x	x)(pi(x	NOUN
ejpam-6171	199	17	?	?	PUNCT
ejpam-6171	200	1	y	y	X
ejpam-6171	200	2	)	)	PUNCT
ejpam-6171	200	3	≤	≤	NUM
ejpam-6171	200	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	200	5	)	)	PUNCT
ejpam-6171	200	6	}	}	PUNCT
ejpam-6171	200	7	)	)	PUNCT
ejpam-6171	200	8	(	(	PUNCT
ejpam-6171	200	9	3.3	3.3	NUM
ejpam-6171	200	10	)	)	PUNCT
ejpam-6171	200	11	(	(	PUNCT
ejpam-6171	200	12	∀x	∀x	X
ejpam-6171	200	13	,	,	PUNCT
ejpam-6171	200	14	y	y	PROPN
ejpam-6171	200	15	∈	∈	PROPN
ejpam-6171	200	16	x)(pf	x)(pf	X
ejpam-6171	201	1	(	(	PUNCT
ejpam-6171	201	2	x	x	X
ejpam-6171	201	3	?	?	PUNCT
ejpam-6171	201	4	y	y	X
ejpam-6171	201	5	)	)	PUNCT
ejpam-6171	201	6	≥	≥	PROPN
ejpam-6171	201	7	min{pf	min{pf	PRON
ejpam-6171	201	8	(	(	PUNCT
ejpam-6171	201	9	x),pf	x),pf	PROPN
ejpam-6171	201	10	(	(	PUNCT
ejpam-6171	201	11	y	y	NOUN
ejpam-6171	201	12	)	)	PUNCT
ejpam-6171	201	13	}	}	PUNCT
ejpam-6171	201	14	)	)	PUNCT
ejpam-6171	201	15	(	(	PUNCT
ejpam-6171	201	16	3.4	3.4	NUM
ejpam-6171	201	17	)	)	PUNCT
ejpam-6171	201	18	k.	k.	NOUN
ejpam-6171	202	1	suayngam	suayngam	INTJ
ejpam-6171	202	2	et	et	PROPN
ejpam-6171	202	3	al	al	PROPN
ejpam-6171	202	4	.	.	PUNCT
ejpam-6171	202	5	/	/	SYM
ejpam-6171	202	6	eur	eur	PROPN
ejpam-6171	202	7	.	.	PUNCT
ejpam-6171	203	1	j.	j.	PROPN
ejpam-6171	203	2	pure	pure	PROPN
ejpam-6171	203	3	appl	appl	PROPN
ejpam-6171	203	4	.	.	PROPN
ejpam-6171	203	5	math	math	PROPN
ejpam-6171	203	6	,	,	PUNCT
ejpam-6171	203	7	18	18	NUM
ejpam-6171	203	8	(	(	PUNCT
ejpam-6171	203	9	3	3	NUM
ejpam-6171	203	10	)	)	PUNCT
ejpam-6171	203	11	(	(	PUNCT
ejpam-6171	203	12	2025	2025	NUM
ejpam-6171	203	13	)	)	PUNCT
ejpam-6171	203	14	,	,	PUNCT
ejpam-6171	203	15	6171	6171	NUM
ejpam-6171	203	16	8	8	NUM
ejpam-6171	203	17	of	of	ADP
ejpam-6171	203	18	28	28	NUM
ejpam-6171	203	19	definition	definition	NOUN
ejpam-6171	203	20	7	7	NUM
ejpam-6171	203	21	.	.	PUNCT
ejpam-6171	204	1	a	a	DET
ejpam-6171	204	2	pns	pns	NOUN
ejpam-6171	204	3	p	p	NOUN
ejpam-6171	204	4	in	in	ADP
ejpam-6171	204	5	x	x	PROPN
ejpam-6171	204	6	is	be	AUX
ejpam-6171	204	7	called	call	VERB
ejpam-6171	204	8	a	a	DET
ejpam-6171	204	9	pythagorean	pythagorean	PROPN
ejpam-6171	204	10	neutrosophic	neutrosophic	ADJ
ejpam-6171	204	11	iup	iup	PROPN
ejpam-6171	204	12	-	-	PUNCT
ejpam-6171	204	13	ideal	ideal	NOUN
ejpam-6171	204	14	of	of	ADP
ejpam-6171	204	15	x	x	PRON
ejpam-6171	204	16	if	if	SCONJ
ejpam-6171	204	17	it	it	PRON
ejpam-6171	204	18	satisfies	satisfy	VERB
ejpam-6171	204	19	the	the	DET
ejpam-6171	204	20	following	follow	VERB
ejpam-6171	204	21	properties	property	NOUN
ejpam-6171	204	22	:	:	PUNCT
ejpam-6171	204	23	(	(	PUNCT
ejpam-6171	204	24	∀x	∀x	X
ejpam-6171	204	25	∈	∈	X
ejpam-6171	204	26	x)(pt	x)(pt	X
ejpam-6171	204	27	(	(	PUNCT
ejpam-6171	204	28	0	0	NUM
ejpam-6171	204	29	)	)	PUNCT
ejpam-6171	204	30	≥	≥	NOUN
ejpam-6171	204	31	pt	pt	INTJ
ejpam-6171	204	32	(	(	PUNCT
ejpam-6171	204	33	x	x	NOUN
ejpam-6171	204	34	)	)	PUNCT
ejpam-6171	204	35	)	)	PUNCT
ejpam-6171	204	36	(	(	PUNCT
ejpam-6171	204	37	3.5	3.5	NUM
ejpam-6171	204	38	)	)	PUNCT
ejpam-6171	204	39	(	(	PUNCT
ejpam-6171	204	40	∀x	∀x	X
ejpam-6171	204	41	∈	∈	PROPN
ejpam-6171	204	42	x)(pi(0	x)(pi(0	PROPN
ejpam-6171	204	43	)	)	PUNCT
ejpam-6171	204	44	≤	≤	NUM
ejpam-6171	204	45	pi(x	pi(x	NOUN
ejpam-6171	204	46	)	)	PUNCT
ejpam-6171	204	47	)	)	PUNCT
ejpam-6171	204	48	(	(	PUNCT
ejpam-6171	204	49	3.6	3.6	NUM
ejpam-6171	204	50	)	)	PUNCT
ejpam-6171	204	51	(	(	PUNCT
ejpam-6171	204	52	∀x	∀x	X
ejpam-6171	204	53	∈	∈	PROPN
ejpam-6171	204	54	x)(pf	x)(pf	X
ejpam-6171	204	55	(	(	PUNCT
ejpam-6171	204	56	0	0	NUM
ejpam-6171	204	57	)	)	PUNCT
ejpam-6171	204	58	≥	≥	NOUN
ejpam-6171	204	59	pf	pf	X
ejpam-6171	204	60	(	(	PUNCT
ejpam-6171	204	61	x	x	NOUN
ejpam-6171	204	62	)	)	PUNCT
ejpam-6171	204	63	)	)	PUNCT
ejpam-6171	204	64	(	(	PUNCT
ejpam-6171	204	65	3.7	3.7	NUM
ejpam-6171	204	66	)	)	PUNCT
ejpam-6171	204	67	(	(	PUNCT
ejpam-6171	204	68	∀x	∀x	X
ejpam-6171	204	69	,	,	PUNCT
ejpam-6171	204	70	y	y	PROPN
ejpam-6171	204	71	,	,	PUNCT
ejpam-6171	204	72	z	z	NOUN
ejpam-6171	204	73	∈	∈	PROPN
ejpam-6171	204	74	x)(pt	x)(pt	PUNCT
ejpam-6171	205	1	(	(	PUNCT
ejpam-6171	205	2	x	x	X
ejpam-6171	205	3	?	?	PUNCT
ejpam-6171	205	4	z	z	X
ejpam-6171	205	5	)	)	PUNCT
ejpam-6171	205	6	≥	≥	NOUN
ejpam-6171	205	7	min{pt	min{pt	X
ejpam-6171	206	1	(	(	PUNCT
ejpam-6171	206	2	x	x	X
ejpam-6171	206	3	?	?	PUNCT
ejpam-6171	207	1	(	(	PUNCT
ejpam-6171	207	2	y	y	PROPN
ejpam-6171	207	3	?	?	PUNCT
ejpam-6171	208	1	z)),pt	z)),pt	PROPN
ejpam-6171	208	2	(	(	PUNCT
ejpam-6171	208	3	y	y	NOUN
ejpam-6171	208	4	)	)	PUNCT
ejpam-6171	208	5	}	}	PUNCT
ejpam-6171	208	6	)	)	PUNCT
ejpam-6171	208	7	(	(	PUNCT
ejpam-6171	208	8	3.8	3.8	NUM
ejpam-6171	208	9	)	)	PUNCT
ejpam-6171	208	10	(	(	PUNCT
ejpam-6171	208	11	∀x	∀x	X
ejpam-6171	208	12	,	,	PUNCT
ejpam-6171	208	13	y	y	PROPN
ejpam-6171	208	14	,	,	PUNCT
ejpam-6171	208	15	z	z	PROPN
ejpam-6171	208	16	∈	∈	PROPN
ejpam-6171	209	1	x)(pi(x	x)(pi(x	NOUN
ejpam-6171	209	2	?	?	PUNCT
ejpam-6171	210	1	z	z	X
ejpam-6171	210	2	)	)	PUNCT
ejpam-6171	210	3	≤	≤	NUM
ejpam-6171	210	4	max{pi(x	max{pi(x	NOUN
ejpam-6171	210	5	?	?	PUNCT
ejpam-6171	211	1	(	(	PUNCT
ejpam-6171	211	2	y	y	NOUN
ejpam-6171	211	3	?	?	PUNCT
ejpam-6171	211	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	211	5	)	)	PUNCT
ejpam-6171	211	6	}	}	PUNCT
ejpam-6171	211	7	)	)	PUNCT
ejpam-6171	211	8	(	(	PUNCT
ejpam-6171	211	9	3.9	3.9	NUM
ejpam-6171	211	10	)	)	PUNCT
ejpam-6171	211	11	(	(	PUNCT
ejpam-6171	211	12	∀x	∀x	X
ejpam-6171	211	13	,	,	PUNCT
ejpam-6171	211	14	y	y	PROPN
ejpam-6171	211	15	,	,	PUNCT
ejpam-6171	211	16	z	z	PROPN
ejpam-6171	211	17	∈	∈	PROPN
ejpam-6171	211	18	x)(pf	x)(pf	X
ejpam-6171	211	19	(	(	PUNCT
ejpam-6171	211	20	x	x	X
ejpam-6171	211	21	?	?	PUNCT
ejpam-6171	212	1	z	z	X
ejpam-6171	212	2	)	)	PUNCT
ejpam-6171	212	3	≥	≥	NOUN
ejpam-6171	212	4	min{pf	min{pf	X
ejpam-6171	212	5	(	(	PUNCT
ejpam-6171	212	6	x	x	X
ejpam-6171	212	7	?	?	PUNCT
ejpam-6171	213	1	(	(	PUNCT
ejpam-6171	213	2	y	y	NOUN
ejpam-6171	213	3	?	?	PUNCT
ejpam-6171	214	1	z)),pf	z)),pf	PROPN
ejpam-6171	214	2	(	(	PUNCT
ejpam-6171	214	3	y	y	NOUN
ejpam-6171	214	4	)	)	PUNCT
ejpam-6171	214	5	}	}	PUNCT
ejpam-6171	214	6	)	)	PUNCT
ejpam-6171	214	7	(	(	PUNCT
ejpam-6171	214	8	3.10	3.10	NUM
ejpam-6171	214	9	)	)	PUNCT
ejpam-6171	214	10	definition	definition	NOUN
ejpam-6171	214	11	8	8	NUM
ejpam-6171	214	12	.	.	PUNCT
ejpam-6171	215	1	a	a	DET
ejpam-6171	215	2	pns	pns	NOUN
ejpam-6171	215	3	p	p	NOUN
ejpam-6171	215	4	in	in	ADP
ejpam-6171	215	5	x	x	PROPN
ejpam-6171	215	6	is	be	AUX
ejpam-6171	215	7	called	call	VERB
ejpam-6171	215	8	a	a	DET
ejpam-6171	215	9	pythagorean	pythagorean	PROPN
ejpam-6171	215	10	neutrosophic	neutrosophic	ADJ
ejpam-6171	215	11	iup	iup	NOUN
ejpam-6171	215	12	-	-	PUNCT
ejpam-6171	215	13	filter	filter	NOUN
ejpam-6171	215	14	of	of	ADP
ejpam-6171	215	15	x	x	PRON
ejpam-6171	215	16	if	if	SCONJ
ejpam-6171	215	17	it	it	PRON
ejpam-6171	215	18	satisfies	satisfy	VERB
ejpam-6171	215	19	(	(	PUNCT
ejpam-6171	215	20	3.5	3.5	NUM
ejpam-6171	215	21	)	)	PUNCT
ejpam-6171	215	22	,	,	PUNCT
ejpam-6171	215	23	(	(	PUNCT
ejpam-6171	215	24	3.6	3.6	NUM
ejpam-6171	215	25	)	)	PUNCT
ejpam-6171	215	26	,	,	PUNCT
ejpam-6171	215	27	(	(	PUNCT
ejpam-6171	215	28	3.7	3.7	NUM
ejpam-6171	215	29	)	)	PUNCT
ejpam-6171	215	30	,	,	PUNCT
ejpam-6171	215	31	and	and	CCONJ
ejpam-6171	215	32	the	the	DET
ejpam-6171	215	33	following	follow	VERB
ejpam-6171	215	34	properties	property	NOUN
ejpam-6171	215	35	:	:	PUNCT
ejpam-6171	215	36	(	(	PUNCT
ejpam-6171	215	37	∀x	∀x	X
ejpam-6171	215	38	,	,	PUNCT
ejpam-6171	215	39	y	y	PROPN
ejpam-6171	215	40	∈	∈	PROPN
ejpam-6171	215	41	x)(pt	x)(pt	PUNCT
ejpam-6171	215	42	(	(	PUNCT
ejpam-6171	215	43	y	y	X
ejpam-6171	215	44	)	)	PUNCT
ejpam-6171	215	45	≥	≥	NOUN
ejpam-6171	215	46	min{pt	min{pt	X
ejpam-6171	215	47	(	(	PUNCT
ejpam-6171	215	48	x	x	X
ejpam-6171	215	49	?	?	PUNCT
ejpam-6171	215	50	y),pt	y),pt	PROPN
ejpam-6171	215	51	(	(	PUNCT
ejpam-6171	215	52	x	x	NOUN
ejpam-6171	215	53	)	)	PUNCT
ejpam-6171	215	54	}	}	PUNCT
ejpam-6171	215	55	)	)	PUNCT
ejpam-6171	215	56	(	(	PUNCT
ejpam-6171	215	57	3.11	3.11	NUM
ejpam-6171	215	58	)	)	PUNCT
ejpam-6171	215	59	(	(	PUNCT
ejpam-6171	215	60	∀x	∀x	X
ejpam-6171	215	61	,	,	PUNCT
ejpam-6171	215	62	y	y	PROPN
ejpam-6171	215	63	∈	∈	PROPN
ejpam-6171	215	64	x)(pi(y	x)(pi(y	X
ejpam-6171	215	65	)	)	PUNCT
ejpam-6171	215	66	≤	≤	NUM
ejpam-6171	215	67	max{pi(x	max{pi(x	NOUN
ejpam-6171	215	68	?	?	PUNCT
ejpam-6171	216	1	y),pi(x	y),pi(x	NUM
ejpam-6171	216	2	)	)	PUNCT
ejpam-6171	216	3	}	}	PUNCT
ejpam-6171	216	4	)	)	PUNCT
ejpam-6171	217	1	(	(	PUNCT
ejpam-6171	217	2	3.12	3.12	NUM
ejpam-6171	217	3	)	)	PUNCT
ejpam-6171	217	4	(	(	PUNCT
ejpam-6171	217	5	∀x	∀x	X
ejpam-6171	217	6	,	,	PUNCT
ejpam-6171	217	7	y	y	PROPN
ejpam-6171	217	8	∈	∈	PROPN
ejpam-6171	217	9	x)(pf	x)(pf	X
ejpam-6171	217	10	(	(	PUNCT
ejpam-6171	217	11	y	y	NOUN
ejpam-6171	217	12	)	)	PUNCT
ejpam-6171	217	13	≥	≥	NOUN
ejpam-6171	217	14	min{pf	min{pf	X
ejpam-6171	217	15	(	(	PUNCT
ejpam-6171	217	16	x	x	X
ejpam-6171	217	17	?	?	PUNCT
ejpam-6171	218	1	y),pf	y),pf	PROPN
ejpam-6171	218	2	(	(	PUNCT
ejpam-6171	218	3	x	x	NOUN
ejpam-6171	218	4	)	)	PUNCT
ejpam-6171	218	5	}	}	PUNCT
ejpam-6171	218	6	)	)	PUNCT
ejpam-6171	218	7	(	(	PUNCT
ejpam-6171	218	8	3.13	3.13	NUM
ejpam-6171	218	9	)	)	PUNCT
ejpam-6171	218	10	definition	definition	NOUN
ejpam-6171	218	11	9	9	NUM
ejpam-6171	218	12	.	.	PUNCT
ejpam-6171	219	1	a	a	DET
ejpam-6171	219	2	pns	pns	NOUN
ejpam-6171	219	3	p	p	NOUN
ejpam-6171	219	4	in	in	ADP
ejpam-6171	219	5	x	x	PROPN
ejpam-6171	219	6	is	be	AUX
ejpam-6171	219	7	called	call	VERB
ejpam-6171	219	8	a	a	DET
ejpam-6171	219	9	pythagorean	pythagorean	PROPN
ejpam-6171	219	10	neutrosophic	neutrosophic	ADJ
ejpam-6171	219	11	strong	strong	ADJ
ejpam-6171	219	12	iup	iup	NOUN
ejpam-6171	219	13	-	-	PUNCT
ejpam-6171	219	14	ideal	ideal	NOUN
ejpam-6171	219	15	of	of	ADP
ejpam-6171	219	16	x	x	PRON
ejpam-6171	219	17	if	if	SCONJ
ejpam-6171	219	18	it	it	PRON
ejpam-6171	219	19	satisfies	satisfy	VERB
ejpam-6171	219	20	the	the	DET
ejpam-6171	219	21	following	follow	VERB
ejpam-6171	219	22	properties	property	NOUN
ejpam-6171	219	23	:	:	PUNCT
ejpam-6171	219	24	(	(	PUNCT
ejpam-6171	219	25	∀x	∀x	X
ejpam-6171	219	26	,	,	PUNCT
ejpam-6171	219	27	y	y	PROPN
ejpam-6171	219	28	∈	∈	PROPN
ejpam-6171	219	29	x)(pt	x)(pt	PUNCT
ejpam-6171	220	1	(	(	PUNCT
ejpam-6171	220	2	x	x	X
ejpam-6171	220	3	?	?	PUNCT
ejpam-6171	220	4	y	y	X
ejpam-6171	220	5	)	)	PUNCT
ejpam-6171	220	6	≥	≥	PROPN
ejpam-6171	220	7	pt	pt	INTJ
ejpam-6171	220	8	(	(	PUNCT
ejpam-6171	220	9	y	y	NOUN
ejpam-6171	220	10	)	)	PUNCT
ejpam-6171	220	11	)	)	PUNCT
ejpam-6171	221	1	(	(	PUNCT
ejpam-6171	221	2	3.14	3.14	NUM
ejpam-6171	221	3	)	)	PUNCT
ejpam-6171	221	4	(	(	PUNCT
ejpam-6171	221	5	∀x	∀x	X
ejpam-6171	221	6	,	,	PUNCT
ejpam-6171	221	7	y	y	PROPN
ejpam-6171	221	8	∈	∈	PROPN
ejpam-6171	221	9	x)(pi(x	x)(pi(x	NOUN
ejpam-6171	221	10	?	?	PUNCT
ejpam-6171	222	1	y	y	X
ejpam-6171	222	2	)	)	PUNCT
ejpam-6171	222	3	≤	≤	NOUN
ejpam-6171	222	4	pi(y	pi(y	NOUN
ejpam-6171	222	5	)	)	PUNCT
ejpam-6171	222	6	)	)	PUNCT
ejpam-6171	222	7	(	(	PUNCT
ejpam-6171	222	8	3.15	3.15	NUM
ejpam-6171	222	9	)	)	PUNCT
ejpam-6171	222	10	(	(	PUNCT
ejpam-6171	222	11	∀x	∀x	X
ejpam-6171	222	12	,	,	PUNCT
ejpam-6171	222	13	y	y	PROPN
ejpam-6171	222	14	∈	∈	PROPN
ejpam-6171	222	15	x)(pf	x)(pf	X
ejpam-6171	223	1	(	(	PUNCT
ejpam-6171	223	2	x	x	X
ejpam-6171	223	3	?	?	PUNCT
ejpam-6171	223	4	y	y	X
ejpam-6171	223	5	)	)	PUNCT
ejpam-6171	223	6	≥	≥	NOUN
ejpam-6171	223	7	pf	pf	PROPN
ejpam-6171	223	8	(	(	PUNCT
ejpam-6171	223	9	y	y	NOUN
ejpam-6171	223	10	)	)	PUNCT
ejpam-6171	223	11	)	)	PUNCT
ejpam-6171	224	1	(	(	PUNCT
ejpam-6171	224	2	3.16	3.16	NUM
ejpam-6171	224	3	)	)	PUNCT
ejpam-6171	224	4	lemma	lemma	PROPN
ejpam-6171	224	5	1	1	NUM
ejpam-6171	224	6	.	.	PUNCT
ejpam-6171	225	1	every	every	DET
ejpam-6171	225	2	pythagorean	pythagorean	PROPN
ejpam-6171	225	3	neutrosophic	neutrosophic	PROPN
ejpam-6171	225	4	iup	iup	NOUN
ejpam-6171	225	5	-	-	PUNCT
ejpam-6171	225	6	subalgebra	subalgebra	NOUN
ejpam-6171	225	7	of	of	ADP
ejpam-6171	225	8	x	x	PUNCT
ejpam-6171	225	9	satisfies	satisfie	NOUN
ejpam-6171	225	10	(	(	PUNCT
ejpam-6171	225	11	3.5	3.5	NUM
ejpam-6171	225	12	)	)	PUNCT
ejpam-6171	225	13	,	,	PUNCT
ejpam-6171	225	14	(	(	PUNCT
ejpam-6171	225	15	3.6	3.6	NUM
ejpam-6171	225	16	)	)	PUNCT
ejpam-6171	225	17	,	,	PUNCT
ejpam-6171	225	18	and	and	CCONJ
ejpam-6171	225	19	(	(	PUNCT
ejpam-6171	225	20	3.7	3.7	NUM
ejpam-6171	225	21	)	)	PUNCT
ejpam-6171	225	22	.	.	PUNCT
ejpam-6171	226	1	proof	proof	NOUN
ejpam-6171	226	2	.	.	PUNCT
ejpam-6171	227	1	assume	assume	VERB
ejpam-6171	227	2	that	that	SCONJ
ejpam-6171	227	3	p	p	NOUN
ejpam-6171	227	4	is	be	AUX
ejpam-6171	227	5	a	a	DET
ejpam-6171	227	6	pythagorean	pythagorean	PROPN
ejpam-6171	227	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	227	8	iup	iup	NOUN
ejpam-6171	227	9	-	-	PUNCT
ejpam-6171	227	10	subalgebra	subalgebra	NOUN
ejpam-6171	227	11	of	of	ADP
ejpam-6171	227	12	x.	x.	NOUN
ejpam-6171	227	13	let	let	VERB
ejpam-6171	227	14	x	x	SYM
ejpam-6171	227	15	∈	∈	PROPN
ejpam-6171	227	16	x.	x.	NOUN
ejpam-6171	227	17	then	then	ADV
ejpam-6171	227	18	pt	pt	X
ejpam-6171	227	19	(	(	PUNCT
ejpam-6171	227	20	0	0	NUM
ejpam-6171	227	21	)	)	PUNCT
ejpam-6171	228	1	=	=	SYM
ejpam-6171	228	2	pt	pt	X
ejpam-6171	228	3	(	(	PUNCT
ejpam-6171	228	4	x	x	X
ejpam-6171	228	5	?	?	PUNCT
ejpam-6171	229	1	x	x	X
ejpam-6171	229	2	)	)	PUNCT
ejpam-6171	229	3	(	(	PUNCT
ejpam-6171	229	4	by	by	ADP
ejpam-6171	229	5	(	(	PUNCT
ejpam-6171	229	6	iup-2	iup-2	NUM
ejpam-6171	229	7	)	)	PUNCT
ejpam-6171	229	8	)	)	PUNCT
ejpam-6171	229	9	≥	≥	NOUN
ejpam-6171	229	10	min{pt	min{pt	X
ejpam-6171	229	11	(	(	PUNCT
ejpam-6171	229	12	x),pt	x),pt	X
ejpam-6171	229	13	(	(	PUNCT
ejpam-6171	229	14	x	x	NOUN
ejpam-6171	229	15	)	)	PUNCT
ejpam-6171	229	16	}	}	PUNCT
ejpam-6171	229	17	(	(	PUNCT
ejpam-6171	229	18	by	by	ADP
ejpam-6171	229	19	(	(	PUNCT
ejpam-6171	229	20	3.2	3.2	NUM
ejpam-6171	229	21	)	)	PUNCT
ejpam-6171	229	22	)	)	PUNCT
ejpam-6171	230	1	=	=	SYM
ejpam-6171	230	2	pt	pt	X
ejpam-6171	230	3	(	(	PUNCT
ejpam-6171	230	4	x	x	NOUN
ejpam-6171	230	5	)	)	PUNCT
ejpam-6171	230	6	,	,	PUNCT
ejpam-6171	230	7	pi(0	pi(0	PROPN
ejpam-6171	230	8	)	)	PUNCT
ejpam-6171	230	9	=	=	SYM
ejpam-6171	230	10	pi(x	pi(x	NOUN
ejpam-6171	230	11	?	?	PUNCT
ejpam-6171	231	1	x	x	X
ejpam-6171	231	2	)	)	PUNCT
ejpam-6171	231	3	(	(	PUNCT
ejpam-6171	231	4	by	by	ADP
ejpam-6171	231	5	(	(	PUNCT
ejpam-6171	231	6	iup-2	iup-2	NUM
ejpam-6171	231	7	)	)	PUNCT
ejpam-6171	231	8	)	)	PUNCT
ejpam-6171	231	9	≤	≤	NUM
ejpam-6171	232	1	max{pi(x),pi(x	max{pi(x),pi(x	CCONJ
ejpam-6171	232	2	)	)	PUNCT
ejpam-6171	232	3	}	}	PUNCT
ejpam-6171	232	4	(	(	PUNCT
ejpam-6171	232	5	by	by	ADP
ejpam-6171	232	6	(	(	PUNCT
ejpam-6171	232	7	3.3	3.3	NUM
ejpam-6171	232	8	)	)	PUNCT
ejpam-6171	232	9	)	)	PUNCT
ejpam-6171	232	10	=	=	SYM
ejpam-6171	232	11	pi(x	pi(x	NOUN
ejpam-6171	232	12	)	)	PUNCT
ejpam-6171	232	13	,	,	PUNCT
ejpam-6171	232	14	pf	pf	PROPN
ejpam-6171	232	15	(	(	PUNCT
ejpam-6171	232	16	0	0	NUM
ejpam-6171	232	17	)	)	PUNCT
ejpam-6171	232	18	=	=	SYM
ejpam-6171	232	19	pf	pf	X
ejpam-6171	232	20	(	(	PUNCT
ejpam-6171	232	21	x	x	PROPN
ejpam-6171	232	22	?	?	PUNCT
ejpam-6171	232	23	x	x	X
ejpam-6171	232	24	)	)	PUNCT
ejpam-6171	232	25	(	(	PUNCT
ejpam-6171	232	26	by	by	ADP
ejpam-6171	232	27	(	(	PUNCT
ejpam-6171	232	28	iup-2	iup-2	NUM
ejpam-6171	232	29	)	)	PUNCT
ejpam-6171	232	30	)	)	PUNCT
ejpam-6171	232	31	≥	≥	NOUN
ejpam-6171	232	32	min{pf	min{pf	PRON
ejpam-6171	232	33	(	(	PUNCT
ejpam-6171	232	34	x),pf	x),pf	PROPN
ejpam-6171	232	35	(	(	PUNCT
ejpam-6171	232	36	x	x	NOUN
ejpam-6171	232	37	)	)	PUNCT
ejpam-6171	232	38	}	}	PUNCT
ejpam-6171	232	39	(	(	PUNCT
ejpam-6171	232	40	by	by	ADP
ejpam-6171	232	41	(	(	PUNCT
ejpam-6171	232	42	3.2	3.2	NUM
ejpam-6171	232	43	)	)	PUNCT
ejpam-6171	232	44	)	)	PUNCT
ejpam-6171	233	1	=	=	SYM
ejpam-6171	233	2	pf	pf	X
ejpam-6171	233	3	(	(	PUNCT
ejpam-6171	233	4	x	x	NOUN
ejpam-6171	233	5	)	)	PUNCT
ejpam-6171	233	6	.	.	PUNCT
ejpam-6171	234	1	hence	hence	ADV
ejpam-6171	234	2	,	,	PUNCT
ejpam-6171	234	3	it	it	PRON
ejpam-6171	234	4	satisfies	satisfy	VERB
ejpam-6171	234	5	(	(	PUNCT
ejpam-6171	234	6	3.5	3.5	NUM
ejpam-6171	234	7	)	)	PUNCT
ejpam-6171	234	8	,	,	PUNCT
ejpam-6171	234	9	(	(	PUNCT
ejpam-6171	234	10	3.6	3.6	NUM
ejpam-6171	234	11	)	)	PUNCT
ejpam-6171	234	12	,	,	PUNCT
ejpam-6171	234	13	and	and	CCONJ
ejpam-6171	234	14	(	(	PUNCT
ejpam-6171	234	15	3.7	3.7	NUM
ejpam-6171	234	16	)	)	PUNCT
ejpam-6171	234	17	.	.	PUNCT
ejpam-6171	235	1	k.	k.	PROPN
ejpam-6171	235	2	suayngam	suayngam	PROPN
ejpam-6171	235	3	et	et	PROPN
ejpam-6171	235	4	al	al	PROPN
ejpam-6171	235	5	.	.	PUNCT
ejpam-6171	235	6	/	/	SYM
ejpam-6171	235	7	eur	eur	PROPN
ejpam-6171	235	8	.	.	PUNCT
ejpam-6171	236	1	j.	j.	PROPN
ejpam-6171	236	2	pure	pure	PROPN
ejpam-6171	236	3	appl	appl	PROPN
ejpam-6171	236	4	.	.	PROPN
ejpam-6171	236	5	math	math	PROPN
ejpam-6171	236	6	,	,	PUNCT
ejpam-6171	236	7	18	18	NUM
ejpam-6171	236	8	(	(	PUNCT
ejpam-6171	236	9	3	3	NUM
ejpam-6171	236	10	)	)	PUNCT
ejpam-6171	236	11	(	(	PUNCT
ejpam-6171	236	12	2025	2025	NUM
ejpam-6171	236	13	)	)	PUNCT
ejpam-6171	236	14	,	,	PUNCT
ejpam-6171	236	15	6171	6171	NUM
ejpam-6171	236	16	9	9	NUM
ejpam-6171	236	17	of	of	ADP
ejpam-6171	236	18	28	28	NUM
ejpam-6171	236	19	theorem	theorem	NOUN
ejpam-6171	236	20	1	1	NUM
ejpam-6171	236	21	.	.	PUNCT
ejpam-6171	237	1	every	every	DET
ejpam-6171	237	2	pythagorean	pythagorean	PROPN
ejpam-6171	237	3	neutrosophic	neutrosophic	ADJ
ejpam-6171	237	4	strong	strong	ADJ
ejpam-6171	237	5	iup	iup	NOUN
ejpam-6171	237	6	-	-	PUNCT
ejpam-6171	237	7	ideal	ideal	NOUN
ejpam-6171	237	8	of	of	ADP
ejpam-6171	237	9	x	x	SYM
ejpam-6171	237	10	satisfies	satisfie	NOUN
ejpam-6171	237	11	(	(	PUNCT
ejpam-6171	237	12	3.5	3.5	NUM
ejpam-6171	237	13	)	)	PUNCT
ejpam-6171	237	14	,	,	PUNCT
ejpam-6171	237	15	(	(	PUNCT
ejpam-6171	237	16	3.6	3.6	NUM
ejpam-6171	237	17	)	)	PUNCT
ejpam-6171	237	18	,	,	PUNCT
ejpam-6171	237	19	and	and	CCONJ
ejpam-6171	237	20	(	(	PUNCT
ejpam-6171	237	21	3.7	3.7	NUM
ejpam-6171	237	22	)	)	PUNCT
ejpam-6171	237	23	.	.	PUNCT
ejpam-6171	238	1	proof	proof	NOUN
ejpam-6171	238	2	.	.	PUNCT
ejpam-6171	239	1	assume	assume	VERB
ejpam-6171	239	2	that	that	SCONJ
ejpam-6171	239	3	pythagorean	pythagorean	PROPN
ejpam-6171	239	4	neutrosophic	neutrosophic	PROPN
ejpam-6171	239	5	strong	strong	ADJ
ejpam-6171	239	6	iup	iup	NOUN
ejpam-6171	239	7	-	-	PUNCT
ejpam-6171	239	8	ideal	ideal	NOUN
ejpam-6171	239	9	of	of	ADP
ejpam-6171	239	10	x.	x.	NOUN
ejpam-6171	239	11	let	let	VERB
ejpam-6171	239	12	x	x	SYM
ejpam-6171	239	13	∈	∈	PROPN
ejpam-6171	239	14	x.	x.	NOUN
ejpam-6171	239	15	then	then	ADV
ejpam-6171	239	16	pt	pt	X
ejpam-6171	239	17	(	(	PUNCT
ejpam-6171	239	18	0	0	NUM
ejpam-6171	239	19	)	)	PUNCT
ejpam-6171	240	1	=	=	SYM
ejpam-6171	240	2	pt	pt	X
ejpam-6171	240	3	(	(	PUNCT
ejpam-6171	240	4	x	x	X
ejpam-6171	240	5	?	?	PUNCT
ejpam-6171	241	1	x	x	X
ejpam-6171	241	2	)	)	PUNCT
ejpam-6171	241	3	(	(	PUNCT
ejpam-6171	241	4	by	by	ADP
ejpam-6171	241	5	(	(	PUNCT
ejpam-6171	241	6	iup-2	iup-2	NUM
ejpam-6171	241	7	)	)	PUNCT
ejpam-6171	241	8	)	)	PUNCT
ejpam-6171	241	9	≥	≥	PROPN
ejpam-6171	241	10	pt	pt	INTJ
ejpam-6171	241	11	(	(	PUNCT
ejpam-6171	241	12	x	x	NOUN
ejpam-6171	241	13	)	)	PUNCT
ejpam-6171	241	14	,	,	PUNCT
ejpam-6171	241	15	(	(	PUNCT
ejpam-6171	241	16	by	by	ADP
ejpam-6171	241	17	(	(	PUNCT
ejpam-6171	241	18	3.14	3.14	NUM
ejpam-6171	241	19	)	)	PUNCT
ejpam-6171	241	20	)	)	PUNCT
ejpam-6171	241	21	pi(0	pi(0	PROPN
ejpam-6171	241	22	)	)	PUNCT
ejpam-6171	241	23	=	=	SYM
ejpam-6171	241	24	pi(x	pi(x	NOUN
ejpam-6171	241	25	?	?	PUNCT
ejpam-6171	242	1	x	x	X
ejpam-6171	242	2	)	)	PUNCT
ejpam-6171	242	3	(	(	PUNCT
ejpam-6171	242	4	by	by	ADP
ejpam-6171	242	5	(	(	PUNCT
ejpam-6171	242	6	iup-2	iup-2	NUM
ejpam-6171	242	7	)	)	PUNCT
ejpam-6171	242	8	)	)	PUNCT
ejpam-6171	242	9	≤	≤	NUM
ejpam-6171	242	10	pi(x	pi(x	NOUN
ejpam-6171	242	11	)	)	PUNCT
ejpam-6171	242	12	,	,	PUNCT
ejpam-6171	242	13	(	(	PUNCT
ejpam-6171	242	14	by	by	ADP
ejpam-6171	242	15	(	(	PUNCT
ejpam-6171	242	16	3.15	3.15	NUM
ejpam-6171	242	17	)	)	PUNCT
ejpam-6171	242	18	)	)	PUNCT
ejpam-6171	242	19	pf	pf	PROPN
ejpam-6171	242	20	(	(	PUNCT
ejpam-6171	242	21	0	0	NUM
ejpam-6171	242	22	)	)	PUNCT
ejpam-6171	242	23	=	=	SYM
ejpam-6171	242	24	pf	pf	X
ejpam-6171	242	25	(	(	PUNCT
ejpam-6171	242	26	x	x	PROPN
ejpam-6171	242	27	?	?	PUNCT
ejpam-6171	242	28	x	x	X
ejpam-6171	242	29	)	)	PUNCT
ejpam-6171	242	30	(	(	PUNCT
ejpam-6171	242	31	by	by	ADP
ejpam-6171	242	32	(	(	PUNCT
ejpam-6171	242	33	iup-2	iup-2	NUM
ejpam-6171	242	34	)	)	PUNCT
ejpam-6171	242	35	)	)	PUNCT
ejpam-6171	242	36	≥	≥	PROPN
ejpam-6171	242	37	pf	pf	X
ejpam-6171	242	38	(	(	PUNCT
ejpam-6171	242	39	x	x	NOUN
ejpam-6171	242	40	)	)	PUNCT
ejpam-6171	242	41	.	.	PUNCT
ejpam-6171	243	1	(	(	PUNCT
ejpam-6171	243	2	by	by	ADP
ejpam-6171	243	3	(	(	PUNCT
ejpam-6171	243	4	3.16	3.16	NUM
ejpam-6171	243	5	)	)	PUNCT
ejpam-6171	243	6	)	)	PUNCT
ejpam-6171	243	7	hence	hence	ADV
ejpam-6171	243	8	,	,	PUNCT
ejpam-6171	243	9	it	it	PRON
ejpam-6171	243	10	satisfies	satisfy	VERB
ejpam-6171	243	11	(	(	PUNCT
ejpam-6171	243	12	3.5	3.5	NUM
ejpam-6171	243	13	)	)	PUNCT
ejpam-6171	243	14	,	,	PUNCT
ejpam-6171	243	15	(	(	PUNCT
ejpam-6171	243	16	3.6	3.6	NUM
ejpam-6171	243	17	)	)	PUNCT
ejpam-6171	243	18	,	,	PUNCT
ejpam-6171	243	19	and	and	CCONJ
ejpam-6171	243	20	(	(	PUNCT
ejpam-6171	243	21	3.7	3.7	NUM
ejpam-6171	243	22	)	)	PUNCT
ejpam-6171	243	23	.	.	PUNCT
ejpam-6171	244	1	theorem	theorem	NOUN
ejpam-6171	244	2	2	2	NUM
ejpam-6171	244	3	.	.	PUNCT
ejpam-6171	245	1	a	a	DET
ejpam-6171	245	2	pythagorean	pythagorean	PROPN
ejpam-6171	245	3	neutrosophic	neutrosophic	ADJ
ejpam-6171	245	4	strong	strong	ADJ
ejpam-6171	245	5	iup	iup	NOUN
ejpam-6171	245	6	-	-	PUNCT
ejpam-6171	245	7	ideal	ideal	ADJ
ejpam-6171	245	8	and	and	CCONJ
ejpam-6171	245	9	constant	constant	ADJ
ejpam-6171	245	10	pns	pns	NOUN
ejpam-6171	245	11	coincide	coincide	NOUN
ejpam-6171	245	12	.	.	PUNCT
ejpam-6171	246	1	proof	proof	NOUN
ejpam-6171	246	2	.	.	PUNCT
ejpam-6171	247	1	assume	assume	VERB
ejpam-6171	247	2	that	that	SCONJ
ejpam-6171	247	3	p	p	NOUN
ejpam-6171	247	4	is	be	AUX
ejpam-6171	247	5	a	a	DET
ejpam-6171	247	6	pythagorean	pythagorean	PROPN
ejpam-6171	247	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	247	8	strong	strong	ADJ
ejpam-6171	247	9	iup	iup	NOUN
ejpam-6171	247	10	-	-	PUNCT
ejpam-6171	247	11	ideal	ideal	NOUN
ejpam-6171	247	12	of	of	ADP
ejpam-6171	247	13	x.	x.	NOUN
ejpam-6171	247	14	let	let	VERB
ejpam-6171	247	15	x	x	SYM
ejpam-6171	247	16	∈	∈	PROPN
ejpam-6171	247	17	x.	x.	NOUN
ejpam-6171	247	18	then	then	ADV
ejpam-6171	248	1	pt	pt	X
ejpam-6171	248	2	(	(	PUNCT
ejpam-6171	248	3	x	x	NOUN
ejpam-6171	248	4	)	)	PUNCT
ejpam-6171	248	5	=	=	SYM
ejpam-6171	248	6	pt	pt	X
ejpam-6171	248	7	(	(	PUNCT
ejpam-6171	248	8	(	(	PUNCT
ejpam-6171	248	9	x	x	SYM
ejpam-6171	248	10	?	?	PUNCT
ejpam-6171	248	11	0	0	NUM
ejpam-6171	248	12	)	)	PUNCT
ejpam-6171	248	13	?	?	PUNCT
ejpam-6171	249	1	0	0	X
ejpam-6171	249	2	)	)	PUNCT
ejpam-6171	249	3	(	(	PUNCT
ejpam-6171	249	4	by	by	ADP
ejpam-6171	249	5	(	(	PUNCT
ejpam-6171	249	6	2.5	2.5	NUM
ejpam-6171	249	7	)	)	PUNCT
ejpam-6171	249	8	)	)	PUNCT
ejpam-6171	249	9	≥	≥	PROPN
ejpam-6171	249	10	pt	pt	X
ejpam-6171	249	11	(	(	PUNCT
ejpam-6171	249	12	0	0	NUM
ejpam-6171	249	13	)	)	PUNCT
ejpam-6171	249	14	,	,	PUNCT
ejpam-6171	249	15	(	(	PUNCT
ejpam-6171	249	16	by	by	ADP
ejpam-6171	249	17	(	(	PUNCT
ejpam-6171	249	18	3.14	3.14	NUM
ejpam-6171	249	19	)	)	PUNCT
ejpam-6171	249	20	)	)	PUNCT
ejpam-6171	249	21	pi(x	pi(x	NOUN
ejpam-6171	249	22	)	)	PUNCT
ejpam-6171	249	23	=	=	NOUN
ejpam-6171	249	24	pi((x	pi((x	NOUN
ejpam-6171	249	25	?	?	PUNCT
ejpam-6171	249	26	0	0	X
ejpam-6171	249	27	)	)	PUNCT
ejpam-6171	249	28	?	?	PUNCT
ejpam-6171	249	29	0	0	X
ejpam-6171	249	30	)	)	PUNCT
ejpam-6171	249	31	(	(	PUNCT
ejpam-6171	249	32	by	by	ADP
ejpam-6171	249	33	(	(	PUNCT
ejpam-6171	249	34	2.5	2.5	NUM
ejpam-6171	249	35	)	)	PUNCT
ejpam-6171	249	36	)	)	PUNCT
ejpam-6171	249	37	≤	≤	PROPN
ejpam-6171	249	38	pi(0	pi(0	PROPN
ejpam-6171	249	39	)	)	PUNCT
ejpam-6171	249	40	,	,	PUNCT
ejpam-6171	249	41	(	(	PUNCT
ejpam-6171	249	42	by	by	ADP
ejpam-6171	249	43	(	(	PUNCT
ejpam-6171	249	44	3.15	3.15	NUM
ejpam-6171	249	45	)	)	PUNCT
ejpam-6171	249	46	)	)	PUNCT
ejpam-6171	249	47	pf	pf	INTJ
ejpam-6171	249	48	(	(	PUNCT
ejpam-6171	249	49	x	x	NOUN
ejpam-6171	249	50	)	)	PUNCT
ejpam-6171	249	51	=	=	SYM
ejpam-6171	249	52	pf	pf	X
ejpam-6171	249	53	(	(	PUNCT
ejpam-6171	249	54	(	(	PUNCT
ejpam-6171	249	55	x	x	SYM
ejpam-6171	249	56	?	?	PUNCT
ejpam-6171	249	57	0	0	NUM
ejpam-6171	249	58	)	)	PUNCT
ejpam-6171	249	59	?	?	PUNCT
ejpam-6171	249	60	0	0	X
ejpam-6171	249	61	)	)	PUNCT
ejpam-6171	249	62	(	(	PUNCT
ejpam-6171	249	63	by	by	ADP
ejpam-6171	249	64	(	(	PUNCT
ejpam-6171	249	65	2.5	2.5	NUM
ejpam-6171	249	66	)	)	PUNCT
ejpam-6171	249	67	)	)	PUNCT
ejpam-6171	249	68	≥	≥	PROPN
ejpam-6171	249	69	pf	pf	X
ejpam-6171	249	70	(	(	PUNCT
ejpam-6171	249	71	0	0	NUM
ejpam-6171	249	72	)	)	PUNCT
ejpam-6171	249	73	.	.	PUNCT
ejpam-6171	250	1	(	(	PUNCT
ejpam-6171	250	2	by	by	ADP
ejpam-6171	250	3	(	(	PUNCT
ejpam-6171	250	4	3.16	3.16	NUM
ejpam-6171	250	5	)	)	PUNCT
ejpam-6171	250	6	)	)	PUNCT
ejpam-6171	250	7	hence	hence	ADV
ejpam-6171	250	8	,	,	PUNCT
ejpam-6171	250	9	p	p	PROPN
ejpam-6171	250	10	is	be	AUX
ejpam-6171	250	11	a	a	DET
ejpam-6171	250	12	constant	constant	NOUN
ejpam-6171	250	13	of	of	ADP
ejpam-6171	250	14	x.	x.	NOUN
ejpam-6171	250	15	conversely	conversely	ADV
ejpam-6171	250	16	,	,	PUNCT
ejpam-6171	250	17	it	it	PRON
ejpam-6171	250	18	is	be	AUX
ejpam-6171	250	19	obvious	obvious	ADJ
ejpam-6171	250	20	that	that	SCONJ
ejpam-6171	250	21	every	every	DET
ejpam-6171	250	22	constant	constant	ADJ
ejpam-6171	250	23	pnf	pnf	NOUN
ejpam-6171	250	24	is	be	AUX
ejpam-6171	250	25	a	a	DET
ejpam-6171	250	26	pythagorean	pythagorean	PROPN
ejpam-6171	250	27	neutrosophic	neutrosophic	ADJ
ejpam-6171	250	28	strong	strong	ADJ
ejpam-6171	250	29	iup	iup	NOUN
ejpam-6171	250	30	-	-	PUNCT
ejpam-6171	250	31	ideal	ideal	NOUN
ejpam-6171	250	32	.	.	PUNCT
ejpam-6171	251	1	theorem	theorem	NOUN
ejpam-6171	251	2	3	3	NUM
ejpam-6171	251	3	.	.	PUNCT
ejpam-6171	252	1	every	every	DET
ejpam-6171	252	2	pythagorean	pythagorean	PROPN
ejpam-6171	252	3	neutrosophic	neutrosophic	ADJ
ejpam-6171	252	4	strong	strong	ADJ
ejpam-6171	252	5	iup	iup	NOUN
ejpam-6171	252	6	-	-	PUNCT
ejpam-6171	252	7	ideal	ideal	NOUN
ejpam-6171	252	8	of	of	ADP
ejpam-6171	252	9	x	x	PUNCT
ejpam-6171	252	10	is	be	AUX
ejpam-6171	252	11	a	a	DET
ejpam-6171	252	12	pythagorean	pythagorean	PROPN
ejpam-6171	252	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	252	14	iup	iup	NOUN
ejpam-6171	252	15	-	-	PUNCT
ejpam-6171	252	16	subalgebra	subalgebra	NOUN
ejpam-6171	252	17	of	of	ADP
ejpam-6171	252	18	x.	x.	NOUN
ejpam-6171	252	19	proof	proof	NOUN
ejpam-6171	252	20	.	.	PUNCT
ejpam-6171	253	1	it	it	PRON
ejpam-6171	253	2	is	be	AUX
ejpam-6171	253	3	straightforward	straightforward	ADJ
ejpam-6171	253	4	by	by	ADP
ejpam-6171	253	5	theorem	theorem	NOUN
ejpam-6171	253	6	2	2	NUM
ejpam-6171	253	7	.	.	NOUN
ejpam-6171	253	8	example	example	NOUN
ejpam-6171	254	1	5	5	NUM
ejpam-6171	254	2	.	.	PUNCT
ejpam-6171	255	1	let	let	VERB
ejpam-6171	255	2	x	x	PUNCT
ejpam-6171	255	3	=	=	PUNCT
ejpam-6171	255	4	{	{	PUNCT
ejpam-6171	255	5	0	0	NUM
ejpam-6171	255	6	,	,	PUNCT
ejpam-6171	255	7	1	1	NUM
ejpam-6171	255	8	,	,	PUNCT
ejpam-6171	255	9	2	2	NUM
ejpam-6171	255	10	,	,	PUNCT
ejpam-6171	255	11	3	3	NUM
ejpam-6171	255	12	,	,	PUNCT
ejpam-6171	255	13	4	4	NUM
ejpam-6171	255	14	,	,	PUNCT
ejpam-6171	255	15	5	5	NUM
ejpam-6171	255	16	}	}	PUNCT
ejpam-6171	255	17	with	with	ADP
ejpam-6171	255	18	the	the	DET
ejpam-6171	255	19	following	follow	VERB
ejpam-6171	255	20	cayley	cayley	ADJ
ejpam-6171	255	21	table	table	NOUN
ejpam-6171	255	22	:	:	PUNCT
ejpam-6171	255	23	·	·	PUNCT
ejpam-6171	255	24	0	0	NUM
ejpam-6171	256	1	1	1	NUM
ejpam-6171	256	2	2	2	NUM
ejpam-6171	256	3	3	3	NUM
ejpam-6171	256	4	4	4	NUM
ejpam-6171	256	5	5	5	NUM
ejpam-6171	256	6	0	0	NUM
ejpam-6171	256	7	0	0	NUM
ejpam-6171	256	8	1	1	NUM
ejpam-6171	256	9	2	2	NUM
ejpam-6171	256	10	3	3	NUM
ejpam-6171	256	11	4	4	NUM
ejpam-6171	256	12	5	5	NUM
ejpam-6171	256	13	1	1	NUM
ejpam-6171	256	14	5	5	NUM
ejpam-6171	256	15	0	0	NUM
ejpam-6171	256	16	4	4	NUM
ejpam-6171	256	17	2	2	NUM
ejpam-6171	256	18	3	3	NUM
ejpam-6171	256	19	1	1	NUM
ejpam-6171	256	20	2	2	NUM
ejpam-6171	256	21	4	4	NUM
ejpam-6171	256	22	2	2	NUM
ejpam-6171	256	23	0	0	NUM
ejpam-6171	256	24	1	1	NUM
ejpam-6171	256	25	5	5	NUM
ejpam-6171	256	26	3	3	NUM
ejpam-6171	256	27	3	3	NUM
ejpam-6171	256	28	3	3	NUM
ejpam-6171	256	29	4	4	NUM
ejpam-6171	256	30	5	5	NUM
ejpam-6171	256	31	0	0	NUM
ejpam-6171	256	32	1	1	NUM
ejpam-6171	256	33	2	2	NUM
ejpam-6171	256	34	4	4	NUM
ejpam-6171	256	35	2	2	NUM
ejpam-6171	256	36	3	3	NUM
ejpam-6171	256	37	1	1	NUM
ejpam-6171	256	38	5	5	NUM
ejpam-6171	256	39	0	0	NUM
ejpam-6171	256	40	4	4	NUM
ejpam-6171	256	41	5	5	NUM
ejpam-6171	256	42	1	1	NUM
ejpam-6171	256	43	5	5	NUM
ejpam-6171	256	44	3	3	NUM
ejpam-6171	256	45	4	4	NUM
ejpam-6171	256	46	2	2	NUM
ejpam-6171	256	47	0	0	NUM
ejpam-6171	256	48	k.	k.	NOUN
ejpam-6171	256	49	suayngam	suayngam	PROPN
ejpam-6171	256	50	et	et	PROPN
ejpam-6171	256	51	al	al	PROPN
ejpam-6171	256	52	.	.	PUNCT
ejpam-6171	256	53	/	/	SYM
ejpam-6171	256	54	eur	eur	PROPN
ejpam-6171	256	55	.	.	PUNCT
ejpam-6171	257	1	j.	j.	PROPN
ejpam-6171	257	2	pure	pure	PROPN
ejpam-6171	257	3	appl	appl	PROPN
ejpam-6171	257	4	.	.	PROPN
ejpam-6171	257	5	math	math	PROPN
ejpam-6171	257	6	,	,	PUNCT
ejpam-6171	257	7	18	18	NUM
ejpam-6171	257	8	(	(	PUNCT
ejpam-6171	257	9	3	3	NUM
ejpam-6171	257	10	)	)	PUNCT
ejpam-6171	257	11	(	(	PUNCT
ejpam-6171	257	12	2025	2025	NUM
ejpam-6171	257	13	)	)	PUNCT
ejpam-6171	257	14	,	,	PUNCT
ejpam-6171	257	15	6171	6171	NUM
ejpam-6171	257	16	10	10	NUM
ejpam-6171	257	17	of	of	ADP
ejpam-6171	257	18	28	28	NUM
ejpam-6171	258	1	then	then	ADV
ejpam-6171	258	2	x	x	PUNCT
ejpam-6171	258	3	is	be	AUX
ejpam-6171	258	4	an	an	DET
ejpam-6171	258	5	iup	iup	NOUN
ejpam-6171	258	6	-	-	PUNCT
ejpam-6171	258	7	algebra	algebra	NOUN
ejpam-6171	258	8	.	.	PUNCT
ejpam-6171	259	1	we	we	PRON
ejpam-6171	259	2	define	define	VERB
ejpam-6171	259	3	p	p	NOUN
ejpam-6171	259	4	on	on	ADP
ejpam-6171	259	5	x	x	PUNCT
ejpam-6171	259	6	as	as	SCONJ
ejpam-6171	259	7	follows	follow	VERB
ejpam-6171	259	8	:	:	PUNCT
ejpam-6171	259	9	pt	pt	X
ejpam-6171	259	10	=	=	SYM
ejpam-6171	259	11	(	(	PUNCT
ejpam-6171	259	12	0	0	NUM
ejpam-6171	259	13	1	1	NUM
ejpam-6171	259	14	1	1	NUM
ejpam-6171	259	15	0.8	0.8	NUM
ejpam-6171	259	16	2	2	NUM
ejpam-6171	259	17	0.4	0.4	NUM
ejpam-6171	259	18	3	3	NUM
ejpam-6171	259	19	0.4	0.4	NUM
ejpam-6171	259	20	4	4	NUM
ejpam-6171	259	21	0.4	0.4	NUM
ejpam-6171	259	22	5	5	NUM
ejpam-6171	259	23	0.8	0.8	NUM
ejpam-6171	259	24	)	)	PUNCT
ejpam-6171	259	25	pi	pi	NOUN
ejpam-6171	259	26	=	=	PUNCT
ejpam-6171	259	27	(	(	PUNCT
ejpam-6171	259	28	0	0	NUM
ejpam-6171	259	29	0	0	NUM
ejpam-6171	259	30	1	1	NUM
ejpam-6171	259	31	0.2	0.2	NUM
ejpam-6171	259	32	2	2	NUM
ejpam-6171	259	33	0.3	0.3	NUM
ejpam-6171	259	34	3	3	NUM
ejpam-6171	259	35	0.3	0.3	NUM
ejpam-6171	259	36	4	4	NUM
ejpam-6171	259	37	0.3	0.3	NUM
ejpam-6171	259	38	5	5	NUM
ejpam-6171	259	39	0.2	0.2	NUM
ejpam-6171	259	40	)	)	PUNCT
ejpam-6171	259	41	pf	pf	NOUN
ejpam-6171	259	42	=	=	PUNCT
ejpam-6171	259	43	(	(	PUNCT
ejpam-6171	259	44	0	0	NUM
ejpam-6171	259	45	1	1	NUM
ejpam-6171	259	46	1	1	NUM
ejpam-6171	259	47	0.6	0.6	NUM
ejpam-6171	259	48	2	2	NUM
ejpam-6171	259	49	0.2	0.2	NUM
ejpam-6171	259	50	3	3	NUM
ejpam-6171	259	51	0.2	0.2	NUM
ejpam-6171	259	52	4	4	NUM
ejpam-6171	259	53	0.2	0.2	NUM
ejpam-6171	259	54	5	5	NUM
ejpam-6171	259	55	0.6	0.6	NUM
ejpam-6171	259	56	)	)	PUNCT
ejpam-6171	259	57	then	then	ADV
ejpam-6171	259	58	p	p	PROPN
ejpam-6171	259	59	is	be	AUX
ejpam-6171	259	60	a	a	DET
ejpam-6171	259	61	pythagorean	pythagorean	PROPN
ejpam-6171	259	62	neutrosophic	neutrosophic	ADJ
ejpam-6171	259	63	iup	iup	NOUN
ejpam-6171	259	64	-	-	PUNCT
ejpam-6171	259	65	subalgebra	subalgebra	NOUN
ejpam-6171	259	66	of	of	ADP
ejpam-6171	259	67	x.	x.	NOUN
ejpam-6171	259	68	since	since	SCONJ
ejpam-6171	259	69	pt	pt	X
ejpam-6171	259	70	(	(	PUNCT
ejpam-6171	259	71	3	3	NUM
ejpam-6171	259	72	?	?	SYM
ejpam-6171	259	73	1	1	NUM
ejpam-6171	259	74	)	)	PUNCT
ejpam-6171	260	1	=	=	SYM
ejpam-6171	260	2	pt	pt	X
ejpam-6171	260	3	(	(	PUNCT
ejpam-6171	260	4	4	4	NUM
ejpam-6171	260	5	)	)	PUNCT
ejpam-6171	260	6	=	=	SYM
ejpam-6171	260	7	0.4	0.4	NUM
ejpam-6171	260	8	�	�	PROPN
ejpam-6171	260	9	0.8	0.8	NUM
ejpam-6171	260	10	=	=	SYM
ejpam-6171	260	11	pt	pt	X
ejpam-6171	260	12	(	(	PUNCT
ejpam-6171	260	13	1	1	NUM
ejpam-6171	260	14	)	)	PUNCT
ejpam-6171	260	15	,	,	PUNCT
ejpam-6171	260	16	pi(3	pi(3	PROPN
ejpam-6171	260	17	?	?	NOUN
ejpam-6171	260	18	0	0	X
ejpam-6171	260	19	)	)	PUNCT
ejpam-6171	260	20	=	=	PUNCT
ejpam-6171	261	1	pi(3	pi(3	ADJ
ejpam-6171	261	2	)	)	PUNCT
ejpam-6171	261	3	=	=	SYM
ejpam-6171	261	4	0.3	0.3	NUM
ejpam-6171	261	5	�	�	NOUN
ejpam-6171	261	6	0	0	NUM
ejpam-6171	261	7	=	=	SYM
ejpam-6171	261	8	pi(0	pi(0	PROPN
ejpam-6171	261	9	)	)	PUNCT
ejpam-6171	261	10	,	,	PUNCT
ejpam-6171	261	11	and	and	CCONJ
ejpam-6171	261	12	pf	pf	X
ejpam-6171	261	13	(	(	PUNCT
ejpam-6171	261	14	4	4	NUM
ejpam-6171	261	15	?	?	SYM
ejpam-6171	261	16	5	5	NUM
ejpam-6171	261	17	)	)	PUNCT
ejpam-6171	261	18	=	=	SYM
ejpam-6171	261	19	pf	pf	X
ejpam-6171	261	20	(	(	PUNCT
ejpam-6171	261	21	4	4	NUM
ejpam-6171	261	22	)	)	PUNCT
ejpam-6171	261	23	=	=	SYM
ejpam-6171	261	24	0.2	0.2	NUM
ejpam-6171	261	25	�	�	NOUN
ejpam-6171	261	26	0.6	0.6	NUM
ejpam-6171	261	27	=	=	SYM
ejpam-6171	261	28	pf	pf	X
ejpam-6171	261	29	(	(	PUNCT
ejpam-6171	261	30	5	5	NUM
ejpam-6171	261	31	)	)	PUNCT
ejpam-6171	261	32	.	.	PUNCT
ejpam-6171	262	1	hence	hence	ADV
ejpam-6171	262	2	,	,	PUNCT
ejpam-6171	262	3	p	p	PRON
ejpam-6171	262	4	is	be	AUX
ejpam-6171	262	5	not	not	PART
ejpam-6171	262	6	a	a	DET
ejpam-6171	262	7	pythagorean	pythagorean	PROPN
ejpam-6171	262	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	262	9	strong	strong	ADJ
ejpam-6171	262	10	iup	iup	NOUN
ejpam-6171	262	11	-	-	PUNCT
ejpam-6171	262	12	ideal	ideal	NOUN
ejpam-6171	262	13	of	of	ADP
ejpam-6171	262	14	x.	x.	PROPN
ejpam-6171	262	15	theorem	theorem	VERB
ejpam-6171	262	16	4	4	NUM
ejpam-6171	262	17	.	.	PUNCT
ejpam-6171	263	1	every	every	DET
ejpam-6171	263	2	pythagorean	pythagorean	PROPN
ejpam-6171	263	3	neutrosophic	neutrosophic	ADJ
ejpam-6171	263	4	strong	strong	ADJ
ejpam-6171	263	5	iup	iup	NOUN
ejpam-6171	263	6	-	-	PUNCT
ejpam-6171	263	7	ideal	ideal	NOUN
ejpam-6171	263	8	of	of	ADP
ejpam-6171	263	9	x	x	PUNCT
ejpam-6171	263	10	is	be	AUX
ejpam-6171	263	11	a	a	DET
ejpam-6171	263	12	pythagorean	pythagorean	PROPN
ejpam-6171	263	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	263	14	iup	iup	PROPN
ejpam-6171	263	15	-	-	PUNCT
ejpam-6171	263	16	ideal	ideal	NOUN
ejpam-6171	263	17	of	of	ADP
ejpam-6171	263	18	x.	x.	NOUN
ejpam-6171	263	19	proof	proof	NOUN
ejpam-6171	263	20	.	.	PUNCT
ejpam-6171	264	1	it	it	PRON
ejpam-6171	264	2	is	be	AUX
ejpam-6171	264	3	straightforward	straightforward	ADJ
ejpam-6171	264	4	by	by	ADP
ejpam-6171	264	5	theorem	theorem	NOUN
ejpam-6171	264	6	2	2	NUM
ejpam-6171	264	7	.	.	NOUN
ejpam-6171	264	8	example	example	NOUN
ejpam-6171	265	1	6	6	NUM
ejpam-6171	265	2	.	.	PUNCT
ejpam-6171	266	1	let	let	VERB
ejpam-6171	266	2	x	x	PUNCT
ejpam-6171	266	3	=	=	PUNCT
ejpam-6171	266	4	{	{	PUNCT
ejpam-6171	266	5	0	0	NUM
ejpam-6171	266	6	,	,	PUNCT
ejpam-6171	266	7	1	1	NUM
ejpam-6171	266	8	,	,	PUNCT
ejpam-6171	266	9	2	2	NUM
ejpam-6171	266	10	,	,	PUNCT
ejpam-6171	266	11	3	3	NUM
ejpam-6171	266	12	,	,	PUNCT
ejpam-6171	266	13	4	4	NUM
ejpam-6171	266	14	,	,	PUNCT
ejpam-6171	266	15	5	5	NUM
ejpam-6171	266	16	}	}	PUNCT
ejpam-6171	266	17	with	with	ADP
ejpam-6171	266	18	the	the	DET
ejpam-6171	266	19	following	follow	VERB
ejpam-6171	266	20	cayley	cayley	ADJ
ejpam-6171	266	21	table	table	NOUN
ejpam-6171	266	22	:	:	PUNCT
ejpam-6171	266	23	·	·	PUNCT
ejpam-6171	266	24	0	0	NUM
ejpam-6171	266	25	1	1	NUM
ejpam-6171	266	26	2	2	NUM
ejpam-6171	266	27	3	3	NUM
ejpam-6171	266	28	4	4	NUM
ejpam-6171	266	29	5	5	NUM
ejpam-6171	266	30	0	0	NUM
ejpam-6171	266	31	0	0	NUM
ejpam-6171	266	32	1	1	NUM
ejpam-6171	266	33	2	2	NUM
ejpam-6171	266	34	3	3	NUM
ejpam-6171	266	35	4	4	NUM
ejpam-6171	266	36	5	5	NUM
ejpam-6171	266	37	1	1	NUM
ejpam-6171	266	38	5	5	NUM
ejpam-6171	266	39	0	0	NUM
ejpam-6171	266	40	4	4	NUM
ejpam-6171	266	41	1	1	NUM
ejpam-6171	266	42	3	3	NUM
ejpam-6171	266	43	2	2	NUM
ejpam-6171	266	44	2	2	NUM
ejpam-6171	266	45	3	3	NUM
ejpam-6171	266	46	4	4	NUM
ejpam-6171	266	47	0	0	NUM
ejpam-6171	266	48	2	2	NUM
ejpam-6171	266	49	5	5	NUM
ejpam-6171	266	50	1	1	NUM
ejpam-6171	266	51	3	3	NUM
ejpam-6171	266	52	2	2	NUM
ejpam-6171	266	53	5	5	NUM
ejpam-6171	266	54	3	3	NUM
ejpam-6171	266	55	0	0	NUM
ejpam-6171	266	56	1	1	NUM
ejpam-6171	266	57	4	4	NUM
ejpam-6171	266	58	4	4	NUM
ejpam-6171	266	59	4	4	NUM
ejpam-6171	266	60	2	2	NUM
ejpam-6171	266	61	1	1	NUM
ejpam-6171	266	62	5	5	NUM
ejpam-6171	266	63	0	0	NUM
ejpam-6171	266	64	3	3	NUM
ejpam-6171	266	65	5	5	NUM
ejpam-6171	266	66	1	1	NUM
ejpam-6171	266	67	3	3	NUM
ejpam-6171	266	68	5	5	NUM
ejpam-6171	266	69	4	4	NUM
ejpam-6171	266	70	2	2	NUM
ejpam-6171	266	71	0	0	NUM
ejpam-6171	266	72	then	then	ADV
ejpam-6171	266	73	x	x	PUNCT
ejpam-6171	266	74	is	be	AUX
ejpam-6171	266	75	an	an	DET
ejpam-6171	266	76	iup	iup	NOUN
ejpam-6171	266	77	-	-	PUNCT
ejpam-6171	266	78	algebra	algebra	NOUN
ejpam-6171	266	79	.	.	PUNCT
ejpam-6171	267	1	we	we	PRON
ejpam-6171	267	2	define	define	VERB
ejpam-6171	267	3	p	p	NOUN
ejpam-6171	267	4	on	on	ADP
ejpam-6171	267	5	x	x	PUNCT
ejpam-6171	267	6	as	as	SCONJ
ejpam-6171	267	7	follows	follow	VERB
ejpam-6171	267	8	:	:	PUNCT
ejpam-6171	268	1	pt	pt	X
ejpam-6171	268	2	=	=	SYM
ejpam-6171	268	3	(	(	PUNCT
ejpam-6171	268	4	0	0	NUM
ejpam-6171	268	5	0.7	0.7	NUM
ejpam-6171	268	6	1	1	NUM
ejpam-6171	268	7	0.1	0.1	NUM
ejpam-6171	268	8	2	2	NUM
ejpam-6171	268	9	0.1	0.1	NUM
ejpam-6171	268	10	3	3	NUM
ejpam-6171	268	11	0.6	0.6	NUM
ejpam-6171	268	12	4	4	NUM
ejpam-6171	268	13	0.6	0.6	NUM
ejpam-6171	268	14	5	5	NUM
ejpam-6171	268	15	0.1	0.1	NUM
ejpam-6171	268	16	)	)	PUNCT
ejpam-6171	268	17	pi	pi	NOUN
ejpam-6171	268	18	=	=	PUNCT
ejpam-6171	268	19	(	(	PUNCT
ejpam-6171	268	20	0	0	NUM
ejpam-6171	268	21	0.7	0.7	NUM
ejpam-6171	268	22	1	1	NUM
ejpam-6171	268	23	1	1	NUM
ejpam-6171	268	24	2	2	NUM
ejpam-6171	268	25	0.9	0.9	NUM
ejpam-6171	268	26	3	3	NUM
ejpam-6171	268	27	0.9	0.9	NUM
ejpam-6171	268	28	4	4	NUM
ejpam-6171	268	29	1	1	NUM
ejpam-6171	268	30	5	5	NUM
ejpam-6171	268	31	1	1	NUM
ejpam-6171	268	32	)	)	PUNCT
ejpam-6171	268	33	pf	pf	NOUN
ejpam-6171	268	34	=	=	PUNCT
ejpam-6171	268	35	(	(	PUNCT
ejpam-6171	268	36	0	0	NUM
ejpam-6171	268	37	0.5	0.5	NUM
ejpam-6171	268	38	1	1	NUM
ejpam-6171	268	39	0.2	0.2	NUM
ejpam-6171	268	40	2	2	NUM
ejpam-6171	268	41	0.4	0.4	NUM
ejpam-6171	268	42	3	3	NUM
ejpam-6171	268	43	0.4	0.4	NUM
ejpam-6171	268	44	4	4	NUM
ejpam-6171	268	45	0.2	0.2	NUM
ejpam-6171	268	46	5	5	NUM
ejpam-6171	268	47	0.2	0.2	NUM
ejpam-6171	268	48	)	)	PUNCT
ejpam-6171	268	49	then	then	ADV
ejpam-6171	268	50	p	p	NOUN
ejpam-6171	268	51	is	be	AUX
ejpam-6171	268	52	a	a	DET
ejpam-6171	268	53	pythagorean	pythagorean	PROPN
ejpam-6171	268	54	neutrosophic	neutrosophic	ADJ
ejpam-6171	268	55	iup	iup	PROPN
ejpam-6171	268	56	-	-	PUNCT
ejpam-6171	268	57	ideal	ideal	NOUN
ejpam-6171	268	58	of	of	ADP
ejpam-6171	268	59	x.	x.	NOUN
ejpam-6171	268	60	since	since	SCONJ
ejpam-6171	268	61	pt	pt	X
ejpam-6171	268	62	(	(	PUNCT
ejpam-6171	268	63	5	5	NUM
ejpam-6171	268	64	?	?	SYM
ejpam-6171	268	65	0	0	NUM
ejpam-6171	268	66	)	)	PUNCT
ejpam-6171	268	67	=	=	SYM
ejpam-6171	268	68	pt	pt	X
ejpam-6171	268	69	(	(	PUNCT
ejpam-6171	268	70	1	1	NUM
ejpam-6171	268	71	)	)	PUNCT
ejpam-6171	268	72	=	=	SYM
ejpam-6171	268	73	0.1	0.1	NUM
ejpam-6171	268	74	�	�	NOUN
ejpam-6171	268	75	0.6	0.6	NUM
ejpam-6171	268	76	=	=	SYM
ejpam-6171	268	77	pt	pt	X
ejpam-6171	268	78	(	(	PUNCT
ejpam-6171	268	79	0	0	NUM
ejpam-6171	268	80	)	)	PUNCT
ejpam-6171	268	81	,	,	PUNCT
ejpam-6171	268	82	pi(2	pi(2	PROPN
ejpam-6171	268	83	?	?	PUNCT
ejpam-6171	268	84	0	0	X
ejpam-6171	268	85	)	)	PUNCT
ejpam-6171	268	86	=	=	PUNCT
ejpam-6171	269	1	pi(3	pi(3	ADJ
ejpam-6171	269	2	)	)	PUNCT
ejpam-6171	269	3	=	=	SYM
ejpam-6171	269	4	0.9	0.9	NUM
ejpam-6171	269	5	�	�	NOUN
ejpam-6171	269	6	0.7	0.7	NUM
ejpam-6171	269	7	=	=	SYM
ejpam-6171	269	8	pi(0	pi(0	PROPN
ejpam-6171	269	9	)	)	PUNCT
ejpam-6171	269	10	,	,	PUNCT
ejpam-6171	269	11	and	and	CCONJ
ejpam-6171	269	12	pf	pf	X
ejpam-6171	269	13	(	(	PUNCT
ejpam-6171	269	14	4	4	NUM
ejpam-6171	269	15	?	?	SYM
ejpam-6171	269	16	3	3	X
ejpam-6171	269	17	)	)	PUNCT
ejpam-6171	269	18	=	=	SYM
ejpam-6171	269	19	pf	pf	X
ejpam-6171	269	20	(	(	PUNCT
ejpam-6171	269	21	5	5	NUM
ejpam-6171	269	22	)	)	PUNCT
ejpam-6171	269	23	=	=	SYM
ejpam-6171	269	24	0.2	0.2	NUM
ejpam-6171	269	25	�	�	PROPN
ejpam-6171	269	26	0.4	0.4	NUM
ejpam-6171	269	27	=	=	SYM
ejpam-6171	269	28	pf	pf	X
ejpam-6171	269	29	(	(	PUNCT
ejpam-6171	269	30	3	3	NUM
ejpam-6171	269	31	)	)	PUNCT
ejpam-6171	269	32	.	.	PUNCT
ejpam-6171	270	1	hence	hence	ADV
ejpam-6171	270	2	,	,	PUNCT
ejpam-6171	270	3	p	p	PRON
ejpam-6171	270	4	is	be	AUX
ejpam-6171	270	5	not	not	PART
ejpam-6171	270	6	a	a	DET
ejpam-6171	270	7	pythagorean	pythagorean	PROPN
ejpam-6171	270	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	270	9	strong	strong	ADJ
ejpam-6171	270	10	iup	iup	NOUN
ejpam-6171	270	11	-	-	PUNCT
ejpam-6171	270	12	ideal	ideal	NOUN
ejpam-6171	270	13	of	of	ADP
ejpam-6171	270	14	x.	x.	PROPN
ejpam-6171	270	15	theorem	theorem	VERB
ejpam-6171	270	16	5	5	NUM
ejpam-6171	270	17	.	.	PUNCT
ejpam-6171	271	1	every	every	DET
ejpam-6171	271	2	pythagorean	pythagorean	PROPN
ejpam-6171	271	3	neutrosophic	neutrosophic	PROPN
ejpam-6171	271	4	iup	iup	PROPN
ejpam-6171	271	5	-	-	PUNCT
ejpam-6171	271	6	ideal	ideal	NOUN
ejpam-6171	271	7	of	of	ADP
ejpam-6171	271	8	x	x	PUNCT
ejpam-6171	271	9	is	be	AUX
ejpam-6171	271	10	a	a	DET
ejpam-6171	271	11	pythagorean	pythagorean	PROPN
ejpam-6171	271	12	neutrosophic	neutrosophic	ADJ
ejpam-6171	271	13	iup	iup	NOUN
ejpam-6171	271	14	-	-	PUNCT
ejpam-6171	271	15	filter	filter	NOUN
ejpam-6171	271	16	of	of	ADP
ejpam-6171	271	17	x.	x.	PROPN
ejpam-6171	271	18	k.	k.	PROPN
ejpam-6171	272	1	suayngam	suayngam	PROPN
ejpam-6171	272	2	et	et	PROPN
ejpam-6171	272	3	al	al	PROPN
ejpam-6171	272	4	.	.	PUNCT
ejpam-6171	272	5	/	/	SYM
ejpam-6171	272	6	eur	eur	PROPN
ejpam-6171	272	7	.	.	PUNCT
ejpam-6171	273	1	j.	j.	PROPN
ejpam-6171	273	2	pure	pure	PROPN
ejpam-6171	273	3	appl	appl	PROPN
ejpam-6171	273	4	.	.	PROPN
ejpam-6171	273	5	math	math	PROPN
ejpam-6171	273	6	,	,	PUNCT
ejpam-6171	273	7	18	18	NUM
ejpam-6171	273	8	(	(	PUNCT
ejpam-6171	273	9	3	3	NUM
ejpam-6171	273	10	)	)	PUNCT
ejpam-6171	273	11	(	(	PUNCT
ejpam-6171	273	12	2025	2025	NUM
ejpam-6171	273	13	)	)	PUNCT
ejpam-6171	273	14	,	,	PUNCT
ejpam-6171	273	15	6171	6171	NUM
ejpam-6171	273	16	11	11	NUM
ejpam-6171	273	17	of	of	ADP
ejpam-6171	273	18	28	28	NUM
ejpam-6171	273	19	proof	proof	NOUN
ejpam-6171	273	20	.	.	PUNCT
ejpam-6171	274	1	assume	assume	VERB
ejpam-6171	274	2	that	that	SCONJ
ejpam-6171	274	3	p	p	NOUN
ejpam-6171	274	4	is	be	AUX
ejpam-6171	274	5	a	a	DET
ejpam-6171	274	6	pythagorean	pythagorean	PROPN
ejpam-6171	274	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	274	8	iup	iup	PROPN
ejpam-6171	274	9	-	-	PUNCT
ejpam-6171	274	10	ideal	ideal	NOUN
ejpam-6171	274	11	of	of	ADP
ejpam-6171	274	12	x.	x.	NOUN
ejpam-6171	274	13	by	by	ADP
ejpam-6171	274	14	the	the	DET
ejpam-6171	274	15	assumption	assumption	NOUN
ejpam-6171	274	16	,	,	PUNCT
ejpam-6171	274	17	it	it	PRON
ejpam-6171	274	18	satisfies	satisfy	VERB
ejpam-6171	274	19	(	(	PUNCT
ejpam-6171	274	20	3.5	3.5	NUM
ejpam-6171	274	21	)	)	PUNCT
ejpam-6171	274	22	,	,	PUNCT
ejpam-6171	274	23	(	(	PUNCT
ejpam-6171	274	24	3.6	3.6	NUM
ejpam-6171	274	25	)	)	PUNCT
ejpam-6171	274	26	,	,	PUNCT
ejpam-6171	274	27	and	and	CCONJ
ejpam-6171	274	28	(	(	PUNCT
ejpam-6171	274	29	3.7	3.7	NUM
ejpam-6171	274	30	)	)	PUNCT
ejpam-6171	274	31	.	.	PUNCT
ejpam-6171	275	1	let	let	VERB
ejpam-6171	275	2	x	x	PRON
ejpam-6171	275	3	,	,	PUNCT
ejpam-6171	275	4	y	y	PROPN
ejpam-6171	275	5	∈	∈	PROPN
ejpam-6171	275	6	x.	x.	NOUN
ejpam-6171	276	1	then	then	ADV
ejpam-6171	276	2	pt	pt	PROPN
ejpam-6171	276	3	(	(	PUNCT
ejpam-6171	276	4	y	y	NOUN
ejpam-6171	276	5	)	)	PUNCT
ejpam-6171	276	6	=	=	SYM
ejpam-6171	276	7	pt	pt	X
ejpam-6171	276	8	(	(	PUNCT
ejpam-6171	276	9	0	0	NUM
ejpam-6171	276	10	?	?	PUNCT
ejpam-6171	277	1	y	y	X
ejpam-6171	277	2	)	)	PUNCT
ejpam-6171	277	3	(	(	PUNCT
ejpam-6171	277	4	by	by	ADP
ejpam-6171	277	5	(	(	PUNCT
ejpam-6171	277	6	iup-1	iup-1	NOUN
ejpam-6171	277	7	)	)	PUNCT
ejpam-6171	277	8	)	)	PUNCT
ejpam-6171	277	9	≥	≥	NOUN
ejpam-6171	277	10	min{pt	min{pt	X
ejpam-6171	277	11	(	(	PUNCT
ejpam-6171	277	12	0	0	NUM
ejpam-6171	277	13	?	?	PUNCT
ejpam-6171	278	1	(	(	PUNCT
ejpam-6171	278	2	x	x	X
ejpam-6171	278	3	?	?	PUNCT
ejpam-6171	278	4	y)),pt	y)),pt	NOUN
ejpam-6171	278	5	(	(	PUNCT
ejpam-6171	278	6	x	x	NOUN
ejpam-6171	278	7	)	)	PUNCT
ejpam-6171	278	8	}	}	PUNCT
ejpam-6171	278	9	(	(	PUNCT
ejpam-6171	278	10	by	by	ADP
ejpam-6171	278	11	(	(	PUNCT
ejpam-6171	278	12	3.8	3.8	NUM
ejpam-6171	278	13	)	)	PUNCT
ejpam-6171	278	14	)	)	PUNCT
ejpam-6171	279	1	=	=	PRON
ejpam-6171	279	2	min{pt	min{pt	NOUN
ejpam-6171	279	3	(	(	PUNCT
ejpam-6171	279	4	x	x	X
ejpam-6171	279	5	?	?	PUNCT
ejpam-6171	279	6	y),pt	y),pt	PROPN
ejpam-6171	279	7	(	(	PUNCT
ejpam-6171	279	8	x	x	NOUN
ejpam-6171	279	9	)	)	PUNCT
ejpam-6171	279	10	}	}	PUNCT
ejpam-6171	279	11	,	,	PUNCT
ejpam-6171	279	12	(	(	PUNCT
ejpam-6171	279	13	by	by	ADP
ejpam-6171	279	14	(	(	PUNCT
ejpam-6171	279	15	iup-1	iup-1	NOUN
ejpam-6171	279	16	)	)	PUNCT
ejpam-6171	279	17	)	)	PUNCT
ejpam-6171	279	18	pi(y	pi(y	NOUN
ejpam-6171	279	19	)	)	PUNCT
ejpam-6171	280	1	=	=	SYM
ejpam-6171	280	2	pi(0	pi(0	PROPN
ejpam-6171	280	3	?	?	PUNCT
ejpam-6171	281	1	y	y	X
ejpam-6171	281	2	)	)	PUNCT
ejpam-6171	281	3	(	(	PUNCT
ejpam-6171	281	4	by	by	ADP
ejpam-6171	281	5	(	(	PUNCT
ejpam-6171	281	6	iup-1	iup-1	NOUN
ejpam-6171	281	7	)	)	PUNCT
ejpam-6171	281	8	)	)	PUNCT
ejpam-6171	281	9	≤	≤	PUNCT
ejpam-6171	282	1	max{pi(0	max{pi(0	PROPN
ejpam-6171	282	2	?	?	PUNCT
ejpam-6171	283	1	(	(	PUNCT
ejpam-6171	283	2	x	x	X
ejpam-6171	283	3	?	?	PUNCT
ejpam-6171	284	1	y)),pi(x	y)),pi(x	NUM
ejpam-6171	284	2	)	)	PUNCT
ejpam-6171	284	3	}	}	PUNCT
ejpam-6171	284	4	(	(	PUNCT
ejpam-6171	284	5	by	by	ADP
ejpam-6171	284	6	(	(	PUNCT
ejpam-6171	284	7	3.9	3.9	NUM
ejpam-6171	284	8	)	)	PUNCT
ejpam-6171	284	9	)	)	PUNCT
ejpam-6171	285	1	=	=	SYM
ejpam-6171	285	2	max{pi(x	max{pi(x	PROPN
ejpam-6171	285	3	?	?	PUNCT
ejpam-6171	286	1	y),pi(x	y),pi(x	NUM
ejpam-6171	286	2	)	)	PUNCT
ejpam-6171	286	3	}	}	PUNCT
ejpam-6171	286	4	,	,	PUNCT
ejpam-6171	286	5	(	(	PUNCT
ejpam-6171	286	6	by	by	ADP
ejpam-6171	286	7	(	(	PUNCT
ejpam-6171	286	8	iup-1	iup-1	NOUN
ejpam-6171	286	9	)	)	PUNCT
ejpam-6171	286	10	)	)	PUNCT
ejpam-6171	286	11	pf	pf	NOUN
ejpam-6171	286	12	(	(	PUNCT
ejpam-6171	286	13	y	y	NOUN
ejpam-6171	286	14	)	)	PUNCT
ejpam-6171	286	15	=	=	SYM
ejpam-6171	286	16	pf	pf	X
ejpam-6171	286	17	(	(	PUNCT
ejpam-6171	286	18	0	0	NUM
ejpam-6171	286	19	?	?	PUNCT
ejpam-6171	286	20	y	y	X
ejpam-6171	286	21	)	)	PUNCT
ejpam-6171	286	22	(	(	PUNCT
ejpam-6171	286	23	by	by	ADP
ejpam-6171	286	24	(	(	PUNCT
ejpam-6171	286	25	iup-1	iup-1	NOUN
ejpam-6171	286	26	)	)	PUNCT
ejpam-6171	286	27	)	)	PUNCT
ejpam-6171	286	28	≥	≥	NOUN
ejpam-6171	286	29	min{pf	min{pf	PRON
ejpam-6171	286	30	(	(	PUNCT
ejpam-6171	286	31	0	0	NUM
ejpam-6171	286	32	?	?	PUNCT
ejpam-6171	287	1	(	(	PUNCT
ejpam-6171	287	2	x	x	X
ejpam-6171	287	3	?	?	PUNCT
ejpam-6171	288	1	y)),pf	y)),pf	PROPN
ejpam-6171	288	2	(	(	PUNCT
ejpam-6171	288	3	x	x	NOUN
ejpam-6171	288	4	)	)	PUNCT
ejpam-6171	288	5	}	}	PUNCT
ejpam-6171	288	6	(	(	PUNCT
ejpam-6171	288	7	by	by	ADP
ejpam-6171	288	8	(	(	PUNCT
ejpam-6171	288	9	3.10	3.10	NUM
ejpam-6171	288	10	)	)	PUNCT
ejpam-6171	288	11	)	)	PUNCT
ejpam-6171	289	1	=	=	PUNCT
ejpam-6171	289	2	min{pf	min{pf	X
ejpam-6171	289	3	(	(	PUNCT
ejpam-6171	289	4	x	x	X
ejpam-6171	289	5	?	?	PUNCT
ejpam-6171	290	1	y),pf	y),pf	PROPN
ejpam-6171	290	2	(	(	PUNCT
ejpam-6171	290	3	x	x	NOUN
ejpam-6171	290	4	)	)	PUNCT
ejpam-6171	290	5	}	}	PUNCT
ejpam-6171	290	6	.	.	PUNCT
ejpam-6171	291	1	(	(	PUNCT
ejpam-6171	291	2	by	by	ADP
ejpam-6171	291	3	(	(	PUNCT
ejpam-6171	291	4	iup-1	iup-1	NOUN
ejpam-6171	291	5	)	)	PUNCT
ejpam-6171	291	6	)	)	PUNCT
ejpam-6171	291	7	hence	hence	ADV
ejpam-6171	291	8	,	,	PUNCT
ejpam-6171	291	9	p	p	PROPN
ejpam-6171	291	10	is	be	AUX
ejpam-6171	291	11	a	a	DET
ejpam-6171	291	12	pythagorean	pythagorean	PROPN
ejpam-6171	291	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	291	14	iup	iup	NOUN
ejpam-6171	291	15	-	-	PUNCT
ejpam-6171	291	16	filter	filter	NOUN
ejpam-6171	291	17	of	of	ADP
ejpam-6171	291	18	x.	x.	PROPN
ejpam-6171	291	19	example	example	NOUN
ejpam-6171	292	1	7	7	X
ejpam-6171	292	2	.	.	PUNCT
ejpam-6171	293	1	let	let	VERB
ejpam-6171	293	2	x	x	PUNCT
ejpam-6171	293	3	=	=	PUNCT
ejpam-6171	293	4	{	{	PUNCT
ejpam-6171	293	5	0	0	NUM
ejpam-6171	293	6	,	,	PUNCT
ejpam-6171	293	7	1	1	NUM
ejpam-6171	293	8	,	,	PUNCT
ejpam-6171	293	9	2	2	NUM
ejpam-6171	293	10	,	,	PUNCT
ejpam-6171	293	11	3	3	NUM
ejpam-6171	293	12	,	,	PUNCT
ejpam-6171	293	13	4	4	NUM
ejpam-6171	293	14	,	,	PUNCT
ejpam-6171	293	15	5	5	NUM
ejpam-6171	293	16	}	}	PUNCT
ejpam-6171	293	17	with	with	ADP
ejpam-6171	293	18	the	the	DET
ejpam-6171	293	19	following	follow	VERB
ejpam-6171	293	20	cayley	cayley	ADJ
ejpam-6171	293	21	table	table	NOUN
ejpam-6171	293	22	:	:	PUNCT
ejpam-6171	293	23	?	?	PUNCT
ejpam-6171	294	1	0	0	NUM
ejpam-6171	294	2	1	1	NUM
ejpam-6171	294	3	2	2	NUM
ejpam-6171	294	4	3	3	NUM
ejpam-6171	294	5	4	4	NUM
ejpam-6171	294	6	5	5	NUM
ejpam-6171	294	7	0	0	NUM
ejpam-6171	294	8	0	0	NUM
ejpam-6171	294	9	1	1	NUM
ejpam-6171	294	10	2	2	NUM
ejpam-6171	294	11	3	3	NUM
ejpam-6171	294	12	4	4	NUM
ejpam-6171	294	13	5	5	NUM
ejpam-6171	294	14	1	1	NUM
ejpam-6171	294	15	5	5	NUM
ejpam-6171	294	16	0	0	NUM
ejpam-6171	294	17	3	3	NUM
ejpam-6171	294	18	4	4	NUM
ejpam-6171	294	19	2	2	NUM
ejpam-6171	294	20	1	1	NUM
ejpam-6171	294	21	2	2	NUM
ejpam-6171	294	22	2	2	NUM
ejpam-6171	294	23	3	3	NUM
ejpam-6171	294	24	0	0	NUM
ejpam-6171	294	25	1	1	NUM
ejpam-6171	294	26	5	5	NUM
ejpam-6171	294	27	4	4	NUM
ejpam-6171	294	28	3	3	NUM
ejpam-6171	294	29	3	3	NUM
ejpam-6171	294	30	4	4	NUM
ejpam-6171	294	31	5	5	NUM
ejpam-6171	294	32	0	0	NUM
ejpam-6171	294	33	1	1	NUM
ejpam-6171	294	34	2	2	NUM
ejpam-6171	294	35	4	4	NUM
ejpam-6171	294	36	4	4	NUM
ejpam-6171	294	37	2	2	NUM
ejpam-6171	294	38	1	1	NUM
ejpam-6171	294	39	5	5	NUM
ejpam-6171	294	40	0	0	NUM
ejpam-6171	294	41	3	3	NUM
ejpam-6171	294	42	5	5	NUM
ejpam-6171	294	43	1	1	NUM
ejpam-6171	294	44	5	5	NUM
ejpam-6171	294	45	4	4	NUM
ejpam-6171	294	46	2	2	NUM
ejpam-6171	294	47	3	3	NUM
ejpam-6171	294	48	0	0	NUM
ejpam-6171	294	49	then	then	ADV
ejpam-6171	294	50	x	x	PUNCT
ejpam-6171	294	51	is	be	AUX
ejpam-6171	294	52	an	an	DET
ejpam-6171	294	53	iup	iup	NOUN
ejpam-6171	294	54	-	-	PUNCT
ejpam-6171	294	55	algebra	algebra	NOUN
ejpam-6171	294	56	.	.	PUNCT
ejpam-6171	295	1	we	we	PRON
ejpam-6171	295	2	define	define	VERB
ejpam-6171	295	3	p	p	NOUN
ejpam-6171	295	4	on	on	ADP
ejpam-6171	295	5	x	x	PUNCT
ejpam-6171	295	6	as	as	SCONJ
ejpam-6171	295	7	follows	follow	VERB
ejpam-6171	295	8	:	:	PUNCT
ejpam-6171	295	9	pt	pt	X
ejpam-6171	295	10	=	=	SYM
ejpam-6171	295	11	(	(	PUNCT
ejpam-6171	295	12	0	0	NUM
ejpam-6171	295	13	0.7	0.7	NUM
ejpam-6171	295	14	1	1	NUM
ejpam-6171	295	15	0.1	0.1	NUM
ejpam-6171	295	16	2	2	NUM
ejpam-6171	295	17	0.1	0.1	NUM
ejpam-6171	295	18	3	3	NUM
ejpam-6171	295	19	0.3	0.3	NUM
ejpam-6171	295	20	4	4	NUM
ejpam-6171	295	21	0.1	0.1	NUM
ejpam-6171	295	22	5	5	NUM
ejpam-6171	295	23	0.1	0.1	NUM
ejpam-6171	295	24	)	)	PUNCT
ejpam-6171	295	25	pi	pi	NOUN
ejpam-6171	295	26	=	=	PUNCT
ejpam-6171	295	27	(	(	PUNCT
ejpam-6171	295	28	0	0	NUM
ejpam-6171	295	29	0.2	0.2	NUM
ejpam-6171	295	30	1	1	NUM
ejpam-6171	295	31	0.8	0.8	NUM
ejpam-6171	295	32	2	2	NUM
ejpam-6171	295	33	0.8	0.8	NUM
ejpam-6171	295	34	3	3	NUM
ejpam-6171	295	35	0.4	0.4	NUM
ejpam-6171	295	36	4	4	NUM
ejpam-6171	295	37	0.8	0.8	NUM
ejpam-6171	295	38	5	5	NUM
ejpam-6171	295	39	0.8	0.8	NUM
ejpam-6171	295	40	)	)	PUNCT
ejpam-6171	295	41	pf	pf	NOUN
ejpam-6171	295	42	=	=	PUNCT
ejpam-6171	295	43	(	(	PUNCT
ejpam-6171	295	44	0	0	NUM
ejpam-6171	295	45	1	1	NUM
ejpam-6171	295	46	1	1	NUM
ejpam-6171	295	47	0.3	0.3	NUM
ejpam-6171	295	48	2	2	NUM
ejpam-6171	295	49	0.3	0.3	NUM
ejpam-6171	295	50	3	3	NUM
ejpam-6171	295	51	0.7	0.7	NUM
ejpam-6171	295	52	4	4	NUM
ejpam-6171	295	53	0.3	0.3	NUM
ejpam-6171	295	54	5	5	NUM
ejpam-6171	295	55	0.3	0.3	NUM
ejpam-6171	295	56	)	)	PUNCT
ejpam-6171	295	57	then	then	ADV
ejpam-6171	295	58	p	p	PROPN
ejpam-6171	295	59	is	be	AUX
ejpam-6171	295	60	a	a	DET
ejpam-6171	295	61	pythagorean	pythagorean	PROPN
ejpam-6171	295	62	neutrosophic	neutrosophic	ADJ
ejpam-6171	295	63	iup	iup	NOUN
ejpam-6171	295	64	-	-	PUNCT
ejpam-6171	295	65	filter	filter	NOUN
ejpam-6171	295	66	of	of	ADP
ejpam-6171	295	67	x.	x.	NOUN
ejpam-6171	295	68	since	since	SCONJ
ejpam-6171	295	69	pt	pt	X
ejpam-6171	295	70	(	(	PUNCT
ejpam-6171	295	71	5	5	NUM
ejpam-6171	295	72	?	?	SYM
ejpam-6171	295	73	1	1	NUM
ejpam-6171	295	74	)	)	PUNCT
ejpam-6171	295	75	=	=	SYM
ejpam-6171	295	76	pt	pt	X
ejpam-6171	295	77	(	(	PUNCT
ejpam-6171	295	78	5	5	NUM
ejpam-6171	295	79	)	)	PUNCT
ejpam-6171	295	80	=	=	SYM
ejpam-6171	295	81	0.1	0.1	NUM
ejpam-6171	295	82	�	�	PROPN
ejpam-6171	295	83	0.3	0.3	NUM
ejpam-6171	295	84	=	=	SYM
ejpam-6171	295	85	min{0.3	min{0.3	PROPN
ejpam-6171	295	86	,	,	PUNCT
ejpam-6171	295	87	0.3	0.3	NUM
ejpam-6171	295	88	}	}	PUNCT
ejpam-6171	295	89	=	=	SYM
ejpam-6171	295	90	min{pt	min{pt	NOUN
ejpam-6171	295	91	(	(	PUNCT
ejpam-6171	295	92	3),pt	3),pt	NUM
ejpam-6171	295	93	(	(	PUNCT
ejpam-6171	295	94	3	3	NUM
ejpam-6171	295	95	)	)	PUNCT
ejpam-6171	295	96	}	}	PUNCT
ejpam-6171	295	97	=	=	SYM
ejpam-6171	295	98	min{pt	min{pt	NOUN
ejpam-6171	295	99	(	(	PUNCT
ejpam-6171	295	100	5	5	NUM
ejpam-6171	295	101	?	?	PUNCT
ejpam-6171	296	1	4),pt	4),pt	NUM
ejpam-6171	297	1	(	(	PUNCT
ejpam-6171	297	2	3	3	NUM
ejpam-6171	297	3	)	)	PUNCT
ejpam-6171	297	4	}	}	PUNCT
ejpam-6171	298	1	=	=	SYM
ejpam-6171	298	2	min{pt	min{pt	NOUN
ejpam-6171	298	3	(	(	PUNCT
ejpam-6171	298	4	5	5	NUM
ejpam-6171	298	5	?	?	PUNCT
ejpam-6171	298	6	(	(	PUNCT
ejpam-6171	298	7	3	3	NUM
ejpam-6171	298	8	?	?	SYM
ejpam-6171	298	9	1)),pt	1)),pt	NUM
ejpam-6171	298	10	(	(	PUNCT
ejpam-6171	298	11	3	3	NUM
ejpam-6171	298	12	)	)	PUNCT
ejpam-6171	298	13	}	}	PUNCT
ejpam-6171	298	14	,	,	PUNCT
ejpam-6171	298	15	pi(4	pi(4	NOUN
ejpam-6171	298	16	?	?	NOUN
ejpam-6171	298	17	2	2	X
ejpam-6171	298	18	)	)	PUNCT
ejpam-6171	298	19	=	=	SYM
ejpam-6171	298	20	pi(1	pi(1	NOUN
ejpam-6171	298	21	)	)	PUNCT
ejpam-6171	298	22	=	=	PUNCT
ejpam-6171	298	23	0.8	0.8	NUM
ejpam-6171	298	24	�	�	PROPN
ejpam-6171	298	25	0.4	0.4	NUM
ejpam-6171	298	26	=	=	SYM
ejpam-6171	298	27	max{0.4	max{0.4	PROPN
ejpam-6171	298	28	,	,	PUNCT
ejpam-6171	298	29	0.4	0.4	NUM
ejpam-6171	298	30	}	}	PUNCT
ejpam-6171	298	31	=	=	SYM
ejpam-6171	298	32	max{pi(3),pi(3	max{pi(3),pi(3	ADJ
ejpam-6171	298	33	)	)	PUNCT
ejpam-6171	298	34	}	}	PUNCT
ejpam-6171	299	1	=	=	SYM
ejpam-6171	299	2	max{pi(4	max{pi(4	NOUN
ejpam-6171	299	3	?	?	PUNCT
ejpam-6171	300	1	5),pi(3	5),pi(3	NUM
ejpam-6171	300	2	)	)	PUNCT
ejpam-6171	300	3	}	}	PUNCT
ejpam-6171	300	4	=	=	SYM
ejpam-6171	300	5	max{pi(4	max{pi(4	NOUN
ejpam-6171	300	6	?	?	PUNCT
ejpam-6171	301	1	(	(	PUNCT
ejpam-6171	301	2	3	3	X
ejpam-6171	301	3	?	?	SYM
ejpam-6171	301	4	2)),pi(3	2)),pi(3	NUM
ejpam-6171	301	5	)	)	PUNCT
ejpam-6171	301	6	}	}	PUNCT
ejpam-6171	301	7	,	,	PUNCT
ejpam-6171	301	8	and	and	CCONJ
ejpam-6171	301	9	pf	pf	INTJ
ejpam-6171	301	10	(	(	PUNCT
ejpam-6171	301	11	1	1	NUM
ejpam-6171	301	12	?	?	SYM
ejpam-6171	301	13	4	4	X
ejpam-6171	301	14	)	)	PUNCT
ejpam-6171	301	15	=	=	PRON
ejpam-6171	301	16	pf	pf	X
ejpam-6171	301	17	(	(	PUNCT
ejpam-6171	301	18	2	2	NUM
ejpam-6171	301	19	)	)	PUNCT
ejpam-6171	301	20	=	=	SYM
ejpam-6171	301	21	0.3	0.3	NUM
ejpam-6171	301	22	�	�	NOUN
ejpam-6171	301	23	0.7	0.7	NUM
ejpam-6171	301	24	=	=	SYM
ejpam-6171	301	25	min{1	min{1	PROPN
ejpam-6171	301	26	,	,	PUNCT
ejpam-6171	301	27	0.7	0.7	NUM
ejpam-6171	301	28	}	}	PUNCT
ejpam-6171	301	29	=	=	PUNCT
ejpam-6171	301	30	min{pf	min{pf	X
ejpam-6171	301	31	(	(	PUNCT
ejpam-6171	301	32	0),pf	0),pf	NUM
ejpam-6171	301	33	(	(	PUNCT
ejpam-6171	301	34	3	3	NUM
ejpam-6171	301	35	)	)	PUNCT
ejpam-6171	301	36	}	}	PUNCT
ejpam-6171	301	37	=	=	PUNCT
ejpam-6171	301	38	min{pf	min{pf	X
ejpam-6171	301	39	(	(	PUNCT
ejpam-6171	301	40	1	1	NUM
ejpam-6171	301	41	?	?	SYM
ejpam-6171	301	42	1),pf	1),pf	NUM
ejpam-6171	301	43	(	(	PUNCT
ejpam-6171	301	44	3	3	NUM
ejpam-6171	301	45	)	)	PUNCT
ejpam-6171	301	46	}	}	PUNCT
ejpam-6171	301	47	=	=	PUNCT
ejpam-6171	301	48	min{pf	min{pf	X
ejpam-6171	301	49	(	(	PUNCT
ejpam-6171	301	50	1	1	NUM
ejpam-6171	301	51	?	?	PUNCT
ejpam-6171	302	1	(	(	PUNCT
ejpam-6171	302	2	3	3	X
ejpam-6171	302	3	?	?	PUNCT
ejpam-6171	303	1	4)),pf	4)),pf	NUM
ejpam-6171	303	2	(	(	PUNCT
ejpam-6171	303	3	3	3	NUM
ejpam-6171	303	4	)	)	PUNCT
ejpam-6171	303	5	}	}	PUNCT
ejpam-6171	303	6	.	.	PUNCT
ejpam-6171	304	1	hence	hence	ADV
ejpam-6171	304	2	,	,	PUNCT
ejpam-6171	304	3	p	p	PRON
ejpam-6171	304	4	is	be	AUX
ejpam-6171	304	5	not	not	PART
ejpam-6171	304	6	a	a	DET
ejpam-6171	304	7	pythagorean	pythagorean	PROPN
ejpam-6171	304	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	304	9	iup	iup	PROPN
ejpam-6171	304	10	-	-	PUNCT
ejpam-6171	304	11	ideal	ideal	NOUN
ejpam-6171	304	12	of	of	ADP
ejpam-6171	304	13	x.	x.	PROPN
ejpam-6171	304	14	theorem	theorem	VERB
ejpam-6171	304	15	6	6	NUM
ejpam-6171	304	16	.	.	PUNCT
ejpam-6171	305	1	every	every	DET
ejpam-6171	305	2	pythagorean	pythagorean	PROPN
ejpam-6171	305	3	neutrosophic	neutrosophic	PROPN
ejpam-6171	305	4	iup	iup	NOUN
ejpam-6171	305	5	-	-	PUNCT
ejpam-6171	305	6	subalgebra	subalgebra	NOUN
ejpam-6171	305	7	of	of	ADP
ejpam-6171	305	8	x	x	SYM
ejpam-6171	305	9	is	be	AUX
ejpam-6171	305	10	a	a	DET
ejpam-6171	305	11	pythagorean	pythagorean	PROPN
ejpam-6171	305	12	neutrosophic	neutrosophic	ADJ
ejpam-6171	305	13	iup	iup	NOUN
ejpam-6171	305	14	-	-	PUNCT
ejpam-6171	305	15	filter	filter	NOUN
ejpam-6171	305	16	of	of	ADP
ejpam-6171	305	17	x.	x.	PROPN
ejpam-6171	305	18	k.	k.	PROPN
ejpam-6171	306	1	suayngam	suayngam	PROPN
ejpam-6171	306	2	et	et	PROPN
ejpam-6171	306	3	al	al	PROPN
ejpam-6171	306	4	.	.	PUNCT
ejpam-6171	306	5	/	/	SYM
ejpam-6171	306	6	eur	eur	PROPN
ejpam-6171	306	7	.	.	PUNCT
ejpam-6171	307	1	j.	j.	PROPN
ejpam-6171	307	2	pure	pure	PROPN
ejpam-6171	307	3	appl	appl	PROPN
ejpam-6171	307	4	.	.	PROPN
ejpam-6171	307	5	math	math	PROPN
ejpam-6171	307	6	,	,	PUNCT
ejpam-6171	307	7	18	18	NUM
ejpam-6171	307	8	(	(	PUNCT
ejpam-6171	307	9	3	3	NUM
ejpam-6171	307	10	)	)	PUNCT
ejpam-6171	307	11	(	(	PUNCT
ejpam-6171	307	12	2025	2025	NUM
ejpam-6171	307	13	)	)	PUNCT
ejpam-6171	307	14	,	,	PUNCT
ejpam-6171	307	15	6171	6171	NUM
ejpam-6171	307	16	12	12	NUM
ejpam-6171	307	17	of	of	ADP
ejpam-6171	307	18	28	28	NUM
ejpam-6171	307	19	proof	proof	NOUN
ejpam-6171	307	20	.	.	PUNCT
ejpam-6171	308	1	assume	assume	VERB
ejpam-6171	308	2	that	that	SCONJ
ejpam-6171	308	3	p	p	NOUN
ejpam-6171	308	4	is	be	AUX
ejpam-6171	308	5	a	a	DET
ejpam-6171	308	6	pythagorean	pythagorean	PROPN
ejpam-6171	308	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	308	8	iup	iup	NOUN
ejpam-6171	308	9	-	-	PUNCT
ejpam-6171	308	10	subalgebra	subalgebra	NOUN
ejpam-6171	308	11	of	of	ADP
ejpam-6171	308	12	x.	x.	NOUN
ejpam-6171	308	13	by	by	ADP
ejpam-6171	308	14	lemma	lemma	PROPN
ejpam-6171	308	15	1	1	NUM
ejpam-6171	308	16	,	,	PUNCT
ejpam-6171	308	17	it	it	PRON
ejpam-6171	308	18	satisfies	satisfy	VERB
ejpam-6171	308	19	(	(	PUNCT
ejpam-6171	308	20	3.5	3.5	NUM
ejpam-6171	308	21	)	)	PUNCT
ejpam-6171	308	22	,	,	PUNCT
ejpam-6171	308	23	(	(	PUNCT
ejpam-6171	308	24	3.6	3.6	NUM
ejpam-6171	308	25	)	)	PUNCT
ejpam-6171	308	26	,	,	PUNCT
ejpam-6171	308	27	and	and	CCONJ
ejpam-6171	308	28	(	(	PUNCT
ejpam-6171	308	29	3.7	3.7	NUM
ejpam-6171	308	30	)	)	PUNCT
ejpam-6171	308	31	.	.	PUNCT
ejpam-6171	309	1	let	let	VERB
ejpam-6171	309	2	x	x	PRON
ejpam-6171	309	3	,	,	PUNCT
ejpam-6171	309	4	y	y	PROPN
ejpam-6171	309	5	∈	∈	PROPN
ejpam-6171	309	6	x.	x.	NOUN
ejpam-6171	310	1	then	then	ADV
ejpam-6171	310	2	pt	pt	PROPN
ejpam-6171	310	3	(	(	PUNCT
ejpam-6171	310	4	y	y	NOUN
ejpam-6171	310	5	)	)	PUNCT
ejpam-6171	310	6	=	=	SYM
ejpam-6171	310	7	pt	pt	X
ejpam-6171	310	8	(	(	PUNCT
ejpam-6171	310	9	0	0	NUM
ejpam-6171	310	10	?	?	PUNCT
ejpam-6171	311	1	y	y	X
ejpam-6171	311	2	)	)	PUNCT
ejpam-6171	311	3	(	(	PUNCT
ejpam-6171	311	4	by	by	ADP
ejpam-6171	311	5	(	(	PUNCT
ejpam-6171	311	6	iup-1	iup-1	NOUN
ejpam-6171	311	7	)	)	PUNCT
ejpam-6171	311	8	)	)	PUNCT
ejpam-6171	312	1	=	=	SYM
ejpam-6171	312	2	pt	pt	X
ejpam-6171	312	3	(	(	PUNCT
ejpam-6171	312	4	(	(	PUNCT
ejpam-6171	312	5	x	x	SYM
ejpam-6171	312	6	?	?	PUNCT
ejpam-6171	312	7	0	0	NUM
ejpam-6171	312	8	)	)	PUNCT
ejpam-6171	312	9	?	?	PUNCT
ejpam-6171	313	1	(	(	PUNCT
ejpam-6171	313	2	x	x	X
ejpam-6171	313	3	?	?	PUNCT
ejpam-6171	313	4	y	y	NOUN
ejpam-6171	313	5	)	)	PUNCT
ejpam-6171	313	6	)	)	PUNCT
ejpam-6171	313	7	(	(	PUNCT
ejpam-6171	313	8	by	by	ADP
ejpam-6171	313	9	(	(	PUNCT
ejpam-6171	313	10	iup-3	iup-3	NUM
ejpam-6171	313	11	)	)	PUNCT
ejpam-6171	313	12	)	)	PUNCT
ejpam-6171	313	13	≥	≥	NOUN
ejpam-6171	313	14	min{pt	min{pt	X
ejpam-6171	313	15	(	(	PUNCT
ejpam-6171	313	16	x	x	X
ejpam-6171	313	17	?	?	PUNCT
ejpam-6171	313	18	0),pt	0),pt	NUM
ejpam-6171	314	1	(	(	PUNCT
ejpam-6171	314	2	x	x	X
ejpam-6171	314	3	?	?	PUNCT
ejpam-6171	314	4	y	y	X
ejpam-6171	314	5	)	)	PUNCT
ejpam-6171	314	6	}	}	PUNCT
ejpam-6171	314	7	(	(	PUNCT
ejpam-6171	314	8	by	by	ADP
ejpam-6171	314	9	(	(	PUNCT
ejpam-6171	314	10	3.2	3.2	NUM
ejpam-6171	314	11	)	)	PUNCT
ejpam-6171	314	12	)	)	PUNCT
ejpam-6171	314	13	≥	≥	NOUN
ejpam-6171	314	14	min{min{pt	min{min{pt	NOUN
ejpam-6171	314	15	(	(	PUNCT
ejpam-6171	314	16	x),pt	x),pt	X
ejpam-6171	314	17	(	(	PUNCT
ejpam-6171	314	18	0)},pt	0)},pt	NUM
ejpam-6171	314	19	(	(	PUNCT
ejpam-6171	314	20	x	x	PROPN
ejpam-6171	314	21	?	?	PUNCT
ejpam-6171	314	22	y	y	X
ejpam-6171	314	23	)	)	PUNCT
ejpam-6171	314	24	}	}	PUNCT
ejpam-6171	314	25	(	(	PUNCT
ejpam-6171	314	26	by	by	ADP
ejpam-6171	314	27	(	(	PUNCT
ejpam-6171	314	28	3.2	3.2	NUM
ejpam-6171	314	29	)	)	PUNCT
ejpam-6171	314	30	)	)	PUNCT
ejpam-6171	315	1	=	=	PRON
ejpam-6171	315	2	min{pt	min{pt	NOUN
ejpam-6171	315	3	(	(	PUNCT
ejpam-6171	315	4	x),pt	x),pt	PROPN
ejpam-6171	315	5	(	(	PUNCT
ejpam-6171	315	6	x	x	X
ejpam-6171	315	7	?	?	PUNCT
ejpam-6171	315	8	y	y	X
ejpam-6171	315	9	)	)	PUNCT
ejpam-6171	315	10	}	}	PUNCT
ejpam-6171	315	11	,	,	PUNCT
ejpam-6171	315	12	(	(	PUNCT
ejpam-6171	315	13	by	by	ADP
ejpam-6171	315	14	(	(	PUNCT
ejpam-6171	315	15	3.5	3.5	NUM
ejpam-6171	315	16	)	)	PUNCT
ejpam-6171	315	17	)	)	PUNCT
ejpam-6171	315	18	pi(y	pi(y	NOUN
ejpam-6171	315	19	)	)	PUNCT
ejpam-6171	315	20	=	=	SYM
ejpam-6171	316	1	pi(0	pi(0	PROPN
ejpam-6171	316	2	?	?	PUNCT
ejpam-6171	317	1	y	y	X
ejpam-6171	317	2	)	)	PUNCT
ejpam-6171	317	3	(	(	PUNCT
ejpam-6171	317	4	by	by	ADP
ejpam-6171	317	5	(	(	PUNCT
ejpam-6171	317	6	iup-1	iup-1	NOUN
ejpam-6171	317	7	)	)	PUNCT
ejpam-6171	317	8	)	)	PUNCT
ejpam-6171	318	1	=	=	NOUN
ejpam-6171	318	2	pi((x	pi((x	NOUN
ejpam-6171	318	3	?	?	PUNCT
ejpam-6171	318	4	0	0	NUM
ejpam-6171	318	5	)	)	PUNCT
ejpam-6171	318	6	?	?	PUNCT
ejpam-6171	319	1	(	(	PUNCT
ejpam-6171	319	2	x	x	X
ejpam-6171	319	3	?	?	PUNCT
ejpam-6171	319	4	y	y	NOUN
ejpam-6171	319	5	)	)	PUNCT
ejpam-6171	319	6	)	)	PUNCT
ejpam-6171	319	7	(	(	PUNCT
ejpam-6171	319	8	by	by	ADP
ejpam-6171	319	9	(	(	PUNCT
ejpam-6171	319	10	iup-3	iup-3	NUM
ejpam-6171	319	11	)	)	PUNCT
ejpam-6171	319	12	)	)	PUNCT
ejpam-6171	319	13	≤	≤	NUM
ejpam-6171	319	14	max{pi(x	max{pi(x	NOUN
ejpam-6171	319	15	?	?	PUNCT
ejpam-6171	319	16	0),pi(x	0),pi(x	NUM
ejpam-6171	319	17	?	?	PUNCT
ejpam-6171	320	1	y	y	X
ejpam-6171	320	2	)	)	PUNCT
ejpam-6171	320	3	}	}	PUNCT
ejpam-6171	320	4	(	(	PUNCT
ejpam-6171	320	5	by	by	ADP
ejpam-6171	320	6	(	(	PUNCT
ejpam-6171	320	7	3.3	3.3	NUM
ejpam-6171	320	8	)	)	PUNCT
ejpam-6171	320	9	)	)	PUNCT
ejpam-6171	320	10	≤	≤	NUM
ejpam-6171	320	11	max{max{pi(x),pi(0)},pi(x	max{max{pi(x),pi(0)},pi(x	NOUN
ejpam-6171	320	12	?	?	PUNCT
ejpam-6171	321	1	y	y	X
ejpam-6171	321	2	)	)	PUNCT
ejpam-6171	321	3	}	}	PUNCT
ejpam-6171	321	4	(	(	PUNCT
ejpam-6171	321	5	by	by	ADP
ejpam-6171	321	6	(	(	PUNCT
ejpam-6171	321	7	3.3	3.3	NUM
ejpam-6171	321	8	)	)	PUNCT
ejpam-6171	321	9	)	)	PUNCT
ejpam-6171	322	1	=	=	PUNCT
ejpam-6171	323	1	max{pi(x),pi(x	max{pi(x),pi(x	NOUN
ejpam-6171	323	2	?	?	PUNCT
ejpam-6171	324	1	y	y	X
ejpam-6171	324	2	)	)	PUNCT
ejpam-6171	324	3	}	}	PUNCT
ejpam-6171	324	4	,	,	PUNCT
ejpam-6171	324	5	(	(	PUNCT
ejpam-6171	324	6	by	by	ADP
ejpam-6171	324	7	(	(	PUNCT
ejpam-6171	324	8	3.6	3.6	NUM
ejpam-6171	324	9	)	)	PUNCT
ejpam-6171	324	10	)	)	PUNCT
ejpam-6171	325	1	pf	pf	PROPN
ejpam-6171	325	2	(	(	PUNCT
ejpam-6171	325	3	y	y	NOUN
ejpam-6171	325	4	)	)	PUNCT
ejpam-6171	325	5	=	=	SYM
ejpam-6171	325	6	pf	pf	X
ejpam-6171	325	7	(	(	PUNCT
ejpam-6171	325	8	0	0	NUM
ejpam-6171	325	9	?	?	PUNCT
ejpam-6171	326	1	y	y	X
ejpam-6171	326	2	)	)	PUNCT
ejpam-6171	326	3	(	(	PUNCT
ejpam-6171	326	4	by	by	ADP
ejpam-6171	326	5	(	(	PUNCT
ejpam-6171	326	6	iup-1	iup-1	NOUN
ejpam-6171	326	7	)	)	PUNCT
ejpam-6171	326	8	)	)	PUNCT
ejpam-6171	327	1	=	=	SYM
ejpam-6171	327	2	pf	pf	X
ejpam-6171	327	3	(	(	PUNCT
ejpam-6171	327	4	(	(	PUNCT
ejpam-6171	327	5	x	x	SYM
ejpam-6171	327	6	?	?	PUNCT
ejpam-6171	327	7	0	0	NUM
ejpam-6171	327	8	)	)	PUNCT
ejpam-6171	327	9	?	?	PUNCT
ejpam-6171	328	1	(	(	PUNCT
ejpam-6171	328	2	x	x	X
ejpam-6171	328	3	?	?	PUNCT
ejpam-6171	328	4	y	y	NOUN
ejpam-6171	328	5	)	)	PUNCT
ejpam-6171	328	6	)	)	PUNCT
ejpam-6171	328	7	(	(	PUNCT
ejpam-6171	328	8	by	by	ADP
ejpam-6171	328	9	(	(	PUNCT
ejpam-6171	328	10	iup-3	iup-3	NUM
ejpam-6171	328	11	)	)	PUNCT
ejpam-6171	328	12	)	)	PUNCT
ejpam-6171	328	13	≥	≥	NOUN
ejpam-6171	328	14	min{pf	min{pf	X
ejpam-6171	328	15	(	(	PUNCT
ejpam-6171	328	16	x	x	PROPN
ejpam-6171	328	17	?	?	PUNCT
ejpam-6171	328	18	0),pf	0),pf	NUM
ejpam-6171	329	1	(	(	PUNCT
ejpam-6171	329	2	x	x	X
ejpam-6171	329	3	?	?	PUNCT
ejpam-6171	329	4	y	y	X
ejpam-6171	329	5	)	)	PUNCT
ejpam-6171	329	6	}	}	PUNCT
ejpam-6171	329	7	(	(	PUNCT
ejpam-6171	329	8	by	by	ADP
ejpam-6171	329	9	(	(	PUNCT
ejpam-6171	329	10	3.4	3.4	NUM
ejpam-6171	329	11	)	)	PUNCT
ejpam-6171	329	12	)	)	PUNCT
ejpam-6171	329	13	≥	≥	X
ejpam-6171	329	14	min{min{pf	min{min{pf	X
ejpam-6171	329	15	(	(	PUNCT
ejpam-6171	329	16	x),pf	x),pf	X
ejpam-6171	329	17	(	(	PUNCT
ejpam-6171	329	18	0)},pf	0)},pf	NUM
ejpam-6171	329	19	(	(	PUNCT
ejpam-6171	329	20	x	x	X
ejpam-6171	329	21	?	?	PUNCT
ejpam-6171	329	22	y	y	X
ejpam-6171	329	23	)	)	PUNCT
ejpam-6171	329	24	}	}	PUNCT
ejpam-6171	329	25	(	(	PUNCT
ejpam-6171	329	26	by	by	ADP
ejpam-6171	329	27	(	(	PUNCT
ejpam-6171	329	28	3.4	3.4	NUM
ejpam-6171	329	29	)	)	PUNCT
ejpam-6171	329	30	)	)	PUNCT
ejpam-6171	330	1	=	=	PUNCT
ejpam-6171	330	2	min{pf	min{pf	X
ejpam-6171	330	3	(	(	PUNCT
ejpam-6171	330	4	x),pf	x),pf	PROPN
ejpam-6171	330	5	(	(	PUNCT
ejpam-6171	330	6	x	x	X
ejpam-6171	330	7	?	?	PUNCT
ejpam-6171	330	8	y	y	X
ejpam-6171	330	9	)	)	PUNCT
ejpam-6171	330	10	}	}	PUNCT
ejpam-6171	330	11	.	.	PUNCT
ejpam-6171	331	1	(	(	PUNCT
ejpam-6171	331	2	by	by	ADP
ejpam-6171	331	3	(	(	PUNCT
ejpam-6171	331	4	3.7	3.7	NUM
ejpam-6171	331	5	)	)	PUNCT
ejpam-6171	331	6	)	)	PUNCT
ejpam-6171	331	7	hence	hence	ADV
ejpam-6171	331	8	,	,	PUNCT
ejpam-6171	331	9	p	p	PROPN
ejpam-6171	331	10	is	be	AUX
ejpam-6171	331	11	a	a	DET
ejpam-6171	331	12	pythagorean	pythagorean	PROPN
ejpam-6171	331	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	331	14	iup	iup	NOUN
ejpam-6171	331	15	-	-	PUNCT
ejpam-6171	331	16	filter	filter	NOUN
ejpam-6171	331	17	of	of	ADP
ejpam-6171	331	18	x.	x.	NOUN
ejpam-6171	331	19	example	example	NOUN
ejpam-6171	331	20	8	8	NUM
ejpam-6171	331	21	.	.	PUNCT
ejpam-6171	332	1	[	[	X
ejpam-6171	332	2	9	9	NUM
ejpam-6171	332	3	]	]	PUNCT
ejpam-6171	332	4	let	let	VERB
ejpam-6171	332	5	r∗	r∗	PROPN
ejpam-6171	332	6	be	be	AUX
ejpam-6171	332	7	the	the	DET
ejpam-6171	332	8	set	set	NOUN
ejpam-6171	332	9	of	of	ADP
ejpam-6171	332	10	all	all	DET
ejpam-6171	332	11	nonzero	nonzero	ADJ
ejpam-6171	332	12	real	real	ADJ
ejpam-6171	332	13	numbers	number	NOUN
ejpam-6171	332	14	.	.	PUNCT
ejpam-6171	333	1	define	define	VERB
ejpam-6171	333	2	a	a	DET
ejpam-6171	333	3	binary	binary	ADJ
ejpam-6171	333	4	operation	operation	NOUN
ejpam-6171	333	5	?	?	PUNCT
ejpam-6171	334	1	on	on	ADP
ejpam-6171	334	2	r∗	r∗	PROPN
ejpam-6171	334	3	by	by	ADP
ejpam-6171	334	4	:	:	PUNCT
ejpam-6171	334	5	(	(	PUNCT
ejpam-6171	334	6	∀x	∀x	X
ejpam-6171	334	7	,	,	PUNCT
ejpam-6171	334	8	y	y	PROPN
ejpam-6171	334	9	∈	∈	PROPN
ejpam-6171	334	10	r∗)(x	r∗)(x	NOUN
ejpam-6171	334	11	?	?	PUNCT
ejpam-6171	335	1	y	y	NOUN
ejpam-6171	335	2	=	=	SYM
ejpam-6171	335	3	y	y	PROPN
ejpam-6171	335	4	x	x	PROPN
ejpam-6171	335	5	)	)	PUNCT
ejpam-6171	335	6	.	.	PUNCT
ejpam-6171	336	1	thus	thus	ADV
ejpam-6171	336	2	,	,	PUNCT
ejpam-6171	336	3	(	(	PUNCT
ejpam-6171	336	4	r∗	r∗	PROPN
ejpam-6171	336	5	,	,	PUNCT
ejpam-6171	336	6	?	?	PUNCT
ejpam-6171	336	7	,	,	PUNCT
ejpam-6171	336	8	1	1	X
ejpam-6171	336	9	)	)	PUNCT
ejpam-6171	336	10	is	be	AUX
ejpam-6171	336	11	an	an	DET
ejpam-6171	336	12	iup	iup	NOUN
ejpam-6171	336	13	-	-	PUNCT
ejpam-6171	336	14	algebra	algebra	NOUN
ejpam-6171	336	15	.	.	PUNCT
ejpam-6171	336	16	example	example	NOUN
ejpam-6171	337	1	9	9	NUM
ejpam-6171	337	2	.	.	PUNCT
ejpam-6171	338	1	from	from	ADP
ejpam-6171	338	2	example	example	NOUN
ejpam-6171	338	3	8	8	NUM
ejpam-6171	338	4	,	,	PUNCT
ejpam-6171	338	5	let	let	VERB
ejpam-6171	338	6	g	g	NOUN
ejpam-6171	338	7	=	=	PRON
ejpam-6171	338	8	{	{	PUNCT
ejpam-6171	338	9	x	x	PUNCT
ejpam-6171	338	10	∈	∈	PROPN
ejpam-6171	338	11	r∗	r∗	NOUN
ejpam-6171	338	12	|	|	ADV
ejpam-6171	338	13	x	x	NOUN
ejpam-6171	338	14	≥	≥	NOUN
ejpam-6171	338	15	1	1	NUM
ejpam-6171	338	16	}	}	PUNCT
ejpam-6171	338	17	.	.	PUNCT
ejpam-6171	339	1	then	then	ADV
ejpam-6171	339	2	1	1	NUM
ejpam-6171	339	3	∈	∈	NOUN
ejpam-6171	339	4	g.	g.	NOUN
ejpam-6171	339	5	next	next	ADV
ejpam-6171	339	6	,	,	PUNCT
ejpam-6171	339	7	let	let	VERB
ejpam-6171	339	8	x	x	PRON
ejpam-6171	339	9	,	,	PUNCT
ejpam-6171	339	10	y	y	PROPN
ejpam-6171	339	11	,	,	PUNCT
ejpam-6171	339	12	z	z	PROPN
ejpam-6171	339	13	∈	∈	PROPN
ejpam-6171	339	14	r∗	r∗	NOUN
ejpam-6171	339	15	be	be	VERB
ejpam-6171	339	16	such	such	ADJ
ejpam-6171	339	17	that	that	PRON
ejpam-6171	339	18	x	x	PUNCT
ejpam-6171	339	19	?	?	PUNCT
ejpam-6171	340	1	(	(	PUNCT
ejpam-6171	340	2	y	y	NOUN
ejpam-6171	340	3	?	?	PUNCT
ejpam-6171	341	1	z	z	X
ejpam-6171	341	2	)	)	PUNCT
ejpam-6171	341	3	≥	≥	NOUN
ejpam-6171	341	4	1	1	NUM
ejpam-6171	341	5	and	and	CCONJ
ejpam-6171	341	6	y	y	PROPN
ejpam-6171	341	7	≥	≥	NUM
ejpam-6171	341	8	1	1	NUM
ejpam-6171	341	9	.	.	PUNCT
ejpam-6171	342	1	then	then	ADV
ejpam-6171	342	2	z	z	PROPN
ejpam-6171	342	3	yx	yx	PROPN
ejpam-6171	342	4	≥	≥	NUM
ejpam-6171	342	5	1	1	NUM
ejpam-6171	342	6	.	.	PUNCT
ejpam-6171	343	1	thus	thus	ADV
ejpam-6171	343	2	,	,	PUNCT
ejpam-6171	343	3	x	x	PUNCT
ejpam-6171	343	4	?	?	PUNCT
ejpam-6171	344	1	z	z	X
ejpam-6171	344	2	=	=	PUNCT
ejpam-6171	345	1	z	z	NOUN
ejpam-6171	345	2	x	x	SYM
ejpam-6171	345	3	=	=	PUNCT
ejpam-6171	345	4	(	(	PUNCT
ejpam-6171	345	5	z	z	PROPN
ejpam-6171	345	6	yx)y	yx)y	PROPN
ejpam-6171	345	7	≥	≥	NUM
ejpam-6171	345	8	1	1	NUM
ejpam-6171	345	9	,	,	PUNCT
ejpam-6171	345	10	that	that	ADV
ejpam-6171	345	11	is	is	ADV
ejpam-6171	345	12	,	,	PUNCT
ejpam-6171	345	13	x	x	PUNCT
ejpam-6171	345	14	?	?	PUNCT
ejpam-6171	345	15	z	z	PROPN
ejpam-6171	345	16	∈	∈	PROPN
ejpam-6171	345	17	g.	g.	NOUN
ejpam-6171	345	18	hence	hence	ADV
ejpam-6171	345	19	,	,	PUNCT
ejpam-6171	345	20	g	g	PROPN
ejpam-6171	345	21	is	be	AUX
ejpam-6171	345	22	an	an	DET
ejpam-6171	345	23	iup	iup	NOUN
ejpam-6171	345	24	-	-	PUNCT
ejpam-6171	345	25	ideal	ideal	NOUN
ejpam-6171	345	26	of	of	ADP
ejpam-6171	345	27	r∗.	r∗.	NOUN
ejpam-6171	345	28	then	then	ADV
ejpam-6171	345	29	g	g	PROPN
ejpam-6171	345	30	is	be	AUX
ejpam-6171	345	31	an	an	DET
ejpam-6171	345	32	iup	iup	NOUN
ejpam-6171	345	33	-	-	PUNCT
ejpam-6171	345	34	filter	filter	NOUN
ejpam-6171	345	35	of	of	ADP
ejpam-6171	345	36	r∗.	r∗.	NOUN
ejpam-6171	345	37	from	from	ADP
ejpam-6171	345	38	theorems	theorem	NOUN
ejpam-6171	345	39	10	10	NUM
ejpam-6171	345	40	and	and	CCONJ
ejpam-6171	345	41	11	11	NUM
ejpam-6171	345	42	,	,	PUNCT
ejpam-6171	345	43	pg[α	pg[α	PROPN
ejpam-6171	345	44	+	+	PROPN
ejpam-6171	345	45	,	,	PUNCT
ejpam-6171	345	46	β−,γ+	β−,γ+	X
ejpam-6171	345	47	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	345	48	]	]	PUNCT
ejpam-6171	345	49	is	be	AUX
ejpam-6171	345	50	a	a	DET
ejpam-6171	345	51	pythagorean	pythagorean	PROPN
ejpam-6171	345	52	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	53	iup	iup	PROPN
ejpam-6171	345	54	-	-	PUNCT
ejpam-6171	345	55	ideal	ideal	NOUN
ejpam-6171	345	56	and	and	CCONJ
ejpam-6171	345	57	a	a	DET
ejpam-6171	345	58	pythagorean	pythagorean	PROPN
ejpam-6171	345	59	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	60	iup	iup	NOUN
ejpam-6171	345	61	-	-	PUNCT
ejpam-6171	345	62	filter	filter	NOUN
ejpam-6171	345	63	of	of	ADP
ejpam-6171	345	64	r∗.	r∗.	NOUN
ejpam-6171	345	65	thus	thus	ADV
ejpam-6171	345	66	,	,	PUNCT
ejpam-6171	345	67	p	p	NOUN
ejpam-6171	345	68	is	be	AUX
ejpam-6171	345	69	a	a	DET
ejpam-6171	345	70	pythagorean	pythagorean	PROPN
ejpam-6171	345	71	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	72	iup	iup	PROPN
ejpam-6171	345	73	-	-	PUNCT
ejpam-6171	345	74	ideal	ideal	NOUN
ejpam-6171	345	75	and	and	CCONJ
ejpam-6171	345	76	a	a	DET
ejpam-6171	345	77	pythagorean	pythagorean	PROPN
ejpam-6171	345	78	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	79	iup	iup	NOUN
ejpam-6171	345	80	-	-	PUNCT
ejpam-6171	345	81	filter	filter	NOUN
ejpam-6171	345	82	of	of	ADP
ejpam-6171	345	83	r∗.	r∗.	NOUN
ejpam-6171	345	84	since	since	SCONJ
ejpam-6171	345	85	1	1	NUM
ejpam-6171	345	86	,	,	PUNCT
ejpam-6171	345	87	3	3	NUM
ejpam-6171	345	88	∈	∈	NOUN
ejpam-6171	345	89	g	g	NOUN
ejpam-6171	345	90	but	but	CCONJ
ejpam-6171	345	91	3	3	NUM
ejpam-6171	345	92	?	?	SYM
ejpam-6171	345	93	1	1	NUM
ejpam-6171	345	94	=	=	SYM
ejpam-6171	345	95	1	1	NUM
ejpam-6171	345	96	3	3	NUM
ejpam-6171	345	97	∈	∈	NOUN
ejpam-6171	345	98	g	g	NOUN
ejpam-6171	345	99	,	,	PUNCT
ejpam-6171	345	100	we	we	PRON
ejpam-6171	345	101	have	have	AUX
ejpam-6171	345	102	g	g	PROPN
ejpam-6171	345	103	is	be	AUX
ejpam-6171	345	104	not	not	PART
ejpam-6171	345	105	an	an	DET
ejpam-6171	345	106	iup	iup	NOUN
ejpam-6171	345	107	-	-	PUNCT
ejpam-6171	345	108	subalgebra	subalgebra	NOUN
ejpam-6171	345	109	of	of	ADP
ejpam-6171	345	110	r∗.	r∗.	NOUN
ejpam-6171	345	111	from	from	ADP
ejpam-6171	345	112	theorem	theorem	ADJ
ejpam-6171	345	113	9	9	NUM
ejpam-6171	345	114	,	,	PUNCT
ejpam-6171	345	115	pg[α	pg[α	PROPN
ejpam-6171	345	116	+	+	PROPN
ejpam-6171	345	117	,	,	PUNCT
ejpam-6171	345	118	β−,γ+	β−,γ+	X
ejpam-6171	345	119	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	345	120	]	]	PUNCT
ejpam-6171	345	121	is	be	AUX
ejpam-6171	345	122	not	not	PART
ejpam-6171	345	123	a	a	DET
ejpam-6171	345	124	pythagorean	pythagorean	PROPN
ejpam-6171	345	125	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	126	iup	iup	NOUN
ejpam-6171	345	127	-	-	PUNCT
ejpam-6171	345	128	subalgebra	subalgebra	NOUN
ejpam-6171	345	129	of	of	ADP
ejpam-6171	345	130	r∗.	r∗.	NOUN
ejpam-6171	345	131	hence	hence	ADV
ejpam-6171	345	132	,	,	PUNCT
ejpam-6171	345	133	p	p	NOUN
ejpam-6171	345	134	is	be	AUX
ejpam-6171	345	135	not	not	PART
ejpam-6171	345	136	a	a	DET
ejpam-6171	345	137	pythagorean	pythagorean	PROPN
ejpam-6171	345	138	neutrosophic	neutrosophic	ADJ
ejpam-6171	345	139	iup	iup	NOUN
ejpam-6171	345	140	-	-	PUNCT
ejpam-6171	345	141	subalgebra	subalgebra	NOUN
ejpam-6171	345	142	of	of	ADP
ejpam-6171	345	143	r∗.	r∗.	NOUN
ejpam-6171	345	144	k.	k.	X
ejpam-6171	345	145	suayngam	suayngam	INTJ
ejpam-6171	345	146	et	et	PROPN
ejpam-6171	345	147	al	al	PROPN
ejpam-6171	345	148	.	.	PUNCT
ejpam-6171	345	149	/	/	SYM
ejpam-6171	345	150	eur	eur	PROPN
ejpam-6171	345	151	.	.	PUNCT
ejpam-6171	346	1	j.	j.	PROPN
ejpam-6171	346	2	pure	pure	PROPN
ejpam-6171	346	3	appl	appl	PROPN
ejpam-6171	346	4	.	.	PROPN
ejpam-6171	346	5	math	math	PROPN
ejpam-6171	346	6	,	,	PUNCT
ejpam-6171	346	7	18	18	NUM
ejpam-6171	346	8	(	(	PUNCT
ejpam-6171	346	9	3	3	NUM
ejpam-6171	346	10	)	)	PUNCT
ejpam-6171	346	11	(	(	PUNCT
ejpam-6171	346	12	2025	2025	NUM
ejpam-6171	346	13	)	)	PUNCT
ejpam-6171	346	14	,	,	PUNCT
ejpam-6171	346	15	6171	6171	NUM
ejpam-6171	346	16	13	13	NUM
ejpam-6171	346	17	of	of	ADP
ejpam-6171	346	18	28	28	NUM
ejpam-6171	346	19	example	example	NOUN
ejpam-6171	346	20	10	10	NUM
ejpam-6171	346	21	.	.	PUNCT
ejpam-6171	347	1	let	let	VERB
ejpam-6171	347	2	x	x	PUNCT
ejpam-6171	347	3	=	=	PUNCT
ejpam-6171	347	4	{	{	PUNCT
ejpam-6171	347	5	0	0	NUM
ejpam-6171	347	6	,	,	PUNCT
ejpam-6171	347	7	1	1	NUM
ejpam-6171	347	8	,	,	PUNCT
ejpam-6171	347	9	2	2	NUM
ejpam-6171	347	10	,	,	PUNCT
ejpam-6171	347	11	3	3	NUM
ejpam-6171	347	12	,	,	PUNCT
ejpam-6171	347	13	4	4	NUM
ejpam-6171	347	14	,	,	PUNCT
ejpam-6171	347	15	5	5	NUM
ejpam-6171	347	16	}	}	PUNCT
ejpam-6171	347	17	with	with	ADP
ejpam-6171	347	18	the	the	DET
ejpam-6171	347	19	following	follow	VERB
ejpam-6171	347	20	cayley	cayley	ADJ
ejpam-6171	347	21	table	table	NOUN
ejpam-6171	347	22	:	:	PUNCT
ejpam-6171	347	23	·	·	PUNCT
ejpam-6171	347	24	0	0	NUM
ejpam-6171	347	25	1	1	NUM
ejpam-6171	347	26	2	2	NUM
ejpam-6171	347	27	3	3	NUM
ejpam-6171	347	28	4	4	NUM
ejpam-6171	347	29	5	5	NUM
ejpam-6171	347	30	0	0	NUM
ejpam-6171	347	31	0	0	NUM
ejpam-6171	347	32	1	1	NUM
ejpam-6171	347	33	2	2	NUM
ejpam-6171	347	34	3	3	NUM
ejpam-6171	347	35	4	4	NUM
ejpam-6171	347	36	5	5	NUM
ejpam-6171	347	37	1	1	NUM
ejpam-6171	347	38	5	5	NUM
ejpam-6171	347	39	0	0	NUM
ejpam-6171	347	40	4	4	NUM
ejpam-6171	347	41	2	2	NUM
ejpam-6171	347	42	3	3	NUM
ejpam-6171	347	43	1	1	NUM
ejpam-6171	347	44	2	2	NUM
ejpam-6171	347	45	2	2	NUM
ejpam-6171	347	46	4	4	NUM
ejpam-6171	347	47	0	0	NUM
ejpam-6171	347	48	5	5	NUM
ejpam-6171	347	49	1	1	NUM
ejpam-6171	347	50	3	3	NUM
ejpam-6171	347	51	3	3	NUM
ejpam-6171	347	52	3	3	NUM
ejpam-6171	347	53	2	2	NUM
ejpam-6171	347	54	1	1	NUM
ejpam-6171	347	55	0	0	NUM
ejpam-6171	347	56	5	5	NUM
ejpam-6171	347	57	4	4	NUM
ejpam-6171	347	58	4	4	NUM
ejpam-6171	347	59	4	4	NUM
ejpam-6171	347	60	3	3	NUM
ejpam-6171	347	61	5	5	NUM
ejpam-6171	347	62	1	1	NUM
ejpam-6171	347	63	0	0	NUM
ejpam-6171	347	64	2	2	NUM
ejpam-6171	347	65	5	5	NUM
ejpam-6171	347	66	1	1	NUM
ejpam-6171	347	67	5	5	NUM
ejpam-6171	347	68	3	3	NUM
ejpam-6171	347	69	4	4	NUM
ejpam-6171	347	70	2	2	NUM
ejpam-6171	347	71	0	0	NUM
ejpam-6171	347	72	then	then	ADV
ejpam-6171	347	73	x	x	PUNCT
ejpam-6171	347	74	is	be	AUX
ejpam-6171	347	75	an	an	DET
ejpam-6171	347	76	iup	iup	NOUN
ejpam-6171	347	77	-	-	PUNCT
ejpam-6171	347	78	algebra	algebra	NOUN
ejpam-6171	347	79	.	.	PUNCT
ejpam-6171	348	1	we	we	PRON
ejpam-6171	348	2	define	define	VERB
ejpam-6171	348	3	p	p	NOUN
ejpam-6171	348	4	on	on	ADP
ejpam-6171	348	5	x	x	PUNCT
ejpam-6171	348	6	as	as	SCONJ
ejpam-6171	348	7	follows	follow	VERB
ejpam-6171	348	8	:	:	PUNCT
ejpam-6171	349	1	pt	pt	X
ejpam-6171	349	2	=	=	SYM
ejpam-6171	349	3	(	(	PUNCT
ejpam-6171	349	4	0	0	NUM
ejpam-6171	349	5	0.8	0.8	NUM
ejpam-6171	349	6	1	1	NUM
ejpam-6171	349	7	0.2	0.2	NUM
ejpam-6171	349	8	2	2	NUM
ejpam-6171	349	9	0.2	0.2	NUM
ejpam-6171	349	10	3	3	NUM
ejpam-6171	349	11	0.2	0.2	NUM
ejpam-6171	349	12	4	4	NUM
ejpam-6171	349	13	0.6	0.6	NUM
ejpam-6171	349	14	5	5	NUM
ejpam-6171	349	15	0.2	0.2	NUM
ejpam-6171	349	16	)	)	PUNCT
ejpam-6171	349	17	pi	pi	NOUN
ejpam-6171	349	18	=	=	PUNCT
ejpam-6171	349	19	(	(	PUNCT
ejpam-6171	349	20	0	0	NUM
ejpam-6171	349	21	0	0	NUM
ejpam-6171	349	22	1	1	NUM
ejpam-6171	349	23	0.3	0.3	NUM
ejpam-6171	349	24	2	2	NUM
ejpam-6171	349	25	0.3	0.3	NUM
ejpam-6171	349	26	3	3	NUM
ejpam-6171	349	27	0.3	0.3	NUM
ejpam-6171	349	28	4	4	NUM
ejpam-6171	349	29	0.1	0.1	NUM
ejpam-6171	349	30	5	5	NUM
ejpam-6171	349	31	0.3	0.3	NUM
ejpam-6171	349	32	)	)	PUNCT
ejpam-6171	349	33	pf	pf	NOUN
ejpam-6171	350	1	=	=	PUNCT
ejpam-6171	351	1	(	(	PUNCT
ejpam-6171	351	2	0	0	NUM
ejpam-6171	351	3	0.6	0.6	NUM
ejpam-6171	351	4	1	1	NUM
ejpam-6171	351	5	0.3	0.3	NUM
ejpam-6171	351	6	2	2	NUM
ejpam-6171	351	7	0.3	0.3	NUM
ejpam-6171	351	8	3	3	NUM
ejpam-6171	351	9	0.3	0.3	NUM
ejpam-6171	351	10	4	4	NUM
ejpam-6171	351	11	0.5	0.5	NUM
ejpam-6171	351	12	5	5	NUM
ejpam-6171	351	13	0.3	0.3	NUM
ejpam-6171	351	14	)	)	PUNCT
ejpam-6171	352	1	then	then	ADV
ejpam-6171	352	2	p	p	PROPN
ejpam-6171	352	3	is	be	AUX
ejpam-6171	352	4	a	a	DET
ejpam-6171	352	5	pythagorean	pythagorean	PROPN
ejpam-6171	352	6	neutrosophic	neutrosophic	ADJ
ejpam-6171	352	7	iup	iup	NOUN
ejpam-6171	352	8	-	-	PUNCT
ejpam-6171	352	9	subalgebra	subalgebra	NOUN
ejpam-6171	352	10	of	of	ADP
ejpam-6171	352	11	x.	x.	NOUN
ejpam-6171	352	12	since	since	SCONJ
ejpam-6171	352	13	pt	pt	X
ejpam-6171	352	14	(	(	PUNCT
ejpam-6171	352	15	5	5	NUM
ejpam-6171	352	16	?	?	SYM
ejpam-6171	352	17	2	2	X
ejpam-6171	352	18	)	)	PUNCT
ejpam-6171	353	1	=	=	SYM
ejpam-6171	353	2	pt	pt	X
ejpam-6171	353	3	(	(	PUNCT
ejpam-6171	353	4	3	3	NUM
ejpam-6171	353	5	)	)	PUNCT
ejpam-6171	353	6	=	=	SYM
ejpam-6171	353	7	0.2	0.2	NUM
ejpam-6171	353	8	�	�	NOUN
ejpam-6171	353	9	0.6	0.6	NUM
ejpam-6171	353	10	=	=	SYM
ejpam-6171	353	11	min{0.8	min{0.8	PROPN
ejpam-6171	353	12	,	,	PUNCT
ejpam-6171	353	13	0.6	0.6	NUM
ejpam-6171	353	14	}	}	PUNCT
ejpam-6171	353	15	=	=	NUM
ejpam-6171	353	16	min{pt	min{pt	NOUN
ejpam-6171	353	17	(	(	PUNCT
ejpam-6171	353	18	0),pt	0),pt	NUM
ejpam-6171	353	19	(	(	PUNCT
ejpam-6171	353	20	4	4	NUM
ejpam-6171	353	21	)	)	PUNCT
ejpam-6171	353	22	}	}	PUNCT
ejpam-6171	354	1	=	=	SYM
ejpam-6171	354	2	min{pt	min{pt	NOUN
ejpam-6171	354	3	(	(	PUNCT
ejpam-6171	354	4	5	5	NUM
ejpam-6171	354	5	?	?	PUNCT
ejpam-6171	354	6	(	(	PUNCT
ejpam-6171	354	7	4	4	NUM
ejpam-6171	354	8	?	?	PUNCT
ejpam-6171	354	9	2)),pt	2)),pt	NUM
ejpam-6171	354	10	(	(	PUNCT
ejpam-6171	354	11	4	4	NUM
ejpam-6171	354	12	)	)	PUNCT
ejpam-6171	354	13	}	}	PUNCT
ejpam-6171	354	14	,	,	PUNCT
ejpam-6171	354	15	pi(3	pi(3	PROPN
ejpam-6171	354	16	?	?	NOUN
ejpam-6171	354	17	1	1	X
ejpam-6171	354	18	)	)	PUNCT
ejpam-6171	354	19	=	=	SYM
ejpam-6171	354	20	pi(2	pi(2	PROPN
ejpam-6171	354	21	)	)	PUNCT
ejpam-6171	354	22	=	=	SYM
ejpam-6171	354	23	0.3	0.3	NUM
ejpam-6171	354	24	�	�	NOUN
ejpam-6171	354	25	0	0	NUM
ejpam-6171	354	26	=	=	SYM
ejpam-6171	354	27	max{pi(0),pi(4	max{pi(0),pi(4	NOUN
ejpam-6171	354	28	)	)	PUNCT
ejpam-6171	354	29	}	}	PUNCT
ejpam-6171	354	30	=	=	PUNCT
ejpam-6171	354	31	max{pi(3	max{pi(3	PROPN
ejpam-6171	354	32	?	?	PUNCT
ejpam-6171	355	1	(	(	PUNCT
ejpam-6171	355	2	4	4	NUM
ejpam-6171	355	3	?	?	SYM
ejpam-6171	355	4	1)),pi(4	1)),pi(4	NUM
ejpam-6171	355	5	)	)	PUNCT
ejpam-6171	355	6	}	}	PUNCT
ejpam-6171	355	7	,	,	PUNCT
ejpam-6171	355	8	and	and	CCONJ
ejpam-6171	355	9	pf	pf	INTJ
ejpam-6171	355	10	(	(	PUNCT
ejpam-6171	355	11	5	5	NUM
ejpam-6171	355	12	?	?	SYM
ejpam-6171	355	13	1	1	NUM
ejpam-6171	355	14	)	)	PUNCT
ejpam-6171	355	15	=	=	PRON
ejpam-6171	355	16	pf	pf	X
ejpam-6171	355	17	(	(	PUNCT
ejpam-6171	355	18	5	5	NUM
ejpam-6171	355	19	)	)	PUNCT
ejpam-6171	355	20	=	=	SYM
ejpam-6171	355	21	0.3	0.3	NUM
ejpam-6171	355	22	�	�	PROPN
ejpam-6171	355	23	0.5	0.5	NUM
ejpam-6171	355	24	=	=	SYM
ejpam-6171	355	25	min{0.5	min{0.5	PROPN
ejpam-6171	355	26	,	,	PUNCT
ejpam-6171	355	27	0.5	0.5	NUM
ejpam-6171	355	28	}	}	PUNCT
ejpam-6171	355	29	=	=	PUNCT
ejpam-6171	355	30	min{pf	min{pf	NOUN
ejpam-6171	355	31	(	(	PUNCT
ejpam-6171	355	32	4),pf	4),pf	NUM
ejpam-6171	355	33	(	(	PUNCT
ejpam-6171	355	34	4	4	NUM
ejpam-6171	355	35	)	)	PUNCT
ejpam-6171	355	36	}	}	PUNCT
ejpam-6171	355	37	=	=	PUNCT
ejpam-6171	355	38	min{pf	min{pf	X
ejpam-6171	355	39	(	(	PUNCT
ejpam-6171	355	40	5	5	NUM
ejpam-6171	355	41	?	?	PUNCT
ejpam-6171	355	42	(	(	PUNCT
ejpam-6171	355	43	4	4	NUM
ejpam-6171	355	44	?	?	PUNCT
ejpam-6171	356	1	1)),pf	1)),pf	NUM
ejpam-6171	356	2	(	(	PUNCT
ejpam-6171	356	3	4	4	NUM
ejpam-6171	356	4	)	)	PUNCT
ejpam-6171	356	5	}	}	PUNCT
ejpam-6171	356	6	.	.	PUNCT
ejpam-6171	357	1	hence	hence	ADV
ejpam-6171	357	2	,	,	PUNCT
ejpam-6171	357	3	p	p	PRON
ejpam-6171	357	4	is	be	AUX
ejpam-6171	357	5	not	not	PART
ejpam-6171	357	6	a	a	DET
ejpam-6171	357	7	pythagorean	pythagorean	PROPN
ejpam-6171	357	8	neutrosophic	neutrosophic	ADJ
ejpam-6171	357	9	iup	iup	PROPN
ejpam-6171	357	10	-	-	PUNCT
ejpam-6171	357	11	ideal	ideal	NOUN
ejpam-6171	357	12	of	of	ADP
ejpam-6171	357	13	x.	x.	NOUN
ejpam-6171	357	14	the	the	DET
ejpam-6171	357	15	study	study	NOUN
ejpam-6171	357	16	identified	identify	VERB
ejpam-6171	357	17	a	a	DET
ejpam-6171	357	18	relationship	relationship	NOUN
ejpam-6171	357	19	among	among	ADP
ejpam-6171	357	20	the	the	DET
ejpam-6171	357	21	four	four	NUM
ejpam-6171	357	22	concepts	concept	NOUN
ejpam-6171	357	23	:	:	PUNCT
ejpam-6171	357	24	pythagorean	pythagorean	PROPN
ejpam-6171	357	25	neutrosophic	neutrosophic	PROPN
ejpam-6171	357	26	iup	iup	PROPN
ejpam-6171	357	27	-	-	PUNCT
ejpam-6171	357	28	ideals	ideal	NOUN
ejpam-6171	357	29	and	and	CCONJ
ejpam-6171	357	30	pythagorean	pythagorean	PROPN
ejpam-6171	357	31	neutrosophic	neutrosophic	PROPN
ejpam-6171	357	32	iup	iup	PROPN
ejpam-6171	357	33	-	-	PUNCT
ejpam-6171	357	34	subalgebras	subalgebra	NOUN
ejpam-6171	357	35	are	be	AUX
ejpam-6171	357	36	generalizations	generalization	NOUN
ejpam-6171	357	37	of	of	ADP
ejpam-6171	357	38	pythagorean	pythagorean	PROPN
ejpam-6171	357	39	neutrosophic	neutrosophic	PROPN
ejpam-6171	357	40	strong	strong	ADJ
ejpam-6171	357	41	iup	iup	NOUN
ejpam-6171	357	42	-	-	PUNCT
ejpam-6171	357	43	ideals	ideal	NOUN
ejpam-6171	357	44	within	within	ADP
ejpam-6171	357	45	iup	iup	NOUN
ejpam-6171	357	46	-	-	PUNCT
ejpam-6171	357	47	algebras	algebra	NOUN
ejpam-6171	357	48	,	,	PUNCT
ejpam-6171	357	49	where	where	SCONJ
ejpam-6171	357	50	pythagorean	pythagorean	PROPN
ejpam-6171	357	51	neutrosophic	neutrosophic	PROPN
ejpam-6171	357	52	strong	strong	ADJ
ejpam-6171	357	53	iup	iup	NOUN
ejpam-6171	357	54	-	-	PUNCT
ejpam-6171	357	55	ideals	ideal	NOUN
ejpam-6171	357	56	can	can	AUX
ejpam-6171	357	57	only	only	ADV
ejpam-6171	357	58	be	be	AUX
ejpam-6171	357	59	a	a	DET
ejpam-6171	357	60	constant	constant	ADJ
ejpam-6171	357	61	pns	pns	NOUN
ejpam-6171	357	62	.	.	PUNCT
ejpam-6171	358	1	pythagorean	pythagorean	PROPN
ejpam-6171	358	2	neutrosophic	neutrosophic	PROPN
ejpam-6171	358	3	iup	iup	PROPN
ejpam-6171	358	4	-	-	PUNCT
ejpam-6171	358	5	filters	filter	NOUN
ejpam-6171	358	6	extend	extend	VERB
ejpam-6171	358	7	the	the	DET
ejpam-6171	358	8	generalization	generalization	NOUN
ejpam-6171	358	9	to	to	PART
ejpam-6171	358	10	include	include	VERB
ejpam-6171	358	11	pythagorean	pythagorean	PROPN
ejpam-6171	358	12	neutrosophic	neutrosophic	PROPN
ejpam-6171	358	13	iup	iup	PROPN
ejpam-6171	358	14	-	-	PUNCT
ejpam-6171	358	15	ideals	ideal	NOUN
ejpam-6171	358	16	and	and	CCONJ
ejpam-6171	358	17	pythagorean	pythagorean	PROPN
ejpam-6171	358	18	neutrosophic	neutrosophic	PROPN
ejpam-6171	358	19	iup	iup	PROPN
ejpam-6171	358	20	-	-	PUNCT
ejpam-6171	358	21	subalgebras	subalgebras	PROPN
ejpam-6171	358	22	.	.	PUNCT
ejpam-6171	359	1	the	the	DET
ejpam-6171	359	2	relationships	relationship	NOUN
ejpam-6171	359	3	among	among	ADP
ejpam-6171	359	4	these	these	DET
ejpam-6171	359	5	four	four	NUM
ejpam-6171	359	6	concepts	concept	NOUN
ejpam-6171	359	7	are	be	AUX
ejpam-6171	359	8	summarized	summarize	VERB
ejpam-6171	359	9	in	in	ADP
ejpam-6171	359	10	figure	figure	NOUN
ejpam-6171	359	11	2	2	NUM
ejpam-6171	359	12	.	.	PUNCT
ejpam-6171	359	13	figure	figure	NOUN
ejpam-6171	359	14	2	2	NUM
ejpam-6171	359	15	:	:	PUNCT
ejpam-6171	359	16	pythagorean	pythagorean	PROPN
ejpam-6171	359	17	neutrosophic	neutrosophic	ADJ
ejpam-6171	359	18	sets	set	NOUN
ejpam-6171	359	19	in	in	ADP
ejpam-6171	359	20	iup	iup	NOUN
ejpam-6171	359	21	-	-	PUNCT
ejpam-6171	359	22	algebras	algebras	PROPN
ejpam-6171	359	23	k.	k.	PROPN
ejpam-6171	359	24	suayngam	suayngam	PROPN
ejpam-6171	360	1	et	et	PROPN
ejpam-6171	360	2	al	al	PROPN
ejpam-6171	360	3	.	.	PUNCT
ejpam-6171	360	4	/	/	SYM
ejpam-6171	360	5	eur	eur	PROPN
ejpam-6171	360	6	.	.	PUNCT
ejpam-6171	361	1	j.	j.	PROPN
ejpam-6171	361	2	pure	pure	PROPN
ejpam-6171	361	3	appl	appl	PROPN
ejpam-6171	361	4	.	.	PROPN
ejpam-6171	361	5	math	math	PROPN
ejpam-6171	361	6	,	,	PUNCT
ejpam-6171	361	7	18	18	NUM
ejpam-6171	361	8	(	(	PUNCT
ejpam-6171	361	9	3	3	NUM
ejpam-6171	361	10	)	)	PUNCT
ejpam-6171	361	11	(	(	PUNCT
ejpam-6171	361	12	2025	2025	NUM
ejpam-6171	361	13	)	)	PUNCT
ejpam-6171	361	14	,	,	PUNCT
ejpam-6171	361	15	6171	6171	NUM
ejpam-6171	361	16	14	14	NUM
ejpam-6171	361	17	of	of	ADP
ejpam-6171	361	18	28	28	NUM
ejpam-6171	361	19	theorem	theorem	NOUN
ejpam-6171	361	20	7	7	NUM
ejpam-6171	361	21	.	.	PUNCT
ejpam-6171	362	1	if	if	SCONJ
ejpam-6171	362	2	p	p	NOUN
ejpam-6171	362	3	is	be	AUX
ejpam-6171	362	4	a	a	DET
ejpam-6171	362	5	pythagorean	pythagorean	PROPN
ejpam-6171	362	6	neutrosophic	neutrosophic	ADJ
ejpam-6171	362	7	iup	iup	NOUN
ejpam-6171	362	8	-	-	PUNCT
ejpam-6171	362	9	subalgebra	subalgebra	NOUN
ejpam-6171	362	10	of	of	ADP
ejpam-6171	362	11	x	x	PUNCT
ejpam-6171	362	12	satisfying	satisfy	VERB
ejpam-6171	362	13	the	the	DET
ejpam-6171	362	14	following	follow	VERB
ejpam-6171	362	15	condition	condition	NOUN
ejpam-6171	362	16	:	:	PUNCT
ejpam-6171	362	17	(	(	PUNCT
ejpam-6171	362	18	∀x	∀x	X
ejpam-6171	362	19	,	,	PUNCT
ejpam-6171	362	20	y	y	PROPN
ejpam-6171	362	21	∈	∈	PROPN
ejpam-6171	362	22	x	x	X
ejpam-6171	362	23	)	)	PUNCT
ejpam-6171	362	24	x	x	NOUN
ejpam-6171	362	25	?	?	PUNCT
ejpam-6171	363	1	y	y	PROPN
ejpam-6171	363	2	6=	6=	ADP
ejpam-6171	363	3	0	0	NUM
ejpam-6171	363	4	⇒	⇒	NOUN
ejpam-6171	363	5			PROPN
ejpam-6171	363	6	pt	pt	X
ejpam-6171	363	7	(	(	PUNCT
ejpam-6171	363	8	x	x	NOUN
ejpam-6171	363	9	)	)	PUNCT
ejpam-6171	363	10	≥	≥	PROPN
ejpam-6171	363	11	pt	pt	INTJ
ejpam-6171	363	12	(	(	PUNCT
ejpam-6171	363	13	y	y	NOUN
ejpam-6171	363	14	)	)	PUNCT
ejpam-6171	363	15	pi(x	pi(x	NOUN
ejpam-6171	363	16	)	)	PUNCT
ejpam-6171	363	17	≤	≤	NOUN
ejpam-6171	363	18	pi(y	pi(y	NOUN
ejpam-6171	363	19	)	)	PUNCT
ejpam-6171	363	20	pf	pf	NOUN
ejpam-6171	363	21	(	(	PUNCT
ejpam-6171	363	22	x	x	NOUN
ejpam-6171	363	23	)	)	PUNCT
ejpam-6171	363	24	≥	≥	NOUN
ejpam-6171	363	25	pf	pf	PROPN
ejpam-6171	363	26	(	(	PUNCT
ejpam-6171	363	27	y	y	NOUN
ejpam-6171	363	28	)	)	PUNCT
ejpam-6171	363	29			PROPN
ejpam-6171	363	30	(	(	PUNCT
ejpam-6171	363	31	3.17	3.17	NUM
ejpam-6171	363	32	)	)	PUNCT
ejpam-6171	363	33	then	then	ADV
ejpam-6171	363	34	p	p	NOUN
ejpam-6171	363	35	is	be	AUX
ejpam-6171	363	36	a	a	DET
ejpam-6171	363	37	pythagorean	pythagorean	PROPN
ejpam-6171	363	38	neutrosophic	neutrosophic	ADJ
ejpam-6171	363	39	strong	strong	ADJ
ejpam-6171	363	40	iup	iup	NOUN
ejpam-6171	363	41	-	-	PUNCT
ejpam-6171	363	42	ideal	ideal	NOUN
ejpam-6171	363	43	of	of	ADP
ejpam-6171	363	44	x.	x.	NOUN
ejpam-6171	363	45	proof	proof	PROPN
ejpam-6171	363	46	.	.	PUNCT
ejpam-6171	364	1	assume	assume	VERB
ejpam-6171	364	2	that	that	SCONJ
ejpam-6171	364	3	p	p	NOUN
ejpam-6171	364	4	is	be	AUX
ejpam-6171	364	5	a	a	DET
ejpam-6171	364	6	pythagorean	pythagorean	PROPN
ejpam-6171	364	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	364	8	iup	iup	NOUN
ejpam-6171	364	9	-	-	PUNCT
ejpam-6171	364	10	subalgebra	subalgebra	NOUN
ejpam-6171	364	11	of	of	ADP
ejpam-6171	364	12	x	x	PUNCT
ejpam-6171	364	13	satisfying	satisfy	VERB
ejpam-6171	364	14	the	the	DET
ejpam-6171	364	15	condition	condition	NOUN
ejpam-6171	364	16	(	(	PUNCT
ejpam-6171	364	17	3.17	3.17	NUM
ejpam-6171	364	18	)	)	PUNCT
ejpam-6171	364	19	.	.	PUNCT
ejpam-6171	365	1	let	let	VERB
ejpam-6171	365	2	x	x	PRON
ejpam-6171	365	3	,	,	PUNCT
ejpam-6171	365	4	y	y	PROPN
ejpam-6171	365	5	∈	∈	PROPN
ejpam-6171	365	6	x.	x.	NOUN
ejpam-6171	365	7	case	case	NOUN
ejpam-6171	365	8	1	1	NUM
ejpam-6171	365	9	:	:	PUNCT
ejpam-6171	365	10	suppose	suppose	VERB
ejpam-6171	365	11	x	x	X
ejpam-6171	365	12	?	?	PUNCT
ejpam-6171	366	1	y	y	PROPN
ejpam-6171	366	2	=	=	PUNCT
ejpam-6171	367	1	0	0	X
ejpam-6171	367	2	.	.	PUNCT
ejpam-6171	368	1	then	then	ADV
ejpam-6171	368	2	pt	pt	X
ejpam-6171	368	3	(	(	PUNCT
ejpam-6171	368	4	x	x	X
ejpam-6171	368	5	?	?	PUNCT
ejpam-6171	369	1	y	y	X
ejpam-6171	369	2	)	)	PUNCT
ejpam-6171	370	1	=	=	SYM
ejpam-6171	370	2	pt	pt	X
ejpam-6171	370	3	(	(	PUNCT
ejpam-6171	370	4	0	0	NUM
ejpam-6171	370	5	)	)	PUNCT
ejpam-6171	370	6	≥	≥	NOUN
ejpam-6171	370	7	pt	pt	INTJ
ejpam-6171	370	8	(	(	PUNCT
ejpam-6171	370	9	y	y	NOUN
ejpam-6171	370	10	)	)	PUNCT
ejpam-6171	370	11	,	,	PUNCT
ejpam-6171	370	12	(	(	PUNCT
ejpam-6171	370	13	by	by	ADP
ejpam-6171	370	14	(	(	PUNCT
ejpam-6171	370	15	3.5	3.5	NUM
ejpam-6171	370	16	)	)	PUNCT
ejpam-6171	370	17	)	)	PUNCT
ejpam-6171	370	18	pi(x	pi(x	NOUN
ejpam-6171	370	19	?	?	PUNCT
ejpam-6171	371	1	y	y	X
ejpam-6171	371	2	)	)	PUNCT
ejpam-6171	371	3	=	=	SYM
ejpam-6171	371	4	pi(0	pi(0	PROPN
ejpam-6171	371	5	)	)	PUNCT
ejpam-6171	371	6	≤	≤	NOUN
ejpam-6171	371	7	pi(y	pi(y	NOUN
ejpam-6171	371	8	)	)	PUNCT
ejpam-6171	371	9	,	,	PUNCT
ejpam-6171	371	10	(	(	PUNCT
ejpam-6171	371	11	by	by	ADP
ejpam-6171	371	12	(	(	PUNCT
ejpam-6171	371	13	3.6	3.6	NUM
ejpam-6171	371	14	)	)	PUNCT
ejpam-6171	371	15	)	)	PUNCT
ejpam-6171	372	1	pf	pf	INTJ
ejpam-6171	372	2	(	(	PUNCT
ejpam-6171	372	3	x	x	PROPN
ejpam-6171	372	4	?	?	PUNCT
ejpam-6171	373	1	y	y	X
ejpam-6171	373	2	)	)	PUNCT
ejpam-6171	374	1	=	=	SYM
ejpam-6171	374	2	pf	pf	X
ejpam-6171	374	3	(	(	PUNCT
ejpam-6171	374	4	0	0	NUM
ejpam-6171	374	5	)	)	PUNCT
ejpam-6171	374	6	≥	≥	NOUN
ejpam-6171	374	7	pf	pf	PROPN
ejpam-6171	374	8	(	(	PUNCT
ejpam-6171	374	9	y	y	NOUN
ejpam-6171	374	10	)	)	PUNCT
ejpam-6171	374	11	.	.	PUNCT
ejpam-6171	375	1	(	(	PUNCT
ejpam-6171	375	2	by	by	ADP
ejpam-6171	375	3	(	(	PUNCT
ejpam-6171	375	4	3.7	3.7	NUM
ejpam-6171	375	5	)	)	PUNCT
ejpam-6171	375	6	)	)	PUNCT
ejpam-6171	375	7	case	case	NOUN
ejpam-6171	375	8	2	2	NUM
ejpam-6171	375	9	:	:	PUNCT
ejpam-6171	375	10	suppose	suppose	VERB
ejpam-6171	375	11	x	x	X
ejpam-6171	375	12	?	?	PUNCT
ejpam-6171	376	1	y	y	PROPN
ejpam-6171	376	2	6=	6=	NUM
ejpam-6171	376	3	0	0	X
ejpam-6171	376	4	.	.	PUNCT
ejpam-6171	377	1	then	then	ADV
ejpam-6171	377	2	pt	pt	X
ejpam-6171	377	3	(	(	PUNCT
ejpam-6171	377	4	x	x	X
ejpam-6171	377	5	?	?	PUNCT
ejpam-6171	377	6	y	y	X
ejpam-6171	377	7	)	)	PUNCT
ejpam-6171	377	8	≥	≥	NOUN
ejpam-6171	377	9	min{pt	min{pt	X
ejpam-6171	378	1	(	(	PUNCT
ejpam-6171	378	2	x),pt	x),pt	PROPN
ejpam-6171	378	3	(	(	PUNCT
ejpam-6171	378	4	y	y	NOUN
ejpam-6171	378	5	)	)	PUNCT
ejpam-6171	378	6	}	}	PUNCT
ejpam-6171	378	7	(	(	PUNCT
ejpam-6171	378	8	by	by	ADP
ejpam-6171	378	9	(	(	PUNCT
ejpam-6171	378	10	3.2	3.2	NUM
ejpam-6171	378	11	)	)	PUNCT
ejpam-6171	378	12	)	)	PUNCT
ejpam-6171	379	1	=	=	SYM
ejpam-6171	379	2	pt	pt	X
ejpam-6171	379	3	(	(	PUNCT
ejpam-6171	379	4	y	y	NOUN
ejpam-6171	379	5	)	)	PUNCT
ejpam-6171	379	6	,	,	PUNCT
ejpam-6171	379	7	pi(x	pi(x	NOUN
ejpam-6171	379	8	?	?	PUNCT
ejpam-6171	380	1	y	y	X
ejpam-6171	380	2	)	)	PUNCT
ejpam-6171	380	3	≤	≤	NUM
ejpam-6171	380	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	380	5	)	)	PUNCT
ejpam-6171	380	6	}	}	PUNCT
ejpam-6171	380	7	(	(	PUNCT
ejpam-6171	380	8	by	by	ADP
ejpam-6171	380	9	(	(	PUNCT
ejpam-6171	380	10	3.3	3.3	NUM
ejpam-6171	380	11	)	)	PUNCT
ejpam-6171	380	12	)	)	PUNCT
ejpam-6171	381	1	=	=	NOUN
ejpam-6171	381	2	pi(y	pi(y	NOUN
ejpam-6171	381	3	)	)	PUNCT
ejpam-6171	381	4	,	,	PUNCT
ejpam-6171	381	5	pf	pf	PROPN
ejpam-6171	381	6	(	(	PUNCT
ejpam-6171	381	7	x	x	PROPN
ejpam-6171	381	8	?	?	PUNCT
ejpam-6171	381	9	y	y	X
ejpam-6171	381	10	)	)	PUNCT
ejpam-6171	381	11	≥	≥	PROPN
ejpam-6171	381	12	min{pf	min{pf	PRON
ejpam-6171	381	13	(	(	PUNCT
ejpam-6171	381	14	x),pf	x),pf	PROPN
ejpam-6171	381	15	(	(	PUNCT
ejpam-6171	381	16	y	y	NOUN
ejpam-6171	381	17	)	)	PUNCT
ejpam-6171	381	18	}	}	PUNCT
ejpam-6171	381	19	(	(	PUNCT
ejpam-6171	381	20	by	by	ADP
ejpam-6171	381	21	(	(	PUNCT
ejpam-6171	381	22	3.4	3.4	NUM
ejpam-6171	381	23	)	)	PUNCT
ejpam-6171	381	24	)	)	PUNCT
ejpam-6171	382	1	=	=	SYM
ejpam-6171	382	2	pf	pf	PROPN
ejpam-6171	382	3	(	(	PUNCT
ejpam-6171	382	4	y	y	NOUN
ejpam-6171	382	5	)	)	PUNCT
ejpam-6171	382	6	.	.	PUNCT
ejpam-6171	383	1	hence	hence	ADV
ejpam-6171	383	2	,	,	PUNCT
ejpam-6171	383	3	p	p	PROPN
ejpam-6171	383	4	is	be	AUX
ejpam-6171	383	5	a	a	DET
ejpam-6171	383	6	pythagorean	pythagorean	PROPN
ejpam-6171	383	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	383	8	strong	strong	ADJ
ejpam-6171	383	9	iup	iup	NOUN
ejpam-6171	383	10	-	-	PUNCT
ejpam-6171	383	11	ideal	ideal	NOUN
ejpam-6171	383	12	of	of	ADP
ejpam-6171	383	13	x.	x.	PROPN
ejpam-6171	383	14	theorem	theorem	VERB
ejpam-6171	383	15	8	8	NUM
ejpam-6171	383	16	.	.	PUNCT
ejpam-6171	384	1	if	if	SCONJ
ejpam-6171	384	2	p	p	NOUN
ejpam-6171	384	3	is	be	AUX
ejpam-6171	384	4	a	a	DET
ejpam-6171	384	5	pythagorean	pythagorean	PROPN
ejpam-6171	384	6	neutrosophic	neutrosophic	ADJ
ejpam-6171	384	7	iup	iup	NOUN
ejpam-6171	384	8	-	-	PUNCT
ejpam-6171	384	9	filter	filter	NOUN
ejpam-6171	384	10	of	of	ADP
ejpam-6171	384	11	x	x	PUNCT
ejpam-6171	384	12	satisfying	satisfy	VERB
ejpam-6171	384	13	the	the	DET
ejpam-6171	384	14	following	follow	VERB
ejpam-6171	384	15	condition	condition	NOUN
ejpam-6171	384	16	:	:	PUNCT
ejpam-6171	384	17	(	(	PUNCT
ejpam-6171	384	18	∀x	∀x	X
ejpam-6171	384	19	,	,	PUNCT
ejpam-6171	384	20	y	y	PROPN
ejpam-6171	384	21	,	,	PUNCT
ejpam-6171	384	22	z	z	NOUN
ejpam-6171	384	23	∈	∈	PROPN
ejpam-6171	384	24	x	x	X
ejpam-6171	384	25	)	)	PUNCT
ejpam-6171	384	26	pt	pt	NUM
ejpam-6171	384	27	(	(	PUNCT
ejpam-6171	384	28	y	y	PROPN
ejpam-6171	384	29	?	?	PUNCT
ejpam-6171	385	1	(	(	PUNCT
ejpam-6171	385	2	x	x	X
ejpam-6171	385	3	?	?	PUNCT
ejpam-6171	386	1	z	z	X
ejpam-6171	386	2	)	)	PUNCT
ejpam-6171	386	3	)	)	PUNCT
ejpam-6171	387	1	=	=	SYM
ejpam-6171	387	2	pt	pt	INTJ
ejpam-6171	387	3	(	(	PUNCT
ejpam-6171	387	4	x	x	X
ejpam-6171	387	5	?	?	PUNCT
ejpam-6171	388	1	(	(	PUNCT
ejpam-6171	388	2	y	y	NOUN
ejpam-6171	388	3	?	?	PUNCT
ejpam-6171	389	1	z	z	X
ejpam-6171	389	2	)	)	PUNCT
ejpam-6171	389	3	)	)	PUNCT
ejpam-6171	389	4	pi(y	pi(y	NOUN
ejpam-6171	389	5	?	?	PUNCT
ejpam-6171	390	1	(	(	PUNCT
ejpam-6171	390	2	x	x	X
ejpam-6171	390	3	?	?	PUNCT
ejpam-6171	390	4	z	z	X
ejpam-6171	390	5	)	)	PUNCT
ejpam-6171	390	6	)	)	PUNCT
ejpam-6171	391	1	=	=	SYM
ejpam-6171	391	2	pi(x	pi(x	NOUN
ejpam-6171	391	3	?	?	PUNCT
ejpam-6171	392	1	(	(	PUNCT
ejpam-6171	392	2	y	y	NOUN
ejpam-6171	392	3	?	?	PUNCT
ejpam-6171	393	1	z	z	X
ejpam-6171	393	2	)	)	PUNCT
ejpam-6171	393	3	)	)	PUNCT
ejpam-6171	394	1	pf	pf	INTJ
ejpam-6171	394	2	(	(	PUNCT
ejpam-6171	394	3	y	y	PROPN
ejpam-6171	394	4	?	?	PUNCT
ejpam-6171	395	1	(	(	PUNCT
ejpam-6171	395	2	x	x	X
ejpam-6171	395	3	?	?	PUNCT
ejpam-6171	395	4	z	z	X
ejpam-6171	395	5	)	)	PUNCT
ejpam-6171	395	6	)	)	PUNCT
ejpam-6171	396	1	=	=	SYM
ejpam-6171	396	2	pf	pf	INTJ
ejpam-6171	396	3	(	(	PUNCT
ejpam-6171	396	4	x	x	PROPN
ejpam-6171	396	5	?	?	PUNCT
ejpam-6171	397	1	(	(	PUNCT
ejpam-6171	397	2	y	y	NOUN
ejpam-6171	397	3	?	?	PUNCT
ejpam-6171	398	1	z	z	X
ejpam-6171	398	2	)	)	PUNCT
ejpam-6171	398	3	)	)	PUNCT
ejpam-6171	399	1			PROPN
ejpam-6171	399	2	(	(	PUNCT
ejpam-6171	399	3	3.18	3.18	NUM
ejpam-6171	399	4	)	)	PUNCT
ejpam-6171	399	5	then	then	ADV
ejpam-6171	399	6	p	p	NOUN
ejpam-6171	399	7	is	be	AUX
ejpam-6171	399	8	a	a	DET
ejpam-6171	399	9	pythagorean	pythagorean	PROPN
ejpam-6171	399	10	neutrosophic	neutrosophic	ADJ
ejpam-6171	399	11	iup	iup	PROPN
ejpam-6171	399	12	-	-	PUNCT
ejpam-6171	399	13	ideal	ideal	NOUN
ejpam-6171	399	14	of	of	ADP
ejpam-6171	399	15	x.	x.	NOUN
ejpam-6171	399	16	proof	proof	PROPN
ejpam-6171	399	17	.	.	PUNCT
ejpam-6171	400	1	assume	assume	VERB
ejpam-6171	400	2	that	that	SCONJ
ejpam-6171	400	3	p	p	NOUN
ejpam-6171	400	4	is	be	AUX
ejpam-6171	400	5	a	a	DET
ejpam-6171	400	6	pythagorean	pythagorean	PROPN
ejpam-6171	400	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	400	8	iup	iup	NOUN
ejpam-6171	400	9	-	-	PUNCT
ejpam-6171	400	10	filter	filter	NOUN
ejpam-6171	400	11	of	of	ADP
ejpam-6171	400	12	x	x	PUNCT
ejpam-6171	400	13	satisfying	satisfy	VERB
ejpam-6171	400	14	the	the	DET
ejpam-6171	400	15	condition	condition	NOUN
ejpam-6171	400	16	(	(	PUNCT
ejpam-6171	400	17	3.18	3.18	NUM
ejpam-6171	400	18	)	)	PUNCT
ejpam-6171	400	19	.	.	PUNCT
ejpam-6171	401	1	by	by	ADP
ejpam-6171	401	2	the	the	DET
ejpam-6171	401	3	assumption	assumption	NOUN
ejpam-6171	401	4	,	,	PUNCT
ejpam-6171	401	5	it	it	PRON
ejpam-6171	401	6	satisfies	satisfy	VERB
ejpam-6171	401	7	(	(	PUNCT
ejpam-6171	401	8	3.5	3.5	NUM
ejpam-6171	401	9	)	)	PUNCT
ejpam-6171	401	10	,	,	PUNCT
ejpam-6171	401	11	(	(	PUNCT
ejpam-6171	401	12	3.6	3.6	NUM
ejpam-6171	401	13	)	)	PUNCT
ejpam-6171	401	14	,	,	PUNCT
ejpam-6171	401	15	and	and	CCONJ
ejpam-6171	401	16	(	(	PUNCT
ejpam-6171	401	17	3.7	3.7	NUM
ejpam-6171	401	18	)	)	PUNCT
ejpam-6171	401	19	.	.	PUNCT
ejpam-6171	402	1	let	let	VERB
ejpam-6171	402	2	x	x	PRON
ejpam-6171	402	3	,	,	PUNCT
ejpam-6171	402	4	y	y	PROPN
ejpam-6171	402	5	,	,	PUNCT
ejpam-6171	402	6	z	z	PROPN
ejpam-6171	402	7	∈	∈	PROPN
ejpam-6171	402	8	x.	x.	NOUN
ejpam-6171	403	1	then	then	ADV
ejpam-6171	403	2	pt	pt	X
ejpam-6171	403	3	(	(	PUNCT
ejpam-6171	403	4	x	x	PROPN
ejpam-6171	403	5	?	?	PUNCT
ejpam-6171	404	1	z	z	X
ejpam-6171	404	2	)	)	PUNCT
ejpam-6171	404	3	≥	≥	NOUN
ejpam-6171	404	4	min{pt	min{pt	X
ejpam-6171	405	1	(	(	PUNCT
ejpam-6171	405	2	y	y	PROPN
ejpam-6171	405	3	?	?	PUNCT
ejpam-6171	406	1	(	(	PUNCT
ejpam-6171	406	2	x	x	X
ejpam-6171	406	3	?	?	PUNCT
ejpam-6171	406	4	z)),pt	z)),pt	PROPN
ejpam-6171	406	5	(	(	PUNCT
ejpam-6171	406	6	y	y	NOUN
ejpam-6171	406	7	)	)	PUNCT
ejpam-6171	406	8	}	}	PUNCT
ejpam-6171	406	9	(	(	PUNCT
ejpam-6171	406	10	by	by	ADP
ejpam-6171	406	11	(	(	PUNCT
ejpam-6171	406	12	3.11	3.11	NUM
ejpam-6171	406	13	)	)	PUNCT
ejpam-6171	406	14	)	)	PUNCT
ejpam-6171	406	15	k.	k.	PROPN
ejpam-6171	407	1	suayngam	suayngam	INTJ
ejpam-6171	407	2	et	et	PROPN
ejpam-6171	407	3	al	al	PROPN
ejpam-6171	407	4	.	.	PUNCT
ejpam-6171	407	5	/	/	SYM
ejpam-6171	407	6	eur	eur	PROPN
ejpam-6171	407	7	.	.	PUNCT
ejpam-6171	408	1	j.	j.	PROPN
ejpam-6171	408	2	pure	pure	PROPN
ejpam-6171	408	3	appl	appl	PROPN
ejpam-6171	408	4	.	.	PROPN
ejpam-6171	408	5	math	math	PROPN
ejpam-6171	408	6	,	,	PUNCT
ejpam-6171	408	7	18	18	NUM
ejpam-6171	408	8	(	(	PUNCT
ejpam-6171	408	9	3	3	NUM
ejpam-6171	408	10	)	)	PUNCT
ejpam-6171	408	11	(	(	PUNCT
ejpam-6171	408	12	2025	2025	NUM
ejpam-6171	408	13	)	)	PUNCT
ejpam-6171	408	14	,	,	PUNCT
ejpam-6171	408	15	6171	6171	NUM
ejpam-6171	408	16	15	15	NUM
ejpam-6171	408	17	of	of	ADP
ejpam-6171	408	18	28	28	NUM
ejpam-6171	408	19	=	=	SYM
ejpam-6171	408	20	min{pt	min{pt	NOUN
ejpam-6171	408	21	(	(	PUNCT
ejpam-6171	408	22	x	x	X
ejpam-6171	408	23	?	?	PUNCT
ejpam-6171	409	1	(	(	PUNCT
ejpam-6171	409	2	y	y	PROPN
ejpam-6171	409	3	?	?	PUNCT
ejpam-6171	410	1	z)),pt	z)),pt	PROPN
ejpam-6171	410	2	(	(	PUNCT
ejpam-6171	410	3	y	y	NOUN
ejpam-6171	410	4	)	)	PUNCT
ejpam-6171	410	5	}	}	PUNCT
ejpam-6171	410	6	,	,	PUNCT
ejpam-6171	410	7	pi(x	pi(x	NOUN
ejpam-6171	410	8	?	?	PUNCT
ejpam-6171	411	1	z	z	X
ejpam-6171	411	2	)	)	PUNCT
ejpam-6171	411	3	≤	≤	NOUN
ejpam-6171	411	4	max{pi(y	max{pi(y	NOUN
ejpam-6171	411	5	?	?	PUNCT
ejpam-6171	412	1	(	(	PUNCT
ejpam-6171	412	2	x	x	X
ejpam-6171	412	3	?	?	PUNCT
ejpam-6171	412	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	412	5	)	)	PUNCT
ejpam-6171	412	6	}	}	PUNCT
ejpam-6171	412	7	(	(	PUNCT
ejpam-6171	412	8	by	by	ADP
ejpam-6171	412	9	(	(	PUNCT
ejpam-6171	412	10	3.12	3.12	NUM
ejpam-6171	412	11	)	)	PUNCT
ejpam-6171	412	12	)	)	PUNCT
ejpam-6171	413	1	=	=	SYM
ejpam-6171	413	2	max{pi(x	max{pi(x	PROPN
ejpam-6171	413	3	?	?	PUNCT
ejpam-6171	414	1	(	(	PUNCT
ejpam-6171	414	2	y	y	NOUN
ejpam-6171	414	3	?	?	PUNCT
ejpam-6171	414	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	414	5	)	)	PUNCT
ejpam-6171	414	6	}	}	PUNCT
ejpam-6171	414	7	,	,	PUNCT
ejpam-6171	414	8	pf	pf	PROPN
ejpam-6171	414	9	(	(	PUNCT
ejpam-6171	414	10	x	x	PROPN
ejpam-6171	414	11	?	?	PUNCT
ejpam-6171	415	1	z	z	X
ejpam-6171	415	2	)	)	PUNCT
ejpam-6171	415	3	≥	≥	NOUN
ejpam-6171	415	4	min{pf	min{pf	X
ejpam-6171	415	5	(	(	PUNCT
ejpam-6171	415	6	y	y	PROPN
ejpam-6171	415	7	?	?	PUNCT
ejpam-6171	416	1	(	(	PUNCT
ejpam-6171	416	2	x	x	X
ejpam-6171	416	3	?	?	PUNCT
ejpam-6171	417	1	z)),pf	z)),pf	PROPN
ejpam-6171	417	2	(	(	PUNCT
ejpam-6171	417	3	y	y	NOUN
ejpam-6171	417	4	)	)	PUNCT
ejpam-6171	417	5	}	}	PUNCT
ejpam-6171	417	6	(	(	PUNCT
ejpam-6171	417	7	by	by	ADP
ejpam-6171	417	8	(	(	PUNCT
ejpam-6171	417	9	3.13	3.13	NUM
ejpam-6171	417	10	)	)	PUNCT
ejpam-6171	417	11	)	)	PUNCT
ejpam-6171	418	1	=	=	PUNCT
ejpam-6171	418	2	min{pf	min{pf	X
ejpam-6171	418	3	(	(	PUNCT
ejpam-6171	418	4	x	x	X
ejpam-6171	418	5	?	?	PUNCT
ejpam-6171	419	1	(	(	PUNCT
ejpam-6171	419	2	y	y	NOUN
ejpam-6171	419	3	?	?	PUNCT
ejpam-6171	420	1	z)),pf	z)),pf	PROPN
ejpam-6171	420	2	(	(	PUNCT
ejpam-6171	420	3	y	y	NOUN
ejpam-6171	420	4	)	)	PUNCT
ejpam-6171	420	5	}	}	PUNCT
ejpam-6171	420	6	.	.	PUNCT
ejpam-6171	421	1	hence	hence	ADV
ejpam-6171	421	2	,	,	PUNCT
ejpam-6171	421	3	p	p	PROPN
ejpam-6171	421	4	is	be	AUX
ejpam-6171	421	5	a	a	DET
ejpam-6171	421	6	pythagorean	pythagorean	PROPN
ejpam-6171	421	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	421	8	iup	iup	PROPN
ejpam-6171	421	9	-	-	PUNCT
ejpam-6171	421	10	ideal	ideal	NOUN
ejpam-6171	421	11	of	of	ADP
ejpam-6171	421	12	x.	x.	NOUN
ejpam-6171	421	13	for	for	ADP
ejpam-6171	421	14	any	any	DET
ejpam-6171	421	15	fixed	fix	VERB
ejpam-6171	421	16	numbers	number	NOUN
ejpam-6171	421	17	α+	α+	ADP
ejpam-6171	421	18	,	,	PUNCT
ejpam-6171	421	19	α−	α−	PROPN
ejpam-6171	421	20	,	,	PUNCT
ejpam-6171	421	21	β+	β+	NOUN
ejpam-6171	421	22	,	,	PUNCT
ejpam-6171	421	23	β−	β−	PRON
ejpam-6171	421	24	,	,	PUNCT
ejpam-6171	421	25	γ+	γ+	NUM
ejpam-6171	421	26	,	,	PUNCT
ejpam-6171	421	27	γ−	γ−	PROPN
ejpam-6171	421	28	∈	∈	PROPN
ejpam-6171	422	1	[	[	X
ejpam-6171	422	2	0	0	NUM
ejpam-6171	422	3	,	,	PUNCT
ejpam-6171	422	4	1	1	NUM
ejpam-6171	422	5	]	]	PUNCT
ejpam-6171	422	6	such	such	ADJ
ejpam-6171	422	7	that	that	SCONJ
ejpam-6171	422	8	α+	α+	X
ejpam-6171	422	9	>	>	X
ejpam-6171	422	10	α−	α−	PROPN
ejpam-6171	422	11	,	,	PUNCT
ejpam-6171	422	12	β+	β+	PUNCT
ejpam-6171	422	13	>	>	X
ejpam-6171	422	14	β−	β−	PROPN
ejpam-6171	422	15	,	,	PUNCT
ejpam-6171	422	16	γ+	γ+	PUNCT
ejpam-6171	422	17	>	>	X
ejpam-6171	422	18	γ−	γ−	PROPN
ejpam-6171	422	19	and	and	CCONJ
ejpam-6171	422	20	a	a	DET
ejpam-6171	422	21	nonempty	nonempty	NOUN
ejpam-6171	422	22	subset	subset	VERB
ejpam-6171	422	23	g	g	NOUN
ejpam-6171	422	24	of	of	ADP
ejpam-6171	422	25	x	x	PROPN
ejpam-6171	422	26	,	,	PUNCT
ejpam-6171	422	27	a	a	DET
ejpam-6171	422	28	pns	pns	NOUN
ejpam-6171	422	29	pg[α	pg[α	PROPN
ejpam-6171	422	30	+	+	PROPN
ejpam-6171	422	31	,	,	PUNCT
ejpam-6171	422	32	β−,γ+	β−,γ+	X
ejpam-6171	422	33	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	422	34	]	]	PUNCT
ejpam-6171	422	35	=	=	SYM
ejpam-6171	422	36	(	(	PUNCT
ejpam-6171	422	37	x	x	X
ejpam-6171	422	38	,	,	PUNCT
ejpam-6171	422	39	pg	pg	PROPN
ejpam-6171	422	40	t	t	PROPN
ejpam-6171	423	1	[	[	X
ejpam-6171	423	2	α	α	X
ejpam-6171	423	3	+	+	X
ejpam-6171	424	1	α−	α−	ADP
ejpam-6171	424	2	]	]	PUNCT
ejpam-6171	424	3	,	,	PUNCT
ejpam-6171	424	4	pg	pg	VERB
ejpam-6171	424	5	i	i	PRON
ejpam-6171	425	1	[	[	X
ejpam-6171	425	2	β	β	X
ejpam-6171	425	3	−	−	NOUN
ejpam-6171	425	4	β+	β+	PUNCT
ejpam-6171	425	5	]	]	PUNCT
ejpam-6171	425	6	,	,	PUNCT
ejpam-6171	425	7	pg	pg	PROPN
ejpam-6171	425	8	f	f	PROPN
ejpam-6171	426	1	[	[	X
ejpam-6171	426	2	γ	γ	X
ejpam-6171	426	3	+	+	X
ejpam-6171	426	4	γ−	γ−	PROPN
ejpam-6171	426	5	]	]	PUNCT
ejpam-6171	426	6	)	)	PUNCT
ejpam-6171	426	7	in	in	ADP
ejpam-6171	426	8	x	x	NOUN
ejpam-6171	426	9	,	,	PUNCT
ejpam-6171	426	10	where	where	SCONJ
ejpam-6171	426	11	pg	pg	PROPN
ejpam-6171	426	12	t	t	PROPN
ejpam-6171	427	1	[	[	X
ejpam-6171	427	2	α	α	X
ejpam-6171	427	3	+	+	X
ejpam-6171	428	1	α−	α−	ADP
ejpam-6171	428	2	]	]	PUNCT
ejpam-6171	428	3	,	,	PUNCT
ejpam-6171	428	4	pg	pg	VERB
ejpam-6171	428	5	i	i	PRON
ejpam-6171	429	1	[	[	X
ejpam-6171	429	2	β	β	X
ejpam-6171	429	3	−	−	NOUN
ejpam-6171	429	4	β+	β+	PUNCT
ejpam-6171	429	5	]	]	X
ejpam-6171	429	6	,	,	PUNCT
ejpam-6171	429	7	and	and	CCONJ
ejpam-6171	429	8	pg	pg	X
ejpam-6171	429	9	f	f	PROPN
ejpam-6171	430	1	[	[	X
ejpam-6171	430	2	γ	γ	X
ejpam-6171	430	3	+	+	CCONJ
ejpam-6171	430	4	γ−	γ−	PROPN
ejpam-6171	430	5	]	]	PUNCT
ejpam-6171	430	6	are	be	AUX
ejpam-6171	430	7	function	function	NOUN
ejpam-6171	430	8	on	on	ADP
ejpam-6171	430	9	x	x	PUNCT
ejpam-6171	430	10	which	which	PRON
ejpam-6171	430	11	are	be	AUX
ejpam-6171	430	12	given	give	VERB
ejpam-6171	430	13	as	as	SCONJ
ejpam-6171	430	14	follows	follow	VERB
ejpam-6171	430	15	:	:	PUNCT
ejpam-6171	430	16	pg	pg	PROPN
ejpam-6171	430	17	t	t	PROPN
ejpam-6171	431	1	[	[	X
ejpam-6171	431	2	α	α	X
ejpam-6171	431	3	+	+	X
ejpam-6171	432	1	α−	α−	ADP
ejpam-6171	432	2	]	]	PUNCT
ejpam-6171	433	1	=	=	PUNCT
ejpam-6171	434	1	{	{	PUNCT
ejpam-6171	435	1	α+	α+	X
ejpam-6171	435	2	if	if	SCONJ
ejpam-6171	435	3	x	x	SYM
ejpam-6171	435	4	∈	∈	PROPN
ejpam-6171	435	5	g	g	NOUN
ejpam-6171	435	6	α−	α−	INTJ
ejpam-6171	435	7	otherwise	otherwise	ADV
ejpam-6171	435	8	pg	pg	VERB
ejpam-6171	435	9	i	i	PRON
ejpam-6171	436	1	[	[	X
ejpam-6171	436	2	β	β	X
ejpam-6171	436	3	−	−	NOUN
ejpam-6171	436	4	β+	β+	PUNCT
ejpam-6171	436	5	]	]	PUNCT
ejpam-6171	436	6	=	=	X
ejpam-6171	437	1	{	{	PUNCT
ejpam-6171	437	2	β−	β−	INTJ
ejpam-6171	437	3	if	if	SCONJ
ejpam-6171	437	4	x	x	SYM
ejpam-6171	437	5	∈	∈	PROPN
ejpam-6171	437	6	g	g	NOUN
ejpam-6171	437	7	β+	β+	PUNCT
ejpam-6171	437	8	otherwise	otherwise	ADV
ejpam-6171	437	9	pg	pg	VERB
ejpam-6171	437	10	f	f	PROPN
ejpam-6171	438	1	[	[	X
ejpam-6171	438	2	γ	γ	X
ejpam-6171	438	3	+	+	X
ejpam-6171	438	4	γ−	γ−	PROPN
ejpam-6171	438	5	]	]	PUNCT
ejpam-6171	439	1	=	=	PRON
ejpam-6171	439	2	{	{	PUNCT
ejpam-6171	439	3	γ+	γ+	PUNCT
ejpam-6171	439	4	if	if	SCONJ
ejpam-6171	439	5	x	x	PROPN
ejpam-6171	439	6	∈	∈	PROPN
ejpam-6171	439	7	g	g	NOUN
ejpam-6171	439	8	γ−	γ−	PROPN
ejpam-6171	439	9	otherwise	otherwise	ADV
ejpam-6171	439	10	lemma	lemma	PROPN
ejpam-6171	439	11	2	2	X
ejpam-6171	439	12	.	.	PUNCT
ejpam-6171	440	1	let	let	VERB
ejpam-6171	440	2	g	g	PRON
ejpam-6171	440	3	be	be	AUX
ejpam-6171	440	4	a	a	DET
ejpam-6171	440	5	nonempty	nonempty	ADJ
ejpam-6171	440	6	subset	subset	NOUN
ejpam-6171	440	7	of	of	ADP
ejpam-6171	440	8	x.	x.	NOUN
ejpam-6171	440	9	then	then	ADV
ejpam-6171	440	10	the	the	DET
ejpam-6171	440	11	constant	constant	ADJ
ejpam-6171	440	12	0	0	NUM
ejpam-6171	440	13	of	of	ADP
ejpam-6171	440	14	x	x	PRON
ejpam-6171	440	15	is	be	AUX
ejpam-6171	440	16	in	in	ADP
ejpam-6171	440	17	g	g	PROPN
ejpam-6171	440	18	if	if	SCONJ
ejpam-6171	441	1	and	and	CCONJ
ejpam-6171	441	2	only	only	ADV
ejpam-6171	441	3	if	if	SCONJ
ejpam-6171	441	4	the	the	DET
ejpam-6171	441	5	characteristic	characteristic	ADJ
ejpam-6171	441	6	pns	pns	PROPN
ejpam-6171	441	7	pg[α	pg[α	PROPN
ejpam-6171	441	8	+	+	PROPN
ejpam-6171	441	9	,	,	PUNCT
ejpam-6171	441	10	β−,γ+	β−,γ+	X
ejpam-6171	441	11	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	441	12	]	]	PUNCT
ejpam-6171	441	13	satisfies	satisfie	NOUN
ejpam-6171	441	14	(	(	PUNCT
ejpam-6171	441	15	3.5	3.5	NUM
ejpam-6171	441	16	)	)	PUNCT
ejpam-6171	441	17	,	,	PUNCT
ejpam-6171	441	18	(	(	PUNCT
ejpam-6171	441	19	3.6	3.6	NUM
ejpam-6171	441	20	)	)	PUNCT
ejpam-6171	441	21	,	,	PUNCT
ejpam-6171	441	22	and	and	CCONJ
ejpam-6171	441	23	(	(	PUNCT
ejpam-6171	441	24	3.7	3.7	NUM
ejpam-6171	441	25	)	)	PUNCT
ejpam-6171	441	26	.	.	PUNCT
ejpam-6171	442	1	proof	proof	NOUN
ejpam-6171	442	2	.	.	PUNCT
ejpam-6171	443	1	assume	assume	VERB
ejpam-6171	443	2	that	that	SCONJ
ejpam-6171	443	3	the	the	DET
ejpam-6171	443	4	constant	constant	ADJ
ejpam-6171	443	5	0	0	NUM
ejpam-6171	443	6	of	of	ADP
ejpam-6171	443	7	x	x	PRON
ejpam-6171	443	8	is	be	AUX
ejpam-6171	443	9	in	in	ADP
ejpam-6171	443	10	g.	g.	PROPN
ejpam-6171	443	11	then	then	ADV
ejpam-6171	443	12	pg	pg	PROPN
ejpam-6171	443	13	t	t	PROPN
ejpam-6171	444	1	[	[	X
ejpam-6171	444	2	α	α	X
ejpam-6171	444	3	+	+	X
ejpam-6171	445	1	α−	α−	ADP
ejpam-6171	445	2	]	]	X
ejpam-6171	445	3	(	(	PUNCT
ejpam-6171	445	4	0	0	NUM
ejpam-6171	445	5	)	)	PUNCT
ejpam-6171	445	6	=	=	SYM
ejpam-6171	445	7	α+	α+	NOUN
ejpam-6171	445	8	,	,	PUNCT
ejpam-6171	445	9	pg	pg	VERB
ejpam-6171	445	10	i	i	PRON
ejpam-6171	446	1	[	[	X
ejpam-6171	446	2	β	β	X
ejpam-6171	446	3	−	−	NOUN
ejpam-6171	446	4	β+	β+	PUNCT
ejpam-6171	446	5	]	]	X
ejpam-6171	446	6	(	(	PUNCT
ejpam-6171	446	7	0	0	NUM
ejpam-6171	446	8	)	)	PUNCT
ejpam-6171	446	9	=	=	SYM
ejpam-6171	446	10	β−	β−	PROPN
ejpam-6171	446	11	,	,	PUNCT
ejpam-6171	446	12	and	and	CCONJ
ejpam-6171	446	13	pg	pg	X
ejpam-6171	446	14	f	f	PROPN
ejpam-6171	447	1	[	[	X
ejpam-6171	447	2	γ	γ	X
ejpam-6171	447	3	+	+	X
ejpam-6171	447	4	γ−	γ−	PROPN
ejpam-6171	447	5	]	]	PUNCT
ejpam-6171	447	6	(	(	PUNCT
ejpam-6171	447	7	0	0	NUM
ejpam-6171	447	8	)	)	PUNCT
ejpam-6171	447	9	=	=	PRON
ejpam-6171	447	10	γ+	γ+	PROPN
ejpam-6171	447	11	.	.	PUNCT
ejpam-6171	448	1	thus	thus	ADV
ejpam-6171	448	2	,	,	PUNCT
ejpam-6171	448	3	pg	pg	PROPN
ejpam-6171	448	4	t	t	PROPN
ejpam-6171	449	1	[	[	X
ejpam-6171	449	2	α	α	X
ejpam-6171	449	3	+	+	X
ejpam-6171	450	1	α−	α−	ADP
ejpam-6171	450	2	]	]	X
ejpam-6171	450	3	(	(	PUNCT
ejpam-6171	450	4	0	0	NUM
ejpam-6171	450	5	)	)	PUNCT
ejpam-6171	450	6	=	=	PRON
ejpam-6171	450	7	α+	α+	PUNCT
ejpam-6171	450	8	≥	≥	NOUN
ejpam-6171	450	9	pg	pg	X
ejpam-6171	450	10	t	t	PROPN
ejpam-6171	451	1	[	[	X
ejpam-6171	451	2	α	α	X
ejpam-6171	451	3	+	+	X
ejpam-6171	452	1	α−	α−	ADP
ejpam-6171	452	2	]	]	X
ejpam-6171	452	3	(	(	PUNCT
ejpam-6171	452	4	x	x	NOUN
ejpam-6171	452	5	)	)	PUNCT
ejpam-6171	452	6	,	,	PUNCT
ejpam-6171	452	7	pg	pg	VERB
ejpam-6171	452	8	i	i	PRON
ejpam-6171	453	1	[	[	X
ejpam-6171	453	2	β	β	X
ejpam-6171	453	3	−	−	NOUN
ejpam-6171	453	4	β+	β+	PUNCT
ejpam-6171	453	5	]	]	X
ejpam-6171	453	6	(	(	PUNCT
ejpam-6171	453	7	0	0	NUM
ejpam-6171	453	8	)	)	PUNCT
ejpam-6171	453	9	=	=	SYM
ejpam-6171	454	1	β−	β−	PUNCT
ejpam-6171	454	2	≤	≤	NOUN
ejpam-6171	454	3	pg	pg	VERB
ejpam-6171	455	1	i	i	PRON
ejpam-6171	456	1	[	[	X
ejpam-6171	456	2	β	β	X
ejpam-6171	456	3	−	−	NOUN
ejpam-6171	456	4	β+	β+	PUNCT
ejpam-6171	456	5	]	]	X
ejpam-6171	456	6	(	(	PUNCT
ejpam-6171	456	7	x	x	NOUN
ejpam-6171	456	8	)	)	PUNCT
ejpam-6171	456	9	,	,	PUNCT
ejpam-6171	456	10	and	and	CCONJ
ejpam-6171	456	11	pg	pg	X
ejpam-6171	456	12	f	f	PROPN
ejpam-6171	457	1	[	[	X
ejpam-6171	457	2	γ	γ	X
ejpam-6171	457	3	+	+	X
ejpam-6171	457	4	γ−	γ−	PROPN
ejpam-6171	457	5	]	]	PUNCT
ejpam-6171	457	6	(	(	PUNCT
ejpam-6171	457	7	0	0	NUM
ejpam-6171	457	8	)	)	PUNCT
ejpam-6171	457	9	=	=	PRON
ejpam-6171	457	10	γ+	γ+	PUNCT
ejpam-6171	457	11	≥	≥	NOUN
ejpam-6171	457	12	pg	pg	NOUN
ejpam-6171	457	13	f	f	PROPN
ejpam-6171	458	1	[	[	X
ejpam-6171	458	2	γ	γ	X
ejpam-6171	458	3	+	+	X
ejpam-6171	458	4	γ−	γ−	PROPN
ejpam-6171	458	5	]	]	PUNCT
ejpam-6171	458	6	(	(	PUNCT
ejpam-6171	458	7	x	x	X
ejpam-6171	458	8	)	)	PUNCT
ejpam-6171	458	9	for	for	ADP
ejpam-6171	458	10	all	all	PRON
ejpam-6171	458	11	x	x	SYM
ejpam-6171	458	12	∈	∈	PROPN
ejpam-6171	458	13	x	x	NOUN
ejpam-6171	458	14	,	,	PUNCT
ejpam-6171	458	15	that	that	ADV
ejpam-6171	458	16	is	is	ADV
ejpam-6171	458	17	,	,	PUNCT
ejpam-6171	458	18	pg[α	pg[α	PROPN
ejpam-6171	458	19	+	+	PROPN
ejpam-6171	458	20	,	,	PUNCT
ejpam-6171	458	21	β−,γ+	β−,γ+	X
ejpam-6171	458	22	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	458	23	]	]	PUNCT
ejpam-6171	458	24	satisfies	satisfie	NOUN
ejpam-6171	458	25	(	(	PUNCT
ejpam-6171	458	26	3.5	3.5	NUM
ejpam-6171	458	27	)	)	PUNCT
ejpam-6171	458	28	,	,	PUNCT
ejpam-6171	458	29	(	(	PUNCT
ejpam-6171	458	30	3.6	3.6	NUM
ejpam-6171	458	31	)	)	PUNCT
ejpam-6171	458	32	,	,	PUNCT
ejpam-6171	458	33	and	and	CCONJ
ejpam-6171	458	34	(	(	PUNCT
ejpam-6171	458	35	3.7	3.7	NUM
ejpam-6171	458	36	)	)	PUNCT
ejpam-6171	458	37	.	.	PUNCT
ejpam-6171	459	1	conversely	conversely	ADV
ejpam-6171	459	2	,	,	PUNCT
ejpam-6171	459	3	assume	assume	VERB
ejpam-6171	459	4	that	that	SCONJ
ejpam-6171	459	5	pg[α	pg[α	PROPN
ejpam-6171	459	6	+	+	PROPN
ejpam-6171	459	7	,	,	PUNCT
ejpam-6171	459	8	β−,γ+	β−,γ+	X
ejpam-6171	459	9	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	459	10	]	]	PUNCT
ejpam-6171	459	11	satisfies	satisfie	NOUN
ejpam-6171	459	12	(	(	PUNCT
ejpam-6171	459	13	3.5	3.5	NUM
ejpam-6171	459	14	)	)	PUNCT
ejpam-6171	459	15	,	,	PUNCT
ejpam-6171	459	16	(	(	PUNCT
ejpam-6171	459	17	3.6	3.6	NUM
ejpam-6171	459	18	)	)	PUNCT
ejpam-6171	459	19	,	,	PUNCT
ejpam-6171	459	20	and	and	CCONJ
ejpam-6171	459	21	(	(	PUNCT
ejpam-6171	459	22	3.7	3.7	NUM
ejpam-6171	459	23	)	)	PUNCT
ejpam-6171	459	24	.	.	PUNCT
ejpam-6171	460	1	then	then	ADV
ejpam-6171	460	2	pg	pg	VERB
ejpam-6171	460	3	t	t	PROPN
ejpam-6171	461	1	[	[	X
ejpam-6171	461	2	α	α	X
ejpam-6171	461	3	+	+	X
ejpam-6171	462	1	α−	α−	ADP
ejpam-6171	462	2	]	]	X
ejpam-6171	462	3	(	(	PUNCT
ejpam-6171	462	4	0	0	NUM
ejpam-6171	462	5	)	)	PUNCT
ejpam-6171	462	6	≥	≥	NOUN
ejpam-6171	462	7	pg	pg	X
ejpam-6171	462	8	t	t	PROPN
ejpam-6171	463	1	[	[	X
ejpam-6171	463	2	α	α	X
ejpam-6171	463	3	+	+	X
ejpam-6171	464	1	α−	α−	ADP
ejpam-6171	464	2	]	]	X
ejpam-6171	464	3	(	(	PUNCT
ejpam-6171	464	4	x	x	X
ejpam-6171	464	5	)	)	PUNCT
ejpam-6171	464	6	for	for	ADP
ejpam-6171	464	7	all	all	DET
ejpam-6171	464	8	x	x	SYM
ejpam-6171	464	9	∈	∈	PROPN
ejpam-6171	464	10	x.	x.	NOUN
ejpam-6171	464	11	since	since	SCONJ
ejpam-6171	464	12	g	g	PROPN
ejpam-6171	464	13	is	be	AUX
ejpam-6171	464	14	a	a	DET
ejpam-6171	464	15	nonempty	nonempty	ADJ
ejpam-6171	464	16	subset	subset	NOUN
ejpam-6171	464	17	of	of	ADP
ejpam-6171	464	18	x	x	PRON
ejpam-6171	464	19	,	,	PUNCT
ejpam-6171	464	20	we	we	PRON
ejpam-6171	464	21	let	let	VERB
ejpam-6171	464	22	a	a	DET
ejpam-6171	464	23	∈	∈	PROPN
ejpam-6171	464	24	g.	g.	NOUN
ejpam-6171	465	1	then	then	ADV
ejpam-6171	465	2	pg	pg	VERB
ejpam-6171	465	3	t	t	PROPN
ejpam-6171	466	1	[	[	X
ejpam-6171	466	2	α	α	X
ejpam-6171	466	3	+	+	X
ejpam-6171	467	1	α−	α−	ADP
ejpam-6171	467	2	]	]	X
ejpam-6171	467	3	(	(	PUNCT
ejpam-6171	467	4	0	0	NUM
ejpam-6171	467	5	)	)	PUNCT
ejpam-6171	467	6	≥	≥	NOUN
ejpam-6171	467	7	pg	pg	X
ejpam-6171	467	8	t	t	PROPN
ejpam-6171	468	1	[	[	X
ejpam-6171	468	2	α	α	X
ejpam-6171	468	3	+	+	X
ejpam-6171	469	1	α−	α−	ADP
ejpam-6171	469	2	]	]	X
ejpam-6171	469	3	(	(	PUNCT
ejpam-6171	469	4	a	a	X
ejpam-6171	469	5	)	)	PUNCT
ejpam-6171	469	6	=	=	SYM
ejpam-6171	469	7	α+	α+	NOUN
ejpam-6171	469	8	,	,	PUNCT
ejpam-6171	469	9	so	so	ADV
ejpam-6171	469	10	pg	pg	PROPN
ejpam-6171	469	11	t	t	PROPN
ejpam-6171	470	1	[	[	X
ejpam-6171	470	2	α	α	X
ejpam-6171	470	3	+	+	X
ejpam-6171	471	1	α−	α−	ADP
ejpam-6171	471	2	]	]	X
ejpam-6171	471	3	(	(	PUNCT
ejpam-6171	471	4	0	0	NUM
ejpam-6171	471	5	)	)	PUNCT
ejpam-6171	471	6	=	=	SYM
ejpam-6171	471	7	α+	α+	NOUN
ejpam-6171	471	8	.	.	PUNCT
ejpam-6171	472	1	hence	hence	ADV
ejpam-6171	472	2	,	,	PUNCT
ejpam-6171	472	3	the	the	DET
ejpam-6171	472	4	constant	constant	ADJ
ejpam-6171	472	5	0	0	NUM
ejpam-6171	472	6	of	of	ADP
ejpam-6171	472	7	x	x	PRON
ejpam-6171	472	8	is	be	AUX
ejpam-6171	472	9	in	in	ADP
ejpam-6171	472	10	g.	g.	PROPN
ejpam-6171	472	11	theorem	theorem	VERB
ejpam-6171	472	12	9	9	NUM
ejpam-6171	472	13	.	.	PUNCT
ejpam-6171	473	1	a	a	DET
ejpam-6171	473	2	nonempty	nonempty	NOUN
ejpam-6171	473	3	subset	subset	VERB
ejpam-6171	473	4	g	g	NOUN
ejpam-6171	473	5	is	be	AUX
ejpam-6171	473	6	an	an	DET
ejpam-6171	473	7	iup	iup	NOUN
ejpam-6171	473	8	-	-	PUNCT
ejpam-6171	473	9	subalgebra	subalgebra	NOUN
ejpam-6171	473	10	of	of	ADP
ejpam-6171	473	11	x	x	PRON
ejpam-6171	473	12	if	if	SCONJ
ejpam-6171	473	13	and	and	CCONJ
ejpam-6171	473	14	only	only	ADV
ejpam-6171	473	15	if	if	SCONJ
ejpam-6171	473	16	the	the	DET
ejpam-6171	473	17	characteristic	characteristic	ADJ
ejpam-6171	473	18	pns	pns	PROPN
ejpam-6171	473	19	pg[α	pg[α	PROPN
ejpam-6171	473	20	+	+	PROPN
ejpam-6171	473	21	,	,	PUNCT
ejpam-6171	473	22	β−,γ+	β−,γ+	X
ejpam-6171	473	23	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	473	24	]	]	PUNCT
ejpam-6171	473	25	is	be	AUX
ejpam-6171	473	26	a	a	DET
ejpam-6171	473	27	pythagorean	pythagorean	PROPN
ejpam-6171	473	28	neutrosophic	neutrosophic	ADJ
ejpam-6171	473	29	iup	iup	NOUN
ejpam-6171	473	30	-	-	PUNCT
ejpam-6171	473	31	subalgebra	subalgebra	NOUN
ejpam-6171	473	32	of	of	ADP
ejpam-6171	473	33	x.	x.	NOUN
ejpam-6171	473	34	proof	proof	PROPN
ejpam-6171	473	35	.	.	PUNCT
ejpam-6171	474	1	assume	assume	VERB
ejpam-6171	474	2	that	that	SCONJ
ejpam-6171	474	3	g	g	PROPN
ejpam-6171	474	4	is	be	AUX
ejpam-6171	474	5	an	an	DET
ejpam-6171	474	6	iup	iup	NOUN
ejpam-6171	474	7	-	-	PUNCT
ejpam-6171	474	8	subalgebra	subalgebra	NOUN
ejpam-6171	474	9	of	of	ADP
ejpam-6171	474	10	x.	x.	NOUN
ejpam-6171	474	11	let	let	VERB
ejpam-6171	474	12	x	x	PRON
ejpam-6171	474	13	,	,	PUNCT
ejpam-6171	474	14	y	y	PROPN
ejpam-6171	474	15	∈	∈	PROPN
ejpam-6171	474	16	x.	x.	NOUN
ejpam-6171	474	17	then	then	ADV
ejpam-6171	474	18	case	case	NOUN
ejpam-6171	474	19	1	1	NUM
ejpam-6171	474	20	:	:	PUNCT
ejpam-6171	474	21	suppose	suppose	VERB
ejpam-6171	474	22	x	x	PRON
ejpam-6171	474	23	,	,	PUNCT
ejpam-6171	474	24	y	y	PROPN
ejpam-6171	474	25	∈	∈	PROPN
ejpam-6171	474	26	g.	g.	NOUN
ejpam-6171	474	27	then	then	ADV
ejpam-6171	474	28	pg	pg	VERB
ejpam-6171	474	29	t	t	PROPN
ejpam-6171	475	1	[	[	X
ejpam-6171	475	2	α	α	X
ejpam-6171	475	3	+	+	X
ejpam-6171	476	1	α−	α−	ADP
ejpam-6171	476	2	]	]	X
ejpam-6171	476	3	(	(	PUNCT
ejpam-6171	476	4	x	x	X
ejpam-6171	476	5	)	)	PUNCT
ejpam-6171	476	6	=	=	SYM
ejpam-6171	476	7	α+	α+	PUNCT
ejpam-6171	476	8	and	and	CCONJ
ejpam-6171	476	9	pg	pg	X
ejpam-6171	476	10	t	t	PROPN
ejpam-6171	477	1	[	[	X
ejpam-6171	477	2	α	α	X
ejpam-6171	477	3	+	+	X
ejpam-6171	478	1	α−	α−	ADP
ejpam-6171	478	2	]	]	X
ejpam-6171	478	3	(	(	PUNCT
ejpam-6171	478	4	y	y	NOUN
ejpam-6171	478	5	)	)	PUNCT
ejpam-6171	478	6	=	=	SYM
ejpam-6171	478	7	α+	α+	NOUN
ejpam-6171	478	8	.	.	PUNCT
ejpam-6171	479	1	since	since	SCONJ
ejpam-6171	479	2	g	g	PROPN
ejpam-6171	479	3	is	be	AUX
ejpam-6171	479	4	an	an	DET
ejpam-6171	479	5	iup	iup	NOUN
ejpam-6171	479	6	-	-	PUNCT
ejpam-6171	479	7	subalgebra	subalgebra	NOUN
ejpam-6171	479	8	of	of	ADP
ejpam-6171	479	9	x	x	PRON
ejpam-6171	479	10	,	,	PUNCT
ejpam-6171	479	11	we	we	PRON
ejpam-6171	479	12	have	have	VERB
ejpam-6171	479	13	x	x	X
ejpam-6171	479	14	?	?	PUNCT
ejpam-6171	480	1	y	y	PROPN
ejpam-6171	480	2	∈	∈	PROPN
ejpam-6171	480	3	g.	g.	PROPN
ejpam-6171	480	4	thus	thus	ADV
ejpam-6171	480	5	,	,	PUNCT
ejpam-6171	480	6	pg	pg	PROPN
ejpam-6171	480	7	t	t	PROPN
ejpam-6171	481	1	[	[	X
ejpam-6171	481	2	α	α	X
ejpam-6171	481	3	+	+	X
ejpam-6171	482	1	α−	α−	ADP
ejpam-6171	482	2	]	]	X
ejpam-6171	482	3	(	(	PUNCT
ejpam-6171	482	4	x	x	X
ejpam-6171	482	5	?	?	PUNCT
ejpam-6171	482	6	y	y	X
ejpam-6171	482	7	)	)	PUNCT
ejpam-6171	482	8	=	=	PRON
ejpam-6171	482	9	α+	α+	PUNCT
ejpam-6171	482	10	≥	≥	NOUN
ejpam-6171	482	11	min{α+	min{α+	NOUN
ejpam-6171	482	12	,	,	PUNCT
ejpam-6171	482	13	α+	α+	NOUN
ejpam-6171	482	14	}	}	PUNCT
ejpam-6171	482	15	=	=	SYM
ejpam-6171	482	16	min{pg	min{pg	ADP
ejpam-6171	482	17	t	t	X
ejpam-6171	483	1	[	[	X
ejpam-6171	483	2	α	α	X
ejpam-6171	483	3	+	+	X
ejpam-6171	484	1	α−	α−	ADP
ejpam-6171	484	2	]	]	X
ejpam-6171	484	3	(	(	PUNCT
ejpam-6171	484	4	x),pg	x),pg	PROPN
ejpam-6171	484	5	t	t	PROPN
ejpam-6171	485	1	[	[	X
ejpam-6171	485	2	α	α	X
ejpam-6171	485	3	+	+	X
ejpam-6171	486	1	α−	α−	ADP
ejpam-6171	486	2	]	]	X
ejpam-6171	486	3	(	(	PUNCT
ejpam-6171	486	4	y	y	NOUN
ejpam-6171	486	5	)	)	PUNCT
ejpam-6171	486	6	}	}	PUNCT
ejpam-6171	486	7	.	.	PUNCT
ejpam-6171	487	1	case	case	NOUN
ejpam-6171	487	2	2	2	NUM
ejpam-6171	487	3	:	:	PUNCT
ejpam-6171	487	4	suppose	suppose	VERB
ejpam-6171	487	5	x	x	X
ejpam-6171	487	6	/∈	/∈	PUNCT
ejpam-6171	487	7	g	g	NOUN
ejpam-6171	487	8	or	or	CCONJ
ejpam-6171	487	9	y	y	PROPN
ejpam-6171	487	10	/∈	/∈	PUNCT
ejpam-6171	488	1	g.	g.	PROPN
ejpam-6171	489	1	then	then	ADV
ejpam-6171	489	2	pg	pg	VERB
ejpam-6171	489	3	t	t	PROPN
ejpam-6171	490	1	[	[	X
ejpam-6171	490	2	α	α	X
ejpam-6171	490	3	+	+	X
ejpam-6171	491	1	α−	α−	ADP
ejpam-6171	491	2	]	]	X
ejpam-6171	491	3	(	(	PUNCT
ejpam-6171	491	4	x	x	X
ejpam-6171	491	5	)	)	PUNCT
ejpam-6171	491	6	=	=	SYM
ejpam-6171	491	7	α−	α−	ADP
ejpam-6171	491	8	or	or	CCONJ
ejpam-6171	491	9	pg	pg	X
ejpam-6171	491	10	t	t	PROPN
ejpam-6171	492	1	[	[	X
ejpam-6171	492	2	α	α	X
ejpam-6171	492	3	+	+	X
ejpam-6171	493	1	α−	α−	ADP
ejpam-6171	493	2	]	]	X
ejpam-6171	493	3	(	(	PUNCT
ejpam-6171	493	4	y	y	NOUN
ejpam-6171	493	5	)	)	PUNCT
ejpam-6171	493	6	=	=	SYM
ejpam-6171	493	7	α−.	α−.	NOUN
ejpam-6171	493	8	thus	thus	ADV
ejpam-6171	493	9	,	,	PUNCT
ejpam-6171	493	10	pg	pg	PROPN
ejpam-6171	493	11	t	t	PROPN
ejpam-6171	494	1	[	[	X
ejpam-6171	494	2	α	α	X
ejpam-6171	494	3	+	+	X
ejpam-6171	495	1	α−	α−	ADP
ejpam-6171	495	2	]	]	X
ejpam-6171	495	3	(	(	PUNCT
ejpam-6171	495	4	x	x	X
ejpam-6171	495	5	?	?	PUNCT
ejpam-6171	495	6	y	y	X
ejpam-6171	495	7	)	)	PUNCT
ejpam-6171	495	8	≥	≥	NOUN
ejpam-6171	495	9	α−	α−	ADP
ejpam-6171	495	10	=	=	PUNCT
ejpam-6171	495	11	min{pg	min{pg	NOUN
ejpam-6171	495	12	t	t	X
ejpam-6171	496	1	[	[	X
ejpam-6171	496	2	α	α	X
ejpam-6171	496	3	+	+	X
ejpam-6171	497	1	α−	α−	ADP
ejpam-6171	497	2	]	]	X
ejpam-6171	497	3	(	(	PUNCT
ejpam-6171	497	4	x),pg	x),pg	PROPN
ejpam-6171	497	5	t	t	PROPN
ejpam-6171	498	1	[	[	X
ejpam-6171	498	2	α	α	X
ejpam-6171	498	3	+	+	X
ejpam-6171	499	1	α−	α−	ADP
ejpam-6171	499	2	]	]	X
ejpam-6171	499	3	(	(	PUNCT
ejpam-6171	499	4	y	y	NOUN
ejpam-6171	499	5	)	)	PUNCT
ejpam-6171	499	6	}	}	PUNCT
ejpam-6171	499	7	.	.	PUNCT
ejpam-6171	500	1	k.	k.	PROPN
ejpam-6171	501	1	suayngam	suayngam	PROPN
ejpam-6171	501	2	et	et	PROPN
ejpam-6171	501	3	al	al	PROPN
ejpam-6171	501	4	.	.	PUNCT
ejpam-6171	501	5	/	/	SYM
ejpam-6171	501	6	eur	eur	PROPN
ejpam-6171	501	7	.	.	PUNCT
ejpam-6171	502	1	j.	j.	PROPN
ejpam-6171	502	2	pure	pure	PROPN
ejpam-6171	502	3	appl	appl	PROPN
ejpam-6171	502	4	.	.	PROPN
ejpam-6171	502	5	math	math	PROPN
ejpam-6171	502	6	,	,	PUNCT
ejpam-6171	502	7	18	18	NUM
ejpam-6171	502	8	(	(	PUNCT
ejpam-6171	502	9	3	3	NUM
ejpam-6171	502	10	)	)	PUNCT
ejpam-6171	502	11	(	(	PUNCT
ejpam-6171	502	12	2025	2025	NUM
ejpam-6171	502	13	)	)	PUNCT
ejpam-6171	502	14	,	,	PUNCT
ejpam-6171	502	15	6171	6171	NUM
ejpam-6171	502	16	16	16	NUM
ejpam-6171	502	17	of	of	ADP
ejpam-6171	502	18	28	28	NUM
ejpam-6171	502	19	case	case	NOUN
ejpam-6171	502	20	1	1	NUM
ejpam-6171	502	21	’	'	PUNCT
ejpam-6171	502	22	:	:	PUNCT
ejpam-6171	502	23	suppose	suppose	VERB
ejpam-6171	502	24	x	x	PRON
ejpam-6171	502	25	,	,	PUNCT
ejpam-6171	502	26	y	y	PROPN
ejpam-6171	502	27	∈	∈	PROPN
ejpam-6171	502	28	g.	g.	NOUN
ejpam-6171	503	1	then	then	ADV
ejpam-6171	503	2	pg	pg	VERB
ejpam-6171	503	3	i	i	PRON
ejpam-6171	504	1	[	[	X
ejpam-6171	504	2	β	β	X
ejpam-6171	504	3	−	−	NOUN
ejpam-6171	504	4	β+	β+	PUNCT
ejpam-6171	504	5	]	]	X
ejpam-6171	504	6	(	(	PUNCT
ejpam-6171	504	7	x	x	X
ejpam-6171	504	8	)	)	PUNCT
ejpam-6171	504	9	=	=	SYM
ejpam-6171	505	1	β−	β−	PROPN
ejpam-6171	506	1	and	and	CCONJ
ejpam-6171	506	2	pg	pg	VERB
ejpam-6171	506	3	i	i	PRON
ejpam-6171	507	1	[	[	X
ejpam-6171	507	2	β	β	X
ejpam-6171	507	3	−	−	NOUN
ejpam-6171	507	4	β+	β+	PUNCT
ejpam-6171	507	5	]	]	X
ejpam-6171	507	6	(	(	PUNCT
ejpam-6171	507	7	y	y	NOUN
ejpam-6171	507	8	)	)	PUNCT
ejpam-6171	507	9	=	=	SYM
ejpam-6171	507	10	β−.	β−.	NOUN
ejpam-6171	507	11	since	since	SCONJ
ejpam-6171	507	12	g	g	PROPN
ejpam-6171	507	13	is	be	AUX
ejpam-6171	507	14	an	an	DET
ejpam-6171	507	15	iup	iup	NOUN
ejpam-6171	507	16	-	-	PUNCT
ejpam-6171	507	17	subalgebra	subalgebra	NOUN
ejpam-6171	507	18	of	of	ADP
ejpam-6171	507	19	x	x	PRON
ejpam-6171	507	20	,	,	PUNCT
ejpam-6171	507	21	we	we	PRON
ejpam-6171	507	22	have	have	VERB
ejpam-6171	507	23	x	x	X
ejpam-6171	507	24	?	?	PUNCT
ejpam-6171	508	1	y	y	PROPN
ejpam-6171	508	2	∈	∈	PROPN
ejpam-6171	508	3	g.	g.	PROPN
ejpam-6171	508	4	thus	thus	ADV
ejpam-6171	508	5	,	,	PUNCT
ejpam-6171	508	6	pg	pg	VERB
ejpam-6171	508	7	i	i	PRON
ejpam-6171	509	1	[	[	X
ejpam-6171	509	2	β	β	X
ejpam-6171	509	3	−	−	NOUN
ejpam-6171	509	4	β+	β+	PUNCT
ejpam-6171	509	5	]	]	X
ejpam-6171	509	6	(	(	PUNCT
ejpam-6171	509	7	x	x	X
ejpam-6171	509	8	?	?	PUNCT
ejpam-6171	510	1	y	y	X
ejpam-6171	510	2	)	)	PUNCT
ejpam-6171	511	1	=	=	PUNCT
ejpam-6171	511	2	β−	β−	PUNCT
ejpam-6171	511	3	≤	≤	NUM
ejpam-6171	511	4	β−	β−	PUNCT
ejpam-6171	512	1	=	=	PRON
ejpam-6171	512	2	max{pg	max{pg	NOUN
ejpam-6171	513	1	i	i	PRON
ejpam-6171	513	2	[	[	X
ejpam-6171	513	3	β	β	X
ejpam-6171	513	4	−	−	NOUN
ejpam-6171	513	5	β+	β+	PUNCT
ejpam-6171	513	6	]	]	X
ejpam-6171	513	7	(	(	PUNCT
ejpam-6171	513	8	x),pg	x),pg	NOUN
ejpam-6171	513	9	i	i	PRON
ejpam-6171	514	1	[	[	X
ejpam-6171	514	2	β	β	X
ejpam-6171	514	3	−	−	NOUN
ejpam-6171	514	4	β+	β+	PUNCT
ejpam-6171	514	5	]	]	X
ejpam-6171	514	6	(	(	PUNCT
ejpam-6171	514	7	y	y	NOUN
ejpam-6171	514	8	)	)	PUNCT
ejpam-6171	514	9	}	}	PUNCT
ejpam-6171	514	10	.	.	PUNCT
ejpam-6171	515	1	case	case	NOUN
ejpam-6171	515	2	2	2	NUM
ejpam-6171	515	3	’	'	PUNCT
ejpam-6171	515	4	:	:	PUNCT
ejpam-6171	515	5	suppose	suppose	VERB
ejpam-6171	515	6	x	x	X
ejpam-6171	515	7	/∈	/∈	PUNCT
ejpam-6171	515	8	g	g	NOUN
ejpam-6171	515	9	or	or	CCONJ
ejpam-6171	515	10	y	y	PROPN
ejpam-6171	515	11	/∈	/∈	PUNCT
ejpam-6171	516	1	g.	g.	PROPN
ejpam-6171	517	1	then	then	ADV
ejpam-6171	517	2	pg	pg	VERB
ejpam-6171	517	3	i	i	PRON
ejpam-6171	518	1	[	[	X
ejpam-6171	518	2	β	β	X
ejpam-6171	518	3	−	−	NOUN
ejpam-6171	518	4	β+	β+	PUNCT
ejpam-6171	518	5	]	]	X
ejpam-6171	518	6	(	(	PUNCT
ejpam-6171	518	7	x	x	X
ejpam-6171	518	8	)	)	PUNCT
ejpam-6171	518	9	=	=	SYM
ejpam-6171	518	10	β+	β+	PUNCT
ejpam-6171	518	11	or	or	CCONJ
ejpam-6171	518	12	pg	pg	VERB
ejpam-6171	518	13	i	i	PRON
ejpam-6171	519	1	[	[	X
ejpam-6171	519	2	β	β	X
ejpam-6171	519	3	−	−	NOUN
ejpam-6171	519	4	β+	β+	PUNCT
ejpam-6171	519	5	]	]	X
ejpam-6171	519	6	(	(	PUNCT
ejpam-6171	519	7	y	y	NOUN
ejpam-6171	519	8	)	)	PUNCT
ejpam-6171	519	9	=	=	PUNCT
ejpam-6171	519	10	β+	β+	X
ejpam-6171	519	11	.	.	PUNCT
ejpam-6171	520	1	thus	thus	ADV
ejpam-6171	520	2	,	,	PUNCT
ejpam-6171	520	3	pg	pg	VERB
ejpam-6171	520	4	i	i	PRON
ejpam-6171	521	1	[	[	X
ejpam-6171	521	2	β	β	X
ejpam-6171	521	3	−	−	NOUN
ejpam-6171	521	4	β+	β+	PUNCT
ejpam-6171	521	5	]	]	X
ejpam-6171	521	6	(	(	PUNCT
ejpam-6171	521	7	x	x	X
ejpam-6171	521	8	?	?	PUNCT
ejpam-6171	521	9	y	y	X
ejpam-6171	521	10	)	)	PUNCT
ejpam-6171	521	11	≤	≤	NOUN
ejpam-6171	521	12	β+	β+	PUNCT
ejpam-6171	522	1	=	=	PRON
ejpam-6171	522	2	max{pg	max{pg	NOUN
ejpam-6171	522	3	i	i	PRON
ejpam-6171	523	1	[	[	X
ejpam-6171	523	2	β	β	X
ejpam-6171	523	3	−	−	NOUN
ejpam-6171	523	4	β+	β+	PUNCT
ejpam-6171	523	5	]	]	X
ejpam-6171	523	6	(	(	PUNCT
ejpam-6171	523	7	x),pg	x),pg	NOUN
ejpam-6171	524	1	i	i	PRON
ejpam-6171	525	1	[	[	X
ejpam-6171	525	2	β	β	X
ejpam-6171	525	3	−	−	NOUN
ejpam-6171	525	4	β+	β+	PUNCT
ejpam-6171	525	5	]	]	X
ejpam-6171	525	6	(	(	PUNCT
ejpam-6171	525	7	y	y	NOUN
ejpam-6171	525	8	)	)	PUNCT
ejpam-6171	525	9	}	}	PUNCT
ejpam-6171	525	10	.	.	PUNCT
ejpam-6171	526	1	case	case	NOUN
ejpam-6171	526	2	1	1	NUM
ejpam-6171	526	3	”	"	PUNCT
ejpam-6171	526	4	:	:	PUNCT
ejpam-6171	526	5	suppose	suppose	VERB
ejpam-6171	526	6	x	x	PRON
ejpam-6171	526	7	,	,	PUNCT
ejpam-6171	526	8	y	y	PROPN
ejpam-6171	526	9	∈	∈	PROPN
ejpam-6171	526	10	g.	g.	NOUN
ejpam-6171	527	1	then	then	ADV
ejpam-6171	527	2	pg	pg	VERB
ejpam-6171	527	3	f	f	PROPN
ejpam-6171	528	1	[	[	X
ejpam-6171	528	2	γ	γ	X
ejpam-6171	528	3	+	+	X
ejpam-6171	528	4	γ−	γ−	PROPN
ejpam-6171	528	5	]	]	PUNCT
ejpam-6171	528	6	(	(	PUNCT
ejpam-6171	528	7	x	x	X
ejpam-6171	528	8	)	)	PUNCT
ejpam-6171	528	9	=	=	SYM
ejpam-6171	528	10	γ+	γ+	PUNCT
ejpam-6171	528	11	and	and	CCONJ
ejpam-6171	528	12	pg	pg	X
ejpam-6171	528	13	f	f	PROPN
ejpam-6171	529	1	[	[	X
ejpam-6171	529	2	γ	γ	X
ejpam-6171	529	3	+	+	X
ejpam-6171	529	4	γ−	γ−	PROPN
ejpam-6171	529	5	]	]	PUNCT
ejpam-6171	529	6	(	(	PUNCT
ejpam-6171	529	7	y	y	NOUN
ejpam-6171	529	8	)	)	PUNCT
ejpam-6171	529	9	=	=	SYM
ejpam-6171	530	1	γ+	γ+	PROPN
ejpam-6171	530	2	.	.	PUNCT
ejpam-6171	531	1	since	since	SCONJ
ejpam-6171	531	2	g	g	PROPN
ejpam-6171	531	3	is	be	AUX
ejpam-6171	531	4	an	an	DET
ejpam-6171	531	5	iup	iup	NOUN
ejpam-6171	531	6	-	-	PUNCT
ejpam-6171	531	7	subalgebra	subalgebra	NOUN
ejpam-6171	531	8	of	of	ADP
ejpam-6171	531	9	x	x	PRON
ejpam-6171	531	10	,	,	PUNCT
ejpam-6171	531	11	we	we	PRON
ejpam-6171	531	12	have	have	VERB
ejpam-6171	531	13	x	x	X
ejpam-6171	531	14	?	?	PUNCT
ejpam-6171	532	1	y	y	PROPN
ejpam-6171	532	2	∈	∈	PROPN
ejpam-6171	532	3	g.	g.	PROPN
ejpam-6171	532	4	thus	thus	ADV
ejpam-6171	532	5	,	,	PUNCT
ejpam-6171	532	6	pg	pg	PROPN
ejpam-6171	532	7	f	f	PROPN
ejpam-6171	533	1	[	[	X
ejpam-6171	533	2	γ	γ	X
ejpam-6171	533	3	+	+	X
ejpam-6171	533	4	γ−	γ−	PROPN
ejpam-6171	533	5	]	]	PUNCT
ejpam-6171	533	6	(	(	PUNCT
ejpam-6171	533	7	x	x	X
ejpam-6171	533	8	?	?	PUNCT
ejpam-6171	534	1	y	y	X
ejpam-6171	534	2	)	)	PUNCT
ejpam-6171	534	3	=	=	PRON
ejpam-6171	534	4	γ+	γ+	PUNCT
ejpam-6171	534	5	≥	≥	NOUN
ejpam-6171	534	6	min{γ+	min{γ+	VERB
ejpam-6171	534	7	,	,	PUNCT
ejpam-6171	534	8	γ+	γ+	X
ejpam-6171	534	9	}	}	PUNCT
ejpam-6171	534	10	=	=	PUNCT
ejpam-6171	535	1	min{pg	min{pg	ADP
ejpam-6171	535	2	f	f	X
ejpam-6171	536	1	[	[	X
ejpam-6171	536	2	γ	γ	X
ejpam-6171	536	3	+	+	X
ejpam-6171	536	4	γ−	γ−	PROPN
ejpam-6171	536	5	]	]	PUNCT
ejpam-6171	536	6	(	(	PUNCT
ejpam-6171	536	7	x),pg	x),pg	PROPN
ejpam-6171	536	8	f	f	PROPN
ejpam-6171	537	1	[	[	X
ejpam-6171	537	2	γ	γ	X
ejpam-6171	537	3	+	+	X
ejpam-6171	537	4	γ−	γ−	PROPN
ejpam-6171	537	5	]	]	PUNCT
ejpam-6171	537	6	(	(	PUNCT
ejpam-6171	537	7	y	y	NOUN
ejpam-6171	537	8	)	)	PUNCT
ejpam-6171	537	9	}	}	PUNCT
ejpam-6171	537	10	.	.	PUNCT
ejpam-6171	538	1	case	case	NOUN
ejpam-6171	538	2	2	2	NUM
ejpam-6171	538	3	”	"	PUNCT
ejpam-6171	538	4	:	:	PUNCT
ejpam-6171	538	5	suppose	suppose	VERB
ejpam-6171	538	6	x	x	X
ejpam-6171	538	7	/∈	/∈	PUNCT
ejpam-6171	538	8	g	g	NOUN
ejpam-6171	538	9	or	or	CCONJ
ejpam-6171	538	10	y	y	PROPN
ejpam-6171	538	11	/∈	/∈	PUNCT
ejpam-6171	539	1	g.	g.	PROPN
ejpam-6171	540	1	then	then	ADV
ejpam-6171	540	2	pg	pg	VERB
ejpam-6171	540	3	f	f	PROPN
ejpam-6171	541	1	[	[	X
ejpam-6171	541	2	γ	γ	X
ejpam-6171	541	3	+	+	X
ejpam-6171	541	4	γ−	γ−	PROPN
ejpam-6171	541	5	]	]	PUNCT
ejpam-6171	541	6	(	(	PUNCT
ejpam-6171	541	7	x	x	X
ejpam-6171	541	8	)	)	PUNCT
ejpam-6171	541	9	=	=	SYM
ejpam-6171	541	10	γ−	γ−	PROPN
ejpam-6171	541	11	or	or	CCONJ
ejpam-6171	541	12	pg	pg	NOUN
ejpam-6171	541	13	f	f	PROPN
ejpam-6171	542	1	[	[	X
ejpam-6171	542	2	γ	γ	X
ejpam-6171	542	3	+	+	X
ejpam-6171	542	4	γ−	γ−	PROPN
ejpam-6171	542	5	]	]	PUNCT
ejpam-6171	542	6	(	(	PUNCT
ejpam-6171	542	7	y	y	NOUN
ejpam-6171	542	8	)	)	PUNCT
ejpam-6171	542	9	=	=	VERB
ejpam-6171	542	10	γ−.	γ−.	NOUN
ejpam-6171	542	11	thus	thus	ADV
ejpam-6171	542	12	,	,	PUNCT
ejpam-6171	542	13	pg	pg	PROPN
ejpam-6171	542	14	f	f	PROPN
ejpam-6171	543	1	[	[	X
ejpam-6171	543	2	γ	γ	X
ejpam-6171	543	3	+	+	X
ejpam-6171	543	4	γ−	γ−	PROPN
ejpam-6171	543	5	]	]	PUNCT
ejpam-6171	543	6	(	(	PUNCT
ejpam-6171	543	7	x	x	X
ejpam-6171	543	8	?	?	PUNCT
ejpam-6171	544	1	y	y	X
ejpam-6171	544	2	)	)	PUNCT
ejpam-6171	544	3	≥	≥	NOUN
ejpam-6171	544	4	γ−	γ−	NUM
ejpam-6171	544	5	=	=	NOUN
ejpam-6171	544	6	min{pg	min{pg	X
ejpam-6171	545	1	f	f	X
ejpam-6171	546	1	[	[	X
ejpam-6171	546	2	γ	γ	X
ejpam-6171	546	3	+	+	X
ejpam-6171	546	4	γ−	γ−	PROPN
ejpam-6171	546	5	]	]	PUNCT
ejpam-6171	546	6	(	(	PUNCT
ejpam-6171	546	7	x),pg	x),pg	PROPN
ejpam-6171	546	8	f	f	PROPN
ejpam-6171	547	1	[	[	X
ejpam-6171	547	2	γ	γ	X
ejpam-6171	547	3	+	+	X
ejpam-6171	547	4	γ−	γ−	PROPN
ejpam-6171	547	5	]	]	PUNCT
ejpam-6171	547	6	(	(	PUNCT
ejpam-6171	547	7	y	y	NOUN
ejpam-6171	547	8	)	)	PUNCT
ejpam-6171	547	9	}	}	PUNCT
ejpam-6171	547	10	.	.	PUNCT
ejpam-6171	548	1	hence	hence	ADV
ejpam-6171	548	2	,	,	PUNCT
ejpam-6171	548	3	the	the	DET
ejpam-6171	548	4	characteristic	characteristic	ADJ
ejpam-6171	548	5	pns	pns	NOUN
ejpam-6171	548	6	pg[α	pg[α	PROPN
ejpam-6171	548	7	+	+	PROPN
ejpam-6171	548	8	,	,	PUNCT
ejpam-6171	548	9	β−,γ+	β−,γ+	X
ejpam-6171	548	10	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	548	11	]	]	PUNCT
ejpam-6171	548	12	is	be	AUX
ejpam-6171	548	13	a	a	DET
ejpam-6171	548	14	pythagorean	pythagorean	PROPN
ejpam-6171	548	15	neutrosophic	neutrosophic	ADJ
ejpam-6171	548	16	iup	iup	NOUN
ejpam-6171	548	17	-	-	PUNCT
ejpam-6171	548	18	subalgebra	subalgebra	NOUN
ejpam-6171	548	19	of	of	ADP
ejpam-6171	548	20	x.	x.	NOUN
ejpam-6171	548	21	conversely	conversely	ADV
ejpam-6171	548	22	,	,	PUNCT
ejpam-6171	548	23	assume	assume	VERB
ejpam-6171	548	24	that	that	SCONJ
ejpam-6171	548	25	the	the	DET
ejpam-6171	548	26	characteristic	characteristic	ADJ
ejpam-6171	548	27	pns	pns	PROPN
ejpam-6171	548	28	pg[α	pg[α	PROPN
ejpam-6171	548	29	+	+	PROPN
ejpam-6171	548	30	,	,	PUNCT
ejpam-6171	548	31	β−,γ+	β−,γ+	X
ejpam-6171	548	32	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	548	33	]	]	PUNCT
ejpam-6171	548	34	is	be	AUX
ejpam-6171	548	35	a	a	DET
ejpam-6171	548	36	pythagorean	pythagorean	PROPN
ejpam-6171	548	37	neutrosophic	neutrosophic	ADJ
ejpam-6171	548	38	iup	iup	NOUN
ejpam-6171	548	39	-	-	PUNCT
ejpam-6171	548	40	subalgebra	subalgebra	NOUN
ejpam-6171	548	41	of	of	ADP
ejpam-6171	548	42	x.	x.	NOUN
ejpam-6171	548	43	let	let	VERB
ejpam-6171	548	44	x	x	PRON
ejpam-6171	548	45	,	,	PUNCT
ejpam-6171	548	46	y	y	PROPN
ejpam-6171	548	47	∈	∈	PROPN
ejpam-6171	548	48	g.	g.	NOUN
ejpam-6171	548	49	then	then	ADV
ejpam-6171	548	50	pg	pg	VERB
ejpam-6171	548	51	t	t	PROPN
ejpam-6171	549	1	[	[	X
ejpam-6171	549	2	α	α	X
ejpam-6171	549	3	+	+	X
ejpam-6171	550	1	α−	α−	ADP
ejpam-6171	550	2	]	]	X
ejpam-6171	550	3	(	(	PUNCT
ejpam-6171	550	4	x	x	X
ejpam-6171	550	5	)	)	PUNCT
ejpam-6171	550	6	=	=	SYM
ejpam-6171	550	7	α+	α+	PUNCT
ejpam-6171	550	8	and	and	CCONJ
ejpam-6171	550	9	pg	pg	X
ejpam-6171	550	10	t	t	PROPN
ejpam-6171	551	1	[	[	X
ejpam-6171	551	2	α	α	X
ejpam-6171	551	3	+	+	X
ejpam-6171	552	1	α−	α−	ADP
ejpam-6171	552	2	]	]	X
ejpam-6171	552	3	(	(	PUNCT
ejpam-6171	552	4	y	y	NOUN
ejpam-6171	552	5	)	)	PUNCT
ejpam-6171	552	6	=	=	SYM
ejpam-6171	552	7	α+	α+	NOUN
ejpam-6171	552	8	.	.	PUNCT
ejpam-6171	552	9	by	by	ADP
ejpam-6171	552	10	(	(	PUNCT
ejpam-6171	552	11	3.2	3.2	NUM
ejpam-6171	552	12	)	)	PUNCT
ejpam-6171	552	13	,	,	PUNCT
ejpam-6171	552	14	we	we	PRON
ejpam-6171	552	15	have	have	VERB
ejpam-6171	552	16	pg	pg	PROPN
ejpam-6171	552	17	t	t	PROPN
ejpam-6171	553	1	[	[	X
ejpam-6171	553	2	α	α	X
ejpam-6171	553	3	+	+	X
ejpam-6171	554	1	α−	α−	ADP
ejpam-6171	554	2	]	]	X
ejpam-6171	554	3	(	(	PUNCT
ejpam-6171	554	4	x	x	X
ejpam-6171	554	5	?	?	PUNCT
ejpam-6171	554	6	y	y	X
ejpam-6171	554	7	)	)	PUNCT
ejpam-6171	554	8	≥	≥	NOUN
ejpam-6171	554	9	min{pg	min{pg	X
ejpam-6171	554	10	t	t	X
ejpam-6171	555	1	[	[	X
ejpam-6171	555	2	α	α	X
ejpam-6171	555	3	+	+	X
ejpam-6171	556	1	α−	α−	ADP
ejpam-6171	556	2	]	]	X
ejpam-6171	556	3	(	(	PUNCT
ejpam-6171	556	4	x),pg	x),pg	PROPN
ejpam-6171	556	5	t	t	PROPN
ejpam-6171	557	1	[	[	X
ejpam-6171	557	2	α	α	X
ejpam-6171	557	3	+	+	X
ejpam-6171	558	1	α−	α−	ADP
ejpam-6171	558	2	]	]	X
ejpam-6171	558	3	(	(	PUNCT
ejpam-6171	558	4	y	y	NOUN
ejpam-6171	558	5	)	)	PUNCT
ejpam-6171	558	6	}	}	PUNCT
ejpam-6171	558	7	=	=	SYM
ejpam-6171	558	8	min{α+	min{α+	PROPN
ejpam-6171	558	9	,	,	PUNCT
ejpam-6171	558	10	α+	α+	NOUN
ejpam-6171	558	11	}	}	PUNCT
ejpam-6171	558	12	=	=	SYM
ejpam-6171	558	13	α+	α+	NOUN
ejpam-6171	558	14	.	.	PUNCT
ejpam-6171	559	1	thus	thus	ADV
ejpam-6171	559	2	pg	pg	X
ejpam-6171	559	3	t	t	PROPN
ejpam-6171	560	1	[	[	X
ejpam-6171	560	2	α	α	X
ejpam-6171	560	3	+	+	X
ejpam-6171	561	1	α−	α−	ADP
ejpam-6171	561	2	]	]	X
ejpam-6171	561	3	(	(	PUNCT
ejpam-6171	561	4	x	x	X
ejpam-6171	561	5	?	?	PUNCT
ejpam-6171	561	6	y	y	X
ejpam-6171	561	7	)	)	PUNCT
ejpam-6171	561	8	=	=	SYM
ejpam-6171	561	9	α+	α+	NOUN
ejpam-6171	561	10	,	,	PUNCT
ejpam-6171	561	11	that	that	ADV
ejpam-6171	561	12	is	be	AUX
ejpam-6171	561	13	,	,	PUNCT
ejpam-6171	561	14	x	x	PUNCT
ejpam-6171	561	15	?	?	PUNCT
ejpam-6171	562	1	y	y	PROPN
ejpam-6171	562	2	∈	∈	PROPN
ejpam-6171	562	3	g.	g.	NOUN
ejpam-6171	562	4	hence	hence	ADV
ejpam-6171	562	5	,	,	PUNCT
ejpam-6171	562	6	g	g	PROPN
ejpam-6171	562	7	is	be	AUX
ejpam-6171	562	8	an	an	DET
ejpam-6171	562	9	iup	iup	NOUN
ejpam-6171	562	10	-	-	PUNCT
ejpam-6171	562	11	subalgebra	subalgebra	NOUN
ejpam-6171	562	12	of	of	ADP
ejpam-6171	562	13	x.	x.	NOUN
ejpam-6171	562	14	theorem	theorem	VERB
ejpam-6171	562	15	10	10	NUM
ejpam-6171	562	16	.	.	PUNCT
ejpam-6171	563	1	a	a	DET
ejpam-6171	563	2	nonempty	nonempty	NOUN
ejpam-6171	563	3	subset	subset	VERB
ejpam-6171	563	4	g	g	NOUN
ejpam-6171	563	5	is	be	AUX
ejpam-6171	563	6	an	an	DET
ejpam-6171	563	7	iup	iup	NOUN
ejpam-6171	563	8	-	-	PUNCT
ejpam-6171	563	9	ideal	ideal	NOUN
ejpam-6171	563	10	of	of	ADP
ejpam-6171	563	11	x	x	SYM
ejpam-6171	563	12	if	if	SCONJ
ejpam-6171	563	13	and	and	CCONJ
ejpam-6171	563	14	only	only	ADV
ejpam-6171	563	15	if	if	SCONJ
ejpam-6171	563	16	the	the	DET
ejpam-6171	563	17	characteristic	characteristic	ADJ
ejpam-6171	563	18	pns	pns	PROPN
ejpam-6171	563	19	pg[α	pg[α	PROPN
ejpam-6171	563	20	+	+	PROPN
ejpam-6171	563	21	,	,	PUNCT
ejpam-6171	563	22	β−,γ+	β−,γ+	X
ejpam-6171	563	23	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	563	24	]	]	PUNCT
ejpam-6171	563	25	is	be	AUX
ejpam-6171	563	26	a	a	DET
ejpam-6171	563	27	pythagorean	pythagorean	PROPN
ejpam-6171	563	28	neutrosophic	neutrosophic	ADJ
ejpam-6171	563	29	iup	iup	PROPN
ejpam-6171	563	30	-	-	PUNCT
ejpam-6171	563	31	ideal	ideal	NOUN
ejpam-6171	563	32	of	of	ADP
ejpam-6171	563	33	x.	x.	NOUN
ejpam-6171	563	34	proof	proof	PROPN
ejpam-6171	563	35	.	.	PUNCT
ejpam-6171	564	1	assume	assume	VERB
ejpam-6171	564	2	that	that	SCONJ
ejpam-6171	564	3	g	g	PROPN
ejpam-6171	564	4	is	be	AUX
ejpam-6171	564	5	an	an	DET
ejpam-6171	564	6	iup	iup	NOUN
ejpam-6171	564	7	-	-	PUNCT
ejpam-6171	564	8	ideal	ideal	NOUN
ejpam-6171	564	9	of	of	ADP
ejpam-6171	564	10	x.	x.	NOUN
ejpam-6171	564	11	since	since	SCONJ
ejpam-6171	564	12	0	0	NUM
ejpam-6171	564	13	∈	∈	PROPN
ejpam-6171	564	14	g	g	NOUN
ejpam-6171	564	15	,	,	PUNCT
ejpam-6171	564	16	it	it	PRON
ejpam-6171	564	17	follows	follow	VERB
ejpam-6171	564	18	from	from	ADP
ejpam-6171	564	19	lemma	lemma	PROPN
ejpam-6171	564	20	2	2	NUM
ejpam-6171	565	1	that	that	PRON
ejpam-6171	565	2	pg	pg	VERB
ejpam-6171	566	1	t	t	PROPN
ejpam-6171	567	1	[	[	X
ejpam-6171	567	2	α	α	X
ejpam-6171	567	3	+	+	X
ejpam-6171	568	1	α−	α−	ADP
ejpam-6171	568	2	]	]	PUNCT
ejpam-6171	568	3	,	,	PUNCT
ejpam-6171	568	4	pg	pg	VERB
ejpam-6171	568	5	i	i	PRON
ejpam-6171	569	1	[	[	X
ejpam-6171	569	2	β	β	X
ejpam-6171	569	3	−	−	NOUN
ejpam-6171	569	4	β+	β+	PUNCT
ejpam-6171	569	5	]	]	X
ejpam-6171	569	6	,	,	PUNCT
ejpam-6171	569	7	and	and	CCONJ
ejpam-6171	569	8	pg	pg	X
ejpam-6171	569	9	f	f	PROPN
ejpam-6171	570	1	[	[	X
ejpam-6171	570	2	γ	γ	X
ejpam-6171	570	3	+	+	CCONJ
ejpam-6171	570	4	γ−	γ−	PROPN
ejpam-6171	570	5	]	]	PUNCT
ejpam-6171	570	6	satisfy	satisfy	NOUN
ejpam-6171	570	7	(	(	PUNCT
ejpam-6171	570	8	3.5	3.5	NUM
ejpam-6171	570	9	)	)	PUNCT
ejpam-6171	570	10	,	,	PUNCT
ejpam-6171	570	11	(	(	PUNCT
ejpam-6171	570	12	3.6	3.6	NUM
ejpam-6171	570	13	)	)	PUNCT
ejpam-6171	570	14	,	,	PUNCT
ejpam-6171	570	15	and	and	CCONJ
ejpam-6171	570	16	(	(	PUNCT
ejpam-6171	570	17	3.7	3.7	NUM
ejpam-6171	570	18	)	)	PUNCT
ejpam-6171	570	19	,	,	PUNCT
ejpam-6171	570	20	respectively	respectively	ADV
ejpam-6171	570	21	.	.	PUNCT
ejpam-6171	571	1	next	next	ADV
ejpam-6171	571	2	,	,	PUNCT
ejpam-6171	571	3	let	let	VERB
ejpam-6171	571	4	x	x	PRON
ejpam-6171	571	5	,	,	PUNCT
ejpam-6171	571	6	y	y	PROPN
ejpam-6171	571	7	,	,	PUNCT
ejpam-6171	571	8	z	z	PROPN
ejpam-6171	571	9	∈	∈	NOUN
ejpam-6171	571	10	x.	x.	NOUN
ejpam-6171	571	11	case	case	NOUN
ejpam-6171	571	12	1	1	NUM
ejpam-6171	571	13	:	:	PUNCT
ejpam-6171	571	14	suppose	suppose	VERB
ejpam-6171	571	15	x?(y?z	x?(y?z	X
ejpam-6171	571	16	)	)	PUNCT
ejpam-6171	571	17	∈	∈	PROPN
ejpam-6171	571	18	g	g	PROPN
ejpam-6171	571	19	and	and	CCONJ
ejpam-6171	571	20	y	y	PROPN
ejpam-6171	571	21	∈	∈	PROPN
ejpam-6171	571	22	g.	g.	NOUN
ejpam-6171	571	23	since	since	SCONJ
ejpam-6171	571	24	g	g	PROPN
ejpam-6171	571	25	is	be	AUX
ejpam-6171	571	26	an	an	DET
ejpam-6171	571	27	iup	iup	NOUN
ejpam-6171	571	28	-	-	PUNCT
ejpam-6171	571	29	ideal	ideal	NOUN
ejpam-6171	571	30	of	of	ADP
ejpam-6171	571	31	x	x	SYM
ejpam-6171	571	32	,	,	PUNCT
ejpam-6171	571	33	we	we	PRON
ejpam-6171	571	34	have	have	VERB
ejpam-6171	571	35	x?z	x?z	PROPN
ejpam-6171	571	36	∈	∈	PROPN
ejpam-6171	571	37	g.	g.	PROPN
ejpam-6171	571	38	thus	thus	ADV
ejpam-6171	571	39	,	,	PUNCT
ejpam-6171	571	40	pg	pg	PROPN
ejpam-6171	571	41	t	t	PROPN
ejpam-6171	572	1	[	[	X
ejpam-6171	572	2	α	α	X
ejpam-6171	572	3	+	+	X
ejpam-6171	573	1	α−	α−	ADP
ejpam-6171	573	2	]	]	X
ejpam-6171	573	3	(	(	PUNCT
ejpam-6171	573	4	x?z	x?z	PROPN
ejpam-6171	573	5	)	)	PUNCT
ejpam-6171	573	6	=	=	SYM
ejpam-6171	573	7	α+	α+	PUNCT
ejpam-6171	573	8	≥	≥	X
ejpam-6171	573	9	α+	α+	X
ejpam-6171	573	10	=	=	SYM
ejpam-6171	573	11	min{α+	min{α+	PROPN
ejpam-6171	573	12	,	,	PUNCT
ejpam-6171	573	13	α+	α+	NOUN
ejpam-6171	573	14	}	}	PUNCT
ejpam-6171	573	15	=	=	SYM
ejpam-6171	573	16	min{pg	min{pg	ADP
ejpam-6171	573	17	t	t	X
ejpam-6171	574	1	[	[	X
ejpam-6171	574	2	α	α	X
ejpam-6171	574	3	+	+	X
ejpam-6171	575	1	α−	α−	ADP
ejpam-6171	575	2	]	]	X
ejpam-6171	575	3	(	(	PUNCT
ejpam-6171	575	4	x?(y?z)),pg	x?(y?z)),pg	PROPN
ejpam-6171	575	5	t	t	PROPN
ejpam-6171	576	1	[	[	X
ejpam-6171	576	2	α	α	X
ejpam-6171	576	3	+	+	X
ejpam-6171	577	1	α−	α−	ADP
ejpam-6171	577	2	]	]	X
ejpam-6171	577	3	(	(	PUNCT
ejpam-6171	577	4	y	y	NOUN
ejpam-6171	577	5	)	)	PUNCT
ejpam-6171	577	6	}	}	PUNCT
ejpam-6171	577	7	.	.	PUNCT
ejpam-6171	578	1	case	case	NOUN
ejpam-6171	578	2	2	2	NUM
ejpam-6171	578	3	:	:	PUNCT
ejpam-6171	578	4	suppose	suppose	VERB
ejpam-6171	578	5	x	x	X
ejpam-6171	578	6	?	?	PUNCT
ejpam-6171	579	1	(	(	PUNCT
ejpam-6171	579	2	y	y	NOUN
ejpam-6171	579	3	?	?	PUNCT
ejpam-6171	580	1	z	z	X
ejpam-6171	580	2	)	)	PUNCT
ejpam-6171	580	3	/∈	/∈	PUNCT
ejpam-6171	581	1	g	g	NOUN
ejpam-6171	581	2	or	or	CCONJ
ejpam-6171	581	3	y	y	PROPN
ejpam-6171	581	4	/∈	/∈	PUNCT
ejpam-6171	582	1	g.	g.	PROPN
ejpam-6171	583	1	then	then	ADV
ejpam-6171	583	2	pg	pg	VERB
ejpam-6171	583	3	t	t	PROPN
ejpam-6171	584	1	[	[	X
ejpam-6171	584	2	α	α	X
ejpam-6171	584	3	+	+	X
ejpam-6171	585	1	α−	α−	ADP
ejpam-6171	585	2	]	]	X
ejpam-6171	585	3	(	(	PUNCT
ejpam-6171	585	4	x	x	X
ejpam-6171	585	5	?	?	PUNCT
ejpam-6171	586	1	(	(	PUNCT
ejpam-6171	586	2	y	y	NOUN
ejpam-6171	586	3	?	?	PUNCT
ejpam-6171	587	1	z	z	X
ejpam-6171	587	2	)	)	PUNCT
ejpam-6171	587	3	)	)	PUNCT
ejpam-6171	588	1	=	=	PUNCT
ejpam-6171	589	1	α−	α−	ADP
ejpam-6171	589	2	or	or	CCONJ
ejpam-6171	589	3	pg	pg	X
ejpam-6171	589	4	t	t	PROPN
ejpam-6171	590	1	[	[	X
ejpam-6171	590	2	α	α	X
ejpam-6171	590	3	+	+	X
ejpam-6171	591	1	α−	α−	ADP
ejpam-6171	591	2	]	]	X
ejpam-6171	591	3	(	(	PUNCT
ejpam-6171	591	4	y	y	NOUN
ejpam-6171	591	5	)	)	PUNCT
ejpam-6171	591	6	=	=	SYM
ejpam-6171	591	7	α−.	α−.	NOUN
ejpam-6171	591	8	thus	thus	ADV
ejpam-6171	591	9	,	,	PUNCT
ejpam-6171	591	10	pg	pg	PROPN
ejpam-6171	591	11	t	t	PROPN
ejpam-6171	592	1	[	[	X
ejpam-6171	592	2	α	α	X
ejpam-6171	592	3	+	+	X
ejpam-6171	593	1	α−	α−	ADP
ejpam-6171	593	2	]	]	X
ejpam-6171	593	3	(	(	PUNCT
ejpam-6171	593	4	x	x	X
ejpam-6171	593	5	?	?	PUNCT
ejpam-6171	593	6	z	z	X
ejpam-6171	593	7	)	)	PUNCT
ejpam-6171	593	8	≥	≥	NOUN
ejpam-6171	593	9	α−	α−	ADP
ejpam-6171	593	10	=	=	PUNCT
ejpam-6171	593	11	min{pg	min{pg	NOUN
ejpam-6171	593	12	t	t	X
ejpam-6171	594	1	[	[	X
ejpam-6171	594	2	α	α	X
ejpam-6171	594	3	+	+	X
ejpam-6171	595	1	α−	α−	ADP
ejpam-6171	595	2	]	]	X
ejpam-6171	595	3	(	(	PUNCT
ejpam-6171	595	4	x	x	X
ejpam-6171	595	5	?	?	PUNCT
ejpam-6171	596	1	(	(	PUNCT
ejpam-6171	596	2	y	y	NOUN
ejpam-6171	596	3	?	?	PUNCT
ejpam-6171	597	1	z)),pg	z)),pg	NOUN
ejpam-6171	597	2	t	t	NOUN
ejpam-6171	598	1	[	[	X
ejpam-6171	598	2	α	α	X
ejpam-6171	598	3	+	+	X
ejpam-6171	599	1	α−	α−	ADP
ejpam-6171	599	2	]	]	X
ejpam-6171	599	3	(	(	PUNCT
ejpam-6171	599	4	y	y	NOUN
ejpam-6171	599	5	)	)	PUNCT
ejpam-6171	599	6	}	}	PUNCT
ejpam-6171	599	7	.	.	PUNCT
ejpam-6171	600	1	case	case	NOUN
ejpam-6171	600	2	1	1	NUM
ejpam-6171	600	3	’	'	PUNCT
ejpam-6171	600	4	:	:	PUNCT
ejpam-6171	600	5	suppose	suppose	VERB
ejpam-6171	600	6	x?(y?z	x?(y?z	X
ejpam-6171	600	7	)	)	PUNCT
ejpam-6171	600	8	∈	∈	PROPN
ejpam-6171	600	9	g	g	PROPN
ejpam-6171	600	10	and	and	CCONJ
ejpam-6171	600	11	y	y	PROPN
ejpam-6171	600	12	∈	∈	PROPN
ejpam-6171	600	13	g.	g.	NOUN
ejpam-6171	600	14	since	since	SCONJ
ejpam-6171	600	15	g	g	PROPN
ejpam-6171	600	16	is	be	AUX
ejpam-6171	600	17	an	an	DET
ejpam-6171	600	18	iup	iup	NOUN
ejpam-6171	600	19	-	-	PUNCT
ejpam-6171	600	20	ideal	ideal	NOUN
ejpam-6171	600	21	of	of	ADP
ejpam-6171	600	22	x	x	SYM
ejpam-6171	600	23	,	,	PUNCT
ejpam-6171	600	24	we	we	PRON
ejpam-6171	600	25	have	have	VERB
ejpam-6171	600	26	x?z	x?z	PROPN
ejpam-6171	600	27	∈	∈	PROPN
ejpam-6171	600	28	g.	g.	PROPN
ejpam-6171	600	29	thus	thus	ADV
ejpam-6171	600	30	,	,	PUNCT
ejpam-6171	600	31	pg	pg	VERB
ejpam-6171	600	32	i	i	PRON
ejpam-6171	601	1	[	[	X
ejpam-6171	601	2	β	β	X
ejpam-6171	601	3	−	−	NOUN
ejpam-6171	601	4	β+	β+	PUNCT
ejpam-6171	601	5	]	]	X
ejpam-6171	601	6	(	(	PUNCT
ejpam-6171	601	7	x?z	x?z	PROPN
ejpam-6171	601	8	)	)	PUNCT
ejpam-6171	601	9	=	=	PUNCT
ejpam-6171	602	1	β−	β−	PUNCT
ejpam-6171	602	2	≤	≤	NUM
ejpam-6171	602	3	β−	β−	PUNCT
ejpam-6171	603	1	=	=	SYM
ejpam-6171	603	2	max{β−	max{β−	PROPN
ejpam-6171	603	3	,	,	PUNCT
ejpam-6171	603	4	β−	β−	PRON
ejpam-6171	603	5	}	}	PUNCT
ejpam-6171	603	6	=	=	VERB
ejpam-6171	603	7	max{pg	max{pg	NOUN
ejpam-6171	603	8	i	i	PRON
ejpam-6171	604	1	[	[	X
ejpam-6171	604	2	β	β	X
ejpam-6171	604	3	−	−	NOUN
ejpam-6171	604	4	β+	β+	PUNCT
ejpam-6171	604	5	]	]	X
ejpam-6171	604	6	(	(	PUNCT
ejpam-6171	604	7	x?(y?z)),pg	x?(y?z)),pg	NOUN
ejpam-6171	605	1	i	i	PRON
ejpam-6171	605	2	[	[	X
ejpam-6171	605	3	β	β	X
ejpam-6171	605	4	−	−	NOUN
ejpam-6171	605	5	β+	β+	PUNCT
ejpam-6171	605	6	]	]	X
ejpam-6171	605	7	(	(	PUNCT
ejpam-6171	605	8	y	y	NOUN
ejpam-6171	605	9	)	)	PUNCT
ejpam-6171	605	10	}	}	PUNCT
ejpam-6171	605	11	.	.	PUNCT
ejpam-6171	606	1	case	case	NOUN
ejpam-6171	606	2	2	2	NUM
ejpam-6171	606	3	’	'	PUNCT
ejpam-6171	606	4	:	:	PUNCT
ejpam-6171	606	5	suppose	suppose	VERB
ejpam-6171	606	6	x	x	X
ejpam-6171	606	7	?	?	PUNCT
ejpam-6171	607	1	(	(	PUNCT
ejpam-6171	607	2	y	y	NOUN
ejpam-6171	607	3	?	?	PUNCT
ejpam-6171	608	1	z	z	X
ejpam-6171	608	2	)	)	PUNCT
ejpam-6171	608	3	/∈	/∈	PUNCT
ejpam-6171	609	1	g	g	NOUN
ejpam-6171	609	2	or	or	CCONJ
ejpam-6171	609	3	y	y	PROPN
ejpam-6171	609	4	/∈	/∈	PUNCT
ejpam-6171	610	1	g.	g.	PROPN
ejpam-6171	611	1	then	then	ADV
ejpam-6171	611	2	pg	pg	VERB
ejpam-6171	611	3	i	i	PRON
ejpam-6171	612	1	[	[	X
ejpam-6171	612	2	β	β	X
ejpam-6171	612	3	−	−	NOUN
ejpam-6171	612	4	β+	β+	PUNCT
ejpam-6171	612	5	]	]	X
ejpam-6171	612	6	(	(	PUNCT
ejpam-6171	612	7	x	x	X
ejpam-6171	612	8	?	?	PUNCT
ejpam-6171	613	1	(	(	PUNCT
ejpam-6171	613	2	y	y	NOUN
ejpam-6171	613	3	?	?	PUNCT
ejpam-6171	614	1	z	z	X
ejpam-6171	614	2	)	)	PUNCT
ejpam-6171	614	3	)	)	PUNCT
ejpam-6171	615	1	=	=	PRON
ejpam-6171	615	2	β+	β+	PUNCT
ejpam-6171	615	3	or	or	CCONJ
ejpam-6171	615	4	pg	pg	VERB
ejpam-6171	615	5	i	i	PRON
ejpam-6171	616	1	[	[	X
ejpam-6171	616	2	β	β	X
ejpam-6171	616	3	−	−	NOUN
ejpam-6171	616	4	β+	β+	PUNCT
ejpam-6171	616	5	]	]	X
ejpam-6171	616	6	(	(	PUNCT
ejpam-6171	616	7	y	y	NOUN
ejpam-6171	616	8	)	)	PUNCT
ejpam-6171	616	9	=	=	PUNCT
ejpam-6171	616	10	β+	β+	X
ejpam-6171	616	11	.	.	PUNCT
ejpam-6171	617	1	thus	thus	ADV
ejpam-6171	617	2	,	,	PUNCT
ejpam-6171	617	3	pg	pg	VERB
ejpam-6171	617	4	i	i	PRON
ejpam-6171	618	1	[	[	X
ejpam-6171	618	2	β	β	X
ejpam-6171	618	3	−	−	NOUN
ejpam-6171	618	4	β+	β+	PUNCT
ejpam-6171	618	5	]	]	X
ejpam-6171	618	6	(	(	PUNCT
ejpam-6171	618	7	x	x	X
ejpam-6171	618	8	?	?	PUNCT
ejpam-6171	619	1	z	z	X
ejpam-6171	619	2	)	)	PUNCT
ejpam-6171	619	3	≤	≤	NOUN
ejpam-6171	619	4	β+	β+	PUNCT
ejpam-6171	620	1	=	=	PRON
ejpam-6171	620	2	max{pg	max{pg	NOUN
ejpam-6171	620	3	i	i	PRON
ejpam-6171	621	1	[	[	X
ejpam-6171	621	2	β	β	X
ejpam-6171	621	3	−	−	NOUN
ejpam-6171	621	4	β+	β+	PUNCT
ejpam-6171	621	5	]	]	X
ejpam-6171	621	6	(	(	PUNCT
ejpam-6171	621	7	x	x	X
ejpam-6171	621	8	?	?	PUNCT
ejpam-6171	622	1	(	(	PUNCT
ejpam-6171	622	2	y	y	NOUN
ejpam-6171	622	3	?	?	PUNCT
ejpam-6171	623	1	z)),pg	z)),pg	VERB
ejpam-6171	623	2	i	i	PRON
ejpam-6171	624	1	[	[	X
ejpam-6171	624	2	β	β	X
ejpam-6171	624	3	−	−	NOUN
ejpam-6171	624	4	β+	β+	PUNCT
ejpam-6171	624	5	]	]	X
ejpam-6171	624	6	(	(	PUNCT
ejpam-6171	624	7	y	y	NOUN
ejpam-6171	624	8	)	)	PUNCT
ejpam-6171	624	9	}	}	PUNCT
ejpam-6171	624	10	.	.	PUNCT
ejpam-6171	625	1	case	case	NOUN
ejpam-6171	625	2	1	1	NUM
ejpam-6171	625	3	”	"	PUNCT
ejpam-6171	625	4	:	:	PUNCT
ejpam-6171	625	5	suppose	suppose	VERB
ejpam-6171	625	6	x?(y?z	x?(y?z	X
ejpam-6171	625	7	)	)	PUNCT
ejpam-6171	625	8	∈	∈	PROPN
ejpam-6171	625	9	g	g	PROPN
ejpam-6171	625	10	and	and	CCONJ
ejpam-6171	625	11	y	y	PROPN
ejpam-6171	625	12	∈	∈	PROPN
ejpam-6171	625	13	g.	g.	NOUN
ejpam-6171	625	14	since	since	SCONJ
ejpam-6171	625	15	g	g	PROPN
ejpam-6171	625	16	is	be	AUX
ejpam-6171	625	17	an	an	DET
ejpam-6171	625	18	iup	iup	NOUN
ejpam-6171	625	19	-	-	PUNCT
ejpam-6171	625	20	ideal	ideal	NOUN
ejpam-6171	625	21	of	of	ADP
ejpam-6171	625	22	x	x	SYM
ejpam-6171	625	23	,	,	PUNCT
ejpam-6171	625	24	we	we	PRON
ejpam-6171	625	25	have	have	VERB
ejpam-6171	625	26	x?z	x?z	PROPN
ejpam-6171	625	27	∈	∈	PROPN
ejpam-6171	625	28	g.	g.	PROPN
ejpam-6171	625	29	thus	thus	ADV
ejpam-6171	625	30	,	,	PUNCT
ejpam-6171	625	31	pg	pg	PROPN
ejpam-6171	625	32	f	f	PROPN
ejpam-6171	626	1	[	[	X
ejpam-6171	626	2	γ	γ	X
ejpam-6171	626	3	+	+	X
ejpam-6171	626	4	γ−	γ−	PROPN
ejpam-6171	626	5	]	]	PUNCT
ejpam-6171	626	6	(	(	PUNCT
ejpam-6171	626	7	x	x	X
ejpam-6171	626	8	?	?	PUNCT
ejpam-6171	627	1	z	z	X
ejpam-6171	627	2	)	)	PUNCT
ejpam-6171	627	3	=	=	PRON
ejpam-6171	627	4	γ+	γ+	PUNCT
ejpam-6171	627	5	≥	≥	NOUN
ejpam-6171	627	6	γ+	γ+	X
ejpam-6171	627	7	=	=	SYM
ejpam-6171	627	8	min{γ+	min{γ+	NOUN
ejpam-6171	627	9	,	,	PUNCT
ejpam-6171	627	10	γ+	γ+	X
ejpam-6171	627	11	}	}	PUNCT
ejpam-6171	627	12	=	=	PUNCT
ejpam-6171	628	1	min{pg	min{pg	ADP
ejpam-6171	628	2	f	f	X
ejpam-6171	629	1	[	[	X
ejpam-6171	629	2	γ	γ	X
ejpam-6171	629	3	+	+	X
ejpam-6171	629	4	γ−	γ−	PROPN
ejpam-6171	629	5	]	]	PUNCT
ejpam-6171	629	6	(	(	PUNCT
ejpam-6171	629	7	x	x	X
ejpam-6171	629	8	?	?	PUNCT
ejpam-6171	630	1	(	(	PUNCT
ejpam-6171	630	2	y	y	NOUN
ejpam-6171	630	3	?	?	PUNCT
ejpam-6171	631	1	z)),pg	z)),pg	NOUN
ejpam-6171	631	2	f	f	X
ejpam-6171	632	1	[	[	X
ejpam-6171	632	2	γ	γ	X
ejpam-6171	632	3	+	+	X
ejpam-6171	632	4	γ−	γ−	PROPN
ejpam-6171	632	5	]	]	PUNCT
ejpam-6171	632	6	(	(	PUNCT
ejpam-6171	632	7	y	y	NOUN
ejpam-6171	632	8	)	)	PUNCT
ejpam-6171	632	9	}	}	PUNCT
ejpam-6171	632	10	.	.	PUNCT
ejpam-6171	633	1	case	case	NOUN
ejpam-6171	633	2	2	2	NUM
ejpam-6171	633	3	”	"	PUNCT
ejpam-6171	633	4	:	:	PUNCT
ejpam-6171	633	5	suppose	suppose	VERB
ejpam-6171	633	6	x	x	X
ejpam-6171	633	7	?	?	PUNCT
ejpam-6171	634	1	(	(	PUNCT
ejpam-6171	634	2	y	y	NOUN
ejpam-6171	634	3	?	?	PUNCT
ejpam-6171	635	1	z	z	X
ejpam-6171	635	2	)	)	PUNCT
ejpam-6171	635	3	/∈	/∈	PUNCT
ejpam-6171	636	1	g	g	NOUN
ejpam-6171	636	2	or	or	CCONJ
ejpam-6171	636	3	y	y	PROPN
ejpam-6171	636	4	/∈	/∈	PUNCT
ejpam-6171	637	1	g.	g.	PROPN
ejpam-6171	638	1	then	then	ADV
ejpam-6171	638	2	pg	pg	VERB
ejpam-6171	638	3	f	f	PROPN
ejpam-6171	639	1	[	[	X
ejpam-6171	639	2	γ	γ	X
ejpam-6171	639	3	+	+	X
ejpam-6171	639	4	γ−	γ−	PROPN
ejpam-6171	639	5	]	]	PUNCT
ejpam-6171	639	6	(	(	PUNCT
ejpam-6171	639	7	x	x	X
ejpam-6171	639	8	?	?	PUNCT
ejpam-6171	640	1	(	(	PUNCT
ejpam-6171	640	2	y	y	NOUN
ejpam-6171	640	3	?	?	PUNCT
ejpam-6171	641	1	z	z	X
ejpam-6171	641	2	)	)	PUNCT
ejpam-6171	641	3	)	)	PUNCT
ejpam-6171	642	1	=	=	SYM
ejpam-6171	642	2	γ−	γ−	PROPN
ejpam-6171	642	3	or	or	CCONJ
ejpam-6171	642	4	pg	pg	NOUN
ejpam-6171	642	5	f	f	PROPN
ejpam-6171	643	1	[	[	X
ejpam-6171	643	2	γ	γ	X
ejpam-6171	643	3	+	+	X
ejpam-6171	643	4	γ−	γ−	PROPN
ejpam-6171	643	5	]	]	PUNCT
ejpam-6171	643	6	(	(	PUNCT
ejpam-6171	643	7	y	y	NOUN
ejpam-6171	643	8	)	)	PUNCT
ejpam-6171	643	9	=	=	VERB
ejpam-6171	643	10	γ−.	γ−.	NOUN
ejpam-6171	643	11	thus	thus	ADV
ejpam-6171	643	12	,	,	PUNCT
ejpam-6171	643	13	pg	pg	PROPN
ejpam-6171	643	14	f	f	PROPN
ejpam-6171	644	1	[	[	X
ejpam-6171	644	2	γ	γ	X
ejpam-6171	644	3	+	+	X
ejpam-6171	644	4	γ−	γ−	PROPN
ejpam-6171	644	5	]	]	PUNCT
ejpam-6171	644	6	(	(	PUNCT
ejpam-6171	644	7	x	x	X
ejpam-6171	644	8	?	?	PUNCT
ejpam-6171	645	1	z	z	X
ejpam-6171	645	2	)	)	PUNCT
ejpam-6171	645	3	≥	≥	NOUN
ejpam-6171	645	4	γ−	γ−	NOUN
ejpam-6171	645	5	=	=	NOUN
ejpam-6171	645	6	min{pg	min{pg	X
ejpam-6171	645	7	f	f	X
ejpam-6171	646	1	[	[	X
ejpam-6171	646	2	γ	γ	X
ejpam-6171	646	3	+	+	X
ejpam-6171	646	4	γ−	γ−	PROPN
ejpam-6171	646	5	]	]	PUNCT
ejpam-6171	646	6	(	(	PUNCT
ejpam-6171	646	7	x	x	X
ejpam-6171	646	8	?	?	PUNCT
ejpam-6171	647	1	(	(	PUNCT
ejpam-6171	647	2	y	y	NOUN
ejpam-6171	647	3	?	?	PUNCT
ejpam-6171	648	1	z)),pg	z)),pg	NOUN
ejpam-6171	648	2	f	f	X
ejpam-6171	649	1	[	[	X
ejpam-6171	649	2	γ	γ	X
ejpam-6171	649	3	+	+	X
ejpam-6171	649	4	γ−	γ−	PROPN
ejpam-6171	649	5	]	]	PUNCT
ejpam-6171	649	6	(	(	PUNCT
ejpam-6171	649	7	y	y	NOUN
ejpam-6171	649	8	)	)	PUNCT
ejpam-6171	649	9	}	}	PUNCT
ejpam-6171	649	10	.	.	PUNCT
ejpam-6171	650	1	hence	hence	ADV
ejpam-6171	650	2	,	,	PUNCT
ejpam-6171	650	3	pg[α	pg[α	PROPN
ejpam-6171	650	4	+	+	PROPN
ejpam-6171	650	5	,	,	PUNCT
ejpam-6171	650	6	β−,γ+	β−,γ+	X
ejpam-6171	650	7	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	650	8	]	]	PUNCT
ejpam-6171	650	9	is	be	AUX
ejpam-6171	650	10	a	a	DET
ejpam-6171	650	11	pythagorean	pythagorean	PROPN
ejpam-6171	650	12	neutrosophic	neutrosophic	ADJ
ejpam-6171	650	13	iup	iup	PROPN
ejpam-6171	650	14	-	-	PUNCT
ejpam-6171	650	15	ideal	ideal	NOUN
ejpam-6171	650	16	of	of	ADP
ejpam-6171	650	17	x.	x.	PROPN
ejpam-6171	650	18	k.	k.	PROPN
ejpam-6171	650	19	suayngam	suayngam	PROPN
ejpam-6171	651	1	et	et	PROPN
ejpam-6171	651	2	al	al	PROPN
ejpam-6171	651	3	.	.	PUNCT
ejpam-6171	651	4	/	/	SYM
ejpam-6171	651	5	eur	eur	PROPN
ejpam-6171	651	6	.	.	PUNCT
ejpam-6171	652	1	j.	j.	PROPN
ejpam-6171	652	2	pure	pure	PROPN
ejpam-6171	652	3	appl	appl	PROPN
ejpam-6171	652	4	.	.	PROPN
ejpam-6171	652	5	math	math	PROPN
ejpam-6171	652	6	,	,	PUNCT
ejpam-6171	652	7	18	18	NUM
ejpam-6171	652	8	(	(	PUNCT
ejpam-6171	652	9	3	3	NUM
ejpam-6171	652	10	)	)	PUNCT
ejpam-6171	652	11	(	(	PUNCT
ejpam-6171	652	12	2025	2025	NUM
ejpam-6171	652	13	)	)	PUNCT
ejpam-6171	652	14	,	,	PUNCT
ejpam-6171	652	15	6171	6171	NUM
ejpam-6171	652	16	17	17	NUM
ejpam-6171	652	17	of	of	ADP
ejpam-6171	652	18	28	28	NUM
ejpam-6171	652	19	conversely	conversely	ADV
ejpam-6171	652	20	,	,	PUNCT
ejpam-6171	652	21	assume	assume	VERB
ejpam-6171	652	22	that	that	SCONJ
ejpam-6171	652	23	the	the	DET
ejpam-6171	652	24	characteristic	characteristic	ADJ
ejpam-6171	652	25	pns	pns	PROPN
ejpam-6171	652	26	pg[α	pg[α	PROPN
ejpam-6171	652	27	+	+	PROPN
ejpam-6171	652	28	,	,	PUNCT
ejpam-6171	652	29	β−,γ+	β−,γ+	X
ejpam-6171	652	30	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	652	31	]	]	PUNCT
ejpam-6171	652	32	is	be	AUX
ejpam-6171	652	33	a	a	DET
ejpam-6171	652	34	pythagorean	pythagorean	PROPN
ejpam-6171	652	35	neutrosophic	neutrosophic	ADJ
ejpam-6171	652	36	iup	iup	PROPN
ejpam-6171	652	37	-	-	PUNCT
ejpam-6171	652	38	ideal	ideal	NOUN
ejpam-6171	652	39	of	of	ADP
ejpam-6171	652	40	x.	x.	NOUN
ejpam-6171	652	41	since	since	SCONJ
ejpam-6171	652	42	pg	pg	PROPN
ejpam-6171	652	43	t	t	PROPN
ejpam-6171	653	1	[	[	X
ejpam-6171	653	2	α	α	X
ejpam-6171	653	3	+	+	X
ejpam-6171	653	4	α−	α−	ADP
ejpam-6171	653	5	]	]	PUNCT
ejpam-6171	653	6	satisfies	satisfie	NOUN
ejpam-6171	653	7	(	(	PUNCT
ejpam-6171	653	8	3.5	3.5	NUM
ejpam-6171	653	9	)	)	PUNCT
ejpam-6171	653	10	,	,	PUNCT
ejpam-6171	653	11	it	it	PRON
ejpam-6171	653	12	follow	follow	VERB
ejpam-6171	653	13	from	from	ADP
ejpam-6171	653	14	lemma	lemma	PROPN
ejpam-6171	653	15	2	2	NUM
ejpam-6171	653	16	that	that	SCONJ
ejpam-6171	653	17	0	0	NUM
ejpam-6171	653	18	∈	∈	NOUN
ejpam-6171	653	19	g.	g.	NOUN
ejpam-6171	653	20	next	next	ADV
ejpam-6171	653	21	,	,	PUNCT
ejpam-6171	653	22	let	let	VERB
ejpam-6171	653	23	x	x	PRON
ejpam-6171	653	24	,	,	PUNCT
ejpam-6171	653	25	y	y	PROPN
ejpam-6171	653	26	,	,	PUNCT
ejpam-6171	653	27	z	z	NOUN
ejpam-6171	653	28	∈	∈	PROPN
ejpam-6171	653	29	x	x	AUX
ejpam-6171	653	30	be	be	AUX
ejpam-6171	653	31	such	such	ADJ
ejpam-6171	653	32	that	that	PRON
ejpam-6171	653	33	x	x	X
ejpam-6171	653	34	?	?	PUNCT
ejpam-6171	654	1	(	(	PUNCT
ejpam-6171	654	2	y	y	NOUN
ejpam-6171	654	3	?	?	PUNCT
ejpam-6171	654	4	z	z	X
ejpam-6171	654	5	)	)	PUNCT
ejpam-6171	654	6	∈	∈	PROPN
ejpam-6171	654	7	g	g	PROPN
ejpam-6171	655	1	and	and	CCONJ
ejpam-6171	655	2	y	y	PROPN
ejpam-6171	655	3	∈	∈	PROPN
ejpam-6171	655	4	g.	g.	NOUN
ejpam-6171	655	5	then	then	ADV
ejpam-6171	655	6	pg	pg	VERB
ejpam-6171	655	7	t	t	PROPN
ejpam-6171	656	1	[	[	X
ejpam-6171	656	2	α	α	X
ejpam-6171	656	3	+	+	X
ejpam-6171	657	1	α−	α−	ADP
ejpam-6171	657	2	]	]	X
ejpam-6171	657	3	(	(	PUNCT
ejpam-6171	657	4	x	x	X
ejpam-6171	657	5	?	?	PUNCT
ejpam-6171	658	1	(	(	PUNCT
ejpam-6171	658	2	y	y	NOUN
ejpam-6171	658	3	?	?	PUNCT
ejpam-6171	658	4	z	z	NOUN
ejpam-6171	658	5	)	)	PUNCT
ejpam-6171	658	6	)	)	PUNCT
ejpam-6171	659	1	=	=	SYM
ejpam-6171	659	2	α+	α+	PUNCT
ejpam-6171	659	3	and	and	CCONJ
ejpam-6171	659	4	pg	pg	X
ejpam-6171	659	5	t	t	PROPN
ejpam-6171	660	1	[	[	X
ejpam-6171	660	2	α	α	X
ejpam-6171	660	3	+	+	X
ejpam-6171	661	1	α−	α−	ADP
ejpam-6171	661	2	]	]	X
ejpam-6171	661	3	(	(	PUNCT
ejpam-6171	661	4	y	y	NOUN
ejpam-6171	661	5	)	)	PUNCT
ejpam-6171	661	6	=	=	SYM
ejpam-6171	661	7	α+	α+	PROPN
ejpam-6171	661	8	.	.	PUNCT
ejpam-6171	662	1	thus	thus	ADV
ejpam-6171	662	2	,	,	PUNCT
ejpam-6171	662	3	min{pg	min{pg	ADP
ejpam-6171	662	4	t	t	X
ejpam-6171	663	1	[	[	X
ejpam-6171	663	2	α	α	X
ejpam-6171	663	3	+	+	X
ejpam-6171	664	1	α−	α−	ADP
ejpam-6171	664	2	]	]	X
ejpam-6171	664	3	(	(	PUNCT
ejpam-6171	664	4	x	x	X
ejpam-6171	664	5	?	?	PUNCT
ejpam-6171	665	1	(	(	PUNCT
ejpam-6171	665	2	y	y	NOUN
ejpam-6171	665	3	?	?	PUNCT
ejpam-6171	666	1	z)),pg	z)),pg	NOUN
ejpam-6171	666	2	t	t	NOUN
ejpam-6171	667	1	[	[	X
ejpam-6171	667	2	α	α	X
ejpam-6171	667	3	+	+	X
ejpam-6171	668	1	α−	α−	ADP
ejpam-6171	668	2	]	]	X
ejpam-6171	668	3	(	(	PUNCT
ejpam-6171	668	4	y	y	NOUN
ejpam-6171	668	5	)	)	PUNCT
ejpam-6171	668	6	}	}	PUNCT
ejpam-6171	668	7	=	=	SYM
ejpam-6171	668	8	α+	α+	NOUN
ejpam-6171	668	9	.	.	PUNCT
ejpam-6171	668	10	by	by	ADP
ejpam-6171	668	11	(	(	PUNCT
ejpam-6171	668	12	3.8	3.8	NUM
ejpam-6171	668	13	)	)	PUNCT
ejpam-6171	668	14	,	,	PUNCT
ejpam-6171	668	15	we	we	PRON
ejpam-6171	668	16	have	have	VERB
ejpam-6171	668	17	pg	pg	PROPN
ejpam-6171	668	18	t	t	PROPN
ejpam-6171	669	1	[	[	X
ejpam-6171	669	2	α	α	X
ejpam-6171	669	3	+	+	X
ejpam-6171	670	1	α−	α−	ADP
ejpam-6171	670	2	]	]	X
ejpam-6171	670	3	(	(	PUNCT
ejpam-6171	670	4	x	x	X
ejpam-6171	670	5	?	?	PUNCT
ejpam-6171	670	6	z	z	X
ejpam-6171	670	7	)	)	PUNCT
ejpam-6171	670	8	≥	≥	NOUN
ejpam-6171	670	9	min{pg	min{pg	X
ejpam-6171	670	10	t	t	X
ejpam-6171	671	1	[	[	X
ejpam-6171	671	2	α	α	X
ejpam-6171	671	3	+	+	X
ejpam-6171	672	1	α−	α−	ADP
ejpam-6171	672	2	]	]	X
ejpam-6171	672	3	(	(	PUNCT
ejpam-6171	672	4	x	x	X
ejpam-6171	672	5	?	?	PUNCT
ejpam-6171	673	1	(	(	PUNCT
ejpam-6171	673	2	y	y	NOUN
ejpam-6171	673	3	?	?	PUNCT
ejpam-6171	674	1	z)),pg	z)),pg	NOUN
ejpam-6171	674	2	t	t	NOUN
ejpam-6171	675	1	[	[	X
ejpam-6171	675	2	α	α	X
ejpam-6171	675	3	+	+	X
ejpam-6171	676	1	α−	α−	ADP
ejpam-6171	676	2	]	]	X
ejpam-6171	676	3	(	(	PUNCT
ejpam-6171	676	4	y	y	NOUN
ejpam-6171	676	5	)	)	PUNCT
ejpam-6171	676	6	}	}	PUNCT
ejpam-6171	676	7	=	=	SYM
ejpam-6171	676	8	α+	α+	NOUN
ejpam-6171	676	9	,	,	PUNCT
ejpam-6171	676	10	that	that	ADV
ejpam-6171	676	11	is	is	ADV
ejpam-6171	676	12	,	,	PUNCT
ejpam-6171	676	13	pg	pg	PROPN
ejpam-6171	676	14	t	t	PROPN
ejpam-6171	677	1	[	[	X
ejpam-6171	677	2	α	α	X
ejpam-6171	677	3	+	+	X
ejpam-6171	678	1	α−	α−	ADP
ejpam-6171	678	2	]	]	X
ejpam-6171	678	3	(	(	PUNCT
ejpam-6171	678	4	x	x	X
ejpam-6171	678	5	?	?	PUNCT
ejpam-6171	678	6	z	z	X
ejpam-6171	678	7	)	)	PUNCT
ejpam-6171	678	8	=	=	SYM
ejpam-6171	678	9	α+	α+	NOUN
ejpam-6171	678	10	.	.	PUNCT
ejpam-6171	679	1	hence	hence	ADV
ejpam-6171	679	2	,	,	PUNCT
ejpam-6171	679	3	x	x	PUNCT
ejpam-6171	679	4	?	?	PUNCT
ejpam-6171	679	5	z	z	X
ejpam-6171	679	6	∈	∈	PROPN
ejpam-6171	679	7	g	g	NOUN
ejpam-6171	679	8	,	,	PUNCT
ejpam-6171	679	9	so	so	SCONJ
ejpam-6171	679	10	g	g	PROPN
ejpam-6171	679	11	is	be	AUX
ejpam-6171	679	12	an	an	DET
ejpam-6171	679	13	iup	iup	NOUN
ejpam-6171	679	14	-	-	PUNCT
ejpam-6171	679	15	ideal	ideal	NOUN
ejpam-6171	679	16	.	.	PUNCT
ejpam-6171	680	1	theorem	theorem	VERB
ejpam-6171	680	2	11	11	NUM
ejpam-6171	680	3	.	.	PUNCT
ejpam-6171	681	1	a	a	DET
ejpam-6171	681	2	nonempty	nonempty	NOUN
ejpam-6171	681	3	subset	subset	VERB
ejpam-6171	681	4	g	g	NOUN
ejpam-6171	681	5	is	be	AUX
ejpam-6171	681	6	an	an	DET
ejpam-6171	681	7	iup	iup	NOUN
ejpam-6171	681	8	-	-	PUNCT
ejpam-6171	681	9	filter	filter	NOUN
ejpam-6171	681	10	of	of	ADP
ejpam-6171	681	11	x	x	SYM
ejpam-6171	681	12	if	if	SCONJ
ejpam-6171	681	13	and	and	CCONJ
ejpam-6171	681	14	only	only	ADV
ejpam-6171	681	15	if	if	SCONJ
ejpam-6171	681	16	the	the	DET
ejpam-6171	681	17	characteristic	characteristic	ADJ
ejpam-6171	681	18	pns	pns	PROPN
ejpam-6171	681	19	pg[α	pg[α	PROPN
ejpam-6171	681	20	+	+	PROPN
ejpam-6171	681	21	,	,	PUNCT
ejpam-6171	681	22	β−,γ+	β−,γ+	X
ejpam-6171	681	23	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	681	24	]	]	PUNCT
ejpam-6171	681	25	is	be	AUX
ejpam-6171	681	26	a	a	DET
ejpam-6171	681	27	pythagorean	pythagorean	PROPN
ejpam-6171	681	28	neutrosophic	neutrosophic	ADJ
ejpam-6171	681	29	iup	iup	NOUN
ejpam-6171	681	30	-	-	PUNCT
ejpam-6171	681	31	filter	filter	NOUN
ejpam-6171	681	32	of	of	ADP
ejpam-6171	681	33	x.	x.	NOUN
ejpam-6171	681	34	proof	proof	PROPN
ejpam-6171	681	35	.	.	PUNCT
ejpam-6171	682	1	assume	assume	VERB
ejpam-6171	682	2	that	that	SCONJ
ejpam-6171	682	3	g	g	PROPN
ejpam-6171	682	4	is	be	AUX
ejpam-6171	682	5	an	an	DET
ejpam-6171	682	6	iup	iup	NOUN
ejpam-6171	682	7	-	-	PUNCT
ejpam-6171	682	8	filter	filter	NOUN
ejpam-6171	682	9	of	of	ADP
ejpam-6171	682	10	x.	x.	NOUN
ejpam-6171	682	11	since	since	SCONJ
ejpam-6171	682	12	0	0	NUM
ejpam-6171	682	13	∈	∈	PROPN
ejpam-6171	682	14	g	g	NOUN
ejpam-6171	682	15	,	,	PUNCT
ejpam-6171	682	16	it	it	PRON
ejpam-6171	682	17	follows	follow	VERB
ejpam-6171	682	18	from	from	ADP
ejpam-6171	682	19	lemma	lemma	PROPN
ejpam-6171	682	20	2	2	NUM
ejpam-6171	683	1	that	that	PRON
ejpam-6171	683	2	pg	pg	VERB
ejpam-6171	684	1	t	t	PROPN
ejpam-6171	685	1	[	[	X
ejpam-6171	685	2	α	α	X
ejpam-6171	685	3	+	+	X
ejpam-6171	686	1	α−	α−	ADP
ejpam-6171	686	2	]	]	PUNCT
ejpam-6171	686	3	,	,	PUNCT
ejpam-6171	686	4	pg	pg	VERB
ejpam-6171	686	5	i	i	PRON
ejpam-6171	687	1	[	[	X
ejpam-6171	687	2	β	β	X
ejpam-6171	687	3	−	−	NOUN
ejpam-6171	687	4	β+	β+	PUNCT
ejpam-6171	687	5	]	]	X
ejpam-6171	687	6	,	,	PUNCT
ejpam-6171	687	7	and	and	CCONJ
ejpam-6171	687	8	pg	pg	X
ejpam-6171	687	9	f	f	PROPN
ejpam-6171	688	1	[	[	X
ejpam-6171	688	2	γ	γ	X
ejpam-6171	688	3	+	+	CCONJ
ejpam-6171	688	4	γ−	γ−	PROPN
ejpam-6171	688	5	]	]	PUNCT
ejpam-6171	688	6	satisfy	satisfy	NOUN
ejpam-6171	688	7	(	(	PUNCT
ejpam-6171	688	8	3.5	3.5	NUM
ejpam-6171	688	9	)	)	PUNCT
ejpam-6171	688	10	,	,	PUNCT
ejpam-6171	688	11	(	(	PUNCT
ejpam-6171	688	12	3.6	3.6	NUM
ejpam-6171	688	13	)	)	PUNCT
ejpam-6171	688	14	,	,	PUNCT
ejpam-6171	688	15	and	and	CCONJ
ejpam-6171	688	16	(	(	PUNCT
ejpam-6171	688	17	3.7	3.7	NUM
ejpam-6171	688	18	)	)	PUNCT
ejpam-6171	688	19	,	,	PUNCT
ejpam-6171	688	20	respectively	respectively	ADV
ejpam-6171	688	21	.	.	PUNCT
ejpam-6171	689	1	next	next	ADV
ejpam-6171	689	2	,	,	PUNCT
ejpam-6171	689	3	let	let	VERB
ejpam-6171	689	4	x	x	PRON
ejpam-6171	689	5	,	,	PUNCT
ejpam-6171	689	6	y	y	PROPN
ejpam-6171	689	7	∈	∈	PROPN
ejpam-6171	689	8	x.	x.	NOUN
ejpam-6171	689	9	case	case	NOUN
ejpam-6171	689	10	1	1	NUM
ejpam-6171	689	11	:	:	PUNCT
ejpam-6171	689	12	suppose	suppose	VERB
ejpam-6171	689	13	x	x	X
ejpam-6171	689	14	?	?	PUNCT
ejpam-6171	690	1	y	y	PROPN
ejpam-6171	690	2	∈	∈	PROPN
ejpam-6171	690	3	g	g	PROPN
ejpam-6171	690	4	and	and	CCONJ
ejpam-6171	690	5	x	x	PROPN
ejpam-6171	690	6	∈	∈	PROPN
ejpam-6171	690	7	g.	g.	NOUN
ejpam-6171	690	8	since	since	SCONJ
ejpam-6171	690	9	g	g	PROPN
ejpam-6171	690	10	is	be	AUX
ejpam-6171	690	11	an	an	DET
ejpam-6171	690	12	iup	iup	NOUN
ejpam-6171	690	13	-	-	PUNCT
ejpam-6171	690	14	filter	filter	NOUN
ejpam-6171	690	15	of	of	ADP
ejpam-6171	690	16	x	x	PRON
ejpam-6171	690	17	,	,	PUNCT
ejpam-6171	690	18	we	we	PRON
ejpam-6171	690	19	have	have	VERB
ejpam-6171	690	20	y	y	PROPN
ejpam-6171	690	21	∈	∈	PROPN
ejpam-6171	690	22	g.	g.	PROPN
ejpam-6171	691	1	thus	thus	ADV
ejpam-6171	691	2	,	,	PUNCT
ejpam-6171	691	3	pg	pg	PROPN
ejpam-6171	691	4	t	t	PROPN
ejpam-6171	692	1	[	[	X
ejpam-6171	692	2	α	α	X
ejpam-6171	692	3	+	+	X
ejpam-6171	693	1	α−	α−	ADP
ejpam-6171	693	2	]	]	X
ejpam-6171	693	3	(	(	PUNCT
ejpam-6171	693	4	y	y	NOUN
ejpam-6171	693	5	)	)	PUNCT
ejpam-6171	693	6	=	=	SYM
ejpam-6171	693	7	α+	α+	PUNCT
ejpam-6171	693	8	≥	≥	X
ejpam-6171	693	9	α+	α+	X
ejpam-6171	693	10	=	=	SYM
ejpam-6171	693	11	min{α+	min{α+	PROPN
ejpam-6171	693	12	,	,	PUNCT
ejpam-6171	693	13	α+	α+	NOUN
ejpam-6171	693	14	}	}	PUNCT
ejpam-6171	693	15	=	=	SYM
ejpam-6171	693	16	min{pg	min{pg	ADP
ejpam-6171	693	17	t	t	X
ejpam-6171	694	1	[	[	X
ejpam-6171	694	2	α	α	X
ejpam-6171	694	3	+	+	X
ejpam-6171	695	1	α−	α−	ADP
ejpam-6171	695	2	]	]	X
ejpam-6171	695	3	(	(	PUNCT
ejpam-6171	695	4	x	x	X
ejpam-6171	695	5	?	?	PUNCT
ejpam-6171	695	6	y),pg	y),pg	NOUN
ejpam-6171	695	7	t	t	PROPN
ejpam-6171	696	1	[	[	X
ejpam-6171	696	2	α	α	X
ejpam-6171	696	3	+	+	X
ejpam-6171	697	1	α−	α−	ADP
ejpam-6171	697	2	]	]	X
ejpam-6171	697	3	(	(	PUNCT
ejpam-6171	697	4	x	x	NOUN
ejpam-6171	697	5	)	)	PUNCT
ejpam-6171	697	6	}	}	PUNCT
ejpam-6171	697	7	.	.	PUNCT
ejpam-6171	698	1	case	case	NOUN
ejpam-6171	698	2	2	2	NUM
ejpam-6171	698	3	:	:	PUNCT
ejpam-6171	698	4	suppose	suppose	VERB
ejpam-6171	698	5	x	x	X
ejpam-6171	698	6	?	?	PUNCT
ejpam-6171	699	1	y	y	PROPN
ejpam-6171	699	2	/∈	/∈	PUNCT
ejpam-6171	700	1	g	g	NOUN
ejpam-6171	700	2	or	or	CCONJ
ejpam-6171	700	3	x	x	PROPN
ejpam-6171	700	4	/∈	/∈	PROPN
ejpam-6171	701	1	g.	g.	NOUN
ejpam-6171	702	1	then	then	ADV
ejpam-6171	702	2	pg	pg	VERB
ejpam-6171	702	3	t	t	PROPN
ejpam-6171	703	1	[	[	X
ejpam-6171	703	2	α	α	X
ejpam-6171	703	3	+	+	X
ejpam-6171	704	1	α−	α−	ADP
ejpam-6171	704	2	]	]	X
ejpam-6171	704	3	(	(	PUNCT
ejpam-6171	704	4	x	x	X
ejpam-6171	704	5	?	?	PUNCT
ejpam-6171	704	6	y	y	X
ejpam-6171	704	7	)	)	PUNCT
ejpam-6171	704	8	=	=	PUNCT
ejpam-6171	705	1	α−	α−	ADP
ejpam-6171	705	2	or	or	CCONJ
ejpam-6171	705	3	pg	pg	X
ejpam-6171	705	4	t	t	PROPN
ejpam-6171	706	1	[	[	X
ejpam-6171	706	2	α	α	X
ejpam-6171	706	3	+	+	X
ejpam-6171	707	1	α−	α−	ADP
ejpam-6171	707	2	]	]	X
ejpam-6171	707	3	(	(	PUNCT
ejpam-6171	707	4	x	x	X
ejpam-6171	707	5	)	)	PUNCT
ejpam-6171	707	6	=	=	SYM
ejpam-6171	707	7	α−.	α−.	NOUN
ejpam-6171	707	8	thus	thus	ADV
ejpam-6171	707	9	,	,	PUNCT
ejpam-6171	707	10	pg	pg	PROPN
ejpam-6171	707	11	t	t	PROPN
ejpam-6171	708	1	[	[	X
ejpam-6171	708	2	α	α	X
ejpam-6171	708	3	+	+	X
ejpam-6171	709	1	α−	α−	ADP
ejpam-6171	709	2	]	]	X
ejpam-6171	709	3	(	(	PUNCT
ejpam-6171	709	4	y	y	NOUN
ejpam-6171	709	5	)	)	PUNCT
ejpam-6171	709	6	≥	≥	NOUN
ejpam-6171	709	7	α−	α−	ADP
ejpam-6171	709	8	=	=	PUNCT
ejpam-6171	709	9	min{pg	min{pg	NOUN
ejpam-6171	709	10	t	t	X
ejpam-6171	710	1	[	[	X
ejpam-6171	710	2	α	α	X
ejpam-6171	710	3	+	+	X
ejpam-6171	711	1	α−	α−	ADP
ejpam-6171	711	2	]	]	X
ejpam-6171	711	3	(	(	PUNCT
ejpam-6171	711	4	x	x	X
ejpam-6171	711	5	?	?	PUNCT
ejpam-6171	711	6	y),pg	y),pg	NOUN
ejpam-6171	711	7	t	t	PROPN
ejpam-6171	712	1	[	[	X
ejpam-6171	712	2	α	α	X
ejpam-6171	712	3	+	+	X
ejpam-6171	713	1	α−	α−	ADP
ejpam-6171	713	2	]	]	X
ejpam-6171	713	3	(	(	PUNCT
ejpam-6171	713	4	x	x	NOUN
ejpam-6171	713	5	)	)	PUNCT
ejpam-6171	713	6	}	}	PUNCT
ejpam-6171	713	7	.	.	PUNCT
ejpam-6171	714	1	case	case	NOUN
ejpam-6171	714	2	1	1	NUM
ejpam-6171	714	3	’	'	PUNCT
ejpam-6171	714	4	:	:	PUNCT
ejpam-6171	714	5	suppose	suppose	VERB
ejpam-6171	714	6	x	x	X
ejpam-6171	714	7	?	?	PUNCT
ejpam-6171	715	1	y	y	PROPN
ejpam-6171	715	2	∈	∈	PROPN
ejpam-6171	715	3	g	g	PROPN
ejpam-6171	715	4	and	and	CCONJ
ejpam-6171	715	5	x	x	PROPN
ejpam-6171	715	6	∈	∈	PROPN
ejpam-6171	715	7	g.	g.	NOUN
ejpam-6171	715	8	since	since	SCONJ
ejpam-6171	715	9	g	g	PROPN
ejpam-6171	715	10	is	be	AUX
ejpam-6171	715	11	an	an	DET
ejpam-6171	715	12	iup	iup	NOUN
ejpam-6171	715	13	-	-	PUNCT
ejpam-6171	715	14	filter	filter	NOUN
ejpam-6171	715	15	of	of	ADP
ejpam-6171	715	16	x	x	PRON
ejpam-6171	715	17	,	,	PUNCT
ejpam-6171	715	18	we	we	PRON
ejpam-6171	715	19	have	have	VERB
ejpam-6171	715	20	y	y	PROPN
ejpam-6171	715	21	∈	∈	PROPN
ejpam-6171	715	22	g.	g.	PROPN
ejpam-6171	716	1	thus	thus	ADV
ejpam-6171	716	2	,	,	PUNCT
ejpam-6171	716	3	pg	pg	VERB
ejpam-6171	716	4	i	i	PRON
ejpam-6171	717	1	[	[	X
ejpam-6171	717	2	β	β	X
ejpam-6171	717	3	−	−	NOUN
ejpam-6171	717	4	β+	β+	PUNCT
ejpam-6171	717	5	]	]	X
ejpam-6171	717	6	(	(	PUNCT
ejpam-6171	717	7	y	y	NOUN
ejpam-6171	717	8	)	)	PUNCT
ejpam-6171	717	9	=	=	PUNCT
ejpam-6171	718	1	β−	β−	PUNCT
ejpam-6171	718	2	≤	≤	NUM
ejpam-6171	718	3	β−	β−	PUNCT
ejpam-6171	719	1	=	=	SYM
ejpam-6171	719	2	max{β−	max{β−	PROPN
ejpam-6171	719	3	,	,	PUNCT
ejpam-6171	719	4	β−	β−	PRON
ejpam-6171	719	5	}	}	PUNCT
ejpam-6171	719	6	=	=	VERB
ejpam-6171	719	7	max{pg	max{pg	NOUN
ejpam-6171	719	8	i	i	PRON
ejpam-6171	720	1	[	[	X
ejpam-6171	720	2	β	β	X
ejpam-6171	720	3	−	−	NOUN
ejpam-6171	720	4	β+	β+	PUNCT
ejpam-6171	720	5	]	]	X
ejpam-6171	720	6	(	(	PUNCT
ejpam-6171	720	7	x	x	X
ejpam-6171	720	8	?	?	PUNCT
ejpam-6171	721	1	y),pg	y),pg	NOUN
ejpam-6171	722	1	i	i	PRON
ejpam-6171	723	1	[	[	X
ejpam-6171	723	2	β	β	X
ejpam-6171	723	3	−	−	NOUN
ejpam-6171	723	4	β+	β+	PUNCT
ejpam-6171	723	5	]	]	X
ejpam-6171	723	6	(	(	PUNCT
ejpam-6171	723	7	x	x	NOUN
ejpam-6171	723	8	)	)	PUNCT
ejpam-6171	723	9	}	}	PUNCT
ejpam-6171	723	10	.	.	PUNCT
ejpam-6171	724	1	case	case	NOUN
ejpam-6171	724	2	2	2	NUM
ejpam-6171	724	3	’	'	PUNCT
ejpam-6171	724	4	:	:	PUNCT
ejpam-6171	724	5	suppose	suppose	VERB
ejpam-6171	724	6	x	x	X
ejpam-6171	724	7	?	?	PUNCT
ejpam-6171	725	1	y	y	PROPN
ejpam-6171	725	2	/∈	/∈	PUNCT
ejpam-6171	726	1	g	g	NOUN
ejpam-6171	726	2	or	or	CCONJ
ejpam-6171	726	3	x	x	PROPN
ejpam-6171	726	4	/∈	/∈	PUNCT
ejpam-6171	727	1	g.	g.	NOUN
ejpam-6171	728	1	then	then	ADV
ejpam-6171	728	2	pg	pg	VERB
ejpam-6171	728	3	i	i	PRON
ejpam-6171	729	1	[	[	X
ejpam-6171	729	2	β	β	X
ejpam-6171	729	3	−	−	NOUN
ejpam-6171	729	4	β+	β+	PUNCT
ejpam-6171	729	5	]	]	X
ejpam-6171	729	6	(	(	PUNCT
ejpam-6171	729	7	x	x	X
ejpam-6171	729	8	?	?	PUNCT
ejpam-6171	730	1	y	y	X
ejpam-6171	730	2	)	)	PUNCT
ejpam-6171	730	3	=	=	PRON
ejpam-6171	730	4	β+	β+	PUNCT
ejpam-6171	730	5	or	or	CCONJ
ejpam-6171	730	6	pg	pg	VERB
ejpam-6171	730	7	i	i	PRON
ejpam-6171	731	1	[	[	X
ejpam-6171	731	2	β	β	X
ejpam-6171	731	3	−	−	NOUN
ejpam-6171	731	4	β+	β+	PUNCT
ejpam-6171	731	5	]	]	X
ejpam-6171	731	6	(	(	PUNCT
ejpam-6171	731	7	x	x	X
ejpam-6171	731	8	)	)	PUNCT
ejpam-6171	732	1	=	=	SYM
ejpam-6171	732	2	β+	β+	X
ejpam-6171	732	3	.	.	PUNCT
ejpam-6171	733	1	thus	thus	ADV
ejpam-6171	733	2	,	,	PUNCT
ejpam-6171	733	3	pg	pg	VERB
ejpam-6171	733	4	i	i	PRON
ejpam-6171	734	1	[	[	X
ejpam-6171	734	2	β	β	X
ejpam-6171	734	3	−	−	NOUN
ejpam-6171	734	4	β+	β+	PUNCT
ejpam-6171	734	5	]	]	X
ejpam-6171	734	6	(	(	PUNCT
ejpam-6171	734	7	y	y	NOUN
ejpam-6171	734	8	)	)	PUNCT
ejpam-6171	734	9	≤	≤	NOUN
ejpam-6171	734	10	β+	β+	PUNCT
ejpam-6171	735	1	=	=	PRON
ejpam-6171	735	2	max{pg	max{pg	NOUN
ejpam-6171	735	3	i	i	PRON
ejpam-6171	736	1	[	[	X
ejpam-6171	736	2	β	β	X
ejpam-6171	736	3	−	−	NOUN
ejpam-6171	736	4	β+	β+	PUNCT
ejpam-6171	736	5	]	]	X
ejpam-6171	736	6	(	(	PUNCT
ejpam-6171	736	7	x	x	X
ejpam-6171	736	8	?	?	PUNCT
ejpam-6171	737	1	y),pg	y),pg	NOUN
ejpam-6171	738	1	i	i	PRON
ejpam-6171	739	1	[	[	X
ejpam-6171	739	2	β	β	X
ejpam-6171	739	3	−	−	NOUN
ejpam-6171	739	4	β+	β+	PUNCT
ejpam-6171	739	5	]	]	X
ejpam-6171	739	6	(	(	PUNCT
ejpam-6171	739	7	x	x	NOUN
ejpam-6171	739	8	)	)	PUNCT
ejpam-6171	739	9	}	}	PUNCT
ejpam-6171	739	10	.	.	PUNCT
ejpam-6171	740	1	case	case	NOUN
ejpam-6171	740	2	1	1	NUM
ejpam-6171	740	3	”	"	PUNCT
ejpam-6171	740	4	:	:	PUNCT
ejpam-6171	740	5	suppose	suppose	VERB
ejpam-6171	740	6	x	x	X
ejpam-6171	740	7	?	?	PUNCT
ejpam-6171	741	1	y	y	PROPN
ejpam-6171	741	2	∈	∈	PROPN
ejpam-6171	741	3	g	g	PROPN
ejpam-6171	741	4	and	and	CCONJ
ejpam-6171	741	5	x	x	PROPN
ejpam-6171	741	6	∈	∈	PROPN
ejpam-6171	741	7	g.	g.	NOUN
ejpam-6171	741	8	since	since	SCONJ
ejpam-6171	741	9	g	g	PROPN
ejpam-6171	741	10	is	be	AUX
ejpam-6171	741	11	an	an	DET
ejpam-6171	741	12	iup	iup	NOUN
ejpam-6171	741	13	-	-	PUNCT
ejpam-6171	741	14	filter	filter	NOUN
ejpam-6171	741	15	of	of	ADP
ejpam-6171	741	16	x	x	PRON
ejpam-6171	741	17	,	,	PUNCT
ejpam-6171	741	18	we	we	PRON
ejpam-6171	741	19	have	have	VERB
ejpam-6171	741	20	y	y	PROPN
ejpam-6171	741	21	∈	∈	PROPN
ejpam-6171	741	22	g.	g.	PROPN
ejpam-6171	742	1	thus	thus	ADV
ejpam-6171	742	2	,	,	PUNCT
ejpam-6171	742	3	pg	pg	PROPN
ejpam-6171	742	4	f	f	PROPN
ejpam-6171	743	1	[	[	X
ejpam-6171	743	2	γ	γ	X
ejpam-6171	743	3	+	+	X
ejpam-6171	743	4	γ−	γ−	PROPN
ejpam-6171	743	5	]	]	PUNCT
ejpam-6171	743	6	(	(	PUNCT
ejpam-6171	743	7	y	y	NOUN
ejpam-6171	743	8	)	)	PUNCT
ejpam-6171	743	9	=	=	PRON
ejpam-6171	743	10	γ+	γ+	PUNCT
ejpam-6171	743	11	≥	≥	NOUN
ejpam-6171	743	12	γ+	γ+	X
ejpam-6171	743	13	=	=	SYM
ejpam-6171	743	14	min{γ+	min{γ+	NOUN
ejpam-6171	743	15	,	,	PUNCT
ejpam-6171	743	16	γ+	γ+	X
ejpam-6171	743	17	}	}	PUNCT
ejpam-6171	743	18	=	=	PUNCT
ejpam-6171	743	19	min{pg	min{pg	ADP
ejpam-6171	743	20	f	f	X
ejpam-6171	744	1	[	[	X
ejpam-6171	744	2	γ	γ	X
ejpam-6171	744	3	+	+	X
ejpam-6171	744	4	γ−	γ−	PROPN
ejpam-6171	744	5	]	]	PUNCT
ejpam-6171	744	6	(	(	PUNCT
ejpam-6171	744	7	x	x	X
ejpam-6171	744	8	?	?	PUNCT
ejpam-6171	745	1	y),pg	y),pg	X
ejpam-6171	745	2	f	f	X
ejpam-6171	746	1	[	[	X
ejpam-6171	746	2	γ	γ	X
ejpam-6171	746	3	+	+	X
ejpam-6171	746	4	γ−	γ−	PROPN
ejpam-6171	746	5	]	]	PUNCT
ejpam-6171	746	6	(	(	PUNCT
ejpam-6171	746	7	x	x	NOUN
ejpam-6171	746	8	)	)	PUNCT
ejpam-6171	746	9	}	}	PUNCT
ejpam-6171	746	10	.	.	PUNCT
ejpam-6171	747	1	case	case	NOUN
ejpam-6171	747	2	2	2	NUM
ejpam-6171	747	3	”	"	PUNCT
ejpam-6171	747	4	:	:	PUNCT
ejpam-6171	747	5	suppose	suppose	VERB
ejpam-6171	747	6	x	x	X
ejpam-6171	747	7	?	?	PUNCT
ejpam-6171	748	1	y	y	PROPN
ejpam-6171	748	2	/∈	/∈	PUNCT
ejpam-6171	749	1	g	g	NOUN
ejpam-6171	749	2	or	or	CCONJ
ejpam-6171	749	3	x	x	PROPN
ejpam-6171	749	4	/∈	/∈	PROPN
ejpam-6171	750	1	g.	g.	NOUN
ejpam-6171	751	1	then	then	ADV
ejpam-6171	751	2	pg	pg	VERB
ejpam-6171	751	3	f	f	PROPN
ejpam-6171	752	1	[	[	X
ejpam-6171	752	2	γ	γ	X
ejpam-6171	752	3	+	+	X
ejpam-6171	752	4	γ−	γ−	PROPN
ejpam-6171	752	5	]	]	PUNCT
ejpam-6171	752	6	(	(	PUNCT
ejpam-6171	752	7	x	x	X
ejpam-6171	752	8	?	?	PUNCT
ejpam-6171	753	1	y	y	X
ejpam-6171	753	2	)	)	PUNCT
ejpam-6171	754	1	=	=	SYM
ejpam-6171	754	2	γ−	γ−	PROPN
ejpam-6171	754	3	or	or	CCONJ
ejpam-6171	754	4	pg	pg	NOUN
ejpam-6171	754	5	f	f	PROPN
ejpam-6171	755	1	[	[	X
ejpam-6171	755	2	γ	γ	X
ejpam-6171	755	3	+	+	X
ejpam-6171	755	4	γ−	γ−	PROPN
ejpam-6171	755	5	]	]	PUNCT
ejpam-6171	755	6	(	(	PUNCT
ejpam-6171	755	7	x	x	X
ejpam-6171	755	8	)	)	PUNCT
ejpam-6171	755	9	=	=	NOUN
ejpam-6171	755	10	γ−.	γ−.	NOUN
ejpam-6171	755	11	thus	thus	ADV
ejpam-6171	755	12	,	,	PUNCT
ejpam-6171	755	13	pg	pg	PROPN
ejpam-6171	755	14	f	f	PROPN
ejpam-6171	756	1	[	[	X
ejpam-6171	756	2	γ	γ	X
ejpam-6171	756	3	+	+	X
ejpam-6171	756	4	γ−	γ−	PROPN
ejpam-6171	756	5	]	]	PUNCT
ejpam-6171	756	6	(	(	PUNCT
ejpam-6171	756	7	y	y	NOUN
ejpam-6171	756	8	)	)	PUNCT
ejpam-6171	756	9	≥	≥	NOUN
ejpam-6171	756	10	γ−	γ−	NOUN
ejpam-6171	756	11	=	=	NOUN
ejpam-6171	756	12	min{pg	min{pg	X
ejpam-6171	756	13	f	f	X
ejpam-6171	757	1	[	[	X
ejpam-6171	757	2	γ	γ	X
ejpam-6171	757	3	+	+	X
ejpam-6171	757	4	γ−	γ−	PROPN
ejpam-6171	757	5	]	]	PUNCT
ejpam-6171	757	6	(	(	PUNCT
ejpam-6171	757	7	x	x	X
ejpam-6171	757	8	?	?	PUNCT
ejpam-6171	758	1	y),pg	y),pg	X
ejpam-6171	758	2	f	f	X
ejpam-6171	759	1	[	[	X
ejpam-6171	759	2	γ	γ	X
ejpam-6171	759	3	+	+	X
ejpam-6171	759	4	γ−	γ−	PROPN
ejpam-6171	759	5	]	]	PUNCT
ejpam-6171	759	6	(	(	PUNCT
ejpam-6171	759	7	x	x	NOUN
ejpam-6171	759	8	)	)	PUNCT
ejpam-6171	759	9	}	}	PUNCT
ejpam-6171	759	10	.	.	PUNCT
ejpam-6171	760	1	hence	hence	ADV
ejpam-6171	760	2	,	,	PUNCT
ejpam-6171	760	3	pg[α	pg[α	PROPN
ejpam-6171	760	4	+	+	PROPN
ejpam-6171	760	5	,	,	PUNCT
ejpam-6171	760	6	β−,γ+	β−,γ+	X
ejpam-6171	760	7	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	760	8	]	]	PUNCT
ejpam-6171	760	9	is	be	AUX
ejpam-6171	760	10	a	a	DET
ejpam-6171	760	11	pythagorean	pythagorean	PROPN
ejpam-6171	760	12	neutrosophic	neutrosophic	ADJ
ejpam-6171	760	13	iup	iup	NOUN
ejpam-6171	760	14	-	-	PUNCT
ejpam-6171	760	15	filter	filter	NOUN
ejpam-6171	760	16	of	of	ADP
ejpam-6171	760	17	x.	x.	NOUN
ejpam-6171	760	18	conversely	conversely	ADV
ejpam-6171	760	19	,	,	PUNCT
ejpam-6171	760	20	assume	assume	VERB
ejpam-6171	760	21	that	that	SCONJ
ejpam-6171	760	22	the	the	DET
ejpam-6171	760	23	characteristic	characteristic	ADJ
ejpam-6171	760	24	pns	pns	PROPN
ejpam-6171	760	25	pg[α	pg[α	PROPN
ejpam-6171	760	26	+	+	PROPN
ejpam-6171	760	27	,	,	PUNCT
ejpam-6171	760	28	β−,γ+	β−,γ+	X
ejpam-6171	760	29	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	760	30	]	]	PUNCT
ejpam-6171	760	31	is	be	AUX
ejpam-6171	760	32	a	a	DET
ejpam-6171	760	33	pythagorean	pythagorean	PROPN
ejpam-6171	760	34	neutrosophic	neutrosophic	ADJ
ejpam-6171	760	35	iup	iup	NOUN
ejpam-6171	760	36	-	-	PUNCT
ejpam-6171	760	37	filter	filter	NOUN
ejpam-6171	760	38	of	of	ADP
ejpam-6171	760	39	x.	x.	NOUN
ejpam-6171	760	40	since	since	SCONJ
ejpam-6171	760	41	pg	pg	PROPN
ejpam-6171	760	42	t	t	PROPN
ejpam-6171	761	1	[	[	X
ejpam-6171	761	2	α	α	X
ejpam-6171	761	3	+	+	X
ejpam-6171	761	4	α−	α−	ADP
ejpam-6171	761	5	]	]	PUNCT
ejpam-6171	761	6	satisfies	satisfie	NOUN
ejpam-6171	761	7	(	(	PUNCT
ejpam-6171	761	8	3.5	3.5	NUM
ejpam-6171	761	9	)	)	PUNCT
ejpam-6171	761	10	,	,	PUNCT
ejpam-6171	761	11	it	it	PRON
ejpam-6171	761	12	follow	follow	VERB
ejpam-6171	761	13	from	from	ADP
ejpam-6171	761	14	lemma	lemma	PROPN
ejpam-6171	761	15	2	2	NUM
ejpam-6171	761	16	that	that	SCONJ
ejpam-6171	761	17	0	0	NUM
ejpam-6171	761	18	∈	∈	NOUN
ejpam-6171	761	19	g.	g.	NOUN
ejpam-6171	761	20	next	next	ADV
ejpam-6171	761	21	,	,	PUNCT
ejpam-6171	761	22	let	let	VERB
ejpam-6171	761	23	x	x	PRON
ejpam-6171	761	24	,	,	PUNCT
ejpam-6171	761	25	y	y	PROPN
ejpam-6171	761	26	∈	∈	PROPN
ejpam-6171	761	27	x	x	AUX
ejpam-6171	761	28	be	be	AUX
ejpam-6171	761	29	such	such	ADJ
ejpam-6171	761	30	that	that	PRON
ejpam-6171	761	31	x	x	PUNCT
ejpam-6171	761	32	?	?	PUNCT
ejpam-6171	762	1	y	y	PROPN
ejpam-6171	762	2	∈	∈	PROPN
ejpam-6171	762	3	g	g	PROPN
ejpam-6171	762	4	and	and	CCONJ
ejpam-6171	762	5	x	x	PROPN
ejpam-6171	762	6	∈	∈	PROPN
ejpam-6171	762	7	g.	g.	NOUN
ejpam-6171	763	1	then	then	ADV
ejpam-6171	763	2	pg	pg	PROPN
ejpam-6171	763	3	t	t	PROPN
ejpam-6171	764	1	[	[	X
ejpam-6171	764	2	α	α	X
ejpam-6171	764	3	+	+	X
ejpam-6171	765	1	α−	α−	ADP
ejpam-6171	765	2	]	]	X
ejpam-6171	765	3	(	(	PUNCT
ejpam-6171	765	4	x	x	X
ejpam-6171	765	5	?	?	PUNCT
ejpam-6171	765	6	y	y	X
ejpam-6171	765	7	)	)	PUNCT
ejpam-6171	765	8	=	=	PRON
ejpam-6171	766	1	α+	α+	PUNCT
ejpam-6171	766	2	and	and	CCONJ
ejpam-6171	766	3	pg	pg	X
ejpam-6171	766	4	t	t	PROPN
ejpam-6171	767	1	[	[	X
ejpam-6171	767	2	α	α	X
ejpam-6171	767	3	+	+	X
ejpam-6171	768	1	α−	α−	ADP
ejpam-6171	768	2	]	]	X
ejpam-6171	768	3	(	(	PUNCT
ejpam-6171	768	4	x	x	X
ejpam-6171	768	5	)	)	PUNCT
ejpam-6171	768	6	=	=	SYM
ejpam-6171	768	7	α+	α+	NOUN
ejpam-6171	768	8	.	.	PUNCT
ejpam-6171	769	1	thus	thus	ADV
ejpam-6171	769	2	,	,	PUNCT
ejpam-6171	769	3	min{pg	min{pg	ADP
ejpam-6171	769	4	t	t	X
ejpam-6171	770	1	[	[	X
ejpam-6171	770	2	α	α	X
ejpam-6171	770	3	+	+	X
ejpam-6171	771	1	α−	α−	ADP
ejpam-6171	771	2	]	]	X
ejpam-6171	771	3	(	(	PUNCT
ejpam-6171	771	4	x	x	X
ejpam-6171	771	5	?	?	PUNCT
ejpam-6171	771	6	y),pg	y),pg	NOUN
ejpam-6171	771	7	t	t	PROPN
ejpam-6171	772	1	[	[	X
ejpam-6171	772	2	α	α	X
ejpam-6171	772	3	+	+	X
ejpam-6171	773	1	α−	α−	ADP
ejpam-6171	773	2	]	]	X
ejpam-6171	773	3	(	(	PUNCT
ejpam-6171	773	4	x	x	NOUN
ejpam-6171	773	5	)	)	PUNCT
ejpam-6171	773	6	}	}	PUNCT
ejpam-6171	773	7	=	=	SYM
ejpam-6171	773	8	α+	α+	NOUN
ejpam-6171	773	9	.	.	PUNCT
ejpam-6171	773	10	by	by	ADP
ejpam-6171	773	11	(	(	PUNCT
ejpam-6171	773	12	3.11	3.11	NUM
ejpam-6171	773	13	)	)	PUNCT
ejpam-6171	773	14	,	,	PUNCT
ejpam-6171	773	15	we	we	PRON
ejpam-6171	773	16	have	have	VERB
ejpam-6171	773	17	pg	pg	PROPN
ejpam-6171	773	18	t	t	PROPN
ejpam-6171	774	1	[	[	X
ejpam-6171	774	2	α	α	X
ejpam-6171	774	3	+	+	X
ejpam-6171	775	1	α−	α−	ADP
ejpam-6171	775	2	]	]	X
ejpam-6171	775	3	(	(	PUNCT
ejpam-6171	775	4	y	y	NOUN
ejpam-6171	775	5	)	)	PUNCT
ejpam-6171	775	6	=	=	PUNCT
ejpam-6171	775	7	min{pg	min{pg	ADP
ejpam-6171	775	8	t	t	X
ejpam-6171	776	1	[	[	X
ejpam-6171	776	2	α	α	X
ejpam-6171	776	3	+	+	X
ejpam-6171	777	1	α−	α−	ADP
ejpam-6171	777	2	]	]	X
ejpam-6171	777	3	(	(	PUNCT
ejpam-6171	777	4	x	x	X
ejpam-6171	777	5	?	?	PUNCT
ejpam-6171	777	6	y),pg	y),pg	NOUN
ejpam-6171	777	7	t	t	PROPN
ejpam-6171	778	1	[	[	X
ejpam-6171	778	2	α	α	X
ejpam-6171	778	3	+	+	X
ejpam-6171	779	1	α−	α−	ADP
ejpam-6171	779	2	]	]	X
ejpam-6171	779	3	(	(	PUNCT
ejpam-6171	779	4	x	x	NOUN
ejpam-6171	779	5	)	)	PUNCT
ejpam-6171	779	6	}	}	PUNCT
ejpam-6171	779	7	=	=	SYM
ejpam-6171	779	8	α+	α+	NOUN
ejpam-6171	779	9	,	,	PUNCT
ejpam-6171	779	10	that	that	ADV
ejpam-6171	779	11	is	is	ADV
ejpam-6171	779	12	,	,	PUNCT
ejpam-6171	779	13	pg	pg	PROPN
ejpam-6171	779	14	t	t	PROPN
ejpam-6171	780	1	[	[	X
ejpam-6171	780	2	α	α	X
ejpam-6171	780	3	+	+	X
ejpam-6171	781	1	α−	α−	ADP
ejpam-6171	781	2	]	]	X
ejpam-6171	781	3	(	(	PUNCT
ejpam-6171	781	4	y	y	NOUN
ejpam-6171	781	5	)	)	PUNCT
ejpam-6171	781	6	=	=	SYM
ejpam-6171	781	7	α+	α+	NOUN
ejpam-6171	781	8	.	.	PUNCT
ejpam-6171	782	1	hence	hence	ADV
ejpam-6171	782	2	,	,	PUNCT
ejpam-6171	782	3	y	y	PROPN
ejpam-6171	782	4	∈	∈	PROPN
ejpam-6171	782	5	g	g	PROPN
ejpam-6171	782	6	,	,	PUNCT
ejpam-6171	782	7	so	so	SCONJ
ejpam-6171	782	8	g	g	PROPN
ejpam-6171	782	9	is	be	AUX
ejpam-6171	782	10	an	an	DET
ejpam-6171	782	11	iup	iup	NOUN
ejpam-6171	782	12	-	-	PUNCT
ejpam-6171	782	13	filter	filter	NOUN
ejpam-6171	782	14	of	of	ADP
ejpam-6171	782	15	x.	x.	PROPN
ejpam-6171	782	16	theorem	theorem	VERB
ejpam-6171	782	17	12	12	NUM
ejpam-6171	782	18	.	.	PUNCT
ejpam-6171	783	1	a	a	DET
ejpam-6171	783	2	nonempty	nonempty	NOUN
ejpam-6171	783	3	subset	subset	VERB
ejpam-6171	783	4	g	g	NOUN
ejpam-6171	783	5	is	be	AUX
ejpam-6171	783	6	a	a	DET
ejpam-6171	783	7	strong	strong	ADJ
ejpam-6171	783	8	iup	iup	NOUN
ejpam-6171	783	9	-	-	PUNCT
ejpam-6171	783	10	ideal	ideal	NOUN
ejpam-6171	783	11	of	of	ADP
ejpam-6171	783	12	x	x	SYM
ejpam-6171	783	13	if	if	SCONJ
ejpam-6171	783	14	and	and	CCONJ
ejpam-6171	783	15	only	only	ADV
ejpam-6171	783	16	if	if	SCONJ
ejpam-6171	783	17	the	the	DET
ejpam-6171	783	18	characteristic	characteristic	ADJ
ejpam-6171	783	19	pns	pns	PROPN
ejpam-6171	783	20	pg[α	pg[α	PROPN
ejpam-6171	783	21	+	+	PROPN
ejpam-6171	783	22	,	,	PUNCT
ejpam-6171	783	23	β−,γ+	β−,γ+	X
ejpam-6171	783	24	α−,β+,γ−	α−,β+,γ−	X
ejpam-6171	783	25	]	]	PUNCT
ejpam-6171	783	26	is	be	AUX
ejpam-6171	783	27	a	a	DET
ejpam-6171	783	28	pythagorean	pythagorean	PROPN
ejpam-6171	783	29	neutrosophic	neutrosophic	ADJ
ejpam-6171	783	30	strong	strong	ADJ
ejpam-6171	783	31	iup	iup	NOUN
ejpam-6171	783	32	-	-	PUNCT
ejpam-6171	783	33	ideal	ideal	NOUN
ejpam-6171	783	34	of	of	ADP
ejpam-6171	783	35	x.	x.	NOUN
ejpam-6171	783	36	proof	proof	NOUN
ejpam-6171	783	37	.	.	PUNCT
ejpam-6171	784	1	it	it	PRON
ejpam-6171	784	2	is	be	AUX
ejpam-6171	784	3	straightforward	straightforward	ADJ
ejpam-6171	784	4	by	by	ADP
ejpam-6171	784	5	theorem	theorem	NOUN
ejpam-6171	784	6	2	2	NUM
ejpam-6171	784	7	.	.	PUNCT
ejpam-6171	784	8	k.	k.	PROPN
ejpam-6171	785	1	suayngam	suayngam	PROPN
ejpam-6171	785	2	et	et	PROPN
ejpam-6171	785	3	al	al	PROPN
ejpam-6171	785	4	.	.	PUNCT
ejpam-6171	785	5	/	/	SYM
ejpam-6171	785	6	eur	eur	PROPN
ejpam-6171	785	7	.	.	PUNCT
ejpam-6171	786	1	j.	j.	PROPN
ejpam-6171	786	2	pure	pure	PROPN
ejpam-6171	786	3	appl	appl	PROPN
ejpam-6171	786	4	.	.	PROPN
ejpam-6171	786	5	math	math	PROPN
ejpam-6171	786	6	,	,	PUNCT
ejpam-6171	786	7	18	18	NUM
ejpam-6171	786	8	(	(	PUNCT
ejpam-6171	786	9	3	3	NUM
ejpam-6171	786	10	)	)	PUNCT
ejpam-6171	786	11	(	(	PUNCT
ejpam-6171	786	12	2025	2025	NUM
ejpam-6171	786	13	)	)	PUNCT
ejpam-6171	786	14	,	,	PUNCT
ejpam-6171	786	15	6171	6171	NUM
ejpam-6171	786	16	18	18	NUM
ejpam-6171	786	17	of	of	ADP
ejpam-6171	786	18	28	28	NUM
ejpam-6171	786	19	lemma	lemma	PROPN
ejpam-6171	786	20	3	3	X
ejpam-6171	786	21	.	.	PUNCT
ejpam-6171	787	1	let	let	VERB
ejpam-6171	787	2	f	f	PRON
ejpam-6171	787	3	be	be	AUX
ejpam-6171	787	4	an	an	DET
ejpam-6171	787	5	fs	fs	NOUN
ejpam-6171	787	6	in	in	ADP
ejpam-6171	787	7	a	a	DET
ejpam-6171	787	8	nonempty	nonempty	ADV
ejpam-6171	787	9	set	set	VERB
ejpam-6171	787	10	x	x	PUNCT
ejpam-6171	787	11	and	and	CCONJ
ejpam-6171	787	12	let	let	VERB
ejpam-6171	787	13	n	n	PRON
ejpam-6171	787	14	be	be	AUX
ejpam-6171	787	15	a	a	DET
ejpam-6171	787	16	positive	positive	ADJ
ejpam-6171	787	17	integer	integer	NOUN
ejpam-6171	787	18	.	.	PUNCT
ejpam-6171	788	1	then	then	ADV
ejpam-6171	788	2	the	the	DET
ejpam-6171	788	3	following	following	ADJ
ejpam-6171	788	4	statements	statement	NOUN
ejpam-6171	788	5	hold	hold	VERB
ejpam-6171	788	6	:	:	PUNCT
ejpam-6171	788	7	(	(	PUNCT
ejpam-6171	788	8	∀x	∀x	X
ejpam-6171	788	9	,	,	PUNCT
ejpam-6171	788	10	y	y	PROPN
ejpam-6171	788	11	∈	∈	PROPN
ejpam-6171	788	12	x	x	X
ejpam-6171	788	13	)	)	PUNCT
ejpam-6171	788	14	(	(	PUNCT
ejpam-6171	788	15	min{f(x	min{f(x	PROPN
ejpam-6171	788	16	)	)	PUNCT
ejpam-6171	788	17	,	,	PUNCT
ejpam-6171	788	18	f(y	f(y	NOUN
ejpam-6171	788	19	)	)	PUNCT
ejpam-6171	788	20	}	}	PUNCT
ejpam-6171	788	21	n	n	PROPN
ejpam-6171	788	22	=	=	SYM
ejpam-6171	788	23	min{f(x	min{f(x	PROPN
ejpam-6171	788	24	)	)	PUNCT
ejpam-6171	788	25	n	n	NOUN
ejpam-6171	788	26	,	,	PUNCT
ejpam-6171	788	27	f(y	f(y	NOUN
ejpam-6171	788	28	)	)	PUNCT
ejpam-6171	788	29	n	n	CCONJ
ejpam-6171	788	30	}	}	PUNCT
ejpam-6171	788	31	)	)	PUNCT
ejpam-6171	788	32	(	(	PUNCT
ejpam-6171	788	33	3.19	3.19	NUM
ejpam-6171	788	34	)	)	PUNCT
ejpam-6171	788	35	(	(	PUNCT
ejpam-6171	788	36	∀x	∀x	X
ejpam-6171	788	37	,	,	PUNCT
ejpam-6171	788	38	y	y	PROPN
ejpam-6171	788	39	∈	∈	PROPN
ejpam-6171	788	40	x	x	X
ejpam-6171	788	41	)	)	PUNCT
ejpam-6171	788	42	(	(	PUNCT
ejpam-6171	788	43	max{f(x	max{f(x	PROPN
ejpam-6171	788	44	)	)	PUNCT
ejpam-6171	788	45	,	,	PUNCT
ejpam-6171	788	46	f(y	f(y	NOUN
ejpam-6171	788	47	)	)	PUNCT
ejpam-6171	788	48	}	}	PUNCT
ejpam-6171	788	49	n	n	NOUN
ejpam-6171	788	50	=	=	SYM
ejpam-6171	788	51	max{f(x	max{f(x	PROPN
ejpam-6171	788	52	)	)	PUNCT
ejpam-6171	788	53	n	n	NOUN
ejpam-6171	788	54	,	,	PUNCT
ejpam-6171	788	55	f(y	f(y	NOUN
ejpam-6171	788	56	)	)	PUNCT
ejpam-6171	788	57	n	n	CCONJ
ejpam-6171	788	58	}	}	PUNCT
ejpam-6171	788	59	)	)	PUNCT
ejpam-6171	788	60	(	(	PUNCT
ejpam-6171	788	61	3.20	3.20	NUM
ejpam-6171	788	62	)	)	PUNCT
ejpam-6171	788	63	lemma	lemma	PROPN
ejpam-6171	788	64	4	4	X
ejpam-6171	788	65	.	.	PUNCT
ejpam-6171	789	1	let	let	VERB
ejpam-6171	789	2	f	f	PRON
ejpam-6171	789	3	be	be	AUX
ejpam-6171	789	4	an	an	DET
ejpam-6171	789	5	fs	fs	NOUN
ejpam-6171	789	6	in	in	ADP
ejpam-6171	789	7	a	a	DET
ejpam-6171	789	8	nonempty	nonempty	ADV
ejpam-6171	789	9	set	set	VERB
ejpam-6171	789	10	x	x	PUNCT
ejpam-6171	789	11	and	and	CCONJ
ejpam-6171	789	12	let	let	VERB
ejpam-6171	789	13	n	n	PRON
ejpam-6171	789	14	be	be	AUX
ejpam-6171	789	15	a	a	DET
ejpam-6171	789	16	positive	positive	ADJ
ejpam-6171	789	17	integer	integer	NOUN
ejpam-6171	789	18	.	.	PUNCT
ejpam-6171	790	1	then	then	ADV
ejpam-6171	790	2	the	the	DET
ejpam-6171	790	3	following	following	ADJ
ejpam-6171	790	4	statements	statement	NOUN
ejpam-6171	790	5	hold	hold	VERB
ejpam-6171	790	6	:	:	PUNCT
ejpam-6171	790	7	(	(	PUNCT
ejpam-6171	790	8	∀x	∀x	X
ejpam-6171	790	9	,	,	PUNCT
ejpam-6171	790	10	y	y	PROPN
ejpam-6171	790	11	,	,	PUNCT
ejpam-6171	790	12	z	z	PROPN
ejpam-6171	790	13	∈	∈	PROPN
ejpam-6171	790	14	x)(f(z	x)(f(z	PROPN
ejpam-6171	790	15	)	)	PUNCT
ejpam-6171	790	16	≥	≥	NOUN
ejpam-6171	790	17	min{f(x	min{f(x	NOUN
ejpam-6171	790	18	)	)	PUNCT
ejpam-6171	790	19	,	,	PUNCT
ejpam-6171	790	20	f(y	f(y	NOUN
ejpam-6171	790	21	)	)	PUNCT
ejpam-6171	790	22	}	}	PUNCT
ejpam-6171	790	23	⇔	⇔	X
ejpam-6171	790	24	fn(z	fn(z	NUM
ejpam-6171	790	25	)	)	PUNCT
ejpam-6171	790	26	≥	≥	NOUN
ejpam-6171	790	27	min{fn(x	min{fn(x	X
ejpam-6171	790	28	)	)	PUNCT
ejpam-6171	790	29	,	,	PUNCT
ejpam-6171	790	30	fn(y	fn(y	PROPN
ejpam-6171	790	31	)	)	PUNCT
ejpam-6171	790	32	}	}	PUNCT
ejpam-6171	790	33	)	)	PUNCT
ejpam-6171	790	34	(	(	PUNCT
ejpam-6171	790	35	3.21	3.21	NUM
ejpam-6171	790	36	)	)	PUNCT
ejpam-6171	790	37	(	(	PUNCT
ejpam-6171	790	38	∀x	∀x	X
ejpam-6171	790	39	,	,	PUNCT
ejpam-6171	790	40	y	y	PROPN
ejpam-6171	790	41	,	,	PUNCT
ejpam-6171	790	42	z	z	PROPN
ejpam-6171	790	43	∈	∈	PROPN
ejpam-6171	790	44	x)(f(z	x)(f(z	PROPN
ejpam-6171	790	45	)	)	PUNCT
ejpam-6171	790	46	≤	≤	NUM
ejpam-6171	791	1	max{f(x	max{f(x	PROPN
ejpam-6171	791	2	)	)	PUNCT
ejpam-6171	791	3	,	,	PUNCT
ejpam-6171	791	4	f(y	f(y	NOUN
ejpam-6171	791	5	)	)	PUNCT
ejpam-6171	791	6	}	}	PUNCT
ejpam-6171	791	7	⇔	⇔	X
ejpam-6171	791	8	fn(z	fn(z	NUM
ejpam-6171	791	9	)	)	PUNCT
ejpam-6171	791	10	≤	≤	NOUN
ejpam-6171	791	11	max{fn(x	max{fn(x	X
ejpam-6171	791	12	)	)	PUNCT
ejpam-6171	791	13	,	,	PUNCT
ejpam-6171	791	14	fn(y	fn(y	PROPN
ejpam-6171	791	15	)	)	PUNCT
ejpam-6171	791	16	}	}	PUNCT
ejpam-6171	791	17	)	)	PUNCT
ejpam-6171	791	18	(	(	PUNCT
ejpam-6171	791	19	3.22	3.22	NUM
ejpam-6171	791	20	)	)	PUNCT
ejpam-6171	791	21	proof	proof	NOUN
ejpam-6171	791	22	.	.	PUNCT
ejpam-6171	792	1	it	it	PRON
ejpam-6171	792	2	is	be	AUX
ejpam-6171	792	3	straightforward	straightforward	ADJ
ejpam-6171	792	4	by	by	ADP
ejpam-6171	792	5	theorem	theorem	ADJ
ejpam-6171	792	6	3	3	NUM
ejpam-6171	792	7	.	.	PUNCT
ejpam-6171	792	8	theorem	theorem	VERB
ejpam-6171	792	9	13	13	NUM
ejpam-6171	792	10	.	.	PUNCT
ejpam-6171	793	1	a	a	DET
ejpam-6171	793	2	pns	pns	PROPN
ejpam-6171	793	3	p	p	PROPN
ejpam-6171	793	4	is	be	AUX
ejpam-6171	793	5	a	a	DET
ejpam-6171	793	6	pythagorean	pythagorean	PROPN
ejpam-6171	793	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	793	8	iup	iup	NOUN
ejpam-6171	793	9	-	-	PUNCT
ejpam-6171	793	10	subalgebra	subalgebra	NOUN
ejpam-6171	793	11	of	of	ADP
ejpam-6171	793	12	x	x	PRON
ejpam-6171	793	13	if	if	SCONJ
ejpam-6171	793	14	and	and	CCONJ
ejpam-6171	793	15	only	only	ADV
ejpam-6171	793	16	if	if	SCONJ
ejpam-6171	793	17	a	a	DET
ejpam-6171	793	18	pns	pns	NOUN
ejpam-6171	793	19	pn	pn	X
ejpam-6171	793	20	=	=	SYM
ejpam-6171	793	21	(	(	PUNCT
ejpam-6171	793	22	x	x	PROPN
ejpam-6171	793	23	,	,	PUNCT
ejpam-6171	793	24	pt	pt	PROPN
ejpam-6171	793	25	n	n	CCONJ
ejpam-6171	793	26	,	,	PUNCT
ejpam-6171	793	27	pin	pin	NOUN
ejpam-6171	793	28	,	,	PUNCT
ejpam-6171	793	29	pf	pf	PROPN
ejpam-6171	793	30	n	n	CCONJ
ejpam-6171	793	31	)	)	PUNCT
ejpam-6171	793	32	is	be	AUX
ejpam-6171	793	33	pythagorean	pythagorean	PROPN
ejpam-6171	793	34	neutrosophic	neutrosophic	PROPN
ejpam-6171	793	35	iup	iup	PROPN
ejpam-6171	793	36	-	-	PUNCT
ejpam-6171	793	37	subalgebra	subalgebra	NOUN
ejpam-6171	793	38	of	of	ADP
ejpam-6171	793	39	x.	x.	NOUN
ejpam-6171	793	40	proof	proof	PROPN
ejpam-6171	793	41	.	.	PUNCT
ejpam-6171	794	1	assume	assume	VERB
ejpam-6171	794	2	that	that	SCONJ
ejpam-6171	794	3	p	p	NOUN
ejpam-6171	794	4	is	be	AUX
ejpam-6171	794	5	a	a	DET
ejpam-6171	794	6	pythagorean	pythagorean	PROPN
ejpam-6171	794	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	794	8	iup	iup	NOUN
ejpam-6171	794	9	-	-	PUNCT
ejpam-6171	794	10	subalgebra	subalgebra	NOUN
ejpam-6171	794	11	of	of	ADP
ejpam-6171	794	12	x.	x.	NOUN
ejpam-6171	794	13	then	then	ADV
ejpam-6171	794	14	pt	pt	PROPN
ejpam-6171	794	15	(	(	PUNCT
ejpam-6171	794	16	x	x	X
ejpam-6171	794	17	?	?	PUNCT
ejpam-6171	795	1	y	y	X
ejpam-6171	795	2	)	)	PUNCT
ejpam-6171	795	3	≥	≥	NOUN
ejpam-6171	795	4	min{pt	min{pt	X
ejpam-6171	796	1	(	(	PUNCT
ejpam-6171	796	2	x),pt	x),pt	PROPN
ejpam-6171	796	3	(	(	PUNCT
ejpam-6171	796	4	y	y	NOUN
ejpam-6171	796	5	)	)	PUNCT
ejpam-6171	796	6	}	}	PUNCT
ejpam-6171	796	7	,	,	PUNCT
ejpam-6171	796	8	pi(x	pi(x	NOUN
ejpam-6171	796	9	?	?	PUNCT
ejpam-6171	797	1	y	y	X
ejpam-6171	797	2	)	)	PUNCT
ejpam-6171	797	3	≤	≤	NUM
ejpam-6171	797	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	797	5	)	)	PUNCT
ejpam-6171	797	6	}	}	PUNCT
ejpam-6171	797	7	,	,	PUNCT
ejpam-6171	797	8	pf	pf	PROPN
ejpam-6171	797	9	(	(	PUNCT
ejpam-6171	797	10	x	x	PROPN
ejpam-6171	797	11	?	?	PUNCT
ejpam-6171	798	1	y	y	X
ejpam-6171	798	2	)	)	PUNCT
ejpam-6171	798	3	≥	≥	PROPN
ejpam-6171	798	4	min{pf	min{pf	PRON
ejpam-6171	798	5	(	(	PUNCT
ejpam-6171	798	6	x),pf	x),pf	PROPN
ejpam-6171	798	7	(	(	PUNCT
ejpam-6171	798	8	y	y	NOUN
ejpam-6171	798	9	)	)	PUNCT
ejpam-6171	798	10	}	}	PUNCT
ejpam-6171	798	11	.	.	PUNCT
ejpam-6171	799	1	thus	thus	ADV
ejpam-6171	799	2	,	,	PUNCT
ejpam-6171	799	3	pt	pt	INTJ
ejpam-6171	799	4	n(x	n(x	PROPN
ejpam-6171	799	5	?	?	PUNCT
ejpam-6171	800	1	y	y	X
ejpam-6171	800	2	)	)	PUNCT
ejpam-6171	800	3	≥	≥	NOUN
ejpam-6171	800	4	min{pt	min{pt	PART
ejpam-6171	800	5	n(x),pt	n(x),pt	PROPN
ejpam-6171	800	6	n(y	n(y	PROPN
ejpam-6171	800	7	)	)	PUNCT
ejpam-6171	800	8	}	}	PUNCT
ejpam-6171	800	9	,	,	PUNCT
ejpam-6171	800	10	(	(	PUNCT
ejpam-6171	800	11	by	by	ADP
ejpam-6171	800	12	(	(	PUNCT
ejpam-6171	800	13	3.21	3.21	NUM
ejpam-6171	800	14	)	)	PUNCT
ejpam-6171	800	15	)	)	PUNCT
ejpam-6171	801	1	pin(x	pin(x	PROPN
ejpam-6171	801	2	?	?	PUNCT
ejpam-6171	802	1	y	y	X
ejpam-6171	802	2	)	)	PUNCT
ejpam-6171	802	3	≤	≤	NOUN
ejpam-6171	802	4	max{pin(x),pin(y	max{pin(x),pin(y	NOUN
ejpam-6171	802	5	)	)	PUNCT
ejpam-6171	802	6	}	}	PUNCT
ejpam-6171	802	7	,	,	PUNCT
ejpam-6171	802	8	(	(	PUNCT
ejpam-6171	802	9	by	by	ADP
ejpam-6171	802	10	(	(	PUNCT
ejpam-6171	802	11	3.22	3.22	NUM
ejpam-6171	802	12	)	)	PUNCT
ejpam-6171	802	13	)	)	PUNCT
ejpam-6171	802	14	pf	pf	ADP
ejpam-6171	802	15	n(x	n(x	PROPN
ejpam-6171	802	16	?	?	PUNCT
ejpam-6171	803	1	y	y	X
ejpam-6171	803	2	)	)	PUNCT
ejpam-6171	803	3	≥	≥	NOUN
ejpam-6171	803	4	min{pf	min{pf	X
ejpam-6171	803	5	n(x),pf	n(x),pf	PROPN
ejpam-6171	803	6	n(y	n(y	PROPN
ejpam-6171	803	7	)	)	PUNCT
ejpam-6171	803	8	}	}	PUNCT
ejpam-6171	803	9	.	.	PUNCT
ejpam-6171	804	1	(	(	PUNCT
ejpam-6171	804	2	by	by	ADP
ejpam-6171	804	3	(	(	PUNCT
ejpam-6171	804	4	3.21	3.21	NUM
ejpam-6171	804	5	)	)	PUNCT
ejpam-6171	804	6	)	)	PUNCT
ejpam-6171	804	7	hence	hence	ADV
ejpam-6171	804	8	,	,	PUNCT
ejpam-6171	804	9	a	a	DET
ejpam-6171	804	10	pns	pns	NOUN
ejpam-6171	804	11	pn	pn	X
ejpam-6171	804	12	=	=	SYM
ejpam-6171	804	13	(	(	PUNCT
ejpam-6171	804	14	x	x	PROPN
ejpam-6171	804	15	,	,	PUNCT
ejpam-6171	804	16	pt	pt	PROPN
ejpam-6171	804	17	n	n	CCONJ
ejpam-6171	804	18	,	,	PUNCT
ejpam-6171	804	19	pin	pin	NOUN
ejpam-6171	804	20	,	,	PUNCT
ejpam-6171	804	21	pf	pf	PROPN
ejpam-6171	804	22	n	n	CCONJ
ejpam-6171	804	23	)	)	PUNCT
ejpam-6171	804	24	is	be	AUX
ejpam-6171	804	25	a	a	DET
ejpam-6171	804	26	pythagorean	pythagorean	PROPN
ejpam-6171	804	27	neutrosophic	neutrosophic	ADJ
ejpam-6171	804	28	iup	iup	NOUN
ejpam-6171	804	29	-	-	PUNCT
ejpam-6171	804	30	subalgebra	subalgebra	NOUN
ejpam-6171	804	31	of	of	ADP
ejpam-6171	804	32	x.	x.	NOUN
ejpam-6171	804	33	conversely	conversely	ADV
ejpam-6171	804	34	,	,	PUNCT
ejpam-6171	804	35	it	it	PRON
ejpam-6171	804	36	is	be	AUX
ejpam-6171	804	37	obvious	obvious	ADJ
ejpam-6171	804	38	to	to	PART
ejpam-6171	804	39	prove	prove	VERB
ejpam-6171	804	40	that	that	SCONJ
ejpam-6171	804	41	p	p	NOUN
ejpam-6171	804	42	is	be	AUX
ejpam-6171	804	43	a	a	DET
ejpam-6171	804	44	pythagorean	pythagorean	PROPN
ejpam-6171	804	45	neutrosophic	neutrosophic	ADJ
ejpam-6171	804	46	iup	iup	NOUN
ejpam-6171	804	47	-	-	PUNCT
ejpam-6171	804	48	subalgebra	subalgebra	NOUN
ejpam-6171	804	49	of	of	ADP
ejpam-6171	804	50	x.	x.	PROPN
ejpam-6171	804	51	theorem	theorem	VERB
ejpam-6171	804	52	14	14	NUM
ejpam-6171	804	53	.	.	PUNCT
ejpam-6171	805	1	a	a	DET
ejpam-6171	805	2	pns	pns	PROPN
ejpam-6171	805	3	p	p	PROPN
ejpam-6171	805	4	is	be	AUX
ejpam-6171	805	5	a	a	DET
ejpam-6171	805	6	pythagorean	pythagorean	PROPN
ejpam-6171	805	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	805	8	iup	iup	PROPN
ejpam-6171	805	9	-	-	PUNCT
ejpam-6171	805	10	ideal	ideal	NOUN
ejpam-6171	805	11	of	of	ADP
ejpam-6171	805	12	x	x	SYM
ejpam-6171	805	13	if	if	SCONJ
ejpam-6171	805	14	and	and	CCONJ
ejpam-6171	805	15	only	only	ADV
ejpam-6171	805	16	if	if	SCONJ
ejpam-6171	805	17	a	a	DET
ejpam-6171	805	18	pns	pns	NOUN
ejpam-6171	805	19	pn	pn	X
ejpam-6171	805	20	=	=	SYM
ejpam-6171	805	21	(	(	PUNCT
ejpam-6171	805	22	x	x	PROPN
ejpam-6171	805	23	,	,	PUNCT
ejpam-6171	805	24	pt	pt	PROPN
ejpam-6171	805	25	n	n	CCONJ
ejpam-6171	805	26	,	,	PUNCT
ejpam-6171	805	27	pin	pin	NOUN
ejpam-6171	805	28	,	,	PUNCT
ejpam-6171	805	29	pf	pf	PROPN
ejpam-6171	805	30	n	n	CCONJ
ejpam-6171	805	31	)	)	PUNCT
ejpam-6171	805	32	is	be	AUX
ejpam-6171	805	33	pythagorean	pythagorean	PROPN
ejpam-6171	805	34	neutrosophic	neutrosophic	PROPN
ejpam-6171	805	35	iup	iup	PROPN
ejpam-6171	805	36	-	-	PUNCT
ejpam-6171	805	37	ideal	ideal	NOUN
ejpam-6171	805	38	of	of	ADP
ejpam-6171	805	39	x.	x.	NOUN
ejpam-6171	805	40	proof	proof	PROPN
ejpam-6171	805	41	.	.	PUNCT
ejpam-6171	806	1	assume	assume	VERB
ejpam-6171	806	2	that	that	SCONJ
ejpam-6171	806	3	p	p	NOUN
ejpam-6171	806	4	is	be	AUX
ejpam-6171	806	5	a	a	DET
ejpam-6171	806	6	pythagorean	pythagorean	PROPN
ejpam-6171	806	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	806	8	iup	iup	PROPN
ejpam-6171	806	9	-	-	PUNCT
ejpam-6171	806	10	ideal	ideal	NOUN
ejpam-6171	806	11	of	of	ADP
ejpam-6171	806	12	x.	x.	NOUN
ejpam-6171	806	13	then	then	ADV
ejpam-6171	806	14	pt	pt	PROPN
ejpam-6171	806	15	(	(	PUNCT
ejpam-6171	806	16	0	0	NUM
ejpam-6171	806	17	)	)	PUNCT
ejpam-6171	806	18	≥	≥	NOUN
ejpam-6171	806	19	pt	pt	INTJ
ejpam-6171	806	20	(	(	PUNCT
ejpam-6171	806	21	x	x	NOUN
ejpam-6171	806	22	)	)	PUNCT
ejpam-6171	806	23	,	,	PUNCT
ejpam-6171	806	24	pi(0	pi(0	PROPN
ejpam-6171	806	25	)	)	PUNCT
ejpam-6171	806	26	≤	≤	NOUN
ejpam-6171	806	27	pi(x	pi(x	NOUN
ejpam-6171	806	28	)	)	PUNCT
ejpam-6171	806	29	,	,	PUNCT
ejpam-6171	806	30	pf	pf	PROPN
ejpam-6171	806	31	(	(	PUNCT
ejpam-6171	806	32	0	0	NUM
ejpam-6171	806	33	)	)	PUNCT
ejpam-6171	806	34	≥	≥	NOUN
ejpam-6171	806	35	pf	pf	X
ejpam-6171	806	36	(	(	PUNCT
ejpam-6171	806	37	x	x	NOUN
ejpam-6171	806	38	)	)	PUNCT
ejpam-6171	806	39	,	,	PUNCT
ejpam-6171	806	40	pt	pt	X
ejpam-6171	806	41	(	(	PUNCT
ejpam-6171	806	42	x	x	NOUN
ejpam-6171	806	43	?	?	PUNCT
ejpam-6171	807	1	z	z	X
ejpam-6171	807	2	)	)	PUNCT
ejpam-6171	807	3	≥	≥	NOUN
ejpam-6171	807	4	min{pt	min{pt	X
ejpam-6171	808	1	(	(	PUNCT
ejpam-6171	808	2	x	x	X
ejpam-6171	808	3	?	?	PUNCT
ejpam-6171	809	1	(	(	PUNCT
ejpam-6171	809	2	y	y	PROPN
ejpam-6171	809	3	?	?	PUNCT
ejpam-6171	810	1	z)),pt	z)),pt	PROPN
ejpam-6171	810	2	(	(	PUNCT
ejpam-6171	810	3	y	y	NOUN
ejpam-6171	810	4	)	)	PUNCT
ejpam-6171	810	5	}	}	PUNCT
ejpam-6171	810	6	,	,	PUNCT
ejpam-6171	810	7	k.	k.	PROPN
ejpam-6171	810	8	suayngam	suayngam	PROPN
ejpam-6171	810	9	et	et	PROPN
ejpam-6171	810	10	al	al	PROPN
ejpam-6171	810	11	.	.	PUNCT
ejpam-6171	810	12	/	/	SYM
ejpam-6171	810	13	eur	eur	PROPN
ejpam-6171	810	14	.	.	PUNCT
ejpam-6171	811	1	j.	j.	PROPN
ejpam-6171	811	2	pure	pure	PROPN
ejpam-6171	811	3	appl	appl	PROPN
ejpam-6171	811	4	.	.	PROPN
ejpam-6171	811	5	math	math	PROPN
ejpam-6171	811	6	,	,	PUNCT
ejpam-6171	811	7	18	18	NUM
ejpam-6171	811	8	(	(	PUNCT
ejpam-6171	811	9	3	3	NUM
ejpam-6171	811	10	)	)	PUNCT
ejpam-6171	811	11	(	(	PUNCT
ejpam-6171	811	12	2025	2025	NUM
ejpam-6171	811	13	)	)	PUNCT
ejpam-6171	811	14	,	,	PUNCT
ejpam-6171	811	15	6171	6171	NUM
ejpam-6171	811	16	19	19	NUM
ejpam-6171	811	17	of	of	ADP
ejpam-6171	811	18	28	28	NUM
ejpam-6171	811	19	pi(x	pi(x	NOUN
ejpam-6171	811	20	?	?	PUNCT
ejpam-6171	812	1	z	z	X
ejpam-6171	812	2	)	)	PUNCT
ejpam-6171	812	3	≤	≤	NUM
ejpam-6171	812	4	max{pi(x	max{pi(x	NOUN
ejpam-6171	812	5	?	?	PUNCT
ejpam-6171	813	1	(	(	PUNCT
ejpam-6171	813	2	y	y	NOUN
ejpam-6171	813	3	?	?	PUNCT
ejpam-6171	813	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	813	5	)	)	PUNCT
ejpam-6171	813	6	}	}	PUNCT
ejpam-6171	813	7	,	,	PUNCT
ejpam-6171	813	8	pf	pf	PROPN
ejpam-6171	813	9	(	(	PUNCT
ejpam-6171	813	10	x	x	PROPN
ejpam-6171	813	11	?	?	PUNCT
ejpam-6171	814	1	z	z	X
ejpam-6171	814	2	)	)	PUNCT
ejpam-6171	814	3	≥	≥	NOUN
ejpam-6171	814	4	min{pf	min{pf	X
ejpam-6171	814	5	(	(	PUNCT
ejpam-6171	814	6	x	x	X
ejpam-6171	814	7	?	?	PUNCT
ejpam-6171	815	1	(	(	PUNCT
ejpam-6171	815	2	y	y	NOUN
ejpam-6171	815	3	?	?	PUNCT
ejpam-6171	816	1	z)),pf	z)),pf	PROPN
ejpam-6171	816	2	(	(	PUNCT
ejpam-6171	816	3	y	y	NOUN
ejpam-6171	816	4	)	)	PUNCT
ejpam-6171	816	5	}	}	PUNCT
ejpam-6171	816	6	.	.	PUNCT
ejpam-6171	817	1	thus	thus	ADV
ejpam-6171	817	2	,	,	PUNCT
ejpam-6171	817	3	pt	pt	PROPN
ejpam-6171	817	4	n(0	n(0	PROPN
ejpam-6171	817	5	)	)	PUNCT
ejpam-6171	817	6	≥	≥	PROPN
ejpam-6171	817	7	pt	pt	X
ejpam-6171	817	8	n(x	n(x	PROPN
ejpam-6171	817	9	)	)	PUNCT
ejpam-6171	817	10	,	,	PUNCT
ejpam-6171	817	11	pin(0	pin(0	PROPN
ejpam-6171	817	12	)	)	PUNCT
ejpam-6171	817	13	≤	≤	NOUN
ejpam-6171	817	14	pin(x	pin(x	NUM
ejpam-6171	817	15	)	)	PUNCT
ejpam-6171	817	16	,	,	PUNCT
ejpam-6171	817	17	pf	pf	PROPN
ejpam-6171	817	18	n(0	n(0	PROPN
ejpam-6171	817	19	)	)	PUNCT
ejpam-6171	817	20	≥	≥	NOUN
ejpam-6171	817	21	pf	pf	X
ejpam-6171	817	22	n(x	n(x	PROPN
ejpam-6171	817	23	)	)	PUNCT
ejpam-6171	817	24	,	,	PUNCT
ejpam-6171	817	25	pt	pt	INTJ
ejpam-6171	817	26	n(x	n(x	PROPN
ejpam-6171	817	27	?	?	PUNCT
ejpam-6171	818	1	z	z	X
ejpam-6171	818	2	)	)	PUNCT
ejpam-6171	818	3	≥	≥	NOUN
ejpam-6171	818	4	min{pt	min{pt	NOUN
ejpam-6171	819	1	n(x	n(x	PROPN
ejpam-6171	819	2	?	?	PUNCT
ejpam-6171	820	1	(	(	PUNCT
ejpam-6171	820	2	y	y	PROPN
ejpam-6171	820	3	?	?	PUNCT
ejpam-6171	820	4	z)),pt	z)),pt	PROPN
ejpam-6171	820	5	n(y	n(y	PROPN
ejpam-6171	820	6	)	)	PUNCT
ejpam-6171	820	7	}	}	PUNCT
ejpam-6171	820	8	,	,	PUNCT
ejpam-6171	820	9	(	(	PUNCT
ejpam-6171	820	10	by	by	ADP
ejpam-6171	820	11	(	(	PUNCT
ejpam-6171	820	12	3.21	3.21	NUM
ejpam-6171	820	13	)	)	PUNCT
ejpam-6171	820	14	)	)	PUNCT
ejpam-6171	821	1	pin(x	pin(x	ADV
ejpam-6171	821	2	?	?	PUNCT
ejpam-6171	822	1	z	z	X
ejpam-6171	822	2	)	)	PUNCT
ejpam-6171	822	3	≤	≤	NOUN
ejpam-6171	822	4	max{pin(x	max{pin(x	PROPN
ejpam-6171	822	5	?	?	PUNCT
ejpam-6171	823	1	(	(	PUNCT
ejpam-6171	823	2	y	y	PROPN
ejpam-6171	823	3	?	?	PUNCT
ejpam-6171	824	1	z)),pin(y	z)),pin(y	NUM
ejpam-6171	824	2	)	)	PUNCT
ejpam-6171	824	3	}	}	PUNCT
ejpam-6171	824	4	,	,	PUNCT
ejpam-6171	824	5	(	(	PUNCT
ejpam-6171	824	6	by	by	ADP
ejpam-6171	824	7	(	(	PUNCT
ejpam-6171	824	8	3.22	3.22	NUM
ejpam-6171	824	9	)	)	PUNCT
ejpam-6171	824	10	)	)	PUNCT
ejpam-6171	824	11	pf	pf	ADP
ejpam-6171	824	12	n(x	n(x	PROPN
ejpam-6171	824	13	?	?	PUNCT
ejpam-6171	825	1	z	z	X
ejpam-6171	825	2	)	)	PUNCT
ejpam-6171	825	3	≥	≥	NOUN
ejpam-6171	825	4	min{pf	min{pf	X
ejpam-6171	825	5	n(x	n(x	PROPN
ejpam-6171	825	6	?	?	PUNCT
ejpam-6171	826	1	(	(	PUNCT
ejpam-6171	826	2	y	y	PROPN
ejpam-6171	826	3	?	?	PUNCT
ejpam-6171	827	1	z)),pf	z)),pf	PROPN
ejpam-6171	827	2	n(y	n(y	PROPN
ejpam-6171	827	3	)	)	PUNCT
ejpam-6171	827	4	}	}	PUNCT
ejpam-6171	827	5	.	.	PUNCT
ejpam-6171	828	1	(	(	PUNCT
ejpam-6171	828	2	by	by	ADP
ejpam-6171	828	3	(	(	PUNCT
ejpam-6171	828	4	3.21	3.21	NUM
ejpam-6171	828	5	)	)	PUNCT
ejpam-6171	828	6	)	)	PUNCT
ejpam-6171	828	7	hence	hence	ADV
ejpam-6171	828	8	,	,	PUNCT
ejpam-6171	828	9	a	a	DET
ejpam-6171	828	10	pns	pns	NOUN
ejpam-6171	828	11	pn	pn	X
ejpam-6171	828	12	=	=	SYM
ejpam-6171	828	13	(	(	PUNCT
ejpam-6171	828	14	x	x	PROPN
ejpam-6171	828	15	,	,	PUNCT
ejpam-6171	828	16	pt	pt	PROPN
ejpam-6171	828	17	n	n	CCONJ
ejpam-6171	828	18	,	,	PUNCT
ejpam-6171	828	19	pin	pin	NOUN
ejpam-6171	828	20	,	,	PUNCT
ejpam-6171	828	21	pf	pf	PROPN
ejpam-6171	828	22	n	n	CCONJ
ejpam-6171	828	23	)	)	PUNCT
ejpam-6171	828	24	is	be	AUX
ejpam-6171	828	25	pythagorean	pythagorean	PROPN
ejpam-6171	828	26	neutrosophic	neutrosophic	PROPN
ejpam-6171	828	27	iup	iup	PROPN
ejpam-6171	828	28	-	-	PUNCT
ejpam-6171	828	29	ideal	ideal	NOUN
ejpam-6171	828	30	of	of	ADP
ejpam-6171	828	31	x.	x.	NOUN
ejpam-6171	828	32	conversely	conversely	ADV
ejpam-6171	828	33	,	,	PUNCT
ejpam-6171	828	34	it	it	PRON
ejpam-6171	828	35	is	be	AUX
ejpam-6171	828	36	obvious	obvious	ADJ
ejpam-6171	828	37	to	to	PART
ejpam-6171	828	38	prove	prove	VERB
ejpam-6171	828	39	that	that	SCONJ
ejpam-6171	828	40	p	p	NOUN
ejpam-6171	828	41	is	be	AUX
ejpam-6171	828	42	a	a	DET
ejpam-6171	828	43	pythagorean	pythagorean	PROPN
ejpam-6171	828	44	neutrosophic	neutrosophic	ADJ
ejpam-6171	828	45	iup	iup	PROPN
ejpam-6171	828	46	-	-	PUNCT
ejpam-6171	828	47	ideal	ideal	NOUN
ejpam-6171	828	48	of	of	ADP
ejpam-6171	828	49	x.	x.	PROPN
ejpam-6171	828	50	theorem	theorem	VERB
ejpam-6171	828	51	15	15	NUM
ejpam-6171	828	52	.	.	PUNCT
ejpam-6171	829	1	a	a	DET
ejpam-6171	829	2	pns	pns	PROPN
ejpam-6171	829	3	p	p	PROPN
ejpam-6171	829	4	is	be	AUX
ejpam-6171	829	5	a	a	DET
ejpam-6171	829	6	pythagorean	pythagorean	PROPN
ejpam-6171	829	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	829	8	iup	iup	NOUN
ejpam-6171	829	9	-	-	PUNCT
ejpam-6171	829	10	filter	filter	NOUN
ejpam-6171	829	11	of	of	ADP
ejpam-6171	829	12	x	x	SYM
ejpam-6171	829	13	if	if	SCONJ
ejpam-6171	829	14	and	and	CCONJ
ejpam-6171	829	15	only	only	ADV
ejpam-6171	829	16	if	if	SCONJ
ejpam-6171	829	17	a	a	DET
ejpam-6171	829	18	pns	pns	NOUN
ejpam-6171	829	19	pn	pn	X
ejpam-6171	829	20	=	=	SYM
ejpam-6171	829	21	(	(	PUNCT
ejpam-6171	829	22	x	x	PROPN
ejpam-6171	829	23	,	,	PUNCT
ejpam-6171	829	24	pt	pt	PROPN
ejpam-6171	829	25	n	n	CCONJ
ejpam-6171	829	26	,	,	PUNCT
ejpam-6171	829	27	pin	pin	NOUN
ejpam-6171	829	28	,	,	PUNCT
ejpam-6171	829	29	pf	pf	PROPN
ejpam-6171	829	30	n	n	CCONJ
ejpam-6171	829	31	)	)	PUNCT
ejpam-6171	829	32	is	be	AUX
ejpam-6171	829	33	pythagorean	pythagorean	PROPN
ejpam-6171	829	34	neutrosophic	neutrosophic	PROPN
ejpam-6171	829	35	iup	iup	PROPN
ejpam-6171	829	36	-	-	PUNCT
ejpam-6171	829	37	filter	filter	NOUN
ejpam-6171	829	38	of	of	ADP
ejpam-6171	829	39	x.	x.	NOUN
ejpam-6171	829	40	proof	proof	PROPN
ejpam-6171	829	41	.	.	PUNCT
ejpam-6171	830	1	assume	assume	VERB
ejpam-6171	830	2	that	that	SCONJ
ejpam-6171	830	3	p	p	NOUN
ejpam-6171	830	4	is	be	AUX
ejpam-6171	830	5	a	a	DET
ejpam-6171	830	6	pythagorean	pythagorean	PROPN
ejpam-6171	830	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	830	8	iup	iup	NOUN
ejpam-6171	830	9	-	-	PUNCT
ejpam-6171	830	10	filter	filter	NOUN
ejpam-6171	830	11	of	of	ADP
ejpam-6171	830	12	x.	x.	NOUN
ejpam-6171	830	13	then	then	ADV
ejpam-6171	830	14	pt	pt	PROPN
ejpam-6171	830	15	(	(	PUNCT
ejpam-6171	830	16	0	0	NUM
ejpam-6171	830	17	)	)	PUNCT
ejpam-6171	830	18	≥	≥	NOUN
ejpam-6171	830	19	pt	pt	INTJ
ejpam-6171	830	20	(	(	PUNCT
ejpam-6171	830	21	x	x	NOUN
ejpam-6171	830	22	)	)	PUNCT
ejpam-6171	830	23	,	,	PUNCT
ejpam-6171	830	24	pi(0	pi(0	PROPN
ejpam-6171	830	25	)	)	PUNCT
ejpam-6171	830	26	≤	≤	NOUN
ejpam-6171	830	27	pi(x	pi(x	NOUN
ejpam-6171	830	28	)	)	PUNCT
ejpam-6171	830	29	,	,	PUNCT
ejpam-6171	830	30	pf	pf	PROPN
ejpam-6171	830	31	(	(	PUNCT
ejpam-6171	830	32	0	0	NUM
ejpam-6171	830	33	)	)	PUNCT
ejpam-6171	830	34	≥	≥	NOUN
ejpam-6171	830	35	pf	pf	X
ejpam-6171	830	36	(	(	PUNCT
ejpam-6171	830	37	x	x	NOUN
ejpam-6171	830	38	)	)	PUNCT
ejpam-6171	830	39	,	,	PUNCT
ejpam-6171	830	40	pt	pt	X
ejpam-6171	830	41	(	(	PUNCT
ejpam-6171	830	42	y	y	NOUN
ejpam-6171	830	43	)	)	PUNCT
ejpam-6171	830	44	≥	≥	NOUN
ejpam-6171	830	45	min{pt	min{pt	X
ejpam-6171	831	1	(	(	PUNCT
ejpam-6171	831	2	x	x	X
ejpam-6171	831	3	?	?	PUNCT
ejpam-6171	831	4	y),pt	y),pt	PROPN
ejpam-6171	831	5	(	(	PUNCT
ejpam-6171	831	6	x	x	NOUN
ejpam-6171	831	7	)	)	PUNCT
ejpam-6171	831	8	}	}	PUNCT
ejpam-6171	831	9	,	,	PUNCT
ejpam-6171	831	10	pi(y	pi(y	NOUN
ejpam-6171	831	11	)	)	PUNCT
ejpam-6171	831	12	≤	≤	NUM
ejpam-6171	831	13	max{pi(x	max{pi(x	NOUN
ejpam-6171	831	14	?	?	PUNCT
ejpam-6171	832	1	y),pi(x	y),pi(x	NUM
ejpam-6171	832	2	)	)	PUNCT
ejpam-6171	832	3	}	}	PUNCT
ejpam-6171	832	4	,	,	PUNCT
ejpam-6171	832	5	pf	pf	PROPN
ejpam-6171	832	6	(	(	PUNCT
ejpam-6171	832	7	y	y	NOUN
ejpam-6171	832	8	)	)	PUNCT
ejpam-6171	832	9	≥	≥	NOUN
ejpam-6171	832	10	min{pf	min{pf	X
ejpam-6171	832	11	(	(	PUNCT
ejpam-6171	832	12	x	x	X
ejpam-6171	832	13	?	?	PUNCT
ejpam-6171	833	1	y),pf	y),pf	PROPN
ejpam-6171	833	2	(	(	PUNCT
ejpam-6171	833	3	x	x	NOUN
ejpam-6171	833	4	)	)	PUNCT
ejpam-6171	833	5	}	}	PUNCT
ejpam-6171	833	6	.	.	PUNCT
ejpam-6171	834	1	thus	thus	ADV
ejpam-6171	834	2	,	,	PUNCT
ejpam-6171	834	3	pt	pt	PROPN
ejpam-6171	834	4	n(0	n(0	PROPN
ejpam-6171	834	5	)	)	PUNCT
ejpam-6171	834	6	≥	≥	PROPN
ejpam-6171	834	7	pt	pt	X
ejpam-6171	834	8	n(x	n(x	PROPN
ejpam-6171	834	9	)	)	PUNCT
ejpam-6171	834	10	,	,	PUNCT
ejpam-6171	834	11	pin(0	pin(0	PROPN
ejpam-6171	834	12	)	)	PUNCT
ejpam-6171	834	13	≤	≤	NOUN
ejpam-6171	834	14	pin(x	pin(x	NUM
ejpam-6171	834	15	)	)	PUNCT
ejpam-6171	834	16	,	,	PUNCT
ejpam-6171	834	17	pf	pf	PROPN
ejpam-6171	834	18	n(0	n(0	PROPN
ejpam-6171	834	19	)	)	PUNCT
ejpam-6171	834	20	≥	≥	NOUN
ejpam-6171	834	21	pf	pf	X
ejpam-6171	834	22	n(x	n(x	PROPN
ejpam-6171	834	23	)	)	PUNCT
ejpam-6171	834	24	,	,	PUNCT
ejpam-6171	834	25	pt	pt	X
ejpam-6171	834	26	n(y	n(y	PROPN
ejpam-6171	834	27	)	)	PUNCT
ejpam-6171	834	28	≥	≥	NOUN
ejpam-6171	834	29	min{pt	min{pt	PUNCT
ejpam-6171	835	1	n(x	n(x	PROPN
ejpam-6171	835	2	?	?	PUNCT
ejpam-6171	835	3	y),pt	y),pt	PROPN
ejpam-6171	835	4	n(x	n(x	PROPN
ejpam-6171	835	5	)	)	PUNCT
ejpam-6171	835	6	}	}	PUNCT
ejpam-6171	835	7	,	,	PUNCT
ejpam-6171	835	8	(	(	PUNCT
ejpam-6171	835	9	by	by	ADP
ejpam-6171	835	10	(	(	PUNCT
ejpam-6171	835	11	3.21	3.21	NUM
ejpam-6171	835	12	)	)	PUNCT
ejpam-6171	835	13	)	)	PUNCT
ejpam-6171	835	14	pin(y	pin(y	X
ejpam-6171	835	15	)	)	PUNCT
ejpam-6171	835	16	≤	≤	NOUN
ejpam-6171	836	1	max{pin(x	max{pin(x	PROPN
ejpam-6171	836	2	?	?	PUNCT
ejpam-6171	837	1	y),pin(x	y),pin(x	PROPN
ejpam-6171	837	2	)	)	PUNCT
ejpam-6171	837	3	}	}	PUNCT
ejpam-6171	837	4	,	,	PUNCT
ejpam-6171	837	5	(	(	PUNCT
ejpam-6171	837	6	by	by	ADP
ejpam-6171	837	7	(	(	PUNCT
ejpam-6171	837	8	3.22	3.22	NUM
ejpam-6171	837	9	)	)	PUNCT
ejpam-6171	837	10	)	)	PUNCT
ejpam-6171	837	11	pf	pf	ADP
ejpam-6171	837	12	n(y	n(y	PROPN
ejpam-6171	837	13	)	)	PUNCT
ejpam-6171	837	14	≥	≥	NOUN
ejpam-6171	837	15	min{pf	min{pf	X
ejpam-6171	837	16	n(x	n(x	PROPN
ejpam-6171	837	17	?	?	PUNCT
ejpam-6171	837	18	y),pf	y),pf	PROPN
ejpam-6171	838	1	n(x	n(x	NOUN
ejpam-6171	838	2	)	)	PUNCT
ejpam-6171	838	3	}	}	PUNCT
ejpam-6171	838	4	.	.	PUNCT
ejpam-6171	839	1	(	(	PUNCT
ejpam-6171	839	2	by	by	ADP
ejpam-6171	839	3	(	(	PUNCT
ejpam-6171	839	4	3.21	3.21	NUM
ejpam-6171	839	5	)	)	PUNCT
ejpam-6171	839	6	)	)	PUNCT
ejpam-6171	839	7	hence	hence	ADV
ejpam-6171	839	8	,	,	PUNCT
ejpam-6171	839	9	a	a	DET
ejpam-6171	839	10	pns	pns	NOUN
ejpam-6171	839	11	pn	pn	X
ejpam-6171	839	12	=	=	SYM
ejpam-6171	839	13	(	(	PUNCT
ejpam-6171	839	14	x	x	PROPN
ejpam-6171	839	15	,	,	PUNCT
ejpam-6171	839	16	pt	pt	PROPN
ejpam-6171	839	17	n	n	CCONJ
ejpam-6171	839	18	,	,	PUNCT
ejpam-6171	839	19	pin	pin	NOUN
ejpam-6171	839	20	,	,	PUNCT
ejpam-6171	839	21	pf	pf	PROPN
ejpam-6171	839	22	n	n	CCONJ
ejpam-6171	839	23	)	)	PUNCT
ejpam-6171	839	24	is	be	AUX
ejpam-6171	839	25	a	a	DET
ejpam-6171	839	26	pythagorean	pythagorean	PROPN
ejpam-6171	839	27	neutrosophic	neutrosophic	ADJ
ejpam-6171	839	28	iup	iup	NOUN
ejpam-6171	839	29	-	-	PUNCT
ejpam-6171	839	30	filter	filter	NOUN
ejpam-6171	839	31	of	of	ADP
ejpam-6171	839	32	x.	x.	NOUN
ejpam-6171	839	33	conversely	conversely	ADV
ejpam-6171	839	34	,	,	PUNCT
ejpam-6171	839	35	it	it	PRON
ejpam-6171	839	36	is	be	AUX
ejpam-6171	839	37	obvious	obvious	ADJ
ejpam-6171	839	38	to	to	PART
ejpam-6171	839	39	prove	prove	VERB
ejpam-6171	839	40	that	that	SCONJ
ejpam-6171	839	41	p	p	NOUN
ejpam-6171	839	42	is	be	AUX
ejpam-6171	839	43	a	a	DET
ejpam-6171	839	44	pythagorean	pythagorean	PROPN
ejpam-6171	839	45	neutrosophic	neutrosophic	ADJ
ejpam-6171	839	46	iup	iup	NOUN
ejpam-6171	839	47	-	-	PUNCT
ejpam-6171	839	48	filter	filter	NOUN
ejpam-6171	839	49	of	of	ADP
ejpam-6171	839	50	x.	x.	PROPN
ejpam-6171	839	51	k.	k.	PROPN
ejpam-6171	840	1	suayngam	suayngam	PROPN
ejpam-6171	840	2	et	et	PROPN
ejpam-6171	840	3	al	al	PROPN
ejpam-6171	840	4	.	.	PUNCT
ejpam-6171	840	5	/	/	SYM
ejpam-6171	840	6	eur	eur	PROPN
ejpam-6171	840	7	.	.	PUNCT
ejpam-6171	841	1	j.	j.	PROPN
ejpam-6171	841	2	pure	pure	PROPN
ejpam-6171	841	3	appl	appl	PROPN
ejpam-6171	841	4	.	.	PROPN
ejpam-6171	841	5	math	math	PROPN
ejpam-6171	841	6	,	,	PUNCT
ejpam-6171	841	7	18	18	NUM
ejpam-6171	841	8	(	(	PUNCT
ejpam-6171	841	9	3	3	NUM
ejpam-6171	841	10	)	)	PUNCT
ejpam-6171	841	11	(	(	PUNCT
ejpam-6171	841	12	2025	2025	NUM
ejpam-6171	841	13	)	)	PUNCT
ejpam-6171	841	14	,	,	PUNCT
ejpam-6171	841	15	6171	6171	NUM
ejpam-6171	841	16	20	20	NUM
ejpam-6171	841	17	of	of	ADP
ejpam-6171	841	18	28	28	NUM
ejpam-6171	841	19	theorem	theorem	NOUN
ejpam-6171	841	20	16	16	NUM
ejpam-6171	841	21	.	.	PUNCT
ejpam-6171	842	1	a	a	DET
ejpam-6171	842	2	pns	pns	PROPN
ejpam-6171	842	3	p	p	PROPN
ejpam-6171	842	4	is	be	AUX
ejpam-6171	842	5	a	a	DET
ejpam-6171	842	6	pythagorean	pythagorean	PROPN
ejpam-6171	842	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	842	8	strong	strong	ADJ
ejpam-6171	842	9	iup	iup	NOUN
ejpam-6171	842	10	-	-	PUNCT
ejpam-6171	842	11	ideal	ideal	NOUN
ejpam-6171	842	12	of	of	ADP
ejpam-6171	842	13	x	x	SYM
ejpam-6171	842	14	if	if	SCONJ
ejpam-6171	842	15	and	and	CCONJ
ejpam-6171	842	16	only	only	ADV
ejpam-6171	842	17	if	if	SCONJ
ejpam-6171	842	18	a	a	DET
ejpam-6171	842	19	pns	pns	NOUN
ejpam-6171	842	20	pn	pn	X
ejpam-6171	842	21	=	=	SYM
ejpam-6171	842	22	(	(	PUNCT
ejpam-6171	842	23	x	x	PROPN
ejpam-6171	842	24	,	,	PUNCT
ejpam-6171	842	25	pt	pt	PROPN
ejpam-6171	842	26	n	n	CCONJ
ejpam-6171	842	27	,	,	PUNCT
ejpam-6171	842	28	pin	pin	NOUN
ejpam-6171	842	29	,	,	PUNCT
ejpam-6171	842	30	pf	pf	PROPN
ejpam-6171	842	31	n	n	CCONJ
ejpam-6171	842	32	)	)	PUNCT
ejpam-6171	842	33	is	be	AUX
ejpam-6171	842	34	pythagorean	pythagorean	PROPN
ejpam-6171	842	35	neutrosophic	neutrosophic	ADJ
ejpam-6171	842	36	strong	strong	ADJ
ejpam-6171	842	37	iup	iup	NOUN
ejpam-6171	842	38	-	-	PUNCT
ejpam-6171	842	39	ideal	ideal	NOUN
ejpam-6171	842	40	of	of	ADP
ejpam-6171	842	41	x.	x.	NOUN
ejpam-6171	842	42	proof	proof	NOUN
ejpam-6171	842	43	.	.	PUNCT
ejpam-6171	843	1	it	it	PRON
ejpam-6171	843	2	is	be	AUX
ejpam-6171	843	3	straightforward	straightforward	ADJ
ejpam-6171	843	4	by	by	ADP
ejpam-6171	843	5	theorem	theorem	NOUN
ejpam-6171	843	6	2	2	NUM
ejpam-6171	843	7	.	.	PUNCT
ejpam-6171	843	8	definition	definition	NOUN
ejpam-6171	843	9	10	10	NUM
ejpam-6171	843	10	.	.	PUNCT
ejpam-6171	844	1	[	[	X
ejpam-6171	844	2	17	17	NUM
ejpam-6171	844	3	]	]	PUNCT
ejpam-6171	844	4	let	let	VERB
ejpam-6171	844	5	f	f	PRON
ejpam-6171	844	6	be	be	AUX
ejpam-6171	844	7	an	an	DET
ejpam-6171	844	8	fs	fs	NOUN
ejpam-6171	844	9	in	in	ADP
ejpam-6171	844	10	a	a	DET
ejpam-6171	844	11	nonempty	nonempty	ADV
ejpam-6171	844	12	set	set	VERB
ejpam-6171	844	13	x.	x.	NOUN
ejpam-6171	844	14	for	for	ADP
ejpam-6171	844	15	any	any	DET
ejpam-6171	844	16	t	t	NOUN
ejpam-6171	844	17	∈	∈	PROPN
ejpam-6171	845	1	[	[	X
ejpam-6171	845	2	0	0	NUM
ejpam-6171	845	3	,	,	PUNCT
ejpam-6171	845	4	1	1	NUM
ejpam-6171	845	5	]	]	PUNCT
ejpam-6171	845	6	,	,	PUNCT
ejpam-6171	845	7	the	the	DET
ejpam-6171	845	8	sets	set	VERB
ejpam-6171	845	9	u(f	u(f	PROPN
ejpam-6171	845	10	;	;	PUNCT
ejpam-6171	845	11	t	t	X
ejpam-6171	845	12	)	)	PUNCT
ejpam-6171	845	13	=	=	PRON
ejpam-6171	846	1	{	{	PUNCT
ejpam-6171	846	2	x	x	PUNCT
ejpam-6171	846	3	∈	∈	PROPN
ejpam-6171	846	4	x	x	X
ejpam-6171	846	5	|	|	ADV
ejpam-6171	846	6	f(x	f(x	PROPN
ejpam-6171	846	7	)	)	PUNCT
ejpam-6171	846	8	≥	≥	NOUN
ejpam-6171	846	9	t	t	PROPN
ejpam-6171	846	10	}	}	PUNCT
ejpam-6171	846	11	,	,	PUNCT
ejpam-6171	846	12	(	(	PUNCT
ejpam-6171	846	13	3.23	3.23	NUM
ejpam-6171	846	14	)	)	PUNCT
ejpam-6171	846	15	l(f	l(f	PROPN
ejpam-6171	846	16	;	;	PUNCT
ejpam-6171	846	17	t	t	X
ejpam-6171	846	18	)	)	PUNCT
ejpam-6171	846	19	=	=	PRON
ejpam-6171	846	20	{	{	PUNCT
ejpam-6171	846	21	x	x	PUNCT
ejpam-6171	846	22	∈	∈	PROPN
ejpam-6171	846	23	x	x	X
ejpam-6171	846	24	|	|	ADV
ejpam-6171	846	25	f(x	f(x	PROPN
ejpam-6171	846	26	)	)	PUNCT
ejpam-6171	846	27	≤	≤	NOUN
ejpam-6171	846	28	t	t	PROPN
ejpam-6171	846	29	}	}	PUNCT
ejpam-6171	846	30	,	,	PUNCT
ejpam-6171	846	31	(	(	PUNCT
ejpam-6171	846	32	3.24	3.24	NUM
ejpam-6171	846	33	)	)	PUNCT
ejpam-6171	846	34	e(f	e(f	PROPN
ejpam-6171	846	35	;	;	PUNCT
ejpam-6171	846	36	t	t	PROPN
ejpam-6171	846	37	)	)	PUNCT
ejpam-6171	846	38	=	=	PRON
ejpam-6171	846	39	{	{	PUNCT
ejpam-6171	846	40	x	x	PUNCT
ejpam-6171	846	41	∈	∈	PROPN
ejpam-6171	846	42	x	x	X
ejpam-6171	846	43	|	|	ADV
ejpam-6171	846	44	f(x	f(x	PROPN
ejpam-6171	846	45	)	)	PUNCT
ejpam-6171	846	46	=	=	SYM
ejpam-6171	846	47	t	t	PROPN
ejpam-6171	846	48	}	}	PUNCT
ejpam-6171	846	49	(	(	PUNCT
ejpam-6171	846	50	3.25	3.25	NUM
ejpam-6171	846	51	)	)	PUNCT
ejpam-6171	846	52	are	be	AUX
ejpam-6171	846	53	called	call	VERB
ejpam-6171	846	54	an	an	DET
ejpam-6171	846	55	upper	upper	ADJ
ejpam-6171	846	56	t	t	NOUN
ejpam-6171	846	57	-	-	PUNCT
ejpam-6171	846	58	level	level	NOUN
ejpam-6171	846	59	subset	subset	NOUN
ejpam-6171	846	60	and	and	CCONJ
ejpam-6171	846	61	a	a	DET
ejpam-6171	846	62	lower	low	ADJ
ejpam-6171	846	63	t	t	NOUN
ejpam-6171	846	64	-	-	PUNCT
ejpam-6171	846	65	level	level	NOUN
ejpam-6171	846	66	subset	subset	NOUN
ejpam-6171	846	67	of	of	ADP
ejpam-6171	846	68	f	f	PROPN
ejpam-6171	846	69	,	,	PUNCT
ejpam-6171	846	70	respectively	respectively	ADV
ejpam-6171	846	71	.	.	PUNCT
ejpam-6171	847	1	the	the	DET
ejpam-6171	847	2	sets	set	VERB
ejpam-6171	847	3	u	u	NOUN
ejpam-6171	847	4	+	+	X
ejpam-6171	847	5	(	(	PUNCT
ejpam-6171	847	6	f	f	PROPN
ejpam-6171	847	7	;	;	PUNCT
ejpam-6171	847	8	t	t	PROPN
ejpam-6171	847	9	)	)	PUNCT
ejpam-6171	847	10	=	=	PRON
ejpam-6171	848	1	{	{	PUNCT
ejpam-6171	848	2	x	x	PUNCT
ejpam-6171	848	3	∈	∈	PROPN
ejpam-6171	848	4	x	x	X
ejpam-6171	848	5	|	|	ADV
ejpam-6171	848	6	f(x	f(x	PROPN
ejpam-6171	848	7	)	)	PUNCT
ejpam-6171	848	8	>	>	X
ejpam-6171	848	9	t	t	PROPN
ejpam-6171	848	10	}	}	PUNCT
ejpam-6171	848	11	,	,	PUNCT
ejpam-6171	848	12	(	(	PUNCT
ejpam-6171	848	13	3.26	3.26	NUM
ejpam-6171	848	14	)	)	PUNCT
ejpam-6171	848	15	l	l	NOUN
ejpam-6171	848	16	−	−	PROPN
ejpam-6171	848	17	(	(	PUNCT
ejpam-6171	848	18	f	f	PROPN
ejpam-6171	848	19	;	;	PUNCT
ejpam-6171	848	20	t	t	PROPN
ejpam-6171	848	21	)	)	PUNCT
ejpam-6171	848	22	=	=	PRON
ejpam-6171	848	23	{	{	PUNCT
ejpam-6171	848	24	x	x	PUNCT
ejpam-6171	848	25	∈	∈	PROPN
ejpam-6171	848	26	x	x	X
ejpam-6171	848	27	|	|	ADV
ejpam-6171	848	28	f(x	f(x	PROPN
ejpam-6171	848	29	)	)	PUNCT
ejpam-6171	848	30	<	<	X
ejpam-6171	848	31	t	t	PROPN
ejpam-6171	848	32	}	}	PUNCT
ejpam-6171	848	33	(	(	PUNCT
ejpam-6171	848	34	3.27	3.27	NUM
ejpam-6171	848	35	)	)	PUNCT
ejpam-6171	848	36	are	be	AUX
ejpam-6171	848	37	called	call	VERB
ejpam-6171	848	38	an	an	DET
ejpam-6171	848	39	upper	upper	ADJ
ejpam-6171	848	40	t	t	NOUN
ejpam-6171	848	41	-	-	PUNCT
ejpam-6171	848	42	strong	strong	ADJ
ejpam-6171	848	43	level	level	NOUN
ejpam-6171	848	44	subset	subset	NOUN
ejpam-6171	848	45	and	and	CCONJ
ejpam-6171	848	46	a	a	DET
ejpam-6171	848	47	lower	low	ADJ
ejpam-6171	848	48	t	t	NOUN
ejpam-6171	848	49	-	-	PUNCT
ejpam-6171	848	50	strong	strong	ADJ
ejpam-6171	848	51	level	level	NOUN
ejpam-6171	848	52	subset	subset	NOUN
ejpam-6171	848	53	of	of	ADP
ejpam-6171	848	54	f	f	PROPN
ejpam-6171	848	55	,	,	PUNCT
ejpam-6171	848	56	respectively	respectively	ADV
ejpam-6171	848	57	.	.	PUNCT
ejpam-6171	849	1	theorem	theorem	VERB
ejpam-6171	849	2	17	17	NUM
ejpam-6171	849	3	.	.	PUNCT
ejpam-6171	850	1	a	a	DET
ejpam-6171	850	2	pns	pns	PROPN
ejpam-6171	850	3	p	p	PROPN
ejpam-6171	850	4	is	be	AUX
ejpam-6171	850	5	a	a	DET
ejpam-6171	850	6	pythagorean	pythagorean	PROPN
ejpam-6171	850	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	850	8	iup	iup	NOUN
ejpam-6171	850	9	-	-	PUNCT
ejpam-6171	850	10	subalgebra	subalgebra	NOUN
ejpam-6171	850	11	of	of	ADP
ejpam-6171	850	12	x	x	PRON
ejpam-6171	850	13	if	if	SCONJ
ejpam-6171	850	14	and	and	CCONJ
ejpam-6171	850	15	only	only	ADV
ejpam-6171	850	16	if	if	SCONJ
ejpam-6171	850	17	for	for	ADP
ejpam-6171	850	18	all	all	DET
ejpam-6171	850	19	α	α	NOUN
ejpam-6171	850	20	,	,	PUNCT
ejpam-6171	850	21	β	β	X
ejpam-6171	850	22	,	,	PUNCT
ejpam-6171	850	23	γ	γ	PROPN
ejpam-6171	850	24	∈	∈	PROPN
ejpam-6171	851	1	[	[	X
ejpam-6171	851	2	0	0	NUM
ejpam-6171	851	3	,	,	PUNCT
ejpam-6171	851	4	1	1	NUM
ejpam-6171	851	5	]	]	PUNCT
ejpam-6171	851	6	,	,	PUNCT
ejpam-6171	851	7	the	the	DET
ejpam-6171	851	8	sets	set	NOUN
ejpam-6171	851	9	u(pt	u(pt	PROPN
ejpam-6171	851	10	;	;	PUNCT
ejpam-6171	851	11	α	α	X
ejpam-6171	851	12	)	)	PUNCT
ejpam-6171	851	13	,	,	PUNCT
ejpam-6171	851	14	l(pi	l(pi	X
ejpam-6171	851	15	;	;	PUNCT
ejpam-6171	851	16	β	β	X
ejpam-6171	851	17	)	)	PUNCT
ejpam-6171	851	18	,	,	PUNCT
ejpam-6171	851	19	and	and	CCONJ
ejpam-6171	851	20	u(pf	u(pf	PROPN
ejpam-6171	851	21	;	;	PUNCT
ejpam-6171	851	22	γ	γ	X
ejpam-6171	851	23	)	)	PUNCT
ejpam-6171	851	24	are	be	AUX
ejpam-6171	851	25	either	either	CCONJ
ejpam-6171	851	26	empty	empty	ADJ
ejpam-6171	851	27	or	or	CCONJ
ejpam-6171	851	28	iup	iup	NOUN
ejpam-6171	851	29	-	-	PUNCT
ejpam-6171	851	30	subalgebras	subalgebras	PROPN
ejpam-6171	851	31	of	of	ADP
ejpam-6171	851	32	x.	x.	PROPN
ejpam-6171	851	33	proof	proof	PROPN
ejpam-6171	851	34	.	.	PUNCT
ejpam-6171	852	1	assume	assume	VERB
ejpam-6171	852	2	that	that	SCONJ
ejpam-6171	852	3	p	p	NOUN
ejpam-6171	852	4	is	be	AUX
ejpam-6171	852	5	a	a	DET
ejpam-6171	852	6	pythagorean	pythagorean	PROPN
ejpam-6171	852	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	852	8	iup	iup	NOUN
ejpam-6171	852	9	-	-	PUNCT
ejpam-6171	852	10	subalgebra	subalgebra	NOUN
ejpam-6171	852	11	of	of	ADP
ejpam-6171	852	12	x.	x.	NOUN
ejpam-6171	852	13	let	let	VERB
ejpam-6171	852	14	α	α	PRON
ejpam-6171	852	15	∈	∈	PROPN
ejpam-6171	853	1	[	[	X
ejpam-6171	853	2	0	0	NUM
ejpam-6171	853	3	,	,	PUNCT
ejpam-6171	853	4	1	1	NUM
ejpam-6171	853	5	]	]	PUNCT
ejpam-6171	853	6	be	be	AUX
ejpam-6171	853	7	such	such	ADJ
ejpam-6171	853	8	that	that	SCONJ
ejpam-6171	853	9	u(pt	u(pt	PROPN
ejpam-6171	853	10	;	;	PUNCT
ejpam-6171	853	11	α	α	X
ejpam-6171	853	12	)	)	PUNCT
ejpam-6171	853	13	6=	6=	ADP
ejpam-6171	853	14	∅.	∅.	AUX
ejpam-6171	853	15	let	let	VERB
ejpam-6171	853	16	x	x	PRON
ejpam-6171	853	17	,	,	PUNCT
ejpam-6171	853	18	y	y	PROPN
ejpam-6171	853	19	∈	∈	PROPN
ejpam-6171	853	20	u(pt	u(pt	PROPN
ejpam-6171	853	21	;	;	PUNCT
ejpam-6171	853	22	α	α	X
ejpam-6171	853	23	)	)	PUNCT
ejpam-6171	853	24	.	.	PUNCT
ejpam-6171	854	1	then	then	ADV
ejpam-6171	854	2	pt	pt	X
ejpam-6171	854	3	(	(	PUNCT
ejpam-6171	854	4	x	x	NOUN
ejpam-6171	854	5	)	)	PUNCT
ejpam-6171	854	6	≥	≥	PROPN
ejpam-6171	854	7	α	α	NOUN
ejpam-6171	854	8	and	and	CCONJ
ejpam-6171	854	9	pt	pt	PROPN
ejpam-6171	854	10	(	(	PUNCT
ejpam-6171	854	11	y	y	NOUN
ejpam-6171	854	12	)	)	PUNCT
ejpam-6171	854	13	≥	≥	PROPN
ejpam-6171	854	14	α	α	NOUN
ejpam-6171	854	15	.	.	PUNCT
ejpam-6171	855	1	thus	thus	ADV
ejpam-6171	855	2	,	,	PUNCT
ejpam-6171	855	3	min{pt	min{pt	PRON
ejpam-6171	855	4	(	(	PUNCT
ejpam-6171	855	5	x),pt	x),pt	PROPN
ejpam-6171	855	6	(	(	PUNCT
ejpam-6171	855	7	y	y	NOUN
ejpam-6171	855	8	)	)	PUNCT
ejpam-6171	855	9	}	}	PUNCT
ejpam-6171	855	10	≥	≥	PROPN
ejpam-6171	855	11	α	α	NOUN
ejpam-6171	855	12	.	.	PUNCT
ejpam-6171	855	13	by	by	ADP
ejpam-6171	855	14	(	(	PUNCT
ejpam-6171	855	15	3.2	3.2	NUM
ejpam-6171	855	16	)	)	PUNCT
ejpam-6171	855	17	,	,	PUNCT
ejpam-6171	855	18	we	we	PRON
ejpam-6171	855	19	have	have	VERB
ejpam-6171	855	20	pt	pt	INTJ
ejpam-6171	855	21	(	(	PUNCT
ejpam-6171	855	22	x	x	NOUN
ejpam-6171	855	23	?	?	PUNCT
ejpam-6171	856	1	y	y	X
ejpam-6171	856	2	)	)	PUNCT
ejpam-6171	856	3	≥	≥	NOUN
ejpam-6171	856	4	min{pt	min{pt	X
ejpam-6171	857	1	(	(	PUNCT
ejpam-6171	857	2	x),pt	x),pt	PROPN
ejpam-6171	857	3	(	(	PUNCT
ejpam-6171	857	4	y	y	NOUN
ejpam-6171	857	5	)	)	PUNCT
ejpam-6171	857	6	}	}	PUNCT
ejpam-6171	857	7	≥	≥	NUM
ejpam-6171	857	8	α	α	NOUN
ejpam-6171	857	9	,	,	PUNCT
ejpam-6171	857	10	that	that	ADV
ejpam-6171	857	11	is	is	ADV
ejpam-6171	857	12	,	,	PUNCT
ejpam-6171	857	13	pt	pt	X
ejpam-6171	857	14	(	(	PUNCT
ejpam-6171	857	15	x	x	X
ejpam-6171	857	16	?	?	PUNCT
ejpam-6171	857	17	y	y	X
ejpam-6171	857	18	)	)	PUNCT
ejpam-6171	857	19	≥	≥	PROPN
ejpam-6171	857	20	α	α	NOUN
ejpam-6171	857	21	.	.	PUNCT
ejpam-6171	858	1	thus	thus	ADV
ejpam-6171	858	2	,	,	PUNCT
ejpam-6171	858	3	x	x	PUNCT
ejpam-6171	858	4	?	?	PUNCT
ejpam-6171	859	1	y	y	PROPN
ejpam-6171	859	2	∈	∈	PROPN
ejpam-6171	859	3	u(pt	u(pt	PROPN
ejpam-6171	859	4	;	;	PUNCT
ejpam-6171	859	5	α	α	X
ejpam-6171	859	6	)	)	PUNCT
ejpam-6171	859	7	.	.	PUNCT
ejpam-6171	860	1	hence	hence	ADV
ejpam-6171	860	2	,	,	PUNCT
ejpam-6171	860	3	u(pt	u(pt	PROPN
ejpam-6171	860	4	;	;	PUNCT
ejpam-6171	860	5	α	α	X
ejpam-6171	860	6	)	)	PUNCT
ejpam-6171	860	7	is	be	AUX
ejpam-6171	860	8	an	an	DET
ejpam-6171	860	9	iup	iup	NOUN
ejpam-6171	860	10	-	-	PUNCT
ejpam-6171	860	11	subalgebra	subalgebra	NOUN
ejpam-6171	860	12	of	of	ADP
ejpam-6171	860	13	x.	x.	NOUN
ejpam-6171	860	14	let	let	VERB
ejpam-6171	860	15	β	β	X
ejpam-6171	860	16	∈	∈	PROPN
ejpam-6171	861	1	[	[	X
ejpam-6171	861	2	0	0	NUM
ejpam-6171	861	3	,	,	PUNCT
ejpam-6171	861	4	1	1	NUM
ejpam-6171	861	5	]	]	PUNCT
ejpam-6171	861	6	be	be	AUX
ejpam-6171	861	7	such	such	ADJ
ejpam-6171	861	8	that	that	SCONJ
ejpam-6171	861	9	l(pi	l(pi	X
ejpam-6171	861	10	;	;	PUNCT
ejpam-6171	861	11	β	β	X
ejpam-6171	861	12	)	)	PUNCT
ejpam-6171	861	13	6=	6=	ADP
ejpam-6171	861	14	∅.	∅.	AUX
ejpam-6171	861	15	let	let	VERB
ejpam-6171	861	16	x	x	PRON
ejpam-6171	861	17	,	,	PUNCT
ejpam-6171	861	18	y	y	PROPN
ejpam-6171	861	19	∈	∈	PROPN
ejpam-6171	861	20	l(pi	l(pi	PROPN
ejpam-6171	861	21	;	;	PUNCT
ejpam-6171	861	22	β	β	X
ejpam-6171	861	23	)	)	PUNCT
ejpam-6171	861	24	.	.	PUNCT
ejpam-6171	862	1	then	then	ADV
ejpam-6171	862	2	pi(x	pi(x	NUM
ejpam-6171	862	3	)	)	PUNCT
ejpam-6171	862	4	≤	≤	NUM
ejpam-6171	862	5	β	β	X
ejpam-6171	862	6	and	and	CCONJ
ejpam-6171	862	7	pi(y	pi(y	NOUN
ejpam-6171	862	8	)	)	PUNCT
ejpam-6171	862	9	≤	≤	NOUN
ejpam-6171	862	10	β	β	X
ejpam-6171	862	11	.	.	PUNCT
ejpam-6171	863	1	thus	thus	ADV
ejpam-6171	863	2	,	,	PUNCT
ejpam-6171	863	3	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	863	4	)	)	PUNCT
ejpam-6171	863	5	}	}	PUNCT
ejpam-6171	863	6	≤	≤	NOUN
ejpam-6171	863	7	β	β	X
ejpam-6171	863	8	.	.	PUNCT
ejpam-6171	864	1	by	by	ADP
ejpam-6171	864	2	(	(	PUNCT
ejpam-6171	864	3	3.3	3.3	NUM
ejpam-6171	864	4	)	)	PUNCT
ejpam-6171	864	5	,	,	PUNCT
ejpam-6171	864	6	we	we	PRON
ejpam-6171	864	7	have	have	VERB
ejpam-6171	864	8	pi(x	pi(x	NUM
ejpam-6171	864	9	?	?	PUNCT
ejpam-6171	865	1	y	y	X
ejpam-6171	865	2	)	)	PUNCT
ejpam-6171	865	3	≤	≤	NUM
ejpam-6171	865	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	865	5	)	)	PUNCT
ejpam-6171	865	6	}	}	PUNCT
ejpam-6171	865	7	≤	≤	NOUN
ejpam-6171	866	1	β	β	NOUN
ejpam-6171	866	2	,	,	PUNCT
ejpam-6171	866	3	that	that	ADV
ejpam-6171	866	4	is	is	ADV
ejpam-6171	866	5	,	,	PUNCT
ejpam-6171	866	6	pi(x	pi(x	ADJ
ejpam-6171	866	7	?	?	PUNCT
ejpam-6171	867	1	y	y	X
ejpam-6171	867	2	)	)	PUNCT
ejpam-6171	867	3	≤	≤	NOUN
ejpam-6171	868	1	β	β	X
ejpam-6171	868	2	.	.	PUNCT
ejpam-6171	869	1	thus	thus	ADV
ejpam-6171	869	2	,	,	PUNCT
ejpam-6171	869	3	x	x	PUNCT
ejpam-6171	869	4	?	?	PUNCT
ejpam-6171	870	1	y	y	PROPN
ejpam-6171	870	2	∈	∈	PROPN
ejpam-6171	870	3	l(pi	l(pi	PROPN
ejpam-6171	870	4	;	;	PUNCT
ejpam-6171	870	5	β	β	X
ejpam-6171	870	6	)	)	PUNCT
ejpam-6171	870	7	.	.	PUNCT
ejpam-6171	871	1	hence	hence	ADV
ejpam-6171	871	2	,	,	PUNCT
ejpam-6171	871	3	l(pi	l(pi	X
ejpam-6171	871	4	;	;	PUNCT
ejpam-6171	871	5	β	β	X
ejpam-6171	871	6	)	)	PUNCT
ejpam-6171	871	7	is	be	AUX
ejpam-6171	871	8	an	an	DET
ejpam-6171	871	9	iup	iup	NOUN
ejpam-6171	871	10	-	-	PUNCT
ejpam-6171	871	11	subalgebra	subalgebra	NOUN
ejpam-6171	871	12	of	of	ADP
ejpam-6171	871	13	x.	x.	NOUN
ejpam-6171	871	14	let	let	VERB
ejpam-6171	871	15	γ	γ	X
ejpam-6171	871	16	∈	∈	PROPN
ejpam-6171	872	1	[	[	X
ejpam-6171	872	2	0	0	NUM
ejpam-6171	872	3	,	,	PUNCT
ejpam-6171	872	4	1	1	NUM
ejpam-6171	872	5	]	]	PUNCT
ejpam-6171	872	6	be	be	AUX
ejpam-6171	872	7	such	such	ADJ
ejpam-6171	872	8	that	that	SCONJ
ejpam-6171	872	9	u(pf	u(pf	PROPN
ejpam-6171	872	10	;	;	PUNCT
ejpam-6171	872	11	γ	γ	X
ejpam-6171	872	12	)	)	PUNCT
ejpam-6171	872	13	6=	6=	AUX
ejpam-6171	872	14	∅.	∅.	ADV
ejpam-6171	872	15	let	let	VERB
ejpam-6171	872	16	x	x	PRON
ejpam-6171	872	17	,	,	PUNCT
ejpam-6171	872	18	y	y	PROPN
ejpam-6171	872	19	∈	∈	PROPN
ejpam-6171	872	20	u(pf	u(pf	PROPN
ejpam-6171	872	21	;	;	PUNCT
ejpam-6171	872	22	γ	γ	X
ejpam-6171	872	23	)	)	PUNCT
ejpam-6171	872	24	.	.	PUNCT
ejpam-6171	873	1	then	then	ADV
ejpam-6171	873	2	pf	pf	PROPN
ejpam-6171	873	3	(	(	PUNCT
ejpam-6171	873	4	x	x	PROPN
ejpam-6171	873	5	)	)	PUNCT
ejpam-6171	873	6	≥	≥	PROPN
ejpam-6171	873	7	γ	γ	PROPN
ejpam-6171	873	8	and	and	CCONJ
ejpam-6171	873	9	pf	pf	PROPN
ejpam-6171	873	10	(	(	PUNCT
ejpam-6171	873	11	y	y	PROPN
ejpam-6171	873	12	)	)	PUNCT
ejpam-6171	873	13	≥	≥	PROPN
ejpam-6171	873	14	γ	γ	PROPN
ejpam-6171	873	15	.	.	PUNCT
ejpam-6171	873	16	thus	thus	ADV
ejpam-6171	873	17	,	,	PUNCT
ejpam-6171	873	18	min{pf	min{pf	PRON
ejpam-6171	873	19	(	(	PUNCT
ejpam-6171	873	20	x),pf	x),pf	PROPN
ejpam-6171	873	21	(	(	PUNCT
ejpam-6171	873	22	y	y	NOUN
ejpam-6171	873	23	)	)	PUNCT
ejpam-6171	873	24	}	}	PUNCT
ejpam-6171	873	25	≥	≥	PROPN
ejpam-6171	873	26	γ	γ	X
ejpam-6171	873	27	.	.	PUNCT
ejpam-6171	873	28	by	by	ADP
ejpam-6171	873	29	(	(	PUNCT
ejpam-6171	873	30	3.4	3.4	NUM
ejpam-6171	873	31	)	)	PUNCT
ejpam-6171	873	32	,	,	PUNCT
ejpam-6171	873	33	we	we	PRON
ejpam-6171	873	34	have	have	VERB
ejpam-6171	873	35	pf	pf	PROPN
ejpam-6171	873	36	(	(	PUNCT
ejpam-6171	873	37	x	x	PROPN
ejpam-6171	873	38	?	?	PUNCT
ejpam-6171	874	1	y	y	X
ejpam-6171	874	2	)	)	PUNCT
ejpam-6171	874	3	≥	≥	PROPN
ejpam-6171	874	4	min{pf	min{pf	PRON
ejpam-6171	874	5	(	(	PUNCT
ejpam-6171	874	6	x),pf	x),pf	PROPN
ejpam-6171	874	7	(	(	PUNCT
ejpam-6171	874	8	y	y	NOUN
ejpam-6171	874	9	)	)	PUNCT
ejpam-6171	874	10	}	}	PUNCT
ejpam-6171	874	11	≥	≥	X
ejpam-6171	874	12	γ	γ	X
ejpam-6171	874	13	,	,	PUNCT
ejpam-6171	874	14	that	that	ADV
ejpam-6171	874	15	is	is	ADV
ejpam-6171	874	16	,	,	PUNCT
ejpam-6171	874	17	pf	pf	PROPN
ejpam-6171	874	18	(	(	PUNCT
ejpam-6171	874	19	x	x	PROPN
ejpam-6171	874	20	?	?	PUNCT
ejpam-6171	875	1	y	y	X
ejpam-6171	875	2	)	)	PUNCT
ejpam-6171	875	3	≥	≥	PROPN
ejpam-6171	875	4	γ	γ	PROPN
ejpam-6171	875	5	.	.	PUNCT
ejpam-6171	875	6	thus	thus	ADV
ejpam-6171	875	7	,	,	PUNCT
ejpam-6171	875	8	x	x	PUNCT
ejpam-6171	875	9	?	?	PUNCT
ejpam-6171	876	1	y	y	PROPN
ejpam-6171	876	2	∈	∈	PROPN
ejpam-6171	876	3	u(pf	u(pf	PROPN
ejpam-6171	876	4	;	;	PUNCT
ejpam-6171	876	5	γ	γ	X
ejpam-6171	876	6	)	)	PUNCT
ejpam-6171	876	7	.	.	PUNCT
ejpam-6171	877	1	hence	hence	ADV
ejpam-6171	877	2	,	,	PUNCT
ejpam-6171	877	3	u(pf	u(pf	PROPN
ejpam-6171	877	4	;	;	PUNCT
ejpam-6171	877	5	γ	γ	X
ejpam-6171	877	6	)	)	PUNCT
ejpam-6171	877	7	is	be	AUX
ejpam-6171	877	8	an	an	DET
ejpam-6171	877	9	iup	iup	NOUN
ejpam-6171	877	10	-	-	PUNCT
ejpam-6171	877	11	subalgebra	subalgebra	NOUN
ejpam-6171	877	12	of	of	ADP
ejpam-6171	877	13	x.	x.	NOUN
ejpam-6171	877	14	conversely	conversely	ADV
ejpam-6171	877	15	,	,	PUNCT
ejpam-6171	877	16	assume	assume	VERB
ejpam-6171	877	17	that	that	SCONJ
ejpam-6171	877	18	for	for	ADP
ejpam-6171	877	19	all	all	DET
ejpam-6171	877	20	α	α	NOUN
ejpam-6171	877	21	,	,	PUNCT
ejpam-6171	877	22	β	β	X
ejpam-6171	877	23	,	,	PUNCT
ejpam-6171	877	24	γ	γ	PROPN
ejpam-6171	877	25	∈	∈	PROPN
ejpam-6171	878	1	[	[	X
ejpam-6171	878	2	0	0	NUM
ejpam-6171	878	3	,	,	PUNCT
ejpam-6171	878	4	1	1	NUM
ejpam-6171	878	5	]	]	PUNCT
ejpam-6171	878	6	,	,	PUNCT
ejpam-6171	878	7	the	the	DET
ejpam-6171	878	8	sets	set	NOUN
ejpam-6171	878	9	u(pt	u(pt	PROPN
ejpam-6171	878	10	;	;	PUNCT
ejpam-6171	878	11	α	α	X
ejpam-6171	878	12	)	)	PUNCT
ejpam-6171	878	13	,	,	PUNCT
ejpam-6171	878	14	l(pi	l(pi	X
ejpam-6171	878	15	;	;	PUNCT
ejpam-6171	878	16	β	β	X
ejpam-6171	878	17	)	)	PUNCT
ejpam-6171	878	18	,	,	PUNCT
ejpam-6171	878	19	and	and	CCONJ
ejpam-6171	878	20	u(pf	u(pf	PROPN
ejpam-6171	878	21	;	;	PUNCT
ejpam-6171	878	22	γ	γ	X
ejpam-6171	878	23	)	)	PUNCT
ejpam-6171	878	24	are	be	AUX
ejpam-6171	878	25	either	either	CCONJ
ejpam-6171	878	26	empty	empty	ADJ
ejpam-6171	878	27	or	or	CCONJ
ejpam-6171	878	28	iup	iup	NOUN
ejpam-6171	878	29	-	-	PUNCT
ejpam-6171	878	30	subalgebras	subalgebras	PROPN
ejpam-6171	878	31	of	of	ADP
ejpam-6171	878	32	x.	x.	PROPN
ejpam-6171	878	33	let	let	VERB
ejpam-6171	878	34	x	x	PRON
ejpam-6171	878	35	,	,	PUNCT
ejpam-6171	878	36	y	y	PROPN
ejpam-6171	878	37	∈	∈	PROPN
ejpam-6171	878	38	x.	x.	NOUN
ejpam-6171	878	39	let	let	VERB
ejpam-6171	878	40	α	α	NOUN
ejpam-6171	878	41	=	=	PUNCT
ejpam-6171	878	42	min{pt	min{pt	NOUN
ejpam-6171	878	43	(	(	PUNCT
ejpam-6171	878	44	x),pt	x),pt	PROPN
ejpam-6171	878	45	(	(	PUNCT
ejpam-6171	878	46	y	y	NOUN
ejpam-6171	878	47	)	)	PUNCT
ejpam-6171	878	48	}	}	PUNCT
ejpam-6171	878	49	.	.	PUNCT
ejpam-6171	879	1	then	then	ADV
ejpam-6171	879	2	pt	pt	X
ejpam-6171	879	3	(	(	PUNCT
ejpam-6171	879	4	x	x	NOUN
ejpam-6171	879	5	)	)	PUNCT
ejpam-6171	879	6	≥	≥	PROPN
ejpam-6171	879	7	α	α	NOUN
ejpam-6171	879	8	and	and	CCONJ
ejpam-6171	879	9	pt	pt	PROPN
ejpam-6171	879	10	(	(	PUNCT
ejpam-6171	879	11	y	y	NOUN
ejpam-6171	879	12	)	)	PUNCT
ejpam-6171	879	13	≥	≥	PROPN
ejpam-6171	879	14	α	α	NOUN
ejpam-6171	879	15	.	.	PUNCT
ejpam-6171	880	1	thus	thus	ADV
ejpam-6171	880	2	,	,	PUNCT
ejpam-6171	880	3	x	x	PRON
ejpam-6171	880	4	,	,	PUNCT
ejpam-6171	880	5	y	y	PROPN
ejpam-6171	880	6	∈	∈	PROPN
ejpam-6171	880	7	u(pt	u(pt	PROPN
ejpam-6171	880	8	;	;	PUNCT
ejpam-6171	880	9	α	α	X
ejpam-6171	880	10	)	)	PUNCT
ejpam-6171	880	11	6=	6=	ADP
ejpam-6171	880	12	∅.	∅.	ADP
ejpam-6171	880	13	by	by	ADP
ejpam-6171	880	14	the	the	DET
ejpam-6171	880	15	assumption	assumption	NOUN
ejpam-6171	880	16	,	,	PUNCT
ejpam-6171	880	17	we	we	PRON
ejpam-6171	880	18	have	have	VERB
ejpam-6171	880	19	u(pt	u(pt	NOUN
ejpam-6171	880	20	;	;	PUNCT
ejpam-6171	880	21	α	α	X
ejpam-6171	880	22	)	)	PUNCT
ejpam-6171	880	23	is	be	AUX
ejpam-6171	880	24	an	an	DET
ejpam-6171	880	25	iup	iup	NOUN
ejpam-6171	880	26	-	-	PUNCT
ejpam-6171	880	27	subalgebra	subalgebra	NOUN
ejpam-6171	880	28	of	of	ADP
ejpam-6171	880	29	x.	x.	NOUN
ejpam-6171	880	30	by	by	ADP
ejpam-6171	880	31	(	(	PUNCT
ejpam-6171	880	32	2.17	2.17	NUM
ejpam-6171	880	33	)	)	PUNCT
ejpam-6171	880	34	,	,	PUNCT
ejpam-6171	880	35	we	we	PRON
ejpam-6171	880	36	have	have	VERB
ejpam-6171	880	37	x	x	X
ejpam-6171	880	38	?	?	PUNCT
ejpam-6171	881	1	y	y	PROPN
ejpam-6171	881	2	∈	∈	PROPN
ejpam-6171	881	3	u(pt	u(pt	PROPN
ejpam-6171	881	4	;	;	PUNCT
ejpam-6171	881	5	α	α	X
ejpam-6171	881	6	)	)	PUNCT
ejpam-6171	881	7	.	.	PUNCT
ejpam-6171	882	1	thus	thus	ADV
ejpam-6171	882	2	,	,	PUNCT
ejpam-6171	882	3	pt	pt	X
ejpam-6171	882	4	(	(	PUNCT
ejpam-6171	882	5	x	x	X
ejpam-6171	882	6	?	?	PUNCT
ejpam-6171	882	7	y	y	X
ejpam-6171	882	8	)	)	PUNCT
ejpam-6171	882	9	≥	≥	NOUN
ejpam-6171	882	10	α	α	NOUN
ejpam-6171	882	11	=	=	X
ejpam-6171	882	12	min{pt	min{pt	NOUN
ejpam-6171	882	13	(	(	PUNCT
ejpam-6171	882	14	x),pt	x),pt	PROPN
ejpam-6171	882	15	(	(	PUNCT
ejpam-6171	882	16	y	y	NOUN
ejpam-6171	882	17	)	)	PUNCT
ejpam-6171	882	18	}	}	PUNCT
ejpam-6171	882	19	.	.	PUNCT
ejpam-6171	883	1	let	let	VERB
ejpam-6171	883	2	x	x	PRON
ejpam-6171	883	3	,	,	PUNCT
ejpam-6171	883	4	y	y	PROPN
ejpam-6171	883	5	∈	∈	PROPN
ejpam-6171	883	6	x.	x.	NOUN
ejpam-6171	883	7	let	let	VERB
ejpam-6171	883	8	β	β	X
ejpam-6171	883	9	=	=	PUNCT
ejpam-6171	883	10	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	883	11	)	)	PUNCT
ejpam-6171	883	12	}	}	PUNCT
ejpam-6171	883	13	.	.	PUNCT
ejpam-6171	884	1	then	then	ADV
ejpam-6171	884	2	pi(x	pi(x	NUM
ejpam-6171	884	3	)	)	PUNCT
ejpam-6171	884	4	≤	≤	NUM
ejpam-6171	884	5	β	β	X
ejpam-6171	884	6	and	and	CCONJ
ejpam-6171	884	7	pi(y	pi(y	NOUN
ejpam-6171	884	8	)	)	PUNCT
ejpam-6171	884	9	≤	≤	NOUN
ejpam-6171	884	10	β	β	X
ejpam-6171	884	11	.	.	PUNCT
ejpam-6171	885	1	thus	thus	ADV
ejpam-6171	885	2	,	,	PUNCT
ejpam-6171	885	3	x	x	PRON
ejpam-6171	885	4	,	,	PUNCT
ejpam-6171	885	5	y	y	PROPN
ejpam-6171	885	6	∈	∈	PROPN
ejpam-6171	885	7	l(pi	l(pi	X
ejpam-6171	885	8	;	;	PUNCT
ejpam-6171	885	9	β	β	X
ejpam-6171	885	10	)	)	PUNCT
ejpam-6171	885	11	6=	6=	ADP
ejpam-6171	885	12	∅.	∅.	ADP
ejpam-6171	885	13	by	by	ADP
ejpam-6171	885	14	the	the	DET
ejpam-6171	885	15	assumption	assumption	NOUN
ejpam-6171	885	16	,	,	PUNCT
ejpam-6171	885	17	we	we	PRON
ejpam-6171	885	18	have	have	VERB
ejpam-6171	885	19	l(pi	l(pi	X
ejpam-6171	885	20	;	;	PUNCT
ejpam-6171	885	21	β	β	X
ejpam-6171	885	22	)	)	PUNCT
ejpam-6171	885	23	is	be	AUX
ejpam-6171	885	24	an	an	DET
ejpam-6171	885	25	iup	iup	NOUN
ejpam-6171	885	26	-	-	PUNCT
ejpam-6171	885	27	subalgebra	subalgebra	NOUN
ejpam-6171	885	28	of	of	ADP
ejpam-6171	885	29	x.	x.	NOUN
ejpam-6171	885	30	by	by	ADP
ejpam-6171	885	31	(	(	PUNCT
ejpam-6171	885	32	2.17	2.17	NUM
ejpam-6171	885	33	)	)	PUNCT
ejpam-6171	885	34	,	,	PUNCT
ejpam-6171	885	35	we	we	PRON
ejpam-6171	885	36	have	have	VERB
ejpam-6171	885	37	x	x	X
ejpam-6171	885	38	?	?	PUNCT
ejpam-6171	886	1	y	y	PROPN
ejpam-6171	886	2	∈	∈	PROPN
ejpam-6171	886	3	l(pi	l(pi	PROPN
ejpam-6171	886	4	;	;	PUNCT
ejpam-6171	886	5	β	β	X
ejpam-6171	886	6	)	)	PUNCT
ejpam-6171	886	7	.	.	PUNCT
ejpam-6171	887	1	thus	thus	ADV
ejpam-6171	887	2	,	,	PUNCT
ejpam-6171	887	3	pi(x	pi(x	ADJ
ejpam-6171	887	4	?	?	PUNCT
ejpam-6171	888	1	y	y	X
ejpam-6171	888	2	)	)	PUNCT
ejpam-6171	888	3	≤	≤	NOUN
ejpam-6171	888	4	β	β	X
ejpam-6171	888	5	=	=	SYM
ejpam-6171	888	6	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	888	7	)	)	PUNCT
ejpam-6171	888	8	}	}	PUNCT
ejpam-6171	888	9	.	.	PUNCT
ejpam-6171	889	1	k.	k.	PROPN
ejpam-6171	890	1	suayngam	suayngam	PROPN
ejpam-6171	890	2	et	et	PROPN
ejpam-6171	890	3	al	al	PROPN
ejpam-6171	890	4	.	.	PUNCT
ejpam-6171	890	5	/	/	SYM
ejpam-6171	890	6	eur	eur	PROPN
ejpam-6171	890	7	.	.	PUNCT
ejpam-6171	891	1	j.	j.	PROPN
ejpam-6171	891	2	pure	pure	PROPN
ejpam-6171	891	3	appl	appl	PROPN
ejpam-6171	891	4	.	.	PROPN
ejpam-6171	891	5	math	math	PROPN
ejpam-6171	891	6	,	,	PUNCT
ejpam-6171	891	7	18	18	NUM
ejpam-6171	891	8	(	(	PUNCT
ejpam-6171	891	9	3	3	NUM
ejpam-6171	891	10	)	)	PUNCT
ejpam-6171	891	11	(	(	PUNCT
ejpam-6171	891	12	2025	2025	NUM
ejpam-6171	891	13	)	)	PUNCT
ejpam-6171	891	14	,	,	PUNCT
ejpam-6171	891	15	6171	6171	NUM
ejpam-6171	891	16	21	21	NUM
ejpam-6171	891	17	of	of	ADP
ejpam-6171	891	18	28	28	NUM
ejpam-6171	891	19	let	let	VERB
ejpam-6171	891	20	x	x	PRON
ejpam-6171	891	21	,	,	PUNCT
ejpam-6171	891	22	y	y	PROPN
ejpam-6171	891	23	∈	∈	PROPN
ejpam-6171	891	24	x.	x.	NOUN
ejpam-6171	891	25	let	let	VERB
ejpam-6171	891	26	γ	γ	X
ejpam-6171	891	27	=	=	VERB
ejpam-6171	891	28	min{pf	min{pf	X
ejpam-6171	891	29	(	(	PUNCT
ejpam-6171	891	30	x),pf	x),pf	PROPN
ejpam-6171	891	31	(	(	PUNCT
ejpam-6171	891	32	y	y	NOUN
ejpam-6171	891	33	)	)	PUNCT
ejpam-6171	891	34	}	}	PUNCT
ejpam-6171	891	35	.	.	PUNCT
ejpam-6171	892	1	then	then	ADV
ejpam-6171	892	2	pf	pf	PROPN
ejpam-6171	892	3	(	(	PUNCT
ejpam-6171	892	4	x	x	PROPN
ejpam-6171	892	5	)	)	PUNCT
ejpam-6171	892	6	≥	≥	PROPN
ejpam-6171	892	7	γ	γ	PROPN
ejpam-6171	892	8	and	and	CCONJ
ejpam-6171	892	9	pf	pf	PROPN
ejpam-6171	892	10	(	(	PUNCT
ejpam-6171	892	11	y	y	PROPN
ejpam-6171	892	12	)	)	PUNCT
ejpam-6171	892	13	≥	≥	PROPN
ejpam-6171	892	14	γ	γ	PROPN
ejpam-6171	892	15	.	.	PROPN
ejpam-6171	892	16	thus	thus	ADV
ejpam-6171	892	17	,	,	PUNCT
ejpam-6171	892	18	x	x	PRON
ejpam-6171	892	19	,	,	PUNCT
ejpam-6171	892	20	y	y	PROPN
ejpam-6171	892	21	∈	∈	PROPN
ejpam-6171	892	22	u(pf	u(pf	PROPN
ejpam-6171	892	23	;	;	PUNCT
ejpam-6171	892	24	γ	γ	X
ejpam-6171	892	25	)	)	PUNCT
ejpam-6171	892	26	6=	6=	ADP
ejpam-6171	892	27	∅.	∅.	ADP
ejpam-6171	892	28	by	by	ADP
ejpam-6171	892	29	the	the	DET
ejpam-6171	892	30	assumption	assumption	NOUN
ejpam-6171	892	31	,	,	PUNCT
ejpam-6171	892	32	we	we	PRON
ejpam-6171	892	33	have	have	VERB
ejpam-6171	892	34	u(pf	u(pf	NOUN
ejpam-6171	892	35	;	;	PUNCT
ejpam-6171	892	36	γ	γ	X
ejpam-6171	892	37	)	)	PUNCT
ejpam-6171	892	38	is	be	AUX
ejpam-6171	892	39	an	an	DET
ejpam-6171	892	40	iup	iup	NOUN
ejpam-6171	892	41	-	-	PUNCT
ejpam-6171	892	42	subalgebra	subalgebra	NOUN
ejpam-6171	892	43	of	of	ADP
ejpam-6171	892	44	x.	x.	NOUN
ejpam-6171	892	45	by	by	ADP
ejpam-6171	892	46	(	(	PUNCT
ejpam-6171	892	47	2.17	2.17	NUM
ejpam-6171	892	48	)	)	PUNCT
ejpam-6171	892	49	,	,	PUNCT
ejpam-6171	892	50	we	we	PRON
ejpam-6171	892	51	have	have	VERB
ejpam-6171	892	52	x	x	X
ejpam-6171	892	53	?	?	PUNCT
ejpam-6171	893	1	y	y	PROPN
ejpam-6171	893	2	∈	∈	PROPN
ejpam-6171	893	3	u(pf	u(pf	PROPN
ejpam-6171	893	4	;	;	PUNCT
ejpam-6171	893	5	γ	γ	X
ejpam-6171	893	6	)	)	PUNCT
ejpam-6171	893	7	.	.	PUNCT
ejpam-6171	894	1	thus	thus	ADV
ejpam-6171	894	2	,	,	PUNCT
ejpam-6171	894	3	pf	pf	PROPN
ejpam-6171	894	4	(	(	PUNCT
ejpam-6171	894	5	x	x	PROPN
ejpam-6171	894	6	?	?	PUNCT
ejpam-6171	894	7	y	y	X
ejpam-6171	894	8	)	)	PUNCT
ejpam-6171	894	9	≥	≥	PROPN
ejpam-6171	894	10	γ	γ	X
ejpam-6171	894	11	=	=	PUNCT
ejpam-6171	894	12	min{pf	min{pf	X
ejpam-6171	894	13	(	(	PUNCT
ejpam-6171	894	14	x),pf	x),pf	PROPN
ejpam-6171	894	15	(	(	PUNCT
ejpam-6171	894	16	y	y	NOUN
ejpam-6171	894	17	)	)	PUNCT
ejpam-6171	894	18	}	}	PUNCT
ejpam-6171	894	19	.	.	PUNCT
ejpam-6171	895	1	hence	hence	ADV
ejpam-6171	895	2	,	,	PUNCT
ejpam-6171	895	3	p	p	PROPN
ejpam-6171	895	4	is	be	AUX
ejpam-6171	895	5	a	a	DET
ejpam-6171	895	6	pythagorean	pythagorean	PROPN
ejpam-6171	895	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	895	8	iup	iup	NOUN
ejpam-6171	895	9	-	-	PUNCT
ejpam-6171	895	10	subalgebra	subalgebra	NOUN
ejpam-6171	895	11	of	of	ADP
ejpam-6171	895	12	x.	x.	PROPN
ejpam-6171	895	13	theorem	theorem	VERB
ejpam-6171	895	14	18	18	NUM
ejpam-6171	895	15	.	.	PUNCT
ejpam-6171	896	1	a	a	DET
ejpam-6171	896	2	pns	pns	NOUN
ejpam-6171	896	3	p	p	NOUN
ejpam-6171	896	4	in	in	ADP
ejpam-6171	896	5	x	x	PROPN
ejpam-6171	896	6	is	be	AUX
ejpam-6171	896	7	a	a	DET
ejpam-6171	896	8	pythagorean	pythagorean	PROPN
ejpam-6171	896	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	896	10	iup	iup	PROPN
ejpam-6171	896	11	-	-	PUNCT
ejpam-6171	896	12	ideal	ideal	NOUN
ejpam-6171	896	13	of	of	ADP
ejpam-6171	896	14	x	x	SYM
ejpam-6171	896	15	if	if	SCONJ
ejpam-6171	896	16	and	and	CCONJ
ejpam-6171	896	17	only	only	ADV
ejpam-6171	896	18	if	if	SCONJ
ejpam-6171	896	19	for	for	ADP
ejpam-6171	896	20	all	all	DET
ejpam-6171	896	21	α	α	NOUN
ejpam-6171	896	22	,	,	PUNCT
ejpam-6171	896	23	β	β	X
ejpam-6171	896	24	,	,	PUNCT
ejpam-6171	896	25	γ	γ	PROPN
ejpam-6171	896	26	∈	∈	PROPN
ejpam-6171	897	1	[	[	X
ejpam-6171	897	2	0	0	NUM
ejpam-6171	897	3	,	,	PUNCT
ejpam-6171	897	4	1	1	NUM
ejpam-6171	897	5	]	]	PUNCT
ejpam-6171	897	6	,	,	PUNCT
ejpam-6171	897	7	the	the	DET
ejpam-6171	897	8	sets	set	NOUN
ejpam-6171	897	9	u(pt	u(pt	PROPN
ejpam-6171	897	10	;	;	PUNCT
ejpam-6171	897	11	α	α	X
ejpam-6171	897	12	)	)	PUNCT
ejpam-6171	897	13	,	,	PUNCT
ejpam-6171	897	14	l(pi	l(pi	X
ejpam-6171	897	15	;	;	PUNCT
ejpam-6171	897	16	β	β	X
ejpam-6171	897	17	)	)	PUNCT
ejpam-6171	897	18	,	,	PUNCT
ejpam-6171	897	19	and	and	CCONJ
ejpam-6171	897	20	u(pf	u(pf	PROPN
ejpam-6171	897	21	;	;	PUNCT
ejpam-6171	897	22	γ	γ	X
ejpam-6171	897	23	)	)	PUNCT
ejpam-6171	897	24	are	be	AUX
ejpam-6171	897	25	either	either	CCONJ
ejpam-6171	897	26	empty	empty	ADJ
ejpam-6171	897	27	or	or	CCONJ
ejpam-6171	897	28	iup	iup	NOUN
ejpam-6171	897	29	-	-	PUNCT
ejpam-6171	897	30	ideals	ideal	NOUN
ejpam-6171	897	31	of	of	ADP
ejpam-6171	897	32	x.	x.	NOUN
ejpam-6171	897	33	proof	proof	NOUN
ejpam-6171	897	34	.	.	PUNCT
ejpam-6171	898	1	assume	assume	VERB
ejpam-6171	898	2	that	that	SCONJ
ejpam-6171	898	3	p	p	NOUN
ejpam-6171	898	4	in	in	ADP
ejpam-6171	898	5	x	x	PROPN
ejpam-6171	898	6	is	be	AUX
ejpam-6171	898	7	a	a	DET
ejpam-6171	898	8	pythagorean	pythagorean	PROPN
ejpam-6171	898	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	898	10	iup	iup	PROPN
ejpam-6171	898	11	-	-	PUNCT
ejpam-6171	898	12	ideal	ideal	NOUN
ejpam-6171	898	13	of	of	ADP
ejpam-6171	898	14	x.	x.	NOUN
ejpam-6171	898	15	let	let	VERB
ejpam-6171	898	16	α	α	PRON
ejpam-6171	898	17	∈	∈	PROPN
ejpam-6171	899	1	[	[	X
ejpam-6171	899	2	0	0	NUM
ejpam-6171	899	3	,	,	PUNCT
ejpam-6171	899	4	1	1	NUM
ejpam-6171	899	5	]	]	PUNCT
ejpam-6171	899	6	be	be	AUX
ejpam-6171	899	7	such	such	ADJ
ejpam-6171	899	8	that	that	SCONJ
ejpam-6171	899	9	u(pt	u(pt	PROPN
ejpam-6171	899	10	;	;	PUNCT
ejpam-6171	899	11	α	α	X
ejpam-6171	899	12	)	)	PUNCT
ejpam-6171	899	13	6=	6=	ADP
ejpam-6171	899	14	∅.	∅.	ADV
ejpam-6171	899	15	let	let	VERB
ejpam-6171	899	16	a	a	DET
ejpam-6171	899	17	∈	∈	NOUN
ejpam-6171	899	18	u(pt	u(pt	NOUN
ejpam-6171	899	19	;	;	PUNCT
ejpam-6171	899	20	α	α	X
ejpam-6171	899	21	)	)	PUNCT
ejpam-6171	899	22	.	.	PUNCT
ejpam-6171	900	1	then	then	ADV
ejpam-6171	900	2	pt	pt	X
ejpam-6171	900	3	(	(	PUNCT
ejpam-6171	900	4	a	a	NOUN
ejpam-6171	900	5	)	)	PUNCT
ejpam-6171	900	6	≥	≥	NOUN
ejpam-6171	900	7	α	α	NOUN
ejpam-6171	900	8	.	.	PUNCT
ejpam-6171	900	9	by	by	ADP
ejpam-6171	900	10	(	(	PUNCT
ejpam-6171	900	11	3.5	3.5	NUM
ejpam-6171	900	12	)	)	PUNCT
ejpam-6171	900	13	,	,	PUNCT
ejpam-6171	900	14	we	we	PRON
ejpam-6171	900	15	have	have	VERB
ejpam-6171	900	16	pt	pt	X
ejpam-6171	900	17	(	(	PUNCT
ejpam-6171	900	18	0	0	NUM
ejpam-6171	900	19	)	)	PUNCT
ejpam-6171	900	20	≥	≥	NOUN
ejpam-6171	900	21	pt	pt	INTJ
ejpam-6171	900	22	(	(	PUNCT
ejpam-6171	900	23	a	a	NOUN
ejpam-6171	900	24	)	)	PUNCT
ejpam-6171	900	25	≥	≥	NOUN
ejpam-6171	900	26	α	α	NOUN
ejpam-6171	900	27	.	.	PUNCT
ejpam-6171	901	1	thus	thus	ADV
ejpam-6171	901	2	,	,	PUNCT
ejpam-6171	901	3	0	0	NUM
ejpam-6171	901	4	∈	∈	PROPN
ejpam-6171	901	5	u(pt	u(pt	PROPN
ejpam-6171	901	6	;	;	PUNCT
ejpam-6171	901	7	α	α	X
ejpam-6171	901	8	)	)	PUNCT
ejpam-6171	901	9	.	.	PUNCT
ejpam-6171	902	1	let	let	VERB
ejpam-6171	902	2	x	x	PRON
ejpam-6171	902	3	,	,	PUNCT
ejpam-6171	902	4	y	y	PROPN
ejpam-6171	902	5	,	,	PUNCT
ejpam-6171	902	6	z	z	NOUN
ejpam-6171	902	7	∈	∈	PROPN
ejpam-6171	902	8	x	x	AUX
ejpam-6171	902	9	be	be	AUX
ejpam-6171	902	10	such	such	ADJ
ejpam-6171	902	11	that	that	PRON
ejpam-6171	902	12	x	x	PUNCT
ejpam-6171	902	13	?	?	PUNCT
ejpam-6171	903	1	(	(	PUNCT
ejpam-6171	903	2	y	y	NOUN
ejpam-6171	903	3	?	?	PUNCT
ejpam-6171	904	1	z	z	X
ejpam-6171	904	2	)	)	PUNCT
ejpam-6171	904	3	∈	∈	PROPN
ejpam-6171	904	4	u(pt	u(pt	PROPN
ejpam-6171	904	5	;	;	PUNCT
ejpam-6171	904	6	α	α	X
ejpam-6171	904	7	)	)	PUNCT
ejpam-6171	904	8	and	and	CCONJ
ejpam-6171	904	9	y	y	PROPN
ejpam-6171	904	10	∈	∈	PROPN
ejpam-6171	904	11	u(pt	u(pt	PROPN
ejpam-6171	904	12	;	;	PUNCT
ejpam-6171	904	13	α	α	X
ejpam-6171	904	14	)	)	PUNCT
ejpam-6171	904	15	.	.	PUNCT
ejpam-6171	905	1	then	then	ADV
ejpam-6171	905	2	pt	pt	INTJ
ejpam-6171	905	3	(	(	PUNCT
ejpam-6171	905	4	x	x	X
ejpam-6171	905	5	?	?	PUNCT
ejpam-6171	906	1	(	(	PUNCT
ejpam-6171	906	2	y	y	NOUN
ejpam-6171	906	3	?	?	PUNCT
ejpam-6171	907	1	z	z	X
ejpam-6171	907	2	)	)	PUNCT
ejpam-6171	907	3	)	)	PUNCT
ejpam-6171	908	1	≥	≥	PROPN
ejpam-6171	908	2	α	α	NOUN
ejpam-6171	908	3	and	and	CCONJ
ejpam-6171	908	4	pt	pt	PROPN
ejpam-6171	908	5	(	(	PUNCT
ejpam-6171	908	6	y	y	NOUN
ejpam-6171	908	7	)	)	PUNCT
ejpam-6171	908	8	≥	≥	PROPN
ejpam-6171	908	9	α	α	NOUN
ejpam-6171	908	10	.	.	PUNCT
ejpam-6171	909	1	thus	thus	ADV
ejpam-6171	909	2	,	,	PUNCT
ejpam-6171	909	3	min{pt	min{pt	PUNCT
ejpam-6171	909	4	(	(	PUNCT
ejpam-6171	909	5	x	x	X
ejpam-6171	909	6	?	?	PUNCT
ejpam-6171	910	1	(	(	PUNCT
ejpam-6171	910	2	y	y	PROPN
ejpam-6171	910	3	?	?	PUNCT
ejpam-6171	911	1	z)),pt	z)),pt	PROPN
ejpam-6171	911	2	(	(	PUNCT
ejpam-6171	911	3	y	y	NOUN
ejpam-6171	911	4	)	)	PUNCT
ejpam-6171	911	5	}	}	PUNCT
ejpam-6171	911	6	≥	≥	PROPN
ejpam-6171	911	7	α	α	NOUN
ejpam-6171	911	8	.	.	PUNCT
ejpam-6171	912	1	by	by	ADP
ejpam-6171	912	2	(	(	PUNCT
ejpam-6171	912	3	3.8	3.8	NUM
ejpam-6171	912	4	)	)	PUNCT
ejpam-6171	912	5	,	,	PUNCT
ejpam-6171	912	6	we	we	PRON
ejpam-6171	912	7	have	have	VERB
ejpam-6171	912	8	pt	pt	INTJ
ejpam-6171	912	9	(	(	PUNCT
ejpam-6171	912	10	x	x	NOUN
ejpam-6171	912	11	?	?	PUNCT
ejpam-6171	913	1	z	z	X
ejpam-6171	913	2	)	)	PUNCT
ejpam-6171	913	3	≥	≥	NOUN
ejpam-6171	913	4	min{pt	min{pt	X
ejpam-6171	914	1	(	(	PUNCT
ejpam-6171	914	2	x	x	X
ejpam-6171	914	3	?	?	PUNCT
ejpam-6171	915	1	(	(	PUNCT
ejpam-6171	915	2	y	y	PROPN
ejpam-6171	915	3	?	?	PUNCT
ejpam-6171	916	1	z)),pt	z)),pt	PROPN
ejpam-6171	916	2	(	(	PUNCT
ejpam-6171	916	3	y	y	NOUN
ejpam-6171	916	4	)	)	PUNCT
ejpam-6171	916	5	}	}	PUNCT
ejpam-6171	916	6	≥	≥	PROPN
ejpam-6171	916	7	α	α	NOUN
ejpam-6171	916	8	.	.	PUNCT
ejpam-6171	917	1	thus	thus	ADV
ejpam-6171	917	2	,	,	PUNCT
ejpam-6171	917	3	x	x	PUNCT
ejpam-6171	917	4	?	?	PUNCT
ejpam-6171	918	1	z	z	PUNCT
ejpam-6171	918	2	∈	∈	PROPN
ejpam-6171	918	3	u(pt	u(pt	PROPN
ejpam-6171	918	4	;	;	PUNCT
ejpam-6171	918	5	α	α	X
ejpam-6171	918	6	)	)	PUNCT
ejpam-6171	918	7	.	.	PUNCT
ejpam-6171	919	1	hence	hence	ADV
ejpam-6171	919	2	,	,	PUNCT
ejpam-6171	919	3	u(pt	u(pt	PROPN
ejpam-6171	919	4	;	;	PUNCT
ejpam-6171	919	5	α	α	X
ejpam-6171	919	6	)	)	PUNCT
ejpam-6171	919	7	is	be	AUX
ejpam-6171	919	8	an	an	DET
ejpam-6171	919	9	iup	iup	NOUN
ejpam-6171	919	10	-	-	PUNCT
ejpam-6171	919	11	ideal	ideal	NOUN
ejpam-6171	919	12	of	of	ADP
ejpam-6171	919	13	x.	x.	NOUN
ejpam-6171	919	14	let	let	VERB
ejpam-6171	919	15	β	β	X
ejpam-6171	919	16	∈	∈	PROPN
ejpam-6171	920	1	[	[	X
ejpam-6171	920	2	0	0	NUM
ejpam-6171	920	3	,	,	PUNCT
ejpam-6171	920	4	1	1	NUM
ejpam-6171	920	5	]	]	PUNCT
ejpam-6171	920	6	be	be	AUX
ejpam-6171	920	7	such	such	ADJ
ejpam-6171	920	8	that	that	SCONJ
ejpam-6171	920	9	l(pi	l(pi	X
ejpam-6171	920	10	;	;	PUNCT
ejpam-6171	920	11	β	β	X
ejpam-6171	920	12	)	)	PUNCT
ejpam-6171	920	13	6=	6=	ADP
ejpam-6171	920	14	∅.	∅.	AUX
ejpam-6171	920	15	let	let	VERB
ejpam-6171	920	16	b	b	NOUN
ejpam-6171	920	17	∈	∈	NOUN
ejpam-6171	920	18	l(pi	l(pi	X
ejpam-6171	920	19	;	;	PUNCT
ejpam-6171	920	20	β	β	X
ejpam-6171	920	21	)	)	PUNCT
ejpam-6171	920	22	.	.	PUNCT
ejpam-6171	921	1	then	then	ADV
ejpam-6171	921	2	pi(b	pi(b	PUNCT
ejpam-6171	921	3	)	)	PUNCT
ejpam-6171	921	4	≤	≤	PUNCT
ejpam-6171	922	1	β	β	X
ejpam-6171	922	2	.	.	PUNCT
ejpam-6171	923	1	by	by	ADP
ejpam-6171	923	2	(	(	PUNCT
ejpam-6171	923	3	3.6	3.6	NUM
ejpam-6171	923	4	)	)	PUNCT
ejpam-6171	923	5	,	,	PUNCT
ejpam-6171	923	6	we	we	PRON
ejpam-6171	923	7	have	have	VERB
ejpam-6171	923	8	pi(0	pi(0	PROPN
ejpam-6171	923	9	)	)	PUNCT
ejpam-6171	923	10	≤	≤	NOUN
ejpam-6171	923	11	pi(b	pi(b	PUNCT
ejpam-6171	923	12	)	)	PUNCT
ejpam-6171	923	13	≤	≤	PUNCT
ejpam-6171	924	1	β	β	X
ejpam-6171	924	2	.	.	PUNCT
ejpam-6171	925	1	thus	thus	ADV
ejpam-6171	925	2	,	,	PUNCT
ejpam-6171	925	3	0	0	NUM
ejpam-6171	925	4	∈	∈	PROPN
ejpam-6171	925	5	l(pi	l(pi	NOUN
ejpam-6171	925	6	;	;	PUNCT
ejpam-6171	925	7	β	β	X
ejpam-6171	925	8	)	)	PUNCT
ejpam-6171	925	9	.	.	PUNCT
ejpam-6171	926	1	let	let	VERB
ejpam-6171	926	2	x	x	PRON
ejpam-6171	926	3	,	,	PUNCT
ejpam-6171	926	4	y	y	PROPN
ejpam-6171	926	5	,	,	PUNCT
ejpam-6171	926	6	z	z	NOUN
ejpam-6171	926	7	∈	∈	PROPN
ejpam-6171	926	8	x	x	AUX
ejpam-6171	926	9	be	be	AUX
ejpam-6171	926	10	such	such	ADJ
ejpam-6171	926	11	that	that	PRON
ejpam-6171	926	12	x	x	PUNCT
ejpam-6171	926	13	?	?	PUNCT
ejpam-6171	927	1	(	(	PUNCT
ejpam-6171	927	2	y	y	NOUN
ejpam-6171	927	3	?	?	PUNCT
ejpam-6171	928	1	z	z	X
ejpam-6171	928	2	)	)	PUNCT
ejpam-6171	928	3	∈	∈	PROPN
ejpam-6171	928	4	l(pi	l(pi	NOUN
ejpam-6171	928	5	;	;	PUNCT
ejpam-6171	928	6	β	β	X
ejpam-6171	928	7	)	)	PUNCT
ejpam-6171	928	8	and	and	CCONJ
ejpam-6171	928	9	y	y	PROPN
ejpam-6171	928	10	∈	∈	PROPN
ejpam-6171	928	11	l(pi	l(pi	PROPN
ejpam-6171	928	12	;	;	PUNCT
ejpam-6171	928	13	β	β	X
ejpam-6171	928	14	)	)	PUNCT
ejpam-6171	928	15	.	.	PUNCT
ejpam-6171	929	1	then	then	ADV
ejpam-6171	929	2	pi(x	pi(x	PROPN
ejpam-6171	929	3	?	?	PUNCT
ejpam-6171	930	1	(	(	PUNCT
ejpam-6171	930	2	y	y	NOUN
ejpam-6171	930	3	?	?	PUNCT
ejpam-6171	931	1	z	z	X
ejpam-6171	931	2	)	)	PUNCT
ejpam-6171	931	3	)	)	PUNCT
ejpam-6171	932	1	≤	≤	NUM
ejpam-6171	932	2	β	β	NOUN
ejpam-6171	932	3	and	and	CCONJ
ejpam-6171	932	4	pi(y	pi(y	NOUN
ejpam-6171	932	5	)	)	PUNCT
ejpam-6171	932	6	≤	≤	NOUN
ejpam-6171	932	7	β	β	X
ejpam-6171	932	8	.	.	PUNCT
ejpam-6171	933	1	thus	thus	ADV
ejpam-6171	933	2	,	,	PUNCT
ejpam-6171	933	3	max{pi(x?(y?z)),pi(y	max{pi(x?(y?z)),pi(y	NUM
ejpam-6171	933	4	)	)	PUNCT
ejpam-6171	933	5	}	}	PUNCT
ejpam-6171	933	6	≤	≤	NOUN
ejpam-6171	933	7	β	β	X
ejpam-6171	933	8	.	.	PUNCT
ejpam-6171	934	1	by	by	ADP
ejpam-6171	934	2	(	(	PUNCT
ejpam-6171	934	3	3.9	3.9	NUM
ejpam-6171	934	4	)	)	PUNCT
ejpam-6171	934	5	,	,	PUNCT
ejpam-6171	934	6	we	we	PRON
ejpam-6171	934	7	have	have	VERB
ejpam-6171	934	8	pi(x?z	pi(x?z	NOUN
ejpam-6171	934	9	)	)	PUNCT
ejpam-6171	934	10	≤	≤	NUM
ejpam-6171	934	11	max{pi(x?(y?z)),pi(y	max{pi(x?(y?z)),pi(y	NUM
ejpam-6171	934	12	)	)	PUNCT
ejpam-6171	934	13	}	}	PUNCT
ejpam-6171	934	14	≤	≤	NOUN
ejpam-6171	934	15	β	β	X
ejpam-6171	934	16	.	.	PUNCT
ejpam-6171	935	1	thus	thus	ADV
ejpam-6171	935	2	,	,	PUNCT
ejpam-6171	935	3	x	x	PUNCT
ejpam-6171	935	4	?	?	PUNCT
ejpam-6171	936	1	z	z	NOUN
ejpam-6171	936	2	∈	∈	PROPN
ejpam-6171	936	3	l(pi	l(pi	NOUN
ejpam-6171	936	4	;	;	PUNCT
ejpam-6171	936	5	β	β	X
ejpam-6171	936	6	)	)	PUNCT
ejpam-6171	936	7	.	.	PUNCT
ejpam-6171	937	1	hence	hence	ADV
ejpam-6171	937	2	,	,	PUNCT
ejpam-6171	937	3	l(pi	l(pi	X
ejpam-6171	937	4	;	;	PUNCT
ejpam-6171	937	5	β	β	X
ejpam-6171	937	6	)	)	PUNCT
ejpam-6171	937	7	is	be	AUX
ejpam-6171	937	8	an	an	DET
ejpam-6171	937	9	iup	iup	NOUN
ejpam-6171	937	10	-	-	PUNCT
ejpam-6171	937	11	ideal	ideal	NOUN
ejpam-6171	937	12	of	of	ADP
ejpam-6171	937	13	x.	x.	NOUN
ejpam-6171	937	14	let	let	VERB
ejpam-6171	937	15	γ	γ	X
ejpam-6171	937	16	∈	∈	PROPN
ejpam-6171	938	1	[	[	X
ejpam-6171	938	2	0	0	NUM
ejpam-6171	938	3	,	,	PUNCT
ejpam-6171	938	4	1	1	NUM
ejpam-6171	938	5	]	]	PUNCT
ejpam-6171	938	6	be	be	AUX
ejpam-6171	938	7	such	such	ADJ
ejpam-6171	938	8	that	that	SCONJ
ejpam-6171	938	9	u(pf	u(pf	PROPN
ejpam-6171	938	10	;	;	PUNCT
ejpam-6171	938	11	γ	γ	X
ejpam-6171	938	12	)	)	PUNCT
ejpam-6171	938	13	6=	6=	ADP
ejpam-6171	938	14	∅.	∅.	ADV
ejpam-6171	938	15	let	let	VERB
ejpam-6171	938	16	c	c	PROPN
ejpam-6171	938	17	∈	∈	PROPN
ejpam-6171	938	18	u(pf	u(pf	PROPN
ejpam-6171	938	19	;	;	PUNCT
ejpam-6171	938	20	γ	γ	X
ejpam-6171	938	21	)	)	PUNCT
ejpam-6171	938	22	.	.	PUNCT
ejpam-6171	939	1	then	then	ADV
ejpam-6171	939	2	pf	pf	PROPN
ejpam-6171	939	3	(	(	PUNCT
ejpam-6171	939	4	c	c	NOUN
ejpam-6171	939	5	)	)	PUNCT
ejpam-6171	939	6	≥	≥	PROPN
ejpam-6171	939	7	γ	γ	X
ejpam-6171	939	8	.	.	PUNCT
ejpam-6171	939	9	by	by	ADP
ejpam-6171	939	10	(	(	PUNCT
ejpam-6171	939	11	3.7	3.7	NUM
ejpam-6171	939	12	)	)	PUNCT
ejpam-6171	939	13	,	,	PUNCT
ejpam-6171	939	14	we	we	PRON
ejpam-6171	939	15	have	have	VERB
ejpam-6171	939	16	pf	pf	PROPN
ejpam-6171	939	17	(	(	PUNCT
ejpam-6171	939	18	0	0	NUM
ejpam-6171	939	19	)	)	PUNCT
ejpam-6171	939	20	≥	≥	NOUN
ejpam-6171	939	21	pf	pf	X
ejpam-6171	939	22	(	(	PUNCT
ejpam-6171	939	23	c	c	NOUN
ejpam-6171	939	24	)	)	PUNCT
ejpam-6171	939	25	≥	≥	PROPN
ejpam-6171	939	26	γ	γ	PROPN
ejpam-6171	939	27	.	.	PROPN
ejpam-6171	940	1	thus	thus	ADV
ejpam-6171	940	2	,	,	PUNCT
ejpam-6171	940	3	0	0	NUM
ejpam-6171	940	4	∈	∈	PROPN
ejpam-6171	940	5	u(pf	u(pf	PROPN
ejpam-6171	940	6	;	;	PUNCT
ejpam-6171	940	7	γ	γ	X
ejpam-6171	940	8	)	)	PUNCT
ejpam-6171	940	9	.	.	PUNCT
ejpam-6171	941	1	let	let	VERB
ejpam-6171	941	2	x	x	PRON
ejpam-6171	941	3	,	,	PUNCT
ejpam-6171	941	4	y	y	PROPN
ejpam-6171	941	5	,	,	PUNCT
ejpam-6171	941	6	z	z	NOUN
ejpam-6171	941	7	∈	∈	PROPN
ejpam-6171	941	8	x	x	AUX
ejpam-6171	941	9	be	be	AUX
ejpam-6171	941	10	such	such	ADJ
ejpam-6171	941	11	that	that	PRON
ejpam-6171	941	12	x	x	PUNCT
ejpam-6171	941	13	?	?	PUNCT
ejpam-6171	942	1	(	(	PUNCT
ejpam-6171	942	2	y	y	NOUN
ejpam-6171	942	3	?	?	PUNCT
ejpam-6171	943	1	z	z	X
ejpam-6171	943	2	)	)	PUNCT
ejpam-6171	943	3	∈	∈	PROPN
ejpam-6171	943	4	u(pf	u(pf	PROPN
ejpam-6171	943	5	;	;	PUNCT
ejpam-6171	943	6	γ	γ	X
ejpam-6171	943	7	)	)	PUNCT
ejpam-6171	943	8	and	and	CCONJ
ejpam-6171	943	9	y	y	PROPN
ejpam-6171	943	10	∈	∈	PROPN
ejpam-6171	943	11	u(pf	u(pf	PROPN
ejpam-6171	943	12	;	;	PUNCT
ejpam-6171	943	13	γ	γ	X
ejpam-6171	943	14	)	)	PUNCT
ejpam-6171	943	15	.	.	PUNCT
ejpam-6171	944	1	then	then	ADV
ejpam-6171	944	2	pf	pf	INTJ
ejpam-6171	944	3	(	(	PUNCT
ejpam-6171	944	4	x	x	PROPN
ejpam-6171	944	5	?	?	PUNCT
ejpam-6171	945	1	(	(	PUNCT
ejpam-6171	945	2	y	y	NOUN
ejpam-6171	945	3	?	?	PUNCT
ejpam-6171	946	1	z	z	X
ejpam-6171	946	2	)	)	PUNCT
ejpam-6171	946	3	)	)	PUNCT
ejpam-6171	947	1	≥	≥	PROPN
ejpam-6171	947	2	γ	γ	PROPN
ejpam-6171	947	3	and	and	CCONJ
ejpam-6171	947	4	pf	pf	PROPN
ejpam-6171	947	5	(	(	PUNCT
ejpam-6171	947	6	y	y	PROPN
ejpam-6171	947	7	)	)	PUNCT
ejpam-6171	947	8	≥	≥	PROPN
ejpam-6171	947	9	γ	γ	PROPN
ejpam-6171	947	10	.	.	PUNCT
ejpam-6171	947	11	thus	thus	ADV
ejpam-6171	947	12	,	,	PUNCT
ejpam-6171	947	13	min{pf	min{pf	PRON
ejpam-6171	947	14	(	(	PUNCT
ejpam-6171	947	15	x	x	X
ejpam-6171	947	16	?	?	PUNCT
ejpam-6171	948	1	(	(	PUNCT
ejpam-6171	948	2	y	y	NOUN
ejpam-6171	948	3	?	?	PUNCT
ejpam-6171	949	1	z)),pf	z)),pf	PROPN
ejpam-6171	949	2	(	(	PUNCT
ejpam-6171	949	3	y	y	NOUN
ejpam-6171	949	4	)	)	PUNCT
ejpam-6171	949	5	}	}	PUNCT
ejpam-6171	949	6	≥	≥	PROPN
ejpam-6171	949	7	γ	γ	X
ejpam-6171	949	8	.	.	PUNCT
ejpam-6171	949	9	by	by	ADP
ejpam-6171	949	10	(	(	PUNCT
ejpam-6171	949	11	3.10	3.10	NUM
ejpam-6171	949	12	)	)	PUNCT
ejpam-6171	949	13	,	,	PUNCT
ejpam-6171	949	14	we	we	PRON
ejpam-6171	949	15	have	have	VERB
ejpam-6171	949	16	pf	pf	NOUN
ejpam-6171	949	17	(	(	PUNCT
ejpam-6171	949	18	x	x	NOUN
ejpam-6171	949	19	?	?	PUNCT
ejpam-6171	950	1	z	z	X
ejpam-6171	950	2	)	)	PUNCT
ejpam-6171	950	3	≥	≥	NOUN
ejpam-6171	950	4	min{pf	min{pf	X
ejpam-6171	950	5	(	(	PUNCT
ejpam-6171	950	6	x	x	X
ejpam-6171	950	7	?	?	PUNCT
ejpam-6171	951	1	(	(	PUNCT
ejpam-6171	951	2	y	y	NOUN
ejpam-6171	951	3	?	?	PUNCT
ejpam-6171	952	1	z)),pf	z)),pf	PROPN
ejpam-6171	952	2	(	(	PUNCT
ejpam-6171	952	3	y	y	NOUN
ejpam-6171	952	4	)	)	PUNCT
ejpam-6171	952	5	}	}	PUNCT
ejpam-6171	952	6	≥	≥	PROPN
ejpam-6171	952	7	γ	γ	X
ejpam-6171	952	8	.	.	PROPN
ejpam-6171	952	9	thus	thus	ADV
ejpam-6171	952	10	,	,	PUNCT
ejpam-6171	952	11	x	x	PUNCT
ejpam-6171	952	12	?	?	PUNCT
ejpam-6171	952	13	z	z	PROPN
ejpam-6171	952	14	∈	∈	PROPN
ejpam-6171	952	15	u(pf	u(pf	PROPN
ejpam-6171	952	16	;	;	PUNCT
ejpam-6171	952	17	γ	γ	X
ejpam-6171	952	18	)	)	PUNCT
ejpam-6171	952	19	.	.	PUNCT
ejpam-6171	953	1	hence	hence	ADV
ejpam-6171	953	2	,	,	PUNCT
ejpam-6171	953	3	u(pf	u(pf	PROPN
ejpam-6171	953	4	;	;	PUNCT
ejpam-6171	953	5	γ	γ	X
ejpam-6171	953	6	)	)	PUNCT
ejpam-6171	953	7	is	be	AUX
ejpam-6171	953	8	an	an	DET
ejpam-6171	953	9	iup	iup	NOUN
ejpam-6171	953	10	-	-	PUNCT
ejpam-6171	953	11	ideal	ideal	NOUN
ejpam-6171	953	12	of	of	ADP
ejpam-6171	953	13	x.	x.	NOUN
ejpam-6171	953	14	conversely	conversely	ADV
ejpam-6171	953	15	,	,	PUNCT
ejpam-6171	953	16	assume	assume	VERB
ejpam-6171	953	17	that	that	SCONJ
ejpam-6171	953	18	for	for	ADP
ejpam-6171	953	19	all	all	DET
ejpam-6171	953	20	α	α	NOUN
ejpam-6171	953	21	,	,	PUNCT
ejpam-6171	953	22	β	β	X
ejpam-6171	953	23	,	,	PUNCT
ejpam-6171	953	24	γ	γ	PROPN
ejpam-6171	953	25	∈	∈	PROPN
ejpam-6171	954	1	[	[	X
ejpam-6171	954	2	0	0	NUM
ejpam-6171	954	3	,	,	PUNCT
ejpam-6171	954	4	1	1	NUM
ejpam-6171	954	5	]	]	PUNCT
ejpam-6171	954	6	,	,	PUNCT
ejpam-6171	954	7	the	the	DET
ejpam-6171	954	8	sets	set	NOUN
ejpam-6171	954	9	u(pt	u(pt	PROPN
ejpam-6171	954	10	;	;	PUNCT
ejpam-6171	954	11	α	α	X
ejpam-6171	954	12	)	)	PUNCT
ejpam-6171	954	13	,	,	PUNCT
ejpam-6171	954	14	l(pi	l(pi	X
ejpam-6171	954	15	;	;	PUNCT
ejpam-6171	954	16	β	β	X
ejpam-6171	954	17	)	)	PUNCT
ejpam-6171	954	18	,	,	PUNCT
ejpam-6171	954	19	and	and	CCONJ
ejpam-6171	954	20	u(pf	u(pf	PROPN
ejpam-6171	954	21	;	;	PUNCT
ejpam-6171	954	22	γ	γ	X
ejpam-6171	954	23	)	)	PUNCT
ejpam-6171	954	24	are	be	AUX
ejpam-6171	954	25	either	either	CCONJ
ejpam-6171	954	26	empty	empty	ADJ
ejpam-6171	954	27	or	or	CCONJ
ejpam-6171	954	28	iup	iup	NOUN
ejpam-6171	954	29	-	-	PUNCT
ejpam-6171	954	30	ideals	ideal	NOUN
ejpam-6171	954	31	of	of	ADP
ejpam-6171	954	32	x.	x.	NOUN
ejpam-6171	954	33	let	let	VERB
ejpam-6171	954	34	x	x	SYM
ejpam-6171	954	35	∈	∈	PROPN
ejpam-6171	954	36	x.	x.	NOUN
ejpam-6171	954	37	let	let	VERB
ejpam-6171	954	38	α	α	NOUN
ejpam-6171	954	39	=	=	SYM
ejpam-6171	954	40	pt	pt	X
ejpam-6171	954	41	(	(	PUNCT
ejpam-6171	954	42	x	x	NOUN
ejpam-6171	954	43	)	)	PUNCT
ejpam-6171	954	44	.	.	PUNCT
ejpam-6171	955	1	then	then	ADV
ejpam-6171	955	2	pt	pt	X
ejpam-6171	955	3	(	(	PUNCT
ejpam-6171	955	4	x	x	NOUN
ejpam-6171	955	5	)	)	PUNCT
ejpam-6171	955	6	≥	≥	PROPN
ejpam-6171	955	7	α	α	NOUN
ejpam-6171	955	8	.	.	PUNCT
ejpam-6171	956	1	thus	thus	ADV
ejpam-6171	956	2	,	,	PUNCT
ejpam-6171	956	3	x	x	SYM
ejpam-6171	956	4	∈	∈	NOUN
ejpam-6171	956	5	u(pt	u(pt	PROPN
ejpam-6171	956	6	;	;	PUNCT
ejpam-6171	956	7	α	α	X
ejpam-6171	956	8	)	)	PUNCT
ejpam-6171	956	9	6=	6=	ADP
ejpam-6171	956	10	∅.	∅.	ADP
ejpam-6171	956	11	by	by	ADP
ejpam-6171	956	12	the	the	DET
ejpam-6171	956	13	assumption	assumption	NOUN
ejpam-6171	956	14	,	,	PUNCT
ejpam-6171	956	15	we	we	PRON
ejpam-6171	956	16	have	have	VERB
ejpam-6171	956	17	u(pt	u(pt	NOUN
ejpam-6171	956	18	;	;	PUNCT
ejpam-6171	956	19	α	α	X
ejpam-6171	956	20	)	)	PUNCT
ejpam-6171	956	21	is	be	AUX
ejpam-6171	956	22	an	an	DET
ejpam-6171	956	23	iupideal	iupideal	NOUN
ejpam-6171	956	24	of	of	ADP
ejpam-6171	956	25	x.	x.	NOUN
ejpam-6171	956	26	by	by	ADP
ejpam-6171	956	27	(	(	PUNCT
ejpam-6171	956	28	2.18	2.18	NUM
ejpam-6171	956	29	)	)	PUNCT
ejpam-6171	956	30	,	,	PUNCT
ejpam-6171	956	31	we	we	PRON
ejpam-6171	956	32	have	have	VERB
ejpam-6171	956	33	0	0	NUM
ejpam-6171	956	34	∈	∈	NOUN
ejpam-6171	956	35	u(pt	u(pt	PROPN
ejpam-6171	956	36	;	;	PUNCT
ejpam-6171	956	37	α	α	X
ejpam-6171	956	38	)	)	PUNCT
ejpam-6171	956	39	.	.	PUNCT
ejpam-6171	957	1	then	then	ADV
ejpam-6171	957	2	pt	pt	X
ejpam-6171	957	3	(	(	PUNCT
ejpam-6171	957	4	0	0	NUM
ejpam-6171	957	5	)	)	PUNCT
ejpam-6171	957	6	≥	≥	NOUN
ejpam-6171	957	7	α	α	NOUN
ejpam-6171	957	8	=	=	SYM
ejpam-6171	957	9	pt	pt	X
ejpam-6171	957	10	(	(	PUNCT
ejpam-6171	957	11	x	x	NOUN
ejpam-6171	957	12	)	)	PUNCT
ejpam-6171	957	13	.	.	PUNCT
ejpam-6171	958	1	let	let	VERB
ejpam-6171	958	2	x	x	PRON
ejpam-6171	958	3	,	,	PUNCT
ejpam-6171	958	4	y	y	PROPN
ejpam-6171	958	5	,	,	PUNCT
ejpam-6171	958	6	z	z	PROPN
ejpam-6171	958	7	∈	∈	PROPN
ejpam-6171	958	8	x.	x.	NOUN
ejpam-6171	958	9	let	let	VERB
ejpam-6171	958	10	α	α	NOUN
ejpam-6171	958	11	=	=	PUNCT
ejpam-6171	958	12	min{pt	min{pt	NOUN
ejpam-6171	958	13	(	(	PUNCT
ejpam-6171	958	14	x	x	X
ejpam-6171	958	15	?	?	PUNCT
ejpam-6171	959	1	(	(	PUNCT
ejpam-6171	959	2	y	y	PROPN
ejpam-6171	959	3	?	?	PUNCT
ejpam-6171	960	1	z)),pt	z)),pt	PROPN
ejpam-6171	960	2	(	(	PUNCT
ejpam-6171	960	3	y	y	NOUN
ejpam-6171	960	4	)	)	PUNCT
ejpam-6171	960	5	}	}	PUNCT
ejpam-6171	960	6	.	.	PUNCT
ejpam-6171	961	1	then	then	ADV
ejpam-6171	961	2	pt	pt	INTJ
ejpam-6171	961	3	(	(	PUNCT
ejpam-6171	961	4	x	x	X
ejpam-6171	961	5	?	?	PUNCT
ejpam-6171	962	1	(	(	PUNCT
ejpam-6171	962	2	y	y	NOUN
ejpam-6171	962	3	?	?	PUNCT
ejpam-6171	963	1	z	z	X
ejpam-6171	963	2	)	)	PUNCT
ejpam-6171	963	3	)	)	PUNCT
ejpam-6171	964	1	≥	≥	PROPN
ejpam-6171	964	2	α	α	NOUN
ejpam-6171	964	3	and	and	CCONJ
ejpam-6171	964	4	pt	pt	PROPN
ejpam-6171	964	5	(	(	PUNCT
ejpam-6171	964	6	y	y	NOUN
ejpam-6171	964	7	)	)	PUNCT
ejpam-6171	964	8	≥	≥	PROPN
ejpam-6171	964	9	α	α	NOUN
ejpam-6171	964	10	.	.	PUNCT
ejpam-6171	965	1	thus	thus	ADV
ejpam-6171	965	2	,	,	PUNCT
ejpam-6171	965	3	x	x	PUNCT
ejpam-6171	965	4	?	?	PUNCT
ejpam-6171	966	1	(	(	PUNCT
ejpam-6171	966	2	y	y	NOUN
ejpam-6171	966	3	?	?	PUNCT
ejpam-6171	967	1	z	z	X
ejpam-6171	967	2	)	)	PUNCT
ejpam-6171	967	3	,	,	PUNCT
ejpam-6171	967	4	y	y	PROPN
ejpam-6171	967	5	∈	∈	PROPN
ejpam-6171	967	6	u(pt	u(pt	PROPN
ejpam-6171	967	7	;	;	PUNCT
ejpam-6171	967	8	α	α	X
ejpam-6171	967	9	)	)	PUNCT
ejpam-6171	967	10	6=	6=	ADP
ejpam-6171	967	11	∅.	∅.	ADP
ejpam-6171	967	12	by	by	ADP
ejpam-6171	967	13	the	the	DET
ejpam-6171	967	14	assumption	assumption	NOUN
ejpam-6171	967	15	,	,	PUNCT
ejpam-6171	967	16	we	we	PRON
ejpam-6171	967	17	have	have	VERB
ejpam-6171	967	18	u(pt	u(pt	NOUN
ejpam-6171	967	19	;	;	PUNCT
ejpam-6171	967	20	α	α	X
ejpam-6171	967	21	)	)	PUNCT
ejpam-6171	967	22	is	be	AUX
ejpam-6171	967	23	an	an	DET
ejpam-6171	967	24	iup	iup	NOUN
ejpam-6171	967	25	-	-	PUNCT
ejpam-6171	967	26	ideal	ideal	NOUN
ejpam-6171	967	27	of	of	ADP
ejpam-6171	967	28	x.	x.	NOUN
ejpam-6171	967	29	by	by	ADP
ejpam-6171	967	30	(	(	PUNCT
ejpam-6171	967	31	2.20	2.20	NUM
ejpam-6171	967	32	)	)	PUNCT
ejpam-6171	967	33	,	,	PUNCT
ejpam-6171	967	34	we	we	PRON
ejpam-6171	967	35	have	have	VERB
ejpam-6171	967	36	x	x	X
ejpam-6171	967	37	?	?	PUNCT
ejpam-6171	968	1	z	z	X
ejpam-6171	968	2	∈	∈	PROPN
ejpam-6171	968	3	u(pt	u(pt	PROPN
ejpam-6171	968	4	;	;	PUNCT
ejpam-6171	968	5	α	α	X
ejpam-6171	968	6	)	)	PUNCT
ejpam-6171	968	7	.	.	PUNCT
ejpam-6171	969	1	thus	thus	ADV
ejpam-6171	969	2	,	,	PUNCT
ejpam-6171	969	3	pt	pt	INTJ
ejpam-6171	969	4	(	(	PUNCT
ejpam-6171	969	5	x	x	NOUN
ejpam-6171	969	6	?	?	PUNCT
ejpam-6171	969	7	z	z	X
ejpam-6171	969	8	)	)	PUNCT
ejpam-6171	969	9	≥	≥	NOUN
ejpam-6171	969	10	α	α	NOUN
ejpam-6171	969	11	=	=	X
ejpam-6171	969	12	min{pt	min{pt	NOUN
ejpam-6171	969	13	(	(	PUNCT
ejpam-6171	969	14	x	x	X
ejpam-6171	969	15	?	?	PUNCT
ejpam-6171	970	1	(	(	PUNCT
ejpam-6171	970	2	y	y	PROPN
ejpam-6171	970	3	?	?	PUNCT
ejpam-6171	971	1	z)),pt	z)),pt	PROPN
ejpam-6171	971	2	(	(	PUNCT
ejpam-6171	971	3	y	y	NOUN
ejpam-6171	971	4	)	)	PUNCT
ejpam-6171	971	5	}	}	PUNCT
ejpam-6171	971	6	.	.	PUNCT
ejpam-6171	972	1	let	let	VERB
ejpam-6171	972	2	x	x	SYM
ejpam-6171	972	3	∈	∈	PROPN
ejpam-6171	972	4	x.	x.	NOUN
ejpam-6171	972	5	let	let	VERB
ejpam-6171	972	6	β	β	X
ejpam-6171	972	7	=	=	SYM
ejpam-6171	972	8	pi(x	pi(x	PROPN
ejpam-6171	972	9	)	)	PUNCT
ejpam-6171	972	10	.	.	PUNCT
ejpam-6171	973	1	then	then	ADV
ejpam-6171	973	2	pi(x	pi(x	NUM
ejpam-6171	973	3	)	)	PUNCT
ejpam-6171	973	4	≤	≤	NUM
ejpam-6171	974	1	β	β	X
ejpam-6171	974	2	.	.	PUNCT
ejpam-6171	975	1	thus	thus	ADV
ejpam-6171	975	2	,	,	PUNCT
ejpam-6171	975	3	x	x	SYM
ejpam-6171	975	4	∈	∈	NOUN
ejpam-6171	975	5	l(pi	l(pi	NOUN
ejpam-6171	975	6	;	;	PUNCT
ejpam-6171	975	7	β	β	X
ejpam-6171	975	8	)	)	PUNCT
ejpam-6171	975	9	6=	6=	ADP
ejpam-6171	975	10	∅.	∅.	ADP
ejpam-6171	975	11	by	by	ADP
ejpam-6171	975	12	the	the	DET
ejpam-6171	975	13	assumption	assumption	NOUN
ejpam-6171	975	14	,	,	PUNCT
ejpam-6171	975	15	we	we	PRON
ejpam-6171	975	16	have	have	VERB
ejpam-6171	975	17	l(pi	l(pi	X
ejpam-6171	975	18	;	;	PUNCT
ejpam-6171	975	19	β	β	X
ejpam-6171	975	20	)	)	PUNCT
ejpam-6171	975	21	is	be	AUX
ejpam-6171	975	22	an	an	DET
ejpam-6171	975	23	iup	iup	NOUN
ejpam-6171	975	24	-	-	PUNCT
ejpam-6171	975	25	ideal	ideal	NOUN
ejpam-6171	975	26	of	of	ADP
ejpam-6171	975	27	x.	x.	NOUN
ejpam-6171	975	28	by	by	ADP
ejpam-6171	975	29	(	(	PUNCT
ejpam-6171	975	30	2.18	2.18	NUM
ejpam-6171	975	31	)	)	PUNCT
ejpam-6171	975	32	,	,	PUNCT
ejpam-6171	975	33	we	we	PRON
ejpam-6171	975	34	have	have	VERB
ejpam-6171	975	35	0	0	NUM
ejpam-6171	975	36	∈	∈	PROPN
ejpam-6171	975	37	l(pi	l(pi	NOUN
ejpam-6171	975	38	;	;	PUNCT
ejpam-6171	975	39	β	β	X
ejpam-6171	975	40	)	)	PUNCT
ejpam-6171	975	41	.	.	PUNCT
ejpam-6171	976	1	then	then	ADV
ejpam-6171	976	2	pi(0	pi(0	PROPN
ejpam-6171	976	3	)	)	PUNCT
ejpam-6171	976	4	≤	≤	PUNCT
ejpam-6171	976	5	β	β	X
ejpam-6171	976	6	=	=	SYM
ejpam-6171	976	7	pi(x	pi(x	NOUN
ejpam-6171	976	8	)	)	PUNCT
ejpam-6171	976	9	.	.	PUNCT
ejpam-6171	977	1	let	let	VERB
ejpam-6171	977	2	x	x	PRON
ejpam-6171	977	3	,	,	PUNCT
ejpam-6171	977	4	y	y	PROPN
ejpam-6171	977	5	,	,	PUNCT
ejpam-6171	977	6	z	z	PROPN
ejpam-6171	977	7	∈	∈	PROPN
ejpam-6171	977	8	x.	x.	NOUN
ejpam-6171	977	9	let	let	VERB
ejpam-6171	977	10	β	β	X
ejpam-6171	977	11	=	=	SYM
ejpam-6171	977	12	max{pi(x	max{pi(x	PROPN
ejpam-6171	977	13	?	?	PUNCT
ejpam-6171	978	1	(	(	PUNCT
ejpam-6171	978	2	y	y	NOUN
ejpam-6171	978	3	?	?	PUNCT
ejpam-6171	978	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	978	5	)	)	PUNCT
ejpam-6171	978	6	}	}	PUNCT
ejpam-6171	978	7	.	.	PUNCT
ejpam-6171	979	1	then	then	ADV
ejpam-6171	979	2	pi(x	pi(x	NOUN
ejpam-6171	979	3	?	?	PUNCT
ejpam-6171	980	1	(	(	PUNCT
ejpam-6171	980	2	y	y	NOUN
ejpam-6171	980	3	?	?	PUNCT
ejpam-6171	981	1	z	z	X
ejpam-6171	981	2	)	)	PUNCT
ejpam-6171	981	3	)	)	PUNCT
ejpam-6171	982	1	≤	≤	NUM
ejpam-6171	982	2	β	β	NOUN
ejpam-6171	982	3	and	and	CCONJ
ejpam-6171	982	4	pi(y	pi(y	NOUN
ejpam-6171	982	5	)	)	PUNCT
ejpam-6171	982	6	≤	≤	NOUN
ejpam-6171	982	7	β	β	X
ejpam-6171	982	8	.	.	PUNCT
ejpam-6171	983	1	thus	thus	ADV
ejpam-6171	983	2	,	,	PUNCT
ejpam-6171	983	3	x	x	PUNCT
ejpam-6171	983	4	?	?	PUNCT
ejpam-6171	984	1	(	(	PUNCT
ejpam-6171	984	2	y	y	NOUN
ejpam-6171	984	3	?	?	PUNCT
ejpam-6171	985	1	z	z	X
ejpam-6171	985	2	)	)	PUNCT
ejpam-6171	985	3	,	,	PUNCT
ejpam-6171	985	4	y	y	PROPN
ejpam-6171	985	5	∈	∈	PROPN
ejpam-6171	985	6	l(pi	l(pi	X
ejpam-6171	985	7	;	;	PUNCT
ejpam-6171	985	8	β	β	X
ejpam-6171	985	9	)	)	PUNCT
ejpam-6171	985	10	6=	6=	ADP
ejpam-6171	985	11	∅.	∅.	ADP
ejpam-6171	985	12	by	by	ADP
ejpam-6171	985	13	the	the	DET
ejpam-6171	985	14	assumption	assumption	NOUN
ejpam-6171	985	15	,	,	PUNCT
ejpam-6171	985	16	we	we	PRON
ejpam-6171	985	17	have	have	VERB
ejpam-6171	985	18	u(pi	u(pi	NOUN
ejpam-6171	985	19	;	;	PUNCT
ejpam-6171	985	20	β	β	X
ejpam-6171	985	21	)	)	PUNCT
ejpam-6171	985	22	is	be	AUX
ejpam-6171	985	23	an	an	DET
ejpam-6171	985	24	iup	iup	NOUN
ejpam-6171	985	25	-	-	PUNCT
ejpam-6171	985	26	ideal	ideal	NOUN
ejpam-6171	985	27	of	of	ADP
ejpam-6171	985	28	x.	x.	NOUN
ejpam-6171	985	29	by	by	ADP
ejpam-6171	985	30	(	(	PUNCT
ejpam-6171	985	31	2.20	2.20	NUM
ejpam-6171	985	32	)	)	PUNCT
ejpam-6171	985	33	,	,	PUNCT
ejpam-6171	985	34	we	we	PRON
ejpam-6171	985	35	have	have	VERB
ejpam-6171	985	36	x	x	X
ejpam-6171	985	37	?	?	PUNCT
ejpam-6171	986	1	z	z	NOUN
ejpam-6171	986	2	∈	∈	PROPN
ejpam-6171	986	3	l(pi	l(pi	NOUN
ejpam-6171	986	4	;	;	PUNCT
ejpam-6171	986	5	β	β	X
ejpam-6171	986	6	)	)	PUNCT
ejpam-6171	986	7	.	.	PUNCT
ejpam-6171	987	1	thus	thus	ADV
ejpam-6171	987	2	,	,	PUNCT
ejpam-6171	987	3	pi(x	pi(x	ADJ
ejpam-6171	987	4	?	?	PUNCT
ejpam-6171	988	1	z	z	X
ejpam-6171	988	2	)	)	PUNCT
ejpam-6171	988	3	≤	≤	NOUN
ejpam-6171	988	4	β	β	X
ejpam-6171	988	5	=	=	SYM
ejpam-6171	988	6	max{pi(x	max{pi(x	PROPN
ejpam-6171	988	7	?	?	PUNCT
ejpam-6171	989	1	(	(	PUNCT
ejpam-6171	989	2	y	y	NOUN
ejpam-6171	989	3	?	?	PUNCT
ejpam-6171	989	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	989	5	)	)	PUNCT
ejpam-6171	989	6	}	}	PUNCT
ejpam-6171	989	7	.	.	PUNCT
ejpam-6171	990	1	let	let	VERB
ejpam-6171	990	2	x	x	SYM
ejpam-6171	990	3	∈	∈	PROPN
ejpam-6171	990	4	x.	x.	NOUN
ejpam-6171	990	5	let	let	VERB
ejpam-6171	990	6	γ	γ	X
ejpam-6171	990	7	=	=	SYM
ejpam-6171	990	8	pf	pf	X
ejpam-6171	990	9	(	(	PUNCT
ejpam-6171	990	10	x	x	NOUN
ejpam-6171	990	11	)	)	PUNCT
ejpam-6171	990	12	.	.	PUNCT
ejpam-6171	991	1	then	then	ADV
ejpam-6171	991	2	pf	pf	PROPN
ejpam-6171	991	3	(	(	PUNCT
ejpam-6171	991	4	x	x	PROPN
ejpam-6171	991	5	)	)	PUNCT
ejpam-6171	991	6	≥	≥	PROPN
ejpam-6171	991	7	γ	γ	PROPN
ejpam-6171	991	8	.	.	PUNCT
ejpam-6171	991	9	thus	thus	ADV
ejpam-6171	991	10	,	,	PUNCT
ejpam-6171	991	11	x	x	PROPN
ejpam-6171	991	12	∈	∈	PROPN
ejpam-6171	991	13	u(pf	u(pf	PROPN
ejpam-6171	991	14	;	;	PUNCT
ejpam-6171	991	15	γ	γ	X
ejpam-6171	991	16	)	)	PUNCT
ejpam-6171	991	17	6=	6=	ADP
ejpam-6171	991	18	∅.	∅.	ADP
ejpam-6171	991	19	by	by	ADP
ejpam-6171	991	20	the	the	DET
ejpam-6171	991	21	assumption	assumption	NOUN
ejpam-6171	991	22	,	,	PUNCT
ejpam-6171	991	23	we	we	PRON
ejpam-6171	991	24	have	have	VERB
ejpam-6171	991	25	u(pf	u(pf	NOUN
ejpam-6171	991	26	;	;	PUNCT
ejpam-6171	991	27	γ	γ	X
ejpam-6171	991	28	)	)	PUNCT
ejpam-6171	991	29	is	be	AUX
ejpam-6171	991	30	an	an	DET
ejpam-6171	991	31	iup	iup	NOUN
ejpam-6171	991	32	-	-	PUNCT
ejpam-6171	991	33	ideal	ideal	NOUN
ejpam-6171	991	34	of	of	ADP
ejpam-6171	991	35	x.	x.	NOUN
ejpam-6171	991	36	by	by	ADP
ejpam-6171	991	37	(	(	PUNCT
ejpam-6171	991	38	2.18	2.18	NUM
ejpam-6171	991	39	)	)	PUNCT
ejpam-6171	991	40	,	,	PUNCT
ejpam-6171	991	41	we	we	PRON
ejpam-6171	991	42	have	have	VERB
ejpam-6171	991	43	0	0	NUM
ejpam-6171	991	44	∈	∈	PROPN
ejpam-6171	991	45	u(pf	u(pf	PROPN
ejpam-6171	991	46	;	;	PUNCT
ejpam-6171	991	47	γ	γ	X
ejpam-6171	991	48	)	)	PUNCT
ejpam-6171	991	49	.	.	PUNCT
ejpam-6171	992	1	then	then	ADV
ejpam-6171	992	2	pf	pf	PROPN
ejpam-6171	992	3	(	(	PUNCT
ejpam-6171	992	4	0	0	NUM
ejpam-6171	992	5	)	)	PUNCT
ejpam-6171	992	6	≥	≥	NOUN
ejpam-6171	992	7	γ	γ	X
ejpam-6171	992	8	=	=	SYM
ejpam-6171	992	9	pf	pf	PROPN
ejpam-6171	992	10	(	(	PUNCT
ejpam-6171	992	11	x	x	NOUN
ejpam-6171	992	12	)	)	PUNCT
ejpam-6171	992	13	.	.	PUNCT
ejpam-6171	993	1	let	let	VERB
ejpam-6171	993	2	x	x	PRON
ejpam-6171	993	3	,	,	PUNCT
ejpam-6171	993	4	y	y	PROPN
ejpam-6171	993	5	,	,	PUNCT
ejpam-6171	993	6	z	z	PROPN
ejpam-6171	993	7	∈	∈	PROPN
ejpam-6171	993	8	x.	x.	NOUN
ejpam-6171	993	9	let	let	VERB
ejpam-6171	993	10	γ	γ	X
ejpam-6171	993	11	=	=	VERB
ejpam-6171	993	12	min{pf	min{pf	X
ejpam-6171	993	13	(	(	PUNCT
ejpam-6171	993	14	x	x	X
ejpam-6171	993	15	?	?	PUNCT
ejpam-6171	994	1	(	(	PUNCT
ejpam-6171	994	2	y	y	NOUN
ejpam-6171	994	3	?	?	PUNCT
ejpam-6171	995	1	z)),pf	z)),pf	PROPN
ejpam-6171	995	2	(	(	PUNCT
ejpam-6171	995	3	y	y	NOUN
ejpam-6171	995	4	)	)	PUNCT
ejpam-6171	995	5	}	}	PUNCT
ejpam-6171	995	6	.	.	PUNCT
ejpam-6171	996	1	then	then	ADV
ejpam-6171	996	2	pf	pf	INTJ
ejpam-6171	996	3	(	(	PUNCT
ejpam-6171	996	4	x?(y	x?(y	PROPN
ejpam-6171	996	5	?	?	PUNCT
ejpam-6171	996	6	z	z	X
ejpam-6171	996	7	)	)	PUNCT
ejpam-6171	996	8	)	)	PUNCT
ejpam-6171	996	9	≥	≥	PROPN
ejpam-6171	996	10	γ	γ	PROPN
ejpam-6171	996	11	and	and	CCONJ
ejpam-6171	996	12	pf	pf	PROPN
ejpam-6171	996	13	(	(	PUNCT
ejpam-6171	996	14	y	y	PROPN
ejpam-6171	996	15	)	)	PUNCT
ejpam-6171	996	16	≥	≥	PROPN
ejpam-6171	996	17	γ	γ	PROPN
ejpam-6171	996	18	.	.	PUNCT
ejpam-6171	996	19	thus	thus	ADV
ejpam-6171	996	20	,	,	PUNCT
ejpam-6171	996	21	x?(y	x?(y	PUNCT
ejpam-6171	996	22	?	?	PUNCT
ejpam-6171	997	1	z	z	X
ejpam-6171	997	2	)	)	PUNCT
ejpam-6171	997	3	,	,	PUNCT
ejpam-6171	997	4	y	y	PROPN
ejpam-6171	997	5	∈	∈	PROPN
ejpam-6171	997	6	u(pf	u(pf	PROPN
ejpam-6171	997	7	;	;	PUNCT
ejpam-6171	997	8	γ	γ	X
ejpam-6171	997	9	)	)	PUNCT
ejpam-6171	997	10	6=	6=	ADP
ejpam-6171	997	11	∅.	∅.	ADP
ejpam-6171	997	12	by	by	ADP
ejpam-6171	997	13	the	the	DET
ejpam-6171	997	14	assumption	assumption	NOUN
ejpam-6171	997	15	,	,	PUNCT
ejpam-6171	997	16	we	we	PRON
ejpam-6171	997	17	have	have	VERB
ejpam-6171	997	18	u(pf	u(pf	NOUN
ejpam-6171	997	19	;	;	PUNCT
ejpam-6171	997	20	γ	γ	X
ejpam-6171	997	21	)	)	PUNCT
ejpam-6171	997	22	is	be	AUX
ejpam-6171	997	23	an	an	DET
ejpam-6171	997	24	iup	iup	NOUN
ejpam-6171	997	25	-	-	PUNCT
ejpam-6171	997	26	ideal	ideal	NOUN
ejpam-6171	997	27	of	of	ADP
ejpam-6171	997	28	x.	x.	NOUN
ejpam-6171	997	29	by	by	ADP
ejpam-6171	997	30	(	(	PUNCT
ejpam-6171	997	31	2.20	2.20	NUM
ejpam-6171	997	32	)	)	PUNCT
ejpam-6171	997	33	,	,	PUNCT
ejpam-6171	997	34	we	we	PRON
ejpam-6171	997	35	have	have	VERB
ejpam-6171	997	36	x	x	X
ejpam-6171	997	37	?	?	PUNCT
ejpam-6171	998	1	z	z	PROPN
ejpam-6171	998	2	∈	∈	PROPN
ejpam-6171	998	3	u(pf	u(pf	PROPN
ejpam-6171	998	4	;	;	PUNCT
ejpam-6171	998	5	γ	γ	X
ejpam-6171	998	6	)	)	PUNCT
ejpam-6171	998	7	.	.	PUNCT
ejpam-6171	999	1	thus	thus	ADV
ejpam-6171	999	2	,	,	PUNCT
ejpam-6171	999	3	k.	k.	PROPN
ejpam-6171	999	4	suayngam	suayngam	PROPN
ejpam-6171	999	5	et	et	PROPN
ejpam-6171	999	6	al	al	PROPN
ejpam-6171	999	7	.	.	PUNCT
ejpam-6171	999	8	/	/	SYM
ejpam-6171	999	9	eur	eur	PROPN
ejpam-6171	999	10	.	.	PUNCT
ejpam-6171	1000	1	j.	j.	PROPN
ejpam-6171	1000	2	pure	pure	PROPN
ejpam-6171	1000	3	appl	appl	PROPN
ejpam-6171	1000	4	.	.	PROPN
ejpam-6171	1000	5	math	math	PROPN
ejpam-6171	1000	6	,	,	PUNCT
ejpam-6171	1000	7	18	18	NUM
ejpam-6171	1000	8	(	(	PUNCT
ejpam-6171	1000	9	3	3	NUM
ejpam-6171	1000	10	)	)	PUNCT
ejpam-6171	1000	11	(	(	PUNCT
ejpam-6171	1000	12	2025	2025	NUM
ejpam-6171	1000	13	)	)	PUNCT
ejpam-6171	1000	14	,	,	PUNCT
ejpam-6171	1000	15	6171	6171	NUM
ejpam-6171	1000	16	22	22	NUM
ejpam-6171	1000	17	of	of	ADP
ejpam-6171	1000	18	28	28	NUM
ejpam-6171	1000	19	pf	pf	NOUN
ejpam-6171	1000	20	(	(	PUNCT
ejpam-6171	1000	21	x	x	PROPN
ejpam-6171	1000	22	?	?	PUNCT
ejpam-6171	1001	1	z	z	X
ejpam-6171	1001	2	)	)	PUNCT
ejpam-6171	1001	3	≥	≥	PROPN
ejpam-6171	1001	4	γ	γ	X
ejpam-6171	1001	5	=	=	PUNCT
ejpam-6171	1001	6	min{pf	min{pf	X
ejpam-6171	1001	7	(	(	PUNCT
ejpam-6171	1001	8	x	x	X
ejpam-6171	1001	9	?	?	PUNCT
ejpam-6171	1002	1	(	(	PUNCT
ejpam-6171	1002	2	y	y	NOUN
ejpam-6171	1002	3	?	?	PUNCT
ejpam-6171	1003	1	z)),pf	z)),pf	PROPN
ejpam-6171	1003	2	(	(	PUNCT
ejpam-6171	1003	3	y	y	NOUN
ejpam-6171	1003	4	)	)	PUNCT
ejpam-6171	1003	5	}	}	PUNCT
ejpam-6171	1003	6	.	.	PUNCT
ejpam-6171	1004	1	hence	hence	ADV
ejpam-6171	1004	2	,	,	PUNCT
ejpam-6171	1004	3	p	p	PROPN
ejpam-6171	1004	4	is	be	AUX
ejpam-6171	1004	5	a	a	DET
ejpam-6171	1004	6	pythagorean	pythagorean	PROPN
ejpam-6171	1004	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	1004	8	iup	iup	PROPN
ejpam-6171	1004	9	-	-	PUNCT
ejpam-6171	1004	10	ideal	ideal	NOUN
ejpam-6171	1004	11	of	of	ADP
ejpam-6171	1004	12	x.	x.	PROPN
ejpam-6171	1004	13	theorem	theorem	VERB
ejpam-6171	1004	14	19	19	NUM
ejpam-6171	1004	15	.	.	PUNCT
ejpam-6171	1005	1	a	a	DET
ejpam-6171	1005	2	pns	pns	NOUN
ejpam-6171	1005	3	p	p	NOUN
ejpam-6171	1005	4	in	in	ADP
ejpam-6171	1005	5	x	x	PROPN
ejpam-6171	1005	6	is	be	AUX
ejpam-6171	1005	7	a	a	DET
ejpam-6171	1005	8	pythagorean	pythagorean	PROPN
ejpam-6171	1005	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1005	10	iup	iup	NOUN
ejpam-6171	1005	11	-	-	PUNCT
ejpam-6171	1005	12	filter	filter	NOUN
ejpam-6171	1005	13	of	of	ADP
ejpam-6171	1005	14	x	x	SYM
ejpam-6171	1005	15	if	if	SCONJ
ejpam-6171	1005	16	and	and	CCONJ
ejpam-6171	1005	17	only	only	ADV
ejpam-6171	1005	18	if	if	SCONJ
ejpam-6171	1005	19	for	for	ADP
ejpam-6171	1005	20	all	all	DET
ejpam-6171	1005	21	α	α	NOUN
ejpam-6171	1005	22	,	,	PUNCT
ejpam-6171	1005	23	β	β	X
ejpam-6171	1005	24	,	,	PUNCT
ejpam-6171	1005	25	γ	γ	PROPN
ejpam-6171	1005	26	∈	∈	PROPN
ejpam-6171	1006	1	[	[	X
ejpam-6171	1006	2	0	0	NUM
ejpam-6171	1006	3	,	,	PUNCT
ejpam-6171	1006	4	1	1	NUM
ejpam-6171	1006	5	]	]	PUNCT
ejpam-6171	1006	6	,	,	PUNCT
ejpam-6171	1006	7	the	the	DET
ejpam-6171	1006	8	sets	set	NOUN
ejpam-6171	1006	9	u(pt	u(pt	PROPN
ejpam-6171	1006	10	;	;	PUNCT
ejpam-6171	1006	11	α	α	X
ejpam-6171	1006	12	)	)	PUNCT
ejpam-6171	1006	13	,	,	PUNCT
ejpam-6171	1006	14	l(pi	l(pi	X
ejpam-6171	1006	15	;	;	PUNCT
ejpam-6171	1006	16	β	β	X
ejpam-6171	1006	17	)	)	PUNCT
ejpam-6171	1006	18	,	,	PUNCT
ejpam-6171	1006	19	and	and	CCONJ
ejpam-6171	1006	20	u(pf	u(pf	PROPN
ejpam-6171	1006	21	;	;	PUNCT
ejpam-6171	1006	22	γ	γ	X
ejpam-6171	1006	23	)	)	PUNCT
ejpam-6171	1006	24	are	be	AUX
ejpam-6171	1006	25	either	either	CCONJ
ejpam-6171	1006	26	empty	empty	ADJ
ejpam-6171	1006	27	or	or	CCONJ
ejpam-6171	1006	28	iup	iup	NOUN
ejpam-6171	1006	29	-	-	PUNCT
ejpam-6171	1006	30	filters	filter	NOUN
ejpam-6171	1006	31	of	of	ADP
ejpam-6171	1006	32	x.	x.	NOUN
ejpam-6171	1006	33	proof	proof	PROPN
ejpam-6171	1006	34	.	.	PUNCT
ejpam-6171	1007	1	assume	assume	VERB
ejpam-6171	1007	2	that	that	SCONJ
ejpam-6171	1007	3	p	p	NOUN
ejpam-6171	1007	4	in	in	ADP
ejpam-6171	1007	5	x	x	PROPN
ejpam-6171	1007	6	is	be	AUX
ejpam-6171	1007	7	a	a	DET
ejpam-6171	1007	8	pythagorean	pythagorean	PROPN
ejpam-6171	1007	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1007	10	iup	iup	NOUN
ejpam-6171	1007	11	-	-	PUNCT
ejpam-6171	1007	12	filter	filter	NOUN
ejpam-6171	1007	13	of	of	ADP
ejpam-6171	1007	14	x.	x.	NOUN
ejpam-6171	1007	15	let	let	VERB
ejpam-6171	1007	16	α	α	PRON
ejpam-6171	1007	17	∈	∈	PROPN
ejpam-6171	1008	1	[	[	X
ejpam-6171	1008	2	0	0	NUM
ejpam-6171	1008	3	,	,	PUNCT
ejpam-6171	1008	4	1	1	NUM
ejpam-6171	1008	5	]	]	PUNCT
ejpam-6171	1008	6	be	be	AUX
ejpam-6171	1008	7	such	such	ADJ
ejpam-6171	1008	8	that	that	SCONJ
ejpam-6171	1008	9	u(pt	u(pt	PROPN
ejpam-6171	1008	10	;	;	PUNCT
ejpam-6171	1008	11	α	α	X
ejpam-6171	1008	12	)	)	PUNCT
ejpam-6171	1008	13	6=	6=	ADP
ejpam-6171	1008	14	∅.	∅.	ADV
ejpam-6171	1008	15	let	let	VERB
ejpam-6171	1008	16	a	a	DET
ejpam-6171	1008	17	∈	∈	NOUN
ejpam-6171	1008	18	u(pt	u(pt	NOUN
ejpam-6171	1008	19	;	;	PUNCT
ejpam-6171	1008	20	α	α	X
ejpam-6171	1008	21	)	)	PUNCT
ejpam-6171	1008	22	.	.	PUNCT
ejpam-6171	1009	1	then	then	ADV
ejpam-6171	1009	2	pt	pt	X
ejpam-6171	1009	3	(	(	PUNCT
ejpam-6171	1009	4	a	a	NOUN
ejpam-6171	1009	5	)	)	PUNCT
ejpam-6171	1009	6	≥	≥	NOUN
ejpam-6171	1009	7	α	α	NOUN
ejpam-6171	1009	8	.	.	PUNCT
ejpam-6171	1009	9	by	by	ADP
ejpam-6171	1009	10	(	(	PUNCT
ejpam-6171	1009	11	3.5	3.5	NUM
ejpam-6171	1009	12	)	)	PUNCT
ejpam-6171	1009	13	,	,	PUNCT
ejpam-6171	1009	14	we	we	PRON
ejpam-6171	1009	15	have	have	VERB
ejpam-6171	1009	16	pt	pt	X
ejpam-6171	1009	17	(	(	PUNCT
ejpam-6171	1009	18	0	0	NUM
ejpam-6171	1009	19	)	)	PUNCT
ejpam-6171	1009	20	≥	≥	NOUN
ejpam-6171	1009	21	pt	pt	INTJ
ejpam-6171	1009	22	(	(	PUNCT
ejpam-6171	1009	23	a	a	NOUN
ejpam-6171	1009	24	)	)	PUNCT
ejpam-6171	1009	25	≥	≥	NOUN
ejpam-6171	1009	26	α	α	NOUN
ejpam-6171	1009	27	.	.	PUNCT
ejpam-6171	1010	1	thus	thus	ADV
ejpam-6171	1010	2	,	,	PUNCT
ejpam-6171	1010	3	0	0	NUM
ejpam-6171	1010	4	∈	∈	PROPN
ejpam-6171	1010	5	u(pt	u(pt	PROPN
ejpam-6171	1010	6	;	;	PUNCT
ejpam-6171	1010	7	α	α	X
ejpam-6171	1010	8	)	)	PUNCT
ejpam-6171	1010	9	.	.	PUNCT
ejpam-6171	1011	1	let	let	VERB
ejpam-6171	1011	2	x	x	PRON
ejpam-6171	1011	3	,	,	PUNCT
ejpam-6171	1011	4	y	y	PROPN
ejpam-6171	1011	5	∈	∈	PROPN
ejpam-6171	1011	6	x	x	AUX
ejpam-6171	1011	7	be	be	AUX
ejpam-6171	1011	8	such	such	ADJ
ejpam-6171	1011	9	that	that	SCONJ
ejpam-6171	1011	10	x?y	x?y	PROPN
ejpam-6171	1011	11	∈	∈	PROPN
ejpam-6171	1011	12	u(pt	u(pt	PROPN
ejpam-6171	1011	13	;	;	PUNCT
ejpam-6171	1011	14	α	α	X
ejpam-6171	1011	15	)	)	PUNCT
ejpam-6171	1011	16	and	and	CCONJ
ejpam-6171	1011	17	x	x	PUNCT
ejpam-6171	1011	18	∈	∈	NOUN
ejpam-6171	1011	19	u(pt	u(pt	PROPN
ejpam-6171	1011	20	;	;	PUNCT
ejpam-6171	1011	21	α	α	X
ejpam-6171	1011	22	)	)	PUNCT
ejpam-6171	1011	23	.	.	PUNCT
ejpam-6171	1012	1	then	then	ADV
ejpam-6171	1012	2	pt	pt	PROPN
ejpam-6171	1012	3	(	(	PUNCT
ejpam-6171	1012	4	x?y	x?y	PROPN
ejpam-6171	1012	5	)	)	PUNCT
ejpam-6171	1012	6	≥	≥	PROPN
ejpam-6171	1012	7	α	α	NOUN
ejpam-6171	1012	8	and	and	CCONJ
ejpam-6171	1012	9	pt	pt	PROPN
ejpam-6171	1012	10	(	(	PUNCT
ejpam-6171	1012	11	x	x	NOUN
ejpam-6171	1012	12	)	)	PUNCT
ejpam-6171	1012	13	≥	≥	PROPN
ejpam-6171	1012	14	α	α	NOUN
ejpam-6171	1012	15	.	.	PUNCT
ejpam-6171	1013	1	thus	thus	ADV
ejpam-6171	1013	2	,	,	PUNCT
ejpam-6171	1013	3	min{pt	min{pt	PRON
ejpam-6171	1013	4	(	(	PUNCT
ejpam-6171	1013	5	x?y),pt	x?y),pt	INTJ
ejpam-6171	1013	6	(	(	PUNCT
ejpam-6171	1013	7	x	x	NOUN
ejpam-6171	1013	8	)	)	PUNCT
ejpam-6171	1013	9	}	}	PUNCT
ejpam-6171	1013	10	≥	≥	PROPN
ejpam-6171	1013	11	α	α	X
ejpam-6171	1013	12	.	.	PUNCT
ejpam-6171	1014	1	by	by	ADP
ejpam-6171	1014	2	(	(	PUNCT
ejpam-6171	1014	3	3.11	3.11	NUM
ejpam-6171	1014	4	)	)	PUNCT
ejpam-6171	1014	5	,	,	PUNCT
ejpam-6171	1014	6	we	we	PRON
ejpam-6171	1014	7	have	have	VERB
ejpam-6171	1014	8	pt	pt	X
ejpam-6171	1014	9	(	(	PUNCT
ejpam-6171	1014	10	y	y	NOUN
ejpam-6171	1014	11	)	)	PUNCT
ejpam-6171	1014	12	≥	≥	NOUN
ejpam-6171	1014	13	min{pt	min{pt	X
ejpam-6171	1015	1	(	(	PUNCT
ejpam-6171	1015	2	x	x	X
ejpam-6171	1015	3	?	?	PUNCT
ejpam-6171	1015	4	y),pt	y),pt	PROPN
ejpam-6171	1015	5	(	(	PUNCT
ejpam-6171	1015	6	x	x	NOUN
ejpam-6171	1015	7	)	)	PUNCT
ejpam-6171	1015	8	}	}	PUNCT
ejpam-6171	1015	9	≥	≥	PROPN
ejpam-6171	1015	10	α	α	NOUN
ejpam-6171	1015	11	.	.	PUNCT
ejpam-6171	1016	1	thus	thus	ADV
ejpam-6171	1016	2	,	,	PUNCT
ejpam-6171	1016	3	y	y	PROPN
ejpam-6171	1016	4	∈	∈	PROPN
ejpam-6171	1016	5	u(pt	u(pt	PROPN
ejpam-6171	1016	6	;	;	PUNCT
ejpam-6171	1016	7	α	α	X
ejpam-6171	1016	8	)	)	PUNCT
ejpam-6171	1016	9	.	.	PUNCT
ejpam-6171	1017	1	hence	hence	ADV
ejpam-6171	1017	2	,	,	PUNCT
ejpam-6171	1017	3	u(pt	u(pt	PROPN
ejpam-6171	1017	4	;	;	PUNCT
ejpam-6171	1017	5	α	α	X
ejpam-6171	1017	6	)	)	PUNCT
ejpam-6171	1017	7	is	be	AUX
ejpam-6171	1017	8	an	an	DET
ejpam-6171	1017	9	iup	iup	NOUN
ejpam-6171	1017	10	-	-	PUNCT
ejpam-6171	1017	11	filter	filter	NOUN
ejpam-6171	1017	12	of	of	ADP
ejpam-6171	1017	13	x.	x.	NOUN
ejpam-6171	1017	14	let	let	VERB
ejpam-6171	1017	15	β	β	X
ejpam-6171	1017	16	∈	∈	PROPN
ejpam-6171	1018	1	[	[	X
ejpam-6171	1018	2	0	0	NUM
ejpam-6171	1018	3	,	,	PUNCT
ejpam-6171	1018	4	1	1	NUM
ejpam-6171	1018	5	]	]	PUNCT
ejpam-6171	1018	6	be	be	AUX
ejpam-6171	1018	7	such	such	ADJ
ejpam-6171	1018	8	that	that	SCONJ
ejpam-6171	1018	9	l(pi	l(pi	X
ejpam-6171	1018	10	;	;	PUNCT
ejpam-6171	1018	11	β	β	X
ejpam-6171	1018	12	)	)	PUNCT
ejpam-6171	1018	13	6=	6=	ADP
ejpam-6171	1018	14	∅.	∅.	AUX
ejpam-6171	1018	15	let	let	VERB
ejpam-6171	1018	16	b	b	NOUN
ejpam-6171	1018	17	∈	∈	NOUN
ejpam-6171	1018	18	l(pi	l(pi	X
ejpam-6171	1018	19	;	;	PUNCT
ejpam-6171	1018	20	β	β	X
ejpam-6171	1018	21	)	)	PUNCT
ejpam-6171	1018	22	.	.	PUNCT
ejpam-6171	1019	1	then	then	ADV
ejpam-6171	1019	2	pi(b	pi(b	PUNCT
ejpam-6171	1019	3	)	)	PUNCT
ejpam-6171	1019	4	≤	≤	PUNCT
ejpam-6171	1020	1	β	β	X
ejpam-6171	1020	2	.	.	PUNCT
ejpam-6171	1021	1	by	by	ADP
ejpam-6171	1021	2	(	(	PUNCT
ejpam-6171	1021	3	3.6	3.6	NUM
ejpam-6171	1021	4	)	)	PUNCT
ejpam-6171	1021	5	,	,	PUNCT
ejpam-6171	1021	6	we	we	PRON
ejpam-6171	1021	7	have	have	VERB
ejpam-6171	1021	8	pi(0	pi(0	PROPN
ejpam-6171	1021	9	)	)	PUNCT
ejpam-6171	1021	10	≤	≤	NOUN
ejpam-6171	1021	11	pi(b	pi(b	PUNCT
ejpam-6171	1021	12	)	)	PUNCT
ejpam-6171	1021	13	≤	≤	PUNCT
ejpam-6171	1022	1	β	β	X
ejpam-6171	1022	2	.	.	PUNCT
ejpam-6171	1023	1	thus	thus	ADV
ejpam-6171	1023	2	,	,	PUNCT
ejpam-6171	1023	3	0	0	NUM
ejpam-6171	1023	4	∈	∈	PROPN
ejpam-6171	1023	5	l(pi	l(pi	NOUN
ejpam-6171	1023	6	;	;	PUNCT
ejpam-6171	1023	7	β	β	X
ejpam-6171	1023	8	)	)	PUNCT
ejpam-6171	1023	9	.	.	PUNCT
ejpam-6171	1024	1	let	let	VERB
ejpam-6171	1024	2	x	x	PRON
ejpam-6171	1024	3	,	,	PUNCT
ejpam-6171	1024	4	y	y	PROPN
ejpam-6171	1024	5	∈	∈	PROPN
ejpam-6171	1024	6	x	x	AUX
ejpam-6171	1024	7	be	be	AUX
ejpam-6171	1024	8	such	such	ADJ
ejpam-6171	1024	9	that	that	SCONJ
ejpam-6171	1024	10	x?y	x?y	PROPN
ejpam-6171	1024	11	∈	∈	PROPN
ejpam-6171	1024	12	l(pi	l(pi	PROPN
ejpam-6171	1024	13	;	;	PUNCT
ejpam-6171	1024	14	β	β	X
ejpam-6171	1024	15	)	)	PUNCT
ejpam-6171	1024	16	and	and	CCONJ
ejpam-6171	1024	17	x	x	PUNCT
ejpam-6171	1024	18	∈	∈	NOUN
ejpam-6171	1024	19	l(pi	l(pi	NOUN
ejpam-6171	1024	20	;	;	PUNCT
ejpam-6171	1024	21	β	β	X
ejpam-6171	1024	22	)	)	PUNCT
ejpam-6171	1024	23	.	.	PUNCT
ejpam-6171	1025	1	then	then	ADV
ejpam-6171	1025	2	pi(x	pi(x	NOUN
ejpam-6171	1025	3	?	?	PUNCT
ejpam-6171	1026	1	y	y	X
ejpam-6171	1026	2	)	)	PUNCT
ejpam-6171	1026	3	≤	≤	NOUN
ejpam-6171	1026	4	β	β	X
ejpam-6171	1026	5	and	and	CCONJ
ejpam-6171	1026	6	pi(x	pi(x	NUM
ejpam-6171	1026	7	)	)	PUNCT
ejpam-6171	1026	8	≤	≤	NOUN
ejpam-6171	1027	1	β	β	X
ejpam-6171	1027	2	.	.	PUNCT
ejpam-6171	1028	1	thus	thus	ADV
ejpam-6171	1028	2	,	,	PUNCT
ejpam-6171	1028	3	max{pi(x	max{pi(x	PROPN
ejpam-6171	1028	4	?	?	PUNCT
ejpam-6171	1029	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1029	2	)	)	PUNCT
ejpam-6171	1029	3	}	}	PUNCT
ejpam-6171	1029	4	≤	≤	NOUN
ejpam-6171	1030	1	β	β	X
ejpam-6171	1030	2	.	.	PUNCT
ejpam-6171	1031	1	by	by	ADP
ejpam-6171	1031	2	(	(	PUNCT
ejpam-6171	1031	3	3.12	3.12	NUM
ejpam-6171	1031	4	)	)	PUNCT
ejpam-6171	1031	5	,	,	PUNCT
ejpam-6171	1031	6	we	we	PRON
ejpam-6171	1031	7	have	have	VERB
ejpam-6171	1031	8	pi(y	pi(y	NOUN
ejpam-6171	1031	9	)	)	PUNCT
ejpam-6171	1031	10	≤	≤	NUM
ejpam-6171	1031	11	max{pi(x	max{pi(x	NOUN
ejpam-6171	1031	12	?	?	PUNCT
ejpam-6171	1032	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1032	2	)	)	PUNCT
ejpam-6171	1032	3	}	}	PUNCT
ejpam-6171	1032	4	≤	≤	NOUN
ejpam-6171	1033	1	β	β	X
ejpam-6171	1033	2	.	.	PUNCT
ejpam-6171	1034	1	thus	thus	ADV
ejpam-6171	1034	2	,	,	PUNCT
ejpam-6171	1034	3	y	y	PROPN
ejpam-6171	1034	4	∈	∈	PROPN
ejpam-6171	1034	5	l(pi	l(pi	PROPN
ejpam-6171	1034	6	;	;	PUNCT
ejpam-6171	1034	7	β	β	X
ejpam-6171	1034	8	)	)	PUNCT
ejpam-6171	1034	9	.	.	PUNCT
ejpam-6171	1035	1	hence	hence	ADV
ejpam-6171	1035	2	,	,	PUNCT
ejpam-6171	1035	3	l(pi	l(pi	X
ejpam-6171	1035	4	;	;	PUNCT
ejpam-6171	1035	5	β	β	X
ejpam-6171	1035	6	)	)	PUNCT
ejpam-6171	1035	7	is	be	AUX
ejpam-6171	1035	8	an	an	DET
ejpam-6171	1035	9	iup	iup	NOUN
ejpam-6171	1035	10	-	-	PUNCT
ejpam-6171	1035	11	ideal	ideal	NOUN
ejpam-6171	1035	12	of	of	ADP
ejpam-6171	1035	13	x.	x.	NOUN
ejpam-6171	1035	14	let	let	VERB
ejpam-6171	1035	15	γ	γ	X
ejpam-6171	1035	16	∈	∈	PROPN
ejpam-6171	1036	1	[	[	X
ejpam-6171	1036	2	0	0	NUM
ejpam-6171	1036	3	,	,	PUNCT
ejpam-6171	1036	4	1	1	NUM
ejpam-6171	1036	5	]	]	PUNCT
ejpam-6171	1036	6	be	be	AUX
ejpam-6171	1036	7	such	such	ADJ
ejpam-6171	1036	8	that	that	SCONJ
ejpam-6171	1036	9	u(pf	u(pf	PROPN
ejpam-6171	1036	10	;	;	PUNCT
ejpam-6171	1036	11	γ	γ	X
ejpam-6171	1036	12	)	)	PUNCT
ejpam-6171	1036	13	6=	6=	ADP
ejpam-6171	1036	14	∅.	∅.	ADV
ejpam-6171	1036	15	let	let	VERB
ejpam-6171	1036	16	c	c	PROPN
ejpam-6171	1036	17	∈	∈	PROPN
ejpam-6171	1036	18	u(pf	u(pf	PROPN
ejpam-6171	1036	19	;	;	PUNCT
ejpam-6171	1036	20	γ	γ	X
ejpam-6171	1036	21	)	)	PUNCT
ejpam-6171	1036	22	.	.	PUNCT
ejpam-6171	1037	1	then	then	ADV
ejpam-6171	1037	2	pf	pf	PROPN
ejpam-6171	1037	3	(	(	PUNCT
ejpam-6171	1037	4	c	c	NOUN
ejpam-6171	1037	5	)	)	PUNCT
ejpam-6171	1037	6	≥	≥	PROPN
ejpam-6171	1037	7	γ	γ	X
ejpam-6171	1037	8	.	.	PUNCT
ejpam-6171	1037	9	by	by	ADP
ejpam-6171	1037	10	(	(	PUNCT
ejpam-6171	1037	11	3.7	3.7	NUM
ejpam-6171	1037	12	)	)	PUNCT
ejpam-6171	1037	13	,	,	PUNCT
ejpam-6171	1037	14	we	we	PRON
ejpam-6171	1037	15	have	have	VERB
ejpam-6171	1037	16	pf	pf	PROPN
ejpam-6171	1037	17	(	(	PUNCT
ejpam-6171	1037	18	0	0	NUM
ejpam-6171	1037	19	)	)	PUNCT
ejpam-6171	1037	20	≥	≥	NOUN
ejpam-6171	1037	21	pf	pf	X
ejpam-6171	1037	22	(	(	PUNCT
ejpam-6171	1037	23	c	c	NOUN
ejpam-6171	1037	24	)	)	PUNCT
ejpam-6171	1037	25	≥	≥	PROPN
ejpam-6171	1037	26	γ	γ	PROPN
ejpam-6171	1037	27	.	.	PROPN
ejpam-6171	1038	1	thus	thus	ADV
ejpam-6171	1038	2	,	,	PUNCT
ejpam-6171	1038	3	0	0	NUM
ejpam-6171	1038	4	∈	∈	PROPN
ejpam-6171	1038	5	u(pf	u(pf	PROPN
ejpam-6171	1038	6	;	;	PUNCT
ejpam-6171	1038	7	γ	γ	X
ejpam-6171	1038	8	)	)	PUNCT
ejpam-6171	1038	9	.	.	PUNCT
ejpam-6171	1039	1	let	let	VERB
ejpam-6171	1039	2	x	x	PRON
ejpam-6171	1039	3	,	,	PUNCT
ejpam-6171	1039	4	y	y	PROPN
ejpam-6171	1039	5	∈	∈	PROPN
ejpam-6171	1039	6	x	x	AUX
ejpam-6171	1039	7	be	be	AUX
ejpam-6171	1039	8	such	such	ADJ
ejpam-6171	1039	9	that	that	PRON
ejpam-6171	1039	10	x	x	PUNCT
ejpam-6171	1039	11	?	?	PUNCT
ejpam-6171	1040	1	y	y	PROPN
ejpam-6171	1040	2	∈	∈	PROPN
ejpam-6171	1040	3	u(pf	u(pf	PROPN
ejpam-6171	1040	4	;	;	PUNCT
ejpam-6171	1040	5	γ	γ	X
ejpam-6171	1040	6	)	)	PUNCT
ejpam-6171	1040	7	and	and	CCONJ
ejpam-6171	1040	8	x	x	PUNCT
ejpam-6171	1040	9	∈	∈	PROPN
ejpam-6171	1040	10	u(pf	u(pf	PROPN
ejpam-6171	1040	11	;	;	PUNCT
ejpam-6171	1040	12	γ	γ	X
ejpam-6171	1040	13	)	)	PUNCT
ejpam-6171	1040	14	.	.	PUNCT
ejpam-6171	1041	1	then	then	ADV
ejpam-6171	1041	2	pf	pf	INTJ
ejpam-6171	1041	3	(	(	PUNCT
ejpam-6171	1041	4	x	x	PROPN
ejpam-6171	1041	5	?	?	PUNCT
ejpam-6171	1041	6	y	y	X
ejpam-6171	1041	7	)	)	PUNCT
ejpam-6171	1041	8	≥	≥	PROPN
ejpam-6171	1041	9	γ	γ	PROPN
ejpam-6171	1041	10	and	and	CCONJ
ejpam-6171	1041	11	pf	pf	PROPN
ejpam-6171	1041	12	(	(	PUNCT
ejpam-6171	1041	13	x	x	NOUN
ejpam-6171	1041	14	)	)	PUNCT
ejpam-6171	1041	15	≥	≥	PROPN
ejpam-6171	1041	16	γ	γ	PROPN
ejpam-6171	1041	17	.	.	PUNCT
ejpam-6171	1041	18	thus	thus	ADV
ejpam-6171	1041	19	,	,	PUNCT
ejpam-6171	1041	20	min{pf	min{pf	PRON
ejpam-6171	1041	21	(	(	PUNCT
ejpam-6171	1041	22	x	x	X
ejpam-6171	1041	23	?	?	PUNCT
ejpam-6171	1042	1	y),pf	y),pf	PROPN
ejpam-6171	1042	2	(	(	PUNCT
ejpam-6171	1042	3	x	x	X
ejpam-6171	1042	4	)	)	PUNCT
ejpam-6171	1042	5	}	}	PUNCT
ejpam-6171	1042	6	≥	≥	PROPN
ejpam-6171	1042	7	γ	γ	X
ejpam-6171	1042	8	.	.	PUNCT
ejpam-6171	1043	1	by	by	ADP
ejpam-6171	1043	2	(	(	PUNCT
ejpam-6171	1043	3	3.13	3.13	NUM
ejpam-6171	1043	4	)	)	PUNCT
ejpam-6171	1043	5	,	,	PUNCT
ejpam-6171	1043	6	we	we	PRON
ejpam-6171	1043	7	have	have	VERB
ejpam-6171	1043	8	pf	pf	PROPN
ejpam-6171	1043	9	(	(	PUNCT
ejpam-6171	1043	10	y	y	NOUN
ejpam-6171	1043	11	)	)	PUNCT
ejpam-6171	1043	12	≥	≥	NOUN
ejpam-6171	1043	13	min{pf	min{pf	X
ejpam-6171	1043	14	(	(	PUNCT
ejpam-6171	1043	15	x	x	X
ejpam-6171	1043	16	?	?	PUNCT
ejpam-6171	1044	1	y),pf	y),pf	PROPN
ejpam-6171	1044	2	(	(	PUNCT
ejpam-6171	1044	3	x	x	X
ejpam-6171	1044	4	)	)	PUNCT
ejpam-6171	1044	5	}	}	PUNCT
ejpam-6171	1044	6	≥	≥	PROPN
ejpam-6171	1044	7	γ	γ	X
ejpam-6171	1044	8	.	.	PUNCT
ejpam-6171	1045	1	thus	thus	ADV
ejpam-6171	1045	2	,	,	PUNCT
ejpam-6171	1045	3	y	y	PROPN
ejpam-6171	1045	4	∈	∈	PROPN
ejpam-6171	1045	5	u(pf	u(pf	PROPN
ejpam-6171	1045	6	;	;	PUNCT
ejpam-6171	1045	7	γ	γ	X
ejpam-6171	1045	8	)	)	PUNCT
ejpam-6171	1045	9	.	.	PUNCT
ejpam-6171	1046	1	hence	hence	ADV
ejpam-6171	1046	2	,	,	PUNCT
ejpam-6171	1046	3	u(pf	u(pf	PROPN
ejpam-6171	1046	4	;	;	PUNCT
ejpam-6171	1046	5	γ	γ	X
ejpam-6171	1046	6	)	)	PUNCT
ejpam-6171	1046	7	is	be	AUX
ejpam-6171	1046	8	an	an	DET
ejpam-6171	1046	9	iup	iup	NOUN
ejpam-6171	1046	10	-	-	PUNCT
ejpam-6171	1046	11	filter	filter	NOUN
ejpam-6171	1046	12	of	of	ADP
ejpam-6171	1046	13	x.	x.	NOUN
ejpam-6171	1046	14	conversely	conversely	ADV
ejpam-6171	1046	15	,	,	PUNCT
ejpam-6171	1046	16	assume	assume	VERB
ejpam-6171	1046	17	that	that	SCONJ
ejpam-6171	1046	18	for	for	ADP
ejpam-6171	1046	19	all	all	DET
ejpam-6171	1046	20	α	α	NOUN
ejpam-6171	1046	21	,	,	PUNCT
ejpam-6171	1046	22	β	β	X
ejpam-6171	1046	23	,	,	PUNCT
ejpam-6171	1046	24	γ	γ	PROPN
ejpam-6171	1046	25	∈	∈	PROPN
ejpam-6171	1047	1	[	[	X
ejpam-6171	1047	2	0	0	NUM
ejpam-6171	1047	3	,	,	PUNCT
ejpam-6171	1047	4	1	1	NUM
ejpam-6171	1047	5	]	]	PUNCT
ejpam-6171	1047	6	,	,	PUNCT
ejpam-6171	1047	7	the	the	DET
ejpam-6171	1047	8	sets	set	NOUN
ejpam-6171	1047	9	u(pt	u(pt	PROPN
ejpam-6171	1047	10	;	;	PUNCT
ejpam-6171	1047	11	α	α	X
ejpam-6171	1047	12	)	)	PUNCT
ejpam-6171	1047	13	,	,	PUNCT
ejpam-6171	1047	14	l(pi	l(pi	X
ejpam-6171	1047	15	;	;	PUNCT
ejpam-6171	1047	16	β	β	X
ejpam-6171	1047	17	)	)	PUNCT
ejpam-6171	1047	18	,	,	PUNCT
ejpam-6171	1047	19	and	and	CCONJ
ejpam-6171	1047	20	u(pf	u(pf	PROPN
ejpam-6171	1047	21	;	;	PUNCT
ejpam-6171	1047	22	γ	γ	X
ejpam-6171	1047	23	)	)	PUNCT
ejpam-6171	1047	24	are	be	AUX
ejpam-6171	1047	25	either	either	CCONJ
ejpam-6171	1047	26	empty	empty	ADJ
ejpam-6171	1047	27	or	or	CCONJ
ejpam-6171	1047	28	iup	iup	NOUN
ejpam-6171	1047	29	-	-	PUNCT
ejpam-6171	1047	30	filters	filter	NOUN
ejpam-6171	1047	31	of	of	ADP
ejpam-6171	1047	32	x.	x.	NOUN
ejpam-6171	1047	33	let	let	VERB
ejpam-6171	1047	34	x	x	SYM
ejpam-6171	1047	35	∈	∈	PROPN
ejpam-6171	1047	36	x.	x.	NOUN
ejpam-6171	1047	37	let	let	VERB
ejpam-6171	1047	38	α	α	NOUN
ejpam-6171	1047	39	=	=	SYM
ejpam-6171	1047	40	pt	pt	X
ejpam-6171	1047	41	(	(	PUNCT
ejpam-6171	1047	42	x	x	NOUN
ejpam-6171	1047	43	)	)	PUNCT
ejpam-6171	1047	44	.	.	PUNCT
ejpam-6171	1048	1	then	then	ADV
ejpam-6171	1048	2	pt	pt	X
ejpam-6171	1048	3	(	(	PUNCT
ejpam-6171	1048	4	x	x	NOUN
ejpam-6171	1048	5	)	)	PUNCT
ejpam-6171	1048	6	≥	≥	PROPN
ejpam-6171	1048	7	α	α	NOUN
ejpam-6171	1048	8	.	.	PUNCT
ejpam-6171	1049	1	thus	thus	ADV
ejpam-6171	1049	2	,	,	PUNCT
ejpam-6171	1049	3	x	x	SYM
ejpam-6171	1049	4	∈	∈	NOUN
ejpam-6171	1049	5	u(pt	u(pt	PROPN
ejpam-6171	1049	6	;	;	PUNCT
ejpam-6171	1049	7	α	α	X
ejpam-6171	1049	8	)	)	PUNCT
ejpam-6171	1049	9	6=	6=	ADP
ejpam-6171	1049	10	∅.	∅.	ADP
ejpam-6171	1049	11	by	by	ADP
ejpam-6171	1049	12	the	the	DET
ejpam-6171	1049	13	assumption	assumption	NOUN
ejpam-6171	1049	14	,	,	PUNCT
ejpam-6171	1049	15	we	we	PRON
ejpam-6171	1049	16	have	have	VERB
ejpam-6171	1049	17	u(pt	u(pt	NOUN
ejpam-6171	1049	18	;	;	PUNCT
ejpam-6171	1049	19	α	α	X
ejpam-6171	1049	20	)	)	PUNCT
ejpam-6171	1049	21	is	be	AUX
ejpam-6171	1049	22	an	an	DET
ejpam-6171	1049	23	iup	iup	NOUN
ejpam-6171	1049	24	-	-	PUNCT
ejpam-6171	1049	25	filter	filter	NOUN
ejpam-6171	1049	26	of	of	ADP
ejpam-6171	1049	27	x.	x.	NOUN
ejpam-6171	1049	28	by	by	ADP
ejpam-6171	1049	29	(	(	PUNCT
ejpam-6171	1049	30	2.18	2.18	NUM
ejpam-6171	1049	31	)	)	PUNCT
ejpam-6171	1049	32	,	,	PUNCT
ejpam-6171	1049	33	we	we	PRON
ejpam-6171	1049	34	have	have	VERB
ejpam-6171	1049	35	0	0	NUM
ejpam-6171	1049	36	∈	∈	NOUN
ejpam-6171	1049	37	u(pt	u(pt	PROPN
ejpam-6171	1049	38	;	;	PUNCT
ejpam-6171	1049	39	α	α	X
ejpam-6171	1049	40	)	)	PUNCT
ejpam-6171	1049	41	.	.	PUNCT
ejpam-6171	1050	1	then	then	ADV
ejpam-6171	1050	2	pt	pt	X
ejpam-6171	1050	3	(	(	PUNCT
ejpam-6171	1050	4	0	0	NUM
ejpam-6171	1050	5	)	)	PUNCT
ejpam-6171	1050	6	≥	≥	NOUN
ejpam-6171	1050	7	α	α	NOUN
ejpam-6171	1050	8	=	=	SYM
ejpam-6171	1050	9	pt	pt	X
ejpam-6171	1050	10	(	(	PUNCT
ejpam-6171	1050	11	x	x	NOUN
ejpam-6171	1050	12	)	)	PUNCT
ejpam-6171	1050	13	.	.	PUNCT
ejpam-6171	1051	1	let	let	VERB
ejpam-6171	1051	2	x	x	PRON
ejpam-6171	1051	3	,	,	PUNCT
ejpam-6171	1051	4	y	y	PROPN
ejpam-6171	1051	5	∈	∈	PROPN
ejpam-6171	1051	6	x.	x.	NOUN
ejpam-6171	1051	7	let	let	VERB
ejpam-6171	1051	8	α	α	NOUN
ejpam-6171	1051	9	=	=	PUNCT
ejpam-6171	1051	10	min{pt	min{pt	NOUN
ejpam-6171	1051	11	(	(	PUNCT
ejpam-6171	1051	12	x	x	X
ejpam-6171	1051	13	?	?	PUNCT
ejpam-6171	1051	14	y),pt	y),pt	PROPN
ejpam-6171	1051	15	(	(	PUNCT
ejpam-6171	1051	16	x	x	NOUN
ejpam-6171	1051	17	)	)	PUNCT
ejpam-6171	1051	18	}	}	PUNCT
ejpam-6171	1051	19	.	.	PUNCT
ejpam-6171	1052	1	then	then	ADV
ejpam-6171	1052	2	pt	pt	INTJ
ejpam-6171	1052	3	(	(	PUNCT
ejpam-6171	1052	4	x	x	X
ejpam-6171	1052	5	?	?	PUNCT
ejpam-6171	1052	6	y	y	X
ejpam-6171	1052	7	)	)	PUNCT
ejpam-6171	1052	8	≥	≥	NOUN
ejpam-6171	1052	9	α	α	NOUN
ejpam-6171	1052	10	and	and	CCONJ
ejpam-6171	1052	11	pt	pt	PROPN
ejpam-6171	1052	12	(	(	PUNCT
ejpam-6171	1052	13	x	x	NOUN
ejpam-6171	1052	14	)	)	PUNCT
ejpam-6171	1052	15	≥	≥	PROPN
ejpam-6171	1052	16	α	α	NOUN
ejpam-6171	1052	17	.	.	PUNCT
ejpam-6171	1053	1	thus	thus	ADV
ejpam-6171	1053	2	,	,	PUNCT
ejpam-6171	1053	3	x	x	X
ejpam-6171	1053	4	?	?	PUNCT
ejpam-6171	1054	1	y	y	NOUN
ejpam-6171	1054	2	,	,	PUNCT
ejpam-6171	1054	3	x	x	SYM
ejpam-6171	1054	4	∈	∈	NOUN
ejpam-6171	1054	5	u(pt	u(pt	PROPN
ejpam-6171	1054	6	;	;	PUNCT
ejpam-6171	1054	7	α	α	X
ejpam-6171	1054	8	)	)	PUNCT
ejpam-6171	1054	9	6=	6=	ADP
ejpam-6171	1054	10	∅.	∅.	ADP
ejpam-6171	1054	11	by	by	ADP
ejpam-6171	1054	12	the	the	DET
ejpam-6171	1054	13	assumption	assumption	NOUN
ejpam-6171	1054	14	,	,	PUNCT
ejpam-6171	1054	15	we	we	PRON
ejpam-6171	1054	16	have	have	VERB
ejpam-6171	1054	17	u(pt	u(pt	NOUN
ejpam-6171	1054	18	;	;	PUNCT
ejpam-6171	1054	19	α	α	X
ejpam-6171	1054	20	)	)	PUNCT
ejpam-6171	1054	21	is	be	AUX
ejpam-6171	1054	22	an	an	DET
ejpam-6171	1054	23	iup	iup	NOUN
ejpam-6171	1054	24	-	-	PUNCT
ejpam-6171	1054	25	filter	filter	NOUN
ejpam-6171	1054	26	of	of	ADP
ejpam-6171	1054	27	x.	x.	NOUN
ejpam-6171	1054	28	by	by	ADP
ejpam-6171	1054	29	(	(	PUNCT
ejpam-6171	1054	30	2.19	2.19	NUM
ejpam-6171	1054	31	)	)	PUNCT
ejpam-6171	1054	32	,	,	PUNCT
ejpam-6171	1054	33	we	we	PRON
ejpam-6171	1054	34	have	have	VERB
ejpam-6171	1054	35	y	y	PROPN
ejpam-6171	1054	36	∈	∈	PROPN
ejpam-6171	1054	37	u(pt	u(pt	PROPN
ejpam-6171	1054	38	;	;	PUNCT
ejpam-6171	1054	39	α	α	X
ejpam-6171	1054	40	)	)	PUNCT
ejpam-6171	1054	41	.	.	PUNCT
ejpam-6171	1055	1	thus	thus	ADV
ejpam-6171	1055	2	,	,	PUNCT
ejpam-6171	1055	3	pt	pt	X
ejpam-6171	1055	4	(	(	PUNCT
ejpam-6171	1055	5	y	y	NOUN
ejpam-6171	1055	6	)	)	PUNCT
ejpam-6171	1055	7	≥	≥	NOUN
ejpam-6171	1055	8	α	α	NOUN
ejpam-6171	1055	9	=	=	X
ejpam-6171	1055	10	min{pt	min{pt	NOUN
ejpam-6171	1055	11	(	(	PUNCT
ejpam-6171	1055	12	x	x	X
ejpam-6171	1055	13	?	?	PUNCT
ejpam-6171	1055	14	y),pt	y),pt	PROPN
ejpam-6171	1055	15	(	(	PUNCT
ejpam-6171	1055	16	x	x	NOUN
ejpam-6171	1055	17	)	)	PUNCT
ejpam-6171	1055	18	}	}	PUNCT
ejpam-6171	1055	19	.	.	PUNCT
ejpam-6171	1056	1	let	let	VERB
ejpam-6171	1056	2	x	x	SYM
ejpam-6171	1056	3	∈	∈	PROPN
ejpam-6171	1056	4	x.	x.	NOUN
ejpam-6171	1056	5	let	let	VERB
ejpam-6171	1056	6	β	β	X
ejpam-6171	1056	7	=	=	SYM
ejpam-6171	1056	8	pi(x	pi(x	PROPN
ejpam-6171	1056	9	)	)	PUNCT
ejpam-6171	1056	10	.	.	PUNCT
ejpam-6171	1057	1	then	then	ADV
ejpam-6171	1057	2	pi(x	pi(x	NUM
ejpam-6171	1057	3	)	)	PUNCT
ejpam-6171	1057	4	≤	≤	NUM
ejpam-6171	1058	1	β	β	X
ejpam-6171	1058	2	.	.	PUNCT
ejpam-6171	1059	1	thus	thus	ADV
ejpam-6171	1059	2	,	,	PUNCT
ejpam-6171	1059	3	x	x	SYM
ejpam-6171	1059	4	∈	∈	NOUN
ejpam-6171	1059	5	l(pi	l(pi	NOUN
ejpam-6171	1059	6	;	;	PUNCT
ejpam-6171	1059	7	β	β	X
ejpam-6171	1059	8	)	)	PUNCT
ejpam-6171	1059	9	6=	6=	ADP
ejpam-6171	1059	10	∅.	∅.	ADP
ejpam-6171	1059	11	by	by	ADP
ejpam-6171	1059	12	the	the	DET
ejpam-6171	1059	13	assumption	assumption	NOUN
ejpam-6171	1059	14	,	,	PUNCT
ejpam-6171	1059	15	we	we	PRON
ejpam-6171	1059	16	have	have	VERB
ejpam-6171	1059	17	l(pi	l(pi	X
ejpam-6171	1059	18	;	;	PUNCT
ejpam-6171	1059	19	β	β	X
ejpam-6171	1059	20	)	)	PUNCT
ejpam-6171	1059	21	is	be	AUX
ejpam-6171	1059	22	an	an	DET
ejpam-6171	1059	23	iup	iup	NOUN
ejpam-6171	1059	24	-	-	PUNCT
ejpam-6171	1059	25	filter	filter	NOUN
ejpam-6171	1059	26	of	of	ADP
ejpam-6171	1059	27	x.	x.	NOUN
ejpam-6171	1059	28	by	by	ADP
ejpam-6171	1059	29	(	(	PUNCT
ejpam-6171	1059	30	2.18	2.18	NUM
ejpam-6171	1059	31	)	)	PUNCT
ejpam-6171	1059	32	,	,	PUNCT
ejpam-6171	1059	33	we	we	PRON
ejpam-6171	1059	34	have	have	VERB
ejpam-6171	1059	35	0	0	NUM
ejpam-6171	1059	36	∈	∈	PROPN
ejpam-6171	1059	37	l(pi	l(pi	NOUN
ejpam-6171	1059	38	;	;	PUNCT
ejpam-6171	1059	39	β	β	X
ejpam-6171	1059	40	)	)	PUNCT
ejpam-6171	1059	41	.	.	PUNCT
ejpam-6171	1060	1	then	then	ADV
ejpam-6171	1060	2	pi(0	pi(0	PROPN
ejpam-6171	1060	3	)	)	PUNCT
ejpam-6171	1060	4	≤	≤	PUNCT
ejpam-6171	1060	5	β	β	X
ejpam-6171	1060	6	=	=	SYM
ejpam-6171	1060	7	pi(x	pi(x	NOUN
ejpam-6171	1060	8	)	)	PUNCT
ejpam-6171	1060	9	.	.	PUNCT
ejpam-6171	1061	1	let	let	VERB
ejpam-6171	1061	2	x	x	PRON
ejpam-6171	1061	3	,	,	PUNCT
ejpam-6171	1061	4	y	y	PROPN
ejpam-6171	1061	5	∈	∈	PROPN
ejpam-6171	1061	6	x.	x.	NOUN
ejpam-6171	1061	7	let	let	VERB
ejpam-6171	1061	8	β	β	X
ejpam-6171	1061	9	=	=	SYM
ejpam-6171	1061	10	max{pi(x?y),pi(x	max{pi(x?y),pi(x	NOUN
ejpam-6171	1061	11	)	)	PUNCT
ejpam-6171	1061	12	}	}	PUNCT
ejpam-6171	1061	13	.	.	PUNCT
ejpam-6171	1062	1	then	then	ADV
ejpam-6171	1062	2	pi(x?y	pi(x?y	NOUN
ejpam-6171	1062	3	)	)	PUNCT
ejpam-6171	1062	4	≤	≤	NOUN
ejpam-6171	1062	5	β	β	X
ejpam-6171	1062	6	and	and	CCONJ
ejpam-6171	1062	7	pi(x	pi(x	NUM
ejpam-6171	1062	8	)	)	PUNCT
ejpam-6171	1062	9	≤	≤	NOUN
ejpam-6171	1063	1	β	β	X
ejpam-6171	1063	2	.	.	PUNCT
ejpam-6171	1064	1	thus	thus	ADV
ejpam-6171	1064	2	,	,	PUNCT
ejpam-6171	1064	3	x?y	x?y	PROPN
ejpam-6171	1064	4	,	,	PUNCT
ejpam-6171	1064	5	x	x	SYM
ejpam-6171	1064	6	∈	∈	PROPN
ejpam-6171	1064	7	l(pi	l(pi	NOUN
ejpam-6171	1064	8	;	;	PUNCT
ejpam-6171	1064	9	β	β	X
ejpam-6171	1064	10	)	)	PUNCT
ejpam-6171	1064	11	6=	6=	ADP
ejpam-6171	1064	12	∅.	∅.	ADP
ejpam-6171	1064	13	by	by	ADP
ejpam-6171	1064	14	the	the	DET
ejpam-6171	1064	15	assumption	assumption	NOUN
ejpam-6171	1064	16	,	,	PUNCT
ejpam-6171	1064	17	we	we	PRON
ejpam-6171	1064	18	have	have	VERB
ejpam-6171	1064	19	l(pi	l(pi	X
ejpam-6171	1064	20	;	;	PUNCT
ejpam-6171	1064	21	β	β	X
ejpam-6171	1064	22	)	)	PUNCT
ejpam-6171	1064	23	is	be	AUX
ejpam-6171	1064	24	an	an	DET
ejpam-6171	1064	25	iupfilter	iupfilter	NOUN
ejpam-6171	1064	26	of	of	ADP
ejpam-6171	1064	27	x.	x.	NOUN
ejpam-6171	1064	28	by	by	ADP
ejpam-6171	1064	29	(	(	PUNCT
ejpam-6171	1064	30	2.19	2.19	NUM
ejpam-6171	1064	31	)	)	PUNCT
ejpam-6171	1064	32	,	,	PUNCT
ejpam-6171	1064	33	we	we	PRON
ejpam-6171	1064	34	have	have	VERB
ejpam-6171	1064	35	y	y	PROPN
ejpam-6171	1064	36	∈	∈	PROPN
ejpam-6171	1064	37	l(pi	l(pi	PROPN
ejpam-6171	1064	38	;	;	PUNCT
ejpam-6171	1064	39	β	β	X
ejpam-6171	1064	40	)	)	PUNCT
ejpam-6171	1064	41	.	.	PUNCT
ejpam-6171	1065	1	thus	thus	ADV
ejpam-6171	1065	2	,	,	PUNCT
ejpam-6171	1065	3	pi(y	pi(y	NOUN
ejpam-6171	1065	4	)	)	PUNCT
ejpam-6171	1065	5	≤	≤	NOUN
ejpam-6171	1065	6	β	β	X
ejpam-6171	1065	7	=	=	SYM
ejpam-6171	1065	8	max{pi(x	max{pi(x	PROPN
ejpam-6171	1065	9	?	?	PUNCT
ejpam-6171	1066	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1066	2	)	)	PUNCT
ejpam-6171	1066	3	}	}	PUNCT
ejpam-6171	1066	4	.	.	PUNCT
ejpam-6171	1067	1	let	let	VERB
ejpam-6171	1067	2	x	x	SYM
ejpam-6171	1067	3	∈	∈	PROPN
ejpam-6171	1067	4	x.	x.	NOUN
ejpam-6171	1067	5	let	let	VERB
ejpam-6171	1067	6	γ	γ	X
ejpam-6171	1067	7	=	=	SYM
ejpam-6171	1067	8	pf	pf	X
ejpam-6171	1067	9	(	(	PUNCT
ejpam-6171	1067	10	x	x	NOUN
ejpam-6171	1067	11	)	)	PUNCT
ejpam-6171	1067	12	.	.	PUNCT
ejpam-6171	1068	1	then	then	ADV
ejpam-6171	1068	2	pf	pf	PROPN
ejpam-6171	1068	3	(	(	PUNCT
ejpam-6171	1068	4	x	x	PROPN
ejpam-6171	1068	5	)	)	PUNCT
ejpam-6171	1068	6	≥	≥	PROPN
ejpam-6171	1068	7	γ	γ	PROPN
ejpam-6171	1068	8	.	.	PUNCT
ejpam-6171	1068	9	thus	thus	ADV
ejpam-6171	1068	10	,	,	PUNCT
ejpam-6171	1068	11	x	x	PROPN
ejpam-6171	1068	12	∈	∈	PROPN
ejpam-6171	1068	13	u(pf	u(pf	PROPN
ejpam-6171	1068	14	;	;	PUNCT
ejpam-6171	1068	15	γ	γ	X
ejpam-6171	1068	16	)	)	PUNCT
ejpam-6171	1068	17	6=	6=	ADP
ejpam-6171	1068	18	∅.	∅.	ADP
ejpam-6171	1068	19	by	by	ADP
ejpam-6171	1068	20	the	the	DET
ejpam-6171	1068	21	assumption	assumption	NOUN
ejpam-6171	1068	22	,	,	PUNCT
ejpam-6171	1068	23	we	we	PRON
ejpam-6171	1068	24	have	have	VERB
ejpam-6171	1068	25	u(pf	u(pf	NOUN
ejpam-6171	1068	26	;	;	PUNCT
ejpam-6171	1068	27	γ	γ	X
ejpam-6171	1068	28	)	)	PUNCT
ejpam-6171	1068	29	is	be	AUX
ejpam-6171	1068	30	an	an	DET
ejpam-6171	1068	31	iup	iup	NOUN
ejpam-6171	1068	32	-	-	PUNCT
ejpam-6171	1068	33	filter	filter	NOUN
ejpam-6171	1068	34	of	of	ADP
ejpam-6171	1068	35	x.	x.	NOUN
ejpam-6171	1068	36	by	by	ADP
ejpam-6171	1068	37	(	(	PUNCT
ejpam-6171	1068	38	2.18	2.18	NUM
ejpam-6171	1068	39	)	)	PUNCT
ejpam-6171	1068	40	,	,	PUNCT
ejpam-6171	1068	41	we	we	PRON
ejpam-6171	1068	42	have	have	VERB
ejpam-6171	1068	43	0	0	NUM
ejpam-6171	1068	44	∈	∈	PROPN
ejpam-6171	1068	45	u(pf	u(pf	PROPN
ejpam-6171	1068	46	;	;	PUNCT
ejpam-6171	1068	47	γ	γ	X
ejpam-6171	1068	48	)	)	PUNCT
ejpam-6171	1068	49	.	.	PUNCT
ejpam-6171	1069	1	then	then	ADV
ejpam-6171	1069	2	pf	pf	PROPN
ejpam-6171	1069	3	(	(	PUNCT
ejpam-6171	1069	4	0	0	NUM
ejpam-6171	1069	5	)	)	PUNCT
ejpam-6171	1069	6	≥	≥	NOUN
ejpam-6171	1069	7	γ	γ	X
ejpam-6171	1069	8	=	=	SYM
ejpam-6171	1069	9	pf	pf	PROPN
ejpam-6171	1069	10	(	(	PUNCT
ejpam-6171	1069	11	x	x	NOUN
ejpam-6171	1069	12	)	)	PUNCT
ejpam-6171	1069	13	.	.	PUNCT
ejpam-6171	1070	1	let	let	VERB
ejpam-6171	1070	2	x	x	PRON
ejpam-6171	1070	3	,	,	PUNCT
ejpam-6171	1070	4	y	y	PROPN
ejpam-6171	1070	5	∈	∈	PROPN
ejpam-6171	1070	6	x.	x.	NOUN
ejpam-6171	1070	7	let	let	VERB
ejpam-6171	1070	8	γ	γ	X
ejpam-6171	1070	9	=	=	VERB
ejpam-6171	1070	10	min{pf	min{pf	X
ejpam-6171	1070	11	(	(	PUNCT
ejpam-6171	1070	12	x	x	X
ejpam-6171	1070	13	?	?	PUNCT
ejpam-6171	1071	1	y),pf	y),pf	PROPN
ejpam-6171	1071	2	(	(	PUNCT
ejpam-6171	1071	3	x	x	NOUN
ejpam-6171	1071	4	)	)	PUNCT
ejpam-6171	1071	5	}	}	PUNCT
ejpam-6171	1071	6	.	.	PUNCT
ejpam-6171	1072	1	then	then	ADV
ejpam-6171	1072	2	pf	pf	INTJ
ejpam-6171	1072	3	(	(	PUNCT
ejpam-6171	1072	4	x	x	PROPN
ejpam-6171	1072	5	?	?	PUNCT
ejpam-6171	1072	6	y	y	X
ejpam-6171	1072	7	)	)	PUNCT
ejpam-6171	1072	8	≥	≥	PROPN
ejpam-6171	1072	9	γ	γ	PROPN
ejpam-6171	1072	10	and	and	CCONJ
ejpam-6171	1072	11	pf	pf	PROPN
ejpam-6171	1072	12	(	(	PUNCT
ejpam-6171	1072	13	x	x	NOUN
ejpam-6171	1072	14	)	)	PUNCT
ejpam-6171	1072	15	≥	≥	PROPN
ejpam-6171	1072	16	γ	γ	PROPN
ejpam-6171	1072	17	.	.	PUNCT
ejpam-6171	1072	18	thus	thus	ADV
ejpam-6171	1072	19	,	,	PUNCT
ejpam-6171	1072	20	x	x	X
ejpam-6171	1072	21	?	?	PUNCT
ejpam-6171	1073	1	y	y	NOUN
ejpam-6171	1073	2	,	,	PUNCT
ejpam-6171	1073	3	x	x	PROPN
ejpam-6171	1073	4	∈	∈	PROPN
ejpam-6171	1073	5	u(pf	u(pf	PROPN
ejpam-6171	1073	6	;	;	PUNCT
ejpam-6171	1073	7	γ	γ	X
ejpam-6171	1073	8	)	)	PUNCT
ejpam-6171	1073	9	6=	6=	ADP
ejpam-6171	1073	10	∅.	∅.	ADP
ejpam-6171	1073	11	by	by	ADP
ejpam-6171	1073	12	the	the	DET
ejpam-6171	1073	13	assumption	assumption	NOUN
ejpam-6171	1073	14	,	,	PUNCT
ejpam-6171	1073	15	we	we	PRON
ejpam-6171	1073	16	have	have	VERB
ejpam-6171	1073	17	u(pf	u(pf	NOUN
ejpam-6171	1073	18	;	;	PUNCT
ejpam-6171	1073	19	γ	γ	X
ejpam-6171	1073	20	)	)	PUNCT
ejpam-6171	1073	21	is	be	AUX
ejpam-6171	1073	22	an	an	DET
ejpam-6171	1073	23	iup	iup	NOUN
ejpam-6171	1073	24	-	-	PUNCT
ejpam-6171	1073	25	filter	filter	NOUN
ejpam-6171	1073	26	of	of	ADP
ejpam-6171	1073	27	x.	x.	NOUN
ejpam-6171	1073	28	by	by	ADP
ejpam-6171	1073	29	(	(	PUNCT
ejpam-6171	1073	30	2.19	2.19	NUM
ejpam-6171	1073	31	)	)	PUNCT
ejpam-6171	1073	32	,	,	PUNCT
ejpam-6171	1073	33	we	we	PRON
ejpam-6171	1073	34	have	have	VERB
ejpam-6171	1073	35	y	y	PROPN
ejpam-6171	1073	36	∈	∈	PROPN
ejpam-6171	1073	37	u(pf	u(pf	PROPN
ejpam-6171	1073	38	;	;	PUNCT
ejpam-6171	1073	39	γ	γ	X
ejpam-6171	1073	40	)	)	PUNCT
ejpam-6171	1073	41	.	.	PUNCT
ejpam-6171	1074	1	thus	thus	ADV
ejpam-6171	1074	2	,	,	PUNCT
ejpam-6171	1074	3	pf	pf	PROPN
ejpam-6171	1074	4	(	(	PUNCT
ejpam-6171	1074	5	y	y	NOUN
ejpam-6171	1074	6	)	)	PUNCT
ejpam-6171	1074	7	≥	≥	PROPN
ejpam-6171	1074	8	γ	γ	X
ejpam-6171	1074	9	=	=	PUNCT
ejpam-6171	1074	10	min{pf	min{pf	X
ejpam-6171	1074	11	(	(	PUNCT
ejpam-6171	1074	12	x	x	X
ejpam-6171	1074	13	?	?	PUNCT
ejpam-6171	1074	14	y),pf	y),pf	PROPN
ejpam-6171	1074	15	(	(	PUNCT
ejpam-6171	1074	16	x	x	NOUN
ejpam-6171	1074	17	)	)	PUNCT
ejpam-6171	1074	18	}	}	PUNCT
ejpam-6171	1074	19	.	.	PUNCT
ejpam-6171	1075	1	hence	hence	ADV
ejpam-6171	1075	2	,	,	PUNCT
ejpam-6171	1075	3	p	p	PROPN
ejpam-6171	1075	4	is	be	AUX
ejpam-6171	1075	5	a	a	DET
ejpam-6171	1075	6	pythagorean	pythagorean	PROPN
ejpam-6171	1075	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	1075	8	iup	iup	NOUN
ejpam-6171	1075	9	-	-	PUNCT
ejpam-6171	1075	10	filter	filter	NOUN
ejpam-6171	1075	11	of	of	ADP
ejpam-6171	1075	12	x.	x.	PROPN
ejpam-6171	1075	13	k.	k.	PROPN
ejpam-6171	1076	1	suayngam	suayngam	PROPN
ejpam-6171	1076	2	et	et	PROPN
ejpam-6171	1076	3	al	al	PROPN
ejpam-6171	1076	4	.	.	PUNCT
ejpam-6171	1076	5	/	/	SYM
ejpam-6171	1076	6	eur	eur	PROPN
ejpam-6171	1076	7	.	.	PUNCT
ejpam-6171	1077	1	j.	j.	PROPN
ejpam-6171	1077	2	pure	pure	PROPN
ejpam-6171	1077	3	appl	appl	PROPN
ejpam-6171	1077	4	.	.	PROPN
ejpam-6171	1077	5	math	math	PROPN
ejpam-6171	1077	6	,	,	PUNCT
ejpam-6171	1077	7	18	18	NUM
ejpam-6171	1077	8	(	(	PUNCT
ejpam-6171	1077	9	3	3	NUM
ejpam-6171	1077	10	)	)	PUNCT
ejpam-6171	1077	11	(	(	PUNCT
ejpam-6171	1077	12	2025	2025	NUM
ejpam-6171	1077	13	)	)	PUNCT
ejpam-6171	1077	14	,	,	PUNCT
ejpam-6171	1077	15	6171	6171	NUM
ejpam-6171	1077	16	23	23	NUM
ejpam-6171	1077	17	of	of	ADP
ejpam-6171	1077	18	28	28	NUM
ejpam-6171	1077	19	theorem	theorem	NOUN
ejpam-6171	1077	20	20	20	NUM
ejpam-6171	1077	21	.	.	PUNCT
ejpam-6171	1078	1	a	a	DET
ejpam-6171	1078	2	pns	pns	NOUN
ejpam-6171	1078	3	p	p	NOUN
ejpam-6171	1078	4	in	in	ADP
ejpam-6171	1078	5	x	x	PROPN
ejpam-6171	1078	6	is	be	AUX
ejpam-6171	1078	7	a	a	DET
ejpam-6171	1078	8	pythagorean	pythagorean	PROPN
ejpam-6171	1078	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1078	10	strong	strong	ADJ
ejpam-6171	1078	11	iup	iup	NOUN
ejpam-6171	1078	12	-	-	PUNCT
ejpam-6171	1078	13	ideal	ideal	NOUN
ejpam-6171	1078	14	of	of	ADP
ejpam-6171	1078	15	x	x	SYM
ejpam-6171	1078	16	if	if	SCONJ
ejpam-6171	1078	17	and	and	CCONJ
ejpam-6171	1078	18	only	only	ADV
ejpam-6171	1078	19	if	if	SCONJ
ejpam-6171	1078	20	for	for	ADP
ejpam-6171	1078	21	all	all	DET
ejpam-6171	1078	22	α	α	NOUN
ejpam-6171	1078	23	,	,	PUNCT
ejpam-6171	1078	24	β	β	X
ejpam-6171	1078	25	,	,	PUNCT
ejpam-6171	1078	26	γ	γ	PROPN
ejpam-6171	1078	27	∈	∈	PROPN
ejpam-6171	1079	1	[	[	X
ejpam-6171	1079	2	0	0	NUM
ejpam-6171	1079	3	,	,	PUNCT
ejpam-6171	1079	4	1	1	NUM
ejpam-6171	1079	5	]	]	PUNCT
ejpam-6171	1079	6	,	,	PUNCT
ejpam-6171	1079	7	the	the	DET
ejpam-6171	1079	8	sets	set	NOUN
ejpam-6171	1079	9	u(pt	u(pt	PROPN
ejpam-6171	1079	10	;	;	PUNCT
ejpam-6171	1079	11	α	α	X
ejpam-6171	1079	12	)	)	PUNCT
ejpam-6171	1079	13	,	,	PUNCT
ejpam-6171	1079	14	l(pi	l(pi	X
ejpam-6171	1079	15	;	;	PUNCT
ejpam-6171	1079	16	β	β	X
ejpam-6171	1079	17	)	)	PUNCT
ejpam-6171	1079	18	,	,	PUNCT
ejpam-6171	1079	19	and	and	CCONJ
ejpam-6171	1079	20	u(pf	u(pf	PROPN
ejpam-6171	1079	21	;	;	PUNCT
ejpam-6171	1079	22	γ	γ	X
ejpam-6171	1079	23	)	)	PUNCT
ejpam-6171	1079	24	are	be	AUX
ejpam-6171	1079	25	either	either	CCONJ
ejpam-6171	1079	26	empty	empty	ADJ
ejpam-6171	1079	27	or	or	CCONJ
ejpam-6171	1079	28	strong	strong	ADJ
ejpam-6171	1079	29	iup	iup	NOUN
ejpam-6171	1079	30	-	-	PUNCT
ejpam-6171	1079	31	ideals	ideal	NOUN
ejpam-6171	1079	32	of	of	ADP
ejpam-6171	1079	33	x.	x.	NOUN
ejpam-6171	1079	34	proof	proof	NOUN
ejpam-6171	1079	35	.	.	PUNCT
ejpam-6171	1080	1	it	it	PRON
ejpam-6171	1080	2	is	be	AUX
ejpam-6171	1080	3	straightforward	straightforward	ADJ
ejpam-6171	1080	4	by	by	ADP
ejpam-6171	1080	5	theorem	theorem	NOUN
ejpam-6171	1080	6	2	2	NUM
ejpam-6171	1080	7	.	.	PUNCT
ejpam-6171	1080	8	theorem	theorem	NOUN
ejpam-6171	1080	9	21	21	NUM
ejpam-6171	1080	10	.	.	PUNCT
ejpam-6171	1081	1	a	a	DET
ejpam-6171	1081	2	pns	pns	NOUN
ejpam-6171	1081	3	p	p	NOUN
ejpam-6171	1081	4	in	in	ADP
ejpam-6171	1081	5	x	x	PROPN
ejpam-6171	1081	6	is	be	AUX
ejpam-6171	1081	7	a	a	DET
ejpam-6171	1081	8	pythagorean	pythagorean	PROPN
ejpam-6171	1081	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1081	10	strong	strong	ADJ
ejpam-6171	1081	11	iup	iup	NOUN
ejpam-6171	1081	12	-	-	PUNCT
ejpam-6171	1081	13	ideal	ideal	NOUN
ejpam-6171	1081	14	of	of	ADP
ejpam-6171	1081	15	x	x	SYM
ejpam-6171	1081	16	if	if	SCONJ
ejpam-6171	1081	17	and	and	CCONJ
ejpam-6171	1081	18	only	only	ADV
ejpam-6171	1081	19	if	if	SCONJ
ejpam-6171	1081	20	the	the	DET
ejpam-6171	1081	21	sets	set	NOUN
ejpam-6171	1081	22	e(pt	e(pt	PROPN
ejpam-6171	1081	23	;	;	PUNCT
ejpam-6171	1081	24	pt	pt	X
ejpam-6171	1081	25	(	(	PUNCT
ejpam-6171	1081	26	0	0	NUM
ejpam-6171	1081	27	)	)	PUNCT
ejpam-6171	1081	28	)	)	PUNCT
ejpam-6171	1081	29	,	,	PUNCT
ejpam-6171	1081	30	e(pi	e(pi	NUM
ejpam-6171	1081	31	;	;	PUNCT
ejpam-6171	1081	32	pi(0	pi(0	PROPN
ejpam-6171	1081	33	)	)	PUNCT
ejpam-6171	1081	34	)	)	PUNCT
ejpam-6171	1081	35	,	,	PUNCT
ejpam-6171	1081	36	and	and	CCONJ
ejpam-6171	1081	37	e(pf	e(pf	NUM
ejpam-6171	1081	38	;	;	PUNCT
ejpam-6171	1081	39	pf	pf	X
ejpam-6171	1081	40	(	(	PUNCT
ejpam-6171	1081	41	0	0	NUM
ejpam-6171	1081	42	)	)	PUNCT
ejpam-6171	1081	43	)	)	PUNCT
ejpam-6171	1081	44	are	be	AUX
ejpam-6171	1081	45	strong	strong	ADJ
ejpam-6171	1081	46	iup	iup	NOUN
ejpam-6171	1081	47	-	-	PUNCT
ejpam-6171	1081	48	ideals	ideal	NOUN
ejpam-6171	1081	49	of	of	ADP
ejpam-6171	1081	50	x.	x.	NOUN
ejpam-6171	1081	51	proof	proof	NOUN
ejpam-6171	1081	52	.	.	PUNCT
ejpam-6171	1082	1	it	it	PRON
ejpam-6171	1082	2	is	be	AUX
ejpam-6171	1082	3	straightforward	straightforward	ADJ
ejpam-6171	1082	4	by	by	ADP
ejpam-6171	1082	5	theorem	theorem	NOUN
ejpam-6171	1082	6	2	2	NUM
ejpam-6171	1082	7	.	.	PUNCT
ejpam-6171	1082	8	theorem	theorem	NOUN
ejpam-6171	1082	9	22	22	NUM
ejpam-6171	1082	10	.	.	PUNCT
ejpam-6171	1083	1	a	a	DET
ejpam-6171	1083	2	pns	pns	NOUN
ejpam-6171	1083	3	p	p	NOUN
ejpam-6171	1083	4	in	in	ADP
ejpam-6171	1083	5	x	x	PROPN
ejpam-6171	1083	6	is	be	AUX
ejpam-6171	1083	7	a	a	DET
ejpam-6171	1083	8	pythagorean	pythagorean	PROPN
ejpam-6171	1083	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1083	10	iup	iup	NOUN
ejpam-6171	1083	11	-	-	PUNCT
ejpam-6171	1083	12	subalgebra	subalgebra	NOUN
ejpam-6171	1083	13	of	of	ADP
ejpam-6171	1083	14	x	x	PRON
ejpam-6171	1083	15	if	if	SCONJ
ejpam-6171	1083	16	and	and	CCONJ
ejpam-6171	1083	17	only	only	ADV
ejpam-6171	1083	18	if	if	SCONJ
ejpam-6171	1083	19	for	for	ADP
ejpam-6171	1083	20	all	all	DET
ejpam-6171	1083	21	α	α	NOUN
ejpam-6171	1083	22	,	,	PUNCT
ejpam-6171	1083	23	β	β	X
ejpam-6171	1083	24	,	,	PUNCT
ejpam-6171	1083	25	γ	γ	PROPN
ejpam-6171	1083	26	∈	∈	PROPN
ejpam-6171	1084	1	[	[	X
ejpam-6171	1084	2	0	0	NUM
ejpam-6171	1084	3	,	,	PUNCT
ejpam-6171	1084	4	1	1	NUM
ejpam-6171	1084	5	]	]	PUNCT
ejpam-6171	1084	6	,	,	PUNCT
ejpam-6171	1084	7	the	the	DET
ejpam-6171	1084	8	sets	set	NOUN
ejpam-6171	1084	9	u	u	NOUN
ejpam-6171	1084	10	+	+	X
ejpam-6171	1084	11	(	(	PUNCT
ejpam-6171	1084	12	pt	pt	INTJ
ejpam-6171	1084	13	;	;	PUNCT
ejpam-6171	1084	14	α	α	X
ejpam-6171	1084	15	)	)	PUNCT
ejpam-6171	1084	16	,	,	PUNCT
ejpam-6171	1084	17	l	l	NOUN
ejpam-6171	1084	18	−	−	PROPN
ejpam-6171	1084	19	(	(	PUNCT
ejpam-6171	1084	20	pi	pi	NOUN
ejpam-6171	1084	21	;	;	PUNCT
ejpam-6171	1084	22	β	β	X
ejpam-6171	1084	23	)	)	PUNCT
ejpam-6171	1084	24	,	,	PUNCT
ejpam-6171	1084	25	and	and	CCONJ
ejpam-6171	1084	26	u	u	NOUN
ejpam-6171	1084	27	+	+	CCONJ
ejpam-6171	1084	28	(	(	PUNCT
ejpam-6171	1084	29	pf	pf	INTJ
ejpam-6171	1084	30	;	;	PUNCT
ejpam-6171	1084	31	γ	γ	X
ejpam-6171	1084	32	)	)	PUNCT
ejpam-6171	1084	33	are	be	AUX
ejpam-6171	1084	34	either	either	CCONJ
ejpam-6171	1084	35	empty	empty	ADJ
ejpam-6171	1084	36	or	or	CCONJ
ejpam-6171	1084	37	iup	iup	NOUN
ejpam-6171	1084	38	-	-	PUNCT
ejpam-6171	1084	39	subalgebras	subalgebras	PROPN
ejpam-6171	1084	40	of	of	ADP
ejpam-6171	1084	41	x.	x.	PROPN
ejpam-6171	1084	42	proof	proof	PROPN
ejpam-6171	1084	43	.	.	PUNCT
ejpam-6171	1085	1	assume	assume	VERB
ejpam-6171	1085	2	that	that	SCONJ
ejpam-6171	1085	3	p	p	NOUN
ejpam-6171	1085	4	in	in	ADP
ejpam-6171	1085	5	x	x	PROPN
ejpam-6171	1085	6	is	be	AUX
ejpam-6171	1085	7	a	a	DET
ejpam-6171	1085	8	pythagorean	pythagorean	PROPN
ejpam-6171	1085	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1085	10	iup	iup	NOUN
ejpam-6171	1085	11	-	-	PUNCT
ejpam-6171	1085	12	subalgebra	subalgebra	NOUN
ejpam-6171	1085	13	of	of	ADP
ejpam-6171	1085	14	x.	x.	NOUN
ejpam-6171	1085	15	let	let	VERB
ejpam-6171	1085	16	α	α	PRON
ejpam-6171	1085	17	∈	∈	PROPN
ejpam-6171	1086	1	[	[	X
ejpam-6171	1086	2	0	0	NUM
ejpam-6171	1086	3	,	,	PUNCT
ejpam-6171	1086	4	1	1	NUM
ejpam-6171	1086	5	]	]	PUNCT
ejpam-6171	1086	6	be	be	AUX
ejpam-6171	1086	7	such	such	ADJ
ejpam-6171	1086	8	that	that	SCONJ
ejpam-6171	1086	9	u	u	NOUN
ejpam-6171	1086	10	+	+	X
ejpam-6171	1086	11	(	(	PUNCT
ejpam-6171	1086	12	pt	pt	INTJ
ejpam-6171	1086	13	;	;	PUNCT
ejpam-6171	1086	14	α	α	X
ejpam-6171	1086	15	)	)	PUNCT
ejpam-6171	1086	16	6=	6=	ADP
ejpam-6171	1086	17	∅.	∅.	AUX
ejpam-6171	1086	18	let	let	VERB
ejpam-6171	1086	19	x	x	PRON
ejpam-6171	1086	20	,	,	PUNCT
ejpam-6171	1086	21	y	y	PROPN
ejpam-6171	1086	22	∈	∈	PROPN
ejpam-6171	1086	23	u	u	PROPN
ejpam-6171	1086	24	+	+	X
ejpam-6171	1086	25	(	(	PUNCT
ejpam-6171	1086	26	pt	pt	INTJ
ejpam-6171	1086	27	;	;	PUNCT
ejpam-6171	1086	28	α	α	X
ejpam-6171	1086	29	)	)	PUNCT
ejpam-6171	1086	30	.	.	PUNCT
ejpam-6171	1087	1	then	then	ADV
ejpam-6171	1087	2	pt	pt	X
ejpam-6171	1087	3	(	(	PUNCT
ejpam-6171	1087	4	x	x	X
ejpam-6171	1087	5	)	)	PUNCT
ejpam-6171	1087	6	>	>	X
ejpam-6171	1087	7	α	α	PROPN
ejpam-6171	1087	8	and	and	CCONJ
ejpam-6171	1087	9	pt	pt	PROPN
ejpam-6171	1087	10	(	(	PUNCT
ejpam-6171	1087	11	y	y	NOUN
ejpam-6171	1087	12	)	)	PUNCT
ejpam-6171	1087	13	>	>	X
ejpam-6171	1088	1	α	α	X
ejpam-6171	1088	2	.	.	PUNCT
ejpam-6171	1089	1	thus	thus	ADV
ejpam-6171	1089	2	,	,	PUNCT
ejpam-6171	1089	3	min{pt	min{pt	PRON
ejpam-6171	1089	4	(	(	PUNCT
ejpam-6171	1089	5	x),pt	x),pt	PROPN
ejpam-6171	1089	6	(	(	PUNCT
ejpam-6171	1089	7	y	y	NOUN
ejpam-6171	1089	8	)	)	PUNCT
ejpam-6171	1089	9	}	}	PUNCT
ejpam-6171	1089	10	>	>	X
ejpam-6171	1089	11	α	α	X
ejpam-6171	1089	12	.	.	PUNCT
ejpam-6171	1089	13	by	by	ADP
ejpam-6171	1089	14	(	(	PUNCT
ejpam-6171	1089	15	3.2	3.2	NUM
ejpam-6171	1089	16	)	)	PUNCT
ejpam-6171	1089	17	,	,	PUNCT
ejpam-6171	1089	18	we	we	PRON
ejpam-6171	1089	19	have	have	VERB
ejpam-6171	1089	20	pt	pt	INTJ
ejpam-6171	1089	21	(	(	PUNCT
ejpam-6171	1089	22	x	x	NOUN
ejpam-6171	1089	23	?	?	PUNCT
ejpam-6171	1090	1	y	y	X
ejpam-6171	1090	2	)	)	PUNCT
ejpam-6171	1090	3	≥	≥	NOUN
ejpam-6171	1090	4	min{pt	min{pt	X
ejpam-6171	1091	1	(	(	PUNCT
ejpam-6171	1091	2	x),pt	x),pt	PROPN
ejpam-6171	1091	3	(	(	PUNCT
ejpam-6171	1091	4	y	y	NOUN
ejpam-6171	1091	5	)	)	PUNCT
ejpam-6171	1091	6	}	}	PUNCT
ejpam-6171	1091	7	>	>	X
ejpam-6171	1091	8	α	α	X
ejpam-6171	1091	9	.	.	PUNCT
ejpam-6171	1092	1	thus	thus	ADV
ejpam-6171	1092	2	,	,	PUNCT
ejpam-6171	1092	3	x?y	x?y	PROPN
ejpam-6171	1092	4	∈	∈	PROPN
ejpam-6171	1092	5	u	u	PROPN
ejpam-6171	1092	6	+	+	X
ejpam-6171	1092	7	(	(	PUNCT
ejpam-6171	1092	8	pt	pt	INTJ
ejpam-6171	1092	9	;	;	PUNCT
ejpam-6171	1092	10	α	α	X
ejpam-6171	1092	11	)	)	PUNCT
ejpam-6171	1092	12	.	.	PUNCT
ejpam-6171	1093	1	hence	hence	ADV
ejpam-6171	1093	2	,	,	PUNCT
ejpam-6171	1093	3	u+	u+	NUM
ejpam-6171	1093	4	(	(	PUNCT
ejpam-6171	1093	5	pt	pt	X
ejpam-6171	1093	6	;	;	PUNCT
ejpam-6171	1093	7	α	α	X
ejpam-6171	1093	8	)	)	PUNCT
ejpam-6171	1093	9	is	be	AUX
ejpam-6171	1093	10	an	an	DET
ejpam-6171	1093	11	iup	iup	NOUN
ejpam-6171	1093	12	-	-	PUNCT
ejpam-6171	1093	13	subalgebra	subalgebra	NOUN
ejpam-6171	1093	14	of	of	ADP
ejpam-6171	1093	15	x.	x.	NOUN
ejpam-6171	1093	16	let	let	VERB
ejpam-6171	1093	17	β	β	X
ejpam-6171	1093	18	∈	∈	PROPN
ejpam-6171	1094	1	[	[	X
ejpam-6171	1094	2	0	0	NUM
ejpam-6171	1094	3	,	,	PUNCT
ejpam-6171	1094	4	1	1	NUM
ejpam-6171	1094	5	]	]	PUNCT
ejpam-6171	1094	6	be	be	AUX
ejpam-6171	1094	7	such	such	ADJ
ejpam-6171	1094	8	that	that	SCONJ
ejpam-6171	1094	9	l	l	NOUN
ejpam-6171	1094	10	−	−	PROPN
ejpam-6171	1094	11	(	(	PUNCT
ejpam-6171	1094	12	pi	pi	NOUN
ejpam-6171	1094	13	;	;	PUNCT
ejpam-6171	1094	14	β	β	X
ejpam-6171	1094	15	)	)	PUNCT
ejpam-6171	1094	16	6=	6=	ADP
ejpam-6171	1094	17	∅.	∅.	AUX
ejpam-6171	1094	18	let	let	VERB
ejpam-6171	1094	19	x	x	PRON
ejpam-6171	1094	20	,	,	PUNCT
ejpam-6171	1094	21	y	y	PROPN
ejpam-6171	1094	22	∈	∈	PROPN
ejpam-6171	1094	23	l	l	NOUN
ejpam-6171	1095	1	−	−	PROPN
ejpam-6171	1095	2	(	(	PUNCT
ejpam-6171	1095	3	pi	pi	NOUN
ejpam-6171	1095	4	;	;	PUNCT
ejpam-6171	1095	5	β	β	X
ejpam-6171	1095	6	)	)	PUNCT
ejpam-6171	1095	7	.	.	PUNCT
ejpam-6171	1096	1	then	then	ADV
ejpam-6171	1096	2	pi(x	pi(x	NUM
ejpam-6171	1096	3	)	)	PUNCT
ejpam-6171	1096	4	<	<	X
ejpam-6171	1096	5	β	β	X
ejpam-6171	1096	6	and	and	CCONJ
ejpam-6171	1096	7	pi(y	pi(y	NOUN
ejpam-6171	1096	8	)	)	PUNCT
ejpam-6171	1096	9	<	<	X
ejpam-6171	1096	10	β	β	X
ejpam-6171	1096	11	.	.	PUNCT
ejpam-6171	1097	1	thus	thus	ADV
ejpam-6171	1097	2	,	,	PUNCT
ejpam-6171	1097	3	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	1097	4	)	)	PUNCT
ejpam-6171	1097	5	}	}	PUNCT
ejpam-6171	1097	6	<	<	X
ejpam-6171	1097	7	β	β	X
ejpam-6171	1097	8	.	.	PUNCT
ejpam-6171	1098	1	by	by	ADP
ejpam-6171	1098	2	(	(	PUNCT
ejpam-6171	1098	3	3.3	3.3	NUM
ejpam-6171	1098	4	)	)	PUNCT
ejpam-6171	1098	5	,	,	PUNCT
ejpam-6171	1098	6	we	we	PRON
ejpam-6171	1098	7	have	have	VERB
ejpam-6171	1098	8	pi(x	pi(x	NUM
ejpam-6171	1098	9	?	?	PUNCT
ejpam-6171	1099	1	y	y	X
ejpam-6171	1099	2	)	)	PUNCT
ejpam-6171	1099	3	≤	≤	NUM
ejpam-6171	1099	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	1099	5	)	)	PUNCT
ejpam-6171	1099	6	}	}	PUNCT
ejpam-6171	1099	7	<	<	X
ejpam-6171	1099	8	β	β	X
ejpam-6171	1099	9	.	.	PUNCT
ejpam-6171	1100	1	thus	thus	ADV
ejpam-6171	1100	2	,	,	PUNCT
ejpam-6171	1100	3	x	x	PUNCT
ejpam-6171	1100	4	?	?	PUNCT
ejpam-6171	1101	1	y	y	PROPN
ejpam-6171	1101	2	∈	∈	PROPN
ejpam-6171	1101	3	l	l	NOUN
ejpam-6171	1101	4	−	−	PROPN
ejpam-6171	1102	1	(	(	PUNCT
ejpam-6171	1102	2	pi	pi	NOUN
ejpam-6171	1102	3	;	;	PUNCT
ejpam-6171	1102	4	β	β	X
ejpam-6171	1102	5	)	)	PUNCT
ejpam-6171	1102	6	.	.	PUNCT
ejpam-6171	1103	1	hence	hence	ADV
ejpam-6171	1103	2	,	,	PUNCT
ejpam-6171	1103	3	l−	l−	PROPN
ejpam-6171	1103	4	(	(	PUNCT
ejpam-6171	1103	5	pi	pi	NOUN
ejpam-6171	1103	6	;	;	PUNCT
ejpam-6171	1103	7	β	β	X
ejpam-6171	1103	8	)	)	PUNCT
ejpam-6171	1103	9	is	be	AUX
ejpam-6171	1103	10	an	an	DET
ejpam-6171	1103	11	iup	iup	NOUN
ejpam-6171	1103	12	-	-	PUNCT
ejpam-6171	1103	13	subalgebra	subalgebra	NOUN
ejpam-6171	1103	14	of	of	ADP
ejpam-6171	1103	15	x.	x.	NOUN
ejpam-6171	1103	16	let	let	VERB
ejpam-6171	1103	17	γ	γ	X
ejpam-6171	1103	18	∈	∈	PROPN
ejpam-6171	1104	1	[	[	X
ejpam-6171	1104	2	0	0	NUM
ejpam-6171	1104	3	,	,	PUNCT
ejpam-6171	1104	4	1	1	NUM
ejpam-6171	1104	5	]	]	PUNCT
ejpam-6171	1104	6	be	be	AUX
ejpam-6171	1104	7	such	such	ADJ
ejpam-6171	1104	8	that	that	SCONJ
ejpam-6171	1104	9	u	u	NOUN
ejpam-6171	1104	10	+	+	X
ejpam-6171	1104	11	(	(	PUNCT
ejpam-6171	1104	12	pf	pf	INTJ
ejpam-6171	1104	13	;	;	PUNCT
ejpam-6171	1104	14	γ	γ	X
ejpam-6171	1104	15	)	)	PUNCT
ejpam-6171	1104	16	6=	6=	ADP
ejpam-6171	1104	17	∅.	∅.	AUX
ejpam-6171	1104	18	let	let	VERB
ejpam-6171	1104	19	x	x	PRON
ejpam-6171	1104	20	,	,	PUNCT
ejpam-6171	1104	21	y	y	PROPN
ejpam-6171	1104	22	∈	∈	PROPN
ejpam-6171	1104	23	u	u	PROPN
ejpam-6171	1104	24	+	+	X
ejpam-6171	1104	25	(	(	PUNCT
ejpam-6171	1104	26	pf	pf	INTJ
ejpam-6171	1104	27	;	;	PUNCT
ejpam-6171	1104	28	γ	γ	X
ejpam-6171	1104	29	)	)	PUNCT
ejpam-6171	1104	30	.	.	PUNCT
ejpam-6171	1105	1	then	then	ADV
ejpam-6171	1105	2	pf	pf	PROPN
ejpam-6171	1105	3	(	(	PUNCT
ejpam-6171	1105	4	x	x	X
ejpam-6171	1105	5	)	)	PUNCT
ejpam-6171	1105	6	>	>	X
ejpam-6171	1105	7	γ	γ	PROPN
ejpam-6171	1105	8	and	and	CCONJ
ejpam-6171	1105	9	pf	pf	PROPN
ejpam-6171	1105	10	(	(	PUNCT
ejpam-6171	1105	11	y	y	NOUN
ejpam-6171	1105	12	)	)	PUNCT
ejpam-6171	1105	13	>	>	X
ejpam-6171	1105	14	γ	γ	X
ejpam-6171	1105	15	.	.	PUNCT
ejpam-6171	1105	16	thus	thus	ADV
ejpam-6171	1105	17	,	,	PUNCT
ejpam-6171	1105	18	min{pf	min{pf	PRON
ejpam-6171	1105	19	(	(	PUNCT
ejpam-6171	1105	20	x),pf	x),pf	PROPN
ejpam-6171	1105	21	(	(	PUNCT
ejpam-6171	1105	22	y	y	NOUN
ejpam-6171	1105	23	)	)	PUNCT
ejpam-6171	1105	24	}	}	PUNCT
ejpam-6171	1105	25	>	>	X
ejpam-6171	1105	26	γ	γ	X
ejpam-6171	1105	27	.	.	PUNCT
ejpam-6171	1105	28	by	by	ADP
ejpam-6171	1105	29	(	(	PUNCT
ejpam-6171	1105	30	3.2	3.2	NUM
ejpam-6171	1105	31	)	)	PUNCT
ejpam-6171	1105	32	,	,	PUNCT
ejpam-6171	1105	33	we	we	PRON
ejpam-6171	1105	34	have	have	VERB
ejpam-6171	1105	35	pf	pf	PROPN
ejpam-6171	1105	36	(	(	PUNCT
ejpam-6171	1105	37	x	x	PROPN
ejpam-6171	1105	38	?	?	PUNCT
ejpam-6171	1106	1	y	y	X
ejpam-6171	1106	2	)	)	PUNCT
ejpam-6171	1106	3	≥	≥	PROPN
ejpam-6171	1106	4	min{pf	min{pf	PRON
ejpam-6171	1106	5	(	(	PUNCT
ejpam-6171	1106	6	x),pf	x),pf	PROPN
ejpam-6171	1106	7	(	(	PUNCT
ejpam-6171	1106	8	y	y	NOUN
ejpam-6171	1106	9	)	)	PUNCT
ejpam-6171	1106	10	}	}	PUNCT
ejpam-6171	1106	11	>	>	X
ejpam-6171	1107	1	γ	γ	X
ejpam-6171	1107	2	.	.	PUNCT
ejpam-6171	1107	3	thus	thus	ADV
ejpam-6171	1107	4	,	,	PUNCT
ejpam-6171	1107	5	x?y	x?y	PROPN
ejpam-6171	1107	6	∈	∈	PROPN
ejpam-6171	1107	7	u	u	PROPN
ejpam-6171	1107	8	+	+	X
ejpam-6171	1107	9	(	(	PUNCT
ejpam-6171	1107	10	pf	pf	INTJ
ejpam-6171	1107	11	;	;	PUNCT
ejpam-6171	1107	12	γ	γ	X
ejpam-6171	1107	13	)	)	PUNCT
ejpam-6171	1107	14	.	.	PUNCT
ejpam-6171	1108	1	hence	hence	ADV
ejpam-6171	1108	2	,	,	PUNCT
ejpam-6171	1108	3	u+	u+	NUM
ejpam-6171	1108	4	(	(	PUNCT
ejpam-6171	1108	5	pf	pf	X
ejpam-6171	1108	6	;	;	PUNCT
ejpam-6171	1108	7	γ	γ	X
ejpam-6171	1108	8	)	)	PUNCT
ejpam-6171	1108	9	is	be	AUX
ejpam-6171	1108	10	an	an	DET
ejpam-6171	1108	11	iup	iup	NOUN
ejpam-6171	1108	12	-	-	PUNCT
ejpam-6171	1108	13	subalgebra	subalgebra	NOUN
ejpam-6171	1108	14	of	of	ADP
ejpam-6171	1108	15	x.	x.	NOUN
ejpam-6171	1108	16	conversely	conversely	ADV
ejpam-6171	1108	17	,	,	PUNCT
ejpam-6171	1108	18	assume	assume	VERB
ejpam-6171	1108	19	that	that	SCONJ
ejpam-6171	1108	20	for	for	ADP
ejpam-6171	1108	21	all	all	DET
ejpam-6171	1108	22	α	α	NOUN
ejpam-6171	1108	23	,	,	PUNCT
ejpam-6171	1108	24	β	β	X
ejpam-6171	1108	25	,	,	PUNCT
ejpam-6171	1108	26	γ	γ	PROPN
ejpam-6171	1108	27	∈	∈	PROPN
ejpam-6171	1109	1	[	[	X
ejpam-6171	1109	2	0	0	NUM
ejpam-6171	1109	3	,	,	PUNCT
ejpam-6171	1109	4	1	1	NUM
ejpam-6171	1109	5	]	]	PUNCT
ejpam-6171	1109	6	,	,	PUNCT
ejpam-6171	1109	7	the	the	DET
ejpam-6171	1109	8	sets	set	NOUN
ejpam-6171	1109	9	u	u	NOUN
ejpam-6171	1109	10	+	+	X
ejpam-6171	1109	11	(	(	PUNCT
ejpam-6171	1109	12	pt	pt	INTJ
ejpam-6171	1109	13	;	;	PUNCT
ejpam-6171	1109	14	α	α	X
ejpam-6171	1109	15	)	)	PUNCT
ejpam-6171	1109	16	,	,	PUNCT
ejpam-6171	1109	17	l	l	NOUN
ejpam-6171	1109	18	−	−	PROPN
ejpam-6171	1109	19	(	(	PUNCT
ejpam-6171	1109	20	pi	pi	NOUN
ejpam-6171	1109	21	;	;	PUNCT
ejpam-6171	1109	22	β	β	X
ejpam-6171	1109	23	)	)	PUNCT
ejpam-6171	1109	24	,	,	PUNCT
ejpam-6171	1109	25	and	and	CCONJ
ejpam-6171	1109	26	u	u	NOUN
ejpam-6171	1109	27	+	+	CCONJ
ejpam-6171	1109	28	(	(	PUNCT
ejpam-6171	1109	29	pf	pf	INTJ
ejpam-6171	1109	30	;	;	PUNCT
ejpam-6171	1109	31	γ	γ	X
ejpam-6171	1109	32	)	)	PUNCT
ejpam-6171	1109	33	are	be	AUX
ejpam-6171	1109	34	either	either	CCONJ
ejpam-6171	1109	35	empty	empty	ADJ
ejpam-6171	1109	36	or	or	CCONJ
ejpam-6171	1109	37	iup	iup	NOUN
ejpam-6171	1109	38	-	-	PUNCT
ejpam-6171	1109	39	subalgebras	subalgebras	PROPN
ejpam-6171	1109	40	of	of	ADP
ejpam-6171	1109	41	x.	x.	PROPN
ejpam-6171	1109	42	let	let	VERB
ejpam-6171	1109	43	x	x	PRON
ejpam-6171	1109	44	,	,	PUNCT
ejpam-6171	1109	45	y	y	PROPN
ejpam-6171	1109	46	∈	∈	PROPN
ejpam-6171	1109	47	x.	x.	NOUN
ejpam-6171	1109	48	assume	assume	VERB
ejpam-6171	1110	1	that	that	SCONJ
ejpam-6171	1110	2	pt	pt	INTJ
ejpam-6171	1110	3	(	(	PUNCT
ejpam-6171	1110	4	x	x	X
ejpam-6171	1110	5	?	?	PUNCT
ejpam-6171	1111	1	y	y	X
ejpam-6171	1111	2	)	)	PUNCT
ejpam-6171	1111	3	<	<	X
ejpam-6171	1111	4	min{pt	min{pt	X
ejpam-6171	1112	1	(	(	PUNCT
ejpam-6171	1112	2	x),pt	x),pt	PROPN
ejpam-6171	1112	3	(	(	PUNCT
ejpam-6171	1112	4	y	y	NOUN
ejpam-6171	1112	5	)	)	PUNCT
ejpam-6171	1112	6	}	}	PUNCT
ejpam-6171	1112	7	.	.	PUNCT
ejpam-6171	1113	1	let	let	VERB
ejpam-6171	1113	2	α	α	NOUN
ejpam-6171	1113	3	=	=	SYM
ejpam-6171	1113	4	pt	pt	X
ejpam-6171	1113	5	(	(	PUNCT
ejpam-6171	1113	6	x	x	PROPN
ejpam-6171	1113	7	?	?	PUNCT
ejpam-6171	1114	1	y	y	X
ejpam-6171	1114	2	)	)	PUNCT
ejpam-6171	1114	3	.	.	PUNCT
ejpam-6171	1115	1	then	then	ADV
ejpam-6171	1115	2	pt	pt	X
ejpam-6171	1115	3	(	(	PUNCT
ejpam-6171	1115	4	x	x	X
ejpam-6171	1115	5	)	)	PUNCT
ejpam-6171	1115	6	>	>	X
ejpam-6171	1115	7	α	α	PROPN
ejpam-6171	1115	8	and	and	CCONJ
ejpam-6171	1115	9	pt	pt	PROPN
ejpam-6171	1115	10	(	(	PUNCT
ejpam-6171	1115	11	y	y	NOUN
ejpam-6171	1115	12	)	)	PUNCT
ejpam-6171	1115	13	>	>	X
ejpam-6171	1116	1	α	α	X
ejpam-6171	1116	2	.	.	PUNCT
ejpam-6171	1117	1	thus	thus	ADV
ejpam-6171	1117	2	,	,	PUNCT
ejpam-6171	1117	3	x	x	PRON
ejpam-6171	1117	4	,	,	PUNCT
ejpam-6171	1117	5	y	y	PROPN
ejpam-6171	1117	6	∈	∈	PROPN
ejpam-6171	1117	7	u	u	PROPN
ejpam-6171	1117	8	+	+	X
ejpam-6171	1117	9	(	(	PUNCT
ejpam-6171	1117	10	pt	pt	INTJ
ejpam-6171	1117	11	;	;	PUNCT
ejpam-6171	1117	12	α	α	X
ejpam-6171	1117	13	)	)	PUNCT
ejpam-6171	1117	14	.	.	PUNCT
ejpam-6171	1118	1	by	by	ADP
ejpam-6171	1118	2	the	the	DET
ejpam-6171	1118	3	assumption	assumption	NOUN
ejpam-6171	1118	4	,	,	PUNCT
ejpam-6171	1118	5	we	we	PRON
ejpam-6171	1118	6	have	have	VERB
ejpam-6171	1118	7	u	u	NOUN
ejpam-6171	1118	8	+	+	CCONJ
ejpam-6171	1118	9	(	(	PUNCT
ejpam-6171	1118	10	pt	pt	INTJ
ejpam-6171	1118	11	;	;	PUNCT
ejpam-6171	1118	12	α	α	X
ejpam-6171	1118	13	)	)	PUNCT
ejpam-6171	1118	14	is	be	AUX
ejpam-6171	1118	15	an	an	DET
ejpam-6171	1118	16	iup	iup	NOUN
ejpam-6171	1118	17	-	-	PUNCT
ejpam-6171	1118	18	subalgebra	subalgebra	NOUN
ejpam-6171	1118	19	.	.	PUNCT
ejpam-6171	1119	1	by	by	ADP
ejpam-6171	1119	2	(	(	PUNCT
ejpam-6171	1119	3	2.17	2.17	NUM
ejpam-6171	1119	4	)	)	PUNCT
ejpam-6171	1119	5	,	,	PUNCT
ejpam-6171	1119	6	we	we	PRON
ejpam-6171	1119	7	have	have	VERB
ejpam-6171	1119	8	x?y	x?y	PROPN
ejpam-6171	1119	9	∈	∈	PROPN
ejpam-6171	1119	10	u	u	PROPN
ejpam-6171	1119	11	+	+	X
ejpam-6171	1119	12	(	(	PUNCT
ejpam-6171	1119	13	pt	pt	INTJ
ejpam-6171	1119	14	;	;	PUNCT
ejpam-6171	1119	15	α	α	X
ejpam-6171	1119	16	)	)	PUNCT
ejpam-6171	1119	17	.	.	PUNCT
ejpam-6171	1120	1	so	so	ADV
ejpam-6171	1120	2	pt	pt	INTJ
ejpam-6171	1120	3	(	(	PUNCT
ejpam-6171	1120	4	x?y	x?y	PROPN
ejpam-6171	1120	5	)	)	PUNCT
ejpam-6171	1120	6	>	>	X
ejpam-6171	1120	7	α	α	X
ejpam-6171	1120	8	=	=	SYM
ejpam-6171	1120	9	pt	pt	X
ejpam-6171	1120	10	(	(	PUNCT
ejpam-6171	1120	11	x?y	x?y	PROPN
ejpam-6171	1120	12	)	)	PUNCT
ejpam-6171	1120	13	,	,	PUNCT
ejpam-6171	1120	14	which	which	PRON
ejpam-6171	1120	15	is	be	AUX
ejpam-6171	1120	16	a	a	DET
ejpam-6171	1120	17	contradiction	contradiction	NOUN
ejpam-6171	1120	18	.	.	PUNCT
ejpam-6171	1121	1	thus	thus	ADV
ejpam-6171	1121	2	,	,	PUNCT
ejpam-6171	1121	3	pt	pt	INTJ
ejpam-6171	1121	4	(	(	PUNCT
ejpam-6171	1121	5	x	x	X
ejpam-6171	1121	6	?	?	PUNCT
ejpam-6171	1121	7	y	y	X
ejpam-6171	1121	8	)	)	PUNCT
ejpam-6171	1121	9	≥	≥	NOUN
ejpam-6171	1121	10	min{pt	min{pt	X
ejpam-6171	1121	11	(	(	PUNCT
ejpam-6171	1121	12	x),pt	x),pt	PROPN
ejpam-6171	1121	13	(	(	PUNCT
ejpam-6171	1121	14	y	y	NOUN
ejpam-6171	1121	15	)	)	PUNCT
ejpam-6171	1121	16	}	}	PUNCT
ejpam-6171	1121	17	.	.	PUNCT
ejpam-6171	1122	1	let	let	VERB
ejpam-6171	1122	2	x	x	PRON
ejpam-6171	1122	3	,	,	PUNCT
ejpam-6171	1122	4	y	y	PROPN
ejpam-6171	1122	5	∈	∈	PROPN
ejpam-6171	1122	6	x.	x.	NOUN
ejpam-6171	1122	7	assume	assume	VERB
ejpam-6171	1122	8	that	that	SCONJ
ejpam-6171	1122	9	pi(x	pi(x	NOUN
ejpam-6171	1122	10	?	?	PUNCT
ejpam-6171	1123	1	y	y	X
ejpam-6171	1123	2	)	)	PUNCT
ejpam-6171	1123	3	>	>	X
ejpam-6171	1123	4	max{pi(x),pi(y	max{pi(x),pi(y	PROPN
ejpam-6171	1123	5	)	)	PUNCT
ejpam-6171	1123	6	}	}	PUNCT
ejpam-6171	1123	7	.	.	PUNCT
ejpam-6171	1124	1	let	let	VERB
ejpam-6171	1124	2	β	β	X
ejpam-6171	1124	3	=	=	NOUN
ejpam-6171	1124	4	pi(x	pi(x	NOUN
ejpam-6171	1124	5	?	?	PUNCT
ejpam-6171	1125	1	y	y	X
ejpam-6171	1125	2	)	)	PUNCT
ejpam-6171	1125	3	.	.	PUNCT
ejpam-6171	1126	1	then	then	ADV
ejpam-6171	1126	2	pi(x	pi(x	NUM
ejpam-6171	1126	3	)	)	PUNCT
ejpam-6171	1126	4	<	<	X
ejpam-6171	1126	5	β	β	X
ejpam-6171	1126	6	and	and	CCONJ
ejpam-6171	1126	7	pi(y	pi(y	NOUN
ejpam-6171	1126	8	)	)	PUNCT
ejpam-6171	1126	9	<	<	X
ejpam-6171	1126	10	β	β	X
ejpam-6171	1126	11	.	.	PUNCT
ejpam-6171	1127	1	thus	thus	ADV
ejpam-6171	1127	2	,	,	PUNCT
ejpam-6171	1127	3	x	x	PRON
ejpam-6171	1127	4	,	,	PUNCT
ejpam-6171	1127	5	y	y	PROPN
ejpam-6171	1127	6	∈	∈	PROPN
ejpam-6171	1127	7	l	l	NOUN
ejpam-6171	1127	8	−	−	PROPN
ejpam-6171	1127	9	(	(	PUNCT
ejpam-6171	1127	10	pi	pi	NOUN
ejpam-6171	1127	11	;	;	PUNCT
ejpam-6171	1127	12	β	β	X
ejpam-6171	1127	13	)	)	PUNCT
ejpam-6171	1127	14	.	.	PUNCT
ejpam-6171	1128	1	by	by	ADP
ejpam-6171	1128	2	the	the	DET
ejpam-6171	1128	3	assumption	assumption	NOUN
ejpam-6171	1128	4	,	,	PUNCT
ejpam-6171	1128	5	we	we	PRON
ejpam-6171	1128	6	have	have	VERB
ejpam-6171	1128	7	l	l	NOUN
ejpam-6171	1128	8	−	−	PROPN
ejpam-6171	1128	9	(	(	PUNCT
ejpam-6171	1128	10	pi	pi	NOUN
ejpam-6171	1128	11	;	;	PUNCT
ejpam-6171	1128	12	β	β	X
ejpam-6171	1128	13	)	)	PUNCT
ejpam-6171	1128	14	is	be	AUX
ejpam-6171	1128	15	an	an	DET
ejpam-6171	1128	16	iup	iup	NOUN
ejpam-6171	1128	17	-	-	PUNCT
ejpam-6171	1128	18	subalgebra	subalgebra	NOUN
ejpam-6171	1128	19	.	.	PUNCT
ejpam-6171	1129	1	by	by	ADP
ejpam-6171	1129	2	(	(	PUNCT
ejpam-6171	1129	3	2.17	2.17	NUM
ejpam-6171	1129	4	)	)	PUNCT
ejpam-6171	1129	5	,	,	PUNCT
ejpam-6171	1129	6	we	we	PRON
ejpam-6171	1129	7	have	have	VERB
ejpam-6171	1129	8	x	x	X
ejpam-6171	1129	9	?	?	PUNCT
ejpam-6171	1130	1	y	y	PROPN
ejpam-6171	1130	2	∈	∈	PROPN
ejpam-6171	1130	3	l	l	NOUN
ejpam-6171	1130	4	−	−	PROPN
ejpam-6171	1131	1	(	(	PUNCT
ejpam-6171	1131	2	pi	pi	NOUN
ejpam-6171	1131	3	;	;	PUNCT
ejpam-6171	1131	4	β	β	X
ejpam-6171	1131	5	)	)	PUNCT
ejpam-6171	1131	6	.	.	PUNCT
ejpam-6171	1132	1	so	so	ADV
ejpam-6171	1132	2	pi(x	pi(x	ADJ
ejpam-6171	1132	3	?	?	PUNCT
ejpam-6171	1133	1	y	y	X
ejpam-6171	1133	2	)	)	PUNCT
ejpam-6171	1133	3	<	<	X
ejpam-6171	1133	4	β	β	X
ejpam-6171	1133	5	=	=	SYM
ejpam-6171	1133	6	pi(x	pi(x	NOUN
ejpam-6171	1133	7	?	?	PUNCT
ejpam-6171	1134	1	y	y	X
ejpam-6171	1134	2	)	)	PUNCT
ejpam-6171	1134	3	,	,	PUNCT
ejpam-6171	1134	4	which	which	PRON
ejpam-6171	1134	5	is	be	AUX
ejpam-6171	1134	6	a	a	DET
ejpam-6171	1134	7	contradiction	contradiction	NOUN
ejpam-6171	1134	8	.	.	PUNCT
ejpam-6171	1135	1	thus	thus	ADV
ejpam-6171	1135	2	,	,	PUNCT
ejpam-6171	1135	3	pi(x	pi(x	ADJ
ejpam-6171	1135	4	?	?	PUNCT
ejpam-6171	1136	1	y	y	X
ejpam-6171	1136	2	)	)	PUNCT
ejpam-6171	1136	3	≤	≤	NUM
ejpam-6171	1136	4	max{pi(x),pi(y	max{pi(x),pi(y	NOUN
ejpam-6171	1136	5	)	)	PUNCT
ejpam-6171	1136	6	}	}	PUNCT
ejpam-6171	1136	7	.	.	PUNCT
ejpam-6171	1137	1	let	let	VERB
ejpam-6171	1137	2	x	x	PRON
ejpam-6171	1137	3	,	,	PUNCT
ejpam-6171	1137	4	y	y	PROPN
ejpam-6171	1137	5	∈	∈	PROPN
ejpam-6171	1137	6	x.	x.	NOUN
ejpam-6171	1137	7	assume	assume	VERB
ejpam-6171	1137	8	that	that	SCONJ
ejpam-6171	1138	1	pf	pf	PROPN
ejpam-6171	1138	2	(	(	PUNCT
ejpam-6171	1138	3	x	x	PROPN
ejpam-6171	1138	4	?	?	PUNCT
ejpam-6171	1139	1	y	y	X
ejpam-6171	1139	2	)	)	PUNCT
ejpam-6171	1139	3	<	<	X
ejpam-6171	1139	4	min{pf	min{pf	X
ejpam-6171	1139	5	(	(	PUNCT
ejpam-6171	1139	6	x),pf	x),pf	PROPN
ejpam-6171	1139	7	(	(	PUNCT
ejpam-6171	1139	8	y	y	NOUN
ejpam-6171	1139	9	)	)	PUNCT
ejpam-6171	1139	10	}	}	PUNCT
ejpam-6171	1139	11	.	.	PUNCT
ejpam-6171	1140	1	let	let	VERB
ejpam-6171	1140	2	γ	γ	X
ejpam-6171	1140	3	=	=	SYM
ejpam-6171	1140	4	pf	pf	PROPN
ejpam-6171	1140	5	(	(	PUNCT
ejpam-6171	1140	6	x	x	PROPN
ejpam-6171	1140	7	?	?	PUNCT
ejpam-6171	1141	1	y	y	X
ejpam-6171	1141	2	)	)	PUNCT
ejpam-6171	1141	3	.	.	PUNCT
ejpam-6171	1142	1	then	then	ADV
ejpam-6171	1142	2	pf	pf	PROPN
ejpam-6171	1142	3	(	(	PUNCT
ejpam-6171	1142	4	x	x	X
ejpam-6171	1142	5	)	)	PUNCT
ejpam-6171	1142	6	>	>	X
ejpam-6171	1142	7	γ	γ	PROPN
ejpam-6171	1142	8	and	and	CCONJ
ejpam-6171	1142	9	pf	pf	PROPN
ejpam-6171	1142	10	(	(	PUNCT
ejpam-6171	1142	11	y	y	NOUN
ejpam-6171	1142	12	)	)	PUNCT
ejpam-6171	1142	13	>	>	X
ejpam-6171	1142	14	γ	γ	X
ejpam-6171	1142	15	.	.	PUNCT
ejpam-6171	1142	16	thus	thus	ADV
ejpam-6171	1142	17	,	,	PUNCT
ejpam-6171	1142	18	x	x	PRON
ejpam-6171	1142	19	,	,	PUNCT
ejpam-6171	1142	20	y	y	PROPN
ejpam-6171	1142	21	∈	∈	PROPN
ejpam-6171	1142	22	u	u	PROPN
ejpam-6171	1142	23	+	+	X
ejpam-6171	1142	24	(	(	PUNCT
ejpam-6171	1142	25	pf	pf	INTJ
ejpam-6171	1142	26	;	;	PUNCT
ejpam-6171	1142	27	γ	γ	X
ejpam-6171	1142	28	)	)	PUNCT
ejpam-6171	1142	29	.	.	PUNCT
ejpam-6171	1143	1	by	by	ADP
ejpam-6171	1143	2	the	the	DET
ejpam-6171	1143	3	assumption	assumption	NOUN
ejpam-6171	1143	4	,	,	PUNCT
ejpam-6171	1143	5	we	we	PRON
ejpam-6171	1143	6	have	have	VERB
ejpam-6171	1143	7	u	u	NOUN
ejpam-6171	1143	8	+	+	CCONJ
ejpam-6171	1143	9	(	(	PUNCT
ejpam-6171	1143	10	pf	pf	INTJ
ejpam-6171	1143	11	;	;	PUNCT
ejpam-6171	1143	12	γ	γ	X
ejpam-6171	1143	13	)	)	PUNCT
ejpam-6171	1143	14	is	be	AUX
ejpam-6171	1143	15	an	an	DET
ejpam-6171	1143	16	iup	iup	NOUN
ejpam-6171	1143	17	-	-	PUNCT
ejpam-6171	1143	18	subalgebra	subalgebra	NOUN
ejpam-6171	1143	19	.	.	PUNCT
ejpam-6171	1144	1	by	by	ADP
ejpam-6171	1144	2	(	(	PUNCT
ejpam-6171	1144	3	2.17	2.17	NUM
ejpam-6171	1144	4	)	)	PUNCT
ejpam-6171	1144	5	,	,	PUNCT
ejpam-6171	1144	6	we	we	PRON
ejpam-6171	1144	7	have	have	VERB
ejpam-6171	1144	8	x	x	PUNCT
ejpam-6171	1144	9	?	?	PUNCT
ejpam-6171	1145	1	y	y	PROPN
ejpam-6171	1145	2	∈	∈	PROPN
ejpam-6171	1145	3	u	u	PROPN
ejpam-6171	1145	4	+	+	X
ejpam-6171	1145	5	(	(	PUNCT
ejpam-6171	1145	6	pf	pf	INTJ
ejpam-6171	1145	7	;	;	PUNCT
ejpam-6171	1145	8	γ	γ	X
ejpam-6171	1145	9	)	)	PUNCT
ejpam-6171	1145	10	.	.	PUNCT
ejpam-6171	1146	1	so	so	ADV
ejpam-6171	1146	2	pf	pf	INTJ
ejpam-6171	1146	3	(	(	PUNCT
ejpam-6171	1146	4	x	x	PROPN
ejpam-6171	1146	5	?	?	PUNCT
ejpam-6171	1147	1	y	y	X
ejpam-6171	1147	2	)	)	PUNCT
ejpam-6171	1147	3	>	>	X
ejpam-6171	1148	1	γ	γ	X
ejpam-6171	1148	2	=	=	SYM
ejpam-6171	1148	3	pf	pf	PROPN
ejpam-6171	1148	4	(	(	PUNCT
ejpam-6171	1148	5	x	x	PROPN
ejpam-6171	1148	6	?	?	PUNCT
ejpam-6171	1149	1	y	y	X
ejpam-6171	1149	2	)	)	PUNCT
ejpam-6171	1149	3	,	,	PUNCT
ejpam-6171	1149	4	which	which	PRON
ejpam-6171	1149	5	is	be	AUX
ejpam-6171	1149	6	a	a	DET
ejpam-6171	1149	7	contradiction	contradiction	NOUN
ejpam-6171	1149	8	.	.	PUNCT
ejpam-6171	1150	1	thus	thus	ADV
ejpam-6171	1150	2	,	,	PUNCT
ejpam-6171	1150	3	pf	pf	PROPN
ejpam-6171	1150	4	(	(	PUNCT
ejpam-6171	1150	5	x	x	PROPN
ejpam-6171	1150	6	?	?	PUNCT
ejpam-6171	1150	7	y	y	X
ejpam-6171	1150	8	)	)	PUNCT
ejpam-6171	1150	9	≥	≥	PROPN
ejpam-6171	1150	10	min{pf	min{pf	PRON
ejpam-6171	1150	11	(	(	PUNCT
ejpam-6171	1150	12	x),pf	x),pf	PROPN
ejpam-6171	1150	13	(	(	PUNCT
ejpam-6171	1150	14	y	y	NOUN
ejpam-6171	1150	15	)	)	PUNCT
ejpam-6171	1150	16	}	}	PUNCT
ejpam-6171	1150	17	.	.	PUNCT
ejpam-6171	1151	1	hence	hence	ADV
ejpam-6171	1151	2	,	,	PUNCT
ejpam-6171	1151	3	p	p	PROPN
ejpam-6171	1151	4	is	be	AUX
ejpam-6171	1151	5	a	a	DET
ejpam-6171	1151	6	pythagorean	pythagorean	PROPN
ejpam-6171	1151	7	neutrosophic	neutrosophic	ADJ
ejpam-6171	1151	8	iup	iup	NOUN
ejpam-6171	1151	9	-	-	PUNCT
ejpam-6171	1151	10	subalgebra	subalgebra	NOUN
ejpam-6171	1151	11	of	of	ADP
ejpam-6171	1151	12	x.	x.	PROPN
ejpam-6171	1151	13	k.	k.	PROPN
ejpam-6171	1151	14	suayngam	suayngam	PROPN
ejpam-6171	1151	15	et	et	PROPN
ejpam-6171	1151	16	al	al	PROPN
ejpam-6171	1151	17	.	.	PUNCT
ejpam-6171	1151	18	/	/	SYM
ejpam-6171	1151	19	eur	eur	PROPN
ejpam-6171	1151	20	.	.	PUNCT
ejpam-6171	1152	1	j.	j.	PROPN
ejpam-6171	1152	2	pure	pure	PROPN
ejpam-6171	1152	3	appl	appl	PROPN
ejpam-6171	1152	4	.	.	PROPN
ejpam-6171	1152	5	math	math	PROPN
ejpam-6171	1152	6	,	,	PUNCT
ejpam-6171	1152	7	18	18	NUM
ejpam-6171	1152	8	(	(	PUNCT
ejpam-6171	1152	9	3	3	NUM
ejpam-6171	1152	10	)	)	PUNCT
ejpam-6171	1152	11	(	(	PUNCT
ejpam-6171	1152	12	2025	2025	NUM
ejpam-6171	1152	13	)	)	PUNCT
ejpam-6171	1152	14	,	,	PUNCT
ejpam-6171	1152	15	6171	6171	NUM
ejpam-6171	1152	16	24	24	NUM
ejpam-6171	1152	17	of	of	ADP
ejpam-6171	1152	18	28	28	NUM
ejpam-6171	1152	19	theorem	theorem	NOUN
ejpam-6171	1152	20	23	23	NUM
ejpam-6171	1152	21	.	.	PUNCT
ejpam-6171	1153	1	a	a	DET
ejpam-6171	1153	2	pns	pns	NOUN
ejpam-6171	1153	3	p	p	NOUN
ejpam-6171	1153	4	in	in	ADP
ejpam-6171	1153	5	x	x	PROPN
ejpam-6171	1153	6	is	be	AUX
ejpam-6171	1153	7	a	a	DET
ejpam-6171	1153	8	pythagorean	pythagorean	PROPN
ejpam-6171	1153	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1153	10	iup	iup	PROPN
ejpam-6171	1153	11	-	-	PUNCT
ejpam-6171	1153	12	ideal	ideal	NOUN
ejpam-6171	1153	13	of	of	ADP
ejpam-6171	1153	14	x	x	SYM
ejpam-6171	1153	15	if	if	SCONJ
ejpam-6171	1153	16	and	and	CCONJ
ejpam-6171	1153	17	only	only	ADV
ejpam-6171	1153	18	if	if	SCONJ
ejpam-6171	1153	19	for	for	ADP
ejpam-6171	1153	20	all	all	DET
ejpam-6171	1153	21	α	α	NOUN
ejpam-6171	1153	22	,	,	PUNCT
ejpam-6171	1153	23	β	β	X
ejpam-6171	1153	24	,	,	PUNCT
ejpam-6171	1153	25	γ	γ	PROPN
ejpam-6171	1153	26	∈	∈	PROPN
ejpam-6171	1154	1	[	[	X
ejpam-6171	1154	2	0	0	NUM
ejpam-6171	1154	3	,	,	PUNCT
ejpam-6171	1154	4	1	1	NUM
ejpam-6171	1154	5	]	]	PUNCT
ejpam-6171	1154	6	,	,	PUNCT
ejpam-6171	1154	7	the	the	DET
ejpam-6171	1154	8	sets	set	NOUN
ejpam-6171	1154	9	u	u	NOUN
ejpam-6171	1154	10	+	+	X
ejpam-6171	1154	11	(	(	PUNCT
ejpam-6171	1154	12	pt	pt	INTJ
ejpam-6171	1154	13	;	;	PUNCT
ejpam-6171	1154	14	α	α	X
ejpam-6171	1154	15	)	)	PUNCT
ejpam-6171	1154	16	,	,	PUNCT
ejpam-6171	1154	17	l	l	NOUN
ejpam-6171	1154	18	−	−	PROPN
ejpam-6171	1154	19	(	(	PUNCT
ejpam-6171	1154	20	pi	pi	NOUN
ejpam-6171	1154	21	;	;	PUNCT
ejpam-6171	1154	22	β	β	X
ejpam-6171	1154	23	)	)	PUNCT
ejpam-6171	1154	24	,	,	PUNCT
ejpam-6171	1154	25	and	and	CCONJ
ejpam-6171	1154	26	u	u	NOUN
ejpam-6171	1154	27	+	+	CCONJ
ejpam-6171	1154	28	(	(	PUNCT
ejpam-6171	1154	29	pf	pf	INTJ
ejpam-6171	1154	30	;	;	PUNCT
ejpam-6171	1154	31	γ	γ	X
ejpam-6171	1154	32	)	)	PUNCT
ejpam-6171	1154	33	are	be	AUX
ejpam-6171	1154	34	either	either	CCONJ
ejpam-6171	1154	35	empty	empty	ADJ
ejpam-6171	1154	36	or	or	CCONJ
ejpam-6171	1154	37	iup	iup	NOUN
ejpam-6171	1154	38	-	-	PUNCT
ejpam-6171	1154	39	ideals	ideal	NOUN
ejpam-6171	1154	40	of	of	ADP
ejpam-6171	1154	41	x.	x.	NOUN
ejpam-6171	1154	42	proof	proof	NOUN
ejpam-6171	1154	43	.	.	PUNCT
ejpam-6171	1155	1	assume	assume	VERB
ejpam-6171	1155	2	that	that	SCONJ
ejpam-6171	1155	3	p	p	NOUN
ejpam-6171	1155	4	in	in	ADP
ejpam-6171	1155	5	x	x	PROPN
ejpam-6171	1155	6	is	be	AUX
ejpam-6171	1155	7	a	a	DET
ejpam-6171	1155	8	pythagorean	pythagorean	PROPN
ejpam-6171	1155	9	grneutrosophic	grneutrosophic	ADJ
ejpam-6171	1155	10	iup	iup	NOUN
ejpam-6171	1155	11	-	-	PUNCT
ejpam-6171	1155	12	ideal	ideal	NOUN
ejpam-6171	1155	13	of	of	ADP
ejpam-6171	1155	14	x.	x.	NOUN
ejpam-6171	1155	15	let	let	VERB
ejpam-6171	1155	16	α	α	PRON
ejpam-6171	1155	17	∈	∈	PROPN
ejpam-6171	1156	1	[	[	X
ejpam-6171	1156	2	0	0	NUM
ejpam-6171	1156	3	,	,	PUNCT
ejpam-6171	1156	4	1	1	NUM
ejpam-6171	1156	5	]	]	PUNCT
ejpam-6171	1156	6	be	be	AUX
ejpam-6171	1156	7	such	such	ADJ
ejpam-6171	1156	8	that	that	SCONJ
ejpam-6171	1156	9	u	u	NOUN
ejpam-6171	1156	10	+	+	X
ejpam-6171	1156	11	(	(	PUNCT
ejpam-6171	1156	12	pt	pt	INTJ
ejpam-6171	1156	13	;	;	PUNCT
ejpam-6171	1156	14	α	α	X
ejpam-6171	1156	15	)	)	PUNCT
ejpam-6171	1156	16	6=	6=	ADP
ejpam-6171	1156	17	∅.	∅.	ADV
ejpam-6171	1156	18	let	let	VERB
ejpam-6171	1156	19	a	a	DET
ejpam-6171	1156	20	∈	∈	PROPN
ejpam-6171	1156	21	u	u	NOUN
ejpam-6171	1156	22	+	+	X
ejpam-6171	1156	23	(	(	PUNCT
ejpam-6171	1156	24	pt	pt	INTJ
ejpam-6171	1156	25	;	;	PUNCT
ejpam-6171	1156	26	α	α	X
ejpam-6171	1156	27	)	)	PUNCT
ejpam-6171	1156	28	.	.	PUNCT
ejpam-6171	1157	1	then	then	ADV
ejpam-6171	1157	2	pt	pt	X
ejpam-6171	1157	3	(	(	PUNCT
ejpam-6171	1157	4	a	a	PROPN
ejpam-6171	1157	5	)	)	PUNCT
ejpam-6171	1157	6	>	>	X
ejpam-6171	1157	7	α	α	X
ejpam-6171	1157	8	.	.	PUNCT
ejpam-6171	1158	1	by	by	ADP
ejpam-6171	1158	2	(	(	PUNCT
ejpam-6171	1158	3	3.5	3.5	NUM
ejpam-6171	1158	4	)	)	PUNCT
ejpam-6171	1158	5	,	,	PUNCT
ejpam-6171	1158	6	we	we	PRON
ejpam-6171	1158	7	have	have	VERB
ejpam-6171	1158	8	pt	pt	X
ejpam-6171	1158	9	(	(	PUNCT
ejpam-6171	1158	10	0	0	NUM
ejpam-6171	1158	11	)	)	PUNCT
ejpam-6171	1158	12	≥	≥	NOUN
ejpam-6171	1158	13	pt	pt	INTJ
ejpam-6171	1158	14	(	(	PUNCT
ejpam-6171	1158	15	a	a	NOUN
ejpam-6171	1158	16	)	)	PUNCT
ejpam-6171	1158	17	>	>	X
ejpam-6171	1159	1	α	α	X
ejpam-6171	1159	2	.	.	PUNCT
ejpam-6171	1160	1	thus	thus	ADV
ejpam-6171	1160	2	,	,	PUNCT
ejpam-6171	1160	3	0	0	NUM
ejpam-6171	1160	4	∈	∈	PROPN
ejpam-6171	1160	5	u	u	NOUN
ejpam-6171	1160	6	+	+	X
ejpam-6171	1160	7	(	(	PUNCT
ejpam-6171	1160	8	pt	pt	INTJ
ejpam-6171	1160	9	;	;	PUNCT
ejpam-6171	1160	10	α	α	X
ejpam-6171	1160	11	)	)	PUNCT
ejpam-6171	1160	12	.	.	PUNCT
ejpam-6171	1161	1	let	let	VERB
ejpam-6171	1161	2	x	x	PRON
ejpam-6171	1161	3	,	,	PUNCT
ejpam-6171	1161	4	y	y	PROPN
ejpam-6171	1161	5	,	,	PUNCT
ejpam-6171	1161	6	z	z	PROPN
ejpam-6171	1161	7	∈	∈	PROPN
ejpam-6171	1161	8	u	u	NOUN
ejpam-6171	1161	9	+	+	X
ejpam-6171	1161	10	(	(	PUNCT
ejpam-6171	1161	11	pt	pt	INTJ
ejpam-6171	1161	12	;	;	PUNCT
ejpam-6171	1161	13	α	α	X
ejpam-6171	1161	14	)	)	PUNCT
ejpam-6171	1161	15	be	be	VERB
ejpam-6171	1161	16	such	such	ADJ
ejpam-6171	1161	17	that	that	PRON
ejpam-6171	1161	18	x	x	PUNCT
ejpam-6171	1161	19	?	?	PUNCT
ejpam-6171	1162	1	(	(	PUNCT
ejpam-6171	1162	2	y	y	NOUN
ejpam-6171	1162	3	?	?	PUNCT
ejpam-6171	1163	1	z	z	X
ejpam-6171	1163	2	)	)	PUNCT
ejpam-6171	1163	3	,	,	PUNCT
ejpam-6171	1163	4	y	y	PROPN
ejpam-6171	1163	5	∈	∈	PROPN
ejpam-6171	1163	6	u	u	PROPN
ejpam-6171	1163	7	+	+	X
ejpam-6171	1163	8	(	(	PUNCT
ejpam-6171	1163	9	pt	pt	INTJ
ejpam-6171	1163	10	;	;	PUNCT
ejpam-6171	1163	11	α	α	X
ejpam-6171	1163	12	)	)	PUNCT
ejpam-6171	1163	13	.	.	PUNCT
ejpam-6171	1164	1	then	then	ADV
ejpam-6171	1164	2	pt	pt	INTJ
ejpam-6171	1164	3	(	(	PUNCT
ejpam-6171	1164	4	x	x	X
ejpam-6171	1164	5	?	?	PUNCT
ejpam-6171	1165	1	(	(	PUNCT
ejpam-6171	1165	2	y	y	NOUN
ejpam-6171	1165	3	?	?	PUNCT
ejpam-6171	1166	1	z	z	X
ejpam-6171	1166	2	)	)	PUNCT
ejpam-6171	1166	3	)	)	PUNCT
ejpam-6171	1167	1	>	>	X
ejpam-6171	1167	2	α	α	PROPN
ejpam-6171	1167	3	and	and	CCONJ
ejpam-6171	1167	4	pt	pt	PROPN
ejpam-6171	1167	5	(	(	PUNCT
ejpam-6171	1167	6	y	y	NOUN
ejpam-6171	1167	7	)	)	PUNCT
ejpam-6171	1167	8	>	>	X
ejpam-6171	1168	1	α	α	X
ejpam-6171	1168	2	.	.	PUNCT
ejpam-6171	1169	1	thus	thus	ADV
ejpam-6171	1169	2	,	,	PUNCT
ejpam-6171	1169	3	min{pt	min{pt	PUNCT
ejpam-6171	1169	4	(	(	PUNCT
ejpam-6171	1169	5	x?(y?z)),pt	x?(y?z)),pt	PROPN
ejpam-6171	1169	6	(	(	PUNCT
ejpam-6171	1169	7	y	y	NOUN
ejpam-6171	1169	8	)	)	PUNCT
ejpam-6171	1169	9	}	}	PUNCT
ejpam-6171	1169	10	>	>	X
ejpam-6171	1169	11	α	α	X
ejpam-6171	1169	12	.	.	PUNCT
ejpam-6171	1169	13	by	by	ADP
ejpam-6171	1169	14	(	(	PUNCT
ejpam-6171	1169	15	3.8	3.8	NUM
ejpam-6171	1169	16	)	)	PUNCT
ejpam-6171	1169	17	,	,	PUNCT
ejpam-6171	1169	18	we	we	PRON
ejpam-6171	1169	19	have	have	VERB
ejpam-6171	1169	20	pt	pt	X
ejpam-6171	1169	21	(	(	PUNCT
ejpam-6171	1169	22	x?z	x?z	PROPN
ejpam-6171	1169	23	)	)	PUNCT
ejpam-6171	1169	24	≥	≥	NOUN
ejpam-6171	1169	25	min{pt	min{pt	X
ejpam-6171	1169	26	(	(	PUNCT
ejpam-6171	1169	27	x?(y?z)),pt	x?(y?z)),pt	PROPN
ejpam-6171	1169	28	(	(	PUNCT
ejpam-6171	1169	29	y	y	NOUN
ejpam-6171	1169	30	)	)	PUNCT
ejpam-6171	1169	31	}	}	PUNCT
ejpam-6171	1169	32	>	>	X
ejpam-6171	1170	1	α	α	X
ejpam-6171	1170	2	.	.	PUNCT
ejpam-6171	1171	1	thus	thus	ADV
ejpam-6171	1171	2	,	,	PUNCT
ejpam-6171	1171	3	x	x	PUNCT
ejpam-6171	1171	4	?	?	PUNCT
ejpam-6171	1171	5	z	z	PUNCT
ejpam-6171	1171	6	∈	∈	PROPN
ejpam-6171	1171	7	u	u	NOUN
ejpam-6171	1171	8	+	+	X
ejpam-6171	1171	9	(	(	PUNCT
ejpam-6171	1171	10	pt	pt	INTJ
ejpam-6171	1171	11	;	;	PUNCT
ejpam-6171	1171	12	α	α	X
ejpam-6171	1171	13	)	)	PUNCT
ejpam-6171	1171	14	.	.	PUNCT
ejpam-6171	1172	1	hence	hence	ADV
ejpam-6171	1172	2	,	,	PUNCT
ejpam-6171	1172	3	u+	u+	NUM
ejpam-6171	1172	4	(	(	PUNCT
ejpam-6171	1172	5	pt	pt	X
ejpam-6171	1172	6	;	;	PUNCT
ejpam-6171	1172	7	α	α	X
ejpam-6171	1172	8	)	)	PUNCT
ejpam-6171	1172	9	is	be	AUX
ejpam-6171	1172	10	an	an	DET
ejpam-6171	1172	11	iup	iup	NOUN
ejpam-6171	1172	12	-	-	PUNCT
ejpam-6171	1172	13	ideal	ideal	NOUN
ejpam-6171	1172	14	of	of	ADP
ejpam-6171	1172	15	x.	x.	NOUN
ejpam-6171	1172	16	let	let	VERB
ejpam-6171	1172	17	β	β	X
ejpam-6171	1172	18	∈	∈	PROPN
ejpam-6171	1173	1	[	[	X
ejpam-6171	1173	2	0	0	NUM
ejpam-6171	1173	3	,	,	PUNCT
ejpam-6171	1173	4	1	1	NUM
ejpam-6171	1173	5	]	]	PUNCT
ejpam-6171	1173	6	be	be	AUX
ejpam-6171	1173	7	such	such	ADJ
ejpam-6171	1173	8	that	that	SCONJ
ejpam-6171	1173	9	l	l	NOUN
ejpam-6171	1173	10	−	−	PROPN
ejpam-6171	1173	11	(	(	PUNCT
ejpam-6171	1173	12	pi	pi	NOUN
ejpam-6171	1173	13	;	;	PUNCT
ejpam-6171	1173	14	β	β	X
ejpam-6171	1173	15	)	)	PUNCT
ejpam-6171	1173	16	6=	6=	ADP
ejpam-6171	1173	17	∅.	∅.	AUX
ejpam-6171	1173	18	let	let	VERB
ejpam-6171	1173	19	b	b	NOUN
ejpam-6171	1173	20	∈	∈	PROPN
ejpam-6171	1173	21	l	l	NOUN
ejpam-6171	1173	22	−	−	PROPN
ejpam-6171	1174	1	(	(	PUNCT
ejpam-6171	1174	2	pi	pi	NOUN
ejpam-6171	1174	3	;	;	PUNCT
ejpam-6171	1174	4	β	β	X
ejpam-6171	1174	5	)	)	PUNCT
ejpam-6171	1174	6	.	.	PUNCT
ejpam-6171	1175	1	then	then	ADV
ejpam-6171	1175	2	pi(b	pi(b	PUNCT
ejpam-6171	1175	3	)	)	PUNCT
ejpam-6171	1175	4	<	<	X
ejpam-6171	1175	5	β	β	X
ejpam-6171	1175	6	.	.	PUNCT
ejpam-6171	1176	1	by	by	ADP
ejpam-6171	1176	2	(	(	PUNCT
ejpam-6171	1176	3	3.6	3.6	NUM
ejpam-6171	1176	4	)	)	PUNCT
ejpam-6171	1176	5	,	,	PUNCT
ejpam-6171	1176	6	we	we	PRON
ejpam-6171	1176	7	have	have	VERB
ejpam-6171	1176	8	pi(0	pi(0	PROPN
ejpam-6171	1176	9	)	)	PUNCT
ejpam-6171	1176	10	≤	≤	NOUN
ejpam-6171	1176	11	pi(b	pi(b	PUNCT
ejpam-6171	1176	12	)	)	PUNCT
ejpam-6171	1176	13	<	<	X
ejpam-6171	1176	14	β	β	X
ejpam-6171	1176	15	.	.	PUNCT
ejpam-6171	1177	1	thus	thus	ADV
ejpam-6171	1177	2	,	,	PUNCT
ejpam-6171	1177	3	0	0	NUM
ejpam-6171	1177	4	∈	∈	PROPN
ejpam-6171	1177	5	l	l	NOUN
ejpam-6171	1177	6	−	−	PROPN
ejpam-6171	1177	7	(	(	PUNCT
ejpam-6171	1177	8	pi	pi	NOUN
ejpam-6171	1177	9	;	;	PUNCT
ejpam-6171	1177	10	β	β	X
ejpam-6171	1177	11	)	)	PUNCT
ejpam-6171	1177	12	.	.	PUNCT
ejpam-6171	1178	1	let	let	VERB
ejpam-6171	1178	2	x	x	PRON
ejpam-6171	1178	3	,	,	PUNCT
ejpam-6171	1178	4	y	y	PROPN
ejpam-6171	1178	5	,	,	PUNCT
ejpam-6171	1178	6	z	z	NOUN
ejpam-6171	1178	7	∈	∈	PROPN
ejpam-6171	1178	8	l	l	NOUN
ejpam-6171	1178	9	−	−	PROPN
ejpam-6171	1178	10	(	(	PUNCT
ejpam-6171	1178	11	pi	pi	NOUN
ejpam-6171	1178	12	;	;	PUNCT
ejpam-6171	1178	13	β	β	AUX
ejpam-6171	1178	14	)	)	PUNCT
ejpam-6171	1178	15	be	be	AUX
ejpam-6171	1178	16	such	such	ADJ
ejpam-6171	1178	17	that	that	PRON
ejpam-6171	1178	18	x	x	PUNCT
ejpam-6171	1178	19	?	?	PUNCT
ejpam-6171	1179	1	(	(	PUNCT
ejpam-6171	1179	2	y	y	NOUN
ejpam-6171	1179	3	?	?	PUNCT
ejpam-6171	1180	1	z	z	X
ejpam-6171	1180	2	)	)	PUNCT
ejpam-6171	1180	3	,	,	PUNCT
ejpam-6171	1180	4	y	y	PROPN
ejpam-6171	1180	5	∈	∈	PROPN
ejpam-6171	1180	6	l	l	NOUN
ejpam-6171	1181	1	−	−	PROPN
ejpam-6171	1181	2	(	(	PUNCT
ejpam-6171	1181	3	pi	pi	NOUN
ejpam-6171	1181	4	;	;	PUNCT
ejpam-6171	1181	5	β	β	X
ejpam-6171	1181	6	)	)	PUNCT
ejpam-6171	1181	7	.	.	PUNCT
ejpam-6171	1182	1	then	then	ADV
ejpam-6171	1182	2	pi(x	pi(x	PROPN
ejpam-6171	1182	3	?	?	PUNCT
ejpam-6171	1183	1	(	(	PUNCT
ejpam-6171	1183	2	y	y	NOUN
ejpam-6171	1183	3	?	?	PUNCT
ejpam-6171	1184	1	z	z	X
ejpam-6171	1184	2	)	)	PUNCT
ejpam-6171	1184	3	)	)	PUNCT
ejpam-6171	1185	1	<	<	X
ejpam-6171	1185	2	β	β	X
ejpam-6171	1185	3	and	and	CCONJ
ejpam-6171	1185	4	pt	pt	PROPN
ejpam-6171	1185	5	(	(	PUNCT
ejpam-6171	1185	6	y	y	NOUN
ejpam-6171	1185	7	)	)	PUNCT
ejpam-6171	1185	8	<	<	X
ejpam-6171	1185	9	β	β	X
ejpam-6171	1185	10	.	.	PUNCT
ejpam-6171	1186	1	thus	thus	ADV
ejpam-6171	1186	2	,	,	PUNCT
ejpam-6171	1186	3	max{pi(x?(y?z)),pi(y	max{pi(x?(y?z)),pi(y	NUM
ejpam-6171	1186	4	)	)	PUNCT
ejpam-6171	1186	5	}	}	PUNCT
ejpam-6171	1186	6	<	<	X
ejpam-6171	1186	7	β	β	X
ejpam-6171	1186	8	.	.	PUNCT
ejpam-6171	1186	9	by	by	ADP
ejpam-6171	1186	10	(	(	PUNCT
ejpam-6171	1186	11	3.9	3.9	NUM
ejpam-6171	1186	12	)	)	PUNCT
ejpam-6171	1186	13	,	,	PUNCT
ejpam-6171	1186	14	we	we	PRON
ejpam-6171	1186	15	have	have	VERB
ejpam-6171	1186	16	pi(x?z	pi(x?z	NOUN
ejpam-6171	1186	17	)	)	PUNCT
ejpam-6171	1186	18	≤	≤	NOUN
ejpam-6171	1186	19	max{pi(x?(y?z)),pi(y	max{pi(x?(y?z)),pi(y	NUM
ejpam-6171	1186	20	)	)	PUNCT
ejpam-6171	1186	21	}	}	PUNCT
ejpam-6171	1186	22	>	>	X
ejpam-6171	1187	1	β	β	X
ejpam-6171	1187	2	.	.	PUNCT
ejpam-6171	1188	1	thus	thus	ADV
ejpam-6171	1188	2	,	,	PUNCT
ejpam-6171	1188	3	x	x	PUNCT
ejpam-6171	1188	4	?	?	PUNCT
ejpam-6171	1189	1	z	z	PUNCT
ejpam-6171	1189	2	∈	∈	PROPN
ejpam-6171	1189	3	l	l	NOUN
ejpam-6171	1189	4	−	−	PROPN
ejpam-6171	1190	1	(	(	PUNCT
ejpam-6171	1190	2	pi	pi	NOUN
ejpam-6171	1190	3	;	;	PUNCT
ejpam-6171	1190	4	β	β	X
ejpam-6171	1190	5	)	)	PUNCT
ejpam-6171	1190	6	.	.	PUNCT
ejpam-6171	1191	1	hence	hence	ADV
ejpam-6171	1191	2	,	,	PUNCT
ejpam-6171	1191	3	l−	l−	PROPN
ejpam-6171	1191	4	(	(	PUNCT
ejpam-6171	1191	5	pi	pi	NOUN
ejpam-6171	1191	6	;	;	PUNCT
ejpam-6171	1191	7	β	β	X
ejpam-6171	1191	8	)	)	PUNCT
ejpam-6171	1191	9	is	be	AUX
ejpam-6171	1191	10	an	an	DET
ejpam-6171	1191	11	iup	iup	NOUN
ejpam-6171	1191	12	-	-	PUNCT
ejpam-6171	1191	13	ideal	ideal	NOUN
ejpam-6171	1191	14	of	of	ADP
ejpam-6171	1191	15	x.	x.	NOUN
ejpam-6171	1191	16	let	let	VERB
ejpam-6171	1191	17	γ	γ	X
ejpam-6171	1191	18	∈	∈	PROPN
ejpam-6171	1192	1	[	[	X
ejpam-6171	1192	2	0	0	NUM
ejpam-6171	1192	3	,	,	PUNCT
ejpam-6171	1192	4	1	1	NUM
ejpam-6171	1192	5	]	]	PUNCT
ejpam-6171	1192	6	be	be	AUX
ejpam-6171	1192	7	such	such	ADJ
ejpam-6171	1192	8	that	that	SCONJ
ejpam-6171	1192	9	u	u	NOUN
ejpam-6171	1192	10	+	+	X
ejpam-6171	1192	11	(	(	PUNCT
ejpam-6171	1192	12	pf	pf	INTJ
ejpam-6171	1192	13	;	;	PUNCT
ejpam-6171	1192	14	γ	γ	X
ejpam-6171	1192	15	)	)	PUNCT
ejpam-6171	1192	16	6=	6=	ADP
ejpam-6171	1193	1	∅.	∅.	AUX
ejpam-6171	1193	2	let	let	VERB
ejpam-6171	1193	3	c	c	NOUN
ejpam-6171	1193	4	∈	∈	PROPN
ejpam-6171	1193	5	u	u	NOUN
ejpam-6171	1193	6	+	+	X
ejpam-6171	1193	7	(	(	PUNCT
ejpam-6171	1193	8	pf	pf	INTJ
ejpam-6171	1193	9	;	;	PUNCT
ejpam-6171	1193	10	γ	γ	X
ejpam-6171	1193	11	)	)	PUNCT
ejpam-6171	1193	12	.	.	PUNCT
ejpam-6171	1194	1	then	then	ADV
ejpam-6171	1194	2	pf	pf	PROPN
ejpam-6171	1194	3	(	(	PUNCT
ejpam-6171	1194	4	c	c	NOUN
ejpam-6171	1194	5	)	)	PUNCT
ejpam-6171	1194	6	>	>	X
ejpam-6171	1194	7	γ	γ	X
ejpam-6171	1194	8	.	.	PROPN
ejpam-6171	1194	9	by	by	ADP
ejpam-6171	1194	10	(	(	PUNCT
ejpam-6171	1194	11	3.7	3.7	NUM
ejpam-6171	1194	12	)	)	PUNCT
ejpam-6171	1194	13	,	,	PUNCT
ejpam-6171	1194	14	we	we	PRON
ejpam-6171	1194	15	have	have	VERB
ejpam-6171	1194	16	pf	pf	PROPN
ejpam-6171	1194	17	(	(	PUNCT
ejpam-6171	1194	18	0	0	NUM
ejpam-6171	1194	19	)	)	PUNCT
ejpam-6171	1194	20	≥	≥	NOUN
ejpam-6171	1194	21	pf	pf	X
ejpam-6171	1194	22	(	(	PUNCT
ejpam-6171	1194	23	c	c	NOUN
ejpam-6171	1194	24	)	)	PUNCT
ejpam-6171	1194	25	>	>	X
ejpam-6171	1195	1	γ	γ	X
ejpam-6171	1195	2	.	.	PUNCT
ejpam-6171	1195	3	thus	thus	ADV
ejpam-6171	1195	4	,	,	PUNCT
ejpam-6171	1195	5	0	0	NUM
ejpam-6171	1195	6	∈	∈	PROPN
ejpam-6171	1195	7	u	u	NOUN
ejpam-6171	1195	8	+	+	X
ejpam-6171	1195	9	(	(	PUNCT
ejpam-6171	1195	10	pf	pf	INTJ
ejpam-6171	1195	11	;	;	PUNCT
ejpam-6171	1195	12	γ	γ	X
ejpam-6171	1195	13	)	)	PUNCT
ejpam-6171	1195	14	.	.	PUNCT
ejpam-6171	1196	1	let	let	VERB
ejpam-6171	1196	2	x	x	PRON
ejpam-6171	1196	3	,	,	PUNCT
ejpam-6171	1196	4	y	y	PROPN
ejpam-6171	1196	5	,	,	PUNCT
ejpam-6171	1196	6	z	z	PROPN
ejpam-6171	1196	7	∈	∈	PROPN
ejpam-6171	1196	8	u	u	NOUN
ejpam-6171	1196	9	+	+	X
ejpam-6171	1196	10	(	(	PUNCT
ejpam-6171	1196	11	pf	pf	INTJ
ejpam-6171	1196	12	;	;	PUNCT
ejpam-6171	1196	13	γ	γ	X
ejpam-6171	1196	14	)	)	PUNCT
ejpam-6171	1196	15	be	be	VERB
ejpam-6171	1196	16	such	such	ADJ
ejpam-6171	1196	17	that	that	PRON
ejpam-6171	1196	18	x?(y?z	x?(y?z	PROPN
ejpam-6171	1196	19	)	)	PUNCT
ejpam-6171	1196	20	,	,	PUNCT
ejpam-6171	1196	21	y	y	PROPN
ejpam-6171	1196	22	∈	∈	PROPN
ejpam-6171	1196	23	u	u	NOUN
ejpam-6171	1196	24	+	+	X
ejpam-6171	1196	25	(	(	PUNCT
ejpam-6171	1196	26	pf	pf	INTJ
ejpam-6171	1196	27	;	;	PUNCT
ejpam-6171	1196	28	γ	γ	X
ejpam-6171	1196	29	)	)	PUNCT
ejpam-6171	1196	30	.	.	PUNCT
ejpam-6171	1197	1	then	then	ADV
ejpam-6171	1197	2	pf	pf	PROPN
ejpam-6171	1197	3	(	(	PUNCT
ejpam-6171	1197	4	x?(y?z	x?(y?z	NUM
ejpam-6171	1197	5	)	)	PUNCT
ejpam-6171	1197	6	)	)	PUNCT
ejpam-6171	1197	7	>	>	X
ejpam-6171	1197	8	γ	γ	PROPN
ejpam-6171	1197	9	and	and	CCONJ
ejpam-6171	1197	10	pf	pf	PROPN
ejpam-6171	1197	11	(	(	PUNCT
ejpam-6171	1197	12	y	y	NOUN
ejpam-6171	1197	13	)	)	PUNCT
ejpam-6171	1197	14	>	>	X
ejpam-6171	1198	1	γ	γ	X
ejpam-6171	1198	2	.	.	PUNCT
ejpam-6171	1199	1	thus	thus	ADV
ejpam-6171	1199	2	,	,	PUNCT
ejpam-6171	1199	3	min{pf	min{pf	PRON
ejpam-6171	1199	4	(	(	PUNCT
ejpam-6171	1199	5	x	x	X
ejpam-6171	1199	6	?	?	PUNCT
ejpam-6171	1200	1	(	(	PUNCT
ejpam-6171	1200	2	y	y	NOUN
ejpam-6171	1200	3	?	?	PUNCT
ejpam-6171	1201	1	z)),pf	z)),pf	PROPN
ejpam-6171	1201	2	(	(	PUNCT
ejpam-6171	1201	3	y	y	NOUN
ejpam-6171	1201	4	)	)	PUNCT
ejpam-6171	1201	5	}	}	PUNCT
ejpam-6171	1201	6	>	>	X
ejpam-6171	1201	7	γ	γ	X
ejpam-6171	1201	8	.	.	PUNCT
ejpam-6171	1201	9	by	by	ADP
ejpam-6171	1201	10	(	(	PUNCT
ejpam-6171	1201	11	3.10	3.10	NUM
ejpam-6171	1201	12	)	)	PUNCT
ejpam-6171	1201	13	,	,	PUNCT
ejpam-6171	1201	14	we	we	PRON
ejpam-6171	1201	15	have	have	VERB
ejpam-6171	1201	16	pf	pf	NOUN
ejpam-6171	1201	17	(	(	PUNCT
ejpam-6171	1201	18	x	x	NOUN
ejpam-6171	1201	19	?	?	PUNCT
ejpam-6171	1202	1	z	z	X
ejpam-6171	1202	2	)	)	PUNCT
ejpam-6171	1202	3	≥	≥	NOUN
ejpam-6171	1202	4	min{pf	min{pf	X
ejpam-6171	1202	5	(	(	PUNCT
ejpam-6171	1202	6	x	x	X
ejpam-6171	1202	7	?	?	PUNCT
ejpam-6171	1203	1	(	(	PUNCT
ejpam-6171	1203	2	y	y	NOUN
ejpam-6171	1203	3	?	?	PUNCT
ejpam-6171	1204	1	z)),pf	z)),pf	PROPN
ejpam-6171	1204	2	(	(	PUNCT
ejpam-6171	1204	3	y	y	NOUN
ejpam-6171	1204	4	)	)	PUNCT
ejpam-6171	1204	5	}	}	PUNCT
ejpam-6171	1204	6	>	>	X
ejpam-6171	1204	7	γ	γ	X
ejpam-6171	1204	8	.	.	PUNCT
ejpam-6171	1204	9	thus	thus	ADV
ejpam-6171	1204	10	,	,	PUNCT
ejpam-6171	1204	11	x	x	PUNCT
ejpam-6171	1204	12	?	?	PUNCT
ejpam-6171	1204	13	z	z	PUNCT
ejpam-6171	1204	14	∈	∈	PROPN
ejpam-6171	1204	15	u	u	NOUN
ejpam-6171	1204	16	+	+	X
ejpam-6171	1204	17	(	(	PUNCT
ejpam-6171	1204	18	pf	pf	INTJ
ejpam-6171	1204	19	;	;	PUNCT
ejpam-6171	1204	20	γ	γ	X
ejpam-6171	1204	21	)	)	PUNCT
ejpam-6171	1204	22	.	.	PUNCT
ejpam-6171	1205	1	hence	hence	ADV
ejpam-6171	1205	2	,	,	PUNCT
ejpam-6171	1205	3	u+	u+	NUM
ejpam-6171	1205	4	(	(	PUNCT
ejpam-6171	1205	5	pf	pf	X
ejpam-6171	1205	6	;	;	PUNCT
ejpam-6171	1205	7	γ	γ	X
ejpam-6171	1205	8	)	)	PUNCT
ejpam-6171	1205	9	is	be	AUX
ejpam-6171	1205	10	an	an	DET
ejpam-6171	1205	11	iup	iup	NOUN
ejpam-6171	1205	12	-	-	PUNCT
ejpam-6171	1205	13	ideal	ideal	NOUN
ejpam-6171	1205	14	of	of	ADP
ejpam-6171	1205	15	x.	x.	NOUN
ejpam-6171	1205	16	conversely	conversely	ADV
ejpam-6171	1205	17	,	,	PUNCT
ejpam-6171	1205	18	assume	assume	VERB
ejpam-6171	1205	19	that	that	SCONJ
ejpam-6171	1205	20	for	for	ADP
ejpam-6171	1205	21	all	all	DET
ejpam-6171	1205	22	α	α	NOUN
ejpam-6171	1205	23	,	,	PUNCT
ejpam-6171	1205	24	β	β	X
ejpam-6171	1205	25	,	,	PUNCT
ejpam-6171	1205	26	γ	γ	PROPN
ejpam-6171	1205	27	∈	∈	PROPN
ejpam-6171	1206	1	[	[	X
ejpam-6171	1206	2	0	0	NUM
ejpam-6171	1206	3	,	,	PUNCT
ejpam-6171	1206	4	1	1	NUM
ejpam-6171	1206	5	]	]	PUNCT
ejpam-6171	1206	6	,	,	PUNCT
ejpam-6171	1206	7	the	the	DET
ejpam-6171	1206	8	sets	set	NOUN
ejpam-6171	1206	9	u	u	NOUN
ejpam-6171	1206	10	+	+	X
ejpam-6171	1206	11	(	(	PUNCT
ejpam-6171	1206	12	pt	pt	INTJ
ejpam-6171	1206	13	;	;	PUNCT
ejpam-6171	1206	14	α	α	X
ejpam-6171	1206	15	)	)	PUNCT
ejpam-6171	1206	16	,	,	PUNCT
ejpam-6171	1206	17	l	l	NOUN
ejpam-6171	1206	18	−	−	PROPN
ejpam-6171	1206	19	(	(	PUNCT
ejpam-6171	1206	20	pi	pi	NOUN
ejpam-6171	1206	21	;	;	PUNCT
ejpam-6171	1206	22	β	β	X
ejpam-6171	1206	23	)	)	PUNCT
ejpam-6171	1206	24	,	,	PUNCT
ejpam-6171	1206	25	and	and	CCONJ
ejpam-6171	1206	26	u	u	NOUN
ejpam-6171	1206	27	+	+	CCONJ
ejpam-6171	1206	28	(	(	PUNCT
ejpam-6171	1206	29	pf	pf	INTJ
ejpam-6171	1206	30	;	;	PUNCT
ejpam-6171	1206	31	γ	γ	X
ejpam-6171	1206	32	)	)	PUNCT
ejpam-6171	1206	33	are	be	AUX
ejpam-6171	1206	34	either	either	CCONJ
ejpam-6171	1206	35	empty	empty	ADJ
ejpam-6171	1206	36	or	or	CCONJ
ejpam-6171	1206	37	iup	iup	NOUN
ejpam-6171	1206	38	-	-	PUNCT
ejpam-6171	1206	39	ideals	ideal	NOUN
ejpam-6171	1206	40	of	of	ADP
ejpam-6171	1206	41	x.	x.	NOUN
ejpam-6171	1206	42	let	let	VERB
ejpam-6171	1206	43	x	x	SYM
ejpam-6171	1206	44	∈	∈	PROPN
ejpam-6171	1206	45	x.	x.	NOUN
ejpam-6171	1206	46	assume	assume	VERB
ejpam-6171	1206	47	that	that	SCONJ
ejpam-6171	1206	48	pt	pt	X
ejpam-6171	1206	49	(	(	PUNCT
ejpam-6171	1206	50	0	0	NUM
ejpam-6171	1206	51	)	)	PUNCT
ejpam-6171	1206	52	<	<	X
ejpam-6171	1206	53	pt	pt	X
ejpam-6171	1206	54	(	(	PUNCT
ejpam-6171	1206	55	x	x	NOUN
ejpam-6171	1206	56	)	)	PUNCT
ejpam-6171	1206	57	.	.	PUNCT
ejpam-6171	1207	1	let	let	VERB
ejpam-6171	1207	2	α	α	NOUN
ejpam-6171	1207	3	=	=	SYM
ejpam-6171	1207	4	pt	pt	X
ejpam-6171	1207	5	(	(	PUNCT
ejpam-6171	1207	6	0	0	NUM
ejpam-6171	1207	7	)	)	PUNCT
ejpam-6171	1207	8	.	.	PUNCT
ejpam-6171	1208	1	then	then	ADV
ejpam-6171	1208	2	x	x	SYM
ejpam-6171	1208	3	∈	∈	PROPN
ejpam-6171	1208	4	u	u	NOUN
ejpam-6171	1208	5	+	+	X
ejpam-6171	1208	6	(	(	PUNCT
ejpam-6171	1208	7	pt	pt	INTJ
ejpam-6171	1208	8	;	;	PUNCT
ejpam-6171	1208	9	α	α	X
ejpam-6171	1208	10	)	)	PUNCT
ejpam-6171	1208	11	6=	6=	ADP
ejpam-6171	1208	12	∅.	∅.	ADP
ejpam-6171	1208	13	by	by	ADP
ejpam-6171	1208	14	the	the	DET
ejpam-6171	1208	15	assumption	assumption	NOUN
ejpam-6171	1208	16	,	,	PUNCT
ejpam-6171	1208	17	we	we	PRON
ejpam-6171	1208	18	have	have	VERB
ejpam-6171	1208	19	u	u	NOUN
ejpam-6171	1208	20	+	+	CCONJ
ejpam-6171	1208	21	(	(	PUNCT
ejpam-6171	1208	22	pt	pt	INTJ
ejpam-6171	1208	23	;	;	PUNCT
ejpam-6171	1208	24	α	α	X
ejpam-6171	1208	25	)	)	PUNCT
ejpam-6171	1208	26	is	be	AUX
ejpam-6171	1208	27	an	an	DET
ejpam-6171	1208	28	iup	iup	NOUN
ejpam-6171	1208	29	-	-	PUNCT
ejpam-6171	1208	30	ideal	ideal	NOUN
ejpam-6171	1208	31	of	of	ADP
ejpam-6171	1208	32	x.	x.	NOUN
ejpam-6171	1208	33	by	by	ADP
ejpam-6171	1208	34	(	(	PUNCT
ejpam-6171	1208	35	2.18	2.18	NUM
ejpam-6171	1208	36	)	)	PUNCT
ejpam-6171	1208	37	,	,	PUNCT
ejpam-6171	1208	38	we	we	PRON
ejpam-6171	1208	39	have	have	VERB
ejpam-6171	1208	40	0	0	NUM
ejpam-6171	1208	41	∈	∈	PROPN
ejpam-6171	1208	42	u	u	NOUN
ejpam-6171	1208	43	+	+	X
ejpam-6171	1208	44	(	(	PUNCT
ejpam-6171	1208	45	pt	pt	INTJ
ejpam-6171	1208	46	;	;	PUNCT
ejpam-6171	1208	47	α	α	X
ejpam-6171	1208	48	)	)	PUNCT
ejpam-6171	1208	49	.	.	PUNCT
ejpam-6171	1209	1	so	so	ADV
ejpam-6171	1209	2	pt	pt	INTJ
ejpam-6171	1209	3	(	(	PUNCT
ejpam-6171	1209	4	0	0	NUM
ejpam-6171	1209	5	)	)	PUNCT
ejpam-6171	1209	6	>	>	X
ejpam-6171	1210	1	α	α	X
ejpam-6171	1210	2	=	=	SYM
ejpam-6171	1210	3	pt	pt	X
ejpam-6171	1210	4	(	(	PUNCT
ejpam-6171	1210	5	0	0	NUM
ejpam-6171	1210	6	)	)	PUNCT
ejpam-6171	1210	7	,	,	PUNCT
ejpam-6171	1210	8	which	which	PRON
ejpam-6171	1210	9	is	be	AUX
ejpam-6171	1210	10	a	a	DET
ejpam-6171	1210	11	contradiction	contradiction	NOUN
ejpam-6171	1210	12	.	.	PUNCT
ejpam-6171	1211	1	thus	thus	ADV
ejpam-6171	1211	2	,	,	PUNCT
ejpam-6171	1211	3	pt	pt	X
ejpam-6171	1211	4	(	(	PUNCT
ejpam-6171	1211	5	0	0	NUM
ejpam-6171	1211	6	)	)	PUNCT
ejpam-6171	1211	7	≥	≥	NOUN
ejpam-6171	1211	8	pt	pt	INTJ
ejpam-6171	1211	9	(	(	PUNCT
ejpam-6171	1211	10	x	x	NOUN
ejpam-6171	1211	11	)	)	PUNCT
ejpam-6171	1211	12	.	.	PUNCT
ejpam-6171	1212	1	let	let	VERB
ejpam-6171	1212	2	x	x	PRON
ejpam-6171	1212	3	,	,	PUNCT
ejpam-6171	1212	4	y	y	PROPN
ejpam-6171	1212	5	,	,	PUNCT
ejpam-6171	1212	6	z	z	PROPN
ejpam-6171	1212	7	∈	∈	PROPN
ejpam-6171	1212	8	x.	x.	NOUN
ejpam-6171	1212	9	assume	assume	VERB
ejpam-6171	1212	10	that	that	SCONJ
ejpam-6171	1213	1	pt	pt	INTJ
ejpam-6171	1213	2	(	(	PUNCT
ejpam-6171	1213	3	x	x	NOUN
ejpam-6171	1213	4	?	?	PUNCT
ejpam-6171	1214	1	z	z	X
ejpam-6171	1214	2	)	)	PUNCT
ejpam-6171	1214	3	<	<	X
ejpam-6171	1214	4	min{pt	min{pt	X
ejpam-6171	1214	5	(	(	PUNCT
ejpam-6171	1214	6	x	x	X
ejpam-6171	1214	7	?	?	PUNCT
ejpam-6171	1215	1	(	(	PUNCT
ejpam-6171	1215	2	y	y	PROPN
ejpam-6171	1215	3	?	?	PUNCT
ejpam-6171	1216	1	z)),pt	z)),pt	PROPN
ejpam-6171	1216	2	(	(	PUNCT
ejpam-6171	1216	3	y	y	NOUN
ejpam-6171	1216	4	)	)	PUNCT
ejpam-6171	1216	5	}	}	PUNCT
ejpam-6171	1216	6	.	.	PUNCT
ejpam-6171	1217	1	let	let	VERB
ejpam-6171	1217	2	α	α	NOUN
ejpam-6171	1217	3	=	=	SYM
ejpam-6171	1217	4	pt	pt	X
ejpam-6171	1217	5	(	(	PUNCT
ejpam-6171	1217	6	x	x	NOUN
ejpam-6171	1217	7	?	?	PUNCT
ejpam-6171	1218	1	z	z	X
ejpam-6171	1218	2	)	)	PUNCT
ejpam-6171	1218	3	.	.	PUNCT
ejpam-6171	1219	1	then	then	ADV
ejpam-6171	1219	2	x	x	X
ejpam-6171	1219	3	?	?	PUNCT
ejpam-6171	1220	1	(	(	PUNCT
ejpam-6171	1220	2	y	y	NOUN
ejpam-6171	1220	3	?	?	PUNCT
ejpam-6171	1221	1	z	z	X
ejpam-6171	1221	2	)	)	PUNCT
ejpam-6171	1221	3	,	,	PUNCT
ejpam-6171	1221	4	y	y	PROPN
ejpam-6171	1221	5	∈	∈	PROPN
ejpam-6171	1221	6	u	u	PROPN
ejpam-6171	1221	7	+	+	X
ejpam-6171	1221	8	(	(	PUNCT
ejpam-6171	1221	9	pt	pt	INTJ
ejpam-6171	1221	10	;	;	PUNCT
ejpam-6171	1221	11	α	α	X
ejpam-6171	1221	12	)	)	PUNCT
ejpam-6171	1221	13	6=	6=	ADP
ejpam-6171	1221	14	∅.	∅.	ADP
ejpam-6171	1221	15	by	by	ADP
ejpam-6171	1221	16	the	the	DET
ejpam-6171	1221	17	assumption	assumption	NOUN
ejpam-6171	1221	18	,	,	PUNCT
ejpam-6171	1221	19	we	we	PRON
ejpam-6171	1221	20	have	have	VERB
ejpam-6171	1221	21	u	u	NOUN
ejpam-6171	1221	22	+	+	CCONJ
ejpam-6171	1221	23	(	(	PUNCT
ejpam-6171	1221	24	pt	pt	INTJ
ejpam-6171	1221	25	;	;	PUNCT
ejpam-6171	1221	26	α	α	X
ejpam-6171	1221	27	)	)	PUNCT
ejpam-6171	1221	28	is	be	AUX
ejpam-6171	1221	29	an	an	DET
ejpam-6171	1221	30	iup	iup	NOUN
ejpam-6171	1221	31	-	-	PUNCT
ejpam-6171	1221	32	ideal	ideal	NOUN
ejpam-6171	1221	33	of	of	ADP
ejpam-6171	1221	34	x.	x.	NOUN
ejpam-6171	1221	35	by	by	ADP
ejpam-6171	1221	36	(	(	PUNCT
ejpam-6171	1221	37	2.20	2.20	NUM
ejpam-6171	1221	38	)	)	PUNCT
ejpam-6171	1221	39	,	,	PUNCT
ejpam-6171	1221	40	we	we	PRON
ejpam-6171	1221	41	have	have	VERB
ejpam-6171	1221	42	x	x	X
ejpam-6171	1221	43	?	?	PUNCT
ejpam-6171	1222	1	z	z	PUNCT
ejpam-6171	1222	2	∈	∈	PROPN
ejpam-6171	1222	3	u	u	NOUN
ejpam-6171	1222	4	+	+	X
ejpam-6171	1222	5	(	(	PUNCT
ejpam-6171	1222	6	pt	pt	INTJ
ejpam-6171	1222	7	;	;	PUNCT
ejpam-6171	1222	8	α	α	X
ejpam-6171	1222	9	)	)	PUNCT
ejpam-6171	1222	10	.	.	PUNCT
ejpam-6171	1223	1	so	so	ADV
ejpam-6171	1223	2	pt	pt	INTJ
ejpam-6171	1223	3	(	(	PUNCT
ejpam-6171	1223	4	x	x	NOUN
ejpam-6171	1223	5	?	?	PUNCT
ejpam-6171	1224	1	z	z	X
ejpam-6171	1224	2	)	)	PUNCT
ejpam-6171	1224	3	>	>	X
ejpam-6171	1225	1	α	α	X
ejpam-6171	1225	2	=	=	SYM
ejpam-6171	1225	3	pt	pt	X
ejpam-6171	1225	4	(	(	PUNCT
ejpam-6171	1225	5	x	x	NOUN
ejpam-6171	1225	6	?	?	PUNCT
ejpam-6171	1226	1	z	z	X
ejpam-6171	1226	2	)	)	PUNCT
ejpam-6171	1226	3	,	,	PUNCT
ejpam-6171	1226	4	which	which	PRON
ejpam-6171	1226	5	is	be	AUX
ejpam-6171	1226	6	a	a	DET
ejpam-6171	1226	7	contradiction	contradiction	NOUN
ejpam-6171	1226	8	.	.	PUNCT
ejpam-6171	1227	1	thus	thus	ADV
ejpam-6171	1227	2	,	,	PUNCT
ejpam-6171	1227	3	pt	pt	INTJ
ejpam-6171	1227	4	(	(	PUNCT
ejpam-6171	1227	5	x	x	NOUN
ejpam-6171	1227	6	?	?	PUNCT
ejpam-6171	1227	7	z	z	X
ejpam-6171	1227	8	)	)	PUNCT
ejpam-6171	1227	9	≥	≥	NOUN
ejpam-6171	1227	10	min{pt	min{pt	X
ejpam-6171	1227	11	(	(	PUNCT
ejpam-6171	1227	12	x	x	X
ejpam-6171	1227	13	?	?	PUNCT
ejpam-6171	1228	1	(	(	PUNCT
ejpam-6171	1228	2	y	y	PROPN
ejpam-6171	1228	3	?	?	PUNCT
ejpam-6171	1229	1	z)),pt	z)),pt	PROPN
ejpam-6171	1229	2	(	(	PUNCT
ejpam-6171	1229	3	y	y	NOUN
ejpam-6171	1229	4	)	)	PUNCT
ejpam-6171	1229	5	}	}	PUNCT
ejpam-6171	1229	6	.	.	PUNCT
ejpam-6171	1230	1	let	let	VERB
ejpam-6171	1230	2	x	x	SYM
ejpam-6171	1230	3	∈	∈	PROPN
ejpam-6171	1230	4	x.	x.	NOUN
ejpam-6171	1230	5	assume	assume	VERB
ejpam-6171	1230	6	that	that	SCONJ
ejpam-6171	1230	7	pi(0	pi(0	PROPN
ejpam-6171	1230	8	)	)	PUNCT
ejpam-6171	1230	9	>	>	X
ejpam-6171	1230	10	pi(x	pi(x	PROPN
ejpam-6171	1230	11	)	)	PUNCT
ejpam-6171	1230	12	.	.	PUNCT
ejpam-6171	1231	1	let	let	VERB
ejpam-6171	1231	2	β	β	X
ejpam-6171	1231	3	=	=	SYM
ejpam-6171	1231	4	pi(0	pi(0	PROPN
ejpam-6171	1231	5	)	)	PUNCT
ejpam-6171	1231	6	.	.	PUNCT
ejpam-6171	1232	1	then	then	ADV
ejpam-6171	1232	2	x	x	SYM
ejpam-6171	1232	3	∈	∈	PROPN
ejpam-6171	1232	4	l	l	NOUN
ejpam-6171	1232	5	−	−	PROPN
ejpam-6171	1233	1	(	(	PUNCT
ejpam-6171	1233	2	pi	pi	NOUN
ejpam-6171	1233	3	;	;	PUNCT
ejpam-6171	1233	4	β	β	X
ejpam-6171	1233	5	)	)	PUNCT
ejpam-6171	1233	6	6=	6=	ADP
ejpam-6171	1233	7	∅.	∅.	ADP
ejpam-6171	1233	8	by	by	ADP
ejpam-6171	1233	9	the	the	DET
ejpam-6171	1233	10	assumption	assumption	NOUN
ejpam-6171	1233	11	,	,	PUNCT
ejpam-6171	1233	12	we	we	PRON
ejpam-6171	1233	13	have	have	VERB
ejpam-6171	1233	14	l	l	NOUN
ejpam-6171	1233	15	−	−	PROPN
ejpam-6171	1233	16	(	(	PUNCT
ejpam-6171	1233	17	pi	pi	NOUN
ejpam-6171	1233	18	;	;	PUNCT
ejpam-6171	1233	19	β	β	X
ejpam-6171	1233	20	)	)	PUNCT
ejpam-6171	1233	21	is	be	AUX
ejpam-6171	1233	22	an	an	DET
ejpam-6171	1233	23	iup	iup	NOUN
ejpam-6171	1233	24	-	-	PUNCT
ejpam-6171	1233	25	ideal	ideal	NOUN
ejpam-6171	1233	26	of	of	ADP
ejpam-6171	1233	27	x.	x.	NOUN
ejpam-6171	1233	28	by	by	ADP
ejpam-6171	1233	29	(	(	PUNCT
ejpam-6171	1233	30	2.18	2.18	NUM
ejpam-6171	1233	31	)	)	PUNCT
ejpam-6171	1233	32	,	,	PUNCT
ejpam-6171	1233	33	we	we	PRON
ejpam-6171	1233	34	have	have	VERB
ejpam-6171	1233	35	0	0	NUM
ejpam-6171	1233	36	∈	∈	NOUN
ejpam-6171	1233	37	l	l	NOUN
ejpam-6171	1233	38	−	−	PROPN
ejpam-6171	1234	1	(	(	PUNCT
ejpam-6171	1234	2	pi	pi	NOUN
ejpam-6171	1234	3	;	;	PUNCT
ejpam-6171	1234	4	β	β	X
ejpam-6171	1234	5	)	)	PUNCT
ejpam-6171	1234	6	.	.	PUNCT
ejpam-6171	1235	1	so	so	ADV
ejpam-6171	1235	2	pi(0	pi(0	PROPN
ejpam-6171	1235	3	)	)	PUNCT
ejpam-6171	1235	4	<	<	X
ejpam-6171	1235	5	β	β	X
ejpam-6171	1235	6	=	=	SYM
ejpam-6171	1235	7	pi(0	pi(0	PROPN
ejpam-6171	1235	8	)	)	PUNCT
ejpam-6171	1235	9	,	,	PUNCT
ejpam-6171	1235	10	which	which	PRON
ejpam-6171	1235	11	is	be	AUX
ejpam-6171	1235	12	a	a	DET
ejpam-6171	1235	13	contradiction	contradiction	NOUN
ejpam-6171	1235	14	.	.	PUNCT
ejpam-6171	1236	1	thus	thus	ADV
ejpam-6171	1236	2	,	,	PUNCT
ejpam-6171	1236	3	pi(0	pi(0	PROPN
ejpam-6171	1236	4	)	)	PUNCT
ejpam-6171	1236	5	≤	≤	NOUN
ejpam-6171	1236	6	pi(x	pi(x	NOUN
ejpam-6171	1236	7	)	)	PUNCT
ejpam-6171	1236	8	.	.	PUNCT
ejpam-6171	1237	1	let	let	VERB
ejpam-6171	1237	2	x	x	PRON
ejpam-6171	1237	3	,	,	PUNCT
ejpam-6171	1237	4	y	y	PROPN
ejpam-6171	1237	5	,	,	PUNCT
ejpam-6171	1237	6	z	z	PROPN
ejpam-6171	1237	7	∈	∈	PROPN
ejpam-6171	1237	8	x.	x.	NOUN
ejpam-6171	1237	9	assume	assume	VERB
ejpam-6171	1237	10	that	that	SCONJ
ejpam-6171	1237	11	pi(x	pi(x	NOUN
ejpam-6171	1237	12	?	?	PUNCT
ejpam-6171	1238	1	z	z	X
ejpam-6171	1238	2	)	)	PUNCT
ejpam-6171	1238	3	>	>	X
ejpam-6171	1238	4	max{pi(x	max{pi(x	PROPN
ejpam-6171	1238	5	?	?	PUNCT
ejpam-6171	1239	1	(	(	PUNCT
ejpam-6171	1239	2	y	y	NOUN
ejpam-6171	1239	3	?	?	PUNCT
ejpam-6171	1239	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	1239	5	)	)	PUNCT
ejpam-6171	1239	6	}	}	PUNCT
ejpam-6171	1239	7	.	.	PUNCT
ejpam-6171	1240	1	let	let	VERB
ejpam-6171	1240	2	β	β	X
ejpam-6171	1240	3	=	=	NOUN
ejpam-6171	1240	4	pi(x	pi(x	NOUN
ejpam-6171	1240	5	?	?	PUNCT
ejpam-6171	1241	1	z	z	X
ejpam-6171	1241	2	)	)	PUNCT
ejpam-6171	1241	3	.	.	PUNCT
ejpam-6171	1242	1	then	then	ADV
ejpam-6171	1242	2	x	x	X
ejpam-6171	1242	3	?	?	PUNCT
ejpam-6171	1243	1	(	(	PUNCT
ejpam-6171	1243	2	y	y	NOUN
ejpam-6171	1243	3	?	?	PUNCT
ejpam-6171	1243	4	z	z	X
ejpam-6171	1243	5	)	)	PUNCT
ejpam-6171	1243	6	,	,	PUNCT
ejpam-6171	1243	7	y	y	PROPN
ejpam-6171	1243	8	∈	∈	PROPN
ejpam-6171	1243	9	l	l	NOUN
ejpam-6171	1244	1	−	−	PROPN
ejpam-6171	1244	2	(	(	PUNCT
ejpam-6171	1244	3	pi	pi	NOUN
ejpam-6171	1244	4	;	;	PUNCT
ejpam-6171	1244	5	β	β	X
ejpam-6171	1244	6	)	)	PUNCT
ejpam-6171	1244	7	6=	6=	ADP
ejpam-6171	1244	8	∅.	∅.	ADP
ejpam-6171	1244	9	by	by	ADP
ejpam-6171	1244	10	the	the	DET
ejpam-6171	1244	11	assumption	assumption	NOUN
ejpam-6171	1244	12	,	,	PUNCT
ejpam-6171	1244	13	we	we	PRON
ejpam-6171	1244	14	have	have	VERB
ejpam-6171	1244	15	l	l	NOUN
ejpam-6171	1244	16	−	−	PROPN
ejpam-6171	1244	17	(	(	PUNCT
ejpam-6171	1244	18	pi	pi	NOUN
ejpam-6171	1244	19	;	;	PUNCT
ejpam-6171	1244	20	β	β	X
ejpam-6171	1244	21	)	)	PUNCT
ejpam-6171	1244	22	is	be	AUX
ejpam-6171	1244	23	an	an	DET
ejpam-6171	1244	24	iup	iup	NOUN
ejpam-6171	1244	25	-	-	PUNCT
ejpam-6171	1244	26	ideal	ideal	NOUN
ejpam-6171	1244	27	of	of	ADP
ejpam-6171	1244	28	x.	x.	NOUN
ejpam-6171	1244	29	by	by	ADP
ejpam-6171	1244	30	(	(	PUNCT
ejpam-6171	1244	31	2.20	2.20	NUM
ejpam-6171	1244	32	)	)	PUNCT
ejpam-6171	1244	33	,	,	PUNCT
ejpam-6171	1244	34	we	we	PRON
ejpam-6171	1244	35	have	have	VERB
ejpam-6171	1244	36	x	x	X
ejpam-6171	1244	37	?	?	PUNCT
ejpam-6171	1245	1	z	z	PUNCT
ejpam-6171	1245	2	∈	∈	PROPN
ejpam-6171	1245	3	l	l	NOUN
ejpam-6171	1245	4	−	−	PROPN
ejpam-6171	1246	1	(	(	PUNCT
ejpam-6171	1246	2	pi	pi	NOUN
ejpam-6171	1246	3	;	;	PUNCT
ejpam-6171	1246	4	β	β	X
ejpam-6171	1246	5	)	)	PUNCT
ejpam-6171	1246	6	.	.	PUNCT
ejpam-6171	1247	1	so	so	ADV
ejpam-6171	1247	2	pi(x	pi(x	ADJ
ejpam-6171	1247	3	?	?	PUNCT
ejpam-6171	1248	1	z	z	X
ejpam-6171	1248	2	)	)	PUNCT
ejpam-6171	1248	3	<	<	X
ejpam-6171	1248	4	β	β	X
ejpam-6171	1248	5	=	=	SYM
ejpam-6171	1248	6	pi(x	pi(x	NOUN
ejpam-6171	1248	7	?	?	PUNCT
ejpam-6171	1249	1	z	z	X
ejpam-6171	1249	2	)	)	PUNCT
ejpam-6171	1249	3	,	,	PUNCT
ejpam-6171	1249	4	which	which	PRON
ejpam-6171	1249	5	is	be	AUX
ejpam-6171	1249	6	a	a	DET
ejpam-6171	1249	7	contradiction	contradiction	NOUN
ejpam-6171	1249	8	.	.	PUNCT
ejpam-6171	1250	1	thus	thus	ADV
ejpam-6171	1250	2	,	,	PUNCT
ejpam-6171	1250	3	pi(x	pi(x	ADJ
ejpam-6171	1250	4	?	?	PUNCT
ejpam-6171	1251	1	z	z	X
ejpam-6171	1251	2	)	)	PUNCT
ejpam-6171	1251	3	≤	≤	NUM
ejpam-6171	1251	4	max{pi(x	max{pi(x	NOUN
ejpam-6171	1251	5	?	?	PUNCT
ejpam-6171	1252	1	(	(	PUNCT
ejpam-6171	1252	2	y	y	NOUN
ejpam-6171	1252	3	?	?	PUNCT
ejpam-6171	1252	4	z)),pi(y	z)),pi(y	NOUN
ejpam-6171	1252	5	)	)	PUNCT
ejpam-6171	1252	6	}	}	PUNCT
ejpam-6171	1252	7	.	.	PUNCT
ejpam-6171	1253	1	let	let	VERB
ejpam-6171	1253	2	x	x	SYM
ejpam-6171	1253	3	∈	∈	PROPN
ejpam-6171	1253	4	x.	x.	NOUN
ejpam-6171	1253	5	assume	assume	VERB
ejpam-6171	1253	6	that	that	SCONJ
ejpam-6171	1253	7	pf	pf	PROPN
ejpam-6171	1253	8	(	(	PUNCT
ejpam-6171	1253	9	0	0	NUM
ejpam-6171	1253	10	)	)	PUNCT
ejpam-6171	1253	11	<	<	X
ejpam-6171	1253	12	pf	pf	X
ejpam-6171	1253	13	(	(	PUNCT
ejpam-6171	1253	14	x	x	NOUN
ejpam-6171	1253	15	)	)	PUNCT
ejpam-6171	1253	16	.	.	PUNCT
ejpam-6171	1254	1	let	let	VERB
ejpam-6171	1254	2	γ	γ	X
ejpam-6171	1254	3	=	=	SYM
ejpam-6171	1254	4	pf	pf	X
ejpam-6171	1254	5	(	(	PUNCT
ejpam-6171	1254	6	0	0	NUM
ejpam-6171	1254	7	)	)	PUNCT
ejpam-6171	1254	8	.	.	PUNCT
ejpam-6171	1255	1	then	then	ADV
ejpam-6171	1255	2	x	x	SYM
ejpam-6171	1255	3	∈	∈	PROPN
ejpam-6171	1255	4	u	u	NOUN
ejpam-6171	1255	5	+	+	X
ejpam-6171	1255	6	(	(	PUNCT
ejpam-6171	1255	7	pf	pf	INTJ
ejpam-6171	1255	8	;	;	PUNCT
ejpam-6171	1255	9	γ	γ	X
ejpam-6171	1255	10	)	)	PUNCT
ejpam-6171	1255	11	6=	6=	ADP
ejpam-6171	1255	12	∅.	∅.	ADP
ejpam-6171	1255	13	by	by	ADP
ejpam-6171	1255	14	the	the	DET
ejpam-6171	1255	15	assumption	assumption	NOUN
ejpam-6171	1255	16	,	,	PUNCT
ejpam-6171	1255	17	we	we	PRON
ejpam-6171	1255	18	have	have	VERB
ejpam-6171	1255	19	u	u	NOUN
ejpam-6171	1255	20	+	+	CCONJ
ejpam-6171	1255	21	(	(	PUNCT
ejpam-6171	1255	22	pf	pf	INTJ
ejpam-6171	1255	23	;	;	PUNCT
ejpam-6171	1255	24	γ	γ	X
ejpam-6171	1255	25	)	)	PUNCT
ejpam-6171	1255	26	is	be	AUX
ejpam-6171	1255	27	an	an	DET
ejpam-6171	1255	28	iup	iup	NOUN
ejpam-6171	1255	29	-	-	PUNCT
ejpam-6171	1255	30	ideal	ideal	NOUN
ejpam-6171	1255	31	of	of	ADP
ejpam-6171	1255	32	x.	x.	NOUN
ejpam-6171	1255	33	by	by	ADP
ejpam-6171	1255	34	(	(	PUNCT
ejpam-6171	1255	35	2.18	2.18	NUM
ejpam-6171	1255	36	)	)	PUNCT
ejpam-6171	1255	37	,	,	PUNCT
ejpam-6171	1255	38	we	we	PRON
ejpam-6171	1255	39	have	have	VERB
ejpam-6171	1255	40	0	0	NUM
ejpam-6171	1255	41	∈	∈	PROPN
ejpam-6171	1255	42	u	u	NOUN
ejpam-6171	1255	43	+	+	X
ejpam-6171	1255	44	(	(	PUNCT
ejpam-6171	1255	45	pf	pf	INTJ
ejpam-6171	1255	46	;	;	PUNCT
ejpam-6171	1255	47	γ	γ	X
ejpam-6171	1255	48	)	)	PUNCT
ejpam-6171	1255	49	.	.	PUNCT
ejpam-6171	1256	1	so	so	ADV
ejpam-6171	1256	2	pf	pf	PROPN
ejpam-6171	1256	3	(	(	PUNCT
ejpam-6171	1256	4	0	0	NUM
ejpam-6171	1256	5	)	)	PUNCT
ejpam-6171	1256	6	>	>	X
ejpam-6171	1257	1	γ	γ	X
ejpam-6171	1257	2	=	=	SYM
ejpam-6171	1257	3	pf	pf	PROPN
ejpam-6171	1257	4	(	(	PUNCT
ejpam-6171	1257	5	0	0	NUM
ejpam-6171	1257	6	)	)	PUNCT
ejpam-6171	1257	7	,	,	PUNCT
ejpam-6171	1257	8	which	which	PRON
ejpam-6171	1257	9	is	be	AUX
ejpam-6171	1257	10	a	a	DET
ejpam-6171	1257	11	contradiction	contradiction	NOUN
ejpam-6171	1257	12	.	.	PUNCT
ejpam-6171	1258	1	thus	thus	ADV
ejpam-6171	1258	2	,	,	PUNCT
ejpam-6171	1258	3	pf	pf	PROPN
ejpam-6171	1258	4	(	(	PUNCT
ejpam-6171	1258	5	0	0	NUM
ejpam-6171	1258	6	)	)	PUNCT
ejpam-6171	1258	7	≥	≥	NOUN
ejpam-6171	1258	8	pf	pf	X
ejpam-6171	1258	9	(	(	PUNCT
ejpam-6171	1258	10	x	x	NOUN
ejpam-6171	1258	11	)	)	PUNCT
ejpam-6171	1258	12	.	.	PUNCT
ejpam-6171	1259	1	let	let	VERB
ejpam-6171	1259	2	x	x	PRON
ejpam-6171	1259	3	,	,	PUNCT
ejpam-6171	1259	4	y	y	PROPN
ejpam-6171	1259	5	,	,	PUNCT
ejpam-6171	1259	6	z	z	PROPN
ejpam-6171	1259	7	∈	∈	PROPN
ejpam-6171	1259	8	x.	x.	NOUN
ejpam-6171	1259	9	assume	assume	VERB
ejpam-6171	1259	10	that	that	SCONJ
ejpam-6171	1259	11	pf	pf	PROPN
ejpam-6171	1259	12	(	(	PUNCT
ejpam-6171	1259	13	x?z	x?z	PROPN
ejpam-6171	1259	14	)	)	PUNCT
ejpam-6171	1259	15	<	<	X
ejpam-6171	1259	16	min{pf	min{pf	X
ejpam-6171	1259	17	(	(	PUNCT
ejpam-6171	1259	18	x?(y?z)),pf	x?(y?z)),pf	PROPN
ejpam-6171	1259	19	(	(	PUNCT
ejpam-6171	1259	20	y	y	NOUN
ejpam-6171	1259	21	)	)	PUNCT
ejpam-6171	1259	22	}	}	PUNCT
ejpam-6171	1259	23	.	.	PUNCT
ejpam-6171	1260	1	let	let	VERB
ejpam-6171	1260	2	γ	γ	X
ejpam-6171	1260	3	=	=	SYM
ejpam-6171	1260	4	pf	pf	X
ejpam-6171	1260	5	(	(	PUNCT
ejpam-6171	1260	6	x?z	x?z	PROPN
ejpam-6171	1260	7	)	)	PUNCT
ejpam-6171	1260	8	.	.	PUNCT
ejpam-6171	1261	1	then	then	ADV
ejpam-6171	1261	2	x	x	X
ejpam-6171	1261	3	?	?	PUNCT
ejpam-6171	1262	1	(	(	PUNCT
ejpam-6171	1262	2	y	y	NOUN
ejpam-6171	1262	3	?	?	PUNCT
ejpam-6171	1263	1	z	z	X
ejpam-6171	1263	2	)	)	PUNCT
ejpam-6171	1263	3	,	,	PUNCT
ejpam-6171	1263	4	y	y	PROPN
ejpam-6171	1263	5	∈	∈	PROPN
ejpam-6171	1263	6	u	u	NOUN
ejpam-6171	1263	7	+	+	X
ejpam-6171	1263	8	(	(	PUNCT
ejpam-6171	1263	9	pf	pf	INTJ
ejpam-6171	1263	10	;	;	PUNCT
ejpam-6171	1263	11	γ	γ	X
ejpam-6171	1263	12	)	)	PUNCT
ejpam-6171	1263	13	6=	6=	ADP
ejpam-6171	1263	14	∅.	∅.	ADP
ejpam-6171	1263	15	by	by	ADP
ejpam-6171	1263	16	the	the	DET
ejpam-6171	1263	17	assumption	assumption	NOUN
ejpam-6171	1263	18	,	,	PUNCT
ejpam-6171	1263	19	we	we	PRON
ejpam-6171	1263	20	have	have	VERB
ejpam-6171	1263	21	u	u	NOUN
ejpam-6171	1263	22	+	+	CCONJ
ejpam-6171	1263	23	(	(	PUNCT
ejpam-6171	1263	24	pf	pf	INTJ
ejpam-6171	1263	25	;	;	PUNCT
ejpam-6171	1263	26	γ	γ	X
ejpam-6171	1263	27	)	)	PUNCT
ejpam-6171	1263	28	is	be	AUX
ejpam-6171	1263	29	an	an	DET
ejpam-6171	1263	30	iup	iup	NOUN
ejpam-6171	1263	31	-	-	PUNCT
ejpam-6171	1263	32	ideal	ideal	NOUN
ejpam-6171	1263	33	of	of	ADP
ejpam-6171	1263	34	x.	x.	NOUN
ejpam-6171	1263	35	by	by	ADP
ejpam-6171	1263	36	(	(	PUNCT
ejpam-6171	1263	37	2.20	2.20	NUM
ejpam-6171	1263	38	)	)	PUNCT
ejpam-6171	1263	39	,	,	PUNCT
ejpam-6171	1263	40	we	we	PRON
ejpam-6171	1263	41	have	have	VERB
ejpam-6171	1263	42	x	x	X
ejpam-6171	1263	43	?	?	PUNCT
ejpam-6171	1264	1	z	z	PUNCT
ejpam-6171	1264	2	∈	∈	PROPN
ejpam-6171	1264	3	u	u	NOUN
ejpam-6171	1264	4	+	+	X
ejpam-6171	1264	5	(	(	PUNCT
ejpam-6171	1264	6	pf	pf	INTJ
ejpam-6171	1264	7	;	;	PUNCT
ejpam-6171	1264	8	γ	γ	X
ejpam-6171	1264	9	)	)	PUNCT
ejpam-6171	1264	10	.	.	PUNCT
ejpam-6171	1265	1	so	so	ADV
ejpam-6171	1265	2	pf	pf	INTJ
ejpam-6171	1265	3	(	(	PUNCT
ejpam-6171	1265	4	x	x	PROPN
ejpam-6171	1265	5	?	?	PUNCT
ejpam-6171	1266	1	z	z	X
ejpam-6171	1266	2	)	)	PUNCT
ejpam-6171	1266	3	>	>	X
ejpam-6171	1266	4	γ	γ	X
ejpam-6171	1266	5	=	=	SYM
ejpam-6171	1266	6	pf	pf	PROPN
ejpam-6171	1266	7	(	(	PUNCT
ejpam-6171	1266	8	x	x	PROPN
ejpam-6171	1266	9	?	?	PUNCT
ejpam-6171	1267	1	z	z	X
ejpam-6171	1267	2	)	)	PUNCT
ejpam-6171	1267	3	,	,	PUNCT
ejpam-6171	1267	4	which	which	PRON
ejpam-6171	1267	5	is	be	AUX
ejpam-6171	1267	6	a	a	DET
ejpam-6171	1267	7	contradiction	contradiction	NOUN
ejpam-6171	1267	8	.	.	PUNCT
ejpam-6171	1268	1	thus	thus	ADV
ejpam-6171	1268	2	,	,	PUNCT
ejpam-6171	1268	3	pf	pf	PROPN
ejpam-6171	1268	4	(	(	PUNCT
ejpam-6171	1268	5	x	x	PROPN
ejpam-6171	1268	6	?	?	PUNCT
ejpam-6171	1268	7	z	z	X
ejpam-6171	1268	8	)	)	PUNCT
ejpam-6171	1268	9	≥	≥	NOUN
ejpam-6171	1268	10	min{pf	min{pf	X
ejpam-6171	1268	11	(	(	PUNCT
ejpam-6171	1268	12	x	x	X
ejpam-6171	1268	13	?	?	PUNCT
ejpam-6171	1269	1	(	(	PUNCT
ejpam-6171	1269	2	y	y	NOUN
ejpam-6171	1269	3	?	?	PUNCT
ejpam-6171	1270	1	z)),pf	z)),pf	PROPN
ejpam-6171	1270	2	(	(	PUNCT
ejpam-6171	1270	3	y	y	NOUN
ejpam-6171	1270	4	)	)	PUNCT
ejpam-6171	1270	5	}	}	PUNCT
ejpam-6171	1270	6	.	.	PUNCT
ejpam-6171	1271	1	k.	k.	PROPN
ejpam-6171	1272	1	suayngam	suayngam	PROPN
ejpam-6171	1272	2	et	et	PROPN
ejpam-6171	1272	3	al	al	PROPN
ejpam-6171	1272	4	.	.	PUNCT
ejpam-6171	1272	5	/	/	SYM
ejpam-6171	1272	6	eur	eur	PROPN
ejpam-6171	1272	7	.	.	PUNCT
ejpam-6171	1273	1	j.	j.	PROPN
ejpam-6171	1273	2	pure	pure	PROPN
ejpam-6171	1273	3	appl	appl	PROPN
ejpam-6171	1273	4	.	.	PROPN
ejpam-6171	1273	5	math	math	PROPN
ejpam-6171	1273	6	,	,	PUNCT
ejpam-6171	1273	7	18	18	NUM
ejpam-6171	1273	8	(	(	PUNCT
ejpam-6171	1273	9	3	3	NUM
ejpam-6171	1273	10	)	)	PUNCT
ejpam-6171	1273	11	(	(	PUNCT
ejpam-6171	1273	12	2025	2025	NUM
ejpam-6171	1273	13	)	)	PUNCT
ejpam-6171	1273	14	,	,	PUNCT
ejpam-6171	1273	15	6171	6171	NUM
ejpam-6171	1273	16	25	25	NUM
ejpam-6171	1273	17	of	of	ADP
ejpam-6171	1273	18	28	28	NUM
ejpam-6171	1273	19	hence	hence	ADV
ejpam-6171	1273	20	,	,	PUNCT
ejpam-6171	1273	21	p	p	PROPN
ejpam-6171	1273	22	is	be	AUX
ejpam-6171	1273	23	a	a	DET
ejpam-6171	1273	24	neutrosophic	neutrosophic	ADJ
ejpam-6171	1273	25	iup	iup	NOUN
ejpam-6171	1273	26	-	-	PUNCT
ejpam-6171	1273	27	ideal	ideal	NOUN
ejpam-6171	1273	28	of	of	ADP
ejpam-6171	1273	29	x.	x.	PROPN
ejpam-6171	1273	30	theorem	theorem	VERB
ejpam-6171	1273	31	24	24	NUM
ejpam-6171	1273	32	.	.	PUNCT
ejpam-6171	1274	1	a	a	DET
ejpam-6171	1274	2	pns	pns	NOUN
ejpam-6171	1274	3	p	p	NOUN
ejpam-6171	1274	4	in	in	ADP
ejpam-6171	1274	5	x	x	PROPN
ejpam-6171	1274	6	is	be	AUX
ejpam-6171	1274	7	a	a	DET
ejpam-6171	1274	8	pythagorean	pythagorean	PROPN
ejpam-6171	1274	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1274	10	iup	iup	NOUN
ejpam-6171	1274	11	-	-	PUNCT
ejpam-6171	1274	12	filter	filter	NOUN
ejpam-6171	1274	13	of	of	ADP
ejpam-6171	1274	14	x	x	SYM
ejpam-6171	1274	15	if	if	SCONJ
ejpam-6171	1274	16	and	and	CCONJ
ejpam-6171	1274	17	only	only	ADV
ejpam-6171	1274	18	if	if	SCONJ
ejpam-6171	1274	19	for	for	ADP
ejpam-6171	1274	20	all	all	DET
ejpam-6171	1274	21	α	α	NOUN
ejpam-6171	1274	22	,	,	PUNCT
ejpam-6171	1274	23	β	β	X
ejpam-6171	1274	24	,	,	PUNCT
ejpam-6171	1274	25	γ	γ	PROPN
ejpam-6171	1274	26	∈	∈	PROPN
ejpam-6171	1275	1	[	[	X
ejpam-6171	1275	2	0	0	NUM
ejpam-6171	1275	3	,	,	PUNCT
ejpam-6171	1275	4	1	1	NUM
ejpam-6171	1275	5	]	]	PUNCT
ejpam-6171	1275	6	,	,	PUNCT
ejpam-6171	1275	7	the	the	DET
ejpam-6171	1275	8	sets	set	NOUN
ejpam-6171	1275	9	u	u	NOUN
ejpam-6171	1275	10	+	+	X
ejpam-6171	1275	11	(	(	PUNCT
ejpam-6171	1275	12	pt	pt	INTJ
ejpam-6171	1275	13	;	;	PUNCT
ejpam-6171	1275	14	α	α	X
ejpam-6171	1275	15	)	)	PUNCT
ejpam-6171	1275	16	,	,	PUNCT
ejpam-6171	1275	17	l	l	NOUN
ejpam-6171	1275	18	−	−	PROPN
ejpam-6171	1275	19	(	(	PUNCT
ejpam-6171	1275	20	pi	pi	NOUN
ejpam-6171	1275	21	;	;	PUNCT
ejpam-6171	1275	22	β	β	X
ejpam-6171	1275	23	)	)	PUNCT
ejpam-6171	1275	24	,	,	PUNCT
ejpam-6171	1275	25	and	and	CCONJ
ejpam-6171	1275	26	u	u	NOUN
ejpam-6171	1275	27	+	+	CCONJ
ejpam-6171	1275	28	(	(	PUNCT
ejpam-6171	1275	29	pf	pf	INTJ
ejpam-6171	1275	30	;	;	PUNCT
ejpam-6171	1275	31	γ	γ	X
ejpam-6171	1275	32	)	)	PUNCT
ejpam-6171	1275	33	are	be	AUX
ejpam-6171	1275	34	either	either	CCONJ
ejpam-6171	1275	35	empty	empty	ADJ
ejpam-6171	1275	36	or	or	CCONJ
ejpam-6171	1275	37	iup	iup	NOUN
ejpam-6171	1275	38	-	-	PUNCT
ejpam-6171	1275	39	filters	filter	NOUN
ejpam-6171	1275	40	of	of	ADP
ejpam-6171	1275	41	x.	x.	NOUN
ejpam-6171	1275	42	proof	proof	PROPN
ejpam-6171	1275	43	.	.	PUNCT
ejpam-6171	1276	1	assume	assume	VERB
ejpam-6171	1276	2	that	that	SCONJ
ejpam-6171	1276	3	p	p	NOUN
ejpam-6171	1276	4	in	in	ADP
ejpam-6171	1276	5	x	x	PROPN
ejpam-6171	1276	6	is	be	AUX
ejpam-6171	1276	7	a	a	DET
ejpam-6171	1276	8	pythagorean	pythagorean	PROPN
ejpam-6171	1276	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1276	10	iup	iup	NOUN
ejpam-6171	1276	11	-	-	PUNCT
ejpam-6171	1276	12	filter	filter	NOUN
ejpam-6171	1276	13	of	of	ADP
ejpam-6171	1276	14	x.	x.	NOUN
ejpam-6171	1276	15	let	let	VERB
ejpam-6171	1276	16	α	α	PRON
ejpam-6171	1276	17	∈	∈	PROPN
ejpam-6171	1277	1	[	[	X
ejpam-6171	1277	2	0	0	NUM
ejpam-6171	1277	3	,	,	PUNCT
ejpam-6171	1277	4	1	1	NUM
ejpam-6171	1277	5	]	]	PUNCT
ejpam-6171	1277	6	be	be	AUX
ejpam-6171	1277	7	such	such	ADJ
ejpam-6171	1277	8	that	that	SCONJ
ejpam-6171	1277	9	u	u	NOUN
ejpam-6171	1277	10	+	+	X
ejpam-6171	1277	11	(	(	PUNCT
ejpam-6171	1277	12	pt	pt	INTJ
ejpam-6171	1277	13	;	;	PUNCT
ejpam-6171	1277	14	α	α	X
ejpam-6171	1277	15	)	)	PUNCT
ejpam-6171	1277	16	6=	6=	ADP
ejpam-6171	1277	17	∅.	∅.	ADV
ejpam-6171	1277	18	let	let	VERB
ejpam-6171	1277	19	a	a	DET
ejpam-6171	1277	20	∈	∈	PROPN
ejpam-6171	1277	21	u	u	NOUN
ejpam-6171	1277	22	+	+	X
ejpam-6171	1277	23	(	(	PUNCT
ejpam-6171	1277	24	pt	pt	INTJ
ejpam-6171	1277	25	;	;	PUNCT
ejpam-6171	1277	26	α	α	X
ejpam-6171	1277	27	)	)	PUNCT
ejpam-6171	1277	28	.	.	PUNCT
ejpam-6171	1278	1	then	then	ADV
ejpam-6171	1278	2	pt	pt	X
ejpam-6171	1278	3	(	(	PUNCT
ejpam-6171	1278	4	a	a	PROPN
ejpam-6171	1278	5	)	)	PUNCT
ejpam-6171	1278	6	>	>	X
ejpam-6171	1278	7	α	α	X
ejpam-6171	1278	8	.	.	PUNCT
ejpam-6171	1279	1	by	by	ADP
ejpam-6171	1279	2	(	(	PUNCT
ejpam-6171	1279	3	3.5	3.5	NUM
ejpam-6171	1279	4	)	)	PUNCT
ejpam-6171	1279	5	,	,	PUNCT
ejpam-6171	1279	6	we	we	PRON
ejpam-6171	1279	7	have	have	VERB
ejpam-6171	1279	8	pt	pt	X
ejpam-6171	1279	9	(	(	PUNCT
ejpam-6171	1279	10	0	0	NUM
ejpam-6171	1279	11	)	)	PUNCT
ejpam-6171	1279	12	≥	≥	NOUN
ejpam-6171	1279	13	pt	pt	INTJ
ejpam-6171	1279	14	(	(	PUNCT
ejpam-6171	1279	15	a	a	NOUN
ejpam-6171	1279	16	)	)	PUNCT
ejpam-6171	1279	17	>	>	X
ejpam-6171	1280	1	α	α	X
ejpam-6171	1280	2	.	.	PUNCT
ejpam-6171	1281	1	thus	thus	ADV
ejpam-6171	1281	2	,	,	PUNCT
ejpam-6171	1281	3	0	0	NUM
ejpam-6171	1281	4	∈	∈	PROPN
ejpam-6171	1281	5	u	u	NOUN
ejpam-6171	1281	6	+	+	X
ejpam-6171	1281	7	(	(	PUNCT
ejpam-6171	1281	8	pt	pt	INTJ
ejpam-6171	1281	9	;	;	PUNCT
ejpam-6171	1281	10	α	α	X
ejpam-6171	1281	11	)	)	PUNCT
ejpam-6171	1281	12	.	.	PUNCT
ejpam-6171	1282	1	let	let	VERB
ejpam-6171	1282	2	x	x	PRON
ejpam-6171	1282	3	,	,	PUNCT
ejpam-6171	1282	4	y	y	PROPN
ejpam-6171	1282	5	∈	∈	PROPN
ejpam-6171	1282	6	u	u	PROPN
ejpam-6171	1282	7	+	+	X
ejpam-6171	1282	8	(	(	PUNCT
ejpam-6171	1282	9	pt	pt	INTJ
ejpam-6171	1282	10	;	;	PUNCT
ejpam-6171	1282	11	α	α	X
ejpam-6171	1282	12	)	)	PUNCT
ejpam-6171	1282	13	be	be	VERB
ejpam-6171	1282	14	such	such	ADJ
ejpam-6171	1282	15	that	that	SCONJ
ejpam-6171	1282	16	x?y	x?y	PROPN
ejpam-6171	1282	17	,	,	PUNCT
ejpam-6171	1282	18	x	x	PUNCT
ejpam-6171	1282	19	∈	∈	PROPN
ejpam-6171	1282	20	u	u	NOUN
ejpam-6171	1282	21	+	+	X
ejpam-6171	1282	22	(	(	PUNCT
ejpam-6171	1282	23	pt	pt	INTJ
ejpam-6171	1282	24	;	;	PUNCT
ejpam-6171	1282	25	α	α	X
ejpam-6171	1282	26	)	)	PUNCT
ejpam-6171	1282	27	.	.	PUNCT
ejpam-6171	1283	1	then	then	ADV
ejpam-6171	1283	2	pt	pt	PROPN
ejpam-6171	1283	3	(	(	PUNCT
ejpam-6171	1283	4	x?y	x?y	PROPN
ejpam-6171	1283	5	)	)	PUNCT
ejpam-6171	1283	6	>	>	X
ejpam-6171	1283	7	α	α	PROPN
ejpam-6171	1283	8	and	and	CCONJ
ejpam-6171	1283	9	pt	pt	PROPN
ejpam-6171	1283	10	(	(	PUNCT
ejpam-6171	1283	11	x	x	X
ejpam-6171	1283	12	)	)	PUNCT
ejpam-6171	1283	13	>	>	X
ejpam-6171	1284	1	α	α	X
ejpam-6171	1284	2	.	.	PUNCT
ejpam-6171	1285	1	thus	thus	ADV
ejpam-6171	1285	2	,	,	PUNCT
ejpam-6171	1285	3	min{pt	min{pt	PRON
ejpam-6171	1285	4	(	(	PUNCT
ejpam-6171	1285	5	x?y),pt	x?y),pt	INTJ
ejpam-6171	1285	6	(	(	PUNCT
ejpam-6171	1285	7	x	x	NOUN
ejpam-6171	1285	8	)	)	PUNCT
ejpam-6171	1285	9	}	}	PUNCT
ejpam-6171	1285	10	>	>	X
ejpam-6171	1285	11	α	α	X
ejpam-6171	1285	12	.	.	PUNCT
ejpam-6171	1286	1	by	by	ADP
ejpam-6171	1286	2	(	(	PUNCT
ejpam-6171	1286	3	3.11	3.11	NUM
ejpam-6171	1286	4	)	)	PUNCT
ejpam-6171	1286	5	,	,	PUNCT
ejpam-6171	1286	6	we	we	PRON
ejpam-6171	1286	7	have	have	VERB
ejpam-6171	1286	8	pt	pt	X
ejpam-6171	1286	9	(	(	PUNCT
ejpam-6171	1286	10	y	y	NOUN
ejpam-6171	1286	11	)	)	PUNCT
ejpam-6171	1286	12	≥	≥	NOUN
ejpam-6171	1286	13	min{pt	min{pt	X
ejpam-6171	1287	1	(	(	PUNCT
ejpam-6171	1287	2	x	x	X
ejpam-6171	1287	3	?	?	PUNCT
ejpam-6171	1287	4	y),pt	y),pt	PROPN
ejpam-6171	1287	5	(	(	PUNCT
ejpam-6171	1287	6	x	x	NOUN
ejpam-6171	1287	7	)	)	PUNCT
ejpam-6171	1287	8	}	}	PUNCT
ejpam-6171	1287	9	>	>	X
ejpam-6171	1288	1	α	α	X
ejpam-6171	1288	2	.	.	PUNCT
ejpam-6171	1289	1	thus	thus	ADV
ejpam-6171	1289	2	,	,	PUNCT
ejpam-6171	1289	3	y	y	PROPN
ejpam-6171	1289	4	∈	∈	PROPN
ejpam-6171	1289	5	u	u	PROPN
ejpam-6171	1289	6	+	+	X
ejpam-6171	1289	7	(	(	PUNCT
ejpam-6171	1289	8	pt	pt	INTJ
ejpam-6171	1289	9	;	;	PUNCT
ejpam-6171	1289	10	α	α	X
ejpam-6171	1289	11	)	)	PUNCT
ejpam-6171	1289	12	.	.	PUNCT
ejpam-6171	1290	1	hence	hence	ADV
ejpam-6171	1290	2	,	,	PUNCT
ejpam-6171	1290	3	u	u	PROPN
ejpam-6171	1290	4	+	+	X
ejpam-6171	1290	5	(	(	PUNCT
ejpam-6171	1290	6	pt	pt	INTJ
ejpam-6171	1290	7	;	;	PUNCT
ejpam-6171	1290	8	α	α	X
ejpam-6171	1290	9	)	)	PUNCT
ejpam-6171	1290	10	is	be	AUX
ejpam-6171	1290	11	an	an	DET
ejpam-6171	1290	12	iup	iup	NOUN
ejpam-6171	1290	13	-	-	PUNCT
ejpam-6171	1290	14	filter	filter	NOUN
ejpam-6171	1290	15	of	of	ADP
ejpam-6171	1290	16	x.	x.	NOUN
ejpam-6171	1290	17	let	let	VERB
ejpam-6171	1290	18	β	β	X
ejpam-6171	1290	19	∈	∈	PROPN
ejpam-6171	1291	1	[	[	X
ejpam-6171	1291	2	0	0	NUM
ejpam-6171	1291	3	,	,	PUNCT
ejpam-6171	1291	4	1	1	NUM
ejpam-6171	1291	5	]	]	PUNCT
ejpam-6171	1291	6	be	be	AUX
ejpam-6171	1291	7	such	such	ADJ
ejpam-6171	1291	8	that	that	SCONJ
ejpam-6171	1291	9	l	l	NOUN
ejpam-6171	1291	10	−	−	PROPN
ejpam-6171	1291	11	(	(	PUNCT
ejpam-6171	1291	12	pi	pi	NOUN
ejpam-6171	1291	13	;	;	PUNCT
ejpam-6171	1291	14	β	β	X
ejpam-6171	1291	15	)	)	PUNCT
ejpam-6171	1291	16	6=	6=	ADP
ejpam-6171	1291	17	∅.	∅.	AUX
ejpam-6171	1291	18	let	let	VERB
ejpam-6171	1291	19	b	b	NOUN
ejpam-6171	1291	20	∈	∈	PROPN
ejpam-6171	1291	21	l	l	NOUN
ejpam-6171	1291	22	−	−	PROPN
ejpam-6171	1292	1	(	(	PUNCT
ejpam-6171	1292	2	pi	pi	NOUN
ejpam-6171	1292	3	;	;	PUNCT
ejpam-6171	1292	4	β	β	X
ejpam-6171	1292	5	)	)	PUNCT
ejpam-6171	1292	6	.	.	PUNCT
ejpam-6171	1293	1	then	then	ADV
ejpam-6171	1293	2	pi(b	pi(b	PUNCT
ejpam-6171	1293	3	)	)	PUNCT
ejpam-6171	1293	4	<	<	X
ejpam-6171	1293	5	β	β	X
ejpam-6171	1293	6	.	.	PUNCT
ejpam-6171	1294	1	by	by	ADP
ejpam-6171	1294	2	(	(	PUNCT
ejpam-6171	1294	3	3.6	3.6	NUM
ejpam-6171	1294	4	)	)	PUNCT
ejpam-6171	1294	5	,	,	PUNCT
ejpam-6171	1294	6	we	we	PRON
ejpam-6171	1294	7	have	have	VERB
ejpam-6171	1294	8	pi(0	pi(0	PROPN
ejpam-6171	1294	9	)	)	PUNCT
ejpam-6171	1294	10	≤	≤	NOUN
ejpam-6171	1294	11	pi(b	pi(b	PUNCT
ejpam-6171	1294	12	)	)	PUNCT
ejpam-6171	1294	13	<	<	X
ejpam-6171	1294	14	β	β	X
ejpam-6171	1294	15	.	.	PUNCT
ejpam-6171	1295	1	thus	thus	ADV
ejpam-6171	1295	2	,	,	PUNCT
ejpam-6171	1295	3	0	0	NUM
ejpam-6171	1295	4	∈	∈	PROPN
ejpam-6171	1295	5	l	l	NOUN
ejpam-6171	1295	6	−	−	PROPN
ejpam-6171	1295	7	(	(	PUNCT
ejpam-6171	1295	8	pi	pi	NOUN
ejpam-6171	1295	9	;	;	PUNCT
ejpam-6171	1295	10	β	β	X
ejpam-6171	1295	11	)	)	PUNCT
ejpam-6171	1295	12	.	.	PUNCT
ejpam-6171	1296	1	let	let	VERB
ejpam-6171	1296	2	x	x	PRON
ejpam-6171	1296	3	,	,	PUNCT
ejpam-6171	1296	4	y	y	PROPN
ejpam-6171	1296	5	∈	∈	PROPN
ejpam-6171	1296	6	l	l	NOUN
ejpam-6171	1296	7	−	−	PROPN
ejpam-6171	1297	1	(	(	PUNCT
ejpam-6171	1297	2	pi	pi	NOUN
ejpam-6171	1297	3	;	;	PUNCT
ejpam-6171	1297	4	β	β	X
ejpam-6171	1297	5	)	)	PUNCT
ejpam-6171	1297	6	be	be	VERB
ejpam-6171	1297	7	such	such	ADJ
ejpam-6171	1297	8	that	that	SCONJ
ejpam-6171	1297	9	x?y	x?y	PROPN
ejpam-6171	1297	10	,	,	PUNCT
ejpam-6171	1297	11	x	x	PUNCT
ejpam-6171	1297	12	∈	∈	PROPN
ejpam-6171	1297	13	l	l	NOUN
ejpam-6171	1297	14	−	−	PROPN
ejpam-6171	1297	15	(	(	PUNCT
ejpam-6171	1297	16	pi	pi	NOUN
ejpam-6171	1297	17	;	;	PUNCT
ejpam-6171	1297	18	β	β	X
ejpam-6171	1297	19	)	)	PUNCT
ejpam-6171	1297	20	.	.	PUNCT
ejpam-6171	1298	1	then	then	ADV
ejpam-6171	1298	2	pi(x?y	pi(x?y	NOUN
ejpam-6171	1298	3	)	)	PUNCT
ejpam-6171	1298	4	<	<	X
ejpam-6171	1298	5	β	β	X
ejpam-6171	1298	6	and	and	CCONJ
ejpam-6171	1298	7	pt	pt	INTJ
ejpam-6171	1298	8	(	(	PUNCT
ejpam-6171	1298	9	x	x	X
ejpam-6171	1298	10	)	)	PUNCT
ejpam-6171	1298	11	<	<	X
ejpam-6171	1298	12	β	β	X
ejpam-6171	1298	13	.	.	PUNCT
ejpam-6171	1299	1	thus	thus	ADV
ejpam-6171	1299	2	,	,	PUNCT
ejpam-6171	1299	3	max{pi(x?y),pi(x	max{pi(x?y),pi(x	NOUN
ejpam-6171	1299	4	)	)	PUNCT
ejpam-6171	1299	5	}	}	PUNCT
ejpam-6171	1300	1	<	<	X
ejpam-6171	1300	2	β	β	X
ejpam-6171	1300	3	.	.	PUNCT
ejpam-6171	1300	4	by	by	ADP
ejpam-6171	1300	5	(	(	PUNCT
ejpam-6171	1300	6	3.12	3.12	NUM
ejpam-6171	1300	7	)	)	PUNCT
ejpam-6171	1300	8	,	,	PUNCT
ejpam-6171	1300	9	we	we	PRON
ejpam-6171	1300	10	have	have	VERB
ejpam-6171	1300	11	pi(y	pi(y	NOUN
ejpam-6171	1300	12	)	)	PUNCT
ejpam-6171	1300	13	≤	≤	NUM
ejpam-6171	1300	14	max{pi(x	max{pi(x	NOUN
ejpam-6171	1300	15	?	?	PUNCT
ejpam-6171	1301	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1301	2	)	)	PUNCT
ejpam-6171	1301	3	}	}	PUNCT
ejpam-6171	1301	4	>	>	X
ejpam-6171	1302	1	β	β	X
ejpam-6171	1302	2	.	.	PUNCT
ejpam-6171	1303	1	thus	thus	ADV
ejpam-6171	1303	2	,	,	PUNCT
ejpam-6171	1303	3	y	y	PROPN
ejpam-6171	1303	4	∈	∈	PROPN
ejpam-6171	1303	5	l	l	NOUN
ejpam-6171	1303	6	−	−	PROPN
ejpam-6171	1303	7	(	(	PUNCT
ejpam-6171	1303	8	pi	pi	NOUN
ejpam-6171	1303	9	;	;	PUNCT
ejpam-6171	1303	10	β	β	X
ejpam-6171	1303	11	)	)	PUNCT
ejpam-6171	1303	12	.	.	PUNCT
ejpam-6171	1304	1	hence	hence	ADV
ejpam-6171	1304	2	,	,	PUNCT
ejpam-6171	1304	3	l	l	NOUN
ejpam-6171	1304	4	−	−	PROPN
ejpam-6171	1304	5	(	(	PUNCT
ejpam-6171	1304	6	pi	pi	NOUN
ejpam-6171	1304	7	;	;	PUNCT
ejpam-6171	1304	8	β	β	X
ejpam-6171	1304	9	)	)	PUNCT
ejpam-6171	1304	10	is	be	AUX
ejpam-6171	1304	11	an	an	DET
ejpam-6171	1304	12	iup	iup	NOUN
ejpam-6171	1304	13	-	-	PUNCT
ejpam-6171	1304	14	ideal	ideal	NOUN
ejpam-6171	1304	15	of	of	ADP
ejpam-6171	1304	16	x.	x.	NOUN
ejpam-6171	1304	17	let	let	VERB
ejpam-6171	1304	18	γ	γ	X
ejpam-6171	1304	19	∈	∈	PROPN
ejpam-6171	1305	1	[	[	X
ejpam-6171	1305	2	0	0	NUM
ejpam-6171	1305	3	,	,	PUNCT
ejpam-6171	1305	4	1	1	NUM
ejpam-6171	1305	5	]	]	PUNCT
ejpam-6171	1305	6	be	be	AUX
ejpam-6171	1305	7	such	such	ADJ
ejpam-6171	1305	8	that	that	SCONJ
ejpam-6171	1305	9	u	u	NOUN
ejpam-6171	1305	10	+	+	X
ejpam-6171	1305	11	(	(	PUNCT
ejpam-6171	1305	12	pf	pf	INTJ
ejpam-6171	1305	13	;	;	PUNCT
ejpam-6171	1305	14	γ	γ	X
ejpam-6171	1305	15	)	)	PUNCT
ejpam-6171	1305	16	6=	6=	ADP
ejpam-6171	1306	1	∅.	∅.	AUX
ejpam-6171	1306	2	let	let	VERB
ejpam-6171	1306	3	c	c	NOUN
ejpam-6171	1306	4	∈	∈	PROPN
ejpam-6171	1306	5	u	u	NOUN
ejpam-6171	1306	6	+	+	X
ejpam-6171	1306	7	(	(	PUNCT
ejpam-6171	1306	8	pf	pf	INTJ
ejpam-6171	1306	9	;	;	PUNCT
ejpam-6171	1306	10	γ	γ	X
ejpam-6171	1306	11	)	)	PUNCT
ejpam-6171	1306	12	.	.	PUNCT
ejpam-6171	1307	1	then	then	ADV
ejpam-6171	1307	2	pf	pf	PROPN
ejpam-6171	1307	3	(	(	PUNCT
ejpam-6171	1307	4	c	c	NOUN
ejpam-6171	1307	5	)	)	PUNCT
ejpam-6171	1307	6	>	>	X
ejpam-6171	1307	7	γ	γ	X
ejpam-6171	1307	8	.	.	PROPN
ejpam-6171	1307	9	by	by	ADP
ejpam-6171	1307	10	(	(	PUNCT
ejpam-6171	1307	11	3.7	3.7	NUM
ejpam-6171	1307	12	)	)	PUNCT
ejpam-6171	1307	13	,	,	PUNCT
ejpam-6171	1307	14	we	we	PRON
ejpam-6171	1307	15	have	have	VERB
ejpam-6171	1307	16	pf	pf	PROPN
ejpam-6171	1307	17	(	(	PUNCT
ejpam-6171	1307	18	0	0	NUM
ejpam-6171	1307	19	)	)	PUNCT
ejpam-6171	1307	20	≥	≥	NOUN
ejpam-6171	1307	21	pf	pf	X
ejpam-6171	1307	22	(	(	PUNCT
ejpam-6171	1307	23	c	c	NOUN
ejpam-6171	1307	24	)	)	PUNCT
ejpam-6171	1307	25	>	>	X
ejpam-6171	1308	1	γ	γ	X
ejpam-6171	1308	2	.	.	PUNCT
ejpam-6171	1308	3	thus	thus	ADV
ejpam-6171	1308	4	,	,	PUNCT
ejpam-6171	1308	5	0	0	NUM
ejpam-6171	1308	6	∈	∈	PROPN
ejpam-6171	1308	7	u	u	NOUN
ejpam-6171	1308	8	+	+	X
ejpam-6171	1308	9	(	(	PUNCT
ejpam-6171	1308	10	pf	pf	INTJ
ejpam-6171	1308	11	;	;	PUNCT
ejpam-6171	1308	12	γ	γ	X
ejpam-6171	1308	13	)	)	PUNCT
ejpam-6171	1308	14	.	.	PUNCT
ejpam-6171	1309	1	let	let	VERB
ejpam-6171	1309	2	x	x	PRON
ejpam-6171	1309	3	,	,	PUNCT
ejpam-6171	1309	4	y	y	PROPN
ejpam-6171	1309	5	∈	∈	PROPN
ejpam-6171	1309	6	u	u	PROPN
ejpam-6171	1309	7	+	+	X
ejpam-6171	1309	8	(	(	PUNCT
ejpam-6171	1309	9	pf	pf	INTJ
ejpam-6171	1309	10	;	;	PUNCT
ejpam-6171	1309	11	γ	γ	X
ejpam-6171	1309	12	)	)	PUNCT
ejpam-6171	1309	13	be	be	VERB
ejpam-6171	1309	14	such	such	ADJ
ejpam-6171	1309	15	that	that	PRON
ejpam-6171	1309	16	x	x	X
ejpam-6171	1309	17	?	?	PUNCT
ejpam-6171	1310	1	y	y	NOUN
ejpam-6171	1310	2	,	,	PUNCT
ejpam-6171	1310	3	x	x	PUNCT
ejpam-6171	1310	4	∈	∈	PROPN
ejpam-6171	1310	5	u	u	NOUN
ejpam-6171	1310	6	+	+	X
ejpam-6171	1310	7	(	(	PUNCT
ejpam-6171	1310	8	pf	pf	INTJ
ejpam-6171	1310	9	;	;	PUNCT
ejpam-6171	1310	10	γ	γ	X
ejpam-6171	1310	11	)	)	PUNCT
ejpam-6171	1310	12	.	.	PUNCT
ejpam-6171	1311	1	then	then	ADV
ejpam-6171	1311	2	pf	pf	INTJ
ejpam-6171	1311	3	(	(	PUNCT
ejpam-6171	1311	4	x	x	PROPN
ejpam-6171	1311	5	?	?	PUNCT
ejpam-6171	1312	1	y	y	X
ejpam-6171	1312	2	)	)	PUNCT
ejpam-6171	1312	3	>	>	X
ejpam-6171	1312	4	γ	γ	PROPN
ejpam-6171	1312	5	and	and	CCONJ
ejpam-6171	1312	6	pf	pf	PROPN
ejpam-6171	1312	7	(	(	PUNCT
ejpam-6171	1312	8	x	x	NOUN
ejpam-6171	1312	9	)	)	PUNCT
ejpam-6171	1312	10	>	>	X
ejpam-6171	1312	11	γ	γ	X
ejpam-6171	1312	12	.	.	PUNCT
ejpam-6171	1312	13	thus	thus	ADV
ejpam-6171	1312	14	,	,	PUNCT
ejpam-6171	1312	15	min{pf	min{pf	PRON
ejpam-6171	1312	16	(	(	PUNCT
ejpam-6171	1312	17	x	x	X
ejpam-6171	1312	18	?	?	PUNCT
ejpam-6171	1313	1	y),pf	y),pf	PROPN
ejpam-6171	1313	2	(	(	PUNCT
ejpam-6171	1313	3	x	x	X
ejpam-6171	1313	4	)	)	PUNCT
ejpam-6171	1313	5	}	}	PUNCT
ejpam-6171	1313	6	>	>	X
ejpam-6171	1314	1	γ	γ	X
ejpam-6171	1314	2	.	.	PUNCT
ejpam-6171	1314	3	by	by	ADP
ejpam-6171	1314	4	(	(	PUNCT
ejpam-6171	1314	5	3.13	3.13	NUM
ejpam-6171	1314	6	)	)	PUNCT
ejpam-6171	1314	7	,	,	PUNCT
ejpam-6171	1314	8	we	we	PRON
ejpam-6171	1314	9	have	have	VERB
ejpam-6171	1314	10	pf	pf	PROPN
ejpam-6171	1314	11	(	(	PUNCT
ejpam-6171	1314	12	y	y	NOUN
ejpam-6171	1314	13	)	)	PUNCT
ejpam-6171	1314	14	≥	≥	NOUN
ejpam-6171	1314	15	min{pf	min{pf	X
ejpam-6171	1314	16	(	(	PUNCT
ejpam-6171	1314	17	x	x	X
ejpam-6171	1314	18	?	?	PUNCT
ejpam-6171	1315	1	y),pf	y),pf	PROPN
ejpam-6171	1315	2	(	(	PUNCT
ejpam-6171	1315	3	x	x	X
ejpam-6171	1315	4	)	)	PUNCT
ejpam-6171	1315	5	}	}	PUNCT
ejpam-6171	1315	6	>	>	X
ejpam-6171	1316	1	γ	γ	X
ejpam-6171	1316	2	.	.	PUNCT
ejpam-6171	1316	3	thus	thus	ADV
ejpam-6171	1316	4	,	,	PUNCT
ejpam-6171	1316	5	y	y	PROPN
ejpam-6171	1316	6	∈	∈	PROPN
ejpam-6171	1316	7	u	u	PROPN
ejpam-6171	1316	8	+	+	X
ejpam-6171	1316	9	(	(	PUNCT
ejpam-6171	1316	10	pf	pf	INTJ
ejpam-6171	1316	11	;	;	PUNCT
ejpam-6171	1316	12	γ	γ	X
ejpam-6171	1316	13	)	)	PUNCT
ejpam-6171	1316	14	.	.	PUNCT
ejpam-6171	1317	1	hence	hence	ADV
ejpam-6171	1317	2	,	,	PUNCT
ejpam-6171	1317	3	u+	u+	NUM
ejpam-6171	1317	4	(	(	PUNCT
ejpam-6171	1317	5	pf	pf	X
ejpam-6171	1317	6	;	;	PUNCT
ejpam-6171	1317	7	γ	γ	X
ejpam-6171	1317	8	)	)	PUNCT
ejpam-6171	1317	9	is	be	AUX
ejpam-6171	1317	10	an	an	DET
ejpam-6171	1317	11	iup	iup	NOUN
ejpam-6171	1317	12	-	-	PUNCT
ejpam-6171	1317	13	ideal	ideal	NOUN
ejpam-6171	1317	14	of	of	ADP
ejpam-6171	1317	15	x.	x.	NOUN
ejpam-6171	1317	16	conversely	conversely	ADV
ejpam-6171	1317	17	,	,	PUNCT
ejpam-6171	1317	18	assume	assume	VERB
ejpam-6171	1317	19	that	that	SCONJ
ejpam-6171	1317	20	for	for	ADP
ejpam-6171	1317	21	all	all	DET
ejpam-6171	1317	22	α	α	NOUN
ejpam-6171	1317	23	,	,	PUNCT
ejpam-6171	1317	24	β	β	X
ejpam-6171	1317	25	,	,	PUNCT
ejpam-6171	1317	26	γ	γ	PROPN
ejpam-6171	1317	27	∈	∈	PROPN
ejpam-6171	1318	1	[	[	X
ejpam-6171	1318	2	0	0	NUM
ejpam-6171	1318	3	,	,	PUNCT
ejpam-6171	1318	4	1	1	NUM
ejpam-6171	1318	5	]	]	PUNCT
ejpam-6171	1318	6	,	,	PUNCT
ejpam-6171	1318	7	the	the	DET
ejpam-6171	1318	8	sets	set	NOUN
ejpam-6171	1318	9	u	u	NOUN
ejpam-6171	1318	10	+	+	X
ejpam-6171	1318	11	(	(	PUNCT
ejpam-6171	1318	12	pt	pt	INTJ
ejpam-6171	1318	13	;	;	PUNCT
ejpam-6171	1318	14	α	α	X
ejpam-6171	1318	15	)	)	PUNCT
ejpam-6171	1318	16	,	,	PUNCT
ejpam-6171	1318	17	l	l	NOUN
ejpam-6171	1318	18	−	−	PROPN
ejpam-6171	1318	19	(	(	PUNCT
ejpam-6171	1318	20	pi	pi	NOUN
ejpam-6171	1318	21	;	;	PUNCT
ejpam-6171	1318	22	β	β	X
ejpam-6171	1318	23	)	)	PUNCT
ejpam-6171	1318	24	,	,	PUNCT
ejpam-6171	1318	25	and	and	CCONJ
ejpam-6171	1318	26	u	u	NOUN
ejpam-6171	1318	27	+	+	CCONJ
ejpam-6171	1318	28	(	(	PUNCT
ejpam-6171	1318	29	pf	pf	INTJ
ejpam-6171	1318	30	;	;	PUNCT
ejpam-6171	1318	31	γ	γ	X
ejpam-6171	1318	32	)	)	PUNCT
ejpam-6171	1318	33	are	be	AUX
ejpam-6171	1318	34	either	either	CCONJ
ejpam-6171	1318	35	empty	empty	ADJ
ejpam-6171	1318	36	or	or	CCONJ
ejpam-6171	1318	37	iup	iup	NOUN
ejpam-6171	1318	38	-	-	PUNCT
ejpam-6171	1318	39	filters	filter	NOUN
ejpam-6171	1318	40	of	of	ADP
ejpam-6171	1318	41	x.	x.	NOUN
ejpam-6171	1318	42	let	let	VERB
ejpam-6171	1318	43	x	x	SYM
ejpam-6171	1318	44	∈	∈	PROPN
ejpam-6171	1318	45	x.	x.	NOUN
ejpam-6171	1318	46	assume	assume	VERB
ejpam-6171	1318	47	that	that	SCONJ
ejpam-6171	1318	48	pt	pt	X
ejpam-6171	1318	49	(	(	PUNCT
ejpam-6171	1318	50	0	0	NUM
ejpam-6171	1318	51	)	)	PUNCT
ejpam-6171	1318	52	<	<	X
ejpam-6171	1318	53	pt	pt	X
ejpam-6171	1318	54	(	(	PUNCT
ejpam-6171	1318	55	x	x	NOUN
ejpam-6171	1318	56	)	)	PUNCT
ejpam-6171	1318	57	.	.	PUNCT
ejpam-6171	1319	1	let	let	VERB
ejpam-6171	1319	2	α	α	NOUN
ejpam-6171	1319	3	=	=	SYM
ejpam-6171	1319	4	pt	pt	X
ejpam-6171	1319	5	(	(	PUNCT
ejpam-6171	1319	6	0	0	NUM
ejpam-6171	1319	7	)	)	PUNCT
ejpam-6171	1319	8	.	.	PUNCT
ejpam-6171	1320	1	then	then	ADV
ejpam-6171	1320	2	x	x	SYM
ejpam-6171	1320	3	∈	∈	PROPN
ejpam-6171	1320	4	u	u	NOUN
ejpam-6171	1320	5	+	+	X
ejpam-6171	1320	6	(	(	PUNCT
ejpam-6171	1320	7	pt	pt	INTJ
ejpam-6171	1320	8	;	;	PUNCT
ejpam-6171	1320	9	α	α	X
ejpam-6171	1320	10	)	)	PUNCT
ejpam-6171	1320	11	6=	6=	ADP
ejpam-6171	1320	12	∅.	∅.	ADP
ejpam-6171	1320	13	by	by	ADP
ejpam-6171	1320	14	the	the	DET
ejpam-6171	1320	15	assumption	assumption	NOUN
ejpam-6171	1320	16	,	,	PUNCT
ejpam-6171	1320	17	we	we	PRON
ejpam-6171	1320	18	have	have	VERB
ejpam-6171	1320	19	u	u	NOUN
ejpam-6171	1320	20	+	+	CCONJ
ejpam-6171	1320	21	(	(	PUNCT
ejpam-6171	1320	22	pt	pt	INTJ
ejpam-6171	1320	23	;	;	PUNCT
ejpam-6171	1320	24	α	α	X
ejpam-6171	1320	25	)	)	PUNCT
ejpam-6171	1320	26	is	be	AUX
ejpam-6171	1320	27	an	an	DET
ejpam-6171	1320	28	iup	iup	NOUN
ejpam-6171	1320	29	-	-	PUNCT
ejpam-6171	1320	30	ideal	ideal	NOUN
ejpam-6171	1320	31	of	of	ADP
ejpam-6171	1320	32	x.	x.	NOUN
ejpam-6171	1320	33	by	by	ADP
ejpam-6171	1320	34	(	(	PUNCT
ejpam-6171	1320	35	2.18	2.18	NUM
ejpam-6171	1320	36	)	)	PUNCT
ejpam-6171	1320	37	,	,	PUNCT
ejpam-6171	1320	38	we	we	PRON
ejpam-6171	1320	39	have	have	VERB
ejpam-6171	1320	40	0	0	NUM
ejpam-6171	1320	41	∈	∈	PROPN
ejpam-6171	1320	42	u	u	NOUN
ejpam-6171	1320	43	+	+	X
ejpam-6171	1320	44	(	(	PUNCT
ejpam-6171	1320	45	pt	pt	INTJ
ejpam-6171	1320	46	;	;	PUNCT
ejpam-6171	1320	47	α	α	X
ejpam-6171	1320	48	)	)	PUNCT
ejpam-6171	1320	49	.	.	PUNCT
ejpam-6171	1321	1	so	so	ADV
ejpam-6171	1321	2	pt	pt	INTJ
ejpam-6171	1321	3	(	(	PUNCT
ejpam-6171	1321	4	0	0	NUM
ejpam-6171	1321	5	)	)	PUNCT
ejpam-6171	1321	6	>	>	X
ejpam-6171	1322	1	α	α	X
ejpam-6171	1322	2	=	=	SYM
ejpam-6171	1322	3	pt	pt	X
ejpam-6171	1322	4	(	(	PUNCT
ejpam-6171	1322	5	0	0	NUM
ejpam-6171	1322	6	)	)	PUNCT
ejpam-6171	1322	7	,	,	PUNCT
ejpam-6171	1322	8	which	which	PRON
ejpam-6171	1322	9	is	be	AUX
ejpam-6171	1322	10	a	a	DET
ejpam-6171	1322	11	contradiction	contradiction	NOUN
ejpam-6171	1322	12	.	.	PUNCT
ejpam-6171	1323	1	thus	thus	ADV
ejpam-6171	1323	2	,	,	PUNCT
ejpam-6171	1323	3	pt	pt	X
ejpam-6171	1323	4	(	(	PUNCT
ejpam-6171	1323	5	0	0	NUM
ejpam-6171	1323	6	)	)	PUNCT
ejpam-6171	1323	7	≥	≥	NOUN
ejpam-6171	1323	8	pt	pt	INTJ
ejpam-6171	1323	9	(	(	PUNCT
ejpam-6171	1323	10	x	x	NOUN
ejpam-6171	1323	11	)	)	PUNCT
ejpam-6171	1323	12	.	.	PUNCT
ejpam-6171	1324	1	let	let	VERB
ejpam-6171	1324	2	x	x	PRON
ejpam-6171	1324	3	,	,	PUNCT
ejpam-6171	1324	4	y	y	PROPN
ejpam-6171	1324	5	∈	∈	PROPN
ejpam-6171	1324	6	x.	x.	NOUN
ejpam-6171	1324	7	assume	assume	VERB
ejpam-6171	1324	8	that	that	SCONJ
ejpam-6171	1324	9	pt	pt	PROPN
ejpam-6171	1324	10	(	(	PUNCT
ejpam-6171	1324	11	y	y	NOUN
ejpam-6171	1324	12	)	)	PUNCT
ejpam-6171	1324	13	<	<	X
ejpam-6171	1324	14	min{pt	min{pt	X
ejpam-6171	1324	15	(	(	PUNCT
ejpam-6171	1324	16	x?y),pt	x?y),pt	INTJ
ejpam-6171	1324	17	(	(	PUNCT
ejpam-6171	1324	18	x	x	NOUN
ejpam-6171	1324	19	)	)	PUNCT
ejpam-6171	1324	20	}	}	PUNCT
ejpam-6171	1324	21	.	.	PUNCT
ejpam-6171	1325	1	let	let	VERB
ejpam-6171	1325	2	α	α	NOUN
ejpam-6171	1325	3	=	=	SYM
ejpam-6171	1325	4	pt	pt	X
ejpam-6171	1325	5	(	(	PUNCT
ejpam-6171	1325	6	y	y	NOUN
ejpam-6171	1325	7	)	)	PUNCT
ejpam-6171	1325	8	.	.	PUNCT
ejpam-6171	1326	1	then	then	ADV
ejpam-6171	1326	2	x?y	x?y	PROPN
ejpam-6171	1326	3	,	,	PUNCT
ejpam-6171	1326	4	x	x	PUNCT
ejpam-6171	1326	5	∈	∈	PROPN
ejpam-6171	1326	6	u	u	NOUN
ejpam-6171	1326	7	+	+	X
ejpam-6171	1326	8	(	(	PUNCT
ejpam-6171	1326	9	pt	pt	INTJ
ejpam-6171	1326	10	;	;	PUNCT
ejpam-6171	1326	11	α	α	X
ejpam-6171	1326	12	)	)	PUNCT
ejpam-6171	1326	13	6=	6=	ADP
ejpam-6171	1326	14	∅.	∅.	ADP
ejpam-6171	1326	15	by	by	ADP
ejpam-6171	1326	16	the	the	DET
ejpam-6171	1326	17	assumption	assumption	NOUN
ejpam-6171	1326	18	,	,	PUNCT
ejpam-6171	1326	19	we	we	PRON
ejpam-6171	1326	20	have	have	VERB
ejpam-6171	1326	21	u	u	NOUN
ejpam-6171	1326	22	+	+	CCONJ
ejpam-6171	1326	23	(	(	PUNCT
ejpam-6171	1326	24	pt	pt	INTJ
ejpam-6171	1326	25	;	;	PUNCT
ejpam-6171	1326	26	α	α	X
ejpam-6171	1326	27	)	)	PUNCT
ejpam-6171	1326	28	is	be	AUX
ejpam-6171	1326	29	an	an	DET
ejpam-6171	1326	30	iup	iup	NOUN
ejpam-6171	1326	31	-	-	PUNCT
ejpam-6171	1326	32	filter	filter	NOUN
ejpam-6171	1326	33	of	of	ADP
ejpam-6171	1326	34	x.	x.	NOUN
ejpam-6171	1326	35	by	by	ADP
ejpam-6171	1326	36	(	(	PUNCT
ejpam-6171	1326	37	2.19	2.19	NUM
ejpam-6171	1326	38	)	)	PUNCT
ejpam-6171	1326	39	,	,	PUNCT
ejpam-6171	1326	40	we	we	PRON
ejpam-6171	1326	41	have	have	VERB
ejpam-6171	1326	42	y	y	PROPN
ejpam-6171	1326	43	∈	∈	PROPN
ejpam-6171	1326	44	u	u	PROPN
ejpam-6171	1326	45	+	+	X
ejpam-6171	1326	46	(	(	PUNCT
ejpam-6171	1326	47	pt	pt	INTJ
ejpam-6171	1326	48	;	;	PUNCT
ejpam-6171	1326	49	α	α	X
ejpam-6171	1326	50	)	)	PUNCT
ejpam-6171	1326	51	.	.	PUNCT
ejpam-6171	1327	1	so	so	ADV
ejpam-6171	1327	2	pt	pt	INTJ
ejpam-6171	1327	3	(	(	PUNCT
ejpam-6171	1327	4	y	y	NOUN
ejpam-6171	1327	5	)	)	PUNCT
ejpam-6171	1327	6	>	>	X
ejpam-6171	1328	1	α	α	X
ejpam-6171	1328	2	=	=	SYM
ejpam-6171	1328	3	pt	pt	X
ejpam-6171	1328	4	(	(	PUNCT
ejpam-6171	1328	5	y	y	NOUN
ejpam-6171	1328	6	)	)	PUNCT
ejpam-6171	1328	7	,	,	PUNCT
ejpam-6171	1328	8	which	which	PRON
ejpam-6171	1328	9	is	be	AUX
ejpam-6171	1328	10	a	a	DET
ejpam-6171	1328	11	contradiction	contradiction	NOUN
ejpam-6171	1328	12	.	.	PUNCT
ejpam-6171	1329	1	thus	thus	ADV
ejpam-6171	1329	2	,	,	PUNCT
ejpam-6171	1329	3	pt	pt	X
ejpam-6171	1329	4	(	(	PUNCT
ejpam-6171	1329	5	y	y	NOUN
ejpam-6171	1329	6	)	)	PUNCT
ejpam-6171	1329	7	≥	≥	NOUN
ejpam-6171	1329	8	min{pt	min{pt	X
ejpam-6171	1330	1	(	(	PUNCT
ejpam-6171	1330	2	x	x	X
ejpam-6171	1330	3	?	?	PUNCT
ejpam-6171	1330	4	y),pt	y),pt	PROPN
ejpam-6171	1330	5	(	(	PUNCT
ejpam-6171	1330	6	x	x	NOUN
ejpam-6171	1330	7	)	)	PUNCT
ejpam-6171	1330	8	}	}	PUNCT
ejpam-6171	1330	9	.	.	PUNCT
ejpam-6171	1331	1	let	let	VERB
ejpam-6171	1331	2	x	x	SYM
ejpam-6171	1331	3	∈	∈	PROPN
ejpam-6171	1331	4	x.	x.	NOUN
ejpam-6171	1331	5	assume	assume	VERB
ejpam-6171	1331	6	that	that	SCONJ
ejpam-6171	1331	7	pi(0	pi(0	PROPN
ejpam-6171	1331	8	)	)	PUNCT
ejpam-6171	1331	9	>	>	X
ejpam-6171	1331	10	pi(x	pi(x	PROPN
ejpam-6171	1331	11	)	)	PUNCT
ejpam-6171	1331	12	.	.	PUNCT
ejpam-6171	1332	1	let	let	VERB
ejpam-6171	1332	2	β	β	X
ejpam-6171	1332	3	=	=	SYM
ejpam-6171	1332	4	pi(0	pi(0	PROPN
ejpam-6171	1332	5	)	)	PUNCT
ejpam-6171	1332	6	.	.	PUNCT
ejpam-6171	1333	1	then	then	ADV
ejpam-6171	1333	2	x	x	SYM
ejpam-6171	1333	3	∈	∈	PROPN
ejpam-6171	1333	4	l	l	NOUN
ejpam-6171	1333	5	−	−	PROPN
ejpam-6171	1334	1	(	(	PUNCT
ejpam-6171	1334	2	pi	pi	NOUN
ejpam-6171	1334	3	;	;	PUNCT
ejpam-6171	1334	4	β	β	X
ejpam-6171	1334	5	)	)	PUNCT
ejpam-6171	1334	6	6=	6=	ADP
ejpam-6171	1334	7	∅.	∅.	ADP
ejpam-6171	1334	8	by	by	ADP
ejpam-6171	1334	9	the	the	DET
ejpam-6171	1334	10	assumption	assumption	NOUN
ejpam-6171	1334	11	,	,	PUNCT
ejpam-6171	1334	12	we	we	PRON
ejpam-6171	1334	13	have	have	VERB
ejpam-6171	1334	14	l	l	NOUN
ejpam-6171	1334	15	−	−	PROPN
ejpam-6171	1334	16	(	(	PUNCT
ejpam-6171	1334	17	pi	pi	NOUN
ejpam-6171	1334	18	;	;	PUNCT
ejpam-6171	1334	19	β	β	X
ejpam-6171	1334	20	)	)	PUNCT
ejpam-6171	1334	21	is	be	AUX
ejpam-6171	1334	22	an	an	DET
ejpam-6171	1334	23	iup	iup	NOUN
ejpam-6171	1334	24	-	-	PUNCT
ejpam-6171	1334	25	filter	filter	NOUN
ejpam-6171	1334	26	of	of	ADP
ejpam-6171	1334	27	x.	x.	NOUN
ejpam-6171	1334	28	by	by	ADP
ejpam-6171	1334	29	(	(	PUNCT
ejpam-6171	1334	30	2.18	2.18	NUM
ejpam-6171	1334	31	)	)	PUNCT
ejpam-6171	1334	32	,	,	PUNCT
ejpam-6171	1334	33	we	we	PRON
ejpam-6171	1334	34	have	have	VERB
ejpam-6171	1334	35	0	0	NUM
ejpam-6171	1334	36	∈	∈	NOUN
ejpam-6171	1334	37	l	l	NOUN
ejpam-6171	1334	38	−	−	PROPN
ejpam-6171	1335	1	(	(	PUNCT
ejpam-6171	1335	2	pi	pi	NOUN
ejpam-6171	1335	3	;	;	PUNCT
ejpam-6171	1335	4	β	β	X
ejpam-6171	1335	5	)	)	PUNCT
ejpam-6171	1335	6	.	.	PUNCT
ejpam-6171	1336	1	so	so	ADV
ejpam-6171	1336	2	pi(0	pi(0	PROPN
ejpam-6171	1336	3	)	)	PUNCT
ejpam-6171	1336	4	<	<	X
ejpam-6171	1336	5	β	β	X
ejpam-6171	1336	6	=	=	SYM
ejpam-6171	1336	7	pi(0	pi(0	PROPN
ejpam-6171	1336	8	)	)	PUNCT
ejpam-6171	1336	9	,	,	PUNCT
ejpam-6171	1336	10	which	which	PRON
ejpam-6171	1336	11	is	be	AUX
ejpam-6171	1336	12	a	a	DET
ejpam-6171	1336	13	contradiction	contradiction	NOUN
ejpam-6171	1336	14	.	.	PUNCT
ejpam-6171	1337	1	thus	thus	ADV
ejpam-6171	1337	2	,	,	PUNCT
ejpam-6171	1337	3	pi(0	pi(0	PROPN
ejpam-6171	1337	4	)	)	PUNCT
ejpam-6171	1337	5	≤	≤	NOUN
ejpam-6171	1337	6	pi(x	pi(x	NOUN
ejpam-6171	1337	7	)	)	PUNCT
ejpam-6171	1337	8	.	.	PUNCT
ejpam-6171	1338	1	let	let	VERB
ejpam-6171	1338	2	x	x	PRON
ejpam-6171	1338	3	,	,	PUNCT
ejpam-6171	1338	4	y	y	PROPN
ejpam-6171	1338	5	∈	∈	PROPN
ejpam-6171	1338	6	x.	x.	NOUN
ejpam-6171	1338	7	assume	assume	VERB
ejpam-6171	1338	8	that	that	SCONJ
ejpam-6171	1338	9	pi(y	pi(y	NOUN
ejpam-6171	1338	10	)	)	PUNCT
ejpam-6171	1338	11	>	>	X
ejpam-6171	1339	1	max{pi(x	max{pi(x	PROPN
ejpam-6171	1339	2	?	?	PUNCT
ejpam-6171	1340	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1340	2	)	)	PUNCT
ejpam-6171	1340	3	}	}	PUNCT
ejpam-6171	1340	4	.	.	PUNCT
ejpam-6171	1341	1	let	let	VERB
ejpam-6171	1341	2	β	β	X
ejpam-6171	1341	3	=	=	NOUN
ejpam-6171	1341	4	pi(y	pi(y	NOUN
ejpam-6171	1341	5	)	)	PUNCT
ejpam-6171	1341	6	.	.	PUNCT
ejpam-6171	1342	1	then	then	ADV
ejpam-6171	1342	2	x	x	X
ejpam-6171	1342	3	?	?	PUNCT
ejpam-6171	1343	1	y	y	NOUN
ejpam-6171	1343	2	,	,	PUNCT
ejpam-6171	1343	3	x	x	X
ejpam-6171	1343	4	∈	∈	NOUN
ejpam-6171	1343	5	l	l	NOUN
ejpam-6171	1343	6	−	−	PROPN
ejpam-6171	1344	1	(	(	PUNCT
ejpam-6171	1344	2	pi	pi	NOUN
ejpam-6171	1344	3	;	;	PUNCT
ejpam-6171	1344	4	β	β	X
ejpam-6171	1344	5	)	)	PUNCT
ejpam-6171	1344	6	6=	6=	ADP
ejpam-6171	1344	7	∅.	∅.	ADP
ejpam-6171	1344	8	by	by	ADP
ejpam-6171	1344	9	the	the	DET
ejpam-6171	1344	10	assumption	assumption	NOUN
ejpam-6171	1344	11	,	,	PUNCT
ejpam-6171	1344	12	we	we	PRON
ejpam-6171	1344	13	have	have	VERB
ejpam-6171	1344	14	l	l	NOUN
ejpam-6171	1344	15	−	−	PROPN
ejpam-6171	1344	16	(	(	PUNCT
ejpam-6171	1344	17	pi	pi	NOUN
ejpam-6171	1344	18	;	;	PUNCT
ejpam-6171	1344	19	β	β	X
ejpam-6171	1344	20	)	)	PUNCT
ejpam-6171	1344	21	is	be	AUX
ejpam-6171	1344	22	an	an	DET
ejpam-6171	1344	23	iup	iup	NOUN
ejpam-6171	1344	24	-	-	PUNCT
ejpam-6171	1344	25	filter	filter	NOUN
ejpam-6171	1344	26	of	of	ADP
ejpam-6171	1344	27	x.	x.	NOUN
ejpam-6171	1344	28	by	by	ADP
ejpam-6171	1344	29	(	(	PUNCT
ejpam-6171	1344	30	2.19	2.19	NUM
ejpam-6171	1344	31	)	)	PUNCT
ejpam-6171	1344	32	,	,	PUNCT
ejpam-6171	1344	33	we	we	PRON
ejpam-6171	1344	34	have	have	VERB
ejpam-6171	1344	35	y	y	PROPN
ejpam-6171	1344	36	∈	∈	PROPN
ejpam-6171	1344	37	l	l	NOUN
ejpam-6171	1344	38	−	−	PROPN
ejpam-6171	1345	1	(	(	PUNCT
ejpam-6171	1345	2	pi	pi	NOUN
ejpam-6171	1345	3	;	;	PUNCT
ejpam-6171	1345	4	β	β	X
ejpam-6171	1345	5	)	)	PUNCT
ejpam-6171	1345	6	.	.	PUNCT
ejpam-6171	1346	1	so	so	ADV
ejpam-6171	1346	2	pi(y	pi(y	NOUN
ejpam-6171	1346	3	)	)	PUNCT
ejpam-6171	1346	4	<	<	X
ejpam-6171	1346	5	β	β	X
ejpam-6171	1346	6	=	=	SYM
ejpam-6171	1346	7	pi(y	pi(y	NOUN
ejpam-6171	1346	8	)	)	PUNCT
ejpam-6171	1346	9	,	,	PUNCT
ejpam-6171	1346	10	which	which	PRON
ejpam-6171	1346	11	is	be	AUX
ejpam-6171	1346	12	a	a	DET
ejpam-6171	1346	13	contradiction	contradiction	NOUN
ejpam-6171	1346	14	.	.	PUNCT
ejpam-6171	1347	1	thus	thus	ADV
ejpam-6171	1347	2	,	,	PUNCT
ejpam-6171	1347	3	pi(y	pi(y	NOUN
ejpam-6171	1347	4	)	)	PUNCT
ejpam-6171	1347	5	≤	≤	NUM
ejpam-6171	1347	6	max{pi(x	max{pi(x	NOUN
ejpam-6171	1347	7	?	?	PUNCT
ejpam-6171	1348	1	y),pi(x	y),pi(x	NUM
ejpam-6171	1348	2	)	)	PUNCT
ejpam-6171	1348	3	}	}	PUNCT
ejpam-6171	1348	4	.	.	PUNCT
ejpam-6171	1349	1	let	let	VERB
ejpam-6171	1349	2	x	x	SYM
ejpam-6171	1349	3	∈	∈	PROPN
ejpam-6171	1349	4	x.	x.	NOUN
ejpam-6171	1349	5	assume	assume	VERB
ejpam-6171	1349	6	that	that	SCONJ
ejpam-6171	1349	7	pf	pf	PROPN
ejpam-6171	1349	8	(	(	PUNCT
ejpam-6171	1349	9	0	0	NUM
ejpam-6171	1349	10	)	)	PUNCT
ejpam-6171	1349	11	<	<	X
ejpam-6171	1349	12	pf	pf	X
ejpam-6171	1349	13	(	(	PUNCT
ejpam-6171	1349	14	x	x	NOUN
ejpam-6171	1349	15	)	)	PUNCT
ejpam-6171	1349	16	.	.	PUNCT
ejpam-6171	1350	1	let	let	VERB
ejpam-6171	1350	2	γ	γ	X
ejpam-6171	1350	3	=	=	SYM
ejpam-6171	1350	4	pf	pf	X
ejpam-6171	1350	5	(	(	PUNCT
ejpam-6171	1350	6	0	0	NUM
ejpam-6171	1350	7	)	)	PUNCT
ejpam-6171	1350	8	.	.	PUNCT
ejpam-6171	1351	1	then	then	ADV
ejpam-6171	1351	2	x	x	SYM
ejpam-6171	1351	3	∈	∈	PROPN
ejpam-6171	1351	4	u	u	NOUN
ejpam-6171	1351	5	+	+	X
ejpam-6171	1351	6	(	(	PUNCT
ejpam-6171	1351	7	pf	pf	INTJ
ejpam-6171	1351	8	;	;	PUNCT
ejpam-6171	1351	9	γ	γ	X
ejpam-6171	1351	10	)	)	PUNCT
ejpam-6171	1351	11	6=	6=	ADP
ejpam-6171	1351	12	∅.	∅.	ADP
ejpam-6171	1351	13	by	by	ADP
ejpam-6171	1351	14	the	the	DET
ejpam-6171	1351	15	assumption	assumption	NOUN
ejpam-6171	1351	16	,	,	PUNCT
ejpam-6171	1351	17	we	we	PRON
ejpam-6171	1351	18	have	have	VERB
ejpam-6171	1351	19	u	u	NOUN
ejpam-6171	1351	20	+	+	CCONJ
ejpam-6171	1351	21	(	(	PUNCT
ejpam-6171	1351	22	pf	pf	INTJ
ejpam-6171	1351	23	;	;	PUNCT
ejpam-6171	1351	24	γ	γ	X
ejpam-6171	1351	25	)	)	PUNCT
ejpam-6171	1351	26	is	be	AUX
ejpam-6171	1351	27	an	an	DET
ejpam-6171	1351	28	iup	iup	NOUN
ejpam-6171	1351	29	-	-	PUNCT
ejpam-6171	1351	30	filter	filter	NOUN
ejpam-6171	1351	31	of	of	ADP
ejpam-6171	1351	32	x.	x.	NOUN
ejpam-6171	1351	33	by	by	ADP
ejpam-6171	1351	34	(	(	PUNCT
ejpam-6171	1351	35	2.18	2.18	NUM
ejpam-6171	1351	36	)	)	PUNCT
ejpam-6171	1351	37	,	,	PUNCT
ejpam-6171	1351	38	we	we	PRON
ejpam-6171	1351	39	have	have	VERB
ejpam-6171	1351	40	0	0	NUM
ejpam-6171	1351	41	∈	∈	PROPN
ejpam-6171	1351	42	u	u	NOUN
ejpam-6171	1351	43	+	+	X
ejpam-6171	1351	44	(	(	PUNCT
ejpam-6171	1351	45	pf	pf	INTJ
ejpam-6171	1351	46	;	;	PUNCT
ejpam-6171	1351	47	γ	γ	X
ejpam-6171	1351	48	)	)	PUNCT
ejpam-6171	1351	49	.	.	PUNCT
ejpam-6171	1352	1	so	so	ADV
ejpam-6171	1352	2	pf	pf	PROPN
ejpam-6171	1352	3	(	(	PUNCT
ejpam-6171	1352	4	0	0	NUM
ejpam-6171	1352	5	)	)	PUNCT
ejpam-6171	1352	6	>	>	X
ejpam-6171	1353	1	γ	γ	X
ejpam-6171	1353	2	=	=	SYM
ejpam-6171	1353	3	pf	pf	PROPN
ejpam-6171	1353	4	(	(	PUNCT
ejpam-6171	1353	5	0	0	NUM
ejpam-6171	1353	6	)	)	PUNCT
ejpam-6171	1353	7	,	,	PUNCT
ejpam-6171	1353	8	which	which	PRON
ejpam-6171	1353	9	is	be	AUX
ejpam-6171	1353	10	a	a	DET
ejpam-6171	1353	11	contradiction	contradiction	NOUN
ejpam-6171	1353	12	.	.	PUNCT
ejpam-6171	1354	1	thus	thus	ADV
ejpam-6171	1354	2	,	,	PUNCT
ejpam-6171	1354	3	pf	pf	PROPN
ejpam-6171	1354	4	(	(	PUNCT
ejpam-6171	1354	5	0	0	NUM
ejpam-6171	1354	6	)	)	PUNCT
ejpam-6171	1354	7	≥	≥	NOUN
ejpam-6171	1354	8	pf	pf	X
ejpam-6171	1354	9	(	(	PUNCT
ejpam-6171	1354	10	x	x	NOUN
ejpam-6171	1354	11	)	)	PUNCT
ejpam-6171	1354	12	.	.	PUNCT
ejpam-6171	1355	1	let	let	VERB
ejpam-6171	1355	2	x	x	PRON
ejpam-6171	1355	3	,	,	PUNCT
ejpam-6171	1355	4	y	y	PROPN
ejpam-6171	1355	5	∈	∈	PROPN
ejpam-6171	1355	6	x.	x.	NOUN
ejpam-6171	1355	7	assume	assume	VERB
ejpam-6171	1355	8	that	that	SCONJ
ejpam-6171	1355	9	pf	pf	PROPN
ejpam-6171	1355	10	(	(	PUNCT
ejpam-6171	1355	11	y	y	NOUN
ejpam-6171	1355	12	)	)	PUNCT
ejpam-6171	1355	13	<	<	X
ejpam-6171	1355	14	min{pf	min{pf	X
ejpam-6171	1355	15	(	(	PUNCT
ejpam-6171	1355	16	x	x	X
ejpam-6171	1355	17	?	?	PUNCT
ejpam-6171	1356	1	y),pf	y),pf	PROPN
ejpam-6171	1356	2	(	(	PUNCT
ejpam-6171	1356	3	x	x	NOUN
ejpam-6171	1356	4	)	)	PUNCT
ejpam-6171	1356	5	}	}	PUNCT
ejpam-6171	1356	6	.	.	PUNCT
ejpam-6171	1357	1	let	let	VERB
ejpam-6171	1357	2	γ	γ	X
ejpam-6171	1357	3	=	=	SYM
ejpam-6171	1357	4	pf	pf	PROPN
ejpam-6171	1357	5	(	(	PUNCT
ejpam-6171	1357	6	y	y	NOUN
ejpam-6171	1357	7	)	)	PUNCT
ejpam-6171	1357	8	.	.	PUNCT
ejpam-6171	1358	1	then	then	ADV
ejpam-6171	1358	2	x?y	x?y	PROPN
ejpam-6171	1358	3	,	,	PUNCT
ejpam-6171	1358	4	x	x	PUNCT
ejpam-6171	1358	5	∈	∈	PROPN
ejpam-6171	1358	6	u	u	NOUN
ejpam-6171	1358	7	+	+	X
ejpam-6171	1358	8	(	(	PUNCT
ejpam-6171	1358	9	pf	pf	INTJ
ejpam-6171	1358	10	;	;	PUNCT
ejpam-6171	1358	11	γ	γ	X
ejpam-6171	1358	12	)	)	PUNCT
ejpam-6171	1358	13	6=	6=	ADP
ejpam-6171	1358	14	∅.	∅.	ADP
ejpam-6171	1358	15	by	by	ADP
ejpam-6171	1358	16	the	the	DET
ejpam-6171	1358	17	assumption	assumption	NOUN
ejpam-6171	1358	18	,	,	PUNCT
ejpam-6171	1358	19	we	we	PRON
ejpam-6171	1358	20	have	have	VERB
ejpam-6171	1358	21	u	u	NOUN
ejpam-6171	1358	22	+	+	CCONJ
ejpam-6171	1358	23	(	(	PUNCT
ejpam-6171	1358	24	pf	pf	INTJ
ejpam-6171	1358	25	;	;	PUNCT
ejpam-6171	1358	26	γ	γ	X
ejpam-6171	1358	27	)	)	PUNCT
ejpam-6171	1358	28	is	be	AUX
ejpam-6171	1358	29	an	an	DET
ejpam-6171	1358	30	iup	iup	NOUN
ejpam-6171	1358	31	-	-	PUNCT
ejpam-6171	1358	32	filter	filter	NOUN
ejpam-6171	1358	33	of	of	ADP
ejpam-6171	1358	34	x.	x.	NOUN
ejpam-6171	1358	35	by	by	ADP
ejpam-6171	1358	36	(	(	PUNCT
ejpam-6171	1358	37	2.19	2.19	NUM
ejpam-6171	1358	38	)	)	PUNCT
ejpam-6171	1358	39	,	,	PUNCT
ejpam-6171	1358	40	we	we	PRON
ejpam-6171	1358	41	have	have	VERB
ejpam-6171	1358	42	y	y	PROPN
ejpam-6171	1358	43	∈	∈	PROPN
ejpam-6171	1358	44	u	u	NOUN
ejpam-6171	1358	45	+	+	X
ejpam-6171	1358	46	(	(	PUNCT
ejpam-6171	1358	47	pf	pf	INTJ
ejpam-6171	1358	48	;	;	PUNCT
ejpam-6171	1358	49	γ	γ	X
ejpam-6171	1358	50	)	)	PUNCT
ejpam-6171	1358	51	.	.	PUNCT
ejpam-6171	1359	1	so	so	ADV
ejpam-6171	1359	2	pf	pf	PROPN
ejpam-6171	1359	3	(	(	PUNCT
ejpam-6171	1359	4	y	y	PROPN
ejpam-6171	1359	5	)	)	PUNCT
ejpam-6171	1359	6	>	>	X
ejpam-6171	1360	1	γ	γ	X
ejpam-6171	1360	2	=	=	SYM
ejpam-6171	1360	3	pf	pf	PROPN
ejpam-6171	1360	4	(	(	PUNCT
ejpam-6171	1360	5	y	y	NOUN
ejpam-6171	1360	6	)	)	PUNCT
ejpam-6171	1360	7	,	,	PUNCT
ejpam-6171	1360	8	which	which	PRON
ejpam-6171	1360	9	is	be	AUX
ejpam-6171	1360	10	a	a	DET
ejpam-6171	1360	11	contradiction	contradiction	NOUN
ejpam-6171	1360	12	.	.	PUNCT
ejpam-6171	1361	1	thus	thus	ADV
ejpam-6171	1361	2	,	,	PUNCT
ejpam-6171	1361	3	pf	pf	PROPN
ejpam-6171	1361	4	(	(	PUNCT
ejpam-6171	1361	5	y	y	NOUN
ejpam-6171	1361	6	)	)	PUNCT
ejpam-6171	1361	7	≥	≥	NOUN
ejpam-6171	1361	8	min{pf	min{pf	X
ejpam-6171	1361	9	(	(	PUNCT
ejpam-6171	1361	10	x	x	X
ejpam-6171	1361	11	?	?	PUNCT
ejpam-6171	1362	1	y),pf	y),pf	PROPN
ejpam-6171	1362	2	(	(	PUNCT
ejpam-6171	1362	3	x	x	NOUN
ejpam-6171	1362	4	)	)	PUNCT
ejpam-6171	1362	5	}	}	PUNCT
ejpam-6171	1362	6	.	.	PUNCT
ejpam-6171	1363	1	k.	k.	PROPN
ejpam-6171	1364	1	suayngam	suayngam	PROPN
ejpam-6171	1364	2	et	et	PROPN
ejpam-6171	1364	3	al	al	PROPN
ejpam-6171	1364	4	.	.	PUNCT
ejpam-6171	1364	5	/	/	SYM
ejpam-6171	1364	6	eur	eur	PROPN
ejpam-6171	1364	7	.	.	PUNCT
ejpam-6171	1365	1	j.	j.	PROPN
ejpam-6171	1365	2	pure	pure	PROPN
ejpam-6171	1365	3	appl	appl	PROPN
ejpam-6171	1365	4	.	.	PROPN
ejpam-6171	1365	5	math	math	PROPN
ejpam-6171	1365	6	,	,	PUNCT
ejpam-6171	1365	7	18	18	NUM
ejpam-6171	1365	8	(	(	PUNCT
ejpam-6171	1365	9	3	3	NUM
ejpam-6171	1365	10	)	)	PUNCT
ejpam-6171	1365	11	(	(	PUNCT
ejpam-6171	1365	12	2025	2025	NUM
ejpam-6171	1365	13	)	)	PUNCT
ejpam-6171	1365	14	,	,	PUNCT
ejpam-6171	1365	15	6171	6171	NUM
ejpam-6171	1365	16	26	26	NUM
ejpam-6171	1365	17	of	of	ADP
ejpam-6171	1365	18	28	28	NUM
ejpam-6171	1365	19	hence	hence	ADV
ejpam-6171	1365	20	,	,	PUNCT
ejpam-6171	1365	21	p	p	PROPN
ejpam-6171	1365	22	is	be	AUX
ejpam-6171	1365	23	a	a	DET
ejpam-6171	1365	24	pythagorean	pythagorean	PROPN
ejpam-6171	1365	25	neutrosophic	neutrosophic	ADJ
ejpam-6171	1365	26	iup	iup	NOUN
ejpam-6171	1365	27	-	-	PUNCT
ejpam-6171	1365	28	filter	filter	NOUN
ejpam-6171	1365	29	of	of	ADP
ejpam-6171	1365	30	x.	x.	PROPN
ejpam-6171	1365	31	theorem	theorem	VERB
ejpam-6171	1365	32	25	25	NUM
ejpam-6171	1365	33	.	.	PUNCT
ejpam-6171	1366	1	a	a	DET
ejpam-6171	1366	2	pns	pns	NOUN
ejpam-6171	1366	3	p	p	NOUN
ejpam-6171	1366	4	in	in	ADP
ejpam-6171	1366	5	x	x	PROPN
ejpam-6171	1366	6	is	be	AUX
ejpam-6171	1366	7	a	a	DET
ejpam-6171	1366	8	pythagorean	pythagorean	PROPN
ejpam-6171	1366	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1366	10	strong	strong	ADJ
ejpam-6171	1366	11	iup	iup	NOUN
ejpam-6171	1366	12	-	-	PUNCT
ejpam-6171	1366	13	ideal	ideal	NOUN
ejpam-6171	1366	14	of	of	ADP
ejpam-6171	1366	15	x	x	SYM
ejpam-6171	1366	16	if	if	SCONJ
ejpam-6171	1366	17	and	and	CCONJ
ejpam-6171	1366	18	only	only	ADV
ejpam-6171	1366	19	if	if	SCONJ
ejpam-6171	1366	20	for	for	ADP
ejpam-6171	1366	21	all	all	DET
ejpam-6171	1366	22	α	α	NOUN
ejpam-6171	1366	23	,	,	PUNCT
ejpam-6171	1366	24	β	β	X
ejpam-6171	1366	25	,	,	PUNCT
ejpam-6171	1366	26	γ	γ	PROPN
ejpam-6171	1366	27	∈	∈	PROPN
ejpam-6171	1367	1	[	[	X
ejpam-6171	1367	2	0	0	NUM
ejpam-6171	1367	3	,	,	PUNCT
ejpam-6171	1367	4	1	1	NUM
ejpam-6171	1367	5	]	]	PUNCT
ejpam-6171	1367	6	,	,	PUNCT
ejpam-6171	1367	7	the	the	DET
ejpam-6171	1367	8	sets	set	NOUN
ejpam-6171	1367	9	u	u	NOUN
ejpam-6171	1367	10	+	+	X
ejpam-6171	1367	11	(	(	PUNCT
ejpam-6171	1367	12	pt	pt	INTJ
ejpam-6171	1367	13	;	;	PUNCT
ejpam-6171	1367	14	α	α	X
ejpam-6171	1367	15	)	)	PUNCT
ejpam-6171	1367	16	,	,	PUNCT
ejpam-6171	1367	17	l	l	NOUN
ejpam-6171	1367	18	−	−	PROPN
ejpam-6171	1367	19	(	(	PUNCT
ejpam-6171	1367	20	pi	pi	NOUN
ejpam-6171	1367	21	;	;	PUNCT
ejpam-6171	1367	22	β	β	X
ejpam-6171	1367	23	)	)	PUNCT
ejpam-6171	1367	24	,	,	PUNCT
ejpam-6171	1367	25	and	and	CCONJ
ejpam-6171	1367	26	u	u	NOUN
ejpam-6171	1367	27	+	+	CCONJ
ejpam-6171	1367	28	(	(	PUNCT
ejpam-6171	1367	29	pf	pf	INTJ
ejpam-6171	1367	30	;	;	PUNCT
ejpam-6171	1367	31	γ	γ	X
ejpam-6171	1367	32	)	)	PUNCT
ejpam-6171	1367	33	are	be	AUX
ejpam-6171	1367	34	either	either	CCONJ
ejpam-6171	1367	35	empty	empty	ADJ
ejpam-6171	1367	36	or	or	CCONJ
ejpam-6171	1367	37	strong	strong	ADJ
ejpam-6171	1367	38	iup	iup	NOUN
ejpam-6171	1367	39	-	-	PUNCT
ejpam-6171	1367	40	ideals	ideal	NOUN
ejpam-6171	1367	41	of	of	ADP
ejpam-6171	1367	42	x.	x.	NOUN
ejpam-6171	1367	43	proof	proof	NOUN
ejpam-6171	1367	44	.	.	PUNCT
ejpam-6171	1368	1	it	it	PRON
ejpam-6171	1368	2	is	be	AUX
ejpam-6171	1368	3	straightforward	straightforward	ADJ
ejpam-6171	1368	4	by	by	ADP
ejpam-6171	1368	5	theorem	theorem	NOUN
ejpam-6171	1368	6	2	2	NUM
ejpam-6171	1368	7	.	.	PUNCT
ejpam-6171	1368	8	definition	definition	NOUN
ejpam-6171	1368	9	11	11	NUM
ejpam-6171	1368	10	.	.	PUNCT
ejpam-6171	1369	1	[	[	X
ejpam-6171	1369	2	18	18	NUM
ejpam-6171	1369	3	]	]	PUNCT
ejpam-6171	1369	4	let	let	VERB
ejpam-6171	1369	5	p	p	PRON
ejpam-6171	1369	6	be	be	AUX
ejpam-6171	1369	7	a	a	DET
ejpam-6171	1369	8	pns	pns	NOUN
ejpam-6171	1369	9	in	in	ADP
ejpam-6171	1369	10	x.	x.	PROPN
ejpam-6171	1369	11	for	for	ADP
ejpam-6171	1369	12	any	any	DET
ejpam-6171	1369	13	α	α	NOUN
ejpam-6171	1369	14	,	,	PUNCT
ejpam-6171	1369	15	β	β	X
ejpam-6171	1369	16	,	,	PUNCT
ejpam-6171	1369	17	γ	γ	PROPN
ejpam-6171	1369	18	∈	∈	PROPN
ejpam-6171	1370	1	[	[	X
ejpam-6171	1370	2	0	0	NUM
ejpam-6171	1370	3	,	,	PUNCT
ejpam-6171	1370	4	1	1	NUM
ejpam-6171	1370	5	]	]	PUNCT
ejpam-6171	1370	6	,	,	PUNCT
ejpam-6171	1370	7	the	the	DET
ejpam-6171	1370	8	sets	set	NOUN
ejpam-6171	1370	9	ulup(α	ulup(α	PROPN
ejpam-6171	1370	10	,	,	PUNCT
ejpam-6171	1370	11	β	β	X
ejpam-6171	1370	12	,	,	PUNCT
ejpam-6171	1370	13	γ	γ	NOUN
ejpam-6171	1370	14	)	)	PUNCT
ejpam-6171	1370	15	=	=	SYM
ejpam-6171	1370	16	{	{	PUNCT
ejpam-6171	1370	17	x	x	PUNCT
ejpam-6171	1370	18	∈	∈	PROPN
ejpam-6171	1370	19	x	x	INTJ
ejpam-6171	1370	20	|	|	ADV
ejpam-6171	1370	21	pt	pt	X
ejpam-6171	1370	22	(	(	PUNCT
ejpam-6171	1370	23	x	x	NOUN
ejpam-6171	1370	24	)	)	PUNCT
ejpam-6171	1370	25	≥	≥	NOUN
ejpam-6171	1370	26	α	α	NOUN
ejpam-6171	1370	27	,	,	PUNCT
ejpam-6171	1370	28	pi(x	pi(x	NUM
ejpam-6171	1370	29	)	)	PUNCT
ejpam-6171	1370	30	≤	≤	NUM
ejpam-6171	1371	1	β	β	X
ejpam-6171	1371	2	,	,	PUNCT
ejpam-6171	1371	3	pf	pf	PROPN
ejpam-6171	1371	4	(	(	PUNCT
ejpam-6171	1371	5	x	x	NOUN
ejpam-6171	1371	6	)	)	PUNCT
ejpam-6171	1371	7	≥	≥	NOUN
ejpam-6171	1371	8	γ	γ	NOUN
ejpam-6171	1371	9	}	}	PUNCT
ejpam-6171	1371	10	,	,	PUNCT
ejpam-6171	1371	11	(	(	PUNCT
ejpam-6171	1371	12	3.28	3.28	NUM
ejpam-6171	1371	13	)	)	PUNCT
ejpam-6171	1371	14	lulp(α	lulp(α	PROPN
ejpam-6171	1371	15	,	,	PUNCT
ejpam-6171	1371	16	β	β	X
ejpam-6171	1371	17	,	,	PUNCT
ejpam-6171	1371	18	γ	γ	NOUN
ejpam-6171	1371	19	)	)	PUNCT
ejpam-6171	1371	20	=	=	SYM
ejpam-6171	1371	21	{	{	PUNCT
ejpam-6171	1371	22	x	x	PUNCT
ejpam-6171	1371	23	∈	∈	PROPN
ejpam-6171	1371	24	x	x	INTJ
ejpam-6171	1371	25	|	|	ADV
ejpam-6171	1371	26	pt	pt	X
ejpam-6171	1371	27	(	(	PUNCT
ejpam-6171	1371	28	x	x	NOUN
ejpam-6171	1371	29	)	)	PUNCT
ejpam-6171	1371	30	≤	≤	NOUN
ejpam-6171	1371	31	α	α	X
ejpam-6171	1371	32	,	,	PUNCT
ejpam-6171	1371	33	pi(x	pi(x	NUM
ejpam-6171	1371	34	)	)	PUNCT
ejpam-6171	1371	35	≥	≥	X
ejpam-6171	1371	36	β	β	NOUN
ejpam-6171	1371	37	,	,	PUNCT
ejpam-6171	1371	38	pf	pf	PROPN
ejpam-6171	1371	39	(	(	PUNCT
ejpam-6171	1371	40	x	x	NOUN
ejpam-6171	1371	41	)	)	PUNCT
ejpam-6171	1371	42	≤	≤	NOUN
ejpam-6171	1371	43	γ	γ	X
ejpam-6171	1371	44	}	}	PUNCT
ejpam-6171	1371	45	,	,	PUNCT
ejpam-6171	1371	46	(	(	PUNCT
ejpam-6171	1371	47	3.29	3.29	NUM
ejpam-6171	1371	48	)	)	PUNCT
ejpam-6171	1371	49	ep(α	ep(α	NUM
ejpam-6171	1371	50	,	,	PUNCT
ejpam-6171	1371	51	β	β	X
ejpam-6171	1371	52	,	,	PUNCT
ejpam-6171	1371	53	γ	γ	NOUN
ejpam-6171	1371	54	)	)	PUNCT
ejpam-6171	1371	55	=	=	SYM
ejpam-6171	1371	56	{	{	PUNCT
ejpam-6171	1371	57	x	x	PUNCT
ejpam-6171	1371	58	∈	∈	PROPN
ejpam-6171	1371	59	x	x	INTJ
ejpam-6171	1371	60	|	|	ADV
ejpam-6171	1371	61	pt	pt	X
ejpam-6171	1371	62	(	(	PUNCT
ejpam-6171	1371	63	x	x	NOUN
ejpam-6171	1371	64	)	)	PUNCT
ejpam-6171	1371	65	=	=	SYM
ejpam-6171	1371	66	α	α	NOUN
ejpam-6171	1371	67	,	,	PUNCT
ejpam-6171	1371	68	pi(x	pi(x	NUM
ejpam-6171	1371	69	)	)	PUNCT
ejpam-6171	1371	70	=	=	SYM
ejpam-6171	1372	1	β	β	X
ejpam-6171	1372	2	,	,	PUNCT
ejpam-6171	1372	3	pf	pf	PROPN
ejpam-6171	1372	4	(	(	PUNCT
ejpam-6171	1372	5	x	x	NOUN
ejpam-6171	1372	6	)	)	PUNCT
ejpam-6171	1372	7	=	=	SYM
ejpam-6171	1372	8	γ	γ	X
ejpam-6171	1372	9	}	}	PUNCT
ejpam-6171	1372	10	(	(	PUNCT
ejpam-6171	1372	11	3.30	3.30	NUM
ejpam-6171	1372	12	)	)	PUNCT
ejpam-6171	1372	13	are	be	AUX
ejpam-6171	1372	14	called	call	VERB
ejpam-6171	1372	15	a	a	DET
ejpam-6171	1372	16	ulu	ulu	PROPN
ejpam-6171	1372	17	-(α	-(α	PUNCT
ejpam-6171	1372	18	,	,	PUNCT
ejpam-6171	1372	19	β	β	X
ejpam-6171	1372	20	,	,	PUNCT
ejpam-6171	1372	21	γ)-level	γ)-level	PROPN
ejpam-6171	1372	22	subset	subset	NOUN
ejpam-6171	1372	23	,	,	PUNCT
ejpam-6171	1372	24	an	an	DET
ejpam-6171	1372	25	lul-(α	lul-(α	NOUN
ejpam-6171	1372	26	,	,	PUNCT
ejpam-6171	1372	27	β	β	X
ejpam-6171	1372	28	,	,	PUNCT
ejpam-6171	1372	29	γ)-level	γ)-level	PROPN
ejpam-6171	1372	30	subset	subset	NOUN
ejpam-6171	1372	31	,	,	PUNCT
ejpam-6171	1372	32	and	and	CCONJ
ejpam-6171	1372	33	an	an	DET
ejpam-6171	1372	34	e-(α	e-(α	ADJ
ejpam-6171	1372	35	,	,	PUNCT
ejpam-6171	1372	36	β	β	NOUN
ejpam-6171	1372	37	,	,	PUNCT
ejpam-6171	1372	38	γ)level	γ)level	PROPN
ejpam-6171	1372	39	subset	subset	NOUN
ejpam-6171	1372	40	of	of	ADP
ejpam-6171	1372	41	p	p	NOUN
ejpam-6171	1372	42	,	,	PUNCT
ejpam-6171	1372	43	respectively	respectively	ADV
ejpam-6171	1372	44	.	.	PUNCT
ejpam-6171	1373	1	the	the	DET
ejpam-6171	1373	2	following	follow	VERB
ejpam-6171	1373	3	five	five	NUM
ejpam-6171	1373	4	corollaries	corollary	NOUN
ejpam-6171	1373	5	are	be	AUX
ejpam-6171	1373	6	derived	derive	VERB
ejpam-6171	1373	7	directly	directly	ADV
ejpam-6171	1373	8	by	by	ADP
ejpam-6171	1373	9	applying	apply	VERB
ejpam-6171	1373	10	theorems	theorem	NOUN
ejpam-6171	1373	11	17	17	NUM
ejpam-6171	1373	12	,	,	PUNCT
ejpam-6171	1373	13	18	18	NUM
ejpam-6171	1373	14	,	,	PUNCT
ejpam-6171	1373	15	19	19	NUM
ejpam-6171	1373	16	,	,	PUNCT
ejpam-6171	1373	17	20	20	NUM
ejpam-6171	1373	18	,	,	PUNCT
ejpam-6171	1373	19	and	and	CCONJ
ejpam-6171	1373	20	21	21	NUM
ejpam-6171	1373	21	,	,	PUNCT
ejpam-6171	1373	22	respectively	respectively	ADV
ejpam-6171	1373	23	.	.	PUNCT
ejpam-6171	1374	1	corollary	corollary	ADJ
ejpam-6171	1374	2	1	1	NUM
ejpam-6171	1374	3	.	.	PUNCT
ejpam-6171	1375	1	a	a	DET
ejpam-6171	1375	2	pns	pns	NOUN
ejpam-6171	1375	3	p	p	NOUN
ejpam-6171	1375	4	in	in	ADP
ejpam-6171	1375	5	x	x	PROPN
ejpam-6171	1375	6	is	be	AUX
ejpam-6171	1375	7	a	a	DET
ejpam-6171	1375	8	pythagorean	pythagorean	PROPN
ejpam-6171	1375	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1375	10	iup	iup	NOUN
ejpam-6171	1375	11	-	-	PUNCT
ejpam-6171	1375	12	subalgebra	subalgebra	NOUN
ejpam-6171	1375	13	of	of	ADP
ejpam-6171	1375	14	x	x	PRON
ejpam-6171	1375	15	if	if	SCONJ
ejpam-6171	1375	16	and	and	CCONJ
ejpam-6171	1375	17	only	only	ADV
ejpam-6171	1375	18	if	if	SCONJ
ejpam-6171	1375	19	for	for	ADP
ejpam-6171	1375	20	all	all	DET
ejpam-6171	1375	21	α	α	NOUN
ejpam-6171	1375	22	,	,	PUNCT
ejpam-6171	1375	23	β	β	X
ejpam-6171	1375	24	,	,	PUNCT
ejpam-6171	1375	25	γ	γ	PROPN
ejpam-6171	1375	26	∈	∈	PROPN
ejpam-6171	1376	1	[	[	X
ejpam-6171	1376	2	0	0	NUM
ejpam-6171	1376	3	,	,	PUNCT
ejpam-6171	1376	4	1	1	NUM
ejpam-6171	1376	5	]	]	PUNCT
ejpam-6171	1376	6	,	,	PUNCT
ejpam-6171	1376	7	the	the	DET
ejpam-6171	1376	8	set	set	PROPN
ejpam-6171	1376	9	ulup(α	ulup(α	PROPN
ejpam-6171	1376	10	,	,	PUNCT
ejpam-6171	1376	11	β	β	X
ejpam-6171	1376	12	,	,	PUNCT
ejpam-6171	1376	13	γ	γ	PROPN
ejpam-6171	1376	14	)	)	PUNCT
ejpam-6171	1376	15	is	be	AUX
ejpam-6171	1376	16	either	either	CCONJ
ejpam-6171	1376	17	empty	empty	ADJ
ejpam-6171	1376	18	or	or	CCONJ
ejpam-6171	1376	19	an	an	DET
ejpam-6171	1376	20	iup	iup	NOUN
ejpam-6171	1376	21	-	-	PUNCT
ejpam-6171	1376	22	subalgebra	subalgebra	NOUN
ejpam-6171	1376	23	of	of	ADP
ejpam-6171	1376	24	x.	x.	PROPN
ejpam-6171	1376	25	corollary	corollary	PROPN
ejpam-6171	1376	26	2	2	PROPN
ejpam-6171	1376	27	.	.	PUNCT
ejpam-6171	1376	28	a	a	DET
ejpam-6171	1376	29	pns	pns	NOUN
ejpam-6171	1376	30	p	p	NOUN
ejpam-6171	1376	31	in	in	ADP
ejpam-6171	1376	32	x	x	PROPN
ejpam-6171	1376	33	is	be	AUX
ejpam-6171	1376	34	a	a	DET
ejpam-6171	1376	35	pythagorean	pythagorean	PROPN
ejpam-6171	1376	36	neutrosophic	neutrosophic	ADJ
ejpam-6171	1376	37	iup	iup	PROPN
ejpam-6171	1376	38	-	-	PUNCT
ejpam-6171	1376	39	ideal	ideal	NOUN
ejpam-6171	1376	40	of	of	ADP
ejpam-6171	1376	41	x	x	SYM
ejpam-6171	1376	42	if	if	SCONJ
ejpam-6171	1376	43	and	and	CCONJ
ejpam-6171	1376	44	only	only	ADV
ejpam-6171	1376	45	if	if	SCONJ
ejpam-6171	1376	46	for	for	ADP
ejpam-6171	1376	47	all	all	DET
ejpam-6171	1376	48	α	α	NOUN
ejpam-6171	1376	49	,	,	PUNCT
ejpam-6171	1376	50	β	β	X
ejpam-6171	1376	51	,	,	PUNCT
ejpam-6171	1376	52	γ	γ	PROPN
ejpam-6171	1376	53	∈	∈	PROPN
ejpam-6171	1377	1	[	[	X
ejpam-6171	1377	2	0	0	NUM
ejpam-6171	1377	3	,	,	PUNCT
ejpam-6171	1377	4	1	1	NUM
ejpam-6171	1377	5	]	]	PUNCT
ejpam-6171	1377	6	,	,	PUNCT
ejpam-6171	1377	7	the	the	DET
ejpam-6171	1377	8	set	set	PROPN
ejpam-6171	1377	9	ulup(α	ulup(α	PROPN
ejpam-6171	1377	10	,	,	PUNCT
ejpam-6171	1377	11	β	β	X
ejpam-6171	1377	12	,	,	PUNCT
ejpam-6171	1377	13	γ	γ	PROPN
ejpam-6171	1377	14	)	)	PUNCT
ejpam-6171	1377	15	is	be	AUX
ejpam-6171	1377	16	either	either	CCONJ
ejpam-6171	1377	17	empty	empty	ADJ
ejpam-6171	1377	18	or	or	CCONJ
ejpam-6171	1377	19	an	an	DET
ejpam-6171	1377	20	iup	iup	NOUN
ejpam-6171	1377	21	-	-	PUNCT
ejpam-6171	1377	22	ideal	ideal	NOUN
ejpam-6171	1377	23	of	of	ADP
ejpam-6171	1377	24	x.	x.	PROPN
ejpam-6171	1377	25	corollary	corollary	PROPN
ejpam-6171	1377	26	3	3	NUM
ejpam-6171	1377	27	.	.	PUNCT
ejpam-6171	1378	1	a	a	DET
ejpam-6171	1378	2	pns	pns	NOUN
ejpam-6171	1378	3	p	p	NOUN
ejpam-6171	1378	4	in	in	ADP
ejpam-6171	1378	5	x	x	PROPN
ejpam-6171	1378	6	is	be	AUX
ejpam-6171	1378	7	a	a	DET
ejpam-6171	1378	8	pythagorean	pythagorean	PROPN
ejpam-6171	1378	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1378	10	iup	iup	NOUN
ejpam-6171	1378	11	-	-	PUNCT
ejpam-6171	1378	12	filter	filter	NOUN
ejpam-6171	1378	13	of	of	ADP
ejpam-6171	1378	14	x	x	SYM
ejpam-6171	1378	15	if	if	SCONJ
ejpam-6171	1378	16	and	and	CCONJ
ejpam-6171	1378	17	only	only	ADV
ejpam-6171	1378	18	if	if	SCONJ
ejpam-6171	1378	19	for	for	ADP
ejpam-6171	1378	20	all	all	DET
ejpam-6171	1378	21	α	α	NOUN
ejpam-6171	1378	22	,	,	PUNCT
ejpam-6171	1378	23	β	β	X
ejpam-6171	1378	24	,	,	PUNCT
ejpam-6171	1378	25	γ	γ	PROPN
ejpam-6171	1378	26	∈	∈	PROPN
ejpam-6171	1379	1	[	[	X
ejpam-6171	1379	2	0	0	NUM
ejpam-6171	1379	3	,	,	PUNCT
ejpam-6171	1379	4	1	1	NUM
ejpam-6171	1379	5	]	]	PUNCT
ejpam-6171	1379	6	,	,	PUNCT
ejpam-6171	1379	7	the	the	DET
ejpam-6171	1379	8	set	set	PROPN
ejpam-6171	1379	9	ulup(α	ulup(α	PROPN
ejpam-6171	1379	10	,	,	PUNCT
ejpam-6171	1379	11	β	β	X
ejpam-6171	1379	12	,	,	PUNCT
ejpam-6171	1379	13	γ	γ	PROPN
ejpam-6171	1379	14	)	)	PUNCT
ejpam-6171	1379	15	is	be	AUX
ejpam-6171	1379	16	either	either	CCONJ
ejpam-6171	1379	17	empty	empty	ADJ
ejpam-6171	1379	18	or	or	CCONJ
ejpam-6171	1379	19	an	an	DET
ejpam-6171	1379	20	iup	iup	NOUN
ejpam-6171	1379	21	-	-	PUNCT
ejpam-6171	1379	22	filter	filter	NOUN
ejpam-6171	1379	23	of	of	ADP
ejpam-6171	1379	24	x.	x.	NOUN
ejpam-6171	1379	25	corollary	corollary	PROPN
ejpam-6171	1379	26	4	4	NUM
ejpam-6171	1379	27	.	.	PUNCT
ejpam-6171	1380	1	a	a	DET
ejpam-6171	1380	2	pns	pns	NOUN
ejpam-6171	1380	3	p	p	NOUN
ejpam-6171	1380	4	in	in	ADP
ejpam-6171	1380	5	x	x	PROPN
ejpam-6171	1380	6	is	be	AUX
ejpam-6171	1380	7	a	a	DET
ejpam-6171	1380	8	pythagorean	pythagorean	PROPN
ejpam-6171	1380	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1380	10	strong	strong	ADJ
ejpam-6171	1380	11	iup	iup	NOUN
ejpam-6171	1380	12	-	-	PUNCT
ejpam-6171	1380	13	ideal	ideal	NOUN
ejpam-6171	1380	14	of	of	ADP
ejpam-6171	1380	15	x	x	SYM
ejpam-6171	1380	16	if	if	SCONJ
ejpam-6171	1380	17	and	and	CCONJ
ejpam-6171	1380	18	only	only	ADV
ejpam-6171	1380	19	if	if	SCONJ
ejpam-6171	1380	20	for	for	ADP
ejpam-6171	1380	21	all	all	DET
ejpam-6171	1380	22	α	α	NOUN
ejpam-6171	1380	23	,	,	PUNCT
ejpam-6171	1380	24	β	β	X
ejpam-6171	1380	25	,	,	PUNCT
ejpam-6171	1380	26	γ	γ	PROPN
ejpam-6171	1380	27	∈	∈	PROPN
ejpam-6171	1381	1	[	[	X
ejpam-6171	1381	2	0	0	NUM
ejpam-6171	1381	3	,	,	PUNCT
ejpam-6171	1381	4	1	1	NUM
ejpam-6171	1381	5	]	]	PUNCT
ejpam-6171	1381	6	,	,	PUNCT
ejpam-6171	1381	7	the	the	DET
ejpam-6171	1381	8	set	set	PROPN
ejpam-6171	1381	9	ulup(α	ulup(α	PROPN
ejpam-6171	1381	10	,	,	PUNCT
ejpam-6171	1381	11	β	β	X
ejpam-6171	1381	12	,	,	PUNCT
ejpam-6171	1381	13	γ	γ	PROPN
ejpam-6171	1381	14	)	)	PUNCT
ejpam-6171	1381	15	is	be	AUX
ejpam-6171	1381	16	either	either	CCONJ
ejpam-6171	1381	17	empty	empty	ADJ
ejpam-6171	1381	18	or	or	CCONJ
ejpam-6171	1381	19	an	an	DET
ejpam-6171	1381	20	strong	strong	ADJ
ejpam-6171	1381	21	iup	iup	NOUN
ejpam-6171	1381	22	-	-	PUNCT
ejpam-6171	1381	23	ideal	ideal	NOUN
ejpam-6171	1381	24	of	of	ADP
ejpam-6171	1381	25	x.	x.	PROPN
ejpam-6171	1381	26	corollary	corollary	PROPN
ejpam-6171	1381	27	5	5	NUM
ejpam-6171	1381	28	.	.	PUNCT
ejpam-6171	1382	1	a	a	DET
ejpam-6171	1382	2	pns	pns	NOUN
ejpam-6171	1382	3	p	p	NOUN
ejpam-6171	1382	4	in	in	ADP
ejpam-6171	1382	5	x	x	PROPN
ejpam-6171	1382	6	is	be	AUX
ejpam-6171	1382	7	a	a	DET
ejpam-6171	1382	8	pythagorean	pythagorean	PROPN
ejpam-6171	1382	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1382	10	strong	strong	ADJ
ejpam-6171	1382	11	iup	iup	NOUN
ejpam-6171	1382	12	-	-	PUNCT
ejpam-6171	1382	13	ideal	ideal	NOUN
ejpam-6171	1382	14	of	of	ADP
ejpam-6171	1382	15	x	x	SYM
ejpam-6171	1382	16	if	if	SCONJ
ejpam-6171	1382	17	and	and	CCONJ
ejpam-6171	1382	18	only	only	ADV
ejpam-6171	1382	19	if	if	SCONJ
ejpam-6171	1382	20	the	the	DET
ejpam-6171	1382	21	set	set	NOUN
ejpam-6171	1382	22	ep(pt	ep(pt	X
ejpam-6171	1382	23	(	(	PUNCT
ejpam-6171	1382	24	0),pi(0),pf	0),pi(0),pf	NOUN
ejpam-6171	1382	25	(	(	PUNCT
ejpam-6171	1382	26	0	0	NUM
ejpam-6171	1382	27	)	)	PUNCT
ejpam-6171	1382	28	)	)	PUNCT
ejpam-6171	1382	29	is	be	AUX
ejpam-6171	1382	30	a	a	DET
ejpam-6171	1382	31	strong	strong	ADJ
ejpam-6171	1382	32	iup	iup	NOUN
ejpam-6171	1382	33	-	-	PUNCT
ejpam-6171	1382	34	ideal	ideal	NOUN
ejpam-6171	1382	35	of	of	ADP
ejpam-6171	1382	36	x	x	PRON
ejpam-6171	1382	37	,	,	PUNCT
ejpam-6171	1382	38	that	that	ADV
ejpam-6171	1382	39	is	is	ADV
ejpam-6171	1382	40	,	,	PUNCT
ejpam-6171	1382	41	e(pt	e(pt	X
ejpam-6171	1382	42	,	,	PUNCT
ejpam-6171	1382	43	pt	pt	X
ejpam-6171	1382	44	(	(	PUNCT
ejpam-6171	1382	45	0	0	NUM
ejpam-6171	1382	46	)	)	PUNCT
ejpam-6171	1382	47	)	)	PUNCT
ejpam-6171	1383	1	=	=	SYM
ejpam-6171	1383	2	x	x	NOUN
ejpam-6171	1383	3	,	,	PUNCT
ejpam-6171	1383	4	e(pi	e(pi	NUM
ejpam-6171	1383	5	,	,	PUNCT
ejpam-6171	1383	6	pi(0	pi(0	PROPN
ejpam-6171	1383	7	)	)	PUNCT
ejpam-6171	1383	8	)	)	PUNCT
ejpam-6171	1384	1	=	=	PUNCT
ejpam-6171	1385	1	x	x	X
ejpam-6171	1385	2	and	and	CCONJ
ejpam-6171	1385	3	e(pf	e(pf	NUM
ejpam-6171	1385	4	,	,	PUNCT
ejpam-6171	1385	5	pf	pf	PROPN
ejpam-6171	1385	6	(	(	PUNCT
ejpam-6171	1385	7	0	0	NUM
ejpam-6171	1385	8	)	)	PUNCT
ejpam-6171	1385	9	)	)	PUNCT
ejpam-6171	1386	1	=	=	PUNCT
ejpam-6171	1386	2	x.	x.	NOUN
ejpam-6171	1387	1	4	4	X
ejpam-6171	1387	2	.	.	PUNCT
ejpam-6171	1387	3	conclusion	conclusion	NOUN
ejpam-6171	1387	4	in	in	ADP
ejpam-6171	1387	5	this	this	DET
ejpam-6171	1387	6	study	study	NOUN
ejpam-6171	1387	7	,	,	PUNCT
ejpam-6171	1387	8	we	we	PRON
ejpam-6171	1387	9	introduced	introduce	VERB
ejpam-6171	1387	10	and	and	CCONJ
ejpam-6171	1387	11	examined	examine	VERB
ejpam-6171	1387	12	the	the	DET
ejpam-6171	1387	13	concepts	concept	NOUN
ejpam-6171	1387	14	of	of	ADP
ejpam-6171	1387	15	pythagorean	pythagorean	PROPN
ejpam-6171	1387	16	neutrosophic	neutrosophic	PROPN
ejpam-6171	1387	17	iup	iup	PROPN
ejpam-6171	1387	18	-	-	PUNCT
ejpam-6171	1387	19	subalgebras	subalgebras	PROPN
ejpam-6171	1387	20	,	,	PUNCT
ejpam-6171	1387	21	pythagorean	pythagorean	PROPN
ejpam-6171	1387	22	neutrosophic	neutrosophic	PROPN
ejpam-6171	1387	23	iup	iup	PROPN
ejpam-6171	1387	24	-	-	PUNCT
ejpam-6171	1387	25	ideals	ideal	NOUN
ejpam-6171	1387	26	,	,	PUNCT
ejpam-6171	1387	27	pythagorean	pythagorean	PROPN
ejpam-6171	1387	28	neutrosophic	neutrosophic	ADJ
ejpam-6171	1387	29	iupfilters	iupfilter	NOUN
ejpam-6171	1387	30	,	,	PUNCT
ejpam-6171	1387	31	and	and	CCONJ
ejpam-6171	1387	32	pythagorean	pythagorean	PROPN
ejpam-6171	1387	33	neutrosophic	neutrosophic	PROPN
ejpam-6171	1387	34	strong	strong	ADJ
ejpam-6171	1387	35	iup	iup	NOUN
ejpam-6171	1387	36	-	-	PUNCT
ejpam-6171	1387	37	ideals	ideal	NOUN
ejpam-6171	1387	38	within	within	ADP
ejpam-6171	1387	39	the	the	DET
ejpam-6171	1387	40	framework	framework	NOUN
ejpam-6171	1387	41	of	of	ADP
ejpam-6171	1387	42	iupalgebras	iupalgebra	NOUN
ejpam-6171	1387	43	.	.	PUNCT
ejpam-6171	1388	1	we	we	PRON
ejpam-6171	1388	2	established	establish	VERB
ejpam-6171	1388	3	their	their	PRON
ejpam-6171	1388	4	fundamental	fundamental	ADJ
ejpam-6171	1388	5	properties	property	NOUN
ejpam-6171	1388	6	and	and	CCONJ
ejpam-6171	1388	7	provided	provide	VERB
ejpam-6171	1388	8	necessary	necessary	ADJ
ejpam-6171	1388	9	and	and	CCONJ
ejpam-6171	1388	10	sufficient	sufficient	ADJ
ejpam-6171	1388	11	conditions	condition	NOUN
ejpam-6171	1388	12	for	for	ADP
ejpam-6171	1388	13	pythagorean	pythagorean	PROPN
ejpam-6171	1388	14	neutrosophic	neutrosophic	ADJ
ejpam-6171	1388	15	sets	set	NOUN
ejpam-6171	1388	16	to	to	PART
ejpam-6171	1388	17	qualify	qualify	VERB
ejpam-6171	1388	18	as	as	ADP
ejpam-6171	1388	19	these	these	DET
ejpam-6171	1388	20	algebraic	algebraic	ADJ
ejpam-6171	1388	21	subsets	subset	NOUN
ejpam-6171	1388	22	.	.	PUNCT
ejpam-6171	1389	1	k.	k.	PROPN
ejpam-6171	1389	2	suayngam	suayngam	PROPN
ejpam-6171	1389	3	et	et	PROPN
ejpam-6171	1389	4	al	al	PROPN
ejpam-6171	1389	5	.	.	PUNCT
ejpam-6171	1389	6	/	/	SYM
ejpam-6171	1389	7	eur	eur	PROPN
ejpam-6171	1389	8	.	.	PUNCT
ejpam-6171	1390	1	j.	j.	PROPN
ejpam-6171	1390	2	pure	pure	PROPN
ejpam-6171	1390	3	appl	appl	PROPN
ejpam-6171	1390	4	.	.	PROPN
ejpam-6171	1390	5	math	math	PROPN
ejpam-6171	1390	6	,	,	PUNCT
ejpam-6171	1390	7	18	18	NUM
ejpam-6171	1390	8	(	(	PUNCT
ejpam-6171	1390	9	3	3	NUM
ejpam-6171	1390	10	)	)	PUNCT
ejpam-6171	1390	11	(	(	PUNCT
ejpam-6171	1390	12	2025	2025	NUM
ejpam-6171	1390	13	)	)	PUNCT
ejpam-6171	1390	14	,	,	PUNCT
ejpam-6171	1390	15	6171	6171	NUM
ejpam-6171	1390	16	27	27	NUM
ejpam-6171	1390	17	of	of	ADP
ejpam-6171	1390	18	28	28	NUM
ejpam-6171	1390	19	additionally	additionally	ADV
ejpam-6171	1391	1	,	,	PUNCT
ejpam-6171	1391	2	we	we	PRON
ejpam-6171	1391	3	investigated	investigate	VERB
ejpam-6171	1391	4	the	the	DET
ejpam-6171	1391	5	relationships	relationship	NOUN
ejpam-6171	1391	6	between	between	ADP
ejpam-6171	1391	7	these	these	DET
ejpam-6171	1391	8	subsets	subset	NOUN
ejpam-6171	1391	9	and	and	CCONJ
ejpam-6171	1391	10	their	their	PRON
ejpam-6171	1391	11	level	level	NOUN
ejpam-6171	1391	12	subsets	subset	NOUN
ejpam-6171	1391	13	,	,	PUNCT
ejpam-6171	1391	14	revealing	reveal	VERB
ejpam-6171	1391	15	significant	significant	ADJ
ejpam-6171	1391	16	structural	structural	ADJ
ejpam-6171	1391	17	interdependencies	interdependency	NOUN
ejpam-6171	1391	18	.	.	PUNCT
ejpam-6171	1392	1	by	by	ADP
ejpam-6171	1392	2	integrating	integrate	VERB
ejpam-6171	1392	3	pythagorean	pythagorean	PROPN
ejpam-6171	1392	4	neutrosophic	neutrosophic	ADJ
ejpam-6171	1392	5	sets	set	NOUN
ejpam-6171	1392	6	into	into	ADP
ejpam-6171	1392	7	iup	iup	NOUN
ejpam-6171	1392	8	-	-	PUNCT
ejpam-6171	1392	9	algebras	algebra	NOUN
ejpam-6171	1392	10	,	,	PUNCT
ejpam-6171	1392	11	this	this	DET
ejpam-6171	1392	12	research	research	NOUN
ejpam-6171	1392	13	extends	extend	VERB
ejpam-6171	1392	14	the	the	DET
ejpam-6171	1392	15	theoretical	theoretical	ADJ
ejpam-6171	1392	16	foundation	foundation	NOUN
ejpam-6171	1392	17	of	of	ADP
ejpam-6171	1392	18	algebraic	algebraic	ADJ
ejpam-6171	1392	19	structures	structure	NOUN
ejpam-6171	1392	20	that	that	PRON
ejpam-6171	1392	21	incorporate	incorporate	VERB
ejpam-6171	1392	22	uncertainty	uncertainty	NOUN
ejpam-6171	1392	23	and	and	CCONJ
ejpam-6171	1392	24	imprecision	imprecision	NOUN
ejpam-6171	1392	25	.	.	PUNCT
ejpam-6171	1393	1	these	these	DET
ejpam-6171	1393	2	findings	finding	NOUN
ejpam-6171	1393	3	contribute	contribute	VERB
ejpam-6171	1393	4	to	to	ADP
ejpam-6171	1393	5	the	the	DET
ejpam-6171	1393	6	broader	broad	ADJ
ejpam-6171	1393	7	fields	field	NOUN
ejpam-6171	1393	8	of	of	ADP
ejpam-6171	1393	9	algebraic	algebraic	ADJ
ejpam-6171	1393	10	logic	logic	NOUN
ejpam-6171	1393	11	and	and	CCONJ
ejpam-6171	1393	12	fuzzy	fuzzy	ADJ
ejpam-6171	1393	13	set	set	NOUN
ejpam-6171	1393	14	theory	theory	NOUN
ejpam-6171	1393	15	,	,	PUNCT
ejpam-6171	1393	16	offering	offer	VERB
ejpam-6171	1393	17	a	a	DET
ejpam-6171	1393	18	more	more	ADV
ejpam-6171	1393	19	robust	robust	ADJ
ejpam-6171	1393	20	framework	framework	NOUN
ejpam-6171	1393	21	for	for	ADP
ejpam-6171	1393	22	mathematical	mathematical	ADJ
ejpam-6171	1393	23	modeling	modeling	NOUN
ejpam-6171	1393	24	in	in	ADP
ejpam-6171	1393	25	decision	decision	NOUN
ejpam-6171	1393	26	-	-	PUNCT
ejpam-6171	1393	27	making	making	NOUN
ejpam-6171	1393	28	,	,	PUNCT
ejpam-6171	1393	29	artificial	artificial	ADJ
ejpam-6171	1393	30	intelligence	intelligence	NOUN
ejpam-6171	1393	31	,	,	PUNCT
ejpam-6171	1393	32	and	and	CCONJ
ejpam-6171	1393	33	computational	computational	ADJ
ejpam-6171	1393	34	uncertainty	uncertainty	NOUN
ejpam-6171	1393	35	analysis	analysis	NOUN
ejpam-6171	1393	36	.	.	PUNCT
ejpam-6171	1394	1	future	future	ADJ
ejpam-6171	1394	2	research	research	NOUN
ejpam-6171	1394	3	could	could	AUX
ejpam-6171	1394	4	further	far	ADV
ejpam-6171	1394	5	extend	extend	VERB
ejpam-6171	1394	6	this	this	DET
ejpam-6171	1394	7	study	study	NOUN
ejpam-6171	1394	8	by	by	ADP
ejpam-6171	1394	9	incorporating	incorporate	VERB
ejpam-6171	1394	10	soft	soft	ADJ
ejpam-6171	1394	11	set	set	NOUN
ejpam-6171	1394	12	theory	theory	NOUN
ejpam-6171	1394	13	,	,	PUNCT
ejpam-6171	1394	14	which	which	PRON
ejpam-6171	1394	15	provides	provide	VERB
ejpam-6171	1394	16	a	a	DET
ejpam-6171	1394	17	flexible	flexible	ADJ
ejpam-6171	1394	18	mathematical	mathematical	ADJ
ejpam-6171	1394	19	approach	approach	NOUN
ejpam-6171	1394	20	to	to	ADP
ejpam-6171	1394	21	dealing	deal	VERB
ejpam-6171	1394	22	with	with	ADP
ejpam-6171	1394	23	parameterized	parameterized	ADJ
ejpam-6171	1394	24	uncertainties	uncertainty	NOUN
ejpam-6171	1394	25	.	.	PUNCT
ejpam-6171	1395	1	the	the	DET
ejpam-6171	1395	2	combination	combination	NOUN
ejpam-6171	1395	3	of	of	ADP
ejpam-6171	1395	4	pythagorean	pythagorean	PROPN
ejpam-6171	1395	5	neutrosophic	neutrosophic	PROPN
ejpam-6171	1395	6	iup	iup	PROPN
ejpam-6171	1395	7	-	-	PUNCT
ejpam-6171	1395	8	algebras	algebras	PROPN
ejpam-6171	1395	9	with	with	ADP
ejpam-6171	1395	10	soft	soft	ADJ
ejpam-6171	1395	11	sets	set	NOUN
ejpam-6171	1395	12	could	could	AUX
ejpam-6171	1395	13	enhance	enhance	VERB
ejpam-6171	1395	14	decision	decision	NOUN
ejpam-6171	1395	15	-	-	PUNCT
ejpam-6171	1395	16	support	support	NOUN
ejpam-6171	1395	17	systems	system	NOUN
ejpam-6171	1395	18	in	in	ADP
ejpam-6171	1395	19	real	real	ADJ
ejpam-6171	1395	20	-	-	PUNCT
ejpam-6171	1395	21	world	world	NOUN
ejpam-6171	1395	22	applications	application	NOUN
ejpam-6171	1395	23	where	where	SCONJ
ejpam-6171	1395	24	parameter	parameter	NOUN
ejpam-6171	1395	25	dependency	dependency	NOUN
ejpam-6171	1395	26	plays	play	VERB
ejpam-6171	1395	27	a	a	DET
ejpam-6171	1395	28	crucial	crucial	ADJ
ejpam-6171	1395	29	role	role	NOUN
ejpam-6171	1395	30	.	.	PUNCT
ejpam-6171	1396	1	additionally	additionally	ADV
ejpam-6171	1396	2	,	,	PUNCT
ejpam-6171	1396	3	exploring	explore	VERB
ejpam-6171	1396	4	cubic	cubic	ADJ
ejpam-6171	1396	5	set	set	NOUN
ejpam-6171	1396	6	theory	theory	NOUN
ejpam-6171	1396	7	,	,	PUNCT
ejpam-6171	1396	8	which	which	PRON
ejpam-6171	1396	9	generalizes	generalize	VERB
ejpam-6171	1396	10	both	both	CCONJ
ejpam-6171	1396	11	fuzzy	fuzzy	ADJ
ejpam-6171	1396	12	and	and	CCONJ
ejpam-6171	1396	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	1396	14	sets	set	NOUN
ejpam-6171	1396	15	by	by	ADP
ejpam-6171	1396	16	considering	consider	VERB
ejpam-6171	1396	17	interval	interval	NOUN
ejpam-6171	1396	18	-	-	PUNCT
ejpam-6171	1396	19	valued	value	VERB
ejpam-6171	1396	20	membership	membership	NOUN
ejpam-6171	1396	21	and	and	CCONJ
ejpam-6171	1396	22	non	non	ADJ
ejpam-6171	1396	23	-	-	ADJ
ejpam-6171	1396	24	membership	membership	ADJ
ejpam-6171	1396	25	functions	function	NOUN
ejpam-6171	1396	26	,	,	PUNCT
ejpam-6171	1396	27	could	could	AUX
ejpam-6171	1396	28	lead	lead	VERB
ejpam-6171	1396	29	to	to	ADP
ejpam-6171	1396	30	deeper	deep	ADJ
ejpam-6171	1396	31	insights	insight	NOUN
ejpam-6171	1396	32	into	into	ADP
ejpam-6171	1396	33	algebraic	algebraic	ADJ
ejpam-6171	1396	34	structures	structure	NOUN
ejpam-6171	1396	35	that	that	PRON
ejpam-6171	1396	36	handle	handle	VERB
ejpam-6171	1396	37	multi	multi	ADJ
ejpam-6171	1396	38	-	-	ADJ
ejpam-6171	1396	39	dimensional	dimensional	ADJ
ejpam-6171	1396	40	uncertainty	uncertainty	NOUN
ejpam-6171	1396	41	.	.	PUNCT
ejpam-6171	1397	1	investigating	investigate	VERB
ejpam-6171	1397	2	these	these	DET
ejpam-6171	1397	3	extensions	extension	NOUN
ejpam-6171	1397	4	could	could	AUX
ejpam-6171	1397	5	open	open	VERB
ejpam-6171	1397	6	new	new	ADJ
ejpam-6171	1397	7	directions	direction	NOUN
ejpam-6171	1397	8	for	for	ADP
ejpam-6171	1397	9	theoretical	theoretical	ADJ
ejpam-6171	1397	10	advancements	advancement	NOUN
ejpam-6171	1397	11	and	and	CCONJ
ejpam-6171	1397	12	practical	practical	ADJ
ejpam-6171	1397	13	implementations	implementation	NOUN
ejpam-6171	1397	14	in	in	ADP
ejpam-6171	1397	15	fields	field	NOUN
ejpam-6171	1397	16	such	such	ADJ
ejpam-6171	1397	17	as	as	ADP
ejpam-6171	1397	18	machine	machine	NOUN
ejpam-6171	1397	19	learning	learning	NOUN
ejpam-6171	1397	20	,	,	PUNCT
ejpam-6171	1397	21	expert	expert	NOUN
ejpam-6171	1397	22	systems	system	NOUN
ejpam-6171	1397	23	,	,	PUNCT
ejpam-6171	1397	24	and	and	CCONJ
ejpam-6171	1397	25	multi	multi	ADJ
ejpam-6171	1397	26	-	-	NOUN
ejpam-6171	1397	27	criteria	criterion	NOUN
ejpam-6171	1397	28	decision	decision	NOUN
ejpam-6171	1397	29	-	-	PUNCT
ejpam-6171	1397	30	making	making	NOUN
ejpam-6171	1397	31	.	.	PUNCT
ejpam-6171	1398	1	acknowledgements	acknowledgement	NOUN
ejpam-6171	1398	2	this	this	DET
ejpam-6171	1398	3	work	work	NOUN
ejpam-6171	1398	4	was	be	AUX
ejpam-6171	1398	5	supported	support	VERB
ejpam-6171	1398	6	by	by	ADP
ejpam-6171	1398	7	the	the	DET
ejpam-6171	1398	8	revenue	revenue	NOUN
ejpam-6171	1398	9	budget	budget	NOUN
ejpam-6171	1398	10	in	in	ADP
ejpam-6171	1398	11	2025	2025	NUM
ejpam-6171	1398	12	,	,	PUNCT
ejpam-6171	1398	13	school	school	NOUN
ejpam-6171	1398	14	of	of	ADP
ejpam-6171	1398	15	science	science	NOUN
ejpam-6171	1398	16	,	,	PUNCT
ejpam-6171	1398	17	university	university	NOUN
ejpam-6171	1398	18	of	of	ADP
ejpam-6171	1398	19	phayao	phayao	NOUN
ejpam-6171	1398	20	(	(	PUNCT
ejpam-6171	1398	21	grant	grant	VERB
ejpam-6171	1398	22	no	no	INTJ
ejpam-6171	1398	23	.	.	PUNCT
ejpam-6171	1398	24	pbtsc68002	pbtsc68002	NUM
ejpam-6171	1398	25	)	)	PUNCT
ejpam-6171	1398	26	.	.	PUNCT
ejpam-6171	1399	1	references	reference	NOUN
ejpam-6171	1399	2	[	[	X
ejpam-6171	1399	3	1	1	NUM
ejpam-6171	1399	4	]	]	PUNCT
ejpam-6171	1399	5	l.	l.	PROPN
ejpam-6171	1399	6	a.	a.	PROPN
ejpam-6171	1399	7	zadeh	zadeh	PROPN
ejpam-6171	1399	8	.	.	PUNCT
ejpam-6171	1400	1	fuzzy	fuzzy	ADJ
ejpam-6171	1400	2	sets	set	NOUN
ejpam-6171	1400	3	.	.	PUNCT
ejpam-6171	1401	1	inf	inf	PROPN
ejpam-6171	1401	2	.	.	PUNCT
ejpam-6171	1401	3	cont	cont	PROPN
ejpam-6171	1401	4	.	.	PROPN
ejpam-6171	1401	5	,	,	PUNCT
ejpam-6171	1401	6	8(3):338–353	8(3):338–353	NUM
ejpam-6171	1401	7	,	,	PUNCT
ejpam-6171	1401	8	1965	1965	NUM
ejpam-6171	1401	9	.	.	PUNCT
ejpam-6171	1402	1	[	[	X
ejpam-6171	1402	2	2	2	X
ejpam-6171	1402	3	]	]	PUNCT
ejpam-6171	1402	4	k.	k.	PROPN
ejpam-6171	1402	5	t.	t.	PROPN
ejpam-6171	1402	6	atanassov	atanassov	PROPN
ejpam-6171	1402	7	.	.	PUNCT
ejpam-6171	1403	1	intuitionistic	intuitionistic	ADJ
ejpam-6171	1403	2	fuzzy	fuzzy	ADJ
ejpam-6171	1403	3	sets	set	NOUN
ejpam-6171	1403	4	.	.	PUNCT
ejpam-6171	1404	1	fuzzy	fuzzy	ADJ
ejpam-6171	1404	2	sets	set	NOUN
ejpam-6171	1404	3	syst	syst	PROPN
ejpam-6171	1404	4	.	.	PUNCT
ejpam-6171	1404	5	,	,	PUNCT
ejpam-6171	1404	6	20(1):87–96	20(1):87–96	NUM
ejpam-6171	1404	7	,	,	PUNCT
ejpam-6171	1404	8	1986	1986	NUM
ejpam-6171	1404	9	.	.	PUNCT
ejpam-6171	1405	1	[	[	X
ejpam-6171	1405	2	3	3	X
ejpam-6171	1405	3	]	]	X
ejpam-6171	1405	4	f.	f.	PROPN
ejpam-6171	1405	5	smarandache	smarandache	PROPN
ejpam-6171	1405	6	.	.	PUNCT
ejpam-6171	1405	7	neutrosophic	neutrosophic	PROPN
ejpam-6171	1405	8	set	set	VERB
ejpam-6171	1405	9	–	–	PUNCT
ejpam-6171	1405	10	a	a	DET
ejpam-6171	1405	11	generalization	generalization	NOUN
ejpam-6171	1405	12	of	of	ADP
ejpam-6171	1405	13	the	the	DET
ejpam-6171	1405	14	intuitionistic	intuitionistic	ADJ
ejpam-6171	1405	15	fuzzy	fuzzy	ADJ
ejpam-6171	1405	16	set	set	NOUN
ejpam-6171	1405	17	.	.	PUNCT
ejpam-6171	1406	1	ieee	ieee	PROPN
ejpam-6171	1406	2	int	int	PROPN
ejpam-6171	1406	3	.	.	PUNCT
ejpam-6171	1407	1	conf	conf	NOUN
ejpam-6171	1407	2	.	.	PUNCT
ejpam-6171	1408	1	granular	granular	ADJ
ejpam-6171	1408	2	comp	comp	NOUN
ejpam-6171	1408	3	.	.	PUNCT
ejpam-6171	1408	4	,	,	PUNCT
ejpam-6171	1408	5	pages	page	NOUN
ejpam-6171	1408	6	38–42	38–42	NUM
ejpam-6171	1408	7	,	,	PUNCT
ejpam-6171	1408	8	2006	2006	NUM
ejpam-6171	1408	9	.	.	PUNCT
ejpam-6171	1409	1	[	[	X
ejpam-6171	1409	2	4	4	NUM
ejpam-6171	1409	3	]	]	X
ejpam-6171	1409	4	r.	r.	PROPN
ejpam-6171	1409	5	r.	r.	PROPN
ejpam-6171	1409	6	yager	yager	PROPN
ejpam-6171	1409	7	.	.	PUNCT
ejpam-6171	1410	1	pythagorean	pythagorean	PROPN
ejpam-6171	1410	2	fuzzy	fuzzy	ADJ
ejpam-6171	1410	3	sets	set	NOUN
ejpam-6171	1410	4	.	.	PUNCT
ejpam-6171	1411	1	ieee	ieee	PROPN
ejpam-6171	1411	2	trans	trans	PROPN
ejpam-6171	1411	3	.	.	PUNCT
ejpam-6171	1411	4	fuzzy	fuzzy	ADJ
ejpam-6171	1411	5	syst	syst	PROPN
ejpam-6171	1411	6	.	.	PROPN
ejpam-6171	1411	7	,	,	PUNCT
ejpam-6171	1411	8	22(4):958–965	22(4):958–965	PROPN
ejpam-6171	1411	9	,	,	PUNCT
ejpam-6171	1411	10	2013	2013	NUM
ejpam-6171	1411	11	.	.	PUNCT
ejpam-6171	1412	1	[	[	X
ejpam-6171	1412	2	5	5	NUM
ejpam-6171	1412	3	]	]	X
ejpam-6171	1412	4	r.	r.	PROPN
ejpam-6171	1412	5	jansi	jansi	PROPN
ejpam-6171	1412	6	,	,	PUNCT
ejpam-6171	1412	7	k.	k.	PROPN
ejpam-6171	1412	8	mohana	mohana	PROPN
ejpam-6171	1412	9	,	,	PUNCT
ejpam-6171	1412	10	and	and	CCONJ
ejpam-6171	1412	11	f.	f.	PROPN
ejpam-6171	1412	12	smarandache	smarandache	PROPN
ejpam-6171	1412	13	.	.	PUNCT
ejpam-6171	1413	1	correlation	correlation	NOUN
ejpam-6171	1413	2	measure	measure	NOUN
ejpam-6171	1413	3	for	for	ADP
ejpam-6171	1413	4	pythagorean	pythagorean	PROPN
ejpam-6171	1413	5	neutrosophic	neutrosophic	ADJ
ejpam-6171	1413	6	sets	set	NOUN
ejpam-6171	1413	7	with	with	ADP
ejpam-6171	1413	8	t	t	PROPN
ejpam-6171	1413	9	and	and	CCONJ
ejpam-6171	1413	10	f	f	PROPN
ejpam-6171	1413	11	as	as	ADP
ejpam-6171	1413	12	dependent	dependent	ADJ
ejpam-6171	1413	13	neutrosophic	neutrosophic	ADJ
ejpam-6171	1413	14	components	component	NOUN
ejpam-6171	1413	15	.	.	PUNCT
ejpam-6171	1414	1	neutrosophic	neutrosophic	ADJ
ejpam-6171	1414	2	sets	set	VERB
ejpam-6171	1414	3	syst	syst	PROPN
ejpam-6171	1414	4	.	.	PUNCT
ejpam-6171	1414	5	,	,	PUNCT
ejpam-6171	1414	6	30:202–212	30:202–212	NUM
ejpam-6171	1414	7	,	,	PUNCT
ejpam-6171	1414	8	2019	2019	NUM
ejpam-6171	1414	9	.	.	PUNCT
ejpam-6171	1415	1	[	[	X
ejpam-6171	1415	2	6	6	NUM
ejpam-6171	1415	3	]	]	PUNCT
ejpam-6171	1415	4	a.	a.	NOUN
ejpam-6171	1415	5	satirad	satirad	PROPN
ejpam-6171	1415	6	,	,	PUNCT
ejpam-6171	1415	7	r.	r.	PROPN
ejpam-6171	1415	8	chinram	chinram	PROPN
ejpam-6171	1415	9	,	,	PUNCT
ejpam-6171	1415	10	and	and	CCONJ
ejpam-6171	1415	11	a.	a.	NOUN
ejpam-6171	1415	12	iampan	iampan	PROPN
ejpam-6171	1415	13	.	.	PUNCT
ejpam-6171	1416	1	pythagorean	pythagorean	PROPN
ejpam-6171	1416	2	fuzzy	fuzzy	ADJ
ejpam-6171	1416	3	sets	set	NOUN
ejpam-6171	1416	4	in	in	ADP
ejpam-6171	1416	5	up	up	ADV
ejpam-6171	1416	6	-	-	PUNCT
ejpam-6171	1416	7	algebras	algebra	NOUN
ejpam-6171	1416	8	and	and	CCONJ
ejpam-6171	1416	9	approximations	approximation	NOUN
ejpam-6171	1416	10	.	.	PUNCT
ejpam-6171	1417	1	aims	aim	VERB
ejpam-6171	1417	2	math	math	NOUN
ejpam-6171	1417	3	.	.	PUNCT
ejpam-6171	1417	4	,	,	PUNCT
ejpam-6171	1417	5	6(6):6002–6032	6(6):6002–6032	PROPN
ejpam-6171	1417	6	,	,	PUNCT
ejpam-6171	1417	7	2021	2021	NUM
ejpam-6171	1417	8	.	.	PUNCT
ejpam-6171	1418	1	[	[	X
ejpam-6171	1418	2	7	7	X
ejpam-6171	1418	3	]	]	PUNCT
ejpam-6171	1418	4	j.	j.	PROPN
ejpam-6171	1418	5	n.	n.	PROPN
ejpam-6171	1418	6	ismail	ismail	PROPN
ejpam-6171	1418	7	,	,	PUNCT
ejpam-6171	1418	8	z.	z.	PROPN
ejpam-6171	1418	9	rodzi	rodzi	PROPN
ejpam-6171	1418	10	,	,	PUNCT
ejpam-6171	1418	11	f.	f.	PROPN
ejpam-6171	1418	12	al	al	PROPN
ejpam-6171	1418	13	-	-	PUNCT
ejpam-6171	1418	14	sharqi	sharqi	PROPN
ejpam-6171	1418	15	,	,	PUNCT
ejpam-6171	1418	16	a.	a.	PROPN
ejpam-6171	1418	17	al	al	PROPN
ejpam-6171	1418	18	-	-	PUNCT
ejpam-6171	1418	19	quran	quran	PROPN
ejpam-6171	1418	20	,	,	PUNCT
ejpam-6171	1418	21	h.	h.	PROPN
ejpam-6171	1418	22	hashim	hashim	PROPN
ejpam-6171	1418	23	,	,	PUNCT
ejpam-6171	1418	24	and	and	CCONJ
ejpam-6171	1418	25	n.	n.	PROPN
ejpam-6171	1418	26	h.	h.	PROPN
ejpam-6171	1418	27	sulaiman	sulaiman	PROPN
ejpam-6171	1418	28	.	.	PUNCT
ejpam-6171	1419	1	algebraic	algebraic	ADJ
ejpam-6171	1419	2	operations	operation	NOUN
ejpam-6171	1419	3	on	on	ADP
ejpam-6171	1419	4	pythagorean	pythagorean	PROPN
ejpam-6171	1419	5	neutrosophic	neutrosophic	ADJ
ejpam-6171	1419	6	sets	set	NOUN
ejpam-6171	1419	7	(	(	PUNCT
ejpam-6171	1419	8	pns	pns	NOUN
ejpam-6171	1419	9	):	):	PUNCT
ejpam-6171	1419	10	extending	extend	VERB
ejpam-6171	1419	11	applicability	applicability	NOUN
ejpam-6171	1419	12	and	and	CCONJ
ejpam-6171	1419	13	decision	decision	NOUN
ejpam-6171	1419	14	-	-	PUNCT
ejpam-6171	1419	15	making	make	VERB
ejpam-6171	1419	16	capabilities	capability	NOUN
ejpam-6171	1419	17	.	.	PUNCT
ejpam-6171	1420	1	int	int	NOUN
ejpam-6171	1420	2	.	.	PUNCT
ejpam-6171	1421	1	j.	j.	PROPN
ejpam-6171	1421	2	neutrosophic	neutrosophic	PROPN
ejpam-6171	1421	3	sci	sci	PROPN
ejpam-6171	1421	4	.	.	PROPN
ejpam-6171	1421	5	,	,	PUNCT
ejpam-6171	1421	6	21(4):127–134	21(4):127–134	NUM
ejpam-6171	1421	7	,	,	PUNCT
ejpam-6171	1421	8	2023	2023	NUM
ejpam-6171	1421	9	.	.	PUNCT
ejpam-6171	1422	1	[	[	X
ejpam-6171	1422	2	8	8	X
ejpam-6171	1422	3	]	]	PUNCT
ejpam-6171	1422	4	s.	s.	PROPN
ejpam-6171	1422	5	a.	a.	PROPN
ejpam-6171	1422	6	razak	razak	PROPN
ejpam-6171	1422	7	,	,	PUNCT
ejpam-6171	1422	8	z.	z.	PROPN
ejpam-6171	1422	9	m.	m.	PROPN
ejpam-6171	1422	10	rodzi	rodzi	PROPN
ejpam-6171	1422	11	,	,	PUNCT
ejpam-6171	1422	12	n.	n.	PROPN
ejpam-6171	1422	13	ahmad	ahmad	PROPN
ejpam-6171	1422	14	,	,	PUNCT
ejpam-6171	1422	15	and	and	CCONJ
ejpam-6171	1422	16	g.	g.	PROPN
ejpam-6171	1422	17	ahmad	ahmad	PROPN
ejpam-6171	1422	18	.	.	PUNCT
ejpam-6171	1423	1	exploring	explore	VERB
ejpam-6171	1423	2	the	the	DET
ejpam-6171	1423	3	boundaries	boundary	NOUN
ejpam-6171	1423	4	of	of	ADP
ejpam-6171	1423	5	uncertainty	uncertainty	NOUN
ejpam-6171	1423	6	:	:	PUNCT
ejpam-6171	1423	7	interval	interval	NOUN
ejpam-6171	1423	8	valued	value	VERB
ejpam-6171	1423	9	pythagorean	pythagorean	PROPN
ejpam-6171	1423	10	neutrosophic	neutrosophic	PROPN
ejpam-6171	1423	11	set	set	NOUN
ejpam-6171	1423	12	and	and	CCONJ
ejpam-6171	1423	13	their	their	PRON
ejpam-6171	1423	14	properties	property	NOUN
ejpam-6171	1423	15	.	.	PUNCT
ejpam-6171	1424	1	malays	malays	PROPN
ejpam-6171	1424	2	.	.	PUNCT
ejpam-6171	1425	1	j.	j.	PROPN
ejpam-6171	1425	2	fundam	fundam	PROPN
ejpam-6171	1425	3	.	.	PUNCT
ejpam-6171	1426	1	appl	appl	PROPN
ejpam-6171	1426	2	.	.	PUNCT
ejpam-6171	1427	1	sci	sci	PROPN
ejpam-6171	1427	2	.	.	PROPN
ejpam-6171	1427	3	,	,	PUNCT
ejpam-6171	1427	4	20(4):813–824	20(4):813–824	NUM
ejpam-6171	1427	5	,	,	PUNCT
ejpam-6171	1427	6	2024	2024	NUM
ejpam-6171	1427	7	.	.	PUNCT
ejpam-6171	1428	1	[	[	X
ejpam-6171	1428	2	9	9	NUM
ejpam-6171	1428	3	]	]	PUNCT
ejpam-6171	1428	4	a.	a.	NOUN
ejpam-6171	1428	5	iampan	iampan	PROPN
ejpam-6171	1428	6	,	,	PUNCT
ejpam-6171	1428	7	p.	p.	PROPN
ejpam-6171	1428	8	julatha	julatha	PROPN
ejpam-6171	1428	9	,	,	PUNCT
ejpam-6171	1428	10	p.	p.	NOUN
ejpam-6171	1428	11	khamrot	khamrot	NOUN
ejpam-6171	1428	12	,	,	PUNCT
ejpam-6171	1428	13	and	and	CCONJ
ejpam-6171	1428	14	d.	d.	PROPN
ejpam-6171	1428	15	a.	a.	PROPN
ejpam-6171	1428	16	romano	romano	PROPN
ejpam-6171	1428	17	.	.	PUNCT
ejpam-6171	1429	1	independent	independent	ADJ
ejpam-6171	1429	2	up	up	ADP
ejpam-6171	1429	3	-	-	PUNCT
ejpam-6171	1429	4	algebras	algebras	X
ejpam-6171	1429	5	.	.	PUNCT
ejpam-6171	1430	1	j.	j.	PROPN
ejpam-6171	1430	2	math	math	PROPN
ejpam-6171	1430	3	.	.	PUNCT
ejpam-6171	1431	1	comput	comput	NOUN
ejpam-6171	1431	2	.	.	PUNCT
ejpam-6171	1432	1	sci	sci	PROPN
ejpam-6171	1432	2	.	.	PROPN
ejpam-6171	1432	3	,	,	PUNCT
ejpam-6171	1432	4	jmcs	jmcs	NOUN
ejpam-6171	1432	5	,	,	PUNCT
ejpam-6171	1432	6	27(1):65–76	27(1):65–76	NUM
ejpam-6171	1432	7	,	,	PUNCT
ejpam-6171	1432	8	2022	2022	NUM
ejpam-6171	1432	9	.	.	PUNCT
ejpam-6171	1433	1	k.	k.	PROPN
ejpam-6171	1433	2	suayngam	suayngam	PROPN
ejpam-6171	1433	3	et	et	PROPN
ejpam-6171	1433	4	al	al	PROPN
ejpam-6171	1433	5	.	.	PUNCT
ejpam-6171	1433	6	/	/	SYM
ejpam-6171	1433	7	eur	eur	PROPN
ejpam-6171	1433	8	.	.	PUNCT
ejpam-6171	1434	1	j.	j.	PROPN
ejpam-6171	1434	2	pure	pure	PROPN
ejpam-6171	1434	3	appl	appl	PROPN
ejpam-6171	1434	4	.	.	PROPN
ejpam-6171	1434	5	math	math	PROPN
ejpam-6171	1434	6	,	,	PUNCT
ejpam-6171	1434	7	18	18	NUM
ejpam-6171	1434	8	(	(	PUNCT
ejpam-6171	1434	9	3	3	NUM
ejpam-6171	1434	10	)	)	PUNCT
ejpam-6171	1434	11	(	(	PUNCT
ejpam-6171	1434	12	2025	2025	NUM
ejpam-6171	1434	13	)	)	PUNCT
ejpam-6171	1434	14	,	,	PUNCT
ejpam-6171	1434	15	6171	6171	NUM
ejpam-6171	1434	16	28	28	NUM
ejpam-6171	1434	17	of	of	ADP
ejpam-6171	1434	18	28	28	NUM
ejpam-6171	1435	1	[	[	SYM
ejpam-6171	1435	2	10	10	NUM
ejpam-6171	1435	3	]	]	X
ejpam-6171	1435	4	c.	c.	PROPN
ejpam-6171	1435	5	chanmanee	chanmanee	PROPN
ejpam-6171	1435	6	,	,	PUNCT
ejpam-6171	1435	7	r.	r.	PROPN
ejpam-6171	1435	8	prasertpong	prasertpong	PROPN
ejpam-6171	1435	9	,	,	PUNCT
ejpam-6171	1435	10	p.	p.	PROPN
ejpam-6171	1435	11	julatha	julatha	PROPN
ejpam-6171	1435	12	,	,	PUNCT
ejpam-6171	1435	13	n.	n.	PROPN
ejpam-6171	1435	14	lekkoksung	lekkoksung	PROPN
ejpam-6171	1435	15	,	,	PUNCT
ejpam-6171	1435	16	and	and	CCONJ
ejpam-6171	1435	17	a.	a.	NOUN
ejpam-6171	1435	18	iampan	iampan	PROPN
ejpam-6171	1435	19	.	.	PUNCT
ejpam-6171	1436	1	on	on	ADP
ejpam-6171	1436	2	external	external	ADJ
ejpam-6171	1436	3	direct	direct	ADJ
ejpam-6171	1436	4	products	product	NOUN
ejpam-6171	1436	5	of	of	ADP
ejpam-6171	1436	6	iup	iup	NOUN
ejpam-6171	1436	7	-	-	PUNCT
ejpam-6171	1436	8	algebras	algebras	PROPN
ejpam-6171	1436	9	.	.	PUNCT
ejpam-6171	1437	1	int	int	NOUN
ejpam-6171	1437	2	.	.	PUNCT
ejpam-6171	1438	1	j.	j.	PROPN
ejpam-6171	1438	2	innov	innov	PROPN
ejpam-6171	1438	3	.	.	PUNCT
ejpam-6171	1439	1	comput	comput	PROPN
ejpam-6171	1439	2	.	.	PUNCT
ejpam-6171	1440	1	inf	inf	PROPN
ejpam-6171	1440	2	.	.	PUNCT
ejpam-6171	1440	3	control	control	PROPN
ejpam-6171	1440	4	,	,	PUNCT
ejpam-6171	1440	5	19(3):775–787	19(3):775–787	NUM
ejpam-6171	1440	6	,	,	PUNCT
ejpam-6171	1440	7	2023	2023	NUM
ejpam-6171	1440	8	.	.	PUNCT
ejpam-6171	1441	1	[	[	X
ejpam-6171	1441	2	11	11	NUM
ejpam-6171	1441	3	]	]	PUNCT
ejpam-6171	1441	4	k.	k.	PROPN
ejpam-6171	1441	5	kuntama	kuntama	PROPN
ejpam-6171	1441	6	,	,	PUNCT
ejpam-6171	1441	7	p.	p.	NOUN
ejpam-6171	1441	8	krongchai	krongchai	PROPN
ejpam-6171	1441	9	,	,	PUNCT
ejpam-6171	1441	10	r.	r.	PROPN
ejpam-6171	1441	11	prasertpong	prasertpong	PROPN
ejpam-6171	1441	12	,	,	PUNCT
ejpam-6171	1441	13	p.	p.	PROPN
ejpam-6171	1441	14	julatha	julatha	PROPN
ejpam-6171	1441	15	,	,	PUNCT
ejpam-6171	1441	16	and	and	CCONJ
ejpam-6171	1441	17	a.	a.	NOUN
ejpam-6171	1441	18	iampan	iampan	PROPN
ejpam-6171	1441	19	.	.	PUNCT
ejpam-6171	1442	1	fuzzy	fuzzy	ADJ
ejpam-6171	1442	2	set	set	VERB
ejpam-6171	1442	3	theory	theory	NOUN
ejpam-6171	1442	4	applied	apply	VERB
ejpam-6171	1442	5	to	to	ADP
ejpam-6171	1442	6	iup	iup	VERB
ejpam-6171	1442	7	-	-	PUNCT
ejpam-6171	1442	8	algebras	algebras	PROPN
ejpam-6171	1442	9	.	.	PUNCT
ejpam-6171	1443	1	j.	j.	PROPN
ejpam-6171	1443	2	math	math	PROPN
ejpam-6171	1443	3	.	.	PUNCT
ejpam-6171	1444	1	comput	comput	NOUN
ejpam-6171	1444	2	.	.	PUNCT
ejpam-6171	1445	1	sci	sci	PROPN
ejpam-6171	1445	2	.	.	PROPN
ejpam-6171	1445	3	,	,	PUNCT
ejpam-6171	1445	4	jmcs	jmcs	NOUN
ejpam-6171	1445	5	,	,	PUNCT
ejpam-6171	1445	6	34(2):128–143	34(2):128–143	PROPN
ejpam-6171	1445	7	,	,	PUNCT
ejpam-6171	1445	8	2024	2024	NUM
ejpam-6171	1445	9	.	.	PUNCT
ejpam-6171	1446	1	[	[	X
ejpam-6171	1446	2	12	12	NUM
ejpam-6171	1446	3	]	]	PUNCT
ejpam-6171	1446	4	k.	k.	PROPN
ejpam-6171	1446	5	suayngam	suayngam	PROPN
ejpam-6171	1446	6	,	,	PUNCT
ejpam-6171	1446	7	r.	r.	PROPN
ejpam-6171	1446	8	prasertpong	prasertpong	PROPN
ejpam-6171	1446	9	,	,	PUNCT
ejpam-6171	1446	10	n.	n.	PROPN
ejpam-6171	1446	11	lekkoksung	lekkoksung	PROPN
ejpam-6171	1446	12	,	,	PUNCT
ejpam-6171	1446	13	p.	p.	PROPN
ejpam-6171	1446	14	julatha	julatha	PROPN
ejpam-6171	1446	15	,	,	PUNCT
ejpam-6171	1446	16	and	and	CCONJ
ejpam-6171	1446	17	a.	a.	NOUN
ejpam-6171	1446	18	iampan	iampan	PROPN
ejpam-6171	1446	19	.	.	PUNCT
ejpam-6171	1447	1	fermatean	fermatean	PROPN
ejpam-6171	1447	2	fuzzy	fuzzy	ADJ
ejpam-6171	1447	3	set	set	NOUN
ejpam-6171	1447	4	theory	theory	NOUN
ejpam-6171	1447	5	applied	apply	VERB
ejpam-6171	1447	6	to	to	ADP
ejpam-6171	1447	7	iup	iup	VERB
ejpam-6171	1447	8	-	-	PUNCT
ejpam-6171	1447	9	algebras	algebras	PROPN
ejpam-6171	1447	10	.	.	PUNCT
ejpam-6171	1448	1	eur	eur	PROPN
ejpam-6171	1448	2	.	.	PUNCT
ejpam-6171	1449	1	j.	j.	PROPN
ejpam-6171	1449	2	pure	pure	PROPN
ejpam-6171	1449	3	appl	appl	PROPN
ejpam-6171	1449	4	.	.	PUNCT
ejpam-6171	1449	5	math	math	PROPN
ejpam-6171	1449	6	.	.	PUNCT
ejpam-6171	1449	7	,	,	PUNCT
ejpam-6171	1450	1	17(4):3022–3042	17(4):3022–3042	NUM
ejpam-6171	1450	2	,	,	PUNCT
ejpam-6171	1450	3	2024	2024	NUM
ejpam-6171	1450	4	.	.	PUNCT
ejpam-6171	1451	1	[	[	X
ejpam-6171	1451	2	13	13	NUM
ejpam-6171	1451	3	]	]	PUNCT
ejpam-6171	1451	4	k.	k.	NOUN
ejpam-6171	1451	5	suayngam	suayngam	PROPN
ejpam-6171	1451	6	,	,	PUNCT
ejpam-6171	1451	7	t.	t.	PROPN
ejpam-6171	1451	8	suwanklang	suwanklang	PROPN
ejpam-6171	1451	9	,	,	PUNCT
ejpam-6171	1451	10	p.	p.	PROPN
ejpam-6171	1451	11	julatha	julatha	PROPN
ejpam-6171	1451	12	,	,	PUNCT
ejpam-6171	1451	13	r.	r.	PROPN
ejpam-6171	1451	14	prasertpong	prasertpong	PROPN
ejpam-6171	1451	15	,	,	PUNCT
ejpam-6171	1451	16	and	and	CCONJ
ejpam-6171	1451	17	a.	a.	NOUN
ejpam-6171	1451	18	iampan	iampan	PROPN
ejpam-6171	1451	19	.	.	PUNCT
ejpam-6171	1452	1	new	new	ADJ
ejpam-6171	1452	2	results	result	NOUN
ejpam-6171	1452	3	on	on	ADP
ejpam-6171	1452	4	intuitionistic	intuitionistic	ADJ
ejpam-6171	1452	5	fuzzy	fuzzy	ADJ
ejpam-6171	1452	6	sets	set	NOUN
ejpam-6171	1452	7	in	in	ADP
ejpam-6171	1452	8	iup	iup	NOUN
ejpam-6171	1452	9	-	-	PUNCT
ejpam-6171	1452	10	algebras	algebras	PROPN
ejpam-6171	1452	11	.	.	PUNCT
ejpam-6171	1453	1	int	int	NOUN
ejpam-6171	1453	2	.	.	PUNCT
ejpam-6171	1454	1	j.	j.	PROPN
ejpam-6171	1454	2	innov	innov	PROPN
ejpam-6171	1454	3	.	.	PUNCT
ejpam-6171	1455	1	comput	comput	PROPN
ejpam-6171	1455	2	.	.	PUNCT
ejpam-6171	1456	1	inf	inf	PROPN
ejpam-6171	1456	2	.	.	PUNCT
ejpam-6171	1456	3	control	control	PROPN
ejpam-6171	1456	4	,	,	PUNCT
ejpam-6171	1456	5	20(4):1125–1141	20(4):1125–1141	NUM
ejpam-6171	1456	6	,	,	PUNCT
ejpam-6171	1456	7	2024	2024	NUM
ejpam-6171	1456	8	.	.	PUNCT
ejpam-6171	1457	1	[	[	X
ejpam-6171	1457	2	14	14	NUM
ejpam-6171	1457	3	]	]	PUNCT
ejpam-6171	1457	4	k.	k.	PROPN
ejpam-6171	1457	5	suayngam	suayngam	PROPN
ejpam-6171	1457	6	,	,	PUNCT
ejpam-6171	1457	7	p.	p.	NOUN
ejpam-6171	1457	8	julatha	julatha	PROPN
ejpam-6171	1457	9	,	,	PUNCT
ejpam-6171	1457	10	r.	r.	PROPN
ejpam-6171	1457	11	prasertpong	prasertpong	PROPN
ejpam-6171	1457	12	,	,	PUNCT
ejpam-6171	1457	13	and	and	CCONJ
ejpam-6171	1457	14	a.	a.	NOUN
ejpam-6171	1457	15	iampan	iampan	PROPN
ejpam-6171	1457	16	.	.	PUNCT
ejpam-6171	1458	1	neutrosophic	neutrosophic	ADJ
ejpam-6171	1458	2	sets	set	NOUN
ejpam-6171	1458	3	in	in	ADP
ejpam-6171	1458	4	iup	iup	NOUN
ejpam-6171	1458	5	-	-	PUNCT
ejpam-6171	1458	6	algebras	algebras	PROPN
ejpam-6171	1458	7	:	:	PUNCT
ejpam-6171	1458	8	a	a	DET
ejpam-6171	1458	9	new	new	ADJ
ejpam-6171	1458	10	exploration	exploration	NOUN
ejpam-6171	1458	11	.	.	PUNCT
ejpam-6171	1459	1	int	int	NOUN
ejpam-6171	1459	2	.	.	PUNCT
ejpam-6171	1460	1	j.	j.	PROPN
ejpam-6171	1460	2	neutrosophic	neutrosophic	PROPN
ejpam-6171	1460	3	sci	sci	PROPN
ejpam-6171	1460	4	.	.	PROPN
ejpam-6171	1460	5	,	,	PUNCT
ejpam-6171	1460	6	25(3):540–560	25(3):540–560	PROPN
ejpam-6171	1460	7	,	,	PUNCT
ejpam-6171	1460	8	2025	2025	NUM
ejpam-6171	1460	9	.	.	PUNCT
ejpam-6171	1461	1	[	[	X
ejpam-6171	1461	2	15	15	NUM
ejpam-6171	1461	3	]	]	X
ejpam-6171	1461	4	k.	k.	PROPN
ejpam-6171	1461	5	suayngam	suayngam	PROPN
ejpam-6171	1461	6	,	,	PUNCT
ejpam-6171	1461	7	r.	r.	PROPN
ejpam-6171	1461	8	prasertpong	prasertpong	PROPN
ejpam-6171	1461	9	,	,	PUNCT
ejpam-6171	1461	10	w.	w.	PROPN
ejpam-6171	1461	11	nakkhasen	nakkhasen	PROPN
ejpam-6171	1461	12	,	,	PUNCT
ejpam-6171	1461	13	p.	p.	NOUN
ejpam-6171	1461	14	julatha	julatha	PROPN
ejpam-6171	1461	15	,	,	PUNCT
ejpam-6171	1461	16	and	and	CCONJ
ejpam-6171	1461	17	a.	a.	NOUN
ejpam-6171	1461	18	iampan	iampan	PROPN
ejpam-6171	1461	19	.	.	PUNCT
ejpam-6171	1462	1	pythagorean	pythagorean	PROPN
ejpam-6171	1462	2	fuzzy	fuzzy	ADJ
ejpam-6171	1462	3	sets	set	NOUN
ejpam-6171	1462	4	:	:	PUNCT
ejpam-6171	1462	5	a	a	DET
ejpam-6171	1462	6	new	new	ADJ
ejpam-6171	1462	7	perspective	perspective	NOUN
ejpam-6171	1462	8	on	on	ADP
ejpam-6171	1462	9	iup	iup	NOUN
ejpam-6171	1462	10	-	-	PUNCT
ejpam-6171	1462	11	algebras	algebras	PROPN
ejpam-6171	1462	12	.	.	PUNCT
ejpam-6171	1463	1	int	int	NOUN
ejpam-6171	1463	2	.	.	PUNCT
ejpam-6171	1464	1	j.	j.	PROPN
ejpam-6171	1464	2	innov	innov	PROPN
ejpam-6171	1464	3	.	.	PUNCT
ejpam-6171	1465	1	comput	comput	PROPN
ejpam-6171	1465	2	.	.	PUNCT
ejpam-6171	1466	1	inf	inf	PROPN
ejpam-6171	1466	2	.	.	PUNCT
ejpam-6171	1466	3	control	control	PROPN
ejpam-6171	1466	4	,	,	PUNCT
ejpam-6171	1466	5	21(2):339–357	21(2):339–357	PROPN
ejpam-6171	1466	6	,	,	PUNCT
ejpam-6171	1466	7	2025	2025	NUM
ejpam-6171	1466	8	.	.	PUNCT
ejpam-6171	1467	1	[	[	X
ejpam-6171	1467	2	16	16	NUM
ejpam-6171	1467	3	]	]	PUNCT
ejpam-6171	1467	4	k.	k.	PROPN
ejpam-6171	1467	5	suayngam	suayngam	PROPN
ejpam-6171	1467	6	,	,	PUNCT
ejpam-6171	1467	7	p.	p.	NOUN
ejpam-6171	1467	8	julatha	julatha	PROPN
ejpam-6171	1467	9	,	,	PUNCT
ejpam-6171	1467	10	w.	w.	PROPN
ejpam-6171	1467	11	nakkhasen	nakkhasen	PROPN
ejpam-6171	1467	12	,	,	PUNCT
ejpam-6171	1467	13	and	and	CCONJ
ejpam-6171	1467	14	a.	a.	NOUN
ejpam-6171	1467	15	iampan	iampan	PROPN
ejpam-6171	1467	16	.	.	PUNCT
ejpam-6171	1468	1	structural	structural	ADJ
ejpam-6171	1468	2	insights	insight	NOUN
ejpam-6171	1468	3	into	into	ADP
ejpam-6171	1468	4	iup	iup	NOUN
ejpam-6171	1468	5	-	-	PUNCT
ejpam-6171	1468	6	algebras	algebras	PROPN
ejpam-6171	1468	7	via	via	ADP
ejpam-6171	1468	8	intuitionistic	intuitionistic	ADJ
ejpam-6171	1468	9	neutrosophic	neutrosophic	ADJ
ejpam-6171	1468	10	set	set	NOUN
ejpam-6171	1468	11	theory	theory	NOUN
ejpam-6171	1468	12	.	.	PUNCT
ejpam-6171	1469	1	eur	eur	PROPN
ejpam-6171	1469	2	.	.	PUNCT
ejpam-6171	1470	1	j.	j.	PROPN
ejpam-6171	1470	2	pure	pure	PROPN
ejpam-6171	1470	3	appl	appl	PROPN
ejpam-6171	1470	4	.	.	PUNCT
ejpam-6171	1470	5	math	math	PROPN
ejpam-6171	1470	6	.	.	PUNCT
ejpam-6171	1470	7	,	,	PUNCT
ejpam-6171	1470	8	18(2):5857	18(2):5857	NUM
ejpam-6171	1470	9	,	,	PUNCT
ejpam-6171	1470	10	2025	2025	NUM
ejpam-6171	1470	11	.	.	PUNCT
ejpam-6171	1471	1	[	[	X
ejpam-6171	1471	2	17	17	NUM
ejpam-6171	1471	3	]	]	PUNCT
ejpam-6171	1471	4	j.	j.	PROPN
ejpam-6171	1471	5	somjanta	somjanta	PROPN
ejpam-6171	1471	6	,	,	PUNCT
ejpam-6171	1471	7	n.	n.	PROPN
ejpam-6171	1471	8	thuekaew	thuekaew	PROPN
ejpam-6171	1471	9	,	,	PUNCT
ejpam-6171	1471	10	p.	p.	NOUN
ejpam-6171	1471	11	kumpeangkeaw	kumpeangkeaw	PROPN
ejpam-6171	1471	12	,	,	PUNCT
ejpam-6171	1471	13	and	and	CCONJ
ejpam-6171	1471	14	a.	a.	NOUN
ejpam-6171	1471	15	iampan	iampan	PROPN
ejpam-6171	1471	16	.	.	PUNCT
ejpam-6171	1472	1	fuzzy	fuzzy	ADJ
ejpam-6171	1472	2	sets	set	NOUN
ejpam-6171	1472	3	in	in	ADP
ejpam-6171	1472	4	upalgebras	upalgebra	NOUN
ejpam-6171	1472	5	.	.	PUNCT
ejpam-6171	1473	1	ann	ann	PROPN
ejpam-6171	1473	2	.	.	PUNCT
ejpam-6171	1473	3	fuzzy	fuzzy	ADJ
ejpam-6171	1473	4	math	math	NOUN
ejpam-6171	1473	5	.	.	PUNCT
ejpam-6171	1474	1	inform	inform	NOUN
ejpam-6171	1474	2	.	.	PUNCT
ejpam-6171	1474	3	,	,	PUNCT
ejpam-6171	1474	4	12(6):739–756	12(6):739–756	PROPN
ejpam-6171	1474	5	,	,	PUNCT
ejpam-6171	1474	6	2016	2016	NUM
ejpam-6171	1474	7	.	.	PUNCT
ejpam-6171	1475	1	[	[	X
ejpam-6171	1475	2	18	18	NUM
ejpam-6171	1475	3	]	]	PUNCT
ejpam-6171	1475	4	m.	m.	NOUN
ejpam-6171	1475	5	songsaeng	songsaeng	PROPN
ejpam-6171	1475	6	and	and	CCONJ
ejpam-6171	1475	7	a.	a.	NOUN
ejpam-6171	1475	8	iampan	iampan	PROPN
ejpam-6171	1475	9	.	.	PUNCT
ejpam-6171	1476	1	neutrosophic	neutrosophic	PROPN
ejpam-6171	1476	2	set	set	PROPN
ejpam-6171	1476	3	theory	theory	NOUN
ejpam-6171	1476	4	applied	apply	VERB
ejpam-6171	1476	5	to	to	ADP
ejpam-6171	1476	6	up	up	ADV
ejpam-6171	1476	7	-	-	PUNCT
ejpam-6171	1476	8	algebras	algebras	X
ejpam-6171	1476	9	.	.	PUNCT
ejpam-6171	1477	1	eur	eur	PROPN
ejpam-6171	1477	2	.	.	PUNCT
ejpam-6171	1478	1	j.	j.	PROPN
ejpam-6171	1478	2	pure	pure	PROPN
ejpam-6171	1478	3	appl	appl	PROPN
ejpam-6171	1478	4	.	.	PUNCT
ejpam-6171	1478	5	math	math	PROPN
ejpam-6171	1478	6	.	.	PUNCT
ejpam-6171	1478	7	,	,	PUNCT
ejpam-6171	1478	8	12(4):1382–1409	12(4):1382–1409	NUM
ejpam-6171	1478	9	,	,	PUNCT
ejpam-6171	1478	10	2019	2019	NUM
ejpam-6171	1478	11	.	.	PUNCT