id	sid	tid	token	lemma	pos
ejpam-5768	1	1	european	european	PROPN
ejpam-5768	1	2	journal	journal	PROPN
ejpam-5768	1	3	of	of	ADP
ejpam-5768	1	4	pure	pure	ADJ
ejpam-5768	1	5	and	and	CCONJ
ejpam-5768	1	6	applied	applied	ADJ
ejpam-5768	1	7	mathematics	mathematic	NOUN
ejpam-5768	1	8	2025	2025	NUM
ejpam-5768	1	9	,	,	PUNCT
ejpam-5768	1	10	vol	vol	NOUN
ejpam-5768	1	11	.	.	PROPN
ejpam-5768	1	12	18	18	NUM
ejpam-5768	1	13	,	,	PUNCT
ejpam-5768	1	14	issue	issue	NOUN
ejpam-5768	1	15	2	2	NUM
ejpam-5768	1	16	,	,	PUNCT
ejpam-5768	1	17	article	article	NOUN
ejpam-5768	1	18	number	number	NOUN
ejpam-5768	1	19	5768	5768	NUM
ejpam-5768	1	20	issn	issn	VERB
ejpam-5768	1	21	1307	1307	NUM
ejpam-5768	1	22	-	-	SYM
ejpam-5768	1	23	5543	5543	NUM
ejpam-5768	1	24	–	–	PUNCT
ejpam-5768	1	25	ejpam.com	ejpam.com	X
ejpam-5768	1	26	published	publish	VERB
ejpam-5768	1	27	by	by	ADP
ejpam-5768	1	28	new	new	PROPN
ejpam-5768	1	29	york	york	PROPN
ejpam-5768	1	30	business	business	PROPN
ejpam-5768	1	31	global	global	PROPN
ejpam-5768	1	32	pythagorean	pythagorean	PROPN
ejpam-5768	1	33	fuzzy	fuzzy	PROPN
ejpam-5768	1	34	hx	hx	PROPN
ejpam-5768	1	35	-	-	PUNCT
ejpam-5768	1	36	subgroups	subgroup	NOUN
ejpam-5768	1	37	and	and	CCONJ
ejpam-5768	1	38	their	their	PRON
ejpam-5768	1	39	applications	application	NOUN
ejpam-5768	1	40	areej	areej	PROPN
ejpam-5768	1	41	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	1	42	department	department	PROPN
ejpam-5768	1	43	of	of	ADP
ejpam-5768	1	44	mathematics	mathematics	PROPN
ejpam-5768	1	45	,	,	PUNCT
ejpam-5768	1	46	college	college	NOUN
ejpam-5768	1	47	of	of	ADP
ejpam-5768	1	48	science	science	PROPN
ejpam-5768	1	49	,	,	PUNCT
ejpam-5768	1	50	taibah	taibah	PROPN
ejpam-5768	1	51	university	university	PROPN
ejpam-5768	1	52	,	,	PUNCT
ejpam-5768	1	53	madinah	madinah	PROPN
ejpam-5768	1	54	,	,	PUNCT
ejpam-5768	1	55	saudi	saudi	PROPN
ejpam-5768	1	56	arabia	arabia	PROPN
ejpam-5768	1	57	abstract	abstract	NOUN
ejpam-5768	1	58	.	.	PUNCT
ejpam-5768	2	1	in	in	ADP
ejpam-5768	2	2	this	this	DET
ejpam-5768	2	3	paper	paper	NOUN
ejpam-5768	2	4	,	,	PUNCT
ejpam-5768	2	5	we	we	PRON
ejpam-5768	2	6	introduce	introduce	VERB
ejpam-5768	2	7	the	the	DET
ejpam-5768	2	8	notion	notion	NOUN
ejpam-5768	2	9	of	of	ADP
ejpam-5768	2	10	a	a	DET
ejpam-5768	2	11	pythagorean	pythagorean	ADJ
ejpam-5768	2	12	fuzzy	fuzzy	ADJ
ejpam-5768	2	13	hx	hx	PROPN
ejpam-5768	2	14	-	-	PUNCT
ejpam-5768	2	15	subgroup	subgroup	PROPN
ejpam-5768	2	16	and	and	CCONJ
ejpam-5768	2	17	a	a	DET
ejpam-5768	2	18	normal	normal	ADJ
ejpam-5768	2	19	hx	hx	PROPN
ejpam-5768	2	20	-	-	PUNCT
ejpam-5768	2	21	subgroup	subgroup	NOUN
ejpam-5768	2	22	.	.	PUNCT
ejpam-5768	3	1	in	in	ADP
ejpam-5768	3	2	addition	addition	NOUN
ejpam-5768	3	3	,	,	PUNCT
ejpam-5768	3	4	we	we	PRON
ejpam-5768	3	5	prove	prove	VERB
ejpam-5768	3	6	various	various	ADJ
ejpam-5768	3	7	chracterisations	chracterisation	NOUN
ejpam-5768	3	8	for	for	ADP
ejpam-5768	3	9	pythagorean	pythagorean	PROPN
ejpam-5768	3	10	fuzzy	fuzzy	ADJ
ejpam-5768	3	11	hx	hx	PROPN
ejpam-5768	3	12	-	-	PUNCT
ejpam-5768	3	13	subgroups	subgroup	NOUN
ejpam-5768	3	14	and	and	CCONJ
ejpam-5768	3	15	pythagorean	pythagorean	PROPN
ejpam-5768	3	16	normal	normal	ADJ
ejpam-5768	3	17	hx	hx	PROPN
ejpam-5768	3	18	-	-	PUNCT
ejpam-5768	3	19	subgroups	subgroup	NOUN
ejpam-5768	3	20	.	.	PUNCT
ejpam-5768	4	1	moreover	moreover	ADV
ejpam-5768	4	2	,	,	PUNCT
ejpam-5768	4	3	the	the	DET
ejpam-5768	4	4	notations	notation	NOUN
ejpam-5768	4	5	of	of	ADP
ejpam-5768	4	6	pythagorean	pythagorean	PROPN
ejpam-5768	4	7	fuzzy	fuzzy	ADJ
ejpam-5768	4	8	hx	hx	PROPN
ejpam-5768	4	9	-	-	PUNCT
ejpam-5768	4	10	subgroups	subgroup	NOUN
ejpam-5768	4	11	homomorphisms	homomorphism	NOUN
ejpam-5768	4	12	and	and	CCONJ
ejpam-5768	4	13	antihomomorphisms	antihomomorphism	NOUN
ejpam-5768	4	14	are	be	AUX
ejpam-5768	4	15	introduced	introduce	VERB
ejpam-5768	4	16	,	,	PUNCT
ejpam-5768	4	17	and	and	CCONJ
ejpam-5768	4	18	some	some	DET
ejpam-5768	4	19	related	relate	VERB
ejpam-5768	4	20	properties	property	NOUN
ejpam-5768	4	21	regarding	regard	VERB
ejpam-5768	4	22	the	the	DET
ejpam-5768	4	23	relationship	relationship	NOUN
ejpam-5768	4	24	between	between	ADP
ejpam-5768	4	25	a	a	DET
ejpam-5768	4	26	pythagorean	pythagorean	ADJ
ejpam-5768	4	27	fuzzy	fuzzy	ADJ
ejpam-5768	4	28	set	set	NOUN
ejpam-5768	4	29	and	and	CCONJ
ejpam-5768	4	30	its	its	PRON
ejpam-5768	4	31	image	image	NOUN
ejpam-5768	4	32	are	be	AUX
ejpam-5768	4	33	investigated	investigate	VERB
ejpam-5768	4	34	.	.	PUNCT
ejpam-5768	5	1	characterisations	characterisation	NOUN
ejpam-5768	5	2	of	of	ADP
ejpam-5768	5	3	level	level	NOUN
ejpam-5768	5	4	pythagorean	pythagorean	PROPN
ejpam-5768	5	5	fuzzy	fuzzy	ADJ
ejpam-5768	5	6	hx	hx	PROPN
ejpam-5768	5	7	-	-	PUNCT
ejpam-5768	5	8	subgroups	subgroup	NOUN
ejpam-5768	5	9	and	and	CCONJ
ejpam-5768	5	10	normal	normal	ADJ
ejpam-5768	5	11	hx	hx	NOUN
ejpam-5768	5	12	-	-	PUNCT
ejpam-5768	5	13	subgroups	subgroup	NOUN
ejpam-5768	5	14	are	be	AUX
ejpam-5768	5	15	proved	prove	VERB
ejpam-5768	5	16	.	.	PUNCT
ejpam-5768	6	1	these	these	DET
ejpam-5768	6	2	results	result	NOUN
ejpam-5768	6	3	generalised	generalise	VERB
ejpam-5768	6	4	some	some	DET
ejpam-5768	6	5	results	result	NOUN
ejpam-5768	6	6	regarding	regard	VERB
ejpam-5768	6	7	fuzzy	fuzzy	ADJ
ejpam-5768	6	8	hx	hx	NOUN
ejpam-5768	6	9	-	-	PUNCT
ejpam-5768	6	10	subgroups	subgroup	NOUN
ejpam-5768	6	11	.	.	PUNCT
ejpam-5768	7	1	2020	2020	NUM
ejpam-5768	7	2	mathematics	mathematic	NOUN
ejpam-5768	7	3	subject	subject	NOUN
ejpam-5768	7	4	classifications	classification	NOUN
ejpam-5768	7	5	:	:	PUNCT
ejpam-5768	7	6	03e72	03e72	NUM
ejpam-5768	7	7	,	,	PUNCT
ejpam-5768	7	8	20n25	20n25	NOUN
ejpam-5768	7	9	key	key	ADJ
ejpam-5768	7	10	words	word	NOUN
ejpam-5768	7	11	and	and	CCONJ
ejpam-5768	7	12	phrases	phrase	NOUN
ejpam-5768	7	13	:	:	PUNCT
ejpam-5768	7	14	hx	hx	NOUN
ejpam-5768	7	15	-	-	PUNCT
ejpam-5768	7	16	groups	group	NOUN
ejpam-5768	7	17	,	,	PUNCT
ejpam-5768	7	18	pythagorean	pythagorean	ADJ
ejpam-5768	7	19	fuzzy	fuzzy	ADJ
ejpam-5768	7	20	sets	set	NOUN
ejpam-5768	7	21	,	,	PUNCT
ejpam-5768	7	22	pythagorean	pythagorean	ADJ
ejpam-5768	7	23	fuzzy	fuzzy	ADJ
ejpam-5768	7	24	homomorphism	homomorphism	PROPN
ejpam-5768	7	25	,	,	PUNCT
ejpam-5768	7	26	pythagorean	pythagorean	PROPN
ejpam-5768	7	27	fuzzy	fuzzy	ADJ
ejpam-5768	7	28	antihomomorphism	antihomomorphism	NOUN
ejpam-5768	7	29	,	,	PUNCT
ejpam-5768	7	30	normal	normal	ADJ
ejpam-5768	7	31	hx	hx	NOUN
ejpam-5768	7	32	-	-	PUNCT
ejpam-5768	7	33	subgroups	subgroup	NOUN
ejpam-5768	7	34	1	1	NUM
ejpam-5768	7	35	.	.	X
ejpam-5768	7	36	introduction	introduction	NOUN
ejpam-5768	7	37	a	a	DET
ejpam-5768	7	38	generalisation	generalisation	NOUN
ejpam-5768	7	39	of	of	ADP
ejpam-5768	7	40	the	the	DET
ejpam-5768	7	41	classical	classical	ADJ
ejpam-5768	7	42	set	set	NOUN
ejpam-5768	7	43	,	,	PUNCT
ejpam-5768	7	44	the	the	DET
ejpam-5768	7	45	fuzzy	fuzzy	ADJ
ejpam-5768	7	46	set	set	NOUN
ejpam-5768	7	47	notion	notion	NOUN
ejpam-5768	7	48	was	be	AUX
ejpam-5768	7	49	first	first	ADV
ejpam-5768	7	50	presented	present	VERB
ejpam-5768	7	51	by	by	ADP
ejpam-5768	7	52	zadeh	zadeh	PROPN
ejpam-5768	7	53	in	in	ADP
ejpam-5768	7	54	1965	1965	NUM
ejpam-5768	8	1	[	[	X
ejpam-5768	8	2	1	1	NUM
ejpam-5768	8	3	]	]	PUNCT
ejpam-5768	8	4	.	.	PUNCT
ejpam-5768	9	1	this	this	DET
ejpam-5768	9	2	set	set	NOUN
ejpam-5768	9	3	addressed	address	VERB
ejpam-5768	9	4	the	the	DET
ejpam-5768	9	5	relationship	relationship	NOUN
ejpam-5768	9	6	between	between	ADP
ejpam-5768	9	7	elements	element	NOUN
ejpam-5768	9	8	and	and	CCONJ
ejpam-5768	9	9	sets	set	NOUN
ejpam-5768	9	10	and	and	CCONJ
ejpam-5768	9	11	answered	answer	VERB
ejpam-5768	9	12	the	the	DET
ejpam-5768	9	13	question	question	NOUN
ejpam-5768	9	14	:	:	PUNCT
ejpam-5768	9	15	to	to	ADP
ejpam-5768	9	16	what	what	DET
ejpam-5768	9	17	extent	extent	NOUN
ejpam-5768	9	18	can	can	AUX
ejpam-5768	9	19	this	this	DET
ejpam-5768	9	20	object	object	NOUN
ejpam-5768	9	21	belong	belong	VERB
ejpam-5768	9	22	to	to	ADP
ejpam-5768	9	23	a	a	DET
ejpam-5768	9	24	particular	particular	ADJ
ejpam-5768	9	25	set	set	NOUN
ejpam-5768	9	26	,	,	PUNCT
ejpam-5768	9	27	as	as	SCONJ
ejpam-5768	9	28	every	every	DET
ejpam-5768	9	29	object	object	NOUN
ejpam-5768	9	30	x	x	PUNCT
ejpam-5768	9	31	has	have	VERB
ejpam-5768	9	32	a	a	DET
ejpam-5768	9	33	value	value	NOUN
ejpam-5768	9	34	η(x	η(x	NOUN
ejpam-5768	9	35	)	)	PUNCT
ejpam-5768	9	36	,	,	PUNCT
ejpam-5768	9	37	where	where	SCONJ
ejpam-5768	9	38	η	η	PROPN
ejpam-5768	9	39	is	be	AUX
ejpam-5768	9	40	called	call	VERB
ejpam-5768	9	41	a	a	DET
ejpam-5768	9	42	membership	membership	NOUN
ejpam-5768	9	43	function	function	NOUN
ejpam-5768	9	44	η	η	PROPN
ejpam-5768	9	45	:	:	PUNCT
ejpam-5768	9	46	x	x	SYM
ejpam-5768	9	47	→	→	PUNCT
ejpam-5768	10	1	[	[	X
ejpam-5768	10	2	0.1	0.1	NUM
ejpam-5768	10	3	]	]	PUNCT
ejpam-5768	10	4	.	.	PUNCT
ejpam-5768	11	1	many	many	ADJ
ejpam-5768	11	2	ideas	idea	NOUN
ejpam-5768	11	3	and	and	CCONJ
ejpam-5768	11	4	abstractions	abstraction	NOUN
ejpam-5768	11	5	have	have	AUX
ejpam-5768	11	6	been	be	AUX
ejpam-5768	11	7	expanded	expand	VERB
ejpam-5768	11	8	since	since	SCONJ
ejpam-5768	11	9	fuzzy	fuzzy	ADJ
ejpam-5768	11	10	set	set	NOUN
ejpam-5768	11	11	theory	theory	NOUN
ejpam-5768	11	12	’s	’s	PART
ejpam-5768	11	13	inception	inception	NOUN
ejpam-5768	11	14	in	in	ADP
ejpam-5768	11	15	order	order	NOUN
ejpam-5768	11	16	to	to	PART
ejpam-5768	11	17	effectively	effectively	ADV
ejpam-5768	11	18	handle	handle	VERB
ejpam-5768	11	19	ambiguity	ambiguity	NOUN
ejpam-5768	11	20	and	and	CCONJ
ejpam-5768	11	21	uncertainty	uncertainty	NOUN
ejpam-5768	11	22	[	[	X
ejpam-5768	11	23	[	[	X
ejpam-5768	11	24	2	2	NUM
ejpam-5768	11	25	]	]	PUNCT
ejpam-5768	11	26	,	,	PUNCT
ejpam-5768	11	27	[	[	X
ejpam-5768	11	28	3	3	NUM
ejpam-5768	11	29	]	]	PUNCT
ejpam-5768	11	30	,	,	PUNCT
ejpam-5768	11	31	[	[	X
ejpam-5768	11	32	4	4	NUM
ejpam-5768	11	33	]	]	PUNCT
ejpam-5768	11	34	]	]	PUNCT
ejpam-5768	11	35	.	.	PUNCT
ejpam-5768	12	1	after	after	ADP
ejpam-5768	12	2	that	that	PRON
ejpam-5768	12	3	,	,	PUNCT
ejpam-5768	12	4	researchers	researcher	NOUN
ejpam-5768	12	5	found	find	VERB
ejpam-5768	12	6	that	that	SCONJ
ejpam-5768	12	7	the	the	DET
ejpam-5768	12	8	membership	membership	NOUN
ejpam-5768	12	9	function	function	NOUN
ejpam-5768	12	10	is	be	AUX
ejpam-5768	12	11	insufficient	insufficient	ADJ
ejpam-5768	12	12	on	on	ADP
ejpam-5768	12	13	its	its	PRON
ejpam-5768	12	14	own	own	ADJ
ejpam-5768	12	15	to	to	PART
ejpam-5768	12	16	tackle	tackle	VERB
ejpam-5768	12	17	some	some	DET
ejpam-5768	12	18	types	type	NOUN
ejpam-5768	12	19	of	of	ADP
ejpam-5768	12	20	situations	situation	NOUN
ejpam-5768	12	21	.	.	PUNCT
ejpam-5768	13	1	this	this	DET
ejpam-5768	13	2	motivated	motivated	ADJ
ejpam-5768	13	3	atanassov	atanassov	NOUN
ejpam-5768	13	4	[	[	X
ejpam-5768	13	5	5	5	NUM
ejpam-5768	13	6	]	]	PUNCT
ejpam-5768	13	7	and	and	CCONJ
ejpam-5768	13	8	[	[	X
ejpam-5768	13	9	6	6	NUM
ejpam-5768	13	10	]	]	PUNCT
ejpam-5768	13	11	to	to	PART
ejpam-5768	13	12	introduce	introduce	VERB
ejpam-5768	13	13	the	the	DET
ejpam-5768	13	14	idea	idea	NOUN
ejpam-5768	13	15	of	of	ADP
ejpam-5768	13	16	intuitionistic	intuitionistic	ADJ
ejpam-5768	13	17	fuzzy	fuzzy	ADJ
ejpam-5768	13	18	set	set	VERB
ejpam-5768	13	19	by	by	ADP
ejpam-5768	13	20	associating	associate	VERB
ejpam-5768	13	21	a	a	DET
ejpam-5768	13	22	fuzzy	fuzzy	ADJ
ejpam-5768	13	23	set	set	VERB
ejpam-5768	13	24	non	non	NOUN
ejpam-5768	13	25	-	-	NOUN
ejpam-5768	13	26	membership	membership	NOUN
ejpam-5768	13	27	with	with	ADP
ejpam-5768	13	28	its	its	PRON
ejpam-5768	13	29	membership	membership	NOUN
ejpam-5768	13	30	function	function	NOUN
ejpam-5768	13	31	.	.	PUNCT
ejpam-5768	14	1	the	the	DET
ejpam-5768	14	2	non	non	ADJ
ejpam-5768	14	3	-	-	ADJ
ejpam-5768	14	4	membership	membership	ADJ
ejpam-5768	14	5	η̂	η̂	NUM
ejpam-5768	14	6	and	and	CCONJ
ejpam-5768	14	7	membership	membership	PROPN
ejpam-5768	14	8	η	η	PROPN
ejpam-5768	14	9	in	in	ADP
ejpam-5768	14	10	this	this	DET
ejpam-5768	14	11	class	class	NOUN
ejpam-5768	14	12	are	be	AUX
ejpam-5768	14	13	satisfied	satisfied	ADJ
ejpam-5768	14	14	:	:	PUNCT
ejpam-5768	14	15	0	0	NUM
ejpam-5768	14	16	≤	≤	NUM
ejpam-5768	14	17	η(x)+	η(x)+	PROPN
ejpam-5768	14	18	η̂(x	η̂(x	NUM
ejpam-5768	14	19	)	)	PUNCT
ejpam-5768	14	20	≤	≤	NUM
ejpam-5768	14	21	1	1	NUM
ejpam-5768	14	22	.	.	PUNCT
ejpam-5768	15	1	this	this	DET
ejpam-5768	15	2	set	set	NOUN
ejpam-5768	15	3	can	can	AUX
ejpam-5768	15	4	cope	cope	VERB
ejpam-5768	15	5	with	with	ADP
ejpam-5768	15	6	ambiguous	ambiguous	ADJ
ejpam-5768	15	7	and	and	CCONJ
ejpam-5768	15	8	unclear	unclear	ADJ
ejpam-5768	15	9	situations	situation	NOUN
ejpam-5768	15	10	more	more	ADV
ejpam-5768	15	11	effectively	effectively	ADV
ejpam-5768	15	12	than	than	ADP
ejpam-5768	15	13	fuzzy	fuzzy	ADJ
ejpam-5768	15	14	sets	set	NOUN
ejpam-5768	15	15	since	since	SCONJ
ejpam-5768	15	16	it	it	PRON
ejpam-5768	15	17	has	have	VERB
ejpam-5768	15	18	both	both	CCONJ
ejpam-5768	15	19	a	a	DET
ejpam-5768	15	20	non	non	ADJ
ejpam-5768	15	21	-	-	NOUN
ejpam-5768	15	22	membership	membership	NOUN
ejpam-5768	15	23	and	and	CCONJ
ejpam-5768	15	24	a	a	DET
ejpam-5768	15	25	membership	membership	NOUN
ejpam-5768	15	26	functions	function	NOUN
ejpam-5768	15	27	,	,	PUNCT
ejpam-5768	15	28	see	see	VERB
ejpam-5768	15	29	[	[	X
ejpam-5768	15	30	[	[	X
ejpam-5768	15	31	7	7	NUM
ejpam-5768	15	32	]	]	PUNCT
ejpam-5768	15	33	,	,	PUNCT
ejpam-5768	15	34	[	[	X
ejpam-5768	15	35	8	8	NUM
ejpam-5768	15	36	]	]	PUNCT
ejpam-5768	15	37	,	,	PUNCT
ejpam-5768	15	38	[	[	X
ejpam-5768	15	39	9	9	NUM
ejpam-5768	15	40	]	]	SYM
ejpam-5768	15	41	]	]	PUNCT
ejpam-5768	15	42	.	.	PUNCT
ejpam-5768	16	1	however	however	ADV
ejpam-5768	16	2	,	,	PUNCT
ejpam-5768	16	3	if	if	SCONJ
ejpam-5768	16	4	the	the	DET
ejpam-5768	16	5	situation	situation	NOUN
ejpam-5768	16	6	required	require	VERB
ejpam-5768	16	7	η(x	η(x	NOUN
ejpam-5768	16	8	)	)	PUNCT
ejpam-5768	16	9	+	+	CCONJ
ejpam-5768	16	10	η̂(x	η̂(x	NUM
ejpam-5768	16	11	)	)	PUNCT
ejpam-5768	16	12	≥	≥	NOUN
ejpam-5768	16	13	1	1	NUM
ejpam-5768	16	14	,	,	PUNCT
ejpam-5768	16	15	then	then	ADV
ejpam-5768	16	16	intuitionistic	intuitionistic	ADJ
ejpam-5768	16	17	fuzzy	fuzzy	ADJ
ejpam-5768	16	18	set	set	NOUN
ejpam-5768	16	19	theory	theory	NOUN
ejpam-5768	16	20	is	be	AUX
ejpam-5768	16	21	not	not	PART
ejpam-5768	16	22	applicable	applicable	ADJ
ejpam-5768	16	23	.	.	PUNCT
ejpam-5768	17	1	to	to	PART
ejpam-5768	17	2	find	find	VERB
ejpam-5768	17	3	a	a	DET
ejpam-5768	17	4	suitable	suitable	ADJ
ejpam-5768	17	5	answer	answer	NOUN
ejpam-5768	17	6	in	in	ADP
ejpam-5768	17	7	these	these	DET
ejpam-5768	17	8	situations	situation	NOUN
ejpam-5768	17	9	,	,	PUNCT
ejpam-5768	17	10	yager	yager	NOUN
ejpam-5768	18	1	[	[	X
ejpam-5768	18	2	10	10	NUM
ejpam-5768	18	3	]	]	PUNCT
ejpam-5768	18	4	introduced	introduce	VERB
ejpam-5768	18	5	the	the	DET
ejpam-5768	18	6	doi	doi	NOUN
ejpam-5768	18	7	:	:	PUNCT
ejpam-5768	18	8	https://doi.org/10.29020/nybg.ejpam.v18i2.5768	https://doi.org/10.29020/nybg.ejpam.v18i2.5768	DET
ejpam-5768	18	9	email	email	NOUN
ejpam-5768	18	10	address	address	NOUN
ejpam-5768	18	11	:	:	PUNCT
ejpam-5768	18	12	aamuhaimeed@taibahu.edu.sa	aamuhaimeed@taibahu.edu.sa	NOUN
ejpam-5768	18	13	(	(	PUNCT
ejpam-5768	18	14	a.	a.	NOUN
ejpam-5768	18	15	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	18	16	)	)	PUNCT
ejpam-5768	18	17	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5768	19	1	1	1	NUM
ejpam-5768	19	2	copyright	copyright	NOUN
ejpam-5768	19	3	:	:	PUNCT
ejpam-5768	19	4	©	©	PROPN
ejpam-5768	19	5	2025	2025	NUM
ejpam-5768	19	6	the	the	DET
ejpam-5768	19	7	author(s	author(s	NOUN
ejpam-5768	19	8	)	)	PUNCT
ejpam-5768	19	9	.	.	PUNCT
ejpam-5768	20	1	(	(	PUNCT
ejpam-5768	20	2	cc	cc	NOUN
ejpam-5768	20	3	by	by	ADP
ejpam-5768	20	4	-	-	PUNCT
ejpam-5768	20	5	nc	nc	PROPN
ejpam-5768	20	6	4.0	4.0	NUM
ejpam-5768	20	7	)	)	PUNCT
ejpam-5768	20	8	a.	a.	NOUN
ejpam-5768	20	9	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	20	10	/	/	SYM
ejpam-5768	20	11	eur	eur	PROPN
ejpam-5768	20	12	.	.	PUNCT
ejpam-5768	21	1	j.	j.	PROPN
ejpam-5768	21	2	pure	pure	PROPN
ejpam-5768	21	3	appl	appl	PROPN
ejpam-5768	21	4	.	.	PROPN
ejpam-5768	21	5	math	math	PROPN
ejpam-5768	21	6	,	,	PUNCT
ejpam-5768	21	7	18	18	NUM
ejpam-5768	21	8	(	(	PUNCT
ejpam-5768	21	9	2	2	NUM
ejpam-5768	21	10	)	)	PUNCT
ejpam-5768	21	11	(	(	PUNCT
ejpam-5768	21	12	2025	2025	NUM
ejpam-5768	21	13	)	)	PUNCT
ejpam-5768	21	14	,	,	PUNCT
ejpam-5768	21	15	5768	5768	NUM
ejpam-5768	21	16	2	2	NUM
ejpam-5768	21	17	of	of	ADP
ejpam-5768	21	18	17	17	NUM
ejpam-5768	21	19	concept	concept	NOUN
ejpam-5768	21	20	of	of	ADP
ejpam-5768	21	21	pythagorean	pythagorean	PROPN
ejpam-5768	21	22	fuzzy	fuzzy	PROPN
ejpam-5768	21	23	subset	subset	NOUN
ejpam-5768	21	24	.	.	PUNCT
ejpam-5768	22	1	it	it	PRON
ejpam-5768	22	2	assigns	assign	VERB
ejpam-5768	22	3	to	to	ADP
ejpam-5768	22	4	every	every	DET
ejpam-5768	22	5	object	object	NOUN
ejpam-5768	22	6	x	x	PUNCT
ejpam-5768	22	7	in	in	ADP
ejpam-5768	22	8	a	a	DET
ejpam-5768	22	9	universe	universe	NOUN
ejpam-5768	22	10	set	set	VERB
ejpam-5768	22	11	two	two	NUM
ejpam-5768	22	12	memberships	membership	NOUN
ejpam-5768	22	13	:	:	PUNCT
ejpam-5768	22	14	the	the	DET
ejpam-5768	22	15	membership	membership	NOUN
ejpam-5768	22	16	η(x	η(x	NOUN
ejpam-5768	22	17	)	)	PUNCT
ejpam-5768	22	18	and	and	CCONJ
ejpam-5768	22	19	the	the	DET
ejpam-5768	22	20	non	non	ADJ
ejpam-5768	22	21	-	-	NOUN
ejpam-5768	22	22	membership	membership	NOUN
ejpam-5768	22	23	η̂(x	η̂(x	NUM
ejpam-5768	22	24	)	)	PUNCT
ejpam-5768	22	25	in	in	ADP
ejpam-5768	22	26	which	which	PRON
ejpam-5768	22	27	0	0	NUM
ejpam-5768	22	28	≤	≤	NOUN
ejpam-5768	22	29	η(x)2	η(x)2	VERB
ejpam-5768	22	30	+	+	CCONJ
ejpam-5768	22	31	η̂(x)2	η̂(x)2	NOUN
ejpam-5768	22	32	≤	≤	NUM
ejpam-5768	22	33	1	1	NUM
ejpam-5768	22	34	.	.	PUNCT
ejpam-5768	22	35	.	.	PUNCT
ejpam-5768	23	1	consequently	consequently	ADV
ejpam-5768	23	2	,	,	PUNCT
ejpam-5768	23	3	a	a	DET
ejpam-5768	23	4	pythagorean	pythagorean	ADJ
ejpam-5768	23	5	fuzzy	fuzzy	ADJ
ejpam-5768	23	6	set	set	NOUN
ejpam-5768	23	7	could	could	AUX
ejpam-5768	23	8	be	be	AUX
ejpam-5768	23	9	thought	think	VERB
ejpam-5768	23	10	of	of	ADP
ejpam-5768	23	11	as	as	ADP
ejpam-5768	23	12	an	an	DET
ejpam-5768	23	13	extension	extension	NOUN
ejpam-5768	23	14	of	of	ADP
ejpam-5768	23	15	an	an	DET
ejpam-5768	23	16	intuitionistic	intuitionistic	ADJ
ejpam-5768	23	17	fuzzy	fuzzy	ADJ
ejpam-5768	23	18	set	set	NOUN
ejpam-5768	23	19	.	.	PUNCT
ejpam-5768	24	1	furthermore	furthermore	ADV
ejpam-5768	24	2	,	,	PUNCT
ejpam-5768	24	3	because	because	SCONJ
ejpam-5768	24	4	the	the	DET
ejpam-5768	24	5	condition	condition	NOUN
ejpam-5768	24	6	0	0	NUM
ejpam-5768	24	7	≤	≤	NUM
ejpam-5768	24	8	η(x)2	η(x)2	VERB
ejpam-5768	24	9	+	+	CCONJ
ejpam-5768	24	10	η̂(x)2	η̂(x)2	NOUN
ejpam-5768	24	11	≤	≤	NUM
ejpam-5768	24	12	1	1	NUM
ejpam-5768	24	13	provides	provide	VERB
ejpam-5768	24	14	more	more	ADJ
ejpam-5768	24	15	pairs	pair	NOUN
ejpam-5768	24	16	(	(	PUNCT
ejpam-5768	24	17	η	η	NOUN
ejpam-5768	24	18	,	,	PUNCT
ejpam-5768	24	19	η̂	η̂	NUM
ejpam-5768	24	20	)	)	PUNCT
ejpam-5768	24	21	than	than	ADP
ejpam-5768	24	22	the	the	DET
ejpam-5768	24	23	condition	condition	NOUN
ejpam-5768	24	24	0	0	NUM
ejpam-5768	24	25	≤	≤	NOUN
ejpam-5768	24	26	η(x	η(x	NOUN
ejpam-5768	24	27	)	)	PUNCT
ejpam-5768	24	28	+	+	CCONJ
ejpam-5768	24	29	η̂(x	η̂(x	NUM
ejpam-5768	24	30	)	)	PUNCT
ejpam-5768	24	31	≤	≤	NUM
ejpam-5768	24	32	1	1	NUM
ejpam-5768	24	33	,	,	PUNCT
ejpam-5768	24	34	this	this	DET
ejpam-5768	24	35	generalisation	generalisation	NOUN
ejpam-5768	24	36	results	result	VERB
ejpam-5768	24	37	in	in	ADP
ejpam-5768	24	38	a	a	DET
ejpam-5768	24	39	greater	great	ADJ
ejpam-5768	24	40	number	number	NOUN
ejpam-5768	24	41	of	of	ADP
ejpam-5768	24	42	applications	application	NOUN
ejpam-5768	24	43	that	that	PRON
ejpam-5768	24	44	can	can	AUX
ejpam-5768	24	45	be	be	AUX
ejpam-5768	24	46	found	find	VERB
ejpam-5768	24	47	using	use	VERB
ejpam-5768	24	48	pythagorean	pythagorean	PROPN
ejpam-5768	24	49	fuzzy	fuzzy	ADJ
ejpam-5768	24	50	sets	set	NOUN
ejpam-5768	24	51	than	than	ADP
ejpam-5768	24	52	those	those	PRON
ejpam-5768	24	53	that	that	PRON
ejpam-5768	24	54	are	be	AUX
ejpam-5768	24	55	solved	solve	VERB
ejpam-5768	24	56	by	by	ADP
ejpam-5768	24	57	intuitionistic	intuitionistic	ADJ
ejpam-5768	24	58	fuzzy	fuzzy	ADJ
ejpam-5768	24	59	sets	set	NOUN
ejpam-5768	24	60	.	.	PUNCT
ejpam-5768	25	1	pythagorean	pythagorean	PROPN
ejpam-5768	25	2	fuzzy	fuzzy	ADJ
ejpam-5768	25	3	sets	set	NOUN
ejpam-5768	25	4	can	can	AUX
ejpam-5768	25	5	therefore	therefore	ADV
ejpam-5768	25	6	be	be	AUX
ejpam-5768	25	7	used	use	VERB
ejpam-5768	25	8	to	to	PART
ejpam-5768	25	9	solve	solve	VERB
ejpam-5768	25	10	more	more	ADJ
ejpam-5768	25	11	issues	issue	NOUN
ejpam-5768	25	12	and	and	CCONJ
ejpam-5768	25	13	produce	produce	VERB
ejpam-5768	25	14	precise	precise	ADJ
ejpam-5768	25	15	and	and	CCONJ
ejpam-5768	25	16	efficient	efficient	ADJ
ejpam-5768	25	17	algorithms	algorithm	NOUN
ejpam-5768	26	1	[	[	X
ejpam-5768	26	2	[	[	X
ejpam-5768	26	3	11	11	NUM
ejpam-5768	26	4	]	]	PUNCT
ejpam-5768	26	5	,	,	PUNCT
ejpam-5768	26	6	[	[	X
ejpam-5768	26	7	12	12	NUM
ejpam-5768	26	8	]	]	PUNCT
ejpam-5768	26	9	,	,	PUNCT
ejpam-5768	26	10	[	[	X
ejpam-5768	26	11	7	7	NUM
ejpam-5768	26	12	]	]	X
ejpam-5768	26	13	]	]	PUNCT
ejpam-5768	26	14	.	.	PUNCT
ejpam-5768	27	1	the	the	DET
ejpam-5768	27	2	concept	concept	NOUN
ejpam-5768	27	3	of	of	ADP
ejpam-5768	27	4	pythagorean	pythagorean	PROPN
ejpam-5768	27	5	fuzzy	fuzzy	ADJ
ejpam-5768	27	6	sets	set	NOUN
ejpam-5768	27	7	was	be	AUX
ejpam-5768	27	8	applied	apply	VERB
ejpam-5768	27	9	to	to	ADP
ejpam-5768	27	10	groups	group	NOUN
ejpam-5768	27	11	,	,	PUNCT
ejpam-5768	27	12	rings	ring	NOUN
ejpam-5768	27	13	and	and	CCONJ
ejpam-5768	27	14	modules	module	NOUN
ejpam-5768	27	15	,	,	PUNCT
ejpam-5768	27	16	see	see	VERB
ejpam-5768	27	17	[	[	X
ejpam-5768	27	18	[	[	X
ejpam-5768	27	19	13	13	NUM
ejpam-5768	27	20	]	]	PUNCT
ejpam-5768	27	21	,	,	PUNCT
ejpam-5768	27	22	[	[	X
ejpam-5768	27	23	14	14	NUM
ejpam-5768	27	24	]	]	PUNCT
ejpam-5768	27	25	,	,	PUNCT
ejpam-5768	27	26	[	[	X
ejpam-5768	27	27	15	15	NUM
ejpam-5768	27	28	]	]	SYM
ejpam-5768	27	29	]	]	PUNCT
ejpam-5768	27	30	.	.	PUNCT
ejpam-5768	28	1	group	group	NOUN
ejpam-5768	28	2	theory	theory	NOUN
ejpam-5768	28	3	is	be	AUX
ejpam-5768	28	4	an	an	DET
ejpam-5768	28	5	important	important	ADJ
ejpam-5768	28	6	branch	branch	NOUN
ejpam-5768	28	7	of	of	ADP
ejpam-5768	28	8	mathematics	mathematic	NOUN
ejpam-5768	28	9	.	.	PUNCT
ejpam-5768	29	1	it	it	PRON
ejpam-5768	29	2	can	can	AUX
ejpam-5768	29	3	sort	sort	VERB
ejpam-5768	29	4	numerous	numerous	ADJ
ejpam-5768	29	5	problems	problem	NOUN
ejpam-5768	29	6	in	in	ADP
ejpam-5768	29	7	several	several	ADJ
ejpam-5768	29	8	fields	field	NOUN
ejpam-5768	29	9	of	of	ADP
ejpam-5768	29	10	science	science	NOUN
ejpam-5768	29	11	.	.	PUNCT
ejpam-5768	30	1	the	the	DET
ejpam-5768	30	2	applications	application	NOUN
ejpam-5768	30	3	of	of	ADP
ejpam-5768	30	4	group	group	NOUN
ejpam-5768	30	5	theory	theory	NOUN
ejpam-5768	30	6	described	describe	VERB
ejpam-5768	30	7	in	in	ADP
ejpam-5768	30	8	many	many	ADJ
ejpam-5768	30	9	papers	paper	NOUN
ejpam-5768	30	10	[	[	PUNCT
ejpam-5768	30	11	[	[	X
ejpam-5768	30	12	16	16	NUM
ejpam-5768	30	13	]	]	PUNCT
ejpam-5768	30	14	,	,	PUNCT
ejpam-5768	30	15	[	[	X
ejpam-5768	30	16	17	17	NUM
ejpam-5768	30	17	]	]	PUNCT
ejpam-5768	30	18	,	,	PUNCT
ejpam-5768	30	19	[	[	X
ejpam-5768	30	20	18	18	NUM
ejpam-5768	30	21	]	]	X
ejpam-5768	30	22	]	]	PUNCT
ejpam-5768	30	23	.	.	PUNCT
ejpam-5768	31	1	the	the	DET
ejpam-5768	31	2	idea	idea	NOUN
ejpam-5768	31	3	of	of	ADP
ejpam-5768	31	4	applying	apply	VERB
ejpam-5768	31	5	fuzzy	fuzzy	ADJ
ejpam-5768	31	6	settings	setting	NOUN
ejpam-5768	31	7	on	on	ADP
ejpam-5768	31	8	groups	group	NOUN
ejpam-5768	31	9	was	be	AUX
ejpam-5768	31	10	introduced	introduce	VERB
ejpam-5768	31	11	by	by	ADP
ejpam-5768	31	12	rosenfeld	rosenfeld	PROPN
ejpam-5768	31	13	[	[	X
ejpam-5768	31	14	19	19	NUM
ejpam-5768	31	15	]	]	X
ejpam-5768	31	16	,	,	PUNCT
ejpam-5768	31	17	who	who	PRON
ejpam-5768	31	18	expanded	expand	VERB
ejpam-5768	31	19	on	on	ADP
ejpam-5768	31	20	the	the	DET
ejpam-5768	31	21	idea	idea	NOUN
ejpam-5768	31	22	of	of	ADP
ejpam-5768	31	23	classical	classical	ADJ
ejpam-5768	31	24	groups	group	NOUN
ejpam-5768	31	25	.	.	PUNCT
ejpam-5768	32	1	after	after	ADP
ejpam-5768	32	2	that	that	PRON
ejpam-5768	32	3	,	,	PUNCT
ejpam-5768	32	4	numerous	numerous	ADJ
ejpam-5768	32	5	efforts	effort	NOUN
ejpam-5768	32	6	have	have	AUX
ejpam-5768	32	7	been	be	AUX
ejpam-5768	32	8	conducted	conduct	VERB
ejpam-5768	32	9	to	to	PART
ejpam-5768	32	10	study	study	VERB
ejpam-5768	32	11	fuzzy	fuzzy	ADJ
ejpam-5768	32	12	groups	group	NOUN
ejpam-5768	32	13	in	in	ADP
ejpam-5768	32	14	several	several	ADJ
ejpam-5768	32	15	fuzzy	fuzzy	ADJ
ejpam-5768	32	16	environments	environment	NOUN
ejpam-5768	32	17	.	.	PUNCT
ejpam-5768	33	1	the	the	DET
ejpam-5768	33	2	notation	notation	NOUN
ejpam-5768	33	3	of	of	ADP
ejpam-5768	33	4	hx	hx	PROPN
ejpam-5768	33	5	-	-	PUNCT
ejpam-5768	33	6	groups	group	NOUN
ejpam-5768	33	7	was	be	AUX
ejpam-5768	33	8	introduced	introduce	VERB
ejpam-5768	33	9	in	in	ADP
ejpam-5768	33	10	[	[	X
ejpam-5768	33	11	20	20	NUM
ejpam-5768	33	12	]	]	PUNCT
ejpam-5768	33	13	,	,	PUNCT
ejpam-5768	33	14	by	by	ADP
ejpam-5768	33	15	li	li	PROPN
ejpam-5768	33	16	hongxing	hongxing	PROPN
ejpam-5768	33	17	.	.	PUNCT
ejpam-5768	34	1	the	the	DET
ejpam-5768	34	2	concept	concept	NOUN
ejpam-5768	34	3	has	have	AUX
ejpam-5768	34	4	since	since	ADV
ejpam-5768	34	5	been	be	AUX
ejpam-5768	34	6	the	the	DET
ejpam-5768	34	7	subject	subject	NOUN
ejpam-5768	34	8	of	of	ADP
ejpam-5768	34	9	various	various	ADJ
ejpam-5768	34	10	studies	study	NOUN
ejpam-5768	34	11	;	;	PUNCT
ejpam-5768	34	12	see	see	VERB
ejpam-5768	34	13	[	[	X
ejpam-5768	34	14	21	21	NUM
ejpam-5768	34	15	]	]	PUNCT
ejpam-5768	34	16	,	,	PUNCT
ejpam-5768	34	17	[	[	X
ejpam-5768	34	18	22	22	NUM
ejpam-5768	34	19	]	]	PUNCT
ejpam-5768	34	20	,	,	PUNCT
ejpam-5768	34	21	[	[	X
ejpam-5768	34	22	23	23	NUM
ejpam-5768	34	23	]	]	PUNCT
ejpam-5768	34	24	,	,	PUNCT
ejpam-5768	34	25	[	[	X
ejpam-5768	34	26	24	24	NUM
ejpam-5768	34	27	]	]	PUNCT
ejpam-5768	34	28	,	,	PUNCT
ejpam-5768	34	29	[	[	X
ejpam-5768	34	30	25	25	NUM
ejpam-5768	34	31	]	]	PUNCT
ejpam-5768	34	32	,	,	PUNCT
ejpam-5768	34	33	[	[	X
ejpam-5768	34	34	23	23	NUM
ejpam-5768	34	35	]	]	PUNCT
ejpam-5768	34	36	.	.	PUNCT
ejpam-5768	35	1	several	several	ADJ
ejpam-5768	35	2	authors	author	NOUN
ejpam-5768	35	3	have	have	AUX
ejpam-5768	35	4	merged	merge	VERB
ejpam-5768	35	5	the	the	DET
ejpam-5768	35	6	ideas	idea	NOUN
ejpam-5768	35	7	of	of	ADP
ejpam-5768	35	8	hx	hx	PROPN
ejpam-5768	35	9	groups	group	NOUN
ejpam-5768	35	10	with	with	ADP
ejpam-5768	35	11	the	the	DET
ejpam-5768	35	12	concept	concept	NOUN
ejpam-5768	35	13	of	of	ADP
ejpam-5768	35	14	fuzzy	fuzzy	ADJ
ejpam-5768	35	15	sets	set	NOUN
ejpam-5768	35	16	to	to	PART
ejpam-5768	35	17	get	get	VERB
ejpam-5768	35	18	some	some	DET
ejpam-5768	35	19	novel	novel	ADJ
ejpam-5768	35	20	findings	finding	NOUN
ejpam-5768	35	21	.	.	PUNCT
ejpam-5768	36	1	this	this	DET
ejpam-5768	36	2	study	study	NOUN
ejpam-5768	36	3	aims	aim	VERB
ejpam-5768	36	4	to	to	PART
ejpam-5768	36	5	establish	establish	VERB
ejpam-5768	36	6	the	the	DET
ejpam-5768	36	7	groundwork	groundwork	NOUN
ejpam-5768	36	8	for	for	ADP
ejpam-5768	36	9	a	a	DET
ejpam-5768	36	10	novel	novel	ADJ
ejpam-5768	36	11	theory	theory	NOUN
ejpam-5768	36	12	of	of	ADP
ejpam-5768	36	13	pythagorean	pythagorean	PROPN
ejpam-5768	36	14	fuzzy	fuzzy	PROPN
ejpam-5768	36	15	hx	hx	PROPN
ejpam-5768	36	16	-	-	PUNCT
ejpam-5768	36	17	subgroup	subgroup	NOUN
ejpam-5768	36	18	as	as	SCONJ
ejpam-5768	36	19	it	it	PRON
ejpam-5768	36	20	is	be	AUX
ejpam-5768	36	21	the	the	DET
ejpam-5768	36	22	extension	extension	NOUN
ejpam-5768	36	23	of	of	ADP
ejpam-5768	36	24	fuzzy	fuzzy	ADJ
ejpam-5768	36	25	hx	hx	NOUN
ejpam-5768	36	26	groups	group	NOUN
ejpam-5768	36	27	and	and	CCONJ
ejpam-5768	36	28	intuitionistic	intuitionistic	ADJ
ejpam-5768	36	29	fuzzy	fuzzy	ADJ
ejpam-5768	36	30	hxsubgroup	hxsubgroup	NOUN
ejpam-5768	36	31	.	.	PUNCT
ejpam-5768	37	1	in	in	ADP
ejpam-5768	37	2	this	this	DET
ejpam-5768	37	3	paper	paper	NOUN
ejpam-5768	37	4	,	,	PUNCT
ejpam-5768	37	5	we	we	PRON
ejpam-5768	37	6	introduce	introduce	VERB
ejpam-5768	37	7	the	the	DET
ejpam-5768	37	8	notion	notion	NOUN
ejpam-5768	37	9	of	of	ADP
ejpam-5768	37	10	a	a	DET
ejpam-5768	37	11	pythagorean	pythagorean	ADJ
ejpam-5768	37	12	fuzzy	fuzzy	ADJ
ejpam-5768	37	13	hx	hx	PROPN
ejpam-5768	37	14	-	-	PUNCT
ejpam-5768	37	15	subgroups	subgroup	NOUN
ejpam-5768	37	16	and	and	CCONJ
ejpam-5768	37	17	normal	normal	ADJ
ejpam-5768	37	18	hx	hx	NOUN
ejpam-5768	37	19	-	-	PUNCT
ejpam-5768	37	20	subgroups	subgroup	NOUN
ejpam-5768	37	21	.	.	PUNCT
ejpam-5768	38	1	in	in	ADP
ejpam-5768	38	2	addition	addition	NOUN
ejpam-5768	38	3	,	,	PUNCT
ejpam-5768	38	4	we	we	PRON
ejpam-5768	38	5	prove	prove	VERB
ejpam-5768	38	6	various	various	ADJ
ejpam-5768	38	7	chracterisations	chracterisation	NOUN
ejpam-5768	38	8	for	for	ADP
ejpam-5768	38	9	pythagorean	pythagorean	PROPN
ejpam-5768	38	10	fuzzy	fuzzy	ADJ
ejpam-5768	38	11	hx	hx	PROPN
ejpam-5768	38	12	-	-	PUNCT
ejpam-5768	38	13	subgroups	subgroup	NOUN
ejpam-5768	38	14	and	and	CCONJ
ejpam-5768	38	15	pythagorean	pythagorean	PROPN
ejpam-5768	38	16	normal	normal	ADJ
ejpam-5768	38	17	hx	hx	PROPN
ejpam-5768	38	18	-	-	PUNCT
ejpam-5768	38	19	subgroups	subgroup	NOUN
ejpam-5768	38	20	.	.	PUNCT
ejpam-5768	39	1	then	then	ADV
ejpam-5768	39	2	homomorphisms	homomorphism	NOUN
ejpam-5768	39	3	of	of	ADP
ejpam-5768	39	4	pythagorean	pythagorean	PROPN
ejpam-5768	39	5	fuzzy	fuzzy	PROPN
ejpam-5768	39	6	hx	hx	PROPN
ejpam-5768	39	7	-	-	PUNCT
ejpam-5768	39	8	subgroups	subgroup	NOUN
ejpam-5768	39	9	and	and	CCONJ
ejpam-5768	39	10	antihomomorphisms	antihomomorphism	NOUN
ejpam-5768	39	11	of	of	ADP
ejpam-5768	39	12	pythagorean	pythagorean	PROPN
ejpam-5768	39	13	fuzzy	fuzzy	ADJ
ejpam-5768	39	14	hxsubgroups	hxsubgroup	NOUN
ejpam-5768	39	15	are	be	AUX
ejpam-5768	39	16	discussed	discuss	VERB
ejpam-5768	39	17	.	.	PUNCT
ejpam-5768	40	1	several	several	ADJ
ejpam-5768	40	2	related	related	ADJ
ejpam-5768	40	3	properties	property	NOUN
ejpam-5768	40	4	regarding	regard	VERB
ejpam-5768	40	5	the	the	DET
ejpam-5768	40	6	relationship	relationship	NOUN
ejpam-5768	40	7	between	between	ADP
ejpam-5768	40	8	a	a	DET
ejpam-5768	40	9	pythagorean	pythagorean	ADJ
ejpam-5768	40	10	fuzzy	fuzzy	ADJ
ejpam-5768	40	11	set	set	NOUN
ejpam-5768	40	12	and	and	CCONJ
ejpam-5768	40	13	its	its	PRON
ejpam-5768	40	14	image	image	NOUN
ejpam-5768	40	15	are	be	AUX
ejpam-5768	40	16	investigated	investigate	VERB
ejpam-5768	40	17	.	.	PUNCT
ejpam-5768	41	1	moreover	moreover	ADV
ejpam-5768	41	2	,	,	PUNCT
ejpam-5768	41	3	pythagorean	pythagorean	PROPN
ejpam-5768	41	4	fuzzy	fuzzy	ADJ
ejpam-5768	41	5	level	level	NOUN
ejpam-5768	41	6	hx	hx	NOUN
ejpam-5768	41	7	-	-	PUNCT
ejpam-5768	41	8	subgroups	subgroup	NOUN
ejpam-5768	41	9	are	be	AUX
ejpam-5768	41	10	discussed	discuss	VERB
ejpam-5768	41	11	,	,	PUNCT
ejpam-5768	41	12	and	and	CCONJ
ejpam-5768	41	13	characterisations	characterisation	NOUN
ejpam-5768	41	14	of	of	ADP
ejpam-5768	41	15	these	these	DET
ejpam-5768	41	16	level	level	NOUN
ejpam-5768	41	17	pythagorean	pythagorean	NOUN
ejpam-5768	41	18	fuzzy	fuzzy	ADJ
ejpam-5768	41	19	hx	hx	PROPN
ejpam-5768	41	20	-	-	PUNCT
ejpam-5768	41	21	subgroups	subgroup	NOUN
ejpam-5768	41	22	and	and	CCONJ
ejpam-5768	41	23	normal	normal	ADJ
ejpam-5768	41	24	hx	hx	NOUN
ejpam-5768	41	25	-	-	PUNCT
ejpam-5768	41	26	subgroups	subgroup	NOUN
ejpam-5768	41	27	are	be	AUX
ejpam-5768	41	28	presented	present	VERB
ejpam-5768	41	29	.	.	PUNCT
ejpam-5768	42	1	throughout	throughout	ADP
ejpam-5768	42	2	this	this	DET
ejpam-5768	42	3	paper	paper	NOUN
ejpam-5768	42	4	,	,	PUNCT
ejpam-5768	42	5	we	we	PRON
ejpam-5768	42	6	write	write	VERB
ejpam-5768	42	7	pfss	pfss	NOUN
ejpam-5768	42	8	to	to	PART
ejpam-5768	42	9	denote	denote	VERB
ejpam-5768	42	10	a	a	DET
ejpam-5768	42	11	pythagorean	pythagorean	ADJ
ejpam-5768	42	12	fuzzy	fuzzy	NOUN
ejpam-5768	42	13	subset	subset	NOUN
ejpam-5768	42	14	,	,	PUNCT
ejpam-5768	42	15	pf	pf	X
ejpam-5768	42	16	hxsg	hxsg	ADV
ejpam-5768	42	17	to	to	PART
ejpam-5768	42	18	denote	denote	VERB
ejpam-5768	42	19	a	a	DET
ejpam-5768	42	20	pythagorean	pythagorean	ADJ
ejpam-5768	42	21	fuzzy	fuzzy	ADJ
ejpam-5768	42	22	hx	hx	PROPN
ejpam-5768	42	23	-	-	PUNCT
ejpam-5768	42	24	subgroup	subgroup	PROPN
ejpam-5768	42	25	and	and	CCONJ
ejpam-5768	42	26	pf	pf	PROPN
ejpam-5768	42	27	hx	hx	PROPN
ejpam-5768	42	28	-	-	PUNCT
ejpam-5768	42	29	nsg	nsg	PROPN
ejpam-5768	42	30	to	to	PART
ejpam-5768	42	31	denote	denote	VERB
ejpam-5768	42	32	a	a	DET
ejpam-5768	42	33	pythagorean	pythagorean	ADJ
ejpam-5768	42	34	fuzzy	fuzzy	ADJ
ejpam-5768	42	35	hx	hx	PROPN
ejpam-5768	42	36	-	-	PUNCT
ejpam-5768	42	37	normal	normal	ADJ
ejpam-5768	42	38	subgroup	subgroup	NOUN
ejpam-5768	42	39	.	.	PUNCT
ejpam-5768	43	1	2	2	X
ejpam-5768	43	2	.	.	X
ejpam-5768	43	3	pythagorean	pythagorean	PROPN
ejpam-5768	43	4	fuzzy	fuzzy	ADJ
ejpam-5768	43	5	hx	hx	PROPN
ejpam-5768	43	6	-	-	PUNCT
ejpam-5768	43	7	subgroups	subgroup	NOUN
ejpam-5768	43	8	recall	recall	VERB
ejpam-5768	43	9	that	that	SCONJ
ejpam-5768	43	10	[	[	X
ejpam-5768	43	11	26	26	NUM
ejpam-5768	43	12	]	]	X
ejpam-5768	43	13	a	a	DET
ejpam-5768	43	14	non	non	X
ejpam-5768	43	15	empty	empty	ADJ
ejpam-5768	43	16	set	set	NOUN
ejpam-5768	43	17	w	w	NOUN
ejpam-5768	43	18	⊆	⊆	NUM
ejpam-5768	43	19	2	2	NUM
ejpam-5768	43	20	g	g	NOUN
ejpam-5768	43	21	−	−	PROPN
ejpam-5768	43	22	{	{	PUNCT
ejpam-5768	43	23	ϕ	ϕ	NOUN
ejpam-5768	43	24	}	}	PUNCT
ejpam-5768	43	25	is	be	AUX
ejpam-5768	43	26	called	call	VERB
ejpam-5768	43	27	an	an	DET
ejpam-5768	43	28	hx	hx	NOUN
ejpam-5768	43	29	group	group	NOUN
ejpam-5768	43	30	on	on	ADP
ejpam-5768	43	31	g	g	PROPN
ejpam-5768	43	32	if	if	SCONJ
ejpam-5768	43	33	w	w	PROPN
ejpam-5768	43	34	is	be	AUX
ejpam-5768	43	35	a	a	DET
ejpam-5768	43	36	group	group	NOUN
ejpam-5768	43	37	with	with	ADP
ejpam-5768	43	38	respect	respect	NOUN
ejpam-5768	43	39	to	to	ADP
ejpam-5768	43	40	algebraic	algebraic	ADJ
ejpam-5768	43	41	operation	operation	NOUN
ejpam-5768	43	42	defined	define	VERB
ejpam-5768	43	43	by	by	ADP
ejpam-5768	43	44	mn	mn	PROPN
ejpam-5768	44	1	=	=	SYM
ejpam-5768	44	2	{	{	PUNCT
ejpam-5768	44	3	mn	mn	NOUN
ejpam-5768	44	4	:	:	PUNCT
ejpam-5768	44	5	m	m	VERB
ejpam-5768	44	6	∈	∈	PROPN
ejpam-5768	44	7	m	m	PROPN
ejpam-5768	44	8	,	,	PUNCT
ejpam-5768	44	9	n	n	PROPN
ejpam-5768	44	10	∈	∈	PROPN
ejpam-5768	44	11	n	n	CCONJ
ejpam-5768	44	12	}	}	PUNCT
ejpam-5768	44	13	and	and	CCONJ
ejpam-5768	44	14	the	the	DET
ejpam-5768	44	15	identity	identity	NOUN
ejpam-5768	44	16	element	element	NOUN
ejpam-5768	44	17	is	be	AUX
ejpam-5768	44	18	denoted	denote	VERB
ejpam-5768	44	19	by	by	ADP
ejpam-5768	44	20	e.	e.	PROPN
ejpam-5768	44	21	we	we	PRON
ejpam-5768	44	22	present	present	VERB
ejpam-5768	44	23	the	the	DET
ejpam-5768	44	24	following	follow	VERB
ejpam-5768	44	25	example	example	NOUN
ejpam-5768	44	26	:	:	PUNCT
ejpam-5768	44	27	a.	a.	NOUN
ejpam-5768	44	28	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	44	29	/	/	SYM
ejpam-5768	44	30	eur	eur	PROPN
ejpam-5768	44	31	.	.	PUNCT
ejpam-5768	45	1	j.	j.	PROPN
ejpam-5768	45	2	pure	pure	PROPN
ejpam-5768	45	3	appl	appl	PROPN
ejpam-5768	45	4	.	.	PROPN
ejpam-5768	45	5	math	math	PROPN
ejpam-5768	45	6	,	,	PUNCT
ejpam-5768	45	7	18	18	NUM
ejpam-5768	45	8	(	(	PUNCT
ejpam-5768	45	9	2	2	NUM
ejpam-5768	45	10	)	)	PUNCT
ejpam-5768	45	11	(	(	PUNCT
ejpam-5768	45	12	2025	2025	NUM
ejpam-5768	45	13	)	)	PUNCT
ejpam-5768	45	14	,	,	PUNCT
ejpam-5768	45	15	5768	5768	NUM
ejpam-5768	45	16	3	3	NUM
ejpam-5768	45	17	of	of	ADP
ejpam-5768	45	18	17	17	NUM
ejpam-5768	45	19	example	example	NOUN
ejpam-5768	45	20	1	1	NUM
ejpam-5768	45	21	.	.	X
ejpam-5768	45	22	consider	consider	VERB
ejpam-5768	45	23	the	the	DET
ejpam-5768	45	24	multiplicative	multiplicative	ADJ
ejpam-5768	45	25	group	group	NOUN
ejpam-5768	45	26	g	g	PROPN
ejpam-5768	45	27	=	=	PUNCT
ejpam-5768	45	28	{	{	PUNCT
ejpam-5768	45	29	±1,±i	±1,±i	ADV
ejpam-5768	45	30	}	}	PUNCT
ejpam-5768	45	31	.	.	PUNCT
ejpam-5768	46	1	then	then	ADV
ejpam-5768	46	2	the	the	DET
ejpam-5768	46	3	set	set	NOUN
ejpam-5768	46	4	w	w	PROPN
ejpam-5768	46	5	=	=	X
ejpam-5768	46	6	{	{	PUNCT
ejpam-5768	46	7	{	{	PUNCT
ejpam-5768	46	8	1,−1	1,−1	NUM
ejpam-5768	46	9	}	}	PUNCT
ejpam-5768	46	10	,	,	PUNCT
ejpam-5768	46	11	{	{	PUNCT
ejpam-5768	46	12	i,−i	i,−i	NOUN
ejpam-5768	46	13	}	}	PUNCT
ejpam-5768	46	14	}	}	PUNCT
ejpam-5768	46	15	is	be	AUX
ejpam-5768	46	16	an	an	DET
ejpam-5768	46	17	hx	hx	NOUN
ejpam-5768	46	18	-	-	PUNCT
ejpam-5768	46	19	group	group	NOUN
ejpam-5768	46	20	,	,	PUNCT
ejpam-5768	46	21	where	where	SCONJ
ejpam-5768	46	22	its	its	PRON
ejpam-5768	46	23	identity	identity	NOUN
ejpam-5768	46	24	is	be	AUX
ejpam-5768	46	25	{	{	PUNCT
ejpam-5768	46	26	1,−1	1,−1	NUM
ejpam-5768	46	27	}	}	PUNCT
ejpam-5768	46	28	.	.	PUNCT
ejpam-5768	47	1	this	this	DET
ejpam-5768	47	2	definition	definition	NOUN
ejpam-5768	47	3	is	be	AUX
ejpam-5768	47	4	applied	apply	VERB
ejpam-5768	47	5	to	to	ADP
ejpam-5768	47	6	fuzzy	fuzzy	ADJ
ejpam-5768	47	7	settings	setting	NOUN
ejpam-5768	47	8	in	in	ADP
ejpam-5768	47	9	[	[	X
ejpam-5768	47	10	27	27	NUM
ejpam-5768	47	11	]	]	PUNCT
ejpam-5768	47	12	and	and	CCONJ
ejpam-5768	47	13	[	[	X
ejpam-5768	47	14	28	28	NUM
ejpam-5768	47	15	]	]	PUNCT
ejpam-5768	47	16	and	and	CCONJ
ejpam-5768	47	17	provide	provide	VERB
ejpam-5768	47	18	a	a	DET
ejpam-5768	47	19	definition	definition	NOUN
ejpam-5768	47	20	for	for	ADP
ejpam-5768	47	21	a	a	DET
ejpam-5768	47	22	fuzzy	fuzzy	ADJ
ejpam-5768	47	23	hx	hx	PROPN
ejpam-5768	47	24	-	-	PUNCT
ejpam-5768	47	25	sg	sg	PROPN
ejpam-5768	47	26	,	,	PUNCT
ejpam-5768	47	27	which	which	PRON
ejpam-5768	47	28	defined	define	VERB
ejpam-5768	47	29	as	as	SCONJ
ejpam-5768	47	30	follows	follow	VERB
ejpam-5768	47	31	:	:	PUNCT
ejpam-5768	47	32	a	a	DET
ejpam-5768	47	33	fuzzy	fuzzy	ADJ
ejpam-5768	47	34	set	set	NOUN
ejpam-5768	47	35	υ	υ	NOUN
ejpam-5768	47	36	is	be	AUX
ejpam-5768	47	37	a	a	DET
ejpam-5768	47	38	fuzzy	fuzzy	ADJ
ejpam-5768	47	39	hx	hx	PROPN
ejpam-5768	47	40	-	-	PUNCT
ejpam-5768	47	41	sg	sg	PROPN
ejpam-5768	47	42	of	of	ADP
ejpam-5768	47	43	an	an	DET
ejpam-5768	47	44	hx	hx	PROPN
ejpam-5768	47	45	-	-	PUNCT
ejpam-5768	47	46	group	group	NOUN
ejpam-5768	47	47	w	w	NOUN
ejpam-5768	48	1	if	if	SCONJ
ejpam-5768	48	2	,	,	PUNCT
ejpam-5768	48	3	for	for	ADP
ejpam-5768	48	4	any	any	DET
ejpam-5768	48	5	w01	w01	NOUN
ejpam-5768	48	6	,	,	PUNCT
ejpam-5768	48	7	w02	w02	NOUN
ejpam-5768	48	8	∈	∈	PROPN
ejpam-5768	48	9	w	w	NOUN
ejpam-5768	48	10	,	,	PUNCT
ejpam-5768	48	11	we	we	PRON
ejpam-5768	48	12	have	have	VERB
ejpam-5768	48	13	:	:	PUNCT
ejpam-5768	48	14	(	(	PUNCT
ejpam-5768	48	15	1	1	X
ejpam-5768	48	16	)	)	PUNCT
ejpam-5768	48	17	υ	υ	NOUN
ejpam-5768	48	18	(	(	PUNCT
ejpam-5768	48	19	w01w02	w01w02	PROPN
ejpam-5768	48	20	)	)	PUNCT
ejpam-5768	48	21	≥	≥	PROPN
ejpam-5768	48	22	min{υ	min{υ	NUM
ejpam-5768	48	23	(	(	PUNCT
ejpam-5768	48	24	w01	w01	NOUN
ejpam-5768	48	25	)	)	PUNCT
ejpam-5768	48	26	,	,	PUNCT
ejpam-5768	48	27	υ	υ	PROPN
ejpam-5768	48	28	(	(	PUNCT
ejpam-5768	48	29	w02	w02	NOUN
ejpam-5768	48	30	)	)	PUNCT
ejpam-5768	48	31	}	}	PUNCT
ejpam-5768	48	32	.	.	PUNCT
ejpam-5768	49	1	(	(	PUNCT
ejpam-5768	49	2	2	2	X
ejpam-5768	49	3	)	)	PUNCT
ejpam-5768	49	4	υ	υ	NOUN
ejpam-5768	49	5	(	(	PUNCT
ejpam-5768	49	6	w−1	w−1	PROPN
ejpam-5768	49	7	01	01	NUM
ejpam-5768	49	8	)	)	PUNCT
ejpam-5768	50	1	=	=	SYM
ejpam-5768	50	2	υ	υ	PROPN
ejpam-5768	50	3	(	(	PUNCT
ejpam-5768	50	4	w01	w01	NOUN
ejpam-5768	50	5	)	)	PUNCT
ejpam-5768	50	6	.	.	PUNCT
ejpam-5768	51	1	now	now	ADV
ejpam-5768	51	2	,	,	PUNCT
ejpam-5768	51	3	we	we	PRON
ejpam-5768	51	4	are	be	AUX
ejpam-5768	51	5	able	able	ADJ
ejpam-5768	51	6	to	to	PART
ejpam-5768	51	7	present	present	VERB
ejpam-5768	51	8	the	the	DET
ejpam-5768	51	9	main	main	ADJ
ejpam-5768	51	10	definition	definition	NOUN
ejpam-5768	51	11	of	of	ADP
ejpam-5768	51	12	pf	pf	PROPN
ejpam-5768	51	13	hx	hx	PROPN
ejpam-5768	51	14	-	-	PUNCT
ejpam-5768	51	15	sg	sg	PROPN
ejpam-5768	51	16	.	.	PUNCT
ejpam-5768	52	1	definition	definition	NOUN
ejpam-5768	52	2	1	1	NUM
ejpam-5768	52	3	.	.	PUNCT
ejpam-5768	53	1	let	let	VERB
ejpam-5768	53	2	g	g	PRON
ejpam-5768	53	3	be	be	AUX
ejpam-5768	53	4	a	a	DET
ejpam-5768	53	5	group	group	NOUN
ejpam-5768	53	6	,	,	PUNCT
ejpam-5768	53	7	w	w	ADP
ejpam-5768	53	8	⊆	⊆	NUM
ejpam-5768	53	9	2g−{ϕ	2g−{ϕ	NUM
ejpam-5768	53	10	}	}	PUNCT
ejpam-5768	53	11	be	be	AUX
ejpam-5768	53	12	an	an	DET
ejpam-5768	53	13	hx	hx	NOUN
ejpam-5768	53	14	-	-	PUNCT
ejpam-5768	53	15	group	group	NOUN
ejpam-5768	53	16	of	of	ADP
ejpam-5768	53	17	g	g	PROPN
ejpam-5768	53	18	and	and	CCONJ
ejpam-5768	53	19	υ	υ	NOUN
ejpam-5768	54	1	=	=	PRON
ejpam-5768	54	2	{	{	PUNCT
ejpam-5768	54	3	(	(	PUNCT
ejpam-5768	54	4	w	w	NOUN
ejpam-5768	54	5	;	;	PUNCT
ejpam-5768	54	6	ῡ	ῡ	PROPN
ejpam-5768	54	7	(	(	PUNCT
ejpam-5768	54	8	w	w	PROPN
ejpam-5768	54	9	)	)	PUNCT
ejpam-5768	54	10	,	,	PUNCT
ejpam-5768	54	11	υ̂	υ̂	NUM
ejpam-5768	54	12	(	(	PUNCT
ejpam-5768	54	13	w	w	NOUN
ejpam-5768	54	14	)	)	PUNCT
ejpam-5768	54	15	)	)	PUNCT
ejpam-5768	54	16	:	:	PUNCT
ejpam-5768	54	17	w	w	X
ejpam-5768	54	18	∈	∈	PROPN
ejpam-5768	54	19	w	w	AUX
ejpam-5768	54	20	}	}	PUNCT
ejpam-5768	54	21	be	be	AUX
ejpam-5768	54	22	a	a	DET
ejpam-5768	54	23	pythagorean	pythagorean	ADJ
ejpam-5768	54	24	fuzzy	fuzzy	ADJ
ejpam-5768	54	25	subset	subset	NOUN
ejpam-5768	54	26	of	of	ADP
ejpam-5768	54	27	w	w	PROPN
ejpam-5768	54	28	.	.	PUNCT
ejpam-5768	55	1	then	then	ADV
ejpam-5768	55	2	υ	υ	PROPN
ejpam-5768	55	3	is	be	AUX
ejpam-5768	55	4	called	call	VERB
ejpam-5768	55	5	a	a	DET
ejpam-5768	55	6	pf	pf	PROPN
ejpam-5768	55	7	hx	hx	PROPN
ejpam-5768	55	8	-	-	PUNCT
ejpam-5768	55	9	sg	sg	PROPN
ejpam-5768	55	10	of	of	ADP
ejpam-5768	55	11	w	w	NOUN
ejpam-5768	55	12	if	if	SCONJ
ejpam-5768	55	13	:	:	PUNCT
ejpam-5768	55	14	(	(	PUNCT
ejpam-5768	55	15	1	1	X
ejpam-5768	55	16	)	)	PUNCT
ejpam-5768	55	17	ῡ	ῡ	PROPN
ejpam-5768	55	18	2(w01w02	2(w01w02	NUM
ejpam-5768	55	19	)	)	PUNCT
ejpam-5768	55	20	≥	≥	PROPN
ejpam-5768	55	21	min{ῡ	min{ῡ	NOUN
ejpam-5768	55	22	2(w01	2(w01	NUM
ejpam-5768	55	23	)	)	PUNCT
ejpam-5768	55	24	,	,	PUNCT
ejpam-5768	55	25	ῡ	ῡ	PROPN
ejpam-5768	55	26	2(w02	2(w02	NUM
ejpam-5768	55	27	)	)	PUNCT
ejpam-5768	55	28	}	}	PUNCT
ejpam-5768	55	29	.	.	PUNCT
ejpam-5768	56	1	(	(	PUNCT
ejpam-5768	56	2	2	2	X
ejpam-5768	56	3	)	)	PUNCT
ejpam-5768	56	4	υ̂	υ̂	NUM
ejpam-5768	56	5	2(w01w02	2(w01w02	X
ejpam-5768	56	6	)	)	PUNCT
ejpam-5768	56	7	≤	≤	NUM
ejpam-5768	56	8	max{υ̂	max{υ̂	PROPN
ejpam-5768	56	9	2(w01	2(w01	NUM
ejpam-5768	56	10	)	)	PUNCT
ejpam-5768	56	11	,	,	PUNCT
ejpam-5768	56	12	υ̂	υ̂	PROPN
ejpam-5768	56	13	2(w02	2(w02	NUM
ejpam-5768	56	14	)	)	PUNCT
ejpam-5768	56	15	}	}	PUNCT
ejpam-5768	56	16	.	.	PUNCT
ejpam-5768	57	1	(	(	PUNCT
ejpam-5768	57	2	3	3	X
ejpam-5768	57	3	)	)	PUNCT
ejpam-5768	57	4	ῡ	ῡ	PROPN
ejpam-5768	57	5	2(w−1	2(w−1	NOUN
ejpam-5768	57	6	01	01	NUM
ejpam-5768	57	7	)	)	PUNCT
ejpam-5768	58	1	=	=	SYM
ejpam-5768	58	2	ῡ	ῡ	PROPN
ejpam-5768	58	3	2(w01	2(w01	NUM
ejpam-5768	58	4	)	)	PUNCT
ejpam-5768	58	5	,	,	PUNCT
ejpam-5768	58	6	υ̂	υ̂	PROPN
ejpam-5768	58	7	2(w−1	2(w−1	NOUN
ejpam-5768	58	8	01	01	NUM
ejpam-5768	58	9	)	)	PUNCT
ejpam-5768	59	1	=	=	PRON
ejpam-5768	59	2	υ̂	υ̂	NUM
ejpam-5768	59	3	2(w01	2(w01	NUM
ejpam-5768	59	4	)	)	PUNCT
ejpam-5768	59	5	.	.	PUNCT
ejpam-5768	60	1	example	example	NOUN
ejpam-5768	61	1	2	2	NUM
ejpam-5768	61	2	.	.	X
ejpam-5768	61	3	consider	consider	VERB
ejpam-5768	61	4	the	the	DET
ejpam-5768	61	5	klien	klien	PROPN
ejpam-5768	61	6	4	4	PROPN
ejpam-5768	61	7	-	-	PUNCT
ejpam-5768	61	8	group	group	NOUN
ejpam-5768	61	9	g	g	NOUN
ejpam-5768	61	10	=	=	SYM
ejpam-5768	61	11	{	{	PUNCT
ejpam-5768	61	12	e	e	NOUN
ejpam-5768	61	13	,	,	PUNCT
ejpam-5768	61	14	x	x	PROPN
ejpam-5768	61	15	,	,	PUNCT
ejpam-5768	61	16	y	y	PROPN
ejpam-5768	61	17	,	,	PUNCT
ejpam-5768	61	18	z	z	NOUN
ejpam-5768	61	19	}	}	PUNCT
ejpam-5768	61	20	and	and	CCONJ
ejpam-5768	61	21	the	the	DET
ejpam-5768	61	22	xh	xh	PROPN
ejpam-5768	61	23	-	-	PUNCT
ejpam-5768	61	24	group	group	NOUN
ejpam-5768	61	25	w	w	NOUN
ejpam-5768	61	26	=	=	PUNCT
ejpam-5768	61	27	{	{	PUNCT
ejpam-5768	61	28	e	e	NOUN
ejpam-5768	61	29	,	,	PUNCT
ejpam-5768	61	30	m	m	VERB
ejpam-5768	61	31	}	}	PUNCT
ejpam-5768	61	32	=	=	SYM
ejpam-5768	61	33	{	{	PUNCT
ejpam-5768	61	34	{	{	PUNCT
ejpam-5768	61	35	e	e	NOUN
ejpam-5768	61	36	,	,	PUNCT
ejpam-5768	61	37	x	x	NOUN
ejpam-5768	61	38	}	}	PUNCT
ejpam-5768	61	39	,	,	PUNCT
ejpam-5768	61	40	{	{	PUNCT
ejpam-5768	61	41	y	y	NOUN
ejpam-5768	61	42	,	,	PUNCT
ejpam-5768	61	43	z	z	NOUN
ejpam-5768	61	44	}	}	PUNCT
ejpam-5768	61	45	}	}	PUNCT
ejpam-5768	61	46	,	,	PUNCT
ejpam-5768	61	47	such	such	ADJ
ejpam-5768	61	48	that	that	SCONJ
ejpam-5768	61	49	∗	∗	NOUN
ejpam-5768	61	50	e	e	NOUN
ejpam-5768	61	51	m	m	NOUN
ejpam-5768	61	52	e	e	X
ejpam-5768	61	53	e	e	X
ejpam-5768	61	54	m	m	VERB
ejpam-5768	61	55	m	m	VERB
ejpam-5768	61	56	m	m	VERB
ejpam-5768	61	57	e	e	NOUN
ejpam-5768	61	58	let	let	VERB
ejpam-5768	61	59	η	η	PROPN
ejpam-5768	61	60	be	be	AUX
ejpam-5768	61	61	a	a	DET
ejpam-5768	61	62	pythagorean	pythagorean	ADJ
ejpam-5768	61	63	fuzzy	fuzzy	ADJ
ejpam-5768	61	64	sets	set	NOUN
ejpam-5768	61	65	,	,	PUNCT
ejpam-5768	61	66	where	where	SCONJ
ejpam-5768	61	67	η̄(e	η̄(e	ADJ
ejpam-5768	61	68	)	)	PUNCT
ejpam-5768	61	69	=	=	SYM
ejpam-5768	61	70	0.6	0.6	NUM
ejpam-5768	61	71	,	,	PUNCT
ejpam-5768	61	72	η̂(e	η̂(e	PROPN
ejpam-5768	61	73	)	)	PUNCT
ejpam-5768	61	74	=	=	NUM
ejpam-5768	61	75	0.3	0.3	NUM
ejpam-5768	61	76	η̄(x	η̄(x	PROPN
ejpam-5768	61	77	)	)	PUNCT
ejpam-5768	61	78	=	=	SYM
ejpam-5768	61	79	0.5	0.5	NUM
ejpam-5768	61	80	,	,	PUNCT
ejpam-5768	61	81	η̂(x	η̂(x	NUM
ejpam-5768	61	82	)	)	PUNCT
ejpam-5768	61	83	=	=	SYM
ejpam-5768	61	84	0.5	0.5	NUM
ejpam-5768	61	85	η̄(y	η̄(y	NOUN
ejpam-5768	61	86	)	)	PUNCT
ejpam-5768	61	87	=	=	SYM
ejpam-5768	61	88	0.4	0.4	NUM
ejpam-5768	61	89	,	,	PUNCT
ejpam-5768	61	90	η̂(y	η̂(y	PROPN
ejpam-5768	61	91	)	)	PUNCT
ejpam-5768	61	92	=	=	PUNCT
ejpam-5768	61	93	0.3	0.3	NUM
ejpam-5768	61	94	η̄(z	η̄(z	PROPN
ejpam-5768	61	95	)	)	PUNCT
ejpam-5768	61	96	=	=	PUNCT
ejpam-5768	61	97	0.3	0.3	NUM
ejpam-5768	61	98	,	,	PUNCT
ejpam-5768	61	99	η̂(z	η̂(z	NOUN
ejpam-5768	61	100	)	)	PUNCT
ejpam-5768	61	101	=	=	SYM
ejpam-5768	61	102	0.5	0.5	NUM
ejpam-5768	61	103	let	let	VERB
ejpam-5768	61	104	ῡ	ῡ	PROPN
ejpam-5768	61	105	(	(	PUNCT
ejpam-5768	61	106	n	n	CCONJ
ejpam-5768	61	107	)	)	PUNCT
ejpam-5768	61	108	=	=	SYM
ejpam-5768	61	109	max{η̄(n	max{η̄(n	NOUN
ejpam-5768	61	110	)	)	PUNCT
ejpam-5768	61	111	:	:	PUNCT
ejpam-5768	62	1	n	n	X
ejpam-5768	62	2	∈	∈	PROPN
ejpam-5768	62	3	n	n	CCONJ
ejpam-5768	62	4	⊆	⊆	NUM
ejpam-5768	62	5	w	w	NOUN
ejpam-5768	62	6	}	}	PUNCT
ejpam-5768	62	7	and	and	CCONJ
ejpam-5768	62	8	υ̂	υ̂	NUM
ejpam-5768	62	9	(	(	PUNCT
ejpam-5768	62	10	n	n	CCONJ
ejpam-5768	62	11	)	)	PUNCT
ejpam-5768	62	12	=	=	SYM
ejpam-5768	62	13	min{η̂(n	min{η̂(n	PROPN
ejpam-5768	62	14	)	)	PUNCT
ejpam-5768	62	15	:	:	PUNCT
ejpam-5768	63	1	n	n	X
ejpam-5768	63	2	∈	∈	PROPN
ejpam-5768	63	3	n	n	CCONJ
ejpam-5768	63	4	⊆	⊆	NUM
ejpam-5768	63	5	w	w	NOUN
ejpam-5768	63	6	}	}	PUNCT
ejpam-5768	63	7	.	.	PUNCT
ejpam-5768	64	1	thus	thus	ADV
ejpam-5768	64	2	ῡ	ῡ	PROPN
ejpam-5768	64	3	(	(	PUNCT
ejpam-5768	64	4	e	e	NOUN
ejpam-5768	64	5	)	)	PUNCT
ejpam-5768	64	6	=	=	SYM
ejpam-5768	64	7	max{η̄(e	max{η̄(e	NOUN
ejpam-5768	64	8	)	)	PUNCT
ejpam-5768	64	9	,	,	PUNCT
ejpam-5768	64	10	η̄(x	η̄(x	PROPN
ejpam-5768	64	11	)	)	PUNCT
ejpam-5768	64	12	}	}	PUNCT
ejpam-5768	64	13	=	=	SYM
ejpam-5768	64	14	0.6	0.6	NUM
ejpam-5768	64	15	υ̂	υ̂	NUM
ejpam-5768	64	16	(	(	PUNCT
ejpam-5768	64	17	e	e	NOUN
ejpam-5768	64	18	)	)	PUNCT
ejpam-5768	64	19	=	=	SYM
ejpam-5768	64	20	min{η̂(e	min{η̂(e	NUM
ejpam-5768	64	21	)	)	PUNCT
ejpam-5768	64	22	,	,	PUNCT
ejpam-5768	64	23	η̂(x	η̂(x	NUM
ejpam-5768	64	24	)	)	PUNCT
ejpam-5768	64	25	}	}	PUNCT
ejpam-5768	64	26	=	=	SYM
ejpam-5768	64	27	0.3	0.3	NUM
ejpam-5768	64	28	ῡ	ῡ	NOUN
ejpam-5768	64	29	(	(	PUNCT
ejpam-5768	64	30	m	m	PROPN
ejpam-5768	64	31	)	)	PUNCT
ejpam-5768	64	32	=	=	SYM
ejpam-5768	64	33	max{η̄(y	max{η̄(y	NOUN
ejpam-5768	64	34	)	)	PUNCT
ejpam-5768	64	35	,	,	PUNCT
ejpam-5768	64	36	η̄(z	η̄(z	PROPN
ejpam-5768	64	37	)	)	PUNCT
ejpam-5768	64	38	}	}	PUNCT
ejpam-5768	64	39	=	=	SYM
ejpam-5768	64	40	0.4	0.4	NUM
ejpam-5768	64	41	υ̂	υ̂	NUM
ejpam-5768	64	42	(	(	PUNCT
ejpam-5768	64	43	m	m	NOUN
ejpam-5768	64	44	)	)	PUNCT
ejpam-5768	64	45	=	=	SYM
ejpam-5768	64	46	min{η̂(y	min{η̂(y	NOUN
ejpam-5768	64	47	)	)	PUNCT
ejpam-5768	64	48	,	,	PUNCT
ejpam-5768	64	49	η̂(z	η̂(z	NOUN
ejpam-5768	64	50	)	)	PUNCT
ejpam-5768	64	51	}	}	PUNCT
ejpam-5768	64	52	=	=	PUNCT
ejpam-5768	64	53	0.3	0.3	NUM
ejpam-5768	64	54	thus	thus	ADV
ejpam-5768	64	55	υ	υ	NOUN
ejpam-5768	64	56	is	be	AUX
ejpam-5768	64	57	a	a	DET
ejpam-5768	64	58	pf	pf	PROPN
ejpam-5768	64	59	hx	hx	PROPN
ejpam-5768	64	60	-	-	PUNCT
ejpam-5768	64	61	sg	sg	PROPN
ejpam-5768	64	62	.	.	PUNCT
ejpam-5768	64	63	a.	a.	PROPN
ejpam-5768	64	64	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	64	65	/	/	SYM
ejpam-5768	64	66	eur	eur	PROPN
ejpam-5768	64	67	.	.	PUNCT
ejpam-5768	65	1	j.	j.	PROPN
ejpam-5768	65	2	pure	pure	PROPN
ejpam-5768	65	3	appl	appl	PROPN
ejpam-5768	65	4	.	.	PROPN
ejpam-5768	65	5	math	math	PROPN
ejpam-5768	65	6	,	,	PUNCT
ejpam-5768	65	7	18	18	NUM
ejpam-5768	65	8	(	(	PUNCT
ejpam-5768	65	9	2	2	NUM
ejpam-5768	65	10	)	)	PUNCT
ejpam-5768	65	11	(	(	PUNCT
ejpam-5768	65	12	2025	2025	NUM
ejpam-5768	65	13	)	)	PUNCT
ejpam-5768	65	14	,	,	PUNCT
ejpam-5768	65	15	5768	5768	NUM
ejpam-5768	65	16	4	4	NUM
ejpam-5768	65	17	of	of	ADP
ejpam-5768	65	18	17	17	NUM
ejpam-5768	65	19	now	now	ADV
ejpam-5768	65	20	,	,	PUNCT
ejpam-5768	65	21	we	we	PRON
ejpam-5768	65	22	inroduce	inroduce	VERB
ejpam-5768	65	23	the	the	DET
ejpam-5768	65	24	following	follow	VERB
ejpam-5768	65	25	proposition	proposition	NOUN
ejpam-5768	65	26	which	which	PRON
ejpam-5768	65	27	will	will	AUX
ejpam-5768	65	28	be	be	AUX
ejpam-5768	65	29	used	use	VERB
ejpam-5768	65	30	in	in	ADP
ejpam-5768	65	31	later	later	ADJ
ejpam-5768	65	32	results	result	NOUN
ejpam-5768	65	33	.	.	PUNCT
ejpam-5768	66	1	proposition	proposition	NOUN
ejpam-5768	66	2	1	1	NUM
ejpam-5768	66	3	.	.	PUNCT
ejpam-5768	67	1	let	let	VERB
ejpam-5768	67	2	υ	υ	PRON
ejpam-5768	67	3	be	be	AUX
ejpam-5768	67	4	a	a	DET
ejpam-5768	67	5	pf	pf	PROPN
ejpam-5768	67	6	hx	hx	PROPN
ejpam-5768	67	7	-	-	PUNCT
ejpam-5768	67	8	sg	sg	PROPN
ejpam-5768	67	9	of	of	ADP
ejpam-5768	67	10	an	an	DET
ejpam-5768	67	11	hx	hx	NOUN
ejpam-5768	67	12	-	-	PUNCT
ejpam-5768	67	13	group	group	NOUN
ejpam-5768	67	14	w	w	PROPN
ejpam-5768	67	15	.	.	PUNCT
ejpam-5768	68	1	then	then	ADV
ejpam-5768	68	2	ῡ	ῡ	PROPN
ejpam-5768	68	3	2(e	2(e	NUM
ejpam-5768	68	4	)	)	PUNCT
ejpam-5768	68	5	≥	≥	PROPN
ejpam-5768	68	6	ῡ	ῡ	PROPN
ejpam-5768	68	7	2(w01	2(w01	NUM
ejpam-5768	68	8	)	)	PUNCT
ejpam-5768	68	9	,	,	PUNCT
ejpam-5768	68	10	υ̂	υ̂	X
ejpam-5768	68	11	2(e	2(e	NUM
ejpam-5768	68	12	)	)	PUNCT
ejpam-5768	68	13	≤	≤	NUM
ejpam-5768	68	14	υ̂	υ̂	NUM
ejpam-5768	68	15	2(w01	2(w01	NUM
ejpam-5768	68	16	)	)	PUNCT
ejpam-5768	68	17	for	for	ADP
ejpam-5768	68	18	all	all	DET
ejpam-5768	68	19	w01	w01	NOUN
ejpam-5768	68	20	∈	∈	PROPN
ejpam-5768	68	21	w	w	NOUN
ejpam-5768	68	22	proof	proof	NOUN
ejpam-5768	68	23	.	.	PUNCT
ejpam-5768	69	1	ῡ	ῡ	PROPN
ejpam-5768	69	2	2(e	2(e	NUM
ejpam-5768	69	3	)	)	PUNCT
ejpam-5768	70	1	=	=	SYM
ejpam-5768	70	2	ῡ	ῡ	PROPN
ejpam-5768	70	3	2(w01w	2(w01w	NUM
ejpam-5768	70	4	−1	−1	NOUN
ejpam-5768	70	5	01	01	NUM
ejpam-5768	70	6	)	)	PUNCT
ejpam-5768	70	7	≥	≥	PROPN
ejpam-5768	70	8	min{ῡ	min{ῡ	NOUN
ejpam-5768	70	9	2(w01	2(w01	NUM
ejpam-5768	70	10	)	)	PUNCT
ejpam-5768	70	11	,	,	PUNCT
ejpam-5768	70	12	ῡ	ῡ	PROPN
ejpam-5768	70	13	2(w−1	2(w−1	PROPN
ejpam-5768	70	14	01	01	NUM
ejpam-5768	70	15	)	)	PUNCT
ejpam-5768	70	16	}	}	PUNCT
ejpam-5768	71	1	=	=	SYM
ejpam-5768	71	2	min{ῡ	min{ῡ	NOUN
ejpam-5768	71	3	2(w01	2(w01	NUM
ejpam-5768	71	4	)	)	PUNCT
ejpam-5768	71	5	,	,	PUNCT
ejpam-5768	71	6	ῡ	ῡ	PROPN
ejpam-5768	71	7	2(w01	2(w01	NUM
ejpam-5768	71	8	)	)	PUNCT
ejpam-5768	71	9	}	}	PUNCT
ejpam-5768	72	1	=	=	SYM
ejpam-5768	72	2	ῡ	ῡ	PROPN
ejpam-5768	72	3	2(w01	2(w01	NUM
ejpam-5768	72	4	)	)	PUNCT
ejpam-5768	72	5	υ̂	υ̂	NUM
ejpam-5768	72	6	2(e	2(e	NUM
ejpam-5768	72	7	)	)	PUNCT
ejpam-5768	72	8	=	=	PRON
ejpam-5768	72	9	υ̂	υ̂	PROPN
ejpam-5768	73	1	2(w01w	2(w01w	NUM
ejpam-5768	73	2	−1	−1	NOUN
ejpam-5768	73	3	01	01	NUM
ejpam-5768	73	4	)	)	PUNCT
ejpam-5768	73	5	≤	≤	NUM
ejpam-5768	73	6	max{υ̂	max{υ̂	PROPN
ejpam-5768	73	7	2(w01	2(w01	NUM
ejpam-5768	73	8	)	)	PUNCT
ejpam-5768	73	9	,	,	PUNCT
ejpam-5768	73	10	υ̂	υ̂	PROPN
ejpam-5768	73	11	2(w−1	2(w−1	NOUN
ejpam-5768	73	12	01	01	NUM
ejpam-5768	73	13	)	)	PUNCT
ejpam-5768	73	14	}	}	PUNCT
ejpam-5768	74	1	=	=	PUNCT
ejpam-5768	74	2	max{υ̂	max{υ̂	NOUN
ejpam-5768	74	3	2(w01	2(w01	NUM
ejpam-5768	74	4	)	)	PUNCT
ejpam-5768	74	5	,	,	PUNCT
ejpam-5768	74	6	υ̂	υ̂	NUM
ejpam-5768	74	7	2(w01	2(w01	NUM
ejpam-5768	74	8	)	)	PUNCT
ejpam-5768	74	9	}	}	PUNCT
ejpam-5768	74	10	=	=	SYM
ejpam-5768	74	11	υ̂	υ̂	NUM
ejpam-5768	74	12	2(w01	2(w01	NUM
ejpam-5768	74	13	)	)	PUNCT
ejpam-5768	74	14	.	.	PUNCT
ejpam-5768	75	1	using	use	VERB
ejpam-5768	75	2	the	the	DET
ejpam-5768	75	3	above	above	ADJ
ejpam-5768	75	4	proposition	proposition	NOUN
ejpam-5768	75	5	,	,	PUNCT
ejpam-5768	75	6	we	we	PRON
ejpam-5768	75	7	are	be	AUX
ejpam-5768	75	8	able	able	ADJ
ejpam-5768	75	9	to	to	PART
ejpam-5768	75	10	provide	provide	VERB
ejpam-5768	75	11	characterisation	characterisation	NOUN
ejpam-5768	75	12	theorem	theorem	NOUN
ejpam-5768	75	13	of	of	ADP
ejpam-5768	75	14	hxsg	hxsg	ADV
ejpam-5768	75	15	.	.	PUNCT
ejpam-5768	76	1	theorem	theorem	NOUN
ejpam-5768	76	2	1	1	NUM
ejpam-5768	76	3	.	.	PUNCT
ejpam-5768	77	1	let	let	VERB
ejpam-5768	77	2	w	w	NOUN
ejpam-5768	77	3	be	be	AUX
ejpam-5768	77	4	an	an	DET
ejpam-5768	77	5	hx	hx	NOUN
ejpam-5768	77	6	-	-	PUNCT
ejpam-5768	77	7	group	group	NOUN
ejpam-5768	77	8	and	and	CCONJ
ejpam-5768	77	9	υ	υ	NOUN
ejpam-5768	77	10	be	be	AUX
ejpam-5768	77	11	a	a	DET
ejpam-5768	77	12	pf	pf	NOUN
ejpam-5768	77	13	subset	subset	NOUN
ejpam-5768	77	14	of	of	ADP
ejpam-5768	77	15	w	w	PROPN
ejpam-5768	77	16	.	.	PUNCT
ejpam-5768	78	1	then	then	ADV
ejpam-5768	78	2	υ	υ	PROPN
ejpam-5768	78	3	is	be	AUX
ejpam-5768	78	4	a	a	DET
ejpam-5768	78	5	pf	pf	PROPN
ejpam-5768	78	6	hx	hx	PROPN
ejpam-5768	78	7	-	-	PUNCT
ejpam-5768	78	8	sg	sg	PROPN
ejpam-5768	78	9	of	of	ADP
ejpam-5768	78	10	w	w	NOUN
ejpam-5768	78	11	if	if	SCONJ
ejpam-5768	79	1	and	and	CCONJ
ejpam-5768	79	2	only	only	ADV
ejpam-5768	79	3	if	if	SCONJ
ejpam-5768	79	4	ῡ	ῡ	PROPN
ejpam-5768	79	5	2(w01w	2(w01w	NUM
ejpam-5768	79	6	−1	−1	NOUN
ejpam-5768	79	7	02	02	NUM
ejpam-5768	79	8	)	)	PUNCT
ejpam-5768	79	9	≥	≥	PROPN
ejpam-5768	79	10	min{ῡ	min{ῡ	NOUN
ejpam-5768	79	11	2(w01	2(w01	NUM
ejpam-5768	79	12	)	)	PUNCT
ejpam-5768	79	13	,	,	PUNCT
ejpam-5768	79	14	ῡ	ῡ	PROPN
ejpam-5768	79	15	2(w02	2(w02	NUM
ejpam-5768	79	16	)	)	PUNCT
ejpam-5768	79	17	}	}	PUNCT
ejpam-5768	79	18	,	,	PUNCT
ejpam-5768	79	19	υ̂	υ̂	PROPN
ejpam-5768	80	1	2(w01w	2(w01w	NUM
ejpam-5768	80	2	−1	−1	NOUN
ejpam-5768	80	3	02	02	NUM
ejpam-5768	80	4	)	)	PUNCT
ejpam-5768	80	5	≤	≤	NUM
ejpam-5768	80	6	max{υ̂	max{υ̂	PROPN
ejpam-5768	80	7	2(w01	2(w01	NUM
ejpam-5768	80	8	)	)	PUNCT
ejpam-5768	80	9	,	,	PUNCT
ejpam-5768	80	10	υ̂	υ̂	PROPN
ejpam-5768	80	11	2(w02	2(w02	NUM
ejpam-5768	80	12	)	)	PUNCT
ejpam-5768	80	13	}	}	PUNCT
ejpam-5768	80	14	.	.	PUNCT
ejpam-5768	81	1	(	(	PUNCT
ejpam-5768	81	2	1	1	X
ejpam-5768	81	3	)	)	PUNCT
ejpam-5768	81	4	proof	proof	NOUN
ejpam-5768	81	5	.	.	PUNCT
ejpam-5768	82	1	if	if	SCONJ
ejpam-5768	82	2	υ	υ	PROPN
ejpam-5768	82	3	is	be	AUX
ejpam-5768	82	4	a	a	DET
ejpam-5768	82	5	pf	pf	PROPN
ejpam-5768	82	6	hx	hx	PROPN
ejpam-5768	82	7	-	-	PUNCT
ejpam-5768	82	8	sg	sg	PROPN
ejpam-5768	82	9	of	of	ADP
ejpam-5768	82	10	w	w	PROPN
ejpam-5768	82	11	,	,	PUNCT
ejpam-5768	82	12	then	then	ADV
ejpam-5768	82	13	ῡ	ῡ	PROPN
ejpam-5768	83	1	2(w01w	2(w01w	NUM
ejpam-5768	83	2	−1	−1	NOUN
ejpam-5768	83	3	02	02	NUM
ejpam-5768	83	4	)	)	PUNCT
ejpam-5768	83	5	≥	≥	PROPN
ejpam-5768	83	6	min{ῡ	min{ῡ	NOUN
ejpam-5768	83	7	2(w01	2(w01	NUM
ejpam-5768	83	8	)	)	PUNCT
ejpam-5768	83	9	,	,	PUNCT
ejpam-5768	84	1	ῡ	ῡ	PROPN
ejpam-5768	84	2	2(w−1	2(w−1	PROPN
ejpam-5768	84	3	02	02	NUM
ejpam-5768	84	4	)	)	PUNCT
ejpam-5768	84	5	}	}	PUNCT
ejpam-5768	85	1	=	=	SYM
ejpam-5768	85	2	min{ῡ	min{ῡ	NOUN
ejpam-5768	85	3	2(w01	2(w01	NUM
ejpam-5768	85	4	)	)	PUNCT
ejpam-5768	85	5	,	,	PUNCT
ejpam-5768	85	6	ῡ	ῡ	PROPN
ejpam-5768	85	7	2(w02	2(w02	NUM
ejpam-5768	85	8	)	)	PUNCT
ejpam-5768	85	9	}	}	PUNCT
ejpam-5768	85	10	also	also	ADV
ejpam-5768	85	11	,	,	PUNCT
ejpam-5768	85	12	υ̂	υ̂	PROPN
ejpam-5768	85	13	2(w01w	2(w01w	NUM
ejpam-5768	85	14	−1	−1	NOUN
ejpam-5768	85	15	02	02	NUM
ejpam-5768	85	16	)	)	PUNCT
ejpam-5768	86	1	≤	≤	NUM
ejpam-5768	87	1	max{υ̂	max{υ̂	PROPN
ejpam-5768	87	2	2(w01	2(w01	NUM
ejpam-5768	87	3	)	)	PUNCT
ejpam-5768	87	4	,	,	PUNCT
ejpam-5768	87	5	υ̂	υ̂	PROPN
ejpam-5768	87	6	2(w−1	2(w−1	NOUN
ejpam-5768	87	7	02	02	NUM
ejpam-5768	87	8	)	)	PUNCT
ejpam-5768	87	9	}	}	PUNCT
ejpam-5768	88	1	=	=	SYM
ejpam-5768	88	2	max{υ̂	max{υ̂	NOUN
ejpam-5768	88	3	2(w01	2(w01	NUM
ejpam-5768	88	4	)	)	PUNCT
ejpam-5768	88	5	,	,	PUNCT
ejpam-5768	88	6	υ̂	υ̂	PROPN
ejpam-5768	88	7	2(w02	2(w02	NUM
ejpam-5768	88	8	)	)	PUNCT
ejpam-5768	88	9	}	}	PUNCT
ejpam-5768	88	10	.	.	PUNCT
ejpam-5768	89	1	if	if	SCONJ
ejpam-5768	89	2	(	(	PUNCT
ejpam-5768	89	3	1	1	X
ejpam-5768	89	4	)	)	PUNCT
ejpam-5768	89	5	holds	hold	VERB
ejpam-5768	89	6	,	,	PUNCT
ejpam-5768	89	7	then	then	ADV
ejpam-5768	89	8	ῡ	ῡ	PROPN
ejpam-5768	89	9	2(w−1	2(w−1	NOUN
ejpam-5768	89	10	01	01	NUM
ejpam-5768	89	11	)	)	PUNCT
ejpam-5768	90	1	=	=	SYM
ejpam-5768	91	1	ῡ	ῡ	NOUN
ejpam-5768	91	2	2(ew−1	2(ew−1	NUM
ejpam-5768	91	3	01	01	NUM
ejpam-5768	91	4	)	)	PUNCT
ejpam-5768	91	5	≥	≥	PROPN
ejpam-5768	91	6	min{ῡ	min{ῡ	NOUN
ejpam-5768	91	7	2(e	2(e	NUM
ejpam-5768	91	8	)	)	PUNCT
ejpam-5768	91	9	,	,	PUNCT
ejpam-5768	91	10	ῡ	ῡ	PROPN
ejpam-5768	91	11	2(w01	2(w01	NUM
ejpam-5768	91	12	)	)	PUNCT
ejpam-5768	91	13	}	}	PUNCT
ejpam-5768	92	1	=	=	SYM
ejpam-5768	92	2	ῡ	ῡ	PROPN
ejpam-5768	92	3	2(w01	2(w01	NUM
ejpam-5768	92	4	)	)	PUNCT
ejpam-5768	92	5	.	.	PUNCT
ejpam-5768	93	1	on	on	ADP
ejpam-5768	93	2	the	the	DET
ejpam-5768	93	3	other	other	ADJ
ejpam-5768	93	4	hand	hand	NOUN
ejpam-5768	93	5	,	,	PUNCT
ejpam-5768	93	6	ῡ	ῡ	PROPN
ejpam-5768	93	7	2(w01	2(w01	NUM
ejpam-5768	93	8	)	)	PUNCT
ejpam-5768	93	9	=	=	SYM
ejpam-5768	93	10	ῡ	ῡ	NOUN
ejpam-5768	93	11	2(ew01	2(ew01	NUM
ejpam-5768	93	12	)	)	PUNCT
ejpam-5768	93	13	≥	≥	NOUN
ejpam-5768	93	14	min{ῡ	min{ῡ	NOUN
ejpam-5768	93	15	2(e	2(e	NUM
ejpam-5768	93	16	)	)	PUNCT
ejpam-5768	93	17	,	,	PUNCT
ejpam-5768	93	18	ῡ	ῡ	PROPN
ejpam-5768	93	19	2(w−1	2(w−1	PROPN
ejpam-5768	93	20	01	01	NUM
ejpam-5768	93	21	)	)	PUNCT
ejpam-5768	93	22	}	}	PUNCT
ejpam-5768	94	1	=	=	SYM
ejpam-5768	94	2	ῡ	ῡ	PROPN
ejpam-5768	94	3	2(w−1	2(w−1	NOUN
ejpam-5768	94	4	01	01	NUM
ejpam-5768	94	5	)	)	PUNCT
ejpam-5768	94	6	.	.	PUNCT
ejpam-5768	95	1	thus	thus	ADV
ejpam-5768	95	2	ῡ	ῡ	PROPN
ejpam-5768	95	3	2(w−1	2(w−1	NOUN
ejpam-5768	95	4	01	01	NUM
ejpam-5768	95	5	)	)	PUNCT
ejpam-5768	96	1	=	=	SYM
ejpam-5768	96	2	ῡ	ῡ	PROPN
ejpam-5768	96	3	2(w01	2(w01	NUM
ejpam-5768	96	4	)	)	PUNCT
ejpam-5768	96	5	.	.	PUNCT
ejpam-5768	97	1	in	in	ADP
ejpam-5768	97	2	addition	addition	NOUN
ejpam-5768	97	3	,	,	PUNCT
ejpam-5768	97	4	ῡ	ῡ	PROPN
ejpam-5768	97	5	2(w01w02	2(w01w02	NUM
ejpam-5768	97	6	)	)	PUNCT
ejpam-5768	97	7	=	=	SYM
ejpam-5768	97	8	ῡ	ῡ	PROPN
ejpam-5768	97	9	2(w01(w	2(w01(w	NUM
ejpam-5768	97	10	−1	−1	NOUN
ejpam-5768	97	11	02	02	NUM
ejpam-5768	97	12	)	)	PUNCT
ejpam-5768	97	13	−1	−1	NOUN
ejpam-5768	97	14	)	)	PUNCT
ejpam-5768	97	15	≥	≥	PROPN
ejpam-5768	97	16	min{ῡ	min{ῡ	NOUN
ejpam-5768	97	17	2(w01	2(w01	NUM
ejpam-5768	97	18	)	)	PUNCT
ejpam-5768	97	19	,	,	PUNCT
ejpam-5768	97	20	ῡ	ῡ	PROPN
ejpam-5768	97	21	2(w−1	2(w−1	PROPN
ejpam-5768	97	22	02	02	NUM
ejpam-5768	97	23	)	)	PUNCT
ejpam-5768	97	24	}	}	PUNCT
ejpam-5768	98	1	=	=	SYM
ejpam-5768	98	2	min{ῡ	min{ῡ	NOUN
ejpam-5768	98	3	2(w01	2(w01	NUM
ejpam-5768	98	4	)	)	PUNCT
ejpam-5768	98	5	,	,	PUNCT
ejpam-5768	98	6	ῡ	ῡ	PROPN
ejpam-5768	98	7	2(w02	2(w02	NUM
ejpam-5768	98	8	)	)	PUNCT
ejpam-5768	98	9	}	}	PUNCT
ejpam-5768	98	10	.	.	PUNCT
ejpam-5768	99	1	similarly	similarly	ADV
ejpam-5768	99	2	,	,	PUNCT
ejpam-5768	99	3	υ̂	υ̂	PROPN
ejpam-5768	99	4	2(w−1	2(w−1	NOUN
ejpam-5768	99	5	01	01	NUM
ejpam-5768	99	6	)	)	PUNCT
ejpam-5768	100	1	=	=	PRON
ejpam-5768	100	2	υ̂	υ̂	NUM
ejpam-5768	100	3	2(ew−1	2(ew−1	NUM
ejpam-5768	100	4	01	01	NUM
ejpam-5768	100	5	)	)	PUNCT
ejpam-5768	100	6	≤	≤	NOUN
ejpam-5768	100	7	max{υ̂	max{υ̂	NOUN
ejpam-5768	100	8	2(e	2(e	NUM
ejpam-5768	100	9	)	)	PUNCT
ejpam-5768	100	10	,	,	PUNCT
ejpam-5768	100	11	υ̂	υ̂	NUM
ejpam-5768	100	12	2(w01	2(w01	NUM
ejpam-5768	100	13	)	)	PUNCT
ejpam-5768	100	14	}	}	PUNCT
ejpam-5768	100	15	=	=	SYM
ejpam-5768	100	16	υ̂	υ̂	NUM
ejpam-5768	100	17	2(w01	2(w01	NUM
ejpam-5768	100	18	)	)	PUNCT
ejpam-5768	100	19	and	and	CCONJ
ejpam-5768	100	20	υ̂	υ̂	NUM
ejpam-5768	100	21	2(w01	2(w01	NUM
ejpam-5768	100	22	)	)	PUNCT
ejpam-5768	100	23	=	=	PRON
ejpam-5768	100	24	υ̂	υ̂	NUM
ejpam-5768	100	25	2(ew01	2(ew01	NUM
ejpam-5768	100	26	)	)	PUNCT
ejpam-5768	100	27	≤	≤	NOUN
ejpam-5768	100	28	max{υ̂	max{υ̂	NOUN
ejpam-5768	100	29	2(e	2(e	NUM
ejpam-5768	100	30	)	)	PUNCT
ejpam-5768	100	31	,	,	PUNCT
ejpam-5768	100	32	υ̂	υ̂	PROPN
ejpam-5768	100	33	2(w−1	2(w−1	NOUN
ejpam-5768	100	34	01	01	NUM
ejpam-5768	100	35	)	)	PUNCT
ejpam-5768	100	36	}	}	PUNCT
ejpam-5768	100	37	=	=	SYM
ejpam-5768	100	38	υ̂	υ̂	NUM
ejpam-5768	100	39	2(w−1	2(w−1	NOUN
ejpam-5768	100	40	01	01	NUM
ejpam-5768	100	41	)	)	PUNCT
ejpam-5768	100	42	.	.	PUNCT
ejpam-5768	101	1	thus	thus	ADV
ejpam-5768	101	2	υ̂	υ̂	PRON
ejpam-5768	101	3	2(w−1	2(w−1	NOUN
ejpam-5768	101	4	01	01	NUM
ejpam-5768	101	5	)	)	PUNCT
ejpam-5768	101	6	=	=	PRON
ejpam-5768	101	7	υ̂	υ̂	NUM
ejpam-5768	101	8	2(w01	2(w01	NUM
ejpam-5768	101	9	)	)	PUNCT
ejpam-5768	101	10	.	.	PUNCT
ejpam-5768	102	1	in	in	ADP
ejpam-5768	102	2	addition	addition	NOUN
ejpam-5768	102	3	,	,	PUNCT
ejpam-5768	102	4	υ̂	υ̂	NUM
ejpam-5768	102	5	2(w01w02	2(w01w02	NUM
ejpam-5768	102	6	)	)	PUNCT
ejpam-5768	102	7	=	=	SYM
ejpam-5768	102	8	υ	υ	PROPN
ejpam-5768	102	9	2(w01(w	2(w01(w	NUM
ejpam-5768	102	10	−1	−1	NOUN
ejpam-5768	102	11	02	02	NUM
ejpam-5768	102	12	)	)	PUNCT
ejpam-5768	102	13	−1	−1	NOUN
ejpam-5768	102	14	)	)	PUNCT
ejpam-5768	102	15	≤	≤	NUM
ejpam-5768	102	16	max{υ̂	max{υ̂	PROPN
ejpam-5768	102	17	2(w01	2(w01	NUM
ejpam-5768	102	18	)	)	PUNCT
ejpam-5768	102	19	,	,	PUNCT
ejpam-5768	102	20	υ̂	υ̂	PROPN
ejpam-5768	102	21	2(w−1	2(w−1	NOUN
ejpam-5768	102	22	02	02	NUM
ejpam-5768	102	23	)	)	PUNCT
ejpam-5768	102	24	}	}	PUNCT
ejpam-5768	103	1	=	=	SYM
ejpam-5768	103	2	max{υ̂	max{υ̂	NOUN
ejpam-5768	103	3	2(w01	2(w01	NUM
ejpam-5768	103	4	)	)	PUNCT
ejpam-5768	103	5	,	,	PUNCT
ejpam-5768	103	6	υ̂	υ̂	PROPN
ejpam-5768	103	7	2(w02	2(w02	NUM
ejpam-5768	103	8	)	)	PUNCT
ejpam-5768	103	9	}	}	PUNCT
ejpam-5768	103	10	hence	hence	ADV
ejpam-5768	103	11	υ	υ	PROPN
ejpam-5768	103	12	is	be	AUX
ejpam-5768	103	13	a	a	DET
ejpam-5768	103	14	pf	pf	PROPN
ejpam-5768	103	15	hx	hx	PROPN
ejpam-5768	103	16	-	-	PUNCT
ejpam-5768	103	17	sg	sg	PROPN
ejpam-5768	103	18	of	of	ADP
ejpam-5768	103	19	w	w	PROPN
ejpam-5768	103	20	.	.	PUNCT
ejpam-5768	103	21	a.	a.	PROPN
ejpam-5768	103	22	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	103	23	/	/	SYM
ejpam-5768	103	24	eur	eur	PROPN
ejpam-5768	103	25	.	.	PUNCT
ejpam-5768	104	1	j.	j.	PROPN
ejpam-5768	104	2	pure	pure	PROPN
ejpam-5768	104	3	appl	appl	PROPN
ejpam-5768	104	4	.	.	PROPN
ejpam-5768	104	5	math	math	PROPN
ejpam-5768	104	6	,	,	PUNCT
ejpam-5768	104	7	18	18	NUM
ejpam-5768	104	8	(	(	PUNCT
ejpam-5768	104	9	2	2	NUM
ejpam-5768	104	10	)	)	PUNCT
ejpam-5768	104	11	(	(	PUNCT
ejpam-5768	104	12	2025	2025	NUM
ejpam-5768	104	13	)	)	PUNCT
ejpam-5768	104	14	,	,	PUNCT
ejpam-5768	104	15	5768	5768	NUM
ejpam-5768	104	16	5	5	NUM
ejpam-5768	104	17	of	of	ADP
ejpam-5768	104	18	17	17	NUM
ejpam-5768	104	19	proposition	proposition	NOUN
ejpam-5768	104	20	2	2	NUM
ejpam-5768	104	21	.	.	PUNCT
ejpam-5768	105	1	let	let	VERB
ejpam-5768	105	2	w	w	NOUN
ejpam-5768	105	3	be	be	AUX
ejpam-5768	105	4	an	an	DET
ejpam-5768	105	5	hx	hx	NOUN
ejpam-5768	105	6	-	-	PUNCT
ejpam-5768	105	7	group	group	NOUN
ejpam-5768	105	8	and	and	CCONJ
ejpam-5768	105	9	υi	υi	NOUN
ejpam-5768	105	10	be	be	AUX
ejpam-5768	105	11	pf	pf	PROPN
ejpam-5768	105	12	hx	hx	PROPN
ejpam-5768	105	13	-	-	PUNCT
ejpam-5768	105	14	sgs	sgs	PROPN
ejpam-5768	105	15	of	of	ADP
ejpam-5768	105	16	w	w	PROPN
ejpam-5768	105	17	.	.	PUNCT
ejpam-5768	106	1	then	then	ADV
ejpam-5768	106	2	⋂	⋂	PROPN
ejpam-5768	106	3	i	i	PRON
ejpam-5768	106	4	υi	υi	VERB
ejpam-5768	106	5	is	be	AUX
ejpam-5768	106	6	a	a	DET
ejpam-5768	106	7	pf	pf	PROPN
ejpam-5768	106	8	hx	hx	PROPN
ejpam-5768	106	9	-	-	PUNCT
ejpam-5768	106	10	sg	sg	PROPN
ejpam-5768	106	11	of	of	ADP
ejpam-5768	106	12	w	w	PROPN
ejpam-5768	106	13	.	.	PUNCT
ejpam-5768	107	1	proof	proof	NOUN
ejpam-5768	107	2	.	.	PUNCT
ejpam-5768	108	1	clear	clear	ADJ
ejpam-5768	108	2	.	.	PUNCT
ejpam-5768	109	1	we	we	PRON
ejpam-5768	109	2	define	define	VERB
ejpam-5768	109	3	two	two	NUM
ejpam-5768	109	4	well	well	ADV
ejpam-5768	109	5	known	know	VERB
ejpam-5768	109	6	pytharorean	pytharorean	ADJ
ejpam-5768	109	7	fuzzy	fuzzy	ADJ
ejpam-5768	109	8	subsets	subset	NOUN
ejpam-5768	109	9	:	:	PUNCT
ejpam-5768	109	10	definition	definition	NOUN
ejpam-5768	109	11	2	2	NUM
ejpam-5768	109	12	.	.	PUNCT
ejpam-5768	110	1	let	let	VERB
ejpam-5768	110	2	w	w	NOUN
ejpam-5768	110	3	be	be	AUX
ejpam-5768	110	4	an	an	DET
ejpam-5768	110	5	hx	hx	NOUN
ejpam-5768	110	6	-	-	PUNCT
ejpam-5768	110	7	group	group	NOUN
ejpam-5768	110	8	and	and	CCONJ
ejpam-5768	110	9	p	p	NOUN
ejpam-5768	110	10	be	be	AUX
ejpam-5768	110	11	a	a	DET
ejpam-5768	110	12	pfss	pfss	NOUN
ejpam-5768	110	13	of	of	ADP
ejpam-5768	110	14	w	w	PROPN
ejpam-5768	110	15	.	.	PUNCT
ejpam-5768	111	1	then	then	ADV
ejpam-5768	111	2	(	(	PUNCT
ejpam-5768	111	3	1	1	X
ejpam-5768	111	4	)	)	PUNCT
ejpam-5768	111	5	p	p	NOUN
ejpam-5768	111	6	⋆	⋆	NOUN
ejpam-5768	111	7	=	=	SYM
ejpam-5768	111	8	ῡ	ῡ	NOUN
ejpam-5768	111	9	⋆	⋆	VERB
ejpam-5768	111	10	p	p	PROPN
ejpam-5768	111	11	∩	∩	NOUN
ejpam-5768	111	12	υ̂	υ̂	X
ejpam-5768	111	13	⋆	⋆	X
ejpam-5768	111	14	p	p	NOUN
ejpam-5768	111	15	,	,	PUNCT
ejpam-5768	111	16	where	where	SCONJ
ejpam-5768	111	17	ῡ	ῡ	PROPN
ejpam-5768	111	18	⋆	⋆	VERB
ejpam-5768	111	19	p	p	PROPN
ejpam-5768	111	20	=	=	X
ejpam-5768	111	21	{	{	PUNCT
ejpam-5768	111	22	w	w	PROPN
ejpam-5768	111	23	∈	∈	PROPN
ejpam-5768	111	24	w	w	NOUN
ejpam-5768	111	25	:	:	PUNCT
ejpam-5768	111	26	ῡ	ῡ	PROPN
ejpam-5768	111	27	2	2	NUM
ejpam-5768	111	28	p	p	NOUN
ejpam-5768	111	29	(	(	PUNCT
ejpam-5768	111	30	w	w	NOUN
ejpam-5768	111	31	)	)	PUNCT
ejpam-5768	111	32	>	>	X
ejpam-5768	111	33	0	0	NUM
ejpam-5768	111	34	}	}	PUNCT
ejpam-5768	111	35	υ̂	υ̂	PRON
ejpam-5768	111	36	⋆	⋆	NOUN
ejpam-5768	111	37	p	p	NOUN
ejpam-5768	111	38	=	=	X
ejpam-5768	111	39	{	{	PUNCT
ejpam-5768	111	40	w	w	PROPN
ejpam-5768	111	41	∈	∈	PROPN
ejpam-5768	111	42	w	w	PROPN
ejpam-5768	111	43	:	:	PUNCT
ejpam-5768	111	44	υ̂	υ̂	PROPN
ejpam-5768	111	45	2(w	2(w	NUM
ejpam-5768	111	46	)	)	PUNCT
ejpam-5768	111	47	<	<	X
ejpam-5768	111	48	1	1	NUM
ejpam-5768	111	49	}	}	PUNCT
ejpam-5768	111	50	(	(	PUNCT
ejpam-5768	111	51	2	2	NUM
ejpam-5768	111	52	)	)	PUNCT
ejpam-5768	111	53	p⋆	p⋆	NOUN
ejpam-5768	111	54	=	=	PUNCT
ejpam-5768	111	55	ῡ⋆p	ῡ⋆p	NOUN
ejpam-5768	111	56	∩	∩	NOUN
ejpam-5768	111	57	υ̂⋆p	υ̂⋆p	NOUN
ejpam-5768	111	58	,	,	PUNCT
ejpam-5768	111	59	where	where	SCONJ
ejpam-5768	111	60	ῡ⋆p	ῡ⋆p	NOUN
ejpam-5768	111	61	=	=	SYM
ejpam-5768	111	62	{	{	PUNCT
ejpam-5768	111	63	w	w	PROPN
ejpam-5768	111	64	∈	∈	PROPN
ejpam-5768	111	65	w	w	NOUN
ejpam-5768	111	66	:	:	PUNCT
ejpam-5768	111	67	ῡ	ῡ	PROPN
ejpam-5768	111	68	2	2	NUM
ejpam-5768	111	69	p	p	NOUN
ejpam-5768	111	70	(	(	PUNCT
ejpam-5768	111	71	w	w	NOUN
ejpam-5768	111	72	)	)	PUNCT
ejpam-5768	111	73	=	=	SYM
ejpam-5768	111	74	1	1	X
ejpam-5768	111	75	}	}	PUNCT
ejpam-5768	111	76	υ̂⋆p	υ̂⋆p	NOUN
ejpam-5768	111	77	=	=	PUNCT
ejpam-5768	111	78	{	{	PUNCT
ejpam-5768	111	79	w	w	PROPN
ejpam-5768	111	80	∈	∈	PROPN
ejpam-5768	111	81	w	w	PROPN
ejpam-5768	111	82	:	:	PUNCT
ejpam-5768	111	83	υ̂	υ̂	PROPN
ejpam-5768	111	84	2(w	2(w	NUM
ejpam-5768	111	85	)	)	PUNCT
ejpam-5768	111	86	=	=	SYM
ejpam-5768	112	1	0	0	X
ejpam-5768	112	2	}	}	PUNCT
ejpam-5768	112	3	now	now	ADV
ejpam-5768	112	4	,	,	PUNCT
ejpam-5768	112	5	we	we	PRON
ejpam-5768	112	6	prove	prove	VERB
ejpam-5768	112	7	that	that	SCONJ
ejpam-5768	112	8	the	the	DET
ejpam-5768	112	9	above	above	ADJ
ejpam-5768	112	10	pfsss	pfsss	NOUN
ejpam-5768	112	11	are	be	AUX
ejpam-5768	112	12	subgroups	subgroup	NOUN
ejpam-5768	112	13	of	of	ADP
ejpam-5768	112	14	g	g	NOUN
ejpam-5768	112	15	in	in	ADP
ejpam-5768	112	16	the	the	DET
ejpam-5768	112	17	case	case	NOUN
ejpam-5768	112	18	that	that	SCONJ
ejpam-5768	112	19	υp	υp	PROPN
ejpam-5768	112	20	is	be	AUX
ejpam-5768	112	21	a	a	DET
ejpam-5768	112	22	pf	pf	PROPN
ejpam-5768	112	23	hx	hx	PROPN
ejpam-5768	112	24	-	-	PUNCT
ejpam-5768	112	25	sg	sg	PROPN
ejpam-5768	112	26	.	.	PUNCT
ejpam-5768	113	1	theorem	theorem	NOUN
ejpam-5768	113	2	2	2	NUM
ejpam-5768	113	3	.	.	PUNCT
ejpam-5768	114	1	if	if	SCONJ
ejpam-5768	114	2	υp	υp	PROPN
ejpam-5768	114	3	is	be	AUX
ejpam-5768	114	4	a	a	DET
ejpam-5768	114	5	pf	pf	PROPN
ejpam-5768	114	6	hx	hx	PROPN
ejpam-5768	114	7	-	-	PUNCT
ejpam-5768	114	8	sg	sg	PROPN
ejpam-5768	114	9	of	of	ADP
ejpam-5768	114	10	g	g	PROPN
ejpam-5768	114	11	,	,	PUNCT
ejpam-5768	114	12	then	then	ADV
ejpam-5768	114	13	p	p	X
ejpam-5768	114	14	⋆	⋆	NOUN
ejpam-5768	114	15	is	be	AUX
ejpam-5768	114	16	a	a	DET
ejpam-5768	114	17	subgroup	subgroup	NOUN
ejpam-5768	114	18	of	of	ADP
ejpam-5768	114	19	g.	g.	PROPN
ejpam-5768	114	20	proof	proof	PROPN
ejpam-5768	114	21	.	.	PUNCT
ejpam-5768	115	1	suppose	suppose	VERB
ejpam-5768	115	2	that	that	SCONJ
ejpam-5768	115	3	w01	w01	NOUN
ejpam-5768	115	4	,	,	PUNCT
ejpam-5768	115	5	w02	w02	NOUN
ejpam-5768	115	6	∈	∈	PROPN
ejpam-5768	115	7	p	p	NOUN
ejpam-5768	115	8	⋆.	⋆.	X
ejpam-5768	115	9	then	then	ADV
ejpam-5768	115	10	ῡ	ῡ	PROPN
ejpam-5768	115	11	2(w01	2(w01	NUM
ejpam-5768	115	12	)	)	PUNCT
ejpam-5768	115	13	>	>	X
ejpam-5768	115	14	0	0	NUM
ejpam-5768	115	15	,	,	PUNCT
ejpam-5768	115	16	ῡ	ῡ	PROPN
ejpam-5768	115	17	2(w02	2(w02	NUM
ejpam-5768	115	18	)	)	PUNCT
ejpam-5768	115	19	>	>	X
ejpam-5768	115	20	0	0	PUNCT
ejpam-5768	115	21	and	and	CCONJ
ejpam-5768	115	22	υ̂	υ̂	NUM
ejpam-5768	115	23	2(w01	2(w01	NUM
ejpam-5768	115	24	)	)	PUNCT
ejpam-5768	115	25	<	<	X
ejpam-5768	115	26	1	1	NUM
ejpam-5768	115	27	,	,	PUNCT
ejpam-5768	115	28	υ̂	υ̂	PROPN
ejpam-5768	115	29	2(w02	2(w02	NUM
ejpam-5768	115	30	)	)	PUNCT
ejpam-5768	115	31	<	<	X
ejpam-5768	116	1	1	1	X
ejpam-5768	116	2	.	.	PUNCT
ejpam-5768	116	3	by	by	ADP
ejpam-5768	116	4	hypothesis	hypothesis	NOUN
ejpam-5768	116	5	,	,	PUNCT
ejpam-5768	116	6	ῡ	ῡ	PROPN
ejpam-5768	117	1	2(w01w	2(w01w	NUM
ejpam-5768	117	2	−1	−1	NOUN
ejpam-5768	117	3	02	02	NUM
ejpam-5768	117	4	)	)	PUNCT
ejpam-5768	117	5	≥	≥	PROPN
ejpam-5768	117	6	min{ῡ	min{ῡ	NOUN
ejpam-5768	117	7	2(w01	2(w01	NUM
ejpam-5768	117	8	)	)	PUNCT
ejpam-5768	117	9	,	,	PUNCT
ejpam-5768	117	10	ῡ	ῡ	PROPN
ejpam-5768	117	11	2(w02	2(w02	NUM
ejpam-5768	117	12	)	)	PUNCT
ejpam-5768	117	13	}	}	PUNCT
ejpam-5768	117	14	>	>	X
ejpam-5768	118	1	0	0	X
ejpam-5768	118	2	.	.	PUNCT
ejpam-5768	118	3	also	also	ADV
ejpam-5768	118	4	,	,	PUNCT
ejpam-5768	118	5	υ̂	υ̂	PROPN
ejpam-5768	119	1	2(w01w	2(w01w	NUM
ejpam-5768	119	2	−1	−1	NOUN
ejpam-5768	119	3	02	02	NUM
ejpam-5768	119	4	)	)	PUNCT
ejpam-5768	119	5	≤	≤	NUM
ejpam-5768	119	6	max{υ̂	max{υ̂	PROPN
ejpam-5768	119	7	2(w01	2(w01	NUM
ejpam-5768	119	8	)	)	PUNCT
ejpam-5768	119	9	,	,	PUNCT
ejpam-5768	119	10	υ̂	υ̂	PROPN
ejpam-5768	119	11	2(w02	2(w02	NUM
ejpam-5768	119	12	)	)	PUNCT
ejpam-5768	119	13	}	}	PUNCT
ejpam-5768	119	14	<	<	X
ejpam-5768	119	15	1	1	X
ejpam-5768	119	16	.	.	PUNCT
ejpam-5768	119	17	hence	hence	ADV
ejpam-5768	119	18	w01w	w01w	ADP
ejpam-5768	119	19	−1	−1	NOUN
ejpam-5768	119	20	02	02	NUM
ejpam-5768	119	21	∈	∈	PROPN
ejpam-5768	119	22	p	p	NOUN
ejpam-5768	119	23	⋆	⋆	NOUN
ejpam-5768	119	24	and	and	CCONJ
ejpam-5768	119	25	therefore	therefore	ADV
ejpam-5768	119	26	,	,	PUNCT
ejpam-5768	119	27	p	p	NOUN
ejpam-5768	119	28	⋆	⋆	VERB
ejpam-5768	119	29	is	be	AUX
ejpam-5768	119	30	a	a	DET
ejpam-5768	119	31	subgroup	subgroup	NOUN
ejpam-5768	119	32	of	of	ADP
ejpam-5768	119	33	g.	g.	PROPN
ejpam-5768	119	34	theorem	theorem	VERB
ejpam-5768	119	35	3	3	X
ejpam-5768	119	36	.	.	PUNCT
ejpam-5768	120	1	if	if	SCONJ
ejpam-5768	120	2	υp	υp	PROPN
ejpam-5768	120	3	is	be	AUX
ejpam-5768	120	4	a	a	DET
ejpam-5768	120	5	pf	pf	PROPN
ejpam-5768	120	6	hx	hx	PROPN
ejpam-5768	120	7	-	-	PUNCT
ejpam-5768	120	8	sg	sg	PROPN
ejpam-5768	120	9	of	of	ADP
ejpam-5768	120	10	g	g	NOUN
ejpam-5768	120	11	,	,	PUNCT
ejpam-5768	120	12	then	then	ADV
ejpam-5768	120	13	p⋆	p⋆	PRON
ejpam-5768	120	14	is	be	AUX
ejpam-5768	120	15	a	a	DET
ejpam-5768	120	16	subgroup	subgroup	NOUN
ejpam-5768	120	17	of	of	ADP
ejpam-5768	120	18	g.	g.	PROPN
ejpam-5768	120	19	proof	proof	PROPN
ejpam-5768	120	20	.	.	PUNCT
ejpam-5768	121	1	assume	assume	VERB
ejpam-5768	121	2	that	that	SCONJ
ejpam-5768	121	3	w01	w01	NOUN
ejpam-5768	121	4	,	,	PUNCT
ejpam-5768	121	5	w02	w02	NOUN
ejpam-5768	121	6	∈	∈	PROPN
ejpam-5768	121	7	p	p	NOUN
ejpam-5768	121	8	⋆.	⋆.	X
ejpam-5768	121	9	then	then	ADV
ejpam-5768	121	10	ῡ	ῡ	PROPN
ejpam-5768	121	11	2(w01	2(w01	NUM
ejpam-5768	121	12	)	)	PUNCT
ejpam-5768	121	13	=	=	SYM
ejpam-5768	121	14	1	1	NUM
ejpam-5768	121	15	,	,	PUNCT
ejpam-5768	121	16	ῡ	ῡ	PROPN
ejpam-5768	121	17	2(w02	2(w02	NUM
ejpam-5768	121	18	)	)	PUNCT
ejpam-5768	122	1	=	=	SYM
ejpam-5768	122	2	1	1	NUM
ejpam-5768	122	3	and	and	CCONJ
ejpam-5768	122	4	υ̂	υ̂	NUM
ejpam-5768	122	5	2(w01	2(w01	NUM
ejpam-5768	122	6	)	)	PUNCT
ejpam-5768	122	7	=	=	SYM
ejpam-5768	123	1	0	0	NUM
ejpam-5768	123	2	,	,	PUNCT
ejpam-5768	123	3	υ̂	υ̂	PROPN
ejpam-5768	123	4	2(w02	2(w02	NUM
ejpam-5768	123	5	)	)	PUNCT
ejpam-5768	123	6	=	=	NOUN
ejpam-5768	124	1	0	0	X
ejpam-5768	124	2	.	.	PUNCT
ejpam-5768	125	1	by	by	ADP
ejpam-5768	125	2	hypothesis	hypothesis	NOUN
ejpam-5768	125	3	,	,	PUNCT
ejpam-5768	125	4	ῡ	ῡ	PROPN
ejpam-5768	125	5	2(w01w	2(w01w	NUM
ejpam-5768	125	6	−1	−1	NOUN
ejpam-5768	125	7	02	02	NUM
ejpam-5768	125	8	)	)	PUNCT
ejpam-5768	125	9	≥	≥	PROPN
ejpam-5768	125	10	min{ῡ	min{ῡ	NOUN
ejpam-5768	125	11	2(w01	2(w01	NUM
ejpam-5768	125	12	)	)	PUNCT
ejpam-5768	125	13	,	,	PUNCT
ejpam-5768	125	14	ῡ	ῡ	PROPN
ejpam-5768	125	15	2(w02	2(w02	NUM
ejpam-5768	125	16	)	)	PUNCT
ejpam-5768	125	17	}	}	PUNCT
ejpam-5768	125	18	=	=	SYM
ejpam-5768	125	19	1	1	X
ejpam-5768	125	20	.	.	PUNCT
ejpam-5768	125	21	also	also	ADV
ejpam-5768	125	22	,	,	PUNCT
ejpam-5768	125	23	υ̂	υ̂	PROPN
ejpam-5768	125	24	2(w01w	2(w01w	NUM
ejpam-5768	125	25	−1	−1	NOUN
ejpam-5768	125	26	02	02	NUM
ejpam-5768	125	27	)	)	PUNCT
ejpam-5768	125	28	≤	≤	NUM
ejpam-5768	125	29	max{υ̂	max{υ̂	PROPN
ejpam-5768	125	30	2(w01	2(w01	NUM
ejpam-5768	125	31	)	)	PUNCT
ejpam-5768	125	32	,	,	PUNCT
ejpam-5768	125	33	υ̂	υ̂	PROPN
ejpam-5768	125	34	2(w02	2(w02	NUM
ejpam-5768	125	35	)	)	PUNCT
ejpam-5768	125	36	}	}	PUNCT
ejpam-5768	125	37	=	=	SYM
ejpam-5768	125	38	0	0	X
ejpam-5768	125	39	.	.	PUNCT
ejpam-5768	126	1	hence	hence	ADV
ejpam-5768	126	2	w01w	w01w	ADP
ejpam-5768	126	3	−1	−1	NOUN
ejpam-5768	126	4	02	02	NUM
ejpam-5768	126	5	∈	∈	NOUN
ejpam-5768	126	6	p⋆	p⋆	NOUN
ejpam-5768	126	7	and	and	CCONJ
ejpam-5768	126	8	therefore	therefore	ADV
ejpam-5768	126	9	,	,	PUNCT
ejpam-5768	126	10	p⋆	p⋆	X
ejpam-5768	126	11	is	be	AUX
ejpam-5768	126	12	a	a	DET
ejpam-5768	126	13	subgroup	subgroup	NOUN
ejpam-5768	126	14	of	of	ADP
ejpam-5768	126	15	g.	g.	PROPN
ejpam-5768	126	16	a	a	DET
ejpam-5768	126	17	(	(	PUNCT
ejpam-5768	126	18	ζ	ζ	NOUN
ejpam-5768	126	19	,	,	PUNCT
ejpam-5768	126	20	δ)-level	δ)-level	PUNCT
ejpam-5768	126	21	pythagorean	pythagorean	ADJ
ejpam-5768	126	22	fuzzy	fuzzy	PROPN
ejpam-5768	126	23	subset	subset	NOUN
ejpam-5768	126	24	can	can	AUX
ejpam-5768	126	25	be	be	AUX
ejpam-5768	126	26	defined	define	VERB
ejpam-5768	126	27	as	as	SCONJ
ejpam-5768	126	28	follows	follow	VERB
ejpam-5768	126	29	:	:	PUNCT
ejpam-5768	126	30	definition	definition	NOUN
ejpam-5768	126	31	3	3	NUM
ejpam-5768	126	32	.	.	PUNCT
ejpam-5768	127	1	let	let	VERB
ejpam-5768	127	2	υ	υ	PRON
ejpam-5768	127	3	be	be	AUX
ejpam-5768	127	4	a	a	DET
ejpam-5768	127	5	pfss	pfss	NOUN
ejpam-5768	127	6	of	of	ADP
ejpam-5768	127	7	w	w	PROPN
ejpam-5768	127	8	.	.	PUNCT
ejpam-5768	128	1	then	then	ADV
ejpam-5768	128	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	128	3	,	,	PUNCT
ejpam-5768	128	4	δ	δ	PROPN
ejpam-5768	128	5	)	)	PUNCT
ejpam-5768	128	6	=	=	PRON
ejpam-5768	128	7	{	{	PUNCT
ejpam-5768	128	8	w	w	NOUN
ejpam-5768	128	9	∈	∈	PROPN
ejpam-5768	128	10	w	w	NOUN
ejpam-5768	128	11	:	:	PUNCT
ejpam-5768	128	12	ῡ	ῡ	PROPN
ejpam-5768	128	13	2(w	2(w	NUM
ejpam-5768	128	14	)	)	PUNCT
ejpam-5768	128	15	≥	≥	NOUN
ejpam-5768	128	16	ζ	ζ	NOUN
ejpam-5768	128	17	and	and	CCONJ
ejpam-5768	128	18	υ̂	υ̂	NUM
ejpam-5768	128	19	2(w	2(w	NUM
ejpam-5768	128	20	)	)	PUNCT
ejpam-5768	128	21	≤	≤	NUM
ejpam-5768	128	22	δ	δ	X
ejpam-5768	128	23	}	}	PUNCT
ejpam-5768	128	24	,	,	PUNCT
ejpam-5768	128	25	where	where	SCONJ
ejpam-5768	128	26	ζ	ζ	X
ejpam-5768	128	27	,	,	PUNCT
ejpam-5768	128	28	δ	δ	PROPN
ejpam-5768	128	29	∈	∈	PROPN
ejpam-5768	129	1	[	[	X
ejpam-5768	129	2	0	0	NUM
ejpam-5768	129	3	,	,	PUNCT
ejpam-5768	129	4	1	1	NUM
ejpam-5768	129	5	]	]	PUNCT
ejpam-5768	129	6	.	.	PUNCT
ejpam-5768	130	1	an	an	DET
ejpam-5768	130	2	essential	essential	ADJ
ejpam-5768	130	3	question	question	NOUN
ejpam-5768	130	4	arises	arise	VERB
ejpam-5768	130	5	:	:	PUNCT
ejpam-5768	130	6	is	be	AUX
ejpam-5768	130	7	there	there	PRON
ejpam-5768	130	8	a	a	DET
ejpam-5768	130	9	relationship	relationship	NOUN
ejpam-5768	130	10	between	between	ADP
ejpam-5768	130	11	these	these	DET
ejpam-5768	130	12	level	level	NOUN
ejpam-5768	130	13	pfsss	pfsss	NOUN
ejpam-5768	130	14	υ(ζ	υ(ζ	PROPN
ejpam-5768	130	15	,	,	PUNCT
ejpam-5768	130	16	δ	δ	PROPN
ejpam-5768	130	17	)	)	PUNCT
ejpam-5768	130	18	and	and	CCONJ
ejpam-5768	130	19	a	a	DET
ejpam-5768	130	20	pfss	pfss	NOUN
ejpam-5768	130	21	υ	υ	NOUN
ejpam-5768	130	22	?	?	PUNCT
ejpam-5768	131	1	we	we	PRON
ejpam-5768	131	2	present	present	VERB
ejpam-5768	131	3	the	the	DET
ejpam-5768	131	4	following	follow	VERB
ejpam-5768	131	5	theorem	theorem	NOUN
ejpam-5768	131	6	to	to	PART
ejpam-5768	131	7	answer	answer	VERB
ejpam-5768	131	8	this	this	DET
ejpam-5768	131	9	question	question	NOUN
ejpam-5768	131	10	.	.	PUNCT
ejpam-5768	132	1	theorem	theorem	ADJ
ejpam-5768	132	2	4	4	NUM
ejpam-5768	132	3	.	.	PUNCT
ejpam-5768	133	1	let	let	VERB
ejpam-5768	133	2	υ	υ	PRON
ejpam-5768	133	3	be	be	AUX
ejpam-5768	133	4	a	a	DET
ejpam-5768	133	5	pfss	pfss	NOUN
ejpam-5768	133	6	of	of	ADP
ejpam-5768	133	7	w	w	PROPN
ejpam-5768	133	8	.	.	PUNCT
ejpam-5768	134	1	then	then	ADV
ejpam-5768	134	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	134	3	,	,	PUNCT
ejpam-5768	134	4	δ	δ	PROPN
ejpam-5768	134	5	)	)	PUNCT
ejpam-5768	134	6	is	be	AUX
ejpam-5768	134	7	an	an	DET
ejpam-5768	134	8	hx	hx	PROPN
ejpam-5768	134	9	-	-	PUNCT
ejpam-5768	134	10	sg	sg	PROPN
ejpam-5768	134	11	,	,	PUNCT
ejpam-5768	134	12	for	for	ADP
ejpam-5768	134	13	all	all	DET
ejpam-5768	134	14	ζ	ζ	NOUN
ejpam-5768	134	15	,	,	PUNCT
ejpam-5768	134	16	δ	δ	PROPN
ejpam-5768	134	17	∈	∈	PROPN
ejpam-5768	135	1	[	[	X
ejpam-5768	135	2	0	0	NUM
ejpam-5768	135	3	,	,	PUNCT
ejpam-5768	135	4	1	1	NUM
ejpam-5768	135	5	]	]	PUNCT
ejpam-5768	135	6	,	,	PUNCT
ejpam-5768	135	7	if	if	SCONJ
ejpam-5768	135	8	and	and	CCONJ
ejpam-5768	135	9	only	only	ADV
ejpam-5768	135	10	if	if	SCONJ
ejpam-5768	135	11	υ	υ	PRON
ejpam-5768	135	12	is	be	AUX
ejpam-5768	135	13	a	a	DET
ejpam-5768	135	14	pf	pf	PROPN
ejpam-5768	135	15	hx	hx	PROPN
ejpam-5768	135	16	-	-	PUNCT
ejpam-5768	135	17	sg	sg	PROPN
ejpam-5768	135	18	.	.	PUNCT
ejpam-5768	135	19	a.	a.	PROPN
ejpam-5768	135	20	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	135	21	/	/	SYM
ejpam-5768	135	22	eur	eur	PROPN
ejpam-5768	135	23	.	.	PUNCT
ejpam-5768	136	1	j.	j.	PROPN
ejpam-5768	136	2	pure	pure	PROPN
ejpam-5768	136	3	appl	appl	PROPN
ejpam-5768	136	4	.	.	PROPN
ejpam-5768	136	5	math	math	PROPN
ejpam-5768	136	6	,	,	PUNCT
ejpam-5768	136	7	18	18	NUM
ejpam-5768	136	8	(	(	PUNCT
ejpam-5768	136	9	2	2	NUM
ejpam-5768	136	10	)	)	PUNCT
ejpam-5768	136	11	(	(	PUNCT
ejpam-5768	136	12	2025	2025	NUM
ejpam-5768	136	13	)	)	PUNCT
ejpam-5768	136	14	,	,	PUNCT
ejpam-5768	136	15	5768	5768	NUM
ejpam-5768	136	16	6	6	NUM
ejpam-5768	136	17	of	of	ADP
ejpam-5768	136	18	17	17	NUM
ejpam-5768	136	19	proof	proof	NOUN
ejpam-5768	136	20	.	.	PUNCT
ejpam-5768	136	21	suppose	suppose	VERB
ejpam-5768	136	22	that	that	SCONJ
ejpam-5768	136	23	υ(ζ	υ(ζ	PROPN
ejpam-5768	136	24	,	,	PUNCT
ejpam-5768	136	25	δ	δ	PROPN
ejpam-5768	136	26	)	)	PUNCT
ejpam-5768	136	27	is	be	AUX
ejpam-5768	136	28	an	an	DET
ejpam-5768	136	29	hx	hx	PROPN
ejpam-5768	136	30	-	-	PUNCT
ejpam-5768	136	31	sg	sg	PROPN
ejpam-5768	136	32	.	.	PUNCT
ejpam-5768	136	33	assume	assume	VERB
ejpam-5768	136	34	that	that	SCONJ
ejpam-5768	136	35	there	there	PRON
ejpam-5768	136	36	exists	exist	VERB
ejpam-5768	136	37	w01	w01	NOUN
ejpam-5768	136	38	,	,	PUNCT
ejpam-5768	136	39	w02	w02	NOUN
ejpam-5768	136	40	∈	∈	PROPN
ejpam-5768	136	41	w	w	ADP
ejpam-5768	136	42	such	such	ADJ
ejpam-5768	136	43	that	that	SCONJ
ejpam-5768	136	44	ῡ	ῡ	PROPN
ejpam-5768	137	1	2(w01w	2(w01w	NUM
ejpam-5768	137	2	−1	−1	NOUN
ejpam-5768	137	3	02	02	NUM
ejpam-5768	137	4	)	)	PUNCT
ejpam-5768	138	1	<	<	X
ejpam-5768	138	2	min{ῡ	min{ῡ	PROPN
ejpam-5768	138	3	2(w01	2(w01	NUM
ejpam-5768	138	4	)	)	PUNCT
ejpam-5768	138	5	,	,	PUNCT
ejpam-5768	138	6	ῡ	ῡ	PROPN
ejpam-5768	138	7	2(w02	2(w02	NUM
ejpam-5768	138	8	)	)	PUNCT
ejpam-5768	138	9	}	}	PUNCT
ejpam-5768	138	10	.	.	PUNCT
ejpam-5768	139	1	then	then	ADV
ejpam-5768	139	2	ῡ	ῡ	PROPN
ejpam-5768	139	3	2(w01w	2(w01w	NUM
ejpam-5768	139	4	−1	−1	NOUN
ejpam-5768	139	5	02	02	NUM
ejpam-5768	139	6	)	)	PUNCT
ejpam-5768	139	7	<	<	X
ejpam-5768	139	8	ζ	ζ	NOUN
ejpam-5768	139	9	,	,	PUNCT
ejpam-5768	139	10	for	for	ADP
ejpam-5768	139	11	some	some	DET
ejpam-5768	139	12	ζ	ζ	NOUN
ejpam-5768	139	13	∈	∈	NOUN
ejpam-5768	140	1	[	[	X
ejpam-5768	140	2	0	0	NUM
ejpam-5768	140	3	,	,	PUNCT
ejpam-5768	140	4	1	1	NUM
ejpam-5768	140	5	]	]	PUNCT
ejpam-5768	140	6	and	and	CCONJ
ejpam-5768	140	7	this	this	PRON
ejpam-5768	140	8	contradicts	contradict	VERB
ejpam-5768	140	9	that	that	SCONJ
ejpam-5768	140	10	υ(ζ	υ(ζ	PROPN
ejpam-5768	140	11	,	,	PUNCT
ejpam-5768	140	12	δ	δ	PROPN
ejpam-5768	140	13	)	)	PUNCT
ejpam-5768	140	14	is	be	AUX
ejpam-5768	140	15	an	an	DET
ejpam-5768	140	16	hx	hx	PROPN
ejpam-5768	140	17	-	-	PUNCT
ejpam-5768	140	18	sg	sg	PROPN
ejpam-5768	140	19	.	.	PUNCT
ejpam-5768	141	1	thus	thus	ADV
ejpam-5768	141	2	ῡ	ῡ	PROPN
ejpam-5768	141	3	2(w01w	2(w01w	NUM
ejpam-5768	141	4	−1	−1	NOUN
ejpam-5768	141	5	02	02	NUM
ejpam-5768	141	6	)	)	PUNCT
ejpam-5768	141	7	≥	≥	PROPN
ejpam-5768	141	8	min{ῡ	min{ῡ	NOUN
ejpam-5768	141	9	2(w01	2(w01	NUM
ejpam-5768	141	10	)	)	PUNCT
ejpam-5768	141	11	,	,	PUNCT
ejpam-5768	141	12	ῡ	ῡ	PROPN
ejpam-5768	141	13	2(w02	2(w02	NUM
ejpam-5768	141	14	)	)	PUNCT
ejpam-5768	141	15	}	}	PUNCT
ejpam-5768	141	16	.	.	PUNCT
ejpam-5768	142	1	now	now	ADV
ejpam-5768	142	2	,	,	PUNCT
ejpam-5768	142	3	let	let	VERB
ejpam-5768	142	4	υ̂	υ̂	PRON
ejpam-5768	143	1	2(w01w	2(w01w	NUM
ejpam-5768	143	2	−1	−1	NOUN
ejpam-5768	143	3	02	02	NUM
ejpam-5768	143	4	)	)	PUNCT
ejpam-5768	143	5	>	>	PUNCT
ejpam-5768	144	1	max{υ̂	max{υ̂	NOUN
ejpam-5768	144	2	2(w01	2(w01	NUM
ejpam-5768	144	3	)	)	PUNCT
ejpam-5768	144	4	,	,	PUNCT
ejpam-5768	144	5	υ̂	υ̂	PROPN
ejpam-5768	144	6	2(w02	2(w02	NUM
ejpam-5768	144	7	)	)	PUNCT
ejpam-5768	144	8	}	}	PUNCT
ejpam-5768	144	9	.	.	PUNCT
ejpam-5768	145	1	this	this	PRON
ejpam-5768	145	2	implies	imply	VERB
ejpam-5768	145	3	that	that	SCONJ
ejpam-5768	145	4	υ̂	υ̂	PROPN
ejpam-5768	145	5	2(w01w	2(w01w	NUM
ejpam-5768	145	6	−1	−1	NOUN
ejpam-5768	145	7	02	02	NUM
ejpam-5768	145	8	)	)	PUNCT
ejpam-5768	145	9	>	>	PUNCT
ejpam-5768	145	10	δ	δ	PROPN
ejpam-5768	145	11	,	,	PUNCT
ejpam-5768	145	12	for	for	ADP
ejpam-5768	145	13	some	some	DET
ejpam-5768	145	14	δ	δ	NOUN
ejpam-5768	145	15	∈	∈	PROPN
ejpam-5768	146	1	[	[	X
ejpam-5768	146	2	0	0	NUM
ejpam-5768	146	3	,	,	PUNCT
ejpam-5768	146	4	1	1	NUM
ejpam-5768	146	5	]	]	PUNCT
ejpam-5768	146	6	and	and	CCONJ
ejpam-5768	146	7	this	this	PRON
ejpam-5768	146	8	contradicts	contradict	VERB
ejpam-5768	146	9	that	that	SCONJ
ejpam-5768	146	10	υ(ζ	υ(ζ	PROPN
ejpam-5768	146	11	,	,	PUNCT
ejpam-5768	146	12	δ	δ	PROPN
ejpam-5768	146	13	)	)	PUNCT
ejpam-5768	146	14	is	be	AUX
ejpam-5768	146	15	an	an	DET
ejpam-5768	146	16	hx	hx	PROPN
ejpam-5768	146	17	-	-	PUNCT
ejpam-5768	146	18	sg	sg	PROPN
ejpam-5768	146	19	.	.	PUNCT
ejpam-5768	147	1	this	this	DET
ejpam-5768	147	2	menas	menas	PROPN
ejpam-5768	147	3	that	that	PRON
ejpam-5768	147	4	υ̂	υ̂	VERB
ejpam-5768	148	1	2(w01w	2(w01w	NUM
ejpam-5768	148	2	−1	−1	NOUN
ejpam-5768	148	3	02	02	NUM
ejpam-5768	148	4	)	)	PUNCT
ejpam-5768	148	5	≤	≤	NUM
ejpam-5768	148	6	max{υ̂	max{υ̂	PROPN
ejpam-5768	148	7	2(w01	2(w01	NUM
ejpam-5768	148	8	)	)	PUNCT
ejpam-5768	148	9	,	,	PUNCT
ejpam-5768	148	10	υ̂	υ̂	PROPN
ejpam-5768	148	11	2(w02	2(w02	NUM
ejpam-5768	148	12	)	)	PUNCT
ejpam-5768	148	13	}	}	PUNCT
ejpam-5768	148	14	.	.	PUNCT
ejpam-5768	149	1	hence	hence	ADV
ejpam-5768	149	2	υ	υ	PROPN
ejpam-5768	149	3	is	be	AUX
ejpam-5768	149	4	a	a	DET
ejpam-5768	149	5	pf	pf	PROPN
ejpam-5768	149	6	hx	hx	PROPN
ejpam-5768	149	7	-	-	PUNCT
ejpam-5768	149	8	sg	sg	PROPN
ejpam-5768	149	9	.	.	PUNCT
ejpam-5768	150	1	conversely	conversely	ADV
ejpam-5768	150	2	,	,	PUNCT
ejpam-5768	150	3	suppose	suppose	VERB
ejpam-5768	150	4	that	that	SCONJ
ejpam-5768	150	5	υ	υ	PROPN
ejpam-5768	150	6	is	be	AUX
ejpam-5768	150	7	a	a	DET
ejpam-5768	150	8	pf	pf	PROPN
ejpam-5768	150	9	hx	hx	PROPN
ejpam-5768	150	10	-	-	PUNCT
ejpam-5768	150	11	sg	sg	PROPN
ejpam-5768	150	12	.	.	PUNCT
ejpam-5768	151	1	let	let	VERB
ejpam-5768	151	2	w01	w01	NOUN
ejpam-5768	151	3	,	,	PUNCT
ejpam-5768	151	4	w02	w02	NOUN
ejpam-5768	151	5	∈	∈	PROPN
ejpam-5768	151	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	151	7	,	,	PUNCT
ejpam-5768	151	8	δ	δ	PROPN
ejpam-5768	151	9	)	)	PUNCT
ejpam-5768	151	10	,	,	PUNCT
ejpam-5768	151	11	for	for	ADP
ejpam-5768	151	12	some	some	PRON
ejpam-5768	151	13	(	(	PUNCT
ejpam-5768	151	14	ζ	ζ	PROPN
ejpam-5768	151	15	,	,	PUNCT
ejpam-5768	151	16	δ	δ	PROPN
ejpam-5768	151	17	)	)	PUNCT
ejpam-5768	151	18	.	.	PUNCT
ejpam-5768	152	1	then	then	ADV
ejpam-5768	152	2	ῡ	ῡ	PROPN
ejpam-5768	152	3	2(w01	2(w01	NUM
ejpam-5768	152	4	)	)	PUNCT
ejpam-5768	152	5	≥	≥	NOUN
ejpam-5768	152	6	ζ	ζ	NOUN
ejpam-5768	152	7	,	,	PUNCT
ejpam-5768	152	8	ῡ	ῡ	PROPN
ejpam-5768	152	9	2(w02	2(w02	NUM
ejpam-5768	152	10	)	)	PUNCT
ejpam-5768	152	11	≥	≥	PROPN
ejpam-5768	152	12	ζ	ζ	NOUN
ejpam-5768	152	13	,	,	PUNCT
ejpam-5768	152	14	υ̂	υ̂	NUM
ejpam-5768	152	15	2(w01	2(w01	NUM
ejpam-5768	152	16	)	)	PUNCT
ejpam-5768	152	17	≤	≤	NOUN
ejpam-5768	152	18	δ	δ	PROPN
ejpam-5768	152	19	and	and	CCONJ
ejpam-5768	152	20	υ̂	υ̂	NUM
ejpam-5768	152	21	2(w02	2(w02	NUM
ejpam-5768	152	22	)	)	PUNCT
ejpam-5768	152	23	≤	≤	NUM
ejpam-5768	152	24	δ	δ	PROPN
ejpam-5768	152	25	.	.	PUNCT
ejpam-5768	153	1	that	that	PRON
ejpam-5768	153	2	υ	υ	PROPN
ejpam-5768	153	3	is	be	AUX
ejpam-5768	153	4	a	a	DET
ejpam-5768	153	5	pf	pf	PROPN
ejpam-5768	153	6	hx	hx	PROPN
ejpam-5768	153	7	-	-	PUNCT
ejpam-5768	153	8	sg	sg	PROPN
ejpam-5768	153	9	implies	imply	VERB
ejpam-5768	153	10	that	that	SCONJ
ejpam-5768	153	11	ῡ	ῡ	PROPN
ejpam-5768	153	12	2(w01w	2(w01w	NUM
ejpam-5768	153	13	−1	−1	NOUN
ejpam-5768	153	14	02	02	NUM
ejpam-5768	153	15	)	)	PUNCT
ejpam-5768	153	16	≥	≥	PROPN
ejpam-5768	153	17	min{ῡ	min{ῡ	NOUN
ejpam-5768	153	18	2(w01	2(w01	NUM
ejpam-5768	153	19	)	)	PUNCT
ejpam-5768	153	20	,	,	PUNCT
ejpam-5768	153	21	ῡ	ῡ	PROPN
ejpam-5768	153	22	2(w02	2(w02	NUM
ejpam-5768	153	23	)	)	PUNCT
ejpam-5768	153	24	}	}	PUNCT
ejpam-5768	153	25	≥	≥	X
ejpam-5768	153	26	ζ	ζ	NOUN
ejpam-5768	153	27	and	and	CCONJ
ejpam-5768	153	28	υ̂	υ̂	NUM
ejpam-5768	154	1	2(w01w	2(w01w	NUM
ejpam-5768	154	2	−1	−1	NOUN
ejpam-5768	154	3	02	02	NUM
ejpam-5768	154	4	)	)	PUNCT
ejpam-5768	154	5	≤	≤	NUM
ejpam-5768	154	6	max{υ̂	max{υ̂	PROPN
ejpam-5768	154	7	2(w01	2(w01	NUM
ejpam-5768	154	8	)	)	PUNCT
ejpam-5768	154	9	,	,	PUNCT
ejpam-5768	154	10	υ̂	υ̂	PROPN
ejpam-5768	154	11	2(w02	2(w02	NUM
ejpam-5768	154	12	)	)	PUNCT
ejpam-5768	154	13	}	}	PUNCT
ejpam-5768	154	14	≤	≤	NUM
ejpam-5768	154	15	δ	δ	PROPN
ejpam-5768	154	16	.	.	PUNCT
ejpam-5768	155	1	hence	hence	ADV
ejpam-5768	155	2	w01w	w01w	ADP
ejpam-5768	155	3	−1	−1	NOUN
ejpam-5768	155	4	02	02	NUM
ejpam-5768	155	5	∈	∈	PROPN
ejpam-5768	155	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	155	7	,	,	PUNCT
ejpam-5768	155	8	δ	δ	PROPN
ejpam-5768	155	9	)	)	PUNCT
ejpam-5768	155	10	and	and	CCONJ
ejpam-5768	155	11	therefore	therefore	ADV
ejpam-5768	155	12	,	,	PUNCT
ejpam-5768	155	13	υ(ζ	υ(ζ	PROPN
ejpam-5768	155	14	,	,	PUNCT
ejpam-5768	155	15	δ	δ	PROPN
ejpam-5768	155	16	)	)	PUNCT
ejpam-5768	155	17	is	be	AUX
ejpam-5768	155	18	an	an	DET
ejpam-5768	155	19	hx	hx	PROPN
ejpam-5768	155	20	-	-	PUNCT
ejpam-5768	155	21	sg	sg	PROPN
ejpam-5768	155	22	.	.	PROPN
ejpam-5768	156	1	3	3	X
ejpam-5768	156	2	.	.	X
ejpam-5768	156	3	pythagorean	pythagorean	PROPN
ejpam-5768	156	4	fuzzy	fuzzy	ADJ
ejpam-5768	156	5	hx	hx	PROPN
ejpam-5768	156	6	-	-	PUNCT
ejpam-5768	156	7	normal	normal	ADJ
ejpam-5768	156	8	subgroups	subgroup	NOUN
ejpam-5768	156	9	in	in	ADP
ejpam-5768	156	10	this	this	DET
ejpam-5768	156	11	section	section	NOUN
ejpam-5768	156	12	,	,	PUNCT
ejpam-5768	156	13	we	we	PRON
ejpam-5768	156	14	study	study	VERB
ejpam-5768	156	15	pf	pf	PROPN
ejpam-5768	156	16	hx	hx	PROPN
ejpam-5768	156	17	-	-	PUNCT
ejpam-5768	156	18	nsgs	nsgs	PROPN
ejpam-5768	156	19	.	.	PUNCT
ejpam-5768	157	1	we	we	PRON
ejpam-5768	157	2	first	first	ADV
ejpam-5768	157	3	introduce	introduce	VERB
ejpam-5768	157	4	the	the	DET
ejpam-5768	157	5	definition	definition	NOUN
ejpam-5768	157	6	of	of	ADP
ejpam-5768	157	7	left	left	ADJ
ejpam-5768	157	8	cosets	coset	NOUN
ejpam-5768	157	9	:	:	PUNCT
ejpam-5768	157	10	definition	definition	NOUN
ejpam-5768	157	11	4	4	NUM
ejpam-5768	157	12	.	.	PUNCT
ejpam-5768	158	1	let	let	VERB
ejpam-5768	158	2	υ	υ	PRON
ejpam-5768	158	3	be	be	AUX
ejpam-5768	158	4	an	an	DET
ejpam-5768	158	5	hx	hx	PROPN
ejpam-5768	158	6	-	-	PUNCT
ejpam-5768	158	7	sg	sg	PROPN
ejpam-5768	158	8	of	of	ADP
ejpam-5768	158	9	w	w	PROPN
ejpam-5768	158	10	.	.	PUNCT
ejpam-5768	159	1	then	then	ADV
ejpam-5768	159	2	a	a	DET
ejpam-5768	159	3	left	left	ADJ
ejpam-5768	159	4	coset	coset	NOUN
ejpam-5768	159	5	of	of	ADP
ejpam-5768	159	6	υ	υ	NOUN
ejpam-5768	159	7	in	in	ADP
ejpam-5768	159	8	w	w	PROPN
ejpam-5768	159	9	is	be	AUX
ejpam-5768	159	10	defined	define	VERB
ejpam-5768	159	11	by	by	ADP
ejpam-5768	159	12	w01υ	w01υ	PROPN
ejpam-5768	159	13	(	(	PUNCT
ejpam-5768	159	14	w02	w02	NOUN
ejpam-5768	159	15	)	)	PUNCT
ejpam-5768	160	1	=	=	SYM
ejpam-5768	160	2	υ	υ	PROPN
ejpam-5768	160	3	(	(	PUNCT
ejpam-5768	160	4	w−1	w−1	PROPN
ejpam-5768	160	5	01	01	NUM
ejpam-5768	160	6	w02	w02	NOUN
ejpam-5768	160	7	)	)	PUNCT
ejpam-5768	160	8	for	for	ADP
ejpam-5768	160	9	all	all	DET
ejpam-5768	160	10	w01	w01	NOUN
ejpam-5768	160	11	,	,	PUNCT
ejpam-5768	160	12	w02	w02	NOUN
ejpam-5768	160	13	∈	∈	PROPN
ejpam-5768	160	14	w	w	PROPN
ejpam-5768	160	15	,	,	PUNCT
ejpam-5768	160	16	that	that	PRON
ejpam-5768	160	17	is	is	ADV
ejpam-5768	160	18	w01ῡ	w01ῡ	PROPN
ejpam-5768	160	19	2(w02	2(w02	NUM
ejpam-5768	160	20	)	)	PUNCT
ejpam-5768	161	1	=	=	SYM
ejpam-5768	161	2	ῡ	ῡ	PROPN
ejpam-5768	161	3	2(w−1	2(w−1	NOUN
ejpam-5768	161	4	01	01	NUM
ejpam-5768	161	5	w02	w02	NOUN
ejpam-5768	161	6	)	)	PUNCT
ejpam-5768	161	7	and	and	CCONJ
ejpam-5768	161	8	w01υ̂	w01υ̂	VERB
ejpam-5768	161	9	2(w02	2(w02	NUM
ejpam-5768	161	10	)	)	PUNCT
ejpam-5768	162	1	=	=	PRON
ejpam-5768	162	2	υ̂	υ̂	NUM
ejpam-5768	163	1	2(w−1	2(w−1	NOUN
ejpam-5768	163	2	01	01	NUM
ejpam-5768	163	3	w02	w02	NOUN
ejpam-5768	163	4	)	)	PUNCT
ejpam-5768	163	5	for	for	ADP
ejpam-5768	163	6	all	all	DET
ejpam-5768	163	7	w01	w01	NOUN
ejpam-5768	163	8	,	,	PUNCT
ejpam-5768	163	9	w02	w02	NOUN
ejpam-5768	163	10	∈	∈	PROPN
ejpam-5768	163	11	w	w	PROPN
ejpam-5768	163	12	.	.	PUNCT
ejpam-5768	164	1	in	in	ADP
ejpam-5768	164	2	order	order	NOUN
ejpam-5768	164	3	to	to	PART
ejpam-5768	164	4	explain	explain	VERB
ejpam-5768	164	5	the	the	DET
ejpam-5768	164	6	above	above	ADJ
ejpam-5768	164	7	definition	definition	NOUN
ejpam-5768	164	8	,	,	PUNCT
ejpam-5768	164	9	we	we	PRON
ejpam-5768	164	10	present	present	VERB
ejpam-5768	164	11	the	the	DET
ejpam-5768	164	12	following	follow	VERB
ejpam-5768	164	13	example	example	NOUN
ejpam-5768	164	14	:	:	PUNCT
ejpam-5768	164	15	example	example	NOUN
ejpam-5768	164	16	3	3	X
ejpam-5768	164	17	.	.	X
ejpam-5768	165	1	consider	consider	VERB
ejpam-5768	165	2	the	the	DET
ejpam-5768	165	3	group	group	NOUN
ejpam-5768	165	4	(	(	PUNCT
ejpam-5768	165	5	z∗	z∗	NOUN
ejpam-5768	165	6	7	7	NUM
ejpam-5768	165	7	,	,	PUNCT
ejpam-5768	165	8	·	·	PUNCT
ejpam-5768	165	9	7	7	NUM
ejpam-5768	165	10	)	)	PUNCT
ejpam-5768	165	11	and	and	CCONJ
ejpam-5768	165	12	the	the	DET
ejpam-5768	165	13	xh	xh	PROPN
ejpam-5768	165	14	-	-	PUNCT
ejpam-5768	165	15	group	group	NOUN
ejpam-5768	165	16	w	w	NOUN
ejpam-5768	165	17	=	=	PUNCT
ejpam-5768	165	18	{	{	PUNCT
ejpam-5768	165	19	e	e	PROPN
ejpam-5768	165	20	,	,	PUNCT
ejpam-5768	165	21	m	m	PROPN
ejpam-5768	165	22	,	,	PUNCT
ejpam-5768	165	23	n	n	CCONJ
ejpam-5768	165	24	}	}	PUNCT
ejpam-5768	165	25	=	=	PRON
ejpam-5768	165	26	{	{	PUNCT
ejpam-5768	165	27	{	{	PUNCT
ejpam-5768	165	28	1	1	NUM
ejpam-5768	165	29	,	,	PUNCT
ejpam-5768	165	30	6	6	NUM
ejpam-5768	165	31	}	}	PUNCT
ejpam-5768	165	32	,	,	PUNCT
ejpam-5768	165	33	{	{	PUNCT
ejpam-5768	165	34	2	2	NUM
ejpam-5768	165	35	,	,	PUNCT
ejpam-5768	165	36	5	5	NUM
ejpam-5768	165	37	}	}	PUNCT
ejpam-5768	165	38	,	,	PUNCT
ejpam-5768	165	39	{	{	PUNCT
ejpam-5768	165	40	3	3	NUM
ejpam-5768	165	41	,	,	PUNCT
ejpam-5768	165	42	4	4	NUM
ejpam-5768	165	43	}	}	PUNCT
ejpam-5768	165	44	}	}	PUNCT
ejpam-5768	165	45	,	,	PUNCT
ejpam-5768	165	46	such	such	ADJ
ejpam-5768	165	47	that	that	SCONJ
ejpam-5768	165	48	∗	∗	NOUN
ejpam-5768	165	49	e	e	NOUN
ejpam-5768	165	50	m	m	VERB
ejpam-5768	165	51	n	n	NUM
ejpam-5768	165	52	e	e	X
ejpam-5768	165	53	e	e	X
ejpam-5768	165	54	m	m	VERB
ejpam-5768	165	55	n	n	PRON
ejpam-5768	165	56	m	m	VERB
ejpam-5768	165	57	m	m	VERB
ejpam-5768	165	58	n	n	ADP
ejpam-5768	165	59	e	e	NOUN
ejpam-5768	165	60	n	n	CCONJ
ejpam-5768	165	61	n	n	NOUN
ejpam-5768	165	62	e	e	VERB
ejpam-5768	165	63	m	m	AUX
ejpam-5768	165	64	let	let	VERB
ejpam-5768	165	65	η	η	PROPN
ejpam-5768	165	66	be	be	AUX
ejpam-5768	165	67	a	a	DET
ejpam-5768	165	68	pythagorean	pythagorean	ADJ
ejpam-5768	165	69	fuzzy	fuzzy	ADJ
ejpam-5768	165	70	set	set	NOUN
ejpam-5768	165	71	,	,	PUNCT
ejpam-5768	165	72	where	where	SCONJ
ejpam-5768	165	73	η̄(1	η̄(1	NOUN
ejpam-5768	165	74	)	)	PUNCT
ejpam-5768	165	75	=	=	SYM
ejpam-5768	165	76	0.6	0.6	NUM
ejpam-5768	165	77	,	,	PUNCT
ejpam-5768	165	78	η̂(1	η̂(1	NOUN
ejpam-5768	165	79	)	)	PUNCT
ejpam-5768	165	80	=	=	NOUN
ejpam-5768	165	81	0.7	0.7	NUM
ejpam-5768	165	82	η̄(2	η̄(2	NOUN
ejpam-5768	165	83	)	)	PUNCT
ejpam-5768	165	84	=	=	SYM
ejpam-5768	165	85	0.5	0.5	NUM
ejpam-5768	165	86	,	,	PUNCT
ejpam-5768	165	87	η̂(2	η̂(2	NOUN
ejpam-5768	165	88	)	)	PUNCT
ejpam-5768	165	89	=	=	PUNCT
ejpam-5768	165	90	0.6	0.6	NUM
ejpam-5768	165	91	η̄(3	η̄(3	NUM
ejpam-5768	165	92	)	)	PUNCT
ejpam-5768	165	93	=	=	PUNCT
ejpam-5768	165	94	0.3	0.3	NUM
ejpam-5768	165	95	,	,	PUNCT
ejpam-5768	165	96	η̂(3	η̂(3	NOUN
ejpam-5768	165	97	)	)	PUNCT
ejpam-5768	165	98	=	=	NOUN
ejpam-5768	165	99	0.5	0.5	NUM
ejpam-5768	165	100	η̄(4	η̄(4	NUM
ejpam-5768	165	101	)	)	PUNCT
ejpam-5768	165	102	=	=	SYM
ejpam-5768	165	103	0.5	0.5	NUM
ejpam-5768	165	104	,	,	PUNCT
ejpam-5768	165	105	η̂(4	η̂(4	NOUN
ejpam-5768	165	106	)	)	PUNCT
ejpam-5768	165	107	=	=	SYM
ejpam-5768	165	108	0.4	0.4	NUM
ejpam-5768	165	109	η̄(5	η̄(5	NUM
ejpam-5768	165	110	)	)	PUNCT
ejpam-5768	165	111	=	=	SYM
ejpam-5768	165	112	0.4	0.4	NUM
ejpam-5768	165	113	,	,	PUNCT
ejpam-5768	165	114	η̂(5	η̂(5	NUM
ejpam-5768	165	115	)	)	PUNCT
ejpam-5768	165	116	=	=	PUNCT
ejpam-5768	165	117	0.4	0.4	NUM
ejpam-5768	165	118	η̄(6	η̄(6	NUM
ejpam-5768	165	119	)	)	PUNCT
ejpam-5768	165	120	=	=	SYM
ejpam-5768	165	121	0.5	0.5	NUM
ejpam-5768	165	122	,	,	PUNCT
ejpam-5768	165	123	η̂(6	η̂(6	NOUN
ejpam-5768	165	124	)	)	PUNCT
ejpam-5768	165	125	=	=	SYM
ejpam-5768	165	126	0.5	0.5	NUM
ejpam-5768	165	127	let	let	VERB
ejpam-5768	165	128	ῡ	ῡ	PROPN
ejpam-5768	165	129	(	(	PUNCT
ejpam-5768	165	130	n	n	CCONJ
ejpam-5768	165	131	)	)	PUNCT
ejpam-5768	165	132	=	=	SYM
ejpam-5768	165	133	max{η̄(n	max{η̄(n	NOUN
ejpam-5768	165	134	)	)	PUNCT
ejpam-5768	165	135	:	:	PUNCT
ejpam-5768	166	1	n	n	X
ejpam-5768	166	2	∈	∈	PROPN
ejpam-5768	166	3	n	n	CCONJ
ejpam-5768	166	4	⊆	⊆	NUM
ejpam-5768	166	5	w	w	NOUN
ejpam-5768	166	6	}	}	PUNCT
ejpam-5768	166	7	and	and	CCONJ
ejpam-5768	166	8	υ̂	υ̂	NUM
ejpam-5768	166	9	(	(	PUNCT
ejpam-5768	166	10	n	n	CCONJ
ejpam-5768	166	11	)	)	PUNCT
ejpam-5768	166	12	=	=	SYM
ejpam-5768	166	13	min{η̂(n	min{η̂(n	PROPN
ejpam-5768	166	14	)	)	PUNCT
ejpam-5768	166	15	:	:	PUNCT
ejpam-5768	167	1	n	n	X
ejpam-5768	167	2	∈	∈	PROPN
ejpam-5768	167	3	n	n	CCONJ
ejpam-5768	167	4	⊆	⊆	NUM
ejpam-5768	167	5	w	w	NOUN
ejpam-5768	167	6	}	}	PUNCT
ejpam-5768	167	7	.	.	PUNCT
ejpam-5768	168	1	thus	thus	ADV
ejpam-5768	168	2	ῡ	ῡ	PROPN
ejpam-5768	168	3	(	(	PUNCT
ejpam-5768	168	4	e	e	NOUN
ejpam-5768	168	5	)	)	PUNCT
ejpam-5768	168	6	=	=	SYM
ejpam-5768	168	7	max{η̄(1	max{η̄(1	NOUN
ejpam-5768	168	8	)	)	PUNCT
ejpam-5768	168	9	,	,	PUNCT
ejpam-5768	168	10	η̄(6	η̄(6	NUM
ejpam-5768	168	11	)	)	PUNCT
ejpam-5768	168	12	}	}	PUNCT
ejpam-5768	168	13	=	=	PUNCT
ejpam-5768	168	14	0.6	0.6	NUM
ejpam-5768	168	15	a.	a.	NOUN
ejpam-5768	168	16	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	168	17	/	/	SYM
ejpam-5768	168	18	eur	eur	PROPN
ejpam-5768	168	19	.	.	PUNCT
ejpam-5768	169	1	j.	j.	PROPN
ejpam-5768	169	2	pure	pure	PROPN
ejpam-5768	169	3	appl	appl	PROPN
ejpam-5768	169	4	.	.	PROPN
ejpam-5768	169	5	math	math	PROPN
ejpam-5768	169	6	,	,	PUNCT
ejpam-5768	169	7	18	18	NUM
ejpam-5768	169	8	(	(	PUNCT
ejpam-5768	169	9	2	2	NUM
ejpam-5768	169	10	)	)	PUNCT
ejpam-5768	169	11	(	(	PUNCT
ejpam-5768	169	12	2025	2025	NUM
ejpam-5768	169	13	)	)	PUNCT
ejpam-5768	169	14	,	,	PUNCT
ejpam-5768	169	15	5768	5768	NUM
ejpam-5768	169	16	7	7	NUM
ejpam-5768	169	17	of	of	ADP
ejpam-5768	169	18	17	17	NUM
ejpam-5768	169	19	υ̂	υ̂	NUM
ejpam-5768	169	20	(	(	PUNCT
ejpam-5768	169	21	e	e	NOUN
ejpam-5768	169	22	)	)	PUNCT
ejpam-5768	169	23	=	=	SYM
ejpam-5768	169	24	min{η̂(1	min{η̂(1	NOUN
ejpam-5768	169	25	)	)	PUNCT
ejpam-5768	169	26	,	,	PUNCT
ejpam-5768	169	27	η̂(6	η̂(6	NOUN
ejpam-5768	169	28	)	)	PUNCT
ejpam-5768	169	29	}	}	PUNCT
ejpam-5768	169	30	=	=	SYM
ejpam-5768	169	31	0.5	0.5	NUM
ejpam-5768	169	32	ῡ	ῡ	NOUN
ejpam-5768	169	33	(	(	PUNCT
ejpam-5768	169	34	m	m	PROPN
ejpam-5768	169	35	)	)	PUNCT
ejpam-5768	169	36	=	=	SYM
ejpam-5768	169	37	max{η̄(2	max{η̄(2	PROPN
ejpam-5768	169	38	)	)	PUNCT
ejpam-5768	169	39	,	,	PUNCT
ejpam-5768	169	40	η̄(5	η̄(5	NUM
ejpam-5768	169	41	)	)	PUNCT
ejpam-5768	169	42	}	}	PUNCT
ejpam-5768	169	43	=	=	SYM
ejpam-5768	169	44	0.5	0.5	NUM
ejpam-5768	169	45	υ̂	υ̂	NUM
ejpam-5768	169	46	(	(	PUNCT
ejpam-5768	169	47	m	m	NOUN
ejpam-5768	169	48	)	)	PUNCT
ejpam-5768	169	49	=	=	SYM
ejpam-5768	169	50	min{η̂(2	min{η̂(2	NOUN
ejpam-5768	169	51	)	)	PUNCT
ejpam-5768	169	52	,	,	PUNCT
ejpam-5768	169	53	η̂(5	η̂(5	NOUN
ejpam-5768	169	54	)	)	PUNCT
ejpam-5768	169	55	}	}	PUNCT
ejpam-5768	169	56	=	=	SYM
ejpam-5768	169	57	0.4	0.4	NUM
ejpam-5768	169	58	ῡ	ῡ	NOUN
ejpam-5768	169	59	(	(	PUNCT
ejpam-5768	169	60	n	n	CCONJ
ejpam-5768	169	61	)	)	PUNCT
ejpam-5768	169	62	=	=	SYM
ejpam-5768	169	63	max{η̄(3	max{η̄(3	NOUN
ejpam-5768	169	64	)	)	PUNCT
ejpam-5768	169	65	,	,	PUNCT
ejpam-5768	169	66	η̄(4	η̄(4	NUM
ejpam-5768	169	67	)	)	PUNCT
ejpam-5768	169	68	}	}	PUNCT
ejpam-5768	169	69	=	=	SYM
ejpam-5768	169	70	0.5	0.5	NUM
ejpam-5768	169	71	υ̂	υ̂	NUM
ejpam-5768	169	72	(	(	PUNCT
ejpam-5768	169	73	n	n	CCONJ
ejpam-5768	169	74	)	)	PUNCT
ejpam-5768	169	75	=	=	SYM
ejpam-5768	169	76	min{η̂(3	min{η̂(3	NOUN
ejpam-5768	169	77	)	)	PUNCT
ejpam-5768	169	78	,	,	PUNCT
ejpam-5768	169	79	η̂(4	η̂(4	NOUN
ejpam-5768	169	80	)	)	PUNCT
ejpam-5768	169	81	}	}	PUNCT
ejpam-5768	170	1	=	=	SYM
ejpam-5768	170	2	0.4	0.4	NUM
ejpam-5768	170	3	we	we	PRON
ejpam-5768	170	4	find	find	VERB
ejpam-5768	170	5	the	the	DET
ejpam-5768	170	6	left	left	ADJ
ejpam-5768	170	7	coset	coset	NOUN
ejpam-5768	170	8	mυ	mυ	NOUN
ejpam-5768	170	9	(	(	PUNCT
ejpam-5768	170	10	n	n	CCONJ
ejpam-5768	170	11	)	)	PUNCT
ejpam-5768	170	12	.	.	PUNCT
ejpam-5768	171	1	by	by	ADP
ejpam-5768	171	2	definition	definition	NOUN
ejpam-5768	171	3	,	,	PUNCT
ejpam-5768	171	4	mυ	mυ	INTJ
ejpam-5768	171	5	(	(	PUNCT
ejpam-5768	171	6	n	n	CCONJ
ejpam-5768	171	7	)	)	PUNCT
ejpam-5768	171	8	=	=	SYM
ejpam-5768	171	9	υ	υ	PROPN
ejpam-5768	171	10	(	(	PUNCT
ejpam-5768	171	11	m−1n	m−1n	NOUN
ejpam-5768	171	12	)	)	PUNCT
ejpam-5768	171	13	.	.	PUNCT
ejpam-5768	172	1	that	that	PRON
ejpam-5768	172	2	is	be	AUX
ejpam-5768	172	3	:	:	PUNCT
ejpam-5768	172	4	mῡ	mῡ	NOUN
ejpam-5768	172	5	2(n	2(n	NUM
ejpam-5768	172	6	)	)	PUNCT
ejpam-5768	172	7	=	=	SYM
ejpam-5768	172	8	ῡ	ῡ	PROPN
ejpam-5768	172	9	2(m−1n	2(m−1n	NOUN
ejpam-5768	172	10	)	)	PUNCT
ejpam-5768	173	1	=	=	SYM
ejpam-5768	173	2	ῡ	ῡ	PROPN
ejpam-5768	173	3	2(nn	2(nn	NUM
ejpam-5768	173	4	)	)	PUNCT
ejpam-5768	174	1	=	=	SYM
ejpam-5768	174	2	ῡ	ῡ	PROPN
ejpam-5768	174	3	2(m	2(m	NUM
ejpam-5768	174	4	)	)	PUNCT
ejpam-5768	174	5	=	=	SYM
ejpam-5768	175	1	0.36	0.36	NUM
ejpam-5768	175	2	mυ̂	mυ̂	NUM
ejpam-5768	175	3	2(n	2(n	NUM
ejpam-5768	175	4	)	)	PUNCT
ejpam-5768	175	5	=	=	SYM
ejpam-5768	175	6	υ̂	υ̂	NUM
ejpam-5768	175	7	2(m−1n	2(m−1n	NUM
ejpam-5768	175	8	)	)	PUNCT
ejpam-5768	175	9	=	=	SYM
ejpam-5768	176	1	υ̂	υ̂	NUM
ejpam-5768	176	2	2(nn	2(nn	NUM
ejpam-5768	176	3	)	)	PUNCT
ejpam-5768	176	4	=	=	SYM
ejpam-5768	176	5	υ̂	υ̂	NUM
ejpam-5768	176	6	2(m	2(m	NUM
ejpam-5768	176	7	)	)	PUNCT
ejpam-5768	177	1	=	=	SYM
ejpam-5768	177	2	0.16	0.16	NUM
ejpam-5768	177	3	.	.	PUNCT
ejpam-5768	178	1	similarly	similarly	ADV
ejpam-5768	178	2	,	,	PUNCT
ejpam-5768	178	3	we	we	PRON
ejpam-5768	178	4	can	can	AUX
ejpam-5768	178	5	compute	compute	VERB
ejpam-5768	178	6	any	any	DET
ejpam-5768	178	7	left	left	ADJ
ejpam-5768	178	8	coset	coset	NOUN
ejpam-5768	178	9	.	.	PUNCT
ejpam-5768	179	1	now	now	ADV
ejpam-5768	179	2	,	,	PUNCT
ejpam-5768	179	3	we	we	PRON
ejpam-5768	179	4	are	be	AUX
ejpam-5768	179	5	ready	ready	ADJ
ejpam-5768	179	6	to	to	PART
ejpam-5768	179	7	define	define	VERB
ejpam-5768	179	8	a	a	DET
ejpam-5768	179	9	pf	pf	PROPN
ejpam-5768	179	10	hx	hx	PROPN
ejpam-5768	179	11	-	-	PUNCT
ejpam-5768	179	12	nsg	nsg	PROPN
ejpam-5768	179	13	.	.	PUNCT
ejpam-5768	180	1	definition	definition	NOUN
ejpam-5768	180	2	5	5	NUM
ejpam-5768	180	3	.	.	PUNCT
ejpam-5768	181	1	let	let	VERB
ejpam-5768	181	2	g	g	PRON
ejpam-5768	181	3	be	be	AUX
ejpam-5768	181	4	a	a	DET
ejpam-5768	181	5	group	group	NOUN
ejpam-5768	181	6	,	,	PUNCT
ejpam-5768	181	7	w	w	ADP
ejpam-5768	181	8	⊆	⊆	NUM
ejpam-5768	181	9	2g−{ϕ	2g−{ϕ	NUM
ejpam-5768	181	10	}	}	PUNCT
ejpam-5768	181	11	be	be	AUX
ejpam-5768	181	12	an	an	DET
ejpam-5768	181	13	hx	hx	NOUN
ejpam-5768	181	14	-	-	PUNCT
ejpam-5768	181	15	group	group	NOUN
ejpam-5768	181	16	of	of	ADP
ejpam-5768	181	17	g	g	PROPN
ejpam-5768	181	18	and	and	CCONJ
ejpam-5768	181	19	υ	υ	NOUN
ejpam-5768	182	1	=	=	PRON
ejpam-5768	182	2	{	{	PUNCT
ejpam-5768	182	3	(	(	PUNCT
ejpam-5768	182	4	w	w	NOUN
ejpam-5768	182	5	;	;	PUNCT
ejpam-5768	182	6	ῡ	ῡ	PROPN
ejpam-5768	182	7	(	(	PUNCT
ejpam-5768	182	8	w	w	PROPN
ejpam-5768	182	9	)	)	PUNCT
ejpam-5768	182	10	,	,	PUNCT
ejpam-5768	182	11	υ̂	υ̂	NUM
ejpam-5768	182	12	(	(	PUNCT
ejpam-5768	182	13	w	w	NOUN
ejpam-5768	182	14	)	)	PUNCT
ejpam-5768	182	15	)	)	PUNCT
ejpam-5768	182	16	:	:	PUNCT
ejpam-5768	182	17	w	w	X
ejpam-5768	182	18	∈	∈	PROPN
ejpam-5768	182	19	w	w	AUX
ejpam-5768	182	20	}	}	PUNCT
ejpam-5768	182	21	be	be	AUX
ejpam-5768	182	22	a	a	DET
ejpam-5768	182	23	pf	pf	PROPN
ejpam-5768	182	24	hx	hx	PROPN
ejpam-5768	182	25	-	-	PUNCT
ejpam-5768	182	26	sg	sg	PROPN
ejpam-5768	182	27	of	of	ADP
ejpam-5768	182	28	w	w	PROPN
ejpam-5768	182	29	.	.	PUNCT
ejpam-5768	183	1	then	then	ADV
ejpam-5768	183	2	υ	υ	PROPN
ejpam-5768	183	3	is	be	AUX
ejpam-5768	183	4	called	call	VERB
ejpam-5768	183	5	a	a	DET
ejpam-5768	183	6	pf	pf	PROPN
ejpam-5768	183	7	hx	hx	PROPN
ejpam-5768	183	8	-	-	PUNCT
ejpam-5768	183	9	nsg	nsg	PROPN
ejpam-5768	183	10	of	of	ADP
ejpam-5768	183	11	w	w	PROPN
ejpam-5768	183	12	if	if	SCONJ
ejpam-5768	183	13	:	:	PUNCT
ejpam-5768	183	14	ῡ	ῡ	PROPN
ejpam-5768	183	15	2(w01w02	2(w01w02	NUM
ejpam-5768	183	16	)	)	PUNCT
ejpam-5768	184	1	=	=	SYM
ejpam-5768	184	2	ῡ	ῡ	PROPN
ejpam-5768	184	3	2(w02w01	2(w02w01	NUM
ejpam-5768	184	4	)	)	PUNCT
ejpam-5768	184	5	and	and	CCONJ
ejpam-5768	184	6	υ̂	υ̂	NUM
ejpam-5768	184	7	2(w01w02	2(w01w02	X
ejpam-5768	184	8	)	)	PUNCT
ejpam-5768	184	9	=	=	SYM
ejpam-5768	184	10	υ̂	υ̂	PROPN
ejpam-5768	184	11	2(w02w01	2(w02w01	NUM
ejpam-5768	184	12	)	)	PUNCT
ejpam-5768	184	13	.	.	PUNCT
ejpam-5768	185	1	alternatively	alternatively	ADV
ejpam-5768	185	2	,	,	PUNCT
ejpam-5768	185	3	υ	υ	PROPN
ejpam-5768	185	4	is	be	AUX
ejpam-5768	185	5	a	a	DET
ejpam-5768	185	6	pf	pf	PROPN
ejpam-5768	185	7	hx	hx	PROPN
ejpam-5768	185	8	-	-	PUNCT
ejpam-5768	185	9	nsg	nsg	PROPN
ejpam-5768	185	10	of	of	ADP
ejpam-5768	185	11	w	w	PROPN
ejpam-5768	185	12	if	if	SCONJ
ejpam-5768	185	13	w01υ	w01υ	PROPN
ejpam-5768	185	14	(	(	PUNCT
ejpam-5768	185	15	w02	w02	NOUN
ejpam-5768	185	16	)	)	PUNCT
ejpam-5768	186	1	=	=	SYM
ejpam-5768	186	2	υ	υ	PROPN
ejpam-5768	186	3	(	(	PUNCT
ejpam-5768	186	4	w02)w01	w02)w01	PROPN
ejpam-5768	186	5	for	for	ADP
ejpam-5768	186	6	all	all	DET
ejpam-5768	186	7	w01	w01	NOUN
ejpam-5768	186	8	,	,	PUNCT
ejpam-5768	186	9	w02	w02	NOUN
ejpam-5768	186	10	∈	∈	PROPN
ejpam-5768	186	11	w	w	PROPN
ejpam-5768	186	12	.	.	PUNCT
ejpam-5768	186	13	example	example	NOUN
ejpam-5768	187	1	4	4	X
ejpam-5768	187	2	.	.	X
ejpam-5768	187	3	consider	consider	VERB
ejpam-5768	187	4	w	w	PROPN
ejpam-5768	187	5	,	,	PUNCT
ejpam-5768	187	6	η	η	PROPN
ejpam-5768	187	7	and	and	CCONJ
ejpam-5768	187	8	υ	υ	NOUN
ejpam-5768	187	9	as	as	ADP
ejpam-5768	187	10	in	in	ADP
ejpam-5768	187	11	example	example	NOUN
ejpam-5768	187	12	3	3	X
ejpam-5768	187	13	.	.	PUNCT
ejpam-5768	188	1	then	then	ADV
ejpam-5768	188	2	:	:	PUNCT
ejpam-5768	188	3	ῡ	ῡ	PROPN
ejpam-5768	188	4	2(em	2(em	PROPN
ejpam-5768	188	5	)	)	PUNCT
ejpam-5768	189	1	=	=	SYM
ejpam-5768	189	2	ῡ	ῡ	PROPN
ejpam-5768	189	3	2(me	2(me	NUM
ejpam-5768	189	4	)	)	PUNCT
ejpam-5768	189	5	=	=	SYM
ejpam-5768	190	1	0.25	0.25	NUM
ejpam-5768	190	2	υ̂	υ̂	NUM
ejpam-5768	190	3	2(em	2(em	NUM
ejpam-5768	190	4	)	)	PUNCT
ejpam-5768	191	1	=	=	SYM
ejpam-5768	191	2	υ̂	υ̂	NUM
ejpam-5768	191	3	2(me	2(me	NOUN
ejpam-5768	191	4	)	)	PUNCT
ejpam-5768	191	5	=	=	PUNCT
ejpam-5768	192	1	0.16	0.16	NUM
ejpam-5768	192	2	ῡ	ῡ	PROPN
ejpam-5768	192	3	2(en	2(en	NUM
ejpam-5768	192	4	)	)	PUNCT
ejpam-5768	193	1	=	=	SYM
ejpam-5768	193	2	ῡ	ῡ	PROPN
ejpam-5768	193	3	2(ne	2(ne	PROPN
ejpam-5768	193	4	)	)	PUNCT
ejpam-5768	194	1	=	=	SYM
ejpam-5768	195	1	0.25	0.25	NUM
ejpam-5768	195	2	υ̂	υ̂	NUM
ejpam-5768	195	3	2(en	2(en	NUM
ejpam-5768	195	4	)	)	PUNCT
ejpam-5768	196	1	=	=	SYM
ejpam-5768	196	2	υ̂	υ̂	NUM
ejpam-5768	196	3	2(ne	2(ne	NUM
ejpam-5768	196	4	)	)	PUNCT
ejpam-5768	196	5	=	=	PUNCT
ejpam-5768	197	1	0.16	0.16	NUM
ejpam-5768	197	2	ῡ	ῡ	PROPN
ejpam-5768	197	3	2(nm	2(nm	NUM
ejpam-5768	197	4	)	)	PUNCT
ejpam-5768	197	5	=	=	SYM
ejpam-5768	198	1	ῡ	ῡ	PROPN
ejpam-5768	198	2	2(mn	2(mn	NOUN
ejpam-5768	198	3	)	)	PUNCT
ejpam-5768	199	1	=	=	SYM
ejpam-5768	199	2	0.36	0.36	NUM
ejpam-5768	199	3	υ̂	υ̂	NUM
ejpam-5768	199	4	2(nm	2(nm	NUM
ejpam-5768	199	5	)	)	PUNCT
ejpam-5768	199	6	=	=	PRON
ejpam-5768	200	1	υ̂	υ̂	VERB
ejpam-5768	200	2	2(mn	2(mn	NOUN
ejpam-5768	200	3	)	)	PUNCT
ejpam-5768	200	4	=	=	SYM
ejpam-5768	201	1	0.25	0.25	NUM
ejpam-5768	201	2	thus	thus	ADV
ejpam-5768	201	3	υ	υ	NOUN
ejpam-5768	201	4	is	be	AUX
ejpam-5768	201	5	a	a	DET
ejpam-5768	201	6	pf	pf	PROPN
ejpam-5768	201	7	hxn	hxn	NOUN
ejpam-5768	201	8	-	-	PUNCT
ejpam-5768	201	9	sg	sg	NOUN
ejpam-5768	201	10	.	.	PUNCT
ejpam-5768	202	1	turning	turn	VERB
ejpam-5768	202	2	to	to	ADP
ejpam-5768	202	3	pf	pf	PROPN
ejpam-5768	202	4	hx	hx	PROPN
ejpam-5768	202	5	-	-	PUNCT
ejpam-5768	202	6	nsgs	nsg	NOUN
ejpam-5768	202	7	,	,	PUNCT
ejpam-5768	202	8	we	we	PRON
ejpam-5768	202	9	can	can	AUX
ejpam-5768	202	10	say	say	VERB
ejpam-5768	202	11	more	more	ADJ
ejpam-5768	202	12	about	about	ADP
ejpam-5768	202	13	the	the	DET
ejpam-5768	202	14	properties	property	NOUN
ejpam-5768	202	15	of	of	ADP
ejpam-5768	202	16	them	they	PRON
ejpam-5768	202	17	.	.	PUNCT
ejpam-5768	203	1	the	the	DET
ejpam-5768	203	2	following	follow	VERB
ejpam-5768	203	3	theorems	theorem	NOUN
ejpam-5768	203	4	described	describe	VERB
ejpam-5768	203	5	these	these	DET
ejpam-5768	203	6	properties	property	NOUN
ejpam-5768	203	7	:	:	PUNCT
ejpam-5768	203	8	theorem	theorem	NOUN
ejpam-5768	203	9	5	5	NUM
ejpam-5768	203	10	.	.	PUNCT
ejpam-5768	204	1	let	let	VERB
ejpam-5768	204	2	w	w	NOUN
ejpam-5768	204	3	be	be	AUX
ejpam-5768	204	4	an	an	DET
ejpam-5768	204	5	hx	hx	NOUN
ejpam-5768	204	6	-	-	PUNCT
ejpam-5768	204	7	group	group	NOUN
ejpam-5768	204	8	and	and	CCONJ
ejpam-5768	204	9	υ	υ	NOUN
ejpam-5768	204	10	be	be	AUX
ejpam-5768	204	11	a	a	DET
ejpam-5768	204	12	pf	pf	PROPN
ejpam-5768	204	13	hx	hx	PROPN
ejpam-5768	204	14	-	-	PUNCT
ejpam-5768	204	15	sg	sg	PROPN
ejpam-5768	204	16	of	of	ADP
ejpam-5768	204	17	w	w	PROPN
ejpam-5768	204	18	.	.	PUNCT
ejpam-5768	205	1	then	then	ADV
ejpam-5768	205	2	υ	υ	PROPN
ejpam-5768	205	3	is	be	AUX
ejpam-5768	205	4	a	a	DET
ejpam-5768	205	5	pf	pf	PROPN
ejpam-5768	205	6	hx	hx	PROPN
ejpam-5768	205	7	-	-	PUNCT
ejpam-5768	205	8	nsg	nsg	PROPN
ejpam-5768	205	9	of	of	ADP
ejpam-5768	205	10	w	w	PROPN
ejpam-5768	205	11	if	if	SCONJ
ejpam-5768	206	1	and	and	CCONJ
ejpam-5768	206	2	only	only	ADV
ejpam-5768	206	3	if	if	SCONJ
ejpam-5768	206	4	ῡ	ῡ	PROPN
ejpam-5768	206	5	2(w−1	2(w−1	NOUN
ejpam-5768	206	6	01	01	NUM
ejpam-5768	206	7	w02w01	w02w01	NOUN
ejpam-5768	206	8	)	)	PUNCT
ejpam-5768	206	9	=	=	SYM
ejpam-5768	206	10	ῡ	ῡ	PROPN
ejpam-5768	206	11	2(w02	2(w02	NUM
ejpam-5768	206	12	)	)	PUNCT
ejpam-5768	206	13	,	,	PUNCT
ejpam-5768	206	14	υ̂	υ̂	PROPN
ejpam-5768	206	15	2(w−1	2(w−1	NOUN
ejpam-5768	206	16	01	01	NUM
ejpam-5768	206	17	w02w01	w02w01	NOUN
ejpam-5768	206	18	)	)	PUNCT
ejpam-5768	206	19	=	=	PRON
ejpam-5768	206	20	υ̂	υ̂	PROPN
ejpam-5768	206	21	2(w02	2(w02	NUM
ejpam-5768	206	22	)	)	PUNCT
ejpam-5768	206	23	.	.	PUNCT
ejpam-5768	207	1	(	(	PUNCT
ejpam-5768	207	2	2	2	X
ejpam-5768	207	3	)	)	PUNCT
ejpam-5768	207	4	proof	proof	NOUN
ejpam-5768	207	5	.	.	PUNCT
ejpam-5768	208	1	if	if	SCONJ
ejpam-5768	208	2	υ	υ	PROPN
ejpam-5768	208	3	is	be	AUX
ejpam-5768	208	4	a	a	DET
ejpam-5768	208	5	pf	pf	PROPN
ejpam-5768	208	6	hx	hx	PROPN
ejpam-5768	208	7	-	-	PUNCT
ejpam-5768	208	8	nsg	nsg	PROPN
ejpam-5768	208	9	of	of	ADP
ejpam-5768	208	10	w	w	PROPN
ejpam-5768	208	11	,	,	PUNCT
ejpam-5768	208	12	then	then	ADV
ejpam-5768	208	13	ῡ	ῡ	PROPN
ejpam-5768	208	14	2(w−1	2(w−1	NOUN
ejpam-5768	208	15	01	01	NUM
ejpam-5768	208	16	w02w01	w02w01	NOUN
ejpam-5768	208	17	)	)	PUNCT
ejpam-5768	208	18	=	=	SYM
ejpam-5768	209	1	ῡ	ῡ	NOUN
ejpam-5768	210	1	2((w−1	2((w−1	NUM
ejpam-5768	210	2	01	01	NUM
ejpam-5768	211	1	w02)w01	w02)w01	NUM
ejpam-5768	211	2	)	)	PUNCT
ejpam-5768	211	3	a.	a.	NOUN
ejpam-5768	211	4	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	211	5	/	/	SYM
ejpam-5768	211	6	eur	eur	PROPN
ejpam-5768	211	7	.	.	PUNCT
ejpam-5768	212	1	j.	j.	PROPN
ejpam-5768	212	2	pure	pure	PROPN
ejpam-5768	212	3	appl	appl	PROPN
ejpam-5768	212	4	.	.	PROPN
ejpam-5768	212	5	math	math	PROPN
ejpam-5768	212	6	,	,	PUNCT
ejpam-5768	212	7	18	18	NUM
ejpam-5768	212	8	(	(	PUNCT
ejpam-5768	212	9	2	2	NUM
ejpam-5768	212	10	)	)	PUNCT
ejpam-5768	212	11	(	(	PUNCT
ejpam-5768	212	12	2025	2025	NUM
ejpam-5768	212	13	)	)	PUNCT
ejpam-5768	212	14	,	,	PUNCT
ejpam-5768	212	15	5768	5768	NUM
ejpam-5768	212	16	8	8	NUM
ejpam-5768	212	17	of	of	ADP
ejpam-5768	212	18	17	17	NUM
ejpam-5768	212	19	=	=	SYM
ejpam-5768	212	20	ῡ	ῡ	PROPN
ejpam-5768	212	21	2((w02w	2((w02w	NUM
ejpam-5768	212	22	−1	−1	NOUN
ejpam-5768	212	23	01	01	NUM
ejpam-5768	212	24	)	)	PUNCT
ejpam-5768	212	25	w01	w01	NOUN
ejpam-5768	212	26	)	)	PUNCT
ejpam-5768	212	27	=	=	SYM
ejpam-5768	213	1	ῡ	ῡ	ADJ
ejpam-5768	213	2	2(w02(w	2(w02(w	NUM
ejpam-5768	213	3	−1	−1	NOUN
ejpam-5768	213	4	01	01	NUM
ejpam-5768	213	5	w01	w01	NOUN
ejpam-5768	213	6	)	)	PUNCT
ejpam-5768	213	7	)	)	PUNCT
ejpam-5768	214	1	=	=	SYM
ejpam-5768	214	2	ῡ	ῡ	PROPN
ejpam-5768	214	3	2(w02	2(w02	NUM
ejpam-5768	214	4	)	)	PUNCT
ejpam-5768	214	5	and	and	CCONJ
ejpam-5768	214	6	υ̂	υ̂	NUM
ejpam-5768	214	7	2(w−1	2(w−1	NOUN
ejpam-5768	214	8	01	01	NUM
ejpam-5768	214	9	w02w01	w02w01	NOUN
ejpam-5768	214	10	)	)	PUNCT
ejpam-5768	214	11	=	=	PUNCT
ejpam-5768	214	12	υ̂	υ̂	NUM
ejpam-5768	215	1	2((w−1	2((w−1	NUM
ejpam-5768	215	2	01	01	NUM
ejpam-5768	215	3	w02)w01	w02)w01	NUM
ejpam-5768	215	4	)	)	PUNCT
ejpam-5768	215	5	=	=	PRON
ejpam-5768	216	1	υ̂	υ̂	NUM
ejpam-5768	216	2	2((w02w	2((w02w	NUM
ejpam-5768	216	3	−1	−1	NOUN
ejpam-5768	216	4	01	01	NUM
ejpam-5768	216	5	)	)	PUNCT
ejpam-5768	216	6	w01	w01	NOUN
ejpam-5768	216	7	)	)	PUNCT
ejpam-5768	216	8	=	=	SYM
ejpam-5768	217	1	υ̂	υ̂	PROPN
ejpam-5768	217	2	2(w02(w	2(w02(w	NUM
ejpam-5768	217	3	−1	−1	NOUN
ejpam-5768	217	4	01	01	NUM
ejpam-5768	217	5	w01	w01	NOUN
ejpam-5768	217	6	)	)	PUNCT
ejpam-5768	217	7	)	)	PUNCT
ejpam-5768	218	1	=	=	PRON
ejpam-5768	218	2	υ̂	υ̂	PROPN
ejpam-5768	218	3	2(w02	2(w02	NUM
ejpam-5768	218	4	)	)	PUNCT
ejpam-5768	218	5	conversely	conversely	ADV
ejpam-5768	218	6	,	,	PUNCT
ejpam-5768	218	7	if	if	SCONJ
ejpam-5768	218	8	(	(	PUNCT
ejpam-5768	218	9	2	2	X
ejpam-5768	218	10	)	)	PUNCT
ejpam-5768	218	11	holds	hold	VERB
ejpam-5768	218	12	,	,	PUNCT
ejpam-5768	218	13	then	then	ADV
ejpam-5768	218	14	ῡ	ῡ	PROPN
ejpam-5768	218	15	2(w01w02	2(w01w02	NUM
ejpam-5768	218	16	)	)	PUNCT
ejpam-5768	219	1	=	=	SYM
ejpam-5768	219	2	w−1	w−1	PROPN
ejpam-5768	219	3	01	01	NUM
ejpam-5768	219	4	ῡ	ῡ	PROPN
ejpam-5768	219	5	2(w02	2(w02	NUM
ejpam-5768	219	6	)	)	PUNCT
ejpam-5768	220	1	=	=	SYM
ejpam-5768	220	2	w−1	w−1	PROPN
ejpam-5768	220	3	01	01	NUM
ejpam-5768	221	1	ῡ	ῡ	PROPN
ejpam-5768	221	2	2(w−1	2(w−1	PROPN
ejpam-5768	221	3	01	01	NUM
ejpam-5768	221	4	w02w01	w02w01	NOUN
ejpam-5768	221	5	)	)	PUNCT
ejpam-5768	221	6	=	=	SYM
ejpam-5768	222	1	ῡ	ῡ	PROPN
ejpam-5768	222	2	2(w01w	2(w01w	NUM
ejpam-5768	222	3	−1	−1	NOUN
ejpam-5768	222	4	01	01	NUM
ejpam-5768	222	5	w02w01	w02w01	NOUN
ejpam-5768	222	6	)	)	PUNCT
ejpam-5768	223	1	=	=	SYM
ejpam-5768	223	2	ῡ	ῡ	PROPN
ejpam-5768	223	3	2(w02w01	2(w02w01	NUM
ejpam-5768	223	4	)	)	PUNCT
ejpam-5768	223	5	and	and	CCONJ
ejpam-5768	223	6	υ̂	υ̂	NUM
ejpam-5768	223	7	2(w01w02	2(w01w02	NUM
ejpam-5768	223	8	)	)	PUNCT
ejpam-5768	224	1	=	=	SYM
ejpam-5768	225	1	w−1	w−1	PROPN
ejpam-5768	225	2	01	01	NUM
ejpam-5768	225	3	υ̂	υ̂	NUM
ejpam-5768	225	4	2(w02	2(w02	NUM
ejpam-5768	225	5	)	)	PUNCT
ejpam-5768	226	1	=	=	SYM
ejpam-5768	226	2	w−1	w−1	PROPN
ejpam-5768	226	3	01	01	NUM
ejpam-5768	227	1	υ̂	υ̂	NUM
ejpam-5768	228	1	2(w−1	2(w−1	NOUN
ejpam-5768	228	2	01	01	NUM
ejpam-5768	228	3	w02w01	w02w01	NOUN
ejpam-5768	228	4	)	)	PUNCT
ejpam-5768	228	5	=	=	PRON
ejpam-5768	228	6	υ̂	υ̂	PROPN
ejpam-5768	229	1	2(w01w	2(w01w	NUM
ejpam-5768	229	2	−1	−1	NOUN
ejpam-5768	229	3	01	01	NUM
ejpam-5768	229	4	w02w01	w02w01	NOUN
ejpam-5768	229	5	)	)	PUNCT
ejpam-5768	229	6	=	=	PRON
ejpam-5768	230	1	υ̂	υ̂	PROPN
ejpam-5768	230	2	2(w02w01	2(w02w01	NUM
ejpam-5768	230	3	)	)	PUNCT
ejpam-5768	231	1	hence	hence	ADV
ejpam-5768	231	2	υ	υ	PROPN
ejpam-5768	231	3	is	be	AUX
ejpam-5768	231	4	a	a	DET
ejpam-5768	231	5	pf	pf	PROPN
ejpam-5768	231	6	hx	hx	PROPN
ejpam-5768	231	7	-	-	PUNCT
ejpam-5768	231	8	nsg	nsg	PROPN
ejpam-5768	231	9	.	.	PUNCT
ejpam-5768	232	1	proposition	proposition	NOUN
ejpam-5768	232	2	3	3	X
ejpam-5768	232	3	.	.	PUNCT
ejpam-5768	233	1	let	let	VERB
ejpam-5768	233	2	w	w	NOUN
ejpam-5768	233	3	be	be	AUX
ejpam-5768	233	4	an	an	DET
ejpam-5768	233	5	hx	hx	NOUN
ejpam-5768	233	6	-	-	PUNCT
ejpam-5768	233	7	group	group	NOUN
ejpam-5768	233	8	and	and	CCONJ
ejpam-5768	233	9	υi	υi	NOUN
ejpam-5768	233	10	be	be	AUX
ejpam-5768	233	11	pf	pf	PROPN
ejpam-5768	233	12	hx	hx	PROPN
ejpam-5768	233	13	-	-	PUNCT
ejpam-5768	233	14	nsgs	nsg	NOUN
ejpam-5768	233	15	of	of	ADP
ejpam-5768	233	16	w	w	PROPN
ejpam-5768	233	17	.	.	PUNCT
ejpam-5768	234	1	then	then	ADV
ejpam-5768	234	2	⋂	⋂	PROPN
ejpam-5768	234	3	i	i	PRON
ejpam-5768	234	4	υi	υi	VERB
ejpam-5768	234	5	is	be	AUX
ejpam-5768	234	6	a	a	DET
ejpam-5768	234	7	pf	pf	PROPN
ejpam-5768	234	8	hx	hx	PROPN
ejpam-5768	234	9	-	-	PUNCT
ejpam-5768	234	10	nsg	nsg	PROPN
ejpam-5768	234	11	of	of	ADP
ejpam-5768	234	12	w	w	PROPN
ejpam-5768	234	13	.	.	PUNCT
ejpam-5768	235	1	proof	proof	NOUN
ejpam-5768	235	2	.	.	PUNCT
ejpam-5768	236	1	clear	clear	ADJ
ejpam-5768	236	2	.	.	PUNCT
ejpam-5768	237	1	the	the	DET
ejpam-5768	237	2	following	follow	VERB
ejpam-5768	237	3	two	two	NUM
ejpam-5768	237	4	theorems	theorem	NOUN
ejpam-5768	237	5	are	be	AUX
ejpam-5768	237	6	the	the	DET
ejpam-5768	237	7	analogue	analogue	NOUN
ejpam-5768	237	8	of	of	ADP
ejpam-5768	237	9	theorem	theorem	ADJ
ejpam-5768	237	10	2	2	NUM
ejpam-5768	237	11	and	and	CCONJ
ejpam-5768	237	12	theorem	theorem	VERB
ejpam-5768	237	13	3	3	NUM
ejpam-5768	237	14	:	:	PUNCT
ejpam-5768	237	15	theorem	theorem	NOUN
ejpam-5768	237	16	6	6	NUM
ejpam-5768	237	17	.	.	PUNCT
ejpam-5768	238	1	if	if	SCONJ
ejpam-5768	238	2	υp	υp	PROPN
ejpam-5768	238	3	is	be	AUX
ejpam-5768	238	4	a	a	DET
ejpam-5768	238	5	pf	pf	PROPN
ejpam-5768	238	6	hx	hx	PROPN
ejpam-5768	238	7	-	-	PUNCT
ejpam-5768	238	8	nsg	nsg	PROPN
ejpam-5768	238	9	of	of	ADP
ejpam-5768	238	10	g	g	PROPN
ejpam-5768	238	11	,	,	PUNCT
ejpam-5768	238	12	then	then	ADV
ejpam-5768	238	13	p	p	X
ejpam-5768	238	14	⋆	⋆	NOUN
ejpam-5768	238	15	is	be	AUX
ejpam-5768	238	16	a	a	DET
ejpam-5768	238	17	normal	normal	ADJ
ejpam-5768	238	18	subgroup	subgroup	NOUN
ejpam-5768	238	19	of	of	ADP
ejpam-5768	238	20	g.	g.	PROPN
ejpam-5768	238	21	proof	proof	PROPN
ejpam-5768	238	22	.	.	PUNCT
ejpam-5768	239	1	since	since	SCONJ
ejpam-5768	239	2	υp	υp	PROPN
ejpam-5768	239	3	is	be	AUX
ejpam-5768	239	4	a	a	DET
ejpam-5768	239	5	pf	pf	PROPN
ejpam-5768	239	6	hx	hx	PROPN
ejpam-5768	239	7	-	-	PUNCT
ejpam-5768	239	8	nsg	nsg	PROPN
ejpam-5768	239	9	of	of	ADP
ejpam-5768	239	10	g	g	PROPN
ejpam-5768	239	11	,	,	PUNCT
ejpam-5768	239	12	then	then	ADV
ejpam-5768	239	13	υp	υp	PROPN
ejpam-5768	239	14	is	be	AUX
ejpam-5768	239	15	a	a	DET
ejpam-5768	239	16	pf	pf	PROPN
ejpam-5768	239	17	hx	hx	PROPN
ejpam-5768	239	18	-	-	PUNCT
ejpam-5768	239	19	sg	sg	PROPN
ejpam-5768	239	20	of	of	ADP
ejpam-5768	239	21	g	g	NOUN
ejpam-5768	239	22	,	,	PUNCT
ejpam-5768	239	23	thus	thus	ADV
ejpam-5768	239	24	,	,	PUNCT
ejpam-5768	239	25	by	by	ADP
ejpam-5768	239	26	theorem	theorem	NOUN
ejpam-5768	239	27	2	2	NUM
ejpam-5768	239	28	,	,	PUNCT
ejpam-5768	239	29	p	p	NOUN
ejpam-5768	239	30	⋆	⋆	NOUN
ejpam-5768	239	31	is	be	AUX
ejpam-5768	239	32	a	a	DET
ejpam-5768	239	33	subgroup	subgroup	NOUN
ejpam-5768	239	34	of	of	ADP
ejpam-5768	239	35	g.	g.	PROPN
ejpam-5768	239	36	now	now	ADV
ejpam-5768	239	37	,	,	PUNCT
ejpam-5768	239	38	suppose	suppose	VERB
ejpam-5768	239	39	that	that	SCONJ
ejpam-5768	239	40	w01	w01	NOUN
ejpam-5768	239	41	,	,	PUNCT
ejpam-5768	239	42	w02	w02	NOUN
ejpam-5768	239	43	∈	∈	PROPN
ejpam-5768	239	44	p	p	NOUN
ejpam-5768	239	45	⋆.	⋆.	X
ejpam-5768	239	46	then	then	ADV
ejpam-5768	239	47	ῡ	ῡ	PROPN
ejpam-5768	239	48	2(w01	2(w01	NUM
ejpam-5768	239	49	)	)	PUNCT
ejpam-5768	239	50	>	>	X
ejpam-5768	239	51	0	0	NUM
ejpam-5768	239	52	,	,	PUNCT
ejpam-5768	239	53	ῡ	ῡ	PROPN
ejpam-5768	239	54	2(w02	2(w02	NUM
ejpam-5768	239	55	)	)	PUNCT
ejpam-5768	239	56	>	>	X
ejpam-5768	239	57	0	0	PUNCT
ejpam-5768	240	1	and	and	CCONJ
ejpam-5768	240	2	υ̂	υ̂	NUM
ejpam-5768	240	3	2(w01	2(w01	NUM
ejpam-5768	240	4	)	)	PUNCT
ejpam-5768	240	5	<	<	X
ejpam-5768	240	6	1	1	NUM
ejpam-5768	240	7	,	,	PUNCT
ejpam-5768	240	8	υ̂	υ̂	PROPN
ejpam-5768	240	9	2(w02	2(w02	NUM
ejpam-5768	240	10	)	)	PUNCT
ejpam-5768	240	11	<	<	X
ejpam-5768	241	1	1	1	X
ejpam-5768	241	2	.	.	PUNCT
ejpam-5768	241	3	that	that	SCONJ
ejpam-5768	241	4	υp	υp	PROPN
ejpam-5768	241	5	is	be	AUX
ejpam-5768	241	6	a	a	DET
ejpam-5768	241	7	pf	pf	PROPN
ejpam-5768	241	8	hx	hx	PROPN
ejpam-5768	241	9	-	-	PUNCT
ejpam-5768	241	10	nsg	nsg	PROPN
ejpam-5768	241	11	implies	imply	VERB
ejpam-5768	241	12	that	that	SCONJ
ejpam-5768	241	13	ῡ	ῡ	PROPN
ejpam-5768	241	14	2(w−1	2(w−1	PROPN
ejpam-5768	241	15	01	01	NUM
ejpam-5768	241	16	w02w01	w02w01	NOUN
ejpam-5768	241	17	)	)	PUNCT
ejpam-5768	242	1	=	=	SYM
ejpam-5768	242	2	ῡ	ῡ	PROPN
ejpam-5768	242	3	2(w02	2(w02	NUM
ejpam-5768	242	4	)	)	PUNCT
ejpam-5768	242	5	}	}	PUNCT
ejpam-5768	242	6	>	>	X
ejpam-5768	242	7	0	0	PUNCT
ejpam-5768	243	1	and	and	CCONJ
ejpam-5768	243	2	υ̂	υ̂	NUM
ejpam-5768	243	3	2(w−1	2(w−1	NOUN
ejpam-5768	243	4	01	01	NUM
ejpam-5768	243	5	w02w01	w02w01	NOUN
ejpam-5768	243	6	)	)	PUNCT
ejpam-5768	243	7	=	=	PRON
ejpam-5768	244	1	υ̂	υ̂	PROPN
ejpam-5768	244	2	2(w02	2(w02	NUM
ejpam-5768	244	3	)	)	PUNCT
ejpam-5768	244	4	}	}	PUNCT
ejpam-5768	245	1	<	<	X
ejpam-5768	245	2	1	1	X
ejpam-5768	245	3	.	.	PUNCT
ejpam-5768	245	4	hence	hence	ADV
ejpam-5768	245	5	w−1	w−1	PROPN
ejpam-5768	245	6	01	01	NUM
ejpam-5768	245	7	w02w01	w02w01	NOUN
ejpam-5768	245	8	∈	∈	PROPN
ejpam-5768	245	9	p	p	PROPN
ejpam-5768	245	10	⋆	⋆	X
ejpam-5768	245	11	and	and	CCONJ
ejpam-5768	245	12	therefore	therefore	ADV
ejpam-5768	245	13	,	,	PUNCT
ejpam-5768	245	14	p	p	NOUN
ejpam-5768	245	15	⋆	⋆	VERB
ejpam-5768	245	16	is	be	AUX
ejpam-5768	245	17	a	a	DET
ejpam-5768	245	18	normal	normal	ADJ
ejpam-5768	245	19	subgroup	subgroup	NOUN
ejpam-5768	245	20	of	of	ADP
ejpam-5768	245	21	g.	g.	PROPN
ejpam-5768	245	22	a.	a.	PROPN
ejpam-5768	245	23	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	245	24	/	/	SYM
ejpam-5768	245	25	eur	eur	PROPN
ejpam-5768	245	26	.	.	PUNCT
ejpam-5768	246	1	j.	j.	PROPN
ejpam-5768	246	2	pure	pure	PROPN
ejpam-5768	246	3	appl	appl	PROPN
ejpam-5768	246	4	.	.	PROPN
ejpam-5768	246	5	math	math	PROPN
ejpam-5768	246	6	,	,	PUNCT
ejpam-5768	246	7	18	18	NUM
ejpam-5768	246	8	(	(	PUNCT
ejpam-5768	246	9	2	2	NUM
ejpam-5768	246	10	)	)	PUNCT
ejpam-5768	246	11	(	(	PUNCT
ejpam-5768	246	12	2025	2025	NUM
ejpam-5768	246	13	)	)	PUNCT
ejpam-5768	246	14	,	,	PUNCT
ejpam-5768	246	15	5768	5768	NUM
ejpam-5768	246	16	9	9	NUM
ejpam-5768	246	17	of	of	ADP
ejpam-5768	246	18	17	17	NUM
ejpam-5768	246	19	theorem	theorem	NOUN
ejpam-5768	246	20	7	7	NUM
ejpam-5768	246	21	.	.	PUNCT
ejpam-5768	247	1	if	if	SCONJ
ejpam-5768	247	2	υp	υp	PROPN
ejpam-5768	247	3	is	be	AUX
ejpam-5768	247	4	a	a	DET
ejpam-5768	247	5	pf	pf	PROPN
ejpam-5768	247	6	hx	hx	PROPN
ejpam-5768	247	7	-	-	PUNCT
ejpam-5768	247	8	nsg	nsg	PROPN
ejpam-5768	247	9	of	of	ADP
ejpam-5768	247	10	g	g	PROPN
ejpam-5768	247	11	,	,	PUNCT
ejpam-5768	247	12	then	then	ADV
ejpam-5768	247	13	p⋆	p⋆	PRON
ejpam-5768	247	14	is	be	AUX
ejpam-5768	247	15	a	a	DET
ejpam-5768	247	16	normal	normal	ADJ
ejpam-5768	247	17	subgroup	subgroup	NOUN
ejpam-5768	247	18	of	of	ADP
ejpam-5768	247	19	g.	g.	PROPN
ejpam-5768	247	20	proof	proof	PROPN
ejpam-5768	247	21	.	.	PUNCT
ejpam-5768	248	1	since	since	SCONJ
ejpam-5768	248	2	υp	υp	PROPN
ejpam-5768	248	3	is	be	AUX
ejpam-5768	248	4	a	a	DET
ejpam-5768	248	5	pf	pf	PROPN
ejpam-5768	248	6	hx	hx	PROPN
ejpam-5768	248	7	-	-	PUNCT
ejpam-5768	248	8	nsg	nsg	PROPN
ejpam-5768	248	9	of	of	ADP
ejpam-5768	248	10	g	g	PROPN
ejpam-5768	248	11	,	,	PUNCT
ejpam-5768	248	12	then	then	ADV
ejpam-5768	248	13	υp	υp	PROPN
ejpam-5768	248	14	is	be	AUX
ejpam-5768	248	15	a	a	DET
ejpam-5768	248	16	pf	pf	PROPN
ejpam-5768	248	17	hx	hx	PROPN
ejpam-5768	248	18	-	-	PUNCT
ejpam-5768	248	19	sg	sg	PROPN
ejpam-5768	248	20	of	of	ADP
ejpam-5768	248	21	g.	g.	PROPN
ejpam-5768	248	22	thus	thus	ADV
ejpam-5768	248	23	,	,	PUNCT
ejpam-5768	248	24	by	by	ADP
ejpam-5768	248	25	theorem	theorem	NOUN
ejpam-5768	248	26	3	3	NUM
ejpam-5768	248	27	,	,	PUNCT
ejpam-5768	248	28	p⋆	p⋆	X
ejpam-5768	248	29	is	be	AUX
ejpam-5768	248	30	a	a	DET
ejpam-5768	248	31	subgroup	subgroup	NOUN
ejpam-5768	248	32	of	of	ADP
ejpam-5768	248	33	g.	g.	PROPN
ejpam-5768	248	34	now	now	ADV
ejpam-5768	248	35	,	,	PUNCT
ejpam-5768	248	36	let	let	VERB
ejpam-5768	248	37	w01	w01	NOUN
ejpam-5768	248	38	,	,	PUNCT
ejpam-5768	248	39	w02	w02	NOUN
ejpam-5768	248	40	∈	∈	PROPN
ejpam-5768	248	41	p⋆.	p⋆.	NOUN
ejpam-5768	248	42	then	then	ADV
ejpam-5768	248	43	ῡ	ῡ	PROPN
ejpam-5768	248	44	2(w01	2(w01	NUM
ejpam-5768	248	45	)	)	PUNCT
ejpam-5768	248	46	=	=	SYM
ejpam-5768	249	1	1	1	NUM
ejpam-5768	249	2	,	,	PUNCT
ejpam-5768	249	3	ῡ	ῡ	PROPN
ejpam-5768	249	4	2(w02	2(w02	NUM
ejpam-5768	249	5	)	)	PUNCT
ejpam-5768	249	6	=	=	SYM
ejpam-5768	249	7	1	1	NUM
ejpam-5768	249	8	and	and	CCONJ
ejpam-5768	249	9	υ̂	υ̂	NUM
ejpam-5768	249	10	2(w01	2(w01	NUM
ejpam-5768	249	11	)	)	PUNCT
ejpam-5768	249	12	=	=	SYM
ejpam-5768	249	13	0	0	NUM
ejpam-5768	249	14	,	,	PUNCT
ejpam-5768	249	15	υ̂	υ̂	PROPN
ejpam-5768	249	16	2(w02	2(w02	NUM
ejpam-5768	249	17	)	)	PUNCT
ejpam-5768	250	1	=	=	NOUN
ejpam-5768	250	2	0	0	X
ejpam-5768	250	3	.	.	PUNCT
ejpam-5768	251	1	that	that	SCONJ
ejpam-5768	251	2	υp	υp	PROPN
ejpam-5768	251	3	is	be	AUX
ejpam-5768	251	4	a	a	DET
ejpam-5768	251	5	pf	pf	PROPN
ejpam-5768	251	6	hx	hx	PROPN
ejpam-5768	251	7	-	-	PUNCT
ejpam-5768	251	8	nsg	nsg	PROPN
ejpam-5768	251	9	implies	imply	VERB
ejpam-5768	251	10	that	that	SCONJ
ejpam-5768	251	11	ῡ	ῡ	PROPN
ejpam-5768	251	12	2(w−1	2(w−1	PROPN
ejpam-5768	251	13	01	01	NUM
ejpam-5768	251	14	w02w01	w02w01	NOUN
ejpam-5768	251	15	)	)	PUNCT
ejpam-5768	252	1	=	=	SYM
ejpam-5768	252	2	ῡ	ῡ	PROPN
ejpam-5768	252	3	2(w02	2(w02	NUM
ejpam-5768	252	4	)	)	PUNCT
ejpam-5768	252	5	}	}	PUNCT
ejpam-5768	253	1	=	=	SYM
ejpam-5768	253	2	1	1	NUM
ejpam-5768	253	3	and	and	CCONJ
ejpam-5768	253	4	υ̂	υ̂	NUM
ejpam-5768	253	5	2(w−1	2(w−1	NOUN
ejpam-5768	253	6	01	01	NUM
ejpam-5768	253	7	w02w01	w02w01	NOUN
ejpam-5768	253	8	)	)	PUNCT
ejpam-5768	253	9	=	=	PRON
ejpam-5768	254	1	υ̂	υ̂	PROPN
ejpam-5768	254	2	2(w02	2(w02	NUM
ejpam-5768	254	3	)	)	PUNCT
ejpam-5768	254	4	}	}	PUNCT
ejpam-5768	255	1	=	=	SYM
ejpam-5768	255	2	0	0	X
ejpam-5768	255	3	.	.	PUNCT
ejpam-5768	256	1	hence	hence	ADV
ejpam-5768	256	2	w−1	w−1	PROPN
ejpam-5768	256	3	01	01	NUM
ejpam-5768	256	4	w02w01	w02w01	NOUN
ejpam-5768	256	5	∈	∈	NOUN
ejpam-5768	256	6	p⋆	p⋆	NOUN
ejpam-5768	256	7	and	and	CCONJ
ejpam-5768	256	8	therefore	therefore	ADV
ejpam-5768	256	9	,	,	PUNCT
ejpam-5768	256	10	p	p	NOUN
ejpam-5768	256	11	⋆	⋆	VERB
ejpam-5768	256	12	is	be	AUX
ejpam-5768	256	13	a	a	DET
ejpam-5768	256	14	normal	normal	ADJ
ejpam-5768	256	15	subgroup	subgroup	NOUN
ejpam-5768	256	16	of	of	ADP
ejpam-5768	256	17	g.	g.	PROPN
ejpam-5768	256	18	for	for	ADP
ejpam-5768	256	19	(	(	PUNCT
ejpam-5768	256	20	ζ	ζ	NOUN
ejpam-5768	256	21	,	,	PUNCT
ejpam-5768	256	22	δ)-level	δ)-level	PUNCT
ejpam-5768	256	23	pfsss	pfsss	NOUN
ejpam-5768	256	24	,	,	PUNCT
ejpam-5768	256	25	we	we	PRON
ejpam-5768	256	26	prove	prove	VERB
ejpam-5768	256	27	the	the	DET
ejpam-5768	256	28	following	follow	VERB
ejpam-5768	256	29	theorem	theorem	NOUN
ejpam-5768	256	30	:	:	PUNCT
ejpam-5768	256	31	theorem	theorem	NOUN
ejpam-5768	256	32	8	8	NUM
ejpam-5768	256	33	.	.	PUNCT
ejpam-5768	257	1	let	let	VERB
ejpam-5768	257	2	υ	υ	PRON
ejpam-5768	257	3	be	be	AUX
ejpam-5768	257	4	a	a	DET
ejpam-5768	257	5	pfss	pfss	NOUN
ejpam-5768	257	6	of	of	ADP
ejpam-5768	257	7	w	w	PROPN
ejpam-5768	257	8	.	.	PUNCT
ejpam-5768	258	1	then	then	ADV
ejpam-5768	258	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	258	3	,	,	PUNCT
ejpam-5768	258	4	δ	δ	PROPN
ejpam-5768	258	5	)	)	PUNCT
ejpam-5768	258	6	is	be	AUX
ejpam-5768	258	7	a	a	DET
ejpam-5768	258	8	pf	pf	PROPN
ejpam-5768	258	9	hx	hx	PROPN
ejpam-5768	258	10	-	-	PUNCT
ejpam-5768	258	11	nsg	nsg	PROPN
ejpam-5768	259	1	if	if	SCONJ
ejpam-5768	260	1	and	and	CCONJ
ejpam-5768	260	2	only	only	ADV
ejpam-5768	260	3	if	if	SCONJ
ejpam-5768	260	4	υ	υ	PRON
ejpam-5768	260	5	is	be	AUX
ejpam-5768	260	6	a	a	DET
ejpam-5768	260	7	pf	pf	PROPN
ejpam-5768	260	8	hx	hx	PROPN
ejpam-5768	260	9	-	-	PUNCT
ejpam-5768	260	10	nsg	nsg	PROPN
ejpam-5768	260	11	.	.	PUNCT
ejpam-5768	261	1	proof	proof	NOUN
ejpam-5768	261	2	.	.	PUNCT
ejpam-5768	262	1	suppose	suppose	VERB
ejpam-5768	262	2	that	that	SCONJ
ejpam-5768	262	3	υ(ζ	υ(ζ	PROPN
ejpam-5768	262	4	,	,	PUNCT
ejpam-5768	262	5	δ	δ	PROPN
ejpam-5768	262	6	)	)	PUNCT
ejpam-5768	262	7	is	be	AUX
ejpam-5768	262	8	a	a	DET
ejpam-5768	262	9	pf	pf	PROPN
ejpam-5768	262	10	hx	hx	PROPN
ejpam-5768	262	11	-	-	PUNCT
ejpam-5768	262	12	nsg	nsg	PROPN
ejpam-5768	262	13	.	.	PUNCT
ejpam-5768	263	1	then	then	ADV
ejpam-5768	263	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	263	3	,	,	PUNCT
ejpam-5768	263	4	δ	δ	PROPN
ejpam-5768	263	5	)	)	PUNCT
ejpam-5768	263	6	is	be	AUX
ejpam-5768	263	7	a	a	DET
ejpam-5768	263	8	pf	pf	PROPN
ejpam-5768	263	9	hx	hx	PROPN
ejpam-5768	263	10	-	-	PUNCT
ejpam-5768	263	11	sg	sg	PROPN
ejpam-5768	263	12	and	and	CCONJ
ejpam-5768	263	13	hence	hence	ADV
ejpam-5768	263	14	by	by	ADP
ejpam-5768	263	15	theorem	theorem	NOUN
ejpam-5768	263	16	4	4	NUM
ejpam-5768	263	17	,	,	PUNCT
ejpam-5768	263	18	υ	υ	NOUN
ejpam-5768	263	19	is	be	AUX
ejpam-5768	263	20	a	a	DET
ejpam-5768	263	21	pf	pf	PROPN
ejpam-5768	263	22	hx	hx	PROPN
ejpam-5768	263	23	-	-	PUNCT
ejpam-5768	263	24	sg	sg	PROPN
ejpam-5768	263	25	.	.	PUNCT
ejpam-5768	264	1	assume	assume	VERB
ejpam-5768	264	2	that	that	SCONJ
ejpam-5768	264	3	there	there	PRON
ejpam-5768	264	4	exists	exist	VERB
ejpam-5768	264	5	w01	w01	NOUN
ejpam-5768	264	6	,	,	PUNCT
ejpam-5768	264	7	w02	w02	NOUN
ejpam-5768	264	8	∈	∈	PROPN
ejpam-5768	264	9	w	w	ADP
ejpam-5768	264	10	such	such	ADJ
ejpam-5768	264	11	that	that	SCONJ
ejpam-5768	264	12	ῡ	ῡ	PROPN
ejpam-5768	264	13	2(w−1	2(w−1	PROPN
ejpam-5768	264	14	01	01	NUM
ejpam-5768	264	15	w02w01	w02w01	NOUN
ejpam-5768	264	16	)	)	PUNCT
ejpam-5768	264	17	<	<	X
ejpam-5768	264	18	ῡ	ῡ	PROPN
ejpam-5768	264	19	2(w02	2(w02	NUM
ejpam-5768	264	20	)	)	PUNCT
ejpam-5768	265	1	=	=	SYM
ejpam-5768	265	2	ζ	ζ	NOUN
ejpam-5768	265	3	,	,	PUNCT
ejpam-5768	265	4	for	for	ADP
ejpam-5768	265	5	some	some	DET
ejpam-5768	265	6	ζ	ζ	NOUN
ejpam-5768	265	7	∈	∈	NOUN
ejpam-5768	266	1	[	[	X
ejpam-5768	266	2	0	0	NUM
ejpam-5768	266	3	,	,	PUNCT
ejpam-5768	266	4	1	1	NUM
ejpam-5768	266	5	]	]	PUNCT
ejpam-5768	266	6	.	.	PUNCT
ejpam-5768	267	1	then	then	ADV
ejpam-5768	267	2	ῡ	ῡ	PROPN
ejpam-5768	267	3	2(w−1	2(w−1	PROPN
ejpam-5768	267	4	01	01	NUM
ejpam-5768	267	5	w02w01	w02w01	NOUN
ejpam-5768	267	6	<	<	X
ejpam-5768	267	7	ζ	ζ	NOUN
ejpam-5768	267	8	and	and	CCONJ
ejpam-5768	267	9	this	this	PRON
ejpam-5768	267	10	contradicts	contradict	VERB
ejpam-5768	267	11	that	that	SCONJ
ejpam-5768	267	12	υ(ζ	υ(ζ	PROPN
ejpam-5768	267	13	,	,	PUNCT
ejpam-5768	267	14	δ	δ	PROPN
ejpam-5768	267	15	)	)	PUNCT
ejpam-5768	267	16	is	be	AUX
ejpam-5768	267	17	a	a	DET
ejpam-5768	267	18	pf	pf	PROPN
ejpam-5768	267	19	hx	hx	PROPN
ejpam-5768	267	20	-	-	PUNCT
ejpam-5768	267	21	nsg	nsg	PROPN
ejpam-5768	267	22	.	.	PUNCT
ejpam-5768	268	1	in	in	ADP
ejpam-5768	268	2	the	the	DET
ejpam-5768	268	3	case	case	NOUN
ejpam-5768	268	4	that	that	SCONJ
ejpam-5768	268	5	ῡ	ῡ	PROPN
ejpam-5768	268	6	2(w02	2(w02	NUM
ejpam-5768	268	7	)	)	PUNCT
ejpam-5768	268	8	<	<	X
ejpam-5768	268	9	ῡ	ῡ	PROPN
ejpam-5768	268	10	2(w−1	2(w−1	PROPN
ejpam-5768	268	11	01	01	NUM
ejpam-5768	268	12	w02w01	w02w01	NOUN
ejpam-5768	268	13	)	)	PUNCT
ejpam-5768	268	14	,	,	PUNCT
ejpam-5768	268	15	then	then	ADV
ejpam-5768	268	16	ῡ	ῡ	PROPN
ejpam-5768	268	17	2(w02	2(w02	NUM
ejpam-5768	268	18	)	)	PUNCT
ejpam-5768	268	19	<	<	X
ejpam-5768	268	20	ῡ	ῡ	PROPN
ejpam-5768	268	21	2(w−1	2(w−1	NOUN
ejpam-5768	268	22	01	01	NUM
ejpam-5768	268	23	w02w01	w02w01	NOUN
ejpam-5768	268	24	)	)	PUNCT
ejpam-5768	268	25	=	=	SYM
ejpam-5768	268	26	ζ	ζ	NOUN
ejpam-5768	268	27	,	,	PUNCT
ejpam-5768	268	28	for	for	ADP
ejpam-5768	268	29	some	some	DET
ejpam-5768	268	30	ζ	ζ	NOUN
ejpam-5768	268	31	∈	∈	NOUN
ejpam-5768	268	32	[	[	X
ejpam-5768	268	33	0	0	NUM
ejpam-5768	268	34	,	,	PUNCT
ejpam-5768	268	35	1	1	NUM
ejpam-5768	268	36	]	]	PUNCT
ejpam-5768	268	37	and	and	CCONJ
ejpam-5768	268	38	this	this	PRON
ejpam-5768	268	39	contradicts	contradict	VERB
ejpam-5768	268	40	that	that	SCONJ
ejpam-5768	268	41	υ(ζ	υ(ζ	PROPN
ejpam-5768	268	42	,	,	PUNCT
ejpam-5768	268	43	δ	δ	PROPN
ejpam-5768	268	44	)	)	PUNCT
ejpam-5768	268	45	is	be	AUX
ejpam-5768	268	46	a	a	DET
ejpam-5768	268	47	pf	pf	PROPN
ejpam-5768	268	48	hx	hx	PROPN
ejpam-5768	268	49	-	-	PUNCT
ejpam-5768	268	50	nsg	nsg	PROPN
ejpam-5768	268	51	.	.	PUNCT
ejpam-5768	269	1	thus	thus	ADV
ejpam-5768	269	2	ῡ	ῡ	PROPN
ejpam-5768	269	3	2(w−1	2(w−1	PROPN
ejpam-5768	269	4	01	01	NUM
ejpam-5768	269	5	w02w01	w02w01	NOUN
ejpam-5768	269	6	)	)	PUNCT
ejpam-5768	269	7	=	=	SYM
ejpam-5768	270	1	ῡ	ῡ	PROPN
ejpam-5768	270	2	2(w02	2(w02	NUM
ejpam-5768	270	3	)	)	PUNCT
ejpam-5768	270	4	.	.	PUNCT
ejpam-5768	271	1	now	now	ADV
ejpam-5768	271	2	,	,	PUNCT
ejpam-5768	271	3	let	let	VERB
ejpam-5768	271	4	υ̂	υ̂	PROPN
ejpam-5768	271	5	2(w−1	2(w−1	NOUN
ejpam-5768	271	6	01	01	NUM
ejpam-5768	271	7	w02w01	w02w01	NOUN
ejpam-5768	271	8	)	)	PUNCT
ejpam-5768	271	9	>	>	X
ejpam-5768	271	10	υ̂	υ̂	PROPN
ejpam-5768	271	11	2(w02	2(w02	NUM
ejpam-5768	271	12	)	)	PUNCT
ejpam-5768	272	1	=	=	SYM
ejpam-5768	272	2	δ	δ	PROPN
ejpam-5768	272	3	,	,	PUNCT
ejpam-5768	272	4	for	for	ADP
ejpam-5768	272	5	some	some	DET
ejpam-5768	272	6	δ	δ	NOUN
ejpam-5768	272	7	∈	∈	PROPN
ejpam-5768	273	1	[	[	X
ejpam-5768	273	2	0	0	NUM
ejpam-5768	273	3	,	,	PUNCT
ejpam-5768	273	4	1	1	NUM
ejpam-5768	273	5	]	]	PUNCT
ejpam-5768	273	6	.	.	PUNCT
ejpam-5768	274	1	then	then	ADV
ejpam-5768	274	2	υ̂	υ̂	NUM
ejpam-5768	274	3	2(w−1	2(w−1	NOUN
ejpam-5768	274	4	01	01	NUM
ejpam-5768	274	5	w02w01	w02w01	NOUN
ejpam-5768	274	6	)	)	PUNCT
ejpam-5768	274	7	>	>	PUNCT
ejpam-5768	275	1	δ	δ	PROPN
ejpam-5768	275	2	and	and	CCONJ
ejpam-5768	275	3	this	this	PRON
ejpam-5768	275	4	contradicts	contradict	VERB
ejpam-5768	275	5	that	that	SCONJ
ejpam-5768	275	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	275	7	,	,	PUNCT
ejpam-5768	275	8	δ	δ	PROPN
ejpam-5768	275	9	)	)	PUNCT
ejpam-5768	275	10	is	be	AUX
ejpam-5768	275	11	a	a	DET
ejpam-5768	275	12	pf	pf	PROPN
ejpam-5768	275	13	hxnsg	hxnsg	NOUN
ejpam-5768	275	14	.	.	PUNCT
ejpam-5768	276	1	in	in	ADP
ejpam-5768	276	2	the	the	DET
ejpam-5768	276	3	case	case	NOUN
ejpam-5768	276	4	that	that	SCONJ
ejpam-5768	276	5	υ̂	υ̂	PROPN
ejpam-5768	276	6	2(w02	2(w02	NUM
ejpam-5768	276	7	)	)	PUNCT
ejpam-5768	276	8	>	>	X
ejpam-5768	277	1	υ̂	υ̂	PROPN
ejpam-5768	277	2	2(w−1	2(w−1	NOUN
ejpam-5768	277	3	01	01	NUM
ejpam-5768	277	4	w02w01	w02w01	NOUN
ejpam-5768	277	5	)	)	PUNCT
ejpam-5768	277	6	,	,	PUNCT
ejpam-5768	277	7	then	then	ADV
ejpam-5768	277	8	υ̂	υ̂	PROPN
ejpam-5768	277	9	2(w02	2(w02	NUM
ejpam-5768	277	10	)	)	PUNCT
ejpam-5768	277	11	>	>	X
ejpam-5768	278	1	υ̂	υ̂	PROPN
ejpam-5768	278	2	2(w−1	2(w−1	NOUN
ejpam-5768	278	3	01	01	NUM
ejpam-5768	278	4	w02w01	w02w01	NOUN
ejpam-5768	278	5	)	)	PUNCT
ejpam-5768	278	6	=	=	SYM
ejpam-5768	278	7	ζ	ζ	NOUN
ejpam-5768	278	8	,	,	PUNCT
ejpam-5768	278	9	for	for	ADP
ejpam-5768	278	10	some	some	DET
ejpam-5768	278	11	ζ	ζ	NOUN
ejpam-5768	278	12	∈	∈	NOUN
ejpam-5768	279	1	[	[	X
ejpam-5768	279	2	0	0	NUM
ejpam-5768	279	3	,	,	PUNCT
ejpam-5768	279	4	1	1	NUM
ejpam-5768	279	5	]	]	PUNCT
ejpam-5768	279	6	and	and	CCONJ
ejpam-5768	279	7	this	this	PRON
ejpam-5768	279	8	contradicts	contradict	VERB
ejpam-5768	279	9	that	that	SCONJ
ejpam-5768	279	10	υ(ζ	υ(ζ	PROPN
ejpam-5768	279	11	,	,	PUNCT
ejpam-5768	279	12	δ	δ	PROPN
ejpam-5768	279	13	)	)	PUNCT
ejpam-5768	279	14	is	be	AUX
ejpam-5768	279	15	a	a	DET
ejpam-5768	279	16	pf	pf	PROPN
ejpam-5768	279	17	hx	hx	PROPN
ejpam-5768	279	18	-	-	PUNCT
ejpam-5768	279	19	nsg	nsg	PROPN
ejpam-5768	279	20	.	.	PUNCT
ejpam-5768	280	1	thus	thus	ADV
ejpam-5768	280	2	υ̂	υ̂	NUM
ejpam-5768	280	3	2(w−1	2(w−1	NOUN
ejpam-5768	280	4	01	01	NUM
ejpam-5768	280	5	w02w01	w02w01	NOUN
ejpam-5768	280	6	)	)	PUNCT
ejpam-5768	280	7	=	=	PRON
ejpam-5768	281	1	υ̂	υ̂	PROPN
ejpam-5768	281	2	2(w02	2(w02	NUM
ejpam-5768	281	3	)	)	PUNCT
ejpam-5768	281	4	.	.	PUNCT
ejpam-5768	282	1	thus	thus	ADV
ejpam-5768	282	2	υ̂	υ̂	NUM
ejpam-5768	282	3	2(w−1	2(w−1	NOUN
ejpam-5768	282	4	01	01	NUM
ejpam-5768	282	5	w02w01	w02w01	NOUN
ejpam-5768	282	6	)	)	PUNCT
ejpam-5768	282	7	=	=	PRON
ejpam-5768	283	1	υ̂	υ̂	PROPN
ejpam-5768	283	2	2(w02	2(w02	NUM
ejpam-5768	283	3	)	)	PUNCT
ejpam-5768	283	4	.	.	PUNCT
ejpam-5768	284	1	therefore	therefore	ADV
ejpam-5768	284	2	,	,	PUNCT
ejpam-5768	284	3	υ	υ	PROPN
ejpam-5768	284	4	is	be	AUX
ejpam-5768	284	5	a	a	DET
ejpam-5768	284	6	pf	pf	PROPN
ejpam-5768	284	7	hx	hx	PROPN
ejpam-5768	284	8	-	-	PUNCT
ejpam-5768	284	9	nsg	nsg	PROPN
ejpam-5768	284	10	.	.	PUNCT
ejpam-5768	285	1	conversely	conversely	ADV
ejpam-5768	285	2	,	,	PUNCT
ejpam-5768	285	3	assume	assume	VERB
ejpam-5768	285	4	that	that	SCONJ
ejpam-5768	285	5	υ	υ	PROPN
ejpam-5768	285	6	is	be	AUX
ejpam-5768	285	7	a	a	DET
ejpam-5768	285	8	pf	pf	PROPN
ejpam-5768	285	9	hx	hx	PROPN
ejpam-5768	285	10	-	-	PUNCT
ejpam-5768	285	11	nsg	nsg	PROPN
ejpam-5768	285	12	.	.	PUNCT
ejpam-5768	286	1	let	let	VERB
ejpam-5768	286	2	w01	w01	NOUN
ejpam-5768	286	3	,	,	PUNCT
ejpam-5768	286	4	w02	w02	NOUN
ejpam-5768	286	5	∈	∈	PROPN
ejpam-5768	286	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	286	7	,	,	PUNCT
ejpam-5768	286	8	δ	δ	PROPN
ejpam-5768	286	9	)	)	PUNCT
ejpam-5768	286	10	,	,	PUNCT
ejpam-5768	286	11	for	for	ADP
ejpam-5768	286	12	some	some	PRON
ejpam-5768	286	13	(	(	PUNCT
ejpam-5768	286	14	ζ	ζ	PROPN
ejpam-5768	286	15	,	,	PUNCT
ejpam-5768	286	16	δ	δ	PROPN
ejpam-5768	286	17	)	)	PUNCT
ejpam-5768	286	18	.	.	PUNCT
ejpam-5768	287	1	then	then	ADV
ejpam-5768	287	2	ῡ	ῡ	PROPN
ejpam-5768	287	3	2(w01	2(w01	NUM
ejpam-5768	287	4	)	)	PUNCT
ejpam-5768	287	5	≥	≥	NOUN
ejpam-5768	287	6	ζ	ζ	NOUN
ejpam-5768	287	7	,	,	PUNCT
ejpam-5768	287	8	ῡ	ῡ	PROPN
ejpam-5768	287	9	2(w02	2(w02	NUM
ejpam-5768	287	10	)	)	PUNCT
ejpam-5768	287	11	≥	≥	PROPN
ejpam-5768	287	12	ζ	ζ	NOUN
ejpam-5768	287	13	,	,	PUNCT
ejpam-5768	287	14	υ̂	υ̂	NUM
ejpam-5768	287	15	2(w01	2(w01	NUM
ejpam-5768	287	16	)	)	PUNCT
ejpam-5768	287	17	≤	≤	NOUN
ejpam-5768	287	18	δ	δ	PROPN
ejpam-5768	287	19	and	and	CCONJ
ejpam-5768	287	20	υ̂	υ̂	NUM
ejpam-5768	287	21	2(w02	2(w02	NUM
ejpam-5768	287	22	)	)	PUNCT
ejpam-5768	287	23	≤	≤	NUM
ejpam-5768	287	24	δ	δ	PROPN
ejpam-5768	287	25	.	.	PUNCT
ejpam-5768	288	1	that	that	PRON
ejpam-5768	288	2	υ	υ	PROPN
ejpam-5768	288	3	is	be	AUX
ejpam-5768	288	4	a	a	DET
ejpam-5768	288	5	pf	pf	PROPN
ejpam-5768	288	6	hx	hx	PROPN
ejpam-5768	288	7	-	-	PUNCT
ejpam-5768	288	8	nsg	nsg	PROPN
ejpam-5768	288	9	implies	imply	VERB
ejpam-5768	288	10	that	that	SCONJ
ejpam-5768	289	1	ῡ	ῡ	PROPN
ejpam-5768	289	2	2(w−1	2(w−1	PROPN
ejpam-5768	289	3	01	01	NUM
ejpam-5768	289	4	w02w01	w02w01	NOUN
ejpam-5768	289	5	)	)	PUNCT
ejpam-5768	289	6	=	=	SYM
ejpam-5768	289	7	ῡ	ῡ	PROPN
ejpam-5768	289	8	2(w02	2(w02	NUM
ejpam-5768	289	9	)	)	PUNCT
ejpam-5768	289	10	}	}	PUNCT
ejpam-5768	289	11	≥	≥	X
ejpam-5768	289	12	ζ	ζ	PROPN
ejpam-5768	289	13	and	and	CCONJ
ejpam-5768	289	14	υ̂	υ̂	PROPN
ejpam-5768	289	15	2(w−1	2(w−1	NOUN
ejpam-5768	289	16	01	01	NUM
ejpam-5768	289	17	w02w01	w02w01	NOUN
ejpam-5768	289	18	)	)	PUNCT
ejpam-5768	289	19	=	=	PRON
ejpam-5768	289	20	υ̂	υ̂	PROPN
ejpam-5768	289	21	2(w02	2(w02	NUM
ejpam-5768	289	22	)	)	PUNCT
ejpam-5768	289	23	≤	≤	NUM
ejpam-5768	290	1	δ	δ	PROPN
ejpam-5768	290	2	.	.	PUNCT
ejpam-5768	291	1	hence	hence	ADV
ejpam-5768	291	2	w−1	w−1	PROPN
ejpam-5768	291	3	01	01	NUM
ejpam-5768	291	4	w02w01	w02w01	NOUN
ejpam-5768	291	5	∈	∈	PROPN
ejpam-5768	291	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	291	7	,	,	PUNCT
ejpam-5768	291	8	δ	δ	PROPN
ejpam-5768	291	9	)	)	PUNCT
ejpam-5768	291	10	and	and	CCONJ
ejpam-5768	291	11	therefore	therefore	ADV
ejpam-5768	291	12	,	,	PUNCT
ejpam-5768	291	13	υ(ζ	υ(ζ	PROPN
ejpam-5768	291	14	,	,	PUNCT
ejpam-5768	291	15	δ	δ	PROPN
ejpam-5768	291	16	)	)	PUNCT
ejpam-5768	291	17	is	be	AUX
ejpam-5768	291	18	a	a	DET
ejpam-5768	291	19	pf	pf	PROPN
ejpam-5768	291	20	hx	hx	PROPN
ejpam-5768	291	21	-	-	PUNCT
ejpam-5768	291	22	nsg	nsg	PROPN
ejpam-5768	291	23	.	.	PUNCT
ejpam-5768	292	1	4	4	X
ejpam-5768	292	2	.	.	X
ejpam-5768	292	3	pythagorean	pythagorean	PROPN
ejpam-5768	292	4	fuzzy	fuzzy	ADJ
ejpam-5768	292	5	homomorphism	homomorphism	PROPN
ejpam-5768	292	6	of	of	ADP
ejpam-5768	292	7	hx	hx	NOUN
ejpam-5768	292	8	-	-	PUNCT
ejpam-5768	292	9	subgroups	subgroup	NOUN
ejpam-5768	292	10	let	let	VERB
ejpam-5768	292	11	p	p	PRON
ejpam-5768	292	12	,	,	PUNCT
ejpam-5768	292	13	s	s	AUX
ejpam-5768	292	14	be	be	AUX
ejpam-5768	292	15	two	two	NUM
ejpam-5768	292	16	hx	hx	NOUN
ejpam-5768	292	17	-	-	PUNCT
ejpam-5768	292	18	groups	group	NOUN
ejpam-5768	292	19	,	,	PUNCT
ejpam-5768	292	20	l	l	PROPN
ejpam-5768	292	21	a	a	DET
ejpam-5768	292	22	pf	pf	PROPN
ejpam-5768	292	23	hx	hx	PROPN
ejpam-5768	292	24	-	-	PUNCT
ejpam-5768	292	25	sg	sg	PROPN
ejpam-5768	292	26	of	of	ADP
ejpam-5768	292	27	p	p	PROPN
ejpam-5768	292	28	and	and	CCONJ
ejpam-5768	292	29	n	n	CCONJ
ejpam-5768	292	30	a	a	DET
ejpam-5768	292	31	pf	pf	PROPN
ejpam-5768	292	32	-	-	PUNCT
ejpam-5768	292	33	hx	hx	NOUN
ejpam-5768	292	34	-	-	PUNCT
ejpam-5768	292	35	sg	sg	PROPN
ejpam-5768	292	36	of	of	ADP
ejpam-5768	292	37	s.	s.	PROPN
ejpam-5768	292	38	consider	consider	VERB
ejpam-5768	292	39	a	a	DET
ejpam-5768	292	40	homomorphism	homomorphism	NOUN
ejpam-5768	292	41	τ	τ	X
ejpam-5768	292	42	:	:	PUNCT
ejpam-5768	292	43	p	p	X
ejpam-5768	292	44	−→	−→	NOUN
ejpam-5768	292	45	s	s	VERB
ejpam-5768	292	46	for	for	ADP
ejpam-5768	292	47	s	s	PROPN
ejpam-5768	292	48	∈	∈	PROPN
ejpam-5768	292	49	s	s	PART
ejpam-5768	292	50	,	,	PUNCT
ejpam-5768	292	51	we	we	PRON
ejpam-5768	292	52	define	define	VERB
ejpam-5768	292	53	:	:	PUNCT
ejpam-5768	292	54	ητ(l)(s	ητ(l)(s	NUM
ejpam-5768	292	55	)	)	PUNCT
ejpam-5768	293	1	=	=	PRON
ejpam-5768	293	2	{	{	PUNCT
ejpam-5768	293	3	max{ῡl(p	max{ῡl(p	NOUN
ejpam-5768	293	4	)	)	PUNCT
ejpam-5768	293	5	:	:	PUNCT
ejpam-5768	294	1	s	s	X
ejpam-5768	294	2	=	=	PUNCT
ejpam-5768	294	3	τ(p	τ(p	NOUN
ejpam-5768	294	4	)	)	PUNCT
ejpam-5768	294	5	}	}	PUNCT
ejpam-5768	294	6	if	if	SCONJ
ejpam-5768	294	7	s	s	PROPN
ejpam-5768	294	8	∈	∈	PROPN
ejpam-5768	294	9	im(τ	im(τ	NUM
ejpam-5768	294	10	)	)	PUNCT
ejpam-5768	294	11	0	0	PUNCT
ejpam-5768	295	1	otherwise	otherwise	ADV
ejpam-5768	295	2	and	and	CCONJ
ejpam-5768	295	3	η̂τ(l)(s	η̂τ(l)(s	ADJ
ejpam-5768	295	4	)	)	PUNCT
ejpam-5768	296	1	=	=	PRON
ejpam-5768	296	2	{	{	PUNCT
ejpam-5768	296	3	min{υ̂l(p	min{υ̂l(p	NOUN
ejpam-5768	296	4	)	)	PUNCT
ejpam-5768	296	5	:	:	PUNCT
ejpam-5768	297	1	s	s	X
ejpam-5768	297	2	=	=	PUNCT
ejpam-5768	297	3	τ(p	τ(p	NOUN
ejpam-5768	297	4	)	)	PUNCT
ejpam-5768	297	5	}	}	PUNCT
ejpam-5768	297	6	if	if	SCONJ
ejpam-5768	297	7	s	s	PROPN
ejpam-5768	297	8	∈	∈	PROPN
ejpam-5768	297	9	im(τ	im(τ	PRON
ejpam-5768	297	10	)	)	PUNCT
ejpam-5768	297	11	1	1	NUM
ejpam-5768	297	12	otherwise	otherwise	ADV
ejpam-5768	297	13	after	after	ADP
ejpam-5768	297	14	presenting	present	VERB
ejpam-5768	297	15	the	the	DET
ejpam-5768	297	16	definition	definition	NOUN
ejpam-5768	297	17	of	of	ADP
ejpam-5768	297	18	pf	pf	PROPN
ejpam-5768	297	19	homomorphism	homomorphism	NOUN
ejpam-5768	297	20	,	,	PUNCT
ejpam-5768	297	21	we	we	PRON
ejpam-5768	297	22	are	be	AUX
ejpam-5768	297	23	now	now	ADV
ejpam-5768	297	24	ready	ready	ADJ
ejpam-5768	297	25	to	to	PART
ejpam-5768	297	26	provide	provide	VERB
ejpam-5768	297	27	a	a	DET
ejpam-5768	297	28	relationship	relationship	NOUN
ejpam-5768	297	29	between	between	ADP
ejpam-5768	297	30	a	a	DET
ejpam-5768	297	31	pfss	pfss	NOUN
ejpam-5768	297	32	and	and	CCONJ
ejpam-5768	297	33	its	its	PRON
ejpam-5768	297	34	image	image	NOUN
ejpam-5768	297	35	.	.	PUNCT
ejpam-5768	298	1	a.	a.	NOUN
ejpam-5768	298	2	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	298	3	/	/	SYM
ejpam-5768	298	4	eur	eur	PROPN
ejpam-5768	298	5	.	.	PUNCT
ejpam-5768	299	1	j.	j.	PROPN
ejpam-5768	299	2	pure	pure	PROPN
ejpam-5768	299	3	appl	appl	PROPN
ejpam-5768	299	4	.	.	PROPN
ejpam-5768	299	5	math	math	PROPN
ejpam-5768	299	6	,	,	PUNCT
ejpam-5768	299	7	18	18	NUM
ejpam-5768	299	8	(	(	PUNCT
ejpam-5768	299	9	2	2	NUM
ejpam-5768	299	10	)	)	PUNCT
ejpam-5768	299	11	(	(	PUNCT
ejpam-5768	299	12	2025	2025	NUM
ejpam-5768	299	13	)	)	PUNCT
ejpam-5768	299	14	,	,	PUNCT
ejpam-5768	299	15	5768	5768	NUM
ejpam-5768	299	16	10	10	NUM
ejpam-5768	299	17	of	of	ADP
ejpam-5768	299	18	17	17	NUM
ejpam-5768	299	19	theorem	theorem	NOUN
ejpam-5768	299	20	9	9	NUM
ejpam-5768	299	21	.	.	PUNCT
ejpam-5768	300	1	let	let	VERB
ejpam-5768	300	2	τ	τ	PROPN
ejpam-5768	300	3	:	:	PUNCT
ejpam-5768	300	4	p	p	X
ejpam-5768	300	5	−→	−→	NOUN
ejpam-5768	300	6	s	s	AUX
ejpam-5768	300	7	be	be	AUX
ejpam-5768	300	8	a	a	DET
ejpam-5768	300	9	homomorphism	homomorphism	NOUN
ejpam-5768	300	10	of	of	ADP
ejpam-5768	300	11	hx	hx	NOUN
ejpam-5768	300	12	-	-	PUNCT
ejpam-5768	300	13	groups	group	NOUN
ejpam-5768	300	14	.	.	PUNCT
ejpam-5768	301	1	if	if	SCONJ
ejpam-5768	301	2	υ	υ	PROPN
ejpam-5768	301	3	is	be	AUX
ejpam-5768	301	4	a	a	DET
ejpam-5768	301	5	pf	pf	PROPN
ejpam-5768	301	6	-	-	PUNCT
ejpam-5768	301	7	hx	hx	NOUN
ejpam-5768	301	8	-	-	PUNCT
ejpam-5768	301	9	sg	sg	NOUN
ejpam-5768	301	10	of	of	ADP
ejpam-5768	301	11	p	p	NOUN
ejpam-5768	301	12	,	,	PUNCT
ejpam-5768	301	13	then	then	ADV
ejpam-5768	301	14	τ(υ	τ(υ	X
ejpam-5768	301	15	)	)	PUNCT
ejpam-5768	301	16	is	be	AUX
ejpam-5768	301	17	a	a	DET
ejpam-5768	301	18	pf	pf	PROPN
ejpam-5768	301	19	-	-	PUNCT
ejpam-5768	301	20	hx	hx	NOUN
ejpam-5768	301	21	-	-	PUNCT
ejpam-5768	301	22	sg	sg	PROPN
ejpam-5768	301	23	of	of	ADP
ejpam-5768	301	24	s.	s.	PROPN
ejpam-5768	301	25	proof	proof	PROPN
ejpam-5768	301	26	.	.	PUNCT
ejpam-5768	302	1	suppose	suppose	VERB
ejpam-5768	302	2	that	that	SCONJ
ejpam-5768	302	3	τ(υ	τ(υ	PRON
ejpam-5768	302	4	)	)	PUNCT
ejpam-5768	302	5	=	=	SYM
ejpam-5768	302	6	{	{	PUNCT
ejpam-5768	302	7	(	(	PUNCT
ejpam-5768	302	8	τ(p	τ(p	NOUN
ejpam-5768	302	9	)	)	PUNCT
ejpam-5768	302	10	,	,	PUNCT
ejpam-5768	302	11	η	η	PROPN
ejpam-5768	302	12	,	,	PUNCT
ejpam-5768	302	13	η̂	η̂	NUM
ejpam-5768	302	14	)	)	PUNCT
ejpam-5768	302	15	:	:	PUNCT
ejpam-5768	302	16	τ(p	τ(p	NOUN
ejpam-5768	302	17	)	)	PUNCT
ejpam-5768	302	18	∈	∈	PROPN
ejpam-5768	302	19	s	s	PART
ejpam-5768	302	20	}	}	PUNCT
ejpam-5768	302	21	.	.	PUNCT
ejpam-5768	303	1	let	let	VERB
ejpam-5768	303	2	τ(p01	τ(p01	NOUN
ejpam-5768	303	3	)	)	PUNCT
ejpam-5768	303	4	,	,	PUNCT
ejpam-5768	303	5	τ(p01	τ(p01	NOUN
ejpam-5768	303	6	)	)	PUNCT
ejpam-5768	303	7	∈	∈	PROPN
ejpam-5768	303	8	s.	s.	PROPN
ejpam-5768	303	9	then	then	ADV
ejpam-5768	303	10	η2(τ(p01)(τ(p02	η2(τ(p01)(τ(p02	PROPN
ejpam-5768	303	11	)	)	PUNCT
ejpam-5768	303	12	)	)	PUNCT
ejpam-5768	303	13	−1	−1	NOUN
ejpam-5768	303	14	)	)	PUNCT
ejpam-5768	304	1	=	=	PUNCT
ejpam-5768	304	2	η2(τ(p01)τ(p	η2(τ(p01)τ(p	NOUN
ejpam-5768	304	3	−1	−1	NOUN
ejpam-5768	304	4	02	02	NUM
ejpam-5768	304	5	)	)	PUNCT
ejpam-5768	304	6	)	)	PUNCT
ejpam-5768	305	1	=	=	PUNCT
ejpam-5768	305	2	η2(τ(p01p	η2(τ(p01p	NOUN
ejpam-5768	305	3	−1	−1	NOUN
ejpam-5768	305	4	02	02	NUM
ejpam-5768	305	5	)	)	PUNCT
ejpam-5768	305	6	≥	≥	NOUN
ejpam-5768	305	7	ῡ	ῡ	NOUN
ejpam-5768	306	1	2(p01p	2(p01p	NUM
ejpam-5768	306	2	−1	−1	NOUN
ejpam-5768	306	3	02	02	NUM
ejpam-5768	306	4	)	)	PUNCT
ejpam-5768	306	5	≥	≥	NOUN
ejpam-5768	306	6	min{ῡ	min{ῡ	NOUN
ejpam-5768	306	7	2(p01	2(p01	NUM
ejpam-5768	306	8	,	,	PUNCT
ejpam-5768	306	9	ῡ	ῡ	PROPN
ejpam-5768	306	10	2(p02	2(p02	NUM
ejpam-5768	306	11	)	)	PUNCT
ejpam-5768	306	12	}	}	PUNCT
ejpam-5768	306	13	=	=	SYM
ejpam-5768	306	14	min{η2(τ(p01	min{η2(τ(p01	NOUN
ejpam-5768	306	15	)	)	PUNCT
ejpam-5768	306	16	)	)	PUNCT
ejpam-5768	306	17	,	,	PUNCT
ejpam-5768	306	18	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	306	19	)	)	PUNCT
ejpam-5768	306	20	)	)	PUNCT
ejpam-5768	306	21	}	}	PUNCT
ejpam-5768	307	1	also	also	ADV
ejpam-5768	307	2	,	,	PUNCT
ejpam-5768	307	3	η̂2(τ(p01)(τ(p02	η̂2(τ(p01)(τ(p02	PROPN
ejpam-5768	307	4	)	)	PUNCT
ejpam-5768	307	5	)	)	PUNCT
ejpam-5768	307	6	−1	−1	NOUN
ejpam-5768	307	7	)	)	PUNCT
ejpam-5768	308	1	=	=	PUNCT
ejpam-5768	308	2	η̂2(τ(p01)τ(p	η̂2(τ(p01)τ(p	NOUN
ejpam-5768	308	3	−1	−1	NOUN
ejpam-5768	308	4	02	02	NUM
ejpam-5768	308	5	)	)	PUNCT
ejpam-5768	308	6	)	)	PUNCT
ejpam-5768	309	1	=	=	PUNCT
ejpam-5768	309	2	η̂2(τ(p01p	η̂2(τ(p01p	NOUN
ejpam-5768	309	3	−1	−1	NOUN
ejpam-5768	309	4	02	02	NUM
ejpam-5768	309	5	)	)	PUNCT
ejpam-5768	309	6	≤	≤	NOUN
ejpam-5768	309	7	υ̂	υ̂	VERB
ejpam-5768	309	8	2(p01p	2(p01p	NUM
ejpam-5768	309	9	−1	−1	NOUN
ejpam-5768	309	10	02	02	NUM
ejpam-5768	309	11	)	)	PUNCT
ejpam-5768	309	12	≤	≤	NOUN
ejpam-5768	309	13	max{υ̂	max{υ̂	NOUN
ejpam-5768	309	14	2(p01	2(p01	NUM
ejpam-5768	309	15	,	,	PUNCT
ejpam-5768	309	16	υ̂	υ̂	X
ejpam-5768	309	17	2(p02	2(p02	NUM
ejpam-5768	309	18	)	)	PUNCT
ejpam-5768	309	19	}	}	PUNCT
ejpam-5768	309	20	=	=	SYM
ejpam-5768	309	21	max{η̂2(τ(p01	max{η̂2(τ(p01	NOUN
ejpam-5768	309	22	)	)	PUNCT
ejpam-5768	309	23	)	)	PUNCT
ejpam-5768	309	24	,	,	PUNCT
ejpam-5768	309	25	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	309	26	)	)	PUNCT
ejpam-5768	309	27	)	)	PUNCT
ejpam-5768	309	28	}	}	PUNCT
ejpam-5768	309	29	hence	hence	ADV
ejpam-5768	309	30	τ(υ	τ(υ	NOUN
ejpam-5768	309	31	)	)	PUNCT
ejpam-5768	309	32	is	be	AUX
ejpam-5768	309	33	a	a	DET
ejpam-5768	309	34	pf	pf	PROPN
ejpam-5768	309	35	-	-	PUNCT
ejpam-5768	309	36	hx	hx	NOUN
ejpam-5768	309	37	-	-	PUNCT
ejpam-5768	309	38	sg	sg	PROPN
ejpam-5768	309	39	of	of	ADP
ejpam-5768	309	40	s.	s.	PROPN
ejpam-5768	309	41	theorem	theorem	VERB
ejpam-5768	309	42	10	10	NUM
ejpam-5768	309	43	.	.	PUNCT
ejpam-5768	310	1	let	let	VERB
ejpam-5768	310	2	τ	τ	PROPN
ejpam-5768	310	3	:	:	PUNCT
ejpam-5768	310	4	p	p	X
ejpam-5768	310	5	−→	−→	NOUN
ejpam-5768	310	6	s	s	AUX
ejpam-5768	310	7	be	be	AUX
ejpam-5768	310	8	a	a	DET
ejpam-5768	310	9	homomorphism	homomorphism	NOUN
ejpam-5768	310	10	of	of	ADP
ejpam-5768	310	11	hx	hx	NOUN
ejpam-5768	310	12	-	-	PUNCT
ejpam-5768	310	13	groups	group	NOUN
ejpam-5768	310	14	.	.	PUNCT
ejpam-5768	311	1	if	if	SCONJ
ejpam-5768	311	2	υ	υ	PROPN
ejpam-5768	311	3	is	be	AUX
ejpam-5768	311	4	a	a	DET
ejpam-5768	311	5	pf	pf	PROPN
ejpam-5768	311	6	-	-	PUNCT
ejpam-5768	311	7	hx	hx	PROPN
ejpam-5768	311	8	-	-	PUNCT
ejpam-5768	311	9	nsg	nsg	PROPN
ejpam-5768	311	10	of	of	ADP
ejpam-5768	311	11	p	p	PROPN
ejpam-5768	311	12	,	,	PUNCT
ejpam-5768	311	13	then	then	ADV
ejpam-5768	311	14	τ(υ	τ(υ	X
ejpam-5768	311	15	)	)	PUNCT
ejpam-5768	311	16	is	be	AUX
ejpam-5768	311	17	a	a	DET
ejpam-5768	311	18	pf	pf	PROPN
ejpam-5768	311	19	hx	hx	PROPN
ejpam-5768	311	20	-	-	PUNCT
ejpam-5768	311	21	nsg	nsg	PROPN
ejpam-5768	311	22	of	of	ADP
ejpam-5768	311	23	s.	s.	PROPN
ejpam-5768	311	24	proof	proof	PROPN
ejpam-5768	311	25	.	.	PUNCT
ejpam-5768	312	1	suppose	suppose	VERB
ejpam-5768	312	2	that	that	SCONJ
ejpam-5768	312	3	τ(υ	τ(υ	PRON
ejpam-5768	312	4	)	)	PUNCT
ejpam-5768	312	5	=	=	SYM
ejpam-5768	312	6	{	{	PUNCT
ejpam-5768	312	7	(	(	PUNCT
ejpam-5768	312	8	τ(p	τ(p	NOUN
ejpam-5768	312	9	)	)	PUNCT
ejpam-5768	312	10	,	,	PUNCT
ejpam-5768	312	11	η	η	PROPN
ejpam-5768	312	12	,	,	PUNCT
ejpam-5768	312	13	η̂	η̂	NUM
ejpam-5768	312	14	)	)	PUNCT
ejpam-5768	312	15	:	:	PUNCT
ejpam-5768	312	16	τ(p	τ(p	NOUN
ejpam-5768	312	17	)	)	PUNCT
ejpam-5768	312	18	∈	∈	PROPN
ejpam-5768	312	19	s	s	PART
ejpam-5768	312	20	}	}	PUNCT
ejpam-5768	312	21	.	.	PUNCT
ejpam-5768	313	1	since	since	SCONJ
ejpam-5768	313	2	υ	υ	PROPN
ejpam-5768	313	3	is	be	AUX
ejpam-5768	313	4	a	a	DET
ejpam-5768	313	5	pf	pf	PROPN
ejpam-5768	313	6	-	-	PUNCT
ejpam-5768	313	7	hx	hx	PROPN
ejpam-5768	313	8	-	-	PUNCT
ejpam-5768	313	9	nsg	nsg	PROPN
ejpam-5768	313	10	of	of	ADP
ejpam-5768	313	11	p	p	PROPN
ejpam-5768	313	12	,	,	PUNCT
ejpam-5768	313	13	then	then	ADV
ejpam-5768	313	14	it	it	PRON
ejpam-5768	313	15	is	be	AUX
ejpam-5768	313	16	a	a	DET
ejpam-5768	313	17	pf	pf	PROPN
ejpam-5768	313	18	-	-	PUNCT
ejpam-5768	313	19	hx	hx	NOUN
ejpam-5768	313	20	-	-	PUNCT
ejpam-5768	313	21	sg	sg	PROPN
ejpam-5768	313	22	and	and	CCONJ
ejpam-5768	313	23	,	,	PUNCT
ejpam-5768	313	24	by	by	ADP
ejpam-5768	313	25	the	the	DET
ejpam-5768	313	26	previous	previous	ADJ
ejpam-5768	313	27	theorem	theorem	NOUN
ejpam-5768	313	28	,	,	PUNCT
ejpam-5768	313	29	τ(υ	τ(υ	X
ejpam-5768	313	30	)	)	PUNCT
ejpam-5768	313	31	is	be	AUX
ejpam-5768	313	32	a	a	DET
ejpam-5768	313	33	pf	pf	PROPN
ejpam-5768	313	34	-	-	PUNCT
ejpam-5768	313	35	hx	hx	NOUN
ejpam-5768	313	36	-	-	PUNCT
ejpam-5768	313	37	sg	sg	PROPN
ejpam-5768	313	38	of	of	ADP
ejpam-5768	313	39	s.	s.	PROPN
ejpam-5768	313	40	we	we	PRON
ejpam-5768	313	41	need	need	VERB
ejpam-5768	313	42	to	to	PART
ejpam-5768	313	43	prove	prove	VERB
ejpam-5768	313	44	that	that	SCONJ
ejpam-5768	313	45	it	it	PRON
ejpam-5768	313	46	is	be	AUX
ejpam-5768	313	47	normal	normal	ADJ
ejpam-5768	313	48	.	.	PUNCT
ejpam-5768	314	1	let	let	VERB
ejpam-5768	314	2	τ(p01	τ(p01	NOUN
ejpam-5768	314	3	)	)	PUNCT
ejpam-5768	314	4	,	,	PUNCT
ejpam-5768	314	5	τ(p01	τ(p01	NOUN
ejpam-5768	314	6	)	)	PUNCT
ejpam-5768	314	7	∈	∈	PROPN
ejpam-5768	314	8	s.	s.	PROPN
ejpam-5768	314	9	then	then	ADV
ejpam-5768	314	10	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	PROPN
ejpam-5768	314	11	)	)	PUNCT
ejpam-5768	314	12	)	)	PUNCT
ejpam-5768	315	1	=	=	SYM
ejpam-5768	315	2	(	(	PUNCT
ejpam-5768	315	3	τ(p01	τ(p01	NOUN
ejpam-5768	315	4	)	)	PUNCT
ejpam-5768	315	5	)	)	PUNCT
ejpam-5768	315	6	−1η2(τ(p02	−1η2(τ(p02	X
ejpam-5768	315	7	)	)	PUNCT
ejpam-5768	315	8	)	)	PUNCT
ejpam-5768	316	1	=	=	SYM
ejpam-5768	316	2	(	(	PUNCT
ejpam-5768	316	3	τ(p01	τ(p01	NOUN
ejpam-5768	316	4	)	)	PUNCT
ejpam-5768	316	5	)	)	PUNCT
ejpam-5768	316	6	−1ῡ	−1ῡ	NUM
ejpam-5768	317	1	2((p02	2((p02	NUM
ejpam-5768	317	2	)	)	PUNCT
ejpam-5768	317	3	=	=	NOUN
ejpam-5768	317	4	(	(	PUNCT
ejpam-5768	317	5	τ(p01	τ(p01	NOUN
ejpam-5768	317	6	)	)	PUNCT
ejpam-5768	317	7	)	)	PUNCT
ejpam-5768	317	8	−1ῡ	−1ῡ	NUM
ejpam-5768	317	9	2((p01	2((p01	NUM
ejpam-5768	317	10	)	)	PUNCT
ejpam-5768	317	11	−1p02p01	−1p02p01	ADJ
ejpam-5768	317	12	)	)	PUNCT
ejpam-5768	317	13	≤	≤	NOUN
ejpam-5768	317	14	(	(	PUNCT
ejpam-5768	317	15	τ(p01	τ(p01	NOUN
ejpam-5768	317	16	)	)	PUNCT
ejpam-5768	317	17	)	)	PUNCT
ejpam-5768	317	18	−1η2(τ((p01	−1η2(τ((p01	NOUN
ejpam-5768	317	19	)	)	PUNCT
ejpam-5768	317	20	−1p02p01	−1p02p01	PROPN
ejpam-5768	317	21	)	)	PUNCT
ejpam-5768	317	22	)	)	PUNCT
ejpam-5768	318	1	=	=	SYM
ejpam-5768	318	2	η2(τ(p01)τ((p01	η2(τ(p01)τ((p01	NOUN
ejpam-5768	318	3	)	)	PUNCT
ejpam-5768	318	4	−1p02p01	−1p02p01	PROPN
ejpam-5768	318	5	)	)	PUNCT
ejpam-5768	318	6	)	)	PUNCT
ejpam-5768	319	1	=	=	SYM
ejpam-5768	319	2	η2(τ(p01(p01	η2(τ(p01(p01	NOUN
ejpam-5768	319	3	)	)	PUNCT
ejpam-5768	319	4	−1p02p01	−1p02p01	PROPN
ejpam-5768	319	5	)	)	PUNCT
ejpam-5768	319	6	)	)	PUNCT
ejpam-5768	320	1	=	=	SYM
ejpam-5768	320	2	η2(τ(p02p01	η2(τ(p02p01	NUM
ejpam-5768	320	3	)	)	PUNCT
ejpam-5768	320	4	)	)	PUNCT
ejpam-5768	320	5	=	=	SYM
ejpam-5768	320	6	η2(τ(p02τ(p01	η2(τ(p02τ(p01	NOUN
ejpam-5768	320	7	)	)	PUNCT
ejpam-5768	320	8	)	)	PUNCT
ejpam-5768	320	9	)	)	PUNCT
ejpam-5768	321	1	thus	thus	ADV
ejpam-5768	321	2	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	X
ejpam-5768	321	3	)	)	PUNCT
ejpam-5768	321	4	)	)	PUNCT
ejpam-5768	321	5	≤	≤	NUM
ejpam-5768	321	6	η2(τ(p02τ(p01	η2(τ(p02τ(p01	NOUN
ejpam-5768	321	7	)	)	PUNCT
ejpam-5768	321	8	)	)	PUNCT
ejpam-5768	321	9	)	)	PUNCT
ejpam-5768	321	10	.	.	PUNCT
ejpam-5768	322	1	on	on	ADP
ejpam-5768	322	2	the	the	DET
ejpam-5768	322	3	other	other	ADJ
ejpam-5768	322	4	hand	hand	NOUN
ejpam-5768	322	5	,	,	PUNCT
ejpam-5768	322	6	η2(τ(p02)τ(p01	η2(τ(p02)τ(p01	PROPN
ejpam-5768	322	7	)	)	PUNCT
ejpam-5768	322	8	)	)	PUNCT
ejpam-5768	323	1	=	=	SYM
ejpam-5768	323	2	(	(	PUNCT
ejpam-5768	323	3	τ(p02	τ(p02	X
ejpam-5768	323	4	)	)	PUNCT
ejpam-5768	323	5	)	)	PUNCT
ejpam-5768	323	6	−1η2(τ(p01	−1η2(τ(p01	NOUN
ejpam-5768	323	7	)	)	PUNCT
ejpam-5768	323	8	a.	a.	NOUN
ejpam-5768	323	9	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	323	10	/	/	SYM
ejpam-5768	323	11	eur	eur	PROPN
ejpam-5768	323	12	.	.	PUNCT
ejpam-5768	324	1	j.	j.	PROPN
ejpam-5768	324	2	pure	pure	PROPN
ejpam-5768	324	3	appl	appl	PROPN
ejpam-5768	324	4	.	.	PROPN
ejpam-5768	324	5	math	math	PROPN
ejpam-5768	324	6	,	,	PUNCT
ejpam-5768	324	7	18	18	NUM
ejpam-5768	324	8	(	(	PUNCT
ejpam-5768	324	9	2	2	NUM
ejpam-5768	324	10	)	)	PUNCT
ejpam-5768	324	11	(	(	PUNCT
ejpam-5768	324	12	2025	2025	NUM
ejpam-5768	324	13	)	)	PUNCT
ejpam-5768	324	14	,	,	PUNCT
ejpam-5768	324	15	5768	5768	NUM
ejpam-5768	324	16	11	11	NUM
ejpam-5768	324	17	of	of	ADP
ejpam-5768	324	18	17	17	NUM
ejpam-5768	324	19	=	=	SYM
ejpam-5768	324	20	(	(	PUNCT
ejpam-5768	324	21	τ(p02	τ(p02	X
ejpam-5768	324	22	)	)	PUNCT
ejpam-5768	324	23	)	)	PUNCT
ejpam-5768	324	24	−1ῡ	−1ῡ	NUM
ejpam-5768	325	1	2((p01	2((p01	NUM
ejpam-5768	325	2	)	)	PUNCT
ejpam-5768	325	3	=	=	SYM
ejpam-5768	325	4	(	(	PUNCT
ejpam-5768	325	5	τ(p02	τ(p02	X
ejpam-5768	325	6	)	)	PUNCT
ejpam-5768	325	7	)	)	PUNCT
ejpam-5768	326	1	−1ῡ	−1ῡ	NUM
ejpam-5768	326	2	2((p02	2((p02	NUM
ejpam-5768	326	3	)	)	PUNCT
ejpam-5768	326	4	−1p01p02	−1p01p02	NOUN
ejpam-5768	326	5	)	)	PUNCT
ejpam-5768	326	6	≤	≤	NOUN
ejpam-5768	326	7	(	(	PUNCT
ejpam-5768	326	8	τ(p02	τ(p02	NOUN
ejpam-5768	326	9	)	)	PUNCT
ejpam-5768	326	10	)	)	PUNCT
ejpam-5768	327	1	−1η2(τ((p02	−1η2(τ((p02	VERB
ejpam-5768	327	2	)	)	PUNCT
ejpam-5768	327	3	−1p01p02	−1p01p02	NOUN
ejpam-5768	327	4	)	)	PUNCT
ejpam-5768	327	5	)	)	PUNCT
ejpam-5768	328	1	=	=	SYM
ejpam-5768	328	2	η2(τ(p02)τ((p02	η2(τ(p02)τ((p02	X
ejpam-5768	328	3	)	)	PUNCT
ejpam-5768	328	4	−1p01p02	−1p01p02	NOUN
ejpam-5768	328	5	)	)	PUNCT
ejpam-5768	328	6	)	)	PUNCT
ejpam-5768	329	1	=	=	SYM
ejpam-5768	329	2	η2(τ(p02(p02	η2(τ(p02(p02	NOUN
ejpam-5768	329	3	)	)	PUNCT
ejpam-5768	329	4	−1p01p02	−1p01p02	NOUN
ejpam-5768	329	5	)	)	PUNCT
ejpam-5768	329	6	)	)	PUNCT
ejpam-5768	330	1	=	=	SYM
ejpam-5768	330	2	η2(τ(p01p02	η2(τ(p01p02	NUM
ejpam-5768	330	3	)	)	PUNCT
ejpam-5768	330	4	)	)	PUNCT
ejpam-5768	331	1	=	=	SYM
ejpam-5768	331	2	η2(τ(p01τ(p02	η2(τ(p01τ(p02	NOUN
ejpam-5768	331	3	)	)	PUNCT
ejpam-5768	331	4	)	)	PUNCT
ejpam-5768	331	5	)	)	PUNCT
ejpam-5768	331	6	.	.	PUNCT
ejpam-5768	332	1	then	then	ADV
ejpam-5768	332	2	η2(τ(p02)τ(p01	η2(τ(p02)τ(p01	PROPN
ejpam-5768	332	3	)	)	PUNCT
ejpam-5768	332	4	)	)	PUNCT
ejpam-5768	333	1	≤	≤	NUM
ejpam-5768	333	2	η2(τ(p01τ(p02	η2(τ(p01τ(p02	NOUN
ejpam-5768	333	3	)	)	PUNCT
ejpam-5768	333	4	)	)	PUNCT
ejpam-5768	333	5	)	)	PUNCT
ejpam-5768	333	6	.	.	PUNCT
ejpam-5768	334	1	hence	hence	ADV
ejpam-5768	334	2	η2(τ(p01τ(p02	η2(τ(p01τ(p02	NOUN
ejpam-5768	334	3	)	)	PUNCT
ejpam-5768	334	4	)	)	PUNCT
ejpam-5768	334	5	)	)	PUNCT
ejpam-5768	335	1	=	=	SYM
ejpam-5768	335	2	η2(τ(p02τ(p01	η2(τ(p02τ(p01	NOUN
ejpam-5768	335	3	)	)	PUNCT
ejpam-5768	335	4	)	)	PUNCT
ejpam-5768	335	5	)	)	PUNCT
ejpam-5768	335	6	.	.	PUNCT
ejpam-5768	336	1	in	in	ADP
ejpam-5768	336	2	addition	addition	NOUN
ejpam-5768	336	3	,	,	PUNCT
ejpam-5768	336	4	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	NOUN
ejpam-5768	336	5	)	)	PUNCT
ejpam-5768	336	6	)	)	PUNCT
ejpam-5768	337	1	=	=	SYM
ejpam-5768	337	2	(	(	PUNCT
ejpam-5768	337	3	τ(p01	τ(p01	NOUN
ejpam-5768	337	4	)	)	PUNCT
ejpam-5768	337	5	)	)	PUNCT
ejpam-5768	337	6	−1η̂2(τ(p02	−1η̂2(τ(p02	PROPN
ejpam-5768	337	7	)	)	PUNCT
ejpam-5768	337	8	)	)	PUNCT
ejpam-5768	338	1	=	=	SYM
ejpam-5768	338	2	(	(	PUNCT
ejpam-5768	338	3	τ(p01	τ(p01	NOUN
ejpam-5768	338	4	)	)	PUNCT
ejpam-5768	338	5	)	)	PUNCT
ejpam-5768	338	6	−1υ̂	−1υ̂	VERB
ejpam-5768	338	7	2((p02	2((p02	NUM
ejpam-5768	338	8	)	)	PUNCT
ejpam-5768	338	9	=	=	SYM
ejpam-5768	338	10	(	(	PUNCT
ejpam-5768	338	11	τ(p01	τ(p01	NOUN
ejpam-5768	338	12	)	)	PUNCT
ejpam-5768	338	13	)	)	PUNCT
ejpam-5768	339	1	−1υ̂	−1υ̂	VERB
ejpam-5768	339	2	2((p01	2((p01	NUM
ejpam-5768	339	3	)	)	PUNCT
ejpam-5768	339	4	−1p02p01	−1p02p01	PROPN
ejpam-5768	339	5	)	)	PUNCT
ejpam-5768	339	6	≥	≥	NOUN
ejpam-5768	339	7	(	(	PUNCT
ejpam-5768	339	8	τ(p01	τ(p01	NOUN
ejpam-5768	339	9	)	)	PUNCT
ejpam-5768	339	10	)	)	PUNCT
ejpam-5768	340	1	−1η̂2(τ((p01	−1η̂2(τ((p01	PROPN
ejpam-5768	340	2	)	)	PUNCT
ejpam-5768	340	3	−1p02p01	−1p02p01	PROPN
ejpam-5768	340	4	)	)	PUNCT
ejpam-5768	340	5	)	)	PUNCT
ejpam-5768	341	1	=	=	SYM
ejpam-5768	341	2	η̂2(τ(p01)[]τ((p01	η̂2(τ(p01)[]τ((p01	PROPN
ejpam-5768	341	3	)	)	PUNCT
ejpam-5768	341	4	−1p02p01	−1p02p01	PROPN
ejpam-5768	341	5	)	)	PUNCT
ejpam-5768	341	6	)	)	PUNCT
ejpam-5768	342	1	=	=	SYM
ejpam-5768	342	2	η̂2(τ(p01(p01	η̂2(τ(p01(p01	NOUN
ejpam-5768	342	3	)	)	PUNCT
ejpam-5768	342	4	−1p02p01	−1p02p01	NOUN
ejpam-5768	342	5	)	)	PUNCT
ejpam-5768	342	6	)	)	PUNCT
ejpam-5768	343	1	=	=	SYM
ejpam-5768	343	2	η̂2(τ(p02p01	η̂2(τ(p02p01	NOUN
ejpam-5768	343	3	)	)	PUNCT
ejpam-5768	343	4	)	)	PUNCT
ejpam-5768	344	1	=	=	SYM
ejpam-5768	344	2	η̂2(τ(p02τ(p01	η̂2(τ(p02τ(p01	NOUN
ejpam-5768	344	3	)	)	PUNCT
ejpam-5768	344	4	)	)	PUNCT
ejpam-5768	344	5	)	)	PUNCT
ejpam-5768	345	1	this	this	PRON
ejpam-5768	345	2	implies	imply	VERB
ejpam-5768	345	3	that	that	SCONJ
ejpam-5768	345	4	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	NOUN
ejpam-5768	345	5	)	)	PUNCT
ejpam-5768	345	6	)	)	PUNCT
ejpam-5768	345	7	≥	≥	NOUN
ejpam-5768	345	8	η̂2(τ(p02τ(p01	η̂2(τ(p02τ(p01	NOUN
ejpam-5768	345	9	)	)	PUNCT
ejpam-5768	345	10	)	)	PUNCT
ejpam-5768	345	11	)	)	PUNCT
ejpam-5768	345	12	.	.	PUNCT
ejpam-5768	346	1	on	on	ADP
ejpam-5768	346	2	the	the	DET
ejpam-5768	346	3	other	other	ADJ
ejpam-5768	346	4	hand	hand	NOUN
ejpam-5768	346	5	,	,	PUNCT
ejpam-5768	346	6	η̂2(τ(p02)τ(p01	η̂2(τ(p02)τ(p01	NOUN
ejpam-5768	346	7	)	)	PUNCT
ejpam-5768	346	8	)	)	PUNCT
ejpam-5768	347	1	=	=	SYM
ejpam-5768	347	2	(	(	PUNCT
ejpam-5768	347	3	τ(p02	τ(p02	X
ejpam-5768	347	4	)	)	PUNCT
ejpam-5768	347	5	)	)	PUNCT
ejpam-5768	347	6	−1η̂2(τ(p01	−1η̂2(τ(p01	NOUN
ejpam-5768	347	7	)	)	PUNCT
ejpam-5768	347	8	=	=	SYM
ejpam-5768	347	9	(	(	PUNCT
ejpam-5768	347	10	τ(p02	τ(p02	X
ejpam-5768	347	11	)	)	PUNCT
ejpam-5768	347	12	)	)	PUNCT
ejpam-5768	348	1	−1υ̂	−1υ̂	VERB
ejpam-5768	348	2	2((p01	2((p01	NUM
ejpam-5768	348	3	)	)	PUNCT
ejpam-5768	348	4	=	=	SYM
ejpam-5768	348	5	(	(	PUNCT
ejpam-5768	348	6	τ(p02	τ(p02	X
ejpam-5768	348	7	)	)	PUNCT
ejpam-5768	348	8	)	)	PUNCT
ejpam-5768	349	1	−1υ̂	−1υ̂	VERB
ejpam-5768	349	2	2((p02	2((p02	NUM
ejpam-5768	349	3	)	)	PUNCT
ejpam-5768	349	4	−1p01p02	−1p01p02	NOUN
ejpam-5768	349	5	)	)	PUNCT
ejpam-5768	349	6	≥	≥	NOUN
ejpam-5768	349	7	(	(	PUNCT
ejpam-5768	349	8	τ(p02	τ(p02	X
ejpam-5768	349	9	)	)	PUNCT
ejpam-5768	349	10	)	)	PUNCT
ejpam-5768	350	1	−1η̂2(τ((p02	−1η̂2(τ((p02	PROPN
ejpam-5768	350	2	)	)	PUNCT
ejpam-5768	350	3	−1p01p02	−1p01p02	NOUN
ejpam-5768	350	4	)	)	PUNCT
ejpam-5768	350	5	)	)	PUNCT
ejpam-5768	350	6	=	=	SYM
ejpam-5768	350	7	η̂2(τ(p02)τ((p02	η̂2(τ(p02)τ((p02	X
ejpam-5768	350	8	)	)	PUNCT
ejpam-5768	350	9	−1p01p02	−1p01p02	NOUN
ejpam-5768	350	10	)	)	PUNCT
ejpam-5768	350	11	)	)	PUNCT
ejpam-5768	351	1	=	=	SYM
ejpam-5768	351	2	η̂2(τ(p02(p02	η̂2(τ(p02(p02	PROPN
ejpam-5768	351	3	)	)	PUNCT
ejpam-5768	351	4	−1p01p02	−1p01p02	NOUN
ejpam-5768	351	5	)	)	PUNCT
ejpam-5768	351	6	)	)	PUNCT
ejpam-5768	352	1	=	=	PUNCT
ejpam-5768	352	2	η̂2(τ(p01p02	η̂2(τ(p01p02	NOUN
ejpam-5768	352	3	)	)	PUNCT
ejpam-5768	352	4	)	)	PUNCT
ejpam-5768	352	5	=	=	SYM
ejpam-5768	352	6	η̂2(τ(p01τ(p02	η̂2(τ(p01τ(p02	X
ejpam-5768	352	7	)	)	PUNCT
ejpam-5768	352	8	)	)	PUNCT
ejpam-5768	352	9	)	)	PUNCT
ejpam-5768	352	10	then	then	ADV
ejpam-5768	352	11	η̂2(τ(p02)τ(p01	η̂2(τ(p02)τ(p01	NOUN
ejpam-5768	352	12	)	)	PUNCT
ejpam-5768	352	13	)	)	PUNCT
ejpam-5768	352	14	≥	≥	NOUN
ejpam-5768	352	15	η̂2(τ(p01τ(p02	η̂2(τ(p01τ(p02	NOUN
ejpam-5768	352	16	)	)	PUNCT
ejpam-5768	352	17	)	)	PUNCT
ejpam-5768	352	18	)	)	PUNCT
ejpam-5768	352	19	.	.	PUNCT
ejpam-5768	353	1	hence	hence	ADV
ejpam-5768	353	2	η̂2(τ(p01τ(p02	η̂2(τ(p01τ(p02	NOUN
ejpam-5768	353	3	)	)	PUNCT
ejpam-5768	353	4	)	)	PUNCT
ejpam-5768	353	5	)	)	PUNCT
ejpam-5768	354	1	=	=	SYM
ejpam-5768	354	2	η̂2(τ(p02τ(p01	η̂2(τ(p02τ(p01	NOUN
ejpam-5768	354	3	)	)	PUNCT
ejpam-5768	354	4	)	)	PUNCT
ejpam-5768	354	5	)	)	PUNCT
ejpam-5768	354	6	.	.	PUNCT
ejpam-5768	355	1	therefore	therefore	ADV
ejpam-5768	355	2	,	,	PUNCT
ejpam-5768	355	3	τ(υ	τ(υ	X
ejpam-5768	355	4	)	)	PUNCT
ejpam-5768	355	5	is	be	AUX
ejpam-5768	355	6	a	a	DET
ejpam-5768	355	7	pf	pf	VERB
ejpam-5768	355	8	-	-	PUNCT
ejpam-5768	355	9	nhx	nhx	NOUN
ejpam-5768	355	10	-	-	PUNCT
ejpam-5768	355	11	sg	sg	NOUN
ejpam-5768	355	12	of	of	ADP
ejpam-5768	355	13	s.	s.	PROPN
ejpam-5768	355	14	theorem	theorem	VERB
ejpam-5768	355	15	11	11	NUM
ejpam-5768	355	16	.	.	PUNCT
ejpam-5768	356	1	let	let	VERB
ejpam-5768	356	2	τ	τ	PROPN
ejpam-5768	356	3	:	:	PUNCT
ejpam-5768	356	4	p	p	X
ejpam-5768	356	5	−→	−→	NOUN
ejpam-5768	356	6	s	s	AUX
ejpam-5768	356	7	be	be	AUX
ejpam-5768	356	8	an	an	DET
ejpam-5768	356	9	antihomomorphism	antihomomorphism	NOUN
ejpam-5768	356	10	of	of	ADP
ejpam-5768	356	11	hx	hx	NOUN
ejpam-5768	356	12	-	-	PUNCT
ejpam-5768	356	13	groups	group	NOUN
ejpam-5768	356	14	.	.	PUNCT
ejpam-5768	357	1	if	if	SCONJ
ejpam-5768	357	2	υ	υ	PROPN
ejpam-5768	357	3	is	be	AUX
ejpam-5768	357	4	a	a	DET
ejpam-5768	357	5	pf	pf	NOUN
ejpam-5768	357	6	-	-	PUNCT
ejpam-5768	357	7	hxsg	hxsg	NOUN
ejpam-5768	357	8	of	of	ADP
ejpam-5768	357	9	p	p	NOUN
ejpam-5768	357	10	,	,	PUNCT
ejpam-5768	357	11	then	then	ADV
ejpam-5768	357	12	τ(υ	τ(υ	X
ejpam-5768	357	13	)	)	PUNCT
ejpam-5768	357	14	is	be	AUX
ejpam-5768	357	15	a	a	DET
ejpam-5768	357	16	pf	pf	PROPN
ejpam-5768	357	17	-	-	PUNCT
ejpam-5768	357	18	hx	hx	NOUN
ejpam-5768	357	19	-	-	PUNCT
ejpam-5768	357	20	sg	sg	PROPN
ejpam-5768	357	21	of	of	ADP
ejpam-5768	357	22	s.	s.	PROPN
ejpam-5768	357	23	a.	a.	PROPN
ejpam-5768	357	24	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	357	25	/	/	SYM
ejpam-5768	357	26	eur	eur	PROPN
ejpam-5768	357	27	.	.	PUNCT
ejpam-5768	358	1	j.	j.	PROPN
ejpam-5768	358	2	pure	pure	PROPN
ejpam-5768	358	3	appl	appl	PROPN
ejpam-5768	358	4	.	.	PROPN
ejpam-5768	358	5	math	math	PROPN
ejpam-5768	358	6	,	,	PUNCT
ejpam-5768	358	7	18	18	NUM
ejpam-5768	358	8	(	(	PUNCT
ejpam-5768	358	9	2	2	NUM
ejpam-5768	358	10	)	)	PUNCT
ejpam-5768	358	11	(	(	PUNCT
ejpam-5768	358	12	2025	2025	NUM
ejpam-5768	358	13	)	)	PUNCT
ejpam-5768	358	14	,	,	PUNCT
ejpam-5768	358	15	5768	5768	NUM
ejpam-5768	358	16	12	12	NUM
ejpam-5768	358	17	of	of	ADP
ejpam-5768	358	18	17	17	NUM
ejpam-5768	358	19	proof	proof	NOUN
ejpam-5768	358	20	.	.	PUNCT
ejpam-5768	358	21	suppose	suppose	VERB
ejpam-5768	358	22	that	that	SCONJ
ejpam-5768	358	23	τ(υ	τ(υ	PRON
ejpam-5768	358	24	)	)	PUNCT
ejpam-5768	358	25	=	=	SYM
ejpam-5768	358	26	{	{	PUNCT
ejpam-5768	358	27	(	(	PUNCT
ejpam-5768	358	28	τ(p	τ(p	NOUN
ejpam-5768	358	29	)	)	PUNCT
ejpam-5768	358	30	,	,	PUNCT
ejpam-5768	358	31	η	η	PROPN
ejpam-5768	358	32	,	,	PUNCT
ejpam-5768	358	33	η̂	η̂	NUM
ejpam-5768	358	34	)	)	PUNCT
ejpam-5768	358	35	:	:	PUNCT
ejpam-5768	358	36	τ(p	τ(p	NOUN
ejpam-5768	358	37	)	)	PUNCT
ejpam-5768	358	38	∈	∈	PROPN
ejpam-5768	358	39	s	s	PART
ejpam-5768	358	40	}	}	PUNCT
ejpam-5768	358	41	.	.	PUNCT
ejpam-5768	359	1	let	let	VERB
ejpam-5768	359	2	τ(p01	τ(p01	NOUN
ejpam-5768	359	3	)	)	PUNCT
ejpam-5768	359	4	,	,	PUNCT
ejpam-5768	359	5	τ(p01	τ(p01	NOUN
ejpam-5768	359	6	)	)	PUNCT
ejpam-5768	359	7	∈	∈	PROPN
ejpam-5768	359	8	s.	s.	PROPN
ejpam-5768	359	9	then	then	ADV
ejpam-5768	359	10	η2(τ(p01)(τ(p02	η2(τ(p01)(τ(p02	PROPN
ejpam-5768	359	11	)	)	PUNCT
ejpam-5768	359	12	)	)	PUNCT
ejpam-5768	359	13	−1	−1	NOUN
ejpam-5768	359	14	)	)	PUNCT
ejpam-5768	360	1	=	=	PUNCT
ejpam-5768	360	2	η2(τ(p01)τ(p	η2(τ(p01)τ(p	NOUN
ejpam-5768	360	3	−1	−1	NOUN
ejpam-5768	360	4	02	02	NUM
ejpam-5768	360	5	)	)	PUNCT
ejpam-5768	360	6	)	)	PUNCT
ejpam-5768	361	1	=	=	SYM
ejpam-5768	361	2	η2(τ(p−1	η2(τ(p−1	NOUN
ejpam-5768	361	3	02	02	NUM
ejpam-5768	361	4	p01	p01	NOUN
ejpam-5768	361	5	)	)	PUNCT
ejpam-5768	361	6	≥	≥	NOUN
ejpam-5768	361	7	ῡ	ῡ	NOUN
ejpam-5768	361	8	2(p−1	2(p−1	ADJ
ejpam-5768	361	9	02	02	NUM
ejpam-5768	361	10	p01	p01	NOUN
ejpam-5768	361	11	)	)	PUNCT
ejpam-5768	361	12	≥	≥	NOUN
ejpam-5768	361	13	min{ῡ	min{ῡ	NOUN
ejpam-5768	361	14	2(p02	2(p02	NUM
ejpam-5768	361	15	)	)	PUNCT
ejpam-5768	361	16	−1	−1	NOUN
ejpam-5768	361	17	,	,	PUNCT
ejpam-5768	361	18	ῡ	ῡ	PROPN
ejpam-5768	361	19	2(p01	2(p01	NUM
ejpam-5768	361	20	)	)	PUNCT
ejpam-5768	361	21	}	}	PUNCT
ejpam-5768	362	1	=	=	SYM
ejpam-5768	362	2	min{ῡ	min{ῡ	NOUN
ejpam-5768	362	3	2(p01	2(p01	NUM
ejpam-5768	362	4	)	)	PUNCT
ejpam-5768	362	5	,	,	PUNCT
ejpam-5768	362	6	ῡ	ῡ	PROPN
ejpam-5768	362	7	2(p02	2(p02	NUM
ejpam-5768	362	8	)	)	PUNCT
ejpam-5768	362	9	}	}	PUNCT
ejpam-5768	362	10	=	=	SYM
ejpam-5768	362	11	min{η2(τ(p01	min{η2(τ(p01	NOUN
ejpam-5768	362	12	)	)	PUNCT
ejpam-5768	362	13	)	)	PUNCT
ejpam-5768	362	14	,	,	PUNCT
ejpam-5768	362	15	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	362	16	)	)	PUNCT
ejpam-5768	362	17	)	)	PUNCT
ejpam-5768	362	18	}	}	PUNCT
ejpam-5768	363	1	also	also	ADV
ejpam-5768	363	2	,	,	PUNCT
ejpam-5768	363	3	η̂2(τ(p01)(τ(p02	η̂2(τ(p01)(τ(p02	PROPN
ejpam-5768	363	4	)	)	PUNCT
ejpam-5768	363	5	)	)	PUNCT
ejpam-5768	363	6	−1	−1	NOUN
ejpam-5768	363	7	)	)	PUNCT
ejpam-5768	364	1	=	=	PUNCT
ejpam-5768	364	2	η̂2(τ(p01)τ(p	η̂2(τ(p01)τ(p	NOUN
ejpam-5768	364	3	−1	−1	NOUN
ejpam-5768	364	4	02	02	NUM
ejpam-5768	364	5	)	)	PUNCT
ejpam-5768	364	6	)	)	PUNCT
ejpam-5768	365	1	=	=	PRON
ejpam-5768	365	2	η̂2(τ(p−1	η̂2(τ(p−1	VERB
ejpam-5768	365	3	02	02	NUM
ejpam-5768	365	4	p01	p01	NOUN
ejpam-5768	365	5	)	)	PUNCT
ejpam-5768	365	6	≤	≤	NOUN
ejpam-5768	365	7	υ̂	υ̂	VERB
ejpam-5768	365	8	2(p−1	2(p−1	ADJ
ejpam-5768	365	9	02	02	NUM
ejpam-5768	365	10	p01	p01	NOUN
ejpam-5768	365	11	)	)	PUNCT
ejpam-5768	365	12	≤	≤	NOUN
ejpam-5768	366	1	max{υ̂	max{υ̂	NOUN
ejpam-5768	366	2	2(p−1	2(p−1	PROPN
ejpam-5768	366	3	02	02	NUM
ejpam-5768	366	4	,	,	PUNCT
ejpam-5768	366	5	υ̂	υ̂	NUM
ejpam-5768	366	6	2(p01	2(p01	NUM
ejpam-5768	366	7	)	)	PUNCT
ejpam-5768	366	8	}	}	PUNCT
ejpam-5768	367	1	=	=	SYM
ejpam-5768	367	2	max{υ̂	max{υ̂	NOUN
ejpam-5768	367	3	2(p01	2(p01	NUM
ejpam-5768	367	4	,	,	PUNCT
ejpam-5768	367	5	υ̂	υ̂	X
ejpam-5768	367	6	2(p02	2(p02	NUM
ejpam-5768	367	7	)	)	PUNCT
ejpam-5768	367	8	}	}	PUNCT
ejpam-5768	367	9	=	=	SYM
ejpam-5768	367	10	max{η̂2(τ(p01	max{η̂2(τ(p01	NOUN
ejpam-5768	367	11	)	)	PUNCT
ejpam-5768	367	12	)	)	PUNCT
ejpam-5768	367	13	,	,	PUNCT
ejpam-5768	367	14	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	367	15	)	)	PUNCT
ejpam-5768	367	16	)	)	PUNCT
ejpam-5768	367	17	}	}	PUNCT
ejpam-5768	367	18	.	.	PUNCT
ejpam-5768	368	1	hence	hence	ADV
ejpam-5768	368	2	τ(υ	τ(υ	NOUN
ejpam-5768	368	3	)	)	PUNCT
ejpam-5768	368	4	is	be	AUX
ejpam-5768	368	5	a	a	DET
ejpam-5768	368	6	pf	pf	PROPN
ejpam-5768	368	7	-	-	PUNCT
ejpam-5768	368	8	hx	hx	NOUN
ejpam-5768	368	9	-	-	PUNCT
ejpam-5768	368	10	sg	sg	PROPN
ejpam-5768	368	11	of	of	ADP
ejpam-5768	368	12	s.	s.	PROPN
ejpam-5768	368	13	theorem	theorem	VERB
ejpam-5768	368	14	12	12	NUM
ejpam-5768	368	15	.	.	PUNCT
ejpam-5768	369	1	let	let	VERB
ejpam-5768	369	2	τ	τ	PROPN
ejpam-5768	369	3	:	:	PUNCT
ejpam-5768	369	4	p	p	X
ejpam-5768	369	5	−→	−→	NOUN
ejpam-5768	369	6	s	s	AUX
ejpam-5768	369	7	be	be	AUX
ejpam-5768	369	8	an	an	DET
ejpam-5768	369	9	antihomomorphism	antihomomorphism	NOUN
ejpam-5768	369	10	of	of	ADP
ejpam-5768	369	11	hx	hx	NOUN
ejpam-5768	369	12	-	-	PUNCT
ejpam-5768	369	13	groups	group	NOUN
ejpam-5768	369	14	.	.	PUNCT
ejpam-5768	370	1	if	if	SCONJ
ejpam-5768	370	2	υ	υ	PROPN
ejpam-5768	370	3	is	be	AUX
ejpam-5768	370	4	a	a	DET
ejpam-5768	370	5	pf	pf	NOUN
ejpam-5768	370	6	-	-	PUNCT
ejpam-5768	370	7	hxnsg	hxnsg	NOUN
ejpam-5768	370	8	of	of	ADP
ejpam-5768	370	9	p	p	PROPN
ejpam-5768	370	10	,	,	PUNCT
ejpam-5768	370	11	then	then	ADV
ejpam-5768	370	12	τ(υ	τ(υ	X
ejpam-5768	370	13	)	)	PUNCT
ejpam-5768	370	14	is	be	AUX
ejpam-5768	370	15	a	a	DET
ejpam-5768	370	16	pf	pf	VERB
ejpam-5768	370	17	-	-	PUNCT
ejpam-5768	370	18	nhx	nhx	NOUN
ejpam-5768	370	19	-	-	PUNCT
ejpam-5768	370	20	sg	sg	NOUN
ejpam-5768	370	21	of	of	ADP
ejpam-5768	370	22	s.	s.	PROPN
ejpam-5768	370	23	proof	proof	PROPN
ejpam-5768	370	24	.	.	PUNCT
ejpam-5768	371	1	suppose	suppose	VERB
ejpam-5768	371	2	that	that	SCONJ
ejpam-5768	371	3	τ(υ	τ(υ	PRON
ejpam-5768	371	4	)	)	PUNCT
ejpam-5768	371	5	=	=	SYM
ejpam-5768	371	6	{	{	PUNCT
ejpam-5768	371	7	(	(	PUNCT
ejpam-5768	371	8	τ(p	τ(p	NOUN
ejpam-5768	371	9	)	)	PUNCT
ejpam-5768	371	10	,	,	PUNCT
ejpam-5768	371	11	η	η	PROPN
ejpam-5768	371	12	,	,	PUNCT
ejpam-5768	371	13	η̂	η̂	NUM
ejpam-5768	371	14	)	)	PUNCT
ejpam-5768	371	15	:	:	PUNCT
ejpam-5768	371	16	τ(p	τ(p	NOUN
ejpam-5768	371	17	)	)	PUNCT
ejpam-5768	371	18	∈	∈	PROPN
ejpam-5768	371	19	s	s	PART
ejpam-5768	371	20	}	}	PUNCT
ejpam-5768	371	21	.	.	PUNCT
ejpam-5768	372	1	since	since	SCONJ
ejpam-5768	372	2	υ	υ	PROPN
ejpam-5768	372	3	is	be	AUX
ejpam-5768	372	4	a	a	DET
ejpam-5768	372	5	pf	pf	PROPN
ejpam-5768	372	6	-	-	PUNCT
ejpam-5768	372	7	hx	hx	PROPN
ejpam-5768	372	8	-	-	PUNCT
ejpam-5768	372	9	nsg	nsg	PROPN
ejpam-5768	372	10	of	of	ADP
ejpam-5768	372	11	p	p	PROPN
ejpam-5768	372	12	,	,	PUNCT
ejpam-5768	372	13	then	then	ADV
ejpam-5768	372	14	it	it	PRON
ejpam-5768	372	15	is	be	AUX
ejpam-5768	372	16	a	a	DET
ejpam-5768	372	17	pf	pf	PROPN
ejpam-5768	372	18	-	-	PUNCT
ejpam-5768	372	19	hx	hx	NOUN
ejpam-5768	372	20	-	-	PUNCT
ejpam-5768	372	21	sg	sg	PROPN
ejpam-5768	372	22	and	and	CCONJ
ejpam-5768	372	23	,	,	PUNCT
ejpam-5768	372	24	by	by	ADP
ejpam-5768	372	25	the	the	DET
ejpam-5768	372	26	previous	previous	ADJ
ejpam-5768	372	27	theorem	theorem	NOUN
ejpam-5768	372	28	,	,	PUNCT
ejpam-5768	372	29	τ(υ	τ(υ	X
ejpam-5768	372	30	)	)	PUNCT
ejpam-5768	372	31	is	be	AUX
ejpam-5768	372	32	a	a	DET
ejpam-5768	372	33	pf	pf	PROPN
ejpam-5768	372	34	-	-	PUNCT
ejpam-5768	372	35	hx	hx	NOUN
ejpam-5768	372	36	-	-	PUNCT
ejpam-5768	372	37	sg	sg	PROPN
ejpam-5768	372	38	of	of	ADP
ejpam-5768	372	39	s.	s.	PROPN
ejpam-5768	372	40	we	we	PRON
ejpam-5768	372	41	need	need	VERB
ejpam-5768	372	42	to	to	PART
ejpam-5768	372	43	prove	prove	VERB
ejpam-5768	372	44	that	that	SCONJ
ejpam-5768	372	45	it	it	PRON
ejpam-5768	372	46	is	be	AUX
ejpam-5768	372	47	normal	normal	ADJ
ejpam-5768	372	48	.	.	PUNCT
ejpam-5768	373	1	let	let	VERB
ejpam-5768	373	2	τ(p01	τ(p01	NOUN
ejpam-5768	373	3	)	)	PUNCT
ejpam-5768	373	4	,	,	PUNCT
ejpam-5768	373	5	τ(p01	τ(p01	NOUN
ejpam-5768	373	6	)	)	PUNCT
ejpam-5768	373	7	∈	∈	PROPN
ejpam-5768	373	8	s.	s.	PROPN
ejpam-5768	373	9	then	then	ADV
ejpam-5768	373	10	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	PROPN
ejpam-5768	373	11	)	)	PUNCT
ejpam-5768	373	12	)	)	PUNCT
ejpam-5768	374	1	=	=	SYM
ejpam-5768	374	2	(	(	PUNCT
ejpam-5768	374	3	τ(p01	τ(p01	NOUN
ejpam-5768	374	4	)	)	PUNCT
ejpam-5768	374	5	)	)	PUNCT
ejpam-5768	374	6	−1η2(τ(p02	−1η2(τ(p02	X
ejpam-5768	374	7	)	)	PUNCT
ejpam-5768	374	8	)	)	PUNCT
ejpam-5768	375	1	=	=	SYM
ejpam-5768	375	2	(	(	PUNCT
ejpam-5768	375	3	τ(p01	τ(p01	NOUN
ejpam-5768	375	4	)	)	PUNCT
ejpam-5768	375	5	)	)	PUNCT
ejpam-5768	375	6	−1ῡ	−1ῡ	NUM
ejpam-5768	376	1	2((p02	2((p02	NUM
ejpam-5768	376	2	)	)	PUNCT
ejpam-5768	376	3	=	=	NOUN
ejpam-5768	376	4	(	(	PUNCT
ejpam-5768	376	5	τ(p01	τ(p01	NOUN
ejpam-5768	376	6	)	)	PUNCT
ejpam-5768	376	7	)	)	PUNCT
ejpam-5768	376	8	−1ῡ	−1ῡ	NUM
ejpam-5768	376	9	2(p01p02(p01	2(p01p02(p01	NUM
ejpam-5768	376	10	)	)	PUNCT
ejpam-5768	376	11	−1	−1	NOUN
ejpam-5768	376	12	)	)	PUNCT
ejpam-5768	376	13	≤	≤	NOUN
ejpam-5768	376	14	(	(	PUNCT
ejpam-5768	376	15	τ(p01	τ(p01	NOUN
ejpam-5768	376	16	)	)	PUNCT
ejpam-5768	376	17	)	)	PUNCT
ejpam-5768	376	18	−1η2(τ(p01p02(p01	−1η2(τ(p01p02(p01	PROPN
ejpam-5768	376	19	)	)	PUNCT
ejpam-5768	376	20	−1	−1	NOUN
ejpam-5768	376	21	)	)	PUNCT
ejpam-5768	376	22	)	)	PUNCT
ejpam-5768	377	1	=	=	PUNCT
ejpam-5768	377	2	η2(τ(p01)τ(p01p02(p01	η2(τ(p01)τ(p01p02(p01	ADJ
ejpam-5768	377	3	)	)	PUNCT
ejpam-5768	377	4	−1	−1	NOUN
ejpam-5768	377	5	)	)	PUNCT
ejpam-5768	377	6	)	)	PUNCT
ejpam-5768	378	1	=	=	SYM
ejpam-5768	378	2	η2(τ(p01p02(p01	η2(τ(p01p02(p01	ADJ
ejpam-5768	378	3	)	)	PUNCT
ejpam-5768	378	4	−1p01	−1p01	NOUN
ejpam-5768	378	5	)	)	PUNCT
ejpam-5768	378	6	)	)	PUNCT
ejpam-5768	379	1	=	=	SYM
ejpam-5768	379	2	η2(τ(p01p02	η2(τ(p01p02	NUM
ejpam-5768	379	3	)	)	PUNCT
ejpam-5768	379	4	)	)	PUNCT
ejpam-5768	380	1	=	=	PUNCT
ejpam-5768	380	2	η2(τ(p02)τ(p01	η2(τ(p02)τ(p01	NOUN
ejpam-5768	380	3	)	)	PUNCT
ejpam-5768	380	4	)	)	PUNCT
ejpam-5768	380	5	)	)	PUNCT
ejpam-5768	381	1	then	then	ADV
ejpam-5768	381	2	η2(τ(p01τ(p02	η2(τ(p01τ(p02	NOUN
ejpam-5768	381	3	)	)	PUNCT
ejpam-5768	381	4	)	)	PUNCT
ejpam-5768	381	5	)	)	PUNCT
ejpam-5768	381	6	≤	≤	NUM
ejpam-5768	381	7	η2(τ(p02τ(p01	η2(τ(p02τ(p01	NOUN
ejpam-5768	381	8	)	)	PUNCT
ejpam-5768	381	9	)	)	PUNCT
ejpam-5768	381	10	)	)	PUNCT
ejpam-5768	381	11	.	.	PUNCT
ejpam-5768	382	1	on	on	ADP
ejpam-5768	382	2	the	the	DET
ejpam-5768	382	3	other	other	ADJ
ejpam-5768	382	4	hand	hand	NOUN
ejpam-5768	382	5	,	,	PUNCT
ejpam-5768	382	6	η2(τ(p02)τ(p01	η2(τ(p02)τ(p01	PROPN
ejpam-5768	382	7	)	)	PUNCT
ejpam-5768	382	8	)	)	PUNCT
ejpam-5768	383	1	=	=	SYM
ejpam-5768	383	2	(	(	PUNCT
ejpam-5768	383	3	τ(p02	τ(p02	X
ejpam-5768	383	4	)	)	PUNCT
ejpam-5768	383	5	)	)	PUNCT
ejpam-5768	383	6	−1η2(τ(p01	−1η2(τ(p01	NOUN
ejpam-5768	383	7	)	)	PUNCT
ejpam-5768	383	8	)	)	PUNCT
ejpam-5768	383	9	a.	a.	NOUN
ejpam-5768	383	10	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	383	11	/	/	SYM
ejpam-5768	383	12	eur	eur	PROPN
ejpam-5768	383	13	.	.	PUNCT
ejpam-5768	384	1	j.	j.	PROPN
ejpam-5768	384	2	pure	pure	PROPN
ejpam-5768	384	3	appl	appl	PROPN
ejpam-5768	384	4	.	.	PROPN
ejpam-5768	384	5	math	math	PROPN
ejpam-5768	384	6	,	,	PUNCT
ejpam-5768	384	7	18	18	NUM
ejpam-5768	384	8	(	(	PUNCT
ejpam-5768	384	9	2	2	NUM
ejpam-5768	384	10	)	)	PUNCT
ejpam-5768	384	11	(	(	PUNCT
ejpam-5768	384	12	2025	2025	NUM
ejpam-5768	384	13	)	)	PUNCT
ejpam-5768	384	14	,	,	PUNCT
ejpam-5768	384	15	5768	5768	NUM
ejpam-5768	384	16	13	13	NUM
ejpam-5768	384	17	of	of	ADP
ejpam-5768	384	18	17	17	NUM
ejpam-5768	384	19	=	=	SYM
ejpam-5768	384	20	(	(	PUNCT
ejpam-5768	384	21	τ(p02	τ(p02	X
ejpam-5768	384	22	)	)	PUNCT
ejpam-5768	384	23	)	)	PUNCT
ejpam-5768	384	24	−1ῡ	−1ῡ	NUM
ejpam-5768	385	1	2((p01	2((p01	NUM
ejpam-5768	385	2	)	)	PUNCT
ejpam-5768	385	3	=	=	SYM
ejpam-5768	385	4	(	(	PUNCT
ejpam-5768	385	5	τ(p02	τ(p02	X
ejpam-5768	385	6	)	)	PUNCT
ejpam-5768	385	7	)	)	PUNCT
ejpam-5768	386	1	−1ῡ	−1ῡ	NUM
ejpam-5768	386	2	2(p02p01(p02	2(p02p01(p02	NUM
ejpam-5768	386	3	)	)	PUNCT
ejpam-5768	386	4	−1	−1	NOUN
ejpam-5768	386	5	)	)	PUNCT
ejpam-5768	386	6	≤	≤	NOUN
ejpam-5768	386	7	(	(	PUNCT
ejpam-5768	386	8	τ(p02	τ(p02	NOUN
ejpam-5768	386	9	)	)	PUNCT
ejpam-5768	386	10	)	)	PUNCT
ejpam-5768	387	1	−1η2(τ(p02p01(p02	−1η2(τ(p02p01(p02	X
ejpam-5768	387	2	)	)	PUNCT
ejpam-5768	387	3	−1	−1	NOUN
ejpam-5768	387	4	)	)	PUNCT
ejpam-5768	387	5	)	)	PUNCT
ejpam-5768	388	1	=	=	PUNCT
ejpam-5768	388	2	η2(τ(p02)τ(p02p01(p02	η2(τ(p02)τ(p02p01(p02	X
ejpam-5768	388	3	)	)	PUNCT
ejpam-5768	388	4	−1	−1	NOUN
ejpam-5768	388	5	)	)	PUNCT
ejpam-5768	388	6	)	)	PUNCT
ejpam-5768	389	1	=	=	PUNCT
ejpam-5768	389	2	η2(τ(p02p01(p02	η2(τ(p02p01(p02	VERB
ejpam-5768	389	3	)	)	PUNCT
ejpam-5768	389	4	−1p02	−1p02	NOUN
ejpam-5768	389	5	)	)	PUNCT
ejpam-5768	389	6	)	)	PUNCT
ejpam-5768	390	1	=	=	SYM
ejpam-5768	390	2	η2(τ(p02p01	η2(τ(p02p01	NUM
ejpam-5768	390	3	)	)	PUNCT
ejpam-5768	390	4	)	)	PUNCT
ejpam-5768	391	1	=	=	PUNCT
ejpam-5768	391	2	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	PROPN
ejpam-5768	391	3	)	)	PUNCT
ejpam-5768	391	4	)	)	PUNCT
ejpam-5768	391	5	)	)	PUNCT
ejpam-5768	392	1	this	this	DET
ejpam-5768	392	2	impliea	impliea	NOUN
ejpam-5768	392	3	that	that	SCONJ
ejpam-5768	392	4	η2(τ(p02τ(p02	η2(τ(p02τ(p02	NOUN
ejpam-5768	392	5	)	)	PUNCT
ejpam-5768	392	6	)	)	PUNCT
ejpam-5768	392	7	)	)	PUNCT
ejpam-5768	392	8	≤	≤	NUM
ejpam-5768	392	9	η2(τ(p01τ(p02	η2(τ(p01τ(p02	NOUN
ejpam-5768	392	10	)	)	PUNCT
ejpam-5768	392	11	)	)	PUNCT
ejpam-5768	392	12	)	)	PUNCT
ejpam-5768	392	13	.	.	PUNCT
ejpam-5768	393	1	thus	thus	ADV
ejpam-5768	393	2	η	η	X
ejpam-5768	393	3	2(τ(p01τ(p02	2(τ(p01τ(p02	NOUN
ejpam-5768	393	4	)	)	PUNCT
ejpam-5768	393	5	)	)	PUNCT
ejpam-5768	393	6	)	)	PUNCT
ejpam-5768	394	1	=	=	SYM
ejpam-5768	394	2	η2(τ(p02τ(p01	η2(τ(p02τ(p01	NOUN
ejpam-5768	394	3	)	)	PUNCT
ejpam-5768	394	4	)	)	PUNCT
ejpam-5768	394	5	)	)	PUNCT
ejpam-5768	394	6	.	.	PUNCT
ejpam-5768	395	1	in	in	ADP
ejpam-5768	395	2	addition	addition	NOUN
ejpam-5768	395	3	,	,	PUNCT
ejpam-5768	395	4	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	NOUN
ejpam-5768	395	5	)	)	PUNCT
ejpam-5768	395	6	)	)	PUNCT
ejpam-5768	396	1	=	=	SYM
ejpam-5768	396	2	(	(	PUNCT
ejpam-5768	396	3	τ(p01	τ(p01	NOUN
ejpam-5768	396	4	)	)	PUNCT
ejpam-5768	396	5	)	)	PUNCT
ejpam-5768	396	6	−1η̂2(τ(p02	−1η̂2(τ(p02	PROPN
ejpam-5768	396	7	)	)	PUNCT
ejpam-5768	396	8	)	)	PUNCT
ejpam-5768	397	1	=	=	SYM
ejpam-5768	397	2	(	(	PUNCT
ejpam-5768	397	3	τ(p01	τ(p01	NOUN
ejpam-5768	397	4	)	)	PUNCT
ejpam-5768	397	5	)	)	PUNCT
ejpam-5768	397	6	−1υ̂	−1υ̂	VERB
ejpam-5768	397	7	2((p02	2((p02	NUM
ejpam-5768	397	8	)	)	PUNCT
ejpam-5768	397	9	=	=	SYM
ejpam-5768	397	10	(	(	PUNCT
ejpam-5768	397	11	τ(p01	τ(p01	NOUN
ejpam-5768	397	12	)	)	PUNCT
ejpam-5768	397	13	)	)	PUNCT
ejpam-5768	398	1	−1υ̂	−1υ̂	VERB
ejpam-5768	398	2	2(p01p02(p01	2(p01p02(p01	NUM
ejpam-5768	398	3	)	)	PUNCT
ejpam-5768	398	4	−1	−1	NOUN
ejpam-5768	398	5	)	)	PUNCT
ejpam-5768	398	6	≤	≤	NOUN
ejpam-5768	398	7	(	(	PUNCT
ejpam-5768	398	8	τ(p01	τ(p01	NOUN
ejpam-5768	398	9	)	)	PUNCT
ejpam-5768	398	10	)	)	PUNCT
ejpam-5768	398	11	−1η̂2(τ(p01p02(p01	−1η̂2(τ(p01p02(p01	NUM
ejpam-5768	398	12	)	)	PUNCT
ejpam-5768	398	13	−1	−1	NOUN
ejpam-5768	398	14	)	)	PUNCT
ejpam-5768	398	15	)	)	PUNCT
ejpam-5768	399	1	=	=	SYM
ejpam-5768	399	2	η̂2(τ(p01)τ(p01p02(p01	η̂2(τ(p01)τ(p01p02(p01	ADJ
ejpam-5768	399	3	)	)	PUNCT
ejpam-5768	399	4	−1	−1	NOUN
ejpam-5768	399	5	)	)	PUNCT
ejpam-5768	399	6	)	)	PUNCT
ejpam-5768	400	1	=	=	PUNCT
ejpam-5768	400	2	η̂2(τ(p01p02(p01	η̂2(τ(p01p02(p01	NOUN
ejpam-5768	400	3	)	)	PUNCT
ejpam-5768	400	4	−1p01	−1p01	NOUN
ejpam-5768	400	5	)	)	PUNCT
ejpam-5768	400	6	)	)	PUNCT
ejpam-5768	401	1	=	=	PUNCT
ejpam-5768	401	2	η̂2(τ(p01p02	η̂2(τ(p01p02	NOUN
ejpam-5768	401	3	)	)	PUNCT
ejpam-5768	401	4	)	)	PUNCT
ejpam-5768	402	1	=	=	SYM
ejpam-5768	402	2	η̂2(τ(p02)τ(p01	η̂2(τ(p02)τ(p01	NOUN
ejpam-5768	402	3	)	)	PUNCT
ejpam-5768	402	4	)	)	PUNCT
ejpam-5768	402	5	)	)	PUNCT
ejpam-5768	403	1	on	on	ADP
ejpam-5768	403	2	the	the	DET
ejpam-5768	403	3	other	other	ADJ
ejpam-5768	403	4	hand	hand	NOUN
ejpam-5768	403	5	,	,	PUNCT
ejpam-5768	403	6	η̂2(τ(p02)τ(p01	η̂2(τ(p02)τ(p01	NOUN
ejpam-5768	403	7	)	)	PUNCT
ejpam-5768	403	8	)	)	PUNCT
ejpam-5768	403	9	=	=	SYM
ejpam-5768	403	10	(	(	PUNCT
ejpam-5768	403	11	τ(p02	τ(p02	X
ejpam-5768	403	12	)	)	PUNCT
ejpam-5768	403	13	)	)	PUNCT
ejpam-5768	403	14	−1η̂2(τ(p01	−1η̂2(τ(p01	NOUN
ejpam-5768	403	15	)	)	PUNCT
ejpam-5768	403	16	)	)	PUNCT
ejpam-5768	403	17	=	=	SYM
ejpam-5768	403	18	(	(	PUNCT
ejpam-5768	403	19	τ(p02	τ(p02	X
ejpam-5768	403	20	)	)	PUNCT
ejpam-5768	403	21	)	)	PUNCT
ejpam-5768	403	22	−1υ̂	−1υ̂	VERB
ejpam-5768	403	23	2((p01	2((p01	NUM
ejpam-5768	403	24	)	)	PUNCT
ejpam-5768	403	25	=	=	SYM
ejpam-5768	403	26	(	(	PUNCT
ejpam-5768	403	27	τ(p02	τ(p02	X
ejpam-5768	403	28	)	)	PUNCT
ejpam-5768	403	29	)	)	PUNCT
ejpam-5768	404	1	−1υ̂	−1υ̂	VERB
ejpam-5768	404	2	2(p02p01(p02	2(p02p01(p02	NUM
ejpam-5768	404	3	)	)	PUNCT
ejpam-5768	404	4	−1	−1	NOUN
ejpam-5768	404	5	)	)	PUNCT
ejpam-5768	404	6	≤	≤	NOUN
ejpam-5768	404	7	(	(	PUNCT
ejpam-5768	404	8	τ(p02	τ(p02	NOUN
ejpam-5768	404	9	)	)	PUNCT
ejpam-5768	404	10	)	)	PUNCT
ejpam-5768	404	11	−1η̂2(τ(p02p01(p02	−1η̂2(τ(p02p01(p02	NUM
ejpam-5768	404	12	)	)	PUNCT
ejpam-5768	404	13	−1	−1	NOUN
ejpam-5768	404	14	)	)	PUNCT
ejpam-5768	404	15	)	)	PUNCT
ejpam-5768	405	1	=	=	PUNCT
ejpam-5768	405	2	η̂2(τ(p02)τ(p02p01(p02	η̂2(τ(p02)τ(p02p01(p02	PROPN
ejpam-5768	405	3	)	)	PUNCT
ejpam-5768	405	4	−1	−1	NOUN
ejpam-5768	405	5	)	)	PUNCT
ejpam-5768	405	6	)	)	PUNCT
ejpam-5768	406	1	=	=	SYM
ejpam-5768	406	2	η̂2(τ(p02p01(p02	η̂2(τ(p02p01(p02	NOUN
ejpam-5768	406	3	)	)	PUNCT
ejpam-5768	406	4	−1p02	−1p02	NOUN
ejpam-5768	406	5	)	)	PUNCT
ejpam-5768	406	6	)	)	PUNCT
ejpam-5768	407	1	=	=	SYM
ejpam-5768	407	2	η̂2(τ(p02p01	η̂2(τ(p02p01	NOUN
ejpam-5768	407	3	)	)	PUNCT
ejpam-5768	407	4	)	)	PUNCT
ejpam-5768	408	1	=	=	SYM
ejpam-5768	408	2	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	PROPN
ejpam-5768	408	3	)	)	PUNCT
ejpam-5768	408	4	)	)	PUNCT
ejpam-5768	408	5	)	)	PUNCT
ejpam-5768	408	6	thus	thus	ADV
ejpam-5768	408	7	η̂2(τ(p01τ(p02	η̂2(τ(p01τ(p02	NOUN
ejpam-5768	408	8	)	)	PUNCT
ejpam-5768	408	9	)	)	PUNCT
ejpam-5768	408	10	)	)	PUNCT
ejpam-5768	409	1	=	=	SYM
ejpam-5768	409	2	η̂2(τ(p02τ(p01	η̂2(τ(p02τ(p01	NOUN
ejpam-5768	409	3	)	)	PUNCT
ejpam-5768	409	4	)	)	PUNCT
ejpam-5768	409	5	)	)	PUNCT
ejpam-5768	409	6	.	.	PUNCT
ejpam-5768	410	1	therefore	therefore	ADV
ejpam-5768	410	2	,	,	PUNCT
ejpam-5768	410	3	τ(υ	τ(υ	X
ejpam-5768	410	4	)	)	PUNCT
ejpam-5768	410	5	is	be	AUX
ejpam-5768	410	6	a	a	DET
ejpam-5768	410	7	pf	pf	VERB
ejpam-5768	410	8	-	-	PUNCT
ejpam-5768	410	9	nhx	nhx	NOUN
ejpam-5768	410	10	-	-	PUNCT
ejpam-5768	410	11	sg	sg	NOUN
ejpam-5768	410	12	of	of	ADP
ejpam-5768	410	13	s.	s.	PROPN
ejpam-5768	410	14	theorem	theorem	VERB
ejpam-5768	410	15	13	13	NUM
ejpam-5768	410	16	.	.	PUNCT
ejpam-5768	411	1	let	let	VERB
ejpam-5768	411	2	τ	τ	PROPN
ejpam-5768	411	3	:	:	PUNCT
ejpam-5768	411	4	p	p	X
ejpam-5768	411	5	−→	−→	NOUN
ejpam-5768	411	6	s	s	AUX
ejpam-5768	411	7	be	be	AUX
ejpam-5768	411	8	a	a	DET
ejpam-5768	411	9	homomorphism	homomorphism	NOUN
ejpam-5768	411	10	of	of	ADP
ejpam-5768	411	11	hx	hx	NOUN
ejpam-5768	411	12	-	-	PUNCT
ejpam-5768	411	13	groups	group	NOUN
ejpam-5768	411	14	and	and	CCONJ
ejpam-5768	411	15	let	let	VERB
ejpam-5768	411	16	υ	υ	DET
ejpam-5768	411	17	2	2	NUM
ejpam-5768	411	18	be	be	AUX
ejpam-5768	411	19	a	a	DET
ejpam-5768	411	20	pfss	pfss	NOUN
ejpam-5768	411	21	of	of	ADP
ejpam-5768	411	22	w	w	PROPN
ejpam-5768	411	23	.	.	PUNCT
ejpam-5768	412	1	if	if	SCONJ
ejpam-5768	412	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	412	3	,	,	PUNCT
ejpam-5768	412	4	δ	δ	PROPN
ejpam-5768	412	5	)	)	PUNCT
ejpam-5768	412	6	is	be	AUX
ejpam-5768	412	7	an	an	DET
ejpam-5768	412	8	hx	hx	PROPN
ejpam-5768	412	9	-	-	PUNCT
ejpam-5768	412	10	sg	sg	PROPN
ejpam-5768	412	11	,	,	PUNCT
ejpam-5768	412	12	then	then	ADV
ejpam-5768	412	13	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	412	14	,	,	PUNCT
ejpam-5768	412	15	δ	δ	NOUN
ejpam-5768	412	16	)	)	PUNCT
ejpam-5768	412	17	)	)	PUNCT
ejpam-5768	412	18	is	be	AUX
ejpam-5768	412	19	an	an	DET
ejpam-5768	412	20	hx	hx	PROPN
ejpam-5768	412	21	-	-	PUNCT
ejpam-5768	412	22	sg	sg	PROPN
ejpam-5768	412	23	.	.	PUNCT
ejpam-5768	412	24	a.	a.	PROPN
ejpam-5768	412	25	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	412	26	/	/	SYM
ejpam-5768	412	27	eur	eur	PROPN
ejpam-5768	412	28	.	.	PUNCT
ejpam-5768	413	1	j.	j.	PROPN
ejpam-5768	413	2	pure	pure	PROPN
ejpam-5768	413	3	appl	appl	PROPN
ejpam-5768	413	4	.	.	PROPN
ejpam-5768	413	5	math	math	PROPN
ejpam-5768	413	6	,	,	PUNCT
ejpam-5768	413	7	18	18	NUM
ejpam-5768	413	8	(	(	PUNCT
ejpam-5768	413	9	2	2	NUM
ejpam-5768	413	10	)	)	PUNCT
ejpam-5768	413	11	(	(	PUNCT
ejpam-5768	413	12	2025	2025	NUM
ejpam-5768	413	13	)	)	PUNCT
ejpam-5768	413	14	,	,	PUNCT
ejpam-5768	413	15	5768	5768	NUM
ejpam-5768	413	16	14	14	NUM
ejpam-5768	413	17	of	of	ADP
ejpam-5768	413	18	17	17	NUM
ejpam-5768	413	19	proof	proof	NOUN
ejpam-5768	413	20	.	.	PUNCT
ejpam-5768	413	21	suppose	suppose	VERB
ejpam-5768	413	22	that	that	SCONJ
ejpam-5768	413	23	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	413	24	,	,	PUNCT
ejpam-5768	413	25	δ	δ	NOUN
ejpam-5768	413	26	)	)	PUNCT
ejpam-5768	413	27	)	)	PUNCT
ejpam-5768	414	1	=	=	PRON
ejpam-5768	414	2	{	{	PUNCT
ejpam-5768	414	3	(	(	PUNCT
ejpam-5768	414	4	τ(p	τ(p	NOUN
ejpam-5768	414	5	)	)	PUNCT
ejpam-5768	414	6	,	,	PUNCT
ejpam-5768	414	7	η2	η2	PROPN
ejpam-5768	414	8	,	,	PUNCT
ejpam-5768	414	9	η̂2	η̂2	NOUN
ejpam-5768	414	10	)	)	PUNCT
ejpam-5768	414	11	:	:	PUNCT
ejpam-5768	414	12	p	p	X
ejpam-5768	414	13	∈	∈	PROPN
ejpam-5768	414	14	υ(ζ	υ(ζ	PROPN
ejpam-5768	414	15	,	,	PUNCT
ejpam-5768	414	16	δ	δ	PROPN
ejpam-5768	414	17	)	)	PUNCT
ejpam-5768	414	18	}	}	PUNCT
ejpam-5768	414	19	.	.	PUNCT
ejpam-5768	415	1	let	let	VERB
ejpam-5768	415	2	τ(p01	τ(p01	NOUN
ejpam-5768	415	3	)	)	PUNCT
ejpam-5768	415	4	,	,	PUNCT
ejpam-5768	415	5	τ(p02	τ(p02	X
ejpam-5768	415	6	)	)	PUNCT
ejpam-5768	415	7	∈	∈	PROPN
ejpam-5768	415	8	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	415	9	,	,	PUNCT
ejpam-5768	415	10	δ	δ	PROPN
ejpam-5768	415	11	)	)	PUNCT
ejpam-5768	415	12	)	)	PUNCT
ejpam-5768	415	13	.	.	PUNCT
ejpam-5768	416	1	then	then	ADV
ejpam-5768	416	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	416	3	)	)	PUNCT
ejpam-5768	416	4	)	)	PUNCT
ejpam-5768	416	5	≥	≥	PROPN
ejpam-5768	416	6	ζ	ζ	NOUN
ejpam-5768	416	7	,	,	PUNCT
ejpam-5768	416	8	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	416	9	)	)	PUNCT
ejpam-5768	416	10	)	)	PUNCT
ejpam-5768	416	11	≥	≥	PROPN
ejpam-5768	416	12	ζ	ζ	NOUN
ejpam-5768	416	13	,	,	PUNCT
ejpam-5768	416	14	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	416	15	)	)	PUNCT
ejpam-5768	416	16	)	)	PUNCT
ejpam-5768	416	17	≤	≤	NUM
ejpam-5768	416	18	δ	δ	PROPN
ejpam-5768	416	19	and	and	CCONJ
ejpam-5768	416	20	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	416	21	)	)	PUNCT
ejpam-5768	416	22	)	)	PUNCT
ejpam-5768	416	23	≤	≤	NUM
ejpam-5768	417	1	δ	δ	PROPN
ejpam-5768	417	2	.	.	PUNCT
ejpam-5768	418	1	then	then	ADV
ejpam-5768	418	2	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	ADJ
ejpam-5768	418	3	)	)	PUNCT
ejpam-5768	418	4	−1	−1	NOUN
ejpam-5768	418	5	)	)	PUNCT
ejpam-5768	419	1	=	=	PUNCT
ejpam-5768	419	2	η2(τ(p01)τ(p	η2(τ(p01)τ(p	NOUN
ejpam-5768	419	3	−1	−1	NOUN
ejpam-5768	419	4	02	02	NUM
ejpam-5768	419	5	)	)	PUNCT
ejpam-5768	419	6	)	)	PUNCT
ejpam-5768	420	1	=	=	PUNCT
ejpam-5768	420	2	η2(τ(p01p	η2(τ(p01p	NOUN
ejpam-5768	420	3	−1	−1	NOUN
ejpam-5768	420	4	02	02	NUM
ejpam-5768	420	5	)	)	PUNCT
ejpam-5768	420	6	)	)	PUNCT
ejpam-5768	420	7	≥	≥	PROPN
ejpam-5768	420	8	ῡ	ῡ	NOUN
ejpam-5768	421	1	2(p01p	2(p01p	NUM
ejpam-5768	421	2	−1	−1	NOUN
ejpam-5768	421	3	02	02	NUM
ejpam-5768	421	4	)	)	PUNCT
ejpam-5768	421	5	since	since	SCONJ
ejpam-5768	421	6	p01	p01	NOUN
ejpam-5768	421	7	,	,	PUNCT
ejpam-5768	421	8	p02	p02	X
ejpam-5768	421	9	∈	∈	PROPN
ejpam-5768	421	10	υ(ζ	υ(ζ	PROPN
ejpam-5768	421	11	,	,	PUNCT
ejpam-5768	421	12	δ	δ	PROPN
ejpam-5768	421	13	)	)	PUNCT
ejpam-5768	421	14	and	and	CCONJ
ejpam-5768	421	15	υ(ζ	υ(ζ	PROPN
ejpam-5768	421	16	,	,	PUNCT
ejpam-5768	421	17	δ	δ	PROPN
ejpam-5768	421	18	)	)	PUNCT
ejpam-5768	421	19	is	be	AUX
ejpam-5768	421	20	an	an	DET
ejpam-5768	421	21	hx	hx	PROPN
ejpam-5768	421	22	-	-	PUNCT
ejpam-5768	421	23	sg	sg	PROPN
ejpam-5768	421	24	,	,	PUNCT
ejpam-5768	421	25	then	then	ADV
ejpam-5768	421	26	p01p	p01p	VERB
ejpam-5768	421	27	−1	−1	NOUN
ejpam-5768	421	28	02	02	NUM
ejpam-5768	421	29	∈	∈	PROPN
ejpam-5768	421	30	υ(ζ	υ(ζ	PROPN
ejpam-5768	421	31	,	,	PUNCT
ejpam-5768	421	32	δ	δ	PROPN
ejpam-5768	421	33	)	)	PUNCT
ejpam-5768	421	34	which	which	PRON
ejpam-5768	421	35	implies	imply	VERB
ejpam-5768	421	36	that	that	SCONJ
ejpam-5768	421	37	ῡ	ῡ	PROPN
ejpam-5768	421	38	2(p01p	2(p01p	NUM
ejpam-5768	421	39	−1	−1	NOUN
ejpam-5768	421	40	02	02	NUM
ejpam-5768	421	41	)	)	PUNCT
ejpam-5768	421	42	≥	≥	PROPN
ejpam-5768	421	43	ζ	ζ	NOUN
ejpam-5768	421	44	.	.	PUNCT
ejpam-5768	422	1	thus	thus	ADV
ejpam-5768	422	2	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	ADJ
ejpam-5768	422	3	)	)	PUNCT
ejpam-5768	422	4	−1	−1	NOUN
ejpam-5768	422	5	)	)	PUNCT
ejpam-5768	422	6	≥	≥	NOUN
ejpam-5768	423	1	ζ	ζ	NOUN
ejpam-5768	423	2	.	.	PUNCT
ejpam-5768	424	1	in	in	ADP
ejpam-5768	424	2	addition	addition	NOUN
ejpam-5768	424	3	,	,	PUNCT
ejpam-5768	424	4	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	NOUN
ejpam-5768	424	5	)	)	PUNCT
ejpam-5768	424	6	−1	−1	NOUN
ejpam-5768	424	7	)	)	PUNCT
ejpam-5768	425	1	=	=	PUNCT
ejpam-5768	425	2	η̂2(τ(p01)τ(p	η̂2(τ(p01)τ(p	NOUN
ejpam-5768	425	3	−1	−1	NOUN
ejpam-5768	425	4	02	02	NUM
ejpam-5768	425	5	)	)	PUNCT
ejpam-5768	425	6	)	)	PUNCT
ejpam-5768	426	1	=	=	PUNCT
ejpam-5768	426	2	η̂2(τ(p01p	η̂2(τ(p01p	NOUN
ejpam-5768	426	3	−1	−1	NOUN
ejpam-5768	426	4	02	02	NUM
ejpam-5768	426	5	)	)	PUNCT
ejpam-5768	426	6	)	)	PUNCT
ejpam-5768	426	7	≤	≤	NUM
ejpam-5768	426	8	υ̂	υ̂	VERB
ejpam-5768	426	9	2(p01p	2(p01p	NUM
ejpam-5768	426	10	−1	−1	NOUN
ejpam-5768	426	11	02	02	NUM
ejpam-5768	426	12	)	)	PUNCT
ejpam-5768	426	13	since	since	SCONJ
ejpam-5768	426	14	p01	p01	NOUN
ejpam-5768	426	15	,	,	PUNCT
ejpam-5768	426	16	p02	p02	X
ejpam-5768	426	17	∈	∈	PROPN
ejpam-5768	426	18	υ(ζ	υ(ζ	PROPN
ejpam-5768	426	19	,	,	PUNCT
ejpam-5768	426	20	δ	δ	PROPN
ejpam-5768	426	21	)	)	PUNCT
ejpam-5768	426	22	and	and	CCONJ
ejpam-5768	426	23	υ(ζ	υ(ζ	PROPN
ejpam-5768	426	24	,	,	PUNCT
ejpam-5768	426	25	δ	δ	PROPN
ejpam-5768	426	26	)	)	PUNCT
ejpam-5768	426	27	is	be	AUX
ejpam-5768	426	28	an	an	DET
ejpam-5768	426	29	hx	hx	PROPN
ejpam-5768	426	30	-	-	PUNCT
ejpam-5768	426	31	sg	sg	PROPN
ejpam-5768	426	32	,	,	PUNCT
ejpam-5768	426	33	it	it	PRON
ejpam-5768	426	34	implies	imply	VERB
ejpam-5768	426	35	that	that	SCONJ
ejpam-5768	426	36	υ̂	υ̂	PROPN
ejpam-5768	426	37	2(p01p	2(p01p	NUM
ejpam-5768	426	38	−1	−1	NOUN
ejpam-5768	426	39	02	02	NUM
ejpam-5768	426	40	)	)	PUNCT
ejpam-5768	426	41	≤	≤	NUM
ejpam-5768	427	1	δ	δ	PROPN
ejpam-5768	427	2	.	.	PUNCT
ejpam-5768	428	1	thus	thus	ADV
ejpam-5768	428	2	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	PROPN
ejpam-5768	428	3	)	)	PUNCT
ejpam-5768	428	4	−1	−1	NOUN
ejpam-5768	428	5	)	)	PUNCT
ejpam-5768	429	1	≤	≤	NUM
ejpam-5768	429	2	δ	δ	PROPN
ejpam-5768	429	3	.	.	PUNCT
ejpam-5768	429	4	therefore	therefore	ADV
ejpam-5768	429	5	,	,	PUNCT
ejpam-5768	429	6	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	429	7	,	,	PUNCT
ejpam-5768	429	8	δ	δ	NOUN
ejpam-5768	429	9	)	)	PUNCT
ejpam-5768	429	10	)	)	PUNCT
ejpam-5768	429	11	is	be	AUX
ejpam-5768	429	12	an	an	DET
ejpam-5768	429	13	hx	hx	PROPN
ejpam-5768	429	14	-	-	PUNCT
ejpam-5768	429	15	sg	sg	PROPN
ejpam-5768	429	16	.	.	PUNCT
ejpam-5768	430	1	theorem	theorem	PROPN
ejpam-5768	430	2	14	14	NUM
ejpam-5768	430	3	.	.	PUNCT
ejpam-5768	431	1	let	let	VERB
ejpam-5768	431	2	τ	τ	PROPN
ejpam-5768	431	3	:	:	PUNCT
ejpam-5768	431	4	p	p	X
ejpam-5768	431	5	−→	−→	NOUN
ejpam-5768	431	6	s	s	AUX
ejpam-5768	431	7	be	be	AUX
ejpam-5768	431	8	a	a	DET
ejpam-5768	431	9	homomorphism	homomorphism	NOUN
ejpam-5768	431	10	of	of	ADP
ejpam-5768	431	11	hx	hx	NOUN
ejpam-5768	431	12	-	-	PUNCT
ejpam-5768	431	13	groups	group	NOUN
ejpam-5768	431	14	and	and	CCONJ
ejpam-5768	431	15	let	let	VERB
ejpam-5768	431	16	υ	υ	DET
ejpam-5768	431	17	2	2	NUM
ejpam-5768	431	18	be	be	AUX
ejpam-5768	431	19	a	a	DET
ejpam-5768	431	20	pfss	pfss	NOUN
ejpam-5768	431	21	of	of	ADP
ejpam-5768	431	22	w	w	PROPN
ejpam-5768	431	23	.	.	PUNCT
ejpam-5768	432	1	if	if	SCONJ
ejpam-5768	432	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	432	3	,	,	PUNCT
ejpam-5768	432	4	δ	δ	PROPN
ejpam-5768	432	5	)	)	PUNCT
ejpam-5768	432	6	is	be	AUX
ejpam-5768	432	7	an	an	DET
ejpam-5768	432	8	hx	hx	PROPN
ejpam-5768	432	9	-	-	PUNCT
ejpam-5768	432	10	nsg	nsg	PROPN
ejpam-5768	432	11	,	,	PUNCT
ejpam-5768	432	12	then	then	ADV
ejpam-5768	432	13	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	432	14	,	,	PUNCT
ejpam-5768	432	15	δ	δ	NOUN
ejpam-5768	432	16	)	)	PUNCT
ejpam-5768	432	17	)	)	PUNCT
ejpam-5768	432	18	is	be	AUX
ejpam-5768	432	19	an	an	DET
ejpam-5768	432	20	hx	hx	PROPN
ejpam-5768	432	21	-	-	PUNCT
ejpam-5768	432	22	nsg	nsg	PROPN
ejpam-5768	432	23	.	.	PUNCT
ejpam-5768	433	1	proof	proof	NOUN
ejpam-5768	433	2	.	.	PUNCT
ejpam-5768	434	1	since	since	SCONJ
ejpam-5768	434	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	434	3	,	,	PUNCT
ejpam-5768	434	4	δ	δ	PROPN
ejpam-5768	434	5	)	)	PUNCT
ejpam-5768	434	6	is	be	AUX
ejpam-5768	434	7	an	an	DET
ejpam-5768	434	8	hx	hx	PROPN
ejpam-5768	434	9	-	-	PUNCT
ejpam-5768	434	10	nsg	nsg	PROPN
ejpam-5768	434	11	,	,	PUNCT
ejpam-5768	434	12	then	then	ADV
ejpam-5768	434	13	it	it	PRON
ejpam-5768	434	14	is	be	AUX
ejpam-5768	434	15	an	an	DET
ejpam-5768	434	16	hx	hx	NOUN
ejpam-5768	434	17	-	-	PUNCT
ejpam-5768	434	18	sg	sg	PROPN
ejpam-5768	434	19	and	and	CCONJ
ejpam-5768	434	20	,	,	PUNCT
ejpam-5768	434	21	by	by	ADP
ejpam-5768	434	22	the	the	DET
ejpam-5768	434	23	previous	previous	ADJ
ejpam-5768	434	24	theorem	theorem	NOUN
ejpam-5768	434	25	,	,	PUNCT
ejpam-5768	434	26	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	434	27	,	,	PUNCT
ejpam-5768	434	28	δ	δ	NOUN
ejpam-5768	434	29	)	)	PUNCT
ejpam-5768	434	30	)	)	PUNCT
ejpam-5768	435	1	is	be	AUX
ejpam-5768	435	2	an	an	DET
ejpam-5768	435	3	hx	hx	PROPN
ejpam-5768	435	4	-	-	PUNCT
ejpam-5768	435	5	sg	sg	PROPN
ejpam-5768	435	6	.	.	PUNCT
ejpam-5768	436	1	we	we	PRON
ejpam-5768	436	2	need	need	VERB
ejpam-5768	436	3	to	to	PART
ejpam-5768	436	4	prove	prove	VERB
ejpam-5768	436	5	that	that	SCONJ
ejpam-5768	436	6	it	it	PRON
ejpam-5768	436	7	is	be	AUX
ejpam-5768	436	8	normal	normal	ADJ
ejpam-5768	436	9	.	.	PUNCT
ejpam-5768	437	1	suppose	suppose	VERB
ejpam-5768	437	2	that	that	SCONJ
ejpam-5768	437	3	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	437	4	,	,	PUNCT
ejpam-5768	437	5	δ	δ	NOUN
ejpam-5768	437	6	)	)	PUNCT
ejpam-5768	437	7	)	)	PUNCT
ejpam-5768	438	1	=	=	PRON
ejpam-5768	438	2	{	{	PUNCT
ejpam-5768	438	3	(	(	PUNCT
ejpam-5768	438	4	τ(p	τ(p	NOUN
ejpam-5768	438	5	)	)	PUNCT
ejpam-5768	438	6	,	,	PUNCT
ejpam-5768	438	7	η2	η2	PROPN
ejpam-5768	438	8	,	,	PUNCT
ejpam-5768	438	9	η̂2	η̂2	NOUN
ejpam-5768	438	10	)	)	PUNCT
ejpam-5768	438	11	:	:	PUNCT
ejpam-5768	438	12	p	p	X
ejpam-5768	438	13	∈	∈	PROPN
ejpam-5768	438	14	υ(ζ	υ(ζ	PROPN
ejpam-5768	438	15	,	,	PUNCT
ejpam-5768	438	16	δ	δ	PROPN
ejpam-5768	438	17	)	)	PUNCT
ejpam-5768	438	18	}	}	PUNCT
ejpam-5768	438	19	.	.	PUNCT
ejpam-5768	439	1	let	let	VERB
ejpam-5768	439	2	τ(p01	τ(p01	NOUN
ejpam-5768	439	3	)	)	PUNCT
ejpam-5768	439	4	,	,	PUNCT
ejpam-5768	439	5	τ(p02	τ(p02	X
ejpam-5768	439	6	)	)	PUNCT
ejpam-5768	439	7	∈	∈	PROPN
ejpam-5768	439	8	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	439	9	,	,	PUNCT
ejpam-5768	439	10	δ	δ	PROPN
ejpam-5768	439	11	)	)	PUNCT
ejpam-5768	439	12	)	)	PUNCT
ejpam-5768	439	13	.	.	PUNCT
ejpam-5768	440	1	then	then	ADV
ejpam-5768	440	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	440	3	)	)	PUNCT
ejpam-5768	440	4	)	)	PUNCT
ejpam-5768	440	5	≥	≥	PROPN
ejpam-5768	440	6	ζ	ζ	NOUN
ejpam-5768	440	7	,	,	PUNCT
ejpam-5768	440	8	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	440	9	)	)	PUNCT
ejpam-5768	440	10	)	)	PUNCT
ejpam-5768	440	11	≥	≥	PROPN
ejpam-5768	440	12	ζ	ζ	NOUN
ejpam-5768	440	13	,	,	PUNCT
ejpam-5768	440	14	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	440	15	)	)	PUNCT
ejpam-5768	440	16	)	)	PUNCT
ejpam-5768	440	17	≤	≤	NUM
ejpam-5768	440	18	δ	δ	PROPN
ejpam-5768	440	19	and	and	CCONJ
ejpam-5768	440	20	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	440	21	)	)	PUNCT
ejpam-5768	440	22	)	)	PUNCT
ejpam-5768	440	23	≤	≤	NUM
ejpam-5768	441	1	δ	δ	PROPN
ejpam-5768	441	2	.	.	PUNCT
ejpam-5768	441	3	then	then	ADV
ejpam-5768	441	4	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	441	5	)	)	PUNCT
ejpam-5768	441	6	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	441	7	)	)	PUNCT
ejpam-5768	441	8	)	)	PUNCT
ejpam-5768	442	1	=	=	SYM
ejpam-5768	442	2	η2(τ(p−1	η2(τ(p−1	NOUN
ejpam-5768	442	3	01	01	NUM
ejpam-5768	442	4	)	)	PUNCT
ejpam-5768	442	5	τ(p02)τ(p01	τ(p02)τ(p01	NOUN
ejpam-5768	442	6	)	)	PUNCT
ejpam-5768	442	7	)	)	PUNCT
ejpam-5768	443	1	=	=	SYM
ejpam-5768	443	2	η2(τ(p−1	η2(τ(p−1	NOUN
ejpam-5768	443	3	01	01	NUM
ejpam-5768	443	4	p02p01	p02p01	NOUN
ejpam-5768	443	5	)	)	PUNCT
ejpam-5768	443	6	)	)	PUNCT
ejpam-5768	444	1	≥	≥	PROPN
ejpam-5768	444	2	ῡ	ῡ	NOUN
ejpam-5768	444	3	2(p−1	2(p−1	ADJ
ejpam-5768	444	4	01	01	NUM
ejpam-5768	444	5	p02p01	p02p01	NOUN
ejpam-5768	444	6	)	)	PUNCT
ejpam-5768	444	7	since	since	SCONJ
ejpam-5768	444	8	p01	p01	NOUN
ejpam-5768	444	9	,	,	PUNCT
ejpam-5768	444	10	p02	p02	X
ejpam-5768	444	11	∈	∈	PROPN
ejpam-5768	444	12	υ(ζ	υ(ζ	PROPN
ejpam-5768	444	13	,	,	PUNCT
ejpam-5768	444	14	δ	δ	PROPN
ejpam-5768	444	15	)	)	PUNCT
ejpam-5768	444	16	and	and	CCONJ
ejpam-5768	444	17	υ(ζ	υ(ζ	PROPN
ejpam-5768	444	18	,	,	PUNCT
ejpam-5768	444	19	δ	δ	PROPN
ejpam-5768	444	20	)	)	PUNCT
ejpam-5768	444	21	is	be	AUX
ejpam-5768	444	22	an	an	DET
ejpam-5768	444	23	hx	hx	PROPN
ejpam-5768	444	24	-	-	PUNCT
ejpam-5768	444	25	nsg	nsg	PROPN
ejpam-5768	444	26	,	,	PUNCT
ejpam-5768	444	27	then	then	ADV
ejpam-5768	444	28	p−1	p−1	PROPN
ejpam-5768	444	29	01	01	NUM
ejpam-5768	444	30	p02p01	p02p01	PROPN
ejpam-5768	444	31	∈	∈	PROPN
ejpam-5768	444	32	υ(ζ	υ(ζ	PROPN
ejpam-5768	444	33	,	,	PUNCT
ejpam-5768	444	34	δ	δ	PROPN
ejpam-5768	444	35	)	)	PUNCT
ejpam-5768	444	36	.	.	PUNCT
ejpam-5768	445	1	thus	thus	ADV
ejpam-5768	445	2	ῡ	ῡ	PROPN
ejpam-5768	445	3	2(p−1	2(p−1	ADJ
ejpam-5768	445	4	01	01	NUM
ejpam-5768	445	5	p02p01	p02p01	ADJ
ejpam-5768	445	6	)	)	PUNCT
ejpam-5768	445	7	≥	≥	NOUN
ejpam-5768	445	8	ζ	ζ	NOUN
ejpam-5768	445	9	.	.	PUNCT
ejpam-5768	446	1	hence	hence	ADV
ejpam-5768	446	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	446	3	)	)	PUNCT
ejpam-5768	446	4	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	446	5	)	)	PUNCT
ejpam-5768	446	6	)	)	PUNCT
ejpam-5768	446	7	≥	≥	PROPN
ejpam-5768	447	1	ζ	ζ	NOUN
ejpam-5768	447	2	.	.	PUNCT
ejpam-5768	448	1	moreover	moreover	ADV
ejpam-5768	448	2	,	,	PUNCT
ejpam-5768	448	3	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	448	4	)	)	PUNCT
ejpam-5768	448	5	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	448	6	)	)	PUNCT
ejpam-5768	448	7	)	)	PUNCT
ejpam-5768	449	1	=	=	PUNCT
ejpam-5768	449	2	η̂2(τ(p−1	η̂2(τ(p−1	NOUN
ejpam-5768	449	3	01	01	NUM
ejpam-5768	449	4	)	)	PUNCT
ejpam-5768	449	5	τ(p02)τ(p01	τ(p02)τ(p01	NOUN
ejpam-5768	449	6	)	)	PUNCT
ejpam-5768	449	7	)	)	PUNCT
ejpam-5768	450	1	=	=	PUNCT
ejpam-5768	450	2	η̂2(τ(p−1	η̂2(τ(p−1	VERB
ejpam-5768	450	3	01	01	NUM
ejpam-5768	450	4	p02p01	p02p01	ADJ
ejpam-5768	450	5	)	)	PUNCT
ejpam-5768	450	6	)	)	PUNCT
ejpam-5768	450	7	≤	≤	NUM
ejpam-5768	450	8	υ̂	υ̂	VERB
ejpam-5768	450	9	2(p−1	2(p−1	ADJ
ejpam-5768	450	10	01	01	NUM
ejpam-5768	450	11	p02p01	p02p01	NOUN
ejpam-5768	450	12	)	)	PUNCT
ejpam-5768	450	13	since	since	SCONJ
ejpam-5768	450	14	p01	p01	NOUN
ejpam-5768	450	15	,	,	PUNCT
ejpam-5768	450	16	p02	p02	X
ejpam-5768	450	17	∈	∈	PROPN
ejpam-5768	450	18	υ(ζ	υ(ζ	PROPN
ejpam-5768	450	19	,	,	PUNCT
ejpam-5768	450	20	δ	δ	PROPN
ejpam-5768	450	21	)	)	PUNCT
ejpam-5768	450	22	and	and	CCONJ
ejpam-5768	450	23	υ(ζ	υ(ζ	PROPN
ejpam-5768	450	24	,	,	PUNCT
ejpam-5768	450	25	δ	δ	PROPN
ejpam-5768	450	26	)	)	PUNCT
ejpam-5768	450	27	is	be	AUX
ejpam-5768	450	28	an	an	DET
ejpam-5768	450	29	hx	hx	PROPN
ejpam-5768	450	30	-	-	PUNCT
ejpam-5768	450	31	nsg	nsg	PROPN
ejpam-5768	450	32	,	,	PUNCT
ejpam-5768	450	33	then	then	ADV
ejpam-5768	450	34	p−1	p−1	PROPN
ejpam-5768	450	35	01	01	NUM
ejpam-5768	450	36	p02p01	p02p01	PROPN
ejpam-5768	450	37	∈	∈	PROPN
ejpam-5768	450	38	υ(ζ	υ(ζ	PROPN
ejpam-5768	450	39	,	,	PUNCT
ejpam-5768	450	40	δ	δ	PROPN
ejpam-5768	450	41	)	)	PUNCT
ejpam-5768	450	42	.	.	PUNCT
ejpam-5768	451	1	thus	thus	ADV
ejpam-5768	451	2	υ̂	υ̂	PRON
ejpam-5768	451	3	2(p−1	2(p−1	ADJ
ejpam-5768	451	4	01	01	NUM
ejpam-5768	451	5	p02p01	p02p01	ADJ
ejpam-5768	451	6	)	)	PUNCT
ejpam-5768	451	7	≤	≤	NUM
ejpam-5768	452	1	δ	δ	PROPN
ejpam-5768	452	2	.	.	PUNCT
ejpam-5768	453	1	hence	hence	ADV
ejpam-5768	453	2	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	453	3	)	)	PUNCT
ejpam-5768	453	4	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	453	5	)	)	PUNCT
ejpam-5768	453	6	)	)	PUNCT
ejpam-5768	454	1	≤	≤	NUM
ejpam-5768	454	2	δ	δ	PROPN
ejpam-5768	454	3	.	.	PUNCT
ejpam-5768	454	4	therefore	therefore	ADV
ejpam-5768	454	5	,	,	PUNCT
ejpam-5768	454	6	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	454	7	,	,	PUNCT
ejpam-5768	454	8	δ	δ	NOUN
ejpam-5768	454	9	)	)	PUNCT
ejpam-5768	454	10	)	)	PUNCT
ejpam-5768	454	11	is	be	AUX
ejpam-5768	454	12	an	an	DET
ejpam-5768	454	13	hx	hx	PROPN
ejpam-5768	454	14	-	-	PUNCT
ejpam-5768	454	15	nsg	nsg	PROPN
ejpam-5768	454	16	.	.	PUNCT
ejpam-5768	455	1	theorem	theorem	VERB
ejpam-5768	455	2	15	15	NUM
ejpam-5768	455	3	.	.	PUNCT
ejpam-5768	456	1	let	let	VERB
ejpam-5768	456	2	τ	τ	PROPN
ejpam-5768	456	3	:	:	PUNCT
ejpam-5768	456	4	p	p	X
ejpam-5768	456	5	−→	−→	NOUN
ejpam-5768	456	6	s	s	AUX
ejpam-5768	456	7	be	be	AUX
ejpam-5768	456	8	an	an	DET
ejpam-5768	456	9	antihomomorphism	antihomomorphism	NOUN
ejpam-5768	456	10	of	of	ADP
ejpam-5768	456	11	hx	hx	NOUN
ejpam-5768	456	12	-	-	PUNCT
ejpam-5768	456	13	groups	group	NOUN
ejpam-5768	456	14	and	and	CCONJ
ejpam-5768	456	15	let	let	VERB
ejpam-5768	456	16	υ	υ	DET
ejpam-5768	456	17	2	2	NUM
ejpam-5768	456	18	be	be	AUX
ejpam-5768	456	19	a	a	DET
ejpam-5768	456	20	pfss	pfss	NOUN
ejpam-5768	456	21	of	of	ADP
ejpam-5768	456	22	w	w	PROPN
ejpam-5768	456	23	.	.	PUNCT
ejpam-5768	457	1	if	if	SCONJ
ejpam-5768	457	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	457	3	,	,	PUNCT
ejpam-5768	457	4	δ	δ	PROPN
ejpam-5768	457	5	)	)	PUNCT
ejpam-5768	457	6	is	be	AUX
ejpam-5768	457	7	an	an	DET
ejpam-5768	457	8	hx	hx	PROPN
ejpam-5768	457	9	-	-	PUNCT
ejpam-5768	457	10	sg	sg	PROPN
ejpam-5768	457	11	,	,	PUNCT
ejpam-5768	457	12	then	then	ADV
ejpam-5768	457	13	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	457	14	,	,	PUNCT
ejpam-5768	457	15	δ	δ	NOUN
ejpam-5768	457	16	)	)	PUNCT
ejpam-5768	457	17	)	)	PUNCT
ejpam-5768	457	18	is	be	AUX
ejpam-5768	457	19	an	an	DET
ejpam-5768	457	20	hx	hx	PROPN
ejpam-5768	457	21	-	-	PUNCT
ejpam-5768	457	22	sg	sg	PROPN
ejpam-5768	457	23	.	.	PUNCT
ejpam-5768	457	24	a.	a.	PROPN
ejpam-5768	457	25	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	457	26	/	/	SYM
ejpam-5768	457	27	eur	eur	PROPN
ejpam-5768	457	28	.	.	PUNCT
ejpam-5768	458	1	j.	j.	PROPN
ejpam-5768	458	2	pure	pure	PROPN
ejpam-5768	458	3	appl	appl	PROPN
ejpam-5768	458	4	.	.	PROPN
ejpam-5768	458	5	math	math	PROPN
ejpam-5768	458	6	,	,	PUNCT
ejpam-5768	458	7	18	18	NUM
ejpam-5768	458	8	(	(	PUNCT
ejpam-5768	458	9	2	2	NUM
ejpam-5768	458	10	)	)	PUNCT
ejpam-5768	458	11	(	(	PUNCT
ejpam-5768	458	12	2025	2025	NUM
ejpam-5768	458	13	)	)	PUNCT
ejpam-5768	458	14	,	,	PUNCT
ejpam-5768	458	15	5768	5768	NUM
ejpam-5768	458	16	15	15	NUM
ejpam-5768	458	17	of	of	ADP
ejpam-5768	458	18	17	17	NUM
ejpam-5768	458	19	proof	proof	NOUN
ejpam-5768	458	20	.	.	PUNCT
ejpam-5768	458	21	suppose	suppose	VERB
ejpam-5768	458	22	that	that	SCONJ
ejpam-5768	458	23	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	458	24	,	,	PUNCT
ejpam-5768	458	25	δ	δ	NOUN
ejpam-5768	458	26	)	)	PUNCT
ejpam-5768	458	27	)	)	PUNCT
ejpam-5768	459	1	=	=	PRON
ejpam-5768	459	2	{	{	PUNCT
ejpam-5768	459	3	(	(	PUNCT
ejpam-5768	459	4	τ(p	τ(p	NOUN
ejpam-5768	459	5	)	)	PUNCT
ejpam-5768	459	6	,	,	PUNCT
ejpam-5768	459	7	η2	η2	PROPN
ejpam-5768	459	8	,	,	PUNCT
ejpam-5768	459	9	η̂2	η̂2	NOUN
ejpam-5768	459	10	)	)	PUNCT
ejpam-5768	459	11	:	:	PUNCT
ejpam-5768	459	12	p	p	X
ejpam-5768	459	13	∈	∈	PROPN
ejpam-5768	459	14	υ(ζ	υ(ζ	PROPN
ejpam-5768	459	15	,	,	PUNCT
ejpam-5768	459	16	δ	δ	PROPN
ejpam-5768	459	17	)	)	PUNCT
ejpam-5768	459	18	}	}	PUNCT
ejpam-5768	459	19	.	.	PUNCT
ejpam-5768	460	1	let	let	VERB
ejpam-5768	460	2	τ(p01	τ(p01	NOUN
ejpam-5768	460	3	)	)	PUNCT
ejpam-5768	460	4	,	,	PUNCT
ejpam-5768	460	5	τ(p02	τ(p02	X
ejpam-5768	460	6	)	)	PUNCT
ejpam-5768	460	7	∈	∈	PROPN
ejpam-5768	460	8	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	460	9	,	,	PUNCT
ejpam-5768	460	10	δ	δ	PROPN
ejpam-5768	460	11	)	)	PUNCT
ejpam-5768	460	12	)	)	PUNCT
ejpam-5768	460	13	.	.	PUNCT
ejpam-5768	461	1	then	then	ADV
ejpam-5768	461	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	461	3	)	)	PUNCT
ejpam-5768	461	4	)	)	PUNCT
ejpam-5768	461	5	≥	≥	PROPN
ejpam-5768	461	6	ζ	ζ	NOUN
ejpam-5768	461	7	,	,	PUNCT
ejpam-5768	461	8	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	461	9	)	)	PUNCT
ejpam-5768	461	10	)	)	PUNCT
ejpam-5768	461	11	≥	≥	PROPN
ejpam-5768	461	12	ζ	ζ	NOUN
ejpam-5768	461	13	,	,	PUNCT
ejpam-5768	461	14	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	461	15	)	)	PUNCT
ejpam-5768	461	16	)	)	PUNCT
ejpam-5768	461	17	≤	≤	NUM
ejpam-5768	461	18	δ	δ	PROPN
ejpam-5768	461	19	and	and	CCONJ
ejpam-5768	461	20	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	461	21	)	)	PUNCT
ejpam-5768	461	22	)	)	PUNCT
ejpam-5768	461	23	≤	≤	NUM
ejpam-5768	462	1	δ	δ	PROPN
ejpam-5768	462	2	.	.	PUNCT
ejpam-5768	463	1	then	then	ADV
ejpam-5768	463	2	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	ADJ
ejpam-5768	463	3	)	)	PUNCT
ejpam-5768	463	4	−1	−1	NOUN
ejpam-5768	463	5	)	)	PUNCT
ejpam-5768	464	1	=	=	PUNCT
ejpam-5768	464	2	η2(τ(p01)τ(p	η2(τ(p01)τ(p	NOUN
ejpam-5768	464	3	−1	−1	NOUN
ejpam-5768	464	4	02	02	NUM
ejpam-5768	464	5	)	)	PUNCT
ejpam-5768	464	6	)	)	PUNCT
ejpam-5768	465	1	=	=	SYM
ejpam-5768	465	2	η2(τ(p−1	η2(τ(p−1	NOUN
ejpam-5768	465	3	02	02	NUM
ejpam-5768	465	4	p01	p01	NOUN
ejpam-5768	465	5	)	)	PUNCT
ejpam-5768	465	6	)	)	PUNCT
ejpam-5768	465	7	≥	≥	PROPN
ejpam-5768	465	8	ῡ	ῡ	NOUN
ejpam-5768	465	9	2(p−1	2(p−1	ADJ
ejpam-5768	465	10	02	02	NUM
ejpam-5768	465	11	p01	p01	NOUN
ejpam-5768	465	12	)	)	PUNCT
ejpam-5768	465	13	since	since	SCONJ
ejpam-5768	465	14	p01	p01	NOUN
ejpam-5768	465	15	,	,	PUNCT
ejpam-5768	465	16	p02	p02	X
ejpam-5768	465	17	∈	∈	PROPN
ejpam-5768	465	18	υ(ζ	υ(ζ	PROPN
ejpam-5768	465	19	,	,	PUNCT
ejpam-5768	465	20	δ	δ	PROPN
ejpam-5768	465	21	)	)	PUNCT
ejpam-5768	465	22	and	and	CCONJ
ejpam-5768	465	23	υ(ζ	υ(ζ	PROPN
ejpam-5768	465	24	,	,	PUNCT
ejpam-5768	465	25	δ	δ	PROPN
ejpam-5768	465	26	)	)	PUNCT
ejpam-5768	465	27	is	be	AUX
ejpam-5768	465	28	an	an	DET
ejpam-5768	465	29	hx	hx	PROPN
ejpam-5768	465	30	-	-	PUNCT
ejpam-5768	465	31	sg	sg	PROPN
ejpam-5768	465	32	,	,	PUNCT
ejpam-5768	465	33	then	then	ADV
ejpam-5768	465	34	p−1	p−1	PROPN
ejpam-5768	465	35	01	01	NUM
ejpam-5768	465	36	,	,	PUNCT
ejpam-5768	466	1	p	p	NOUN
ejpam-5768	466	2	−1	−1	NOUN
ejpam-5768	466	3	02	02	NUM
ejpam-5768	466	4	∈	∈	PROPN
ejpam-5768	466	5	υ(ζ	υ(ζ	PROPN
ejpam-5768	466	6	,	,	PUNCT
ejpam-5768	466	7	δ	δ	PROPN
ejpam-5768	466	8	)	)	PUNCT
ejpam-5768	466	9	and	and	CCONJ
ejpam-5768	466	10	so	so	ADV
ejpam-5768	466	11	p−1	p−1	PROPN
ejpam-5768	466	12	02	02	NUM
ejpam-5768	466	13	(	(	PUNCT
ejpam-5768	466	14	p	p	NOUN
ejpam-5768	466	15	−1	−1	NOUN
ejpam-5768	466	16	01	01	NUM
ejpam-5768	466	17	)	)	PUNCT
ejpam-5768	466	18	−1	−1	NOUN
ejpam-5768	466	19	∈	∈	PROPN
ejpam-5768	466	20	υ(ζ	υ(ζ	PROPN
ejpam-5768	466	21	,	,	PUNCT
ejpam-5768	466	22	δ	δ	PROPN
ejpam-5768	466	23	)	)	PUNCT
ejpam-5768	466	24	.	.	PUNCT
ejpam-5768	467	1	now	now	ADV
ejpam-5768	467	2	,	,	PUNCT
ejpam-5768	467	3	ῡ	ῡ	PROPN
ejpam-5768	467	4	2(p−1	2(p−1	ADJ
ejpam-5768	467	5	02	02	NUM
ejpam-5768	467	6	p01	p01	NOUN
ejpam-5768	467	7	)	)	PUNCT
ejpam-5768	467	8	=	=	SYM
ejpam-5768	468	1	ῡ	ῡ	NOUN
ejpam-5768	468	2	2(p−1	2(p−1	ADJ
ejpam-5768	468	3	02	02	NUM
ejpam-5768	469	1	(	(	PUNCT
ejpam-5768	469	2	p	p	NOUN
ejpam-5768	469	3	−1	−1	NOUN
ejpam-5768	469	4	01	01	NUM
ejpam-5768	469	5	)	)	PUNCT
ejpam-5768	469	6	−1	−1	NOUN
ejpam-5768	469	7	)	)	PUNCT
ejpam-5768	469	8	≥	≥	PROPN
ejpam-5768	470	1	ζ	ζ	NOUN
ejpam-5768	470	2	.	.	PUNCT
ejpam-5768	470	3	thus	thus	ADV
ejpam-5768	470	4	η2(τ(p01)τ(p02	η2(τ(p01)τ(p02	ADJ
ejpam-5768	470	5	)	)	PUNCT
ejpam-5768	470	6	−1	−1	NOUN
ejpam-5768	470	7	)	)	PUNCT
ejpam-5768	470	8	≥	≥	NOUN
ejpam-5768	470	9	ζ	ζ	NOUN
ejpam-5768	470	10	.	.	PUNCT
ejpam-5768	471	1	in	in	ADP
ejpam-5768	471	2	addition	addition	NOUN
ejpam-5768	471	3	,	,	PUNCT
ejpam-5768	471	4	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	NOUN
ejpam-5768	471	5	)	)	PUNCT
ejpam-5768	471	6	−1	−1	NOUN
ejpam-5768	471	7	)	)	PUNCT
ejpam-5768	472	1	=	=	PUNCT
ejpam-5768	472	2	η̂2(τ(p01)τ(p	η̂2(τ(p01)τ(p	NOUN
ejpam-5768	472	3	−1	−1	NOUN
ejpam-5768	472	4	02	02	NUM
ejpam-5768	472	5	)	)	PUNCT
ejpam-5768	472	6	)	)	PUNCT
ejpam-5768	473	1	=	=	PRON
ejpam-5768	473	2	η̂2(τ(p−1	η̂2(τ(p−1	VERB
ejpam-5768	473	3	02	02	NUM
ejpam-5768	473	4	p01	p01	NOUN
ejpam-5768	473	5	)	)	PUNCT
ejpam-5768	473	6	)	)	PUNCT
ejpam-5768	473	7	≤	≤	NUM
ejpam-5768	473	8	υ̂	υ̂	VERB
ejpam-5768	473	9	2(p−1	2(p−1	ADJ
ejpam-5768	473	10	02	02	NUM
ejpam-5768	473	11	p01	p01	NOUN
ejpam-5768	473	12	)	)	PUNCT
ejpam-5768	473	13	since	since	SCONJ
ejpam-5768	473	14	p−1	p−1	PROPN
ejpam-5768	473	15	02	02	NUM
ejpam-5768	473	16	(	(	PUNCT
ejpam-5768	473	17	p	p	NOUN
ejpam-5768	473	18	−1	−1	NOUN
ejpam-5768	473	19	01	01	NUM
ejpam-5768	473	20	)	)	PUNCT
ejpam-5768	474	1	−1	−1	NOUN
ejpam-5768	474	2	∈	∈	PROPN
ejpam-5768	474	3	υ(ζ	υ(ζ	PROPN
ejpam-5768	474	4	,	,	PUNCT
ejpam-5768	474	5	δ	δ	PROPN
ejpam-5768	474	6	)	)	PUNCT
ejpam-5768	474	7	,	,	PUNCT
ejpam-5768	474	8	it	it	PRON
ejpam-5768	474	9	implies	imply	VERB
ejpam-5768	474	10	that	that	SCONJ
ejpam-5768	474	11	υ̂	υ̂	NUM
ejpam-5768	474	12	2(p−1	2(p−1	ADJ
ejpam-5768	474	13	02	02	NUM
ejpam-5768	474	14	p01	p01	NOUN
ejpam-5768	474	15	)	)	PUNCT
ejpam-5768	474	16	=	=	PUNCT
ejpam-5768	474	17	υ̂	υ̂	NUM
ejpam-5768	474	18	2(p−1	2(p−1	NUM
ejpam-5768	474	19	02	02	NUM
ejpam-5768	474	20	(	(	PUNCT
ejpam-5768	474	21	p	p	NOUN
ejpam-5768	474	22	−1	−1	NOUN
ejpam-5768	474	23	01	01	NUM
ejpam-5768	474	24	)	)	PUNCT
ejpam-5768	474	25	−1	−1	NOUN
ejpam-5768	474	26	)	)	PUNCT
ejpam-5768	474	27	≤	≤	NUM
ejpam-5768	475	1	δ	δ	PROPN
ejpam-5768	475	2	.	.	PUNCT
ejpam-5768	476	1	thus	thus	ADV
ejpam-5768	476	2	η̂2(τ(p01)τ(p02	η̂2(τ(p01)τ(p02	PROPN
ejpam-5768	476	3	)	)	PUNCT
ejpam-5768	476	4	−1	−1	NOUN
ejpam-5768	476	5	)	)	PUNCT
ejpam-5768	477	1	≤	≤	NUM
ejpam-5768	477	2	δ	δ	PROPN
ejpam-5768	477	3	.	.	PUNCT
ejpam-5768	477	4	therefore	therefore	ADV
ejpam-5768	477	5	,	,	PUNCT
ejpam-5768	477	6	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	477	7	,	,	PUNCT
ejpam-5768	477	8	δ	δ	NOUN
ejpam-5768	477	9	)	)	PUNCT
ejpam-5768	477	10	)	)	PUNCT
ejpam-5768	477	11	is	be	AUX
ejpam-5768	477	12	an	an	DET
ejpam-5768	477	13	hx	hx	PROPN
ejpam-5768	477	14	-	-	PUNCT
ejpam-5768	477	15	sg	sg	PROPN
ejpam-5768	477	16	.	.	PUNCT
ejpam-5768	478	1	theorem	theorem	PROPN
ejpam-5768	478	2	16	16	NUM
ejpam-5768	478	3	.	.	PUNCT
ejpam-5768	479	1	let	let	VERB
ejpam-5768	479	2	τ	τ	PROPN
ejpam-5768	479	3	:	:	PUNCT
ejpam-5768	479	4	p	p	X
ejpam-5768	479	5	−→	−→	NOUN
ejpam-5768	479	6	s	s	AUX
ejpam-5768	479	7	be	be	AUX
ejpam-5768	479	8	a	a	DET
ejpam-5768	479	9	homomorphism	homomorphism	NOUN
ejpam-5768	479	10	of	of	ADP
ejpam-5768	479	11	hx	hx	NOUN
ejpam-5768	479	12	-	-	PUNCT
ejpam-5768	479	13	groups	group	NOUN
ejpam-5768	479	14	and	and	CCONJ
ejpam-5768	479	15	let	let	VERB
ejpam-5768	479	16	υ	υ	DET
ejpam-5768	479	17	2	2	NUM
ejpam-5768	479	18	be	be	AUX
ejpam-5768	479	19	a	a	DET
ejpam-5768	479	20	pfss	pfss	NOUN
ejpam-5768	479	21	of	of	ADP
ejpam-5768	479	22	w	w	PROPN
ejpam-5768	479	23	.	.	PUNCT
ejpam-5768	480	1	if	if	SCONJ
ejpam-5768	480	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	480	3	,	,	PUNCT
ejpam-5768	480	4	δ	δ	PROPN
ejpam-5768	480	5	)	)	PUNCT
ejpam-5768	480	6	is	be	AUX
ejpam-5768	480	7	an	an	DET
ejpam-5768	480	8	hx	hx	PROPN
ejpam-5768	480	9	-	-	PUNCT
ejpam-5768	480	10	nsg	nsg	PROPN
ejpam-5768	480	11	,	,	PUNCT
ejpam-5768	480	12	then	then	ADV
ejpam-5768	480	13	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	480	14	,	,	PUNCT
ejpam-5768	480	15	δ	δ	NOUN
ejpam-5768	480	16	)	)	PUNCT
ejpam-5768	480	17	)	)	PUNCT
ejpam-5768	480	18	is	be	AUX
ejpam-5768	480	19	an	an	DET
ejpam-5768	480	20	hx	hx	PROPN
ejpam-5768	480	21	-	-	PUNCT
ejpam-5768	480	22	nsg	nsg	PROPN
ejpam-5768	480	23	.	.	PUNCT
ejpam-5768	481	1	proof	proof	NOUN
ejpam-5768	481	2	.	.	PUNCT
ejpam-5768	482	1	since	since	SCONJ
ejpam-5768	482	2	υ(ζ	υ(ζ	PROPN
ejpam-5768	482	3	,	,	PUNCT
ejpam-5768	482	4	δ	δ	PROPN
ejpam-5768	482	5	)	)	PUNCT
ejpam-5768	482	6	is	be	AUX
ejpam-5768	482	7	an	an	DET
ejpam-5768	482	8	hx	hx	PROPN
ejpam-5768	482	9	-	-	PUNCT
ejpam-5768	482	10	nsg	nsg	PROPN
ejpam-5768	482	11	,	,	PUNCT
ejpam-5768	482	12	then	then	ADV
ejpam-5768	482	13	it	it	PRON
ejpam-5768	482	14	is	be	AUX
ejpam-5768	482	15	an	an	DET
ejpam-5768	482	16	hx	hx	NOUN
ejpam-5768	482	17	-	-	PUNCT
ejpam-5768	482	18	sg	sg	PROPN
ejpam-5768	482	19	and	and	CCONJ
ejpam-5768	482	20	,	,	PUNCT
ejpam-5768	482	21	by	by	ADP
ejpam-5768	482	22	the	the	DET
ejpam-5768	482	23	previous	previous	ADJ
ejpam-5768	482	24	theorem	theorem	NOUN
ejpam-5768	482	25	,	,	PUNCT
ejpam-5768	482	26	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	482	27	,	,	PUNCT
ejpam-5768	482	28	δ	δ	NOUN
ejpam-5768	482	29	)	)	PUNCT
ejpam-5768	482	30	)	)	PUNCT
ejpam-5768	483	1	is	be	AUX
ejpam-5768	483	2	an	an	DET
ejpam-5768	483	3	hx	hx	PROPN
ejpam-5768	483	4	-	-	PUNCT
ejpam-5768	483	5	sg	sg	PROPN
ejpam-5768	483	6	.	.	PUNCT
ejpam-5768	484	1	we	we	PRON
ejpam-5768	484	2	need	need	VERB
ejpam-5768	484	3	to	to	PART
ejpam-5768	484	4	prove	prove	VERB
ejpam-5768	484	5	that	that	SCONJ
ejpam-5768	484	6	it	it	PRON
ejpam-5768	484	7	is	be	AUX
ejpam-5768	484	8	normal	normal	ADJ
ejpam-5768	484	9	.	.	PUNCT
ejpam-5768	485	1	suppose	suppose	VERB
ejpam-5768	485	2	that	that	SCONJ
ejpam-5768	485	3	τ(υ(ζ	τ(υ(ζ	NOUN
ejpam-5768	485	4	,	,	PUNCT
ejpam-5768	485	5	δ	δ	NOUN
ejpam-5768	485	6	)	)	PUNCT
ejpam-5768	485	7	)	)	PUNCT
ejpam-5768	486	1	=	=	PRON
ejpam-5768	486	2	{	{	PUNCT
ejpam-5768	486	3	(	(	PUNCT
ejpam-5768	486	4	τ(p	τ(p	NOUN
ejpam-5768	486	5	)	)	PUNCT
ejpam-5768	486	6	,	,	PUNCT
ejpam-5768	486	7	η2	η2	PROPN
ejpam-5768	486	8	,	,	PUNCT
ejpam-5768	486	9	η̂2	η̂2	NOUN
ejpam-5768	486	10	)	)	PUNCT
ejpam-5768	486	11	:	:	PUNCT
ejpam-5768	486	12	p	p	X
ejpam-5768	486	13	∈	∈	PROPN
ejpam-5768	486	14	υ(ζ	υ(ζ	PROPN
ejpam-5768	486	15	,	,	PUNCT
ejpam-5768	486	16	δ	δ	PROPN
ejpam-5768	486	17	)	)	PUNCT
ejpam-5768	486	18	}	}	PUNCT
ejpam-5768	486	19	.	.	PUNCT
ejpam-5768	487	1	let	let	VERB
ejpam-5768	487	2	τ(p01	τ(p01	NOUN
ejpam-5768	487	3	)	)	PUNCT
ejpam-5768	487	4	,	,	PUNCT
ejpam-5768	487	5	τ(p02	τ(p02	X
ejpam-5768	487	6	)	)	PUNCT
ejpam-5768	487	7	∈	∈	PROPN
ejpam-5768	487	8	τ(υ(ζ	τ(υ(ζ	PROPN
ejpam-5768	487	9	,	,	PUNCT
ejpam-5768	487	10	δ	δ	PROPN
ejpam-5768	487	11	)	)	PUNCT
ejpam-5768	487	12	)	)	PUNCT
ejpam-5768	487	13	.	.	PUNCT
ejpam-5768	488	1	then	then	ADV
ejpam-5768	488	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	488	3	)	)	PUNCT
ejpam-5768	488	4	)	)	PUNCT
ejpam-5768	488	5	≥	≥	PROPN
ejpam-5768	488	6	ζ	ζ	NOUN
ejpam-5768	488	7	,	,	PUNCT
ejpam-5768	488	8	η2(τ(p02	η2(τ(p02	NOUN
ejpam-5768	488	9	)	)	PUNCT
ejpam-5768	488	10	)	)	PUNCT
ejpam-5768	488	11	≥	≥	PROPN
ejpam-5768	488	12	ζ	ζ	NOUN
ejpam-5768	488	13	,	,	PUNCT
ejpam-5768	488	14	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	488	15	)	)	PUNCT
ejpam-5768	488	16	)	)	PUNCT
ejpam-5768	488	17	≤	≤	NUM
ejpam-5768	488	18	δ	δ	PROPN
ejpam-5768	488	19	and	and	CCONJ
ejpam-5768	488	20	η̂2(τ(p02	η̂2(τ(p02	PROPN
ejpam-5768	488	21	)	)	PUNCT
ejpam-5768	488	22	)	)	PUNCT
ejpam-5768	488	23	≤	≤	NUM
ejpam-5768	489	1	δ	δ	PROPN
ejpam-5768	489	2	.	.	PUNCT
ejpam-5768	489	3	then	then	ADV
ejpam-5768	489	4	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	489	5	)	)	PUNCT
ejpam-5768	489	6	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	489	7	)	)	PUNCT
ejpam-5768	489	8	)	)	PUNCT
ejpam-5768	490	1	=	=	SYM
ejpam-5768	490	2	η2(τ(p−1	η2(τ(p−1	NOUN
ejpam-5768	490	3	01	01	NUM
ejpam-5768	490	4	)	)	PUNCT
ejpam-5768	490	5	τ(p02)τ(p01	τ(p02)τ(p01	NOUN
ejpam-5768	490	6	)	)	PUNCT
ejpam-5768	490	7	)	)	PUNCT
ejpam-5768	491	1	=	=	PUNCT
ejpam-5768	492	1	η2(τ(p02p	η2(τ(p02p	PROPN
ejpam-5768	492	2	−1	−1	NOUN
ejpam-5768	492	3	01	01	NUM
ejpam-5768	492	4	)	)	PUNCT
ejpam-5768	492	5	τ(p01	τ(p01	NOUN
ejpam-5768	492	6	)	)	PUNCT
ejpam-5768	492	7	)	)	PUNCT
ejpam-5768	493	1	=	=	PUNCT
ejpam-5768	493	2	η2(τ(p01p02p	η2(τ(p01p02p	ADJ
ejpam-5768	493	3	−1	−1	NOUN
ejpam-5768	493	4	01	01	NUM
ejpam-5768	493	5	)	)	PUNCT
ejpam-5768	493	6	)	)	PUNCT
ejpam-5768	493	7	≥	≥	PROPN
ejpam-5768	494	1	ῡ	ῡ	PROPN
ejpam-5768	494	2	2(p01p02p	2(p01p02p	PROPN
ejpam-5768	494	3	−1	−1	NOUN
ejpam-5768	494	4	01	01	NUM
ejpam-5768	494	5	)	)	PUNCT
ejpam-5768	494	6	.	.	PUNCT
ejpam-5768	495	1	since	since	SCONJ
ejpam-5768	495	2	p01	p01	NOUN
ejpam-5768	495	3	,	,	PUNCT
ejpam-5768	495	4	p02	p02	X
ejpam-5768	495	5	∈	∈	PROPN
ejpam-5768	495	6	υ(ζ	υ(ζ	PROPN
ejpam-5768	495	7	,	,	PUNCT
ejpam-5768	495	8	δ	δ	PROPN
ejpam-5768	495	9	)	)	PUNCT
ejpam-5768	495	10	and	and	CCONJ
ejpam-5768	495	11	υ(ζ	υ(ζ	PROPN
ejpam-5768	495	12	,	,	PUNCT
ejpam-5768	495	13	δ	δ	PROPN
ejpam-5768	495	14	)	)	PUNCT
ejpam-5768	495	15	is	be	AUX
ejpam-5768	495	16	an	an	DET
ejpam-5768	495	17	hx	hx	PROPN
ejpam-5768	495	18	-	-	PUNCT
ejpam-5768	495	19	nsg	nsg	PROPN
ejpam-5768	495	20	,	,	PUNCT
ejpam-5768	495	21	then	then	ADV
ejpam-5768	495	22	p−1	p−1	PROPN
ejpam-5768	495	23	01	01	NUM
ejpam-5768	495	24	,	,	PUNCT
ejpam-5768	495	25	p02	p02	PROPN
ejpam-5768	495	26	∈	∈	PROPN
ejpam-5768	495	27	υ(ζ	υ(ζ	PROPN
ejpam-5768	495	28	,	,	PUNCT
ejpam-5768	495	29	δ	δ	PROPN
ejpam-5768	495	30	)	)	PUNCT
ejpam-5768	495	31	.	.	PUNCT
ejpam-5768	496	1	this	this	PRON
ejpam-5768	496	2	implies	imply	VERB
ejpam-5768	496	3	that	that	SCONJ
ejpam-5768	496	4	p01p02p	p01p02p	PROPN
ejpam-5768	496	5	−1	−1	NOUN
ejpam-5768	496	6	01	01	NUM
ejpam-5768	496	7	=	=	SYM
ejpam-5768	496	8	(	(	PUNCT
ejpam-5768	496	9	p−1	p−1	PROPN
ejpam-5768	496	10	01	01	NUM
ejpam-5768	496	11	)	)	PUNCT
ejpam-5768	496	12	−1p02p	−1p02p	NUM
ejpam-5768	496	13	−1	−1	NOUN
ejpam-5768	496	14	01	01	NUM
ejpam-5768	496	15	∈	∈	PROPN
ejpam-5768	496	16	υ(ζ	υ(ζ	PROPN
ejpam-5768	496	17	,	,	PUNCT
ejpam-5768	496	18	δ	δ	PROPN
ejpam-5768	496	19	)	)	PUNCT
ejpam-5768	496	20	.	.	PUNCT
ejpam-5768	497	1	thus	thus	ADV
ejpam-5768	497	2	ῡ	ῡ	PROPN
ejpam-5768	497	3	2(p01p02p	2(p01p02p	PROPN
ejpam-5768	497	4	−1	−1	NOUN
ejpam-5768	497	5	01	01	NUM
ejpam-5768	497	6	)	)	PUNCT
ejpam-5768	497	7	≥	≥	PROPN
ejpam-5768	497	8	ζ	ζ	NOUN
ejpam-5768	497	9	.	.	PUNCT
ejpam-5768	498	1	hence	hence	ADV
ejpam-5768	498	2	η2(τ(p01	η2(τ(p01	NOUN
ejpam-5768	498	3	)	)	PUNCT
ejpam-5768	498	4	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	498	5	)	)	PUNCT
ejpam-5768	498	6	)	)	PUNCT
ejpam-5768	498	7	≥	≥	PROPN
ejpam-5768	499	1	ζ	ζ	NOUN
ejpam-5768	499	2	.	.	PUNCT
ejpam-5768	500	1	moreover	moreover	ADV
ejpam-5768	500	2	,	,	PUNCT
ejpam-5768	500	3	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	500	4	)	)	PUNCT
ejpam-5768	500	5	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	500	6	)	)	PUNCT
ejpam-5768	500	7	)	)	PUNCT
ejpam-5768	501	1	=	=	PUNCT
ejpam-5768	501	2	η̂2(τ(p−1	η̂2(τ(p−1	NOUN
ejpam-5768	501	3	01	01	NUM
ejpam-5768	501	4	)	)	PUNCT
ejpam-5768	501	5	τ(p02)τ(p01	τ(p02)τ(p01	NOUN
ejpam-5768	501	6	)	)	PUNCT
ejpam-5768	501	7	)	)	PUNCT
ejpam-5768	502	1	=	=	PUNCT
ejpam-5768	502	2	η̂2(τ(p02p	η̂2(τ(p02p	PROPN
ejpam-5768	502	3	−1	−1	NOUN
ejpam-5768	502	4	01	01	NUM
ejpam-5768	502	5	)	)	PUNCT
ejpam-5768	502	6	τ(p01	τ(p01	NOUN
ejpam-5768	502	7	)	)	PUNCT
ejpam-5768	502	8	)	)	PUNCT
ejpam-5768	503	1	=	=	PUNCT
ejpam-5768	503	2	η̂2(τ(p01p02p	η̂2(τ(p01p02p	PROPN
ejpam-5768	503	3	−1	−1	NOUN
ejpam-5768	503	4	01	01	NUM
ejpam-5768	503	5	)	)	PUNCT
ejpam-5768	503	6	)	)	PUNCT
ejpam-5768	503	7	≤	≤	NUM
ejpam-5768	503	8	υ̂	υ̂	VERB
ejpam-5768	503	9	2(p01p02p	2(p01p02p	NUM
ejpam-5768	503	10	−1	−1	NOUN
ejpam-5768	503	11	01	01	NUM
ejpam-5768	503	12	)	)	PUNCT
ejpam-5768	503	13	.	.	PUNCT
ejpam-5768	504	1	since	since	SCONJ
ejpam-5768	504	2	p−1	p−1	PROPN
ejpam-5768	504	3	01	01	NUM
ejpam-5768	504	4	,	,	PUNCT
ejpam-5768	504	5	p02	p02	PROPN
ejpam-5768	504	6	∈	∈	PROPN
ejpam-5768	504	7	υ(ζ	υ(ζ	PROPN
ejpam-5768	504	8	,	,	PUNCT
ejpam-5768	504	9	δ	δ	PROPN
ejpam-5768	504	10	)	)	PUNCT
ejpam-5768	504	11	,	,	PUNCT
ejpam-5768	504	12	then	then	ADV
ejpam-5768	504	13	p01p02p	p01p02p	VERB
ejpam-5768	504	14	−1	−1	NOUN
ejpam-5768	504	15	01	01	NUM
ejpam-5768	504	16	=	=	SYM
ejpam-5768	504	17	(	(	PUNCT
ejpam-5768	504	18	p−1	p−1	PROPN
ejpam-5768	504	19	01	01	NUM
ejpam-5768	504	20	)	)	PUNCT
ejpam-5768	504	21	−1p02p	−1p02p	NUM
ejpam-5768	504	22	−1	−1	NOUN
ejpam-5768	504	23	01	01	NUM
ejpam-5768	504	24	∈	∈	PROPN
ejpam-5768	504	25	υ(ζ	υ(ζ	PROPN
ejpam-5768	504	26	,	,	PUNCT
ejpam-5768	504	27	δ	δ	PROPN
ejpam-5768	504	28	)	)	PUNCT
ejpam-5768	504	29	.	.	PUNCT
ejpam-5768	505	1	thus	thus	ADV
ejpam-5768	505	2	υ̂	υ̂	NUM
ejpam-5768	505	3	2(p01p02p	2(p01p02p	NUM
ejpam-5768	505	4	−1	−1	NOUN
ejpam-5768	505	5	01	01	NUM
ejpam-5768	505	6	)	)	PUNCT
ejpam-5768	505	7	≤	≤	NUM
ejpam-5768	506	1	δ	δ	PROPN
ejpam-5768	506	2	.	.	PUNCT
ejpam-5768	507	1	hence	hence	ADV
ejpam-5768	507	2	η̂2(τ(p01	η̂2(τ(p01	PROPN
ejpam-5768	507	3	)	)	PUNCT
ejpam-5768	507	4	−1τ(p02)τ(p01	−1τ(p02)τ(p01	NOUN
ejpam-5768	507	5	)	)	PUNCT
ejpam-5768	507	6	)	)	PUNCT
ejpam-5768	508	1	≤	≤	NUM
ejpam-5768	509	1	δ	δ	PROPN
ejpam-5768	509	2	.	.	PUNCT
ejpam-5768	509	3	a.	a.	PROPN
ejpam-5768	509	4	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	509	5	/	/	SYM
ejpam-5768	509	6	eur	eur	PROPN
ejpam-5768	509	7	.	.	PUNCT
ejpam-5768	510	1	j.	j.	PROPN
ejpam-5768	510	2	pure	pure	PROPN
ejpam-5768	510	3	appl	appl	PROPN
ejpam-5768	510	4	.	.	PROPN
ejpam-5768	510	5	math	math	PROPN
ejpam-5768	510	6	,	,	PUNCT
ejpam-5768	510	7	18	18	NUM
ejpam-5768	510	8	(	(	PUNCT
ejpam-5768	510	9	2	2	NUM
ejpam-5768	510	10	)	)	PUNCT
ejpam-5768	510	11	(	(	PUNCT
ejpam-5768	510	12	2025	2025	NUM
ejpam-5768	510	13	)	)	PUNCT
ejpam-5768	510	14	,	,	PUNCT
ejpam-5768	510	15	5768	5768	NUM
ejpam-5768	510	16	16	16	NUM
ejpam-5768	510	17	of	of	ADP
ejpam-5768	510	18	17	17	NUM
ejpam-5768	510	19	5	5	NUM
ejpam-5768	510	20	.	.	PUNCT
ejpam-5768	511	1	conclusion	conclusion	NOUN
ejpam-5768	511	2	this	this	DET
ejpam-5768	511	3	article	article	NOUN
ejpam-5768	511	4	introduced	introduce	VERB
ejpam-5768	511	5	the	the	DET
ejpam-5768	511	6	novel	novel	ADJ
ejpam-5768	511	7	concept	concept	NOUN
ejpam-5768	511	8	of	of	ADP
ejpam-5768	511	9	a	a	DET
ejpam-5768	511	10	pythagorean	pythagorean	ADJ
ejpam-5768	511	11	fuzzy	fuzzy	ADJ
ejpam-5768	511	12	hx	hx	PROPN
ejpam-5768	511	13	-	-	PUNCT
ejpam-5768	511	14	subgroup	subgroup	PROPN
ejpam-5768	511	15	and	and	CCONJ
ejpam-5768	511	16	a	a	DET
ejpam-5768	511	17	normal	normal	ADJ
ejpam-5768	511	18	hx	hx	PROPN
ejpam-5768	511	19	-	-	PUNCT
ejpam-5768	511	20	subgroup	subgroup	NOUN
ejpam-5768	511	21	.	.	PUNCT
ejpam-5768	512	1	this	this	DET
ejpam-5768	512	2	study	study	NOUN
ejpam-5768	512	3	aims	aim	VERB
ejpam-5768	512	4	to	to	PART
ejpam-5768	512	5	establish	establish	VERB
ejpam-5768	512	6	the	the	DET
ejpam-5768	512	7	groundwork	groundwork	NOUN
ejpam-5768	512	8	for	for	ADP
ejpam-5768	512	9	a	a	DET
ejpam-5768	512	10	novel	novel	ADJ
ejpam-5768	512	11	theory	theory	NOUN
ejpam-5768	512	12	of	of	ADP
ejpam-5768	512	13	pythagorean	pythagorean	PROPN
ejpam-5768	512	14	fuzzy	fuzzy	ADJ
ejpam-5768	512	15	hx	hx	PROPN
ejpam-5768	512	16	-	-	PUNCT
ejpam-5768	512	17	subgroups	subgroup	NOUN
ejpam-5768	512	18	as	as	SCONJ
ejpam-5768	512	19	it	it	PRON
ejpam-5768	512	20	is	be	AUX
ejpam-5768	512	21	the	the	DET
ejpam-5768	512	22	extension	extension	NOUN
ejpam-5768	512	23	of	of	ADP
ejpam-5768	512	24	fuzzy	fuzzy	ADJ
ejpam-5768	512	25	hx	hx	NOUN
ejpam-5768	512	26	groups	group	NOUN
ejpam-5768	512	27	and	and	CCONJ
ejpam-5768	512	28	intuitionistic	intuitionistic	ADJ
ejpam-5768	512	29	fuzzy	fuzzy	ADJ
ejpam-5768	512	30	hx	hx	PROPN
ejpam-5768	512	31	-	-	PUNCT
ejpam-5768	512	32	subgroup	subgroup	NOUN
ejpam-5768	512	33	.	.	PUNCT
ejpam-5768	513	1	various	various	ADJ
ejpam-5768	513	2	chracterisations	chracterisation	NOUN
ejpam-5768	513	3	for	for	ADP
ejpam-5768	513	4	pythagorean	pythagorean	PROPN
ejpam-5768	513	5	fuzzy	fuzzy	ADJ
ejpam-5768	513	6	hx	hx	PROPN
ejpam-5768	513	7	-	-	PUNCT
ejpam-5768	513	8	subgroups	subgroup	NOUN
ejpam-5768	513	9	and	and	CCONJ
ejpam-5768	513	10	pythagorean	pythagorean	PROPN
ejpam-5768	513	11	normal	normal	ADJ
ejpam-5768	513	12	hx	hx	PROPN
ejpam-5768	513	13	-	-	PUNCT
ejpam-5768	513	14	subgroups	subgroup	NOUN
ejpam-5768	513	15	are	be	AUX
ejpam-5768	513	16	proved	prove	VERB
ejpam-5768	513	17	.	.	PUNCT
ejpam-5768	514	1	moreover	moreover	ADV
ejpam-5768	514	2	,	,	PUNCT
ejpam-5768	514	3	the	the	DET
ejpam-5768	514	4	notations	notation	NOUN
ejpam-5768	514	5	of	of	ADP
ejpam-5768	514	6	pythagorean	pythagorean	PROPN
ejpam-5768	514	7	fuzzy	fuzzy	ADJ
ejpam-5768	514	8	hx	hx	PROPN
ejpam-5768	514	9	-	-	PUNCT
ejpam-5768	514	10	subgroups	subgroup	NOUN
ejpam-5768	514	11	homomorphisms	homomorphism	NOUN
ejpam-5768	514	12	and	and	CCONJ
ejpam-5768	514	13	antihomomorphisms	antihomomorphism	NOUN
ejpam-5768	514	14	are	be	AUX
ejpam-5768	514	15	initiated	initiate	VERB
ejpam-5768	514	16	,	,	PUNCT
ejpam-5768	514	17	and	and	CCONJ
ejpam-5768	514	18	some	some	DET
ejpam-5768	514	19	related	relate	VERB
ejpam-5768	514	20	properties	property	NOUN
ejpam-5768	514	21	regarding	regard	VERB
ejpam-5768	514	22	the	the	DET
ejpam-5768	514	23	relationship	relationship	NOUN
ejpam-5768	514	24	between	between	ADP
ejpam-5768	514	25	a	a	DET
ejpam-5768	514	26	pythagorean	pythagorean	ADJ
ejpam-5768	514	27	fuzzy	fuzzy	ADJ
ejpam-5768	514	28	set	set	NOUN
ejpam-5768	514	29	and	and	CCONJ
ejpam-5768	514	30	its	its	PRON
ejpam-5768	514	31	image	image	NOUN
ejpam-5768	514	32	are	be	AUX
ejpam-5768	514	33	investigated	investigate	VERB
ejpam-5768	514	34	.	.	PUNCT
ejpam-5768	515	1	characterisations	characterisation	NOUN
ejpam-5768	515	2	of	of	ADP
ejpam-5768	515	3	level	level	NOUN
ejpam-5768	515	4	pythagorean	pythagorean	PROPN
ejpam-5768	515	5	fuzzy	fuzzy	ADJ
ejpam-5768	515	6	hx	hx	PROPN
ejpam-5768	515	7	-	-	PUNCT
ejpam-5768	515	8	subgroups	subgroup	NOUN
ejpam-5768	515	9	and	and	CCONJ
ejpam-5768	515	10	normal	normal	ADJ
ejpam-5768	515	11	hx	hx	NOUN
ejpam-5768	515	12	-	-	PUNCT
ejpam-5768	515	13	subgroups	subgroup	NOUN
ejpam-5768	515	14	are	be	AUX
ejpam-5768	515	15	presented	present	VERB
ejpam-5768	515	16	.	.	PUNCT
ejpam-5768	516	1	in	in	ADP
ejpam-5768	516	2	future	future	ADJ
ejpam-5768	516	3	work	work	NOUN
ejpam-5768	516	4	,	,	PUNCT
ejpam-5768	516	5	this	this	DET
ejpam-5768	516	6	study	study	NOUN
ejpam-5768	516	7	can	can	AUX
ejpam-5768	516	8	be	be	AUX
ejpam-5768	516	9	expanded	expand	VERB
ejpam-5768	516	10	to	to	PART
ejpam-5768	516	11	pythagorean	pythagorean	VERB
ejpam-5768	516	12	fuzzy	fuzzy	ADJ
ejpam-5768	516	13	soft	soft	ADJ
ejpam-5768	516	14	hx	hx	NOUN
ejpam-5768	516	15	-	-	PUNCT
ejpam-5768	516	16	groups	group	NOUN
ejpam-5768	516	17	and	and	CCONJ
ejpam-5768	516	18	to	to	PART
ejpam-5768	516	19	apply	apply	VERB
ejpam-5768	516	20	more	more	ADJ
ejpam-5768	516	21	strategies	strategy	NOUN
ejpam-5768	516	22	for	for	ADP
ejpam-5768	516	23	handling	handle	VERB
ejpam-5768	516	24	other	other	ADJ
ejpam-5768	516	25	hybrid	hybrid	ADJ
ejpam-5768	516	26	models	model	NOUN
ejpam-5768	516	27	,	,	PUNCT
ejpam-5768	516	28	such	such	ADJ
ejpam-5768	516	29	as	as	ADP
ejpam-5768	516	30	m	m	NOUN
ejpam-5768	516	31	-	-	ADJ
ejpam-5768	516	32	polar	polar	ADJ
ejpam-5768	516	33	soft	soft	ADJ
ejpam-5768	516	34	hx	hx	NOUN
ejpam-5768	516	35	-	-	PUNCT
ejpam-5768	516	36	groups	group	NOUN
ejpam-5768	516	37	,	,	PUNCT
ejpam-5768	516	38	bipolar	bipolar	ADJ
ejpam-5768	516	39	soft	soft	ADJ
ejpam-5768	516	40	hx	hx	NOUN
ejpam-5768	516	41	-	-	PUNCT
ejpam-5768	516	42	groups	group	NOUN
ejpam-5768	516	43	,	,	PUNCT
ejpam-5768	516	44	and	and	CCONJ
ejpam-5768	516	45	neutrosophic	neutrosophic	ADJ
ejpam-5768	516	46	soft	soft	ADJ
ejpam-5768	516	47	hx	hx	NOUN
ejpam-5768	516	48	-	-	PUNCT
ejpam-5768	516	49	groups	group	NOUN
ejpam-5768	516	50	.	.	PUNCT
ejpam-5768	517	1	references	reference	NOUN
ejpam-5768	517	2	[	[	X
ejpam-5768	517	3	1	1	NUM
ejpam-5768	517	4	]	]	PUNCT
ejpam-5768	517	5	lotfi	lotfi	X
ejpam-5768	517	6	a	a	DET
ejpam-5768	517	7	zadeh	zadeh	PROPN
ejpam-5768	517	8	.	.	PUNCT
ejpam-5768	517	9	fuzzy	fuzzy	ADJ
ejpam-5768	517	10	sets	set	NOUN
ejpam-5768	517	11	.	.	PUNCT
ejpam-5768	518	1	information	information	NOUN
ejpam-5768	518	2	and	and	CCONJ
ejpam-5768	518	3	control	control	NOUN
ejpam-5768	518	4	,	,	PUNCT
ejpam-5768	518	5	8(3):338–353	8(3):338–353	NUM
ejpam-5768	518	6	,	,	PUNCT
ejpam-5768	518	7	1965	1965	NUM
ejpam-5768	518	8	.	.	PUNCT
ejpam-5768	519	1	[	[	X
ejpam-5768	519	2	2	2	X
ejpam-5768	519	3	]	]	PUNCT
ejpam-5768	519	4	radwan	radwan	VERB
ejpam-5768	519	5	abu	abu	PROPN
ejpam-5768	519	6	-	-	PUNCT
ejpam-5768	519	7	gdairi	gdairi	PROPN
ejpam-5768	519	8	and	and	CCONJ
ejpam-5768	519	9	ibrahim	ibrahim	PROPN
ejpam-5768	519	10	noaman	noaman	PROPN
ejpam-5768	519	11	.	.	PUNCT
ejpam-5768	520	1	generating	generate	VERB
ejpam-5768	520	2	fuzzy	fuzzy	ADJ
ejpam-5768	520	3	sets	set	NOUN
ejpam-5768	520	4	and	and	CCONJ
ejpam-5768	520	5	fuzzy	fuzzy	ADJ
ejpam-5768	520	6	relations	relation	NOUN
ejpam-5768	520	7	based	base	VERB
ejpam-5768	520	8	on	on	ADP
ejpam-5768	520	9	information	information	NOUN
ejpam-5768	520	10	.	.	PUNCT
ejpam-5768	521	1	wseas	wseas	VERB
ejpam-5768	521	2	transactions	transaction	NOUN
ejpam-5768	521	3	on	on	ADP
ejpam-5768	521	4	mathematics	mathematic	NOUN
ejpam-5768	521	5	,	,	PUNCT
ejpam-5768	521	6	20:178–185	20:178–185	NUM
ejpam-5768	521	7	,	,	PUNCT
ejpam-5768	521	8	2021	2021	NUM
ejpam-5768	521	9	.	.	PUNCT
ejpam-5768	522	1	[	[	X
ejpam-5768	522	2	3	3	X
ejpam-5768	522	3	]	]	X
ejpam-5768	522	4	michael	michael	PROPN
ejpam-5768	522	5	gr	gr	PROPN
ejpam-5768	522	6	voskoglou	voskoglou	PROPN
ejpam-5768	522	7	.	.	PUNCT
ejpam-5768	523	1	topological	topological	ADJ
ejpam-5768	523	2	spaces	space	NOUN
ejpam-5768	523	3	on	on	ADP
ejpam-5768	523	4	fuzzy	fuzzy	ADJ
ejpam-5768	523	5	structures	structure	NOUN
ejpam-5768	523	6	.	.	PUNCT
ejpam-5768	524	1	wseas	wseas	PROPN
ejpam-5768	524	2	trans	trans	PROPN
ejpam-5768	524	3	.	.	PROPN
ejpam-5768	525	1	math	math	PROPN
ejpam-5768	525	2	,	,	PUNCT
ejpam-5768	525	3	21:624–628	21:624–628	PROPN
ejpam-5768	525	4	,	,	PUNCT
ejpam-5768	525	5	2022	2022	NUM
ejpam-5768	525	6	.	.	PUNCT
ejpam-5768	526	1	[	[	X
ejpam-5768	526	2	4	4	NUM
ejpam-5768	526	3	]	]	PUNCT
ejpam-5768	526	4	asima	asima	NOUN
ejpam-5768	526	5	razzaque	razzaque	NOUN
ejpam-5768	526	6	,	,	PUNCT
ejpam-5768	526	7	abdul	abdul	PROPN
ejpam-5768	526	8	razaq	razaq	PROPN
ejpam-5768	526	9	,	,	PUNCT
ejpam-5768	526	10	ghaliah	ghaliah	PROPN
ejpam-5768	526	11	alhamzi	alhamzi	PROPN
ejpam-5768	526	12	,	,	PUNCT
ejpam-5768	526	13	harish	harish	PROPN
ejpam-5768	526	14	garg	garg	PROPN
ejpam-5768	526	15	,	,	PUNCT
ejpam-5768	526	16	and	and	CCONJ
ejpam-5768	526	17	muhammad	muhammad	PROPN
ejpam-5768	526	18	iftikhar	iftikhar	PROPN
ejpam-5768	526	19	faraz	faraz	PROPN
ejpam-5768	526	20	.	.	PUNCT
ejpam-5768	527	1	a	a	DET
ejpam-5768	527	2	detailed	detailed	ADJ
ejpam-5768	527	3	study	study	NOUN
ejpam-5768	527	4	of	of	ADP
ejpam-5768	527	5	mathematical	mathematical	ADJ
ejpam-5768	527	6	rings	ring	NOUN
ejpam-5768	527	7	in	in	ADP
ejpam-5768	527	8	q	q	ADJ
ejpam-5768	527	9	-	-	PUNCT
ejpam-5768	527	10	rung	rung	ADJ
ejpam-5768	527	11	orthopair	orthopair	ADJ
ejpam-5768	527	12	fuzzy	fuzzy	ADJ
ejpam-5768	527	13	framework	framework	NOUN
ejpam-5768	527	14	.	.	PUNCT
ejpam-5768	528	1	symmetry	symmetry	NOUN
ejpam-5768	528	2	,	,	PUNCT
ejpam-5768	528	3	15(3):697	15(3):697	ADP
ejpam-5768	528	4	,	,	PUNCT
ejpam-5768	528	5	2023	2023	NUM
ejpam-5768	528	6	.	.	PUNCT
ejpam-5768	529	1	[	[	X
ejpam-5768	529	2	5	5	NUM
ejpam-5768	529	3	]	]	PUNCT
ejpam-5768	529	4	k	k	PROPN
ejpam-5768	529	5	atanassov	atanassov	PROPN
ejpam-5768	529	6	.	.	PUNCT
ejpam-5768	530	1	intuitionistic	intuitionistic	ADJ
ejpam-5768	530	2	fuzzy	fuzzy	ADJ
ejpam-5768	530	3	sets	set	NOUN
ejpam-5768	530	4	.	.	PUNCT
ejpam-5768	531	1	fuzzy	fuzzy	ADJ
ejpam-5768	531	2	sets	set	NOUN
ejpam-5768	531	3	syst	syst	NOUN
ejpam-5768	531	4	.	.	PUNCT
ejpam-5768	532	1	1986	1986	NUM
ejpam-5768	532	2	.	.	PUNCT
ejpam-5768	533	1	[	[	X
ejpam-5768	533	2	6	6	NUM
ejpam-5768	533	3	]	]	PUNCT
ejpam-5768	533	4	krassimir	krassimir	PROPN
ejpam-5768	533	5	t	t	PROPN
ejpam-5768	533	6	atanassov	atanassov	NOUN
ejpam-5768	533	7	.	.	PUNCT
ejpam-5768	534	1	intuitionistic	intuitionistic	ADJ
ejpam-5768	534	2	fuzzy	fuzzy	ADJ
ejpam-5768	534	3	sets	set	NOUN
ejpam-5768	534	4	.	.	PUNCT
ejpam-5768	535	1	springer	springer	NOUN
ejpam-5768	535	2	,	,	PUNCT
ejpam-5768	535	3	1999	1999	NUM
ejpam-5768	535	4	.	.	PUNCT
ejpam-5768	536	1	[	[	X
ejpam-5768	536	2	7	7	NUM
ejpam-5768	536	3	]	]	PUNCT
ejpam-5768	536	4	abrar	abrar	PROPN
ejpam-5768	536	5	hussain	hussain	PROPN
ejpam-5768	536	6	,	,	PUNCT
ejpam-5768	536	7	kifayat	kifayat	PROPN
ejpam-5768	536	8	ullah	ullah	PROPN
ejpam-5768	536	9	,	,	PUNCT
ejpam-5768	536	10	mohammed	mohammed	PROPN
ejpam-5768	536	11	nasser	nasser	PROPN
ejpam-5768	536	12	alshahrani	alshahrani	PROPN
ejpam-5768	536	13	,	,	PUNCT
ejpam-5768	536	14	miin	miin	NOUN
ejpam-5768	536	15	-	-	PUNCT
ejpam-5768	536	16	shen	shen	PROPN
ejpam-5768	536	17	yang	yang	PROPN
ejpam-5768	536	18	,	,	PUNCT
ejpam-5768	536	19	and	and	CCONJ
ejpam-5768	536	20	dragan	dragan	VERB
ejpam-5768	536	21	pamucar	pamucar	NOUN
ejpam-5768	536	22	.	.	PUNCT
ejpam-5768	537	1	novel	novel	PROPN
ejpam-5768	537	2	aczel	aczel	PROPN
ejpam-5768	537	3	–	–	PUNCT
ejpam-5768	537	4	alsina	alsina	NOUN
ejpam-5768	537	5	operators	operator	NOUN
ejpam-5768	537	6	for	for	ADP
ejpam-5768	537	7	pythagorean	pythagorean	PROPN
ejpam-5768	537	8	fuzzy	fuzzy	ADJ
ejpam-5768	537	9	sets	set	NOUN
ejpam-5768	537	10	with	with	ADP
ejpam-5768	537	11	application	application	NOUN
ejpam-5768	537	12	in	in	ADP
ejpam-5768	537	13	multi	multi	ADJ
ejpam-5768	537	14	-	-	ADJ
ejpam-5768	537	15	attribute	attribute	NOUN
ejpam-5768	537	16	decision	decision	NOUN
ejpam-5768	537	17	making	making	NOUN
ejpam-5768	537	18	.	.	PUNCT
ejpam-5768	538	1	symmetry	symmetry	NOUN
ejpam-5768	538	2	,	,	PUNCT
ejpam-5768	538	3	14(5):940	14(5):940	NUM
ejpam-5768	538	4	,	,	PUNCT
ejpam-5768	538	5	2022	2022	NUM
ejpam-5768	538	6	.	.	PUNCT
ejpam-5768	539	1	[	[	X
ejpam-5768	539	2	8	8	NUM
ejpam-5768	539	3	]	]	X
ejpam-5768	539	4	hanan	hanan	PROPN
ejpam-5768	539	5	alolaiyan	alolaiyan	PROPN
ejpam-5768	539	6	,	,	PUNCT
ejpam-5768	539	7	abdul	abdul	PROPN
ejpam-5768	539	8	razaq	razaq	PROPN
ejpam-5768	539	9	,	,	PUNCT
ejpam-5768	539	10	humaira	humaira	PROPN
ejpam-5768	539	11	ashfaq	ashfaq	NOUN
ejpam-5768	539	12	,	,	PUNCT
ejpam-5768	539	13	dilshad	dilshad	ADJ
ejpam-5768	539	14	alghazzawi	alghazzawi	PROPN
ejpam-5768	539	15	,	,	PUNCT
ejpam-5768	539	16	umer	umer	PROPN
ejpam-5768	539	17	shuaib	shuaib	PROPN
ejpam-5768	539	18	,	,	PUNCT
ejpam-5768	539	19	and	and	CCONJ
ejpam-5768	539	20	jia	jia	PROPN
ejpam-5768	539	21	-	-	PROPN
ejpam-5768	539	22	bao	bao	PROPN
ejpam-5768	539	23	liu	liu	PROPN
ejpam-5768	539	24	.	.	PUNCT
ejpam-5768	540	1	improving	improve	VERB
ejpam-5768	540	2	similarity	similarity	NOUN
ejpam-5768	540	3	measures	measure	NOUN
ejpam-5768	540	4	for	for	ADP
ejpam-5768	540	5	modeling	model	VERB
ejpam-5768	540	6	real	real	ADJ
ejpam-5768	540	7	-	-	PUNCT
ejpam-5768	540	8	world	world	NOUN
ejpam-5768	540	9	issues	issue	NOUN
ejpam-5768	540	10	with	with	ADP
ejpam-5768	540	11	interval	interval	NOUN
ejpam-5768	540	12	-	-	PUNCT
ejpam-5768	540	13	valued	value	VERB
ejpam-5768	540	14	intuitionistic	intuitionistic	ADJ
ejpam-5768	540	15	fuzzy	fuzzy	ADJ
ejpam-5768	540	16	sets	set	NOUN
ejpam-5768	540	17	.	.	PUNCT
ejpam-5768	541	1	ieee	ieee	NOUN
ejpam-5768	541	2	access	access	NOUN
ejpam-5768	541	3	,	,	PUNCT
ejpam-5768	541	4	12:10482–10496	12:10482–10496	NUM
ejpam-5768	541	5	,	,	PUNCT
ejpam-5768	541	6	2024	2024	NUM
ejpam-5768	541	7	.	.	PUNCT
ejpam-5768	542	1	[	[	X
ejpam-5768	542	2	9	9	NUM
ejpam-5768	542	3	]	]	X
ejpam-5768	542	4	harish	harish	PROPN
ejpam-5768	542	5	garg	garg	PROPN
ejpam-5768	542	6	,	,	PUNCT
ejpam-5768	542	7	dibakar	dibakar	PROPN
ejpam-5768	542	8	dutta	dutta	PROPN
ejpam-5768	542	9	,	,	PUNCT
ejpam-5768	542	10	palash	palash	PROPN
ejpam-5768	542	11	dutta	dutta	PROPN
ejpam-5768	542	12	,	,	PUNCT
ejpam-5768	542	13	and	and	CCONJ
ejpam-5768	542	14	brindaban	brindaban	ADJ
ejpam-5768	542	15	gohain	gohain	NOUN
ejpam-5768	542	16	.	.	PUNCT
ejpam-5768	543	1	an	an	DET
ejpam-5768	543	2	extended	extended	ADJ
ejpam-5768	543	3	group	group	NOUN
ejpam-5768	543	4	decision	decision	NOUN
ejpam-5768	543	5	-	-	PUNCT
ejpam-5768	543	6	making	make	VERB
ejpam-5768	543	7	algorithm	algorithm	NOUN
ejpam-5768	543	8	with	with	ADP
ejpam-5768	543	9	intuitionistic	intuitionistic	ADJ
ejpam-5768	543	10	fuzzy	fuzzy	ADJ
ejpam-5768	543	11	set	set	VERB
ejpam-5768	543	12	information	information	NOUN
ejpam-5768	543	13	distance	distance	NOUN
ejpam-5768	543	14	measures	measure	NOUN
ejpam-5768	543	15	and	and	CCONJ
ejpam-5768	543	16	their	their	PRON
ejpam-5768	543	17	applications	application	NOUN
ejpam-5768	543	18	.	.	PUNCT
ejpam-5768	544	1	computers	computer	NOUN
ejpam-5768	544	2	&	&	CCONJ
ejpam-5768	544	3	industrial	industrial	ADJ
ejpam-5768	544	4	engineering	engineering	PROPN
ejpam-5768	544	5	,	,	PUNCT
ejpam-5768	544	6	197:110537	197:110537	NUM
ejpam-5768	544	7	,	,	PUNCT
ejpam-5768	544	8	2024	2024	NUM
ejpam-5768	544	9	.	.	PUNCT
ejpam-5768	545	1	[	[	X
ejpam-5768	545	2	10	10	NUM
ejpam-5768	545	3	]	]	X
ejpam-5768	545	4	ronald	ronald	PROPN
ejpam-5768	545	5	r	r	NOUN
ejpam-5768	545	6	yager	yager	NOUN
ejpam-5768	545	7	.	.	PUNCT
ejpam-5768	546	1	pythagorean	pythagorean	PROPN
ejpam-5768	546	2	fuzzy	fuzzy	ADJ
ejpam-5768	546	3	subsets	subset	NOUN
ejpam-5768	546	4	.	.	PUNCT
ejpam-5768	547	1	in	in	ADP
ejpam-5768	547	2	2013	2013	NUM
ejpam-5768	547	3	joint	joint	ADJ
ejpam-5768	547	4	ifsa	ifsa	PROPN
ejpam-5768	547	5	world	world	PROPN
ejpam-5768	547	6	congress	congress	PROPN
ejpam-5768	547	7	and	and	CCONJ
ejpam-5768	547	8	nafips	nafip	NOUN
ejpam-5768	547	9	annual	annual	ADJ
ejpam-5768	547	10	meeting	meeting	NOUN
ejpam-5768	547	11	(	(	PUNCT
ejpam-5768	547	12	ifsa	ifsa	PROPN
ejpam-5768	547	13	/	/	SYM
ejpam-5768	547	14	nafips	nafip	NOUN
ejpam-5768	547	15	)	)	PUNCT
ejpam-5768	547	16	,	,	PUNCT
ejpam-5768	547	17	pages	page	NOUN
ejpam-5768	547	18	57–61	57–61	NUM
ejpam-5768	547	19	.	.	PUNCT
ejpam-5768	547	20	ieee	ieee	PROPN
ejpam-5768	547	21	,	,	PUNCT
ejpam-5768	547	22	2013	2013	NUM
ejpam-5768	547	23	.	.	PUNCT
ejpam-5768	548	1	[	[	X
ejpam-5768	548	2	11	11	NUM
ejpam-5768	548	3	]	]	X
ejpam-5768	548	4	ghaliah	ghaliah	PROPN
ejpam-5768	548	5	alhamzi	alhamzi	PROPN
ejpam-5768	548	6	,	,	PUNCT
ejpam-5768	548	7	saman	saman	PROPN
ejpam-5768	548	8	javaid	javaid	PROPN
ejpam-5768	548	9	,	,	PUNCT
ejpam-5768	548	10	umer	umer	PROPN
ejpam-5768	548	11	shuaib	shuaib	PROPN
ejpam-5768	548	12	,	,	PUNCT
ejpam-5768	548	13	abdul	abdul	PROPN
ejpam-5768	548	14	razaq	razaq	PROPN
ejpam-5768	548	15	,	,	PUNCT
ejpam-5768	548	16	harish	harish	PROPN
ejpam-5768	548	17	garg	garg	NOUN
ejpam-5768	548	18	,	,	PUNCT
ejpam-5768	548	19	and	and	CCONJ
ejpam-5768	548	20	asima	asima	NOUN
ejpam-5768	548	21	razzaque	razzaque	NOUN
ejpam-5768	548	22	.	.	PUNCT
ejpam-5768	549	1	enhancing	enhance	VERB
ejpam-5768	549	2	interval	interval	NOUN
ejpam-5768	549	3	-	-	PUNCT
ejpam-5768	549	4	valued	value	VERB
ejpam-5768	549	5	pythagorean	pythagorean	PROPN
ejpam-5768	549	6	fuzzy	fuzzy	ADJ
ejpam-5768	549	7	decision	decision	NOUN
ejpam-5768	549	8	-	-	PUNCT
ejpam-5768	549	9	making	making	NOUN
ejpam-5768	549	10	through	through	ADP
ejpam-5768	549	11	dombi	dombi	NOUN
ejpam-5768	549	12	-	-	PUNCT
ejpam-5768	549	13	based	base	VERB
ejpam-5768	549	14	aggregation	aggregation	NOUN
ejpam-5768	549	15	operators	operator	NOUN
ejpam-5768	549	16	.	.	PUNCT
ejpam-5768	550	1	symmetry	symmetry	PROPN
ejpam-5768	550	2	,	,	PUNCT
ejpam-5768	550	3	15(3):765	15(3):765	NUM
ejpam-5768	550	4	,	,	PUNCT
ejpam-5768	550	5	2023	2023	NUM
ejpam-5768	550	6	.	.	PUNCT
ejpam-5768	551	1	[	[	X
ejpam-5768	551	2	12	12	NUM
ejpam-5768	551	3	]	]	X
ejpam-5768	551	4	hanan	hanan	PROPN
ejpam-5768	551	5	alolaiyan	alolaiyan	PROPN
ejpam-5768	551	6	,	,	PUNCT
ejpam-5768	551	7	umme	umme	ADJ
ejpam-5768	551	8	kalsoom	kalsoom	PROPN
ejpam-5768	551	9	,	,	PUNCT
ejpam-5768	551	10	umer	umer	PROPN
ejpam-5768	551	11	shuaib	shuaib	PROPN
ejpam-5768	551	12	,	,	PUNCT
ejpam-5768	551	13	abdul	abdul	PROPN
ejpam-5768	551	14	razaq	razaq	PROPN
ejpam-5768	551	15	,	,	PUNCT
ejpam-5768	551	16	abdul	abdul	PROPN
ejpam-5768	551	17	wakil	wakil	PROPN
ejpam-5768	551	18	baidar	baidar	PROPN
ejpam-5768	551	19	,	,	PUNCT
ejpam-5768	551	20	a.	a.	NOUN
ejpam-5768	551	21	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	551	22	/	/	SYM
ejpam-5768	551	23	eur	eur	PROPN
ejpam-5768	551	24	.	.	PUNCT
ejpam-5768	552	1	j.	j.	PROPN
ejpam-5768	552	2	pure	pure	PROPN
ejpam-5768	552	3	appl	appl	PROPN
ejpam-5768	552	4	.	.	PROPN
ejpam-5768	552	5	math	math	PROPN
ejpam-5768	552	6	,	,	PUNCT
ejpam-5768	552	7	18	18	NUM
ejpam-5768	552	8	(	(	PUNCT
ejpam-5768	552	9	2	2	NUM
ejpam-5768	552	10	)	)	PUNCT
ejpam-5768	552	11	(	(	PUNCT
ejpam-5768	552	12	2025	2025	NUM
ejpam-5768	552	13	)	)	PUNCT
ejpam-5768	552	14	,	,	PUNCT
ejpam-5768	552	15	5768	5768	NUM
ejpam-5768	552	16	17	17	NUM
ejpam-5768	552	17	of	of	ADP
ejpam-5768	552	18	17	17	NUM
ejpam-5768	552	19	and	and	CCONJ
ejpam-5768	552	20	qin	qin	PROPN
ejpam-5768	552	21	xin	xin	PROPN
ejpam-5768	552	22	.	.	PUNCT
ejpam-5768	553	1	precision	precision	NOUN
ejpam-5768	553	2	measurement	measurement	NOUN
ejpam-5768	553	3	for	for	ADP
ejpam-5768	553	4	effective	effective	ADJ
ejpam-5768	553	5	pollution	pollution	NOUN
ejpam-5768	553	6	mitigation	mitigation	NOUN
ejpam-5768	553	7	by	by	ADP
ejpam-5768	553	8	evaluating	evaluate	VERB
ejpam-5768	553	9	air	air	NOUN
ejpam-5768	553	10	quality	quality	NOUN
ejpam-5768	553	11	monitoring	monitoring	NOUN
ejpam-5768	553	12	systems	system	NOUN
ejpam-5768	553	13	in	in	ADP
ejpam-5768	553	14	linguistic	linguistic	ADJ
ejpam-5768	553	15	pythagorean	pythagorean	PROPN
ejpam-5768	553	16	fuzzy	fuzzy	PROPN
ejpam-5768	553	17	dombi	dombi	PROPN
ejpam-5768	553	18	environment	environment	PROPN
ejpam-5768	553	19	.	.	PUNCT
ejpam-5768	554	1	scientific	scientific	ADJ
ejpam-5768	554	2	reports	report	NOUN
ejpam-5768	554	3	,	,	PUNCT
ejpam-5768	554	4	14(1):31944	14(1):31944	NUM
ejpam-5768	554	5	,	,	PUNCT
ejpam-5768	554	6	2024	2024	NUM
ejpam-5768	554	7	.	.	PUNCT
ejpam-5768	555	1	[	[	X
ejpam-5768	555	2	13	13	NUM
ejpam-5768	555	3	]	]	SYM
ejpam-5768	555	4	m	m	VERB
ejpam-5768	555	5	shazib	shazib	PROPN
ejpam-5768	555	6	hameed	hameed	PROPN
ejpam-5768	555	7	,	,	PUNCT
ejpam-5768	555	8	salman	salman	PROPN
ejpam-5768	555	9	mukhtar	mukhtar	PROPN
ejpam-5768	555	10	,	,	PUNCT
ejpam-5768	555	11	haq	haq	PROPN
ejpam-5768	555	12	nawaz	nawaz	PROPN
ejpam-5768	555	13	khan	khan	PROPN
ejpam-5768	555	14	,	,	PUNCT
ejpam-5768	555	15	shahbaz	shahbaz	PROPN
ejpam-5768	555	16	ali	ali	PROPN
ejpam-5768	555	17	,	,	PUNCT
ejpam-5768	555	18	m	m	PROPN
ejpam-5768	555	19	haris	haris	PROPN
ejpam-5768	555	20	mateen	mateen	PROPN
ejpam-5768	555	21	,	,	PUNCT
ejpam-5768	555	22	and	and	CCONJ
ejpam-5768	555	23	muhammad	muhammad	PROPN
ejpam-5768	555	24	gulzar	gulzar	PROPN
ejpam-5768	555	25	.	.	PUNCT
ejpam-5768	556	1	pythagorean	pythagorean	PROPN
ejpam-5768	556	2	fuzzy	fuzzy	ADJ
ejpam-5768	556	3	n	n	CCONJ
ejpam-5768	556	4	-	-	PUNCT
ejpam-5768	556	5	soft	soft	ADJ
ejpam-5768	556	6	groups	group	NOUN
ejpam-5768	556	7	.	.	PUNCT
ejpam-5768	557	1	int	int	NOUN
ejpam-5768	557	2	.	.	PUNCT
ejpam-5768	558	1	j.	j.	PROPN
ejpam-5768	558	2	electr	electr	PROPN
ejpam-5768	558	3	.	.	PUNCT
ejpam-5768	559	1	comput	comput	NOUN
ejpam-5768	559	2	.	.	PUNCT
ejpam-5768	560	1	eng	eng	PROPN
ejpam-5768	560	2	,	,	PUNCT
ejpam-5768	560	3	21:1030–1038	21:1030–1038	PROPN
ejpam-5768	560	4	,	,	PUNCT
ejpam-5768	560	5	2021	2021	NUM
ejpam-5768	560	6	.	.	PUNCT
ejpam-5768	561	1	[	[	X
ejpam-5768	561	2	14	14	NUM
ejpam-5768	561	3	]	]	PUNCT
ejpam-5768	561	4	abdul	abdul	PROPN
ejpam-5768	561	5	razaq	razaq	PROPN
ejpam-5768	561	6	and	and	CCONJ
ejpam-5768	561	7	ghaliah	ghaliah	PROPN
ejpam-5768	561	8	alhamzi	alhamzi	PROPN
ejpam-5768	561	9	.	.	PUNCT
ejpam-5768	562	1	on	on	ADP
ejpam-5768	562	2	pythagorean	pythagorean	PROPN
ejpam-5768	562	3	fuzzy	fuzzy	ADJ
ejpam-5768	562	4	ideals	ideal	NOUN
ejpam-5768	562	5	of	of	ADP
ejpam-5768	562	6	a	a	DET
ejpam-5768	562	7	classical	classical	ADJ
ejpam-5768	562	8	ring	ring	NOUN
ejpam-5768	562	9	.	.	PUNCT
ejpam-5768	563	1	aims	aim	VERB
ejpam-5768	563	2	math	math	NOUN
ejpam-5768	563	3	,	,	PUNCT
ejpam-5768	563	4	8(2):4280–4303	8(2):4280–4303	NUM
ejpam-5768	563	5	,	,	PUNCT
ejpam-5768	563	6	2023	2023	NUM
ejpam-5768	563	7	.	.	PUNCT
ejpam-5768	564	1	[	[	X
ejpam-5768	564	2	15	15	NUM
ejpam-5768	564	3	]	]	X
ejpam-5768	564	4	areej	areej	NOUN
ejpam-5768	564	5	almuhaimeed	almuhaimeed	PROPN
ejpam-5768	564	6	.	.	PUNCT
ejpam-5768	565	1	pythagorean	pythagorean	PROPN
ejpam-5768	565	2	fuzzy	fuzzy	ADJ
ejpam-5768	565	3	small	small	ADJ
ejpam-5768	565	4	submodules	submodule	NOUN
ejpam-5768	565	5	.	.	PUNCT
ejpam-5768	566	1	european	european	ADJ
ejpam-5768	566	2	journal	journal	PROPN
ejpam-5768	566	3	of	of	ADP
ejpam-5768	566	4	pure	pure	ADJ
ejpam-5768	566	5	and	and	CCONJ
ejpam-5768	566	6	applied	applied	ADJ
ejpam-5768	566	7	mathematics	mathematic	NOUN
ejpam-5768	566	8	,	,	PUNCT
ejpam-5768	566	9	15(1):36–46	15(1):36–46	NUM
ejpam-5768	566	10	,	,	PUNCT
ejpam-5768	566	11	2022	2022	NUM
ejpam-5768	566	12	.	.	PUNCT
ejpam-5768	567	1	[	[	X
ejpam-5768	567	2	16	16	NUM
ejpam-5768	567	3	]	]	PUNCT
ejpam-5768	567	4	fa	fa	NOUN
ejpam-5768	567	5	cotton	cotton	NOUN
ejpam-5768	567	6	.	.	PUNCT
ejpam-5768	568	1	the	the	DET
ejpam-5768	568	2	intuitionistic	intuitionistic	ADJ
ejpam-5768	568	3	fuzzy	fuzzy	ADJ
ejpam-5768	568	4	normal	normal	ADJ
ejpam-5768	568	5	subgroup	subgroup	NOUN
ejpam-5768	568	6	and	and	CCONJ
ejpam-5768	568	7	its	its	PRON
ejpam-5768	568	8	some	some	DET
ejpam-5768	568	9	equivalent	equivalent	ADJ
ejpam-5768	568	10	propositions	proposition	NOUN
ejpam-5768	568	11	,	,	PUNCT
ejpam-5768	568	12	1991	1991	NUM
ejpam-5768	568	13	.	.	PUNCT
ejpam-5768	569	1	[	[	X
ejpam-5768	569	2	17	17	NUM
ejpam-5768	569	3	]	]	PUNCT
ejpam-5768	569	4	delaram	delaram	NOUN
ejpam-5768	569	5	kahrobaei	kahrobaei	PROPN
ejpam-5768	569	6	and	and	CCONJ
ejpam-5768	569	7	michael	michael	PROPN
ejpam-5768	569	8	anshel	anshel	PROPN
ejpam-5768	569	9	.	.	PUNCT
ejpam-5768	570	1	applications	application	NOUN
ejpam-5768	570	2	of	of	ADP
ejpam-5768	570	3	group	group	NOUN
ejpam-5768	570	4	theory	theory	NOUN
ejpam-5768	570	5	in	in	ADP
ejpam-5768	570	6	cryptography	cryptography	NOUN
ejpam-5768	570	7	.	.	PUNCT
ejpam-5768	571	1	international	international	ADJ
ejpam-5768	571	2	journal	journal	NOUN
ejpam-5768	571	3	of	of	ADP
ejpam-5768	571	4	pure	pure	ADJ
ejpam-5768	571	5	and	and	CCONJ
ejpam-5768	571	6	applied	applied	ADJ
ejpam-5768	571	7	mathematics	mathematic	NOUN
ejpam-5768	571	8	,	,	PUNCT
ejpam-5768	571	9	58:21–23	58:21–23	NUM
ejpam-5768	571	10	,	,	PUNCT
ejpam-5768	571	11	2010	2010	NUM
ejpam-5768	571	12	.	.	PUNCT
ejpam-5768	572	1	[	[	X
ejpam-5768	572	2	18	18	NUM
ejpam-5768	572	3	]	]	PUNCT
ejpam-5768	572	4	abdul	abdul	PROPN
ejpam-5768	572	5	razaq	razaq	PROPN
ejpam-5768	572	6	,	,	PUNCT
ejpam-5768	572	7	shumaila	shumaila	NOUN
ejpam-5768	572	8	akhter	akhter	NOUN
ejpam-5768	572	9	,	,	PUNCT
ejpam-5768	572	10	awais	awais	PROPN
ejpam-5768	572	11	yousaf	yousaf	PROPN
ejpam-5768	572	12	,	,	PUNCT
ejpam-5768	572	13	umer	umer	PROPN
ejpam-5768	572	14	shuaib	shuaib	PROPN
ejpam-5768	572	15	,	,	PUNCT
ejpam-5768	572	16	and	and	CCONJ
ejpam-5768	572	17	musheer	musheer	ADJ
ejpam-5768	572	18	ahmad	ahmad	PROPN
ejpam-5768	572	19	.	.	PUNCT
ejpam-5768	573	1	a	a	DET
ejpam-5768	573	2	group	group	NOUN
ejpam-5768	573	3	theoretic	theoretic	NOUN
ejpam-5768	573	4	construction	construction	NOUN
ejpam-5768	573	5	of	of	ADP
ejpam-5768	573	6	highly	highly	ADV
ejpam-5768	573	7	nonlinear	nonlinear	ADJ
ejpam-5768	573	8	substitution	substitution	NOUN
ejpam-5768	573	9	box	box	NOUN
ejpam-5768	573	10	and	and	CCONJ
ejpam-5768	573	11	its	its	PRON
ejpam-5768	573	12	applications	application	NOUN
ejpam-5768	573	13	in	in	ADP
ejpam-5768	573	14	image	image	NOUN
ejpam-5768	573	15	encryption	encryption	NOUN
ejpam-5768	573	16	.	.	PUNCT
ejpam-5768	574	1	multimedia	multimedia	NOUN
ejpam-5768	574	2	tools	tool	NOUN
ejpam-5768	574	3	and	and	CCONJ
ejpam-5768	574	4	applications	application	NOUN
ejpam-5768	574	5	,	,	PUNCT
ejpam-5768	574	6	pages	page	NOUN
ejpam-5768	574	7	1–22	1–22	PROPN
ejpam-5768	574	8	,	,	PUNCT
ejpam-5768	574	9	2022	2022	NUM
ejpam-5768	574	10	.	.	PUNCT
ejpam-5768	575	1	[	[	X
ejpam-5768	575	2	19	19	NUM
ejpam-5768	575	3	]	]	X
ejpam-5768	575	4	azriel	azriel	PROPN
ejpam-5768	575	5	rosenfeld	rosenfeld	PROPN
ejpam-5768	575	6	.	.	PUNCT
ejpam-5768	576	1	fuzzy	fuzzy	ADJ
ejpam-5768	576	2	groups	group	NOUN
ejpam-5768	576	3	.	.	PUNCT
ejpam-5768	577	1	journal	journal	PROPN
ejpam-5768	577	2	of	of	ADP
ejpam-5768	577	3	mathematical	mathematical	ADJ
ejpam-5768	577	4	analysis	analysis	NOUN
ejpam-5768	577	5	and	and	CCONJ
ejpam-5768	577	6	applications	application	NOUN
ejpam-5768	577	7	,	,	PUNCT
ejpam-5768	577	8	35(3):512–517	35(3):512–517	PROPN
ejpam-5768	577	9	,	,	PUNCT
ejpam-5768	577	10	1971	1971	NUM
ejpam-5768	577	11	.	.	PUNCT
ejpam-5768	578	1	[	[	X
ejpam-5768	578	2	20	20	NUM
ejpam-5768	578	3	]	]	PUNCT
ejpam-5768	578	4	li	li	PROPN
ejpam-5768	578	5	hongxing	hongxe	VERB
ejpam-5768	578	6	.	.	PUNCT
ejpam-5768	579	1	hx	hx	PROPN
ejpam-5768	579	2	group	group	PROPN
ejpam-5768	579	3	.	.	PUNCT
ejpam-5768	580	1	busefal	busefal	PROPN
ejpam-5768	580	2	,	,	PUNCT
ejpam-5768	580	3	pages	page	NOUN
ejpam-5768	580	4	31–37	31–37	NUM
ejpam-5768	580	5	,	,	PUNCT
ejpam-5768	580	6	1987	1987	NUM
ejpam-5768	580	7	.	.	PUNCT
ejpam-5768	581	1	[	[	X
ejpam-5768	581	2	21	21	NUM
ejpam-5768	581	3	]	]	PUNCT
ejpam-5768	581	4	m	m	VERB
ejpam-5768	581	5	hongahai	hongahai	NOUN
ejpam-5768	581	6	and	and	CCONJ
ejpam-5768	581	7	z	z	NOUN
ejpam-5768	581	8	wenyi	wenyi	PROPN
ejpam-5768	581	9	.	.	PUNCT
ejpam-5768	582	1	direct	direct	ADJ
ejpam-5768	582	2	product	product	NOUN
ejpam-5768	582	3	of	of	ADP
ejpam-5768	582	4	hx	hx	NOUN
ejpam-5768	582	5	-	-	PUNCT
ejpam-5768	582	6	groups	group	NOUN
ejpam-5768	582	7	and	and	CCONJ
ejpam-5768	582	8	hx	hx	NOUN
ejpam-5768	582	9	-	-	PUNCT
ejpam-5768	582	10	groups	group	NOUN
ejpam-5768	582	11	on	on	ADP
ejpam-5768	582	12	direct	direct	ADJ
ejpam-5768	582	13	product	product	NOUN
ejpam-5768	582	14	groups	group	NOUN
ejpam-5768	582	15	.	.	PUNCT
ejpam-5768	583	1	busefal	busefal	PROPN
ejpam-5768	583	2	,	,	PUNCT
ejpam-5768	583	3	54	54	NUM
ejpam-5768	583	4	,	,	PUNCT
ejpam-5768	583	5	1993	1993	NUM
ejpam-5768	583	6	.	.	PUNCT
ejpam-5768	584	1	[	[	X
ejpam-5768	584	2	22	22	NUM
ejpam-5768	584	3	]	]	X
ejpam-5768	584	4	rabah	rabah	X
ejpam-5768	584	5	kellil	kellil	PROPN
ejpam-5768	584	6	and	and	CCONJ
ejpam-5768	584	7	ferdaous	ferdaous	ADJ
ejpam-5768	584	8	bouaziz	bouaziz	NOUN
ejpam-5768	584	9	.	.	PUNCT
ejpam-5768	585	1	new	new	ADJ
ejpam-5768	585	2	investigations	investigation	NOUN
ejpam-5768	585	3	on	on	ADP
ejpam-5768	585	4	hx	hx	NOUN
ejpam-5768	585	5	-	-	PUNCT
ejpam-5768	585	6	groups	group	NOUN
ejpam-5768	585	7	and	and	CCONJ
ejpam-5768	585	8	soft	soft	ADJ
ejpam-5768	585	9	groups	group	NOUN
ejpam-5768	585	10	.	.	PUNCT
ejpam-5768	586	1	italian	italian	ADJ
ejpam-5768	586	2	journal	journal	NOUN
ejpam-5768	586	3	of	of	ADP
ejpam-5768	586	4	pure	pure	ADJ
ejpam-5768	586	5	and	and	CCONJ
ejpam-5768	586	6	applied	applied	ADJ
ejpam-5768	586	7	mathematics	mathematic	NOUN
ejpam-5768	586	8	on	on	ADP
ejpam-5768	586	9	,	,	PUNCT
ejpam-5768	586	10	(	(	PUNCT
ejpam-5768	586	11	42	42	NUM
ejpam-5768	586	12	)	)	PUNCT
ejpam-5768	586	13	,	,	PUNCT
ejpam-5768	586	14	2017	2017	NUM
ejpam-5768	586	15	.	.	PUNCT
ejpam-5768	587	1	[	[	X
ejpam-5768	587	2	23	23	NUM
ejpam-5768	587	3	]	]	X
ejpam-5768	587	4	piergiulio	piergiulio	ADJ
ejpam-5768	587	5	corsini	corsini	PROPN
ejpam-5768	587	6	.	.	PUNCT
ejpam-5768	588	1	hx	hx	PROPN
ejpam-5768	588	2	-	-	PUNCT
ejpam-5768	588	3	groups	group	NOUN
ejpam-5768	588	4	and	and	CCONJ
ejpam-5768	588	5	hypergroups	hypergroup	NOUN
ejpam-5768	588	6	.	.	PUNCT
ejpam-5768	589	1	analele	analele	ADP
ejpam-5768	589	2	ştiinţifice	ştiinţifice	PROPN
ejpam-5768	589	3	ale	ale	NOUN
ejpam-5768	589	4	universităţii	universităţii	PROPN
ejpam-5768	589	5	”	"	PUNCT
ejpam-5768	589	6	ovidius	ovidius	ADJ
ejpam-5768	589	7	”	"	PUNCT
ejpam-5768	589	8	constanţa	constanţa	NOUN
ejpam-5768	589	9	.	.	PUNCT
ejpam-5768	590	1	seria	seria	PROPN
ejpam-5768	590	2	matematică	matematică	PROPN
ejpam-5768	590	3	,	,	PUNCT
ejpam-5768	590	4	24(3):101–121	24(3):101–121	NUM
ejpam-5768	590	5	,	,	PUNCT
ejpam-5768	590	6	2016	2016	NUM
ejpam-5768	590	7	.	.	PUNCT
ejpam-5768	591	1	[	[	X
ejpam-5768	591	2	24	24	NUM
ejpam-5768	591	3	]	]	PUNCT
ejpam-5768	591	4	irina	irina	PROPN
ejpam-5768	591	5	cristea	cristea	PROPN
ejpam-5768	591	6	,	,	PUNCT
ejpam-5768	591	7	michal	michal	PROPN
ejpam-5768	591	8	novák	novák	PROPN
ejpam-5768	591	9	,	,	PUNCT
ejpam-5768	591	10	and	and	CCONJ
ejpam-5768	591	11	babatunde	babatunde	PROPN
ejpam-5768	591	12	oluwaseun	oluwaseun	PROPN
ejpam-5768	591	13	onasanya	onasanya	PROPN
ejpam-5768	591	14	.	.	PUNCT
ejpam-5768	592	1	links	link	NOUN
ejpam-5768	592	2	between	between	ADP
ejpam-5768	592	3	hx	hx	NOUN
ejpam-5768	592	4	-	-	PUNCT
ejpam-5768	592	5	groups	group	NOUN
ejpam-5768	592	6	and	and	CCONJ
ejpam-5768	592	7	hypergroups	hypergroup	NOUN
ejpam-5768	592	8	.	.	PUNCT
ejpam-5768	593	1	in	in	ADP
ejpam-5768	593	2	algebra	algebra	PROPN
ejpam-5768	593	3	colloquium	colloquium	NOUN
ejpam-5768	593	4	,	,	PUNCT
ejpam-5768	593	5	volume	volume	NOUN
ejpam-5768	593	6	28	28	NUM
ejpam-5768	593	7	,	,	PUNCT
ejpam-5768	593	8	pages	page	VERB
ejpam-5768	593	9	441–452	441–452	NUM
ejpam-5768	593	10	.	.	PUNCT
ejpam-5768	594	1	world	world	NOUN
ejpam-5768	594	2	scientific	scientific	ADJ
ejpam-5768	594	3	,	,	PUNCT
ejpam-5768	594	4	2021	2021	NUM
ejpam-5768	594	5	.	.	PUNCT
ejpam-5768	595	1	[	[	X
ejpam-5768	595	2	25	25	NUM
ejpam-5768	595	3	]	]	X
ejpam-5768	595	4	piergiulio	piergiulio	ADJ
ejpam-5768	595	5	corsini	corsini	PROPN
ejpam-5768	595	6	.	.	PUNCT
ejpam-5768	596	1	hypergroups	hypergroup	NOUN
ejpam-5768	596	2	associated	associate	VERB
ejpam-5768	596	3	with	with	ADP
ejpam-5768	596	4	hx	hx	NOUN
ejpam-5768	596	5	-	-	PUNCT
ejpam-5768	596	6	groups	group	NOUN
ejpam-5768	596	7	.	.	PUNCT
ejpam-5768	597	1	analele	analele	ADP
ejpam-5768	597	2	ştiinţifice	ştiinţifice	PROPN
ejpam-5768	597	3	ale	ale	NOUN
ejpam-5768	597	4	universităţii	universităţii	PROPN
ejpam-5768	597	5	”	"	PUNCT
ejpam-5768	597	6	ovidius	ovidius	ADJ
ejpam-5768	597	7	”	"	PUNCT
ejpam-5768	597	8	constanţa	constanţa	NOUN
ejpam-5768	597	9	.	.	PUNCT
ejpam-5768	598	1	seria	seria	PROPN
ejpam-5768	598	2	matematică	matematică	PROPN
ejpam-5768	598	3	,	,	PUNCT
ejpam-5768	598	4	25(2):49–64	25(2):49–64	NUM
ejpam-5768	598	5	,	,	PUNCT
ejpam-5768	598	6	2017	2017	NUM
ejpam-5768	598	7	.	.	PUNCT
ejpam-5768	599	1	[	[	X
ejpam-5768	599	2	26	26	NUM
ejpam-5768	599	3	]	]	PUNCT
ejpam-5768	599	4	li	li	PROPN
ejpam-5768	599	5	hongxing	hongxe	VERB
ejpam-5768	599	6	.	.	PUNCT
ejpam-5768	600	1	hx	hx	PROPN
ejpam-5768	600	2	group	group	PROPN
ejpam-5768	600	3	.	.	PUNCT
ejpam-5768	601	1	busefal	busefal	PROPN
ejpam-5768	601	2	,	,	PUNCT
ejpam-5768	601	3	33:31–37	33:31–37	PROPN
ejpam-5768	601	4	,	,	PUNCT
ejpam-5768	601	5	1987	1987	NUM
ejpam-5768	601	6	.	.	PUNCT
ejpam-5768	602	1	[	[	X
ejpam-5768	602	2	27	27	NUM
ejpam-5768	602	3	]	]	X
ejpam-5768	602	4	luo	luo	PROPN
ejpam-5768	602	5	chengzhong	chengzhong	PROPN
ejpam-5768	602	6	.	.	PUNCT
ejpam-5768	603	1	fuzzy	fuzzy	ADJ
ejpam-5768	603	2	hx	hx	PROPN
ejpam-5768	603	3	group	group	PROPN
ejpam-5768	603	4	.	.	PUNCT
ejpam-5768	604	1	busefal	busefal	PROPN
ejpam-5768	604	2	,	,	PUNCT
ejpam-5768	604	3	41	41	NUM
ejpam-5768	604	4	-	-	SYM
ejpam-5768	604	5	14:97–106	14:97–106	NUM
ejpam-5768	604	6	,	,	PUNCT
ejpam-5768	604	7	1989	1989	NUM
ejpam-5768	604	8	.	.	PUNCT
ejpam-5768	605	1	[	[	X
ejpam-5768	605	2	28	28	NUM
ejpam-5768	605	3	]	]	X
ejpam-5768	605	4	muthuraman	muthuraman	NOUN
ejpam-5768	605	5	.	.	PUNCT
ejpam-5768	606	1	m.	m.	NOUN
ejpam-5768	606	2	s	s	VERB
ejpam-5768	606	3	muthuraj	muthuraj	ADJ
ejpam-5768	606	4	.	.	PUNCT
ejpam-5768	607	1	r	r	NOUN
ejpam-5768	607	2	,	,	PUNCT
ejpam-5768	607	3	manikandan	manikandan	PROPN
ejpam-5768	607	4	.	.	PUNCT
ejpam-5768	608	1	k.	k.	PROPN
ejpam-5768	608	2	h	h	PROPN
ejpam-5768	608	3	and	and	CCONJ
ejpam-5768	608	4	sithar	sithar	PROPN
ejpam-5768	608	5	selvam	selvam	PROPN
ejpam-5768	608	6	.	.	PUNCT
ejpam-5768	609	1	p.	p.	NOUN
ejpam-5768	609	2	m.	m.	NOUN
ejpam-5768	610	1	anti	anti	ADJ
ejpam-5768	610	2	q	q	ADJ
ejpam-5768	610	3	-	-	ADJ
ejpam-5768	610	4	fuzzy	fuzzy	ADJ
ejpam-5768	610	5	hx	hx	NOUN
ejpam-5768	610	6	group	group	NOUN
ejpam-5768	610	7	and	and	CCONJ
ejpam-5768	610	8	its	its	PRON
ejpam-5768	610	9	lower	low	ADJ
ejpam-5768	610	10	level	level	NOUN
ejpam-5768	610	11	sub	sub	NOUN
ejpam-5768	610	12	hx	hx	NOUN
ejpam-5768	610	13	groups	group	NOUN
ejpam-5768	610	14	.	.	PUNCT
ejpam-5768	611	1	nternationaljournal	nternationaljournal	ADJ
ejpam-5768	611	2	of	of	ADP
ejpam-5768	611	3	computer	computer	NOUN
ejpam-5768	611	4	applications	application	NOUN
ejpam-5768	611	5	,	,	PUNCT
ejpam-5768	611	6	6:16–20	6:16–20	NUM
ejpam-5768	611	7	,	,	PUNCT
ejpam-5768	611	8	2010	2010	NUM
ejpam-5768	611	9	.	.	PUNCT