id	sid	tid	token	lemma	pos
ejpam-4170	1	1	european	european	PROPN
ejpam-4170	1	2	journal	journal	PROPN
ejpam-4170	1	3	of	of	ADP
ejpam-4170	1	4	pure	pure	ADJ
ejpam-4170	1	5	and	and	CCONJ
ejpam-4170	1	6	applied	apply	VERB
ejpam-4170	1	7	mathematics	mathematic	NOUN
ejpam-4170	1	8	vol	vol	NOUN
ejpam-4170	1	9	.	.	PROPN
ejpam-4170	2	1	15	15	NUM
ejpam-4170	2	2	,	,	PUNCT
ejpam-4170	2	3	no	no	INTJ
ejpam-4170	2	4	.	.	NOUN
ejpam-4170	2	5	1	1	NUM
ejpam-4170	2	6	,	,	PUNCT
ejpam-4170	2	7	2022	2022	NUM
ejpam-4170	2	8	,	,	PUNCT
ejpam-4170	2	9	36	36	NUM
ejpam-4170	2	10	-	-	SYM
ejpam-4170	2	11	46	46	NUM
ejpam-4170	2	12	issn	issn	PROPN
ejpam-4170	2	13	1307	1307	NUM
ejpam-4170	2	14	-	-	SYM
ejpam-4170	2	15	5543	5543	NUM
ejpam-4170	2	16	–	–	PUNCT
ejpam-4170	2	17	ejpam.com	ejpam.com	X
ejpam-4170	2	18	published	publish	VERB
ejpam-4170	2	19	by	by	ADP
ejpam-4170	2	20	new	new	PROPN
ejpam-4170	2	21	york	york	PROPN
ejpam-4170	2	22	business	business	PROPN
ejpam-4170	2	23	global	global	PROPN
ejpam-4170	2	24	pythagorean	pythagorean	PROPN
ejpam-4170	2	25	fuzzy	fuzzy	ADJ
ejpam-4170	2	26	small	small	ADJ
ejpam-4170	2	27	submodules	submodule	NOUN
ejpam-4170	2	28	areej	areej	PROPN
ejpam-4170	2	29	almuhaimeed	almuhaimeed	PROPN
ejpam-4170	2	30	department	department	PROPN
ejpam-4170	2	31	of	of	ADP
ejpam-4170	2	32	mathematics	mathematics	PROPN
ejpam-4170	2	33	,	,	PUNCT
ejpam-4170	2	34	college	college	NOUN
ejpam-4170	2	35	of	of	ADP
ejpam-4170	2	36	science	science	PROPN
ejpam-4170	2	37	,	,	PUNCT
ejpam-4170	2	38	taibah	taibah	PROPN
ejpam-4170	2	39	university	university	PROPN
ejpam-4170	2	40	,	,	PUNCT
ejpam-4170	2	41	medina	medina	PROPN
ejpam-4170	2	42	,	,	PUNCT
ejpam-4170	2	43	saudi	saudi	PROPN
ejpam-4170	2	44	arabia	arabia	PROPN
ejpam-4170	2	45	abstract	abstract	NOUN
ejpam-4170	2	46	.	.	PUNCT
ejpam-4170	3	1	in	in	ADP
ejpam-4170	3	2	this	this	DET
ejpam-4170	3	3	paper	paper	NOUN
ejpam-4170	3	4	,	,	PUNCT
ejpam-4170	3	5	we	we	PRON
ejpam-4170	3	6	introduce	introduce	VERB
ejpam-4170	3	7	the	the	DET
ejpam-4170	3	8	notion	notion	NOUN
ejpam-4170	3	9	of	of	ADP
ejpam-4170	3	10	a	a	DET
ejpam-4170	3	11	pythagorean	pythagorean	ADJ
ejpam-4170	3	12	fuzzy	fuzzy	ADJ
ejpam-4170	3	13	small	small	ADJ
ejpam-4170	3	14	submodule	submodule	NOUN
ejpam-4170	3	15	.	.	PUNCT
ejpam-4170	4	1	we	we	PRON
ejpam-4170	4	2	prove	prove	VERB
ejpam-4170	4	3	various	various	ADJ
ejpam-4170	4	4	characterisations	characterisation	NOUN
ejpam-4170	4	5	for	for	ADP
ejpam-4170	4	6	pythagorean	pythagorean	ADJ
ejpam-4170	4	7	fuzzy	fuzzy	ADJ
ejpam-4170	4	8	small	small	ADJ
ejpam-4170	4	9	submodules	submodule	NOUN
ejpam-4170	4	10	.	.	PUNCT
ejpam-4170	5	1	we	we	PRON
ejpam-4170	5	2	provide	provide	VERB
ejpam-4170	5	3	a	a	DET
ejpam-4170	5	4	relation	relation	NOUN
ejpam-4170	5	5	between	between	ADP
ejpam-4170	5	6	a	a	DET
ejpam-4170	5	7	pythagorean	pythagorean	ADJ
ejpam-4170	5	8	fuzzy	fuzzy	ADJ
ejpam-4170	5	9	small	small	ADJ
ejpam-4170	5	10	submodule	submodule	NOUN
ejpam-4170	5	11	and	and	CCONJ
ejpam-4170	5	12	a	a	DET
ejpam-4170	5	13	basic	basic	ADJ
ejpam-4170	5	14	small	small	ADJ
ejpam-4170	5	15	submodule	submodule	NOUN
ejpam-4170	5	16	.	.	PUNCT
ejpam-4170	6	1	in	in	ADP
ejpam-4170	6	2	addition	addition	NOUN
ejpam-4170	6	3	,	,	PUNCT
ejpam-4170	6	4	some	some	DET
ejpam-4170	6	5	important	important	ADJ
ejpam-4170	6	6	properties	property	NOUN
ejpam-4170	6	7	regarding	regard	VERB
ejpam-4170	6	8	pythagorean	pythagorean	PROPN
ejpam-4170	6	9	fuzzy	fuzzy	ADJ
ejpam-4170	6	10	small	small	ADJ
ejpam-4170	6	11	submodules	submodule	NOUN
ejpam-4170	6	12	are	be	AUX
ejpam-4170	6	13	investigated	investigate	VERB
ejpam-4170	6	14	.	.	PUNCT
ejpam-4170	7	1	2020	2020	NUM
ejpam-4170	7	2	mathematics	mathematic	NOUN
ejpam-4170	7	3	subject	subject	NOUN
ejpam-4170	7	4	classifications	classification	NOUN
ejpam-4170	7	5	:	:	PUNCT
ejpam-4170	7	6	03e72	03e72	NUM
ejpam-4170	7	7	,	,	PUNCT
ejpam-4170	7	8	03b52	03b52	NUM
ejpam-4170	7	9	,	,	PUNCT
ejpam-4170	7	10	94d05	94d05	NUM
ejpam-4170	7	11	,	,	PUNCT
ejpam-4170	7	12	08a72	08a72	NOUN
ejpam-4170	7	13	key	key	ADJ
ejpam-4170	7	14	words	word	NOUN
ejpam-4170	7	15	and	and	CCONJ
ejpam-4170	7	16	phrases	phrase	NOUN
ejpam-4170	7	17	:	:	PUNCT
ejpam-4170	7	18	pythagorean	pythagorean	PROPN
ejpam-4170	7	19	fuzzy	fuzzy	ADJ
ejpam-4170	7	20	set	set	NOUN
ejpam-4170	7	21	,	,	PUNCT
ejpam-4170	7	22	pythagorean	pythagorean	PROPN
ejpam-4170	7	23	fuzzy	fuzzy	ADJ
ejpam-4170	7	24	small	small	ADJ
ejpam-4170	7	25	submodule	submodule	NOUN
ejpam-4170	7	26	,	,	PUNCT
ejpam-4170	7	27	homomorphism	homomorphism	NOUN
ejpam-4170	7	28	,	,	PUNCT
ejpam-4170	7	29	1	1	NUM
ejpam-4170	7	30	.	.	PUNCT
ejpam-4170	7	31	introduction	introduction	NOUN
ejpam-4170	7	32	in	in	ADP
ejpam-4170	7	33	1965	1965	NUM
ejpam-4170	7	34	,	,	PUNCT
ejpam-4170	7	35	zadeh	zadeh	PROPN
ejpam-4170	8	1	[	[	X
ejpam-4170	8	2	16	16	NUM
ejpam-4170	8	3	]	]	PUNCT
ejpam-4170	8	4	introduced	introduce	VERB
ejpam-4170	8	5	the	the	DET
ejpam-4170	8	6	concept	concept	NOUN
ejpam-4170	8	7	of	of	ADP
ejpam-4170	8	8	fuzzy	fuzzy	ADJ
ejpam-4170	8	9	set	set	NOUN
ejpam-4170	8	10	which	which	PRON
ejpam-4170	8	11	was	be	AUX
ejpam-4170	8	12	a	a	DET
ejpam-4170	8	13	generalisation	generalisation	NOUN
ejpam-4170	8	14	of	of	ADP
ejpam-4170	8	15	the	the	DET
ejpam-4170	8	16	classical	classical	ADJ
ejpam-4170	8	17	set	set	NOUN
ejpam-4170	8	18	.	.	PUNCT
ejpam-4170	9	1	this	this	PRON
ejpam-4170	9	2	encourages	encourage	VERB
ejpam-4170	9	3	many	many	ADJ
ejpam-4170	9	4	researchers	researcher	NOUN
ejpam-4170	9	5	to	to	PART
ejpam-4170	9	6	investigate	investigate	VERB
ejpam-4170	9	7	set	set	NOUN
ejpam-4170	9	8	theory	theory	NOUN
ejpam-4170	9	9	in	in	ADP
ejpam-4170	9	10	fuzzy	fuzzy	ADJ
ejpam-4170	9	11	setting	setting	NOUN
ejpam-4170	9	12	.	.	PUNCT
ejpam-4170	10	1	pythagorean	pythagorean	PROPN
ejpam-4170	10	2	fuzzy	fuzzy	ADJ
ejpam-4170	10	3	set	set	NOUN
ejpam-4170	10	4	is	be	AUX
ejpam-4170	10	5	one	one	NUM
ejpam-4170	10	6	of	of	ADP
ejpam-4170	10	7	the	the	DET
ejpam-4170	10	8	most	most	ADV
ejpam-4170	10	9	important	important	ADJ
ejpam-4170	10	10	fuzzy	fuzzy	ADJ
ejpam-4170	10	11	sets	set	NOUN
ejpam-4170	10	12	.	.	PUNCT
ejpam-4170	11	1	its	its	PRON
ejpam-4170	11	2	importance	importance	NOUN
ejpam-4170	11	3	lies	lie	VERB
ejpam-4170	11	4	behind	behind	ADP
ejpam-4170	11	5	the	the	DET
ejpam-4170	11	6	fact	fact	NOUN
ejpam-4170	11	7	that	that	SCONJ
ejpam-4170	11	8	this	this	DET
ejpam-4170	11	9	set	set	NOUN
ejpam-4170	11	10	can	can	AUX
ejpam-4170	11	11	be	be	AUX
ejpam-4170	11	12	applied	apply	VERB
ejpam-4170	11	13	in	in	ADP
ejpam-4170	11	14	order	order	NOUN
ejpam-4170	11	15	to	to	PART
ejpam-4170	11	16	characterized	characterize	VERB
ejpam-4170	11	17	uncertain	uncertain	ADJ
ejpam-4170	11	18	data	datum	NOUN
ejpam-4170	11	19	accurately	accurately	ADV
ejpam-4170	11	20	.	.	PUNCT
ejpam-4170	12	1	this	this	DET
ejpam-4170	12	2	kind	kind	NOUN
ejpam-4170	12	3	of	of	ADP
ejpam-4170	12	4	fuzzy	fuzzy	ADJ
ejpam-4170	12	5	sets	set	NOUN
ejpam-4170	12	6	has	have	AUX
ejpam-4170	12	7	been	be	AUX
ejpam-4170	12	8	widely	widely	ADV
ejpam-4170	12	9	investigated	investigate	VERB
ejpam-4170	12	10	.	.	PUNCT
ejpam-4170	13	1	peng	peng	PROPN
ejpam-4170	14	1	[	[	X
ejpam-4170	14	2	11	11	NUM
ejpam-4170	14	3	]	]	PUNCT
ejpam-4170	14	4	introduced	introduce	VERB
ejpam-4170	14	5	several	several	ADJ
ejpam-4170	14	6	operators	operator	NOUN
ejpam-4170	14	7	on	on	ADP
ejpam-4170	14	8	a	a	DET
ejpam-4170	14	9	pythagorean	pythagorean	ADJ
ejpam-4170	14	10	fuzzy	fuzzy	NOUN
ejpam-4170	14	11	set	set	VERB
ejpam-4170	14	12	and	and	CCONJ
ejpam-4170	14	13	discussed	discuss	VERB
ejpam-4170	14	14	its	its	PRON
ejpam-4170	14	15	properties	property	NOUN
ejpam-4170	14	16	.	.	PUNCT
ejpam-4170	15	1	yager	yager	NOUN
ejpam-4170	16	1	[	[	X
ejpam-4170	16	2	15	15	NUM
ejpam-4170	16	3	]	]	PUNCT
ejpam-4170	16	4	introduced	introduce	VERB
ejpam-4170	16	5	the	the	DET
ejpam-4170	16	6	concept	concept	NOUN
ejpam-4170	16	7	of	of	ADP
ejpam-4170	16	8	a	a	DET
ejpam-4170	16	9	pythagorean	pythagorean	ADJ
ejpam-4170	16	10	fuzzy	fuzzy	NOUN
ejpam-4170	16	11	subset	subset	VERB
ejpam-4170	16	12	as	as	ADP
ejpam-4170	16	13	a	a	DET
ejpam-4170	16	14	generalization	generalization	NOUN
ejpam-4170	16	15	of	of	ADP
ejpam-4170	16	16	an	an	DET
ejpam-4170	16	17	intuitionistic	intuitionistic	ADJ
ejpam-4170	16	18	fuzzy	fuzzy	ADJ
ejpam-4170	16	19	subset	subset	NOUN
ejpam-4170	16	20	.	.	PUNCT
ejpam-4170	17	1	in	in	ADP
ejpam-4170	17	2	[	[	X
ejpam-4170	17	3	4	4	NUM
ejpam-4170	17	4	]	]	PUNCT
ejpam-4170	17	5	,	,	PUNCT
ejpam-4170	17	6	lattices	lattice	NOUN
ejpam-4170	17	7	which	which	PRON
ejpam-4170	17	8	have	have	AUX
ejpam-4170	17	9	been	be	AUX
ejpam-4170	17	10	suggested	suggest	VERB
ejpam-4170	17	11	for	for	SCONJ
ejpam-4170	17	12	pythagorean	pythagorean	PROPN
ejpam-4170	17	13	fuzzy	fuzzy	ADJ
ejpam-4170	17	14	sets	set	NOUN
ejpam-4170	17	15	were	be	AUX
ejpam-4170	17	16	characterized	characterize	VERB
ejpam-4170	17	17	and	and	CCONJ
ejpam-4170	17	18	then	then	ADV
ejpam-4170	17	19	the	the	DET
ejpam-4170	17	20	results	result	NOUN
ejpam-4170	17	21	extended	extend	VERB
ejpam-4170	17	22	to	to	ADP
ejpam-4170	17	23	the	the	DET
ejpam-4170	17	24	unit	unit	NOUN
ejpam-4170	17	25	disc	disc	NOUN
ejpam-4170	17	26	of	of	ADP
ejpam-4170	17	27	the	the	DET
ejpam-4170	17	28	complex	complex	ADJ
ejpam-4170	17	29	plane	plane	NOUN
ejpam-4170	17	30	.	.	PUNCT
ejpam-4170	18	1	moreover	moreover	ADV
ejpam-4170	18	2	,	,	PUNCT
ejpam-4170	18	3	it	it	PRON
ejpam-4170	18	4	can	can	AUX
ejpam-4170	18	5	be	be	AUX
ejpam-4170	18	6	applied	apply	VERB
ejpam-4170	18	7	on	on	ADP
ejpam-4170	18	8	many	many	ADJ
ejpam-4170	18	9	areas	area	NOUN
ejpam-4170	18	10	,	,	PUNCT
ejpam-4170	18	11	for	for	ADP
ejpam-4170	18	12	instance	instance	NOUN
ejpam-4170	18	13	,	,	PUNCT
ejpam-4170	18	14	decision	decision	NOUN
ejpam-4170	18	15	making	making	NOUN
ejpam-4170	18	16	,	,	PUNCT
ejpam-4170	18	17	information	information	NOUN
ejpam-4170	18	18	measures	measure	NOUN
ejpam-4170	18	19	and	and	CCONJ
ejpam-4170	18	20	aggregation	aggregation	NOUN
ejpam-4170	18	21	operators	operator	NOUN
ejpam-4170	18	22	.	.	PUNCT
ejpam-4170	19	1	yager	yager	NOUN
ejpam-4170	19	2	used	use	VERB
ejpam-4170	19	3	pythagorean	pythagorean	PROPN
ejpam-4170	19	4	memmbership	memmbership	NOUN
ejpam-4170	19	5	in	in	ADP
ejpam-4170	19	6	decision	decision	NOUN
ejpam-4170	19	7	making	make	VERB
ejpam-4170	19	8	[	[	X
ejpam-4170	19	9	14	14	NUM
ejpam-4170	19	10	]	]	PUNCT
ejpam-4170	19	11	.	.	PUNCT
ejpam-4170	20	1	in	in	ADP
ejpam-4170	20	2	[	[	X
ejpam-4170	20	3	10	10	NUM
ejpam-4170	20	4	]	]	PUNCT
ejpam-4170	20	5	,	,	PUNCT
ejpam-4170	20	6	some	some	DET
ejpam-4170	20	7	algorithms	algorithm	NOUN
ejpam-4170	20	8	in	in	ADP
ejpam-4170	20	9	decision	decision	NOUN
ejpam-4170	20	10	making	make	VERB
ejpam-4170	20	11	problems	problem	NOUN
ejpam-4170	20	12	were	be	AUX
ejpam-4170	20	13	presented	present	VERB
ejpam-4170	20	14	.	.	PUNCT
ejpam-4170	21	1	grag	grag	PROPN
ejpam-4170	22	1	[	[	X
ejpam-4170	22	2	5	5	NUM
ejpam-4170	22	3	]	]	PUNCT
ejpam-4170	22	4	presented	present	VERB
ejpam-4170	22	5	some	some	DET
ejpam-4170	22	6	generalised	generalise	VERB
ejpam-4170	22	7	aggregation	aggregation	NOUN
ejpam-4170	22	8	operators	operator	NOUN
ejpam-4170	22	9	in	in	ADP
ejpam-4170	22	10	order	order	NOUN
ejpam-4170	22	11	to	to	PART
ejpam-4170	22	12	illustrate	illustrate	VERB
ejpam-4170	22	13	a	a	DET
ejpam-4170	22	14	group	group	NOUN
ejpam-4170	22	15	decision	decision	NOUN
ejpam-4170	22	16	making	make	VERB
ejpam-4170	22	17	problem	problem	NOUN
ejpam-4170	22	18	.	.	PUNCT
ejpam-4170	23	1	in	in	ADP
ejpam-4170	23	2	[	[	X
ejpam-4170	23	3	6	6	NUM
ejpam-4170	23	4	]	]	PUNCT
ejpam-4170	23	5	,	,	PUNCT
ejpam-4170	23	6	he	he	PRON
ejpam-4170	23	7	presented	present	VERB
ejpam-4170	23	8	an	an	DET
ejpam-4170	23	9	improved	improved	ADJ
ejpam-4170	23	10	score	score	NOUN
ejpam-4170	23	11	function	function	NOUN
ejpam-4170	23	12	for	for	ADP
ejpam-4170	23	13	solving	solve	VERB
ejpam-4170	23	14	multi	multi	ADJ
ejpam-4170	23	15	-	-	ADJ
ejpam-4170	23	16	criteria	criterion	NOUN
ejpam-4170	23	17	decision	decision	NOUN
ejpam-4170	23	18	-	-	PUNCT
ejpam-4170	23	19	making	making	NOUN
ejpam-4170	23	20	in	in	ADP
ejpam-4170	23	21	the	the	DET
ejpam-4170	23	22	environment	environment	NOUN
ejpam-4170	23	23	of	of	ADP
ejpam-4170	23	24	pythagorean	pythagorean	PROPN
ejpam-4170	23	25	fuzzy	fuzzy	ADJ
ejpam-4170	23	26	set	set	NOUN
ejpam-4170	23	27	.	.	PUNCT
ejpam-4170	24	1	in	in	ADP
ejpam-4170	24	2	[	[	X
ejpam-4170	24	3	8	8	NUM
ejpam-4170	24	4	]	]	PUNCT
ejpam-4170	24	5	,	,	PUNCT
ejpam-4170	24	6	hesitant	hesitant	ADJ
ejpam-4170	24	7	pythagorean	pythagorean	PROPN
ejpam-4170	24	8	fuzzy	fuzzy	ADJ
ejpam-4170	24	9	set	set	NOUN
ejpam-4170	24	10	was	be	AUX
ejpam-4170	24	11	investigated	investigate	VERB
ejpam-4170	24	12	and	and	CCONJ
ejpam-4170	24	13	applied	apply	VERB
ejpam-4170	24	14	to	to	ADP
ejpam-4170	24	15	some	some	DET
ejpam-4170	24	16	methods	method	NOUN
ejpam-4170	24	17	for	for	ADP
ejpam-4170	24	18	multiple	multiple	ADJ
ejpam-4170	24	19	criteria	criterion	NOUN
ejpam-4170	24	20	decision	decision	NOUN
ejpam-4170	24	21	making	make	VERB
ejpam-4170	24	22	.	.	PUNCT
ejpam-4170	25	1	a	a	DET
ejpam-4170	25	2	new	new	ADJ
ejpam-4170	25	3	approach	approach	NOUN
ejpam-4170	25	4	in	in	ADP
ejpam-4170	25	5	computing	compute	VERB
ejpam-4170	25	6	the	the	DET
ejpam-4170	25	7	weight	weight	NOUN
ejpam-4170	25	8	of	of	ADP
ejpam-4170	25	9	decision	decision	NOUN
ejpam-4170	25	10	makers	maker	NOUN
ejpam-4170	25	11	is	be	AUX
ejpam-4170	25	12	presented	present	VERB
ejpam-4170	25	13	in	in	ADP
ejpam-4170	25	14	[	[	X
ejpam-4170	25	15	9	9	NUM
ejpam-4170	25	16	]	]	PUNCT
ejpam-4170	25	17	using	use	VERB
ejpam-4170	25	18	properties	property	NOUN
ejpam-4170	25	19	of	of	ADP
ejpam-4170	25	20	pythagorean	pythagorean	PROPN
ejpam-4170	25	21	fuzzy	fuzzy	ADJ
ejpam-4170	25	22	sets	set	NOUN
ejpam-4170	25	23	.	.	PUNCT
ejpam-4170	26	1	distance	distance	NOUN
ejpam-4170	26	2	and	and	CCONJ
ejpam-4170	26	3	similarity	similarity	NOUN
ejpam-4170	26	4	measures	measure	NOUN
ejpam-4170	26	5	of	of	ADP
ejpam-4170	26	6	pythagorean	pythagorean	PROPN
ejpam-4170	26	7	doi	doi	PROPN
ejpam-4170	26	8	:	:	PUNCT
ejpam-4170	26	9	https://doi.org/10.29020/nybg.ejpam.v15i1.4170	https://doi.org/10.29020/nybg.ejpam.v15i1.4170	DET
ejpam-4170	26	10	email	email	NOUN
ejpam-4170	26	11	address	address	NOUN
ejpam-4170	26	12	:	:	PUNCT
ejpam-4170	26	13	aamuhaimeed@taibahu.edu.sa	aamuhaimeed@taibahu.edu.sa	NOUN
ejpam-4170	26	14	(	(	PUNCT
ejpam-4170	26	15	a.	a.	NOUN
ejpam-4170	26	16	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	26	17	)	)	PUNCT
ejpam-4170	26	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4170	27	1	36	36	NUM
ejpam-4170	27	2	©	©	PROPN
ejpam-4170	27	3	2022	2022	NUM
ejpam-4170	27	4	ejpam	ejpam	VERB
ejpam-4170	27	5	all	all	DET
ejpam-4170	27	6	rights	right	NOUN
ejpam-4170	27	7	reserved	reserve	VERB
ejpam-4170	27	8	.	.	PUNCT
ejpam-4170	28	1	a.	a.	PROPN
ejpam-4170	28	2	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	28	3	/	/	SYM
ejpam-4170	28	4	eur	eur	PROPN
ejpam-4170	28	5	.	.	PUNCT
ejpam-4170	29	1	j.	j.	PROPN
ejpam-4170	29	2	pure	pure	PROPN
ejpam-4170	29	3	appl	appl	PROPN
ejpam-4170	29	4	.	.	PROPN
ejpam-4170	29	5	math	math	PROPN
ejpam-4170	29	6	,	,	PUNCT
ejpam-4170	29	7	15	15	NUM
ejpam-4170	29	8	(	(	PUNCT
ejpam-4170	29	9	1	1	NUM
ejpam-4170	29	10	)	)	PUNCT
ejpam-4170	29	11	(	(	PUNCT
ejpam-4170	29	12	2022	2022	NUM
ejpam-4170	29	13	)	)	PUNCT
ejpam-4170	29	14	,	,	PUNCT
ejpam-4170	29	15	36	36	NUM
ejpam-4170	29	16	-	-	SYM
ejpam-4170	29	17	46	46	NUM
ejpam-4170	29	18	37	37	NUM
ejpam-4170	29	19	fuzzy	fuzzy	ADJ
ejpam-4170	29	20	set	set	NOUN
ejpam-4170	29	21	was	be	AUX
ejpam-4170	29	22	presented	present	VERB
ejpam-4170	29	23	and	and	CCONJ
ejpam-4170	29	24	applied	apply	VERB
ejpam-4170	29	25	to	to	ADP
ejpam-4170	29	26	decision	decision	NOUN
ejpam-4170	29	27	making	making	NOUN
ejpam-4170	29	28	,	,	PUNCT
ejpam-4170	29	29	see	see	VERB
ejpam-4170	29	30	[	[	X
ejpam-4170	29	31	17	17	NUM
ejpam-4170	29	32	]	]	PUNCT
ejpam-4170	29	33	.	.	PUNCT
ejpam-4170	30	1	for	for	ADP
ejpam-4170	30	2	more	more	ADJ
ejpam-4170	30	3	application	application	NOUN
ejpam-4170	30	4	of	of	ADP
ejpam-4170	30	5	this	this	DET
ejpam-4170	30	6	concept	concept	NOUN
ejpam-4170	30	7	in	in	ADP
ejpam-4170	30	8	decision	decision	NOUN
ejpam-4170	30	9	making	making	NOUN
ejpam-4170	30	10	,	,	PUNCT
ejpam-4170	30	11	see	see	VERB
ejpam-4170	30	12	[	[	X
ejpam-4170	30	13	13	13	NUM
ejpam-4170	30	14	]	]	PUNCT
ejpam-4170	30	15	and	and	CCONJ
ejpam-4170	30	16	[	[	X
ejpam-4170	30	17	12	12	NUM
ejpam-4170	30	18	]	]	PUNCT
ejpam-4170	30	19	.	.	PUNCT
ejpam-4170	31	1	the	the	DET
ejpam-4170	31	2	study	study	NOUN
ejpam-4170	31	3	of	of	ADP
ejpam-4170	31	4	pythagorean	pythagorean	PROPN
ejpam-4170	31	5	fuzzy	fuzzy	ADJ
ejpam-4170	31	6	sets	set	NOUN
ejpam-4170	31	7	is	be	AUX
ejpam-4170	31	8	a	a	DET
ejpam-4170	31	9	step	step	NOUN
ejpam-4170	31	10	in	in	ADP
ejpam-4170	31	11	order	order	NOUN
ejpam-4170	31	12	to	to	PART
ejpam-4170	31	13	study	study	VERB
ejpam-4170	31	14	q	q	ADJ
ejpam-4170	31	15	-	-	PUNCT
ejpam-4170	31	16	rung	rung	ADJ
ejpam-4170	31	17	orthopair	orthopair	ADJ
ejpam-4170	31	18	fuzzy	fuzzy	ADJ
ejpam-4170	31	19	sets	set	NOUN
ejpam-4170	31	20	as	as	ADP
ejpam-4170	31	21	a	a	DET
ejpam-4170	31	22	generalization	generalization	NOUN
ejpam-4170	31	23	of	of	ADP
ejpam-4170	31	24	pythagorean	pythagorean	PROPN
ejpam-4170	31	25	fuzzy	fuzzy	ADJ
ejpam-4170	31	26	sets	set	NOUN
ejpam-4170	31	27	see	see	VERB
ejpam-4170	31	28	[	[	X
ejpam-4170	31	29	7	7	NUM
ejpam-4170	31	30	]	]	PUNCT
ejpam-4170	31	31	,	,	PUNCT
ejpam-4170	31	32	[	[	X
ejpam-4170	31	33	1	1	NUM
ejpam-4170	31	34	]	]	PUNCT
ejpam-4170	31	35	and	and	CCONJ
ejpam-4170	31	36	[	[	X
ejpam-4170	31	37	2	2	NUM
ejpam-4170	31	38	]	]	PUNCT
ejpam-4170	31	39	.	.	PUNCT
ejpam-4170	32	1	in	in	ADP
ejpam-4170	32	2	this	this	DET
ejpam-4170	32	3	paper	paper	NOUN
ejpam-4170	32	4	,	,	PUNCT
ejpam-4170	32	5	we	we	PRON
ejpam-4170	32	6	introduce	introduce	VERB
ejpam-4170	32	7	the	the	DET
ejpam-4170	32	8	notion	notion	NOUN
ejpam-4170	32	9	of	of	ADP
ejpam-4170	32	10	pythagorean	pythagorean	PROPN
ejpam-4170	32	11	submodule	submodule	PROPN
ejpam-4170	32	12	.	.	PUNCT
ejpam-4170	33	1	in	in	ADP
ejpam-4170	33	2	addition	addition	NOUN
ejpam-4170	33	3	,	,	PUNCT
ejpam-4170	33	4	we	we	PRON
ejpam-4170	33	5	present	present	VERB
ejpam-4170	33	6	the	the	DET
ejpam-4170	33	7	concept	concept	NOUN
ejpam-4170	33	8	pythagorean	pythagorean	PROPN
ejpam-4170	33	9	small	small	ADJ
ejpam-4170	33	10	submodule	submodule	NOUN
ejpam-4170	33	11	and	and	CCONJ
ejpam-4170	33	12	investigate	investigate	VERB
ejpam-4170	33	13	some	some	DET
ejpam-4170	33	14	results	result	NOUN
ejpam-4170	33	15	regarding	regard	VERB
ejpam-4170	33	16	this	this	DET
ejpam-4170	33	17	concept	concept	NOUN
ejpam-4170	33	18	.	.	PUNCT
ejpam-4170	34	1	moreover	moreover	ADV
ejpam-4170	34	2	,	,	PUNCT
ejpam-4170	34	3	we	we	PRON
ejpam-4170	34	4	find	find	VERB
ejpam-4170	34	5	a	a	DET
ejpam-4170	34	6	relationship	relationship	NOUN
ejpam-4170	34	7	between	between	ADP
ejpam-4170	34	8	small	small	ADJ
ejpam-4170	34	9	submodule	submodule	NOUN
ejpam-4170	34	10	and	and	CCONJ
ejpam-4170	34	11	pythagorean	pythagorean	PROPN
ejpam-4170	34	12	fuzzy	fuzzy	ADJ
ejpam-4170	34	13	small	small	ADJ
ejpam-4170	34	14	submodule	submodule	NOUN
ejpam-4170	34	15	.	.	PUNCT
ejpam-4170	35	1	we	we	PRON
ejpam-4170	35	2	also	also	ADV
ejpam-4170	35	3	study	study	VERB
ejpam-4170	35	4	homomorphism	homomorphism	NOUN
ejpam-4170	35	5	between	between	ADP
ejpam-4170	35	6	pythagorean	pythagorean	PROPN
ejpam-4170	35	7	fuzzy	fuzzy	ADJ
ejpam-4170	35	8	modules	module	NOUN
ejpam-4170	35	9	.	.	PUNCT
ejpam-4170	36	1	2	2	X
ejpam-4170	36	2	.	.	X
ejpam-4170	36	3	preliminaries	preliminary	NOUN
ejpam-4170	36	4	definition	definition	NOUN
ejpam-4170	36	5	1	1	NUM
ejpam-4170	36	6	.	.	PUNCT
ejpam-4170	37	1	a	a	DET
ejpam-4170	37	2	pythagorean	pythagorean	ADJ
ejpam-4170	37	3	fuzzy	fuzzy	ADJ
ejpam-4170	37	4	set	set	NOUN
ejpam-4170	37	5	(	(	PUNCT
ejpam-4170	37	6	pfs	pfs	PROPN
ejpam-4170	37	7	)	)	PUNCT
ejpam-4170	37	8	p	p	NOUN
ejpam-4170	37	9	of	of	ADP
ejpam-4170	37	10	universe	universe	NOUN
ejpam-4170	37	11	of	of	ADP
ejpam-4170	37	12	discourse	discourse	NOUN
ejpam-4170	37	13	x	x	VERB
ejpam-4170	37	14	is	be	AUX
ejpam-4170	37	15	of	of	ADP
ejpam-4170	37	16	the	the	DET
ejpam-4170	37	17	form	form	NOUN
ejpam-4170	37	18	p	p	X
ejpam-4170	37	19	=	=	X
ejpam-4170	37	20	{	{	PUNCT
ejpam-4170	37	21	(	(	PUNCT
ejpam-4170	37	22	a	a	PROPN
ejpam-4170	37	23	,	,	PUNCT
ejpam-4170	37	24	ηp	ηp	PROPN
ejpam-4170	37	25	(	(	PUNCT
ejpam-4170	37	26	a	a	X
ejpam-4170	37	27	)	)	PUNCT
ejpam-4170	37	28	,	,	PUNCT
ejpam-4170	37	29	η̂p	η̂p	NOUN
ejpam-4170	37	30	(	(	PUNCT
ejpam-4170	37	31	a	a	NOUN
ejpam-4170	37	32	)	)	PUNCT
ejpam-4170	37	33	)	)	PUNCT
ejpam-4170	37	34	:	:	PUNCT
ejpam-4170	37	35	a	a	DET
ejpam-4170	37	36	∈	∈	PROPN
ejpam-4170	37	37	x	x	X
ejpam-4170	37	38	}	}	PUNCT
ejpam-4170	37	39	,	,	PUNCT
ejpam-4170	37	40	where	where	SCONJ
ejpam-4170	37	41	ηp	ηp	INTJ
ejpam-4170	37	42	(	(	PUNCT
ejpam-4170	37	43	a	a	X
ejpam-4170	37	44	)	)	PUNCT
ejpam-4170	37	45	and	and	CCONJ
ejpam-4170	37	46	η̂p	η̂p	NOUN
ejpam-4170	37	47	(	(	PUNCT
ejpam-4170	37	48	a	a	X
ejpam-4170	37	49	)	)	PUNCT
ejpam-4170	37	50	are	be	AUX
ejpam-4170	37	51	the	the	DET
ejpam-4170	37	52	membership	membership	NOUN
ejpam-4170	37	53	and	and	CCONJ
ejpam-4170	37	54	non	non	ADJ
ejpam-4170	37	55	-	-	ADJ
ejpam-4170	37	56	membership	membership	ADJ
ejpam-4170	37	57	values	value	NOUN
ejpam-4170	37	58	of	of	ADP
ejpam-4170	37	59	a	a	DET
ejpam-4170	37	60	respectively	respectively	ADV
ejpam-4170	37	61	in	in	ADP
ejpam-4170	37	62	which	which	PRON
ejpam-4170	37	63	0	0	NUM
ejpam-4170	37	64	≤	≤	NUM
ejpam-4170	37	65	ηp	ηp	ADP
ejpam-4170	37	66	(	(	PUNCT
ejpam-4170	37	67	a	a	X
ejpam-4170	37	68	)	)	PUNCT
ejpam-4170	37	69	≤	≤	NUM
ejpam-4170	37	70	1	1	NUM
ejpam-4170	37	71	,	,	PUNCT
ejpam-4170	37	72	0	0	NUM
ejpam-4170	37	73	≤	≤	NUM
ejpam-4170	37	74	η̂p	η̂p	NOUN
ejpam-4170	37	75	(	(	PUNCT
ejpam-4170	37	76	a	a	X
ejpam-4170	37	77	)	)	PUNCT
ejpam-4170	37	78	≤	≤	NUM
ejpam-4170	37	79	1	1	NUM
ejpam-4170	37	80	and	and	CCONJ
ejpam-4170	37	81	0	0	NUM
ejpam-4170	37	82	≤	≤	NUM
ejpam-4170	37	83	ηp	ηp	ADP
ejpam-4170	37	84	(	(	PUNCT
ejpam-4170	37	85	a	a	X
ejpam-4170	37	86	)	)	PUNCT
ejpam-4170	37	87	2	2	NUM
ejpam-4170	37	88	+	+	CCONJ
ejpam-4170	37	89	η̂p	η̂p	X
ejpam-4170	37	90	(	(	PUNCT
ejpam-4170	37	91	a	a	X
ejpam-4170	37	92	)	)	PUNCT
ejpam-4170	37	93	2	2	NUM
ejpam-4170	37	94	≤	≤	NUM
ejpam-4170	37	95	1	1	NUM
ejpam-4170	37	96	,	,	PUNCT
ejpam-4170	37	97	for	for	ADP
ejpam-4170	37	98	every	every	DET
ejpam-4170	37	99	a	a	DET
ejpam-4170	37	100	∈	∈	NOUN
ejpam-4170	37	101	x.	x.	NOUN
ejpam-4170	38	1	we	we	PRON
ejpam-4170	38	2	prsent	prsent	VERB
ejpam-4170	38	3	some	some	DET
ejpam-4170	38	4	basic	basic	ADJ
ejpam-4170	38	5	notions	notion	NOUN
ejpam-4170	38	6	regarding	regard	VERB
ejpam-4170	38	7	pythagorean	pythagorean	PROPN
ejpam-4170	38	8	fuzzy	fuzzy	ADJ
ejpam-4170	38	9	sets	set	NOUN
ejpam-4170	38	10	.	.	PUNCT
ejpam-4170	39	1	definition	definition	NOUN
ejpam-4170	39	2	2	2	NUM
ejpam-4170	39	3	.	.	PUNCT
ejpam-4170	40	1	let	let	VERB
ejpam-4170	40	2	p	p	PRON
ejpam-4170	40	3	,	,	PUNCT
ejpam-4170	40	4	s	s	AUX
ejpam-4170	40	5	be	be	AUX
ejpam-4170	40	6	pythagorean	pythagorean	ADJ
ejpam-4170	40	7	fuzzy	fuzzy	ADJ
ejpam-4170	40	8	sets	set	NOUN
ejpam-4170	40	9	in	in	ADP
ejpam-4170	40	10	a	a	DET
ejpam-4170	40	11	fixed	fix	VERB
ejpam-4170	40	12	set	set	NOUN
ejpam-4170	40	13	x.	x.	NOUN
ejpam-4170	41	1	then	then	ADV
ejpam-4170	41	2	•	•	NOUN
ejpam-4170	41	3	p	p	NOUN
ejpam-4170	41	4	is	be	AUX
ejpam-4170	41	5	a	a	DET
ejpam-4170	41	6	subset	subset	NOUN
ejpam-4170	41	7	of	of	ADP
ejpam-4170	41	8	s	s	PRON
ejpam-4170	41	9	if	if	SCONJ
ejpam-4170	41	10	for	for	ADP
ejpam-4170	41	11	all	all	DET
ejpam-4170	41	12	a	a	DET
ejpam-4170	41	13	∈	∈	NOUN
ejpam-4170	41	14	x	x	NOUN
ejpam-4170	41	15	,	,	PUNCT
ejpam-4170	41	16	we	we	PRON
ejpam-4170	41	17	have	have	VERB
ejpam-4170	41	18	η2p	η2p	NOUN
ejpam-4170	41	19	(	(	PUNCT
ejpam-4170	41	20	a	a	X
ejpam-4170	41	21	)	)	PUNCT
ejpam-4170	41	22	≤	≤	NUM
ejpam-4170	41	23	η2s(a	η2s(a	PROPN
ejpam-4170	41	24	)	)	PUNCT
ejpam-4170	41	25	and	and	CCONJ
ejpam-4170	41	26	η̂2p	η̂2p	NUM
ejpam-4170	41	27	(	(	PUNCT
ejpam-4170	41	28	a	a	NOUN
ejpam-4170	41	29	)	)	PUNCT
ejpam-4170	41	30	≥	≥	NOUN
ejpam-4170	41	31	η̂2s(a	η̂2s(a	NUM
ejpam-4170	41	32	)	)	PUNCT
ejpam-4170	41	33	.	.	PUNCT
ejpam-4170	42	1	•	•	NUM
ejpam-4170	42	2	η2p∩s(a	η2p∩s(a	PROPN
ejpam-4170	42	3	)	)	PUNCT
ejpam-4170	42	4	=	=	SYM
ejpam-4170	43	1	min{η2p	min{η2p	NOUN
ejpam-4170	43	2	(	(	PUNCT
ejpam-4170	43	3	a	a	NOUN
ejpam-4170	43	4	)	)	PUNCT
ejpam-4170	43	5	,	,	PUNCT
ejpam-4170	43	6	η2s(a	η2s(a	PROPN
ejpam-4170	43	7	)	)	PUNCT
ejpam-4170	43	8	:	:	PUNCT
ejpam-4170	43	9	a	a	DET
ejpam-4170	43	10	∈	∈	PROPN
ejpam-4170	43	11	x	x	NOUN
ejpam-4170	43	12	}	}	PUNCT
ejpam-4170	43	13	and	and	CCONJ
ejpam-4170	43	14	η̂2p∩s(a	η̂2p∩s(a	PROPN
ejpam-4170	43	15	)	)	PUNCT
ejpam-4170	43	16	=	=	SYM
ejpam-4170	43	17	max{η̂2p	max{η̂2p	NOUN
ejpam-4170	43	18	(	(	PUNCT
ejpam-4170	43	19	a	a	NOUN
ejpam-4170	43	20	)	)	PUNCT
ejpam-4170	43	21	,	,	PUNCT
ejpam-4170	43	22	η̂2s(a	η̂2s(a	PROPN
ejpam-4170	43	23	)	)	PUNCT
ejpam-4170	43	24	}	}	PUNCT
ejpam-4170	43	25	.	.	PUNCT
ejpam-4170	44	1	•	•	NUM
ejpam-4170	44	2	η2p∪s(a	η2p∪s(a	NOUN
ejpam-4170	44	3	)	)	PUNCT
ejpam-4170	44	4	=	=	SYM
ejpam-4170	45	1	max{η2p	max{η2p	PROPN
ejpam-4170	45	2	(	(	PUNCT
ejpam-4170	45	3	a	a	NOUN
ejpam-4170	45	4	)	)	PUNCT
ejpam-4170	45	5	,	,	PUNCT
ejpam-4170	45	6	η2s(a	η2s(a	PROPN
ejpam-4170	45	7	)	)	PUNCT
ejpam-4170	45	8	}	}	PUNCT
ejpam-4170	45	9	and	and	CCONJ
ejpam-4170	45	10	η̂2p∪s(a	η̂2p∪s(a	PROPN
ejpam-4170	45	11	)	)	PUNCT
ejpam-4170	45	12	=	=	NOUN
ejpam-4170	45	13	min{η̂2p	min{η̂2p	NOUN
ejpam-4170	45	14	(	(	PUNCT
ejpam-4170	45	15	a	a	NOUN
ejpam-4170	45	16	)	)	PUNCT
ejpam-4170	45	17	,	,	PUNCT
ejpam-4170	45	18	η̂2s(a	η̂2s(a	PROPN
ejpam-4170	45	19	)	)	PUNCT
ejpam-4170	45	20	}	}	PUNCT
ejpam-4170	45	21	.	.	PUNCT
ejpam-4170	46	1	•	•	NUM
ejpam-4170	46	2	η2p+s(a	η2p+s(a	PROPN
ejpam-4170	46	3	)	)	PUNCT
ejpam-4170	46	4	=	=	PUNCT
ejpam-4170	46	5	η2p	η2p	X
ejpam-4170	46	6	(	(	PUNCT
ejpam-4170	46	7	a	a	X
ejpam-4170	46	8	)	)	PUNCT
ejpam-4170	46	9	+	+	CCONJ
ejpam-4170	46	10	η2s(a)−	η2s(a)−	NOUN
ejpam-4170	46	11	η2p	η2p	VERB
ejpam-4170	46	12	(	(	PUNCT
ejpam-4170	46	13	a)η	a)η	SYM
ejpam-4170	46	14	2	2	NUM
ejpam-4170	46	15	s(a	s(a	NOUN
ejpam-4170	46	16	)	)	PUNCT
ejpam-4170	46	17	and	and	CCONJ
ejpam-4170	46	18	η̂	η̂	SYM
ejpam-4170	46	19	2	2	NUM
ejpam-4170	46	20	p+s(a	p+s(a	NUM
ejpam-4170	46	21	)	)	PUNCT
ejpam-4170	46	22	=	=	NOUN
ejpam-4170	46	23	η̂2p	η̂2p	NOUN
ejpam-4170	46	24	(	(	PUNCT
ejpam-4170	46	25	a)η̂	a)η̂	NOUN
ejpam-4170	46	26	2	2	NUM
ejpam-4170	46	27	s(a	s(a	NOUN
ejpam-4170	46	28	)	)	PUNCT
ejpam-4170	46	29	.	.	PUNCT
ejpam-4170	47	1	now	now	ADV
ejpam-4170	47	2	,	,	PUNCT
ejpam-4170	47	3	we	we	PRON
ejpam-4170	47	4	are	be	AUX
ejpam-4170	47	5	able	able	ADJ
ejpam-4170	47	6	to	to	PART
ejpam-4170	47	7	introduce	introduce	VERB
ejpam-4170	47	8	the	the	DET
ejpam-4170	47	9	definition	definition	NOUN
ejpam-4170	47	10	of	of	ADP
ejpam-4170	47	11	a	a	DET
ejpam-4170	47	12	pythagorean	pythagorean	ADJ
ejpam-4170	47	13	fuzzy	fuzzy	ADJ
ejpam-4170	47	14	submodule	submodule	NOUN
ejpam-4170	47	15	.	.	PUNCT
ejpam-4170	48	1	definition	definition	NOUN
ejpam-4170	48	2	3	3	X
ejpam-4170	48	3	.	.	PUNCT
ejpam-4170	49	1	let	let	VERB
ejpam-4170	49	2	m	m	PRON
ejpam-4170	49	3	be	be	AUX
ejpam-4170	49	4	an	an	DET
ejpam-4170	49	5	r	r	NOUN
ejpam-4170	49	6	-	-	PUNCT
ejpam-4170	49	7	module	module	NOUN
ejpam-4170	49	8	and	and	CCONJ
ejpam-4170	49	9	p	p	NOUN
ejpam-4170	49	10	a	a	DET
ejpam-4170	49	11	pythagorean	pythagorean	ADJ
ejpam-4170	49	12	fuzzy	fuzzy	ADJ
ejpam-4170	49	13	subset	subset	NOUN
ejpam-4170	49	14	of	of	ADP
ejpam-4170	49	15	m	m	PROPN
ejpam-4170	49	16	.	.	PUNCT
ejpam-4170	50	1	then	then	ADV
ejpam-4170	50	2	p	p	NOUN
ejpam-4170	50	3	is	be	AUX
ejpam-4170	50	4	called	call	VERB
ejpam-4170	50	5	a	a	DET
ejpam-4170	50	6	pythagorean	pythagorean	ADJ
ejpam-4170	50	7	fuzzy	fuzzy	ADJ
ejpam-4170	50	8	submodule	submodule	NOUN
ejpam-4170	50	9	of	of	ADP
ejpam-4170	50	10	m	m	PROPN
ejpam-4170	50	11	,	,	PUNCT
ejpam-4170	50	12	denoted	denote	VERB
ejpam-4170	50	13	by	by	ADP
ejpam-4170	50	14	p	p	PROPN
ejpam-4170	50	15	≤pf	≤pf	PROPN
ejpam-4170	50	16	m	m	PRON
ejpam-4170	50	17	,	,	PUNCT
ejpam-4170	50	18	if	if	SCONJ
ejpam-4170	50	19	the	the	DET
ejpam-4170	50	20	following	follow	VERB
ejpam-4170	50	21	conditions	condition	NOUN
ejpam-4170	50	22	are	be	AUX
ejpam-4170	50	23	satisfied	satisfied	ADJ
ejpam-4170	50	24	:	:	PUNCT
ejpam-4170	50	25	(	(	PUNCT
ejpam-4170	50	26	1	1	X
ejpam-4170	50	27	)	)	PUNCT
ejpam-4170	50	28	η2p	η2p	X
ejpam-4170	50	29	(	(	PUNCT
ejpam-4170	50	30	0	0	NUM
ejpam-4170	50	31	)	)	PUNCT
ejpam-4170	50	32	=	=	SYM
ejpam-4170	50	33	1	1	NUM
ejpam-4170	50	34	and	and	CCONJ
ejpam-4170	50	35	η̂2p	η̂2p	NUM
ejpam-4170	50	36	(	(	PUNCT
ejpam-4170	50	37	1	1	NUM
ejpam-4170	50	38	)	)	PUNCT
ejpam-4170	50	39	=	=	SYM
ejpam-4170	50	40	0	0	X
ejpam-4170	50	41	.	.	PUNCT
ejpam-4170	50	42	a.	a.	PROPN
ejpam-4170	50	43	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	50	44	/	/	SYM
ejpam-4170	50	45	eur	eur	PROPN
ejpam-4170	50	46	.	.	PUNCT
ejpam-4170	51	1	j.	j.	PROPN
ejpam-4170	51	2	pure	pure	PROPN
ejpam-4170	51	3	appl	appl	PROPN
ejpam-4170	51	4	.	.	PROPN
ejpam-4170	51	5	math	math	PROPN
ejpam-4170	51	6	,	,	PUNCT
ejpam-4170	51	7	15	15	NUM
ejpam-4170	51	8	(	(	PUNCT
ejpam-4170	51	9	1	1	NUM
ejpam-4170	51	10	)	)	PUNCT
ejpam-4170	51	11	(	(	PUNCT
ejpam-4170	51	12	2022	2022	NUM
ejpam-4170	51	13	)	)	PUNCT
ejpam-4170	51	14	,	,	PUNCT
ejpam-4170	51	15	36	36	NUM
ejpam-4170	51	16	-	-	SYM
ejpam-4170	51	17	46	46	NUM
ejpam-4170	51	18	38	38	NUM
ejpam-4170	51	19	(	(	PUNCT
ejpam-4170	51	20	2	2	NUM
ejpam-4170	51	21	)	)	PUNCT
ejpam-4170	51	22	η2p	η2p	X
ejpam-4170	51	23	(	(	PUNCT
ejpam-4170	51	24	a+	a+	X
ejpam-4170	51	25	b	b	NOUN
ejpam-4170	51	26	)	)	PUNCT
ejpam-4170	51	27	≥	≥	NOUN
ejpam-4170	51	28	min{η2p	min{η2p	NOUN
ejpam-4170	51	29	(	(	PUNCT
ejpam-4170	51	30	a	a	X
ejpam-4170	51	31	)	)	PUNCT
ejpam-4170	51	32	,	,	PUNCT
ejpam-4170	51	33	η2p	η2p	X
ejpam-4170	51	34	(	(	PUNCT
ejpam-4170	51	35	b	b	NOUN
ejpam-4170	51	36	)	)	PUNCT
ejpam-4170	51	37	}	}	PUNCT
ejpam-4170	51	38	for	for	ADP
ejpam-4170	51	39	all	all	DET
ejpam-4170	51	40	a	a	DET
ejpam-4170	51	41	,	,	PUNCT
ejpam-4170	51	42	b	b	NOUN
ejpam-4170	51	43	∈m	∈m	NOUN
ejpam-4170	51	44	and	and	CCONJ
ejpam-4170	51	45	η̂2p	η̂2p	ADJ
ejpam-4170	51	46	(	(	PUNCT
ejpam-4170	51	47	a+	a+	SYM
ejpam-4170	51	48	b	b	NOUN
ejpam-4170	51	49	)	)	PUNCT
ejpam-4170	51	50	≤	≤	NOUN
ejpam-4170	51	51	max{η̂2p	max{η̂2p	NOUN
ejpam-4170	51	52	(	(	PUNCT
ejpam-4170	51	53	a	a	NOUN
ejpam-4170	51	54	)	)	PUNCT
ejpam-4170	51	55	,	,	PUNCT
ejpam-4170	51	56	η̂2p	η̂2p	NUM
ejpam-4170	51	57	(	(	PUNCT
ejpam-4170	51	58	b	b	NOUN
ejpam-4170	51	59	)	)	PUNCT
ejpam-4170	51	60	}	}	PUNCT
ejpam-4170	51	61	for	for	ADP
ejpam-4170	51	62	all	all	DET
ejpam-4170	51	63	a	a	DET
ejpam-4170	51	64	,	,	PUNCT
ejpam-4170	51	65	b	b	NOUN
ejpam-4170	51	66	∈m	∈m	NOUN
ejpam-4170	51	67	.	.	PUNCT
ejpam-4170	52	1	(	(	PUNCT
ejpam-4170	52	2	3	3	X
ejpam-4170	52	3	)	)	PUNCT
ejpam-4170	52	4	η2p	η2p	X
ejpam-4170	52	5	(	(	PUNCT
ejpam-4170	52	6	ra	ra	NOUN
ejpam-4170	52	7	)	)	PUNCT
ejpam-4170	52	8	≥	≥	NOUN
ejpam-4170	52	9	η2p	η2p	X
ejpam-4170	52	10	(	(	PUNCT
ejpam-4170	52	11	a	a	X
ejpam-4170	52	12	)	)	PUNCT
ejpam-4170	52	13	and	and	CCONJ
ejpam-4170	52	14	η̂	η̂	PROPN
ejpam-4170	52	15	2	2	NUM
ejpam-4170	52	16	p	p	NOUN
ejpam-4170	52	17	(	(	PUNCT
ejpam-4170	52	18	ra	ra	NOUN
ejpam-4170	52	19	)	)	PUNCT
ejpam-4170	52	20	≤	≤	NOUN
ejpam-4170	52	21	η̂2p	η̂2p	NUM
ejpam-4170	52	22	(	(	PUNCT
ejpam-4170	52	23	a	a	NOUN
ejpam-4170	52	24	)	)	PUNCT
ejpam-4170	52	25	for	for	ADP
ejpam-4170	52	26	all	all	DET
ejpam-4170	52	27	a	a	DET
ejpam-4170	52	28	∈m	∈m	NOUN
ejpam-4170	52	29	and	and	CCONJ
ejpam-4170	52	30	r	r	NOUN
ejpam-4170	52	31	∈	∈	NOUN
ejpam-4170	52	32	r	r	NOUN
ejpam-4170	52	33	recall	recall	NOUN
ejpam-4170	52	34	that	that	PRON
ejpam-4170	52	35	for	for	ADP
ejpam-4170	52	36	a	a	DET
ejpam-4170	52	37	module	module	NOUN
ejpam-4170	52	38	m	m	VERB
ejpam-4170	52	39	,	,	PUNCT
ejpam-4170	52	40	we	we	PRON
ejpam-4170	52	41	define	define	VERB
ejpam-4170	52	42	the	the	DET
ejpam-4170	52	43	pythagorean	pythagorean	PROPN
ejpam-4170	52	44	fuzzy	fuzzy	NOUN
ejpam-4170	52	45	set	set	VERB
ejpam-4170	52	46	χpfm	χpfm	NOUN
ejpam-4170	52	47	=	=	SYM
ejpam-4170	52	48	(	(	PUNCT
ejpam-4170	52	49	χm	χm	PROPN
ejpam-4170	52	50	,	,	PUNCT
ejpam-4170	52	51	χ	χ	PROPN
ejpam-4170	52	52	c	c	NOUN
ejpam-4170	52	53	m	m	VERB
ejpam-4170	52	54	)	)	PUNCT
ejpam-4170	52	55	in	in	ADP
ejpam-4170	52	56	which	which	PRON
ejpam-4170	52	57	χm	χm	NOUN
ejpam-4170	52	58	(	(	PUNCT
ejpam-4170	52	59	a	a	NOUN
ejpam-4170	52	60	)	)	PUNCT
ejpam-4170	52	61	=	=	SYM
ejpam-4170	52	62	{	{	PUNCT
ejpam-4170	52	63	1	1	NUM
ejpam-4170	52	64	if	if	SCONJ
ejpam-4170	52	65	a	a	DET
ejpam-4170	52	66	∈m	∈m	NOUN
ejpam-4170	52	67	0	0	NUM
ejpam-4170	52	68	otherwise	otherwise	ADV
ejpam-4170	52	69	and	and	CCONJ
ejpam-4170	52	70	χcm	χcm	PROPN
ejpam-4170	52	71	(	(	PUNCT
ejpam-4170	52	72	a	a	X
ejpam-4170	52	73	)	)	PUNCT
ejpam-4170	52	74	=	=	PRON
ejpam-4170	52	75	{	{	PUNCT
ejpam-4170	52	76	0	0	NUM
ejpam-4170	52	77	if	if	SCONJ
ejpam-4170	52	78	a	a	DET
ejpam-4170	52	79	∈m	∈m	NOUN
ejpam-4170	52	80	1	1	NUM
ejpam-4170	52	81	otherwise	otherwise	ADV
ejpam-4170	52	82	definition	definition	NOUN
ejpam-4170	52	83	4	4	NUM
ejpam-4170	52	84	.	.	PUNCT
ejpam-4170	53	1	let	let	VERB
ejpam-4170	53	2	m	m	PRON
ejpam-4170	53	3	be	be	AUX
ejpam-4170	53	4	a	a	DET
ejpam-4170	53	5	module	module	NOUN
ejpam-4170	53	6	and	and	CCONJ
ejpam-4170	53	7	p	p	NOUN
ejpam-4170	53	8	be	be	AUX
ejpam-4170	53	9	a	a	DET
ejpam-4170	53	10	pythagorean	pythagorean	ADJ
ejpam-4170	53	11	fuzzy	fuzzy	ADJ
ejpam-4170	53	12	subset	subset	NOUN
ejpam-4170	53	13	of	of	ADP
ejpam-4170	53	14	m	m	PROPN
ejpam-4170	53	15	.	.	PUNCT
ejpam-4170	54	1	then	then	ADV
ejpam-4170	54	2	(	(	PUNCT
ejpam-4170	54	3	1	1	X
ejpam-4170	54	4	)	)	PUNCT
ejpam-4170	54	5	p	p	NOUN
ejpam-4170	54	6	⋆	⋆	NOUN
ejpam-4170	54	7	=	=	SYM
ejpam-4170	54	8	η⋆p	η⋆p	NOUN
ejpam-4170	54	9	∩	∩	PROPN
ejpam-4170	54	10	η̂⋆p	η̂⋆p	PROPN
ejpam-4170	54	11	,	,	PUNCT
ejpam-4170	54	12	where	where	SCONJ
ejpam-4170	54	13	η⋆p	η⋆p	NOUN
ejpam-4170	54	14	=	=	X
ejpam-4170	54	15	{	{	PUNCT
ejpam-4170	54	16	a	a	DET
ejpam-4170	54	17	∈m	∈m	NOUN
ejpam-4170	54	18	:	:	PUNCT
ejpam-4170	54	19	ηp	ηp	PROPN
ejpam-4170	54	20	(	(	PUNCT
ejpam-4170	54	21	a	a	X
ejpam-4170	54	22	)	)	PUNCT
ejpam-4170	54	23	>	>	X
ejpam-4170	54	24	0	0	NUM
ejpam-4170	54	25	}	}	PUNCT
ejpam-4170	54	26	η̂⋆p	η̂⋆p	NOUN
ejpam-4170	54	27	=	=	SYM
ejpam-4170	54	28	{	{	PUNCT
ejpam-4170	54	29	a	a	DET
ejpam-4170	54	30	∈m	∈m	NOUN
ejpam-4170	54	31	:	:	PUNCT
ejpam-4170	54	32	η̂(a	η̂(a	NOUN
ejpam-4170	54	33	)	)	PUNCT
ejpam-4170	54	34	<	<	X
ejpam-4170	54	35	1	1	NUM
ejpam-4170	54	36	}	}	PUNCT
ejpam-4170	54	37	(	(	PUNCT
ejpam-4170	54	38	2	2	NUM
ejpam-4170	54	39	)	)	PUNCT
ejpam-4170	54	40	p⋆	p⋆	NOUN
ejpam-4170	54	41	=	=	PUNCT
ejpam-4170	54	42	η⋆p	η⋆p	NOUN
ejpam-4170	54	43	∩	∩	PROPN
ejpam-4170	54	44	η̂⋆p	η̂⋆p	PROPN
ejpam-4170	54	45	,	,	PUNCT
ejpam-4170	54	46	where	where	SCONJ
ejpam-4170	54	47	η⋆p	η⋆p	NOUN
ejpam-4170	54	48	=	=	X
ejpam-4170	54	49	{	{	PUNCT
ejpam-4170	54	50	a	a	DET
ejpam-4170	54	51	∈m	∈m	NOUN
ejpam-4170	54	52	:	:	PUNCT
ejpam-4170	54	53	ηp	ηp	PROPN
ejpam-4170	54	54	(	(	PUNCT
ejpam-4170	54	55	a	a	X
ejpam-4170	54	56	)	)	PUNCT
ejpam-4170	54	57	=	=	SYM
ejpam-4170	54	58	1	1	X
ejpam-4170	54	59	}	}	PUNCT
ejpam-4170	54	60	η̂⋆p	η̂⋆p	NOUN
ejpam-4170	54	61	=	=	SYM
ejpam-4170	54	62	{	{	PUNCT
ejpam-4170	54	63	a	a	DET
ejpam-4170	54	64	∈m	∈m	NOUN
ejpam-4170	54	65	:	:	PUNCT
ejpam-4170	54	66	η̂(a	η̂(a	X
ejpam-4170	54	67	)	)	PUNCT
ejpam-4170	54	68	=	=	SYM
ejpam-4170	54	69	0	0	NUM
ejpam-4170	54	70	}	}	SYM
ejpam-4170	54	71	3	3	NUM
ejpam-4170	54	72	.	.	X
ejpam-4170	55	1	pythagorean	pythagorean	PROPN
ejpam-4170	55	2	fuzzy	fuzzy	ADJ
ejpam-4170	55	3	small	small	ADJ
ejpam-4170	55	4	submodule	submodule	NOUN
ejpam-4170	55	5	recall	recall	NOUN
ejpam-4170	55	6	that	that	SCONJ
ejpam-4170	55	7	a	a	DET
ejpam-4170	55	8	submodule	submodule	NOUN
ejpam-4170	55	9	n	n	PROPN
ejpam-4170	55	10	of	of	ADP
ejpam-4170	55	11	a	a	DET
ejpam-4170	55	12	module	module	NOUN
ejpam-4170	55	13	m	m	VERB
ejpam-4170	55	14	is	be	AUX
ejpam-4170	55	15	called	call	VERB
ejpam-4170	55	16	a	a	DET
ejpam-4170	55	17	small	small	ADJ
ejpam-4170	55	18	submodule	submodule	NOUN
ejpam-4170	55	19	of	of	ADP
ejpam-4170	55	20	m	m	PROPN
ejpam-4170	55	21	,	,	PUNCT
ejpam-4170	55	22	denoted	denote	VERB
ejpam-4170	55	23	byn	byn	PROPN
ejpam-4170	55	24	≪m	≪m	X
ejpam-4170	55	25	,	,	PUNCT
ejpam-4170	55	26	ifn+s	ifn+s	PROPN
ejpam-4170	55	27	̸=m	̸=m	PROPN
ejpam-4170	55	28	for	for	ADP
ejpam-4170	55	29	every	every	DET
ejpam-4170	55	30	proper	proper	ADJ
ejpam-4170	55	31	submodule	submodule	NOUN
ejpam-4170	55	32	s	s	PROPN
ejpam-4170	55	33	ofm	ofm	PROPN
ejpam-4170	55	34	.	.	PUNCT
ejpam-4170	56	1	clearly	clearly	ADV
ejpam-4170	56	2	,	,	PUNCT
ejpam-4170	56	3	the	the	DET
ejpam-4170	56	4	zero	zero	NUM
ejpam-4170	56	5	submodule	submodule	NOUN
ejpam-4170	56	6	is	be	AUX
ejpam-4170	56	7	a	a	DET
ejpam-4170	56	8	small	small	ADJ
ejpam-4170	56	9	submodule	submodule	NOUN
ejpam-4170	56	10	of	of	ADP
ejpam-4170	56	11	any	any	DET
ejpam-4170	56	12	module	module	NOUN
ejpam-4170	56	13	m	m	NOUN
ejpam-4170	56	14	.	.	PUNCT
ejpam-4170	57	1	moreover	moreover	ADV
ejpam-4170	57	2	,	,	PUNCT
ejpam-4170	57	3	a	a	DET
ejpam-4170	57	4	small	small	ADJ
ejpam-4170	57	5	submodule	submodule	NOUN
ejpam-4170	57	6	of	of	ADP
ejpam-4170	57	7	a	a	DET
ejpam-4170	57	8	module	module	NOUN
ejpam-4170	57	9	m	m	NOUN
ejpam-4170	57	10	should	should	AUX
ejpam-4170	57	11	be	be	AUX
ejpam-4170	57	12	a	a	DET
ejpam-4170	57	13	proper	proper	ADJ
ejpam-4170	57	14	submodule	submodule	NOUN
ejpam-4170	57	15	.	.	PUNCT
ejpam-4170	58	1	now	now	ADV
ejpam-4170	58	2	,	,	PUNCT
ejpam-4170	58	3	we	we	PRON
ejpam-4170	58	4	present	present	VERB
ejpam-4170	58	5	some	some	DET
ejpam-4170	58	6	well	well	ADV
ejpam-4170	58	7	-	-	PUNCT
ejpam-4170	58	8	known	know	VERB
ejpam-4170	58	9	properties	property	NOUN
ejpam-4170	58	10	regarding	regard	VERB
ejpam-4170	58	11	the	the	DET
ejpam-4170	58	12	concept	concept	NOUN
ejpam-4170	58	13	of	of	ADP
ejpam-4170	58	14	small	small	ADJ
ejpam-4170	58	15	submodules	submodule	NOUN
ejpam-4170	58	16	.	.	PUNCT
ejpam-4170	59	1	theorem	theorem	NOUN
ejpam-4170	59	2	1	1	NUM
ejpam-4170	59	3	.	.	PUNCT
ejpam-4170	60	1	[	[	X
ejpam-4170	60	2	3	3	X
ejpam-4170	60	3	]	]	PUNCT
ejpam-4170	60	4	suppose	suppose	VERB
ejpam-4170	60	5	that	that	SCONJ
ejpam-4170	60	6	m	m	PROPN
ejpam-4170	60	7	is	be	AUX
ejpam-4170	60	8	a	a	DET
ejpam-4170	60	9	module	module	NOUN
ejpam-4170	60	10	and	and	CCONJ
ejpam-4170	60	11	s	s	PROPN
ejpam-4170	60	12	,	,	PUNCT
ejpam-4170	60	13	t	t	PROPN
ejpam-4170	60	14	,	,	PUNCT
ejpam-4170	60	15	n	n	X
ejpam-4170	60	16	are	be	AUX
ejpam-4170	60	17	submodules	submodule	NOUN
ejpam-4170	60	18	of	of	ADP
ejpam-4170	60	19	m	m	NOUN
ejpam-4170	60	20	such	such	ADJ
ejpam-4170	60	21	that	that	PRON
ejpam-4170	60	22	s	s	VERB
ejpam-4170	60	23	≤	≤	PROPN
ejpam-4170	60	24	t	t	NOUN
ejpam-4170	60	25	.	.	PUNCT
ejpam-4170	61	1	then	then	ADV
ejpam-4170	61	2	(	(	PUNCT
ejpam-4170	61	3	1	1	X
ejpam-4170	61	4	)	)	PUNCT
ejpam-4170	61	5	s	s	VERB
ejpam-4170	61	6	+	+	NOUN
ejpam-4170	61	7	n	n	NOUN
ejpam-4170	61	8	≪m	≪m	X
ejpam-4170	61	9	if	if	SCONJ
ejpam-4170	61	10	and	and	CCONJ
ejpam-4170	61	11	only	only	ADV
ejpam-4170	61	12	if	if	SCONJ
ejpam-4170	61	13	s	s	X
ejpam-4170	61	14	≪m	≪m	X
ejpam-4170	61	15	and	and	CCONJ
ejpam-4170	61	16	n	n	CCONJ
ejpam-4170	61	17	≪m	≪m	PRON
ejpam-4170	61	18	.	.	PUNCT
ejpam-4170	62	1	(	(	PUNCT
ejpam-4170	62	2	2	2	X
ejpam-4170	62	3	)	)	PUNCT
ejpam-4170	62	4	t	t	NOUN
ejpam-4170	62	5	≪m	≪m	X
ejpam-4170	62	6	if	if	SCONJ
ejpam-4170	62	7	and	and	CCONJ
ejpam-4170	62	8	only	only	ADV
ejpam-4170	62	9	if	if	SCONJ
ejpam-4170	62	10	s	s	VERB
ejpam-4170	62	11	≪m	≪m	X
ejpam-4170	62	12	and	and	CCONJ
ejpam-4170	62	13	t	t	PROPN
ejpam-4170	62	14	s	s	PART
ejpam-4170	62	15	≪	≪	PROPN
ejpam-4170	62	16	m	m	PROPN
ejpam-4170	62	17	s	s	NUM
ejpam-4170	62	18	.	.	PUNCT
ejpam-4170	63	1	(	(	PUNCT
ejpam-4170	63	2	3	3	X
ejpam-4170	63	3	)	)	PUNCT
ejpam-4170	63	4	if	if	SCONJ
ejpam-4170	63	5	s	s	VERB
ejpam-4170	63	6	≪	≪	ADJ
ejpam-4170	63	7	t	t	PROPN
ejpam-4170	63	8	,	,	PUNCT
ejpam-4170	63	9	then	then	ADV
ejpam-4170	63	10	s	s	VERB
ejpam-4170	63	11	≪m	≪m	X
ejpam-4170	63	12	.	.	PUNCT
ejpam-4170	64	1	a.	a.	PROPN
ejpam-4170	64	2	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	64	3	/	/	SYM
ejpam-4170	64	4	eur	eur	PROPN
ejpam-4170	64	5	.	.	PUNCT
ejpam-4170	65	1	j.	j.	PROPN
ejpam-4170	65	2	pure	pure	PROPN
ejpam-4170	65	3	appl	appl	PROPN
ejpam-4170	65	4	.	.	PROPN
ejpam-4170	65	5	math	math	PROPN
ejpam-4170	65	6	,	,	PUNCT
ejpam-4170	65	7	15	15	NUM
ejpam-4170	65	8	(	(	PUNCT
ejpam-4170	65	9	1	1	NUM
ejpam-4170	65	10	)	)	PUNCT
ejpam-4170	65	11	(	(	PUNCT
ejpam-4170	65	12	2022	2022	NUM
ejpam-4170	65	13	)	)	PUNCT
ejpam-4170	65	14	,	,	PUNCT
ejpam-4170	65	15	36	36	NUM
ejpam-4170	65	16	-	-	SYM
ejpam-4170	65	17	46	46	NUM
ejpam-4170	65	18	39	39	NUM
ejpam-4170	65	19	now	now	ADV
ejpam-4170	65	20	,	,	PUNCT
ejpam-4170	65	21	we	we	PRON
ejpam-4170	65	22	are	be	AUX
ejpam-4170	65	23	ready	ready	ADJ
ejpam-4170	65	24	to	to	PART
ejpam-4170	65	25	introduce	introduce	VERB
ejpam-4170	65	26	the	the	DET
ejpam-4170	65	27	main	main	ADJ
ejpam-4170	65	28	concept	concept	NOUN
ejpam-4170	65	29	in	in	ADP
ejpam-4170	65	30	this	this	DET
ejpam-4170	65	31	paper	paper	NOUN
ejpam-4170	65	32	.	.	PUNCT
ejpam-4170	66	1	consider	consider	VERB
ejpam-4170	66	2	a	a	DET
ejpam-4170	66	3	module	module	NOUN
ejpam-4170	66	4	m	m	NOUN
ejpam-4170	66	5	.	.	PUNCT
ejpam-4170	67	1	then	then	ADV
ejpam-4170	67	2	a	a	DET
ejpam-4170	67	3	pfs	pfs	PROPN
ejpam-4170	67	4	,	,	PUNCT
ejpam-4170	67	5	p	p	NOUN
ejpam-4170	67	6	=	=	PUNCT
ejpam-4170	67	7	(	(	PUNCT
ejpam-4170	67	8	ηp	ηp	INTJ
ejpam-4170	67	9	,	,	PUNCT
ejpam-4170	67	10	η̂p	η̂p	PROPN
ejpam-4170	67	11	)	)	PUNCT
ejpam-4170	67	12	,	,	PUNCT
ejpam-4170	67	13	is	be	AUX
ejpam-4170	67	14	called	call	VERB
ejpam-4170	67	15	a	a	DET
ejpam-4170	67	16	pythagorean	pythagorean	ADJ
ejpam-4170	67	17	fuzzy	fuzzy	ADJ
ejpam-4170	67	18	small	small	ADJ
ejpam-4170	67	19	submodule	submodule	NOUN
ejpam-4170	67	20	of	of	ADP
ejpam-4170	67	21	m	m	PROPN
ejpam-4170	67	22	,	,	PUNCT
ejpam-4170	67	23	denoted	denote	VERB
ejpam-4170	67	24	by	by	ADP
ejpam-4170	67	25	p	p	PROPN
ejpam-4170	67	26	≪pf	≪pf	PROPN
ejpam-4170	67	27	m	m	PRON
ejpam-4170	67	28	,	,	PUNCT
ejpam-4170	67	29	if	if	SCONJ
ejpam-4170	67	30	p	p	PROPN
ejpam-4170	67	31	+	+	NOUN
ejpam-4170	67	32	s	s	PART
ejpam-4170	67	33	̸=	̸=	PROPN
ejpam-4170	67	34	χpfm	χpfm	NOUN
ejpam-4170	67	35	for	for	ADP
ejpam-4170	67	36	any	any	DET
ejpam-4170	67	37	psf	psf	NOUN
ejpam-4170	67	38	s	s	PART
ejpam-4170	67	39	̸=	̸=	PROPN
ejpam-4170	67	40	χpfm	χpfm	NOUN
ejpam-4170	67	41	.	.	PUNCT
ejpam-4170	68	1	that	that	PRON
ejpam-4170	68	2	is	be	AUX
ejpam-4170	68	3	whenever	whenever	SCONJ
ejpam-4170	68	4	p	p	PROPN
ejpam-4170	68	5	+	+	NUM
ejpam-4170	68	6	s	s	NOUN
ejpam-4170	68	7	=	=	X
ejpam-4170	68	8	χpfm	χpfm	NOUN
ejpam-4170	68	9	,	,	PUNCT
ejpam-4170	68	10	then	then	ADV
ejpam-4170	68	11	s	s	VERB
ejpam-4170	68	12	=	=	NOUN
ejpam-4170	68	13	χpfm	χpfm	NOUN
ejpam-4170	68	14	.	.	PUNCT
ejpam-4170	69	1	theorem	theorem	NOUN
ejpam-4170	69	2	2	2	NUM
ejpam-4170	69	3	.	.	PUNCT
ejpam-4170	70	1	let	let	VERB
ejpam-4170	70	2	m	m	PRON
ejpam-4170	70	3	be	be	AUX
ejpam-4170	70	4	a	a	DET
ejpam-4170	70	5	module	module	NOUN
ejpam-4170	70	6	and	and	CCONJ
ejpam-4170	70	7	p	p	NOUN
ejpam-4170	70	8	be	be	AUX
ejpam-4170	70	9	a	a	DET
ejpam-4170	70	10	submodule	submodule	NOUN
ejpam-4170	70	11	of	of	ADP
ejpam-4170	70	12	m	m	PROPN
ejpam-4170	70	13	.	.	PUNCT
ejpam-4170	71	1	then	then	ADV
ejpam-4170	71	2	p	p	X
ejpam-4170	71	3	≪m	≪m	X
ejpam-4170	71	4	if	if	SCONJ
ejpam-4170	71	5	χpfp	χpfp	PROPN
ejpam-4170	71	6	≪pf	≪pf	PROPN
ejpam-4170	71	7	m	m	VERB
ejpam-4170	71	8	.	.	PUNCT
ejpam-4170	72	1	proof	proof	NOUN
ejpam-4170	72	2	.	.	PUNCT
ejpam-4170	73	1	suppose	suppose	VERB
ejpam-4170	73	2	that	that	SCONJ
ejpam-4170	73	3	χpfp	χpfp	PROPN
ejpam-4170	73	4	≪pf	≪pf	PROPN
ejpam-4170	73	5	m	m	PROPN
ejpam-4170	73	6	and	and	CCONJ
ejpam-4170	73	7	p	p	PRON
ejpam-4170	74	1	+	+	NOUN
ejpam-4170	74	2	s	s	X
ejpam-4170	74	3	=	=	NOUN
ejpam-4170	74	4	m	m	VERB
ejpam-4170	74	5	for	for	ADP
ejpam-4170	74	6	some	some	DET
ejpam-4170	74	7	proper	proper	ADJ
ejpam-4170	74	8	submodule	submodule	NOUN
ejpam-4170	74	9	s	s	PROPN
ejpam-4170	74	10	of	of	ADP
ejpam-4170	74	11	m	m	PROPN
ejpam-4170	74	12	.	.	PUNCT
ejpam-4170	75	1	then	then	ADV
ejpam-4170	75	2	for	for	ADP
ejpam-4170	75	3	any	any	DET
ejpam-4170	75	4	m	m	NOUN
ejpam-4170	75	5	∈m	∈m	NOUN
ejpam-4170	75	6	there	there	ADV
ejpam-4170	75	7	exist	exist	VERB
ejpam-4170	75	8	a	a	DET
ejpam-4170	75	9	∈	∈	PROPN
ejpam-4170	75	10	p	p	NOUN
ejpam-4170	75	11	and	and	CCONJ
ejpam-4170	75	12	b	b	PROPN
ejpam-4170	75	13	∈	∈	NOUN
ejpam-4170	75	14	s	s	VERB
ejpam-4170	76	1	such	such	ADJ
ejpam-4170	76	2	that	that	SCONJ
ejpam-4170	76	3	a+	a+	PRON
ejpam-4170	76	4	b	b	X
ejpam-4170	76	5	=	=	PUNCT
ejpam-4170	76	6	m.	m.	NOUN
ejpam-4170	76	7	we	we	PRON
ejpam-4170	76	8	obtain	obtain	VERB
ejpam-4170	76	9	η2	η2	PROPN
ejpam-4170	76	10	χpf	χpf	VERB
ejpam-4170	76	11	p	p	X
ejpam-4170	76	12	+	+	PROPN
ejpam-4170	76	13	χpf	χpf	ADJ
ejpam-4170	76	14	s	s	X
ejpam-4170	76	15	(	(	PUNCT
ejpam-4170	76	16	m	m	NOUN
ejpam-4170	76	17	)	)	PUNCT
ejpam-4170	76	18	=	=	SYM
ejpam-4170	76	19	χ2	χ2	PROPN
ejpam-4170	76	20	p	p	PROPN
ejpam-4170	76	21	(	(	PUNCT
ejpam-4170	76	22	m	m	NOUN
ejpam-4170	76	23	)	)	PUNCT
ejpam-4170	77	1	+	+	CCONJ
ejpam-4170	77	2	χ2	χ2	PROPN
ejpam-4170	77	3	s(m)−	s(m)−	PROPN
ejpam-4170	77	4	χ2	χ2	PROPN
ejpam-4170	77	5	p	p	PROPN
ejpam-4170	77	6	(	(	PUNCT
ejpam-4170	77	7	m)χ2	m)χ2	PROPN
ejpam-4170	77	8	s(m	s(m	PROPN
ejpam-4170	77	9	)	)	PUNCT
ejpam-4170	77	10	≥	≥	PROPN
ejpam-4170	77	11	min{χ2	min{χ2	NOUN
ejpam-4170	77	12	p	p	X
ejpam-4170	77	13	(	(	PUNCT
ejpam-4170	77	14	a	a	NOUN
ejpam-4170	77	15	)	)	PUNCT
ejpam-4170	77	16	+	+	CCONJ
ejpam-4170	77	17	χ2	χ2	PROPN
ejpam-4170	77	18	s(a)−	s(a)−	PROPN
ejpam-4170	77	19	χ2	χ2	PROPN
ejpam-4170	77	20	p	p	PROPN
ejpam-4170	77	21	(	(	PUNCT
ejpam-4170	77	22	a)χ	a)χ	X
ejpam-4170	77	23	2	2	NUM
ejpam-4170	77	24	s(a	s(a	NOUN
ejpam-4170	77	25	)	)	PUNCT
ejpam-4170	77	26	,	,	PUNCT
ejpam-4170	77	27	χ	χ	X
ejpam-4170	77	28	2	2	NUM
ejpam-4170	77	29	p	p	NOUN
ejpam-4170	77	30	(	(	PUNCT
ejpam-4170	77	31	b	b	NOUN
ejpam-4170	77	32	)	)	PUNCT
ejpam-4170	77	33	+	+	CCONJ
ejpam-4170	77	34	χ2	χ2	PROPN
ejpam-4170	77	35	s(b)−	s(b)−	PROPN
ejpam-4170	77	36	χ2	χ2	PROPN
ejpam-4170	77	37	p	p	NOUN
ejpam-4170	77	38	(	(	PUNCT
ejpam-4170	77	39	b)χ	b)χ	ADJ
ejpam-4170	77	40	2	2	NUM
ejpam-4170	77	41	s(b	s(b	NOUN
ejpam-4170	77	42	)	)	PUNCT
ejpam-4170	77	43	}	}	PUNCT
ejpam-4170	77	44	=	=	PUNCT
ejpam-4170	77	45	1	1	NUM
ejpam-4170	77	46	this	this	PRON
ejpam-4170	77	47	means	mean	VERB
ejpam-4170	77	48	that	that	SCONJ
ejpam-4170	77	49	η2	η2	ADJ
ejpam-4170	77	50	χpf	χpf	VERB
ejpam-4170	77	51	p	p	X
ejpam-4170	77	52	+	+	PROPN
ejpam-4170	77	53	χpf	χpf	ADJ
ejpam-4170	77	54	s	s	PART
ejpam-4170	77	55	=	=	SYM
ejpam-4170	77	56	χ2	χ2	PROPN
ejpam-4170	77	57	m	m	VERB
ejpam-4170	77	58	.	.	PUNCT
ejpam-4170	78	1	similarly	similarly	ADV
ejpam-4170	78	2	,	,	PUNCT
ejpam-4170	78	3	η̂2	η̂2	NOUN
ejpam-4170	78	4	χpf	χpf	NOUN
ejpam-4170	78	5	p	p	PROPN
ejpam-4170	78	6	+	+	PROPN
ejpam-4170	78	7	χpf	χpf	ADJ
ejpam-4170	78	8	s	s	X
ejpam-4170	78	9	(	(	PUNCT
ejpam-4170	78	10	m	m	NOUN
ejpam-4170	78	11	)	)	PUNCT
ejpam-4170	79	1	=	=	SYM
ejpam-4170	79	2	χc	χc	ADP
ejpam-4170	79	3	2	2	NUM
ejpam-4170	79	4	p	p	NOUN
ejpam-4170	79	5	(	(	PUNCT
ejpam-4170	79	6	m)χc	m)χc	NOUN
ejpam-4170	79	7	2	2	NUM
ejpam-4170	79	8	s	s	PART
ejpam-4170	79	9	(	(	PUNCT
ejpam-4170	79	10	m	m	NOUN
ejpam-4170	79	11	)	)	PUNCT
ejpam-4170	79	12	≤	≤	NOUN
ejpam-4170	79	13	max{χc2p	max{χc2p	NOUN
ejpam-4170	79	14	(	(	PUNCT
ejpam-4170	79	15	a)χc	a)χc	PROPN
ejpam-4170	79	16	2	2	NUM
ejpam-4170	79	17	s	s	PART
ejpam-4170	79	18	(	(	PUNCT
ejpam-4170	79	19	a	a	NOUN
ejpam-4170	79	20	)	)	PUNCT
ejpam-4170	79	21	,	,	PUNCT
ejpam-4170	79	22	χc	χc	PROPN
ejpam-4170	79	23	2	2	NUM
ejpam-4170	79	24	p	p	NOUN
ejpam-4170	79	25	(	(	PUNCT
ejpam-4170	79	26	b)χc	b)χc	PROPN
ejpam-4170	79	27	2	2	NUM
ejpam-4170	79	28	s	s	PART
ejpam-4170	79	29	(	(	PUNCT
ejpam-4170	79	30	b	b	NOUN
ejpam-4170	79	31	)	)	PUNCT
ejpam-4170	79	32	}	}	PUNCT
ejpam-4170	79	33	=	=	SYM
ejpam-4170	79	34	0	0	NUM
ejpam-4170	80	1	this	this	PRON
ejpam-4170	80	2	means	mean	VERB
ejpam-4170	80	3	that	that	SCONJ
ejpam-4170	80	4	η̂2	η̂2	NOUN
ejpam-4170	80	5	χpf	χpf	NOUN
ejpam-4170	80	6	p	p	PROPN
ejpam-4170	80	7	+	+	PROPN
ejpam-4170	80	8	χpf	χpf	NOUN
ejpam-4170	80	9	s	s	X
ejpam-4170	80	10	=	=	SYM
ejpam-4170	80	11	χc	χc	PROPN
ejpam-4170	80	12	2	2	NUM
ejpam-4170	80	13	m	m	NOUN
ejpam-4170	80	14	.	.	PUNCT
ejpam-4170	81	1	thus	thus	ADV
ejpam-4170	81	2	χpfp	χpfp	VERB
ejpam-4170	81	3	+	+	SYM
ejpam-4170	81	4	χpfs	χpf	NOUN
ejpam-4170	81	5	=	=	SYM
ejpam-4170	81	6	χpfm	χpfm	NOUN
ejpam-4170	81	7	,	,	PUNCT
ejpam-4170	81	8	but	but	CCONJ
ejpam-4170	81	9	this	this	PRON
ejpam-4170	81	10	contradicts	contradict	VERB
ejpam-4170	81	11	the	the	DET
ejpam-4170	81	12	facts	fact	NOUN
ejpam-4170	81	13	that	that	PRON
ejpam-4170	81	14	χpfp	χpfp	VERB
ejpam-4170	81	15	≪pf	≪pf	PROPN
ejpam-4170	81	16	m	m	PROPN
ejpam-4170	81	17	and	and	CCONJ
ejpam-4170	81	18	χpfs	χpfs	PROPN
ejpam-4170	81	19	̸=	̸=	PROPN
ejpam-4170	81	20	χpfm	χpfm	NOUN
ejpam-4170	81	21	as	as	SCONJ
ejpam-4170	81	22	s	s	PRON
ejpam-4170	81	23	is	be	AUX
ejpam-4170	81	24	a	a	DET
ejpam-4170	81	25	proper	proper	ADJ
ejpam-4170	81	26	submodule	submodule	NOUN
ejpam-4170	81	27	of	of	ADP
ejpam-4170	81	28	m	m	PROPN
ejpam-4170	81	29	.	.	PUNCT
ejpam-4170	82	1	therefore	therefore	ADV
ejpam-4170	82	2	,	,	PUNCT
ejpam-4170	82	3	p	p	PRON
ejpam-4170	82	4	is	be	AUX
ejpam-4170	82	5	a	a	DET
ejpam-4170	82	6	small	small	ADJ
ejpam-4170	82	7	submodule	submodule	NOUN
ejpam-4170	82	8	of	of	ADP
ejpam-4170	82	9	m	m	PROPN
ejpam-4170	82	10	.	.	PUNCT
ejpam-4170	83	1	theorem	theorem	ADJ
ejpam-4170	83	2	3	3	X
ejpam-4170	83	3	.	.	PUNCT
ejpam-4170	84	1	let	let	VERB
ejpam-4170	84	2	m	m	PRON
ejpam-4170	84	3	be	be	AUX
ejpam-4170	84	4	a	a	DET
ejpam-4170	84	5	module	module	NOUN
ejpam-4170	84	6	and	and	CCONJ
ejpam-4170	84	7	p	p	NOUN
ejpam-4170	84	8	be	be	AUX
ejpam-4170	84	9	a	a	DET
ejpam-4170	84	10	pythagorean	pythagorean	ADJ
ejpam-4170	84	11	fuzzy	fuzzy	ADJ
ejpam-4170	84	12	submodule	submodule	NOUN
ejpam-4170	84	13	of	of	ADP
ejpam-4170	84	14	m	m	PROPN
ejpam-4170	84	15	.	.	PUNCT
ejpam-4170	85	1	if	if	SCONJ
ejpam-4170	85	2	p	p	PROPN
ejpam-4170	85	3	≪pf	≪pf	PROPN
ejpam-4170	85	4	m	m	PRON
ejpam-4170	85	5	,	,	PUNCT
ejpam-4170	85	6	then	then	ADV
ejpam-4170	85	7	p⋆	p⋆	X
ejpam-4170	85	8	≪m	≪m	X
ejpam-4170	85	9	.	.	PUNCT
ejpam-4170	86	1	proof	proof	NOUN
ejpam-4170	86	2	.	.	PUNCT
ejpam-4170	87	1	assume	assume	VERB
ejpam-4170	87	2	that	that	SCONJ
ejpam-4170	87	3	p	p	PROPN
ejpam-4170	87	4	≪pf	≪pf	PROPN
ejpam-4170	87	5	m	m	VERB
ejpam-4170	87	6	.	.	PUNCT
ejpam-4170	88	1	in	in	ADP
ejpam-4170	88	2	order	order	NOUN
ejpam-4170	88	3	to	to	PART
ejpam-4170	88	4	see	see	VERB
ejpam-4170	88	5	that	that	SCONJ
ejpam-4170	88	6	p⋆	p⋆	NOUN
ejpam-4170	88	7	≪m	≪m	X
ejpam-4170	88	8	,	,	PUNCT
ejpam-4170	88	9	suppose	suppose	VERB
ejpam-4170	88	10	that	that	SCONJ
ejpam-4170	88	11	p⋆+s	p⋆+s	PRON
ejpam-4170	88	12	=	=	NOUN
ejpam-4170	88	13	m	m	VERB
ejpam-4170	88	14	for	for	ADP
ejpam-4170	88	15	a	a	DET
ejpam-4170	88	16	submodule	submodule	NOUN
ejpam-4170	88	17	s	s	PROPN
ejpam-4170	88	18	of	of	ADP
ejpam-4170	88	19	m	m	PROPN
ejpam-4170	88	20	.	.	PUNCT
ejpam-4170	89	1	we	we	PRON
ejpam-4170	89	2	aim	aim	VERB
ejpam-4170	89	3	to	to	PART
ejpam-4170	89	4	prove	prove	VERB
ejpam-4170	89	5	that	that	SCONJ
ejpam-4170	89	6	p	p	PROPN
ejpam-4170	89	7	+	+	NUM
ejpam-4170	89	8	χpfs	χpf	NOUN
ejpam-4170	89	9	=	=	SYM
ejpam-4170	89	10	χpfm	χpfm	NOUN
ejpam-4170	89	11	.	.	PUNCT
ejpam-4170	90	1	let	let	VERB
ejpam-4170	90	2	m	m	PRON
ejpam-4170	90	3	∈	∈	VERB
ejpam-4170	90	4	m	m	NOUN
ejpam-4170	90	5	.	.	PUNCT
ejpam-4170	91	1	then	then	ADV
ejpam-4170	91	2	m	m	VERB
ejpam-4170	91	3	=	=	SYM
ejpam-4170	91	4	a+	a+	PUNCT
ejpam-4170	91	5	b	b	NOUN
ejpam-4170	91	6	,	,	PUNCT
ejpam-4170	91	7	for	for	ADP
ejpam-4170	91	8	some	some	PRON
ejpam-4170	91	9	a	a	DET
ejpam-4170	91	10	∈	∈	NOUN
ejpam-4170	91	11	p⋆	p⋆	NOUN
ejpam-4170	91	12	and	and	CCONJ
ejpam-4170	91	13	b	b	PROPN
ejpam-4170	91	14	∈	∈	PROPN
ejpam-4170	91	15	s.	s.	PROPN
ejpam-4170	91	16	then	then	ADV
ejpam-4170	91	17	ηp+χpf	ηp+χpf	VERB
ejpam-4170	91	18	s	s	VERB
ejpam-4170	91	19	(	(	PUNCT
ejpam-4170	91	20	m	m	NOUN
ejpam-4170	91	21	)	)	PUNCT
ejpam-4170	92	1	=	=	VERB
ejpam-4170	92	2	η2p	η2p	X
ejpam-4170	92	3	(	(	PUNCT
ejpam-4170	92	4	m	m	NOUN
ejpam-4170	92	5	)	)	PUNCT
ejpam-4170	92	6	+	+	CCONJ
ejpam-4170	92	7	χ2	χ2	PROPN
ejpam-4170	92	8	s(m)−	s(m)−	PROPN
ejpam-4170	92	9	η2p	η2p	X
ejpam-4170	92	10	(	(	PUNCT
ejpam-4170	92	11	m)χ2	m)χ2	PROPN
ejpam-4170	92	12	s(m	s(m	PROPN
ejpam-4170	92	13	)	)	PUNCT
ejpam-4170	92	14	≥min{η2p	≥min{η2p	PUNCT
ejpam-4170	92	15	(	(	PUNCT
ejpam-4170	92	16	a	a	X
ejpam-4170	92	17	)	)	PUNCT
ejpam-4170	92	18	+	+	CCONJ
ejpam-4170	92	19	χ2	χ2	NOUN
ejpam-4170	92	20	s(a)−	s(a)−	PROPN
ejpam-4170	92	21	η2p	η2p	ADV
ejpam-4170	92	22	(	(	PUNCT
ejpam-4170	92	23	a)χ	a)χ	X
ejpam-4170	92	24	2	2	NUM
ejpam-4170	92	25	s(a	s(a	NOUN
ejpam-4170	92	26	)	)	PUNCT
ejpam-4170	92	27	,	,	PUNCT
ejpam-4170	92	28	η	η	PROPN
ejpam-4170	92	29	2	2	NUM
ejpam-4170	92	30	p	p	NOUN
ejpam-4170	92	31	(	(	PUNCT
ejpam-4170	92	32	b	b	NOUN
ejpam-4170	92	33	)	)	PUNCT
ejpam-4170	92	34	+	+	CCONJ
ejpam-4170	92	35	χ2	χ2	PROPN
ejpam-4170	92	36	s(b)−	s(b)−	PROPN
ejpam-4170	92	37	η2p	η2p	NOUN
ejpam-4170	92	38	(	(	PUNCT
ejpam-4170	92	39	b)χ	b)χ	ADJ
ejpam-4170	92	40	2	2	NUM
ejpam-4170	92	41	s(b	s(b	NOUN
ejpam-4170	92	42	)	)	PUNCT
ejpam-4170	92	43	}	}	PUNCT
ejpam-4170	93	1	=	=	SYM
ejpam-4170	93	2	1	1	NUM
ejpam-4170	93	3	moreover	moreover	ADV
ejpam-4170	93	4	,	,	PUNCT
ejpam-4170	93	5	η̂p+χpf	η̂p+χpf	ADP
ejpam-4170	93	6	s	s	X
ejpam-4170	93	7	(	(	PUNCT
ejpam-4170	93	8	m	m	NOUN
ejpam-4170	93	9	)	)	PUNCT
ejpam-4170	93	10	=	=	NOUN
ejpam-4170	93	11	η̂2p	η̂2p	NOUN
ejpam-4170	93	12	(	(	PUNCT
ejpam-4170	93	13	m)χc	m)χc	NOUN
ejpam-4170	93	14	2	2	NUM
ejpam-4170	93	15	s	s	PART
ejpam-4170	93	16	(	(	PUNCT
ejpam-4170	93	17	m	m	NOUN
ejpam-4170	93	18	)	)	PUNCT
ejpam-4170	93	19	≤max{η̂2p	≤max{η̂2p	NOUN
ejpam-4170	93	20	(	(	PUNCT
ejpam-4170	93	21	a)χc	a)χc	PROPN
ejpam-4170	93	22	2	2	NUM
ejpam-4170	93	23	s	s	PART
ejpam-4170	93	24	(	(	PUNCT
ejpam-4170	93	25	a	a	NOUN
ejpam-4170	93	26	)	)	PUNCT
ejpam-4170	93	27	,	,	PUNCT
ejpam-4170	93	28	η̂2p	η̂2p	NUM
ejpam-4170	93	29	(	(	PUNCT
ejpam-4170	93	30	b)χ	b)χ	ADJ
ejpam-4170	93	31	c2	c2	PROPN
ejpam-4170	93	32	s	s	X
ejpam-4170	93	33	(	(	PUNCT
ejpam-4170	93	34	b	b	NOUN
ejpam-4170	93	35	)	)	PUNCT
ejpam-4170	93	36	}	}	PUNCT
ejpam-4170	93	37	=	=	SYM
ejpam-4170	93	38	0	0	X
ejpam-4170	93	39	thus	thus	ADV
ejpam-4170	93	40	p	p	X
ejpam-4170	93	41	+	+	NUM
ejpam-4170	93	42	χpfs	χpf	NOUN
ejpam-4170	93	43	=	=	SYM
ejpam-4170	93	44	χpfm	χpfm	NOUN
ejpam-4170	93	45	.	.	PUNCT
ejpam-4170	94	1	by	by	ADP
ejpam-4170	94	2	hypothesis	hypothesis	NOUN
ejpam-4170	94	3	,	,	PUNCT
ejpam-4170	94	4	χpfs	χpf	NOUN
ejpam-4170	94	5	=	=	SYM
ejpam-4170	94	6	χpfm	χpfm	PROPN
ejpam-4170	94	7	.	.	PUNCT
ejpam-4170	95	1	therefore	therefore	ADV
ejpam-4170	95	2	,	,	PUNCT
ejpam-4170	95	3	s	s	VERB
ejpam-4170	95	4	=	=	NOUN
ejpam-4170	95	5	m	m	NOUN
ejpam-4170	95	6	.	.	PUNCT
ejpam-4170	96	1	a.	a.	PROPN
ejpam-4170	96	2	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	96	3	/	/	SYM
ejpam-4170	96	4	eur	eur	PROPN
ejpam-4170	96	5	.	.	PUNCT
ejpam-4170	97	1	j.	j.	PROPN
ejpam-4170	97	2	pure	pure	PROPN
ejpam-4170	97	3	appl	appl	PROPN
ejpam-4170	97	4	.	.	PROPN
ejpam-4170	97	5	math	math	PROPN
ejpam-4170	97	6	,	,	PUNCT
ejpam-4170	97	7	15	15	NUM
ejpam-4170	97	8	(	(	PUNCT
ejpam-4170	97	9	1	1	NUM
ejpam-4170	97	10	)	)	PUNCT
ejpam-4170	97	11	(	(	PUNCT
ejpam-4170	97	12	2022	2022	NUM
ejpam-4170	97	13	)	)	PUNCT
ejpam-4170	97	14	,	,	PUNCT
ejpam-4170	97	15	36	36	NUM
ejpam-4170	97	16	-	-	SYM
ejpam-4170	97	17	46	46	NUM
ejpam-4170	97	18	40	40	NUM
ejpam-4170	97	19	example	example	NOUN
ejpam-4170	97	20	1	1	NUM
ejpam-4170	97	21	.	.	X
ejpam-4170	97	22	consider	consider	VERB
ejpam-4170	97	23	the	the	DET
ejpam-4170	97	24	z	z	NOUN
ejpam-4170	97	25	-	-	PUNCT
ejpam-4170	97	26	module	module	NOUN
ejpam-4170	97	27	z10	z10	NOUN
ejpam-4170	97	28	and	and	CCONJ
ejpam-4170	97	29	the	the	DET
ejpam-4170	97	30	submodule	submodule	NOUN
ejpam-4170	97	31	s	s	PART
ejpam-4170	97	32	=	=	NOUN
ejpam-4170	97	33	⟨5̄⟩.	⟨5̄⟩.	NOUN
ejpam-4170	97	34	let	let	VERB
ejpam-4170	97	35	p	p	PRON
ejpam-4170	97	36	be	be	AUX
ejpam-4170	97	37	a	a	DET
ejpam-4170	97	38	pythagorean	pythagorean	ADJ
ejpam-4170	97	39	fuzzy	fuzzy	ADJ
ejpam-4170	97	40	submodule	submodule	NOUN
ejpam-4170	97	41	of	of	ADP
ejpam-4170	97	42	z10	z10	NOUN
ejpam-4170	97	43	defined	define	VERB
ejpam-4170	97	44	as	as	SCONJ
ejpam-4170	97	45	follows	follow	VERB
ejpam-4170	97	46	ηp	ηp	PROPN
ejpam-4170	97	47	(	(	PUNCT
ejpam-4170	97	48	m	m	NOUN
ejpam-4170	97	49	)	)	PUNCT
ejpam-4170	98	1	=	=	PRON
ejpam-4170	98	2	{	{	PUNCT
ejpam-4170	98	3	1	1	NUM
ejpam-4170	98	4	if	if	SCONJ
ejpam-4170	98	5	m	m	VERB
ejpam-4170	98	6	∈	∈	NOUN
ejpam-4170	98	7	s	s	PART
ejpam-4170	98	8	1	1	NUM
ejpam-4170	98	9	4	4	NUM
ejpam-4170	98	10	otherwise	otherwise	ADV
ejpam-4170	98	11	and	and	CCONJ
ejpam-4170	98	12	η̂p	η̂p	VERB
ejpam-4170	98	13	(	(	PUNCT
ejpam-4170	98	14	m	m	NOUN
ejpam-4170	98	15	)	)	PUNCT
ejpam-4170	98	16	=	=	PRON
ejpam-4170	98	17	{	{	PUNCT
ejpam-4170	98	18	0	0	NUM
ejpam-4170	98	19	if	if	SCONJ
ejpam-4170	98	20	m	m	VERB
ejpam-4170	98	21	∈	∈	NOUN
ejpam-4170	98	22	s	s	PART
ejpam-4170	98	23	1	1	NUM
ejpam-4170	98	24	6	6	NUM
ejpam-4170	98	25	otherwise	otherwise	ADV
ejpam-4170	98	26	it	it	PRON
ejpam-4170	98	27	is	be	AUX
ejpam-4170	98	28	clear	clear	ADJ
ejpam-4170	98	29	that	that	SCONJ
ejpam-4170	98	30	p⋆	p⋆	NOUN
ejpam-4170	98	31	is	be	AUX
ejpam-4170	98	32	not	not	PART
ejpam-4170	98	33	a	a	DET
ejpam-4170	98	34	small	small	ADJ
ejpam-4170	98	35	submodule	submodule	NOUN
ejpam-4170	98	36	of	of	ADP
ejpam-4170	98	37	z10	z10	NOUN
ejpam-4170	98	38	as	as	ADP
ejpam-4170	98	39	p⋆	p⋆	NOUN
ejpam-4170	98	40	+	+	CCONJ
ejpam-4170	98	41	⟨2̄⟩	⟨2̄⟩	X
ejpam-4170	98	42	=	=	NOUN
ejpam-4170	98	43	z10	z10	NOUN
ejpam-4170	98	44	.	.	PUNCT
ejpam-4170	99	1	thus	thus	ADV
ejpam-4170	99	2	p	p	X
ejpam-4170	99	3	is	be	AUX
ejpam-4170	99	4	not	not	PART
ejpam-4170	99	5	a	a	DET
ejpam-4170	99	6	pythagorean	pythagorean	ADJ
ejpam-4170	99	7	fuzzy	fuzzy	ADJ
ejpam-4170	99	8	small	small	ADJ
ejpam-4170	99	9	submodule	submodule	NOUN
ejpam-4170	99	10	of	of	ADP
ejpam-4170	99	11	z10	z10	PROPN
ejpam-4170	99	12	.	.	PUNCT
ejpam-4170	100	1	corollary	corollary	ADJ
ejpam-4170	100	2	1	1	NUM
ejpam-4170	100	3	.	.	PUNCT
ejpam-4170	101	1	let	let	VERB
ejpam-4170	101	2	p	p	PRON
ejpam-4170	101	3	,	,	PUNCT
ejpam-4170	101	4	s	s	AUX
ejpam-4170	101	5	be	be	AUX
ejpam-4170	101	6	two	two	NUM
ejpam-4170	101	7	pythagorean	pythagorean	ADJ
ejpam-4170	101	8	fuzzy	fuzzy	ADJ
ejpam-4170	101	9	submodules	submodule	NOUN
ejpam-4170	101	10	of	of	ADP
ejpam-4170	101	11	a	a	DET
ejpam-4170	101	12	module	module	NOUN
ejpam-4170	101	13	m	m	NOUN
ejpam-4170	101	14	in	in	ADP
ejpam-4170	101	15	which	which	PRON
ejpam-4170	101	16	p	p	NOUN
ejpam-4170	101	17	⊆	⊆	NUM
ejpam-4170	101	18	s.	s.	PROPN
ejpam-4170	101	19	then	then	ADV
ejpam-4170	101	20	p	p	PROPN
ejpam-4170	101	21	≪pf	≪pf	PROPN
ejpam-4170	101	22	s	s	VERB
ejpam-4170	101	23	if	if	SCONJ
ejpam-4170	101	24	and	and	CCONJ
ejpam-4170	101	25	only	only	ADV
ejpam-4170	101	26	if	if	SCONJ
ejpam-4170	101	27	p⋆	p⋆	NOUN
ejpam-4170	101	28	≪	≪	ADJ
ejpam-4170	101	29	s⋆.	s⋆.	NOUN
ejpam-4170	101	30	proof	proof	NOUN
ejpam-4170	101	31	.	.	PUNCT
ejpam-4170	102	1	clear	clear	ADJ
ejpam-4170	102	2	.	.	PUNCT
ejpam-4170	103	1	theorem	theorem	ADJ
ejpam-4170	103	2	4	4	NUM
ejpam-4170	103	3	.	.	PUNCT
ejpam-4170	104	1	let	let	VERB
ejpam-4170	104	2	m	m	PRON
ejpam-4170	104	3	be	be	AUX
ejpam-4170	104	4	a	a	DET
ejpam-4170	104	5	module	module	NOUN
ejpam-4170	104	6	,	,	PUNCT
ejpam-4170	104	7	s	s	AUX
ejpam-4170	104	8	be	be	AUX
ejpam-4170	104	9	a	a	DET
ejpam-4170	104	10	submodule	submodule	NOUN
ejpam-4170	104	11	of	of	ADP
ejpam-4170	104	12	m	m	PROPN
ejpam-4170	104	13	and	and	CCONJ
ejpam-4170	104	14	p	p	PROPN
ejpam-4170	104	15	is	be	AUX
ejpam-4170	104	16	a	a	DET
ejpam-4170	104	17	pythagorean	pythagorean	ADJ
ejpam-4170	104	18	fuzzy	fuzzy	ADJ
ejpam-4170	104	19	submodule	submodule	NOUN
ejpam-4170	104	20	of	of	ADP
ejpam-4170	104	21	m	m	PROPN
ejpam-4170	104	22	in	in	ADP
ejpam-4170	104	23	which	which	PRON
ejpam-4170	104	24	p	p	NOUN
ejpam-4170	104	25	⊆	⊆	NUM
ejpam-4170	104	26	χpfs	χpf	NOUN
ejpam-4170	104	27	.	.	PUNCT
ejpam-4170	105	1	if	if	SCONJ
ejpam-4170	105	2	p	p	PROPN
ejpam-4170	105	3	|s	|s	PROPN
ejpam-4170	105	4	is	be	AUX
ejpam-4170	105	5	a	a	DET
ejpam-4170	105	6	pythagorean	pythagorean	ADJ
ejpam-4170	105	7	fuzzy	fuzzy	ADJ
ejpam-4170	105	8	small	small	ADJ
ejpam-4170	105	9	submodule	submodule	NOUN
ejpam-4170	105	10	of	of	ADP
ejpam-4170	105	11	s	s	PROPN
ejpam-4170	105	12	,	,	PUNCT
ejpam-4170	105	13	then	then	ADV
ejpam-4170	105	14	p	p	PROPN
ejpam-4170	105	15	is	be	AUX
ejpam-4170	105	16	pythagorean	pythagorean	ADJ
ejpam-4170	105	17	fuzzy	fuzzy	ADJ
ejpam-4170	105	18	small	small	ADJ
ejpam-4170	105	19	submodule	submodule	NOUN
ejpam-4170	105	20	of	of	ADP
ejpam-4170	105	21	m	m	PROPN
ejpam-4170	105	22	.	.	PUNCT
ejpam-4170	106	1	proof	proof	NOUN
ejpam-4170	106	2	.	.	PUNCT
ejpam-4170	107	1	assume	assume	VERB
ejpam-4170	107	2	that	that	SCONJ
ejpam-4170	107	3	t	t	PROPN
ejpam-4170	107	4	is	be	AUX
ejpam-4170	107	5	a	a	DET
ejpam-4170	107	6	pythagorean	pythagorean	ADJ
ejpam-4170	107	7	fuzzy	fuzzy	ADJ
ejpam-4170	107	8	submodule	submodule	PROPN
ejpam-4170	107	9	ofm	ofm	PROPN
ejpam-4170	107	10	such	such	ADJ
ejpam-4170	107	11	that	that	SCONJ
ejpam-4170	107	12	p	p	PROPN
ejpam-4170	107	13	+	+	NOUN
ejpam-4170	107	14	t	t	NOUN
ejpam-4170	107	15	=	=	SYM
ejpam-4170	107	16	χpfm	χpfm	NOUN
ejpam-4170	107	17	.	.	PUNCT
ejpam-4170	108	1	in	in	ADP
ejpam-4170	108	2	order	order	NOUN
ejpam-4170	108	3	to	to	PART
ejpam-4170	108	4	see	see	VERB
ejpam-4170	108	5	that	that	SCONJ
ejpam-4170	108	6	p	p	PROPN
ejpam-4170	108	7	|s	|s	PROPN
ejpam-4170	108	8	+	+	CCONJ
ejpam-4170	108	9	(	(	PUNCT
ejpam-4170	108	10	t	t	PROPN
ejpam-4170	108	11	|s	|s	PROPN
ejpam-4170	108	12	∩	∩	PROPN
ejpam-4170	108	13	χpfs	χpf	NOUN
ejpam-4170	108	14	)	)	PUNCT
ejpam-4170	108	15	,	,	PUNCT
ejpam-4170	108	16	let	let	VERB
ejpam-4170	108	17	a	a	DET
ejpam-4170	108	18	∈	∈	NOUN
ejpam-4170	108	19	s.	s.	PROPN
ejpam-4170	109	1	then	then	ADV
ejpam-4170	109	2	we	we	PRON
ejpam-4170	109	3	obtain	obtain	VERB
ejpam-4170	109	4	η2	η2	NOUN
ejpam-4170	109	5	p	p	PROPN
ejpam-4170	109	6	|s+(t	|s+(t	PROPN
ejpam-4170	109	7	|s∩χpf	|s∩χpf	PROPN
ejpam-4170	109	8	s	s	PART
ejpam-4170	109	9	)	)	PUNCT
ejpam-4170	109	10	(	(	PUNCT
ejpam-4170	109	11	a	a	X
ejpam-4170	109	12	)	)	PUNCT
ejpam-4170	109	13	=	=	NOUN
ejpam-4170	109	14	η2p	η2p	X
ejpam-4170	109	15	|s	|s	PROPN
ejpam-4170	109	16	(	(	PUNCT
ejpam-4170	109	17	a	a	NOUN
ejpam-4170	109	18	)	)	PUNCT
ejpam-4170	109	19	+	+	CCONJ
ejpam-4170	109	20	η2	η2	ADJ
ejpam-4170	109	21	t	t	PROPN
ejpam-4170	109	22	|s∩χpf	|s∩χpf	PROPN
ejpam-4170	109	23	s	s	X
ejpam-4170	109	24	(	(	PUNCT
ejpam-4170	109	25	a)−	a)−	PROPN
ejpam-4170	109	26	η2p	η2p	ADP
ejpam-4170	109	27	|s	|s	PROPN
ejpam-4170	109	28	(	(	PUNCT
ejpam-4170	109	29	a)η	a)η	SYM
ejpam-4170	109	30	2	2	NUM
ejpam-4170	109	31	t	t	PROPN
ejpam-4170	109	32	|s∩χpf	|s∩χpf	PROPN
ejpam-4170	109	33	s	s	X
ejpam-4170	109	34	(	(	PUNCT
ejpam-4170	109	35	a	a	NOUN
ejpam-4170	109	36	)	)	PUNCT
ejpam-4170	109	37	=	=	NOUN
ejpam-4170	109	38	η2p	η2p	X
ejpam-4170	109	39	|s	|s	PROPN
ejpam-4170	109	40	(	(	PUNCT
ejpam-4170	109	41	a	a	NOUN
ejpam-4170	109	42	)	)	PUNCT
ejpam-4170	109	43	+	+	ADP
ejpam-4170	109	44	min{η2	min{η2	X
ejpam-4170	109	45	t	t	PRON
ejpam-4170	109	46	|s	|s	PROPN
ejpam-4170	109	47	(	(	PUNCT
ejpam-4170	109	48	a	a	NOUN
ejpam-4170	109	49	)	)	PUNCT
ejpam-4170	109	50	,	,	PUNCT
ejpam-4170	109	51	χ	χ	X
ejpam-4170	109	52	2	2	NUM
ejpam-4170	109	53	s(a	s(a	NOUN
ejpam-4170	109	54	)	)	PUNCT
ejpam-4170	109	55	}	}	PUNCT
ejpam-4170	109	56	−	−	ADP
ejpam-4170	109	57	η2p	η2p	X
ejpam-4170	109	58	|s	|s	PROPN
ejpam-4170	109	59	(	(	PUNCT
ejpam-4170	109	60	a)min{η2	a)min{η2	PROPN
ejpam-4170	109	61	t	t	PROPN
ejpam-4170	109	62	|s	|s	PROPN
ejpam-4170	109	63	(	(	PUNCT
ejpam-4170	109	64	a	a	NOUN
ejpam-4170	109	65	)	)	PUNCT
ejpam-4170	109	66	,	,	PUNCT
ejpam-4170	109	67	χ	χ	X
ejpam-4170	109	68	2	2	NUM
ejpam-4170	109	69	s(a	s(a	NOUN
ejpam-4170	109	70	)	)	PUNCT
ejpam-4170	109	71	}	}	PUNCT
ejpam-4170	110	1	=	=	X
ejpam-4170	110	2	min{η2p	min{η2p	NOUN
ejpam-4170	110	3	(	(	PUNCT
ejpam-4170	110	4	a	a	NOUN
ejpam-4170	110	5	)	)	PUNCT
ejpam-4170	110	6	,	,	PUNCT
ejpam-4170	110	7	χ2	χ2	PROPN
ejpam-4170	110	8	s(a)}+min{η2	s(a)}+min{η2	ADP
ejpam-4170	110	9	t	t	PROPN
ejpam-4170	110	10	(	(	PUNCT
ejpam-4170	110	11	a	a	NOUN
ejpam-4170	110	12	)	)	PUNCT
ejpam-4170	110	13	,	,	PUNCT
ejpam-4170	110	14	χ2	χ2	PROPN
ejpam-4170	110	15	s(a	s(a	PROPN
ejpam-4170	110	16	)	)	PUNCT
ejpam-4170	110	17	}	}	PUNCT
ejpam-4170	110	18	−min{η2p	−min{η2p	PRON
ejpam-4170	110	19	(	(	PUNCT
ejpam-4170	110	20	a	a	NOUN
ejpam-4170	110	21	)	)	PUNCT
ejpam-4170	110	22	,	,	PUNCT
ejpam-4170	110	23	χ2	χ2	PROPN
ejpam-4170	110	24	s(a)}min{η2	s(a)}min{η2	PROPN
ejpam-4170	110	25	t	t	PROPN
ejpam-4170	110	26	(	(	PUNCT
ejpam-4170	110	27	a	a	NOUN
ejpam-4170	110	28	)	)	PUNCT
ejpam-4170	110	29	,	,	PUNCT
ejpam-4170	110	30	χ2	χ2	PROPN
ejpam-4170	110	31	s(a	s(a	PROPN
ejpam-4170	110	32	)	)	PUNCT
ejpam-4170	110	33	}	}	PUNCT
ejpam-4170	111	1	=	=	X
ejpam-4170	111	2	η2p	η2p	X
ejpam-4170	111	3	(	(	PUNCT
ejpam-4170	111	4	a	a	X
ejpam-4170	111	5	)	)	PUNCT
ejpam-4170	111	6	+	+	CCONJ
ejpam-4170	111	7	η2	η2	ADJ
ejpam-4170	111	8	t	t	NOUN
ejpam-4170	111	9	(	(	PUNCT
ejpam-4170	111	10	a)−	a)−	ADV
ejpam-4170	111	11	η2p	η2p	ADV
ejpam-4170	111	12	(	(	PUNCT
ejpam-4170	111	13	a)η	a)η	NUM
ejpam-4170	111	14	2	2	NUM
ejpam-4170	111	15	t	t	NOUN
ejpam-4170	111	16	(	(	PUNCT
ejpam-4170	111	17	a	a	NOUN
ejpam-4170	111	18	)	)	PUNCT
ejpam-4170	111	19	=	=	NOUN
ejpam-4170	111	20	η2p+t	η2p+t	NOUN
ejpam-4170	111	21	(	(	PUNCT
ejpam-4170	111	22	a	a	X
ejpam-4170	111	23	)	)	PUNCT
ejpam-4170	111	24	=	=	NOUN
ejpam-4170	111	25	χ2	χ2	PROPN
ejpam-4170	111	26	m	m	NOUN
ejpam-4170	111	27	(	(	PUNCT
ejpam-4170	111	28	a	a	X
ejpam-4170	111	29	)	)	PUNCT
ejpam-4170	111	30	=	=	SYM
ejpam-4170	111	31	1	1	NUM
ejpam-4170	111	32	=	=	ADJ
ejpam-4170	111	33	χ2	χ2	PROPN
ejpam-4170	111	34	s(a	s(a	PROPN
ejpam-4170	111	35	)	)	PUNCT
ejpam-4170	111	36	and	and	CCONJ
ejpam-4170	111	37	η̂2	η̂2	NOUN
ejpam-4170	111	38	p	p	PRON
ejpam-4170	111	39	|s+(t	|s+(t	PROPN
ejpam-4170	111	40	|s∩χpf	|s∩χpf	PROPN
ejpam-4170	111	41	s	s	PART
ejpam-4170	111	42	)	)	PUNCT
ejpam-4170	111	43	(	(	PUNCT
ejpam-4170	111	44	a	a	X
ejpam-4170	111	45	)	)	PUNCT
ejpam-4170	111	46	=	=	NOUN
ejpam-4170	111	47	η̂2p	η̂2p	NUM
ejpam-4170	111	48	|s	|s	PROPN
ejpam-4170	111	49	(	(	PUNCT
ejpam-4170	111	50	a)η̂	a)η̂	NOUN
ejpam-4170	111	51	2	2	NUM
ejpam-4170	111	52	t	t	NOUN
ejpam-4170	111	53	|s∩χpf	|s∩χpf	PROPN
ejpam-4170	111	54	s	s	X
ejpam-4170	111	55	(	(	PUNCT
ejpam-4170	111	56	a	a	NOUN
ejpam-4170	111	57	)	)	PUNCT
ejpam-4170	111	58	=	=	PROPN
ejpam-4170	111	59	η̂2p	η̂2p	NUM
ejpam-4170	111	60	|s	|s	PROPN
ejpam-4170	111	61	(	(	PUNCT
ejpam-4170	111	62	a)max{η̂2	a)max{η̂2	PROPN
ejpam-4170	111	63	t	t	PROPN
ejpam-4170	111	64	|s	|s	PROPN
ejpam-4170	111	65	(	(	PUNCT
ejpam-4170	111	66	a	a	NOUN
ejpam-4170	111	67	)	)	PUNCT
ejpam-4170	111	68	,	,	PUNCT
ejpam-4170	111	69	χ	χ	PROPN
ejpam-4170	111	70	c2	c2	PROPN
ejpam-4170	111	71	spf	spf	PROPN
ejpam-4170	111	72	(	(	PUNCT
ejpam-4170	111	73	a	a	NOUN
ejpam-4170	111	74	)	)	PUNCT
ejpam-4170	111	75	}	}	PUNCT
ejpam-4170	111	76	=	=	NOUN
ejpam-4170	111	77	max{η̂2p	max{η̂2p	NOUN
ejpam-4170	111	78	(	(	PUNCT
ejpam-4170	111	79	a	a	NOUN
ejpam-4170	111	80	)	)	PUNCT
ejpam-4170	111	81	,	,	PUNCT
ejpam-4170	111	82	χc	χc	PROPN
ejpam-4170	111	83	2	2	NUM
ejpam-4170	111	84	spf	spf	NOUN
ejpam-4170	111	85	(	(	PUNCT
ejpam-4170	111	86	a)}max{η̂2	a)}max{η̂2	NOUN
ejpam-4170	111	87	t	t	PROPN
ejpam-4170	111	88	(	(	PUNCT
ejpam-4170	111	89	a	a	X
ejpam-4170	111	90	)	)	PUNCT
ejpam-4170	111	91	,	,	PUNCT
ejpam-4170	111	92	χc	χc	PROPN
ejpam-4170	111	93	2	2	NUM
ejpam-4170	111	94	spf	spf	NOUN
ejpam-4170	111	95	(	(	PUNCT
ejpam-4170	111	96	a	a	NOUN
ejpam-4170	111	97	)	)	PUNCT
ejpam-4170	111	98	}	}	PUNCT
ejpam-4170	111	99	a.	a.	NOUN
ejpam-4170	111	100	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	111	101	/	/	SYM
ejpam-4170	111	102	eur	eur	PROPN
ejpam-4170	111	103	.	.	PUNCT
ejpam-4170	112	1	j.	j.	PROPN
ejpam-4170	112	2	pure	pure	PROPN
ejpam-4170	112	3	appl	appl	PROPN
ejpam-4170	112	4	.	.	PROPN
ejpam-4170	112	5	math	math	PROPN
ejpam-4170	112	6	,	,	PUNCT
ejpam-4170	112	7	15	15	NUM
ejpam-4170	112	8	(	(	PUNCT
ejpam-4170	112	9	1	1	NUM
ejpam-4170	112	10	)	)	PUNCT
ejpam-4170	112	11	(	(	PUNCT
ejpam-4170	112	12	2022	2022	NUM
ejpam-4170	112	13	)	)	PUNCT
ejpam-4170	112	14	,	,	PUNCT
ejpam-4170	112	15	36	36	NUM
ejpam-4170	112	16	-	-	SYM
ejpam-4170	112	17	46	46	NUM
ejpam-4170	112	18	41	41	NUM
ejpam-4170	113	1	=	=	NOUN
ejpam-4170	113	2	η̂2p	η̂2p	NOUN
ejpam-4170	113	3	(	(	PUNCT
ejpam-4170	113	4	a)η̂	a)η̂	NOUN
ejpam-4170	113	5	2	2	NUM
ejpam-4170	113	6	t	t	NOUN
ejpam-4170	113	7	(	(	PUNCT
ejpam-4170	113	8	a	a	NOUN
ejpam-4170	113	9	)	)	PUNCT
ejpam-4170	113	10	=	=	NOUN
ejpam-4170	113	11	η̂2p+t	η̂2p+t	X
ejpam-4170	113	12	(	(	PUNCT
ejpam-4170	113	13	a	a	X
ejpam-4170	113	14	)	)	PUNCT
ejpam-4170	114	1	=	=	NOUN
ejpam-4170	114	2	χc	χc	PROPN
ejpam-4170	114	3	2	2	NUM
ejpam-4170	114	4	m	m	NOUN
ejpam-4170	114	5	(	(	PUNCT
ejpam-4170	114	6	a	a	X
ejpam-4170	114	7	)	)	PUNCT
ejpam-4170	114	8	=	=	SYM
ejpam-4170	114	9	0	0	SYM
ejpam-4170	114	10	=	=	NOUN
ejpam-4170	114	11	χc	χc	ADP
ejpam-4170	114	12	2	2	NUM
ejpam-4170	114	13	s	s	PART
ejpam-4170	114	14	(	(	PUNCT
ejpam-4170	114	15	a	a	X
ejpam-4170	114	16	)	)	PUNCT
ejpam-4170	114	17	this	this	PRON
ejpam-4170	114	18	implies	imply	VERB
ejpam-4170	114	19	that	that	SCONJ
ejpam-4170	114	20	p	p	PROPN
ejpam-4170	114	21	|s+(t	|s+(t	PROPN
ejpam-4170	114	22	|s	|s	PROPN
ejpam-4170	114	23	∩χpfs	∩χpfs	PROPN
ejpam-4170	114	24	)	)	PUNCT
ejpam-4170	114	25	=	=	NOUN
ejpam-4170	114	26	χpfs	χpf	NOUN
ejpam-4170	114	27	.	.	PUNCT
ejpam-4170	115	1	by	by	ADP
ejpam-4170	115	2	hypothesis	hypothesis	NOUN
ejpam-4170	115	3	,	,	PUNCT
ejpam-4170	115	4	we	we	PRON
ejpam-4170	115	5	conclude	conclude	VERB
ejpam-4170	115	6	that	that	SCONJ
ejpam-4170	115	7	t	t	PROPN
ejpam-4170	115	8	|s	|s	PROPN
ejpam-4170	115	9	∩χpfs	∩χpfs	PROPN
ejpam-4170	115	10	=	=	PROPN
ejpam-4170	115	11	χpfs	χpf	NOUN
ejpam-4170	115	12	.	.	PUNCT
ejpam-4170	116	1	thus	thus	ADV
ejpam-4170	116	2	χpfs	χpf	VERB
ejpam-4170	116	3	⊆	⊆	NUM
ejpam-4170	116	4	t	t	PROPN
ejpam-4170	116	5	|s	|s	PROPN
ejpam-4170	116	6	.	.	PUNCT
ejpam-4170	117	1	then	then	ADV
ejpam-4170	117	2	χpfm	χpfm	VERB
ejpam-4170	118	1	=	=	PUNCT
ejpam-4170	118	2	p	p	X
ejpam-4170	119	1	+	+	X
ejpam-4170	119	2	t	t	PROPN
ejpam-4170	119	3	⊆	⊆	NUM
ejpam-4170	119	4	t	t	NOUN
ejpam-4170	119	5	⊆	⊆	NUM
ejpam-4170	119	6	χpfm	χpfm	NOUN
ejpam-4170	119	7	.	.	PUNCT
ejpam-4170	120	1	therefore	therefore	ADV
ejpam-4170	120	2	,	,	PUNCT
ejpam-4170	120	3	t	t	PROPN
ejpam-4170	120	4	=	=	SYM
ejpam-4170	120	5	χpfm	χpfm	NOUN
ejpam-4170	120	6	and	and	CCONJ
ejpam-4170	120	7	p	p	NOUN
ejpam-4170	120	8	is	be	AUX
ejpam-4170	120	9	pythagorean	pythagorean	ADJ
ejpam-4170	120	10	fuzzy	fuzzy	ADJ
ejpam-4170	120	11	small	small	ADJ
ejpam-4170	120	12	submodule	submodule	NOUN
ejpam-4170	120	13	of	of	ADP
ejpam-4170	120	14	m	m	PROPN
ejpam-4170	120	15	.	.	PUNCT
ejpam-4170	121	1	as	as	ADP
ejpam-4170	121	2	a	a	DET
ejpam-4170	121	3	consequence	consequence	NOUN
ejpam-4170	121	4	of	of	ADP
ejpam-4170	121	5	the	the	DET
ejpam-4170	121	6	above	above	ADJ
ejpam-4170	121	7	theorem	theorem	NOUN
ejpam-4170	121	8	,	,	PUNCT
ejpam-4170	121	9	we	we	PRON
ejpam-4170	121	10	have	have	VERB
ejpam-4170	121	11	:	:	PUNCT
ejpam-4170	121	12	corollary	corollary	ADJ
ejpam-4170	121	13	2	2	X
ejpam-4170	121	14	.	.	PUNCT
ejpam-4170	122	1	let	let	VERB
ejpam-4170	122	2	m	m	PRON
ejpam-4170	122	3	be	be	AUX
ejpam-4170	122	4	a	a	DET
ejpam-4170	122	5	module	module	NOUN
ejpam-4170	122	6	and	and	CCONJ
ejpam-4170	122	7	,	,	PUNCT
ejpam-4170	122	8	p	p	PROPN
ejpam-4170	122	9	and	and	CCONJ
ejpam-4170	122	10	s	s	NOUN
ejpam-4170	122	11	are	be	AUX
ejpam-4170	122	12	pythagorean	pythagorean	ADJ
ejpam-4170	122	13	fuzzy	fuzzy	ADJ
ejpam-4170	122	14	submodules	submodule	NOUN
ejpam-4170	122	15	of	of	ADP
ejpam-4170	122	16	m	m	NOUN
ejpam-4170	122	17	in	in	ADP
ejpam-4170	122	18	which	which	PRON
ejpam-4170	122	19	p	p	NOUN
ejpam-4170	122	20	⊆	⊆	NUM
ejpam-4170	122	21	s.	s.	PROPN
ejpam-4170	122	22	if	if	SCONJ
ejpam-4170	122	23	p	p	NOUN
ejpam-4170	122	24	is	be	AUX
ejpam-4170	122	25	pythagorean	pythagorean	ADJ
ejpam-4170	122	26	fuzzy	fuzzy	ADJ
ejpam-4170	122	27	small	small	ADJ
ejpam-4170	122	28	submodule	submodule	NOUN
ejpam-4170	122	29	of	of	ADP
ejpam-4170	122	30	s	s	PROPN
ejpam-4170	122	31	,	,	PUNCT
ejpam-4170	122	32	then	then	ADV
ejpam-4170	122	33	p	p	PROPN
ejpam-4170	122	34	is	be	AUX
ejpam-4170	122	35	pythagorean	pythagorean	ADJ
ejpam-4170	122	36	fuzzy	fuzzy	ADJ
ejpam-4170	122	37	small	small	ADJ
ejpam-4170	122	38	submodule	submodule	NOUN
ejpam-4170	122	39	of	of	ADP
ejpam-4170	122	40	m	m	PROPN
ejpam-4170	122	41	.	.	PUNCT
ejpam-4170	123	1	proof	proof	NOUN
ejpam-4170	123	2	.	.	PUNCT
ejpam-4170	124	1	clear	clear	ADJ
ejpam-4170	124	2	.	.	PUNCT
ejpam-4170	125	1	remark	remark	PROPN
ejpam-4170	125	2	1	1	NUM
ejpam-4170	125	3	.	.	PUNCT
ejpam-4170	126	1	the	the	DET
ejpam-4170	126	2	converse	converse	NOUN
ejpam-4170	126	3	of	of	ADP
ejpam-4170	126	4	theorem	theorem	NOUN
ejpam-4170	126	5	4	4	NUM
ejpam-4170	126	6	need	need	AUX
ejpam-4170	126	7	not	not	PART
ejpam-4170	126	8	be	be	AUX
ejpam-4170	126	9	true	true	ADJ
ejpam-4170	126	10	in	in	ADP
ejpam-4170	126	11	general	general	ADJ
ejpam-4170	126	12	.	.	PUNCT
ejpam-4170	127	1	that	that	PRON
ejpam-4170	127	2	is	be	AUX
ejpam-4170	127	3	if	if	SCONJ
ejpam-4170	127	4	m	m	NOUN
ejpam-4170	127	5	is	be	AUX
ejpam-4170	127	6	a	a	DET
ejpam-4170	127	7	module	module	NOUN
ejpam-4170	127	8	,	,	PUNCT
ejpam-4170	127	9	s	s	AUX
ejpam-4170	127	10	be	be	AUX
ejpam-4170	127	11	a	a	DET
ejpam-4170	127	12	submodule	submodule	NOUN
ejpam-4170	127	13	of	of	ADP
ejpam-4170	127	14	m	m	PROPN
ejpam-4170	127	15	and	and	CCONJ
ejpam-4170	127	16	p	p	PROPN
ejpam-4170	127	17	is	be	AUX
ejpam-4170	127	18	a	a	DET
ejpam-4170	127	19	pythagorean	pythagorean	ADJ
ejpam-4170	127	20	fuzzy	fuzzy	ADJ
ejpam-4170	127	21	small	small	ADJ
ejpam-4170	127	22	submodule	submodule	NOUN
ejpam-4170	127	23	of	of	ADP
ejpam-4170	127	24	m	m	PROPN
ejpam-4170	127	25	in	in	ADP
ejpam-4170	127	26	which	which	PRON
ejpam-4170	127	27	p	p	NOUN
ejpam-4170	127	28	⊆	⊆	NUM
ejpam-4170	127	29	χpfs	χpf	NOUN
ejpam-4170	127	30	,	,	PUNCT
ejpam-4170	127	31	then	then	ADV
ejpam-4170	127	32	it	it	PRON
ejpam-4170	127	33	is	be	AUX
ejpam-4170	127	34	not	not	PART
ejpam-4170	127	35	true	true	ADJ
ejpam-4170	127	36	in	in	ADP
ejpam-4170	127	37	general	general	ADJ
ejpam-4170	127	38	that	that	SCONJ
ejpam-4170	127	39	p	p	PROPN
ejpam-4170	127	40	|s	|s	PROPN
ejpam-4170	127	41	is	be	AUX
ejpam-4170	127	42	a	a	DET
ejpam-4170	127	43	pythagorean	pythagorean	ADJ
ejpam-4170	127	44	fuzzy	fuzzy	ADJ
ejpam-4170	127	45	small	small	ADJ
ejpam-4170	127	46	submodule	submodule	NOUN
ejpam-4170	127	47	of	of	ADP
ejpam-4170	127	48	s.	s.	PROPN
ejpam-4170	127	49	for	for	ADP
ejpam-4170	127	50	example	example	NOUN
ejpam-4170	127	51	take	take	VERB
ejpam-4170	127	52	p	p	PROPN
ejpam-4170	127	53	|s	|s	PROPN
ejpam-4170	127	54	=	=	SYM
ejpam-4170	127	55	s.	s.	PROPN
ejpam-4170	127	56	proposition	proposition	NOUN
ejpam-4170	127	57	1	1	X
ejpam-4170	127	58	.	.	PUNCT
ejpam-4170	128	1	let	let	VERB
ejpam-4170	128	2	m	m	PRON
ejpam-4170	128	3	be	be	AUX
ejpam-4170	128	4	a	a	DET
ejpam-4170	128	5	module	module	NOUN
ejpam-4170	128	6	and	and	CCONJ
ejpam-4170	128	7	p	p	NOUN
ejpam-4170	128	8	,	,	PUNCT
ejpam-4170	128	9	s	s	PROPN
ejpam-4170	128	10	,	,	PUNCT
ejpam-4170	128	11	t	t	AUX
ejpam-4170	128	12	be	be	AUX
ejpam-4170	128	13	pythagorean	pythagorean	PROPN
ejpam-4170	128	14	fuzzy	fuzzy	ADJ
ejpam-4170	128	15	submodules	submodule	NOUN
ejpam-4170	128	16	of	of	ADP
ejpam-4170	128	17	m	m	PROPN
ejpam-4170	128	18	.	.	PUNCT
ejpam-4170	129	1	then	then	ADV
ejpam-4170	129	2	:	:	PUNCT
ejpam-4170	129	3	(	(	PUNCT
ejpam-4170	129	4	p	p	X
ejpam-4170	129	5	∩	∩	X
ejpam-4170	129	6	s	s	PART
ejpam-4170	129	7	)	)	PUNCT
ejpam-4170	129	8	+	+	CCONJ
ejpam-4170	129	9	(	(	PUNCT
ejpam-4170	129	10	p	p	NOUN
ejpam-4170	129	11	∩	∩	ADJ
ejpam-4170	129	12	t	t	NOUN
ejpam-4170	129	13	)	)	PUNCT
ejpam-4170	129	14	⊆	⊆	NUM
ejpam-4170	129	15	p	p	NOUN
ejpam-4170	129	16	∩	∩	NOUN
ejpam-4170	129	17	(	(	PUNCT
ejpam-4170	129	18	s	s	VERB
ejpam-4170	129	19	+	+	X
ejpam-4170	129	20	t	t	NOUN
ejpam-4170	129	21	)	)	PUNCT
ejpam-4170	129	22	.	.	PUNCT
ejpam-4170	130	1	proof	proof	NOUN
ejpam-4170	130	2	.	.	PUNCT
ejpam-4170	131	1	let	let	VERB
ejpam-4170	131	2	m	m	PRON
ejpam-4170	131	3	∈m	∈m	VERB
ejpam-4170	131	4	.	.	PUNCT
ejpam-4170	132	1	then	then	ADV
ejpam-4170	132	2	η2(p∩s)+(p∩t	η2(p∩s)+(p∩t	PROPN
ejpam-4170	132	3	)	)	PUNCT
ejpam-4170	132	4	(	(	PUNCT
ejpam-4170	132	5	m	m	NOUN
ejpam-4170	132	6	)	)	PUNCT
ejpam-4170	132	7	=	=	SYM
ejpam-4170	132	8	η2p∩s(m	η2p∩s(m	PROPN
ejpam-4170	132	9	)	)	PUNCT
ejpam-4170	132	10	+	+	CCONJ
ejpam-4170	132	11	η2p∩t	η2p∩t	PROPN
ejpam-4170	132	12	(	(	PUNCT
ejpam-4170	132	13	m)−	m)−	PROPN
ejpam-4170	132	14	η2p∩s(m)η2p∩t	η2p∩s(m)η2p∩t	PROPN
ejpam-4170	132	15	(	(	PUNCT
ejpam-4170	132	16	m	m	NOUN
ejpam-4170	132	17	)	)	PUNCT
ejpam-4170	133	1	=	=	NOUN
ejpam-4170	133	2	min{η2p	min{η2p	NOUN
ejpam-4170	133	3	(	(	PUNCT
ejpam-4170	133	4	m	m	NOUN
ejpam-4170	133	5	)	)	PUNCT
ejpam-4170	133	6	,	,	PUNCT
ejpam-4170	133	7	η2s(m)}+min{η2p	η2s(m)}+min{η2p	PROPN
ejpam-4170	133	8	(	(	PUNCT
ejpam-4170	133	9	m	m	NOUN
ejpam-4170	133	10	)	)	PUNCT
ejpam-4170	133	11	,	,	PUNCT
ejpam-4170	133	12	η2	η2	X
ejpam-4170	133	13	t	t	PROPN
ejpam-4170	133	14	(	(	PUNCT
ejpam-4170	133	15	m	m	NOUN
ejpam-4170	133	16	)	)	PUNCT
ejpam-4170	133	17	}	}	PUNCT
ejpam-4170	133	18	−min{η2p	−min{η2p	PROPN
ejpam-4170	133	19	(	(	PUNCT
ejpam-4170	133	20	m	m	NOUN
ejpam-4170	133	21	)	)	PUNCT
ejpam-4170	133	22	,	,	PUNCT
ejpam-4170	133	23	η2s(m)}min{η2p	η2s(m)}min{η2p	PROPN
ejpam-4170	133	24	(	(	PUNCT
ejpam-4170	133	25	m	m	NOUN
ejpam-4170	133	26	)	)	PUNCT
ejpam-4170	133	27	,	,	PUNCT
ejpam-4170	133	28	η2	η2	X
ejpam-4170	133	29	t	t	PROPN
ejpam-4170	133	30	(	(	PUNCT
ejpam-4170	133	31	m	m	NOUN
ejpam-4170	133	32	)	)	PUNCT
ejpam-4170	133	33	}	}	PUNCT
ejpam-4170	133	34	≤min{η2p	≤min{η2p	PROPN
ejpam-4170	133	35	(	(	PUNCT
ejpam-4170	133	36	m	m	NOUN
ejpam-4170	133	37	)	)	PUNCT
ejpam-4170	133	38	,	,	PUNCT
ejpam-4170	133	39	η2s(m	η2s(m	PROPN
ejpam-4170	133	40	)	)	PUNCT
ejpam-4170	134	1	+	+	NUM
ejpam-4170	134	2	η2	η2	ADJ
ejpam-4170	134	3	t	t	NOUN
ejpam-4170	134	4	(	(	PUNCT
ejpam-4170	134	5	m)−	m)−	PROPN
ejpam-4170	134	6	η2s(m)η2	η2s(m)η2	PROPN
ejpam-4170	134	7	t	t	PROPN
ejpam-4170	134	8	(	(	PUNCT
ejpam-4170	134	9	m	m	NOUN
ejpam-4170	134	10	)	)	PUNCT
ejpam-4170	134	11	}	}	PUNCT
ejpam-4170	135	1	=	=	X
ejpam-4170	135	2	min{η2p	min{η2p	NOUN
ejpam-4170	135	3	(	(	PUNCT
ejpam-4170	135	4	m	m	NOUN
ejpam-4170	135	5	)	)	PUNCT
ejpam-4170	135	6	,	,	PUNCT
ejpam-4170	135	7	η2s+t	η2s+t	PROPN
ejpam-4170	135	8	(	(	PUNCT
ejpam-4170	135	9	m	m	NOUN
ejpam-4170	135	10	)	)	PUNCT
ejpam-4170	136	1	=	=	NOUN
ejpam-4170	136	2	η2p∩(s+t	η2p∩(s+t	NUM
ejpam-4170	136	3	)	)	PUNCT
ejpam-4170	136	4	(	(	PUNCT
ejpam-4170	136	5	m	m	NOUN
ejpam-4170	136	6	)	)	PUNCT
ejpam-4170	136	7	moreover	moreover	ADV
ejpam-4170	136	8	,	,	PUNCT
ejpam-4170	136	9	η̂2(p∩s)+(p∩t	η̂2(p∩s)+(p∩t	ADJ
ejpam-4170	136	10	)	)	PUNCT
ejpam-4170	136	11	(	(	PUNCT
ejpam-4170	136	12	m	m	NOUN
ejpam-4170	136	13	)	)	PUNCT
ejpam-4170	137	1	=	=	NOUN
ejpam-4170	137	2	η̂2p∩s(m)η̂2p∩t	η̂2p∩s(m)η̂2p∩t	NOUN
ejpam-4170	137	3	(	(	PUNCT
ejpam-4170	137	4	m	m	NOUN
ejpam-4170	137	5	)	)	PUNCT
ejpam-4170	138	1	=	=	NOUN
ejpam-4170	138	2	max{η̂2p	max{η̂2p	NOUN
ejpam-4170	138	3	(	(	PUNCT
ejpam-4170	138	4	m	m	NOUN
ejpam-4170	138	5	)	)	PUNCT
ejpam-4170	138	6	,	,	PUNCT
ejpam-4170	138	7	η̂2s(m)}max{η̂2p	η̂2s(m)}max{η̂2p	X
ejpam-4170	138	8	(	(	PUNCT
ejpam-4170	138	9	m	m	NOUN
ejpam-4170	138	10	)	)	PUNCT
ejpam-4170	138	11	,	,	PUNCT
ejpam-4170	138	12	η̂2	η̂2	PROPN
ejpam-4170	138	13	t	t	PROPN
ejpam-4170	138	14	(	(	PUNCT
ejpam-4170	138	15	m	m	NOUN
ejpam-4170	138	16	)	)	PUNCT
ejpam-4170	138	17	}	}	PUNCT
ejpam-4170	138	18	≥max{η̂2p	≥max{η̂2p	NOUN
ejpam-4170	138	19	(	(	PUNCT
ejpam-4170	138	20	m	m	PROPN
ejpam-4170	138	21	)	)	PUNCT
ejpam-4170	138	22	,	,	PUNCT
ejpam-4170	138	23	η̂2s(m)η̂2	η̂2s(m)η̂2	PROPN
ejpam-4170	138	24	t	t	PROPN
ejpam-4170	138	25	(	(	PUNCT
ejpam-4170	138	26	m	m	NOUN
ejpam-4170	138	27	)	)	PUNCT
ejpam-4170	138	28	}	}	PUNCT
ejpam-4170	138	29	=	=	NUM
ejpam-4170	138	30	η̂2p∩(s+t	η̂2p∩(s+t	NOUN
ejpam-4170	138	31	)	)	PUNCT
ejpam-4170	138	32	(	(	PUNCT
ejpam-4170	138	33	m	m	NOUN
ejpam-4170	138	34	)	)	PUNCT
ejpam-4170	138	35	a.	a.	NOUN
ejpam-4170	138	36	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	138	37	/	/	SYM
ejpam-4170	138	38	eur	eur	PROPN
ejpam-4170	138	39	.	.	PUNCT
ejpam-4170	139	1	j.	j.	PROPN
ejpam-4170	139	2	pure	pure	PROPN
ejpam-4170	139	3	appl	appl	PROPN
ejpam-4170	139	4	.	.	PROPN
ejpam-4170	139	5	math	math	PROPN
ejpam-4170	139	6	,	,	PUNCT
ejpam-4170	139	7	15	15	NUM
ejpam-4170	139	8	(	(	PUNCT
ejpam-4170	139	9	1	1	NUM
ejpam-4170	139	10	)	)	PUNCT
ejpam-4170	139	11	(	(	PUNCT
ejpam-4170	139	12	2022	2022	NUM
ejpam-4170	139	13	)	)	PUNCT
ejpam-4170	139	14	,	,	PUNCT
ejpam-4170	139	15	36	36	NUM
ejpam-4170	139	16	-	-	SYM
ejpam-4170	139	17	46	46	NUM
ejpam-4170	139	18	42	42	NUM
ejpam-4170	139	19	proposition	proposition	NOUN
ejpam-4170	139	20	2	2	NUM
ejpam-4170	139	21	.	.	PUNCT
ejpam-4170	140	1	let	let	VERB
ejpam-4170	140	2	m	m	PRON
ejpam-4170	140	3	be	be	AUX
ejpam-4170	140	4	a	a	DET
ejpam-4170	140	5	module	module	NOUN
ejpam-4170	140	6	and	and	CCONJ
ejpam-4170	140	7	,	,	PUNCT
ejpam-4170	140	8	p	p	PROPN
ejpam-4170	140	9	and	and	CCONJ
ejpam-4170	140	10	s	s	NOUN
ejpam-4170	140	11	are	be	AUX
ejpam-4170	140	12	pythagorean	pythagorean	ADJ
ejpam-4170	140	13	fuzzy	fuzzy	ADJ
ejpam-4170	140	14	submodules	submodule	NOUN
ejpam-4170	140	15	of	of	ADP
ejpam-4170	140	16	m	m	NOUN
ejpam-4170	140	17	in	in	ADP
ejpam-4170	140	18	which	which	PRON
ejpam-4170	140	19	χpfm	χpfm	NOUN
ejpam-4170	141	1	=	=	PRON
ejpam-4170	141	2	p	p	PROPN
ejpam-4170	141	3	⊕	⊕	PROPN
ejpam-4170	141	4	pf	pf	PROPN
ejpam-4170	141	5	s.	s.	PROPN
ejpam-4170	141	6	then	then	ADV
ejpam-4170	141	7	m	m	VERB
ejpam-4170	141	8	=	=	ADJ
ejpam-4170	141	9	p	p	X
ejpam-4170	141	10	⋆	⋆	PUNCT
ejpam-4170	141	11	⊕	⊕	PROPN
ejpam-4170	141	12	s⋆	s⋆	X
ejpam-4170	141	13	=	=	SYM
ejpam-4170	141	14	p⋆	p⋆	PRON
ejpam-4170	141	15	⊕	⊕	PROPN
ejpam-4170	141	16	s⋆.	s⋆.	NOUN
ejpam-4170	141	17	proof	proof	NOUN
ejpam-4170	141	18	.	.	PUNCT
ejpam-4170	142	1	let	let	VERB
ejpam-4170	142	2	m	m	PRON
ejpam-4170	142	3	∈m	∈m	VERB
ejpam-4170	142	4	.	.	PUNCT
ejpam-4170	143	1	then	then	ADV
ejpam-4170	143	2	1	1	NUM
ejpam-4170	143	3	=	=	NUM
ejpam-4170	143	4	χ2	χ2	NUM
ejpam-4170	143	5	m	m	NOUN
ejpam-4170	143	6	(	(	PUNCT
ejpam-4170	143	7	m	m	NOUN
ejpam-4170	143	8	)	)	PUNCT
ejpam-4170	143	9	=	=	SYM
ejpam-4170	143	10	η2p+s(m	η2p+s(m	PROPN
ejpam-4170	143	11	)	)	PUNCT
ejpam-4170	144	1	=	=	VERB
ejpam-4170	144	2	η2p	η2p	X
ejpam-4170	144	3	(	(	PUNCT
ejpam-4170	144	4	m	m	NOUN
ejpam-4170	144	5	)	)	PUNCT
ejpam-4170	144	6	+	+	CCONJ
ejpam-4170	144	7	η2s(m)−	η2s(m)−	VERB
ejpam-4170	144	8	η2p	η2p	X
ejpam-4170	144	9	(	(	PUNCT
ejpam-4170	144	10	m)η2s(m	m)η2s(m	NOUN
ejpam-4170	144	11	)	)	PUNCT
ejpam-4170	144	12	=	=	VERB
ejpam-4170	144	13	η2p	η2p	X
ejpam-4170	144	14	(	(	PUNCT
ejpam-4170	144	15	m)(1−	m)(1−	PROPN
ejpam-4170	144	16	η2s(m	η2s(m	PROPN
ejpam-4170	144	17	)	)	PUNCT
ejpam-4170	144	18	)	)	PUNCT
ejpam-4170	145	1	+	+	CCONJ
ejpam-4170	145	2	η2s(m	η2s(m	PROPN
ejpam-4170	145	3	)	)	PUNCT
ejpam-4170	145	4	this	this	PRON
ejpam-4170	145	5	implies	imply	VERB
ejpam-4170	145	6	that	that	SCONJ
ejpam-4170	145	7	η2p	η2p	ADJ
ejpam-4170	145	8	(	(	PUNCT
ejpam-4170	145	9	m	m	NOUN
ejpam-4170	145	10	)	)	PUNCT
ejpam-4170	145	11	=	=	SYM
ejpam-4170	145	12	1	1	NUM
ejpam-4170	145	13	or	or	CCONJ
ejpam-4170	145	14	η2s(m	η2s(m	PROPN
ejpam-4170	145	15	)	)	PUNCT
ejpam-4170	145	16	=	=	SYM
ejpam-4170	146	1	1	1	NUM
ejpam-4170	146	2	,	,	PUNCT
ejpam-4170	146	3	so	so	SCONJ
ejpam-4170	146	4	that	that	SCONJ
ejpam-4170	146	5	η̂2p	η̂2p	ADJ
ejpam-4170	146	6	(	(	PUNCT
ejpam-4170	146	7	m	m	NOUN
ejpam-4170	146	8	)	)	PUNCT
ejpam-4170	146	9	=	=	SYM
ejpam-4170	146	10	0	0	NUM
ejpam-4170	146	11	or	or	CCONJ
ejpam-4170	146	12	η̂2s(m	η̂2s(m	PROPN
ejpam-4170	146	13	)	)	PUNCT
ejpam-4170	147	1	=	=	PUNCT
ejpam-4170	148	1	0	0	X
ejpam-4170	148	2	.	.	PUNCT
ejpam-4170	149	1	hence	hence	ADV
ejpam-4170	149	2	m	m	VERB
ejpam-4170	149	3	∈	∈	ADJ
ejpam-4170	149	4	p⋆	p⋆	NOUN
ejpam-4170	149	5	or	or	CCONJ
ejpam-4170	149	6	m	m	PROPN
ejpam-4170	149	7	∈	∈	NOUN
ejpam-4170	149	8	s⋆	s⋆	NOUN
ejpam-4170	149	9	,	,	PUNCT
ejpam-4170	149	10	so	so	SCONJ
ejpam-4170	149	11	that	that	SCONJ
ejpam-4170	149	12	m	m	VERB
ejpam-4170	149	13	=	=	ADJ
ejpam-4170	149	14	p⋆	p⋆	NOUN
ejpam-4170	149	15	+	+	CCONJ
ejpam-4170	149	16	s⋆.	s⋆.	NOUN
ejpam-4170	149	17	hence	hence	ADV
ejpam-4170	149	18	m	m	VERB
ejpam-4170	149	19	=	=	ADJ
ejpam-4170	150	1	p	p	X
ejpam-4170	150	2	⋆	⋆	X
ejpam-4170	151	1	+	+	CCONJ
ejpam-4170	151	2	s⋆.	s⋆.	NOUN
ejpam-4170	151	3	we	we	PRON
ejpam-4170	151	4	aim	aim	VERB
ejpam-4170	151	5	now	now	ADV
ejpam-4170	151	6	to	to	PART
ejpam-4170	151	7	show	show	VERB
ejpam-4170	151	8	that	that	SCONJ
ejpam-4170	151	9	the	the	DET
ejpam-4170	151	10	intersection	intersection	NOUN
ejpam-4170	151	11	p	p	X
ejpam-4170	151	12	⋆	⋆	NOUN
ejpam-4170	151	13	∩	∩	NOUN
ejpam-4170	151	14	s⋆	s⋆	ADJ
ejpam-4170	151	15	=	=	SYM
ejpam-4170	151	16	0	0	X
ejpam-4170	151	17	.	.	PUNCT
ejpam-4170	151	18	assume	assume	VERB
ejpam-4170	151	19	that	that	SCONJ
ejpam-4170	151	20	m	m	PROPN
ejpam-4170	151	21	∈	∈	PROPN
ejpam-4170	151	22	p	p	X
ejpam-4170	151	23	⋆	⋆	X
ejpam-4170	151	24	∩	∩	ADJ
ejpam-4170	151	25	s⋆.	s⋆.	NOUN
ejpam-4170	151	26	then	then	ADV
ejpam-4170	151	27	η2p	η2p	ADJ
ejpam-4170	151	28	(	(	PUNCT
ejpam-4170	151	29	m	m	NOUN
ejpam-4170	151	30	)	)	PUNCT
ejpam-4170	151	31	,	,	PUNCT
ejpam-4170	151	32	η2s(m	η2s(m	PROPN
ejpam-4170	151	33	)	)	PUNCT
ejpam-4170	151	34	>	>	X
ejpam-4170	152	1	0	0	X
ejpam-4170	152	2	.	.	PUNCT
ejpam-4170	153	1	since	since	SCONJ
ejpam-4170	153	2	χpfm	χpfm	NOUN
ejpam-4170	153	3	=	=	PROPN
ejpam-4170	153	4	p	p	PROPN
ejpam-4170	153	5	⊕	⊕	PROPN
ejpam-4170	153	6	pf	pf	PROPN
ejpam-4170	153	7	s	s	PROPN
ejpam-4170	153	8	,	,	PUNCT
ejpam-4170	153	9	we	we	PRON
ejpam-4170	153	10	obtain	obtain	VERB
ejpam-4170	153	11	0	0	NUM
ejpam-4170	153	12	<	<	X
ejpam-4170	153	13	min{η2p	min{η2p	NOUN
ejpam-4170	153	14	(	(	PUNCT
ejpam-4170	153	15	m	m	NOUN
ejpam-4170	153	16	)	)	PUNCT
ejpam-4170	153	17	,	,	PUNCT
ejpam-4170	153	18	η2s(m	η2s(m	PROPN
ejpam-4170	153	19	)	)	PUNCT
ejpam-4170	153	20	}	}	PUNCT
ejpam-4170	154	1	=	=	X
ejpam-4170	154	2	χ2	χ2	PROPN
ejpam-4170	154	3	0(m	0(m	NUM
ejpam-4170	154	4	)	)	PUNCT
ejpam-4170	154	5	,	,	PUNCT
ejpam-4170	154	6	which	which	PRON
ejpam-4170	154	7	means	mean	VERB
ejpam-4170	154	8	that	that	SCONJ
ejpam-4170	154	9	m	m	VERB
ejpam-4170	154	10	=	=	SYM
ejpam-4170	154	11	0	0	NUM
ejpam-4170	154	12	and	and	CCONJ
ejpam-4170	154	13	hence	hence	ADV
ejpam-4170	154	14	,	,	PUNCT
ejpam-4170	154	15	p⋆	p⋆	DET
ejpam-4170	154	16	∩	∩	NOUN
ejpam-4170	154	17	s⋆	s⋆	ADJ
ejpam-4170	154	18	⊆	⊆	NUM
ejpam-4170	154	19	p	p	NOUN
ejpam-4170	154	20	⋆	⋆	NOUN
ejpam-4170	154	21	∩	∩	NOUN
ejpam-4170	154	22	s⋆	s⋆	X
ejpam-4170	154	23	=	=	SYM
ejpam-4170	154	24	0	0	X
ejpam-4170	154	25	.	.	PUNCT
ejpam-4170	155	1	therefore	therefore	ADV
ejpam-4170	155	2	,	,	PUNCT
ejpam-4170	155	3	the	the	DET
ejpam-4170	155	4	result	result	NOUN
ejpam-4170	155	5	holds	hold	VERB
ejpam-4170	155	6	.	.	PUNCT
ejpam-4170	156	1	now	now	ADV
ejpam-4170	156	2	,	,	PUNCT
ejpam-4170	156	3	we	we	PRON
ejpam-4170	156	4	are	be	AUX
ejpam-4170	156	5	able	able	ADJ
ejpam-4170	156	6	to	to	PART
ejpam-4170	156	7	show	show	VERB
ejpam-4170	156	8	that	that	SCONJ
ejpam-4170	156	9	the	the	DET
ejpam-4170	156	10	convers	conver	NOUN
ejpam-4170	156	11	of	of	ADP
ejpam-4170	156	12	corollary	corollary	ADJ
ejpam-4170	156	13	2	2	NUM
ejpam-4170	156	14	is	be	AUX
ejpam-4170	156	15	true	true	ADJ
ejpam-4170	156	16	if	if	SCONJ
ejpam-4170	156	17	s	s	NOUN
ejpam-4170	156	18	is	be	AUX
ejpam-4170	156	19	a	a	DET
ejpam-4170	156	20	pythagorean	pythagorean	ADJ
ejpam-4170	156	21	fuzzy	fuzzy	ADJ
ejpam-4170	156	22	direct	direct	ADJ
ejpam-4170	156	23	summand	summand	NOUN
ejpam-4170	156	24	of	of	ADP
ejpam-4170	156	25	m	m	PROPN
ejpam-4170	156	26	as	as	SCONJ
ejpam-4170	156	27	follows	follow	VERB
ejpam-4170	156	28	:	:	PUNCT
ejpam-4170	156	29	theorem	theorem	NOUN
ejpam-4170	156	30	5	5	NUM
ejpam-4170	156	31	.	.	PUNCT
ejpam-4170	157	1	let	let	VERB
ejpam-4170	157	2	m	m	PRON
ejpam-4170	157	3	be	be	AUX
ejpam-4170	157	4	a	a	DET
ejpam-4170	157	5	module	module	NOUN
ejpam-4170	157	6	and	and	CCONJ
ejpam-4170	157	7	,	,	PUNCT
ejpam-4170	157	8	p	p	PROPN
ejpam-4170	157	9	and	and	CCONJ
ejpam-4170	157	10	s	s	NOUN
ejpam-4170	157	11	are	be	AUX
ejpam-4170	157	12	pythagorean	pythagorean	ADJ
ejpam-4170	157	13	fuzzy	fuzzy	ADJ
ejpam-4170	157	14	submodules	submodule	NOUN
ejpam-4170	157	15	of	of	ADP
ejpam-4170	157	16	m	m	NOUN
ejpam-4170	157	17	in	in	ADP
ejpam-4170	157	18	which	which	PRON
ejpam-4170	157	19	p	p	VERB
ejpam-4170	157	20	⊆	⊆	NUM
ejpam-4170	157	21	s	s	NOUN
ejpam-4170	157	22	and	and	CCONJ
ejpam-4170	157	23	s	s	VERB
ejpam-4170	157	24	is	be	AUX
ejpam-4170	157	25	a	a	DET
ejpam-4170	157	26	pythagorean	pythagorean	ADJ
ejpam-4170	157	27	fuzzy	fuzzy	ADJ
ejpam-4170	157	28	direct	direct	ADJ
ejpam-4170	157	29	summand	summand	NOUN
ejpam-4170	157	30	of	of	ADP
ejpam-4170	157	31	m	m	PROPN
ejpam-4170	157	32	.	.	PUNCT
ejpam-4170	158	1	then	then	ADV
ejpam-4170	158	2	p	p	PROPN
ejpam-4170	158	3	is	be	AUX
ejpam-4170	158	4	pythagorean	pythagorean	ADJ
ejpam-4170	158	5	fuzzy	fuzzy	ADJ
ejpam-4170	158	6	small	small	ADJ
ejpam-4170	158	7	submodule	submodule	NOUN
ejpam-4170	158	8	of	of	ADP
ejpam-4170	158	9	s	s	PRON
ejpam-4170	158	10	if	if	SCONJ
ejpam-4170	159	1	and	and	CCONJ
ejpam-4170	159	2	only	only	ADV
ejpam-4170	159	3	if	if	SCONJ
ejpam-4170	159	4	p	p	NOUN
ejpam-4170	159	5	is	be	AUX
ejpam-4170	159	6	pythagorean	pythagorean	ADJ
ejpam-4170	159	7	fuzzy	fuzzy	ADJ
ejpam-4170	159	8	small	small	ADJ
ejpam-4170	159	9	submodule	submodule	NOUN
ejpam-4170	159	10	of	of	ADP
ejpam-4170	159	11	m	m	PROPN
ejpam-4170	159	12	.	.	PUNCT
ejpam-4170	160	1	proof	proof	NOUN
ejpam-4170	160	2	.	.	PUNCT
ejpam-4170	161	1	assume	assume	VERB
ejpam-4170	161	2	that	that	SCONJ
ejpam-4170	161	3	p	p	NOUN
ejpam-4170	161	4	is	be	AUX
ejpam-4170	161	5	a	a	DET
ejpam-4170	161	6	pythagorean	pythagorean	ADJ
ejpam-4170	161	7	fuzzy	fuzzy	ADJ
ejpam-4170	161	8	small	small	ADJ
ejpam-4170	161	9	submodule	submodule	NOUN
ejpam-4170	161	10	of	of	ADP
ejpam-4170	161	11	m	m	PROPN
ejpam-4170	161	12	.	.	PUNCT
ejpam-4170	162	1	applying	apply	VERB
ejpam-4170	162	2	theorem	theorem	NOUN
ejpam-4170	162	3	3	3	NUM
ejpam-4170	162	4	,	,	PUNCT
ejpam-4170	162	5	p⋆	p⋆	X
ejpam-4170	162	6	is	be	AUX
ejpam-4170	162	7	a	a	DET
ejpam-4170	162	8	small	small	ADJ
ejpam-4170	162	9	submodule	submodule	NOUN
ejpam-4170	162	10	of	of	ADP
ejpam-4170	162	11	m	m	PROPN
ejpam-4170	162	12	.	.	PUNCT
ejpam-4170	163	1	that	that	PRON
ejpam-4170	163	2	s	s	VERB
ejpam-4170	163	3	is	be	AUX
ejpam-4170	163	4	a	a	DET
ejpam-4170	163	5	pythagorean	pythagorean	ADJ
ejpam-4170	163	6	fuzzy	fuzzy	ADJ
ejpam-4170	163	7	direct	direct	ADJ
ejpam-4170	163	8	summand	summand	NOUN
ejpam-4170	163	9	of	of	ADP
ejpam-4170	163	10	m	m	PROPN
ejpam-4170	163	11	and	and	CCONJ
ejpam-4170	163	12	p⋆	p⋆	NOUN
ejpam-4170	163	13	⊆	⊆	NUM
ejpam-4170	163	14	s⋆	s⋆	NOUN
ejpam-4170	163	15	,	,	PUNCT
ejpam-4170	163	16	implies	imply	VERB
ejpam-4170	163	17	that	that	SCONJ
ejpam-4170	163	18	p⋆	p⋆	NOUN
ejpam-4170	163	19	is	be	AUX
ejpam-4170	163	20	a	a	DET
ejpam-4170	163	21	small	small	ADJ
ejpam-4170	163	22	submodule	submodule	NOUN
ejpam-4170	163	23	of	of	ADP
ejpam-4170	163	24	s⋆.	s⋆.	NOUN
ejpam-4170	163	25	applying	apply	VERB
ejpam-4170	163	26	corollary	corollary	NOUN
ejpam-4170	163	27	2	2	NUM
ejpam-4170	163	28	,	,	PUNCT
ejpam-4170	163	29	the	the	DET
ejpam-4170	163	30	result	result	NOUN
ejpam-4170	163	31	hold	hold	VERB
ejpam-4170	163	32	.	.	PUNCT
ejpam-4170	164	1	theorem	theorem	ADJ
ejpam-4170	164	2	6	6	NUM
ejpam-4170	164	3	.	.	PUNCT
ejpam-4170	165	1	let	let	VERB
ejpam-4170	165	2	m	m	PRON
ejpam-4170	165	3	be	be	AUX
ejpam-4170	165	4	a	a	DET
ejpam-4170	165	5	module	module	NOUN
ejpam-4170	165	6	and	and	CCONJ
ejpam-4170	165	7	,	,	PUNCT
ejpam-4170	165	8	p	p	NOUN
ejpam-4170	165	9	and	and	CCONJ
ejpam-4170	165	10	s	s	VERB
ejpam-4170	165	11	be	be	AUX
ejpam-4170	165	12	pythagorean	pythagorean	ADJ
ejpam-4170	165	13	fuzzy	fuzzy	ADJ
ejpam-4170	165	14	submodules	submodule	NOUN
ejpam-4170	165	15	of	of	ADP
ejpam-4170	165	16	m	m	NOUN
ejpam-4170	165	17	such	such	ADJ
ejpam-4170	165	18	that	that	SCONJ
ejpam-4170	165	19	p	p	PROPN
ejpam-4170	165	20	∩	∩	NOUN
ejpam-4170	165	21	s	s	PART
ejpam-4170	165	22	=	=	PUNCT
ejpam-4170	165	23	χpf0	χpf0	PROPN
ejpam-4170	165	24	.	.	PUNCT
ejpam-4170	166	1	then	then	ADV
ejpam-4170	166	2	(	(	PUNCT
ejpam-4170	166	3	1	1	X
ejpam-4170	166	4	)	)	PUNCT
ejpam-4170	166	5	(	(	PUNCT
ejpam-4170	166	6	p	p	PROPN
ejpam-4170	166	7	⊕	⊕	PROPN
ejpam-4170	166	8	pf	pf	NOUN
ejpam-4170	166	9	s	s	PART
ejpam-4170	166	10	)	)	PUNCT
ejpam-4170	166	11	⋆	⋆	X
ejpam-4170	167	1	=	=	SYM
ejpam-4170	167	2	p	p	X
ejpam-4170	167	3	⋆	⋆	PUNCT
ejpam-4170	167	4	⊕	⊕	PROPN
ejpam-4170	167	5	s⋆.	s⋆.	NOUN
ejpam-4170	167	6	(	(	PUNCT
ejpam-4170	167	7	2	2	NUM
ejpam-4170	167	8	)	)	PUNCT
ejpam-4170	167	9	(	(	PUNCT
ejpam-4170	167	10	p	p	PROPN
ejpam-4170	167	11	⊕	⊕	PROPN
ejpam-4170	167	12	pf	pf	PROPN
ejpam-4170	167	13	s)⋆	s)⋆	PROPN
ejpam-4170	167	14	=	=	PUNCT
ejpam-4170	167	15	p⋆	p⋆	PRON
ejpam-4170	167	16	⊕	⊕	PROPN
ejpam-4170	167	17	s⋆.	s⋆.	NOUN
ejpam-4170	167	18	proof	proof	NOUN
ejpam-4170	167	19	.	.	PUNCT
ejpam-4170	168	1	(	(	PUNCT
ejpam-4170	168	2	1	1	X
ejpam-4170	168	3	)	)	PUNCT
ejpam-4170	168	4	since	since	SCONJ
ejpam-4170	168	5	p	p	NOUN
ejpam-4170	168	6	∩	∩	NOUN
ejpam-4170	168	7	s	s	PART
ejpam-4170	168	8	=	=	PUNCT
ejpam-4170	168	9	χpf0	χpf0	NOUN
ejpam-4170	168	10	,	,	PUNCT
ejpam-4170	168	11	we	we	PRON
ejpam-4170	168	12	need	need	VERB
ejpam-4170	168	13	to	to	PART
ejpam-4170	168	14	prove	prove	VERB
ejpam-4170	168	15	that	that	SCONJ
ejpam-4170	168	16	(	(	PUNCT
ejpam-4170	168	17	p	p	X
ejpam-4170	168	18	+	+	NOUN
ejpam-4170	168	19	s)⋆	s)⋆	PROPN
ejpam-4170	168	20	=	=	PUNCT
ejpam-4170	169	1	p	p	X
ejpam-4170	169	2	⋆	⋆	X
ejpam-4170	169	3	+	+	CCONJ
ejpam-4170	169	4	s⋆.	s⋆.	NOUN
ejpam-4170	169	5	suppose	suppose	VERB
ejpam-4170	169	6	that	that	SCONJ
ejpam-4170	169	7	m	m	VERB
ejpam-4170	169	8	∈	∈	PROPN
ejpam-4170	169	9	(	(	PUNCT
ejpam-4170	169	10	p	p	NOUN
ejpam-4170	169	11	+	+	X
ejpam-4170	169	12	s)⋆.	s)⋆.	VERB
ejpam-4170	169	13	by	by	ADP
ejpam-4170	169	14	definition	definition	NOUN
ejpam-4170	169	15	,	,	PUNCT
ejpam-4170	169	16	η2p+s(m	η2p+s(m	PROPN
ejpam-4170	169	17	)	)	PUNCT
ejpam-4170	169	18	>	>	X
ejpam-4170	169	19	0	0	X
ejpam-4170	169	20	.	.	PUNCT
ejpam-4170	170	1	this	this	PRON
ejpam-4170	170	2	implies	imply	VERB
ejpam-4170	170	3	that	that	SCONJ
ejpam-4170	170	4	0	0	PUNCT
ejpam-4170	170	5	<	<	X
ejpam-4170	170	6	η2p	η2p	X
ejpam-4170	170	7	(	(	PUNCT
ejpam-4170	170	8	m	m	NOUN
ejpam-4170	170	9	)	)	PUNCT
ejpam-4170	171	1	+	+	CCONJ
ejpam-4170	172	1	η2s(m)−	η2s(m)−	VERB
ejpam-4170	172	2	η2p	η2p	X
ejpam-4170	172	3	(	(	PUNCT
ejpam-4170	172	4	m)η2s(m	m)η2s(m	NOUN
ejpam-4170	172	5	)	)	PUNCT
ejpam-4170	172	6	=	=	PUNCT
ejpam-4170	172	7	η2p	η2p	X
ejpam-4170	172	8	(	(	PUNCT
ejpam-4170	172	9	m)(1−	m)(1−	PROPN
ejpam-4170	172	10	η2s(m	η2s(m	PROPN
ejpam-4170	172	11	)	)	PUNCT
ejpam-4170	172	12	)	)	PUNCT
ejpam-4170	173	1	+	+	CCONJ
ejpam-4170	173	2	η2s(m	η2s(m	PROPN
ejpam-4170	173	3	)	)	PUNCT
ejpam-4170	173	4	a.	a.	NOUN
ejpam-4170	173	5	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	173	6	/	/	SYM
ejpam-4170	173	7	eur	eur	PROPN
ejpam-4170	173	8	.	.	PUNCT
ejpam-4170	174	1	j.	j.	PROPN
ejpam-4170	174	2	pure	pure	PROPN
ejpam-4170	174	3	appl	appl	PROPN
ejpam-4170	174	4	.	.	PROPN
ejpam-4170	174	5	math	math	PROPN
ejpam-4170	174	6	,	,	PUNCT
ejpam-4170	174	7	15	15	NUM
ejpam-4170	174	8	(	(	PUNCT
ejpam-4170	174	9	1	1	NUM
ejpam-4170	174	10	)	)	PUNCT
ejpam-4170	174	11	(	(	PUNCT
ejpam-4170	174	12	2022	2022	NUM
ejpam-4170	174	13	)	)	PUNCT
ejpam-4170	174	14	,	,	PUNCT
ejpam-4170	174	15	36	36	NUM
ejpam-4170	174	16	-	-	SYM
ejpam-4170	174	17	46	46	NUM
ejpam-4170	174	18	43	43	NUM
ejpam-4170	174	19	which	which	PRON
ejpam-4170	174	20	means	mean	VERB
ejpam-4170	174	21	that	that	SCONJ
ejpam-4170	174	22	η2p	η2p	ADJ
ejpam-4170	174	23	(	(	PUNCT
ejpam-4170	174	24	m	m	NOUN
ejpam-4170	174	25	)	)	PUNCT
ejpam-4170	174	26	̸=	̸=	NOUN
ejpam-4170	174	27	0	0	NUM
ejpam-4170	174	28	or	or	CCONJ
ejpam-4170	174	29	η2s(m	η2s(m	PROPN
ejpam-4170	174	30	)	)	PUNCT
ejpam-4170	174	31	̸=	̸=	PROPN
ejpam-4170	174	32	0	0	NUM
ejpam-4170	174	33	.	.	PUNCT
ejpam-4170	175	1	moreover	moreover	ADV
ejpam-4170	175	2	,	,	PUNCT
ejpam-4170	175	3	1	1	NUM
ejpam-4170	175	4	>	>	X
ejpam-4170	175	5	η̂2p+s(m	η̂2p+s(m	PROPN
ejpam-4170	175	6	)	)	PUNCT
ejpam-4170	175	7	=	=	NOUN
ejpam-4170	175	8	η̂2p	η̂2p	ADJ
ejpam-4170	175	9	(	(	PUNCT
ejpam-4170	175	10	m)η̂2s(m	m)η̂2s(m	PROPN
ejpam-4170	175	11	)	)	PUNCT
ejpam-4170	175	12	which	which	PRON
ejpam-4170	175	13	implies	imply	VERB
ejpam-4170	175	14	that	that	SCONJ
ejpam-4170	175	15	η̂2p	η̂2p	ADJ
ejpam-4170	175	16	(	(	PUNCT
ejpam-4170	175	17	m	m	NOUN
ejpam-4170	175	18	)	)	PUNCT
ejpam-4170	175	19	<	<	X
ejpam-4170	175	20	1	1	NUM
ejpam-4170	175	21	or	or	CCONJ
ejpam-4170	175	22	η̂2s(m	η̂2s(m	NUM
ejpam-4170	175	23	)	)	PUNCT
ejpam-4170	175	24	<	<	X
ejpam-4170	175	25	1	1	X
ejpam-4170	175	26	.	.	PUNCT
ejpam-4170	175	27	thus	thus	ADV
ejpam-4170	175	28	m	m	NOUN
ejpam-4170	175	29	∈	∈	NOUN
ejpam-4170	175	30	p	p	NOUN
ejpam-4170	175	31	⋆	⋆	NOUN
ejpam-4170	175	32	or	or	CCONJ
ejpam-4170	175	33	m	m	PROPN
ejpam-4170	175	34	∈	∈	NOUN
ejpam-4170	175	35	s⋆	s⋆	NOUN
ejpam-4170	175	36	,	,	PUNCT
ejpam-4170	175	37	so	so	SCONJ
ejpam-4170	175	38	that	that	SCONJ
ejpam-4170	175	39	m	m	VERB
ejpam-4170	176	1	∈	∈	NOUN
ejpam-4170	176	2	p	p	X
ejpam-4170	176	3	⋆	⋆	X
ejpam-4170	176	4	+	+	CCONJ
ejpam-4170	176	5	s⋆.	s⋆.	NOUN
ejpam-4170	176	6	and	and	CCONJ
ejpam-4170	176	7	(	(	PUNCT
ejpam-4170	176	8	p	p	X
ejpam-4170	176	9	+	+	CCONJ
ejpam-4170	176	10	s)⋆	s)⋆	PROPN
ejpam-4170	176	11	⊆	⊆	NUM
ejpam-4170	176	12	p	p	NOUN
ejpam-4170	176	13	⋆	⋆	VERB
ejpam-4170	176	14	+	+	CCONJ
ejpam-4170	176	15	s⋆.	s⋆.	ADV
ejpam-4170	176	16	now	now	ADV
ejpam-4170	176	17	,	,	PUNCT
ejpam-4170	176	18	suppose	suppose	VERB
ejpam-4170	176	19	that	that	SCONJ
ejpam-4170	176	20	m	m	PROPN
ejpam-4170	176	21	=	=	SYM
ejpam-4170	176	22	a1	a1	NOUN
ejpam-4170	176	23	+	+	CCONJ
ejpam-4170	176	24	b1	b1	NOUN
ejpam-4170	176	25	∈	∈	PROPN
ejpam-4170	176	26	p	p	X
ejpam-4170	176	27	⋆	⋆	X
ejpam-4170	176	28	+	+	CCONJ
ejpam-4170	176	29	s⋆	s⋆	X
ejpam-4170	176	30	,	,	PUNCT
ejpam-4170	176	31	where	where	SCONJ
ejpam-4170	176	32	a1	a1	NOUN
ejpam-4170	176	33	∈	∈	PROPN
ejpam-4170	176	34	p	p	X
ejpam-4170	176	35	⋆	⋆	NOUN
ejpam-4170	176	36	and	and	CCONJ
ejpam-4170	176	37	b1	b1	PROPN
ejpam-4170	176	38	∈	∈	PROPN
ejpam-4170	176	39	s⋆.	s⋆.	NOUN
ejpam-4170	176	40	by	by	ADP
ejpam-4170	176	41	definition	definition	NOUN
ejpam-4170	176	42	,	,	PUNCT
ejpam-4170	176	43	η2p	η2p	X
ejpam-4170	176	44	(	(	PUNCT
ejpam-4170	176	45	a1	a1	PROPN
ejpam-4170	176	46	)	)	PUNCT
ejpam-4170	176	47	,	,	PUNCT
ejpam-4170	176	48	η	η	PROPN
ejpam-4170	176	49	2	2	NUM
ejpam-4170	176	50	s(b1	s(b1	NOUN
ejpam-4170	176	51	)	)	PUNCT
ejpam-4170	176	52	>	>	X
ejpam-4170	177	1	0	0	X
ejpam-4170	177	2	.	.	PUNCT
ejpam-4170	178	1	thus	thus	ADV
ejpam-4170	178	2	0	0	NUM
ejpam-4170	178	3	<	<	X
ejpam-4170	178	4	min{η2p	min{η2p	NOUN
ejpam-4170	178	5	(	(	PUNCT
ejpam-4170	178	6	a1)(1−	a1)(1−	NOUN
ejpam-4170	178	7	η2s(a1	η2s(a1	NUM
ejpam-4170	178	8	)	)	PUNCT
ejpam-4170	178	9	)	)	PUNCT
ejpam-4170	179	1	+	+	CCONJ
ejpam-4170	179	2	η2s(a1	η2s(a1	NOUN
ejpam-4170	179	3	)	)	PUNCT
ejpam-4170	179	4	,	,	PUNCT
ejpam-4170	179	5	η	η	PROPN
ejpam-4170	179	6	2	2	NUM
ejpam-4170	179	7	p	p	NOUN
ejpam-4170	179	8	(	(	PUNCT
ejpam-4170	179	9	b1)(1−	b1)(1−	ADJ
ejpam-4170	179	10	η2s(b1	η2s(b1	NOUN
ejpam-4170	179	11	)	)	PUNCT
ejpam-4170	179	12	)	)	PUNCT
ejpam-4170	180	1	+	+	CCONJ
ejpam-4170	180	2	η2s(b1	η2s(b1	NOUN
ejpam-4170	180	3	)	)	PUNCT
ejpam-4170	180	4	}	}	PUNCT
ejpam-4170	181	1	=	=	NOUN
ejpam-4170	181	2	min{η2p	min{η2p	NOUN
ejpam-4170	181	3	(	(	PUNCT
ejpam-4170	181	4	a1	a1	NOUN
ejpam-4170	181	5	)	)	PUNCT
ejpam-4170	181	6	+	+	NUM
ejpam-4170	181	7	η2s(a1)−	η2s(a1)−	NUM
ejpam-4170	181	8	η2p	η2p	VERB
ejpam-4170	181	9	(	(	PUNCT
ejpam-4170	181	10	a1)η	a1)η	ADJ
ejpam-4170	181	11	2	2	NUM
ejpam-4170	181	12	s(a1	s(a1	NOUN
ejpam-4170	181	13	)	)	PUNCT
ejpam-4170	181	14	,	,	PUNCT
ejpam-4170	181	15	η	η	PROPN
ejpam-4170	181	16	2	2	NUM
ejpam-4170	181	17	p	p	NOUN
ejpam-4170	181	18	(	(	PUNCT
ejpam-4170	181	19	b1	b1	NOUN
ejpam-4170	181	20	)	)	PUNCT
ejpam-4170	181	21	+	+	NUM
ejpam-4170	181	22	η2s(b1)−	η2s(b1)−	NOUN
ejpam-4170	181	23	η2p	η2p	NOUN
ejpam-4170	181	24	(	(	PUNCT
ejpam-4170	181	25	b1)η	b1)η	PROPN
ejpam-4170	181	26	2	2	NUM
ejpam-4170	181	27	s(b1	s(b1	NOUN
ejpam-4170	181	28	)	)	PUNCT
ejpam-4170	181	29	}	}	PUNCT
ejpam-4170	181	30	≤η2p	≤η2p	PUNCT
ejpam-4170	181	31	(	(	PUNCT
ejpam-4170	181	32	m	m	NOUN
ejpam-4170	181	33	)	)	PUNCT
ejpam-4170	181	34	+	+	CCONJ
ejpam-4170	181	35	η2s(m)−	η2s(m)−	VERB
ejpam-4170	181	36	η2p	η2p	X
ejpam-4170	181	37	(	(	PUNCT
ejpam-4170	181	38	m)η2s(m	m)η2s(m	NOUN
ejpam-4170	181	39	)	)	PUNCT
ejpam-4170	181	40	=	=	SYM
ejpam-4170	181	41	η2p+s(m	η2p+s(m	PROPN
ejpam-4170	181	42	)	)	PUNCT
ejpam-4170	181	43	moreover	moreover	ADV
ejpam-4170	181	44	,	,	PUNCT
ejpam-4170	181	45	η̂2p	η̂2p	ADJ
ejpam-4170	181	46	(	(	PUNCT
ejpam-4170	181	47	a1	a1	NOUN
ejpam-4170	181	48	)	)	PUNCT
ejpam-4170	181	49	,	,	PUNCT
ejpam-4170	181	50	η̂	η̂	NUM
ejpam-4170	181	51	2	2	NUM
ejpam-4170	181	52	s(b1	s(b1	NOUN
ejpam-4170	181	53	)	)	PUNCT
ejpam-4170	181	54	<	<	X
ejpam-4170	181	55	1	1	NUM
ejpam-4170	181	56	which	which	PRON
ejpam-4170	181	57	implies	imply	VERB
ejpam-4170	181	58	that	that	SCONJ
ejpam-4170	181	59	1	1	X
ejpam-4170	181	60	>	>	X
ejpam-4170	181	61	max{η̂2p	max{η̂2p	PROPN
ejpam-4170	181	62	(	(	PUNCT
ejpam-4170	181	63	a1)η̂2s(a1	a1)η̂2s(a1	PROPN
ejpam-4170	181	64	)	)	PUNCT
ejpam-4170	181	65	,	,	PUNCT
ejpam-4170	181	66	η̂2p	η̂2p	NUM
ejpam-4170	181	67	(	(	PUNCT
ejpam-4170	181	68	b1)η̂2s(b1	b1)η̂2s(b1	NOUN
ejpam-4170	181	69	)	)	PUNCT
ejpam-4170	181	70	}	}	PUNCT
ejpam-4170	181	71	≥η̂2p	≥η̂2p	NOUN
ejpam-4170	181	72	(	(	PUNCT
ejpam-4170	181	73	m)η̂2s(m	m)η̂2s(m	PROPN
ejpam-4170	181	74	)	)	PUNCT
ejpam-4170	181	75	=	=	SYM
ejpam-4170	181	76	η̂2p+s(m	η̂2p+s(m	PROPN
ejpam-4170	181	77	)	)	PUNCT
ejpam-4170	181	78	thus	thus	ADV
ejpam-4170	181	79	m	m	NOUN
ejpam-4170	181	80	∈	∈	ADJ
ejpam-4170	181	81	(	(	PUNCT
ejpam-4170	181	82	p	p	NOUN
ejpam-4170	181	83	+	+	X
ejpam-4170	181	84	s)⋆.	s)⋆.	PROPN
ejpam-4170	181	85	then	then	ADV
ejpam-4170	181	86	p	p	X
ejpam-4170	181	87	⋆	⋆	VERB
ejpam-4170	181	88	+	+	CCONJ
ejpam-4170	181	89	s⋆	s⋆	NUM
ejpam-4170	181	90	⊆	⊆	NUM
ejpam-4170	181	91	(	(	PUNCT
ejpam-4170	181	92	p	p	NOUN
ejpam-4170	181	93	+	+	CCONJ
ejpam-4170	181	94	s)⋆	s)⋆	PROPN
ejpam-4170	181	95	and	and	CCONJ
ejpam-4170	181	96	therefore	therefore	ADV
ejpam-4170	181	97	,	,	PUNCT
ejpam-4170	181	98	the	the	DET
ejpam-4170	181	99	equality	equality	NOUN
ejpam-4170	181	100	holds	hold	VERB
ejpam-4170	181	101	.	.	PUNCT
ejpam-4170	182	1	(	(	PUNCT
ejpam-4170	182	2	2	2	NUM
ejpam-4170	182	3	)	)	PUNCT
ejpam-4170	182	4	since	since	SCONJ
ejpam-4170	182	5	p	p	NOUN
ejpam-4170	182	6	∩	∩	NOUN
ejpam-4170	182	7	s	s	PART
ejpam-4170	182	8	=	=	PUNCT
ejpam-4170	182	9	χpf0	χpf0	NOUN
ejpam-4170	182	10	,	,	PUNCT
ejpam-4170	182	11	we	we	PRON
ejpam-4170	182	12	need	need	VERB
ejpam-4170	182	13	to	to	PART
ejpam-4170	182	14	prove	prove	VERB
ejpam-4170	182	15	that	that	SCONJ
ejpam-4170	182	16	(	(	PUNCT
ejpam-4170	182	17	p	p	X
ejpam-4170	182	18	+	+	NOUN
ejpam-4170	182	19	s)⋆	s)⋆	NOUN
ejpam-4170	182	20	=	=	PUNCT
ejpam-4170	182	21	p⋆	p⋆	NOUN
ejpam-4170	182	22	+	+	CCONJ
ejpam-4170	182	23	s⋆.	s⋆.	NOUN
ejpam-4170	182	24	suppose	suppose	VERB
ejpam-4170	182	25	that	that	SCONJ
ejpam-4170	182	26	m	m	VERB
ejpam-4170	182	27	∈	∈	PROPN
ejpam-4170	182	28	(	(	PUNCT
ejpam-4170	182	29	p	p	NOUN
ejpam-4170	182	30	+	+	X
ejpam-4170	182	31	s)⋆.	s)⋆.	VERB
ejpam-4170	182	32	by	by	ADP
ejpam-4170	182	33	definition	definition	NOUN
ejpam-4170	182	34	,	,	PUNCT
ejpam-4170	182	35	η2p+s(m	η2p+s(m	PROPN
ejpam-4170	182	36	)	)	PUNCT
ejpam-4170	183	1	=	=	PUNCT
ejpam-4170	184	1	1	1	X
ejpam-4170	184	2	.	.	PUNCT
ejpam-4170	184	3	this	this	PRON
ejpam-4170	184	4	implies	imply	VERB
ejpam-4170	184	5	that	that	SCONJ
ejpam-4170	184	6	1	1	X
ejpam-4170	184	7	=	=	SYM
ejpam-4170	184	8	η2p	η2p	X
ejpam-4170	184	9	(	(	PUNCT
ejpam-4170	184	10	m	m	NOUN
ejpam-4170	184	11	)	)	PUNCT
ejpam-4170	184	12	+	+	CCONJ
ejpam-4170	184	13	η2s(m)−	η2s(m)−	VERB
ejpam-4170	184	14	η2p	η2p	X
ejpam-4170	184	15	(	(	PUNCT
ejpam-4170	184	16	m)η2s(m	m)η2s(m	NOUN
ejpam-4170	184	17	)	)	PUNCT
ejpam-4170	184	18	=	=	PUNCT
ejpam-4170	184	19	η2p	η2p	X
ejpam-4170	184	20	(	(	PUNCT
ejpam-4170	184	21	m)(1−	m)(1−	PROPN
ejpam-4170	184	22	η2s(m	η2s(m	PROPN
ejpam-4170	184	23	)	)	PUNCT
ejpam-4170	184	24	)	)	PUNCT
ejpam-4170	185	1	+	+	CCONJ
ejpam-4170	185	2	η2s(m	η2s(m	PROPN
ejpam-4170	185	3	)	)	PUNCT
ejpam-4170	185	4	which	which	PRON
ejpam-4170	185	5	means	mean	VERB
ejpam-4170	185	6	that	that	SCONJ
ejpam-4170	185	7	η2p	η2p	ADJ
ejpam-4170	185	8	(	(	PUNCT
ejpam-4170	185	9	m	m	NOUN
ejpam-4170	185	10	)	)	PUNCT
ejpam-4170	185	11	=	=	SYM
ejpam-4170	185	12	1	1	NUM
ejpam-4170	185	13	or	or	CCONJ
ejpam-4170	185	14	η2s(m	η2s(m	PROPN
ejpam-4170	185	15	)	)	PUNCT
ejpam-4170	185	16	=	=	SYM
ejpam-4170	186	1	1	1	X
ejpam-4170	186	2	.	.	PUNCT
ejpam-4170	186	3	moreover	moreover	ADV
ejpam-4170	186	4	,	,	PUNCT
ejpam-4170	186	5	0	0	NUM
ejpam-4170	186	6	=	=	SYM
ejpam-4170	186	7	η̂2p+s(m	η̂2p+s(m	PROPN
ejpam-4170	186	8	)	)	PUNCT
ejpam-4170	186	9	=	=	NOUN
ejpam-4170	186	10	η̂2p	η̂2p	ADJ
ejpam-4170	186	11	(	(	PUNCT
ejpam-4170	186	12	m)η̂2s(m	m)η̂2s(m	PROPN
ejpam-4170	186	13	)	)	PUNCT
ejpam-4170	186	14	which	which	PRON
ejpam-4170	186	15	implies	imply	VERB
ejpam-4170	186	16	that	that	SCONJ
ejpam-4170	186	17	η̂2p	η̂2p	ADJ
ejpam-4170	186	18	(	(	PUNCT
ejpam-4170	186	19	m	m	NOUN
ejpam-4170	186	20	)	)	PUNCT
ejpam-4170	186	21	=	=	SYM
ejpam-4170	186	22	0	0	NUM
ejpam-4170	186	23	or	or	CCONJ
ejpam-4170	186	24	η̂2s(m	η̂2s(m	PROPN
ejpam-4170	186	25	)	)	PUNCT
ejpam-4170	187	1	=	=	SYM
ejpam-4170	187	2	0	0	X
ejpam-4170	187	3	.	.	PUNCT
ejpam-4170	188	1	thus	thus	ADV
ejpam-4170	188	2	m	m	ADP
ejpam-4170	188	3	∈	∈	NOUN
ejpam-4170	188	4	p⋆	p⋆	NOUN
ejpam-4170	188	5	or	or	CCONJ
ejpam-4170	188	6	m	m	PROPN
ejpam-4170	188	7	∈	∈	NOUN
ejpam-4170	188	8	s⋆	s⋆	NOUN
ejpam-4170	188	9	,	,	PUNCT
ejpam-4170	188	10	so	so	SCONJ
ejpam-4170	188	11	that	that	SCONJ
ejpam-4170	188	12	m	m	VERB
ejpam-4170	188	13	∈	∈	NOUN
ejpam-4170	188	14	p⋆	p⋆	NOUN
ejpam-4170	188	15	+	+	CCONJ
ejpam-4170	188	16	s⋆	s⋆	ADJ
ejpam-4170	188	17	and	and	CCONJ
ejpam-4170	188	18	(	(	PUNCT
ejpam-4170	188	19	p	p	X
ejpam-4170	188	20	+	+	X
ejpam-4170	188	21	s)⋆	s)⋆	PROPN
ejpam-4170	188	22	⊆	⊆	NUM
ejpam-4170	188	23	p⋆	p⋆	NOUN
ejpam-4170	188	24	+	+	CCONJ
ejpam-4170	188	25	s⋆.	s⋆.	ADV
ejpam-4170	188	26	now	now	ADV
ejpam-4170	188	27	,	,	PUNCT
ejpam-4170	188	28	suppose	suppose	VERB
ejpam-4170	188	29	that	that	SCONJ
ejpam-4170	188	30	m	m	PROPN
ejpam-4170	188	31	=	=	SYM
ejpam-4170	188	32	a1	a1	NOUN
ejpam-4170	188	33	+	+	CCONJ
ejpam-4170	188	34	b1	b1	NOUN
ejpam-4170	188	35	∈	∈	NOUN
ejpam-4170	188	36	p⋆	p⋆	NOUN
ejpam-4170	188	37	+	+	CCONJ
ejpam-4170	188	38	s⋆	s⋆	X
ejpam-4170	188	39	,	,	PUNCT
ejpam-4170	188	40	where	where	SCONJ
ejpam-4170	188	41	a1	a1	NOUN
ejpam-4170	188	42	∈	∈	PROPN
ejpam-4170	188	43	p⋆	p⋆	NOUN
ejpam-4170	188	44	and	and	CCONJ
ejpam-4170	188	45	b1	b1	PROPN
ejpam-4170	188	46	∈	∈	PROPN
ejpam-4170	188	47	s⋆.	s⋆.	NOUN
ejpam-4170	188	48	by	by	ADP
ejpam-4170	188	49	definition	definition	NOUN
ejpam-4170	188	50	,	,	PUNCT
ejpam-4170	188	51	η2p	η2p	X
ejpam-4170	188	52	(	(	PUNCT
ejpam-4170	188	53	a1	a1	PROPN
ejpam-4170	188	54	)	)	PUNCT
ejpam-4170	188	55	,	,	PUNCT
ejpam-4170	188	56	η	η	PROPN
ejpam-4170	188	57	2	2	NUM
ejpam-4170	188	58	s(b1	s(b1	NOUN
ejpam-4170	188	59	)	)	PUNCT
ejpam-4170	188	60	=	=	SYM
ejpam-4170	189	1	1	1	X
ejpam-4170	189	2	.	.	PUNCT
ejpam-4170	189	3	thus	thus	ADV
ejpam-4170	189	4	1	1	NUM
ejpam-4170	189	5	=	=	NOUN
ejpam-4170	189	6	min{η2p	min{η2p	NOUN
ejpam-4170	189	7	(	(	PUNCT
ejpam-4170	189	8	a1	a1	NOUN
ejpam-4170	189	9	)	)	PUNCT
ejpam-4170	189	10	+	+	NUM
ejpam-4170	189	11	η2s(a1)−	η2s(a1)−	NUM
ejpam-4170	189	12	η2p	η2p	VERB
ejpam-4170	189	13	(	(	PUNCT
ejpam-4170	189	14	a1)η	a1)η	ADJ
ejpam-4170	189	15	2	2	NUM
ejpam-4170	189	16	s(a1	s(a1	NOUN
ejpam-4170	189	17	)	)	PUNCT
ejpam-4170	189	18	,	,	PUNCT
ejpam-4170	189	19	η	η	PROPN
ejpam-4170	189	20	2	2	NUM
ejpam-4170	189	21	p	p	NOUN
ejpam-4170	189	22	(	(	PUNCT
ejpam-4170	189	23	b1	b1	NOUN
ejpam-4170	189	24	)	)	PUNCT
ejpam-4170	189	25	+	+	NUM
ejpam-4170	189	26	η2s(b1)−	η2s(b1)−	NOUN
ejpam-4170	189	27	η2p	η2p	NOUN
ejpam-4170	189	28	(	(	PUNCT
ejpam-4170	189	29	b1)η	b1)η	PROPN
ejpam-4170	189	30	2	2	NUM
ejpam-4170	189	31	s(b1	s(b1	NOUN
ejpam-4170	189	32	)	)	PUNCT
ejpam-4170	189	33	}	}	PUNCT
ejpam-4170	189	34	≤η2p	≤η2p	PUNCT
ejpam-4170	189	35	(	(	PUNCT
ejpam-4170	189	36	m	m	NOUN
ejpam-4170	189	37	)	)	PUNCT
ejpam-4170	190	1	+	+	CCONJ
ejpam-4170	190	2	η2s(m)−	η2s(m)−	VERB
ejpam-4170	190	3	η2p	η2p	X
ejpam-4170	190	4	(	(	PUNCT
ejpam-4170	190	5	m)η2s(m	m)η2s(m	NOUN
ejpam-4170	190	6	)	)	PUNCT
ejpam-4170	190	7	=	=	SYM
ejpam-4170	190	8	η2p+s(m	η2p+s(m	PROPN
ejpam-4170	190	9	)	)	PUNCT
ejpam-4170	190	10	moreover	moreover	ADV
ejpam-4170	190	11	,	,	PUNCT
ejpam-4170	190	12	η̂2p	η̂2p	ADJ
ejpam-4170	190	13	(	(	PUNCT
ejpam-4170	190	14	a1	a1	NOUN
ejpam-4170	190	15	)	)	PUNCT
ejpam-4170	190	16	,	,	PUNCT
ejpam-4170	190	17	η̂	η̂	NUM
ejpam-4170	190	18	2	2	NUM
ejpam-4170	190	19	s(b1	s(b1	NOUN
ejpam-4170	190	20	)	)	PUNCT
ejpam-4170	190	21	=	=	SYM
ejpam-4170	190	22	0	0	NUM
ejpam-4170	190	23	which	which	PRON
ejpam-4170	190	24	implies	imply	VERB
ejpam-4170	190	25	that	that	SCONJ
ejpam-4170	190	26	0	0	NUM
ejpam-4170	191	1	=	=	ADJ
ejpam-4170	191	2	max{η̂2p	max{η̂2p	NOUN
ejpam-4170	191	3	(	(	PUNCT
ejpam-4170	191	4	a1)η̂2s(a1	a1)η̂2s(a1	PROPN
ejpam-4170	191	5	)	)	PUNCT
ejpam-4170	191	6	,	,	PUNCT
ejpam-4170	191	7	η̂2p	η̂2p	NUM
ejpam-4170	191	8	(	(	PUNCT
ejpam-4170	191	9	b1)η̂2s(b1	b1)η̂2s(b1	NOUN
ejpam-4170	191	10	)	)	PUNCT
ejpam-4170	191	11	}	}	PUNCT
ejpam-4170	191	12	≥η̂2p	≥η̂2p	NOUN
ejpam-4170	191	13	(	(	PUNCT
ejpam-4170	191	14	m)η̂2s(m	m)η̂2s(m	PROPN
ejpam-4170	191	15	)	)	PUNCT
ejpam-4170	191	16	=	=	SYM
ejpam-4170	191	17	η̂2p+s(m	η̂2p+s(m	PROPN
ejpam-4170	191	18	)	)	PUNCT
ejpam-4170	191	19	thus	thus	ADV
ejpam-4170	191	20	m	m	NOUN
ejpam-4170	191	21	∈	∈	ADJ
ejpam-4170	191	22	(	(	PUNCT
ejpam-4170	191	23	p	p	NOUN
ejpam-4170	191	24	+	+	X
ejpam-4170	191	25	s)⋆.	s)⋆.	PROPN
ejpam-4170	191	26	then	then	ADV
ejpam-4170	191	27	p⋆	p⋆	NOUN
ejpam-4170	191	28	+	+	CCONJ
ejpam-4170	191	29	s⋆	s⋆	NUM
ejpam-4170	191	30	⊆	⊆	NUM
ejpam-4170	191	31	(	(	PUNCT
ejpam-4170	191	32	p	p	NOUN
ejpam-4170	191	33	+	+	CCONJ
ejpam-4170	191	34	s)⋆	s)⋆	PROPN
ejpam-4170	191	35	and	and	CCONJ
ejpam-4170	191	36	therefore	therefore	ADV
ejpam-4170	191	37	,	,	PUNCT
ejpam-4170	191	38	the	the	DET
ejpam-4170	191	39	equality	equality	NOUN
ejpam-4170	191	40	holds	hold	VERB
ejpam-4170	191	41	.	.	PUNCT
ejpam-4170	192	1	a.	a.	PROPN
ejpam-4170	192	2	alhumaimeed	alhumaimeed	PROPN
ejpam-4170	192	3	/	/	SYM
ejpam-4170	192	4	eur	eur	PROPN
ejpam-4170	192	5	.	.	PUNCT
ejpam-4170	193	1	j.	j.	PROPN
ejpam-4170	193	2	pure	pure	PROPN
ejpam-4170	193	3	appl	appl	PROPN
ejpam-4170	193	4	.	.	PROPN
ejpam-4170	193	5	math	math	PROPN
ejpam-4170	193	6	,	,	PUNCT
ejpam-4170	193	7	15	15	NUM
ejpam-4170	193	8	(	(	PUNCT
ejpam-4170	193	9	1	1	NUM
ejpam-4170	193	10	)	)	PUNCT
ejpam-4170	193	11	(	(	PUNCT
ejpam-4170	193	12	2022	2022	NUM
ejpam-4170	193	13	)	)	PUNCT
ejpam-4170	193	14	,	,	PUNCT
ejpam-4170	193	15	36	36	NUM
ejpam-4170	193	16	-	-	SYM
ejpam-4170	193	17	46	46	NUM
ejpam-4170	193	18	44	44	NUM
ejpam-4170	193	19	4	4	NUM
ejpam-4170	193	20	.	.	PUNCT
ejpam-4170	194	1	homomorphism	homomorphism	NOUN
ejpam-4170	194	2	let	let	VERB
ejpam-4170	194	3	p	p	PRON
ejpam-4170	194	4	,	,	PUNCT
ejpam-4170	194	5	s	s	AUX
ejpam-4170	194	6	be	be	AUX
ejpam-4170	194	7	two	two	NUM
ejpam-4170	194	8	r	r	NOUN
ejpam-4170	194	9	-	-	PUNCT
ejpam-4170	194	10	modules	module	NOUN
ejpam-4170	194	11	,	,	PUNCT
ejpam-4170	194	12	l	l	NOUN
ejpam-4170	194	13	≤pf	≤pf	NOUN
ejpam-4170	194	14	p	p	NOUN
ejpam-4170	194	15	and	and	CCONJ
ejpam-4170	194	16	n	n	PRON
ejpam-4170	194	17	≤pf	≤pf	NOUN
ejpam-4170	194	18	s.	s.	PROPN
ejpam-4170	194	19	consider	consider	VERB
ejpam-4170	194	20	an	an	DET
ejpam-4170	194	21	r	r	NOUN
ejpam-4170	194	22	-	-	PUNCT
ejpam-4170	194	23	homomorphism	homomorphism	NOUN
ejpam-4170	194	24	ψ	ψ	X
ejpam-4170	194	25	:	:	PUNCT
ejpam-4170	194	26	p	p	X
ejpam-4170	194	27	−→	−→	NOUN
ejpam-4170	194	28	s	s	VERB
ejpam-4170	194	29	for	for	ADP
ejpam-4170	194	30	s	s	PROPN
ejpam-4170	194	31	∈	∈	PROPN
ejpam-4170	194	32	s	s	PART
ejpam-4170	194	33	,	,	PUNCT
ejpam-4170	194	34	we	we	PRON
ejpam-4170	194	35	define	define	VERB
ejpam-4170	194	36	:	:	PUNCT
ejpam-4170	194	37	ηψ(l)(s	ηψ(l)(s	NUM
ejpam-4170	194	38	)	)	PUNCT
ejpam-4170	195	1	=	=	PRON
ejpam-4170	195	2	{	{	PUNCT
ejpam-4170	195	3	max{ηl(p	max{ηl(p	PROPN
ejpam-4170	195	4	)	)	PUNCT
ejpam-4170	195	5	:	:	PUNCT
ejpam-4170	195	6	s	s	X
ejpam-4170	195	7	=	=	PUNCT
ejpam-4170	195	8	ψ(p	ψ(p	NOUN
ejpam-4170	195	9	)	)	PUNCT
ejpam-4170	195	10	}	}	PUNCT
ejpam-4170	195	11	if	if	SCONJ
ejpam-4170	195	12	s	s	X
ejpam-4170	195	13	∈	∈	PROPN
ejpam-4170	195	14	im(ψ	im(ψ	NOUN
ejpam-4170	195	15	)	)	PUNCT
ejpam-4170	195	16	0	0	PUNCT
ejpam-4170	196	1	otherwise	otherwise	ADV
ejpam-4170	196	2	and	and	CCONJ
ejpam-4170	196	3	η̂ψ(l)(s	η̂ψ(l)(s	NOUN
ejpam-4170	196	4	)	)	PUNCT
ejpam-4170	196	5	=	=	PRON
ejpam-4170	196	6	{	{	PUNCT
ejpam-4170	196	7	min{ηl(p	min{ηl(p	NOUN
ejpam-4170	196	8	)	)	PUNCT
ejpam-4170	196	9	:	:	PUNCT
ejpam-4170	197	1	s	s	X
ejpam-4170	197	2	=	=	PUNCT
ejpam-4170	197	3	ψ(p	ψ(p	NOUN
ejpam-4170	197	4	)	)	PUNCT
ejpam-4170	197	5	}	}	PUNCT
ejpam-4170	197	6	if	if	SCONJ
ejpam-4170	197	7	s	s	X
ejpam-4170	197	8	∈	∈	PROPN
ejpam-4170	197	9	im(ψ	im(ψ	NOUN
ejpam-4170	197	10	)	)	PUNCT
ejpam-4170	197	11	1	1	NUM
ejpam-4170	197	12	otherwise	otherwise	ADV
ejpam-4170	197	13	now	now	ADV
ejpam-4170	197	14	,	,	PUNCT
ejpam-4170	197	15	we	we	PRON
ejpam-4170	197	16	are	be	AUX
ejpam-4170	197	17	ready	ready	ADJ
ejpam-4170	197	18	to	to	PART
ejpam-4170	197	19	prove	prove	VERB
ejpam-4170	197	20	the	the	DET
ejpam-4170	197	21	following	following	NOUN
ejpam-4170	197	22	:	:	PUNCT
ejpam-4170	197	23	theorem	theorem	NOUN
ejpam-4170	197	24	7	7	NUM
ejpam-4170	197	25	.	.	PUNCT
ejpam-4170	198	1	let	let	VERB
ejpam-4170	198	2	ψ	ψ	X
ejpam-4170	198	3	:	:	PUNCT
ejpam-4170	198	4	p	p	X
ejpam-4170	198	5	−→	−→	NOUN
ejpam-4170	198	6	s	s	AUX
ejpam-4170	198	7	be	be	AUX
ejpam-4170	198	8	a	a	DET
ejpam-4170	198	9	monomorphism	monomorphism	NOUN
ejpam-4170	198	10	of	of	ADP
ejpam-4170	198	11	modules	module	NOUN
ejpam-4170	198	12	.	.	PUNCT
ejpam-4170	199	1	if	if	SCONJ
ejpam-4170	199	2	t	t	PROPN
ejpam-4170	199	3	is	be	AUX
ejpam-4170	199	4	a	a	DET
ejpam-4170	199	5	pythagorean	pythagorean	ADJ
ejpam-4170	199	6	fuzzy	fuzzy	ADJ
ejpam-4170	199	7	small	small	ADJ
ejpam-4170	199	8	submodule	submodule	NOUN
ejpam-4170	199	9	of	of	ADP
ejpam-4170	199	10	p	p	PROPN
ejpam-4170	199	11	,	,	PUNCT
ejpam-4170	199	12	then	then	ADV
ejpam-4170	199	13	ψ(t	ψ(t	PROPN
ejpam-4170	199	14	)	)	PUNCT
ejpam-4170	199	15	is	be	AUX
ejpam-4170	199	16	a	a	DET
ejpam-4170	199	17	pythagorean	pythagorean	ADJ
ejpam-4170	199	18	fuzzy	fuzzy	ADJ
ejpam-4170	199	19	small	small	ADJ
ejpam-4170	199	20	submodule	submodule	NOUN
ejpam-4170	199	21	of	of	ADP
ejpam-4170	199	22	s.	s.	PROPN
ejpam-4170	199	23	proof	proof	PROPN
ejpam-4170	199	24	.	.	PUNCT
ejpam-4170	200	1	suppose	suppose	VERB
ejpam-4170	200	2	that	that	SCONJ
ejpam-4170	200	3	ψ(t	ψ(t	PROPN
ejpam-4170	200	4	)	)	PUNCT
ejpam-4170	201	1	+	+	NUM
ejpam-4170	201	2	l	l	NOUN
ejpam-4170	201	3	=	=	NOUN
ejpam-4170	201	4	χpfs	χpf	NOUN
ejpam-4170	201	5	.	.	PUNCT
ejpam-4170	202	1	we	we	PRON
ejpam-4170	202	2	aim	aim	VERB
ejpam-4170	202	3	to	to	PART
ejpam-4170	202	4	prove	prove	VERB
ejpam-4170	202	5	that	that	PRON
ejpam-4170	202	6	l	l	NOUN
ejpam-4170	202	7	=	=	NOUN
ejpam-4170	202	8	χpfs	χpf	NOUN
ejpam-4170	202	9	.	.	PUNCT
ejpam-4170	203	1	let	let	VERB
ejpam-4170	203	2	s	s	PRON
ejpam-4170	203	3	∈	∈	VERB
ejpam-4170	203	4	s	s	NOUN
ejpam-4170	203	5	,	,	PUNCT
ejpam-4170	203	6	then	then	ADV
ejpam-4170	203	7	1	1	NUM
ejpam-4170	203	8	=	=	NOUN
ejpam-4170	203	9	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	203	10	)	)	PUNCT
ejpam-4170	204	1	+	+	SYM
ejpam-4170	204	2	l(s	l(s	X
ejpam-4170	204	3	)	)	PUNCT
ejpam-4170	205	1	=	=	VERB
ejpam-4170	205	2	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	205	3	)	)	PUNCT
ejpam-4170	205	4	(	(	PUNCT
ejpam-4170	205	5	s	s	X
ejpam-4170	205	6	)	)	PUNCT
ejpam-4170	205	7	+	+	CCONJ
ejpam-4170	205	8	η2l(s)−	η2l(s)−	PROPN
ejpam-4170	205	9	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	205	10	)	)	PUNCT
ejpam-4170	205	11	(	(	PUNCT
ejpam-4170	205	12	s)η	s)η	NOUN
ejpam-4170	205	13	2	2	NUM
ejpam-4170	205	14	l(s	l(s	PROPN
ejpam-4170	205	15	)	)	PUNCT
ejpam-4170	205	16	in	in	ADP
ejpam-4170	205	17	the	the	DET
ejpam-4170	205	18	case	case	NOUN
ejpam-4170	205	19	that	that	PRON
ejpam-4170	205	20	s	s	VERB
ejpam-4170	205	21	/∈	/∈	PUNCT
ejpam-4170	205	22	im(ψ	im(ψ	NOUN
ejpam-4170	205	23	)	)	PUNCT
ejpam-4170	205	24	,	,	PUNCT
ejpam-4170	205	25	we	we	PRON
ejpam-4170	205	26	obtain	obtain	VERB
ejpam-4170	205	27	1	1	NUM
ejpam-4170	205	28	=	=	NOUN
ejpam-4170	205	29	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	205	30	)	)	PUNCT
ejpam-4170	205	31	(	(	PUNCT
ejpam-4170	205	32	s	s	X
ejpam-4170	205	33	)	)	PUNCT
ejpam-4170	205	34	+	+	CCONJ
ejpam-4170	205	35	η2l(s)−	η2l(s)−	PROPN
ejpam-4170	205	36	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	205	37	)	)	PUNCT
ejpam-4170	205	38	(	(	PUNCT
ejpam-4170	205	39	s)η	s)η	NOUN
ejpam-4170	205	40	2	2	NUM
ejpam-4170	205	41	l(s	l(s	PROPN
ejpam-4170	205	42	)	)	PUNCT
ejpam-4170	205	43	=	=	SYM
ejpam-4170	205	44	η2l(s	η2l(s	PROPN
ejpam-4170	205	45	)	)	PUNCT
ejpam-4170	205	46	and	and	CCONJ
ejpam-4170	205	47	so	so	ADV
ejpam-4170	205	48	1	1	NUM
ejpam-4170	205	49	=	=	SYM
ejpam-4170	205	50	η2l(s	η2l(s	PROPN
ejpam-4170	205	51	)	)	PUNCT
ejpam-4170	205	52	and	and	CCONJ
ejpam-4170	205	53	η̂	η̂	SYM
ejpam-4170	205	54	2	2	NUM
ejpam-4170	205	55	l(s	l(s	PROPN
ejpam-4170	205	56	)	)	PUNCT
ejpam-4170	205	57	=	=	SYM
ejpam-4170	206	1	0	0	X
ejpam-4170	206	2	.	.	PUNCT
ejpam-4170	207	1	if	if	SCONJ
ejpam-4170	207	2	s	s	X
ejpam-4170	207	3	∈	∈	PROPN
ejpam-4170	207	4	im(ψ	im(ψ	NOUN
ejpam-4170	207	5	)	)	PUNCT
ejpam-4170	207	6	,	,	PUNCT
ejpam-4170	207	7	we	we	PRON
ejpam-4170	207	8	have	have	VERB
ejpam-4170	207	9	1	1	NUM
ejpam-4170	207	10	=	=	NOUN
ejpam-4170	207	11	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	207	12	)	)	PUNCT
ejpam-4170	207	13	(	(	PUNCT
ejpam-4170	207	14	s	s	X
ejpam-4170	207	15	)	)	PUNCT
ejpam-4170	208	1	+	+	CCONJ
ejpam-4170	208	2	η2l(s)−	η2l(s)−	PROPN
ejpam-4170	208	3	η2ψ(t	η2ψ(t	PROPN
ejpam-4170	208	4	)	)	PUNCT
ejpam-4170	208	5	(	(	PUNCT
ejpam-4170	208	6	s)η	s)η	NOUN
ejpam-4170	208	7	2	2	NUM
ejpam-4170	208	8	l(s	l(s	PROPN
ejpam-4170	208	9	)	)	PUNCT
ejpam-4170	209	1	=	=	SYM
ejpam-4170	209	2	max{η2	max{η2	X
ejpam-4170	209	3	t	t	NOUN
ejpam-4170	209	4	(	(	PUNCT
ejpam-4170	209	5	p	p	NOUN
ejpam-4170	209	6	)	)	PUNCT
ejpam-4170	209	7	:	:	PUNCT
ejpam-4170	209	8	ψ(p	ψ(p	NOUN
ejpam-4170	209	9	)	)	PUNCT
ejpam-4170	209	10	=	=	PUNCT
ejpam-4170	209	11	s}+	s}+	VERB
ejpam-4170	209	12	η2l(s)−max{η2	η2l(s)−max{η2	PROPN
ejpam-4170	209	13	t	t	PROPN
ejpam-4170	209	14	(	(	PUNCT
ejpam-4170	209	15	p	p	NOUN
ejpam-4170	209	16	)	)	PUNCT
ejpam-4170	209	17	:	:	PUNCT
ejpam-4170	209	18	ψ(p	ψ(p	PROPN
ejpam-4170	209	19	)	)	PUNCT
ejpam-4170	209	20	=	=	SYM
ejpam-4170	209	21	s}η2l(s	s}η2l(	NOUN
ejpam-4170	209	22	)	)	PUNCT
ejpam-4170	209	23	=	=	NOUN
ejpam-4170	209	24	η2	η2	X
ejpam-4170	209	25	t	t	NOUN
ejpam-4170	209	26	(	(	PUNCT
ejpam-4170	209	27	p	p	NOUN
ejpam-4170	209	28	)	)	PUNCT
ejpam-4170	209	29	+	+	CCONJ
ejpam-4170	209	30	η2l(s)−	η2l(s)−	NUM
ejpam-4170	209	31	η2	η2	PROPN
ejpam-4170	209	32	t	t	NOUN
ejpam-4170	209	33	(	(	PUNCT
ejpam-4170	209	34	p)η	p)η	NOUN
ejpam-4170	209	35	2	2	NUM
ejpam-4170	209	36	l(s	l(s	PROPN
ejpam-4170	209	37	)	)	PUNCT
ejpam-4170	209	38	,	,	PUNCT
ejpam-4170	209	39	for	for	ADP
ejpam-4170	209	40	some	some	DET
ejpam-4170	209	41	p	p	NOUN
ejpam-4170	209	42	in	in	ADP
ejpam-4170	209	43	which	which	PRON
ejpam-4170	209	44	ψ(p	ψ(p	VERB
ejpam-4170	209	45	)	)	PUNCT
ejpam-4170	209	46	=	=	SYM
ejpam-4170	209	47	s	s	PART
ejpam-4170	209	48	=	=	NOUN
ejpam-4170	209	49	η2	η2	X
ejpam-4170	209	50	t	t	NOUN
ejpam-4170	209	51	(	(	PUNCT
ejpam-4170	209	52	p)(1−	p)(1−	PROPN
ejpam-4170	209	53	η2l(s	η2l(s	PROPN
ejpam-4170	209	54	)	)	PUNCT
ejpam-4170	209	55	)	)	PUNCT
ejpam-4170	210	1	+	+	CCONJ
ejpam-4170	210	2	η2l(s	η2l(s	PROPN
ejpam-4170	210	3	)	)	PUNCT
ejpam-4170	210	4	if	if	SCONJ
ejpam-4170	210	5	η2	η2	ADJ
ejpam-4170	210	6	t	t	NOUN
ejpam-4170	210	7	(	(	PUNCT
ejpam-4170	210	8	p	p	NOUN
ejpam-4170	210	9	)	)	PUNCT
ejpam-4170	210	10	=	=	SYM
ejpam-4170	210	11	1	1	NUM
ejpam-4170	210	12	,	,	PUNCT
ejpam-4170	210	13	then	then	ADV
ejpam-4170	210	14	t	t	PROPN
ejpam-4170	210	15	=	=	PUNCT
ejpam-4170	210	16	χp	χp	NOUN
ejpam-4170	210	17	and	and	CCONJ
ejpam-4170	210	18	this	this	PRON
ejpam-4170	210	19	is	be	AUX
ejpam-4170	210	20	a	a	DET
ejpam-4170	210	21	contradiction	contradiction	NOUN
ejpam-4170	210	22	with	with	ADP
ejpam-4170	210	23	the	the	DET
ejpam-4170	210	24	fact	fact	NOUN
ejpam-4170	210	25	that	that	SCONJ
ejpam-4170	210	26	t	t	PROPN
ejpam-4170	210	27	is	be	AUX
ejpam-4170	210	28	a	a	DET
ejpam-4170	210	29	pythagorean	pythagorean	ADJ
ejpam-4170	210	30	fuzzy	fuzzy	ADJ
ejpam-4170	210	31	small	small	ADJ
ejpam-4170	210	32	submodule	submodule	NOUN
ejpam-4170	210	33	of	of	ADP
ejpam-4170	210	34	p	p	PROPN
ejpam-4170	210	35	.	.	PUNCT
ejpam-4170	211	1	thus	thus	ADV
ejpam-4170	211	2	η2l(s	η2l(s	PROPN
ejpam-4170	211	3	)	)	PUNCT
ejpam-4170	211	4	=	=	SYM
ejpam-4170	211	5	1	1	NUM
ejpam-4170	211	6	and	and	CCONJ
ejpam-4170	211	7	l	l	NOUN
ejpam-4170	212	1	=	=	NOUN
ejpam-4170	212	2	χs	χs	PROPN
ejpam-4170	212	3	.	.	PUNCT
ejpam-4170	213	1	moreover	moreover	ADV
ejpam-4170	213	2	,	,	PUNCT
ejpam-4170	213	3	0	0	PUNCT
ejpam-4170	213	4	=	=	SYM
ejpam-4170	213	5	η̂2ψ(t	η̂2ψ(t	X
ejpam-4170	213	6	)	)	PUNCT
ejpam-4170	214	1	+	+	PROPN
ejpam-4170	214	2	l(s	l(s	PROPN
ejpam-4170	214	3	)	)	PUNCT
ejpam-4170	214	4	references	reference	VERB
ejpam-4170	214	5	45	45	NUM
ejpam-4170	214	6	=	=	SYM
ejpam-4170	214	7	η̂2ψ(t	η̂2ψ(t	X
ejpam-4170	214	8	)	)	PUNCT
ejpam-4170	214	9	(	(	PUNCT
ejpam-4170	214	10	s)η̂	s)η̂	NOUN
ejpam-4170	214	11	2	2	NUM
ejpam-4170	214	12	l(s	l(s	PROPN
ejpam-4170	214	13	)	)	PUNCT
ejpam-4170	215	1	=	=	SYM
ejpam-4170	215	2	η̂2	η̂2	NOUN
ejpam-4170	215	3	t	t	PROPN
ejpam-4170	215	4	(	(	PUNCT
ejpam-4170	215	5	p)η̂	p)η̂	NOUN
ejpam-4170	215	6	2	2	NUM
ejpam-4170	215	7	l(s	l(s	PROPN
ejpam-4170	215	8	)	)	PUNCT
ejpam-4170	215	9	for	for	ADP
ejpam-4170	215	10	some	some	DET
ejpam-4170	215	11	p	p	NOUN
ejpam-4170	215	12	in	in	ADP
ejpam-4170	215	13	which	which	PRON
ejpam-4170	215	14	ψ(p	ψ(p	VERB
ejpam-4170	215	15	)	)	PUNCT
ejpam-4170	215	16	=	=	PUNCT
ejpam-4170	215	17	s	s	VERB
ejpam-4170	215	18	note	note	NOUN
ejpam-4170	215	19	that	that	SCONJ
ejpam-4170	215	20	ψ	ψ	NOUN
ejpam-4170	215	21	is	be	AUX
ejpam-4170	215	22	one	one	NUM
ejpam-4170	215	23	to	to	ADP
ejpam-4170	215	24	one	one	NUM
ejpam-4170	215	25	and	and	CCONJ
ejpam-4170	215	26	so	so	ADV
ejpam-4170	215	27	p	p	PRON
ejpam-4170	215	28	is	be	AUX
ejpam-4170	215	29	unique	unique	ADJ
ejpam-4170	215	30	.	.	PUNCT
ejpam-4170	216	1	by	by	ADP
ejpam-4170	216	2	hypothesis	hypothesis	NOUN
ejpam-4170	216	3	η̂2	η̂2	PROPN
ejpam-4170	216	4	t	t	PROPN
ejpam-4170	216	5	(	(	PUNCT
ejpam-4170	216	6	p	p	X
ejpam-4170	216	7	)	)	PUNCT
ejpam-4170	216	8	̸=	̸=	PROPN
ejpam-4170	216	9	0	0	NUM
ejpam-4170	216	10	,	,	PUNCT
ejpam-4170	216	11	so	so	SCONJ
ejpam-4170	216	12	that	that	SCONJ
ejpam-4170	216	13	η̂2l(s	η̂2l(s	X
ejpam-4170	216	14	)	)	PUNCT
ejpam-4170	216	15	=	=	SYM
ejpam-4170	217	1	0	0	X
ejpam-4170	217	2	.	.	PUNCT
ejpam-4170	217	3	hence	hence	ADV
ejpam-4170	217	4	l	l	NOUN
ejpam-4170	217	5	=	=	NOUN
ejpam-4170	217	6	χpfs	χpf	NOUN
ejpam-4170	217	7	.	.	PUNCT
ejpam-4170	218	1	remark	remark	PROPN
ejpam-4170	218	2	2	2	NUM
ejpam-4170	218	3	.	.	PUNCT
ejpam-4170	219	1	(	(	PUNCT
ejpam-4170	219	2	1	1	X
ejpam-4170	219	3	)	)	PUNCT
ejpam-4170	219	4	if	if	SCONJ
ejpam-4170	219	5	ψ	ψ	NOUN
ejpam-4170	219	6	is	be	AUX
ejpam-4170	219	7	not	not	PART
ejpam-4170	219	8	one	one	NUM
ejpam-4170	219	9	to	to	ADP
ejpam-4170	219	10	one	one	NUM
ejpam-4170	219	11	,	,	PUNCT
ejpam-4170	219	12	then	then	ADV
ejpam-4170	219	13	the	the	DET
ejpam-4170	219	14	above	above	ADJ
ejpam-4170	219	15	theorem	theorem	NOUN
ejpam-4170	219	16	need	need	AUX
ejpam-4170	219	17	not	not	PART
ejpam-4170	219	18	be	be	AUX
ejpam-4170	219	19	true	true	ADJ
ejpam-4170	219	20	.	.	PUNCT
ejpam-4170	220	1	for	for	ADP
ejpam-4170	220	2	instance	instance	NOUN
ejpam-4170	220	3	,	,	PUNCT
ejpam-4170	220	4	take	take	VERB
ejpam-4170	220	5	s	s	VERB
ejpam-4170	220	6	the	the	DET
ejpam-4170	220	7	zero	zero	NUM
ejpam-4170	220	8	module	module	NOUN
ejpam-4170	220	9	and	and	CCONJ
ejpam-4170	220	10	ψ	ψ	NOUN
ejpam-4170	220	11	the	the	DET
ejpam-4170	220	12	zero	zero	NUM
ejpam-4170	220	13	homomorphism	homomorphism	NOUN
ejpam-4170	220	14	.	.	PUNCT
ejpam-4170	221	1	(	(	PUNCT
ejpam-4170	221	2	2	2	X
ejpam-4170	221	3	)	)	PUNCT
ejpam-4170	221	4	the	the	DET
ejpam-4170	221	5	converse	converse	NOUN
ejpam-4170	221	6	of	of	ADP
ejpam-4170	221	7	the	the	DET
ejpam-4170	221	8	above	above	ADJ
ejpam-4170	221	9	theorem	theorem	NOUN
ejpam-4170	221	10	need	need	AUX
ejpam-4170	221	11	not	not	PART
ejpam-4170	221	12	be	be	AUX
ejpam-4170	221	13	true	true	ADJ
ejpam-4170	221	14	.	.	PUNCT
ejpam-4170	222	1	that	that	PRON
ejpam-4170	222	2	is	be	AUX
ejpam-4170	222	3	if	if	SCONJ
ejpam-4170	222	4	ψ	ψ	X
ejpam-4170	222	5	:	:	PUNCT
ejpam-4170	222	6	p	p	X
ejpam-4170	222	7	−→	−→	NOUN
ejpam-4170	222	8	s	s	PART
ejpam-4170	222	9	is	be	AUX
ejpam-4170	222	10	a	a	DET
ejpam-4170	222	11	monomorphism	monomorphism	NOUN
ejpam-4170	222	12	of	of	ADP
ejpam-4170	222	13	modules	module	NOUN
ejpam-4170	222	14	,	,	PUNCT
ejpam-4170	222	15	t	t	PROPN
ejpam-4170	222	16	is	be	AUX
ejpam-4170	222	17	a	a	DET
ejpam-4170	222	18	pythagorean	pythagorean	ADJ
ejpam-4170	222	19	fuzzy	fuzzy	ADJ
ejpam-4170	222	20	submodule	submodule	NOUN
ejpam-4170	222	21	of	of	ADP
ejpam-4170	222	22	p	p	PROPN
ejpam-4170	222	23	and	and	CCONJ
ejpam-4170	222	24	ψ(t	ψ(t	PROPN
ejpam-4170	222	25	)	)	PUNCT
ejpam-4170	223	1	is	be	AUX
ejpam-4170	223	2	a	a	DET
ejpam-4170	223	3	pythagorean	pythagorean	ADJ
ejpam-4170	223	4	fuzzy	fuzzy	ADJ
ejpam-4170	223	5	small	small	ADJ
ejpam-4170	223	6	submodule	submodule	NOUN
ejpam-4170	223	7	of	of	ADP
ejpam-4170	223	8	s	s	PROPN
ejpam-4170	223	9	,	,	PUNCT
ejpam-4170	223	10	then	then	ADV
ejpam-4170	223	11	it	it	PRON
ejpam-4170	223	12	is	be	AUX
ejpam-4170	223	13	not	not	PART
ejpam-4170	223	14	true	true	ADJ
ejpam-4170	223	15	in	in	ADP
ejpam-4170	223	16	general	general	ADJ
ejpam-4170	223	17	that	that	SCONJ
ejpam-4170	223	18	t	t	PROPN
ejpam-4170	223	19	is	be	AUX
ejpam-4170	223	20	a	a	DET
ejpam-4170	223	21	pythagorean	pythagorean	ADJ
ejpam-4170	223	22	fuzzy	fuzzy	ADJ
ejpam-4170	223	23	small	small	ADJ
ejpam-4170	223	24	submodule	submodule	NOUN
ejpam-4170	223	25	of	of	ADP
ejpam-4170	223	26	p	p	PROPN
ejpam-4170	223	27	.	.	PUNCT
ejpam-4170	224	1	for	for	ADP
ejpam-4170	224	2	example	example	NOUN
ejpam-4170	224	3	,	,	PUNCT
ejpam-4170	224	4	let	let	VERB
ejpam-4170	224	5	p	p	PRON
ejpam-4170	224	6	be	be	AUX
ejpam-4170	224	7	a	a	DET
ejpam-4170	224	8	pythagorean	pythagorean	ADJ
ejpam-4170	224	9	fuzzy	fuzzy	ADJ
ejpam-4170	224	10	small	small	ADJ
ejpam-4170	224	11	submodule	submodule	NOUN
ejpam-4170	224	12	of	of	ADP
ejpam-4170	224	13	s	s	PRON
ejpam-4170	224	14	and	and	CCONJ
ejpam-4170	224	15	consider	consider	VERB
ejpam-4170	224	16	the	the	DET
ejpam-4170	224	17	inclusion	inclusion	NOUN
ejpam-4170	224	18	p	p	PROPN
ejpam-4170	224	19	↪	↪	PROPN
ejpam-4170	224	20	→	→	SYM
ejpam-4170	224	21	s.	s.	PROPN
ejpam-4170	224	22	then	then	ADV
ejpam-4170	224	23	ψ(p	ψ(p	PROPN
ejpam-4170	224	24	)	)	PUNCT
ejpam-4170	225	1	=	=	PUNCT
ejpam-4170	225	2	p	p	NOUN
ejpam-4170	225	3	is	be	AUX
ejpam-4170	225	4	a	a	DET
ejpam-4170	225	5	pythagorean	pythagorean	ADJ
ejpam-4170	225	6	fuzzy	fuzzy	ADJ
ejpam-4170	225	7	small	small	ADJ
ejpam-4170	225	8	submodule	submodule	NOUN
ejpam-4170	225	9	of	of	ADP
ejpam-4170	225	10	s	s	PROPN
ejpam-4170	225	11	but	but	CCONJ
ejpam-4170	225	12	p	p	NOUN
ejpam-4170	225	13	is	be	AUX
ejpam-4170	225	14	not	not	PART
ejpam-4170	225	15	a	a	DET
ejpam-4170	225	16	pythagorean	pythagorean	ADJ
ejpam-4170	225	17	fuzzy	fuzzy	ADJ
ejpam-4170	225	18	small	small	ADJ
ejpam-4170	225	19	submodule	submodule	NOUN
ejpam-4170	225	20	of	of	ADP
ejpam-4170	225	21	p	p	PROPN
ejpam-4170	225	22	.	.	PUNCT
ejpam-4170	226	1	5	5	X
ejpam-4170	226	2	.	.	X
ejpam-4170	226	3	conclusion	conclusion	NOUN
ejpam-4170	226	4	and	and	CCONJ
ejpam-4170	226	5	future	future	ADJ
ejpam-4170	226	6	directions	direction	NOUN
ejpam-4170	226	7	in	in	ADP
ejpam-4170	226	8	this	this	DET
ejpam-4170	226	9	paper	paper	NOUN
ejpam-4170	226	10	,	,	PUNCT
ejpam-4170	226	11	we	we	PRON
ejpam-4170	226	12	introduce	introduce	VERB
ejpam-4170	226	13	the	the	DET
ejpam-4170	226	14	notion	notion	NOUN
ejpam-4170	226	15	of	of	ADP
ejpam-4170	226	16	pythagorean	pythagorean	PROPN
ejpam-4170	226	17	submodule	submodule	PROPN
ejpam-4170	226	18	.	.	PUNCT
ejpam-4170	227	1	in	in	ADP
ejpam-4170	227	2	addition	addition	NOUN
ejpam-4170	227	3	,	,	PUNCT
ejpam-4170	227	4	we	we	PRON
ejpam-4170	227	5	present	present	VERB
ejpam-4170	227	6	the	the	DET
ejpam-4170	227	7	concept	concept	NOUN
ejpam-4170	227	8	pythagorean	pythagorean	PROPN
ejpam-4170	227	9	small	small	ADJ
ejpam-4170	227	10	submodule	submodule	NOUN
ejpam-4170	227	11	and	and	CCONJ
ejpam-4170	227	12	investigate	investigate	VERB
ejpam-4170	227	13	some	some	DET
ejpam-4170	227	14	results	result	NOUN
ejpam-4170	227	15	regarding	regard	VERB
ejpam-4170	227	16	this	this	DET
ejpam-4170	227	17	concept	concept	NOUN
ejpam-4170	227	18	.	.	PUNCT
ejpam-4170	228	1	moreover	moreover	ADV
ejpam-4170	228	2	,	,	PUNCT
ejpam-4170	228	3	we	we	PRON
ejpam-4170	228	4	find	find	VERB
ejpam-4170	228	5	a	a	DET
ejpam-4170	228	6	relationship	relationship	NOUN
ejpam-4170	228	7	between	between	ADP
ejpam-4170	228	8	small	small	ADJ
ejpam-4170	228	9	submodule	submodule	NOUN
ejpam-4170	228	10	and	and	CCONJ
ejpam-4170	228	11	pythagorean	pythagorean	PROPN
ejpam-4170	228	12	fuzzy	fuzzy	ADJ
ejpam-4170	228	13	small	small	ADJ
ejpam-4170	228	14	submodule	submodule	NOUN
ejpam-4170	228	15	.	.	PUNCT
ejpam-4170	229	1	we	we	PRON
ejpam-4170	229	2	also	also	ADV
ejpam-4170	229	3	study	study	VERB
ejpam-4170	229	4	homomorphism	homomorphism	NOUN
ejpam-4170	229	5	between	between	ADP
ejpam-4170	229	6	pythagorean	pythagorean	PROPN
ejpam-4170	229	7	fuzzy	fuzzy	ADJ
ejpam-4170	229	8	modules	module	NOUN
ejpam-4170	229	9	.	.	PUNCT
ejpam-4170	230	1	this	this	DET
ejpam-4170	230	2	work	work	NOUN
ejpam-4170	230	3	can	can	AUX
ejpam-4170	230	4	be	be	AUX
ejpam-4170	230	5	extended	extend	VERB
ejpam-4170	230	6	and	and	CCONJ
ejpam-4170	230	7	generalised	generalise	VERB
ejpam-4170	230	8	in	in	ADP
ejpam-4170	230	9	the	the	DET
ejpam-4170	230	10	environment	environment	NOUN
ejpam-4170	230	11	of	of	ADP
ejpam-4170	230	12	q	q	ADJ
ejpam-4170	230	13	-	-	PUNCT
ejpam-4170	230	14	rung	rung	ADJ
ejpam-4170	230	15	orthopair	orthopair	ADJ
ejpam-4170	230	16	fuzzy	fuzzy	ADJ
ejpam-4170	230	17	sets	set	NOUN
ejpam-4170	230	18	.	.	PUNCT
ejpam-4170	231	1	it	it	PRON
ejpam-4170	231	2	can	can	AUX
ejpam-4170	231	3	be	be	AUX
ejpam-4170	231	4	applied	apply	VERB
ejpam-4170	231	5	in	in	ADP
ejpam-4170	231	6	order	order	NOUN
ejpam-4170	231	7	to	to	PART
ejpam-4170	231	8	solve	solve	VERB
ejpam-4170	231	9	multi	multi	ADJ
ejpam-4170	231	10	-	-	ADJ
ejpam-4170	231	11	criteria	criterion	NOUN
ejpam-4170	231	12	decision	decision	NOUN
ejpam-4170	231	13	making	make	VERB
ejpam-4170	231	14	problems	problem	NOUN
ejpam-4170	231	15	.	.	PUNCT
ejpam-4170	232	1	references	reference	NOUN
ejpam-4170	232	2	[	[	X
ejpam-4170	232	3	1	1	X
ejpam-4170	232	4	]	]	PUNCT
ejpam-4170	232	5	muhammad	muhammad	PROPN
ejpam-4170	232	6	irfan	irfan	PROPN
ejpam-4170	232	7	ali	ali	PROPN
ejpam-4170	232	8	.	.	PUNCT
ejpam-4170	233	1	another	another	DET
ejpam-4170	233	2	view	view	NOUN
ejpam-4170	233	3	on	on	ADP
ejpam-4170	233	4	q	q	ADJ
ejpam-4170	233	5	-	-	PUNCT
ejpam-4170	233	6	rung	rung	ADJ
ejpam-4170	233	7	orthopair	orthopair	ADJ
ejpam-4170	233	8	fuzzy	fuzzy	ADJ
ejpam-4170	233	9	sets	set	NOUN
ejpam-4170	233	10	.	.	PUNCT
ejpam-4170	234	1	international	international	ADJ
ejpam-4170	234	2	journal	journal	NOUN
ejpam-4170	234	3	of	of	ADP
ejpam-4170	234	4	intelligent	intelligent	ADJ
ejpam-4170	234	5	systems	system	NOUN
ejpam-4170	234	6	,	,	PUNCT
ejpam-4170	234	7	33(11):2139–2153	33(11):2139–2153	NUM
ejpam-4170	234	8	,	,	PUNCT
ejpam-4170	234	9	2018	2018	NUM
ejpam-4170	234	10	.	.	PUNCT
ejpam-4170	235	1	[	[	X
ejpam-4170	235	2	2	2	NUM
ejpam-4170	235	3	]	]	PUNCT
ejpam-4170	235	4	zeeshan	zeeshan	PROPN
ejpam-4170	235	5	ali	ali	PROPN
ejpam-4170	235	6	and	and	CCONJ
ejpam-4170	235	7	tahir	tahir	PROPN
ejpam-4170	235	8	mahmood	mahmood	PROPN
ejpam-4170	235	9	.	.	PUNCT
ejpam-4170	236	1	maclaurin	maclaurin	PROPN
ejpam-4170	236	2	symmetric	symmetric	ADJ
ejpam-4170	236	3	mean	mean	NOUN
ejpam-4170	236	4	operators	operator	NOUN
ejpam-4170	236	5	and	and	CCONJ
ejpam-4170	236	6	their	their	PRON
ejpam-4170	236	7	applications	application	NOUN
ejpam-4170	236	8	in	in	ADP
ejpam-4170	236	9	the	the	DET
ejpam-4170	236	10	environment	environment	NOUN
ejpam-4170	236	11	of	of	ADP
ejpam-4170	236	12	complex	complex	ADJ
ejpam-4170	236	13	q	q	ADJ
ejpam-4170	236	14	-	-	PUNCT
ejpam-4170	236	15	rung	rung	ADJ
ejpam-4170	236	16	orthopair	orthopair	ADJ
ejpam-4170	236	17	fuzzy	fuzzy	ADJ
ejpam-4170	236	18	sets	set	NOUN
ejpam-4170	236	19	.	.	PUNCT
ejpam-4170	237	1	computational	computational	ADJ
ejpam-4170	237	2	and	and	CCONJ
ejpam-4170	237	3	applied	applied	ADJ
ejpam-4170	237	4	mathematics	mathematic	NOUN
ejpam-4170	237	5	,	,	PUNCT
ejpam-4170	237	6	39:1–27	39:1–27	NUM
ejpam-4170	237	7	,	,	PUNCT
ejpam-4170	237	8	2020	2020	NUM
ejpam-4170	237	9	.	.	PUNCT
ejpam-4170	238	1	[	[	X
ejpam-4170	238	2	3	3	NUM
ejpam-4170	238	3	]	]	X
ejpam-4170	238	4	frank	frank	PROPN
ejpam-4170	238	5	w	w	PROPN
ejpam-4170	238	6	anderson	anderson	PROPN
ejpam-4170	238	7	and	and	CCONJ
ejpam-4170	238	8	kent	kent	PROPN
ejpam-4170	238	9	r	r	PROPN
ejpam-4170	238	10	fuller	full	ADJ
ejpam-4170	238	11	.	.	PUNCT
ejpam-4170	239	1	rings	ring	NOUN
ejpam-4170	239	2	and	and	CCONJ
ejpam-4170	239	3	categories	category	NOUN
ejpam-4170	239	4	of	of	ADP
ejpam-4170	239	5	modules	module	NOUN
ejpam-4170	239	6	,	,	PUNCT
ejpam-4170	239	7	volume	volume	NOUN
ejpam-4170	239	8	13	13	NUM
ejpam-4170	239	9	.	.	PUNCT
ejpam-4170	240	1	springer	springer	PROPN
ejpam-4170	240	2	science	science	PROPN
ejpam-4170	240	3	&	&	CCONJ
ejpam-4170	240	4	business	business	NOUN
ejpam-4170	240	5	media	medium	NOUN
ejpam-4170	240	6	,	,	PUNCT
ejpam-4170	240	7	2012	2012	NUM
ejpam-4170	240	8	.	.	PUNCT
ejpam-4170	241	1	[	[	X
ejpam-4170	241	2	4	4	NUM
ejpam-4170	241	3	]	]	X
ejpam-4170	241	4	scott	scott	PROPN
ejpam-4170	241	5	dick	dick	PROPN
ejpam-4170	241	6	,	,	PUNCT
ejpam-4170	241	7	ronald	ronald	PROPN
ejpam-4170	241	8	r	r	NOUN
ejpam-4170	241	9	yager	yager	NOUN
ejpam-4170	241	10	,	,	PUNCT
ejpam-4170	241	11	and	and	CCONJ
ejpam-4170	241	12	omolbanin	omolbanin	PROPN
ejpam-4170	241	13	yazdanbakhsh	yazdanbakhsh	NOUN
ejpam-4170	241	14	.	.	PUNCT
ejpam-4170	242	1	on	on	ADP
ejpam-4170	242	2	pythagorean	pythagorean	PROPN
ejpam-4170	242	3	and	and	CCONJ
ejpam-4170	242	4	complex	complex	ADJ
ejpam-4170	242	5	fuzzy	fuzzy	ADJ
ejpam-4170	242	6	set	set	NOUN
ejpam-4170	242	7	operations	operation	NOUN
ejpam-4170	242	8	.	.	PUNCT
ejpam-4170	243	1	ieee	ieee	NOUN
ejpam-4170	243	2	transactions	transaction	NOUN
ejpam-4170	243	3	on	on	ADP
ejpam-4170	243	4	fuzzy	fuzzy	ADJ
ejpam-4170	243	5	systems	system	NOUN
ejpam-4170	243	6	,	,	PUNCT
ejpam-4170	243	7	24(5):1009–1021	24(5):1009–1021	NUM
ejpam-4170	243	8	,	,	PUNCT
ejpam-4170	243	9	2015	2015	NUM
ejpam-4170	243	10	.	.	PUNCT
ejpam-4170	244	1	references	reference	NOUN
ejpam-4170	244	2	46	46	NUM
ejpam-4170	245	1	[	[	X
ejpam-4170	245	2	5	5	NUM
ejpam-4170	245	3	]	]	X
ejpam-4170	245	4	harish	harish	PROPN
ejpam-4170	245	5	garg	garg	PROPN
ejpam-4170	245	6	.	.	PUNCT
ejpam-4170	246	1	generalised	generalise	VERB
ejpam-4170	246	2	pythagorean	pythagorean	PROPN
ejpam-4170	246	3	fuzzy	fuzzy	ADJ
ejpam-4170	246	4	geometric	geometric	ADJ
ejpam-4170	246	5	interactive	interactive	ADJ
ejpam-4170	246	6	aggregation	aggregation	NOUN
ejpam-4170	246	7	operators	operator	NOUN
ejpam-4170	246	8	using	use	VERB
ejpam-4170	246	9	einstein	einstein	ADJ
ejpam-4170	246	10	operations	operation	NOUN
ejpam-4170	246	11	and	and	CCONJ
ejpam-4170	246	12	their	their	PRON
ejpam-4170	246	13	application	application	NOUN
ejpam-4170	246	14	to	to	ADP
ejpam-4170	246	15	decision	decision	NOUN
ejpam-4170	246	16	making	making	NOUN
ejpam-4170	246	17	.	.	PUNCT
ejpam-4170	247	1	journal	journal	NOUN
ejpam-4170	247	2	of	of	ADP
ejpam-4170	247	3	experimental	experimental	PROPN
ejpam-4170	247	4	&	&	CCONJ
ejpam-4170	247	5	theoretical	theoretical	ADJ
ejpam-4170	247	6	artificial	artificial	ADJ
ejpam-4170	247	7	intelligence	intelligence	NOUN
ejpam-4170	247	8	,	,	PUNCT
ejpam-4170	247	9	30(6):763–794	30(6):763–794	PROPN
ejpam-4170	247	10	,	,	PUNCT
ejpam-4170	247	11	2018	2018	NUM
ejpam-4170	247	12	.	.	PUNCT
ejpam-4170	248	1	[	[	X
ejpam-4170	248	2	6	6	NUM
ejpam-4170	248	3	]	]	PUNCT
ejpam-4170	248	4	harish	harish	PROPN
ejpam-4170	248	5	garg	garg	PROPN
ejpam-4170	248	6	.	.	PUNCT
ejpam-4170	249	1	a	a	DET
ejpam-4170	249	2	linear	linear	ADJ
ejpam-4170	249	3	programming	programming	NOUN
ejpam-4170	249	4	method	method	NOUN
ejpam-4170	249	5	based	base	VERB
ejpam-4170	249	6	on	on	ADP
ejpam-4170	249	7	an	an	DET
ejpam-4170	249	8	improved	improved	ADJ
ejpam-4170	249	9	score	score	NOUN
ejpam-4170	249	10	function	function	NOUN
ejpam-4170	249	11	for	for	ADP
ejpam-4170	249	12	interval	interval	NOUN
ejpam-4170	249	13	-	-	PUNCT
ejpam-4170	249	14	valued	value	VERB
ejpam-4170	249	15	pythagorean	pythagorean	ADJ
ejpam-4170	249	16	fuzzy	fuzzy	ADJ
ejpam-4170	249	17	numbers	number	NOUN
ejpam-4170	249	18	and	and	CCONJ
ejpam-4170	249	19	its	its	PRON
ejpam-4170	249	20	application	application	NOUN
ejpam-4170	249	21	to	to	ADP
ejpam-4170	249	22	decisionmaking	decisionmake	VERB
ejpam-4170	249	23	.	.	PUNCT
ejpam-4170	250	1	international	international	ADJ
ejpam-4170	250	2	journal	journal	NOUN
ejpam-4170	250	3	of	of	ADP
ejpam-4170	250	4	uncertainty	uncertainty	NOUN
ejpam-4170	250	5	,	,	PUNCT
ejpam-4170	250	6	fuzziness	fuzziness	NOUN
ejpam-4170	250	7	and	and	CCONJ
ejpam-4170	250	8	knowledge	knowledge	NOUN
ejpam-4170	250	9	-	-	PUNCT
ejpam-4170	250	10	based	base	VERB
ejpam-4170	250	11	systems	system	NOUN
ejpam-4170	250	12	,	,	PUNCT
ejpam-4170	250	13	26(01):67–80	26(01):67–80	NUM
ejpam-4170	250	14	,	,	PUNCT
ejpam-4170	250	15	2018	2018	NUM
ejpam-4170	250	16	.	.	PUNCT
ejpam-4170	251	1	[	[	X
ejpam-4170	251	2	7	7	X
ejpam-4170	251	3	]	]	X
ejpam-4170	251	4	muhammad	muhammad	PROPN
ejpam-4170	251	5	jabir	jabir	PROPN
ejpam-4170	251	6	khan	khan	PROPN
ejpam-4170	251	7	,	,	PUNCT
ejpam-4170	251	8	poom	poom	NOUN
ejpam-4170	251	9	kumam	kumam	NOUN
ejpam-4170	251	10	,	,	PUNCT
ejpam-4170	251	11	and	and	CCONJ
ejpam-4170	251	12	meshal	meshal	PROPN
ejpam-4170	251	13	shutaywi	shutaywi	NOUN
ejpam-4170	251	14	.	.	PUNCT
ejpam-4170	252	1	knowledge	knowledge	NOUN
ejpam-4170	252	2	measure	measure	NOUN
ejpam-4170	252	3	for	for	ADP
ejpam-4170	252	4	the	the	DET
ejpam-4170	252	5	q	q	ADJ
ejpam-4170	252	6	-	-	PUNCT
ejpam-4170	252	7	rung	rung	ADJ
ejpam-4170	252	8	orthopair	orthopair	ADJ
ejpam-4170	252	9	fuzzy	fuzzy	ADJ
ejpam-4170	252	10	sets	set	NOUN
ejpam-4170	252	11	.	.	PUNCT
ejpam-4170	253	1	international	international	ADJ
ejpam-4170	253	2	journal	journal	NOUN
ejpam-4170	253	3	of	of	ADP
ejpam-4170	253	4	intelligent	intelligent	ADJ
ejpam-4170	253	5	systems	system	NOUN
ejpam-4170	253	6	,	,	PUNCT
ejpam-4170	253	7	36(2):628–655	36(2):628–655	PROPN
ejpam-4170	253	8	,	,	PUNCT
ejpam-4170	253	9	2021	2021	NUM
ejpam-4170	253	10	.	.	PUNCT
ejpam-4170	254	1	[	[	X
ejpam-4170	254	2	8	8	NUM
ejpam-4170	254	3	]	]	PUNCT
ejpam-4170	254	4	decui	decui	NOUN
ejpam-4170	254	5	liang	liang	PROPN
ejpam-4170	254	6	and	and	CCONJ
ejpam-4170	254	7	zeshui	zeshui	PROPN
ejpam-4170	254	8	xu	xu	PROPN
ejpam-4170	254	9	.	.	PUNCT
ejpam-4170	255	1	the	the	DET
ejpam-4170	255	2	new	new	ADJ
ejpam-4170	255	3	extension	extension	NOUN
ejpam-4170	255	4	of	of	ADP
ejpam-4170	255	5	topsis	topsis	NOUN
ejpam-4170	255	6	method	method	NOUN
ejpam-4170	255	7	for	for	ADP
ejpam-4170	255	8	multiple	multiple	ADJ
ejpam-4170	255	9	criteria	criterion	NOUN
ejpam-4170	255	10	decision	decision	NOUN
ejpam-4170	255	11	making	make	VERB
ejpam-4170	255	12	with	with	ADP
ejpam-4170	255	13	hesitant	hesitant	ADJ
ejpam-4170	255	14	pythagorean	pythagorean	PROPN
ejpam-4170	255	15	fuzzy	fuzzy	ADJ
ejpam-4170	255	16	sets	set	NOUN
ejpam-4170	255	17	.	.	PUNCT
ejpam-4170	255	18	applied	apply	VERB
ejpam-4170	255	19	soft	soft	ADJ
ejpam-4170	255	20	computing	computing	NOUN
ejpam-4170	255	21	,	,	PUNCT
ejpam-4170	255	22	60:167–179	60:167–179	PROPN
ejpam-4170	255	23	,	,	PUNCT
ejpam-4170	255	24	2017	2017	NUM
ejpam-4170	255	25	.	.	PUNCT
ejpam-4170	256	1	[	[	X
ejpam-4170	256	2	9	9	NUM
ejpam-4170	256	3	]	]	X
ejpam-4170	256	4	vahid	vahid	PROPN
ejpam-4170	256	5	mohagheghi	mohagheghi	PROPN
ejpam-4170	256	6	,	,	PUNCT
ejpam-4170	256	7	s	s	PART
ejpam-4170	256	8	meysam	meysam	NOUN
ejpam-4170	256	9	mousavi	mousavi	PROPN
ejpam-4170	256	10	,	,	PUNCT
ejpam-4170	256	11	and	and	CCONJ
ejpam-4170	256	12	behnam	behnam	NOUN
ejpam-4170	256	13	vahdani	vahdani	PROPN
ejpam-4170	256	14	.	.	PUNCT
ejpam-4170	257	1	enhancing	enhance	VERB
ejpam-4170	257	2	decisionmaking	decisionmake	VERB
ejpam-4170	257	3	flexibility	flexibility	NOUN
ejpam-4170	257	4	by	by	ADP
ejpam-4170	257	5	introducing	introduce	VERB
ejpam-4170	257	6	a	a	DET
ejpam-4170	257	7	new	new	ADJ
ejpam-4170	257	8	last	last	ADJ
ejpam-4170	257	9	aggregation	aggregation	NOUN
ejpam-4170	257	10	evaluating	evaluate	VERB
ejpam-4170	257	11	approach	approach	NOUN
ejpam-4170	257	12	based	base	VERB
ejpam-4170	257	13	on	on	ADP
ejpam-4170	257	14	multi	multi	ADJ
ejpam-4170	257	15	-	-	ADJ
ejpam-4170	257	16	criteria	criterion	NOUN
ejpam-4170	257	17	group	group	NOUN
ejpam-4170	257	18	decision	decision	NOUN
ejpam-4170	257	19	making	make	VERB
ejpam-4170	257	20	and	and	CCONJ
ejpam-4170	257	21	pythagorean	pythagorean	VERB
ejpam-4170	257	22	fuzzy	fuzzy	ADJ
ejpam-4170	257	23	sets	set	NOUN
ejpam-4170	257	24	.	.	PUNCT
ejpam-4170	258	1	applied	apply	VERB
ejpam-4170	258	2	soft	soft	ADJ
ejpam-4170	258	3	computing	computing	NOUN
ejpam-4170	258	4	,	,	PUNCT
ejpam-4170	258	5	61:527–535	61:527–535	PROPN
ejpam-4170	258	6	,	,	PUNCT
ejpam-4170	258	7	2017	2017	NUM
ejpam-4170	258	8	.	.	PUNCT
ejpam-4170	259	1	[	[	X
ejpam-4170	259	2	10	10	NUM
ejpam-4170	259	3	]	]	X
ejpam-4170	259	4	xindong	xindong	PROPN
ejpam-4170	259	5	peng	peng	PROPN
ejpam-4170	259	6	and	and	CCONJ
ejpam-4170	259	7	ganeshsree	ganeshsree	PROPN
ejpam-4170	259	8	selvachandran	selvachandran	ADJ
ejpam-4170	259	9	.	.	PUNCT
ejpam-4170	260	1	pythagorean	pythagorean	PROPN
ejpam-4170	260	2	fuzzy	fuzzy	ADJ
ejpam-4170	260	3	set	set	NOUN
ejpam-4170	260	4	:	:	PUNCT
ejpam-4170	260	5	state	state	NOUN
ejpam-4170	260	6	of	of	ADP
ejpam-4170	260	7	the	the	DET
ejpam-4170	260	8	art	art	NOUN
ejpam-4170	260	9	and	and	CCONJ
ejpam-4170	260	10	future	future	ADJ
ejpam-4170	260	11	directions	direction	NOUN
ejpam-4170	260	12	.	.	PUNCT
ejpam-4170	261	1	artificial	artificial	ADJ
ejpam-4170	261	2	intelligence	intelligence	NOUN
ejpam-4170	261	3	review	review	NOUN
ejpam-4170	261	4	,	,	PUNCT
ejpam-4170	261	5	52(3):1873–1927	52(3):1873–1927	NUM
ejpam-4170	261	6	,	,	PUNCT
ejpam-4170	261	7	2019	2019	NUM
ejpam-4170	261	8	.	.	PUNCT
ejpam-4170	262	1	[	[	X
ejpam-4170	262	2	11	11	NUM
ejpam-4170	262	3	]	]	PUNCT
ejpam-4170	262	4	xindong	xindong	PROPN
ejpam-4170	262	5	peng	peng	PROPN
ejpam-4170	262	6	and	and	CCONJ
ejpam-4170	262	7	yong	yong	PROPN
ejpam-4170	262	8	yang	yang	PROPN
ejpam-4170	262	9	.	.	PUNCT
ejpam-4170	263	1	some	some	DET
ejpam-4170	263	2	results	result	NOUN
ejpam-4170	263	3	for	for	ADP
ejpam-4170	263	4	pythagorean	pythagorean	ADJ
ejpam-4170	263	5	fuzzy	fuzzy	ADJ
ejpam-4170	263	6	sets	set	NOUN
ejpam-4170	263	7	.	.	PUNCT
ejpam-4170	264	1	international	international	ADJ
ejpam-4170	264	2	journal	journal	NOUN
ejpam-4170	264	3	of	of	ADP
ejpam-4170	264	4	intelligent	intelligent	ADJ
ejpam-4170	264	5	systems	system	NOUN
ejpam-4170	264	6	,	,	PUNCT
ejpam-4170	264	7	30(11):1133–1160	30(11):1133–1160	NUM
ejpam-4170	264	8	,	,	PUNCT
ejpam-4170	264	9	2015	2015	NUM
ejpam-4170	264	10	.	.	PUNCT
ejpam-4170	265	1	[	[	X
ejpam-4170	265	2	12	12	NUM
ejpam-4170	265	3	]	]	X
ejpam-4170	265	4	luis	luis	PROPN
ejpam-4170	265	5	pérez	pérez	PROPN
ejpam-4170	265	6	-	-	PUNCT
ejpam-4170	265	7	domı́nguez	domı́nguez	PROPN
ejpam-4170	265	8	,	,	PUNCT
ejpam-4170	265	9	luis	luis	PROPN
ejpam-4170	265	10	alberto	alberto	PROPN
ejpam-4170	265	11	rodŕıguez	rodŕıguez	PROPN
ejpam-4170	265	12	-	-	PUNCT
ejpam-4170	265	13	picón	picón	NOUN
ejpam-4170	265	14	,	,	PUNCT
ejpam-4170	265	15	alejandro	alejandro	PROPN
ejpam-4170	265	16	alvarado	alvarado	PROPN
ejpam-4170	265	17	-	-	PUNCT
ejpam-4170	265	18	iniesta	iniesta	PROPN
ejpam-4170	265	19	,	,	PUNCT
ejpam-4170	265	20	david	david	PROPN
ejpam-4170	265	21	luviano	luviano	PROPN
ejpam-4170	265	22	cruz	cruz	PROPN
ejpam-4170	265	23	,	,	PUNCT
ejpam-4170	265	24	and	and	CCONJ
ejpam-4170	265	25	zeshui	zeshui	PROPN
ejpam-4170	265	26	xu	xu	PROPN
ejpam-4170	265	27	.	.	PUNCT
ejpam-4170	266	1	moora	moora	PROPN
ejpam-4170	266	2	under	under	ADP
ejpam-4170	266	3	pythagorean	pythagorean	PROPN
ejpam-4170	266	4	fuzzy	fuzzy	ADJ
ejpam-4170	266	5	set	set	NOUN
ejpam-4170	266	6	for	for	ADP
ejpam-4170	266	7	multiple	multiple	ADJ
ejpam-4170	266	8	criteria	criterion	NOUN
ejpam-4170	266	9	decision	decision	NOUN
ejpam-4170	266	10	making	making	NOUN
ejpam-4170	266	11	.	.	PUNCT
ejpam-4170	267	1	complexity	complexity	NOUN
ejpam-4170	267	2	,	,	PUNCT
ejpam-4170	267	3	2018	2018	NUM
ejpam-4170	267	4	,	,	PUNCT
ejpam-4170	267	5	2018	2018	NUM
ejpam-4170	267	6	.	.	PUNCT
ejpam-4170	268	1	[	[	X
ejpam-4170	268	2	13	13	NUM
ejpam-4170	268	3	]	]	X
ejpam-4170	268	4	k	k	PROPN
ejpam-4170	268	5	rahman	rahman	PROPN
ejpam-4170	268	6	,	,	PUNCT
ejpam-4170	268	7	s	s	PROPN
ejpam-4170	268	8	abdullah	abdullah	PROPN
ejpam-4170	268	9	,	,	PUNCT
ejpam-4170	268	10	m	m	PROPN
ejpam-4170	268	11	shakeel	shakeel	PROPN
ejpam-4170	268	12	,	,	PUNCT
ejpam-4170	268	13	m	m	PROPN
ejpam-4170	268	14	sajjad	sajjad	PROPN
ejpam-4170	268	15	ali	ali	PROPN
ejpam-4170	268	16	khan	khan	PROPN
ejpam-4170	268	17	,	,	PUNCT
ejpam-4170	268	18	and	and	CCONJ
ejpam-4170	268	19	murad	murad	PROPN
ejpam-4170	268	20	ullah	ullah	PROPN
ejpam-4170	268	21	.	.	PROPN
ejpam-4170	269	1	intervalvalued	intervalvalue	VERB
ejpam-4170	269	2	pythagorean	pythagorean	PROPN
ejpam-4170	269	3	fuzzy	fuzzy	ADJ
ejpam-4170	269	4	geometric	geometric	ADJ
ejpam-4170	269	5	aggregation	aggregation	NOUN
ejpam-4170	269	6	operators	operator	NOUN
ejpam-4170	269	7	and	and	CCONJ
ejpam-4170	269	8	their	their	PRON
ejpam-4170	269	9	application	application	NOUN
ejpam-4170	269	10	to	to	ADP
ejpam-4170	269	11	group	group	NOUN
ejpam-4170	269	12	decision	decision	NOUN
ejpam-4170	269	13	making	make	VERB
ejpam-4170	269	14	problem	problem	NOUN
ejpam-4170	269	15	.	.	PUNCT
ejpam-4170	270	1	cogent	cogent	NOUN
ejpam-4170	270	2	mathematics	mathematic	NOUN
ejpam-4170	270	3	,	,	PUNCT
ejpam-4170	270	4	4(1):1338638	4(1):1338638	NUM
ejpam-4170	270	5	,	,	PUNCT
ejpam-4170	270	6	2017	2017	NUM
ejpam-4170	270	7	.	.	PUNCT
ejpam-4170	271	1	[	[	X
ejpam-4170	271	2	14	14	NUM
ejpam-4170	271	3	]	]	X
ejpam-4170	271	4	ronald	ronald	PROPN
ejpam-4170	271	5	r	r	NOUN
ejpam-4170	271	6	yager	yager	NOUN
ejpam-4170	271	7	.	.	PUNCT
ejpam-4170	272	1	pythagorean	pythagorean	PROPN
ejpam-4170	272	2	membership	membership	NOUN
ejpam-4170	272	3	grades	grade	NOUN
ejpam-4170	272	4	in	in	ADP
ejpam-4170	272	5	multicriteria	multicriteria	PROPN
ejpam-4170	272	6	decision	decision	NOUN
ejpam-4170	272	7	making	making	NOUN
ejpam-4170	272	8	.	.	PUNCT
ejpam-4170	273	1	ieee	ieee	NOUN
ejpam-4170	273	2	transactions	transaction	NOUN
ejpam-4170	273	3	on	on	ADP
ejpam-4170	273	4	fuzzy	fuzzy	ADJ
ejpam-4170	273	5	systems	system	NOUN
ejpam-4170	273	6	,	,	PUNCT
ejpam-4170	273	7	22(4):958–965	22(4):958–965	PROPN
ejpam-4170	273	8	,	,	PUNCT
ejpam-4170	273	9	2013	2013	NUM
ejpam-4170	273	10	.	.	PUNCT
ejpam-4170	274	1	[	[	X
ejpam-4170	274	2	15	15	NUM
ejpam-4170	274	3	]	]	X
ejpam-4170	274	4	ronald	ronald	PROPN
ejpam-4170	274	5	r	r	NOUN
ejpam-4170	274	6	yager	yager	NOUN
ejpam-4170	274	7	.	.	PUNCT
ejpam-4170	275	1	properties	property	NOUN
ejpam-4170	275	2	and	and	CCONJ
ejpam-4170	275	3	applications	application	NOUN
ejpam-4170	275	4	of	of	ADP
ejpam-4170	275	5	pythagorean	pythagorean	ADJ
ejpam-4170	275	6	fuzzy	fuzzy	ADJ
ejpam-4170	275	7	sets	set	NOUN
ejpam-4170	275	8	.	.	PUNCT
ejpam-4170	276	1	in	in	ADP
ejpam-4170	276	2	imprecision	imprecision	NOUN
ejpam-4170	276	3	and	and	CCONJ
ejpam-4170	276	4	uncertainty	uncertainty	NOUN
ejpam-4170	276	5	in	in	ADP
ejpam-4170	276	6	information	information	NOUN
ejpam-4170	276	7	representation	representation	NOUN
ejpam-4170	276	8	and	and	CCONJ
ejpam-4170	276	9	processing	processing	NOUN
ejpam-4170	276	10	,	,	PUNCT
ejpam-4170	276	11	pages	page	NOUN
ejpam-4170	276	12	119–136	119–136	NUM
ejpam-4170	276	13	.	.	PUNCT
ejpam-4170	276	14	springer	springer	NOUN
ejpam-4170	276	15	,	,	PUNCT
ejpam-4170	276	16	2016	2016	NUM
ejpam-4170	276	17	.	.	PUNCT
ejpam-4170	277	1	[	[	X
ejpam-4170	277	2	16	16	NUM
ejpam-4170	277	3	]	]	PUNCT
ejpam-4170	277	4	lotfi	lotfi	X
ejpam-4170	277	5	a	a	DET
ejpam-4170	277	6	zadeh	zadeh	PROPN
ejpam-4170	277	7	.	.	PUNCT
ejpam-4170	277	8	fuzzy	fuzzy	ADJ
ejpam-4170	277	9	sets	set	NOUN
ejpam-4170	277	10	.	.	PUNCT
ejpam-4170	278	1	information	information	NOUN
ejpam-4170	278	2	and	and	CCONJ
ejpam-4170	278	3	control	control	NOUN
ejpam-4170	278	4	,	,	PUNCT
ejpam-4170	278	5	8(3):338–353	8(3):338–353	NUM
ejpam-4170	278	6	,	,	PUNCT
ejpam-4170	278	7	1965	1965	NUM
ejpam-4170	278	8	.	.	PUNCT
ejpam-4170	279	1	[	[	X
ejpam-4170	279	2	17	17	NUM
ejpam-4170	279	3	]	]	PUNCT
ejpam-4170	279	4	wenyi	wenyi	PROPN
ejpam-4170	279	5	zeng	zeng	PROPN
ejpam-4170	279	6	,	,	PUNCT
ejpam-4170	279	7	deqing	deqing	ADJ
ejpam-4170	279	8	li	li	NOUN
ejpam-4170	279	9	,	,	PUNCT
ejpam-4170	279	10	and	and	CCONJ
ejpam-4170	279	11	qian	qian	PROPN
ejpam-4170	279	12	yin	yin	PROPN
ejpam-4170	279	13	.	.	PUNCT
ejpam-4170	280	1	distance	distance	NOUN
ejpam-4170	280	2	and	and	CCONJ
ejpam-4170	280	3	similarity	similarity	NOUN
ejpam-4170	280	4	measures	measure	NOUN
ejpam-4170	280	5	of	of	ADP
ejpam-4170	280	6	pythagorean	pythagorean	ADJ
ejpam-4170	280	7	fuzzy	fuzzy	ADJ
ejpam-4170	280	8	sets	set	NOUN
ejpam-4170	280	9	and	and	CCONJ
ejpam-4170	280	10	their	their	PRON
ejpam-4170	280	11	applications	application	NOUN
ejpam-4170	280	12	to	to	ADP
ejpam-4170	280	13	multiple	multiple	ADJ
ejpam-4170	280	14	criteria	criterion	NOUN
ejpam-4170	280	15	group	group	NOUN
ejpam-4170	280	16	decision	decision	NOUN
ejpam-4170	280	17	making	making	NOUN
ejpam-4170	280	18	.	.	PUNCT
ejpam-4170	281	1	international	international	ADJ
ejpam-4170	281	2	journal	journal	NOUN
ejpam-4170	281	3	of	of	ADP
ejpam-4170	281	4	intelligent	intelligent	ADJ
ejpam-4170	281	5	systems	system	NOUN
ejpam-4170	281	6	,	,	PUNCT
ejpam-4170	281	7	33(11):2236–2254	33(11):2236–2254	NUM
ejpam-4170	281	8	,	,	PUNCT
ejpam-4170	281	9	2018	2018	NUM
ejpam-4170	281	10	.	.	PUNCT