id	sid	tid	token	lemma	pos
iajs-2731	1	1	84	84	NUM
iajs-2731	1	2	this	this	DET
iajs-2731	1	3	work	work	NOUN
iajs-2731	1	4	is	be	AUX
iajs-2731	1	5	licensed	license	VERB
iajs-2731	1	6	under	under	ADP
iajs-2731	1	7	a	a	DET
iajs-2731	1	8	creative	creative	ADJ
iajs-2731	1	9	commons	common	NOUN
iajs-2731	1	10	attribution	attribution	NOUN
iajs-2731	1	11	4.0	4.0	NUM
iajs-2731	1	12	international	international	ADJ
iajs-2731	1	13	license	license	NOUN
iajs-2731	1	14	.	.	PUNCT
iajs-2731	2	1	loc	loc	ADJ
iajs-2731	2	2	-	-	ADJ
iajs-2731	2	3	hollow	hollow	ADJ
iajs-2731	2	4	fuzzy	fuzzy	ADJ
iajs-2731	2	5	modules	module	NOUN
iajs-2731	2	6	with	with	ADP
iajs-2731	2	7	related	related	ADJ
iajs-2731	2	8	modules	module	NOUN
iajs-2731	2	9	abstract	abstract	ADJ
iajs-2731	2	10	the	the	DET
iajs-2731	2	11	concept	concept	NOUN
iajs-2731	2	12	of	of	ADP
iajs-2731	2	13	a	a	DET
iajs-2731	2	14	small	small	ADJ
iajs-2731	2	15	fsubm	fsubm	NOUN
iajs-2731	2	16	was	be	AUX
iajs-2731	2	17	presented	present	VERB
iajs-2731	2	18	in	in	ADP
iajs-2731	2	19	a	a	DET
iajs-2731	2	20	previous	previous	ADJ
iajs-2731	2	21	study	study	NOUN
iajs-2731	2	22	.	.	PUNCT
iajs-2731	3	1	this	this	DET
iajs-2731	3	2	work	work	NOUN
iajs-2731	3	3	introduced	introduce	VERB
iajs-2731	3	4	a	a	DET
iajs-2731	3	5	concept	concept	NOUN
iajs-2731	3	6	of	of	ADP
iajs-2731	3	7	a	a	DET
iajs-2731	3	8	hollow	hollow	ADJ
iajs-2731	3	9	fmodule	fmodule	NOUN
iajs-2731	3	10	,	,	PUNCT
iajs-2731	3	11	where	where	SCONJ
iajs-2731	3	12	a	a	DET
iajs-2731	3	13	module	module	NOUN
iajs-2731	3	14	is	be	AUX
iajs-2731	3	15	said	say	VERB
iajs-2731	3	16	to	to	PART
iajs-2731	3	17	be	be	AUX
iajs-2731	3	18	hollow	hollow	ADJ
iajs-2731	3	19	fuzzy	fuzzy	ADJ
iajs-2731	3	20	when	when	SCONJ
iajs-2731	3	21	every	every	DET
iajs-2731	3	22	subm	subm	PROPN
iajs-2731	3	23	of	of	ADP
iajs-2731	3	24	it	it	PRON
iajs-2731	3	25	is	be	AUX
iajs-2731	3	26	a	a	DET
iajs-2731	3	27	small	small	ADJ
iajs-2731	3	28	f	f	PROPN
iajs-2731	3	29	subm	subm	PROPN
iajs-2731	3	30	.	.	PUNCT
iajs-2731	4	1	some	some	DET
iajs-2731	4	2	new	new	ADJ
iajs-2731	4	3	types	type	NOUN
iajs-2731	4	4	of	of	ADP
iajs-2731	4	5	hollow	hollow	ADJ
iajs-2731	4	6	modules	module	NOUN
iajs-2731	4	7	are	be	AUX
iajs-2731	4	8	provided	provide	VERB
iajs-2731	4	9	namely	namely	ADV
iajs-2731	4	10	,	,	PUNCT
iajs-2731	4	11	lochollow	lochollow	NOUN
iajs-2731	4	12	fmodules	fmodule	NOUN
iajs-2731	4	13	as	as	ADP
iajs-2731	4	14	a	a	DET
iajs-2731	4	15	strength	strength	NOUN
iajs-2731	4	16	of	of	ADP
iajs-2731	4	17	the	the	DET
iajs-2731	4	18	hollow	hollow	ADJ
iajs-2731	4	19	module	module	NOUN
iajs-2731	4	20	,	,	PUNCT
iajs-2731	4	21	where	where	SCONJ
iajs-2731	4	22	every	every	DET
iajs-2731	4	23	lochollow	lochollow	NOUN
iajs-2731	4	24	fmodule	fmodule	NOUN
iajs-2731	4	25	is	be	AUX
iajs-2731	4	26	a	a	DET
iajs-2731	4	27	hollow	hollow	ADJ
iajs-2731	4	28	module	module	NOUN
iajs-2731	4	29	,	,	PUNCT
iajs-2731	4	30	but	but	CCONJ
iajs-2731	4	31	the	the	DET
iajs-2731	4	32	converse	converse	NOUN
iajs-2731	4	33	is	be	AUX
iajs-2731	4	34	not	not	PART
iajs-2731	4	35	true	true	ADJ
iajs-2731	4	36	.	.	PUNCT
iajs-2731	5	1	many	many	ADJ
iajs-2731	5	2	properties	property	NOUN
iajs-2731	5	3	and	and	CCONJ
iajs-2731	5	4	characterizations	characterization	NOUN
iajs-2731	5	5	of	of	ADP
iajs-2731	5	6	these	these	DET
iajs-2731	5	7	concepts	concept	NOUN
iajs-2731	5	8	are	be	AUX
iajs-2731	5	9	proved	prove	VERB
iajs-2731	5	10	,	,	PUNCT
iajs-2731	5	11	also	also	ADV
iajs-2731	5	12	the	the	DET
iajs-2731	5	13	relationship	relationship	NOUN
iajs-2731	5	14	between	between	ADP
iajs-2731	5	15	all	all	DET
iajs-2731	5	16	these	these	DET
iajs-2731	5	17	types	type	NOUN
iajs-2731	5	18	is	be	AUX
iajs-2731	5	19	researched	research	VERB
iajs-2731	5	20	.	.	PUNCT
iajs-2731	6	1	many	many	ADJ
iajs-2731	6	2	important	important	ADJ
iajs-2731	6	3	results	result	NOUN
iajs-2731	6	4	that	that	PRON
iajs-2731	6	5	explain	explain	VERB
iajs-2731	6	6	this	this	DET
iajs-2731	6	7	relationship	relationship	NOUN
iajs-2731	6	8	are	be	AUX
iajs-2731	6	9	demonstrated	demonstrate	VERB
iajs-2731	6	10	also	also	ADV
iajs-2731	6	11	several	several	ADJ
iajs-2731	6	12	characterizations	characterization	NOUN
iajs-2731	6	13	and	and	CCONJ
iajs-2731	6	14	properties	property	NOUN
iajs-2731	6	15	related	relate	VERB
iajs-2731	6	16	to	to	ADP
iajs-2731	6	17	these	these	DET
iajs-2731	6	18	concepts	concept	NOUN
iajs-2731	6	19	are	be	AUX
iajs-2731	6	20	given	give	VERB
iajs-2731	6	21	.	.	PUNCT
iajs-2731	7	1	keywords	keyword	NOUN
iajs-2731	7	2	:	:	PUNCT
iajs-2731	7	3	fmodules	fmodule	NOUN
iajs-2731	7	4	,	,	PUNCT
iajs-2731	7	5	small	small	ADJ
iajs-2731	7	6	fsubm	fsubm	NOUN
iajs-2731	7	7	,	,	PUNCT
iajs-2731	7	8	maximal	maximal	ADJ
iajs-2731	7	9	f	f	PROPN
iajs-2731	7	10	-	-	PUNCT
iajs-2731	7	11	subm	subm	PROPN
iajs-2731	7	12	,	,	PUNCT
iajs-2731	7	13	hollow	hollow	ADJ
iajs-2731	7	14	fmodules	fmodule	NOUN
iajs-2731	7	15	,	,	PUNCT
iajs-2731	7	16	lochollow	lochollow	NOUN
iajs-2731	7	17	fmodules	fmodule	NOUN
iajs-2731	7	18	.	.	PUNCT
iajs-2731	8	1	𝟏.	𝟏.	X
iajs-2731	8	2	𝐈ntroduction	𝐈ntroduction	NOUN
iajs-2731	8	3	call	call	NOUN
iajs-2731	8	4	m	m	VERB
iajs-2731	8	5	is	be	AUX
iajs-2731	8	6	l	l	ADJ
iajs-2731	8	7	-	-	ADJ
iajs-2731	8	8	hollow	hollow	ADJ
iajs-2731	8	9	module	module	NOUN
iajs-2731	8	10	where	where	SCONJ
iajs-2731	8	11	a	a	DET
iajs-2731	8	12	module	module	NOUN
iajs-2731	8	13	has	have	VERB
iajs-2731	8	14	a	a	DET
iajs-2731	8	15	unique	unique	ADJ
iajs-2731	8	16	maximal	maximal	ADJ
iajs-2731	8	17	submodule	submodule	NOUN
iajs-2731	8	18	that	that	PRON
iajs-2731	8	19	contains	contain	VERB
iajs-2731	8	20	all	all	DET
iajs-2731	8	21	small	small	ADJ
iajs-2731	8	22	submodules	submodule	NOUN
iajs-2731	8	23	of	of	ADP
iajs-2731	8	24	m	m	PROPN
iajs-2731	8	25	[	[	X
iajs-2731	8	26	10	10	NUM
iajs-2731	8	27	]	]	PUNCT
iajs-2731	8	28	.	.	PUNCT
iajs-2731	9	1	in	in	ADP
iajs-2731	9	2	this	this	DET
iajs-2731	9	3	paper	paper	NOUN
iajs-2731	9	4	,	,	PUNCT
iajs-2731	9	5	we	we	PRON
iajs-2731	9	6	fuzzify	fuzzify	VERB
iajs-2731	9	7	these	these	DET
iajs-2731	9	8	concepts	concept	NOUN
iajs-2731	9	9	from	from	ADP
iajs-2731	9	10	l	l	ADJ
iajs-2731	9	11	-	-	ADJ
iajs-2731	9	12	hollow	hollow	ADJ
iajs-2731	9	13	module	module	NOUN
iajs-2731	9	14	to	to	AUX
iajs-2731	9	15	loc	loc	VERB
iajs-2731	9	16	hollow	hollow	ADJ
iajs-2731	9	17	fmodules	fmodule	NOUN
iajs-2731	9	18	.	.	PUNCT
iajs-2731	10	1	moreover	moreover	ADV
iajs-2731	10	2	,	,	PUNCT
iajs-2731	10	3	we	we	PRON
iajs-2731	10	4	generalize	generalize	VERB
iajs-2731	10	5	numerous	numerous	ADJ
iajs-2731	10	6	properties	property	NOUN
iajs-2731	10	7	of	of	ADP
iajs-2731	10	8	lochollow	lochollow	NOUN
iajs-2731	10	9	fmodules	fmodule	NOUN
iajs-2731	10	10	.	.	PUNCT
iajs-2731	11	1	this	this	DET
iajs-2731	11	2	work	work	NOUN
iajs-2731	11	3	contains	contain	VERB
iajs-2731	11	4	four	four	NUM
iajs-2731	11	5	parts	part	NOUN
iajs-2731	11	6	.	.	PUNCT
iajs-2731	12	1	𝜤n	𝜤n	DET
iajs-2731	12	2	part	part	NOUN
iajs-2731	12	3	one	one	NUM
iajs-2731	12	4	,	,	PUNCT
iajs-2731	12	5	we	we	PRON
iajs-2731	12	6	recollect	recollect	VERB
iajs-2731	12	7	several	several	ADJ
iajs-2731	12	8	𝑑efinitions	𝑑efinition	NOUN
iajs-2731	12	9	and	and	CCONJ
iajs-2731	12	10	𝜌roperties	𝜌ropertie	NOUN
iajs-2731	12	11	that	that	PRON
iajs-2731	12	12	are	be	AUX
iajs-2731	12	13	useful	useful	ADJ
iajs-2731	12	14	are	be	AUX
iajs-2731	12	15	needed	need	VERB
iajs-2731	12	16	later	later	ADV
iajs-2731	12	17	.	.	PUNCT
iajs-2731	13	1	𝜤n	𝜤n	DET
iajs-2731	13	2	part	part	NOUN
iajs-2731	13	3	two	two	NUM
iajs-2731	13	4	,	,	PUNCT
iajs-2731	13	5	several	several	ADJ
iajs-2731	13	6	fundamental	fundamental	ADJ
iajs-2731	13	7	ρroperties	ρropertie	NOUN
iajs-2731	13	8	of	of	ADP
iajs-2731	13	9	l	l	NOUN
iajs-2731	13	10	-	-	ADJ
iajs-2731	13	11	hollow	hollow	ADJ
iajs-2731	13	12	fmodules	fmodule	NOUN
iajs-2731	13	13	are	be	AUX
iajs-2731	13	14	argued	argue	VERB
iajs-2731	13	15	.	.	PUNCT
iajs-2731	14	1	part	part	NOUN
iajs-2731	14	2	three	three	NUM
iajs-2731	14	3	includes	include	VERB
iajs-2731	14	4	the	the	DET
iajs-2731	14	5	relation	relation	NOUN
iajs-2731	14	6	between	between	ADP
iajs-2731	14	7	hollow	hollow	ADJ
iajs-2731	14	8	fmodules	fmodule	NOUN
iajs-2731	14	9	,	,	PUNCT
iajs-2731	14	10	and	and	CCONJ
iajs-2731	14	11	lochollow	lochollow	NOUN
iajs-2731	14	12	fmodules	fmodule	NOUN
iajs-2731	14	13	.	.	PUNCT
iajs-2731	15	1	finally	finally	ADV
iajs-2731	15	2	part	part	NOUN
iajs-2731	15	3	four	four	NUM
iajs-2731	15	4	,	,	PUNCT
iajs-2731	15	5	we	we	PRON
iajs-2731	15	6	shall	shall	AUX
iajs-2731	15	7	give	give	VERB
iajs-2731	15	8	the	the	DET
iajs-2731	15	9	relation	relation	NOUN
iajs-2731	15	10	between	between	ADP
iajs-2731	15	11	loc	loc	NOUN
iajs-2731	15	12	-	-	ADJ
iajs-2731	15	13	hollow	hollow	ADJ
iajs-2731	15	14	fmodules	fmodule	NOUN
iajs-2731	15	15	and	and	CCONJ
iajs-2731	15	16	different	different	ADJ
iajs-2731	15	17	modules	module	NOUN
iajs-2731	15	18	like	like	VERB
iajs-2731	15	19	as	as	SCONJ
iajs-2731	15	20	amply	amply	ADV
iajs-2731	15	21	supplemented	supplement	VERB
iajs-2731	15	22	fmodules	fmodule	NOUN
iajs-2731	15	23	,	,	PUNCT
iajs-2731	15	24	indecomposable	indecomposable	ADJ
iajs-2731	15	25	fmodules	fmodule	NOUN
iajs-2731	15	26	,	,	PUNCT
iajs-2731	15	27	and	and	CCONJ
iajs-2731	15	28	lifting	lift	VERB
iajs-2731	15	29	fmodules	fmodule	NOUN
iajs-2731	15	30	1	1	NUM
iajs-2731	15	31	.	.	PUNCT
iajs-2731	15	32	preliminaries	preliminary	NOUN
iajs-2731	15	33	:	:	PUNCT
iajs-2731	15	34	1.1	1.1	NUM
iajs-2731	15	35	definition	definition	NOUN
iajs-2731	15	36	[	[	X
iajs-2731	15	37	1	1	X
iajs-2731	15	38	]	]	X
iajs-2731	15	39	let	let	VERB
iajs-2731	15	40	m	m	PRON
iajs-2731	15	41	≠	≠	PROPN
iajs-2731	15	42	∅	∅	NOUN
iajs-2731	15	43	,	,	PUNCT
iajs-2731	15	44	let	let	VERB
iajs-2731	15	45	ɨ	ɨ	PRON
iajs-2731	15	46	be	be	AUX
iajs-2731	15	47	the	the	DET
iajs-2731	15	48	closed	closed	ADJ
iajs-2731	15	49	interval	interval	NOUN
iajs-2731	15	50	[	[	X
iajs-2731	15	51	0	0	NUM
iajs-2731	15	52	,	,	PUNCT
iajs-2731	15	53	1	1	NUM
iajs-2731	15	54	]	]	PUNCT
iajs-2731	15	55	on	on	ADP
iajs-2731	15	56	the	the	DET
iajs-2731	15	57	real	real	ADJ
iajs-2731	15	58	line	line	NOUN
iajs-2731	15	59	(	(	PUNCT
iajs-2731	15	60	real	real	ADJ
iajs-2731	15	61	number	number	NOUN
iajs-2731	15	62	)	)	PUNCT
iajs-2731	15	63	,	,	PUNCT
iajs-2731	15	64	f	f	PROPN
iajs-2731	15	65	set	set	VERB
iajs-2731	15	66	a	a	PRON
iajs-2731	15	67	in	in	ADP
iajs-2731	15	68	m	m	PROPN
iajs-2731	15	69	(	(	PUNCT
iajs-2731	15	70	a	a	DET
iajs-2731	15	71	fsubset	fsubset	NOUN
iajs-2731	15	72	a	a	PRON
iajs-2731	15	73	of	of	ADP
iajs-2731	15	74	m	m	PRON
iajs-2731	15	75	)	)	PUNCT
iajs-2731	15	76	is	be	AUX
iajs-2731	15	77	fun	fun	ADJ
iajs-2731	15	78	from	from	ADP
iajs-2731	15	79	x	x	X
iajs-2731	15	80	to	to	ADP
iajs-2731	15	81	ɨ	ɨ	PRON
iajs-2731	15	82	.	.	PUNCT
iajs-2731	15	83	''	''	PUNCT
iajs-2731	16	1	the	the	DET
iajs-2731	16	2	following	follow	VERB
iajs-2731	16	3	example	example	NOUN
iajs-2731	16	4	describes	describe	VERB
iajs-2731	16	5	the	the	DET
iajs-2731	16	6	above	above	ADJ
iajs-2731	16	7	definition	definition	NOUN
iajs-2731	16	8	:	:	PUNCT
iajs-2731	16	9	ibn	ibn	PROPN
iajs-2731	16	10	al	al	PROPN
iajs-2731	16	11	haitham	haitham	PROPN
iajs-2731	16	12	journal	journal	PROPN
iajs-2731	16	13	for	for	ADP
iajs-2731	16	14	pure	pure	ADJ
iajs-2731	16	15	and	and	CCONJ
iajs-2731	16	16	applied	applied	ADJ
iajs-2731	16	17	sciences	sciences	PROPN
iajs-2731	16	18	journal	journal	PROPN
iajs-2731	16	19	homepage	homepage	NOUN
iajs-2731	16	20	:	:	PUNCT
iajs-2731	16	21	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2731	16	22	doi	doi	NOUN
iajs-2731	16	23	:	:	PUNCT
iajs-2731	16	24	10.30526/35.2.2731	10.30526/35.2.2731	PROPN
iajs-2731	16	25	article	article	NOUN
iajs-2731	16	26	history	history	NOUN
iajs-2731	16	27	:	:	PUNCT
iajs-2731	16	28	received	receive	VERB
iajs-2731	16	29	21	21	NUM
iajs-2731	16	30	,	,	PUNCT
iajs-2731	16	31	november	november	PROPN
iajs-2731	16	32	,	,	PUNCT
iajs-2731	16	33	2021	2021	NUM
iajs-2731	16	34	,	,	PUNCT
iajs-2731	16	35	accepted,25	accepted,25	PROPN
iajs-2731	16	36	,	,	PUNCT
iajs-2731	16	37	january	january	PROPN
iajs-2731	16	38	,	,	PUNCT
iajs-2731	16	39	2022	2022	NUM
iajs-2731	16	40	,	,	PUNCT
iajs-2731	16	41	published	publish	VERB
iajs-2731	16	42	in	in	ADP
iajs-2731	16	43	april	april	PROPN
iajs-2731	16	44	2022	2022	NUM
iajs-2731	16	45	.	.	PUNCT
iajs-2731	17	1	shaheed	shaheed	PROPN
iajs-2731	17	2	jameel	jameel	PROPN
iajs-2731	17	3	kharbeet	kharbeet	PROPN
iajs-2731	17	4	depatment	depatment	PROPN
iajs-2731	17	5	of	of	ADP
iajs-2731	17	6	mthmatics	mthmatic	NOUN
iajs-2731	17	7	,	,	PUNCT
iajs-2731	17	8	college	college	NOUN
iajs-2731	17	9	of	of	ADP
iajs-2731	17	10	education	education	NOUN
iajs-2731	17	11	of	of	ADP
iajs-2731	17	12	pure	pure	ADJ
iajs-2731	17	13	science	science	NOUN
iajs-2731	17	14	,	,	PUNCT
iajs-2731	17	15	ibn	ibn	PROPN
iajs-2731	17	16	alhaitham	alhaitham	NOUN
iajs-2731	17	17	,	,	PUNCT
iajs-2731	17	18	university	university	NOUN
iajs-2731	17	19	of	of	ADP
iajs-2731	17	20	baghdad	baghdad	PROPN
iajs-2731	17	21	,	,	PUNCT
iajs-2731	17	22	baghdad	baghdad	PROPN
iajs-2731	17	23	–	–	PUNCT
iajs-2731	17	24	iraq	iraq	PROPN
iajs-2731	17	25	.	.	PUNCT
iajs-2731	18	1	almaarif	almaarif	PROPN
iajs-2731	18	2	university	university	PROPN
iajs-2731	18	3	college	college	PROPN
iajs-2731	18	4	,	,	PUNCT
iajs-2731	18	5	ramadi	ramadi	PROPN
iajs-2731	18	6	–	–	PUNCT
iajs-2731	18	7	iraq	iraq	PROPN
iajs-2731	18	8	.	.	PUNCT
iajs-2731	19	1	jkharbeet@uoa.edu.iq	jkharbeet@uoa.edu.iq	ADJ
iajs-2731	19	2	hatam	hatam	PROPN
iajs-2731	19	3	yahya	yahya	PROPN
iajs-2731	19	4	khalf	khalf	PROPN
iajs-2731	19	5	depatment	depatment	NOUN
iajs-2731	19	6	of	of	ADP
iajs-2731	19	7	mthmatics	mthmatic	NOUN
iajs-2731	19	8	,	,	PUNCT
iajs-2731	19	9	college	college	NOUN
iajs-2731	19	10	of	of	ADP
iajs-2731	19	11	education	education	NOUN
iajs-2731	19	12	of	of	ADP
iajs-2731	19	13	pure	pure	ADJ
iajs-2731	19	14	science	science	NOUN
iajs-2731	19	15	,	,	PUNCT
iajs-2731	19	16	ibn	ibn	PROPN
iajs-2731	19	17	alhaitham	alhaitham	NOUN
iajs-2731	19	18	,	,	PUNCT
iajs-2731	19	19	university	university	NOUN
iajs-2731	19	20	of	of	ADP
iajs-2731	19	21	baghdad	baghdad	PROPN
iajs-2731	19	22	,	,	PUNCT
iajs-2731	19	23	baghdad	baghdad	PROPN
iajs-2731	19	24	–	–	PUNCT
iajs-2731	19	25	iraq	iraq	PROPN
iajs-2731	19	26	.	.	PUNCT
iajs-2731	20	1	dr.hatamyahya@yahoo.com	dr.hatamyahya@yahoo.com	PROPN
iajs-2731	20	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2731	20	3	mailto:kharbeet@uoa.edu.iq	mailto:kharbeet@uoa.edu.iq	PROPN
iajs-2731	20	4	mailto:dr.hatamyahya@yahoo.com	mailto:dr.hatamyahya@yahoo.com	PROPN
iajs-2731	20	5	ibn	ibn	PROPN
iajs-2731	20	6	al	al	PROPN
iajs-2731	20	7	-	-	PUNCT
iajs-2731	20	8	haitham	haitham	PROPN
iajs-2731	20	9	jour	jour	X
iajs-2731	20	10	.	.	PROPN
iajs-2731	20	11	for	for	ADP
iajs-2731	20	12	pure	pure	ADJ
iajs-2731	20	13	&	&	CCONJ
iajs-2731	20	14	appl	appl	PROPN
iajs-2731	20	15	.	.	PUNCT
iajs-2731	21	1	sci	sci	PROPN
iajs-2731	21	2	.	.	PROPN
iajs-2731	22	1	53	53	NUM
iajs-2731	22	2	(	(	PUNCT
iajs-2731	22	3	2)2022	2)2022	NUM
iajs-2731	22	4	85	85	NUM
iajs-2731	22	5	1.2	1.2	NUM
iajs-2731	22	6	example	example	NOUN
iajs-2731	22	7	[	[	X
iajs-2731	22	8	1	1	X
iajs-2731	22	9	]	]	PUNCT
iajs-2731	22	10	let	let	AUX
iajs-2731	22	11	m	m	PRON
iajs-2731	22	12	be	be	AUX
iajs-2731	22	13	the	the	DET
iajs-2731	22	14	real	real	ADJ
iajs-2731	22	15	line	line	NOUN
iajs-2731	22	16	r	r	NOUN
iajs-2731	22	17	and	and	CCONJ
iajs-2731	22	18	a	a	DET
iajs-2731	22	19	be	be	AUX
iajs-2731	22	20	a	a	DET
iajs-2731	22	21	fuzzy	fuzzy	ADJ
iajs-2731	22	22	set	set	NOUN
iajs-2731	22	23	of	of	ADP
iajs-2731	22	24	numbers	number	NOUN
iajs-2731	22	25	that	that	PRON
iajs-2731	22	26	is	be	AUX
iajs-2731	22	27	much	much	ADV
iajs-2731	22	28	greater	great	ADJ
iajs-2731	22	29	than	than	ADP
iajs-2731	22	30	i.	i.	NOUN
iajs-2731	22	31	then	then	ADV
iajs-2731	22	32	one	one	PRON
iajs-2731	22	33	can	can	AUX
iajs-2731	22	34	accord	accord	VERB
iajs-2731	22	35	an	an	DET
iajs-2731	22	36	accurate	accurate	ADJ
iajs-2731	22	37	characterization	characterization	NOUN
iajs-2731	22	38	of	of	ADP
iajs-2731	22	39	a	a	DET
iajs-2731	22	40	specifying	specifying	NOUN
iajs-2731	22	41	:	:	PUNCT
iajs-2731	22	42	a(x	a(x	NOUN
iajs-2731	22	43	)	)	PUNCT
iajs-2731	22	44	=	=	NOUN
iajs-2731	22	45	{	{	PUNCT
iajs-2731	22	46	1	1	NUM
iajs-2731	22	47	−	−	NUM
iajs-2731	22	48	1	1	NUM
iajs-2731	22	49	𝑥2	𝑥2	NOUN
iajs-2731	22	50	𝑖𝑓	𝑖𝑓	VERB
iajs-2731	22	51	𝑥	𝑥	NOUN
iajs-2731	22	52	>	>	X
iajs-2731	22	53	1	1	NUM
iajs-2731	22	54	0	0	NUM
iajs-2731	22	55	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	22	56	𝑥	𝑥	NOUN
iajs-2731	22	57	≤	≤	NUM
iajs-2731	22	58	1	1	NUM
iajs-2731	22	59	''	''	SYM
iajs-2731	22	60	1.3''definition	1.3''definition	NUM
iajs-2731	23	1	[	[	X
iajs-2731	23	2	2	2	NUM
iajs-2731	23	3	]	]	PUNCT
iajs-2731	23	4	let	let	VERB
iajs-2731	23	5	xt	xt	NUM
iajs-2731	23	6	:	:	PUNCT
iajs-2731	23	7	m	m	VERB
iajs-2731	23	8	→	→	SYM
iajs-2731	23	9	[	[	X
iajs-2731	23	10	0	0	NUM
iajs-2731	23	11	,	,	PUNCT
iajs-2731	23	12	1	1	NUM
iajs-2731	23	13	]	]	PUNCT
iajs-2731	23	14	be	be	AUX
iajs-2731	23	15	a	a	DET
iajs-2731	23	16	fuzzy	fuzzy	ADJ
iajs-2731	23	17	set	set	NOUN
iajs-2731	23	18	in	in	ADP
iajs-2731	23	19	m	m	PROPN
iajs-2731	23	20	,	,	PUNCT
iajs-2731	23	21	where	where	SCONJ
iajs-2731	23	22	x	x	SYM
iajs-2731	23	23	∈m	∈m	NOUN
iajs-2731	23	24	,	,	PUNCT
iajs-2731	23	25	t	t	PROPN
iajs-2731	23	26	∈	∈	PROPN
iajs-2731	23	27	(	(	PUNCT
iajs-2731	23	28	0,1	0,1	NOUN
iajs-2731	23	29	]	]	PUNCT
iajs-2731	23	30	defined	define	VERB
iajs-2731	23	31	by	by	ADP
iajs-2731	23	32	xt	xt	PROPN
iajs-2731	23	33	(	(	PUNCT
iajs-2731	23	34	y	y	NOUN
iajs-2731	23	35	)	)	PUNCT
iajs-2731	23	36	=	=	PRON
iajs-2731	23	37	{	{	PUNCT
iajs-2731	24	1	1	1	NUM
iajs-2731	24	2	𝑖𝑓	𝑖𝑓	NUM
iajs-2731	24	3	𝑡	𝑡	PROPN
iajs-2731	24	4	=	=	PROPN
iajs-2731	24	5	𝑦	𝑦	SYM
iajs-2731	24	6	0	0	NUM
iajs-2731	24	7	𝑖𝑓	𝑖𝑓	NUM
iajs-2731	24	8	𝑡	𝑡	PROPN
iajs-2731	24	9	≠	≠	PROPN
iajs-2731	24	10	𝑦	𝑦	NUM
iajs-2731	24	11	∀y∈	∀y∈	ADJ
iajs-2731	24	12	m.	m.	NOUN
iajs-2731	24	13	xt	xt	PROPN
iajs-2731	24	14	is	be	AUX
iajs-2731	24	15	said	say	VERB
iajs-2731	24	16	a	a	DET
iajs-2731	24	17	fuzzy	fuzzy	ADJ
iajs-2731	24	18	singleton	singleton	NOUN
iajs-2731	24	19	where	where	SCONJ
iajs-2731	24	20	x=	x=	PROPN
iajs-2731	24	21	0	0	NUM
iajs-2731	24	22	,	,	PUNCT
iajs-2731	24	23	t	t	NOUN
iajs-2731	24	24	=	=	SYM
iajs-2731	24	25	1	1	NUM
iajs-2731	24	26	,	,	PUNCT
iajs-2731	24	27	therefor	therefor	ADP
iajs-2731	24	28	01	01	NUM
iajs-2731	24	29	(	(	PUNCT
iajs-2731	24	30	y	y	NOUN
iajs-2731	24	31	)	)	PUNCT
iajs-2731	24	32	=	=	PRON
iajs-2731	24	33	{	{	PUNCT
iajs-2731	25	1	1	1	NUM
iajs-2731	25	2	𝑖𝑓	𝑖𝑓	NUM
iajs-2731	25	3	𝑦	𝑦	NOUN
iajs-2731	25	4	=	=	NOUN
iajs-2731	25	5	0	0	NUM
iajs-2731	25	6	0	0	NUM
iajs-2731	25	7	𝑖𝑓	𝑖𝑓	PROPN
iajs-2731	25	8	𝑦	𝑦	NOUN
iajs-2731	25	9	≠	≠	PROPN
iajs-2731	25	10	0	0	NUM
iajs-2731	25	11	,	,	PUNCT
iajs-2731	25	12	is	be	AUX
iajs-2731	25	13	fuzzy	fuzzy	ADJ
iajs-2731	25	14	singleton	singleton	NOUN
iajs-2731	25	15	fuzzy	fuzzy	ADJ
iajs-2731	25	16	zero	zero	NUM
iajs-2731	25	17	singleton	singleton	NOUN
iajs-2731	25	18	''	''	PUNCT
iajs-2731	25	19	1.4	1.4	NUM
iajs-2731	25	20	definition	definition	NOUN
iajs-2731	25	21	[	[	X
iajs-2731	25	22	3	3	X
iajs-2731	25	23	]	]	X
iajs-2731	25	24	let	let	VERB
iajs-2731	25	25	u	u	PRON
iajs-2731	25	26	and	and	CCONJ
iajs-2731	25	27	p	p	NOUN
iajs-2731	25	28	be	be	AUX
iajs-2731	25	29	two	two	NUM
iajs-2731	25	30	fsets	fset	NOUN
iajs-2731	25	31	inμ	inμ	NOUN
iajs-2731	25	32	,	,	PUNCT
iajs-2731	25	33	then	then	ADV
iajs-2731	25	34	:	:	PUNCT
iajs-2731	25	35	1	1	NUM
iajs-2731	25	36	(	(	PUNCT
iajs-2731	25	37	u	u	NOUN
iajs-2731	25	38	⋃	⋃	PROPN
iajs-2731	25	39	p	p	NOUN
iajs-2731	25	40	)	)	PUNCT
iajs-2731	25	41	(	(	PUNCT
iajs-2731	25	42	x)=	x)=	PROPN
iajs-2731	25	43	max	max	PROPN
iajs-2731	25	44	{	{	PUNCT
iajs-2731	25	45	u(x	u(x	PROPN
iajs-2731	25	46	)	)	PUNCT
iajs-2731	25	47	,	,	PUNCT
iajs-2731	25	48	p(x	p(x	PROPN
iajs-2731	25	49	)	)	PUNCT
iajs-2731	25	50	}	}	PUNCT
iajs-2731	25	51	,	,	PUNCT
iajs-2731	25	52	∀x	∀x	X
iajs-2731	25	53	∈	∈	PROPN
iajs-2731	25	54	m.	m.	NOUN
iajs-2731	25	55	2	2	NUM
iajs-2731	25	56	(	(	PUNCT
iajs-2731	25	57	u	u	NOUN
iajs-2731	25	58	⋂	⋂	PROPN
iajs-2731	25	59	p	p	NOUN
iajs-2731	25	60	)	)	PUNCT
iajs-2731	25	61	(	(	PUNCT
iajs-2731	25	62	x)=	x)=	PROPN
iajs-2731	25	63	min	min	PROPN
iajs-2731	25	64	{	{	PUNCT
iajs-2731	25	65	u(x	u(x	PROPN
iajs-2731	25	66	)	)	PUNCT
iajs-2731	25	67	,	,	PUNCT
iajs-2731	25	68	p(x	p(x	PROPN
iajs-2731	25	69	)	)	PUNCT
iajs-2731	25	70	}	}	PUNCT
iajs-2731	25	71	,	,	PUNCT
iajs-2731	25	72	∀	∀	PUNCT
iajs-2731	25	73	x	x	SYM
iajs-2731	25	74	∈	∈	NOUN
iajs-2731	25	75	m.	m.	NOUN
iajs-2731	25	76	(	(	PUNCT
iajs-2731	25	77	u	u	NOUN
iajs-2731	25	78	⋃	⋃	PROPN
iajs-2731	25	79	p	p	NOUN
iajs-2731	25	80	)	)	PUNCT
iajs-2731	25	81	a	a	NOUN
iajs-2731	25	82	,	,	PUNCT
iajs-2731	25	83	(	(	PUNCT
iajs-2731	25	84	u	u	NOUN
iajs-2731	25	85	⋂	⋂	PROPN
iajs-2731	25	86	p	p	NOUN
iajs-2731	25	87	)	)	PUNCT
iajs-2731	25	88	are	be	AUX
iajs-2731	25	89	fuzzy	fuzzy	ADJ
iajs-2731	25	90	sets	set	NOUN
iajs-2731	25	91	in	in	ADP
iajs-2731	25	92	m	m	PROPN
iajs-2731	25	93	in	in	ADP
iajs-2731	25	94	general	general	ADJ
iajs-2731	25	95	if	if	SCONJ
iajs-2731	25	96	{	{	PUNCT
iajs-2731	25	97	ua	ua	X
iajs-2731	25	98	,	,	PUNCT
iajs-2731	25	99	a	a	DET
iajs-2731	25	100	∈	∈	PROPN
iajs-2731	25	101	a	a	PRON
iajs-2731	25	102	}	}	PUNCT
iajs-2731	25	103	,	,	PUNCT
iajs-2731	25	104	is	be	AUX
iajs-2731	25	105	u	u	PROPN
iajs-2731	25	106	family	family	NOUN
iajs-2731	25	107	of	of	ADP
iajs-2731	25	108	fuzzy	fuzzy	ADJ
iajs-2731	25	109	sets	set	NOUN
iajs-2731	25	110	in	in	ADP
iajs-2731	25	111	m	m	PROPN
iajs-2731	25	112	,	,	PUNCT
iajs-2731	25	113	then	then	ADV
iajs-2731	25	114	(	(	PUNCT
iajs-2731	25	115	(	(	PUNCT
iajs-2731	25	116	⋂	⋂	PROPN
iajs-2731	25	117	a	a	PRON
iajs-2731	25	118	∈	∈	PROPN
iajs-2731	25	119	a	a	DET
iajs-2731	25	120	)	)	PUNCT
iajs-2731	25	121	𝑣𝑎	𝑣𝑎	NOUN
iajs-2731	25	122	)	)	PUNCT
iajs-2731	25	123	(	(	PUNCT
iajs-2731	25	124	x	x	X
iajs-2731	25	125	)	)	PUNCT
iajs-2731	25	126	=	=	SYM
iajs-2731	25	127	inf	inf	PROPN
iajs-2731	25	128	{	{	PUNCT
iajs-2731	25	129	ua	ua	PROPN
iajs-2731	25	130	(	(	PUNCT
iajs-2731	25	131	x	x	NOUN
iajs-2731	25	132	)	)	PUNCT
iajs-2731	25	133	,	,	PUNCT
iajs-2731	25	134	a	a	DET
iajs-2731	25	135	∈	∈	PROPN
iajs-2731	25	136	a	a	PRON
iajs-2731	25	137	}	}	PUNCT
iajs-2731	25	138	,	,	PUNCT
iajs-2731	25	139	∀	∀	PUNCT
iajs-2731	25	140	x	x	SYM
iajs-2731	25	141	∈	∈	PROPN
iajs-2731	25	142	s.	s.	PROPN
iajs-2731	25	143	(	(	PUNCT
iajs-2731	25	144	(	(	PUNCT
iajs-2731	25	145	⋃	⋃	ADP
iajs-2731	25	146	a	a	DET
iajs-2731	25	147	∈	∈	PROPN
iajs-2731	25	148	a	a	DET
iajs-2731	25	149	)	)	PUNCT
iajs-2731	25	150	𝑣𝑎	𝑣𝑎	NOUN
iajs-2731	25	151	)	)	PUNCT
iajs-2731	25	152	(	(	PUNCT
iajs-2731	25	153	x	x	X
iajs-2731	25	154	)	)	PUNCT
iajs-2731	25	155	=	=	SYM
iajs-2731	25	156	sup	sup	X
iajs-2731	25	157	{	{	PUNCT
iajs-2731	25	158	ua	ua	PROPN
iajs-2731	25	159	(	(	PUNCT
iajs-2731	25	160	x	x	NOUN
iajs-2731	25	161	)	)	PUNCT
iajs-2731	25	162	,	,	PUNCT
iajs-2731	25	163	a	a	DET
iajs-2731	25	164	∈	∈	PROPN
iajs-2731	25	165	a	a	PRON
iajs-2731	25	166	}	}	PUNCT
iajs-2731	25	167	,	,	PUNCT
iajs-2731	25	168	∀x	∀x	X
iajs-2731	25	169	∈	∈	PROPN
iajs-2731	25	170	s.	s.	PROPN
iajs-2731	25	171	''	''	PUNCT
iajs-2731	25	172	1.5	1.5	NUM
iajs-2731	25	173	definition	definition	NOUN
iajs-2731	25	174	[	[	X
iajs-2731	25	175	5	5	NUM
iajs-2731	25	176	]	]	X
iajs-2731	25	177	[	[	X
iajs-2731	25	178	4	4	X
iajs-2731	25	179	]	]	X
iajs-2731	25	180	if	if	SCONJ
iajs-2731	25	181	u	u	NOUN
iajs-2731	25	182	is	be	AUX
iajs-2731	25	183	fset	fset	VERB
iajs-2731	25	184	in	in	ADP
iajs-2731	25	185	m	m	PROPN
iajs-2731	25	186	,	,	PUNCT
iajs-2731	25	187	for	for	ADP
iajs-2731	25	188	all	all	DET
iajs-2731	25	189	t	t	NOUN
iajs-2731	25	190	∈	∈	PROPN
iajs-2731	25	191	(	(	PUNCT
iajs-2731	25	192	0,1].the	0,1].the	PRON
iajs-2731	25	193	set	set	NOUN
iajs-2731	25	194	ut	ut	PROPN
iajs-2731	25	195	=	=	PRON
iajs-2731	25	196	{	{	PUNCT
iajs-2731	25	197	x	x	PUNCT
iajs-2731	25	198	∈	∈	PROPN
iajs-2731	25	199	m.	m.	NOUN
iajs-2731	25	200	,	,	PUNCT
iajs-2731	25	201	u(x	u(x	PROPN
iajs-2731	25	202	)	)	PUNCT
iajs-2731	25	203	≥	≥	NOUN
iajs-2731	25	204	t	t	PROPN
iajs-2731	25	205	}	}	PUNCT
iajs-2731	25	206	,	,	PUNCT
iajs-2731	25	207	is	be	AUX
iajs-2731	25	208	named	name	VERB
iajs-2731	25	209	a	a	DET
iajs-2731	25	210	level	level	NOUN
iajs-2731	25	211	subset	subset	NOUN
iajs-2731	25	212	of	of	ADP
iajs-2731	25	213	u.	u.	PROPN
iajs-2731	25	214	note	note	VERB
iajs-2731	25	215	that	that	SCONJ
iajs-2731	25	216	,	,	PUNCT
iajs-2731	25	217	ut	ut	PROPN
iajs-2731	25	218	is	be	AUX
iajs-2731	25	219	a	a	DET
iajs-2731	25	220	subset	subset	NOUN
iajs-2731	25	221	of	of	ADP
iajs-2731	25	222	m	m	PROPN
iajs-2731	25	223	in	in	ADP
iajs-2731	25	224	the	the	DET
iajs-2731	25	225	ordinary	ordinary	ADJ
iajs-2731	25	226	sense	sense	NOUN
iajs-2731	25	227	.	.	PUNCT
iajs-2731	25	228	''	''	PUNCT
iajs-2731	26	1	1.6	1.6	NUM
iajs-2731	26	2	remark	remark	NOUN
iajs-2731	26	3	[	[	X
iajs-2731	26	4	1	1	NUM
iajs-2731	26	5	]	]	PUNCT
iajs-2731	26	6	let	let	NOUN
iajs-2731	26	7	v	v	NOUN
iajs-2731	26	8	,	,	PUNCT
iajs-2731	26	9	p	p	NOUN
iajs-2731	26	10	are	be	AUX
iajs-2731	26	11	two	two	NUM
iajs-2731	26	12	fsets	fset	NOUN
iajs-2731	26	13	inμ	inμ	NOUN
iajs-2731	26	14	,	,	PUNCT
iajs-2731	26	15	then	then	ADV
iajs-2731	26	16	:	:	PUNCT
iajs-2731	26	17	1-(u	1-(u	NUM
iajs-2731	26	18	⋃	⋃	NOUN
iajs-2731	26	19	p	p	NOUN
iajs-2731	26	20	)	)	PUNCT
iajs-2731	26	21	t	t	PROPN
iajs-2731	26	22	=	=	SYM
iajs-2731	26	23	ut	ut	PROPN
iajs-2731	26	24	⋃	⋃	PROPN
iajs-2731	26	25	pt	pt	X
iajs-2731	26	26	}	}	PUNCT
iajs-2731	26	27	,	,	PUNCT
iajs-2731	26	28	for	for	ADP
iajs-2731	26	29	any	any	DET
iajs-2731	26	30	t	t	NOUN
iajs-2731	26	31	∈	∈	PROPN
iajs-2731	26	32	(	(	PUNCT
iajs-2731	26	33	0	0	NUM
iajs-2731	26	34	,	,	PUNCT
iajs-2731	26	35	1	1	NUM
iajs-2731	26	36	]	]	PUNCT
iajs-2731	26	37	.	.	PUNCT
iajs-2731	27	1	2-(u	2-(u	NUM
iajs-2731	28	1	⋂	⋂	PROPN
iajs-2731	28	2	p	p	NOUN
iajs-2731	28	3	)	)	PUNCT
iajs-2731	28	4	t	t	PROPN
iajs-2731	28	5	=	=	SYM
iajs-2731	28	6	ut	ut	PROPN
iajs-2731	28	7	⋂	⋂	PROPN
iajs-2731	28	8	pt	pt	PROPN
iajs-2731	28	9	for	for	ADP
iajs-2731	28	10	any	any	DET
iajs-2731	28	11	t	t	NOUN
iajs-2731	28	12	∈	∈	PROPN
iajs-2731	28	13	(	(	PUNCT
iajs-2731	28	14	0	0	NUM
iajs-2731	28	15	,	,	PUNCT
iajs-2731	28	16	1	1	NUM
iajs-2731	28	17	]	]	PUNCT
iajs-2731	28	18	.	.	PUNCT
iajs-2731	29	1	3u	3u	NUM
iajs-2731	29	2	=	=	SYM
iajs-2731	30	1	p	p	X
iajs-2731	30	2	if	if	SCONJ
iajs-2731	30	3	and	and	CCONJ
iajs-2731	30	4	only	only	ADV
iajs-2731	30	5	if	if	SCONJ
iajs-2731	30	6	ut	ut	PROPN
iajs-2731	30	7	=	=	PROPN
iajs-2731	30	8	pt	pt	PROPN
iajs-2731	30	9	for	for	ADP
iajs-2731	30	10	any	any	DET
iajs-2731	30	11	t	t	NOUN
iajs-2731	30	12	∈	∈	PROPN
iajs-2731	30	13	(	(	PUNCT
iajs-2731	30	14	0	0	NUM
iajs-2731	30	15	,	,	PUNCT
iajs-2731	30	16	1	1	NUM
iajs-2731	30	17	]	]	PUNCT
iajs-2731	30	18	.	.	PUNCT
iajs-2731	30	19	''	''	PUNCT
iajs-2731	31	1	1.7	1.7	NUM
iajs-2731	31	2	definition	definition	NOUN
iajs-2731	31	3	[	[	X
iajs-2731	31	4	1	1	X
iajs-2731	31	5	]	]	PUNCT
iajs-2731	31	6	let	let	VERB
iajs-2731	31	7	u	u	NOUN
iajs-2731	31	8	is	be	AUX
iajs-2731	31	9	fsets	fset	NOUN
iajs-2731	31	10	inμ	inμ	NOUN
iajs-2731	31	11	,	,	PUNCT
iajs-2731	31	12	u	u	PROPN
iajs-2731	31	13	is	be	AUX
iajs-2731	31	14	named	name	VERB
iajs-2731	31	15	empty	empty	ADJ
iajs-2731	31	16	fuzzy	fuzzy	ADJ
iajs-2731	31	17	set	set	NOUN
iajs-2731	31	18	,	,	PUNCT
iajs-2731	31	19	denoted	denote	VERB
iajs-2731	31	20	by	by	ADP
iajs-2731	31	21	𝜃	𝜃	PRON
iajs-2731	31	22	if	if	SCONJ
iajs-2731	31	23	⟺	⟺	PRON
iajs-2731	31	24	u(x	u(x	VERB
iajs-2731	31	25	)	)	PUNCT
iajs-2731	31	26	=	=	SYM
iajs-2731	31	27	0	0	NUM
iajs-2731	31	28	,	,	PUNCT
iajs-2731	31	29	∀	∀	X
iajs-2731	31	30	x∈	x∈	NOUN
iajs-2731	31	31	𝑀.	𝑀.	PROPN
iajs-2731	31	32	''	''	PUNCT
iajs-2731	31	33	1.8	1.8	NUM
iajs-2731	31	34	d𝒅efinition	d𝒅efinition	NOUN
iajs-2731	31	35	[	[	X
iajs-2731	31	36	2	2	X
iajs-2731	31	37	]	]	X
iajs-2731	31	38	[	[	X
iajs-2731	31	39	4	4	X
iajs-2731	31	40	]	]	PUNCT
iajs-2731	31	41	let	let	VERB
iajs-2731	31	42	m	m	PRON
iajs-2731	31	43	be	be	AUX
iajs-2731	31	44	an	an	DET
iajs-2731	31	45	řmodule	řmodule	NOUN
iajs-2731	31	46	,	,	PUNCT
iajs-2731	31	47	a	a	DET
iajs-2731	31	48	fsets	fset	NOUN
iajs-2731	31	49	x	x	PUNCT
iajs-2731	31	50	of	of	ADP
iajs-2731	31	51	m	m	PROPN
iajs-2731	31	52	is	be	AUX
iajs-2731	31	53	named	name	VERB
iajs-2731	31	54	is	be	AUX
iajs-2731	31	55	fmodule	fmodule	ADJ
iajs-2731	31	56	where	where	SCONJ
iajs-2731	31	57	:	:	PUNCT
iajs-2731	31	58	1a	1a	X
iajs-2731	31	59	(	(	PUNCT
iajs-2731	31	60	0	0	NUM
iajs-2731	31	61	)	)	PUNCT
iajs-2731	31	62	=	=	SYM
iajs-2731	32	1	1	1	NUM
iajs-2731	32	2	.	.	NOUN
iajs-2731	32	3	2	2	NUM
iajs-2731	32	4	-	-	PUNCT
iajs-2731	32	5	a(x	a(x	PRON
iajs-2731	32	6	–	–	PUNCT
iajs-2731	32	7	y	y	NOUN
iajs-2731	32	8	)	)	PUNCT
iajs-2731	32	9	≥	≥	PROPN
iajs-2731	32	10	min	min	NOUN
iajs-2731	32	11	{	{	PUNCT
iajs-2731	32	12	a	a	DET
iajs-2731	32	13	(	(	PUNCT
iajs-2731	32	14	x	x	NOUN
iajs-2731	32	15	)	)	PUNCT
iajs-2731	32	16	,	,	PUNCT
iajs-2731	32	17	a(y	a(y	PROPN
iajs-2731	32	18	)	)	PUNCT
iajs-2731	32	19	}	}	PUNCT
iajs-2731	32	20	.	.	PUNCT
iajs-2731	33	1	3	3	NUM
iajs-2731	33	2	-	-	SYM
iajs-2731	33	3	a	a	DET
iajs-2731	33	4	(	(	PUNCT
iajs-2731	33	5	rx	rx	NOUN
iajs-2731	33	6	)	)	PUNCT
iajs-2731	33	7	≥	≥	NOUN
iajs-2731	33	8	a	a	DET
iajs-2731	33	9	(	(	PUNCT
iajs-2731	33	10	x	x	NOUN
iajs-2731	33	11	)	)	PUNCT
iajs-2731	33	12	,	,	PUNCT
iajs-2731	33	13	∀x	∀x	VERB
iajs-2731	33	14	∈	∈	PROPN
iajs-2731	33	15	m	m	NOUN
iajs-2731	33	16	,	,	PUNCT
iajs-2731	33	17	𝛾	𝛾	PROPN
iajs-2731	33	18	∈	∈	PROPN
iajs-2731	33	19	ř	ř	X
iajs-2731	33	20	.	.	PUNCT
iajs-2731	33	21	''	''	PUNCT
iajs-2731	34	1	1.9	1.9	NUM
iajs-2731	34	2	definition	definition	NOUN
iajs-2731	34	3	[	[	X
iajs-2731	34	4	5][4	5][4	NUM
iajs-2731	34	5	]	]	X
iajs-2731	34	6	let	let	VERB
iajs-2731	34	7	a	a	DET
iajs-2731	34	8	,	,	PUNCT
iajs-2731	34	9	b	b	NOUN
iajs-2731	34	10	be	be	AUX
iajs-2731	34	11	two	two	NUM
iajs-2731	34	12	fmodules	fmodule	NOUN
iajs-2731	34	13	of	of	ADP
iajs-2731	34	14	an	an	DET
iajs-2731	34	15	rmodule	rmodule	NOUN
iajs-2731	34	16	m.	m.	NOUN
iajs-2731	34	17	b	b	PROPN
iajs-2731	34	18	is	be	AUX
iajs-2731	34	19	named	name	VERB
iajs-2731	34	20	fmodule	fmodule	ADV
iajs-2731	34	21	of	of	ADP
iajs-2731	34	22	a	a	PRON
iajs-2731	34	23	,	,	PUNCT
iajs-2731	34	24	if	if	SCONJ
iajs-2731	34	25	a	a	DET
iajs-2731	34	26	⊆	⊆	NUM
iajs-2731	34	27	b.	b.	NOUN
iajs-2731	34	28	''	''	PUNCT
iajs-2731	34	29	1.10	1.10	NUM
iajs-2731	34	30	definition	definition	NOUN
iajs-2731	34	31	[	[	X
iajs-2731	34	32	2	2	NUM
iajs-2731	34	33	]	]	PUNCT
iajs-2731	34	34	let	let	VERB
iajs-2731	34	35	a	a	PRON
iajs-2731	34	36	,	,	PUNCT
iajs-2731	34	37	c	c	PROPN
iajs-2731	34	38	are	be	AUX
iajs-2731	34	39	fsubm	fsubm	NOUN
iajs-2731	34	40	of	of	ADP
iajs-2731	34	41	an	an	DET
iajs-2731	34	42	fmodule	fmodule	ADJ
iajs-2731	34	43	x	x	NOUN
iajs-2731	34	44	,	,	PUNCT
iajs-2731	34	45	then	then	ADV
iajs-2731	34	46	a+c	a+c	PROPN
iajs-2731	34	47	is	be	AUX
iajs-2731	34	48	fmodule	fmodule	ADJ
iajs-2731	34	49	m.	m.	NOUN
iajs-2731	34	50	''	''	PUNCT
iajs-2731	34	51	1.11	1.11	NUM
iajs-2731	34	52	definition	definition	NOUN
iajs-2731	34	53	[	[	X
iajs-2731	34	54	7	7	X
iajs-2731	34	55	]	]	X
iajs-2731	34	56	let	let	VERB
iajs-2731	34	57	x	x	PRON
iajs-2731	34	58	be	be	AUX
iajs-2731	34	59	fmodule	fmodule	ADJ
iajs-2731	34	60	,	,	PUNCT
iajs-2731	34	61	x	x	VERB
iajs-2731	34	62	is	be	AUX
iajs-2731	34	63	named	name	VERB
iajs-2731	34	64	simple	simple	ADJ
iajs-2731	34	65	fmodule	fmodule	ADV
iajs-2731	34	66	if	if	SCONJ
iajs-2731	34	67	x	x	PRON
iajs-2731	34	68	has	have	VERB
iajs-2731	34	69	only	only	ADV
iajs-2731	34	70	one	one	NUM
iajs-2731	34	71	proper	proper	ADJ
iajs-2731	34	72	fsubm	fsubm	NOUN
iajs-2731	34	73	,	,	PUNCT
iajs-2731	34	74	which	which	PRON
iajs-2731	34	75	is	be	AUX
iajs-2731	34	76	01	01	NUM
iajs-2731	34	77	.	.	PUNCT
iajs-2731	34	78	''	''	PUNCT
iajs-2731	35	1	ibn	ibn	PROPN
iajs-2731	35	2	al	al	PROPN
iajs-2731	35	3	-	-	PUNCT
iajs-2731	35	4	haitham	haitham	PROPN
iajs-2731	35	5	jour	jour	X
iajs-2731	35	6	.	.	PROPN
iajs-2731	35	7	for	for	ADP
iajs-2731	35	8	pure	pure	ADJ
iajs-2731	35	9	&	&	CCONJ
iajs-2731	35	10	appl	appl	PROPN
iajs-2731	35	11	.	.	PUNCT
iajs-2731	36	1	sci	sci	PROPN
iajs-2731	36	2	.	.	PROPN
iajs-2731	37	1	53	53	NUM
iajs-2731	37	2	(	(	PUNCT
iajs-2731	37	3	2)2022	2)2022	VERB
iajs-2731	37	4	86	86	NUM
iajs-2731	37	5	1.12	1.12	NUM
iajs-2731	37	6	definition	definition	NOUN
iajs-2731	37	7	[	[	X
iajs-2731	37	8	12	12	NUM
iajs-2731	37	9	]	]	PUNCT
iajs-2731	37	10	let	let	VERB
iajs-2731	37	11	a	a	PRON
iajs-2731	37	12	and	and	CCONJ
iajs-2731	37	13	b	b	NOUN
iajs-2731	37	14	be	be	AUX
iajs-2731	37	15	two	two	NUM
iajs-2731	37	16	fmodules	fmodule	NOUN
iajs-2731	37	17	of	of	ADP
iajs-2731	37	18	an	an	DET
iajs-2731	37	19	r	r	NOUN
iajs-2731	37	20	-	-	PUNCT
iajs-2731	37	21	module	module	NOUN
iajs-2731	37	22	x1	x1	NOUN
iajs-2731	37	23	and	and	CCONJ
iajs-2731	37	24	x2	x2	PROPN
iajs-2731	37	25	.	.	PUNCT
iajs-2731	38	1	let	let	VERB
iajs-2731	38	2	𝑓	𝑓	PRON
iajs-2731	38	3	:	:	PUNCT
iajs-2731	38	4	a→b	a→b	NUM
iajs-2731	38	5	be	be	AUX
iajs-2731	38	6	a	a	DET
iajs-2731	38	7	fuzzy	fuzzy	ADJ
iajs-2731	38	8	homomorphism	homomorphism	NOUN
iajs-2731	38	9	.	.	PUNCT
iajs-2731	39	1	if	if	SCONJ
iajs-2731	39	2	𝜐	𝜐	PROPN
iajs-2731	39	3	and	and	CCONJ
iajs-2731	39	4	𝜎	𝜎	PROPN
iajs-2731	39	5	are	be	AUX
iajs-2731	39	6	two	two	NUM
iajs-2731	39	7	fsubms	fsubms	NOUN
iajs-2731	39	8	of	of	ADP
iajs-2731	39	9	a	a	DET
iajs-2731	39	10	,	,	PUNCT
iajs-2731	39	11	b	b	NOUN
iajs-2731	39	12	,	,	PUNCT
iajs-2731	39	13	therefore	therefore	ADV
iajs-2731	39	14	1	1	NUM
iajs-2731	39	15	𝑓(𝜐	𝑓(𝜐	ADJ
iajs-2731	39	16	)	)	PUNCT
iajs-2731	39	17	is	be	AUX
iajs-2731	39	18	fuzzy	fuzzy	ADJ
iajs-2731	39	19	submodules	submodule	NOUN
iajs-2731	39	20	of	of	ADP
iajs-2731	39	21	b	b	NOUN
iajs-2731	39	22	,	,	PUNCT
iajs-2731	39	23	whenever	whenever	SCONJ
iajs-2731	39	24	𝑓	𝑓	PRON
iajs-2731	39	25	is	be	AUX
iajs-2731	39	26	an	an	DET
iajs-2731	39	27	epimorphism	epimorphism	NOUN
iajs-2731	39	28	.	.	PUNCT
iajs-2731	40	1	2-𝑓-1(𝜎	2-𝑓-1(𝜎	X
iajs-2731	40	2	)	)	PUNCT
iajs-2731	40	3	is	be	AUX
iajs-2731	40	4	fuzzy	fuzzy	ADJ
iajs-2731	40	5	submodules	submodule	NOUN
iajs-2731	40	6	of	of	ADP
iajs-2731	40	7	a	a	DET
iajs-2731	40	8	''	''	PUNCT
iajs-2731	40	9	1.13	1.13	NUM
iajs-2731	40	10	definition	definition	NOUN
iajs-2731	40	11	[	[	X
iajs-2731	40	12	11	11	NUM
iajs-2731	40	13	]	]	PUNCT
iajs-2731	40	14	let	let	VERB
iajs-2731	40	15	𝐴	𝐴	PROPN
iajs-2731	40	16	be	be	AUX
iajs-2731	40	17	a	a	DET
iajs-2731	40	18	proper	proper	ADJ
iajs-2731	40	19	fsub	fsub	NOUN
iajs-2731	40	20	of	of	ADP
iajs-2731	40	21	m.	m.	NOUN
iajs-2731	40	22	then	then	ADV
iajs-2731	40	23	𝐴	𝐴	PROPN
iajs-2731	40	24	is	be	AUX
iajs-2731	40	25	named	name	VERB
iajs-2731	40	26	a	a	DET
iajs-2731	40	27	maximal	maximal	ADJ
iajs-2731	40	28	fuzzy	fuzzy	ADJ
iajs-2731	40	29	submodule	submodule	NOUN
iajs-2731	40	30	of	of	ADP
iajs-2731	40	31	m.	m.	NOUN
iajs-2731	40	32	whether	whether	SCONJ
iajs-2731	40	33	to	to	ADP
iajs-2731	40	34	any	any	DET
iajs-2731	40	35	other	other	ADJ
iajs-2731	40	36	proper	proper	ADJ
iajs-2731	40	37	fuzzy	fuzzy	ADJ
iajs-2731	40	38	submodule	submodule	NOUN
iajs-2731	40	39	𝛽	𝛽	PROPN
iajs-2731	40	40	of	of	ADP
iajs-2731	40	41	m	m	PROPN
iajs-2731	40	42	containing	contain	VERB
iajs-2731	40	43	𝐴	𝐴	PROPN
iajs-2731	40	44	then	then	ADV
iajs-2731	40	45	𝐴	𝐴	PROPN
iajs-2731	40	46	=	=	SYM
iajs-2731	40	47	𝛽.	𝛽.	NOUN
iajs-2731	40	48	''	''	PUNCT
iajs-2731	40	49	1.14	1.14	NUM
iajs-2731	40	50	definition	definition	NOUN
iajs-2731	40	51	[	[	X
iajs-2731	40	52	8	8	X
iajs-2731	40	53	]	]	X
iajs-2731	40	54	a	a	DET
iajs-2731	40	55	proper	proper	ADJ
iajs-2731	40	56	fuzzy	fuzzy	ADJ
iajs-2731	40	57	subm	subm	PROPN
iajs-2731	40	58	𝐴	𝐴	PROPN
iajs-2731	40	59	of	of	ADP
iajs-2731	40	60	an	an	DET
iajs-2731	40	61	r	r	NOUN
iajs-2731	40	62	-	-	PUNCT
iajs-2731	40	63	module	module	NOUN
iajs-2731	40	64	x	x	NOUN
iajs-2731	40	65	,	,	PUNCT
iajs-2731	40	66	is	be	AUX
iajs-2731	40	67	named	name	VERB
iajs-2731	40	68	a	a	DET
iajs-2731	40	69	small	small	ADJ
iajs-2731	40	70	fuzzy	fuzzy	NOUN
iajs-2731	40	71	if	if	SCONJ
iajs-2731	40	72	𝐴	𝐴	PROPN
iajs-2731	40	73	is	be	AUX
iajs-2731	40	74	a	a	DET
iajs-2731	40	75	fuzzy	fuzzy	ADJ
iajs-2731	40	76	subm	subm	NOUN
iajs-2731	40	77	of	of	ADP
iajs-2731	40	78	𝜒	𝜒	PROPN
iajs-2731	40	79	,	,	PUNCT
iajs-2731	40	80	then	then	ADV
iajs-2731	40	81	𝐴	𝐴	PROPN
iajs-2731	40	82	is	be	AUX
iajs-2731	40	83	named	name	VERB
iajs-2731	40	84	a	a	DET
iajs-2731	40	85	small	small	ADJ
iajs-2731	40	86	fuzzy	fuzzy	NOUN
iajs-2731	40	87	in	in	ADP
iajs-2731	40	88	𝜒	𝜒	PRON
iajs-2731	40	89	if	if	SCONJ
iajs-2731	40	90	every	every	DET
iajs-2731	40	91	fuzzy	fuzzy	ADJ
iajs-2731	40	92	submodule	submodule	NOUN
iajs-2731	40	93	𝛽	𝛽	PROPN
iajs-2731	40	94	of	of	ADP
iajs-2731	40	95	𝜒	𝜒	NUM
iajs-2731	40	96	,	,	PUNCT
iajs-2731	40	97	s.t	s.t	PROPN
iajs-2731	40	98	𝐴	𝐴	PROPN
iajs-2731	40	99	+	+	CCONJ
iajs-2731	40	100	𝛽	𝛽	NOUN
iajs-2731	40	101	=	=	SYM
iajs-2731	40	102	𝜒.	𝜒.	NOUN
iajs-2731	40	103	implies	imply	VERB
iajs-2731	40	104	𝛽	𝛽	NOUN
iajs-2731	40	105	=	=	SYM
iajs-2731	40	106	𝜒	𝜒	X
iajs-2731	40	107	''	''	PUNCT
iajs-2731	40	108	1.15	1.15	NUM
iajs-2731	40	109	definition	definition	NOUN
iajs-2731	41	1	[	[	X
iajs-2731	41	2	13	13	NUM
iajs-2731	41	3	]	]	PUNCT
iajs-2731	41	4	let	let	VERB
iajs-2731	41	5	a	a	PRON
iajs-2731	41	6	be	be	AUX
iajs-2731	41	7	fmodule	fmodule	ADJ
iajs-2731	41	8	of	of	ADP
iajs-2731	41	9	an	an	DET
iajs-2731	41	10	rmodule	rmodule	NOUN
iajs-2731	41	11	m.	m.	NOUN
iajs-2731	41	12	a	a	PRON
iajs-2731	41	13	is	be	AUX
iajs-2731	41	14	named	name	VERB
iajs-2731	41	15	finitely	finitely	ADV
iajs-2731	41	16	generated	generate	VERB
iajs-2731	41	17	fm	fm	PROPN
iajs-2731	41	18	odule	odule	NOUN
iajs-2731	41	19	if	if	SCONJ
iajs-2731	41	20	there	there	PRON
iajs-2731	41	21	exists	exist	VERB
iajs-2731	41	22	xt1	xt1	PROPN
iajs-2731	41	23	,	,	PUNCT
iajs-2731	41	24	xt2	xt2	PROPN
iajs-2731	41	25	,	,	PUNCT
iajs-2731	41	26	,	,	PUNCT
iajs-2731	41	27	,	,	PUNCT
iajs-2731	41	28	,	,	PUNCT
iajs-2731	41	29	,	,	PUNCT
iajs-2731	41	30	,	,	PUNCT
iajs-2731	41	31	xt	xt	PROPN
iajs-2731	41	32	n	n	PRON
iajs-2731	41	33	⊆	⊆	NUM
iajs-2731	41	34	a	a	DET
iajs-2731	41	35	such	such	ADJ
iajs-2731	41	36	that	that	DET
iajs-2731	41	37	a=	a=	NOUN
iajs-2731	41	38	{	{	PUNCT
iajs-2731	41	39	a1	a1	NOUN
iajs-2731	41	40	(	(	PUNCT
iajs-2731	41	41	xt)t1	xt)t1	PROPN
iajs-2731	41	42	+	+	NUM
iajs-2731	41	43	a2	a2	PROPN
iajs-2731	41	44	(	(	PUNCT
iajs-2731	41	45	xt)t2	xt)t2	PROPN
iajs-2731	41	46	+	+	PUNCT
iajs-2731	41	47	.	.	PUNCT
iajs-2731	41	48	.	.	PUNCT
iajs-2731	41	49	.	.	PUNCT
iajs-2731	42	1	+	+	CCONJ
iajs-2731	42	2	an	an	DET
iajs-2731	42	3	(	(	PUNCT
iajs-2731	42	4	xn)tn	xn)tn	PROPN
iajs-2731	42	5	}	}	PUNCT
iajs-2731	42	6	,	,	PUNCT
iajs-2731	42	7	where	where	SCONJ
iajs-2731	42	8	ai	ai	VERB
iajs-2731	42	9	∈r	∈r	NOUN
iajs-2731	42	10	,	,	PUNCT
iajs-2731	42	11	a(x)t	a(x)t	PROPN
iajs-2731	42	12	=	=	SYM
iajs-2731	42	13	(	(	PUNCT
iajs-2731	42	14	ax)t	ax)t	PROPN
iajs-2731	42	15	,	,	PUNCT
iajs-2731	42	16	∀	∀	NUM
iajs-2731	42	17	t	t	NOUN
iajs-2731	42	18	∈	∈	PROPN
iajs-2731	42	19	(	(	PUNCT
iajs-2731	42	20	0,1	0,1	NOUN
iajs-2731	42	21	]	]	PUNCT
iajs-2731	42	22	,	,	PUNCT
iajs-2731	42	23	(	(	PUNCT
iajs-2731	42	24	ax	ax	NOUN
iajs-2731	42	25	)	)	PUNCT
iajs-2731	42	26	t	t	NOUN
iajs-2731	42	27	(	(	PUNCT
iajs-2731	42	28	y	y	NOUN
iajs-2731	42	29	)	)	PUNCT
iajs-2731	42	30	=	=	PRON
iajs-2731	42	31	{	{	PUNCT
iajs-2731	43	1	1	1	NUM
iajs-2731	43	2	𝑖𝑓	𝑖𝑓	NUM
iajs-2731	43	3	𝑦	𝑦	NOUN
iajs-2731	43	4	=	=	SYM
iajs-2731	43	5	𝑎𝑥	𝑎𝑥	X
iajs-2731	43	6	0	0	NUM
iajs-2731	43	7	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	43	8	''	''	PUNCT
iajs-2731	43	9	1.16	1.16	NUM
iajs-2731	43	10	definition	definition	NOUN
iajs-2731	43	11	[	[	X
iajs-2731	43	12	5	5	NUM
iajs-2731	43	13	]	]	PUNCT
iajs-2731	43	14	let	let	VERB
iajs-2731	43	15	µ	µ	NUM
iajs-2731	43	16	,	,	PUNCT
iajs-2731	43	17	𝛶be	𝛶be	PROPN
iajs-2731	43	18	two	two	NUM
iajs-2731	43	19	fmodules	fmodule	NOUN
iajs-2731	43	20	,	,	PUNCT
iajs-2731	43	21	𝑓	𝑓	X
iajs-2731	43	22	:	:	PUNCT
iajs-2731	43	23	x1	x1	PROPN
iajs-2731	43	24	→	→	SYM
iajs-2731	43	25	x2	x2	PROPN
iajs-2731	43	26	is	be	AUX
iajs-2731	43	27	a	a	DET
iajs-2731	43	28	homomorphism	homomorphism	NOUN
iajs-2731	43	29	between	between	ADP
iajs-2731	43	30	x1	x1	PROPN
iajs-2731	43	31	and	and	CCONJ
iajs-2731	43	32	x2	x2	NOUN
iajs-2731	43	33	respectively.then	respectively.then	ADP
iajs-2731	43	34	fkernel	fkernel	NOUN
iajs-2731	43	35	of	of	ADP
iajs-2731	43	36	𝑓	𝑓	PRON
iajs-2731	43	37	,	,	PUNCT
iajs-2731	43	38	f	f	X
iajs-2731	43	39	-	-	PUNCT
iajs-2731	43	40	kernel	kernel	PROPN
iajs-2731	43	41	𝑓	𝑓	PRON
iajs-2731	43	42	is	be	AUX
iajs-2731	43	43	the	the	DET
iajs-2731	43	44	fuzzy	fuzzy	ADJ
iajs-2731	43	45	subset	subset	NOUN
iajs-2731	43	46	of	of	ADP
iajs-2731	43	47	m1	m1	NOUN
iajs-2731	43	48	,	,	PUNCT
iajs-2731	43	49	defined	define	VERB
iajs-2731	43	50	by	by	ADP
iajs-2731	43	51	?	?	PUNCT
iajs-2731	44	1	f	f	X
iajs-2731	44	2	-	-	PUNCT
iajs-2731	44	3	ker	ker	NOUN
iajs-2731	44	4	𝑓(x	𝑓(x	PROPN
iajs-2731	44	5	)	)	PUNCT
iajs-2731	45	1	=	=	PRON
iajs-2731	45	2	{	{	PUNCT
iajs-2731	45	3	µ(0	µ(0	NOUN
iajs-2731	45	4	)	)	PUNCT
iajs-2731	45	5	𝑖𝑓	𝑖𝑓	VERB
iajs-2731	45	6	𝑥	𝑥	PRON
iajs-2731	45	7	∈	∈	PROPN
iajs-2731	45	8	𝑘𝑒𝑟	𝑘𝑒𝑟	NOUN
iajs-2731	46	1	𝑓	𝑓	ADV
iajs-2731	46	2	0	0	NUM
iajs-2731	46	3	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	46	4	𝑥	𝑥	PROPN
iajs-2731	46	5	∉	∉	PROPN
iajs-2731	46	6	𝑘𝑒𝑟	𝑘𝑒𝑟	PROPN
iajs-2731	46	7	𝑓	𝑓	PRON
iajs-2731	46	8	''	''	PUNCT
iajs-2731	46	9	1	1	NUM
iajs-2731	46	10	.	.	PUNCT
iajs-2731	46	11	characterization	characterization	NOUN
iajs-2731	46	12	of	of	ADP
iajs-2731	46	13	lochollow	lochollow	ADJ
iajs-2731	46	14	fuzzy	fuzzy	ADJ
iajs-2731	46	15	module	module	NOUN
iajs-2731	46	16	throughout	throughout	ADP
iajs-2731	46	17	the	the	DET
iajs-2731	46	18	part	part	NOUN
iajs-2731	46	19	,	,	PUNCT
iajs-2731	46	20	we	we	PRON
iajs-2731	46	21	introduce	introduce	VERB
iajs-2731	46	22	the	the	DET
iajs-2731	46	23	definition	definition	NOUN
iajs-2731	46	24	of	of	ADP
iajs-2731	46	25	lochollow	lochollow	NOUN
iajs-2731	46	26	fmodules	fmodule	NOUN
iajs-2731	46	27	and	and	CCONJ
iajs-2731	46	28	study	study	VERB
iajs-2731	46	29	the	the	DET
iajs-2731	46	30	basic	basic	ADJ
iajs-2731	46	31	properties	property	NOUN
iajs-2731	46	32	of	of	ADP
iajs-2731	46	33	these	these	DET
iajs-2731	46	34	kinds	kind	NOUN
iajs-2731	46	35	of	of	ADP
iajs-2731	46	36	modules	module	NOUN
iajs-2731	46	37	.	.	PUNCT
iajs-2731	47	1	2.1	2.1	NUM
iajs-2731	47	2	definition	definition	NOUN
iajs-2731	47	3	an	an	DET
iajs-2731	47	4	r	r	NOUN
iajs-2731	47	5	-	-	PUNCT
iajs-2731	47	6	module	module	NOUN
iajs-2731	47	7	x	x	PRON
iajs-2731	47	8	is	be	AUX
iajs-2731	47	9	lochollow	lochollow	ADJ
iajs-2731	47	10	fuzzy	fuzzy	ADJ
iajs-2731	47	11	module	module	NOUN
iajs-2731	47	12	if	if	SCONJ
iajs-2731	47	13	x	x	PRON
iajs-2731	47	14	has	have	VERB
iajs-2731	47	15	a	a	DET
iajs-2731	47	16	unique	unique	ADJ
iajs-2731	47	17	maximal	maximal	ADJ
iajs-2731	47	18	fuzzy	fuzzy	ADJ
iajs-2731	47	19	submodule	submodule	NOUN
iajs-2731	47	20	that	that	PRON
iajs-2731	47	21	contains	contain	VERB
iajs-2731	47	22	each	each	PRON
iajs-2731	47	23	a	a	DET
iajs-2731	47	24	small	small	ADJ
iajs-2731	47	25	fuzzy	fuzzy	ADJ
iajs-2731	47	26	submodule	submodule	NOUN
iajs-2731	47	27	of	of	ADP
iajs-2731	47	28	x.	x.	PROPN
iajs-2731	47	29	2.2	2.2	NUM
iajs-2731	47	30	example	example	NOUN
iajs-2731	47	31	let	let	VERB
iajs-2731	47	32	m=	m=	X
iajs-2731	47	33	z4	z4	NOUN
iajs-2731	47	34	,	,	PUNCT
iajs-2731	47	35	r	r	NOUN
iajs-2731	47	36	=	=	SYM
iajs-2731	47	37	z	z	NOUN
iajs-2731	47	38	,	,	PUNCT
iajs-2731	47	39	define	define	VERB
iajs-2731	47	40	x	x	NOUN
iajs-2731	47	41	:	:	PUNCT
iajs-2731	47	42	m	m	VERB
iajs-2731	47	43	→	→	SYM
iajs-2731	47	44	[	[	PUNCT
iajs-2731	47	45	0,1	0,1	NUM
iajs-2731	47	46	]	]	PUNCT
iajs-2731	47	47	𝑎𝑠	𝑎𝑠	AUX
iajs-2731	47	48	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	PROPN
iajs-2731	47	49	x(x	x(x	PROPN
iajs-2731	47	50	)	)	PUNCT
iajs-2731	48	1	=	=	PRON
iajs-2731	48	2	{	{	PUNCT
iajs-2731	48	3	1	1	NUM
iajs-2731	48	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	48	5	𝑥	𝑥	PRON
iajs-2731	48	6	∈	∈	PROPN
iajs-2731	48	7	𝑀	𝑀	PROPN
iajs-2731	48	8	0	0	NUM
iajs-2731	48	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	48	10	and	and	CCONJ
iajs-2731	48	11	define	define	VERB
iajs-2731	48	12	a	a	DET
iajs-2731	48	13	:	:	PUNCT
iajs-2731	48	14	m	m	NOUN
iajs-2731	48	15	→	→	SYM
iajs-2731	48	16	[	[	PUNCT
iajs-2731	48	17	0,1	0,1	NUM
iajs-2731	48	18	]	]	PUNCT
iajs-2731	48	19	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2731	48	20	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	48	21	a(x	a(x	PROPN
iajs-2731	48	22	)	)	PUNCT
iajs-2731	48	23	=	=	PRON
iajs-2731	48	24	{	{	PUNCT
iajs-2731	48	25	𝑡	𝑡	X
iajs-2731	48	26	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	48	27	𝑥	𝑥	PRON
iajs-2731	48	28	∈	∈	NOUN
iajs-2731	48	29	𝑁	𝑁	PROPN
iajs-2731	48	30	0	0	NUM
iajs-2731	48	31	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	48	32	,	,	PUNCT
iajs-2731	48	33	t	t	PROPN
iajs-2731	48	34	∈	∈	PROPN
iajs-2731	48	35	(	(	PUNCT
iajs-2731	48	36	0	0	NUM
iajs-2731	48	37	,	,	PUNCT
iajs-2731	48	38	1	1	NUM
iajs-2731	48	39	]	]	PUNCT
iajs-2731	48	40	,	,	PUNCT
iajs-2731	48	41	where	where	SCONJ
iajs-2731	48	42	n=	n=	ADJ
iajs-2731	48	43	2z4	2z4	NUM
iajs-2731	48	44	clear	clear	ADJ
iajs-2731	48	45	that	that	SCONJ
iajs-2731	48	46	x	x	PRON
iajs-2731	48	47	is	be	AUX
iajs-2731	48	48	a	a	DET
iajs-2731	48	49	fuzzy	fuzzy	ADJ
iajs-2731	48	50	module	module	NOUN
iajs-2731	48	51	and	and	CCONJ
iajs-2731	48	52	at	at	ADP
iajs-2731	48	53	=	=	NOUN
iajs-2731	48	54	n	n	PART
iajs-2731	48	55	is	be	AUX
iajs-2731	48	56	a	a	DET
iajs-2731	48	57	submodule	submodule	NOUN
iajs-2731	48	58	in	in	ADP
iajs-2731	48	59	xt	xt	PROPN
iajs-2731	48	60	,	,	PUNCT
iajs-2731	48	61	since	since	SCONJ
iajs-2731	48	62	at	at	ADV
iajs-2731	48	63	is	be	AUX
iajs-2731	48	64	only	only	ADV
iajs-2731	48	65	the	the	DET
iajs-2731	48	66	maximal	maximal	ADJ
iajs-2731	48	67	submodule	submodule	NOUN
iajs-2731	48	68	of	of	ADP
iajs-2731	48	69	xt	xt	NUM
iajs-2731	48	70	by	by	ADP
iajs-2731	48	71	[	[	PUNCT
iajs-2731	48	72	9	9	NUM
iajs-2731	48	73	]	]	PUNCT
iajs-2731	48	74	,	,	PUNCT
iajs-2731	48	75	therefore	therefore	ADV
iajs-2731	48	76	a	a	PRON
iajs-2731	48	77	is	be	AUX
iajs-2731	48	78	only	only	ADV
iajs-2731	48	79	the	the	DET
iajs-2731	48	80	maximal	maximal	ADJ
iajs-2731	48	81	fuzzy	fuzzy	ADJ
iajs-2731	48	82	submodule	submodule	NOUN
iajs-2731	48	83	of	of	ADP
iajs-2731	48	84	x	x	PUNCT
iajs-2731	49	1	and	and	CCONJ
iajs-2731	49	2	it	it	PRON
iajs-2731	49	3	is	be	AUX
iajs-2731	49	4	a	a	DET
iajs-2731	49	5	fuzzy	fuzzy	ADJ
iajs-2731	49	6	small	small	ADJ
iajs-2731	49	7	submodule	submodule	NOUN
iajs-2731	49	8	in	in	ADP
iajs-2731	49	9	x	x	PUNCT
iajs-2731	49	10	by	by	ADP
iajs-2731	49	11	[	[	PUNCT
iajs-2731	49	12	8	8	NUM
iajs-2731	49	13	]	]	PUNCT
iajs-2731	49	14	,	,	PUNCT
iajs-2731	49	15	which	which	PRON
iajs-2731	49	16	contains	contain	VERB
iajs-2731	49	17	all	all	DET
iajs-2731	49	18	fuzzy	fuzzy	ADJ
iajs-2731	49	19	small	small	ADJ
iajs-2731	49	20	submodule	submodule	NOUN
iajs-2731	49	21	.	.	PUNCT
iajs-2731	50	1	on	on	ADP
iajs-2731	50	2	the	the	DET
iajs-2731	50	3	other	other	ADJ
iajs-2731	50	4	side	side	NOUN
iajs-2731	50	5	1	1	NUM
iajs-2731	50	6	let	let	VERB
iajs-2731	50	7	m=𝑍6	m=𝑍6	ADJ
iajs-2731	50	8	as	as	ADP
iajs-2731	50	9	z	z	NOUN
iajs-2731	50	10	-	-	PUNCT
iajs-2731	50	11	module	module	NOUN
iajs-2731	50	12	,	,	PUNCT
iajs-2731	50	13	r	r	NOUN
iajs-2731	50	14	=	=	NOUN
iajs-2731	50	15	z.	z.	NOUN
iajs-2731	50	16	define	define	VERB
iajs-2731	50	17	x	x	X
iajs-2731	50	18	:	:	PUNCT
iajs-2731	50	19	m⟶	m⟶	PROPN
iajs-2731	50	20	[	[	X
iajs-2731	50	21	0	0	NUM
iajs-2731	50	22	,	,	PUNCT
iajs-2731	50	23	1	1	NUM
iajs-2731	50	24	]	]	PUNCT
iajs-2731	50	25	by	by	ADP
iajs-2731	50	26	xx	xx	NUM
iajs-2731	50	27	)	)	PUNCT
iajs-2731	51	1	=	=	PRON
iajs-2731	51	2	{	{	PUNCT
iajs-2731	51	3	1	1	NUM
iajs-2731	51	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	51	5	𝑥	𝑥	PRON
iajs-2731	51	6	∈	∈	PROPN
iajs-2731	51	7	𝑀	𝑀	PROPN
iajs-2731	51	8	0	0	PROPN
iajs-2731	51	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2731	51	10	a	a	PRON
iajs-2731	51	11	:	:	PUNCT
iajs-2731	51	12	m⟶	m⟶	PROPN
iajs-2731	51	13	[	[	X
iajs-2731	51	14	0	0	NUM
iajs-2731	51	15	,	,	PUNCT
iajs-2731	51	16	1	1	NUM
iajs-2731	51	17	]	]	PUNCT
iajs-2731	51	18	by	by	ADP
iajs-2731	51	19	a(x	a(x	NOUN
iajs-2731	51	20	)	)	PUNCT
iajs-2731	51	21	=	=	NOUN
iajs-2731	51	22	{	{	PUNCT
iajs-2731	51	23	𝑡	𝑡	X
iajs-2731	51	24	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	51	25	𝑥	𝑥	PRON
iajs-2731	51	26	∈	∈	PROPN
iajs-2731	51	27	n	n	PRON
iajs-2731	51	28	0.75	0.75	NUM
iajs-2731	51	29	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	51	30	∀	∀	X
iajs-2731	51	31	t	t	NOUN
iajs-2731	51	32	∈	∈	PROPN
iajs-2731	51	33	(	(	PUNCT
iajs-2731	51	34	0,1	0,1	NUM
iajs-2731	51	35	]	]	PUNCT
iajs-2731	51	36	,	,	PUNCT
iajs-2731	51	37	n=2z	n=2z	PROPN
iajs-2731	51	38	ibn	ibn	PROPN
iajs-2731	51	39	al	al	PROPN
iajs-2731	51	40	-	-	PUNCT
iajs-2731	51	41	haitham	haitham	PROPN
iajs-2731	51	42	jour	jour	X
iajs-2731	51	43	.	.	PROPN
iajs-2731	51	44	for	for	ADP
iajs-2731	51	45	pure	pure	ADJ
iajs-2731	51	46	&	&	CCONJ
iajs-2731	51	47	appl	appl	PROPN
iajs-2731	51	48	.	.	PUNCT
iajs-2731	52	1	sci	sci	PROPN
iajs-2731	52	2	.	.	PROPN
iajs-2731	53	1	53	53	NUM
iajs-2731	53	2	(	(	PUNCT
iajs-2731	53	3	2)2022	2)2022	VERB
iajs-2731	53	4	87	87	NUM
iajs-2731	53	5	b	b	NOUN
iajs-2731	53	6	:	:	PUNCT
iajs-2731	53	7	m⟶	m⟶	PROPN
iajs-2731	53	8	[	[	X
iajs-2731	53	9	0	0	NUM
iajs-2731	53	10	,	,	PUNCT
iajs-2731	53	11	1	1	NUM
iajs-2731	53	12	]	]	PUNCT
iajs-2731	53	13	by	by	ADP
iajs-2731	53	14	b	b	PROPN
iajs-2731	53	15	(	(	PUNCT
iajs-2731	53	16	t	t	PROPN
iajs-2731	53	17	)	)	PUNCT
iajs-2731	53	18	=	=	PRON
iajs-2731	53	19	{	{	PUNCT
iajs-2731	53	20	𝑡	𝑡	X
iajs-2731	53	21	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	53	22	𝑥	𝑥	PRON
iajs-2731	53	23	∈	∈	PROPN
iajs-2731	53	24	𝐿	𝐿	PROPN
iajs-2731	53	25	0,25	0,25	NOUN
iajs-2731	53	26	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	53	27	∀	∀	X
iajs-2731	53	28	t	t	PROPN
iajs-2731	53	29	∈	∈	PROPN
iajs-2731	53	30	(	(	PUNCT
iajs-2731	53	31	0,1	0,1	NUM
iajs-2731	53	32	]	]	PUNCT
iajs-2731	53	33	,	,	PUNCT
iajs-2731	53	34	l=3z	l=3z	VERB
iajs-2731	53	35	clear	clear	ADJ
iajs-2731	53	36	that	that	SCONJ
iajs-2731	53	37	x	x	PRON
iajs-2731	53	38	is	be	AUX
iajs-2731	53	39	fuzzy	fuzzy	ADJ
iajs-2731	53	40	module	module	NOUN
iajs-2731	53	41	and	and	CCONJ
iajs-2731	53	42	xt	xt	NOUN
iajs-2731	53	43	=	=	PROPN
iajs-2731	53	44	m	m	PROPN
iajs-2731	53	45	,	,	PUNCT
iajs-2731	53	46	at=⟨2̅⟩	at=⟨2̅⟩	PROPN
iajs-2731	53	47	,	,	PUNCT
iajs-2731	53	48	bt	bt	X
iajs-2731	54	1	=	=	ADJ
iajs-2731	54	2	⟨3̅⟩	⟨3̅⟩	PROPN
iajs-2731	54	3	aare	aare	VERB
iajs-2731	54	4	submodules	submodule	NOUN
iajs-2731	54	5	in	in	ADP
iajs-2731	54	6	xt	xt	PROPN
iajs-2731	54	7	.	.	PUNCT
iajs-2731	55	1	but	but	CCONJ
iajs-2731	55	2	at	at	ADP
iajs-2731	55	3	,	,	PUNCT
iajs-2731	55	4	bt	bt	PROPN
iajs-2731	55	5	are	be	AUX
iajs-2731	55	6	two	two	NUM
iajs-2731	55	7	maximal	maximal	ADJ
iajs-2731	55	8	submodules	submodule	NOUN
iajs-2731	55	9	in	in	ADP
iajs-2731	55	10	xt	xt	ADP
iajs-2731	55	11	by	by	ADP
iajs-2731	55	12	[	[	X
iajs-2731	55	13	9	9	NUM
iajs-2731	55	14	]	]	PUNCT
iajs-2731	55	15	,	,	PUNCT
iajs-2731	55	16	therefore	therefore	ADV
iajs-2731	55	17	a	a	PRON
iajs-2731	55	18	,	,	PUNCT
iajs-2731	55	19	b	b	NOUN
iajs-2731	55	20	are	be	AUX
iajs-2731	55	21	two	two	NUM
iajs-2731	55	22	fuzzy	fuzzy	ADJ
iajs-2731	55	23	maximal	maximal	ADJ
iajs-2731	55	24	submodules	submodule	NOUN
iajs-2731	55	25	in	in	ADP
iajs-2731	55	26	x	x	X
iajs-2731	55	27	.	.	PUNCT
iajs-2731	56	1	thus	thus	ADV
iajs-2731	56	2	x	x	X
iajs-2731	56	3	=	=	NOUN
iajs-2731	56	4	z6	z6	PROPN
iajs-2731	56	5	is	be	AUX
iajs-2731	56	6	not	not	PART
iajs-2731	56	7	a	a	DET
iajs-2731	56	8	loc	loc	ADJ
iajs-2731	56	9	-	-	ADJ
iajs-2731	56	10	hollow	hollow	ADJ
iajs-2731	56	11	fuzzy	fuzzy	ADJ
iajs-2731	56	12	module	module	NOUN
iajs-2731	56	13	.	.	PUNCT
iajs-2731	57	1	2.3	2.3	NUM
iajs-2731	57	2	𝝆roposition	𝝆roposition	NOUN
iajs-2731	57	3	let	let	VERB
iajs-2731	57	4	x	x	PRON
iajs-2731	57	5	be	be	AUX
iajs-2731	57	6	a	a	DET
iajs-2731	57	7	fmodule	fmodule	NOUN
iajs-2731	57	8	.	.	PUNCT
iajs-2731	58	1	then	then	ADV
iajs-2731	58	2	x	x	X
iajs-2731	58	3	is	be	AUX
iajs-2731	58	4	loc	loc	ADJ
iajs-2731	58	5	-	-	ADJ
iajs-2731	58	6	hollow	hollow	ADJ
iajs-2731	58	7	fmodules	fmodule	NOUN
iajs-2731	58	8	⇔xt	⇔xt	NOUN
iajs-2731	58	9	is	be	AUX
iajs-2731	58	10	loc	loc	ADJ
iajs-2731	58	11	-	-	ADJ
iajs-2731	58	12	hollow	hollow	ADJ
iajs-2731	58	13	fmodules	fmodule	NOUN
iajs-2731	59	1	∀	∀	X
iajs-2731	59	2	t	t	NOUN
iajs-2731	59	3	∈	∈	PROPN
iajs-2731	59	4	(	(	PUNCT
iajs-2731	59	5	0,1	0,1	NOUN
iajs-2731	59	6	]	]	PUNCT
iajs-2731	59	7	.	.	PUNCT
iajs-2731	60	1	proof	proof	NOUN
iajs-2731	60	2	:	:	PUNCT
iajs-2731	60	3	⟹	⟹	PROPN
iajs-2731	60	4	suppose	suppose	VERB
iajs-2731	60	5	that	that	SCONJ
iajs-2731	60	6	x	x	PRON
iajs-2731	60	7	is	be	AUX
iajs-2731	60	8	a	a	DET
iajs-2731	60	9	loc	loc	ADJ
iajs-2731	60	10	-	-	ADJ
iajs-2731	60	11	hollow	hollow	ADJ
iajs-2731	60	12	fmodules	fmodule	NOUN
iajs-2731	60	13	,	,	PUNCT
iajs-2731	60	14	then	then	ADV
iajs-2731	60	15	there	there	PRON
iajs-2731	60	16	exists	exist	VERB
iajs-2731	60	17	a	a	DET
iajs-2731	60	18	unique	unique	ADJ
iajs-2731	60	19	maximal	maximal	ADJ
iajs-2731	60	20	fuzzy	fuzzy	ADJ
iajs-2731	60	21	submodule	submodule	NOUN
iajs-2731	60	22	a	a	PRON
iajs-2731	60	23	which	which	PRON
iajs-2731	60	24	contains	contain	VERB
iajs-2731	60	25	all	all	DET
iajs-2731	60	26	a	a	DET
iajs-2731	60	27	small	small	ADJ
iajs-2731	60	28	f	f	PROPN
iajs-2731	60	29	-subm	-subm	NOUN
iajs-2731	60	30	in	in	ADP
iajs-2731	60	31	x.	x.	NOUN
iajs-2731	60	32	suppose	suppose	VERB
iajs-2731	60	33	that	that	SCONJ
iajs-2731	60	34	𝑨	𝑨	PROPN
iajs-2731	60	35	is	be	AUX
iajs-2731	60	36	a	a	DET
iajs-2731	60	37	f	f	PROPN
iajs-2731	60	38	-subm	-subm	PROPN
iajs-2731	60	39	in	in	ADP
iajs-2731	60	40	𝜒	𝜒	NOUN
iajs-2731	60	41	,	,	PUNCT
iajs-2731	60	42	therefore	therefore	ADV
iajs-2731	60	43	a+b	a+b	NOUN
iajs-2731	60	44	=	=	NOUN
iajs-2731	60	45	x	x	NOUN
iajs-2731	60	46	for	for	ADP
iajs-2731	60	47	some	some	DET
iajs-2731	60	48	f	f	PROPN
iajs-2731	60	49	-subm	-subm	PROPN
iajs-2731	60	50	b	b	PROPN
iajs-2731	60	51	in	in	ADP
iajs-2731	60	52	a	a	PRON
iajs-2731	60	53	of	of	ADP
iajs-2731	60	54	x	x	NOUN
iajs-2731	60	55	,	,	PUNCT
iajs-2731	60	56	and	and	CCONJ
iajs-2731	60	57	(	(	PUNCT
iajs-2731	60	58	a+b)t	a+b)t	PROPN
iajs-2731	60	59	=	=	SYM
iajs-2731	60	60	xt	xt	X
iajs-2731	60	61	∀	∀	NOUN
iajs-2731	60	62	t	t	PROPN
iajs-2731	60	63	∈	∈	PROPN
iajs-2731	60	64	(	(	PUNCT
iajs-2731	60	65	0,1	0,1	NUM
iajs-2731	60	66	]	]	PUNCT
iajs-2731	60	67	by	by	ADP
iajs-2731	60	68	[	[	PUNCT
iajs-2731	60	69	6	6	NUM
iajs-2731	60	70	]	]	PUNCT
iajs-2731	60	71	.	.	PUNCT
iajs-2731	61	1	but	but	CCONJ
iajs-2731	61	2	at	at	SCONJ
iajs-2731	61	3	is	be	AUX
iajs-2731	61	4	a	a	DET
iajs-2731	61	5	unique	unique	ADJ
iajs-2731	61	6	maximal	maximal	ADJ
iajs-2731	61	7	f	f	PROPN
iajs-2731	61	8	-subm	-subm	PROPN
iajs-2731	61	9	in	in	ADP
iajs-2731	61	10	xt	xt	ADP
iajs-2731	61	11	,	,	PUNCT
iajs-2731	61	12	by	by	ADP
iajs-2731	61	13	[	[	X
iajs-2731	61	14	10	10	NUM
iajs-2731	61	15	]	]	PUNCT
iajs-2731	61	16	.	.	PUNCT
iajs-2731	62	1	therefore	therefore	ADV
iajs-2731	62	2	at	at	ADP
iajs-2731	62	3	is	be	AUX
iajs-2731	62	4	a	a	DET
iajs-2731	62	5	unique	unique	ADJ
iajs-2731	62	6	maximal	maximal	ADJ
iajs-2731	62	7	submodule	submodule	NOUN
iajs-2731	62	8	in	in	ADP
iajs-2731	62	9	xt	xt	PROPN
iajs-2731	62	10	.	.	PUNCT
iajs-2731	63	1	thus	thus	ADV
iajs-2731	63	2	xt	xt	PROPN
iajs-2731	63	3	is	be	AUX
iajs-2731	63	4	a	a	DET
iajs-2731	63	5	loc	loc	ADJ
iajs-2731	63	6	-	-	ADJ
iajs-2731	63	7	hollow	hollow	ADJ
iajs-2731	63	8	module	module	NOUN
iajs-2731	63	9	∀	∀	X
iajs-2731	63	10	t	t	X
iajs-2731	63	11	∈	∈	PROPN
iajs-2731	63	12	(	(	PUNCT
iajs-2731	63	13	0,1	0,1	NOUN
iajs-2731	63	14	]	]	PUNCT
iajs-2731	63	15	.	.	PUNCT
iajs-2731	64	1	⟸	⟸	PROPN
iajs-2731	64	2	let	let	VERB
iajs-2731	64	3	xt	xt	PART
iajs-2731	64	4	be	be	AUX
iajs-2731	64	5	a	a	DET
iajs-2731	64	6	loc	loc	ADJ
iajs-2731	64	7	-	-	ADJ
iajs-2731	64	8	hollow	hollow	ADJ
iajs-2731	64	9	module	module	NOUN
iajs-2731	64	10	,	,	PUNCT
iajs-2731	64	11	suppose	suppose	VERB
iajs-2731	64	12	that	that	SCONJ
iajs-2731	64	13	bt	bt	PROPN
iajs-2731	64	14	be	be	AUX
iajs-2731	64	15	a	a	DET
iajs-2731	64	16	submodule	submodule	NOUN
iajs-2731	64	17	of	of	ADP
iajs-2731	64	18	xt	xt	PROPN
iajs-2731	64	19	and	and	CCONJ
iajs-2731	64	20	(	(	PUNCT
iajs-2731	64	21	a+b)t	a+b)t	PROPN
iajs-2731	64	22	=	=	PROPN
iajs-2731	64	23	xt	xt	NOUN
iajs-2731	64	24	for	for	ADP
iajs-2731	64	25	some	some	DET
iajs-2731	64	26	f	f	PROPN
iajs-2731	64	27	-subm	-subm	PROPN
iajs-2731	64	28	bt	bt	VERB
iajs-2731	64	29	in	in	ADP
iajs-2731	64	30	at	at	ADP
iajs-2731	64	31	of	of	ADP
iajs-2731	64	32	xt	xt	ADP
iajs-2731	64	33	∀	∀	X
iajs-2731	64	34	t	t	PROPN
iajs-2731	64	35	∈	∈	PROPN
iajs-2731	64	36	(	(	PUNCT
iajs-2731	64	37	0,1	0,1	NUM
iajs-2731	64	38	]	]	PUNCT
iajs-2731	64	39	,	,	PUNCT
iajs-2731	64	40	hence	hence	ADV
iajs-2731	64	41	a+b	a+b	ADJ
iajs-2731	64	42	=	=	NOUN
iajs-2731	64	43	x	x	X
iajs-2731	64	44	or	or	CCONJ
iajs-2731	64	45	some	some	DET
iajs-2731	64	46	f	f	PROPN
iajs-2731	64	47	-subm	-subm	PROPN
iajs-2731	64	48	b	b	PROPN
iajs-2731	64	49	in	in	ADP
iajs-2731	64	50	a	a	PRON
iajs-2731	64	51	of	of	ADP
iajs-2731	64	52	x.	x.	NOUN
iajs-2731	64	53	and	and	CCONJ
iajs-2731	64	54	bt	bt	PROPN
iajs-2731	64	55	is	be	AUX
iajs-2731	64	56	a	a	DET
iajs-2731	64	57	f	f	PROPN
iajs-2731	64	58	subm	subm	PROPN
iajs-2731	64	59	in	in	ADP
iajs-2731	64	60	xt	xt	PROPN
iajs-2731	64	61	by	by	ADP
iajs-2731	64	62	[	[	X
iajs-2731	64	63	8	8	NUM
iajs-2731	64	64	]	]	PUNCT
iajs-2731	64	65	.	.	PUNCT
iajs-2731	65	1	but	but	CCONJ
iajs-2731	65	2	a	a	PRON
iajs-2731	65	3	is	be	AUX
iajs-2731	65	4	a	a	DET
iajs-2731	65	5	unique	unique	ADJ
iajs-2731	65	6	maximal	maximal	ADJ
iajs-2731	65	7	fsubm	fsubm	NOUN
iajs-2731	65	8	of	of	ADP
iajs-2731	65	9	x	x	PRON
iajs-2731	65	10	which	which	PRON
iajs-2731	65	11	contains	contain	VERB
iajs-2731	65	12	all	all	DET
iajs-2731	65	13	a	a	DET
iajs-2731	65	14	small	small	ADJ
iajs-2731	65	15	f	f	X
iajs-2731	65	16	-	-	PUNCT
iajs-2731	65	17	subm	subm	PROPN
iajs-2731	65	18	by	by	ADP
iajs-2731	65	19	our	our	PRON
iajs-2731	65	20	assumption	assumption	NOUN
iajs-2731	65	21	.	.	PUNCT
iajs-2731	66	1	therefore	therefore	ADV
iajs-2731	66	2	x	x	X
iajs-2731	66	3	is	be	AUX
iajs-2731	66	4	lochollow	lochollow	NOUN
iajs-2731	66	5	fmodules	fmodule	NOUN
iajs-2731	66	6	.	.	PUNCT
iajs-2731	67	1	(	(	PUNCT
iajs-2731	67	2	2.4	2.4	NUM
iajs-2731	67	3	)	)	PUNCT
iajs-2731	67	4	remark	remark	NOUN
iajs-2731	67	5	and	and	CCONJ
iajs-2731	67	6	example	example	NOUN
iajs-2731	67	7	1every	1every	NUM
iajs-2731	67	8	lochollow	lochollow	NOUN
iajs-2731	67	9	fmodule	fmodule	NOUN
iajs-2731	67	10	is	be	AUX
iajs-2731	67	11	hollow	hollow	ADJ
iajs-2731	67	12	fmodule	fmodule	ADV
iajs-2731	67	13	.	.	PUNCT
iajs-2731	68	1	proof	proof	NOUN
iajs-2731	68	2	suppose	suppose	VERB
iajs-2731	68	3	that	that	SCONJ
iajs-2731	68	4	x	x	PRON
iajs-2731	68	5	is	be	AUX
iajs-2731	68	6	a	a	DET
iajs-2731	68	7	loc	loc	ADJ
iajs-2731	68	8	-	-	ADJ
iajs-2731	68	9	hollow	hollow	ADJ
iajs-2731	68	10	fmodule	fmodule	NOUN
iajs-2731	68	11	,	,	PUNCT
iajs-2731	68	12	then	then	ADV
iajs-2731	68	13	there	there	PRON
iajs-2731	68	14	exists	exist	VERB
iajs-2731	68	15	a	a	DET
iajs-2731	68	16	unique	unique	ADJ
iajs-2731	68	17	maximal	maximal	ADJ
iajs-2731	68	18	fsubm	fsubm	NOUN
iajs-2731	68	19	containing	contain	VERB
iajs-2731	68	20	all	all	DET
iajs-2731	68	21	a	a	DET
iajs-2731	68	22	small	small	ADJ
iajs-2731	68	23	f	f	X
iajs-2731	68	24	-	-	PUNCT
iajs-2731	68	25	subm	subm	PROPN
iajs-2731	68	26	say	say	VERB
iajs-2731	68	27	b	b	X
iajs-2731	68	28	in	in	ADP
iajs-2731	68	29	x.	x.	NOUN
iajs-2731	68	30	and	and	CCONJ
iajs-2731	68	31	since	since	SCONJ
iajs-2731	68	32	b	b	PROPN
iajs-2731	68	33	is	be	AUX
iajs-2731	68	34	f	f	PROPN
iajs-2731	68	35	-subm	-subm	PROPN
iajs-2731	68	36	of	of	ADP
iajs-2731	68	37	x.	x.	NOUN
iajs-2731	68	38	therefore	therefore	ADV
iajs-2731	68	39	b	b	AUX
iajs-2731	68	40	contain	contain	VERB
iajs-2731	68	41	in	in	ADP
iajs-2731	68	42	x.	x.	NOUN
iajs-2731	68	43	by	by	ADP
iajs-2731	68	44	definition	definition	NOUN
iajs-2731	68	45	of	of	ADP
iajs-2731	68	46	hollow	hollow	ADJ
iajs-2731	68	47	fmodule	fmodule	NOUN
iajs-2731	68	48	.	.	PUNCT
iajs-2731	69	1	this	this	PRON
iajs-2731	69	2	implies	imply	VERB
iajs-2731	69	3	that	that	SCONJ
iajs-2731	69	4	x	x	PRON
iajs-2731	69	5	is	be	AUX
iajs-2731	69	6	a	a	DET
iajs-2731	69	7	hollow	hollow	ADJ
iajs-2731	69	8	fmodule	fmodule	NOUN
iajs-2731	69	9	.	.	PUNCT
iajs-2731	70	1	2the	2the	PRON
iajs-2731	70	2	convers	conver	NOUN
iajs-2731	70	3	remark	remark	VERB
iajs-2731	70	4	(	(	PUNCT
iajs-2731	70	5	2.4	2.4	NUM
iajs-2731	70	6	)	)	PUNCT
iajs-2731	70	7	(	(	PUNCT
iajs-2731	70	8	1	1	X
iajs-2731	70	9	)	)	PUNCT
iajs-2731	70	10	is	be	AUX
iajs-2731	70	11	not	not	PART
iajs-2731	70	12	true	true	ADJ
iajs-2731	70	13	in	in	ADP
iajs-2731	70	14	general	general	ADJ
iajs-2731	70	15	for	for	ADP
iajs-2731	70	16	instant	instant	NOUN
iajs-2731	70	17	let	let	NOUN
iajs-2731	70	18	m	m	NOUN
iajs-2731	70	19	=	=	PROPN
iajs-2731	70	20	z	z	PROPN
iajs-2731	70	21	p∞	p∞	PROPN
iajs-2731	70	22	.	.	PUNCT
iajs-2731	71	1	r	r	X
iajs-2731	71	2	=	=	NOUN
iajs-2731	71	3	z	z	NOUN
iajs-2731	71	4	,	,	PUNCT
iajs-2731	71	5	define	define	VERB
iajs-2731	71	6	x	x	NOUN
iajs-2731	71	7	:	:	PUNCT
iajs-2731	71	8	m	m	VERB
iajs-2731	71	9	→	→	SYM
iajs-2731	71	10	[	[	PUNCT
iajs-2731	71	11	0,1	0,1	NUM
iajs-2731	71	12	]	]	PUNCT
iajs-2731	71	13	𝑎𝑠	𝑎𝑠	AUX
iajs-2731	71	14	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	PROPN
iajs-2731	71	15	x(x	x(x	PROPN
iajs-2731	71	16	)	)	PUNCT
iajs-2731	72	1	=	=	PRON
iajs-2731	72	2	{	{	PUNCT
iajs-2731	72	3	1	1	NUM
iajs-2731	72	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	72	5	𝑥	𝑥	PRON
iajs-2731	72	6	∈	∈	PROPN
iajs-2731	72	7	𝑀	𝑀	PROPN
iajs-2731	72	8	0	0	PUNCT
iajs-2731	72	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2731	72	10	clear	clear	ADJ
iajs-2731	72	11	that	that	SCONJ
iajs-2731	72	12	𝜒	𝜒	NOUN
iajs-2731	72	13	is	be	AUX
iajs-2731	72	14	fmodule	fmodule	ADJ
iajs-2731	72	15	and	and	CCONJ
iajs-2731	72	16	xt	xt	X
iajs-2731	72	17	=	=	SYM
iajs-2731	72	18	z	z	PROPN
iajs-2731	72	19	p∞	p∞	NOUN
iajs-2731	72	20	=	=	NOUN
iajs-2731	73	1	m.	m.	NOUN
iajs-2731	73	2	but	but	CCONJ
iajs-2731	73	3	z	z	PROPN
iajs-2731	73	4	p∞	p∞	PROPN
iajs-2731	73	5	has	have	VERB
iajs-2731	73	6	a	a	DET
iajs-2731	73	7	small	small	ADJ
iajs-2731	73	8	submodule	submodule	NOUN
iajs-2731	73	9	by	by	ADP
iajs-2731	73	10	[	[	X
iajs-2731	73	11	5	5	NUM
iajs-2731	73	12	]	]	PUNCT
iajs-2731	73	13	,	,	PUNCT
iajs-2731	73	14	and	and	CCONJ
iajs-2731	73	15	it	it	PRON
iajs-2731	73	16	is	be	AUX
iajs-2731	73	17	a	a	DET
iajs-2731	73	18	fuzzy	fuzzy	ADJ
iajs-2731	73	19	small	small	ADJ
iajs-2731	73	20	by	by	ADP
iajs-2731	73	21	[	[	X
iajs-2731	73	22	1].then	1].then	NUM
iajs-2731	73	23	xt	xt	PUNCT
iajs-2731	74	1	=	=	NOUN
iajs-2731	74	2	𝑍𝑝	𝑍𝑝	PROPN
iajs-2731	74	3	∞	∞	PROPN
iajs-2731	74	4	is	be	AUX
iajs-2731	74	5	a	a	DET
iajs-2731	74	6	hollow	hollow	ADJ
iajs-2731	74	7	module	module	NOUN
iajs-2731	74	8	.	.	PUNCT
iajs-2731	75	1	but	but	CCONJ
iajs-2731	75	2	not	not	PART
iajs-2731	75	3	lochollow	lochollow	NOUN
iajs-2731	75	4	module	module	NOUN
iajs-2731	75	5	.	.	PUNCT
iajs-2731	76	1	then	then	ADV
iajs-2731	76	2	x	x	PRON
iajs-2731	76	3	is	be	AUX
iajs-2731	76	4	not	not	PART
iajs-2731	76	5	loc	loc	NOUN
iajs-2731	76	6	hollow	hollow	ADJ
iajs-2731	76	7	module	module	NOUN
iajs-2731	76	8	by	by	ADP
iajs-2731	76	9	proposition	proposition	NOUN
iajs-2731	76	10	:	:	PUNCT
iajs-2731	76	11	(	(	PUNCT
iajs-2731	76	12	2.3	2.3	NUM
iajs-2731	76	13	)	)	PUNCT
iajs-2731	76	14	the	the	DET
iajs-2731	76	15	3	3	NUM
iajs-2731	76	16	-	-	PUNCT
iajs-2731	76	17	every	every	DET
iajs-2731	76	18	local	local	ADJ
iajs-2731	76	19	fuzzy	fuzzy	ADJ
iajs-2731	76	20	module	module	NOUN
iajs-2731	76	21	is	be	AUX
iajs-2731	76	22	lochollow	lochollow	NOUN
iajs-2731	76	23	fmodule	fmodule	ADJ
iajs-2731	76	24	,	,	PUNCT
iajs-2731	76	25	while	while	SCONJ
iajs-2731	76	26	the	the	DET
iajs-2731	76	27	converse	converse	NOUN
iajs-2731	76	28	is	be	AUX
iajs-2731	76	29	not	not	PART
iajs-2731	76	30	true	true	ADJ
iajs-2731	76	31	in	in	ADP
iajs-2731	76	32	general	general	ADJ
iajs-2731	76	33	.	.	PUNCT
iajs-2731	77	1	proof	proof	NOUN
iajs-2731	77	2	:	:	PUNCT
iajs-2731	77	3	let	let	VERB
iajs-2731	77	4	m=	m=	AUX
iajs-2731	77	5	z2	z2	PROPN
iajs-2731	77	6	⊕	⊕	PROPN
iajs-2731	77	7	q	q	PROPN
iajs-2731	77	8	,	,	PUNCT
iajs-2731	77	9	r	r	NOUN
iajs-2731	77	10	=	=	SYM
iajs-2731	77	11	z	z	NOUN
iajs-2731	77	12	,	,	PUNCT
iajs-2731	77	13	define	define	VERB
iajs-2731	77	14	by	by	ADP
iajs-2731	77	15	x	x	NOUN
iajs-2731	77	16	:	:	PUNCT
iajs-2731	77	17	m	m	VERB
iajs-2731	77	18	→	→	SYM
iajs-2731	77	19	[	[	PUNCT
iajs-2731	77	20	0,1	0,1	NUM
iajs-2731	77	21	]	]	PUNCT
iajs-2731	77	22	𝑎𝑠	𝑎𝑠	ADP
iajs-2731	77	23	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	PROPN
iajs-2731	77	24	x(x	x(x	PROPN
iajs-2731	77	25	,	,	PUNCT
iajs-2731	77	26	y	y	NOUN
iajs-2731	77	27	)	)	PUNCT
iajs-2731	77	28	=	=	PRON
iajs-2731	77	29	{	{	PUNCT
iajs-2731	77	30	1	1	NUM
iajs-2731	77	31	∀(𝑥	∀(𝑥	NUM
iajs-2731	77	32	,	,	PUNCT
iajs-2731	77	33	𝑦	𝑦	NOUN
iajs-2731	77	34	)	)	PUNCT
iajs-2731	77	35	𝑥	𝑥	PRON
iajs-2731	77	36	∈	∈	PROPN
iajs-2731	77	37	𝑀	𝑀	PROPN
iajs-2731	77	38	0	0	NUM
iajs-2731	77	39	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2731	77	40	such	such	ADJ
iajs-2731	77	41	that	that	SCONJ
iajs-2731	77	42	x	x	SYM
iajs-2731	77	43	∈	∈	PROPN
iajs-2731	77	44	0̅	0̅	PROPN
iajs-2731	77	45	,	,	PUNCT
iajs-2731	77	46	y	y	PROPN
iajs-2731	77	47	∈	∈	PROPN
iajs-2731	77	48	q	q	X
iajs-2731	77	49	∀	∀	X
iajs-2731	77	50	t	t	NOUN
iajs-2731	77	51	∈	∈	PROPN
iajs-2731	77	52	(	(	PUNCT
iajs-2731	77	53	0,1	0,1	NOUN
iajs-2731	77	54	]	]	PUNCT
iajs-2731	77	55	.	.	PUNCT
iajs-2731	78	1	clear	clear	ADJ
iajs-2731	78	2	that	that	SCONJ
iajs-2731	78	3	𝜒	𝜒	NOUN
iajs-2731	78	4	is	be	AUX
iajs-2731	78	5	fmodule	fmodule	ADJ
iajs-2731	78	6	and	and	CCONJ
iajs-2731	78	7	m	m	NOUN
iajs-2731	78	8	=	=	NOUN
iajs-2731	78	9	xt	xt	X
iajs-2731	78	10	,	,	PUNCT
iajs-2731	78	11	z2	z2	PROPN
iajs-2731	78	12	⊕	⊕	PROPN
iajs-2731	78	13	q	q	PROPN
iajs-2731	78	14	is	be	AUX
iajs-2731	78	15	lochollow	lochollow	NOUN
iajs-2731	78	16	module	module	NOUN
iajs-2731	78	17	.	.	PUNCT
iajs-2731	79	1	but	but	CCONJ
iajs-2731	79	2	not	not	PART
iajs-2731	79	3	local	local	ADJ
iajs-2731	79	4	module	module	NOUN
iajs-2731	79	5	since	since	SCONJ
iajs-2731	79	6	xt	xt	PROPN
iajs-2731	80	1	=	=	SYM
iajs-2731	80	2	{	{	PUNCT
iajs-2731	80	3	0̅1	0̅1	PROPN
iajs-2731	80	4	}	}	PUNCT
iajs-2731	80	5	⊕q	⊕q	PROPN
iajs-2731	80	6	≅	≅	PROPN
iajs-2731	80	7	q	q	PROPN
iajs-2731	80	8	is	be	AUX
iajs-2731	80	9	a	a	DET
iajs-2731	80	10	unique	unique	ADJ
iajs-2731	80	11	maximal	maximal	ADJ
iajs-2731	80	12	submodule	submodule	NOUN
iajs-2731	80	13	of	of	ADP
iajs-2731	80	14	z2	z2	PROPN
iajs-2731	80	15	⊕	⊕	PROPN
iajs-2731	80	16	q	q	PROPN
iajs-2731	81	1	and	and	CCONJ
iajs-2731	81	2	{	{	PUNCT
iajs-2731	81	3	01	01	NUM
iajs-2731	81	4	}	}	PUNCT
iajs-2731	81	5	⊕	⊕	PROPN
iajs-2731	81	6	{	{	PUNCT
iajs-2731	81	7	01	01	NUM
iajs-2731	81	8	}	}	PUNCT
iajs-2731	81	9	is	be	AUX
iajs-2731	81	10	a	a	DET
iajs-2731	81	11	small	small	ADJ
iajs-2731	81	12	submodule	submodule	NOUN
iajs-2731	81	13	of	of	ADP
iajs-2731	81	14	z2	z2	PROPN
iajs-2731	81	15	⊕	⊕	PROPN
iajs-2731	81	16	q	q	PROPN
iajs-2731	81	17	and	and	CCONJ
iajs-2731	81	18	contained	contain	VERB
iajs-2731	81	19	in	in	ADP
iajs-2731	81	20	{	{	PUNCT
iajs-2731	81	21	01	01	NUM
iajs-2731	81	22	}	}	PUNCT
iajs-2731	81	23	⊕	⊕	PROPN
iajs-2731	81	24	q	q	PROPN
iajs-2731	82	1	≅	≅	PROPN
iajs-2731	82	2	q.	q.	PROPN
iajs-2731	82	3	but	but	CCONJ
iajs-2731	82	4	z2	z2	PROPN
iajs-2731	82	5	⊕	⊕	PROPN
iajs-2731	82	6	{	{	PUNCT
iajs-2731	82	7	01	01	NUM
iajs-2731	82	8	}	}	PUNCT
iajs-2731	82	9	is	be	AUX
iajs-2731	82	10	a	a	DET
iajs-2731	82	11	proper	proper	ADJ
iajs-2731	82	12	submodule	submodule	NOUN
iajs-2731	82	13	of	of	ADP
iajs-2731	82	14	z2	z2	PROPN
iajs-2731	82	15	⊕	⊕	PROPN
iajs-2731	82	16	q	q	PROPN
iajs-2731	82	17	,	,	PUNCT
iajs-2731	82	18	also	also	ADV
iajs-2731	82	19	z2	z2	PROPN
iajs-2731	82	20	⊕	⊕	PROPN
iajs-2731	82	21	{	{	PUNCT
iajs-2731	82	22	01	01	NUM
iajs-2731	82	23	}	}	PUNCT
iajs-2731	82	24	is	be	AUX
iajs-2731	82	25	not	not	PART
iajs-2731	82	26	contained	contain	VERB
iajs-2731	82	27	in	in	ADP
iajs-2731	82	28	{	{	PUNCT
iajs-2731	82	29	01	01	NUM
iajs-2731	82	30	}	}	PUNCT
iajs-2731	82	31	⊕	⊕	PROPN
iajs-2731	82	32	q.	q.	PROPN
iajs-2731	82	33	thus	thus	ADV
iajs-2731	82	34	x	x	AUX
iajs-2731	82	35	is	be	AUX
iajs-2731	82	36	lochollow	lochollow	ADJ
iajs-2731	82	37	fuzzy	fuzzy	ADJ
iajs-2731	82	38	module	module	NOUN
iajs-2731	82	39	but	but	CCONJ
iajs-2731	82	40	a	a	DET
iajs-2731	82	41	local	local	ADJ
iajs-2731	82	42	fuzzy	fuzzy	ADJ
iajs-2731	82	43	module	module	NOUN
iajs-2731	82	44	.	.	PUNCT
iajs-2731	83	1	4	4	NUM
iajs-2731	83	2	-	-	X
iajs-2731	83	3	every	every	DET
iajs-2731	83	4	simple	simple	ADJ
iajs-2731	83	5	fuzzy	fuzzy	ADJ
iajs-2731	83	6	submodule	submodule	NOUN
iajs-2731	83	7	is	be	AUX
iajs-2731	83	8	not	not	PART
iajs-2731	83	9	lochollow	lochollow	ADJ
iajs-2731	83	10	fuzzy	fuzzy	ADJ
iajs-2731	83	11	module	module	NOUN
iajs-2731	83	12	.	.	PUNCT
iajs-2731	84	1	ibn	ibn	PROPN
iajs-2731	84	2	al	al	PROPN
iajs-2731	84	3	-	-	PUNCT
iajs-2731	84	4	haitham	haitham	PROPN
iajs-2731	84	5	jour	jour	X
iajs-2731	84	6	.	.	PROPN
iajs-2731	84	7	for	for	ADP
iajs-2731	84	8	pure	pure	ADJ
iajs-2731	84	9	&	&	CCONJ
iajs-2731	84	10	appl	appl	PROPN
iajs-2731	84	11	.	.	PUNCT
iajs-2731	85	1	sci	sci	PROPN
iajs-2731	85	2	.	.	PROPN
iajs-2731	86	1	53	53	NUM
iajs-2731	86	2	(	(	PUNCT
iajs-2731	86	3	2)2022	2)2022	NOUN
iajs-2731	86	4	88	88	NUM
iajs-2731	86	5	proof	proof	NOUN
iajs-2731	86	6	:	:	PUNCT
iajs-2731	86	7	let	let	VERB
iajs-2731	86	8	m	m	PRON
iajs-2731	86	9	=	=	NOUN
iajs-2731	86	10	z5	z5	X
iajs-2731	86	11	,	,	PUNCT
iajs-2731	86	12	r	r	NOUN
iajs-2731	86	13	=	=	SYM
iajs-2731	86	14	z	z	PROPN
iajs-2731	86	15	,	,	PUNCT
iajs-2731	86	16	𝜒	𝜒	X
iajs-2731	86	17	:	:	PUNCT
iajs-2731	86	18	m	m	VERB
iajs-2731	86	19	→	→	SYM
iajs-2731	86	20	[	[	PUNCT
iajs-2731	86	21	0,1	0,1	NUM
iajs-2731	86	22	]	]	PUNCT
iajs-2731	86	23	𝑎𝑠	𝑎𝑠	AUX
iajs-2731	86	24	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	VERB
iajs-2731	86	25	𝜒	𝜒	X
iajs-2731	86	26	(	(	PUNCT
iajs-2731	86	27	x	x	NOUN
iajs-2731	86	28	)	)	PUNCT
iajs-2731	86	29	=	=	NOUN
iajs-2731	86	30	{	{	PUNCT
iajs-2731	86	31	1	1	NUM
iajs-2731	86	32	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	86	33	𝑥	𝑥	PRON
iajs-2731	86	34	∈	∈	PROPN
iajs-2731	86	35	𝑀	𝑀	PROPN
iajs-2731	86	36	0	0	NUM
iajs-2731	86	37	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	86	38	,	,	PUNCT
iajs-2731	86	39	t	t	PROPN
iajs-2731	86	40	∈	∈	PROPN
iajs-2731	86	41	(	(	PUNCT
iajs-2731	86	42	0	0	NUM
iajs-2731	86	43	,	,	PUNCT
iajs-2731	86	44	1	1	NUM
iajs-2731	86	45	]	]	PUNCT
iajs-2731	86	46	,	,	PUNCT
iajs-2731	86	47	a(x	a(x	PROPN
iajs-2731	86	48	)	)	PUNCT
iajs-2731	86	49	=	=	PRON
iajs-2731	86	50	{	{	PUNCT
iajs-2731	86	51	𝑡	𝑡	X
iajs-2731	86	52	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	86	53	𝑥	𝑥	PRON
iajs-2731	86	54	∈	∈	NOUN
iajs-2731	86	55	𝑁	𝑁	NOUN
iajs-2731	86	56	0	0	NUM
iajs-2731	86	57	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	86	58	t	t	PROPN
iajs-2731	86	59	∈	∈	PROPN
iajs-2731	86	60	(	(	PUNCT
iajs-2731	86	61	0	0	NUM
iajs-2731	86	62	,	,	PUNCT
iajs-2731	86	63	1	1	NUM
iajs-2731	86	64	]	]	PUNCT
iajs-2731	86	65	,	,	PUNCT
iajs-2731	86	66	n=0̅.	n=0̅.	NOUN
iajs-2731	86	67	clear	clear	ADJ
iajs-2731	86	68	that	that	SCONJ
iajs-2731	86	69	x	x	PRON
iajs-2731	86	70	is	be	AUX
iajs-2731	86	71	a	a	DET
iajs-2731	86	72	simple	simple	ADJ
iajs-2731	86	73	fuzzy	fuzzy	ADJ
iajs-2731	86	74	module	module	NOUN
iajs-2731	86	75	.	.	PUNCT
iajs-2731	87	1	but	but	CCONJ
iajs-2731	87	2	not	not	PART
iajs-2731	87	3	local	local	ADJ
iajs-2731	87	4	fuzzy	fuzzy	ADJ
iajs-2731	87	5	module	module	NOUN
iajs-2731	87	6	since	since	SCONJ
iajs-2731	87	7	m=	m=	X
iajs-2731	87	8	{	{	PUNCT
iajs-2731	87	9	0̅1	0̅1	NOUN
iajs-2731	87	10	}	}	PUNCT
iajs-2731	87	11	⊕z5	⊕z5	ADP
iajs-2731	88	1	=	=	VERB
iajs-2731	88	2	z5	z5	X
iajs-2731	88	3	is	be	AUX
iajs-2731	88	4	a	a	DET
iajs-2731	88	5	unique	unique	ADJ
iajs-2731	88	6	maximal	maximal	ADJ
iajs-2731	88	7	fuzzy	fuzzy	ADJ
iajs-2731	88	8	submodule	submodule	NOUN
iajs-2731	88	9	of	of	ADP
iajs-2731	88	10	z5	z5	PROPN
iajs-2731	88	11	and	and	CCONJ
iajs-2731	88	12	{	{	PUNCT
iajs-2731	88	13	01	01	NUM
iajs-2731	88	14	}	}	PUNCT
iajs-2731	88	15	is	be	AUX
iajs-2731	88	16	only	only	ADV
iajs-2731	88	17	one	one	NUM
iajs-2731	88	18	proper	proper	ADJ
iajs-2731	88	19	fuzzy	fuzzy	ADJ
iajs-2731	88	20	submodule	submodule	NOUN
iajs-2731	88	21	,	,	PUNCT
iajs-2731	88	22	which	which	PRON
iajs-2731	88	23	is	be	AUX
iajs-2731	88	24	a	a	DET
iajs-2731	88	25	small	small	ADJ
iajs-2731	88	26	submodule	submodule	NOUN
iajs-2731	88	27	of	of	ADP
iajs-2731	88	28	z5	z5	PROPN
iajs-2731	88	29	and	and	CCONJ
iajs-2731	88	30	contained	contain	VERB
iajs-2731	88	31	in	in	ADP
iajs-2731	88	32	{	{	PUNCT
iajs-2731	88	33	01	01	NUM
iajs-2731	88	34	}	}	PUNCT
iajs-2731	88	35	⊕	⊕	PROPN
iajs-2731	88	36	z5	z5	PROPN
iajs-2731	88	37	=	=	SYM
iajs-2731	88	38	z5	z5	X
iajs-2731	88	39	.	.	PUNCT
iajs-2731	89	1	also	also	ADV
iajs-2731	89	2	z5	z5	PROPN
iajs-2731	89	3	⊕	⊕	PROPN
iajs-2731	89	4	{	{	PUNCT
iajs-2731	89	5	01	01	NUM
iajs-2731	89	6	}	}	PUNCT
iajs-2731	89	7	is	be	AUX
iajs-2731	89	8	not	not	PART
iajs-2731	89	9	contained	contain	VERB
iajs-2731	89	10	in	in	ADP
iajs-2731	89	11	{	{	PUNCT
iajs-2731	89	12	01	01	NUM
iajs-2731	89	13	}	}	PUNCT
iajs-2731	89	14	⊕	⊕	PROPN
iajs-2731	89	15	z5.that	z5.that	X
iajs-2731	89	16	is	be	AUX
iajs-2731	89	17	x	x	PUNCT
iajs-2731	89	18	is	be	AUX
iajs-2731	89	19	a	a	DET
iajs-2731	89	20	simple	simple	ADJ
iajs-2731	89	21	fuzzy	fuzzy	ADJ
iajs-2731	89	22	module	module	NOUN
iajs-2731	89	23	and	and	CCONJ
iajs-2731	89	24	not	not	PART
iajs-2731	89	25	lochollow	lochollow	VERB
iajs-2731	89	26	fuzzy	fuzzy	ADJ
iajs-2731	89	27	module	module	NOUN
iajs-2731	89	28	.	.	PUNCT
iajs-2731	90	1	5	5	NUM
iajs-2731	90	2	-	-	PUNCT
iajs-2731	90	3	every	every	DET
iajs-2731	90	4	lochollow	lochollow	NOUN
iajs-2731	90	5	fmodule	fmodule	NOUN
iajs-2731	90	6	is	be	AUX
iajs-2731	90	7	not	not	PART
iajs-2731	90	8	a	a	DET
iajs-2731	90	9	simple	simple	ADJ
iajs-2731	90	10	fuzzy	fuzzy	ADJ
iajs-2731	90	11	submodule	submodule	NOUN
iajs-2731	90	12	:	:	PUNCT
iajs-2731	90	13	proof	proof	NOUN
iajs-2731	90	14	:	:	PUNCT
iajs-2731	90	15	let	let	VERB
iajs-2731	90	16	m=	m=	X
iajs-2731	90	17	z8	z8	NOUN
iajs-2731	90	18	,	,	PUNCT
iajs-2731	90	19	r	r	NOUN
iajs-2731	90	20	=	=	SYM
iajs-2731	90	21	z	z	NOUN
iajs-2731	90	22	,	,	PUNCT
iajs-2731	90	23	define	define	VERB
iajs-2731	90	24	x	x	NOUN
iajs-2731	90	25	:	:	PUNCT
iajs-2731	90	26	m	m	VERB
iajs-2731	90	27	→	→	SYM
iajs-2731	90	28	[	[	PUNCT
iajs-2731	90	29	0,1	0,1	NUM
iajs-2731	90	30	]	]	PUNCT
iajs-2731	90	31	𝑎𝑠	𝑎𝑠	AUX
iajs-2731	90	32	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	PROPN
iajs-2731	90	33	x(x	x(x	PROPN
iajs-2731	90	34	)	)	PUNCT
iajs-2731	91	1	=	=	PRON
iajs-2731	91	2	{	{	PUNCT
iajs-2731	91	3	1	1	NUM
iajs-2731	91	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	91	5	𝑥	𝑥	PRON
iajs-2731	91	6	∈	∈	PROPN
iajs-2731	91	7	𝑀	𝑀	PROPN
iajs-2731	91	8	0	0	NUM
iajs-2731	91	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	91	10	and	and	CCONJ
iajs-2731	91	11	define	define	VERB
iajs-2731	91	12	a	a	DET
iajs-2731	91	13	:	:	PUNCT
iajs-2731	91	14	m	m	NOUN
iajs-2731	91	15	→	→	SYM
iajs-2731	91	16	[	[	PUNCT
iajs-2731	91	17	0,1	0,1	NUM
iajs-2731	91	18	]	]	PUNCT
iajs-2731	91	19	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2731	91	20	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	91	21	a(x	a(x	PROPN
iajs-2731	91	22	)	)	PUNCT
iajs-2731	91	23	=	=	PRON
iajs-2731	91	24	{	{	PUNCT
iajs-2731	91	25	𝑡	𝑡	X
iajs-2731	91	26	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	91	27	𝑥	𝑥	PRON
iajs-2731	91	28	∈	∈	NOUN
iajs-2731	91	29	𝑁	𝑁	PROPN
iajs-2731	91	30	0	0	NUM
iajs-2731	91	31	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	91	32	,	,	PUNCT
iajs-2731	91	33	t	t	PROPN
iajs-2731	91	34	∈	∈	PROPN
iajs-2731	91	35	(	(	PUNCT
iajs-2731	91	36	0	0	NUM
iajs-2731	91	37	,	,	PUNCT
iajs-2731	91	38	1	1	NUM
iajs-2731	91	39	]	]	PUNCT
iajs-2731	91	40	.	.	PUNCT
iajs-2731	92	1	n=	n=	ADJ
iajs-2731	93	1	2z8	2z8	NUM
iajs-2731	93	2	b	b	NOUN
iajs-2731	93	3	:	:	PUNCT
iajs-2731	93	4	m	m	VERB
iajs-2731	93	5	→	→	SYM
iajs-2731	93	6	[	[	PUNCT
iajs-2731	93	7	0,1	0,1	NUM
iajs-2731	93	8	]	]	PUNCT
iajs-2731	93	9	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	X
iajs-2731	93	10	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	93	11	b(x	b(x	NOUN
iajs-2731	93	12	)	)	PUNCT
iajs-2731	94	1	=	=	PRON
iajs-2731	94	2	{	{	PUNCT
iajs-2731	94	3	𝑡	𝑡	X
iajs-2731	94	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	94	5	𝑥	𝑥	DET
iajs-2731	94	6	∈	∈	PROPN
iajs-2731	94	7	𝐿	𝐿	PROPN
iajs-2731	94	8	0	0	PROPN
iajs-2731	94	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	94	10	,	,	PUNCT
iajs-2731	94	11	t	t	PROPN
iajs-2731	94	12	∈	∈	PROPN
iajs-2731	94	13	(	(	PUNCT
iajs-2731	94	14	0	0	NUM
iajs-2731	94	15	,	,	PUNCT
iajs-2731	94	16	1	1	NUM
iajs-2731	94	17	]	]	PUNCT
iajs-2731	94	18	.	.	PUNCT
iajs-2731	95	1	l=	l=	ADJ
iajs-2731	95	2	4z8	4z8	ADV
iajs-2731	95	3	consequently	consequently	ADV
iajs-2731	95	4	,	,	PUNCT
iajs-2731	95	5	x	x	SYM
iajs-2731	95	6	fuzzy	fuzzy	ADJ
iajs-2731	95	7	module	module	NOUN
iajs-2731	95	8	.	.	PUNCT
iajs-2731	96	1	and	and	CCONJ
iajs-2731	96	2	at	at	ADP
iajs-2731	96	3	=	=	NOUN
iajs-2731	96	4	n	n	X
iajs-2731	96	5	,	,	PUNCT
iajs-2731	96	6	bt	bt	PROPN
iajs-2731	96	7	=	=	NOUN
iajs-2731	96	8	l	l	NOUN
iajs-2731	96	9	are	be	AUX
iajs-2731	96	10	two	two	NUM
iajs-2731	96	11	submodules	submodule	NOUN
iajs-2731	96	12	of	of	ADP
iajs-2731	96	13	xt	xt	PROPN
iajs-2731	96	14	and	and	CCONJ
iajs-2731	96	15	at	at	ADP
iajs-2731	96	16	a	a	DET
iajs-2731	96	17	small	small	ADJ
iajs-2731	96	18	submodule	submodule	NOUN
iajs-2731	96	19	of	of	ADP
iajs-2731	96	20	xt	xt	NUM
iajs-2731	96	21	by	by	ADP
iajs-2731	96	22	[	[	X
iajs-2731	96	23	5	5	NUM
iajs-2731	96	24	]	]	PUNCT
iajs-2731	96	25	,	,	PUNCT
iajs-2731	96	26	then	then	ADV
iajs-2731	96	27	a	a	PRON
iajs-2731	96	28	is	be	AUX
iajs-2731	96	29	a	a	DET
iajs-2731	96	30	small	small	ADJ
iajs-2731	96	31	submodule	submodule	NOUN
iajs-2731	96	32	of	of	ADP
iajs-2731	96	33	x	x	PUNCT
iajs-2731	96	34	by	by	ADP
iajs-2731	96	35	[	[	PUNCT
iajs-2731	96	36	9	9	NUM
iajs-2731	96	37	]	]	PUNCT
iajs-2731	96	38	,	,	PUNCT
iajs-2731	96	39	which	which	PRON
iajs-2731	96	40	is	be	AUX
iajs-2731	96	41	a	a	DET
iajs-2731	96	42	maximal	maximal	ADJ
iajs-2731	96	43	submodule	submodule	NOUN
iajs-2731	96	44	of	of	ADP
iajs-2731	96	45	𝜒	𝜒	NOUN
iajs-2731	96	46	.	.	PUNCT
iajs-2731	97	1	but	but	CCONJ
iajs-2731	97	2	(	(	PUNCT
iajs-2731	97	3	⊕	⊕	NOUN
iajs-2731	97	4	𝑩)t	𝑩)t	NOUN
iajs-2731	98	1	=	=	VERB
iajs-2731	98	2	xt	xt	X
iajs-2731	98	3	by	by	ADP
iajs-2731	98	4	[	[	X
iajs-2731	98	5	10	10	NUM
iajs-2731	98	6	]	]	PUNCT
iajs-2731	98	7	are	be	AUX
iajs-2731	98	8	two	two	NUM
iajs-2731	98	9	direct	direct	ADJ
iajs-2731	98	10	sum	sum	NOUN
iajs-2731	98	11	submodules	submodule	NOUN
iajs-2731	98	12	of	of	ADP
iajs-2731	98	13	xt	xt	PROPN
iajs-2731	98	14	.	.	PROPN
iajs-2731	99	1	and	and	CCONJ
iajs-2731	99	2	a	a	DET
iajs-2731	99	3	⊕	⊕	PROPN
iajs-2731	99	4	b	b	X
iajs-2731	99	5	=	=	PROPN
iajs-2731	99	6	𝜒	𝜒	X
iajs-2731	100	1	.this	.this	PRON
iajs-2731	100	2	implies	imply	VERB
iajs-2731	100	3	that	that	SCONJ
iajs-2731	100	4	x	x	PRON
iajs-2731	100	5	is	be	AUX
iajs-2731	100	6	not	not	PART
iajs-2731	100	7	a	a	DET
iajs-2731	100	8	simple	simple	ADJ
iajs-2731	100	9	fmodule	fmodule	NOUN
iajs-2731	100	10	.	.	PUNCT
iajs-2731	101	1	2.5	2.5	NUM
iajs-2731	101	2	𝝆roposition	𝝆roposition	NOUN
iajs-2731	101	3	:	:	PUNCT
iajs-2731	101	4	epimorphic	epimorphic	ADJ
iajs-2731	101	5	image	image	NOUN
iajs-2731	101	6	of	of	ADP
iajs-2731	101	7	lochollow	lochollow	NOUN
iajs-2731	101	8	fmodule	fmodule	NOUN
iajs-2731	101	9	is	be	AUX
iajs-2731	101	10	lochollow	lochollow	NOUN
iajs-2731	101	11	fmodule	fmodule	ADV
iajs-2731	101	12	.	.	PUNCT
iajs-2731	102	1	𝝆roof	𝝆roof	PRON
iajs-2731	102	2	:	:	PUNCT
iajs-2731	102	3	let	let	VERB
iajs-2731	102	4	x1	x1	PRON
iajs-2731	102	5	be	be	AUX
iajs-2731	102	6	lochollow	lochollow	NOUN
iajs-2731	102	7	fmodule	fmodule	ADV
iajs-2731	102	8	and	and	CCONJ
iajs-2731	102	9	let	let	VERB
iajs-2731	102	10	f	f	X
iajs-2731	102	11	:	:	PUNCT
iajs-2731	102	12	x1	x1	PROPN
iajs-2731	102	13	→	→	PUNCT
iajs-2731	102	14	x2	x2	PROPN
iajs-2731	102	15	be	be	AUX
iajs-2731	102	16	an	an	DET
iajs-2731	102	17	epimorphism	epimorphism	NOUN
iajs-2731	102	18	with	with	ADP
iajs-2731	102	19	x2	x2	PROPN
iajs-2731	102	20	is	be	AUX
iajs-2731	102	21	a	a	DET
iajs-2731	102	22	fmodule	fmodule	NOUN
iajs-2731	102	23	[	[	X
iajs-2731	102	24	9	9	NUM
iajs-2731	102	25	]	]	PUNCT
iajs-2731	102	26	.	.	PUNCT
iajs-2731	103	1	suppose	suppose	VERB
iajs-2731	103	2	that	that	SCONJ
iajs-2731	103	3	at	at	AUX
iajs-2731	103	4	be	be	AUX
iajs-2731	103	5	a	a	DET
iajs-2731	103	6	unique	unique	ADJ
iajs-2731	103	7	maximal	maximal	ADJ
iajs-2731	103	8	submodule	submodule	NOUN
iajs-2731	103	9	of	of	ADP
iajs-2731	103	10	x2	x2	PROPN
iajs-2731	103	11	with	with	ADP
iajs-2731	103	12	at	at	ADP
iajs-2731	103	13	+	+	NOUN
iajs-2731	103	14	ct	ct	NUM
iajs-2731	103	15	=(	=(	NOUN
iajs-2731	103	16	x2)t	x2)t	PROPN
iajs-2731	103	17	where	where	SCONJ
iajs-2731	103	18	ct	ct	PROPN
iajs-2731	103	19	proper	proper	ADJ
iajs-2731	103	20	submodule	submodule	NOUN
iajs-2731	103	21	of	of	ADP
iajs-2731	103	22	(	(	PUNCT
iajs-2731	103	23	x2)t	x2)t	PROPN
iajs-2731	103	24	=	=	SYM
iajs-2731	103	25	xt	xt	PROPN
iajs-2731	103	26	,	,	PUNCT
iajs-2731	103	27	∀	∀	X
iajs-2731	103	28	t	t	NOUN
iajs-2731	103	29	∈	∈	PROPN
iajs-2731	103	30	(	(	PUNCT
iajs-2731	103	31	0	0	NUM
iajs-2731	103	32	,	,	PUNCT
iajs-2731	103	33	1	1	NUM
iajs-2731	103	34	]	]	PUNCT
iajs-2731	103	35	,	,	PUNCT
iajs-2731	103	36	but	but	CCONJ
iajs-2731	103	37	a	a	PRON
iajs-2731	103	38	is	be	AUX
iajs-2731	103	39	a	a	DET
iajs-2731	103	40	unique	unique	ADJ
iajs-2731	103	41	maximal	maximal	ADJ
iajs-2731	103	42	submodule	submodule	NOUN
iajs-2731	104	1	[	[	X
iajs-2731	104	2	10	10	NUM
iajs-2731	104	3	]	]	X
iajs-2731	104	4	.then	.then	X
iajs-2731	105	1	a	a	DET
iajs-2731	105	2	+	+	X
iajs-2731	105	3	c	c	NOUN
iajs-2731	105	4	=	=	SYM
iajs-2731	105	5	x2	x2	PROPN
iajs-2731	105	6	.	.	PUNCT
iajs-2731	106	1	hence	hence	ADV
iajs-2731	106	2	at+	at+	VERB
iajs-2731	106	3	ct	ct	PROPN
iajs-2731	106	4	=(	=(	PROPN
iajs-2731	106	5	x1)t	x1)t	PROPN
iajs-2731	106	6	.now	.now	PROPN
iajs-2731	106	7	.	.	PUNCT
iajs-2731	107	1	f-1	f-1	NOUN
iajs-2731	107	2	(	(	PUNCT
iajs-2731	107	3	a	a	NOUN
iajs-2731	107	4	)	)	PUNCT
iajs-2731	107	5	is	be	AUX
iajs-2731	107	6	a	a	DET
iajs-2731	107	7	unique	unique	ADJ
iajs-2731	107	8	maximal	maximal	ADJ
iajs-2731	107	9	fuzzy	fuzzy	ADJ
iajs-2731	107	10	submodule	submodule	NOUN
iajs-2731	107	11	of	of	ADP
iajs-2731	107	12	x1	x1	PROPN
iajs-2731	107	13	since	since	SCONJ
iajs-2731	107	14	otherwise	otherwise	ADV
iajs-2731	107	15	f-1	f-1	PROPN
iajs-2731	107	16	(	(	PUNCT
iajs-2731	107	17	a)=	a)=	PROPN
iajs-2731	107	18	x1=	x1=	PROPN
iajs-2731	107	19	xt	xt	PROPN
iajs-2731	107	20	,	,	PUNCT
iajs-2731	107	21	∀	∀	X
iajs-2731	107	22	t	t	NOUN
iajs-2731	107	23	∈	∈	PROPN
iajs-2731	107	24	(	(	PUNCT
iajs-2731	107	25	0	0	NUM
iajs-2731	107	26	,	,	PUNCT
iajs-2731	107	27	1	1	NUM
iajs-2731	107	28	]	]	PUNCT
iajs-2731	107	29	and	and	CCONJ
iajs-2731	107	30	hence	hence	ADV
iajs-2731	107	31	f	f	PROPN
iajs-2731	107	32	(	(	PUNCT
iajs-2731	107	33	ωf-1	ωf-1	NUM
iajs-2731	107	34	(	(	PUNCT
iajs-2731	107	35	a	a	NOUN
iajs-2731	107	36	)	)	PUNCT
iajs-2731	107	37	)	)	PUNCT
iajs-2731	108	1	=	=	SYM
iajs-2731	108	2	f	f	PROPN
iajs-2731	108	3	(	(	PUNCT
iajs-2731	108	4	x1	x1	PROPN
iajs-2731	108	5	)	)	PUNCT
iajs-2731	108	6	=	=	SYM
iajs-2731	109	1	x2	x2	PROPN
iajs-2731	109	2	implies	imply	VERB
iajs-2731	109	3	that	that	SCONJ
iajs-2731	109	4	a=	a=	PROPN
iajs-2731	109	5	x2	x2	X
iajs-2731	109	6	,	,	PUNCT
iajs-2731	109	7	which	which	PRON
iajs-2731	109	8	is	be	AUX
iajs-2731	109	9	a	a	DET
iajs-2731	109	10	contradiction	contradiction	NOUN
iajs-2731	109	11	,	,	PUNCT
iajs-2731	109	12	with	with	ADP
iajs-2731	109	13	a	a	PRON
iajs-2731	109	14	is	be	AUX
iajs-2731	109	15	a	a	DET
iajs-2731	109	16	unique	unique	ADJ
iajs-2731	109	17	maximal	maximal	ADJ
iajs-2731	109	18	fuzzy	fuzzy	ADJ
iajs-2731	109	19	submodule	submodule	NOUN
iajs-2731	109	20	of	of	ADP
iajs-2731	109	21	x2	x2	PROPN
iajs-2731	109	22	,	,	PUNCT
iajs-2731	109	23	thus	thus	ADV
iajs-2731	109	24	f-1	f-1	NOUN
iajs-2731	109	25	(	(	PUNCT
iajs-2731	109	26	a	a	NOUN
iajs-2731	109	27	)	)	PUNCT
iajs-2731	109	28	is	be	AUX
iajs-2731	109	29	a	a	DET
iajs-2731	109	30	unique	unique	ADJ
iajs-2731	109	31	maximal	maximal	ADJ
iajs-2731	109	32	fuzzy	fuzzy	ADJ
iajs-2731	109	33	submodule	submodule	NOUN
iajs-2731	109	34	of	of	ADP
iajs-2731	109	35	x1	x1	PROPN
iajs-2731	109	36	and	and	CCONJ
iajs-2731	109	37	since	since	SCONJ
iajs-2731	109	38	x1	x1	PROPN
iajs-2731	109	39	is	be	AUX
iajs-2731	109	40	lochollow	lochollow	ADJ
iajs-2731	109	41	fuzzy	fuzzy	ADJ
iajs-2731	109	42	module	module	NOUN
iajs-2731	109	43	,	,	PUNCT
iajs-2731	109	44	therefore	therefore	ADV
iajs-2731	109	45	f-1	f-1	PROPN
iajs-2731	109	46	(	(	PUNCT
iajs-2731	109	47	a	a	PRON
iajs-2731	109	48	)	)	PUNCT
iajs-2731	109	49	contains	contain	VERB
iajs-2731	109	50	all	all	DET
iajs-2731	109	51	a	a	DET
iajs-2731	109	52	fuzzy	fuzzy	ADJ
iajs-2731	109	53	small	small	ADJ
iajs-2731	109	54	submodule	submodule	NOUN
iajs-2731	109	55	of	of	ADP
iajs-2731	109	56	x1	x1	PROPN
iajs-2731	109	57	and	and	CCONJ
iajs-2731	109	58	hence	hence	ADV
iajs-2731	109	59	f	f	PROPN
iajs-2731	109	60	(	(	PUNCT
iajs-2731	109	61	f-1	f-1	NOUN
iajs-2731	109	62	(	(	PUNCT
iajs-2731	109	63	a	a	NOUN
iajs-2731	109	64	)	)	PUNCT
iajs-2731	109	65	)	)	PUNCT
iajs-2731	109	66	is	be	AUX
iajs-2731	109	67	a	a	DET
iajs-2731	109	68	fuzzy	fuzzy	ADJ
iajs-2731	109	69	small	small	ADJ
iajs-2731	109	70	submodule	submodule	NOUN
iajs-2731	109	71	of	of	ADP
iajs-2731	109	72	f	f	PROPN
iajs-2731	109	73	(	(	PUNCT
iajs-2731	109	74	x1	x1	PROPN
iajs-2731	109	75	)	)	PUNCT
iajs-2731	109	76	.	.	PUNCT
iajs-2731	110	1	this	this	DET
iajs-2731	110	2	a	a	PRON
iajs-2731	110	3	is	be	AUX
iajs-2731	110	4	a	a	DET
iajs-2731	110	5	fuzzy	fuzzy	ADJ
iajs-2731	110	6	small	small	ADJ
iajs-2731	110	7	submodule	submodule	NOUN
iajs-2731	110	8	of	of	ADP
iajs-2731	110	9	x2	x2	PROPN
iajs-2731	110	10	,	,	PUNCT
iajs-2731	110	11	where	where	SCONJ
iajs-2731	110	12	at	at	ADP
iajs-2731	110	13	is	be	AUX
iajs-2731	110	14	an	an	DET
iajs-2731	110	15	f	f	PROPN
iajs-2731	110	16	small	small	ADJ
iajs-2731	110	17	submodule	submodule	NOUN
iajs-2731	110	18	of	of	ADP
iajs-2731	110	19	(	(	PUNCT
iajs-2731	110	20	x2	x2	PROPN
iajs-2731	110	21	)	)	PUNCT
iajs-2731	110	22	t	t	PROPN
iajs-2731	110	23	,	,	PUNCT
iajs-2731	110	24	∀	∀	X
iajs-2731	110	25	t	t	NOUN
iajs-2731	110	26	∈	∈	PROPN
iajs-2731	110	27	)	)	PUNCT
iajs-2731	110	28	0	0	NUM
iajs-2731	110	29	,	,	PUNCT
iajs-2731	110	30	1	1	NUM
iajs-2731	110	31	]	]	PUNCT
iajs-2731	110	32	.	.	PUNCT
iajs-2731	111	1	therefore	therefore	ADV
iajs-2731	111	2	x2	x2	PROPN
iajs-2731	111	3	is	be	AUX
iajs-2731	111	4	lochollow	lochollow	ADJ
iajs-2731	111	5	fuzzy	fuzzy	ADJ
iajs-2731	111	6	module	module	NOUN
iajs-2731	111	7	.	.	PUNCT
iajs-2731	112	1	the	the	DET
iajs-2731	112	2	next	next	ADJ
iajs-2731	112	3	proposition	proposition	NOUN
iajs-2731	112	4	appears	appear	VERB
iajs-2731	112	5	more	more	ADJ
iajs-2731	112	6	particulars	particular	NOUN
iajs-2731	112	7	of	of	ADP
iajs-2731	112	8	lochollow	lochollow	ADJ
iajs-2731	112	9	fuzzy	fuzzy	ADJ
iajs-2731	112	10	modules	module	NOUN
iajs-2731	112	11	.	.	PUNCT
iajs-2731	113	1	2.6	2.6	NUM
iajs-2731	113	2	proposition	proposition	NOUN
iajs-2731	113	3	:	:	PUNCT
iajs-2731	113	4	let	let	VERB
iajs-2731	113	5	c	c	PART
iajs-2731	113	6	be	be	AUX
iajs-2731	113	7	a	a	DET
iajs-2731	113	8	small	small	ADJ
iajs-2731	113	9	ƒuzzy	ƒuzzy	NOUN
iajs-2731	113	10	submodule	submodule	NOUN
iajs-2731	113	11	of	of	ADP
iajs-2731	113	12	fmodules	fmodule	NOUN
iajs-2731	113	13	x.	x.	NOUN
iajs-2731	113	14	whether	whether	SCONJ
iajs-2731	113	15	𝜒/c	𝜒/c	PRON
iajs-2731	113	16	is	be	AUX
iajs-2731	113	17	loc	loc	ADJ
iajs-2731	113	18	-	-	ADJ
iajs-2731	113	19	hollow	hollow	ADJ
iajs-2731	113	20	fmodules	fmodule	NOUN
iajs-2731	113	21	then	then	ADV
iajs-2731	113	22	x	x	PUNCT
iajs-2731	113	23	is	be	AUX
iajs-2731	113	24	lochollow	lochollow	NOUN
iajs-2731	113	25	fmodules	fmodule	NOUN
iajs-2731	113	26	.	.	PUNCT
iajs-2731	114	1	ibn	ibn	PROPN
iajs-2731	114	2	al	al	PROPN
iajs-2731	114	3	-	-	PUNCT
iajs-2731	114	4	haitham	haitham	PROPN
iajs-2731	114	5	jour	jour	X
iajs-2731	114	6	.	.	PROPN
iajs-2731	114	7	for	for	ADP
iajs-2731	114	8	pure	pure	ADJ
iajs-2731	114	9	&	&	CCONJ
iajs-2731	114	10	appl	appl	PROPN
iajs-2731	114	11	.	.	PUNCT
iajs-2731	115	1	sci	sci	PROPN
iajs-2731	115	2	.	.	PROPN
iajs-2731	116	1	53	53	NUM
iajs-2731	116	2	(	(	PUNCT
iajs-2731	116	3	2)2022	2)2022	NOUN
iajs-2731	116	4	89	89	NUM
iajs-2731	116	5	proof	proof	NOUN
iajs-2731	116	6	let	let	VERB
iajs-2731	116	7	x	x	PRON
iajs-2731	116	8	/	/	SYM
iajs-2731	116	9	c	c	PROPN
iajs-2731	116	10	is	be	AUX
iajs-2731	116	11	lochollow	lochollow	ADJ
iajs-2731	116	12	fmodules	fmodule	NOUN
iajs-2731	116	13	,	,	PUNCT
iajs-2731	116	14	where	where	SCONJ
iajs-2731	116	15	c	c	PROPN
iajs-2731	116	16	is	be	AUX
iajs-2731	116	17	a	a	DET
iajs-2731	116	18	small	small	ADJ
iajs-2731	116	19	ƒuzzy	ƒuzzy	NOUN
iajs-2731	116	20	submodule	submodule	NOUN
iajs-2731	116	21	of	of	ADP
iajs-2731	116	22	x.	x.	NOUN
iajs-2731	116	23	then	then	ADV
iajs-2731	116	24	there	there	PRON
iajs-2731	116	25	exists	exist	VERB
iajs-2731	116	26	a	a	DET
iajs-2731	116	27	unique	unique	ADJ
iajs-2731	116	28	maximal	maximal	ADJ
iajs-2731	116	29	small	small	ADJ
iajs-2731	116	30	ƒuzzy	ƒuzzy	NOUN
iajs-2731	116	31	submodule	submodule	NOUN
iajs-2731	116	32	a	a	PROPN
iajs-2731	116	33	/	/	SYM
iajs-2731	116	34	c	c	NOUN
iajs-2731	116	35	in	in	ADP
iajs-2731	116	36	𝜒	𝜒	NOUN
iajs-2731	116	37	/c	/c	NOUN
iajs-2731	116	38	,	,	PUNCT
iajs-2731	116	39	with	with	ADP
iajs-2731	116	40	x	x	NOUN
iajs-2731	116	41	=	=	NOUN
iajs-2731	116	42	b+	b+	NOUN
iajs-2731	116	43	d	d	NOUN
iajs-2731	116	44	and	and	CCONJ
iajs-2731	116	45	xt	xt	NOUN
iajs-2731	116	46	=	=	NOUN
iajs-2731	116	47	bt+	bt+	NOUN
iajs-2731	116	48	dt	dt	PROPN
iajs-2731	116	49	,	,	PUNCT
iajs-2731	116	50	∀	∀	X
iajs-2731	116	51	t	t	NOUN
iajs-2731	116	52	∈	∈	PROPN
iajs-2731	116	53	(	(	PUNCT
iajs-2731	116	54	0,1	0,1	NUM
iajs-2731	116	55	]	]	PUNCT
iajs-2731	116	56	by	by	ADP
iajs-2731	116	57	[	[	X
iajs-2731	116	58	6	6	NUM
iajs-2731	116	59	]	]	PUNCT
iajs-2731	116	60	.	.	PUNCT
iajs-2731	117	1	where	where	SCONJ
iajs-2731	117	2	b	b	NOUN
iajs-2731	117	3	is	be	AUX
iajs-2731	117	4	a	a	DET
iajs-2731	117	5	small	small	ADJ
iajs-2731	117	6	ƒuzzy	ƒuzzy	NOUN
iajs-2731	117	7	submodule	submodule	NOUN
iajs-2731	117	8	of	of	ADP
iajs-2731	117	9	𝜒	𝜒	PROPN
iajs-2731	117	10	and	and	CCONJ
iajs-2731	117	11	d	d	NOUN
iajs-2731	117	12	is	be	AUX
iajs-2731	117	13	a	a	DET
iajs-2731	117	14	proper	proper	ADJ
iajs-2731	117	15	ƒuzzy	ƒuzzy	NOUN
iajs-2731	117	16	submodule	submodule	NOUN
iajs-2731	117	17	of	of	ADP
iajs-2731	117	18	𝜒x	𝜒x	PROPN
iajs-2731	117	19	.	.	PUNCT
iajs-2731	118	1	then	then	ADV
iajs-2731	118	2	(	(	PUNCT
iajs-2731	118	3	d+𝑩)/c=	d+𝑩)/c=	NOUN
iajs-2731	118	4	𝜒	𝜒	X
iajs-2731	118	5	/c	/c	PUNCT
iajs-2731	118	6	implies	imply	VERB
iajs-2731	118	7	that	that	SCONJ
iajs-2731	118	8	(	(	PUNCT
iajs-2731	118	9	(	(	PUNCT
iajs-2731	118	10	d+c	d+c	PROPN
iajs-2731	118	11	/	/	SYM
iajs-2731	118	12	c	c	NOUN
iajs-2731	118	13	)	)	PUNCT
iajs-2731	118	14	)	)	PUNCT
iajs-2731	119	1	+	+	CCONJ
iajs-2731	119	2	(	(	PUNCT
iajs-2731	119	3	b+c)/c=	b+c)/c=	NOUN
iajs-2731	119	4	𝜒	𝜒	NOUN
iajs-2731	119	5	/c	/c	PRON
iajs-2731	119	6	,	,	PUNCT
iajs-2731	119	7	since	since	SCONJ
iajs-2731	119	8	(	(	PUNCT
iajs-2731	119	9	d+c	d+c	PROPN
iajs-2731	119	10	/	/	SYM
iajs-2731	119	11	c	c	NOUN
iajs-2731	119	12	)	)	PUNCT
iajs-2731	119	13	is	be	AUX
iajs-2731	119	14	a	a	DET
iajs-2731	119	15	ƒuzzy	ƒuzzy	ADJ
iajs-2731	119	16	submodule	submodule	NOUN
iajs-2731	119	17	of	of	ADP
iajs-2731	119	18	a	a	PRON
iajs-2731	119	19	/	/	SYM
iajs-2731	119	20	c	c	NOUN
iajs-2731	119	21	,	,	PUNCT
iajs-2731	119	22	𝜒	𝜒	X
iajs-2731	119	23	/c	/c	PUNCT
iajs-2731	119	24	is	be	AUX
iajs-2731	119	25	loc	loc	ADJ
iajs-2731	119	26	-	-	ADJ
iajs-2731	119	27	hollow	hollow	ADJ
iajs-2731	119	28	fmodules	fmodule	NOUN
iajs-2731	119	29	,	,	PUNCT
iajs-2731	119	30	then	then	ADV
iajs-2731	119	31	(	(	PUNCT
iajs-2731	119	32	d+c	d+c	PROPN
iajs-2731	119	33	/	/	SYM
iajs-2731	119	34	c	c	NOUN
iajs-2731	119	35	)	)	PUNCT
iajs-2731	119	36	is	be	AUX
iajs-2731	119	37	a	a	DET
iajs-2731	119	38	small	small	ADJ
iajs-2731	119	39	fuzzy	fuzzy	NOUN
iajs-2731	119	40	of	of	ADP
iajs-2731	119	41	x	x	PROPN
iajs-2731	119	42	/	/	SYM
iajs-2731	119	43	c.	c.	NOUN
iajs-2731	119	44	thus	thus	ADV
iajs-2731	119	45	(	(	PUNCT
iajs-2731	119	46	b+c)/c	b+c)/c	NOUN
iajs-2731	119	47	=	=	SYM
iajs-2731	119	48	x	x	NOUN
iajs-2731	119	49	/	/	SYM
iajs-2731	119	50	c	c	NOUN
iajs-2731	119	51	,	,	PUNCT
iajs-2731	119	52	so	so	ADV
iajs-2731	119	53	b+c	b+c	PRON
iajs-2731	119	54	=	=	NOUN
iajs-2731	119	55	x.	x.	NOUN
iajs-2731	119	56	since	since	SCONJ
iajs-2731	119	57	ct	ct	PROPN
iajs-2731	119	58	is	be	AUX
iajs-2731	119	59	a	a	DET
iajs-2731	119	60	submodule	submodule	NOUN
iajs-2731	119	61	of	of	ADP
iajs-2731	119	62	xt	xt	PROPN
iajs-2731	119	63	,	,	PUNCT
iajs-2731	119	64	bt	bt	PROPN
iajs-2731	119	65	=	=	PRON
iajs-2731	119	66	xt	xt	NOUN
iajs-2731	119	67	by	by	ADP
iajs-2731	119	68	[	[	X
iajs-2731	119	69	9	9	NUM
iajs-2731	119	70	]	]	PUNCT
iajs-2731	119	71	.	.	PUNCT
iajs-2731	120	1	therefore	therefore	ADV
iajs-2731	120	2	c	c	PROPN
iajs-2731	120	3	is	be	AUX
iajs-2731	120	4	a	a	DET
iajs-2731	120	5	fsubm	fsubm	NOUN
iajs-2731	120	6	of	of	ADP
iajs-2731	120	7	x	x	NOUN
iajs-2731	120	8	,	,	PUNCT
iajs-2731	120	9	and	and	CCONJ
iajs-2731	120	10	b	b	X
iajs-2731	120	11	=	=	NOUN
iajs-2731	120	12	x.	x.	NOUN
iajs-2731	120	13	thus	thus	ADV
iajs-2731	120	14	,	,	PUNCT
iajs-2731	120	15	x	x	PRON
iajs-2731	120	16	is	be	AUX
iajs-2731	120	17	lochollow	lochollow	NOUN
iajs-2731	120	18	fmodules	fmodule	NOUN
iajs-2731	120	19	.	.	PUNCT
iajs-2731	121	1	2.7	2.7	NUM
iajs-2731	121	2	corollary	corollary	NOUN
iajs-2731	121	3	:	:	PUNCT
iajs-2731	121	4	suppose	suppose	VERB
iajs-2731	121	5	that	that	SCONJ
iajs-2731	121	6	𝜒	𝜒	NOUN
iajs-2731	121	7	be	be	AUX
iajs-2731	121	8	fmodules	fmodule	NOUN
iajs-2731	121	9	.	.	PUNCT
iajs-2731	122	1	if	if	SCONJ
iajs-2731	122	2	x	x	PRON
iajs-2731	122	3	be	be	AUX
iajs-2731	122	4	a	a	DET
iajs-2731	122	5	loc	loc	ADJ
iajs-2731	122	6	-	-	ADJ
iajs-2731	122	7	hollow	hollow	ADJ
iajs-2731	122	8	fmodules	fmodule	NOUN
iajs-2731	122	9	then	then	ADV
iajs-2731	122	10	𝜒	𝜒	X
iajs-2731	122	11	/𝑨	/𝑨	PUNCT
iajs-2731	122	12	is	be	AUX
iajs-2731	122	13	lochollow	lochollow	ADJ
iajs-2731	122	14	f	f	PROPN
iajs-2731	122	15	modules	module	NOUN
iajs-2731	122	16	for	for	ADP
iajs-2731	122	17	every	every	DET
iajs-2731	122	18	proper	proper	ADJ
iajs-2731	122	19	fsubm	fsubm	PROPN
iajs-2731	122	20	𝑨	𝑨	PROPN
iajs-2731	122	21	of𝜒.	of𝜒.	PRON
iajs-2731	122	22	proof	proof	NOUN
iajs-2731	122	23	:	:	PUNCT
iajs-2731	122	24	suppose	suppose	VERB
iajs-2731	122	25	that	that	SCONJ
iajs-2731	122	26	𝜒	𝜒	NOUN
iajs-2731	122	27	is	be	AUX
iajs-2731	122	28	loc	loc	ADJ
iajs-2731	122	29	-	-	ADJ
iajs-2731	122	30	hollow	hollow	ADJ
iajs-2731	122	31	fmodules	fmodule	NOUN
iajs-2731	122	32	.	.	PUNCT
iajs-2731	123	1	then	then	ADV
iajs-2731	123	2	there	there	PRON
iajs-2731	123	3	exists	exist	VERB
iajs-2731	123	4	a	a	DET
iajs-2731	123	5	unique	unique	ADJ
iajs-2731	123	6	maximal	maximal	ADJ
iajs-2731	123	7	fsubm	fsubm	NOUN
iajs-2731	123	8	a	a	PRON
iajs-2731	123	9	contains	contain	VERB
iajs-2731	123	10	all	all	DET
iajs-2731	123	11	a	a	DET
iajs-2731	123	12	small	small	ADJ
iajs-2731	123	13	fsubm	fsubm	NOUN
iajs-2731	123	14	.	.	PUNCT
iajs-2731	124	1	let	let	VERB
iajs-2731	124	2	𝑨	𝑨	PRON
iajs-2731	124	3	be	be	AUX
iajs-2731	124	4	a	a	DET
iajs-2731	124	5	proper	proper	ADJ
iajs-2731	124	6	ƒuzzy	ƒuzzy	NOUN
iajs-2731	124	7	submodule	submodule	NOUN
iajs-2731	124	8	of	of	ADP
iajs-2731	124	9	loc	loc	NOUN
iajs-2731	124	10	-	-	ADJ
iajs-2731	124	11	hollow	hollow	ADJ
iajs-2731	124	12	fmodules	fmodule	NOUN
iajs-2731	124	13	𝜒	𝜒	NOUN
iajs-2731	124	14	and	and	CCONJ
iajs-2731	124	15	let	let	VERB
iajs-2731	124	16	𝜋	𝜋	NOUN
iajs-2731	124	17	:	:	PUNCT
iajs-2731	124	18	𝜒	𝜒	X
iajs-2731	124	19	→	→	SYM
iajs-2731	124	20	𝜒	𝜒	X
iajs-2731	124	21	/𝑨	/𝑨	PUNCT
iajs-2731	124	22	be	be	VERB
iajs-2731	124	23	nature	nature	NOUN
iajs-2731	124	24	epimorphism	epimorphism	NOUN
iajs-2731	124	25	then	then	ADV
iajs-2731	124	26	𝜒	𝜒	X
iajs-2731	124	27	/𝑨	/𝑨	PUNCT
iajs-2731	124	28	is	be	AUX
iajs-2731	124	29	loc	loc	ADJ
iajs-2731	124	30	-	-	ADJ
iajs-2731	124	31	hollow	hollow	ADJ
iajs-2731	124	32	fmodules	fmodule	NOUN
iajs-2731	124	33	by	by	ADP
iajs-2731	124	34	proposition	proposition	NOUN
iajs-2731	124	35	(	(	PUNCT
iajs-2731	124	36	2.5	2.5	NUM
iajs-2731	124	37	)	)	PUNCT
iajs-2731	124	38	.	.	PUNCT
iajs-2731	125	1	2.8	2.8	NUM
iajs-2731	125	2	𝝆roposition	𝝆roposition	NOUN
iajs-2731	125	3	:	:	PUNCT
iajs-2731	125	4	let	let	VERB
iajs-2731	125	5	a	a	PRON
iajs-2731	125	6	be	be	AUX
iajs-2731	125	7	a	a	DET
iajs-2731	125	8	proper	proper	ADJ
iajs-2731	125	9	fsubm	fsubm	NOUN
iajs-2731	125	10	of	of	ADP
iajs-2731	125	11	an	an	DET
iajs-2731	125	12	r	r	NOUN
iajs-2731	125	13	-	-	PUNCT
iajs-2731	125	14	module	module	NOUN
iajs-2731	125	15	x.	x.	NOUN
iajs-2731	125	16	if	if	SCONJ
iajs-2731	125	17	x	x	PROPN
iajs-2731	125	18	is	be	AUX
iajs-2731	125	19	loc	loc	ADJ
iajs-2731	125	20	-	-	ADJ
iajs-2731	125	21	hollow	hollow	ADJ
iajs-2731	125	22	fmodule	fmodule	NOUN
iajs-2731	125	23	and	and	CCONJ
iajs-2731	125	24	x	x	X
iajs-2731	125	25	/	/	SYM
iajs-2731	125	26	a	a	PRON
iajs-2731	125	27	is	be	AUX
iajs-2731	125	28	finitely	finitely	ADV
iajs-2731	125	29	generated	generate	VERB
iajs-2731	125	30	then	then	ADV
iajs-2731	125	31	x	x	PUNCT
iajs-2731	125	32	is	be	AUX
iajs-2731	125	33	finitely	finitely	ADV
iajs-2731	125	34	generated	generate	VERB
iajs-2731	125	35	fmodule	fmodule	ADJ
iajs-2731	125	36	.	.	PUNCT
iajs-2731	126	1	proof	proof	NOUN
iajs-2731	126	2	:	:	PUNCT
iajs-2731	126	3	let	let	VERB
iajs-2731	126	4	a	a	PRON
iajs-2731	126	5	be	be	AUX
iajs-2731	126	6	a	a	DET
iajs-2731	126	7	proper	proper	ADJ
iajs-2731	126	8	fuzzy	fuzzy	ADJ
iajs-2731	126	9	submodule	submodule	NOUN
iajs-2731	126	10	of	of	ADP
iajs-2731	126	11	loc	loc	ADJ
iajs-2731	126	12	-	-	ADJ
iajs-2731	126	13	hollow	hollow	ADJ
iajs-2731	126	14	fuzzy	fuzzy	ADJ
iajs-2731	126	15	module	module	NOUN
iajs-2731	126	16	x	x	PUNCT
iajs-2731	126	17	with	with	SCONJ
iajs-2731	126	18	x	x	X
iajs-2731	126	19	/	/	SYM
iajs-2731	126	20	a	a	PRON
iajs-2731	126	21	is	be	AUX
iajs-2731	126	22	finitely	finitely	ADV
iajs-2731	126	23	generated	generate	VERB
iajs-2731	126	24	then	then	ADV
iajs-2731	126	25	x	x	X
iajs-2731	126	26	/	/	SYM
iajs-2731	126	27	a	a	DET
iajs-2731	126	28	=	=	NOUN
iajs-2731	126	29	a1	a1	NOUN
iajs-2731	126	30	(	(	PUNCT
iajs-2731	126	31	(	(	PUNCT
iajs-2731	126	32	xt)t1	xt)t1	PROPN
iajs-2731	126	33	)	)	PUNCT
iajs-2731	127	1	+	+	ADP
iajs-2731	127	2	a	a	X
iajs-2731	127	3	)	)	PUNCT
iajs-2731	127	4	+	+	NUM
iajs-2731	127	5	a2	a2	PROPN
iajs-2731	127	6	(	(	PUNCT
iajs-2731	127	7	(	(	PUNCT
iajs-2731	127	8	xt)t2	xt)t2	PROPN
iajs-2731	127	9	+	+	ADV
iajs-2731	127	10	a	a	X
iajs-2731	127	11	)	)	PUNCT
iajs-2731	127	12	+	+	CCONJ
iajs-2731	127	13	.	.	PUNCT
iajs-2731	127	14	.	.	PUNCT
iajs-2731	127	15	.	.	PUNCT
iajs-2731	128	1	+	+	CCONJ
iajs-2731	129	1	an	an	DET
iajs-2731	129	2	(	(	PUNCT
iajs-2731	129	3	(	(	PUNCT
iajs-2731	129	4	xn)tn	xn)tn	PUNCT
iajs-2731	129	5	+	+	NOUN
iajs-2731	129	6	a	a	X
iajs-2731	129	7	)	)	PUNCT
iajs-2731	129	8	}	}	PUNCT
iajs-2731	129	9	,	,	PUNCT
iajs-2731	129	10	where	where	SCONJ
iajs-2731	129	11	ai	ai	VERB
iajs-2731	129	12	∈r	∈r	NOUN
iajs-2731	129	13	and	and	CCONJ
iajs-2731	129	14	a(x)t	a(x)t	PROPN
iajs-2731	129	15	=	=	SYM
iajs-2731	129	16	(	(	PUNCT
iajs-2731	129	17	ax)t	ax)t	PROPN
iajs-2731	129	18	,	,	PUNCT
iajs-2731	129	19	∀	∀	NUM
iajs-2731	129	20	t	t	NOUN
iajs-2731	129	21	∈	∈	PROPN
iajs-2731	129	22	(	(	PUNCT
iajs-2731	129	23	0,1	0,1	NUM
iajs-2731	129	24	]	]	PUNCT
iajs-2731	129	25	,	,	PUNCT
iajs-2731	129	26	xi	xi	X
iajs-2731	129	27	∈	∈	PROPN
iajs-2731	129	28	x	x	NOUN
iajs-2731	129	29	,	,	PUNCT
iajs-2731	129	30	for	for	ADP
iajs-2731	129	31	all	all	DET
iajs-2731	129	32	i=	i=	ADJ
iajs-2731	129	33	1,2	1,2	NUM
iajs-2731	129	34	,	,	PUNCT
iajs-2731	129	35	,	,	PUNCT
iajs-2731	129	36	,	,	PUNCT
iajs-2731	129	37	n	n	CCONJ
iajs-2731	129	38	,	,	PUNCT
iajs-2731	129	39	where	where	SCONJ
iajs-2731	129	40	(	(	PUNCT
iajs-2731	129	41	ax)t	ax)t	PROPN
iajs-2731	129	42	(	(	PUNCT
iajs-2731	129	43	y	y	NOUN
iajs-2731	129	44	)	)	PUNCT
iajs-2731	129	45	=	=	PRON
iajs-2731	129	46	{	{	PUNCT
iajs-2731	129	47	1	1	NUM
iajs-2731	129	48	𝑖𝑓	𝑖𝑓	NUM
iajs-2731	129	49	𝑦	𝑦	NOUN
iajs-2731	129	50	=	=	SYM
iajs-2731	129	51	𝑎𝑥	𝑎𝑥	X
iajs-2731	129	52	0	0	NUM
iajs-2731	129	53	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2731	129	54	∀	∀	PUNCT
iajs-2731	129	55	t∈	t∈	X
iajs-2731	129	56	(	(	PUNCT
iajs-2731	129	57	0,1	0,1	NUM
iajs-2731	129	58	]	]	PUNCT
iajs-2731	129	59	.	.	PUNCT
iajs-2731	130	1	we	we	PRON
iajs-2731	130	2	claim	claim	VERB
iajs-2731	130	3	that	that	SCONJ
iajs-2731	130	4	x	x	PUNCT
iajs-2731	130	5	=	=	NOUN
iajs-2731	130	6	a1	a1	NOUN
iajs-2731	130	7	(	(	PUNCT
iajs-2731	130	8	(	(	PUNCT
iajs-2731	130	9	xt)t1	xt)t1	PROPN
iajs-2731	130	10	)	)	PUNCT
iajs-2731	131	1	+	+	SYM
iajs-2731	131	2	a2	a2	PROPN
iajs-2731	131	3	(	(	PUNCT
iajs-2731	131	4	(	(	PUNCT
iajs-2731	131	5	xt)t2	xt)t2	PROPN
iajs-2731	131	6	+	+	PUNCT
iajs-2731	131	7	.	.	PUNCT
iajs-2731	131	8	.	.	PUNCT
iajs-2731	131	9	.	.	PUNCT
iajs-2731	132	1	+	+	CCONJ
iajs-2731	132	2	an	an	DET
iajs-2731	132	3	(	(	PUNCT
iajs-2731	132	4	(	(	PUNCT
iajs-2731	132	5	xn)tn	xn)tn	NUM
iajs-2731	132	6	∀	∀	NOUN
iajs-2731	132	7	ai	ai	VERB
iajs-2731	132	8	∈r	∈r	NOUN
iajs-2731	132	9	.	.	PUNCT
iajs-2731	133	1	let	let	VERB
iajs-2731	133	2	x	x	PRON
iajs-2731	133	3	⊆x	⊆x	ADV
iajs-2731	133	4	,	,	PUNCT
iajs-2731	133	5	then	then	ADV
iajs-2731	133	6	x	x	X
iajs-2731	134	1	+	+	ADP
iajs-2731	134	2	a	a	DET
iajs-2731	134	3	∈	∈	ADJ
iajs-2731	134	4	x	x	SYM
iajs-2731	134	5	/	/	SYM
iajs-2731	134	6	a	a	PRON
iajs-2731	134	7	,	,	PUNCT
iajs-2731	134	8	implies	imply	VERB
iajs-2731	134	9	that	that	PRON
iajs-2731	134	10	c+	c+	VERB
iajs-2731	134	11	a=	a=	ADV
iajs-2731	134	12	a1(x1+a)+	a1(x1+a)+	VERB
iajs-2731	134	13	a2(x2)+a	a2(x2)+a	PROPN
iajs-2731	134	14	+	+	PROPN
iajs-2731	134	15	.	.	PUNCT
iajs-2731	134	16	.	.	PUNCT
iajs-2731	134	17	.	.	PUNCT
iajs-2731	135	1	+	+	CCONJ
iajs-2731	135	2	an(xn	an(xn	NOUN
iajs-2731	136	1	+	+	ADJ
iajs-2731	136	2	a)=	a)=	PROPN
iajs-2731	136	3	a1(x1)+	a1(x1)+	PROPN
iajs-2731	136	4	a2(x2	a2(x2	PROPN
iajs-2731	136	5	)	)	PUNCT
iajs-2731	136	6	+	+	PROPN
iajs-2731	136	7	.	.	PUNCT
iajs-2731	136	8	.	.	PUNCT
iajs-2731	136	9	.	.	PUNCT
iajs-2731	137	1	+	+	CCONJ
iajs-2731	137	2	an(xn	an(xn	NOUN
iajs-2731	137	3	)	)	PUNCT
iajs-2731	138	1	+	+	NOUN
iajs-2731	138	2	a	a	PRON
iajs-2731	138	3	.	.	PUNCT
iajs-2731	139	1	thus	thus	ADV
iajs-2731	139	2	implies	imply	VERB
iajs-2731	139	3	that	that	SCONJ
iajs-2731	139	4	c=	c=	VERB
iajs-2731	139	5	a1(x1	a1(x1	NOUN
iajs-2731	139	6	)	)	PUNCT
iajs-2731	139	7	+	+	NUM
iajs-2731	139	8	a2(x2	a2(x2	NOUN
iajs-2731	139	9	)	)	PUNCT
iajs-2731	139	10	+	+	CCONJ
iajs-2731	139	11	.	.	PUNCT
iajs-2731	139	12	.	.	PUNCT
iajs-2731	139	13	.	.	PUNCT
iajs-2731	140	1	+	+	CCONJ
iajs-2731	140	2	an	an	DET
iajs-2731	140	3	(	(	PUNCT
iajs-2731	140	4	xn	xn	PROPN
iajs-2731	140	5	)	)	PUNCT
iajs-2731	140	6	+	+	NOUN
iajs-2731	140	7	u	u	NOUN
iajs-2731	140	8	for	for	ADP
iajs-2731	140	9	some	some	DET
iajs-2731	140	10	u	u	NOUN
iajs-2731	140	11	⊆	⊆	NUM
iajs-2731	140	12	a.	a.	NOUN
iajs-2731	140	13	thus	thus	ADV
iajs-2731	140	14	x	x	X
iajs-2731	140	15	=	=	SYM
iajs-2731	140	16	a1(x1)+	a1(x1)+	PROPN
iajs-2731	140	17	a2(x2	a2(x2	NOUN
iajs-2731	140	18	)	)	PUNCT
iajs-2731	140	19	+	+	PROPN
iajs-2731	140	20	.	.	PUNCT
iajs-2731	140	21	.	.	PUNCT
iajs-2731	140	22	.	.	PUNCT
iajs-2731	141	1	+	+	CCONJ
iajs-2731	141	2	an(xn	an(xn	NOUN
iajs-2731	141	3	)	)	PUNCT
iajs-2731	142	1	+	+	NOUN
iajs-2731	142	2	u	u	NOUN
iajs-2731	142	3	,	,	PUNCT
iajs-2731	142	4	and	and	CCONJ
iajs-2731	142	5	since	since	SCONJ
iajs-2731	142	6	x	x	PROPN
iajs-2731	142	7	is	be	AUX
iajs-2731	142	8	loc	loc	ADJ
iajs-2731	142	9	-	-	ADJ
iajs-2731	142	10	hollow	hollow	ADJ
iajs-2731	142	11	fmodules	fmodule	NOUN
iajs-2731	142	12	.	.	PUNCT
iajs-2731	143	1	,	,	PUNCT
iajs-2731	143	2	therefore	therefore	ADV
iajs-2731	143	3	x	x	X
iajs-2731	143	4	is	be	AUX
iajs-2731	143	5	hollow	hollow	ADJ
iajs-2731	143	6	fuzzy	fuzzy	ADJ
iajs-2731	143	7	module	module	NOUN
iajs-2731	143	8	by	by	ADP
iajs-2731	143	9	remark	remark	NOUN
iajs-2731	143	10	(	(	PUNCT
iajs-2731	143	11	(	(	PUNCT
iajs-2731	143	12	2.1.4)(1	2.1.4)(1	NOUN
iajs-2731	143	13	)	)	PUNCT
iajs-2731	143	14	)	)	PUNCT
iajs-2731	143	15	.	.	PUNCT
iajs-2731	144	1	then	then	ADV
iajs-2731	144	2	a	a	PRON
iajs-2731	144	3	is	be	AUX
iajs-2731	144	4	a	a	DET
iajs-2731	144	5	small	small	ADJ
iajs-2731	144	6	fsubm	fsubm	NOUN
iajs-2731	144	7	of	of	ADP
iajs-2731	144	8	x	x	PRON
iajs-2731	144	9	which	which	PRON
iajs-2731	144	10	implies	imply	VERB
iajs-2731	144	11	that	that	SCONJ
iajs-2731	144	12	x=	x=	PROPN
iajs-2731	144	13	a1(x1	a1(x1	PROPN
iajs-2731	144	14	)	)	PUNCT
iajs-2731	144	15	+	+	NUM
iajs-2731	144	16	a2(x2	a2(x2	NOUN
iajs-2731	144	17	)	)	PUNCT
iajs-2731	144	18	+	+	CCONJ
iajs-2731	144	19	.	.	PUNCT
iajs-2731	144	20	.	.	PUNCT
iajs-2731	144	21	.	.	PUNCT
iajs-2731	145	1	+	+	CCONJ
iajs-2731	145	2	an	an	DET
iajs-2731	145	3	(	(	PUNCT
iajs-2731	145	4	xn	xn	PROPN
iajs-2731	145	5	)	)	PUNCT
iajs-2731	145	6	where	where	SCONJ
iajs-2731	145	7	ai	ai	VERB
iajs-2731	145	8	∈r	∈r	NOUN
iajs-2731	145	9	.	.	PUNCT
iajs-2731	146	1	thus	thus	ADV
iajs-2731	146	2	x	x	VERB
iajs-2731	146	3	is	be	AUX
iajs-2731	146	4	finitely	finitely	ADV
iajs-2731	146	5	generated	generate	VERB
iajs-2731	146	6	fmodules	fmodule	NOUN
iajs-2731	146	7	.	.	PUNCT
iajs-2731	147	1	3	3	X
iajs-2731	147	2	.	.	X
iajs-2731	147	3	relation	relation	NOUN
iajs-2731	147	4	between	between	ADP
iajs-2731	147	5	hollow	hollow	ADJ
iajs-2731	147	6	fuzzy	fuzzy	ADJ
iajs-2731	147	7	module	module	NOUN
iajs-2731	147	8	and	and	CCONJ
iajs-2731	147	9	lochollow	lochollow	NOUN
iajs-2731	147	10	fuzzy	fuzzy	ADJ
iajs-2731	147	11	modules	module	NOUN
iajs-2731	147	12	we	we	PRON
iajs-2731	147	13	introduce	introduce	VERB
iajs-2731	147	14	the	the	DET
iajs-2731	147	15	following	follow	VERB
iajs-2731	147	16	definition	definition	NOUN
iajs-2731	147	17	let	let	VERB
iajs-2731	147	18	𝜒	𝜒	PRON
iajs-2731	147	19	be	be	AUX
iajs-2731	147	20	an	an	DET
iajs-2731	147	21	fmodule	fmodule	NOUN
iajs-2731	147	22	of	of	ADP
iajs-2731	147	23	an	an	DET
iajs-2731	147	24	r	r	NOUN
iajs-2731	147	25	-	-	PUNCT
iajs-2731	147	26	module	module	NOUN
iajs-2731	147	27	x.	x.	NOUN
iajs-2731	147	28	x	x	PUNCT
iajs-2731	147	29	is	be	AUX
iajs-2731	147	30	named	name	VERB
iajs-2731	147	31	hollow	hollow	ADJ
iajs-2731	147	32	f	f	NOUN
iajs-2731	147	33	module	module	NOUN
iajs-2731	147	34	if	if	SCONJ
iajs-2731	147	35	every	every	DET
iajs-2731	147	36	proper	proper	ADJ
iajs-2731	147	37	fsubmodule	fsubmodule	NOUN
iajs-2731	147	38	of	of	ADP
iajs-2731	147	39	x	x	PUNCT
iajs-2731	147	40	is	be	AUX
iajs-2731	147	41	a	a	DET
iajs-2731	147	42	small	small	ADJ
iajs-2731	147	43	fmodule	fmodule	NOUN
iajs-2731	147	44	of	of	ADP
iajs-2731	147	45	x.	x.	NOUN
iajs-2731	147	46	hence	hence	ADV
iajs-2731	147	47	we	we	PRON
iajs-2731	147	48	can	can	AUX
iajs-2731	147	49	say	say	VERB
iajs-2731	147	50	every	every	DET
iajs-2731	147	51	loc	loc	NOUN
iajs-2731	147	52	hollow	hollow	ADJ
iajs-2731	147	53	fuzzy	fuzzy	ADJ
iajs-2731	147	54	module	module	NOUN
iajs-2731	147	55	is	be	AUX
iajs-2731	147	56	hollow	hollow	ADJ
iajs-2731	147	57	fuzzy	fuzzy	ADJ
iajs-2731	147	58	module	module	NOUN
iajs-2731	147	59	,	,	PUNCT
iajs-2731	147	60	and	and	CCONJ
iajs-2731	147	61	we	we	PRON
iajs-2731	147	62	introduced	introduce	VERB
iajs-2731	147	63	an	an	DET
iajs-2731	147	64	examples	example	NOUN
iajs-2731	147	65	display	display	NOUN
iajs-2731	147	66	that	that	SCONJ
iajs-2731	147	67	the	the	DET
iajs-2731	147	68	converse	converse	NOUN
iajs-2731	147	69	is	be	AUX
iajs-2731	147	70	not	not	PART
iajs-2731	147	71	true	true	ADJ
iajs-2731	147	72	.	.	PUNCT
iajs-2731	148	1	in	in	ADP
iajs-2731	148	2	this	this	DET
iajs-2731	148	3	section	section	NOUN
iajs-2731	148	4	,	,	PUNCT
iajs-2731	148	5	we	we	PRON
iajs-2731	148	6	discuss	discuss	VERB
iajs-2731	148	7	conditions	condition	NOUN
iajs-2731	148	8	under	under	ADP
iajs-2731	148	9	which	which	PRON
iajs-2731	148	10	hollow	hollow	ADJ
iajs-2731	148	11	fuzzy	fuzzy	ADJ
iajs-2731	148	12	modules	module	NOUN
iajs-2731	148	13	could	could	AUX
iajs-2731	148	14	be	be	AUX
iajs-2731	148	15	lochollow	lochollow	ADJ
iajs-2731	148	16	fuzzy	fuzzy	ADJ
iajs-2731	148	17	modules	module	NOUN
iajs-2731	148	18	.	.	PUNCT
iajs-2731	149	1	ibn	ibn	PROPN
iajs-2731	149	2	al	al	PROPN
iajs-2731	149	3	-	-	PUNCT
iajs-2731	149	4	haitham	haitham	PROPN
iajs-2731	149	5	jour	jour	X
iajs-2731	149	6	.	.	PROPN
iajs-2731	149	7	for	for	ADP
iajs-2731	149	8	pure	pure	ADJ
iajs-2731	149	9	&	&	CCONJ
iajs-2731	149	10	appl	appl	PROPN
iajs-2731	149	11	.	.	PUNCT
iajs-2731	150	1	sci	sci	PROPN
iajs-2731	150	2	.	.	PROPN
iajs-2731	151	1	53	53	NUM
iajs-2731	151	2	(	(	PUNCT
iajs-2731	151	3	2)2022	2)2022	VERB
iajs-2731	151	4	90	90	NUM
iajs-2731	151	5	3.1	3.1	NUM
iajs-2731	151	6	proposition	proposition	NOUN
iajs-2731	151	7	:	:	PUNCT
iajs-2731	151	8	let	let	VERB
iajs-2731	151	9	x	x	PRON
iajs-2731	151	10	be	be	AUX
iajs-2731	151	11	a	a	DET
iajs-2731	151	12	fmodule	fmodule	NOUN
iajs-2731	151	13	of	of	ADP
iajs-2731	151	14	an	an	DET
iajs-2731	151	15	r	r	NOUN
iajs-2731	151	16	-	-	PUNCT
iajs-2731	151	17	module	module	NOUN
iajs-2731	151	18	,	,	PUNCT
iajs-2731	151	19	x	x	X
iajs-2731	151	20	is	be	AUX
iajs-2731	151	21	a	a	DET
iajs-2731	151	22	loc	loc	NOUN
iajs-2731	151	23	–	–	PUNCT
iajs-2731	151	24	hollow	hollow	ADJ
iajs-2731	151	25	fmodule	fmodule	NOUN
iajs-2731	151	26	⟺x	⟺x	PROPN
iajs-2731	151	27	is	be	AUX
iajs-2731	151	28	a	a	DET
iajs-2731	151	29	hollow	hollow	ADJ
iajs-2731	151	30	and	and	CCONJ
iajs-2731	151	31	cyclic	cyclic	ADJ
iajs-2731	151	32	fmodule	fmodule	NOUN
iajs-2731	151	33	.	.	PUNCT
iajs-2731	152	1	proof	proof	NOUN
iajs-2731	152	2	;	;	PUNCT
iajs-2731	152	3	⟹	⟹	NOUN
iajs-2731	152	4	suppose	suppose	VERB
iajs-2731	152	5	that	that	SCONJ
iajs-2731	152	6	x	x	PRON
iajs-2731	152	7	is	be	AUX
iajs-2731	152	8	loc	loc	NOUN
iajs-2731	152	9	–	–	PUNCT
iajs-2731	152	10	hollow	hollow	ADJ
iajs-2731	152	11	fmodules	fmodule	NOUN
iajs-2731	152	12	then	then	ADV
iajs-2731	152	13	there	there	PRON
iajs-2731	152	14	exists	exist	VERB
iajs-2731	152	15	a	a	DET
iajs-2731	152	16	unique	unique	ADJ
iajs-2731	152	17	maximal	maximal	ADJ
iajs-2731	152	18	fuzzy	fuzzy	ADJ
iajs-2731	152	19	submodule	submodule	NOUN
iajs-2731	152	20	a	a	PRON
iajs-2731	152	21	which	which	PRON
iajs-2731	152	22	contains	contain	VERB
iajs-2731	152	23	all	all	DET
iajs-2731	152	24	a	a	DET
iajs-2731	152	25	small	small	ADJ
iajs-2731	152	26	fsubmodule	fsubmodule	NOUN
iajs-2731	152	27	of	of	ADP
iajs-2731	152	28	x.	x.	NOUN
iajs-2731	152	29	this	this	PRON
iajs-2731	152	30	mean	mean	VERB
iajs-2731	152	31	there	there	PRON
iajs-2731	152	32	exists	exist	VERB
iajs-2731	152	33	at	at	ADP
iajs-2731	152	34	is	be	AUX
iajs-2731	152	35	a	a	DET
iajs-2731	152	36	small	small	ADJ
iajs-2731	152	37	submodule	submodule	NOUN
iajs-2731	152	38	of	of	ADP
iajs-2731	152	39	xt	xt	PROPN
iajs-2731	152	40	,	,	PUNCT
iajs-2731	152	41	∀	∀	X
iajs-2731	152	42	t∈	t∈	X
iajs-2731	152	43	(	(	PUNCT
iajs-2731	152	44	0,1	0,1	NUM
iajs-2731	152	45	]	]	PUNCT
iajs-2731	152	46	.	.	PUNCT
iajs-2731	153	1	let	let	VERB
iajs-2731	153	2	xt	xt	PUNCT
iajs-2731	153	3	⊆x	⊆x	VERB
iajs-2731	153	4	with	with	ADP
iajs-2731	153	5	xt	xt	ADP
iajs-2731	153	6	⊈	⊈	PROPN
iajs-2731	153	7	a	a	DET
iajs-2731	153	8	,	,	PUNCT
iajs-2731	153	9	(	(	PUNCT
iajs-2731	153	10	xt	xt	X
iajs-2731	153	11	)	)	PUNCT
iajs-2731	153	12	is	be	AUX
iajs-2731	153	13	a	a	DET
iajs-2731	153	14	submodule	submodule	NOUN
iajs-2731	153	15	of	of	ADP
iajs-2731	153	16	xt	xt	PROPN
iajs-2731	153	17	.	.	PUNCT
iajs-2731	154	1	then	then	ADV
iajs-2731	154	2	rx	rx	VERB
iajs-2731	154	3	is	be	AUX
iajs-2731	154	4	a	a	DET
iajs-2731	154	5	f	f	PROPN
iajs-2731	154	6	submodule	submodule	NOUN
iajs-2731	154	7	of	of	ADP
iajs-2731	154	8	x.	x.	NOUN
iajs-2731	154	9	we	we	PRON
iajs-2731	154	10	claim	claim	VERB
iajs-2731	154	11	that	that	SCONJ
iajs-2731	154	12	x	x	X
iajs-2731	154	13	=	=	SYM
iajs-2731	154	14	(	(	PUNCT
iajs-2731	154	15	xt	xt	NOUN
iajs-2731	154	16	)	)	PUNCT
iajs-2731	154	17	such	such	ADJ
iajs-2731	154	18	that	that	DET
iajs-2731	154	19	ys	ys	PROPN
iajs-2731	154	20	⊆	⊆	NUM
iajs-2731	154	21	x	x	PROPN
iajs-2731	154	22	has	have	AUX
iajs-2731	154	23	written	write	VERB
iajs-2731	154	24	ys=	ys=	PROPN
iajs-2731	154	25	xt	xt	ADP
iajs-2731	154	26	𝑎ℓ	𝑎ℓ	ADP
iajs-2731	154	27	for	for	ADP
iajs-2731	154	28	some	some	DET
iajs-2731	154	29	fuzzy	fuzzy	ADJ
iajs-2731	154	30	singleton	singleton	NOUN
iajs-2731	154	31	𝑎ℓ	𝑎ℓ	ADP
iajs-2731	154	32	of	of	ADP
iajs-2731	154	33	r	r	NOUN
iajs-2731	154	34	where	where	SCONJ
iajs-2731	154	35	t	t	PROPN
iajs-2731	154	36	,	,	PUNCT
iajs-2731	154	37	s	s	PART
iajs-2731	154	38	,	,	PUNCT
iajs-2731	154	39	ℓ	ℓ	PROPN
iajs-2731	154	40	∈	∈	PROPN
iajs-2731	154	41	(	(	PUNCT
iajs-2731	154	42	0	0	NUM
iajs-2731	154	43	,	,	PUNCT
iajs-2731	154	44	1	1	NUM
iajs-2731	154	45	]	]	PUNCT
iajs-2731	154	46	,	,	PUNCT
iajs-2731	154	47	rx	rx	VERB
iajs-2731	154	48	=	=	SYM
iajs-2731	154	49	x	x	SYM
iajs-2731	154	50	,	,	PUNCT
iajs-2731	154	51	so	so	ADV
iajs-2731	154	52	rxt	rxt	PROPN
iajs-2731	154	53	=	=	SYM
iajs-2731	154	54	xt	xt	PROPN
iajs-2731	154	55	,	,	PUNCT
iajs-2731	154	56	∀	∀	X
iajs-2731	154	57	t∈	t∈	X
iajs-2731	154	58	(	(	PUNCT
iajs-2731	154	59	0,1].if	0,1].if	PART
iajs-2731	154	60	rx	rx	VERB
iajs-2731	154	61	≠x	≠x	ADV
iajs-2731	154	62	then	then	ADV
iajs-2731	154	63	rx	rx	VERB
iajs-2731	154	64	is	be	AUX
iajs-2731	154	65	proper	proper	ADJ
iajs-2731	154	66	a	a	DET
iajs-2731	154	67	small	small	ADJ
iajs-2731	154	68	fsubmodule	fsubmodule	NOUN
iajs-2731	154	69	of	of	ADP
iajs-2731	154	70	x	x	PUNCT
iajs-2731	154	71	and	and	CCONJ
iajs-2731	154	72	hence	hence	ADV
iajs-2731	154	73	rx	rx	VERB
iajs-2731	154	74	is	be	AUX
iajs-2731	154	75	a	a	DET
iajs-2731	154	76	small	small	ADJ
iajs-2731	154	77	fuzzy	fuzzy	ADJ
iajs-2731	154	78	submodule	submodule	NOUN
iajs-2731	154	79	of	of	ADP
iajs-2731	154	80	a	a	PRON
iajs-2731	154	81	which	which	PRON
iajs-2731	154	82	implies	imply	VERB
iajs-2731	154	83	that	that	SCONJ
iajs-2731	154	84	x	x	SYM
iajs-2731	154	85	∈a	∈a	NUM
iajs-2731	154	86	,	,	PUNCT
iajs-2731	154	87	which	which	DET
iajs-2731	154	88	contradiction	contradiction	NOUN
iajs-2731	154	89	.	.	PUNCT
iajs-2731	155	1	thus	thus	ADV
iajs-2731	155	2	rx	rx	VERB
iajs-2731	155	3	=	=	NOUN
iajs-2731	155	4	x	x	X
iajs-2731	155	5	then	then	ADV
iajs-2731	155	6	x	x	X
iajs-2731	155	7	is	be	AUX
iajs-2731	155	8	cyclic	cyclic	ADJ
iajs-2731	155	9	modules	module	NOUN
iajs-2731	155	10	.	.	PUNCT
iajs-2731	156	1	now	now	ADV
iajs-2731	156	2	since	since	SCONJ
iajs-2731	156	3	x	x	X
iajs-2731	156	4	loc	loc	X
iajs-2731	156	5	–	–	PUNCT
iajs-2731	156	6	hollow	hollow	ADJ
iajs-2731	156	7	fmodules	fmodule	NOUN
iajs-2731	156	8	then	then	ADV
iajs-2731	156	9	x	x	PUNCT
iajs-2731	156	10	is	be	AUX
iajs-2731	156	11	hollow	hollow	ADJ
iajs-2731	156	12	fmodules	fmodule	NOUN
iajs-2731	156	13	by	by	ADP
iajs-2731	156	14	remark	remark	NOUN
iajs-2731	156	15	(	(	PUNCT
iajs-2731	156	16	2.4	2.4	NUM
iajs-2731	156	17	)	)	PUNCT
iajs-2731	156	18	(	(	PUNCT
iajs-2731	156	19	1	1	NUM
iajs-2731	156	20	)	)	PUNCT
iajs-2731	156	21	.	.	PUNCT
iajs-2731	157	1	⟸	⟸	PROPN
iajs-2731	157	2	suppose	suppose	VERB
iajs-2731	157	3	that	that	SCONJ
iajs-2731	157	4	x	x	PRON
iajs-2731	157	5	is	be	AUX
iajs-2731	157	6	hollow	hollow	ADJ
iajs-2731	157	7	cyclic	cyclic	ADJ
iajs-2731	157	8	fmodules	fmodule	NOUN
iajs-2731	157	9	and	and	CCONJ
iajs-2731	157	10	then	then	ADV
iajs-2731	157	11	it	it	PRON
iajs-2731	157	12	is	be	AUX
iajs-2731	157	13	finitely	finitely	ADV
iajs-2731	157	14	generated	generate	VERB
iajs-2731	157	15	fmodules	fmodule	NOUN
iajs-2731	157	16	and	and	CCONJ
iajs-2731	157	17	hence	hence	ADV
iajs-2731	157	18	x	x	PUNCT
iajs-2731	157	19	has	have	AUX
iajs-2731	157	20	maximal	maximal	ADJ
iajs-2731	157	21	fsubmodule	fsubmodule	NOUN
iajs-2731	157	22	suppose	suppose	VERB
iajs-2731	157	23	that	that	SCONJ
iajs-2731	157	24	x	x	PRON
iajs-2731	157	25	is	be	AUX
iajs-2731	157	26	hollow	hollow	ADJ
iajs-2731	157	27	cyclic	cyclic	ADJ
iajs-2731	157	28	fmodules	fmodule	NOUN
iajs-2731	157	29	and	and	CCONJ
iajs-2731	157	30	then	then	ADV
iajs-2731	157	31	it	it	PRON
iajs-2731	157	32	is	be	AUX
iajs-2731	157	33	fuzzy	fuzzy	ADJ
iajs-2731	157	34	finitely	finitely	ADV
iajs-2731	157	35	generated	generate	VERB
iajs-2731	157	36	module	module	NOUN
iajs-2731	157	37	and	and	CCONJ
iajs-2731	157	38	hence	hence	ADV
iajs-2731	157	39	x	x	PUNCT
iajs-2731	157	40	has	have	AUX
iajs-2731	157	41	maximal	maximal	ADJ
iajs-2731	157	42	fsubmodule	fsubmodule	NOUN
iajs-2731	157	43	contained	contain	VERB
iajs-2731	157	44	all	all	PRON
iajs-2731	157	45	proper	proper	ADJ
iajs-2731	157	46	a	a	DET
iajs-2731	157	47	small	small	ADJ
iajs-2731	157	48	fsubmodule	fsubmodule	NOUN
iajs-2731	157	49	say	say	VERB
iajs-2731	157	50	a	a	PRON
iajs-2731	157	51	,	,	PUNCT
iajs-2731	157	52	let	let	VERB
iajs-2731	157	53	b	b	NOUN
iajs-2731	157	54	fsubmodule	fsubmodule	NOUN
iajs-2731	157	55	of	of	ADP
iajs-2731	157	56	x.	x.	NOUN
iajs-2731	157	57	if	if	SCONJ
iajs-2731	157	58	b	b	PROPN
iajs-2731	157	59	is	be	AUX
iajs-2731	157	60	not	not	PART
iajs-2731	157	61	contained	contain	VERB
iajs-2731	157	62	in	in	ADP
iajs-2731	157	63	a	a	PRON
iajs-2731	157	64	,	,	PUNCT
iajs-2731	157	65	then	then	ADV
iajs-2731	157	66	b+a	b+a	PROPN
iajs-2731	157	67	=	=	SYM
iajs-2731	157	68	x	x	PROPN
iajs-2731	157	69	,	,	PUNCT
iajs-2731	157	70	but	but	CCONJ
iajs-2731	157	71	x	x	X
iajs-2731	157	72	is	be	AUX
iajs-2731	157	73	a	a	DET
iajs-2731	157	74	loc	loc	NOUN
iajs-2731	157	75	–	–	PUNCT
iajs-2731	157	76	hollow	hollow	ADJ
iajs-2731	157	77	f	f	NOUN
iajs-2731	157	78	modules	module	NOUN
iajs-2731	157	79	.	.	PUNCT
iajs-2731	158	1	thus	thus	ADV
iajs-2731	158	2	a	a	DET
iajs-2731	158	3	=	=	NOUN
iajs-2731	158	4	x	x	NOUN
iajs-2731	158	5	hence	hence	ADV
iajs-2731	158	6	at	at	ADP
iajs-2731	158	7	=	=	NOUN
iajs-2731	158	8	xt	xt	X
iajs-2731	158	9	where	where	SCONJ
iajs-2731	158	10	t	t	PROPN
iajs-2731	158	11	∈	∈	PROPN
iajs-2731	158	12	(	(	PUNCT
iajs-2731	158	13	0	0	NUM
iajs-2731	158	14	,	,	PUNCT
iajs-2731	158	15	1	1	NUM
iajs-2731	158	16	]	]	PUNCT
iajs-2731	158	17	by	by	ADP
iajs-2731	158	18	[	[	X
iajs-2731	158	19	8	8	NUM
iajs-2731	158	20	]	]	PUNCT
iajs-2731	158	21	,	,	PUNCT
iajs-2731	158	22	which	which	PRON
iajs-2731	158	23	contradiction	contradiction	NOUN
iajs-2731	158	24	.	.	PUNCT
iajs-2731	159	1	this	this	PRON
iajs-2731	159	2	implies	imply	VERB
iajs-2731	159	3	that	that	SCONJ
iajs-2731	159	4	every	every	DET
iajs-2731	159	5	proper	proper	ADJ
iajs-2731	159	6	fsubmodule	fsubmodule	NOUN
iajs-2731	159	7	of	of	ADP
iajs-2731	159	8	x	x	PROPN
iajs-2731	159	9	is	be	AUX
iajs-2731	159	10	contained	contain	VERB
iajs-2731	159	11	in	in	ADP
iajs-2731	159	12	a	a	PRON
iajs-2731	159	13	,	,	PUNCT
iajs-2731	159	14	which	which	PRON
iajs-2731	159	15	implies	imply	VERB
iajs-2731	159	16	that	that	SCONJ
iajs-2731	159	17	x	x	PRON
iajs-2731	159	18	has	have	VERB
iajs-2731	159	19	a	a	DET
iajs-2731	159	20	unique	unique	ADJ
iajs-2731	159	21	maximal	maximal	ADJ
iajs-2731	159	22	f	f	PROPN
iajs-2731	159	23	submodule	submodule	NOUN
iajs-2731	159	24	that	that	PRON
iajs-2731	159	25	contained	contain	VERB
iajs-2731	159	26	all	all	DET
iajs-2731	159	27	proper	proper	ADJ
iajs-2731	159	28	fsubmodules	fsubmodule	NOUN
iajs-2731	159	29	of	of	ADP
iajs-2731	159	30	x.	x.	PROPN
iajs-2731	159	31	therefore	therefore	ADV
iajs-2731	159	32	x	x	X
iajs-2731	159	33	is	be	AUX
iajs-2731	159	34	a	a	DET
iajs-2731	159	35	local	local	ADJ
iajs-2731	159	36	fmodule	fmodule	NOUN
iajs-2731	159	37	.	.	PUNCT
iajs-2731	160	1	3.2	3.2	NUM
iajs-2731	160	2	corollary	corollary	NOUN
iajs-2731	160	3	:	:	PUNCT
iajs-2731	160	4	let	let	VERB
iajs-2731	160	5	x	x	PRON
iajs-2731	160	6	be	be	AUX
iajs-2731	160	7	a	a	DET
iajs-2731	160	8	fmodules	fmodule	NOUN
iajs-2731	160	9	an	an	DET
iajs-2731	160	10	ř	ř	ADJ
iajs-2731	160	11	–	–	PUNCT
iajs-2731	160	12	module	module	NOUN
iajs-2731	160	13	.	.	PUNCT
iajs-2731	161	1	x	x	PRON
iajs-2731	161	2	is	be	AUX
iajs-2731	161	3	a	a	DET
iajs-2731	161	4	lochollow	lochollow	NOUN
iajs-2731	161	5	fmodule	fmodule	NOUN
iajs-2731	161	6	⟺	⟺	PROPN
iajs-2731	161	7	x	x	X
iajs-2731	161	8	is	be	AUX
iajs-2731	161	9	a	a	DET
iajs-2731	161	10	hollow	hollow	ADJ
iajs-2731	161	11	and	and	CCONJ
iajs-2731	161	12	finitely	finitely	ADV
iajs-2731	161	13	generated	generate	VERB
iajs-2731	161	14	fmodule	fmodule	ADJ
iajs-2731	161	15	.	.	PUNCT
iajs-2731	162	1	proof	proof	NOUN
iajs-2731	162	2	:	:	PUNCT
iajs-2731	162	3	⟹	⟹	PROPN
iajs-2731	162	4	suppose	suppose	VERB
iajs-2731	162	5	that	that	SCONJ
iajs-2731	162	6	x	x	PRON
iajs-2731	162	7	is	be	AUX
iajs-2731	162	8	lochollow	lochollow	NOUN
iajs-2731	162	9	fmodules	fmodule	NOUN
iajs-2731	162	10	,	,	PUNCT
iajs-2731	162	11	then	then	ADV
iajs-2731	162	12	x	x	PUNCT
iajs-2731	162	13	is	be	AUX
iajs-2731	162	14	hollow	hollow	ADJ
iajs-2731	162	15	fmodules	fmodule	NOUN
iajs-2731	162	16	and	and	CCONJ
iajs-2731	162	17	cyclic	cyclic	NOUN
iajs-2731	162	18	by	by	ADP
iajs-2731	162	19	proposition	proposition	NOUN
iajs-2731	162	20	(	(	PUNCT
iajs-2731	162	21	3.1	3.1	NUM
iajs-2731	162	22	)	)	PUNCT
iajs-2731	162	23	,	,	PUNCT
iajs-2731	162	24	and	and	CCONJ
iajs-2731	162	25	since	since	SCONJ
iajs-2731	162	26	x	x	PRON
iajs-2731	162	27	is	be	AUX
iajs-2731	162	28	cyclic	cyclic	ADJ
iajs-2731	162	29	fmodules	fmodule	NOUN
iajs-2731	162	30	,	,	PUNCT
iajs-2731	162	31	.	.	PUNCT
iajs-2731	163	1	thus	thus	ADV
iajs-2731	163	2	x	x	X
iajs-2731	163	3	is	be	AUX
iajs-2731	163	4	finitely	finitely	ADV
iajs-2731	163	5	generated	generate	VERB
iajs-2731	163	6	⟸	⟸	ADJ
iajs-2731	163	7	let	let	VERB
iajs-2731	163	8	x	x	PRON
iajs-2731	163	9	finitely	finitely	ADV
iajs-2731	163	10	generated	generate	VERB
iajs-2731	163	11	hollow	hollow	ADJ
iajs-2731	163	12	module	module	NOUN
iajs-2731	163	13	the	the	DET
iajs-2731	163	14	x	x	SYM
iajs-2731	163	15	=	=	X
iajs-2731	163	16	{	{	PUNCT
iajs-2731	163	17	ra1	ra1	PROPN
iajs-2731	163	18	(	(	PUNCT
iajs-2731	163	19	xt)t1	xt)t1	PROPN
iajs-2731	163	20	+	+	CCONJ
iajs-2731	163	21	ra2	ra2	PROPN
iajs-2731	163	22	(	(	PUNCT
iajs-2731	163	23	xt)t2	xt)t2	PROPN
iajs-2731	163	24	+	+	PUNCT
iajs-2731	163	25	.	.	PUNCT
iajs-2731	163	26	.	.	PUNCT
iajs-2731	163	27	.	.	PUNCT
iajs-2731	164	1	+	+	CCONJ
iajs-2731	164	2	ran	run	VERB
iajs-2731	164	3	(	(	PUNCT
iajs-2731	164	4	xn)tn	xn)tn	PUNCT
iajs-2731	164	5	}	}	PUNCT
iajs-2731	164	6	,	,	PUNCT
iajs-2731	164	7	where	where	SCONJ
iajs-2731	164	8	ai	ai	VERB
iajs-2731	164	9	∈r	∈r	NOUN
iajs-2731	164	10	,	,	PUNCT
iajs-2731	164	11	a(x)t	a(x)t	PROPN
iajs-2731	164	12	=	=	SYM
iajs-2731	164	13	(	(	PUNCT
iajs-2731	164	14	ax)t	ax)t	PROPN
iajs-2731	164	15	,	,	PUNCT
iajs-2731	164	16	∀	∀	NUM
iajs-2731	164	17	t	t	NOUN
iajs-2731	164	18	∈	∈	PROPN
iajs-2731	164	19	(	(	PUNCT
iajs-2731	164	20	0,1	0,1	NOUN
iajs-2731	164	21	]	]	PUNCT
iajs-2731	164	22	.if	.if	PUNCT
iajs-2731	165	1	x	x	SYM
iajs-2731	165	2	≠	≠	PROPN
iajs-2731	165	3	ra1	ra1	PROPN
iajs-2731	165	4	(	(	PUNCT
iajs-2731	165	5	xt	xt	PROPN
iajs-2731	165	6	)	)	PUNCT
iajs-2731	165	7	t1	t1	NOUN
iajs-2731	165	8	then	then	ADV
iajs-2731	165	9	rxt1	rxt1	PROPN
iajs-2731	165	10	is	be	AUX
iajs-2731	165	11	proper	proper	ADJ
iajs-2731	165	12	fuzzy	fuzzy	ADJ
iajs-2731	165	13	submodule	submodule	NOUN
iajs-2731	165	14	of	of	ADP
iajs-2731	165	15	x	x	PUNCT
iajs-2731	165	16	which	which	PRON
iajs-2731	165	17	implies	imply	VERB
iajs-2731	165	18	that	that	SCONJ
iajs-2731	165	19	r	r	NOUN
iajs-2731	165	20	xt1	xt1	X
iajs-2731	165	21	is	be	AUX
iajs-2731	165	22	fuzzy	fuzzy	ADJ
iajs-2731	165	23	small	small	ADJ
iajs-2731	165	24	submodule	submodule	NOUN
iajs-2731	165	25	of	of	ADP
iajs-2731	165	26	x.	x.	NOUN
iajs-2731	165	27	hence	hence	ADV
iajs-2731	165	28	x	x	PUNCT
iajs-2731	166	1	=	=	PRON
iajs-2731	166	2	{	{	PUNCT
iajs-2731	166	3	ra1	ra1	PROPN
iajs-2731	166	4	(	(	PUNCT
iajs-2731	166	5	xt	xt	PROPN
iajs-2731	166	6	)	)	PUNCT
iajs-2731	166	7	t1	t1	NOUN
iajs-2731	166	8	+	+	CCONJ
iajs-2731	166	9	ra2	ra2	PROPN
iajs-2731	166	10	(	(	PUNCT
iajs-2731	166	11	xt	xt	NOUN
iajs-2731	166	12	)	)	PUNCT
iajs-2731	166	13	t2	t2	NOUN
iajs-2731	166	14	+	+	PUNCT
iajs-2731	166	15	.	.	PUNCT
iajs-2731	166	16	.	.	PUNCT
iajs-2731	166	17	.	.	PUNCT
iajs-2731	167	1	+	+	CCONJ
iajs-2731	167	2	ran	run	VERB
iajs-2731	167	3	(	(	PUNCT
iajs-2731	167	4	xn	xn	PROPN
iajs-2731	167	5	)	)	PUNCT
iajs-2731	167	6	tn}.so	tn}.so	ADP
iajs-2731	167	7	,	,	PUNCT
iajs-2731	167	8	we	we	PRON
iajs-2731	167	9	delete	delete	VERB
iajs-2731	167	10	the	the	DET
iajs-2731	167	11	summand	summand	NOUN
iajs-2731	167	12	one	one	NUM
iajs-2731	167	13	by	by	ADV
iajs-2731	167	14	until	until	SCONJ
iajs-2731	167	15	we	we	PRON
iajs-2731	167	16	have	have	VERB
iajs-2731	167	17	x=	x=	PROPN
iajs-2731	167	18	ra1	ra1	PROPN
iajs-2731	167	19	(	(	PUNCT
iajs-2731	167	20	xt	xt	ADJ
iajs-2731	167	21	)	)	PUNCT
iajs-2731	167	22	ti	ti	NOUN
iajs-2731	167	23	for	for	ADP
iajs-2731	167	24	some	some	DET
iajs-2731	167	25	i	i	PRON
iajs-2731	167	26	.thus	.thu	NOUN
iajs-2731	167	27	x	x	VERB
iajs-2731	167	28	is	be	AUX
iajs-2731	167	29	cyclic	cyclic	ADJ
iajs-2731	167	30	fmodules	fmodule	NOUN
iajs-2731	167	31	and	and	CCONJ
iajs-2731	167	32	by	by	ADP
iajs-2731	167	33	proposition	proposition	NOUN
iajs-2731	167	34	(	(	PUNCT
iajs-2731	167	35	3.1	3.1	NUM
iajs-2731	167	36	)	)	PUNCT
iajs-2731	167	37	.	.	PUNCT
iajs-2731	168	1	therefore	therefore	ADV
iajs-2731	168	2	x	x	X
iajs-2731	168	3	is	be	AUX
iajs-2731	168	4	lochollow	lochollow	ADJ
iajs-2731	168	5	f	f	PROPN
iajs-2731	168	6	modules	module	NOUN
iajs-2731	168	7	.	.	PUNCT
iajs-2731	169	1	3.3	3.3	NUM
iajs-2731	169	2	𝝆roposition	𝝆roposition	NOUN
iajs-2731	169	3	:	:	PUNCT
iajs-2731	169	4	suppose	suppose	VERB
iajs-2731	169	5	that	that	SCONJ
iajs-2731	169	6	𝜒	𝜒	NOUN
iajs-2731	169	7	be	be	AUX
iajs-2731	169	8	fmodules	fmodule	NOUN
iajs-2731	169	9	of	of	ADP
iajs-2731	169	10	an	an	DET
iajs-2731	169	11	𝑅	𝑅	PROPN
iajs-2731	169	12	–	–	PUNCT
iajs-2731	169	13	module	module	NOUN
iajs-2731	169	14	m	m	NOUN
iajs-2731	169	15	,	,	PUNCT
iajs-2731	169	16	𝜒	𝜒	PROPN
iajs-2731	169	17	is	be	AUX
iajs-2731	169	18	a	a	DET
iajs-2731	169	19	lochollow	lochollow	NOUN
iajs-2731	169	20	fmodule⟺	fmodule⟺	NOUN
iajs-2731	169	21	x	x	PRON
iajs-2731	169	22	is	be	AUX
iajs-2731	169	23	hollow	hollow	ADJ
iajs-2731	169	24	and	and	CCONJ
iajs-2731	169	25	has	have	VERB
iajs-2731	169	26	a	a	DET
iajs-2731	169	27	unique	unique	ADJ
iajs-2731	169	28	maximal	maximal	ADJ
iajs-2731	169	29	fuzzy	fuzzy	ADJ
iajs-2731	169	30	submodule	submodule	NOUN
iajs-2731	169	31	.	.	PUNCT
iajs-2731	170	1	proof	proof	NOUN
iajs-2731	170	2	⟹	⟹	ADV
iajs-2731	170	3	suppose	suppose	VERB
iajs-2731	170	4	that	that	SCONJ
iajs-2731	170	5	x	x	PRON
iajs-2731	170	6	is	be	AUX
iajs-2731	170	7	lochollow	lochollow	ADJ
iajs-2731	170	8	fuzzy	fuzzy	ADJ
iajs-2731	170	9	module	module	NOUN
iajs-2731	170	10	then	then	ADV
iajs-2731	170	11	x	x	PRON
iajs-2731	170	12	is	be	AUX
iajs-2731	170	13	hollow	hollow	ADJ
iajs-2731	170	14	a	a	DET
iajs-2731	170	15	fmodules	fmodule	NOUN
iajs-2731	170	16	by	by	ADP
iajs-2731	170	17	remark	remark	NOUN
iajs-2731	170	18	(	(	PUNCT
iajs-2731	170	19	2.4	2.4	NUM
iajs-2731	170	20	)	)	PUNCT
iajs-2731	170	21	(	(	PUNCT
iajs-2731	170	22	1	1	NUM
iajs-2731	170	23	)	)	PUNCT
iajs-2731	170	24	.	.	PUNCT
iajs-2731	171	1	and	and	CCONJ
iajs-2731	171	2	by	by	ADP
iajs-2731	171	3	definition	definition	NOUN
iajs-2731	171	4	(	(	PUNCT
iajs-2731	171	5	2.1	2.1	NUM
iajs-2731	171	6	)	)	PUNCT
iajs-2731	171	7	,	,	PUNCT
iajs-2731	171	8	then	then	ADV
iajs-2731	171	9	x	x	PUNCT
iajs-2731	171	10	has	have	VERB
iajs-2731	171	11	a	a	DET
iajs-2731	171	12	unique	unique	ADJ
iajs-2731	171	13	maximal	maximal	ADJ
iajs-2731	171	14	fuzzy	fuzzy	ADJ
iajs-2731	171	15	submodule	submodule	NOUN
iajs-2731	171	16	fuzzy	fuzzy	ADJ
iajs-2731	171	17	module	module	NOUN
iajs-2731	171	18	.	.	PUNCT
iajs-2731	172	1	⟸	⟸	PROPN
iajs-2731	172	2	let	let	VERB
iajs-2731	172	3	x	x	PRON
iajs-2731	172	4	be	be	AUX
iajs-2731	172	5	t	t	PROPN
iajs-2731	172	6	hollow	hollow	ADJ
iajs-2731	172	7	fmodules	fmodule	NOUN
iajs-2731	172	8	which	which	PRON
iajs-2731	172	9	have	have	VERB
iajs-2731	172	10	unique	unique	ADJ
iajs-2731	172	11	maximal	maximal	ADJ
iajs-2731	172	12	fuzzy	fuzzy	ADJ
iajs-2731	172	13	submodule	submodule	NOUN
iajs-2731	172	14	fuzzy	fuzzy	ADJ
iajs-2731	172	15	module	module	NOUN
iajs-2731	172	16	,	,	PUNCT
iajs-2731	172	17	say	say	VERB
iajs-2731	172	18	a	a	PRON
iajs-2731	172	19	,	,	PUNCT
iajs-2731	172	20	we	we	PRON
iajs-2731	172	21	only	only	ADV
iajs-2731	172	22	have	have	VERB
iajs-2731	172	23	to	to	PART
iajs-2731	172	24	show	show	VERB
iajs-2731	172	25	that	that	SCONJ
iajs-2731	172	26	x	x	PRON
iajs-2731	172	27	is	be	AUX
iajs-2731	172	28	a	a	DET
iajs-2731	172	29	cyclic	cyclic	ADJ
iajs-2731	172	30	fuzzy	fuzzy	ADJ
iajs-2731	172	31	module	module	NOUN
iajs-2731	172	32	,	,	PUNCT
iajs-2731	172	33	let	let	VERB
iajs-2731	172	34	xt	xt	ADP
iajs-2731	172	35	⊆	⊆	NUM
iajs-2731	172	36	x	x	NOUN
iajs-2731	172	37	and	and	CCONJ
iajs-2731	172	38	xt	xt	ADP
iajs-2731	172	39	⊆	⊆	NUM
iajs-2731	172	40	a	a	DET
iajs-2731	172	41	clear	clear	ADJ
iajs-2731	172	42	that	that	SCONJ
iajs-2731	172	43	x	x	PUNCT
iajs-2731	172	44	∈	∈	NOUN
iajs-2731	172	45	xt	xt	X
iajs-2731	172	46	and	and	CCONJ
iajs-2731	172	47	x	x	PROPN
iajs-2731	172	48	∉	∉	PROPN
iajs-2731	172	49	at	at	ADP
iajs-2731	172	50	,	,	PUNCT
iajs-2731	172	51	∀	∀	X
iajs-2731	172	52	∈	∈	NOUN
iajs-2731	172	53	(	(	PUNCT
iajs-2731	172	54	0,1	0,1	NOUN
iajs-2731	172	55	]	]	PUNCT
iajs-2731	172	56	,	,	PUNCT
iajs-2731	172	57	then	then	ADV
iajs-2731	172	58	rxt	rxt	PROPN
iajs-2731	172	59	+	+	CCONJ
iajs-2731	172	60	at	at	ADP
iajs-2731	172	61	=	=	NOUN
iajs-2731	172	62	xt	xt	X
iajs-2731	173	1	و	و	PRON
iajs-2731	173	2	∀	∀	X
iajs-2731	173	3	t	t	NOUN
iajs-2731	173	4	∈	∈	PROPN
iajs-2731	173	5	(	(	PUNCT
iajs-2731	173	6	0,1	0,1	NOUN
iajs-2731	173	7	]	]	PUNCT
iajs-2731	173	8	.	.	PUNCT
iajs-2731	174	1	but	but	CCONJ
iajs-2731	174	2	𝜒	𝜒	NOUN
iajs-2731	174	3	is	be	AUX
iajs-2731	174	4	a	a	DET
iajs-2731	174	5	hollow	hollow	ADJ
iajs-2731	174	6	fmodule	fmodule	NOUN
iajs-2731	175	1	s	s	VERB
iajs-2731	175	2	then	then	ADV
iajs-2731	175	3	𝑨	𝑨	PROPN
iajs-2731	175	4	small	small	ADJ
iajs-2731	175	5	fuzz	fuzz	NOUN
iajs-2731	175	6	ibn	ibn	PROPN
iajs-2731	175	7	al	al	PROPN
iajs-2731	175	8	-	-	PUNCT
iajs-2731	175	9	haitham	haitham	PROPN
iajs-2731	175	10	jour	jour	X
iajs-2731	175	11	.	.	PROPN
iajs-2731	176	1	for	for	ADP
iajs-2731	176	2	pure	pure	ADJ
iajs-2731	176	3	&	&	CCONJ
iajs-2731	176	4	appl	appl	PROPN
iajs-2731	176	5	.	.	PUNCT
iajs-2731	177	1	sci	sci	PROPN
iajs-2731	177	2	.	.	PROPN
iajs-2731	178	1	53	53	NUM
iajs-2731	178	2	(	(	PUNCT
iajs-2731	178	3	2)2022	2)2022	NUM
iajs-2731	178	4	91	91	NUM
iajs-2731	178	5	submodule	submodule	NOUN
iajs-2731	178	6	of	of	ADP
iajs-2731	178	7	𝜒	𝜒	NOUN
iajs-2731	178	8	and	and	CCONJ
iajs-2731	178	9	hence	hence	ADV
iajs-2731	178	10	x=	x=	PUNCT
iajs-2731	179	1	rx	rx	VERB
iajs-2731	179	2	.	.	PUNCT
iajs-2731	180	1	therefore	therefore	ADV
iajs-2731	180	2	x	x	VERB
iajs-2731	180	3	is	be	AUX
iajs-2731	180	4	cyclic	cyclic	ADJ
iajs-2731	180	5	fuzzy	fuzzy	ADJ
iajs-2731	180	6	module	module	NOUN
iajs-2731	180	7	,	,	PUNCT
iajs-2731	180	8	and	and	CCONJ
iajs-2731	180	9	by	by	ADP
iajs-2731	180	10	proposition	proposition	NOUN
iajs-2731	180	11	(	(	PUNCT
iajs-2731	180	12	3.1	3.1	NUM
iajs-2731	180	13	)	)	PUNCT
iajs-2731	180	14	.	.	PUNCT
iajs-2731	181	1	therefore	therefore	ADV
iajs-2731	181	2	x	x	X
iajs-2731	181	3	is	be	AUX
iajs-2731	181	4	lochollow	lochollow	NOUN
iajs-2731	181	5	fmodules	fmodule	NOUN
iajs-2731	181	6	.	.	PUNCT
iajs-2731	182	1	3	3	X
iajs-2731	182	2	.	.	SYM
iajs-2731	182	3	4	4	NUM
iajs-2731	182	4	𝝆roposition	𝝆roposition	NOUN
iajs-2731	182	5	let	let	VERB
iajs-2731	182	6	x	x	PRON
iajs-2731	182	7	be	be	AUX
iajs-2731	182	8	fmodule	fmodule	ADJ
iajs-2731	182	9	,	,	PUNCT
iajs-2731	182	10	x	x	PUNCT
iajs-2731	182	11	is	be	AUX
iajs-2731	182	12	a	a	DET
iajs-2731	182	13	lochollow	lochollow	NOUN
iajs-2731	182	14	fmodule	fmodule	NOUN
iajs-2731	182	15	⟺x	⟺x	PROPN
iajs-2731	182	16	is	be	AUX
iajs-2731	182	17	cyclic	cyclic	ADJ
iajs-2731	182	18	and	and	CCONJ
iajs-2731	182	19	x	x	SYM
iajs-2731	182	20	/	/	SYM
iajs-2731	182	21	a	a	PRON
iajs-2731	182	22	is	be	AUX
iajs-2731	182	23	indecomposable	indecomposable	ADJ
iajs-2731	182	24	.	.	PUNCT
iajs-2731	183	1	proof	proof	NOUN
iajs-2731	183	2	:	:	PUNCT
iajs-2731	183	3	⟹	⟹	PROPN
iajs-2731	183	4	suppose	suppose	VERB
iajs-2731	183	5	that	that	SCONJ
iajs-2731	183	6	x	x	PRON
iajs-2731	183	7	is	be	AUX
iajs-2731	183	8	lochollow	lochollow	ADJ
iajs-2731	183	9	fmodule	fmodule	ADV
iajs-2731	183	10	then	then	ADV
iajs-2731	183	11	x	x	PRON
iajs-2731	183	12	is	be	AUX
iajs-2731	183	13	hollow	hollow	ADJ
iajs-2731	183	14	and	and	CCONJ
iajs-2731	183	15	cyclic	cyclic	ADJ
iajs-2731	183	16	fmodule	fmodule	NOUN
iajs-2731	183	17	by	by	ADP
iajs-2731	183	18	proposition	proposition	NOUN
iajs-2731	183	19	(	(	PUNCT
iajs-2731	183	20	2.1	2.1	NUM
iajs-2731	183	21	)	)	PUNCT
iajs-2731	183	22	.	.	PUNCT
iajs-2731	184	1	then	then	ADV
iajs-2731	184	2	x	x	X
iajs-2731	184	3	/	/	SYM
iajs-2731	184	4	a	a	DET
iajs-2731	184	5	indecomposable	indecomposable	ADJ
iajs-2731	184	6	.	.	PUNCT
iajs-2731	185	1	⟸	⟸	PROPN
iajs-2731	185	2	let	let	VERB
iajs-2731	185	3	x	x	PRON
iajs-2731	185	4	be	be	AUX
iajs-2731	185	5	a	a	DET
iajs-2731	185	6	cyclic	cyclic	ADJ
iajs-2731	185	7	fmodule	fmodule	NOUN
iajs-2731	185	8	and	and	CCONJ
iajs-2731	185	9	every	every	DET
iajs-2731	185	10	x	x	NOUN
iajs-2731	185	11	/	/	X
iajs-2731	185	12	a	a	PRON
iajs-2731	185	13	is	be	AUX
iajs-2731	185	14	indecomposable	indecomposable	ADJ
iajs-2731	185	15	,	,	PUNCT
iajs-2731	185	16	then	then	ADV
iajs-2731	185	17	by	by	ADP
iajs-2731	185	18	part	part	NOUN
iajs-2731	185	19	one	one	NUM
iajs-2731	185	20	.	.	PUNCT
iajs-2731	186	1	x	x	PRON
iajs-2731	186	2	is	be	AUX
iajs-2731	186	3	hollow	hollow	ADJ
iajs-2731	186	4	f	f	NOUN
iajs-2731	186	5	module	module	NOUN
iajs-2731	186	6	and	and	CCONJ
iajs-2731	186	7	proposition	proposition	NOUN
iajs-2731	186	8	(	(	PUNCT
iajs-2731	186	9	2.1	2.1	NUM
iajs-2731	186	10	)	)	PUNCT
iajs-2731	186	11	.	.	PUNCT
iajs-2731	187	1	thus	thus	ADV
iajs-2731	187	2	x	x	SYM
iajs-2731	187	3	lochollow	lochollow	NOUN
iajs-2731	187	4	fmodule	fmodule	ADV
iajs-2731	187	5	.	.	PUNCT
iajs-2731	188	1	3.5	3.5	NUM
iajs-2731	188	2	proposition	proposition	NOUN
iajs-2731	188	3	let	let	VERB
iajs-2731	188	4	x	x	PRON
iajs-2731	188	5	be	be	AUX
iajs-2731	188	6	a	a	DET
iajs-2731	188	7	fuzzy	fuzzy	ADJ
iajs-2731	188	8	module	module	NOUN
iajs-2731	188	9	of	of	ADP
iajs-2731	188	10	an	an	DET
iajs-2731	188	11	r	r	NOUN
iajs-2731	188	12	–	–	PUNCT
iajs-2731	188	13	module	module	NOUN
iajs-2731	188	14	m	m	NOUN
iajs-2731	188	15	,	,	PUNCT
iajs-2731	188	16	x	x	VERB
iajs-2731	188	17	is	be	AUX
iajs-2731	188	18	a	a	DET
iajs-2731	188	19	lochollow	lochollow	NOUN
iajs-2731	188	20	fmodule⟺	fmodule⟺	NOUN
iajs-2731	188	21	x	x	PRON
iajs-2731	188	22	is	be	AUX
iajs-2731	188	23	a	a	DET
iajs-2731	188	24	hollow	hollow	ADJ
iajs-2731	188	25	f	f	NOUN
iajs-2731	188	26	module	module	NOUN
iajs-2731	188	27	,	,	PUNCT
iajs-2731	188	28	and	and	CCONJ
iajs-2731	188	29	f	f	X
iajs-2731	188	30	-	-	PUNCT
iajs-2731	188	31	rad(x)≠	rad(x)≠	ADJ
iajs-2731	188	32	x.	x.	NOUN
iajs-2731	188	33	proof	proof	NOUN
iajs-2731	188	34	:	:	PUNCT
iajs-2731	188	35	⟹	⟹	PROPN
iajs-2731	188	36	suppose	suppose	VERB
iajs-2731	188	37	that	that	SCONJ
iajs-2731	188	38	x	x	PRON
iajs-2731	188	39	is	be	AUX
iajs-2731	188	40	lochollow	lochollow	ADJ
iajs-2731	188	41	fmodule	fmodule	ADV
iajs-2731	188	42	,	,	PUNCT
iajs-2731	188	43	then	then	ADV
iajs-2731	188	44	x	x	PUNCT
iajs-2731	188	45	is	be	AUX
iajs-2731	188	46	x	x	PUNCT
iajs-2731	188	47	hollow	hollow	ADJ
iajs-2731	188	48	and	and	CCONJ
iajs-2731	188	49	cyclic	cyclic	ADJ
iajs-2731	188	50	fmodule	fmodule	NOUN
iajs-2731	188	51	by	by	ADP
iajs-2731	188	52	𝜌proposition	𝜌proposition	NOUN
iajs-2731	188	53	(	(	PUNCT
iajs-2731	188	54	2.1	2.1	NUM
iajs-2731	188	55	)	)	PUNCT
iajs-2731	188	56	.	.	PUNCT
iajs-2731	189	1	xt	xt	PROPN
iajs-2731	189	2	is	be	AUX
iajs-2731	189	3	cyclic	cyclic	ADJ
iajs-2731	189	4	module	module	NOUN
iajs-2731	189	5	and	and	CCONJ
iajs-2731	189	6	it	it	PRON
iajs-2731	189	7	’s	’s	AUX
iajs-2731	189	8	finitely	finitely	ADV
iajs-2731	189	9	generated	generate	VERB
iajs-2731	189	10	,	,	PUNCT
iajs-2731	189	11	∀	∀	NOUN
iajs-2731	189	12	t	t	NOUN
iajs-2731	189	13	∈	∈	PROPN
iajs-2731	189	14	(	(	PUNCT
iajs-2731	189	15	0,1	0,1	NUM
iajs-2731	189	16	]	]	PUNCT
iajs-2731	189	17	by	by	ADP
iajs-2731	189	18	[	[	X
iajs-2731	189	19	9	9	NUM
iajs-2731	189	20	]	]	PUNCT
iajs-2731	189	21	.	.	PUNCT
iajs-2731	190	1	hence	hence	ADV
iajs-2731	190	2	frad(x)≠	frad(x)≠	X
iajs-2731	190	3	x.	x.	NOUN
iajs-2731	190	4	⟸	⟸	ADJ
iajs-2731	190	5	let	let	VERB
iajs-2731	190	6	x	x	PRON
iajs-2731	190	7	hollow	hollow	ADJ
iajs-2731	190	8	fuzzy	fuzzy	ADJ
iajs-2731	190	9	module	module	NOUN
iajs-2731	190	10	and	and	CCONJ
iajs-2731	190	11	f	f	NOUN
iajs-2731	190	12	-	-	PUNCT
iajs-2731	190	13	rad(x)≠	rad(x)≠	PROPN
iajs-2731	190	14	x	x	PUNCT
iajs-2731	190	15	is	be	AUX
iajs-2731	190	16	a	a	DET
iajs-2731	190	17	small	small	ADJ
iajs-2731	190	18	fsubmodule	fsubmodule	NOUN
iajs-2731	190	19	of	of	ADP
iajs-2731	190	20	x.	x.	NOUN
iajs-2731	190	21	f	f	X
iajs-2731	190	22	-	-	PUNCT
iajs-2731	190	23	rad(x	rad(x	NOUN
iajs-2731	190	24	)	)	PUNCT
iajs-2731	190	25	is	be	AUX
iajs-2731	190	26	unique	unique	ADJ
iajs-2731	190	27	maximal	maximal	ADJ
iajs-2731	190	28	fsubmodule	fsubmodule	NOUN
iajs-2731	190	29	of	of	ADP
iajs-2731	190	30	x	x	PUNCT
iajs-2731	190	31	and	and	CCONJ
iajs-2731	190	32	this	this	DET
iajs-2731	190	33	x/	x/	NOUN
iajs-2731	190	34	rad(x	rad(x	PROPN
iajs-2731	190	35	)	)	PUNCT
iajs-2731	190	36	is	be	AUX
iajs-2731	190	37	a	a	DET
iajs-2731	190	38	simple	simple	ADJ
iajs-2731	190	39	fuzzy	fuzzy	ADJ
iajs-2731	190	40	module	module	NOUN
iajs-2731	190	41	and	and	CCONJ
iajs-2731	190	42	hence	hence	ADV
iajs-2731	190	43	cyclic	cyclic	ADJ
iajs-2731	190	44	.	.	PUNCT
iajs-2731	191	1	implies	imply	VERB
iajs-2731	191	2	that	that	SCONJ
iajs-2731	191	3	x/	x/	PROPN
iajs-2731	191	4	rad(x	rad(x	X
iajs-2731	191	5	)	)	PUNCT
iajs-2731	191	6	=	=	PUNCT
iajs-2731	192	1	<	<	X
iajs-2731	192	2	x	x	X
iajs-2731	192	3	+	+	NOUN
iajs-2731	192	4	rad(x	rad(x	NOUN
iajs-2731	192	5	)	)	PUNCT
iajs-2731	192	6	>	>	X
iajs-2731	192	7	for	for	ADP
iajs-2731	192	8	some	some	DET
iajs-2731	192	9	x	x	SYM
iajs-2731	192	10	⊆	⊆	NUM
iajs-2731	192	11	x.	x.	NOUN
iajs-2731	192	12	we	we	PRON
iajs-2731	192	13	clam	clam	VERB
iajs-2731	192	14	that	that	SCONJ
iajs-2731	192	15	x=	x=	PUNCT
iajs-2731	193	1	rx	rx	VERB
iajs-2731	193	2	.	.	PUNCT
iajs-2731	194	1	let	let	VERB
iajs-2731	194	2	u	u	PRON
iajs-2731	194	3	∈	∈	PROPN
iajs-2731	194	4	x	x	X
iajs-2731	194	5	then	then	ADV
iajs-2731	194	6	u+	u+	ADJ
iajs-2731	194	7	rad(x	rad(x	NOUN
iajs-2731	194	8	)	)	PUNCT
iajs-2731	194	9	∈	∈	NOUN
iajs-2731	194	10	x/	x/	NOUN
iajs-2731	194	11	rad(x	rad(x	PROPN
iajs-2731	194	12	)	)	PUNCT
iajs-2731	194	13	,	,	PUNCT
iajs-2731	194	14	therefore	therefore	ADV
iajs-2731	194	15	a	a	DET
iajs-2731	194	16	∈	∈	NOUN
iajs-2731	194	17	r	r	NOUN
iajs-2731	194	18	such	such	ADJ
iajs-2731	194	19	that	that	DET
iajs-2731	194	20	u+	u+	NOUN
iajs-2731	194	21	rad(x	rad(x	NOUN
iajs-2731	194	22	)	)	PUNCT
iajs-2731	194	23	=	=	SYM
iajs-2731	194	24	a	a	PRON
iajs-2731	194	25	(	(	PUNCT
iajs-2731	194	26	u+	u+	NOUN
iajs-2731	194	27	rad(x	rad(x	NOUN
iajs-2731	194	28	)	)	PUNCT
iajs-2731	194	29	)	)	PUNCT
iajs-2731	195	1	=	=	NUM
iajs-2731	195	2	au+	au+	NOUN
iajs-2731	195	3	rad(x	rad(x	PROPN
iajs-2731	195	4	)	)	PUNCT
iajs-2731	195	5	,	,	PUNCT
iajs-2731	195	6	implies	imply	VERB
iajs-2731	195	7	that	that	SCONJ
iajs-2731	195	8	uax	uax	PROPN
iajs-2731	195	9	∈	∈	PRON
iajs-2731	195	10	rad(x	rad(x	PROPN
iajs-2731	195	11	)	)	PUNCT
iajs-2731	195	12	which	which	PRON
iajs-2731	195	13	implies	imply	VERB
iajs-2731	195	14	that	that	SCONJ
iajs-2731	195	15	uax	uax	PROPN
iajs-2731	195	16	=	=	SYM
iajs-2731	195	17	b	b	NOUN
iajs-2731	195	18	for	for	ADP
iajs-2731	195	19	some	some	DET
iajs-2731	195	20	b	b	NOUN
iajs-2731	195	21	∈	∈	NOUN
iajs-2731	195	22	rad(x	rad(x	NOUN
iajs-2731	195	23	)	)	PUNCT
iajs-2731	195	24	.	.	PUNCT
iajs-2731	196	1	thus	thus	ADV
iajs-2731	196	2	u=	u=	ADJ
iajs-2731	196	3	ax	ax	NOUN
iajs-2731	196	4	+	+	CCONJ
iajs-2731	196	5	b	b	X
iajs-2731	196	6	∈	∈	NOUN
iajs-2731	196	7	rx	rx	VERB
iajs-2731	196	8	+	+	NOUN
iajs-2731	196	9	rad(x	rad(x	ADJ
iajs-2731	196	10	)	)	PUNCT
iajs-2731	196	11	,	,	PUNCT
iajs-2731	196	12	hence	hence	ADV
iajs-2731	196	13	x=	x=	PUNCT
iajs-2731	196	14	rx	rx	VERB
iajs-2731	196	15	+	+	NOUN
iajs-2731	196	16	rad(x	rad(x	ADJ
iajs-2731	196	17	)	)	PUNCT
iajs-2731	196	18	.	.	PUNCT
iajs-2731	197	1	but	but	CCONJ
iajs-2731	197	2	f	f	X
iajs-2731	197	3	-	-	PUNCT
iajs-2731	197	4	rad(x	rad(x	NOUN
iajs-2731	197	5	)	)	PUNCT
iajs-2731	197	6	is	be	AUX
iajs-2731	197	7	fuzzy	fuzzy	ADJ
iajs-2731	197	8	small	small	ADJ
iajs-2731	197	9	submodule	submodule	NOUN
iajs-2731	197	10	of	of	ADP
iajs-2731	197	11	x	x	PROPN
iajs-2731	197	12	implies	imply	VERB
iajs-2731	197	13	that	that	SCONJ
iajs-2731	197	14	x=	x=	PROPN
iajs-2731	198	1	rx	rx	PROPN
iajs-2731	198	2	.	.	PUNCT
iajs-2731	199	1	thus	thus	ADV
iajs-2731	199	2	x	x	VERB
iajs-2731	199	3	is	be	AUX
iajs-2731	199	4	cyclic	cyclic	ADJ
iajs-2731	199	5	fmodule	fmodule	ADV
iajs-2731	199	6	by	by	ADP
iajs-2731	199	7	𝜌roposition	𝜌roposition	NOUN
iajs-2731	199	8	(	(	PUNCT
iajs-2731	199	9	2.1).we	2.1).we	NUM
iajs-2731	199	10	get	get	VERB
iajs-2731	199	11	x	x	SYM
iajs-2731	199	12	is	be	AUX
iajs-2731	199	13	a	a	DET
iajs-2731	199	14	lochollow	lochollow	NOUN
iajs-2731	199	15	fmodule	fmodule	NOUN
iajs-2731	199	16	.	.	PUNCT
iajs-2731	200	1	4	4	X
iajs-2731	200	2	.	.	X
iajs-2731	200	3	the	the	DET
iajs-2731	200	4	relationships	relationship	NOUN
iajs-2731	200	5	between	between	ADP
iajs-2731	200	6	lochollow	lochollow	NOUN
iajs-2731	200	7	fuzzy	fuzzy	ADJ
iajs-2731	200	8	module	module	NOUN
iajs-2731	200	9	and	and	CCONJ
iajs-2731	200	10	other	other	ADJ
iajs-2731	200	11	types	type	NOUN
iajs-2731	200	12	of	of	ADP
iajs-2731	200	13	modules	module	NOUN
iajs-2731	200	14	in	in	ADP
iajs-2731	200	15	this	this	DET
iajs-2731	200	16	section	section	NOUN
iajs-2731	200	17	,	,	PUNCT
iajs-2731	200	18	we	we	PRON
iajs-2731	200	19	shall	shall	AUX
iajs-2731	200	20	give	give	VERB
iajs-2731	200	21	the	the	DET
iajs-2731	200	22	relation	relation	NOUN
iajs-2731	200	23	between	between	ADP
iajs-2731	200	24	the	the	DET
iajs-2731	200	25	loc	loc	NOUN
iajs-2731	200	26	-	-	ADJ
iajs-2731	200	27	hollow	hollow	ADJ
iajs-2731	200	28	fuzzy	fuzzy	ADJ
iajs-2731	200	29	module	module	NOUN
iajs-2731	200	30	and	and	CCONJ
iajs-2731	200	31	different	different	ADJ
iajs-2731	200	32	modules	module	NOUN
iajs-2731	200	33	like	like	VERB
iajs-2731	200	34	as	as	SCONJ
iajs-2731	200	35	amply	amply	ADV
iajs-2731	200	36	supplemented	supplement	VERB
iajs-2731	200	37	modules	module	NOUN
iajs-2731	200	38	,	,	PUNCT
iajs-2731	200	39	indecomposable	indecomposable	ADJ
iajs-2731	200	40	modules	module	NOUN
iajs-2731	200	41	and	and	CCONJ
iajs-2731	200	42	lifting	lifting	NOUN
iajs-2731	200	43	modules	module	NOUN
iajs-2731	200	44	.	.	PUNCT
iajs-2731	201	1	we	we	PRON
iajs-2731	201	2	shall	shall	AUX
iajs-2731	201	3	fuzzify	fuzzify	VERB
iajs-2731	201	4	the	the	DET
iajs-2731	201	5	following	follow	VERB
iajs-2731	201	6	definitions	definition	NOUN
iajs-2731	201	7	:	:	PUNCT
iajs-2731	201	8	4.1	4.1	NUM
iajs-2731	201	9	definition	definition	NOUN
iajs-2731	201	10	:	:	PUNCT
iajs-2731	201	11	let	let	VERB
iajs-2731	201	12	a	a	DET
iajs-2731	201	13	,	,	PUNCT
iajs-2731	201	14	b	b	NOUN
iajs-2731	201	15	are	be	AUX
iajs-2731	201	16	fsubmodules	fsubmodule	NOUN
iajs-2731	201	17	of	of	ADP
iajs-2731	201	18	fuzzy	fuzzy	ADJ
iajs-2731	201	19	module	module	NOUN
iajs-2731	201	20	x	x	PUNCT
iajs-2731	201	21	then	then	ADV
iajs-2731	201	22	a	a	PRON
iajs-2731	201	23	is	be	AUX
iajs-2731	201	24	named	name	VERB
iajs-2731	201	25	a	a	DET
iajs-2731	201	26	fuzzy	fuzzy	ADJ
iajs-2731	201	27	supplement	supplement	NOUN
iajs-2731	201	28	of	of	ADP
iajs-2731	201	29	b	b	NOUN
iajs-2731	201	30	in	in	ADP
iajs-2731	201	31	x	x	SYM
iajs-2731	201	32	,	,	PUNCT
iajs-2731	201	33	if	if	SCONJ
iajs-2731	201	34	a	a	PRON
iajs-2731	201	35	is	be	AUX
iajs-2731	201	36	minimal	minimal	ADJ
iajs-2731	201	37	with	with	ADP
iajs-2731	201	38	a	a	DET
iajs-2731	201	39	+	+	NOUN
iajs-2731	201	40	b	b	NOUN
iajs-2731	201	41	=	=	SYM
iajs-2731	201	42	x.	x.	NOUN
iajs-2731	201	43	equivalently	equivalently	ADV
iajs-2731	201	44	,	,	PUNCT
iajs-2731	201	45	a	a	PRON
iajs-2731	201	46	is	be	AUX
iajs-2731	201	47	named	name	VERB
iajs-2731	201	48	fsupplement	fsupplement	NOUN
iajs-2731	201	49	of	of	ADP
iajs-2731	201	50	b	b	PROPN
iajs-2731	201	51	⟺	⟺	PROPN
iajs-2731	201	52	𝑨	𝑨	PROPN
iajs-2731	201	53	+	+	CCONJ
iajs-2731	201	54	𝑩=	𝑩=	NOUN
iajs-2731	201	55	𝜒	𝜒	X
iajs-2731	201	56	and	and	CCONJ
iajs-2731	201	57	𝑨	𝑨	PROPN
iajs-2731	201	58	∩	∩	ADJ
iajs-2731	201	59	𝑩	𝑩	NOUN
iajs-2731	201	60	is	be	AUX
iajs-2731	201	61	a	a	DET
iajs-2731	201	62	small	small	ADJ
iajs-2731	201	63	fsubm	fsubm	NOUN
iajs-2731	201	64	of	of	ADP
iajs-2731	201	65	𝑨.	𝑨.	PROPN
iajs-2731	201	66	an	an	DET
iajs-2731	201	67	fsubm	fsubm	NOUN
iajs-2731	201	68	a	a	PRON
iajs-2731	201	69	of	of	ADP
iajs-2731	201	70	x	x	SYM
iajs-2731	201	71	is	be	AUX
iajs-2731	201	72	named	name	VERB
iajs-2731	201	73	fsupplement	fsupplement	ADJ
iajs-2731	201	74	,	,	PUNCT
iajs-2731	201	75	if	if	SCONJ
iajs-2731	201	76	there	there	PRON
iajs-2731	201	77	is	be	VERB
iajs-2731	201	78	fsubm	fsubm	PROPN
iajs-2731	201	79	b	b	PROPN
iajs-2731	201	80	of	of	ADP
iajs-2731	201	81	x	x	SYM
iajs-2731	201	82	such	such	ADJ
iajs-2731	201	83	that	that	SCONJ
iajs-2731	201	84	a	a	PRON
iajs-2731	201	85	is	be	AUX
iajs-2731	201	86	f	f	NOUN
iajs-2731	201	87	-	-	PUNCT
iajs-2731	201	88	supplement	supplement	NOUN
iajs-2731	201	89	of	of	ADP
iajs-2731	201	90	b.	b.	PROPN
iajs-2731	201	91	4.2	4.2	NUM
iajs-2731	201	92	example	example	NOUN
iajs-2731	201	93	let	let	VERB
iajs-2731	201	94	m	m	VERB
iajs-2731	201	95	=	=	X
iajs-2731	201	96	z4	z4	X
iajs-2731	201	97	,	,	PUNCT
iajs-2731	201	98	define	define	VERB
iajs-2731	201	99	x	x	NOUN
iajs-2731	201	100	:	:	PUNCT
iajs-2731	201	101	m	m	VERB
iajs-2731	201	102	→	→	SYM
iajs-2731	201	103	[	[	X
iajs-2731	201	104	0	0	NUM
iajs-2731	201	105	,	,	PUNCT
iajs-2731	201	106	1	1	NUM
iajs-2731	201	107	]	]	PUNCT
iajs-2731	201	108	and	and	CCONJ
iajs-2731	201	109	define	define	VERB
iajs-2731	201	110	:	:	PUNCT
iajs-2731	201	111	a	a	X
iajs-2731	201	112	:	:	PUNCT
iajs-2731	201	113	m	m	VERB
iajs-2731	201	114	→	→	SYM
iajs-2731	202	1	[	[	X
iajs-2731	202	2	0	0	NUM
iajs-2731	202	3	,	,	PUNCT
iajs-2731	202	4	1	1	NUM
iajs-2731	202	5	]	]	PUNCT
iajs-2731	202	6	,	,	PUNCT
iajs-2731	202	7	define	define	VERB
iajs-2731	202	8	by	by	ADP
iajs-2731	202	9	a	a	DET
iajs-2731	202	10	(	(	PUNCT
iajs-2731	202	11	t	t	NOUN
iajs-2731	202	12	)	)	PUNCT
iajs-2731	202	13	=	=	PRON
iajs-2731	202	14	{	{	PUNCT
iajs-2731	202	15	𝑡	𝑡	X
iajs-2731	202	16	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	202	17	𝑥	𝑥	PRON
iajs-2731	202	18	∈	∈	NOUN
iajs-2731	202	19	𝑁	𝑁	NOUN
iajs-2731	202	20	0	0	NUM
iajs-2731	202	21	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	202	22	∀	∀	X
iajs-2731	202	23	t	t	NOUN
iajs-2731	202	24	∈	∈	PROPN
iajs-2731	202	25	(	(	PUNCT
iajs-2731	202	26	0	0	NUM
iajs-2731	202	27	,	,	PUNCT
iajs-2731	202	28	1	1	NUM
iajs-2731	202	29	]	]	PUNCT
iajs-2731	202	30	,	,	PUNCT
iajs-2731	202	31	n	n	NOUN
iajs-2731	202	32	=	=	NOUN
iajs-2731	202	33	2z	2z	NUM
iajs-2731	202	34	and	and	CCONJ
iajs-2731	202	35	let	let	VERB
iajs-2731	202	36	b	b	X
iajs-2731	202	37	:	:	PUNCT
iajs-2731	202	38	m	m	VERB
iajs-2731	202	39	→	→	SYM
iajs-2731	203	1	[	[	X
iajs-2731	203	2	0	0	NUM
iajs-2731	203	3	,	,	PUNCT
iajs-2731	203	4	1	1	NUM
iajs-2731	203	5	]	]	PUNCT
iajs-2731	203	6	,	,	PUNCT
iajs-2731	203	7	define	define	VERB
iajs-2731	203	8	by	by	ADP
iajs-2731	203	9	ibn	ibn	PROPN
iajs-2731	203	10	al	al	PROPN
iajs-2731	203	11	-	-	PUNCT
iajs-2731	203	12	haitham	haitham	PROPN
iajs-2731	203	13	jour	jour	X
iajs-2731	203	14	.	.	PROPN
iajs-2731	203	15	for	for	ADP
iajs-2731	203	16	pure	pure	ADJ
iajs-2731	203	17	&	&	CCONJ
iajs-2731	203	18	appl	appl	PROPN
iajs-2731	203	19	.	.	PUNCT
iajs-2731	204	1	sci	sci	PROPN
iajs-2731	204	2	.	.	PROPN
iajs-2731	205	1	53	53	NUM
iajs-2731	205	2	(	(	PUNCT
iajs-2731	205	3	2)2022	2)2022	VERB
iajs-2731	205	4	92	92	NUM
iajs-2731	205	5	b	b	NOUN
iajs-2731	205	6	(	(	PUNCT
iajs-2731	205	7	t	t	PROPN
iajs-2731	205	8	)	)	PUNCT
iajs-2731	205	9	=	=	PRON
iajs-2731	205	10	{	{	PUNCT
iajs-2731	205	11	𝑡	𝑡	X
iajs-2731	205	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	205	13	𝑥	𝑥	DET
iajs-2731	205	14	∈	∈	PROPN
iajs-2731	205	15	𝐿	𝐿	PROPN
iajs-2731	205	16	0	0	PROPN
iajs-2731	205	17	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	205	18	∀	∀	X
iajs-2731	205	19	t	t	NOUN
iajs-2731	205	20	∈	∈	PROPN
iajs-2731	205	21	(	(	PUNCT
iajs-2731	205	22	0	0	NUM
iajs-2731	205	23	,	,	PUNCT
iajs-2731	205	24	1	1	NUM
iajs-2731	205	25	]	]	PUNCT
iajs-2731	205	26	,	,	PUNCT
iajs-2731	205	27	l=⟨0̅⟩	l=⟨0̅⟩	PROPN
iajs-2731	205	28	clear	clear	ADJ
iajs-2731	205	29	that	that	SCONJ
iajs-2731	205	30	at	at	ADP
iajs-2731	205	31	=	=	SYM
iajs-2731	205	32	⟨2̅⟩	⟨2̅⟩	PROPN
iajs-2731	205	33	,	,	PUNCT
iajs-2731	205	34	bt	bt	X
iajs-2731	205	35	=	=	SYM
iajs-2731	205	36	⟨4̅⟩	⟨4̅⟩	PROPN
iajs-2731	205	37	are	be	AUX
iajs-2731	205	38	two	two	NUM
iajs-2731	205	39	submodules	submodule	NOUN
iajs-2731	205	40	of	of	ADP
iajs-2731	205	41	𝜒	𝜒	NOUN
iajs-2731	205	42	t	t	NOUN
iajs-2731	205	43	,	,	PUNCT
iajs-2731	205	44	𝜒	𝜒	PROPN
iajs-2731	205	45	t	t	NOUN
iajs-2731	205	46	=	=	PUNCT
iajs-2731	206	1	m	m	PROPN
iajs-2731	206	2	=	=	NOUN
iajs-2731	206	3	z4	z4	PROPN
iajs-2731	206	4	.	.	PUNCT
iajs-2731	207	1	hence	hence	ADV
iajs-2731	207	2	(	(	PUNCT
iajs-2731	207	3	a+b)t	a+b)t	PROPN
iajs-2731	207	4	=	=	SYM
iajs-2731	207	5	(	(	PUNCT
iajs-2731	207	6	𝜒)t	𝜒)t	NOUN
iajs-2731	207	7	of	of	ADP
iajs-2731	207	8	𝜒	𝜒	PROPN
iajs-2731	207	9	t	t	NOUN
iajs-2731	207	10	,	,	PUNCT
iajs-2731	207	11	∀	∀	NOUN
iajs-2731	207	12	t	t	NOUN
iajs-2731	207	13	∈	∈	PROPN
iajs-2731	207	14	(	(	PUNCT
iajs-2731	207	15	0	0	NUM
iajs-2731	207	16	,	,	PUNCT
iajs-2731	207	17	1	1	NUM
iajs-2731	207	18	]	]	PUNCT
iajs-2731	207	19	by[3	by[3	X
iajs-2731	207	20	]	]	PUNCT
iajs-2731	207	21	,	,	PUNCT
iajs-2731	207	22	therefor	therefor	ADP
iajs-2731	207	23	a+b	a+b	NUM
iajs-2731	207	24	=	=	SYM
iajs-2731	207	25	𝜒	𝜒	X
iajs-2731	207	26	∀	∀	NOUN
iajs-2731	207	27	a	a	PRON
iajs-2731	207	28	,	,	PUNCT
iajs-2731	207	29	b	b	PROPN
iajs-2731	207	30	𝛼re	𝛼re	NOUN
iajs-2731	207	31	a	a	DET
iajs-2731	207	32	fsubm	fsubm	NOUN
iajs-2731	207	33	of	of	ADP
iajs-2731	207	34	𝜒.	𝜒.	NOUN
iajs-2731	207	35	but	but	CCONJ
iajs-2731	207	36	bt	bt	NOUN
iajs-2731	207	37	is	be	AUX
iajs-2731	207	38	a	a	DET
iajs-2731	207	39	supplement	supplement	NOUN
iajs-2731	207	40	in	in	ADP
iajs-2731	207	41	at	at	ADP
iajs-2731	207	42	of	of	ADP
iajs-2731	207	43	𝜒	𝜒	NOUN
iajs-2731	207	44	t	t	NOUN
iajs-2731	207	45	by	by	ADP
iajs-2731	207	46	[	[	X
iajs-2731	207	47	9	9	NUM
iajs-2731	207	48	]	]	PUNCT
iajs-2731	207	49	,	,	PUNCT
iajs-2731	207	50	hence	hence	ADV
iajs-2731	207	51	b=	b=	VERB
iajs-2731	207	52	01	01	NUM
iajs-2731	207	53	of	of	ADP
iajs-2731	207	54	𝜒	𝜒	NOUN
iajs-2731	207	55	is	be	AUX
iajs-2731	207	56	fsupplement	fsupplement	ADJ
iajs-2731	207	57	in	in	ADP
iajs-2731	207	58	a	a	DET
iajs-2731	207	59	of𝜒.	of𝜒.	NOUN
iajs-2731	207	60	on	on	ADP
iajs-2731	207	61	the	the	DET
iajs-2731	207	62	other	other	ADJ
iajs-2731	207	63	side	side	NOUN
iajs-2731	207	64	,	,	PUNCT
iajs-2731	207	65	a	a	PRON
iajs-2731	207	66	is	be	AUX
iajs-2731	207	67	not	not	PART
iajs-2731	207	68	fsupplement	fsupplement	ADJ
iajs-2731	207	69	because	because	SCONJ
iajs-2731	207	70	01	01	NUM
iajs-2731	207	71	is	be	AUX
iajs-2731	207	72	a	a	DET
iajs-2731	207	73	minimal	minimal	ADJ
iajs-2731	207	74	in	in	ADP
iajs-2731	207	75	𝜒	𝜒	NOUN
iajs-2731	207	76	=	=	NOUN
iajs-2731	207	77	z4	z4	X
iajs-2731	207	78	.	.	PUNCT
iajs-2731	208	1	4.3	4.3	NUM
iajs-2731	208	2	definition	definition	NOUN
iajs-2731	208	3	:	:	PUNCT
iajs-2731	208	4	let	let	VERB
iajs-2731	208	5	a	a	DET
iajs-2731	208	6	,	,	PUNCT
iajs-2731	208	7	b	b	NOUN
iajs-2731	208	8	are	be	AUX
iajs-2731	208	9	fuzzy	fuzzy	ADJ
iajs-2731	208	10	submodules	submodule	NOUN
iajs-2731	208	11	of	of	ADP
iajs-2731	208	12	fmodule	fmodule	ADJ
iajs-2731	208	13	x.	x.	NOUN
iajs-2731	209	1	then	then	ADV
iajs-2731	209	2	x	x	VERB
iajs-2731	209	3	is	be	AUX
iajs-2731	209	4	called	call	VERB
iajs-2731	209	5	amply	amply	ADV
iajs-2731	209	6	fsupplemented	fsupplemente	VERB
iajs-2731	209	7	with	with	ADP
iajs-2731	209	8	a	a	DET
iajs-2731	209	9	+	+	NOUN
iajs-2731	209	10	b	b	NOUN
iajs-2731	209	11	=	=	SYM
iajs-2731	209	12	x	x	NOUN
iajs-2731	209	13	if	if	SCONJ
iajs-2731	209	14	there	there	PRON
iajs-2731	209	15	is	be	VERB
iajs-2731	209	16	a	a	DET
iajs-2731	209	17	supplement	supplement	NOUN
iajs-2731	209	18	u	u	NOUN
iajs-2731	209	19	of	of	ADP
iajs-2731	209	20	a	a	DET
iajs-2731	209	21	such	such	ADJ
iajs-2731	209	22	that	that	SCONJ
iajs-2731	209	23	u	u	PROPN
iajs-2731	209	24	⊆	⊆	NUM
iajs-2731	209	25	b	b	NOUN
iajs-2731	209	26	in	in	ADP
iajs-2731	209	27	x.	x.	PROPN
iajs-2731	209	28	4.4	4.4	NUM
iajs-2731	209	29	example	example	NOUN
iajs-2731	209	30	let	let	VERB
iajs-2731	209	31	m	m	PRON
iajs-2731	209	32	=	=	X
iajs-2731	209	33	z12	z12	NUM
iajs-2731	209	34	as	as	ADP
iajs-2731	209	35	z	z	NOUN
iajs-2731	209	36	-	-	PUNCT
iajs-2731	209	37	module	module	NOUN
iajs-2731	209	38	,	,	PUNCT
iajs-2731	209	39	define	define	VERB
iajs-2731	209	40	x	x	NOUN
iajs-2731	209	41	:	:	PUNCT
iajs-2731	209	42	m	m	VERB
iajs-2731	209	43	→	→	SYM
iajs-2731	209	44	[	[	PUNCT
iajs-2731	209	45	0,1	0,1	NUM
iajs-2731	209	46	]	]	PUNCT
iajs-2731	209	47	𝑎𝑠	𝑎𝑠	PART
iajs-2731	209	48	𝑓𝑜𝑙𝑙𝑜𝑤	𝑓𝑜𝑙𝑙𝑜𝑤	VERB
iajs-2731	209	49	,	,	PUNCT
iajs-2731	209	50	x(x	x(x	PROPN
iajs-2731	209	51	)	)	PUNCT
iajs-2731	210	1	=	=	PRON
iajs-2731	210	2	{	{	PUNCT
iajs-2731	210	3	1	1	NUM
iajs-2731	210	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	210	5	𝑥	𝑥	PRON
iajs-2731	210	6	∈	∈	PROPN
iajs-2731	210	7	𝑀	𝑀	PROPN
iajs-2731	210	8	0	0	NUM
iajs-2731	210	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	PROPN
iajs-2731	210	10	define	define	VERB
iajs-2731	210	11	a	a	DET
iajs-2731	210	12	:	:	PUNCT
iajs-2731	210	13	m	m	NOUN
iajs-2731	210	14	→	→	SYM
iajs-2731	210	15	[	[	PUNCT
iajs-2731	210	16	0,1	0,1	NUM
iajs-2731	210	17	]	]	PUNCT
iajs-2731	210	18	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2731	210	19	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	210	20	a(x	a(x	PROPN
iajs-2731	210	21	)	)	PUNCT
iajs-2731	210	22	=	=	PRON
iajs-2731	210	23	{	{	PUNCT
iajs-2731	210	24	𝑡	𝑡	X
iajs-2731	210	25	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	210	26	𝑥	𝑥	PRON
iajs-2731	210	27	∈	∈	NOUN
iajs-2731	210	28	𝑁	𝑁	PROPN
iajs-2731	210	29	0	0	NUM
iajs-2731	210	30	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	210	31	,	,	PUNCT
iajs-2731	210	32	t	t	PROPN
iajs-2731	210	33	∈	∈	PROPN
iajs-2731	210	34	(	(	PUNCT
iajs-2731	210	35	0	0	NUM
iajs-2731	210	36	,	,	PUNCT
iajs-2731	210	37	1	1	NUM
iajs-2731	210	38	]	]	PUNCT
iajs-2731	210	39	.	.	PUNCT
iajs-2731	211	1	n=	n=	ADJ
iajs-2731	211	2	3z	3z	NUM
iajs-2731	211	3	also	also	ADV
iajs-2731	211	4	b	b	NOUN
iajs-2731	211	5	:	:	PUNCT
iajs-2731	211	6	m	m	AUX
iajs-2731	211	7	→	→	SYM
iajs-2731	211	8	[	[	PUNCT
iajs-2731	211	9	0,1	0,1	NUM
iajs-2731	211	10	]	]	PUNCT
iajs-2731	211	11	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	X
iajs-2731	211	12	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	211	13	b(x	b(x	NOUN
iajs-2731	211	14	)	)	PUNCT
iajs-2731	212	1	=	=	PRON
iajs-2731	212	2	{	{	PUNCT
iajs-2731	212	3	𝑡	𝑡	X
iajs-2731	212	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	212	5	𝑥	𝑥	DET
iajs-2731	212	6	∈	∈	PROPN
iajs-2731	212	7	𝐿	𝐿	PROPN
iajs-2731	212	8	0	0	PROPN
iajs-2731	212	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	212	10	,	,	PUNCT
iajs-2731	212	11	t	t	PROPN
iajs-2731	212	12	∈	∈	PROPN
iajs-2731	212	13	(	(	PUNCT
iajs-2731	212	14	0	0	NUM
iajs-2731	212	15	,	,	PUNCT
iajs-2731	212	16	1	1	NUM
iajs-2731	212	17	]	]	PUNCT
iajs-2731	212	18	,	,	PUNCT
iajs-2731	212	19	l=	l=	ADJ
iajs-2731	212	20	2z	2z	NUM
iajs-2731	212	21	c	c	X
iajs-2731	212	22	:	:	PUNCT
iajs-2731	212	23	m	m	AUX
iajs-2731	212	24	→	→	SYM
iajs-2731	212	25	[	[	PUNCT
iajs-2731	212	26	0,1	0,1	NUM
iajs-2731	212	27	]	]	PUNCT
iajs-2731	212	28	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	X
iajs-2731	212	29	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2731	212	30	b(x	b(x	NOUN
iajs-2731	212	31	)	)	PUNCT
iajs-2731	213	1	=	=	PRON
iajs-2731	213	2	{	{	PUNCT
iajs-2731	213	3	𝑡	𝑡	X
iajs-2731	213	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	213	5	𝑥	𝑥	DET
iajs-2731	213	6	∈	∈	PROPN
iajs-2731	213	7	𝐾	𝐾	NOUN
iajs-2731	213	8	0	0	NUM
iajs-2731	213	9	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	213	10	,	,	PUNCT
iajs-2731	213	11	t	t	PROPN
iajs-2731	213	12	∈	∈	PROPN
iajs-2731	213	13	(	(	PUNCT
iajs-2731	213	14	0	0	NUM
iajs-2731	213	15	,	,	PUNCT
iajs-2731	213	16	1	1	NUM
iajs-2731	213	17	]	]	PUNCT
iajs-2731	213	18	,	,	PUNCT
iajs-2731	213	19	k=	k=	X
iajs-2731	213	20	4z	4z	X
iajs-2731	213	21	clear	clear	ADJ
iajs-2731	213	22	that	that	SCONJ
iajs-2731	213	23	x	x	PRON
iajs-2731	213	24	is	be	AUX
iajs-2731	213	25	fuzzy	fuzzy	ADJ
iajs-2731	213	26	module	module	NOUN
iajs-2731	213	27	and	and	CCONJ
iajs-2731	213	28	xt	xt	NOUN
iajs-2731	214	1	=	=	NOUN
iajs-2731	214	2	m	m	PROPN
iajs-2731	214	3	,	,	PUNCT
iajs-2731	214	4	at	at	ADP
iajs-2731	214	5	=	=	NOUN
iajs-2731	214	6	3z	3z	NUM
iajs-2731	214	7	,	,	PUNCT
iajs-2731	214	8	bt=2z	bt=2z	NOUN
iajs-2731	214	9	and	and	CCONJ
iajs-2731	214	10	ct=4z	ct=4z	PROPN
iajs-2731	214	11	are	be	AUX
iajs-2731	214	12	submodules	submodule	NOUN
iajs-2731	214	13	in	in	ADP
iajs-2731	214	14	xt	xt	PROPN
iajs-2731	214	15	with	with	ADP
iajs-2731	214	16	(	(	PUNCT
iajs-2731	214	17	a	a	DET
iajs-2731	214	18	+	+	X
iajs-2731	214	19	b)t=	b)t=	ADJ
iajs-2731	214	20	(	(	PUNCT
iajs-2731	214	21	𝜒)t	𝜒)t	X
iajs-2731	214	22	,	,	PUNCT
iajs-2731	214	23	∀	∀	VERB
iajs-2731	214	24	t	t	NOUN
iajs-2731	214	25	∈	∈	PROPN
iajs-2731	214	26	(	(	PUNCT
iajs-2731	214	27	0	0	NUM
iajs-2731	214	28	,	,	PUNCT
iajs-2731	214	29	1	1	NUM
iajs-2731	214	30	]	]	PUNCT
iajs-2731	214	31	,	,	PUNCT
iajs-2731	214	32	therefore	therefore	ADV
iajs-2731	214	33	𝑨	𝑨	PROPN
iajs-2731	214	34	+	+	CCONJ
iajs-2731	214	35	𝑩=	𝑩=	VERB
iajs-2731	214	36	𝜒	𝜒	X
iajs-2731	214	37	where	where	SCONJ
iajs-2731	214	38	𝑨	𝑨	PROPN
iajs-2731	214	39	,	,	PUNCT
iajs-2731	214	40	𝑩	𝑩	PROPN
iajs-2731	214	41	are	be	AUX
iajs-2731	214	42	fuzzy	fuzzy	ADJ
iajs-2731	214	43	submodules	submodule	NOUN
iajs-2731	214	44	of	of	ADP
iajs-2731	214	45	x	x	X
iajs-2731	214	46	.	.	PUNCT
iajs-2731	215	1	but	but	CCONJ
iajs-2731	215	2	𝑨	𝑨	PROPN
iajs-2731	215	3	is	be	AUX
iajs-2731	215	4	fuzzy	fuzzy	ADJ
iajs-2731	215	5	supplement	supplement	NOUN
iajs-2731	215	6	of	of	ADP
iajs-2731	215	7	b	b	NOUN
iajs-2731	215	8	in	in	ADP
iajs-2731	215	9	x	x	X
iajs-2731	215	10	.also	.also	PUNCT
iajs-2731	215	11	we	we	PRON
iajs-2731	215	12	have	have	VERB
iajs-2731	215	13	a	a	DET
iajs-2731	215	14	+	+	NOUN
iajs-2731	215	15	c	c	NOUN
iajs-2731	215	16	=	=	SYM
iajs-2731	215	17	x	x	INTJ
iajs-2731	215	18	where	where	SCONJ
iajs-2731	215	19	c	c	NOUN
iajs-2731	215	20	is	be	AUX
iajs-2731	215	21	a	a	DET
iajs-2731	215	22	fsubmodule	fsubmodule	NOUN
iajs-2731	215	23	in	in	ADP
iajs-2731	215	24	x	x	NOUN
iajs-2731	215	25	,	,	PUNCT
iajs-2731	215	26	again	again	ADV
iajs-2731	215	27	at	at	ADP
iajs-2731	215	28	is	be	AUX
iajs-2731	215	29	a	a	DET
iajs-2731	215	30	supplement	supplement	NOUN
iajs-2731	215	31	of	of	ADP
iajs-2731	215	32	ct	ct	NUM
iajs-2731	215	33	in	in	ADP
iajs-2731	215	34	xt	xt	ADP
iajs-2731	215	35	∀	∀	PUNCT
iajs-2731	215	36	t	t	PROPN
iajs-2731	215	37	∈	∈	PROPN
iajs-2731	215	38	(	(	PUNCT
iajs-2731	215	39	0	0	NUM
iajs-2731	215	40	,	,	PUNCT
iajs-2731	215	41	1	1	NUM
iajs-2731	215	42	]	]	PUNCT
iajs-2731	215	43	therefore	therefore	ADV
iajs-2731	215	44	a	a	PRON
iajs-2731	215	45	is	be	AUX
iajs-2731	215	46	fsupplement	fsupplement	NOUN
iajs-2731	215	47	of	of	ADP
iajs-2731	215	48	c	c	PROPN
iajs-2731	215	49	in	in	ADP
iajs-2731	215	50	x.	x.	NOUN
iajs-2731	215	51	thus	thus	ADV
iajs-2731	215	52	x	x	X
iajs-2731	215	53	is	be	AUX
iajs-2731	215	54	amply	amply	ADV
iajs-2731	215	55	f	f	PROPN
iajs-2731	215	56	supplemented	supplement	VERB
iajs-2731	215	57	.	.	PUNCT
iajs-2731	216	1	4.5	4.5	NUM
iajs-2731	216	2	remark	remark	VERB
iajs-2731	216	3	each	each	DET
iajs-2731	216	4	a	a	DET
iajs-2731	216	5	direct	direct	ADJ
iajs-2731	216	6	summand	summand	NOUN
iajs-2731	216	7	of	of	ADP
iajs-2731	216	8	fmodule	fmodule	NOUN
iajs-2731	216	9	is	be	AUX
iajs-2731	216	10	fsupplement	fsupplement	ADJ
iajs-2731	216	11	submodule	submodule	NOUN
iajs-2731	216	12	of	of	ADP
iajs-2731	216	13	x.	x.	PROPN
iajs-2731	216	14	proof	proof	NOUN
iajs-2731	216	15	let	let	VERB
iajs-2731	216	16	a	a	PRON
iajs-2731	216	17	be	be	AUX
iajs-2731	216	18	fsubmodule	fsubmodule	NOUN
iajs-2731	216	19	of	of	ADP
iajs-2731	216	20	x	x	PRON
iajs-2731	216	21	,	,	PUNCT
iajs-2731	216	22	then	then	ADV
iajs-2731	216	23	there	there	PRON
iajs-2731	216	24	exist	exist	VERB
iajs-2731	216	25	b	b	NOUN
iajs-2731	216	26	of	of	ADP
iajs-2731	216	27	x	x	SYM
iajs-2731	216	28	such	such	ADJ
iajs-2731	216	29	that	that	SCONJ
iajs-2731	216	30	(	(	PUNCT
iajs-2731	216	31	a	a	DET
iajs-2731	216	32	⨁	⨁	PROPN
iajs-2731	216	33	b)t=	b)t=	NOUN
iajs-2731	216	34	xt	xt	ADP
iajs-2731	216	35	then	then	ADV
iajs-2731	216	36	(	(	PUNCT
iajs-2731	216	37	a+b	a+b	NUM
iajs-2731	216	38	)	)	PUNCT
iajs-2731	216	39	t=	t=	PRON
iajs-2731	216	40	xt	xt	PUNCT
iajs-2731	217	1	by	by	ADP
iajs-2731	217	2	[	[	PUNCT
iajs-2731	217	3	6	6	NUM
iajs-2731	217	4	]	]	PUNCT
iajs-2731	217	5	,	,	PUNCT
iajs-2731	217	6	therefore	therefore	ADV
iajs-2731	217	7	a	a	DET
iajs-2731	217	8	⨁	⨁	PROPN
iajs-2731	217	9	b	b	PROPN
iajs-2731	217	10	=	=	PUNCT
iajs-2731	217	11	x	x	X
iajs-2731	217	12	then	then	ADV
iajs-2731	217	13	a	a	DET
iajs-2731	217	14	+	+	X
iajs-2731	217	15	b	b	NOUN
iajs-2731	217	16	=	=	SYM
iajs-2731	217	17	x	x	PROPN
iajs-2731	217	18	and	and	CCONJ
iajs-2731	217	19	b∩	b∩	PROPN
iajs-2731	217	20	a	a	PRON
iajs-2731	217	21	is	be	AUX
iajs-2731	217	22	a	a	DET
iajs-2731	217	23	proper	proper	ADJ
iajs-2731	217	24	fsubmodule	fsubmodule	NOUN
iajs-2731	217	25	of	of	ADP
iajs-2731	217	26	a.	a.	NOUN
iajs-2731	217	27	to	to	PART
iajs-2731	217	28	prove	prove	VERB
iajs-2731	217	29	that	that	SCONJ
iajs-2731	217	30	a	a	PRON
iajs-2731	217	31	is	be	AUX
iajs-2731	217	32	supplement	supplement	NOUN
iajs-2731	217	33	of	of	ADP
iajs-2731	217	34	b	b	NOUN
iajs-2731	217	35	in	in	ADP
iajs-2731	217	36	x	x	PUNCT
iajs-2731	217	37	.suppose	.suppose	ADP
iajs-2731	217	38	that	that	SCONJ
iajs-2731	217	39	there	there	PRON
iajs-2731	217	40	exist	exist	VERB
iajs-2731	217	41	c	c	AUX
iajs-2731	217	42	be	be	AUX
iajs-2731	217	43	a	a	DET
iajs-2731	217	44	fsubmodule	fsubmodule	NOUN
iajs-2731	217	45	in	in	ADP
iajs-2731	217	46	a	a	DET
iajs-2731	217	47	such	such	ADJ
iajs-2731	217	48	that	that	DET
iajs-2731	217	49	c+b=	c+b=	NOUN
iajs-2731	217	50	x.	x.	NOUN
iajs-2731	217	51	then	then	ADV
iajs-2731	217	52	a=	a=	VERB
iajs-2731	217	53	x∩	x∩	PROPN
iajs-2731	217	54	a=	a=	PROPN
iajs-2731	217	55	(	(	PUNCT
iajs-2731	217	56	c+b	c+b	PROPN
iajs-2731	217	57	)	)	PUNCT
iajs-2731	217	58	∩	∩	NOUN
iajs-2731	217	59	a	a	X
iajs-2731	217	60	but	but	CCONJ
iajs-2731	217	61	(	(	PUNCT
iajs-2731	217	62	(	(	PUNCT
iajs-2731	217	63	c+b	c+b	NOUN
iajs-2731	217	64	)	)	PUNCT
iajs-2731	217	65	∩	∩	NOUN
iajs-2731	217	66	a)t	a)t	PUNCT
iajs-2731	218	1	=	=	SYM
iajs-2731	218	2	(	(	PUNCT
iajs-2731	218	3	ct+	ct+	PROPN
iajs-2731	218	4	bt	bt	NOUN
iajs-2731	218	5	)	)	PUNCT
iajs-2731	218	6	∩	∩	NOUN
iajs-2731	218	7	at	at	ADP
iajs-2731	218	8	∀	∀	NOUN
iajs-2731	218	9	t	t	NOUN
iajs-2731	218	10	∈	∈	PROPN
iajs-2731	218	11	(	(	PUNCT
iajs-2731	218	12	0	0	NUM
iajs-2731	218	13	,	,	PUNCT
iajs-2731	218	14	1	1	NUM
iajs-2731	218	15	]	]	PUNCT
iajs-2731	218	16	implies	imply	VERB
iajs-2731	218	17	that	that	SCONJ
iajs-2731	218	18	c+	c+	X
iajs-2731	218	19	(	(	PUNCT
iajs-2731	218	20	b∩	b∩	PROPN
iajs-2731	218	21	a	a	X
iajs-2731	218	22	)	)	PUNCT
iajs-2731	218	23	by	by	ADP
iajs-2731	218	24	[	[	X
iajs-2731	218	25	14	14	NUM
iajs-2731	218	26	]	]	PUNCT
iajs-2731	218	27	.but	.but	PUNCT
iajs-2731	219	1	by	by	ADP
iajs-2731	219	2	our	our	PRON
iajs-2731	219	3	assumption	assumption	NOUN
iajs-2731	219	4	(	(	PUNCT
iajs-2731	219	5	b∩	b∩	PROPN
iajs-2731	219	6	a	a	X
iajs-2731	219	7	)	)	PUNCT
iajs-2731	219	8	is	be	AUX
iajs-2731	219	9	a	a	DET
iajs-2731	219	10	proper	proper	ADJ
iajs-2731	219	11	fsubmodule	fsubmodule	NOUN
iajs-2731	219	12	in	in	ADP
iajs-2731	219	13	a	a	DET
iajs-2731	219	14	implies	implie	NOUN
iajs-2731	219	15	that	that	SCONJ
iajs-2731	219	16	a	a	DET
iajs-2731	219	17	=	=	NOUN
iajs-2731	219	18	c	c	NOUN
iajs-2731	219	19	.	.	PUNCT
iajs-2731	220	1	thus	thus	ADV
iajs-2731	220	2	a	a	PRON
iajs-2731	220	3	is	be	AUX
iajs-2731	220	4	supplement	supplement	NOUN
iajs-2731	220	5	of	of	ADP
iajs-2731	220	6	b	b	NOUN
iajs-2731	220	7	in	in	ADP
iajs-2731	220	8	x	x	PROPN
iajs-2731	220	9	.	.	PUNCT
iajs-2731	221	1	4.6	4.6	NUM
iajs-2731	221	2	proposition	proposition	NOUN
iajs-2731	221	3	every	every	DET
iajs-2731	221	4	lochollow	lochollow	ADJ
iajs-2731	221	5	fuzzy	fuzzy	ADJ
iajs-2731	221	6	module	module	NOUN
iajs-2731	221	7	is	be	AUX
iajs-2731	221	8	an	an	DET
iajs-2731	221	9	amply	amply	ADV
iajs-2731	221	10	supplemented	supplement	VERB
iajs-2731	221	11	(	(	PUNCT
iajs-2731	221	12	supplement	supplement	NOUN
iajs-2731	221	13	fuzzy	fuzzy	ADJ
iajs-2731	221	14	)	)	PUNCT
iajs-2731	221	15	is	be	AUX
iajs-2731	221	16	a	a	DET
iajs-2731	221	17	ƒuzzy	ƒuzzy	NOUN
iajs-2731	221	18	submodule	submodule	NOUN
iajs-2731	221	19	.	.	PUNCT
iajs-2731	222	1	ibn	ibn	PROPN
iajs-2731	222	2	al	al	PROPN
iajs-2731	222	3	-	-	PUNCT
iajs-2731	222	4	haitham	haitham	PROPN
iajs-2731	222	5	jour	jour	X
iajs-2731	222	6	.	.	PROPN
iajs-2731	222	7	for	for	ADP
iajs-2731	222	8	pure	pure	ADJ
iajs-2731	222	9	&	&	CCONJ
iajs-2731	222	10	appl	appl	PROPN
iajs-2731	222	11	.	.	PUNCT
iajs-2731	223	1	sci	sci	PROPN
iajs-2731	223	2	.	.	PROPN
iajs-2731	224	1	53	53	NUM
iajs-2731	224	2	(	(	PUNCT
iajs-2731	224	3	2)2022	2)2022	VERB
iajs-2731	224	4	93	93	NUM
iajs-2731	224	5	proof	proof	NOUN
iajs-2731	224	6	:	:	PUNCT
iajs-2731	224	7	suppose	suppose	VERB
iajs-2731	224	8	that	that	SCONJ
iajs-2731	224	9	𝜒	𝜒	NOUN
iajs-2731	224	10	is	be	AUX
iajs-2731	224	11	a	a	DET
iajs-2731	224	12	lochollow	lochollow	NOUN
iajs-2731	224	13	fmodule	fmodule	NOUN
iajs-2731	224	14	,	,	PUNCT
iajs-2731	224	15	is	be	AUX
iajs-2731	224	16	a	a	DET
iajs-2731	224	17	unique	unique	ADJ
iajs-2731	224	18	maximal	maximal	ADJ
iajs-2731	224	19	ƒuzzy	ƒuzzy	NOUN
iajs-2731	224	20	submodule	submodule	NOUN
iajs-2731	224	21	of	of	ADP
iajs-2731	224	22	𝜒.	𝜒.	NOUN
iajs-2731	224	23	since	since	SCONJ
iajs-2731	224	24	𝜒	𝜒	NOUN
iajs-2731	224	25	is	be	AUX
iajs-2731	224	26	lochollow	lochollow	NOUN
iajs-2731	224	27	fmodule	fmodule	NOUN
iajs-2731	224	28	,	,	PUNCT
iajs-2731	224	29	then	then	ADV
iajs-2731	224	30	we	we	PRON
iajs-2731	224	31	have	have	AUX
iajs-2731	224	32	at+	at+	VERB
iajs-2731	224	33	xt=)x)t	xt=)x)t	NOUN
iajs-2731	224	34	,	,	PUNCT
iajs-2731	224	35	t∈	t∈	X
iajs-2731	224	36	(	(	PUNCT
iajs-2731	224	37	0,1	0,1	NUM
iajs-2731	224	38	]	]	PUNCT
iajs-2731	224	39	.this	.this	PRON
iajs-2731	224	40	leads	lead	VERB
iajs-2731	224	41	a+x	a+x	ADV
iajs-2731	224	42	=	=	SYM
iajs-2731	224	43	x	x	NOUN
iajs-2731	224	44	and	and	CCONJ
iajs-2731	224	45	a∩x=	a∩x=	PROPN
iajs-2731	224	46	a	a	PRON
iajs-2731	224	47	is	be	AUX
iajs-2731	224	48	a	a	DET
iajs-2731	224	49	small	small	ADJ
iajs-2731	224	50	fuzzy	fuzzy	ADJ
iajs-2731	224	51	submodule	submodule	NOUN
iajs-2731	224	52	of	of	ADP
iajs-2731	224	53	x	x	PUNCT
iajs-2731	224	54	by	by	ADP
iajs-2731	224	55	[	[	PUNCT
iajs-2731	224	56	9	9	NUM
iajs-2731	224	57	]	]	PUNCT
iajs-2731	224	58	.	.	PUNCT
iajs-2731	225	1	therefore	therefore	ADV
iajs-2731	225	2	x	x	X
iajs-2731	225	3	is	be	AUX
iajs-2731	225	4	an	an	DET
iajs-2731	225	5	amply	amply	ADV
iajs-2731	225	6	supplemented	supplement	VERB
iajs-2731	225	7	module	module	NOUN
iajs-2731	225	8	.	.	PUNCT
iajs-2731	226	1	𝜍onverse	𝜍onverse	NOUN
iajs-2731	226	2	of	of	ADP
iajs-2731	226	3	𝜌roposition	𝜌roposition	NOUN
iajs-2731	226	4	(	(	PUNCT
iajs-2731	226	5	4.6	4.6	NUM
iajs-2731	226	6	)	)	PUNCT
iajs-2731	226	7	is	be	AUX
iajs-2731	226	8	not	not	PART
iajs-2731	226	9	satisfied	satisfied	ADJ
iajs-2731	226	10	by	by	ADP
iajs-2731	226	11	the	the	DET
iajs-2731	226	12	following	follow	VERB
iajs-2731	226	13	4.7	4.7	NUM
iajs-2731	226	14	example	example	NOUN
iajs-2731	226	15	let	let	VERB
iajs-2731	226	16	m	m	PROPN
iajs-2731	226	17	=	=	NOUN
iajs-2731	226	18	z6	z6	PROPN
iajs-2731	226	19	,	,	PUNCT
iajs-2731	226	20	define	define	VERB
iajs-2731	226	21	𝜒	𝜒	NUM
iajs-2731	226	22	:	:	PUNCT
iajs-2731	226	23	m	m	VERB
iajs-2731	226	24	→	→	SYM
iajs-2731	227	1	[	[	X
iajs-2731	227	2	0	0	NUM
iajs-2731	227	3	,	,	PUNCT
iajs-2731	227	4	1	1	NUM
iajs-2731	227	5	]	]	PUNCT
iajs-2731	227	6	and	and	CCONJ
iajs-2731	227	7	define	define	VERB
iajs-2731	227	8	by	by	ADP
iajs-2731	227	9	:	:	PUNCT
iajs-2731	227	10	a	a	X
iajs-2731	227	11	:	:	PUNCT
iajs-2731	227	12	m	m	VERB
iajs-2731	227	13	→	→	SYM
iajs-2731	227	14	[	[	X
iajs-2731	227	15	0	0	NUM
iajs-2731	227	16	,	,	PUNCT
iajs-2731	227	17	1	1	NUM
iajs-2731	227	18	]	]	PUNCT
iajs-2731	227	19	,	,	PUNCT
iajs-2731	227	20	define	define	VERB
iajs-2731	227	21	by	by	ADP
iajs-2731	227	22	a	a	DET
iajs-2731	227	23	(	(	PUNCT
iajs-2731	227	24	t	t	NOUN
iajs-2731	227	25	)	)	PUNCT
iajs-2731	227	26	=	=	PRON
iajs-2731	227	27	{	{	PUNCT
iajs-2731	227	28	𝑡	𝑡	X
iajs-2731	227	29	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	227	30	𝑥	𝑥	PRON
iajs-2731	227	31	∈	∈	NOUN
iajs-2731	227	32	𝑁	𝑁	NOUN
iajs-2731	227	33	0	0	NUM
iajs-2731	227	34	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	227	35	∀	∀	X
iajs-2731	227	36	t	t	NOUN
iajs-2731	227	37	∈	∈	PROPN
iajs-2731	227	38	(	(	PUNCT
iajs-2731	227	39	0	0	NUM
iajs-2731	227	40	,	,	PUNCT
iajs-2731	227	41	1	1	NUM
iajs-2731	227	42	]	]	PUNCT
iajs-2731	227	43	,	,	PUNCT
iajs-2731	227	44	n	n	NOUN
iajs-2731	227	45	=	=	NOUN
iajs-2731	227	46	2z	2z	NUM
iajs-2731	227	47	and	and	CCONJ
iajs-2731	227	48	let	let	VERB
iajs-2731	227	49	b	b	X
iajs-2731	227	50	:	:	PUNCT
iajs-2731	227	51	m	m	VERB
iajs-2731	227	52	→	→	SYM
iajs-2731	228	1	[	[	X
iajs-2731	228	2	0	0	NUM
iajs-2731	228	3	,	,	PUNCT
iajs-2731	228	4	1	1	NUM
iajs-2731	228	5	]	]	PUNCT
iajs-2731	228	6	,	,	PUNCT
iajs-2731	228	7	define	define	VERB
iajs-2731	228	8	by	by	ADP
iajs-2731	228	9	b	b	PROPN
iajs-2731	228	10	(	(	PUNCT
iajs-2731	228	11	t	t	PROPN
iajs-2731	228	12	)	)	PUNCT
iajs-2731	228	13	=	=	PRON
iajs-2731	228	14	{	{	PUNCT
iajs-2731	228	15	𝑡	𝑡	X
iajs-2731	228	16	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	228	17	𝑥	𝑥	DET
iajs-2731	228	18	∈	∈	PROPN
iajs-2731	228	19	𝐿	𝐿	PROPN
iajs-2731	228	20	0	0	PROPN
iajs-2731	228	21	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	228	22	∀	∀	X
iajs-2731	228	23	t	t	NOUN
iajs-2731	228	24	∈	∈	PROPN
iajs-2731	228	25	(	(	PUNCT
iajs-2731	228	26	0	0	NUM
iajs-2731	228	27	,	,	PUNCT
iajs-2731	228	28	1	1	NUM
iajs-2731	228	29	]	]	PUNCT
iajs-2731	228	30	,	,	PUNCT
iajs-2731	228	31	l=3z	l=3z	VERB
iajs-2731	228	32	clear	clear	ADJ
iajs-2731	228	33	that	that	SCONJ
iajs-2731	228	34	at	at	ADP
iajs-2731	228	35	=	=	SYM
iajs-2731	228	36	⟨2̅⟩	⟨2̅⟩	PROPN
iajs-2731	228	37	,	,	PUNCT
iajs-2731	228	38	bt	bt	PRON
iajs-2731	228	39	=	=	PUNCT
iajs-2731	228	40	⟨3̅⟩	⟨3̅⟩	PROPN
iajs-2731	228	41	are	be	AUX
iajs-2731	228	42	two	two	NUM
iajs-2731	228	43	submodule	submodule	NOUN
iajs-2731	228	44	of	of	ADP
iajs-2731	228	45	x	x	PROPN
iajs-2731	228	46	and	and	CCONJ
iajs-2731	228	47	xt	xt	X
iajs-2731	228	48	=	=	NOUN
iajs-2731	228	49	m	m	PROPN
iajs-2731	228	50	=	=	PROPN
iajs-2731	228	51	z6	z6	PROPN
iajs-2731	228	52	.	.	PUNCT
iajs-2731	229	1	hence	hence	ADV
iajs-2731	229	2	(	(	PUNCT
iajs-2731	229	3	a+b)t	a+b)t	PROPN
iajs-2731	229	4	=	=	SYM
iajs-2731	229	5	(	(	PUNCT
iajs-2731	229	6	x)t	x)t	PUNCT
iajs-2731	229	7	direct	direct	ADJ
iajs-2731	229	8	summand	summand	NOUN
iajs-2731	229	9	of	of	ADP
iajs-2731	229	10	xt	xt	PROPN
iajs-2731	229	11	is	be	AUX
iajs-2731	229	12	fuzzy	fuzzy	ADJ
iajs-2731	229	13	submodule	submodule	NOUN
iajs-2731	229	14	of	of	ADP
iajs-2731	229	15	x	x	X
iajs-2731	229	16	by[3	by[3	X
iajs-2731	229	17	]	]	PUNCT
iajs-2731	229	18	and	and	CCONJ
iajs-2731	229	19	there	there	PRON
iajs-2731	229	20	exists	exist	VERB
iajs-2731	229	21	a	a	DET
iajs-2731	229	22	supplement	supplement	NOUN
iajs-2731	229	23	xt	xt	ADP
iajs-2731	229	24	of	of	ADP
iajs-2731	229	25	a	a	PRON
iajs-2731	229	26	in	in	ADP
iajs-2731	229	27	xt	xt	ADP
iajs-2731	229	28	such	such	ADJ
iajs-2731	229	29	that	that	PRON
iajs-2731	229	30	xt	xt	PROPN
iajs-2731	230	1	⊆bt	⊆bt	INTJ
iajs-2731	230	2	,	,	PUNCT
iajs-2731	230	3	∀	∀	NOUN
iajs-2731	230	4	t	t	NOUN
iajs-2731	230	5	∈	∈	PROPN
iajs-2731	230	6	(	(	PUNCT
iajs-2731	230	7	0	0	NUM
iajs-2731	230	8	,	,	PUNCT
iajs-2731	230	9	1	1	NUM
iajs-2731	230	10	]	]	PUNCT
iajs-2731	230	11	.	.	PUNCT
iajs-2731	231	1	then	then	ADV
iajs-2731	231	2	x	x	X
iajs-2731	231	3	=	=	NOUN
iajs-2731	231	4	z6	z6	PROPN
iajs-2731	231	5	amply	amply	ADV
iajs-2731	231	6	supplemented	supplement	VERB
iajs-2731	231	7	module	module	NOUN
iajs-2731	231	8	.	.	PUNCT
iajs-2731	232	1	but	but	CCONJ
iajs-2731	232	2	z6	z6	PROPN
iajs-2731	232	3	is	be	AUX
iajs-2731	232	4	not	not	PART
iajs-2731	232	5	lochollow	lochollow	ADJ
iajs-2731	232	6	y	y	PROPN
iajs-2731	232	7	module	module	NOUN
iajs-2731	232	8	.	.	PUNCT
iajs-2731	233	1	since	since	SCONJ
iajs-2731	233	2	z6	z6	PROPN
iajs-2731	233	3	has	have	VERB
iajs-2731	233	4	no	no	DET
iajs-2731	233	5	unique	unique	ADJ
iajs-2731	233	6	maximal	maximal	ADJ
iajs-2731	233	7	submodule	submodule	NOUN
iajs-2731	233	8	.	.	PUNCT
iajs-2731	234	1	thus	thus	ADV
iajs-2731	234	2	x	x	PRON
iajs-2731	234	3	is	be	AUX
iajs-2731	234	4	supplemented	supplement	VERB
iajs-2731	234	5	fuzzy	fuzzy	ADJ
iajs-2731	234	6	module	module	NOUN
iajs-2731	234	7	but	but	CCONJ
iajs-2731	234	8	not	not	PART
iajs-2731	234	9	loc	loc	NOUN
iajs-2731	234	10	hollow	hollow	ADJ
iajs-2731	234	11	fuzzy	fuzzy	ADJ
iajs-2731	234	12	module	module	NOUN
iajs-2731	234	13	4.8	4.8	NUM
iajs-2731	234	14	proposition	proposition	NOUN
iajs-2731	234	15	every	every	DET
iajs-2731	234	16	lochollow	lochollow	NOUN
iajs-2731	234	17	fmodule	fmodule	NOUN
iajs-2731	234	18	is	be	AUX
iajs-2731	234	19	an	an	DET
iajs-2731	234	20	indecomposable	indecomposable	ADJ
iajs-2731	234	21	fmodule	fmodule	NOUN
iajs-2731	234	22	.	.	PUNCT
iajs-2731	235	1	proof	proof	NOUN
iajs-2731	235	2	:	:	PUNCT
iajs-2731	235	3	suppose	suppose	VERB
iajs-2731	235	4	that	that	SCONJ
iajs-2731	235	5	𝜒	𝜒	PRON
iajs-2731	235	6	be	be	AUX
iajs-2731	235	7	a	a	DET
iajs-2731	235	8	lochollow	lochollow	NOUN
iajs-2731	235	9	fmodule	fmodule	ADV
iajs-2731	235	10	.	.	PUNCT
iajs-2731	236	1	∃	∃	PROPN
iajs-2731	236	2	a	a	DET
iajs-2731	236	3	unique	unique	ADJ
iajs-2731	236	4	maximal	maximal	ADJ
iajs-2731	236	5	fuzzy	fuzzy	ADJ
iajs-2731	236	6	submodule	submodule	NOUN
iajs-2731	236	7	say	say	VERB
iajs-2731	236	8	a	a	DET
iajs-2731	236	9	which	which	PRON
iajs-2731	236	10	contains	contain	VERB
iajs-2731	236	11	all	all	DET
iajs-2731	236	12	fuzzy	fuzzy	ADJ
iajs-2731	236	13	small	small	ADJ
iajs-2731	236	14	submodule	submodule	NOUN
iajs-2731	236	15	of	of	ADP
iajs-2731	236	16	x	x	PRON
iajs-2731	236	17	,	,	PUNCT
iajs-2731	236	18	let	let	VERB
iajs-2731	236	19	x	x	PRON
iajs-2731	236	20	is	be	AUX
iajs-2731	236	21	decomposable	decomposable	ADJ
iajs-2731	236	22	,	,	PUNCT
iajs-2731	236	23	then	then	ADV
iajs-2731	236	24	there	there	PRON
iajs-2731	236	25	exists	exist	VERB
iajs-2731	236	26	c	c	NOUN
iajs-2731	236	27	,	,	PUNCT
iajs-2731	236	28	b	b	NOUN
iajs-2731	236	29	are	be	AUX
iajs-2731	236	30	a	a	DET
iajs-2731	236	31	proper	proper	ADJ
iajs-2731	236	32	fuzzy	fuzzy	ADJ
iajs-2731	236	33	submodules	submodule	NOUN
iajs-2731	236	34	of	of	ADP
iajs-2731	236	35	x	x	X
iajs-2731	236	36	and	and	CCONJ
iajs-2731	236	37	a	a	PRON
iajs-2731	236	38	,	,	PUNCT
iajs-2731	236	39	b≠	b≠	PROPN
iajs-2731	236	40	01	01	NUM
iajs-2731	236	41	,	,	PUNCT
iajs-2731	236	42	such	such	ADJ
iajs-2731	236	43	that	that	SCONJ
iajs-2731	236	44	c	c	NOUN
iajs-2731	236	45	,	,	PUNCT
iajs-2731	236	46	𝑩	𝑩	PROPN
iajs-2731	236	47	are	be	AUX
iajs-2731	236	48	ƒsubms	ƒsubms	ADJ
iajs-2731	236	49	of	of	ADP
iajs-2731	236	50	a	a	DET
iajs-2731	236	51	and	and	CCONJ
iajs-2731	236	52	𝜒=	𝜒=	NOUN
iajs-2731	236	53	c	c	PROPN
iajs-2731	236	54	⊕	⊕	PROPN
iajs-2731	236	55	𝑩	𝑩	PROPN
iajs-2731	236	56	hence	hence	ADV
iajs-2731	236	57	(	(	PUNCT
iajs-2731	236	58	x)t=	x)t=	PROPN
iajs-2731	236	59	ct	ct	PROPN
iajs-2731	236	60	⊕	⊕	PROPN
iajs-2731	236	61	bt	bt	PROPN
iajs-2731	236	62	,	,	PUNCT
iajs-2731	237	1	∀	∀	X
iajs-2731	237	2	t	t	NOUN
iajs-2731	237	3	∈	∈	PROPN
iajs-2731	237	4	(	(	PUNCT
iajs-2731	237	5	0	0	NUM
iajs-2731	237	6	,	,	PUNCT
iajs-2731	237	7	1],but	1],but	NUM
iajs-2731	237	8	x	x	X
iajs-2731	237	9	is	be	AUX
iajs-2731	237	10	hollow	hollow	ADJ
iajs-2731	237	11	then	then	ADV
iajs-2731	237	12	either	either	CCONJ
iajs-2731	237	13	b	b	X
iajs-2731	237	14	is	be	AUX
iajs-2731	237	15	a	a	DET
iajs-2731	237	16	small	small	ADJ
iajs-2731	237	17	fuzzy	fuzzy	NOUN
iajs-2731	237	18	of	of	ADP
iajs-2731	237	19	x	x	PUNCT
iajs-2731	237	20	with	with	ADP
iajs-2731	237	21	b	b	PROPN
iajs-2731	237	22	is	be	AUX
iajs-2731	237	23	fsubm	fsubm	NOUN
iajs-2731	237	24	of	of	ADP
iajs-2731	237	25	a	a	DET
iajs-2731	237	26	implies	implie	NOUN
iajs-2731	237	27	that	that	SCONJ
iajs-2731	237	28	x	x	NOUN
iajs-2731	237	29	=	=	NOUN
iajs-2731	237	30	c	c	NOUN
iajs-2731	237	31	or	or	CCONJ
iajs-2731	237	32	c	c	PROPN
iajs-2731	237	33	is	be	AUX
iajs-2731	237	34	a	a	DET
iajs-2731	237	35	small	small	ADJ
iajs-2731	237	36	fuzzy	fuzzy	NOUN
iajs-2731	237	37	of	of	ADP
iajs-2731	237	38	x	x	PUNCT
iajs-2731	237	39	with	with	ADP
iajs-2731	237	40	c	c	PROPN
iajs-2731	237	41	is	be	AUX
iajs-2731	237	42	fsubmodule	fsubmodule	NOUN
iajs-2731	237	43	of	of	ADP
iajs-2731	237	44	a	a	DET
iajs-2731	237	45	implies	implie	NOUN
iajs-2731	237	46	that	that	SCONJ
iajs-2731	237	47	x	x	PROPN
iajs-2731	237	48	=	=	SYM
iajs-2731	237	49	b	b	PROPN
iajs-2731	237	50	,	,	PUNCT
iajs-2731	237	51	which	which	PRON
iajs-2731	237	52	contradiction	contradiction	NOUN
iajs-2731	237	53	.	.	PUNCT
iajs-2731	238	1	then	then	ADV
iajs-2731	238	2	x	x	PRON
iajs-2731	238	3	is	be	AUX
iajs-2731	238	4	indecomposable	indecomposable	ADJ
iajs-2731	238	5	module	module	NOUN
iajs-2731	238	6	.	.	PUNCT
iajs-2731	239	1	4.9	4.9	NUM
iajs-2731	239	2	remark	remark	NOUN
iajs-2731	239	3	:	:	PUNCT
iajs-2731	239	4	the	the	DET
iajs-2731	239	5	convers	conver	NOUN
iajs-2731	239	6	of	of	ADP
iajs-2731	239	7	proposition	proposition	NOUN
iajs-2731	239	8	(	(	PUNCT
iajs-2731	239	9	4.8	4.8	NUM
iajs-2731	239	10	)	)	PUNCT
iajs-2731	239	11	it	it	PRON
iajs-2731	239	12	is	be	AUX
iajs-2731	239	13	not	not	PART
iajs-2731	239	14	always	always	ADV
iajs-2731	239	15	true	true	ADJ
iajs-2731	239	16	,	,	PUNCT
iajs-2731	239	17	as	as	ADV
iajs-2731	239	18	well	well	ADV
iajs-2731	239	19	as	as	ADP
iajs-2731	239	20	,	,	PUNCT
iajs-2731	239	21	in	in	ADP
iajs-2731	239	22	the	the	DET
iajs-2731	239	23	next	next	ADJ
iajs-2731	239	24	example	example	NOUN
iajs-2731	239	25	.	.	PUNCT
iajs-2731	240	1	4.10	4.10	NUM
iajs-2731	240	2	example	example	NOUN
iajs-2731	240	3	let	let	VERB
iajs-2731	240	4	m	m	NOUN
iajs-2731	240	5	=	=	ADJ
iajs-2731	240	6	z	z	NOUN
iajs-2731	240	7	–	–	PUNCT
iajs-2731	240	8	module	module	NOUN
iajs-2731	240	9	z	z	NOUN
iajs-2731	240	10	,	,	PUNCT
iajs-2731	240	11	and	and	CCONJ
iajs-2731	240	12	a	a	DET
iajs-2731	240	13	=	=	X
iajs-2731	240	14	5z	5z	NOUN
iajs-2731	240	15	x	x	NOUN
iajs-2731	240	16	:	:	PUNCT
iajs-2731	240	17	m	m	VERB
iajs-2731	240	18	→	→	SYM
iajs-2731	241	1	[	[	X
iajs-2731	241	2	0	0	NUM
iajs-2731	241	3	,	,	PUNCT
iajs-2731	241	4	1	1	NUM
iajs-2731	241	5	]	]	X
iajs-2731	241	6	s.t	s.t	PROPN
iajs-2731	241	7	x	x	SYM
iajs-2731	241	8	(	(	PUNCT
iajs-2731	241	9	x	x	NOUN
iajs-2731	241	10	)	)	PUNCT
iajs-2731	241	11	=	=	SYM
iajs-2731	241	12	{	{	PUNCT
iajs-2731	241	13	𝑡	𝑡	X
iajs-2731	241	14	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	241	15	𝑥	𝑥	X
iajs-2731	241	16	∈	∈	PROPN
iajs-2731	241	17	m	m	VERB
iajs-2731	241	18	0	0	NUM
iajs-2731	241	19	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	241	20	,	,	PUNCT
iajs-2731	241	21	∀	∀	X
iajs-2731	241	22	t	t	NOUN
iajs-2731	241	23	∈	∈	PROPN
iajs-2731	241	24	(	(	PUNCT
iajs-2731	241	25	0	0	NUM
iajs-2731	241	26	,	,	PUNCT
iajs-2731	241	27	1	1	NUM
iajs-2731	241	28	]	]	PUNCT
iajs-2731	241	29	.	.	PUNCT
iajs-2731	242	1	clear	clear	ADJ
iajs-2731	242	2	that	that	SCONJ
iajs-2731	242	3	xt	xt	PUNCT
iajs-2731	243	1	=	=	NOUN
iajs-2731	243	2	m	m	NOUN
iajs-2731	243	3	and	and	CCONJ
iajs-2731	243	4	m	m	VERB
iajs-2731	243	5	is	be	AUX
iajs-2731	243	6	indecomposable	indecomposable	ADJ
iajs-2731	243	7	module	module	NOUN
iajs-2731	243	8	but	but	CCONJ
iajs-2731	243	9	lochollow	lochollow	NOUN
iajs-2731	243	10	module	module	NOUN
iajs-2731	243	11	[	[	PUNCT
iajs-2731	243	12	5	5	NUM
iajs-2731	243	13	]	]	PUNCT
iajs-2731	243	14	.	.	PUNCT
iajs-2731	244	1	therefore	therefore	ADV
iajs-2731	244	2	x	x	X
iajs-2731	244	3	is	be	AUX
iajs-2731	244	4	indecomposable	indecomposable	ADJ
iajs-2731	244	5	module	module	NOUN
iajs-2731	244	6	but	but	CCONJ
iajs-2731	244	7	not	not	PART
iajs-2731	244	8	lochollow	lochollow	VERB
iajs-2731	244	9	fuzzy	fuzzy	ADJ
iajs-2731	244	10	module	module	NOUN
iajs-2731	244	11	.	.	PUNCT
iajs-2731	245	1	ibn	ibn	PROPN
iajs-2731	245	2	al	al	PROPN
iajs-2731	245	3	-	-	PUNCT
iajs-2731	245	4	haitham	haitham	PROPN
iajs-2731	245	5	jour	jour	X
iajs-2731	245	6	.	.	PROPN
iajs-2731	245	7	for	for	ADP
iajs-2731	245	8	pure	pure	ADJ
iajs-2731	245	9	&	&	CCONJ
iajs-2731	245	10	appl	appl	PROPN
iajs-2731	245	11	.	.	PUNCT
iajs-2731	246	1	sci	sci	PROPN
iajs-2731	246	2	.	.	PROPN
iajs-2731	247	1	53	53	NUM
iajs-2731	247	2	(	(	PUNCT
iajs-2731	247	3	2)2022	2)2022	NOUN
iajs-2731	247	4	94	94	NUM
iajs-2731	247	5	recall	recall	NOUN
iajs-2731	247	6	that	that	SCONJ
iajs-2731	247	7	if	if	SCONJ
iajs-2731	247	8	x	x	PRON
iajs-2731	247	9	finitely	finitely	ADV
iajs-2731	247	10	generated	generate	VERB
iajs-2731	247	11	fm	fm	PROPN
iajs-2731	247	12	odule	odule	PROPN
iajs-2731	247	13	then	then	ADV
iajs-2731	247	14	x	x	X
iajs-2731	247	15	/	/	SYM
iajs-2731	247	16	c	c	PROPN
iajs-2731	247	17	is	be	AUX
iajs-2731	247	18	finitely	finitely	ADV
iajs-2731	247	19	generated	generate	VERB
iajs-2731	247	20	fmodule	fmodule	NOUN
iajs-2731	247	21	for	for	ADP
iajs-2731	247	22	every	every	DET
iajs-2731	247	23	c	c	NOUN
iajs-2731	247	24	a	a	DET
iajs-2731	247	25	fsubm	fsubm	NOUN
iajs-2731	247	26	x.	x.	NOUN
iajs-2731	247	27	but	but	CCONJ
iajs-2731	247	28	the	the	DET
iajs-2731	247	29	converse	converse	NOUN
iajs-2731	247	30	is	be	AUX
iajs-2731	247	31	not	not	PART
iajs-2731	247	32	true	true	ADJ
iajs-2731	247	33	.	.	PUNCT
iajs-2731	248	1	the	the	DET
iajs-2731	248	2	following	follow	VERB
iajs-2731	248	3	proposition	proposition	NOUN
iajs-2731	248	4	shows	show	VERB
iajs-2731	248	5	if	if	SCONJ
iajs-2731	248	6	x	x	PRON
iajs-2731	248	7	is	be	AUX
iajs-2731	248	8	a	a	DET
iajs-2731	248	9	hollow	hollow	ADJ
iajs-2731	248	10	f	f	NOUN
iajs-2731	248	11	module	module	NOUN
iajs-2731	248	12	and	and	CCONJ
iajs-2731	248	13	x	x	SYM
iajs-2731	248	14	/	/	SYM
iajs-2731	248	15	c	c	VERB
iajs-2731	248	16	a	a	DET
iajs-2731	248	17	finitely	finitely	ADV
iajs-2731	248	18	generated	generate	VERB
iajs-2731	248	19	fmodule	fmodule	NOUN
iajs-2731	248	20	.	.	PUNCT
iajs-2731	249	1	then	then	ADV
iajs-2731	249	2	x	x	PUNCT
iajs-2731	249	3	also	also	ADV
iajs-2731	249	4	a	a	DET
iajs-2731	249	5	finitely	finitely	ADV
iajs-2731	249	6	generated	generate	VERB
iajs-2731	249	7	fmodule	fmodule	NOUN
iajs-2731	249	8	.	.	PUNCT
iajs-2731	250	1	4.11	4.11	NUM
iajs-2731	250	2	proposition	proposition	NOUN
iajs-2731	250	3	every	every	DET
iajs-2731	250	4	lochollow	lochollow	NOUN
iajs-2731	250	5	fmodule	fmodule	NOUN
iajs-2731	250	6	is	be	AUX
iajs-2731	250	7	lifting	lift	VERB
iajs-2731	250	8	fmodule	fmodule	ADV
iajs-2731	250	9	.	.	PUNCT
iajs-2731	251	1	𝝆roof	𝝆roof	PRON
iajs-2731	251	2	:	:	PUNCT
iajs-2731	251	3	let	let	VERB
iajs-2731	251	4	x	x	PRON
iajs-2731	251	5	be	be	AUX
iajs-2731	251	6	a	a	DET
iajs-2731	251	7	lochollow	lochollow	ADJ
iajs-2731	251	8	fuzzy	fuzzy	ADJ
iajs-2731	251	9	module	module	NOUN
iajs-2731	251	10	.	.	PUNCT
iajs-2731	252	1	∃	∃	PROPN
iajs-2731	252	2	a	a	DET
iajs-2731	252	3	unique	unique	ADJ
iajs-2731	252	4	maximal	maximal	ADJ
iajs-2731	252	5	fuzzy	fuzzy	ADJ
iajs-2731	252	6	submodule	submodule	NOUN
iajs-2731	252	7	says	say	VERB
iajs-2731	252	8	a	a	DET
iajs-2731	252	9	which	which	PRON
iajs-2731	252	10	contains	contain	VERB
iajs-2731	252	11	all	all	DET
iajs-2731	252	12	fuzzy	fuzzy	ADJ
iajs-2731	252	13	small	small	ADJ
iajs-2731	252	14	submodule	submodule	NOUN
iajs-2731	252	15	of	of	ADP
iajs-2731	252	16	x.	x.	PROPN
iajs-2731	253	1	so	so	ADV
iajs-2731	253	2	x=	x=	PROPN
iajs-2731	254	1	x	x	PUNCT
iajs-2731	254	2	⊕	⊕	NOUN
iajs-2731	254	3	{	{	PUNCT
iajs-2731	254	4	01	01	NUM
iajs-2731	254	5	}	}	PUNCT
iajs-2731	254	6	where	where	SCONJ
iajs-2731	254	7	{	{	PUNCT
iajs-2731	254	8	01	01	NUM
iajs-2731	254	9	}	}	PUNCT
iajs-2731	254	10	is	be	AUX
iajs-2731	254	11	a	a	DET
iajs-2731	254	12	fuzzy	fuzzy	ADJ
iajs-2731	254	13	submodule	submodule	NOUN
iajs-2731	254	14	of	of	ADP
iajs-2731	254	15	a	a	PRON
iajs-2731	254	16	,	,	PUNCT
iajs-2731	254	17	𝑨	𝑨	PROPN
iajs-2731	254	18	∩	∩	NOUN
iajs-2731	254	19	𝜒=	𝜒=	NOUN
iajs-2731	254	20	𝑨	𝑨	NOUN
iajs-2731	254	21	.	.	PUNCT
iajs-2731	255	1	but	but	CCONJ
iajs-2731	255	2	x	x	SYM
iajs-2731	255	3	lochollow	lochollow	NOUN
iajs-2731	255	4	fmodules	fmodule	VERB
iajs-2731	255	5	.	.	PUNCT
iajs-2731	256	1	then	then	ADV
iajs-2731	256	2	𝑨	𝑨	PROPN
iajs-2731	256	3	𝜒	𝜒	NOUN
iajs-2731	256	4	=	=	X
iajs-2731	256	5	𝑨	𝑨	NOUN
iajs-2731	256	6	is	be	AUX
iajs-2731	256	7	a	a	DET
iajs-2731	256	8	ƒuzzy	ƒuzzy	ADJ
iajs-2731	256	9	small	small	ADJ
iajs-2731	256	10	submodule	submodule	NOUN
iajs-2731	256	11	by	by	ADP
iajs-2731	256	12	[	[	X
iajs-2731	256	13	8	8	NUM
iajs-2731	256	14	]	]	PUNCT
iajs-2731	256	15	implies	imply	VERB
iajs-2731	256	16	that	that	SCONJ
iajs-2731	256	17	x	x	PRON
iajs-2731	256	18	is	be	AUX
iajs-2731	256	19	lifting	lift	VERB
iajs-2731	256	20	a	a	DET
iajs-2731	256	21	fuzzy	fuzzy	ADJ
iajs-2731	256	22	module	module	NOUN
iajs-2731	256	23	.	.	PUNCT
iajs-2731	257	1	4.12	4.12	NUM
iajs-2731	257	2	remark	remark	NOUN
iajs-2731	257	3	:	:	PUNCT
iajs-2731	257	4	the	the	DET
iajs-2731	257	5	convers	conver	NOUN
iajs-2731	257	6	of	of	ADP
iajs-2731	257	7	proposition	proposition	NOUN
iajs-2731	257	8	(	(	PUNCT
iajs-2731	257	9	4.11	4.11	NUM
iajs-2731	257	10	)	)	PUNCT
iajs-2731	257	11	it	it	PRON
iajs-2731	257	12	is	be	AUX
iajs-2731	257	13	not	not	PART
iajs-2731	257	14	satisfy	satisfy	ADJ
iajs-2731	257	15	we	we	PRON
iajs-2731	257	16	can	can	AUX
iajs-2731	257	17	show	show	VERB
iajs-2731	257	18	that	that	SCONJ
iajs-2731	257	19	by	by	ADP
iajs-2731	257	20	the	the	DET
iajs-2731	257	21	following	follow	VERB
iajs-2731	257	22	4.13	4.13	NUM
iajs-2731	257	23	example	example	NOUN
iajs-2731	257	24	:	:	PUNCT
iajs-2731	257	25	let	let	VERB
iajs-2731	257	26	x	x	PUNCT
iajs-2731	257	27	=	=	NOUN
iajs-2731	257	28	z10	z10	NOUN
iajs-2731	257	29	,	,	PUNCT
iajs-2731	257	30	define	define	VERB
iajs-2731	257	31	x	x	NOUN
iajs-2731	257	32	:	:	PUNCT
iajs-2731	257	33	m	m	VERB
iajs-2731	257	34	→	→	SYM
iajs-2731	257	35	[	[	X
iajs-2731	257	36	0	0	NUM
iajs-2731	257	37	,	,	PUNCT
iajs-2731	257	38	1	1	NUM
iajs-2731	257	39	]	]	PUNCT
iajs-2731	257	40	define	define	NOUN
iajs-2731	257	41	by	by	ADP
iajs-2731	257	42	:	:	PUNCT
iajs-2731	257	43	𝑨	𝑨	PROPN
iajs-2731	257	44	:	:	PUNCT
iajs-2731	257	45	m	m	VERB
iajs-2731	257	46	→	→	SYM
iajs-2731	257	47	[	[	X
iajs-2731	257	48	0	0	NUM
iajs-2731	257	49	,	,	PUNCT
iajs-2731	257	50	1	1	NUM
iajs-2731	257	51	]	]	PUNCT
iajs-2731	257	52	,	,	PUNCT
iajs-2731	257	53	s.t	s.t	PROPN
iajs-2731	257	54	a	a	PROPN
iajs-2731	257	55	(	(	PUNCT
iajs-2731	257	56	t	t	NOUN
iajs-2731	257	57	)	)	PUNCT
iajs-2731	257	58	=	=	PRON
iajs-2731	257	59	{	{	PUNCT
iajs-2731	257	60	𝑡	𝑡	X
iajs-2731	257	61	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	257	62	𝑥	𝑥	PRON
iajs-2731	257	63	∈	∈	NOUN
iajs-2731	257	64	𝑁	𝑁	PROPN
iajs-2731	257	65	0.5	0.5	NUM
iajs-2731	257	66	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	257	67	∀	∀	X
iajs-2731	257	68	t	t	NOUN
iajs-2731	257	69	∈	∈	PROPN
iajs-2731	257	70	(	(	PUNCT
iajs-2731	257	71	0	0	NUM
iajs-2731	257	72	,	,	PUNCT
iajs-2731	257	73	1	1	NUM
iajs-2731	257	74	]	]	PUNCT
iajs-2731	257	75	,	,	PUNCT
iajs-2731	257	76	n=	n=	ADV
iajs-2731	257	77	⟨0̅⟩	⟨0̅⟩	PROPN
iajs-2731	257	78	and	and	CCONJ
iajs-2731	257	79	let	let	VERB
iajs-2731	257	80	b	b	X
iajs-2731	257	81	:	:	PUNCT
iajs-2731	257	82	m	m	VERB
iajs-2731	257	83	→	→	SYM
iajs-2731	257	84	[	[	X
iajs-2731	257	85	0	0	NUM
iajs-2731	257	86	,	,	PUNCT
iajs-2731	257	87	1	1	NUM
iajs-2731	257	88	]	]	PUNCT
iajs-2731	257	89	,	,	PUNCT
iajs-2731	257	90	define	define	VERB
iajs-2731	257	91	by	by	ADP
iajs-2731	257	92	b	b	PROPN
iajs-2731	257	93	(	(	PUNCT
iajs-2731	257	94	t	t	PROPN
iajs-2731	257	95	)	)	PUNCT
iajs-2731	257	96	=	=	PRON
iajs-2731	257	97	{	{	PUNCT
iajs-2731	257	98	𝑡	𝑡	X
iajs-2731	257	99	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	257	100	𝑥	𝑥	DET
iajs-2731	257	101	∈	∈	PROPN
iajs-2731	257	102	𝐿	𝐿	PROPN
iajs-2731	257	103	0.3	0.3	NUM
iajs-2731	257	104	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	257	105	∀	∀	X
iajs-2731	257	106	t	t	NOUN
iajs-2731	257	107	∈	∈	PROPN
iajs-2731	257	108	(	(	PUNCT
iajs-2731	257	109	0	0	NUM
iajs-2731	257	110	,	,	PUNCT
iajs-2731	257	111	1	1	NUM
iajs-2731	257	112	]	]	PUNCT
iajs-2731	257	113	,	,	PUNCT
iajs-2731	257	114	l=	l=	ADJ
iajs-2731	257	115	2z	2z	NUM
iajs-2731	257	116	c	c	X
iajs-2731	257	117	(	(	PUNCT
iajs-2731	257	118	t	t	NOUN
iajs-2731	257	119	)	)	PUNCT
iajs-2731	257	120	=	=	PRON
iajs-2731	257	121	{	{	PUNCT
iajs-2731	257	122	𝑡	𝑡	X
iajs-2731	257	123	𝑖𝑓	𝑖𝑓	ADP
iajs-2731	257	124	𝑥	𝑥	PRON
iajs-2731	257	125	∈	∈	PROPN
iajs-2731	257	126	𝐾	𝐾	NOUN
iajs-2731	257	127	0.25	0.25	NUM
iajs-2731	257	128	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2731	257	129	∀	∀	X
iajs-2731	257	130	t	t	NOUN
iajs-2731	257	131	∈	∈	PROPN
iajs-2731	257	132	(	(	PUNCT
iajs-2731	257	133	0	0	NUM
iajs-2731	257	134	,	,	PUNCT
iajs-2731	257	135	1	1	NUM
iajs-2731	257	136	]	]	PUNCT
iajs-2731	257	137	,	,	PUNCT
iajs-2731	257	138	k=	k=	X
iajs-2731	257	139	5z	5z	NOUN
iajs-2731	257	140	clear	clear	ADJ
iajs-2731	257	141	that	that	SCONJ
iajs-2731	257	142	xt	xt	PUNCT
iajs-2731	258	1	=	=	SYM
iajs-2731	258	2	m	m	PROPN
iajs-2731	258	3	=	=	NOUN
iajs-2731	258	4	z10	z10	NOUN
iajs-2731	258	5	is	be	AUX
iajs-2731	258	6	fuzzy	fuzzy	ADJ
iajs-2731	258	7	module	module	NOUN
iajs-2731	258	8	,	,	PUNCT
iajs-2731	258	9	at	at	SCONJ
iajs-2731	258	10	=	=	SYM
iajs-2731	258	11	𝑁	𝑁	PROPN
iajs-2731	258	12	,	,	PUNCT
iajs-2731	258	13	bt	bt	NOUN
iajs-2731	258	14	=	=	SYM
iajs-2731	258	15	l	l	PROPN
iajs-2731	258	16	and	and	CCONJ
iajs-2731	258	17	ct	ct	NOUN
iajs-2731	258	18	=	=	SYM
iajs-2731	258	19	𝐾	𝐾	PROPN
iajs-2731	258	20	are	be	AUX
iajs-2731	258	21	submodules	submodule	NOUN
iajs-2731	258	22	in	in	ADP
iajs-2731	258	23	xt	xt	PROPN
iajs-2731	259	1	.hence	.hence	PROPN
iajs-2731	259	2	(	(	PUNCT
iajs-2731	259	3	a	a	DET
iajs-2731	259	4	⨁	⨁	PROPN
iajs-2731	259	5	x)t=(x)t	x)t=(x)t	NUM
iajs-2731	259	6	is	be	AUX
iajs-2731	259	7	only	only	ADV
iajs-2731	259	8	direct	direct	ADJ
iajs-2731	259	9	summand	summand	NOUN
iajs-2731	259	10	of	of	ADP
iajs-2731	259	11	xt	xt	PROPN
iajs-2731	259	12	is	be	AUX
iajs-2731	259	13	fuzzy	fuzzy	ADJ
iajs-2731	259	14	submodule	submodule	NOUN
iajs-2731	259	15	of	of	ADP
iajs-2731	259	16	xt	xt	PROPN
iajs-2731	259	17	,	,	PUNCT
iajs-2731	259	18	and	and	CCONJ
iajs-2731	259	19	xt	xt	X
iajs-2731	259	20	∩	∩	NOUN
iajs-2731	259	21	at	at	ADP
iajs-2731	259	22	=	=	SYM
iajs-2731	259	23	01	01	NUM
iajs-2731	259	24	is	be	AUX
iajs-2731	259	25	a	a	DET
iajs-2731	259	26	small	small	ADJ
iajs-2731	259	27	submodule	submodule	NOUN
iajs-2731	259	28	in	in	ADP
iajs-2731	259	29	bt	bt	PROPN
iajs-2731	259	30	=	=	PROPN
iajs-2731	259	31	⟨2̅⟩	⟨2̅⟩	PROPN
iajs-2731	259	32	.	.	PUNCT
iajs-2731	260	1	but	but	CCONJ
iajs-2731	260	2	z10	z10	NOUN
iajs-2731	260	3	has	have	VERB
iajs-2731	260	4	no	no	DET
iajs-2731	260	5	unique	unique	ADJ
iajs-2731	260	6	maximal	maximal	ADJ
iajs-2731	260	7	fuzzy	fuzzy	ADJ
iajs-2731	260	8	submodule	submodule	NOUN
iajs-2731	260	9	.	.	PUNCT
iajs-2731	261	1	the	the	DET
iajs-2731	261	2	convers	conver	NOUN
iajs-2731	261	3	of	of	ADP
iajs-2731	261	4	proposition	proposition	NOUN
iajs-2731	261	5	(	(	PUNCT
iajs-2731	261	6	4.12	4.12	NUM
iajs-2731	261	7	)	)	PUNCT
iajs-2731	261	8	is	be	AUX
iajs-2731	261	9	achieved	achieve	VERB
iajs-2731	261	10	if	if	SCONJ
iajs-2731	261	11	the	the	DET
iajs-2731	261	12	following	follow	VERB
iajs-2731	261	13	condition	condition	NOUN
iajs-2731	261	14	is	be	AUX
iajs-2731	261	15	hold	hold	VERB
iajs-2731	261	16	4.13	4.13	NUM
iajs-2731	261	17	proposition	proposition	NOUN
iajs-2731	261	18	:	:	PUNCT
iajs-2731	261	19	let	let	VERB
iajs-2731	261	20	x	x	PRON
iajs-2731	261	21	be	be	AUX
iajs-2731	261	22	a	a	DET
iajs-2731	261	23	cyclic	cyclic	ADJ
iajs-2731	261	24	fuzzy	fuzzy	ADJ
iajs-2731	261	25	indecomposable	indecomposable	ADJ
iajs-2731	261	26	module	module	NOUN
iajs-2731	261	27	of	of	ADP
iajs-2731	261	28	an	an	DET
iajs-2731	261	29	r	r	NOUN
iajs-2731	261	30	-	-	PUNCT
iajs-2731	261	31	module	module	NOUN
iajs-2731	261	32	m	m	NOUN
iajs-2731	261	33	.if	.if	PUNCT
iajs-2731	262	1	x	x	PRON
iajs-2731	262	2	is	be	AUX
iajs-2731	262	3	lifting	lift	VERB
iajs-2731	262	4	fuzzy	fuzzy	ADJ
iajs-2731	262	5	module	module	NOUN
iajs-2731	262	6	,	,	PUNCT
iajs-2731	262	7	then	then	ADV
iajs-2731	262	8	x	x	PUNCT
iajs-2731	262	9	is	be	AUX
iajs-2731	262	10	lochollow	lochollow	ADJ
iajs-2731	262	11	fuzzy	fuzzy	ADJ
iajs-2731	262	12	module	module	NOUN
iajs-2731	262	13	.	.	PUNCT
iajs-2731	263	1	proof	proof	NOUN
iajs-2731	263	2	:	:	PUNCT
iajs-2731	263	3	let	let	VERB
iajs-2731	263	4	a	a	PRON
iajs-2731	263	5	be	be	AUX
iajs-2731	263	6	a	a	DET
iajs-2731	263	7	proper	proper	ADJ
iajs-2731	263	8	fsubmodule	fsubmodule	NOUN
iajs-2731	263	9	of	of	ADP
iajs-2731	263	10	x	x	PRON
iajs-2731	263	11	,	,	PUNCT
iajs-2731	263	12	but	but	CCONJ
iajs-2731	263	13	𝜒	𝜒	NOUN
iajs-2731	263	14	is	be	AUX
iajs-2731	263	15	lifting	lift	VERB
iajs-2731	263	16	fmodule	fmodule	ADV
iajs-2731	263	17	.	.	PUNCT
iajs-2731	264	1	then	then	ADV
iajs-2731	264	2	𝜒	𝜒	X
iajs-2731	264	3	=	=	SYM
iajs-2731	264	4	𝑨+b	𝑨+b	PROPN
iajs-2731	264	5	,	,	PUNCT
iajs-2731	264	6	where	where	SCONJ
iajs-2731	264	7	b	b	NOUN
iajs-2731	264	8	is	be	AUX
iajs-2731	264	9	f	f	PROPN
iajs-2731	264	10	submodule	submodule	NOUN
iajs-2731	264	11	and	and	CCONJ
iajs-2731	264	12	a	a	DET
iajs-2731	264	13	∩	∩	ADJ
iajs-2731	264	14	b	b	NOUN
iajs-2731	264	15	is	be	AUX
iajs-2731	264	16	a	a	DET
iajs-2731	264	17	small	small	ADJ
iajs-2731	264	18	fsubmodule	fsubmodule	NOUN
iajs-2731	264	19	of	of	ADP
iajs-2731	264	20	a.	a.	NOUN
iajs-2731	264	21	but	but	CCONJ
iajs-2731	264	22	x	x	PRON
iajs-2731	264	23	is	be	AUX
iajs-2731	264	24	an	an	DET
iajs-2731	264	25	indecomposable	indecomposable	ADJ
iajs-2731	264	26	fuzzy	fuzzy	ADJ
iajs-2731	264	27	module	module	NOUN
iajs-2731	264	28	,	,	PUNCT
iajs-2731	264	29	implies	imply	VERB
iajs-2731	264	30	that	that	SCONJ
iajs-2731	265	1	b=01	b=01	NOUN
iajs-2731	265	2	and	and	CCONJ
iajs-2731	265	3	hence	hence	ADV
iajs-2731	265	4	at	at	ADP
iajs-2731	265	5	=	=	SYM
iajs-2731	265	6	xt	xt	PROPN
iajs-2731	265	7	,	,	PUNCT
iajs-2731	265	8	∀	∀	X
iajs-2731	265	9	t	t	NOUN
iajs-2731	265	10	∈	∈	PROPN
iajs-2731	265	11	(	(	PUNCT
iajs-2731	265	12	0	0	NUM
iajs-2731	265	13	,	,	PUNCT
iajs-2731	265	14	1	1	NUM
iajs-2731	265	15	]	]	PUNCT
iajs-2731	265	16	by	by	ADP
iajs-2731	265	17	[	[	X
iajs-2731	265	18	9	9	NUM
iajs-2731	265	19	]	]	PUNCT
iajs-2731	265	20	,	,	PUNCT
iajs-2731	265	21	since	since	SCONJ
iajs-2731	265	22	xt	xt	PROPN
iajs-2731	265	23	is	be	AUX
iajs-2731	265	24	a	a	DET
iajs-2731	265	25	hollow	hollow	ADJ
iajs-2731	265	26	fuzzy	fuzzy	ADJ
iajs-2731	265	27	module	module	NOUN
iajs-2731	265	28	and	and	CCONJ
iajs-2731	265	29	hence	hence	ADV
iajs-2731	265	30	a=	a=	VERB
iajs-2731	265	31	x.	x.	NOUN
iajs-2731	265	32	which	which	PRON
iajs-2731	265	33	implies	imply	VERB
iajs-2731	265	34	that	that	SCONJ
iajs-2731	265	35	a	a	DET
iajs-2731	265	36	∩	∩	NOUN
iajs-2731	265	37	x	x	X
iajs-2731	265	38	=	=	SYM
iajs-2731	265	39	a	a	NOUN
iajs-2731	265	40	,	,	PUNCT
iajs-2731	265	41	clear	clear	ADJ
iajs-2731	265	42	that	that	SCONJ
iajs-2731	265	43	we	we	PRON
iajs-2731	265	44	have	have	VERB
iajs-2731	265	45	at	at	ADP
iajs-2731	265	46	∩	∩	NOUN
iajs-2731	265	47	xt=	xt=	VERB
iajs-2731	265	48	at	at	ADP
iajs-2731	265	49	,	,	PUNCT
iajs-2731	265	50	∀	∀	X
iajs-2731	265	51	t	t	NOUN
iajs-2731	265	52	∈	∈	PROPN
iajs-2731	265	53	(	(	PUNCT
iajs-2731	265	54	0	0	NUM
iajs-2731	265	55	,	,	PUNCT
iajs-2731	265	56	1	1	NUM
iajs-2731	265	57	]	]	PUNCT
iajs-2731	265	58	.	.	PUNCT
iajs-2731	266	1	then	then	ADV
iajs-2731	266	2	a	a	PRON
iajs-2731	266	3	is	be	AUX
iajs-2731	266	4	a	a	DET
iajs-2731	266	5	small	small	ADJ
iajs-2731	266	6	fsubmodule	fsubmodule	NOUN
iajs-2731	266	7	of	of	ADP
iajs-2731	266	8	x.	x.	NOUN
iajs-2731	267	1	so	so	ADV
iajs-2731	267	2	x	x	VERB
iajs-2731	267	3	is	be	AUX
iajs-2731	267	4	hollow	hollow	ADJ
iajs-2731	267	5	fuzzy	fuzzy	ADJ
iajs-2731	267	6	module	module	NOUN
iajs-2731	267	7	and	and	CCONJ
iajs-2731	267	8	since	since	SCONJ
iajs-2731	267	9	x	x	PRON
iajs-2731	267	10	is	be	AUX
iajs-2731	267	11	cyclic	cyclic	ADJ
iajs-2731	267	12	fuzzy	fuzzy	ADJ
iajs-2731	267	13	module	module	NOUN
iajs-2731	267	14	.	.	PUNCT
iajs-2731	268	1	then	then	ADV
iajs-2731	268	2	x	x	PRON
iajs-2731	268	3	is	be	AUX
iajs-2731	268	4	lochollow	lochollow	ADJ
iajs-2731	268	5	fuzzy	fuzzy	ADJ
iajs-2731	268	6	module	module	NOUN
iajs-2731	268	7	through	through	ADP
iajs-2731	268	8	proposition	proposition	NOUN
iajs-2731	268	9	(	(	PUNCT
iajs-2731	268	10	3.1	3.1	NUM
iajs-2731	268	11	)	)	PUNCT
iajs-2731	268	12	.	.	PUNCT
iajs-2731	269	1	ibn	ibn	PROPN
iajs-2731	269	2	al	al	PROPN
iajs-2731	269	3	-	-	PUNCT
iajs-2731	269	4	haitham	haitham	PROPN
iajs-2731	269	5	jour	jour	X
iajs-2731	269	6	.	.	PROPN
iajs-2731	269	7	for	for	ADP
iajs-2731	269	8	pure	pure	ADJ
iajs-2731	269	9	&	&	CCONJ
iajs-2731	269	10	appl	appl	PROPN
iajs-2731	269	11	.	.	PUNCT
iajs-2731	270	1	sci	sci	PROPN
iajs-2731	270	2	.	.	PROPN
iajs-2731	271	1	53	53	NUM
iajs-2731	271	2	(	(	PUNCT
iajs-2731	271	3	2)2022	2)2022	VERB
iajs-2731	271	4	95	95	NUM
iajs-2731	271	5	4.15	4.15	NUM
iajs-2731	271	6	proposition	proposition	NOUN
iajs-2731	271	7	let	let	VERB
iajs-2731	271	8	c	c	NOUN
iajs-2731	271	9	be	be	AUX
iajs-2731	271	10	a	a	DET
iajs-2731	271	11	maximal	maximal	ADJ
iajs-2731	271	12	fsubmodule	fsubmodule	NOUN
iajs-2731	271	13	of	of	ADP
iajs-2731	271	14	a	a	DET
iajs-2731	271	15	fuzzy	fuzzy	ADJ
iajs-2731	271	16	module	module	NOUN
iajs-2731	271	17	x.	x.	NOUN
iajs-2731	272	1	if	if	SCONJ
iajs-2731	272	2	b	b	PROPN
iajs-2731	272	3	is	be	AUX
iajs-2731	272	4	a	a	DET
iajs-2731	272	5	fsupplement	fsupplement	NOUN
iajs-2731	272	6	of	of	ADP
iajs-2731	272	7	c	c	PROPN
iajs-2731	272	8	in	in	ADP
iajs-2731	272	9	x	x	SYM
iajs-2731	272	10	,	,	PUNCT
iajs-2731	272	11	therefore	therefore	ADV
iajs-2731	272	12	b	b	NOUN
iajs-2731	272	13	is	be	AUX
iajs-2731	272	14	a	a	DET
iajs-2731	272	15	lochollow	lochollow	ADJ
iajs-2731	272	16	fuzzy	fuzzy	ADJ
iajs-2731	272	17	module	module	NOUN
iajs-2731	272	18	.	.	PUNCT
iajs-2731	273	1	proof	proof	NOUN
iajs-2731	273	2	:	:	PUNCT
iajs-2731	273	3	let	let	VERB
iajs-2731	273	4	b	b	NOUN
iajs-2731	273	5	is	be	AUX
iajs-2731	273	6	fsupplement	fsupplement	NOUN
iajs-2731	273	7	of	of	ADP
iajs-2731	273	8	c	c	NOUN
iajs-2731	273	9	,	,	PUNCT
iajs-2731	273	10	let	let	VERB
iajs-2731	273	11	b1	b1	NOUN
iajs-2731	273	12	be	be	AUX
iajs-2731	273	13	a	a	DET
iajs-2731	273	14	proper	proper	ADJ
iajs-2731	273	15	fsubm	fsubm	NOUN
iajs-2731	273	16	of	of	ADP
iajs-2731	273	17	b	b	NOUN
iajs-2731	273	18	with	with	ADP
iajs-2731	273	19	b1+b2=	b1+b2=	NOUN
iajs-2731	273	20	b	b	X
iajs-2731	273	21	for	for	ADP
iajs-2731	273	22	some	some	DET
iajs-2731	273	23	f	f	NOUN
iajs-2731	273	24	-	-	PUNCT
iajs-2731	273	25	submodule	submodule	NOUN
iajs-2731	273	26	b2	b2	NOUN
iajs-2731	273	27	of	of	ADP
iajs-2731	273	28	b	b	PROPN
iajs-2731	273	29	and	and	CCONJ
iajs-2731	273	30	(	(	PUNCT
iajs-2731	273	31	b1+b2)t=	b1+b2)t=	ADJ
iajs-2731	273	32	(	(	PUNCT
iajs-2731	273	33	b)t	b)t	NOUN
iajs-2731	273	34	,	,	PUNCT
iajs-2731	274	1	∀	∀	X
iajs-2731	274	2	t	t	NOUN
iajs-2731	274	3	∈	∈	PROPN
iajs-2731	274	4	(	(	PUNCT
iajs-2731	274	5	0	0	NUM
iajs-2731	274	6	,	,	PUNCT
iajs-2731	274	7	1	1	NUM
iajs-2731	274	8	]	]	PUNCT
iajs-2731	274	9	.	.	PUNCT
iajs-2731	275	1	let	let	VERB
iajs-2731	275	2	c+b	c+b	X
iajs-2731	276	1	=	=	PUNCT
iajs-2731	277	1	x	x	PUNCT
iajs-2731	278	1	=	=	PRON
iajs-2731	279	1	c+	c+	NOUN
iajs-2731	279	2	b1+b2=	b1+b2=	NOUN
iajs-2731	279	3	x	x	X
iajs-2731	279	4	and	and	CCONJ
iajs-2731	279	5	b1	b1	NOUN
iajs-2731	279	6	is	be	AUX
iajs-2731	279	7	a	a	DET
iajs-2731	279	8	fsubmodule	fsubmodule	NOUN
iajs-2731	279	9	of	of	ADP
iajs-2731	279	10	c	c	NOUN
iajs-2731	279	11	,	,	PUNCT
iajs-2731	279	12	b1=	b1=	NOUN
iajs-2731	279	13	x	x	INTJ
iajs-2731	279	14	,	,	PUNCT
iajs-2731	279	15	by	by	ADP
iajs-2731	279	16	[	[	PUNCT
iajs-2731	279	17	1],also	1],also	NUM
iajs-2731	279	18	c	c	NOUN
iajs-2731	279	19	is	be	AUX
iajs-2731	279	20	a	a	DET
iajs-2731	279	21	maximal	maximal	ADJ
iajs-2731	279	22	fsubm	fsubm	NOUN
iajs-2731	279	23	of	of	ADP
iajs-2731	279	24	x	x	SYM
iajs-2731	279	25	therefore	therefore	ADV
iajs-2731	279	26	b1	b1	PROPN
iajs-2731	279	27	=	=	SYM
iajs-2731	279	28	b	b	PROPN
iajs-2731	279	29	,	,	PUNCT
iajs-2731	279	30	which	which	PRON
iajs-2731	279	31	contradiction	contradiction	NOUN
iajs-2731	279	32	with	with	ADP
iajs-2731	279	33	our	our	PRON
iajs-2731	279	34	assumption	assumption	NOUN
iajs-2731	279	35	.thus	.thus	PRON
iajs-2731	279	36	c	c	PROPN
iajs-2731	280	1	+	+	PROPN
iajs-2731	280	2	b	b	X
iajs-2731	280	3	=	=	NOUN
iajs-2731	280	4	x	x	X
iajs-2731	280	5	and	and	CCONJ
iajs-2731	280	6	since	since	SCONJ
iajs-2731	280	7	c	c	PROPN
iajs-2731	280	8	is	be	AUX
iajs-2731	280	9	a	a	DET
iajs-2731	280	10	maximal	maximal	ADJ
iajs-2731	280	11	fsubmodule	fsubmodule	NOUN
iajs-2731	280	12	of	of	ADP
iajs-2731	280	13	x	x	DET
iajs-2731	280	14	so	so	ADV
iajs-2731	280	15	we	we	PRON
iajs-2731	280	16	have	have	AUX
iajs-2731	280	17	b2	b2	NOUN
iajs-2731	280	18	=	=	NOUN
iajs-2731	280	19	b	b	NOUN
iajs-2731	280	20	implies	imply	VERB
iajs-2731	280	21	that	that	SCONJ
iajs-2731	280	22	c	c	PROPN
iajs-2731	280	23	is	be	AUX
iajs-2731	280	24	a	a	DET
iajs-2731	280	25	hollow	hollow	ADJ
iajs-2731	280	26	fuzzy	fuzzy	ADJ
iajs-2731	280	27	module	module	NOUN
iajs-2731	280	28	.	.	PUNCT
iajs-2731	281	1	to	to	PART
iajs-2731	281	2	prove	prove	VERB
iajs-2731	281	3	that	that	SCONJ
iajs-2731	281	4	c	c	PROPN
iajs-2731	281	5	is	be	AUX
iajs-2731	281	6	a	a	DET
iajs-2731	281	7	cyclic	cyclic	ADJ
iajs-2731	281	8	fuzzy	fuzzy	ADJ
iajs-2731	281	9	module	module	NOUN
iajs-2731	281	10	,	,	PUNCT
iajs-2731	281	11	let	let	VERB
iajs-2731	281	12	xt	xt	PUNCT
iajs-2731	281	13	⊆x	⊆x	NOUN
iajs-2731	281	14	and	and	CCONJ
iajs-2731	281	15	xt	xt	ADP
iajs-2731	281	16	⊆	⊆	NUM
iajs-2731	281	17	c	c	NOUN
iajs-2731	281	18	,	,	PUNCT
iajs-2731	281	19	so	so	SCONJ
iajs-2731	281	20	there	there	PRON
iajs-2731	281	21	exists	exist	VERB
iajs-2731	281	22	(	(	PUNCT
iajs-2731	281	23	xt	xt	X
iajs-2731	281	24	)	)	PUNCT
iajs-2731	281	25	a	a	DET
iajs-2731	281	26	small	small	ADJ
iajs-2731	281	27	submodule	submodule	NOUN
iajs-2731	281	28	of	of	ADP
iajs-2731	281	29	xt	xt	PROPN
iajs-2731	281	30	,	,	PUNCT
iajs-2731	281	31	∀	∀	X
iajs-2731	281	32	t∈	t∈	X
iajs-2731	281	33	(	(	PUNCT
iajs-2731	281	34	0,1	0,1	NUM
iajs-2731	281	35	]	]	PUNCT
iajs-2731	281	36	.	.	PUNCT
iajs-2731	282	1	rx	rx	VERB
iajs-2731	282	2	+	+	ADP
iajs-2731	282	3	c	c	NOUN
iajs-2731	282	4	=	=	NOUN
iajs-2731	282	5	x	x	NOUN
iajs-2731	282	6	and	and	CCONJ
iajs-2731	282	7	this	this	PRON
iajs-2731	282	8	implies	imply	VERB
iajs-2731	282	9	that	that	SCONJ
iajs-2731	282	10	rx=	rx=	PROPN
iajs-2731	282	11	c	c	X
iajs-2731	282	12	,	,	PUNCT
iajs-2731	282	13	where	where	SCONJ
iajs-2731	282	14	(	(	PUNCT
iajs-2731	282	15	rx)t	rx)t	NOUN
iajs-2731	282	16	=	=	SYM
iajs-2731	282	17	ct	ct	NUM
iajs-2731	282	18	∀	∀	NOUN
iajs-2731	282	19	t∈	t∈	X
iajs-2731	282	20	(	(	PUNCT
iajs-2731	282	21	0,1	0,1	NUM
iajs-2731	282	22	]	]	PUNCT
iajs-2731	282	23	through	through	ADP
iajs-2731	282	24	dependent	dependent	NOUN
iajs-2731	282	25	on	on	ADP
iajs-2731	282	26	minimality	minimality	NOUN
iajs-2731	282	27	of	of	ADP
iajs-2731	282	28	c	c	PROPN
iajs-2731	282	29	and	and	CCONJ
iajs-2731	282	30	by	by	ADP
iajs-2731	282	31	proposition	proposition	NOUN
iajs-2731	282	32	(	(	PUNCT
iajs-2731	282	33	3.1	3.1	NUM
iajs-2731	282	34	)	)	PUNCT
iajs-2731	282	35	.	.	PUNCT
iajs-2731	283	1	thus	thus	ADV
iajs-2731	283	2	b	b	X
iajs-2731	283	3	is	be	AUX
iajs-2731	283	4	lochollow	lochollow	ADJ
iajs-2731	283	5	fuzzy	fuzzy	ADJ
iajs-2731	283	6	module	module	NOUN
iajs-2731	283	7	.	.	PUNCT
iajs-2731	283	8	.	.	PUNCT
iajs-2731	284	1	conclusion	conclusion	NOUN
iajs-2731	284	2	:	:	PUNCT
iajs-2731	284	3	5	5	NUM
iajs-2731	284	4	in	in	ADP
iajs-2731	284	5	this	this	DET
iajs-2731	284	6	work	work	NOUN
iajs-2731	284	7	,	,	PUNCT
iajs-2731	284	8	we	we	PRON
iajs-2731	284	9	introduced	introduce	VERB
iajs-2731	284	10	a	a	DET
iajs-2731	284	11	concept	concept	NOUN
iajs-2731	284	12	of	of	ADP
iajs-2731	284	13	a	a	DET
iajs-2731	284	14	hollow	hollow	ADJ
iajs-2731	284	15	fmodule	fmodule	NOUN
iajs-2731	284	16	,	,	PUNCT
iajs-2731	284	17	where	where	SCONJ
iajs-2731	284	18	a	a	DET
iajs-2731	284	19	module	module	NOUN
iajs-2731	284	20	is	be	AUX
iajs-2731	284	21	said	say	VERB
iajs-2731	284	22	to	to	PART
iajs-2731	284	23	be	be	AUX
iajs-2731	284	24	a	a	DET
iajs-2731	284	25	hollow	hollow	ADJ
iajs-2731	284	26	fuzzy	fuzzy	NOUN
iajs-2731	284	27	when	when	SCONJ
iajs-2731	284	28	every	every	DET
iajs-2731	284	29	subm	subm	PROPN
iajs-2731	284	30	is	be	AUX
iajs-2731	284	31	a	a	DET
iajs-2731	284	32	small	small	ADJ
iajs-2731	284	33	f	f	PROPN
iajs-2731	284	34	-	-	PUNCT
iajs-2731	284	35	subm	subm	PROPN
iajs-2731	284	36	.	.	PUNCT
iajs-2731	285	1	also	also	ADV
iajs-2731	285	2	,	,	PUNCT
iajs-2731	285	3	fuzzify	fuzzify	VERB
iajs-2731	285	4	these	these	DET
iajs-2731	285	5	concepts	concept	NOUN
iajs-2731	285	6	l	l	ADJ
iajs-2731	285	7	-	-	ADJ
iajs-2731	285	8	hollow	hollow	ADJ
iajs-2731	285	9	module	module	NOUN
iajs-2731	285	10	to	to	PART
iajs-2731	285	11	lochollow	lochollow	NOUN
iajs-2731	285	12	fmodules	fmodule	NOUN
iajs-2731	285	13	.	.	PUNCT
iajs-2731	286	1	moreover	moreover	ADV
iajs-2731	286	2	,	,	PUNCT
iajs-2731	286	3	we	we	PRON
iajs-2731	286	4	generalized	generalize	VERB
iajs-2731	286	5	numerous	numerous	ADJ
iajs-2731	286	6	properties	property	NOUN
iajs-2731	286	7	of	of	ADP
iajs-2731	286	8	loc	loc	NOUN
iajs-2731	286	9	-	-	ADJ
iajs-2731	286	10	hollow	hollow	ADJ
iajs-2731	286	11	fmodules	fmodule	NOUN
iajs-2731	286	12	and	and	CCONJ
iajs-2731	286	13	got	get	VERB
iajs-2731	286	14	several	several	ADJ
iajs-2731	286	15	results	result	NOUN
iajs-2731	286	16	and	and	CCONJ
iajs-2731	286	17	fundamental	fundamental	ADJ
iajs-2731	286	18	properties	property	NOUN
iajs-2731	286	19	of	of	ADP
iajs-2731	286	20	l	l	NOUN
iajs-2731	286	21	-	-	ADJ
iajs-2731	286	22	hollow	hollow	ADJ
iajs-2731	286	23	fmodules	fmodule	NOUN
iajs-2731	286	24	which	which	PRON
iajs-2731	286	25	are	be	AUX
iajs-2731	286	26	argued	argue	VERB
iajs-2731	286	27	.	.	PUNCT
iajs-2731	287	1	includes	include	VERB
iajs-2731	287	2	the	the	DET
iajs-2731	287	3	relation	relation	NOUN
iajs-2731	287	4	between	between	ADP
iajs-2731	287	5	hollow	hollow	ADJ
iajs-2731	287	6	fmodules	fmodule	NOUN
iajs-2731	287	7	and	and	CCONJ
iajs-2731	287	8	loc	loc	NOUN
iajs-2731	287	9	-	-	ADJ
iajs-2731	287	10	hollow	hollow	ADJ
iajs-2731	287	11	f	f	NOUN
iajs-2731	287	12	-	-	PUNCT
iajs-2731	287	13	modules	module	NOUN
iajs-2731	287	14	.	.	PUNCT
iajs-2731	288	1	however	however	ADV
iajs-2731	288	2	,	,	PUNCT
iajs-2731	288	3	allocated	allocate	VERB
iajs-2731	288	4	the	the	DET
iajs-2731	288	5	relation	relation	NOUN
iajs-2731	288	6	between	between	ADP
iajs-2731	288	7	loc	loc	NOUN
iajs-2731	288	8	-	-	ADJ
iajs-2731	288	9	hollow	hollow	ADJ
iajs-2731	288	10	fmodules	fmodule	NOUN
iajs-2731	288	11	and	and	CCONJ
iajs-2731	288	12	different	different	ADJ
iajs-2731	288	13	modules	module	NOUN
iajs-2731	288	14	like	like	VERB
iajs-2731	288	15	amply	amply	ADV
iajs-2731	288	16	supplemented	supplement	VERB
iajs-2731	288	17	fmodules	fmodule	NOUN
iajs-2731	288	18	,	,	PUNCT
iajs-2731	288	19	indecomposable	indecomposable	ADJ
iajs-2731	288	20	fmodules	fmodule	NOUN
iajs-2731	288	21	,	,	PUNCT
iajs-2731	288	22	and	and	CCONJ
iajs-2731	288	23	lifting	lift	VERB
iajs-2731	288	24	fmodules	fmodule	NOUN
iajs-2731	288	25	.	.	PUNCT
iajs-2731	289	1	finally	finally	ADV
iajs-2731	289	2	,	,	PUNCT
iajs-2731	289	3	it	it	PRON
iajs-2731	289	4	is	be	AUX
iajs-2731	289	5	significant	significant	ADJ
iajs-2731	289	6	to	to	PART
iajs-2731	289	7	remark	remark	VERB
iajs-2731	289	8	that	that	SCONJ
iajs-2731	289	9	the	the	DET
iajs-2731	289	10	characterizations	characterization	NOUN
iajs-2731	289	11	and	and	CCONJ
iajs-2731	289	12	properties	property	NOUN
iajs-2731	289	13	that	that	PRON
iajs-2731	289	14	are	be	AUX
iajs-2731	289	15	fundamentally	fundamentally	ADV
iajs-2731	289	16	related	relate	VERB
iajs-2731	289	17	to	to	ADP
iajs-2731	289	18	these	these	DET
iajs-2731	289	19	concepts	concept	NOUN
iajs-2731	289	20	are	be	AUX
iajs-2731	289	21	introduced	introduce	VERB
iajs-2731	289	22	.	.	PUNCT
iajs-2731	290	1	references	reference	NOUN
iajs-2731	290	2	1	1	NUM
iajs-2731	290	3	.	.	PUNCT
iajs-2731	291	1	zadehi	zadehi	PROPN
iajs-2731	291	2	,	,	PUNCT
iajs-2731	291	3	l.a	l.a	PROPN
iajs-2731	291	4	.fuzzy	.fuzzy	PRON
iajs-2731	291	5	sets	set	VERB
iajs-2731	291	6	information	information	NOUN
iajs-2731	291	7	and	and	CCONJ
iajs-2731	291	8	control	control	NOUN
iajs-2731	291	9	.	.	PUNCT
iajs-2731	292	1	1965	1965	NUM
iajs-2731	292	2	,	,	PUNCT
iajs-2731	292	3	8	8	NUM
iajs-2731	292	4	,	,	PUNCT
iajs-2731	292	5	333	333	NUM
iajs-2731	292	6	-	-	SYM
iajs-2731	292	7	353	353	NUM
iajs-2731	292	8	.	.	NOUN
iajs-2731	293	1	2	2	NUM
iajs-2731	293	2	.	.	X
iajs-2731	293	3	zadehi	zadehi	PROPN
iajs-2731	293	4	,	,	PUNCT
iajs-2731	293	5	m.	m.	NOUN
iajs-2731	293	6	m.	m.	NOUN
iajs-2731	293	7	on	on	ADP
iajs-2731	293	8	lfuzzy	lfuzzy	ADJ
iajs-2731	293	9	residual	residual	ADJ
iajs-2731	293	10	quotient	quotient	NOUN
iajs-2731	293	11	modules	module	NOUN
iajs-2731	293	12	and	and	CCONJ
iajs-2731	293	13	p.	p.	NOUN
iajs-2731	293	14	primary	primary	ADJ
iajs-2731	293	15	submodules	submodule	NOUN
iajs-2731	293	16	,	,	PUNCT
iajs-2731	293	17	fuzzy	fuzzy	ADJ
iajs-2731	293	18	sets	set	NOUN
iajs-2731	293	19	and	and	CCONJ
iajs-2731	293	20	systems	system	NOUN
iajs-2731	293	21	.	.	PUNCT
iajs-2731	294	1	1992,5	1992,5	NUM
iajs-2731	294	2	,	,	PUNCT
iajs-2731	294	3	331	331	NUM
iajs-2731	294	4	-	-	SYM
iajs-2731	294	5	344	344	NUM
iajs-2731	294	6	.	.	PUNCT
iajs-2731	295	1	3	3	NUM
iajs-2731	295	2	.	.	X
iajs-2731	295	3	zadehi	zadehi	PROPN
iajs-2731	295	4	,	,	PUNCT
iajs-2731	295	5	m.m	m.m	PROPN
iajs-2731	295	6	.	.	PROPN
iajs-2731	295	7	a	a	DET
iajs-2731	295	8	characterization	characterization	NOUN
iajs-2731	295	9	of	of	ADP
iajs-2731	295	10	lfuzzy	lfuzzy	ADJ
iajs-2731	295	11	ideals	ideal	NOUN
iajs-2731	295	12	,	,	PUNCT
iajs-2731	295	13	f	f	X
iajs-2731	295	14	-	-	PUNCT
iajs-2731	295	15	sets	set	NOUN
iajs-2731	295	16	and	and	CCONJ
iajs-2731	295	17	systems	system	NOUN
iajs-2731	295	18	.	.	PUNCT
iajs-2731	296	1	1991	1991	NUM
iajs-2731	296	2	,	,	PUNCT
iajs-2731	296	3	44,147	44,147	NUM
iajs-2731	296	4	-160	-160	NOUN
iajs-2731	296	5	.	.	PROPN
iajs-2731	297	1	4	4	NUM
iajs-2731	297	2	.	.	X
iajs-2731	297	3	mashinchi	mashinchi	PROPN
iajs-2731	297	4	,	,	PUNCT
iajs-2731	297	5	m.	m.	NOUN
iajs-2731	297	6	;	;	PUNCT
iajs-2731	297	7	zadehi	zadehi	X
iajs-2731	297	8	,	,	PUNCT
iajs-2731	297	9	m.	m.	NOUN
iajs-2731	297	10	m.	m.	NOUN
iajs-2731	297	11	on	on	ADP
iajs-2731	297	12	l	l	ADJ
iajs-2731	297	13	-	-	ADJ
iajs-2731	297	14	fuzzy	fuzzy	ADJ
iajs-2731	297	15	primary	primary	ADJ
iajs-2731	297	16	submodule	submodule	NOUN
iajs-2731	297	17	,	,	PUNCT
iajs-2731	297	18	fsets	fset	NOUN
iajs-2731	297	19	and	and	CCONJ
iajs-2731	297	20	system	system	NOUN
iajs-2731	297	21	,	,	PUNCT
iajs-2731	297	22	1995,49	1995,49	PROPN
iajs-2731	297	23	,	,	PUNCT
iajs-2731	297	24	231236	231236	NUM
iajs-2731	297	25	5	5	NUM
iajs-2731	297	26	.	.	PUNCT
iajs-2731	298	1	martinez	martinez	PROPN
iajs-2731	298	2	,	,	PUNCT
iajs-2731	298	3	l.	l.	PROPN
iajs-2731	298	4	fuzzy	fuzzy	ADJ
iajs-2731	298	5	module	module	NOUN
iajs-2731	298	6	over	over	ADP
iajs-2731	298	7	fuzzy	fuzzy	ADJ
iajs-2731	298	8	rings	ring	NOUN
iajs-2731	298	9	in	in	ADP
iajs-2731	298	10	connection	connection	NOUN
iajs-2731	298	11	with	with	ADP
iajs-2731	298	12	fuzzy	fuzzy	ADJ
iajs-2731	298	13	ideals	ideal	NOUN
iajs-2731	298	14	of	of	ADP
iajs-2731	298	15	rings	ring	NOUN
iajs-2731	298	16	,	,	PUNCT
iajs-2731	298	17	fuzzysets	fuzzyset	NOUN
iajs-2731	298	18	and	and	CCONJ
iajs-2731	298	19	system	system	NOUN
iajs-2731	298	20	.	.	PUNCT
iajs-2731	299	1	1996,4	1996,4	NUM
iajs-2731	299	2	,	,	PUNCT
iajs-2731	299	3	843	843	NUM
iajs-2731	299	4	-	-	NOUN
iajs-2731	299	5	857	857	NUM
iajs-2731	299	6	.	.	PUNCT
iajs-2731	300	1	6	6	NUM
iajs-2731	300	2	.	.	PUNCT
iajs-2731	300	3	rabi	rabi	NOUN
iajs-2731	300	4	,	,	PUNCT
iajs-2731	300	5	h.	h.	PROPN
iajs-2731	300	6	j.	j.	PROPN
iajs-2731	300	7	prime	prime	PROPN
iajs-2731	300	8	fuzzy	fuzzy	ADJ
iajs-2731	300	9	submodules	submodule	NOUN
iajs-2731	300	10	and	and	CCONJ
iajs-2731	300	11	prime	prime	ADJ
iajs-2731	300	12	fuzzy	fuzzy	ADJ
iajs-2731	301	1	modules.2001	modules.2001	PROPN
iajs-2731	301	2	,	,	PUNCT
iajs-2731	301	3	m.sc	m.sc	PROPN
iajs-2731	301	4	.	.	PUNCT
iajs-2731	302	1	thesis	thesis	NOUN
iajs-2731	302	2	,	,	PUNCT
iajs-2731	302	3	university	university	NOUN
iajs-2731	302	4	of	of	ADP
iajs-2731	302	5	baghdad	baghdad	PROPN
iajs-2731	302	6	.	.	PUNCT
iajs-2731	303	1	7	7	X
iajs-2731	303	2	.	.	X
iajs-2731	303	3	mayson	mayson	PROPN
iajs-2731	303	4	,	,	PUNCT
iajs-2731	303	5	a.h	a.h	PROPN
iajs-2731	303	6	;	;	PUNCT
iajs-2731	303	7	hatam	hatam	NOUN
iajs-2731	303	8	,	,	PUNCT
iajs-2731	303	9	y.	y.	PROPN
iajs-2731	303	10	khalaf	khalaf	PROPN
iajs-2731	303	11	.	.	PUNCT
iajs-2731	304	1	fuzzy	fuzzy	ADJ
iajs-2731	304	2	regular	regular	ADJ
iajs-2731	304	3	fmodules	fmodule	NOUN
iajs-2731	304	4	.	.	PUNCT
iajs-2731	305	1	2002	2002	NUM
iajs-2731	305	2	.	.	PUNCT
iajs-2731	306	1	m.sc	m.sc	NOUN
iajs-2731	306	2	thesis	thesis	NOUN
iajs-2731	306	3	of	of	ADP
iajs-2731	306	4	university	university	NOUN
iajs-2731	306	5	baghdad	baghdad	PROPN
iajs-2731	306	6	.	.	PUNCT
iajs-2731	307	1	8	8	NUM
iajs-2731	307	2	.	.	X
iajs-2731	307	3	buthyna	buthyna	PROPN
iajs-2731	307	4	,	,	PUNCT
iajs-2731	307	5	n.sh	n.sh	PROPN
iajs-2731	307	6	.	.	PUNCT
iajs-2731	307	7	;	;	PUNCT
iajs-2731	307	8	hatam	hatam	NOUN
iajs-2731	307	9	,	,	PUNCT
iajs-2731	307	10	y.kh	y.kh	PROPN
iajs-2731	307	11	.	.	PUNCT
iajs-2731	307	12	fuzzy	fuzzy	ADJ
iajs-2731	307	13	-	-	PUNCT
iajs-2731	307	14	small	small	ADJ
iajs-2731	307	15	submodules	submodule	NOUN
iajs-2731	307	16	with	with	ADP
iajs-2731	307	17	study	study	NOUN
iajs-2731	307	18	of	of	ADP
iajs-2731	307	19	most	most	ADV
iajs-2731	307	20	important	important	ADJ
iajs-2731	307	21	relevant	relevant	ADJ
iajs-2731	307	22	results	result	NOUN
iajs-2731	307	23	.	.	PUNCT
iajs-2731	308	1	(	(	PUNCT
iajs-2731	308	2	2020),2nd	2020),2nd	NUM
iajs-2731	308	3	alnoor	alnoor	ADJ
iajs-2731	308	4	international	international	ADJ
iajs-2731	308	5	conference	conference	NOUN
iajs-2731	308	6	for	for	ADP
iajs-2731	308	7	science	science	NOUN
iajs-2731	308	8	and	and	CCONJ
iajs-2731	308	9	technology	technology	NOUN
iajs-2731	308	10	,	,	PUNCT
iajs-2731	308	11	august	august	PROPN
iajs-2731	308	12	28	28	NUM
iajs-2731	308	13	-	-	SYM
iajs-2731	308	14	29	29	NUM
iajs-2731	308	15	,	,	PUNCT
iajs-2731	308	16	baku	baku	PROPN
iajs-2731	308	17	,	,	PUNCT
iajs-2731	308	18	azerbaijan	azerbaijan	PROPN
iajs-2731	308	19	,	,	PUNCT
iajs-2731	308	20	137	137	NUM
iajs-2731	308	21	-	-	SYM
iajs-2731	308	22	140	140	NUM
iajs-2731	308	23	.	.	NOUN
iajs-2731	309	1	9	9	NUM
iajs-2731	309	2	.	.	X
iajs-2731	309	3	thear	thear	VERB
iajs-2731	309	4	z.kh	z.kh	PROPN
iajs-2731	309	5	;	;	PUNCT
iajs-2731	309	6	nada	nada	PROPN
iajs-2731	309	7	,	,	PUNCT
iajs-2731	309	8	k	k	PROPN
iajs-2731	309	9	,	,	PUNCT
iajs-2731	309	10	a	a	PRON
iajs-2731	309	11	,	,	PUNCT
iajs-2731	309	12	some	some	DET
iajs-2731	309	13	generalization	generalization	NOUN
iajs-2731	309	14	of	of	ADP
iajs-2731	309	15	l	l	NOUN
iajs-2731	309	16	–	–	PUNCT
iajs-2731	309	17	hollow	hollow	ADJ
iajs-2731	309	18	modules	module	NOUN
iajs-2731	309	19	.	.	PUNCT
iajs-2731	310	1	2019	2019	NUM
iajs-2731	310	2	.	.	PUNCT
iajs-2731	311	1	m.scthesis	m.scthesis	ADJ
iajs-2731	311	2	university	university	PROPN
iajs-2731	311	3	of	of	ADP
iajs-2731	311	4	tikrit	tikrit	NOUN
iajs-2731	311	5	.	.	PUNCT
iajs-2731	312	1	10	10	NUM
iajs-2731	312	2	.	.	PUNCT
iajs-2731	313	1	hassan	hassan	PROPN
iajs-2731	313	2	,	,	PUNCT
iajs-2731	313	3	k.	k.	PROPN
iajs-2731	313	4	marhon	marhon	PROPN
iajs-2731	313	5	;	;	PUNCT
iajs-2731	313	6	hatam	hatam	NOUN
iajs-2731	313	7	,	,	PUNCT
iajs-2731	313	8	y.	y.	PROPN
iajs-2731	313	9	khalaf	khalaf	PROPN
iajs-2731	313	10	,	,	PUNCT
iajs-2731	313	11	.	.	PUNCT
iajs-2731	314	1	fuzzy	fuzzy	PROPN
iajs-2731	314	2	closed	close	VERB
iajs-2731	314	3	submodule	submodule	NOUN
iajs-2731	314	4	and	and	CCONJ
iajs-2731	314	5	fuzzy	fuzzy	ADJ
iajs-2731	314	6	w	w	NOUN
iajs-2731	314	7	-	-	PUNCT
iajs-2731	314	8	closed	closed	ADJ
iajs-2731	314	9	submodules	submodule	NOUN
iajs-2731	314	10	with	with	ADP
iajs-2731	314	11	of	of	ADP
iajs-2731	314	12	their	their	PRON
iajs-2731	314	13	generalization	generalization	NOUN
iajs-2731	314	14	.	.	PUNCT
iajs-2731	315	1	2020	2020	NUM
iajs-2731	315	2	.	.	PUNCT
iajs-2731	316	1	ph.d	ph.d	PROPN
iajs-2731	316	2	.	.	PUNCT
iajs-2731	317	1	thesis	thesis	NOUN
iajs-2731	317	2	of	of	ADP
iajs-2731	317	3	baghdad	baghdad	PROPN
iajs-2731	317	4	11	11	NUM
iajs-2731	317	5	.	.	PUNCT
iajs-2731	318	1	yahya	yahya	PROPN
iajs-2731	318	2	,	,	PUNCT
iajs-2731	318	3	t.a	t.a	PROPN
iajs-2731	318	4	.	.	PROPN
iajs-2731	318	5	darzi	darzi	PROPN
iajs-2731	318	6	,	,	PUNCT
iajs-2731	318	7	on	on	ADP
iajs-2731	318	8	fuzzy	fuzzy	ADJ
iajs-2731	318	9	supplement	supplement	NOUN
iajs-2731	318	10	submodules	submodule	NOUN
iajs-2731	318	11	.	.	PUNCT
iajs-2731	319	1	february	february	PROPN
iajs-2731	319	2	.	.	PUNCT
iajs-2731	320	1	annals	annal	NOUN
iajs-2731	320	2	of	of	ADP
iajs-2731	320	3	uzzymathematics	uzzymathematic	NOUN
iajs-2731	320	4	and	and	CCONJ
iajs-2731	320	5	informatics	informatic	NOUN
iajs-2731	320	6	,	,	PUNCT
iajs-2731	320	7	2015	2015	NUM
iajs-2731	320	8	,	,	PUNCT
iajs-2731	320	9	9	9	NUM
iajs-2731	320	10	,	,	PUNCT
iajs-2731	320	11	2	2	NUM
iajs-2731	320	12	,	,	PUNCT
iajs-2731	320	13	205	205	NUM
iajs-2731	320	14	-	-	SYM
iajs-2731	320	15	213	213	NUM
iajs-2731	320	16	.	.	PUNCT
iajs-2731	321	1	ibn	ibn	PROPN
iajs-2731	321	2	al	al	PROPN
iajs-2731	321	3	-	-	PUNCT
iajs-2731	321	4	haitham	haitham	PROPN
iajs-2731	321	5	jour	jour	X
iajs-2731	321	6	.	.	PROPN
iajs-2731	321	7	for	for	ADP
iajs-2731	321	8	pure	pure	ADJ
iajs-2731	321	9	&	&	CCONJ
iajs-2731	321	10	appl	appl	PROPN
iajs-2731	321	11	.	.	PUNCT
iajs-2731	322	1	sci	sci	PROPN
iajs-2731	322	2	.	.	PROPN
iajs-2731	323	1	53	53	NUM
iajs-2731	323	2	(	(	PUNCT
iajs-2731	323	3	2)2022	2)2022	VERB
iajs-2731	323	4	96	96	NUM
iajs-2731	323	5	12	12	NUM
iajs-2731	323	6	.	.	PUNCT
iajs-2731	324	1	kumar	kumar	PROPN
iajs-2731	324	2	,	,	PUNCT
iajs-2731	324	3	r	r	PROPN
iajs-2731	324	4	,	,	PUNCT
iajs-2731	324	5	s.	s.	PROPN
iajs-2731	324	6	k.	k.	PROPN
iajs-2731	324	7	bhambir	bhambir	PROPN
iajs-2731	324	8	,	,	PUNCT
iajs-2731	324	9	kumar	kumar	PROPN
iajs-2731	324	10	p.	p.	PROPN
iajs-2731	324	11	fuzzy	fuzzy	PROPN
iajs-2731	324	12	submodule	submodule	NOUN
iajs-2731	324	13	of	of	ADP
iajs-2731	324	14	some	some	DET
iajs-2731	324	15	analogous	analogous	ADJ
iajs-2731	324	16	and	and	CCONJ
iajs-2731	324	17	deviation	deviation	NOUN
iajs-2731	324	18	,	,	PUNCT
iajs-2731	324	19	fsets	fset	NOUN
iajs-2731	324	20	and	and	CCONJ
iajs-2731	324	21	system	system	NOUN
iajs-2731	324	22	.	.	PUNCT
iajs-2731	325	1	1995	1995	NUM
iajs-2731	325	2	,	,	PUNCT
iajs-2731	325	3	70	70	NUM
iajs-2731	325	4	,	,	PUNCT
iajs-2731	325	5	125	125	NUM
iajs-2731	325	6	-	-	SYM
iajs-2731	325	7	130	130	NUM
iajs-2731	325	8	.	.	PUNCT
iajs-2731	326	1	13	13	NUM
iajs-2731	326	2	.	.	PUNCT
iajs-2731	326	3	inaam	inaam	PROPN
iajs-2731	326	4	,	,	PUNCT
iajs-2731	326	5	m.	m.	NOUN
iajs-2731	326	6	hadi	hadi	PROPN
iajs-2731	326	7	,	,	PUNCT
iajs-2731	326	8	myson	myson	PROPN
iajs-2731	326	9	,	,	PUNCT
iajs-2731	326	10	a	a	PRON
iajs-2731	326	11	,	,	PUNCT
iajs-2731	327	1	hamill	hamill	NOUN
iajs-2731	327	2	.	.	PUNCT
iajs-2731	327	3	cancellation	cancellation	NOUN
iajs-2731	327	4	and	and	CCONJ
iajs-2731	327	5	weakly	weakly	ADJ
iajs-2731	327	6	cancellation	cancellation	NOUN
iajs-2731	327	7	fuzzymodules	fuzzymodule	NOUN
iajs-2731	327	8	.	.	PUNCT
iajs-2731	328	1	basrah	basrah	PROPN
iajs-2731	328	2	research	research	PROPN
iajs-2731	328	3	's	's	PART
iajs-2731	328	4	(	(	PUNCT
iajs-2731	328	5	sciences	science	NOUN
iajs-2731	328	6	)	)	PUNCT
iajs-2731	328	7	,	,	PUNCT
iajs-2731	328	8	2011,37	2011,37	NUM
iajs-2731	328	9	,	,	PUNCT
iajs-2731	328	10	4.d	4.d	NUM
iajs-2731	328	11	.	.	PROPN
iajs-2731	329	1	14	14	NUM
iajs-2731	329	2	.	.	X
iajs-2731	329	3	hamel	hamel	PROPN
iajs-2731	329	4	,	,	PUNCT
iajs-2731	329	5	m.a	m.a	PROPN
iajs-2731	329	6	.	.	PROPN
iajs-2731	329	7	and	and	CCONJ
iajs-2731	329	8	khalaf	khalaf	PROPN
iajs-2731	329	9	,	,	PUNCT
iajs-2731	329	10	h.y	h.y	PROPN
iajs-2731	329	11	.	.	PROPN
iajs-2731	329	12	,	,	PUNCT
iajs-2731	329	13	fuzzy	fuzzy	ADJ
iajs-2731	329	14	semimaximal	semimaximal	ADJ
iajs-2731	329	15	submodules	submodule	NOUN
iajs-2731	329	16	.	.	PUNCT
iajs-2731	330	1	ibn	ibn	PROPN
iajs-2731	330	2	al	al	PROPN
iajs-2731	330	3	-	-	PUNCT
iajs-2731	330	4	haitham	haitham	PROPN
iajs-2731	330	5	journal	journal	PROPN
iajs-2731	330	6	for	for	ADP
iajs-2731	330	7	pure	pure	ADJ
iajs-2731	330	8	and	and	CCONJ
iajs-2731	330	9	applied	applied	ADJ
iajs-2731	330	10	sciences	science	NOUN
iajs-2731	330	11	,	,	PUNCT
iajs-2731	330	12	2020,33(4	2020,33(4	NOUN
iajs-2731	330	13	)	)	PUNCT
iajs-2731	330	14	,	,	PUNCT
iajs-2731	330	15	137	137	NUM
iajs-2731	330	16	-	-	SYM
iajs-2731	330	17	147	147	NUM
iajs-2731	330	18	.	.	PUNCT