id	sid	tid	token	lemma	pos
iajs-909	1	1	ibn	ibn	PROPN
iajs-909	1	2	alhaitham	alhaitham	NOUN
iajs-909	1	3	j.	j.	PROPN
iajs-909	2	1	fo	fo	ADP
iajs-909	2	2	r	r	NOUN
iajs-909	2	3	pure	pure	ADJ
iajs-909	2	4	&	&	CCONJ
iajs-909	2	5	appl	appl	PROPN
iajs-909	2	6	.	.	PUNCT
iajs-909	3	1	sc	sc	PROPN
iajs-909	3	2	i	i	INTJ
iajs-909	3	3	.	.	PUNCT
iajs-909	4	1	vo	vo	INTJ
iajs-909	5	1	l.23	l.23	ADV
iajs-909	5	2	(	(	PUNCT
iajs-909	5	3	2	2	NUM
iajs-909	5	4	)	)	PUNCT
iajs-909	5	5	2010	2010	NUM
iajs-909	5	6	chained	chain	VERB
iajs-909	5	7	fuzzy	fuzzy	ADJ
iajs-909	5	8	modules	module	NOUN
iajs-909	5	9	s.	s.	PROPN
iajs-909	5	10	b.semeein	b.semeein	PROPN
iajs-909	5	11	department	department	PROPN
iajs-909	5	12	of	of	ADP
iajs-909	5	13	mathematics	mathematics	PROPN
iajs-909	5	14	,	,	PUNCT
iajs-909	5	15	college	college	NOUN
iajs-909	5	16	of	of	ADP
iajs-909	5	17	education	education	PROPN
iajs-909	5	18	ibn	ibn	PROPN
iajs-909	5	19	-	-	PUNCT
iajs-909	5	20	al	al	PROPN
iajs-909	5	21	-	-	PUNCT
iajs-909	5	22	haitham	haitham	PROPN
iajs-909	5	23	,	,	PUNCT
iajs-909	5	24	university	university	PROPN
iajs-909	5	25	of	of	ADP
iajs-909	5	26	baghdad	baghdad	PROPN
iajs-909	5	27	abstract	abstract	ADV
iajs-909	5	28	let	let	VERB
iajs-909	5	29	r	r	PRON
iajs-909	5	30	be	be	AUX
iajs-909	5	31	a	a	DET
iajs-909	5	32	commutative	commutative	ADJ
iajs-909	5	33	ring	ring	NOUN
iajs-909	5	34	with	with	ADP
iajs-909	5	35	unity	unity	NOUN
iajs-909	5	36	.	.	PUNCT
iajs-909	6	1	in	in	ADP
iajs-909	6	2	this	this	DET
iajs-909	6	3	paper	paper	NOUN
iajs-909	6	4	we	we	PRON
iajs-909	6	5	introduce	introduce	VERB
iajs-909	6	6	the	the	DET
iajs-909	6	7	notion	notion	NOUN
iajs-909	6	8	of	of	ADP
iajs-909	6	9	chained	chain	VERB
iajs-909	6	10	fuzzy	fuzzy	ADJ
iajs-909	6	11	modules	module	NOUN
iajs-909	6	12	as	as	ADP
iajs-909	6	13	a	a	DET
iajs-909	6	14	generalization	generalization	NOUN
iajs-909	6	15	of	of	ADP
iajs-909	6	16	chained	chain	VERB
iajs-909	6	17	modules	module	NOUN
iajs-909	6	18	.	.	PUNCT
iajs-909	7	1	we	we	PRON
iajs-909	7	2	investigate	investigate	VERB
iajs-909	7	3	several	several	ADJ
iajs-909	7	4	characterizations	characterization	NOUN
iajs-909	7	5	and	and	CCONJ
iajs-909	7	6	properties	property	NOUN
iajs-909	7	7	of	of	ADP
iajs-909	7	8	this	this	DET
iajs-909	7	9	concept	concept	NOUN
iajs-909	7	10	introduction	introduction	NOUN
iajs-909	7	11	in	in	ADP
iajs-909	7	12	this	this	DET
iajs-909	7	13	paper	paper	NOUN
iajs-909	7	14	we	we	PRON
iajs-909	7	15	introduce	introduce	VERB
iajs-909	7	16	the	the	DET
iajs-909	7	17	concept	concept	NOUN
iajs-909	7	18	of	of	ADP
iajs-909	7	19	chained	chain	VERB
iajs-909	7	20	fuzzy	fuzzy	ADJ
iajs-909	7	21	modules	module	NOUN
iajs-909	7	22	as	as	ADP
iajs-909	7	23	a	a	DET
iajs-909	7	24	generalization	generalization	NOUN
iajs-909	7	25	of	of	ADP
iajs-909	7	26	the	the	DET
iajs-909	7	27	concept	concept	NOUN
iajs-909	7	28	(	(	PUNCT
iajs-909	7	29	chained	chain	VERB
iajs-909	7	30	modules	module	NOUN
iajs-909	7	31	)	)	PUNCT
iajs-909	7	32	in	in	ADP
iajs-909	7	33	ordinary	ordinary	ADJ
iajs-909	7	34	algebra	algebra	NOUN
iajs-909	7	35	.this	.this	PRON
iajs-909	7	36	paper	paper	NOUN
iajs-909	7	37	consists	consist	VERB
iajs-909	7	38	of	of	ADP
iajs-909	7	39	three	three	NUM
iajs-909	7	40	sections	section	NOUN
iajs-909	7	41	in	in	ADP
iajs-909	7	42	section	section	NOUN
iajs-909	7	43	one	one	NUM
iajs-909	7	44	,	,	PUNCT
iajs-909	7	45	we	we	PRON
iajs-909	7	46	recall	recall	VERB
iajs-909	7	47	some	some	DET
iajs-909	7	48	basic	basic	ADJ
iajs-909	7	49	definitions	definition	NOUN
iajs-909	7	50	and	and	CCONJ
iajs-909	7	51	results	result	NOUN
iajs-909	7	52	which	which	PRON
iajs-909	7	53	we	we	PRON
iajs-909	7	54	needed	need	VERB
iajs-909	7	55	later	later	ADV
iajs-909	7	56	.	.	PUNCT
iajs-909	8	1	in	in	ADP
iajs-909	8	2	section	section	NOUN
iajs-909	8	3	two	two	NUM
iajs-909	8	4	,	,	PUNCT
iajs-909	8	5	we	we	PRON
iajs-909	8	6	give	give	VERB
iajs-909	8	7	some	some	DET
iajs-909	8	8	results	result	NOUN
iajs-909	8	9	about	about	ADP
iajs-909	8	10	chained	chain	VERB
iajs-909	8	11	fuzzy	fuzzy	ADJ
iajs-909	8	12	modules	module	NOUN
iajs-909	8	13	such	such	ADJ
iajs-909	8	14	as	as	SCONJ
iajs-909	8	15	it	it	PRON
iajs-909	8	16	's	be	AUX
iajs-909	8	17	relationship	relationship	NOUN
iajs-909	8	18	with	with	ADP
iajs-909	8	19	it	it	PRON
iajs-909	8	20	levels	level	NOUN
iajs-909	8	21	.	.	PUNCT
iajs-909	9	1	section	section	NOUN
iajs-909	9	2	three	three	NUM
iajs-909	9	3	is	be	AUX
iajs-909	9	4	devoted	devote	VERB
iajs-909	9	5	for	for	ADP
iajs-909	9	6	studying	study	VERB
iajs-909	9	7	the	the	DET
iajs-909	9	8	direct	direct	ADJ
iajs-909	9	9	sum	sum	NOUN
iajs-909	9	10	of	of	ADP
iajs-909	9	11	chained	chain	VERB
iajs-909	9	12	fuzzy	fuzzy	ADJ
iajs-909	9	13	modules	module	NOUN
iajs-909	9	14	.	.	PUNCT
iajs-909	10	1	finally	finally	ADV
iajs-909	10	2	,	,	PUNCT
iajs-909	10	3	we	we	PRON
iajs-909	10	4	study	study	VERB
iajs-909	10	5	the	the	DET
iajs-909	10	6	homomorphic	homomorphic	ADJ
iajs-909	10	7	image	image	NOUN
iajs-909	10	8	and	and	CCONJ
iajs-909	10	9	inverse	inverse	NOUN
iajs-909	10	10	of	of	ADP
iajs-909	10	11	chained	chain	VERB
iajs-909	10	12	fuzzy	fuzzy	ADJ
iajs-909	10	13	modules	module	NOUN
iajs-909	10	14	.	.	PUNCT
iajs-909	11	1	1	1	X
iajs-909	11	2	.	.	X
iajs-909	11	3	preliminaries	preliminary	NOUN
iajs-909	11	4	the	the	DET
iajs-909	11	5	following	follow	VERB
iajs-909	11	6	definitions	definition	NOUN
iajs-909	11	7	and	and	CCONJ
iajs-909	11	8	results	result	NOUN
iajs-909	11	9	are	be	AUX
iajs-909	11	10	needed	need	VERB
iajs-909	11	11	later	later	ADV
iajs-909	11	12	.	.	PUNCT
iajs-909	12	1	1.1	1.1	NUM
iajs-909	12	2	definition	definition	NOUN
iajs-909	12	3	,	,	PUNCT
iajs-909	12	4	[	[	X
iajs-909	12	5	1	1	X
iajs-909	12	6	]	]	PUNCT
iajs-909	12	7	let	let	VERB
iajs-909	12	8	m	m	PRON
iajs-909	12	9	be	be	AUX
iajs-909	12	10	a	a	DET
iajs-909	12	11	nonempty	nonempty	ADV
iajs-909	12	12	set	set	VERB
iajs-909	13	1	and	and	CCONJ
iajs-909	13	2	i	i	PRON
iajs-909	13	3	be	be	VERB
iajs-909	13	4	the	the	DET
iajs-909	13	5	closed	closed	ADJ
iajs-909	13	6	interval	interval	NOUN
iajs-909	13	7	[	[	X
iajs-909	13	8	0,1	0,1	NUM
iajs-909	13	9	]	]	PUNCT
iajs-909	13	10	of	of	ADP
iajs-909	13	11	the	the	DET
iajs-909	13	12	real	real	ADJ
iajs-909	13	13	line	line	NOUN
iajs-909	13	14	(	(	PUNCT
iajs-909	13	15	numbers	number	NOUN
iajs-909	13	16	)	)	PUNCT
iajs-909	13	17	.	.	PUNCT
iajs-909	14	1	a	a	DET
iajs-909	14	2	fuzzy	fuzzy	ADJ
iajs-909	14	3	set	set	VERB
iajs-909	14	4	a	a	PRON
iajs-909	14	5	in	in	ADP
iajs-909	14	6	m	m	PROPN
iajs-909	14	7	(	(	PUNCT
iajs-909	14	8	a	a	DET
iajs-909	14	9	fuzzy	fuzzy	NOUN
iajs-909	14	10	subset	subset	VERB
iajs-909	14	11	a	a	PRON
iajs-909	14	12	of	of	ADP
iajs-909	14	13	m	m	PROPN
iajs-909	14	14	)	)	PUNCT
iajs-909	14	15	is	be	AUX
iajs-909	14	16	a	a	DET
iajs-909	14	17	function	function	NOUN
iajs-909	14	18	from	from	ADP
iajs-909	14	19	m	m	PROPN
iajs-909	14	20	into	into	ADP
iajs-909	14	21	i.	i.	PROPN
iajs-909	14	22	1.2	1.2	NUM
iajs-909	14	23	definition	definition	NOUN
iajs-909	14	24	,	,	PUNCT
iajs-909	14	25	[	[	X
iajs-909	14	26	2	2	NUM
iajs-909	14	27	]	]	PUNCT
iajs-909	14	28	let	let	VERB
iajs-909	14	29	xt	xt	NUM
iajs-909	14	30	:	:	PUNCT
iajs-909	14	31	m	m	PROPN
iajs-909	14	32			NOUN
iajs-909	15	1	[	[	X
iajs-909	15	2	0,1	0,1	NUM
iajs-909	15	3	]	]	PUNCT
iajs-909	15	4	be	be	VERB
iajs-909	15	5	a	a	DET
iajs-909	15	6	fuzzy	fuzzy	ADJ
iajs-909	15	7	set	set	NOUN
iajs-909	15	8	in	in	ADP
iajs-909	15	9	m	m	PROPN
iajs-909	15	10	,	,	PUNCT
iajs-909	15	11	where	where	SCONJ
iajs-909	15	12	xm	xm	NOUN
iajs-909	15	13	,	,	PUNCT
iajs-909	15	14	t[0,1	t[0,1	NOUN
iajs-909	15	15	]	]	PUNCT
iajs-909	15	16	defined	define	VERB
iajs-909	15	17	by	by	ADP
iajs-909	15	18	:	:	PUNCT
iajs-909	15	19	t	t	PROPN
iajs-909	15	20	t	t	PROPN
iajs-909	15	21	if	if	SCONJ
iajs-909	15	22	x	x	PROPN
iajs-909	15	23	y	y	NOUN
iajs-909	15	24	x	x	X
iajs-909	15	25	(	(	PUNCT
iajs-909	15	26	y	y	NOUN
iajs-909	15	27	)	)	PUNCT
iajs-909	15	28	0	0	PUNCT
iajs-909	16	1	if	if	SCONJ
iajs-909	16	2	x	x	PRON
iajs-909	16	3	y	y	PROPN
iajs-909	16	4			NOUN
iajs-909	16	5			PROPN
iajs-909	16	6			NUM
iajs-909	16	7			VERB
iajs-909	16	8	for	for	ADP
iajs-909	16	9	all	all	DET
iajs-909	16	10	ym	ym	NOUN
iajs-909	16	11	,	,	PUNCT
iajs-909	16	12	xt	xt	PROPN
iajs-909	16	13	is	be	AUX
iajs-909	16	14	called	call	VERB
iajs-909	16	15	a	a	DET
iajs-909	16	16	fuzzy	fuzzy	ADJ
iajs-909	16	17	singleton	singleton	NOUN
iajs-909	16	18	or	or	CCONJ
iajs-909	16	19	fuzzy	fuzzy	ADJ
iajs-909	16	20	point	point	NOUN
iajs-909	16	21	in	in	ADP
iajs-909	16	22	m	m	PROPN
iajs-909	16	23	,	,	PUNCT
iajs-909	16	24	if	if	SCONJ
iajs-909	16	25	x=0	x=0	PROPN
iajs-909	16	26	and	and	CCONJ
iajs-909	16	27	t=1	t=1	PROPN
iajs-909	16	28	,	,	PUNCT
iajs-909	16	29	then	then	ADV
iajs-909	16	30	1	1	NUM
iajs-909	16	31	1	1	NUM
iajs-909	16	32	if	if	SCONJ
iajs-909	16	33	y	y	PROPN
iajs-909	16	34	0	0	NUM
iajs-909	16	35	0	0	NUM
iajs-909	16	36	(	(	PUNCT
iajs-909	16	37	y	y	NOUN
iajs-909	16	38	)	)	PUNCT
iajs-909	16	39	0	0	PUNCT
iajs-909	17	1	if	if	SCONJ
iajs-909	17	2	y	y	PROPN
iajs-909	17	3	0	0	NUM
iajs-909	17	4			NOUN
iajs-909	17	5			PROPN
iajs-909	17	6			NOUN
iajs-909	17	7			VERB
iajs-909	17	8	we	we	PRON
iajs-909	17	9	shall	shall	AUX
iajs-909	17	10	call	call	VERB
iajs-909	17	11	such	such	ADJ
iajs-909	17	12	fuzzy	fuzzy	ADJ
iajs-909	17	13	singleton	singleton	NOUN
iajs-909	17	14	the	the	DET
iajs-909	17	15	fuzzy	fuzzy	ADJ
iajs-909	17	16	zero	zero	NUM
iajs-909	17	17	singleton	singleton	NOUN
iajs-909	17	18	.	.	PUNCT
iajs-909	18	1	1.3	1.3	NUM
iajs-909	18	2	definition	definition	NOUN
iajs-909	18	3	,	,	PUNCT
iajs-909	18	4	[	[	X
iajs-909	18	5	2	2	NUM
iajs-909	18	6	]	]	PUNCT
iajs-909	18	7	let	let	VERB
iajs-909	18	8	a	a	PRON
iajs-909	18	9	and	and	CCONJ
iajs-909	18	10	b	b	NOUN
iajs-909	18	11	be	be	AUX
iajs-909	18	12	two	two	NUM
iajs-909	18	13	fuzzy	fuzzy	ADJ
iajs-909	18	14	sets	set	NOUN
iajs-909	18	15	in	in	ADP
iajs-909	18	16	m	m	PROPN
iajs-909	18	17	,	,	PUNCT
iajs-909	18	18	then	then	ADV
iajs-909	18	19	1	1	X
iajs-909	18	20	.	.	PUNCT
iajs-909	19	1	a	a	PRON
iajs-909	19	2	=	=	SYM
iajs-909	19	3	b	b	PROPN
iajs-909	20	1	i	i	PRON
iajs-909	20	2	f	f	PROPN
iajs-909	20	3	a	a	PRON
iajs-909	20	4	n	n	NOUN
iajs-909	20	5	d	d	X
iajs-909	20	6	o	o	NOUN
iajs-909	20	7	n	n	X
iajs-909	20	8	l	l	NOUN
iajs-909	20	9	y	y	NOUN
iajs-909	21	1	i	i	PRON
iajs-909	21	2	f	f	PROPN
iajs-909	21	3	a	a	PRON
iajs-909	21	4	(	(	PUNCT
iajs-909	21	5	x	x	SYM
iajs-909	21	6	)	)	PUNCT
iajs-909	21	7	=	=	SYM
iajs-909	21	8	b	b	X
iajs-909	21	9	(	(	PUNCT
iajs-909	21	10	x	x	PROPN
iajs-909	21	11	)	)	PUNCT
iajs-909	21	12	,	,	PUNCT
iajs-909	22	1	f	f	X
iajs-909	22	2	o	o	NOUN
iajs-909	22	3	r	r	NOUN
iajs-909	22	4	a	a	DET
iajs-909	22	5	l	l	NOUN
iajs-909	22	6	l	l	NOUN
iajs-909	22	7	x	x	SYM
iajs-909	22	8			NOUN
iajs-909	22	9	m	m	VERB
iajs-909	22	10	.	.	PUNCT
iajs-909	23	1	2	2	X
iajs-909	23	2	.	.	X
iajs-909	23	3	ab	ab	PROPN
iajs-909	24	1	if	if	SCONJ
iajs-909	24	2	and	and	CCONJ
iajs-909	24	3	only	only	ADV
iajs-909	24	4	if	if	SCONJ
iajs-909	24	5	a(x	a(x	NOUN
iajs-909	24	6	)	)	PUNCT
iajs-909	24	7			NOUN
iajs-909	24	8	b(x	b(x	NOUN
iajs-909	24	9	)	)	PUNCT
iajs-909	24	10	,	,	PUNCT
iajs-909	24	11	for	for	ADP
iajs-909	24	12	all	all	DET
iajs-909	24	13	xm	xm	NOUN
iajs-909	24	14	.	.	PROPN
iajs-909	24	15	3	3	NUM
iajs-909	24	16	.	.	PUNCT
iajs-909	24	17	(	(	PUNCT
iajs-909	24	18	ab)(x)=min{a(x),b(x	ab)(x)=min{a(x),b(x	ADJ
iajs-909	24	19	)	)	PUNCT
iajs-909	24	20	}	}	PUNCT
iajs-909	24	21	for	for	ADP
iajs-909	24	22	all	all	DET
iajs-909	24	23	xm	xm	NOUN
iajs-909	24	24	.	.	PROPN
iajs-909	24	25	4	4	NUM
iajs-909	24	26	.	.	X
iajs-909	24	27	(	(	PUNCT
iajs-909	24	28	ab)(x)=max{a(x),b(x	ab)(x)=max{a(x),b(x	NUM
iajs-909	24	29	)	)	PUNCT
iajs-909	24	30	}	}	PUNCT
iajs-909	24	31	for	for	ADP
iajs-909	24	32	all	all	DET
iajs-909	24	33	xm	xm	NOUN
iajs-909	24	34	.	.	NOUN
iajs-909	24	35	1.4	1.4	NUM
iajs-909	24	36	definition	definition	NOUN
iajs-909	24	37	,	,	PUNCT
iajs-909	24	38	[	[	X
iajs-909	24	39	3	3	X
iajs-909	24	40	]	]	PUNCT
iajs-909	24	41	let	let	VERB
iajs-909	24	42	a	a	DET
iajs-909	24	43	be	be	AUX
iajs-909	24	44	a	a	DET
iajs-909	24	45	fuzzy	fuzzy	ADJ
iajs-909	24	46	set	set	NOUN
iajs-909	24	47	in	in	ADP
iajs-909	24	48	m	m	PROPN
iajs-909	24	49	and	and	CCONJ
iajs-909	24	50	t[0,1	t[0,1	NOUN
iajs-909	24	51	]	]	PUNCT
iajs-909	24	52	.	.	PUNCT
iajs-909	25	1	the	the	DET
iajs-909	25	2	set	set	NOUN
iajs-909	25	3	at={xm	at={xm	NOUN
iajs-909	25	4	,	,	PUNCT
iajs-909	25	5	a(x)t	a(x)t	NOUN
iajs-909	25	6	}	}	PUNCT
iajs-909	25	7	is	be	AUX
iajs-909	25	8	called	call	VERB
iajs-909	25	9	level	level	NOUN
iajs-909	25	10	subset	subset	NOUN
iajs-909	25	11	of	of	ADP
iajs-909	25	12	a.	a.	NOUN
iajs-909	25	13	1.5	1.5	NUM
iajs-909	25	14	remark	remark	NOUN
iajs-909	25	15	(	(	PUNCT
iajs-909	25	16	1	1	X
iajs-909	25	17	)	)	PUNCT
iajs-909	25	18	the	the	DET
iajs-909	25	19	following	follow	VERB
iajs-909	25	20	properties	property	NOUN
iajs-909	25	21	of	of	ADP
iajs-909	25	22	level	level	NOUN
iajs-909	25	23	subsets	subset	NOUN
iajs-909	25	24	hold	hold	VERB
iajs-909	25	25	for	for	ADP
iajs-909	25	26	each	each	DET
iajs-909	25	27	t[0,1	t[0,1	NOUN
iajs-909	25	28	]	]	PUNCT
iajs-909	26	1	1	1	X
iajs-909	26	2	.	.	PUNCT
iajs-909	26	3	(	(	PUNCT
iajs-909	26	4	ab)t	ab)t	PROPN
iajs-909	26	5	=	=	SYM
iajs-909	26	6	at	at	ADP
iajs-909	26	7			NOUN
iajs-909	26	8	bt	bt	NOUN
iajs-909	26	9	2	2	NUM
iajs-909	26	10	.	.	PUNCT
iajs-909	27	1	a	a	DET
iajs-909	27	2	=	=	NOUN
iajs-909	27	3	b	b	NOUN
iajs-909	27	4	if	if	SCONJ
iajs-909	27	5	and	and	CCONJ
iajs-909	27	6	only	only	ADV
iajs-909	27	7	if	if	SCONJ
iajs-909	27	8	at	at	ADP
iajs-909	27	9	=	=	NOUN
iajs-909	27	10	bt	bt	NOUN
iajs-909	27	11	,	,	PUNCT
iajs-909	27	12	for	for	ADP
iajs-909	27	13	all	all	DET
iajs-909	27	14	t[0,1	t[0,1	NOUN
iajs-909	27	15	]	]	PUNCT
iajs-909	27	16	.	.	PUNCT
iajs-909	28	1	ihjpas	ihjpa	VERB
iajs-909	28	2	ibn	ibn	PROPN
iajs-909	28	3	alhaitham	alhaitham	PROPN
iajs-909	29	1	j.	j.	PROPN
iajs-909	30	1	fo	fo	ADP
iajs-909	30	2	r	r	NOUN
iajs-909	30	3	pure	pure	ADJ
iajs-909	30	4	&	&	CCONJ
iajs-909	30	5	appl	appl	PROPN
iajs-909	30	6	.	.	PUNCT
iajs-909	31	1	sc	sc	PROPN
iajs-909	31	2	i.	i.	PROPN
iajs-909	31	3	vo	vo	PROPN
iajs-909	31	4	l.23	l.23	PROPN
iajs-909	31	5	(	(	PUNCT
iajs-909	31	6	2	2	NUM
iajs-909	31	7	)	)	PUNCT
iajs-909	31	8	2010	2010	NUM
iajs-909	31	9	where	where	SCONJ
iajs-909	31	10	a	a	PRON
iajs-909	31	11	and	and	CCONJ
iajs-909	31	12	b	b	NOUN
iajs-909	31	13	are	be	AUX
iajs-909	31	14	fuzzy	fuzzy	ADJ
iajs-909	31	15	sets	set	NOUN
iajs-909	31	16	.	.	PUNCT
iajs-909	32	1	now	now	ADV
iajs-909	32	2	,	,	PUNCT
iajs-909	32	3	we	we	PRON
iajs-909	32	4	can	can	AUX
iajs-909	32	5	give	give	VERB
iajs-909	32	6	the	the	DET
iajs-909	32	7	definition	definition	NOUN
iajs-909	32	8	of	of	ADP
iajs-909	32	9	image	image	NOUN
iajs-909	32	10	and	and	CCONJ
iajs-909	32	11	inverse	inverse	NOUN
iajs-909	32	12	image	image	NOUN
iajs-909	32	13	of	of	ADP
iajs-909	32	14	a	a	DET
iajs-909	32	15	fuzzy	fuzzy	ADJ
iajs-909	32	16	set	set	NOUN
iajs-909	32	17	.	.	PUNCT
iajs-909	33	1	1.6	1.6	NUM
iajs-909	33	2	definition	definition	NOUN
iajs-909	33	3	,	,	PUNCT
iajs-909	33	4	[	[	X
iajs-909	33	5	4	4	X
iajs-909	33	6	]	]	PUNCT
iajs-909	33	7	let	let	VERB
iajs-909	33	8	f	f	PRON
iajs-909	33	9	be	be	AUX
iajs-909	33	10	a	a	DET
iajs-909	33	11	mapping	mapping	NOUN
iajs-909	33	12	from	from	ADP
iajs-909	33	13	a	a	DET
iajs-909	33	14	set	set	NOUN
iajs-909	33	15	m	m	NOUN
iajs-909	33	16	into	into	ADP
iajs-909	33	17	a	a	DET
iajs-909	33	18	set	set	NOUN
iajs-909	33	19	n	n	CCONJ
iajs-909	33	20	,	,	PUNCT
iajs-909	33	21	a	a	DET
iajs-909	33	22	be	be	AUX
iajs-909	33	23	a	a	DET
iajs-909	33	24	fuzzy	fuzzy	ADJ
iajs-909	33	25	set	set	NOUN
iajs-909	33	26	in	in	ADP
iajs-909	33	27	m	m	PROPN
iajs-909	33	28	and	and	CCONJ
iajs-909	33	29	b	b	AUX
iajs-909	33	30	be	be	AUX
iajs-909	33	31	a	a	DET
iajs-909	33	32	fuzzy	fuzzy	ADJ
iajs-909	33	33	set	set	NOUN
iajs-909	33	34	in	in	ADP
iajs-909	33	35	n.	n.	NOUN
iajs-909	33	36	the	the	DET
iajs-909	33	37	image	image	NOUN
iajs-909	33	38	defined	define	VERB
iajs-909	33	39	by	by	ADP
iajs-909	33	40	:	:	PUNCT
iajs-909	33	41	1	1	NUM
iajs-909	33	42	1sup{a(z	1sup{a(z	NUM
iajs-909	33	43	)	)	PUNCT
iajs-909	34	1	z	z	NOUN
iajs-909	34	2	f	f	PROPN
iajs-909	34	3	(	(	PUNCT
iajs-909	34	4	y	y	NOUN
iajs-909	34	5	)	)	PUNCT
iajs-909	34	6	}	}	PUNCT
iajs-909	34	7	if	if	SCONJ
iajs-909	34	8	f	f	PROPN
iajs-909	34	9	(	(	PUNCT
iajs-909	34	10	y	y	PROPN
iajs-909	34	11	)	)	PUNCT
iajs-909	34	12	f	f	PROPN
iajs-909	34	13	(	(	PUNCT
iajs-909	34	14	)	)	PUNCT
iajs-909	34	15	for	for	ADP
iajs-909	34	16	all	all	DET
iajs-909	34	17	y	y	PROPN
iajs-909	34	18	n	n	ADV
iajs-909	34	19	0	0	NUM
iajs-909	34	20	otherwise	otherwise	ADV
iajs-909	34	21			PROPN
iajs-909	34	22			PROPN
iajs-909	34	23			PROPN
iajs-909	34	24			NOUN
iajs-909	34	25			VERB
iajs-909	34	26			NUM
iajs-909	34	27			PROPN
iajs-909	34	28			PROPN
iajs-909	34	29			NOUN
iajs-909	35	1			NUM
iajs-909	35	2			NUM
iajs-909	35	3	where	where	SCONJ
iajs-909	35	4	f	f	X
iajs-909	35	5	–	–	PUNCT
iajs-909	35	6	1	1	NUM
iajs-909	35	7	(	(	PUNCT
iajs-909	35	8	y)={x	y)={x	NOUN
iajs-909	35	9	:	:	PUNCT
iajs-909	35	10	f(x)=y	f(x)=y	NOUN
iajs-909	35	11	}	}	PUNCT
iajs-909	35	12	and	and	CCONJ
iajs-909	35	13	the	the	DET
iajs-909	35	14	inverse	inverse	ADJ
iajs-909	35	15	image	image	NOUN
iajs-909	35	16	of	of	ADP
iajs-909	35	17	b	b	NOUN
iajs-909	35	18	,	,	PUNCT
iajs-909	35	19	denoted	denote	VERB
iajs-909	35	20	by	by	ADP
iajs-909	35	21	f	f	PROPN
iajs-909	35	22	–	–	PUNCT
iajs-909	35	23	1	1	NUM
iajs-909	35	24	(	(	PUNCT
iajs-909	35	25	b	b	NOUN
iajs-909	35	26	)	)	PUNCT
iajs-909	35	27	,	,	PUNCT
iajs-909	35	28	is	be	AUX
iajs-909	35	29	the	the	DET
iajs-909	35	30	fuzzy	fuzzy	ADJ
iajs-909	35	31	set	set	NOUN
iajs-909	35	32	in	in	ADP
iajs-909	35	33	m	m	PROPN
iajs-909	35	34	defined	define	VERB
iajs-909	35	35	by	by	ADP
iajs-909	35	36	:	:	PUNCT
iajs-909	35	37	f	f	X
iajs-909	35	38	–	–	PUNCT
iajs-909	35	39	1	1	NUM
iajs-909	35	40	(	(	PUNCT
iajs-909	35	41	b)(x)=b(f(x	b)(x)=b(f(x	NOUN
iajs-909	35	42	)	)	PUNCT
iajs-909	35	43	)	)	PUNCT
iajs-909	35	44	,	,	PUNCT
iajs-909	35	45	for	for	ADP
iajs-909	35	46	all	all	DET
iajs-909	35	47	xm	xm	NOUN
iajs-909	35	48	.	.	PUNCT
iajs-909	35	49	1.7	1.7	NUM
iajs-909	35	50	definition	definition	NOUN
iajs-909	35	51	,	,	PUNCT
iajs-909	35	52	[	[	X
iajs-909	35	53	5	5	NUM
iajs-909	35	54	]	]	PUNCT
iajs-909	35	55	let	let	VERB
iajs-909	35	56	f	f	PRON
iajs-909	35	57	be	be	AUX
iajs-909	35	58	a	a	DET
iajs-909	35	59	mapping	mapping	NOUN
iajs-909	35	60	from	from	ADP
iajs-909	35	61	a	a	DET
iajs-909	35	62	set	set	NOUN
iajs-909	35	63	m	m	NOUN
iajs-909	35	64	into	into	ADP
iajs-909	35	65	a	a	DET
iajs-909	35	66	set	set	NOUN
iajs-909	35	67	m	m	NOUN
iajs-909	35	68	'	'	NUM
iajs-909	35	69	.	.	PUNCT
iajs-909	36	1	a	a	DET
iajs-909	36	2	fuzzy	fuzzy	NOUN
iajs-909	36	3	subset	subset	VERB
iajs-909	36	4	a	a	PRON
iajs-909	36	5	of	of	ADP
iajs-909	36	6	m	m	PROPN
iajs-909	36	7	is	be	AUX
iajs-909	36	8	called	call	VERB
iajs-909	36	9	finvariant	finvariant	ADJ
iajs-909	36	10	if	if	SCONJ
iajs-909	36	11	a(x)=a(y	a(x)=a(y	PROPN
iajs-909	36	12	)	)	PUNCT
iajs-909	36	13	whenever	whenever	SCONJ
iajs-909	36	14	f(x)=f(y	f(x)=f(y	ADV
iajs-909	36	15	)	)	PUNCT
iajs-909	36	16	,	,	PUNCT
iajs-909	36	17	where	where	SCONJ
iajs-909	36	18	x	x	X
iajs-909	36	19	,	,	PUNCT
iajs-909	36	20	y	y	PROPN
iajs-909	36	21	m	m	PROPN
iajs-909	36	22	.	.	PUNCT
iajs-909	37	1	the	the	DET
iajs-909	37	2	following	follow	VERB
iajs-909	37	3	lemma	lemma	PROPN
iajs-909	37	4	is	be	AUX
iajs-909	37	5	needed	need	VERB
iajs-909	37	6	in	in	ADP
iajs-909	37	7	section	section	NOUN
iajs-909	37	8	three	three	NUM
iajs-909	37	9	.	.	PUNCT
iajs-909	38	1	1.8	1.8	NUM
iajs-909	38	2	lemma	lemma	PROPN
iajs-909	38	3	,	,	PUNCT
iajs-909	38	4	[	[	X
iajs-909	38	5	5	5	X
iajs-909	38	6	]	]	PUNCT
iajs-909	38	7	if	if	SCONJ
iajs-909	38	8	f	f	PROPN
iajs-909	38	9	is	be	AUX
iajs-909	38	10	a	a	DET
iajs-909	38	11	function	function	NOUN
iajs-909	38	12	defined	define	VERB
iajs-909	38	13	on	on	ADP
iajs-909	38	14	a	a	DET
iajs-909	38	15	set	set	NOUN
iajs-909	38	16	m	m	NOUN
iajs-909	38	17	,	,	PUNCT
iajs-909	38	18	a1	a1	NOUN
iajs-909	38	19	and	and	CCONJ
iajs-909	38	20	a2	a2	PROPN
iajs-909	38	21	are	be	AUX
iajs-909	38	22	fuzzy	fuzzy	ADJ
iajs-909	38	23	subset	subset	NOUN
iajs-909	38	24	of	of	ADP
iajs-909	38	25	m	m	PROPN
iajs-909	38	26	,	,	PUNCT
iajs-909	38	27	b1	b1	NOUN
iajs-909	38	28	and	and	CCONJ
iajs-909	38	29	b2	b2	NOUN
iajs-909	38	30	are	be	AUX
iajs-909	38	31	fuzzy	fuzzy	ADJ
iajs-909	38	32	subset	subset	NOUN
iajs-909	38	33	of	of	ADP
iajs-909	38	34	f(m	f(m	PROPN
iajs-909	38	35	)	)	PUNCT
iajs-909	38	36	.	.	PUNCT
iajs-909	39	1	then	then	ADV
iajs-909	39	2	the	the	DET
iajs-909	39	3	following	follow	VERB
iajs-909	39	4	are	be	AUX
iajs-909	39	5	true	true	ADJ
iajs-909	39	6	:	:	PUNCT
iajs-909	39	7	1	1	X
iajs-909	39	8	.	.	X
iajs-909	39	9	a1	a1	PROPN
iajs-909	39	10	=	=	PROPN
iajs-909	39	11	f	f	PROPN
iajs-909	39	12	–	–	PUNCT
iajs-909	39	13	1	1	NUM
iajs-909	39	14	(	(	PUNCT
iajs-909	39	15	f(a	f(a	NOUN
iajs-909	39	16	)	)	PUNCT
iajs-909	39	17	)	)	PUNCT
iajs-909	39	18	,	,	PUNCT
iajs-909	39	19	whenever	whenever	SCONJ
iajs-909	39	20	a1	a1	NOUN
iajs-909	39	21	is	be	AUX
iajs-909	39	22	f	f	NOUN
iajs-909	39	23	-	-	PUNCT
iajs-909	39	24	invariant	invariant	ADJ
iajs-909	39	25	.	.	PUNCT
iajs-909	40	1	2	2	X
iajs-909	40	2	.	.	X
iajs-909	41	1	f(f	f(f	PROPN
iajs-909	41	2	–	–	PUNCT
iajs-909	41	3	1	1	NUM
iajs-909	41	4	(	(	PUNCT
iajs-909	41	5	b1))=b1	b1))=b1	ADP
iajs-909	41	6	3	3	X
iajs-909	41	7	.	.	X
iajs-909	42	1	if	if	SCONJ
iajs-909	42	2	a1a2	a1a2	PROPN
iajs-909	42	3	,	,	PUNCT
iajs-909	42	4	then	then	ADV
iajs-909	42	5	f(a1)f(a2	f(a1)f(a2	PROPN
iajs-909	42	6	)	)	PUNCT
iajs-909	42	7	4	4	NUM
iajs-909	42	8	.	.	PUNCT
iajs-909	43	1	if	if	SCONJ
iajs-909	43	2	b1b2	b1b2	PROPN
iajs-909	43	3	,	,	PUNCT
iajs-909	43	4	then	then	ADV
iajs-909	43	5	f	f	PROPN
iajs-909	43	6	–	–	PUNCT
iajs-909	43	7	1(b1)f	1(b1)f	NUM
iajs-909	43	8	–	–	PUNCT
iajs-909	43	9	1(b2	1(b2	NOUN
iajs-909	43	10	)	)	PUNCT
iajs-909	43	11	.	.	PUNCT
iajs-909	44	1	1.9	1.9	NUM
iajs-909	44	2	definition	definition	NOUN
iajs-909	44	3	,	,	PUNCT
iajs-909	44	4	[	[	X
iajs-909	44	5	6	6	NUM
iajs-909	44	6	]	]	X
iajs-909	44	7	let	let	ADJ
iajs-909	44	8	(	(	PUNCT
iajs-909	44	9	r,+,	r,+,	NOUN
iajs-909	44	10	)	)	PUNCT
iajs-909	44	11	be	be	VERB
iajs-909	44	12	a	a	DET
iajs-909	44	13	ring	ring	NOUN
iajs-909	44	14	and	and	CCONJ
iajs-909	44	15	let	let	VERB
iajs-909	44	16	x	x	PRON
iajs-909	44	17	be	be	AUX
iajs-909	44	18	a	a	DET
iajs-909	44	19	fuzzy	fuzzy	ADJ
iajs-909	44	20	set	set	NOUN
iajs-909	44	21	in	in	ADP
iajs-909	44	22	r.	r.	PROPN
iajs-909	44	23	then	then	ADV
iajs-909	44	24	x	x	VERB
iajs-909	44	25	is	be	AUX
iajs-909	44	26	called	call	VERB
iajs-909	44	27	a	a	DET
iajs-909	44	28	fuzzy	fuzzy	ADJ
iajs-909	44	29	ring	ring	NOUN
iajs-909	44	30	in	in	ADP
iajs-909	44	31	ring	ring	NOUN
iajs-909	44	32	(	(	PUNCT
iajs-909	44	33	r,+,	r,+,	NOUN
iajs-909	44	34	)	)	PUNCT
iajs-909	44	35	if	if	SCONJ
iajs-909	44	36	and	and	CCONJ
iajs-909	44	37	only	only	ADV
iajs-909	44	38	if	if	SCONJ
iajs-909	44	39	,	,	PUNCT
iajs-909	44	40	for	for	ADP
iajs-909	44	41	each	each	DET
iajs-909	44	42	x	x	NOUN
iajs-909	44	43	,	,	PUNCT
iajs-909	44	44	y	y	PROPN
iajs-909	44	45			NOUN
iajs-909	44	46	r	r	NOUN
iajs-909	44	47	1	1	NUM
iajs-909	44	48	.	.	PUNCT
iajs-909	44	49	x(x+y	x(x+y	NUM
iajs-909	44	50	)	)	PUNCT
iajs-909	44	51			NUM
iajs-909	44	52	min{x(x	min{x(x	NOUN
iajs-909	44	53	)	)	PUNCT
iajs-909	44	54	,	,	PUNCT
iajs-909	44	55	x(y	x(y	PROPN
iajs-909	44	56	)	)	PUNCT
iajs-909	44	57	}	}	PUNCT
iajs-909	44	58	2	2	NUM
iajs-909	44	59	.	.	PUNCT
iajs-909	44	60	x(x	x(x	NOUN
iajs-909	44	61	)	)	PUNCT
iajs-909	45	1	=	=	SYM
iajs-909	45	2	x	x	X
iajs-909	45	3	(	(	PUNCT
iajs-909	45	4	–	–	PUNCT
iajs-909	45	5	x	x	X
iajs-909	45	6	)	)	PUNCT
iajs-909	45	7	3	3	NUM
iajs-909	45	8	.	.	PUNCT
iajs-909	45	9	x(xy	x(xy	NUM
iajs-909	45	10	)	)	PUNCT
iajs-909	45	11			NUM
iajs-909	45	12	min{x(x	min{x(x	NOUN
iajs-909	45	13	)	)	PUNCT
iajs-909	45	14	,	,	PUNCT
iajs-909	45	15	x(y	x(y	PROPN
iajs-909	45	16	)	)	PUNCT
iajs-909	45	17	}	}	PUNCT
iajs-909	45	18	.	.	PUNCT
iajs-909	46	1	1.10	1.10	NUM
iajs-909	46	2	definition	definition	NOUN
iajs-909	46	3	[	[	X
iajs-909	46	4	7	7	X
iajs-909	46	5	]	]	X
iajs-909	46	6	a	a	DET
iajs-909	46	7	fuzzy	fuzzy	ADJ
iajs-909	46	8	subset	subset	NOUN
iajs-909	46	9	x	x	PUNCT
iajs-909	46	10	of	of	ADP
iajs-909	46	11	a	a	DET
iajs-909	46	12	ring	ring	NOUN
iajs-909	46	13	r	r	NOUN
iajs-909	46	14	is	be	AUX
iajs-909	46	15	called	call	VERB
iajs-909	46	16	a	a	DET
iajs-909	46	17	fuzzy	fuzzy	ADJ
iajs-909	46	18	ideal	ideal	NOUN
iajs-909	46	19	of	of	ADP
iajs-909	46	20	r	r	NOUN
iajs-909	46	21	,	,	PUNCT
iajs-909	46	22	if	if	SCONJ
iajs-909	46	23	for	for	ADP
iajs-909	46	24	each	each	DET
iajs-909	46	25	x	x	NOUN
iajs-909	46	26	,	,	PUNCT
iajs-909	46	27	y	y	PROPN
iajs-909	46	28			NOUN
iajs-909	46	29	r	r	NOUN
iajs-909	46	30	1	1	NUM
iajs-909	46	31	.	.	PUNCT
iajs-909	46	32	x(x	x(x	PROPN
iajs-909	46	33	–	–	PUNCT
iajs-909	46	34	y	y	NOUN
iajs-909	46	35	)	)	PUNCT
iajs-909	46	36			NUM
iajs-909	46	37	min{x(x	min{x(x	PROPN
iajs-909	46	38	)	)	PUNCT
iajs-909	46	39	,	,	PUNCT
iajs-909	46	40	x(y	x(y	PROPN
iajs-909	46	41	)	)	PUNCT
iajs-909	46	42	}	}	PUNCT
iajs-909	46	43	2	2	NUM
iajs-909	46	44	.	.	PUNCT
iajs-909	46	45	x(xy	x(xy	NUM
iajs-909	46	46	)	)	PUNCT
iajs-909	46	47			NUM
iajs-909	46	48	max{x(x	max{x(x	PROPN
iajs-909	46	49	)	)	PUNCT
iajs-909	46	50	,	,	PUNCT
iajs-909	46	51	x(y	x(y	PROPN
iajs-909	46	52	)	)	PUNCT
iajs-909	46	53	}	}	PUNCT
iajs-909	46	54	.	.	PUNCT
iajs-909	47	1	1.11	1.11	NUM
iajs-909	47	2	definition	definition	NOUN
iajs-909	47	3	[	[	X
iajs-909	47	4	2	2	X
iajs-909	47	5	]	]	PUNCT
iajs-909	47	6	let	let	VERB
iajs-909	47	7	m	m	PRON
iajs-909	47	8	be	be	AUX
iajs-909	47	9	an	an	DET
iajs-909	47	10	r	r	NOUN
iajs-909	47	11	-	-	PUNCT
iajs-909	47	12	module	module	NOUN
iajs-909	47	13	.	.	PUNCT
iajs-909	48	1	a	a	DET
iajs-909	48	2	fuzzy	fuzzy	ADJ
iajs-909	48	3	set	set	NOUN
iajs-909	48	4	x	x	PUNCT
iajs-909	48	5	of	of	ADP
iajs-909	48	6	m	m	PROPN
iajs-909	48	7	is	be	AUX
iajs-909	48	8	called	call	VERB
iajs-909	48	9	a	a	DET
iajs-909	48	10	fuzzy	fuzzy	ADJ
iajs-909	48	11	module	module	NOUN
iajs-909	48	12	of	of	ADP
iajs-909	48	13	m	m	NOUN
iajs-909	48	14	if	if	SCONJ
iajs-909	48	15	1	1	NUM
iajs-909	48	16	.	.	PUNCT
iajs-909	48	17	x(x	x(x	PROPN
iajs-909	48	18	–	–	PUNCT
iajs-909	48	19	y	y	NOUN
iajs-909	48	20	)	)	PUNCT
iajs-909	48	21			NUM
iajs-909	48	22	min{x(x	min{x(x	PROPN
iajs-909	48	23	)	)	PUNCT
iajs-909	48	24	,	,	PUNCT
iajs-909	48	25	x(y	x(y	PROPN
iajs-909	48	26	)	)	PUNCT
iajs-909	48	27	}	}	PUNCT
iajs-909	48	28	,	,	PUNCT
iajs-909	48	29	for	for	ADP
iajs-909	48	30	all	all	DET
iajs-909	48	31	x	x	NOUN
iajs-909	48	32	,	,	PUNCT
iajs-909	48	33	y	y	PROPN
iajs-909	48	34	m	m	PROPN
iajs-909	48	35	.	.	PROPN
iajs-909	48	36	2	2	NUM
iajs-909	48	37	.	.	X
iajs-909	48	38	x(rx	x(rx	NUM
iajs-909	48	39	)	)	PUNCT
iajs-909	48	40	x(x	x(x	NOUN
iajs-909	48	41	)	)	PUNCT
iajs-909	48	42	,	,	PUNCT
iajs-909	48	43	for	for	ADP
iajs-909	48	44	all	all	DET
iajs-909	48	45	xm	xm	PROPN
iajs-909	48	46	and	and	CCONJ
iajs-909	48	47	rr	rr	PRON
iajs-909	48	48	.	.	NOUN
iajs-909	48	49	3	3	NUM
iajs-909	48	50	.	.	X
iajs-909	48	51	x(0)=1	x(0)=1	NUM
iajs-909	48	52	.	.	PUNCT
iajs-909	49	1	1.12	1.12	NUM
iajs-909	49	2	definition	definition	NOUN
iajs-909	49	3	[	[	X
iajs-909	49	4	6	6	NUM
iajs-909	49	5	]	]	PUNCT
iajs-909	49	6	let	let	VERB
iajs-909	49	7	x	x	PUNCT
iajs-909	49	8	and	and	CCONJ
iajs-909	49	9	a	a	DET
iajs-909	49	10	be	be	AUX
iajs-909	49	11	two	two	NUM
iajs-909	49	12	fuzzy	fuzzy	ADJ
iajs-909	49	13	modules	module	NOUN
iajs-909	49	14	of	of	ADP
iajs-909	49	15	an	an	DET
iajs-909	49	16	r	r	NOUN
iajs-909	49	17	-	-	PUNCT
iajs-909	49	18	module	module	NOUN
iajs-909	49	19	m.	m.	NOUN
iajs-909	49	20	a	a	PRON
iajs-909	49	21	is	be	AUX
iajs-909	49	22	called	call	VERB
iajs-909	49	23	a	a	DET
iajs-909	49	24	fuzzy	fuzzy	ADJ
iajs-909	49	25	submodule	submodule	NOUN
iajs-909	49	26	of	of	ADP
iajs-909	49	27	x	x	PRON
iajs-909	49	28	if	if	SCONJ
iajs-909	49	29	ax	ax	NOUN
iajs-909	49	30	.	.	PUNCT
iajs-909	50	1	1.13	1.13	NUM
iajs-909	50	2	proposition	proposition	NOUN
iajs-909	50	3	[	[	X
iajs-909	50	4	7	7	NUM
iajs-909	50	5	]	]	PUNCT
iajs-909	50	6	let	let	VERB
iajs-909	50	7	a	a	PRON
iajs-909	50	8	be	be	AUX
iajs-909	50	9	a	a	DET
iajs-909	50	10	fuzzy	fuzzy	ADJ
iajs-909	50	11	set	set	NOUN
iajs-909	50	12	of	of	ADP
iajs-909	50	13	m.	m.	NOUN
iajs-909	50	14	then	then	ADV
iajs-909	50	15	the	the	DET
iajs-909	50	16	level	level	NOUN
iajs-909	50	17	subset	subset	VERB
iajs-909	50	18	at	at	ADP
iajs-909	50	19	,	,	PUNCT
iajs-909	50	20	t(0,1	t(0,1	NOUN
iajs-909	50	21	]	]	PUNCT
iajs-909	50	22	is	be	AUX
iajs-909	50	23	a	a	DET
iajs-909	50	24	submodule	submodule	NOUN
iajs-909	50	25	of	of	ADP
iajs-909	50	26	m	m	PROPN
iajs-909	50	27	if	if	SCONJ
iajs-909	51	1	and	and	CCONJ
iajs-909	51	2	only	only	ADV
iajs-909	51	3	if	if	SCONJ
iajs-909	51	4	a	a	PRON
iajs-909	51	5	is	be	AUX
iajs-909	51	6	a	a	DET
iajs-909	51	7	fuzzy	fuzzy	ADJ
iajs-909	51	8	submodule	submodule	NOUN
iajs-909	51	9	of	of	ADP
iajs-909	51	10	x	x	SYM
iajs-909	51	11	where	where	SCONJ
iajs-909	51	12	x	x	PRON
iajs-909	51	13	is	be	AUX
iajs-909	51	14	a	a	DET
iajs-909	51	15	fuzzy	fuzzy	ADJ
iajs-909	51	16	module	module	NOUN
iajs-909	51	17	of	of	ADP
iajs-909	51	18	m	m	NOUN
iajs-909	51	19	such	such	ADJ
iajs-909	51	20	that	that	SCONJ
iajs-909	51	21	a(x)x(x	a(x)x(x	NOUN
iajs-909	51	22	)	)	PUNCT
iajs-909	51	23	,	,	PUNCT
iajs-909	51	24	xm	xm	PROPN
iajs-909	51	25	.	.	PUNCT
iajs-909	52	1	1.14	1.14	NUM
iajs-909	52	2	definition	definition	NOUN
iajs-909	52	3	[	[	X
iajs-909	52	4	8	8	X
iajs-909	52	5	]	]	X
iajs-909	52	6	a	a	DET
iajs-909	52	7	fuzzy	fuzzy	ADJ
iajs-909	52	8	module	module	NOUN
iajs-909	52	9	x	x	PUNCT
iajs-909	52	10	of	of	ADP
iajs-909	52	11	an	an	DET
iajs-909	52	12	r	r	NOUN
iajs-909	52	13	-	-	PUNCT
iajs-909	52	14	module	module	NOUN
iajs-909	52	15	m	m	NOUN
iajs-909	52	16	is	be	AUX
iajs-909	52	17	called	call	VERB
iajs-909	52	18	fuzzy	fuzzy	ADJ
iajs-909	52	19	simple	simple	ADJ
iajs-909	52	20	if	if	SCONJ
iajs-909	52	21	and	and	CCONJ
iajs-909	52	22	only	only	ADV
iajs-909	52	23	if	if	SCONJ
iajs-909	52	24	x	x	PRON
iajs-909	52	25	has	have	VERB
iajs-909	52	26	no	no	DET
iajs-909	52	27	fuzzy	fuzzy	ADJ
iajs-909	52	28	proper	proper	ADJ
iajs-909	52	29	submodules	submodule	NOUN
iajs-909	52	30	.	.	PUNCT
iajs-909	53	1	1.15	1.15	NUM
iajs-909	53	2	definition	definition	NOUN
iajs-909	53	3	[	[	X
iajs-909	53	4	8	8	X
iajs-909	53	5	]	]	PUNCT
iajs-909	53	6	a	a	DET
iajs-909	53	7	fuzzy	fuzzy	ADJ
iajs-909	53	8	module	module	NOUN
iajs-909	53	9	x	x	PUNCT
iajs-909	53	10	of	of	ADP
iajs-909	53	11	an	an	DET
iajs-909	53	12	r	r	NOUN
iajs-909	53	13	-	-	PUNCT
iajs-909	53	14	module	module	NOUN
iajs-909	53	15	m	m	NOUN
iajs-909	53	16	is	be	AUX
iajs-909	53	17	called	call	VERB
iajs-909	53	18	fuzzy	fuzzy	ADJ
iajs-909	53	19	cyclic	cyclic	NOUN
iajs-909	53	20	module	module	NOUN
iajs-909	53	21	,	,	PUNCT
iajs-909	53	22	if	if	SCONJ
iajs-909	53	23	there	there	PRON
iajs-909	53	24	exists	exist	VERB
iajs-909	53	25	xtx	xtx	PROPN
iajs-909	54	1	such	such	ADJ
iajs-909	54	2	that	that	SCONJ
iajs-909	54	3	each	each	DET
iajs-909	54	4	ykx	ykx	ADV
iajs-909	54	5	written	write	VERB
iajs-909	54	6	as	as	ADP
iajs-909	54	7	y	y	PROPN
iajs-909	54	8	k	k	PROPN
iajs-909	54	9	=	=	NOUN
iajs-909	54	10	rℓxt	rℓxt	VERB
iajs-909	54	11	for	for	ADP
iajs-909	54	12	some	some	DET
iajs-909	54	13	fuzzy	fuzzy	ADJ
iajs-909	54	14	ihjpas	ihjpa	VERB
iajs-909	54	15	ibn	ibn	PROPN
iajs-909	54	16	alhaitham	alhaitham	PROPN
iajs-909	54	17	j.	j.	PROPN
iajs-909	55	1	fo	fo	ADP
iajs-909	55	2	r	r	NOUN
iajs-909	55	3	pure	pure	ADJ
iajs-909	55	4	&	&	CCONJ
iajs-909	55	5	appl	appl	PROPN
iajs-909	55	6	.	.	PUNCT
iajs-909	56	1	sc	sc	PROPN
iajs-909	56	2	i.	i.	PROPN
iajs-909	56	3	vo	vo	PROPN
iajs-909	56	4	l.23	l.23	PROPN
iajs-909	56	5	(	(	PUNCT
iajs-909	56	6	2	2	NUM
iajs-909	56	7	)	)	PUNCT
iajs-909	56	8	2010	2010	NUM
iajs-909	56	9	singleton	singleton	NOUN
iajs-909	56	10	rℓ	rℓ	NOUN
iajs-909	56	11	of	of	ADP
iajs-909	56	12	r	r	NOUN
iajs-909	56	13	where	where	SCONJ
iajs-909	56	14	k	k	NOUN
iajs-909	56	15	,	,	PUNCT
iajs-909	56	16	ℓ	ℓ	PROPN
iajs-909	56	17	,	,	PUNCT
iajs-909	56	18	t[0,1	t[0,1	NOUN
iajs-909	56	19	]	]	PUNCT
iajs-909	56	20	.	.	PUNCT
iajs-909	57	1	in	in	ADP
iajs-909	57	2	this	this	DET
iajs-909	57	3	case	case	NOUN
iajs-909	57	4	,	,	PUNCT
iajs-909	57	5	we	we	PRON
iajs-909	57	6	shall	shall	AUX
iajs-909	57	7	write	write	VERB
iajs-909	57	8	x=(xt	x=(xt	NOUN
iajs-909	57	9	)	)	PUNCT
iajs-909	57	10	to	to	PART
iajs-909	57	11	denoted	denote	VERB
iajs-909	57	12	the	the	DET
iajs-909	57	13	fuzzy	fuzzy	ADJ
iajs-909	57	14	cyclic	cyclic	NOUN
iajs-909	57	15	module	module	NOUN
iajs-909	57	16	generated	generate	VERB
iajs-909	57	17	by	by	ADP
iajs-909	57	18	xt	xt	PROPN
iajs-909	57	19	.	.	PROPN
iajs-909	58	1	1.16	1.16	NUM
iajs-909	58	2	definition	definition	NOUN
iajs-909	58	3	[	[	X
iajs-909	58	4	6	6	NUM
iajs-909	58	5	]	]	PUNCT
iajs-909	58	6	let	let	VERB
iajs-909	58	7	x	x	PRON
iajs-909	58	8	and	and	CCONJ
iajs-909	58	9	y	y	PROPN
iajs-909	58	10	be	be	AUX
iajs-909	58	11	two	two	NUM
iajs-909	58	12	fuzzy	fuzzy	ADJ
iajs-909	58	13	modules	module	NOUN
iajs-909	58	14	of	of	ADP
iajs-909	58	15	r	r	NOUN
iajs-909	58	16	-	-	PUNCT
iajs-909	58	17	modules	module	NOUN
iajs-909	58	18	m	m	NOUN
iajs-909	58	19	1	1	NUM
iajs-909	58	20	and	and	CCONJ
iajs-909	58	21	m	m	PROPN
iajs-909	58	22	2	2	NUM
iajs-909	58	23	respectively	respectively	ADV
iajs-909	58	24	,	,	PUNCT
iajs-909	58	25	f	f	X
iajs-909	58	26	:	:	PUNCT
iajs-909	58	27	xy	xy	PROPN
iajs-909	58	28	is	be	AUX
iajs-909	58	29	called	call	VERB
iajs-909	58	30	a	a	DET
iajs-909	58	31	fuzzy	fuzzy	ADJ
iajs-909	58	32	homomorphism	homomorphism	NOUN
iajs-909	58	33	if	if	SCONJ
iajs-909	58	34	f	f	X
iajs-909	58	35	:	:	PUNCT
iajs-909	58	36	m	m	VERB
iajs-909	58	37	1m	1m	NUM
iajs-909	58	38	2	2	NUM
iajs-909	58	39	is	be	AUX
iajs-909	58	40	r	r	NOUN
iajs-909	58	41	-	-	PUNCT
iajs-909	58	42	homomorphism	homomorphism	NOUN
iajs-909	58	43	and	and	CCONJ
iajs-909	58	44	y(f(x))=x(x	y(f(x))=x(x	NOUN
iajs-909	58	45	)	)	PUNCT
iajs-909	58	46	for	for	ADP
iajs-909	58	47	each	each	PRON
iajs-909	58	48	xm	xm	NOUN
iajs-909	59	1	1	1	X
iajs-909	59	2	.	.	X
iajs-909	59	3	1.17	1.17	NUM
iajs-909	59	4	remark	remark	NOUN
iajs-909	59	5	[	[	X
iajs-909	59	6	9	9	NUM
iajs-909	59	7	]	]	SYM
iajs-909	59	8	1let	1let	NUM
iajs-909	59	9	m	m	NOUN
iajs-909	59	10	and	and	CCONJ
iajs-909	59	11	m	m	PROPN
iajs-909	59	12	'	'	PUNCT
iajs-909	59	13	be	be	VERB
iajs-909	59	14	two	two	NUM
iajs-909	59	15	r	r	NOUN
iajs-909	59	16	-	-	PUNCT
iajs-909	59	17	modules	module	NOUN
iajs-909	59	18	,	,	PUNCT
iajs-909	59	19	f	f	X
iajs-909	59	20	:	:	PUNCT
iajs-909	59	21	mm	mm	PROPN
iajs-909	59	22	'	'	PUNCT
iajs-909	59	23	be	be	AUX
iajs-909	59	24	an	an	DET
iajs-909	59	25	epimorphism	epimorphism	NOUN
iajs-909	59	26	.	.	PUNCT
iajs-909	60	1	if	if	SCONJ
iajs-909	60	2	a	a	PRON
iajs-909	60	3	is	be	AUX
iajs-909	60	4	a	a	DET
iajs-909	60	5	fuzzy	fuzzy	ADJ
iajs-909	60	6	submodule	submodule	NOUN
iajs-909	60	7	of	of	ADP
iajs-909	60	8	m	m	PROPN
iajs-909	60	9	,	,	PUNCT
iajs-909	60	10	then	then	ADV
iajs-909	60	11	f(a	f(a	PROPN
iajs-909	60	12	)	)	PUNCT
iajs-909	60	13	is	be	AUX
iajs-909	60	14	a	a	DET
iajs-909	60	15	fuzzy	fuzzy	ADJ
iajs-909	60	16	submodule	submodule	NOUN
iajs-909	60	17	of	of	ADP
iajs-909	60	18	m	m	NOUN
iajs-909	60	19	'	'	PUNCT
iajs-909	60	20	.	.	PUNCT
iajs-909	61	1	2let	2let	NUM
iajs-909	61	2	m	m	NOUN
iajs-909	61	3	and	and	CCONJ
iajs-909	61	4	m	m	PROPN
iajs-909	61	5	'	'	PUNCT
iajs-909	61	6	be	be	VERB
iajs-909	61	7	two	two	NUM
iajs-909	61	8	r	r	NOUN
iajs-909	61	9	-	-	PUNCT
iajs-909	61	10	modules	module	NOUN
iajs-909	61	11	,	,	PUNCT
iajs-909	61	12	f	f	X
iajs-909	61	13	:	:	PUNCT
iajs-909	61	14	mm	mm	PROPN
iajs-909	61	15	'	'	PUNCT
iajs-909	61	16	be	be	AUX
iajs-909	61	17	a	a	DET
iajs-909	61	18	homomorphism	homomorphism	NOUN
iajs-909	61	19	.	.	PUNCT
iajs-909	62	1	if	if	SCONJ
iajs-909	62	2	b	b	PROPN
iajs-909	62	3	is	be	AUX
iajs-909	62	4	a	a	DET
iajs-909	62	5	fuzzy	fuzzy	ADJ
iajs-909	62	6	submodule	submodule	NOUN
iajs-909	62	7	of	of	ADP
iajs-909	62	8	m	m	PROPN
iajs-909	62	9	'	'	PUNCT
iajs-909	62	10	,	,	PUNCT
iajs-909	62	11	then	then	ADV
iajs-909	62	12	f	f	PROPN
iajs-909	62	13	–	–	PUNCT
iajs-909	62	14	1(b	1(b	NUM
iajs-909	62	15	)	)	PUNCT
iajs-909	62	16	is	be	AUX
iajs-909	62	17	a	a	DET
iajs-909	62	18	fuzzy	fuzzy	ADJ
iajs-909	62	19	submodule	submodule	NOUN
iajs-909	62	20	of	of	ADP
iajs-909	62	21	m.	m.	NOUN
iajs-909	62	22	1.18	1.18	NUM
iajs-909	62	23	definition	definition	NOUN
iajs-909	62	24	[	[	X
iajs-909	62	25	2	2	X
iajs-909	62	26	]	]	PUNCT
iajs-909	62	27	suppose	suppose	VERB
iajs-909	62	28	a	a	PRON
iajs-909	62	29	and	and	CCONJ
iajs-909	62	30	b	b	NOUN
iajs-909	62	31	be	be	AUX
iajs-909	62	32	two	two	NUM
iajs-909	62	33	fuzzy	fuzzy	ADJ
iajs-909	62	34	modules	module	NOUN
iajs-909	62	35	of	of	ADP
iajs-909	62	36	r	r	NOUN
iajs-909	62	37	-	-	PUNCT
iajs-909	62	38	module	module	NOUN
iajs-909	62	39	m.	m.	NOUN
iajs-909	62	40	we	we	PRON
iajs-909	62	41	define	define	VERB
iajs-909	62	42	(	(	PUNCT
iajs-909	62	43	a	a	DET
iajs-909	62	44	:	:	SYM
iajs-909	62	45	b	b	NOUN
iajs-909	62	46	)	)	PUNCT
iajs-909	62	47	by	by	ADP
iajs-909	62	48	:	:	PUNCT
iajs-909	62	49	(	(	PUNCT
iajs-909	62	50	a	a	X
iajs-909	62	51	:	:	PUNCT
iajs-909	62	52	b)={rt	b)={rt	NOUN
iajs-909	62	53	:	:	PUNCT
iajs-909	62	54	rt	rt	PROPN
iajs-909	62	55	is	be	AUX
iajs-909	62	56	a	a	DET
iajs-909	62	57	fuzzy	fuzzy	ADJ
iajs-909	62	58	singleton	singleton	NOUN
iajs-909	62	59	of	of	ADP
iajs-909	62	60	r	r	NOUN
iajs-909	62	61	such	such	ADJ
iajs-909	62	62	that	that	PRON
iajs-909	62	63	rtba	rtba	NOUN
iajs-909	62	64	}	}	PUNCT
iajs-909	62	65	and	and	CCONJ
iajs-909	62	66	(	(	PUNCT
iajs-909	62	67	a	a	PRON
iajs-909	62	68	:	:	PUNCT
iajs-909	62	69	b)(r)=sup{t[0,1]rtba	b)(r)=sup{t[0,1]rtba	NOUN
iajs-909	62	70	,	,	PUNCT
iajs-909	62	71	for	for	ADP
iajs-909	62	72	all	all	DET
iajs-909	62	73	rr	rr	NUM
iajs-909	62	74	}	}	PUNCT
iajs-909	62	75	.	.	PUNCT
iajs-909	63	1	if	if	SCONJ
iajs-909	63	2	b=(bk	b=(bk	NOUN
iajs-909	63	3	)	)	PUNCT
iajs-909	63	4	,	,	PUNCT
iajs-909	63	5	(	(	PUNCT
iajs-909	63	6	a:(bk))={rtrtbka	a:(bk))={rtrtbka	PROPN
iajs-909	63	7	,	,	PUNCT
iajs-909	63	8	rt	rt	PROPN
iajs-909	63	9	is	be	AUX
iajs-909	63	10	a	a	DET
iajs-909	63	11	fuzzy	fuzzy	ADJ
iajs-909	63	12	singleton	singleton	NOUN
iajs-909	63	13	of	of	ADP
iajs-909	63	14	r	r	NOUN
iajs-909	63	15	}	}	PUNCT
iajs-909	63	16	.	.	PUNCT
iajs-909	64	1	1.19	1.19	NUM
iajs-909	64	2	definition	definition	NOUN
iajs-909	64	3	[	[	X
iajs-909	64	4	10	10	NUM
iajs-909	64	5	]	]	PUNCT
iajs-909	64	6	let	let	VERB
iajs-909	64	7	x	x	PRON
iajs-909	64	8	and	and	CCONJ
iajs-909	64	9	y	y	PROPN
iajs-909	64	10	be	be	AUX
iajs-909	64	11	two	two	NUM
iajs-909	64	12	fuzzy	fuzzy	ADJ
iajs-909	64	13	modules	module	NOUN
iajs-909	64	14	of	of	ADP
iajs-909	64	15	m	m	PROPN
iajs-909	64	16	1	1	NUM
iajs-909	64	17	,	,	PUNCT
iajs-909	64	18	m	m	VERB
iajs-909	64	19	2	2	NUM
iajs-909	64	20	respectively	respectively	ADV
iajs-909	64	21	.	.	PUNCT
iajs-909	65	1	define	define	VERB
iajs-909	65	2	xy	xy	PROPN
iajs-909	65	3	:	:	PUNCT
iajs-909	65	4	m1m	m1m	PROPN
iajs-909	65	5	2[0,1	2[0,1	NOUN
iajs-909	65	6	]	]	PUNCT
iajs-909	65	7	by	by	ADP
iajs-909	65	8	(	(	PUNCT
iajs-909	65	9	xy)(a	xy)(a	PROPN
iajs-909	65	10	,	,	PUNCT
iajs-909	65	11	b)=min{x(a),y(b	b)=min{x(a),y(b	PROPN
iajs-909	65	12	)	)	PUNCT
iajs-909	65	13	}	}	PUNCT
iajs-909	65	14	for	for	ADP
iajs-909	65	15	all	all	PRON
iajs-909	65	16	(	(	PUNCT
iajs-909	65	17	a	a	PRON
iajs-909	65	18	,	,	PUNCT
iajs-909	65	19	b)	b)	NOUN
iajs-909	66	1	m	m	VERB
iajs-909	67	1	1m	1m	NUM
iajs-909	67	2	2	2	NUM
iajs-909	67	3	.	.	PUNCT
iajs-909	68	1	xy	xy	PROPN
iajs-909	68	2	is	be	AUX
iajs-909	68	3	called	call	VERB
iajs-909	68	4	a	a	DET
iajs-909	68	5	fuzzy	fuzzy	ADJ
iajs-909	68	6	external	external	ADJ
iajs-909	68	7	direct	direct	ADJ
iajs-909	68	8	sum	sum	NOUN
iajs-909	68	9	of	of	ADP
iajs-909	68	10	x	x	PUNCT
iajs-909	68	11	and	and	CCONJ
iajs-909	68	12	y.	y.	PROPN
iajs-909	68	13	1.20	1.20	NUM
iajs-909	68	14	proposition	proposition	NOUN
iajs-909	68	15	[	[	X
iajs-909	68	16	10	10	NUM
iajs-909	68	17	]	]	PUNCT
iajs-909	68	18	let	let	VERB
iajs-909	68	19	x	x	PRON
iajs-909	68	20	and	and	CCONJ
iajs-909	68	21	y	y	PROPN
iajs-909	68	22	are	be	AUX
iajs-909	68	23	fuzzy	fuzzy	ADJ
iajs-909	68	24	modules	module	NOUN
iajs-909	68	25	of	of	ADP
iajs-909	68	26	m	m	PROPN
iajs-909	68	27	1	1	NUM
iajs-909	68	28	and	and	CCONJ
iajs-909	68	29	m2	m2	PROPN
iajs-909	68	30	respectively	respectively	ADV
iajs-909	68	31	,	,	PUNCT
iajs-909	68	32	then	then	ADV
iajs-909	68	33	xy	xy	PROPN
iajs-909	68	34	is	be	AUX
iajs-909	68	35	a	a	DET
iajs-909	68	36	fuzzy	fuzzy	ADJ
iajs-909	68	37	module	module	NOUN
iajs-909	68	38	of	of	ADP
iajs-909	68	39	m	m	PROPN
iajs-909	68	40	1m	1m	PROPN
iajs-909	68	41	2	2	NUM
iajs-909	68	42	.	.	PUNCT
iajs-909	68	43	1.21	1.21	NUM
iajs-909	68	44	proposition	proposition	NOUN
iajs-909	68	45	[	[	X
iajs-909	68	46	10	10	NUM
iajs-909	68	47	]	]	PUNCT
iajs-909	68	48	let	let	VERB
iajs-909	68	49	a	a	PRON
iajs-909	68	50	and	and	CCONJ
iajs-909	68	51	b	b	NOUN
iajs-909	68	52	be	be	AUX
iajs-909	68	53	two	two	NUM
iajs-909	68	54	fuzzy	fuzzy	ADJ
iajs-909	68	55	submodules	submodule	NOUN
iajs-909	68	56	of	of	ADP
iajs-909	68	57	a	a	DET
iajs-909	68	58	fuzzy	fuzzy	ADJ
iajs-909	68	59	module	module	NOUN
iajs-909	68	60	x	x	ADP
iajs-909	68	61	such	such	ADJ
iajs-909	68	62	that	that	SCONJ
iajs-909	68	63	x	x	NOUN
iajs-909	68	64	=	=	NOUN
iajs-909	68	65	ab	ab	PROPN
iajs-909	68	66	,	,	PUNCT
iajs-909	68	67	then	then	ADV
iajs-909	68	68	xs	xs	PROPN
iajs-909	68	69	=	=	PROPN
iajs-909	68	70	asbs	asbs	PROPN
iajs-909	68	71	,	,	PUNCT
iajs-909	68	72	for	for	ADP
iajs-909	68	73	all	all	PRON
iajs-909	68	74	s	s	VERB
iajs-909	68	75	(0,1	(0,1	NUM
iajs-909	68	76	]	]	PUNCT
iajs-909	68	77	.	.	PUNCT
iajs-909	69	1	2	2	X
iajs-909	69	2	.	.	NUM
iajs-909	69	3	chained	chain	VERB
iajs-909	69	4	fuzzy	fuzzy	ADJ
iajs-909	69	5	module	module	NOUN
iajs-909	69	6	in	in	ADP
iajs-909	69	7	this	this	DET
iajs-909	69	8	section	section	NOUN
iajs-909	69	9	we	we	PRON
iajs-909	69	10	introduce	introduce	VERB
iajs-909	69	11	the	the	DET
iajs-909	69	12	concept	concept	NOUN
iajs-909	69	13	of	of	ADP
iajs-909	69	14	chained	chain	VERB
iajs-909	69	15	fuzzy	fuzzy	ADJ
iajs-909	69	16	module	module	NOUN
iajs-909	69	17	.	.	PUNCT
iajs-909	70	1	some	some	DET
iajs-909	70	2	basic	basic	ADJ
iajs-909	70	3	results	result	NOUN
iajs-909	70	4	of	of	ADP
iajs-909	70	5	this	this	DET
iajs-909	70	6	concept	concept	NOUN
iajs-909	70	7	are	be	AUX
iajs-909	70	8	considerate	considerate	ADJ
iajs-909	70	9	2.1	2.1	NUM
iajs-909	70	10	definition	definition	NOUN
iajs-909	70	11	,	,	PUNCT
iajs-909	70	12	[	[	X
iajs-909	70	13	11	11	NUM
iajs-909	70	14	]	]	PUNCT
iajs-909	70	15	an	an	DET
iajs-909	70	16	r	r	NOUN
iajs-909	70	17	-	-	PUNCT
iajs-909	70	18	module	module	NOUN
iajs-909	70	19	m	m	NOUN
iajs-909	70	20	is	be	AUX
iajs-909	70	21	called	call	VERB
iajs-909	70	22	chained	chain	VERB
iajs-909	70	23	module	module	NOUN
iajs-909	70	24	if	if	SCONJ
iajs-909	70	25	for	for	ADP
iajs-909	70	26	each	each	DET
iajs-909	70	27	submodules	submodule	NOUN
iajs-909	70	28	a	a	PRON
iajs-909	70	29	,	,	PUNCT
iajs-909	70	30	b	b	PROPN
iajs-909	70	31	of	of	ADP
iajs-909	70	32	m	m	PROPN
iajs-909	70	33	,	,	PUNCT
iajs-909	70	34	either	either	CCONJ
iajs-909	70	35	a	a	DET
iajs-909	70	36			PROPN
iajs-909	70	37	b	b	PROPN
iajs-909	70	38	or	or	CCONJ
iajs-909	70	39	b	b	PROPN
iajs-909	70	40			PROPN
iajs-909	70	41	a.	a.	NOUN
iajs-909	70	42	we	we	PRON
iajs-909	70	43	fuzzify	fuzzify	VERB
iajs-909	70	44	this	this	DET
iajs-909	70	45	definition	definition	NOUN
iajs-909	70	46	as	as	SCONJ
iajs-909	70	47	follows	follow	VERB
iajs-909	70	48	:	:	PUNCT
iajs-909	70	49	2.2	2.2	NUM
iajs-909	70	50	definition	definition	NOUN
iajs-909	70	51	let	let	VERB
iajs-909	70	52	x	x	PRON
iajs-909	70	53	be	be	AUX
iajs-909	70	54	a	a	DET
iajs-909	70	55	fuzzy	fuzzy	ADJ
iajs-909	70	56	module	module	NOUN
iajs-909	70	57	of	of	ADP
iajs-909	70	58	an	an	DET
iajs-909	70	59	r	r	NOUN
iajs-909	70	60	-	-	PUNCT
iajs-909	70	61	module	module	NOUN
iajs-909	71	1	m	m	NOUN
iajs-909	71	2	then	then	ADV
iajs-909	71	3	x	x	VERB
iajs-909	71	4	is	be	AUX
iajs-909	71	5	called	call	VERB
iajs-909	71	6	a	a	DET
iajs-909	71	7	chained	chain	VERB
iajs-909	71	8	fuzzy	fuzzy	ADJ
iajs-909	71	9	module	module	NOUN
iajs-909	71	10	if	if	SCONJ
iajs-909	71	11	for	for	SCONJ
iajs-909	71	12	each	each	DET
iajs-909	71	13	fuzzy	fuzzy	ADJ
iajs-909	71	14	submodules	submodule	NOUN
iajs-909	71	15	of	of	ADP
iajs-909	71	16	x	x	PUNCT
iajs-909	71	17	either	either	CCONJ
iajs-909	71	18	a	a	DET
iajs-909	71	19			PROPN
iajs-909	71	20	b	b	PROPN
iajs-909	71	21	or	or	CCONJ
iajs-909	71	22	b	b	PROPN
iajs-909	71	23			PROPN
iajs-909	71	24	a.	a.	NOUN
iajs-909	71	25	to	to	PART
iajs-909	71	26	prove	prove	VERB
iajs-909	71	27	our	our	PRON
iajs-909	71	28	next	next	ADJ
iajs-909	71	29	theorem	theorem	NOUN
iajs-909	71	30	,	,	PUNCT
iajs-909	71	31	first	first	ADV
iajs-909	71	32	we	we	PRON
iajs-909	71	33	prove	prove	VERB
iajs-909	71	34	the	the	DET
iajs-909	71	35	following	follow	VERB
iajs-909	71	36	lemma	lemma	PROPN
iajs-909	71	37	:	:	PUNCT
iajs-909	71	38	2.3	2.3	NUM
iajs-909	71	39	lemma	lemma	PROPN
iajs-909	71	40	let	let	VERB
iajs-909	71	41	a	a	PRON
iajs-909	71	42	and	and	CCONJ
iajs-909	71	43	b	b	NOUN
iajs-909	71	44	be	be	AUX
iajs-909	71	45	two	two	NUM
iajs-909	71	46	fuzzy	fuzzy	ADJ
iajs-909	71	47	subset	subset	NOUN
iajs-909	71	48	of	of	ADP
iajs-909	71	49	r	r	NOUN
iajs-909	71	50	then	then	ADV
iajs-909	71	51	a	a	DET
iajs-909	71	52			PROPN
iajs-909	71	53	b	b	PROPN
iajs-909	72	1	if	if	SCONJ
iajs-909	73	1	and	and	CCONJ
iajs-909	73	2	only	only	ADV
iajs-909	73	3	if	if	SCONJ
iajs-909	73	4	at	at	ADP
iajs-909	73	5			PROPN
iajs-909	73	6	bt	bt	PROPN
iajs-909	73	7	,	,	PUNCT
iajs-909	73	8	for	for	ADP
iajs-909	73	9	each	each	DET
iajs-909	73	10	t[0,1	t[0,1	NOUN
iajs-909	73	11	]	]	PUNCT
iajs-909	73	12	.	.	PUNCT
iajs-909	74	1	proof	proof	NOUN
iajs-909	74	2	:	:	PUNCT
iajs-909	74	3	it	it	PRON
iajs-909	74	4	is	be	AUX
iajs-909	74	5	easy	easy	ADJ
iajs-909	74	6	so	so	SCONJ
iajs-909	74	7	it	it	PRON
iajs-909	74	8	is	be	AUX
iajs-909	74	9	omitted	omit	VERB
iajs-909	74	10	.	.	PUNCT
iajs-909	75	1	the	the	DET
iajs-909	75	2	following	follow	VERB
iajs-909	75	3	theorem	theorem	NOUN
iajs-909	75	4	characterizes	characterize	VERB
iajs-909	75	5	chained	chain	VERB
iajs-909	75	6	fuzzy	fuzzy	ADJ
iajs-909	75	7	module	module	NOUN
iajs-909	75	8	in	in	ADP
iajs-909	75	9	terms	term	NOUN
iajs-909	75	10	of	of	ADP
iajs-909	75	11	it	it	PRON
iajs-909	75	12	is	be	AUX
iajs-909	75	13	level	level	NOUN
iajs-909	75	14	module	module	NOUN
iajs-909	75	15	.	.	PUNCT
iajs-909	76	1	2.4	2.4	NUM
iajs-909	76	2	theorem	theorem	VERB
iajs-909	76	3	a	a	DET
iajs-909	76	4	fuzzy	fuzzy	ADJ
iajs-909	76	5	module	module	NOUN
iajs-909	76	6	x	x	PUNCT
iajs-909	76	7	of	of	ADP
iajs-909	76	8	an	an	DET
iajs-909	76	9	r	r	NOUN
iajs-909	76	10	-	-	PUNCT
iajs-909	76	11	module	module	NOUN
iajs-909	76	12	m	m	NOUN
iajs-909	76	13	is	be	AUX
iajs-909	76	14	a	a	DET
iajs-909	76	15	chained	chain	VERB
iajs-909	76	16	if	if	SCONJ
iajs-909	76	17	and	and	CCONJ
iajs-909	76	18	only	only	ADV
iajs-909	76	19	if	if	SCONJ
iajs-909	76	20	xt	xt	PROPN
iajs-909	76	21	is	be	AUX
iajs-909	76	22	a	a	DET
iajs-909	76	23	chained	chain	VERB
iajs-909	76	24	module	module	NOUN
iajs-909	76	25	,	,	PUNCT
iajs-909	76	26			NOUN
iajs-909	76	27	t(0,1	t(0,1	NOUN
iajs-909	76	28	]	]	PUNCT
iajs-909	76	29	.	.	PUNCT
iajs-909	77	1	proof	proof	NOUN
iajs-909	77	2	:	:	PUNCT
iajs-909	77	3	if	if	SCONJ
iajs-909	77	4	x	x	PRON
iajs-909	77	5	is	be	AUX
iajs-909	77	6	chained	chain	VERB
iajs-909	77	7	fuzzy	fuzzy	ADJ
iajs-909	77	8	module	module	NOUN
iajs-909	77	9	.	.	PUNCT
iajs-909	78	1	to	to	PART
iajs-909	78	2	prove	prove	VERB
iajs-909	78	3	xt	xt	PROPN
iajs-909	78	4	is	be	AUX
iajs-909	78	5	chained	chain	VERB
iajs-909	78	6	module	module	NOUN
iajs-909	78	7			NOUN
iajs-909	78	8	t(0,1	t(0,1	NOUN
iajs-909	78	9	]	]	PUNCT
iajs-909	78	10	.	.	PUNCT
iajs-909	79	1	let	let	VERB
iajs-909	79	2	i	i	PRON
iajs-909	79	3	,	,	PUNCT
iajs-909	79	4	j	j	PROPN
iajs-909	79	5	be	be	VERB
iajs-909	79	6	submodules	submodule	NOUN
iajs-909	79	7	of	of	ADP
iajs-909	79	8	xt	xt	PROPN
iajs-909	79	9	.	.	PUNCT
iajs-909	80	1	define	define	VERB
iajs-909	80	2	:	:	PUNCT
iajs-909	80	3	t	t	NOUN
iajs-909	80	4	x	x	SYM
iajs-909	80	5	(	(	PUNCT
iajs-909	80	6	x	x	X
iajs-909	80	7	)	)	PUNCT
iajs-909	80	8	0	0	NUM
iajs-909	81	1	x	x	SYM
iajs-909	81	2			PUNCT
iajs-909	82	1			NOUN
iajs-909	82	2			NUM
iajs-909	83	1			NUM
iajs-909	83	2			NOUN
iajs-909	83	3	,	,	PUNCT
iajs-909	83	4	t	t	PROPN
iajs-909	83	5	x	x	X
iajs-909	83	6	j	j	PROPN
iajs-909	83	7	b(x	b(x	NOUN
iajs-909	83	8	)	)	PUNCT
iajs-909	83	9	0	0	NUM
iajs-909	84	1	x	x	SYM
iajs-909	84	2	j	j	PROPN
iajs-909	84	3			NUM
iajs-909	85	1			NUM
iajs-909	86	1			NOUN
iajs-909	86	2			NOUN
iajs-909	86	3	ihjpas	ihjpa	VERB
iajs-909	86	4	ibn	ibn	PROPN
iajs-909	86	5	alhaitham	alhaitham	PROPN
iajs-909	87	1	j.	j.	PROPN
iajs-909	88	1	fo	fo	ADP
iajs-909	88	2	r	r	NOUN
iajs-909	88	3	pure	pure	ADJ
iajs-909	88	4	&	&	CCONJ
iajs-909	88	5	appl	appl	PROPN
iajs-909	88	6	.	.	PUNCT
iajs-909	89	1	sc	sc	PROPN
iajs-909	89	2	i.	i.	PROPN
iajs-909	89	3	vo	vo	PROPN
iajs-909	89	4	l.23	l.23	PROPN
iajs-909	89	5	(	(	PUNCT
iajs-909	89	6	2	2	NUM
iajs-909	89	7	)	)	PUNCT
iajs-909	89	8	2010	2010	NUM
iajs-909	89	9	a	a	PRON
iajs-909	89	10	,	,	PUNCT
iajs-909	89	11	b	b	NOUN
iajs-909	89	12	are	be	AUX
iajs-909	89	13	fuzzy	fuzzy	ADJ
iajs-909	89	14	submodules	submodule	NOUN
iajs-909	89	15	of	of	ADP
iajs-909	89	16	x.	x.	NOUN
iajs-909	89	17	but	but	CCONJ
iajs-909	89	18	at	at	ADP
iajs-909	89	19	=	=	PROPN
iajs-909	89	20	i	i	PROPN
iajs-909	89	21	,	,	PUNCT
iajs-909	89	22	bt	bt	PROPN
iajs-909	89	23	=	=	PROPN
iajs-909	89	24	j	j	NOUN
iajs-909	89	25	since	since	SCONJ
iajs-909	89	26	x	x	PRON
iajs-909	89	27	is	be	AUX
iajs-909	89	28	chained	chain	VERB
iajs-909	89	29	fuzzy	fuzzy	ADJ
iajs-909	89	30	module	module	NOUN
iajs-909	89	31	,	,	PUNCT
iajs-909	89	32	then	then	ADV
iajs-909	89	33	either	either	CCONJ
iajs-909	89	34	a	a	DET
iajs-909	89	35			PROPN
iajs-909	89	36	b	b	PROPN
iajs-909	89	37	or	or	CCONJ
iajs-909	89	38	b	b	NOUN
iajs-909	89	39			PROPN
iajs-909	89	40	a.	a.	NOUN
iajs-909	89	41	hence	hence	ADV
iajs-909	89	42	at	at	ADP
iajs-909	89	43			PROPN
iajs-909	89	44	bt	bt	PROPN
iajs-909	89	45	or	or	CCONJ
iajs-909	89	46	bt	bt	NOUN
iajs-909	89	47			PROPN
iajs-909	89	48	at	at	ADP
iajs-909	89	49	(	(	PUNCT
iajs-909	89	50	by	by	ADP
iajs-909	89	51	lemma	lemma	PROPN
iajs-909	89	52	(	(	PUNCT
iajs-909	89	53	2.3	2.3	NUM
iajs-909	89	54	)	)	PUNCT
iajs-909	89	55	)	)	PUNCT
iajs-909	89	56	.	.	PUNCT
iajs-909	90	1	thus	thus	ADV
iajs-909	90	2	i	i	PRON
iajs-909	90	3			VERB
iajs-909	90	4	j	j	PROPN
iajs-909	90	5	or	or	CCONJ
iajs-909	90	6	j	j	PROPN
iajs-909	90	7			PROPN
iajs-909	90	8	i.	i.	PROPN
iajs-909	90	9	conversely	conversely	ADV
iajs-909	90	10	,	,	PUNCT
iajs-909	90	11	if	if	SCONJ
iajs-909	90	12	xt	xt	PROPN
iajs-909	90	13	is	be	AUX
iajs-909	90	14	chained	chain	VERB
iajs-909	90	15	module	module	NOUN
iajs-909	90	16	,	,	PUNCT
iajs-909	90	17	to	to	PART
iajs-909	90	18	prove	prove	VERB
iajs-909	90	19	x	x	PUNCT
iajs-909	90	20	is	be	AUX
iajs-909	90	21	a	a	DET
iajs-909	90	22	chained	chain	VERB
iajs-909	90	23	fuzzy	fuzzy	ADJ
iajs-909	90	24	module	module	NOUN
iajs-909	90	25	,	,	PUNCT
iajs-909	90	26	let	let	VERB
iajs-909	90	27	a	a	DET
iajs-909	90	28	,	,	PUNCT
iajs-909	90	29	b	b	NOUN
iajs-909	90	30	fuzzy	fuzzy	ADJ
iajs-909	90	31	submodules	submodule	NOUN
iajs-909	90	32	in	in	ADP
iajs-909	90	33	x.	x.	NOUN
iajs-909	90	34	then	then	ADV
iajs-909	90	35	at	at	ADP
iajs-909	90	36	,	,	PUNCT
iajs-909	90	37	bt	bt	PROPN
iajs-909	90	38	are	be	AUX
iajs-909	90	39	submodules	submodule	NOUN
iajs-909	90	40	in	in	ADP
iajs-909	90	41	xt	xt	PROPN
iajs-909	90	42	,	,	PUNCT
iajs-909	90	43	for	for	ADP
iajs-909	90	44	all	all	DET
iajs-909	90	45	t(0,1	t(0,1	NOUN
iajs-909	90	46	]	]	PUNCT
iajs-909	90	47	since	since	SCONJ
iajs-909	90	48	xt	xt	PROPN
iajs-909	90	49	is	be	AUX
iajs-909	90	50	chained	chain	VERB
iajs-909	90	51	rmodule	rmodule	NOUN
iajs-909	90	52	then	then	ADV
iajs-909	90	53	at	at	ADP
iajs-909	90	54			PROPN
iajs-909	90	55	bt	bt	PROPN
iajs-909	90	56	or	or	CCONJ
iajs-909	90	57	bt	bt	NOUN
iajs-909	90	58			PROPN
iajs-909	90	59	at	at	ADP
iajs-909	90	60	which	which	PRON
iajs-909	90	61	implies	imply	VERB
iajs-909	90	62	a	a	DET
iajs-909	90	63			PROPN
iajs-909	90	64	b	b	PROPN
iajs-909	90	65	or	or	CCONJ
iajs-909	90	66	b	b	PROPN
iajs-909	90	67			PROPN
iajs-909	90	68	a	a	PRON
iajs-909	90	69	(	(	PUNCT
iajs-909	90	70	lemma	lemma	PROPN
iajs-909	90	71	(	(	PUNCT
iajs-909	90	72	2.3	2.3	NUM
iajs-909	90	73	)	)	PUNCT
iajs-909	90	74	)	)	PUNCT
iajs-909	90	75	.	.	PUNCT
iajs-909	91	1	2.5	2.5	NUM
iajs-909	91	2	examples	example	NOUN
iajs-909	91	3	1	1	NUM
iajs-909	91	4	.	.	PUNCT
iajs-909	91	5	let	let	VERB
iajs-909	91	6	x(x)=1	x(x)=1	PRON
iajs-909	91	7	for	for	ADP
iajs-909	91	8	all	all	DET
iajs-909	91	9	xz8	xz8	PROPN
iajs-909	91	10	xt	xt	PROPN
iajs-909	91	11	=	=	PROPN
iajs-909	91	12	z8	z8	PROPN
iajs-909	91	13	for	for	ADP
iajs-909	91	14	all	all	DET
iajs-909	91	15	t[0,1	t[0,1	NOUN
iajs-909	91	16	]	]	PUNCT
iajs-909	91	17	.	.	PUNCT
iajs-909	92	1	but	but	CCONJ
iajs-909	92	2	z8	z8	NOUN
iajs-909	92	3	is	be	AUX
iajs-909	92	4	chained	chain	VERB
iajs-909	92	5	.	.	PUNCT
iajs-909	93	1	hence	hence	ADV
iajs-909	93	2	by	by	ADP
iajs-909	93	3	theorem	theorem	NOUN
iajs-909	93	4	(	(	PUNCT
iajs-909	93	5	2.4	2.4	NUM
iajs-909	93	6	)	)	PUNCT
iajs-909	93	7	x	x	X
iajs-909	93	8	is	be	AUX
iajs-909	93	9	a	a	DET
iajs-909	93	10	chained	chain	VERB
iajs-909	93	11	fuzzy	fuzzy	ADJ
iajs-909	93	12	module	module	NOUN
iajs-909	93	13	.	.	PUNCT
iajs-909	94	1	2	2	X
iajs-909	94	2	.	.	X
iajs-909	94	3	every	every	DET
iajs-909	94	4	fuzzy	fuzzy	ADJ
iajs-909	94	5	simple	simple	ADJ
iajs-909	94	6	module	module	NOUN
iajs-909	94	7	is	be	AUX
iajs-909	94	8	a	a	DET
iajs-909	94	9	chained	chain	VERB
iajs-909	94	10	fuzzy	fuzzy	ADJ
iajs-909	94	11	module	module	NOUN
iajs-909	94	12	.	.	PUNCT
iajs-909	95	1	2.6	2.6	NUM
iajs-909	95	2	remark	remark	NOUN
iajs-909	95	3	if	if	SCONJ
iajs-909	95	4	yx	yx	NOUN
iajs-909	95	5	and	and	CCONJ
iajs-909	95	6	x	x	X
iajs-909	95	7	is	be	AUX
iajs-909	95	8	a	a	DET
iajs-909	95	9	chained	chain	VERB
iajs-909	95	10	fuzzy	fuzzy	ADJ
iajs-909	95	11	module	module	NOUN
iajs-909	95	12	then	then	ADV
iajs-909	95	13	y	y	PROPN
iajs-909	95	14	is	be	AUX
iajs-909	95	15	chained	chain	VERB
iajs-909	95	16	fuzzy	fuzzy	ADJ
iajs-909	95	17	module	module	NOUN
iajs-909	95	18	.	.	PUNCT
iajs-909	96	1	proof	proof	NOUN
iajs-909	96	2	:	:	PUNCT
iajs-909	96	3	let	let	VERB
iajs-909	96	4	a	a	DET
iajs-909	96	5	,	,	PUNCT
iajs-909	96	6	b	b	NOUN
iajs-909	96	7	be	be	AUX
iajs-909	96	8	two	two	NUM
iajs-909	96	9	fuzzy	fuzzy	ADJ
iajs-909	96	10	submodules	submodule	NOUN
iajs-909	96	11	of	of	ADP
iajs-909	96	12	y	y	PROPN
iajs-909	96	13	then	then	ADV
iajs-909	96	14	a	a	PRON
iajs-909	96	15	,	,	PUNCT
iajs-909	96	16	b	b	NOUN
iajs-909	96	17	are	be	AUX
iajs-909	96	18	fuzzy	fuzzy	ADJ
iajs-909	96	19	submodules	submodule	NOUN
iajs-909	96	20	of	of	ADP
iajs-909	96	21	x	x	PRON
iajs-909	96	22	,	,	PUNCT
iajs-909	96	23	since	since	SCONJ
iajs-909	96	24	x	x	PRON
iajs-909	96	25	is	be	AUX
iajs-909	96	26	chained	chain	VERB
iajs-909	96	27	fuzzy	fuzzy	ADJ
iajs-909	96	28	module	module	NOUN
iajs-909	96	29	.	.	PUNCT
iajs-909	97	1	then	then	ADV
iajs-909	97	2	ab	ab	PROPN
iajs-909	97	3	or	or	CCONJ
iajs-909	97	4	ba	ba	NOUN
iajs-909	97	5	which	which	PRON
iajs-909	97	6	implies	imply	VERB
iajs-909	97	7	y	y	PROPN
iajs-909	97	8	is	be	AUX
iajs-909	97	9	a	a	DET
iajs-909	97	10	chained	chain	VERB
iajs-909	97	11	fuzzy	fuzzy	ADJ
iajs-909	97	12	module	module	NOUN
iajs-909	97	13	.	.	PUNCT
iajs-909	98	1	2.7	2.7	NUM
iajs-909	98	2	definition	definition	NOUN
iajs-909	98	3	,	,	PUNCT
iajs-909	98	4	[	[	X
iajs-909	98	5	10	10	NUM
iajs-909	98	6	]	]	X
iajs-909	98	7	a	a	DET
iajs-909	98	8	fuzzy	fuzzy	ADJ
iajs-909	98	9	module	module	NOUN
iajs-909	98	10	x	x	PRON
iajs-909	98	11	is	be	AUX
iajs-909	98	12	called	call	VERB
iajs-909	98	13	uniform	uniform	ADJ
iajs-909	98	14	fuzzy	fuzzy	ADJ
iajs-909	98	15	module	module	NOUN
iajs-909	98	16	if	if	SCONJ
iajs-909	98	17	ab01	ab01	PROPN
iajs-909	98	18	for	for	ADP
iajs-909	98	19	any	any	DET
iajs-909	98	20	nontrivial	nontrivial	ADJ
iajs-909	98	21	fuzzy	fuzzy	ADJ
iajs-909	98	22	submodules	submodule	NOUN
iajs-909	98	23	a	a	DET
iajs-909	98	24	and	and	CCONJ
iajs-909	98	25	b	b	NOUN
iajs-909	98	26	of	of	ADP
iajs-909	98	27	x.	x.	NOUN
iajs-909	98	28	2.8	2.8	NUM
iajs-909	98	29	proposition	proposition	NOUN
iajs-909	98	30	a	a	DET
iajs-909	98	31	fuzzy	fuzzy	ADJ
iajs-909	98	32	module	module	NOUN
iajs-909	98	33	x	x	PUNCT
iajs-909	98	34	of	of	ADP
iajs-909	98	35	an	an	DET
iajs-909	98	36	r	r	NOUN
iajs-909	98	37	-	-	PUNCT
iajs-909	98	38	module	module	NOUN
iajs-909	98	39	m	m	NOUN
iajs-909	98	40	is	be	AUX
iajs-909	98	41	uniform	uniform	ADJ
iajs-909	98	42	if	if	SCONJ
iajs-909	98	43	and	and	CCONJ
iajs-909	98	44	only	only	ADV
iajs-909	98	45	if	if	SCONJ
iajs-909	98	46	xt	xt	PROPN
iajs-909	98	47	is	be	AUX
iajs-909	98	48	a	a	DET
iajs-909	98	49	uniform	uniform	ADJ
iajs-909	98	50	module	module	NOUN
iajs-909	98	51	,	,	PUNCT
iajs-909	98	52			NOUN
iajs-909	98	53	t(0,1	t(0,1	NOUN
iajs-909	98	54	]	]	PUNCT
iajs-909	98	55	.	.	PUNCT
iajs-909	99	1	proof	proof	NOUN
iajs-909	99	2	:	:	PUNCT
iajs-909	99	3	if	if	SCONJ
iajs-909	99	4	x	x	PRON
iajs-909	99	5	is	be	AUX
iajs-909	99	6	uniform	uniform	ADJ
iajs-909	99	7	fuzzy	fuzzy	ADJ
iajs-909	99	8	module	module	NOUN
iajs-909	99	9	,	,	PUNCT
iajs-909	99	10	to	to	PART
iajs-909	99	11	prove	prove	VERB
iajs-909	99	12	xt	xt	PROPN
iajs-909	99	13	is	be	AUX
iajs-909	99	14	uniform	uniform	ADJ
iajs-909	99	15	module	module	NOUN
iajs-909	99	16			NOUN
iajs-909	99	17	t(0,1	t(0,1	NOUN
iajs-909	99	18	]	]	PUNCT
iajs-909	99	19	.	.	PUNCT
iajs-909	100	1	let	let	VERB
iajs-909	100	2	i	i	PRON
iajs-909	100	3	,	,	PUNCT
iajs-909	100	4	j	j	PROPN
iajs-909	100	5	be	be	VERB
iajs-909	100	6	submodules	submodule	NOUN
iajs-909	100	7	of	of	ADP
iajs-909	100	8	xt	xt	PROPN
iajs-909	100	9	.	.	PUNCT
iajs-909	101	1	define	define	VERB
iajs-909	101	2	t	t	PROPN
iajs-909	101	3	x	x	SYM
iajs-909	101	4	(	(	PUNCT
iajs-909	101	5	x	x	X
iajs-909	101	6	)	)	PUNCT
iajs-909	101	7	0	0	NUM
iajs-909	102	1	x	x	SYM
iajs-909	102	2			PUNCT
iajs-909	103	1			NOUN
iajs-909	103	2			NUM
iajs-909	104	1			NUM
iajs-909	104	2			NOUN
iajs-909	104	3	,	,	PUNCT
iajs-909	104	4	t	t	PROPN
iajs-909	104	5	x	x	X
iajs-909	104	6	j	j	PROPN
iajs-909	104	7	b(x	b(x	NOUN
iajs-909	104	8	)	)	PUNCT
iajs-909	104	9	0	0	NUM
iajs-909	105	1	x	x	SYM
iajs-909	105	2	j	j	PROPN
iajs-909	105	3			NUM
iajs-909	106	1			NUM
iajs-909	107	1			NUM
iajs-909	107	2			NOUN
iajs-909	108	1	a	a	PRON
iajs-909	108	2	,	,	PUNCT
iajs-909	108	3	b	b	NOUN
iajs-909	108	4	are	be	AUX
iajs-909	108	5	fuzzy	fuzzy	ADJ
iajs-909	108	6	submodules	submodule	NOUN
iajs-909	108	7	of	of	ADP
iajs-909	108	8	x.	x.	NOUN
iajs-909	108	9	but	but	CCONJ
iajs-909	108	10	at	at	ADP
iajs-909	108	11	=	=	PROPN
iajs-909	108	12	i	i	PROPN
iajs-909	108	13	,	,	PUNCT
iajs-909	108	14	bt	bt	PROPN
iajs-909	108	15	=	=	PROPN
iajs-909	108	16	j	j	NOUN
iajs-909	108	17	since	since	SCONJ
iajs-909	108	18	x	x	PROPN
iajs-909	108	19	is	be	AUX
iajs-909	108	20	uniform	uniform	ADJ
iajs-909	108	21	fuzzy	fuzzy	ADJ
iajs-909	108	22	module	module	NOUN
iajs-909	108	23	,	,	PUNCT
iajs-909	108	24	then	then	ADV
iajs-909	108	25	ab01	ab01	PROPN
iajs-909	108	26	.	.	PUNCT
iajs-909	109	1	hence	hence	ADV
iajs-909	109	2	(	(	PUNCT
iajs-909	109	3	ab)t01	ab)t01	NUM
iajs-909	109	4	which	which	PRON
iajs-909	109	5	implies	imply	VERB
iajs-909	109	6	atbt01	atbt01	PROPN
iajs-909	109	7	(	(	PUNCT
iajs-909	109	8	by	by	ADP
iajs-909	109	9	remark	remark	NOUN
iajs-909	109	10	1.5	1.5	NUM
iajs-909	109	11	)	)	PUNCT
iajs-909	109	12	.	.	PUNCT
iajs-909	110	1	this	this	DET
iajs-909	110	2	ij01	ij01	NOUN
iajs-909	110	3	.	.	PUNCT
iajs-909	111	1	conversely	conversely	ADV
iajs-909	111	2	,	,	PUNCT
iajs-909	111	3	if	if	SCONJ
iajs-909	111	4	xt	xt	PROPN
iajs-909	111	5	is	be	AUX
iajs-909	111	6	uniform	uniform	ADJ
iajs-909	111	7	module	module	NOUN
iajs-909	111	8	,	,	PUNCT
iajs-909	111	9	to	to	PART
iajs-909	111	10	prove	prove	VERB
iajs-909	111	11	x	x	PUNCT
iajs-909	111	12	is	be	AUX
iajs-909	111	13	a	a	DET
iajs-909	111	14	uniform	uniform	ADJ
iajs-909	111	15	fuzzy	fuzzy	ADJ
iajs-909	111	16	module	module	NOUN
iajs-909	111	17	,	,	PUNCT
iajs-909	111	18	let	let	VERB
iajs-909	111	19	a	a	DET
iajs-909	111	20	,	,	PUNCT
iajs-909	111	21	b	b	NOUN
iajs-909	111	22	fuzzy	fuzzy	ADJ
iajs-909	111	23	submodules	submodule	NOUN
iajs-909	111	24	in	in	ADP
iajs-909	111	25	x.	x.	NOUN
iajs-909	111	26	then	then	ADV
iajs-909	111	27	at	at	ADP
iajs-909	111	28	,	,	PUNCT
iajs-909	111	29	bt	bt	PROPN
iajs-909	111	30	are	be	AUX
iajs-909	111	31	submodules	submodule	NOUN
iajs-909	111	32	in	in	ADP
iajs-909	111	33	xt	xt	PROPN
iajs-909	111	34	,	,	PUNCT
iajs-909	111	35	for	for	ADP
iajs-909	111	36	all	all	DET
iajs-909	111	37	t(0,1	t(0,1	NOUN
iajs-909	111	38	]	]	PUNCT
iajs-909	111	39	,	,	PUNCT
iajs-909	111	40	since	since	SCONJ
iajs-909	111	41	xt	xt	PROPN
iajs-909	111	42	is	be	AUX
iajs-909	111	43	uniform	uniform	ADJ
iajs-909	111	44	rmodule	rmodule	NOUN
iajs-909	111	45	then	then	ADV
iajs-909	111	46	atbt01	atbt01	PROPN
iajs-909	111	47	which	which	PRON
iajs-909	111	48	implies	imply	VERB
iajs-909	111	49	(	(	PUNCT
iajs-909	111	50	ab)t01	ab)t01	NUM
iajs-909	111	51	(	(	PUNCT
iajs-909	111	52	by	by	ADP
iajs-909	111	53	remark	remark	NOUN
iajs-909	111	54	1.5	1.5	NUM
iajs-909	111	55	)	)	PUNCT
iajs-909	111	56	.	.	PUNCT
iajs-909	112	1	thus	thus	ADV
iajs-909	112	2	ab01	ab01	PROPN
iajs-909	112	3	.	.	PUNCT
iajs-909	113	1	now	now	ADV
iajs-909	113	2	,	,	PUNCT
iajs-909	113	3	we	we	PRON
iajs-909	113	4	shall	shall	AUX
iajs-909	113	5	show	show	VERB
iajs-909	113	6	the	the	DET
iajs-909	113	7	relationship	relationship	NOUN
iajs-909	113	8	between	between	ADP
iajs-909	113	9	uniform	uniform	ADJ
iajs-909	113	10	fuzzy	fuzzy	ADJ
iajs-909	113	11	module	module	NOUN
iajs-909	113	12	and	and	CCONJ
iajs-909	113	13	chained	chain	VERB
iajs-909	113	14	fuzzy	fuzzy	ADJ
iajs-909	113	15	module	module	NOUN
iajs-909	113	16	as	as	ADP
iajs-909	113	17	the	the	DET
iajs-909	113	18	following	follow	VERB
iajs-909	113	19	proposition	proposition	NOUN
iajs-909	113	20	:	:	PUNCT
iajs-909	113	21	2.9	2.9	NUM
iajs-909	113	22	proposition	proposition	NOUN
iajs-909	113	23	every	every	DET
iajs-909	113	24	chained	chain	VERB
iajs-909	113	25	fuzzy	fuzzy	ADJ
iajs-909	113	26	module	module	NOUN
iajs-909	113	27	is	be	AUX
iajs-909	113	28	a	a	DET
iajs-909	113	29	uniform	uniform	ADJ
iajs-909	113	30	fuzzy	fuzzy	ADJ
iajs-909	113	31	module	module	NOUN
iajs-909	113	32	.	.	PUNCT
iajs-909	114	1	proof	proof	NOUN
iajs-909	114	2	:	:	PUNCT
iajs-909	114	3	let	let	VERB
iajs-909	114	4	x	x	PRON
iajs-909	114	5	be	be	AUX
iajs-909	114	6	a	a	DET
iajs-909	114	7	chained	chain	VERB
iajs-909	114	8	fuzzy	fuzzy	ADJ
iajs-909	114	9	module	module	NOUN
iajs-909	114	10	of	of	ADP
iajs-909	114	11	an	an	DET
iajs-909	114	12	r	r	NOUN
iajs-909	114	13	-	-	PUNCT
iajs-909	114	14	module	module	NOUN
iajs-909	114	15	m	m	NOUN
iajs-909	114	16	then	then	ADV
iajs-909	114	17	ab	ab	PROPN
iajs-909	114	18	or	or	CCONJ
iajs-909	114	19	ba	ba	NOUN
iajs-909	114	20	if	if	SCONJ
iajs-909	114	21	ab	ab	PROPN
iajs-909	114	22	then	then	ADV
iajs-909	114	23	ab=	ab=	PROPN
iajs-909	114	24	a	a	DET
iajs-909	114	25	if	if	SCONJ
iajs-909	114	26	ba	ba	NOUN
iajs-909	114	27	then	then	ADV
iajs-909	114	28	ab	ab	NOUN
iajs-909	114	29	=	=	PROPN
iajs-909	114	30	b	b	NOUN
iajs-909	114	31	which	which	PRON
iajs-909	114	32	implies	imply	VERB
iajs-909	114	33	ab=01	ab=01	NOUN
iajs-909	114	34	.	.	PUNCT
iajs-909	115	1	2.10	2.10	NUM
iajs-909	115	2	remark	remark	NOUN
iajs-909	115	3	the	the	DET
iajs-909	115	4	converse	converse	NOUN
iajs-909	115	5	of	of	ADP
iajs-909	115	6	proposition	proposition	NOUN
iajs-909	115	7	(	(	PUNCT
iajs-909	115	8	2.9	2.9	NUM
iajs-909	115	9	)	)	PUNCT
iajs-909	115	10	is	be	AUX
iajs-909	115	11	not	not	PART
iajs-909	115	12	true	true	ADJ
iajs-909	115	13	for	for	SCONJ
iajs-909	115	14	the	the	DET
iajs-909	115	15	following	follow	VERB
iajs-909	115	16	example	example	NOUN
iajs-909	115	17	shows	show	VERB
iajs-909	115	18	:	:	PUNCT
iajs-909	115	19	2.11	2.11	NUM
iajs-909	115	20	example	example	NOUN
iajs-909	115	21	let	let	VERB
iajs-909	115	22	m	m	PRON
iajs-909	115	23	=	=	NOUN
iajs-909	115	24	z	z	NOUN
iajs-909	115	25	as	as	ADP
iajs-909	115	26	a	a	DET
iajs-909	115	27	z	z	NOUN
iajs-909	115	28	-	-	PUNCT
iajs-909	115	29	module	module	NOUN
iajs-909	115	30	x	x	NOUN
iajs-909	115	31	:	:	PUNCT
iajs-909	115	32	m[0,1	m[0,1	NOUN
iajs-909	115	33	]	]	X
iajs-909	115	34	such	such	ADJ
iajs-909	115	35	that	that	SCONJ
iajs-909	115	36	x(x)=1	x(x)=1	PROPN
iajs-909	115	37	xm	xm	PROPN
iajs-909	115	38	xt	xt	PROPN
iajs-909	115	39	=	=	PROPN
iajs-909	115	40	z	z	PROPN
iajs-909	115	41	for	for	ADP
iajs-909	115	42	all	all	DET
iajs-909	115	43	t[0,1	t[0,1	NOUN
iajs-909	115	44	]	]	PUNCT
iajs-909	115	45	.	.	PUNCT
iajs-909	116	1	but	but	CCONJ
iajs-909	116	2	z	z	NOUN
iajs-909	116	3	is	be	AUX
iajs-909	116	4	uniform	uniform	ADJ
iajs-909	116	5	.	.	PUNCT
iajs-909	117	1	hence	hence	ADV
iajs-909	117	2	by	by	ADP
iajs-909	117	3	proposition	proposition	NOUN
iajs-909	117	4	(	(	PUNCT
iajs-909	117	5	2.9	2.9	NUM
iajs-909	117	6	)	)	PUNCT
iajs-909	117	7	x	x	X
iajs-909	117	8	is	be	AUX
iajs-909	117	9	a	a	DET
iajs-909	117	10	uniform	uniform	ADJ
iajs-909	117	11	fuzzy	fuzzy	ADJ
iajs-909	117	12	module	module	NOUN
iajs-909	117	13	.	.	PUNCT
iajs-909	118	1	but	but	CCONJ
iajs-909	118	2	x	x	X
iajs-909	118	3	is	be	AUX
iajs-909	118	4	not	not	PART
iajs-909	118	5	chained	chain	VERB
iajs-909	118	6	fuzzy	fuzzy	ADJ
iajs-909	118	7	module	module	NOUN
iajs-909	118	8	since	since	SCONJ
iajs-909	118	9			PROPN
iajs-909	118	10	a	a	PRON
iajs-909	118	11	,	,	PUNCT
iajs-909	118	12	b	b	NOUN
iajs-909	118	13	fuzzy	fuzzy	ADJ
iajs-909	118	14	submodules	submodule	NOUN
iajs-909	118	15	of	of	ADP
iajs-909	118	16	x	x	PUNCT
iajs-909	118	17	defined	define	VERB
iajs-909	118	18	by	by	ADP
iajs-909	118	19	1	1	NUM
iajs-909	118	20	x	x	SYM
iajs-909	118	21	(	(	PUNCT
iajs-909	118	22	2	2	NUM
iajs-909	118	23	)	)	PUNCT
iajs-909	118	24	(	(	PUNCT
iajs-909	118	25	x	x	X
iajs-909	118	26	)	)	PUNCT
iajs-909	118	27	1	1	NUM
iajs-909	118	28	x	x	SYM
iajs-909	118	29	(	(	PUNCT
iajs-909	118	30	2	2	NUM
iajs-909	118	31	)	)	PUNCT
iajs-909	118	32	4	4	NUM
iajs-909	118	33			NOUN
iajs-909	118	34			NUM
iajs-909	118	35			PROPN
iajs-909	118	36			PROPN
iajs-909	118	37			NUM
iajs-909	118	38			X
iajs-909	118	39	,	,	PUNCT
iajs-909	118	40	1	1	NUM
iajs-909	118	41	x	x	SYM
iajs-909	118	42	(	(	PUNCT
iajs-909	118	43	5	5	NUM
iajs-909	118	44	)	)	PUNCT
iajs-909	118	45	b(x	b(x	NOUN
iajs-909	118	46	)	)	PUNCT
iajs-909	118	47	1	1	NUM
iajs-909	118	48	x	x	SYM
iajs-909	118	49	(	(	PUNCT
iajs-909	118	50	5	5	NUM
iajs-909	118	51	)	)	PUNCT
iajs-909	118	52	4	4	NUM
iajs-909	118	53			NOUN
iajs-909	118	54			NUM
iajs-909	118	55			NOUN
iajs-909	118	56			NUM
iajs-909	118	57			NUM
iajs-909	118	58	and	and	CCONJ
iajs-909	118	59	a	a	DET
iajs-909	118	60			PROPN
iajs-909	118	61	b	b	PROPN
iajs-909	118	62	and	and	CCONJ
iajs-909	118	63	b	b	PROPN
iajs-909	118	64			PROPN
iajs-909	118	65	a.	a.	NOUN
iajs-909	118	66	recall	recall	NOUN
iajs-909	118	67	that	that	SCONJ
iajs-909	118	68	if	if	SCONJ
iajs-909	118	69	a	a	PRON
iajs-909	118	70	and	and	CCONJ
iajs-909	118	71	b	b	NOUN
iajs-909	118	72	are	be	AUX
iajs-909	118	73	two	two	NUM
iajs-909	118	74	submodules	submodule	NOUN
iajs-909	118	75	of	of	ADP
iajs-909	118	76	an	an	DET
iajs-909	118	77	r	r	NOUN
iajs-909	118	78	-	-	PUNCT
iajs-909	118	79	module	module	NOUN
iajs-909	118	80	m	m	NOUN
iajs-909	118	81	,	,	PUNCT
iajs-909	118	82	then	then	ADV
iajs-909	118	83	a	a	PRON
iajs-909	118	84	and	and	CCONJ
iajs-909	118	85	b	b	NOUN
iajs-909	118	86	are	be	AUX
iajs-909	118	87	called	call	VERB
iajs-909	118	88	comparable	comparable	ADJ
iajs-909	118	89	if	if	SCONJ
iajs-909	118	90	ab	ab	PROPN
iajs-909	118	91	and	and	CCONJ
iajs-909	118	92	ba	ba	NOUN
iajs-909	118	93	.	.	PUNCT
iajs-909	119	1	we	we	PRON
iajs-909	119	2	shall	shall	AUX
iajs-909	119	3	fuzzify	fuzzify	VERB
iajs-909	119	4	this	this	DET
iajs-909	119	5	concept	concept	NOUN
iajs-909	119	6	as	as	SCONJ
iajs-909	119	7	follows	follow	VERB
iajs-909	119	8	:	:	PUNCT
iajs-909	119	9	ihjpas	ihjpas	PROPN
iajs-909	119	10	ibn	ibn	PROPN
iajs-909	119	11	alhaitham	alhaitham	PROPN
iajs-909	120	1	j.	j.	PROPN
iajs-909	121	1	fo	fo	ADP
iajs-909	121	2	r	r	NOUN
iajs-909	121	3	pure	pure	ADJ
iajs-909	121	4	&	&	CCONJ
iajs-909	121	5	appl	appl	PROPN
iajs-909	121	6	.	.	PUNCT
iajs-909	122	1	sc	sc	PROPN
iajs-909	122	2	i.	i.	PROPN
iajs-909	122	3	vo	vo	PROPN
iajs-909	122	4	l.23	l.23	PROPN
iajs-909	122	5	(	(	PUNCT
iajs-909	122	6	2	2	NUM
iajs-909	122	7	)	)	PUNCT
iajs-909	122	8	2010	2010	NUM
iajs-909	122	9	2.12	2.12	NUM
iajs-909	122	10	definition	definition	NOUN
iajs-909	122	11	let	let	VERB
iajs-909	122	12	a	a	DET
iajs-909	122	13	,	,	PUNCT
iajs-909	122	14	b	b	NOUN
iajs-909	122	15	be	be	AUX
iajs-909	122	16	two	two	NUM
iajs-909	122	17	fuzzy	fuzzy	ADJ
iajs-909	122	18	submodules	submodule	NOUN
iajs-909	122	19	of	of	ADP
iajs-909	122	20	a	a	DET
iajs-909	122	21	fuzzy	fuzzy	ADJ
iajs-909	122	22	module	module	NOUN
iajs-909	122	23	x	x	PUNCT
iajs-909	122	24	of	of	ADP
iajs-909	122	25	an	an	DET
iajs-909	122	26	r	r	NOUN
iajs-909	122	27	-	-	PUNCT
iajs-909	122	28	module	module	NOUN
iajs-909	122	29	m	m	NOUN
iajs-909	122	30	,	,	PUNCT
iajs-909	122	31	then	then	ADV
iajs-909	122	32	a	a	PRON
iajs-909	122	33	and	and	CCONJ
iajs-909	122	34	b	b	NOUN
iajs-909	122	35	are	be	AUX
iajs-909	122	36	called	call	VERB
iajs-909	122	37	comparable	comparable	ADJ
iajs-909	122	38	if	if	SCONJ
iajs-909	122	39	ab	ab	PROPN
iajs-909	122	40	and	and	CCONJ
iajs-909	122	41	ba	ba	NOUN
iajs-909	122	42	.	.	PUNCT
iajs-909	123	1	2.13	2.13	NUM
iajs-909	123	2	proposition	proposition	NOUN
iajs-909	123	3	a	a	DET
iajs-909	123	4	fuzzy	fuzzy	ADJ
iajs-909	123	5	module	module	NOUN
iajs-909	123	6	x	x	PUNCT
iajs-909	123	7	of	of	ADP
iajs-909	123	8	an	an	DET
iajs-909	123	9	r	r	NOUN
iajs-909	123	10	-	-	PUNCT
iajs-909	123	11	module	module	NOUN
iajs-909	123	12	m	m	NOUN
iajs-909	123	13	is	be	AUX
iajs-909	123	14	chained	chain	VERB
iajs-909	123	15	iff	iff	PROPN
iajs-909	123	16	every	every	DET
iajs-909	123	17	two	two	NUM
iajs-909	123	18	cyclic	cyclic	ADJ
iajs-909	123	19	fuzzy	fuzzy	ADJ
iajs-909	123	20	submodules	submodule	NOUN
iajs-909	123	21	of	of	ADP
iajs-909	123	22	x	x	SYM
iajs-909	123	23	are	be	AUX
iajs-909	123	24	comparable	comparable	ADJ
iajs-909	123	25	.	.	PUNCT
iajs-909	124	1	proof	proof	NOUN
iajs-909	124	2	:	:	PUNCT
iajs-909	124	3	let	let	VERB
iajs-909	124	4	a	a	PRON
iajs-909	124	5	and	and	CCONJ
iajs-909	124	6	b	b	NOUN
iajs-909	124	7	be	be	AUX
iajs-909	124	8	fuzzy	fuzzy	ADJ
iajs-909	124	9	submodules	submodule	NOUN
iajs-909	124	10	of	of	ADP
iajs-909	124	11	x.	x.	NOUN
iajs-909	124	12	suppose	suppose	VERB
iajs-909	124	13	a	a	DET
iajs-909	124	14			PROPN
iajs-909	124	15	b	b	PROPN
iajs-909	124	16	,	,	PUNCT
iajs-909	124	17	we	we	PRON
iajs-909	124	18	show	show	VERB
iajs-909	124	19	ba	ba	NOUN
iajs-909	124	20	since	since	SCONJ
iajs-909	124	21	a	a	DET
iajs-909	124	22			PROPN
iajs-909	124	23	b	b	PROPN
iajs-909	124	24	,	,	PUNCT
iajs-909	124	25	there	there	PRON
iajs-909	124	26	exists	exist	VERB
iajs-909	124	27	xta	xta	NOUN
iajs-909	125	1	and	and	CCONJ
iajs-909	125	2	xtb	xtb	PROPN
iajs-909	126	1	<	<	X
iajs-909	126	2	xt>a	xt>a	PROPN
iajs-909	126	3	and	and	CCONJ
iajs-909	126	4	<	<	X
iajs-909	126	5	xt	xt	X
iajs-909	126	6	>	>	X
iajs-909	126	7			PROPN
iajs-909	126	8	b.	b.	PROPN
iajs-909	126	9	let	let	VERB
iajs-909	126	10	ykb	ykb	PROPN
iajs-909	126	11	,	,	PUNCT
iajs-909	126	12	then	then	ADV
iajs-909	126	13	<	<	X
iajs-909	126	14	yk>b	yk>b	NOUN
iajs-909	126	15	,	,	PUNCT
iajs-909	126	16	<	<	X
iajs-909	126	17	xt	xt	X
iajs-909	126	18	>	>	X
iajs-909	126	19	,	,	PUNCT
iajs-909	126	20	<	<	X
iajs-909	126	21	yk	yk	X
iajs-909	126	22	>	>	X
iajs-909	126	23	are	be	AUX
iajs-909	126	24	cyclic	cyclic	ADJ
iajs-909	126	25	fuzzy	fuzzy	ADJ
iajs-909	126	26	submodules	submodule	NOUN
iajs-909	126	27	of	of	ADP
iajs-909	126	28	x	x	NOUN
iajs-909	126	29	,	,	PUNCT
iajs-909	126	30	then	then	ADV
iajs-909	126	31	either	either	CCONJ
iajs-909	126	32	<	<	X
iajs-909	126	33	xt	xt	X
iajs-909	126	34	>	>	X
iajs-909	126	35			PROPN
iajs-909	126	36	<	<	X
iajs-909	126	37	yk	yk	X
iajs-909	126	38	>	>	X
iajs-909	126	39	or	or	CCONJ
iajs-909	126	40	<	<	X
iajs-909	126	41	y	y	PROPN
iajs-909	126	42	k	k	PROPN
iajs-909	126	43	>	>	X
iajs-909	126	44			PROPN
iajs-909	126	45	<	<	X
iajs-909	126	46	xt	xt	X
iajs-909	126	47	>	>	X
iajs-909	126	48	.	.	PUNCT
iajs-909	127	1	if	if	SCONJ
iajs-909	127	2	<	<	X
iajs-909	127	3	xt	xt	X
iajs-909	127	4	>	>	X
iajs-909	127	5			PROPN
iajs-909	127	6	<	<	X
iajs-909	127	7	yk	yk	PROPN
iajs-909	127	8	>	>	X
iajs-909	127	9	implies	imply	VERB
iajs-909	127	10	<	<	X
iajs-909	127	11	xt	xt	X
iajs-909	127	12	>	>	X
iajs-909	127	13			PROPN
iajs-909	127	14	b	b	PROPN
iajs-909	127	15	(	(	PUNCT
iajs-909	127	16	since	since	SCONJ
iajs-909	127	17	<	<	X
iajs-909	127	18	yk>b	yk>b	NOUN
iajs-909	127	19	)	)	PUNCT
iajs-909	127	20	.	.	PUNCT
iajs-909	128	1	thus	thus	ADV
iajs-909	128	2	xtb	xtb	PRON
iajs-909	128	3	is	be	AUX
iajs-909	128	4	a	a	DET
iajs-909	128	5	contradiction	contradiction	NOUN
iajs-909	128	6	.	.	PUNCT
iajs-909	129	1	if	if	SCONJ
iajs-909	129	2	<	<	X
iajs-909	129	3	yk	yk	X
iajs-909	129	4	>	>	X
iajs-909	129	5			PROPN
iajs-909	129	6	<	<	X
iajs-909	129	7	xt	xt	X
iajs-909	129	8	>	>	X
iajs-909	129	9	implies	imply	VERB
iajs-909	129	10	that	that	SCONJ
iajs-909	129	11	<	<	X
iajs-909	129	12	yk	yk	X
iajs-909	129	13	>	>	X
iajs-909	129	14			PROPN
iajs-909	129	15	a	a	PRON
iajs-909	129	16	(	(	PUNCT
iajs-909	129	17	since	since	SCONJ
iajs-909	129	18	<	<	X
iajs-909	129	19	xt	xt	X
iajs-909	129	20	>	>	X
iajs-909	129	21			PROPN
iajs-909	129	22	a	a	PRON
iajs-909	129	23	)	)	PUNCT
iajs-909	129	24	.	.	PUNCT
iajs-909	130	1	thus	thus	ADV
iajs-909	130	2	ba	ba	VERB
iajs-909	130	3	so	so	ADV
iajs-909	130	4	x	x	VERB
iajs-909	130	5	is	be	AUX
iajs-909	130	6	chained	chain	VERB
iajs-909	130	7	.	.	PUNCT
iajs-909	131	1	the	the	DET
iajs-909	131	2	converse	converse	NOUN
iajs-909	131	3	is	be	AUX
iajs-909	131	4	obvious	obvious	ADJ
iajs-909	131	5	.	.	PUNCT
iajs-909	132	1	2.14	2.14	NUM
iajs-909	132	2	remark	remark	VERB
iajs-909	132	3	a	a	DET
iajs-909	132	4	chained	chain	VERB
iajs-909	132	5	fuzzy	fuzzy	ADJ
iajs-909	132	6	module	module	NOUN
iajs-909	132	7	is	be	AUX
iajs-909	132	8	indecomposable	indecomposable	ADJ
iajs-909	132	9	.	.	PUNCT
iajs-909	133	1	proof	proof	NOUN
iajs-909	133	2	:	:	PUNCT
iajs-909	133	3	suppose	suppose	VERB
iajs-909	133	4	x	x	PRON
iajs-909	133	5	is	be	AUX
iajs-909	133	6	decomposable	decomposable	ADJ
iajs-909	133	7	,	,	PUNCT
iajs-909	133	8	then	then	ADV
iajs-909	133	9	x	x	X
iajs-909	133	10	=	=	NOUN
iajs-909	133	11	ab	ab	NOUN
iajs-909	133	12	for	for	ADP
iajs-909	133	13	some	some	DET
iajs-909	133	14	fuzzy	fuzzy	ADJ
iajs-909	133	15	submodule	submodule	NOUN
iajs-909	133	16	a	a	PRON
iajs-909	133	17	and	and	CCONJ
iajs-909	133	18	b	b	NOUN
iajs-909	133	19	of	of	ADP
iajs-909	133	20	x.	x.	NOUN
iajs-909	133	21	thus	thus	ADV
iajs-909	133	22	ab	ab	NOUN
iajs-909	133	23	=	=	NOUN
iajs-909	133	24	o1	o1	NOUN
iajs-909	133	25	is	be	AUX
iajs-909	133	26	a	a	DET
iajs-909	133	27	contradiction	contradiction	NOUN
iajs-909	133	28	(	(	PUNCT
iajs-909	133	29	proposition	proposition	NOUN
iajs-909	133	30	2.9	2.9	NUM
iajs-909	133	31	)	)	PUNCT
iajs-909	133	32	.	.	PUNCT
iajs-909	134	1	now	now	ADV
iajs-909	134	2	,	,	PUNCT
iajs-909	134	3	we	we	PRON
iajs-909	134	4	introduce	introduce	VERB
iajs-909	134	5	the	the	DET
iajs-909	134	6	notion	notion	NOUN
iajs-909	134	7	of	of	ADP
iajs-909	134	8	chained	chain	VERB
iajs-909	134	9	fuzzy	fuzzy	ADJ
iajs-909	134	10	ring	ring	NOUN
iajs-909	134	11	.	.	PUNCT
iajs-909	135	1	first	first	ADV
iajs-909	135	2	we	we	PRON
iajs-909	135	3	have	have	VERB
iajs-909	135	4	the	the	DET
iajs-909	135	5	following	follow	VERB
iajs-909	135	6	definition	definition	NOUN
iajs-909	135	7	.	.	PUNCT
iajs-909	136	1	2.15	2.15	NUM
iajs-909	136	2	definition	definition	NOUN
iajs-909	136	3	,	,	PUNCT
iajs-909	136	4	[	[	X
iajs-909	136	5	11	11	NUM
iajs-909	136	6	]	]	X
iajs-909	136	7	a	a	DET
iajs-909	136	8	ring	ring	NOUN
iajs-909	136	9	r	r	NOUN
iajs-909	136	10	is	be	AUX
iajs-909	136	11	called	call	VERB
iajs-909	136	12	chained	chain	VERB
iajs-909	136	13	if	if	SCONJ
iajs-909	136	14	and	and	CCONJ
iajs-909	136	15	only	only	ADV
iajs-909	136	16	if	if	SCONJ
iajs-909	136	17	for	for	ADP
iajs-909	136	18	each	each	DET
iajs-909	136	19	fuzzy	fuzzy	ADJ
iajs-909	136	20	ideals	ideal	NOUN
iajs-909	136	21	i	i	PRON
iajs-909	136	22	,	,	PUNCT
iajs-909	136	23	j	j	PROPN
iajs-909	136	24	of	of	ADP
iajs-909	136	25	r	r	NOUN
iajs-909	136	26	either	either	CCONJ
iajs-909	136	27	ij	ij	PROPN
iajs-909	136	28	or	or	CCONJ
iajs-909	136	29	ji	ji	NOUN
iajs-909	136	30	.	.	PUNCT
iajs-909	137	1	2.16	2.16	NUM
iajs-909	137	2	definition	definition	NOUN
iajs-909	137	3	a	a	DET
iajs-909	137	4	fuzzy	fuzzy	ADJ
iajs-909	137	5	ring	ring	NOUN
iajs-909	137	6	x	x	INTJ
iajs-909	137	7	of	of	ADP
iajs-909	137	8	a	a	DET
iajs-909	137	9	ring	ring	NOUN
iajs-909	137	10	r	r	NOUN
iajs-909	137	11	is	be	AUX
iajs-909	137	12	called	call	VERB
iajs-909	137	13	chained	chain	VERB
iajs-909	137	14	if	if	SCONJ
iajs-909	137	15	and	and	CCONJ
iajs-909	137	16	only	only	ADV
iajs-909	137	17	if	if	SCONJ
iajs-909	137	18	for	for	ADP
iajs-909	137	19	each	each	DET
iajs-909	137	20	fuzzy	fuzzy	ADJ
iajs-909	137	21	ideals	ideal	NOUN
iajs-909	137	22	i	i	PRON
iajs-909	137	23	,	,	PUNCT
iajs-909	137	24	j	j	PROPN
iajs-909	137	25	of	of	ADP
iajs-909	137	26	x	x	SYM
iajs-909	137	27	either	either	CCONJ
iajs-909	137	28	ij	ij	PROPN
iajs-909	137	29	or	or	CCONJ
iajs-909	137	30	ji	ji	NOUN
iajs-909	137	31	.	.	PUNCT
iajs-909	138	1	2.17	2.17	NUM
iajs-909	138	2	remark	remark	NOUN
iajs-909	138	3	a	a	DET
iajs-909	138	4	fuzzy	fuzzy	ADJ
iajs-909	138	5	ring	ring	NOUN
iajs-909	138	6	x	x	VERB
iajs-909	138	7	is	be	AUX
iajs-909	138	8	chained	chain	VERB
iajs-909	138	9	if	if	SCONJ
iajs-909	138	10	and	and	CCONJ
iajs-909	138	11	only	only	ADV
iajs-909	138	12	if	if	SCONJ
iajs-909	138	13	xt	xt	PROPN
iajs-909	138	14	is	be	AUX
iajs-909	138	15	chained	chain	VERB
iajs-909	138	16	ring	ring	NOUN
iajs-909	138	17			ADJ
iajs-909	138	18	t(0,1	t(0,1	NOUN
iajs-909	138	19	]	]	PUNCT
iajs-909	138	20	.	.	PUNCT
iajs-909	139	1	proof	proof	NOUN
iajs-909	139	2	:	:	PUNCT
iajs-909	139	3	it	it	PRON
iajs-909	139	4	is	be	AUX
iajs-909	139	5	easy	easy	ADJ
iajs-909	139	6	so	so	SCONJ
iajs-909	139	7	it	it	PRON
iajs-909	139	8	is	be	AUX
iajs-909	139	9	omitted	omit	VERB
iajs-909	139	10	.	.	PUNCT
iajs-909	140	1	2.18	2.18	NUM
iajs-909	140	2	definition	definition	NOUN
iajs-909	140	3	,	,	PUNCT
iajs-909	140	4	[	[	X
iajs-909	140	5	8	8	NUM
iajs-909	140	6	]	]	PUNCT
iajs-909	140	7	a	a	DET
iajs-909	140	8	fuzzy	fuzzy	ADJ
iajs-909	140	9	module	module	NOUN
iajs-909	140	10	x	x	PUNCT
iajs-909	140	11	of	of	ADP
iajs-909	140	12	an	an	DET
iajs-909	140	13	r	r	NOUN
iajs-909	140	14	-	-	PUNCT
iajs-909	140	15	module	module	NOUN
iajs-909	140	16	m	m	NOUN
iajs-909	140	17	is	be	AUX
iajs-909	140	18	called	call	VERB
iajs-909	140	19	multiplication	multiplication	NOUN
iajs-909	140	20	fuzzy	fuzzy	ADJ
iajs-909	140	21	module	module	NOUN
iajs-909	140	22	if	if	SCONJ
iajs-909	140	23	for	for	ADP
iajs-909	140	24	each	each	DET
iajs-909	140	25	nonempty	nonempty	ADJ
iajs-909	140	26	fuzzy	fuzzy	ADJ
iajs-909	140	27	submodule	submodule	NOUN
iajs-909	140	28	a	a	PRON
iajs-909	140	29	of	of	ADP
iajs-909	140	30	x	x	PRON
iajs-909	140	31	,	,	PUNCT
iajs-909	140	32	there	there	PRON
iajs-909	140	33	exists	exist	VERB
iajs-909	140	34	a	a	DET
iajs-909	140	35	fuzzy	fuzzy	ADJ
iajs-909	140	36	ideal	ideal	NOUN
iajs-909	140	37	i	i	PRON
iajs-909	140	38	of	of	ADP
iajs-909	140	39	r	r	NOUN
iajs-909	140	40	such	such	ADJ
iajs-909	140	41	that	that	SCONJ
iajs-909	140	42	a	a	DET
iajs-909	140	43	=	=	NOUN
iajs-909	140	44	ix	ix	ADJ
iajs-909	140	45	.	.	PUNCT
iajs-909	141	1	2.19	2.19	NUM
iajs-909	141	2	proposition	proposition	NOUN
iajs-909	141	3	let	let	VERB
iajs-909	141	4	x	x	PRON
iajs-909	141	5	be	be	AUX
iajs-909	141	6	a	a	DET
iajs-909	141	7	multiplication	multiplication	NOUN
iajs-909	141	8	module	module	NOUN
iajs-909	141	9	of	of	ADP
iajs-909	141	10	an	an	DET
iajs-909	141	11	r	r	NOUN
iajs-909	141	12	-	-	PUNCT
iajs-909	141	13	module	module	NOUN
iajs-909	141	14	m	m	NOUN
iajs-909	141	15	if	if	SCONJ
iajs-909	141	16	r	r	NOUN
iajs-909	141	17	is	be	AUX
iajs-909	141	18	a	a	DET
iajs-909	141	19	chained	chain	VERB
iajs-909	141	20	ring	ring	NOUN
iajs-909	141	21	then	then	ADV
iajs-909	141	22	x	x	PUNCT
iajs-909	141	23	is	be	AUX
iajs-909	141	24	a	a	DET
iajs-909	141	25	chained	chain	VERB
iajs-909	141	26	fuzzy	fuzzy	ADJ
iajs-909	141	27	module	module	NOUN
iajs-909	141	28	.	.	PUNCT
iajs-909	142	1	proof	proof	NOUN
iajs-909	142	2	:	:	PUNCT
iajs-909	142	3	let	let	VERB
iajs-909	142	4	a	a	PRON
iajs-909	142	5	and	and	CCONJ
iajs-909	142	6	b	b	NOUN
iajs-909	142	7	be	be	AUX
iajs-909	142	8	fuzzy	fuzzy	ADJ
iajs-909	142	9	submodules	submodule	NOUN
iajs-909	142	10	of	of	ADP
iajs-909	142	11	x.	x.	NOUN
iajs-909	142	12	then	then	ADV
iajs-909	142	13	there	there	PRON
iajs-909	142	14	exists	exist	VERB
iajs-909	142	15	fuzzy	fuzzy	ADJ
iajs-909	142	16	ideals	ideal	NOUN
iajs-909	143	1	i	i	PRON
iajs-909	143	2	and	and	CCONJ
iajs-909	143	3	j	j	PROPN
iajs-909	143	4	of	of	ADP
iajs-909	143	5	r	r	NOUN
iajs-909	143	6	such	such	ADJ
iajs-909	143	7	that	that	SCONJ
iajs-909	143	8	a	a	DET
iajs-909	143	9	=	=	NOUN
iajs-909	143	10	ix	ix	NOUN
iajs-909	143	11	and	and	CCONJ
iajs-909	143	12	b	b	X
iajs-909	143	13	=	=	PROPN
iajs-909	143	14	jx	jx	PROPN
iajs-909	143	15	,	,	PUNCT
iajs-909	143	16	since	since	SCONJ
iajs-909	143	17	it	it	PRON
iajs-909	143	18	and	and	CCONJ
iajs-909	143	19	jt	jt	PROPN
iajs-909	143	20	ideals	ideal	NOUN
iajs-909	143	21	of	of	ADP
iajs-909	143	22	r	r	NOUN
iajs-909	143	23	and	and	CCONJ
iajs-909	143	24	r	r	NOUN
iajs-909	143	25	is	be	AUX
iajs-909	143	26	chained	chain	VERB
iajs-909	143	27	,	,	PUNCT
iajs-909	143	28	therefore	therefore	ADV
iajs-909	143	29	it	it	PRON
iajs-909	143	30			PROPN
iajs-909	143	31	jt	jt	PROPN
iajs-909	143	32	or	or	CCONJ
iajs-909	143	33	jt	jt	VERB
iajs-909	143	34	it	it	PRON
iajs-909	143	35	.	.	PUNCT
iajs-909	144	1	thus	thus	ADV
iajs-909	144	2	ij	ij	PROPN
iajs-909	144	3	or	or	CCONJ
iajs-909	144	4	ji	ji	PROPN
iajs-909	144	5	(	(	PUNCT
iajs-909	144	6	by	by	ADP
iajs-909	144	7	remark	remark	NOUN
iajs-909	144	8	2.3	2.3	NUM
iajs-909	144	9	)	)	PUNCT
iajs-909	144	10	implies	imply	VERB
iajs-909	144	11	that	that	SCONJ
iajs-909	144	12	ixjx	ixjx	NOUN
iajs-909	144	13	or	or	CCONJ
iajs-909	144	14	jxix	jxix	INTJ
iajs-909	144	15	.	.	PUNCT
iajs-909	145	1	thus	thus	ADV
iajs-909	145	2	ab	ab	PROPN
iajs-909	145	3	or	or	CCONJ
iajs-909	145	4	ba	ba	NOUN
iajs-909	145	5	.	.	PUNCT
iajs-909	146	1	2.20	2.20	NUM
iajs-909	146	2	definition	definition	NOUN
iajs-909	146	3	let	let	VERB
iajs-909	146	4	x	x	PRON
iajs-909	146	5	be	be	AUX
iajs-909	146	6	a	a	DET
iajs-909	146	7	chained	chain	VERB
iajs-909	146	8	fuzzy	fuzzy	ADJ
iajs-909	146	9	module	module	NOUN
iajs-909	146	10	of	of	ADP
iajs-909	146	11	an	an	DET
iajs-909	146	12	r	r	NOUN
iajs-909	146	13	-	-	PUNCT
iajs-909	146	14	module	module	NOUN
iajs-909	146	15	m	m	NOUN
iajs-909	146	16	and	and	CCONJ
iajs-909	146	17	let	let	VERB
iajs-909	146	18	v(x)={(o1	v(x)={(o1	VERB
iajs-909	146	19	:	:	PUNCT
iajs-909	146	20	xt)xtx	xt)xtx	NOUN
iajs-909	146	21	}	}	PUNCT
iajs-909	146	22	.	.	PUNCT
iajs-909	147	1	2.21	2.21	NUM
iajs-909	147	2	definition	definition	NOUN
iajs-909	147	3	,	,	PUNCT
iajs-909	147	4	[	[	X
iajs-909	147	5	10	10	NUM
iajs-909	147	6	]	]	PUNCT
iajs-909	147	7	let	let	VERB
iajs-909	147	8	x	x	PRON
iajs-909	147	9	be	be	AUX
iajs-909	147	10	a	a	DET
iajs-909	147	11	non	non	X
iajs-909	147	12	empty	empty	ADJ
iajs-909	147	13	fuzzy	fuzzy	ADJ
iajs-909	147	14	module	module	NOUN
iajs-909	147	15	of	of	ADP
iajs-909	147	16	r	r	NOUN
iajs-909	147	17	-	-	PUNCT
iajs-909	147	18	module	module	NOUN
iajs-909	147	19	m.	m.	NOUN
iajs-909	147	20	the	the	DET
iajs-909	147	21	fuzzy	fuzzy	ADJ
iajs-909	147	22	annihilator	annihilator	NOUN
iajs-909	147	23	of	of	ADP
iajs-909	147	24	a	a	DET
iajs-909	147	25	denoted	denote	VERB
iajs-909	147	26	by	by	ADP
iajs-909	147	27	(	(	PUNCT
iajs-909	147	28	f	f	NOUN
iajs-909	147	29	-	-	PUNCT
iajs-909	147	30	anna	anna	NOUN
iajs-909	147	31	)	)	PUNCT
iajs-909	147	32	is	be	AUX
iajs-909	147	33	defined	define	VERB
iajs-909	147	34	by	by	ADP
iajs-909	147	35	{	{	PUNCT
iajs-909	147	36	xt	xt	ADP
iajs-909	147	37	:	:	PUNCT
iajs-909	147	38	xr	xr	NOUN
iajs-909	147	39	,	,	PUNCT
iajs-909	147	40	xtao1},t[0,1	xtao1},t[0,1	NOUN
iajs-909	147	41	]	]	X
iajs-909	147	42	,	,	PUNCT
iajs-909	147	43	where	where	SCONJ
iajs-909	147	44	a	a	PRON
iajs-909	147	45	is	be	AUX
iajs-909	147	46	a	a	DET
iajs-909	147	47	proper	proper	ADJ
iajs-909	147	48	fuzzy	fuzzy	ADJ
iajs-909	147	49	submodule	submodule	NOUN
iajs-909	147	50	of	of	ADP
iajs-909	147	51	x.	x.	PROPN
iajs-909	147	52	note	note	VERB
iajs-909	147	53	that	that	SCONJ
iajs-909	147	54	:	:	PUNCT
iajs-909	147	55	(	(	PUNCT
iajs-909	147	56	f	f	X
iajs-909	147	57	-	-	PUNCT
iajs-909	147	58	anna)(a)=sup{t	anna)(a)=sup{t	NOUN
iajs-909	147	59	:	:	PUNCT
iajs-909	147	60	t[0,1],atao1	t[0,1],atao1	NOUN
iajs-909	147	61	}	}	PUNCT
iajs-909	147	62	,	,	PUNCT
iajs-909	147	63	for	for	ADP
iajs-909	147	64	all	all	DET
iajs-909	147	65	ar	ar	ADP
iajs-909	147	66	;	;	PUNCT
iajs-909	147	67	that	that	PRON
iajs-909	147	68	is	be	AUX
iajs-909	147	69	f	f	NOUN
iajs-909	147	70	-	-	PUNCT
iajs-909	147	71	anna=(o1	anna=(o1	NUM
iajs-909	147	72	:	:	PUNCT
iajs-909	147	73	a	a	NOUN
iajs-909	147	74	)	)	PUNCT
iajs-909	147	75	.	.	PUNCT
iajs-909	148	1	2.22	2.22	NUM
iajs-909	148	2	definition	definition	NOUN
iajs-909	148	3	,	,	PUNCT
iajs-909	148	4	[	[	X
iajs-909	148	5	10	10	NUM
iajs-909	148	6	]	]	X
iajs-909	148	7	a	a	DET
iajs-909	148	8	fuzzy	fuzzy	ADJ
iajs-909	148	9	module	module	NOUN
iajs-909	148	10	x	x	PRON
iajs-909	148	11	is	be	AUX
iajs-909	148	12	called	call	VERB
iajs-909	148	13	faithful	faithful	ADJ
iajs-909	148	14	if	if	SCONJ
iajs-909	148	15	f	f	PROPN
iajs-909	148	16	-	-	PUNCT
iajs-909	148	17	annx	annx	PROPN
iajs-909	148	18	=	=	NOUN
iajs-909	148	19	o1	o1	PROPN
iajs-909	148	20	.	.	PUNCT
iajs-909	149	1	2.23	2.23	NUM
iajs-909	149	2	remark	remark	NOUN
iajs-909	149	3	if	if	SCONJ
iajs-909	149	4	x	x	PRON
iajs-909	149	5	is	be	AUX
iajs-909	149	6	chained	chain	VERB
iajs-909	149	7	faithful	faithful	ADJ
iajs-909	149	8	fuzzy	fuzzy	ADJ
iajs-909	149	9	module	module	NOUN
iajs-909	149	10	then	then	ADV
iajs-909	149	11	1	1	NUM
iajs-909	149	12	to	to	ADP
iajs-909	149	13	x	x	PROPN
iajs-909	149	14	x	x	PROPN
iajs-909	149	15			NOUN
iajs-909	149	16			PUNCT
iajs-909	149	17	(	(	PUNCT
iajs-909	149	18	o1	o1	NOUN
iajs-909	149	19	:	:	PUNCT
iajs-909	149	20	xt)=o1	xt)=o1	PROPN
iajs-909	149	21	,	,	PUNCT
iajs-909	149	22	xtx	xtx	X
iajs-909	149	23	.	.	PUNCT
iajs-909	149	24	ihjpas	ihjpa	VERB
iajs-909	149	25	ibn	ibn	PROPN
iajs-909	149	26	alhaitham	alhaitham	PROPN
iajs-909	150	1	j.	j.	PROPN
iajs-909	151	1	fo	fo	ADP
iajs-909	151	2	r	r	NOUN
iajs-909	151	3	pure	pure	ADJ
iajs-909	151	4	&	&	CCONJ
iajs-909	151	5	appl	appl	PROPN
iajs-909	151	6	.	.	PUNCT
iajs-909	152	1	sc	sc	PROPN
iajs-909	152	2	i.	i.	PROPN
iajs-909	152	3	vo	vo	PROPN
iajs-909	152	4	l.23	l.23	PROPN
iajs-909	152	5	(	(	PUNCT
iajs-909	152	6	2	2	NUM
iajs-909	152	7	)	)	PUNCT
iajs-909	152	8	2010	2010	NUM
iajs-909	152	9	proof	proof	NOUN
iajs-909	152	10	:	:	PUNCT
iajs-909	152	11	if	if	SCONJ
iajs-909	152	12	1	1	NUM
iajs-909	152	13	to	to	PART
iajs-909	152	14	x	x	PROPN
iajs-909	152	15	x	x	PROPN
iajs-909	152	16			NOUN
iajs-909	152	17			PUNCT
iajs-909	152	18	(	(	PUNCT
iajs-909	152	19	o1	o1	NOUN
iajs-909	152	20	:	:	PUNCT
iajs-909	152	21	xt)o1	xt)o1	PROPN
iajs-909	152	22	then	then	ADV
iajs-909	152	23	there	there	PRON
iajs-909	152	24	is	be	VERB
iajs-909	152	25	rℓr	rℓr	VERB
iajs-909	152	26	,	,	PUNCT
iajs-909	152	27	rℓo1	rℓo1	VERB
iajs-909	152	28	such	such	ADJ
iajs-909	152	29	that	that	DET
iajs-909	152	30	rℓxt	rℓxt	NOUN
iajs-909	152	31	=	=	SYM
iajs-909	152	32	o1	o1	PROPN
iajs-909	152	33	,	,	PUNCT
iajs-909	152	34	xtx	xtx	NOUN
iajs-909	152	35	then	then	ADV
iajs-909	152	36	rℓx	rℓx	VERB
iajs-909	152	37	=	=	NOUN
iajs-909	152	38	o1	o1	NOUN
iajs-909	152	39	a	a	DET
iajs-909	152	40	contradiction	contradiction	NOUN
iajs-909	152	41	.	.	PUNCT
iajs-909	153	1	2.24	2.24	NUM
iajs-909	153	2	definition	definition	NOUN
iajs-909	153	3	,	,	PUNCT
iajs-909	153	4	[	[	X
iajs-909	153	5	13	13	NUM
iajs-909	153	6	]	]	PUNCT
iajs-909	153	7	a	a	DET
iajs-909	153	8	fuzzy	fuzzy	ADJ
iajs-909	153	9	ideal	ideal	NOUN
iajs-909	153	10	a	a	PRON
iajs-909	153	11	of	of	ADP
iajs-909	153	12	a	a	DET
iajs-909	153	13	ring	ring	NOUN
iajs-909	153	14	r	r	NOUN
iajs-909	153	15	is	be	AUX
iajs-909	153	16	called	call	VERB
iajs-909	153	17	fuzzy	fuzzy	ADJ
iajs-909	153	18	prime	prime	ADJ
iajs-909	153	19	ideal	ideal	NOUN
iajs-909	153	20	,	,	PUNCT
iajs-909	153	21	if	if	SCONJ
iajs-909	153	22	a	a	PRON
iajs-909	153	23	is	be	AUX
iajs-909	153	24	non	non	ADJ
iajs-909	153	25	-	-	ADJ
iajs-909	153	26	constant	constant	ADJ
iajs-909	153	27	and	and	CCONJ
iajs-909	153	28	for	for	ADP
iajs-909	153	29	any	any	DET
iajs-909	153	30	fuzzy	fuzzy	ADJ
iajs-909	153	31	ideals	ideal	NOUN
iajs-909	153	32	b	b	NOUN
iajs-909	153	33	and	and	CCONJ
iajs-909	153	34	c	c	PROPN
iajs-909	153	35	of	of	ADP
iajs-909	153	36	r	r	NOUN
iajs-909	153	37	such	such	ADJ
iajs-909	153	38	that	that	SCONJ
iajs-909	153	39	b	b	ADV
iajs-909	153	40	ca	ca	NOUN
iajs-909	153	41	,	,	PUNCT
iajs-909	153	42	then	then	ADV
iajs-909	153	43	either	either	CCONJ
iajs-909	153	44	ba	ba	NOUN
iajs-909	153	45	or	or	CCONJ
iajs-909	153	46	ca	ca	NOUN
iajs-909	153	47	.	.	PUNCT
iajs-909	154	1	equivalently	equivalently	ADV
iajs-909	154	2	,	,	PUNCT
iajs-909	154	3	a	a	DET
iajs-909	154	4	fuzzy	fuzzy	ADJ
iajs-909	154	5	ideal	ideal	NOUN
iajs-909	154	6	a	a	PRON
iajs-909	154	7	of	of	ADP
iajs-909	154	8	a	a	DET
iajs-909	154	9	ring	ring	NOUN
iajs-909	154	10	r	r	NOUN
iajs-909	154	11	is	be	AUX
iajs-909	154	12	called	call	VERB
iajs-909	154	13	fuzzy	fuzzy	ADJ
iajs-909	154	14	prime	prime	ADJ
iajs-909	154	15	ideal	ideal	NOUN
iajs-909	154	16	if	if	SCONJ
iajs-909	154	17	a	a	PRON
iajs-909	154	18	is	be	AUX
iajs-909	154	19	a	a	DET
iajs-909	154	20	non	non	ADJ
iajs-909	154	21	-	-	ADJ
iajs-909	154	22	constant	constant	ADJ
iajs-909	154	23	and	and	CCONJ
iajs-909	154	24	for	for	ADP
iajs-909	154	25	all	all	DET
iajs-909	154	26	aℓ	aℓ	ADJ
iajs-909	154	27	,	,	PUNCT
iajs-909	154	28	bh	bh	NOUN
iajs-909	154	29	fuzzy	fuzzy	ADJ
iajs-909	154	30	singletons	singleton	NOUN
iajs-909	154	31	of	of	ADP
iajs-909	154	32	r	r	NOUN
iajs-909	154	33	such	such	ADJ
iajs-909	154	34	that	that	SCONJ
iajs-909	154	35	aℓbha	aℓbha	ADV
iajs-909	154	36	implies	imply	VERB
iajs-909	154	37	that	that	SCONJ
iajs-909	154	38	either	either	CCONJ
iajs-909	154	39	aℓ	aℓ	NOUN
iajs-909	154	40	a	a	NOUN
iajs-909	154	41	or	or	CCONJ
iajs-909	154	42	bha	bha	NOUN
iajs-909	154	43	,	,	PUNCT
iajs-909	154	44	ℓ	ℓ	PROPN
iajs-909	154	45	,	,	PUNCT
iajs-909	154	46	h[0,1	h[0,1	PROPN
iajs-909	154	47	]	]	PUNCT
iajs-909	154	48	.	.	PUNCT
iajs-909	155	1	2.25	2.25	NUM
iajs-909	155	2	remark	remark	NOUN
iajs-909	155	3	if	if	SCONJ
iajs-909	155	4	x	x	PRON
iajs-909	155	5	is	be	AUX
iajs-909	155	6	a	a	DET
iajs-909	155	7	chained	chain	VERB
iajs-909	155	8	fuzzy	fuzzy	ADJ
iajs-909	155	9	module	module	NOUN
iajs-909	155	10	of	of	ADP
iajs-909	155	11	an	an	DET
iajs-909	155	12	r	r	NOUN
iajs-909	155	13	-	-	PUNCT
iajs-909	155	14	module	module	NOUN
iajs-909	155	15	m	m	NOUN
iajs-909	155	16	then	then	ADV
iajs-909	155	17	,	,	PUNCT
iajs-909	155	18	1	1	X
iajs-909	155	19	.	.	X
iajs-909	156	1	v(x	v(x	NOUN
iajs-909	156	2	)	)	PUNCT
iajs-909	156	3	is	be	AUX
iajs-909	156	4	a	a	DET
iajs-909	156	5	linearly	linearly	ADV
iajs-909	156	6	ordered	order	VERB
iajs-909	156	7	set	set	NOUN
iajs-909	156	8	of	of	ADP
iajs-909	156	9	fuzzy	fuzzy	ADJ
iajs-909	156	10	ideals	ideal	NOUN
iajs-909	156	11	of	of	ADP
iajs-909	156	12	r.	r.	PROPN
iajs-909	156	13	2	2	NUM
iajs-909	156	14	.	.	PUNCT
iajs-909	156	15	p=	p=	NOUN
iajs-909	156	16	1	1	NUM
iajs-909	156	17	to	to	ADP
iajs-909	156	18	x	x	PROPN
iajs-909	156	19	x	x	PROPN
iajs-909	156	20			PROPN
iajs-909	156	21			PROPN
iajs-909	156	22	(	(	PUNCT
iajs-909	156	23	o1	o1	PROPN
iajs-909	156	24	:	:	PUNCT
iajs-909	156	25	xt	xt	NUM
iajs-909	156	26	)	)	PUNCT
iajs-909	156	27	is	be	AUX
iajs-909	156	28	a	a	DET
iajs-909	156	29	fuzzy	fuzzy	ADJ
iajs-909	156	30	prime	prime	ADJ
iajs-909	156	31	ideal	ideal	NOUN
iajs-909	156	32	of	of	ADP
iajs-909	156	33	r.	r.	PROPN
iajs-909	156	34	proof	proof	NOUN
iajs-909	156	35	:	:	PUNCT
iajs-909	156	36	(	(	PUNCT
iajs-909	156	37	1	1	X
iajs-909	156	38	)	)	PUNCT
iajs-909	156	39	let	let	VERB
iajs-909	156	40	a	a	DET
iajs-909	156	41	,	,	PUNCT
iajs-909	156	42	b	b	NOUN
iajs-909	156	43			NOUN
iajs-909	156	44	v(x	v(x	PROPN
iajs-909	156	45	)	)	PUNCT
iajs-909	156	46	then	then	ADV
iajs-909	156	47	a=(o1	a=(o1	VERB
iajs-909	156	48	:	:	PUNCT
iajs-909	156	49	xt	xt	NUM
iajs-909	156	50	)	)	PUNCT
iajs-909	156	51	and	and	CCONJ
iajs-909	156	52	b=(o1	b=(o1	NOUN
iajs-909	156	53	:	:	PUNCT
iajs-909	156	54	yt	yt	NOUN
iajs-909	156	55	)	)	PUNCT
iajs-909	156	56	for	for	ADP
iajs-909	156	57	some	some	DET
iajs-909	156	58	xto1	xto1	PROPN
iajs-909	156	59	,	,	PUNCT
iajs-909	156	60	yt	yt	PROPN
iajs-909	156	61	=	=	NOUN
iajs-909	156	62	o1	o1	PROPN
iajs-909	156	63	and	and	CCONJ
iajs-909	156	64	xt	xt	NOUN
iajs-909	156	65	,	,	PUNCT
iajs-909	156	66	ytx	ytx	NOUN
iajs-909	156	67	since	since	SCONJ
iajs-909	156	68	x	x	PRON
iajs-909	156	69	is	be	AUX
iajs-909	156	70	chained	chain	VERB
iajs-909	156	71	fuzzy	fuzzy	ADJ
iajs-909	156	72	module	module	NOUN
iajs-909	156	73	then	then	ADV
iajs-909	156	74	xt	xt	PROPN
iajs-909	156	75	is	be	AUX
iajs-909	156	76	chained	chain	VERB
iajs-909	156	77	module	module	NOUN
iajs-909	156	78	(	(	PUNCT
iajs-909	156	79	by	by	ADP
iajs-909	156	80	theorem	theorem	NOUN
iajs-909	156	81	(	(	PUNCT
iajs-909	156	82	2.4	2.4	NUM
iajs-909	156	83	)	)	PUNCT
iajs-909	156	84	)	)	PUNCT
iajs-909	157	1	implies	imply	VERB
iajs-909	157	2	that	that	SCONJ
iajs-909	157	3	v(xt	v(xt	NOUN
iajs-909	157	4	)	)	PUNCT
iajs-909	157	5	is	be	AUX
iajs-909	157	6	a	a	DET
iajs-909	157	7	linearly	linearly	ADV
iajs-909	157	8	ordered	order	VERB
iajs-909	157	9	set	set	NOUN
iajs-909	157	10	of	of	ADP
iajs-909	157	11	ideals	ideal	NOUN
iajs-909	157	12	of	of	ADP
iajs-909	157	13	r	r	NOUN
iajs-909	157	14	(	(	PUNCT
iajs-909	157	15	see	see	VERB
iajs-909	157	16	[	[	X
iajs-909	157	17	12,remark	12,remark	NUM
iajs-909	157	18	(	(	PUNCT
iajs-909	157	19	1.9	1.9	NUM
iajs-909	157	20	)	)	PUNCT
iajs-909	157	21	)	)	PUNCT
iajs-909	157	22	.	.	PUNCT
iajs-909	158	1	thus	thus	ADV
iajs-909	158	2	v(x	v(x	PROPN
iajs-909	158	3	)	)	PUNCT
iajs-909	158	4	is	be	AUX
iajs-909	158	5	a	a	DET
iajs-909	158	6	linearly	linearly	ADV
iajs-909	158	7	ordered	order	VERB
iajs-909	158	8	set	set	NOUN
iajs-909	158	9	of	of	ADP
iajs-909	158	10	fuzzy	fuzzy	ADJ
iajs-909	158	11	ideals	ideal	NOUN
iajs-909	158	12	of	of	ADP
iajs-909	158	13	r.	r.	PROPN
iajs-909	158	14	(	(	PUNCT
iajs-909	158	15	2	2	NUM
iajs-909	158	16	)	)	PUNCT
iajs-909	158	17	p=	p=	NOUN
iajs-909	158	18	1	1	NUM
iajs-909	158	19	to	to	ADP
iajs-909	158	20	x	x	PROPN
iajs-909	158	21	x	x	PROPN
iajs-909	158	22			PROPN
iajs-909	158	23			PROPN
iajs-909	158	24	(	(	PUNCT
iajs-909	158	25	o1	o1	PROPN
iajs-909	158	26	:	:	PUNCT
iajs-909	158	27	xt	xt	NUM
iajs-909	158	28	)	)	PUNCT
iajs-909	158	29	is	be	AUX
iajs-909	158	30	a	a	DET
iajs-909	158	31	fuzzy	fuzzy	ADJ
iajs-909	158	32	ideal	ideal	NOUN
iajs-909	158	33	of	of	ADP
iajs-909	158	34	r.	r.	PROPN
iajs-909	158	35	to	to	PART
iajs-909	158	36	show	show	VERB
iajs-909	158	37	that	that	SCONJ
iajs-909	158	38	p	p	NOUN
iajs-909	158	39	is	be	AUX
iajs-909	158	40	a	a	DET
iajs-909	158	41	fuzzy	fuzzy	ADJ
iajs-909	158	42	prime	prime	ADJ
iajs-909	158	43	ideal	ideal	NOUN
iajs-909	158	44	,	,	PUNCT
iajs-909	158	45	let	let	VERB
iajs-909	158	46	aℓ	aℓ	NOUN
iajs-909	158	47	,	,	PUNCT
iajs-909	158	48	bhr	bhr	VERB
iajs-909	158	49	such	such	ADJ
iajs-909	158	50	that	that	SCONJ
iajs-909	158	51	aℓbhp	aℓbhp	PROPN
iajs-909	158	52	,	,	PUNCT
iajs-909	158	53	then	then	ADV
iajs-909	158	54	there	there	PRON
iajs-909	158	55	is	be	VERB
iajs-909	158	56	o1	o1	ADP
iajs-909	158	57	xtx	xtx	NOUN
iajs-909	158	58	such	such	ADJ
iajs-909	158	59	that	that	DET
iajs-909	158	60	aℓbh(o1	aℓbh(o1	PROPN
iajs-909	158	61	:	:	PUNCT
iajs-909	158	62	xt	xt	NUM
iajs-909	158	63	)	)	PUNCT
iajs-909	158	64	.	.	PUNCT
iajs-909	159	1	then	then	ADV
iajs-909	159	2	aℓbh	aℓbh	PROPN
iajs-909	159	3	xt	xt	PROPN
iajs-909	159	4	=	=	NOUN
iajs-909	159	5	o1	o1	PROPN
iajs-909	159	6	.	.	PUNCT
iajs-909	160	1	this	this	PRON
iajs-909	160	2	implies	imply	VERB
iajs-909	160	3	that	that	SCONJ
iajs-909	160	4	aℓ(o1	aℓ(o1	VERB
iajs-909	160	5	:	:	PUNCT
iajs-909	160	6	bhxt	bhxt	NOUN
iajs-909	160	7	)	)	PUNCT
iajs-909	160	8	.	.	PUNCT
iajs-909	161	1	now	now	ADV
iajs-909	161	2	if	if	SCONJ
iajs-909	161	3	bhxt=	bhxt=	NUM
iajs-909	161	4	o1	o1	PROPN
iajs-909	161	5	then	then	ADV
iajs-909	161	6	bh(o1	bh(o1	PROPN
iajs-909	161	7	:	:	PUNCT
iajs-909	161	8	xt	xt	NUM
iajs-909	161	9	)	)	PUNCT
iajs-909	161	10	.	.	PUNCT
iajs-909	162	1	thus	thus	ADV
iajs-909	162	2	bhp	bhp	VERB
iajs-909	162	3	and	and	CCONJ
iajs-909	162	4	if	if	SCONJ
iajs-909	162	5	bhxt	bhxt	NOUN
iajs-909	162	6	o1	o1	NOUN
iajs-909	162	7	then	then	ADV
iajs-909	162	8	bhxt	bhxt	NOUN
iajs-909	162	9	o1x	o1x	NOUN
iajs-909	162	10	,	,	PUNCT
iajs-909	162	11	and	and	CCONJ
iajs-909	162	12	hence	hence	ADV
iajs-909	162	13	aℓp	aℓp	NOUN
iajs-909	162	14	.	.	PUNCT
iajs-909	163	1	3	3	X
iajs-909	163	2	.	.	X
iajs-909	163	3	direct	direct	ADJ
iajs-909	163	4	sum	sum	NOUN
iajs-909	163	5	of	of	ADP
iajs-909	163	6	chained	chain	VERB
iajs-909	163	7	fuzzy	fuzzy	ADJ
iajs-909	163	8	module	module	NOUN
iajs-909	163	9	we	we	PRON
iajs-909	163	10	turn	turn	VERB
iajs-909	163	11	attention	attention	NOUN
iajs-909	163	12	to	to	ADP
iajs-909	163	13	the	the	DET
iajs-909	163	14	direct	direct	ADJ
iajs-909	163	15	sum	sum	NOUN
iajs-909	163	16	of	of	ADP
iajs-909	163	17	chained	chain	VERB
iajs-909	163	18	fuzzy	fuzzy	ADJ
iajs-909	163	19	modules	module	NOUN
iajs-909	163	20	.	.	PUNCT
iajs-909	164	1	3.1	3.1	NUM
iajs-909	164	2	remark	remark	NOUN
iajs-909	164	3	if	if	SCONJ
iajs-909	164	4	x	x	PRON
iajs-909	164	5	and	and	CCONJ
iajs-909	164	6	y	y	PROPN
iajs-909	164	7	are	be	AUX
iajs-909	164	8	two	two	NUM
iajs-909	164	9	chained	chain	VERB
iajs-909	164	10	fuzzy	fuzzy	ADJ
iajs-909	164	11	modules	module	NOUN
iajs-909	164	12	of	of	ADP
iajs-909	164	13	an	an	DET
iajs-909	164	14	r	r	NOUN
iajs-909	164	15	-	-	PUNCT
iajs-909	164	16	module	module	NOUN
iajs-909	164	17	m1	m1	NOUN
iajs-909	164	18	and	and	CCONJ
iajs-909	164	19	m	m	PROPN
iajs-909	164	20	2	2	NUM
iajs-909	164	21	respectively	respectively	ADV
iajs-909	164	22	then	then	ADV
iajs-909	164	23	xy	xy	PROPN
iajs-909	164	24	is	be	AUX
iajs-909	164	25	not	not	PART
iajs-909	164	26	necessary	necessary	ADJ
iajs-909	164	27	chained	chain	VERB
iajs-909	164	28	fuzzy	fuzzy	ADJ
iajs-909	164	29	module	module	NOUN
iajs-909	164	30	of	of	ADP
iajs-909	164	31	m1	m1	PROPN
iajs-909	164	32			PROPN
iajs-909	164	33	m	m	VERB
iajs-909	164	34	2	2	NUM
iajs-909	164	35	as	as	SCONJ
iajs-909	164	36	the	the	DET
iajs-909	164	37	following	follow	VERB
iajs-909	164	38	example	example	NOUN
iajs-909	164	39	shows	show	VERB
iajs-909	164	40	:	:	PUNCT
iajs-909	164	41	3.2	3.2	NUM
iajs-909	164	42	example	example	NOUN
iajs-909	164	43	let	let	VERB
iajs-909	164	44	x	x	PRON
iajs-909	164	45	:	:	PUNCT
iajs-909	164	46	z6{0	z6{0	SYM
iajs-909	164	47	,	,	PUNCT
iajs-909	164	48	1	1	NUM
iajs-909	164	49	3	3	NUM
iajs-909	164	50	}	}	PUNCT
iajs-909	164	51	such	such	ADJ
iajs-909	164	52	that	that	SCONJ
iajs-909	164	53	1	1	NUM
iajs-909	164	54	if	if	SCONJ
iajs-909	164	55	a	a	DET
iajs-909	164	56	2	2	NUM
iajs-909	164	57	x(a	x(a	NOUN
iajs-909	164	58	)	)	PUNCT
iajs-909	164	59	3	3	NUM
iajs-909	164	60	0	0	NUM
iajs-909	164	61	if	if	SCONJ
iajs-909	164	62	a	a	DET
iajs-909	164	63	,	,	PUNCT
iajs-909	164	64	2	2	NUM
iajs-909	164	65			NOUN
iajs-909	164	66			NOUN
iajs-909	164	67			PUNCT
iajs-909	165	1			NUM
iajs-909	165	2			NUM
iajs-909	165	3			NOUN
iajs-909	165	4			NUM
iajs-909	165	5			PROPN
iajs-909	165	6			PROPN
iajs-909	165	7	let	let	VERB
iajs-909	165	8	y	y	PRON
iajs-909	165	9	:	:	PUNCT
iajs-909	165	10	z6{0	z6{0	NUM
iajs-909	165	11	,	,	PUNCT
iajs-909	165	12	1	1	NUM
iajs-909	165	13	3	3	NUM
iajs-909	165	14	}	}	PUNCT
iajs-909	165	15	such	such	ADJ
iajs-909	165	16	that	that	SCONJ
iajs-909	165	17	1	1	NUM
iajs-909	165	18	if	if	SCONJ
iajs-909	165	19	a	a	DET
iajs-909	165	20	3	3	NUM
iajs-909	165	21	y(a	y(a	NOUN
iajs-909	165	22	)	)	PUNCT
iajs-909	165	23	3	3	NUM
iajs-909	165	24	0	0	NUM
iajs-909	165	25	if	if	SCONJ
iajs-909	165	26	a	a	PRON
iajs-909	165	27	,	,	PUNCT
iajs-909	165	28	3	3	NUM
iajs-909	165	29			NOUN
iajs-909	165	30			NOUN
iajs-909	165	31			PUNCT
iajs-909	166	1			NUM
iajs-909	166	2			NUM
iajs-909	166	3			NUM
iajs-909	166	4			NOUN
iajs-909	166	5			PROPN
iajs-909	166	6			NOUN
iajs-909	166	7	it	it	PRON
iajs-909	166	8	is	be	AUX
iajs-909	166	9	clear	clear	ADJ
iajs-909	166	10	that	that	SCONJ
iajs-909	166	11	x	x	PRON
iajs-909	166	12	and	and	CCONJ
iajs-909	166	13	y	y	PROPN
iajs-909	166	14	are	be	AUX
iajs-909	166	15	chained	chain	VERB
iajs-909	166	16	fuzzy	fuzzy	ADJ
iajs-909	166	17	modules	module	NOUN
iajs-909	166	18	of	of	ADP
iajs-909	166	19	z6	z6	PROPN
iajs-909	166	20	.	.	PUNCT
iajs-909	167	1	hence	hence	ADV
iajs-909	167	2	xy	xy	PROPN
iajs-909	167	3	is	be	AUX
iajs-909	167	4	not	not	PART
iajs-909	167	5	a	a	DET
iajs-909	167	6	chained	chain	VERB
iajs-909	167	7	fuzzy	fuzzy	ADJ
iajs-909	167	8	module	module	NOUN
iajs-909	167	9	of	of	ADP
iajs-909	167	10	z6z6	z6z6	PROPN
iajs-909	167	11	.	.	PUNCT
iajs-909	168	1	since	since	SCONJ
iajs-909	168	2			NUM
iajs-909	168	3	a	a	DET
iajs-909	168	4	,	,	PUNCT
iajs-909	168	5	b	b	NOUN
iajs-909	168	6	fuzzy	fuzzy	ADJ
iajs-909	168	7	submodules	submodule	NOUN
iajs-909	168	8	of	of	ADP
iajs-909	168	9	xy	xy	PROPN
iajs-909	168	10	,	,	PUNCT
iajs-909	168	11	where	where	SCONJ
iajs-909	168	12	1	1	NUM
iajs-909	168	13	if	if	SCONJ
iajs-909	168	14	(	(	PUNCT
iajs-909	168	15	a	a	DET
iajs-909	168	16	,	,	PUNCT
iajs-909	168	17	b	b	NOUN
iajs-909	168	18	)	)	PUNCT
iajs-909	168	19	2	2	NUM
iajs-909	168	20	0	0	NUM
iajs-909	168	21	a(a	a(a	PROPN
iajs-909	168	22	,	,	PUNCT
iajs-909	168	23	b	b	X
iajs-909	168	24	)	)	PUNCT
iajs-909	168	25	3	3	NUM
iajs-909	168	26	0	0	NUM
iajs-909	168	27	if	if	SCONJ
iajs-909	168	28	(	(	PUNCT
iajs-909	168	29	a	a	DET
iajs-909	168	30	,	,	PUNCT
iajs-909	168	31	b	b	NOUN
iajs-909	168	32	)	)	PUNCT
iajs-909	168	33	,	,	PUNCT
iajs-909	168	34	2	2	NUM
iajs-909	168	35	0	0	NUM
iajs-909	168	36			NOUN
iajs-909	168	37			VERB
iajs-909	168	38			PRON
iajs-909	168	39			PROPN
iajs-909	168	40			PUNCT
iajs-909	169	1			ADJ
iajs-909	169	2			NUM
iajs-909	169	3			NOUN
iajs-909	169	4			NOUN
iajs-909	169	5			PROPN
iajs-909	169	6			INTJ
iajs-909	169	7			PROPN
iajs-909	169	8			PROPN
iajs-909	169	9	and	and	CCONJ
iajs-909	169	10	1	1	NUM
iajs-909	169	11	if	if	SCONJ
iajs-909	169	12	(	(	PUNCT
iajs-909	169	13	a	a	DET
iajs-909	169	14	,	,	PUNCT
iajs-909	169	15	b	b	NOUN
iajs-909	169	16	)	)	PUNCT
iajs-909	169	17	0	0	NUM
iajs-909	169	18	3	3	NUM
iajs-909	169	19	b(a	b(a	NOUN
iajs-909	169	20	,	,	PUNCT
iajs-909	169	21	b	b	X
iajs-909	169	22	)	)	PUNCT
iajs-909	169	23	3	3	NUM
iajs-909	169	24	0	0	NUM
iajs-909	170	1	if	if	SCONJ
iajs-909	170	2	(	(	PUNCT
iajs-909	170	3	a	a	DET
iajs-909	170	4	,	,	PUNCT
iajs-909	170	5	b	b	NOUN
iajs-909	170	6	)	)	PUNCT
iajs-909	170	7	,	,	PUNCT
iajs-909	170	8	0	0	NUM
iajs-909	170	9	3	3	NUM
iajs-909	170	10			NOUN
iajs-909	170	11			VERB
iajs-909	170	12			PRON
iajs-909	170	13			PROPN
iajs-909	170	14			PUNCT
iajs-909	171	1			ADJ
iajs-909	171	2			NUM
iajs-909	171	3			NOUN
iajs-909	171	4			NOUN
iajs-909	171	5			PROPN
iajs-909	171	6			INTJ
iajs-909	171	7			PROPN
iajs-909	171	8			PROPN
iajs-909	172	1			PROPN
iajs-909	172	2	ihjpas	ihjpa	VERB
iajs-909	172	3	ibn	ibn	PROPN
iajs-909	172	4	alhaitham	alhaitham	PROPN
iajs-909	173	1	j.	j.	PROPN
iajs-909	174	1	fo	fo	ADP
iajs-909	174	2	r	r	NOUN
iajs-909	174	3	pure	pure	ADJ
iajs-909	174	4	&	&	CCONJ
iajs-909	174	5	appl	appl	PROPN
iajs-909	174	6	.	.	PUNCT
iajs-909	175	1	sc	sc	PROPN
iajs-909	175	2	i.	i.	PROPN
iajs-909	175	3	vo	vo	PROPN
iajs-909	175	4	l.23	l.23	PROPN
iajs-909	175	5	(	(	PUNCT
iajs-909	175	6	2	2	NUM
iajs-909	175	7	)	)	PUNCT
iajs-909	175	8	2010	2010	NUM
iajs-909	175	9	but	but	CCONJ
iajs-909	175	10	a(2,0)=	a(2,0)=	NOUN
iajs-909	175	11	1	1	NUM
iajs-909	175	12	3	3	NUM
iajs-909	175	13	,	,	PUNCT
iajs-909	175	14	b(2,0)=0	b(2,0)=0	PROPN
iajs-909	175	15	,	,	PUNCT
iajs-909	175	16	that	that	PRON
iajs-909	175	17	is	be	AUX
iajs-909	175	18	a	a	DET
iajs-909	175	19			PROPN
iajs-909	175	20	b.	b.	PROPN
iajs-909	175	21	alaso	alaso	PROPN
iajs-909	175	22	a(0,3)=0	a(0,3)=0	PROPN
iajs-909	175	23	,	,	PUNCT
iajs-909	175	24	b(0,3)=	b(0,3)=	NUM
iajs-909	175	25	1	1	NUM
iajs-909	175	26	3	3	NUM
iajs-909	175	27	,	,	PUNCT
iajs-909	175	28	that	that	PRON
iajs-909	175	29	is	be	AUX
iajs-909	175	30	b	b	PROPN
iajs-909	175	31			PROPN
iajs-909	175	32	a.	a.	NOUN
iajs-909	175	33	thus	thus	ADV
iajs-909	175	34	xy	xy	PROPN
iajs-909	175	35	is	be	AUX
iajs-909	175	36	not	not	PART
iajs-909	175	37	a	a	DET
iajs-909	175	38	chained	chain	VERB
iajs-909	175	39	fuzzy	fuzzy	ADJ
iajs-909	175	40	module	module	NOUN
iajs-909	175	41	of	of	ADP
iajs-909	175	42	z6z6	z6z6	PROPN
iajs-909	175	43	.	.	PUNCT
iajs-909	176	1	3.3	3.3	NUM
iajs-909	176	2	theorem	theorem	NOUN
iajs-909	176	3	let	let	VERB
iajs-909	176	4	x	x	PRON
iajs-909	176	5	and	and	CCONJ
iajs-909	176	6	y	y	PROPN
iajs-909	176	7	be	be	AUX
iajs-909	176	8	a	a	DET
iajs-909	176	9	fuzzy	fuzzy	ADJ
iajs-909	176	10	modules	module	NOUN
iajs-909	176	11	of	of	ADP
iajs-909	176	12	an	an	DET
iajs-909	176	13	r	r	NOUN
iajs-909	176	14	-	-	PUNCT
iajs-909	176	15	modules	module	NOUN
iajs-909	176	16	m	m	NOUN
iajs-909	176	17	1	1	NUM
iajs-909	176	18	and	and	CCONJ
iajs-909	176	19	m2	m2	PROPN
iajs-909	176	20	respectively	respectively	ADV
iajs-909	176	21	,	,	PUNCT
iajs-909	176	22	if	if	SCONJ
iajs-909	176	23	xy	xy	PROPN
iajs-909	176	24	is	be	AUX
iajs-909	176	25	a	a	DET
iajs-909	176	26	chained	chain	VERB
iajs-909	176	27	fuzzy	fuzzy	ADJ
iajs-909	176	28	module	module	NOUN
iajs-909	176	29	of	of	ADP
iajs-909	176	30	m	m	PROPN
iajs-909	176	31	1m	1m	NUM
iajs-909	176	32	2	2	NUM
iajs-909	176	33	then	then	ADV
iajs-909	176	34	x	x	PUNCT
iajs-909	176	35	is	be	AUX
iajs-909	176	36	a	a	DET
iajs-909	176	37	chained	chain	VERB
iajs-909	176	38	fuzzy	fuzzy	ADJ
iajs-909	176	39	module	module	NOUN
iajs-909	176	40	of	of	ADP
iajs-909	176	41	m	m	PROPN
iajs-909	176	42	1	1	NUM
iajs-909	176	43	and	and	CCONJ
iajs-909	176	44	y	y	PROPN
iajs-909	176	45	is	be	AUX
iajs-909	176	46	a	a	DET
iajs-909	176	47	chained	chain	VERB
iajs-909	176	48	fuzzy	fuzzy	ADJ
iajs-909	176	49	module	module	NOUN
iajs-909	176	50	of	of	ADP
iajs-909	176	51	m2	m2	PROPN
iajs-909	176	52	.	.	PUNCT
iajs-909	177	1	proof	proof	NOUN
iajs-909	177	2	:	:	PUNCT
iajs-909	177	3	by	by	ADP
iajs-909	177	4	similar	similar	ADJ
iajs-909	177	5	proof	proof	NOUN
iajs-909	177	6	of	of	ADP
iajs-909	177	7	theorem	theorem	NOUN
iajs-909	177	8	(	(	PUNCT
iajs-909	177	9	4.10	4.10	NUM
iajs-909	177	10	)	)	PUNCT
iajs-909	177	11	in	in	ADP
iajs-909	177	12	[	[	X
iajs-909	177	13	14	14	NUM
iajs-909	177	14	]	]	PUNCT
iajs-909	177	15	.	.	PUNCT
iajs-909	178	1	next	next	ADV
iajs-909	178	2	,	,	PUNCT
iajs-909	178	3	we	we	PRON
iajs-909	178	4	shall	shall	AUX
iajs-909	178	5	indicate	indicate	VERB
iajs-909	178	6	the	the	DET
iajs-909	178	7	behaviors	behavior	NOUN
iajs-909	178	8	of	of	ADP
iajs-909	178	9	chained	chain	VERB
iajs-909	178	10	fuzzy	fuzzy	ADJ
iajs-909	178	11	modules	module	NOUN
iajs-909	178	12	under	under	ADP
iajs-909	178	13	homomorphism	homomorphism	NOUN
iajs-909	178	14	.	.	PUNCT
iajs-909	179	1	3.4	3.4	NUM
iajs-909	179	2	theorem	theorem	NOUN
iajs-909	179	3	let	let	VERB
iajs-909	179	4	x	x	PRON
iajs-909	179	5	and	and	CCONJ
iajs-909	179	6	y	y	PROPN
iajs-909	179	7	be	be	AUX
iajs-909	179	8	a	a	DET
iajs-909	179	9	fuzzy	fuzzy	ADJ
iajs-909	179	10	modules	module	NOUN
iajs-909	179	11	of	of	ADP
iajs-909	179	12	an	an	DET
iajs-909	179	13	r	r	NOUN
iajs-909	179	14	-	-	PUNCT
iajs-909	179	15	modules	module	NOUN
iajs-909	179	16	m	m	NOUN
iajs-909	179	17	1	1	NUM
iajs-909	179	18	and	and	CCONJ
iajs-909	179	19	m2	m2	PROPN
iajs-909	179	20	respectively	respectively	ADV
iajs-909	179	21	.	.	PUNCT
iajs-909	180	1	let	let	VERB
iajs-909	180	2	f	f	X
iajs-909	180	3	:	:	PUNCT
iajs-909	180	4	xy	xy	PROPN
iajs-909	180	5	be	be	AUX
iajs-909	180	6	a	a	DET
iajs-909	180	7	fuzzy	fuzzy	ADJ
iajs-909	180	8	epimorphism	epimorphism	NOUN
iajs-909	180	9	.	.	PUNCT
iajs-909	181	1	if	if	SCONJ
iajs-909	181	2	x	x	PRON
iajs-909	181	3	is	be	AUX
iajs-909	181	4	a	a	DET
iajs-909	181	5	chained	chain	VERB
iajs-909	181	6	fuzzy	fuzzy	ADJ
iajs-909	181	7	module	module	NOUN
iajs-909	181	8	,	,	PUNCT
iajs-909	181	9	then	then	ADV
iajs-909	181	10	y	y	PROPN
iajs-909	181	11	is	be	AUX
iajs-909	181	12	a	a	DET
iajs-909	181	13	chained	chain	VERB
iajs-909	181	14	fuzzy	fuzzy	ADJ
iajs-909	181	15	module	module	NOUN
iajs-909	181	16	.	.	PUNCT
iajs-909	182	1	proof	proof	NOUN
iajs-909	182	2	:	:	PUNCT
iajs-909	182	3	let	let	VERB
iajs-909	182	4	a	a	DET
iajs-909	182	5	,	,	PUNCT
iajs-909	182	6	b	b	NOUN
iajs-909	182	7	are	be	AUX
iajs-909	182	8	fuzzy	fuzzy	ADJ
iajs-909	182	9	submodules	submodule	NOUN
iajs-909	182	10	in	in	ADP
iajs-909	182	11	y.	y.	PROPN
iajs-909	183	1	then	then	ADV
iajs-909	183	2	f	f	PROPN
iajs-909	183	3	–	–	PUNCT
iajs-909	183	4	1(a	1(a	NUM
iajs-909	183	5	)	)	PUNCT
iajs-909	183	6	,	,	PUNCT
iajs-909	183	7	f	f	PROPN
iajs-909	183	8	–	–	PUNCT
iajs-909	183	9	1(b	1(b	NUM
iajs-909	183	10	)	)	PUNCT
iajs-909	183	11	are	be	AUX
iajs-909	183	12	fuzzy	fuzzy	ADJ
iajs-909	183	13	submodules	submodule	NOUN
iajs-909	183	14	in	in	ADP
iajs-909	183	15	x	x	PROPN
iajs-909	183	16	(	(	PUNCT
iajs-909	183	17	remark	remark	NOUN
iajs-909	183	18	(	(	PUNCT
iajs-909	183	19	1.17),(2	1.17),(2	NUM
iajs-909	183	20	)	)	PUNCT
iajs-909	183	21	)	)	PUNCT
iajs-909	183	22	,	,	PUNCT
iajs-909	183	23	since	since	SCONJ
iajs-909	183	24	x	x	PRON
iajs-909	183	25	is	be	AUX
iajs-909	183	26	chained	chain	VERB
iajs-909	183	27	fuzzy	fuzzy	ADJ
iajs-909	183	28	module	module	NOUN
iajs-909	183	29	,	,	PUNCT
iajs-909	183	30	then	then	ADV
iajs-909	183	31	either	either	CCONJ
iajs-909	183	32	f	f	PROPN
iajs-909	183	33	–	–	PUNCT
iajs-909	183	34	1(a)	1(a)	NUM
iajs-909	183	35	f	f	X
iajs-909	183	36	–	–	PUNCT
iajs-909	183	37	1(b	1(b	NUM
iajs-909	183	38	)	)	PUNCT
iajs-909	183	39	or	or	CCONJ
iajs-909	183	40	f	f	X
iajs-909	183	41	–	–	PUNCT
iajs-909	183	42	1(b)	1(b)	PROPN
iajs-909	183	43	f	f	PROPN
iajs-909	183	44	–	–	PUNCT
iajs-909	183	45	1(a	1(a	NUM
iajs-909	183	46	)	)	PUNCT
iajs-909	183	47	.	.	PUNCT
iajs-909	184	1	now	now	ADV
iajs-909	184	2	,	,	PUNCT
iajs-909	184	3	if	if	SCONJ
iajs-909	184	4	f	f	PROPN
iajs-909	184	5	–	–	PUNCT
iajs-909	184	6	1(a)	1(a)	NUM
iajs-909	184	7	f	f	X
iajs-909	184	8	–	–	PUNCT
iajs-909	184	9	1(b	1(b	NUM
iajs-909	184	10	)	)	PUNCT
iajs-909	184	11	,	,	PUNCT
iajs-909	184	12	then	then	ADV
iajs-909	184	13	f(f	f(f	PROPN
iajs-909	184	14	–	–	PUNCT
iajs-909	184	15	1(a))	1(a))	NUM
iajs-909	184	16	f(f	f(f	PROPN
iajs-909	184	17	–	–	PUNCT
iajs-909	184	18	1(b	1(b	NUM
iajs-909	184	19	)	)	PUNCT
iajs-909	184	20	)	)	PUNCT
iajs-909	184	21	(	(	PUNCT
iajs-909	184	22	by	by	ADP
iajs-909	184	23	lemma	lemma	PROPN
iajs-909	184	24	(	(	PUNCT
iajs-909	184	25	1.8),(2	1.8),(2	NUM
iajs-909	184	26	)	)	PUNCT
iajs-909	184	27	)	)	PUNCT
iajs-909	184	28	.	.	PUNCT
iajs-909	185	1	similarly	similarly	ADV
iajs-909	185	2	,	,	PUNCT
iajs-909	185	3	if	if	SCONJ
iajs-909	185	4	f	f	PROPN
iajs-909	185	5	–	–	PUNCT
iajs-909	185	6	1(b)	1(b)	PROPN
iajs-909	185	7	f	f	PROPN
iajs-909	185	8	–	–	PUNCT
iajs-909	185	9	1(a	1(a	NUM
iajs-909	185	10	)	)	PUNCT
iajs-909	185	11	,	,	PUNCT
iajs-909	185	12	then	then	ADV
iajs-909	185	13	ba	ba	VERB
iajs-909	185	14	.	.	PUNCT
iajs-909	186	1	therefore	therefore	ADV
iajs-909	186	2	y	y	PROPN
iajs-909	186	3	is	be	AUX
iajs-909	186	4	a	a	DET
iajs-909	186	5	chained	chain	VERB
iajs-909	186	6	fuzzy	fuzzy	ADJ
iajs-909	186	7	module	module	NOUN
iajs-909	186	8	.	.	PUNCT
iajs-909	187	1	3.5	3.5	NUM
iajs-909	187	2	proposition	proposition	NOUN
iajs-909	187	3	let	let	VERB
iajs-909	187	4	x	x	PRON
iajs-909	187	5	and	and	CCONJ
iajs-909	187	6	y	y	PROPN
iajs-909	187	7	be	be	AUX
iajs-909	187	8	two	two	NUM
iajs-909	187	9	fuzzy	fuzzy	ADJ
iajs-909	187	10	modules	module	NOUN
iajs-909	187	11	of	of	ADP
iajs-909	187	12	an	an	DET
iajs-909	187	13	r	r	NOUN
iajs-909	187	14	-	-	PUNCT
iajs-909	187	15	modules	module	NOUN
iajs-909	187	16	m	m	NOUN
iajs-909	187	17	1	1	NUM
iajs-909	187	18	and	and	CCONJ
iajs-909	187	19	m	m	PROPN
iajs-909	187	20	2	2	NUM
iajs-909	187	21	respectively	respectively	ADV
iajs-909	187	22	.	.	PUNCT
iajs-909	188	1	let	let	VERB
iajs-909	188	2	f	f	X
iajs-909	188	3	:	:	PUNCT
iajs-909	188	4	xy	xy	PROPN
iajs-909	188	5	be	be	AUX
iajs-909	188	6	a	a	DET
iajs-909	188	7	fuzzy	fuzzy	ADJ
iajs-909	188	8	homomorphism	homomorphism	NOUN
iajs-909	188	9	and	and	CCONJ
iajs-909	188	10	every	every	DET
iajs-909	188	11	submodule	submodule	NOUN
iajs-909	188	12	of	of	ADP
iajs-909	188	13	y	y	PROPN
iajs-909	188	14	is	be	AUX
iajs-909	188	15	f	f	NOUN
iajs-909	188	16	-	-	PUNCT
iajs-909	188	17	invariant	invariant	ADJ
iajs-909	188	18	.	.	PUNCT
iajs-909	189	1	if	if	SCONJ
iajs-909	189	2	y	y	PROPN
iajs-909	189	3	is	be	AUX
iajs-909	189	4	a	a	DET
iajs-909	189	5	chained	chain	VERB
iajs-909	189	6	fuzzy	fuzzy	ADJ
iajs-909	189	7	module	module	NOUN
iajs-909	189	8	,	,	PUNCT
iajs-909	189	9	then	then	ADV
iajs-909	189	10	x	x	PUNCT
iajs-909	189	11	is	be	AUX
iajs-909	189	12	a	a	DET
iajs-909	189	13	chained	chain	VERB
iajs-909	189	14	fuzzy	fuzzy	ADJ
iajs-909	189	15	module	module	NOUN
iajs-909	189	16	.	.	PUNCT
iajs-909	190	1	proof	proof	NOUN
iajs-909	190	2	:	:	PUNCT
iajs-909	190	3	let	let	VERB
iajs-909	190	4	a	a	DET
iajs-909	190	5	,	,	PUNCT
iajs-909	190	6	b	b	NOUN
iajs-909	190	7	are	be	AUX
iajs-909	190	8	fuzzy	fuzzy	ADJ
iajs-909	190	9	submodules	submodule	NOUN
iajs-909	190	10	in	in	ADP
iajs-909	190	11	x.	x.	PROPN
iajs-909	190	12	hence	hence	ADV
iajs-909	190	13	f(a	f(a	PROPN
iajs-909	190	14	)	)	PUNCT
iajs-909	190	15	,	,	PUNCT
iajs-909	190	16	f(b	f(b	PROPN
iajs-909	190	17	)	)	PUNCT
iajs-909	190	18	are	be	AUX
iajs-909	190	19	fuzzy	fuzzy	ADJ
iajs-909	190	20	submodules	submodule	NOUN
iajs-909	190	21	in	in	ADP
iajs-909	190	22	y	y	PROPN
iajs-909	190	23	(	(	PUNCT
iajs-909	190	24	remark	remark	NOUN
iajs-909	190	25	(	(	PUNCT
iajs-909	190	26	1.17),(1	1.17),(1	NUM
iajs-909	190	27	)	)	PUNCT
iajs-909	190	28	)	)	PUNCT
iajs-909	190	29	,	,	PUNCT
iajs-909	190	30	since	since	SCONJ
iajs-909	190	31	y	y	PROPN
iajs-909	190	32	is	be	AUX
iajs-909	190	33	a	a	DET
iajs-909	190	34	chained	chain	VERB
iajs-909	190	35	fuzzy	fuzzy	ADJ
iajs-909	190	36	module	module	NOUN
iajs-909	190	37	then	then	ADV
iajs-909	190	38	f	f	PROPN
iajs-909	190	39	(	(	PUNCT
iajs-909	190	40	a)	a)	PROPN
iajs-909	190	41	f	f	PROPN
iajs-909	190	42	(	(	PUNCT
iajs-909	190	43	b	b	NOUN
iajs-909	190	44	)	)	PUNCT
iajs-909	190	45	or	or	CCONJ
iajs-909	190	46	f	f	X
iajs-909	190	47	(	(	PUNCT
iajs-909	190	48	b)	b)	NOUN
iajs-909	190	49	f	f	X
iajs-909	190	50	(	(	PUNCT
iajs-909	190	51	a	a	NOUN
iajs-909	190	52	)	)	PUNCT
iajs-909	190	53	.	.	PUNCT
iajs-909	191	1	now	now	ADV
iajs-909	191	2	,	,	PUNCT
iajs-909	191	3	if	if	SCONJ
iajs-909	191	4	f	f	PROPN
iajs-909	191	5	(	(	PUNCT
iajs-909	191	6	a)	a)	PROPN
iajs-909	191	7	f	f	PROPN
iajs-909	191	8	(	(	PUNCT
iajs-909	191	9	b	b	NOUN
iajs-909	191	10	)	)	PUNCT
iajs-909	191	11	,	,	PUNCT
iajs-909	191	12	then	then	ADV
iajs-909	191	13	f	f	X
iajs-909	191	14	–	–	PUNCT
iajs-909	191	15	1(f	1(f	NUM
iajs-909	191	16	(	(	PUNCT
iajs-909	191	17	a))f	a))f	X
iajs-909	191	18	–	–	PUNCT
iajs-909	191	19	1(f	1(f	NUM
iajs-909	191	20	(	(	PUNCT
iajs-909	191	21	b	b	NOUN
iajs-909	191	22	)	)	PUNCT
iajs-909	191	23	)	)	PUNCT
iajs-909	191	24	(	(	PUNCT
iajs-909	191	25	by	by	ADP
iajs-909	191	26	lemma	lemma	PROPN
iajs-909	191	27	(	(	PUNCT
iajs-909	191	28	1.8),(4	1.8),(4	NUM
iajs-909	191	29	)	)	PUNCT
iajs-909	191	30	)	)	PUNCT
iajs-909	191	31	.	.	PUNCT
iajs-909	192	1	hence	hence	ADV
iajs-909	192	2	ab	ab	PROPN
iajs-909	192	3	(	(	PUNCT
iajs-909	192	4	by	by	ADP
iajs-909	192	5	lemma	lemma	PROPN
iajs-909	192	6	(	(	PUNCT
iajs-909	192	7	1.8),(1	1.8),(1	NUM
iajs-909	192	8	)	)	PUNCT
iajs-909	192	9	)	)	PUNCT
iajs-909	192	10	.	.	PUNCT
iajs-909	193	1	similarly	similarly	ADV
iajs-909	193	2	,	,	PUNCT
iajs-909	193	3	if	if	SCONJ
iajs-909	193	4	f	f	PROPN
iajs-909	193	5	(	(	PUNCT
iajs-909	193	6	b)	b)	NOUN
iajs-909	193	7	f	f	X
iajs-909	193	8	(	(	PUNCT
iajs-909	193	9	a	a	NOUN
iajs-909	193	10	)	)	PUNCT
iajs-909	193	11	,	,	PUNCT
iajs-909	193	12	then	then	ADV
iajs-909	193	13	ba	ba	VERB
iajs-909	193	14	.	.	PUNCT
iajs-909	194	1	therefore	therefore	ADV
iajs-909	194	2	x	x	X
iajs-909	194	3	is	be	AUX
iajs-909	194	4	a	a	DET
iajs-909	194	5	chained	chain	VERB
iajs-909	194	6	fuzzy	fuzzy	ADJ
iajs-909	194	7	module	module	NOUN
iajs-909	194	8	.	.	PUNCT
iajs-909	195	1	references	reference	NOUN
iajs-909	195	2	1	1	NUM
iajs-909	195	3	.	.	X
iajs-909	195	4	zahdi	zahdi	PROPN
iajs-909	195	5	,	,	PUNCT
iajs-909	195	6	l.a	l.a	PROPN
iajs-909	195	7	.	.	PROPN
iajs-909	195	8	,	,	PUNCT
iajs-909	195	9	(	(	PUNCT
iajs-909	195	10	1965	1965	NUM
iajs-909	195	11	)	)	PUNCT
iajs-909	195	12	,	,	PUNCT
iajs-909	195	13	fuzzy	fuzzy	ADJ
iajs-909	195	14	sets	set	NOUN
iajs-909	195	15	,	,	PUNCT
iajs-909	195	16	information	information	NOUN
iajs-909	195	17	and	and	CCONJ
iajs-909	195	18	control	control	NOUN
iajs-909	195	19	,	,	PUNCT
iajs-909	195	20	8	8	NUM
iajs-909	195	21	,	,	PUNCT
iajs-909	195	22	338	338	NUM
iajs-909	195	23	-	-	SYM
iajs-909	195	24	353	353	NUM
iajs-909	195	25	.	.	NOUN
iajs-909	196	1	2	2	NUM
iajs-909	196	2	.	.	X
iajs-909	196	3	zadehi	zadehi	NOUN
iajs-909	196	4	,	,	PUNCT
iajs-909	196	5	m.	m.	NOUN
iajs-909	196	6	m.	m.	NOUN
iajs-909	196	7	,	,	PUNCT
iajs-909	196	8	(	(	PUNCT
iajs-909	196	9	1992	1992	NUM
iajs-909	196	10	)	)	PUNCT
iajs-909	196	11	,	,	PUNCT
iajs-909	196	12	on	on	ADP
iajs-909	196	13	l	l	ADJ
iajs-909	196	14	-	-	ADJ
iajs-909	196	15	fuzzy	fuzzy	ADJ
iajs-909	196	16	residual	residual	ADJ
iajs-909	196	17	quotient	quotient	NOUN
iajs-909	196	18	modules	module	NOUN
iajs-909	196	19	and	and	CCONJ
iajs-909	196	20	p	p	NOUN
iajs-909	196	21	-	-	PUNCT
iajs-909	196	22	primary	primary	ADJ
iajs-909	196	23	submodules	submodule	NOUN
iajs-909	196	24	,	,	PUNCT
iajs-909	196	25	fuzzy	fuzzy	ADJ
iajs-909	196	26	sets	set	NOUN
iajs-909	196	27	and	and	CCONJ
iajs-909	196	28	systems	system	NOUN
iajs-909	196	29	,	,	PUNCT
iajs-909	196	30	51	51	NUM
iajs-909	196	31	:	:	PUNCT
iajs-909	196	32	,	,	PUNCT
iajs-909	196	33	331	331	NUM
iajs-909	196	34	-	-	SYM
iajs-909	196	35	344	344	NUM
iajs-909	196	36	.	.	PUNCT
iajs-909	197	1	3	3	X
iajs-909	197	2	.	.	X
iajs-909	197	3	mashinchi	mashinchi	PROPN
iajs-909	197	4	,	,	PUNCT
iajs-909	197	5	m.	m.	NOUN
iajs-909	197	6	and	and	CCONJ
iajs-909	197	7	zahedi	zahedi	PROPN
iajs-909	197	8	,	,	PUNCT
iajs-909	197	9	m.	m.	NOUN
iajs-909	197	10	m.	m.	NOUN
iajs-909	197	11	,	,	PUNCT
iajs-909	197	12	“	"	PUNCT
iajs-909	197	13	on	on	ADP
iajs-909	197	14	l	l	ADJ
iajs-909	197	15	-	-	ADJ
iajs-909	197	16	fuzzy	fuzzy	ADJ
iajs-909	197	17	primary	primary	ADJ
iajs-909	197	18	submodules	submodule	NOUN
iajs-909	197	19	”	"	PUNCT
iajs-909	197	20	,	,	PUNCT
iajs-909	197	21	fuzzy	fuzzy	ADJ
iajs-909	197	22	sets	set	NOUN
iajs-909	197	23	and	and	CCONJ
iajs-909	197	24	systems,.49	systems,.49	PROPN
iajs-909	197	25	:	:	PUNCT
iajs-909	197	26	,	,	PUNCT
iajs-909	197	27	pp.231	pp.231	PROPN
iajs-909	197	28	-	-	PUNCT
iajs-909	197	29	236	236	NUM
iajs-909	197	30	,	,	PUNCT
iajs-909	197	31	(	(	PUNCT
iajs-909	197	32	1996	1996	NUM
iajs-909	197	33	)	)	PUNCT
iajs-909	197	34	.	.	PUNCT
iajs-909	198	1	4	4	X
iajs-909	198	2	.	.	X
iajs-909	198	3	zadehi	zadehi	NOUN
iajs-909	198	4	,	,	PUNCT
iajs-909	198	5	m.	m.	NOUN
iajs-909	198	6	m.	m.	NOUN
iajs-909	198	7	,	,	PUNCT
iajs-909	198	8	(	(	PUNCT
iajs-909	198	9	1991	1991	NUM
iajs-909	198	10	)	)	PUNCT
iajs-909	198	11	,	,	PUNCT
iajs-909	198	12	a	a	DET
iajs-909	198	13	characterization	characterization	NOUN
iajs-909	198	14	of	of	ADP
iajs-909	198	15	l	l	ADJ
iajs-909	198	16	-	-	ADJ
iajs-909	198	17	fuzzy	fuzzy	ADJ
iajs-909	198	18	prime	prime	ADJ
iajs-909	198	19	ideals	ideal	NOUN
iajs-909	198	20	,	,	PUNCT
iajs-909	198	21	fuzzy	fuzzy	ADJ
iajs-909	198	22	sets	set	NOUN
iajs-909	198	23	and	and	CCONJ
iajs-909	198	24	systems	system	NOUN
iajs-909	198	25	,	,	PUNCT
iajs-909	198	26	44	44	NUM
iajs-909	198	27	:	:	PUNCT
iajs-909	198	28	,	,	PUNCT
iajs-909	198	29	147	147	NUM
iajs-909	198	30	-	-	SYM
iajs-909	198	31	160	160	NUM
iajs-909	198	32	.	.	NOUN
iajs-909	199	1	5	5	NUM
iajs-909	199	2	.	.	X
iajs-909	199	3	kumar	kumar	PROPN
iajs-909	199	4	,	,	PUNCT
iajs-909	199	5	r.	r.	PROPN
iajs-909	199	6	,	,	PUNCT
iajs-909	199	7	(	(	PUNCT
iajs-909	199	8	1991	1991	NUM
iajs-909	199	9	)	)	PUNCT
iajs-909	199	10	,	,	PUNCT
iajs-909	199	11	fuzzy	fuzzy	ADJ
iajs-909	199	12	semi	semi	ADJ
iajs-909	199	13	-	-	ADJ
iajs-909	199	14	primary	primary	ADJ
iajs-909	199	15	ideals	ideal	NOUN
iajs-909	199	16	of	of	ADP
iajs-909	199	17	rings	ring	NOUN
iajs-909	199	18	,	,	PUNCT
iajs-909	199	19	fuzzy	fuzzy	ADJ
iajs-909	199	20	sets	set	NOUN
iajs-909	199	21	and	and	CCONJ
iajs-909	199	22	systems	system	NOUN
iajs-909	199	23	,	,	PUNCT
iajs-909	199	24	42	42	NUM
iajs-909	199	25	:	:	PUNCT
iajs-909	199	26	,	,	PUNCT
iajs-909	199	27	263	263	NUM
iajs-909	199	28	-	-	SYM
iajs-909	199	29	272	272	NUM
iajs-909	199	30	.	.	NOUN
iajs-909	200	1	6	6	NUM
iajs-909	200	2	.	.	X
iajs-909	201	1	martinez	martinez	PROPN
iajs-909	201	2	,	,	PUNCT
iajs-909	201	3	l.	l.	PROPN
iajs-909	201	4	,	,	PUNCT
iajs-909	201	5	(	(	PUNCT
iajs-909	201	6	1996	1996	NUM
iajs-909	201	7	)	)	PUNCT
iajs-909	201	8	,	,	PUNCT
iajs-909	201	9	fuzzy	fuzzy	ADJ
iajs-909	201	10	modules	module	NOUN
iajs-909	201	11	over	over	ADP
iajs-909	201	12	fuzzy	fuzzy	ADJ
iajs-909	201	13	rings	ring	NOUN
iajs-909	201	14	in	in	ADP
iajs-909	201	15	connection	connection	NOUN
iajs-909	201	16	with	with	ADP
iajs-909	201	17	fuzzy	fuzzy	ADJ
iajs-909	201	18	ideal	ideal	NOUN
iajs-909	201	19	of	of	ADP
iajs-909	201	20	fuzzy	fuzzy	ADJ
iajs-909	201	21	ring	ring	NOUN
iajs-909	201	22	,	,	PUNCT
iajs-909	201	23	j.	j.	PROPN
iajs-909	201	24	fuzzy	fuzzy	PROPN
iajs-909	201	25	math	math	PROPN
iajs-909	201	26	.	.	PUNCT
iajs-909	201	27	,	,	PUNCT
iajs-909	201	28	4	4	NUM
iajs-909	201	29	:	:	PUNCT
iajs-909	201	30	,	,	PUNCT
iajs-909	201	31	843	843	NUM
iajs-909	201	32	-	-	SYM
iajs-909	201	33	857	857	NUM
iajs-909	201	34	.	.	PUNCT
iajs-909	202	1	7	7	X
iajs-909	202	2	.	.	X
iajs-909	202	3	qaid	qaid	PROPN
iajs-909	202	4	,	,	PUNCT
iajs-909	202	5	a.	a.	PROPN
iajs-909	202	6	,	,	PUNCT
iajs-909	202	7	(	(	PUNCT
iajs-909	202	8	1999	1999	NUM
iajs-909	202	9	)	)	PUNCT
iajs-909	202	10	,	,	PUNCT
iajs-909	202	11	some	some	PRON
iajs-909	202	12	results	result	VERB
iajs-909	202	13	on	on	ADP
iajs-909	202	14	fuzzy	fuzzy	ADJ
iajs-909	202	15	modules	module	NOUN
iajs-909	202	16	,	,	PUNCT
iajs-909	202	17	m	m	PROPN
iajs-909	202	18	.sc	.sc	PROPN
iajs-909	202	19	.	.	PUNCT
iajs-909	202	20	,	,	PUNCT
iajs-909	202	21	thesis	thesis	NOUN
iajs-909	202	22	,	,	PUNCT
iajs-909	202	23	university	university	NOUN
iajs-909	202	24	of	of	ADP
iajs-909	202	25	baghdad	baghdad	PROPN
iajs-909	202	26	.	.	PUNCT
iajs-909	203	1	8	8	NUM
iajs-909	203	2	.	.	PUNCT
iajs-909	204	1	khalaf	khalaf	PROPN
iajs-909	204	2	,	,	PUNCT
iajs-909	204	3	y.k	y.k	PROPN
iajs-909	204	4	.	.	PROPN
iajs-909	204	5	,	,	PUNCT
iajs-909	204	6	(	(	PUNCT
iajs-909	204	7	2001	2001	NUM
iajs-909	204	8	)	)	PUNCT
iajs-909	204	9	,	,	PUNCT
iajs-909	204	10	fuzzy	fuzzy	ADJ
iajs-909	204	11	quasi	quasi	ADJ
iajs-909	204	12	-	-	ADJ
iajs-909	204	13	prime	prime	ADJ
iajs-909	204	14	modules	module	NOUN
iajs-909	204	15	and	and	CCONJ
iajs-909	204	16	fuzzy	fuzzy	ADJ
iajs-909	204	17	quasi	quasi	ADJ
iajs-909	204	18	-	-	ADJ
iajs-909	204	19	prime	prime	ADJ
iajs-909	204	20	submodules	submodule	NOUN
iajs-909	204	21	,	,	PUNCT
iajs-909	204	22	m.sc	m.sc	PROPN
iajs-909	204	23	.	.	PUNCT
iajs-909	205	1	thesis	thesis	NOUN
iajs-909	205	2	,	,	PUNCT
iajs-909	205	3	university	university	NOUN
iajs-909	205	4	of	of	ADP
iajs-909	205	5	baghdad	baghdad	PROPN
iajs-909	205	6	.	.	PUNCT
iajs-909	206	1	9	9	X
iajs-909	206	2	.	.	X
iajs-909	206	3	gada	gada	PROPN
iajs-909	206	4	,	,	PUNCT
iajs-909	206	5	a.a	a.a	PROPN
iajs-909	206	6	.	.	PROPN
iajs-909	206	7	,	,	PUNCT
iajs-909	206	8	(	(	PUNCT
iajs-909	206	9	2000	2000	NUM
iajs-909	206	10	)	)	PUNCT
iajs-909	206	11	,	,	PUNCT
iajs-909	206	12	fuzzy	fuzzy	ADJ
iajs-909	206	13	spectrum	spectrum	NOUN
iajs-909	206	14	of	of	ADP
iajs-909	206	15	a	a	DET
iajs-909	206	16	modules	module	NOUN
iajs-909	206	17	over	over	ADP
iajs-909	206	18	commutative	commutative	ADJ
iajs-909	206	19	ring	ring	NOUN
iajs-909	206	20	,	,	PUNCT
iajs-909	206	21	msc	msc	PROPN
iajs-909	206	22	.	.	PROPN
iajs-909	206	23	thesis	thesis	PROPN
iajs-909	206	24	,	,	PUNCT
iajs-909	206	25	university	university	NOUN
iajs-909	206	26	of	of	ADP
iajs-909	206	27	baghdad	baghdad	PROPN
iajs-909	206	28	.	.	PUNCT
iajs-909	207	1	10	10	NUM
iajs-909	207	2	.	.	PUNCT
iajs-909	208	1	jari	jari	PROPN
iajs-909	208	2	.	.	PROPN
iajs-909	209	1	h.r	h.r	PROPN
iajs-909	209	2	.	.	PUNCT
iajs-909	209	3	,	,	PUNCT
iajs-909	209	4	prime	prime	ADJ
iajs-909	209	5	fuzzy	fuzzy	ADJ
iajs-909	209	6	submodules	submodule	NOUN
iajs-909	209	7	and	and	CCONJ
iajs-909	209	8	prime	prime	ADJ
iajs-909	209	9	fuzzy	fuzzy	ADJ
iajs-909	209	10	modules	module	NOUN
iajs-909	209	11	,	,	PUNCT
iajs-909	209	12	msc	msc	PROPN
iajs-909	209	13	.	.	PROPN
iajs-909	210	1	thesis	thesis	PROPN
iajs-909	210	2	,	,	PUNCT
iajs-909	210	3	university	university	NOUN
iajs-909	210	4	of	of	ADP
iajs-909	210	5	baghdad	baghdad	PROPN
iajs-909	210	6	.	.	PUNCT
iajs-909	211	1	11	11	NUM
iajs-909	211	2	.	.	PUNCT
iajs-909	211	3	shores	shores	PROPN
iajs-909	212	1	t.s	t.s	PROPN
iajs-909	212	2	.	.	PROPN
iajs-909	212	3	,	,	PUNCT
iajs-909	212	4	and	and	CCONJ
iajs-909	212	5	.iewis	.iewis	ADP
iajs-909	212	6	w.j	w.j	PROPN
iajs-909	212	7	.	.	PROPN
iajs-909	212	8	,	,	PUNCT
iajs-909	212	9	(	(	PUNCT
iajs-909	212	10	1974	1974	NUM
iajs-909	212	11	)	)	PUNCT
iajs-909	212	12	,	,	PUNCT
iajs-909	212	13	serial	serial	ADJ
iajs-909	212	14	modules	module	NOUN
iajs-909	212	15	and	and	CCONJ
iajs-909	212	16	endomorphism	endomorphism	PROPN
iajs-909	212	17	rings	ring	NOUN
iajs-909	212	18	,	,	PUNCT
iajs-909	212	19	duke	duke	PROPN
iajs-909	212	20	math	math	PROPN
iajs-909	212	21	.	.	PUNCT
iajs-909	213	1	j.	j.	PROPN
iajs-909	213	2	,	,	PUNCT
iajs-909	213	3	41	41	NUM
iajs-909	213	4	:	:	PUNCT
iajs-909	213	5	,	,	PUNCT
iajs-909	213	6	889	889	NUM
iajs-909	213	7	-	-	SYM
iajs-909	213	8	909	909	NUM
iajs-909	213	9	.	.	PUNCT
iajs-909	213	10	12	12	NUM
iajs-909	213	11	.	.	PUNCT
iajs-909	214	1	mohmad	mohmad	PROPN
iajs-909	214	2	,	,	PUNCT
iajs-909	214	3	a.a	a.a	PROPN
iajs-909	214	4	.	.	PROPN
iajs-909	214	5	,	,	PUNCT
iajs-909	214	6	(	(	PUNCT
iajs-909	214	7	1997	1997	NUM
iajs-909	214	8	)	)	PUNCT
iajs-909	214	9	,	,	PUNCT
iajs-909	214	10	chained	chain	VERB
iajs-909	214	11	modules	module	NOUN
iajs-909	214	12	,	,	PUNCT
iajs-909	214	13	m	m	PROPN
iajs-909	214	14	.sc	.sc	NOUN
iajs-909	214	15	.	.	PUNCT
iajs-909	215	1	thesis	thesis	NOUN
iajs-909	215	2	,	,	PUNCT
iajs-909	215	3	university	university	NOUN
iajs-909	215	4	of	of	ADP
iajs-909	215	5	bghdad	bghdad	PROPN
iajs-909	215	6	.	.	PUNCT
iajs-909	216	1	ihjpas	ihjpas	PROPN
iajs-909	216	2	13	13	NUM
iajs-909	216	3	.	.	PUNCT
iajs-909	217	1	mukhrjee	mukhrjee	PROPN
iajs-909	217	2	,	,	PUNCT
iajs-909	217	3	t.k	t.k	PROPN
iajs-909	217	4	.	.	PROPN
iajs-909	217	5	,	,	PUNCT
iajs-909	217	6	(	(	PUNCT
iajs-909	217	7	1989	1989	NUM
iajs-909	217	8	)	)	PUNCT
iajs-909	217	9	,	,	PUNCT
iajs-909	217	10	prime	prime	ADJ
iajs-909	217	11	fuzzy	fuzzy	ADJ
iajs-909	217	12	ideals	ideal	NOUN
iajs-909	217	13	in	in	ADP
iajs-909	217	14	rings	ring	NOUN
iajs-909	217	15	,	,	PUNCT
iajs-909	217	16	fuzzy	fuzzy	ADJ
iajs-909	217	17	sets	set	NOUN
iajs-909	217	18	and	and	CCONJ
iajs-909	217	19	systems	system	NOUN
iajs-909	217	20	,	,	PUNCT
iajs-909	217	21	32	32	NUM
iajs-909	217	22	:	:	PUNCT
iajs-909	217	23	,	,	PUNCT
iajs-909	217	24	337	337	NUM
iajs-909	217	25	-	-	SYM
iajs-909	217	26	341	341	NUM
iajs-909	217	27	.	.	PUNCT
iajs-909	217	28	14	14	NUM
iajs-909	217	29	.	.	PUNCT
iajs-909	218	1	hadi	hadi	PROPN
iajs-909	218	2	,	,	PUNCT
iajs-909	218	3	i.m.a	i.m.a	NOUN
iajs-909	218	4	.	.	PROPN
iajs-909	218	5	,	,	PUNCT
iajs-909	218	6	(	(	PUNCT
iajs-909	218	7	2002	2002	NUM
iajs-909	218	8	)	)	PUNCT
iajs-909	218	9	,	,	PUNCT
iajs-909	218	10	some	some	DET
iajs-909	218	11	types	type	NOUN
iajs-909	218	12	of	of	ADP
iajs-909	218	13	fuzzy	fuzzy	ADJ
iajs-909	218	14	rings	ring	NOUN
iajs-909	218	15	,	,	PUNCT
iajs-909	218	16	mathematics	mathematics	PROPN
iajs-909	218	17	and	and	CCONJ
iajs-909	218	18	physics	physics	PROPN
iajs-909	218	19	j.	j.	PROPN
iajs-909	218	20	i(17)(1	i(17)(1	PROPN
iajs-909	218	21	)	)	PUNCT
iajs-909	218	22	,	,	PUNCT
iajs-909	218	23	1	1	NUM
iajs-909	218	24	-	-	SYM
iajs-909	218	25	17	17	NUM
iajs-909	218	26	.	.	PUNCT
iajs-909	219	1	ihjpas	ihjpas	PROPN
iajs-909	219	2	2010	2010	NUM
iajs-909	219	3	)	)	PUNCT
iajs-909	219	4	2	2	NUM
iajs-909	219	5	(	(	PUNCT
iajs-909	219	6	23مجلة	23مجلة	NUM
iajs-909	219	7	ابن	ابن	PROPN
iajs-909	219	8	الھیثم	الھیثم	PROPN
iajs-909	219	9	للعلوم	للعلوم	PROPN
iajs-909	219	10	الصرفة	الصرفة	PROPN
iajs-909	219	11	والتطبیقیة	والتطبیقیة	PROPN
iajs-909	219	12	المجلد	المجلد	PROPN
iajs-909	219	13	المودیوالت	المودیوالت	PROPN
iajs-909	219	14	الضبابیة	الضبابیة	PROPN
iajs-909	219	15	المسلسلة	المسلسلة	PROPN
iajs-909	219	16	شروق	شروق	PROPN
iajs-909	219	17	بهجت	بهجت	PROPN
iajs-909	219	18	جامعة	جامعة	PROPN
iajs-909	219	19	بغداد	بغداد	PROPN
iajs-909	219	20	،	،	PROPN
iajs-909	219	21	ابن	ابن	PROPN
iajs-909	219	22	الهیثم	الهیثم	PROPN
iajs-909	219	23	-كلیة	-كلیة	PROPN
iajs-909	219	24	التربیة	التربیة	NOUN
iajs-909	219	25	،	،	NOUN
iajs-909	219	26	قسم	قسم	PROPN
iajs-909	219	27	الریاضیات	الریاضیات	PROPN
iajs-909	219	28	الخالصة	الخالصة	PROPN
iajs-909	219	29	حلقة	حلقة	PROPN
iajs-909	219	30	أبدالیة	أبدالیة	PROPN
iajs-909	220	1	ذا	ذا	PROPN
iajs-909	220	2	عنصر	عنصر	PROPN
iajs-909	220	3	محاید	محاید	PROPN
iajs-909	220	4	rلتكن	rلتكن	PROPN
iajs-909	220	5	لقـد	لقـد	PROPN
iajs-909	220	6	أعطینـا	أعطینـا	PROPN
iajs-909	220	7	العدیـد	العدیـد	PROPN
iajs-909	220	8	.	.	PUNCT
iajs-909	221	1	في	في	PRON
iajs-909	221	2	هذا	هذا	NOUN
iajs-909	221	3	البحث	البحث	NOUN
iajs-909	221	4	قـدمنا	قـدمنا	PROPN
iajs-909	221	5	مفهـوم	مفهـوم	PROPN
iajs-909	222	1	المودیـوالت	المودیـوالت	ADJ
iajs-909	222	2	الضـبابیة	الضـبابیة	NOUN
iajs-909	222	3	المسلسـلة	المسلسـلة	PROPN
iajs-909	222	4	تعمیمـا	تعمیمـا	PROPN
iajs-909	222	5	لمفهـوم	لمفهـوم	PROPN
iajs-909	222	6	المودیـوالت	المودیـوالت	ADJ
iajs-909	222	7	المسلسـلة	المسلسـلة	NOUN
iajs-909	222	8	.من	.من	PUNCT
iajs-909	222	9	التمیزات	التمیزات	ADJ
iajs-909	222	10	والخواص	والخواص	PROPN
iajs-909	222	11	األساسیة	األساسیة	PROPN
iajs-909	222	12	لهذا	لهذا	PROPN
iajs-909	222	13	المفهوم	المفهوم	PROPN
iajs-909	222	14	ihjpas	ihjpa	VERB