id	sid	tid	token	lemma	pos
iajs-830	1	1	2011	2011	NUM
iajs-830	1	2	)	)	PUNCT
iajs-830	1	3	1	1	NUM
iajs-830	1	4	(	(	PUNCT
iajs-830	1	5	24مجلة	24مجلة	NUM
iajs-830	1	6	ابن	ابن	VERB
iajs-830	1	7	الهیثم	الهیثم	ADJ
iajs-830	1	8	للعلوم	للعلوم	PROPN
iajs-830	1	9	الصرفة	الصرفة	PROPN
iajs-830	1	10	والتطبیقیة	والتطبیقیة	PROPN
iajs-830	1	11	المجلد	المجلد	PROPN
iajs-830	1	12	المقاسات	المقاسات	PROPN
iajs-830	1	13	التوزیعیة	التوزیعیة	VERB
iajs-830	1	14	الضبابیة	الضبابیة	NOUN
iajs-830	1	15	شروق	شروق	VERB
iajs-830	1	16	بهجت	بهجت	PROPN
iajs-830	1	17	،	،	PROPN
iajs-830	1	18	أنعام	أنعام	PROPN
iajs-830	1	19	محمد	محمد	PROPN
iajs-830	1	20	علي	علي	NOUN
iajs-830	1	21	هادي	هادي	NOUN
iajs-830	1	22	جامعة	جامعة	NOUN
iajs-830	1	23	بغداد،ابن	بغداد،ابن	PROPN
iajs-830	1	24	الهیثم	الهیثم	VERB
iajs-830	1	25	-كلیة	-كلیة	PROPN
iajs-830	1	26	التربیة	التربیة	NOUN
iajs-830	1	27	،	،	NOUN
iajs-830	1	28	قسم	قسم	PROPN
iajs-830	1	29	الریاضیات	الریاضیات	VERB
iajs-830	1	30	2009	2009	NUM
iajs-830	1	31	كانون	كانون	NOUN
iajs-830	1	32	االول	االول	PROPN
iajs-830	1	33	13استلم	13استلم	NUM
iajs-830	1	34	البحث	البحث	NOUN
iajs-830	1	35	في	في	ADP
iajs-830	1	36	2010	2010	NUM
iajs-830	1	37	اذار	اذار	NOUN
iajs-830	1	38	9قبل	9قبل	NUM
iajs-830	1	39	البحث	البحث	NOUN
iajs-830	1	40	في	في	ADP
iajs-830	1	41	الصةالخ	الصةالخ	PROPN
iajs-830	1	42	ــة	ــة	PROPN
iajs-830	1	43	ابدالیــة	ابدالیــة	PROPN
iajs-830	1	44	rلــتكن	rلــتكن	NOUN
iajs-830	2	1	ـبابیةودرســنا	ـبابیةودرســنا	VERB
iajs-830	2	2	قـــدمنافـــي	قـــدمنافـــي	PROPN
iajs-830	2	3	هــذا	هــذا	PROPN
iajs-830	2	4	البحـــث	البحـــث	PROPN
iajs-830	2	5	.	.	PUNCT
iajs-830	3	1	محایـــدبحلقـ	محایـــدبحلقـ	VERB
iajs-830	3	2	ـات	ـات	PROPN
iajs-830	3	3	التوزیعیـــة	التوزیعیـــة	ADJ
iajs-830	3	4	الضـ	الضـ	NOUN
iajs-830	3	5	ـابیة	ـابیة	NOUN
iajs-830	3	6	المقاســ	المقاســ	VERB
iajs-830	3	7	ـات	ـات	PROPN
iajs-830	3	8	الحســ	الحســ	ADJ
iajs-830	3	9	والحلقــ	والحلقــ	NOUN
iajs-830	3	10	.المفاهیموأعطینا	.المفاهیموأعطینا	VERB
iajs-830	3	11	بعض	بعض	NOUN
iajs-830	3	12	الخواص	الخواص	PROPN
iajs-830	3	13	االساسیة	االساسیة	PROPN
iajs-830	3	14	حول	حول	PROPN
iajs-830	3	15	هذه	هذه	PROPN
iajs-830	3	16	.للمقاسات	.للمقاسات	PROPN
iajs-830	3	17	التوزیعیة	التوزیعیة	VERB
iajs-830	3	18	والحلقات	والحلقات	NOUN
iajs-830	3	19	الحسابیة	الحسابیة	NOUN
iajs-830	3	20	)	)	PUNCT
iajs-830	3	21	ناعتیادی(ن	ناعتیادی(ن	PROPN
iajs-830	3	22	ییمتعمالضبابیة	ییمتعمالضبابیة	PROPN
iajs-830	3	23	ibn	ibn	PROPN
iajs-830	3	24	alhaitham	alhaitham	PROPN
iajs-830	3	25	j.	j.	PROPN
iajs-830	3	26	for	for	ADP
iajs-830	3	27	pure	pure	ADJ
iajs-830	3	28	&	&	CCONJ
iajs-830	3	29	appl	appl	PROPN
iajs-830	3	30	.	.	PUNCT
iajs-830	4	1	sci	sci	PROPN
iajs-830	4	2	.	.	PUNCT
iajs-830	4	3	vol.24	vol.24	NOUN
iajs-830	4	4	(	(	PUNCT
iajs-830	4	5	1	1	NUM
iajs-830	4	6	)	)	PUNCT
iajs-830	4	7	2011	2011	NUM
iajs-830	4	8	fuzzy	fuzzy	ADJ
iajs-830	4	9	distributive	distributive	ADJ
iajs-830	4	10	modules	module	NOUN
iajs-830	4	11	i.m.a.hadi	i.m.a.hadi	PROPN
iajs-830	4	12	,	,	PUNCT
iajs-830	4	13	sh	sh	PROPN
iajs-830	4	14	.	.	PROPN
iajs-830	4	15	b.semeein	b.semeein	PROPN
iajs-830	4	16	department	department	NOUN
iajs-830	4	17	of	of	ADP
iajs-830	4	18	mathematics	mathematics	PROPN
iajs-830	4	19	,	,	PUNCT
iajs-830	4	20	college	college	NOUN
iajs-830	4	21	of	of	ADP
iajs-830	4	22	education	education	NOUN
iajs-830	4	23	,	,	PUNCT
iajs-830	4	24	ibn	ibn	PROPN
iajs-830	4	25	-	-	PUNCT
iajs-830	4	26	al	al	PROPN
iajs-830	4	27	-	-	PUNCT
iajs-830	4	28	haitham	haitham	PROPN
iajs-830	4	29	,	,	PUNCT
iajs-830	4	30	university	university	PROPN
iajs-830	4	31	of	of	ADP
iajs-830	4	32	baghdad	baghdad	PROPN
iajs-830	4	33	received	receive	VERB
iajs-830	4	34	in	in	ADP
iajs-830	4	35	december,13,2009	december,13,2009	NOUN
iajs-830	4	36	accepted	accept	VERB
iajs-830	4	37	in	in	ADP
iajs-830	4	38	march	march	PROPN
iajs-830	4	39	,	,	PUNCT
iajs-830	4	40	9,2010	9,2010	PROPN
iajs-830	4	41	abstract	abstract	NOUN
iajs-830	4	42	let	let	VERB
iajs-830	4	43	r	r	PRON
iajs-830	4	44	be	be	AUX
iajs-830	4	45	a	a	DET
iajs-830	4	46	commutative	commutative	ADJ
iajs-830	4	47	ring	ring	NOUN
iajs-830	4	48	with	with	ADP
iajs-830	4	49	unity	unity	NOUN
iajs-830	4	50	.	.	PUNCT
iajs-830	5	1	in	in	ADP
iajs-830	5	2	this	this	DET
iajs-830	5	3	paper	paper	NOUN
iajs-830	5	4	we	we	PRON
iajs-830	5	5	introduce	introduce	VERB
iajs-830	5	6	and	and	CCONJ
iajs-830	5	7	study	study	VERB
iajs-830	5	8	fuzzy	fuzzy	ADJ
iajs-830	5	9	distributive	distributive	ADJ
iajs-830	5	10	modules	module	NOUN
iajs-830	5	11	and	and	CCONJ
iajs-830	5	12	fuzzy	fuzzy	ADJ
iajs-830	5	13	arithmetical	arithmetical	ADJ
iajs-830	5	14	rings	ring	NOUN
iajs-830	5	15	as	as	ADP
iajs-830	5	16	generalizations	generalization	NOUN
iajs-830	5	17	of	of	ADP
iajs-830	5	18	(	(	PUNCT
iajs-830	5	19	ordinary	ordinary	ADJ
iajs-830	5	20	)	)	PUNCT
iajs-830	5	21	distributive	distributive	ADJ
iajs-830	5	22	modules	module	NOUN
iajs-830	5	23	and	and	CCONJ
iajs-830	5	24	arithmetical	arithmetical	ADJ
iajs-830	5	25	ring	ring	NOUN
iajs-830	5	26	.	.	PUNCT
iajs-830	6	1	we	we	PRON
iajs-830	6	2	give	give	VERB
iajs-830	6	3	some	some	DET
iajs-830	6	4	basic	basic	ADJ
iajs-830	6	5	properties	property	NOUN
iajs-830	6	6	about	about	ADP
iajs-830	6	7	these	these	DET
iajs-830	6	8	concepts	concept	NOUN
iajs-830	6	9	.	.	PUNCT
iajs-830	7	1	introduction	introduction	NOUN
iajs-830	7	2	in	in	ADP
iajs-830	7	3	this	this	DET
iajs-830	7	4	paper	paper	NOUN
iajs-830	7	5	we	we	PRON
iajs-830	7	6	introduce	introduce	VERB
iajs-830	7	7	and	and	CCONJ
iajs-830	7	8	study	study	VERB
iajs-830	7	9	fuzzy	fuzzy	ADJ
iajs-830	7	10	distributive	distributive	ADJ
iajs-830	7	11	modules	module	NOUN
iajs-830	7	12	as	as	ADP
iajs-830	7	13	a	a	DET
iajs-830	7	14	generalization	generalization	NOUN
iajs-830	7	15	of	of	ADP
iajs-830	7	16	the	the	DET
iajs-830	7	17	concept	concept	NOUN
iajs-830	7	18	(	(	PUNCT
iajs-830	7	19	distributive	distributive	ADJ
iajs-830	7	20	modules	module	NOUN
iajs-830	7	21	)	)	PUNCT
iajs-830	7	22	in	in	ADP
iajs-830	7	23	ordinary	ordinary	ADJ
iajs-830	7	24	algebra	algebra	NOUN
iajs-830	7	25	.	.	PUNCT
iajs-830	8	1	in	in	ADP
iajs-830	8	2	section	section	NOUN
iajs-830	8	3	one	one	NUM
iajs-830	8	4	,	,	PUNCT
iajs-830	8	5	we	we	PRON
iajs-830	8	6	recall	recall	VERB
iajs-830	8	7	some	some	DET
iajs-830	8	8	basic	basic	ADJ
iajs-830	8	9	definitions	definition	NOUN
iajs-830	8	10	and	and	CCONJ
iajs-830	8	11	results	result	NOUN
iajs-830	8	12	which	which	PRON
iajs-830	8	13	we	we	PRON
iajs-830	8	14	will	will	AUX
iajs-830	8	15	be	be	AUX
iajs-830	8	16	needed	need	VERB
iajs-830	8	17	later	later	ADV
iajs-830	8	18	.	.	PUNCT
iajs-830	9	1	in	in	ADP
iajs-830	9	2	section	section	NOUN
iajs-830	9	3	two	two	NUM
iajs-830	9	4	,	,	PUNCT
iajs-830	9	5	we	we	PRON
iajs-830	9	6	give	give	VERB
iajs-830	9	7	some	some	DET
iajs-830	9	8	basic	basic	ADJ
iajs-830	9	9	results	result	NOUN
iajs-830	9	10	about	about	ADP
iajs-830	9	11	fuzzy	fuzzy	ADJ
iajs-830	9	12	distributive	distributive	ADJ
iajs-830	9	13	modules	module	NOUN
iajs-830	9	14	.	.	PUNCT
iajs-830	10	1	also	also	ADV
iajs-830	10	2	we	we	PRON
iajs-830	10	3	study	study	VERB
iajs-830	10	4	the	the	DET
iajs-830	10	5	direct	direct	ADJ
iajs-830	10	6	sum	sum	NOUN
iajs-830	10	7	of	of	ADP
iajs-830	10	8	fuzzy	fuzzy	ADJ
iajs-830	10	9	distributive	distributive	ADJ
iajs-830	10	10	modules	module	NOUN
iajs-830	10	11	.	.	PUNCT
iajs-830	11	1	in	in	ADP
iajs-830	11	2	section	section	NOUN
iajs-830	11	3	three	three	NUM
iajs-830	11	4	,	,	PUNCT
iajs-830	11	5	we	we	PRON
iajs-830	11	6	study	study	VERB
iajs-830	11	7	the	the	DET
iajs-830	11	8	homomorphic	homomorphic	ADJ
iajs-830	11	9	image	image	NOUN
iajs-830	11	10	and	and	CCONJ
iajs-830	11	11	inverse	inverse	NOUN
iajs-830	11	12	image	image	NOUN
iajs-830	11	13	of	of	ADP
iajs-830	11	14	fuzzy	fuzzy	ADJ
iajs-830	11	15	distributive	distributive	ADJ
iajs-830	11	16	modules	module	NOUN
iajs-830	11	17	.	.	PUNCT
iajs-830	12	1	in	in	ADP
iajs-830	12	2	section	section	NOUN
iajs-830	12	3	four	four	NUM
iajs-830	12	4	,	,	PUNCT
iajs-830	12	5	we	we	PRON
iajs-830	12	6	introduce	introduce	VERB
iajs-830	12	7	and	and	CCONJ
iajs-830	12	8	study	study	VERB
iajs-830	12	9	fuzzy	fuzzy	ADJ
iajs-830	12	10	arithmetical	arithmetical	ADJ
iajs-830	12	11	rings	ring	NOUN
iajs-830	12	12	as	as	ADP
iajs-830	12	13	a	a	DET
iajs-830	12	14	generalization	generalization	NOUN
iajs-830	12	15	of	of	ADP
iajs-830	12	16	the	the	DET
iajs-830	12	17	concept	concept	NOUN
iajs-830	12	18	(	(	PUNCT
iajs-830	12	19	arithmetical	arithmetical	ADJ
iajs-830	12	20	rings	ring	NOUN
iajs-830	12	21	)	)	PUNCT
iajs-830	12	22	in	in	ADP
iajs-830	12	23	ordinary	ordinary	ADJ
iajs-830	12	24	algebra	algebra	NOUN
iajs-830	12	25	.	.	PUNCT
iajs-830	13	1	1	1	X
iajs-830	13	2	.	.	X
iajs-830	13	3	preliminaries	preliminary	NOUN
iajs-830	13	4	in	in	ADP
iajs-830	13	5	this	this	DET
iajs-830	13	6	section	section	NOUN
iajs-830	13	7	,	,	PUNCT
iajs-830	13	8	some	some	DET
iajs-830	13	9	basic	basic	ADJ
iajs-830	13	10	definitions	definition	NOUN
iajs-830	13	11	and	and	CCONJ
iajs-830	13	12	results	result	NOUN
iajs-830	13	13	are	be	AUX
iajs-830	13	14	collected	collect	VERB
iajs-830	13	15	.	.	PUNCT
iajs-830	14	1	1.1	1.1	NUM
iajs-830	14	2	definition	definition	NOUN
iajs-830	14	3	[	[	X
iajs-830	14	4	1	1	X
iajs-830	14	5	]	]	PUNCT
iajs-830	14	6	let	let	VERB
iajs-830	14	7	s	s	PRON
iajs-830	14	8	be	be	AUX
iajs-830	14	9	a	a	DET
iajs-830	14	10	non	non	ADJ
iajs-830	14	11	-	-	ADJ
iajs-830	14	12	empty	empty	ADJ
iajs-830	14	13	set	set	NOUN
iajs-830	14	14	and	and	CCONJ
iajs-830	14	15	i	i	PRON
iajs-830	14	16	be	be	VERB
iajs-830	14	17	the	the	DET
iajs-830	14	18	closed	closed	ADJ
iajs-830	14	19	interval	interval	NOUN
iajs-830	15	1	[	[	X
iajs-830	15	2	0,1	0,1	NUM
iajs-830	15	3	]	]	PUNCT
iajs-830	15	4	of	of	ADP
iajs-830	15	5	the	the	DET
iajs-830	15	6	real	real	ADJ
iajs-830	15	7	line	line	NOUN
iajs-830	15	8	(	(	PUNCT
iajs-830	15	9	real	real	ADJ
iajs-830	15	10	numbers	number	NOUN
iajs-830	15	11	)	)	PUNCT
iajs-830	15	12	.	.	PUNCT
iajs-830	16	1	a	a	DET
iajs-830	16	2	fuzzy	fuzzy	ADJ
iajs-830	16	3	set	set	VERB
iajs-830	16	4	a	a	PRON
iajs-830	16	5	in	in	ADP
iajs-830	16	6	s	s	PROPN
iajs-830	16	7	(	(	PUNCT
iajs-830	16	8	a	a	DET
iajs-830	16	9	fuzzy	fuzzy	ADJ
iajs-830	16	10	subset	subset	NOUN
iajs-830	16	11	of	of	ADP
iajs-830	16	12	s	s	NOUN
iajs-830	16	13	)	)	PUNCT
iajs-830	16	14	is	be	AUX
iajs-830	16	15	a	a	DET
iajs-830	16	16	function	function	NOUN
iajs-830	16	17	from	from	ADP
iajs-830	16	18	s	s	PROPN
iajs-830	16	19	into	into	ADP
iajs-830	16	20	i.	i.	PROPN
iajs-830	16	21	1.2	1.2	NUM
iajs-830	16	22	definition	definition	NOUN
iajs-830	16	23	[	[	X
iajs-830	16	24	2	2	NUM
iajs-830	16	25	]	]	PUNCT
iajs-830	16	26	let	let	VERB
iajs-830	16	27	xt	xt	X
iajs-830	16	28	:	:	PUNCT
iajs-830	16	29	s	s	PART
iajs-830	16	30			X
iajs-830	17	1	[	[	X
iajs-830	17	2	0,1	0,1	NUM
iajs-830	17	3	]	]	PUNCT
iajs-830	17	4	be	be	VERB
iajs-830	17	5	a	a	DET
iajs-830	17	6	fuzzy	fuzzy	ADJ
iajs-830	17	7	set	set	NOUN
iajs-830	17	8	in	in	ADP
iajs-830	17	9	s	s	PROPN
iajs-830	17	10	,	,	PUNCT
iajs-830	17	11	where	where	SCONJ
iajs-830	17	12	xs	xs	PROPN
iajs-830	17	13	,	,	PUNCT
iajs-830	17	14	t[0,1	t[0,1	NOUN
iajs-830	17	15	]	]	PUNCT
iajs-830	17	16	defined	define	VERB
iajs-830	17	17	by	by	ADP
iajs-830	17	18	:	:	PUNCT
iajs-830	17	19	xt(y)=t	xt(y)=t	PROPN
iajs-830	17	20	if	if	SCONJ
iajs-830	17	21	x	x	X
iajs-830	17	22	=	=	NOUN
iajs-830	17	23	y	y	PROPN
iajs-830	17	24	,	,	PUNCT
iajs-830	17	25	and	and	CCONJ
iajs-830	17	26	xt(y	xt(y	NUM
iajs-830	17	27	)	)	PUNCT
iajs-830	18	1	=	=	SYM
iajs-830	18	2	0	0	PUNCT
iajs-830	19	1	if	if	SCONJ
iajs-830	19	2	xy	xy	PROPN
iajs-830	19	3	ys	ys	PROPN
iajs-830	19	4	.	.	PUNCT
iajs-830	20	1	xt	xt	PROPN
iajs-830	20	2	is	be	AUX
iajs-830	20	3	called	call	VERB
iajs-830	20	4	a	a	DET
iajs-830	20	5	fuzzy	fuzzy	ADJ
iajs-830	20	6	singleton	singleton	NOUN
iajs-830	20	7	or	or	CCONJ
iajs-830	20	8	fuzzy	fuzzy	ADJ
iajs-830	20	9	point	point	NOUN
iajs-830	20	10	in	in	ADP
iajs-830	20	11	s.	s.	PROPN
iajs-830	20	12	1.3	1.3	NUM
iajs-830	20	13	definition	definition	NOUN
iajs-830	20	14	[	[	X
iajs-830	20	15	3	3	X
iajs-830	20	16	]	]	PUNCT
iajs-830	20	17	let	let	VERB
iajs-830	20	18	a	a	PRON
iajs-830	20	19	and	and	CCONJ
iajs-830	20	20	b	b	NOUN
iajs-830	20	21	be	be	AUX
iajs-830	20	22	two	two	NUM
iajs-830	20	23	fuzzy	fuzzy	ADJ
iajs-830	20	24	sets	set	NOUN
iajs-830	20	25	in	in	ADP
iajs-830	20	26	s	s	PROPN
iajs-830	20	27	,	,	PUNCT
iajs-830	20	28	then	then	ADV
iajs-830	20	29	1	1	NUM
iajs-830	20	30	.	.	PUNCT
iajs-830	21	1	a	a	DET
iajs-830	21	2	=	=	NOUN
iajs-830	21	3	b	b	NOUN
iajs-830	21	4	if	if	SCONJ
iajs-830	21	5	and	and	CCONJ
iajs-830	21	6	only	only	ADV
iajs-830	21	7	if	if	SCONJ
iajs-830	21	8	a(x)=b(x	a(x)=b(x	PROPN
iajs-830	21	9	)	)	PUNCT
iajs-830	21	10	,	,	PUNCT
iajs-830	21	11	for	for	ADP
iajs-830	21	12	all	all	DET
iajs-830	21	13	xs	xs	PROPN
iajs-830	21	14	.	.	PUNCT
iajs-830	22	1	2	2	NUM
iajs-830	22	2	.	.	X
iajs-830	22	3	ab	ab	PROPN
iajs-830	23	1	if	if	SCONJ
iajs-830	23	2	and	and	CCONJ
iajs-830	23	3	only	only	ADV
iajs-830	23	4	if	if	SCONJ
iajs-830	23	5	a(x	a(x	NOUN
iajs-830	23	6	)	)	PUNCT
iajs-830	23	7			NOUN
iajs-830	23	8	b(x	b(x	NOUN
iajs-830	23	9	)	)	PUNCT
iajs-830	23	10	,	,	PUNCT
iajs-830	23	11	for	for	ADP
iajs-830	23	12	all	all	DET
iajs-830	23	13	xs	xs	PROPN
iajs-830	23	14	.	.	PUNCT
iajs-830	24	1	3	3	X
iajs-830	24	2	.	.	X
iajs-830	24	3	(	(	PUNCT
iajs-830	24	4	ab)(x)=min{a(x),b(x	ab)(x)=min{a(x),b(x	ADJ
iajs-830	24	5	)	)	PUNCT
iajs-830	24	6	}	}	PUNCT
iajs-830	24	7	for	for	ADP
iajs-830	24	8	all	all	DET
iajs-830	24	9	xs	xs	PROPN
iajs-830	24	10	.	.	PUNCT
iajs-830	25	1	1.4	1.4	NUM
iajs-830	25	2	definition	definition	NOUN
iajs-830	25	3	[	[	X
iajs-830	25	4	4	4	X
iajs-830	25	5	]	]	PUNCT
iajs-830	25	6	let	let	VERB
iajs-830	25	7	a	a	DET
iajs-830	25	8	be	be	AUX
iajs-830	25	9	a	a	DET
iajs-830	25	10	fuzzy	fuzzy	ADJ
iajs-830	25	11	set	set	NOUN
iajs-830	25	12	in	in	ADP
iajs-830	25	13	s	s	PROPN
iajs-830	25	14	,	,	PUNCT
iajs-830	25	15	for	for	ADP
iajs-830	25	16	all	all	DET
iajs-830	25	17	t[0,1	t[0,1	NOUN
iajs-830	25	18	]	]	PUNCT
iajs-830	25	19	,	,	PUNCT
iajs-830	25	20	the	the	DET
iajs-830	25	21	set	set	NOUN
iajs-830	25	22	at={xs	at={xs	NOUN
iajs-830	25	23	,	,	PUNCT
iajs-830	25	24	a(x)t	a(x)t	PROPN
iajs-830	25	25	}	}	PUNCT
iajs-830	25	26	is	be	AUX
iajs-830	25	27	called	call	VERB
iajs-830	25	28	level	level	NOUN
iajs-830	25	29	subset	subset	NOUN
iajs-830	25	30	of	of	ADP
iajs-830	25	31	a.	a.	NOUN
iajs-830	25	32	1.5	1.5	NUM
iajs-830	25	33	remark	remark	NOUN
iajs-830	25	34	[	[	X
iajs-830	25	35	1	1	X
iajs-830	25	36	]	]	X
iajs-830	25	37	the	the	DET
iajs-830	25	38	following	follow	VERB
iajs-830	25	39	properties	property	NOUN
iajs-830	25	40	of	of	ADP
iajs-830	25	41	level	level	NOUN
iajs-830	25	42	subsets	subset	NOUN
iajs-830	25	43	hold	hold	VERB
iajs-830	25	44	for	for	ADP
iajs-830	25	45	each	each	DET
iajs-830	25	46	t(0,1	t(0,1	NOUN
iajs-830	25	47	]	]	PUNCT
iajs-830	26	1	1	1	NUM
iajs-830	26	2	.	.	PUNCT
iajs-830	27	1	(	(	PUNCT
iajs-830	27	2	ab)t	ab)t	PROPN
iajs-830	27	3	=	=	SYM
iajs-830	27	4	at	at	ADP
iajs-830	27	5			NOUN
iajs-830	27	6	bt	bt	NOUN
iajs-830	27	7	2	2	NUM
iajs-830	27	8	.	.	PUNCT
iajs-830	28	1	a	a	DET
iajs-830	28	2	=	=	NOUN
iajs-830	28	3	b	b	NOUN
iajs-830	28	4	if	if	SCONJ
iajs-830	28	5	and	and	CCONJ
iajs-830	28	6	only	only	ADV
iajs-830	28	7	if	if	SCONJ
iajs-830	28	8	at	at	ADP
iajs-830	28	9	=	=	NOUN
iajs-830	28	10	bt	bt	NOUN
iajs-830	28	11	,	,	PUNCT
iajs-830	28	12	for	for	ADP
iajs-830	28	13	all	all	DET
iajs-830	28	14	t(0,1	t(0,1	NOUN
iajs-830	28	15	]	]	PUNCT
iajs-830	28	16	.	.	PUNCT
iajs-830	29	1	1.6	1.6	NUM
iajs-830	29	2	definition	definition	NOUN
iajs-830	29	3	[	[	X
iajs-830	29	4	5	5	NUM
iajs-830	29	5	]	]	X
iajs-830	29	6	let	let	ADJ
iajs-830	29	7	(	(	PUNCT
iajs-830	29	8	r,+,	r,+,	NOUN
iajs-830	29	9	)	)	PUNCT
iajs-830	29	10	be	be	VERB
iajs-830	29	11	a	a	DET
iajs-830	29	12	ring	ring	NOUN
iajs-830	29	13	and	and	CCONJ
iajs-830	29	14	let	let	VERB
iajs-830	29	15	x	x	PRON
iajs-830	29	16	be	be	AUX
iajs-830	29	17	a	a	DET
iajs-830	29	18	fuzzy	fuzzy	ADJ
iajs-830	29	19	set	set	NOUN
iajs-830	29	20	in	in	ADP
iajs-830	29	21	r.	r.	PROPN
iajs-830	29	22	then	then	ADV
iajs-830	29	23	x	x	VERB
iajs-830	29	24	is	be	AUX
iajs-830	29	25	called	call	VERB
iajs-830	29	26	a	a	DET
iajs-830	29	27	fuzzy	fuzzy	ADJ
iajs-830	29	28	ring	ring	NOUN
iajs-830	29	29	in	in	ADP
iajs-830	29	30	ring	ring	NOUN
iajs-830	29	31	(	(	PUNCT
iajs-830	29	32	r,+,	r,+,	NOUN
iajs-830	29	33	)	)	PUNCT
iajs-830	29	34	if	if	SCONJ
iajs-830	29	35	and	and	CCONJ
iajs-830	29	36	only	only	ADV
iajs-830	29	37	if	if	SCONJ
iajs-830	29	38	,	,	PUNCT
iajs-830	29	39	for	for	ADP
iajs-830	29	40	each	each	DET
iajs-830	29	41	x	x	NOUN
iajs-830	29	42	,	,	PUNCT
iajs-830	29	43	y	y	PROPN
iajs-830	29	44			PROPN
iajs-830	29	45	r	r	PROPN
iajs-830	29	46	ibn	ibn	PROPN
iajs-830	29	47	alhaitham	alhaitham	NOUN
iajs-830	29	48	j.	j.	PROPN
iajs-830	30	1	fo	fo	ADP
iajs-830	30	2	r	r	NOUN
iajs-830	30	3	pure	pure	ADJ
iajs-830	30	4	&	&	CCONJ
iajs-830	30	5	appl	appl	PROPN
iajs-830	30	6	.	.	PUNCT
iajs-830	31	1	sc	sc	PROPN
iajs-830	31	2	i.	i.	PROPN
iajs-830	31	3	vo	vo	PROPN
iajs-830	32	1	l.24	l.24	PROPN
iajs-830	32	2	(	(	PUNCT
iajs-830	32	3	1	1	NUM
iajs-830	32	4	)	)	PUNCT
iajs-830	32	5	2011	2011	NUM
iajs-830	32	6	1	1	NUM
iajs-830	32	7	.	.	PUNCT
iajs-830	32	8	x(x+y	x(x+y	NUM
iajs-830	32	9	)	)	PUNCT
iajs-830	32	10			NUM
iajs-830	32	11	min{x(x	min{x(x	NOUN
iajs-830	32	12	)	)	PUNCT
iajs-830	32	13	,	,	PUNCT
iajs-830	32	14	x(y	x(y	PROPN
iajs-830	32	15	)	)	PUNCT
iajs-830	32	16	}	}	PUNCT
iajs-830	32	17	2	2	NUM
iajs-830	32	18	.	.	PUNCT
iajs-830	32	19	x(x	x(x	NOUN
iajs-830	32	20	)	)	PUNCT
iajs-830	33	1	=	=	SYM
iajs-830	33	2	x	x	X
iajs-830	33	3	(	(	PUNCT
iajs-830	33	4	–	–	PUNCT
iajs-830	33	5	x	x	X
iajs-830	33	6	)	)	PUNCT
iajs-830	33	7	3	3	NUM
iajs-830	33	8	.	.	PUNCT
iajs-830	33	9	x(xy	x(xy	NUM
iajs-830	33	10	)	)	PUNCT
iajs-830	33	11			NUM
iajs-830	33	12	min{x(x	min{x(x	NOUN
iajs-830	33	13	)	)	PUNCT
iajs-830	33	14	,	,	PUNCT
iajs-830	33	15	x(y	x(y	PROPN
iajs-830	33	16	)	)	PUNCT
iajs-830	33	17	}	}	PUNCT
iajs-830	33	18	.	.	PUNCT
iajs-830	34	1	1.7	1.7	NUM
iajs-830	34	2	definition	definition	NOUN
iajs-830	34	3	[	[	X
iajs-830	34	4	6	6	NUM
iajs-830	34	5	]	]	PUNCT
iajs-830	34	6	a	a	DET
iajs-830	34	7	fuzzy	fuzzy	ADJ
iajs-830	34	8	subset	subset	NOUN
iajs-830	34	9	x	x	PUNCT
iajs-830	34	10	of	of	ADP
iajs-830	34	11	a	a	DET
iajs-830	34	12	ring	ring	NOUN
iajs-830	34	13	r	r	NOUN
iajs-830	34	14	is	be	AUX
iajs-830	34	15	called	call	VERB
iajs-830	34	16	a	a	DET
iajs-830	34	17	fuzzy	fuzzy	ADJ
iajs-830	34	18	ideal	ideal	NOUN
iajs-830	34	19	of	of	ADP
iajs-830	34	20	r	r	NOUN
iajs-830	34	21	,	,	PUNCT
iajs-830	34	22	if	if	SCONJ
iajs-830	34	23	for	for	ADP
iajs-830	34	24	each	each	DET
iajs-830	34	25	x	x	NOUN
iajs-830	34	26	,	,	PUNCT
iajs-830	34	27	y	y	PROPN
iajs-830	34	28			NOUN
iajs-830	34	29	r	r	NOUN
iajs-830	34	30	1	1	NUM
iajs-830	34	31	.	.	PUNCT
iajs-830	34	32	x(x	x(x	PROPN
iajs-830	34	33	–	–	PUNCT
iajs-830	34	34	y	y	NOUN
iajs-830	34	35	)	)	PUNCT
iajs-830	34	36			NUM
iajs-830	34	37	min{x(x	min{x(x	PROPN
iajs-830	34	38	)	)	PUNCT
iajs-830	34	39	,	,	PUNCT
iajs-830	34	40	x(y	x(y	PROPN
iajs-830	34	41	)	)	PUNCT
iajs-830	34	42	}	}	PUNCT
iajs-830	34	43	2	2	NUM
iajs-830	34	44	.	.	PUNCT
iajs-830	34	45	x(xy	x(xy	NUM
iajs-830	34	46	)	)	PUNCT
iajs-830	34	47			NUM
iajs-830	34	48	max{x(x	max{x(x	PROPN
iajs-830	34	49	)	)	PUNCT
iajs-830	34	50	,	,	PUNCT
iajs-830	34	51	x(y	x(y	PROPN
iajs-830	34	52	)	)	PUNCT
iajs-830	34	53	}	}	PUNCT
iajs-830	34	54	.	.	PUNCT
iajs-830	35	1	1.8	1.8	NUM
iajs-830	35	2	definition	definition	NOUN
iajs-830	35	3	[	[	X
iajs-830	35	4	2	2	X
iajs-830	35	5	]	]	PUNCT
iajs-830	35	6	let	let	VERB
iajs-830	35	7	m	m	PRON
iajs-830	35	8	be	be	AUX
iajs-830	35	9	an	an	DET
iajs-830	35	10	r	r	NOUN
iajs-830	35	11	-	-	PUNCT
iajs-830	35	12	module	module	NOUN
iajs-830	35	13	.	.	PUNCT
iajs-830	36	1	a	a	DET
iajs-830	36	2	fuzzy	fuzzy	ADJ
iajs-830	36	3	set	set	NOUN
iajs-830	36	4	x	x	PUNCT
iajs-830	36	5	of	of	ADP
iajs-830	36	6	m	m	PROPN
iajs-830	36	7	is	be	AUX
iajs-830	36	8	called	call	VERB
iajs-830	36	9	a	a	DET
iajs-830	36	10	fuzzy	fuzzy	ADJ
iajs-830	36	11	module	module	NOUN
iajs-830	36	12	of	of	ADP
iajs-830	36	13	m	m	NOUN
iajs-830	36	14	if	if	SCONJ
iajs-830	36	15	1	1	NUM
iajs-830	36	16	.	.	PUNCT
iajs-830	36	17	x(x	x(x	PROPN
iajs-830	36	18	–	–	PUNCT
iajs-830	36	19	y	y	NOUN
iajs-830	36	20	)	)	PUNCT
iajs-830	36	21			NUM
iajs-830	36	22	min{x(x	min{x(x	PROPN
iajs-830	36	23	)	)	PUNCT
iajs-830	36	24	,	,	PUNCT
iajs-830	36	25	x(y	x(y	PROPN
iajs-830	36	26	)	)	PUNCT
iajs-830	36	27	}	}	PUNCT
iajs-830	36	28	,	,	PUNCT
iajs-830	36	29	for	for	ADP
iajs-830	36	30	all	all	DET
iajs-830	36	31	x	x	NOUN
iajs-830	36	32	,	,	PUNCT
iajs-830	36	33	y	y	PROPN
iajs-830	36	34	m	m	PROPN
iajs-830	36	35	.	.	PROPN
iajs-830	36	36	2	2	NUM
iajs-830	36	37	.	.	X
iajs-830	36	38	x(rx	x(rx	NUM
iajs-830	36	39	)	)	PUNCT
iajs-830	36	40	x(x	x(x	NOUN
iajs-830	36	41	)	)	PUNCT
iajs-830	36	42	,	,	PUNCT
iajs-830	36	43	for	for	ADP
iajs-830	36	44	all	all	DET
iajs-830	36	45	xm	xm	PROPN
iajs-830	36	46	and	and	CCONJ
iajs-830	36	47	rr	rr	PRON
iajs-830	36	48	.	.	NOUN
iajs-830	36	49	3	3	NUM
iajs-830	36	50	.	.	X
iajs-830	36	51	x(0)=1	x(0)=1	NUM
iajs-830	36	52	.	.	PUNCT
iajs-830	37	1	1.9	1.9	NUM
iajs-830	37	2	definition	definition	NOUN
iajs-830	37	3	[	[	X
iajs-830	37	4	4	4	X
iajs-830	37	5	]	]	X
iajs-830	37	6	let	let	VERB
iajs-830	37	7	x	x	PRON
iajs-830	37	8	and	and	CCONJ
iajs-830	37	9	a	a	DET
iajs-830	37	10	be	be	AUX
iajs-830	37	11	two	two	NUM
iajs-830	37	12	fuzzy	fuzzy	ADJ
iajs-830	37	13	modules	module	NOUN
iajs-830	37	14	of	of	ADP
iajs-830	37	15	an	an	DET
iajs-830	37	16	r	r	NOUN
iajs-830	37	17	-	-	PUNCT
iajs-830	37	18	module	module	NOUN
iajs-830	37	19	m.	m.	NOUN
iajs-830	37	20	a	a	PRON
iajs-830	37	21	is	be	AUX
iajs-830	37	22	called	call	VERB
iajs-830	37	23	a	a	DET
iajs-830	37	24	fuzzy	fuzzy	ADJ
iajs-830	37	25	submodule	submodule	NOUN
iajs-830	37	26	of	of	ADP
iajs-830	37	27	x	x	PRON
iajs-830	37	28	if	if	SCONJ
iajs-830	37	29	ax	ax	NOUN
iajs-830	37	30	.	.	PUNCT
iajs-830	38	1	1.10	1.10	NUM
iajs-830	38	2	proposition	proposition	NOUN
iajs-830	38	3	[	[	X
iajs-830	38	4	7	7	NUM
iajs-830	38	5	]	]	PUNCT
iajs-830	38	6	let	let	VERB
iajs-830	38	7	a	a	PRON
iajs-830	38	8	be	be	AUX
iajs-830	38	9	a	a	DET
iajs-830	38	10	fuzzy	fuzzy	ADJ
iajs-830	38	11	set	set	NOUN
iajs-830	38	12	of	of	ADP
iajs-830	38	13	an	an	DET
iajs-830	38	14	r	r	NOUN
iajs-830	38	15	-	-	PUNCT
iajs-830	38	16	module	module	NOUN
iajs-830	38	17	m.	m.	NOUN
iajs-830	38	18	then	then	ADV
iajs-830	38	19	the	the	DET
iajs-830	38	20	level	level	NOUN
iajs-830	38	21	subset	subset	VERB
iajs-830	38	22	at	at	ADP
iajs-830	38	23	,	,	PUNCT
iajs-830	38	24	t[0,1	t[0,1	NOUN
iajs-830	38	25	]	]	PUNCT
iajs-830	38	26	is	be	AUX
iajs-830	38	27	a	a	DET
iajs-830	38	28	submodule	submodule	NOUN
iajs-830	38	29	of	of	ADP
iajs-830	38	30	m	m	PROPN
iajs-830	38	31	if	if	SCONJ
iajs-830	39	1	and	and	CCONJ
iajs-830	39	2	only	only	ADV
iajs-830	39	3	if	if	SCONJ
iajs-830	39	4	a	a	PRON
iajs-830	39	5	is	be	AUX
iajs-830	39	6	a	a	DET
iajs-830	39	7	fuzzy	fuzzy	ADJ
iajs-830	39	8	submodule	submodule	NOUN
iajs-830	39	9	of	of	ADP
iajs-830	39	10	x	x	SYM
iajs-830	39	11	where	where	SCONJ
iajs-830	39	12	x	x	PRON
iajs-830	39	13	is	be	AUX
iajs-830	39	14	a	a	DET
iajs-830	39	15	fuzzy	fuzzy	ADJ
iajs-830	39	16	module	module	NOUN
iajs-830	39	17	of	of	ADP
iajs-830	39	18	an	an	DET
iajs-830	39	19	r	r	NOUN
iajs-830	39	20	-	-	PUNCT
iajs-830	39	21	module	module	NOUN
iajs-830	39	22	m.	m.	NOUN
iajs-830	39	23	1.11	1.11	NUM
iajs-830	39	24	definition	definition	NOUN
iajs-830	39	25	[	[	X
iajs-830	39	26	8	8	NUM
iajs-830	39	27	]	]	PUNCT
iajs-830	39	28	let	let	VERB
iajs-830	39	29	x	x	PRON
iajs-830	39	30	:	:	PUNCT
iajs-830	39	31	r[0,1	r[0,1	NOUN
iajs-830	39	32	]	]	PUNCT
iajs-830	39	33	be	be	AUX
iajs-830	39	34	a	a	DET
iajs-830	39	35	fuzzy	fuzzy	ADJ
iajs-830	39	36	ring	ring	NOUN
iajs-830	39	37	,	,	PUNCT
iajs-830	39	38	let	let	VERB
iajs-830	39	39	a	a	DET
iajs-830	39	40	:	:	PUNCT
iajs-830	39	41	r[0,1	r[0,1	NOUN
iajs-830	39	42	]	]	X
iajs-830	39	43	.	.	PUNCT
iajs-830	40	1	a	a	PRON
iajs-830	40	2	is	be	AUX
iajs-830	40	3	called	call	VERB
iajs-830	40	4	a	a	DET
iajs-830	40	5	fuzzy	fuzzy	ADJ
iajs-830	40	6	ideal	ideal	NOUN
iajs-830	40	7	of	of	ADP
iajs-830	40	8	x	x	PRON
iajs-830	40	9	if	if	SCONJ
iajs-830	40	10	a	a	DET
iajs-830	40	11	satisfies	satisfie	NOUN
iajs-830	40	12	the	the	DET
iajs-830	40	13	following	following	NOUN
iajs-830	40	14	:	:	PUNCT
iajs-830	41	1	1	1	X
iajs-830	41	2	.	.	X
iajs-830	41	3	a≠	a≠	NOUN
iajs-830	41	4	2	2	NUM
iajs-830	41	5	.	.	PUNCT
iajs-830	41	6	a(x	a(x	PROPN
iajs-830	41	7	–	–	PUNCT
iajs-830	41	8	y	y	NOUN
iajs-830	41	9	)	)	PUNCT
iajs-830	41	10			NUM
iajs-830	41	11	min{a(x	min{a(x	PROPN
iajs-830	41	12	)	)	PUNCT
iajs-830	41	13	,	,	PUNCT
iajs-830	41	14	a(y	a(y	PROPN
iajs-830	41	15	)	)	PUNCT
iajs-830	41	16	}	}	PUNCT
iajs-830	41	17	,	,	PUNCT
iajs-830	41	18	for	for	ADP
iajs-830	41	19	all	all	DET
iajs-830	41	20	x	x	NOUN
iajs-830	41	21	,	,	PUNCT
iajs-830	41	22	y	y	PROPN
iajs-830	41	23	r	r	PROPN
iajs-830	41	24	.	.	PUNCT
iajs-830	42	1	3	3	X
iajs-830	42	2	.	.	X
iajs-830	42	3	a(xy	a(xy	NOUN
iajs-830	42	4	)	)	PUNCT
iajs-830	43	1			NUM
iajs-830	43	2	min{x(x	min{x(x	PROPN
iajs-830	43	3	)	)	PUNCT
iajs-830	43	4	,	,	PUNCT
iajs-830	43	5	a(y	a(y	PROPN
iajs-830	43	6	)	)	PUNCT
iajs-830	43	7	}	}	PUNCT
iajs-830	43	8	,	,	PUNCT
iajs-830	43	9	for	for	ADP
iajs-830	43	10	all	all	DET
iajs-830	43	11	x	x	NOUN
iajs-830	43	12	,	,	PUNCT
iajs-830	43	13	y	y	PROPN
iajs-830	43	14	r	r	PROPN
iajs-830	43	15	.	.	PUNCT
iajs-830	44	1	4	4	NUM
iajs-830	44	2	.	.	X
iajs-830	44	3	a(x	a(x	NOUN
iajs-830	44	4	)	)	PUNCT
iajs-830	44	5			NOUN
iajs-830	44	6	x(x	x(x	NOUN
iajs-830	44	7	)	)	PUNCT
iajs-830	44	8	,	,	PUNCT
iajs-830	44	9			NOUN
iajs-830	44	10	x	x	SYM
iajs-830	44	11			PROPN
iajs-830	44	12	r.	r.	PROPN
iajs-830	44	13	1.12	1.12	NUM
iajs-830	44	14	definition	definition	NOUN
iajs-830	44	15	[	[	X
iajs-830	44	16	9	9	NUM
iajs-830	44	17	]	]	PUNCT
iajs-830	44	18	let	let	VERB
iajs-830	44	19	a	a	DET
iajs-830	44	20	,	,	PUNCT
iajs-830	44	21	b	b	NOUN
iajs-830	44	22	be	be	AUX
iajs-830	44	23	two	two	NUM
iajs-830	44	24	fuzzy	fuzzy	ADJ
iajs-830	44	25	ideals	ideal	NOUN
iajs-830	44	26	of	of	ADP
iajs-830	44	27	a	a	DET
iajs-830	44	28	fuzzy	fuzzy	ADJ
iajs-830	44	29	ring	ring	NOUN
iajs-830	44	30	x.	x.	NOUN
iajs-830	44	31	then	then	ADV
iajs-830	44	32	1	1	X
iajs-830	44	33	.	.	PUNCT
iajs-830	45	1	the	the	DET
iajs-830	45	2	sum	sum	NOUN
iajs-830	45	3	a+b	a+b	NUM
iajs-830	45	4	of	of	ADP
iajs-830	45	5	a	a	PRON
iajs-830	45	6	and	and	CCONJ
iajs-830	45	7	b	b	NOUN
iajs-830	45	8	is	be	AUX
iajs-830	45	9	defined	define	VERB
iajs-830	45	10	as	as	ADP
iajs-830	45	11	:	:	PUNCT
iajs-830	45	12	(	(	PUNCT
iajs-830	45	13	a+b)(x	a+b)(x	NOUN
iajs-830	45	14	)	)	PUNCT
iajs-830	45	15	=	=	PUNCT
iajs-830	46	1	a	a	DET
iajs-830	46	2	b	b	NOUN
iajs-830	46	3	x	x	SYM
iajs-830	46	4	sup	sup	NOUN
iajs-830	46	5			PROPN
iajs-830	46	6			NUM
iajs-830	46	7	{	{	PUNCT
iajs-830	46	8	min{a(a),b(b	min{a(a),b(b	PROPN
iajs-830	46	9	)	)	PUNCT
iajs-830	46	10	}	}	PUNCT
iajs-830	46	11	,	,	PUNCT
iajs-830	46	12	xr	xr	PROPN
iajs-830	46	13	.	.	PUNCT
iajs-830	47	1	2	2	NUM
iajs-830	47	2	.	.	X
iajs-830	47	3	the	the	DET
iajs-830	47	4	product	product	NOUN
iajs-830	47	5	ab	ab	PROPN
iajs-830	47	6	of	of	ADP
iajs-830	47	7	a	a	PRON
iajs-830	47	8	and	and	CCONJ
iajs-830	47	9	b	b	NOUN
iajs-830	47	10	is	be	AUX
iajs-830	47	11	defined	define	VERB
iajs-830	47	12	as	as	ADP
iajs-830	47	13	(	(	PUNCT
iajs-830	47	14	ab)(x	ab)(x	PROPN
iajs-830	47	15	)	)	PUNCT
iajs-830	48	1	=	=	SYM
iajs-830	49	1	n	n	CCONJ
iajs-830	49	2	i	i	PRON
iajs-830	50	1	i	i	PRON
iajs-830	50	2	i	i	VERB
iajs-830	50	3	1	1	NUM
iajs-830	50	4	x	x	SYM
iajs-830	50	5	a	a	DET
iajs-830	50	6	b	b	NOUN
iajs-830	50	7	sup	sup	NOUN
iajs-830	50	8			NOUN
iajs-830	50	9			PROPN
iajs-830	50	10	{	{	PUNCT
iajs-830	50	11	inf{min{a(ai),b(bi	inf{min{a(ai),b(bi	PROPN
iajs-830	50	12	)	)	PUNCT
iajs-830	50	13	}	}	PUNCT
iajs-830	50	14	}	}	PUNCT
iajs-830	50	15	.	.	PUNCT
iajs-830	51	1	1.13	1.13	NUM
iajs-830	51	2	proposition	proposition	NOUN
iajs-830	51	3	let	let	VERB
iajs-830	51	4	a	a	PRON
iajs-830	51	5	and	and	CCONJ
iajs-830	51	6	b	b	NOUN
iajs-830	51	7	be	be	AUX
iajs-830	51	8	two	two	NUM
iajs-830	51	9	fuzzy	fuzzy	ADJ
iajs-830	51	10	submodules	submodule	NOUN
iajs-830	51	11	of	of	ADP
iajs-830	51	12	a	a	DET
iajs-830	51	13	fuzzy	fuzzy	ADJ
iajs-830	51	14	module	module	NOUN
iajs-830	51	15	x	x	X
iajs-830	51	16	.then	.then	X
iajs-830	51	17	(	(	PUNCT
iajs-830	51	18	ab)t	ab)t	PROPN
iajs-830	51	19	=	=	PUNCT
iajs-830	51	20	atbt	atbt	PROPN
iajs-830	51	21	,	,	PUNCT
iajs-830	51	22			NOUN
iajs-830	51	23	t(0,1	t(0,1	NOUN
iajs-830	51	24	]	]	PUNCT
iajs-830	51	25	.	.	PUNCT
iajs-830	52	1	proof	proof	NOUN
iajs-830	52	2	:	:	PUNCT
iajs-830	52	3	by	by	ADP
iajs-830	52	4	similar	similar	ADJ
iajs-830	52	5	proof	proof	NOUN
iajs-830	52	6	in	in	ADP
iajs-830	52	7	[	[	X
iajs-830	52	8	10,theorem	10,theorem	NUM
iajs-830	52	9	2.4	2.4	NUM
iajs-830	52	10	]	]	PUNCT
iajs-830	52	11	.	.	PUNCT
iajs-830	53	1	1.14	1.14	NUM
iajs-830	53	2	proposition	proposition	NOUN
iajs-830	53	3	let	let	VERB
iajs-830	53	4	a	a	PRON
iajs-830	53	5	and	and	CCONJ
iajs-830	53	6	b	b	NOUN
iajs-830	53	7	be	be	AUX
iajs-830	53	8	two	two	NUM
iajs-830	53	9	fuzzy	fuzzy	ADJ
iajs-830	53	10	submodules	submodule	NOUN
iajs-830	53	11	of	of	ADP
iajs-830	53	12	fuzzy	fuzzy	ADJ
iajs-830	53	13	module	module	NOUN
iajs-830	53	14	.	.	PUNCT
iajs-830	54	1	then	then	ADV
iajs-830	54	2	(	(	PUNCT
iajs-830	54	3	a+b)t	a+b)t	PROPN
iajs-830	54	4	=	=	SYM
iajs-830	54	5	at+bt	at+bt	PROPN
iajs-830	54	6	,	,	PUNCT
iajs-830	54	7			NOUN
iajs-830	54	8	t(0,1	t(0,1	NOUN
iajs-830	54	9	]	]	PUNCT
iajs-830	54	10	.	.	PUNCT
iajs-830	55	1	proof	proof	NOUN
iajs-830	55	2	:	:	PUNCT
iajs-830	55	3	let	let	VERB
iajs-830	55	4	x(a+b)t	x(a+b)t	PROPN
iajs-830	55	5	.	.	PUNCT
iajs-830	56	1	then	then	ADV
iajs-830	56	2	(	(	PUNCT
iajs-830	56	3	a+b)(x)=sup{min{a(a),b(b	a+b)(x)=sup{min{a(a),b(b	PROPN
iajs-830	56	4	)	)	PUNCT
iajs-830	56	5	}	}	PUNCT
iajs-830	56	6	,	,	PUNCT
iajs-830	56	7	x	x	NOUN
iajs-830	56	8	=	=	NOUN
iajs-830	56	9	a+b}t	a+b}t	NOUN
iajs-830	56	10	but	but	CCONJ
iajs-830	56	11	a+b	a+b	NUM
iajs-830	56	12	has	have	VERB
iajs-830	56	13	a	a	DET
iajs-830	56	14	suprimum	suprimum	ADJ
iajs-830	56	15	property	property	NOUN
iajs-830	56	16	,	,	PUNCT
iajs-830	56	17	so	so	SCONJ
iajs-830	56	18	there	there	PRON
iajs-830	56	19	exist	exist	VERB
iajs-830	56	20	a	a	DET
iajs-830	56	21	,	,	PUNCT
iajs-830	56	22	b	b	NOUN
iajs-830	56	23	m	m	ADP
iajs-830	56	24	such	such	ADJ
iajs-830	56	25	that	that	DET
iajs-830	56	26	sup{min{a(a),b(b	sup{min{a(a),b(b	NOUN
iajs-830	56	27	)	)	PUNCT
iajs-830	56	28	}	}	PUNCT
iajs-830	56	29	,	,	PUNCT
iajs-830	56	30	x	x	X
iajs-830	56	31	=	=	AUX
iajs-830	56	32	a+b}=min{a(a),b(b)}t	a+b}=min{a(a),b(b)}t	NOUN
iajs-830	56	33	consequently	consequently	ADV
iajs-830	56	34	,	,	PUNCT
iajs-830	56	35	a(a)t	a(a)t	ADV
iajs-830	56	36	,	,	PUNCT
iajs-830	56	37	b(b)t	b(b)t	PROPN
iajs-830	56	38	.	.	PUNCT
iajs-830	57	1	thus	thus	ADV
iajs-830	57	2	,	,	PUNCT
iajs-830	57	3	aat	aat	VERB
iajs-830	57	4	and	and	CCONJ
iajs-830	57	5	bbt	bbt	NOUN
iajs-830	57	6	,	,	PUNCT
iajs-830	57	7	it	it	PRON
iajs-830	57	8	follows	follow	VERB
iajs-830	57	9	that	that	SCONJ
iajs-830	57	10	x	x	PROPN
iajs-830	57	11	=	=	NOUN
iajs-830	57	12	a+bat+bt	a+bat+bt	NOUN
iajs-830	57	13	which	which	PRON
iajs-830	57	14	means	mean	VERB
iajs-830	57	15	(	(	PUNCT
iajs-830	57	16	a+b)t	a+b)t	PROPN
iajs-830	57	17			VERB
iajs-830	57	18	at+bt	at+bt	PROPN
iajs-830	57	19	now	now	ADV
iajs-830	57	20	,	,	PUNCT
iajs-830	57	21	let	let	VERB
iajs-830	57	22	xat+bt	xat+bt	PROPN
iajs-830	57	23	,	,	PUNCT
iajs-830	57	24	then	then	ADV
iajs-830	57	25			PROPN
iajs-830	57	26	!	!	PROPN
iajs-830	57	27	aat	aat	VERB
iajs-830	57	28	and	and	CCONJ
iajs-830	57	29			PROPN
iajs-830	57	30	!	!	PROPN
iajs-830	57	31	bbt	bbt	NOUN
iajs-830	57	32	such	such	ADJ
iajs-830	57	33	that	that	PRON
iajs-830	57	34	x=	x=	PROPN
iajs-830	58	1	a+b	a+b	NUM
iajs-830	58	2	.	.	PUNCT
iajs-830	59	1	thus	thus	ADV
iajs-830	59	2	(	(	PUNCT
iajs-830	59	3	a+b)(x)=sup{min{a(a),b(b)},x	a+b)(x)=sup{min{a(a),b(b)},x	VERB
iajs-830	59	4	=	=	SYM
iajs-830	59	5	a+b	a+b	PROPN
iajs-830	59	6	}	}	PUNCT
iajs-830	59	7	ibn	ibn	NOUN
iajs-830	59	8	alhaitham	alhaitham	NOUN
iajs-830	59	9	j.	j.	PROPN
iajs-830	60	1	fo	fo	ADP
iajs-830	60	2	r	r	NOUN
iajs-830	60	3	pure	pure	ADJ
iajs-830	60	4	&	&	CCONJ
iajs-830	60	5	appl	appl	PROPN
iajs-830	60	6	.	.	PUNCT
iajs-830	61	1	sc	sc	PROPN
iajs-830	61	2	i.	i.	PROPN
iajs-830	61	3	vo	vo	PROPN
iajs-830	62	1	l.24	l.24	PROPN
iajs-830	62	2	(	(	PUNCT
iajs-830	62	3	1	1	NUM
iajs-830	62	4	)	)	PUNCT
iajs-830	62	5	2011	2011	NUM
iajs-830	62	6	=	=	SYM
iajs-830	62	7	min{a(a),b(b)}t	min{a(a),b(b)}t	NOUN
iajs-830	62	8	since	since	SCONJ
iajs-830	62	9	the	the	DET
iajs-830	62	10	representation	representation	NOUN
iajs-830	62	11	of	of	ADP
iajs-830	62	12	any	any	DET
iajs-830	62	13	element	element	NOUN
iajs-830	62	14	of	of	ADP
iajs-830	62	15	m	m	PROPN
iajs-830	62	16	is	be	AUX
iajs-830	62	17	unique	unique	ADJ
iajs-830	62	18	.	.	PUNCT
iajs-830	63	1	1.15	1.15	NUM
iajs-830	63	2	definition	definition	NOUN
iajs-830	63	3	[	[	X
iajs-830	63	4	2	2	X
iajs-830	63	5	]	]	PUNCT
iajs-830	63	6	suppose	suppose	VERB
iajs-830	63	7	that	that	SCONJ
iajs-830	63	8	a	a	PRON
iajs-830	63	9	and	and	CCONJ
iajs-830	63	10	b	b	NOUN
iajs-830	63	11	be	be	AUX
iajs-830	63	12	two	two	NUM
iajs-830	63	13	fuzzy	fuzzy	ADJ
iajs-830	63	14	modules	module	NOUN
iajs-830	63	15	of	of	ADP
iajs-830	63	16	r	r	NOUN
iajs-830	63	17	-	-	PUNCT
iajs-830	63	18	modules	module	NOUN
iajs-830	63	19	m.	m.	NOUN
iajs-830	63	20	we	we	PRON
iajs-830	63	21	define	define	VERB
iajs-830	63	22	(	(	PUNCT
iajs-830	63	23	a	a	DET
iajs-830	63	24	:	:	SYM
iajs-830	63	25	b	b	NOUN
iajs-830	63	26	)	)	PUNCT
iajs-830	63	27	by	by	ADP
iajs-830	63	28	:	:	PUNCT
iajs-830	63	29	(	(	PUNCT
iajs-830	63	30	a	a	X
iajs-830	63	31	:	:	PUNCT
iajs-830	63	32	b)={r1	b)={r1	NOUN
iajs-830	63	33	:	:	PUNCT
iajs-830	63	34	r1	r1	PROPN
iajs-830	63	35	is	be	AUX
iajs-830	63	36	a	a	DET
iajs-830	63	37	fuzzy	fuzzy	ADJ
iajs-830	63	38	singleton	singleton	NOUN
iajs-830	63	39	of	of	ADP
iajs-830	63	40	r	r	NOUN
iajs-830	63	41	such	such	ADJ
iajs-830	63	42	that	that	DET
iajs-830	63	43	r1ba	r1ba	NOUN
iajs-830	63	44	}	}	PUNCT
iajs-830	63	45	and	and	CCONJ
iajs-830	63	46	(	(	PUNCT
iajs-830	63	47	a	a	X
iajs-830	63	48	:	:	PUNCT
iajs-830	63	49	b)(r)=sup	b)(r)=sup	PROPN
iajs-830	63	50	{	{	PUNCT
iajs-830	63	51	t[0,1	t[0,1	NOUN
iajs-830	63	52	]	]	PUNCT
iajs-830	63	53			DET
iajs-830	63	54	rtba	rtba	NOUN
iajs-830	63	55	,	,	PUNCT
iajs-830	63	56	for	for	ADP
iajs-830	63	57	all	all	DET
iajs-830	63	58	rr	rr	NOUN
iajs-830	63	59	}	}	PUNCT
iajs-830	63	60	if	if	SCONJ
iajs-830	63	61	b=(bk	b=(bk	NOUN
iajs-830	63	62	)	)	PUNCT
iajs-830	63	63	,	,	PUNCT
iajs-830	63	64	then	then	ADV
iajs-830	63	65	:	:	PUNCT
iajs-830	63	66	(	(	PUNCT
iajs-830	63	67	a:(bk))={rtrtbk	a:(bk))={rtrtbk	PROPN
iajs-830	63	68	a	a	PROPN
iajs-830	63	69	,	,	PUNCT
iajs-830	63	70	rt	rt	PROPN
iajs-830	63	71	is	be	AUX
iajs-830	63	72	a	a	DET
iajs-830	63	73	fuzzy	fuzzy	ADJ
iajs-830	63	74	singleton	singleton	NOUN
iajs-830	63	75	of	of	ADP
iajs-830	63	76	r	r	NOUN
iajs-830	63	77	}	}	PUNCT
iajs-830	63	78	1.16	1.16	NUM
iajs-830	63	79	definition	definition	NOUN
iajs-830	63	80	[	[	X
iajs-830	63	81	11	11	NUM
iajs-830	63	82	]	]	PUNCT
iajs-830	63	83	let	let	VERB
iajs-830	63	84	x	x	PRON
iajs-830	63	85	and	and	CCONJ
iajs-830	63	86	y	y	PROPN
iajs-830	63	87	be	be	AUX
iajs-830	63	88	two	two	NUM
iajs-830	63	89	fuzzy	fuzzy	ADJ
iajs-830	63	90	modules	module	NOUN
iajs-830	63	91	of	of	ADP
iajs-830	63	92	m	m	PROPN
iajs-830	63	93	1	1	NUM
iajs-830	63	94	,	,	PUNCT
iajs-830	63	95	m	m	VERB
iajs-830	63	96	2	2	NUM
iajs-830	63	97	respectively	respectively	ADV
iajs-830	63	98	.	.	PUNCT
iajs-830	64	1	define	define	VERB
iajs-830	64	2	xy	xy	PROPN
iajs-830	64	3	:	:	PUNCT
iajs-830	64	4	m	m	VERB
iajs-830	64	5	1m	1m	NUM
iajs-830	64	6	2[0,1	2[0,1	NOUN
iajs-830	64	7	]	]	PUNCT
iajs-830	64	8	by	by	ADP
iajs-830	64	9	(	(	PUNCT
iajs-830	64	10	xy)(a	xy)(a	PROPN
iajs-830	64	11	,	,	PUNCT
iajs-830	64	12	b)=min{x(a),y(b	b)=min{x(a),y(b	PROPN
iajs-830	64	13	)	)	PUNCT
iajs-830	64	14	}	}	PUNCT
iajs-830	64	15	for	for	ADP
iajs-830	64	16	all	all	PRON
iajs-830	64	17	(	(	PUNCT
iajs-830	64	18	a	a	DET
iajs-830	64	19	,	,	PUNCT
iajs-830	64	20	b	b	NOUN
iajs-830	64	21	)	)	PUNCT
iajs-830	64	22			NOUN
iajs-830	64	23	m	m	VERB
iajs-830	64	24	1m	1m	NUM
iajs-830	64	25	2	2	NUM
iajs-830	64	26	xy	xy	PROPN
iajs-830	64	27	is	be	AUX
iajs-830	64	28	called	call	VERB
iajs-830	64	29	a	a	DET
iajs-830	64	30	fuzzy	fuzzy	ADJ
iajs-830	64	31	external	external	ADJ
iajs-830	64	32	direct	direct	ADJ
iajs-830	64	33	sum	sum	NOUN
iajs-830	64	34	of	of	ADP
iajs-830	64	35	x	x	PUNCT
iajs-830	64	36	and	and	CCONJ
iajs-830	64	37	y.	y.	PROPN
iajs-830	64	38	1.17	1.17	NUM
iajs-830	64	39	proposition	proposition	NOUN
iajs-830	64	40	[	[	X
iajs-830	64	41	11	11	NUM
iajs-830	64	42	]	]	PUNCT
iajs-830	64	43	let	let	VERB
iajs-830	64	44	x	x	PRON
iajs-830	64	45	and	and	CCONJ
iajs-830	64	46	y	y	PROPN
iajs-830	64	47	are	be	AUX
iajs-830	64	48	fuzzy	fuzzy	ADJ
iajs-830	64	49	modules	module	NOUN
iajs-830	64	50	of	of	ADP
iajs-830	64	51	m	m	PROPN
iajs-830	64	52	1	1	NUM
iajs-830	64	53	and	and	CCONJ
iajs-830	64	54	m	m	PROPN
iajs-830	64	55	2	2	NUM
iajs-830	64	56	respectively	respectively	ADV
iajs-830	64	57	,	,	PUNCT
iajs-830	64	58	then	then	ADV
iajs-830	64	59	xy	xy	PROPN
iajs-830	64	60	is	be	AUX
iajs-830	64	61	a	a	DET
iajs-830	64	62	fuzzy	fuzzy	ADJ
iajs-830	64	63	module	module	NOUN
iajs-830	64	64	of	of	ADP
iajs-830	64	65	m1m	m1m	PROPN
iajs-830	64	66	2	2	NUM
iajs-830	64	67	.	.	NOUN
iajs-830	64	68	1.18	1.18	NUM
iajs-830	64	69	proposition	proposition	NOUN
iajs-830	64	70	[	[	X
iajs-830	64	71	11	11	NUM
iajs-830	64	72	]	]	PUNCT
iajs-830	64	73	let	let	VERB
iajs-830	64	74	a	a	PRON
iajs-830	64	75	and	and	CCONJ
iajs-830	64	76	b	b	NOUN
iajs-830	64	77	be	be	AUX
iajs-830	64	78	two	two	NUM
iajs-830	64	79	fuzzy	fuzzy	ADJ
iajs-830	64	80	submodules	submodule	NOUN
iajs-830	64	81	of	of	ADP
iajs-830	64	82	a	a	DET
iajs-830	64	83	fuzzy	fuzzy	ADJ
iajs-830	64	84	module	module	NOUN
iajs-830	64	85	x	x	NOUN
iajs-830	64	86	,	,	PUNCT
iajs-830	64	87	such	such	ADJ
iajs-830	64	88	that	that	SCONJ
iajs-830	64	89	x	x	NOUN
iajs-830	64	90	=	=	NOUN
iajs-830	64	91	ab	ab	PROPN
iajs-830	64	92	,	,	PUNCT
iajs-830	64	93	then	then	ADV
iajs-830	64	94	xs	xs	PROPN
iajs-830	64	95	=	=	PUNCT
iajs-830	64	96	as	as	ADP
iajs-830	64	97	bs	bs	PROPN
iajs-830	64	98	for	for	ADP
iajs-830	64	99	all	all	DET
iajs-830	64	100	s(0,1	s(0,1	NOUN
iajs-830	64	101	]	]	PUNCT
iajs-830	64	102	.	.	PUNCT
iajs-830	65	1	2	2	X
iajs-830	65	2	.	.	X
iajs-830	65	3	fuzzy	fuzzy	ADJ
iajs-830	65	4	distributive	distributive	ADJ
iajs-830	65	5	module	module	NOUN
iajs-830	65	6	in	in	ADP
iajs-830	65	7	this	this	DET
iajs-830	65	8	section	section	NOUN
iajs-830	65	9	we	we	PRON
iajs-830	65	10	fuzzyify	fuzzyify	VERB
iajs-830	65	11	the	the	DET
iajs-830	65	12	concept	concept	NOUN
iajs-830	65	13	of	of	ADP
iajs-830	65	14	distributive	distributive	ADJ
iajs-830	65	15	modules	module	NOUN
iajs-830	65	16	into	into	ADP
iajs-830	65	17	fuzzy	fuzzy	ADJ
iajs-830	65	18	distributive	distributive	ADJ
iajs-830	65	19	modules	module	NOUN
iajs-830	65	20	.	.	PUNCT
iajs-830	66	1	then	then	ADV
iajs-830	66	2	we	we	PRON
iajs-830	66	3	study	study	VERB
iajs-830	66	4	some	some	PRON
iajs-830	66	5	of	of	ADP
iajs-830	66	6	their	their	PRON
iajs-830	66	7	basic	basic	ADJ
iajs-830	66	8	properties	property	NOUN
iajs-830	66	9	.	.	PUNCT
iajs-830	67	1	recall	recall	VERB
iajs-830	67	2	that	that	SCONJ
iajs-830	67	3	an	an	DET
iajs-830	67	4	r	r	NOUN
iajs-830	67	5	-	-	PUNCT
iajs-830	67	6	module	module	NOUN
iajs-830	67	7	m	m	NOUN
iajs-830	67	8	is	be	AUX
iajs-830	67	9	said	say	VERB
iajs-830	67	10	to	to	PART
iajs-830	67	11	be	be	AUX
iajs-830	67	12	distributive	distributive	ADJ
iajs-830	67	13	if	if	SCONJ
iajs-830	67	14	for	for	ADP
iajs-830	67	15	any	any	DET
iajs-830	67	16	r	r	NOUN
iajs-830	67	17	-	-	PUNCT
iajs-830	67	18	submodules	submodules	NOUN
iajs-830	67	19	a	a	PRON
iajs-830	67	20	,	,	PUNCT
iajs-830	67	21	b	b	PROPN
iajs-830	67	22	and	and	CCONJ
iajs-830	67	23	c	c	PROPN
iajs-830	67	24	of	of	ADP
iajs-830	67	25	m	m	PRON
iajs-830	67	26	,	,	PUNCT
iajs-830	67	27	a(b+c	a(b+c	ADV
iajs-830	67	28	)	)	PUNCT
iajs-830	67	29	=	=	SYM
iajs-830	67	30	(	(	PUNCT
iajs-830	67	31	ab)+(ac	ab)+(ac	PROPN
iajs-830	67	32	)	)	PUNCT
iajs-830	68	1	[	[	X
iajs-830	68	2	12	12	NUM
iajs-830	68	3	]	]	PUNCT
iajs-830	68	4	.	.	PUNCT
iajs-830	69	1	2.1	2.1	NUM
iajs-830	69	2	definition	definition	NOUN
iajs-830	69	3	let	let	VERB
iajs-830	69	4	m	m	PRON
iajs-830	69	5	be	be	AUX
iajs-830	69	6	an	an	DET
iajs-830	69	7	r	r	NOUN
iajs-830	69	8	-	-	PUNCT
iajs-830	69	9	module	module	NOUN
iajs-830	69	10	,	,	PUNCT
iajs-830	69	11	let	let	VERB
iajs-830	69	12	x	x	PRON
iajs-830	69	13	be	be	AUX
iajs-830	69	14	a	a	DET
iajs-830	69	15	fuzzy	fuzzy	ADJ
iajs-830	69	16	module	module	NOUN
iajs-830	69	17	over	over	ADP
iajs-830	69	18	m.	m.	NOUN
iajs-830	69	19	x	x	PUNCT
iajs-830	69	20	is	be	AUX
iajs-830	69	21	called	call	VERB
iajs-830	69	22	distributive	distributive	ADJ
iajs-830	69	23	if	if	SCONJ
iajs-830	69	24	for	for	ADP
iajs-830	69	25	any	any	DET
iajs-830	69	26	fuzzy	fuzzy	ADJ
iajs-830	69	27	submodules	submodule	NOUN
iajs-830	69	28	a	a	PRON
iajs-830	69	29	,	,	PUNCT
iajs-830	69	30	b	b	PROPN
iajs-830	69	31	and	and	CCONJ
iajs-830	69	32	c	c	PROPN
iajs-830	69	33	of	of	ADP
iajs-830	69	34	x	x	NOUN
iajs-830	69	35	,	,	PUNCT
iajs-830	69	36	a(b+c	a(b+c	ADV
iajs-830	69	37	)	)	PUNCT
iajs-830	69	38	=	=	SYM
iajs-830	70	1	(	(	PUNCT
iajs-830	70	2	ab)+(ac	ab)+(ac	PROPN
iajs-830	70	3	)	)	PUNCT
iajs-830	70	4	the	the	DET
iajs-830	70	5	following	following	ADJ
iajs-830	70	6	result	result	NOUN
iajs-830	70	7	explains	explain	VERB
iajs-830	70	8	the	the	DET
iajs-830	70	9	relationship	relationship	NOUN
iajs-830	70	10	between	between	ADP
iajs-830	70	11	fuzzy	fuzzy	ADJ
iajs-830	70	12	distributive	distributive	ADJ
iajs-830	70	13	modules	module	NOUN
iajs-830	70	14	and	and	CCONJ
iajs-830	70	15	its	its	PRON
iajs-830	70	16	level	level	NOUN
iajs-830	70	17	.	.	PUNCT
iajs-830	71	1	2.2	2.2	NUM
iajs-830	71	2	theorem	theorem	VERB
iajs-830	71	3	a	a	DET
iajs-830	71	4	fuzzy	fuzzy	ADJ
iajs-830	71	5	module	module	NOUN
iajs-830	71	6	x	x	PUNCT
iajs-830	71	7	of	of	ADP
iajs-830	71	8	an	an	DET
iajs-830	71	9	r	r	NOUN
iajs-830	71	10	-	-	PUNCT
iajs-830	71	11	module	module	NOUN
iajs-830	71	12	m	m	NOUN
iajs-830	71	13	is	be	AUX
iajs-830	71	14	a	a	DET
iajs-830	71	15	fuzzy	fuzzy	ADJ
iajs-830	71	16	distributive	distributive	ADJ
iajs-830	71	17	if	if	SCONJ
iajs-830	71	18	and	and	CCONJ
iajs-830	71	19	only	only	ADV
iajs-830	71	20	if	if	SCONJ
iajs-830	71	21	xt	xt	PROPN
iajs-830	71	22	is	be	AUX
iajs-830	71	23	a	a	DET
iajs-830	71	24	distributive	distributive	ADJ
iajs-830	71	25	module	module	NOUN
iajs-830	71	26	,	,	PUNCT
iajs-830	71	27	t(0,1	t(0,1	NOUN
iajs-830	71	28	]	]	PUNCT
iajs-830	71	29	.	.	PUNCT
iajs-830	72	1	proof	proof	NOUN
iajs-830	72	2	:	:	PUNCT
iajs-830	72	3	if	if	SCONJ
iajs-830	72	4	x	x	PRON
iajs-830	72	5	is	be	AUX
iajs-830	72	6	fuzzy	fuzzy	ADJ
iajs-830	72	7	distributive	distributive	ADJ
iajs-830	72	8	module	module	NOUN
iajs-830	72	9	.	.	PUNCT
iajs-830	73	1	to	to	PART
iajs-830	73	2	prove	prove	VERB
iajs-830	73	3	xt	xt	PROPN
iajs-830	73	4	is	be	AUX
iajs-830	73	5	distributive	distributive	ADJ
iajs-830	73	6	module	module	NOUN
iajs-830	73	7	.	.	PUNCT
iajs-830	74	1			PROPN
iajs-830	74	2	t(0,1	t(0,1	PROPN
iajs-830	74	3	]	]	PUNCT
iajs-830	74	4	,	,	PUNCT
iajs-830	74	5	let	let	VERB
iajs-830	74	6	i	i	PRON
iajs-830	74	7	,	,	PUNCT
iajs-830	74	8	j	j	PROPN
iajs-830	74	9	,	,	PUNCT
iajs-830	74	10	k	k	X
iajs-830	74	11	be	be	VERB
iajs-830	74	12	submodules	submodule	NOUN
iajs-830	74	13	of	of	ADP
iajs-830	74	14	xt	xt	PROPN
iajs-830	74	15	.	.	PUNCT
iajs-830	74	16	define	define	VERB
iajs-830	74	17	t	t	PROPN
iajs-830	74	18	x	x	SYM
iajs-830	74	19	(	(	PUNCT
iajs-830	74	20	x	x	X
iajs-830	74	21	)	)	PUNCT
iajs-830	74	22	0	0	NUM
iajs-830	75	1	x	x	SYM
iajs-830	75	2			PUNCT
iajs-830	76	1			NOUN
iajs-830	76	2			NUM
iajs-830	76	3			NUM
iajs-830	76	4			NOUN
iajs-830	76	5			NOUN
iajs-830	76	6	,	,	PUNCT
iajs-830	76	7	t	t	PROPN
iajs-830	76	8	x	x	X
iajs-830	76	9	j	j	PROPN
iajs-830	76	10	b(x	b(x	NOUN
iajs-830	76	11	)	)	PUNCT
iajs-830	76	12	0	0	NUM
iajs-830	77	1	x	x	SYM
iajs-830	77	2	j	j	PROPN
iajs-830	77	3			NUM
iajs-830	77	4			NUM
iajs-830	77	5			NUM
iajs-830	77	6			NOUN
iajs-830	77	7	,	,	PUNCT
iajs-830	77	8	t	t	PROPN
iajs-830	77	9	x	x	PUNCT
iajs-830	78	1	k	k	X
iajs-830	78	2	c(x	c(x	NOUN
iajs-830	78	3	)	)	PUNCT
iajs-830	78	4	0	0	NUM
iajs-830	79	1	x	x	SYM
iajs-830	79	2	k	k	NOUN
iajs-830	79	3			VERB
iajs-830	79	4			NUM
iajs-830	80	1			NUM
iajs-830	80	2			NOUN
iajs-830	81	1	it	it	PRON
iajs-830	81	2	is	be	AUX
iajs-830	81	3	clear	clear	ADJ
iajs-830	81	4	that	that	SCONJ
iajs-830	81	5	a	a	DET
iajs-830	81	6	,	,	PUNCT
iajs-830	81	7	b	b	NOUN
iajs-830	81	8	,	,	PUNCT
iajs-830	81	9	c	c	PROPN
iajs-830	81	10	are	be	AUX
iajs-830	81	11	fuzzy	fuzzy	ADJ
iajs-830	81	12	submodules	submodule	NOUN
iajs-830	81	13	of	of	ADP
iajs-830	81	14	x	x	X
iajs-830	81	15	and	and	CCONJ
iajs-830	81	16	at	at	ADP
iajs-830	81	17	=	=	PROPN
iajs-830	81	18	i	i	PROPN
iajs-830	81	19	,	,	PUNCT
iajs-830	81	20	bt	bt	PROPN
iajs-830	81	21	=	=	PROPN
iajs-830	81	22	j	j	PROPN
iajs-830	81	23	,	,	PUNCT
iajs-830	81	24	ct	ct	PROPN
iajs-830	81	25	=	=	PROPN
iajs-830	81	26	k.	k.	PROPN
iajs-830	81	27	since	since	SCONJ
iajs-830	81	28	x	x	PRON
iajs-830	81	29	is	be	AUX
iajs-830	81	30	fuzzy	fuzzy	ADJ
iajs-830	81	31	distributive	distributive	ADJ
iajs-830	81	32	,	,	PUNCT
iajs-830	81	33	a(b+c	a(b+c	ADV
iajs-830	81	34	)	)	PUNCT
iajs-830	81	35	=	=	SYM
iajs-830	81	36	(	(	PUNCT
iajs-830	81	37	ab)+(ac	ab)+(ac	PROPN
iajs-830	81	38	)	)	PUNCT
iajs-830	81	39	.	.	PUNCT
iajs-830	82	1	hence	hence	ADV
iajs-830	83	1	[	[	X
iajs-830	83	2	a(b+c)]t	a(b+c)]t	X
iajs-830	83	3	=	=	PUNCT
iajs-830	84	1	[	[	X
iajs-830	84	2	(	(	PUNCT
iajs-830	84	3	ab)+(ac)]t	ab)+(ac)]t	ADJ
iajs-830	84	4	,	,	PUNCT
iajs-830	84	5			NOUN
iajs-830	84	6	t(0,1	t(0,1	NOUN
iajs-830	84	7	]	]	PUNCT
iajs-830	84	8	.	.	PUNCT
iajs-830	85	1	at(b+c)t	at(b+c)t	X
iajs-830	86	1	=	=	PUNCT
iajs-830	86	2	(	(	PUNCT
iajs-830	86	3	ab)t+(ac)t	ab)t+(ac)t	NUM
iajs-830	86	4	(	(	PUNCT
iajs-830	86	5	remark	remark	NOUN
iajs-830	86	6	1.5	1.5	NUM
iajs-830	86	7	and	and	CCONJ
iajs-830	86	8	proposition	proposition	NOUN
iajs-830	86	9	1.13	1.13	NUM
iajs-830	86	10	)	)	PUNCT
iajs-830	86	11	at(bt+c	at(bt+c	PROPN
iajs-830	86	12	t	t	PROPN
iajs-830	86	13	)	)	PUNCT
iajs-830	86	14	=	=	PUNCT
iajs-830	87	1	(	(	PUNCT
iajs-830	87	2	atb	atb	NOUN
iajs-830	87	3	t	t	PROPN
iajs-830	87	4	)	)	PUNCT
iajs-830	88	1	+	+	PROPN
iajs-830	88	2	(	(	PUNCT
iajs-830	88	3	atc	atc	PROPN
iajs-830	88	4	t	t	PROPN
iajs-830	88	5	)	)	PUNCT
iajs-830	88	6	(	(	PUNCT
iajs-830	88	7	remark	remark	VERB
iajs-830	88	8	1.5	1.5	NUM
iajs-830	88	9	and	and	CCONJ
iajs-830	88	10	proposition	proposition	NOUN
iajs-830	88	11	1.13	1.13	NUM
iajs-830	88	12	)	)	PUNCT
iajs-830	88	13	this	this	DET
iajs-830	88	14	i(j+k	i(j+k	NOUN
iajs-830	88	15	)	)	PUNCT
iajs-830	88	16	=	=	SYM
iajs-830	88	17	(	(	PUNCT
iajs-830	88	18	ij)+(ik	ij)+(ik	NOUN
iajs-830	88	19	)	)	PUNCT
iajs-830	88	20	conversely	conversely	ADV
iajs-830	88	21	,	,	PUNCT
iajs-830	88	22	if	if	SCONJ
iajs-830	88	23	xt	xt	PROPN
iajs-830	88	24	is	be	AUX
iajs-830	88	25	a	a	DET
iajs-830	88	26	distributive	distributive	ADJ
iajs-830	88	27	module	module	NOUN
iajs-830	88	28	,	,	PUNCT
iajs-830	88	29	for	for	ADP
iajs-830	88	30	all	all	DET
iajs-830	88	31	t(0,1	t(0,1	NOUN
iajs-830	88	32	]	]	PUNCT
iajs-830	88	33	.	.	PUNCT
iajs-830	89	1	to	to	PART
iajs-830	89	2	prove	prove	VERB
iajs-830	89	3	x	x	PUNCT
iajs-830	89	4	is	be	AUX
iajs-830	89	5	a	a	DET
iajs-830	89	6	fuzzy	fuzzy	ADJ
iajs-830	89	7	distributive	distributive	ADJ
iajs-830	89	8	module	module	NOUN
iajs-830	89	9	.	.	PUNCT
iajs-830	90	1	ibn	ibn	PROPN
iajs-830	90	2	alhaitham	alhaitham	PROPN
iajs-830	90	3	j.	j.	PROPN
iajs-830	90	4	for	for	ADP
iajs-830	90	5	pure	pure	ADJ
iajs-830	90	6	&	&	CCONJ
iajs-830	90	7	appl	appl	PROPN
iajs-830	90	8	.	.	PUNCT
iajs-830	91	1	sci	sci	PROPN
iajs-830	91	2	.	.	PUNCT
iajs-830	91	3	vol.24	vol.24	NOUN
iajs-830	91	4	(	(	PUNCT
iajs-830	91	5	1	1	NUM
iajs-830	91	6	)	)	PUNCT
iajs-830	91	7	2011	2011	NUM
iajs-830	91	8	let	let	VERB
iajs-830	91	9	a	a	DET
iajs-830	91	10	,	,	PUNCT
iajs-830	91	11	b	b	NOUN
iajs-830	91	12	and	and	CCONJ
iajs-830	91	13	c	c	PROPN
iajs-830	91	14	fuzzy	fuzzy	ADJ
iajs-830	91	15	submodules	submodule	NOUN
iajs-830	91	16	in	in	ADP
iajs-830	91	17	x.	x.	NOUN
iajs-830	91	18	then	then	ADV
iajs-830	91	19	at	at	ADP
iajs-830	91	20	,	,	PUNCT
iajs-830	91	21	bt	bt	PROPN
iajs-830	91	22	,	,	PUNCT
iajs-830	91	23	ct	ct	PROPN
iajs-830	91	24	are	be	AUX
iajs-830	91	25	submodules	submodule	NOUN
iajs-830	91	26	in	in	ADP
iajs-830	91	27	xt	xt	PROPN
iajs-830	91	28	,	,	PUNCT
iajs-830	91	29	for	for	ADP
iajs-830	91	30	all	all	DET
iajs-830	91	31	t(0,1	t(0,1	NOUN
iajs-830	91	32	]	]	PUNCT
iajs-830	91	33	.	.	PUNCT
iajs-830	92	1	since	since	SCONJ
iajs-830	92	2	xt	xt	PROPN
iajs-830	92	3	is	be	AUX
iajs-830	92	4	a	a	DET
iajs-830	92	5	distributive	distributive	ADJ
iajs-830	92	6	r	r	NOUN
iajs-830	92	7	-	-	PUNCT
iajs-830	92	8	module	module	NOUN
iajs-830	92	9	then	then	ADV
iajs-830	92	10	at(bt+c	at(bt+c	PROPN
iajs-830	92	11	t	t	PROPN
iajs-830	92	12	)	)	PUNCT
iajs-830	92	13	=	=	PUNCT
iajs-830	93	1	(	(	PUNCT
iajs-830	93	2	atb	atb	NOUN
iajs-830	93	3	t	t	PROPN
iajs-830	93	4	)	)	PUNCT
iajs-830	94	1	+	+	PROPN
iajs-830	94	2	(	(	PUNCT
iajs-830	94	3	atc	atc	PROPN
iajs-830	94	4	t	t	PROPN
iajs-830	94	5	)	)	PUNCT
iajs-830	94	6	at(b+c)t	at(b+c)t	PUNCT
iajs-830	95	1	=	=	PUNCT
iajs-830	95	2	(	(	PUNCT
iajs-830	95	3	ab)t+(ac)t	ab)t+(ac)t	NUM
iajs-830	95	4	(	(	PUNCT
iajs-830	95	5	remark	remark	NOUN
iajs-830	95	6	1.5	1.5	NUM
iajs-830	95	7	and	and	CCONJ
iajs-830	95	8	proposition	proposition	NOUN
iajs-830	95	9	1.13	1.13	NUM
iajs-830	95	10	)	)	PUNCT
iajs-830	96	1	[	[	X
iajs-830	96	2	a(b+c)]t	a(b+c)]t	X
iajs-830	96	3	=	=	PUNCT
iajs-830	97	1	[	[	X
iajs-830	97	2	(	(	PUNCT
iajs-830	97	3	ab)+(ac)]t	ab)+(ac)]t	INTJ
iajs-830	97	4	(	(	PUNCT
iajs-830	97	5	remark	remark	VERB
iajs-830	97	6	1.5	1.5	NUM
iajs-830	97	7	and	and	CCONJ
iajs-830	97	8	proposition	proposition	NOUN
iajs-830	97	9	1.13	1.13	NUM
iajs-830	97	10	)	)	PUNCT
iajs-830	97	11	then	then	ADV
iajs-830	97	12	a(b+c	a(b+c	ADV
iajs-830	97	13	)	)	PUNCT
iajs-830	97	14	=	=	SYM
iajs-830	97	15	(	(	PUNCT
iajs-830	97	16	ab)+(ac	ab)+(ac	PROPN
iajs-830	97	17	)	)	PUNCT
iajs-830	97	18	.	.	PUNCT
iajs-830	98	1	(	(	PUNCT
iajs-830	98	2	remark	remark	NOUN
iajs-830	98	3	1.5	1.5	NUM
iajs-830	98	4	,	,	PUNCT
iajs-830	98	5	(	(	PUNCT
iajs-830	98	6	2	2	NUM
iajs-830	98	7	)	)	PUNCT
iajs-830	98	8	)	)	PUNCT
iajs-830	99	1			ADP
iajs-830	99	2	2.3	2.3	NUM
iajs-830	99	3	example	example	NOUN
iajs-830	99	4	let	let	VERB
iajs-830	99	5	m	m	PRON
iajs-830	99	6	=	=	VERB
iajs-830	99	7	rr	rr	NOUN
iajs-830	99	8	where	where	SCONJ
iajs-830	99	9	r	r	NOUN
iajs-830	99	10	is	be	AUX
iajs-830	99	11	any	any	DET
iajs-830	99	12	ring	ring	NOUN
iajs-830	99	13	,	,	PUNCT
iajs-830	99	14	m	m	VERB
iajs-830	99	15	is	be	AUX
iajs-830	99	16	an	an	DET
iajs-830	99	17	r	r	NOUN
iajs-830	99	18	-	-	PUNCT
iajs-830	99	19	module	module	NOUN
iajs-830	99	20	,	,	PUNCT
iajs-830	99	21	let	let	VERB
iajs-830	99	22	x	x	PRON
iajs-830	99	23	:	:	PUNCT
iajs-830	99	24	m	m	VERB
iajs-830	99	25	[0,1	[0,1	NOUN
iajs-830	99	26	]	]	PUNCT
iajs-830	99	27	defined	define	VERB
iajs-830	99	28	by	by	ADP
iajs-830	99	29	x(x)=1	x(x)=1	PROPN
iajs-830	99	30	,	,	PUNCT
iajs-830	99	31	let	let	VERB
iajs-830	99	32	1	1	NUM
iajs-830	99	33	(	(	PUNCT
iajs-830	99	34	x	x	NOUN
iajs-830	99	35	,	,	PUNCT
iajs-830	99	36	y	y	NOUN
iajs-830	99	37	)	)	PUNCT
iajs-830	99	38	r(1,1	r(1,1	NOUN
iajs-830	99	39	)	)	PUNCT
iajs-830	99	40	(	(	PUNCT
iajs-830	99	41	x	x	X
iajs-830	99	42	,	,	PUNCT
iajs-830	99	43	y	y	PROPN
iajs-830	99	44	)	)	PUNCT
iajs-830	99	45	0	0	PUNCT
iajs-830	100	1	otherwise	otherwise	ADV
iajs-830	100	2			PUNCT
iajs-830	100	3			PROPN
iajs-830	100	4			NUM
iajs-830	101	1			NUM
iajs-830	101	2			INTJ
iajs-830	101	3	,	,	PUNCT
iajs-830	101	4	1	1	NUM
iajs-830	101	5	(	(	PUNCT
iajs-830	101	6	x	x	NOUN
iajs-830	101	7	,	,	PUNCT
iajs-830	101	8	y	y	NOUN
iajs-830	101	9	)	)	PUNCT
iajs-830	101	10	r(0,1	r(0,1	NOUN
iajs-830	101	11	)	)	PUNCT
iajs-830	101	12	(	(	PUNCT
iajs-830	101	13	x	x	X
iajs-830	101	14	,	,	PUNCT
iajs-830	101	15	y	y	PROPN
iajs-830	101	16	)	)	PUNCT
iajs-830	101	17	0	0	NUM
iajs-830	102	1	otherwise	otherwise	ADV
iajs-830	102	2			NUM
iajs-830	102	3			NOUN
iajs-830	102	4			NOUN
iajs-830	103	1			NUM
iajs-830	103	2			INTJ
iajs-830	103	3	,	,	PUNCT
iajs-830	103	4	1	1	NUM
iajs-830	103	5	(	(	PUNCT
iajs-830	103	6	x	x	NOUN
iajs-830	103	7	,	,	PUNCT
iajs-830	103	8	y	y	PROPN
iajs-830	103	9	)	)	PUNCT
iajs-830	103	10	r(1,0	r(1,0	NOUN
iajs-830	103	11	)	)	PUNCT
iajs-830	103	12	c(x	c(x	PROPN
iajs-830	103	13	,	,	PUNCT
iajs-830	103	14	y	y	NOUN
iajs-830	103	15	)	)	PUNCT
iajs-830	103	16	0	0	NUM
iajs-830	104	1	otherwise	otherwise	ADV
iajs-830	104	2			NUM
iajs-830	104	3			NUM
iajs-830	104	4			NOUN
iajs-830	104	5			NOUN
iajs-830	104	6	at	at	ADP
iajs-830	104	7	=	=	NOUN
iajs-830	104	8	r(1,1	r(1,1	NOUN
iajs-830	104	9	)	)	PUNCT
iajs-830	104	10	,	,	PUNCT
iajs-830	104	11	bt	bt	NOUN
iajs-830	104	12	=	=	NOUN
iajs-830	104	13	r(0,1	r(0,1	NOUN
iajs-830	104	14	)	)	PUNCT
iajs-830	104	15	,	,	PUNCT
iajs-830	104	16	ct	ct	PROPN
iajs-830	104	17	=	=	NOUN
iajs-830	104	18	r(1,0	r(1,0	NUM
iajs-830	104	19	)	)	PUNCT
iajs-830	104	20	,	,	PUNCT
iajs-830	104	21			NOUN
iajs-830	104	22	t(0,1	t(0,1	NOUN
iajs-830	104	23	]	]	PUNCT
iajs-830	104	24	.	.	PUNCT
iajs-830	105	1	at(bt+c	at(bt+c	PROPN
iajs-830	105	2	t	t	PROPN
iajs-830	105	3	)	)	PUNCT
iajs-830	105	4	=	=	SYM
iajs-830	105	5	r(1,1	r(1,1	NOUN
iajs-830	105	6	)	)	PUNCT
iajs-830	105	7	,	,	PUNCT
iajs-830	105	8	(	(	PUNCT
iajs-830	105	9	atb	atb	X
iajs-830	105	10	t)+(atct)=(r(1,1)r(0,1))+(r(1,1)r(1,0))=(0)+(0)=(0	t)+(atct)=(r(1,1)r(0,1))+(r(1,1)r(1,0))=(0)+(0)=(0	ADV
iajs-830	105	11	)	)	PUNCT
iajs-830	105	12	thus	thus	ADV
iajs-830	105	13	at(b+c)t	at(b+c)t	ADJ
iajs-830	105	14			PROPN
iajs-830	105	15	(	(	PUNCT
iajs-830	105	16	ab)t+(ac)t	ab)t+(ac)t	NUM
iajs-830	105	17	,	,	PUNCT
iajs-830	105	18	which	which	PRON
iajs-830	105	19	implies	imply	VERB
iajs-830	105	20	xt	xt	PROPN
iajs-830	105	21	is	be	AUX
iajs-830	105	22	not	not	PART
iajs-830	105	23	a	a	DET
iajs-830	105	24	distributive	distributive	ADJ
iajs-830	105	25	module	module	NOUN
iajs-830	105	26	.	.	PUNCT
iajs-830	106	1	thus	thus	ADV
iajs-830	106	2	x	x	PRON
iajs-830	106	3	is	be	AUX
iajs-830	106	4	not	not	PART
iajs-830	106	5	a	a	DET
iajs-830	106	6	fuzzy	fuzzy	ADJ
iajs-830	106	7	distributive	distributive	ADJ
iajs-830	106	8	module	module	NOUN
iajs-830	106	9	.	.	PUNCT
iajs-830	107	1	2.4	2.4	NUM
iajs-830	107	2	definition	definition	NOUN
iajs-830	107	3	[	[	X
iajs-830	107	4	13	13	NUM
iajs-830	107	5	]	]	PUNCT
iajs-830	107	6	an	an	DET
iajs-830	107	7	r	r	NOUN
iajs-830	107	8	-	-	PUNCT
iajs-830	107	9	module	module	NOUN
iajs-830	107	10	m	m	NOUN
iajs-830	107	11	is	be	AUX
iajs-830	107	12	called	call	VERB
iajs-830	107	13	chained	chain	VERB
iajs-830	107	14	if	if	SCONJ
iajs-830	107	15	for	for	ADP
iajs-830	107	16	each	each	DET
iajs-830	107	17	submodules	submodule	NOUN
iajs-830	107	18	a	a	PRON
iajs-830	107	19	,	,	PUNCT
iajs-830	107	20	b	b	PROPN
iajs-830	107	21	of	of	ADP
iajs-830	107	22	m	m	PROPN
iajs-830	107	23	,	,	PUNCT
iajs-830	107	24	either	either	CCONJ
iajs-830	107	25	a	a	DET
iajs-830	107	26			PROPN
iajs-830	107	27	b	b	PROPN
iajs-830	107	28	or	or	CCONJ
iajs-830	107	29	b	b	PROPN
iajs-830	107	30			PROPN
iajs-830	107	31	a.	a.	NOUN
iajs-830	107	32	we	we	PRON
iajs-830	107	33	fuzzified	fuzzifie	VERB
iajs-830	107	34	this	this	DET
iajs-830	107	35	concept	concept	NOUN
iajs-830	107	36	as	as	SCONJ
iajs-830	107	37	follows	follow	VERB
iajs-830	107	38	.	.	PUNCT
iajs-830	108	1	2.5	2.5	NUM
iajs-830	108	2	definition	definition	NOUN
iajs-830	108	3	let	let	VERB
iajs-830	108	4	x	x	PRON
iajs-830	108	5	be	be	AUX
iajs-830	108	6	a	a	DET
iajs-830	108	7	fuzzy	fuzzy	ADJ
iajs-830	108	8	module	module	NOUN
iajs-830	108	9	of	of	ADP
iajs-830	108	10	an	an	DET
iajs-830	108	11	r	r	NOUN
iajs-830	108	12	-	-	PUNCT
iajs-830	108	13	module	module	NOUN
iajs-830	108	14	m	m	NOUN
iajs-830	108	15	then	then	ADV
iajs-830	108	16	x	x	VERB
iajs-830	108	17	is	be	AUX
iajs-830	108	18	called	call	VERB
iajs-830	108	19	a	a	DET
iajs-830	108	20	fuzzy	fuzzy	ADJ
iajs-830	108	21	chained	chain	VERB
iajs-830	108	22	module	module	NOUN
iajs-830	108	23	if	if	SCONJ
iajs-830	108	24	for	for	ADP
iajs-830	108	25	each	each	DET
iajs-830	108	26	fuzzy	fuzzy	ADJ
iajs-830	108	27	submodules	submodule	NOUN
iajs-830	108	28	a	a	PRON
iajs-830	108	29	,	,	PUNCT
iajs-830	108	30	b	b	PROPN
iajs-830	108	31	of	of	ADP
iajs-830	108	32	x	x	PROPN
iajs-830	108	33	,	,	PUNCT
iajs-830	108	34	either	either	CCONJ
iajs-830	108	35	a	a	DET
iajs-830	108	36			PROPN
iajs-830	108	37	b	b	PROPN
iajs-830	108	38	or	or	CCONJ
iajs-830	108	39	b	b	NOUN
iajs-830	108	40			PROPN
iajs-830	108	41	a.	a.	NOUN
iajs-830	108	42	now	now	ADV
iajs-830	108	43	,	,	PUNCT
iajs-830	108	44	we	we	PRON
iajs-830	108	45	shall	shall	AUX
iajs-830	108	46	give	give	VERB
iajs-830	108	47	a	a	DET
iajs-830	108	48	relationship	relationship	NOUN
iajs-830	108	49	between	between	ADP
iajs-830	108	50	fuzzy	fuzzy	ADJ
iajs-830	108	51	distributive	distributive	ADJ
iajs-830	108	52	module	module	NOUN
iajs-830	108	53	and	and	CCONJ
iajs-830	108	54	fuzzy	fuzzy	ADJ
iajs-830	108	55	chained	chain	VERB
iajs-830	108	56	module	module	NOUN
iajs-830	108	57	.	.	PUNCT
iajs-830	109	1	2.6	2.6	NUM
iajs-830	109	2	proposition	proposition	NOUN
iajs-830	109	3	let	let	VERB
iajs-830	109	4	x	x	PRON
iajs-830	109	5	be	be	AUX
iajs-830	109	6	a	a	DET
iajs-830	109	7	fuzzy	fuzzy	ADJ
iajs-830	109	8	chained	chain	VERB
iajs-830	109	9	module	module	NOUN
iajs-830	109	10	of	of	ADP
iajs-830	109	11	an	an	DET
iajs-830	109	12	r	r	NOUN
iajs-830	109	13	-	-	PUNCT
iajs-830	109	14	module	module	NOUN
iajs-830	109	15	m.	m.	NOUN
iajs-830	109	16	then	then	ADV
iajs-830	109	17	x	x	PRON
iajs-830	109	18	is	be	AUX
iajs-830	109	19	a	a	DET
iajs-830	109	20	fuzzy	fuzzy	ADJ
iajs-830	109	21	distributive	distributive	ADJ
iajs-830	109	22	module	module	NOUN
iajs-830	109	23	.	.	PUNCT
iajs-830	110	1	proof	proof	NOUN
iajs-830	110	2	:	:	PUNCT
iajs-830	110	3	let	let	VERB
iajs-830	110	4	a	a	DET
iajs-830	110	5	,	,	PUNCT
iajs-830	110	6	b	b	NOUN
iajs-830	110	7	,	,	PUNCT
iajs-830	110	8	c	c	X
iajs-830	110	9	fuzzy	fuzzy	ADJ
iajs-830	110	10	submodules	submodule	NOUN
iajs-830	110	11	of	of	ADP
iajs-830	110	12	x	x	PRON
iajs-830	110	13	,	,	PUNCT
iajs-830	110	14	we	we	PRON
iajs-830	110	15	can	can	AUX
iajs-830	110	16	assume	assume	VERB
iajs-830	110	17	that	that	SCONJ
iajs-830	110	18	abc	abc	PROPN
iajs-830	110	19	.	.	PUNCT
iajs-830	111	1	hence	hence	ADV
iajs-830	111	2	a(b+c)=ab	a(b+c)=ab	VERB
iajs-830	111	3	=	=	NOUN
iajs-830	111	4	a.	a.	NOUN
iajs-830	111	5	but	but	CCONJ
iajs-830	111	6	(	(	PUNCT
iajs-830	111	7	ab)+(ac)=a+a	ab)+(ac)=a+a	NOUN
iajs-830	111	8	=	=	NOUN
iajs-830	111	9	a.	a.	NOUN
iajs-830	111	10	thus	thus	ADV
iajs-830	111	11	a(b+c	a(b+c	ADV
iajs-830	111	12	)	)	PUNCT
iajs-830	112	1	=	=	SYM
iajs-830	112	2	(	(	PUNCT
iajs-830	112	3	ab)+(ac	ab)+(ac	PROPN
iajs-830	112	4	)	)	PUNCT
iajs-830	112	5	.	.	PUNCT
iajs-830	113	1			NUM
iajs-830	113	2	2.7	2.7	NUM
iajs-830	113	3	remark	remark	VERB
iajs-830	113	4	the	the	DET
iajs-830	113	5	converse	converse	NOUN
iajs-830	113	6	of	of	ADP
iajs-830	113	7	proposition	proposition	NOUN
iajs-830	113	8	(	(	PUNCT
iajs-830	113	9	2.6	2.6	NUM
iajs-830	113	10	)	)	PUNCT
iajs-830	113	11	is	be	AUX
iajs-830	113	12	not	not	PART
iajs-830	113	13	true	true	ADJ
iajs-830	113	14	in	in	ADP
iajs-830	113	15	general	general	ADJ
iajs-830	113	16	as	as	SCONJ
iajs-830	113	17	the	the	DET
iajs-830	113	18	following	follow	VERB
iajs-830	113	19	example	example	NOUN
iajs-830	113	20	shows	show	VERB
iajs-830	113	21	.	.	PUNCT
iajs-830	114	1	2.8	2.8	NUM
iajs-830	114	2	example	example	NOUN
iajs-830	114	3	let	let	VERB
iajs-830	114	4	x(x)=1	x(x)=1	PRON
iajs-830	114	5	for	for	ADP
iajs-830	114	6	all	all	DET
iajs-830	114	7	xz	xz	NOUN
iajs-830	114	8	,	,	PUNCT
iajs-830	114	9	xt	xt	X
iajs-830	114	10	=	=	PROPN
iajs-830	114	11	z	z	PROPN
iajs-830	114	12	,	,	PUNCT
iajs-830	114	13	t[0,1	t[0,1	NOUN
iajs-830	114	14	]	]	PUNCT
iajs-830	114	15	.	.	PUNCT
iajs-830	115	1	but	but	CCONJ
iajs-830	115	2	z	z	NOUN
iajs-830	115	3	is	be	AUX
iajs-830	115	4	distributive	distributive	ADJ
iajs-830	115	5	.	.	PUNCT
iajs-830	116	1	hence	hence	ADV
iajs-830	116	2	by	by	ADP
iajs-830	116	3	theorem	theorem	NOUN
iajs-830	116	4	(	(	PUNCT
iajs-830	116	5	2.2	2.2	NUM
iajs-830	116	6	)	)	PUNCT
iajs-830	116	7	,	,	PUNCT
iajs-830	116	8	x	x	X
iajs-830	116	9	is	be	AUX
iajs-830	116	10	a	a	DET
iajs-830	116	11	fuzzy	fuzzy	ADJ
iajs-830	116	12	distributive	distributive	ADJ
iajs-830	116	13	.	.	PUNCT
iajs-830	117	1	however	however	ADV
iajs-830	117	2	x	x	PRON
iajs-830	117	3	is	be	AUX
iajs-830	117	4	not	not	PART
iajs-830	117	5	chained	chain	VERB
iajs-830	117	6	since	since	SCONJ
iajs-830	117	7	there	there	PRON
iajs-830	117	8	exists	exist	VERB
iajs-830	117	9	fuzzy	fuzzy	ADJ
iajs-830	117	10	submodules	submodule	NOUN
iajs-830	117	11	a	a	PRON
iajs-830	117	12	,	,	PUNCT
iajs-830	117	13	b	b	X
iajs-830	117	14	such	such	ADJ
iajs-830	117	15	that	that	DET
iajs-830	117	16	1	1	NUM
iajs-830	117	17	x	x	SYM
iajs-830	117	18	2z	2z	NUM
iajs-830	117	19	(	(	PUNCT
iajs-830	117	20	x	x	X
iajs-830	117	21	)	)	PUNCT
iajs-830	117	22	0	0	NUM
iajs-830	118	1	x	x	SYM
iajs-830	118	2	2z	2z	NUM
iajs-830	118	3			PUNCT
iajs-830	119	1			PROPN
iajs-830	119	2			PROPN
iajs-830	119	3			PROPN
iajs-830	119	4	,	,	PUNCT
iajs-830	119	5	1	1	NUM
iajs-830	119	6	x	x	SYM
iajs-830	119	7	3z	3z	NUM
iajs-830	119	8	(	(	PUNCT
iajs-830	119	9	x	x	X
iajs-830	119	10	)	)	PUNCT
iajs-830	119	11	0	0	NUM
iajs-830	120	1	x	x	SYM
iajs-830	120	2	3z	3z	NUM
iajs-830	120	3			ADJ
iajs-830	120	4			NOUN
iajs-830	120	5			NOUN
iajs-830	121	1			NUM
iajs-830	121	2			NOUN
iajs-830	121	3	and	and	CCONJ
iajs-830	121	4	a	a	DET
iajs-830	121	5			PROPN
iajs-830	121	6	b	b	PROPN
iajs-830	121	7	and	and	CCONJ
iajs-830	121	8	b	b	X
iajs-830	122	1			PROPN
iajs-830	122	2	a.	a.	NOUN
iajs-830	122	3	now	now	ADV
iajs-830	122	4	,	,	PUNCT
iajs-830	122	5	we	we	PRON
iajs-830	122	6	can	can	AUX
iajs-830	122	7	give	give	VERB
iajs-830	122	8	the	the	DET
iajs-830	122	9	following	following	NOUN
iajs-830	122	10	.	.	PUNCT
iajs-830	123	1	2.9	2.9	NUM
iajs-830	123	2	theorem	theorem	NOUN
iajs-830	123	3	let	let	VERB
iajs-830	123	4	x	x	PRON
iajs-830	123	5	be	be	AUX
iajs-830	123	6	a	a	DET
iajs-830	123	7	fuzzy	fuzzy	ADJ
iajs-830	123	8	distributive	distributive	ADJ
iajs-830	123	9	module	module	NOUN
iajs-830	123	10	of	of	ADP
iajs-830	123	11	an	an	DET
iajs-830	123	12	r	r	NOUN
iajs-830	123	13	-	-	PUNCT
iajs-830	123	14	module	module	NOUN
iajs-830	123	15	m	m	NOUN
iajs-830	123	16	,	,	PUNCT
iajs-830	123	17	then	then	ADV
iajs-830	123	18	for	for	ADP
iajs-830	123	19	all	all	PRON
iajs-830	123	20	at	at	ADP
iajs-830	123	21	,	,	PUNCT
iajs-830	123	22	bk	bk	ADV
iajs-830	123	23	x	x	NOUN
iajs-830	123	24	,	,	PUNCT
iajs-830	123	25	<	<	X
iajs-830	123	26	1j>=(at	1j>=(at	NUM
iajs-830	123	27	:	:	PUNCT
iajs-830	123	28	bk)+(bk	bk)+(bk	PROPN
iajs-830	123	29	:	:	PUNCT
iajs-830	123	30	at	at	ADP
iajs-830	123	31	)	)	PUNCT
iajs-830	123	32	for	for	ADP
iajs-830	123	33	all	all	DET
iajs-830	123	34	j(0,1	j(0,1	NOUN
iajs-830	123	35	]	]	PUNCT
iajs-830	123	36	.	.	PUNCT
iajs-830	124	1	proof	proof	NOUN
iajs-830	124	2	:	:	PUNCT
iajs-830	124	3	let	let	AUX
iajs-830	124	4	at	at	ADP
iajs-830	124	5	,	,	PUNCT
iajs-830	124	6	bk	bk	ADV
iajs-830	124	7	x	x	NOUN
iajs-830	124	8	,	,	PUNCT
iajs-830	124	9	then	then	ADV
iajs-830	124	10	axt	axt	PROPN
iajs-830	124	11	,	,	PUNCT
iajs-830	124	12	bxk	bxk	PUNCT
iajs-830	124	13	.	.	PUNCT
iajs-830	125	1	assume	assume	VERB
iajs-830	125	2	kt	kt	PROPN
iajs-830	125	3	.	.	PUNCT
iajs-830	126	1	hence	hence	ADV
iajs-830	126	2	xk	xk	PROPN
iajs-830	126	3			PROPN
iajs-830	126	4	xt	xt	PROPN
iajs-830	126	5	and	and	CCONJ
iajs-830	126	6	so	so	ADV
iajs-830	126	7	axk	axk	NOUN
iajs-830	126	8	.	.	PUNCT
iajs-830	127	1	thus	thus	ADV
iajs-830	127	2	a	a	DET
iajs-830	127	3	,	,	PUNCT
iajs-830	127	4	b	b	NOUN
iajs-830	127	5			PROPN
iajs-830	127	6	xk	xk	PROPN
iajs-830	127	7	.	.	PROPN
iajs-830	128	1	but	but	CCONJ
iajs-830	128	2	xk	xk	PROPN
iajs-830	128	3	is	be	AUX
iajs-830	128	4	distributive	distributive	ADJ
iajs-830	128	5	r	r	NOUN
iajs-830	128	6	-	-	PUNCT
iajs-830	128	7	module	module	NOUN
iajs-830	129	1	so	so	CCONJ
iajs-830	129	2	(	(	PUNCT
iajs-830	129	3	a	a	DET
iajs-830	129	4	:	:	PUNCT
iajs-830	129	5	b)+(b	b)+(b	NOUN
iajs-830	129	6	:	:	PUNCT
iajs-830	129	7	a)=r	a)=r	X
iajs-830	129	8	(	(	PUNCT
iajs-830	129	9	by	by	ADP
iajs-830	129	10	[	[	X
iajs-830	129	11	12,theorem	12,theorem	NUM
iajs-830	129	12	(	(	PUNCT
iajs-830	129	13	1.3),p.54	1.3),p.54	NUM
iajs-830	129	14	)	)	PUNCT
iajs-830	129	15	.	.	PUNCT
iajs-830	130	1	it	it	PRON
iajs-830	130	2	follows	follow	VERB
iajs-830	130	3	that	that	SCONJ
iajs-830	130	4	1	1	X
iajs-830	130	5	=	=	SYM
iajs-830	130	6	r1+r2	r1+r2	PROPN
iajs-830	130	7	where	where	SCONJ
iajs-830	130	8	r1(a	r1(a	NOUN
iajs-830	130	9	:	:	PUNCT
iajs-830	130	10	b	b	NOUN
iajs-830	130	11	)	)	PUNCT
iajs-830	130	12	,	,	PUNCT
iajs-830	130	13	r2(b	r2(b	NOUN
iajs-830	130	14	:	:	PUNCT
iajs-830	130	15	a	a	X
iajs-830	130	16	)	)	PUNCT
iajs-830	130	17	for	for	ADP
iajs-830	130	18	some	some	DET
iajs-830	130	19	r1	r1	NOUN
iajs-830	130	20	,	,	PUNCT
iajs-830	130	21	r2	r2	PROPN
iajs-830	130	22	.	.	PUNCT
iajs-830	131	1	hence	hence	ADV
iajs-830	131	2	1j=	1j=	NUM
iajs-830	131	3	(	(	PUNCT
iajs-830	131	4	r1)j	r1)j	NOUN
iajs-830	131	5	+	+	CCONJ
iajs-830	131	6	(	(	PUNCT
iajs-830	131	7	r2)j	r2)j	VERB
iajs-830	131	8	for	for	ADP
iajs-830	131	9	all	all	DET
iajs-830	131	10	j(0,1	j(0,1	NOUN
iajs-830	131	11	]	]	X
iajs-830	131	12	.	.	PUNCT
iajs-830	132	1	but	but	CCONJ
iajs-830	132	2	ibn	ibn	PROPN
iajs-830	132	3	alhaitham	alhaitham	NOUN
iajs-830	132	4	j.	j.	PROPN
iajs-830	133	1	fo	fo	ADP
iajs-830	133	2	r	r	NOUN
iajs-830	133	3	pure	pure	ADJ
iajs-830	133	4	&	&	CCONJ
iajs-830	133	5	appl	appl	PROPN
iajs-830	133	6	.	.	PUNCT
iajs-830	134	1	sc	sc	PROPN
iajs-830	134	2	i.	i.	PROPN
iajs-830	134	3	vo	vo	PROPN
iajs-830	134	4	l.24	l.24	PROPN
iajs-830	134	5	(	(	PUNCT
iajs-830	134	6	1	1	NUM
iajs-830	134	7	)	)	PUNCT
iajs-830	134	8	2011	2011	NUM
iajs-830	134	9	(	(	PUNCT
iajs-830	134	10	r1)jbk	r1)jbk	X
iajs-830	134	11	=	=	SYM
iajs-830	134	12	(	(	PUNCT
iajs-830	134	13	r1b)s	r1b)s	NOUN
iajs-830	134	14	,	,	PUNCT
iajs-830	134	15	where	where	SCONJ
iajs-830	134	16	s	s	NOUN
iajs-830	134	17	=	=	SYM
iajs-830	134	18	min{j	min{j	PROPN
iajs-830	134	19	,	,	PUNCT
iajs-830	134	20	k	k	NOUN
iajs-830	134	21	}	}	PUNCT
iajs-830	134	22	=	=	SYM
iajs-830	134	23	(	(	PUNCT
iajs-830	134	24	r'a)s	r'a)s	PROPN
iajs-830	134	25	,	,	PUNCT
iajs-830	134	26	since	since	SCONJ
iajs-830	134	27	r1(a	r1(a	NOUN
iajs-830	134	28	:	:	PUNCT
iajs-830	134	29	b	b	X
iajs-830	134	30	)	)	PUNCT
iajs-830	134	31			PROPN
iajs-830	134	32	<	<	X
iajs-830	134	33	as	as	ADP
iajs-830	134	34	>	>	X
iajs-830	134	35			PROPN
iajs-830	134	36	<	<	X
iajs-830	134	37	ak	ak	PROPN
iajs-830	134	38	>	>	X
iajs-830	134	39	hence	hence	ADV
iajs-830	134	40	(	(	PUNCT
iajs-830	134	41	r1)j(at	r1)j(at	NOUN
iajs-830	134	42	:	:	PUNCT
iajs-830	134	43	bk	bk	NOUN
iajs-830	134	44	)	)	PUNCT
iajs-830	134	45	,	,	PUNCT
iajs-830	134	46			NOUN
iajs-830	134	47	j(0,1	j(0,1	PROPN
iajs-830	134	48	]	]	PUNCT
iajs-830	134	49	(	(	PUNCT
iajs-830	134	50	r2)jat	r2)jat	X
iajs-830	134	51	=	=	SYM
iajs-830	134	52	(	(	PUNCT
iajs-830	134	53	r2a)f	r2a)f	PROPN
iajs-830	134	54	,	,	PUNCT
iajs-830	134	55	where	where	SCONJ
iajs-830	134	56	f	f	PROPN
iajs-830	134	57	=	=	SYM
iajs-830	134	58	min{j	min{j	PROPN
iajs-830	134	59	,	,	PUNCT
iajs-830	134	60	t	t	PROPN
iajs-830	134	61	}	}	PUNCT
iajs-830	134	62	=	=	SYM
iajs-830	134	63	(	(	PUNCT
iajs-830	134	64	r''b)f	r''b)f	PROPN
iajs-830	134	65	,	,	PUNCT
iajs-830	134	66	since	since	SCONJ
iajs-830	134	67	r2(b	r2(b	NOUN
iajs-830	134	68	:	:	PUNCT
iajs-830	134	69	a	a	X
iajs-830	134	70	)	)	PUNCT
iajs-830	134	71			PROPN
iajs-830	134	72	<	<	X
iajs-830	134	73	bt	bt	PROPN
iajs-830	134	74	>	>	X
iajs-830	134	75			PROPN
iajs-830	134	76	<	<	X
iajs-830	134	77	bk	bk	INTJ
iajs-830	134	78	>	>	X
iajs-830	134	79	hence	hence	ADV
iajs-830	134	80	(	(	PUNCT
iajs-830	134	81	r2)j(bk	r2)j(bk	NOUN
iajs-830	134	82	:	:	PUNCT
iajs-830	134	83	at	at	ADP
iajs-830	134	84	)	)	PUNCT
iajs-830	134	85	,	,	PUNCT
iajs-830	134	86			NOUN
iajs-830	134	87	j(0,1	j(0,1	NOUN
iajs-830	134	88	]	]	X
iajs-830	134	89	so	so	ADV
iajs-830	134	90	(	(	PUNCT
iajs-830	134	91	r1)j+(r2)j(at	r1)j+(r2)j(at	NOUN
iajs-830	134	92	:	:	PUNCT
iajs-830	134	93	bk	bk	NOUN
iajs-830	134	94	)	)	PUNCT
iajs-830	134	95	+	+	CCONJ
iajs-830	134	96	(	(	PUNCT
iajs-830	134	97	bk	bk	NOUN
iajs-830	134	98	:	:	PUNCT
iajs-830	134	99	at	at	ADP
iajs-830	134	100	)	)	PUNCT
iajs-830	134	101	and	and	CCONJ
iajs-830	134	102	hence	hence	ADV
iajs-830	134	103	(	(	PUNCT
iajs-830	134	104	r1+r2)j(at	r1+r2)j(at	NOUN
iajs-830	134	105	:	:	PUNCT
iajs-830	134	106	bk	bk	NOUN
iajs-830	134	107	)	)	PUNCT
iajs-830	134	108	+	+	CCONJ
iajs-830	134	109	(	(	PUNCT
iajs-830	134	110	bk	bk	NOUN
iajs-830	134	111	:	:	PUNCT
iajs-830	134	112	at	at	ADP
iajs-830	134	113	)	)	PUNCT
iajs-830	134	114	.	.	PUNCT
iajs-830	135	1	thus	thus	ADV
iajs-830	135	2	1j	1j	NUM
iajs-830	135	3			NOUN
iajs-830	135	4	(	(	PUNCT
iajs-830	135	5	at	at	ADP
iajs-830	135	6	:	:	PUNCT
iajs-830	135	7	bk	bk	VERB
iajs-830	135	8	)	)	PUNCT
iajs-830	135	9	+	+	CCONJ
iajs-830	135	10	(	(	PUNCT
iajs-830	135	11	bk	bk	NOUN
iajs-830	135	12	:	:	PUNCT
iajs-830	135	13	at	at	ADP
iajs-830	135	14	)	)	PUNCT
iajs-830	135	15	.	.	PUNCT
iajs-830	136	1			NUM
iajs-830	136	2	2.10	2.10	NUM
iajs-830	136	3	remark	remark	NOUN
iajs-830	136	4	if	if	SCONJ
iajs-830	136	5	yx	yx	NOUN
iajs-830	136	6	and	and	CCONJ
iajs-830	136	7	x	x	NOUN
iajs-830	136	8	is	be	AUX
iajs-830	136	9	fuzzy	fuzzy	ADJ
iajs-830	136	10	distributive	distributive	ADJ
iajs-830	136	11	module	module	NOUN
iajs-830	136	12	then	then	ADV
iajs-830	136	13	y	y	PROPN
iajs-830	136	14	is	be	AUX
iajs-830	136	15	fuzzy	fuzzy	ADJ
iajs-830	136	16	distributive	distributive	ADJ
iajs-830	136	17	module	module	NOUN
iajs-830	136	18	.	.	PUNCT
iajs-830	137	1	proof	proof	NOUN
iajs-830	137	2	:	:	PUNCT
iajs-830	137	3	let	let	VERB
iajs-830	137	4	a	a	DET
iajs-830	137	5	,	,	PUNCT
iajs-830	137	6	b	b	NOUN
iajs-830	137	7	,	,	PUNCT
iajs-830	137	8	c	c	PROPN
iajs-830	137	9	are	be	AUX
iajs-830	137	10	fuzzy	fuzzy	ADJ
iajs-830	137	11	submodules	submodule	NOUN
iajs-830	137	12	of	of	ADP
iajs-830	137	13	y	y	PROPN
iajs-830	137	14	,	,	PUNCT
iajs-830	137	15	then	then	ADV
iajs-830	137	16	a	a	DET
iajs-830	137	17	,	,	PUNCT
iajs-830	137	18	b	b	NOUN
iajs-830	137	19	,	,	PUNCT
iajs-830	137	20	c	c	PROPN
iajs-830	137	21	are	be	AUX
iajs-830	137	22	fuzzy	fuzzy	ADJ
iajs-830	137	23	submodules	submodule	NOUN
iajs-830	137	24	of	of	ADP
iajs-830	137	25	x	x	X
iajs-830	137	26	(	(	PUNCT
iajs-830	137	27	since	since	SCONJ
iajs-830	137	28	yx	yx	NOUN
iajs-830	137	29	)	)	PUNCT
iajs-830	137	30	.	.	PUNCT
iajs-830	138	1	but	but	CCONJ
iajs-830	138	2	x	x	PRON
iajs-830	138	3	is	be	AUX
iajs-830	138	4	fuzzy	fuzzy	ADJ
iajs-830	138	5	distributive	distributive	ADJ
iajs-830	138	6	module	module	NOUN
iajs-830	138	7	,	,	PUNCT
iajs-830	138	8	so	so	ADV
iajs-830	138	9	a(b+c	a(b+c	ADV
iajs-830	138	10	)	)	PUNCT
iajs-830	139	1	=	=	SYM
iajs-830	139	2	(	(	PUNCT
iajs-830	139	3	ab)+(ac	ab)+(ac	PROPN
iajs-830	139	4	)	)	PUNCT
iajs-830	139	5	which	which	PRON
iajs-830	139	6	implies	imply	VERB
iajs-830	139	7	y	y	PROPN
iajs-830	139	8	is	be	AUX
iajs-830	139	9	a	a	DET
iajs-830	139	10	fuzzy	fuzzy	ADJ
iajs-830	139	11	distributive	distributive	ADJ
iajs-830	139	12	.	.	PUNCT
iajs-830	140	1			NUM
iajs-830	140	2	now	now	ADV
iajs-830	140	3	,	,	PUNCT
iajs-830	140	4	we	we	PRON
iajs-830	140	5	study	study	VERB
iajs-830	140	6	the	the	DET
iajs-830	140	7	direct	direct	ADJ
iajs-830	140	8	sum	sum	NOUN
iajs-830	140	9	of	of	ADP
iajs-830	140	10	fuzzy	fuzzy	ADJ
iajs-830	140	11	distributive	distributive	ADJ
iajs-830	140	12	modules	module	NOUN
iajs-830	140	13	.	.	PUNCT
iajs-830	141	1	but	but	CCONJ
iajs-830	141	2	first	first	ADV
iajs-830	141	3	we	we	PRON
iajs-830	141	4	state	state	VERB
iajs-830	141	5	and	and	CCONJ
iajs-830	141	6	prove	prove	VERB
iajs-830	141	7	the	the	DET
iajs-830	141	8	following	follow	VERB
iajs-830	141	9	lemma	lemma	PROPN
iajs-830	141	10	.	.	PROPN
iajs-830	142	1	2.11	2.11	NUM
iajs-830	142	2	lemma	lemma	PROPN
iajs-830	142	3	if	if	SCONJ
iajs-830	142	4	m	m	PROPN
iajs-830	142	5	1	1	NUM
iajs-830	142	6	,	,	PUNCT
iajs-830	142	7	m	m	VERB
iajs-830	142	8	2	2	NUM
iajs-830	142	9	are	be	AUX
iajs-830	142	10	distributive	distributive	ADJ
iajs-830	142	11	r	r	NOUN
iajs-830	142	12	-	-	PUNCT
iajs-830	142	13	modules	module	NOUN
iajs-830	142	14	such	such	ADJ
iajs-830	142	15	that	that	DET
iajs-830	142	16	annm1	annm1	NOUN
iajs-830	142	17	+	+	CCONJ
iajs-830	142	18	annm2	annm2	PROPN
iajs-830	142	19	=	=	SYM
iajs-830	142	20	r	r	NOUN
iajs-830	142	21	,	,	PUNCT
iajs-830	142	22	then	then	ADV
iajs-830	142	23	m1m	m1m	PROPN
iajs-830	142	24	2	2	NUM
iajs-830	142	25	=	=	NOUN
iajs-830	142	26	m	m	VERB
iajs-830	142	27	is	be	AUX
iajs-830	142	28	a	a	DET
iajs-830	142	29	distributive	distributive	ADJ
iajs-830	142	30	r	r	NOUN
iajs-830	142	31	-	-	PUNCT
iajs-830	142	32	module	module	NOUN
iajs-830	142	33	.	.	PUNCT
iajs-830	143	1	proof	proof	NOUN
iajs-830	143	2	:	:	PUNCT
iajs-830	143	3	let	let	VERB
iajs-830	143	4	a	a	DET
iajs-830	143	5	,	,	PUNCT
iajs-830	143	6	b	b	NOUN
iajs-830	143	7	,	,	PUNCT
iajs-830	143	8	c	c	AUX
iajs-830	143	9	be	be	AUX
iajs-830	143	10	submodules	submodule	NOUN
iajs-830	143	11	of	of	ADP
iajs-830	143	12	m.	m.	NOUN
iajs-830	143	13	since	since	SCONJ
iajs-830	143	14	annm	annm	NOUN
iajs-830	143	15	1	1	NUM
iajs-830	143	16	+	+	NUM
iajs-830	143	17	annm2	annm2	NOUN
iajs-830	143	18	=	=	SYM
iajs-830	143	19	r	r	NOUN
iajs-830	143	20	,	,	PUNCT
iajs-830	143	21	a	a	DET
iajs-830	143	22	=	=	NOUN
iajs-830	143	23	a1b1	a1b1	ADJ
iajs-830	143	24	,	,	PUNCT
iajs-830	143	25	b=	b=	NOUN
iajs-830	143	26	a2b2	a2b2	PROPN
iajs-830	143	27	,	,	PUNCT
iajs-830	143	28	c=	c=	VERB
iajs-830	143	29	a3b3	a3b3	ADJ
iajs-830	143	30	for	for	ADP
iajs-830	143	31	some	some	DET
iajs-830	143	32	submodules	submodule	NOUN
iajs-830	143	33	a1	a1	NOUN
iajs-830	143	34	,	,	PUNCT
iajs-830	143	35	a2	a2	PROPN
iajs-830	143	36	,	,	PUNCT
iajs-830	143	37	a3	a3	NOUN
iajs-830	143	38	of	of	ADP
iajs-830	143	39	m	m	PROPN
iajs-830	143	40	1	1	NUM
iajs-830	143	41	and	and	CCONJ
iajs-830	143	42	some	some	DET
iajs-830	143	43	submodules	submodule	NOUN
iajs-830	143	44	b1	b1	NOUN
iajs-830	143	45	,	,	PUNCT
iajs-830	143	46	b2	b2	NOUN
iajs-830	143	47	,	,	PUNCT
iajs-830	143	48	b3	b3	PROPN
iajs-830	143	49	of	of	ADP
iajs-830	143	50	m	m	PROPN
iajs-830	143	51	2	2	NUM
iajs-830	143	52	.	.	PUNCT
iajs-830	143	53	to	to	PART
iajs-830	143	54	prove	prove	VERB
iajs-830	143	55	a(b+c	a(b+c	ADV
iajs-830	143	56	)	)	PUNCT
iajs-830	143	57	=	=	SYM
iajs-830	143	58	(	(	PUNCT
iajs-830	143	59	ab)+(ac	ab)+(ac	NOUN
iajs-830	143	60	)	)	PUNCT
iajs-830	143	61	a(b+c	a(b+c	ADV
iajs-830	143	62	)	)	PUNCT
iajs-830	143	63	=	=	SYM
iajs-830	143	64	(	(	PUNCT
iajs-830	143	65	a1b1)[(a2b2)+(a3b3	a1b1)[(a2b2)+(a3b3	PROPN
iajs-830	143	66	)	)	PUNCT
iajs-830	143	67	]	]	PUNCT
iajs-830	144	1	=	=	SYM
iajs-830	144	2	(	(	PUNCT
iajs-830	144	3	a1b1)[(a2+a3)+	a1b1)[(a2+a3)+	PROPN
iajs-830	144	4	(	(	PUNCT
iajs-830	144	5	b2+b3	b2+b3	PROPN
iajs-830	144	6	)	)	PUNCT
iajs-830	144	7	]	]	PUNCT
iajs-830	144	8	=	=	PUNCT
iajs-830	145	1	[	[	X
iajs-830	145	2	a1(a2+a3)][b1(b2+b3	a1(a2+a3)][b1(b2+b3	ADJ
iajs-830	145	3	)	)	PUNCT
iajs-830	145	4	]	]	PUNCT
iajs-830	146	1	=	=	PUNCT
iajs-830	147	1	[	[	X
iajs-830	147	2	(	(	PUNCT
iajs-830	147	3	a1a2)+(a1a3)][(b1b2)+(b1b3	a1a2)+(a1a3)][(b1b2)+(b1b3	NOUN
iajs-830	147	4	)	)	PUNCT
iajs-830	147	5	]	]	PUNCT
iajs-830	148	1	(	(	PUNCT
iajs-830	148	2	m	m	NOUN
iajs-830	148	3	1	1	NUM
iajs-830	148	4	and	and	CCONJ
iajs-830	148	5	m2	m2	PROPN
iajs-830	148	6	are	be	AUX
iajs-830	148	7	distributive	distributive	ADJ
iajs-830	148	8	modules	module	NOUN
iajs-830	148	9	)	)	PUNCT
iajs-830	149	1	=	=	PUNCT
iajs-830	150	1	[	[	X
iajs-830	150	2	(	(	PUNCT
iajs-830	150	3	a1a2)(b1b2	a1a2)(b1b2	PROPN
iajs-830	150	4	)	)	PUNCT
iajs-830	150	5	]	]	PUNCT
iajs-830	151	1	+	+	CCONJ
iajs-830	151	2	[	[	X
iajs-830	151	3	(	(	PUNCT
iajs-830	151	4	a1a3)(b1b3	a1a3)(b1b3	PROPN
iajs-830	151	5	)	)	PUNCT
iajs-830	151	6	]	]	PUNCT
iajs-830	152	1	=	=	PUNCT
iajs-830	153	1	[	[	X
iajs-830	153	2	(	(	PUNCT
iajs-830	153	3	a1b1)(a2	a1b1)(a2	PROPN
iajs-830	153	4	b2)]+[(a1b1)(a3	b2)]+[(a1b1)(a3	PROPN
iajs-830	153	5	b3	b3	PROPN
iajs-830	153	6	)	)	PUNCT
iajs-830	153	7	]	]	PUNCT
iajs-830	154	1	=	=	PUNCT
iajs-830	154	2	(	(	PUNCT
iajs-830	154	3	ab)+(ac	ab)+(ac	PROPN
iajs-830	154	4	)	)	PUNCT
iajs-830	154	5	.	.	PUNCT
iajs-830	155	1			NUM
iajs-830	155	2	2.12	2.12	NUM
iajs-830	155	3	proposition	proposition	NOUN
iajs-830	155	4	let	let	VERB
iajs-830	155	5	x	x	PRON
iajs-830	155	6	and	and	CCONJ
iajs-830	155	7	y	y	PROPN
iajs-830	155	8	be	be	AUX
iajs-830	155	9	fuzzy	fuzzy	ADJ
iajs-830	155	10	distributive	distributive	ADJ
iajs-830	155	11	modules	module	NOUN
iajs-830	155	12	of	of	ADP
iajs-830	155	13	r	r	NOUN
iajs-830	155	14	-	-	PUNCT
iajs-830	155	15	modules	module	NOUN
iajs-830	155	16	m	m	NOUN
iajs-830	155	17	1	1	NUM
iajs-830	155	18	,	,	PUNCT
iajs-830	155	19	m	m	VERB
iajs-830	155	20	2	2	NUM
iajs-830	155	21	respectively	respectively	ADV
iajs-830	155	22	,	,	PUNCT
iajs-830	155	23	then	then	ADV
iajs-830	155	24	xy	xy	PROPN
iajs-830	155	25	is	be	AUX
iajs-830	155	26	a	a	DET
iajs-830	155	27	fuzzy	fuzzy	ADJ
iajs-830	155	28	distributive	distributive	ADJ
iajs-830	155	29	module	module	NOUN
iajs-830	155	30	of	of	ADP
iajs-830	155	31	m	m	PROPN
iajs-830	155	32	1m	1m	NUM
iajs-830	155	33	2	2	NUM
iajs-830	155	34	,	,	PUNCT
iajs-830	155	35	provided	provide	VERB
iajs-830	155	36	annm1	annm1	NOUN
iajs-830	156	1	+	+	X
iajs-830	157	1	annm2	annm2	NOUN
iajs-830	157	2	=	=	PUNCT
iajs-830	157	3	r.	r.	NOUN
iajs-830	157	4	proof	proof	NOUN
iajs-830	157	5	:	:	PUNCT
iajs-830	157	6	by	by	ADP
iajs-830	157	7	theorem	theorem	NOUN
iajs-830	157	8	(	(	PUNCT
iajs-830	157	9	2.2	2.2	NUM
iajs-830	157	10	)	)	PUNCT
iajs-830	157	11	,	,	PUNCT
iajs-830	157	12	xt	xt	PUNCT
iajs-830	157	13	and	and	CCONJ
iajs-830	157	14	yt	yt	PROPN
iajs-830	157	15	are	be	AUX
iajs-830	157	16	distributive	distributive	ADJ
iajs-830	157	17	submodules	submodule	NOUN
iajs-830	157	18	of	of	ADP
iajs-830	157	19	m	m	PROPN
iajs-830	157	20	1	1	NUM
iajs-830	157	21	and	and	CCONJ
iajs-830	157	22	m2	m2	PROPN
iajs-830	157	23	respectively	respectively	ADV
iajs-830	157	24	,	,	PUNCT
iajs-830	157	25	for	for	ADP
iajs-830	157	26	all	all	DET
iajs-830	157	27	t	t	NOUN
iajs-830	157	28	(0,1	(0,1	NUM
iajs-830	157	29	]	]	PUNCT
iajs-830	157	30	.	.	PUNCT
iajs-830	158	1	hence	hence	ADV
iajs-830	158	2	by	by	ADP
iajs-830	158	3	lemma	lemma	PROPN
iajs-830	158	4	(	(	PUNCT
iajs-830	158	5	2.11	2.11	NUM
iajs-830	158	6	)	)	PUNCT
iajs-830	158	7	(	(	PUNCT
iajs-830	158	8	xtyt	xtyt	X
iajs-830	158	9	)	)	PUNCT
iajs-830	158	10	is	be	AUX
iajs-830	158	11	a	a	DET
iajs-830	158	12	distributive	distributive	ADJ
iajs-830	158	13	submodule	submodule	NOUN
iajs-830	158	14	of	of	ADP
iajs-830	158	15	m	m	PROPN
iajs-830	158	16	1m	1m	PROPN
iajs-830	158	17	2	2	NUM
iajs-830	158	18	.	.	PUNCT
iajs-830	159	1	but	but	CCONJ
iajs-830	159	2	(	(	PUNCT
iajs-830	159	3	xy)t	xy)t	NOUN
iajs-830	159	4	=	=	SYM
iajs-830	159	5	(	(	PUNCT
iajs-830	159	6	xtyt	xtyt	X
iajs-830	159	7	)	)	PUNCT
iajs-830	159	8	by	by	ADP
iajs-830	159	9	(	(	PUNCT
iajs-830	159	10	(	(	PUNCT
iajs-830	159	11	11	11	NUM
iajs-830	159	12	)	)	PUNCT
iajs-830	159	13	,	,	PUNCT
iajs-830	159	14	lemma	lemma	PROPN
iajs-830	159	15	(	(	PUNCT
iajs-830	159	16	2.2.4	2.2.4	NUM
iajs-830	159	17	)	)	PUNCT
iajs-830	159	18	)	)	PUNCT
iajs-830	159	19	.	.	PUNCT
iajs-830	160	1	thus	thus	ADV
iajs-830	160	2	xy	xy	PROPN
iajs-830	160	3	is	be	AUX
iajs-830	160	4	a	a	DET
iajs-830	160	5	fuzzy	fuzzy	ADJ
iajs-830	160	6	distributive	distributive	ADJ
iajs-830	160	7	module	module	NOUN
iajs-830	160	8	by	by	ADP
iajs-830	160	9	theorem	theorem	NOUN
iajs-830	160	10	(	(	PUNCT
iajs-830	160	11	2.2	2.2	NUM
iajs-830	160	12	)	)	PUNCT
iajs-830	160	13	.	.	PUNCT
iajs-830	161	1			NUM
iajs-830	161	2	3	3	NUM
iajs-830	161	3	.	.	PUNCT
iajs-830	162	1	the	the	DET
iajs-830	162	2	image	image	NOUN
iajs-830	162	3	and	and	CCONJ
iajs-830	162	4	inverse	inverse	NOUN
iajs-830	162	5	image	image	NOUN
iajs-830	162	6	of	of	ADP
iajs-830	162	7	fuzzy	fuzzy	ADJ
iajs-830	162	8	distributive	distributive	ADJ
iajs-830	162	9	modules	module	NOUN
iajs-830	162	10	in	in	ADP
iajs-830	162	11	this	this	DET
iajs-830	162	12	section	section	NOUN
iajs-830	162	13	,	,	PUNCT
iajs-830	162	14	we	we	PRON
iajs-830	162	15	shall	shall	AUX
iajs-830	162	16	indicate	indicate	VERB
iajs-830	162	17	the	the	DET
iajs-830	162	18	behaviour	behaviour	NOUN
iajs-830	162	19	of	of	ADP
iajs-830	162	20	fuzzy	fuzzy	ADJ
iajs-830	162	21	distributive	distributive	ADJ
iajs-830	162	22	modules	module	NOUN
iajs-830	162	23	under	under	ADP
iajs-830	162	24	homomorphisms	homomorphism	NOUN
iajs-830	162	25	.	.	PUNCT
iajs-830	163	1	to	to	PART
iajs-830	163	2	do	do	VERB
iajs-830	163	3	this	this	PRON
iajs-830	163	4	we	we	PRON
iajs-830	163	5	need	need	VERB
iajs-830	163	6	some	some	DET
iajs-830	163	7	definitions	definition	NOUN
iajs-830	163	8	and	and	CCONJ
iajs-830	163	9	propositions	proposition	NOUN
iajs-830	163	10	.	.	PUNCT
iajs-830	164	1	3.1	3.1	NUM
iajs-830	164	2	definition	definition	NOUN
iajs-830	164	3	(	(	PUNCT
iajs-830	164	4	5	5	X
iajs-830	164	5	)	)	PUNCT
iajs-830	164	6	let	let	VERB
iajs-830	164	7	f	f	PRON
iajs-830	164	8	be	be	AUX
iajs-830	164	9	a	a	DET
iajs-830	164	10	mapping	mapping	NOUN
iajs-830	164	11	from	from	ADP
iajs-830	164	12	a	a	DET
iajs-830	164	13	set	set	NOUN
iajs-830	164	14	m	m	NOUN
iajs-830	164	15	into	into	ADP
iajs-830	164	16	a	a	DET
iajs-830	164	17	set	set	NOUN
iajs-830	164	18	n	n	CCONJ
iajs-830	164	19	,	,	PUNCT
iajs-830	164	20	let	let	VERB
iajs-830	164	21	a	a	PRON
iajs-830	164	22	be	be	AUX
iajs-830	164	23	a	a	DET
iajs-830	164	24	fuzzy	fuzzy	ADJ
iajs-830	164	25	set	set	NOUN
iajs-830	164	26	in	in	ADP
iajs-830	164	27	m	m	PROPN
iajs-830	164	28	and	and	CCONJ
iajs-830	164	29	b	b	X
iajs-830	164	30	be	be	AUX
iajs-830	164	31	a	a	DET
iajs-830	164	32	fuzzy	fuzzy	ADJ
iajs-830	164	33	set	set	NOUN
iajs-830	164	34	in	in	ADP
iajs-830	164	35	n.	n.	NOUN
iajs-830	164	36	the	the	DET
iajs-830	164	37	image	image	NOUN
iajs-830	164	38	of	of	ADP
iajs-830	164	39	a	a	DET
iajs-830	164	40	denoted	denote	VERB
iajs-830	164	41	by	by	ADP
iajs-830	164	42	f(a	f(a	PROPN
iajs-830	164	43	)	)	PUNCT
iajs-830	164	44	is	be	AUX
iajs-830	164	45	the	the	DET
iajs-830	164	46	set	set	NOUN
iajs-830	164	47	in	in	ADP
iajs-830	164	48	n	n	CCONJ
iajs-830	164	49	defined	define	VERB
iajs-830	164	50	by	by	ADP
iajs-830	164	51	1	1	NUM
iajs-830	164	52	1sup{a(z	1sup{a(z	NUM
iajs-830	164	53	)	)	PUNCT
iajs-830	165	1	z	z	NOUN
iajs-830	165	2	f	f	PROPN
iajs-830	165	3	(	(	PUNCT
iajs-830	165	4	y	y	NOUN
iajs-830	165	5	)	)	PUNCT
iajs-830	165	6	}	}	PUNCT
iajs-830	165	7	if	if	SCONJ
iajs-830	165	8	f	f	PROPN
iajs-830	165	9	(	(	PUNCT
iajs-830	165	10	y	y	PROPN
iajs-830	165	11	)	)	PUNCT
iajs-830	165	12	,	,	PUNCT
iajs-830	165	13	y	y	PROPN
iajs-830	165	14	a	a	DET
iajs-830	165	15	f(a)(y	f(a)(y	NOUN
iajs-830	165	16	)	)	PUNCT
iajs-830	165	17	0	0	PUNCT
iajs-830	165	18	otherwise	otherwise	ADV
iajs-830	165	19			NOUN
iajs-830	165	20			PROPN
iajs-830	165	21			PROPN
iajs-830	165	22			VERB
iajs-830	165	23			NOUN
iajs-830	165	24			NOUN
iajs-830	165	25			PROPN
iajs-830	165	26			NUM
iajs-830	165	27			NOUN
iajs-830	165	28	and	and	CCONJ
iajs-830	165	29	the	the	DET
iajs-830	165	30	inverse	inverse	ADJ
iajs-830	165	31	image	image	NOUN
iajs-830	165	32	of	of	ADP
iajs-830	165	33	f	f	PROPN
iajs-830	165	34	denoted	denote	VERB
iajs-830	165	35	by	by	ADP
iajs-830	165	36	f	f	PROPN
iajs-830	165	37	–	–	PUNCT
iajs-830	165	38	1	1	NUM
iajs-830	165	39	(	(	PUNCT
iajs-830	165	40	b	b	NOUN
iajs-830	165	41	)	)	PUNCT
iajs-830	165	42	,	,	PUNCT
iajs-830	165	43	where	where	SCONJ
iajs-830	165	44	f	f	PROPN
iajs-830	165	45	–	–	PUNCT
iajs-830	165	46	1(b)(x)=b(f(x	1(b)(x)=b(f(x	NOUN
iajs-830	165	47	)	)	PUNCT
iajs-830	165	48	)	)	PUNCT
iajs-830	165	49	,	,	PUNCT
iajs-830	165	50	for	for	ADP
iajs-830	165	51	all	all	DET
iajs-830	165	52	xm	xm	NOUN
iajs-830	165	53	.	.	PUNCT
iajs-830	165	54	recall	recall	VERB
iajs-830	165	55	the	the	DET
iajs-830	165	56	following	following	NOUN
iajs-830	165	57	.	.	PUNCT
iajs-830	166	1	ibn	ibn	PROPN
iajs-830	166	2	alhaitham	alhaitham	NOUN
iajs-830	167	1	j.	j.	PROPN
iajs-830	168	1	fo	fo	ADP
iajs-830	168	2	r	r	NOUN
iajs-830	168	3	pure	pure	ADJ
iajs-830	168	4	&	&	CCONJ
iajs-830	168	5	appl	appl	PROPN
iajs-830	168	6	.	.	PUNCT
iajs-830	169	1	sc	sc	PROPN
iajs-830	169	2	i.	i.	PROPN
iajs-830	169	3	vo	vo	PROPN
iajs-830	169	4	l.24	l.24	PROPN
iajs-830	169	5	(	(	PUNCT
iajs-830	169	6	1	1	NUM
iajs-830	169	7	)	)	PUNCT
iajs-830	169	8	2011	2011	NUM
iajs-830	169	9	3.2	3.2	NUM
iajs-830	169	10	definition	definition	NOUN
iajs-830	169	11	(	(	PUNCT
iajs-830	169	12	14	14	NUM
iajs-830	169	13	)	)	PUNCT
iajs-830	169	14	let	let	VERB
iajs-830	169	15	f	f	PRON
iajs-830	169	16	be	be	AUX
iajs-830	169	17	a	a	DET
iajs-830	169	18	function	function	NOUN
iajs-830	169	19	from	from	ADP
iajs-830	169	20	a	a	DET
iajs-830	169	21	set	set	NOUN
iajs-830	169	22	m	m	NOUN
iajs-830	169	23	into	into	ADP
iajs-830	169	24	a	a	DET
iajs-830	169	25	set	set	NOUN
iajs-830	169	26	m	m	NOUN
iajs-830	169	27	'	'	NUM
iajs-830	169	28	.	.	PUNCT
iajs-830	170	1	a	a	DET
iajs-830	170	2	fuzzy	fuzzy	NOUN
iajs-830	170	3	subset	subset	VERB
iajs-830	170	4	a	a	PRON
iajs-830	170	5	of	of	ADP
iajs-830	170	6	m	m	PROPN
iajs-830	170	7	is	be	AUX
iajs-830	170	8	called	call	VERB
iajs-830	170	9	f	f	X
iajs-830	170	10	-	-	PUNCT
iajs-830	170	11	invariant	invariant	ADJ
iajs-830	170	12	if	if	SCONJ
iajs-830	170	13	a(x)=a(y	a(x)=a(y	PROPN
iajs-830	170	14	)	)	PUNCT
iajs-830	170	15	whenever	whenever	SCONJ
iajs-830	170	16	f(x)=f(y	f(x)=f(y	ADV
iajs-830	170	17	)	)	PUNCT
iajs-830	170	18	,	,	PUNCT
iajs-830	170	19	where	where	SCONJ
iajs-830	170	20	x	x	X
iajs-830	170	21	,	,	PUNCT
iajs-830	170	22	y	y	PROPN
iajs-830	170	23	m	m	PROPN
iajs-830	170	24	.	.	PROPN
iajs-830	170	25	3.3	3.3	NUM
iajs-830	170	26	definition	definition	NOUN
iajs-830	170	27	(	(	PUNCT
iajs-830	170	28	4	4	X
iajs-830	170	29	)	)	PUNCT
iajs-830	170	30	let	let	VERB
iajs-830	170	31	x	x	PRON
iajs-830	170	32	and	and	CCONJ
iajs-830	170	33	y	y	PROPN
iajs-830	170	34	be	be	AUX
iajs-830	170	35	two	two	NUM
iajs-830	170	36	fuzzy	fuzzy	ADJ
iajs-830	170	37	modules	module	NOUN
iajs-830	170	38	of	of	ADP
iajs-830	170	39	r	r	NOUN
iajs-830	170	40	-	-	PUNCT
iajs-830	170	41	modules	module	NOUN
iajs-830	170	42	m	m	NOUN
iajs-830	170	43	1	1	NUM
iajs-830	170	44	and	and	CCONJ
iajs-830	170	45	m2	m2	PROPN
iajs-830	170	46	respectively	respectively	ADV
iajs-830	170	47	.	.	PUNCT
iajs-830	171	1	f	f	X
iajs-830	171	2	:	:	PUNCT
iajs-830	171	3	xy	xy	PROPN
iajs-830	171	4	is	be	AUX
iajs-830	171	5	called	call	VERB
iajs-830	171	6	a	a	DET
iajs-830	171	7	fuzzy	fuzzy	ADJ
iajs-830	171	8	homomorphism	homomorphism	NOUN
iajs-830	171	9	if	if	SCONJ
iajs-830	171	10	f	f	X
iajs-830	171	11	:	:	PUNCT
iajs-830	171	12	m	m	VERB
iajs-830	171	13	1	1	NUM
iajs-830	171	14			PROPN
iajs-830	171	15	m	m	VERB
iajs-830	171	16	2	2	NUM
iajs-830	171	17	is	be	AUX
iajs-830	171	18	r	r	NOUN
iajs-830	171	19	-	-	PUNCT
iajs-830	171	20	homomorphism	homomorphism	NOUN
iajs-830	171	21	and	and	CCONJ
iajs-830	171	22	y(f(x	y(f(x	NOUN
iajs-830	171	23	)	)	PUNCT
iajs-830	171	24	)	)	PUNCT
iajs-830	172	1	=	=	SYM
iajs-830	172	2	x(x	x(x	PROPN
iajs-830	172	3	)	)	PUNCT
iajs-830	172	4	,	,	PUNCT
iajs-830	172	5	for	for	ADP
iajs-830	172	6	each	each	DET
iajs-830	172	7	xm	xm	NOUN
iajs-830	172	8	1	1	X
iajs-830	172	9	.	.	X
iajs-830	172	10	3.4	3.4	NUM
iajs-830	172	11	proposition	proposition	NOUN
iajs-830	172	12	let	let	VERB
iajs-830	172	13	x	x	PRON
iajs-830	172	14	and	and	CCONJ
iajs-830	172	15	y	y	PROPN
iajs-830	172	16	be	be	AUX
iajs-830	172	17	two	two	NUM
iajs-830	172	18	fuzzy	fuzzy	ADJ
iajs-830	172	19	modules	module	NOUN
iajs-830	172	20	of	of	ADP
iajs-830	172	21	r	r	NOUN
iajs-830	172	22	-	-	PUNCT
iajs-830	172	23	modules	module	NOUN
iajs-830	172	24	m	m	NOUN
iajs-830	172	25	1	1	NUM
iajs-830	172	26	and	and	CCONJ
iajs-830	172	27	m	m	PROPN
iajs-830	172	28	2	2	NUM
iajs-830	172	29	respectively	respectively	ADV
iajs-830	172	30	.	.	PUNCT
iajs-830	173	1	f	f	X
iajs-830	173	2	:	:	PUNCT
iajs-830	173	3	xy	xy	PROPN
iajs-830	173	4	be	be	AUX
iajs-830	173	5	a	a	DET
iajs-830	173	6	fuzzy	fuzzy	ADJ
iajs-830	173	7	homomorphism	homomorphism	NOUN
iajs-830	173	8	if	if	SCONJ
iajs-830	173	9	a	a	PRON
iajs-830	173	10	and	and	CCONJ
iajs-830	173	11	b	b	NOUN
iajs-830	173	12	are	be	AUX
iajs-830	173	13	two	two	NUM
iajs-830	173	14	fuzzy	fuzzy	ADJ
iajs-830	173	15	submodules	submodule	NOUN
iajs-830	173	16	of	of	ADP
iajs-830	173	17	x	x	X
iajs-830	173	18	and	and	CCONJ
iajs-830	173	19	y	y	PROPN
iajs-830	173	20	respectively	respectively	ADV
iajs-830	173	21	,	,	PUNCT
iajs-830	173	22	then	then	ADV
iajs-830	173	23	1	1	NUM
iajs-830	173	24	.	.	X
iajs-830	173	25	f(a	f(a	NOUN
iajs-830	173	26	)	)	PUNCT
iajs-830	173	27	is	be	AUX
iajs-830	173	28	a	a	DET
iajs-830	173	29	fuzzy	fuzzy	ADJ
iajs-830	173	30	submodule	submodule	NOUN
iajs-830	173	31	of	of	ADP
iajs-830	173	32	y	y	PROPN
iajs-830	173	33	,	,	PUNCT
iajs-830	173	34	(	(	PUNCT
iajs-830	173	35	14	14	NUM
iajs-830	173	36	)	)	PUNCT
iajs-830	173	37	.	.	PUNCT
iajs-830	174	1	2	2	X
iajs-830	174	2	.	.	X
iajs-830	174	3	f	f	PROPN
iajs-830	174	4	–	–	PUNCT
iajs-830	174	5	1(a	1(a	NUM
iajs-830	174	6	)	)	PUNCT
iajs-830	174	7	is	be	AUX
iajs-830	174	8	a	a	DET
iajs-830	174	9	fuzzy	fuzzy	ADJ
iajs-830	174	10	submodule	submodule	NOUN
iajs-830	174	11	of	of	ADP
iajs-830	174	12	y	y	PROPN
iajs-830	174	13	,	,	PUNCT
iajs-830	174	14	(	(	PUNCT
iajs-830	174	15	14	14	NUM
iajs-830	174	16	)	)	PUNCT
iajs-830	174	17	.	.	PUNCT
iajs-830	175	1	3	3	X
iajs-830	175	2	.	.	X
iajs-830	175	3	f(ab)=f(a)f(b	f(ab)=f(a)f(b	NOUN
iajs-830	175	4	)	)	PUNCT
iajs-830	175	5	,	,	PUNCT
iajs-830	175	6	whenever	whenever	SCONJ
iajs-830	175	7	a	a	DET
iajs-830	175	8	,	,	PUNCT
iajs-830	175	9	b	b	NOUN
iajs-830	175	10	are	be	AUX
iajs-830	175	11	f	f	NOUN
iajs-830	175	12	-	-	PUNCT
iajs-830	175	13	invariant	invariant	ADJ
iajs-830	175	14	,	,	PUNCT
iajs-830	175	15	(	(	PUNCT
iajs-830	175	16	15	15	NUM
iajs-830	175	17	)	)	PUNCT
iajs-830	175	18	.	.	PUNCT
iajs-830	176	1	4	4	X
iajs-830	176	2	.	.	X
iajs-830	176	3	f	f	X
iajs-830	176	4	–	–	PUNCT
iajs-830	176	5	1	1	NUM
iajs-830	176	6	(	(	PUNCT
iajs-830	176	7	ab)=f	ab)=f	X
iajs-830	176	8	–	–	PUNCT
iajs-830	176	9	1	1	NUM
iajs-830	176	10	(	(	PUNCT
iajs-830	176	11	a)f	a)f	X
iajs-830	176	12	–	–	PUNCT
iajs-830	176	13	1	1	NUM
iajs-830	176	14	(	(	PUNCT
iajs-830	176	15	b	b	NOUN
iajs-830	176	16	)	)	PUNCT
iajs-830	176	17	,	,	PUNCT
iajs-830	176	18	where	where	SCONJ
iajs-830	176	19	f	f	PROPN
iajs-830	176	20	is	be	AUX
iajs-830	176	21	monomorphism	monomorphism	NOUN
iajs-830	176	22	,	,	PUNCT
iajs-830	176	23	(	(	PUNCT
iajs-830	176	24	15	15	NUM
iajs-830	176	25	)	)	PUNCT
iajs-830	176	26	.	.	PUNCT
iajs-830	177	1	5	5	X
iajs-830	177	2	.	.	X
iajs-830	177	3	f(a+b)=f(a)+f(b	f(a+b)=f(a)+f(b	NOUN
iajs-830	177	4	)	)	PUNCT
iajs-830	177	5	,	,	PUNCT
iajs-830	177	6	(	(	PUNCT
iajs-830	177	7	15	15	NUM
iajs-830	177	8	)	)	PUNCT
iajs-830	177	9	.	.	PUNCT
iajs-830	178	1	6	6	X
iajs-830	178	2	.	.	X
iajs-830	179	1	f(f	f(f	PROPN
iajs-830	179	2	–	–	PUNCT
iajs-830	179	3	1(a))=a	1(a))=a	NUM
iajs-830	179	4	,	,	PUNCT
iajs-830	179	5	(	(	PUNCT
iajs-830	179	6	15	15	NUM
iajs-830	179	7	)	)	PUNCT
iajs-830	179	8	.	.	PUNCT
iajs-830	180	1	7	7	X
iajs-830	180	2	.	.	X
iajs-830	180	3	f	f	X
iajs-830	180	4	–	–	PUNCT
iajs-830	180	5	1(f(a))=a	1(f(a))=a	NUM
iajs-830	180	6	whenever	whenever	SCONJ
iajs-830	180	7	a	a	PRON
iajs-830	180	8	is	be	AUX
iajs-830	180	9	f	f	NOUN
iajs-830	180	10	-	-	PUNCT
iajs-830	180	11	invariant	invariant	ADJ
iajs-830	180	12	,	,	PUNCT
iajs-830	180	13	(	(	PUNCT
iajs-830	180	14	15	15	NUM
iajs-830	180	15	)	)	PUNCT
iajs-830	180	16	.	.	PUNCT
iajs-830	181	1	first	first	ADV
iajs-830	181	2	we	we	PRON
iajs-830	181	3	have	have	VERB
iajs-830	181	4	the	the	DET
iajs-830	181	5	following	follow	VERB
iajs-830	181	6	result	result	NOUN
iajs-830	181	7	.	.	PUNCT
iajs-830	182	1	3.5	3.5	NUM
iajs-830	182	2	proposition	proposition	NOUN
iajs-830	182	3	let	let	VERB
iajs-830	182	4	x	x	PRON
iajs-830	182	5	and	and	CCONJ
iajs-830	182	6	y	y	PROPN
iajs-830	182	7	be	be	AUX
iajs-830	182	8	two	two	NUM
iajs-830	182	9	fuzzy	fuzzy	ADJ
iajs-830	182	10	modules	module	NOUN
iajs-830	182	11	of	of	ADP
iajs-830	182	12	r	r	NOUN
iajs-830	182	13	-	-	PUNCT
iajs-830	182	14	modules	module	NOUN
iajs-830	182	15	m	m	NOUN
iajs-830	182	16	1	1	NUM
iajs-830	182	17	and	and	CCONJ
iajs-830	182	18	m2	m2	PROPN
iajs-830	182	19	respectively	respectively	ADV
iajs-830	182	20	.	.	PUNCT
iajs-830	183	1	let	let	VERB
iajs-830	183	2	f	f	X
iajs-830	183	3	:	:	PUNCT
iajs-830	183	4	xy	xy	PROPN
iajs-830	183	5	be	be	AUX
iajs-830	183	6	a	a	DET
iajs-830	183	7	fuzzy	fuzzy	ADJ
iajs-830	183	8	epimorphism	epimorphism	NOUN
iajs-830	183	9	,	,	PUNCT
iajs-830	183	10	and	and	CCONJ
iajs-830	183	11	every	every	DET
iajs-830	183	12	fuzzy	fuzzy	ADJ
iajs-830	183	13	submodule	submodule	NOUN
iajs-830	183	14	of	of	ADP
iajs-830	183	15	x	x	PROPN
iajs-830	183	16	is	be	AUX
iajs-830	183	17	f	f	NOUN
iajs-830	183	18	-	-	PUNCT
iajs-830	183	19	invariant	invariant	ADJ
iajs-830	183	20	.	.	PUNCT
iajs-830	184	1	if	if	SCONJ
iajs-830	184	2	x	x	PRON
iajs-830	184	3	is	be	AUX
iajs-830	184	4	a	a	DET
iajs-830	184	5	fuzzy	fuzzy	ADJ
iajs-830	184	6	distributive	distributive	ADJ
iajs-830	184	7	module	module	NOUN
iajs-830	184	8	,	,	PUNCT
iajs-830	184	9	then	then	ADV
iajs-830	184	10	y	y	PROPN
iajs-830	184	11	is	be	AUX
iajs-830	184	12	a	a	DET
iajs-830	184	13	fuzzy	fuzzy	ADJ
iajs-830	184	14	distributive	distributive	ADJ
iajs-830	184	15	module	module	NOUN
iajs-830	184	16	.	.	PUNCT
iajs-830	185	1	proof	proof	NOUN
iajs-830	185	2	:	:	PUNCT
iajs-830	185	3	let	let	VERB
iajs-830	185	4	a	a	DET
iajs-830	185	5	,	,	PUNCT
iajs-830	185	6	b	b	NOUN
iajs-830	185	7	,	,	PUNCT
iajs-830	185	8	c	c	AUX
iajs-830	185	9	be	be	AUX
iajs-830	185	10	fuzzy	fuzzy	ADJ
iajs-830	185	11	submodules	submodule	NOUN
iajs-830	185	12	in	in	ADP
iajs-830	185	13	y.	y.	PROPN
iajs-830	185	14	f	f	PROPN
iajs-830	185	15	–	–	PUNCT
iajs-830	185	16	1(a	1(a	NUM
iajs-830	185	17	)	)	PUNCT
iajs-830	185	18	,	,	PUNCT
iajs-830	185	19	f	f	X
iajs-830	185	20	–	–	PUNCT
iajs-830	185	21	1(b	1(b	NUM
iajs-830	185	22	)	)	PUNCT
iajs-830	185	23	,	,	PUNCT
iajs-830	185	24	f	f	X
iajs-830	185	25	–	–	PUNCT
iajs-830	185	26	1(c	1(c	NUM
iajs-830	185	27	)	)	PUNCT
iajs-830	185	28	are	be	AUX
iajs-830	185	29	fuzzy	fuzzy	ADJ
iajs-830	185	30	submodules	submodule	NOUN
iajs-830	185	31	in	in	ADP
iajs-830	185	32	x	x	PUNCT
iajs-830	185	33	by	by	ADP
iajs-830	185	34	proposition	proposition	NOUN
iajs-830	185	35	3.4	3.4	NUM
iajs-830	185	36	,	,	PUNCT
iajs-830	185	37	(	(	PUNCT
iajs-830	185	38	2	2	NUM
iajs-830	185	39	)	)	PUNCT
iajs-830	185	40	.	.	PUNCT
iajs-830	186	1	since	since	SCONJ
iajs-830	186	2	x	x	PRON
iajs-830	186	3	is	be	AUX
iajs-830	186	4	a	a	DET
iajs-830	186	5	fuzzy	fuzzy	ADJ
iajs-830	186	6	distributive	distributive	ADJ
iajs-830	186	7	,	,	PUNCT
iajs-830	186	8	then	then	ADV
iajs-830	186	9	f	f	X
iajs-830	186	10	–	–	PUNCT
iajs-830	186	11	1(a)(f	1(a)(f	NUM
iajs-830	186	12	–	–	PUNCT
iajs-830	186	13	1(b)+	1(b)+	NUM
iajs-830	186	14	f	f	NUM
iajs-830	186	15	–	–	PUNCT
iajs-830	186	16	1(c	1(c	NUM
iajs-830	186	17	)	)	PUNCT
iajs-830	186	18	)	)	PUNCT
iajs-830	187	1	=	=	PUNCT
iajs-830	187	2	(	(	PUNCT
iajs-830	187	3	f	f	X
iajs-830	187	4	–	–	PUNCT
iajs-830	187	5	1(a)f	1(a)f	NUM
iajs-830	187	6	–	–	PUNCT
iajs-830	187	7	1(b	1(b	NUM
iajs-830	187	8	)	)	PUNCT
iajs-830	187	9	)	)	PUNCT
iajs-830	188	1	+	+	CCONJ
iajs-830	188	2	(	(	PUNCT
iajs-830	188	3	f	f	X
iajs-830	188	4	–	–	PUNCT
iajs-830	188	5	1(a)f	1(a)f	NUM
iajs-830	188	6	–	–	PUNCT
iajs-830	188	7	1(c	1(c	NUM
iajs-830	188	8	)	)	PUNCT
iajs-830	188	9	)	)	PUNCT
iajs-830	189	1	f	f	X
iajs-830	189	2	–	–	PUNCT
iajs-830	189	3	1	1	NUM
iajs-830	189	4	(	(	PUNCT
iajs-830	189	5	a)(f	a)(f	NOUN
iajs-830	189	6	–	–	PUNCT
iajs-830	189	7	1	1	NUM
iajs-830	189	8	(	(	PUNCT
iajs-830	189	9	b)+	b)+	PROPN
iajs-830	189	10	f	f	PROPN
iajs-830	189	11	–	–	PUNCT
iajs-830	189	12	1	1	NUM
iajs-830	189	13	(	(	PUNCT
iajs-830	189	14	c	c	NOUN
iajs-830	189	15	)	)	PUNCT
iajs-830	189	16	)	)	PUNCT
iajs-830	190	1	=	=	PUNCT
iajs-830	190	2	(	(	PUNCT
iajs-830	190	3	f	f	X
iajs-830	190	4	–	–	PUNCT
iajs-830	190	5	1	1	NUM
iajs-830	190	6	(	(	PUNCT
iajs-830	190	7	ab	ab	NOUN
iajs-830	190	8	)	)	PUNCT
iajs-830	190	9	)	)	PUNCT
iajs-830	191	1	+	+	CCONJ
iajs-830	191	2	(	(	PUNCT
iajs-830	191	3	f	f	X
iajs-830	191	4	–	–	PUNCT
iajs-830	191	5	1	1	NUM
iajs-830	191	6	(	(	PUNCT
iajs-830	191	7	ac	ac	NOUN
iajs-830	191	8	)	)	PUNCT
iajs-830	191	9	)	)	PUNCT
iajs-830	191	10	,	,	PUNCT
iajs-830	191	11	proposition	proposition	NOUN
iajs-830	191	12	3.4,(4	3.4,(4	NUM
iajs-830	191	13	)	)	PUNCT
iajs-830	191	14	)	)	PUNCT
iajs-830	191	15	f[f	f[f	PROPN
iajs-830	191	16	–	–	PUNCT
iajs-830	191	17	1	1	NUM
iajs-830	191	18	(	(	PUNCT
iajs-830	191	19	a)(f	a)(f	NOUN
iajs-830	191	20	–	–	PUNCT
iajs-830	191	21	1	1	NUM
iajs-830	191	22	(	(	PUNCT
iajs-830	191	23	b)+	b)+	PROPN
iajs-830	191	24	f	f	PROPN
iajs-830	191	25	–	–	PUNCT
iajs-830	191	26	1	1	NUM
iajs-830	191	27	(	(	PUNCT
iajs-830	191	28	c))]=f[(f	c))]=f[(f	NUM
iajs-830	191	29	–	–	PUNCT
iajs-830	191	30	1	1	NUM
iajs-830	191	31	(	(	PUNCT
iajs-830	191	32	ab	ab	NOUN
iajs-830	191	33	)	)	PUNCT
iajs-830	191	34	)	)	PUNCT
iajs-830	192	1	+	+	CCONJ
iajs-830	192	2	(	(	PUNCT
iajs-830	192	3	f	f	X
iajs-830	192	4	–	–	PUNCT
iajs-830	192	5	1	1	NUM
iajs-830	192	6	(	(	PUNCT
iajs-830	192	7	ac	ac	NOUN
iajs-830	192	8	)	)	PUNCT
iajs-830	192	9	)	)	PUNCT
iajs-830	192	10	]	]	PUNCT
iajs-830	193	1	f(f	f(f	PROPN
iajs-830	193	2	–	–	PUNCT
iajs-830	193	3	1	1	NUM
iajs-830	193	4	(	(	PUNCT
iajs-830	193	5	a))f(f	a))f(f	PROPN
iajs-830	193	6	–	–	PUNCT
iajs-830	193	7	1	1	NUM
iajs-830	193	8	(	(	PUNCT
iajs-830	193	9	b)+	b)+	PROPN
iajs-830	193	10	f	f	PROPN
iajs-830	193	11	–	–	PUNCT
iajs-830	193	12	1	1	NUM
iajs-830	193	13	(	(	PUNCT
iajs-830	193	14	c	c	NOUN
iajs-830	193	15	)	)	PUNCT
iajs-830	193	16	)	)	PUNCT
iajs-830	194	1	=	=	SYM
iajs-830	194	2	f(f	f(f	PROPN
iajs-830	194	3	–	–	PUNCT
iajs-830	194	4	1	1	NUM
iajs-830	194	5	(	(	PUNCT
iajs-830	194	6	ab	ab	NOUN
iajs-830	194	7	)	)	PUNCT
iajs-830	194	8	)	)	PUNCT
iajs-830	195	1	+	+	CCONJ
iajs-830	195	2	f(f	f(f	PROPN
iajs-830	195	3	–	–	PUNCT
iajs-830	195	4	1	1	NUM
iajs-830	195	5	(	(	PUNCT
iajs-830	195	6	ac	ac	NOUN
iajs-830	195	7	)	)	PUNCT
iajs-830	195	8	)	)	PUNCT
iajs-830	195	9	,	,	PUNCT
iajs-830	195	10	proposition	proposition	NOUN
iajs-830	195	11	3.4,(3),(4	3.4,(3),(4	NUM
iajs-830	195	12	)	)	PUNCT
iajs-830	195	13	)	)	PUNCT
iajs-830	195	14	a(b+	a(b+	NUM
iajs-830	195	15	c	c	X
iajs-830	195	16	)	)	PUNCT
iajs-830	195	17	=	=	NOUN
iajs-830	195	18	(	(	PUNCT
iajs-830	195	19	ab	ab	NOUN
iajs-830	195	20	)	)	PUNCT
iajs-830	195	21	+	+	CCONJ
iajs-830	195	22	(	(	PUNCT
iajs-830	195	23	ac	ac	ADJ
iajs-830	195	24	)	)	PUNCT
iajs-830	195	25	,	,	PUNCT
iajs-830	195	26	proposition	proposition	NOUN
iajs-830	195	27	3.4,(6	3.4,(6	NUM
iajs-830	195	28	)	)	PUNCT
iajs-830	195	29	)	)	PUNCT
iajs-830	195	30	.	.	PUNCT
iajs-830	196	1			NUM
iajs-830	196	2	3.6	3.6	NUM
iajs-830	196	3	proposition	proposition	NOUN
iajs-830	196	4	let	let	VERB
iajs-830	196	5	x	x	PRON
iajs-830	196	6	and	and	CCONJ
iajs-830	196	7	y	y	PROPN
iajs-830	196	8	be	be	AUX
iajs-830	196	9	two	two	NUM
iajs-830	196	10	fuzzy	fuzzy	ADJ
iajs-830	196	11	modules	module	NOUN
iajs-830	196	12	over	over	ADP
iajs-830	196	13	r	r	NOUN
iajs-830	196	14	-	-	PUNCT
iajs-830	196	15	modules	module	NOUN
iajs-830	196	16	m	m	NOUN
iajs-830	196	17	1	1	NUM
iajs-830	196	18	and	and	CCONJ
iajs-830	196	19	m	m	PROPN
iajs-830	196	20	2	2	NUM
iajs-830	196	21	respectively	respectively	ADV
iajs-830	196	22	.	.	PUNCT
iajs-830	197	1	let	let	VERB
iajs-830	197	2	f	f	X
iajs-830	197	3	:	:	PUNCT
iajs-830	197	4	xy	xy	PROPN
iajs-830	197	5	be	be	AUX
iajs-830	197	6	a	a	DET
iajs-830	197	7	fuzzy	fuzzy	ADJ
iajs-830	197	8	homomorphism	homomorphism	NOUN
iajs-830	197	9	,	,	PUNCT
iajs-830	197	10	and	and	CCONJ
iajs-830	197	11	every	every	DET
iajs-830	197	12	fuzzy	fuzzy	ADJ
iajs-830	197	13	submodule	submodule	NOUN
iajs-830	197	14	of	of	ADP
iajs-830	197	15	y	y	PROPN
iajs-830	197	16	is	be	AUX
iajs-830	197	17	f	f	NOUN
iajs-830	197	18	-	-	PUNCT
iajs-830	197	19	invariant	invariant	ADJ
iajs-830	197	20	.	.	PUNCT
iajs-830	198	1	if	if	SCONJ
iajs-830	198	2	y	y	PROPN
iajs-830	198	3	is	be	AUX
iajs-830	198	4	a	a	DET
iajs-830	198	5	fuzzy	fuzzy	ADJ
iajs-830	198	6	distributive	distributive	ADJ
iajs-830	198	7	module	module	NOUN
iajs-830	198	8	,	,	PUNCT
iajs-830	198	9	then	then	ADV
iajs-830	198	10	x	x	PUNCT
iajs-830	198	11	is	be	AUX
iajs-830	198	12	a	a	DET
iajs-830	198	13	fuzzy	fuzzy	ADJ
iajs-830	198	14	distributive	distributive	ADJ
iajs-830	198	15	module	module	NOUN
iajs-830	198	16	.	.	PUNCT
iajs-830	199	1	proof	proof	NOUN
iajs-830	199	2	:	:	PUNCT
iajs-830	199	3	let	let	VERB
iajs-830	199	4	a	a	DET
iajs-830	199	5	,	,	PUNCT
iajs-830	199	6	b	b	NOUN
iajs-830	199	7	,	,	PUNCT
iajs-830	199	8	c	c	PROPN
iajs-830	199	9	are	be	AUX
iajs-830	199	10	fuzzy	fuzzy	ADJ
iajs-830	199	11	submodules	submodule	NOUN
iajs-830	199	12	in	in	ADP
iajs-830	199	13	x.	x.	PROPN
iajs-830	199	14	hence	hence	ADV
iajs-830	199	15	f(a	f(a	PROPN
iajs-830	199	16	)	)	PUNCT
iajs-830	199	17	,	,	PUNCT
iajs-830	199	18	f(b	f(b	PROPN
iajs-830	199	19	)	)	PUNCT
iajs-830	199	20	,	,	PUNCT
iajs-830	199	21	f(c	f(c	PROPN
iajs-830	199	22	)	)	PUNCT
iajs-830	199	23	are	be	AUX
iajs-830	199	24	fuzzy	fuzzy	ADJ
iajs-830	199	25	submodules	submodule	NOUN
iajs-830	199	26	in	in	ADP
iajs-830	199	27	y	y	PROPN
iajs-830	199	28	,	,	PUNCT
iajs-830	199	29	by	by	ADP
iajs-830	199	30	proposition	proposition	NOUN
iajs-830	199	31	3.4,(1	3.4,(1	NUM
iajs-830	199	32	)	)	PUNCT
iajs-830	199	33	.	.	PUNCT
iajs-830	200	1	since	since	SCONJ
iajs-830	200	2	y	y	PROPN
iajs-830	200	3	is	be	AUX
iajs-830	200	4	a	a	DET
iajs-830	200	5	fuzzy	fuzzy	ADJ
iajs-830	200	6	distributive	distributive	ADJ
iajs-830	200	7	module	module	NOUN
iajs-830	200	8	,	,	PUNCT
iajs-830	200	9	then	then	ADV
iajs-830	200	10	f(a)(f(b)+	f(a)(f(b)+	NUM
iajs-830	200	11	f(c	f(c	NOUN
iajs-830	200	12	)	)	PUNCT
iajs-830	200	13	)	)	PUNCT
iajs-830	201	1	=	=	SYM
iajs-830	201	2	(	(	PUNCT
iajs-830	201	3	f(a)f(b	f(a)f(b	NOUN
iajs-830	201	4	)	)	PUNCT
iajs-830	201	5	)	)	PUNCT
iajs-830	202	1	+	+	CCONJ
iajs-830	202	2	(	(	PUNCT
iajs-830	202	3	f(a)f(c	f(a)f(c	PROPN
iajs-830	202	4	)	)	PUNCT
iajs-830	202	5	)	)	PUNCT
iajs-830	202	6	f(a)(f(b)+	f(a)(f(b)+	NUM
iajs-830	202	7	f(c	f(c	NOUN
iajs-830	202	8	)	)	PUNCT
iajs-830	202	9	)	)	PUNCT
iajs-830	203	1	=	=	SYM
iajs-830	203	2	f(ab	f(ab	ADJ
iajs-830	203	3	)	)	PUNCT
iajs-830	204	1	+	+	CCONJ
iajs-830	204	2	f(ac	f(ac	NOUN
iajs-830	204	3	)	)	PUNCT
iajs-830	204	4	)	)	PUNCT
iajs-830	204	5	,	,	PUNCT
iajs-830	204	6	proposition	proposition	NOUN
iajs-830	204	7	3.4,(3),(5	3.4,(3),(5	NUM
iajs-830	204	8	)	)	PUNCT
iajs-830	204	9	)	)	PUNCT
iajs-830	205	1	f(a(b+	f(a(b+	PUNCT
iajs-830	205	2	c	c	X
iajs-830	205	3	)	)	PUNCT
iajs-830	205	4	)	)	PUNCT
iajs-830	206	1	=	=	SYM
iajs-830	206	2	f((ab	f((ab	NOUN
iajs-830	206	3	)	)	PUNCT
iajs-830	207	1	+	+	CCONJ
iajs-830	207	2	(	(	PUNCT
iajs-830	207	3	ac	ac	NOUN
iajs-830	207	4	)	)	PUNCT
iajs-830	207	5	)	)	PUNCT
iajs-830	207	6	,	,	PUNCT
iajs-830	207	7	proposition	proposition	NOUN
iajs-830	207	8	3.4,(3),(5	3.4,(3),(5	NUM
iajs-830	207	9	)	)	PUNCT
iajs-830	207	10	)	)	PUNCT
iajs-830	207	11	.	.	PUNCT
iajs-830	208	1	f	f	PROPN
iajs-830	208	2	–	–	PUNCT
iajs-830	209	1	1(f(a(b+	1(f(a(b+	NUM
iajs-830	209	2	c	c	NOUN
iajs-830	209	3	)	)	PUNCT
iajs-830	209	4	)	)	PUNCT
iajs-830	209	5	)	)	PUNCT
iajs-830	210	1	=	=	SYM
iajs-830	210	2	f	f	X
iajs-830	210	3	–	–	PUNCT
iajs-830	210	4	1(f((ab	1(f((ab	NUM
iajs-830	210	5	)	)	PUNCT
iajs-830	211	1	+	+	CCONJ
iajs-830	211	2	(	(	PUNCT
iajs-830	211	3	ac	ac	NOUN
iajs-830	211	4	)	)	PUNCT
iajs-830	211	5	)	)	PUNCT
iajs-830	211	6	)	)	PUNCT
iajs-830	211	7	.	.	PUNCT
iajs-830	212	1	then	then	ADV
iajs-830	212	2	a(b+	a(b+	NUM
iajs-830	212	3	c	c	NOUN
iajs-830	212	4	)	)	PUNCT
iajs-830	212	5	=	=	NOUN
iajs-830	212	6	(	(	PUNCT
iajs-830	212	7	ab	ab	NOUN
iajs-830	212	8	)	)	PUNCT
iajs-830	212	9	+	+	CCONJ
iajs-830	212	10	(	(	PUNCT
iajs-830	212	11	ac	ac	ADJ
iajs-830	212	12	)	)	PUNCT
iajs-830	212	13	,	,	PUNCT
iajs-830	212	14	proposition	proposition	NOUN
iajs-830	212	15	3.4,(7	3.4,(7	NUM
iajs-830	212	16	)	)	PUNCT
iajs-830	212	17	)	)	PUNCT
iajs-830	212	18	.	.	PUNCT
iajs-830	213	1			NUM
iajs-830	213	2	4	4	NUM
iajs-830	213	3	.	.	PUNCT
iajs-830	213	4	fuzzy	fuzzy	ADJ
iajs-830	213	5	arithmetical	arithmetical	ADJ
iajs-830	213	6	rings	ring	NOUN
iajs-830	213	7	in	in	ADP
iajs-830	213	8	this	this	DET
iajs-830	213	9	section	section	NOUN
iajs-830	213	10	,	,	PUNCT
iajs-830	213	11	we	we	PRON
iajs-830	213	12	introduce	introduce	VERB
iajs-830	213	13	the	the	DET
iajs-830	213	14	notion	notion	NOUN
iajs-830	213	15	of	of	ADP
iajs-830	213	16	arithmetical	arithmetical	ADJ
iajs-830	213	17	fuzzy	fuzzy	ADJ
iajs-830	213	18	ring	ring	NOUN
iajs-830	213	19	.	.	PUNCT
iajs-830	214	1	first	first	ADV
iajs-830	214	2	we	we	PRON
iajs-830	214	3	have	have	VERB
iajs-830	214	4	the	the	DET
iajs-830	214	5	following	follow	VERB
iajs-830	214	6	definition	definition	NOUN
iajs-830	214	7	.	.	PUNCT
iajs-830	215	1	4.1	4.1	NUM
iajs-830	215	2	definition[12	definition[12	NOUN
iajs-830	215	3	]	]	X
iajs-830	215	4	a	a	DET
iajs-830	215	5	ring	ring	NOUN
iajs-830	215	6	r	r	NOUN
iajs-830	215	7	is	be	AUX
iajs-830	215	8	said	say	VERB
iajs-830	215	9	to	to	PART
iajs-830	215	10	be	be	AUX
iajs-830	215	11	an	an	DET
iajs-830	215	12	arithmetical	arithmetical	ADJ
iajs-830	215	13	ring	ring	NOUN
iajs-830	215	14	if	if	SCONJ
iajs-830	215	15	r	r	NOUN
iajs-830	215	16	,	,	PUNCT
iajs-830	215	17	considered	consider	VERB
iajs-830	215	18	as	as	ADP
iajs-830	215	19	r	r	NOUN
iajs-830	215	20	-	-	PUNCT
iajs-830	215	21	module	module	NOUN
iajs-830	215	22	over	over	ADP
iajs-830	215	23	it	it	PRON
iajs-830	215	24	self	self	NOUN
iajs-830	215	25	,	,	PUNCT
iajs-830	215	26	is	be	AUX
iajs-830	215	27	distributive	distributive	ADJ
iajs-830	215	28	that	that	PRON
iajs-830	215	29	is	be	AUX
iajs-830	215	30	r	r	NOUN
iajs-830	215	31	is	be	AUX
iajs-830	215	32	arithmetical	arithmetical	ADJ
iajs-830	215	33	if	if	SCONJ
iajs-830	215	34	i(j+k)=(ij)+(ik	i(j+k)=(ij)+(ik	X
iajs-830	215	35	)	)	PUNCT
iajs-830	215	36	for	for	ADP
iajs-830	215	37	all	all	DET
iajs-830	215	38	ideals	ideal	NOUN
iajs-830	215	39	i	i	PRON
iajs-830	215	40	,	,	PUNCT
iajs-830	215	41	j	j	PROPN
iajs-830	215	42	,	,	PUNCT
iajs-830	215	43	k	k	PROPN
iajs-830	215	44	of	of	ADP
iajs-830	215	45	r.	r.	PROPN
iajs-830	215	46	we	we	PRON
iajs-830	215	47	fuzzify	fuzzify	VERB
iajs-830	215	48	this	this	DET
iajs-830	215	49	definition	definition	NOUN
iajs-830	215	50	as	as	SCONJ
iajs-830	215	51	follows	follow	VERB
iajs-830	215	52	:	:	PUNCT
iajs-830	215	53	4.2	4.2	NUM
iajs-830	215	54	definition	definition	NOUN
iajs-830	215	55	a	a	DET
iajs-830	215	56	fuzzy	fuzzy	ADJ
iajs-830	215	57	ring	ring	NOUN
iajs-830	215	58	x	x	INTJ
iajs-830	215	59	of	of	ADP
iajs-830	215	60	a	a	DET
iajs-830	215	61	ring	ring	NOUN
iajs-830	215	62	r	r	NOUN
iajs-830	215	63	is	be	AUX
iajs-830	215	64	called	call	VERB
iajs-830	215	65	arithmetical	arithmetical	ADJ
iajs-830	215	66	if	if	SCONJ
iajs-830	215	67	and	and	CCONJ
iajs-830	215	68	only	only	ADV
iajs-830	215	69	if	if	SCONJ
iajs-830	215	70	a(b+	a(b+	NUM
iajs-830	215	71	c	c	NOUN
iajs-830	215	72	)	)	PUNCT
iajs-830	215	73	=	=	NOUN
iajs-830	215	74	(	(	PUNCT
iajs-830	215	75	ab	ab	NOUN
iajs-830	215	76	)	)	PUNCT
iajs-830	216	1	+	+	CCONJ
iajs-830	216	2	(	(	PUNCT
iajs-830	216	3	ac	ac	NOUN
iajs-830	216	4	)	)	PUNCT
iajs-830	216	5	for	for	ADP
iajs-830	216	6	all	all	DET
iajs-830	216	7	a	a	DET
iajs-830	216	8	,	,	PUNCT
iajs-830	216	9	b	b	NOUN
iajs-830	216	10	,	,	PUNCT
iajs-830	216	11	c	c	X
iajs-830	216	12	fuzzy	fuzzy	ADJ
iajs-830	216	13	ideals	ideal	NOUN
iajs-830	216	14	of	of	ADP
iajs-830	216	15	x.	x.	PROPN
iajs-830	216	16	ibn	ibn	PROPN
iajs-830	216	17	alhaitham	alhaitham	PROPN
iajs-830	217	1	j.	j.	PROPN
iajs-830	218	1	fo	fo	ADP
iajs-830	218	2	r	r	NOUN
iajs-830	218	3	pure	pure	ADJ
iajs-830	218	4	&	&	CCONJ
iajs-830	218	5	appl	appl	PROPN
iajs-830	218	6	.	.	PUNCT
iajs-830	219	1	sc	sc	PROPN
iajs-830	219	2	i.	i.	PROPN
iajs-830	219	3	vo	vo	PROPN
iajs-830	220	1	l.24	l.24	PROPN
iajs-830	220	2	(	(	PUNCT
iajs-830	220	3	1	1	NUM
iajs-830	220	4	)	)	PUNCT
iajs-830	220	5	2011	2011	NUM
iajs-830	220	6	4.3	4.3	NUM
iajs-830	220	7	note	note	VERB
iajs-830	220	8	a	a	DET
iajs-830	220	9	fuzzy	fuzzy	ADJ
iajs-830	220	10	ring	ring	NOUN
iajs-830	220	11	x	x	VERB
iajs-830	220	12	is	be	AUX
iajs-830	220	13	arithmetical	arithmetical	ADJ
iajs-830	220	14	if	if	SCONJ
iajs-830	220	15	and	and	CCONJ
iajs-830	220	16	only	only	ADV
iajs-830	220	17	xt	xt	PROPN
iajs-830	220	18	is	be	AUX
iajs-830	220	19	arithmetical	arithmetical	ADJ
iajs-830	220	20	ring	ring	NOUN
iajs-830	220	21	t(0,1	t(0,1	NOUN
iajs-830	220	22	]	]	X
iajs-830	220	23	.	.	PUNCT
iajs-830	221	1	4.4	4.4	NUM
iajs-830	221	2	example	example	NOUN
iajs-830	221	3	let	let	VERB
iajs-830	221	4	x(x)=1	x(x)=1	PRON
iajs-830	221	5	for	for	ADP
iajs-830	221	6	all	all	PRON
iajs-830	221	7	xϵ	xϵ	NOUN
iajs-830	221	8	z4	z4	PROPN
iajs-830	221	9	,	,	PUNCT
iajs-830	221	10	xt	xt	X
iajs-830	221	11	=	=	PROPN
iajs-830	221	12	z4	z4	PROPN
iajs-830	221	13	,	,	PUNCT
iajs-830	221	14			NOUN
iajs-830	221	15	xϵ	xϵ	ADP
iajs-830	221	16	z4	z4	PROPN
iajs-830	221	17	.but	.but	PUNCT
iajs-830	221	18	z4	z4	PROPN
iajs-830	221	19	is	be	AUX
iajs-830	221	20	arithmetical	arithmetical	ADJ
iajs-830	221	21	ring	ring	NOUN
iajs-830	221	22	.	.	PUNCT
iajs-830	222	1	hence	hence	ADV
iajs-830	222	2	by	by	ADP
iajs-830	222	3	note	note	NOUN
iajs-830	222	4	(	(	PUNCT
iajs-830	222	5	4.3	4.3	NUM
iajs-830	222	6	)	)	PUNCT
iajs-830	222	7	,	,	PUNCT
iajs-830	222	8	x	x	X
iajs-830	222	9	is	be	AUX
iajs-830	222	10	fuzzy	fuzzy	ADJ
iajs-830	222	11	arithmetical	arithmetical	ADJ
iajs-830	222	12	ring	ring	NOUN
iajs-830	222	13	now	now	ADV
iajs-830	222	14	,	,	PUNCT
iajs-830	222	15	we	we	PRON
iajs-830	222	16	can	can	AUX
iajs-830	222	17	give	give	VERB
iajs-830	222	18	the	the	DET
iajs-830	222	19	following	following	NOUN
iajs-830	222	20	theorem	theorem	VERB
iajs-830	222	21	.	.	PROPN
iajs-830	222	22	4.5	4.5	NUM
iajs-830	222	23	theorem	theorem	VERB
iajs-830	222	24	a	a	DET
iajs-830	222	25	fuzzy	fuzzy	ADJ
iajs-830	222	26	ring	ring	NOUN
iajs-830	222	27	x	x	INTJ
iajs-830	222	28	of	of	ADP
iajs-830	222	29	a	a	DET
iajs-830	222	30	ring	ring	NOUN
iajs-830	222	31	r	r	NOUN
iajs-830	222	32	is	be	AUX
iajs-830	222	33	arithmetical	arithmetical	ADJ
iajs-830	222	34	if	if	SCONJ
iajs-830	222	35	and	and	CCONJ
iajs-830	222	36	only	only	ADV
iajs-830	222	37	if	if	SCONJ
iajs-830	222	38	a	a	DET
iajs-830	222	39	+	+	X
iajs-830	222	40	(	(	PUNCT
iajs-830	222	41	b	b	NOUN
iajs-830	222	42			X
iajs-830	222	43	c	c	NOUN
iajs-830	222	44	)	)	PUNCT
iajs-830	222	45	=	=	SYM
iajs-830	222	46	(	(	PUNCT
iajs-830	222	47	a+b	a+b	NUM
iajs-830	222	48	)	)	PUNCT
iajs-830	222	49			NOUN
iajs-830	222	50	(	(	PUNCT
iajs-830	222	51	a+c	a+c	NOUN
iajs-830	222	52	)	)	PUNCT
iajs-830	222	53	proof	proof	NOUN
iajs-830	222	54	:	:	PUNCT
iajs-830	222	55	let	let	VERB
iajs-830	222	56	x	x	PRON
iajs-830	222	57	be	be	AUX
iajs-830	222	58	a	a	DET
iajs-830	222	59	fuzzy	fuzzy	ADJ
iajs-830	222	60	arithmetical	arithmetical	ADJ
iajs-830	222	61	ring	ring	NOUN
iajs-830	222	62	,	,	PUNCT
iajs-830	222	63	let	let	VERB
iajs-830	222	64	a	a	DET
iajs-830	222	65	,	,	PUNCT
iajs-830	222	66	b	b	NOUN
iajs-830	222	67	,	,	PUNCT
iajs-830	222	68	c	c	AUX
iajs-830	222	69	be	be	AUX
iajs-830	222	70	fuzzy	fuzzy	ADJ
iajs-830	222	71	ideals	ideal	NOUN
iajs-830	222	72	of	of	ADP
iajs-830	222	73	x.	x.	NOUN
iajs-830	222	74	hence	hence	ADV
iajs-830	222	75	at	at	ADP
iajs-830	222	76	,	,	PUNCT
iajs-830	222	77	bt	bt	PROPN
iajs-830	222	78	,	,	PUNCT
iajs-830	222	79	ct	ct	PROPN
iajs-830	222	80	are	be	AUX
iajs-830	222	81	ideals	ideal	NOUN
iajs-830	222	82	of	of	ADP
iajs-830	222	83	xt	xt	PROPN
iajs-830	222	84	.	.	PUNCT
iajs-830	223	1	since	since	SCONJ
iajs-830	223	2	x	x	PRON
iajs-830	223	3	is	be	AUX
iajs-830	223	4	fuzzy	fuzzy	ADJ
iajs-830	223	5	arithmetical	arithmetical	ADJ
iajs-830	223	6	ring	ring	NOUN
iajs-830	223	7	then	then	ADV
iajs-830	223	8	xt	xt	PROPN
iajs-830	223	9	is	be	AUX
iajs-830	223	10	arithmetical	arithmetical	ADJ
iajs-830	223	11	ring	ring	NOUN
iajs-830	223	12	(	(	PUNCT
iajs-830	223	13	by	by	ADP
iajs-830	223	14	note	note	NOUN
iajs-830	223	15	(	(	PUNCT
iajs-830	223	16	4.3	4.3	NUM
iajs-830	223	17	)	)	PUNCT
iajs-830	223	18	)	)	PUNCT
iajs-830	223	19	.	.	PUNCT
iajs-830	224	1	hence	hence	ADV
iajs-830	224	2	,	,	PUNCT
iajs-830	224	3	at	at	ADP
iajs-830	224	4	+	+	CCONJ
iajs-830	224	5	(	(	PUNCT
iajs-830	224	6	bt	bt	PROPN
iajs-830	224	7			PROPN
iajs-830	224	8	c	c	PROPN
iajs-830	224	9	t	t	PROPN
iajs-830	224	10	)	)	PUNCT
iajs-830	224	11	=	=	PUNCT
iajs-830	224	12	(	(	PUNCT
iajs-830	224	13	at	at	ADP
iajs-830	224	14	+	+	ADV
iajs-830	224	15	bt)	bt)	PROPN
iajs-830	224	16	(	(	PUNCT
iajs-830	224	17	at	at	ADP
iajs-830	224	18	+	+	NOUN
iajs-830	224	19	ct	ct	NUM
iajs-830	224	20	)	)	PUNCT
iajs-830	224	21	(	(	PUNCT
iajs-830	224	22	(	(	PUNCT
iajs-830	224	23	16),exc.18	16),exc.18	NUM
iajs-830	224	24	)	)	PUNCT
iajs-830	224	25	at	at	ADP
iajs-830	224	26	+	+	X
iajs-830	224	27	(	(	PUNCT
iajs-830	224	28	b	b	NOUN
iajs-830	224	29			PUNCT
iajs-830	224	30	c)t	c)t	NOUN
iajs-830	225	1	=	=	SYM
iajs-830	225	2	(	(	PUNCT
iajs-830	225	3	a+b)t	a+b)t	X
iajs-830	225	4			NOUN
iajs-830	225	5	(	(	PUNCT
iajs-830	225	6	a+c	a+c	NOUN
iajs-830	225	7	)	)	PUNCT
iajs-830	225	8	t	t	PROPN
iajs-830	225	9	,	,	PUNCT
iajs-830	225	10	(	(	PUNCT
iajs-830	225	11	remark	remark	NOUN
iajs-830	225	12	(	(	PUNCT
iajs-830	225	13	1.5	1.5	NUM
iajs-830	225	14	)	)	PUNCT
iajs-830	225	15	,	,	PUNCT
iajs-830	225	16	proposition	proposition	NOUN
iajs-830	225	17	(	(	PUNCT
iajs-830	225	18	1.13	1.13	NUM
iajs-830	225	19	)	)	PUNCT
iajs-830	225	20	)	)	PUNCT
iajs-830	225	21	(	(	PUNCT
iajs-830	225	22	a	a	PRON
iajs-830	225	23	+	+	X
iajs-830	225	24	(	(	PUNCT
iajs-830	225	25	b	b	NOUN
iajs-830	225	26			X
iajs-830	225	27	c))t=((a+b)t	c))t=((a+b)t	X
iajs-830	225	28			NOUN
iajs-830	225	29	(	(	PUNCT
iajs-830	225	30	a+c))t	a+c))t	NOUN
iajs-830	225	31	(	(	PUNCT
iajs-830	225	32	remark	remark	NOUN
iajs-830	225	33	(	(	PUNCT
iajs-830	225	34	1.5	1.5	NUM
iajs-830	225	35	)	)	PUNCT
iajs-830	225	36	,	,	PUNCT
iajs-830	225	37	proposition	proposition	NOUN
iajs-830	225	38	(	(	PUNCT
iajs-830	225	39	1.13	1.13	NUM
iajs-830	225	40	)	)	PUNCT
iajs-830	225	41	)	)	PUNCT
iajs-830	225	42	thus	thus	ADV
iajs-830	225	43	a	a	DET
iajs-830	225	44	+	+	X
iajs-830	225	45	(	(	PUNCT
iajs-830	225	46	b	b	NOUN
iajs-830	225	47			X
iajs-830	225	48	c	c	NOUN
iajs-830	225	49	)	)	PUNCT
iajs-830	225	50	=	=	SYM
iajs-830	225	51	(	(	PUNCT
iajs-830	225	52	a+b	a+b	NUM
iajs-830	225	53	)	)	PUNCT
iajs-830	225	54			NOUN
iajs-830	225	55	(	(	PUNCT
iajs-830	225	56	a+c	a+c	NOUN
iajs-830	225	57	)	)	PUNCT
iajs-830	225	58	.	.	PUNCT
iajs-830	226	1	conversely	conversely	ADV
iajs-830	226	2	,	,	PUNCT
iajs-830	226	3	to	to	PART
iajs-830	226	4	prove	prove	VERB
iajs-830	226	5	x	x	PUNCT
iajs-830	226	6	is	be	AUX
iajs-830	226	7	a	a	DET
iajs-830	226	8	fuzzy	fuzzy	ADJ
iajs-830	226	9	arithmetical	arithmetical	ADJ
iajs-830	226	10	ring	ring	NOUN
iajs-830	226	11	.	.	PUNCT
iajs-830	227	1	we	we	PRON
iajs-830	227	2	shall	shall	AUX
iajs-830	227	3	prove	prove	VERB
iajs-830	227	4	xt	xt	PROPN
iajs-830	227	5	is	be	AUX
iajs-830	227	6	an	an	DET
iajs-830	227	7	arithmetical	arithmetical	ADJ
iajs-830	227	8	ring	ring	NOUN
iajs-830	227	9	for	for	ADP
iajs-830	227	10	all	all	DET
iajs-830	227	11	t(0,1	t(0,1	NOUN
iajs-830	227	12	]	]	PUNCT
iajs-830	227	13	.	.	PUNCT
iajs-830	228	1	let	let	VERB
iajs-830	228	2	i	i	PRON
iajs-830	228	3	,	,	PUNCT
iajs-830	228	4	j	j	PROPN
iajs-830	228	5	,	,	PUNCT
iajs-830	228	6	k	k	X
iajs-830	228	7	be	be	VERB
iajs-830	228	8	ideals	ideal	NOUN
iajs-830	228	9	in	in	ADP
iajs-830	228	10	xt	xt	PROPN
iajs-830	228	11	.	.	PUNCT
iajs-830	229	1	it	it	PRON
iajs-830	229	2	follows	follow	VERB
iajs-830	229	3	that	that	SCONJ
iajs-830	229	4	there	there	PRON
iajs-830	229	5	exist	exist	VERB
iajs-830	229	6	a	a	DET
iajs-830	229	7	,	,	PUNCT
iajs-830	229	8	b	b	NOUN
iajs-830	229	9	,	,	PUNCT
iajs-830	229	10	c	c	X
iajs-830	229	11	fuzzy	fuzzy	ADJ
iajs-830	229	12	ideals	ideal	NOUN
iajs-830	229	13	of	of	ADP
iajs-830	229	14	x	x	NOUN
iajs-830	229	15	,	,	PUNCT
iajs-830	229	16	where	where	SCONJ
iajs-830	229	17	t	t	NOUN
iajs-830	229	18	x	x	PROPN
iajs-830	229	19	i	i	PRON
iajs-830	229	20	(	(	PUNCT
iajs-830	229	21	x	x	X
iajs-830	229	22	)	)	PUNCT
iajs-830	229	23	0	0	NUM
iajs-830	230	1	x	x	SYM
iajs-830	230	2	i	i	PRON
iajs-830	230	3			VERB
iajs-830	231	1			PROPN
iajs-830	231	2			NUM
iajs-830	232	1			NUM
iajs-830	232	2			NOUN
iajs-830	232	3	,	,	PUNCT
iajs-830	232	4	t	t	PROPN
iajs-830	232	5	x	x	X
iajs-830	232	6	j	j	PROPN
iajs-830	232	7	(	(	PUNCT
iajs-830	232	8	x	x	X
iajs-830	232	9	)	)	PUNCT
iajs-830	232	10	0	0	NUM
iajs-830	233	1	x	x	SYM
iajs-830	233	2	j	j	PROPN
iajs-830	233	3			NUM
iajs-830	233	4			NOUN
iajs-830	233	5			NOUN
iajs-830	234	1			NUM
iajs-830	234	2			NOUN
iajs-830	234	3	,	,	PUNCT
iajs-830	234	4	t	t	PROPN
iajs-830	234	5	x	x	PUNCT
iajs-830	234	6	k	k	X
iajs-830	234	7	c(x	c(x	NOUN
iajs-830	234	8	)	)	PUNCT
iajs-830	234	9	0	0	NUM
iajs-830	235	1	x	x	SYM
iajs-830	235	2	k	k	NOUN
iajs-830	235	3			VERB
iajs-830	235	4			NUM
iajs-830	235	5			NUM
iajs-830	235	6			PUNCT
iajs-830	235	7	but	but	CCONJ
iajs-830	235	8	by	by	ADP
iajs-830	235	9	hypothesis	hypothesis	NOUN
iajs-830	235	10	,	,	PUNCT
iajs-830	235	11	a	a	DET
iajs-830	235	12	+	+	X
iajs-830	235	13	(	(	PUNCT
iajs-830	235	14	b	b	NOUN
iajs-830	235	15			X
iajs-830	235	16	c	c	NOUN
iajs-830	235	17	)	)	PUNCT
iajs-830	235	18	=	=	SYM
iajs-830	235	19	(	(	PUNCT
iajs-830	235	20	a+b	a+b	NUM
iajs-830	235	21	)	)	PUNCT
iajs-830	235	22			NOUN
iajs-830	235	23	(	(	PUNCT
iajs-830	235	24	a+c	a+c	NOUN
iajs-830	235	25	)	)	PUNCT
iajs-830	235	26	.	.	PUNCT
iajs-830	236	1	hence	hence	ADV
iajs-830	236	2	[	[	X
iajs-830	236	3	a	a	X
iajs-830	236	4	+	+	NOUN
iajs-830	236	5	(	(	PUNCT
iajs-830	236	6	b	b	NOUN
iajs-830	236	7			PUNCT
iajs-830	236	8	c)]t	c)]t	NOUN
iajs-830	237	1	=	=	X
iajs-830	237	2	[	[	X
iajs-830	237	3	(	(	PUNCT
iajs-830	237	4	a+b	a+b	NUM
iajs-830	237	5	)	)	PUNCT
iajs-830	237	6			NOUN
iajs-830	237	7	(	(	PUNCT
iajs-830	237	8	a+c)]t	a+c)]t	ADV
iajs-830	237	9	for	for	ADP
iajs-830	237	10	all	all	DET
iajs-830	237	11	t(0,1	t(0,1	NOUN
iajs-830	237	12	]	]	PUNCT
iajs-830	237	13	.	.	PUNCT
iajs-830	238	1	it	it	PRON
iajs-830	238	2	follows	follow	VERB
iajs-830	238	3	that	that	SCONJ
iajs-830	238	4	;	;	PUNCT
iajs-830	238	5	at	at	ADP
iajs-830	238	6	+	+	CCONJ
iajs-830	238	7	(	(	PUNCT
iajs-830	238	8	bt	bt	PROPN
iajs-830	238	9			PROPN
iajs-830	238	10	ct	ct	PROPN
iajs-830	238	11	)	)	PUNCT
iajs-830	238	12	=	=	SYM
iajs-830	238	13	(	(	PUNCT
iajs-830	238	14	at+bt	at+bt	NOUN
iajs-830	238	15	)	)	PUNCT
iajs-830	238	16			NOUN
iajs-830	238	17	(	(	PUNCT
iajs-830	238	18	at+ct	at+ct	NOUN
iajs-830	238	19	)	)	PUNCT
iajs-830	238	20	,	,	PUNCT
iajs-830	238	21	(	(	PUNCT
iajs-830	238	22	remark	remark	NOUN
iajs-830	238	23	(	(	PUNCT
iajs-830	238	24	1.5	1.5	NUM
iajs-830	238	25	)	)	PUNCT
iajs-830	238	26	,	,	PUNCT
iajs-830	238	27	proposition	proposition	NOUN
iajs-830	238	28	(	(	PUNCT
iajs-830	238	29	1.13	1.13	NUM
iajs-830	238	30	)	)	PUNCT
iajs-830	238	31	)	)	PUNCT
iajs-830	238	32	.	.	PUNCT
iajs-830	239	1	but	but	CCONJ
iajs-830	239	2	at	at	ADP
iajs-830	239	3	=	=	PROPN
iajs-830	239	4	i	i	PROPN
iajs-830	239	5	,	,	PUNCT
iajs-830	239	6	bt	bt	PROPN
iajs-830	239	7	=	=	PROPN
iajs-830	239	8	j	j	PROPN
iajs-830	239	9	,	,	PUNCT
iajs-830	239	10	ct	ct	PROPN
iajs-830	239	11	=	=	NOUN
iajs-830	239	12	k	k	NOUN
iajs-830	239	13	,	,	PUNCT
iajs-830	239	14	hence	hence	ADV
iajs-830	239	15	i(j+k)=(ij)+(ik	i(j+k)=(ij)+(ik	NOUN
iajs-830	239	16	)	)	PUNCT
iajs-830	239	17	,	,	PUNCT
iajs-830	239	18	which	which	PRON
iajs-830	239	19	implies	imply	VERB
iajs-830	239	20	that	that	SCONJ
iajs-830	239	21	xt	xt	PROPN
iajs-830	239	22	is	be	AUX
iajs-830	239	23	an	an	DET
iajs-830	239	24	arithmetical	arithmetical	ADJ
iajs-830	239	25	ring	ring	NOUN
iajs-830	239	26	,	,	PUNCT
iajs-830	239	27	by	by	ADP
iajs-830	239	28	(	(	PUNCT
iajs-830	239	29	(	(	PUNCT
iajs-830	239	30	16	16	NUM
iajs-830	239	31	)	)	PUNCT
iajs-830	239	32	,	,	PUNCT
iajs-830	239	33	exc.18	exc.18	ADJ
iajs-830	239	34	)	)	PUNCT
iajs-830	239	35	.	.	PUNCT
iajs-830	240	1	thus	thus	ADV
iajs-830	240	2	x	x	PRON
iajs-830	240	3	is	be	AUX
iajs-830	240	4	an	an	DET
iajs-830	240	5	arithmetical	arithmetical	ADJ
iajs-830	240	6	ring	ring	NOUN
iajs-830	240	7	,	,	PUNCT
iajs-830	240	8	by	by	ADP
iajs-830	240	9	note	note	NOUN
iajs-830	240	10	(	(	PUNCT
iajs-830	240	11	4.3	4.3	NUM
iajs-830	240	12	)	)	PUNCT
iajs-830	240	13	.	.	PUNCT
iajs-830	241	1			NUM
iajs-830	241	2	4.6	4.6	NUM
iajs-830	241	3	theorem	theorem	NOUN
iajs-830	241	4	let	let	VERB
iajs-830	241	5	r	r	PRON
iajs-830	241	6	be	be	AUX
iajs-830	241	7	an	an	DET
iajs-830	241	8	integral	integral	ADJ
iajs-830	241	9	domain	domain	NOUN
iajs-830	241	10	,	,	PUNCT
iajs-830	241	11	let	let	VERB
iajs-830	241	12	x	x	PRON
iajs-830	241	13	be	be	AUX
iajs-830	241	14	a	a	DET
iajs-830	241	15	fuzzy	fuzzy	ADJ
iajs-830	241	16	ring	ring	NOUN
iajs-830	241	17	such	such	ADJ
iajs-830	241	18	that	that	SCONJ
iajs-830	241	19	x(a)=1	x(a)=1	ADJ
iajs-830	241	20	ar	ar	NOUN
iajs-830	241	21	.	.	PUNCT
iajs-830	242	1	then	then	ADV
iajs-830	242	2	the	the	DET
iajs-830	242	3	following	following	NOUN
iajs-830	242	4	are	be	AUX
iajs-830	242	5	equivalent	equivalent	ADJ
iajs-830	242	6	1	1	NUM
iajs-830	242	7	.	.	PUNCT
iajs-830	243	1	x	x	PRON
iajs-830	243	2	is	be	AUX
iajs-830	243	3	arithmetical	arithmetical	ADJ
iajs-830	243	4	2	2	NUM
iajs-830	243	5	.	.	PUNCT
iajs-830	243	6	a(bc)=ab	a(bc)=ab	PROPN
iajs-830	244	1			PUNCT
iajs-830	244	2	ac	ac	ADJ
iajs-830	244	3	for	for	ADP
iajs-830	244	4	all	all	DET
iajs-830	244	5	fuzzy	fuzzy	ADJ
iajs-830	244	6	ideals	ideal	NOUN
iajs-830	244	7	a	a	PRON
iajs-830	244	8	,	,	PUNCT
iajs-830	244	9	b	b	NOUN
iajs-830	244	10	,	,	PUNCT
iajs-830	244	11	c	c	NOUN
iajs-830	244	12	of	of	ADP
iajs-830	244	13	x.	x.	NOUN
iajs-830	244	14	3	3	NUM
iajs-830	244	15	.	.	PUNCT
iajs-830	245	1	(	(	PUNCT
iajs-830	245	2	a+b)(ab)=ab	a+b)(ab)=ab	NOUN
iajs-830	245	3	for	for	ADP
iajs-830	245	4	all	all	DET
iajs-830	245	5	fuzzy	fuzzy	ADJ
iajs-830	245	6	ideals	ideal	NOUN
iajs-830	245	7	a	a	PRON
iajs-830	245	8	,	,	PUNCT
iajs-830	245	9	b	b	PROPN
iajs-830	245	10	of	of	ADP
iajs-830	245	11	x.	x.	NOUN
iajs-830	245	12	proof	proof	NOUN
iajs-830	245	13	:	:	PUNCT
iajs-830	245	14	(	(	PUNCT
iajs-830	245	15	1	1	X
iajs-830	245	16	)	)	PUNCT
iajs-830	245	17			NOUN
iajs-830	245	18	(	(	PUNCT
iajs-830	245	19	2	2	NUM
iajs-830	245	20	):	):	PUNCT
iajs-830	245	21	to	to	PART
iajs-830	245	22	prove	prove	VERB
iajs-830	245	23	a(bc	a(bc	NOUN
iajs-830	245	24	)	)	PUNCT
iajs-830	246	1	=	=	SYM
iajs-830	246	2	ab	ab	PROPN
iajs-830	246	3			PUNCT
iajs-830	246	4	ac	ac	PROPN
iajs-830	246	5	for	for	ADP
iajs-830	246	6	all	all	DET
iajs-830	246	7	fuzzy	fuzzy	ADJ
iajs-830	246	8	ideals	ideal	NOUN
iajs-830	246	9	a	a	PRON
iajs-830	246	10	,	,	PUNCT
iajs-830	246	11	b	b	NOUN
iajs-830	246	12	,	,	PUNCT
iajs-830	246	13	c	c	PROPN
iajs-830	246	14	of	of	ADP
iajs-830	246	15	x.	x.	NOUN
iajs-830	246	16	since	since	SCONJ
iajs-830	246	17	x	x	PRON
iajs-830	246	18	is	be	AUX
iajs-830	246	19	a	a	DET
iajs-830	246	20	fuzzy	fuzzy	ADJ
iajs-830	246	21	arithmetical	arithmetical	NOUN
iajs-830	246	22	then	then	ADV
iajs-830	246	23	xt	xt	PROPN
iajs-830	246	24	is	be	AUX
iajs-830	246	25	an	an	DET
iajs-830	246	26	arithmetical	arithmetical	ADJ
iajs-830	246	27	ring	ring	NOUN
iajs-830	246	28	for	for	ADP
iajs-830	246	29	all	all	DET
iajs-830	246	30	t	t	NOUN
iajs-830	247	1	(0,1	(0,1	NUM
iajs-830	247	2	]	]	PUNCT
iajs-830	247	3	and	and	CCONJ
iajs-830	247	4	since	since	SCONJ
iajs-830	247	5	at	at	ADP
iajs-830	247	6	,	,	PUNCT
iajs-830	247	7	bt	bt	PROPN
iajs-830	247	8	,	,	PUNCT
iajs-830	247	9	ct	ct	PROPN
iajs-830	247	10	are	be	AUX
iajs-830	247	11	ideals	ideal	NOUN
iajs-830	247	12	of	of	ADP
iajs-830	247	13	xt	xt	PROPN
iajs-830	247	14	,	,	PUNCT
iajs-830	247	15	t	t	PROPN
iajs-830	248	1	(0,1	(0,1	NUM
iajs-830	248	2	]	]	PUNCT
iajs-830	248	3	we	we	PRON
iajs-830	248	4	get	get	VERB
iajs-830	248	5	at(btct	at(btct	PUNCT
iajs-830	248	6	)	)	PUNCT
iajs-830	249	1	=	=	SYM
iajs-830	249	2	atbt	atbt	NOUN
iajs-830	249	3			PUNCT
iajs-830	249	4	atct	atct	NOUN
iajs-830	249	5	,	,	PUNCT
iajs-830	249	6	(	(	PUNCT
iajs-830	249	7	[	[	X
iajs-830	249	8	14],theorem	14],theorem	NUM
iajs-830	249	9	(	(	PUNCT
iajs-830	249	10	6.6	6.6	NUM
iajs-830	249	11	)	)	PUNCT
iajs-830	249	12	)	)	PUNCT
iajs-830	249	13	.	.	PUNCT
iajs-830	250	1	hence	hence	ADV
iajs-830	250	2	at(bc)t	at(bc)t	ADV
iajs-830	250	3	=	=	SYM
iajs-830	250	4	(	(	PUNCT
iajs-830	250	5	ab)t	ab)t	PROPN
iajs-830	250	6			X
iajs-830	250	7	(	(	PUNCT
iajs-830	250	8	ac)t	ac)t	PROPN
iajs-830	250	9	,	,	PUNCT
iajs-830	250	10	(	(	PUNCT
iajs-830	250	11	proposition	proposition	NOUN
iajs-830	250	12	(	(	PUNCT
iajs-830	250	13	1.12),(2	1.12),(2	NUM
iajs-830	250	14	)	)	PUNCT
iajs-830	250	15	,	,	PUNCT
iajs-830	250	16	remark	remark	NOUN
iajs-830	250	17	(	(	PUNCT
iajs-830	250	18	1.5	1.5	NUM
iajs-830	250	19	)	)	PUNCT
iajs-830	250	20	)	)	PUNCT
iajs-830	250	21	(	(	PUNCT
iajs-830	250	22	a(bc))t	a(bc))t	NOUN
iajs-830	250	23	=	=	SYM
iajs-830	250	24	(	(	PUNCT
iajs-830	250	25	abac)t	abac)t	NOUN
iajs-830	250	26	,	,	PUNCT
iajs-830	250	27	for	for	ADP
iajs-830	250	28	all	all	DET
iajs-830	250	29	t	t	NOUN
iajs-830	250	30	(0,1	(0,1	NUM
iajs-830	250	31	]	]	PUNCT
iajs-830	250	32	(	(	PUNCT
iajs-830	250	33	proposition	proposition	NOUN
iajs-830	250	34	1.12,(2	1.12,(2	NUM
iajs-830	250	35	)	)	PUNCT
iajs-830	250	36	)	)	PUNCT
iajs-830	250	37	thus	thus	ADV
iajs-830	250	38	a(bc)=ab	a(bc)=ab	X
iajs-830	250	39			PUNCT
iajs-830	250	40	ac	ac	PROPN
iajs-830	250	41	.	.	PUNCT
iajs-830	251	1	(	(	PUNCT
iajs-830	251	2	2	2	X
iajs-830	251	3	)	)	PUNCT
iajs-830	251	4			NOUN
iajs-830	251	5	(	(	PUNCT
iajs-830	251	6	3	3	NUM
iajs-830	251	7	):	):	PUNCT
iajs-830	251	8	if	if	SCONJ
iajs-830	251	9	a(bc)=ab	a(bc)=ab	PROPN
iajs-830	251	10			PUNCT
iajs-830	251	11	ac	ac	VERB
iajs-830	251	12	for	for	ADP
iajs-830	251	13	all	all	DET
iajs-830	251	14	fuzzy	fuzzy	ADJ
iajs-830	251	15	ideals	ideal	NOUN
iajs-830	251	16	a	a	DET
iajs-830	251	17	,	,	PUNCT
iajs-830	251	18	b	b	NOUN
iajs-830	251	19	,	,	PUNCT
iajs-830	251	20	c	c	NOUN
iajs-830	251	21	of	of	ADP
iajs-830	251	22	x	x	PRON
iajs-830	251	23	,	,	PUNCT
iajs-830	251	24	let	let	VERB
iajs-830	251	25	t	t	PROPN
iajs-830	251	26	(0,1	(0,1	NUM
iajs-830	251	27	]	]	PUNCT
iajs-830	251	28	,	,	PUNCT
iajs-830	251	29	let	let	VERB
iajs-830	251	30	i	i	PRON
iajs-830	251	31	,	,	PUNCT
iajs-830	251	32	j	j	PROPN
iajs-830	251	33	,	,	PUNCT
iajs-830	251	34	k	k	X
iajs-830	251	35	be	be	VERB
iajs-830	251	36	ideals	ideal	NOUN
iajs-830	251	37	of	of	ADP
iajs-830	251	38	xt	xt	PROPN
iajs-830	251	39	.	.	PUNCT
iajs-830	252	1	then	then	ADV
iajs-830	252	2	there	there	PRON
iajs-830	252	3	exists	exist	VERB
iajs-830	252	4	fuzzy	fuzzy	ADJ
iajs-830	252	5	ideals	ideal	NOUN
iajs-830	252	6	a	a	PRON
iajs-830	252	7	,	,	PUNCT
iajs-830	252	8	b	b	NOUN
iajs-830	252	9	,	,	PUNCT
iajs-830	252	10	c	c	NOUN
iajs-830	252	11	of	of	ADP
iajs-830	252	12	x	x	INTJ
iajs-830	252	13	such	such	ADJ
iajs-830	252	14	that	that	SCONJ
iajs-830	252	15	at	at	ADP
iajs-830	252	16	=	=	PROPN
iajs-830	252	17	i	i	PROPN
iajs-830	252	18	,	,	PUNCT
iajs-830	252	19	bt	bt	PROPN
iajs-830	252	20	=	=	PROPN
iajs-830	252	21	j	j	PROPN
iajs-830	252	22	,	,	PUNCT
iajs-830	252	23	ct	ct	PROPN
iajs-830	252	24	=	=	NOUN
iajs-830	252	25	k	k	PROPN
iajs-830	252	26	,	,	PUNCT
iajs-830	252	27	where	where	SCONJ
iajs-830	252	28	t	t	NOUN
iajs-830	252	29	x	x	PROPN
iajs-830	252	30	i	i	PRON
iajs-830	252	31	(	(	PUNCT
iajs-830	252	32	x	x	X
iajs-830	252	33	)	)	PUNCT
iajs-830	252	34	0	0	NUM
iajs-830	253	1	x	x	SYM
iajs-830	253	2	i	i	PRON
iajs-830	253	3			VERB
iajs-830	254	1			PROPN
iajs-830	254	2			NUM
iajs-830	255	1			NUM
iajs-830	255	2			NOUN
iajs-830	255	3	,	,	PUNCT
iajs-830	255	4	t	t	PROPN
iajs-830	255	5	x	x	X
iajs-830	255	6	j	j	PROPN
iajs-830	255	7	(	(	PUNCT
iajs-830	255	8	x	x	X
iajs-830	255	9	)	)	PUNCT
iajs-830	255	10	0	0	NUM
iajs-830	256	1	x	x	SYM
iajs-830	256	2	j	j	PROPN
iajs-830	256	3			NUM
iajs-830	256	4			NOUN
iajs-830	256	5			NOUN
iajs-830	257	1			NUM
iajs-830	257	2			NOUN
iajs-830	257	3	,	,	PUNCT
iajs-830	257	4	t	t	PROPN
iajs-830	257	5	x	x	PUNCT
iajs-830	257	6	k	k	X
iajs-830	257	7	c(x	c(x	NOUN
iajs-830	257	8	)	)	PUNCT
iajs-830	257	9	0	0	NUM
iajs-830	258	1	x	x	SYM
iajs-830	258	2	k	k	NOUN
iajs-830	258	3			VERB
iajs-830	258	4			NUM
iajs-830	258	5			NOUN
iajs-830	258	6			PUNCT
iajs-830	258	7	by	by	ADP
iajs-830	258	8	(	(	PUNCT
iajs-830	258	9	2	2	NUM
iajs-830	258	10	)	)	PUNCT
iajs-830	258	11	,	,	PUNCT
iajs-830	258	12	a(bc)=(ab)(ac	a(bc)=(ab)(ac	PROPN
iajs-830	258	13	)	)	PUNCT
iajs-830	258	14	,	,	PUNCT
iajs-830	258	15	which	which	PRON
iajs-830	258	16	implies	imply	VERB
iajs-830	258	17	that	that	SCONJ
iajs-830	258	18	(	(	PUNCT
iajs-830	258	19	a(bc))t	a(bc))t	NOUN
iajs-830	258	20	=	=	SYM
iajs-830	258	21	(	(	PUNCT
iajs-830	258	22	abac)t	abac)t	NOUN
iajs-830	258	23	,	,	PUNCT
iajs-830	258	24	for	for	ADP
iajs-830	258	25	all	all	DET
iajs-830	258	26	t	t	NOUN
iajs-830	259	1	(0,1	(0,1	NUM
iajs-830	259	2	]	]	PUNCT
iajs-830	259	3	.	.	PUNCT
iajs-830	260	1	hence	hence	ADV
iajs-830	260	2	at(bc)t	at(bc)t	ADV
iajs-830	260	3	=	=	SYM
iajs-830	260	4	(	(	PUNCT
iajs-830	260	5	ab)t	ab)t	PROPN
iajs-830	260	6			X
iajs-830	260	7	(	(	PUNCT
iajs-830	260	8	ac)t	ac)t	PROPN
iajs-830	260	9	,	,	PUNCT
iajs-830	260	10	(	(	PUNCT
iajs-830	260	11	remark	remark	NOUN
iajs-830	260	12	(	(	PUNCT
iajs-830	260	13	1.5),proposition	1.5),proposition	NUM
iajs-830	260	14	(	(	PUNCT
iajs-830	260	15	1.12	1.12	NUM
iajs-830	260	16	)	)	PUNCT
iajs-830	260	17	)	)	PUNCT
iajs-830	260	18	,	,	PUNCT
iajs-830	260	19	so	so	SCONJ
iajs-830	260	20	that	that	SCONJ
iajs-830	260	21	at(btct	at(btct	ADV
iajs-830	260	22	)	)	PUNCT
iajs-830	261	1	=	=	SYM
iajs-830	261	2	atbt	atbt	NOUN
iajs-830	261	3			PUNCT
iajs-830	261	4	atct	atct	NOUN
iajs-830	261	5	,	,	PUNCT
iajs-830	261	6	(	(	PUNCT
iajs-830	261	7	remark	remark	NOUN
iajs-830	261	8	(	(	PUNCT
iajs-830	261	9	1.5),proposition	1.5),proposition	NUM
iajs-830	261	10	(	(	PUNCT
iajs-830	261	11	1.12	1.12	NUM
iajs-830	261	12	)	)	PUNCT
iajs-830	261	13	)	)	PUNCT
iajs-830	261	14	.	.	PUNCT
iajs-830	262	1	i	i	PRON
iajs-830	262	2	(	(	PUNCT
iajs-830	262	3	jk)=(ij)+(ik	jk)=(ij)+(ik	NOUN
iajs-830	262	4	)	)	PUNCT
iajs-830	262	5	.	.	PUNCT
iajs-830	263	1	then	then	ADV
iajs-830	263	2	by	by	ADP
iajs-830	263	3	(	(	PUNCT
iajs-830	263	4	(	(	PUNCT
iajs-830	263	5	16),exc	16),exc	PROPN
iajs-830	263	6	.	.	NOUN
iajs-830	263	7	18	18	NUM
iajs-830	263	8	)	)	PUNCT
iajs-830	263	9	(	(	PUNCT
iajs-830	263	10	i+j)(ij)=ij	i+j)(ij)=ij	PROPN
iajs-830	263	11	.	.	PUNCT
iajs-830	264	1	thus	thus	ADV
iajs-830	264	2	(	(	PUNCT
iajs-830	264	3	at+bt	at+bt	NOUN
iajs-830	264	4	)	)	PUNCT
iajs-830	264	5	(	(	PUNCT
iajs-830	264	6	atbt	atbt	PROPN
iajs-830	264	7	)	)	PUNCT
iajs-830	264	8	=	=	VERB
iajs-830	264	9	atbt	atbt	NOUN
iajs-830	264	10	,	,	PUNCT
iajs-830	264	11	(	(	PUNCT
iajs-830	264	12	a+b)t	a+b)t	PROPN
iajs-830	264	13	(	(	PUNCT
iajs-830	264	14	ab)t	ab)t	PROPN
iajs-830	264	15	=	=	PUNCT
iajs-830	264	16	(	(	PUNCT
iajs-830	264	17	ab)t	ab)t	PROPN
iajs-830	264	18	which	which	PRON
iajs-830	264	19	implies	imply	VERB
iajs-830	264	20	that	that	SCONJ
iajs-830	264	21	(	(	PUNCT
iajs-830	264	22	a+b)(ab)=ab	a+b)(ab)=ab	NOUN
iajs-830	264	23	.	.	PUNCT
iajs-830	265	1	(	(	PUNCT
iajs-830	265	2	3	3	X
iajs-830	265	3	)	)	PUNCT
iajs-830	265	4			NOUN
iajs-830	265	5	(	(	PUNCT
iajs-830	265	6	1):if	1):if	NUM
iajs-830	265	7	(	(	PUNCT
iajs-830	265	8	a+b)(ab)=ab	a+b)(ab)=ab	NOUN
iajs-830	265	9	for	for	ADP
iajs-830	265	10	all	all	DET
iajs-830	265	11	fuzzy	fuzzy	ADJ
iajs-830	265	12	ideals	ideal	NOUN
iajs-830	265	13	a	a	PRON
iajs-830	265	14	,	,	PUNCT
iajs-830	265	15	b	b	PROPN
iajs-830	265	16	of	of	ADP
iajs-830	265	17	x.	x.	NOUN
iajs-830	265	18	let	let	VERB
iajs-830	265	19	t	t	PROPN
iajs-830	265	20	(0,1	(0,1	NUM
iajs-830	265	21	]	]	PUNCT
iajs-830	265	22	,	,	PUNCT
iajs-830	265	23	let	let	VERB
iajs-830	265	24	i	i	PRON
iajs-830	265	25	,	,	PUNCT
iajs-830	265	26	j	j	PROPN
iajs-830	265	27	,	,	PUNCT
iajs-830	265	28	k	k	PROPN
iajs-830	265	29	be	be	AUX
iajs-830	265	30	ibn	ibn	PROPN
iajs-830	265	31	alhaitham	alhaitham	NOUN
iajs-830	265	32	j.	j.	PROPN
iajs-830	266	1	fo	fo	ADP
iajs-830	266	2	r	r	NOUN
iajs-830	266	3	pure	pure	ADJ
iajs-830	266	4	&	&	CCONJ
iajs-830	266	5	appl	appl	PROPN
iajs-830	266	6	.	.	PUNCT
iajs-830	267	1	sc	sc	PROPN
iajs-830	267	2	i.	i.	PROPN
iajs-830	267	3	vo	vo	PROPN
iajs-830	267	4	l.24	l.24	PROPN
iajs-830	267	5	(	(	PUNCT
iajs-830	267	6	1	1	NUM
iajs-830	267	7	)	)	PUNCT
iajs-830	267	8	2011	2011	NUM
iajs-830	267	9	ideals	ideal	NOUN
iajs-830	267	10	of	of	ADP
iajs-830	267	11	xt	xt	PROPN
iajs-830	267	12	.	.	PUNCT
iajs-830	268	1	then	then	ADV
iajs-830	268	2	there	there	PRON
iajs-830	268	3	exists	exist	VERB
iajs-830	268	4	fuzzy	fuzzy	ADJ
iajs-830	268	5	ideals	ideal	NOUN
iajs-830	268	6	a	a	PRON
iajs-830	268	7	,	,	PUNCT
iajs-830	268	8	b	b	NOUN
iajs-830	268	9	,	,	PUNCT
iajs-830	268	10	c	c	NOUN
iajs-830	268	11	of	of	ADP
iajs-830	268	12	x	x	INTJ
iajs-830	268	13	such	such	ADJ
iajs-830	268	14	that	that	SCONJ
iajs-830	268	15	at	at	ADP
iajs-830	268	16	=	=	PROPN
iajs-830	268	17	i	i	PROPN
iajs-830	268	18	,	,	PUNCT
iajs-830	268	19	bt	bt	PROPN
iajs-830	268	20	=	=	PROPN
iajs-830	268	21	j	j	PROPN
iajs-830	268	22	,	,	PUNCT
iajs-830	268	23	ct	ct	PROPN
iajs-830	268	24	=	=	NOUN
iajs-830	268	25	k	k	PROPN
iajs-830	268	26	,	,	PUNCT
iajs-830	268	27	where	where	SCONJ
iajs-830	268	28	t	t	NOUN
iajs-830	268	29	x	x	PROPN
iajs-830	268	30	i	i	PRON
iajs-830	268	31	(	(	PUNCT
iajs-830	268	32	x	x	X
iajs-830	268	33	)	)	PUNCT
iajs-830	268	34	0	0	NUM
iajs-830	269	1	x	x	SYM
iajs-830	269	2	i	i	PRON
iajs-830	269	3			VERB
iajs-830	270	1			PROPN
iajs-830	270	2			NUM
iajs-830	271	1			NUM
iajs-830	271	2			NOUN
iajs-830	271	3	,	,	PUNCT
iajs-830	271	4	t	t	PROPN
iajs-830	271	5	x	x	X
iajs-830	271	6	j	j	PROPN
iajs-830	271	7	(	(	PUNCT
iajs-830	271	8	x	x	X
iajs-830	271	9	)	)	PUNCT
iajs-830	271	10	0	0	NUM
iajs-830	272	1	x	x	SYM
iajs-830	272	2	j	j	PROPN
iajs-830	272	3			NUM
iajs-830	272	4			NOUN
iajs-830	272	5			NOUN
iajs-830	273	1			NUM
iajs-830	273	2			NOUN
iajs-830	273	3	,	,	PUNCT
iajs-830	273	4	t	t	PROPN
iajs-830	273	5	x	x	PUNCT
iajs-830	273	6	k	k	X
iajs-830	273	7	c(x	c(x	NOUN
iajs-830	273	8	)	)	PUNCT
iajs-830	273	9	0	0	NUM
iajs-830	274	1	x	x	SYM
iajs-830	274	2	k	k	NOUN
iajs-830	274	3			VERB
iajs-830	274	4			NUM
iajs-830	274	5			NOUN
iajs-830	274	6			PUNCT
iajs-830	274	7	by	by	ADP
iajs-830	274	8	(	(	PUNCT
iajs-830	274	9	3	3	NUM
iajs-830	274	10	)	)	PUNCT
iajs-830	274	11	,	,	PUNCT
iajs-830	274	12	(	(	PUNCT
iajs-830	274	13	a+b)(ab)=ab	a+b)(ab)=ab	NOUN
iajs-830	274	14	,	,	PUNCT
iajs-830	274	15	which	which	PRON
iajs-830	274	16	implies	imply	VERB
iajs-830	274	17	that	that	SCONJ
iajs-830	274	18	(	(	PUNCT
iajs-830	274	19	(	(	PUNCT
iajs-830	274	20	a+b)(ab))t=(ab)t	a+b)(ab))t=(ab)t	PROPN
iajs-830	274	21	for	for	ADP
iajs-830	274	22	all	all	DET
iajs-830	274	23	t	t	NOUN
iajs-830	274	24	(0,1	(0,1	NUM
iajs-830	274	25	]	]	PUNCT
iajs-830	274	26	.	.	PUNCT
iajs-830	275	1	hence	hence	ADV
iajs-830	275	2	(	(	PUNCT
iajs-830	275	3	a+b)t(ab)t	a+b)t(ab)t	PROPN
iajs-830	275	4	=	=	SYM
iajs-830	275	5	atbt	atbt	NOUN
iajs-830	275	6	,	,	PUNCT
iajs-830	275	7	(	(	PUNCT
iajs-830	275	8	proposition	proposition	NOUN
iajs-830	275	9	(	(	PUNCT
iajs-830	275	10	1.12),(2	1.12),(2	NUM
iajs-830	275	11	)	)	PUNCT
iajs-830	275	12	)	)	PUNCT
iajs-830	275	13	,	,	PUNCT
iajs-830	275	14	so	so	SCONJ
iajs-830	275	15	that	that	SCONJ
iajs-830	275	16	(	(	PUNCT
iajs-830	275	17	at+bt	at+bt	NOUN
iajs-830	275	18	)	)	PUNCT
iajs-830	275	19	(	(	PUNCT
iajs-830	275	20	atbt	atbt	ADJ
iajs-830	275	21	)	)	PUNCT
iajs-830	275	22	=	=	NOUN
iajs-830	275	23	atbt	atbt	NOUN
iajs-830	275	24	,	,	PUNCT
iajs-830	275	25	(	(	PUNCT
iajs-830	275	26	remark	remark	NOUN
iajs-830	275	27	(	(	PUNCT
iajs-830	275	28	1.5	1.5	NUM
iajs-830	275	29	)	)	PUNCT
iajs-830	275	30	,	,	PUNCT
iajs-830	275	31	proposition	proposition	NOUN
iajs-830	275	32	(	(	PUNCT
iajs-830	275	33	1.13	1.13	NUM
iajs-830	275	34	)	)	PUNCT
iajs-830	275	35	)	)	PUNCT
iajs-830	275	36	.	.	PUNCT
iajs-830	276	1	(	(	PUNCT
iajs-830	276	2	j+k	j+k	NUM
iajs-830	276	3	)	)	PUNCT
iajs-830	276	4	(	(	PUNCT
iajs-830	276	5	ij)=ij	ij)=ij	PROPN
iajs-830	276	6	.	.	PUNCT
iajs-830	277	1	then	then	ADV
iajs-830	277	2	by	by	ADP
iajs-830	277	3	(	(	PUNCT
iajs-830	277	4	[	[	X
iajs-830	277	5	14	14	NUM
iajs-830	277	6	]	]	PUNCT
iajs-830	277	7	,	,	PUNCT
iajs-830	277	8	theorem	theorem	ADJ
iajs-830	277	9	(	(	PUNCT
iajs-830	277	10	6.6	6.6	NUM
iajs-830	277	11	)	)	PUNCT
iajs-830	277	12	)	)	PUNCT
iajs-830	278	1	xt	xt	PROPN
iajs-830	278	2	is	be	AUX
iajs-830	278	3	arithmetical	arithmetical	ADJ
iajs-830	278	4	ring	ring	NOUN
iajs-830	278	5	for	for	ADP
iajs-830	278	6	all	all	DET
iajs-830	278	7	t	t	NOUN
iajs-830	279	1	(0,1	(0,1	NUM
iajs-830	279	2	]	]	PUNCT
iajs-830	279	3	.	.	PUNCT
iajs-830	280	1	thus	thus	ADV
iajs-830	280	2	x	x	PRON
iajs-830	280	3	is	be	AUX
iajs-830	280	4	a	a	DET
iajs-830	280	5	fuzzy	fuzzy	ADJ
iajs-830	280	6	arithmetical	arithmetical	ADJ
iajs-830	280	7	ring	ring	NOUN
iajs-830	280	8	(	(	PUNCT
iajs-830	280	9	by	by	ADP
iajs-830	280	10	note	note	NOUN
iajs-830	280	11	4.3	4.3	NUM
iajs-830	280	12	)	)	PUNCT
iajs-830	280	13	.	.	PUNCT
iajs-830	281	1			PROPN
iajs-830	281	2	4.7	4.7	NUM
iajs-830	281	3	theorem	theorem	NOUN
iajs-830	281	4	let	let	VERB
iajs-830	281	5	r	r	PRON
iajs-830	281	6	be	be	AUX
iajs-830	281	7	a	a	DET
iajs-830	281	8	noetherian	noetherian	ADJ
iajs-830	281	9	integral	integral	ADJ
iajs-830	281	10	domain	domain	NOUN
iajs-830	281	11	,	,	PUNCT
iajs-830	281	12	let	let	VERB
iajs-830	281	13	x	x	PRON
iajs-830	281	14	be	be	AUX
iajs-830	281	15	a	a	DET
iajs-830	281	16	fuzzy	fuzzy	ADJ
iajs-830	281	17	ring	ring	NOUN
iajs-830	281	18	such	such	ADJ
iajs-830	281	19	that	that	SCONJ
iajs-830	281	20	x(a)=1	x(a)=1	ADJ
iajs-830	281	21	ar	ar	NOUN
iajs-830	281	22	.	.	PUNCT
iajs-830	282	1	then	then	ADV
iajs-830	282	2	the	the	DET
iajs-830	282	3	following	following	NOUN
iajs-830	282	4	are	be	AUX
iajs-830	282	5	equivalent	equivalent	ADJ
iajs-830	282	6	1	1	NUM
iajs-830	282	7	.	.	PUNCT
iajs-830	283	1	x	x	PRON
iajs-830	283	2	is	be	AUX
iajs-830	283	3	arithmetical	arithmetical	ADJ
iajs-830	283	4	2	2	NUM
iajs-830	283	5	.	.	PUNCT
iajs-830	283	6	a(bc)=ab	a(bc)=ab	PROPN
iajs-830	284	1			PUNCT
iajs-830	284	2	ac	ac	ADJ
iajs-830	284	3	for	for	ADP
iajs-830	284	4	all	all	DET
iajs-830	284	5	fuzzy	fuzzy	ADJ
iajs-830	284	6	ideals	ideal	NOUN
iajs-830	284	7	a	a	PRON
iajs-830	284	8	,	,	PUNCT
iajs-830	284	9	b	b	NOUN
iajs-830	284	10	,	,	PUNCT
iajs-830	284	11	c	c	NOUN
iajs-830	284	12	of	of	ADP
iajs-830	284	13	x.	x.	NOUN
iajs-830	284	14	3	3	NUM
iajs-830	284	15	.	.	PUNCT
iajs-830	285	1	(	(	PUNCT
iajs-830	285	2	a+b)(ab)=ab	a+b)(ab)=ab	NOUN
iajs-830	285	3	for	for	ADP
iajs-830	285	4	all	all	DET
iajs-830	285	5	fuzzy	fuzzy	ADJ
iajs-830	285	6	ideals	ideal	NOUN
iajs-830	285	7	a	a	DET
iajs-830	285	8	,	,	PUNCT
iajs-830	285	9	b	b	PROPN
iajs-830	285	10	of	of	ADP
iajs-830	285	11	x.	x.	NOUN
iajs-830	285	12	4	4	NUM
iajs-830	285	13	.	.	PUNCT
iajs-830	286	1	if	if	SCONJ
iajs-830	286	2	a	a	PRON
iajs-830	286	3	,	,	PUNCT
iajs-830	286	4	c	c	PROPN
iajs-830	286	5	are	be	AUX
iajs-830	286	6	fuzzy	fuzzy	ADJ
iajs-830	286	7	ideals	ideal	NOUN
iajs-830	286	8	of	of	ADP
iajs-830	286	9	x	x	PUNCT
iajs-830	286	10	and	and	CCONJ
iajs-830	286	11	if	if	SCONJ
iajs-830	286	12	ca	ca	PROPN
iajs-830	286	13	,	,	PUNCT
iajs-830	286	14	then	then	ADV
iajs-830	286	15	there	there	PRON
iajs-830	286	16	exists	exist	VERB
iajs-830	286	17	fuzzy	fuzzy	ADJ
iajs-830	286	18	ideal	ideal	NOUN
iajs-830	287	1	b	b	ADP
iajs-830	287	2	such	such	ADJ
iajs-830	287	3	that	that	SCONJ
iajs-830	287	4	a	a	DET
iajs-830	287	5	=	=	SYM
iajs-830	287	6	bc	bc	PROPN
iajs-830	287	7	.	.	PUNCT
iajs-830	288	1	proof	proof	NOUN
iajs-830	288	2	:	:	PUNCT
iajs-830	288	3	(	(	PUNCT
iajs-830	288	4	1	1	X
iajs-830	288	5	)	)	PUNCT
iajs-830	288	6			NOUN
iajs-830	288	7	(	(	PUNCT
iajs-830	288	8	2	2	NUM
iajs-830	288	9	)	)	PUNCT
iajs-830	288	10	and	and	CCONJ
iajs-830	288	11	(	(	PUNCT
iajs-830	288	12	2	2	X
iajs-830	288	13	)	)	PUNCT
iajs-830	288	14			NOUN
iajs-830	288	15	(	(	PUNCT
iajs-830	288	16	3	3	X
iajs-830	288	17	)	)	PUNCT
iajs-830	288	18	follows	follow	VERB
iajs-830	288	19	directly	directly	ADV
iajs-830	288	20	by	by	ADP
iajs-830	288	21	theorem	theorem	NOUN
iajs-830	288	22	(	(	PUNCT
iajs-830	288	23	4.5	4.5	NUM
iajs-830	288	24	)	)	PUNCT
iajs-830	288	25	.	.	PUNCT
iajs-830	289	1	(	(	PUNCT
iajs-830	289	2	3	3	X
iajs-830	289	3	)	)	PUNCT
iajs-830	289	4			NOUN
iajs-830	289	5	(	(	PUNCT
iajs-830	289	6	4	4	NUM
iajs-830	289	7	)	)	PUNCT
iajs-830	289	8	,	,	PUNCT
iajs-830	289	9	let	let	VERB
iajs-830	289	10	a	a	PRON
iajs-830	289	11	,	,	PUNCT
iajs-830	289	12	c	c	AUX
iajs-830	289	13	be	be	AUX
iajs-830	289	14	fuzzy	fuzzy	ADJ
iajs-830	289	15	ideals	ideal	NOUN
iajs-830	289	16	of	of	ADP
iajs-830	289	17	x	x	PUNCT
iajs-830	289	18	such	such	ADJ
iajs-830	289	19	that	that	DET
iajs-830	289	20	ca	ca	NOUN
iajs-830	289	21	,	,	PUNCT
iajs-830	289	22	implies	imply	VERB
iajs-830	289	23	at	at	ADP
iajs-830	289	24			PROPN
iajs-830	289	25	ct	ct	PROPN
iajs-830	289	26	,	,	PUNCT
iajs-830	289	27	then	then	ADV
iajs-830	289	28	at	at	ADP
iajs-830	289	29	=	=	ADJ
iajs-830	289	30	btct	btct	NOUN
iajs-830	289	31	(	(	PUNCT
iajs-830	289	32	[	[	X
iajs-830	289	33	14	14	NUM
iajs-830	289	34	]	]	PUNCT
iajs-830	289	35	,	,	PUNCT
iajs-830	289	36	theorem	theorem	ADJ
iajs-830	289	37	(	(	PUNCT
iajs-830	289	38	6.26	6.26	NUM
iajs-830	289	39	)	)	PUNCT
iajs-830	289	40	)	)	PUNCT
iajs-830	289	41	,	,	PUNCT
iajs-830	289	42	at=(bc)t	at=(bc)t	PROPN
iajs-830	289	43	then	then	ADV
iajs-830	289	44	a	a	DET
iajs-830	289	45	=	=	SYM
iajs-830	289	46	bc	bc	PROPN
iajs-830	289	47	.	.	PROPN
iajs-830	289	48	(	(	PUNCT
iajs-830	289	49	4	4	NUM
iajs-830	289	50	)	)	PUNCT
iajs-830	289	51			NOUN
iajs-830	289	52	(	(	PUNCT
iajs-830	289	53	1	1	NUM
iajs-830	289	54	)	)	PUNCT
iajs-830	289	55	,	,	PUNCT
iajs-830	289	56	let	let	VERB
iajs-830	289	57	a	a	PRON
iajs-830	289	58	,	,	PUNCT
iajs-830	289	59	c	c	PROPN
iajs-830	289	60	are	be	AUX
iajs-830	289	61	fuzzy	fuzzy	ADJ
iajs-830	289	62	ideals	ideal	NOUN
iajs-830	289	63	of	of	ADP
iajs-830	289	64	x	x	NOUN
iajs-830	289	65	,	,	PUNCT
iajs-830	289	66	if	if	SCONJ
iajs-830	289	67	ca	ca	PROPN
iajs-830	289	68	,	,	PUNCT
iajs-830	289	69	then	then	ADV
iajs-830	289	70	there	there	PRON
iajs-830	289	71	exists	exist	VERB
iajs-830	289	72	fuzzy	fuzzy	ADJ
iajs-830	289	73	ideal	ideal	ADJ
iajs-830	289	74	b	b	PROPN
iajs-830	289	75	of	of	ADP
iajs-830	289	76	x	x	SYM
iajs-830	289	77	such	such	ADJ
iajs-830	289	78	that	that	SCONJ
iajs-830	289	79	a	a	DET
iajs-830	289	80	=	=	SYM
iajs-830	289	81	bc	bc	PROPN
iajs-830	289	82	,	,	PUNCT
iajs-830	289	83	then	then	ADV
iajs-830	289	84	at=(bc)t	at=(bc)t	PROPN
iajs-830	289	85	so	so	ADV
iajs-830	289	86	w	w	NOUN
iajs-830	289	87	get	get	VERB
iajs-830	289	88	at	at	ADP
iajs-830	289	89	=	=	NOUN
iajs-830	289	90	btct	btct	NOUN
iajs-830	289	91	,	,	PUNCT
iajs-830	289	92	xt	xt	X
iajs-830	289	93	is	be	AUX
iajs-830	289	94	arithmetical	arithmetical	ADJ
iajs-830	289	95	ring	ring	NOUN
iajs-830	289	96	(	(	PUNCT
iajs-830	289	97	[	[	X
iajs-830	289	98	14	14	NUM
iajs-830	289	99	]	]	PUNCT
iajs-830	289	100	,	,	PUNCT
iajs-830	289	101	theorem	theorem	ADJ
iajs-830	289	102	(	(	PUNCT
iajs-830	289	103	6.26	6.26	NUM
iajs-830	289	104	)	)	PUNCT
iajs-830	289	105	)	)	PUNCT
iajs-830	289	106	.	.	PUNCT
iajs-830	290	1	thus	thus	ADV
iajs-830	290	2	x	x	PRON
iajs-830	290	3	is	be	AUX
iajs-830	290	4	a	a	DET
iajs-830	290	5	fuzzy	fuzzy	ADJ
iajs-830	290	6	arithmetical	arithmetical	ADJ
iajs-830	290	7	ring	ring	NOUN
iajs-830	290	8	.	.	PUNCT
iajs-830	291	1	references	reference	NOUN
iajs-830	291	2	1	1	NUM
iajs-830	291	3	.	.	X
iajs-830	291	4	zahdi	zahdi	PROPN
iajs-830	291	5	,	,	PUNCT
iajs-830	291	6	l.a	l.a	PROPN
iajs-830	291	7	.	.	PROPN
iajs-830	291	8	(	(	PUNCT
iajs-830	291	9	1965	1965	NUM
iajs-830	291	10	)	)	PUNCT
iajs-830	291	11	,	,	PUNCT
iajs-830	291	12	fuzzy	fuzzy	ADJ
iajs-830	291	13	sets	set	NOUN
iajs-830	291	14	,	,	PUNCT
iajs-830	291	15	information	information	NOUN
iajs-830	291	16	and	and	CCONJ
iajs-830	291	17	control	control	NOUN
iajs-830	291	18	,	,	PUNCT
iajs-830	291	19	8	8	NUM
iajs-830	291	20	,	,	PUNCT
iajs-830	291	21	338	338	NUM
iajs-830	291	22	-	-	SYM
iajs-830	291	23	353	353	NUM
iajs-830	291	24	.	.	NOUN
iajs-830	292	1	2	2	NUM
iajs-830	292	2	.	.	X
iajs-830	292	3	zadehi	zadehi	PROPN
iajs-830	292	4	,	,	PUNCT
iajs-830	292	5	m.	m.	NOUN
iajs-830	292	6	m.(1992	m.(1992	PROPN
iajs-830	292	7	)	)	PUNCT
iajs-830	292	8	,	,	PUNCT
iajs-830	292	9	on	on	ADP
iajs-830	292	10	l	l	ADJ
iajs-830	292	11	-	-	ADJ
iajs-830	292	12	fuzzy	fuzzy	ADJ
iajs-830	292	13	residual	residual	ADJ
iajs-830	292	14	quotient	quotient	NOUN
iajs-830	292	15	modules	module	NOUN
iajs-830	292	16	and	and	CCONJ
iajs-830	292	17	p	p	NOUN
iajs-830	292	18	-	-	PUNCT
iajs-830	292	19	primary	primary	ADJ
iajs-830	292	20	submodules	submodule	NOUN
iajs-830	292	21	,	,	PUNCT
iajs-830	292	22	fuzzy	fuzzy	ADJ
iajs-830	292	23	sets	set	NOUN
iajs-830	292	24	and	and	CCONJ
iajs-830	292	25	systems	system	NOUN
iajs-830	292	26	,	,	PUNCT
iajs-830	292	27	51	51	NUM
iajs-830	292	28	,	,	PUNCT
iajs-830	292	29	331	331	NUM
iajs-830	292	30	-	-	SYM
iajs-830	292	31	344	344	NUM
iajs-830	292	32	.	.	PUNCT
iajs-830	293	1	3	3	NUM
iajs-830	293	2	.	.	X
iajs-830	293	3	zadehi	zadehi	PROPN
iajs-830	293	4	,	,	PUNCT
iajs-830	293	5	m.	m.	NOUN
iajs-830	293	6	m.	m.	NOUN
iajs-830	293	7	(	(	PUNCT
iajs-830	293	8	1991	1991	NUM
iajs-830	293	9	)	)	PUNCT
iajs-830	293	10	,	,	PUNCT
iajs-830	293	11	a	a	DET
iajs-830	293	12	characterization	characterization	NOUN
iajs-830	293	13	of	of	ADP
iajs-830	293	14	l	l	ADJ
iajs-830	293	15	-	-	ADJ
iajs-830	293	16	fuzzy	fuzzy	ADJ
iajs-830	293	17	prime	prime	ADJ
iajs-830	293	18	ideals	ideal	NOUN
iajs-830	293	19	,	,	PUNCT
iajs-830	293	20	fuzzy	fuzzy	ADJ
iajs-830	293	21	sets	set	NOUN
iajs-830	293	22	and	and	CCONJ
iajs-830	293	23	systems	system	NOUN
iajs-830	293	24	,	,	PUNCT
iajs-830	293	25	44	44	NUM
iajs-830	293	26	,	,	PUNCT
iajs-830	293	27	147	147	NUM
iajs-830	293	28	-	-	SYM
iajs-830	293	29	160	160	NUM
iajs-830	293	30	.	.	NOUN
iajs-830	294	1	4	4	NUM
iajs-830	294	2	.	.	X
iajs-830	294	3	martinez	martinez	PROPN
iajs-830	294	4	,	,	PUNCT
iajs-830	294	5	l.	l.	PROPN
iajs-830	294	6	(	(	PUNCT
iajs-830	294	7	1996	1996	NUM
iajs-830	294	8	)	)	PUNCT
iajs-830	294	9	,	,	PUNCT
iajs-830	294	10	fuzzy	fuzzy	ADJ
iajs-830	294	11	modules	module	NOUN
iajs-830	294	12	over	over	ADP
iajs-830	294	13	fuzzy	fuzzy	ADJ
iajs-830	294	14	rings	ring	NOUN
iajs-830	294	15	in	in	ADP
iajs-830	294	16	connection	connection	NOUN
iajs-830	294	17	with	with	ADP
iajs-830	294	18	fuzzy	fuzzy	ADJ
iajs-830	294	19	ideal	ideal	NOUN
iajs-830	294	20	of	of	ADP
iajs-830	294	21	fuzzy	fuzzy	ADJ
iajs-830	294	22	ring	ring	NOUN
iajs-830	294	23	,	,	PUNCT
iajs-830	294	24	j.	j.	PROPN
iajs-830	294	25	fuzzy	fuzzy	PROPN
iajs-830	294	26	math	math	PROPN
iajs-830	294	27	.	.	PUNCT
iajs-830	294	28	,	,	PUNCT
iajs-830	294	29	4	4	NUM
iajs-830	294	30	,	,	PUNCT
iajs-830	294	31	843	843	NUM
iajs-830	294	32	-	-	NOUN
iajs-830	294	33	857	857	NUM
iajs-830	294	34	.	.	PUNCT
iajs-830	295	1	5	5	NUM
iajs-830	295	2	.	.	X
iajs-830	295	3	nanda	nanda	PROPN
iajs-830	295	4	,	,	PUNCT
iajs-830	295	5	s.	s.	PROPN
iajs-830	295	6	(	(	PUNCT
iajs-830	295	7	1989	1989	NUM
iajs-830	295	8	)	)	PUNCT
iajs-830	295	9	,	,	PUNCT
iajs-830	295	10	fuzzy	fuzzy	ADJ
iajs-830	295	11	modules	module	NOUN
iajs-830	295	12	over	over	ADP
iajs-830	295	13	fuzzy	fuzzy	ADJ
iajs-830	295	14	rings	ring	NOUN
iajs-830	295	15	,	,	PUNCT
iajs-830	295	16	bull	bull	NOUN
iajs-830	295	17	.	.	PUNCT
iajs-830	296	1	col.math	col.math	PROPN
iajs-830	296	2	.	.	PUNCT
iajs-830	297	1	soc	soc	PROPN
iajs-830	297	2	.	.	PROPN
iajs-830	297	3	,	,	PUNCT
iajs-830	297	4	81	81	NUM
iajs-830	297	5	,	,	PUNCT
iajs-830	297	6	197	197	NUM
iajs-830	297	7	-	-	SYM
iajs-830	297	8	200	200	NUM
iajs-830	297	9	.	.	NOUN
iajs-830	298	1	6	6	NUM
iajs-830	298	2	.	.	X
iajs-830	299	1	bhambert	bhambert	ADJ
iajs-830	299	2	,	,	PUNCT
iajs-830	299	3	s.k	s.k	PROPN
iajs-830	299	4	.	.	PROPN
iajs-830	299	5	;	;	PUNCT
iajs-830	300	1	kumar	kumar	PROPN
iajs-830	300	2	and	and	CCONJ
iajs-830	300	3	kumar	kumar	PROPN
iajs-830	300	4	p.	p.	PROPN
iajs-830	300	5	(	(	PUNCT
iajs-830	300	6	1995	1995	NUM
iajs-830	300	7	)	)	PUNCT
iajs-830	300	8	,	,	PUNCT
iajs-830	300	9	fuzzy	fuzzy	ADJ
iajs-830	300	10	prime	prime	ADJ
iajs-830	300	11	submodule	submodule	NOUN
iajs-830	300	12	and	and	CCONJ
iajs-830	300	13	radical	radical	ADJ
iajs-830	300	14	of	of	ADP
iajs-830	300	15	fuzzy	fuzzy	ADJ
iajs-830	300	16	submodules	submodule	NOUN
iajs-830	300	17	,	,	PUNCT
iajs-830	300	18	bull	bull	NOUN
iajs-830	300	19	.	.	PUNCT
iajs-830	300	20	soc	soc	PROPN
iajs-830	300	21	.	.	PROPN
iajs-830	300	22	,	,	PUNCT
iajs-830	300	23	87	87	NUM
iajs-830	300	24	,	,	PUNCT
iajs-830	300	25	163	163	NUM
iajs-830	300	26	-	-	SYM
iajs-830	300	27	168	168	NUM
iajs-830	300	28	.	.	PUNCT
iajs-830	301	1	7	7	X
iajs-830	301	2	.	.	X
iajs-830	301	3	mukherjee	mukherjee	NOUN
iajs-830	301	4	,	,	PUNCT
iajs-830	301	5	t.	t.	PROPN
iajs-830	301	6	k.	k.	PROPN
iajs-830	301	7	;	;	PUNCT
iajs-830	301	8	sen	sen	PROPN
iajs-830	301	9	,	,	PUNCT
iajs-830	301	10	m.	m.	PROPN
iajs-830	301	11	k.	k.	PROPN
iajs-830	301	12	and	and	CCONJ
iajs-830	301	13	roy	roy	PROPN
iajs-830	301	14	,	,	PUNCT
iajs-830	301	15	d.	d.	PROPN
iajs-830	301	16	(	(	PUNCT
iajs-830	301	17	1996	1996	NUM
iajs-830	301	18	)	)	PUNCT
iajs-830	301	19	,	,	PUNCT
iajs-830	301	20	n	n	X
iajs-830	301	21	fuzzy	fuzzy	ADJ
iajs-830	301	22	submodules	submodule	NOUN
iajs-830	301	23	and	and	CCONJ
iajs-830	301	24	their	their	PRON
iajs-830	301	25	radicals	radical	NOUN
iajs-830	301	26	,	,	PUNCT
iajs-830	301	27	j.	j.	PROPN
iajs-830	301	28	fuzzy	fuzzy	PROPN
iajs-830	301	29	math	math	PROPN
iajs-830	301	30	.	.	PUNCT
iajs-830	301	31	,	,	PUNCT
iajs-830	301	32	4	4	NUM
iajs-830	301	33	,	,	PUNCT
iajs-830	301	34	549	549	NUM
iajs-830	301	35	-	-	SYM
iajs-830	301	36	558	558	NUM
iajs-830	301	37	.	.	NOUN
iajs-830	301	38	8	8	NUM
iajs-830	301	39	.	.	X
iajs-830	302	1	liu	liu	PROPN
iajs-830	302	2	,	,	PUNCT
iajs-830	302	3	w.j	w.j	PROPN
iajs-830	302	4	.	.	PROPN
iajs-830	302	5	(	(	PUNCT
iajs-830	302	6	1982	1982	NUM
iajs-830	302	7	)	)	PUNCT
iajs-830	302	8	,	,	PUNCT
iajs-830	302	9	fuzzy	fuzzy	ADJ
iajs-830	302	10	invariant	invariant	ADJ
iajs-830	302	11	subgroups	subgroup	NOUN
iajs-830	302	12	and	and	CCONJ
iajs-830	302	13	fuzzy	fuzzy	ADJ
iajs-830	302	14	ideals	ideal	NOUN
iajs-830	302	15	,	,	PUNCT
iajs-830	302	16	fuzzy	fuzzy	ADJ
iajs-830	302	17	sets	set	NOUN
iajs-830	302	18	and	and	CCONJ
iajs-830	302	19	systems	system	NOUN
iajs-830	302	20	,	,	PUNCT
iajs-830	302	21	8	8	NUM
iajs-830	302	22	,	,	PUNCT
iajs-830	302	23	133	133	NUM
iajs-830	302	24	-	-	SYM
iajs-830	302	25	139	139	NUM
iajs-830	302	26	.	.	PUNCT
iajs-830	303	1	9	9	X
iajs-830	303	2	.	.	X
iajs-830	303	3	martine	martine	PROPN
iajs-830	303	4	,	,	PUNCT
iajs-830	303	5	l.	l.	PROPN
iajs-830	303	6	(	(	PUNCT
iajs-830	303	7	1995	1995	NUM
iajs-830	303	8	)	)	PUNCT
iajs-830	303	9	,	,	PUNCT
iajs-830	303	10	fuzzy	fuzzy	ADJ
iajs-830	303	11	subgroups	subgroup	NOUN
iajs-830	303	12	of	of	ADP
iajs-830	303	13	fuzzy	fuzzy	ADJ
iajs-830	303	14	groups	group	NOUN
iajs-830	303	15	and	and	CCONJ
iajs-830	303	16	fuzzy	fuzzy	ADJ
iajs-830	303	17	ideals	ideal	NOUN
iajs-830	303	18	of	of	ADP
iajs-830	303	19	fuzzy	fuzzy	ADJ
iajs-830	303	20	ring	ring	NOUN
iajs-830	303	21	,	,	PUNCT
iajs-830	303	22	the	the	DET
iajs-830	303	23	journal	journal	NOUN
iajs-830	303	24	of	of	ADP
iajs-830	303	25	fuzzy	fuzzy	ADJ
iajs-830	303	26	math	math	NOUN
iajs-830	303	27	.	.	PUNCT
iajs-830	303	28	,	,	PUNCT
iajs-830	303	29	3:(4	3:(4	NUM
iajs-830	303	30	)	)	PUNCT
iajs-830	303	31	,	,	PUNCT
iajs-830	303	32	883	883	NUM
iajs-830	303	33	-	-	SYM
iajs-830	303	34	849	849	NUM
iajs-830	303	35	.	.	NOUN
iajs-830	303	36	10	10	NUM
iajs-830	303	37	.	.	PUNCT
iajs-830	304	1	hadi	hadi	PROPN
iajs-830	304	2	,	,	PUNCT
iajs-830	304	3	m.a	m.a	PROPN
iajs-830	304	4	.	.	PROPN
iajs-830	304	5	(	(	PUNCT
iajs-830	304	6	2001	2001	NUM
iajs-830	304	7	)	)	PUNCT
iajs-830	304	8	,	,	PUNCT
iajs-830	304	9	on	on	ADP
iajs-830	304	10	fuzzy	fuzzy	ADJ
iajs-830	304	11	ideals	ideal	NOUN
iajs-830	304	12	of	of	ADP
iajs-830	304	13	fuzzy	fuzzy	ADJ
iajs-830	304	14	rings	ring	NOUN
iajs-830	304	15	,	,	PUNCT
iajs-830	304	16	16:(4	16:(4	NUM
iajs-830	304	17	)	)	PUNCT
iajs-830	304	18	,	,	PUNCT
iajs-830	304	19	17	17	NUM
iajs-830	304	20	-	-	SYM
iajs-830	304	21	33	33	NUM
iajs-830	304	22	.	.	PUNCT
iajs-830	304	23	11	11	NUM
iajs-830	304	24	.	.	PUNCT
iajs-830	305	1	rabi	rabi	NOUN
iajs-830	305	2	,	,	PUNCT
iajs-830	305	3	h.	h.	PROPN
iajs-830	305	4	j.	j.	PROPN
iajs-830	305	5	(	(	PUNCT
iajs-830	305	6	2001	2001	NUM
iajs-830	305	7	)	)	PUNCT
iajs-830	305	8	,	,	PUNCT
iajs-830	305	9	prime	prime	ADJ
iajs-830	305	10	fuzzy	fuzzy	ADJ
iajs-830	305	11	submodules	submodule	NOUN
iajs-830	305	12	and	and	CCONJ
iajs-830	305	13	prime	prime	ADJ
iajs-830	305	14	fuzzy	fuzzy	ADJ
iajs-830	305	15	modules	module	NOUN
iajs-830	305	16	,	,	PUNCT
iajs-830	305	17	m	m	PROPN
iajs-830	305	18	.sc	.sc	NOUN
iajs-830	305	19	.	.	PUNCT
iajs-830	306	1	thesis	thesis	NOUN
iajs-830	306	2	,	,	PUNCT
iajs-830	306	3	university	university	NOUN
iajs-830	306	4	of	of	ADP
iajs-830	306	5	baghdad	baghdad	PROPN
iajs-830	306	6	.	.	PUNCT
iajs-830	307	1	12	12	NUM
iajs-830	307	2	.	.	PUNCT
iajs-830	308	1	mohmad	mohmad	PROPN
iajs-830	308	2	,	,	PUNCT
iajs-830	308	3	a.a	a.a	PROPN
iajs-830	308	4	.	.	PROPN
iajs-830	308	5	(	(	PUNCT
iajs-830	308	6	1997	1997	NUM
iajs-830	308	7	)	)	PUNCT
iajs-830	308	8	,	,	PUNCT
iajs-830	308	9	chained	chain	VERB
iajs-830	308	10	modules	module	NOUN
iajs-830	308	11	,	,	PUNCT
iajs-830	308	12	m	m	PROPN
iajs-830	308	13	.sc	.sc	NOUN
iajs-830	308	14	.	.	PUNCT
iajs-830	309	1	thesis	thesis	PROPN
iajs-830	309	2	university	university	PROPN
iajs-830	309	3	of	of	ADP
iajs-830	309	4	baghdad	baghdad	PROPN
iajs-830	309	5	.	.	PUNCT
iajs-830	310	1	13	13	NUM
iajs-830	310	2	.	.	PUNCT
iajs-830	310	3	shores	shore	NOUN
iajs-830	310	4	,	,	PUNCT
iajs-830	310	5	t.s	t.s	PROPN
iajs-830	310	6	.	.	PROPN
iajs-830	310	7	and	and	CCONJ
iajs-830	310	8	lewis	lewis	PROPN
iajs-830	310	9	,	,	PUNCT
iajs-830	310	10	w.j	w.j	PROPN
iajs-830	310	11	.	.	PROPN
iajs-830	310	12	(	(	PUNCT
iajs-830	310	13	1974	1974	NUM
iajs-830	310	14	)	)	PUNCT
iajs-830	310	15	,	,	PUNCT
iajs-830	310	16	serial	serial	ADJ
iajs-830	310	17	modules	module	NOUN
iajs-830	310	18	and	and	CCONJ
iajs-830	310	19	endomorphism	endomorphism	PROPN
iajs-830	310	20	rings	ring	NOUN
iajs-830	310	21	,	,	PUNCT
iajs-830	310	22	duke	duke	PROPN
iajs-830	310	23	math	math	PROPN
iajs-830	310	24	.	.	PUNCT
iajs-830	311	1	j.	j.	PROPN
iajs-830	311	2	,	,	PUNCT
iajs-830	311	3	41	41	NUM
iajs-830	311	4	,	,	PUNCT
iajs-830	311	5	889	889	NUM
iajs-830	311	6	-	-	SYM
iajs-830	311	7	909	909	NUM
iajs-830	311	8	.	.	PUNCT
iajs-830	311	9	14	14	NUM
iajs-830	311	10	.	.	PUNCT
iajs-830	312	1	kumar	kumar	PROPN
iajs-830	312	2	,	,	PUNCT
iajs-830	312	3	r.	r.	PROPN
iajs-830	312	4	(	(	PUNCT
iajs-830	312	5	1991	1991	NUM
iajs-830	312	6	)	)	PUNCT
iajs-830	312	7	,	,	PUNCT
iajs-830	312	8	fuzzy	fuzzy	ADJ
iajs-830	312	9	semi	semi	ADJ
iajs-830	312	10	-	-	ADJ
iajs-830	312	11	primary	primary	ADJ
iajs-830	312	12	ideals	ideal	NOUN
iajs-830	312	13	of	of	ADP
iajs-830	312	14	rings	ring	NOUN
iajs-830	312	15	,	,	PUNCT
iajs-830	312	16	fuzzy	fuzzy	ADJ
iajs-830	312	17	sets	set	NOUN
iajs-830	312	18	and	and	CCONJ
iajs-830	312	19	systems	system	NOUN
iajs-830	312	20	,	,	PUNCT
iajs-830	312	21	42	42	NUM
iajs-830	312	22	,	,	PUNCT
iajs-830	312	23	263	263	NUM
iajs-830	312	24	-	-	SYM
iajs-830	312	25	272	272	NUM
iajs-830	312	26	.	.	PUNCT
iajs-830	312	27	15	15	NUM
iajs-830	312	28	.	.	PUNCT
iajs-830	313	1	zhao	zhao	PROPN
iajs-830	313	2	,	,	PUNCT
iajs-830	313	3	jiandi	jiandi	PROPN
iajs-830	313	4	,	,	PUNCT
iajs-830	313	5	shik	shik	PROPN
iajs-830	313	6	.	.	PUNCT
iajs-830	314	1	yue	yue	PROPN
iajs-830	314	2	m.	m.	PROPN
iajs-830	314	3	(	(	PUNCT
iajs-830	314	4	1993	1993	NUM
iajs-830	314	5	)	)	PUNCT
iajs-830	314	6	,	,	PUNCT
iajs-830	314	7	fuzzy	fuzzy	ADJ
iajs-830	314	8	modules	module	NOUN
iajs-830	314	9	over	over	ADP
iajs-830	314	10	fuzzy	fuzzy	ADJ
iajs-830	314	11	rings	ring	NOUN
iajs-830	314	12	,	,	PUNCT
iajs-830	314	13	the	the	DET
iajs-830	314	14	j.	j.	NOUN
iajs-830	314	15	of	of	ADP
iajs-830	314	16	fuzzy	fuzzy	ADJ
iajs-830	314	17	math	math	NOUN
iajs-830	314	18	.	.	PUNCT
iajs-830	314	19	,	,	PUNCT
iajs-830	314	20	3	3	NUM
iajs-830	314	21	,	,	PUNCT
iajs-830	314	22	531	531	NUM
iajs-830	314	23	-	-	SYM
iajs-830	314	24	540	540	NUM
iajs-830	314	25	.	.	PUNCT
iajs-830	315	1	16	16	NUM
iajs-830	315	2	.	.	PUNCT
iajs-830	316	1	larson	larson	PROPN
iajs-830	316	2	,	,	PUNCT
iajs-830	316	3	m	m	VERB
iajs-830	316	4	.d	.d	ADJ
iajs-830	316	5	.	.	PUNCT
iajs-830	317	1	and	and	CCONJ
iajs-830	317	2	mccarthy	mccarthy	PROPN
iajs-830	317	3	,	,	PUNCT
iajs-830	317	4	p.j	p.j	PROPN
iajs-830	317	5	.	.	PROPN
iajs-830	317	6	(	(	PUNCT
iajs-830	317	7	1971	1971	NUM
iajs-830	317	8	)	)	PUNCT
iajs-830	317	9	,	,	PUNCT
iajs-830	317	10	multiplicutive	multiplicutive	ADJ
iajs-830	317	11	theory	theory	NOUN
iajs-830	317	12	of	of	ADP
iajs-830	317	13	ideals	ideal	NOUN
iajs-830	317	14	,	,	PUNCT
iajs-830	317	15	academic	academic	ADJ
iajs-830	317	16	press	press	NOUN
iajs-830	317	17	,	,	PUNCT
iajs-830	317	18	london	london	PROPN
iajs-830	317	19	,	,	PUNCT
iajs-830	317	20	new	new	ADJ
iajs-830	317	21	yourk	yourk	PROPN
iajs-830	317	22	.	.	PUNCT