id	sid	tid	token	lemma	pos
iajs-1045	1	1	ibn	ibn	PROPN
iajs-1045	1	2	alhaitham	alhaitham	NOUN
iajs-1045	1	3	j.	j.	PROPN
iajs-1045	1	4	for	for	ADP
iajs-1045	1	5	pure	pure	ADJ
iajs-1045	1	6	&	&	CCONJ
iajs-1045	1	7	appl	appl	PROPN
iajs-1045	1	8	.	.	PUNCT
iajs-1045	2	1	sci	sci	PROPN
iajs-1045	2	2	.	.	PUNCT
iajs-1045	3	1	vol.22	vol.22	PROPN
iajs-1045	3	2	(	(	PUNCT
iajs-1045	3	3	4	4	NUM
iajs-1045	3	4	)	)	PUNCT
iajs-1045	3	5	2009	2009	NUM
iajs-1045	3	6	a	a	DET
iajs-1045	3	7	space	space	NOUN
iajs-1045	3	8	of	of	ADP
iajs-1045	3	9	fuzzy	fuzzy	ADJ
iajs-1045	3	10	orderings	ordering	NOUN
iajs-1045	3	11	l.n.m.tawfiq	l.n.m.tawfiq	ADP
iajs-1045	3	12	department	department	NOUN
iajs-1045	3	13	of	of	ADP
iajs-1045	3	14	mathematics	mathematics	PROPN
iajs-1045	3	15	,	,	PUNCT
iajs-1045	3	16	college	college	NOUN
iajs-1045	3	17	of	of	ADP
iajs-1045	3	18	education	education	PROPN
iajs-1045	3	19	ibn	ibn	PROPN
iajs-1045	3	20	-	-	PUNCT
iajs-1045	3	21	al	al	PROPN
iajs-1045	3	22	-	-	PUNCT
iajs-1045	3	23	haitham	haitham	PROPN
iajs-1045	3	24	,	,	PUNCT
iajs-1045	3	25	university	university	PROPN
iajs-1045	3	26	of	of	ADP
iajs-1045	3	27	baghdad	baghdad	PROPN
iajs-1045	3	28	abstract	abstract	ADV
iajs-1045	3	29	in	in	ADP
iajs-1045	3	30	this	this	DET
iajs-1045	3	31	paper	paper	NOUN
iajs-1045	3	32	the	the	DET
iajs-1045	3	33	chain	chain	NOUN
iajs-1045	3	34	length	length	NOUN
iajs-1045	3	35	of	of	ADP
iajs-1045	3	36	a	a	DET
iajs-1045	3	37	space	space	NOUN
iajs-1045	3	38	of	of	ADP
iajs-1045	3	39	fuzzy	fuzzy	ADJ
iajs-1045	3	40	orderings	ordering	NOUN
iajs-1045	3	41	is	be	AUX
iajs-1045	3	42	defined	define	VERB
iajs-1045	3	43	,	,	PUNCT
iajs-1045	3	44	and	and	CCONJ
iajs-1045	3	45	various	various	ADJ
iajs-1045	3	46	properties	property	NOUN
iajs-1045	3	47	of	of	ADP
iajs-1045	3	48	this	this	DET
iajs-1045	3	49	invariant	invariant	NOUN
iajs-1045	3	50	are	be	AUX
iajs-1045	3	51	proved	prove	VERB
iajs-1045	3	52	.	.	PUNCT
iajs-1045	4	1	the	the	DET
iajs-1045	4	2	structure	structure	NOUN
iajs-1045	4	3	theorem	theorem	VERB
iajs-1045	4	4	for	for	ADP
iajs-1045	4	5	spaces	space	NOUN
iajs-1045	4	6	of	of	ADP
iajs-1045	4	7	finite	finite	PROPN
iajs-1045	4	8	chain	chain	NOUN
iajs-1045	4	9	length	length	NOUN
iajs-1045	4	10	is	be	AUX
iajs-1045	4	11	proved	prove	VERB
iajs-1045	4	12	.	.	PUNCT
iajs-1045	5	1	spaces	space	NOUN
iajs-1045	5	2	of	of	ADP
iajs-1045	5	3	fuzzy	fuzzy	ADJ
iajs-1045	5	4	orderings	ordering	NOUN
iajs-1045	5	5	throughout	throughout	ADP
iajs-1045	5	6	x	x	X
iajs-1045	5	7	=	=	SYM
iajs-1045	5	8	(	(	PUNCT
iajs-1045	5	9	x	x	X
iajs-1045	5	10	,	,	PUNCT
iajs-1045	5	11	a	a	PRON
iajs-1045	5	12	)	)	PUNCT
iajs-1045	5	13	denoted	denote	VERB
iajs-1045	5	14	a	a	DET
iajs-1045	5	15	space	space	NOUN
iajs-1045	5	16	of	of	ADP
iajs-1045	5	17	fuzzy	fuzzy	ADJ
iajs-1045	5	18	orderings	ordering	NOUN
iajs-1045	5	19	.	.	PUNCT
iajs-1045	6	1	that	that	PRON
iajs-1045	6	2	is	is	ADV
iajs-1045	6	3	,	,	PUNCT
iajs-1045	6	4	a	a	PRON
iajs-1045	6	5	is	be	AUX
iajs-1045	6	6	a	a	DET
iajs-1045	6	7	fuzzy	fuzzy	ADJ
iajs-1045	6	8	subgroup	subgroup	NOUN
iajs-1045	6	9	of	of	ADP
iajs-1045	6	10	abelian	abelian	PROPN
iajs-1045	6	11	group	group	PROPN
iajs-1045	6	12	g	g	PROPN
iajs-1045	6	13	of	of	ADP
iajs-1045	6	14	exponent	exponent	PROPN
iajs-1045	6	15	2	2	NUM
iajs-1045	6	16	.	.	PUNCT
iajs-1045	7	1	(	(	PUNCT
iajs-1045	7	2	see	see	VERB
iajs-1045	7	3	[	[	X
iajs-1045	7	4	1	1	X
iajs-1045	7	5	]	]	X
iajs-1045	7	6	(	(	PUNCT
iajs-1045	7	7	i.e.	i.e.	X
iajs-1045	7	8	x2	x2	X
iajs-1045	7	9	=	=	SYM
iajs-1045	7	10	1	1	NUM
iajs-1045	7	11	,	,	PUNCT
iajs-1045	7	12			NOUN
iajs-1045	7	13	x	x	SYM
iajs-1045	7	14			NOUN
iajs-1045	7	15	g	g	PROPN
iajs-1045	7	16	)	)	PUNCT
iajs-1045	7	17	,	,	PUNCT
iajs-1045	7	18	and	and	CCONJ
iajs-1045	7	19	x	x	X
iajs-1045	7	20	is	be	AUX
iajs-1045	7	21	a	a	DET
iajs-1045	7	22	(	(	PUNCT
iajs-1045	7	23	non	non	X
iajs-1045	7	24	empty	empty	ADJ
iajs-1045	7	25	)	)	PUNCT
iajs-1045	7	26	fuzzy	fuzzy	ADJ
iajs-1045	7	27	subset	subset	NOUN
iajs-1045	7	28	of	of	ADP
iajs-1045	7	29	the	the	DET
iajs-1045	7	30	character	character	NOUN
iajs-1045	7	31	group	group	NOUN
iajs-1045	7	32			NOUN
iajs-1045	7	33	(	(	PUNCT
iajs-1045	7	34	a	a	X
iajs-1045	7	35	)	)	PUNCT
iajs-1045	7	36	=	=	SYM
iajs-1045	7	37	hom(a,{1,–1	hom(a,{1,–1	PROPN
iajs-1045	7	38	}	}	PUNCT
iajs-1045	7	39	)	)	PUNCT
iajs-1045	8	1	satisfying	satisfy	VERB
iajs-1045	8	2	:	:	PUNCT
iajs-1045	8	3	1	1	X
iajs-1045	8	4	.	.	X
iajs-1045	8	5	x	x	PRON
iajs-1045	8	6	is	be	AUX
iajs-1045	8	7	a	a	DET
iajs-1045	8	8	fuzzy	fuzzy	ADJ
iajs-1045	8	9	closed	closed	ADJ
iajs-1045	8	10	subset	subset	NOUN
iajs-1045	8	11	of	of	ADP
iajs-1045	8	12			NOUN
iajs-1045	8	13	(	(	PUNCT
iajs-1045	8	14	a	a	X
iajs-1045	8	15	)	)	PUNCT
iajs-1045	8	16	.	.	PUNCT
iajs-1045	9	1	2	2	X
iajs-1045	9	2	.	.	X
iajs-1045	9	3			ADP
iajs-1045	9	4	an	an	DET
iajs-1045	9	5	element	element	NOUN
iajs-1045	9	6	e	e	NOUN
iajs-1045	9	7			NOUN
iajs-1045	9	8	a	a	DET
iajs-1045	9	9	such	such	ADJ
iajs-1045	9	10	that	that	DET
iajs-1045	9	11	(e	(e	PROPN
iajs-1045	9	12	)	)	PUNCT
iajs-1045	9	13	=	=	SYM
iajs-1045	9	14	–	–	PUNCT
iajs-1045	9	15	1	1	NUM
iajs-1045	9	16			NOUN
iajs-1045	9	17			PROPN
iajs-1045	9	18			NOUN
iajs-1045	9	19	x.	x.	NOUN
iajs-1045	9	20	3	3	X
iajs-1045	9	21	.	.	PUNCT
iajs-1045	10	1	x	x	PROPN
iajs-1045	11	1	:	:	PUNCT
iajs-1045	11	2	=	=	X
iajs-1045	11	3	{	{	PUNCT
iajs-1045	11	4	a	a	DET
iajs-1045	11	5			NOUN
iajs-1045	11	6	a\	a\	PROPN
iajs-1045	11	7	(a	(a	ADJ
iajs-1045	11	8	)	)	PUNCT
iajs-1045	11	9	=	=	SYM
iajs-1045	11	10	1	1	NUM
iajs-1045	11	11			NOUN
iajs-1045	11	12			PROPN
iajs-1045	11	13			NOUN
iajs-1045	11	14	x	x	NOUN
iajs-1045	11	15	}	}	PUNCT
iajs-1045	11	16	=	=	SYM
iajs-1045	11	17	1	1	NUM
iajs-1045	11	18	.	.	NOUN
iajs-1045	11	19	4	4	NUM
iajs-1045	11	20	.	.	X
iajs-1045	12	1	if	if	SCONJ
iajs-1045	12	2	f	f	PROPN
iajs-1045	12	3	and	and	CCONJ
iajs-1045	12	4	g	g	PROPN
iajs-1045	12	5	are	be	AUX
iajs-1045	12	6	forms	form	NOUN
iajs-1045	12	7	over	over	ADP
iajs-1045	12	8	a	a	PRON
iajs-1045	12	9	and	and	CCONJ
iajs-1045	13	1	if	if	SCONJ
iajs-1045	13	2	x	x	X
iajs-1045	13	3			NOUN
iajs-1045	13	4	d	d	X
iajs-1045	13	5	(	(	PUNCT
iajs-1045	13	6	f	f	PROPN
iajs-1045	13	7			PROPN
iajs-1045	13	8	g	g	PROPN
iajs-1045	13	9	)	)	PUNCT
iajs-1045	13	10	then	then	ADV
iajs-1045	13	11			VERB
iajs-1045	13	12	y	y	PROPN
iajs-1045	13	13			NOUN
iajs-1045	13	14	d	d	PROPN
iajs-1045	13	15	(	(	PUNCT
iajs-1045	13	16	f	f	PROPN
iajs-1045	13	17	)	)	PUNCT
iajs-1045	13	18	and	and	CCONJ
iajs-1045	13	19	z	z	NOUN
iajs-1045	13	20			PROPN
iajs-1045	13	21	d(g	d(g	PROPN
iajs-1045	13	22	)	)	PUNCT
iajs-1045	13	23	such	such	ADJ
iajs-1045	13	24	that	that	SCONJ
iajs-1045	13	25	x	x	SYM
iajs-1045	13	26			NOUN
iajs-1045	13	27	d	d	X
iajs-1045	13	28	<	<	X
iajs-1045	13	29	y	y	PROPN
iajs-1045	13	30	,	,	PUNCT
iajs-1045	13	31	z	z	NOUN
iajs-1045	13	32	>	>	X
iajs-1045	13	33	.	.	PUNCT
iajs-1045	14	1	observe	observe	VERB
iajs-1045	14	2	,	,	PUNCT
iajs-1045	14	3	by	by	ADP
iajs-1045	14	4	3	3	NUM
iajs-1045	14	5	,	,	PUNCT
iajs-1045	14	6	that	that	SCONJ
iajs-1045	14	7	the	the	DET
iajs-1045	14	8	element	element	NOUN
iajs-1045	14	9	e	e	PROPN
iajs-1045	14	10			NOUN
iajs-1045	14	11	a	a	PRON
iajs-1045	14	12	whose	whose	DET
iajs-1045	14	13	existence	existence	NOUN
iajs-1045	14	14	is	be	AUX
iajs-1045	14	15	asserted	assert	VERB
iajs-1045	14	16	by	by	ADP
iajs-1045	14	17	2	2	NUM
iajs-1045	14	18	is	be	AUX
iajs-1045	14	19	unique	unique	ADJ
iajs-1045	14	20	.	.	PUNCT
iajs-1045	15	1	also	also	ADV
iajs-1045	15	2	,	,	PUNCT
iajs-1045	15	3	e	e	X
iajs-1045	15	4			NOUN
iajs-1045	15	5	1	1	NUM
iajs-1045	15	6	(	(	PUNCT
iajs-1045	15	7	since	since	SCONJ
iajs-1045	15	8	(1	(1	VERB
iajs-1045	15	9	)	)	PUNCT
iajs-1045	15	10	=	=	SYM
iajs-1045	15	11	1	1	NUM
iajs-1045	15	12			NOUN
iajs-1045	15	13			PROPN
iajs-1045	15	14			NOUN
iajs-1045	15	15	x	x	NOUN
iajs-1045	15	16	)	)	PUNCT
iajs-1045	15	17	.	.	PUNCT
iajs-1045	16	1	notice	notice	VERB
iajs-1045	16	2	that	that	SCONJ
iajs-1045	16	3	for	for	ADP
iajs-1045	16	4	a	a	DET
iajs-1045	16	5			NOUN
iajs-1045	16	6	a	a	PRON
iajs-1045	16	7	,	,	PUNCT
iajs-1045	16	8	the	the	DET
iajs-1045	16	9	set	set	NOUN
iajs-1045	16	10	x(a):=	x(a):=	X
iajs-1045	16	11	{	{	PUNCT
iajs-1045	16	12			PROPN
iajs-1045	16	13			NOUN
iajs-1045	16	14	x(a	x(a	PUNCT
iajs-1045	16	15	)	)	PUNCT
iajs-1045	17	1	=	=	SYM
iajs-1045	17	2	1	1	X
iajs-1045	17	3	}	}	PUNCT
iajs-1045	17	4	is	be	AUX
iajs-1045	17	5	clopen	clopen	ADJ
iajs-1045	17	6	(	(	PUNCT
iajs-1045	17	7	i.e.	i.e.	X
iajs-1045	17	8	both	both	PRON
iajs-1045	17	9	closed	closed	ADJ
iajs-1045	17	10	and	and	CCONJ
iajs-1045	17	11	open	open	ADJ
iajs-1045	17	12	)	)	PUNCT
iajs-1045	17	13	in	in	ADP
iajs-1045	17	14	x.	x.	NOUN
iajs-1045	17	15	moreover	moreover	ADV
iajs-1045	17	16	,	,	PUNCT
iajs-1045	17	17	(a	(a	PROPN
iajs-1045	17	18	)	)	PUNCT
iajs-1045	17	19	=	=	PUNCT
iajs-1045	17	20	–	–	PUNCT
iajs-1045	17	21	1	1	NUM
iajs-1045	17	22			X
iajs-1045	17	23			PROPN
iajs-1045	17	24	(	(	PUNCT
iajs-1045	17	25	–	–	PUNCT
iajs-1045	17	26	a	a	X
iajs-1045	17	27	)	)	PUNCT
iajs-1045	17	28	=	=	SYM
iajs-1045	17	29	1	1	NUM
iajs-1045	17	30	holds	hold	VERB
iajs-1045	17	31	for	for	ADP
iajs-1045	17	32	any	any	DET
iajs-1045	17	33			PROPN
iajs-1045	17	34			NOUN
iajs-1045	17	35	x	x	X
iajs-1045	17	36	(	(	PUNCT
iajs-1045	17	37	by	by	ADP
iajs-1045	17	38	2	2	NUM
iajs-1045	17	39	)	)	PUNCT
iajs-1045	17	40	.	.	PUNCT
iajs-1045	18	1	definition	definition	NOUN
iajs-1045	18	2	1	1	NUM
iajs-1045	18	3	a	a	DET
iajs-1045	18	4	forms	form	NOUN
iajs-1045	18	5	f	f	PROPN
iajs-1045	18	6	and	and	CCONJ
iajs-1045	18	7	g	g	PROPN
iajs-1045	18	8	are	be	AUX
iajs-1045	18	9	said	say	VERB
iajs-1045	18	10	to	to	PART
iajs-1045	18	11	be	be	AUX
iajs-1045	18	12	isometric	isometric	ADJ
iajs-1045	18	13	(	(	PUNCT
iajs-1045	18	14	over	over	ADP
iajs-1045	18	15	x	x	X
iajs-1045	18	16	)	)	PUNCT
iajs-1045	18	17	if	if	SCONJ
iajs-1045	18	18	they	they	PRON
iajs-1045	18	19	have	have	VERB
iajs-1045	18	20	the	the	DET
iajs-1045	18	21	same	same	ADJ
iajs-1045	18	22	dimension	dimension	NOUN
iajs-1045	18	23	and	and	CCONJ
iajs-1045	18	24	(f	(f	NOUN
iajs-1045	18	25	)	)	PUNCT
iajs-1045	19	1	=	=	SYM
iajs-1045	19	2	(g	(g	NOUN
iajs-1045	19	3	)	)	PUNCT
iajs-1045	19	4			NOUN
iajs-1045	19	5			PROPN
iajs-1045	19	6			NOUN
iajs-1045	19	7	x.	x.	NOUN
iajs-1045	20	1	this	this	PRON
iajs-1045	20	2	is	be	AUX
iajs-1045	20	3	denoted	denote	VERB
iajs-1045	20	4	by	by	ADP
iajs-1045	20	5	writing	write	VERB
iajs-1045	20	6	f	f	PRON
iajs-1045	20	7			PROPN
iajs-1045	20	8	g	g	PROPN
iajs-1045	20	9	or	or	CCONJ
iajs-1045	20	10	g	g	PROPN
iajs-1045	20	11			PROPN
iajs-1045	20	12	f	f	PROPN
iajs-1045	20	13	(	(	PUNCT
iajs-1045	20	14	over	over	ADP
iajs-1045	20	15	x	x	NOUN
iajs-1045	20	16	)	)	PUNCT
iajs-1045	20	17	.	.	PUNCT
iajs-1045	21	1	note	note	VERB
iajs-1045	21	2	a	a	DET
iajs-1045	21	3	form	form	NOUN
iajs-1045	21	4	f	f	NOUN
iajs-1045	21	5	is	be	AUX
iajs-1045	21	6	said	say	VERB
iajs-1045	21	7	to	to	PART
iajs-1045	21	8	represent	represent	VERB
iajs-1045	21	9	the	the	DET
iajs-1045	21	10	element	element	NOUN
iajs-1045	21	11	x	x	PROPN
iajs-1045	21	12			PROPN
iajs-1045	21	13	a	a	X
iajs-1045	21	14	(	(	PUNCT
iajs-1045	21	15	over	over	ADP
iajs-1045	21	16	x	x	NOUN
iajs-1045	21	17	)	)	PUNCT
iajs-1045	21	18	if	if	SCONJ
iajs-1045	21	19			ADP
iajs-1045	21	20	elements	element	NOUN
iajs-1045	21	21	x1,	x1,	NOUN
iajs-1045	21	22	…	…	PUNCT
iajs-1045	21	23	,xn	,xn	PUNCT
iajs-1045	21	24			NOUN
iajs-1045	21	25	a	a	DET
iajs-1045	21	26	such	such	ADJ
iajs-1045	21	27	that	that	SCONJ
iajs-1045	21	28	f	f	X
iajs-1045	21	29			X
iajs-1045	21	30	<	<	X
iajs-1045	21	31	x	x	SYM
iajs-1045	21	32	,	,	PUNCT
iajs-1045	21	33	x2	x2	PROPN
iajs-1045	21	34	,	,	PUNCT
iajs-1045	21	35	…	…	PUNCT
iajs-1045	21	36	,	,	PUNCT
iajs-1045	21	37	xn	xn	PROPN
iajs-1045	21	38	>	>	X
iajs-1045	21	39			ADJ
iajs-1045	21	40	d(f	d(f	NOUN
iajs-1045	21	41	)	)	PUNCT
iajs-1045	21	42	or	or	CCONJ
iajs-1045	21	43	d(f	d(f	NOUN
iajs-1045	21	44	,	,	PUNCT
iajs-1045	21	45	x	x	X
iajs-1045	21	46	)	)	PUNCT
iajs-1045	21	47	will	will	AUX
iajs-1045	21	48	be	be	AUX
iajs-1045	21	49	used	use	VERB
iajs-1045	21	50	to	to	PART
iajs-1045	21	51	denote	denote	VERB
iajs-1045	21	52	the	the	DET
iajs-1045	21	53	set	set	NOUN
iajs-1045	21	54	of	of	ADP
iajs-1045	21	55	elements	element	NOUN
iajs-1045	21	56	of	of	ADP
iajs-1045	21	57	a	a	PRON
iajs-1045	21	58	which	which	PRON
iajs-1045	21	59	are	be	AUX
iajs-1045	21	60	represented	represent	VERB
iajs-1045	21	61	by	by	ADP
iajs-1045	21	62	f	f	PROPN
iajs-1045	21	63	in	in	ADP
iajs-1045	21	64	this	this	DET
iajs-1045	21	65	sense	sense	NOUN
iajs-1045	21	66	.	.	PUNCT
iajs-1045	22	1	definition	definition	NOUN
iajs-1045	22	2	2	2	NUM
iajs-1045	22	3	a	a	DET
iajs-1045	22	4	form	form	NOUN
iajs-1045	22	5	f	f	PROPN
iajs-1045	22	6	is	be	AUX
iajs-1045	22	7	said	say	VERB
iajs-1045	22	8	to	to	PART
iajs-1045	22	9	be	be	AUX
iajs-1045	22	10	isotropic	isotropic	ADJ
iajs-1045	22	11	if	if	SCONJ
iajs-1045	22	12			PROPN
iajs-1045	22	13	x3	x3	ADJ
iajs-1045	22	14	,	,	PUNCT
iajs-1045	22	15	…	…	PUNCT
iajs-1045	22	16	,	,	PUNCT
iajs-1045	22	17	xn	xn	PROPN
iajs-1045	23	1			PROPN
iajs-1045	23	2	a	a	PROPN
iajs-1045	23	3	,	,	PUNCT
iajs-1045	23	4	such	such	ADJ
iajs-1045	23	5	that	that	SCONJ
iajs-1045	23	6	f	f	X
iajs-1045	23	7			X
iajs-1045	23	8	<	<	X
iajs-1045	23	9	1,–1	1,–1	NUM
iajs-1045	23	10	,	,	PUNCT
iajs-1045	23	11	x3	x3	ADJ
iajs-1045	23	12	,	,	PUNCT
iajs-1045	23	13	…	…	PUNCT
iajs-1045	23	14	,	,	PUNCT
iajs-1045	23	15	xn	xn	PROPN
iajs-1045	23	16	>	>	PUNCT
iajs-1045	23	17	.	.	PUNCT
iajs-1045	24	1	notice	notice	PROPN
iajs-1045	24	2	,	,	PUNCT
iajs-1045	24	3	in	in	ADP
iajs-1045	24	4	particular	particular	ADJ
iajs-1045	24	5	,	,	PUNCT
iajs-1045	24	6	this	this	PRON
iajs-1045	24	7	implies	imply	VERB
iajs-1045	24	8	dim(f	dim(f	PROPN
iajs-1045	24	9	)	)	PUNCT
iajs-1045	24	10			PROPN
iajs-1045	24	11	2	2	NUM
iajs-1045	24	12	.	.	PUNCT
iajs-1045	24	13	a	a	DET
iajs-1045	24	14	form	form	NOUN
iajs-1045	24	15	which	which	PRON
iajs-1045	24	16	is	be	AUX
iajs-1045	24	17	not	not	PART
iajs-1045	24	18	isotropic	isotropic	NOUN
iajs-1045	24	19	is	be	AUX
iajs-1045	24	20	said	say	VERB
iajs-1045	24	21	to	to	PART
iajs-1045	24	22	be	be	AUX
iajs-1045	24	23	anisotropic	anisotropic	NOUN
iajs-1045	24	24	,	,	PUNCT
iajs-1045	24	25	for	for	ADP
iajs-1045	24	26	any	any	DET
iajs-1045	24	27	x	x	NOUN
iajs-1045	24	28			NOUN
iajs-1045	24	29	a	a	PROPN
iajs-1045	24	30	,	,	PUNCT
iajs-1045	24	31	<	<	X
iajs-1045	24	32	x	x	X
iajs-1045	24	33	,	,	PUNCT
iajs-1045	24	34	–	–	PUNCT
iajs-1045	24	35	x	x	X
iajs-1045	24	36	>	>	X
iajs-1045	24	37			PROPN
iajs-1045	24	38	<	<	X
iajs-1045	24	39	1,–1	1,–1	NUM
iajs-1045	24	40	>	>	X
iajs-1045	24	41	.	.	PUNCT
iajs-1045	25	1	any	any	DET
iajs-1045	25	2	such	such	ADJ
iajs-1045	25	3	form	form	NOUN
iajs-1045	25	4	will	will	AUX
iajs-1045	25	5	be	be	AUX
iajs-1045	25	6	called	call	VERB
iajs-1045	25	7	a	a	DET
iajs-1045	25	8	hyperbolic	hyperbolic	ADJ
iajs-1045	25	9	plane	plane	NOUN
iajs-1045	25	10	.	.	PUNCT
iajs-1045	26	1	theorem	theorem	VERB
iajs-1045	26	2	1	1	NUM
iajs-1045	26	3	the	the	DET
iajs-1045	26	4	following	following	NOUN
iajs-1045	26	5	are	be	AUX
iajs-1045	26	6	equivalent	equivalent	ADJ
iajs-1045	26	7	(	(	PUNCT
iajs-1045	26	8	i	i	NOUN
iajs-1045	26	9	)	)	PUNCT
iajs-1045	26	10			NOUN
iajs-1045	26	11	x	x	PUNCT
iajs-1045	26	12			NOUN
iajs-1045	26	13	g	g	NOUN
iajs-1045	26	14	,	,	PUNCT
iajs-1045	26	15	x	x	SYM
iajs-1045	26	16			NOUN
iajs-1045	26	17	–	–	PUNCT
iajs-1045	26	18	1	1	NUM
iajs-1045	26	19			NOUN
iajs-1045	26	20	d<1	d<1	VERB
iajs-1045	26	21	,	,	PUNCT
iajs-1045	26	22	x	x	X
iajs-1045	26	23	>	>	X
iajs-1045	26	24	=	=	PUNCT
iajs-1045	26	25	{	{	PUNCT
iajs-1045	26	26	1	1	NUM
iajs-1045	26	27	,	,	PUNCT
iajs-1045	26	28	x	x	NOUN
iajs-1045	26	29	}	}	PUNCT
iajs-1045	26	30	.	.	PUNCT
iajs-1045	27	1	(	(	PUNCT
iajs-1045	27	2	ii	ii	NOUN
iajs-1045	27	3	)	)	PUNCT
iajs-1045	27	4	x	x	X
iajs-1045	28	1	=	=	PRON
iajs-1045	28	2	{	{	PUNCT
iajs-1045	28	3			X
iajs-1045	28	4			NOUN
iajs-1045	28	5			NOUN
iajs-1045	28	6	(	(	PUNCT
iajs-1045	28	7	a)(–1	a)(–1	ADJ
iajs-1045	28	8	)	)	PUNCT
iajs-1045	28	9	=	=	PUNCT
iajs-1045	28	10	–	–	PUNCT
iajs-1045	28	11	1	1	NUM
iajs-1045	28	12	}	}	PUNCT
iajs-1045	28	13	.	.	PUNCT
iajs-1045	29	1	proof	proof	NOUN
iajs-1045	29	2	:	:	PUNCT
iajs-1045	29	3	see	see	VERB
iajs-1045	29	4	[	[	X
iajs-1045	29	5	3	3	NUM
iajs-1045	29	6	]	]	PUNCT
iajs-1045	29	7	.	.	PUNCT
iajs-1045	30	1	a	a	DET
iajs-1045	30	2	space	space	NOUN
iajs-1045	30	3	of	of	ADP
iajs-1045	30	4	fuzzy	fuzzy	ADJ
iajs-1045	30	5	ordering	ordering	NOUN
iajs-1045	30	6	satisfying	satisfy	VERB
iajs-1045	30	7	either	either	PRON
iajs-1045	30	8	of	of	ADP
iajs-1045	30	9	the	the	DET
iajs-1045	30	10	equivalent	equivalent	ADJ
iajs-1045	30	11	conditions	condition	NOUN
iajs-1045	30	12	in	in	ADP
iajs-1045	30	13	theorem	theorem	NOUN
iajs-1045	30	14	1	1	NUM
iajs-1045	30	15	will	will	AUX
iajs-1045	30	16	be	be	AUX
iajs-1045	30	17	referred	refer	VERB
iajs-1045	30	18	to	to	ADP
iajs-1045	30	19	as	as	ADP
iajs-1045	30	20	a	a	DET
iajs-1045	30	21	fan	fan	NOUN
iajs-1045	30	22	.	.	PUNCT
iajs-1045	31	1	ibn	ibn	PROPN
iajs-1045	31	2	alhaitham	alhaitham	PROPN
iajs-1045	31	3	j.	j.	PROPN
iajs-1045	31	4	for	for	ADP
iajs-1045	31	5	pure	pure	ADJ
iajs-1045	31	6	&	&	CCONJ
iajs-1045	31	7	appl	appl	PROPN
iajs-1045	31	8	.	.	PUNCT
iajs-1045	32	1	sci	sci	PROPN
iajs-1045	32	2	.	.	PUNCT
iajs-1045	33	1	vol.22	vol.22	PROPN
iajs-1045	33	2	(	(	PUNCT
iajs-1045	33	3	4	4	NUM
iajs-1045	33	4	)	)	PUNCT
iajs-1045	33	5	2009	2009	NUM
iajs-1045	33	6	corollary	corollary	NOUN
iajs-1045	33	7	1	1	NUM
iajs-1045	33	8	suppose	suppose	VERB
iajs-1045	33	9	x	x	PRON
iajs-1045	33	10	is	be	AUX
iajs-1045	33	11	a	a	DET
iajs-1045	33	12	fan	fan	NOUN
iajs-1045	33	13	.	.	PUNCT
iajs-1045	34	1	then	then	ADV
iajs-1045	34	2	every	every	DET
iajs-1045	34	3	subspace	subspace	NOUN
iajs-1045	34	4	of	of	ADP
iajs-1045	34	5	x	x	PUNCT
iajs-1045	34	6	is	be	AUX
iajs-1045	34	7	also	also	ADV
iajs-1045	34	8	a	a	DET
iajs-1045	34	9	fan	fan	NOUN
iajs-1045	34	10	.	.	PUNCT
iajs-1045	35	1	proof	proof	NOUN
iajs-1045	35	2	:	:	PUNCT
iajs-1045	35	3	compare	compare	VERB
iajs-1045	35	4	[	[	X
iajs-1045	35	5	3	3	NUM
iajs-1045	35	6	]	]	PUNCT
iajs-1045	35	7	.	.	PUNCT
iajs-1045	36	1	recall	recall	PROPN
iajs-1045	36	2	,	,	PUNCT
iajs-1045	36	3	a	a	DET
iajs-1045	36	4	space	space	NOUN
iajs-1045	36	5	of	of	ADP
iajs-1045	36	6	fuzzy	fuzzy	ADJ
iajs-1045	36	7	orderings	ordering	NOUN
iajs-1045	36	8	(	(	PUNCT
iajs-1045	36	9	x	x	X
iajs-1045	36	10	,	,	PUNCT
iajs-1045	36	11	a	a	PRON
iajs-1045	36	12	)	)	PUNCT
iajs-1045	36	13	is	be	AUX
iajs-1045	36	14	said	say	VERB
iajs-1045	36	15	to	to	PART
iajs-1045	36	16	be	be	AUX
iajs-1045	36	17	finite	finite	ADJ
iajs-1045	36	18	if	if	SCONJ
iajs-1045	36	19	x	x	X
iajs-1045	36	20	(	(	PUNCT
iajs-1045	36	21	or	or	CCONJ
iajs-1045	36	22	equivalently	equivalently	ADV
iajs-1045	36	23	a	a	PRON
iajs-1045	36	24	)	)	PUNCT
iajs-1045	36	25	is	be	AUX
iajs-1045	36	26	finite	finite	ADJ
iajs-1045	36	27	fuzzy	fuzzy	ADJ
iajs-1045	36	28	set	set	NOUN
iajs-1045	36	29	;	;	PUNCT
iajs-1045	36	30	and	and	CCONJ
iajs-1045	36	31	two	two	NUM
iajs-1045	36	32	spaces	space	NOUN
iajs-1045	36	33	of	of	ADP
iajs-1045	36	34	fuzzy	fuzzy	ADJ
iajs-1045	36	35	orderings	ordering	NOUN
iajs-1045	36	36	(	(	PUNCT
iajs-1045	36	37	x	x	X
iajs-1045	36	38	,	,	PUNCT
iajs-1045	36	39	a	a	PRON
iajs-1045	36	40	)	)	PUNCT
iajs-1045	36	41	and	and	CCONJ
iajs-1045	36	42	(	(	PUNCT
iajs-1045	36	43	x,a	x,a	PROPN
iajs-1045	36	44	)	)	PUNCT
iajs-1045	36	45	are	be	AUX
iajs-1045	36	46	said	say	VERB
iajs-1045	36	47	to	to	PART
iajs-1045	36	48	be	be	AUX
iajs-1045	36	49	isomorphic	isomorphic	ADJ
iajs-1045	36	50	if	if	SCONJ
iajs-1045	36	51	there	there	PRON
iajs-1045	36	52	exists	exist	VERB
iajs-1045	36	53	a	a	DET
iajs-1045	36	54	group	group	NOUN
iajs-1045	36	55	isomorphism	isomorphism	NOUN
iajs-1045	36	56	:a	:a	NOUN
iajs-1045	36	57			PROPN
iajs-1045	36	58	a	a	PROPN
iajs-1045	36	59	such	such	ADJ
iajs-1045	36	60	that	that	SCONJ
iajs-1045	36	61	the	the	DET
iajs-1045	36	62	dual	dual	ADJ
iajs-1045	36	63	isomorphism	isomorphism	NOUN
iajs-1045	36	64	*:(a	*:(a	PRON
iajs-1045	36	65	)	)	PUNCT
iajs-1045	36	66			PROPN
iajs-1045	36	67	(a	(a	PROPN
iajs-1045	36	68	)	)	PUNCT
iajs-1045	36	69	maps	map	VERB
iajs-1045	36	70	x	x	PUNCT
iajs-1045	37	1	on	on	ADP
iajs-1045	37	2	to	to	PART
iajs-1045	37	3	x.	x.	NOUN
iajs-1045	37	4	definition	definition	NOUN
iajs-1045	37	5	3	3	NUM
iajs-1045	37	6	the	the	DET
iajs-1045	37	7	chain	chain	NOUN
iajs-1045	37	8	length	length	NOUN
iajs-1045	37	9	of	of	ADP
iajs-1045	37	10	x	x	PROPN
iajs-1045	37	11	(	(	PUNCT
iajs-1045	37	12	denoted	denote	VERB
iajs-1045	37	13	c1(x	c1(x	NOUN
iajs-1045	37	14	)	)	PUNCT
iajs-1045	37	15	)	)	PUNCT
iajs-1045	37	16	is	be	AUX
iajs-1045	37	17	the	the	DET
iajs-1045	37	18	maximum	maximum	ADJ
iajs-1045	37	19	integer	integer	NOUN
iajs-1045	37	20	k	k	PROPN
iajs-1045	37	21			NUM
iajs-1045	37	22	1	1	NUM
iajs-1045	37	23	such	such	ADJ
iajs-1045	37	24	that	that	DET
iajs-1045	37	25			PROPN
iajs-1045	37	26	a0	a0	PROPN
iajs-1045	37	27	,	,	PUNCT
iajs-1045	37	28			PROPN
iajs-1045	37	29	,	,	PUNCT
iajs-1045	37	30	ak	ak	PROPN
iajs-1045	37	31			PROPN
iajs-1045	37	32	a	a	DET
iajs-1045	37	33	satisfy	satisfy	NOUN
iajs-1045	37	34	ing	ing	ADJ
iajs-1045	37	35	:	:	PUNCT
iajs-1045	37	36	x(ai	x(ai	PROPN
iajs-1045	37	37	–	–	PUNCT
iajs-1045	37	38	1	1	X
iajs-1045	37	39	)	)	PUNCT
iajs-1045	37	40			PROPN
iajs-1045	37	41	x(ai	x(ai	PROPN
iajs-1045	37	42	)	)	PUNCT
iajs-1045	37	43	,	,	PUNCT
iajs-1045	37	44	i	i	PRON
iajs-1045	37	45	=	=	NOUN
iajs-1045	37	46	1	1	NUM
iajs-1045	37	47	,	,	PUNCT
iajs-1045	37	48			PROPN
iajs-1045	37	49	,	,	PUNCT
iajs-1045	37	50	k	k	PROPN
iajs-1045	37	51	(	(	PUNCT
iajs-1045	37	52	or	or	CCONJ
iajs-1045	37	53	c1(x	c1(x	NOUN
iajs-1045	37	54	)	)	PUNCT
iajs-1045	37	55	=	=	NOUN
iajs-1045	37	56			VERB
iajs-1045	37	57	if	if	SCONJ
iajs-1045	37	58	no	no	DET
iajs-1045	37	59	such	such	ADJ
iajs-1045	37	60	maximum	maximum	ADJ
iajs-1045	37	61	exists	exist	NOUN
iajs-1045	37	62	)	)	PUNCT
iajs-1045	37	63	.	.	PUNCT
iajs-1045	38	1	remark	remark	NOUN
iajs-1045	38	2	1	1	NUM
iajs-1045	38	3	it	it	PRON
iajs-1045	38	4	is	be	AUX
iajs-1045	38	5	easily	easily	ADV
iajs-1045	38	6	verified	verify	VERB
iajs-1045	38	7	that	that	SCONJ
iajs-1045	38	8	c1(x	c1(x	NOUN
iajs-1045	38	9	)	)	PUNCT
iajs-1045	38	10	=	=	SYM
iajs-1045	38	11	1	1	NUM
iajs-1045	38	12	if	if	SCONJ
iajs-1045	38	13	and	and	CCONJ
iajs-1045	38	14	only	only	ADV
iajs-1045	38	15	if	if	SCONJ
iajs-1045	38	16	x	x	NOUN
iajs-1045	38	17	=	=	SYM
iajs-1045	38	18	1	1	NUM
iajs-1045	38	19	,	,	PUNCT
iajs-1045	38	20	and	and	CCONJ
iajs-1045	38	21	c1(x	c1(x	NOUN
iajs-1045	38	22	)	)	PUNCT
iajs-1045	38	23			NOUN
iajs-1045	38	24	2	2	NUM
iajs-1045	38	25	if	if	SCONJ
iajs-1045	38	26	and	and	CCONJ
iajs-1045	38	27	only	only	ADV
iajs-1045	38	28	if	if	SCONJ
iajs-1045	38	29	x	x	PRON
iajs-1045	38	30	is	be	AUX
iajs-1045	38	31	a	a	DET
iajs-1045	38	32	fan	fan	NOUN
iajs-1045	38	33	.	.	PUNCT
iajs-1045	39	1	recall	recall	VERB
iajs-1045	39	2	that	that	PRON
iajs-1045	39	3	x	x	PRON
iajs-1045	39	4	is	be	AUX
iajs-1045	39	5	said	say	VERB
iajs-1045	39	6	to	to	PART
iajs-1045	39	7	be	be	AUX
iajs-1045	39	8	decomposable	decomposable	ADJ
iajs-1045	39	9	if	if	SCONJ
iajs-1045	39	10	there	there	PRON
iajs-1045	39	11	exist	exist	VERB
iajs-1045	39	12	non	non	ADJ
iajs-1045	39	13	-	-	ADJ
iajs-1045	39	14	empty	empty	ADJ
iajs-1045	39	15	subspaces	subspace	NOUN
iajs-1045	39	16	xi	xi	ADP
iajs-1045	39	17	of	of	ADP
iajs-1045	39	18	x	x	PRON
iajs-1045	39	19	,	,	PUNCT
iajs-1045	39	20	i	i	NOUN
iajs-1045	39	21	=	=	NOUN
iajs-1045	39	22	1,2	1,2	NUM
iajs-1045	40	1	such	such	ADJ
iajs-1045	40	2	that	that	SCONJ
iajs-1045	40	3	x	x	X
iajs-1045	40	4	=	=	SYM
iajs-1045	40	5	x1	x1	PROPN
iajs-1045	40	6			ADJ
iajs-1045	40	7	x2	x2	INTJ
iajs-1045	40	8	.	.	PUNCT
iajs-1045	41	1	let	let	VERB
iajs-1045	41	2	us	we	PRON
iajs-1045	41	3	denote	denote	VERB
iajs-1045	41	4	by	by	ADP
iajs-1045	41	5	gr(x	gr(x	NOUN
iajs-1045	41	6	)	)	PUNCT
iajs-1045	41	7	the	the	DET
iajs-1045	41	8	translation	translation	NOUN
iajs-1045	41	9	fuzzy	fuzzy	ADJ
iajs-1045	41	10	group	group	NOUN
iajs-1045	41	11	of	of	ADP
iajs-1045	41	12	x	x	PROPN
iajs-1045	41	13	,	,	PUNCT
iajs-1045	41	14	i.e.	i.e.	X
iajs-1045	41	15	,	,	PUNCT
iajs-1045	41	16	gr(x	gr(x	X
iajs-1045	41	17	)	)	PUNCT
iajs-1045	42	1	=	=	SYM
iajs-1045	42	2	{	{	PUNCT
iajs-1045	42	3	t	t	PROPN
iajs-1045	42	4			PROPN
iajs-1045	42	5	(x	(x	PROPN
iajs-1045	42	6	)	)	PUNCT
iajs-1045	42	7			NUM
iajs-1045	42	8	tx	tx	NOUN
iajs-1045	42	9	=	=	SYM
iajs-1045	42	10	x	x	NOUN
iajs-1045	42	11	}	}	PUNCT
iajs-1045	42	12	.	.	PUNCT
iajs-1045	43	1	thus	thus	ADV
iajs-1045	43	2	gr(x	gr(x	X
iajs-1045	43	3	)	)	PUNCT
iajs-1045	43	4	is	be	AUX
iajs-1045	43	5	a	a	DET
iajs-1045	43	6	closed	closed	ADJ
iajs-1045	43	7	fuzzy	fuzzy	ADJ
iajs-1045	43	8	subgroup	subgroup	NOUN
iajs-1045	43	9	of	of	ADP
iajs-1045	43	10	(a	(a	PROPN
iajs-1045	43	11	)	)	PUNCT
iajs-1045	43	12	.	.	PUNCT
iajs-1045	44	1	let	let	VERB
iajs-1045	44	2	the	the	DET
iajs-1045	44	3	residue	residue	NOUN
iajs-1045	44	4	space	space	NOUN
iajs-1045	44	5	of	of	ADP
iajs-1045	44	6	x	x	AUX
iajs-1045	44	7	be	be	AUX
iajs-1045	44	8	defined	define	VERB
iajs-1045	44	9	to	to	PART
iajs-1045	44	10	be	be	AUX
iajs-1045	44	11	x	x	X
iajs-1045	44	12	=	=	SYM
iajs-1045	44	13	(	(	PUNCT
iajs-1045	44	14	x,a	x,a	PROPN
iajs-1045	44	15	)	)	PUNCT
iajs-1045	44	16	where	where	SCONJ
iajs-1045	44	17	a=gr(x	a=gr(x	PROPN
iajs-1045	44	18	)	)	PUNCT
iajs-1045	44	19			NOUN
iajs-1045	44	20	a	a	VERB
iajs-1045	44	21	,	,	PUNCT
iajs-1045	44	22	and	and	CCONJ
iajs-1045	44	23	where	where	SCONJ
iajs-1045	44	24	x	x	PROPN
iajs-1045	44	25	denotes	denote	VERB
iajs-1045	44	26	the	the	DET
iajs-1045	44	27	image	image	NOUN
iajs-1045	44	28	of	of	ADP
iajs-1045	44	29	x	x	PUNCT
iajs-1045	44	30	in	in	ADP
iajs-1045	44	31	(a	(a	NUM
iajs-1045	44	32	)	)	PUNCT
iajs-1045	44	33	via	via	ADP
iajs-1045	44	34	restriction	restriction	NOUN
iajs-1045	44	35	,	,	PUNCT
iajs-1045	44	36	x	x	PROPN
iajs-1045	44	37	is	be	AUX
iajs-1045	44	38	a	a	DET
iajs-1045	44	39	space	space	NOUN
iajs-1045	44	40	of	of	ADP
iajs-1045	44	41	fuzzy	fuzzy	ADJ
iajs-1045	44	42	orderings	ordering	NOUN
iajs-1045	44	43	.	.	PUNCT
iajs-1045	45	1	moreover	moreover	ADV
iajs-1045	45	2	gr(x	gr(x	PROPN
iajs-1045	45	3	)	)	PUNCT
iajs-1045	45	4	=	=	SYM
iajs-1045	45	5	1	1	NUM
iajs-1045	45	6	,	,	PUNCT
iajs-1045	45	7	and	and	CCONJ
iajs-1045	45	8	x	x	X
iajs-1045	45	9	is	be	AUX
iajs-1045	45	10	a	a	DET
iajs-1045	45	11	fuzzy	fuzzy	ADJ
iajs-1045	45	12	group	group	NOUN
iajs-1045	45	13	extension	extension	NOUN
iajs-1045	45	14	of	of	ADP
iajs-1045	45	15	x.	x.	PROPN
iajs-1045	46	1	we	we	PRON
iajs-1045	46	2	can	can	AUX
iajs-1045	46	3	state	state	VERB
iajs-1045	46	4	the	the	DET
iajs-1045	46	5	main	main	ADJ
iajs-1045	46	6	theorem	theorem	NOUN
iajs-1045	46	7	concerning	concern	VERB
iajs-1045	46	8	spaces	space	NOUN
iajs-1045	46	9	of	of	ADP
iajs-1045	46	10	finite	finite	ADJ
iajs-1045	46	11	chain	chain	NOUN
iajs-1045	46	12	length	length	NOUN
iajs-1045	46	13	.	.	PUNCT
iajs-1045	47	1	theorem	theorem	ADJ
iajs-1045	47	2	2	2	NUM
iajs-1045	47	3	suppose	suppose	VERB
iajs-1045	47	4	c1(x	c1(x	NOUN
iajs-1045	47	5	)	)	PUNCT
iajs-1045	47	6	<	<	X
iajs-1045	47	7	.	.	X
iajs-1045	47	8	then	then	ADV
iajs-1045	47	9	either	either	PRON
iajs-1045	47	10	x	x	X
iajs-1045	47	11	=	=	SYM
iajs-1045	47	12	1	1	NUM
iajs-1045	47	13	,	,	PUNCT
iajs-1045	47	14	or	or	CCONJ
iajs-1045	47	15	gr(x	gr(x	NUM
iajs-1045	47	16	)	)	PUNCT
iajs-1045	47	17			NOUN
iajs-1045	47	18	1	1	NUM
iajs-1045	47	19	,	,	PUNCT
iajs-1045	47	20	or	or	CCONJ
iajs-1045	47	21	x	x	NOUN
iajs-1045	47	22	is	be	AUX
iajs-1045	47	23	decomposable	decomposable	ADJ
iajs-1045	47	24	.	.	PUNCT
iajs-1045	48	1	the	the	DET
iajs-1045	48	2	proof	proof	NOUN
iajs-1045	48	3	of	of	ADP
iajs-1045	48	4	this	this	DET
iajs-1045	48	5	key	key	ADJ
iajs-1045	48	6	result	result	NOUN
iajs-1045	48	7	is	be	AUX
iajs-1045	48	8	found	find	VERB
iajs-1045	48	9	in	in	ADP
iajs-1045	48	10	[	[	X
iajs-1045	48	11	4	4	NUM
iajs-1045	48	12	]	]	PUNCT
iajs-1045	48	13	.	.	PUNCT
iajs-1045	49	1	for	for	ADP
iajs-1045	49	2	now	now	ADV
iajs-1045	49	3	we	we	PRON
iajs-1045	49	4	concentrate	concentrate	VERB
iajs-1045	49	5	on	on	ADP
iajs-1045	49	6	giving	give	VERB
iajs-1045	49	7	two	two	NUM
iajs-1045	49	8	important	important	ADJ
iajs-1045	49	9	applications	application	NOUN
iajs-1045	49	10	.	.	PUNCT
iajs-1045	50	1	theorem	theorem	NOUN
iajs-1045	50	2	3	3	NUM
iajs-1045	50	3	suppose	suppose	VERB
iajs-1045	50	4	a	a	DET
iajs-1045	50	5	form	form	NOUN
iajs-1045	50	6	f	f	PROPN
iajs-1045	50	7	is	be	AUX
iajs-1045	50	8	anisotropic	anisotropic	NOUN
iajs-1045	50	9	over	over	ADP
iajs-1045	50	10	a	a	DET
iajs-1045	50	11	space	space	NOUN
iajs-1045	50	12	of	of	ADP
iajs-1045	50	13	fuzzy	fuzzy	ADJ
iajs-1045	50	14	ordering	ordering	NOUN
iajs-1045	50	15	x0	x0	PROPN
iajs-1045	50	16	.	.	PUNCT
iajs-1045	51	1	then	then	ADV
iajs-1045	51	2	there	there	PRON
iajs-1045	51	3	exists	exist	VERB
iajs-1045	51	4	a	a	DET
iajs-1045	51	5	finite	finite	ADJ
iajs-1045	51	6	subspace	subspace	NOUN
iajs-1045	51	7	x	x	PUNCT
iajs-1045	51	8			PROPN
iajs-1045	51	9	x0	x0	PROPN
iajs-1045	51	10	such	such	ADJ
iajs-1045	51	11	that	that	SCONJ
iajs-1045	51	12	f	f	PROPN
iajs-1045	51	13	is	be	AUX
iajs-1045	51	14	an	an	DET
iajs-1045	51	15	isotropic	isotropic	NOUN
iajs-1045	51	16	over	over	ADP
iajs-1045	51	17	x.	x.	NOUN
iajs-1045	51	18	proof	proof	NOUN
iajs-1045	51	19	:	:	PUNCT
iajs-1045	51	20	let	let	VERB
iajs-1045	51	21	x=(x	x=(x	PROPN
iajs-1045	51	22	,	,	PUNCT
iajs-1045	51	23	a	a	PRON
iajs-1045	51	24	)	)	PUNCT
iajs-1045	51	25	be	be	AUX
iajs-1045	51	26	a	a	DET
iajs-1045	51	27	subspace	subspace	NOUN
iajs-1045	51	28	of	of	ADP
iajs-1045	51	29	x0	x0	PROPN
iajs-1045	51	30	chosen	choose	VERB
iajs-1045	51	31	minimal	minimal	ADJ
iajs-1045	51	32	subject	subject	NOUN
iajs-1045	51	33	to	to	ADP
iajs-1045	51	34	f	f	PROPN
iajs-1045	51	35	is	be	AUX
iajs-1045	51	36	anisotropic	anisotropic	NOUN
iajs-1045	51	37	over	over	ADP
iajs-1045	52	1	x.	x.	NOUN
iajs-1045	52	2	let	let	VERB
iajs-1045	52	3	a0	a0	PROPN
iajs-1045	52	4	,	,	PUNCT
iajs-1045	52	5			PROPN
iajs-1045	52	6	,	,	PUNCT
iajs-1045	52	7	ak	ak	PROPN
iajs-1045	52	8			PROPN
iajs-1045	52	9	a	a	DET
iajs-1045	52	10	satisfy	satisfy	NOUN
iajs-1045	52	11	:	:	PUNCT
iajs-1045	52	12	d<1,ai	d<1,ai	NOUN
iajs-1045	52	13	–	–	PUNCT
iajs-1045	52	14	1	1	NUM
iajs-1045	52	15	>	>	PUNCT
iajs-1045	52	16			PROPN
iajs-1045	52	17	d<1,ai	d<1,ai	NOUN
iajs-1045	53	1	>	>	X
iajs-1045	53	2	,	,	PUNCT
iajs-1045	53	3	i	i	PRON
iajs-1045	53	4	=	=	PUNCT
iajs-1045	53	5	1,,k	1,,k	NUM
iajs-1045	53	6	.	.	PUNCT
iajs-1045	54	1	thus	thus	ADV
iajs-1045	54	2	<	<	X
iajs-1045	54	3	1,ai	1,ai	X
iajs-1045	54	4	>	>	X
iajs-1045	54	5			PROPN
iajs-1045	54	6	<	<	X
iajs-1045	54	7	ai	ai	INTJ
iajs-1045	54	8	–	–	PUNCT
iajs-1045	54	9	1	1	NUM
iajs-1045	54	10	,	,	PUNCT
iajs-1045	54	11	ai	ai	VERB
iajs-1045	54	12	–	–	PUNCT
iajs-1045	54	13	1	1	NUM
iajs-1045	54	14	ai	ai	VERB
iajs-1045	54	15	>	>	X
iajs-1045	54	16	and	and	CCONJ
iajs-1045	54	17	ai	ai	INTJ
iajs-1045	54	18	–	–	PUNCT
iajs-1045	54	19	1	1	NUM
iajs-1045	54	20			NOUN
iajs-1045	54	21	ai	ai	VERB
iajs-1045	54	22	for	for	ADP
iajs-1045	54	23	i	i	PRON
iajs-1045	54	24	=	=	PUNCT
iajs-1045	54	25	1,,k	1,,k	NUM
iajs-1045	54	26	.	.	PUNCT
iajs-1045	55	1	we	we	PRON
iajs-1045	55	2	may	may	AUX
iajs-1045	55	3	assume	assume	VERB
iajs-1045	55	4	a0	a0	NOUN
iajs-1045	55	5	=	=	SYM
iajs-1045	55	6	1	1	NUM
iajs-1045	55	7	,	,	PUNCT
iajs-1045	55	8	ak	ak	PROPN
iajs-1045	55	9	=	=	ADJ
iajs-1045	55	10	1	1	PROPN
iajs-1045	55	11	.	.	PUNCT
iajs-1045	56	1	let	let	VERB
iajs-1045	56	2	bi	bi	NOUN
iajs-1045	56	3	=	=	PRON
iajs-1045	56	4	ai	ai	INTJ
iajs-1045	56	5	–	–	PUNCT
iajs-1045	56	6	1	1	NUM
iajs-1045	56	7	ai	ai	NOUN
iajs-1045	56	8	.	.	PUNCT
iajs-1045	57	1	thus	thus	ADV
iajs-1045	57	2	bi	bi	PROPN
iajs-1045	57	3			PROPN
iajs-1045	57	4	1	1	NUM
iajs-1045	57	5	,	,	PUNCT
iajs-1045	57	6	so	so	ADV
iajs-1045	57	7	x(bi	x(bi	PROPN
iajs-1045	57	8	)	)	PUNCT
iajs-1045	57	9	is	be	AUX
iajs-1045	57	10	a	a	DET
iajs-1045	57	11	proper	proper	ADJ
iajs-1045	57	12	subspace	subspace	NOUN
iajs-1045	57	13	of	of	ADP
iajs-1045	57	14	x.	x.	NOUN
iajs-1045	57	15	thus	thus	ADV
iajs-1045	57	16	f	f	PROPN
iajs-1045	57	17	is	be	AUX
iajs-1045	57	18	isotropic	isotropic	ADJ
iajs-1045	57	19	over	over	ADP
iajs-1045	57	20	x(bi	x(bi	PROPN
iajs-1045	57	21	)	)	PUNCT
iajs-1045	57	22	,	,	PUNCT
iajs-1045	57	23	i.e.	i.e.	X
iajs-1045	57	24	there	there	PRON
iajs-1045	57	25	exists	exist	VERB
iajs-1045	57	26	a	a	DET
iajs-1045	57	27	form	form	NOUN
iajs-1045	57	28	gi	gi	NOUN
iajs-1045	57	29	of	of	ADP
iajs-1045	57	30	dimension	dimension	NOUN
iajs-1045	57	31	n	n	CCONJ
iajs-1045	57	32	–	–	PUNCT
iajs-1045	57	33	2	2	NUM
iajs-1045	57	34	(	(	PUNCT
iajs-1045	57	35	where	where	SCONJ
iajs-1045	57	36	n	n	PRON
iajs-1045	57	37	denotes	denote	VERB
iajs-1045	57	38	the	the	DET
iajs-1045	57	39	dimension	dimension	NOUN
iajs-1045	57	40	of	of	ADP
iajs-1045	57	41	f	f	NOUN
iajs-1045	57	42	)	)	PUNCT
iajs-1045	57	43	such	such	ADJ
iajs-1045	57	44	that	that	SCONJ
iajs-1045	57	45	f	f	PROPN
iajs-1045	57	46			NOUN
iajs-1045	57	47	gi	gi	NOUN
iajs-1045	57	48	over	over	ADP
iajs-1045	57	49	x(bi	x(bi	PROPN
iajs-1045	57	50	)	)	PUNCT
iajs-1045	57	51	.	.	PUNCT
iajs-1045	58	1	thus	thus	ADV
iajs-1045	58	2	:	:	PUNCT
iajs-1045	58	3	f	f	PROPN
iajs-1045	58	4			X
iajs-1045	58	5	<	<	X
iajs-1045	58	6	1,bi	1,bi	NUM
iajs-1045	58	7	>	>	X
iajs-1045	58	8			NOUN
iajs-1045	58	9	gi	gi	NOUN
iajs-1045	58	10			ADJ
iajs-1045	58	11	<	<	X
iajs-1045	58	12	1,bi	1,bi	NUM
iajs-1045	58	13	>	>	X
iajs-1045	58	14	over	over	ADP
iajs-1045	58	15	x	x	NOUN
iajs-1045	58	16	,	,	PUNCT
iajs-1045	58	17	so	so	CCONJ
iajs-1045	58	18	by	by	ADP
iajs-1045	58	19	addition	addition	NOUN
iajs-1045	58	20	k	k	PROPN
iajs-1045	59	1	k	k	PROPN
iajs-1045	59	2	i	i	PRON
iajs-1045	59	3	1	1	NUM
iajs-1045	59	4	i	i	NOUN
iajs-1045	59	5	1	1	NUM
iajs-1045	59	6	1	1	NUM
iajs-1045	59	7	,	,	PUNCT
iajs-1045	59	8	1	1	NUM
iajs-1045	59	9	,	,	PUNCT
iajs-1045	59	10	i	i	PRON
iajs-1045	59	11	i	i	PRON
iajs-1045	60	1	i	i	PRON
iajs-1045	61	1	f	f	PROPN
iajs-1045	61	2	b	b	PROPN
iajs-1045	61	3	g	g	PROPN
iajs-1045	61	4	b	b	PROPN
iajs-1045	62	1			NUM
iajs-1045	62	2			PROPN
iajs-1045	62	3			NOUN
iajs-1045	62	4			PROPN
iajs-1045	62	5			PROPN
iajs-1045	62	6			VERB
iajs-1045	62	7			PROPN
iajs-1045	62	8			X
iajs-1045	62	9			PROPN
iajs-1045	62	10			PROPN
iajs-1045	62	11			NOUN
iajs-1045	62	12			X
iajs-1045	62	13			NUM
iajs-1045	62	14	:	:	PUNCT
iajs-1045	62	15	(	(	PUNCT
iajs-1045	62	16	over	over	ADP
iajs-1045	62	17	x	x	NOUN
iajs-1045	62	18	)	)	PUNCT
iajs-1045	62	19	(1	(1	NUM
iajs-1045	62	20	)	)	PUNCT
iajs-1045	62	21	but	but	CCONJ
iajs-1045	62	22	using	use	VERB
iajs-1045	62	23	the	the	DET
iajs-1045	62	24	assumptions	assumption	NOUN
iajs-1045	62	25	on	on	ADP
iajs-1045	62	26	a0	a0	PROPN
iajs-1045	62	27	,	,	PUNCT
iajs-1045	62	28			PROPN
iajs-1045	62	29	,	,	PUNCT
iajs-1045	62	30	ak	ak	PROPN
iajs-1045	62	31	we	we	PRON
iajs-1045	62	32	see	see	VERB
iajs-1045	62	33	that	that	SCONJ
iajs-1045	62	34	(	(	PUNCT
iajs-1045	62	35	over	over	ADP
iajs-1045	62	36	x	x	NOUN
iajs-1045	62	37	)	)	PUNCT
iajs-1045	62	38	<	<	X
iajs-1045	62	39	b0	b0	PROPN
iajs-1045	62	40	,	,	PUNCT
iajs-1045	62	41			PROPN
iajs-1045	62	42	,	,	PUNCT
iajs-1045	62	43	bk	bk	ADP
iajs-1045	62	44	>	>	X
iajs-1045	62	45			PROPN
iajs-1045	62	46	<	<	X
iajs-1045	62	47	a0	a0	PROPN
iajs-1045	62	48	a1	a1	PROPN
iajs-1045	62	49	,	,	PUNCT
iajs-1045	62	50	a1	a1	NOUN
iajs-1045	62	51	a2,	a2,	PROPN
iajs-1045	62	52	,	,	PUNCT
iajs-1045	62	53	ak	ak	PROPN
iajs-1045	62	54	–	–	PUNCT
iajs-1045	62	55	1	1	NUM
iajs-1045	62	56	ak	ak	PROPN
iajs-1045	62	57	>	>	X
iajs-1045	62	58			PROPN
iajs-1045	62	59	<	<	X
iajs-1045	62	60	a1	a1	PROPN
iajs-1045	62	61	,	,	PUNCT
iajs-1045	62	62	a1	a1	NOUN
iajs-1045	62	63	a2,	a2,	PROPN
iajs-1045	62	64	,	,	PUNCT
iajs-1045	62	65	ak	ak	PROPN
iajs-1045	62	66	–	–	PUNCT
iajs-1045	62	67	1	1	NUM
iajs-1045	62	68	ak	ak	PROPN
iajs-1045	62	69	>	>	X
iajs-1045	62	70			PROPN
iajs-1045	62	71	<	<	X
iajs-1045	62	72	1	1	NUM
iajs-1045	62	73	,	,	PUNCT
iajs-1045	62	74	a2	a2	PROPN
iajs-1045	62	75	,	,	PUNCT
iajs-1045	62	76	a2	a2	PROPN
iajs-1045	62	77	a3,	a3,	PROPN
iajs-1045	62	78	,	,	PUNCT
iajs-1045	62	79	ak	ak	PROPN
iajs-1045	62	80	–	–	PUNCT
iajs-1045	62	81	1	1	NUM
iajs-1045	62	82	ak	ak	PROPN
iajs-1045	62	83	>	>	X
iajs-1045	62	84			PROPN
iajs-1045	62	85	<	<	X
iajs-1045	62	86	1,	1,	NUM
iajs-1045	62	87	,	,	PUNCT
iajs-1045	62	88	1	1	NUM
iajs-1045	62	89	,	,	PUNCT
iajs-1045	62	90	ak	ak	PROPN
iajs-1045	62	91	>	>	X
iajs-1045	62	92			PROPN
iajs-1045	62	93	<	<	X
iajs-1045	62	94	1,,1	1,,1	X
iajs-1045	62	95	,	,	PUNCT
iajs-1045	62	96	1	1	NUM
iajs-1045	62	97	>	>	PUNCT
iajs-1045	62	98	.	.	PUNCT
iajs-1045	63	1	substituting	substitute	VERB
iajs-1045	63	2	this	this	PRON
iajs-1045	63	3	in	in	ADP
iajs-1045	63	4	(	(	PUNCT
iajs-1045	63	5	1	1	X
iajs-1045	63	6	)	)	PUNCT
iajs-1045	63	7	yields	yield	NOUN
iajs-1045	63	8	k	k	PROPN
iajs-1045	64	1	i	i	PRON
iajs-1045	64	2	1	1	NUM
iajs-1045	64	3	(	(	PUNCT
iajs-1045	64	4	2	2	NUM
iajs-1045	64	5	2	2	NUM
iajs-1045	64	6	)	)	PUNCT
iajs-1045	64	7	1	1	NUM
iajs-1045	64	8	,	,	PUNCT
iajs-1045	64	9	i	i	PRON
iajs-1045	64	10	i	i	VERB
iajs-1045	64	11	k	k	NOUN
iajs-1045	65	1	f	f	PROPN
iajs-1045	65	2	g	g	PROPN
iajs-1045	65	3	b	b	PROPN
iajs-1045	65	4			PROPN
iajs-1045	65	5			PROPN
iajs-1045	66	1			NOUN
iajs-1045	66	2	:	:	PUNCT
iajs-1045	66	3	ibn	ibn	NOUN
iajs-1045	66	4	alhaitham	alhaitham	NOUN
iajs-1045	66	5	j.	j.	PROPN
iajs-1045	66	6	for	for	ADP
iajs-1045	66	7	pure	pure	ADJ
iajs-1045	66	8	&	&	CCONJ
iajs-1045	66	9	appl	appl	PROPN
iajs-1045	66	10	.	.	PUNCT
iajs-1045	67	1	sci	sci	PROPN
iajs-1045	67	2	.	.	PUNCT
iajs-1045	68	1	vol.22	vol.22	PROPN
iajs-1045	68	2	(	(	PUNCT
iajs-1045	68	3	4	4	NUM
iajs-1045	68	4	)	)	PUNCT
iajs-1045	68	5	2009	2009	NUM
iajs-1045	69	1	now	now	ADV
iajs-1045	69	2	f	f	X
iajs-1045	69	3	(	(	PUNCT
iajs-1045	69	4	and	and	CCONJ
iajs-1045	69	5	hence	hence	ADV
iajs-1045	69	6	(	(	PUNCT
iajs-1045	69	7	2k	2k	NUM
iajs-1045	69	8	–	–	PUNCT
iajs-1045	69	9	2	2	NUM
iajs-1045	69	10	)	)	PUNCT
iajs-1045	69	11	f	f	NOUN
iajs-1045	69	12	,	,	PUNCT
iajs-1045	69	13	by	by	ADP
iajs-1045	69	14	(	(	PUNCT
iajs-1045	69	15	3	3	NUM
iajs-1045	69	16	,	,	PUNCT
iajs-1045	69	17	corollary	corollary	NOUN
iajs-1045	69	18	3.5(ii	3.5(ii	NUM
iajs-1045	69	19	)	)	PUNCT
iajs-1045	69	20	)	)	PUNCT
iajs-1045	69	21	is	be	AUX
iajs-1045	69	22	anisotropic	anisotropic	NOUN
iajs-1045	69	23	over	over	ADP
iajs-1045	69	24	x	x	NOUN
iajs-1045	69	25	,	,	PUNCT
iajs-1045	69	26	so	so	ADV
iajs-1045	69	27	comparing	compare	VERB
iajs-1045	69	28	dimensions	dimension	NOUN
iajs-1045	69	29	,	,	PUNCT
iajs-1045	69	30	and	and	CCONJ
iajs-1045	69	31	using	use	VERB
iajs-1045	69	32	(	(	PUNCT
iajs-1045	69	33	3	3	NUM
iajs-1045	69	34	,	,	PUNCT
iajs-1045	69	35	lemma	lemma	PROPN
iajs-1045	69	36	2.4	2.4	NUM
iajs-1045	69	37	)	)	PUNCT
iajs-1045	69	38	,	,	PUNCT
iajs-1045	69	39	(	(	PUNCT
iajs-1045	69	40	2k	2k	NUM
iajs-1045	69	41	–	–	PUNCT
iajs-1045	69	42	2	2	NUM
iajs-1045	69	43	)	)	PUNCT
iajs-1045	69	44	n	n	PRON
iajs-1045	69	45			NUM
iajs-1045	69	46	k	k	NOUN
iajs-1045	69	47	(	(	PUNCT
iajs-1045	69	48	n	n	X
iajs-1045	69	49	–	–	PUNCT
iajs-1045	69	50	2)(2	2)(2	NUM
iajs-1045	69	51	)	)	PUNCT
iajs-1045	69	52	,	,	PUNCT
iajs-1045	69	53	i.e.	i.e.	X
iajs-1045	69	54	,	,	PUNCT
iajs-1045	69	55	k	k	PROPN
iajs-1045	69	56			NOUN
iajs-1045	69	57	1	1	NUM
iajs-1045	69	58	2	2	NUM
iajs-1045	69	59	n.	n.	NOUN
iajs-1045	69	60	this	this	PRON
iajs-1045	69	61	proves	prove	VERB
iajs-1045	69	62	c1(x	c1(x	NOUN
iajs-1045	69	63	)	)	PUNCT
iajs-1045	69	64	<	<	X
iajs-1045	69	65	.	.	X
iajs-1045	69	66	now	now	ADV
iajs-1045	69	67	,	,	PUNCT
iajs-1045	69	68	we	we	PRON
iajs-1045	69	69	apply	apply	VERB
iajs-1045	69	70	theorem	theorem	NOUN
iajs-1045	69	71	2	2	NUM
iajs-1045	69	72	.	.	PUNCT
iajs-1045	70	1	if	if	SCONJ
iajs-1045	70	2	x	x	NOUN
iajs-1045	70	3	=	=	SYM
iajs-1045	70	4	1	1	NUM
iajs-1045	70	5	we	we	PRON
iajs-1045	70	6	are	be	AUX
iajs-1045	70	7	done	do	VERB
iajs-1045	70	8	.	.	PUNCT
iajs-1045	71	1	suppose	suppose	VERB
iajs-1045	71	2	x	x	NOUN
iajs-1045	71	3	=	=	SYM
iajs-1045	71	4	x1	x1	PROPN
iajs-1045	71	5			PROPN
iajs-1045	71	6	x2	x2	INTJ
iajs-1045	71	7	where	where	SCONJ
iajs-1045	71	8	xi	xi	X
iajs-1045	71	9	=	=	SYM
iajs-1045	71	10	(	(	PUNCT
iajs-1045	71	11	xi	xi	PROPN
iajs-1045	71	12	,	,	PUNCT
iajs-1045	71	13	a	a	DET
iajs-1045	71	14	/i	/i	NOUN
iajs-1045	71	15	)	)	PUNCT
iajs-1045	71	16	is	be	AUX
iajs-1045	71	17	a	a	DET
iajs-1045	71	18	non	non	ADJ
iajs-1045	71	19	-	-	ADJ
iajs-1045	71	20	empty	empty	ADJ
iajs-1045	71	21	subspace	subspace	NOUN
iajs-1045	71	22	of	of	ADP
iajs-1045	71	23	x	x	PRON
iajs-1045	71	24	,	,	PUNCT
iajs-1045	71	25	i	i	NOUN
iajs-1045	71	26	=	=	NOUN
iajs-1045	71	27	1,2	1,2	NUM
iajs-1045	71	28	.	.	PUNCT
iajs-1045	72	1	thus	thus	ADV
iajs-1045	72	2	there	there	PRON
iajs-1045	72	3	exist	exist	VERB
iajs-1045	72	4	elements	element	NOUN
iajs-1045	72	5	ai	ai	VERB
iajs-1045	72	6	3	3	NUM
iajs-1045	72	7	,	,	PUNCT
iajs-1045	72	8			PROPN
iajs-1045	72	9	,	,	PUNCT
iajs-1045	72	10	ai	ai	VERB
iajs-1045	72	11	n	n	ADV
iajs-1045	72	12			NOUN
iajs-1045	72	13	a	a	DET
iajs-1045	72	14	such	such	ADJ
iajs-1045	72	15	that	that	SCONJ
iajs-1045	72	16	f	f	X
iajs-1045	72	17			X
iajs-1045	72	18	<	<	X
iajs-1045	72	19	–	–	PUNCT
iajs-1045	72	20	1,1	1,1	NUM
iajs-1045	72	21	,	,	PUNCT
iajs-1045	72	22	ai	ai	VERB
iajs-1045	72	23	3	3	NUM
iajs-1045	72	24	,	,	PUNCT
iajs-1045	72	25			PROPN
iajs-1045	72	26	,	,	PUNCT
iajs-1045	72	27	ai	ai	VERB
iajs-1045	72	28	n	n	ADV
iajs-1045	72	29	>	>	X
iajs-1045	72	30	over	over	ADP
iajs-1045	72	31	xi	xi	PROPN
iajs-1045	72	32	,	,	PUNCT
iajs-1045	72	33	i	i	NOUN
iajs-1045	72	34	=	=	NOUN
iajs-1045	72	35	1,2	1,2	NUM
iajs-1045	72	36	.	.	PUNCT
iajs-1045	73	1	since	since	SCONJ
iajs-1045	73	2	x	x	PROPN
iajs-1045	73	3	=	=	SYM
iajs-1045	73	4	x1	x1	PROPN
iajs-1045	73	5			PROPN
iajs-1045	73	6	x2	x2	PROPN
iajs-1045	73	7	,	,	PUNCT
iajs-1045	73	8	the	the	DET
iajs-1045	73	9	natural	natural	ADJ
iajs-1045	73	10	injection	injection	NOUN
iajs-1045	73	11	a	a	DET
iajs-1045	73	12			NOUN
iajs-1045	73	13	a	a	DET
iajs-1045	73	14	/1	/1	PROPN
iajs-1045	73	15			NOUN
iajs-1045	73	16	a	a	DET
iajs-1045	73	17	/2	/2	NOUN
iajs-1045	73	18	is	be	AUX
iajs-1045	73	19	surjective	surjective	ADJ
iajs-1045	73	20	,	,	PUNCT
iajs-1045	73	21	so	so	SCONJ
iajs-1045	73	22	there	there	PRON
iajs-1045	73	23	exist	exist	VERB
iajs-1045	73	24	a3	a3	NOUN
iajs-1045	73	25	,	,	PUNCT
iajs-1045	73	26			PROPN
iajs-1045	73	27	,	,	PUNCT
iajs-1045	73	28	an	an	DET
iajs-1045	73	29			NOUN
iajs-1045	73	30	a	a	PRON
iajs-1045	73	31	such	such	ADJ
iajs-1045	73	32	that	that	SCONJ
iajs-1045	73	33	aj	aj	PROPN
iajs-1045	73	34			PROPN
iajs-1045	73	35	aij	aij	PROPN
iajs-1045	73	36	(	(	PUNCT
iajs-1045	73	37	mod	mod	PROPN
iajs-1045	73	38	i	i	PROPN
iajs-1045	73	39	)	)	PUNCT
iajs-1045	73	40	,	,	PUNCT
iajs-1045	73	41	3	3	NUM
iajs-1045	73	42			NUM
iajs-1045	73	43	j	j	PROPN
iajs-1045	73	44			PROPN
iajs-1045	73	45	n	n	CCONJ
iajs-1045	73	46	,	,	PUNCT
iajs-1045	73	47	i	i	PRON
iajs-1045	73	48	=	=	NOUN
iajs-1045	73	49	1,2	1,2	NUM
iajs-1045	73	50	.	.	PUNCT
iajs-1045	74	1	then	then	ADV
iajs-1045	74	2	clearly	clearly	ADV
iajs-1045	74	3	f	f	PROPN
iajs-1045	74	4			PROPN
iajs-1045	74	5	<	<	PROPN
iajs-1045	74	6	1,–1	1,–1	NUM
iajs-1045	74	7	,	,	PUNCT
iajs-1045	74	8	a3,,an	a3,,an	PROPN
iajs-1045	74	9	>	>	X
iajs-1045	74	10	over	over	ADP
iajs-1045	74	11	x	x	PROPN
iajs-1045	74	12	,	,	PUNCT
iajs-1045	74	13	a	a	DET
iajs-1045	74	14	contradiction	contradiction	NOUN
iajs-1045	74	15	.	.	PUNCT
iajs-1045	75	1	thus	thus	ADV
iajs-1045	75	2	x	x	PRON
iajs-1045	75	3	is	be	AUX
iajs-1045	75	4	indecomposable	indecomposable	ADJ
iajs-1045	75	5	,	,	PUNCT
iajs-1045	75	6	so	so	ADV
iajs-1045	75	7	gr(x	gr(x	NUM
iajs-1045	75	8	)	)	PUNCT
iajs-1045	75	9			NOUN
iajs-1045	76	1	1	1	X
iajs-1045	76	2	.	.	PUNCT
iajs-1045	77	1	let	let	AUX
iajs-1045	77	2	x=(x,a	x=(x,a	PROPN
iajs-1045	77	3	)	)	PUNCT
iajs-1045	77	4	denote	denote	VERB
iajs-1045	77	5	the	the	DET
iajs-1045	77	6	residue	residue	NOUN
iajs-1045	77	7	space	space	NOUN
iajs-1045	77	8	of	of	ADP
iajs-1045	77	9	x	x	PUNCT
iajs-1045	77	10	and	and	CCONJ
iajs-1045	77	11	decompose	decompose	NOUN
iajs-1045	77	12	f	f	PROPN
iajs-1045	77	13	as	as	ADP
iajs-1045	77	14	f	f	PROPN
iajs-1045	77	15	1	1	ADP
iajs-1045	77	16	f1	f1	PROPN
iajs-1045	77	17			PROPN
iajs-1045	77	18	s	s	PROPN
iajs-1045	77	19	fs	fs	ADP
iajs-1045	77	20	where	where	SCONJ
iajs-1045	77	21	f1,,fs	f1,,fs	PROPN
iajs-1045	77	22	are	be	AUX
iajs-1045	77	23	forms	form	NOUN
iajs-1045	77	24	over	over	ADP
iajs-1045	77	25	a	a	PROPN
iajs-1045	77	26	,	,	PUNCT
iajs-1045	77	27	and	and	CCONJ
iajs-1045	77	28	1,,sa	1,,sa	NOUN
iajs-1045	77	29	are	be	AUX
iajs-1045	77	30	distinct	distinct	ADJ
iajs-1045	77	31	modulo	modulo	NOUN
iajs-1045	77	32	a.	a.	VERB
iajs-1045	77	33	the	the	DET
iajs-1045	77	34	assertion	assertion	NOUN
iajs-1045	77	35	that	that	SCONJ
iajs-1045	77	36	f	f	PROPN
iajs-1045	77	37	is	be	AUX
iajs-1045	77	38	anisotropic	anisotropic	ADJ
iajs-1045	77	39	over	over	ADP
iajs-1045	77	40	x	x	VERB
iajs-1045	77	41	is	be	AUX
iajs-1045	77	42	equivalent	equivalent	ADJ
iajs-1045	77	43	to	to	ADP
iajs-1045	77	44	the	the	DET
iajs-1045	77	45	assertion	assertion	NOUN
iajs-1045	77	46	that	that	SCONJ
iajs-1045	77	47	each	each	DET
iajs-1045	77	48	f1,,fs	f1,,fs	NOUN
iajs-1045	77	49	is	be	AUX
iajs-1045	77	50	anisotropic	anisotropic	NOUN
iajs-1045	77	51	over	over	ADP
iajs-1045	77	52	x.	x.	PROPN
iajs-1045	78	1	there	there	PRON
iajs-1045	78	2	are	be	VERB
iajs-1045	78	3	two	two	NUM
iajs-1045	78	4	cases	case	NOUN
iajs-1045	78	5	to	to	PART
iajs-1045	78	6	be	be	AUX
iajs-1045	78	7	considered	consider	VERB
iajs-1045	78	8	.	.	PUNCT
iajs-1045	79	1	suppose	suppose	VERB
iajs-1045	79	2	s	s	X
iajs-1045	79	3	=	=	NOUN
iajs-1045	79	4	1	1	X
iajs-1045	79	5	.	.	PUNCT
iajs-1045	80	1	let	let	VERB
iajs-1045	80	2			NOUN
iajs-1045	80	3	be	be	AUX
iajs-1045	80	4	any	any	DET
iajs-1045	80	5	fuzzy	fuzzy	ADJ
iajs-1045	80	6	subgroup	subgroup	NOUN
iajs-1045	80	7	of	of	ADP
iajs-1045	80	8	a	a	DET
iajs-1045	80	9	such	such	ADJ
iajs-1045	80	10	that	that	SCONJ
iajs-1045	80	11	a	a	PRON
iajs-1045	80	12	is	be	AUX
iajs-1045	80	13	the	the	DET
iajs-1045	80	14	direct	direct	ADJ
iajs-1045	80	15	product	product	NOUN
iajs-1045	80	16	a	a	PRON
iajs-1045	80	17	=	=	PUNCT
iajs-1045	80	18			X
iajs-1045	80	19			PROPN
iajs-1045	80	20	a	a	PROPN
iajs-1045	80	21	,	,	PUNCT
iajs-1045	80	22	and	and	CCONJ
iajs-1045	80	23	let	let	VERB
iajs-1045	80	24	y	y	PROPN
iajs-1045	80	25	=	=	PRON
iajs-1045	80	26			PUNCT
iajs-1045	81	1			NOUN
iajs-1045	81	2			X
iajs-1045	81	3	x.	x.	NOUN
iajs-1045	82	1	then	then	ADV
iajs-1045	82	2	one	one	NUM
iajs-1045	82	3	verifies	verifie	NOUN
iajs-1045	82	4	easily	easily	ADV
iajs-1045	82	5	that	that	SCONJ
iajs-1045	82	6	y	y	PROPN
iajs-1045	82	7	=	=	PRON
iajs-1045	82	8	(	(	PUNCT
iajs-1045	82	9	y	y	NOUN
iajs-1045	82	10	,	,	PUNCT
iajs-1045	82	11	a/	a/	NOUN
iajs-1045	82	12	)	)	PUNCT
iajs-1045	82	13	is	be	AUX
iajs-1045	82	14	a	a	DET
iajs-1045	82	15	subspace	subspace	NOUN
iajs-1045	82	16	of	of	ADP
iajs-1045	82	17	x	x	PUNCT
iajs-1045	82	18	and	and	CCONJ
iajs-1045	82	19	that	that	SCONJ
iajs-1045	82	20	(	(	PUNCT
iajs-1045	82	21	y	y	NOUN
iajs-1045	82	22	,	,	PUNCT
iajs-1045	82	23	a/	a/	NOUN
iajs-1045	82	24	)	)	PUNCT
iajs-1045	82	25			NOUN
iajs-1045	82	26	(	(	PUNCT
iajs-1045	82	27	x,a	x,a	PROPN
iajs-1045	82	28	)	)	PUNCT
iajs-1045	82	29	,	,	PUNCT
iajs-1045	82	30	this	this	DET
iajs-1045	82	31	equivalence	equivalence	NOUN
iajs-1045	82	32	being	be	AUX
iajs-1045	82	33	induced	induce	VERB
iajs-1045	82	34	by	by	ADP
iajs-1045	82	35	the	the	DET
iajs-1045	82	36	natural	natural	ADJ
iajs-1045	82	37	isomorphism	isomorphism	NOUN
iajs-1045	82	38	a/	a/	PROPN
iajs-1045	82	39			PROPN
iajs-1045	82	40	a	a	VERB
iajs-1045	82	41	thus	thus	ADV
iajs-1045	82	42	,	,	PUNCT
iajs-1045	82	43	since	since	SCONJ
iajs-1045	82	44	f1	f1	NOUN
iajs-1045	82	45	is	be	AUX
iajs-1045	82	46	anisotropic	anisotropic	NOUN
iajs-1045	82	47	over	over	ADP
iajs-1045	82	48	x	x	PROPN
iajs-1045	82	49	,	,	PUNCT
iajs-1045	82	50	it	it	PRON
iajs-1045	82	51	(	(	PUNCT
iajs-1045	82	52	and	and	CCONJ
iajs-1045	82	53	then	then	ADV
iajs-1045	82	54	f	f	PROPN
iajs-1045	82	55			PROPN
iajs-1045	82	56	1	1	PROPN
iajs-1045	82	57	f1	f1	PROPN
iajs-1045	82	58	)	)	PUNCT
iajs-1045	82	59	is	be	AUX
iajs-1045	82	60	anisotropic	anisotropic	NOUN
iajs-1045	82	61	over	over	ADP
iajs-1045	82	62	y.	y.	PROPN
iajs-1045	82	63	but	but	CCONJ
iajs-1045	82	64	,	,	PUNCT
iajs-1045	82	65	on	on	ADP
iajs-1045	82	66	the	the	DET
iajs-1045	82	67	other	other	ADJ
iajs-1045	82	68	hand	hand	NOUN
iajs-1045	82	69	gr(x	gr(x	X
iajs-1045	82	70	)	)	PUNCT
iajs-1045	82	71			NOUN
iajs-1045	82	72	1	1	NUM
iajs-1045	82	73	,	,	PUNCT
iajs-1045	82	74	i.e.	i.e.	X
iajs-1045	82	75	a	a	PROPN
iajs-1045	82	76			PROPN
iajs-1045	82	77	a	a	DET
iajs-1045	82	78	,	,	PUNCT
iajs-1045	82	79	i.e.	i.e.	X
iajs-1045	82	80			X
iajs-1045	82	81			NOUN
iajs-1045	82	82	1	1	NUM
iajs-1045	82	83	,	,	PUNCT
iajs-1045	82	84	i.e.	i.e.	X
iajs-1045	82	85	,	,	PUNCT
iajs-1045	82	86	y	y	PROPN
iajs-1045	82	87			PROPN
iajs-1045	82	88	x.	x.	NOUN
iajs-1045	83	1	this	this	PRON
iajs-1045	83	2	contradicts	contradict	VERB
iajs-1045	83	3	the	the	DET
iajs-1045	83	4	minimal	minimal	ADJ
iajs-1045	83	5	choice	choice	NOUN
iajs-1045	83	6	of	of	ADP
iajs-1045	83	7	x.	x.	NOUN
iajs-1045	83	8	thus	thus	ADV
iajs-1045	83	9	s	s	X
iajs-1045	83	10			NUM
iajs-1045	83	11	2	2	NUM
iajs-1045	83	12	.	.	PUNCT
iajs-1045	84	1	it	it	PRON
iajs-1045	84	2	follows	follow	VERB
iajs-1045	84	3	that	that	SCONJ
iajs-1045	84	4	each	each	DET
iajs-1045	84	5	fi	fi	NOUN
iajs-1045	84	6	has	have	VERB
iajs-1045	84	7	strictly	strictly	ADV
iajs-1045	84	8	lower	low	ADJ
iajs-1045	84	9	dimension	dimension	NOUN
iajs-1045	84	10	than	than	ADP
iajs-1045	84	11	f	f	PROPN
iajs-1045	84	12	so	so	ADV
iajs-1045	84	13	by	by	ADP
iajs-1045	84	14	induction	induction	NOUN
iajs-1045	84	15	on	on	ADP
iajs-1045	84	16	the	the	DET
iajs-1045	84	17	dimension	dimension	NOUN
iajs-1045	84	18	,	,	PUNCT
iajs-1045	84	19	there	there	PRON
iajs-1045	84	20	exist	exist	VERB
iajs-1045	84	21	finite	finite	ADJ
iajs-1045	84	22	subspaces	subspace	NOUN
iajs-1045	84	23	z1	z1	PROPN
iajs-1045	84	24	,	,	PUNCT
iajs-1045	84	25			PROPN
iajs-1045	84	26	,	,	PUNCT
iajs-1045	84	27	zs	zs	PROPN
iajs-1045	84	28			ADJ
iajs-1045	84	29			PROPN
iajs-1045	84	30	x	x	PROPN
iajs-1045	84	31	such	such	ADJ
iajs-1045	84	32	that	that	DET
iajs-1045	84	33	fi	fi	NOUN
iajs-1045	84	34	is	be	AUX
iajs-1045	84	35	anisotropic	anisotropic	NOUN
iajs-1045	84	36	over	over	ADP
iajs-1045	84	37	z	z	NOUN
iajs-1045	85	1	i	i	PRON
iajs-1045	85	2			ADJ
iajs-1045	85	3	.	.	PUNCT
iajs-1045	86	1	thus	thus	ADV
iajs-1045	86	2	f1,,fs	f1,,fs	PROPN
iajs-1045	86	3	are	be	AUX
iajs-1045	86	4	all	all	ADV
iajs-1045	86	5	anisotropic	anisotropic	NOUN
iajs-1045	86	6	over	over	ADP
iajs-1045	86	7	the	the	DET
iajs-1045	86	8	subspace	subspace	NOUN
iajs-1045	86	9	of	of	ADP
iajs-1045	86	10	x	x	PROPN
iajs-1045	86	11	generated	generate	VERB
iajs-1045	86	12	by	by	ADP
iajs-1045	86	13	z1	z1	PROPN
iajs-1045	86	14	,	,	PUNCT
iajs-1045	86	15			PROPN
iajs-1045	86	16	,	,	PUNCT
iajs-1045	86	17	zs	zs	PROPN
iajs-1045	86	18	.	.	PROPN
iajs-1045	86	19	denote	denote	VERB
iajs-1045	86	20	this	this	DET
iajs-1045	86	21	space	space	NOUN
iajs-1045	86	22	by	by	ADP
iajs-1045	86	23	z	z	NOUN
iajs-1045	86	24	=	=	SYM
iajs-1045	86	25	(	(	PUNCT
iajs-1045	86	26	z	z	NOUN
iajs-1045	86	27	,a/	,a/	ADJ
iajs-1045	86	28	)	)	PUNCT
iajs-1045	86	29	.	.	PUNCT
iajs-1045	87	1	note	note	VERB
iajs-1045	87	2	z	z	NOUN
iajs-1045	87	3	is	be	AUX
iajs-1045	87	4	still	still	ADV
iajs-1045	87	5	finite	finite	ADJ
iajs-1045	87	6	z	z	NOUN
iajs-1045	87	7	=	=	PUNCT
iajs-1045	87	8			PUNCT
iajs-1045	88	1			NOUN
iajs-1045	88	2			X
iajs-1045	88	3	x.	x.	NOUN
iajs-1045	89	1	then	then	ADV
iajs-1045	89	2	z	z	PROPN
iajs-1045	89	3	=	=	SYM
iajs-1045	89	4	(	(	PUNCT
iajs-1045	89	5	z	z	NOUN
iajs-1045	89	6	,	,	PUNCT
iajs-1045	89	7	a/	a/	NUM
iajs-1045	89	8	)	)	PUNCT
iajs-1045	89	9	is	be	AUX
iajs-1045	89	10	a	a	DET
iajs-1045	89	11	subspace	subspace	NOUN
iajs-1045	89	12	of	of	ADP
iajs-1045	89	13	x	x	X
iajs-1045	89	14	,	,	PUNCT
iajs-1045	89	15	and	and	CCONJ
iajs-1045	89	16	a	a	DET
iajs-1045	89	17	fuzzy	fuzzy	ADJ
iajs-1045	89	18	group	group	NOUN
iajs-1045	89	19	extension	extension	NOUN
iajs-1045	89	20	of	of	ADP
iajs-1045	89	21	z	z	NOUN
iajs-1045	89	22	=	=	SYM
iajs-1045	89	23	(	(	PUNCT
iajs-1045	89	24	z,a/	z,a/	NOUN
iajs-1045	89	25	)	)	PUNCT
iajs-1045	89	26	.	.	PUNCT
iajs-1045	90	1	moreover	moreover	ADV
iajs-1045	90	2	,	,	PUNCT
iajs-1045	90	3	since	since	SCONJ
iajs-1045	90	4	1,,s	1,,	NOUN
iajs-1045	90	5	are	be	AUX
iajs-1045	90	6	distinct	distinct	ADJ
iajs-1045	90	7	modulo	modulo	PROPN
iajs-1045	90	8	a	a	PROPN
iajs-1045	90	9	,	,	PUNCT
iajs-1045	90	10	f	f	PROPN
iajs-1045	90	11	is	be	AUX
iajs-1045	90	12	anisotropic	anisotropic	NOUN
iajs-1045	90	13	over	over	ADP
iajs-1045	90	14	z.	z.	PROPN
iajs-1045	91	1	thus	thus	ADV
iajs-1045	91	2	,	,	PUNCT
iajs-1045	91	3	by	by	ADP
iajs-1045	91	4	minimal	minimal	ADJ
iajs-1045	91	5	choice	choice	NOUN
iajs-1045	91	6	of	of	ADP
iajs-1045	91	7	x	x	X
iajs-1045	91	8	,	,	PUNCT
iajs-1045	91	9	z=	z=	NOUN
iajs-1045	91	10	x	x	X
iajs-1045	91	11	,	,	PUNCT
iajs-1045	91	12	i.e.	i.e.	X
iajs-1045	91	13			PUNCT
iajs-1045	91	14	=	=	SYM
iajs-1045	91	15	1	1	NUM
iajs-1045	91	16	,	,	PUNCT
iajs-1045	91	17	i.e.	i.e.	X
iajs-1045	91	18	,	,	PUNCT
iajs-1045	91	19	z	z	NUM
iajs-1045	91	20	=	=	SYM
iajs-1045	91	21	x	x	PROPN
iajs-1045	91	22	is	be	AUX
iajs-1045	91	23	finite	finite	ADJ
iajs-1045	91	24	.	.	PUNCT
iajs-1045	92	1	however	however	ADV
iajs-1045	92	2	,	,	PUNCT
iajs-1045	92	3	x	x	PRON
iajs-1045	92	4	itself	itself	PRON
iajs-1045	92	5	could	could	AUX
iajs-1045	92	6	be	be	AUX
iajs-1045	92	7	infinite	infinite	ADJ
iajs-1045	92	8	(	(	PUNCT
iajs-1045	92	9	since	since	ADV
iajs-1045	92	10	,	,	PUNCT
iajs-1045	92	11	a	a	PRON
iajs-1045	92	12	priori	priori	X
iajs-1045	92	13	,	,	PUNCT
iajs-1045	92	14	gr(x	gr(x	X
iajs-1045	92	15	)	)	PUNCT
iajs-1045	92	16	could	could	AUX
iajs-1045	92	17	be	be	AUX
iajs-1045	92	18	infinite	infinite	ADJ
iajs-1045	92	19	)	)	PUNCT
iajs-1045	92	20	.	.	PUNCT
iajs-1045	93	1	define	define	VERB
iajs-1045	93	2	a	a	PROPN
iajs-1045	93	3	to	to	PART
iajs-1045	93	4	be	be	AUX
iajs-1045	93	5	the	the	DET
iajs-1045	93	6	fuzzy	fuzzy	ADJ
iajs-1045	93	7	subgroup	subgroup	NOUN
iajs-1045	93	8	of	of	ADP
iajs-1045	93	9	a	a	DET
iajs-1045	93	10	generated	generate	VERB
iajs-1045	93	11	by	by	ADP
iajs-1045	93	12	a	a	PROPN
iajs-1045	93	13	and	and	CCONJ
iajs-1045	93	14	1,,s	1,,	NOUN
iajs-1045	93	15	,	,	PUNCT
iajs-1045	93	16	and	and	CCONJ
iajs-1045	93	17	let	let	VERB
iajs-1045	93	18	x	x	NOUN
iajs-1045	93	19	denote	denote	VERB
iajs-1045	93	20	the	the	DET
iajs-1045	93	21	restriction	restriction	NOUN
iajs-1045	93	22	of	of	ADP
iajs-1045	93	23	x	x	PUNCT
iajs-1045	93	24	to	to	ADP
iajs-1045	93	25	a	a	PROPN
iajs-1045	93	26	.	.	PUNCT
iajs-1045	94	1	thus	thus	ADV
iajs-1045	94	2	(	(	PUNCT
iajs-1045	94	3	x	x	X
iajs-1045	94	4	,	,	PUNCT
iajs-1045	94	5	a	a	PRON
iajs-1045	94	6	)	)	PUNCT
iajs-1045	94	7	is	be	AUX
iajs-1045	94	8	a	a	DET
iajs-1045	94	9	fuzzy	fuzzy	ADJ
iajs-1045	94	10	group	group	NOUN
iajs-1045	94	11	extension	extension	NOUN
iajs-1045	94	12	see	see	NOUN
iajs-1045	94	13	[	[	PUNCT
iajs-1045	94	14	2	2	NUM
iajs-1045	94	15	]	]	PUNCT
iajs-1045	94	16	of	of	ADP
iajs-1045	94	17	(	(	PUNCT
iajs-1045	94	18	x	x	PROPN
iajs-1045	94	19	,	,	PUNCT
iajs-1045	94	20	a	a	PROPN
iajs-1045	94	21	)	)	PUNCT
iajs-1045	94	22	which	which	PRON
iajs-1045	94	23	,	,	PUNCT
iajs-1045	94	24	inturn	inturn	NOUN
iajs-1045	94	25	,	,	PUNCT
iajs-1045	94	26	is	be	AUX
iajs-1045	94	27	a	a	DET
iajs-1045	94	28	fuzzy	fuzzy	ADJ
iajs-1045	94	29	group	group	NOUN
iajs-1045	94	30	extension	extension	NOUN
iajs-1045	94	31	of	of	ADP
iajs-1045	94	32	(	(	PUNCT
iajs-1045	94	33	x,a	x,a	PROPN
iajs-1045	94	34	)	)	PUNCT
iajs-1045	94	35	.	.	PUNCT
iajs-1045	95	1	moreover	moreover	ADV
iajs-1045	95	2	(	(	PUNCT
iajs-1045	95	3	x	x	PROPN
iajs-1045	95	4	,	,	PUNCT
iajs-1045	95	5	a	a	PROPN
iajs-1045	95	6	)	)	PUNCT
iajs-1045	95	7	is	be	AUX
iajs-1045	95	8	finite	finite	ADJ
iajs-1045	95	9	,	,	PUNCT
iajs-1045	95	10	and	and	CCONJ
iajs-1045	95	11	f	f	PROPN
iajs-1045	95	12	is	be	AUX
iajs-1045	95	13	anisotropic	anisotropic	NOUN
iajs-1045	95	14	over	over	ADP
iajs-1045	95	15	x	x	PROPN
iajs-1045	95	16	.	.	PUNCT
iajs-1045	96	1	finally	finally	ADV
iajs-1045	96	2	,	,	PUNCT
iajs-1045	96	3	let	let	VERB
iajs-1045	96	4			NOUN
iajs-1045	96	5	be	be	AUX
iajs-1045	96	6	fuzzy	fuzzy	ADJ
iajs-1045	96	7	subgroup	subgroup	NOUN
iajs-1045	96	8	of	of	ADP
iajs-1045	96	9	a	a	DET
iajs-1045	96	10	so	so	ADV
iajs-1045	96	11	that	that	SCONJ
iajs-1045	96	12	a	a	DET
iajs-1045	96	13	=	=	NOUN
iajs-1045	96	14	a	a	X
iajs-1045	96	15	,	,	PUNCT
iajs-1045	96	16	and	and	CCONJ
iajs-1045	96	17	let	let	VERB
iajs-1045	96	18	y=	y=	PROPN
iajs-1045	96	19			NOUN
iajs-1045	96	20	x	x	VERB
iajs-1045	96	21	.	.	PUNCT
iajs-1045	97	1	then	then	ADV
iajs-1045	97	2	y	y	PROPN
iajs-1045	97	3	=	=	SYM
iajs-1045	97	4	(	(	PUNCT
iajs-1045	97	5	y	y	NOUN
iajs-1045	97	6	,	,	PUNCT
iajs-1045	97	7	a/	a/	NOUN
iajs-1045	97	8	)	)	PUNCT
iajs-1045	97	9	is	be	AUX
iajs-1045	97	10	a	a	DET
iajs-1045	97	11	subspace	subspace	NOUN
iajs-1045	97	12	of	of	ADP
iajs-1045	97	13	x	x	PRON
iajs-1045	97	14	naturally	naturally	ADV
iajs-1045	97	15	equivalent	equivalent	ADJ
iajs-1045	97	16	to	to	ADP
iajs-1045	97	17	(	(	PUNCT
iajs-1045	97	18	x	x	PROPN
iajs-1045	97	19	,	,	PUNCT
iajs-1045	97	20	a	a	PROPN
iajs-1045	97	21	)	)	PUNCT
iajs-1045	97	22	.	.	PUNCT
iajs-1045	98	1	thus	thus	ADV
iajs-1045	98	2	y	y	PROPN
iajs-1045	98	3	is	be	AUX
iajs-1045	98	4	finite	finite	ADJ
iajs-1045	98	5	,	,	PUNCT
iajs-1045	98	6	and	and	CCONJ
iajs-1045	98	7	f	f	PROPN
iajs-1045	98	8	is	be	AUX
iajs-1045	98	9	anisotropic	anisotropic	NOUN
iajs-1045	98	10	over	over	ADP
iajs-1045	98	11	y.	y.	NOUN
iajs-1045	98	12	thus	thus	ADV
iajs-1045	98	13	y	y	PROPN
iajs-1045	99	1	=	=	PUNCT
iajs-1045	99	2	x	x	X
iajs-1045	99	3	is	be	AUX
iajs-1045	99	4	finite	finite	PROPN
iajs-1045	99	5	.	.	PUNCT
iajs-1045	100	1	notice	notice	NOUN
iajs-1045	100	2	,	,	PUNCT
iajs-1045	100	3	the	the	DET
iajs-1045	100	4	condition	condition	NOUN
iajs-1045	100	5	x(ai	x(ai	PROPN
iajs-1045	100	6	–	–	PUNCT
iajs-1045	100	7	1	1	X
iajs-1045	100	8	)	)	PUNCT
iajs-1045	100	9			PROPN
iajs-1045	100	10	x(ai	x(ai	PROPN
iajs-1045	100	11	)	)	PUNCT
iajs-1045	100	12	is	be	AUX
iajs-1045	100	13	equivalent	equivalent	ADJ
iajs-1045	100	14	to	to	PART
iajs-1045	100	15	d<1,ai	d<1,ai	VERB
iajs-1045	100	16	>	>	PUNCT
iajs-1045	101	1			PROPN
iajs-1045	101	2	d<1,ai	d<1,ai	NOUN
iajs-1045	101	3	–	–	PUNCT
iajs-1045	101	4	1	1	NUM
iajs-1045	101	5	>	>	PUNCT
iajs-1045	101	6	.	.	PUNCT
iajs-1045	102	1	theorem	theorem	ADJ
iajs-1045	102	2	4	4	NUM
iajs-1045	102	3	(	(	PUNCT
iajs-1045	102	4	i	i	NOUN
iajs-1045	102	5	)	)	PUNCT
iajs-1045	102	6	suppose	suppose	VERB
iajs-1045	102	7	xi	xi	X
iajs-1045	102	8	=	=	SYM
iajs-1045	102	9	(	(	PUNCT
iajs-1045	102	10	xi	xi	PROPN
iajs-1045	102	11	,	,	PUNCT
iajs-1045	102	12	a/i	a/i	PROPN
iajs-1045	102	13	)	)	PUNCT
iajs-1045	102	14	,	,	PUNCT
iajs-1045	102	15	i	i	PRON
iajs-1045	102	16	=	=	NOUN
iajs-1045	102	17	1,,n	1,,n	NUM
iajs-1045	102	18	are	be	AUX
iajs-1045	102	19	subspaces	subspace	NOUN
iajs-1045	102	20	of	of	ADP
iajs-1045	102	21	x	x	PUNCT
iajs-1045	102	22	generating	generate	VERB
iajs-1045	102	23	x.	x.	NOUN
iajs-1045	102	24	then	then	ADV
iajs-1045	102	25	:	:	PUNCT
iajs-1045	102	26	cl(x	cl(x	X
iajs-1045	102	27	)	)	PUNCT
iajs-1045	102	28	=	=	SYM
iajs-1045	103	1	n	n	CCONJ
iajs-1045	103	2	i	i	PROPN
iajs-1045	103	3	1	1	NUM
iajs-1045	103	4	cl(x	cl(x	NUM
iajs-1045	103	5	)	)	PUNCT
iajs-1045	104	1	i	i	PRON
iajs-1045	104	2			VERB
iajs-1045	104	3			X
iajs-1045	104	4	.	.	PUNCT
iajs-1045	105	1	(	(	PUNCT
iajs-1045	105	2	ii	ii	NOUN
iajs-1045	105	3	)	)	PUNCT
iajs-1045	105	4	if	if	SCONJ
iajs-1045	105	5	,	,	PUNCT
iajs-1045	105	6	in	in	ADP
iajs-1045	105	7	addition	addition	NOUN
iajs-1045	105	8	,	,	PUNCT
iajs-1045	105	9	x	x	PUNCT
iajs-1045	105	10	=	=	SYM
iajs-1045	105	11	x1	x1	PROPN
iajs-1045	105	12			PROPN
iajs-1045	105	13	xn	xn	PROPN
iajs-1045	105	14	,	,	PUNCT
iajs-1045	105	15	then	then	ADV
iajs-1045	105	16	:	:	PUNCT
iajs-1045	105	17	c1(x	c1(x	X
iajs-1045	105	18	)	)	PUNCT
iajs-1045	105	19	=	=	SYM
iajs-1045	106	1	n	n	CCONJ
iajs-1045	106	2	i	i	NOUN
iajs-1045	106	3	1	1	NUM
iajs-1045	106	4	c1(x	c1(x	VERB
iajs-1045	106	5	)	)	PUNCT
iajs-1045	107	1	i	i	PRON
iajs-1045	107	2			VERB
iajs-1045	107	3			X
iajs-1045	107	4	.	.	PUNCT
iajs-1045	108	1	(	(	PUNCT
iajs-1045	108	2	iii	iii	X
iajs-1045	108	3	)	)	PUNCT
iajs-1045	108	4	if	if	SCONJ
iajs-1045	108	5	x	x	PRON
iajs-1045	108	6	is	be	AUX
iajs-1045	108	7	a	a	DET
iajs-1045	108	8	fuzzy	fuzzy	ADJ
iajs-1045	108	9	group	group	NOUN
iajs-1045	108	10	extension	extension	NOUN
iajs-1045	108	11	of	of	ADP
iajs-1045	108	12	x	x	PROPN
iajs-1045	108	13	,	,	PUNCT
iajs-1045	108	14	then	then	ADV
iajs-1045	108	15	cl(x	cl(x	X
iajs-1045	108	16	)	)	PUNCT
iajs-1045	108	17	=	=	SYM
iajs-1045	108	18	cl(x	cl(x	PROPN
iajs-1045	108	19	)	)	PUNCT
iajs-1045	108	20	,	,	PUNCT
iajs-1045	108	21	except	except	SCONJ
iajs-1045	108	22	in	in	ADP
iajs-1045	108	23	the	the	DET
iajs-1045	108	24	case	case	NOUN
iajs-1045	108	25	x	x	NUM
iajs-1045	108	26	=	=	SYM
iajs-1045	108	27	1	1	NUM
iajs-1045	108	28	(	(	PUNCT
iajs-1045	108	29	in	in	ADP
iajs-1045	108	30	which	which	DET
iajs-1045	108	31	case	case	NOUN
iajs-1045	108	32	x	x	X
iajs-1045	108	33	is	be	AUX
iajs-1045	108	34	a	a	DET
iajs-1045	108	35	fan	fan	NOUN
iajs-1045	108	36	)	)	PUNCT
iajs-1045	108	37	.	.	PUNCT
iajs-1045	109	1	ibn	ibn	PROPN
iajs-1045	109	2	alhaitham	alhaitham	PROPN
iajs-1045	109	3	j.	j.	PROPN
iajs-1045	109	4	for	for	ADP
iajs-1045	109	5	pure	pure	ADJ
iajs-1045	109	6	&	&	CCONJ
iajs-1045	109	7	appl	appl	PROPN
iajs-1045	109	8	.	.	PUNCT
iajs-1045	110	1	sci	sci	PROPN
iajs-1045	110	2	.	.	PUNCT
iajs-1045	111	1	vol.22	vol.22	PROPN
iajs-1045	111	2	(	(	PUNCT
iajs-1045	111	3	4	4	NUM
iajs-1045	111	4	)	)	PUNCT
iajs-1045	111	5	2009	2009	NUM
iajs-1045	111	6	proof	proof	NOUN
iajs-1045	111	7	:	:	PUNCT
iajs-1045	111	8	(	(	PUNCT
iajs-1045	111	9	i	i	NOUN
iajs-1045	111	10	)	)	PUNCT
iajs-1045	111	11	suppose	suppose	VERB
iajs-1045	111	12	x(aj	x(aj	PROPN
iajs-1045	111	13	–	–	PUNCT
iajs-1045	111	14	1	1	NUM
iajs-1045	111	15	)	)	PUNCT
iajs-1045	111	16			PROPN
iajs-1045	111	17	x(aj	x(aj	PROPN
iajs-1045	111	18	)	)	PUNCT
iajs-1045	111	19	,	,	PUNCT
iajs-1045	111	20	j	j	X
iajs-1045	111	21	=	=	PUNCT
iajs-1045	111	22	1,,k	1,,k	NUM
iajs-1045	111	23	.	.	PUNCT
iajs-1045	112	1	then	then	ADV
iajs-1045	112	2	for	for	ADP
iajs-1045	112	3	each	each	DET
iajs-1045	112	4	i	i	PRON
iajs-1045	112	5	,	,	PUNCT
iajs-1045	112	6	1	1	NUM
iajs-1045	112	7			NOUN
iajs-1045	112	8	i	i	PRON
iajs-1045	112	9			NOUN
iajs-1045	112	10	n	n	CCONJ
iajs-1045	112	11	,	,	PUNCT
iajs-1045	112	12	xi(aj	xi(aj	PROPN
iajs-1045	112	13	–	–	PUNCT
iajs-1045	112	14	1	1	X
iajs-1045	112	15	)	)	PUNCT
iajs-1045	112	16			PROPN
iajs-1045	112	17	xi(aj	xi(aj	PROPN
iajs-1045	112	18	)	)	PUNCT
iajs-1045	112	19	.	.	PUNCT
iajs-1045	113	1	moreover	moreover	ADV
iajs-1045	113	2	,	,	PUNCT
iajs-1045	113	3	since	since	SCONJ
iajs-1045	113	4	x(aj	x(aj	PROPN
iajs-1045	113	5	–	–	PUNCT
iajs-1045	113	6	1	1	X
iajs-1045	113	7	)	)	PUNCT
iajs-1045	113	8			NOUN
iajs-1045	113	9	x(aj	x(aj	PROPN
iajs-1045	113	10	)	)	PUNCT
iajs-1045	113	11	,	,	PUNCT
iajs-1045	113	12	there	there	PRON
iajs-1045	113	13	exists	exist	VERB
iajs-1045	113	14	i	i	PRON
iajs-1045	113	15	,	,	PUNCT
iajs-1045	113	16	1	1	NUM
iajs-1045	113	17			NOUN
iajs-1045	113	18	i	i	PRON
iajs-1045	113	19			NOUN
iajs-1045	113	20	n	n	CCONJ
iajs-1045	113	21	such	such	ADJ
iajs-1045	113	22	that	that	SCONJ
iajs-1045	113	23	x	x	SYM
iajs-1045	113	24	i(aj	i(aj	PROPN
iajs-1045	113	25	–	–	PUNCT
iajs-1045	113	26	1	1	X
iajs-1045	113	27	)	)	PUNCT
iajs-1045	113	28			PROPN
iajs-1045	113	29	xi(aj	xi(aj	PROPN
iajs-1045	113	30	)	)	PUNCT
iajs-1045	113	31	.	.	PUNCT
iajs-1045	114	1	(	(	PUNCT
iajs-1045	114	2	for	for	ADP
iajs-1045	114	3	if	if	SCONJ
iajs-1045	114	4	xi(aj	xi(aj	PROPN
iajs-1045	114	5	–	–	PUNCT
iajs-1045	114	6	1	1	X
iajs-1045	114	7	)	)	PUNCT
iajs-1045	114	8	=	=	SYM
iajs-1045	114	9	xi(aj	xi(aj	PROPN
iajs-1045	114	10	)	)	PUNCT
iajs-1045	114	11	for	for	ADP
iajs-1045	114	12	all	all	DET
iajs-1045	114	13	i	i	PRON
iajs-1045	114	14			NOUN
iajs-1045	114	15	n	n	CCONJ
iajs-1045	114	16	,	,	PUNCT
iajs-1045	114	17	then	then	ADV
iajs-1045	114	18	aj	aj	PROPN
iajs-1045	114	19	aj	aj	PROPN
iajs-1045	114	20	–	–	PUNCT
iajs-1045	114	21	1	1	NUM
iajs-1045	114	22			NOUN
iajs-1045	114	23	n	n	CCONJ
iajs-1045	114	24	i	i	NOUN
iajs-1045	114	25	1	1	NUM
iajs-1045	114	26	1	1	NUM
iajs-1045	115	1	i	i	PRON
iajs-1045	115	2			VERB
iajs-1045	115	3			ADV
iajs-1045	116	1			PROPN
iajs-1045	116	2	,	,	PUNCT
iajs-1045	116	3	i.e.	i.e.	X
iajs-1045	116	4	,	,	PUNCT
iajs-1045	116	5	aj	aj	PROPN
iajs-1045	116	6	=	=	PROPN
iajs-1045	116	7	aj	aj	PROPN
iajs-1045	116	8	–	–	PUNCT
iajs-1045	116	9	1	1	NUM
iajs-1045	116	10	a	a	DET
iajs-1045	116	11	contradiction	contradiction	NOUN
iajs-1045	116	12	)	)	PUNCT
iajs-1045	116	13	.	.	PUNCT
iajs-1045	117	1	this	this	PRON
iajs-1045	117	2	holds	hold	VERB
iajs-1045	117	3	for	for	ADP
iajs-1045	117	4	j	j	PROPN
iajs-1045	117	5	=	=	SYM
iajs-1045	117	6	1,,k	1,,k	NUM
iajs-1045	117	7	.	.	PUNCT
iajs-1045	118	1	simple	simple	ADJ
iajs-1045	118	2	counting	counting	NOUN
iajs-1045	118	3	yields	yield	NOUN
iajs-1045	118	4	n	n	CCONJ
iajs-1045	118	5	n	n	NOUN
iajs-1045	118	6	i	i	NOUN
iajs-1045	118	7	1	1	NUM
iajs-1045	118	8	i	i	NOUN
iajs-1045	118	9	1	1	NUM
iajs-1045	118	10	k	k	NOUN
iajs-1045	118	11	cl(x	cl(x	X
iajs-1045	118	12	)	)	PUNCT
iajs-1045	118	13	,	,	PUNCT
iajs-1045	118	14	i.e.	i.e.	X
iajs-1045	118	15	,cl(x	,cl(x	PUNCT
iajs-1045	118	16	)	)	PUNCT
iajs-1045	118	17	cl(x	cl(x	X
iajs-1045	118	18	)	)	PUNCT
iajs-1045	119	1	i	i	PRON
iajs-1045	119	2	i	i	PRON
iajs-1045	119	3			VERB
iajs-1045	119	4			NUM
iajs-1045	119	5			NOUN
iajs-1045	119	6			X
iajs-1045	119	7			X
iajs-1045	119	8	.	.	PUNCT
iajs-1045	120	1	(	(	PUNCT
iajs-1045	120	2	ii	ii	X
iajs-1045	120	3	)	)	PUNCT
iajs-1045	120	4	we	we	PRON
iajs-1045	120	5	are	be	AUX
iajs-1045	120	6	assuming	assume	VERB
iajs-1045	120	7	x	x	X
iajs-1045	120	8	=	=	SYM
iajs-1045	120	9	ui	ui	PROPN
iajs-1045	120	10	xi	xi	X
iajs-1045	120	11	and	and	CCONJ
iajs-1045	120	12	the	the	DET
iajs-1045	120	13	natural	natural	ADJ
iajs-1045	120	14	homomorphism	homomorphism	NOUN
iajs-1045	120	15	from	from	ADP
iajs-1045	120	16	a	a	PRON
iajs-1045	120	17	into	into	ADP
iajs-1045	120	18	ia	ia	PROPN
iajs-1045	120	19	/i	/i	PROPN
iajs-1045	120	20	is	be	AUX
iajs-1045	120	21	an	an	DET
iajs-1045	120	22	isomorphism	isomorphism	NOUN
iajs-1045	120	23	.	.	PUNCT
iajs-1045	121	1	suppose	suppose	VERB
iajs-1045	122	1	xi(ai	xi(ai	PROPN
iajs-1045	122	2	,	,	PUNCT
iajs-1045	122	3	j	j	PROPN
iajs-1045	122	4	–	–	PUNCT
iajs-1045	122	5	1	1	NUM
iajs-1045	122	6	)	)	PUNCT
iajs-1045	122	7			PROPN
iajs-1045	122	8	xi	xi	X
iajs-1045	122	9	(	(	PUNCT
iajs-1045	122	10	ai	ai	PROPN
iajs-1045	122	11	,	,	PUNCT
iajs-1045	122	12	j	j	PROPN
iajs-1045	122	13	)	)	PUNCT
iajs-1045	122	14	,	,	PUNCT
iajs-1045	122	15	j	j	X
iajs-1045	122	16	=	=	PUNCT
iajs-1045	123	1	1,,ki	1,,ki	NUM
iajs-1045	123	2	,	,	PUNCT
iajs-1045	123	3	i	i	PRON
iajs-1045	123	4	=	=	NOUN
iajs-1045	123	5	1,,n	1,,n	NUM
iajs-1045	123	6	.	.	PUNCT
iajs-1045	124	1	we	we	PRON
iajs-1045	124	2	may	may	AUX
iajs-1045	124	3	as	as	ADV
iajs-1045	124	4	well	well	ADV
iajs-1045	124	5	assume	assume	VERB
iajs-1045	124	6	ai,0	ai,0	PROPN
iajs-1045	124	7	=	=	SYM
iajs-1045	124	8	–	–	PUNCT
iajs-1045	124	9	1	1	NUM
iajs-1045	124	10	,	,	PUNCT
iajs-1045	124	11	and	and	CCONJ
iajs-1045	124	12	a	a	DET
iajs-1045	124	13	1	1	NUM
iajs-1045	124	14	ii	ii	NOUN
iajs-1045	124	15	,	,	PUNCT
iajs-1045	124	16	k	k	PROPN
iajs-1045	124	17			PROPN
iajs-1045	124	18	.	.	PUNCT
iajs-1045	125	1	choose	choose	VERB
iajs-1045	125	2	elements	element	NOUN
iajs-1045	125	3	bij	bij	VERB
iajs-1045	125	4			PROPN
iajs-1045	125	5	a	a	DET
iajs-1045	125	6	such	such	ADJ
iajs-1045	125	7	that	that	PRON
iajs-1045	125	8	:	:	PUNCT
iajs-1045	125	9	bij	bij	VERB
iajs-1045	125	10	=	=	SYM
iajs-1045	125	11	1	1	NUM
iajs-1045	125	12	(	(	PUNCT
iajs-1045	125	13	mod	mod	NOUN
iajs-1045	125	14	k	k	NOUN
iajs-1045	125	15	)	)	PUNCT
iajs-1045	125	16	for	for	ADP
iajs-1045	125	17	k	k	PROPN
iajs-1045	125	18	<	<	X
iajs-1045	125	19	i.	i.	PROPN
iajs-1045	125	20	bij	bij	PROPN
iajs-1045	125	21			PROPN
iajs-1045	125	22	aij	aij	PROPN
iajs-1045	125	23	(	(	PUNCT
iajs-1045	125	24	mod	mod	PROPN
iajs-1045	125	25	i	i	PROPN
iajs-1045	125	26	)	)	PUNCT
iajs-1045	125	27	,	,	PUNCT
iajs-1045	125	28	and	and	CCONJ
iajs-1045	125	29	bij	bij	VERB
iajs-1045	125	30			PROPN
iajs-1045	125	31	–	–	PUNCT
iajs-1045	125	32	1	1	NUM
iajs-1045	125	33	(	(	PUNCT
iajs-1045	125	34	mod	mod	PROPN
iajs-1045	125	35	k	k	NOUN
iajs-1045	125	36	)	)	PUNCT
iajs-1045	125	37	,	,	PUNCT
iajs-1045	125	38	for	for	SCONJ
iajs-1045	125	39	k	k	PROPN
iajs-1045	125	40	>	>	X
iajs-1045	125	41	i.	i.	PROPN
iajs-1045	125	42	notice	notice	VERB
iajs-1045	125	43	that	that	SCONJ
iajs-1045	125	44	x(bij	x(bij	PROPN
iajs-1045	125	45	)	)	PUNCT
iajs-1045	125	46	=	=	PUNCT
iajs-1045	126	1	(	(	PUNCT
iajs-1045	126	2	us	we	PRON
iajs-1045	126	3	<	<	X
iajs-1045	126	4	i	i	PROPN
iajs-1045	126	5	xs	xs	NOUN
iajs-1045	126	6	)	)	PUNCT
iajs-1045	126	7	u	u	NOUN
iajs-1045	126	8	x	x	X
iajs-1045	126	9	i(aij	i(aij	PROPN
iajs-1045	126	10	)	)	PUNCT
iajs-1045	126	11	.	.	PUNCT
iajs-1045	127	1	it	it	PRON
iajs-1045	127	2	follows	follow	VERB
iajs-1045	127	3	that	that	SCONJ
iajs-1045	127	4	x(b10	x(b10	ADV
iajs-1045	127	5	)	)	PUNCT
iajs-1045	127	6			NUM
iajs-1045	127	7	11x	11x	NOUN
iajs-1045	127	8	(	(	PUNCT
iajs-1045	127	9	)	)	PUNCT
iajs-1045	127	10	kb	kb	PROPN
iajs-1045	127	11	=	=	SYM
iajs-1045	127	12	x(b20	x(b20	X
iajs-1045	127	13	)	)	PUNCT
iajs-1045	128	1			PROPN
iajs-1045	128	2	n	n	NUM
iajs-1045	128	3	x	x	NOUN
iajs-1045	128	4	(	(	PUNCT
iajs-1045	128	5	)	)	PUNCT
iajs-1045	128	6	n	n	CCONJ
iajs-1045	128	7	kb	kb	PROPN
iajs-1045	128	8	.	.	PUNCT
iajs-1045	129	1	there	there	PRON
iajs-1045	129	2	are	be	VERB
iajs-1045	129	3			PROPN
iajs-1045	129	4	ki	ki	PROPN
iajs-1045	129	5	inequalities	inequality	NOUN
iajs-1045	129	6	in	in	ADP
iajs-1045	129	7	this	this	DET
iajs-1045	129	8	chain	chain	NOUN
iajs-1045	129	9	,	,	PUNCT
iajs-1045	129	10	so	so	ADV
iajs-1045	129	11	cl(x	cl(x	NOUN
iajs-1045	129	12	)	)	PUNCT
iajs-1045	129	13			NUM
iajs-1045	129	14			X
iajs-1045	130	1	ki	ki	INTJ
iajs-1045	130	2	,	,	PUNCT
iajs-1045	130	3	and	and	CCONJ
iajs-1045	130	4	hence	hence	ADV
iajs-1045	130	5	cl(x	cl(x	NUM
iajs-1045	130	6	)	)	PUNCT
iajs-1045	130	7			NUM
iajs-1045	130	8			X
iajs-1045	130	9	cl(xi	cl(xi	NOUN
iajs-1045	130	10	)	)	PUNCT
iajs-1045	130	11	.	.	PUNCT
iajs-1045	131	1	the	the	DET
iajs-1045	131	2	other	other	ADJ
iajs-1045	131	3	inequality	inequality	NOUN
iajs-1045	131	4	follows	follow	VERB
iajs-1045	131	5	from	from	ADP
iajs-1045	131	6	(	(	PUNCT
iajs-1045	131	7	i	i	NOUN
iajs-1045	131	8	)	)	PUNCT
iajs-1045	131	9	.	.	PUNCT
iajs-1045	132	1	(	(	PUNCT
iajs-1045	132	2	iii	iii	X
iajs-1045	132	3	)	)	PUNCT
iajs-1045	132	4	suppose	suppose	VERB
iajs-1045	132	5	x	x	SCONJ
iajs-1045	132	6	1	1	X
iajs-1045	132	7	.	.	PUNCT
iajs-1045	132	8	suppose	suppose	VERB
iajs-1045	132	9	x(ai	x(ai	PROPN
iajs-1045	132	10	–	–	PUNCT
iajs-1045	132	11	1	1	NUM
iajs-1045	132	12	)	)	PUNCT
iajs-1045	132	13			PROPN
iajs-1045	132	14	x(ai	x(ai	PROPN
iajs-1045	132	15	)	)	PUNCT
iajs-1045	132	16	,	,	PUNCT
iajs-1045	132	17	i	i	NOUN
iajs-1045	132	18	=	=	PUNCT
iajs-1045	132	19	1,,k	1,,k	NUM
iajs-1045	132	20	,	,	PUNCT
iajs-1045	132	21	with	with	ADP
iajs-1045	132	22	ai	ai	PROPN
iajs-1045	132	23			PROPN
iajs-1045	132	24	a.	a.	PROPN
iajs-1045	133	1	then	then	ADV
iajs-1045	133	2	clearly	clearly	ADV
iajs-1045	133	3	x(ai	x(ai	PROPN
iajs-1045	133	4	–	–	PUNCT
iajs-1045	133	5	1	1	X
iajs-1045	133	6	)	)	PUNCT
iajs-1045	133	7			PROPN
iajs-1045	133	8	x(ai	x(ai	PROPN
iajs-1045	133	9	)	)	PUNCT
iajs-1045	133	10	,	,	PUNCT
iajs-1045	133	11	i	i	NOUN
iajs-1045	133	12	=	=	PUNCT
iajs-1045	133	13	1,,k	1,,k	NUM
iajs-1045	133	14	.	.	PUNCT
iajs-1045	134	1	thus	thus	ADV
iajs-1045	134	2	cl(x	cl(x	NOUN
iajs-1045	134	3	)	)	PUNCT
iajs-1045	134	4			NUM
iajs-1045	134	5	cl(x	cl(x	PROPN
iajs-1045	134	6	)	)	PUNCT
iajs-1045	134	7	.	.	PUNCT
iajs-1045	135	1	now	now	ADV
iajs-1045	135	2	suppose	suppose	VERB
iajs-1045	135	3	d<1,ai	d<1,ai	NOUN
iajs-1045	135	4	>	>	PUNCT
iajs-1045	135	5			PROPN
iajs-1045	135	6	d<1	d<1	PROPN
iajs-1045	135	7	,	,	PUNCT
iajs-1045	135	8	ai	ai	INTJ
iajs-1045	135	9	–	–	PUNCT
iajs-1045	135	10	1	1	NUM
iajs-1045	135	11	>	>	PUNCT
iajs-1045	135	12	,	,	PUNCT
iajs-1045	135	13	i	i	PRON
iajs-1045	135	14	=	=	PUNCT
iajs-1045	135	15	1,,k	1,,k	NUM
iajs-1045	135	16	,	,	PUNCT
iajs-1045	135	17	with	with	ADP
iajs-1045	135	18	a1	a1	NOUN
iajs-1045	135	19	,	,	PUNCT
iajs-1045	135	20			PROPN
iajs-1045	135	21	,	,	PUNCT
iajs-1045	135	22	ak	ak	PROPN
iajs-1045	135	23			PROPN
iajs-1045	135	24	a.	a.	NOUN
iajs-1045	136	1	we	we	PRON
iajs-1045	136	2	may	may	AUX
iajs-1045	136	3	assume	assume	VERB
iajs-1045	136	4	a0	a0	PROPN
iajs-1045	136	5	=	=	SYM
iajs-1045	136	6	–	–	PUNCT
iajs-1045	136	7	1	1	NUM
iajs-1045	136	8	,	,	PUNCT
iajs-1045	136	9	ak	ak	PROPN
iajs-1045	136	10	=	=	ADJ
iajs-1045	136	11	1	1	PROPN
iajs-1045	136	12	.	.	PUNCT
iajs-1045	137	1	then	then	ADV
iajs-1045	137	2	a1	a1	PROPN
iajs-1045	137	3			PROPN
iajs-1045	137	4	–	–	PUNCT
iajs-1045	137	5	1	1	X
iajs-1045	137	6	.	.	X
iajs-1045	138	1	there	there	PRON
iajs-1045	138	2	are	be	VERB
iajs-1045	138	3	two	two	NUM
iajs-1045	138	4	cases	case	NOUN
iajs-1045	138	5	to	to	PART
iajs-1045	138	6	be	be	AUX
iajs-1045	138	7	considered	consider	VERB
iajs-1045	138	8	1	1	NUM
iajs-1045	138	9	st	st	NOUN
iajs-1045	138	10	case	case	NOUN
iajs-1045	138	11	:	:	PUNCT
iajs-1045	138	12	suppose	suppose	VERB
iajs-1045	138	13	a1	a1	PROPN
iajs-1045	138	14			PUNCT
iajs-1045	138	15	a.	a.	NOUN
iajs-1045	138	16	it	it	PRON
iajs-1045	138	17	follows	follow	VERB
iajs-1045	138	18	(	(	PUNCT
iajs-1045	138	19	from	from	ADP
iajs-1045	138	20	the	the	DET
iajs-1045	138	21	definition	definition	NOUN
iajs-1045	138	22	of	of	ADP
iajs-1045	138	23	fuzzy	fuzzy	ADJ
iajs-1045	138	24	group	group	NOUN
iajs-1045	138	25	extension	extension	NOUN
iajs-1045	138	26	)	)	PUNCT
iajs-1045	138	27	that	that	SCONJ
iajs-1045	138	28	d<1,a1	d<1,a1	VERB
iajs-1045	138	29	>	>	X
iajs-1045	138	30	=	=	SYM
iajs-1045	138	31	{	{	PUNCT
iajs-1045	138	32	1,a1	1,a1	NUM
iajs-1045	138	33	}	}	PUNCT
iajs-1045	138	34	.	.	PUNCT
iajs-1045	139	1	thus	thus	ADV
iajs-1045	139	2	k	k	X
iajs-1045	139	3			NOUN
iajs-1045	139	4	2	2	NUM
iajs-1045	139	5	in	in	ADP
iajs-1045	139	6	this	this	DET
iajs-1045	139	7	case	case	NOUN
iajs-1045	139	8	.	.	PUNCT
iajs-1045	140	1	thus	thus	ADV
iajs-1045	140	2	,	,	PUNCT
iajs-1045	140	3	since	since	SCONJ
iajs-1045	140	4	x	x	ADP
iajs-1045	140	5	1	1	NUM
iajs-1045	140	6	,	,	PUNCT
iajs-1045	140	7	cl(x	cl(x	PROPN
iajs-1045	140	8	)	)	PUNCT
iajs-1045	140	9			NUM
iajs-1045	140	10	2	2	NUM
iajs-1045	140	11			NUM
iajs-1045	140	12	k.	k.	NOUN
iajs-1045	140	13	2nd	2nd	PROPN
iajs-1045	140	14	case	case	NOUN
iajs-1045	140	15	:	:	PUNCT
iajs-1045	140	16	suppose	suppose	VERB
iajs-1045	140	17	a1	a1	PROPN
iajs-1045	140	18			NOUN
iajs-1045	140	19	a.	a.	ADV
iajs-1045	140	20	then	then	ADV
iajs-1045	140	21	d<1,a1	d<1,a1	PROPN
iajs-1045	140	22	>	>	X
iajs-1045	140	23			PROPN
iajs-1045	140	24	a	a	PROPN
iajs-1045	140	25	(	(	PUNCT
iajs-1045	140	26	e.g.	e.g.	ADV
iajs-1045	140	27	by	by	ADP
iajs-1045	140	28	(	(	PUNCT
iajs-1045	140	29	5	5	NUM
iajs-1045	140	30	,	,	PUNCT
iajs-1045	140	31	lemma	lemma	PROPN
iajs-1045	140	32	4.9	4.9	NUM
iajs-1045	140	33	)	)	PUNCT
iajs-1045	140	34	;	;	PUNCT
iajs-1045	140	35	notice	notice	VERB
iajs-1045	140	36	a1	a1	PROPN
iajs-1045	140	37			NOUN
iajs-1045	140	38	–	–	PUNCT
iajs-1045	140	39	1	1	X
iajs-1045	140	40	.	.	PUNCT
iajs-1045	140	41	thus	thus	ADV
iajs-1045	140	42	a1	a1	PROPN
iajs-1045	140	43	,	,	PUNCT
iajs-1045	140	44			PROPN
iajs-1045	140	45	,	,	PUNCT
iajs-1045	140	46	ak	ak	PROPN
iajs-1045	140	47	are	be	AUX
iajs-1045	140	48	all	all	PRON
iajs-1045	140	49	in	in	ADP
iajs-1045	140	50	a	a	PROPN
iajs-1045	140	51	,	,	PUNCT
iajs-1045	140	52	and	and	CCONJ
iajs-1045	140	53	x(ai	x(ai	PROPN
iajs-1045	140	54	–	–	PUNCT
iajs-1045	140	55	1	1	NUM
iajs-1045	140	56	)	)	PUNCT
iajs-1045	140	57			PROPN
iajs-1045	140	58	x(ai	x(ai	PROPN
iajs-1045	140	59	)	)	PUNCT
iajs-1045	140	60	,	,	PUNCT
iajs-1045	140	61	i	i	NOUN
iajs-1045	140	62	=	=	PUNCT
iajs-1045	140	63	1,,k	1,,k	NUM
iajs-1045	140	64	.	.	PUNCT
iajs-1045	141	1	thus	thus	ADV
iajs-1045	141	2	cl(x	cl(x	PROPN
iajs-1045	141	3	)	)	PUNCT
iajs-1045	141	4			PROPN
iajs-1045	141	5	k.	k.	PROPN
iajs-1045	142	1	thus	thus	ADV
iajs-1045	142	2	,	,	PUNCT
iajs-1045	142	3	in	in	ADP
iajs-1045	142	4	any	any	DET
iajs-1045	142	5	case	case	NOUN
iajs-1045	142	6	cl(x	cl(x	NOUN
iajs-1045	142	7	)	)	PUNCT
iajs-1045	142	8			NUM
iajs-1045	142	9	k	k	PROPN
iajs-1045	142	10	,	,	PUNCT
iajs-1045	142	11	so	so	ADV
iajs-1045	142	12	cl(x	cl(x	PROPN
iajs-1045	142	13	)	)	PUNCT
iajs-1045	142	14			NUM
iajs-1045	142	15	cl(x	cl(x	NOUN
iajs-1045	142	16	)	)	PUNCT
iajs-1045	142	17	.	.	PUNCT
iajs-1045	143	1	lemma	lemma	PROPN
iajs-1045	143	2	1	1	NUM
iajs-1045	143	3	suppose	suppose	VERB
iajs-1045	143	4	b	b	NUM
iajs-1045	143	5	,	,	PUNCT
iajs-1045	143	6	a0	a0	PROPN
iajs-1045	143	7	,	,	PUNCT
iajs-1045	143	8			PROPN
iajs-1045	143	9	,	,	PUNCT
iajs-1045	143	10	ak	ak	PROPN
iajs-1045	143	11			PROPN
iajs-1045	143	12	a	a	DET
iajs-1045	143	13	satisfy	satisfy	NOUN
iajs-1045	143	14	d<1,b	d<1,b	NOUN
iajs-1045	143	15	>	>	X
iajs-1045	143	16	=	=	PUNCT
iajs-1045	143	17	{	{	PUNCT
iajs-1045	143	18	1,b	1,b	NUM
iajs-1045	143	19	}	}	PUNCT
iajs-1045	143	20	,	,	PUNCT
iajs-1045	143	21	and	and	CCONJ
iajs-1045	143	22	d<1,ai	d<1,ai	NOUN
iajs-1045	143	23	–	–	PUNCT
iajs-1045	143	24	1	1	NUM
iajs-1045	143	25	>	>	X
iajs-1045	143	26	<	<	X
iajs-1045	143	27	1,b	1,b	NUM
iajs-1045	143	28	>	>	X
iajs-1045	143	29			PROPN
iajs-1045	143	30	d<1,ai	d<1,ai	PROPN
iajs-1045	143	31	>	>	X
iajs-1045	143	32	<	<	X
iajs-1045	143	33	1,b	1,b	NOUN
iajs-1045	143	34	>	>	X
iajs-1045	143	35	,	,	PUNCT
iajs-1045	143	36	i	i	PRON
iajs-1045	143	37	=	=	PUNCT
iajs-1045	143	38	1,,k	1,,k	NUM
iajs-1045	143	39	.	.	PUNCT
iajs-1045	144	1	then	then	ADV
iajs-1045	144	2	there	there	PRON
iajs-1045	144	3	exists	exist	VERB
iajs-1045	144	4	ia	ia	NOUN
iajs-1045	144	5			NOUN
iajs-1045	144	6	d	d	X
iajs-1045	144	7	<	<	X
iajs-1045	144	8	ai	ai	PROPN
iajs-1045	144	9	,	,	PUNCT
iajs-1045	144	10	aib	aib	PROPN
iajs-1045	144	11	>	>	X
iajs-1045	145	1	=	=	PUNCT
iajs-1045	145	2	{	{	PUNCT
iajs-1045	145	3	ai	ai	PROPN
iajs-1045	145	4	,	,	PUNCT
iajs-1045	145	5	aib	aib	PROPN
iajs-1045	145	6	}	}	PUNCT
iajs-1045	145	7	such	such	ADJ
iajs-1045	145	8	that	that	SCONJ
iajs-1045	145	9	d<1	d<1	NOUN
iajs-1045	145	10	,	,	PUNCT
iajs-1045	145	11	i	i	PRON
iajs-1045	145	12	1a	1a	VERB
iajs-1045	145	13			PROPN
iajs-1045	145	14			CCONJ
iajs-1045	145	15	>	>	X
iajs-1045	145	16			PROPN
iajs-1045	145	17	d<1	d<1	PROPN
iajs-1045	145	18	,	,	PUNCT
iajs-1045	145	19	ia	ia	VERB
iajs-1045	145	20	>	>	PUNCT
iajs-1045	145	21	,	,	PUNCT
iajs-1045	145	22	i	i	PRON
iajs-1045	145	23	=	=	PUNCT
iajs-1045	145	24	1,,k	1,,k	NUM
iajs-1045	145	25	.	.	PUNCT
iajs-1045	146	1	proof	proof	NOUN
iajs-1045	146	2	:	:	PUNCT
iajs-1045	146	3	compare	compare	VERB
iajs-1045	146	4	[	[	X
iajs-1045	146	5	6	6	NUM
iajs-1045	146	6	]	]	PUNCT
iajs-1045	146	7	.	.	PUNCT
iajs-1045	147	1	we	we	PRON
iajs-1045	147	2	now	now	ADV
iajs-1045	147	3	proceed	proceed	VERB
iajs-1045	147	4	to	to	PART
iajs-1045	147	5	prove	prove	VERB
iajs-1045	147	6	a	a	DET
iajs-1045	147	7	deeper	deep	ADJ
iajs-1045	147	8	property	property	NOUN
iajs-1045	147	9	of	of	ADP
iajs-1045	147	10	chain	chain	NOUN
iajs-1045	147	11	length	length	NOUN
iajs-1045	147	12	.	.	PUNCT
iajs-1045	148	1	theorem	theorem	ADJ
iajs-1045	148	2	5	5	NUM
iajs-1045	148	3	suppose	suppose	VERB
iajs-1045	148	4	y	y	PROPN
iajs-1045	148	5	is	be	AUX
iajs-1045	148	6	a	a	DET
iajs-1045	148	7	subspace	subspace	NOUN
iajs-1045	148	8	of	of	ADP
iajs-1045	148	9	x.	x.	PROPN
iajs-1045	148	10	then	then	ADV
iajs-1045	148	11	c1(y	c1(y	PROPN
iajs-1045	148	12	)	)	PUNCT
iajs-1045	148	13			NOUN
iajs-1045	148	14	c1(x	c1(x	NOUN
iajs-1045	148	15	)	)	PUNCT
iajs-1045	148	16	.	.	PUNCT
iajs-1045	149	1	proof	proof	NOUN
iajs-1045	149	2	:	:	PUNCT
iajs-1045	149	3	suppose	suppose	VERB
iajs-1045	149	4	,	,	PUNCT
iajs-1045	149	5	to	to	ADP
iajs-1045	149	6	the	the	DET
iajs-1045	149	7	contrary	contrary	NOUN
iajs-1045	149	8	,	,	PUNCT
iajs-1045	149	9	cl(y	cl(y	PUNCT
iajs-1045	149	10	)	)	PUNCT
iajs-1045	149	11	>	>	X
iajs-1045	149	12	cl(x	cl(x	PROPN
iajs-1045	149	13	)	)	PUNCT
iajs-1045	149	14	.	.	PUNCT
iajs-1045	150	1	then	then	ADV
iajs-1045	150	2	,	,	PUNCT
iajs-1045	150	3	in	in	ADP
iajs-1045	150	4	particular	particular	ADJ
iajs-1045	150	5	,	,	PUNCT
iajs-1045	150	6	cl(x	cl(x	X
iajs-1045	150	7	)	)	PUNCT
iajs-1045	150	8	<	<	X
iajs-1045	150	9	.	.	X
iajs-1045	150	10	choose	choose	VERB
iajs-1045	150	11	a	a	DET
iajs-1045	150	12	subspace	subspace	NOUN
iajs-1045	150	13	z	z	PROPN
iajs-1045	150	14			PROPN
iajs-1045	150	15	x	x	SYM
iajs-1045	150	16	minimal	minimal	ADJ
iajs-1045	150	17	subject	subject	NOUN
iajs-1045	150	18	to	to	ADP
iajs-1045	150	19	(	(	PUNCT
iajs-1045	150	20	1)z	1)z	NUM
iajs-1045	150	21			PROPN
iajs-1045	150	22	y	y	PROPN
iajs-1045	150	23	and	and	CCONJ
iajs-1045	150	24	(	(	PUNCT
iajs-1045	150	25	2	2	X
iajs-1045	150	26	)	)	PUNCT
iajs-1045	150	27	c1(z	c1(z	NOUN
iajs-1045	150	28	)	)	PUNCT
iajs-1045	150	29			NOUN
iajs-1045	150	30	c1(x	c1(x	NOUN
iajs-1045	150	31	)	)	PUNCT
iajs-1045	150	32	.	.	PUNCT
iajs-1045	151	1	to	to	PART
iajs-1045	151	2	show	show	VERB
iajs-1045	151	3	such	such	ADJ
iajs-1045	151	4	z	z	NOUN
iajs-1045	151	5	exists	exist	VERB
iajs-1045	151	6	.	.	PUNCT
iajs-1045	152	1	suppose	suppose	VERB
iajs-1045	152	2	{	{	PUNCT
iajs-1045	152	3	z	z	NOUN
iajs-1045	152	4	i	i	PRON
iajs-1045	152	5	}	}	PUNCT
iajs-1045	152	6	is	be	AUX
iajs-1045	152	7	a	a	DET
iajs-1045	152	8	collection	collection	NOUN
iajs-1045	152	9	of	of	ADP
iajs-1045	152	10	subspaces	subspace	NOUN
iajs-1045	152	11	of	of	ADP
iajs-1045	152	12	x	x	PUNCT
iajs-1045	152	13	satisfying	satisfy	VERB
iajs-1045	152	14	(	(	PUNCT
iajs-1045	152	15	1	1	NUM
iajs-1045	152	16	)	)	PUNCT
iajs-1045	152	17	and	and	CCONJ
iajs-1045	152	18	(	(	PUNCT
iajs-1045	152	19	2	2	NUM
iajs-1045	152	20	)	)	PUNCT
iajs-1045	152	21	and	and	CCONJ
iajs-1045	152	22	linearly	linearly	ADV
iajs-1045	152	23	ordered	order	VERB
iajs-1045	152	24	by	by	ADP
iajs-1045	152	25	inclusion	inclusion	NOUN
iajs-1045	152	26	.	.	PUNCT
iajs-1045	153	1	let	let	VERB
iajs-1045	153	2	z	z	NOUN
iajs-1045	153	3	=	=	PUNCT
iajs-1045	154	1	i	i	PRON
iajs-1045	154	2	z	z	PROPN
iajs-1045	154	3	i.	i.	NOUN
iajs-1045	154	4	then	then	ADV
iajs-1045	154	5	z	z	PROPN
iajs-1045	154	6	is	be	AUX
iajs-1045	154	7	a	a	DET
iajs-1045	154	8	subspace	subspace	NOUN
iajs-1045	154	9	of	of	ADP
iajs-1045	154	10	x	x	PUNCT
iajs-1045	154	11	satisfying	satisfy	VERB
iajs-1045	154	12	(	(	PUNCT
iajs-1045	154	13	1	1	NUM
iajs-1045	154	14	)	)	PUNCT
iajs-1045	154	15	.	.	PUNCT
iajs-1045	155	1	to	to	PART
iajs-1045	155	2	show	show	VERB
iajs-1045	155	3	z	z	NOUN
iajs-1045	155	4	satisfies	satisfie	NOUN
iajs-1045	155	5	(	(	PUNCT
iajs-1045	155	6	2	2	X
iajs-1045	155	7	)	)	PUNCT
iajs-1045	155	8	suppose	suppose	VERB
iajs-1045	155	9	a0	a0	PROPN
iajs-1045	155	10	,	,	PUNCT
iajs-1045	155	11			PROPN
iajs-1045	155	12	,	,	PUNCT
iajs-1045	155	13	ak	ak	PROPN
iajs-1045	155	14	a	a	PROPN
iajs-1045	155	15	satisfy	satisfy	PROPN
iajs-1045	155	16	z(aj)z(a	z(aj)z(a	X
iajs-1045	155	17	j	j	X
iajs-1045	155	18	–	–	PUNCT
iajs-1045	155	19	1	1	NUM
iajs-1045	155	20	)	)	PUNCT
iajs-1045	155	21	,	,	PUNCT
iajs-1045	155	22	j	j	X
iajs-1045	155	23	=	=	PUNCT
iajs-1045	155	24	1,,k	1,,k	NUM
iajs-1045	155	25	.	.	PUNCT
iajs-1045	156	1	thus	thus	ADV
iajs-1045	156	2	the	the	DET
iajs-1045	156	3	set	set	NOUN
iajs-1045	156	4	m	m	NOUN
iajs-1045	156	5	=	=	NOUN
iajs-1045	156	6	{	{	PUNCT
iajs-1045	156	7	x	x	NOUN
iajs-1045	156	8	<	<	X
iajs-1045	156	9	1	1	NUM
iajs-1045	156	10	,	,	PUNCT
iajs-1045	156	11	aj	aj	PROPN
iajs-1045	156	12			PROPN
iajs-1045	156	13			PROPN
iajs-1045	156	14	<	<	X
iajs-1045	156	15	a	a	DET
iajs-1045	156	16	j	j	PROPN
iajs-1045	156	17	–	–	PUNCT
iajs-1045	156	18	1	1	NUM
iajs-1045	156	19	,	,	PUNCT
iajs-1045	156	20	a	a	DET
iajs-1045	156	21	j	j	PROPN
iajs-1045	156	22	–	–	PUNCT
iajs-1045	156	23	1	1	NUM
iajs-1045	156	24	aj	aj	PROPN
iajs-1045	156	25	>	>	PROPN
iajs-1045	156	26	,	,	PUNCT
iajs-1045	156	27	j	j	PROPN
iajs-1045	156	28	=	=	PUNCT
iajs-1045	157	1	1,,k	1,,k	NUM
iajs-1045	157	2	}	}	PUNCT
iajs-1045	157	3	is	be	AUX
iajs-1045	157	4	open	open	ADJ
iajs-1045	157	5	in	in	ADP
iajs-1045	157	6	x	x	PUNCT
iajs-1045	157	7	and	and	CCONJ
iajs-1045	157	8	contains	contain	VERB
iajs-1045	157	9	z.	z.	NOUN
iajs-1045	157	10	by	by	ADP
iajs-1045	157	11	compactness	compactness	NOUN
iajs-1045	157	12	,	,	PUNCT
iajs-1045	158	1	z	z	NOUN
iajs-1045	159	1	i	i	PRON
iajs-1045	159	2			VERB
iajs-1045	159	3	m	m	VERB
iajs-1045	159	4	for	for	ADP
iajs-1045	159	5	some	some	DET
iajs-1045	159	6	i	i	PRON
iajs-1045	159	7	,	,	PUNCT
iajs-1045	159	8	so	so	ADV
iajs-1045	159	9	zi(aj	zi(aj	PROPN
iajs-1045	159	10	)	)	PUNCT
iajs-1045	159	11	z	z	PROPN
iajs-1045	159	12	i(a	i(a	PROPN
iajs-1045	159	13	j	j	PROPN
iajs-1045	159	14	–	–	PUNCT
iajs-1045	159	15	1	1	NUM
iajs-1045	159	16	)	)	PUNCT
iajs-1045	159	17	,	,	PUNCT
iajs-1045	159	18	j	j	X
iajs-1045	160	1	=	=	PUNCT
iajs-1045	160	2	1,,k	1,,k	NUM
iajs-1045	160	3	.	.	PUNCT
iajs-1045	161	1	these	these	DET
iajs-1045	161	2	inclusions	inclusion	NOUN
iajs-1045	161	3	must	must	AUX
iajs-1045	161	4	be	be	AUX
iajs-1045	161	5	strict	strict	ADJ
iajs-1045	161	6	,	,	PUNCT
iajs-1045	161	7	since	since	SCONJ
iajs-1045	161	8	z	z	NUM
iajs-1045	161	9			PROPN
iajs-1045	161	10	zi	zi	PROPN
iajs-1045	161	11	.	.	PUNCT
iajs-1045	162	1	thus	thus	ADV
iajs-1045	162	2	k	k	X
iajs-1045	162	3			NUM
iajs-1045	162	4	cl(zi	cl(zi	NOUN
iajs-1045	162	5	)	)	PUNCT
iajs-1045	162	6			NOUN
iajs-1045	162	7	cl(x	cl(x	NOUN
iajs-1045	162	8	)	)	PUNCT
iajs-1045	162	9	,	,	PUNCT
iajs-1045	162	10	so	so	ADV
iajs-1045	162	11	cl(z	cl(z	NOUN
iajs-1045	162	12	)	)	PUNCT
iajs-1045	162	13			NOUN
iajs-1045	162	14	cl(x	cl(x	NOUN
iajs-1045	162	15	)	)	PUNCT
iajs-1045	162	16	.	.	PUNCT
iajs-1045	163	1	so	so	ADV
iajs-1045	163	2	z	z	PROPN
iajs-1045	163	3	exists	exist	VERB
iajs-1045	163	4	as	as	SCONJ
iajs-1045	163	5	asserted	assert	VERB
iajs-1045	163	6	.	.	PUNCT
iajs-1045	164	1	to	to	PART
iajs-1045	164	2	simplify	simplify	VERB
iajs-1045	164	3	notation	notation	NOUN
iajs-1045	164	4	,	,	PUNCT
iajs-1045	164	5	we	we	PRON
iajs-1045	164	6	may	may	AUX
iajs-1045	164	7	assume	assume	VERB
iajs-1045	164	8	x	x	X
iajs-1045	164	9	=	=	PUNCT
iajs-1045	164	10	z.	z.	PROPN
iajs-1045	164	11	let	let	VERB
iajs-1045	164	12	y	y	PROPN
iajs-1045	164	13	=	=	SYM
iajs-1045	164	14	(	(	PUNCT
iajs-1045	164	15	y	y	NOUN
iajs-1045	164	16	,	,	PUNCT
iajs-1045	164	17	a/	a/	NOUN
iajs-1045	164	18	)	)	PUNCT
iajs-1045	164	19	,	,	PUNCT
iajs-1045	164	20	since	since	SCONJ
iajs-1045	164	21	y	y	PROPN
iajs-1045	164	22			PROPN
iajs-1045	164	23	x(cl(y	x(cl(y	PROPN
iajs-1045	164	24	)	)	PUNCT
iajs-1045	164	25	>	>	X
iajs-1045	164	26	cl(x	cl(x	PROPN
iajs-1045	164	27	)	)	PUNCT
iajs-1045	164	28	)	)	PUNCT
iajs-1045	164	29	.	.	PUNCT
iajs-1045	165	1	it	it	PRON
iajs-1045	165	2	follows	follow	VERB
iajs-1045	165	3	that	that	SCONJ
iajs-1045	165	4			PUNCT
iajs-1045	165	5			NOUN
iajs-1045	165	6	1	1	NUM
iajs-1045	165	7	,	,	PUNCT
iajs-1045	165	8	so	so	SCONJ
iajs-1045	165	9	there	there	PRON
iajs-1045	165	10	exists	exist	VERB
iajs-1045	165	11	a	a	DET
iajs-1045	165	12			NOUN
iajs-1045	165	13			PROPN
iajs-1045	165	14	,	,	PUNCT
iajs-1045	165	15	a	a	DET
iajs-1045	165	16			PROPN
iajs-1045	165	17	1	1	NUM
iajs-1045	165	18	.	.	PUNCT
iajs-1045	166	1	thus	thus	ADV
iajs-1045	166	2	y	y	PROPN
iajs-1045	166	3			PROPN
iajs-1045	166	4	x(a	x(a	NOUN
iajs-1045	166	5	)	)	PUNCT
iajs-1045	166	6			PROPN
iajs-1045	166	7	x.	x.	NOUN
iajs-1045	166	8	since	since	SCONJ
iajs-1045	166	9	cl(x	cl(x	NOUN
iajs-1045	166	10	)	)	PUNCT
iajs-1045	166	11	<	<	X
iajs-1045	166	12			NOUN
iajs-1045	166	13	,	,	PUNCT
iajs-1045	166	14	there	there	PRON
iajs-1045	166	15	exists	exist	VERB
iajs-1045	166	16	b	b	PROPN
iajs-1045	166	17	a	a	PRON
iajs-1045	166	18	,	,	PUNCT
iajs-1045	166	19	b	b	PROPN
iajs-1045	166	20			PROPN
iajs-1045	166	21	1	1	NUM
iajs-1045	166	22	,	,	PUNCT
iajs-1045	166	23	such	such	ADJ
iajs-1045	166	24	that	that	SCONJ
iajs-1045	166	25	ibn	ibn	PROPN
iajs-1045	166	26	alhaitham	alhaitham	NOUN
iajs-1045	166	27	j.	j.	PROPN
iajs-1045	166	28	for	for	ADP
iajs-1045	166	29	pure	pure	ADJ
iajs-1045	166	30	&	&	CCONJ
iajs-1045	166	31	appl	appl	PROPN
iajs-1045	166	32	.	.	PUNCT
iajs-1045	167	1	sci	sci	PROPN
iajs-1045	167	2	.	.	PUNCT
iajs-1045	168	1	vol.22	vol.22	PROPN
iajs-1045	168	2	(	(	PUNCT
iajs-1045	168	3	4	4	NUM
iajs-1045	168	4	)	)	PUNCT
iajs-1045	168	5	2009	2009	NUM
iajs-1045	168	6	x(a	x(a	NOUN
iajs-1045	168	7	)	)	PUNCT
iajs-1045	168	8			PROPN
iajs-1045	168	9	x(b	x(b	PROPN
iajs-1045	168	10	)	)	PUNCT
iajs-1045	168	11			PROPN
iajs-1045	168	12	x	x	SYM
iajs-1045	168	13	,	,	PUNCT
iajs-1045	168	14	x(b	x(b	PROPN
iajs-1045	168	15	)	)	PUNCT
iajs-1045	168	16	maximal	maximal	ADJ
iajs-1045	168	17	.	.	PUNCT
iajs-1045	169	1	thus	thus	ADV
iajs-1045	169	2	d<1,b	d<1,b	NOUN
iajs-1045	169	3	>	>	X
iajs-1045	169	4	is	be	AUX
iajs-1045	169	5	minimal	minimal	ADJ
iajs-1045	169	6	,	,	PUNCT
iajs-1045	169	7	i.e.	i.e.	X
iajs-1045	169	8	,	,	PUNCT
iajs-1045	169	9	d<1,b	d<1,b	NOUN
iajs-1045	169	10	>	>	X
iajs-1045	169	11	=	=	PUNCT
iajs-1045	169	12	{	{	PUNCT
iajs-1045	169	13	1,b	1,b	NUM
iajs-1045	169	14	}	}	PUNCT
iajs-1045	169	15	.	.	PUNCT
iajs-1045	170	1	by	by	ADP
iajs-1045	170	2	the	the	DET
iajs-1045	170	3	minimal	minimal	ADJ
iajs-1045	170	4	choice	choice	NOUN
iajs-1045	170	5	of	of	ADP
iajs-1045	170	6	x	x	X
iajs-1045	170	7	(=	(=	X
iajs-1045	170	8	z	z	NOUN
iajs-1045	170	9	)	)	PUNCT
iajs-1045	170	10	,	,	PUNCT
iajs-1045	170	11	it	it	PRON
iajs-1045	170	12	follows	follow	VERB
iajs-1045	170	13	that	that	SCONJ
iajs-1045	170	14	cl(x(b	cl(x(b	NOUN
iajs-1045	170	15	)	)	PUNCT
iajs-1045	170	16	)	)	PUNCT
iajs-1045	170	17	>	>	X
iajs-1045	170	18	cl(x	cl(x	PROPN
iajs-1045	170	19	)	)	PUNCT
iajs-1045	170	20	.	.	PUNCT
iajs-1045	171	1	on	on	ADP
iajs-1045	171	2	the	the	DET
iajs-1045	171	3	other	other	ADJ
iajs-1045	171	4	hand	hand	NOUN
iajs-1045	171	5	it	it	PRON
iajs-1045	171	6	follows	follow	VERB
iajs-1045	171	7	from	from	ADP
iajs-1045	171	8	lemma	lemma	PROPN
iajs-1045	171	9	(	(	PUNCT
iajs-1045	171	10	1	1	NUM
iajs-1045	171	11	)	)	PUNCT
iajs-1045	171	12	that	that	PRON
iajs-1045	171	13	cl(x(b	cl(x(b	X
iajs-1045	171	14	)	)	PUNCT
iajs-1045	171	15	)	)	PUNCT
iajs-1045	171	16			NOUN
iajs-1045	171	17	cl(x	cl(x	NOUN
iajs-1045	171	18	)	)	PUNCT
iajs-1045	171	19	.	.	PUNCT
iajs-1045	172	1	this	this	PRON
iajs-1045	172	2	is	be	AUX
iajs-1045	172	3	a	a	DET
iajs-1045	172	4	contradiction	contradiction	NOUN
iajs-1045	172	5	.	.	PUNCT
iajs-1045	173	1	references	reference	NOUN
iajs-1045	173	2	1	1	NUM
iajs-1045	173	3	.	.	PUNCT
iajs-1045	174	1	malik	malik	PROPN
iajs-1045	174	2	,	,	PUNCT
iajs-1045	174	3	d.s	d.s	PROPN
iajs-1045	174	4	.	.	PROPN
iajs-1045	174	5	and	and	CCONJ
iajs-1045	174	6	mordeson	mordeson	NOUN
iajs-1045	174	7	,	,	PUNCT
iajs-1045	174	8	j.n	j.n	PROPN
iajs-1045	174	9	.	.	PUNCT
iajs-1045	175	1	(	(	PUNCT
iajs-1045	175	2	1991),fuzzy	1991),fuzzy	NUM
iajs-1045	175	3	subgroups	subgroup	NOUN
iajs-1045	175	4	of	of	ADP
iajs-1045	175	5	abelian	abelian	PROPN
iajs-1045	175	6	group	group	NOUN
iajs-1045	175	7	,	,	PUNCT
iajs-1045	175	8	chinese	chinese	PROPN
iajs-1045	175	9	j.m	j.m	PROPN
iajs-1045	175	10	ath	ath	NOUN
iajs-1045	175	11	.	.	PROPN
iajs-1045	175	12	,	,	PUNCT
iajs-1045	175	13	19(2	19(2	NUM
iajs-1045	175	14	)	)	PUNCT
iajs-1045	175	15	.	.	PUNCT
iajs-1045	176	1	2	2	X
iajs-1045	176	2	.	.	X
iajs-1045	176	3	mordeson	mordeson	NOUN
iajs-1045	176	4	,	,	PUNCT
iajs-1045	176	5	j.n	j.n	PROPN
iajs-1045	176	6	.	.	PROPN
iajs-1045	176	7	and	and	CCONJ
iajs-1045	176	8	sen	sen	PROPN
iajs-1045	176	9	,	,	PUNCT
iajs-1045	176	10	m.k	m.k	PROPN
iajs-1045	176	11	.	.	PUNCT
iajs-1045	176	12	(	(	PUNCT
iajs-1045	176	13	1995	1995	NUM
iajs-1045	176	14	)	)	PUNCT
iajs-1045	176	15	,	,	PUNCT
iajs-1045	176	16	basic	basic	ADJ
iajs-1045	176	17	fuzzy	fuzzy	ADJ
iajs-1045	176	18	subgroups	subgroup	NOUN
iajs-1045	176	19	,	,	PUNCT
iajs-1045	176	20	inform	inform	VERB
iajs-1045	176	21	sci	sci	PROPN
iajs-1045	176	22	.	.	PROPN
iajs-1045	176	23	,	,	PUNCT
iajs-1045	176	24	82	82	NUM
iajs-1045	176	25	,	,	PUNCT
iajs-1045	176	26	167	167	NUM
iajs-1045	176	27	-	-	SYM
iajs-1045	176	28	179	179	NUM
iajs-1045	176	29	.	.	PUNCT
iajs-1045	177	1	3	3	X
iajs-1045	177	2	.	.	X
iajs-1045	177	3	marshall	marshall	PROPN
iajs-1045	177	4	,	,	PUNCT
iajs-1045	177	5	m.	m.	NOUN
iajs-1045	177	6	(	(	PUNCT
iajs-1045	177	7	1980	1980	NUM
iajs-1045	177	8	)	)	PUNCT
iajs-1045	177	9	,	,	PUNCT
iajs-1045	177	10	the	the	DET
iajs-1045	177	11	wittring	wittring	NOUN
iajs-1045	177	12	of	of	ADP
iajs-1045	177	13	a	a	DET
iajs-1045	177	14	space	space	NOUN
iajs-1045	177	15	of	of	ADP
iajs-1045	177	16	ordeeerings	ordeeering	NOUN
iajs-1045	177	17	,	,	PUNCT
iajs-1045	177	18	trans	trans	PROPN
iajs-1045	177	19	.	.	PROPN
iajs-1045	178	1	amer	amer	PROPN
iajs-1045	178	2	.	.	PUNCT
iajs-1045	178	3	math	math	PROPN
iajs-1045	178	4	.	.	PUNCT
iajs-1045	179	1	soc.258	soc.258	PROPN
iajs-1045	179	2	.	.	PUNCT
iajs-1045	180	1	4	4	X
iajs-1045	180	2	.	.	X
iajs-1045	180	3	marshall	marshall	PROPN
iajs-1045	180	4	,	,	PUNCT
iajs-1045	180	5	m.	m.	NOUN
iajs-1045	180	6	(	(	PUNCT
iajs-1045	180	7	1989	1989	NUM
iajs-1045	180	8	)	)	PUNCT
iajs-1045	180	9	,	,	PUNCT
iajs-1045	180	10	ouotients	ouotient	NOUN
iajs-1045	180	11	and	and	CCONJ
iajs-1045	180	12	inverse	inverse	NOUN
iajs-1045	180	13	limits	limit	NOUN
iajs-1045	180	14	of	of	ADP
iajs-1045	180	15	spaces	space	NOUN
iajs-1045	180	16	of	of	ADP
iajs-1045	180	17	orderings	ordering	NOUN
iajs-1045	180	18	,	,	PUNCT
iajs-1045	180	19	can	can	AUX
iajs-1045	180	20	.	.	PUNCT
iajs-1045	181	1	j.m	j.m	PROPN
iajs-1045	181	2	ath	ath	NOUN
iajs-1045	181	3	.	.	PROPN
iajs-1045	182	1	31,604	31,604	NUM
iajs-1045	182	2	-	-	SYM
iajs-1045	182	3	616	616	NUM
iajs-1045	182	4	.	.	PUNCT
iajs-1045	183	1	5	5	X
iajs-1045	183	2	.	.	X
iajs-1045	183	3	marshall	marshall	PROPN
iajs-1045	183	4	,	,	PUNCT
iajs-1045	183	5	m.	m.	NOUN
iajs-1045	183	6	(	(	PUNCT
iajs-1045	183	7	1989	1989	NUM
iajs-1045	183	8	)	)	PUNCT
iajs-1045	183	9	,	,	PUNCT
iajs-1045	183	10	classification	classification	NOUN
iajs-1045	183	11	of	of	ADP
iajs-1045	183	12	finite	finite	ADJ
iajs-1045	183	13	space	space	NOUN
iajs-1045	183	14	of	of	ADP
iajs-1045	183	15	orderings	ordering	NOUN
iajs-1045	183	16	,	,	PUNCT
iajs-1045	183	17	can	can	AUX
iajs-1045	183	18	.	.	PUNCT
iajs-1045	184	1	j.m	j.m	PROPN
iajs-1045	184	2	ath	ath	NOUN
iajs-1045	184	3	.	.	PROPN
iajs-1045	185	1	31	31	NUM
iajs-1045	185	2	,	,	PUNCT
iajs-1045	185	3	320	320	NUM
iajs-1045	185	4	-	-	SYM
iajs-1045	185	5	330	330	NUM
iajs-1045	185	6	.	.	NOUN
iajs-1045	186	1	6	6	NUM
iajs-1045	186	2	.	.	X
iajs-1045	186	3	marshall	marshall	PROPN
iajs-1045	186	4	,	,	PUNCT
iajs-1045	186	5	m.	m.	NOUN
iajs-1045	186	6	(	(	PUNCT
iajs-1045	186	7	1990	1990	NUM
iajs-1045	186	8	)	)	PUNCT
iajs-1045	186	9	,	,	PUNCT
iajs-1045	186	10	spaces	space	NOUN
iajs-1045	186	11	of	of	ADP
iajs-1045	186	12	orde	orde	PROPN
iajs-1045	186	13	ngs	ngs	PROPN
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iajs-1045	186	15	,	,	PUNCT
iajs-1045	186	16	can	can	AUX
iajs-1045	186	17	.	.	PUNCT
iajs-1045	187	1	j.m	j.m	PROPN
iajs-1045	187	2	ath	ath	NOUN
iajs-1045	187	3	.	.	PROPN
iajs-1045	187	4	,	,	PUNCT
iajs-1045	187	5	xxxii(3	xxxii(3	PROPN
iajs-1045	187	6	):	):	PUNCT
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iajs-1045	187	8	-	-	SYM
iajs-1045	187	9	627	627	NUM
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iajs-1045	187	11	2009	2009	NUM
iajs-1045	187	12	)	)	PUNCT
iajs-1045	187	13	4	4	NUM
iajs-1045	187	14	(	(	PUNCT
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iajs-1045	187	16	ابن	ابن	PROPN
iajs-1045	187	17	الھیثم	الھیثم	PROPN
iajs-1045	187	18	للعلوم	للعلوم	PROPN
iajs-1045	187	19	الصرفة	الصرفة	PROPN
iajs-1045	187	20	والتطبیقیة	والتطبیقیة	PROPN
iajs-1045	187	21	المجلد	المجلد	PROPN
iajs-1045	187	22	الفضاء	الفضاء	PROPN
iajs-1045	187	23	الضبابي	الضبابي	PROPN
iajs-1045	187	24	الترتیب	الترتیب	PROPN
iajs-1045	187	25	لمى	لمى	PROPN
iajs-1045	188	1	ناجي	ناجي	PROPN
iajs-1045	188	2	محمد	محمد	ADJ
iajs-1045	188	3	توفیق	توفیق	PROPN
iajs-1045	188	4	جامعة	جامعة	PROPN
iajs-1045	188	5	بغداد	بغداد	PROPN
iajs-1045	188	6	،	،	PROPN
iajs-1045	188	7	ابن	ابن	PROPN
iajs-1045	188	8	الهیثم	الهیثم	PROPN
iajs-1045	188	9	-كلیة	-كلیة	PROPN
iajs-1045	188	10	التربیة	التربیة	NOUN
iajs-1045	188	11	،	،	NOUN
iajs-1045	188	12	قسم	قسم	PROPN
iajs-1045	188	13	الریاضیات	الریاضیات	PROPN
iajs-1045	188	14	خالصةال	خالصةال	PROPN
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iajs-1045	188	17	تعریــف	تعریــف	NOUN
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iajs-1045	188	19	سلــسلة	سلــسلة	NOUN
iajs-1045	188	20	فـي	فـي	NOUN
iajs-1045	188	21	فـضاء	فـضاء	NOUN
iajs-1045	189	1	ضــبابي	ضــبابي	PROPN
iajs-1045	190	1	الترتیـب	الترتیـب	PROPN
iajs-1045	191	1	ومــن	ومــن	PROPN
iajs-1045	191	2	ثـم	ثـم	NOUN
iajs-1045	191	3	عـرض	عـرض	ADJ
iajs-1045	191	4	خــواص	خــواص	ADJ
iajs-1045	191	5	وبرهنتهـا	وبرهنتهـا	PROPN
iajs-1045	191	6	،	،	PROPN
iajs-1045	191	7	ولقــد	ولقــد	PROPN
iajs-1045	191	8	تـم	تـم	PROPN
iajs-1045	191	9	برهــان	برهــان	PROPN
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iajs-1045	191	11	األساسیة	األساسیة	PROPN
iajs-1045	191	12	لطول	لطول	PROPN
iajs-1045	191	13	السلسلة	السلسلة	PROPN
iajs-1045	191	14	المنتهیة	المنتهیة	PROPN
iajs-1045	191	15	وعرض	وعرض	PROPN
iajs-1045	191	16	بعض	بعض	NOUN
iajs-1045	191	17	النتائج	النتائج	PROPN
iajs-1045	191	18	المتعلقة	المتعلقة	PROPN
iajs-1045	191	19	بالموضوع	بالموضوع	NOUN