id	sid	tid	token	lemma	pos
ejpam-65	1	1	european	european	PROPN
ejpam-65	1	2	journal	journal	PROPN
ejpam-65	1	3	of	of	ADP
ejpam-65	1	4	pure	pure	ADJ
ejpam-65	1	5	and	and	CCONJ
ejpam-65	1	6	applied	apply	VERB
ejpam-65	1	7	mathematics	mathematic	NOUN
ejpam-65	1	8	vol	vol	NOUN
ejpam-65	1	9	.	.	PROPN
ejpam-65	2	1	1	1	NUM
ejpam-65	2	2	,	,	PUNCT
ejpam-65	2	3	no	no	INTJ
ejpam-65	2	4	.	.	NOUN
ejpam-65	2	5	2	2	NUM
ejpam-65	2	6	,	,	PUNCT
ejpam-65	2	7	2008	2008	NUM
ejpam-65	2	8	,	,	PUNCT
ejpam-65	2	9	(	(	PUNCT
ejpam-65	2	10	32	32	NUM
ejpam-65	2	11	-	-	SYM
ejpam-65	2	12	50	50	NUM
ejpam-65	2	13	)	)	PUNCT
ejpam-65	2	14	issn	issn	PROPN
ejpam-65	2	15	1307	1307	NUM
ejpam-65	2	16	-	-	SYM
ejpam-65	2	17	5543	5543	NUM
ejpam-65	2	18	–	–	PUNCT
ejpam-65	2	19	www.ejpam.com	www.ejpam.com	X
ejpam-65	2	20	on	on	ADP
ejpam-65	2	21	dimension	dimension	NOUN
ejpam-65	2	22	of	of	ADP
ejpam-65	2	23	hypervector	hypervector	NOUN
ejpam-65	2	24	spaces	space	NOUN
ejpam-65	2	25	r.	r.	PROPN
ejpam-65	2	26	ameri∗	ameri∗	PROPN
ejpam-65	2	27	,	,	PUNCT
ejpam-65	2	28	o.r	o.r	PROPN
ejpam-65	2	29	.	.	PROPN
ejpam-65	2	30	dehghan	dehghan	PROPN
ejpam-65	2	31	department	department	PROPN
ejpam-65	2	32	of	of	ADP
ejpam-65	2	33	mathematics	mathematic	NOUN
ejpam-65	2	34	,	,	PUNCT
ejpam-65	2	35	faculty	faculty	NOUN
ejpam-65	2	36	of	of	ADP
ejpam-65	2	37	basic	basic	ADJ
ejpam-65	2	38	sciences	science	NOUN
ejpam-65	2	39	,	,	PUNCT
ejpam-65	2	40	university	university	NOUN
ejpam-65	2	41	of	of	ADP
ejpam-65	2	42	mazandaran	mazandaran	PROPN
ejpam-65	2	43	,	,	PUNCT
ejpam-65	2	44	babolsar	babolsar	PROPN
ejpam-65	2	45	,	,	PUNCT
ejpam-65	2	46	iran	iran	PROPN
ejpam-65	2	47	.	.	PUNCT
ejpam-65	3	1	abstract	abstract	ADJ
ejpam-65	3	2	.	.	PUNCT
ejpam-65	4	1	the	the	DET
ejpam-65	4	2	purpose	purpose	NOUN
ejpam-65	4	3	of	of	ADP
ejpam-65	4	4	this	this	DET
ejpam-65	4	5	paper	paper	NOUN
ejpam-65	4	6	is	be	AUX
ejpam-65	4	7	the	the	DET
ejpam-65	4	8	study	study	NOUN
ejpam-65	4	9	of	of	ADP
ejpam-65	4	10	dimension	dimension	NOUN
ejpam-65	4	11	of	of	ADP
ejpam-65	4	12	hypervector	hypervector	NOUN
ejpam-65	4	13	spaces	space	NOUN
ejpam-65	4	14	.	.	PUNCT
ejpam-65	5	1	in	in	ADP
ejpam-65	5	2	this	this	DET
ejpam-65	5	3	regard	regard	NOUN
ejpam-65	5	4	first	first	ADV
ejpam-65	5	5	we	we	PRON
ejpam-65	5	6	introduce	introduce	VERB
ejpam-65	5	7	the	the	DET
ejpam-65	5	8	notions	notion	NOUN
ejpam-65	5	9	of	of	ADP
ejpam-65	5	10	linear	linear	ADJ
ejpam-65	5	11	independent	independent	ADJ
ejpam-65	5	12	(	(	PUNCT
ejpam-65	5	13	resp	resp	NOUN
ejpam-65	5	14	.	.	PUNCT
ejpam-65	6	1	dependent	dependent	ADJ
ejpam-65	6	2	)	)	PUNCT
ejpam-65	6	3	and	and	CCONJ
ejpam-65	6	4	basis	basis	NOUN
ejpam-65	6	5	of	of	ADP
ejpam-65	6	6	hypervector	hypervector	NOUN
ejpam-65	6	7	spaces	space	NOUN
ejpam-65	6	8	.	.	PUNCT
ejpam-65	7	1	then	then	ADV
ejpam-65	7	2	we	we	PRON
ejpam-65	7	3	study	study	VERB
ejpam-65	7	4	the	the	DET
ejpam-65	7	5	properties	property	NOUN
ejpam-65	7	6	of	of	ADP
ejpam-65	7	7	hypervector	hypervector	NOUN
ejpam-65	7	8	spaces	space	NOUN
ejpam-65	7	9	and	and	CCONJ
ejpam-65	7	10	prove	prove	VERB
ejpam-65	7	11	that	that	SCONJ
ejpam-65	7	12	under	under	ADP
ejpam-65	7	13	certain	certain	ADJ
ejpam-65	7	14	conditions	condition	NOUN
ejpam-65	7	15	dimension	dimension	NOUN
ejpam-65	7	16	for	for	ADP
ejpam-65	7	17	such	such	ADJ
ejpam-65	7	18	spaces	space	NOUN
ejpam-65	7	19	there	there	PRON
ejpam-65	7	20	exist	exist	VERB
ejpam-65	7	21	.	.	PUNCT
ejpam-65	8	1	finally	finally	ADV
ejpam-65	8	2	,	,	PUNCT
ejpam-65	8	3	we	we	PRON
ejpam-65	8	4	use	use	VERB
ejpam-65	8	5	the	the	DET
ejpam-65	8	6	fundamental	fundamental	ADJ
ejpam-65	8	7	relation	relation	NOUN
ejpam-65	8	8	on	on	ADP
ejpam-65	8	9	hypervector	hypervector	NOUN
ejpam-65	8	10	spaces	space	NOUN
ejpam-65	8	11	to	to	PART
ejpam-65	8	12	construct	construct	VERB
ejpam-65	8	13	a	a	DET
ejpam-65	8	14	functor	functor	NOUN
ejpam-65	8	15	from	from	ADP
ejpam-65	8	16	the	the	DET
ejpam-65	8	17	category	category	NOUN
ejpam-65	8	18	of	of	ADP
ejpam-65	8	19	hypervector	hypervector	NOUN
ejpam-65	8	20	spaces	space	NOUN
ejpam-65	8	21	over	over	ADP
ejpam-65	8	22	a	a	DET
ejpam-65	8	23	fixed	fix	VERB
ejpam-65	8	24	field	field	NOUN
ejpam-65	8	25	k	k	PROPN
ejpam-65	8	26	and	and	CCONJ
ejpam-65	8	27	the	the	DET
ejpam-65	8	28	category	category	NOUN
ejpam-65	8	29	of	of	ADP
ejpam-65	8	30	classical	classical	ADJ
ejpam-65	8	31	vector	vector	NOUN
ejpam-65	8	32	spaces	space	NOUN
ejpam-65	8	33	over	over	ADP
ejpam-65	8	34	k	k	PROPN
ejpam-65	8	35	,	,	PUNCT
ejpam-65	8	36	and	and	CCONJ
ejpam-65	8	37	we	we	PRON
ejpam-65	8	38	will	will	AUX
ejpam-65	8	39	prove	prove	VERB
ejpam-65	8	40	that	that	SCONJ
ejpam-65	8	41	this	this	DET
ejpam-65	8	42	functor	functor	PROPN
ejpam-65	8	43	preserves	preserve	VERB
ejpam-65	8	44	dimension	dimension	NOUN
ejpam-65	8	45	.	.	PUNCT
ejpam-65	9	1	ams	am	NOUN
ejpam-65	9	2	subject	subject	ADJ
ejpam-65	9	3	classifications	classification	NOUN
ejpam-65	9	4	:	:	PUNCT
ejpam-65	9	5	20n20	20n20	NUM
ejpam-65	9	6	,	,	PUNCT
ejpam-65	9	7	20n25	20n25	NUM
ejpam-65	9	8	.	.	PUNCT
ejpam-65	10	1	key	key	ADJ
ejpam-65	10	2	words	word	NOUN
ejpam-65	10	3	:	:	PUNCT
ejpam-65	10	4	hypervector	hypervector	NOUN
ejpam-65	10	5	spaces	space	NOUN
ejpam-65	10	6	,	,	PUNCT
ejpam-65	10	7	linearly	linearly	ADV
ejpam-65	10	8	independent	independent	ADJ
ejpam-65	10	9	,	,	PUNCT
ejpam-65	10	10	basis	basis	NOUN
ejpam-65	10	11	,	,	PUNCT
ejpam-65	10	12	dimension	dimension	NOUN
ejpam-65	10	13	,	,	PUNCT
ejpam-65	10	14	transformation	transformation	NOUN
ejpam-65	10	15	,	,	PUNCT
ejpam-65	10	16	fundamental	fundamental	ADJ
ejpam-65	10	17	relation	relation	NOUN
ejpam-65	10	18	.	.	PUNCT
ejpam-65	11	1	1	1	X
ejpam-65	11	2	.	.	X
ejpam-65	11	3	introduction	introduction	NOUN
ejpam-65	11	4	hyperstructures	hyperstructure	NOUN
ejpam-65	11	5	theory	theory	NOUN
ejpam-65	11	6	was	be	AUX
ejpam-65	11	7	born	bear	VERB
ejpam-65	11	8	in	in	ADP
ejpam-65	11	9	1934	1934	NUM
ejpam-65	11	10	,	,	PUNCT
ejpam-65	11	11	when	when	SCONJ
ejpam-65	11	12	marty	marty	PROPN
ejpam-65	11	13	[	[	X
ejpam-65	11	14	5	5	NUM
ejpam-65	11	15	]	]	PUNCT
ejpam-65	11	16	defined	define	VERB
ejpam-65	11	17	hypergroups	hypergroup	NOUN
ejpam-65	11	18	,	,	PUNCT
ejpam-65	11	19	began	begin	VERB
ejpam-65	11	20	to	to	PART
ejpam-65	11	21	analysis	analysis	VERB
ejpam-65	11	22	their	their	PRON
ejpam-65	11	23	properties	property	NOUN
ejpam-65	11	24	and	and	CCONJ
ejpam-65	11	25	applied	apply	VERB
ejpam-65	11	26	them	they	PRON
ejpam-65	11	27	to	to	ADP
ejpam-65	11	28	groups	group	NOUN
ejpam-65	11	29	,	,	PUNCT
ejpam-65	11	30	rational	rational	ADJ
ejpam-65	11	31	algebraic	algebraic	ADJ
ejpam-65	11	32	functions	function	NOUN
ejpam-65	11	33	.	.	PUNCT
ejpam-65	12	1	now	now	ADV
ejpam-65	12	2	they	they	PRON
ejpam-65	12	3	are	be	AUX
ejpam-65	12	4	widely	widely	ADV
ejpam-65	12	5	studied	study	VERB
ejpam-65	12	6	from	from	ADP
ejpam-65	12	7	theoretical	theoretical	ADJ
ejpam-65	12	8	point	point	NOUN
ejpam-65	12	9	of	of	ADP
ejpam-65	12	10	view	view	NOUN
ejpam-65	12	11	and	and	CCONJ
ejpam-65	12	12	for	for	ADP
ejpam-65	12	13	their	their	PRON
ejpam-65	12	14	applications	application	NOUN
ejpam-65	12	15	to	to	ADP
ejpam-65	12	16	many	many	ADJ
ejpam-65	12	17	subjects	subject	NOUN
ejpam-65	12	18	of	of	ADP
ejpam-65	12	19	pure	pure	ADJ
ejpam-65	12	20	and	and	CCONJ
ejpam-65	12	21	applied	applied	ADJ
ejpam-65	12	22	properties	property	NOUN
ejpam-65	12	23	.	.	PUNCT
ejpam-65	13	1	since	since	SCONJ
ejpam-65	13	2	then	then	ADV
ejpam-65	13	3	many	many	ADJ
ejpam-65	13	4	researchers	researcher	NOUN
ejpam-65	13	5	have	have	AUX
ejpam-65	13	6	worked	work	VERB
ejpam-65	13	7	on	on	ADP
ejpam-65	13	8	hyperalgebraic	hyperalgebraic	PROPN
ejpam-65	13	9	structures	structure	NOUN
ejpam-65	13	10	and	and	CCONJ
ejpam-65	13	11	developed	develop	VERB
ejpam-65	13	12	this	this	DET
ejpam-65	13	13	theory	theory	NOUN
ejpam-65	13	14	(	(	PUNCT
ejpam-65	13	15	ref	ref	NOUN
ejpam-65	13	16	.	.	PUNCT
ejpam-65	14	1	[	[	X
ejpam-65	14	2	3	3	NUM
ejpam-65	14	3	]	]	PUNCT
ejpam-65	14	4	,	,	PUNCT
ejpam-65	14	5	[	[	X
ejpam-65	14	6	4	4	X
ejpam-65	14	7	]	]	PUNCT
ejpam-65	14	8	and	and	CCONJ
ejpam-65	14	9	[	[	X
ejpam-65	14	10	10	10	NUM
ejpam-65	14	11	]	]	PUNCT
ejpam-65	14	12	)	)	PUNCT
ejpam-65	14	13	.	.	PUNCT
ejpam-65	15	1	m.s	m.s	PROPN
ejpam-65	15	2	.	.	PROPN
ejpam-65	15	3	tallini	tallini	PROPN
ejpam-65	15	4	introduced	introduce	VERB
ejpam-65	15	5	the	the	DET
ejpam-65	15	6	notion	notion	NOUN
ejpam-65	15	7	of	of	ADP
ejpam-65	15	8	hypervector	hypervector	NOUN
ejpam-65	15	9	spaces	space	NOUN
ejpam-65	15	10	(	(	PUNCT
ejpam-65	15	11	[	[	X
ejpam-65	15	12	6	6	NUM
ejpam-65	15	13	]	]	PUNCT
ejpam-65	15	14	,	,	PUNCT
ejpam-65	15	15	[	[	X
ejpam-65	15	16	7	7	NUM
ejpam-65	15	17	]	]	PUNCT
ejpam-65	15	18	)	)	PUNCT
ejpam-65	15	19	and	and	CCONJ
ejpam-65	15	20	studied	study	VERB
ejpam-65	15	21	basic	basic	ADJ
ejpam-65	15	22	properties	property	NOUN
ejpam-65	15	23	of	of	ADP
ejpam-65	15	24	them	they	PRON
ejpam-65	15	25	.	.	PUNCT
ejpam-65	16	1	in	in	ADP
ejpam-65	16	2	[	[	X
ejpam-65	16	3	8	8	NUM
ejpam-65	16	4	]	]	X
ejpam-65	16	5	the	the	DET
ejpam-65	16	6	notion	notion	NOUN
ejpam-65	16	7	of	of	ADP
ejpam-65	16	8	matroidal	matroidal	ADJ
ejpam-65	16	9	hypervector	hypervector	NOUN
ejpam-65	16	10	space	space	NOUN
ejpam-65	16	11	was	be	AUX
ejpam-65	16	12	introduced	introduce	VERB
ejpam-65	16	13	and	and	CCONJ
ejpam-65	16	14	the	the	DET
ejpam-65	16	15	basic	basic	ADJ
ejpam-65	16	16	properties	property	NOUN
ejpam-65	16	17	of	of	ADP
ejpam-65	16	18	such	such	ADJ
ejpam-65	16	19	space	space	NOUN
ejpam-65	16	20	studied	study	VERB
ejpam-65	16	21	.	.	PUNCT
ejpam-65	17	1	in	in	ADP
ejpam-65	17	2	this	this	DET
ejpam-65	17	3	paper	paper	NOUN
ejpam-65	17	4	we	we	PRON
ejpam-65	17	5	study	study	VERB
ejpam-65	17	6	the	the	DET
ejpam-65	17	7	properties	property	NOUN
ejpam-65	17	8	of	of	ADP
ejpam-65	17	9	dimension	dimension	NOUN
ejpam-65	17	10	of	of	ADP
ejpam-65	17	11	hypervector	hypervector	NOUN
ejpam-65	17	12	spaces	space	NOUN
ejpam-65	17	13	.	.	PUNCT
ejpam-65	18	1	in	in	ADP
ejpam-65	18	2	§	§	NOUN
ejpam-65	18	3	3	3	NUM
ejpam-65	18	4	we	we	PRON
ejpam-65	18	5	introduce	introduce	VERB
ejpam-65	18	6	the	the	DET
ejpam-65	18	7	notions	notion	NOUN
ejpam-65	18	8	of	of	ADP
ejpam-65	18	9	linearly	linearly	ADV
ejpam-65	18	10	independent	independent	ADJ
ejpam-65	18	11	(	(	PUNCT
ejpam-65	18	12	resp	resp	NOUN
ejpam-65	18	13	.	.	PUNCT
ejpam-65	19	1	dependent	dependent	ADJ
ejpam-65	19	2	)	)	PUNCT
ejpam-65	19	3	,	,	PUNCT
ejpam-65	19	4	generator	generator	NOUN
ejpam-65	19	5	,	,	PUNCT
ejpam-65	19	6	and	and	CCONJ
ejpam-65	19	7	basis	basis	NOUN
ejpam-65	19	8	of	of	ADP
ejpam-65	19	9	a	a	DET
ejpam-65	19	10	hypervector	hypervector	NOUN
ejpam-65	19	11	space	space	NOUN
ejpam-65	19	12	.	.	PUNCT
ejpam-65	20	1	we	we	PRON
ejpam-65	20	2	show	show	VERB
ejpam-65	20	3	that	that	SCONJ
ejpam-65	20	4	in	in	ADP
ejpam-65	20	5	contrast	contrast	NOUN
ejpam-65	20	6	of	of	ADP
ejpam-65	20	7	the	the	DET
ejpam-65	20	8	classical	classical	ADJ
ejpam-65	20	9	vector	vector	NOUN
ejpam-65	20	10	spaces	space	VERB
ejpam-65	20	11	a	a	DET
ejpam-65	20	12	hypervector	hypervector	NOUN
ejpam-65	20	13	space	space	NOUN
ejpam-65	20	14	has	have	VERB
ejpam-65	20	15	not	not	PART
ejpam-65	20	16	necessarily	necessarily	ADV
ejpam-65	20	17	a	a	DET
ejpam-65	20	18	basis	basis	NOUN
ejpam-65	20	19	.	.	PUNCT
ejpam-65	21	1	we	we	PRON
ejpam-65	21	2	will	will	AUX
ejpam-65	21	3	prove	prove	VERB
ejpam-65	21	4	that	that	SCONJ
ejpam-65	21	5	under	under	ADP
ejpam-65	21	6	the	the	DET
ejpam-65	21	7	certain	certain	ADJ
ejpam-65	21	8	conditions	condition	NOUN
ejpam-65	21	9	a	a	DET
ejpam-65	21	10	hypervector	hypervector	NOUN
ejpam-65	21	11	space	space	NOUN
ejpam-65	21	12	has	have	VERB
ejpam-65	21	13	a	a	DET
ejpam-65	21	14	basis	basis	NOUN
ejpam-65	21	15	and	and	CCONJ
ejpam-65	21	16	then	then	ADV
ejpam-65	21	17	we	we	PRON
ejpam-65	21	18	investigate	investigate	VERB
ejpam-65	21	19	the	the	DET
ejpam-65	21	20	basic	basic	ADJ
ejpam-65	21	21	properties	property	NOUN
ejpam-65	21	22	of	of	ADP
ejpam-65	21	23	dimension	dimension	NOUN
ejpam-65	21	24	of	of	ADP
ejpam-65	21	25	such	such	ADJ
ejpam-65	21	26	spaces	space	NOUN
ejpam-65	21	27	.	.	PUNCT
ejpam-65	22	1	in	in	ADP
ejpam-65	22	2	§	§	NOUN
ejpam-65	22	3	4	4	NUM
ejpam-65	22	4	for	for	ADP
ejpam-65	22	5	a	a	DET
ejpam-65	22	6	given	give	VERB
ejpam-65	22	7	hypervector	hypervector	NOUN
ejpam-65	22	8	space	space	NOUN
ejpam-65	22	9	v	v	NOUN
ejpam-65	22	10	over	over	ADP
ejpam-65	22	11	a	a	DET
ejpam-65	22	12	classical	classical	ADJ
ejpam-65	22	13	field	field	NOUN
ejpam-65	22	14	k	k	NOUN
ejpam-65	22	15	,	,	PUNCT
ejpam-65	22	16	the	the	DET
ejpam-65	22	17	fundamental	fundamental	ADJ
ejpam-65	22	18	relation	relation	NOUN
ejpam-65	22	19	on	on	ADP
ejpam-65	22	20	v	v	NUM
ejpam-65	22	21	,	,	PUNCT
ejpam-65	22	22	ε∗	ε∗	PROPN
ejpam-65	22	23	,	,	PUNCT
ejpam-65	22	24	is	be	AUX
ejpam-65	22	25	defined	define	VERB
ejpam-65	22	26	as	as	ADP
ejpam-65	22	27	the	the	DET
ejpam-65	22	28	smallest	small	ADJ
ejpam-65	22	29	equivalence	equivalence	NOUN
ejpam-65	22	30	relation	relation	NOUN
ejpam-65	22	31	on	on	ADP
ejpam-65	22	32	v	v	ADP
ejpam-65	22	33	such	such	ADJ
ejpam-65	22	34	that	that	DET
ejpam-65	22	35	v	v	NOUN
ejpam-65	22	36	/	/	SYM
ejpam-65	22	37	ε∗	ε∗	PROPN
ejpam-65	22	38	is	be	AUX
ejpam-65	22	39	a	a	DET
ejpam-65	22	40	classical	classical	ADJ
ejpam-65	22	41	vector	vector	NOUN
ejpam-65	22	42	spaces	space	NOUN
ejpam-65	22	43	over	over	ADP
ejpam-65	22	44	k.	k.	PROPN
ejpam-65	23	1	then	then	ADV
ejpam-65	23	2	it	it	PRON
ejpam-65	23	3	is	be	AUX
ejpam-65	23	4	proved	prove	VERB
ejpam-65	23	5	that	that	SCONJ
ejpam-65	23	6	dimkv	dimkv	VERB
ejpam-65	23	7	=	=	SYM
ejpam-65	23	8	dimkv	dimkv	PROPN
ejpam-65	23	9	/	/	SYM
ejpam-65	23	10	ε	ε	PROPN
ejpam-65	23	11	∗.	∗.	PROPN
ejpam-65	23	12	in	in	ADP
ejpam-65	23	13	§	§	NOUN
ejpam-65	23	14	5	5	NUM
ejpam-65	23	15	we	we	PRON
ejpam-65	23	16	form	form	VERB
ejpam-65	23	17	the	the	DET
ejpam-65	23	18	category	category	NOUN
ejpam-65	23	19	of	of	ADP
ejpam-65	23	20	hypervector	hypervector	NOUN
ejpam-65	23	21	spaces	space	NOUN
ejpam-65	23	22	,	,	PUNCT
ejpam-65	23	23	and	and	CCONJ
ejpam-65	23	24	then	then	ADV
ejpam-65	23	25	we	we	PRON
ejpam-65	23	26	use	use	VERB
ejpam-65	23	27	the	the	DET
ejpam-65	23	28	fundamental	fundamental	ADJ
ejpam-65	23	29	relation	relation	NOUN
ejpam-65	23	30	to	to	PART
ejpam-65	23	31	construct	construct	VERB
ejpam-65	23	32	a	a	DET
ejpam-65	23	33	functor	functor	NOUN
ejpam-65	23	34	between	between	ADP
ejpam-65	23	35	the	the	DET
ejpam-65	23	36	category	category	NOUN
ejpam-65	23	37	of	of	ADP
ejpam-65	23	38	hypervector	hypervector	NOUN
ejpam-65	23	39	spaces	space	NOUN
ejpam-65	23	40	over	over	ADP
ejpam-65	23	41	k	k	PROPN
ejpam-65	23	42	and	and	CCONJ
ejpam-65	23	43	the	the	DET
ejpam-65	23	44	category	category	NOUN
ejpam-65	23	45	of	of	ADP
ejpam-65	23	46	vector	vector	NOUN
ejpam-65	23	47	spaces	space	NOUN
ejpam-65	23	48	over	over	ADP
ejpam-65	23	49	k	k	PROPN
ejpam-65	23	50	,	,	PUNCT
ejpam-65	23	51	and	and	CCONJ
ejpam-65	23	52	prove	prove	VERB
ejpam-65	23	53	that	that	SCONJ
ejpam-65	23	54	this	this	DET
ejpam-65	23	55	functor	functor	PROPN
ejpam-65	23	56	preserves	preserve	VERB
ejpam-65	23	57	dimension	dimension	NOUN
ejpam-65	23	58	.	.	PUNCT
ejpam-65	24	1	∗corresponding	∗corresponde	VERB
ejpam-65	24	2	author	author	NOUN
ejpam-65	24	3	.	.	PUNCT
ejpam-65	25	1	email	email	NOUN
ejpam-65	25	2	addresses	address	NOUN
ejpam-65	25	3	:	:	PUNCT
ejpam-65	25	4	ameri@umz.ac.ir	ameri@umz.ac.ir	NOUN
ejpam-65	25	5	(	(	PUNCT
ejpam-65	25	6	r.	r.	PROPN
ejpam-65	25	7	ameri	ameri	PROPN
ejpam-65	25	8	)	)	PUNCT
ejpam-65	25	9	,	,	PUNCT
ejpam-65	26	1	dehghan@umz.ac.ir	dehghan@umz.ac.ir	ADV
ejpam-65	26	2	(	(	PUNCT
ejpam-65	26	3	o.r	o.r	PROPN
ejpam-65	26	4	.	.	PROPN
ejpam-65	26	5	dehghan	dehghan	PROPN
ejpam-65	26	6	)	)	PUNCT
ejpam-65	26	7	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-65	27	1	32	32	NUM
ejpam-65	28	1	c	c	X
ejpam-65	28	2	©	©	PROPN
ejpam-65	28	3	2007	2007	NUM
ejpam-65	28	4	ejpam	ejpam	NOUN
ejpam-65	28	5	all	all	DET
ejpam-65	28	6	rights	right	NOUN
ejpam-65	28	7	reserved	reserve	VERB
ejpam-65	28	8	.	.	PUNCT
ejpam-65	29	1	r.	r.	PROPN
ejpam-65	29	2	ameri	ameri	PROPN
ejpam-65	29	3	,	,	PUNCT
ejpam-65	29	4	o.r	o.r	PROPN
ejpam-65	29	5	.	.	PROPN
ejpam-65	29	6	dehghan	dehghan	PROPN
ejpam-65	29	7	/	/	SYM
ejpam-65	29	8	eur	eur	PROPN
ejpam-65	29	9	.	.	PUNCT
ejpam-65	30	1	j.	j.	PROPN
ejpam-65	30	2	pure	pure	PROPN
ejpam-65	30	3	appl	appl	PROPN
ejpam-65	30	4	.	.	PROPN
ejpam-65	30	5	math	math	PROPN
ejpam-65	30	6	,	,	PUNCT
ejpam-65	30	7	1	1	NUM
ejpam-65	30	8	(	(	PUNCT
ejpam-65	30	9	2008	2008	NUM
ejpam-65	30	10	)	)	PUNCT
ejpam-65	30	11	,	,	PUNCT
ejpam-65	30	12	(	(	PUNCT
ejpam-65	30	13	32	32	NUM
ejpam-65	30	14	-	-	SYM
ejpam-65	30	15	50	50	NUM
ejpam-65	30	16	)	)	PUNCT
ejpam-65	30	17	33	33	NUM
ejpam-65	30	18	2	2	NUM
ejpam-65	30	19	.	.	PUNCT
ejpam-65	30	20	preliminaries	preliminary	NOUN
ejpam-65	30	21	a	a	DET
ejpam-65	30	22	map	map	NOUN
ejpam-65	30	23	◦	◦	NOUN
ejpam-65	30	24	:	:	PUNCT
ejpam-65	30	25	h	h	NOUN
ejpam-65	30	26	×h	×h	PROPN
ejpam-65	30	27	−→	−→	ADJ
ejpam-65	30	28	p∗(h	p∗(h	NOUN
ejpam-65	30	29	)	)	PUNCT
ejpam-65	30	30	is	be	AUX
ejpam-65	30	31	called	call	VERB
ejpam-65	30	32	hyperoperation	hyperoperation	NOUN
ejpam-65	30	33	or	or	CCONJ
ejpam-65	30	34	join	join	VERB
ejpam-65	30	35	operation	operation	NOUN
ejpam-65	30	36	,	,	PUNCT
ejpam-65	30	37	where	where	SCONJ
ejpam-65	30	38	p∗(h	p∗(h	NOUN
ejpam-65	30	39	)	)	PUNCT
ejpam-65	30	40	is	be	AUX
ejpam-65	30	41	the	the	DET
ejpam-65	30	42	set	set	NOUN
ejpam-65	30	43	of	of	ADP
ejpam-65	30	44	all	all	DET
ejpam-65	30	45	non	non	ADJ
ejpam-65	30	46	-	-	ADJ
ejpam-65	30	47	empty	empty	ADJ
ejpam-65	30	48	subsets	subset	NOUN
ejpam-65	30	49	of	of	ADP
ejpam-65	30	50	h	h	NOUN
ejpam-65	30	51	.	.	PUNCT
ejpam-65	31	1	the	the	DET
ejpam-65	31	2	join	join	NOUN
ejpam-65	31	3	operation	operation	NOUN
ejpam-65	31	4	is	be	AUX
ejpam-65	31	5	extended	extend	VERB
ejpam-65	31	6	to	to	ADP
ejpam-65	31	7	subsets	subset	NOUN
ejpam-65	31	8	of	of	ADP
ejpam-65	31	9	h	h	NOUN
ejpam-65	31	10	in	in	ADP
ejpam-65	31	11	natural	natural	ADJ
ejpam-65	31	12	way	way	NOUN
ejpam-65	31	13	,	,	PUNCT
ejpam-65	31	14	so	so	SCONJ
ejpam-65	31	15	that	that	SCONJ
ejpam-65	31	16	a	a	DET
ejpam-65	31	17	◦	◦	NOUN
ejpam-65	31	18	b	b	NOUN
ejpam-65	31	19	is	be	AUX
ejpam-65	31	20	given	give	VERB
ejpam-65	31	21	by	by	ADP
ejpam-65	31	22	a	a	DET
ejpam-65	31	23	◦	◦	NOUN
ejpam-65	31	24	b	b	NOUN
ejpam-65	31	25	=	=	SYM
ejpam-65	31	26	⋃	⋃	NOUN
ejpam-65	31	27	{	{	PUNCT
ejpam-65	31	28	a	a	DET
ejpam-65	31	29	◦	◦	NOUN
ejpam-65	31	30	b	b	NOUN
ejpam-65	31	31	:	:	PUNCT
ejpam-65	31	32	a	a	DET
ejpam-65	31	33	∈	∈	PROPN
ejpam-65	31	34	a	a	PRON
ejpam-65	31	35	and	and	CCONJ
ejpam-65	31	36	b	b	NOUN
ejpam-65	31	37	∈	∈	PROPN
ejpam-65	31	38	b	b	PROPN
ejpam-65	31	39	}	}	PUNCT
ejpam-65	31	40	.	.	PUNCT
ejpam-65	32	1	the	the	DET
ejpam-65	32	2	notations	notation	NOUN
ejpam-65	32	3	a	a	DET
ejpam-65	32	4	◦	◦	NOUN
ejpam-65	32	5	a	a	PRON
ejpam-65	32	6	and	and	CCONJ
ejpam-65	32	7	a	a	DET
ejpam-65	32	8	◦	◦	NOUN
ejpam-65	32	9	a	a	PRON
ejpam-65	32	10	are	be	AUX
ejpam-65	32	11	used	use	VERB
ejpam-65	32	12	for	for	ADP
ejpam-65	32	13	{	{	PUNCT
ejpam-65	32	14	a	a	DET
ejpam-65	32	15	}	}	PUNCT
ejpam-65	32	16	◦	◦	NOUN
ejpam-65	32	17	a	a	PRON
ejpam-65	32	18	and	and	CCONJ
ejpam-65	32	19	a	a	DET
ejpam-65	32	20	◦	◦	NOUN
ejpam-65	32	21	{	{	PUNCT
ejpam-65	32	22	a	a	PRON
ejpam-65	32	23	}	}	PUNCT
ejpam-65	32	24	respectively	respectively	ADV
ejpam-65	32	25	.	.	PUNCT
ejpam-65	33	1	generally	generally	ADV
ejpam-65	33	2	,	,	PUNCT
ejpam-65	33	3	the	the	DET
ejpam-65	33	4	singleton	singleton	NOUN
ejpam-65	33	5	{	{	PUNCT
ejpam-65	33	6	a	a	PRON
ejpam-65	33	7	}	}	PUNCT
ejpam-65	33	8	is	be	AUX
ejpam-65	33	9	identified	identify	VERB
ejpam-65	33	10	by	by	ADP
ejpam-65	33	11	its	its	PRON
ejpam-65	33	12	element	element	NOUN
ejpam-65	33	13	a.	a.	NOUN
ejpam-65	33	14	definition	definition	NOUN
ejpam-65	33	15	2.1	2.1	NUM
ejpam-65	33	16	.	.	PUNCT
ejpam-65	34	1	[	[	X
ejpam-65	34	2	6	6	NUM
ejpam-65	34	3	]	]	PUNCT
ejpam-65	34	4	let	let	VERB
ejpam-65	34	5	k	k	PRON
ejpam-65	34	6	be	be	AUX
ejpam-65	34	7	a	a	DET
ejpam-65	34	8	field	field	NOUN
ejpam-65	34	9	and	and	CCONJ
ejpam-65	34	10	(	(	PUNCT
ejpam-65	34	11	v,+	v,+	NUM
ejpam-65	34	12	)	)	PUNCT
ejpam-65	34	13	be	be	AUX
ejpam-65	34	14	an	an	DET
ejpam-65	34	15	abelian	abelian	ADJ
ejpam-65	34	16	group	group	NOUN
ejpam-65	34	17	.	.	PUNCT
ejpam-65	35	1	we	we	PRON
ejpam-65	35	2	define	define	VERB
ejpam-65	35	3	a	a	DET
ejpam-65	35	4	hypervector	hypervector	NOUN
ejpam-65	35	5	space	space	NOUN
ejpam-65	35	6	over	over	ADP
ejpam-65	35	7	k	k	PROPN
ejpam-65	35	8	to	to	PART
ejpam-65	35	9	be	be	AUX
ejpam-65	35	10	the	the	DET
ejpam-65	35	11	quadrupled	quadruple	VERB
ejpam-65	35	12	(	(	PUNCT
ejpam-65	35	13	v,+	v,+	NUM
ejpam-65	35	14	,	,	PUNCT
ejpam-65	35	15	◦	◦	NOUN
ejpam-65	35	16	,	,	PUNCT
ejpam-65	35	17	k	k	NOUN
ejpam-65	35	18	)	)	PUNCT
ejpam-65	35	19	,	,	PUNCT
ejpam-65	35	20	where	where	SCONJ
ejpam-65	35	21	”	"	PUNCT
ejpam-65	35	22	◦	◦	NOUN
ejpam-65	35	23	”	"	PUNCT
ejpam-65	35	24	is	be	AUX
ejpam-65	35	25	a	a	DET
ejpam-65	35	26	mapping	mapping	NOUN
ejpam-65	35	27	◦	◦	NOUN
ejpam-65	35	28	:	:	PUNCT
ejpam-65	35	29	k	k	X
ejpam-65	35	30	×	×	NOUN
ejpam-65	35	31	v	v	INTJ
ejpam-65	35	32	−→	−→	ADJ
ejpam-65	35	33	p∗(v	p∗(v	PROPN
ejpam-65	35	34	)	)	PUNCT
ejpam-65	35	35	,	,	PUNCT
ejpam-65	35	36	such	such	ADJ
ejpam-65	35	37	that	that	SCONJ
ejpam-65	35	38	the	the	DET
ejpam-65	35	39	following	follow	VERB
ejpam-65	35	40	conditions	condition	NOUN
ejpam-65	35	41	hold	hold	VERB
ejpam-65	35	42	:	:	PUNCT
ejpam-65	35	43	(	(	PUNCT
ejpam-65	35	44	h1	h1	PROPN
ejpam-65	35	45	)	)	PUNCT
ejpam-65	35	46	∀a	∀a	NOUN
ejpam-65	35	47	∈	∈	PROPN
ejpam-65	35	48	k	k	NOUN
ejpam-65	35	49	,	,	PUNCT
ejpam-65	35	50	∀x	∀x	X
ejpam-65	35	51	,	,	PUNCT
ejpam-65	35	52	y	y	PROPN
ejpam-65	35	53	∈	∈	PROPN
ejpam-65	35	54	v	v	PROPN
ejpam-65	35	55	,	,	PUNCT
ejpam-65	35	56	a	a	DET
ejpam-65	35	57	◦	◦	NOUN
ejpam-65	35	58	(	(	PUNCT
ejpam-65	35	59	x+	x+	ADJ
ejpam-65	35	60	y	y	NOUN
ejpam-65	35	61	)	)	PUNCT
ejpam-65	35	62	⊆	⊆	PROPN
ejpam-65	35	63	a	a	DET
ejpam-65	35	64	◦	◦	NOUN
ejpam-65	35	65	x+	x+	PUNCT
ejpam-65	35	66	a	a	DET
ejpam-65	35	67	◦	◦	NOUN
ejpam-65	35	68	y	y	NOUN
ejpam-65	35	69	,	,	PUNCT
ejpam-65	35	70	right	right	ADJ
ejpam-65	35	71	distributive	distributive	ADJ
ejpam-65	35	72	law	law	NOUN
ejpam-65	35	73	,	,	PUNCT
ejpam-65	35	74	(	(	PUNCT
ejpam-65	35	75	h2	h2	NOUN
ejpam-65	35	76	)	)	PUNCT
ejpam-65	35	77	∀a	∀a	NOUN
ejpam-65	35	78	,	,	PUNCT
ejpam-65	35	79	b	b	X
ejpam-65	35	80	∈	∈	PROPN
ejpam-65	35	81	k	k	NOUN
ejpam-65	35	82	,	,	PUNCT
ejpam-65	35	83	∀x	∀x	X
ejpam-65	35	84	∈	∈	PROPN
ejpam-65	35	85	v	v	NOUN
ejpam-65	35	86	,	,	PUNCT
ejpam-65	35	87	(	(	PUNCT
ejpam-65	35	88	a+	a+	X
ejpam-65	35	89	b	b	X
ejpam-65	35	90	)	)	PUNCT
ejpam-65	35	91	◦	◦	NOUN
ejpam-65	35	92	x	x	SYM
ejpam-65	35	93	⊆	⊆	NUM
ejpam-65	35	94	a	a	DET
ejpam-65	35	95	◦	◦	NOUN
ejpam-65	35	96	x+	x+	X
ejpam-65	35	97	b	b	NOUN
ejpam-65	35	98	◦	◦	NOUN
ejpam-65	35	99	x	x	NOUN
ejpam-65	35	100	,	,	PUNCT
ejpam-65	35	101	left	leave	VERB
ejpam-65	35	102	distributive	distributive	ADJ
ejpam-65	35	103	law	law	NOUN
ejpam-65	35	104	,	,	PUNCT
ejpam-65	35	105	(	(	PUNCT
ejpam-65	35	106	h3	h3	NOUN
ejpam-65	35	107	)	)	PUNCT
ejpam-65	35	108	∀a	∀a	NOUN
ejpam-65	35	109	,	,	PUNCT
ejpam-65	35	110	b	b	X
ejpam-65	35	111	∈	∈	PROPN
ejpam-65	35	112	k	k	NOUN
ejpam-65	35	113	,	,	PUNCT
ejpam-65	35	114	∀x	∀x	X
ejpam-65	35	115	∈	∈	PROPN
ejpam-65	35	116	v	v	NOUN
ejpam-65	35	117	,	,	PUNCT
ejpam-65	35	118	a	a	DET
ejpam-65	35	119	◦	◦	NOUN
ejpam-65	35	120	(	(	PUNCT
ejpam-65	35	121	b	b	X
ejpam-65	35	122	◦	◦	NOUN
ejpam-65	35	123	x	x	NOUN
ejpam-65	35	124	)	)	PUNCT
ejpam-65	35	125	=	=	SYM
ejpam-65	35	126	(	(	PUNCT
ejpam-65	35	127	ab	ab	NOUN
ejpam-65	35	128	)	)	PUNCT
ejpam-65	35	129	◦	◦	NOUN
ejpam-65	35	130	x	x	SYM
ejpam-65	35	131	,	,	PUNCT
ejpam-65	35	132	associative	associative	ADJ
ejpam-65	35	133	law	law	NOUN
ejpam-65	35	134	,	,	PUNCT
ejpam-65	35	135	(	(	PUNCT
ejpam-65	35	136	h4	h4	NOUN
ejpam-65	35	137	)	)	PUNCT
ejpam-65	35	138	∀a	∀a	NOUN
ejpam-65	35	139	∈	∈	PROPN
ejpam-65	35	140	k	k	NOUN
ejpam-65	35	141	,	,	PUNCT
ejpam-65	35	142	∀x	∀x	X
ejpam-65	35	143	∈	∈	PROPN
ejpam-65	35	144	v	v	NOUN
ejpam-65	35	145	,	,	PUNCT
ejpam-65	35	146	a	a	DET
ejpam-65	35	147	◦	◦	NOUN
ejpam-65	35	148	(	(	PUNCT
ejpam-65	35	149	−x	−x	NOUN
ejpam-65	35	150	)	)	PUNCT
ejpam-65	35	151	=	=	PUNCT
ejpam-65	35	152	(	(	PUNCT
ejpam-65	35	153	−a	−a	ADJ
ejpam-65	35	154	)	)	PUNCT
ejpam-65	35	155	◦	◦	NOUN
ejpam-65	35	156	x	x	X
ejpam-65	35	157	=	=	SYM
ejpam-65	35	158	−(a	−(a	ADJ
ejpam-65	35	159	◦	◦	NOUN
ejpam-65	35	160	x	x	X
ejpam-65	35	161	)	)	PUNCT
ejpam-65	35	162	,	,	PUNCT
ejpam-65	35	163	(	(	PUNCT
ejpam-65	35	164	h5	h5	PROPN
ejpam-65	35	165	)	)	PUNCT
ejpam-65	35	166	∀x	∀x	VERB
ejpam-65	35	167	∈	∈	PROPN
ejpam-65	35	168	v	v	NOUN
ejpam-65	35	169	,	,	PUNCT
ejpam-65	35	170	x	x	SYM
ejpam-65	35	171	∈	∈	NOUN
ejpam-65	35	172	1	1	NUM
ejpam-65	35	173	◦	◦	NOUN
ejpam-65	35	174	x.	x.	NOUN
ejpam-65	35	175	remark	remark	VERB
ejpam-65	35	176	2.1	2.1	NUM
ejpam-65	35	177	.	.	PUNCT
ejpam-65	36	1	(	(	PUNCT
ejpam-65	36	2	i	i	NOUN
ejpam-65	36	3	)	)	PUNCT
ejpam-65	36	4	in	in	ADP
ejpam-65	36	5	the	the	DET
ejpam-65	36	6	right	right	ADJ
ejpam-65	36	7	hand	hand	NOUN
ejpam-65	36	8	side	side	NOUN
ejpam-65	36	9	of	of	ADP
ejpam-65	36	10	(	(	PUNCT
ejpam-65	36	11	h1	h1	PROPN
ejpam-65	36	12	)	)	PUNCT
ejpam-65	36	13	the	the	DET
ejpam-65	36	14	sum	sum	NOUN
ejpam-65	36	15	is	be	AUX
ejpam-65	36	16	meant	mean	VERB
ejpam-65	36	17	in	in	ADP
ejpam-65	36	18	the	the	DET
ejpam-65	36	19	sense	sense	NOUN
ejpam-65	36	20	of	of	ADP
ejpam-65	36	21	frobenius	frobenius	NOUN
ejpam-65	36	22	,	,	PUNCT
ejpam-65	36	23	that	that	PRON
ejpam-65	36	24	is	is	ADV
ejpam-65	36	25	we	we	PRON
ejpam-65	36	26	consider	consider	VERB
ejpam-65	36	27	the	the	DET
ejpam-65	36	28	set	set	NOUN
ejpam-65	36	29	of	of	ADP
ejpam-65	36	30	all	all	DET
ejpam-65	36	31	sums	sum	NOUN
ejpam-65	36	32	of	of	ADP
ejpam-65	36	33	an	an	DET
ejpam-65	36	34	element	element	NOUN
ejpam-65	36	35	of	of	ADP
ejpam-65	36	36	a	a	DET
ejpam-65	36	37	◦	◦	NOUN
ejpam-65	36	38	x	x	PUNCT
ejpam-65	36	39	with	with	ADP
ejpam-65	36	40	an	an	DET
ejpam-65	36	41	element	element	NOUN
ejpam-65	36	42	of	of	ADP
ejpam-65	36	43	a	a	DET
ejpam-65	36	44	◦	◦	NOUN
ejpam-65	37	1	y.	y.	NOUN
ejpam-65	38	1	similarly	similarly	ADV
ejpam-65	38	2	we	we	PRON
ejpam-65	38	3	have	have	VERB
ejpam-65	38	4	in	in	ADP
ejpam-65	38	5	(	(	PUNCT
ejpam-65	38	6	h2	h2	NOUN
ejpam-65	38	7	)	)	PUNCT
ejpam-65	38	8	.	.	PUNCT
ejpam-65	39	1	(	(	PUNCT
ejpam-65	39	2	ii	ii	X
ejpam-65	39	3	)	)	PUNCT
ejpam-65	39	4	we	we	PRON
ejpam-65	39	5	say	say	VERB
ejpam-65	39	6	that	that	SCONJ
ejpam-65	39	7	(	(	PUNCT
ejpam-65	39	8	v,+	v,+	NUM
ejpam-65	39	9	,	,	PUNCT
ejpam-65	39	10	◦	◦	NOUN
ejpam-65	39	11	,	,	PUNCT
ejpam-65	39	12	k	k	NOUN
ejpam-65	39	13	)	)	PUNCT
ejpam-65	39	14	is	be	AUX
ejpam-65	39	15	anti	anti	ADJ
ejpam-65	39	16	-	-	ADJ
ejpam-65	39	17	left	left	ADJ
ejpam-65	39	18	distributive	distributive	ADJ
ejpam-65	39	19	,	,	PUNCT
ejpam-65	39	20	if	if	SCONJ
ejpam-65	39	21	∀a	∀a	NOUN
ejpam-65	39	22	,	,	PUNCT
ejpam-65	39	23	b	b	PROPN
ejpam-65	39	24	∈	∈	PROPN
ejpam-65	39	25	k	k	NOUN
ejpam-65	39	26	,	,	PUNCT
ejpam-65	39	27	∀x	∀x	X
ejpam-65	39	28	∈	∈	PROPN
ejpam-65	39	29	v	v	NOUN
ejpam-65	39	30	,	,	PUNCT
ejpam-65	39	31	(	(	PUNCT
ejpam-65	39	32	a+	a+	X
ejpam-65	39	33	b	b	X
ejpam-65	39	34	)	)	PUNCT
ejpam-65	39	35	◦	◦	NOUN
ejpam-65	39	36	x	x	SYM
ejpam-65	39	37	⊇	⊇	NOUN
ejpam-65	39	38	a	a	DET
ejpam-65	39	39	◦	◦	NOUN
ejpam-65	39	40	x+	x+	X
ejpam-65	39	41	b	b	X
ejpam-65	39	42	◦	◦	NOUN
ejpam-65	39	43	x	x	NOUN
ejpam-65	39	44	,	,	PUNCT
ejpam-65	39	45	and	and	CCONJ
ejpam-65	39	46	strongly	strongly	ADV
ejpam-65	39	47	left	leave	VERB
ejpam-65	39	48	distributive	distributive	ADJ
ejpam-65	39	49	,	,	PUNCT
ejpam-65	39	50	if	if	SCONJ
ejpam-65	39	51	∀a	∀a	NOUN
ejpam-65	39	52	,	,	PUNCT
ejpam-65	40	1	b	b	PROPN
ejpam-65	40	2	∈	∈	PROPN
ejpam-65	40	3	k	k	NOUN
ejpam-65	40	4	,	,	PUNCT
ejpam-65	40	5	∀x	∀x	X
ejpam-65	40	6	∈	∈	PROPN
ejpam-65	40	7	v	v	NOUN
ejpam-65	40	8	,	,	PUNCT
ejpam-65	40	9	(	(	PUNCT
ejpam-65	40	10	a+	a+	X
ejpam-65	40	11	b	b	X
ejpam-65	40	12	)	)	PUNCT
ejpam-65	40	13	◦	◦	NOUN
ejpam-65	40	14	x	x	X
ejpam-65	40	15	=	=	PUNCT
ejpam-65	40	16	a	a	DET
ejpam-65	40	17	◦	◦	NOUN
ejpam-65	40	18	x+	x+	X
ejpam-65	40	19	b	b	X
ejpam-65	40	20	◦	◦	NOUN
ejpam-65	40	21	x	x	SYM
ejpam-65	40	22	,	,	PUNCT
ejpam-65	40	23	in	in	ADP
ejpam-65	40	24	a	a	DET
ejpam-65	40	25	similar	similar	ADJ
ejpam-65	40	26	way	way	NOUN
ejpam-65	40	27	we	we	PRON
ejpam-65	40	28	define	define	VERB
ejpam-65	40	29	the	the	DET
ejpam-65	40	30	anti	anti	ADJ
ejpam-65	40	31	-	-	ADJ
ejpam-65	40	32	right	right	ADJ
ejpam-65	40	33	distributive	distributive	ADJ
ejpam-65	40	34	and	and	CCONJ
ejpam-65	40	35	strongly	strongly	ADV
ejpam-65	40	36	right	right	ADJ
ejpam-65	40	37	distributive	distributive	ADJ
ejpam-65	40	38	hypervector	hypervector	NOUN
ejpam-65	40	39	spaces	space	NOUN
ejpam-65	40	40	,	,	PUNCT
ejpam-65	40	41	respectively	respectively	ADV
ejpam-65	40	42	.	.	PUNCT
ejpam-65	41	1	v	v	NOUN
ejpam-65	41	2	is	be	AUX
ejpam-65	41	3	called	call	VERB
ejpam-65	41	4	strongly	strongly	ADV
ejpam-65	41	5	distributive	distributive	ADJ
ejpam-65	41	6	if	if	SCONJ
ejpam-65	41	7	it	it	PRON
ejpam-65	41	8	is	be	AUX
ejpam-65	41	9	both	both	PRON
ejpam-65	41	10	strongly	strongly	ADV
ejpam-65	41	11	left	left	ADJ
ejpam-65	41	12	and	and	CCONJ
ejpam-65	41	13	strongly	strongly	ADV
ejpam-65	41	14	right	right	ADV
ejpam-65	41	15	distributive	distributive	ADJ
ejpam-65	41	16	.	.	PUNCT
ejpam-65	42	1	(	(	PUNCT
ejpam-65	42	2	iii	iii	X
ejpam-65	42	3	)	)	PUNCT
ejpam-65	42	4	the	the	DET
ejpam-65	42	5	left	left	ADJ
ejpam-65	42	6	hand	hand	NOUN
ejpam-65	42	7	side	side	NOUN
ejpam-65	42	8	of	of	ADP
ejpam-65	42	9	(	(	PUNCT
ejpam-65	42	10	h3	h3	NOUN
ejpam-65	42	11	)	)	PUNCT
ejpam-65	42	12	means	mean	VERB
ejpam-65	42	13	the	the	DET
ejpam-65	42	14	set	set	VERB
ejpam-65	42	15	-	-	PUNCT
ejpam-65	42	16	theoretical	theoretical	ADJ
ejpam-65	42	17	union	union	NOUN
ejpam-65	42	18	of	of	ADP
ejpam-65	42	19	all	all	DET
ejpam-65	42	20	the	the	DET
ejpam-65	42	21	sets	set	NOUN
ejpam-65	42	22	a	a	DET
ejpam-65	42	23	◦	◦	NOUN
ejpam-65	42	24	y	y	PROPN
ejpam-65	42	25	,	,	PUNCT
ejpam-65	42	26	where	where	SCONJ
ejpam-65	42	27	y	y	PROPN
ejpam-65	42	28	runs	run	VERB
ejpam-65	42	29	over	over	ADP
ejpam-65	42	30	the	the	DET
ejpam-65	42	31	set	set	NOUN
ejpam-65	42	32	b	b	PROPN
ejpam-65	42	33	◦	◦	NOUN
ejpam-65	42	34	x	x	SYM
ejpam-65	42	35	,	,	PUNCT
ejpam-65	42	36	i.e.	i.e.	X
ejpam-65	42	37	a	a	DET
ejpam-65	42	38	◦	◦	NOUN
ejpam-65	42	39	(	(	PUNCT
ejpam-65	42	40	b	b	X
ejpam-65	42	41	◦	◦	NOUN
ejpam-65	42	42	x	x	NOUN
ejpam-65	42	43	)	)	PUNCT
ejpam-65	42	44	=	=	PUNCT
ejpam-65	43	1	⋃	⋃	NOUN
ejpam-65	43	2	y∈b	y∈b	NOUN
ejpam-65	43	3	◦	◦	NOUN
ejpam-65	43	4	x	x	SYM
ejpam-65	43	5	a	a	DET
ejpam-65	43	6	◦	◦	NOUN
ejpam-65	43	7	y.	y.	NOUN
ejpam-65	43	8	(	(	PUNCT
ejpam-65	43	9	iv	iv	X
ejpam-65	43	10	)	)	PUNCT
ejpam-65	43	11	let	let	VERB
ejpam-65	43	12	ω	ω	NOUN
ejpam-65	43	13	=	=	SYM
ejpam-65	43	14	0	0	NUM
ejpam-65	43	15	◦	◦	NOUN
ejpam-65	43	16	0	0	NUM
ejpam-65	43	17	,	,	PUNCT
ejpam-65	43	18	where	where	SCONJ
ejpam-65	43	19	0	0	NUM
ejpam-65	43	20	is	be	AUX
ejpam-65	43	21	the	the	DET
ejpam-65	43	22	zero	zero	NUM
ejpam-65	43	23	of	of	ADP
ejpam-65	43	24	(	(	PUNCT
ejpam-65	43	25	v,+	v,+	NUM
ejpam-65	43	26	)	)	PUNCT
ejpam-65	43	27	.	.	PUNCT
ejpam-65	44	1	in	in	ADP
ejpam-65	44	2	[	[	X
ejpam-65	44	3	6	6	NUM
ejpam-65	44	4	]	]	PUNCT
ejpam-65	44	5	it	it	PRON
ejpam-65	44	6	is	be	AUX
ejpam-65	44	7	shown	show	VERB
ejpam-65	44	8	if	if	SCONJ
ejpam-65	44	9	v	v	NOUN
ejpam-65	44	10	is	be	AUX
ejpam-65	44	11	either	either	CCONJ
ejpam-65	44	12	strongly	strongly	ADV
ejpam-65	44	13	right	right	ADJ
ejpam-65	44	14	or	or	CCONJ
ejpam-65	44	15	left	leave	VERB
ejpam-65	44	16	distributive	distributive	ADJ
ejpam-65	44	17	,	,	PUNCT
ejpam-65	44	18	then	then	ADV
ejpam-65	44	19	ω	ω	PROPN
ejpam-65	44	20	is	be	AUX
ejpam-65	44	21	a	a	DET
ejpam-65	44	22	subgroup	subgroup	NOUN
ejpam-65	44	23	of	of	ADP
ejpam-65	44	24	(	(	PUNCT
ejpam-65	44	25	v,+	v,+	NUM
ejpam-65	44	26	)	)	PUNCT
ejpam-65	44	27	.	.	PUNCT
ejpam-65	45	1	example	example	NOUN
ejpam-65	45	2	2.1	2.1	NUM
ejpam-65	45	3	.	.	PUNCT
ejpam-65	46	1	in	in	ADP
ejpam-65	46	2	(	(	PUNCT
ejpam-65	46	3	rn,+	rn,+	X
ejpam-65	46	4	)	)	PUNCT
ejpam-65	46	5	we	we	PRON
ejpam-65	46	6	define	define	VERB
ejpam-65	46	7	,	,	PUNCT
ejpam-65	46	8	∀a	∀a	NOUN
ejpam-65	46	9	∈	∈	NOUN
ejpam-65	46	10	r	r	NOUN
ejpam-65	46	11	and	and	CCONJ
ejpam-65	46	12	∀x	∀x	NUM
ejpam-65	46	13	∈	∈	PROPN
ejpam-65	46	14	rn	rn	PROPN
ejpam-65	46	15	,	,	PUNCT
ejpam-65	46	16	a	a	DET
ejpam-65	46	17	◦	◦	NOUN
ejpam-65	46	18	x	x	PUNCT
ejpam-65	46	19	as	as	ADP
ejpam-65	46	20	the	the	DET
ejpam-65	46	21	set	set	NOUN
ejpam-65	46	22	of	of	ADP
ejpam-65	46	23	vectors	vector	NOUN
ejpam-65	46	24	in	in	ADP
ejpam-65	46	25	rn	rn	PROPN
ejpam-65	46	26	belonging	belong	VERB
ejpam-65	46	27	to	to	ADP
ejpam-65	46	28	the	the	DET
ejpam-65	46	29	closed	closed	ADJ
ejpam-65	46	30	segment	segment	NOUN
ejpam-65	46	31	whose	whose	DET
ejpam-65	46	32	vertices	vertex	NOUN
ejpam-65	46	33	are	be	AUX
ejpam-65	46	34	the	the	DET
ejpam-65	46	35	origin	origin	NOUN
ejpam-65	46	36	,	,	PUNCT
ejpam-65	46	37	0	0	NUM
ejpam-65	46	38	,	,	PUNCT
ejpam-65	46	39	and	and	CCONJ
ejpam-65	46	40	the	the	DET
ejpam-65	46	41	point	point	NOUN
ejpam-65	46	42	ax	ax	NOUN
ejpam-65	46	43	in	in	ADP
ejpam-65	46	44	rn	rn	PROPN
ejpam-65	46	45	.	.	PUNCT
ejpam-65	47	1	then	then	ADV
ejpam-65	47	2	(	(	PUNCT
ejpam-65	47	3	rn,+	rn,+	NOUN
ejpam-65	47	4	,	,	PUNCT
ejpam-65	47	5	◦	◦	NOUN
ejpam-65	47	6	,	,	PUNCT
ejpam-65	47	7	r	r	NOUN
ejpam-65	47	8	)	)	PUNCT
ejpam-65	47	9	is	be	AUX
ejpam-65	47	10	a	a	DET
ejpam-65	47	11	hypervector	hypervector	NOUN
ejpam-65	47	12	space	space	NOUN
ejpam-65	47	13	.	.	PUNCT
ejpam-65	48	1	r.	r.	PROPN
ejpam-65	48	2	ameri	ameri	PROPN
ejpam-65	48	3	,	,	PUNCT
ejpam-65	48	4	o.r	o.r	PROPN
ejpam-65	48	5	.	.	PROPN
ejpam-65	48	6	dehghan	dehghan	PROPN
ejpam-65	48	7	/	/	SYM
ejpam-65	48	8	eur	eur	PROPN
ejpam-65	48	9	.	.	PUNCT
ejpam-65	49	1	j.	j.	PROPN
ejpam-65	49	2	pure	pure	PROPN
ejpam-65	49	3	appl	appl	PROPN
ejpam-65	49	4	.	.	PROPN
ejpam-65	49	5	math	math	PROPN
ejpam-65	49	6	,	,	PUNCT
ejpam-65	49	7	1	1	NUM
ejpam-65	49	8	(	(	PUNCT
ejpam-65	49	9	2008	2008	NUM
ejpam-65	49	10	)	)	PUNCT
ejpam-65	49	11	,	,	PUNCT
ejpam-65	49	12	(	(	PUNCT
ejpam-65	49	13	32	32	NUM
ejpam-65	49	14	-	-	SYM
ejpam-65	49	15	50	50	NUM
ejpam-65	49	16	)	)	PUNCT
ejpam-65	49	17	34	34	NUM
ejpam-65	49	18	example	example	NOUN
ejpam-65	49	19	2.2	2.2	NUM
ejpam-65	49	20	.	.	PUNCT
ejpam-65	50	1	in	in	ADP
ejpam-65	50	2	(	(	PUNCT
ejpam-65	50	3	r2,+	r2,+	NOUN
ejpam-65	50	4	)	)	PUNCT
ejpam-65	50	5	we	we	PRON
ejpam-65	50	6	define	define	VERB
ejpam-65	50	7	{	{	PUNCT
ejpam-65	50	8	◦	◦	NOUN
ejpam-65	50	9	:	:	PUNCT
ejpam-65	50	10	r×	r×	NOUN
ejpam-65	50	11	r2	r2	PROPN
ejpam-65	50	12	−→	−→	NOUN
ejpam-65	50	13	p∗(r2	p∗(r2	PROPN
ejpam-65	50	14	)	)	PUNCT
ejpam-65	50	15	a	a	DET
ejpam-65	50	16	◦	◦	NOUN
ejpam-65	50	17	(	(	PUNCT
ejpam-65	50	18	x	x	NOUN
ejpam-65	50	19	,	,	PUNCT
ejpam-65	50	20	y	y	PROPN
ejpam-65	50	21	)	)	PUNCT
ejpam-65	50	22	=	=	NOUN
ejpam-65	50	23	ax×	ax×	NOUN
ejpam-65	50	24	r	r	NOUN
ejpam-65	50	25	or	or	CCONJ
ejpam-65	50	26	{	{	PUNCT
ejpam-65	50	27	◦	◦	NOUN
ejpam-65	50	28	:	:	PUNCT
ejpam-65	50	29	r×	r×	NOUN
ejpam-65	50	30	r2	r2	PROPN
ejpam-65	50	31	−→	−→	NOUN
ejpam-65	50	32	p∗(r2	p∗(r2	PROPN
ejpam-65	50	33	)	)	PUNCT
ejpam-65	50	34	a	a	DET
ejpam-65	50	35	◦	◦	NOUN
ejpam-65	50	36	(	(	PUNCT
ejpam-65	50	37	x	x	NOUN
ejpam-65	50	38	,	,	PUNCT
ejpam-65	50	39	y	y	NOUN
ejpam-65	50	40	)	)	PUNCT
ejpam-65	50	41	=	=	PUNCT
ejpam-65	50	42	r×	r×	NOUN
ejpam-65	50	43	ay	ay	NOUN
ejpam-65	50	44	then	then	ADV
ejpam-65	50	45	(	(	PUNCT
ejpam-65	50	46	r2,+	r2,+	NOUN
ejpam-65	50	47	,	,	PUNCT
ejpam-65	50	48	◦	◦	NOUN
ejpam-65	50	49	,	,	PUNCT
ejpam-65	50	50	r	r	NOUN
ejpam-65	50	51	)	)	PUNCT
ejpam-65	50	52	is	be	AUX
ejpam-65	50	53	a	a	DET
ejpam-65	50	54	strongly	strongly	ADV
ejpam-65	50	55	distributive	distributive	ADJ
ejpam-65	50	56	hypervector	hypervector	NOUN
ejpam-65	50	57	space	space	NOUN
ejpam-65	50	58	.	.	PUNCT
ejpam-65	51	1	example	example	NOUN
ejpam-65	51	2	2.3	2.3	NUM
ejpam-65	51	3	.	.	PUNCT
ejpam-65	52	1	let	let	AUX
ejpam-65	52	2	(	(	PUNCT
ejpam-65	52	3	v,+	v,+	NUM
ejpam-65	52	4	,	,	PUNCT
ejpam-65	52	5	.,k	.,k	NUM
ejpam-65	52	6	)	)	PUNCT
ejpam-65	52	7	be	be	AUX
ejpam-65	52	8	a	a	DET
ejpam-65	52	9	classical	classical	ADJ
ejpam-65	52	10	vector	vector	NOUN
ejpam-65	52	11	space	space	NOUN
ejpam-65	52	12	and	and	CCONJ
ejpam-65	52	13	p	p	NOUN
ejpam-65	52	14	be	be	AUX
ejpam-65	52	15	a	a	DET
ejpam-65	52	16	subspace	subspace	NOUN
ejpam-65	52	17	of	of	ADP
ejpam-65	52	18	v	v	NOUN
ejpam-65	52	19	and	and	CCONJ
ejpam-65	52	20	{	{	PUNCT
ejpam-65	52	21	◦	◦	NOUN
ejpam-65	52	22	:	:	PUNCT
ejpam-65	52	23	k	k	X
ejpam-65	52	24	×	×	NOUN
ejpam-65	52	25	v	v	INTJ
ejpam-65	52	26	−→	−→	ADJ
ejpam-65	52	27	p∗(v	p∗(v	PROPN
ejpam-65	52	28	)	)	PUNCT
ejpam-65	52	29	a	a	DET
ejpam-65	52	30	◦	◦	NOUN
ejpam-65	52	31	x	x	SYM
ejpam-65	53	1	=	=	NOUN
ejpam-65	53	2	a.x+	a.x+	X
ejpam-65	53	3	p	p	X
ejpam-65	53	4	then	then	ADV
ejpam-65	53	5	(	(	PUNCT
ejpam-65	53	6	v,+	v,+	NUM
ejpam-65	53	7	,	,	PUNCT
ejpam-65	53	8	◦	◦	NOUN
ejpam-65	53	9	,	,	PUNCT
ejpam-65	53	10	k	k	NOUN
ejpam-65	53	11	)	)	PUNCT
ejpam-65	53	12	is	be	AUX
ejpam-65	53	13	a	a	DET
ejpam-65	53	14	strongly	strongly	ADV
ejpam-65	53	15	distributive	distributive	ADJ
ejpam-65	53	16	hypervector	hypervector	NOUN
ejpam-65	53	17	space	space	NOUN
ejpam-65	53	18	.	.	PUNCT
ejpam-65	54	1	theorem	theorem	VERB
ejpam-65	54	2	2.1	2.1	NUM
ejpam-65	54	3	.	.	PUNCT
ejpam-65	55	1	[	[	X
ejpam-65	55	2	6	6	NUM
ejpam-65	55	3	]	]	PUNCT
ejpam-65	55	4	every	every	DET
ejpam-65	55	5	strongly	strongly	ADV
ejpam-65	55	6	right	right	ADJ
ejpam-65	55	7	distributive	distributive	ADJ
ejpam-65	55	8	hypervector	hypervector	NOUN
ejpam-65	55	9	space	space	NOUN
ejpam-65	55	10	is	be	AUX
ejpam-65	55	11	strongly	strongly	ADV
ejpam-65	55	12	left	leave	VERB
ejpam-65	55	13	distributive	distributive	ADJ
ejpam-65	55	14	hypervector	hypervector	NOUN
ejpam-65	55	15	space	space	NOUN
ejpam-65	55	16	.	.	PUNCT
ejpam-65	56	1	let	let	AUX
ejpam-65	56	2	(	(	PUNCT
ejpam-65	56	3	v,+	v,+	NUM
ejpam-65	56	4	)	)	PUNCT
ejpam-65	56	5	be	be	AUX
ejpam-65	56	6	an	an	DET
ejpam-65	56	7	abelian	abelian	ADJ
ejpam-65	56	8	group	group	NOUN
ejpam-65	56	9	,	,	PUNCT
ejpam-65	56	10	ω	ω	PROPN
ejpam-65	56	11	a	a	DET
ejpam-65	56	12	subgroup	subgroup	NOUN
ejpam-65	56	13	of	of	ADP
ejpam-65	56	14	v	v	NOUN
ejpam-65	56	15	and	and	CCONJ
ejpam-65	56	16	k	k	NOUN
ejpam-65	56	17	a	a	DET
ejpam-65	56	18	field	field	NOUN
ejpam-65	56	19	such	such	ADJ
ejpam-65	56	20	that	that	PRON
ejpam-65	56	21	w	w	PROPN
ejpam-65	56	22	=	=	SYM
ejpam-65	56	23	v	v	NOUN
ejpam-65	56	24	/	/	SYM
ejpam-65	56	25	ω	ω	PROPN
ejpam-65	56	26	is	be	AUX
ejpam-65	56	27	a	a	DET
ejpam-65	56	28	classical	classical	ADJ
ejpam-65	56	29	vector	vector	NOUN
ejpam-65	56	30	space	space	NOUN
ejpam-65	56	31	over	over	ADP
ejpam-65	56	32	k.	k.	PROPN
ejpam-65	57	1	if	if	SCONJ
ejpam-65	57	2	p	p	X
ejpam-65	57	3	:	:	PUNCT
ejpam-65	57	4	v	v	ADP
ejpam-65	57	5	−→	−→	NOUN
ejpam-65	57	6	w	w	NOUN
ejpam-65	57	7	is	be	AUX
ejpam-65	57	8	the	the	DET
ejpam-65	57	9	canonical	canonical	ADJ
ejpam-65	57	10	projection	projection	NOUN
ejpam-65	57	11	of	of	ADP
ejpam-65	57	12	(	(	PUNCT
ejpam-65	57	13	v,+	v,+	NUM
ejpam-65	57	14	)	)	PUNCT
ejpam-65	57	15	onto	onto	ADP
ejpam-65	57	16	(	(	PUNCT
ejpam-65	57	17	w,+	w,+	NOUN
ejpam-65	57	18	)	)	PUNCT
ejpam-65	57	19	and	and	CCONJ
ejpam-65	57	20	we	we	PRON
ejpam-65	57	21	set	set	VERB
ejpam-65	57	22	:	:	PUNCT
ejpam-65	57	23	{	{	PUNCT
ejpam-65	57	24	◦	◦	NOUN
ejpam-65	57	25	:	:	PUNCT
ejpam-65	57	26	k	k	X
ejpam-65	57	27	×	×	NOUN
ejpam-65	57	28	v	v	INTJ
ejpam-65	57	29	−→	−→	ADJ
ejpam-65	57	30	p∗(v	p∗(v	PROPN
ejpam-65	57	31	)	)	PUNCT
ejpam-65	57	32	a	a	DET
ejpam-65	57	33	◦	◦	NOUN
ejpam-65	57	34	x	x	X
ejpam-65	57	35	=	=	SYM
ejpam-65	57	36	p−1	p−1	PROPN
ejpam-65	57	37	(	(	PUNCT
ejpam-65	57	38	a.p	a.p	PROPN
ejpam-65	57	39	(	(	PUNCT
ejpam-65	57	40	x	x	NOUN
ejpam-65	57	41	)	)	PUNCT
ejpam-65	57	42	)	)	PUNCT
ejpam-65	57	43	then	then	ADV
ejpam-65	57	44	(	(	PUNCT
ejpam-65	57	45	v,+	v,+	NUM
ejpam-65	57	46	,	,	PUNCT
ejpam-65	57	47	◦	◦	NOUN
ejpam-65	57	48	,	,	PUNCT
ejpam-65	57	49	k	k	NOUN
ejpam-65	57	50	)	)	PUNCT
ejpam-65	57	51	is	be	AUX
ejpam-65	57	52	a	a	DET
ejpam-65	57	53	strongly	strongly	ADV
ejpam-65	57	54	distributive	distributive	ADJ
ejpam-65	57	55	hypervector	hypervector	NOUN
ejpam-65	57	56	space	space	NOUN
ejpam-65	57	57	over	over	ADP
ejpam-65	57	58	k.	k.	NOUN
ejpam-65	58	1	moreover	moreover	ADV
ejpam-65	58	2	every	every	DET
ejpam-65	58	3	strongly	strongly	ADV
ejpam-65	58	4	distributive	distributive	ADJ
ejpam-65	58	5	hypervector	hypervector	NOUN
ejpam-65	58	6	space	space	NOUN
ejpam-65	58	7	can	can	AUX
ejpam-65	58	8	be	be	AUX
ejpam-65	58	9	obtained	obtain	VERB
ejpam-65	58	10	in	in	ADP
ejpam-65	58	11	such	such	DET
ejpam-65	58	12	a	a	DET
ejpam-65	58	13	way	way	NOUN
ejpam-65	58	14	.	.	PUNCT
ejpam-65	59	1	example	example	NOUN
ejpam-65	59	2	2.4	2.4	NUM
ejpam-65	59	3	.	.	PUNCT
ejpam-65	60	1	[	[	X
ejpam-65	60	2	6	6	NUM
ejpam-65	60	3	]	]	PUNCT
ejpam-65	60	4	(	(	PUNCT
ejpam-65	60	5	i	i	NOUN
ejpam-65	60	6	)	)	PUNCT
ejpam-65	60	7	in	in	ADP
ejpam-65	60	8	(	(	PUNCT
ejpam-65	60	9	r2,+	r2,+	NOUN
ejpam-65	60	10	)	)	PUNCT
ejpam-65	60	11	we	we	PRON
ejpam-65	60	12	define	define	VERB
ejpam-65	60	13	the	the	DET
ejpam-65	60	14	product	product	NOUN
ejpam-65	60	15	times	time	NOUN
ejpam-65	60	16	a	a	DET
ejpam-65	60	17	scalar	scalar	NOUN
ejpam-65	60	18	in	in	ADP
ejpam-65	60	19	r	r	NOUN
ejpam-65	60	20	by	by	ADP
ejpam-65	60	21	setting	set	VERB
ejpam-65	60	22	:	:	PUNCT
ejpam-65	60	23	∀a	∀a	NOUN
ejpam-65	60	24	∈	∈	PROPN
ejpam-65	60	25	r	r	NOUN
ejpam-65	60	26	,	,	PUNCT
ejpam-65	60	27	∀x	∀x	X
ejpam-65	60	28	∈	∈	PROPN
ejpam-65	60	29	r2	r2	NOUN
ejpam-65	60	30	:	:	PUNCT
ejpam-65	60	31	a	a	DET
ejpam-65	60	32	◦	◦	NOUN
ejpam-65	60	33	x	x	SYM
ejpam-65	61	1	=	=	NOUN
ejpam-65	61	2	{	{	PUNCT
ejpam-65	61	3	line	line	NOUN
ejpam-65	61	4	ox	ox	NOUN
ejpam-65	61	5	if	if	SCONJ
ejpam-65	61	6	x	x	PROPN
ejpam-65	61	7	6=	6=	ADP
ejpam-65	61	8	0	0	NUM
ejpam-65	61	9	{	{	PUNCT
ejpam-65	61	10	0	0	NUM
ejpam-65	61	11	}	}	PUNCT
ejpam-65	61	12	if	if	SCONJ
ejpam-65	61	13	x	x	PROPN
ejpam-65	61	14	6=	6=	ADP
ejpam-65	61	15	0	0	NUM
ejpam-65	61	16	,	,	PUNCT
ejpam-65	61	17	where	where	SCONJ
ejpam-65	61	18	0	0	X
ejpam-65	61	19	=	=	SYM
ejpam-65	61	20	(	(	PUNCT
ejpam-65	61	21	0	0	NUM
ejpam-65	61	22	,	,	PUNCT
ejpam-65	61	23	0	0	NUM
ejpam-65	61	24	)	)	PUNCT
ejpam-65	61	25	.	.	PUNCT
ejpam-65	62	1	then	then	ADV
ejpam-65	62	2	(	(	PUNCT
ejpam-65	62	3	r2,+	r2,+	NOUN
ejpam-65	62	4	,	,	PUNCT
ejpam-65	62	5	◦	◦	NOUN
ejpam-65	62	6	,	,	PUNCT
ejpam-65	62	7	r	r	NOUN
ejpam-65	62	8	)	)	PUNCT
ejpam-65	62	9	is	be	AUX
ejpam-65	62	10	a	a	DET
ejpam-65	62	11	strongly	strongly	ADV
ejpam-65	62	12	left	leave	VERB
ejpam-65	62	13	,	,	PUNCT
ejpam-65	62	14	but	but	CCONJ
ejpam-65	62	15	not	not	PART
ejpam-65	62	16	right	right	ADJ
ejpam-65	62	17	,	,	PUNCT
ejpam-65	62	18	distributive	distributive	ADJ
ejpam-65	62	19	hypervector	hypervector	NOUN
ejpam-65	62	20	space	space	NOUN
ejpam-65	62	21	.	.	PUNCT
ejpam-65	63	1	from	from	ADP
ejpam-65	63	2	now	now	ADV
ejpam-65	63	3	on	on	ADV
ejpam-65	63	4	,	,	PUNCT
ejpam-65	63	5	in	in	ADP
ejpam-65	63	6	every	every	PRON
ejpam-65	63	7	strongly	strongly	ADV
ejpam-65	63	8	left	leave	VERB
ejpam-65	63	9	distributive	distributive	ADJ
ejpam-65	63	10	hypervector	hypervector	NOUN
ejpam-65	63	11	space	space	NOUN
ejpam-65	63	12	we	we	PRON
ejpam-65	63	13	set	set	VERB
ejpam-65	63	14	:	:	PUNCT
ejpam-65	63	15	t	t	PROPN
ejpam-65	63	16	=	=	SYM
ejpam-65	63	17	{	{	PUNCT
ejpam-65	63	18	x	x	PUNCT
ejpam-65	63	19	∈	∈	PROPN
ejpam-65	63	20	v	v	NOUN
ejpam-65	63	21	:	:	PUNCT
ejpam-65	63	22	x	x	SYM
ejpam-65	63	23	∈	∈	NOUN
ejpam-65	63	24	0	0	PUNCT
ejpam-65	64	1	◦	◦	NOUN
ejpam-65	64	2	x	x	X
ejpam-65	64	3	}	}	PUNCT
ejpam-65	64	4	=	=	SYM
ejpam-65	64	5	{	{	PUNCT
ejpam-65	64	6	x	x	PUNCT
ejpam-65	64	7	∈	∈	PROPN
ejpam-65	64	8	v	v	NOUN
ejpam-65	64	9	:	:	PUNCT
ejpam-65	64	10	1	1	NUM
ejpam-65	64	11	◦	◦	NOUN
ejpam-65	64	12	x	x	SYM
ejpam-65	64	13	=	=	SYM
ejpam-65	64	14	0	0	NUM
ejpam-65	64	15	◦	◦	NOUN
ejpam-65	64	16	x	x	NOUN
ejpam-65	64	17	}	}	PUNCT
ejpam-65	64	18	=	=	SYM
ejpam-65	64	19	{	{	PUNCT
ejpam-65	64	20	x	x	PUNCT
ejpam-65	64	21	∈	∈	PROPN
ejpam-65	64	22	v	v	NOUN
ejpam-65	64	23	:	:	PUNCT
ejpam-65	64	24	∀a	∀a	X
ejpam-65	64	25	∈	∈	PROPN
ejpam-65	64	26	k	k	NOUN
ejpam-65	64	27	,	,	PUNCT
ejpam-65	64	28	a	a	DET
ejpam-65	64	29	◦	◦	NOUN
ejpam-65	64	30	x	x	SYM
ejpam-65	64	31	=	=	SYM
ejpam-65	64	32	0	0	NUM
ejpam-65	64	33	◦	◦	NOUN
ejpam-65	64	34	x	x	NOUN
ejpam-65	64	35	}	}	PUNCT
ejpam-65	64	36	example	example	NOUN
ejpam-65	64	37	2.5	2.5	NUM
ejpam-65	64	38	.	.	PUNCT
ejpam-65	65	1	let	let	AUX
ejpam-65	65	2	(	(	PUNCT
ejpam-65	65	3	v,+	v,+	NUM
ejpam-65	65	4	)	)	PUNCT
ejpam-65	65	5	be	be	AUX
ejpam-65	65	6	an	an	DET
ejpam-65	65	7	abelian	abelian	ADJ
ejpam-65	65	8	group	group	NOUN
ejpam-65	65	9	and	and	CCONJ
ejpam-65	65	10	ω	ω	DET
ejpam-65	65	11	a	a	DET
ejpam-65	65	12	proper	proper	ADJ
ejpam-65	65	13	subgroup	subgroup	NOUN
ejpam-65	65	14	of	of	ADP
ejpam-65	65	15	(	(	PUNCT
ejpam-65	65	16	v,+	v,+	NUM
ejpam-65	65	17	)	)	PUNCT
ejpam-65	65	18	.	.	PUNCT
ejpam-65	66	1	for	for	ADP
ejpam-65	66	2	any	any	DET
ejpam-65	66	3	field	field	NOUN
ejpam-65	66	4	k	k	PROPN
ejpam-65	66	5	set	set	VERB
ejpam-65	66	6	:	:	PUNCT
ejpam-65	66	7	{	{	PUNCT
ejpam-65	66	8	◦	◦	NOUN
ejpam-65	66	9	:	:	PUNCT
ejpam-65	66	10	k	k	X
ejpam-65	66	11	×	×	NOUN
ejpam-65	66	12	v	v	INTJ
ejpam-65	66	13	−→	−→	ADJ
ejpam-65	66	14	p∗(v	p∗(v	PROPN
ejpam-65	66	15	)	)	PUNCT
ejpam-65	66	16	a	a	DET
ejpam-65	66	17	◦	◦	NOUN
ejpam-65	66	18	x	x	X
ejpam-65	66	19	=	=	SYM
ejpam-65	66	20	〈	〈	PROPN
ejpam-65	66	21	x	x	X
ejpam-65	66	22	,	,	PUNCT
ejpam-65	66	23	ω	ω	NOUN
ejpam-65	66	24	〉	〉	NUM
ejpam-65	66	25	where	where	SCONJ
ejpam-65	66	26	〈	〈	PROPN
ejpam-65	66	27	x	x	X
ejpam-65	66	28	,	,	PUNCT
ejpam-65	66	29	ω	ω	PROPN
ejpam-65	66	30	〉	〉	PROPN
ejpam-65	66	31	is	be	AUX
ejpam-65	66	32	the	the	DET
ejpam-65	66	33	subgroup	subgroup	NOUN
ejpam-65	66	34	of	of	ADP
ejpam-65	66	35	(	(	PUNCT
ejpam-65	66	36	v,+	v,+	NOUN
ejpam-65	66	37	)	)	PUNCT
ejpam-65	66	38	spanned	span	VERB
ejpam-65	66	39	by	by	ADP
ejpam-65	66	40	x	x	SYM
ejpam-65	66	41	and	and	CCONJ
ejpam-65	66	42	ω	ω	X
ejpam-65	66	43	.	.	PUNCT
ejpam-65	67	1	then	then	ADV
ejpam-65	67	2	(	(	PUNCT
ejpam-65	67	3	v,+	v,+	NUM
ejpam-65	67	4	,	,	PUNCT
ejpam-65	67	5	◦	◦	NOUN
ejpam-65	67	6	,	,	PUNCT
ejpam-65	67	7	k	k	NOUN
ejpam-65	67	8	)	)	PUNCT
ejpam-65	67	9	is	be	AUX
ejpam-65	67	10	a	a	DET
ejpam-65	67	11	strongly	strongly	ADV
ejpam-65	67	12	left	leave	VERB
ejpam-65	67	13	,	,	PUNCT
ejpam-65	67	14	but	but	CCONJ
ejpam-65	67	15	not	not	PART
ejpam-65	67	16	right	right	ADJ
ejpam-65	67	17	,	,	PUNCT
ejpam-65	67	18	distributive	distributive	ADJ
ejpam-65	67	19	hypervector	hypervector	NOUN
ejpam-65	67	20	space	space	NOUN
ejpam-65	67	21	such	such	ADJ
ejpam-65	67	22	that	that	DET
ejpam-65	67	23	t	t	NOUN
ejpam-65	67	24	=	=	SYM
ejpam-65	67	25	v	v	PROPN
ejpam-65	67	26	.	.	PUNCT
ejpam-65	68	1	r.	r.	PROPN
ejpam-65	68	2	ameri	ameri	PROPN
ejpam-65	68	3	,	,	PUNCT
ejpam-65	68	4	o.r	o.r	PROPN
ejpam-65	68	5	.	.	PROPN
ejpam-65	68	6	dehghan	dehghan	PROPN
ejpam-65	68	7	/	/	SYM
ejpam-65	68	8	eur	eur	PROPN
ejpam-65	68	9	.	.	PUNCT
ejpam-65	69	1	j.	j.	PROPN
ejpam-65	69	2	pure	pure	PROPN
ejpam-65	69	3	appl	appl	PROPN
ejpam-65	69	4	.	.	PROPN
ejpam-65	69	5	math	math	PROPN
ejpam-65	69	6	,	,	PUNCT
ejpam-65	69	7	1	1	NUM
ejpam-65	69	8	(	(	PUNCT
ejpam-65	69	9	2008	2008	NUM
ejpam-65	69	10	)	)	PUNCT
ejpam-65	69	11	,	,	PUNCT
ejpam-65	69	12	(	(	PUNCT
ejpam-65	69	13	32	32	NUM
ejpam-65	69	14	-	-	SYM
ejpam-65	69	15	50	50	NUM
ejpam-65	69	16	)	)	PUNCT
ejpam-65	69	17	35	35	NUM
ejpam-65	69	18	3	3	NUM
ejpam-65	69	19	.	.	PUNCT
ejpam-65	70	1	basis	basis	NOUN
ejpam-65	70	2	of	of	ADP
ejpam-65	70	3	hypervector	hypervector	NOUN
ejpam-65	70	4	spaces	space	NOUN
ejpam-65	70	5	in	in	ADP
ejpam-65	70	6	[	[	X
ejpam-65	70	7	8	8	NUM
ejpam-65	70	8	]	]	PUNCT
ejpam-65	70	9	,	,	PUNCT
ejpam-65	70	10	the	the	DET
ejpam-65	70	11	notions	notion	NOUN
ejpam-65	70	12	of	of	ADP
ejpam-65	70	13	generator	generator	NOUN
ejpam-65	70	14	,	,	PUNCT
ejpam-65	70	15	dependent	dependent	ADJ
ejpam-65	70	16	(	(	PUNCT
ejpam-65	70	17	resp	resp	NOUN
ejpam-65	70	18	.	.	PUNCT
ejpam-65	71	1	independent	independent	ADJ
ejpam-65	71	2	)	)	PUNCT
ejpam-65	71	3	set	set	NOUN
ejpam-65	71	4	,	,	PUNCT
ejpam-65	71	5	and	and	CCONJ
ejpam-65	71	6	basis	basis	NOUN
ejpam-65	71	7	was	be	AUX
ejpam-65	71	8	defined	define	VERB
ejpam-65	71	9	,	,	PUNCT
ejpam-65	71	10	in	in	ADP
ejpam-65	71	11	the	the	DET
ejpam-65	71	12	sense	sense	NOUN
ejpam-65	71	13	of	of	ADP
ejpam-65	71	14	universal	universal	ADJ
ejpam-65	71	15	algebra	algebra	NOUN
ejpam-65	71	16	.	.	PUNCT
ejpam-65	72	1	in	in	ADP
ejpam-65	72	2	the	the	DET
ejpam-65	72	3	following	following	NOUN
ejpam-65	72	4	we	we	PRON
ejpam-65	72	5	introduce	introduce	VERB
ejpam-65	72	6	and	and	CCONJ
ejpam-65	72	7	characterize	characterize	VERB
ejpam-65	72	8	these	these	DET
ejpam-65	72	9	notions	notion	NOUN
ejpam-65	72	10	based	base	VERB
ejpam-65	72	11	on	on	ADP
ejpam-65	72	12	the	the	DET
ejpam-65	72	13	theory	theory	NOUN
ejpam-65	72	14	of	of	ADP
ejpam-65	72	15	linear	linear	PROPN
ejpam-65	72	16	algebra	algebra	NOUN
ejpam-65	72	17	and	and	CCONJ
ejpam-65	72	18	study	study	VERB
ejpam-65	72	19	the	the	DET
ejpam-65	72	20	basic	basic	ADJ
ejpam-65	72	21	properties	property	NOUN
ejpam-65	72	22	of	of	ADP
ejpam-65	72	23	them	they	PRON
ejpam-65	72	24	.	.	PUNCT
ejpam-65	73	1	in	in	ADP
ejpam-65	73	2	the	the	DET
ejpam-65	73	3	sequel	sequel	NOUN
ejpam-65	73	4	by	by	ADP
ejpam-65	73	5	v	v	PRON
ejpam-65	73	6	we	we	PRON
ejpam-65	73	7	mean	mean	VERB
ejpam-65	73	8	a	a	DET
ejpam-65	73	9	hypervector	hypervector	NOUN
ejpam-65	73	10	space	space	NOUN
ejpam-65	73	11	over	over	ADP
ejpam-65	73	12	the	the	DET
ejpam-65	73	13	field	field	NOUN
ejpam-65	73	14	k.	k.	PROPN
ejpam-65	73	15	definition	definition	NOUN
ejpam-65	73	16	3.1	3.1	NUM
ejpam-65	73	17	.	.	PUNCT
ejpam-65	74	1	a	a	DET
ejpam-65	74	2	nonempty	nonempty	NOUN
ejpam-65	74	3	subset	subset	VERB
ejpam-65	74	4	w	w	NOUN
ejpam-65	74	5	of	of	ADP
ejpam-65	74	6	v	v	NOUN
ejpam-65	74	7	is	be	AUX
ejpam-65	74	8	called	call	VERB
ejpam-65	74	9	a	a	DET
ejpam-65	74	10	subhyperspace	subhyperspace	NOUN
ejpam-65	74	11	if	if	SCONJ
ejpam-65	74	12	w	w	PROPN
ejpam-65	74	13	is	be	AUX
ejpam-65	74	14	itself	itself	PRON
ejpam-65	74	15	a	a	DET
ejpam-65	74	16	hypervector	hypervector	NOUN
ejpam-65	74	17	space	space	NOUN
ejpam-65	74	18	with	with	ADP
ejpam-65	74	19	the	the	DET
ejpam-65	74	20	hyperoperation	hyperoperation	NOUN
ejpam-65	74	21	on	on	ADP
ejpam-65	74	22	v	v	NUM
ejpam-65	74	23	,	,	PUNCT
ejpam-65	74	24	i.e.	i.e.	PROPN
ejpam-65	74	25	w	w	PROPN
ejpam-65	74	26	6=	6=	ADP
ejpam-65	74	27	∅	∅	NOUN
ejpam-65	74	28	,	,	PUNCT
ejpam-65	74	29	∀x	∀x	NUM
ejpam-65	74	30	,	,	PUNCT
ejpam-65	74	31	y	y	PROPN
ejpam-65	74	32	∈w	∈w	VERB
ejpam-65	74	33	=	=	PRON
ejpam-65	74	34	⇒	⇒	NOUN
ejpam-65	74	35	x−	x−	PROPN
ejpam-65	74	36	y	y	PROPN
ejpam-65	74	37	∈w	∈w	PROPN
ejpam-65	74	38	,	,	PUNCT
ejpam-65	74	39	∀a	∀a	NOUN
ejpam-65	74	40	∈	∈	PROPN
ejpam-65	74	41	k	k	NOUN
ejpam-65	74	42	,	,	PUNCT
ejpam-65	74	43	∀x	∀x	X
ejpam-65	74	44	∈w	∈w	VERB
ejpam-65	74	45	=	=	PRON
ejpam-65	74	46	⇒	⇒	VERB
ejpam-65	74	47	a	a	DET
ejpam-65	74	48	◦	◦	NOUN
ejpam-65	74	49	x	x	SYM
ejpam-65	74	50	⊆w	⊆w	NOUN
ejpam-65	74	51	.	.	PUNCT
ejpam-65	75	1	in	in	ADP
ejpam-65	75	2	this	this	DET
ejpam-65	75	3	case	case	NOUN
ejpam-65	75	4	we	we	PRON
ejpam-65	75	5	write	write	VERB
ejpam-65	75	6	w	w	PROPN
ejpam-65	75	7	6	6	NUM
ejpam-65	75	8	v	v	NOUN
ejpam-65	75	9	.	.	PUNCT
ejpam-65	76	1	lemma	lemma	PROPN
ejpam-65	76	2	3.1	3.1	NUM
ejpam-65	76	3	.	.	PUNCT
ejpam-65	77	1	a	a	DET
ejpam-65	77	2	nonempty	nonempty	NOUN
ejpam-65	77	3	subset	subset	VERB
ejpam-65	77	4	w	w	NOUN
ejpam-65	77	5	of	of	ADP
ejpam-65	77	6	v	v	NOUN
ejpam-65	77	7	is	be	AUX
ejpam-65	77	8	a	a	DET
ejpam-65	77	9	subhyperspace	subhyperspace	NOUN
ejpam-65	77	10	if	if	SCONJ
ejpam-65	77	11	and	and	CCONJ
ejpam-65	77	12	only	only	ADV
ejpam-65	77	13	if	if	SCONJ
ejpam-65	77	14	a	a	DET
ejpam-65	77	15	◦	◦	NOUN
ejpam-65	77	16	u	u	NOUN
ejpam-65	77	17	+	+	CCONJ
ejpam-65	77	18	b	b	NOUN
ejpam-65	78	1	◦	◦	NOUN
ejpam-65	78	2	v	v	NUM
ejpam-65	78	3	⊆	⊆	NUM
ejpam-65	78	4	w	w	NOUN
ejpam-65	78	5	,	,	PUNCT
ejpam-65	78	6	∀a	∀a	PROPN
ejpam-65	78	7	,	,	PUNCT
ejpam-65	78	8	b	b	PROPN
ejpam-65	78	9	∈	∈	PROPN
ejpam-65	78	10	k	k	PROPN
ejpam-65	78	11	,	,	PUNCT
ejpam-65	78	12	∀u	∀u	NOUN
ejpam-65	78	13	,	,	PUNCT
ejpam-65	78	14	v	v	NOUN
ejpam-65	78	15	∈w	∈w	NOUN
ejpam-65	78	16	.	.	PUNCT
ejpam-65	79	1	proof	proof	NOUN
ejpam-65	79	2	.	.	PUNCT
ejpam-65	80	1	let	let	VERB
ejpam-65	80	2	w	w	NOUN
ejpam-65	80	3	be	be	AUX
ejpam-65	80	4	a	a	DET
ejpam-65	80	5	subhyperspace	subhyperspace	NOUN
ejpam-65	80	6	of	of	ADP
ejpam-65	80	7	v	v	NOUN
ejpam-65	80	8	.	.	PUNCT
ejpam-65	81	1	then	then	ADV
ejpam-65	81	2	for	for	ADP
ejpam-65	81	3	every	every	DET
ejpam-65	81	4	a	a	PROPN
ejpam-65	81	5	,	,	PUNCT
ejpam-65	81	6	b	b	PROPN
ejpam-65	81	7	∈	∈	PROPN
ejpam-65	81	8	k	k	PROPN
ejpam-65	81	9	and	and	CCONJ
ejpam-65	81	10	u	u	PROPN
ejpam-65	81	11	,	,	PUNCT
ejpam-65	81	12	v	v	ADP
ejpam-65	81	13	∈	∈	NOUN
ejpam-65	82	1	w	w	NOUN
ejpam-65	82	2	we	we	PRON
ejpam-65	82	3	have	have	VERB
ejpam-65	82	4	a	a	DET
ejpam-65	82	5	◦	◦	NOUN
ejpam-65	82	6	u	u	NOUN
ejpam-65	82	7	⊆	⊆	NUM
ejpam-65	82	8	w	w	NOUN
ejpam-65	82	9	and	and	CCONJ
ejpam-65	82	10	b	b	NOUN
ejpam-65	82	11	◦	◦	NOUN
ejpam-65	82	12	v	v	NUM
ejpam-65	82	13	⊆	⊆	NUM
ejpam-65	82	14	w	w	NOUN
ejpam-65	82	15	.	.	PUNCT
ejpam-65	83	1	thus	thus	ADV
ejpam-65	83	2	a	a	DET
ejpam-65	83	3	◦	◦	NOUN
ejpam-65	83	4	u	u	NOUN
ejpam-65	83	5	+	+	CCONJ
ejpam-65	83	6	b	b	NOUN
ejpam-65	84	1	◦	◦	NOUN
ejpam-65	84	2	v	v	NUM
ejpam-65	84	3	⊆	⊆	NUM
ejpam-65	84	4	w	w	NOUN
ejpam-65	84	5	.	.	PUNCT
ejpam-65	85	1	conversely	conversely	ADV
ejpam-65	85	2	if	if	SCONJ
ejpam-65	85	3	u	u	NOUN
ejpam-65	85	4	,	,	PUNCT
ejpam-65	85	5	v	v	ADP
ejpam-65	85	6	∈	∈	NOUN
ejpam-65	85	7	w	w	NOUN
ejpam-65	85	8	,	,	PUNCT
ejpam-65	85	9	then	then	ADV
ejpam-65	85	10	u	u	NOUN
ejpam-65	85	11	+	+	PROPN
ejpam-65	85	12	v	v	NUM
ejpam-65	85	13	∈	∈	NOUN
ejpam-65	85	14	1	1	NUM
ejpam-65	85	15	◦	◦	NOUN
ejpam-65	85	16	u	u	NOUN
ejpam-65	85	17	+	+	NOUN
ejpam-65	85	18	1	1	NUM
ejpam-65	85	19	◦	◦	NOUN
ejpam-65	85	20	v	v	NUM
ejpam-65	85	21	⊆	⊆	NUM
ejpam-65	85	22	w	w	NOUN
ejpam-65	85	23	,	,	PUNCT
ejpam-65	85	24	hence	hence	ADV
ejpam-65	85	25	u	u	NOUN
ejpam-65	85	26	+	+	PROPN
ejpam-65	85	27	v	v	PROPN
ejpam-65	85	28	∈	∈	PROPN
ejpam-65	85	29	w.	w.	NOUN
ejpam-65	85	30	also	also	ADV
ejpam-65	85	31	0	0	X
ejpam-65	85	32	∈	∈	NOUN
ejpam-65	85	33	1	1	NUM
ejpam-65	85	34	◦	◦	NOUN
ejpam-65	85	35	0	0	NUM
ejpam-65	85	36	,	,	PUNCT
ejpam-65	85	37	implies	imply	VERB
ejpam-65	85	38	that	that	SCONJ
ejpam-65	85	39	a	a	DET
ejpam-65	85	40	◦	◦	NOUN
ejpam-65	85	41	u	u	NOUN
ejpam-65	85	42	⊆	⊆	NUM
ejpam-65	85	43	a	a	DET
ejpam-65	85	44	◦	◦	NOUN
ejpam-65	85	45	u	u	NOUN
ejpam-65	85	46	+	+	CCONJ
ejpam-65	85	47	1	1	NUM
ejpam-65	85	48	◦	◦	NOUN
ejpam-65	85	49	0	0	NUM
ejpam-65	85	50	.	.	PUNCT
ejpam-65	86	1	thus	thus	ADV
ejpam-65	86	2	a	a	DET
ejpam-65	86	3	◦	◦	NOUN
ejpam-65	86	4	u	u	NOUN
ejpam-65	86	5	⊆w	⊆w	NOUN
ejpam-65	86	6	and	and	CCONJ
ejpam-65	86	7	hence	hence	ADV
ejpam-65	86	8	w	w	PROPN
ejpam-65	86	9	is	be	AUX
ejpam-65	86	10	a	a	DET
ejpam-65	86	11	subhyperspace	subhyperspace	NOUN
ejpam-65	86	12	.	.	PUNCT
ejpam-65	87	1	remark	remark	PROPN
ejpam-65	87	2	3.1	3.1	NUM
ejpam-65	87	3	.	.	PUNCT
ejpam-65	88	1	(	(	PUNCT
ejpam-65	88	2	i)we	i)we	PROPN
ejpam-65	88	3	denote	denote	VERB
ejpam-65	88	4	by	by	ADP
ejpam-65	88	5	s	s	PROPN
ejpam-65	88	6	the	the	DET
ejpam-65	88	7	family	family	NOUN
ejpam-65	88	8	of	of	ADP
ejpam-65	88	9	all	all	DET
ejpam-65	88	10	subhyperspaces	subhyperspace	NOUN
ejpam-65	88	11	of	of	ADP
ejpam-65	88	12	v.	v.	ADV
ejpam-65	88	13	we	we	PRON
ejpam-65	88	14	easily	easily	ADV
ejpam-65	88	15	prove	prove	VERB
ejpam-65	88	16	that	that	SCONJ
ejpam-65	88	17	:	:	PUNCT
ejpam-65	88	18	{	{	PUNCT
ejpam-65	88	19	v	v	NUM
ejpam-65	88	20	∈	∈	NOUN
ejpam-65	88	21	s	s	NOUN
ejpam-65	88	22	,	,	PUNCT
ejpam-65	88	23	{	{	PUNCT
ejpam-65	88	24	wi}i∈i	wi}i∈i	X
ejpam-65	88	25	,	,	PUNCT
ejpam-65	88	26	wi	wi	PROPN
ejpam-65	88	27	∈	∈	PROPN
ejpam-65	88	28	s	s	PART
ejpam-65	89	1	=	=	NOUN
ejpam-65	89	2	⇒	⇒	X
ejpam-65	89	3	⋂	⋂	PROPN
ejpam-65	89	4	i∈i	i∈i	ADJ
ejpam-65	89	5	wi	wi	PROPN
ejpam-65	89	6	∈	∈	PROPN
ejpam-65	89	7	s.	s.	PROPN
ejpam-65	90	1	it	it	PRON
ejpam-65	90	2	follows	follow	VERB
ejpam-65	90	3	that	that	SCONJ
ejpam-65	90	4	s	s	VERB
ejpam-65	90	5	is	be	AUX
ejpam-65	90	6	a	a	DET
ejpam-65	90	7	closure	closure	NOUN
ejpam-65	90	8	system	system	NOUN
ejpam-65	90	9	in	in	ADP
ejpam-65	90	10	v	v	NUM
ejpam-65	90	11	.	.	PUNCT
ejpam-65	91	1	(	(	PUNCT
ejpam-65	91	2	ii	ii	NOUN
ejpam-65	91	3	)	)	PUNCT
ejpam-65	91	4	if	if	SCONJ
ejpam-65	91	5	w1	w1	NOUN
ejpam-65	91	6	and	and	CCONJ
ejpam-65	91	7	w2	w2	NOUN
ejpam-65	91	8	are	be	AUX
ejpam-65	91	9	any	any	DET
ejpam-65	91	10	two	two	NUM
ejpam-65	91	11	subhyperspaces	subhyperspace	NOUN
ejpam-65	91	12	of	of	ADP
ejpam-65	91	13	v	v	NOUN
ejpam-65	91	14	,	,	PUNCT
ejpam-65	91	15	then	then	ADV
ejpam-65	91	16	w1	w1	NOUN
ejpam-65	91	17	∪w2	∪w2	NOUN
ejpam-65	91	18	is	be	AUX
ejpam-65	91	19	a	a	DET
ejpam-65	91	20	subhyperspace	subhyperspace	NOUN
ejpam-65	91	21	of	of	ADP
ejpam-65	91	22	v	v	NOUN
ejpam-65	91	23	if	if	SCONJ
ejpam-65	91	24	and	and	CCONJ
ejpam-65	91	25	only	only	ADV
ejpam-65	91	26	if	if	SCONJ
ejpam-65	91	27	w1	w1	NOUN
ejpam-65	91	28	⊆w2	⊆w2	NOUN
ejpam-65	91	29	or	or	CCONJ
ejpam-65	91	30	w2	w2	PROPN
ejpam-65	91	31	⊆w1	⊆w1	PROPN
ejpam-65	91	32	.	.	PUNCT
ejpam-65	92	1	definition	definition	NOUN
ejpam-65	92	2	3.2	3.2	NUM
ejpam-65	92	3	.	.	PUNCT
ejpam-65	93	1	a	a	DET
ejpam-65	93	2	subset	subset	NOUN
ejpam-65	93	3	s	s	NOUN
ejpam-65	93	4	of	of	ADP
ejpam-65	93	5	v	v	NOUN
ejpam-65	93	6	is	be	AUX
ejpam-65	93	7	called	call	VERB
ejpam-65	93	8	linearly	linearly	ADV
ejpam-65	93	9	independent	independent	ADJ
ejpam-65	93	10	if	if	SCONJ
ejpam-65	93	11	for	for	ADP
ejpam-65	93	12	every	every	DET
ejpam-65	93	13	vectors	vector	NOUN
ejpam-65	93	14	v1	v1	NOUN
ejpam-65	93	15	,	,	PUNCT
ejpam-65	93	16	v2	v2	PROPN
ejpam-65	93	17	,	,	PUNCT
ejpam-65	93	18	.	.	PUNCT
ejpam-65	93	19	.	.	PUNCT
ejpam-65	94	1	.	.	PUNCT
ejpam-65	95	1	,	,	PUNCT
ejpam-65	95	2	vn	vn	VERB
ejpam-65	95	3	in	in	ADP
ejpam-65	95	4	s	s	PROPN
ejpam-65	95	5	,	,	PUNCT
ejpam-65	95	6	and	and	CCONJ
ejpam-65	95	7	c1	c1	PROPN
ejpam-65	95	8	,	,	PUNCT
ejpam-65	95	9	.	.	PUNCT
ejpam-65	95	10	.	.	PUNCT
ejpam-65	96	1	.	.	PUNCT
ejpam-65	97	1	,	,	PUNCT
ejpam-65	97	2	cn	cn	PROPN
ejpam-65	97	3	∈	∈	PROPN
ejpam-65	97	4	k	k	PROPN
ejpam-65	97	5	,	,	PUNCT
ejpam-65	97	6	0	0	NUM
ejpam-65	97	7	∈	∈	PROPN
ejpam-65	97	8	c1	c1	PROPN
ejpam-65	97	9	◦	◦	NOUN
ejpam-65	97	10	v1	v1	PROPN
ejpam-65	97	11	+	+	X
ejpam-65	97	12	·	·	PUNCT
ejpam-65	97	13	·	·	PUNCT
ejpam-65	97	14	·	·	PUNCT
ejpam-65	98	1	+	+	CCONJ
ejpam-65	98	2	cn	cn	PROPN
ejpam-65	98	3	◦	◦	PROPN
ejpam-65	98	4	vn	vn	PROPN
ejpam-65	98	5	,	,	PUNCT
ejpam-65	98	6	implies	imply	VERB
ejpam-65	98	7	that	that	DET
ejpam-65	98	8	c1	c1	PROPN
ejpam-65	98	9	=	=	PROPN
ejpam-65	98	10	c2	c2	PROPN
ejpam-65	98	11	=	=	PUNCT
ejpam-65	98	12	·	·	PUNCT
ejpam-65	98	13	·	·	PUNCT
ejpam-65	98	14	·	·	PUNCT
ejpam-65	99	1	=	=	PUNCT
ejpam-65	99	2	cn	cn	PROPN
ejpam-65	99	3	=	=	NOUN
ejpam-65	99	4	0	0	PROPN
ejpam-65	99	5	.	.	PUNCT
ejpam-65	100	1	a	a	DET
ejpam-65	100	2	subset	subset	NOUN
ejpam-65	100	3	s	s	NOUN
ejpam-65	100	4	of	of	ADP
ejpam-65	100	5	v	v	NOUN
ejpam-65	100	6	is	be	AUX
ejpam-65	100	7	called	call	VERB
ejpam-65	100	8	linearly	linearly	ADV
ejpam-65	100	9	dependent	dependent	ADJ
ejpam-65	100	10	if	if	SCONJ
ejpam-65	100	11	it	it	PRON
ejpam-65	100	12	is	be	AUX
ejpam-65	100	13	not	not	PART
ejpam-65	100	14	linearly	linearly	ADV
ejpam-65	100	15	independent	independent	ADJ
ejpam-65	100	16	.	.	PUNCT
ejpam-65	101	1	lemma	lemma	PROPN
ejpam-65	101	2	3.2	3.2	NUM
ejpam-65	101	3	.	.	PUNCT
ejpam-65	102	1	if	if	SCONJ
ejpam-65	102	2	v	v	NOUN
ejpam-65	102	3	is	be	AUX
ejpam-65	102	4	strongly	strongly	ADV
ejpam-65	102	5	left	leave	VERB
ejpam-65	102	6	distributive	distributive	ADJ
ejpam-65	102	7	,	,	PUNCT
ejpam-65	102	8	then	then	ADV
ejpam-65	102	9	a	a	DET
ejpam-65	102	10	subset	subset	NOUN
ejpam-65	102	11	s	s	NOUN
ejpam-65	102	12	of	of	ADP
ejpam-65	102	13	v	v	NOUN
ejpam-65	102	14	is	be	AUX
ejpam-65	102	15	linearly	linearly	ADV
ejpam-65	102	16	independent	independent	ADJ
ejpam-65	102	17	if	if	SCONJ
ejpam-65	102	18	and	and	CCONJ
ejpam-65	102	19	only	only	ADV
ejpam-65	102	20	if	if	SCONJ
ejpam-65	102	21	for	for	ADP
ejpam-65	102	22	every	every	DET
ejpam-65	102	23	vectors	vector	NOUN
ejpam-65	102	24	v1	v1	NOUN
ejpam-65	102	25	,	,	PUNCT
ejpam-65	102	26	v2	v2	PROPN
ejpam-65	102	27	,	,	PUNCT
ejpam-65	102	28	.	.	PUNCT
ejpam-65	102	29	.	.	PUNCT
ejpam-65	103	1	.	.	PUNCT
ejpam-65	104	1	,	,	PUNCT
ejpam-65	104	2	vn	vn	VERB
ejpam-65	104	3	in	in	ADP
ejpam-65	104	4	s	s	PROPN
ejpam-65	104	5	,	,	PUNCT
ejpam-65	104	6	and	and	CCONJ
ejpam-65	104	7	c1	c1	PROPN
ejpam-65	104	8	,	,	PUNCT
ejpam-65	104	9	.	.	PUNCT
ejpam-65	104	10	.	.	PUNCT
ejpam-65	105	1	.	.	PUNCT
ejpam-65	106	1	,	,	PUNCT
ejpam-65	106	2	cn	cn	PROPN
ejpam-65	106	3	∈	∈	PROPN
ejpam-65	106	4	k	k	PROPN
ejpam-65	106	5	,	,	PUNCT
ejpam-65	106	6	ω∩	ω∩	PROPN
ejpam-65	106	7	c1	c1	PROPN
ejpam-65	106	8	◦	◦	NOUN
ejpam-65	106	9	v1	v1	PROPN
ejpam-65	106	10	+	+	X
ejpam-65	106	11	·	·	PUNCT
ejpam-65	106	12	·	·	PUNCT
ejpam-65	106	13	·	·	PUNCT
ejpam-65	106	14	+	+	NUM
ejpam-65	106	15	cn	cn	VERB
ejpam-65	106	16	◦	◦	PROPN
ejpam-65	106	17	vn	vn	PROPN
ejpam-65	106	18	6=	6=	PUNCT
ejpam-65	106	19	∅	∅	NOUN
ejpam-65	106	20	,	,	PUNCT
ejpam-65	106	21	implies	imply	VERB
ejpam-65	106	22	that	that	DET
ejpam-65	106	23	c1	c1	PROPN
ejpam-65	106	24	=	=	PROPN
ejpam-65	106	25	c2	c2	PROPN
ejpam-65	106	26	=	=	PUNCT
ejpam-65	106	27	·	·	PUNCT
ejpam-65	106	28	·	·	PUNCT
ejpam-65	106	29	·	·	PUNCT
ejpam-65	107	1	=	=	PUNCT
ejpam-65	107	2	cn	cn	PROPN
ejpam-65	107	3	=	=	NOUN
ejpam-65	107	4	0	0	PROPN
ejpam-65	107	5	.	.	PUNCT
ejpam-65	107	6	proof	proof	NOUN
ejpam-65	107	7	.	.	PUNCT
ejpam-65	108	1	let	let	VERB
ejpam-65	108	2	v	v	PART
ejpam-65	108	3	be	be	AUX
ejpam-65	108	4	linearly	linearly	ADV
ejpam-65	108	5	independent	independent	ADJ
ejpam-65	108	6	and	and	CCONJ
ejpam-65	108	7	for	for	ADP
ejpam-65	108	8	vectors	vector	NOUN
ejpam-65	108	9	v1	v1	NOUN
ejpam-65	108	10	,	,	PUNCT
ejpam-65	108	11	v2	v2	PROPN
ejpam-65	108	12	,	,	PUNCT
ejpam-65	108	13	.	.	PUNCT
ejpam-65	108	14	.	.	PUNCT
ejpam-65	109	1	.	.	PUNCT
ejpam-65	110	1	,	,	PUNCT
ejpam-65	110	2	vn	vn	VERB
ejpam-65	110	3	in	in	ADP
ejpam-65	110	4	s	s	PROPN
ejpam-65	110	5	,	,	PUNCT
ejpam-65	110	6	and	and	CCONJ
ejpam-65	110	7	c1	c1	PROPN
ejpam-65	110	8	,	,	PUNCT
ejpam-65	110	9	.	.	PUNCT
ejpam-65	110	10	.	.	PUNCT
ejpam-65	111	1	.	.	PUNCT
ejpam-65	112	1	,	,	PUNCT
ejpam-65	112	2	cn	cn	PROPN
ejpam-65	112	3	∈	∈	PROPN
ejpam-65	112	4	k	k	PROPN
ejpam-65	112	5	,	,	PUNCT
ejpam-65	112	6	x	x	SYM
ejpam-65	112	7	∈	∈	PROPN
ejpam-65	112	8	ω	ω	NUM
ejpam-65	112	9	∩	∩	PROPN
ejpam-65	112	10	c1	c1	PROPN
ejpam-65	112	11	◦	◦	PROPN
ejpam-65	112	12	v1	v1	PROPN
ejpam-65	112	13	+	+	X
ejpam-65	112	14	·	·	PUNCT
ejpam-65	112	15	·	·	PUNCT
ejpam-65	112	16	·	·	PUNCT
ejpam-65	112	17	+	+	NUM
ejpam-65	112	18	cn	cn	VERB
ejpam-65	112	19	◦	◦	PROPN
ejpam-65	112	20	vn	vn	PROPN
ejpam-65	112	21	.	.	PUNCT
ejpam-65	113	1	then	then	ADV
ejpam-65	113	2	0	0	NUM
ejpam-65	113	3	=	=	SYM
ejpam-65	113	4	x−	x−	PROPN
ejpam-65	113	5	x	x	SYM
ejpam-65	113	6	∈	∈	NOUN
ejpam-65	113	7	0	0	PUNCT
ejpam-65	113	8	◦	◦	NOUN
ejpam-65	113	9	0−	0−	NUM
ejpam-65	113	10	c1	c1	PROPN
ejpam-65	113	11	◦	◦	NOUN
ejpam-65	113	12	v1	v1	PROPN
ejpam-65	113	13	+	+	X
ejpam-65	113	14	·	·	PUNCT
ejpam-65	113	15	·	·	PUNCT
ejpam-65	113	16	·	·	PUNCT
ejpam-65	114	1	−	−	PUNCT
ejpam-65	114	2	cn	cn	INTJ
ejpam-65	114	3	◦	◦	NOUN
ejpam-65	114	4	vn	vn	PROPN
ejpam-65	114	5	=	=	NOUN
ejpam-65	114	6	⇒	⇒	PROPN
ejpam-65	114	7	c1	c1	NOUN
ejpam-65	114	8	=	=	PROPN
ejpam-65	114	9	c2	c2	PROPN
ejpam-65	114	10	=	=	PUNCT
ejpam-65	114	11	·	·	PUNCT
ejpam-65	114	12	·	·	PUNCT
ejpam-65	114	13	·	·	PUNCT
ejpam-65	115	1	=	=	PUNCT
ejpam-65	115	2	cn	cn	PROPN
ejpam-65	115	3	=	=	NOUN
ejpam-65	115	4	0	0	PROPN
ejpam-65	115	5	.	.	PUNCT
ejpam-65	115	6	r.	r.	PROPN
ejpam-65	115	7	ameri	ameri	PROPN
ejpam-65	115	8	,	,	PUNCT
ejpam-65	115	9	o.r	o.r	PROPN
ejpam-65	115	10	.	.	PROPN
ejpam-65	115	11	dehghan	dehghan	PROPN
ejpam-65	115	12	/	/	SYM
ejpam-65	115	13	eur	eur	PROPN
ejpam-65	115	14	.	.	PUNCT
ejpam-65	116	1	j.	j.	PROPN
ejpam-65	116	2	pure	pure	PROPN
ejpam-65	116	3	appl	appl	PROPN
ejpam-65	116	4	.	.	PROPN
ejpam-65	116	5	math	math	PROPN
ejpam-65	116	6	,	,	PUNCT
ejpam-65	116	7	1	1	NUM
ejpam-65	116	8	(	(	PUNCT
ejpam-65	116	9	2008	2008	NUM
ejpam-65	116	10	)	)	PUNCT
ejpam-65	116	11	,	,	PUNCT
ejpam-65	116	12	(	(	PUNCT
ejpam-65	116	13	32	32	NUM
ejpam-65	116	14	-	-	SYM
ejpam-65	116	15	50	50	NUM
ejpam-65	116	16	)	)	PUNCT
ejpam-65	116	17	36	36	NUM
ejpam-65	116	18	conversely	conversely	ADV
ejpam-65	116	19	,	,	PUNCT
ejpam-65	116	20	if	if	SCONJ
ejpam-65	116	21	for	for	ADP
ejpam-65	116	22	vectors	vector	NOUN
ejpam-65	116	23	v1	v1	NOUN
ejpam-65	116	24	,	,	PUNCT
ejpam-65	116	25	v2	v2	PROPN
ejpam-65	116	26	,	,	PUNCT
ejpam-65	116	27	.	.	PUNCT
ejpam-65	116	28	.	.	PUNCT
ejpam-65	116	29	.	.	PUNCT
ejpam-65	117	1	,	,	PUNCT
ejpam-65	117	2	vn	vn	VERB
ejpam-65	117	3	in	in	ADP
ejpam-65	117	4	s	s	PROPN
ejpam-65	117	5	,	,	PUNCT
ejpam-65	117	6	and	and	CCONJ
ejpam-65	117	7	c1	c1	PROPN
ejpam-65	117	8	,	,	PUNCT
ejpam-65	117	9	.	.	PUNCT
ejpam-65	117	10	.	.	PUNCT
ejpam-65	118	1	.	.	PUNCT
ejpam-65	119	1	,	,	PUNCT
ejpam-65	119	2	cn	cn	PROPN
ejpam-65	119	3	∈	∈	PROPN
ejpam-65	119	4	k	k	PROPN
ejpam-65	119	5	,	,	PUNCT
ejpam-65	119	6	0	0	NUM
ejpam-65	119	7	∈	∈	PROPN
ejpam-65	119	8	c1	c1	PROPN
ejpam-65	119	9	◦	◦	NOUN
ejpam-65	119	10	v1	v1	PROPN
ejpam-65	119	11	+	+	X
ejpam-65	119	12	·	·	PUNCT
ejpam-65	119	13	·	·	PUNCT
ejpam-65	119	14	·	·	PUNCT
ejpam-65	120	1	+	+	CCONJ
ejpam-65	120	2	cn	cn	PROPN
ejpam-65	120	3	◦	◦	PROPN
ejpam-65	120	4	vn	vn	PROPN
ejpam-65	120	5	,	,	PUNCT
ejpam-65	120	6	then	then	ADV
ejpam-65	120	7	by	by	ADP
ejpam-65	120	8	remark	remark	NOUN
ejpam-65	120	9	2.1	2.1	NUM
ejpam-65	120	10	0	0	NUM
ejpam-65	120	11	∈	∈	PROPN
ejpam-65	120	12	ω	ω	PROPN
ejpam-65	120	13	∩	∩	ADJ
ejpam-65	120	14	c1	c1	PROPN
ejpam-65	120	15	◦	◦	PROPN
ejpam-65	120	16	v1	v1	PROPN
ejpam-65	120	17	+	+	X
ejpam-65	120	18	·	·	PUNCT
ejpam-65	120	19	·	·	PUNCT
ejpam-65	120	20	·	·	PUNCT
ejpam-65	120	21	+	+	NUM
ejpam-65	120	22	cn	cn	VERB
ejpam-65	120	23	◦	◦	PROPN
ejpam-65	120	24	vn	vn	PROPN
ejpam-65	120	25	.	.	PUNCT
ejpam-65	121	1	thus	thus	ADV
ejpam-65	121	2	c1	c1	PROPN
ejpam-65	121	3	=	=	PROPN
ejpam-65	121	4	c2	c2	PROPN
ejpam-65	121	5	=	=	PUNCT
ejpam-65	121	6	·	·	PUNCT
ejpam-65	121	7	·	·	PUNCT
ejpam-65	121	8	·	·	PUNCT
ejpam-65	122	1	=	=	PUNCT
ejpam-65	122	2	cn	cn	PROPN
ejpam-65	122	3	=	=	NOUN
ejpam-65	122	4	0	0	PROPN
ejpam-65	122	5	.	.	PUNCT
ejpam-65	123	1	definition	definition	NOUN
ejpam-65	123	2	3.3	3.3	NUM
ejpam-65	123	3	.	.	PUNCT
ejpam-65	124	1	a	a	DET
ejpam-65	124	2	basis	basis	NOUN
ejpam-65	124	3	for	for	ADP
ejpam-65	124	4	v	v	NOUN
ejpam-65	124	5	is	be	AUX
ejpam-65	124	6	a	a	DET
ejpam-65	124	7	linearly	linearly	ADV
ejpam-65	124	8	independent	independent	ADJ
ejpam-65	124	9	subset	subset	NOUN
ejpam-65	124	10	of	of	ADP
ejpam-65	124	11	v	v	NOUN
ejpam-65	124	12	such	such	ADJ
ejpam-65	124	13	that	that	DET
ejpam-65	124	14	span	span	NOUN
ejpam-65	124	15	v	v	NOUN
ejpam-65	124	16	.	.	PUNCT
ejpam-65	125	1	we	we	PRON
ejpam-65	125	2	say	say	VERB
ejpam-65	125	3	that	that	SCONJ
ejpam-65	125	4	v	v	NOUN
ejpam-65	125	5	has	have	AUX
ejpam-65	125	6	finite	finite	VERB
ejpam-65	125	7	dimensional	dimensional	ADJ
ejpam-65	125	8	if	if	SCONJ
ejpam-65	125	9	it	it	PRON
ejpam-65	125	10	has	have	VERB
ejpam-65	125	11	a	a	DET
ejpam-65	125	12	finite	finite	ADJ
ejpam-65	125	13	basis	basis	NOUN
ejpam-65	125	14	.	.	PUNCT
ejpam-65	126	1	remark	remark	PROPN
ejpam-65	126	2	3.2	3.2	NUM
ejpam-65	126	3	.	.	PUNCT
ejpam-65	127	1	note	note	VERB
ejpam-65	127	2	that	that	SCONJ
ejpam-65	127	3	some	some	DET
ejpam-65	127	4	hypervector	hypervector	NOUN
ejpam-65	127	5	spaces	space	VERB
ejpam-65	127	6	v	v	ADP
ejpam-65	127	7	(	(	PUNCT
ejpam-65	127	8	some	some	DET
ejpam-65	127	9	set	set	NOUN
ejpam-65	127	10	w	w	NOUN
ejpam-65	127	11	of	of	ADP
ejpam-65	127	12	vectors	vector	NOUN
ejpam-65	127	13	)	)	PUNCT
ejpam-65	127	14	may	may	AUX
ejpam-65	127	15	not	not	PART
ejpam-65	127	16	have	have	VERB
ejpam-65	127	17	any	any	DET
ejpam-65	127	18	collection	collection	NOUN
ejpam-65	127	19	of	of	ADP
ejpam-65	127	20	linearly	linearly	ADV
ejpam-65	127	21	independent	independent	ADJ
ejpam-65	127	22	vectors	vector	NOUN
ejpam-65	127	23	.	.	PUNCT
ejpam-65	128	1	such	such	ADJ
ejpam-65	128	2	hypervector	hypervector	NOUN
ejpam-65	128	3	space	space	NOUN
ejpam-65	128	4	(	(	PUNCT
ejpam-65	128	5	set	set	NOUN
ejpam-65	128	6	)	)	PUNCT
ejpam-65	128	7	is	be	AUX
ejpam-65	128	8	called	call	VERB
ejpam-65	128	9	independentless	independentless	NOUN
ejpam-65	128	10	.	.	PUNCT
ejpam-65	129	1	clearly	clearly	ADV
ejpam-65	129	2	if	if	SCONJ
ejpam-65	129	3	v	v	NOUN
ejpam-65	129	4	is	be	AUX
ejpam-65	129	5	independentless	independentless	NOUN
ejpam-65	129	6	,	,	PUNCT
ejpam-65	129	7	then	then	ADV
ejpam-65	129	8	v	v	NOUN
ejpam-65	129	9	has	have	VERB
ejpam-65	129	10	not	not	PART
ejpam-65	129	11	any	any	DET
ejpam-65	129	12	basis	basis	NOUN
ejpam-65	129	13	and	and	CCONJ
ejpam-65	129	14	for	for	SCONJ
ejpam-65	129	15	such	such	ADJ
ejpam-65	129	16	hypervector	hypervector	NOUN
ejpam-65	129	17	spaces	space	NOUN
ejpam-65	129	18	dimension	dimension	NOUN
ejpam-65	129	19	is	be	AUX
ejpam-65	129	20	not	not	PART
ejpam-65	129	21	defined	define	VERB
ejpam-65	129	22	.	.	PUNCT
ejpam-65	130	1	in	in	ADP
ejpam-65	130	2	this	this	DET
ejpam-65	130	3	case	case	NOUN
ejpam-65	130	4	we	we	PRON
ejpam-65	130	5	say	say	VERB
ejpam-65	130	6	that	that	SCONJ
ejpam-65	130	7	v	v	NOUN
ejpam-65	130	8	is	be	AUX
ejpam-65	130	9	dimensionless	dimensionless	NOUN
ejpam-65	130	10	.	.	PUNCT
ejpam-65	131	1	recall	recall	VERB
ejpam-65	131	2	that	that	DET
ejpam-65	131	3	t	t	NOUN
ejpam-65	131	4	=	=	PRON
ejpam-65	131	5	{	{	PUNCT
ejpam-65	131	6	x	x	PUNCT
ejpam-65	131	7	∈	∈	PROPN
ejpam-65	131	8	v	v	NOUN
ejpam-65	131	9	:	:	PUNCT
ejpam-65	131	10	x	x	SYM
ejpam-65	131	11	∈	∈	NOUN
ejpam-65	131	12	0	0	PUNCT
ejpam-65	132	1	◦	◦	NOUN
ejpam-65	132	2	x	x	X
ejpam-65	132	3	}	}	PUNCT
ejpam-65	132	4	=	=	SYM
ejpam-65	132	5	{	{	PUNCT
ejpam-65	132	6	x	x	PUNCT
ejpam-65	132	7	∈	∈	PROPN
ejpam-65	132	8	v	v	NOUN
ejpam-65	132	9	:	:	PUNCT
ejpam-65	132	10	1	1	NUM
ejpam-65	132	11	◦	◦	NOUN
ejpam-65	132	12	x	x	SYM
ejpam-65	132	13	=	=	SYM
ejpam-65	132	14	0	0	NUM
ejpam-65	132	15	◦	◦	NOUN
ejpam-65	132	16	x	x	NOUN
ejpam-65	132	17	}	}	PUNCT
ejpam-65	132	18	=	=	SYM
ejpam-65	132	19	{	{	PUNCT
ejpam-65	132	20	x	x	PUNCT
ejpam-65	132	21	∈	∈	PROPN
ejpam-65	132	22	v	v	NOUN
ejpam-65	132	23	:	:	PUNCT
ejpam-65	132	24	∀a	∀a	X
ejpam-65	132	25	∈	∈	PROPN
ejpam-65	132	26	k	k	NOUN
ejpam-65	132	27	,	,	PUNCT
ejpam-65	132	28	a	a	DET
ejpam-65	132	29	◦	◦	NOUN
ejpam-65	132	30	x	x	SYM
ejpam-65	132	31	=	=	SYM
ejpam-65	132	32	0	0	NUM
ejpam-65	132	33	◦	◦	NOUN
ejpam-65	132	34	x	x	NOUN
ejpam-65	132	35	}	}	PUNCT
ejpam-65	132	36	corollary	corollary	ADJ
ejpam-65	132	37	3.1	3.1	NUM
ejpam-65	132	38	.	.	PUNCT
ejpam-65	133	1	every	every	PRON
ejpam-65	133	2	strongly	strongly	ADV
ejpam-65	133	3	left	leave	VERB
ejpam-65	133	4	distributive	distributive	ADJ
ejpam-65	133	5	hypervector	hypervector	NOUN
ejpam-65	133	6	space	space	NOUN
ejpam-65	133	7	with	with	ADP
ejpam-65	133	8	t	t	PROPN
ejpam-65	133	9	=	=	SYM
ejpam-65	133	10	v	v	NOUN
ejpam-65	133	11	is	be	AUX
ejpam-65	133	12	independentless	independentless	NOUN
ejpam-65	133	13	.	.	PUNCT
ejpam-65	134	1	example	example	NOUN
ejpam-65	134	2	3.1	3.1	NUM
ejpam-65	134	3	.	.	PUNCT
ejpam-65	135	1	the	the	DET
ejpam-65	135	2	hypervector	hypervector	NOUN
ejpam-65	135	3	space	space	NOUN
ejpam-65	135	4	(	(	PUNCT
ejpam-65	135	5	r2,+	r2,+	NOUN
ejpam-65	135	6	,	,	PUNCT
ejpam-65	135	7	◦	◦	NOUN
ejpam-65	135	8	,	,	PUNCT
ejpam-65	135	9	r	r	NOUN
ejpam-65	135	10	)	)	PUNCT
ejpam-65	135	11	in	in	ADP
ejpam-65	135	12	example	example	NOUN
ejpam-65	135	13	2.4	2.4	NUM
ejpam-65	135	14	is	be	AUX
ejpam-65	135	15	a	a	DET
ejpam-65	135	16	nontrivial	nontrivial	ADJ
ejpam-65	135	17	example	example	NOUN
ejpam-65	135	18	of	of	ADP
ejpam-65	135	19	an	an	DET
ejpam-65	135	20	independentless	independentless	NOUN
ejpam-65	135	21	hypervector	hypervector	NOUN
ejpam-65	135	22	space	space	NOUN
ejpam-65	135	23	,	,	PUNCT
ejpam-65	135	24	since	since	SCONJ
ejpam-65	135	25	0	0	NUM
ejpam-65	135	26	belongs	belong	VERB
ejpam-65	135	27	to	to	ADP
ejpam-65	135	28	every	every	DET
ejpam-65	135	29	line	line	NOUN
ejpam-65	135	30	through	through	ADP
ejpam-65	135	31	the	the	DET
ejpam-65	135	32	0	0	NUM
ejpam-65	135	33	.	.	PUNCT
ejpam-65	135	34	definition	definition	NOUN
ejpam-65	135	35	3.4	3.4	NUM
ejpam-65	135	36	.	.	PUNCT
ejpam-65	136	1	if	if	SCONJ
ejpam-65	136	2	s	s	NOUN
ejpam-65	136	3	is	be	AUX
ejpam-65	136	4	a	a	DET
ejpam-65	136	5	nonempty	nonempty	ADJ
ejpam-65	136	6	subset	subset	NOUN
ejpam-65	136	7	of	of	ADP
ejpam-65	136	8	v	v	NOUN
ejpam-65	136	9	,	,	PUNCT
ejpam-65	136	10	then	then	ADV
ejpam-65	136	11	the	the	DET
ejpam-65	136	12	linear	linear	ADJ
ejpam-65	136	13	span	span	NOUN
ejpam-65	136	14	of	of	ADP
ejpam-65	136	15	s	s	PRON
ejpam-65	136	16	is	be	AUX
ejpam-65	136	17	defined	define	VERB
ejpam-65	136	18	by	by	ADP
ejpam-65	136	19	:	:	PUNCT
ejpam-65	136	20	l(s	l(s	PROPN
ejpam-65	136	21	)	)	PUNCT
ejpam-65	137	1	=	=	PRON
ejpam-65	137	2	{	{	PUNCT
ejpam-65	137	3	t	t	PROPN
ejpam-65	137	4	∈	∈	PROPN
ejpam-65	137	5	v	v	X
ejpam-65	137	6	:	:	PUNCT
ejpam-65	137	7	t	t	PROPN
ejpam-65	137	8	∈	∈	PROPN
ejpam-65	138	1	n∑	n∑	INTJ
ejpam-65	138	2	i=1	i=1	PROPN
ejpam-65	138	3	ai	ai	VERB
ejpam-65	138	4	◦	◦	NOUN
ejpam-65	138	5	si	si	X
ejpam-65	138	6	,	,	PUNCT
ejpam-65	138	7	ai	ai	INTJ
ejpam-65	138	8	∈	∈	PROPN
ejpam-65	138	9	k	k	PROPN
ejpam-65	138	10	,	,	PUNCT
ejpam-65	138	11	si	si	PROPN
ejpam-65	138	12	∈	∈	PROPN
ejpam-65	138	13	s	s	X
ejpam-65	138	14	,	,	PUNCT
ejpam-65	138	15	n	n	PROPN
ejpam-65	138	16	∈	∈	PROPN
ejpam-65	138	17	n	n	CCONJ
ejpam-65	138	18	}	}	PUNCT
ejpam-65	138	19	(	(	PUNCT
ejpam-65	138	20	3.1	3.1	NUM
ejpam-65	138	21	)	)	PUNCT
ejpam-65	138	22	=	=	PRON
ejpam-65	138	23	{	{	PUNCT
ejpam-65	138	24	t1	t1	NOUN
ejpam-65	138	25	+	+	CCONJ
ejpam-65	138	26	t2	t2	PROPN
ejpam-65	138	27	+	+	CCONJ
ejpam-65	138	28	·	·	PUNCT
ejpam-65	138	29	·	·	PUNCT
ejpam-65	138	30	·	·	PUNCT
ejpam-65	138	31	+	+	NUM
ejpam-65	138	32	tn	tn	NOUN
ejpam-65	138	33	:	:	PUNCT
ejpam-65	138	34	ti	ti	PROPN
ejpam-65	138	35	∈	∈	PROPN
ejpam-65	138	36	ai	ai	VERB
ejpam-65	138	37	◦	◦	NOUN
ejpam-65	138	38	si	si	X
ejpam-65	138	39	,	,	PUNCT
ejpam-65	138	40	ai	ai	INTJ
ejpam-65	138	41	∈	∈	PROPN
ejpam-65	138	42	k	k	PROPN
ejpam-65	138	43	,	,	PUNCT
ejpam-65	138	44	si	si	PROPN
ejpam-65	138	45	∈	∈	PROPN
ejpam-65	138	46	s	s	X
ejpam-65	138	47	,	,	PUNCT
ejpam-65	138	48	n	n	PROPN
ejpam-65	138	49	∈	∈	PROPN
ejpam-65	138	50	n	n	CCONJ
ejpam-65	138	51	}	}	PUNCT
ejpam-65	138	52	.	.	PUNCT
ejpam-65	139	1	lemma	lemma	PROPN
ejpam-65	139	2	3.3	3.3	NUM
ejpam-65	139	3	.	.	PUNCT
ejpam-65	140	1	l(s	l(s	PROPN
ejpam-65	140	2	)	)	PUNCT
ejpam-65	140	3	is	be	AUX
ejpam-65	140	4	the	the	DET
ejpam-65	140	5	smallest	small	ADJ
ejpam-65	140	6	subhyperspace	subhyperspace	NOUN
ejpam-65	140	7	of	of	ADP
ejpam-65	140	8	v	v	NOUN
ejpam-65	140	9	containing	contain	VERB
ejpam-65	140	10	s.	s.	PROPN
ejpam-65	140	11	proof	proof	NOUN
ejpam-65	140	12	.	.	PUNCT
ejpam-65	141	1	let	let	VERB
ejpam-65	141	2	t1	t1	NOUN
ejpam-65	141	3	,	,	PUNCT
ejpam-65	141	4	t2	t2	PROPN
ejpam-65	141	5	∈	∈	PROPN
ejpam-65	141	6	l(s	l(s	PROPN
ejpam-65	141	7	)	)	PUNCT
ejpam-65	141	8	,	,	PUNCT
ejpam-65	141	9	then	then	ADV
ejpam-65	141	10	t1	t1	PROPN
ejpam-65	141	11	∈	∈	PROPN
ejpam-65	142	1	n∑	n∑	INTJ
ejpam-65	142	2	i=1	i=1	PROPN
ejpam-65	142	3	ai	ai	VERB
ejpam-65	142	4	◦	◦	NOUN
ejpam-65	142	5	si	si	NOUN
ejpam-65	142	6	and	and	CCONJ
ejpam-65	142	7	t2	t2	PROPN
ejpam-65	142	8	∈	∈	PROPN
ejpam-65	142	9	m∑	m∑	NOUN
ejpam-65	143	1	i=1	i=1	PROPN
ejpam-65	143	2	ái	ái	PROPN
ejpam-65	143	3	◦	◦	NOUN
ejpam-65	143	4	śi	śi	NOUN
ejpam-65	143	5	,	,	PUNCT
ejpam-65	143	6	where	where	SCONJ
ejpam-65	143	7	ai	ai	VERB
ejpam-65	143	8	,	,	PUNCT
ejpam-65	143	9	ái	ái	NOUN
ejpam-65	143	10	∈	∈	PROPN
ejpam-65	144	1	k	k	PROPN
ejpam-65	144	2	and	and	CCONJ
ejpam-65	144	3	si	si	PROPN
ejpam-65	144	4	,	,	PUNCT
ejpam-65	144	5	śi	śi	NOUN
ejpam-65	144	6	∈	∈	PROPN
ejpam-65	144	7	s.	s.	PROPN
ejpam-65	144	8	thus	thus	ADV
ejpam-65	144	9	t1	t1	VERB
ejpam-65	144	10	−	−	PROPN
ejpam-65	144	11	t2	t2	PROPN
ejpam-65	144	12	∈	∈	PROPN
ejpam-65	145	1	n∑	n∑	NOUN
ejpam-65	146	1	i=1	i=1	PROPN
ejpam-65	146	2	ai	ai	VERB
ejpam-65	146	3	◦	◦	NOUN
ejpam-65	146	4	si−	si−	PUNCT
ejpam-65	146	5	m∑	m∑	NOUN
ejpam-65	147	1	i=1	i=1	NOUN
ejpam-65	147	2	ái	ái	PROPN
ejpam-65	147	3	◦	◦	NOUN
ejpam-65	147	4	śi	śi	NOUN
ejpam-65	148	1	=	=	PUNCT
ejpam-65	148	2	n∑	n∑	INTJ
ejpam-65	148	3	i=1	i=1	PROPN
ejpam-65	148	4	ai	ai	VERB
ejpam-65	148	5	◦	◦	NOUN
ejpam-65	148	6	si+	si+	ADJ
ejpam-65	148	7	m∑	m∑	ADP
ejpam-65	148	8	i=1	i=1	PROPN
ejpam-65	148	9	(	(	PUNCT
ejpam-65	148	10	−ái	−ái	NOUN
ejpam-65	148	11	)	)	PUNCT
ejpam-65	148	12	◦	◦	NOUN
ejpam-65	148	13	śi	śi	NOUN
ejpam-65	148	14	,	,	PUNCT
ejpam-65	148	15	(	(	PUNCT
ejpam-65	148	16	by	by	ADP
ejpam-65	148	17	h4	h4	PROPN
ejpam-65	148	18	)	)	PUNCT
ejpam-65	148	19	r.	r.	PROPN
ejpam-65	148	20	ameri	ameri	PROPN
ejpam-65	148	21	,	,	PUNCT
ejpam-65	148	22	o.r	o.r	PROPN
ejpam-65	148	23	.	.	PROPN
ejpam-65	148	24	dehghan	dehghan	PROPN
ejpam-65	148	25	/	/	SYM
ejpam-65	148	26	eur	eur	PROPN
ejpam-65	148	27	.	.	PUNCT
ejpam-65	149	1	j.	j.	PROPN
ejpam-65	149	2	pure	pure	PROPN
ejpam-65	149	3	appl	appl	PROPN
ejpam-65	149	4	.	.	PROPN
ejpam-65	149	5	math	math	PROPN
ejpam-65	149	6	,	,	PUNCT
ejpam-65	149	7	1	1	NUM
ejpam-65	149	8	(	(	PUNCT
ejpam-65	149	9	2008	2008	NUM
ejpam-65	149	10	)	)	PUNCT
ejpam-65	149	11	,	,	PUNCT
ejpam-65	149	12	(	(	PUNCT
ejpam-65	149	13	32	32	NUM
ejpam-65	149	14	-	-	SYM
ejpam-65	149	15	50	50	NUM
ejpam-65	149	16	)	)	PUNCT
ejpam-65	149	17	37	37	NUM
ejpam-65	149	18	then	then	ADV
ejpam-65	149	19	t1	t1	NOUN
ejpam-65	149	20	−	−	PROPN
ejpam-65	150	1	t2	t2	PROPN
ejpam-65	150	2	∈	∈	PROPN
ejpam-65	150	3	l(s	l(s	PROPN
ejpam-65	150	4	)	)	PUNCT
ejpam-65	150	5	.	.	PUNCT
ejpam-65	151	1	also	also	ADV
ejpam-65	151	2	if	if	SCONJ
ejpam-65	151	3	t	t	PROPN
ejpam-65	151	4	∈	∈	PROPN
ejpam-65	151	5	l(s	l(s	PROPN
ejpam-65	151	6	)	)	PUNCT
ejpam-65	151	7	and	and	CCONJ
ejpam-65	151	8	k	k	PROPN
ejpam-65	151	9	∈	∈	PROPN
ejpam-65	151	10	k	k	NOUN
ejpam-65	151	11	,	,	PUNCT
ejpam-65	151	12	then	then	ADV
ejpam-65	151	13	k	k	PROPN
ejpam-65	151	14	◦	◦	NOUN
ejpam-65	151	15	t	t	PROPN
ejpam-65	152	1	⊆	⊆	NUM
ejpam-65	152	2	k	k	PROPN
ejpam-65	152	3	◦	◦	NOUN
ejpam-65	152	4	n∑	n∑	PROPN
ejpam-65	152	5	i=1	i=1	PROPN
ejpam-65	152	6	ai	ai	VERB
ejpam-65	152	7	◦	◦	NOUN
ejpam-65	152	8	si	si	NOUN
ejpam-65	152	9	⊆	⊆	NUM
ejpam-65	152	10	n∑	n∑	NOUN
ejpam-65	152	11	i=1	i=1	PROPN
ejpam-65	153	1	k	k	PROPN
ejpam-65	154	1	◦	◦	NOUN
ejpam-65	154	2	(	(	PUNCT
ejpam-65	154	3	ai	ai	VERB
ejpam-65	154	4	◦	◦	NOUN
ejpam-65	154	5	si	si	NOUN
ejpam-65	154	6	)	)	PUNCT
ejpam-65	154	7	(	(	PUNCT
ejpam-65	154	8	by	by	ADP
ejpam-65	154	9	h1	h1	PROPN
ejpam-65	154	10	)	)	PUNCT
ejpam-65	154	11	=	=	PUNCT
ejpam-65	155	1	n∑	n∑	NOUN
ejpam-65	155	2	i=1	i=1	PROPN
ejpam-65	155	3	(	(	PUNCT
ejpam-65	155	4	kai	kai	PROPN
ejpam-65	155	5	)	)	PUNCT
ejpam-65	155	6	◦	◦	NOUN
ejpam-65	155	7	si	si	X
ejpam-65	155	8	(	(	PUNCT
ejpam-65	155	9	by	by	ADP
ejpam-65	155	10	h3	h3	NOUN
ejpam-65	155	11	)	)	PUNCT
ejpam-65	156	1	so	so	CCONJ
ejpam-65	156	2	k	k	PROPN
ejpam-65	156	3	◦	◦	NOUN
ejpam-65	156	4	t	t	PROPN
ejpam-65	156	5	⊆	⊆	NUM
ejpam-65	156	6	l(s	l(s	PROPN
ejpam-65	156	7	)	)	PUNCT
ejpam-65	156	8	,	,	PUNCT
ejpam-65	156	9	therefore	therefore	ADV
ejpam-65	156	10	l(s	l(s	PROPN
ejpam-65	156	11	)	)	PUNCT
ejpam-65	156	12	≤	≤	NOUN
ejpam-65	156	13	v	v	NOUN
ejpam-65	156	14	.	.	PUNCT
ejpam-65	157	1	now	now	ADV
ejpam-65	157	2	suppose	suppose	VERB
ejpam-65	157	3	that	that	SCONJ
ejpam-65	157	4	w	w	NOUN
ejpam-65	157	5	is	be	AUX
ejpam-65	157	6	a	a	DET
ejpam-65	157	7	subhyperspace	subhyperspace	NOUN
ejpam-65	157	8	of	of	ADP
ejpam-65	157	9	v	v	NOUN
ejpam-65	157	10	containing	contain	VERB
ejpam-65	157	11	s.	s.	PROPN
ejpam-65	157	12	then	then	ADV
ejpam-65	157	13	for	for	ADP
ejpam-65	157	14	every	every	DET
ejpam-65	157	15	t	t	NOUN
ejpam-65	157	16	∈	∈	PROPN
ejpam-65	157	17	l(s	l(s	PROPN
ejpam-65	157	18	)	)	PUNCT
ejpam-65	157	19	we	we	PRON
ejpam-65	157	20	have	have	VERB
ejpam-65	157	21	t	t	PROPN
ejpam-65	157	22	∈	∈	PROPN
ejpam-65	158	1	n∑	n∑	PROPN
ejpam-65	159	1	i=1	i=1	PROPN
ejpam-65	159	2	ai	ai	VERB
ejpam-65	159	3	◦	◦	NOUN
ejpam-65	159	4	si	si	PROPN
ejpam-65	159	5	,	,	PUNCT
ejpam-65	159	6	for	for	ADP
ejpam-65	159	7	some	some	DET
ejpam-65	159	8	ai	ai	NOUN
ejpam-65	159	9	∈	∈	PROPN
ejpam-65	159	10	k	k	NOUN
ejpam-65	159	11	,	,	PUNCT
ejpam-65	159	12	si	si	PROPN
ejpam-65	159	13	∈	∈	PROPN
ejpam-65	159	14	s	s	X
ejpam-65	159	15	and	and	CCONJ
ejpam-65	159	16	n	n	PRON
ejpam-65	159	17	∈	∈	PROPN
ejpam-65	159	18	n.	n.	NOUN
ejpam-65	159	19	since	since	SCONJ
ejpam-65	159	20	w	w	PROPN
ejpam-65	159	21	is	be	AUX
ejpam-65	159	22	a	a	DET
ejpam-65	159	23	subhyperspace	subhyperspace	NOUN
ejpam-65	159	24	and	and	CCONJ
ejpam-65	159	25	s	s	NOUN
ejpam-65	159	26	⊆w	⊆w	NOUN
ejpam-65	159	27	,	,	PUNCT
ejpam-65	159	28	thus	thus	ADV
ejpam-65	159	29	si	si	PROPN
ejpam-65	159	30	∈w	∈w	NOUN
ejpam-65	159	31	,	,	PUNCT
ejpam-65	160	1	so	so	CCONJ
ejpam-65	161	1	n∑	n∑	PROPN
ejpam-65	161	2	i=1	i=1	PROPN
ejpam-65	161	3	ai	ai	VERB
ejpam-65	161	4	◦	◦	NOUN
ejpam-65	161	5	si	si	INTJ
ejpam-65	161	6	⊆w	⊆w	NOUN
ejpam-65	161	7	,	,	PUNCT
ejpam-65	161	8	therefore	therefore	ADV
ejpam-65	161	9	t	t	PROPN
ejpam-65	161	10	∈w	∈w	PROPN
ejpam-65	161	11	.	.	PUNCT
ejpam-65	162	1	consequently	consequently	ADV
ejpam-65	162	2	l(s	l(s	PROPN
ejpam-65	162	3	)	)	PUNCT
ejpam-65	163	1	⊆	⊆	NUM
ejpam-65	163	2	w	w	NOUN
ejpam-65	163	3	.	.	PUNCT
ejpam-65	164	1	thus	thus	ADV
ejpam-65	164	2	l(s	l(s	PROPN
ejpam-65	164	3	)	)	PUNCT
ejpam-65	164	4	is	be	AUX
ejpam-65	164	5	the	the	DET
ejpam-65	164	6	smallest	small	ADJ
ejpam-65	164	7	subhyperspace	subhyperspace	NOUN
ejpam-65	164	8	of	of	ADP
ejpam-65	164	9	v	v	NOUN
ejpam-65	164	10	.	.	PUNCT
ejpam-65	165	1	also	also	ADV
ejpam-65	165	2	for	for	ADP
ejpam-65	165	3	all	all	DET
ejpam-65	165	4	s	s	PART
ejpam-65	165	5	∈	∈	PROPN
ejpam-65	165	6	s	s	NOUN
ejpam-65	165	7	,	,	PUNCT
ejpam-65	165	8	s	s	PART
ejpam-65	165	9	∈	∈	NOUN
ejpam-65	165	10	1k	1k	NUM
ejpam-65	165	11	◦	◦	NOUN
ejpam-65	165	12	s	s	PART
ejpam-65	165	13	,	,	PUNCT
ejpam-65	165	14	so	so	PRON
ejpam-65	165	15	s	s	X
ejpam-65	165	16	∈	∈	PROPN
ejpam-65	165	17	l(s	l(s	PROPN
ejpam-65	165	18	)	)	PUNCT
ejpam-65	166	1	,	,	PUNCT
ejpam-65	166	2	therefore	therefore	ADV
ejpam-65	166	3	s	s	VERB
ejpam-65	166	4	⊆	⊆	NUM
ejpam-65	166	5	l(s	l(s	PROPN
ejpam-65	166	6	)	)	PUNCT
ejpam-65	166	7	.	.	PUNCT
ejpam-65	167	1	sometimes	sometimes	ADV
ejpam-65	167	2	we	we	PRON
ejpam-65	167	3	denote	denote	VERB
ejpam-65	167	4	l(s	l(s	PROPN
ejpam-65	167	5	)	)	PUNCT
ejpam-65	167	6	,	,	PUNCT
ejpam-65	167	7	by	by	ADP
ejpam-65	167	8	〈	〈	PROPN
ejpam-65	167	9	s	s	PROPN
ejpam-65	167	10	〉	〉	PROPN
ejpam-65	167	11	,	,	PUNCT
ejpam-65	167	12	which	which	PRON
ejpam-65	167	13	is	be	AUX
ejpam-65	167	14	called	call	VERB
ejpam-65	167	15	the	the	DET
ejpam-65	167	16	subhyperspace	subhyperspace	NOUN
ejpam-65	167	17	generated	generate	VERB
ejpam-65	167	18	by	by	ADP
ejpam-65	167	19	s.	s.	PROPN
ejpam-65	167	20	proposition	proposition	PROPN
ejpam-65	167	21	3.1	3.1	NUM
ejpam-65	167	22	.	.	PUNCT
ejpam-65	168	1	let	let	VERB
ejpam-65	168	2	v	v	PART
ejpam-65	168	3	be	be	AUX
ejpam-65	168	4	strongly	strongly	ADV
ejpam-65	168	5	left	leave	VERB
ejpam-65	168	6	distributive	distributive	ADJ
ejpam-65	168	7	.	.	PUNCT
ejpam-65	169	1	then	then	ADV
ejpam-65	169	2	∀x	∀x	VERB
ejpam-65	169	3	∈	∈	PROPN
ejpam-65	169	4	v	v	NOUN
ejpam-65	169	5	,	,	PUNCT
ejpam-65	169	6	〈	〈	PROPN
ejpam-65	169	7	x	x	X
ejpam-65	169	8	〉	〉	NOUN
ejpam-65	169	9	=	=	SYM
ejpam-65	169	10	⋃	⋃	NOUN
ejpam-65	169	11	a∈k	a∈k	NOUN
ejpam-65	169	12	a	a	DET
ejpam-65	169	13	◦	◦	NOUN
ejpam-65	169	14	x.	x.	NOUN
ejpam-65	169	15	proof	proof	NOUN
ejpam-65	169	16	.	.	PUNCT
ejpam-65	170	1	let	let	VERB
ejpam-65	170	2	t	t	PROPN
ejpam-65	170	3	∈	∈	PROPN
ejpam-65	170	4	〈	〈	PROPN
ejpam-65	170	5	x	x	PROPN
ejpam-65	170	6	〉	〉	PROPN
ejpam-65	170	7	.	.	PUNCT
ejpam-65	171	1	then	then	ADV
ejpam-65	171	2	there	there	PRON
ejpam-65	171	3	exist	exist	VERB
ejpam-65	171	4	a1	a1	NOUN
ejpam-65	171	5	,	,	PUNCT
ejpam-65	171	6	.	.	PUNCT
ejpam-65	171	7	.	.	PUNCT
ejpam-65	172	1	.	.	PUNCT
ejpam-65	173	1	,	,	PUNCT
ejpam-65	173	2	an	an	DET
ejpam-65	173	3	∈	∈	NOUN
ejpam-65	173	4	k	k	NOUN
ejpam-65	173	5	such	such	ADJ
ejpam-65	173	6	that	that	SCONJ
ejpam-65	173	7	t	t	PROPN
ejpam-65	173	8	∈	∈	PROPN
ejpam-65	173	9	a1	a1	NOUN
ejpam-65	173	10	◦	◦	NOUN
ejpam-65	173	11	x+	x+	X
ejpam-65	173	12	·	·	PUNCT
ejpam-65	173	13	·	·	PUNCT
ejpam-65	173	14	·	·	PUNCT
ejpam-65	174	1	+	+	CCONJ
ejpam-65	174	2	an	an	DET
ejpam-65	174	3	◦	◦	NOUN
ejpam-65	174	4	x.	x.	NOUN
ejpam-65	175	1	so	so	ADV
ejpam-65	175	2	t	t	PROPN
ejpam-65	175	3	∈	∈	PROPN
ejpam-65	175	4	a1	a1	NOUN
ejpam-65	175	5	◦	◦	NOUN
ejpam-65	175	6	x+	x+	X
ejpam-65	175	7	·	·	PUNCT
ejpam-65	175	8	·	·	PUNCT
ejpam-65	175	9	·	·	PUNCT
ejpam-65	176	1	+	+	CCONJ
ejpam-65	176	2	an	an	DET
ejpam-65	176	3	◦	◦	NOUN
ejpam-65	176	4	x	x	X
ejpam-65	177	1	=	=	SYM
ejpam-65	177	2	(	(	PUNCT
ejpam-65	177	3	a1	a1	NOUN
ejpam-65	177	4	+	+	X
ejpam-65	177	5	·	·	PUNCT
ejpam-65	177	6	·	·	PUNCT
ejpam-65	177	7	·	·	PUNCT
ejpam-65	177	8	+	+	NUM
ejpam-65	177	9	an	an	X
ejpam-65	177	10	)	)	PUNCT
ejpam-65	177	11	◦	◦	NOUN
ejpam-65	177	12	x	x	SYM
ejpam-65	177	13	,	,	PUNCT
ejpam-65	177	14	thus	thus	ADV
ejpam-65	177	15	t	t	X
ejpam-65	177	16	∈	∈	PROPN
ejpam-65	177	17	⋃	⋃	PROPN
ejpam-65	177	18	a∈k	a∈k	NOUN
ejpam-65	177	19	a	a	DET
ejpam-65	177	20	◦	◦	NOUN
ejpam-65	177	21	x.	x.	NOUN
ejpam-65	177	22	conversely	conversely	ADV
ejpam-65	177	23	,	,	PUNCT
ejpam-65	177	24	if	if	SCONJ
ejpam-65	177	25	t	t	PROPN
ejpam-65	177	26	∈	∈	PROPN
ejpam-65	177	27	⋃	⋃	PROPN
ejpam-65	177	28	a∈k	a∈k	NOUN
ejpam-65	177	29	a	a	DET
ejpam-65	177	30	◦	◦	NOUN
ejpam-65	177	31	x	x	SYM
ejpam-65	177	32	,	,	PUNCT
ejpam-65	177	33	then	then	ADV
ejpam-65	177	34	∃a	∃a	NOUN
ejpam-65	177	35	∈	∈	PROPN
ejpam-65	177	36	k	k	NOUN
ejpam-65	177	37	,	,	PUNCT
ejpam-65	177	38	t	t	PROPN
ejpam-65	177	39	∈	∈	PROPN
ejpam-65	177	40	a	a	DET
ejpam-65	177	41	◦	◦	NOUN
ejpam-65	177	42	x	x	PUNCT
ejpam-65	178	1	=	=	NOUN
ejpam-65	178	2	⇒	⇒	NOUN
ejpam-65	178	3	t	t	PROPN
ejpam-65	178	4	∈	∈	PROPN
ejpam-65	178	5	〈	〈	PROPN
ejpam-65	178	6	x	x	PROPN
ejpam-65	178	7	〉	〉	PROPN
ejpam-65	178	8	.	.	PUNCT
ejpam-65	179	1	proposition	proposition	NOUN
ejpam-65	179	2	3.2	3.2	NUM
ejpam-65	179	3	.	.	PUNCT
ejpam-65	180	1	if	if	SCONJ
ejpam-65	180	2	w1	w1	NOUN
ejpam-65	180	3	and	and	CCONJ
ejpam-65	180	4	w2	w2	NOUN
ejpam-65	180	5	are	be	AUX
ejpam-65	180	6	two	two	NUM
ejpam-65	180	7	subhyperspaces	subhyperspace	NOUN
ejpam-65	180	8	of	of	ADP
ejpam-65	180	9	v	v	NOUN
ejpam-65	180	10	,	,	PUNCT
ejpam-65	180	11	then	then	ADV
ejpam-65	180	12	l(w1	l(w1	PROPN
ejpam-65	180	13	∪w2	∪w2	NOUN
ejpam-65	180	14	)	)	PUNCT
ejpam-65	181	1	=	=	SYM
ejpam-65	181	2	w1	w1	NOUN
ejpam-65	181	3	+	+	NOUN
ejpam-65	181	4	w2	w2	NOUN
ejpam-65	181	5	.	.	PUNCT
ejpam-65	182	1	proof	proof	NOUN
ejpam-65	182	2	.	.	PUNCT
ejpam-65	183	1	obvious	obvious	ADJ
ejpam-65	183	2	.	.	PUNCT
ejpam-65	184	1	lemma	lemma	PROPN
ejpam-65	184	2	3.4	3.4	NUM
ejpam-65	184	3	.	.	PUNCT
ejpam-65	185	1	let	let	VERB
ejpam-65	185	2	v	v	PART
ejpam-65	185	3	be	be	AUX
ejpam-65	185	4	anti	anti	ADJ
ejpam-65	185	5	-	-	ADJ
ejpam-65	185	6	left	left	ADJ
ejpam-65	185	7	distributive	distributive	ADJ
ejpam-65	185	8	and	and	CCONJ
ejpam-65	185	9	v1	v1	NOUN
ejpam-65	185	10	,	,	PUNCT
ejpam-65	185	11	v2	v2	PROPN
ejpam-65	185	12	,	,	PUNCT
ejpam-65	185	13	...	...	PUNCT
ejpam-65	185	14	,	,	PUNCT
ejpam-65	185	15	vn	vn	INTJ
ejpam-65	185	16	be	be	AUX
ejpam-65	185	17	linearly	linearly	ADV
ejpam-65	185	18	independent	independent	ADJ
ejpam-65	185	19	in	in	ADP
ejpam-65	185	20	v	v	NOUN
ejpam-65	185	21	.	.	PUNCT
ejpam-65	186	1	then	then	ADV
ejpam-65	186	2	every	every	DET
ejpam-65	186	3	element	element	NOUN
ejpam-65	186	4	in	in	ADP
ejpam-65	186	5	their	their	PRON
ejpam-65	186	6	linear	linear	ADJ
ejpam-65	186	7	span	span	NOUN
ejpam-65	186	8	belongs	belong	VERB
ejpam-65	186	9	to	to	ADP
ejpam-65	186	10	a	a	DET
ejpam-65	186	11	unique	unique	ADJ
ejpam-65	186	12	sum	sum	NOUN
ejpam-65	186	13	in	in	ADP
ejpam-65	186	14	the	the	DET
ejpam-65	186	15	form	form	NOUN
ejpam-65	186	16	c1	c1	NOUN
ejpam-65	186	17	◦	◦	NOUN
ejpam-65	186	18	v1+c2	v1+c2	PROPN
ejpam-65	186	19	◦	◦	NOUN
ejpam-65	186	20	v2+	v2+	NUM
ejpam-65	186	21	...	...	PUNCT
ejpam-65	186	22	+cn	+cn	PROPN
ejpam-65	186	23	◦	◦	NOUN
ejpam-65	186	24	vn	vn	NOUN
ejpam-65	186	25	with	with	ADP
ejpam-65	186	26	ci	ci	PROPN
ejpam-65	186	27	∈	∈	PROPN
ejpam-65	186	28	k.	k.	PROPN
ejpam-65	186	29	r.	r.	PROPN
ejpam-65	186	30	ameri	ameri	PROPN
ejpam-65	186	31	,	,	PUNCT
ejpam-65	186	32	o.r	o.r	PROPN
ejpam-65	186	33	.	.	PROPN
ejpam-65	186	34	dehghan	dehghan	PROPN
ejpam-65	186	35	/	/	SYM
ejpam-65	186	36	eur	eur	PROPN
ejpam-65	186	37	.	.	PUNCT
ejpam-65	187	1	j.	j.	PROPN
ejpam-65	187	2	pure	pure	PROPN
ejpam-65	187	3	appl	appl	PROPN
ejpam-65	187	4	.	.	PROPN
ejpam-65	187	5	math	math	PROPN
ejpam-65	187	6	,	,	PUNCT
ejpam-65	187	7	1	1	NUM
ejpam-65	187	8	(	(	PUNCT
ejpam-65	187	9	2008	2008	NUM
ejpam-65	187	10	)	)	PUNCT
ejpam-65	187	11	,	,	PUNCT
ejpam-65	187	12	(	(	PUNCT
ejpam-65	187	13	32	32	NUM
ejpam-65	187	14	-	-	SYM
ejpam-65	187	15	50	50	NUM
ejpam-65	187	16	)	)	PUNCT
ejpam-65	187	17	38	38	NUM
ejpam-65	187	18	proof	proof	NOUN
ejpam-65	187	19	.	.	PUNCT
ejpam-65	188	1	by	by	ADP
ejpam-65	188	2	definition	definition	NOUN
ejpam-65	188	3	3.4	3.4	NUM
ejpam-65	188	4	,	,	PUNCT
ejpam-65	188	5	every	every	DET
ejpam-65	188	6	element	element	NOUN
ejpam-65	188	7	in	in	ADP
ejpam-65	188	8	the	the	DET
ejpam-65	188	9	linear	linear	ADJ
ejpam-65	188	10	span	span	NOUN
ejpam-65	188	11	is	be	AUX
ejpam-65	188	12	belong	belong	ADJ
ejpam-65	188	13	to	to	ADP
ejpam-65	188	14	a	a	DET
ejpam-65	188	15	set	set	NOUN
ejpam-65	188	16	of	of	ADP
ejpam-65	188	17	the	the	DET
ejpam-65	188	18	form	form	NOUN
ejpam-65	188	19	c1	c1	PROPN
ejpam-65	188	20	◦	◦	NOUN
ejpam-65	188	21	v1	v1	PROPN
ejpam-65	188	22	+	+	CCONJ
ejpam-65	188	23	c2	c2	PROPN
ejpam-65	188	24	◦	◦	VERB
ejpam-65	188	25	v2	v2	PROPN
ejpam-65	188	26	+	+	CCONJ
ejpam-65	188	27	...	...	PUNCT
ejpam-65	188	28	+	+	CCONJ
ejpam-65	188	29	cn	cn	ADJ
ejpam-65	188	30	◦	◦	NOUN
ejpam-65	188	31	vn	vn	PROPN
ejpam-65	188	32	.	.	PUNCT
ejpam-65	189	1	to	to	PART
ejpam-65	189	2	prove	prove	VERB
ejpam-65	189	3	uniqueness	uniqueness	NOUN
ejpam-65	189	4	we	we	PRON
ejpam-65	189	5	must	must	AUX
ejpam-65	189	6	demonstrate	demonstrate	VERB
ejpam-65	189	7	if	if	SCONJ
ejpam-65	189	8	for	for	ADP
ejpam-65	189	9	some	some	DET
ejpam-65	189	10	u	u	PROPN
ejpam-65	189	11	∈	∈	PROPN
ejpam-65	189	12	v	v	NOUN
ejpam-65	189	13	,	,	PUNCT
ejpam-65	189	14	u	u	PROPN
ejpam-65	189	15	∈	∈	PROPN
ejpam-65	189	16	c1	c1	PROPN
ejpam-65	189	17	◦	◦	NOUN
ejpam-65	189	18	v1	v1	PROPN
ejpam-65	190	1	+	+	CCONJ
ejpam-65	190	2	c2	c2	PROPN
ejpam-65	190	3	◦	◦	VERB
ejpam-65	190	4	v2	v2	PROPN
ejpam-65	190	5	+	+	CCONJ
ejpam-65	190	6	...	...	PUNCT
ejpam-65	191	1	+	+	CCONJ
ejpam-65	191	2	cn	cn	ADJ
ejpam-65	191	3	◦	◦	NOUN
ejpam-65	191	4	vn	vn	NOUN
ejpam-65	191	5	and	and	CCONJ
ejpam-65	191	6	u	u	PROPN
ejpam-65	191	7	∈	∈	PROPN
ejpam-65	191	8	d1	d1	PROPN
ejpam-65	191	9	◦	◦	NOUN
ejpam-65	191	10	v1	v1	PROPN
ejpam-65	191	11	+	+	CCONJ
ejpam-65	191	12	d2	d2	PROPN
ejpam-65	191	13	◦	◦	NOUN
ejpam-65	191	14	v2	v2	PROPN
ejpam-65	191	15	+	+	CCONJ
ejpam-65	191	16	...	...	PUNCT
ejpam-65	191	17	+	+	CCONJ
ejpam-65	191	18	dn	dn	PROPN
ejpam-65	191	19	◦	◦	PROPN
ejpam-65	191	20	vn	vn	PROPN
ejpam-65	191	21	,	,	PUNCT
ejpam-65	191	22	then	then	ADV
ejpam-65	191	23	c1	c1	PROPN
ejpam-65	191	24	=	=	PUNCT
ejpam-65	191	25	d1	d1	PROPN
ejpam-65	191	26	,	,	PUNCT
ejpam-65	191	27	...	...	PUNCT
ejpam-65	191	28	,	,	PUNCT
ejpam-65	191	29	dn	dn	PROPN
ejpam-65	191	30	=	=	SYM
ejpam-65	191	31	cn	cn	PROPN
ejpam-65	191	32	.	.	PUNCT
ejpam-65	192	1	but	but	CCONJ
ejpam-65	192	2	by	by	ADP
ejpam-65	192	3	(	(	PUNCT
ejpam-65	192	4	h4	h4	NOUN
ejpam-65	192	5	)	)	PUNCT
ejpam-65	192	6	we	we	PRON
ejpam-65	192	7	obtain	obtain	VERB
ejpam-65	192	8	that	that	PRON
ejpam-65	192	9	0	0	NUM
ejpam-65	193	1	=	=	SYM
ejpam-65	193	2	u−	u−	PROPN
ejpam-65	193	3	u	u	NOUN
ejpam-65	193	4	∈	∈	PROPN
ejpam-65	193	5	c1	c1	PROPN
ejpam-65	193	6	◦	◦	NOUN
ejpam-65	193	7	v1	v1	PROPN
ejpam-65	193	8	+	+	CCONJ
ejpam-65	193	9	c2	c2	PROPN
ejpam-65	193	10	◦	◦	VERB
ejpam-65	193	11	v2	v2	PROPN
ejpam-65	193	12	+	+	X
ejpam-65	193	13	·	·	PUNCT
ejpam-65	193	14	·	·	PUNCT
ejpam-65	193	15	·	·	PUNCT
ejpam-65	194	1	+	+	NUM
ejpam-65	194	2	cn	cn	ADJ
ejpam-65	194	3	◦	◦	NOUN
ejpam-65	194	4	vn	vn	ADP
ejpam-65	194	5	−	−	NOUN
ejpam-65	194	6	−	−	PROPN
ejpam-65	195	1	(	(	PUNCT
ejpam-65	195	2	d1	d1	NOUN
ejpam-65	195	3	◦	◦	NOUN
ejpam-65	195	4	v1	v1	NOUN
ejpam-65	195	5	+	+	CCONJ
ejpam-65	195	6	d2	d2	PROPN
ejpam-65	195	7	◦	◦	NOUN
ejpam-65	195	8	v2	v2	PROPN
ejpam-65	195	9	+	+	X
ejpam-65	195	10	·	·	PUNCT
ejpam-65	195	11	·	·	PUNCT
ejpam-65	195	12	·	·	PUNCT
ejpam-65	195	13	+	+	NUM
ejpam-65	195	14	dn	dn	PROPN
ejpam-65	195	15	◦	◦	PROPN
ejpam-65	195	16	vn	vn	X
ejpam-65	195	17	)	)	PUNCT
ejpam-65	195	18	=	=	SYM
ejpam-65	195	19	c1	c1	PROPN
ejpam-65	195	20	◦	◦	NOUN
ejpam-65	195	21	v1	v1	PROPN
ejpam-65	195	22	+	+	CCONJ
ejpam-65	195	23	c2	c2	PROPN
ejpam-65	195	24	◦	◦	VERB
ejpam-65	195	25	v2	v2	PROPN
ejpam-65	195	26	+	+	X
ejpam-65	195	27	·	·	PUNCT
ejpam-65	195	28	·	·	PUNCT
ejpam-65	195	29	·	·	PUNCT
ejpam-65	195	30	+	+	NUM
ejpam-65	195	31	cn	cn	ADJ
ejpam-65	195	32	◦	◦	NOUN
ejpam-65	195	33	vn	vn	ADP
ejpam-65	195	34	−	−	NOUN
ejpam-65	195	35	−	−	PROPN
ejpam-65	196	1	(	(	PUNCT
ejpam-65	196	2	d1	d1	PROPN
ejpam-65	196	3	◦	◦	NOUN
ejpam-65	196	4	v1)−	v1)−	PROPN
ejpam-65	196	5	(	(	PUNCT
ejpam-65	196	6	d2	d2	VERB
ejpam-65	196	7	◦	◦	NOUN
ejpam-65	196	8	v2)−	v2)−	X
ejpam-65	196	9	·	·	PUNCT
ejpam-65	196	10	·	·	PUNCT
ejpam-65	196	11	·	·	PUNCT
ejpam-65	197	1	−	−	PUNCT
ejpam-65	197	2	(	(	PUNCT
ejpam-65	197	3	dn	dn	ADP
ejpam-65	197	4	◦	◦	PROPN
ejpam-65	197	5	vn	vn	PROPN
ejpam-65	197	6	)	)	PUNCT
ejpam-65	197	7	=	=	SYM
ejpam-65	197	8	c1	c1	PROPN
ejpam-65	197	9	◦	◦	NOUN
ejpam-65	197	10	v1	v1	PROPN
ejpam-65	197	11	+	+	CCONJ
ejpam-65	197	12	c2	c2	PROPN
ejpam-65	197	13	◦	◦	VERB
ejpam-65	197	14	v2	v2	PROPN
ejpam-65	197	15	+	+	X
ejpam-65	197	16	·	·	PUNCT
ejpam-65	197	17	·	·	PUNCT
ejpam-65	197	18	·	·	PUNCT
ejpam-65	198	1	+	+	NUM
ejpam-65	198	2	cn	cn	VERB
ejpam-65	198	3	◦	◦	NOUN
ejpam-65	198	4	vn	vn	NOUN
ejpam-65	198	5	+	+	X
ejpam-65	199	1	+	+	ADJ
ejpam-65	199	2	(	(	PUNCT
ejpam-65	199	3	−d1	−d1	NOUN
ejpam-65	199	4	)	)	PUNCT
ejpam-65	199	5	◦	◦	NOUN
ejpam-65	199	6	v1	v1	NOUN
ejpam-65	199	7	+	+	CCONJ
ejpam-65	199	8	(	(	PUNCT
ejpam-65	199	9	−d2	−d2	NOUN
ejpam-65	199	10	)	)	PUNCT
ejpam-65	199	11	◦	◦	VERB
ejpam-65	199	12	v2	v2	PROPN
ejpam-65	199	13	+	+	X
ejpam-65	199	14	·	·	PUNCT
ejpam-65	199	15	·	·	PUNCT
ejpam-65	199	16	·	·	PUNCT
ejpam-65	199	17	+	+	CCONJ
ejpam-65	199	18	(	(	PUNCT
ejpam-65	199	19	−dn	−dn	NOUN
ejpam-65	199	20	)	)	PUNCT
ejpam-65	199	21	◦	◦	NOUN
ejpam-65	199	22	vn	vn	NOUN
ejpam-65	199	23	,	,	PUNCT
ejpam-65	199	24	now	now	ADV
ejpam-65	199	25	since	since	SCONJ
ejpam-65	199	26	v	v	NOUN
ejpam-65	199	27	is	be	AUX
ejpam-65	199	28	anti	anti	ADJ
ejpam-65	199	29	-	-	ADJ
ejpam-65	199	30	left	left	ADJ
ejpam-65	199	31	distributive	distributive	ADJ
ejpam-65	199	32	,	,	PUNCT
ejpam-65	199	33	then	then	ADV
ejpam-65	199	34	we	we	PRON
ejpam-65	199	35	have	have	VERB
ejpam-65	199	36	0	0	NUM
ejpam-65	199	37	∈	∈	NOUN
ejpam-65	199	38	(	(	PUNCT
ejpam-65	199	39	c1	c1	NOUN
ejpam-65	199	40	−	−	PROPN
ejpam-65	199	41	d1	d1	PROPN
ejpam-65	199	42	)	)	PUNCT
ejpam-65	199	43	◦	◦	NOUN
ejpam-65	199	44	v1	v1	NOUN
ejpam-65	199	45	+	+	X
ejpam-65	199	46	·	·	PUNCT
ejpam-65	199	47	·	·	PUNCT
ejpam-65	199	48	·	·	PUNCT
ejpam-65	199	49	+	+	CCONJ
ejpam-65	199	50	(	(	PUNCT
ejpam-65	199	51	cn	cn	INTJ
ejpam-65	199	52	−	−	PROPN
ejpam-65	199	53	dn	dn	PROPN
ejpam-65	199	54	)	)	PUNCT
ejpam-65	199	55	◦	◦	NOUN
ejpam-65	199	56	vn	vn	PROPN
ejpam-65	199	57	,	,	PUNCT
ejpam-65	199	58	which	which	PRON
ejpam-65	199	59	implies	imply	VERB
ejpam-65	199	60	that	that	DET
ejpam-65	199	61	c1	c1	NOUN
ejpam-65	199	62	=	=	PUNCT
ejpam-65	199	63	d1	d1	PROPN
ejpam-65	199	64	,	,	PUNCT
ejpam-65	199	65	.	.	PUNCT
ejpam-65	199	66	.	.	PUNCT
ejpam-65	200	1	.	.	PUNCT
ejpam-65	201	1	,	,	PUNCT
ejpam-65	201	2	cn	cn	PROPN
ejpam-65	201	3	=	=	PUNCT
ejpam-65	201	4	dn	dn	PROPN
ejpam-65	201	5	,	,	PUNCT
ejpam-65	201	6	by	by	ADP
ejpam-65	201	7	linearly	linearly	ADV
ejpam-65	201	8	independently	independently	ADV
ejpam-65	201	9	of	of	ADP
ejpam-65	201	10	v1	v1	NOUN
ejpam-65	201	11	,	,	PUNCT
ejpam-65	201	12	v2	v2	NOUN
ejpam-65	201	13	,	,	PUNCT
ejpam-65	201	14	.	.	PUNCT
ejpam-65	201	15	.	.	PUNCT
ejpam-65	202	1	.	.	PUNCT
ejpam-65	203	1	,	,	PUNCT
ejpam-65	203	2	vn	vn	PROPN
ejpam-65	203	3	.	.	PROPN
ejpam-65	203	4	remark	remark	PROPN
ejpam-65	203	5	3.3	3.3	NUM
ejpam-65	203	6	.	.	PUNCT
ejpam-65	204	1	clearly	clearly	ADV
ejpam-65	204	2	lemma	lemma	PROPN
ejpam-65	204	3	3.4	3.4	NUM
ejpam-65	204	4	is	be	AUX
ejpam-65	204	5	satisfied	satisfied	ADJ
ejpam-65	204	6	for	for	ADP
ejpam-65	204	7	every	every	DET
ejpam-65	204	8	strongly	strongly	ADV
ejpam-65	204	9	distributive	distributive	ADJ
ejpam-65	204	10	hypervector	hypervector	NOUN
ejpam-65	204	11	space	space	NOUN
ejpam-65	204	12	.	.	PUNCT
ejpam-65	205	1	definition	definition	NOUN
ejpam-65	205	2	3.5	3.5	NUM
ejpam-65	205	3	.	.	PUNCT
ejpam-65	206	1	a	a	DET
ejpam-65	206	2	hypervector	hypervector	NOUN
ejpam-65	206	3	space	space	NOUN
ejpam-65	206	4	v	v	NOUN
ejpam-65	206	5	over	over	ADP
ejpam-65	206	6	k	k	PROPN
ejpam-65	206	7	is	be	AUX
ejpam-65	206	8	said	say	VERB
ejpam-65	206	9	to	to	PART
ejpam-65	206	10	be	be	AUX
ejpam-65	206	11	k	k	NOUN
ejpam-65	206	12	-	-	ADJ
ejpam-65	206	13	invertible	invertible	ADJ
ejpam-65	206	14	or	or	CCONJ
ejpam-65	206	15	shortly	shortly	ADV
ejpam-65	206	16	invertible	invertible	ADJ
ejpam-65	206	17	if	if	SCONJ
ejpam-65	206	18	and	and	CCONJ
ejpam-65	206	19	only	only	ADV
ejpam-65	206	20	if	if	SCONJ
ejpam-65	206	21	u	u	PROPN
ejpam-65	206	22	∈	∈	VERB
ejpam-65	206	23	a	a	DET
ejpam-65	206	24	◦	◦	NOUN
ejpam-65	206	25	v	v	NOUN
ejpam-65	206	26	implies	imply	VERB
ejpam-65	206	27	that	that	SCONJ
ejpam-65	206	28	v	v	NUM
ejpam-65	206	29	∈	∈	PROPN
ejpam-65	206	30	a−1	a−1	PROPN
ejpam-65	206	31	◦	◦	NOUN
ejpam-65	206	32	u.	u.	NOUN
ejpam-65	206	33	theorem	theorem	VERB
ejpam-65	206	34	3.1	3.1	NUM
ejpam-65	206	35	.	.	PUNCT
ejpam-65	207	1	let	let	VERB
ejpam-65	207	2	v	v	PART
ejpam-65	207	3	be	be	AUX
ejpam-65	207	4	invertible	invertible	ADJ
ejpam-65	207	5	.	.	PUNCT
ejpam-65	208	1	then	then	ADV
ejpam-65	208	2	for	for	ADP
ejpam-65	208	3	every	every	DET
ejpam-65	208	4	v1	v1	NOUN
ejpam-65	208	5	,	,	PUNCT
ejpam-65	208	6	...	...	PUNCT
ejpam-65	208	7	,	,	PUNCT
ejpam-65	208	8	vn	vn	X
ejpam-65	208	9	in	in	ADP
ejpam-65	208	10	v	v	NUM
ejpam-65	208	11	,	,	PUNCT
ejpam-65	208	12	either	either	CCONJ
ejpam-65	208	13	v1	v1	NOUN
ejpam-65	208	14	,	,	PUNCT
ejpam-65	208	15	...	...	PUNCT
ejpam-65	208	16	,	,	PUNCT
ejpam-65	208	17	vn	vn	PROPN
ejpam-65	208	18	are	be	AUX
ejpam-65	208	19	linearly	linearly	ADV
ejpam-65	208	20	independent	independent	ADJ
ejpam-65	208	21	or	or	CCONJ
ejpam-65	208	22	for	for	ADP
ejpam-65	208	23	some	some	DET
ejpam-65	208	24	1	1	NUM
ejpam-65	208	25	≤	≤	NUM
ejpam-65	208	26	j	j	PROPN
ejpam-65	208	27	≤	≤	NUM
ejpam-65	208	28	n	n	CCONJ
ejpam-65	208	29	,	,	PUNCT
ejpam-65	208	30	vj	vj	X
ejpam-65	208	31	is	be	AUX
ejpam-65	208	32	in	in	ADP
ejpam-65	208	33	a	a	DET
ejpam-65	208	34	linear	linear	ADJ
ejpam-65	208	35	combination	combination	NOUN
ejpam-65	208	36	of	of	ADP
ejpam-65	208	37	the	the	DET
ejpam-65	208	38	others	other	NOUN
ejpam-65	208	39	.	.	PUNCT
ejpam-65	209	1	proof	proof	NOUN
ejpam-65	209	2	.	.	PUNCT
ejpam-65	210	1	suppose	suppose	VERB
ejpam-65	210	2	that	that	SCONJ
ejpam-65	210	3	v1	v1	NOUN
ejpam-65	210	4	,	,	PUNCT
ejpam-65	210	5	.	.	PUNCT
ejpam-65	210	6	.	.	PUNCT
ejpam-65	210	7	.	.	PUNCT
ejpam-65	211	1	,	,	PUNCT
ejpam-65	211	2	vn	vn	PROPN
ejpam-65	211	3	are	be	AUX
ejpam-65	211	4	not	not	PART
ejpam-65	211	5	linearly	linearly	ADV
ejpam-65	211	6	independent	independent	ADJ
ejpam-65	211	7	.	.	PUNCT
ejpam-65	212	1	then	then	ADV
ejpam-65	212	2	0	0	NUM
ejpam-65	212	3	∈	∈	PROPN
ejpam-65	212	4	c1	c1	PROPN
ejpam-65	212	5	◦	◦	NOUN
ejpam-65	212	6	v1	v1	PROPN
ejpam-65	212	7	+	+	X
ejpam-65	212	8	·	·	PUNCT
ejpam-65	212	9	·	·	PUNCT
ejpam-65	212	10	·	·	PUNCT
ejpam-65	212	11	+	+	NUM
ejpam-65	212	12	cn	cn	VERB
ejpam-65	212	13	◦	◦	NOUN
ejpam-65	212	14	vn	vn	NOUN
ejpam-65	212	15	for	for	ADP
ejpam-65	212	16	some	some	DET
ejpam-65	212	17	c1	c1	NOUN
ejpam-65	212	18	,	,	PUNCT
ejpam-65	212	19	.	.	PUNCT
ejpam-65	212	20	.	.	PUNCT
ejpam-65	212	21	.	.	PUNCT
ejpam-65	213	1	,	,	PUNCT
ejpam-65	213	2	cn	cn	INTJ
ejpam-65	213	3	,	,	PUNCT
ejpam-65	213	4	such	such	ADJ
ejpam-65	213	5	that	that	SCONJ
ejpam-65	213	6	cj	cj	PROPN
ejpam-65	213	7	6=	6=	ADP
ejpam-65	213	8	0	0	NUM
ejpam-65	213	9	for	for	ADP
ejpam-65	213	10	some	some	DET
ejpam-65	213	11	j	j	NOUN
ejpam-65	213	12	,	,	PUNCT
ejpam-65	213	13	1	1	NUM
ejpam-65	213	14	≤	≤	NUM
ejpam-65	213	15	j	j	PROPN
ejpam-65	213	16	≤	≤	PROPN
ejpam-65	213	17	n.	n.	NOUN
ejpam-65	213	18	thus	thus	ADV
ejpam-65	213	19	0	0	X
ejpam-65	214	1	=	=	SYM
ejpam-65	214	2	t1	t1	NOUN
ejpam-65	214	3	+	+	X
ejpam-65	214	4	·	·	PUNCT
ejpam-65	214	5	·	·	PUNCT
ejpam-65	214	6	·	·	PUNCT
ejpam-65	215	1	+	+	NUM
ejpam-65	215	2	tn	tn	NOUN
ejpam-65	215	3	for	for	ADP
ejpam-65	215	4	some	some	DET
ejpam-65	215	5	ti	ti	NOUN
ejpam-65	215	6	∈	∈	PROPN
ejpam-65	215	7	ci	ci	PROPN
ejpam-65	215	8	◦	◦	PROPN
ejpam-65	215	9	vi	vi	PROPN
ejpam-65	215	10	.	.	PUNCT
ejpam-65	215	11	by	by	ADP
ejpam-65	215	12	invertibility	invertibility	NOUN
ejpam-65	215	13	of	of	ADP
ejpam-65	215	14	v	v	NOUN
ejpam-65	215	15	,	,	PUNCT
ejpam-65	215	16	it	it	PRON
ejpam-65	215	17	conclude	conclude	VERB
ejpam-65	215	18	that	that	SCONJ
ejpam-65	215	19	vj	vj	PROPN
ejpam-65	215	20	∈	∈	PROPN
ejpam-65	215	21	cj−1	cj−1	PROPN
ejpam-65	215	22	◦	◦	NOUN
ejpam-65	215	23	tj	tj	NOUN
ejpam-65	215	24	and	and	CCONJ
ejpam-65	215	25	hence	hence	ADV
ejpam-65	215	26	vj	vj	PROPN
ejpam-65	215	27	∈	∈	PROPN
ejpam-65	215	28	cj	cj	X
ejpam-65	215	29	−1	−1	NOUN
ejpam-65	215	30	◦	◦	NOUN
ejpam-65	215	31	(	(	PUNCT
ejpam-65	215	32	−t1	−t1	NOUN
ejpam-65	215	33	−	−	PROPN
ejpam-65	215	34	·	·	PUNCT
ejpam-65	215	35	·	·	PUNCT
ejpam-65	215	36	·	·	PUNCT
ejpam-65	216	1	−	−	NOUN
ejpam-65	216	2	tj−1	tj−1	VERB
ejpam-65	216	3	−	−	PROPN
ejpam-65	216	4	tj+1	tj+1	NOUN
ejpam-65	216	5	−	−	X
ejpam-65	216	6	·	·	PUNCT
ejpam-65	216	7	·	·	PUNCT
ejpam-65	216	8	·	·	PUNCT
ejpam-65	217	1	−	−	NUM
ejpam-65	217	2	tn	tn	NOUN
ejpam-65	217	3	)	)	PUNCT
ejpam-65	218	1	⊆	⊆	NUM
ejpam-65	218	2	cj	cj	NOUN
ejpam-65	218	3	−1	−1	NOUN
ejpam-65	218	4	◦	◦	NOUN
ejpam-65	218	5	(	(	PUNCT
ejpam-65	218	6	−t1	−t1	PROPN
ejpam-65	218	7	)	)	PUNCT
ejpam-65	218	8	+	+	CCONJ
ejpam-65	218	9	·	·	PUNCT
ejpam-65	218	10	·	·	PUNCT
ejpam-65	218	11	·	·	PUNCT
ejpam-65	218	12	+	+	NUM
ejpam-65	218	13	cj	cj	NUM
ejpam-65	218	14	−1	−1	ADV
ejpam-65	218	15	◦	◦	NOUN
ejpam-65	218	16	(	(	PUNCT
ejpam-65	218	17	−tj−1	−tj−1	NUM
ejpam-65	218	18	)	)	PUNCT
ejpam-65	218	19	+	+	PUNCT
ejpam-65	219	1	+	+	PUNCT
ejpam-65	219	2	cj−1	cj−1	PROPN
ejpam-65	219	3	◦	◦	NOUN
ejpam-65	219	4	(	(	PUNCT
ejpam-65	219	5	−tj+1	−tj+1	PROPN
ejpam-65	219	6	)	)	PUNCT
ejpam-65	219	7	+	+	NUM
ejpam-65	219	8	·	·	PUNCT
ejpam-65	219	9	·	·	PUNCT
ejpam-65	219	10	·	·	PUNCT
ejpam-65	219	11	+	+	NUM
ejpam-65	219	12	cj	cj	NUM
ejpam-65	219	13	−1	−1	ADV
ejpam-65	219	14	◦	◦	NOUN
ejpam-65	219	15	(	(	PUNCT
ejpam-65	219	16	−tn	−tn	NOUN
ejpam-65	219	17	)	)	PUNCT
ejpam-65	219	18	=	=	NOUN
ejpam-65	219	19	−cj−1	−cj−1	NUM
ejpam-65	219	20	◦	◦	NOUN
ejpam-65	219	21	t1	t1	NOUN
ejpam-65	219	22	−	−	PROPN
ejpam-65	219	23	·	·	PUNCT
ejpam-65	219	24	·	·	PUNCT
ejpam-65	219	25	·	·	PUNCT
ejpam-65	220	1	−	−	NOUN
ejpam-65	220	2	cj−1	cj−1	NOUN
ejpam-65	220	3	◦	◦	VERB
ejpam-65	220	4	tj−1	tj−1	NOUN
ejpam-65	220	5	−	−	PROPN
ejpam-65	220	6	cj−1	cj−1	NOUN
ejpam-65	220	7	◦	◦	VERB
ejpam-65	220	8	tj+1	tj+1	ADJ
ejpam-65	220	9	−	−	PROPN
ejpam-65	220	10	·	·	PUNCT
ejpam-65	220	11	·	·	PUNCT
ejpam-65	220	12	·	·	PUNCT
ejpam-65	221	1	−	−	NOUN
ejpam-65	222	1	cj−1	cj−1	NOUN
ejpam-65	222	2	◦	◦	VERB
ejpam-65	222	3	tn	tn	NOUN
ejpam-65	222	4	⊆	⊆	NUM
ejpam-65	222	5	−cj−1	−cj−1	NOUN
ejpam-65	222	6	◦	◦	NOUN
ejpam-65	222	7	(	(	PUNCT
ejpam-65	222	8	c1	c1	NOUN
ejpam-65	222	9	◦	◦	NOUN
ejpam-65	222	10	v1)−	v1)−	PROPN
ejpam-65	222	11	·	·	PUNCT
ejpam-65	222	12	·	·	PUNCT
ejpam-65	222	13	·	·	PUNCT
ejpam-65	223	1	−	−	PROPN
ejpam-65	223	2	cj−1	cj−1	NOUN
ejpam-65	223	3	◦	◦	NOUN
ejpam-65	223	4	(	(	PUNCT
ejpam-65	223	5	cj−1	cj−1	NOUN
ejpam-65	223	6	◦	◦	NOUN
ejpam-65	223	7	vj−1)−	vj−1)−	NOUN
ejpam-65	223	8	−cj−1	−cj−1	NUM
ejpam-65	223	9	◦	◦	NOUN
ejpam-65	223	10	(	(	PUNCT
ejpam-65	223	11	cj+1	cj+1	NUM
ejpam-65	223	12	◦	◦	VERB
ejpam-65	223	13	vj+1)−	vj+1)−	PROPN
ejpam-65	223	14	·	·	PUNCT
ejpam-65	223	15	·	·	PUNCT
ejpam-65	223	16	·	·	PUNCT
ejpam-65	224	1	−	−	NUM
ejpam-65	224	2	cj−1	cj−1	NOUN
ejpam-65	224	3	◦	◦	NOUN
ejpam-65	224	4	(	(	PUNCT
ejpam-65	224	5	cn	cn	PROPN
ejpam-65	224	6	◦	◦	PROPN
ejpam-65	224	7	vn	vn	PROPN
ejpam-65	224	8	)	)	PUNCT
ejpam-65	224	9	=	=	SYM
ejpam-65	224	10	(	(	PUNCT
ejpam-65	224	11	−cj−1c1	−cj−1c1	NOUN
ejpam-65	224	12	)	)	PUNCT
ejpam-65	224	13	◦	◦	NOUN
ejpam-65	224	14	v1	v1	NOUN
ejpam-65	224	15	+	+	X
ejpam-65	224	16	·	·	PUNCT
ejpam-65	224	17	·	·	PUNCT
ejpam-65	224	18	·	·	PUNCT
ejpam-65	224	19	+	+	CCONJ
ejpam-65	224	20	(	(	PUNCT
ejpam-65	224	21	−cj−1cj−1	−cj−1cj−1	NUM
ejpam-65	224	22	)	)	PUNCT
ejpam-65	224	23	◦	◦	NOUN
ejpam-65	224	24	vj−1	vj−1	PROPN
ejpam-65	224	25	+	+	CCONJ
ejpam-65	224	26	+	+	PROPN
ejpam-65	224	27	(	(	PUNCT
ejpam-65	224	28	−cj−1cj+1	−cj−1cj+1	PRON
ejpam-65	224	29	)	)	PUNCT
ejpam-65	224	30	◦	◦	NOUN
ejpam-65	224	31	vj+1	vj+1	NOUN
ejpam-65	224	32	+	+	NOUN
ejpam-65	224	33	·	·	PUNCT
ejpam-65	224	34	·	·	PUNCT
ejpam-65	224	35	·	·	PUNCT
ejpam-65	224	36	+	+	CCONJ
ejpam-65	224	37	(	(	PUNCT
ejpam-65	224	38	−cj−1cn	−cj−1cn	NOUN
ejpam-65	224	39	)	)	PUNCT
ejpam-65	224	40	◦	◦	NOUN
ejpam-65	224	41	vn	vn	NOUN
ejpam-65	224	42	,	,	PUNCT
ejpam-65	224	43	thus	thus	ADV
ejpam-65	224	44	vj	vj	X
ejpam-65	224	45	is	be	AUX
ejpam-65	224	46	in	in	ADP
ejpam-65	224	47	a	a	DET
ejpam-65	224	48	linear	linear	ADJ
ejpam-65	224	49	combination	combination	NOUN
ejpam-65	224	50	of	of	ADP
ejpam-65	224	51	v1	v1	NOUN
ejpam-65	224	52	,	,	PUNCT
ejpam-65	224	53	.	.	PUNCT
ejpam-65	224	54	.	.	PUNCT
ejpam-65	225	1	.	.	PUNCT
ejpam-65	226	1	,	,	PUNCT
ejpam-65	226	2	vj−1	vj−1	PROPN
ejpam-65	226	3	,	,	PUNCT
ejpam-65	226	4	vj+1	vj+1	PROPN
ejpam-65	226	5	,	,	PUNCT
ejpam-65	226	6	.	.	PUNCT
ejpam-65	226	7	.	.	PUNCT
ejpam-65	226	8	.	.	PUNCT
ejpam-65	227	1	,	,	PUNCT
ejpam-65	227	2	vn	vn	INTJ
ejpam-65	227	3	,	,	PUNCT
ejpam-65	227	4	as	as	SCONJ
ejpam-65	227	5	desired	desire	VERB
ejpam-65	227	6	.	.	PUNCT
ejpam-65	228	1	r.	r.	PROPN
ejpam-65	228	2	ameri	ameri	PROPN
ejpam-65	228	3	,	,	PUNCT
ejpam-65	228	4	o.r	o.r	PROPN
ejpam-65	228	5	.	.	PROPN
ejpam-65	228	6	dehghan	dehghan	PROPN
ejpam-65	228	7	/	/	SYM
ejpam-65	228	8	eur	eur	PROPN
ejpam-65	228	9	.	.	PUNCT
ejpam-65	229	1	j.	j.	PROPN
ejpam-65	229	2	pure	pure	PROPN
ejpam-65	229	3	appl	appl	PROPN
ejpam-65	229	4	.	.	PROPN
ejpam-65	229	5	math	math	PROPN
ejpam-65	229	6	,	,	PUNCT
ejpam-65	229	7	1	1	NUM
ejpam-65	229	8	(	(	PUNCT
ejpam-65	229	9	2008	2008	NUM
ejpam-65	229	10	)	)	PUNCT
ejpam-65	229	11	,	,	PUNCT
ejpam-65	229	12	(	(	PUNCT
ejpam-65	229	13	32	32	NUM
ejpam-65	229	14	-	-	SYM
ejpam-65	229	15	50	50	NUM
ejpam-65	229	16	)	)	PUNCT
ejpam-65	229	17	39	39	NUM
ejpam-65	229	18	lemma	lemma	PROPN
ejpam-65	229	19	3.5	3.5	NUM
ejpam-65	229	20	.	.	PUNCT
ejpam-65	230	1	let	let	VERB
ejpam-65	230	2	v	v	PART
ejpam-65	230	3	be	be	AUX
ejpam-65	230	4	strongly	strongly	ADV
ejpam-65	230	5	left	leave	VERB
ejpam-65	230	6	distributive	distributive	ADJ
ejpam-65	230	7	and	and	CCONJ
ejpam-65	230	8	invertible	invertible	ADJ
ejpam-65	230	9	.	.	PUNCT
ejpam-65	231	1	if	if	SCONJ
ejpam-65	231	2	w	w	NOUN
ejpam-65	231	3	is	be	AUX
ejpam-65	231	4	a	a	DET
ejpam-65	231	5	subhyperspace	subhyperspace	NOUN
ejpam-65	231	6	of	of	ADP
ejpam-65	231	7	v	v	NUM
ejpam-65	231	8	span	span	NOUN
ejpam-65	231	9	by	by	ADP
ejpam-65	231	10	β	β	X
ejpam-65	231	11	=	=	SYM
ejpam-65	231	12	{	{	PUNCT
ejpam-65	231	13	v1	v1	PROPN
ejpam-65	231	14	,	,	PUNCT
ejpam-65	231	15	.	.	PUNCT
ejpam-65	231	16	.	.	PUNCT
ejpam-65	232	1	.	.	PUNCT
ejpam-65	233	1	,	,	PUNCT
ejpam-65	233	2	vn	vn	PROPN
ejpam-65	233	3	}	}	PUNCT
ejpam-65	233	4	,	,	PUNCT
ejpam-65	233	5	such	such	ADJ
ejpam-65	233	6	that	that	SCONJ
ejpam-65	233	7	β	β	NOUN
ejpam-65	233	8	is	be	AUX
ejpam-65	233	9	not	not	PART
ejpam-65	233	10	independentless	independentless	NOUN
ejpam-65	233	11	,	,	PUNCT
ejpam-65	233	12	then	then	ADV
ejpam-65	233	13	β	β	PROPN
ejpam-65	233	14	has	have	VERB
ejpam-65	233	15	a	a	DET
ejpam-65	233	16	linearly	linearly	ADV
ejpam-65	233	17	independent	independent	ADJ
ejpam-65	233	18	subset	subset	NOUN
ejpam-65	233	19	such	such	ADJ
ejpam-65	233	20	that	that	SCONJ
ejpam-65	233	21	spanning	span	VERB
ejpam-65	233	22	w	w	X
ejpam-65	233	23	.	.	PUNCT
ejpam-65	234	1	proof	proof	NOUN
ejpam-65	234	2	.	.	PUNCT
ejpam-65	235	1	since	since	SCONJ
ejpam-65	235	2	β	β	PROPN
ejpam-65	235	3	is	be	AUX
ejpam-65	235	4	not	not	PART
ejpam-65	235	5	independentless	independentless	ADJ
ejpam-65	235	6	,	,	PUNCT
ejpam-65	235	7	so	so	SCONJ
ejpam-65	235	8	β	β	PROPN
ejpam-65	235	9	has	have	VERB
ejpam-65	235	10	a	a	DET
ejpam-65	235	11	linearly	linearly	ADV
ejpam-65	235	12	independent	independent	ADJ
ejpam-65	235	13	subset	subset	NOUN
ejpam-65	235	14	{	{	PUNCT
ejpam-65	235	15	v1	v1	NOUN
ejpam-65	235	16	,	,	PUNCT
ejpam-65	235	17	.	.	PUNCT
ejpam-65	235	18	.	.	PUNCT
ejpam-65	236	1	.	.	PUNCT
ejpam-65	237	1	,	,	PUNCT
ejpam-65	237	2	vk	vk	ADP
ejpam-65	237	3	}	}	PUNCT
ejpam-65	237	4	.	.	PUNCT
ejpam-65	238	1	now	now	ADV
ejpam-65	238	2	if	if	SCONJ
ejpam-65	238	3	k	k	PROPN
ejpam-65	238	4	=	=	SYM
ejpam-65	238	5	n	n	CCONJ
ejpam-65	238	6	,	,	PUNCT
ejpam-65	238	7	we	we	PRON
ejpam-65	238	8	are	be	AUX
ejpam-65	238	9	done	do	VERB
ejpam-65	238	10	.	.	PUNCT
ejpam-65	239	1	if	if	SCONJ
ejpam-65	239	2	not	not	PART
ejpam-65	239	3	,	,	PUNCT
ejpam-65	239	4	weed	weed	VERB
ejpam-65	239	5	out	out	ADP
ejpam-65	239	6	from	from	ADP
ejpam-65	239	7	this	this	DET
ejpam-65	239	8	set	set	NOUN
ejpam-65	239	9	the	the	DET
ejpam-65	239	10	first	first	ADJ
ejpam-65	239	11	vj	vj	NOUN
ejpam-65	239	12	,	,	PUNCT
ejpam-65	239	13	which	which	PRON
ejpam-65	239	14	is	be	AUX
ejpam-65	239	15	in	in	ADP
ejpam-65	239	16	a	a	DET
ejpam-65	239	17	linear	linear	ADJ
ejpam-65	239	18	combination	combination	NOUN
ejpam-65	239	19	of	of	ADP
ejpam-65	239	20	others	other	NOUN
ejpam-65	239	21	.	.	PUNCT
ejpam-65	240	1	it	it	PRON
ejpam-65	240	2	is	be	AUX
ejpam-65	240	3	clearly	clearly	ADV
ejpam-65	240	4	that	that	SCONJ
ejpam-65	240	5	j	j	PROPN
ejpam-65	240	6	>	>	X
ejpam-65	240	7	k.	k.	PROPN
ejpam-65	240	8	let	let	VERB
ejpam-65	240	9	vj	vj	PROPN
ejpam-65	240	10	∈	∈	PROPN
ejpam-65	240	11	b1	b1	PROPN
ejpam-65	240	12	◦	◦	NOUN
ejpam-65	240	13	v1	v1	PROPN
ejpam-65	240	14	+	+	X
ejpam-65	240	15	·	·	PUNCT
ejpam-65	240	16	·	·	PUNCT
ejpam-65	240	17	·	·	PUNCT
ejpam-65	241	1	+	+	NUM
ejpam-65	241	2	bj−1	bj−1	PROPN
ejpam-65	241	3	◦	◦	NOUN
ejpam-65	241	4	vj−1	vj−1	PROPN
ejpam-65	241	5	+	+	CCONJ
ejpam-65	241	6	bj+1	bj+1	NUM
ejpam-65	241	7	◦	◦	NOUN
ejpam-65	241	8	vj+1	vj+1	PRON
ejpam-65	241	9	+	+	NOUN
ejpam-65	241	10	·	·	PUNCT
ejpam-65	241	11	·	·	PUNCT
ejpam-65	241	12	·	·	PUNCT
ejpam-65	241	13	+	+	NUM
ejpam-65	241	14	bn	bn	NUM
ejpam-65	241	15	◦	◦	NOUN
ejpam-65	241	16	vn	vn	NOUN
ejpam-65	241	17	.	.	PUNCT
ejpam-65	242	1	(	(	PUNCT
ejpam-65	242	2	3.2	3.2	NUM
ejpam-65	242	3	)	)	PUNCT
ejpam-65	242	4	the	the	DET
ejpam-65	242	5	subset	subset	NOUN
ejpam-65	242	6	so	so	ADV
ejpam-65	242	7	constructed	construct	VERB
ejpam-65	242	8	,	,	PUNCT
ejpam-65	242	9	v1	v1	NOUN
ejpam-65	242	10	,	,	PUNCT
ejpam-65	242	11	.	.	PUNCT
ejpam-65	242	12	.	.	PUNCT
ejpam-65	242	13	.	.	PUNCT
ejpam-65	243	1	,	,	PUNCT
ejpam-65	243	2	vk	vk	VERB
ejpam-65	243	3	,	,	PUNCT
ejpam-65	243	4	.	.	PUNCT
ejpam-65	243	5	.	.	PUNCT
ejpam-65	244	1	.	.	PUNCT
ejpam-65	245	1	,	,	PUNCT
ejpam-65	245	2	vj−1	vj−1	PROPN
ejpam-65	245	3	,	,	PUNCT
ejpam-65	245	4	vj+1	vj+1	PROPN
ejpam-65	245	5	,	,	PUNCT
ejpam-65	245	6	.	.	PUNCT
ejpam-65	245	7	.	.	PUNCT
ejpam-65	245	8	.	.	PUNCT
ejpam-65	246	1	,	,	PUNCT
ejpam-65	246	2	vn	vn	PROPN
ejpam-65	246	3	has	have	VERB
ejpam-65	246	4	n	n	CCONJ
ejpam-65	246	5	−	−	NUM
ejpam-65	246	6	1	1	NUM
ejpam-65	246	7	elements	element	NOUN
ejpam-65	246	8	and	and	CCONJ
ejpam-65	246	9	its	its	PRON
ejpam-65	246	10	linear	linear	ADJ
ejpam-65	246	11	span	span	NOUN
ejpam-65	246	12	is	be	AUX
ejpam-65	246	13	contained	contain	VERB
ejpam-65	246	14	in	in	ADP
ejpam-65	246	15	w	w	PROPN
ejpam-65	246	16	.	.	PUNCT
ejpam-65	247	1	because	because	SCONJ
ejpam-65	247	2	if	if	SCONJ
ejpam-65	247	3	v	v	NUM
ejpam-65	247	4	∈	∈	PROPN
ejpam-65	248	1	n∑	n∑	NOUN
ejpam-65	249	1	i=1	i=1	X
ejpam-65	250	1	i	i	PRON
ejpam-65	250	2	6	6	NUM
ejpam-65	250	3	=	=	X
ejpam-65	250	4	j	j	NOUN
ejpam-65	250	5	ai	ai	VERB
ejpam-65	250	6	◦	◦	NOUN
ejpam-65	250	7	vi	vi	PROPN
ejpam-65	250	8	,	,	PUNCT
ejpam-65	250	9	then	then	ADV
ejpam-65	250	10	since	since	SCONJ
ejpam-65	250	11	0	0	NUM
ejpam-65	250	12	∈	∈	NOUN
ejpam-65	250	13	0	0	NUM
ejpam-65	250	14	◦	◦	NOUN
ejpam-65	250	15	vj	vj	NOUN
ejpam-65	250	16	,	,	PUNCT
ejpam-65	250	17	it	it	PRON
ejpam-65	250	18	follows	follow	VERB
ejpam-65	250	19	that	that	SCONJ
ejpam-65	250	20	v	v	ADP
ejpam-65	250	21	∈w	∈w	NOUN
ejpam-65	250	22	.	.	PUNCT
ejpam-65	251	1	however	however	ADV
ejpam-65	251	2	,	,	PUNCT
ejpam-65	251	3	we	we	PRON
ejpam-65	251	4	claim	claim	VERB
ejpam-65	251	5	that	that	SCONJ
ejpam-65	251	6	it	it	PRON
ejpam-65	251	7	is	be	AUX
ejpam-65	251	8	actually	actually	ADV
ejpam-65	251	9	equal	equal	ADJ
ejpam-65	251	10	to	to	ADP
ejpam-65	251	11	w	w	NOUN
ejpam-65	251	12	:	:	PUNCT
ejpam-65	251	13	for	for	ADP
ejpam-65	251	14	every	every	DET
ejpam-65	251	15	w	w	PROPN
ejpam-65	251	16	∈	∈	PROPN
ejpam-65	251	17	w	w	NOUN
ejpam-65	251	18	,	,	PUNCT
ejpam-65	251	19	there	there	PRON
ejpam-65	251	20	exist	exist	VERB
ejpam-65	251	21	a1	a1	NOUN
ejpam-65	251	22	,	,	PUNCT
ejpam-65	251	23	a2	a2	PROPN
ejpam-65	251	24	,	,	PUNCT
ejpam-65	251	25	.	.	PUNCT
ejpam-65	251	26	.	.	PUNCT
ejpam-65	252	1	.	.	PUNCT
ejpam-65	253	1	,	,	PUNCT
ejpam-65	253	2	an	an	PRON
ejpam-65	253	3	in	in	ADP
ejpam-65	253	4	k	k	PROPN
ejpam-65	253	5	such	such	ADJ
ejpam-65	253	6	that	that	SCONJ
ejpam-65	253	7	w	w	PROPN
ejpam-65	253	8	∈	∈	PROPN
ejpam-65	254	1	n∑	n∑	NOUN
ejpam-65	255	1	i=1	i=1	PROPN
ejpam-65	255	2	ai	ai	VERB
ejpam-65	255	3	◦	◦	NOUN
ejpam-65	255	4	vi	vi	NOUN
ejpam-65	255	5	,	,	PUNCT
ejpam-65	255	6	thus	thus	ADV
ejpam-65	255	7	by	by	ADP
ejpam-65	255	8	(	(	PUNCT
ejpam-65	255	9	2	2	X
ejpam-65	255	10	)	)	PUNCT
ejpam-65	255	11	we	we	PRON
ejpam-65	255	12	have	have	VERB
ejpam-65	255	13	w	w	PROPN
ejpam-65	255	14	∈	∈	NOUN
ejpam-65	255	15	a1	a1	NOUN
ejpam-65	255	16	◦	◦	NOUN
ejpam-65	255	17	v1	v1	NOUN
ejpam-65	255	18	+	+	X
ejpam-65	255	19	·	·	PUNCT
ejpam-65	255	20	·	·	PUNCT
ejpam-65	256	1	·	·	PUNCT
ejpam-65	256	2	+	+	NUM
ejpam-65	256	3	aj	aj	PROPN
ejpam-65	256	4	◦	◦	PROPN
ejpam-65	256	5	(	(	PUNCT
ejpam-65	256	6	b1	b1	VERB
ejpam-65	256	7	◦	◦	NOUN
ejpam-65	256	8	v1	v1	PROPN
ejpam-65	256	9	+	+	X
ejpam-65	256	10	·	·	PUNCT
ejpam-65	256	11	·	·	PUNCT
ejpam-65	256	12	·	·	PUNCT
ejpam-65	257	1	+	+	NUM
ejpam-65	257	2	bj−1	bj−1	PROPN
ejpam-65	257	3	◦	◦	NOUN
ejpam-65	257	4	vj−1	vj−1	PROPN
ejpam-65	257	5	+	+	CCONJ
ejpam-65	257	6	+	+	ADJ
ejpam-65	257	7	bj+1	bj+1	ADJ
ejpam-65	257	8	◦	◦	NOUN
ejpam-65	257	9	vj+1	vj+1	NUM
ejpam-65	257	10	+	+	PUNCT
ejpam-65	257	11	·	·	PUNCT
ejpam-65	257	12	·	·	PUNCT
ejpam-65	257	13	·	·	PUNCT
ejpam-65	257	14	+	+	NUM
ejpam-65	257	15	bn	bn	NUM
ejpam-65	257	16	◦	◦	NOUN
ejpam-65	257	17	vn	vn	NOUN
ejpam-65	257	18	)	)	PUNCT
ejpam-65	257	19	+	+	CCONJ
ejpam-65	257	20	·	·	PUNCT
ejpam-65	257	21	·	·	PUNCT
ejpam-65	257	22	·	·	PUNCT
ejpam-65	257	23	+	+	CCONJ
ejpam-65	257	24	an	an	DET
ejpam-65	257	25	◦	◦	NOUN
ejpam-65	257	26	vn	vn	X
ejpam-65	257	27	=	=	SYM
ejpam-65	257	28	a1	a1	PROPN
ejpam-65	257	29	◦	◦	NOUN
ejpam-65	257	30	v1	v1	NOUN
ejpam-65	257	31	+	+	X
ejpam-65	257	32	·	·	PUNCT
ejpam-65	257	33	·	·	PUNCT
ejpam-65	257	34	·	·	PUNCT
ejpam-65	257	35	+	+	NUM
ejpam-65	257	36	ajb1	ajb1	ADJ
ejpam-65	257	37	◦	◦	NOUN
ejpam-65	257	38	v1	v1	NOUN
ejpam-65	257	39	+	+	X
ejpam-65	257	40	·	·	PUNCT
ejpam-65	257	41	·	·	PUNCT
ejpam-65	257	42	·	·	PUNCT
ejpam-65	257	43	+	+	NUM
ejpam-65	257	44	ajbj−1	ajbj−1	CCONJ
ejpam-65	257	45	◦	◦	NOUN
ejpam-65	257	46	vj−1	vj−1	PROPN
ejpam-65	257	47	+	+	CCONJ
ejpam-65	257	48	+	+	ADJ
ejpam-65	257	49	ajbj+1	ajbj+1	ADJ
ejpam-65	257	50	◦	◦	NOUN
ejpam-65	257	51	vj+1	vj+1	PRON
ejpam-65	257	52	+	+	CCONJ
ejpam-65	257	53	·	·	PUNCT
ejpam-65	257	54	·	·	PUNCT
ejpam-65	257	55	·	·	PUNCT
ejpam-65	257	56	+	+	NUM
ejpam-65	257	57	ajbn	ajbn	PROPN
ejpam-65	257	58	◦	◦	PROPN
ejpam-65	257	59	vn	vn	PROPN
ejpam-65	257	60	+	+	X
ejpam-65	257	61	·	·	PUNCT
ejpam-65	257	62	·	·	PUNCT
ejpam-65	257	63	·	·	PUNCT
ejpam-65	257	64	+	+	CCONJ
ejpam-65	257	65	an	an	DET
ejpam-65	257	66	◦	◦	NOUN
ejpam-65	257	67	vn	vn	NOUN
ejpam-65	257	68	=	=	SYM
ejpam-65	257	69	(	(	PUNCT
ejpam-65	257	70	a1	a1	NOUN
ejpam-65	257	71	+	+	CCONJ
ejpam-65	257	72	ajb1	ajb1	ADJ
ejpam-65	257	73	)	)	PUNCT
ejpam-65	257	74	◦	◦	NOUN
ejpam-65	257	75	v1	v1	NOUN
ejpam-65	257	76	+	+	X
ejpam-65	257	77	·	·	PUNCT
ejpam-65	257	78	·	·	PUNCT
ejpam-65	257	79	·	·	PUNCT
ejpam-65	257	80	+	+	CCONJ
ejpam-65	257	81	(	(	PUNCT
ejpam-65	257	82	aj−1	aj−1	NOUN
ejpam-65	257	83	+	+	NUM
ejpam-65	257	84	ajbj−1	ajbj−1	PROPN
ejpam-65	257	85	)	)	PUNCT
ejpam-65	257	86	◦	◦	NOUN
ejpam-65	257	87	vj−1	vj−1	PROPN
ejpam-65	257	88	+	+	CCONJ
ejpam-65	257	89	+	+	PROPN
ejpam-65	257	90	(	(	PUNCT
ejpam-65	257	91	aj+1	aj+1	PRON
ejpam-65	257	92	+	+	ADJ
ejpam-65	257	93	ajbj+1	ajbj+1	PROPN
ejpam-65	257	94	)	)	PUNCT
ejpam-65	257	95	◦	◦	NOUN
ejpam-65	257	96	vj+1	vj+1	PRON
ejpam-65	257	97	+	+	NOUN
ejpam-65	257	98	·	·	PUNCT
ejpam-65	257	99	·	·	PUNCT
ejpam-65	257	100	·	·	PUNCT
ejpam-65	258	1	+	+	CCONJ
ejpam-65	258	2	(	(	PUNCT
ejpam-65	258	3	an	an	DET
ejpam-65	258	4	+	+	NOUN
ejpam-65	258	5	ajbn	ajbn	NOUN
ejpam-65	258	6	)	)	PUNCT
ejpam-65	258	7	◦	◦	NOUN
ejpam-65	258	8	vn	vn	NOUN
ejpam-65	258	9	,	,	PUNCT
ejpam-65	258	10	that	that	ADV
ejpam-65	258	11	is	is	ADV
ejpam-65	258	12	,	,	PUNCT
ejpam-65	258	13	w	w	NOUN
ejpam-65	258	14	is	be	AUX
ejpam-65	258	15	in	in	ADP
ejpam-65	258	16	a	a	DET
ejpam-65	258	17	linear	linear	ADJ
ejpam-65	258	18	combination	combination	NOUN
ejpam-65	258	19	of	of	ADP
ejpam-65	258	20	v1	v1	NOUN
ejpam-65	258	21	,	,	PUNCT
ejpam-65	258	22	.	.	PUNCT
ejpam-65	258	23	.	.	PUNCT
ejpam-65	258	24	.	.	PUNCT
ejpam-65	259	1	,	,	PUNCT
ejpam-65	259	2	vk	vk	VERB
ejpam-65	259	3	,	,	PUNCT
ejpam-65	259	4	.	.	PUNCT
ejpam-65	259	5	.	.	PUNCT
ejpam-65	260	1	.	.	PUNCT
ejpam-65	261	1	,	,	PUNCT
ejpam-65	261	2	vj−1	vj−1	PROPN
ejpam-65	261	3	,	,	PUNCT
ejpam-65	261	4	vj+1	vj+1	PROPN
ejpam-65	261	5	,	,	PUNCT
ejpam-65	261	6	.	.	PUNCT
ejpam-65	261	7	.	.	PUNCT
ejpam-65	261	8	.	.	PUNCT
ejpam-65	262	1	,	,	PUNCT
ejpam-65	262	2	vn	vn	X
ejpam-65	262	3	.	.	PUNCT
ejpam-65	263	1	continuing	continue	VERB
ejpam-65	263	2	this	this	PRON
ejpam-65	263	3	weeding	weed	VERB
ejpam-65	263	4	out	out	ADP
ejpam-65	263	5	process	process	NOUN
ejpam-65	263	6	,	,	PUNCT
ejpam-65	263	7	we	we	PRON
ejpam-65	263	8	reach	reach	VERB
ejpam-65	263	9	a	a	DET
ejpam-65	263	10	subset	subset	NOUN
ejpam-65	263	11	v1	v1	NOUN
ejpam-65	263	12	,	,	PUNCT
ejpam-65	263	13	.	.	PUNCT
ejpam-65	263	14	.	.	PUNCT
ejpam-65	264	1	.	.	PUNCT
ejpam-65	265	1	,	,	PUNCT
ejpam-65	265	2	vk	vk	INTJ
ejpam-65	265	3	,	,	PUNCT
ejpam-65	265	4	vi1	vi1	INTJ
ejpam-65	265	5	,	,	PUNCT
ejpam-65	265	6	.	.	PUNCT
ejpam-65	265	7	.	.	PUNCT
ejpam-65	266	1	.	.	PUNCT
ejpam-65	267	1	,	,	PUNCT
ejpam-65	267	2	vir	vir	X
ejpam-65	267	3	whose	whose	DET
ejpam-65	267	4	linear	linear	ADJ
ejpam-65	267	5	span	span	NOUN
ejpam-65	267	6	is	be	AUX
ejpam-65	267	7	still	still	ADV
ejpam-65	267	8	w	w	NOUN
ejpam-65	267	9	but	but	CCONJ
ejpam-65	267	10	in	in	ADP
ejpam-65	267	11	which	which	PRON
ejpam-65	267	12	no	no	DET
ejpam-65	267	13	element	element	NOUN
ejpam-65	267	14	is	be	AUX
ejpam-65	267	15	in	in	ADP
ejpam-65	267	16	a	a	DET
ejpam-65	267	17	linear	linear	ADJ
ejpam-65	267	18	combination	combination	NOUN
ejpam-65	267	19	of	of	ADP
ejpam-65	267	20	others	other	NOUN
ejpam-65	267	21	.	.	PUNCT
ejpam-65	268	1	by	by	ADP
ejpam-65	268	2	theorem	theorem	NOUN
ejpam-65	268	3	3.1	3.1	NUM
ejpam-65	268	4	the	the	DET
ejpam-65	268	5	elements	element	NOUN
ejpam-65	268	6	v1	v1	NOUN
ejpam-65	268	7	,	,	PUNCT
ejpam-65	268	8	.	.	PUNCT
ejpam-65	268	9	.	.	PUNCT
ejpam-65	268	10	.	.	PUNCT
ejpam-65	269	1	,	,	PUNCT
ejpam-65	269	2	vk	vk	INTJ
ejpam-65	269	3	,	,	PUNCT
ejpam-65	269	4	vi1	vi1	INTJ
ejpam-65	269	5	,	,	PUNCT
ejpam-65	269	6	.	.	PUNCT
ejpam-65	269	7	.	.	PUNCT
ejpam-65	270	1	.	.	PUNCT
ejpam-65	271	1	,	,	PUNCT
ejpam-65	271	2	vir	vir	PROPN
ejpam-65	271	3	must	must	AUX
ejpam-65	271	4	be	be	AUX
ejpam-65	271	5	linearly	linearly	ADV
ejpam-65	271	6	independent	independent	ADJ
ejpam-65	271	7	.	.	PUNCT
ejpam-65	272	1	corollary	corollary	ADJ
ejpam-65	272	2	3.2	3.2	NUM
ejpam-65	272	3	.	.	PUNCT
ejpam-65	273	1	let	let	VERB
ejpam-65	273	2	v	v	PART
ejpam-65	273	3	be	be	AUX
ejpam-65	273	4	strongly	strongly	ADV
ejpam-65	273	5	left	leave	VERB
ejpam-65	273	6	distributive	distributive	ADJ
ejpam-65	273	7	and	and	CCONJ
ejpam-65	273	8	invertible	invertible	ADJ
ejpam-65	273	9	.	.	PUNCT
ejpam-65	274	1	then	then	ADV
ejpam-65	274	2	every	every	DET
ejpam-65	274	3	non	non	ADJ
ejpam-65	274	4	independentless	independentless	NOUN
ejpam-65	274	5	spanning	span	VERB
ejpam-65	274	6	subset	subset	NOUN
ejpam-65	274	7	of	of	ADP
ejpam-65	274	8	v	v	NOUN
ejpam-65	274	9	containing	contain	VERB
ejpam-65	274	10	a	a	DET
ejpam-65	274	11	basis	basis	NOUN
ejpam-65	274	12	.	.	PUNCT
ejpam-65	275	1	corollary	corollary	ADJ
ejpam-65	275	2	3.3	3.3	NUM
ejpam-65	275	3	.	.	PUNCT
ejpam-65	276	1	let	let	VERB
ejpam-65	276	2	v	v	PART
ejpam-65	276	3	be	be	AUX
ejpam-65	276	4	strongly	strongly	ADV
ejpam-65	276	5	left	leave	VERB
ejpam-65	276	6	distributive	distributive	ADJ
ejpam-65	276	7	and	and	CCONJ
ejpam-65	276	8	invertible	invertible	ADJ
ejpam-65	276	9	.	.	PUNCT
ejpam-65	277	1	if	if	SCONJ
ejpam-65	277	2	v	v	NOUN
ejpam-65	277	3	containing	contain	VERB
ejpam-65	277	4	a	a	DET
ejpam-65	277	5	finite	finite	NOUN
ejpam-65	277	6	spanning	span	VERB
ejpam-65	277	7	set	set	NOUN
ejpam-65	277	8	which	which	PRON
ejpam-65	277	9	is	be	AUX
ejpam-65	277	10	not	not	PART
ejpam-65	277	11	independentless	independentless	NOUN
ejpam-65	277	12	,	,	PUNCT
ejpam-65	277	13	then	then	ADV
ejpam-65	277	14	v	v	NOUN
ejpam-65	277	15	is	be	AUX
ejpam-65	277	16	finite	finite	ADJ
ejpam-65	277	17	dimensional	dimensional	ADJ
ejpam-65	277	18	.	.	PUNCT
ejpam-65	278	1	proof	proof	NOUN
ejpam-65	278	2	.	.	PUNCT
ejpam-65	279	1	suppose	suppose	VERB
ejpam-65	279	2	v	v	X
ejpam-65	279	3	=	=	SYM
ejpam-65	279	4	〈	〈	PROPN
ejpam-65	279	5	v1	v1	NOUN
ejpam-65	279	6	,	,	PUNCT
ejpam-65	279	7	v2	v2	PROPN
ejpam-65	279	8	,	,	PUNCT
ejpam-65	279	9	...	...	PUNCT
ejpam-65	279	10	,	,	PUNCT
ejpam-65	279	11	vn	vn	VERB
ejpam-65	279	12	〉	〉	NUM
ejpam-65	279	13	such	such	ADJ
ejpam-65	279	14	that	that	DET
ejpam-65	279	15	v1	v1	NOUN
ejpam-65	279	16	,	,	PUNCT
ejpam-65	279	17	...	...	PUNCT
ejpam-65	279	18	,	,	PUNCT
ejpam-65	279	19	vk	vk	NOUN
ejpam-65	279	20	are	be	AUX
ejpam-65	279	21	not	not	PART
ejpam-65	279	22	independentless	independentless	NOUN
ejpam-65	279	23	.	.	PUNCT
ejpam-65	280	1	then	then	ADV
ejpam-65	280	2	by	by	ADP
ejpam-65	280	3	lemma	lemma	PROPN
ejpam-65	280	4	3.5	3.5	NUM
ejpam-65	280	5	there	there	ADV
ejpam-65	280	6	exist	exist	VERB
ejpam-65	280	7	linearly	linearly	ADV
ejpam-65	280	8	independent	independent	ADJ
ejpam-65	280	9	vectors	vector	NOUN
ejpam-65	280	10	v1	v1	NOUN
ejpam-65	280	11	,	,	PUNCT
ejpam-65	280	12	.	.	PUNCT
ejpam-65	280	13	.	.	PUNCT
ejpam-65	281	1	.	.	PUNCT
ejpam-65	282	1	,	,	PUNCT
ejpam-65	282	2	vk	vk	INTJ
ejpam-65	282	3	,	,	PUNCT
ejpam-65	282	4	vi1	vi1	INTJ
ejpam-65	282	5	,	,	PUNCT
ejpam-65	282	6	.	.	PUNCT
ejpam-65	282	7	.	.	PUNCT
ejpam-65	283	1	.	.	PUNCT
ejpam-65	284	1	,	,	PUNCT
ejpam-65	284	2	vir	vir	X
ejpam-65	284	3	such	such	ADJ
ejpam-65	284	4	that	that	DET
ejpam-65	284	5	v	v	NOUN
ejpam-65	284	6	=	=	SYM
ejpam-65	284	7	〈	〈	NOUN
ejpam-65	284	8	v1	v1	NOUN
ejpam-65	284	9	,	,	PUNCT
ejpam-65	284	10	.	.	PUNCT
ejpam-65	284	11	.	.	PUNCT
ejpam-65	285	1	.	.	PUNCT
ejpam-65	286	1	,	,	PUNCT
ejpam-65	286	2	vk	vk	INTJ
ejpam-65	286	3	,	,	PUNCT
ejpam-65	286	4	vi1	vi1	INTJ
ejpam-65	286	5	,	,	PUNCT
ejpam-65	286	6	.	.	PUNCT
ejpam-65	286	7	.	.	PUNCT
ejpam-65	286	8	.	.	PUNCT
ejpam-65	287	1	,	,	PUNCT
ejpam-65	287	2	vir	vir	PROPN
ejpam-65	287	3	〉	〉	PROPN
ejpam-65	287	4	.	.	PUNCT
ejpam-65	288	1	theorem	theorem	VERB
ejpam-65	288	2	3.2	3.2	NUM
ejpam-65	288	3	.	.	PUNCT
ejpam-65	289	1	let	let	VERB
ejpam-65	289	2	v	v	PART
ejpam-65	289	3	be	be	AUX
ejpam-65	289	4	strongly	strongly	ADV
ejpam-65	289	5	left	leave	VERB
ejpam-65	289	6	distributive	distributive	ADJ
ejpam-65	289	7	and	and	CCONJ
ejpam-65	289	8	invertible	invertible	ADJ
ejpam-65	289	9	.	.	PUNCT
ejpam-65	290	1	if	if	SCONJ
ejpam-65	290	2	v	v	NOUN
ejpam-65	290	3	has	have	VERB
ejpam-65	290	4	a	a	DET
ejpam-65	290	5	finite	finite	ADJ
ejpam-65	290	6	basis	basis	NOUN
ejpam-65	290	7	with	with	ADP
ejpam-65	290	8	n	n	ADP
ejpam-65	290	9	elements	element	NOUN
ejpam-65	290	10	,	,	PUNCT
ejpam-65	290	11	then	then	ADV
ejpam-65	290	12	every	every	DET
ejpam-65	290	13	linearly	linearly	ADV
ejpam-65	290	14	independent	independent	ADJ
ejpam-65	290	15	subset	subset	NOUN
ejpam-65	290	16	of	of	ADP
ejpam-65	290	17	v	v	NOUN
ejpam-65	290	18	has	have	VERB
ejpam-65	290	19	no	no	PRON
ejpam-65	290	20	more	more	ADJ
ejpam-65	290	21	than	than	ADP
ejpam-65	290	22	n	n	PRON
ejpam-65	290	23	elements	element	NOUN
ejpam-65	290	24	.	.	PUNCT
ejpam-65	291	1	r.	r.	PROPN
ejpam-65	291	2	ameri	ameri	PROPN
ejpam-65	291	3	,	,	PUNCT
ejpam-65	291	4	o.r	o.r	PROPN
ejpam-65	291	5	.	.	PROPN
ejpam-65	291	6	dehghan	dehghan	PROPN
ejpam-65	291	7	/	/	SYM
ejpam-65	291	8	eur	eur	PROPN
ejpam-65	291	9	.	.	PUNCT
ejpam-65	292	1	j.	j.	PROPN
ejpam-65	292	2	pure	pure	PROPN
ejpam-65	292	3	appl	appl	PROPN
ejpam-65	292	4	.	.	PROPN
ejpam-65	292	5	math	math	PROPN
ejpam-65	292	6	,	,	PUNCT
ejpam-65	292	7	1	1	NUM
ejpam-65	292	8	(	(	PUNCT
ejpam-65	292	9	2008	2008	NUM
ejpam-65	292	10	)	)	PUNCT
ejpam-65	292	11	,	,	PUNCT
ejpam-65	292	12	(	(	PUNCT
ejpam-65	292	13	32	32	NUM
ejpam-65	292	14	-	-	SYM
ejpam-65	292	15	50	50	NUM
ejpam-65	292	16	)	)	PUNCT
ejpam-65	292	17	40	40	NUM
ejpam-65	292	18	proof	proof	NOUN
ejpam-65	292	19	.	.	PUNCT
ejpam-65	293	1	let	let	VERB
ejpam-65	293	2	{	{	PUNCT
ejpam-65	293	3	v1	v1	VERB
ejpam-65	293	4	,	,	PUNCT
ejpam-65	293	5	.	.	PUNCT
ejpam-65	293	6	.	.	PUNCT
ejpam-65	293	7	.	.	PUNCT
ejpam-65	294	1	,	,	PUNCT
ejpam-65	294	2	vn	vn	AUX
ejpam-65	294	3	}	}	PUNCT
ejpam-65	294	4	be	be	AUX
ejpam-65	294	5	a	a	DET
ejpam-65	294	6	basis	basis	NOUN
ejpam-65	294	7	of	of	ADP
ejpam-65	294	8	v	v	NOUN
ejpam-65	294	9	and	and	CCONJ
ejpam-65	294	10	{	{	PUNCT
ejpam-65	294	11	w1	w1	NOUN
ejpam-65	294	12	,	,	PUNCT
ejpam-65	294	13	.	.	PUNCT
ejpam-65	294	14	.	.	PUNCT
ejpam-65	295	1	.	.	PUNCT
ejpam-65	296	1	,	,	PUNCT
ejpam-65	296	2	wm	wm	AUX
ejpam-65	296	3	}	}	PUNCT
ejpam-65	296	4	be	be	AUX
ejpam-65	296	5	a	a	DET
ejpam-65	296	6	linearly	linearly	ADV
ejpam-65	296	7	independent	independent	ADJ
ejpam-65	296	8	subset	subset	NOUN
ejpam-65	296	9	of	of	ADP
ejpam-65	296	10	v	v	NOUN
ejpam-65	296	11	.	.	PUNCT
ejpam-65	297	1	we	we	PRON
ejpam-65	297	2	show	show	VERB
ejpam-65	297	3	that	that	SCONJ
ejpam-65	297	4	m	m	VERB
ejpam-65	297	5	≤	≤	ADJ
ejpam-65	297	6	n	n	CCONJ
ejpam-65	297	7	:	:	PUNCT
ejpam-65	297	8	every	every	DET
ejpam-65	297	9	vector	vector	NOUN
ejpam-65	297	10	in	in	ADP
ejpam-65	297	11	v	v	NUM
ejpam-65	297	12	,	,	PUNCT
ejpam-65	297	13	so	so	CCONJ
ejpam-65	297	14	in	in	ADP
ejpam-65	297	15	particular	particular	ADJ
ejpam-65	297	16	wm	wm	PROPN
ejpam-65	297	17	,	,	PUNCT
ejpam-65	297	18	is	be	AUX
ejpam-65	297	19	in	in	ADP
ejpam-65	297	20	a	a	DET
ejpam-65	297	21	linear	linear	ADJ
ejpam-65	297	22	combination	combination	NOUN
ejpam-65	297	23	of	of	ADP
ejpam-65	297	24	v1	v1	NOUN
ejpam-65	297	25	,	,	PUNCT
ejpam-65	297	26	.	.	PUNCT
ejpam-65	297	27	.	.	PUNCT
ejpam-65	298	1	.	.	PUNCT
ejpam-65	299	1	,	,	PUNCT
ejpam-65	299	2	vn	vn	PROPN
ejpam-65	299	3	.	.	PUNCT
ejpam-65	300	1	therefore	therefore	ADV
ejpam-65	300	2	the	the	DET
ejpam-65	300	3	vectors	vector	NOUN
ejpam-65	300	4	wm	wm	PROPN
ejpam-65	300	5	,	,	PUNCT
ejpam-65	300	6	v1	v1	PROPN
ejpam-65	300	7	,	,	PUNCT
ejpam-65	300	8	.	.	PUNCT
ejpam-65	300	9	.	.	PUNCT
ejpam-65	301	1	.	.	PUNCT
ejpam-65	302	1	,	,	PUNCT
ejpam-65	302	2	vn	vn	PROPN
ejpam-65	302	3	are	be	AUX
ejpam-65	302	4	linearly	linearly	ADV
ejpam-65	302	5	dependent	dependent	ADJ
ejpam-65	302	6	.	.	PUNCT
ejpam-65	303	1	moreover	moreover	ADV
ejpam-65	303	2	they	they	PRON
ejpam-65	303	3	span	span	VERB
ejpam-65	303	4	v	v	ADP
ejpam-65	303	5	,	,	PUNCT
ejpam-65	303	6	since	since	SCONJ
ejpam-65	303	7	v1	v1	NOUN
ejpam-65	303	8	,	,	PUNCT
ejpam-65	303	9	.	.	PUNCT
ejpam-65	303	10	.	.	PUNCT
ejpam-65	304	1	.	.	PUNCT
ejpam-65	305	1	,	,	PUNCT
ejpam-65	305	2	vn	vn	AUX
ejpam-65	305	3	already	already	ADV
ejpam-65	305	4	do	do	VERB
ejpam-65	305	5	so	so	ADV
ejpam-65	305	6	and	and	CCONJ
ejpam-65	305	7	0	0	NUM
ejpam-65	305	8	∈	∈	NOUN
ejpam-65	305	9	0	0	NUM
ejpam-65	306	1	◦	◦	NOUN
ejpam-65	306	2	wm	wm	PROPN
ejpam-65	306	3	.	.	PUNCT
ejpam-65	306	4	thus	thus	ADV
ejpam-65	306	5	by	by	ADP
ejpam-65	306	6	lemma	lemma	PROPN
ejpam-65	306	7	3.5	3.5	NUM
ejpam-65	306	8	there	there	ADV
ejpam-65	306	9	exists	exist	VERB
ejpam-65	306	10	some	some	DET
ejpam-65	306	11	proper	proper	ADJ
ejpam-65	306	12	subset	subset	NOUN
ejpam-65	306	13	{	{	PUNCT
ejpam-65	306	14	wm	wm	PROPN
ejpam-65	306	15	,	,	PUNCT
ejpam-65	306	16	vi1	vi1	INTJ
ejpam-65	306	17	,	,	PUNCT
ejpam-65	306	18	.	.	PUNCT
ejpam-65	306	19	.	.	PUNCT
ejpam-65	307	1	.	.	PUNCT
ejpam-65	308	1	,	,	PUNCT
ejpam-65	308	2	vir	vir	PROPN
ejpam-65	308	3	}	}	PUNCT
ejpam-65	308	4	of	of	ADP
ejpam-65	308	5	{	{	PUNCT
ejpam-65	308	6	wm	wm	PROPN
ejpam-65	308	7	,	,	PUNCT
ejpam-65	308	8	v1	v1	PROPN
ejpam-65	308	9	,	,	PUNCT
ejpam-65	308	10	.	.	PUNCT
ejpam-65	308	11	.	.	PUNCT
ejpam-65	308	12	.	.	PUNCT
ejpam-65	309	1	,	,	PUNCT
ejpam-65	309	2	vn}with	vn}with	NOUN
ejpam-65	309	3	k	k	PROPN
ejpam-65	309	4	≤	≤	PROPN
ejpam-65	309	5	n−1	n−1	PROPN
ejpam-65	309	6	,	,	PUNCT
ejpam-65	309	7	such	such	ADJ
ejpam-65	309	8	that	that	SCONJ
ejpam-65	309	9	forms	form	VERB
ejpam-65	309	10	a	a	DET
ejpam-65	309	11	basis	basis	NOUN
ejpam-65	309	12	for	for	ADP
ejpam-65	309	13	v	v	NOUN
ejpam-65	309	14	.	.	PUNCT
ejpam-65	310	1	we	we	PRON
ejpam-65	310	2	have	have	AUX
ejpam-65	310	3	traded	trade	VERB
ejpam-65	310	4	off	off	ADP
ejpam-65	310	5	onew	onew	NOUN
ejpam-65	310	6	,	,	PUNCT
ejpam-65	310	7	in	in	ADP
ejpam-65	310	8	forming	form	VERB
ejpam-65	310	9	this	this	DET
ejpam-65	310	10	new	new	ADJ
ejpam-65	310	11	basis	basis	NOUN
ejpam-65	310	12	,	,	PUNCT
ejpam-65	310	13	for	for	ADP
ejpam-65	310	14	at	at	ADV
ejpam-65	310	15	least	least	ADV
ejpam-65	310	16	one	one	NUM
ejpam-65	310	17	vi	vi	NOUN
ejpam-65	310	18	.	.	PROPN
ejpam-65	310	19	repeat	repeat	VERB
ejpam-65	310	20	this	this	DET
ejpam-65	310	21	procedure	procedure	NOUN
ejpam-65	310	22	with	with	ADP
ejpam-65	310	23	the	the	DET
ejpam-65	310	24	set	set	NOUN
ejpam-65	310	25	{	{	PUNCT
ejpam-65	310	26	wm−1	wm−1	NOUN
ejpam-65	310	27	,	,	PUNCT
ejpam-65	310	28	wm	wm	PROPN
ejpam-65	310	29	,	,	PUNCT
ejpam-65	310	30	vi1	vi1	INTJ
ejpam-65	310	31	,	,	PUNCT
ejpam-65	310	32	.	.	PUNCT
ejpam-65	310	33	.	.	PUNCT
ejpam-65	311	1	.	.	PUNCT
ejpam-65	312	1	,	,	PUNCT
ejpam-65	312	2	vir	vir	PROPN
ejpam-65	312	3	}	}	PUNCT
ejpam-65	312	4	.	.	PUNCT
ejpam-65	313	1	from	from	ADP
ejpam-65	313	2	this	this	DET
ejpam-65	313	3	linearly	linearly	ADV
ejpam-65	313	4	dependent	dependent	ADJ
ejpam-65	313	5	set	set	NOUN
ejpam-65	313	6	,	,	PUNCT
ejpam-65	313	7	by	by	ADP
ejpam-65	313	8	lemma	lemma	PROPN
ejpam-65	313	9	3.5	3.5	NUM
ejpam-65	313	10	we	we	PRON
ejpam-65	313	11	can	can	AUX
ejpam-65	313	12	extract	extract	VERB
ejpam-65	313	13	a	a	DET
ejpam-65	313	14	basis	basis	NOUN
ejpam-65	313	15	of	of	ADP
ejpam-65	313	16	the	the	DET
ejpam-65	313	17	form	form	NOUN
ejpam-65	313	18	{	{	PUNCT
ejpam-65	313	19	wm−1	wm−1	NOUN
ejpam-65	313	20	,	,	PUNCT
ejpam-65	313	21	wm	wm	PROPN
ejpam-65	313	22	,	,	PUNCT
ejpam-65	313	23	vi1	vi1	INTJ
ejpam-65	313	24	,	,	PUNCT
ejpam-65	313	25	.	.	PUNCT
ejpam-65	313	26	.	.	PUNCT
ejpam-65	314	1	.	.	PUNCT
ejpam-65	315	1	,	,	PUNCT
ejpam-65	315	2	vis	vis	X
ejpam-65	315	3	}	}	PUNCT
ejpam-65	315	4	such	such	ADJ
ejpam-65	315	5	that	that	PRON
ejpam-65	315	6	s	s	VERB
ejpam-65	315	7	≤	≤	NUM
ejpam-65	315	8	n	n	CCONJ
ejpam-65	315	9	−	−	PROPN
ejpam-65	315	10	2	2	X
ejpam-65	315	11	.	.	X
ejpam-65	316	1	keeping	keep	VERB
ejpam-65	316	2	up	up	ADP
ejpam-65	316	3	this	this	DET
ejpam-65	316	4	procedure	procedure	NOUN
ejpam-65	316	5	we	we	PRON
ejpam-65	316	6	eventually	eventually	ADV
ejpam-65	316	7	get	get	VERB
ejpam-65	316	8	down	down	ADP
ejpam-65	316	9	to	to	ADP
ejpam-65	316	10	a	a	DET
ejpam-65	316	11	basis	basis	NOUN
ejpam-65	316	12	of	of	ADP
ejpam-65	316	13	v	v	NOUN
ejpam-65	316	14	of	of	ADP
ejpam-65	316	15	the	the	DET
ejpam-65	316	16	form	form	NOUN
ejpam-65	316	17	{	{	PUNCT
ejpam-65	316	18	w2	w2	NOUN
ejpam-65	316	19	,	,	PUNCT
ejpam-65	316	20	.	.	PUNCT
ejpam-65	316	21	.	.	PUNCT
ejpam-65	317	1	.	.	PUNCT
ejpam-65	318	1	,	,	PUNCT
ejpam-65	318	2	wm−1	wm−1	NOUN
ejpam-65	318	3	,	,	PUNCT
ejpam-65	318	4	wm	wm	PROPN
ejpam-65	318	5	,	,	PUNCT
ejpam-65	318	6	vα	vα	PROPN
ejpam-65	318	7	,	,	PUNCT
ejpam-65	318	8	vβ	vβ	NOUN
ejpam-65	318	9	,	,	PUNCT
ejpam-65	318	10	.	.	PUNCT
ejpam-65	318	11	.	.	PUNCT
ejpam-65	318	12	.	.	PUNCT
ejpam-65	318	13	}	}	PUNCT
ejpam-65	318	14	.	.	PUNCT
ejpam-65	319	1	since	since	SCONJ
ejpam-65	319	2	w1	w1	NOUN
ejpam-65	319	3	is	be	AUX
ejpam-65	319	4	not	not	PART
ejpam-65	319	5	in	in	ADP
ejpam-65	319	6	a	a	DET
ejpam-65	319	7	linear	linear	ADJ
ejpam-65	319	8	combination	combination	NOUN
ejpam-65	319	9	of	of	ADP
ejpam-65	319	10	w2	w2	NOUN
ejpam-65	319	11	,	,	PUNCT
ejpam-65	319	12	.	.	PUNCT
ejpam-65	319	13	.	.	PUNCT
ejpam-65	320	1	.	.	PUNCT
ejpam-65	321	1	,	,	PUNCT
ejpam-65	321	2	wm−1	wm−1	NOUN
ejpam-65	321	3	,	,	PUNCT
ejpam-65	321	4	the	the	DET
ejpam-65	321	5	above	above	ADJ
ejpam-65	321	6	basis	basis	NOUN
ejpam-65	321	7	must	must	AUX
ejpam-65	321	8	actually	actually	ADV
ejpam-65	321	9	include	include	VERB
ejpam-65	321	10	some	some	DET
ejpam-65	321	11	v.	v.	NOUN
ejpam-65	321	12	to	to	PART
ejpam-65	321	13	get	get	VERB
ejpam-65	321	14	to	to	ADP
ejpam-65	321	15	this	this	DET
ejpam-65	321	16	basis	basis	NOUN
ejpam-65	321	17	we	we	PRON
ejpam-65	321	18	have	have	AUX
ejpam-65	321	19	introduced	introduce	VERB
ejpam-65	321	20	m−	m−	PROPN
ejpam-65	321	21	1	1	NUM
ejpam-65	321	22	w‘s	w‘s	PROPN
ejpam-65	321	23	,	,	PUNCT
ejpam-65	321	24	each	each	DET
ejpam-65	321	25	such	such	ADJ
ejpam-65	321	26	introduction	introduction	NOUN
ejpam-65	321	27	having	having	AUX
ejpam-65	321	28	cost	cost	VERB
ejpam-65	321	29	us	we	PRON
ejpam-65	321	30	at	at	ADV
ejpam-65	321	31	least	least	ADV
ejpam-65	321	32	one	one	NUM
ejpam-65	321	33	v	v	NOUN
ejpam-65	321	34	,	,	PUNCT
ejpam-65	321	35	and	and	CCONJ
ejpam-65	321	36	yet	yet	ADV
ejpam-65	321	37	there	there	PRON
ejpam-65	321	38	is	be	VERB
ejpam-65	321	39	a	a	DET
ejpam-65	321	40	v	v	NOUN
ejpam-65	321	41	left	left	NOUN
ejpam-65	321	42	.	.	PUNCT
ejpam-65	322	1	thus	thus	ADV
ejpam-65	322	2	m−	m−	PROPN
ejpam-65	322	3	1	1	NUM
ejpam-65	322	4	≤	≤	NUM
ejpam-65	322	5	n−	n−	NOUN
ejpam-65	322	6	1	1	NUM
ejpam-65	322	7	and	and	CCONJ
ejpam-65	322	8	so	so	ADV
ejpam-65	322	9	m	m	VERB
ejpam-65	322	10	≤	≤	ADJ
ejpam-65	322	11	n.	n.	NOUN
ejpam-65	322	12	corollary	corollary	NOUN
ejpam-65	322	13	3.4	3.4	NUM
ejpam-65	322	14	.	.	PUNCT
ejpam-65	323	1	let	let	VERB
ejpam-65	323	2	v	v	PART
ejpam-65	323	3	be	be	AUX
ejpam-65	323	4	strongly	strongly	ADV
ejpam-65	323	5	left	leave	VERB
ejpam-65	323	6	distributive	distributive	ADJ
ejpam-65	323	7	and	and	CCONJ
ejpam-65	323	8	invertible	invertible	ADJ
ejpam-65	323	9	.	.	PUNCT
ejpam-65	324	1	if	if	SCONJ
ejpam-65	324	2	v	v	NOUN
ejpam-65	324	3	is	be	AUX
ejpam-65	324	4	finite	finite	ADJ
ejpam-65	324	5	dimensional	dimensional	ADJ
ejpam-65	324	6	then	then	ADV
ejpam-65	324	7	every	every	DET
ejpam-65	324	8	two	two	NUM
ejpam-65	324	9	basis	basis	NOUN
ejpam-65	324	10	of	of	ADP
ejpam-65	324	11	v	v	NOUN
ejpam-65	324	12	have	have	VERB
ejpam-65	324	13	the	the	DET
ejpam-65	324	14	same	same	ADJ
ejpam-65	324	15	elements	element	NOUN
ejpam-65	324	16	.	.	PUNCT
ejpam-65	325	1	proof	proof	NOUN
ejpam-65	325	2	.	.	PUNCT
ejpam-65	326	1	let	let	VERB
ejpam-65	326	2	{	{	PUNCT
ejpam-65	326	3	v1	v1	VERB
ejpam-65	326	4	,	,	PUNCT
ejpam-65	326	5	.	.	PUNCT
ejpam-65	326	6	.	.	PUNCT
ejpam-65	326	7	.	.	PUNCT
ejpam-65	327	1	,	,	PUNCT
ejpam-65	327	2	vn	vn	PROPN
ejpam-65	327	3	}	}	PUNCT
ejpam-65	327	4	and	and	CCONJ
ejpam-65	327	5	{	{	PUNCT
ejpam-65	327	6	w1	w1	NOUN
ejpam-65	327	7	,	,	PUNCT
ejpam-65	327	8	.	.	PUNCT
ejpam-65	327	9	.	.	PUNCT
ejpam-65	328	1	.	.	PUNCT
ejpam-65	329	1	,	,	PUNCT
ejpam-65	329	2	wm	wm	AUX
ejpam-65	329	3	}	}	PUNCT
ejpam-65	329	4	be	be	AUX
ejpam-65	329	5	two	two	NUM
ejpam-65	329	6	basis	basis	NOUN
ejpam-65	329	7	of	of	ADP
ejpam-65	329	8	v	v	NOUN
ejpam-65	329	9	over	over	ADP
ejpam-65	329	10	k.	k.	PROPN
ejpam-65	329	11	then	then	ADV
ejpam-65	329	12	by	by	ADP
ejpam-65	329	13	theorem	theorem	NOUN
ejpam-65	329	14	3.2	3.2	NUM
ejpam-65	329	15	we	we	PRON
ejpam-65	329	16	have	have	VERB
ejpam-65	329	17	m	m	NOUN
ejpam-65	329	18	≤	≤	NOUN
ejpam-65	329	19	n	n	CCONJ
ejpam-65	329	20	,	,	PUNCT
ejpam-65	329	21	since	since	SCONJ
ejpam-65	329	22	w1	w1	NOUN
ejpam-65	329	23	,	,	PUNCT
ejpam-65	329	24	.	.	PUNCT
ejpam-65	329	25	.	.	PUNCT
ejpam-65	330	1	.	.	PUNCT
ejpam-65	331	1	,	,	PUNCT
ejpam-65	331	2	wm	wm	PROPN
ejpam-65	331	3	are	be	AUX
ejpam-65	331	4	linearly	linearly	ADV
ejpam-65	331	5	independent	independent	ADJ
ejpam-65	331	6	.	.	PUNCT
ejpam-65	332	1	now	now	ADV
ejpam-65	332	2	interchange	interchange	VERB
ejpam-65	332	3	the	the	DET
ejpam-65	332	4	roles	role	NOUN
ejpam-65	332	5	of	of	ADP
ejpam-65	332	6	the	the	DET
ejpam-65	332	7	v‘s	v‘s	NOUN
ejpam-65	332	8	and	and	CCONJ
ejpam-65	332	9	w‘s	w‘s	PROPN
ejpam-65	332	10	and	and	CCONJ
ejpam-65	332	11	obtain	obtain	VERB
ejpam-65	332	12	that	that	DET
ejpam-65	332	13	n	n	NOUN
ejpam-65	332	14	≤	≤	ADJ
ejpam-65	332	15	m.	m.	NOUN
ejpam-65	332	16	together	together	ADV
ejpam-65	332	17	these	these	PRON
ejpam-65	332	18	say	say	VERB
ejpam-65	332	19	that	that	SCONJ
ejpam-65	332	20	n	n	NOUN
ejpam-65	332	21	=	=	VERB
ejpam-65	332	22	m.	m.	NOUN
ejpam-65	332	23	lemma	lemma	PROPN
ejpam-65	332	24	3.6	3.6	NUM
ejpam-65	332	25	.	.	PUNCT
ejpam-65	333	1	let	let	VERB
ejpam-65	333	2	v	v	PART
ejpam-65	333	3	be	be	AUX
ejpam-65	333	4	strongly	strongly	ADV
ejpam-65	333	5	left	leave	VERB
ejpam-65	333	6	distributive	distributive	ADJ
ejpam-65	333	7	and	and	CCONJ
ejpam-65	333	8	invertible	invertible	ADJ
ejpam-65	333	9	.	.	PUNCT
ejpam-65	334	1	if	if	SCONJ
ejpam-65	334	2	v	v	NOUN
ejpam-65	334	3	is	be	AUX
ejpam-65	334	4	finite	finite	ADJ
ejpam-65	334	5	dimensional	dimensional	ADJ
ejpam-65	334	6	,	,	PUNCT
ejpam-65	334	7	then	then	ADV
ejpam-65	334	8	every	every	DET
ejpam-65	334	9	linearly	linearly	ADV
ejpam-65	334	10	independent	independent	ADJ
ejpam-65	334	11	subset	subset	NOUN
ejpam-65	334	12	of	of	ADP
ejpam-65	334	13	v	v	NOUN
ejpam-65	334	14	is	be	AUX
ejpam-65	334	15	contained	contain	VERB
ejpam-65	334	16	in	in	ADP
ejpam-65	334	17	a	a	DET
ejpam-65	334	18	finite	finite	ADJ
ejpam-65	334	19	basis	basis	NOUN
ejpam-65	334	20	.	.	PUNCT
ejpam-65	335	1	proof	proof	NOUN
ejpam-65	335	2	.	.	PUNCT
ejpam-65	336	1	let	let	VERB
ejpam-65	336	2	{	{	PUNCT
ejpam-65	336	3	v1	v1	VERB
ejpam-65	336	4	,	,	PUNCT
ejpam-65	336	5	.	.	PUNCT
ejpam-65	336	6	.	.	PUNCT
ejpam-65	336	7	.	.	PUNCT
ejpam-65	337	1	,	,	PUNCT
ejpam-65	337	2	vn	vn	AUX
ejpam-65	337	3	}	}	PUNCT
ejpam-65	337	4	be	be	AUX
ejpam-65	337	5	a	a	DET
ejpam-65	337	6	basis	basis	NOUN
ejpam-65	337	7	of	of	ADP
ejpam-65	337	8	v	v	NOUN
ejpam-65	337	9	and	and	CCONJ
ejpam-65	337	10	{	{	PUNCT
ejpam-65	337	11	u1	u1	NOUN
ejpam-65	337	12	,	,	PUNCT
ejpam-65	337	13	.	.	PUNCT
ejpam-65	337	14	.	.	PUNCT
ejpam-65	338	1	.	.	PUNCT
ejpam-65	339	1	,	,	PUNCT
ejpam-65	339	2	um	um	INTJ
ejpam-65	339	3	}	}	PUNCT
ejpam-65	339	4	be	be	AUX
ejpam-65	339	5	a	a	DET
ejpam-65	339	6	linearly	linearly	ADV
ejpam-65	339	7	independent	independent	ADJ
ejpam-65	339	8	subset	subset	NOUN
ejpam-65	339	9	of	of	ADP
ejpam-65	339	10	v	v	NOUN
ejpam-65	339	11	.	.	PUNCT
ejpam-65	340	1	then	then	ADV
ejpam-65	340	2	vectors	vector	NOUN
ejpam-65	340	3	u1	u1	NOUN
ejpam-65	340	4	,	,	PUNCT
ejpam-65	340	5	.	.	PUNCT
ejpam-65	340	6	.	.	PUNCT
ejpam-65	340	7	.	.	PUNCT
ejpam-65	341	1	,	,	PUNCT
ejpam-65	341	2	um	um	INTJ
ejpam-65	341	3	,	,	PUNCT
ejpam-65	341	4	v1	v1	PROPN
ejpam-65	341	5	,	,	PUNCT
ejpam-65	341	6	.	.	PUNCT
ejpam-65	341	7	.	.	PUNCT
ejpam-65	342	1	.	.	PUNCT
ejpam-65	343	1	,	,	PUNCT
ejpam-65	343	2	vn	vn	PROPN
ejpam-65	343	3	span	span	PROPN
ejpam-65	343	4	v	v	PROPN
ejpam-65	343	5	(	(	PUNCT
ejpam-65	343	6	since	since	SCONJ
ejpam-65	343	7	v1	v1	NOUN
ejpam-65	343	8	,	,	PUNCT
ejpam-65	343	9	.	.	PUNCT
ejpam-65	343	10	.	.	PUNCT
ejpam-65	344	1	.	.	PUNCT
ejpam-65	345	1	,	,	PUNCT
ejpam-65	345	2	vn	vn	PROPN
ejpam-65	345	3	span	span	NOUN
ejpam-65	345	4	v	v	ADP
ejpam-65	345	5	and	and	CCONJ
ejpam-65	345	6	0	0	NUM
ejpam-65	345	7	∈	∈	NOUN
ejpam-65	345	8	m∑	m∑	VERB
ejpam-65	346	1	j=1	j=1	PROPN
ejpam-65	346	2	0	0	NUM
ejpam-65	346	3	◦	◦	NOUN
ejpam-65	346	4	uj	uj	NUM
ejpam-65	346	5	)	)	PUNCT
ejpam-65	346	6	.	.	PUNCT
ejpam-65	347	1	by	by	ADP
ejpam-65	347	2	lemma	lemma	PROPN
ejpam-65	347	3	3.5	3.5	NUM
ejpam-65	347	4	there	there	PRON
ejpam-65	347	5	is	be	VERB
ejpam-65	347	6	a	a	DET
ejpam-65	347	7	subset	subset	NOUN
ejpam-65	347	8	of	of	ADP
ejpam-65	347	9	these	these	PRON
ejpam-65	347	10	of	of	ADP
ejpam-65	347	11	the	the	DET
ejpam-65	347	12	form	form	NOUN
ejpam-65	347	13	u1	u1	NOUN
ejpam-65	347	14	,	,	PUNCT
ejpam-65	347	15	.	.	PUNCT
ejpam-65	347	16	.	.	PUNCT
ejpam-65	347	17	.	.	PUNCT
ejpam-65	348	1	,	,	PUNCT
ejpam-65	348	2	um	um	INTJ
ejpam-65	348	3	,	,	PUNCT
ejpam-65	348	4	vi1	vi1	INTJ
ejpam-65	348	5	,	,	PUNCT
ejpam-65	348	6	.	.	PUNCT
ejpam-65	348	7	.	.	PUNCT
ejpam-65	349	1	.	.	PUNCT
ejpam-65	350	1	,	,	PUNCT
ejpam-65	350	2	vir	vir	X
ejpam-65	350	3	which	which	PRON
ejpam-65	350	4	consist	consist	VERB
ejpam-65	350	5	of	of	ADP
ejpam-65	350	6	linearly	linearly	ADV
ejpam-65	350	7	independent	independent	ADJ
ejpam-65	350	8	elements	element	NOUN
ejpam-65	350	9	which	which	PRON
ejpam-65	350	10	span	span	VERB
ejpam-65	350	11	v	v	X
ejpam-65	350	12	.	.	PUNCT
ejpam-65	350	13	remark	remark	PROPN
ejpam-65	350	14	3.4	3.4	NUM
ejpam-65	350	15	.	.	PUNCT
ejpam-65	351	1	let	let	AUX
ejpam-65	351	2	(	(	PUNCT
ejpam-65	351	3	v,+	v,+	NUM
ejpam-65	351	4	,	,	PUNCT
ejpam-65	351	5	◦	◦	NOUN
ejpam-65	351	6	,	,	PUNCT
ejpam-65	351	7	k	k	NOUN
ejpam-65	351	8	)	)	PUNCT
ejpam-65	351	9	be	be	AUX
ejpam-65	351	10	a	a	DET
ejpam-65	351	11	(	(	PUNCT
ejpam-65	351	12	resp	resp	NOUN
ejpam-65	351	13	.	.	PUNCT
ejpam-65	352	1	strongly	strongly	ADV
ejpam-65	352	2	left	leave	VERB
ejpam-65	352	3	distributive	distributive	ADJ
ejpam-65	352	4	)	)	PUNCT
ejpam-65	352	5	hypervector	hypervector	NOUN
ejpam-65	352	6	space	space	NOUN
ejpam-65	352	7	and	and	CCONJ
ejpam-65	352	8	w	w	AUX
ejpam-65	352	9	be	be	AUX
ejpam-65	352	10	a	a	DET
ejpam-65	352	11	subhyperspace	subhyperspace	NOUN
ejpam-65	352	12	of	of	ADP
ejpam-65	352	13	v	v	NOUN
ejpam-65	352	14	.	.	PUNCT
ejpam-65	353	1	consider	consider	VERB
ejpam-65	353	2	the	the	DET
ejpam-65	353	3	quotient	quotient	NOUN
ejpam-65	353	4	abelian	abelian	PROPN
ejpam-65	353	5	group	group	PROPN
ejpam-65	353	6	(	(	PUNCT
ejpam-65	353	7	v	v	NOUN
ejpam-65	353	8	/	/	SYM
ejpam-65	353	9	w,+	w,+	NOUN
ejpam-65	353	10	)	)	PUNCT
ejpam-65	353	11	.	.	PUNCT
ejpam-65	354	1	define	define	VERB
ejpam-65	354	2	the	the	DET
ejpam-65	354	3	rule	rule	NOUN
ejpam-65	354	4	{	{	PUNCT
ejpam-65	354	5	∗	∗	NOUN
ejpam-65	354	6	:	:	PUNCT
ejpam-65	355	1	k	k	PROPN
ejpam-65	355	2	×	×	PROPN
ejpam-65	355	3	v	v	NOUN
ejpam-65	355	4	/	/	SYM
ejpam-65	355	5	w	w	NOUN
ejpam-65	355	6	−→	−→	ADJ
ejpam-65	355	7	p∗(v	p∗(v	PROPN
ejpam-65	355	8	/	/	SYM
ejpam-65	355	9	w	w	NOUN
ejpam-65	355	10	)	)	PUNCT
ejpam-65	355	11	(	(	PUNCT
ejpam-65	355	12	a	a	PRON
ejpam-65	355	13	,	,	PUNCT
ejpam-65	355	14	v	v	ADP
ejpam-65	355	15	+	+	PROPN
ejpam-65	355	16	w	w	NOUN
ejpam-65	355	17	)	)	PUNCT
ejpam-65	355	18	7−→	7−→	NOUN
ejpam-65	355	19	a	a	DET
ejpam-65	355	20	◦	◦	NOUN
ejpam-65	355	21	v	v	ADP
ejpam-65	356	1	+	+	NOUN
ejpam-65	356	2	w	w	ADV
ejpam-65	356	3	then	then	ADV
ejpam-65	356	4	it	it	PRON
ejpam-65	356	5	is	be	AUX
ejpam-65	356	6	easy	easy	ADJ
ejpam-65	356	7	to	to	PART
ejpam-65	356	8	verify	verify	VERB
ejpam-65	356	9	that	that	SCONJ
ejpam-65	356	10	(	(	PUNCT
ejpam-65	356	11	v	v	NOUN
ejpam-65	356	12	/	/	SYM
ejpam-65	356	13	w,+	w,+	NOUN
ejpam-65	356	14	,	,	PUNCT
ejpam-65	356	15	∗,k	∗,k	NOUN
ejpam-65	356	16	)	)	PUNCT
ejpam-65	356	17	is	be	AUX
ejpam-65	356	18	a	a	DET
ejpam-65	356	19	(	(	PUNCT
ejpam-65	356	20	resp	resp	NOUN
ejpam-65	356	21	.	.	PUNCT
ejpam-65	357	1	strongly	strongly	ADV
ejpam-65	357	2	left	leave	VERB
ejpam-65	357	3	distributive	distributive	ADJ
ejpam-65	357	4	)	)	PUNCT
ejpam-65	357	5	hypervector	hypervector	NOUN
ejpam-65	357	6	space	space	NOUN
ejpam-65	357	7	over	over	ADP
ejpam-65	357	8	k	k	PROPN
ejpam-65	358	1	and	and	CCONJ
ejpam-65	358	2	it	it	PRON
ejpam-65	358	3	is	be	AUX
ejpam-65	358	4	called	call	VERB
ejpam-65	358	5	the	the	DET
ejpam-65	358	6	quotient	quotient	NOUN
ejpam-65	358	7	hypervector	hypervector	NOUN
ejpam-65	358	8	space	space	NOUN
ejpam-65	358	9	of	of	ADP
ejpam-65	358	10	v	v	NOUN
ejpam-65	358	11	over	over	ADP
ejpam-65	358	12	w	w	PROPN
ejpam-65	358	13	.	.	PUNCT
ejpam-65	359	1	theorem	theorem	VERB
ejpam-65	359	2	3.3	3.3	NUM
ejpam-65	359	3	.	.	PUNCT
ejpam-65	360	1	let	let	VERB
ejpam-65	360	2	v	v	PART
ejpam-65	360	3	be	be	AUX
ejpam-65	360	4	strongly	strongly	ADV
ejpam-65	360	5	left	leave	VERB
ejpam-65	360	6	distributive	distributive	ADJ
ejpam-65	360	7	and	and	CCONJ
ejpam-65	360	8	invertible	invertible	ADJ
ejpam-65	360	9	.	.	PUNCT
ejpam-65	361	1	if	if	SCONJ
ejpam-65	361	2	v	v	NOUN
ejpam-65	361	3	is	be	AUX
ejpam-65	361	4	finite	finite	ADJ
ejpam-65	361	5	dimensional	dimensional	ADJ
ejpam-65	361	6	and	and	CCONJ
ejpam-65	361	7	w	w	NOUN
ejpam-65	361	8	is	be	AUX
ejpam-65	361	9	subhyperspace	subhyperspace	NOUN
ejpam-65	361	10	of	of	ADP
ejpam-65	361	11	v	v	NOUN
ejpam-65	361	12	,	,	PUNCT
ejpam-65	361	13	then	then	ADV
ejpam-65	361	14	the	the	DET
ejpam-65	361	15	following	follow	VERB
ejpam-65	361	16	hold	hold	NOUN
ejpam-65	361	17	:	:	PUNCT
ejpam-65	361	18	(	(	PUNCT
ejpam-65	361	19	i	i	NOUN
ejpam-65	361	20	)	)	PUNCT
ejpam-65	362	1	w	w	PROPN
ejpam-65	362	2	is	be	AUX
ejpam-65	362	3	finite	finite	ADJ
ejpam-65	362	4	dimensional	dimensional	ADJ
ejpam-65	362	5	and	and	CCONJ
ejpam-65	362	6	dimw	dimw	VERB
ejpam-65	362	7	≤	≤	NUM
ejpam-65	362	8	dimv	dimv	NOUN
ejpam-65	362	9	.	.	PUNCT
ejpam-65	363	1	(	(	PUNCT
ejpam-65	363	2	ii	ii	NOUN
ejpam-65	363	3	)	)	PUNCT
ejpam-65	363	4	dimv	dimv	NOUN
ejpam-65	363	5	/	/	SYM
ejpam-65	363	6	w	w	NOUN
ejpam-65	363	7	=	=	SYM
ejpam-65	363	8	dimv	dimv	NOUN
ejpam-65	363	9	−	−	NOUN
ejpam-65	363	10	dimw	dimw	NOUN
ejpam-65	363	11	.	.	PUNCT
ejpam-65	364	1	r.	r.	PROPN
ejpam-65	364	2	ameri	ameri	PROPN
ejpam-65	364	3	,	,	PUNCT
ejpam-65	364	4	o.r	o.r	PROPN
ejpam-65	364	5	.	.	PROPN
ejpam-65	364	6	dehghan	dehghan	PROPN
ejpam-65	364	7	/	/	SYM
ejpam-65	364	8	eur	eur	PROPN
ejpam-65	364	9	.	.	PUNCT
ejpam-65	365	1	j.	j.	PROPN
ejpam-65	365	2	pure	pure	PROPN
ejpam-65	365	3	appl	appl	PROPN
ejpam-65	365	4	.	.	PROPN
ejpam-65	365	5	math	math	PROPN
ejpam-65	365	6	,	,	PUNCT
ejpam-65	365	7	1	1	NUM
ejpam-65	365	8	(	(	PUNCT
ejpam-65	365	9	2008	2008	NUM
ejpam-65	365	10	)	)	PUNCT
ejpam-65	365	11	,	,	PUNCT
ejpam-65	365	12	(	(	PUNCT
ejpam-65	365	13	32	32	NUM
ejpam-65	365	14	-	-	SYM
ejpam-65	365	15	50	50	NUM
ejpam-65	365	16	)	)	PUNCT
ejpam-65	365	17	41	41	NUM
ejpam-65	365	18	proof	proof	NOUN
ejpam-65	365	19	.	.	PUNCT
ejpam-65	366	1	(	(	PUNCT
ejpam-65	366	2	i	i	NOUN
ejpam-65	366	3	)	)	PUNCT
ejpam-65	366	4	let	let	VERB
ejpam-65	366	5	y	y	PRON
ejpam-65	366	6	be	be	AUX
ejpam-65	366	7	a	a	DET
ejpam-65	366	8	basis	basis	NOUN
ejpam-65	366	9	of	of	ADP
ejpam-65	366	10	w	w	PROPN
ejpam-65	366	11	.	.	PUNCT
ejpam-65	367	1	then	then	ADV
ejpam-65	367	2	by	by	ADP
ejpam-65	367	3	lemma	lemma	PROPN
ejpam-65	367	4	3.6	3.6	NUM
ejpam-65	367	5	there	there	ADV
ejpam-65	367	6	exists	exist	VERB
ejpam-65	367	7	a	a	DET
ejpam-65	367	8	basis	basis	NOUN
ejpam-65	367	9	x	x	PUNCT
ejpam-65	367	10	of	of	ADP
ejpam-65	367	11	v	v	NOUN
ejpam-65	367	12	such	such	ADJ
ejpam-65	367	13	that	that	PRON
ejpam-65	367	14	contains	contain	VERB
ejpam-65	367	15	y	y	PROPN
ejpam-65	367	16	.	.	PUNCT
ejpam-65	368	1	thus	thus	ADV
ejpam-65	368	2	by	by	ADP
ejpam-65	368	3	corollary	corollary	ADJ
ejpam-65	368	4	3.4	3.4	NUM
ejpam-65	368	5	we	we	PRON
ejpam-65	368	6	have	have	VERB
ejpam-65	368	7	dimw	dimw	NOUN
ejpam-65	368	8	=	=	NOUN
ejpam-65	368	9	|y	|y	NOUN
ejpam-65	368	10	|	|	ADV
ejpam-65	368	11	≤	≤	PUNCT
ejpam-65	369	1	|x|	|x|	PROPN
ejpam-65	370	1	=	=	SYM
ejpam-65	370	2	dimv	dimv	NOUN
ejpam-65	370	3	.	.	PUNCT
ejpam-65	371	1	(	(	PUNCT
ejpam-65	371	2	ii	ii	NOUN
ejpam-65	371	3	)	)	PUNCT
ejpam-65	371	4	let	let	VERB
ejpam-65	371	5	{	{	PUNCT
ejpam-65	371	6	w1	w1	NOUN
ejpam-65	371	7	,	,	PUNCT
ejpam-65	371	8	.	.	PUNCT
ejpam-65	371	9	.	.	PUNCT
ejpam-65	372	1	.	.	PUNCT
ejpam-65	373	1	,	,	PUNCT
ejpam-65	373	2	wm	wm	AUX
ejpam-65	373	3	}	}	PUNCT
ejpam-65	373	4	be	be	AUX
ejpam-65	373	5	a	a	DET
ejpam-65	373	6	basis	basis	NOUN
ejpam-65	373	7	of	of	ADP
ejpam-65	373	8	w	w	PROPN
ejpam-65	373	9	.	.	PUNCT
ejpam-65	374	1	by	by	ADP
ejpam-65	374	2	lemma	lemma	PROPN
ejpam-65	374	3	3.6	3.6	NUM
ejpam-65	374	4	we	we	PRON
ejpam-65	374	5	can	can	AUX
ejpam-65	374	6	fill	fill	VERB
ejpam-65	374	7	this	this	PRON
ejpam-65	374	8	out	out	ADP
ejpam-65	374	9	to	to	ADP
ejpam-65	374	10	a	a	DET
ejpam-65	374	11	basis	basis	NOUN
ejpam-65	374	12	,	,	PUNCT
ejpam-65	374	13	{	{	PUNCT
ejpam-65	374	14	w1	w1	NOUN
ejpam-65	374	15	,	,	PUNCT
ejpam-65	374	16	.	.	PUNCT
ejpam-65	374	17	.	.	PUNCT
ejpam-65	375	1	.	.	PUNCT
ejpam-65	376	1	,	,	PUNCT
ejpam-65	376	2	wm	wm	PROPN
ejpam-65	376	3	,	,	PUNCT
ejpam-65	376	4	v1	v1	PROPN
ejpam-65	376	5	,	,	PUNCT
ejpam-65	376	6	.	.	PUNCT
ejpam-65	376	7	.	.	PUNCT
ejpam-65	377	1	.	.	PUNCT
ejpam-65	378	1	,	,	PUNCT
ejpam-65	378	2	vr	vr	NOUN
ejpam-65	378	3	}	}	PUNCT
ejpam-65	378	4	of	of	ADP
ejpam-65	378	5	v	v	NOUN
ejpam-65	378	6	,	,	PUNCT
ejpam-65	378	7	where	where	SCONJ
ejpam-65	378	8	m	m	VERB
ejpam-65	378	9	+	+	NOUN
ejpam-65	378	10	r	r	NOUN
ejpam-65	378	11	=	=	SYM
ejpam-65	378	12	dimv	dimv	NOUN
ejpam-65	378	13	and	and	CCONJ
ejpam-65	378	14	m	m	NOUN
ejpam-65	378	15	=	=	NOUN
ejpam-65	378	16	dimw	dimw	NOUN
ejpam-65	378	17	.	.	PUNCT
ejpam-65	379	1	let	let	VERB
ejpam-65	379	2	v̄1	v̄1	ADV
ejpam-65	379	3	,	,	PUNCT
ejpam-65	379	4	.	.	PUNCT
ejpam-65	379	5	.	.	PUNCT
ejpam-65	380	1	.	.	PUNCT
ejpam-65	381	1	,	,	PUNCT
ejpam-65	381	2	v̄r	v̄r	NOUN
ejpam-65	381	3	be	be	VERB
ejpam-65	381	4	the	the	DET
ejpam-65	381	5	images	image	NOUN
ejpam-65	381	6	in	in	ADP
ejpam-65	381	7	v̄	v̄	NOUN
ejpam-65	381	8	=	=	SYM
ejpam-65	381	9	v	v	NOUN
ejpam-65	381	10	/	/	SYM
ejpam-65	381	11	w	w	NOUN
ejpam-65	381	12	,	,	PUNCT
ejpam-65	381	13	of	of	ADP
ejpam-65	381	14	v1	v1	NOUN
ejpam-65	381	15	,	,	PUNCT
ejpam-65	381	16	.	.	PUNCT
ejpam-65	381	17	.	.	PUNCT
ejpam-65	382	1	.	.	PUNCT
ejpam-65	383	1	,	,	PUNCT
ejpam-65	383	2	vr	vr	PROPN
ejpam-65	383	3	.	.	PROPN
ejpam-65	384	1	since	since	SCONJ
ejpam-65	384	2	any	any	DET
ejpam-65	384	3	vector	vector	NOUN
ejpam-65	384	4	v	v	ADP
ejpam-65	384	5	∈	∈	NOUN
ejpam-65	384	6	v	v	NOUN
ejpam-65	384	7	is	be	AUX
ejpam-65	384	8	in	in	ADP
ejpam-65	384	9	a	a	DET
ejpam-65	384	10	linear	linear	ADJ
ejpam-65	384	11	combination	combination	NOUN
ejpam-65	384	12	of	of	ADP
ejpam-65	384	13	w1	w1	NOUN
ejpam-65	384	14	,	,	PUNCT
ejpam-65	384	15	.	.	PUNCT
ejpam-65	384	16	.	.	PUNCT
ejpam-65	385	1	.	.	PUNCT
ejpam-65	386	1	,	,	PUNCT
ejpam-65	386	2	wm	wm	PROPN
ejpam-65	386	3	,	,	PUNCT
ejpam-65	386	4	v1	v1	PROPN
ejpam-65	386	5	,	,	PUNCT
ejpam-65	386	6	.	.	PUNCT
ejpam-65	386	7	.	.	PUNCT
ejpam-65	387	1	.	.	PUNCT
ejpam-65	388	1	,	,	PUNCT
ejpam-65	388	2	vr	vr	PROPN
ejpam-65	388	3	,	,	PUNCT
ejpam-65	388	4	so	so	ADV
ejpam-65	388	5	v	v	ADP
ejpam-65	388	6	∈	∈	PROPN
ejpam-65	388	7	α1	α1	PROPN
ejpam-65	388	8	◦	◦	NOUN
ejpam-65	388	9	w1	w1	NOUN
ejpam-65	388	10	+	+	X
ejpam-65	388	11	·	·	PUNCT
ejpam-65	388	12	·	·	PUNCT
ejpam-65	388	13	·	·	PUNCT
ejpam-65	389	1	+	+	NUM
ejpam-65	389	2	αm	αm	NOUN
ejpam-65	389	3	◦	◦	NOUN
ejpam-65	389	4	wm	wm	X
ejpam-65	389	5	+	+	CCONJ
ejpam-65	389	6	β1	β1	VERB
ejpam-65	389	7	◦	◦	NOUN
ejpam-65	389	8	v1	v1	NOUN
ejpam-65	389	9	+	+	X
ejpam-65	389	10	·	·	PUNCT
ejpam-65	389	11	·	·	PUNCT
ejpam-65	389	12	·	·	PUNCT
ejpam-65	389	13	+	+	NUM
ejpam-65	389	14	βr	βr	ADP
ejpam-65	389	15	◦	◦	NOUN
ejpam-65	389	16	vr	vr	PROPN
ejpam-65	389	17	,	,	PUNCT
ejpam-65	389	18	then	then	ADV
ejpam-65	389	19	v	v	ADP
ejpam-65	389	20	∈	∈	PROPN
ejpam-65	389	21	α1	α1	PROPN
ejpam-65	389	22	◦	◦	NOUN
ejpam-65	389	23	w1	w1	NOUN
ejpam-65	389	24	+	+	X
ejpam-65	389	25	·	·	PUNCT
ejpam-65	389	26	·	·	PUNCT
ejpam-65	389	27	·	·	PUNCT
ejpam-65	390	1	+	+	NUM
ejpam-65	390	2	αm	αm	NOUN
ejpam-65	390	3	◦	◦	NOUN
ejpam-65	390	4	wm	wm	X
ejpam-65	390	5	+	+	CCONJ
ejpam-65	390	6	β1	β1	VERB
ejpam-65	390	7	◦	◦	NOUN
ejpam-65	390	8	v1	v1	NOUN
ejpam-65	390	9	+	+	X
ejpam-65	390	10	·	·	PUNCT
ejpam-65	390	11	·	·	PUNCT
ejpam-65	390	12	·	·	PUNCT
ejpam-65	390	13	+	+	NUM
ejpam-65	390	14	βr	βr	AUX
ejpam-65	390	15	◦	◦	NOUN
ejpam-65	390	16	vr	vr	PROPN
ejpam-65	390	17	⊆	⊆	NUM
ejpam-65	390	18	β1	β1	PROPN
ejpam-65	390	19	◦	◦	NOUN
ejpam-65	390	20	v1	v1	NOUN
ejpam-65	390	21	+	+	X
ejpam-65	390	22	·	·	PUNCT
ejpam-65	390	23	·	·	PUNCT
ejpam-65	390	24	·	·	PUNCT
ejpam-65	390	25	+	+	NUM
ejpam-65	390	26	βr	βr	ADP
ejpam-65	390	27	◦	◦	NOUN
ejpam-65	390	28	vr	vr	X
ejpam-65	390	29	=	=	SYM
ejpam-65	390	30	β1	β1	PROPN
ejpam-65	390	31	∗	∗	NOUN
ejpam-65	390	32	v̄1	v̄1	X
ejpam-65	391	1	+	+	X
ejpam-65	391	2	·	·	PUNCT
ejpam-65	391	3	·	·	PUNCT
ejpam-65	391	4	·	·	PUNCT
ejpam-65	391	5	+	+	NUM
ejpam-65	391	6	βr	βr	NUM
ejpam-65	391	7	∗	∗	NOUN
ejpam-65	391	8	v̄r	v̄r	NOUN
ejpam-65	391	9	,	,	PUNCT
ejpam-65	391	10	(	(	PUNCT
ejpam-65	391	11	since	since	SCONJ
ejpam-65	391	12	αi	αi	PRON
ejpam-65	391	13	◦	◦	VERB
ejpam-65	391	14	wi	wi	PROPN
ejpam-65	391	15	=	=	PUNCT
ejpam-65	391	16	αi	αi	PROPN
ejpam-65	391	17	◦	◦	PROPN
ejpam-65	391	18	wi	wi	PROPN
ejpam-65	391	19	+	+	PROPN
ejpam-65	391	20	w	w	PROPN
ejpam-65	391	21	⊆w	⊆w	NOUN
ejpam-65	391	22	)	)	PUNCT
ejpam-65	391	23	.	.	PUNCT
ejpam-65	392	1	thus	thus	ADV
ejpam-65	392	2	v̄1	v̄1	PRON
ejpam-65	392	3	,	,	PUNCT
ejpam-65	392	4	.	.	PUNCT
ejpam-65	392	5	.	.	PUNCT
ejpam-65	392	6	.	.	PUNCT
ejpam-65	393	1	,	,	PUNCT
ejpam-65	393	2	v̄r	v̄r	PRON
ejpam-65	393	3	span	span	VERB
ejpam-65	393	4	v	v	PROPN
ejpam-65	393	5	/	/	SYM
ejpam-65	393	6	w	w	NOUN
ejpam-65	393	7	.	.	PUNCT
ejpam-65	394	1	we	we	PRON
ejpam-65	394	2	claim	claim	VERB
ejpam-65	394	3	that	that	SCONJ
ejpam-65	394	4	they	they	PRON
ejpam-65	394	5	are	be	AUX
ejpam-65	394	6	linearly	linearly	ADV
ejpam-65	394	7	independent	independent	ADJ
ejpam-65	394	8	,	,	PUNCT
ejpam-65	394	9	for	for	ADP
ejpam-65	394	10	if	if	SCONJ
ejpam-65	394	11	0	0	NUM
ejpam-65	394	12	∈	∈	PROPN
ejpam-65	394	13	γ1	γ1	NOUN
ejpam-65	394	14	∗	∗	NOUN
ejpam-65	394	15	v̄1	v̄1	X
ejpam-65	394	16	+	+	X
ejpam-65	394	17	·	·	PUNCT
ejpam-65	394	18	·	·	PUNCT
ejpam-65	394	19	·	·	PUNCT
ejpam-65	395	1	+	+	CCONJ
ejpam-65	395	2	γr	γr	PROPN
ejpam-65	395	3	∗	∗	NOUN
ejpam-65	395	4	v̄r	v̄r	PROPN
ejpam-65	395	5	,	,	PUNCT
ejpam-65	395	6	then	then	ADV
ejpam-65	395	7	0	0	NUM
ejpam-65	395	8	∈	∈	PROPN
ejpam-65	395	9	γ1	γ1	NOUN
ejpam-65	395	10	◦	◦	NOUN
ejpam-65	395	11	v1	v1	PROPN
ejpam-65	395	12	+	+	X
ejpam-65	395	13	·	·	PUNCT
ejpam-65	395	14	·	·	PUNCT
ejpam-65	395	15	·	·	PUNCT
ejpam-65	395	16	+	+	NUM
ejpam-65	395	17	γr	γr	AUX
ejpam-65	395	18	◦	◦	NOUN
ejpam-65	395	19	vr	vr	PROPN
ejpam-65	395	20	+	+	NOUN
ejpam-65	395	21	w	w	PROPN
ejpam-65	395	22	⊆	⊆	NUM
ejpam-65	395	23	γ1	γ1	NOUN
ejpam-65	395	24	◦	◦	NOUN
ejpam-65	395	25	v1	v1	PROPN
ejpam-65	395	26	+	+	X
ejpam-65	395	27	·	·	PUNCT
ejpam-65	395	28	·	·	PUNCT
ejpam-65	395	29	·	·	PUNCT
ejpam-65	396	1	+	+	NUM
ejpam-65	396	2	γr	γr	X
ejpam-65	396	3	◦	◦	NOUN
ejpam-65	396	4	vr	vr	NOUN
ejpam-65	396	5	+	+	CCONJ
ejpam-65	396	6	λ1	λ1	PROPN
ejpam-65	396	7	◦	◦	NOUN
ejpam-65	396	8	w1	w1	NOUN
ejpam-65	396	9	+	+	CCONJ
ejpam-65	396	10	·	·	PUNCT
ejpam-65	396	11	·	·	PUNCT
ejpam-65	396	12	·	·	PUNCT
ejpam-65	396	13	+	+	NUM
ejpam-65	396	14	λm	λm	ADP
ejpam-65	396	15	◦	◦	NOUN
ejpam-65	396	16	wm	wm	PROPN
ejpam-65	396	17	,	,	PUNCT
ejpam-65	396	18	which	which	PRON
ejpam-65	396	19	by	by	ADP
ejpam-65	396	20	the	the	DET
ejpam-65	396	21	linear	linear	ADJ
ejpam-65	396	22	independence	independence	NOUN
ejpam-65	396	23	of	of	ADP
ejpam-65	396	24	the	the	DET
ejpam-65	396	25	set	set	NOUN
ejpam-65	396	26	{	{	PUNCT
ejpam-65	396	27	w1	w1	NOUN
ejpam-65	396	28	,	,	PUNCT
ejpam-65	396	29	.	.	PUNCT
ejpam-65	396	30	.	.	PUNCT
ejpam-65	396	31	.	.	PUNCT
ejpam-65	397	1	,	,	PUNCT
ejpam-65	397	2	wm	wm	PROPN
ejpam-65	397	3	,	,	PUNCT
ejpam-65	397	4	v1	v1	PROPN
ejpam-65	397	5	,	,	PUNCT
ejpam-65	397	6	.	.	PUNCT
ejpam-65	397	7	.	.	PUNCT
ejpam-65	398	1	.	.	PUNCT
ejpam-65	399	1	,	,	PUNCT
ejpam-65	399	2	vr	vr	NOUN
ejpam-65	399	3	}	}	PUNCT
ejpam-65	399	4	forces	force	NOUN
ejpam-65	399	5	γ1	γ1	NOUN
ejpam-65	399	6	=	=	SYM
ejpam-65	399	7	·	·	PUNCT
ejpam-65	399	8	·	·	PUNCT
ejpam-65	399	9	·	·	PUNCT
ejpam-65	400	1	=	=	PUNCT
ejpam-65	400	2	γr	γr	X
ejpam-65	400	3	=	=	PUNCT
ejpam-65	400	4	λ1	λ1	PROPN
ejpam-65	400	5	=	=	SYM
ejpam-65	400	6	·	·	PUNCT
ejpam-65	400	7	·	·	PUNCT
ejpam-65	400	8	·	·	PUNCT
ejpam-65	401	1	=	=	PUNCT
ejpam-65	401	2	λm	λm	NOUN
ejpam-65	401	3	=	=	SYM
ejpam-65	401	4	0	0	X
ejpam-65	401	5	.	.	PUNCT
ejpam-65	402	1	we	we	PRON
ejpam-65	402	2	have	have	AUX
ejpam-65	402	3	shown	show	VERB
ejpam-65	402	4	that	that	SCONJ
ejpam-65	402	5	v	v	NOUN
ejpam-65	402	6	/	/	SYM
ejpam-65	402	7	w	w	PROPN
ejpam-65	402	8	has	have	VERB
ejpam-65	402	9	a	a	DET
ejpam-65	402	10	basis	basis	NOUN
ejpam-65	402	11	of	of	ADP
ejpam-65	402	12	r	r	NOUN
ejpam-65	402	13	elements	element	NOUN
ejpam-65	402	14	,	,	PUNCT
ejpam-65	402	15	and	and	CCONJ
ejpam-65	402	16	dimv	dimv	NOUN
ejpam-65	402	17	/	/	SYM
ejpam-65	402	18	w	w	NOUN
ejpam-65	402	19	=	=	SYM
ejpam-65	402	20	dimv	dimv	NOUN
ejpam-65	402	21	−m	−m	NOUN
ejpam-65	402	22	=	=	SYM
ejpam-65	402	23	dimv	dimv	NOUN
ejpam-65	402	24	−	−	NOUN
ejpam-65	402	25	dimw	dimw	NOUN
ejpam-65	402	26	.	.	PUNCT
ejpam-65	402	27	example	example	NOUN
ejpam-65	403	1	3.2	3.2	NUM
ejpam-65	403	2	.	.	PUNCT
ejpam-65	404	1	let	let	VERB
ejpam-65	404	2	β	β	X
ejpam-65	404	3	=	=	PUNCT
ejpam-65	404	4	{	{	PUNCT
ejpam-65	404	5	e1	e1	PROPN
ejpam-65	404	6	,	,	PUNCT
ejpam-65	404	7	.	.	PUNCT
ejpam-65	404	8	.	.	PUNCT
ejpam-65	404	9	.	.	PUNCT
ejpam-65	405	1	,	,	PUNCT
ejpam-65	405	2	en	en	AUX
ejpam-65	405	3	}	}	PUNCT
ejpam-65	405	4	be	be	AUX
ejpam-65	405	5	a	a	DET
ejpam-65	405	6	basis	basis	NOUN
ejpam-65	405	7	for	for	ADP
ejpam-65	405	8	a	a	DET
ejpam-65	405	9	classical	classical	ADJ
ejpam-65	405	10	vector	vector	NOUN
ejpam-65	405	11	space	space	NOUN
ejpam-65	405	12	(	(	PUNCT
ejpam-65	405	13	v,+	v,+	NUM
ejpam-65	405	14	,	,	PUNCT
ejpam-65	405	15	.,k	.,k	NUM
ejpam-65	405	16	)	)	PUNCT
ejpam-65	405	17	and	and	CCONJ
ejpam-65	405	18	let	let	VERB
ejpam-65	405	19	β́	β́	NOUN
ejpam-65	405	20	=	=	PUNCT
ejpam-65	405	21	{	{	PUNCT
ejpam-65	405	22	e1	e1	PROPN
ejpam-65	405	23	,	,	PUNCT
ejpam-65	405	24	.	.	PUNCT
ejpam-65	405	25	.	.	PUNCT
ejpam-65	406	1	.	.	PUNCT
ejpam-65	407	1	,	,	PUNCT
ejpam-65	407	2	ek	ek	PROPN
ejpam-65	407	3	}	}	PUNCT
ejpam-65	407	4	be	be	AUX
ejpam-65	407	5	a	a	DET
ejpam-65	407	6	basis	basis	NOUN
ejpam-65	407	7	for	for	ADP
ejpam-65	407	8	subspace	subspace	NOUN
ejpam-65	407	9	p	p	NOUN
ejpam-65	407	10	of	of	ADP
ejpam-65	407	11	v	v	NOUN
ejpam-65	407	12	.	.	PUNCT
ejpam-65	408	1	then	then	ADV
ejpam-65	408	2	β∗	β∗	NOUN
ejpam-65	408	3	=	=	SYM
ejpam-65	408	4	{	{	PUNCT
ejpam-65	408	5	ek+1	ek+1	PROPN
ejpam-65	408	6	,	,	PUNCT
ejpam-65	408	7	.	.	PUNCT
ejpam-65	408	8	.	.	PUNCT
ejpam-65	408	9	.	.	PUNCT
ejpam-65	409	1	,	,	PUNCT
ejpam-65	409	2	en	en	ADP
ejpam-65	409	3	}	}	PUNCT
ejpam-65	409	4	is	be	AUX
ejpam-65	409	5	a	a	DET
ejpam-65	409	6	basis	basis	NOUN
ejpam-65	409	7	for	for	ADP
ejpam-65	409	8	hypervector	hypervector	NOUN
ejpam-65	409	9	space	space	NOUN
ejpam-65	409	10	(	(	PUNCT
ejpam-65	409	11	v,+	v,+	NUM
ejpam-65	409	12	,	,	PUNCT
ejpam-65	409	13	◦	◦	NOUN
ejpam-65	409	14	,	,	PUNCT
ejpam-65	409	15	k	k	NOUN
ejpam-65	409	16	)	)	PUNCT
ejpam-65	409	17	in	in	ADP
ejpam-65	409	18	example	example	NOUN
ejpam-65	409	19	2.3	2.3	NUM
ejpam-65	409	20	,	,	PUNCT
ejpam-65	409	21	because	because	SCONJ
ejpam-65	409	22	for	for	SCONJ
ejpam-65	409	23	every	every	DET
ejpam-65	409	24	x	x	SYM
ejpam-65	409	25	∈	∈	PROPN
ejpam-65	409	26	v	v	NOUN
ejpam-65	409	27	we	we	PRON
ejpam-65	409	28	have	have	VERB
ejpam-65	409	29	:	:	PUNCT
ejpam-65	410	1	x	x	SYM
ejpam-65	410	2	=	=	PUNCT
ejpam-65	410	3	a1.e1	a1.e1	PROPN
ejpam-65	410	4	+	+	CCONJ
ejpam-65	410	5	·	·	PUNCT
ejpam-65	410	6	·	·	PUNCT
ejpam-65	410	7	·	·	PUNCT
ejpam-65	410	8	+	+	NUM
ejpam-65	411	1	ak.ek	ak.ek	NOUN
ejpam-65	411	2	+	+	CCONJ
ejpam-65	411	3	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	411	4	+	+	CCONJ
ejpam-65	411	5	·	·	PUNCT
ejpam-65	411	6	·	·	PUNCT
ejpam-65	411	7	·	·	PUNCT
ejpam-65	411	8	+	+	NUM
ejpam-65	411	9	an.en	an.en	PROPN
ejpam-65	411	10	∈	∈	NOUN
ejpam-65	411	11	a1.e1	a1.e1	PROPN
ejpam-65	411	12	+	+	CCONJ
ejpam-65	411	13	·	·	PUNCT
ejpam-65	411	14	·	·	PUNCT
ejpam-65	411	15	·	·	PUNCT
ejpam-65	411	16	+	+	NUM
ejpam-65	412	1	ak.ek	ak.ek	NOUN
ejpam-65	412	2	+	+	CCONJ
ejpam-65	412	3	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	412	4	+	+	CCONJ
ejpam-65	412	5	·	·	PUNCT
ejpam-65	412	6	·	·	PUNCT
ejpam-65	412	7	·	·	PUNCT
ejpam-65	412	8	+	+	NUM
ejpam-65	413	1	an.en	an.en	ADV
ejpam-65	413	2	+	+	NOUN
ejpam-65	413	3	p	p	X
ejpam-65	413	4	=	=	PUNCT
ejpam-65	413	5	ak+1.ek+1	ak+1.ek+1	PROPN
ejpam-65	413	6	+	+	CCONJ
ejpam-65	413	7	·	·	PUNCT
ejpam-65	413	8	·	·	PUNCT
ejpam-65	413	9	·	·	PUNCT
ejpam-65	414	1	+	+	NUM
ejpam-65	414	2	an.en	an.en	ADV
ejpam-65	414	3	+	+	NOUN
ejpam-65	414	4	p	p	NOUN
ejpam-65	414	5	=	=	PUNCT
ejpam-65	414	6	ak+1	ak+1	VERB
ejpam-65	414	7	◦	◦	NOUN
ejpam-65	414	8	ek+1	ek+1	NOUN
ejpam-65	414	9	+	+	X
ejpam-65	414	10	·	·	PUNCT
ejpam-65	414	11	·	·	PUNCT
ejpam-65	414	12	·	·	PUNCT
ejpam-65	414	13	+	+	CCONJ
ejpam-65	414	14	an	an	DET
ejpam-65	414	15	◦	◦	NOUN
ejpam-65	414	16	en	en	X
ejpam-65	414	17	,	,	PUNCT
ejpam-65	414	18	thus	thus	ADV
ejpam-65	414	19	β∗	β∗	NOUN
ejpam-65	414	20	span	span	NOUN
ejpam-65	414	21	(	(	PUNCT
ejpam-65	414	22	v,+	v,+	NUM
ejpam-65	414	23	,	,	PUNCT
ejpam-65	414	24	◦	◦	NOUN
ejpam-65	414	25	,	,	PUNCT
ejpam-65	414	26	k	k	NOUN
ejpam-65	414	27	)	)	PUNCT
ejpam-65	414	28	.	.	PUNCT
ejpam-65	415	1	moreover	moreover	ADV
ejpam-65	415	2	β∗	β∗	NOUN
ejpam-65	415	3	is	be	AUX
ejpam-65	415	4	linearly	linearly	ADV
ejpam-65	415	5	independent	independent	ADJ
ejpam-65	415	6	,	,	PUNCT
ejpam-65	415	7	because	because	SCONJ
ejpam-65	415	8	0	0	NUM
ejpam-65	415	9	∈	∈	NOUN
ejpam-65	415	10	ak+1	ak+1	VERB
ejpam-65	415	11	◦	◦	NOUN
ejpam-65	415	12	ek+1	ek+1	NOUN
ejpam-65	415	13	+	+	X
ejpam-65	415	14	·	·	PUNCT
ejpam-65	415	15	·	·	PUNCT
ejpam-65	415	16	·	·	PUNCT
ejpam-65	416	1	+	+	CCONJ
ejpam-65	416	2	an	an	DET
ejpam-65	416	3	◦	◦	NOUN
ejpam-65	416	4	en	en	X
ejpam-65	416	5	=	=	NOUN
ejpam-65	416	6	⇒	⇒	NOUN
ejpam-65	416	7	0	0	NUM
ejpam-65	416	8	∈	∈	PROPN
ejpam-65	416	9	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	416	10	+	+	CCONJ
ejpam-65	416	11	·	·	PUNCT
ejpam-65	416	12	·	·	PUNCT
ejpam-65	416	13	·	·	PUNCT
ejpam-65	417	1	+	+	NUM
ejpam-65	417	2	an.en	an.en	ADV
ejpam-65	417	3	+	+	NOUN
ejpam-65	417	4	p	p	NOUN
ejpam-65	417	5	=	=	NOUN
ejpam-65	417	6	⇒	⇒	NOUN
ejpam-65	417	7	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	417	8	+	+	CCONJ
ejpam-65	417	9	·	·	PUNCT
ejpam-65	417	10	·	·	PUNCT
ejpam-65	417	11	·	·	PUNCT
ejpam-65	417	12	+	+	NUM
ejpam-65	417	13	an.en	an.en	PROPN
ejpam-65	417	14	∈	∈	NOUN
ejpam-65	417	15	p	p	NOUN
ejpam-65	417	16	=	=	NOUN
ejpam-65	417	17	⇒	⇒	NOUN
ejpam-65	417	18	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	417	19	+	+	CCONJ
ejpam-65	417	20	·	·	PUNCT
ejpam-65	417	21	·	·	PUNCT
ejpam-65	417	22	·	·	PUNCT
ejpam-65	418	1	+	+	NUM
ejpam-65	418	2	an.en	an.en	ADJ
ejpam-65	418	3	=	=	SYM
ejpam-65	418	4	a1.e1	a1.e1	PROPN
ejpam-65	418	5	+	+	CCONJ
ejpam-65	418	6	·	·	PUNCT
ejpam-65	418	7	·	·	PUNCT
ejpam-65	418	8	·	·	PUNCT
ejpam-65	418	9	+	+	NUM
ejpam-65	418	10	ak.ek	ak.ek	NOUN
ejpam-65	418	11	=	=	NOUN
ejpam-65	418	12	⇒	⇒	VERB
ejpam-65	418	13	a1.e1	a1.e1	PROPN
ejpam-65	418	14	+	+	CCONJ
ejpam-65	418	15	·	·	PUNCT
ejpam-65	418	16	·	·	PUNCT
ejpam-65	418	17	·	·	PUNCT
ejpam-65	418	18	+	+	NUM
ejpam-65	418	19	ak.ek	ak.ek	NOUN
ejpam-65	418	20	−	−	PROPN
ejpam-65	418	21	ak+1.ek+1	ak+1.ek+1	NOUN
ejpam-65	418	22	−	−	PROPN
ejpam-65	418	23	·	·	PUNCT
ejpam-65	418	24	·	·	PUNCT
ejpam-65	418	25	·	·	PUNCT
ejpam-65	419	1	−	−	PUNCT
ejpam-65	419	2	an.en	an.en	ADV
ejpam-65	419	3	=	=	SYM
ejpam-65	419	4	0	0	PUNCT
ejpam-65	420	1	=	=	NOUN
ejpam-65	420	2	⇒	⇒	NOUN
ejpam-65	420	3	a1	a1	NOUN
ejpam-65	420	4	=	=	SYM
ejpam-65	420	5	·	·	PUNCT
ejpam-65	420	6	·	·	PUNCT
ejpam-65	420	7	·	·	PUNCT
ejpam-65	421	1	=	=	SYM
ejpam-65	421	2	ak	ak	PROPN
ejpam-65	421	3	=	=	PUNCT
ejpam-65	421	4	ak+1	ak+1	NOUN
ejpam-65	421	5	=	=	SYM
ejpam-65	421	6	·	·	PUNCT
ejpam-65	421	7	·	·	PUNCT
ejpam-65	421	8	·	·	PUNCT
ejpam-65	422	1	=	=	PUNCT
ejpam-65	422	2	an	an	PRON
ejpam-65	422	3	=	=	NOUN
ejpam-65	422	4	0	0	PROPN
ejpam-65	422	5	.	.	PUNCT
ejpam-65	422	6	therefor	therefor	ADP
ejpam-65	422	7	β∗	β∗	PROPN
ejpam-65	422	8	is	be	AUX
ejpam-65	422	9	a	a	DET
ejpam-65	422	10	basis	basis	NOUN
ejpam-65	422	11	.	.	PUNCT
ejpam-65	423	1	r.	r.	PROPN
ejpam-65	423	2	ameri	ameri	PROPN
ejpam-65	423	3	,	,	PUNCT
ejpam-65	423	4	o.r	o.r	PROPN
ejpam-65	423	5	.	.	PROPN
ejpam-65	423	6	dehghan	dehghan	PROPN
ejpam-65	423	7	/	/	SYM
ejpam-65	423	8	eur	eur	PROPN
ejpam-65	423	9	.	.	PUNCT
ejpam-65	424	1	j.	j.	PROPN
ejpam-65	424	2	pure	pure	PROPN
ejpam-65	424	3	appl	appl	PROPN
ejpam-65	424	4	.	.	PROPN
ejpam-65	424	5	math	math	PROPN
ejpam-65	424	6	,	,	PUNCT
ejpam-65	424	7	1	1	NUM
ejpam-65	424	8	(	(	PUNCT
ejpam-65	424	9	2008	2008	NUM
ejpam-65	424	10	)	)	PUNCT
ejpam-65	424	11	,	,	PUNCT
ejpam-65	424	12	(	(	PUNCT
ejpam-65	424	13	32	32	NUM
ejpam-65	424	14	-	-	SYM
ejpam-65	424	15	50	50	NUM
ejpam-65	424	16	)	)	PUNCT
ejpam-65	425	1	42	42	NUM
ejpam-65	425	2	example	example	NOUN
ejpam-65	425	3	3.3	3.3	NUM
ejpam-65	425	4	.	.	PUNCT
ejpam-65	426	1	let	let	VERB
ejpam-65	426	2	(	(	PUNCT
ejpam-65	426	3	k[x],+	k[x],+	NUM
ejpam-65	426	4	,	,	PUNCT
ejpam-65	426	5	·	·	PUNCT
ejpam-65	426	6	,	,	PUNCT
ejpam-65	426	7	k	k	NOUN
ejpam-65	426	8	)	)	PUNCT
ejpam-65	426	9	be	be	VERB
ejpam-65	426	10	the	the	DET
ejpam-65	426	11	vector	vector	NOUN
ejpam-65	426	12	space	space	NOUN
ejpam-65	426	13	of	of	ADP
ejpam-65	426	14	all	all	DET
ejpam-65	426	15	polynomials	polynomial	NOUN
ejpam-65	426	16	of	of	ADP
ejpam-65	426	17	degree	degree	NOUN
ejpam-65	426	18	less	less	ADJ
ejpam-65	426	19	than	than	ADP
ejpam-65	426	20	n	n	X
ejpam-65	426	21	in	in	ADP
ejpam-65	426	22	x.	x.	NOUN
ejpam-65	426	23	define	define	VERB
ejpam-65	426	24	the	the	DET
ejpam-65	426	25	external	external	ADJ
ejpam-65	426	26	operation	operation	NOUN
ejpam-65	426	27	”	"	PUNCT
ejpam-65	426	28	◦	◦	NOUN
ejpam-65	426	29	”	"	PUNCT
ejpam-65	426	30	on	on	ADP
ejpam-65	426	31	k[x	k[x	PROPN
ejpam-65	426	32	]	]	PUNCT
ejpam-65	426	33	by	by	ADP
ejpam-65	426	34	{	{	PUNCT
ejpam-65	426	35	◦	◦	NOUN
ejpam-65	426	36	:	:	PUNCT
ejpam-65	426	37	k	k	PROPN
ejpam-65	426	38	×k[x	×k[x	PROPN
ejpam-65	426	39	]	]	X
ejpam-65	426	40	−→	−→	NOUN
ejpam-65	426	41	p∗	p∗	NOUN
ejpam-65	426	42	(	(	PUNCT
ejpam-65	426	43	k[x	k[x	PROPN
ejpam-65	426	44	]	]	X
ejpam-65	426	45	)	)	PUNCT
ejpam-65	426	46	a	a	DET
ejpam-65	426	47	◦	◦	NOUN
ejpam-65	426	48	f(x	f(x	PROPN
ejpam-65	426	49	)	)	PUNCT
ejpam-65	426	50	=	=	PUNCT
ejpam-65	427	1	a	a	DET
ejpam-65	427	2	·	·	PUNCT
ejpam-65	427	3	f(x	f(x	PROPN
ejpam-65	427	4	)	)	PUNCT
ejpam-65	428	1	+	+	CCONJ
ejpam-65	429	1	〈	〈	PROPN
ejpam-65	429	2	x	x	SYM
ejpam-65	429	3	〉	〉	PROPN
ejpam-65	429	4	,	,	PUNCT
ejpam-65	429	5	where	where	SCONJ
ejpam-65	429	6	〈	〈	PROPN
ejpam-65	429	7	x	x	PROPN
ejpam-65	429	8	〉	〉	NOUN
ejpam-65	429	9	=	=	SYM
ejpam-65	429	10	{	{	PUNCT
ejpam-65	429	11	kx	kx	X
ejpam-65	429	12	:	:	PUNCT
ejpam-65	429	13	k	k	PROPN
ejpam-65	429	14	∈	∈	PROPN
ejpam-65	429	15	k	k	X
ejpam-65	429	16	}	}	PUNCT
ejpam-65	429	17	.then	.then	VERB
ejpam-65	430	1	it	it	PRON
ejpam-65	430	2	is	be	AUX
ejpam-65	430	3	easy	easy	ADJ
ejpam-65	430	4	to	to	PART
ejpam-65	430	5	verify	verify	VERB
ejpam-65	430	6	that	that	SCONJ
ejpam-65	430	7	(	(	PUNCT
ejpam-65	430	8	k[x],+	k[x],+	NUM
ejpam-65	430	9	,	,	PUNCT
ejpam-65	430	10	◦	◦	NOUN
ejpam-65	430	11	,	,	PUNCT
ejpam-65	430	12	k	k	NOUN
ejpam-65	430	13	)	)	PUNCT
ejpam-65	430	14	is	be	AUX
ejpam-65	430	15	a	a	DET
ejpam-65	430	16	strongly	strongly	ADV
ejpam-65	430	17	distributive	distributive	ADJ
ejpam-65	430	18	hypervector	hypervector	NOUN
ejpam-65	430	19	space	space	NOUN
ejpam-65	430	20	and	and	CCONJ
ejpam-65	430	21	β	β	X
ejpam-65	430	22	=	=	PUNCT
ejpam-65	430	23	{	{	PUNCT
ejpam-65	430	24	1	1	NUM
ejpam-65	430	25	,	,	PUNCT
ejpam-65	430	26	x2	x2	PROPN
ejpam-65	430	27	,	,	PUNCT
ejpam-65	430	28	x3	x3	ADJ
ejpam-65	430	29	,	,	PUNCT
ejpam-65	430	30	.	.	PUNCT
ejpam-65	430	31	.	.	PUNCT
ejpam-65	431	1	.	.	PUNCT
ejpam-65	432	1	,	,	PUNCT
ejpam-65	432	2	xn−1	xn−1	PROPN
ejpam-65	432	3	}	}	PUNCT
ejpam-65	432	4	is	be	AUX
ejpam-65	432	5	a	a	DET
ejpam-65	432	6	basis	basis	NOUN
ejpam-65	432	7	for	for	ADP
ejpam-65	432	8	(	(	PUNCT
ejpam-65	432	9	k[x],+	k[x],+	PROPN
ejpam-65	432	10	,	,	PUNCT
ejpam-65	432	11	◦	◦	NOUN
ejpam-65	432	12	,	,	PUNCT
ejpam-65	432	13	k	k	NOUN
ejpam-65	432	14	)	)	PUNCT
ejpam-65	432	15	.	.	PUNCT
ejpam-65	433	1	definition	definition	NOUN
ejpam-65	433	2	3.6	3.6	NUM
ejpam-65	433	3	.	.	PUNCT
ejpam-65	434	1	let	let	VERB
ejpam-65	434	2	v	v	NOUN
ejpam-65	434	3	and	and	CCONJ
ejpam-65	434	4	w	w	NOUN
ejpam-65	434	5	be	be	AUX
ejpam-65	434	6	hypervector	hypervector	NOUN
ejpam-65	434	7	spaces	space	NOUN
ejpam-65	434	8	over	over	ADP
ejpam-65	434	9	k.	k.	PROPN
ejpam-65	435	1	a	a	DET
ejpam-65	435	2	mapping	mapping	NOUN
ejpam-65	435	3	t	t	NOUN
ejpam-65	435	4	:	:	PUNCT
ejpam-65	435	5	v	v	NOUN
ejpam-65	435	6	−→w	−→w	NOUN
ejpam-65	435	7	is	be	AUX
ejpam-65	435	8	called	call	VERB
ejpam-65	435	9	(	(	PUNCT
ejpam-65	435	10	i	i	NOUN
ejpam-65	435	11	)	)	PUNCT
ejpam-65	435	12	weak	weak	ADJ
ejpam-65	435	13	linear	linear	ADJ
ejpam-65	435	14	transformation	transformation	NOUN
ejpam-65	435	15	iff	iff	PROPN
ejpam-65	435	16	t	t	PROPN
ejpam-65	435	17	(	(	PUNCT
ejpam-65	435	18	x+	x+	PROPN
ejpam-65	435	19	y	y	NOUN
ejpam-65	435	20	)	)	PUNCT
ejpam-65	436	1	=	=	SYM
ejpam-65	436	2	t	t	PROPN
ejpam-65	436	3	(	(	PUNCT
ejpam-65	436	4	x	x	X
ejpam-65	436	5	)	)	PUNCT
ejpam-65	437	1	+	+	NUM
ejpam-65	437	2	t	t	PROPN
ejpam-65	437	3	(	(	PUNCT
ejpam-65	437	4	y	y	NOUN
ejpam-65	437	5	)	)	PUNCT
ejpam-65	437	6	and	and	CCONJ
ejpam-65	437	7	t	t	PROPN
ejpam-65	437	8	(	(	PUNCT
ejpam-65	437	9	a	a	DET
ejpam-65	437	10	◦	◦	NOUN
ejpam-65	437	11	x	x	SYM
ejpam-65	437	12	)	)	PUNCT
ejpam-65	437	13	∩	∩	NOUN
ejpam-65	437	14	a	a	DET
ejpam-65	437	15	◦	◦	NOUN
ejpam-65	437	16	t	t	X
ejpam-65	437	17	(	(	PUNCT
ejpam-65	437	18	x	x	X
ejpam-65	437	19	)	)	PUNCT
ejpam-65	437	20	6=	6=	ADP
ejpam-65	437	21	∅	∅	NOUN
ejpam-65	437	22	,	,	PUNCT
ejpam-65	437	23	∀x	∀x	NUM
ejpam-65	437	24	,	,	PUNCT
ejpam-65	437	25	y	y	PROPN
ejpam-65	437	26	∈	∈	PROPN
ejpam-65	437	27	v	v	NOUN
ejpam-65	437	28	,	,	PUNCT
ejpam-65	437	29	a	a	DET
ejpam-65	437	30	∈	∈	PROPN
ejpam-65	437	31	k	k	NOUN
ejpam-65	437	32	,	,	PUNCT
ejpam-65	437	33	(	(	PUNCT
ejpam-65	437	34	ii	ii	NOUN
ejpam-65	437	35	)	)	PUNCT
ejpam-65	437	36	linear	linear	PROPN
ejpam-65	437	37	transformation	transformation	NOUN
ejpam-65	437	38	iff	iff	PROPN
ejpam-65	437	39	t	t	PROPN
ejpam-65	437	40	(	(	PUNCT
ejpam-65	437	41	x+	x+	PROPN
ejpam-65	437	42	y	y	NOUN
ejpam-65	437	43	)	)	PUNCT
ejpam-65	437	44	=	=	SYM
ejpam-65	437	45	t	t	PROPN
ejpam-65	437	46	(	(	PUNCT
ejpam-65	437	47	x	x	X
ejpam-65	437	48	)	)	PUNCT
ejpam-65	438	1	+	+	NUM
ejpam-65	438	2	t	t	PROPN
ejpam-65	438	3	(	(	PUNCT
ejpam-65	438	4	y	y	NOUN
ejpam-65	438	5	)	)	PUNCT
ejpam-65	438	6	and	and	CCONJ
ejpam-65	438	7	t	t	PROPN
ejpam-65	438	8	(	(	PUNCT
ejpam-65	438	9	a	a	DET
ejpam-65	438	10	◦	◦	NOUN
ejpam-65	438	11	x	x	SYM
ejpam-65	438	12	)	)	PUNCT
ejpam-65	438	13	⊆	⊆	NUM
ejpam-65	438	14	a	a	DET
ejpam-65	438	15	◦	◦	NOUN
ejpam-65	438	16	t	t	X
ejpam-65	438	17	(	(	PUNCT
ejpam-65	438	18	x	x	NOUN
ejpam-65	438	19	)	)	PUNCT
ejpam-65	438	20	,	,	PUNCT
ejpam-65	438	21	∀x	∀x	NUM
ejpam-65	438	22	,	,	PUNCT
ejpam-65	438	23	y	y	PROPN
ejpam-65	438	24	∈	∈	PROPN
ejpam-65	438	25	v	v	NOUN
ejpam-65	438	26	,	,	PUNCT
ejpam-65	438	27	a	a	DET
ejpam-65	438	28	∈	∈	PROPN
ejpam-65	438	29	k	k	NOUN
ejpam-65	438	30	,	,	PUNCT
ejpam-65	438	31	(	(	PUNCT
ejpam-65	438	32	iii	iii	NOUN
ejpam-65	438	33	)	)	PUNCT
ejpam-65	438	34	good	good	ADJ
ejpam-65	438	35	linear	linear	PROPN
ejpam-65	438	36	transformation	transformation	NOUN
ejpam-65	438	37	iff	iff	PROPN
ejpam-65	438	38	t	t	PROPN
ejpam-65	438	39	(	(	PUNCT
ejpam-65	438	40	x+	x+	PROPN
ejpam-65	438	41	y	y	NOUN
ejpam-65	438	42	)	)	PUNCT
ejpam-65	439	1	=	=	SYM
ejpam-65	439	2	t	t	PROPN
ejpam-65	439	3	(	(	PUNCT
ejpam-65	439	4	x	x	X
ejpam-65	439	5	)	)	PUNCT
ejpam-65	440	1	+	+	NUM
ejpam-65	440	2	t	t	PROPN
ejpam-65	440	3	(	(	PUNCT
ejpam-65	440	4	y	y	NOUN
ejpam-65	440	5	)	)	PUNCT
ejpam-65	440	6	and	and	CCONJ
ejpam-65	440	7	t	t	PROPN
ejpam-65	440	8	(	(	PUNCT
ejpam-65	440	9	a	a	DET
ejpam-65	440	10	◦	◦	NOUN
ejpam-65	440	11	x	x	X
ejpam-65	440	12	)	)	PUNCT
ejpam-65	440	13	=	=	SYM
ejpam-65	440	14	a	a	DET
ejpam-65	440	15	◦	◦	NOUN
ejpam-65	440	16	t	t	X
ejpam-65	440	17	(	(	PUNCT
ejpam-65	440	18	x	x	NOUN
ejpam-65	440	19	)	)	PUNCT
ejpam-65	440	20	,	,	PUNCT
ejpam-65	440	21	∀x	∀x	NUM
ejpam-65	440	22	,	,	PUNCT
ejpam-65	440	23	y	y	PROPN
ejpam-65	440	24	∈	∈	PROPN
ejpam-65	440	25	v	v	NOUN
ejpam-65	440	26	,	,	PUNCT
ejpam-65	440	27	a	a	DET
ejpam-65	440	28	∈	∈	PROPN
ejpam-65	440	29	k.	k.	PROPN
ejpam-65	441	1	a	a	PROPN
ejpam-65	441	2	(	(	PUNCT
ejpam-65	441	3	resp	resp	NOUN
ejpam-65	441	4	.	.	PUNCT
ejpam-65	442	1	weak	weak	ADJ
ejpam-65	442	2	,	,	PUNCT
ejpam-65	442	3	good	good	ADJ
ejpam-65	442	4	)	)	PUNCT
ejpam-65	442	5	linear	linear	PROPN
ejpam-65	442	6	isomorphism	isomorphism	NOUN
ejpam-65	442	7	is	be	AUX
ejpam-65	442	8	defined	define	VERB
ejpam-65	442	9	as	as	ADP
ejpam-65	442	10	usual	usual	ADJ
ejpam-65	442	11	.	.	PUNCT
ejpam-65	443	1	if	if	SCONJ
ejpam-65	443	2	t	t	NOUN
ejpam-65	443	3	:	:	PUNCT
ejpam-65	443	4	v	v	NOUN
ejpam-65	443	5	−→w	−→w	NOUN
ejpam-65	443	6	is	be	AUX
ejpam-65	443	7	a	a	DET
ejpam-65	443	8	(	(	PUNCT
ejpam-65	443	9	resp	resp	NOUN
ejpam-65	443	10	.	.	PUNCT
ejpam-65	444	1	weak	weak	ADJ
ejpam-65	444	2	,	,	PUNCT
ejpam-65	444	3	good	good	ADJ
ejpam-65	444	4	)	)	PUNCT
ejpam-65	444	5	linear	linear	PROPN
ejpam-65	444	6	isomorphism	isomorphism	NOUN
ejpam-65	444	7	,	,	PUNCT
ejpam-65	444	8	then	then	ADV
ejpam-65	444	9	it	it	PRON
ejpam-65	444	10	is	be	AUX
ejpam-65	444	11	denoted	denote	VERB
ejpam-65	444	12	by	by	ADP
ejpam-65	444	13	(	(	PUNCT
ejpam-65	444	14	resp	resp	NOUN
ejpam-65	444	15	.	.	PUNCT
ejpam-65	445	1	v	v	NUM
ejpam-65	445	2	∼=w	∼=w	NOUN
ejpam-65	445	3	w	w	PROPN
ejpam-65	445	4	,	,	PUNCT
ejpam-65	445	5	v	v	NOUN
ejpam-65	445	6	∼=g	∼=g	X
ejpam-65	445	7	w	w	NOUN
ejpam-65	445	8	)	)	PUNCT
ejpam-65	445	9	v	v	ADP
ejpam-65	445	10	∼=	∼=	PROPN
ejpam-65	445	11	w.	w.	NOUN
ejpam-65	445	12	definition	definition	NOUN
ejpam-65	445	13	3.7	3.7	NUM
ejpam-65	445	14	.	.	PUNCT
ejpam-65	446	1	let	let	VERB
ejpam-65	446	2	t	t	NOUN
ejpam-65	446	3	:	:	PUNCT
ejpam-65	446	4	v	v	AUX
ejpam-65	446	5	−→w	−→w	NOUN
ejpam-65	446	6	be	be	AUX
ejpam-65	446	7	a	a	DET
ejpam-65	446	8	linear	linear	ADJ
ejpam-65	446	9	transformation	transformation	NOUN
ejpam-65	446	10	.	.	PUNCT
ejpam-65	447	1	the	the	DET
ejpam-65	447	2	kernel	kernel	NOUN
ejpam-65	447	3	of	of	ADP
ejpam-65	447	4	t	t	PROPN
ejpam-65	447	5	is	be	AUX
ejpam-65	447	6	denoted	denote	VERB
ejpam-65	447	7	by	by	ADP
ejpam-65	447	8	kert	kert	PROPN
ejpam-65	447	9	and	and	CCONJ
ejpam-65	447	10	defined	define	VERB
ejpam-65	447	11	by	by	ADP
ejpam-65	447	12	kert	kert	PROPN
ejpam-65	447	13	=	=	PUNCT
ejpam-65	447	14	{	{	PUNCT
ejpam-65	447	15	x	x	PROPN
ejpam-65	447	16	∈	∈	PROPN
ejpam-65	447	17	v	v	NOUN
ejpam-65	447	18	:	:	PUNCT
ejpam-65	447	19	t	t	PROPN
ejpam-65	447	20	(	(	PUNCT
ejpam-65	447	21	x	x	X
ejpam-65	447	22	)	)	PUNCT
ejpam-65	447	23	∈	∈	PROPN
ejpam-65	447	24	ω	ω	PROPN
ejpam-65	447	25	}	}	PUNCT
ejpam-65	447	26	,	,	PUNCT
ejpam-65	447	27	where	where	SCONJ
ejpam-65	447	28	ω	ω	X
ejpam-65	447	29	=	=	SYM
ejpam-65	447	30	0	0	NUM
ejpam-65	447	31	◦	◦	NOUN
ejpam-65	447	32	0w	0w	NUM
ejpam-65	447	33	.	.	PUNCT
ejpam-65	448	1	proposition	proposition	NOUN
ejpam-65	448	2	3.3	3.3	NUM
ejpam-65	448	3	.	.	PUNCT
ejpam-65	449	1	let	let	VERB
ejpam-65	449	2	w	w	NOUN
ejpam-65	449	3	be	be	AUX
ejpam-65	449	4	a	a	DET
ejpam-65	449	5	subhyperspace	subhyperspace	NOUN
ejpam-65	449	6	of	of	ADP
ejpam-65	449	7	v	v	NOUN
ejpam-65	449	8	.	.	PUNCT
ejpam-65	450	1	then	then	ADV
ejpam-65	450	2	the	the	DET
ejpam-65	450	3	mapping	mapping	NOUN
ejpam-65	450	4	{	{	PUNCT
ejpam-65	450	5	π	π	NOUN
ejpam-65	450	6	:	:	PUNCT
ejpam-65	450	7	v	v	ADP
ejpam-65	450	8	−→	−→	NOUN
ejpam-65	450	9	v	v	NOUN
ejpam-65	450	10	/	/	SYM
ejpam-65	450	11	w	w	NOUN
ejpam-65	450	12	x	x	SYM
ejpam-65	450	13	7−→	7−→	PROPN
ejpam-65	450	14	x+w	x+w	NUM
ejpam-65	450	15	is	be	AUX
ejpam-65	450	16	an	an	PRON
ejpam-65	450	17	onto	onto	ADP
ejpam-65	450	18	good	good	ADJ
ejpam-65	450	19	linear	linear	ADJ
ejpam-65	450	20	transformation	transformation	NOUN
ejpam-65	450	21	.	.	PUNCT
ejpam-65	451	1	the	the	DET
ejpam-65	451	2	mapping	mapping	NOUN
ejpam-65	451	3	π	π	PROPN
ejpam-65	451	4	is	be	AUX
ejpam-65	451	5	called	call	VERB
ejpam-65	451	6	canonical	canonical	ADJ
ejpam-65	451	7	transformation	transformation	NOUN
ejpam-65	451	8	.	.	PUNCT
ejpam-65	452	1	proof	proof	NOUN
ejpam-65	452	2	.	.	PUNCT
ejpam-65	453	1	obvious	obvious	ADJ
ejpam-65	453	2	.	.	PUNCT
ejpam-65	454	1	proposition	proposition	NOUN
ejpam-65	454	2	3.4	3.4	NUM
ejpam-65	454	3	.	.	PUNCT
ejpam-65	455	1	let	let	VERB
ejpam-65	455	2	t	t	NOUN
ejpam-65	455	3	:	:	PUNCT
ejpam-65	455	4	v	v	AUX
ejpam-65	455	5	−→	−→	NOUN
ejpam-65	455	6	u	u	NOUN
ejpam-65	455	7	be	be	VERB
ejpam-65	455	8	a	a	DET
ejpam-65	455	9	good	good	ADJ
ejpam-65	455	10	linear	linear	ADJ
ejpam-65	455	11	transformation	transformation	NOUN
ejpam-65	455	12	.	.	PUNCT
ejpam-65	456	1	(	(	PUNCT
ejpam-65	456	2	i	i	NOUN
ejpam-65	456	3	)	)	PUNCT
ejpam-65	456	4	if	if	SCONJ
ejpam-65	456	5	w	w	NOUN
ejpam-65	456	6	is	be	AUX
ejpam-65	456	7	a	a	DET
ejpam-65	456	8	subhyperspace	subhyperspace	NOUN
ejpam-65	456	9	of	of	ADP
ejpam-65	456	10	v	v	NOUN
ejpam-65	456	11	,	,	PUNCT
ejpam-65	456	12	then	then	ADV
ejpam-65	456	13	the	the	DET
ejpam-65	456	14	image	image	NOUN
ejpam-65	456	15	of	of	ADP
ejpam-65	456	16	w	w	PROPN
ejpam-65	456	17	,	,	PUNCT
ejpam-65	456	18	t	t	PROPN
ejpam-65	456	19	(	(	PUNCT
ejpam-65	456	20	w	w	NOUN
ejpam-65	456	21	)	)	PUNCT
ejpam-65	456	22	is	be	AUX
ejpam-65	456	23	a	a	DET
ejpam-65	456	24	subhyperspace	subhyperspace	NOUN
ejpam-65	456	25	of	of	ADP
ejpam-65	456	26	u	u	NOUN
ejpam-65	456	27	.	.	PUNCT
ejpam-65	457	1	(	(	PUNCT
ejpam-65	457	2	ii	ii	NOUN
ejpam-65	457	3	)	)	PUNCT
ejpam-65	457	4	if	if	SCONJ
ejpam-65	457	5	l	l	NOUN
ejpam-65	457	6	be	be	VERB
ejpam-65	457	7	a	a	DET
ejpam-65	457	8	subhyperspace	subhyperspace	NOUN
ejpam-65	457	9	of	of	ADP
ejpam-65	457	10	u	u	NOUN
ejpam-65	457	11	,	,	PUNCT
ejpam-65	457	12	then	then	ADV
ejpam-65	457	13	the	the	DET
ejpam-65	457	14	preimage	preimage	NOUN
ejpam-65	457	15	of	of	ADP
ejpam-65	457	16	l	l	PROPN
ejpam-65	457	17	,	,	PUNCT
ejpam-65	457	18	t−1(l	t−1(l	PROPN
ejpam-65	457	19	)	)	PUNCT
ejpam-65	457	20	is	be	AUX
ejpam-65	457	21	a	a	DET
ejpam-65	457	22	subhyperspace	subhyperspace	NOUN
ejpam-65	457	23	of	of	ADP
ejpam-65	457	24	v	v	NOUN
ejpam-65	457	25	containing	contain	VERB
ejpam-65	457	26	kert	kert	PROPN
ejpam-65	457	27	.	.	PUNCT
ejpam-65	458	1	r.	r.	PROPN
ejpam-65	458	2	ameri	ameri	PROPN
ejpam-65	458	3	,	,	PUNCT
ejpam-65	458	4	o.r	o.r	PROPN
ejpam-65	458	5	.	.	PROPN
ejpam-65	458	6	dehghan	dehghan	PROPN
ejpam-65	458	7	/	/	SYM
ejpam-65	458	8	eur	eur	PROPN
ejpam-65	458	9	.	.	PUNCT
ejpam-65	459	1	j.	j.	PROPN
ejpam-65	459	2	pure	pure	PROPN
ejpam-65	459	3	appl	appl	PROPN
ejpam-65	459	4	.	.	PROPN
ejpam-65	459	5	math	math	PROPN
ejpam-65	459	6	,	,	PUNCT
ejpam-65	459	7	1	1	NUM
ejpam-65	459	8	(	(	PUNCT
ejpam-65	459	9	2008	2008	NUM
ejpam-65	459	10	)	)	PUNCT
ejpam-65	459	11	,	,	PUNCT
ejpam-65	459	12	(	(	PUNCT
ejpam-65	459	13	32	32	NUM
ejpam-65	459	14	-	-	SYM
ejpam-65	459	15	50	50	NUM
ejpam-65	459	16	)	)	PUNCT
ejpam-65	459	17	43	43	NUM
ejpam-65	459	18	proof	proof	NOUN
ejpam-65	459	19	.	.	PUNCT
ejpam-65	460	1	(	(	PUNCT
ejpam-65	460	2	i	i	NOUN
ejpam-65	460	3	)	)	PUNCT
ejpam-65	460	4	let	let	VERB
ejpam-65	460	5	a	a	DET
ejpam-65	460	6	∈	∈	PROPN
ejpam-65	460	7	k	k	PROPN
ejpam-65	460	8	and	and	CCONJ
ejpam-65	460	9	x́	x́	PROPN
ejpam-65	460	10	,	,	PUNCT
ejpam-65	460	11	ý	ý	PROPN
ejpam-65	460	12	∈	∈	PROPN
ejpam-65	460	13	t	t	PROPN
ejpam-65	460	14	(	(	PUNCT
ejpam-65	460	15	w	w	PROPN
ejpam-65	460	16	)	)	PUNCT
ejpam-65	460	17	,	,	PUNCT
ejpam-65	460	18	such	such	ADJ
ejpam-65	460	19	that	that	PRON
ejpam-65	460	20	x́	x́	PROPN
ejpam-65	461	1	=	=	PROPN
ejpam-65	461	2	t	t	PROPN
ejpam-65	461	3	(	(	PUNCT
ejpam-65	461	4	x	x	NOUN
ejpam-65	461	5	)	)	PUNCT
ejpam-65	461	6	,	,	PUNCT
ejpam-65	461	7	ý	ý	PROPN
ejpam-65	461	8	=	=	SYM
ejpam-65	461	9	t	t	PROPN
ejpam-65	461	10	(	(	PUNCT
ejpam-65	461	11	y	y	NOUN
ejpam-65	461	12	)	)	PUNCT
ejpam-65	461	13	for	for	ADP
ejpam-65	461	14	some	some	DET
ejpam-65	461	15	x	x	NOUN
ejpam-65	461	16	,	,	PUNCT
ejpam-65	461	17	y	y	PROPN
ejpam-65	461	18	∈	∈	PROPN
ejpam-65	461	19	w	w	PROPN
ejpam-65	461	20	.	.	PUNCT
ejpam-65	462	1	thus	thus	ADV
ejpam-65	462	2	x+	x+	ADJ
ejpam-65	462	3	y	y	PROPN
ejpam-65	462	4	∈w	∈w	NOUN
ejpam-65	462	5	and	and	CCONJ
ejpam-65	462	6	a	a	DET
ejpam-65	462	7	◦	◦	NOUN
ejpam-65	462	8	x	x	SYM
ejpam-65	462	9	⊆w	⊆w	NOUN
ejpam-65	462	10	.	.	PUNCT
ejpam-65	463	1	so	so	ADV
ejpam-65	463	2	x́−	x́−	PUNCT
ejpam-65	464	1	ý	ý	PROPN
ejpam-65	464	2	=	=	SYM
ejpam-65	464	3	t	t	PROPN
ejpam-65	464	4	(	(	PUNCT
ejpam-65	464	5	x)−	x)−	PROPN
ejpam-65	464	6	t	t	PROPN
ejpam-65	464	7	(	(	PUNCT
ejpam-65	464	8	y	y	NOUN
ejpam-65	464	9	)	)	PUNCT
ejpam-65	464	10	=	=	SYM
ejpam-65	464	11	t	t	PROPN
ejpam-65	464	12	(	(	PUNCT
ejpam-65	464	13	x−	x−	PROPN
ejpam-65	464	14	y	y	PROPN
ejpam-65	464	15	)	)	PUNCT
ejpam-65	464	16	∈	∈	PROPN
ejpam-65	464	17	t	t	PROPN
ejpam-65	464	18	(	(	PUNCT
ejpam-65	464	19	w	w	NOUN
ejpam-65	464	20	)	)	PUNCT
ejpam-65	464	21	,	,	PUNCT
ejpam-65	464	22	and	and	CCONJ
ejpam-65	464	23	a	a	DET
ejpam-65	464	24	◦	◦	NOUN
ejpam-65	464	25	x́	x́	X
ejpam-65	464	26	=	=	PUNCT
ejpam-65	465	1	a	a	DET
ejpam-65	465	2	◦	◦	NOUN
ejpam-65	465	3	t	t	X
ejpam-65	465	4	(	(	PUNCT
ejpam-65	465	5	x	x	X
ejpam-65	465	6	)	)	PUNCT
ejpam-65	465	7	=	=	SYM
ejpam-65	465	8	t	t	PROPN
ejpam-65	465	9	(	(	PUNCT
ejpam-65	465	10	a	a	DET
ejpam-65	465	11	◦	◦	NOUN
ejpam-65	465	12	x	x	SYM
ejpam-65	465	13	)	)	PUNCT
ejpam-65	465	14	⊆	⊆	NUM
ejpam-65	465	15	t	t	NOUN
ejpam-65	465	16	(	(	PUNCT
ejpam-65	465	17	w	w	PROPN
ejpam-65	465	18	)	)	PUNCT
ejpam-65	465	19	,	,	PUNCT
ejpam-65	465	20	therefore	therefore	ADV
ejpam-65	465	21	t	t	PROPN
ejpam-65	465	22	(	(	PUNCT
ejpam-65	465	23	w	w	NOUN
ejpam-65	465	24	)	)	PUNCT
ejpam-65	465	25	6	6	NUM
ejpam-65	465	26	u	u	NOUN
ejpam-65	465	27	.	.	PUNCT
ejpam-65	466	1	(	(	PUNCT
ejpam-65	466	2	ii	ii	NOUN
ejpam-65	466	3	)	)	PUNCT
ejpam-65	466	4	let	let	VERB
ejpam-65	466	5	a	a	DET
ejpam-65	466	6	∈	∈	PROPN
ejpam-65	466	7	k	k	PROPN
ejpam-65	466	8	and	and	CCONJ
ejpam-65	466	9	x	x	PROPN
ejpam-65	466	10	,	,	PUNCT
ejpam-65	466	11	y	y	PROPN
ejpam-65	466	12	∈	∈	PROPN
ejpam-65	466	13	t−1(l	t−1(l	PROPN
ejpam-65	466	14	)	)	PUNCT
ejpam-65	466	15	,	,	PUNCT
ejpam-65	467	1	such	such	ADJ
ejpam-65	467	2	that	that	SCONJ
ejpam-65	467	3	t	t	PROPN
ejpam-65	467	4	(	(	PUNCT
ejpam-65	467	5	x	x	NOUN
ejpam-65	467	6	)	)	PUNCT
ejpam-65	467	7	=	=	SYM
ejpam-65	467	8	x́	x́	PROPN
ejpam-65	467	9	,	,	PUNCT
ejpam-65	467	10	t	t	PROPN
ejpam-65	467	11	(	(	PUNCT
ejpam-65	467	12	y	y	NOUN
ejpam-65	467	13	)	)	PUNCT
ejpam-65	467	14	=	=	SYM
ejpam-65	468	1	ý	ý	ADJ
ejpam-65	468	2	,	,	PUNCT
ejpam-65	468	3	for	for	ADP
ejpam-65	468	4	some	some	DET
ejpam-65	468	5	x́	x́	PROPN
ejpam-65	468	6	,	,	PUNCT
ejpam-65	468	7	ý	ý	PROPN
ejpam-65	468	8	∈	∈	PROPN
ejpam-65	468	9	l.	l.	NOUN
ejpam-65	468	10	thus	thus	ADV
ejpam-65	468	11	x́+	x́+	NUM
ejpam-65	468	12	ý	ý	ADJ
ejpam-65	468	13	∈	∈	PROPN
ejpam-65	468	14	l	l	NOUN
ejpam-65	468	15	and	and	CCONJ
ejpam-65	468	16	a	a	DET
ejpam-65	468	17	◦	◦	NOUN
ejpam-65	468	18	x́	x́	X
ejpam-65	469	1	⊆	⊆	NUM
ejpam-65	469	2	l.	l.	PROPN
ejpam-65	469	3	so	so	ADV
ejpam-65	469	4	x́−	x́−	PUNCT
ejpam-65	470	1	ý	ý	PROPN
ejpam-65	470	2	=	=	SYM
ejpam-65	470	3	t	t	PROPN
ejpam-65	470	4	(	(	PUNCT
ejpam-65	470	5	x)−	x)−	PROPN
ejpam-65	470	6	t	t	PROPN
ejpam-65	470	7	(	(	PUNCT
ejpam-65	470	8	y	y	NOUN
ejpam-65	470	9	)	)	PUNCT
ejpam-65	470	10	=	=	SYM
ejpam-65	470	11	t	t	PROPN
ejpam-65	470	12	(	(	PUNCT
ejpam-65	470	13	x−	x−	PROPN
ejpam-65	470	14	y	y	PROPN
ejpam-65	470	15	)	)	PUNCT
ejpam-65	470	16	,	,	PUNCT
ejpam-65	470	17	=	=	PRON
ejpam-65	470	18	⇒	⇒	VERB
ejpam-65	470	19	x−	x−	PROPN
ejpam-65	470	20	y	y	PROPN
ejpam-65	470	21	∈	∈	PROPN
ejpam-65	470	22	t−1(l	t−1(l	PROPN
ejpam-65	470	23	)	)	PUNCT
ejpam-65	470	24	,	,	PUNCT
ejpam-65	470	25	and	and	CCONJ
ejpam-65	470	26	a	a	DET
ejpam-65	470	27	◦	◦	NOUN
ejpam-65	470	28	x́	x́	X
ejpam-65	471	1	=	=	PUNCT
ejpam-65	471	2	a	a	DET
ejpam-65	471	3	◦	◦	NOUN
ejpam-65	471	4	t	t	X
ejpam-65	471	5	(	(	PUNCT
ejpam-65	471	6	x	x	X
ejpam-65	471	7	)	)	PUNCT
ejpam-65	471	8	=	=	SYM
ejpam-65	471	9	t	t	PROPN
ejpam-65	471	10	(	(	PUNCT
ejpam-65	471	11	a	a	DET
ejpam-65	471	12	◦	◦	NOUN
ejpam-65	471	13	x	x	NOUN
ejpam-65	471	14	)	)	PUNCT
ejpam-65	471	15	,	,	PUNCT
ejpam-65	471	16	=	=	PRON
ejpam-65	471	17	⇒	⇒	VERB
ejpam-65	471	18	a	a	DET
ejpam-65	471	19	◦	◦	NOUN
ejpam-65	471	20	x	x	SYM
ejpam-65	471	21	⊆	⊆	NUM
ejpam-65	471	22	t−1(a	t−1(a	NOUN
ejpam-65	471	23	◦	◦	PROPN
ejpam-65	471	24	x́	x́	PROPN
ejpam-65	471	25	)	)	PUNCT
ejpam-65	472	1	⊆	⊆	NUM
ejpam-65	472	2	t−1(l	t−1(l	PROPN
ejpam-65	472	3	)	)	PUNCT
ejpam-65	472	4	,	,	PUNCT
ejpam-65	472	5	therefor	therefor	ADP
ejpam-65	472	6	t−1(l	t−1(l	PROPN
ejpam-65	472	7	)	)	PUNCT
ejpam-65	472	8	6	6	NUM
ejpam-65	472	9	v	v	NOUN
ejpam-65	472	10	.	.	PUNCT
ejpam-65	473	1	also	also	ADV
ejpam-65	473	2	if	if	SCONJ
ejpam-65	473	3	x	x	PROPN
ejpam-65	473	4	∈	∈	PROPN
ejpam-65	473	5	kert	kert	PROPN
ejpam-65	473	6	,	,	PUNCT
ejpam-65	473	7	then	then	ADV
ejpam-65	473	8	t	t	PROPN
ejpam-65	473	9	(	(	PUNCT
ejpam-65	473	10	x	x	X
ejpam-65	473	11	)	)	PUNCT
ejpam-65	473	12	∈	∈	NOUN
ejpam-65	473	13	0	0	NUM
ejpam-65	473	14	◦	◦	NOUN
ejpam-65	473	15	0u	0u	ADJ
ejpam-65	474	1	⊆	⊆	SYM
ejpam-65	474	2	0	0	NUM
ejpam-65	474	3	◦	◦	NOUN
ejpam-65	474	4	l	l	NOUN
ejpam-65	474	5	⊆	⊆	NUM
ejpam-65	474	6	l	l	NOUN
ejpam-65	474	7	,	,	PUNCT
ejpam-65	474	8	=	=	SYM
ejpam-65	474	9	⇒	⇒	NOUN
ejpam-65	474	10	x	x	SYM
ejpam-65	474	11	∈	∈	PROPN
ejpam-65	474	12	t−1(l	t−1(l	PROPN
ejpam-65	474	13	)	)	PUNCT
ejpam-65	474	14	,	,	PUNCT
ejpam-65	474	15	thus	thus	ADV
ejpam-65	474	16	kert	kert	PROPN
ejpam-65	474	17	⊆	⊆	NUM
ejpam-65	474	18	t−1(l	t−1(l	PROPN
ejpam-65	474	19	)	)	PUNCT
ejpam-65	474	20	.	.	PUNCT
ejpam-65	475	1	proposition	proposition	NOUN
ejpam-65	475	2	3.5	3.5	NUM
ejpam-65	475	3	.	.	PUNCT
ejpam-65	476	1	let	let	VERB
ejpam-65	476	2	v	v	NOUN
ejpam-65	476	3	and	and	CCONJ
ejpam-65	476	4	u	u	PRON
ejpam-65	476	5	be	be	AUX
ejpam-65	476	6	strongly	strongly	ADV
ejpam-65	476	7	left	leave	VERB
ejpam-65	476	8	distributive	distributive	ADJ
ejpam-65	476	9	hypervector	hypervector	NOUN
ejpam-65	476	10	spaces	space	NOUN
ejpam-65	476	11	over	over	ADP
ejpam-65	476	12	the	the	DET
ejpam-65	476	13	field	field	NOUN
ejpam-65	477	1	k	k	NOUN
ejpam-65	477	2	,	,	PUNCT
ejpam-65	477	3	and	and	CCONJ
ejpam-65	477	4	t	t	PROPN
ejpam-65	477	5	:	:	PUNCT
ejpam-65	477	6	v	v	X
ejpam-65	477	7	−→	−→	NOUN
ejpam-65	477	8	u	u	NOUN
ejpam-65	477	9	be	be	VERB
ejpam-65	477	10	a	a	DET
ejpam-65	477	11	linear	linear	ADJ
ejpam-65	477	12	transformation	transformation	NOUN
ejpam-65	477	13	.	.	PUNCT
ejpam-65	478	1	then	then	ADV
ejpam-65	478	2	kert	kert	PROPN
ejpam-65	478	3	is	be	AUX
ejpam-65	478	4	a	a	DET
ejpam-65	478	5	subhyperspace	subhyperspace	NOUN
ejpam-65	478	6	of	of	ADP
ejpam-65	478	7	v	v	NOUN
ejpam-65	478	8	.	.	PUNCT
ejpam-65	479	1	moreover	moreover	ADV
ejpam-65	479	2	,	,	PUNCT
ejpam-65	479	3	ω	ω	PROPN
ejpam-65	479	4	⊆	⊆	NUM
ejpam-65	479	5	kert	kert	NOUN
ejpam-65	479	6	.	.	PUNCT
ejpam-65	480	1	proof	proof	NOUN
ejpam-65	480	2	.	.	PUNCT
ejpam-65	481	1	since	since	SCONJ
ejpam-65	481	2	t	t	PROPN
ejpam-65	481	3	(	(	PUNCT
ejpam-65	481	4	ω	ω	NOUN
ejpam-65	481	5	)	)	PUNCT
ejpam-65	481	6	=	=	SYM
ejpam-65	481	7	t	t	PROPN
ejpam-65	481	8	(	(	PUNCT
ejpam-65	481	9	0	0	NUM
ejpam-65	481	10	◦	◦	NOUN
ejpam-65	481	11	0	0	NUM
ejpam-65	481	12	)	)	PUNCT
ejpam-65	481	13	⊆	⊆	NUM
ejpam-65	481	14	0	0	NUM
ejpam-65	481	15	◦	◦	NOUN
ejpam-65	481	16	t	t	PROPN
ejpam-65	481	17	(	(	PUNCT
ejpam-65	481	18	0	0	NUM
ejpam-65	481	19	)	)	PUNCT
ejpam-65	481	20	=	=	SYM
ejpam-65	481	21	0	0	PUNCT
ejpam-65	481	22	◦	◦	NOUN
ejpam-65	481	23	0	0	NUM
ejpam-65	481	24	=	=	SYM
ejpam-65	481	25	ω	ω	PROPN
ejpam-65	481	26	,	,	PUNCT
ejpam-65	481	27	thus	thus	ADV
ejpam-65	481	28	∅	∅	NOUN
ejpam-65	481	29	6=	6=	NOUN
ejpam-65	481	30	kert	kert	PROPN
ejpam-65	481	31	⊆	⊆	NUM
ejpam-65	481	32	v	v	NOUN
ejpam-65	481	33	.	.	PUNCT
ejpam-65	482	1	also	also	ADV
ejpam-65	482	2	∀a	∀a	NUM
ejpam-65	482	3	,	,	PUNCT
ejpam-65	482	4	b	b	PROPN
ejpam-65	482	5	∈	∈	PROPN
ejpam-65	482	6	k	k	PROPN
ejpam-65	482	7	and	and	CCONJ
ejpam-65	482	8	∀x	∀x	NUM
ejpam-65	482	9	,	,	PUNCT
ejpam-65	482	10	y	y	PROPN
ejpam-65	482	11	∈	∈	PROPN
ejpam-65	482	12	kert	kert	PROPN
ejpam-65	482	13	,	,	PUNCT
ejpam-65	482	14	t	t	PROPN
ejpam-65	482	15	(	(	PUNCT
ejpam-65	482	16	x	x	X
ejpam-65	482	17	)	)	PUNCT
ejpam-65	482	18	∈	∈	PROPN
ejpam-65	482	19	ω	ω	PROPN
ejpam-65	482	20	,	,	PUNCT
ejpam-65	482	21	t	t	PROPN
ejpam-65	482	22	(	(	PUNCT
ejpam-65	482	23	y	y	NOUN
ejpam-65	482	24	)	)	PUNCT
ejpam-65	482	25	∈	∈	PROPN
ejpam-65	482	26	ω	ω	PROPN
ejpam-65	482	27	.	.	PUNCT
ejpam-65	483	1	so	so	ADV
ejpam-65	483	2	t	t	PROPN
ejpam-65	483	3	(	(	PUNCT
ejpam-65	483	4	a	a	DET
ejpam-65	483	5	◦	◦	NOUN
ejpam-65	483	6	x+	x+	X
ejpam-65	483	7	b	b	X
ejpam-65	483	8	◦	◦	NOUN
ejpam-65	483	9	y	y	NOUN
ejpam-65	483	10	)	)	PUNCT
ejpam-65	483	11	=	=	SYM
ejpam-65	483	12	t	t	PROPN
ejpam-65	483	13	(	(	PUNCT
ejpam-65	483	14	a	a	DET
ejpam-65	483	15	◦	◦	NOUN
ejpam-65	483	16	x	x	X
ejpam-65	483	17	)	)	PUNCT
ejpam-65	484	1	+	+	NUM
ejpam-65	484	2	t	t	PROPN
ejpam-65	484	3	(	(	PUNCT
ejpam-65	484	4	b	b	X
ejpam-65	484	5	◦	◦	VERB
ejpam-65	484	6	y	y	PROPN
ejpam-65	484	7	)	)	PUNCT
ejpam-65	484	8	⊆	⊆	PROPN
ejpam-65	484	9	a	a	DET
ejpam-65	484	10	◦	◦	NOUN
ejpam-65	484	11	t	t	X
ejpam-65	484	12	(	(	PUNCT
ejpam-65	484	13	x	x	X
ejpam-65	484	14	)	)	PUNCT
ejpam-65	485	1	+	+	NUM
ejpam-65	485	2	b	b	X
ejpam-65	485	3	◦	◦	NOUN
ejpam-65	485	4	t	t	PROPN
ejpam-65	485	5	(	(	PUNCT
ejpam-65	485	6	y	y	PROPN
ejpam-65	485	7	)	)	PUNCT
ejpam-65	485	8	⊆	⊆	PROPN
ejpam-65	485	9	a	a	DET
ejpam-65	485	10	◦	◦	NOUN
ejpam-65	485	11	ω	ω	NOUN
ejpam-65	485	12	+	+	CCONJ
ejpam-65	485	13	b	b	X
ejpam-65	485	14	◦	◦	NOUN
ejpam-65	485	15	ω	ω	X
ejpam-65	485	16	=	=	SYM
ejpam-65	485	17	ω	ω	PROPN
ejpam-65	485	18	+	+	PROPN
ejpam-65	485	19	ω	ω	PROPN
ejpam-65	485	20	=	=	SYM
ejpam-65	485	21	ω	ω	PROPN
ejpam-65	485	22	,	,	PUNCT
ejpam-65	485	23	=	=	PRON
ejpam-65	485	24	⇒	⇒	VERB
ejpam-65	485	25	a	a	DET
ejpam-65	485	26	◦	◦	NOUN
ejpam-65	485	27	x+	x+	X
ejpam-65	485	28	b	b	X
ejpam-65	485	29	◦	◦	NOUN
ejpam-65	485	30	y	y	PROPN
ejpam-65	485	31	⊆	⊆	NUM
ejpam-65	485	32	kert	kert	PROPN
ejpam-65	485	33	.	.	PUNCT
ejpam-65	486	1	therefore	therefore	ADV
ejpam-65	486	2	kert	kert	PROPN
ejpam-65	486	3	6	6	NUM
ejpam-65	486	4	v	v	NOUN
ejpam-65	486	5	.	.	PUNCT
ejpam-65	487	1	r.	r.	PROPN
ejpam-65	487	2	ameri	ameri	PROPN
ejpam-65	487	3	,	,	PUNCT
ejpam-65	487	4	o.r	o.r	PROPN
ejpam-65	487	5	.	.	PROPN
ejpam-65	487	6	dehghan	dehghan	PROPN
ejpam-65	487	7	/	/	SYM
ejpam-65	487	8	eur	eur	PROPN
ejpam-65	487	9	.	.	PUNCT
ejpam-65	488	1	j.	j.	PROPN
ejpam-65	488	2	pure	pure	PROPN
ejpam-65	488	3	appl	appl	PROPN
ejpam-65	488	4	.	.	PROPN
ejpam-65	488	5	math	math	PROPN
ejpam-65	488	6	,	,	PUNCT
ejpam-65	488	7	1	1	NUM
ejpam-65	488	8	(	(	PUNCT
ejpam-65	488	9	2008	2008	NUM
ejpam-65	488	10	)	)	PUNCT
ejpam-65	488	11	,	,	PUNCT
ejpam-65	488	12	(	(	PUNCT
ejpam-65	488	13	32	32	NUM
ejpam-65	488	14	-	-	SYM
ejpam-65	488	15	50	50	NUM
ejpam-65	488	16	)	)	PUNCT
ejpam-65	488	17	44	44	NUM
ejpam-65	488	18	proposition	proposition	NOUN
ejpam-65	488	19	3.6	3.6	NUM
ejpam-65	488	20	.	.	PUNCT
ejpam-65	489	1	let	let	VERB
ejpam-65	489	2	v	v	NOUN
ejpam-65	489	3	and	and	CCONJ
ejpam-65	489	4	u	u	PRON
ejpam-65	489	5	be	be	AUX
ejpam-65	489	6	strongly	strongly	ADV
ejpam-65	489	7	left	leave	VERB
ejpam-65	489	8	distributive	distributive	ADJ
ejpam-65	489	9	hypervector	hypervector	NOUN
ejpam-65	489	10	spaces	space	NOUN
ejpam-65	489	11	over	over	ADP
ejpam-65	489	12	the	the	DET
ejpam-65	489	13	field	field	NOUN
ejpam-65	490	1	k	k	NOUN
ejpam-65	490	2	,	,	PUNCT
ejpam-65	490	3	and	and	CCONJ
ejpam-65	490	4	t	t	PROPN
ejpam-65	490	5	:	:	PUNCT
ejpam-65	490	6	v	v	X
ejpam-65	490	7	−→	−→	NOUN
ejpam-65	490	8	u	u	NOUN
ejpam-65	490	9	be	be	VERB
ejpam-65	490	10	a	a	DET
ejpam-65	490	11	good	good	ADJ
ejpam-65	490	12	linear	linear	ADJ
ejpam-65	490	13	transformation	transformation	NOUN
ejpam-65	490	14	.	.	PUNCT
ejpam-65	491	1	then	then	ADV
ejpam-65	491	2	there	there	PRON
ejpam-65	491	3	is	be	VERB
ejpam-65	491	4	a	a	DET
ejpam-65	491	5	one	one	NUM
ejpam-65	491	6	-	-	PUNCT
ejpam-65	491	7	to	to	ADP
ejpam-65	491	8	-	-	PUNCT
ejpam-65	491	9	one	one	NUM
ejpam-65	491	10	correspondence	correspondence	NOUN
ejpam-65	491	11	between	between	ADP
ejpam-65	491	12	subhyperspaces	subhyperspace	NOUN
ejpam-65	491	13	of	of	ADP
ejpam-65	491	14	v	v	NOUN
ejpam-65	491	15	containing	contain	VERB
ejpam-65	491	16	kert	kert	NOUN
ejpam-65	491	17	and	and	CCONJ
ejpam-65	491	18	subhyperspaces	subhyperspace	NOUN
ejpam-65	491	19	of	of	ADP
ejpam-65	491	20	u	u	NOUN
ejpam-65	491	21	.	.	PUNCT
ejpam-65	492	1	proof	proof	NOUN
ejpam-65	492	2	.	.	PUNCT
ejpam-65	493	1	let	let	VERB
ejpam-65	493	2	a	a	DET
ejpam-65	493	3	=	=	X
ejpam-65	493	4	{	{	PUNCT
ejpam-65	493	5	w	w	NOUN
ejpam-65	493	6	:	:	PUNCT
ejpam-65	493	7	w	w	PROPN
ejpam-65	493	8	6	6	NUM
ejpam-65	493	9	v	v	NOUN
ejpam-65	493	10	and	and	CCONJ
ejpam-65	493	11	w	w	PROPN
ejpam-65	493	12	⊇	⊇	PROPN
ejpam-65	493	13	kert	kert	PROPN
ejpam-65	493	14	}	}	PUNCT
ejpam-65	493	15	,	,	PUNCT
ejpam-65	493	16	and	and	CCONJ
ejpam-65	493	17	b	b	X
ejpam-65	493	18	=	=	SYM
ejpam-65	493	19	{	{	PUNCT
ejpam-65	493	20	l	l	NOUN
ejpam-65	493	21	:	:	PUNCT
ejpam-65	493	22	l	l	NOUN
ejpam-65	493	23	6	6	NUM
ejpam-65	493	24	u	u	NOUN
ejpam-65	493	25	}	}	PUNCT
ejpam-65	493	26	,	,	PUNCT
ejpam-65	493	27	then	then	ADV
ejpam-65	493	28	we	we	PRON
ejpam-65	493	29	show	show	VERB
ejpam-65	493	30	that	that	SCONJ
ejpam-65	493	31	the	the	DET
ejpam-65	493	32	mapping	mapping	NOUN
ejpam-65	493	33	{	{	PUNCT
ejpam-65	493	34	ϕ	ϕ	NOUN
ejpam-65	493	35	:	:	PUNCT
ejpam-65	493	36	a	a	DET
ejpam-65	493	37	−→	−→	NOUN
ejpam-65	493	38	b	b	PROPN
ejpam-65	493	39	w	w	PROPN
ejpam-65	493	40	7−→	7−→	PROPN
ejpam-65	493	41	t	t	NOUN
ejpam-65	493	42	(	(	PUNCT
ejpam-65	493	43	w	w	NOUN
ejpam-65	493	44	)	)	PUNCT
ejpam-65	493	45	is	be	AUX
ejpam-65	493	46	a	a	DET
ejpam-65	493	47	bijection	bijection	NOUN
ejpam-65	493	48	.	.	PUNCT
ejpam-65	494	1	by	by	ADP
ejpam-65	494	2	proposition	proposition	NOUN
ejpam-65	494	3	3.4	3.4	NUM
ejpam-65	494	4	,	,	PUNCT
ejpam-65	494	5	ϕ	ϕ	NOUN
ejpam-65	494	6	is	be	AUX
ejpam-65	494	7	well	well	ADV
ejpam-65	494	8	defined	define	VERB
ejpam-65	494	9	.	.	PUNCT
ejpam-65	495	1	now	now	ADV
ejpam-65	495	2	if	if	SCONJ
ejpam-65	495	3	w1	w1	NOUN
ejpam-65	495	4	and	and	CCONJ
ejpam-65	495	5	w2	w2	NOUN
ejpam-65	495	6	are	be	AUX
ejpam-65	495	7	two	two	NUM
ejpam-65	495	8	elements	element	NOUN
ejpam-65	495	9	of	of	ADP
ejpam-65	495	10	a	a	DET
ejpam-65	495	11	such	such	ADJ
ejpam-65	495	12	that	that	DET
ejpam-65	495	13	w1	w1	PROPN
ejpam-65	495	14	6=	6=	PROPN
ejpam-65	495	15	w2	w2	NOUN
ejpam-65	495	16	.	.	PUNCT
ejpam-65	496	1	without	without	ADP
ejpam-65	496	2	loss	loss	NOUN
ejpam-65	496	3	of	of	ADP
ejpam-65	496	4	generality	generality	NOUN
ejpam-65	496	5	,	,	PUNCT
ejpam-65	496	6	suppose	suppose	VERB
ejpam-65	496	7	that	that	SCONJ
ejpam-65	496	8	w2	w2	PROPN
ejpam-65	496	9	*	*	PROPN
ejpam-65	496	10	w1	w1	PROPN
ejpam-65	496	11	then	then	ADV
ejpam-65	496	12	∃w1	∃w1	VERB
ejpam-65	496	13	∈w1	∈w1	PROPN
ejpam-65	496	14	−w2	−w2	PROPN
ejpam-65	496	15	or	or	CCONJ
ejpam-65	496	16	∃w2	∃w2	PROPN
ejpam-65	496	17	∈w2	∈w2	PROPN
ejpam-65	496	18	−w1	−w1	PROPN
ejpam-65	496	19	.	.	PUNCT
ejpam-65	497	1	if	if	SCONJ
ejpam-65	497	2	w1	w1	PROPN
ejpam-65	497	3	∈	∈	PROPN
ejpam-65	497	4	w1	w1	PROPN
ejpam-65	497	5	−w2	−w2	PROPN
ejpam-65	497	6	,	,	PUNCT
ejpam-65	497	7	then	then	ADV
ejpam-65	497	8	t	t	PROPN
ejpam-65	497	9	(	(	PUNCT
ejpam-65	497	10	w1	w1	NOUN
ejpam-65	497	11	)	)	PUNCT
ejpam-65	497	12	∈	∈	PROPN
ejpam-65	497	13	t	t	PROPN
ejpam-65	497	14	(	(	PUNCT
ejpam-65	497	15	w1	w1	NOUN
ejpam-65	497	16	)	)	PUNCT
ejpam-65	497	17	−	−	PROPN
ejpam-65	497	18	t	t	PROPN
ejpam-65	497	19	(	(	PUNCT
ejpam-65	497	20	w2	w2	NOUN
ejpam-65	497	21	)	)	PUNCT
ejpam-65	497	22	,	,	PUNCT
ejpam-65	497	23	so	so	SCONJ
ejpam-65	497	24	t	t	PROPN
ejpam-65	497	25	(	(	PUNCT
ejpam-65	497	26	w1	w1	PROPN
ejpam-65	497	27	)	)	PUNCT
ejpam-65	497	28	6=	6=	ADP
ejpam-65	497	29	t	t	PROPN
ejpam-65	497	30	(	(	PUNCT
ejpam-65	497	31	w2	w2	NOUN
ejpam-65	497	32	)	)	PUNCT
ejpam-65	497	33	.	.	PUNCT
ejpam-65	498	1	if	if	SCONJ
ejpam-65	498	2	w2	w2	PROPN
ejpam-65	498	3	∈	∈	PROPN
ejpam-65	498	4	w2	w2	PROPN
ejpam-65	498	5	−w1	−w1	PROPN
ejpam-65	498	6	,	,	PUNCT
ejpam-65	498	7	then	then	ADV
ejpam-65	498	8	similarly	similarly	ADV
ejpam-65	498	9	t	t	PROPN
ejpam-65	498	10	(	(	PUNCT
ejpam-65	498	11	w1	w1	PROPN
ejpam-65	498	12	)	)	PUNCT
ejpam-65	498	13	6=	6=	ADP
ejpam-65	498	14	t	t	PROPN
ejpam-65	498	15	(	(	PUNCT
ejpam-65	498	16	w2	w2	PROPN
ejpam-65	498	17	)	)	PUNCT
ejpam-65	498	18	.	.	PUNCT
ejpam-65	499	1	therefore	therefore	ADV
ejpam-65	499	2	ϕ	ϕ	PROPN
ejpam-65	499	3	is	be	AUX
ejpam-65	499	4	a	a	DET
ejpam-65	499	5	welldefined	welldefine	VERB
ejpam-65	499	6	one	one	NOUN
ejpam-65	499	7	to	to	ADP
ejpam-65	499	8	one	one	NUM
ejpam-65	499	9	map	map	NOUN
ejpam-65	499	10	.	.	PUNCT
ejpam-65	500	1	now	now	ADV
ejpam-65	500	2	if	if	SCONJ
ejpam-65	500	3	l	l	PROPN
ejpam-65	500	4	∈	∈	PROPN
ejpam-65	500	5	b	b	X
ejpam-65	500	6	,	,	PUNCT
ejpam-65	500	7	let	let	VERB
ejpam-65	500	8	w	w	PROPN
ejpam-65	500	9	=	=	SYM
ejpam-65	500	10	t−1(l	t−1(l	PROPN
ejpam-65	500	11	)	)	PUNCT
ejpam-65	500	12	.	.	PUNCT
ejpam-65	501	1	then	then	ADV
ejpam-65	501	2	by	by	ADP
ejpam-65	501	3	proposition	proposition	NOUN
ejpam-65	501	4	3.5	3.5	NUM
ejpam-65	501	5	,	,	PUNCT
ejpam-65	501	6	we	we	PRON
ejpam-65	501	7	obtain	obtain	VERB
ejpam-65	501	8	that	that	DET
ejpam-65	501	9	w	w	PROPN
ejpam-65	501	10	∈	∈	PROPN
ejpam-65	501	11	a	a	PRON
ejpam-65	501	12	and	and	CCONJ
ejpam-65	501	13	t	t	PROPN
ejpam-65	501	14	(	(	PUNCT
ejpam-65	501	15	w	w	PROPN
ejpam-65	501	16	)	)	PUNCT
ejpam-65	501	17	=	=	VERB
ejpam-65	502	1	l.	l.	PROPN
ejpam-65	502	2	consequently	consequently	ADV
ejpam-65	502	3	ϕ	ϕ	PROPN
ejpam-65	502	4	is	be	AUX
ejpam-65	502	5	a	a	DET
ejpam-65	502	6	bijection	bijection	NOUN
ejpam-65	502	7	.	.	PUNCT
ejpam-65	503	1	corollary	corollary	ADJ
ejpam-65	503	2	3.5	3.5	NUM
ejpam-65	503	3	.	.	PUNCT
ejpam-65	504	1	every	every	DET
ejpam-65	504	2	subhyperspace	subhyperspace	NOUN
ejpam-65	504	3	of	of	ADP
ejpam-65	504	4	v	v	NOUN
ejpam-65	504	5	/	/	SYM
ejpam-65	504	6	w	w	NOUN
ejpam-65	504	7	is	be	AUX
ejpam-65	504	8	of	of	ADP
ejpam-65	504	9	the	the	DET
ejpam-65	504	10	form	form	NOUN
ejpam-65	504	11	l	l	PROPN
ejpam-65	504	12	/	/	SYM
ejpam-65	504	13	w	w	PROPN
ejpam-65	504	14	,	,	PUNCT
ejpam-65	504	15	such	such	ADJ
ejpam-65	504	16	that	that	SCONJ
ejpam-65	504	17	l	l	NOUN
ejpam-65	504	18	is	be	AUX
ejpam-65	504	19	a	a	DET
ejpam-65	504	20	subhyperspace	subhyperspace	NOUN
ejpam-65	504	21	of	of	ADP
ejpam-65	504	22	v	v	NOUN
ejpam-65	504	23	containing	contain	VERB
ejpam-65	504	24	w	w	NOUN
ejpam-65	504	25	.	.	PUNCT
ejpam-65	505	1	proof	proof	NOUN
ejpam-65	505	2	.	.	PUNCT
ejpam-65	506	1	by	by	ADP
ejpam-65	506	2	proposition	proposition	NOUN
ejpam-65	506	3	3.3	3.3	NUM
ejpam-65	506	4	the	the	DET
ejpam-65	506	5	mapping	mapping	NOUN
ejpam-65	506	6	π	π	NOUN
ejpam-65	506	7	:	:	PUNCT
ejpam-65	506	8	v	v	ADP
ejpam-65	506	9	−→	−→	NOUN
ejpam-65	506	10	v	v	NOUN
ejpam-65	506	11	/	/	SYM
ejpam-65	506	12	w	w	NOUN
ejpam-65	506	13	is	be	AUX
ejpam-65	506	14	a	a	DET
ejpam-65	506	15	good	good	ADJ
ejpam-65	506	16	linear	linear	ADJ
ejpam-65	506	17	transformation	transformation	NOUN
ejpam-65	506	18	.	.	PUNCT
ejpam-65	507	1	it	it	PRON
ejpam-65	507	2	is	be	AUX
ejpam-65	507	3	easy	easy	ADJ
ejpam-65	507	4	to	to	PART
ejpam-65	507	5	verify	verify	VERB
ejpam-65	507	6	that	that	DET
ejpam-65	507	7	kerπ	kerπ	PROPN
ejpam-65	507	8	=	=	SYM
ejpam-65	507	9	w.	w.	PROPN
ejpam-65	507	10	thus	thus	ADV
ejpam-65	507	11	by	by	ADP
ejpam-65	507	12	proposition	proposition	NOUN
ejpam-65	507	13	3.6	3.6	NUM
ejpam-65	507	14	,	,	PUNCT
ejpam-65	507	15	for	for	ADP
ejpam-65	507	16	an	an	DET
ejpam-65	507	17	arbitrary	arbitrary	ADJ
ejpam-65	507	18	subhyperspace	subhyperspace	NOUN
ejpam-65	507	19	l̄	l̄	NOUN
ejpam-65	507	20	of	of	ADP
ejpam-65	507	21	v	v	NOUN
ejpam-65	507	22	/	/	SYM
ejpam-65	507	23	w	w	PROPN
ejpam-65	507	24	,	,	PUNCT
ejpam-65	507	25	there	there	PRON
ejpam-65	507	26	exists	exist	VERB
ejpam-65	507	27	a	a	DET
ejpam-65	507	28	subhyperspace	subhyperspace	NOUN
ejpam-65	507	29	l	l	NOUN
ejpam-65	507	30	of	of	ADP
ejpam-65	507	31	v	v	NOUN
ejpam-65	507	32	,	,	PUNCT
ejpam-65	507	33	such	such	ADJ
ejpam-65	507	34	that	that	SCONJ
ejpam-65	507	35	w	w	ADP
ejpam-65	507	36	⊆	⊆	NUM
ejpam-65	507	37	l	l	NOUN
ejpam-65	507	38	and	and	CCONJ
ejpam-65	507	39	ϕ(l	ϕ(l	NOUN
ejpam-65	507	40	)	)	PUNCT
ejpam-65	508	1	=	=	NOUN
ejpam-65	508	2	l̄.	l̄.	PUNCT
ejpam-65	508	3	moreover	moreover	ADV
ejpam-65	508	4	,	,	PUNCT
ejpam-65	508	5	ϕ(l	ϕ(l	PUNCT
ejpam-65	508	6	)	)	PUNCT
ejpam-65	508	7	=	=	PRON
ejpam-65	508	8	{	{	PUNCT
ejpam-65	508	9	l	l	NOUN
ejpam-65	509	1	+	+	NOUN
ejpam-65	509	2	w	w	NOUN
ejpam-65	509	3	:	:	PUNCT
ejpam-65	509	4	l	l	NOUN
ejpam-65	509	5	∈	∈	PROPN
ejpam-65	509	6	l	l	NOUN
ejpam-65	509	7	}	}	PUNCT
ejpam-65	509	8	=	=	SYM
ejpam-65	509	9	l	l	NOUN
ejpam-65	509	10	/	/	SYM
ejpam-65	509	11	w	w	PROPN
ejpam-65	509	12	.	.	PUNCT
ejpam-65	510	1	this	this	PRON
ejpam-65	510	2	complete	complete	ADJ
ejpam-65	510	3	the	the	DET
ejpam-65	510	4	proof	proof	NOUN
ejpam-65	510	5	.	.	PUNCT
ejpam-65	511	1	theorem	theorem	VERB
ejpam-65	511	2	3.4	3.4	NUM
ejpam-65	511	3	.	.	PUNCT
ejpam-65	512	1	let	let	VERB
ejpam-65	512	2	v	v	NOUN
ejpam-65	512	3	and	and	CCONJ
ejpam-65	512	4	u	u	PRON
ejpam-65	512	5	be	be	AUX
ejpam-65	512	6	strongly	strongly	ADV
ejpam-65	512	7	left	leave	VERB
ejpam-65	512	8	distributive	distributive	ADJ
ejpam-65	512	9	hypervector	hypervector	NOUN
ejpam-65	512	10	spaces	space	NOUN
ejpam-65	512	11	over	over	ADP
ejpam-65	512	12	the	the	DET
ejpam-65	512	13	field	field	NOUN
ejpam-65	513	1	k	k	NOUN
ejpam-65	513	2	,	,	PUNCT
ejpam-65	513	3	and	and	CCONJ
ejpam-65	513	4	t	t	PROPN
ejpam-65	513	5	:	:	PUNCT
ejpam-65	513	6	v	v	X
ejpam-65	513	7	−→	−→	NOUN
ejpam-65	513	8	u	u	NOUN
ejpam-65	513	9	be	be	VERB
ejpam-65	513	10	a	a	DET
ejpam-65	513	11	linear	linear	ADJ
ejpam-65	513	12	transformation	transformation	NOUN
ejpam-65	513	13	.	.	PUNCT
ejpam-65	514	1	then	then	ADV
ejpam-65	514	2	v/	v/	VERB
ejpam-65	514	3	kert	kert	PROPN
ejpam-65	514	4	∼=	∼=	PROPN
ejpam-65	514	5	t	t	PROPN
ejpam-65	514	6	(	(	PUNCT
ejpam-65	514	7	v	v	NOUN
ejpam-65	514	8	)	)	PUNCT
ejpam-65	514	9	/ω	/ω	PUNCT
ejpam-65	514	10	,	,	PUNCT
ejpam-65	514	11	moreover	moreover	ADV
ejpam-65	514	12	if	if	SCONJ
ejpam-65	514	13	t	t	PROPN
ejpam-65	514	14	is	be	AUX
ejpam-65	514	15	onto	onto	ADP
ejpam-65	514	16	,	,	PUNCT
ejpam-65	514	17	then	then	ADV
ejpam-65	514	18	v/	v/	VERB
ejpam-65	514	19	kert	kert	PROPN
ejpam-65	514	20	∼=	∼=	PROPN
ejpam-65	514	21	u	u	PROPN
ejpam-65	514	22	/	/	SYM
ejpam-65	514	23	ω	ω	NOUN
ejpam-65	514	24	.	.	PUNCT
ejpam-65	515	1	proof	proof	NOUN
ejpam-65	515	2	.	.	PUNCT
ejpam-65	516	1	we	we	PRON
ejpam-65	516	2	show	show	VERB
ejpam-65	516	3	that	that	SCONJ
ejpam-65	516	4	the	the	DET
ejpam-65	516	5	mapping	mapping	NOUN
ejpam-65	516	6	{	{	PUNCT
ejpam-65	516	7	ϕ	ϕ	NOUN
ejpam-65	516	8	:	:	PUNCT
ejpam-65	516	9	v/	v/	PROPN
ejpam-65	516	10	kert	kert	PROPN
ejpam-65	516	11	−→	−→	PROPN
ejpam-65	516	12	t	t	PROPN
ejpam-65	516	13	(	(	PUNCT
ejpam-65	516	14	v	v	NOUN
ejpam-65	516	15	)	)	PUNCT
ejpam-65	516	16	/ω	/ω	PUNCT
ejpam-65	516	17	ϕ(x+	ϕ(x+	INTJ
ejpam-65	516	18	kert	kert	PROPN
ejpam-65	516	19	)	)	PUNCT
ejpam-65	517	1	=	=	SYM
ejpam-65	517	2	t	t	PROPN
ejpam-65	517	3	(	(	PUNCT
ejpam-65	517	4	x	x	X
ejpam-65	517	5	)	)	PUNCT
ejpam-65	518	1	+	+	CCONJ
ejpam-65	518	2	ω	ω	NOUN
ejpam-65	518	3	is	be	AUX
ejpam-65	518	4	an	an	DET
ejpam-65	518	5	isomorphism	isomorphism	NOUN
ejpam-65	518	6	.	.	PUNCT
ejpam-65	519	1	for	for	ADP
ejpam-65	519	2	this	this	PRON
ejpam-65	519	3	let	let	NOUN
ejpam-65	519	4	x+	x+	PROPN
ejpam-65	519	5	kert	kert	PROPN
ejpam-65	519	6	and	and	CCONJ
ejpam-65	519	7	y	y	PROPN
ejpam-65	520	1	+	+	CCONJ
ejpam-65	520	2	kert	kert	PROPN
ejpam-65	520	3	be	be	VERB
ejpam-65	520	4	two	two	NUM
ejpam-65	520	5	elements	element	NOUN
ejpam-65	520	6	of	of	ADP
ejpam-65	520	7	v/	v/	NOUN
ejpam-65	520	8	kert	kert	PROPN
ejpam-65	520	9	.	.	PUNCT
ejpam-65	521	1	then	then	ADV
ejpam-65	521	2	x+	x+	PROPN
ejpam-65	521	3	kert	kert	PROPN
ejpam-65	521	4	=	=	PUNCT
ejpam-65	521	5	y	y	PROPN
ejpam-65	521	6	+	+	CCONJ
ejpam-65	521	7	kert	kert	PROPN
ejpam-65	521	8	⇐	⇐	PROPN
ejpam-65	521	9	⇒	⇒	PROPN
ejpam-65	521	10	x−	x−	PROPN
ejpam-65	521	11	y	y	PROPN
ejpam-65	521	12	∈	∈	PROPN
ejpam-65	521	13	kert	kert	PROPN
ejpam-65	521	14	⇐	⇐	PROPN
ejpam-65	521	15	⇒	⇒	PROPN
ejpam-65	521	16	t	t	PROPN
ejpam-65	521	17	(	(	PUNCT
ejpam-65	521	18	x−	x−	PROPN
ejpam-65	521	19	y	y	PROPN
ejpam-65	521	20	)	)	PUNCT
ejpam-65	521	21	∈	∈	PROPN
ejpam-65	521	22	ω	ω	NUM
ejpam-65	521	23	⇐	⇐	PROPN
ejpam-65	521	24	⇒	⇒	PROPN
ejpam-65	521	25	t	t	PROPN
ejpam-65	521	26	(	(	PUNCT
ejpam-65	521	27	x)−	x)−	PROPN
ejpam-65	521	28	t	t	PROPN
ejpam-65	521	29	(	(	PUNCT
ejpam-65	521	30	y	y	NOUN
ejpam-65	521	31	)	)	PUNCT
ejpam-65	521	32	∈	∈	PROPN
ejpam-65	521	33	ω	ω	NUM
ejpam-65	521	34	⇐	⇐	PROPN
ejpam-65	521	35	⇒	⇒	PROPN
ejpam-65	521	36	t	t	PROPN
ejpam-65	521	37	(	(	PUNCT
ejpam-65	521	38	x	x	X
ejpam-65	521	39	)	)	PUNCT
ejpam-65	522	1	+	+	CCONJ
ejpam-65	522	2	ω	ω	X
ejpam-65	522	3	=	=	SYM
ejpam-65	522	4	t	t	PROPN
ejpam-65	522	5	(	(	PUNCT
ejpam-65	522	6	y	y	NOUN
ejpam-65	522	7	)	)	PUNCT
ejpam-65	523	1	+	+	CCONJ
ejpam-65	523	2	ω	ω	NUM
ejpam-65	523	3	⇐	⇐	ADJ
ejpam-65	523	4	⇒	⇒	PROPN
ejpam-65	523	5	ϕ(x+	ϕ(x+	INTJ
ejpam-65	523	6	kert	kert	PROPN
ejpam-65	523	7	)	)	PUNCT
ejpam-65	524	1	=	=	PUNCT
ejpam-65	525	1	ϕ(y	ϕ(y	PROPN
ejpam-65	525	2	+	+	CCONJ
ejpam-65	525	3	kert	kert	PROPN
ejpam-65	525	4	)	)	PUNCT
ejpam-65	525	5	,	,	PUNCT
ejpam-65	525	6	r.	r.	PROPN
ejpam-65	525	7	ameri	ameri	PROPN
ejpam-65	525	8	,	,	PUNCT
ejpam-65	525	9	o.r	o.r	PROPN
ejpam-65	525	10	.	.	PROPN
ejpam-65	525	11	dehghan	dehghan	PROPN
ejpam-65	525	12	/	/	SYM
ejpam-65	525	13	eur	eur	PROPN
ejpam-65	525	14	.	.	PUNCT
ejpam-65	526	1	j.	j.	PROPN
ejpam-65	526	2	pure	pure	PROPN
ejpam-65	526	3	appl	appl	PROPN
ejpam-65	526	4	.	.	PROPN
ejpam-65	526	5	math	math	PROPN
ejpam-65	526	6	,	,	PUNCT
ejpam-65	526	7	1	1	NUM
ejpam-65	526	8	(	(	PUNCT
ejpam-65	526	9	2008	2008	NUM
ejpam-65	526	10	)	)	PUNCT
ejpam-65	526	11	,	,	PUNCT
ejpam-65	526	12	(	(	PUNCT
ejpam-65	526	13	32	32	NUM
ejpam-65	526	14	-	-	SYM
ejpam-65	526	15	50	50	NUM
ejpam-65	526	16	)	)	PUNCT
ejpam-65	526	17	45	45	NUM
ejpam-65	526	18	thus	thus	ADV
ejpam-65	526	19	ϕ	ϕ	NOUN
ejpam-65	526	20	is	be	AUX
ejpam-65	526	21	welldefined	welldefine	VERB
ejpam-65	526	22	and	and	CCONJ
ejpam-65	526	23	one	one	NUM
ejpam-65	526	24	to	to	ADP
ejpam-65	526	25	one	one	NUM
ejpam-65	526	26	.	.	PUNCT
ejpam-65	527	1	it	it	PRON
ejpam-65	527	2	is	be	AUX
ejpam-65	527	3	clearly	clearly	ADV
ejpam-65	527	4	that	that	SCONJ
ejpam-65	527	5	ϕ	ϕ	NOUN
ejpam-65	527	6	is	be	AUX
ejpam-65	527	7	onto	onto	ADP
ejpam-65	527	8	.	.	PUNCT
ejpam-65	528	1	now	now	ADV
ejpam-65	528	2	for	for	ADP
ejpam-65	528	3	every	every	PRON
ejpam-65	528	4	x	x	SYM
ejpam-65	528	5	+	+	CCONJ
ejpam-65	528	6	kert	kert	PROPN
ejpam-65	528	7	and	and	CCONJ
ejpam-65	528	8	y	y	PROPN
ejpam-65	528	9	+	+	CCONJ
ejpam-65	528	10	kert	kert	PROPN
ejpam-65	528	11	belong	belong	VERB
ejpam-65	528	12	to	to	ADP
ejpam-65	528	13	v/	v/	PROPN
ejpam-65	528	14	kert	kert	PROPN
ejpam-65	528	15	and	and	CCONJ
ejpam-65	528	16	a	a	DET
ejpam-65	528	17	∈	∈	NOUN
ejpam-65	528	18	k	k	NOUN
ejpam-65	528	19	we	we	PRON
ejpam-65	528	20	have	have	VERB
ejpam-65	528	21	:	:	PUNCT
ejpam-65	528	22	ϕ	ϕ	X
ejpam-65	528	23	(	(	PUNCT
ejpam-65	528	24	(	(	PUNCT
ejpam-65	528	25	x+	x+	X
ejpam-65	528	26	kert	kert	PROPN
ejpam-65	528	27	)	)	PUNCT
ejpam-65	529	1	+	+	CCONJ
ejpam-65	529	2	(	(	PUNCT
ejpam-65	529	3	y	y	PROPN
ejpam-65	529	4	+	+	CCONJ
ejpam-65	529	5	kert	kert	PROPN
ejpam-65	529	6	)	)	PUNCT
ejpam-65	529	7	)	)	PUNCT
ejpam-65	530	1	=	=	SYM
ejpam-65	530	2	ϕ	ϕ	PROPN
ejpam-65	530	3	(	(	PUNCT
ejpam-65	530	4	x+	x+	PROPN
ejpam-65	530	5	y	y	PROPN
ejpam-65	530	6	+	+	CCONJ
ejpam-65	530	7	kert	kert	PROPN
ejpam-65	530	8	)	)	PUNCT
ejpam-65	531	1	=	=	SYM
ejpam-65	531	2	t	t	PROPN
ejpam-65	531	3	(	(	PUNCT
ejpam-65	531	4	x+	x+	PROPN
ejpam-65	531	5	y	y	NOUN
ejpam-65	531	6	)	)	PUNCT
ejpam-65	532	1	+	+	CCONJ
ejpam-65	532	2	ω	ω	X
ejpam-65	532	3	=	=	SYM
ejpam-65	532	4	t	t	PROPN
ejpam-65	532	5	(	(	PUNCT
ejpam-65	532	6	x	x	X
ejpam-65	532	7	)	)	PUNCT
ejpam-65	533	1	+	+	NUM
ejpam-65	533	2	t	t	PROPN
ejpam-65	533	3	(	(	PUNCT
ejpam-65	533	4	y	y	NOUN
ejpam-65	533	5	)	)	PUNCT
ejpam-65	533	6	+	+	CCONJ
ejpam-65	533	7	ω	ω	X
ejpam-65	533	8	=	=	SYM
ejpam-65	533	9	t	t	PROPN
ejpam-65	533	10	(	(	PUNCT
ejpam-65	533	11	x	x	X
ejpam-65	533	12	)	)	PUNCT
ejpam-65	534	1	+	+	CCONJ
ejpam-65	534	2	ω	ω	PROPN
ejpam-65	534	3	+	+	NUM
ejpam-65	534	4	t	t	PROPN
ejpam-65	534	5	(	(	PUNCT
ejpam-65	534	6	y	y	NOUN
ejpam-65	534	7	)	)	PUNCT
ejpam-65	535	1	+	+	NOUN
ejpam-65	535	2	ω	ω	X
ejpam-65	535	3	=	=	SYM
ejpam-65	535	4	ϕ(x+	ϕ(x+	X
ejpam-65	535	5	kert	kert	PROPN
ejpam-65	535	6	)	)	PUNCT
ejpam-65	536	1	+	+	CCONJ
ejpam-65	536	2	ϕ(y	ϕ(y	PROPN
ejpam-65	536	3	+	+	CCONJ
ejpam-65	536	4	kert	kert	PROPN
ejpam-65	536	5	)	)	PUNCT
ejpam-65	536	6	,	,	PUNCT
ejpam-65	536	7	and	and	CCONJ
ejpam-65	536	8	ϕ	ϕ	X
ejpam-65	536	9	(	(	PUNCT
ejpam-65	536	10	a	a	DET
ejpam-65	536	11	∗	∗	NOUN
ejpam-65	536	12	(	(	PUNCT
ejpam-65	536	13	x+	x+	X
ejpam-65	536	14	kert	kert	PROPN
ejpam-65	536	15	)	)	PUNCT
ejpam-65	536	16	)	)	PUNCT
ejpam-65	537	1	=	=	SYM
ejpam-65	537	2	ϕ	ϕ	X
ejpam-65	537	3	(	(	PUNCT
ejpam-65	537	4	a	a	DET
ejpam-65	537	5	◦	◦	NOUN
ejpam-65	537	6	x+	x+	VERB
ejpam-65	537	7	kert	kert	PROPN
ejpam-65	537	8	)	)	PUNCT
ejpam-65	537	9	=	=	PRON
ejpam-65	537	10	{	{	PUNCT
ejpam-65	537	11	ϕ	ϕ	NOUN
ejpam-65	537	12	(	(	PUNCT
ejpam-65	537	13	t+	t+	NOUN
ejpam-65	537	14	kert	kert	PROPN
ejpam-65	537	15	)	)	PUNCT
ejpam-65	537	16	:	:	PUNCT
ejpam-65	538	1	t	t	PROPN
ejpam-65	538	2	∈	∈	PROPN
ejpam-65	538	3	a	a	DET
ejpam-65	538	4	◦	◦	NOUN
ejpam-65	538	5	x	x	NOUN
ejpam-65	538	6	}	}	PUNCT
ejpam-65	538	7	=	=	SYM
ejpam-65	538	8	{	{	PUNCT
ejpam-65	538	9	t	t	PROPN
ejpam-65	538	10	(	(	PUNCT
ejpam-65	538	11	t	t	PROPN
ejpam-65	538	12	)	)	PUNCT
ejpam-65	539	1	+	+	NUM
ejpam-65	540	1	ω	ω	NUM
ejpam-65	540	2	:	:	PUNCT
ejpam-65	540	3	t	t	PROPN
ejpam-65	540	4	∈	∈	PROPN
ejpam-65	540	5	a	a	DET
ejpam-65	540	6	◦	◦	NOUN
ejpam-65	540	7	x	x	NOUN
ejpam-65	540	8	}	}	PUNCT
ejpam-65	540	9	=	=	SYM
ejpam-65	540	10	t	t	PROPN
ejpam-65	540	11	(	(	PUNCT
ejpam-65	540	12	a	a	DET
ejpam-65	540	13	◦	◦	NOUN
ejpam-65	540	14	x	x	X
ejpam-65	540	15	)	)	PUNCT
ejpam-65	541	1	+	+	CCONJ
ejpam-65	541	2	ω	ω	NUM
ejpam-65	541	3	⊆	⊆	NUM
ejpam-65	541	4	a	a	DET
ejpam-65	541	5	◦	◦	NOUN
ejpam-65	541	6	t	t	X
ejpam-65	541	7	(	(	PUNCT
ejpam-65	541	8	x	x	X
ejpam-65	541	9	)	)	PUNCT
ejpam-65	542	1	+	+	CCONJ
ejpam-65	542	2	ω	ω	X
ejpam-65	542	3	=	=	PUNCT
ejpam-65	542	4	a	a	PRON
ejpam-65	542	5	•	•	NOUN
ejpam-65	542	6	(	(	PUNCT
ejpam-65	542	7	t	t	PROPN
ejpam-65	542	8	(	(	PUNCT
ejpam-65	542	9	x	x	NOUN
ejpam-65	542	10	)	)	PUNCT
ejpam-65	542	11	+	+	CCONJ
ejpam-65	542	12	ω	ω	X
ejpam-65	542	13	)	)	PUNCT
ejpam-65	542	14	=	=	NOUN
ejpam-65	543	1	a	a	DET
ejpam-65	543	2	•	•	INTJ
ejpam-65	543	3	ϕ(x+	ϕ(x+	INTJ
ejpam-65	543	4	kert	kert	PROPN
ejpam-65	543	5	)	)	PUNCT
ejpam-65	543	6	.	.	PUNCT
ejpam-65	544	1	therefore	therefore	ADV
ejpam-65	544	2	ϕ	ϕ	PROPN
ejpam-65	544	3	is	be	AUX
ejpam-65	544	4	an	an	DET
ejpam-65	544	5	isomorphism	isomorphism	NOUN
ejpam-65	544	6	.	.	PUNCT
ejpam-65	545	1	corollary	corollary	ADJ
ejpam-65	545	2	3.6	3.6	NUM
ejpam-65	545	3	.	.	PUNCT
ejpam-65	546	1	let	let	VERB
ejpam-65	546	2	v	v	PART
ejpam-65	546	3	be	be	AUX
ejpam-65	546	4	a	a	DET
ejpam-65	546	5	strongly	strongly	ADV
ejpam-65	546	6	left	leave	VERB
ejpam-65	546	7	distributive	distributive	ADJ
ejpam-65	546	8	,	,	PUNCT
ejpam-65	546	9	and	and	CCONJ
ejpam-65	546	10	let	let	VERB
ejpam-65	546	11	β	β	X
ejpam-65	546	12	=	=	PUNCT
ejpam-65	546	13	{	{	PUNCT
ejpam-65	546	14	x1	x1	PROPN
ejpam-65	546	15	,	,	PUNCT
ejpam-65	546	16	.	.	PUNCT
ejpam-65	546	17	.	.	PUNCT
ejpam-65	547	1	.	.	PUNCT
ejpam-65	548	1	,	,	PUNCT
ejpam-65	548	2	xn	xn	X
ejpam-65	548	3	}	}	PUNCT
ejpam-65	548	4	be	be	VERB
ejpam-65	548	5	a	a	DET
ejpam-65	548	6	basis	basis	NOUN
ejpam-65	548	7	for	for	ADP
ejpam-65	548	8	v	v	NOUN
ejpam-65	548	9	.	.	PUNCT
ejpam-65	549	1	then	then	ADV
ejpam-65	549	2	v/0	v/0	DET
ejpam-65	549	3	◦	◦	NOUN
ejpam-65	549	4	ω	ω	PUNCT
ejpam-65	549	5	∼=	∼=	PROPN
ejpam-65	549	6	kn	kn	NOUN
ejpam-65	549	7	,	,	PUNCT
ejpam-65	549	8	where	where	SCONJ
ejpam-65	549	9	ω	ω	PROPN
ejpam-65	549	10	=	=	SYM
ejpam-65	549	11	n∑	n∑	PROPN
ejpam-65	549	12	i=1	i=1	X
ejpam-65	549	13	xi	xi	PROPN
ejpam-65	549	14	.	.	PUNCT
ejpam-65	550	1	proof	proof	NOUN
ejpam-65	550	2	.	.	PUNCT
ejpam-65	551	1	note	note	VERB
ejpam-65	551	2	that	that	SCONJ
ejpam-65	551	3	(	(	PUNCT
ejpam-65	551	4	kn,+	kn,+	PROPN
ejpam-65	551	5	,	,	PUNCT
ejpam-65	551	6	◦	◦	PROPN
ejpam-65	551	7	k	k	PROPN
ejpam-65	551	8	,	,	PUNCT
ejpam-65	551	9	k	k	PROPN
ejpam-65	551	10	)	)	PUNCT
ejpam-65	551	11	is	be	AUX
ejpam-65	551	12	a	a	DET
ejpam-65	551	13	strongly	strongly	ADV
ejpam-65	551	14	distributive	distributive	ADJ
ejpam-65	551	15	hypervector	hypervector	NOUN
ejpam-65	551	16	space	space	NOUN
ejpam-65	551	17	with	with	ADP
ejpam-65	551	18	trivial	trivial	ADJ
ejpam-65	551	19	external	external	ADJ
ejpam-65	551	20	operation	operation	NOUN
ejpam-65	551	21	◦	◦	PROPN
ejpam-65	551	22	k	k	PROPN
ejpam-65	551	23	,	,	PUNCT
ejpam-65	551	24	that	that	PRON
ejpam-65	551	25	is	be	AUX
ejpam-65	551	26	:	:	PUNCT
ejpam-65	551	27	{	{	PUNCT
ejpam-65	551	28	◦	◦	NOUN
ejpam-65	551	29	k	k	NOUN
ejpam-65	551	30	:	:	PUNCT
ejpam-65	552	1	k	k	X
ejpam-65	552	2	×kn	×kn	PROPN
ejpam-65	552	3	−→	−→	NOUN
ejpam-65	552	4	p∗	p∗	PROPN
ejpam-65	552	5	(	(	PUNCT
ejpam-65	552	6	kn	kn	PROPN
ejpam-65	552	7	)	)	PUNCT
ejpam-65	552	8	a	a	DET
ejpam-65	552	9	◦	◦	NOUN
ejpam-65	552	10	k	k	X
ejpam-65	552	11	(	(	PUNCT
ejpam-65	552	12	a1	a1	PROPN
ejpam-65	552	13	,	,	PUNCT
ejpam-65	552	14	.	.	PUNCT
ejpam-65	552	15	.	.	PUNCT
ejpam-65	552	16	.	.	PUNCT
ejpam-65	553	1	,	,	PUNCT
ejpam-65	553	2	an	an	X
ejpam-65	553	3	)	)	PUNCT
ejpam-65	553	4	=	=	SYM
ejpam-65	553	5	{	{	PUNCT
ejpam-65	553	6	(	(	PUNCT
ejpam-65	553	7	aa1	aa1	PROPN
ejpam-65	553	8	,	,	PUNCT
ejpam-65	553	9	.	.	PUNCT
ejpam-65	553	10	.	.	PUNCT
ejpam-65	553	11	.	.	PUNCT
ejpam-65	554	1	,	,	PUNCT
ejpam-65	554	2	aan	aan	PROPN
ejpam-65	554	3	)	)	PUNCT
ejpam-65	554	4	}	}	PUNCT
ejpam-65	554	5	.	.	PUNCT
ejpam-65	555	1	define	define	VERB
ejpam-65	555	2	the	the	DET
ejpam-65	555	3	mapping	mapping	NOUN
ejpam-65	555	4			PUNCT
ejpam-65	555	5	t	t	PROPN
ejpam-65	555	6	:	:	PUNCT
ejpam-65	555	7	v	v	ADP
ejpam-65	555	8	−→	−→	NOUN
ejpam-65	556	1	kn	kn	PROPN
ejpam-65	556	2	x	x	PUNCT
ejpam-65	556	3	∈	∈	PROPN
ejpam-65	556	4	n∑	n∑	INTJ
ejpam-65	556	5	i=1	i=1	PROPN
ejpam-65	556	6	ai	ai	VERB
ejpam-65	556	7	◦	◦	NOUN
ejpam-65	556	8	xi	xi	X
ejpam-65	556	9	7−→	7−→	PROPN
ejpam-65	556	10	(	(	PUNCT
ejpam-65	556	11	a1	a1	NOUN
ejpam-65	556	12	,	,	PUNCT
ejpam-65	556	13	.	.	PUNCT
ejpam-65	556	14	.	.	PUNCT
ejpam-65	557	1	.	.	PUNCT
ejpam-65	558	1	,	,	PUNCT
ejpam-65	558	2	an	an	X
ejpam-65	558	3	)	)	PUNCT
ejpam-65	558	4	,	,	PUNCT
ejpam-65	558	5	it	it	PRON
ejpam-65	558	6	is	be	AUX
ejpam-65	558	7	easy	easy	ADJ
ejpam-65	558	8	to	to	PART
ejpam-65	558	9	see	see	VERB
ejpam-65	558	10	that	that	PRON
ejpam-65	558	11	t	t	PROPN
ejpam-65	558	12	is	be	AUX
ejpam-65	558	13	an	an	PRON
ejpam-65	558	14	onto	onto	ADP
ejpam-65	558	15	linear	linear	ADJ
ejpam-65	558	16	transformation	transformation	NOUN
ejpam-65	558	17	,	,	PUNCT
ejpam-65	558	18	such	such	ADJ
ejpam-65	558	19	that	that	SCONJ
ejpam-65	558	20	kert	kert	PROPN
ejpam-65	558	21	=	=	PROPN
ejpam-65	558	22	=	=	SYM
ejpam-65	558	23	0	0	NUM
ejpam-65	558	24	◦	◦	NOUN
ejpam-65	558	25	ω	ω	PROPN
ejpam-65	558	26	,	,	PUNCT
ejpam-65	558	27	since	since	SCONJ
ejpam-65	558	28	v	v	NOUN
ejpam-65	558	29	is	be	AUX
ejpam-65	558	30	strongly	strongly	ADV
ejpam-65	558	31	left	leave	VERB
ejpam-65	558	32	distributive	distributive	ADJ
ejpam-65	558	33	.	.	PUNCT
ejpam-65	559	1	then	then	ADV
ejpam-65	559	2	by	by	ADP
ejpam-65	559	3	theorem	theorem	NOUN
ejpam-65	559	4	3.4	3.4	NUM
ejpam-65	559	5	it	it	PRON
ejpam-65	559	6	follows	follow	VERB
ejpam-65	559	7	that	that	SCONJ
ejpam-65	559	8	:	:	PUNCT
ejpam-65	559	9	v/0	v/0	PUNCT
ejpam-65	559	10	◦	◦	NOUN
ejpam-65	559	11	ω	ω	SYM
ejpam-65	559	12	∼=	∼=	NOUN
ejpam-65	559	13	v/	v/	NOUN
ejpam-65	559	14	kert	kert	PROPN
ejpam-65	559	15	∼=	∼=	PROPN
ejpam-65	559	16	kn/0	kn/0	ADJ
ejpam-65	559	17	∼=	∼=	PROPN
ejpam-65	559	18	kn	kn	PROPN
ejpam-65	559	19	.	.	PROPN
ejpam-65	559	20	r.	r.	PROPN
ejpam-65	559	21	ameri	ameri	PROPN
ejpam-65	559	22	,	,	PUNCT
ejpam-65	559	23	o.r	o.r	PROPN
ejpam-65	559	24	.	.	PROPN
ejpam-65	559	25	dehghan	dehghan	PROPN
ejpam-65	559	26	/	/	SYM
ejpam-65	559	27	eur	eur	PROPN
ejpam-65	559	28	.	.	PUNCT
ejpam-65	560	1	j.	j.	PROPN
ejpam-65	560	2	pure	pure	PROPN
ejpam-65	560	3	appl	appl	PROPN
ejpam-65	560	4	.	.	PROPN
ejpam-65	560	5	math	math	PROPN
ejpam-65	560	6	,	,	PUNCT
ejpam-65	560	7	1	1	NUM
ejpam-65	560	8	(	(	PUNCT
ejpam-65	560	9	2008	2008	NUM
ejpam-65	560	10	)	)	PUNCT
ejpam-65	560	11	,	,	PUNCT
ejpam-65	560	12	(	(	PUNCT
ejpam-65	560	13	32	32	NUM
ejpam-65	560	14	-	-	SYM
ejpam-65	560	15	50	50	NUM
ejpam-65	560	16	)	)	PUNCT
ejpam-65	560	17	46	46	NUM
ejpam-65	560	18	4	4	NUM
ejpam-65	560	19	.	.	PUNCT
ejpam-65	560	20	fundamental	fundamental	ADJ
ejpam-65	560	21	relation	relation	NOUN
ejpam-65	560	22	of	of	ADP
ejpam-65	560	23	hypervector	hypervector	NOUN
ejpam-65	560	24	spaces	space	NOUN
ejpam-65	560	25	let	let	VERB
ejpam-65	560	26	(	(	PUNCT
ejpam-65	560	27	v,+	v,+	NUM
ejpam-65	560	28	,	,	PUNCT
ejpam-65	560	29	◦	◦	NOUN
ejpam-65	560	30	,	,	PUNCT
ejpam-65	560	31	k	k	NOUN
ejpam-65	560	32	)	)	PUNCT
ejpam-65	560	33	be	be	VERB
ejpam-65	560	34	a	a	DET
ejpam-65	560	35	hypervector	hypervector	NOUN
ejpam-65	560	36	space	space	NOUN
ejpam-65	560	37	over	over	ADP
ejpam-65	560	38	k.	k.	PROPN
ejpam-65	560	39	the	the	DET
ejpam-65	560	40	smallest	small	ADJ
ejpam-65	560	41	equivalence	equivalence	NOUN
ejpam-65	560	42	relation	relation	NOUN
ejpam-65	560	43	ε∗	ε∗	VERB
ejpam-65	560	44	on	on	ADP
ejpam-65	560	45	v	v	NUM
ejpam-65	560	46	,	,	PUNCT
ejpam-65	560	47	such	such	ADJ
ejpam-65	560	48	that	that	SCONJ
ejpam-65	560	49	the	the	DET
ejpam-65	560	50	quotient	quotient	NOUN
ejpam-65	560	51	v	v	NOUN
ejpam-65	560	52	/	/	SYM
ejpam-65	560	53	ε∗	ε∗	PROPN
ejpam-65	560	54	is	be	AUX
ejpam-65	560	55	a	a	DET
ejpam-65	560	56	vector	vector	NOUN
ejpam-65	560	57	space	space	NOUN
ejpam-65	560	58	over	over	ADP
ejpam-65	560	59	k	k	PROPN
ejpam-65	560	60	is	be	AUX
ejpam-65	560	61	called	call	VERB
ejpam-65	560	62	the	the	DET
ejpam-65	560	63	fundamental	fundamental	ADJ
ejpam-65	560	64	relation	relation	NOUN
ejpam-65	560	65	of	of	ADP
ejpam-65	560	66	v	v	NOUN
ejpam-65	560	67	.	.	PUNCT
ejpam-65	561	1	t.	t.	PROPN
ejpam-65	561	2	vougiouklis	vougioukli	NOUN
ejpam-65	561	3	in	in	ADP
ejpam-65	561	4	[	[	X
ejpam-65	561	5	10	10	NUM
ejpam-65	561	6	]	]	PUNCT
ejpam-65	561	7	introduced	introduce	VERB
ejpam-65	561	8	and	and	CCONJ
ejpam-65	561	9	studied	study	VERB
ejpam-65	561	10	the	the	DET
ejpam-65	561	11	fundamental	fundamental	ADJ
ejpam-65	561	12	relation	relation	NOUN
ejpam-65	561	13	of	of	ADP
ejpam-65	561	14	hv	hv	PROPN
ejpam-65	561	15	-	-	PUNCT
ejpam-65	561	16	vector	vector	NOUN
ejpam-65	561	17	space	space	NOUN
ejpam-65	561	18	(	(	PUNCT
ejpam-65	561	19	a	a	DET
ejpam-65	561	20	general	general	ADJ
ejpam-65	561	21	class	class	NOUN
ejpam-65	561	22	of	of	ADP
ejpam-65	561	23	hypervector	hypervector	NOUN
ejpam-65	561	24	spaces	space	NOUN
ejpam-65	561	25	)	)	PUNCT
ejpam-65	561	26	.	.	PUNCT
ejpam-65	562	1	in	in	ADP
ejpam-65	562	2	the	the	DET
ejpam-65	562	3	following	following	NOUN
ejpam-65	562	4	we	we	PRON
ejpam-65	562	5	characterize	characterize	VERB
ejpam-65	562	6	the	the	DET
ejpam-65	562	7	fundamental	fundamental	ADJ
ejpam-65	562	8	relation	relation	NOUN
ejpam-65	562	9	on	on	ADP
ejpam-65	562	10	hypervector	hypervector	NOUN
ejpam-65	562	11	spaces	space	NOUN
ejpam-65	562	12	(	(	PUNCT
ejpam-65	562	13	in	in	ADP
ejpam-65	562	14	the	the	DET
ejpam-65	562	15	sense	sense	NOUN
ejpam-65	562	16	of	of	ADP
ejpam-65	562	17	tallini	tallini	NOUN
ejpam-65	562	18	)	)	PUNCT
ejpam-65	562	19	and	and	CCONJ
ejpam-65	562	20	study	study	VERB
ejpam-65	562	21	the	the	DET
ejpam-65	562	22	relationship	relationship	NOUN
ejpam-65	562	23	between	between	ADP
ejpam-65	562	24	v	v	NOUN
ejpam-65	562	25	and	and	CCONJ
ejpam-65	562	26	v	v	NOUN
ejpam-65	562	27	/	/	SYM
ejpam-65	562	28	ε∗.	ε∗.	NOUN
ejpam-65	562	29	let	let	VERB
ejpam-65	562	30	u	u	PRON
ejpam-65	562	31	be	be	AUX
ejpam-65	562	32	the	the	DET
ejpam-65	562	33	set	set	NOUN
ejpam-65	562	34	of	of	ADP
ejpam-65	562	35	all	all	DET
ejpam-65	562	36	finite	finite	PROPN
ejpam-65	562	37	linear	linear	ADJ
ejpam-65	562	38	combinations	combination	NOUN
ejpam-65	562	39	of	of	ADP
ejpam-65	562	40	elements	element	NOUN
ejpam-65	562	41	of	of	ADP
ejpam-65	562	42	v	v	NOUN
ejpam-65	562	43	with	with	ADP
ejpam-65	562	44	coefficient	coefficient	NOUN
ejpam-65	562	45	in	in	ADP
ejpam-65	562	46	k	k	PROPN
ejpam-65	562	47	,	,	PUNCT
ejpam-65	562	48	that	that	PRON
ejpam-65	562	49	is	be	AUX
ejpam-65	562	50	u	u	NOUN
ejpam-65	562	51	=	=	PUNCT
ejpam-65	562	52	{	{	PUNCT
ejpam-65	563	1	n∑	n∑	INTJ
ejpam-65	563	2	i=1	i=1	PROPN
ejpam-65	563	3	ai	ai	VERB
ejpam-65	563	4	◦	◦	NOUN
ejpam-65	563	5	xi	xi	X
ejpam-65	563	6	:	:	PUNCT
ejpam-65	563	7	ai	ai	VERB
ejpam-65	563	8	∈	∈	PROPN
ejpam-65	563	9	k	k	PROPN
ejpam-65	563	10	and	and	CCONJ
ejpam-65	563	11	xi	xi	ADP
ejpam-65	563	12	∈	∈	PROPN
ejpam-65	563	13	v	v	NOUN
ejpam-65	563	14	,	,	PUNCT
ejpam-65	563	15	n	n	NOUN
ejpam-65	563	16	∈	∈	PROPN
ejpam-65	563	17	n	n	CCONJ
ejpam-65	563	18	}	}	PUNCT
ejpam-65	563	19	.	.	PUNCT
ejpam-65	564	1	define	define	VERB
ejpam-65	564	2	the	the	DET
ejpam-65	564	3	relation	relation	NOUN
ejpam-65	564	4	ε	ε	PROPN
ejpam-65	564	5	over	over	ADP
ejpam-65	564	6	v	v	NOUN
ejpam-65	564	7	by	by	ADP
ejpam-65	564	8	xεy	xεy	PROPN
ejpam-65	564	9	⇐	⇐	PROPN
ejpam-65	564	10	⇒	⇒	PROPN
ejpam-65	564	11	∃u	∃u	PROPN
ejpam-65	564	12	∈	∈	PROPN
ejpam-65	564	13	u	u	NOUN
ejpam-65	564	14	:	:	PUNCT
ejpam-65	564	15	{	{	PUNCT
ejpam-65	564	16	x	x	NOUN
ejpam-65	564	17	,	,	PUNCT
ejpam-65	564	18	y	y	PROPN
ejpam-65	564	19	}	}	PUNCT
ejpam-65	564	20	⊆	⊆	NUM
ejpam-65	564	21	u	u	NOUN
ejpam-65	564	22	,	,	PUNCT
ejpam-65	564	23	∀x	∀x	NUM
ejpam-65	564	24	,	,	PUNCT
ejpam-65	564	25	y	y	PROPN
ejpam-65	564	26	∈	∈	PROPN
ejpam-65	564	27	v	v	NOUN
ejpam-65	564	28	.	.	PUNCT
ejpam-65	565	1	then	then	ADV
ejpam-65	565	2	ε∗	ε∗	PROPN
ejpam-65	565	3	is	be	AUX
ejpam-65	565	4	the	the	DET
ejpam-65	565	5	transitive	transitive	ADJ
ejpam-65	565	6	closure	closure	NOUN
ejpam-65	565	7	of	of	ADP
ejpam-65	565	8	ε	ε	PROPN
ejpam-65	565	9	.	.	PUNCT
ejpam-65	566	1	define	define	VERB
ejpam-65	566	2	addition	addition	NOUN
ejpam-65	566	3	operation	operation	NOUN
ejpam-65	566	4	and	and	CCONJ
ejpam-65	566	5	scalar	scalar	ADJ
ejpam-65	566	6	multiplication	multiplication	NOUN
ejpam-65	566	7	on	on	ADP
ejpam-65	566	8	v	v	NOUN
ejpam-65	566	9	/	/	SYM
ejpam-65	566	10	ε∗	ε∗	VERB
ejpam-65	566	11	by	by	ADP
ejpam-65	566	12	{	{	PUNCT
ejpam-65	566	13	⊕	⊕	NOUN
ejpam-65	566	14	:	:	PUNCT
ejpam-65	567	1	v	v	X
ejpam-65	567	2	/	/	SYM
ejpam-65	567	3	ε∗	ε∗	PROPN
ejpam-65	567	4	×	×	PROPN
ejpam-65	567	5	v	v	NOUN
ejpam-65	567	6	/	/	SYM
ejpam-65	567	7	ε∗	ε∗	VERB
ejpam-65	567	8	−→	−→	NOUN
ejpam-65	567	9	v	v	NOUN
ejpam-65	567	10	/	/	SYM
ejpam-65	567	11	ε∗	ε∗	PROPN
ejpam-65	567	12	ε∗(x)⊕	ε∗(x)⊕	PUNCT
ejpam-65	567	13	ε∗(y	ε∗(y	PROPN
ejpam-65	567	14	)	)	PUNCT
ejpam-65	567	15	=	=	PRON
ejpam-65	567	16	{	{	PUNCT
ejpam-65	567	17	ε∗(t	ε∗(t	NOUN
ejpam-65	567	18	)	)	PUNCT
ejpam-65	567	19	:	:	PUNCT
ejpam-65	568	1	t	t	PROPN
ejpam-65	568	2	∈	∈	PROPN
ejpam-65	568	3	ε∗(x	ε∗(x	PROPN
ejpam-65	568	4	)	)	PUNCT
ejpam-65	568	5	+	+	CCONJ
ejpam-65	568	6	ε∗(y	ε∗(y	PROPN
ejpam-65	568	7	)	)	PUNCT
ejpam-65	568	8	}	}	PUNCT
ejpam-65	568	9	,	,	PUNCT
ejpam-65	568	10	and	and	CCONJ
ejpam-65	568	11	{	{	PUNCT
ejpam-65	568	12	�	�	PROPN
ejpam-65	568	13	:	:	PUNCT
ejpam-65	568	14	k	k	PROPN
ejpam-65	568	15	×	×	PROPN
ejpam-65	568	16	v	v	NOUN
ejpam-65	568	17	/	/	SYM
ejpam-65	568	18	ε∗	ε∗	VERB
ejpam-65	568	19	−→	−→	NOUN
ejpam-65	568	20	v	v	NOUN
ejpam-65	568	21	/	/	SYM
ejpam-65	568	22	ε∗	ε∗	PROPN
ejpam-65	568	23	a	a	DET
ejpam-65	568	24	�	�	PROPN
ejpam-65	568	25	ε∗(x	ε∗(x	NOUN
ejpam-65	568	26	)	)	PUNCT
ejpam-65	568	27	=	=	SYM
ejpam-65	568	28	{	{	PUNCT
ejpam-65	568	29	ε∗(z	ε∗(z	PROPN
ejpam-65	568	30	)	)	PUNCT
ejpam-65	568	31	:	:	PUNCT
ejpam-65	569	1	z	z	X
ejpam-65	569	2	∈	∈	PROPN
ejpam-65	569	3	a	a	DET
ejpam-65	569	4	◦	◦	NOUN
ejpam-65	569	5	ε∗(x	ε∗(x	NOUN
ejpam-65	569	6	)	)	PUNCT
ejpam-65	569	7	}	}	PUNCT
ejpam-65	569	8	,	,	PUNCT
ejpam-65	569	9	lemma	lemma	PROPN
ejpam-65	569	10	4.1	4.1	NUM
ejpam-65	569	11	.	.	PUNCT
ejpam-65	570	1	the	the	DET
ejpam-65	570	2	following	follow	VERB
ejpam-65	570	3	statement	statement	NOUN
ejpam-65	570	4	are	be	AUX
ejpam-65	570	5	satisfied	satisfied	ADJ
ejpam-65	570	6	:	:	PUNCT
ejpam-65	570	7	(	(	PUNCT
ejpam-65	570	8	i	i	NOUN
ejpam-65	570	9	)	)	PUNCT
ejpam-65	570	10	ε∗(a	ε∗(a	PROPN
ejpam-65	570	11	◦	◦	NOUN
ejpam-65	570	12	x	x	X
ejpam-65	570	13	)	)	PUNCT
ejpam-65	570	14	=	=	SYM
ejpam-65	570	15	ε∗(y	ε∗(y	PROPN
ejpam-65	570	16	)	)	PUNCT
ejpam-65	570	17	for	for	ADP
ejpam-65	570	18	all	all	DET
ejpam-65	570	19	y	y	PROPN
ejpam-65	570	20	∈	∈	PROPN
ejpam-65	570	21	a	a	DET
ejpam-65	570	22	◦	◦	NOUN
ejpam-65	570	23	x	x	SYM
ejpam-65	570	24	,	,	PUNCT
ejpam-65	570	25	∀a	∀a	NOUN
ejpam-65	570	26	∈	∈	PROPN
ejpam-65	570	27	k	k	NOUN
ejpam-65	570	28	,	,	PUNCT
ejpam-65	570	29	∀x	∀x	X
ejpam-65	570	30	∈	∈	PROPN
ejpam-65	570	31	v	v	NOUN
ejpam-65	570	32	,	,	PUNCT
ejpam-65	570	33	where	where	SCONJ
ejpam-65	570	34	ε∗(a	ε∗(a	PROPN
ejpam-65	570	35	◦	◦	NOUN
ejpam-65	570	36	x	x	X
ejpam-65	570	37	)	)	PUNCT
ejpam-65	570	38	=	=	PUNCT
ejpam-65	570	39	⋃	⋃	NOUN
ejpam-65	570	40	b∈a	b∈a	NOUN
ejpam-65	570	41	◦	◦	NOUN
ejpam-65	570	42	x	x	SYM
ejpam-65	570	43	ε∗(b	ε∗(b	PROPN
ejpam-65	570	44	)	)	PUNCT
ejpam-65	570	45	.	.	PUNCT
ejpam-65	571	1	(	(	PUNCT
ejpam-65	571	2	ii	ii	NOUN
ejpam-65	571	3	)	)	PUNCT
ejpam-65	571	4	ε∗(x)⊕	ε∗(x)⊕	PUNCT
ejpam-65	571	5	ε∗(y	ε∗(y	PROPN
ejpam-65	571	6	)	)	PUNCT
ejpam-65	571	7	=	=	PUNCT
ejpam-65	571	8	ε∗(x+	ε∗(x+	PROPN
ejpam-65	571	9	y	y	PROPN
ejpam-65	571	10	)	)	PUNCT
ejpam-65	571	11	.	.	PUNCT
ejpam-65	572	1	(	(	PUNCT
ejpam-65	572	2	iii	iii	X
ejpam-65	572	3	)	)	PUNCT
ejpam-65	572	4	ε∗(0	ε∗(0	NOUN
ejpam-65	572	5	)	)	PUNCT
ejpam-65	572	6	is	be	AUX
ejpam-65	572	7	the	the	DET
ejpam-65	572	8	identity	identity	NOUN
ejpam-65	572	9	element	element	NOUN
ejpam-65	572	10	of	of	ADP
ejpam-65	572	11	(	(	PUNCT
ejpam-65	572	12	v	v	NOUN
ejpam-65	572	13	/	/	SYM
ejpam-65	572	14	ε∗,⊕	ε∗,⊕	NOUN
ejpam-65	572	15	)	)	PUNCT
ejpam-65	572	16	.	.	PUNCT
ejpam-65	573	1	(	(	PUNCT
ejpam-65	573	2	iv	iv	X
ejpam-65	573	3	)	)	PUNCT
ejpam-65	573	4	(	(	PUNCT
ejpam-65	573	5	v	v	NOUN
ejpam-65	573	6	/	/	SYM
ejpam-65	573	7	ε∗,⊕,	ε∗,⊕,	PROPN
ejpam-65	573	8	�	�	PROPN
ejpam-65	573	9	,k	,k	PUNCT
ejpam-65	573	10	)	)	PUNCT
ejpam-65	573	11	is	be	AUX
ejpam-65	573	12	a	a	DET
ejpam-65	573	13	vector	vector	NOUN
ejpam-65	573	14	space	space	NOUN
ejpam-65	573	15	over	over	ADP
ejpam-65	573	16	k.	k.	PROPN
ejpam-65	573	17	the	the	DET
ejpam-65	573	18	vector	vector	NOUN
ejpam-65	573	19	space	space	NOUN
ejpam-65	573	20	(	(	PUNCT
ejpam-65	573	21	v	v	NOUN
ejpam-65	573	22	/	/	SYM
ejpam-65	573	23	ε∗,⊕,	ε∗,⊕,	PROPN
ejpam-65	573	24	�	�	PROPN
ejpam-65	573	25	,k	,k	PUNCT
ejpam-65	573	26	)	)	PUNCT
ejpam-65	573	27	is	be	AUX
ejpam-65	573	28	called	call	VERB
ejpam-65	573	29	the	the	DET
ejpam-65	573	30	fundamental	fundamental	ADJ
ejpam-65	573	31	vector	vector	NOUN
ejpam-65	573	32	space	space	NOUN
ejpam-65	573	33	of	of	ADP
ejpam-65	573	34	v	v	NOUN
ejpam-65	573	35	.	.	PUNCT
ejpam-65	574	1	proof	proof	NOUN
ejpam-65	574	2	.	.	PUNCT
ejpam-65	575	1	the	the	DET
ejpam-65	575	2	proof	proof	NOUN
ejpam-65	575	3	is	be	AUX
ejpam-65	575	4	similar	similar	ADJ
ejpam-65	575	5	to	to	ADP
ejpam-65	575	6	the	the	DET
ejpam-65	575	7	proof	proof	NOUN
ejpam-65	575	8	of	of	ADP
ejpam-65	575	9	[	[	X
ejpam-65	575	10	11	11	NUM
ejpam-65	575	11	,	,	PUNCT
ejpam-65	575	12	thm	thm	PROPN
ejpam-65	575	13	.	.	PUNCT
ejpam-65	576	1	2.4	2.4	NUM
ejpam-65	576	2	]	]	PUNCT
ejpam-65	576	3	.	.	PUNCT
ejpam-65	577	1	theorem	theorem	VERB
ejpam-65	577	2	4.1	4.1	NUM
ejpam-65	577	3	.	.	PUNCT
ejpam-65	578	1	let	let	AUX
ejpam-65	578	2	(	(	PUNCT
ejpam-65	578	3	v,+	v,+	NUM
ejpam-65	578	4	,	,	PUNCT
ejpam-65	578	5	◦	◦	NOUN
ejpam-65	578	6	,	,	PUNCT
ejpam-65	578	7	k	k	NOUN
ejpam-65	578	8	)	)	PUNCT
ejpam-65	578	9	be	be	VERB
ejpam-65	578	10	a	a	DET
ejpam-65	578	11	hypervector	hypervector	NOUN
ejpam-65	578	12	space	space	NOUN
ejpam-65	578	13	and	and	CCONJ
ejpam-65	578	14	(	(	PUNCT
ejpam-65	578	15	v	v	NOUN
ejpam-65	578	16	/	/	SYM
ejpam-65	578	17	ε∗,⊕,	ε∗,⊕,	PROPN
ejpam-65	578	18	�	�	PROPN
ejpam-65	578	19	,k	,k	PUNCT
ejpam-65	578	20	)	)	PUNCT
ejpam-65	578	21	be	be	VERB
ejpam-65	578	22	the	the	DET
ejpam-65	578	23	fundamental	fundamental	ADJ
ejpam-65	578	24	vector	vector	NOUN
ejpam-65	578	25	space	space	NOUN
ejpam-65	578	26	of	of	ADP
ejpam-65	578	27	v	v	NOUN
ejpam-65	578	28	.	.	PUNCT
ejpam-65	579	1	then	then	ADV
ejpam-65	579	2	dimv	dimv	NOUN
ejpam-65	579	3	=	=	SYM
ejpam-65	579	4	dimv	dimv	NOUN
ejpam-65	579	5	/	/	SYM
ejpam-65	579	6	ε∗.	ε∗.	NOUN
ejpam-65	579	7	proof	proof	NOUN
ejpam-65	579	8	.	.	PUNCT
ejpam-65	580	1	let	let	VERB
ejpam-65	580	2	s	s	PRON
ejpam-65	580	3	=	=	PUNCT
ejpam-65	580	4	{	{	PUNCT
ejpam-65	580	5	x1	x1	PROPN
ejpam-65	580	6	,	,	PUNCT
ejpam-65	580	7	x2	x2	PROPN
ejpam-65	580	8	,	,	PUNCT
ejpam-65	580	9	.	.	PUNCT
ejpam-65	580	10	.	.	PUNCT
ejpam-65	581	1	.	.	PUNCT
ejpam-65	582	1	,	,	PUNCT
ejpam-65	582	2	xn	xn	X
ejpam-65	582	3	}	}	PUNCT
ejpam-65	582	4	be	be	VERB
ejpam-65	582	5	a	a	DET
ejpam-65	582	6	basis	basis	NOUN
ejpam-65	582	7	for	for	ADP
ejpam-65	582	8	v	v	NOUN
ejpam-65	582	9	.	.	PUNCT
ejpam-65	583	1	then	then	ADV
ejpam-65	583	2	we	we	PRON
ejpam-65	583	3	show	show	VERB
ejpam-65	583	4	that	that	SCONJ
ejpam-65	583	5	s∗	s∗	PROPN
ejpam-65	583	6	=	=	SYM
ejpam-65	583	7	{	{	PUNCT
ejpam-65	583	8	ε∗(x1	ε∗(x1	PROPN
ejpam-65	583	9	)	)	PUNCT
ejpam-65	583	10	,	,	PUNCT
ejpam-65	583	11	ε∗(x2	ε∗(x2	PROPN
ejpam-65	583	12	)	)	PUNCT
ejpam-65	583	13	,	,	PUNCT
ejpam-65	583	14	.	.	PUNCT
ejpam-65	583	15	.	.	PUNCT
ejpam-65	584	1	.	.	PUNCT
ejpam-65	585	1	,	,	PUNCT
ejpam-65	585	2	ε∗(xn	ε∗(xn	NOUN
ejpam-65	585	3	)	)	PUNCT
ejpam-65	585	4	}	}	PUNCT
ejpam-65	585	5	is	be	AUX
ejpam-65	585	6	a	a	DET
ejpam-65	585	7	basis	basis	NOUN
ejpam-65	585	8	for	for	ADP
ejpam-65	585	9	v	v	NOUN
ejpam-65	585	10	/	/	SYM
ejpam-65	585	11	ε∗.	ε∗.	NOUN
ejpam-65	585	12	for	for	ADP
ejpam-65	585	13	this	this	DET
ejpam-65	585	14	let	let	VERB
ejpam-65	585	15	ε∗(x	ε∗(x	NOUN
ejpam-65	585	16	)	)	PUNCT
ejpam-65	585	17	∈	∈	PROPN
ejpam-65	585	18	v	v	NOUN
ejpam-65	585	19	/	/	SYM
ejpam-65	585	20	ε∗	ε∗	PROPN
ejpam-65	585	21	,	,	PUNCT
ejpam-65	585	22	then	then	ADV
ejpam-65	585	23	:	:	PUNCT
ejpam-65	585	24	x	x	X
ejpam-65	585	25	∈	∈	NOUN
ejpam-65	585	26	v	v	NOUN
ejpam-65	585	27	=	=	NOUN
ejpam-65	585	28	⇒	⇒	PROPN
ejpam-65	585	29	∃	∃	PROPN
ejpam-65	585	30	a1	a1	PROPN
ejpam-65	585	31	,	,	PUNCT
ejpam-65	585	32	.	.	PUNCT
ejpam-65	585	33	.	.	PUNCT
ejpam-65	586	1	.	.	PUNCT
ejpam-65	587	1	,	,	PUNCT
ejpam-65	587	2	an	an	DET
ejpam-65	587	3	∈	∈	PROPN
ejpam-65	587	4	k	k	NOUN
ejpam-65	587	5	;	;	PUNCT
ejpam-65	587	6	x	x	SYM
ejpam-65	587	7	∈	∈	PROPN
ejpam-65	588	1	n∑	n∑	NOUN
ejpam-65	588	2	i=1	i=1	PROPN
ejpam-65	588	3	ai	ai	VERB
ejpam-65	588	4	◦	◦	NOUN
ejpam-65	588	5	xi	xi	NOUN
ejpam-65	588	6	,	,	PUNCT
ejpam-65	588	7	=	=	PRON
ejpam-65	588	8	⇒	⇒	NOUN
ejpam-65	588	9	x	x	PUNCT
ejpam-65	589	1	=	=	SYM
ejpam-65	589	2	t1	t1	NOUN
ejpam-65	589	3	+	+	X
ejpam-65	589	4	·	·	PUNCT
ejpam-65	589	5	·	·	PUNCT
ejpam-65	589	6	·	·	PUNCT
ejpam-65	590	1	+	+	NUM
ejpam-65	590	2	tn	tn	NOUN
ejpam-65	590	3	;	;	PUNCT
ejpam-65	590	4	for	for	ADP
ejpam-65	590	5	some	some	DET
ejpam-65	590	6	ti	ti	NOUN
ejpam-65	590	7	∈	∈	NOUN
ejpam-65	590	8	ai	ai	VERB
ejpam-65	590	9	◦	◦	NOUN
ejpam-65	590	10	xi	xi	X
ejpam-65	590	11	,	,	PUNCT
ejpam-65	590	12	1	1	NUM
ejpam-65	590	13	6	6	NUM
ejpam-65	590	14	i	i	PRON
ejpam-65	590	15	6	6	NUM
ejpam-65	590	16	n.	n.	PROPN
ejpam-65	590	17	r.	r.	PROPN
ejpam-65	590	18	ameri	ameri	PROPN
ejpam-65	590	19	,	,	PUNCT
ejpam-65	590	20	o.r	o.r	PROPN
ejpam-65	590	21	.	.	PROPN
ejpam-65	590	22	dehghan	dehghan	PROPN
ejpam-65	590	23	/	/	SYM
ejpam-65	590	24	eur	eur	PROPN
ejpam-65	590	25	.	.	PUNCT
ejpam-65	591	1	j.	j.	PROPN
ejpam-65	591	2	pure	pure	PROPN
ejpam-65	591	3	appl	appl	PROPN
ejpam-65	591	4	.	.	PROPN
ejpam-65	591	5	math	math	PROPN
ejpam-65	591	6	,	,	PUNCT
ejpam-65	591	7	1	1	NUM
ejpam-65	591	8	(	(	PUNCT
ejpam-65	591	9	2008	2008	NUM
ejpam-65	591	10	)	)	PUNCT
ejpam-65	591	11	,	,	PUNCT
ejpam-65	591	12	(	(	PUNCT
ejpam-65	591	13	32	32	NUM
ejpam-65	591	14	-	-	SYM
ejpam-65	591	15	50	50	NUM
ejpam-65	591	16	)	)	PUNCT
ejpam-65	591	17	47	47	NUM
ejpam-65	591	18	then	then	ADV
ejpam-65	591	19	by	by	ADP
ejpam-65	591	20	lemma	lemma	PROPN
ejpam-65	591	21	4.1	4.1	NUM
ejpam-65	591	22	,	,	PUNCT
ejpam-65	591	23	we	we	PRON
ejpam-65	591	24	obtain	obtain	VERB
ejpam-65	591	25	ε∗(ai	ε∗(ai	PROPN
ejpam-65	591	26	◦	◦	NOUN
ejpam-65	591	27	xi	xi	NUM
ejpam-65	591	28	)	)	PUNCT
ejpam-65	591	29	=	=	SYM
ejpam-65	591	30	ε∗(ti	ε∗(ti	NOUN
ejpam-65	591	31	)	)	PUNCT
ejpam-65	591	32	.	.	PUNCT
ejpam-65	592	1	and	and	CCONJ
ejpam-65	592	2	ε∗(x	ε∗(x	NOUN
ejpam-65	592	3	)	)	PUNCT
ejpam-65	592	4	=	=	PUNCT
ejpam-65	593	1	ε∗(t1	ε∗(t1	NOUN
ejpam-65	593	2	+	+	CCONJ
ejpam-65	593	3	·	·	PUNCT
ejpam-65	593	4	·	·	PUNCT
ejpam-65	593	5	·	·	PUNCT
ejpam-65	593	6	+	+	NUM
ejpam-65	593	7	tn	tn	NOUN
ejpam-65	593	8	)	)	PUNCT
ejpam-65	593	9	=	=	SYM
ejpam-65	593	10	ε∗(t1)⊕	ε∗(t1)⊕	X
ejpam-65	593	11	·	·	PUNCT
ejpam-65	593	12	·	·	PUNCT
ejpam-65	593	13	·	·	PUNCT
ejpam-65	593	14	⊕	⊕	PROPN
ejpam-65	593	15	ε∗(tn	ε∗(tn	NOUN
ejpam-65	593	16	)	)	PUNCT
ejpam-65	593	17	=	=	SYM
ejpam-65	594	1	ε∗(a1	ε∗(a1	PROPN
ejpam-65	594	2	◦	◦	NOUN
ejpam-65	594	3	x1)⊕	x1)⊕	X
ejpam-65	594	4	·	·	PUNCT
ejpam-65	594	5	·	·	PUNCT
ejpam-65	594	6	·	·	PUNCT
ejpam-65	595	1	⊕	⊕	NOUN
ejpam-65	595	2	ε∗(an	ε∗(an	VERB
ejpam-65	595	3	◦	◦	NOUN
ejpam-65	595	4	xn	xn	PUNCT
ejpam-65	595	5	)	)	PUNCT
ejpam-65	596	1	=	=	NOUN
ejpam-65	596	2	a1	a1	NOUN
ejpam-65	596	3	�	�	PROPN
ejpam-65	596	4	ε∗(x1)⊕	ε∗(x1)⊕	X
ejpam-65	596	5	·	·	PUNCT
ejpam-65	596	6	·	·	PUNCT
ejpam-65	596	7	·	·	PUNCT
ejpam-65	596	8	⊕	⊕	PROPN
ejpam-65	596	9	an	an	DET
ejpam-65	596	10	�	�	PROPN
ejpam-65	596	11	ε∗(xn	ε∗(xn	NOUN
ejpam-65	596	12	)	)	PUNCT
ejpam-65	596	13	,	,	PUNCT
ejpam-65	596	14	therefore	therefore	ADV
ejpam-65	596	15	v	v	ADJ
ejpam-65	596	16	/	/	SYM
ejpam-65	596	17	ε∗	ε∗	PROPN
ejpam-65	596	18	is	be	AUX
ejpam-65	596	19	generated	generate	VERB
ejpam-65	596	20	by	by	ADP
ejpam-65	596	21	s∗.	s∗.	ADJ
ejpam-65	596	22	now	now	ADV
ejpam-65	596	23	we	we	PRON
ejpam-65	596	24	show	show	VERB
ejpam-65	596	25	that	that	SCONJ
ejpam-65	596	26	s∗	s∗	PROPN
ejpam-65	596	27	is	be	AUX
ejpam-65	596	28	linearly	linearly	ADV
ejpam-65	596	29	independent	independent	ADJ
ejpam-65	596	30	.	.	PUNCT
ejpam-65	597	1	suppose	suppose	VERB
ejpam-65	597	2	that	that	SCONJ
ejpam-65	597	3	a1	a1	PROPN
ejpam-65	597	4	�	�	PROPN
ejpam-65	597	5	ε∗(x1)⊕	ε∗(x1)⊕	X
ejpam-65	597	6	·	·	PUNCT
ejpam-65	597	7	·	·	PUNCT
ejpam-65	597	8	·	·	PUNCT
ejpam-65	597	9	⊕	⊕	PROPN
ejpam-65	597	10	an	an	DET
ejpam-65	597	11	�	�	PROPN
ejpam-65	597	12	ε∗(xn	ε∗(xn	NOUN
ejpam-65	597	13	)	)	PUNCT
ejpam-65	597	14	=	=	SYM
ejpam-65	597	15	ε∗(0	ε∗(0	NOUN
ejpam-65	597	16	)	)	PUNCT
ejpam-65	597	17	,	,	PUNCT
ejpam-65	597	18	then	then	ADV
ejpam-65	597	19	:	:	PUNCT
ejpam-65	597	20	ε∗(a1	ε∗(a1	PROPN
ejpam-65	597	21	◦	◦	NOUN
ejpam-65	597	22	x1)⊕	x1)⊕	X
ejpam-65	597	23	·	·	PUNCT
ejpam-65	597	24	·	·	PUNCT
ejpam-65	597	25	·	·	PUNCT
ejpam-65	598	1	⊕	⊕	NOUN
ejpam-65	598	2	ε∗(an	ε∗(an	VERB
ejpam-65	598	3	◦	◦	NOUN
ejpam-65	598	4	xn	xn	PUNCT
ejpam-65	598	5	)	)	PUNCT
ejpam-65	599	1	=	=	SYM
ejpam-65	599	2	ε∗(0	ε∗(0	NOUN
ejpam-65	599	3	)	)	PUNCT
ejpam-65	599	4	=	=	SYM
ejpam-65	599	5	⇒	⇒	X
ejpam-65	599	6	ε∗(a1	ε∗(a1	PROPN
ejpam-65	599	7	◦	◦	NOUN
ejpam-65	599	8	x1	x1	PROPN
ejpam-65	599	9	+	+	X
ejpam-65	599	10	·	·	PUNCT
ejpam-65	599	11	·	·	PUNCT
ejpam-65	599	12	·	·	PUNCT
ejpam-65	599	13	+	+	CCONJ
ejpam-65	599	14	an	an	DET
ejpam-65	599	15	◦	◦	NOUN
ejpam-65	599	16	xn	xn	PUNCT
ejpam-65	599	17	)	)	PUNCT
ejpam-65	599	18	=	=	SYM
ejpam-65	599	19	ε∗(0	ε∗(0	NOUN
ejpam-65	599	20	)	)	PUNCT
ejpam-65	599	21	=	=	NOUN
ejpam-65	599	22	⇒	⇒	NOUN
ejpam-65	599	23	0	0	NUM
ejpam-65	599	24	∈	∈	NOUN
ejpam-65	599	25	a1	a1	NOUN
ejpam-65	599	26	◦	◦	NOUN
ejpam-65	599	27	x1	x1	PROPN
ejpam-65	600	1	+	+	X
ejpam-65	600	2	·	·	PUNCT
ejpam-65	600	3	·	·	PUNCT
ejpam-65	600	4	·	·	PUNCT
ejpam-65	600	5	+	+	CCONJ
ejpam-65	600	6	an	an	DET
ejpam-65	600	7	◦	◦	NOUN
ejpam-65	600	8	xn	xn	PROPN
ejpam-65	600	9	.	.	PUNCT
ejpam-65	601	1	since	since	SCONJ
ejpam-65	601	2	s	s	NOUN
ejpam-65	601	3	is	be	AUX
ejpam-65	601	4	linearly	linearly	ADV
ejpam-65	601	5	independent	independent	ADJ
ejpam-65	601	6	,	,	PUNCT
ejpam-65	601	7	so	so	ADV
ejpam-65	601	8	a1	a1	NOUN
ejpam-65	601	9	=	=	SYM
ejpam-65	601	10	·	·	PUNCT
ejpam-65	601	11	·	·	PUNCT
ejpam-65	601	12	·	·	PUNCT
ejpam-65	602	1	=	=	PUNCT
ejpam-65	602	2	an	an	DET
ejpam-65	602	3	=	=	NOUN
ejpam-65	602	4	0	0	NUM
ejpam-65	602	5	.	.	PUNCT
ejpam-65	603	1	consequently	consequently	ADV
ejpam-65	603	2	,	,	PUNCT
ejpam-65	603	3	s∗	s∗	PROPN
ejpam-65	603	4	is	be	AUX
ejpam-65	603	5	linearly	linearly	ADV
ejpam-65	603	6	independent	independent	ADJ
ejpam-65	603	7	.	.	PUNCT
ejpam-65	604	1	5	5	X
ejpam-65	604	2	.	.	X
ejpam-65	604	3	category	category	NOUN
ejpam-65	604	4	of	of	ADP
ejpam-65	604	5	hypervector	hypervector	NOUN
ejpam-65	604	6	spaces	space	NOUN
ejpam-65	604	7	for	for	ADP
ejpam-65	604	8	hypervector	hypervector	NOUN
ejpam-65	604	9	spaces	space	NOUN
ejpam-65	604	10	v	v	ADP
ejpam-65	604	11	and	and	CCONJ
ejpam-65	604	12	w	w	NOUN
ejpam-65	604	13	on	on	ADP
ejpam-65	604	14	the	the	DET
ejpam-65	604	15	field	field	NOUN
ejpam-65	604	16	k	k	NOUN
ejpam-65	604	17	,	,	PUNCT
ejpam-65	604	18	by	by	ADP
ejpam-65	604	19	homw	homw	PROPN
ejpam-65	604	20	k(v	k(v	PROPN
ejpam-65	604	21	,	,	PUNCT
ejpam-65	604	22	w	w	PROPN
ejpam-65	604	23	)	)	PUNCT
ejpam-65	604	24	,	,	PUNCT
ejpam-65	604	25	homi	homi	PROPN
ejpam-65	604	26	k(v	k(v	PROPN
ejpam-65	604	27	,	,	PUNCT
ejpam-65	604	28	w	w	PROPN
ejpam-65	604	29	)	)	PUNCT
ejpam-65	604	30	and	and	CCONJ
ejpam-65	604	31	homg	homg	PROPN
ejpam-65	604	32	k(v	k(v	PROPN
ejpam-65	604	33	,	,	PUNCT
ejpam-65	604	34	w	w	PROPN
ejpam-65	604	35	)	)	PUNCT
ejpam-65	604	36	we	we	PRON
ejpam-65	604	37	mean	mean	VERB
ejpam-65	604	38	the	the	DET
ejpam-65	604	39	set	set	NOUN
ejpam-65	604	40	of	of	ADP
ejpam-65	604	41	all	all	DET
ejpam-65	604	42	week	week	NOUN
ejpam-65	604	43	linear	linear	PROPN
ejpam-65	604	44	transformation	transformation	NOUN
ejpam-65	604	45	,	,	PUNCT
ejpam-65	604	46	linear	linear	ADJ
ejpam-65	604	47	transformation	transformation	NOUN
ejpam-65	604	48	and	and	CCONJ
ejpam-65	604	49	good	good	ADJ
ejpam-65	604	50	linear	linear	ADJ
ejpam-65	604	51	transformation	transformation	NOUN
ejpam-65	604	52	,	,	PUNCT
ejpam-65	604	53	respectively	respectively	ADV
ejpam-65	604	54	.	.	PUNCT
ejpam-65	605	1	definition	definition	NOUN
ejpam-65	605	2	5.1	5.1	NUM
ejpam-65	605	3	.	.	PUNCT
ejpam-65	606	1	the	the	DET
ejpam-65	606	2	category	category	NOUN
ejpam-65	606	3	of	of	ADP
ejpam-65	606	4	hypervector	hypervector	NOUN
ejpam-65	606	5	spaces	space	NOUN
ejpam-65	606	6	overk	overk	PROPN
ejpam-65	606	7	denoted	denote	VERB
ejpam-65	606	8	byhvwk	byhvwk	NOUN
ejpam-65	606	9	is	be	AUX
ejpam-65	606	10	defined	define	VERB
ejpam-65	606	11	as	as	SCONJ
ejpam-65	606	12	follows	follow	VERB
ejpam-65	606	13	:	:	PUNCT
ejpam-65	606	14	(	(	PUNCT
ejpam-65	606	15	i	i	NOUN
ejpam-65	606	16	)	)	PUNCT
ejpam-65	606	17	the	the	DET
ejpam-65	606	18	objects	object	NOUN
ejpam-65	606	19	ofhvwk	ofhvwk	VERB
ejpam-65	606	20	are	be	AUX
ejpam-65	606	21	all	all	PRON
ejpam-65	606	22	hypervector	hypervector	NOUN
ejpam-65	606	23	spaces	space	NOUN
ejpam-65	606	24	over	over	ADP
ejpam-65	606	25	k	k	PROPN
ejpam-65	606	26	;	;	PUNCT
ejpam-65	606	27	(	(	PUNCT
ejpam-65	606	28	ii	ii	NOUN
ejpam-65	606	29	)	)	PUNCT
ejpam-65	606	30	for	for	ADP
ejpam-65	606	31	the	the	DET
ejpam-65	606	32	objects	object	NOUN
ejpam-65	606	33	v	v	ADP
ejpam-65	606	34	and	and	CCONJ
ejpam-65	606	35	w	w	NOUN
ejpam-65	606	36	of	of	ADP
ejpam-65	606	37	hvwk	hvwk	NOUN
ejpam-65	606	38	,	,	PUNCT
ejpam-65	606	39	the	the	DET
ejpam-65	606	40	set	set	NOUN
ejpam-65	606	41	of	of	ADP
ejpam-65	606	42	all	all	DET
ejpam-65	606	43	morphisms	morphism	NOUN
ejpam-65	606	44	from	from	ADP
ejpam-65	606	45	v	v	NUM
ejpam-65	606	46	to	to	ADP
ejpam-65	606	47	w	w	PROPN
ejpam-65	606	48	is	be	AUX
ejpam-65	606	49	the	the	DET
ejpam-65	606	50	set	set	ADJ
ejpam-65	606	51	homw	homw	PROPN
ejpam-65	606	52	k(v	k(v	PROPN
ejpam-65	606	53	,	,	PUNCT
ejpam-65	606	54	w	w	PROPN
ejpam-65	606	55	)	)	PUNCT
ejpam-65	606	56	;	;	PUNCT
ejpam-65	606	57	(	(	PUNCT
ejpam-65	606	58	iii	iii	X
ejpam-65	606	59	)	)	PUNCT
ejpam-65	606	60	the	the	DET
ejpam-65	606	61	composition	composition	NOUN
ejpam-65	606	62	st	st	PROPN
ejpam-65	606	63	of	of	ADP
ejpam-65	606	64	morphisms	morphisms	PROPN
ejpam-65	606	65	t	t	PROPN
ejpam-65	606	66	:	:	PUNCT
ejpam-65	606	67	v	v	ADP
ejpam-65	606	68	−→	−→	NOUN
ejpam-65	606	69	l	l	NOUN
ejpam-65	606	70	and	and	CCONJ
ejpam-65	606	71	s	s	VERB
ejpam-65	606	72	:	:	PUNCT
ejpam-65	607	1	l	l	NOUN
ejpam-65	607	2	−→w	−→w	PUNCT
ejpam-65	607	3	is	be	AUX
ejpam-65	607	4	defined	define	VERB
ejpam-65	607	5	as	as	ADP
ejpam-65	607	6	usual	usual	ADJ
ejpam-65	607	7	;	;	PUNCT
ejpam-65	607	8	(	(	PUNCT
ejpam-65	607	9	iv	iv	X
ejpam-65	607	10	)	)	PUNCT
ejpam-65	607	11	for	for	ADP
ejpam-65	607	12	any	any	DET
ejpam-65	607	13	object	object	NOUN
ejpam-65	607	14	v	v	NOUN
ejpam-65	607	15	,	,	PUNCT
ejpam-65	607	16	the	the	DET
ejpam-65	607	17	morphism	morphism	NOUN
ejpam-65	607	18	1v	1v	NUM
ejpam-65	607	19	:	:	PUNCT
ejpam-65	607	20	v	v	ADP
ejpam-65	607	21	−→	−→	NOUN
ejpam-65	607	22	v	v	NOUN
ejpam-65	607	23	,	,	PUNCT
ejpam-65	607	24	is	be	AUX
ejpam-65	607	25	the	the	DET
ejpam-65	607	26	identity	identity	NOUN
ejpam-65	607	27	map	map	NOUN
ejpam-65	607	28	from	from	ADP
ejpam-65	607	29	v	v	NUM
ejpam-65	607	30	to	to	ADP
ejpam-65	607	31	v	v	NOUN
ejpam-65	607	32	.	.	PUNCT
ejpam-65	608	1	note	note	VERB
ejpam-65	608	2	that	that	SCONJ
ejpam-65	608	3	in	in	ADP
ejpam-65	608	4	the	the	DET
ejpam-65	608	5	definition	definition	NOUN
ejpam-65	608	6	5.1	5.1	NUM
ejpam-65	608	7	part	part	NOUN
ejpam-65	608	8	(	(	PUNCT
ejpam-65	608	9	ii	ii	NOUN
ejpam-65	608	10	)	)	PUNCT
ejpam-65	608	11	if	if	SCONJ
ejpam-65	608	12	we	we	PRON
ejpam-65	608	13	replace	replace	VERB
ejpam-65	608	14	homw	homw	PROPN
ejpam-65	608	15	k(v	k(v	PROPN
ejpam-65	608	16	,	,	PUNCT
ejpam-65	608	17	w	w	NOUN
ejpam-65	608	18	)	)	PUNCT
ejpam-65	608	19	by	by	ADP
ejpam-65	608	20	homi	homi	PROPN
ejpam-65	608	21	k(v	k(v	PROPN
ejpam-65	608	22	,	,	PUNCT
ejpam-65	608	23	w	w	NOUN
ejpam-65	608	24	)	)	PUNCT
ejpam-65	608	25	or	or	CCONJ
ejpam-65	608	26	homg	homg	NOUN
ejpam-65	608	27	k(v	k(v	PROPN
ejpam-65	608	28	,	,	PUNCT
ejpam-65	608	29	w	w	PROPN
ejpam-65	608	30	)	)	PUNCT
ejpam-65	608	31	,	,	PUNCT
ejpam-65	608	32	then	then	ADV
ejpam-65	608	33	we	we	PRON
ejpam-65	608	34	will	will	AUX
ejpam-65	608	35	obtain	obtain	VERB
ejpam-65	608	36	some	some	DET
ejpam-65	608	37	new	new	ADJ
ejpam-65	608	38	categories	category	NOUN
ejpam-65	608	39	,	,	PUNCT
ejpam-65	608	40	which	which	PRON
ejpam-65	608	41	we	we	PRON
ejpam-65	608	42	denote	denote	VERB
ejpam-65	608	43	them	they	PRON
ejpam-65	608	44	by	by	ADP
ejpam-65	608	45	hv	hv	PROPN
ejpam-65	608	46	ik	ik	PROPN
ejpam-65	608	47	and	and	CCONJ
ejpam-65	608	48	hvgk	hvgk	PROPN
ejpam-65	608	49	,	,	PUNCT
ejpam-65	608	50	respectively	respectively	ADV
ejpam-65	608	51	.	.	PUNCT
ejpam-65	609	1	in	in	ADP
ejpam-65	609	2	fact	fact	NOUN
ejpam-65	609	3	,	,	PUNCT
ejpam-65	609	4	hvgk	hvgk	PROPN
ejpam-65	609	5	�	�	PROPN
ejpam-65	609	6	hv	hv	PROPN
ejpam-65	609	7	i	i	PROPN
ejpam-65	609	8	k	k	PROPN
ejpam-65	609	9	�	�	PROPN
ejpam-65	609	10	hvwk	hvwk	NOUN
ejpam-65	609	11	(	(	PUNCT
ejpam-65	609	12	by	by	ADP
ejpam-65	609	13	a	a	DET
ejpam-65	609	14	�	�	PROPN
ejpam-65	609	15	b	b	PROPN
ejpam-65	609	16	read	read	VERB
ejpam-65	609	17	a	a	PRON
ejpam-65	609	18	is	be	AUX
ejpam-65	609	19	a	a	DET
ejpam-65	609	20	subcategory	subcategory	NOUN
ejpam-65	609	21	of	of	ADP
ejpam-65	609	22	b	b	PROPN
ejpam-65	609	23	)	)	PUNCT
ejpam-65	609	24	.	.	PUNCT
ejpam-65	610	1	we	we	PRON
ejpam-65	610	2	denote	denote	VERB
ejpam-65	610	3	the	the	DET
ejpam-65	610	4	category	category	NOUN
ejpam-65	610	5	of	of	ADP
ejpam-65	610	6	all	all	DET
ejpam-65	610	7	vector	vector	NOUN
ejpam-65	610	8	spaces	space	NOUN
ejpam-65	610	9	over	over	ADP
ejpam-65	610	10	the	the	DET
ejpam-65	610	11	field	field	NOUN
ejpam-65	610	12	k	k	X
ejpam-65	610	13	by	by	ADP
ejpam-65	610	14	vk	vk	NOUN
ejpam-65	610	15	.	.	PUNCT
ejpam-65	611	1	in	in	ADP
ejpam-65	611	2	fact	fact	NOUN
ejpam-65	611	3	,	,	PUNCT
ejpam-65	611	4	vk	vk	PROPN
ejpam-65	611	5	�	�	PROPN
ejpam-65	611	6	hvgk	hvgk	PROPN
ejpam-65	611	7	.	.	PUNCT
ejpam-65	612	1	(	(	PUNCT
ejpam-65	612	2	see	see	VERB
ejpam-65	612	3	[	[	X
ejpam-65	612	4	2	2	NUM
ejpam-65	612	5	]	]	PUNCT
ejpam-65	612	6	)	)	PUNCT
ejpam-65	612	7	lemma	lemma	PROPN
ejpam-65	612	8	5.1	5.1	NUM
ejpam-65	612	9	.	.	PUNCT
ejpam-65	613	1	let	let	VERB
ejpam-65	613	2	v	v	NOUN
ejpam-65	613	3	and	and	CCONJ
ejpam-65	613	4	w	w	NOUN
ejpam-65	613	5	be	be	AUX
ejpam-65	613	6	two	two	NUM
ejpam-65	613	7	hypervector	hypervector	NOUN
ejpam-65	613	8	spaces	space	NOUN
ejpam-65	613	9	and	and	CCONJ
ejpam-65	613	10	t	t	NOUN
ejpam-65	613	11	:	:	PUNCT
ejpam-65	613	12	v	v	AUX
ejpam-65	613	13	−→	−→	NOUN
ejpam-65	613	14	w	w	NOUN
ejpam-65	613	15	be	be	AUX
ejpam-65	613	16	a	a	DET
ejpam-65	613	17	good	good	ADJ
ejpam-65	613	18	linear	linear	ADJ
ejpam-65	613	19	transformation	transformation	NOUN
ejpam-65	613	20	.	.	PUNCT
ejpam-65	614	1	then	then	ADV
ejpam-65	614	2	(	(	PUNCT
ejpam-65	614	3	i	i	NOUN
ejpam-65	614	4	)	)	PUNCT
ejpam-65	614	5	∀x	∀x	VERB
ejpam-65	614	6	∈	∈	PROPN
ejpam-65	614	7	v	v	NOUN
ejpam-65	614	8	,	,	PUNCT
ejpam-65	614	9	t	t	PROPN
ejpam-65	614	10	(	(	PUNCT
ejpam-65	614	11	ε∗(x	ε∗(x	NOUN
ejpam-65	614	12	)	)	PUNCT
ejpam-65	614	13	)	)	PUNCT
ejpam-65	615	1	⊆	⊆	NUM
ejpam-65	615	2	ε∗(t	ε∗(t	NOUN
ejpam-65	615	3	(	(	PUNCT
ejpam-65	615	4	x	x	NOUN
ejpam-65	615	5	)	)	PUNCT
ejpam-65	615	6	)	)	PUNCT
ejpam-65	615	7	;	;	PUNCT
ejpam-65	615	8	(	(	PUNCT
ejpam-65	615	9	ii	ii	NOUN
ejpam-65	615	10	)	)	PUNCT
ejpam-65	615	11	the	the	DET
ejpam-65	615	12	map	map	NOUN
ejpam-65	615	13	{	{	PUNCT
ejpam-65	615	14	t	t	NOUN
ejpam-65	615	15	∗	∗	NOUN
ejpam-65	615	16	:	:	PUNCT
ejpam-65	615	17	v	v	X
ejpam-65	615	18	/	/	SYM
ejpam-65	615	19	ε∗	ε∗	PROPN
ejpam-65	615	20	−→w	−→w	NUM
ejpam-65	615	21	/	/	SYM
ejpam-65	615	22	ε∗	ε∗	PROPN
ejpam-65	615	23	t	t	NOUN
ejpam-65	615	24	∗(ε∗(x	∗(ε∗(x	NOUN
ejpam-65	615	25	)	)	PUNCT
ejpam-65	615	26	)	)	PUNCT
ejpam-65	616	1	=	=	SYM
ejpam-65	616	2	ε∗(t	ε∗(t	NOUN
ejpam-65	616	3	(	(	PUNCT
ejpam-65	616	4	x	x	NOUN
ejpam-65	616	5	)	)	PUNCT
ejpam-65	616	6	)	)	PUNCT
ejpam-65	616	7	is	be	AUX
ejpam-65	616	8	a	a	DET
ejpam-65	616	9	linear	linear	ADJ
ejpam-65	616	10	transformation	transformation	NOUN
ejpam-65	616	11	.	.	PUNCT
ejpam-65	617	1	r.	r.	PROPN
ejpam-65	617	2	ameri	ameri	PROPN
ejpam-65	617	3	,	,	PUNCT
ejpam-65	617	4	o.r	o.r	PROPN
ejpam-65	617	5	.	.	PROPN
ejpam-65	617	6	dehghan	dehghan	PROPN
ejpam-65	617	7	/	/	SYM
ejpam-65	617	8	eur	eur	PROPN
ejpam-65	617	9	.	.	PUNCT
ejpam-65	618	1	j.	j.	PROPN
ejpam-65	618	2	pure	pure	PROPN
ejpam-65	618	3	appl	appl	PROPN
ejpam-65	618	4	.	.	PROPN
ejpam-65	618	5	math	math	PROPN
ejpam-65	618	6	,	,	PUNCT
ejpam-65	618	7	1	1	NUM
ejpam-65	618	8	(	(	PUNCT
ejpam-65	618	9	2008	2008	NUM
ejpam-65	618	10	)	)	PUNCT
ejpam-65	618	11	,	,	PUNCT
ejpam-65	618	12	(	(	PUNCT
ejpam-65	618	13	32	32	NUM
ejpam-65	618	14	-	-	SYM
ejpam-65	618	15	50	50	NUM
ejpam-65	618	16	)	)	PUNCT
ejpam-65	618	17	48	48	NUM
ejpam-65	618	18	proof	proof	NOUN
ejpam-65	618	19	.	.	PUNCT
ejpam-65	619	1	straightforward	straightforward	ADJ
ejpam-65	619	2	.	.	PUNCT
ejpam-65	620	1	in	in	ADP
ejpam-65	620	2	[	[	X
ejpam-65	620	3	7	7	X
ejpam-65	620	4	]	]	X
ejpam-65	620	5	the	the	DET
ejpam-65	620	6	notions	notion	NOUN
ejpam-65	620	7	of	of	ADP
ejpam-65	620	8	pseudonorm	pseudonorm	NOUN
ejpam-65	620	9	and	and	CCONJ
ejpam-65	620	10	norm	norm	NOUN
ejpam-65	620	11	in	in	ADP
ejpam-65	620	12	hypervector	hypervector	NOUN
ejpam-65	620	13	spaces	space	NOUN
ejpam-65	620	14	was	be	AUX
ejpam-65	620	15	introduced	introduce	VERB
ejpam-65	620	16	.	.	PUNCT
ejpam-65	621	1	let	let	VERB
ejpam-65	621	2	(	(	PUNCT
ejpam-65	621	3	v,+	v,+	NUM
ejpam-65	621	4	,	,	PUNCT
ejpam-65	621	5	◦	◦	NOUN
ejpam-65	621	6	,	,	PUNCT
ejpam-65	621	7	k	k	NOUN
ejpam-65	621	8	)	)	PUNCT
ejpam-65	621	9	be	be	VERB
ejpam-65	621	10	a	a	DET
ejpam-65	621	11	hypervector	hypervector	NOUN
ejpam-65	621	12	space	space	NOUN
ejpam-65	621	13	over	over	ADP
ejpam-65	621	14	a	a	DET
ejpam-65	621	15	valued	value	VERB
ejpam-65	621	16	field	field	NOUN
ejpam-65	621	17	k	k	NOUN
ejpam-65	621	18	,	,	PUNCT
ejpam-65	621	19	(	(	PUNCT
ejpam-65	621	20	for	for	ADP
ejpam-65	621	21	a	a	DET
ejpam-65	621	22	∈	∈	PROPN
ejpam-65	621	23	k	k	NOUN
ejpam-65	621	24	,	,	PUNCT
ejpam-65	621	25	we	we	PRON
ejpam-65	621	26	denote	denote	VERB
ejpam-65	621	27	by	by	ADP
ejpam-65	621	28	/a/	/a/	PUNCT
ejpam-65	622	1	the	the	DET
ejpam-65	622	2	valuation	valuation	NOUN
ejpam-65	622	3	of	of	ADP
ejpam-65	622	4	a	a	PRON
ejpam-65	622	5	in	in	ADP
ejpam-65	622	6	k	k	NOUN
ejpam-65	622	7	)	)	PUNCT
ejpam-65	622	8	.	.	PUNCT
ejpam-65	623	1	a	a	DET
ejpam-65	623	2	pseudonorm	pseudonorm	NOUN
ejpam-65	623	3	in	in	ADP
ejpam-65	623	4	v	v	NOUN
ejpam-65	623	5	is	be	AUX
ejpam-65	623	6	a	a	DET
ejpam-65	623	7	mapping	mapping	NOUN
ejpam-65	623	8	‖	‖	ADJ
ejpam-65	623	9	‖	‖	ADJ
ejpam-65	623	10	:	:	PUNCT
ejpam-65	623	11	v	v	ADP
ejpam-65	623	12	−→	−→	NOUN
ejpam-65	623	13	r	r	NOUN
ejpam-65	623	14	,	,	PUNCT
ejpam-65	623	15	such	such	ADJ
ejpam-65	623	16	that	that	SCONJ
ejpam-65	623	17	the	the	DET
ejpam-65	623	18	following	follow	VERB
ejpam-65	623	19	conditions	condition	NOUN
ejpam-65	623	20	hold	hold	VERB
ejpam-65	623	21	:	:	PUNCT
ejpam-65	623	22	(	(	PUNCT
ejpam-65	623	23	i	i	NOUN
ejpam-65	623	24	)	)	PUNCT
ejpam-65	623	25	‖0‖	‖0‖	PROPN
ejpam-65	624	1	=	=	SYM
ejpam-65	624	2	0	0	PROPN
ejpam-65	624	3	,	,	PUNCT
ejpam-65	624	4	(	(	PUNCT
ejpam-65	624	5	ii	ii	NOUN
ejpam-65	624	6	)	)	PUNCT
ejpam-65	624	7	∀x	∀x	NUM
ejpam-65	624	8	,	,	PUNCT
ejpam-65	624	9	y	y	PROPN
ejpam-65	624	10	∈	∈	PROPN
ejpam-65	624	11	v	v	NOUN
ejpam-65	624	12	,	,	PUNCT
ejpam-65	624	13	‖x+	‖x+	PROPN
ejpam-65	624	14	y‖	y‖	PROPN
ejpam-65	625	1	≤	≤	ADV
ejpam-65	625	2	‖x‖+	‖x‖+	NUM
ejpam-65	625	3	‖y‖	‖y‖	PROPN
ejpam-65	625	4	,	,	PUNCT
ejpam-65	625	5	(	(	PUNCT
ejpam-65	625	6	iii	iii	NOUN
ejpam-65	625	7	)	)	PUNCT
ejpam-65	625	8	∀a	∀a	NOUN
ejpam-65	625	9	∈	∈	PROPN
ejpam-65	625	10	k	k	NOUN
ejpam-65	625	11	,	,	PUNCT
ejpam-65	625	12	∀x	∀x	X
ejpam-65	625	13	,	,	PUNCT
ejpam-65	625	14	y	y	PROPN
ejpam-65	625	15	∈	∈	PROPN
ejpam-65	625	16	v	v	NOUN
ejpam-65	625	17	,	,	PUNCT
ejpam-65	625	18	sup	sup	NOUN
ejpam-65	625	19	‖a	‖a	NOUN
ejpam-65	625	20	◦	◦	VERB
ejpam-65	625	21	x‖	x‖	X
ejpam-65	626	1	=	=	SYM
ejpam-65	626	2	/a/	/a/	PUNCT
ejpam-65	626	3	‖x‖	‖x‖	PROPN
ejpam-65	626	4	.	.	PUNCT
ejpam-65	627	1	a	a	DET
ejpam-65	627	2	pseudonorm	pseudonorm	NOUN
ejpam-65	627	3	in	in	ADP
ejpam-65	627	4	v	v	NOUN
ejpam-65	627	5	is	be	AUX
ejpam-65	627	6	called	call	VERB
ejpam-65	627	7	norm	norm	NOUN
ejpam-65	627	8	if	if	SCONJ
ejpam-65	627	9	:	:	PUNCT
ejpam-65	627	10	(	(	PUNCT
ejpam-65	627	11	iv	iv	X
ejpam-65	627	12	)	)	PUNCT
ejpam-65	627	13	‖x‖	‖x‖	PROPN
ejpam-65	627	14	=	=	SYM
ejpam-65	627	15	0	0	NUM
ejpam-65	627	16	⇐	⇐	ADJ
ejpam-65	627	17	⇒	⇒	NOUN
ejpam-65	627	18	x	x	PUNCT
ejpam-65	628	1	=	=	SYM
ejpam-65	628	2	0	0	X
ejpam-65	628	3	.	.	PUNCT
ejpam-65	629	1	now	now	ADV
ejpam-65	629	2	let	let	VERB
ejpam-65	629	3	‖	‖	PROPN
ejpam-65	629	4	‖	‖	VERB
ejpam-65	629	5	be	be	AUX
ejpam-65	629	6	a	a	DET
ejpam-65	629	7	norm	norm	NOUN
ejpam-65	629	8	on	on	ADP
ejpam-65	629	9	the	the	DET
ejpam-65	629	10	fundamental	fundamental	ADJ
ejpam-65	629	11	vector	vector	NOUN
ejpam-65	629	12	space	space	NOUN
ejpam-65	629	13	v	v	NOUN
ejpam-65	629	14	/	/	SYM
ejpam-65	629	15	ε∗	ε∗	PROPN
ejpam-65	629	16	of	of	ADP
ejpam-65	629	17	v	v	NOUN
ejpam-65	629	18	.	.	PUNCT
ejpam-65	630	1	define	define	VERB
ejpam-65	630	2	the	the	DET
ejpam-65	630	3	mapping	mapping	NOUN
ejpam-65	630	4	‖	‖	PROPN
ejpam-65	630	5	‖∗	‖∗	PROPN
ejpam-65	630	6	:	:	PUNCT
ejpam-65	630	7	v	v	X
ejpam-65	630	8	−→	−→	NOUN
ejpam-65	630	9	r	r	NOUN
ejpam-65	630	10	by	by	ADP
ejpam-65	630	11	‖x‖∗	‖x‖∗	PROPN
ejpam-65	630	12	=	=	SYM
ejpam-65	630	13	‖ε∗(x)‖	‖ε∗(x)‖	PROPN
ejpam-65	630	14	.	.	PUNCT
ejpam-65	631	1	the	the	DET
ejpam-65	631	2	next	next	ADJ
ejpam-65	631	3	result	result	NOUN
ejpam-65	631	4	follows	follow	VERB
ejpam-65	631	5	:	:	PUNCT
ejpam-65	631	6	theorem	theorem	VERB
ejpam-65	631	7	5.1	5.1	NUM
ejpam-65	631	8	.	.	PUNCT
ejpam-65	632	1	the	the	DET
ejpam-65	632	2	mapping	mapping	NOUN
ejpam-65	632	3	‖	‖	PROPN
ejpam-65	632	4	‖∗	‖∗	PROPN
ejpam-65	632	5	is	be	AUX
ejpam-65	632	6	a	a	DET
ejpam-65	632	7	pseudonorm	pseudonorm	NOUN
ejpam-65	632	8	on	on	ADP
ejpam-65	632	9	v.	v.	ADP
ejpam-65	632	10	proof	proof	NOUN
ejpam-65	632	11	.	.	PUNCT
ejpam-65	633	1	(	(	PUNCT
ejpam-65	633	2	i	i	NOUN
ejpam-65	633	3	)	)	PUNCT
ejpam-65	633	4	‖0‖∗	‖0‖∗	PROPN
ejpam-65	634	1	=	=	SYM
ejpam-65	634	2	‖ε∗(0)‖	‖ε∗(0)‖	PROPN
ejpam-65	634	3	=	=	SYM
ejpam-65	634	4	0	0	PROPN
ejpam-65	634	5	,	,	PUNCT
ejpam-65	634	6	(	(	PUNCT
ejpam-65	634	7	ii	ii	NOUN
ejpam-65	634	8	)	)	PUNCT
ejpam-65	634	9	∀x	∀x	NUM
ejpam-65	634	10	,	,	PUNCT
ejpam-65	634	11	y	y	PROPN
ejpam-65	634	12	∈	∈	PROPN
ejpam-65	634	13	v	v	PROPN
ejpam-65	634	14	,	,	PUNCT
ejpam-65	634	15	‖x+	‖x+	PROPN
ejpam-65	634	16	y‖∗	y‖∗	X
ejpam-65	634	17	=	=	SYM
ejpam-65	635	1	‖ε∗(x+	‖ε∗(x+	VERB
ejpam-65	635	2	y)‖	y)‖	NOUN
ejpam-65	635	3	=	=	SYM
ejpam-65	635	4	‖ε∗(x)⊕	‖ε∗(x)⊕	X
ejpam-65	635	5	ε∗(y)‖	ε∗(y)‖	PROPN
ejpam-65	635	6	≤	≤	PROPN
ejpam-65	635	7	‖ε∗(x)‖+	‖ε∗(x)‖+	PROPN
ejpam-65	635	8	‖ε∗(y)‖	‖ε∗(y)‖	PROPN
ejpam-65	635	9	=	=	PUNCT
ejpam-65	636	1	‖x‖∗	‖x‖∗	PROPN
ejpam-65	636	2	+	+	NUM
ejpam-65	636	3	‖y‖∗	‖y‖∗	NUM
ejpam-65	636	4	,	,	PUNCT
ejpam-65	636	5	(	(	PUNCT
ejpam-65	636	6	iii	iii	NOUN
ejpam-65	636	7	)	)	PUNCT
ejpam-65	636	8	∀a	∀a	NOUN
ejpam-65	636	9	∈	∈	PROPN
ejpam-65	636	10	k	k	NOUN
ejpam-65	636	11	,	,	PUNCT
ejpam-65	636	12	∀x	∀x	X
ejpam-65	636	13	,	,	PUNCT
ejpam-65	636	14	y	y	PROPN
ejpam-65	636	15	∈	∈	PROPN
ejpam-65	636	16	v	v	NOUN
ejpam-65	636	17	,	,	PUNCT
ejpam-65	636	18	sup	sup	NOUN
ejpam-65	636	19	‖a	‖a	NOUN
ejpam-65	636	20	◦	◦	NOUN
ejpam-65	636	21	x‖∗	x‖∗	PUNCT
ejpam-65	637	1	=	=	PUNCT
ejpam-65	637	2	sup	sup	NOUN
ejpam-65	637	3	‖ε∗(a	‖ε∗(a	NUM
ejpam-65	637	4	◦	◦	NOUN
ejpam-65	637	5	x)‖	x)‖	NOUN
ejpam-65	638	1	=	=	PUNCT
ejpam-65	638	2	‖a	‖a	NOUN
ejpam-65	638	3	�	�	X
ejpam-65	638	4	ε∗(x)‖	ε∗(x)‖	X
ejpam-65	638	5	=	=	PUNCT
ejpam-65	639	1	/a/	/a/	PUNCT
ejpam-65	639	2	‖ε∗(x)‖	‖ε∗(x)‖	NOUN
ejpam-65	639	3	=	=	PUNCT
ejpam-65	639	4	/a/	/a/	PUNCT
ejpam-65	640	1	‖x‖∗	‖x‖∗	PROPN
ejpam-65	640	2	.	.	PUNCT
ejpam-65	640	3	remark	remark	PROPN
ejpam-65	640	4	5.1	5.1	NUM
ejpam-65	640	5	.	.	PUNCT
ejpam-65	641	1	the	the	DET
ejpam-65	641	2	‖	‖	PROPN
ejpam-65	641	3	‖∗	‖∗	PROPN
ejpam-65	641	4	is	be	AUX
ejpam-65	641	5	called	call	VERB
ejpam-65	641	6	the	the	DET
ejpam-65	641	7	fundamental	fundamental	ADJ
ejpam-65	641	8	pseudonorm	pseudonorm	NOUN
ejpam-65	641	9	associated	associate	VERB
ejpam-65	641	10	to	to	ADP
ejpam-65	641	11	‖	‖	PROPN
ejpam-65	641	12	‖	‖	PROPN
ejpam-65	641	13	.	.	PUNCT
ejpam-65	642	1	theorem	theorem	VERB
ejpam-65	642	2	5.2	5.2	NUM
ejpam-65	642	3	.	.	PUNCT
ejpam-65	643	1	the	the	DET
ejpam-65	643	2	mapping	mapping	NOUN
ejpam-65	643	3	f	f	NOUN
ejpam-65	643	4	:	:	PUNCT
ejpam-65	643	5	hvgk	hvgk	VERB
ejpam-65	643	6	−→	−→	ADJ
ejpam-65	643	7	vk	vk	NOUN
ejpam-65	643	8	is	be	AUX
ejpam-65	643	9	defined	define	VERB
ejpam-65	643	10	by	by	ADP
ejpam-65	643	11	f	f	PROPN
ejpam-65	643	12	(	(	PUNCT
ejpam-65	643	13	v	v	NOUN
ejpam-65	643	14	)	)	PUNCT
ejpam-65	643	15	=	=	SYM
ejpam-65	644	1	v	v	NOUN
ejpam-65	644	2	/	/	SYM
ejpam-65	644	3	ε∗	ε∗	PROPN
ejpam-65	644	4	is	be	AUX
ejpam-65	644	5	a	a	DET
ejpam-65	644	6	functor	functor	NOUN
ejpam-65	644	7	.	.	PUNCT
ejpam-65	645	1	moreover	moreover	ADV
ejpam-65	645	2	,	,	PUNCT
ejpam-65	645	3	the	the	DET
ejpam-65	645	4	functor	functor	PROPN
ejpam-65	645	5	f	f	PROPN
ejpam-65	645	6	preserves	preserve	VERB
ejpam-65	645	7	the	the	DET
ejpam-65	645	8	dimension	dimension	NOUN
ejpam-65	645	9	.	.	PUNCT
ejpam-65	646	1	proof	proof	NOUN
ejpam-65	646	2	.	.	PUNCT
ejpam-65	647	1	by	by	ADP
ejpam-65	647	2	lemma	lemma	PROPN
ejpam-65	647	3	5.1	5.1	NUM
ejpam-65	647	4	,	,	PUNCT
ejpam-65	647	5	f	f	PROPN
ejpam-65	647	6	is	be	AUX
ejpam-65	647	7	well	well	ADV
ejpam-65	647	8	-	-	PUNCT
ejpam-65	647	9	defined	define	VERB
ejpam-65	647	10	.	.	PUNCT
ejpam-65	648	1	let	let	VERB
ejpam-65	648	2	t	t	NOUN
ejpam-65	648	3	:	:	PUNCT
ejpam-65	648	4	v	v	ADP
ejpam-65	648	5	−→	−→	NOUN
ejpam-65	648	6	w	w	PROPN
ejpam-65	648	7	and	and	CCONJ
ejpam-65	648	8	s	s	VERB
ejpam-65	648	9	:	:	PUNCT
ejpam-65	648	10	w	w	X
ejpam-65	648	11	−→	−→	NOUN
ejpam-65	648	12	l	l	NOUN
ejpam-65	648	13	be	be	AUX
ejpam-65	648	14	good	good	ADJ
ejpam-65	648	15	linear	linear	ADJ
ejpam-65	648	16	transformations	transformation	NOUN
ejpam-65	648	17	.	.	PUNCT
ejpam-65	649	1	then	then	ADV
ejpam-65	649	2	we	we	PRON
ejpam-65	649	3	have	have	VERB
ejpam-65	649	4	f	f	X
ejpam-65	649	5	(	(	PUNCT
ejpam-65	649	6	sot	sot	ADJ
ejpam-65	649	7	)	)	PUNCT
ejpam-65	649	8	=	=	SYM
ejpam-65	649	9	(	(	PUNCT
ejpam-65	649	10	sot	sot	ADJ
ejpam-65	649	11	)	)	PUNCT
ejpam-65	649	12	∗	∗	NOUN
ejpam-65	649	13	.	.	PUNCT
ejpam-65	650	1	also	also	ADV
ejpam-65	650	2	(	(	PUNCT
ejpam-65	650	3	sot	sot	ADJ
ejpam-65	650	4	)	)	PUNCT
ejpam-65	650	5	∗	∗	NOUN
ejpam-65	650	6	(	(	PUNCT
ejpam-65	650	7	ε∗(x	ε∗(x	NOUN
ejpam-65	650	8	)	)	PUNCT
ejpam-65	650	9	)	)	PUNCT
ejpam-65	651	1	=	=	SYM
ejpam-65	651	2	ε∗	ε∗	X
ejpam-65	651	3	(	(	PUNCT
ejpam-65	651	4	(	(	PUNCT
ejpam-65	651	5	sot	sot	ADJ
ejpam-65	651	6	)	)	PUNCT
ejpam-65	651	7	(	(	PUNCT
ejpam-65	651	8	x	x	NOUN
ejpam-65	651	9	)	)	PUNCT
ejpam-65	651	10	)	)	PUNCT
ejpam-65	651	11	=	=	SYM
ejpam-65	652	1	ε∗	ε∗	PROPN
ejpam-65	652	2	(	(	PUNCT
ejpam-65	652	3	s(t	s(t	PROPN
ejpam-65	652	4	(	(	PUNCT
ejpam-65	652	5	x	x	NOUN
ejpam-65	652	6	)	)	PUNCT
ejpam-65	652	7	)	)	PUNCT
ejpam-65	652	8	)	)	PUNCT
ejpam-65	653	1	=	=	PRON
ejpam-65	653	2	s∗ε∗(t	s∗ε∗(t	NOUN
ejpam-65	653	3	(	(	PUNCT
ejpam-65	653	4	x	x	NOUN
ejpam-65	653	5	)	)	PUNCT
ejpam-65	653	6	)	)	PUNCT
ejpam-65	653	7	=	=	SYM
ejpam-65	654	1	s∗t	s∗t	X
ejpam-65	654	2	∗(ε∗(x	∗(ε∗(x	NOUN
ejpam-65	654	3	)	)	PUNCT
ejpam-65	654	4	)	)	PUNCT
ejpam-65	655	1	=	=	SYM
ejpam-65	655	2	f	f	PROPN
ejpam-65	655	3	(	(	PUNCT
ejpam-65	655	4	s)f	s)f	X
ejpam-65	655	5	(	(	PUNCT
ejpam-65	655	6	t	t	NOUN
ejpam-65	655	7	)	)	PUNCT
ejpam-65	655	8	(	(	PUNCT
ejpam-65	655	9	ε∗(x	ε∗(x	NOUN
ejpam-65	655	10	)	)	PUNCT
ejpam-65	655	11	)	)	PUNCT
ejpam-65	655	12	,	,	PUNCT
ejpam-65	655	13	references	reference	NOUN
ejpam-65	655	14	49	49	NUM
ejpam-65	655	15	for	for	ADP
ejpam-65	655	16	all	all	DET
ejpam-65	655	17	x	x	SYM
ejpam-65	655	18	∈	∈	NOUN
ejpam-65	655	19	v	v	NOUN
ejpam-65	655	20	.	.	PUNCT
ejpam-65	656	1	hence	hence	ADV
ejpam-65	656	2	f	f	PROPN
ejpam-65	656	3	(	(	PUNCT
ejpam-65	656	4	sot	sot	ADJ
ejpam-65	656	5	)	)	PUNCT
ejpam-65	656	6	=	=	SYM
ejpam-65	656	7	f	f	PROPN
ejpam-65	656	8	(	(	PUNCT
ejpam-65	656	9	s)f	s)f	X
ejpam-65	656	10	(	(	PUNCT
ejpam-65	656	11	t	t	NOUN
ejpam-65	656	12	)	)	PUNCT
ejpam-65	657	1	.also	.also	PUNCT
ejpam-65	657	2	,	,	PUNCT
ejpam-65	657	3	f	f	PROPN
ejpam-65	657	4	(	(	PUNCT
ejpam-65	657	5	1∗v	1∗v	NUM
ejpam-65	657	6	)	)	PUNCT
ejpam-65	657	7	:	:	PUNCT
ejpam-65	657	8	v	v	X
ejpam-65	657	9	/	/	SYM
ejpam-65	657	10	ε∗	ε∗	VERB
ejpam-65	657	11	−→	−→	NOUN
ejpam-65	657	12	v	v	NOUN
ejpam-65	657	13	/	/	SYM
ejpam-65	657	14	ε∗	ε∗	PROPN
ejpam-65	657	15	,	,	PUNCT
ejpam-65	657	16	is	be	AUX
ejpam-65	657	17	obtained	obtain	VERB
ejpam-65	657	18	by	by	ADP
ejpam-65	657	19	1∗v	1∗v	NUM
ejpam-65	657	20	(	(	PUNCT
ejpam-65	657	21	ε∗(x	ε∗(x	NOUN
ejpam-65	657	22	)	)	PUNCT
ejpam-65	657	23	)	)	PUNCT
ejpam-65	658	1	=	=	SYM
ejpam-65	658	2	ε∗(x	ε∗(x	NOUN
ejpam-65	658	3	)	)	PUNCT
ejpam-65	658	4	,	,	PUNCT
ejpam-65	658	5	is	be	AUX
ejpam-65	658	6	the	the	DET
ejpam-65	658	7	identity	identity	NOUN
ejpam-65	658	8	morphism	morphism	NOUN
ejpam-65	658	9	.	.	PUNCT
ejpam-65	659	1	therefore	therefore	ADV
ejpam-65	659	2	,	,	PUNCT
ejpam-65	659	3	f	f	PROPN
ejpam-65	659	4	is	be	AUX
ejpam-65	659	5	a	a	DET
ejpam-65	659	6	functor	functor	NOUN
ejpam-65	659	7	.	.	PUNCT
ejpam-65	660	1	also	also	ADV
ejpam-65	660	2	by	by	ADP
ejpam-65	660	3	theorem	theorem	NOUN
ejpam-65	660	4	4.1	4.1	NUM
ejpam-65	660	5	,	,	PUNCT
ejpam-65	660	6	dim(f	dim(f	PROPN
ejpam-65	660	7	(	(	PUNCT
ejpam-65	660	8	v	v	NOUN
ejpam-65	660	9	)	)	PUNCT
ejpam-65	660	10	)	)	PUNCT
ejpam-65	661	1	=	=	PUNCT
ejpam-65	661	2	dim(v	dim(v	PROPN
ejpam-65	661	3	/	/	SYM
ejpam-65	661	4	ε∗	ε∗	PROPN
ejpam-65	661	5	)	)	PUNCT
ejpam-65	661	6	=	=	SYM
ejpam-65	661	7	dim(v	dim(v	PROPN
ejpam-65	661	8	)	)	PUNCT
ejpam-65	661	9	.	.	PUNCT
ejpam-65	662	1	corollary	corollary	ADJ
ejpam-65	662	2	5.1	5.1	NUM
ejpam-65	662	3	.	.	PUNCT
ejpam-65	663	1	let	let	VERB
ejpam-65	663	2	t	t	NOUN
ejpam-65	663	3	:	:	PUNCT
ejpam-65	663	4	v	v	AUX
ejpam-65	663	5	−→	−→	NOUN
ejpam-65	663	6	w	w	NOUN
ejpam-65	663	7	be	be	AUX
ejpam-65	663	8	a	a	DET
ejpam-65	663	9	morphism	morphism	NOUN
ejpam-65	663	10	in	in	ADP
ejpam-65	663	11	hvgk	hvgk	PROPN
ejpam-65	663	12	.	.	PUNCT
ejpam-65	664	1	then	then	ADV
ejpam-65	664	2	the	the	DET
ejpam-65	664	3	following	follow	VERB
ejpam-65	664	4	diagram	diagram	NOUN
ejpam-65	664	5	is	be	AUX
ejpam-65	664	6	commutative	commutative	ADJ
ejpam-65	664	7	:	:	PUNCT
ejpam-65	664	8	v	v	NUM
ejpam-65	664	9	t−→	t−→	NOUN
ejpam-65	664	10	w	w	NOUN
ejpam-65	664	11	ϕv	ϕv	ADP
ejpam-65	664	12	↓	↓	NOUN
ejpam-65	664	13	↓ϕw	↓ϕw	PROPN
ejpam-65	664	14	v	v	NOUN
ejpam-65	664	15	/	/	SYM
ejpam-65	664	16	ε∗	ε∗	PROPN
ejpam-65	664	17	t	t	X
ejpam-65	664	18	∗−→	∗−→	PRON
ejpam-65	664	19	w	w	PROPN
ejpam-65	664	20	/	/	SYM
ejpam-65	664	21	ε∗	ε∗	PROPN
ejpam-65	665	1	where	where	SCONJ
ejpam-65	665	2	ϕv	ϕv	ADV
ejpam-65	666	1	and	and	CCONJ
ejpam-65	666	2	ϕw	ϕw	PRON
ejpam-65	666	3	are	be	AUX
ejpam-65	666	4	the	the	DET
ejpam-65	666	5	canonical	canonical	ADJ
ejpam-65	666	6	projections	projection	NOUN
ejpam-65	666	7	of	of	ADP
ejpam-65	666	8	v	v	NOUN
ejpam-65	666	9	and	and	CCONJ
ejpam-65	666	10	w	w	NOUN
ejpam-65	666	11	,	,	PUNCT
ejpam-65	666	12	respectively	respectively	ADV
ejpam-65	666	13	.	.	PUNCT
ejpam-65	667	1	proof	proof	NOUN
ejpam-65	667	2	.	.	PUNCT
ejpam-65	668	1	let	let	VERB
ejpam-65	668	2	x	x	SYM
ejpam-65	668	3	∈	∈	NOUN
ejpam-65	668	4	v	v	NOUN
ejpam-65	668	5	.	.	PUNCT
ejpam-65	669	1	then	then	ADV
ejpam-65	669	2	ϕw	ϕw	INTJ
ejpam-65	669	3	(	(	PUNCT
ejpam-65	669	4	t	t	PROPN
ejpam-65	669	5	(	(	PUNCT
ejpam-65	669	6	x	x	NOUN
ejpam-65	669	7	)	)	PUNCT
ejpam-65	669	8	)	)	PUNCT
ejpam-65	670	1	=	=	SYM
ejpam-65	670	2	ε∗(t	ε∗(t	NOUN
ejpam-65	670	3	(	(	PUNCT
ejpam-65	670	4	x	x	NOUN
ejpam-65	670	5	)	)	PUNCT
ejpam-65	670	6	)	)	PUNCT
ejpam-65	671	1	=	=	SYM
ejpam-65	671	2	t	t	NOUN
ejpam-65	671	3	∗(ε∗(x	∗(ε∗(x	NOUN
ejpam-65	671	4	)	)	PUNCT
ejpam-65	671	5	)	)	PUNCT
ejpam-65	672	1	=	=	SYM
ejpam-65	672	2	t	t	PROPN
ejpam-65	672	3	∗(ϕv	∗(ϕv	X
ejpam-65	672	4	(	(	PUNCT
ejpam-65	672	5	x	x	NOUN
ejpam-65	672	6	)	)	PUNCT
ejpam-65	672	7	)	)	PUNCT
ejpam-65	673	1	=	=	SYM
ejpam-65	673	2	t	t	PROPN
ejpam-65	673	3	∗ϕv	∗ϕv	PROPN
ejpam-65	673	4	(	(	PUNCT
ejpam-65	673	5	x	x	NOUN
ejpam-65	673	6	)	)	PUNCT
ejpam-65	673	7	.	.	PUNCT
ejpam-65	674	1	6	6	X
ejpam-65	674	2	.	.	X
ejpam-65	674	3	acknowledgement	acknowledgement	NOUN
ejpam-65	674	4	this	this	DET
ejpam-65	674	5	research	research	NOUN
ejpam-65	674	6	is	be	AUX
ejpam-65	674	7	partially	partially	ADV
ejpam-65	674	8	supported	support	VERB
ejpam-65	674	9	by	by	ADP
ejpam-65	674	10	the	the	DET
ejpam-65	674	11	”	"	PUNCT
ejpam-65	674	12	fuzzy	fuzzy	ADJ
ejpam-65	674	13	systems	system	NOUN
ejpam-65	674	14	and	and	CCONJ
ejpam-65	674	15	it‘s	it‘s	NOUN
ejpam-65	674	16	applications	application	NOUN
ejpam-65	674	17	center	center	NOUN
ejpam-65	674	18	of	of	ADP
ejpam-65	674	19	excellence	excellence	PROPN
ejpam-65	674	20	,	,	PUNCT
ejpam-65	674	21	shahid	shahid	PROPN
ejpam-65	674	22	bahonar	bahonar	VERB
ejpam-65	674	23	university	university	PROPN
ejpam-65	674	24	of	of	ADP
ejpam-65	674	25	kerman	kerman	PROPN
ejpam-65	674	26	,	,	PUNCT
ejpam-65	674	27	iran	iran	PROPN
ejpam-65	674	28	”	"	PUNCT
ejpam-65	674	29	and	and	CCONJ
ejpam-65	674	30	”	"	PUNCT
ejpam-65	674	31	research	research	NOUN
ejpam-65	674	32	center	center	NOUN
ejpam-65	674	33	in	in	ADP
ejpam-65	674	34	algebraic	algebraic	ADJ
ejpam-65	674	35	hyperstructures	hyperstructure	NOUN
ejpam-65	674	36	and	and	CCONJ
ejpam-65	674	37	fuzzy	fuzzy	ADJ
ejpam-65	674	38	mathematics	mathematic	NOUN
ejpam-65	674	39	,	,	PUNCT
ejpam-65	674	40	university	university	NOUN
ejpam-65	674	41	of	of	ADP
ejpam-65	674	42	mazandaran	mazandaran	PROPN
ejpam-65	674	43	,	,	PUNCT
ejpam-65	674	44	babolsar	babolsar	PROPN
ejpam-65	674	45	,	,	PUNCT
ejpam-65	674	46	iran	iran	PROPN
ejpam-65	674	47	”	"	PUNCT
ejpam-65	674	48	.	.	PUNCT
ejpam-65	675	1	references	reference	NOUN
ejpam-65	675	2	[	[	X
ejpam-65	675	3	1	1	NUM
ejpam-65	675	4	]	]	X
ejpam-65	675	5	r.	r.	PROPN
ejpam-65	675	6	ameri	ameri	PROPN
ejpam-65	675	7	,	,	PUNCT
ejpam-65	675	8	fuzzy	fuzzy	ADJ
ejpam-65	675	9	hypervector	hypervector	NOUN
ejpam-65	675	10	spaces	space	NOUN
ejpam-65	675	11	over	over	ADP
ejpam-65	675	12	valued	value	VERB
ejpam-65	675	13	fields	field	NOUN
ejpam-65	675	14	,	,	PUNCT
ejpam-65	675	15	iranian	iranian	ADJ
ejpam-65	675	16	journal	journal	NOUN
ejpam-65	675	17	of	of	ADP
ejpam-65	675	18	fuzzy	fuzzy	ADJ
ejpam-65	675	19	systems	system	NOUN
ejpam-65	675	20	,	,	PUNCT
ejpam-65	675	21	2	2	NUM
ejpam-65	675	22	:	:	SYM
ejpam-65	675	23	37	37	NUM
ejpam-65	675	24	-	-	SYM
ejpam-65	675	25	47	47	NUM
ejpam-65	675	26	(	(	PUNCT
ejpam-65	675	27	2005	2005	NUM
ejpam-65	675	28	)	)	PUNCT
ejpam-65	675	29	.	.	PUNCT
ejpam-65	676	1	[	[	X
ejpam-65	676	2	2	2	NUM
ejpam-65	676	3	]	]	X
ejpam-65	676	4	r.	r.	PROPN
ejpam-65	676	5	ameri	ameri	PROPN
ejpam-65	676	6	,	,	PUNCT
ejpam-65	676	7	on	on	ADP
ejpam-65	676	8	categories	category	NOUN
ejpam-65	676	9	of	of	ADP
ejpam-65	676	10	hypergroups	hypergroup	NOUN
ejpam-65	676	11	and	and	CCONJ
ejpam-65	676	12	hypermodules	hypermodule	NOUN
ejpam-65	676	13	,	,	PUNCT
ejpam-65	676	14	journal	journal	NOUN
ejpam-65	676	15	of	of	ADP
ejpam-65	676	16	discrete	discrete	ADJ
ejpam-65	676	17	mathematical	mathematical	ADJ
ejpam-65	676	18	science	science	NOUN
ejpam-65	676	19	and	and	CCONJ
ejpam-65	676	20	cryptography	cryptography	NOUN
ejpam-65	676	21	,	,	PUNCT
ejpam-65	676	22	6	6	NUM
ejpam-65	676	23	,	,	PUNCT
ejpam-65	676	24	2	2	NUM
ejpam-65	676	25	-	-	SYM
ejpam-65	676	26	3	3	NUM
ejpam-65	676	27	:	:	SYM
ejpam-65	676	28	121	121	NUM
ejpam-65	676	29	-	-	SYM
ejpam-65	676	30	132	132	NUM
ejpam-65	676	31	(	(	PUNCT
ejpam-65	676	32	2003	2003	NUM
ejpam-65	676	33	)	)	PUNCT
ejpam-65	676	34	.	.	PUNCT
ejpam-65	677	1	[	[	X
ejpam-65	677	2	3	3	X
ejpam-65	677	3	]	]	X
ejpam-65	677	4	p.	p.	NOUN
ejpam-65	677	5	corsini	corsini	PROPN
ejpam-65	677	6	,	,	PUNCT
ejpam-65	677	7	prolegomena	prolegomena	NOUN
ejpam-65	677	8	of	of	ADP
ejpam-65	677	9	hypergroup	hypergroup	PROPN
ejpam-65	677	10	theory	theory	NOUN
ejpam-65	677	11	,	,	PUNCT
ejpam-65	677	12	second	second	ADJ
ejpam-65	677	13	edition	edition	NOUN
ejpam-65	677	14	,	,	PUNCT
ejpam-65	677	15	aviani	aviani	PROPN
ejpam-65	677	16	editor	editor	NOUN
ejpam-65	677	17	,	,	PUNCT
ejpam-65	677	18	1993	1993	NUM
ejpam-65	677	19	.	.	PUNCT
ejpam-65	678	1	[	[	X
ejpam-65	678	2	4	4	X
ejpam-65	678	3	]	]	X
ejpam-65	678	4	p.	p.	NOUN
ejpam-65	678	5	corsini	corsini	PROPN
ejpam-65	678	6	,	,	PUNCT
ejpam-65	678	7	v.	v.	CCONJ
ejpam-65	678	8	leoreanu	leoreanu	PROPN
ejpam-65	678	9	,	,	PUNCT
ejpam-65	678	10	applications	application	NOUN
ejpam-65	678	11	of	of	ADP
ejpam-65	678	12	hyperstructure	hyperstructure	PROPN
ejpam-65	678	13	theory	theory	NOUN
ejpam-65	678	14	,	,	PUNCT
ejpam-65	678	15	kluwer	kluwer	PROPN
ejpam-65	678	16	academic	academic	ADJ
ejpam-65	678	17	publications	publication	NOUN
ejpam-65	678	18	,	,	PUNCT
ejpam-65	678	19	2003	2003	NUM
ejpam-65	678	20	.	.	PUNCT
ejpam-65	679	1	[	[	X
ejpam-65	679	2	5	5	X
ejpam-65	679	3	]	]	PUNCT
ejpam-65	679	4	f.	f.	PROPN
ejpam-65	679	5	marty	marty	PROPN
ejpam-65	679	6	,	,	PUNCT
ejpam-65	679	7	sur	sur	PROPN
ejpam-65	679	8	une	une	PROPN
ejpam-65	679	9	generalization	generalization	NOUN
ejpam-65	679	10	de	de	X
ejpam-65	679	11	la	la	PROPN
ejpam-65	679	12	notion	notion	PROPN
ejpam-65	679	13	de	de	X
ejpam-65	679	14	groupe	groupe	PROPN
ejpam-65	679	15	,	,	PUNCT
ejpam-65	679	16	8th	8th	ADJ
ejpam-65	679	17	congress	congress	PROPN
ejpam-65	679	18	des	des	PROPN
ejpam-65	679	19	mathematiciens	mathematiciens	PROPN
ejpam-65	679	20	scandinaves	scandinaves	PROPN
ejpam-65	679	21	,	,	PUNCT
ejpam-65	679	22	stockholm	stockholm	PROPN
ejpam-65	679	23	45	45	NUM
ejpam-65	679	24	-	-	SYM
ejpam-65	679	25	49	49	NUM
ejpam-65	679	26	(	(	PUNCT
ejpam-65	679	27	1934	1934	NUM
ejpam-65	679	28	)	)	PUNCT
ejpam-65	679	29	.	.	PUNCT
ejpam-65	680	1	[	[	X
ejpam-65	680	2	6	6	NUM
ejpam-65	680	3	]	]	PUNCT
ejpam-65	680	4	m.	m.	NOUN
ejpam-65	680	5	s.	s.	PROPN
ejpam-65	680	6	tallini	tallini	PROPN
ejpam-65	680	7	,	,	PUNCT
ejpam-65	680	8	hypervector	hypervector	NOUN
ejpam-65	680	9	spaces	space	NOUN
ejpam-65	680	10	,	,	PUNCT
ejpam-65	680	11	proceedings	proceeding	NOUN
ejpam-65	680	12	of	of	ADP
ejpam-65	680	13	the	the	DET
ejpam-65	680	14	fourth	fourth	PROPN
ejpam-65	680	15	international	international	ADJ
ejpam-65	680	16	congress	congress	PROPN
ejpam-65	680	17	on	on	ADP
ejpam-65	680	18	algebraic	algebraic	PROPN
ejpam-65	680	19	hyperstructures	hyperstructure	NOUN
ejpam-65	680	20	and	and	CCONJ
ejpam-65	680	21	applications	application	NOUN
ejpam-65	680	22	,	,	PUNCT
ejpam-65	680	23	xanthi	xanthi	PROPN
ejpam-65	680	24	,	,	PUNCT
ejpam-65	680	25	greece	greece	PROPN
ejpam-65	680	26	,	,	PUNCT
ejpam-65	680	27	167	167	NUM
ejpam-65	680	28	-	-	SYM
ejpam-65	680	29	174	174	NUM
ejpam-65	680	30	(	(	PUNCT
ejpam-65	680	31	1990	1990	NUM
ejpam-65	680	32	)	)	PUNCT
ejpam-65	680	33	.	.	PUNCT
ejpam-65	681	1	[	[	X
ejpam-65	681	2	7	7	X
ejpam-65	681	3	]	]	PUNCT
ejpam-65	681	4	m.	m.	NOUN
ejpam-65	681	5	s.	s.	PROPN
ejpam-65	681	6	tallini	tallini	PROPN
ejpam-65	681	7	,	,	PUNCT
ejpam-65	681	8	weak	weak	ADJ
ejpam-65	681	9	hypervector	hypervector	NOUN
ejpam-65	681	10	spaces	space	NOUN
ejpam-65	681	11	and	and	CCONJ
ejpam-65	681	12	norms	norm	NOUN
ejpam-65	681	13	in	in	ADP
ejpam-65	681	14	such	such	ADJ
ejpam-65	681	15	spaces	space	NOUN
ejpam-65	681	16	,	,	PUNCT
ejpam-65	681	17	proceedings	proceeding	NOUN
ejpam-65	681	18	of	of	ADP
ejpam-65	681	19	the	the	DET
ejpam-65	681	20	fifth	fifth	ADJ
ejpam-65	681	21	international	international	ADJ
ejpam-65	681	22	congress	congress	PROPN
ejpam-65	681	23	on	on	ADP
ejpam-65	681	24	algebraic	algebraic	PROPN
ejpam-65	681	25	hyperstructures	hyperstructure	NOUN
ejpam-65	681	26	and	and	CCONJ
ejpam-65	681	27	applications	application	NOUN
ejpam-65	681	28	,	,	PUNCT
ejpam-65	681	29	jasi	jasi	PROPN
ejpam-65	681	30	,	,	PUNCT
ejpam-65	681	31	rumania	rumania	PROPN
ejpam-65	681	32	,	,	PUNCT
ejpam-65	681	33	hadronic	hadronic	ADJ
ejpam-65	681	34	press	press	NOUN
ejpam-65	681	35	,	,	PUNCT
ejpam-65	681	36	199	199	NUM
ejpam-65	681	37	-	-	SYM
ejpam-65	681	38	206	206	NUM
ejpam-65	681	39	(	(	PUNCT
ejpam-65	681	40	1994	1994	NUM
ejpam-65	681	41	)	)	PUNCT
ejpam-65	681	42	.	.	PUNCT
ejpam-65	682	1	[	[	X
ejpam-65	682	2	8	8	NUM
ejpam-65	682	3	]	]	PUNCT
ejpam-65	682	4	m.	m.	NOUN
ejpam-65	682	5	s.	s.	PROPN
ejpam-65	682	6	tallini	tallini	PROPN
ejpam-65	682	7	,	,	PUNCT
ejpam-65	682	8	matroidal	matroidal	ADJ
ejpam-65	682	9	hypervector	hypervector	NOUN
ejpam-65	682	10	spaces	space	NOUN
ejpam-65	682	11	,	,	PUNCT
ejpam-65	682	12	journal	journal	NOUN
ejpam-65	682	13	of	of	ADP
ejpam-65	682	14	geometry	geometry	NOUN
ejpam-65	682	15	,	,	PUNCT
ejpam-65	682	16	42	42	NUM
ejpam-65	682	17	:	:	PUNCT
ejpam-65	682	18	132	132	NUM
ejpam-65	682	19	-	-	SYM
ejpam-65	682	20	140	140	NUM
ejpam-65	682	21	(	(	PUNCT
ejpam-65	682	22	1991	1991	NUM
ejpam-65	682	23	)	)	PUNCT
ejpam-65	682	24	.	.	PUNCT
ejpam-65	683	1	references	reference	NOUN
ejpam-65	683	2	50	50	NUM
ejpam-65	683	3	[	[	X
ejpam-65	683	4	9	9	NUM
ejpam-65	683	5	]	]	X
ejpam-65	683	6	g.	g.	PROPN
ejpam-65	683	7	tallini	tallini	PROPN
ejpam-65	683	8	,	,	PUNCT
ejpam-65	683	9	dimentions	dimention	NOUN
ejpam-65	683	10	in	in	ADP
ejpam-65	683	11	multivalued	multivalued	ADJ
ejpam-65	683	12	algebraic	algebraic	ADJ
ejpam-65	683	13	structures	structure	NOUN
ejpam-65	683	14	,	,	PUNCT
ejpam-65	683	15	italian	italian	ADJ
ejpam-65	683	16	journal	journal	NOUN
ejpam-65	683	17	of	of	ADP
ejpam-65	683	18	pure	pure	ADJ
ejpam-65	683	19	and	and	CCONJ
ejpam-65	683	20	applied	applied	ADJ
ejpam-65	683	21	mathematics	mathematic	NOUN
ejpam-65	683	22	,	,	PUNCT
ejpam-65	683	23	1	1	NUM
ejpam-65	683	24	:	:	SYM
ejpam-65	683	25	51	51	NUM
ejpam-65	683	26	-	-	SYM
ejpam-65	683	27	64	64	NUM
ejpam-65	683	28	(	(	PUNCT
ejpam-65	683	29	1997	1997	NUM
ejpam-65	683	30	)	)	PUNCT
ejpam-65	683	31	.	.	PUNCT
ejpam-65	684	1	[	[	X
ejpam-65	684	2	10	10	NUM
ejpam-65	684	3	]	]	X
ejpam-65	684	4	t.	t.	NOUN
ejpam-65	684	5	vougiuklis	vougiuklis	PROPN
ejpam-65	684	6	,	,	PUNCT
ejpam-65	684	7	hyperstructures	hyperstructure	NOUN
ejpam-65	684	8	and	and	CCONJ
ejpam-65	684	9	their	their	PRON
ejpam-65	684	10	representations	representation	NOUN
ejpam-65	684	11	,	,	PUNCT
ejpam-65	684	12	hadronic	hadronic	ADJ
ejpam-65	684	13	press	press	NOUN
ejpam-65	684	14	,	,	PUNCT
ejpam-65	684	15	inc	inc	PROPN
ejpam-65	684	16	.	.	PROPN
ejpam-65	684	17	,	,	PUNCT
ejpam-65	684	18	1994	1994	NUM
ejpam-65	684	19	.	.	PUNCT
ejpam-65	685	1	[	[	X
ejpam-65	685	2	11	11	NUM
ejpam-65	685	3	]	]	PUNCT
ejpam-65	685	4	t.	t.	NOUN
ejpam-65	685	5	vougiuklis	vougiuklis	PROPN
ejpam-65	685	6	,	,	PUNCT
ejpam-65	685	7	hv	hv	PROPN
ejpam-65	685	8	-vector	-vector	PROPN
ejpam-65	685	9	spaces	space	NOUN
ejpam-65	685	10	,	,	PUNCT
ejpam-65	685	11	proceedings	proceeding	NOUN
ejpam-65	685	12	of	of	ADP
ejpam-65	685	13	the	the	DET
ejpam-65	685	14	fifth	fifth	ADJ
ejpam-65	685	15	international	international	ADJ
ejpam-65	685	16	congress	congress	PROPN
ejpam-65	685	17	on	on	ADP
ejpam-65	685	18	algebraic	algebraic	PROPN
ejpam-65	685	19	hyperstructures	hyperstructure	NOUN
ejpam-65	685	20	and	and	CCONJ
ejpam-65	685	21	applications	application	NOUN
ejpam-65	685	22	,	,	PUNCT
ejpam-65	685	23	jasi	jasi	PROPN
ejpam-65	685	24	,	,	PUNCT
ejpam-65	685	25	rumania	rumania	PROPN
ejpam-65	685	26	,	,	PUNCT
ejpam-65	685	27	hadronic	hadronic	ADJ
ejpam-65	685	28	press	press	NOUN
ejpam-65	685	29	,	,	PUNCT
ejpam-65	685	30	181	181	NUM
ejpam-65	685	31	-	-	SYM
ejpam-65	685	32	190	190	NUM
ejpam-65	685	33	(	(	PUNCT
ejpam-65	685	34	1994	1994	NUM
ejpam-65	685	35	)	)	PUNCT
ejpam-65	685	36	.	.	PUNCT
ejpam-65	686	1	[	[	X
ejpam-65	686	2	12	12	NUM
ejpam-65	686	3	]	]	PUNCT
ejpam-65	686	4	m.	m.	NOUN
ejpam-65	686	5	m.	m.	PROPN
ejpam-65	686	6	zahedi	zahedi	PROPN
ejpam-65	686	7	,	,	PUNCT
ejpam-65	686	8	r.	r.	PROPN
ejpam-65	686	9	ameri	ameri	PROPN
ejpam-65	686	10	,	,	PUNCT
ejpam-65	686	11	on	on	ADP
ejpam-65	686	12	the	the	DET
ejpam-65	686	13	prime	prime	ADJ
ejpam-65	686	14	,	,	PUNCT
ejpam-65	686	15	primary	primary	ADJ
ejpam-65	686	16	and	and	CCONJ
ejpam-65	686	17	maximal	maximal	ADJ
ejpam-65	686	18	subhypermodules	subhypermodule	NOUN
ejpam-65	686	19	,	,	PUNCT
ejpam-65	686	20	italian	italian	ADJ
ejpam-65	686	21	journal	journal	NOUN
ejpam-65	686	22	of	of	ADP
ejpam-65	686	23	pure	pure	ADJ
ejpam-65	686	24	and	and	CCONJ
ejpam-65	686	25	applied	applied	ADJ
ejpam-65	686	26	mathematics	mathematic	NOUN
ejpam-65	686	27	,	,	PUNCT
ejpam-65	686	28	5	5	NUM
ejpam-65	686	29	:	:	SYM
ejpam-65	686	30	61	61	NUM
ejpam-65	686	31	-	-	SYM
ejpam-65	686	32	80	80	NUM
ejpam-65	686	33	(	(	PUNCT
ejpam-65	686	34	1999	1999	NUM
ejpam-65	686	35	)	)	PUNCT
ejpam-65	686	36	.	.	PUNCT