id	sid	tid	token	lemma	pos
ejpam-1851	1	1	compile	compile	NOUN
ejpam-1851	1	2	/	/	SYM
ejpam-1851	1	3	output.dvi	output.dvi	NOUN
ejpam-1851	1	4	european	european	ADJ
ejpam-1851	1	5	journal	journal	NOUN
ejpam-1851	1	6	of	of	ADP
ejpam-1851	1	7	pure	pure	ADJ
ejpam-1851	1	8	and	and	CCONJ
ejpam-1851	1	9	applied	apply	VERB
ejpam-1851	1	10	mathematics	mathematic	NOUN
ejpam-1851	1	11	vol	vol	NOUN
ejpam-1851	1	12	.	.	PUNCT
ejpam-1851	2	1	7	7	NUM
ejpam-1851	2	2	,	,	PUNCT
ejpam-1851	2	3	no	no	INTJ
ejpam-1851	2	4	.	.	NOUN
ejpam-1851	2	5	4	4	NUM
ejpam-1851	2	6	,	,	PUNCT
ejpam-1851	2	7	2014	2014	NUM
ejpam-1851	2	8	,	,	PUNCT
ejpam-1851	2	9	472	472	NUM
ejpam-1851	2	10	-	-	SYM
ejpam-1851	2	11	485	485	NUM
ejpam-1851	2	12	issn	issn	PROPN
ejpam-1851	2	13	1307	1307	NUM
ejpam-1851	2	14	-	-	SYM
ejpam-1851	2	15	5543	5543	NUM
ejpam-1851	2	16	–	–	PUNCT
ejpam-1851	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1851	2	18	the	the	DET
ejpam-1851	2	19	linear	linear	ADJ
ejpam-1851	2	20	span	span	NOUN
ejpam-1851	2	21	of	of	ADP
ejpam-1851	2	22	four	four	NUM
ejpam-1851	2	23	points	point	NOUN
ejpam-1851	2	24	in	in	ADP
ejpam-1851	2	25	the	the	DET
ejpam-1851	2	26	plücker	plücker	NOUN
ejpam-1851	2	27	’s	’s	NOUN
ejpam-1851	2	28	quadric	quadric	ADJ
ejpam-1851	2	29	in	in	ADP
ejpam-1851	2	30	p5	p5	ADJ
ejpam-1851	2	31	jacqueline	jacqueline	PROPN
ejpam-1851	2	32	rojas1,∗	rojas1,∗	NOUN
ejpam-1851	2	33	,	,	PUNCT
ejpam-1851	2	34	ramón	ramón	ADJ
ejpam-1851	2	35	mendoza2	mendoza2	NOUN
ejpam-1851	2	36	1	1	NUM
ejpam-1851	2	37	ccen	ccen	PROPN
ejpam-1851	2	38	departamento	departamento	PROPN
ejpam-1851	2	39	de	de	PROPN
ejpam-1851	2	40	matemática	matemática	PROPN
ejpam-1851	2	41	ufpb	ufpb	PROPN
ejpam-1851	2	42	cidade	cidade	PROPN
ejpam-1851	2	43	universitária	universitária	PROPN
ejpam-1851	2	44	,	,	PUNCT
ejpam-1851	2	45	58051	58051	NUM
ejpam-1851	2	46	-	-	SYM
ejpam-1851	2	47	900	900	NUM
ejpam-1851	2	48	,	,	PUNCT
ejpam-1851	2	49	joão	joão	PROPN
ejpam-1851	2	50	pessoa	pessoa	NOUN
ejpam-1851	2	51	pb	pb	ADP
ejpam-1851	2	52	brasil	brasil	PROPN
ejpam-1851	2	53	2	2	NUM
ejpam-1851	2	54	ccen	ccen	NOUN
ejpam-1851	2	55	departamento	departamento	PROPN
ejpam-1851	2	56	de	de	PROPN
ejpam-1851	2	57	matemática	matemática	PROPN
ejpam-1851	2	58	ufpe	ufpe	PROPN
ejpam-1851	2	59	cidade	cidade	PROPN
ejpam-1851	2	60	universitária	universitária	PROPN
ejpam-1851	2	61	,	,	PUNCT
ejpam-1851	2	62	50740	50740	NUM
ejpam-1851	2	63	-	-	SYM
ejpam-1851	2	64	540	540	NUM
ejpam-1851	2	65	,	,	PUNCT
ejpam-1851	2	66	recife	recife	PROPN
ejpam-1851	2	67	pe	pe	PROPN
ejpam-1851	2	68	brasil	brasil	PROPN
ejpam-1851	2	69	abstract	abstract	NOUN
ejpam-1851	2	70	.	.	PUNCT
ejpam-1851	3	1	given	give	VERB
ejpam-1851	3	2	four	four	NUM
ejpam-1851	3	3	(	(	PUNCT
ejpam-1851	3	4	distinct	distinct	ADJ
ejpam-1851	3	5	)	)	PUNCT
ejpam-1851	3	6	lines	line	NOUN
ejpam-1851	3	7	ℓ1	ℓ1	NOUN
ejpam-1851	3	8	,	,	PUNCT
ejpam-1851	3	9	ℓ2	ℓ2	NOUN
ejpam-1851	3	10	,	,	PUNCT
ejpam-1851	3	11	ℓ3	ℓ3	PROPN
ejpam-1851	3	12	,	,	PUNCT
ejpam-1851	3	13	ℓ4	ℓ4	NOUN
ejpam-1851	3	14	in	in	ADP
ejpam-1851	3	15	p3	p3	PROPN
ejpam-1851	3	16	.	.	PUNCT
ejpam-1851	4	1	let	let	VERB
ejpam-1851	4	2	pi	pi	NOUN
ejpam-1851	4	3	(	(	PUNCT
ejpam-1851	4	4	i	i	NOUN
ejpam-1851	4	5	=	=	NOUN
ejpam-1851	4	6	1	1	NUM
ejpam-1851	4	7	,	,	PUNCT
ejpam-1851	4	8	.	.	PUNCT
ejpam-1851	4	9	.	.	PUNCT
ejpam-1851	5	1	.	.	PUNCT
ejpam-1851	6	1	,	,	PUNCT
ejpam-1851	6	2	4	4	X
ejpam-1851	6	3	)	)	PUNCT
ejpam-1851	6	4	be	be	VERB
ejpam-1851	6	5	the	the	DET
ejpam-1851	6	6	image	image	NOUN
ejpam-1851	6	7	of	of	ADP
ejpam-1851	6	8	ℓi	ℓi	PROPN
ejpam-1851	6	9	in	in	ADP
ejpam-1851	6	10	the	the	DET
ejpam-1851	6	11	plücker	plücker	NOUN
ejpam-1851	6	12	’s	’s	PART
ejpam-1851	6	13	quadric	quadric	ADJ
ejpam-1851	6	14	q	q	X
ejpam-1851	6	15	⊂	⊂	PUNCT
ejpam-1851	6	16	p5	p5	ADJ
ejpam-1851	6	17	under	under	ADP
ejpam-1851	6	18	the	the	DET
ejpam-1851	6	19	plücker	plücker	NOUN
ejpam-1851	6	20	embedding	embed	VERB
ejpam-1851	6	21	p	p	X
ejpam-1851	6	22	(	(	PUNCT
ejpam-1851	6	23	in	in	ADP
ejpam-1851	6	24	(	(	PUNCT
ejpam-1851	6	25	1	1	NUM
ejpam-1851	6	26	)	)	PUNCT
ejpam-1851	6	27	)	)	PUNCT
ejpam-1851	6	28	.	.	PUNCT
ejpam-1851	7	1	set	set	VERB
ejpam-1851	7	2	λ	λ	PROPN
ejpam-1851	7	3	=	=	SYM
ejpam-1851	7	4	p1	p1	PROPN
ejpam-1851	7	5	,	,	PUNCT
ejpam-1851	7	6	.	.	PUNCT
ejpam-1851	7	7	.	.	PUNCT
ejpam-1851	8	1	.	.	PUNCT
ejpam-1851	9	1	,	,	PUNCT
ejpam-1851	9	2	p4	p4	PROPN
ejpam-1851	9	3	�	�	PROPN
ejpam-1851	9	4	be	be	AUX
ejpam-1851	9	5	the	the	DET
ejpam-1851	9	6	linear	linear	ADJ
ejpam-1851	9	7	span	span	NOUN
ejpam-1851	9	8	of	of	ADP
ejpam-1851	9	9	those	those	DET
ejpam-1851	9	10	four	four	NUM
ejpam-1851	9	11	points	point	NOUN
ejpam-1851	9	12	in	in	ADP
ejpam-1851	9	13	p5	p5	PROPN
ejpam-1851	9	14	.	.	PUNCT
ejpam-1851	10	1	the	the	DET
ejpam-1851	10	2	purpose	purpose	NOUN
ejpam-1851	10	3	of	of	ADP
ejpam-1851	10	4	this	this	DET
ejpam-1851	10	5	article	article	NOUN
ejpam-1851	10	6	is	be	AUX
ejpam-1851	10	7	to	to	PART
ejpam-1851	10	8	write	write	VERB
ejpam-1851	10	9	specifically	specifically	ADV
ejpam-1851	10	10	what	what	PRON
ejpam-1851	10	11	kind	kind	NOUN
ejpam-1851	10	12	of	of	ADP
ejpam-1851	10	13	quadric	quadric	ADJ
ejpam-1851	10	14	λ∩q	λ∩q	PROPN
ejpam-1851	10	15	can	can	AUX
ejpam-1851	10	16	be	be	AUX
ejpam-1851	10	17	,	,	PUNCT
ejpam-1851	10	18	taking	take	VERB
ejpam-1851	10	19	under	under	ADP
ejpam-1851	10	20	considerations	consideration	NOUN
ejpam-1851	10	21	all	all	DET
ejpam-1851	10	22	possible	possible	ADJ
ejpam-1851	10	23	configurations	configuration	NOUN
ejpam-1851	10	24	of	of	ADP
ejpam-1851	10	25	these	these	DET
ejpam-1851	10	26	four	four	NUM
ejpam-1851	10	27	lines	line	NOUN
ejpam-1851	10	28	in	in	ADP
ejpam-1851	10	29	p3	p3	PROPN
ejpam-1851	10	30	.	.	PUNCT
ejpam-1851	11	1	in	in	ADP
ejpam-1851	11	2	particular	particular	ADJ
ejpam-1851	11	3	,	,	PUNCT
ejpam-1851	11	4	having	have	VERB
ejpam-1851	11	5	in	in	ADP
ejpam-1851	11	6	mind	mind	NOUN
ejpam-1851	11	7	the	the	DET
ejpam-1851	11	8	classical	classical	ADJ
ejpam-1851	11	9	problem	problem	NOUN
ejpam-1851	11	10	in	in	ADP
ejpam-1851	11	11	schubert	schubert	PROPN
ejpam-1851	11	12	calculus	calculus	PROPN
ejpam-1851	11	13	:	:	PUNCT
ejpam-1851	11	14	how	how	SCONJ
ejpam-1851	11	15	many	many	ADJ
ejpam-1851	11	16	lines	line	NOUN
ejpam-1851	11	17	in	in	ADP
ejpam-1851	11	18	3	3	NUM
ejpam-1851	11	19	-	-	PUNCT
ejpam-1851	11	20	space	space	NOUN
ejpam-1851	11	21	meet	meet	NOUN
ejpam-1851	11	22	four	four	NUM
ejpam-1851	11	23	given	give	VERB
ejpam-1851	11	24	lines	line	NOUN
ejpam-1851	11	25	in	in	ADP
ejpam-1851	11	26	general	general	ADJ
ejpam-1851	11	27	position	position	NOUN
ejpam-1851	11	28	?	?	PUNCT
ejpam-1851	12	1	whose	whose	DET
ejpam-1851	12	2	answer	answer	NOUN
ejpam-1851	12	3	is	be	AUX
ejpam-1851	12	4	2	2	NUM
ejpam-1851	12	5	(	(	PUNCT
ejpam-1851	12	6	see	see	VERB
ejpam-1851	12	7	p.	p.	NOUN
ejpam-1851	12	8	272	272	NUM
ejpam-1851	12	9	in	in	ADP
ejpam-1851	12	10	[	[	X
ejpam-1851	12	11	3	3	NUM
ejpam-1851	12	12	]	]	PUNCT
ejpam-1851	12	13	or	or	CCONJ
ejpam-1851	12	14	p.	p.	NOUN
ejpam-1851	12	15	746	746	NUM
ejpam-1851	12	16	in	in	ADP
ejpam-1851	12	17	[	[	X
ejpam-1851	12	18	4	4	NUM
ejpam-1851	12	19	]	]	NUM
ejpam-1851	12	20	)	)	PUNCT
ejpam-1851	12	21	.	.	PUNCT
ejpam-1851	13	1	we	we	PRON
ejpam-1851	13	2	verified	verify	VERB
ejpam-1851	13	3	that	that	SCONJ
ejpam-1851	13	4	four	four	NUM
ejpam-1851	13	5	lines	line	NOUN
ejpam-1851	13	6	in	in	ADP
ejpam-1851	13	7	p3	p3	PROPN
ejpam-1851	13	8	are	be	AUX
ejpam-1851	13	9	in	in	ADP
ejpam-1851	13	10	general	general	ADJ
ejpam-1851	13	11	position	position	NOUN
ejpam-1851	13	12	if	if	SCONJ
ejpam-1851	13	13	and	and	CCONJ
ejpam-1851	13	14	only	only	ADV
ejpam-1851	13	15	if	if	SCONJ
ejpam-1851	13	16	λ	λ	NOUN
ejpam-1851	13	17	is	be	AUX
ejpam-1851	13	18	a	a	DET
ejpam-1851	13	19	3	3	NUM
ejpam-1851	13	20	-	-	PUNCT
ejpam-1851	13	21	plane	plane	NOUN
ejpam-1851	13	22	and	and	CCONJ
ejpam-1851	13	23	λ	λ	PROPN
ejpam-1851	13	24	∩q	∩q	PROPN
ejpam-1851	13	25	is	be	AUX
ejpam-1851	13	26	an	an	DET
ejpam-1851	13	27	irreducible	irreducible	ADJ
ejpam-1851	13	28	quadric	quadric	ADJ
ejpam-1851	13	29	surface	surface	NOUN
ejpam-1851	13	30	.	.	PUNCT
ejpam-1851	14	1	in	in	ADP
ejpam-1851	14	2	fact	fact	NOUN
ejpam-1851	14	3	,	,	PUNCT
ejpam-1851	14	4	we	we	PRON
ejpam-1851	14	5	prove	prove	VERB
ejpam-1851	14	6	that	that	SCONJ
ejpam-1851	14	7	there	there	PRON
ejpam-1851	14	8	are	be	VERB
ejpam-1851	14	9	exactly	exactly	ADV
ejpam-1851	14	10	two	two	NUM
ejpam-1851	14	11	solutions	solution	NOUN
ejpam-1851	14	12	if	if	SCONJ
ejpam-1851	14	13	and	and	CCONJ
ejpam-1851	14	14	only	only	ADV
ejpam-1851	14	15	if	if	SCONJ
ejpam-1851	14	16	λ	λ	NOUN
ejpam-1851	14	17	is	be	AUX
ejpam-1851	14	18	a	a	DET
ejpam-1851	14	19	3	3	NUM
ejpam-1851	14	20	-	-	PUNCT
ejpam-1851	14	21	plane	plane	NOUN
ejpam-1851	14	22	and	and	CCONJ
ejpam-1851	14	23	λ∩q	λ∩q	PROPN
ejpam-1851	14	24	is	be	AUX
ejpam-1851	14	25	a	a	DET
ejpam-1851	14	26	nonsingular	nonsingular	ADJ
ejpam-1851	14	27	quadric	quadric	NOUN
ejpam-1851	14	28	.	.	PUNCT
ejpam-1851	15	1	2010	2010	NUM
ejpam-1851	15	2	mathematics	mathematic	NOUN
ejpam-1851	15	3	subject	subject	NOUN
ejpam-1851	15	4	classifications	classification	NOUN
ejpam-1851	15	5	:	:	PUNCT
ejpam-1851	15	6	ams	am	NOUN
ejpam-1851	15	7	14n05	14n05	NUM
ejpam-1851	15	8	,	,	PUNCT
ejpam-1851	15	9	14n20	14n20	NUM
ejpam-1851	15	10	key	key	ADJ
ejpam-1851	15	11	words	word	NOUN
ejpam-1851	15	12	and	and	CCONJ
ejpam-1851	15	13	phrases	phrase	NOUN
ejpam-1851	15	14	:	:	PUNCT
ejpam-1851	15	15	plücker	plücker	NOUN
ejpam-1851	15	16	’s	’	VERB
ejpam-1851	15	17	quadric	quadric	ADJ
ejpam-1851	15	18	,	,	PUNCT
ejpam-1851	15	19	linear	linear	ADJ
ejpam-1851	15	20	span	span	NOUN
ejpam-1851	15	21	,	,	PUNCT
ejpam-1851	15	22	4	4	NUM
ejpam-1851	15	23	-	-	PUNCT
ejpam-1851	15	24	line	line	NOUN
ejpam-1851	15	25	problem	problem	NOUN
ejpam-1851	15	26	.	.	PUNCT
ejpam-1851	16	1	1	1	X
ejpam-1851	16	2	.	.	X
ejpam-1851	16	3	introduction	introduction	NOUN
ejpam-1851	16	4	plücker	plücker	NOUN
ejpam-1851	16	5	’s	’s	PART
ejpam-1851	16	6	coordinates	coordinate	NOUN
ejpam-1851	16	7	were	be	AUX
ejpam-1851	16	8	introduced	introduce	VERB
ejpam-1851	16	9	by	by	ADP
ejpam-1851	16	10	the	the	DET
ejpam-1851	16	11	german	german	ADJ
ejpam-1851	16	12	geometer	geometer	NOUN
ejpam-1851	16	13	julius	julius	PROPN
ejpam-1851	16	14	plücker	plücker	PROPN
ejpam-1851	16	15	(	(	PUNCT
ejpam-1851	16	16	18011868	18011868	NUM
ejpam-1851	16	17	)	)	PUNCT
ejpam-1851	16	18	in	in	ADP
ejpam-1851	16	19	the	the	DET
ejpam-1851	16	20	19th	19th	ADJ
ejpam-1851	16	21	century	century	NOUN
ejpam-1851	16	22	,	,	PUNCT
ejpam-1851	16	23	as	as	ADP
ejpam-1851	16	24	a	a	DET
ejpam-1851	16	25	way	way	NOUN
ejpam-1851	16	26	to	to	PART
ejpam-1851	16	27	assign	assign	VERB
ejpam-1851	16	28	six	six	NUM
ejpam-1851	16	29	homogenous	homogenous	ADJ
ejpam-1851	16	30	coordinates	coordinate	NOUN
ejpam-1851	16	31	to	to	ADP
ejpam-1851	16	32	each	each	DET
ejpam-1851	16	33	line	line	NOUN
ejpam-1851	16	34	in	in	ADP
ejpam-1851	16	35	the	the	DET
ejpam-1851	16	36	complex	complex	ADJ
ejpam-1851	16	37	projective	projective	ADJ
ejpam-1851	16	38	3	3	NUM
ejpam-1851	16	39	-	-	PUNCT
ejpam-1851	16	40	space	space	NOUN
ejpam-1851	16	41	p3	p3	NOUN
ejpam-1851	16	42	.	.	PUNCT
ejpam-1851	17	1	since	since	SCONJ
ejpam-1851	17	2	they	they	PRON
ejpam-1851	17	3	satisfy	satisfy	VERB
ejpam-1851	17	4	a	a	DET
ejpam-1851	17	5	homogeneous	homogeneous	ADJ
ejpam-1851	17	6	quadratic	quadratic	ADJ
ejpam-1851	17	7	equation	equation	NOUN
ejpam-1851	17	8	,	,	PUNCT
ejpam-1851	17	9	it	it	PRON
ejpam-1851	17	10	follows	follow	VERB
ejpam-1851	17	11	an	an	DET
ejpam-1851	17	12	embedding	embedding	NOUN
ejpam-1851	17	13	of	of	ADP
ejpam-1851	17	14	the	the	DET
ejpam-1851	17	15	4	4	NUM
ejpam-1851	17	16	-	-	PUNCT
ejpam-1851	17	17	dimensional	dimensional	ADJ
ejpam-1851	17	18	space	space	NOUN
ejpam-1851	17	19	of	of	ADP
ejpam-1851	17	20	lines	line	NOUN
ejpam-1851	17	21	in	in	ADP
ejpam-1851	17	22	p3	p3	PROPN
ejpam-1851	17	23	(	(	PUNCT
ejpam-1851	17	24	denoted	denote	VERB
ejpam-1851	17	25	by	by	ADP
ejpam-1851	17	26	g1(p	g1(p	PROPN
ejpam-1851	17	27	3	3	NUM
ejpam-1851	17	28	)	)	PUNCT
ejpam-1851	17	29	and	and	CCONJ
ejpam-1851	17	30	called	call	VERB
ejpam-1851	17	31	grassmannian	grassmannian	PROPN
ejpam-1851	17	32	of	of	ADP
ejpam-1851	17	33	lines	line	NOUN
ejpam-1851	17	34	in	in	ADP
ejpam-1851	17	35	p3	p3	PROPN
ejpam-1851	17	36	)	)	PUNCT
ejpam-1851	17	37	onto	onto	ADP
ejpam-1851	17	38	a	a	DET
ejpam-1851	17	39	nonsingular	nonsingular	ADJ
ejpam-1851	17	40	quadric	quadric	ADJ
ejpam-1851	17	41	hypersurface	hypersurface	NOUN
ejpam-1851	17	42	q	q	PUNCT
ejpam-1851	17	43	in	in	ADP
ejpam-1851	17	44	p5	p5	ADJ
ejpam-1851	17	45	(	(	PUNCT
ejpam-1851	17	46	see	see	VERB
ejpam-1851	17	47	proposition	proposition	NOUN
ejpam-1851	17	48	3	3	NUM
ejpam-1851	17	49	)	)	PUNCT
ejpam-1851	17	50	.	.	PUNCT
ejpam-1851	18	1	thus	thus	ADV
ejpam-1851	18	2	,	,	PUNCT
ejpam-1851	18	3	reminding	remind	VERB
ejpam-1851	18	4	the	the	DET
ejpam-1851	18	5	schubert	schubert	PROPN
ejpam-1851	18	6	’s	’s	PART
ejpam-1851	18	7	classical	classical	ADJ
ejpam-1851	18	8	enumerative	enumerative	ADJ
ejpam-1851	18	9	problem	problem	NOUN
ejpam-1851	18	10	:	:	PUNCT
ejpam-1851	18	11	how	how	SCONJ
ejpam-1851	18	12	many	many	ADJ
ejpam-1851	18	13	lines	line	NOUN
ejpam-1851	18	14	in	in	ADP
ejpam-1851	18	15	3	3	NUM
ejpam-1851	18	16	-	-	PUNCT
ejpam-1851	18	17	space	space	NOUN
ejpam-1851	18	18	meet	meet	NOUN
ejpam-1851	18	19	four	four	NUM
ejpam-1851	18	20	given	give	VERB
ejpam-1851	18	21	lines	line	NOUN
ejpam-1851	18	22	in	in	ADP
ejpam-1851	18	23	general	general	ADJ
ejpam-1851	18	24	position	position	NOUN
ejpam-1851	18	25	?	?	PUNCT
ejpam-1851	19	1	whose	whose	DET
ejpam-1851	19	2	answer	answer	NOUN
ejpam-1851	19	3	can	can	AUX
ejpam-1851	19	4	be	be	AUX
ejpam-1851	19	5	found	find	VERB
ejpam-1851	19	6	in	in	ADP
ejpam-1851	19	7	many	many	ADJ
ejpam-1851	19	8	texts	text	NOUN
ejpam-1851	19	9	and	and	CCONJ
ejpam-1851	19	10	is	be	AUX
ejpam-1851	19	11	given	give	VERB
ejpam-1851	19	12	by	by	ADP
ejpam-1851	19	13	:	:	PUNCT
ejpam-1851	19	14	“	"	PUNCT
ejpam-1851	19	15	there	there	PRON
ejpam-1851	19	16	are	be	VERB
ejpam-1851	19	17	two	two	NUM
ejpam-1851	19	18	lines	line	NOUN
ejpam-1851	19	19	in	in	ADP
ejpam-1851	19	20	p3	p3	PROPN
ejpam-1851	19	21	which	which	PRON
ejpam-1851	19	22	meet	meet	VERB
ejpam-1851	19	23	4	4	NUM
ejpam-1851	19	24	given	give	VERB
ejpam-1851	19	25	lines	line	NOUN
ejpam-1851	19	26	in	in	ADP
ejpam-1851	19	27	general	general	ADJ
ejpam-1851	19	28	position	position	NOUN
ejpam-1851	19	29	”	"	PUNCT
ejpam-1851	19	30	(	(	PUNCT
ejpam-1851	19	31	see	see	VERB
ejpam-1851	19	32	example	example	NOUN
ejpam-1851	19	33	14.7.2	14.7.2	NUM
ejpam-1851	19	34	at	at	ADP
ejpam-1851	19	35	p.	p.	NOUN
ejpam-1851	19	36	272	272	NUM
ejpam-1851	19	37	in	in	ADP
ejpam-1851	19	38	[	[	X
ejpam-1851	19	39	3	3	NUM
ejpam-1851	19	40	]	]	NUM
ejpam-1851	19	41	)	)	PUNCT
ejpam-1851	19	42	.	.	PUNCT
ejpam-1851	20	1	in	in	ADP
ejpam-1851	20	2	fact	fact	NOUN
ejpam-1851	20	3	,	,	PUNCT
ejpam-1851	20	4	if	if	SCONJ
ejpam-1851	20	5	you	you	PRON
ejpam-1851	20	6	want	want	VERB
ejpam-1851	20	7	to	to	PART
ejpam-1851	20	8	know	know	VERB
ejpam-1851	20	9	an	an	DET
ejpam-1851	20	10	algorithm	algorithm	NOUN
ejpam-1851	20	11	to	to	PART
ejpam-1851	20	12	determine	determine	VERB
ejpam-1851	20	13	explicit	explicit	ADJ
ejpam-1851	20	14	solutions	solution	NOUN
ejpam-1851	20	15	,	,	PUNCT
ejpam-1851	20	16	then	then	ADV
ejpam-1851	20	17	see	see	VERB
ejpam-1851	20	18	[	[	X
ejpam-1851	20	19	7	7	NUM
ejpam-1851	20	20	]	]	PUNCT
ejpam-1851	20	21	.	.	PUNCT
ejpam-1851	21	1	on	on	ADP
ejpam-1851	21	2	the	the	DET
ejpam-1851	21	3	other	other	ADJ
ejpam-1851	21	4	hand	hand	NOUN
ejpam-1851	21	5	,	,	PUNCT
ejpam-1851	21	6	any	any	DET
ejpam-1851	21	7	beginner	beginner	NOUN
ejpam-1851	21	8	in	in	ADP
ejpam-1851	21	9	the	the	DET
ejpam-1851	21	10	art	art	NOUN
ejpam-1851	21	11	of	of	ADP
ejpam-1851	21	12	solving	solve	VERB
ejpam-1851	21	13	enumerative	enumerative	ADJ
ejpam-1851	21	14	problems	problem	NOUN
ejpam-1851	21	15	will	will	AUX
ejpam-1851	21	16	ask	ask	VERB
ejpam-1851	21	17	:	:	PUNCT
ejpam-1851	21	18	what	what	PRON
ejpam-1851	21	19	does	do	AUX
ejpam-1851	21	20	general	general	ADJ
ejpam-1851	21	21	position	position	NOUN
ejpam-1851	21	22	means	mean	VERB
ejpam-1851	21	23	?	?	PUNCT
ejpam-1851	22	1	in	in	ADP
ejpam-1851	22	2	algebraic	algebraic	ADJ
ejpam-1851	22	3	geometry	geometry	NOUN
ejpam-1851	22	4	,	,	PUNCT
ejpam-1851	22	5	general	general	ADJ
ejpam-1851	22	6	position	position	NOUN
ejpam-1851	22	7	is	be	AUX
ejpam-1851	22	8	a	a	DET
ejpam-1851	22	9	notion	notion	NOUN
ejpam-1851	22	10	of	of	ADP
ejpam-1851	22	11	genericity	genericity	NOUN
ejpam-1851	22	12	for	for	ADP
ejpam-1851	22	13	a	a	DET
ejpam-1851	22	14	set	set	NOUN
ejpam-1851	22	15	of	of	ADP
ejpam-1851	22	16	points	point	NOUN
ejpam-1851	22	17	,	,	PUNCT
ejpam-1851	22	18	or	or	CCONJ
ejpam-1851	22	19	other	other	ADJ
ejpam-1851	22	20	geometric	geometric	ADJ
ejpam-1851	22	21	objects	object	NOUN
ejpam-1851	22	22	.	.	PUNCT
ejpam-1851	23	1	it	it	PRON
ejpam-1851	23	2	means	mean	VERB
ejpam-1851	23	3	the	the	DET
ejpam-1851	23	4	∗corresponding	∗corresponde	VERB
ejpam-1851	23	5	author	author	NOUN
ejpam-1851	23	6	.	.	PUNCT
ejpam-1851	24	1	email	email	NOUN
ejpam-1851	24	2	addresses	address	NOUN
ejpam-1851	24	3	:	:	PUNCT
ejpam-1851	24	4	jacq@mat.ufpb.br	jacq@mat.ufpb.br	PROPN
ejpam-1851	24	5	(	(	PUNCT
ejpam-1851	24	6	j.	j.	PROPN
ejpam-1851	24	7	rojas	rojas	PROPN
ejpam-1851	24	8	)	)	PUNCT
ejpam-1851	24	9	,	,	PUNCT
ejpam-1851	24	10	ramon@dmat.ufpe.br	ramon@dmat.ufpe.br	PROPN
ejpam-1851	24	11	(	(	PUNCT
ejpam-1851	24	12	r.	r.	PROPN
ejpam-1851	24	13	mendoza	mendoza	PROPN
ejpam-1851	24	14	)	)	PUNCT
ejpam-1851	24	15	,	,	PUNCT
ejpam-1851	24	16	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1851	25	1	472	472	NUM
ejpam-1851	26	1	c	c	X
ejpam-1851	26	2	©	©	NOUN
ejpam-1851	26	3	2014	2014	NUM
ejpam-1851	26	4	ejpam	ejpam	NOUN
ejpam-1851	26	5	all	all	DET
ejpam-1851	26	6	rights	right	NOUN
ejpam-1851	26	7	reserved	reserve	VERB
ejpam-1851	26	8	.	.	PUNCT
ejpam-1851	27	1	j.	j.	PROPN
ejpam-1851	27	2	rojas	rojas	PROPN
ejpam-1851	27	3	,	,	PUNCT
ejpam-1851	27	4	r.	r.	PROPN
ejpam-1851	27	5	mendoza	mendoza	PROPN
ejpam-1851	27	6	/	/	SYM
ejpam-1851	27	7	eur	eur	PROPN
ejpam-1851	27	8	.	.	PUNCT
ejpam-1851	28	1	j.	j.	PROPN
ejpam-1851	28	2	pure	pure	PROPN
ejpam-1851	28	3	appl	appl	PROPN
ejpam-1851	28	4	.	.	PROPN
ejpam-1851	28	5	math	math	PROPN
ejpam-1851	28	6	,	,	PUNCT
ejpam-1851	28	7	7	7	NUM
ejpam-1851	28	8	(	(	PUNCT
ejpam-1851	28	9	2014	2014	NUM
ejpam-1851	28	10	)	)	PUNCT
ejpam-1851	28	11	,	,	PUNCT
ejpam-1851	28	12	472	472	NUM
ejpam-1851	28	13	-	-	SYM
ejpam-1851	28	14	485	485	NUM
ejpam-1851	28	15	473	473	NUM
ejpam-1851	28	16	general	general	ADJ
ejpam-1851	28	17	case	case	NOUN
ejpam-1851	28	18	situation	situation	NOUN
ejpam-1851	28	19	,	,	PUNCT
ejpam-1851	28	20	as	as	ADP
ejpam-1851	28	21	opposite	opposite	ADJ
ejpam-1851	28	22	to	to	ADP
ejpam-1851	28	23	some	some	DET
ejpam-1851	28	24	more	more	ADV
ejpam-1851	28	25	special	special	ADJ
ejpam-1851	28	26	or	or	CCONJ
ejpam-1851	28	27	coincident	coincident	ADJ
ejpam-1851	28	28	cases	case	NOUN
ejpam-1851	28	29	that	that	PRON
ejpam-1851	28	30	are	be	AUX
ejpam-1851	28	31	possible	possible	ADJ
ejpam-1851	28	32	.	.	PUNCT
ejpam-1851	29	1	its	its	PRON
ejpam-1851	29	2	precise	precise	ADJ
ejpam-1851	29	3	meaning	meaning	NOUN
ejpam-1851	29	4	differs	differ	VERB
ejpam-1851	29	5	in	in	ADP
ejpam-1851	29	6	different	different	ADJ
ejpam-1851	29	7	settings	setting	NOUN
ejpam-1851	29	8	.	.	PUNCT
ejpam-1851	30	1	for	for	ADP
ejpam-1851	30	2	example	example	NOUN
ejpam-1851	30	3	,	,	PUNCT
ejpam-1851	30	4	in	in	ADP
ejpam-1851	30	5	[	[	X
ejpam-1851	30	6	8	8	NUM
ejpam-1851	30	7	]	]	PUNCT
ejpam-1851	30	8	the	the	DET
ejpam-1851	30	9	authors	author	NOUN
ejpam-1851	30	10	imposed	impose	VERB
ejpam-1851	30	11	the	the	DET
ejpam-1851	30	12	condition	condition	NOUN
ejpam-1851	30	13	ℓi	ℓi	PROPN
ejpam-1851	30	14	∩	∩	PROPN
ejpam-1851	30	15	ℓ	ℓ	PROPN
ejpam-1851	30	16	j	j	PROPN
ejpam-1851	30	17	=	=	X
ejpam-1851	30	18	;	;	PUNCT
ejpam-1851	30	19	(	(	PUNCT
ejpam-1851	30	20	1≤	1≤	INTJ
ejpam-1851	31	1	i	i	PRON
ejpam-1851	31	2	<	<	X
ejpam-1851	31	3	j	j	PROPN
ejpam-1851	31	4	≤	≤	ADV
ejpam-1851	31	5	4	4	NUM
ejpam-1851	31	6	)	)	PUNCT
ejpam-1851	31	7	to	to	ADP
ejpam-1851	31	8	the	the	DET
ejpam-1851	31	9	four	four	NUM
ejpam-1851	31	10	given	give	VERB
ejpam-1851	31	11	lines	line	NOUN
ejpam-1851	31	12	ℓ1	ℓ1	NOUN
ejpam-1851	31	13	,	,	PUNCT
ejpam-1851	31	14	ℓ2	ℓ2	NOUN
ejpam-1851	31	15	,	,	PUNCT
ejpam-1851	31	16	ℓ3	ℓ3	PROPN
ejpam-1851	31	17	,	,	PUNCT
ejpam-1851	31	18	ℓ4	ℓ4	NOUN
ejpam-1851	31	19	in	in	ADP
ejpam-1851	31	20	p3	p3	PROPN
ejpam-1851	31	21	and	and	CCONJ
ejpam-1851	31	22	,	,	PUNCT
ejpam-1851	31	23	even	even	ADV
ejpam-1851	31	24	under	under	ADP
ejpam-1851	31	25	this	this	DET
ejpam-1851	31	26	assumption	assumption	NOUN
ejpam-1851	31	27	they	they	PRON
ejpam-1851	31	28	found	find	VERB
ejpam-1851	31	29	(	(	PUNCT
ejpam-1851	31	30	in	in	ADP
ejpam-1851	31	31	one	one	NUM
ejpam-1851	31	32	case	case	NOUN
ejpam-1851	31	33	)	)	PUNCT
ejpam-1851	31	34	infinitely	infinitely	ADV
ejpam-1851	31	35	many	many	ADJ
ejpam-1851	31	36	solutions	solution	NOUN
ejpam-1851	31	37	for	for	ADP
ejpam-1851	31	38	the	the	DET
ejpam-1851	31	39	4	4	NUM
ejpam-1851	31	40	-	-	PUNCT
ejpam-1851	31	41	lines	line	NOUN
ejpam-1851	31	42	problem	problem	NOUN
ejpam-1851	31	43	(	(	PUNCT
ejpam-1851	31	44	cf	cf	NOUN
ejpam-1851	31	45	.	.	X
ejpam-1851	31	46	3.1	3.1	NUM
ejpam-1851	31	47	in	in	ADP
ejpam-1851	31	48	the	the	DET
ejpam-1851	31	49	last	last	ADJ
ejpam-1851	31	50	subsection	subsection	NOUN
ejpam-1851	31	51	)	)	PUNCT
ejpam-1851	31	52	.	.	PUNCT
ejpam-1851	32	1	so	so	ADV
ejpam-1851	32	2	,	,	PUNCT
ejpam-1851	32	3	this	this	DET
ejpam-1851	32	4	condition	condition	NOUN
ejpam-1851	32	5	is	be	AUX
ejpam-1851	32	6	not	not	PART
ejpam-1851	32	7	enough	enough	ADJ
ejpam-1851	32	8	for	for	SCONJ
ejpam-1851	32	9	the	the	DET
ejpam-1851	32	10	four	four	NUM
ejpam-1851	32	11	given	give	VERB
ejpam-1851	32	12	lines	line	NOUN
ejpam-1851	32	13	to	to	PART
ejpam-1851	32	14	be	be	AUX
ejpam-1851	32	15	in	in	ADP
ejpam-1851	32	16	general	general	ADJ
ejpam-1851	32	17	position	position	NOUN
ejpam-1851	32	18	.	.	PUNCT
ejpam-1851	33	1	in	in	ADP
ejpam-1851	33	2	this	this	DET
ejpam-1851	33	3	article	article	NOUN
ejpam-1851	33	4	,	,	PUNCT
ejpam-1851	33	5	we	we	PRON
ejpam-1851	33	6	use	use	VERB
ejpam-1851	33	7	the	the	DET
ejpam-1851	33	8	identification	identification	NOUN
ejpam-1851	33	9	between	between	ADP
ejpam-1851	33	10	lines	line	NOUN
ejpam-1851	33	11	in	in	ADP
ejpam-1851	33	12	p3	p3	PROPN
ejpam-1851	33	13	and	and	CCONJ
ejpam-1851	33	14	points	point	NOUN
ejpam-1851	33	15	in	in	ADP
ejpam-1851	33	16	the	the	DET
ejpam-1851	33	17	quadric	quadric	ADJ
ejpam-1851	33	18	hypersurface	hypersurface	NOUN
ejpam-1851	33	19	q	q	PUNCT
ejpam-1851	33	20	to	to	PART
ejpam-1851	33	21	explain	explain	VERB
ejpam-1851	33	22	what	what	PRON
ejpam-1851	33	23	is	be	AUX
ejpam-1851	33	24	the	the	DET
ejpam-1851	33	25	precise	precise	ADJ
ejpam-1851	33	26	meaning	meaning	NOUN
ejpam-1851	33	27	of	of	ADP
ejpam-1851	33	28	general	general	ADJ
ejpam-1851	33	29	position	position	NOUN
ejpam-1851	33	30	for	for	ADP
ejpam-1851	33	31	that	that	DET
ejpam-1851	33	32	problem	problem	NOUN
ejpam-1851	33	33	.	.	PUNCT
ejpam-1851	34	1	nevertheless	nevertheless	ADV
ejpam-1851	34	2	,	,	PUNCT
ejpam-1851	34	3	the	the	DET
ejpam-1851	34	4	emphasis	emphasis	NOUN
ejpam-1851	34	5	in	in	ADP
ejpam-1851	34	6	our	our	PRON
ejpam-1851	34	7	work	work	NOUN
ejpam-1851	34	8	lies	lie	VERB
ejpam-1851	34	9	on	on	ADP
ejpam-1851	34	10	to	to	PART
ejpam-1851	34	11	take	take	VERB
ejpam-1851	34	12	under	under	ADP
ejpam-1851	34	13	considerations	consideration	NOUN
ejpam-1851	34	14	all	all	DET
ejpam-1851	34	15	possible	possible	ADJ
ejpam-1851	34	16	configurations	configuration	NOUN
ejpam-1851	34	17	of	of	ADP
ejpam-1851	34	18	these	these	DET
ejpam-1851	34	19	four	four	NUM
ejpam-1851	34	20	lines	line	NOUN
ejpam-1851	34	21	in	in	ADP
ejpam-1851	34	22	p3	p3	PROPN
ejpam-1851	34	23	and	and	CCONJ
ejpam-1851	34	24	write	write	VERB
ejpam-1851	34	25	specifically	specifically	ADV
ejpam-1851	34	26	what	what	PRON
ejpam-1851	34	27	kind	kind	NOUN
ejpam-1851	34	28	of	of	ADP
ejpam-1851	34	29	quadric	quadric	ADJ
ejpam-1851	34	30	λ∩q	λ∩q	PROPN
ejpam-1851	34	31	can	can	AUX
ejpam-1851	34	32	be	be	AUX
ejpam-1851	34	33	.	.	PUNCT
ejpam-1851	35	1	one	one	NUM
ejpam-1851	35	2	key	key	ADJ
ejpam-1851	35	3	ingredient	ingredient	NOUN
ejpam-1851	35	4	in	in	ADP
ejpam-1851	35	5	this	this	DET
ejpam-1851	35	6	work	work	NOUN
ejpam-1851	35	7	lies	lie	VERB
ejpam-1851	35	8	on	on	ADP
ejpam-1851	35	9	the	the	DET
ejpam-1851	35	10	well	well	ADV
ejpam-1851	35	11	known	know	VERB
ejpam-1851	35	12	description	description	NOUN
ejpam-1851	35	13	of	of	ADP
ejpam-1851	35	14	all	all	DET
ejpam-1851	35	15	linear	linear	ADJ
ejpam-1851	35	16	subspaces	subspace	NOUN
ejpam-1851	35	17	contained	contain	VERB
ejpam-1851	35	18	in	in	ADP
ejpam-1851	35	19	a	a	DET
ejpam-1851	35	20	quadric	quadric	ADJ
ejpam-1851	35	21	hypersurface	hypersurface	NOUN
ejpam-1851	35	22	in	in	ADP
ejpam-1851	35	23	p3	p3	PROPN
ejpam-1851	35	24	and	and	CCONJ
ejpam-1851	35	25	p5	p5	ADJ
ejpam-1851	35	26	(	(	PUNCT
ejpam-1851	35	27	see	see	VERB
ejpam-1851	35	28	subsection	subsection	NOUN
ejpam-1851	35	29	2.1	2.1	NUM
ejpam-1851	35	30	and	and	CCONJ
ejpam-1851	35	31	3.1	3.1	NUM
ejpam-1851	35	32	)	)	PUNCT
ejpam-1851	35	33	.	.	PUNCT
ejpam-1851	36	1	finally	finally	ADV
ejpam-1851	36	2	,	,	PUNCT
ejpam-1851	36	3	we	we	PRON
ejpam-1851	36	4	note	note	VERB
ejpam-1851	36	5	that	that	SCONJ
ejpam-1851	36	6	plücker	plücker	NOUN
ejpam-1851	36	7	was	be	AUX
ejpam-1851	36	8	a	a	DET
ejpam-1851	36	9	geometer	geometer	NOUN
ejpam-1851	36	10	that	that	PRON
ejpam-1851	36	11	firmly	firmly	ADV
ejpam-1851	36	12	believed	believe	VERB
ejpam-1851	36	13	in	in	ADP
ejpam-1851	36	14	the	the	DET
ejpam-1851	36	15	importance	importance	NOUN
ejpam-1851	36	16	of	of	ADP
ejpam-1851	36	17	the	the	DET
ejpam-1851	36	18	applications	application	NOUN
ejpam-1851	36	19	of	of	ADP
ejpam-1851	36	20	mathematics	mathematic	NOUN
ejpam-1851	36	21	to	to	ADP
ejpam-1851	36	22	the	the	DET
ejpam-1851	36	23	physical	physical	ADJ
ejpam-1851	36	24	sciences	science	NOUN
ejpam-1851	36	25	.	.	PUNCT
ejpam-1851	37	1	so	so	ADV
ejpam-1851	37	2	,	,	PUNCT
ejpam-1851	37	3	in	in	ADP
ejpam-1851	37	4	1847	1847	NUM
ejpam-1851	37	5	he	he	PRON
ejpam-1851	37	6	turned	turn	VERB
ejpam-1851	37	7	to	to	ADP
ejpam-1851	37	8	physics	physics	NOUN
ejpam-1851	37	9	,	,	PUNCT
ejpam-1851	37	10	accepting	accept	VERB
ejpam-1851	37	11	the	the	DET
ejpam-1851	37	12	chair	chair	NOUN
ejpam-1851	37	13	of	of	ADP
ejpam-1851	37	14	physics	physics	PROPN
ejpam-1851	37	15	at	at	ADP
ejpam-1851	37	16	bonn	bonn	PROPN
ejpam-1851	37	17	and	and	CCONJ
ejpam-1851	37	18	working	work	VERB
ejpam-1851	37	19	on	on	ADP
ejpam-1851	37	20	magnetism	magnetism	NOUN
ejpam-1851	37	21	,	,	PUNCT
ejpam-1851	37	22	electronics	electronic	NOUN
ejpam-1851	37	23	and	and	CCONJ
ejpam-1851	37	24	atomic	atomic	ADJ
ejpam-1851	37	25	physics	physic	NOUN
ejpam-1851	37	26	.	.	PUNCT
ejpam-1851	38	1	he	he	PRON
ejpam-1851	38	2	anticipated	anticipate	VERB
ejpam-1851	38	3	gustav	gustav	PROPN
ejpam-1851	38	4	kirchhoff	kirchhoff	PROPN
ejpam-1851	38	5	and	and	CCONJ
ejpam-1851	38	6	robert	robert	PROPN
ejpam-1851	38	7	wilhelm	wilhelm	PROPN
ejpam-1851	38	8	bunsen	bunsen	PROPN
ejpam-1851	38	9	in	in	ADP
ejpam-1851	38	10	announcing	announce	VERB
ejpam-1851	38	11	that	that	SCONJ
ejpam-1851	38	12	the	the	DET
ejpam-1851	38	13	lines	line	NOUN
ejpam-1851	38	14	of	of	ADP
ejpam-1851	38	15	the	the	DET
ejpam-1851	38	16	spectrum	spectrum	NOUN
ejpam-1851	38	17	were	be	AUX
ejpam-1851	38	18	characteristic	characteristic	ADJ
ejpam-1851	38	19	of	of	ADP
ejpam-1851	38	20	the	the	DET
ejpam-1851	38	21	chemical	chemical	ADJ
ejpam-1851	38	22	substance	substance	NOUN
ejpam-1851	38	23	which	which	PRON
ejpam-1851	38	24	emitted	emit	VERB
ejpam-1851	38	25	them	they	PRON
ejpam-1851	38	26	,	,	PUNCT
ejpam-1851	38	27	and	and	CCONJ
ejpam-1851	38	28	in	in	ADP
ejpam-1851	38	29	indicating	indicate	VERB
ejpam-1851	38	30	the	the	DET
ejpam-1851	38	31	value	value	NOUN
ejpam-1851	38	32	of	of	ADP
ejpam-1851	38	33	this	this	DET
ejpam-1851	38	34	discovery	discovery	NOUN
ejpam-1851	38	35	in	in	ADP
ejpam-1851	38	36	chemical	chemical	ADJ
ejpam-1851	38	37	analysis	analysis	NOUN
ejpam-1851	38	38	.	.	PUNCT
ejpam-1851	39	1	according	accord	VERB
ejpam-1851	39	2	to	to	ADP
ejpam-1851	39	3	johann	johann	PROPN
ejpam-1851	39	4	hittorf	hittorf	PROPN
ejpam-1851	39	5	he	he	PRON
ejpam-1851	39	6	was	be	AUX
ejpam-1851	39	7	the	the	DET
ejpam-1851	39	8	first	first	ADJ
ejpam-1851	39	9	who	who	PRON
ejpam-1851	39	10	saw	see	VERB
ejpam-1851	39	11	the	the	DET
ejpam-1851	39	12	three	three	NUM
ejpam-1851	39	13	lines	line	NOUN
ejpam-1851	39	14	of	of	ADP
ejpam-1851	39	15	the	the	DET
ejpam-1851	39	16	hydrogen	hydrogen	NOUN
ejpam-1851	39	17	spectrum	spectrum	NOUN
ejpam-1851	39	18	,	,	PUNCT
ejpam-1851	39	19	which	which	PRON
ejpam-1851	39	20	a	a	DET
ejpam-1851	39	21	few	few	ADJ
ejpam-1851	39	22	months	month	NOUN
ejpam-1851	39	23	after	after	SCONJ
ejpam-1851	39	24	his	his	PRON
ejpam-1851	39	25	death	death	NOUN
ejpam-1851	39	26	were	be	AUX
ejpam-1851	39	27	recognized	recognize	VERB
ejpam-1851	39	28	in	in	ADP
ejpam-1851	39	29	the	the	DET
ejpam-1851	39	30	spectrum	spectrum	NOUN
ejpam-1851	39	31	of	of	ADP
ejpam-1851	39	32	the	the	DET
ejpam-1851	39	33	solar	solar	ADJ
ejpam-1851	39	34	protuberances	protuberance	NOUN
ejpam-1851	39	35	.	.	PUNCT
ejpam-1851	40	1	2	2	X
ejpam-1851	40	2	.	.	NUM
ejpam-1851	40	3	notations	notation	NOUN
ejpam-1851	40	4	and	and	CCONJ
ejpam-1851	40	5	preliminary	preliminary	ADJ
ejpam-1851	40	6	results	result	NOUN
ejpam-1851	40	7	we	we	PRON
ejpam-1851	40	8	denote	denote	VERB
ejpam-1851	40	9	by	by	ADP
ejpam-1851	40	10	c	c	PROPN
ejpam-1851	40	11	the	the	DET
ejpam-1851	40	12	field	field	NOUN
ejpam-1851	40	13	of	of	ADP
ejpam-1851	40	14	complex	complex	ADJ
ejpam-1851	40	15	numbers	number	NOUN
ejpam-1851	40	16	.	.	PUNCT
ejpam-1851	41	1	let	let	VERB
ejpam-1851	41	2	v	v	PART
ejpam-1851	41	3	be	be	AUX
ejpam-1851	41	4	an	an	DET
ejpam-1851	41	5	n	n	ADV
ejpam-1851	41	6	-	-	PUNCT
ejpam-1851	41	7	dimensional	dimensional	ADJ
ejpam-1851	41	8	vector	vector	NOUN
ejpam-1851	41	9	space	space	NOUN
ejpam-1851	41	10	over	over	ADP
ejpam-1851	41	11	c.	c.	PROPN
ejpam-1851	41	12	denote	denote	VERB
ejpam-1851	41	13	by	by	ADP
ejpam-1851	41	14	[	[	X
ejpam-1851	41	15	v1	v1	NOUN
ejpam-1851	41	16	,	,	PUNCT
ejpam-1851	41	17	.	.	PUNCT
ejpam-1851	41	18	.	.	PUNCT
ejpam-1851	42	1	.	.	PUNCT
ejpam-1851	43	1	,	,	PUNCT
ejpam-1851	43	2	vk	vk	ADP
ejpam-1851	43	3	]	]	X
ejpam-1851	43	4	the	the	DET
ejpam-1851	43	5	subspace	subspace	NOUN
ejpam-1851	43	6	of	of	ADP
ejpam-1851	43	7	v	v	NUM
ejpam-1851	43	8	generated	generate	VERB
ejpam-1851	43	9	by	by	ADP
ejpam-1851	43	10	the	the	DET
ejpam-1851	43	11	vectors	vector	NOUN
ejpam-1851	43	12	v1	v1	NOUN
ejpam-1851	43	13	,	,	PUNCT
ejpam-1851	43	14	.	.	PUNCT
ejpam-1851	43	15	.	.	PUNCT
ejpam-1851	44	1	.	.	PUNCT
ejpam-1851	45	1	,	,	PUNCT
ejpam-1851	45	2	vk	vk	ADP
ejpam-1851	45	3	∈	∈	PROPN
ejpam-1851	45	4	v	v	NOUN
ejpam-1851	45	5	.	.	PUNCT
ejpam-1851	46	1	the	the	DET
ejpam-1851	46	2	k	k	ADJ
ejpam-1851	46	3	-	-	ADJ
ejpam-1851	46	4	grassmannian	grassmannian	ADJ
ejpam-1851	46	5	associated	associate	VERB
ejpam-1851	46	6	to	to	ADP
ejpam-1851	46	7	the	the	DET
ejpam-1851	46	8	vector	vector	NOUN
ejpam-1851	46	9	space	space	NOUN
ejpam-1851	46	10	v	v	NOUN
ejpam-1851	46	11	.	.	PUNCT
ejpam-1851	47	1	for	for	ADP
ejpam-1851	47	2	each	each	DET
ejpam-1851	47	3	integer	integer	PROPN
ejpam-1851	47	4	k	k	PROPN
ejpam-1851	47	5	,	,	PUNCT
ejpam-1851	47	6	0≤	0≤	NUM
ejpam-1851	47	7	k	k	PROPN
ejpam-1851	47	8	≤	≤	ADJ
ejpam-1851	47	9	n=	n=	ADJ
ejpam-1851	47	10	dim	dim	ADJ
ejpam-1851	47	11	v	v	NOUN
ejpam-1851	47	12	,	,	PUNCT
ejpam-1851	47	13	we	we	PRON
ejpam-1851	47	14	denote	denote	VERB
ejpam-1851	47	15	by	by	ADP
ejpam-1851	47	16	gk(v	gk(v	PROPN
ejpam-1851	47	17	)	)	PUNCT
ejpam-1851	47	18	the	the	DET
ejpam-1851	47	19	set	set	NOUN
ejpam-1851	47	20	of	of	ADP
ejpam-1851	47	21	all	all	DET
ejpam-1851	47	22	k	k	ADJ
ejpam-1851	47	23	-	-	ADJ
ejpam-1851	47	24	dimensional	dimensional	ADJ
ejpam-1851	47	25	linear	linear	ADJ
ejpam-1851	47	26	subspaces	subspace	NOUN
ejpam-1851	47	27	of	of	ADP
ejpam-1851	47	28	v	v	NOUN
ejpam-1851	47	29	and	and	CCONJ
ejpam-1851	47	30	call	call	VERB
ejpam-1851	47	31	it	it	PRON
ejpam-1851	47	32	the	the	DET
ejpam-1851	47	33	k	k	ADJ
ejpam-1851	47	34	-	-	ADJ
ejpam-1851	47	35	grassmannian	grassmannian	ADJ
ejpam-1851	47	36	associated	associate	VERB
ejpam-1851	47	37	to	to	ADP
ejpam-1851	47	38	v	v	NOUN
ejpam-1851	47	39	.	.	PUNCT
ejpam-1851	48	1	in	in	ADP
ejpam-1851	48	2	the	the	DET
ejpam-1851	48	3	particular	particular	ADJ
ejpam-1851	48	4	case	case	NOUN
ejpam-1851	48	5	k	k	NOUN
ejpam-1851	48	6	=	=	SYM
ejpam-1851	48	7	1	1	NUM
ejpam-1851	48	8	,	,	PUNCT
ejpam-1851	48	9	the	the	DET
ejpam-1851	48	10	1	1	NUM
ejpam-1851	48	11	-	-	PUNCT
ejpam-1851	48	12	grassmannian	grassmannian	ADJ
ejpam-1851	48	13	associated	associate	VERB
ejpam-1851	48	14	to	to	ADP
ejpam-1851	48	15	v	v	NOUN
ejpam-1851	48	16	it	it	PRON
ejpam-1851	48	17	is	be	AUX
ejpam-1851	48	18	also	also	ADV
ejpam-1851	48	19	called	call	VERB
ejpam-1851	48	20	projective	projective	ADJ
ejpam-1851	48	21	space	space	NOUN
ejpam-1851	48	22	associated	associate	VERB
ejpam-1851	48	23	to	to	ADP
ejpam-1851	48	24	v	v	NOUN
ejpam-1851	48	25	and	and	CCONJ
ejpam-1851	48	26	it	it	PRON
ejpam-1851	48	27	is	be	AUX
ejpam-1851	48	28	denoted	denote	VERB
ejpam-1851	48	29	by	by	ADP
ejpam-1851	48	30	p(v	p(v	NOUN
ejpam-1851	48	31	)	)	PUNCT
ejpam-1851	48	32	(	(	PUNCT
ejpam-1851	48	33	i.e.	i.e.	X
ejpam-1851	48	34	p(v	p(v	NOUN
ejpam-1851	48	35	)	)	PUNCT
ejpam-1851	48	36	:	:	PUNCT
ejpam-1851	49	1	=	=	PUNCT
ejpam-1851	49	2	g1(v	g1(v	NOUN
ejpam-1851	49	3	)	)	PUNCT
ejpam-1851	49	4	)	)	PUNCT
ejpam-1851	49	5	.	.	PUNCT
ejpam-1851	50	1	we	we	PRON
ejpam-1851	50	2	use	use	VERB
ejpam-1851	50	3	the	the	DET
ejpam-1851	50	4	notation	notation	NOUN
ejpam-1851	50	5	pn	pn	PROPN
ejpam-1851	50	6	instead	instead	ADV
ejpam-1851	50	7	of	of	ADP
ejpam-1851	50	8	p(cn+1	p(cn+1	PROPN
ejpam-1851	50	9	)	)	PUNCT
ejpam-1851	50	10	and	and	CCONJ
ejpam-1851	50	11	p	p	NOUN
ejpam-1851	50	12	=	=	PUNCT
ejpam-1851	51	1	[	[	X
ejpam-1851	51	2	a0	a0	NOUN
ejpam-1851	51	3	:	:	PUNCT
ejpam-1851	51	4	.	.	PUNCT
ejpam-1851	51	5	.	.	PUNCT
ejpam-1851	51	6	.	.	PUNCT
ejpam-1851	52	1	:	:	PUNCT
ejpam-1851	52	2	an	an	X
ejpam-1851	52	3	]	]	X
ejpam-1851	52	4	for	for	ADP
ejpam-1851	52	5	p	p	NOUN
ejpam-1851	52	6	=	=	PUNCT
ejpam-1851	53	1	[	[	X
ejpam-1851	53	2	(	(	PUNCT
ejpam-1851	53	3	a0	a0	NOUN
ejpam-1851	53	4	,	,	PUNCT
ejpam-1851	53	5	.	.	PUNCT
ejpam-1851	53	6	.	.	PUNCT
ejpam-1851	54	1	.	.	PUNCT
ejpam-1851	55	1	,	,	PUNCT
ejpam-1851	55	2	an	an	X
ejpam-1851	55	3	)	)	PUNCT
ejpam-1851	55	4	]	]	PUNCT
ejpam-1851	55	5	∈	∈	PROPN
ejpam-1851	55	6	p	p	NOUN
ejpam-1851	55	7	n	n	NOUN
ejpam-1851	55	8	,	,	PUNCT
ejpam-1851	55	9	just	just	ADV
ejpam-1851	55	10	for	for	ADP
ejpam-1851	55	11	the	the	DET
ejpam-1851	55	12	sake	sake	NOUN
ejpam-1851	55	13	of	of	ADP
ejpam-1851	55	14	simplicity	simplicity	NOUN
ejpam-1851	55	15	.	.	PUNCT
ejpam-1851	56	1	if	if	SCONJ
ejpam-1851	56	2	w	w	PROPN
ejpam-1851	56	3	∈	∈	PROPN
ejpam-1851	56	4	gk+1(v	gk+1(v	PROPN
ejpam-1851	56	5	)	)	PUNCT
ejpam-1851	56	6	then	then	ADV
ejpam-1851	56	7	p(w	p(w	PROPN
ejpam-1851	56	8	)	)	PUNCT
ejpam-1851	56	9	⊆	⊆	NUM
ejpam-1851	56	10	p(v	p(v	NOUN
ejpam-1851	56	11	)	)	PUNCT
ejpam-1851	56	12	will	will	AUX
ejpam-1851	56	13	be	be	AUX
ejpam-1851	56	14	called	call	VERB
ejpam-1851	56	15	k	k	ADJ
ejpam-1851	56	16	-	-	ADJ
ejpam-1851	56	17	linear	linear	ADJ
ejpam-1851	56	18	subspace	subspace	NOUN
ejpam-1851	56	19	of	of	ADP
ejpam-1851	56	20	p(v	p(v	NOUN
ejpam-1851	56	21	)	)	PUNCT
ejpam-1851	56	22	.	.	PUNCT
ejpam-1851	57	1	the	the	DET
ejpam-1851	57	2	set	set	NOUN
ejpam-1851	57	3	of	of	ADP
ejpam-1851	57	4	all	all	DET
ejpam-1851	57	5	klinear	klinear	ADJ
ejpam-1851	57	6	subspaces	subspace	NOUN
ejpam-1851	57	7	of	of	ADP
ejpam-1851	57	8	p(v	p(v	NOUN
ejpam-1851	57	9	)	)	PUNCT
ejpam-1851	57	10	will	will	AUX
ejpam-1851	57	11	be	be	AUX
ejpam-1851	57	12	denoted	denote	VERB
ejpam-1851	57	13	by	by	ADP
ejpam-1851	57	14	gk(p(v	gk(p(v	NOUN
ejpam-1851	57	15	)	)	PUNCT
ejpam-1851	57	16	)	)	PUNCT
ejpam-1851	57	17	,	,	PUNCT
ejpam-1851	57	18	the	the	DET
ejpam-1851	57	19	grassmannian	grassmannian	NOUN
ejpam-1851	57	20	of	of	ADP
ejpam-1851	57	21	k	k	ADJ
ejpam-1851	57	22	-	-	ADJ
ejpam-1851	57	23	linear	linear	ADJ
ejpam-1851	57	24	subspaces	subspace	NOUN
ejpam-1851	57	25	of	of	ADP
ejpam-1851	57	26	p(v	p(v	NOUN
ejpam-1851	57	27	)	)	PUNCT
ejpam-1851	57	28	.	.	PUNCT
ejpam-1851	58	1	moreover	moreover	ADV
ejpam-1851	58	2	,	,	PUNCT
ejpam-1851	58	3	we	we	PRON
ejpam-1851	58	4	shall	shall	AUX
ejpam-1851	58	5	call	call	VERB
ejpam-1851	58	6	g1(p(v	g1(p(v	NOUN
ejpam-1851	58	7	)	)	PUNCT
ejpam-1851	58	8	)	)	PUNCT
ejpam-1851	58	9	,	,	PUNCT
ejpam-1851	58	10	g2(p(v	g2(p(v	PROPN
ejpam-1851	58	11	)	)	PUNCT
ejpam-1851	58	12	)	)	PUNCT
ejpam-1851	58	13	and	and	CCONJ
ejpam-1851	58	14	gn−1(p(v	gn−1(p(v	NOUN
ejpam-1851	58	15	)	)	PUNCT
ejpam-1851	58	16	)	)	PUNCT
ejpam-1851	59	1	the	the	DET
ejpam-1851	59	2	grassmannian	grassmannian	NOUN
ejpam-1851	59	3	of	of	ADP
ejpam-1851	59	4	lines	line	NOUN
ejpam-1851	59	5	,	,	PUNCT
ejpam-1851	59	6	planes	plane	NOUN
ejpam-1851	59	7	and	and	CCONJ
ejpam-1851	59	8	hyperplanes	hyperplane	NOUN
ejpam-1851	59	9	in	in	ADP
ejpam-1851	59	10	p(v	p(v	NOUN
ejpam-1851	59	11	)	)	PUNCT
ejpam-1851	59	12	,	,	PUNCT
ejpam-1851	59	13	respectively	respectively	ADV
ejpam-1851	59	14	.	.	PUNCT
ejpam-1851	60	1	so	so	ADV
ejpam-1851	60	2	,	,	PUNCT
ejpam-1851	60	3	since	since	SCONJ
ejpam-1851	60	4	a	a	DET
ejpam-1851	60	5	line	line	NOUN
ejpam-1851	60	6	ℓ	ℓ	PROPN
ejpam-1851	60	7	in	in	ADP
ejpam-1851	60	8	p3	p3	PROPN
ejpam-1851	60	9	is	be	AUX
ejpam-1851	60	10	equal	equal	ADJ
ejpam-1851	60	11	to	to	ADP
ejpam-1851	60	12	p(w	p(w	PROPN
ejpam-1851	60	13	)	)	PUNCT
ejpam-1851	60	14	for	for	ADP
ejpam-1851	60	15	some	some	DET
ejpam-1851	60	16	w	w	PROPN
ejpam-1851	60	17	∈	∈	PROPN
ejpam-1851	60	18	g2(c	g2(c	PROPN
ejpam-1851	60	19	4	4	NUM
ejpam-1851	60	20	)	)	PUNCT
ejpam-1851	60	21	,	,	PUNCT
ejpam-1851	60	22	we	we	PRON
ejpam-1851	60	23	have	have	VERB
ejpam-1851	60	24	the	the	DET
ejpam-1851	60	25	correspondence	correspondence	NOUN
ejpam-1851	60	26	g2(c	g2(c	NOUN
ejpam-1851	60	27	4	4	NUM
ejpam-1851	60	28	)	)	PUNCT
ejpam-1851	60	29	−→	−→	NOUN
ejpam-1851	60	30	g1(p	g1(p	NOUN
ejpam-1851	60	31	3	3	X
ejpam-1851	60	32	)	)	PUNCT
ejpam-1851	60	33	w	w	PROPN
ejpam-1851	60	34	7−→	7−→	PROPN
ejpam-1851	60	35	p(w	p(w	PROPN
ejpam-1851	60	36	)	)	PUNCT
ejpam-1851	60	37	,	,	PUNCT
ejpam-1851	60	38	between	between	ADP
ejpam-1851	60	39	the	the	DET
ejpam-1851	60	40	2	2	NUM
ejpam-1851	60	41	-	-	PUNCT
ejpam-1851	60	42	grassmannian	grassmannian	ADJ
ejpam-1851	60	43	associated	associate	VERB
ejpam-1851	60	44	to	to	AUX
ejpam-1851	60	45	c4	c4	VERB
ejpam-1851	60	46	and	and	CCONJ
ejpam-1851	60	47	the	the	DET
ejpam-1851	60	48	grassmannian	grassmannian	NOUN
ejpam-1851	60	49	of	of	ADP
ejpam-1851	60	50	lines	line	NOUN
ejpam-1851	60	51	in	in	ADP
ejpam-1851	60	52	p3	p3	PROPN
ejpam-1851	60	53	.	.	PUNCT
ejpam-1851	61	1	thus	thus	ADV
ejpam-1851	61	2	,	,	PUNCT
ejpam-1851	61	3	all	all	DET
ejpam-1851	61	4	assertions	assertion	NOUN
ejpam-1851	61	5	involving	involve	VERB
ejpam-1851	61	6	g2(c	g2(c	NOUN
ejpam-1851	61	7	4	4	NUM
ejpam-1851	61	8	)	)	PUNCT
ejpam-1851	61	9	can	can	AUX
ejpam-1851	61	10	be	be	AUX
ejpam-1851	61	11	translated	translate	VERB
ejpam-1851	61	12	into	into	ADP
ejpam-1851	61	13	g1(p	g1(p	PROPN
ejpam-1851	61	14	3	3	NUM
ejpam-1851	61	15	)	)	PUNCT
ejpam-1851	61	16	.	.	PUNCT
ejpam-1851	62	1	j.	j.	PROPN
ejpam-1851	62	2	rojas	rojas	PROPN
ejpam-1851	62	3	,	,	PUNCT
ejpam-1851	62	4	r.	r.	PROPN
ejpam-1851	62	5	mendoza	mendoza	PROPN
ejpam-1851	62	6	/	/	SYM
ejpam-1851	62	7	eur	eur	PROPN
ejpam-1851	62	8	.	.	PUNCT
ejpam-1851	63	1	j.	j.	PROPN
ejpam-1851	63	2	pure	pure	PROPN
ejpam-1851	63	3	appl	appl	PROPN
ejpam-1851	63	4	.	.	PROPN
ejpam-1851	63	5	math	math	PROPN
ejpam-1851	63	6	,	,	PUNCT
ejpam-1851	63	7	7	7	NUM
ejpam-1851	63	8	(	(	PUNCT
ejpam-1851	63	9	2014	2014	NUM
ejpam-1851	63	10	)	)	PUNCT
ejpam-1851	63	11	,	,	PUNCT
ejpam-1851	63	12	472	472	NUM
ejpam-1851	63	13	-	-	SYM
ejpam-1851	63	14	485	485	NUM
ejpam-1851	63	15	474	474	NUM
ejpam-1851	63	16	next	next	ADV
ejpam-1851	63	17	we	we	PRON
ejpam-1851	63	18	introduce	introduce	VERB
ejpam-1851	63	19	the	the	DET
ejpam-1851	63	20	notion	notion	NOUN
ejpam-1851	63	21	of	of	ADP
ejpam-1851	63	22	algebraic	algebraic	PROPN
ejpam-1851	63	23	projective	projective	NOUN
ejpam-1851	63	24	set	set	VERB
ejpam-1851	63	25	in	in	ADP
ejpam-1851	63	26	pn	pn	PROPN
ejpam-1851	63	27	.	.	PUNCT
ejpam-1851	64	1	we	we	PRON
ejpam-1851	64	2	will	will	AUX
ejpam-1851	64	3	see	see	VERB
ejpam-1851	64	4	in	in	ADP
ejpam-1851	64	5	proposition	proposition	NOUN
ejpam-1851	64	6	1	1	NUM
ejpam-1851	64	7	that	that	SCONJ
ejpam-1851	64	8	k	k	ADJ
ejpam-1851	64	9	-	-	ADJ
ejpam-1851	64	10	linear	linear	ADJ
ejpam-1851	64	11	subspaces	subspace	NOUN
ejpam-1851	64	12	of	of	ADP
ejpam-1851	64	13	pn	pn	PROPN
ejpam-1851	64	14	are	be	AUX
ejpam-1851	64	15	examples	example	NOUN
ejpam-1851	64	16	of	of	ADP
ejpam-1851	64	17	algebraic	algebraic	ADJ
ejpam-1851	64	18	sets	set	NOUN
ejpam-1851	64	19	.	.	PUNCT
ejpam-1851	65	1	algebraic	algebraic	PROPN
ejpam-1851	65	2	projective	projective	ADJ
ejpam-1851	65	3	sets	set	NOUN
ejpam-1851	65	4	in	in	ADP
ejpam-1851	65	5	pn	pn	PROPN
ejpam-1851	65	6	.	.	PROPN
ejpam-1851	65	7	let	let	VERB
ejpam-1851	65	8	c[x	c[x	VERB
ejpam-1851	65	9	]	]	PUNCT
ejpam-1851	66	1	=	=	X
ejpam-1851	66	2	c[x0	c[x0	ADV
ejpam-1851	66	3	,	,	PUNCT
ejpam-1851	66	4	.	.	PUNCT
ejpam-1851	66	5	.	.	PUNCT
ejpam-1851	66	6	.	.	PUNCT
ejpam-1851	67	1	,	,	PUNCT
ejpam-1851	67	2	xn	xn	X
ejpam-1851	67	3	]	]	X
ejpam-1851	67	4	be	be	VERB
ejpam-1851	67	5	the	the	DET
ejpam-1851	67	6	polynomial	polynomial	ADJ
ejpam-1851	67	7	ring	ring	NOUN
ejpam-1851	67	8	over	over	ADP
ejpam-1851	67	9	c	c	PROPN
ejpam-1851	67	10	in	in	ADP
ejpam-1851	67	11	the	the	DET
ejpam-1851	67	12	variables	variable	NOUN
ejpam-1851	67	13	x0	x0	PROPN
ejpam-1851	67	14	,	,	PUNCT
ejpam-1851	67	15	.	.	PUNCT
ejpam-1851	67	16	.	.	PUNCT
ejpam-1851	68	1	.	.	PUNCT
ejpam-1851	69	1	,	,	PUNCT
ejpam-1851	69	2	xn	xn	X
ejpam-1851	69	3	.	.	PUNCT
ejpam-1851	70	1	now	now	ADV
ejpam-1851	70	2	,	,	PUNCT
ejpam-1851	70	3	for	for	ADP
ejpam-1851	70	4	each	each	DET
ejpam-1851	70	5	integer	integer	NOUN
ejpam-1851	70	6	d	d	PROPN
ejpam-1851	70	7	≥	≥	NOUN
ejpam-1851	70	8	0	0	NUM
ejpam-1851	70	9	consider	consider	VERB
ejpam-1851	70	10	the	the	DET
ejpam-1851	70	11	vector	vector	NOUN
ejpam-1851	70	12	subspace	subspace	NOUN
ejpam-1851	70	13	c[x	c[x	NOUN
ejpam-1851	70	14	]	]	X
ejpam-1851	70	15	d	d	X
ejpam-1851	70	16	,	,	PUNCT
ejpam-1851	70	17	generated	generate	VERB
ejpam-1851	70	18	by	by	ADP
ejpam-1851	70	19	all	all	DET
ejpam-1851	70	20	monomials	monomial	NOUN
ejpam-1851	70	21	in	in	ADP
ejpam-1851	70	22	x0	x0	PROPN
ejpam-1851	70	23	,	,	PUNCT
ejpam-1851	70	24	.	.	PUNCT
ejpam-1851	70	25	.	.	PUNCT
ejpam-1851	71	1	.	.	PUNCT
ejpam-1851	72	1	,	,	PUNCT
ejpam-1851	72	2	xn	xn	PROPN
ejpam-1851	72	3	of	of	ADP
ejpam-1851	72	4	degree	degree	NOUN
ejpam-1851	72	5	d.	d.	PROPN
ejpam-1851	72	6	each	each	DET
ejpam-1851	72	7	element	element	NOUN
ejpam-1851	72	8	in	in	ADP
ejpam-1851	72	9	c[x	c[x	NOUN
ejpam-1851	72	10	]	]	X
ejpam-1851	72	11	d	d	NOUN
ejpam-1851	72	12	will	will	AUX
ejpam-1851	72	13	be	be	AUX
ejpam-1851	72	14	called	call	VERB
ejpam-1851	72	15	an	an	DET
ejpam-1851	72	16	homogeneous	homogeneous	ADJ
ejpam-1851	72	17	polynomial	polynomial	NOUN
ejpam-1851	72	18	of	of	ADP
ejpam-1851	72	19	degree	degree	NOUN
ejpam-1851	72	20	d.	d.	PROPN
ejpam-1851	72	21	if	if	SCONJ
ejpam-1851	72	22	f	f	PROPN
ejpam-1851	72	23	∈	∈	PROPN
ejpam-1851	72	24	c[x	c[x	NOUN
ejpam-1851	72	25	]	]	X
ejpam-1851	73	1	d	d	X
ejpam-1851	73	2	,	,	PUNCT
ejpam-1851	73	3	then	then	ADV
ejpam-1851	73	4	we	we	PRON
ejpam-1851	73	5	define	define	VERB
ejpam-1851	73	6	,	,	PUNCT
ejpam-1851	73	7	z	z	PROPN
ejpam-1851	73	8	(	(	PUNCT
ejpam-1851	73	9	f	f	PROPN
ejpam-1851	73	10	)	)	PUNCT
ejpam-1851	73	11	,	,	PUNCT
ejpam-1851	73	12	the	the	DET
ejpam-1851	73	13	zero	zero	NUM
ejpam-1851	73	14	set	set	NOUN
ejpam-1851	73	15	of	of	ADP
ejpam-1851	73	16	f	f	PROPN
ejpam-1851	73	17	in	in	ADP
ejpam-1851	73	18	pn	pn	PROPN
ejpam-1851	73	19	by	by	ADP
ejpam-1851	73	20	z	z	PROPN
ejpam-1851	73	21	(	(	PUNCT
ejpam-1851	73	22	f	f	X
ejpam-1851	73	23	)	)	PUNCT
ejpam-1851	73	24	=	=	SYM
ejpam-1851	74	1	¦	¦	PROPN
ejpam-1851	75	1	[	[	X
ejpam-1851	75	2	v	v	X
ejpam-1851	75	3	]	]	X
ejpam-1851	75	4	∈	∈	PROPN
ejpam-1851	75	5	pn	pn	NOUN
ejpam-1851	75	6	|	|	ADV
ejpam-1851	75	7	f(v	f(v	VERB
ejpam-1851	75	8	)	)	PUNCT
ejpam-1851	76	1	=	=	PUNCT
ejpam-1851	76	2	0	0	PUNCT
ejpam-1851	77	1	©	©	NOUN
ejpam-1851	77	2	.	.	PUNCT
ejpam-1851	78	1	for	for	ADP
ejpam-1851	78	2	example	example	NOUN
ejpam-1851	78	3	,	,	PUNCT
ejpam-1851	78	4	if	if	SCONJ
ejpam-1851	78	5	l	l	PROPN
ejpam-1851	78	6	∈	∈	PROPN
ejpam-1851	78	7	c[x	c[x	NOUN
ejpam-1851	78	8	]	]	X
ejpam-1851	78	9	1	1	NUM
ejpam-1851	78	10	,	,	PUNCT
ejpam-1851	78	11	then	then	ADV
ejpam-1851	78	12	z	z	PROPN
ejpam-1851	78	13	(	(	PUNCT
ejpam-1851	78	14	l	l	NOUN
ejpam-1851	78	15	)	)	PUNCT
ejpam-1851	78	16	=	=	SYM
ejpam-1851	78	17	p(w	p(w	PROPN
ejpam-1851	78	18	)	)	PUNCT
ejpam-1851	78	19	with	with	ADP
ejpam-1851	78	20	w	w	NOUN
ejpam-1851	78	21	=	=	SYM
ejpam-1851	78	22	{	{	PUNCT
ejpam-1851	78	23	v	v	NUM
ejpam-1851	78	24	∈	∈	NOUN
ejpam-1851	78	25	cn+1	cn+1	VERB
ejpam-1851	78	26	|	|	NOUN
ejpam-1851	78	27	l(v	l(v	NOUN
ejpam-1851	78	28	)	)	PUNCT
ejpam-1851	78	29	=	=	PUNCT
ejpam-1851	78	30	0	0	NUM
ejpam-1851	78	31	}	}	PUNCT
ejpam-1851	78	32	.	.	PUNCT
ejpam-1851	79	1	therefore	therefore	ADV
ejpam-1851	79	2	,	,	PUNCT
ejpam-1851	79	3	z	z	PROPN
ejpam-1851	79	4	(	(	PUNCT
ejpam-1851	79	5	l	l	NOUN
ejpam-1851	79	6	)	)	PUNCT
ejpam-1851	79	7	is	be	AUX
ejpam-1851	79	8	a	a	DET
ejpam-1851	79	9	hyperplane	hyperplane	NOUN
ejpam-1851	79	10	of	of	ADP
ejpam-1851	79	11	pn	pn	PROPN
ejpam-1851	79	12	,	,	PUNCT
ejpam-1851	79	13	if	if	SCONJ
ejpam-1851	79	14	l	l	PROPN
ejpam-1851	79	15	6=	6=	ADP
ejpam-1851	79	16	0	0	NUM
ejpam-1851	79	17	else	else	ADV
ejpam-1851	79	18	z	z	PROPN
ejpam-1851	79	19	(	(	PUNCT
ejpam-1851	79	20	l	l	NOUN
ejpam-1851	79	21	)	)	PUNCT
ejpam-1851	80	1	=	=	SYM
ejpam-1851	80	2	pn	pn	PROPN
ejpam-1851	80	3	.	.	PUNCT
ejpam-1851	81	1	if	if	SCONJ
ejpam-1851	81	2	d	d	PROPN
ejpam-1851	81	3	≥	≥	NUM
ejpam-1851	81	4	1	1	NUM
ejpam-1851	81	5	,	,	PUNCT
ejpam-1851	81	6	then	then	ADV
ejpam-1851	81	7	an	an	DET
ejpam-1851	81	8	element	element	NOUN
ejpam-1851	81	9	[	[	X
ejpam-1851	81	10	f	f	X
ejpam-1851	81	11	]	]	X
ejpam-1851	81	12	in	in	ADP
ejpam-1851	81	13	the	the	DET
ejpam-1851	81	14	projectivization	projectivization	NOUN
ejpam-1851	81	15	of	of	ADP
ejpam-1851	81	16	c[x	c[x	NOUN
ejpam-1851	81	17	]	]	X
ejpam-1851	81	18	d	d	NOUN
ejpam-1851	81	19	will	will	AUX
ejpam-1851	81	20	be	be	AUX
ejpam-1851	81	21	called	call	VERB
ejpam-1851	81	22	hypersurface	hypersurface	NOUN
ejpam-1851	81	23	of	of	ADP
ejpam-1851	81	24	degree	degree	NOUN
ejpam-1851	81	25	d	d	PROPN
ejpam-1851	81	26	in	in	ADP
ejpam-1851	81	27	pn	pn	PROPN
ejpam-1851	81	28	.	.	PROPN
ejpam-1851	82	1	from	from	ADP
ejpam-1851	82	2	here	here	ADV
ejpam-1851	82	3	on	on	ADV
ejpam-1851	82	4	,	,	PUNCT
ejpam-1851	82	5	when	when	SCONJ
ejpam-1851	82	6	we	we	PRON
ejpam-1851	82	7	say	say	VERB
ejpam-1851	82	8	:	:	PUNCT
ejpam-1851	82	9	let	let	VERB
ejpam-1851	82	10	x	x	PRON
ejpam-1851	82	11	⊂	⊂	PROPN
ejpam-1851	82	12	pn	pn	AUX
ejpam-1851	82	13	be	be	AUX
ejpam-1851	82	14	the	the	DET
ejpam-1851	82	15	reduced	reduce	VERB
ejpam-1851	82	16	hypersurface	hypersurface	NOUN
ejpam-1851	82	17	defined	define	VERB
ejpam-1851	82	18	by	by	ADP
ejpam-1851	82	19	f	f	PROPN
ejpam-1851	82	20	∈	∈	PROPN
ejpam-1851	82	21	c[x	c[x	NOUN
ejpam-1851	82	22	]	]	X
ejpam-1851	83	1	d	d	X
ejpam-1851	83	2	,	,	PUNCT
ejpam-1851	83	3	that	that	PRON
ejpam-1851	83	4	means	mean	VERB
ejpam-1851	83	5	that	that	SCONJ
ejpam-1851	83	6	x	x	PROPN
ejpam-1851	83	7	=	=	SYM
ejpam-1851	83	8	z	z	X
ejpam-1851	83	9	(	(	PUNCT
ejpam-1851	83	10	f	f	X
ejpam-1851	83	11	)	)	PUNCT
ejpam-1851	84	1	⊂	⊂	PROPN
ejpam-1851	84	2	pn	pn	PROPN
ejpam-1851	84	3	and	and	CCONJ
ejpam-1851	84	4	f	f	PROPN
ejpam-1851	84	5	is	be	AUX
ejpam-1851	84	6	square	square	ADV
ejpam-1851	84	7	-	-	PUNCT
ejpam-1851	84	8	free	free	ADJ
ejpam-1851	84	9	.	.	PUNCT
ejpam-1851	85	1	a	a	DET
ejpam-1851	85	2	subset	subset	NOUN
ejpam-1851	85	3	x	x	PUNCT
ejpam-1851	85	4	of	of	ADP
ejpam-1851	85	5	pn	pn	PROPN
ejpam-1851	85	6	will	will	AUX
ejpam-1851	85	7	be	be	AUX
ejpam-1851	85	8	called	call	VERB
ejpam-1851	85	9	algebraic	algebraic	PROPN
ejpam-1851	85	10	projective	projective	NOUN
ejpam-1851	85	11	set	set	NOUN
ejpam-1851	85	12	,	,	PUNCT
ejpam-1851	85	13	if	if	SCONJ
ejpam-1851	85	14	there	there	PRON
ejpam-1851	85	15	exist	exist	VERB
ejpam-1851	85	16	homogeneous	homogeneous	ADJ
ejpam-1851	85	17	polynomials	polynomial	NOUN
ejpam-1851	85	18	f1	f1	NOUN
ejpam-1851	85	19	,	,	PUNCT
ejpam-1851	85	20	.	.	PUNCT
ejpam-1851	85	21	.	.	PUNCT
ejpam-1851	86	1	.	.	PUNCT
ejpam-1851	87	1	,	,	PUNCT
ejpam-1851	87	2	fk	fk	INTJ
ejpam-1851	87	3	in	in	ADP
ejpam-1851	87	4	c[x	c[x	NOUN
ejpam-1851	87	5	]	]	PUNCT
ejpam-1851	87	6	such	such	ADJ
ejpam-1851	87	7	that	that	SCONJ
ejpam-1851	87	8	x	x	X
ejpam-1851	87	9	=	=	SYM
ejpam-1851	87	10	z	z	X
ejpam-1851	87	11	(	(	PUNCT
ejpam-1851	87	12	f1)∩	f1)∩	X
ejpam-1851	87	13	·	·	PUNCT
ejpam-1851	87	14	·	·	PUNCT
ejpam-1851	87	15	·	·	PUNCT
ejpam-1851	88	1	∩z	∩z	INTJ
ejpam-1851	88	2	(	(	PUNCT
ejpam-1851	88	3	fk	fk	INTJ
ejpam-1851	88	4	)	)	PUNCT
ejpam-1851	88	5	.	.	PUNCT
ejpam-1851	89	1	next	next	ADV
ejpam-1851	89	2	we	we	PRON
ejpam-1851	89	3	show	show	VERB
ejpam-1851	89	4	that	that	SCONJ
ejpam-1851	89	5	any	any	DET
ejpam-1851	89	6	r	r	NOUN
ejpam-1851	89	7	-	-	PUNCT
ejpam-1851	89	8	linear	linear	NOUN
ejpam-1851	89	9	subspace	subspace	NOUN
ejpam-1851	89	10	in	in	ADP
ejpam-1851	89	11	pn	pn	PROPN
ejpam-1851	89	12	is	be	AUX
ejpam-1851	89	13	the	the	DET
ejpam-1851	89	14	intersection	intersection	NOUN
ejpam-1851	89	15	of	of	ADP
ejpam-1851	89	16	exactly	exactly	ADV
ejpam-1851	89	17	n−	n−	PROPN
ejpam-1851	89	18	r	r	NOUN
ejpam-1851	89	19	hyperplanes	hyperplane	NOUN
ejpam-1851	89	20	.	.	PUNCT
ejpam-1851	90	1	proposition	proposition	NOUN
ejpam-1851	90	2	1	1	NUM
ejpam-1851	90	3	.	.	PUNCT
ejpam-1851	91	1	let	let	VERB
ejpam-1851	91	2	λ	λ	NOUN
ejpam-1851	91	3	be	be	AUX
ejpam-1851	91	4	an	an	DET
ejpam-1851	91	5	r	r	NOUN
ejpam-1851	91	6	-	-	PUNCT
ejpam-1851	91	7	linear	linear	NOUN
ejpam-1851	91	8	subspace	subspace	NOUN
ejpam-1851	91	9	in	in	ADP
ejpam-1851	91	10	pn	pn	PROPN
ejpam-1851	91	11	with	with	ADP
ejpam-1851	91	12	n	n	PROPN
ejpam-1851	91	13	>	>	X
ejpam-1851	91	14	r.	r.	PROPN
ejpam-1851	91	15	then	then	ADV
ejpam-1851	91	16	,	,	PUNCT
ejpam-1851	91	17	there	there	PRON
ejpam-1851	91	18	are	be	VERB
ejpam-1851	91	19	exactly	exactly	ADV
ejpam-1851	91	20	n	n	ADV
ejpam-1851	91	21	−	−	NOUN
ejpam-1851	91	22	r	r	NOUN
ejpam-1851	91	23	linearly	linearly	ADV
ejpam-1851	91	24	independent	independent	ADJ
ejpam-1851	91	25	linear	linear	ADJ
ejpam-1851	91	26	forms	form	NOUN
ejpam-1851	91	27	l1	l1	PROPN
ejpam-1851	91	28	,	,	PUNCT
ejpam-1851	91	29	.	.	PUNCT
ejpam-1851	91	30	.	.	PUNCT
ejpam-1851	92	1	.	.	PUNCT
ejpam-1851	93	1	,	,	PUNCT
ejpam-1851	93	2	ln−r	ln−r	VERB
ejpam-1851	93	3	in	in	ADP
ejpam-1851	93	4	c[x	c[x	NOUN
ejpam-1851	93	5	]	]	PUNCT
ejpam-1851	93	6	such	such	ADJ
ejpam-1851	93	7	that	that	SCONJ
ejpam-1851	93	8	λ	λ	NOUN
ejpam-1851	93	9	=	=	SYM
ejpam-1851	93	10	z	z	X
ejpam-1851	93	11	(	(	PUNCT
ejpam-1851	93	12	l1)∩	l1)∩	X
ejpam-1851	93	13	·	·	PUNCT
ejpam-1851	93	14	·	·	PUNCT
ejpam-1851	93	15	·	·	PUNCT
ejpam-1851	94	1	∩z	∩z	INTJ
ejpam-1851	94	2	(	(	PUNCT
ejpam-1851	94	3	ln−r	ln−r	NOUN
ejpam-1851	94	4	)	)	PUNCT
ejpam-1851	94	5	.	.	PUNCT
ejpam-1851	95	1	proof	proof	NOUN
ejpam-1851	95	2	.	.	PUNCT
ejpam-1851	96	1	assume	assume	VERB
ejpam-1851	96	2	that	that	SCONJ
ejpam-1851	96	3	λ	λ	NOUN
ejpam-1851	96	4	=	=	SYM
ejpam-1851	96	5	p(w	p(w	PROPN
ejpam-1851	96	6	)	)	PUNCT
ejpam-1851	96	7	with	with	ADP
ejpam-1851	96	8	w	w	PROPN
ejpam-1851	96	9	∈	∈	PROPN
ejpam-1851	96	10	gr+1(c	gr+1(c	PROPN
ejpam-1851	96	11	n+1	n+1	PROPN
ejpam-1851	96	12	)	)	PUNCT
ejpam-1851	96	13	.	.	PUNCT
ejpam-1851	97	1	let	let	VERB
ejpam-1851	97	2	α	α	NOUN
ejpam-1851	97	3	=	=	SYM
ejpam-1851	97	4	{	{	PUNCT
ejpam-1851	97	5	e1	e1	PROPN
ejpam-1851	97	6	,	,	PUNCT
ejpam-1851	97	7	.	.	PUNCT
ejpam-1851	97	8	.	.	PUNCT
ejpam-1851	98	1	.	.	PUNCT
ejpam-1851	99	1	,	,	PUNCT
ejpam-1851	99	2	en+1	en+1	X
ejpam-1851	99	3	}	}	PUNCT
ejpam-1851	99	4	be	be	AUX
ejpam-1851	99	5	the	the	DET
ejpam-1851	99	6	canonical	canonical	ADJ
ejpam-1851	99	7	base	base	NOUN
ejpam-1851	99	8	of	of	ADP
ejpam-1851	99	9	cn+1	cn+1	NUM
ejpam-1851	99	10	and	and	CCONJ
ejpam-1851	99	11	let	let	VERB
ejpam-1851	99	12	β	β	X
ejpam-1851	99	13	=	=	PUNCT
ejpam-1851	99	14	{	{	PUNCT
ejpam-1851	99	15	e∗1	e∗1	ADJ
ejpam-1851	99	16	,	,	PUNCT
ejpam-1851	99	17	.	.	PUNCT
ejpam-1851	99	18	.	.	PUNCT
ejpam-1851	100	1	.	.	PUNCT
ejpam-1851	101	1	,	,	PUNCT
ejpam-1851	101	2	e∗n+1	e∗n+1	PROPN
ejpam-1851	101	3	}	}	PUNCT
ejpam-1851	101	4	be	be	VERB
ejpam-1851	101	5	the	the	DET
ejpam-1851	101	6	associated	associated	ADJ
ejpam-1851	101	7	dual	dual	ADJ
ejpam-1851	101	8	base	base	NOUN
ejpam-1851	101	9	of	of	ADP
ejpam-1851	101	10	(	(	PUNCT
ejpam-1851	101	11	cn+1)∗.	cn+1)∗.	PROPN
ejpam-1851	101	12	next	next	ADV
ejpam-1851	101	13	we	we	PRON
ejpam-1851	101	14	consider	consider	VERB
ejpam-1851	101	15	the	the	DET
ejpam-1851	101	16	linear	linear	ADJ
ejpam-1851	101	17	isomorphism	isomorphism	PROPN
ejpam-1851	101	18	ϕ	ϕ	NOUN
ejpam-1851	101	19	:	:	PUNCT
ejpam-1851	101	20	(	(	PUNCT
ejpam-1851	101	21	cn+1)∗	cn+1)∗	NOUN
ejpam-1851	101	22	→	→	SYM
ejpam-1851	101	23	c[x	c[x	NOUN
ejpam-1851	101	24	]	]	SYM
ejpam-1851	101	25	1	1	NUM
ejpam-1851	101	26	e∗	e∗	NOUN
ejpam-1851	102	1	i	i	PRON
ejpam-1851	102	2	7→	7→	NUM
ejpam-1851	102	3	x	x	X
ejpam-1851	102	4	i−1	i−1	PROPN
ejpam-1851	102	5	.	.	PUNCT
ejpam-1851	103	1	now	now	ADV
ejpam-1851	103	2	,	,	PUNCT
ejpam-1851	103	3	let	let	VERB
ejpam-1851	103	4	w	w	PROPN
ejpam-1851	103	5	0	0	PROPN
ejpam-1851	103	6	be	be	AUX
ejpam-1851	103	7	the	the	DET
ejpam-1851	103	8	annihilator	annihilator	NOUN
ejpam-1851	103	9	of	of	ADP
ejpam-1851	103	10	w	w	PROPN
ejpam-1851	103	11	,	,	PUNCT
ejpam-1851	103	12	then	then	ADV
ejpam-1851	103	13	ϕ(w	ϕ(w	PROPN
ejpam-1851	103	14	0	0	NUM
ejpam-1851	103	15	)	)	PUNCT
ejpam-1851	103	16	is	be	AUX
ejpam-1851	103	17	an	an	DET
ejpam-1851	103	18	(	(	PUNCT
ejpam-1851	103	19	n−	n−	PROPN
ejpam-1851	103	20	r)-dimensional	r)-dimensional	ADJ
ejpam-1851	103	21	linear	linear	ADJ
ejpam-1851	103	22	subspace	subspace	NOUN
ejpam-1851	103	23	of	of	ADP
ejpam-1851	103	24	c[x	c[x	NOUN
ejpam-1851	103	25	]	]	X
ejpam-1851	103	26	1	1	X
ejpam-1851	103	27	.	.	PUNCT
ejpam-1851	104	1	finally	finally	ADV
ejpam-1851	104	2	,	,	PUNCT
ejpam-1851	104	3	an	an	DET
ejpam-1851	104	4	easy	easy	ADJ
ejpam-1851	104	5	verification	verification	NOUN
ejpam-1851	104	6	show	show	NOUN
ejpam-1851	104	7	that	that	SCONJ
ejpam-1851	104	8	any	any	DET
ejpam-1851	104	9	base	base	NOUN
ejpam-1851	104	10	{	{	PUNCT
ejpam-1851	104	11	l1	l1	PROPN
ejpam-1851	104	12	,	,	PUNCT
ejpam-1851	104	13	.	.	PUNCT
ejpam-1851	104	14	.	.	PUNCT
ejpam-1851	104	15	.	.	PUNCT
ejpam-1851	105	1	,	,	PUNCT
ejpam-1851	105	2	ln−r	ln−r	NOUN
ejpam-1851	105	3	}	}	PUNCT
ejpam-1851	105	4	of	of	ADP
ejpam-1851	105	5	ϕ(w	ϕ(w	PROPN
ejpam-1851	105	6	0	0	NUM
ejpam-1851	105	7	)	)	PUNCT
ejpam-1851	105	8	verified	verify	VERB
ejpam-1851	105	9	that	that	SCONJ
ejpam-1851	105	10	λ	λ	NOUN
ejpam-1851	105	11	=	=	PUNCT
ejpam-1851	105	12	z	z	X
ejpam-1851	105	13	(	(	PUNCT
ejpam-1851	105	14	l1)∩	l1)∩	X
ejpam-1851	105	15	·	·	PUNCT
ejpam-1851	105	16	·	·	PUNCT
ejpam-1851	105	17	·	·	PUNCT
ejpam-1851	106	1	∩z	∩z	INTJ
ejpam-1851	106	2	(	(	PUNCT
ejpam-1851	106	3	ln−r	ln−r	NOUN
ejpam-1851	106	4	)	)	PUNCT
ejpam-1851	106	5	.	.	PUNCT
ejpam-1851	107	1	incidence	incidence	NOUN
ejpam-1851	107	2	of	of	ADP
ejpam-1851	107	3	r	r	NOUN
ejpam-1851	107	4	-	-	PUNCT
ejpam-1851	107	5	linear	linear	NOUN
ejpam-1851	107	6	subspaces	subspace	NOUN
ejpam-1851	107	7	with	with	ADP
ejpam-1851	107	8	reduced	reduce	VERB
ejpam-1851	107	9	hypersurfaces	hypersurface	NOUN
ejpam-1851	107	10	in	in	ADP
ejpam-1851	107	11	pn	pn	PROPN
ejpam-1851	107	12	.	.	PUNCT
ejpam-1851	108	1	the	the	DET
ejpam-1851	108	2	following	follow	VERB
ejpam-1851	108	3	proposition	proposition	NOUN
ejpam-1851	108	4	will	will	AUX
ejpam-1851	108	5	play	play	VERB
ejpam-1851	108	6	an	an	DET
ejpam-1851	108	7	important	important	ADJ
ejpam-1851	108	8	role	role	NOUN
ejpam-1851	108	9	in	in	ADP
ejpam-1851	108	10	our	our	PRON
ejpam-1851	108	11	investigations	investigation	NOUN
ejpam-1851	108	12	.	.	PUNCT
ejpam-1851	109	1	proposition	proposition	NOUN
ejpam-1851	109	2	2	2	NUM
ejpam-1851	109	3	.	.	PUNCT
ejpam-1851	110	1	let	let	VERB
ejpam-1851	110	2	z	z	PRON
ejpam-1851	110	3	⊂	⊂	PROPN
ejpam-1851	110	4	pn	pn	AUX
ejpam-1851	110	5	be	be	AUX
ejpam-1851	110	6	the	the	DET
ejpam-1851	110	7	reduced	reduce	VERB
ejpam-1851	110	8	hypersurface	hypersurface	NOUN
ejpam-1851	110	9	defined	define	VERB
ejpam-1851	110	10	by	by	ADP
ejpam-1851	110	11	the	the	DET
ejpam-1851	110	12	homogeneous	homogeneous	ADJ
ejpam-1851	110	13	polynomial	polynomial	ADJ
ejpam-1851	110	14	f	f	PROPN
ejpam-1851	110	15	of	of	ADP
ejpam-1851	110	16	degree	degree	NOUN
ejpam-1851	110	17	d	d	NOUN
ejpam-1851	110	18	and	and	CCONJ
ejpam-1851	110	19	λ	λ	PROPN
ejpam-1851	110	20	be	be	AUX
ejpam-1851	110	21	an	an	DET
ejpam-1851	110	22	r	r	NOUN
ejpam-1851	110	23	-	-	PUNCT
ejpam-1851	110	24	linear	linear	ADJ
ejpam-1851	110	25	subspace	subspace	NOUN
ejpam-1851	110	26	of	of	ADP
ejpam-1851	110	27	pn	pn	PROPN
ejpam-1851	110	28	with	with	ADP
ejpam-1851	110	29	r	r	PROPN
ejpam-1851	110	30	≥	≥	NUM
ejpam-1851	110	31	1	1	NUM
ejpam-1851	110	32	.	.	PUNCT
ejpam-1851	111	1	then	then	ADV
ejpam-1851	111	2	the	the	DET
ejpam-1851	111	3	following	follow	VERB
ejpam-1851	111	4	conditions	condition	NOUN
ejpam-1851	111	5	are	be	AUX
ejpam-1851	111	6	verified	verify	VERB
ejpam-1851	111	7	.	.	PUNCT
ejpam-1851	112	1	j.	j.	PROPN
ejpam-1851	112	2	rojas	rojas	PROPN
ejpam-1851	112	3	,	,	PUNCT
ejpam-1851	112	4	r.	r.	PROPN
ejpam-1851	112	5	mendoza	mendoza	PROPN
ejpam-1851	112	6	/	/	SYM
ejpam-1851	112	7	eur	eur	PROPN
ejpam-1851	112	8	.	.	PUNCT
ejpam-1851	113	1	j.	j.	PROPN
ejpam-1851	113	2	pure	pure	PROPN
ejpam-1851	113	3	appl	appl	PROPN
ejpam-1851	113	4	.	.	PROPN
ejpam-1851	113	5	math	math	PROPN
ejpam-1851	113	6	,	,	PUNCT
ejpam-1851	113	7	7	7	NUM
ejpam-1851	113	8	(	(	PUNCT
ejpam-1851	113	9	2014	2014	NUM
ejpam-1851	113	10	)	)	PUNCT
ejpam-1851	113	11	,	,	PUNCT
ejpam-1851	113	12	472	472	NUM
ejpam-1851	113	13	-	-	SYM
ejpam-1851	113	14	485	485	NUM
ejpam-1851	113	15	475	475	NUM
ejpam-1851	113	16	(	(	PUNCT
ejpam-1851	113	17	1	1	NUM
ejpam-1851	113	18	)	)	PUNCT
ejpam-1851	113	19	z	z	NOUN
ejpam-1851	113	20	∩λ	∩λ	PROPN
ejpam-1851	113	21	6=	6=	PROPN
ejpam-1851	113	22	;	;	PUNCT
ejpam-1851	113	23	;	;	PUNCT
ejpam-1851	113	24	(	(	PUNCT
ejpam-1851	113	25	2	2	X
ejpam-1851	113	26	)	)	PUNCT
ejpam-1851	113	27	z	z	NOUN
ejpam-1851	113	28	∩λ	∩λ	PROPN
ejpam-1851	113	29	consists	consist	VERB
ejpam-1851	113	30	of	of	ADP
ejpam-1851	113	31	infinitely	infinitely	ADV
ejpam-1851	113	32	many	many	ADJ
ejpam-1851	113	33	points	point	NOUN
ejpam-1851	113	34	,	,	PUNCT
ejpam-1851	113	35	if	if	SCONJ
ejpam-1851	113	36	λ	λ	PROPN
ejpam-1851	113	37	⊂	⊂	PROPN
ejpam-1851	113	38	z	z	PROPN
ejpam-1851	113	39	or	or	CCONJ
ejpam-1851	113	40	r	r	NOUN
ejpam-1851	113	41	≥	≥	NOUN
ejpam-1851	113	42	2	2	NUM
ejpam-1851	113	43	,	,	PUNCT
ejpam-1851	113	44	else	else	ADV
ejpam-1851	113	45	z	z	NOUN
ejpam-1851	113	46	∩λ	∩λ	PROPN
ejpam-1851	113	47	consists	consist	VERB
ejpam-1851	113	48	of	of	ADP
ejpam-1851	113	49	at	at	ADP
ejpam-1851	113	50	most	most	ADJ
ejpam-1851	113	51	d	d	NUM
ejpam-1851	113	52	points	point	NOUN
ejpam-1851	113	53	.	.	PUNCT
ejpam-1851	114	1	proof	proof	NOUN
ejpam-1851	114	2	.	.	PUNCT
ejpam-1851	115	1	to	to	PART
ejpam-1851	115	2	arrive	arrive	VERB
ejpam-1851	115	3	at	at	ADP
ejpam-1851	115	4	statements	statement	NOUN
ejpam-1851	115	5	(	(	PUNCT
ejpam-1851	115	6	1	1	NUM
ejpam-1851	115	7	)	)	PUNCT
ejpam-1851	115	8	and	and	CCONJ
ejpam-1851	115	9	first	first	ADJ
ejpam-1851	115	10	part	part	NOUN
ejpam-1851	115	11	of	of	ADP
ejpam-1851	115	12	(	(	PUNCT
ejpam-1851	115	13	2	2	X
ejpam-1851	115	14	)	)	PUNCT
ejpam-1851	115	15	have	have	VERB
ejpam-1851	115	16	in	in	ADP
ejpam-1851	115	17	mind	mind	NOUN
ejpam-1851	115	18	that	that	SCONJ
ejpam-1851	115	19	dim	dim	ADJ
ejpam-1851	115	20	z	z	NOUN
ejpam-1851	115	21	=	=	SYM
ejpam-1851	115	22	n	n	CCONJ
ejpam-1851	115	23	−	−	NUM
ejpam-1851	115	24	1	1	NUM
ejpam-1851	115	25	,	,	PUNCT
ejpam-1851	115	26	dimλ	dimλ	NOUN
ejpam-1851	115	27	=	=	SYM
ejpam-1851	115	28	r	r	NOUN
ejpam-1851	115	29	and	and	CCONJ
ejpam-1851	115	30	apply	apply	VERB
ejpam-1851	115	31	theorem	theorem	VERB
ejpam-1851	115	32	7.2	7.2	NUM
ejpam-1851	115	33	at	at	ADP
ejpam-1851	115	34	p.	p.	NOUN
ejpam-1851	115	35	48	48	NUM
ejpam-1851	115	36	in	in	ADP
ejpam-1851	115	37	[	[	X
ejpam-1851	115	38	5	5	NUM
ejpam-1851	115	39	]	]	PUNCT
ejpam-1851	115	40	.	.	PUNCT
ejpam-1851	116	1	already	already	ADV
ejpam-1851	116	2	,	,	PUNCT
ejpam-1851	116	3	the	the	DET
ejpam-1851	116	4	last	last	ADJ
ejpam-1851	116	5	part	part	NOUN
ejpam-1851	116	6	of	of	ADP
ejpam-1851	116	7	statement	statement	NOUN
ejpam-1851	116	8	(	(	PUNCT
ejpam-1851	116	9	2	2	X
ejpam-1851	116	10	)	)	PUNCT
ejpam-1851	116	11	follows	follow	VERB
ejpam-1851	116	12	easily	easily	ADV
ejpam-1851	116	13	from	from	ADP
ejpam-1851	116	14	the	the	DET
ejpam-1851	116	15	fundamental	fundamental	ADJ
ejpam-1851	116	16	theorem	theorem	NOUN
ejpam-1851	116	17	of	of	ADP
ejpam-1851	116	18	algebra	algebra	PROPN
ejpam-1851	116	19	.	.	PUNCT
ejpam-1851	117	1	projective	projective	PROPN
ejpam-1851	117	2	tangent	tangent	NOUN
ejpam-1851	117	3	space	space	NOUN
ejpam-1851	117	4	and	and	CCONJ
ejpam-1851	117	5	nonsigular	nonsigular	ADJ
ejpam-1851	117	6	reduced	reduce	VERB
ejpam-1851	117	7	hypersurface	hypersurface	NOUN
ejpam-1851	117	8	.	.	PUNCT
ejpam-1851	118	1	let	let	VERB
ejpam-1851	118	2	z	z	PRON
ejpam-1851	118	3	⊂	⊂	PROPN
ejpam-1851	118	4	pn	pn	AUX
ejpam-1851	118	5	be	be	AUX
ejpam-1851	118	6	the	the	DET
ejpam-1851	118	7	reduced	reduce	VERB
ejpam-1851	118	8	hypersurface	hypersurface	NOUN
ejpam-1851	118	9	defined	define	VERB
ejpam-1851	118	10	by	by	ADP
ejpam-1851	118	11	f	f	PROPN
ejpam-1851	118	12	∈	∈	PROPN
ejpam-1851	118	13	c[x	c[x	NOUN
ejpam-1851	118	14	]	]	X
ejpam-1851	119	1	d	d	NOUN
ejpam-1851	119	2	and	and	CCONJ
ejpam-1851	119	3	p	p	NOUN
ejpam-1851	119	4	=	=	PUNCT
ejpam-1851	120	1	[	[	X
ejpam-1851	120	2	v	v	X
ejpam-1851	120	3	]	]	X
ejpam-1851	120	4	∈	∈	PROPN
ejpam-1851	120	5	z.	z.	PROPN
ejpam-1851	120	6	let	let	VERB
ejpam-1851	120	7	f	f	PROPN
ejpam-1851	120	8	′v	′v	PROPN
ejpam-1851	120	9	:	:	PUNCT
ejpam-1851	120	10	cn+1	cn+1	VERB
ejpam-1851	120	11	−→	−→	NOUN
ejpam-1851	120	12	c	c	AUX
ejpam-1851	120	13	be	be	AUX
ejpam-1851	120	14	the	the	DET
ejpam-1851	120	15	differential	differential	NOUN
ejpam-1851	120	16	of	of	ADP
ejpam-1851	120	17	f	f	PROPN
ejpam-1851	120	18	at	at	ADP
ejpam-1851	120	19	v.	v.	ADV
ejpam-1851	120	20	we	we	PRON
ejpam-1851	120	21	define	define	VERB
ejpam-1851	120	22	the	the	DET
ejpam-1851	120	23	projective	projective	ADJ
ejpam-1851	120	24	tangent	tangent	NOUN
ejpam-1851	120	25	space	space	NOUN
ejpam-1851	120	26	,	,	PUNCT
ejpam-1851	120	27	tpz	tpz	NOUN
ejpam-1851	120	28	,	,	PUNCT
ejpam-1851	120	29	to	to	ADP
ejpam-1851	120	30	the	the	DET
ejpam-1851	120	31	hypersurface	hypersurface	NOUN
ejpam-1851	120	32	z	z	PROPN
ejpam-1851	120	33	at	at	ADP
ejpam-1851	120	34	p	p	NOUN
ejpam-1851	120	35	by	by	ADP
ejpam-1851	120	36	tpz	tpz	NOUN
ejpam-1851	120	37	=	=	SYM
ejpam-1851	120	38	p(ker(f	p(ker(f	PROPN
ejpam-1851	120	39	′v	′v	PROPN
ejpam-1851	120	40	)	)	PUNCT
ejpam-1851	120	41	)	)	PUNCT
ejpam-1851	121	1	=	=	SYM
ejpam-1851	121	2	¦	¦	PROPN
ejpam-1851	122	1	[	[	X
ejpam-1851	122	2	u0	u0	X
ejpam-1851	122	3	:	:	PUNCT
ejpam-1851	122	4	.	.	PUNCT
ejpam-1851	122	5	.	.	PUNCT
ejpam-1851	122	6	.	.	PUNCT
ejpam-1851	123	1	:	:	PUNCT
ejpam-1851	123	2	un	un	PROPN
ejpam-1851	123	3	]	]	X
ejpam-1851	123	4	∈	∈	PROPN
ejpam-1851	123	5	p	p	NOUN
ejpam-1851	123	6	n	n	CCONJ
ejpam-1851	123	7	|	|	ADV
ejpam-1851	123	8	n	n	CCONJ
ejpam-1851	123	9	∑	∑	ADP
ejpam-1851	123	10	i=0	i=0	PROPN
ejpam-1851	123	11	∂	∂	X
ejpam-1851	123	12	f	f	NOUN
ejpam-1851	123	13	∂	∂	NOUN
ejpam-1851	123	14	x	x	VERB
ejpam-1851	123	15	i	i	PROPN
ejpam-1851	123	16	(	(	PUNCT
ejpam-1851	123	17	v	v	NOUN
ejpam-1851	123	18	)	)	PUNCT
ejpam-1851	123	19	·	·	PUNCT
ejpam-1851	123	20	ui	ui	NOUN
ejpam-1851	124	1	=	=	NOUN
ejpam-1851	124	2	0	0	PUNCT
ejpam-1851	125	1	©	©	NOUN
ejpam-1851	125	2	.	.	PUNCT
ejpam-1851	126	1	now	now	ADV
ejpam-1851	126	2	,	,	PUNCT
ejpam-1851	126	3	note	note	VERB
ejpam-1851	126	4	that	that	SCONJ
ejpam-1851	126	5	:	:	PUNCT
ejpam-1851	126	6	•	•	NUM
ejpam-1851	126	7	follows	follow	VERB
ejpam-1851	126	8	from	from	ADP
ejpam-1851	126	9	the	the	DET
ejpam-1851	126	10	euler	euler	PROPN
ejpam-1851	126	11	relation	relation	NOUN
ejpam-1851	126	12	∑n	∑n	PROPN
ejpam-1851	126	13	i=0	i=0	PROPN
ejpam-1851	126	14	∂	∂	NOUN
ejpam-1851	126	15	f	f	NOUN
ejpam-1851	126	16	∂	∂	NOUN
ejpam-1851	126	17	x	x	NOUN
ejpam-1851	126	18	i	i	NOUN
ejpam-1851	126	19	·	·	PUNCT
ejpam-1851	127	1	x	x	X
ejpam-1851	127	2	i	i	PRON
ejpam-1851	127	3	=	=	PUNCT
ejpam-1851	127	4	df	df	VERB
ejpam-1851	127	5	that	that	PRON
ejpam-1851	127	6	p	p	PROPN
ejpam-1851	127	7	∈	∈	PROPN
ejpam-1851	127	8	tpz	tpz	NOUN
ejpam-1851	127	9	.	.	PUNCT
ejpam-1851	128	1	•	•	NUM
ejpam-1851	128	2	tpz	tpz	NOUN
ejpam-1851	128	3	is	be	AUX
ejpam-1851	128	4	a	a	DET
ejpam-1851	128	5	hyperplane	hyperplane	NOUN
ejpam-1851	128	6	in	in	ADP
ejpam-1851	128	7	pn	pn	PROPN
ejpam-1851	128	8	if	if	SCONJ
ejpam-1851	129	1	and	and	CCONJ
ejpam-1851	129	2	only	only	ADV
ejpam-1851	129	3	if	if	SCONJ
ejpam-1851	129	4	p	p	PROPN
ejpam-1851	129	5	6∈	6∈	PROPN
ejpam-1851	129	6	∩n	∩n	PROPN
ejpam-1851	129	7	i=0z	i=0z	PROPN
ejpam-1851	129	8	(	(	PUNCT
ejpam-1851	129	9	∂	∂	NUM
ejpam-1851	129	10	f	f	PROPN
ejpam-1851	129	11	∂	∂	NOUN
ejpam-1851	129	12	x	x	NOUN
ejpam-1851	129	13	i	i	PROPN
ejpam-1851	129	14	)	)	PUNCT
ejpam-1851	129	15	.	.	PUNCT
ejpam-1851	130	1	a	a	DET
ejpam-1851	130	2	point	point	NOUN
ejpam-1851	130	3	p	p	X
ejpam-1851	130	4	∈	∈	PROPN
ejpam-1851	130	5	z	z	NOUN
ejpam-1851	130	6	satisfying	satisfy	VERB
ejpam-1851	130	7	the	the	DET
ejpam-1851	130	8	last	last	ADJ
ejpam-1851	130	9	condition	condition	NOUN
ejpam-1851	130	10	above	above	ADV
ejpam-1851	130	11	will	will	AUX
ejpam-1851	130	12	be	be	AUX
ejpam-1851	130	13	called	call	VERB
ejpam-1851	130	14	a	a	DET
ejpam-1851	130	15	nonsingular	nonsingular	ADJ
ejpam-1851	130	16	point	point	NOUN
ejpam-1851	130	17	of	of	ADP
ejpam-1851	130	18	z	z	PROPN
ejpam-1851	130	19	,	,	PUNCT
ejpam-1851	130	20	else	else	ADV
ejpam-1851	130	21	p	p	NOUN
ejpam-1851	130	22	will	will	AUX
ejpam-1851	130	23	be	be	AUX
ejpam-1851	130	24	called	call	VERB
ejpam-1851	130	25	a	a	DET
ejpam-1851	130	26	singular	singular	ADJ
ejpam-1851	130	27	point	point	NOUN
ejpam-1851	130	28	of	of	ADP
ejpam-1851	130	29	z	z	NOUN
ejpam-1851	130	30	.	.	PUNCT
ejpam-1851	131	1	if	if	SCONJ
ejpam-1851	131	2	all	all	DET
ejpam-1851	131	3	the	the	DET
ejpam-1851	131	4	points	point	NOUN
ejpam-1851	131	5	in	in	ADP
ejpam-1851	131	6	z	z	NOUN
ejpam-1851	131	7	are	be	AUX
ejpam-1851	131	8	nonsingular	nonsingular	ADJ
ejpam-1851	131	9	,	,	PUNCT
ejpam-1851	131	10	then	then	ADV
ejpam-1851	131	11	z	z	NOUN
ejpam-1851	131	12	will	will	AUX
ejpam-1851	131	13	be	be	AUX
ejpam-1851	131	14	called	call	VERB
ejpam-1851	131	15	a	a	DET
ejpam-1851	131	16	nonsingular	nonsingular	ADJ
ejpam-1851	131	17	hypersurface	hypersurface	NOUN
ejpam-1851	131	18	in	in	ADP
ejpam-1851	131	19	pn	pn	PROPN
ejpam-1851	131	20	.	.	PROPN
ejpam-1851	132	1	for	for	ADP
ejpam-1851	132	2	example	example	NOUN
ejpam-1851	132	3	,	,	PUNCT
ejpam-1851	132	4	after	after	ADP
ejpam-1851	132	5	a	a	DET
ejpam-1851	132	6	linear	linear	ADJ
ejpam-1851	132	7	change	change	NOUN
ejpam-1851	132	8	of	of	ADP
ejpam-1851	132	9	coordinates	coordinate	NOUN
ejpam-1851	132	10	(	(	PUNCT
ejpam-1851	132	11	see	see	VERB
ejpam-1851	132	12	theorem	theorem	NOUN
ejpam-1851	132	13	4	4	NUM
ejpam-1851	132	14	at	at	ADP
ejpam-1851	132	15	p.	p.	NOUN
ejpam-1851	132	16	411	411	NUM
ejpam-1851	132	17	in	in	ADP
ejpam-1851	132	18	[	[	X
ejpam-1851	132	19	2	2	NUM
ejpam-1851	132	20	]	]	PUNCT
ejpam-1851	132	21	)	)	PUNCT
ejpam-1851	132	22	we	we	PRON
ejpam-1851	132	23	concluded	conclude	VERB
ejpam-1851	132	24	that	that	SCONJ
ejpam-1851	132	25	z	z	NOUN
ejpam-1851	132	26	(	(	PUNCT
ejpam-1851	132	27	x	x	SYM
ejpam-1851	132	28	2	2	NUM
ejpam-1851	132	29	0+x	0+x	NUM
ejpam-1851	132	30	2	2	NUM
ejpam-1851	132	31	1	1	NUM
ejpam-1851	132	32	+	+	NOUN
ejpam-1851	132	33	x	x	SYM
ejpam-1851	132	34	2	2	NUM
ejpam-1851	132	35	2	2	NUM
ejpam-1851	132	36	+	+	NOUN
ejpam-1851	132	37	x	x	PROPN
ejpam-1851	132	38	2	2	NUM
ejpam-1851	132	39	3	3	NUM
ejpam-1851	132	40	)	)	PUNCT
ejpam-1851	132	41	is	be	AUX
ejpam-1851	132	42	the	the	DET
ejpam-1851	132	43	unique	unique	ADJ
ejpam-1851	132	44	nonsingular	nonsingular	ADJ
ejpam-1851	132	45	quadric	quadric	ADJ
ejpam-1851	132	46	surface	surface	NOUN
ejpam-1851	132	47	in	in	ADP
ejpam-1851	132	48	p3	p3	PROPN
ejpam-1851	132	49	,	,	PUNCT
ejpam-1851	132	50	whereas	whereas	SCONJ
ejpam-1851	132	51	the	the	DET
ejpam-1851	132	52	singular	singular	NOUN
ejpam-1851	132	53	reduced	reduce	VERB
ejpam-1851	132	54	quadric	quadric	ADJ
ejpam-1851	132	55	surfaces	surface	NOUN
ejpam-1851	132	56	in	in	ADP
ejpam-1851	132	57	p3	p3	PROPN
ejpam-1851	132	58	correspond	correspond	VERB
ejpam-1851	132	59	either	either	CCONJ
ejpam-1851	132	60	to	to	ADP
ejpam-1851	132	61	the	the	DET
ejpam-1851	132	62	union	union	NOUN
ejpam-1851	132	63	of	of	ADP
ejpam-1851	132	64	two	two	NUM
ejpam-1851	132	65	planes	plane	NOUN
ejpam-1851	132	66	or	or	CCONJ
ejpam-1851	132	67	a	a	DET
ejpam-1851	132	68	quadric	quadric	ADJ
ejpam-1851	132	69	cone	cone	NOUN
ejpam-1851	132	70	.	.	PUNCT
ejpam-1851	133	1	moreover	moreover	ADV
ejpam-1851	133	2	,	,	PUNCT
ejpam-1851	133	3	it	it	PRON
ejpam-1851	133	4	is	be	AUX
ejpam-1851	133	5	straightforward	straightforward	ADJ
ejpam-1851	133	6	to	to	PART
ejpam-1851	133	7	show	show	VERB
ejpam-1851	133	8	that	that	SCONJ
ejpam-1851	133	9	the	the	DET
ejpam-1851	133	10	vertex	vertex	NOUN
ejpam-1851	133	11	of	of	ADP
ejpam-1851	133	12	a	a	DET
ejpam-1851	133	13	quadric	quadric	ADJ
ejpam-1851	133	14	cone	cone	NOUN
ejpam-1851	133	15	is	be	AUX
ejpam-1851	133	16	its	its	PRON
ejpam-1851	133	17	unique	unique	ADJ
ejpam-1851	133	18	singular	singular	ADJ
ejpam-1851	133	19	point	point	NOUN
ejpam-1851	133	20	.	.	PUNCT
ejpam-1851	134	1	in	in	ADP
ejpam-1851	134	2	the	the	DET
ejpam-1851	134	3	case	case	NOUN
ejpam-1851	134	4	of	of	ADP
ejpam-1851	134	5	the	the	DET
ejpam-1851	134	6	union	union	NOUN
ejpam-1851	134	7	of	of	ADP
ejpam-1851	134	8	two	two	NUM
ejpam-1851	134	9	(	(	PUNCT
ejpam-1851	134	10	distinct	distinct	ADJ
ejpam-1851	134	11	)	)	PUNCT
ejpam-1851	134	12	planes	plane	NOUN
ejpam-1851	134	13	,	,	PUNCT
ejpam-1851	134	14	it	it	PRON
ejpam-1851	134	15	is	be	AUX
ejpam-1851	134	16	verified	verify	VERB
ejpam-1851	134	17	that	that	SCONJ
ejpam-1851	134	18	the	the	DET
ejpam-1851	134	19	points	point	NOUN
ejpam-1851	134	20	in	in	ADP
ejpam-1851	134	21	the	the	DET
ejpam-1851	134	22	line	line	NOUN
ejpam-1851	134	23	where	where	SCONJ
ejpam-1851	134	24	the	the	DET
ejpam-1851	134	25	two	two	NUM
ejpam-1851	134	26	planes	plane	NOUN
ejpam-1851	134	27	meet	meet	NOUN
ejpam-1851	134	28	are	be	AUX
ejpam-1851	134	29	their	their	PRON
ejpam-1851	134	30	singular	singular	ADJ
ejpam-1851	134	31	points	point	NOUN
ejpam-1851	134	32	.	.	PUNCT
ejpam-1851	135	1	2.1	2.1	NUM
ejpam-1851	135	2	.	.	PUNCT
ejpam-1851	135	3	lines	line	NOUN
ejpam-1851	135	4	on	on	ADP
ejpam-1851	135	5	reduced	reduced	ADJ
ejpam-1851	135	6	quadrics	quadric	NOUN
ejpam-1851	135	7	surfaces	surface	NOUN
ejpam-1851	135	8	in	in	ADP
ejpam-1851	135	9	p3	p3	PROPN
ejpam-1851	135	10	next	next	ADV
ejpam-1851	135	11	,	,	PUNCT
ejpam-1851	135	12	we	we	PRON
ejpam-1851	135	13	present	present	VERB
ejpam-1851	135	14	some	some	DET
ejpam-1851	135	15	observations	observation	NOUN
ejpam-1851	135	16	about	about	ADP
ejpam-1851	135	17	lines	line	NOUN
ejpam-1851	135	18	contained	contain	VERB
ejpam-1851	135	19	on	on	ADP
ejpam-1851	135	20	reduced	reduced	ADJ
ejpam-1851	135	21	quadrics	quadric	NOUN
ejpam-1851	135	22	surfaces	surface	NOUN
ejpam-1851	135	23	in	in	ADP
ejpam-1851	135	24	p	p	PROPN
ejpam-1851	135	25	3	3	NUM
ejpam-1851	135	26	.	.	NOUN
ejpam-1851	135	27	•	•	NUM
ejpam-1851	135	28	union	union	NOUN
ejpam-1851	135	29	of	of	ADP
ejpam-1851	135	30	two	two	NUM
ejpam-1851	135	31	planes	plane	NOUN
ejpam-1851	135	32	.	.	PUNCT
ejpam-1851	136	1	of	of	ADP
ejpam-1851	136	2	course	course	ADV
ejpam-1851	136	3	any	any	DET
ejpam-1851	136	4	line	line	NOUN
ejpam-1851	136	5	in	in	ADP
ejpam-1851	136	6	this	this	DET
ejpam-1851	136	7	surface	surface	NOUN
ejpam-1851	136	8	will	will	AUX
ejpam-1851	136	9	be	be	AUX
ejpam-1851	136	10	contained	contain	VERB
ejpam-1851	136	11	in	in	ADP
ejpam-1851	136	12	one	one	NUM
ejpam-1851	136	13	of	of	ADP
ejpam-1851	136	14	the	the	DET
ejpam-1851	136	15	planes	plane	NOUN
ejpam-1851	136	16	.	.	PUNCT
ejpam-1851	137	1	thus	thus	ADV
ejpam-1851	137	2	this	this	DET
ejpam-1851	137	3	surface	surface	NOUN
ejpam-1851	137	4	could	could	AUX
ejpam-1851	137	5	have	have	VERB
ejpam-1851	137	6	at	at	ADP
ejpam-1851	137	7	most	most	ADJ
ejpam-1851	137	8	two	two	NUM
ejpam-1851	137	9	disjoint	disjoint	ADJ
ejpam-1851	137	10	lines	line	NOUN
ejpam-1851	137	11	.	.	PUNCT
ejpam-1851	138	1	•	•	NUM
ejpam-1851	138	2	quadric	quadric	ADJ
ejpam-1851	138	3	cone	cone	NOUN
ejpam-1851	138	4	.	.	PUNCT
ejpam-1851	139	1	any	any	DET
ejpam-1851	139	2	line	line	NOUN
ejpam-1851	139	3	in	in	ADP
ejpam-1851	139	4	this	this	DET
ejpam-1851	139	5	surface	surface	NOUN
ejpam-1851	139	6	passes	pass	VERB
ejpam-1851	139	7	through	through	ADP
ejpam-1851	139	8	its	its	PRON
ejpam-1851	139	9	vertex	vertex	NOUN
ejpam-1851	139	10	.	.	PUNCT
ejpam-1851	140	1	thus	thus	ADV
ejpam-1851	140	2	this	this	DET
ejpam-1851	140	3	surface	surface	NOUN
ejpam-1851	140	4	does	do	AUX
ejpam-1851	140	5	not	not	PART
ejpam-1851	140	6	contain	contain	VERB
ejpam-1851	140	7	disjoint	disjoint	NOUN
ejpam-1851	140	8	lines	line	NOUN
ejpam-1851	140	9	.	.	PUNCT
ejpam-1851	141	1	•	•	NUM
ejpam-1851	141	2	the	the	DET
ejpam-1851	141	3	nonsingular	nonsingular	ADJ
ejpam-1851	141	4	quadric	quadric	ADJ
ejpam-1851	141	5	surface	surface	NOUN
ejpam-1851	141	6	.	.	PUNCT
ejpam-1851	142	1	as	as	SCONJ
ejpam-1851	142	2	we	we	PRON
ejpam-1851	142	3	shall	shall	AUX
ejpam-1851	142	4	describe	describe	VERB
ejpam-1851	142	5	in	in	ADP
ejpam-1851	142	6	the	the	DET
ejpam-1851	142	7	next	next	ADJ
ejpam-1851	142	8	lemma	lemma	PROPN
ejpam-1851	142	9	,	,	PUNCT
ejpam-1851	142	10	a	a	DET
ejpam-1851	142	11	nonsingular	nonsingular	ADJ
ejpam-1851	142	12	quadric	quadric	ADJ
ejpam-1851	142	13	surface	surface	NOUN
ejpam-1851	142	14	in	in	ADP
ejpam-1851	142	15	p3	p3	PROPN
ejpam-1851	142	16	contains	contain	VERB
ejpam-1851	142	17	exactly	exactly	ADV
ejpam-1851	142	18	two	two	NUM
ejpam-1851	142	19	families	family	NOUN
ejpam-1851	142	20	of	of	ADP
ejpam-1851	142	21	lines	line	NOUN
ejpam-1851	142	22	parametrize	parametrize	ADJ
ejpam-1851	142	23	by	by	ADP
ejpam-1851	142	24	p1	p1	PROPN
ejpam-1851	142	25	.	.	PUNCT
ejpam-1851	143	1	in	in	ADP
ejpam-1851	143	2	fact	fact	NOUN
ejpam-1851	143	3	,	,	PUNCT
ejpam-1851	143	4	this	this	DET
ejpam-1851	143	5	lemma	lemma	PROPN
ejpam-1851	143	6	is	be	AUX
ejpam-1851	143	7	part	part	NOUN
ejpam-1851	143	8	of	of	ADP
ejpam-1851	143	9	exercise	exercise	NOUN
ejpam-1851	143	10	2.15	2.15	NUM
ejpam-1851	143	11	in	in	ADP
ejpam-1851	143	12	hartshorne	hartshorne	PROPN
ejpam-1851	143	13	’s	’s	PART
ejpam-1851	143	14	book	book	NOUN
ejpam-1851	144	1	[	[	X
ejpam-1851	144	2	5	5	NUM
ejpam-1851	144	3	]	]	PUNCT
ejpam-1851	144	4	.	.	PUNCT
ejpam-1851	145	1	see	see	VERB
ejpam-1851	145	2	the	the	DET
ejpam-1851	145	3	proof	proof	NOUN
ejpam-1851	145	4	at	at	ADP
ejpam-1851	145	5	p.	p.	PROPN
ejpam-1851	145	6	478	478	NUM
ejpam-1851	145	7	-	-	SYM
ejpam-1851	145	8	479	479	NUM
ejpam-1851	145	9	in	in	ADP
ejpam-1851	145	10	[	[	X
ejpam-1851	145	11	4	4	NUM
ejpam-1851	145	12	]	]	PUNCT
ejpam-1851	145	13	or	or	CCONJ
ejpam-1851	145	14	[	[	X
ejpam-1851	145	15	8	8	NUM
ejpam-1851	145	16	]	]	PUNCT
ejpam-1851	145	17	.	.	PUNCT
ejpam-1851	146	1	j.	j.	PROPN
ejpam-1851	146	2	rojas	rojas	PROPN
ejpam-1851	146	3	,	,	PUNCT
ejpam-1851	146	4	r.	r.	PROPN
ejpam-1851	146	5	mendoza	mendoza	PROPN
ejpam-1851	146	6	/	/	SYM
ejpam-1851	146	7	eur	eur	PROPN
ejpam-1851	146	8	.	.	PUNCT
ejpam-1851	147	1	j.	j.	PROPN
ejpam-1851	147	2	pure	pure	PROPN
ejpam-1851	147	3	appl	appl	PROPN
ejpam-1851	147	4	.	.	PROPN
ejpam-1851	147	5	math	math	PROPN
ejpam-1851	147	6	,	,	PUNCT
ejpam-1851	147	7	7	7	NUM
ejpam-1851	147	8	(	(	PUNCT
ejpam-1851	147	9	2014	2014	NUM
ejpam-1851	147	10	)	)	PUNCT
ejpam-1851	147	11	,	,	PUNCT
ejpam-1851	147	12	472	472	NUM
ejpam-1851	147	13	-	-	SYM
ejpam-1851	147	14	485	485	NUM
ejpam-1851	147	15	476	476	NUM
ejpam-1851	147	16	lemma	lemma	PROPN
ejpam-1851	147	17	1	1	X
ejpam-1851	147	18	.	.	PUNCT
ejpam-1851	148	1	let	let	VERB
ejpam-1851	148	2	q	q	PART
ejpam-1851	148	3	be	be	AUX
ejpam-1851	148	4	a	a	DET
ejpam-1851	148	5	nonsingular	nonsingular	ADJ
ejpam-1851	148	6	quadric	quadric	ADJ
ejpam-1851	148	7	surface	surface	NOUN
ejpam-1851	148	8	in	in	ADP
ejpam-1851	148	9	p3	p3	PROPN
ejpam-1851	148	10	.	.	PUNCT
ejpam-1851	149	1	then	then	ADV
ejpam-1851	149	2	there	there	PRON
ejpam-1851	149	3	exist	exist	VERB
ejpam-1851	149	4	two	two	NUM
ejpam-1851	149	5	families	family	NOUN
ejpam-1851	149	6	of	of	ADP
ejpam-1851	149	7	lines	line	NOUN
ejpam-1851	149	8	l	l	NOUN
ejpam-1851	150	1	=	=	PUNCT
ejpam-1851	150	2	{	{	PUNCT
ejpam-1851	150	3	lp}p∈p1	lp}p∈p1	NOUN
ejpam-1851	150	4	andm	andm	NOUN
ejpam-1851	150	5	=	=	PUNCT
ejpam-1851	150	6	{	{	PUNCT
ejpam-1851	150	7	mp}p∈p1	mp}p∈p1	PROPN
ejpam-1851	150	8	in	in	ADP
ejpam-1851	150	9	q	q	NOUN
ejpam-1851	150	10	such	such	ADJ
ejpam-1851	150	11	that	that	SCONJ
ejpam-1851	150	12	(	(	PUNCT
ejpam-1851	150	13	1	1	X
ejpam-1851	150	14	)	)	PUNCT
ejpam-1851	150	15	lp	lp	NOUN
ejpam-1851	150	16	∩	∩	NOUN
ejpam-1851	150	17	lq	lq	VERB
ejpam-1851	150	18	=	=	X
ejpam-1851	150	19	;	;	PUNCT
ejpam-1851	150	20	and	and	CCONJ
ejpam-1851	150	21	mp	mp	PROPN
ejpam-1851	150	22	∩mq	∩mq	NOUN
ejpam-1851	150	23	=	=	X
ejpam-1851	150	24	;	;	PUNCT
ejpam-1851	150	25	for	for	ADP
ejpam-1851	150	26	all	all	DET
ejpam-1851	150	27	lp	lp	ADJ
ejpam-1851	150	28	,	,	PUNCT
ejpam-1851	150	29	lq	lq	NOUN
ejpam-1851	150	30	∈	∈	PROPN
ejpam-1851	150	31	l	l	NOUN
ejpam-1851	150	32	,	,	PUNCT
ejpam-1851	150	33	mp	mp	PROPN
ejpam-1851	150	34	,	,	PUNCT
ejpam-1851	150	35	mq	mq	PROPN
ejpam-1851	150	36	∈m	∈m	NOUN
ejpam-1851	150	37	and	and	CCONJ
ejpam-1851	150	38	p	p	NOUN
ejpam-1851	150	39	6=	6=	ADP
ejpam-1851	150	40	q	q	PROPN
ejpam-1851	150	41	∈	∈	PROPN
ejpam-1851	150	42	p1	p1	NOUN
ejpam-1851	150	43	.	.	PUNCT
ejpam-1851	151	1	(	(	PUNCT
ejpam-1851	151	2	2	2	X
ejpam-1851	151	3	)	)	PUNCT
ejpam-1851	151	4	lp	lp	NOUN
ejpam-1851	151	5	∩mq	∩mq	NOUN
ejpam-1851	151	6	6=	6=	X
ejpam-1851	151	7	;	;	PUNCT
ejpam-1851	151	8	for	for	ADP
ejpam-1851	151	9	all	all	DET
ejpam-1851	151	10	lp	lp	NOUN
ejpam-1851	151	11	∈	∈	PROPN
ejpam-1851	151	12	l	l	NOUN
ejpam-1851	151	13	,	,	PUNCT
ejpam-1851	151	14	mq	mq	NOUN
ejpam-1851	151	15	∈m	∈m	NOUN
ejpam-1851	151	16	and	and	CCONJ
ejpam-1851	151	17	p	p	NOUN
ejpam-1851	151	18	,	,	PUNCT
ejpam-1851	151	19	q	q	PROPN
ejpam-1851	151	20	∈	∈	PROPN
ejpam-1851	151	21	p1	p1	NOUN
ejpam-1851	151	22	.	.	PUNCT
ejpam-1851	152	1	(	(	PUNCT
ejpam-1851	152	2	3	3	X
ejpam-1851	152	3	)	)	PUNCT
ejpam-1851	152	4	if	if	SCONJ
ejpam-1851	152	5	ℓ	ℓ	PROPN
ejpam-1851	152	6	is	be	AUX
ejpam-1851	152	7	a	a	DET
ejpam-1851	152	8	line	line	NOUN
ejpam-1851	152	9	contained	contain	VERB
ejpam-1851	152	10	in	in	ADP
ejpam-1851	152	11	q	q	PROPN
ejpam-1851	152	12	then	then	ADV
ejpam-1851	152	13	ℓ	ℓ	PROPN
ejpam-1851	152	14	∈	∈	PROPN
ejpam-1851	152	15	l	l	NOUN
ejpam-1851	152	16	or	or	CCONJ
ejpam-1851	152	17	ℓ	ℓ	NOUN
ejpam-1851	152	18	∈m	∈m	NOUN
ejpam-1851	152	19	.	.	PUNCT
ejpam-1851	153	1	(	(	PUNCT
ejpam-1851	153	2	4	4	X
ejpam-1851	153	3	)	)	PUNCT
ejpam-1851	153	4	given	give	VERB
ejpam-1851	153	5	x	x	PUNCT
ejpam-1851	153	6	∈q	∈q	NOUN
ejpam-1851	153	7	there	there	ADV
ejpam-1851	153	8	exist	exist	VERB
ejpam-1851	153	9	unique	unique	ADJ
ejpam-1851	153	10	lines	line	NOUN
ejpam-1851	153	11	lp(x	lp(x	PUNCT
ejpam-1851	153	12	)	)	PUNCT
ejpam-1851	153	13	∈	∈	PROPN
ejpam-1851	153	14	l	l	NOUN
ejpam-1851	153	15	and	and	CCONJ
ejpam-1851	153	16	mq(x	mq(x	NOUN
ejpam-1851	153	17	)	)	PUNCT
ejpam-1851	153	18	∈m	∈m	NOUN
ejpam-1851	153	19	such	such	ADJ
ejpam-1851	153	20	that	that	SCONJ
ejpam-1851	153	21	{	{	PUNCT
ejpam-1851	153	22	x	x	NOUN
ejpam-1851	153	23	}	}	PUNCT
ejpam-1851	153	24	=	=	SYM
ejpam-1851	153	25	lp(x)∩mq(x	lp(x)∩mq(x	PROPN
ejpam-1851	153	26	)	)	PUNCT
ejpam-1851	153	27	.	.	PUNCT
ejpam-1851	154	1	one	one	NUM
ejpam-1851	154	2	other	other	ADJ
ejpam-1851	154	3	simple	simple	ADJ
ejpam-1851	154	4	but	but	CCONJ
ejpam-1851	154	5	important	important	ADJ
ejpam-1851	154	6	fact	fact	NOUN
ejpam-1851	154	7	which	which	PRON
ejpam-1851	154	8	will	will	AUX
ejpam-1851	154	9	help	help	VERB
ejpam-1851	154	10	us	we	PRON
ejpam-1851	154	11	to	to	PART
ejpam-1851	154	12	prove	prove	VERB
ejpam-1851	154	13	our	our	PRON
ejpam-1851	154	14	main	main	ADJ
ejpam-1851	154	15	result	result	NOUN
ejpam-1851	154	16	(	(	PUNCT
ejpam-1851	154	17	theorem	theorem	NOUN
ejpam-1851	154	18	1	1	NUM
ejpam-1851	154	19	)	)	PUNCT
ejpam-1851	154	20	is	be	AUX
ejpam-1851	154	21	the	the	DET
ejpam-1851	154	22	next	next	ADJ
ejpam-1851	154	23	lemma	lemma	PROPN
ejpam-1851	154	24	.	.	PUNCT
ejpam-1851	155	1	lemma	lemma	PROPN
ejpam-1851	155	2	2	2	NUM
ejpam-1851	155	3	.	.	PUNCT
ejpam-1851	156	1	given	give	VERB
ejpam-1851	156	2	the	the	DET
ejpam-1851	156	3	lines	line	NOUN
ejpam-1851	156	4	ℓ1	ℓ1	NOUN
ejpam-1851	156	5	,	,	PUNCT
ejpam-1851	156	6	ℓ2	ℓ2	NOUN
ejpam-1851	156	7	and	and	CCONJ
ejpam-1851	156	8	ℓ3	ℓ3	PROPN
ejpam-1851	156	9	in	in	ADP
ejpam-1851	156	10	p3	p3	PROPN
ejpam-1851	156	11	such	such	ADJ
ejpam-1851	156	12	that	that	SCONJ
ejpam-1851	156	13	ℓi	ℓi	PROPN
ejpam-1851	156	14	∩	∩	PROPN
ejpam-1851	156	15	ℓ	ℓ	PROPN
ejpam-1851	156	16	j	j	PROPN
ejpam-1851	156	17	=	=	X
ejpam-1851	156	18	;	;	PUNCT
ejpam-1851	156	19	for	for	ADP
ejpam-1851	156	20	1≤	1≤	NUM
ejpam-1851	157	1	i	i	PRON
ejpam-1851	157	2	<	<	X
ejpam-1851	157	3	j	j	PROPN
ejpam-1851	157	4	≤	≤	ADV
ejpam-1851	157	5	3	3	NUM
ejpam-1851	157	6	,	,	PUNCT
ejpam-1851	157	7	there	there	PRON
ejpam-1851	157	8	exists	exist	VERB
ejpam-1851	157	9	a	a	DET
ejpam-1851	157	10	nonsingular	nonsingular	ADJ
ejpam-1851	157	11	quadric	quadric	ADJ
ejpam-1851	157	12	surface	surface	NOUN
ejpam-1851	157	13	q	q	PUNCT
ejpam-1851	157	14	in	in	ADP
ejpam-1851	157	15	p3	p3	NOUN
ejpam-1851	157	16	containing	contain	VERB
ejpam-1851	157	17	ℓ1	ℓ1	NOUN
ejpam-1851	157	18	,	,	PUNCT
ejpam-1851	157	19	ℓ2	ℓ2	NOUN
ejpam-1851	157	20	and	and	CCONJ
ejpam-1851	157	21	ℓ3	ℓ3	PROPN
ejpam-1851	157	22	.	.	PUNCT
ejpam-1851	158	1	proof	proof	NOUN
ejpam-1851	158	2	.	.	PUNCT
ejpam-1851	159	1	take	take	VERB
ejpam-1851	159	2	3	3	NUM
ejpam-1851	159	3	points	point	NOUN
ejpam-1851	159	4	p1i	p1i	NOUN
ejpam-1851	159	5	,	,	PUNCT
ejpam-1851	159	6	p2i	p2i	ADV
ejpam-1851	159	7	,	,	PUNCT
ejpam-1851	159	8	p3i	p3i	VERB
ejpam-1851	159	9	on	on	ADP
ejpam-1851	159	10	each	each	DET
ejpam-1851	159	11	line	line	NOUN
ejpam-1851	159	12	ℓi	ℓi	PROPN
ejpam-1851	159	13	(	(	PUNCT
ejpam-1851	159	14	i	i	NOUN
ejpam-1851	159	15	=	=	NOUN
ejpam-1851	159	16	1,2,3	1,2,3	NUM
ejpam-1851	159	17	)	)	PUNCT
ejpam-1851	159	18	and	and	CCONJ
ejpam-1851	159	19	note	note	VERB
ejpam-1851	159	20	that	that	DET
ejpam-1851	159	21	w	w	NOUN
ejpam-1851	159	22	=	=	PUNCT
ejpam-1851	159	23	{	{	PUNCT
ejpam-1851	159	24	g	g	PROPN
ejpam-1851	159	25	∈	∈	PROPN
ejpam-1851	159	26	c[x	c[x	NOUN
ejpam-1851	159	27	]	]	X
ejpam-1851	159	28	2	2	NUM
ejpam-1851	159	29	|	|	NOUN
ejpam-1851	159	30	g(pi	g(pi	X
ejpam-1851	159	31	j	j	NOUN
ejpam-1851	159	32	)	)	PUNCT
ejpam-1851	159	33	=	=	SYM
ejpam-1851	159	34	0	0	NUM
ejpam-1851	160	1	for	for	ADP
ejpam-1851	160	2	all	all	DET
ejpam-1851	160	3	1≤	1≤	NUM
ejpam-1851	160	4	i	i	PRON
ejpam-1851	160	5	,	,	PUNCT
ejpam-1851	160	6	j	j	PROPN
ejpam-1851	160	7	≤	≤	PROPN
ejpam-1851	160	8	3	3	NUM
ejpam-1851	160	9	}	}	PUNCT
ejpam-1851	160	10	(	(	PUNCT
ejpam-1851	160	11	here	here	ADV
ejpam-1851	160	12	n=	n=	ADJ
ejpam-1851	160	13	3	3	X
ejpam-1851	160	14	)	)	PUNCT
ejpam-1851	160	15	is	be	AUX
ejpam-1851	160	16	a	a	DET
ejpam-1851	160	17	subspace	subspace	NOUN
ejpam-1851	160	18	of	of	ADP
ejpam-1851	160	19	c[x	c[x	NOUN
ejpam-1851	160	20	]	]	X
ejpam-1851	160	21	2	2	NUM
ejpam-1851	160	22	of	of	ADP
ejpam-1851	160	23	dimension	dimension	NOUN
ejpam-1851	160	24	greater	great	ADJ
ejpam-1851	160	25	than	than	ADP
ejpam-1851	160	26	or	or	CCONJ
ejpam-1851	160	27	equal	equal	ADJ
ejpam-1851	160	28	to	to	ADP
ejpam-1851	160	29	1	1	NUM
ejpam-1851	160	30	.	.	PUNCT
ejpam-1851	161	1	so	so	ADV
ejpam-1851	161	2	we	we	PRON
ejpam-1851	161	3	can	can	AUX
ejpam-1851	161	4	choose	choose	VERB
ejpam-1851	161	5	g	g	PROPN
ejpam-1851	161	6	6=	6=	ADP
ejpam-1851	161	7	0	0	NUM
ejpam-1851	161	8	in	in	ADP
ejpam-1851	161	9	w	w	NOUN
ejpam-1851	161	10	and	and	CCONJ
ejpam-1851	161	11	take	take	VERB
ejpam-1851	161	12	q	q	NOUN
ejpam-1851	161	13	=	=	PUNCT
ejpam-1851	161	14	z	z	NOUN
ejpam-1851	161	15	(	(	PUNCT
ejpam-1851	161	16	g	g	NOUN
ejpam-1851	161	17	)	)	PUNCT
ejpam-1851	161	18	.	.	PUNCT
ejpam-1851	162	1	now	now	ADV
ejpam-1851	162	2	follows	follow	VERB
ejpam-1851	162	3	from	from	ADP
ejpam-1851	162	4	proposition	proposition	NOUN
ejpam-1851	162	5	2	2	NUM
ejpam-1851	162	6	that	that	PRON
ejpam-1851	162	7	each	each	DET
ejpam-1851	162	8	line	line	NOUN
ejpam-1851	162	9	ℓi	ℓi	PROPN
ejpam-1851	162	10	is	be	AUX
ejpam-1851	162	11	contained	contain	VERB
ejpam-1851	162	12	in	in	ADP
ejpam-1851	162	13	q.	q.	NOUN
ejpam-1851	162	14	finally	finally	ADV
ejpam-1851	162	15	,	,	PUNCT
ejpam-1851	162	16	since	since	SCONJ
ejpam-1851	162	17	q	q	NOUN
ejpam-1851	162	18	contains	contain	VERB
ejpam-1851	162	19	three	three	NUM
ejpam-1851	162	20	pairwise	pairwise	NOUN
ejpam-1851	162	21	disjoint	disjoint	NOUN
ejpam-1851	162	22	lines	line	NOUN
ejpam-1851	162	23	,	,	PUNCT
ejpam-1851	162	24	then	then	ADV
ejpam-1851	162	25	q	q	X
ejpam-1851	162	26	must	must	AUX
ejpam-1851	162	27	be	be	AUX
ejpam-1851	162	28	a	a	DET
ejpam-1851	162	29	nonsingular	nonsingular	ADJ
ejpam-1851	162	30	surface	surface	NOUN
ejpam-1851	162	31	in	in	ADP
ejpam-1851	162	32	p	p	PROPN
ejpam-1851	162	33	3	3	NUM
ejpam-1851	162	34	.	.	NOUN
ejpam-1851	162	35	3	3	NUM
ejpam-1851	162	36	.	.	X
ejpam-1851	163	1	the	the	DET
ejpam-1851	163	2	plücker	plücker	NOUN
ejpam-1851	163	3	’s	’s	PART
ejpam-1851	163	4	quadric	quadric	ADJ
ejpam-1851	163	5	q	q	NOUN
ejpam-1851	163	6	in	in	ADP
ejpam-1851	163	7	p5	p5	ADJ
ejpam-1851	163	8	the	the	DET
ejpam-1851	163	9	plücker	plücker	NOUN
ejpam-1851	163	10	embedding	embed	VERB
ejpam-1851	163	11	p	p	X
ejpam-1851	163	12	:	:	PUNCT
ejpam-1851	163	13	gk(p(v	gk(p(v	NOUN
ejpam-1851	163	14	)	)	PUNCT
ejpam-1851	163	15	)	)	PUNCT
ejpam-1851	164	1	−→	−→	NOUN
ejpam-1851	164	2	p(λ	p(λ	NOUN
ejpam-1851	164	3	k+1v	k+1v	PROPN
ejpam-1851	164	4	)	)	PUNCT
ejpam-1851	164	5	is	be	AUX
ejpam-1851	164	6	the	the	DET
ejpam-1851	164	7	map	map	NOUN
ejpam-1851	164	8	that	that	PRON
ejpam-1851	164	9	enable	enable	VERB
ejpam-1851	164	10	us	we	PRON
ejpam-1851	164	11	to	to	PART
ejpam-1851	164	12	identify	identify	VERB
ejpam-1851	164	13	the	the	DET
ejpam-1851	164	14	grassmannian	grassmannian	ADJ
ejpam-1851	164	15	gk(p(v	gk(p(v	NOUN
ejpam-1851	164	16	)	)	PUNCT
ejpam-1851	164	17	)	)	PUNCT
ejpam-1851	164	18	with	with	ADP
ejpam-1851	164	19	a	a	DET
ejpam-1851	164	20	projective	projective	ADJ
ejpam-1851	164	21	variety	variety	NOUN
ejpam-1851	164	22	in	in	ADP
ejpam-1851	164	23	p(λk+1v	p(λk+1v	NOUN
ejpam-1851	164	24	)	)	PUNCT
ejpam-1851	164	25	(	(	PUNCT
ejpam-1851	164	26	λk+1v	λk+1v	ADV
ejpam-1851	164	27	denotes	denote	VERB
ejpam-1851	164	28	the	the	DET
ejpam-1851	164	29	(	(	PUNCT
ejpam-1851	164	30	k+1)th	k+1)th	NOUN
ejpam-1851	164	31	exterior	exterior	ADJ
ejpam-1851	164	32	power	power	NOUN
ejpam-1851	164	33	of	of	ADP
ejpam-1851	164	34	v	v	NOUN
ejpam-1851	164	35	)	)	PUNCT
ejpam-1851	164	36	.	.	PUNCT
ejpam-1851	165	1	it	it	PRON
ejpam-1851	165	2	is	be	AUX
ejpam-1851	165	3	defined	define	VERB
ejpam-1851	165	4	by	by	ADP
ejpam-1851	165	5	p(w	p(w	PROPN
ejpam-1851	165	6	)	)	PUNCT
ejpam-1851	165	7	7→	7→	PROPN
ejpam-1851	166	1	[	[	X
ejpam-1851	166	2	u0	u0	ADJ
ejpam-1851	166	3	∧	∧	PROPN
ejpam-1851	166	4	.	.	PUNCT
ejpam-1851	166	5	.	.	PUNCT
ejpam-1851	166	6	.∧	.∧	PUNCT
ejpam-1851	167	1	uk	uk	PROPN
ejpam-1851	167	2	]	]	PUNCT
ejpam-1851	167	3	,	,	PUNCT
ejpam-1851	167	4	if	if	SCONJ
ejpam-1851	167	5	w	w	NOUN
ejpam-1851	167	6	=	=	PUNCT
ejpam-1851	168	1	[	[	X
ejpam-1851	168	2	u0	u0	X
ejpam-1851	168	3	,	,	PUNCT
ejpam-1851	168	4	.	.	PUNCT
ejpam-1851	168	5	.	.	PUNCT
ejpam-1851	168	6	.	.	PUNCT
ejpam-1851	169	1	,	,	PUNCT
ejpam-1851	169	2	uk	uk	PROPN
ejpam-1851	169	3	]	]	PUNCT
ejpam-1851	169	4	.	.	PUNCT
ejpam-1851	170	1	the	the	DET
ejpam-1851	170	2	plücker	plücker	NOUN
ejpam-1851	170	3	’s	’s	PART
ejpam-1851	170	4	quadric	quadric	ADJ
ejpam-1851	170	5	q	q	NOUN
ejpam-1851	170	6	in	in	ADP
ejpam-1851	170	7	p5	p5	PROPN
ejpam-1851	170	8	.	.	PUNCT
ejpam-1851	171	1	in	in	ADP
ejpam-1851	171	2	what	what	PRON
ejpam-1851	171	3	follows	follow	VERB
ejpam-1851	171	4	we	we	PRON
ejpam-1851	171	5	will	will	AUX
ejpam-1851	171	6	consider	consider	VERB
ejpam-1851	171	7	v	v	NOUN
ejpam-1851	171	8	=	=	SYM
ejpam-1851	171	9	c4	c4	NOUN
ejpam-1851	171	10	.	.	PUNCT
ejpam-1851	172	1	now	now	ADV
ejpam-1851	172	2	,	,	PUNCT
ejpam-1851	172	3	fix	fix	VERB
ejpam-1851	172	4	the	the	DET
ejpam-1851	172	5	base	base	NOUN
ejpam-1851	172	6	{	{	PUNCT
ejpam-1851	172	7	ei	ei	NOUN
ejpam-1851	172	8	∧	∧	PROPN
ejpam-1851	172	9	e	e	NOUN
ejpam-1851	172	10	j}1≤i	j}1≤i	X
ejpam-1851	172	11	<	<	X
ejpam-1851	172	12	j≤4	j≤4	PROPN
ejpam-1851	172	13	of	of	ADP
ejpam-1851	172	14	λ2	λ2	PROPN
ejpam-1851	172	15	c	c	NOUN
ejpam-1851	172	16	4	4	NUM
ejpam-1851	172	17	where	where	SCONJ
ejpam-1851	172	18	{	{	PUNCT
ejpam-1851	172	19	ei	ei	NOUN
ejpam-1851	172	20	}	}	PUNCT
ejpam-1851	172	21	4	4	NUM
ejpam-1851	172	22	i=1	i=1	X
ejpam-1851	172	23	is	be	AUX
ejpam-1851	172	24	the	the	DET
ejpam-1851	172	25	canonical	canonical	ADJ
ejpam-1851	172	26	basis	basis	NOUN
ejpam-1851	172	27	of	of	ADP
ejpam-1851	172	28	c4	c4	NOUN
ejpam-1851	172	29	.	.	PUNCT
ejpam-1851	173	1	let	let	VERB
ejpam-1851	173	2	us	we	PRON
ejpam-1851	173	3	consider	consider	VERB
ejpam-1851	173	4	w	w	NOUN
ejpam-1851	173	5	=	=	PUNCT
ejpam-1851	174	1	[	[	X
ejpam-1851	174	2	u	u	NOUN
ejpam-1851	174	3	,	,	PUNCT
ejpam-1851	174	4	v	v	NOUN
ejpam-1851	174	5	]	]	X
ejpam-1851	174	6	∈	∈	PROPN
ejpam-1851	174	7	g2(c	g2(c	NOUN
ejpam-1851	174	8	4	4	NUM
ejpam-1851	174	9	)	)	PUNCT
ejpam-1851	174	10	with	with	ADP
ejpam-1851	174	11	u=	u=	PROPN
ejpam-1851	174	12	(	(	PUNCT
ejpam-1851	174	13	u0,u1,u2,u3	u0,u1,u2,u3	NOUN
ejpam-1851	174	14	)	)	PUNCT
ejpam-1851	174	15	and	and	CCONJ
ejpam-1851	174	16	v=	v=	NOUN
ejpam-1851	174	17	(	(	PUNCT
ejpam-1851	174	18	v0	v0	NOUN
ejpam-1851	174	19	,	,	PUNCT
ejpam-1851	174	20	v1	v1	NOUN
ejpam-1851	174	21	,	,	PUNCT
ejpam-1851	174	22	v2	v2	PROPN
ejpam-1851	174	23	,	,	PUNCT
ejpam-1851	174	24	v3	v3	PROPN
ejpam-1851	174	25	)	)	PUNCT
ejpam-1851	174	26	,	,	PUNCT
ejpam-1851	174	27	then	then	ADV
ejpam-1851	174	28	we	we	PRON
ejpam-1851	174	29	see	see	VERB
ejpam-1851	174	30	that	that	SCONJ
ejpam-1851	174	31	u∧	u∧	PROPN
ejpam-1851	174	32	v=	v=	PROPN
ejpam-1851	174	33	∑	∑	PROPN
ejpam-1851	174	34	1≤k	1≤k	NUM
ejpam-1851	174	35	<	<	X
ejpam-1851	174	36	l≤4	l≤4	PROPN
ejpam-1851	174	37	wk−1,l−1ek	wk−1,l−1ek	PROPN
ejpam-1851	174	38	∧	∧	PROPN
ejpam-1851	174	39	el	el	PROPN
ejpam-1851	174	40	where	where	SCONJ
ejpam-1851	174	41	wi	wi	PROPN
ejpam-1851	174	42	j	j	PROPN
ejpam-1851	174	43	=	=	PRON
ejpam-1851	175	1	ui	ui	PROPN
ejpam-1851	175	2	v	v	NUM
ejpam-1851	175	3	j	j	PROPN
ejpam-1851	175	4	−	−	PROPN
ejpam-1851	176	1	u	u	PROPN
ejpam-1851	176	2	j	j	PROPN
ejpam-1851	176	3	vi	vi	PROPN
ejpam-1851	176	4	for	for	ADP
ejpam-1851	176	5	0≤	0≤	NUM
ejpam-1851	177	1	i	i	PRON
ejpam-1851	177	2	<	<	X
ejpam-1851	177	3	j	j	PROPN
ejpam-1851	177	4	≤	≤	ADV
ejpam-1851	177	5	3	3	NUM
ejpam-1851	177	6	.	.	PUNCT
ejpam-1851	178	1	thus	thus	ADV
ejpam-1851	178	2	the	the	DET
ejpam-1851	178	3	plücker	plücker	NOUN
ejpam-1851	178	4	embedding	embed	VERB
ejpam-1851	178	5	p	p	NOUN
ejpam-1851	178	6	above	above	ADV
ejpam-1851	178	7	in	in	ADP
ejpam-1851	178	8	coordinates	coordinate	NOUN
ejpam-1851	178	9	is	be	AUX
ejpam-1851	178	10	given	give	VERB
ejpam-1851	178	11	by	by	ADP
ejpam-1851	178	12	p	p	X
ejpam-1851	178	13	:	:	PUNCT
ejpam-1851	178	14	g1(p	g1(p	ADJ
ejpam-1851	178	15	3	3	X
ejpam-1851	178	16	)	)	PUNCT
ejpam-1851	178	17	−→p5	−→p5	NOUN
ejpam-1851	178	18	p([u	p([u	VERB
ejpam-1851	178	19	,	,	PUNCT
ejpam-1851	178	20	v	v	NOUN
ejpam-1851	178	21	]	]	X
ejpam-1851	178	22	)	)	PUNCT
ejpam-1851	179	1	7−→[w01	7−→[w01	NOUN
ejpam-1851	179	2	:	:	PUNCT
ejpam-1851	179	3	w02	w02	NOUN
ejpam-1851	179	4	:	:	PUNCT
ejpam-1851	179	5	w03	w03	NOUN
ejpam-1851	179	6	:	:	PUNCT
ejpam-1851	179	7	w12	w12	NOUN
ejpam-1851	179	8	:	:	PUNCT
ejpam-1851	179	9	w13	w13	PROPN
ejpam-1851	179	10	:	:	PUNCT
ejpam-1851	179	11	w23	w23	VERB
ejpam-1851	179	12	]	]	PUNCT
ejpam-1851	179	13	.	.	PUNCT
ejpam-1851	180	1	(	(	PUNCT
ejpam-1851	180	2	1	1	X
ejpam-1851	180	3	)	)	PUNCT
ejpam-1851	180	4	proposition	proposition	NOUN
ejpam-1851	180	5	3	3	NUM
ejpam-1851	180	6	.	.	PUNCT
ejpam-1851	181	1	the	the	DET
ejpam-1851	181	2	plücker	plücker	NOUN
ejpam-1851	181	3	map	map	NOUN
ejpam-1851	181	4	p	p	X
ejpam-1851	181	5	:	:	PUNCT
ejpam-1851	181	6	g1(p	g1(p	ADJ
ejpam-1851	181	7	3	3	X
ejpam-1851	181	8	)	)	PUNCT
ejpam-1851	181	9	−→	−→	ADJ
ejpam-1851	181	10	p5	p5	NOUN
ejpam-1851	181	11	defined	define	VERB
ejpam-1851	181	12	in	in	ADP
ejpam-1851	181	13	(	(	PUNCT
ejpam-1851	181	14	1	1	NUM
ejpam-1851	181	15	)	)	PUNCT
ejpam-1851	181	16	is	be	AUX
ejpam-1851	181	17	an	an	DET
ejpam-1851	181	18	embedding	embedding	NOUN
ejpam-1851	181	19	of	of	ADP
ejpam-1851	181	20	the	the	DET
ejpam-1851	181	21	grassmannian	grassmannian	ADJ
ejpam-1851	181	22	g1(p	g1(p	PROPN
ejpam-1851	181	23	3	3	X
ejpam-1851	181	24	)	)	PUNCT
ejpam-1851	181	25	over	over	ADP
ejpam-1851	181	26	the	the	DET
ejpam-1851	181	27	nonsingular	nonsingular	ADJ
ejpam-1851	181	28	quadric	quadric	ADJ
ejpam-1851	181	29	hypersurface	hypersurface	NOUN
ejpam-1851	181	30	q	q	PROPN
ejpam-1851	182	1	=	=	SYM
ejpam-1851	182	2	z	z	NOUN
ejpam-1851	182	3	(	(	PUNCT
ejpam-1851	182	4	f	f	X
ejpam-1851	182	5	)	)	PUNCT
ejpam-1851	182	6	⊂	⊂	PROPN
ejpam-1851	182	7	p5	p5	ADJ
ejpam-1851	182	8	where	where	SCONJ
ejpam-1851	182	9	f	f	PROPN
ejpam-1851	182	10	=	=	X
ejpam-1851	182	11	x0x5	x0x5	PROPN
ejpam-1851	182	12	−	−	NOUN
ejpam-1851	182	13	x1x4	x1x4	X
ejpam-1851	183	1	+	+	NUM
ejpam-1851	183	2	x2x3	x2x3	X
ejpam-1851	183	3	∈	∈	NOUN
ejpam-1851	183	4	c[x	c[x	NOUN
ejpam-1851	183	5	]	]	X
ejpam-1851	183	6	(	(	PUNCT
ejpam-1851	183	7	n=	n=	ADJ
ejpam-1851	183	8	5	5	NUM
ejpam-1851	183	9	)	)	PUNCT
ejpam-1851	183	10	.	.	PUNCT
ejpam-1851	184	1	j.	j.	PROPN
ejpam-1851	184	2	rojas	rojas	PROPN
ejpam-1851	184	3	,	,	PUNCT
ejpam-1851	184	4	r.	r.	PROPN
ejpam-1851	184	5	mendoza	mendoza	PROPN
ejpam-1851	184	6	/	/	SYM
ejpam-1851	184	7	eur	eur	PROPN
ejpam-1851	184	8	.	.	PUNCT
ejpam-1851	185	1	j.	j.	PROPN
ejpam-1851	185	2	pure	pure	PROPN
ejpam-1851	185	3	appl	appl	PROPN
ejpam-1851	185	4	.	.	PROPN
ejpam-1851	185	5	math	math	PROPN
ejpam-1851	185	6	,	,	PUNCT
ejpam-1851	185	7	7	7	NUM
ejpam-1851	185	8	(	(	PUNCT
ejpam-1851	185	9	2014	2014	NUM
ejpam-1851	185	10	)	)	PUNCT
ejpam-1851	185	11	,	,	PUNCT
ejpam-1851	185	12	472	472	NUM
ejpam-1851	185	13	-	-	SYM
ejpam-1851	185	14	485	485	NUM
ejpam-1851	185	15	477	477	NUM
ejpam-1851	185	16	proof	proof	NOUN
ejpam-1851	185	17	.	.	PUNCT
ejpam-1851	186	1	see	see	VERB
ejpam-1851	186	2	theorem	theorem	VERB
ejpam-1851	186	3	11	11	NUM
ejpam-1851	186	4	at	at	ADP
ejpam-1851	186	5	p.	p.	NOUN
ejpam-1851	186	6	409	409	NUM
ejpam-1851	186	7	in	in	ADP
ejpam-1851	186	8	[	[	X
ejpam-1851	186	9	2	2	NUM
ejpam-1851	186	10	]	]	PUNCT
ejpam-1851	186	11	or	or	CCONJ
ejpam-1851	186	12	p.	p.	NOUN
ejpam-1851	186	13	209	209	NUM
ejpam-1851	186	14	-	-	SYM
ejpam-1851	186	15	211	211	NUM
ejpam-1851	186	16	in	in	ADP
ejpam-1851	186	17	[	[	X
ejpam-1851	186	18	4	4	NUM
ejpam-1851	186	19	]	]	PUNCT
ejpam-1851	186	20	.	.	PUNCT
ejpam-1851	187	1	we	we	PRON
ejpam-1851	187	2	will	will	AUX
ejpam-1851	187	3	refer	refer	VERB
ejpam-1851	187	4	to	to	ADP
ejpam-1851	187	5	q	q	NOUN
ejpam-1851	187	6	in	in	ADP
ejpam-1851	187	7	proposition	proposition	NOUN
ejpam-1851	187	8	3	3	NUM
ejpam-1851	187	9	as	as	ADP
ejpam-1851	187	10	the	the	DET
ejpam-1851	187	11	plücker	plücker	NOUN
ejpam-1851	187	12	’s	’s	PART
ejpam-1851	187	13	quadric	quadric	ADJ
ejpam-1851	187	14	q	q	X
ejpam-1851	187	15	(	(	PUNCT
ejpam-1851	187	16	in	in	ADP
ejpam-1851	187	17	p5	p5	ADJ
ejpam-1851	187	18	)	)	PUNCT
ejpam-1851	187	19	.	.	PUNCT
ejpam-1851	188	1	next	next	ADV
ejpam-1851	188	2	,	,	PUNCT
ejpam-1851	188	3	we	we	PRON
ejpam-1851	188	4	will	will	AUX
ejpam-1851	188	5	make	make	VERB
ejpam-1851	188	6	a	a	DET
ejpam-1851	188	7	remark	remark	NOUN
ejpam-1851	188	8	that	that	PRON
ejpam-1851	188	9	tells	tell	VERB
ejpam-1851	188	10	us	we	PRON
ejpam-1851	188	11	who	who	PRON
ejpam-1851	188	12	is	be	AUX
ejpam-1851	188	13	the	the	DET
ejpam-1851	188	14	image	image	NOUN
ejpam-1851	188	15	of	of	ADP
ejpam-1851	188	16	the	the	DET
ejpam-1851	188	17	families	family	NOUN
ejpam-1851	188	18	of	of	ADP
ejpam-1851	188	19	lines	line	NOUN
ejpam-1851	188	20	l	l	PROPN
ejpam-1851	188	21	and	and	CCONJ
ejpam-1851	188	22	m	m	PROPN
ejpam-1851	188	23	of	of	ADP
ejpam-1851	188	24	a	a	DET
ejpam-1851	188	25	nonsingular	nonsingular	ADJ
ejpam-1851	188	26	quadric	quadric	ADJ
ejpam-1851	188	27	surface	surface	NOUN
ejpam-1851	188	28	in	in	ADP
ejpam-1851	188	29	p3	p3	PROPN
ejpam-1851	188	30	,	,	PUNCT
ejpam-1851	188	31	under	under	ADP
ejpam-1851	188	32	the	the	DET
ejpam-1851	188	33	plücker	plücker	NOUN
ejpam-1851	188	34	embedding	embed	VERB
ejpam-1851	188	35	p	p	X
ejpam-1851	188	36	(	(	PUNCT
ejpam-1851	188	37	cf	cf	NOUN
ejpam-1851	188	38	.	.	PUNCT
ejpam-1851	189	1	lemma	lemma	PROPN
ejpam-1851	189	2	1	1	NUM
ejpam-1851	189	3	)	)	PUNCT
ejpam-1851	189	4	.	.	PUNCT
ejpam-1851	190	1	remark	remark	NOUN
ejpam-1851	190	2	1	1	NUM
ejpam-1851	190	3	.	.	PUNCT
ejpam-1851	191	1	it	it	PRON
ejpam-1851	191	2	can	can	AUX
ejpam-1851	191	3	be	be	AUX
ejpam-1851	191	4	shown	show	VERB
ejpam-1851	191	5	that	that	SCONJ
ejpam-1851	192	1	p	p	X
ejpam-1851	192	2	(	(	PUNCT
ejpam-1851	192	3	l	l	NOUN
ejpam-1851	192	4	)	)	PUNCT
ejpam-1851	193	1	=	=	SYM
ejpam-1851	193	2	z	z	NOUN
ejpam-1851	193	3	(	(	PUNCT
ejpam-1851	193	4	x0	x0	PROPN
ejpam-1851	193	5	+	+	CCONJ
ejpam-1851	193	6	x5	x5	PROPN
ejpam-1851	193	7	,	,	PUNCT
ejpam-1851	193	8	x1	x1	PROPN
ejpam-1851	193	9	−	−	PROPN
ejpam-1851	193	10	x4	x4	PROPN
ejpam-1851	193	11	,	,	PUNCT
ejpam-1851	193	12	x2	x2	PROPN
ejpam-1851	194	1	+	+	CCONJ
ejpam-1851	194	2	x3	x3	ADJ
ejpam-1851	194	3	,	,	PUNCT
ejpam-1851	194	4	f	f	X
ejpam-1851	194	5	)	)	PUNCT
ejpam-1851	194	6	and	and	CCONJ
ejpam-1851	194	7	p	p	X
ejpam-1851	194	8	(	(	PUNCT
ejpam-1851	194	9	m	m	PROPN
ejpam-1851	194	10	)	)	PUNCT
ejpam-1851	195	1	=	=	SYM
ejpam-1851	195	2	z	z	PROPN
ejpam-1851	195	3	(	(	PUNCT
ejpam-1851	195	4	x0−	x0−	PROPN
ejpam-1851	195	5	x5	x5	PROPN
ejpam-1851	195	6	,	,	PUNCT
ejpam-1851	195	7	x1	x1	PROPN
ejpam-1851	195	8	+	+	PROPN
ejpam-1851	195	9	x4	x4	PROPN
ejpam-1851	195	10	,	,	PUNCT
ejpam-1851	195	11	x2−	x2−	PROPN
ejpam-1851	195	12	x3	x3	PROPN
ejpam-1851	195	13	,	,	PUNCT
ejpam-1851	195	14	f	f	X
ejpam-1851	195	15	)	)	PUNCT
ejpam-1851	195	16	,	,	PUNCT
ejpam-1851	195	17	so	so	CCONJ
ejpam-1851	195	18	the	the	DET
ejpam-1851	195	19	image	image	NOUN
ejpam-1851	195	20	of	of	ADP
ejpam-1851	195	21	the	the	DET
ejpam-1851	195	22	families	family	NOUN
ejpam-1851	195	23	of	of	ADP
ejpam-1851	195	24	lines	line	NOUN
ejpam-1851	195	25	l	l	PROPN
ejpam-1851	195	26	andm	andm	NOUN
ejpam-1851	195	27	,	,	PUNCT
ejpam-1851	195	28	under	under	ADP
ejpam-1851	195	29	the	the	DET
ejpam-1851	195	30	plücker	plücker	NOUN
ejpam-1851	195	31	embeddingp	embeddingp	NOUN
ejpam-1851	195	32	(	(	PUNCT
ejpam-1851	195	33	in	in	ADP
ejpam-1851	195	34	(	(	PUNCT
ejpam-1851	195	35	1	1	NUM
ejpam-1851	195	36	)	)	PUNCT
ejpam-1851	195	37	)	)	PUNCT
ejpam-1851	195	38	are	be	AUX
ejpam-1851	195	39	nonsingular	nonsingular	ADJ
ejpam-1851	195	40	conics	conic	NOUN
ejpam-1851	195	41	lying	lie	VERB
ejpam-1851	195	42	in	in	ADP
ejpam-1851	195	43	complementary	complementary	ADJ
ejpam-1851	195	44	planes	plane	NOUN
ejpam-1851	195	45	(	(	PUNCT
ejpam-1851	195	46	also	also	ADV
ejpam-1851	195	47	see	see	VERB
ejpam-1851	195	48	p.	p.	NOUN
ejpam-1851	195	49	196	196	NUM
ejpam-1851	195	50	in	in	ADP
ejpam-1851	195	51	[	[	X
ejpam-1851	195	52	6	6	NUM
ejpam-1851	195	53	]	]	NUM
ejpam-1851	195	54	)	)	PUNCT
ejpam-1851	195	55	.	.	PUNCT
ejpam-1851	196	1	3.1	3.1	NUM
ejpam-1851	196	2	.	.	PUNCT
ejpam-1851	197	1	linear	linear	ADJ
ejpam-1851	197	2	subspaces	subspace	NOUN
ejpam-1851	197	3	in	in	ADP
ejpam-1851	197	4	the	the	DET
ejpam-1851	197	5	plücker	plücker	NOUN
ejpam-1851	197	6	’s	’s	NOUN
ejpam-1851	197	7	quadric	quadric	ADJ
ejpam-1851	197	8	q	q	NOUN
ejpam-1851	198	1	we	we	PRON
ejpam-1851	198	2	begin	begin	VERB
ejpam-1851	198	3	by	by	ADP
ejpam-1851	198	4	proving	prove	VERB
ejpam-1851	198	5	a	a	DET
ejpam-1851	198	6	simple	simple	ADJ
ejpam-1851	198	7	result	result	NOUN
ejpam-1851	198	8	,	,	PUNCT
ejpam-1851	198	9	which	which	PRON
ejpam-1851	198	10	allows	allow	VERB
ejpam-1851	198	11	us	we	PRON
ejpam-1851	198	12	to	to	PART
ejpam-1851	198	13	conclude	conclude	VERB
ejpam-1851	198	14	that	that	SCONJ
ejpam-1851	198	15	the	the	DET
ejpam-1851	198	16	plücker	plücker	NOUN
ejpam-1851	198	17	’s	’s	PART
ejpam-1851	198	18	quadric	quadric	ADJ
ejpam-1851	198	19	q	q	PROPN
ejpam-1851	198	20	does	do	AUX
ejpam-1851	198	21	not	not	PART
ejpam-1851	198	22	contain	contain	VERB
ejpam-1851	198	23	3	3	NUM
ejpam-1851	198	24	-	-	PUNCT
ejpam-1851	198	25	planes	plane	NOUN
ejpam-1851	198	26	or	or	CCONJ
ejpam-1851	198	27	hyperplanes	hyperplane	NOUN
ejpam-1851	198	28	.	.	PUNCT
ejpam-1851	199	1	in	in	ADP
ejpam-1851	199	2	fact	fact	NOUN
ejpam-1851	199	3	,	,	PUNCT
ejpam-1851	199	4	it	it	PRON
ejpam-1851	199	5	is	be	AUX
ejpam-1851	199	6	verified	verify	VERB
ejpam-1851	199	7	that	that	SCONJ
ejpam-1851	199	8	q	q	PRON
ejpam-1851	199	9	only	only	ADV
ejpam-1851	199	10	contains	contain	VERB
ejpam-1851	199	11	lines	line	NOUN
ejpam-1851	199	12	and	and	CCONJ
ejpam-1851	199	13	planes	plane	NOUN
ejpam-1851	199	14	,	,	PUNCT
ejpam-1851	199	15	and	and	CCONJ
ejpam-1851	199	16	this	this	PRON
ejpam-1851	199	17	will	will	AUX
ejpam-1851	199	18	be	be	AUX
ejpam-1851	199	19	described	describe	VERB
ejpam-1851	199	20	in	in	ADP
ejpam-1851	199	21	propositions	proposition	NOUN
ejpam-1851	199	22	5	5	NUM
ejpam-1851	199	23	and	and	CCONJ
ejpam-1851	199	24	6	6	NUM
ejpam-1851	199	25	,	,	PUNCT
ejpam-1851	199	26	respectively	respectively	ADV
ejpam-1851	199	27	(	(	PUNCT
ejpam-1851	199	28	see	see	VERB
ejpam-1851	199	29	[	[	X
ejpam-1851	199	30	1	1	NUM
ejpam-1851	199	31	]	]	NUM
ejpam-1851	199	32	)	)	PUNCT
ejpam-1851	199	33	.	.	PUNCT
ejpam-1851	200	1	proposition	proposition	NOUN
ejpam-1851	200	2	4	4	NUM
ejpam-1851	200	3	.	.	PUNCT
ejpam-1851	201	1	let	let	VERB
ejpam-1851	201	2	z	z	PRON
ejpam-1851	201	3	⊂	⊂	PROPN
ejpam-1851	201	4	pn	pn	AUX
ejpam-1851	201	5	be	be	AUX
ejpam-1851	201	6	a	a	DET
ejpam-1851	201	7	nonsingular	nonsingular	ADJ
ejpam-1851	201	8	quadric	quadric	ADJ
ejpam-1851	201	9	hypersurface	hypersurface	NOUN
ejpam-1851	201	10	defined	define	VERB
ejpam-1851	201	11	by	by	ADP
ejpam-1851	201	12	g	g	PROPN
ejpam-1851	201	13	∈	∈	PROPN
ejpam-1851	201	14	c[x	c[x	NOUN
ejpam-1851	201	15	]	]	X
ejpam-1851	201	16	2	2	NUM
ejpam-1851	201	17	and	and	CCONJ
ejpam-1851	201	18	λ	λ	X
ejpam-1851	201	19	a	a	DET
ejpam-1851	201	20	r	r	NOUN
ejpam-1851	201	21	-	-	PUNCT
ejpam-1851	201	22	linear	linear	ADJ
ejpam-1851	201	23	subspaces	subspace	NOUN
ejpam-1851	201	24	of	of	ADP
ejpam-1851	201	25	pn	pn	PROPN
ejpam-1851	201	26	.	.	PUNCT
ejpam-1851	202	1	if	if	SCONJ
ejpam-1851	202	2	λ	λ	PROPN
ejpam-1851	202	3	⊂	⊂	PROPN
ejpam-1851	202	4	z	z	PROPN
ejpam-1851	202	5	,	,	PUNCT
ejpam-1851	202	6	then	then	ADV
ejpam-1851	202	7	we	we	PRON
ejpam-1851	202	8	have	have	VERB
ejpam-1851	202	9	2r	2r	NUM
ejpam-1851	202	10	<	<	X
ejpam-1851	202	11	n.	n.	NOUN
ejpam-1851	202	12	proof	proof	NOUN
ejpam-1851	202	13	.	.	PUNCT
ejpam-1851	203	1	set	set	VERB
ejpam-1851	203	2	λ	λ	PROPN
ejpam-1851	203	3	=	=	SYM
ejpam-1851	203	4	p(w	p(w	PROPN
ejpam-1851	203	5	)	)	PUNCT
ejpam-1851	203	6	where	where	SCONJ
ejpam-1851	203	7	w	w	NOUN
ejpam-1851	203	8	=	=	PUNCT
ejpam-1851	204	1	[	[	X
ejpam-1851	204	2	w0	w0	NOUN
ejpam-1851	204	3	,	,	PUNCT
ejpam-1851	204	4	.	.	PUNCT
ejpam-1851	204	5	.	.	PUNCT
ejpam-1851	205	1	.	.	PUNCT
ejpam-1851	206	1	,	,	PUNCT
ejpam-1851	206	2	wr	wr	X
ejpam-1851	206	3	]	]	PUNCT
ejpam-1851	206	4	.	.	PUNCT
ejpam-1851	207	1	assume	assume	VERB
ejpam-1851	207	2	that	that	SCONJ
ejpam-1851	207	3	g	g	PROPN
ejpam-1851	207	4	=	=	PUNCT
ejpam-1851	207	5	∑n	∑n	PROPN
ejpam-1851	207	6	i=0	i=0	PROPN
ejpam-1851	207	7	x	x	SYM
ejpam-1851	207	8	2	2	X
ejpam-1851	207	9	i	i	NOUN
ejpam-1851	207	10	(	(	PUNCT
ejpam-1851	207	11	otherwise	otherwise	ADV
ejpam-1851	207	12	make	make	VERB
ejpam-1851	207	13	a	a	DET
ejpam-1851	207	14	linear	linear	ADJ
ejpam-1851	207	15	change	change	NOUN
ejpam-1851	207	16	of	of	ADP
ejpam-1851	207	17	coordinates	coordinate	NOUN
ejpam-1851	207	18	)	)	PUNCT
ejpam-1851	207	19	.	.	PUNCT
ejpam-1851	208	1	so	so	ADV
ejpam-1851	208	2	g	g	PROPN
ejpam-1851	208	3	induz	induz	VERB
ejpam-1851	208	4	the	the	DET
ejpam-1851	208	5	bilinear	bilinear	NOUN
ejpam-1851	208	6	form	form	NOUN
ejpam-1851	208	7	b	b	X
ejpam-1851	208	8	:	:	PUNCT
ejpam-1851	208	9	cn+1	cn+1	VERB
ejpam-1851	208	10	×cn+1	×cn+1	NOUN
ejpam-1851	208	11	→	→	SYM
ejpam-1851	209	1	c	c	NOUN
ejpam-1851	209	2	given	give	VERB
ejpam-1851	209	3	by	by	ADP
ejpam-1851	209	4	b(v	b(v	NOUN
ejpam-1851	209	5	,	,	PUNCT
ejpam-1851	209	6	w	w	NOUN
ejpam-1851	209	7	)	)	PUNCT
ejpam-1851	209	8	=	=	SYM
ejpam-1851	210	1	v0w0	v0w0	PROPN
ejpam-1851	210	2	+	+	X
ejpam-1851	210	3	·	·	PUNCT
ejpam-1851	210	4	·	·	PUNCT
ejpam-1851	210	5	·	·	PUNCT
ejpam-1851	211	1	+	+	CCONJ
ejpam-1851	211	2	vnwn	vnwn	ADJ
ejpam-1851	211	3	,	,	PUNCT
ejpam-1851	211	4	if	if	SCONJ
ejpam-1851	211	5	v=	v=	X
ejpam-1851	211	6	(	(	PUNCT
ejpam-1851	211	7	v0	v0	NOUN
ejpam-1851	211	8	,	,	PUNCT
ejpam-1851	211	9	.	.	PUNCT
ejpam-1851	211	10	.	.	PUNCT
ejpam-1851	211	11	.	.	PUNCT
ejpam-1851	212	1	,	,	PUNCT
ejpam-1851	212	2	vn	vn	PROPN
ejpam-1851	212	3	)	)	PUNCT
ejpam-1851	212	4	and	and	CCONJ
ejpam-1851	212	5	w=	w=	PROPN
ejpam-1851	212	6	(	(	PUNCT
ejpam-1851	212	7	w0	w0	PROPN
ejpam-1851	212	8	,	,	PUNCT
ejpam-1851	212	9	.	.	PUNCT
ejpam-1851	212	10	.	.	PUNCT
ejpam-1851	212	11	.	.	PUNCT
ejpam-1851	213	1	,	,	PUNCT
ejpam-1851	213	2	wn	wn	PROPN
ejpam-1851	213	3	)	)	PUNCT
ejpam-1851	213	4	.	.	PUNCT
ejpam-1851	214	1	let	let	VERB
ejpam-1851	214	2	w⊥	w⊥	NOUN
ejpam-1851	214	3	=	=	PRON
ejpam-1851	214	4	{	{	PUNCT
ejpam-1851	214	5	v	v	NUM
ejpam-1851	214	6	∈	∈	NOUN
ejpam-1851	214	7	cn+1	cn+1	NOUN
ejpam-1851	214	8	|	|	ADV
ejpam-1851	214	9	b(v	b(v	NOUN
ejpam-1851	214	10	,	,	PUNCT
ejpam-1851	214	11	w	w	NOUN
ejpam-1851	214	12	)	)	PUNCT
ejpam-1851	214	13	=	=	SYM
ejpam-1851	214	14	0	0	NUM
ejpam-1851	214	15	for	for	ADP
ejpam-1851	214	16	all	all	DET
ejpam-1851	214	17	w	w	PROPN
ejpam-1851	214	18	∈w	∈w	NOUN
ejpam-1851	214	19	}	}	PUNCT
ejpam-1851	214	20	be	be	AUX
ejpam-1851	214	21	the	the	DET
ejpam-1851	214	22	the	the	DET
ejpam-1851	214	23	orthogonal	orthogonal	ADJ
ejpam-1851	214	24	subspace	subspace	NOUN
ejpam-1851	214	25	associated	associate	VERB
ejpam-1851	214	26	to	to	ADP
ejpam-1851	214	27	w	w	PROPN
ejpam-1851	214	28	.	.	PUNCT
ejpam-1851	215	1	note	note	VERB
ejpam-1851	215	2	that	that	SCONJ
ejpam-1851	215	3	w⊥	w⊥	NOUN
ejpam-1851	215	4	=	=	SYM
ejpam-1851	215	5	ker(t0	ker(t0	NOUN
ejpam-1851	215	6	)	)	PUNCT
ejpam-1851	215	7	∩	∩	NOUN
ejpam-1851	215	8	·	·	PUNCT
ejpam-1851	215	9	·	·	PUNCT
ejpam-1851	215	10	·	·	PUNCT
ejpam-1851	215	11	∩	∩	X
ejpam-1851	215	12	ker(tr	ker(tr	PROPN
ejpam-1851	215	13	)	)	PUNCT
ejpam-1851	215	14	where	where	SCONJ
ejpam-1851	215	15	ti	ti	PROPN
ejpam-1851	215	16	is	be	AUX
ejpam-1851	215	17	the	the	DET
ejpam-1851	215	18	linear	linear	ADJ
ejpam-1851	215	19	functional	functional	ADJ
ejpam-1851	215	20	over	over	ADP
ejpam-1851	215	21	cn+1	cn+1	NOUN
ejpam-1851	215	22	given	give	VERB
ejpam-1851	215	23	by	by	ADP
ejpam-1851	215	24	ti(v	ti(v	NUM
ejpam-1851	215	25	)	)	PUNCT
ejpam-1851	215	26	=	=	SYM
ejpam-1851	215	27	b(v	b(v	PROPN
ejpam-1851	215	28	,	,	PUNCT
ejpam-1851	215	29	wi	wi	PROPN
ejpam-1851	215	30	)	)	PUNCT
ejpam-1851	215	31	,	,	PUNCT
ejpam-1851	215	32	i	i	PRON
ejpam-1851	215	33	∈	∈	PROPN
ejpam-1851	215	34	{	{	PUNCT
ejpam-1851	215	35	0	0	NUM
ejpam-1851	215	36	,	,	PUNCT
ejpam-1851	215	37	.	.	PUNCT
ejpam-1851	215	38	.	.	PUNCT
ejpam-1851	216	1	.	.	PUNCT
ejpam-1851	217	1	,	,	PUNCT
ejpam-1851	217	2	r	r	NOUN
ejpam-1851	217	3	}	}	PUNCT
ejpam-1851	217	4	.	.	PUNCT
ejpam-1851	218	1	since	since	SCONJ
ejpam-1851	218	2	these	these	DET
ejpam-1851	218	3	linear	linear	ADJ
ejpam-1851	218	4	functionals	functional	NOUN
ejpam-1851	218	5	are	be	AUX
ejpam-1851	218	6	linearly	linearly	ADV
ejpam-1851	218	7	independent	independent	ADJ
ejpam-1851	218	8	we	we	PRON
ejpam-1851	218	9	conclude	conclude	VERB
ejpam-1851	218	10	that	that	PRON
ejpam-1851	218	11	dim	dim	ADJ
ejpam-1851	218	12	w⊥	w⊥	NOUN
ejpam-1851	218	13	=	=	SYM
ejpam-1851	218	14	n+	n+	X
ejpam-1851	218	15	1−	1−	NUM
ejpam-1851	219	1	dim	dim	ADJ
ejpam-1851	219	2	w	w	NOUN
ejpam-1851	219	3	=	=	SYM
ejpam-1851	219	4	n−	n−	PROPN
ejpam-1851	219	5	r.	r.	VERB
ejpam-1851	219	6	on	on	ADP
ejpam-1851	219	7	the	the	DET
ejpam-1851	219	8	other	other	ADJ
ejpam-1851	219	9	hand	hand	NOUN
ejpam-1851	219	10	,	,	PUNCT
ejpam-1851	219	11	λ	λ	X
ejpam-1851	219	12	⊂	⊂	PROPN
ejpam-1851	219	13	z	z	NOUN
ejpam-1851	220	1	if	if	SCONJ
ejpam-1851	221	1	and	and	CCONJ
ejpam-1851	221	2	only	only	ADV
ejpam-1851	221	3	if	if	SCONJ
ejpam-1851	221	4	g(w	g(w	ADJ
ejpam-1851	221	5	)	)	PUNCT
ejpam-1851	221	6	=	=	SYM
ejpam-1851	221	7	b(w	b(w	PROPN
ejpam-1851	221	8	,	,	PUNCT
ejpam-1851	221	9	w	w	NOUN
ejpam-1851	221	10	)	)	PUNCT
ejpam-1851	221	11	=	=	SYM
ejpam-1851	221	12	0	0	NUM
ejpam-1851	221	13	for	for	ADP
ejpam-1851	221	14	all	all	DET
ejpam-1851	221	15	w	w	PROPN
ejpam-1851	221	16	∈	∈	PROPN
ejpam-1851	221	17	w	w	NOUN
ejpam-1851	221	18	.	.	PUNCT
ejpam-1851	222	1	thus	thus	ADV
ejpam-1851	222	2	λ	λ	X
ejpam-1851	222	3	⊂	⊂	PROPN
ejpam-1851	222	4	z	z	NOUN
ejpam-1851	223	1	if	if	SCONJ
ejpam-1851	224	1	and	and	CCONJ
ejpam-1851	224	2	only	only	ADV
ejpam-1851	224	3	if	if	SCONJ
ejpam-1851	224	4	w	w	PROPN
ejpam-1851	224	5	⊆	⊆	NUM
ejpam-1851	224	6	w⊥.	w⊥.	NOUN
ejpam-1851	224	7	so	so	ADV
ejpam-1851	224	8	,	,	PUNCT
ejpam-1851	224	9	if	if	SCONJ
ejpam-1851	224	10	λ	λ	PROPN
ejpam-1851	224	11	⊂	⊂	PROPN
ejpam-1851	224	12	z	z	AUX
ejpam-1851	224	13	then	then	ADV
ejpam-1851	224	14	dim	dim	VERB
ejpam-1851	224	15	w	w	PROPN
ejpam-1851	224	16	≤	≤	X
ejpam-1851	224	17	dim	dim	ADJ
ejpam-1851	224	18	w⊥	w⊥	NOUN
ejpam-1851	224	19	=	=	SYM
ejpam-1851	224	20	n+	n+	ADP
ejpam-1851	224	21	1	1	NUM
ejpam-1851	224	22	−	−	NOUN
ejpam-1851	224	23	dim	dim	ADJ
ejpam-1851	224	24	w	w	NOUN
ejpam-1851	224	25	.	.	PUNCT
ejpam-1851	225	1	therefore	therefore	ADV
ejpam-1851	225	2	,	,	PUNCT
ejpam-1851	225	3	2	2	NUM
ejpam-1851	225	4	dim	dim	NOUN
ejpam-1851	225	5	w	w	NOUN
ejpam-1851	225	6	≤	≤	NUM
ejpam-1851	225	7	n+	n+	NUM
ejpam-1851	225	8	1	1	NUM
ejpam-1851	225	9	which	which	PRON
ejpam-1851	225	10	implies	imply	VERB
ejpam-1851	225	11	2r	2r	NUM
ejpam-1851	225	12	<	<	X
ejpam-1851	225	13	n.	n.	NOUN
ejpam-1851	225	14	in	in	ADP
ejpam-1851	225	15	what	what	PRON
ejpam-1851	225	16	follows	follow	VERB
ejpam-1851	225	17	,	,	PUNCT
ejpam-1851	225	18	we	we	PRON
ejpam-1851	225	19	denoted	denote	VERB
ejpam-1851	225	20	by	by	ADP
ejpam-1851	225	21	ℓ1	ℓ1	NOUN
ejpam-1851	225	22	,	,	PUNCT
ejpam-1851	225	23	.	.	PUNCT
ejpam-1851	225	24	.	.	PUNCT
ejpam-1851	226	1	.	.	PUNCT
ejpam-1851	227	1	,	,	PUNCT
ejpam-1851	227	2	ℓk	ℓk	CCONJ
ejpam-1851	227	3	�	�	PROPN
ejpam-1851	227	4	the	the	DET
ejpam-1851	227	5	smallest	small	ADJ
ejpam-1851	227	6	linear	linear	ADJ
ejpam-1851	227	7	subspace	subspace	NOUN
ejpam-1851	227	8	of	of	ADP
ejpam-1851	227	9	pn	pn	PROPN
ejpam-1851	227	10	containing	contain	VERB
ejpam-1851	227	11	the	the	DET
ejpam-1851	227	12	lines	line	NOUN
ejpam-1851	227	13	ℓ1	ℓ1	NOUN
ejpam-1851	227	14	,	,	PUNCT
ejpam-1851	227	15	.	.	PUNCT
ejpam-1851	227	16	.	.	PUNCT
ejpam-1851	227	17	.	.	PUNCT
ejpam-1851	228	1	,	,	PUNCT
ejpam-1851	228	2	ℓk	ℓk	INTJ
ejpam-1851	228	3	of	of	ADP
ejpam-1851	228	4	pn	pn	PROPN
ejpam-1851	228	5	.	.	PROPN
ejpam-1851	229	1	for	for	ADP
ejpam-1851	229	2	example	example	NOUN
ejpam-1851	229	3	,	,	PUNCT
ejpam-1851	229	4	if	if	SCONJ
ejpam-1851	229	5	ℓ1	ℓ1	VERB
ejpam-1851	229	6	and	and	CCONJ
ejpam-1851	229	7	ℓ2	ℓ2	NOUN
ejpam-1851	229	8	are	be	AUX
ejpam-1851	229	9	two	two	NUM
ejpam-1851	229	10	distinct	distinct	ADJ
ejpam-1851	229	11	lines	line	NOUN
ejpam-1851	229	12	in	in	ADP
ejpam-1851	229	13	pn	pn	PROPN
ejpam-1851	229	14	having	have	VERB
ejpam-1851	229	15	a	a	DET
ejpam-1851	229	16	common	common	ADJ
ejpam-1851	229	17	point	point	NOUN
ejpam-1851	229	18	,	,	PUNCT
ejpam-1851	229	19	then	then	ADV
ejpam-1851	229	20	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	229	21	�	�	PROPN
ejpam-1851	229	22	is	be	AUX
ejpam-1851	229	23	a	a	DET
ejpam-1851	229	24	plane	plane	NOUN
ejpam-1851	229	25	,	,	PUNCT
ejpam-1851	229	26	else	else	ADV
ejpam-1851	229	27	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	229	28	�	�	PROPN
ejpam-1851	229	29	is	be	AUX
ejpam-1851	229	30	a	a	DET
ejpam-1851	229	31	3	3	NUM
ejpam-1851	229	32	-	-	PUNCT
ejpam-1851	229	33	plane	plane	NOUN
ejpam-1851	229	34	.	.	PUNCT
ejpam-1851	230	1	next	next	ADV
ejpam-1851	230	2	we	we	PRON
ejpam-1851	230	3	give	give	VERB
ejpam-1851	230	4	the	the	DET
ejpam-1851	230	5	description	description	NOUN
ejpam-1851	230	6	of	of	ADP
ejpam-1851	230	7	lines	line	NOUN
ejpam-1851	230	8	and	and	CCONJ
ejpam-1851	230	9	planes	plane	NOUN
ejpam-1851	230	10	in	in	ADP
ejpam-1851	230	11	q.	q.	NOUN
ejpam-1851	230	12	lines	line	NOUN
ejpam-1851	230	13	in	in	ADP
ejpam-1851	230	14	the	the	DET
ejpam-1851	230	15	plücker	plücker	NOUN
ejpam-1851	230	16	’s	’s	PART
ejpam-1851	230	17	quadric	quadric	PROPN
ejpam-1851	230	18	q.	q.	NOUN
ejpam-1851	230	19	the	the	DET
ejpam-1851	230	20	lines	line	NOUN
ejpam-1851	230	21	in	in	ADP
ejpam-1851	230	22	the	the	DET
ejpam-1851	230	23	plücker	plücker	NOUN
ejpam-1851	230	24	’s	’s	PART
ejpam-1851	230	25	quadric	quadric	ADJ
ejpam-1851	230	26	q	q	NOUN
ejpam-1851	230	27	are	be	AUX
ejpam-1851	230	28	parametrized	parametrize	VERB
ejpam-1851	230	29	by	by	ADP
ejpam-1851	230	30	the	the	DET
ejpam-1851	230	31	incidence	incidence	ADJ
ejpam-1851	230	32	variety	variety	NOUN
ejpam-1851	230	33	γ	γ	X
ejpam-1851	230	34	=	=	PRON
ejpam-1851	230	35	{	{	PUNCT
ejpam-1851	230	36	(	(	PUNCT
ejpam-1851	230	37	p	p	X
ejpam-1851	230	38	,	,	PUNCT
ejpam-1851	230	39	π	π	NOUN
ejpam-1851	230	40	)	)	PUNCT
ejpam-1851	230	41	|	|	ADV
ejpam-1851	230	42	p	p	NOUN
ejpam-1851	230	43	∈	∈	PROPN
ejpam-1851	230	44	π	π	X
ejpam-1851	230	45	}	}	PUNCT
ejpam-1851	230	46	⊂	⊂	PROPN
ejpam-1851	230	47	p3	p3	PROPN
ejpam-1851	230	48	×g2(p	×g2(p	ADP
ejpam-1851	230	49	3	3	NUM
ejpam-1851	230	50	)	)	PUNCT
ejpam-1851	230	51	.	.	PUNCT
ejpam-1851	231	1	in	in	ADP
ejpam-1851	231	2	fact	fact	NOUN
ejpam-1851	231	3	,	,	PUNCT
ejpam-1851	231	4	let	let	VERB
ejpam-1851	231	5	p	p	PROPN
ejpam-1851	231	6	∈	∈	PROPN
ejpam-1851	231	7	p3	p3	NOUN
ejpam-1851	231	8	and	and	CCONJ
ejpam-1851	231	9	π	π	PROPN
ejpam-1851	231	10	⊂	⊂	PROPN
ejpam-1851	231	11	p3	p3	PROPN
ejpam-1851	231	12	be	be	VERB
ejpam-1851	231	13	a	a	DET
ejpam-1851	231	14	plane	plane	NOUN
ejpam-1851	231	15	through	through	ADP
ejpam-1851	231	16	p.	p.	NOUN
ejpam-1851	231	17	set	set	PROPN
ejpam-1851	231	18	ωp(π	ωp(π	NUM
ejpam-1851	231	19	)	)	PUNCT
ejpam-1851	231	20	=	=	SYM
ejpam-1851	232	1	¦	¦	PROPN
ejpam-1851	232	2	ℓ	ℓ	PROPN
ejpam-1851	232	3	∈	∈	PROPN
ejpam-1851	232	4	g1(p	g1(p	NOUN
ejpam-1851	232	5	3	3	X
ejpam-1851	232	6	)	)	PUNCT
ejpam-1851	232	7	|	|	ADV
ejpam-1851	232	8	p	p	PROPN
ejpam-1851	232	9	∈	∈	PROPN
ejpam-1851	232	10	ℓ	ℓ	PROPN
ejpam-1851	232	11	⊂	⊂	PROPN
ejpam-1851	233	1	π	π	PROPN
ejpam-1851	233	2	©	©	PROPN
ejpam-1851	233	3	.	.	PUNCT
ejpam-1851	234	1	j.	j.	PROPN
ejpam-1851	234	2	rojas	rojas	PROPN
ejpam-1851	234	3	,	,	PUNCT
ejpam-1851	234	4	r.	r.	PROPN
ejpam-1851	234	5	mendoza	mendoza	PROPN
ejpam-1851	234	6	/	/	SYM
ejpam-1851	234	7	eur	eur	PROPN
ejpam-1851	234	8	.	.	PUNCT
ejpam-1851	235	1	j.	j.	PROPN
ejpam-1851	235	2	pure	pure	PROPN
ejpam-1851	235	3	appl	appl	PROPN
ejpam-1851	235	4	.	.	PROPN
ejpam-1851	235	5	math	math	PROPN
ejpam-1851	235	6	,	,	PUNCT
ejpam-1851	235	7	7	7	NUM
ejpam-1851	235	8	(	(	PUNCT
ejpam-1851	235	9	2014	2014	NUM
ejpam-1851	235	10	)	)	PUNCT
ejpam-1851	235	11	,	,	PUNCT
ejpam-1851	235	12	472	472	NUM
ejpam-1851	235	13	-	-	SYM
ejpam-1851	235	14	485	485	NUM
ejpam-1851	235	15	478	478	NUM
ejpam-1851	235	16	proposition	proposition	NOUN
ejpam-1851	235	17	5	5	NUM
ejpam-1851	235	18	.	.	PUNCT
ejpam-1851	236	1	let	let	VERB
ejpam-1851	236	2	p	p	NOUN
ejpam-1851	236	3	:	:	PUNCT
ejpam-1851	236	4	g1(p	g1(p	ADJ
ejpam-1851	236	5	3	3	X
ejpam-1851	236	6	)	)	PUNCT
ejpam-1851	236	7	→	→	PUNCT
ejpam-1851	236	8	p5	p5	ADJ
ejpam-1851	236	9	be	be	AUX
ejpam-1851	236	10	the	the	DET
ejpam-1851	236	11	plücker	plücker	NOUN
ejpam-1851	236	12	embedding	embed	VERB
ejpam-1851	236	13	in	in	ADP
ejpam-1851	236	14	(	(	PUNCT
ejpam-1851	236	15	1	1	NUM
ejpam-1851	236	16	)	)	PUNCT
ejpam-1851	236	17	.	.	PUNCT
ejpam-1851	237	1	if	if	SCONJ
ejpam-1851	237	2	l	l	NOUN
ejpam-1851	237	3	is	be	AUX
ejpam-1851	237	4	a	a	DET
ejpam-1851	237	5	line	line	NOUN
ejpam-1851	237	6	in	in	ADP
ejpam-1851	237	7	p5	p5	ADJ
ejpam-1851	237	8	contained	contain	VERB
ejpam-1851	237	9	in	in	ADP
ejpam-1851	237	10	q	q	NOUN
ejpam-1851	237	11	,	,	PUNCT
ejpam-1851	237	12	then	then	ADV
ejpam-1851	237	13	we	we	PRON
ejpam-1851	237	14	have	have	VERB
ejpam-1851	237	15	:	:	PUNCT
ejpam-1851	237	16	(	(	PUNCT
ejpam-1851	237	17	i	i	NOUN
ejpam-1851	237	18	)	)	PUNCT
ejpam-1851	237	19	p	p	X
ejpam-1851	237	20	(	(	PUNCT
ejpam-1851	237	21	ωp(π	ωp(π	NUM
ejpam-1851	237	22	)	)	PUNCT
ejpam-1851	237	23	)	)	PUNCT
ejpam-1851	238	1	is	be	AUX
ejpam-1851	238	2	a	a	DET
ejpam-1851	238	3	line	line	NOUN
ejpam-1851	238	4	in	in	ADP
ejpam-1851	238	5	p5	p5	ADJ
ejpam-1851	238	6	contained	contain	VERB
ejpam-1851	238	7	in	in	ADP
ejpam-1851	238	8	q	q	NOUN
ejpam-1851	238	9	for	for	ADP
ejpam-1851	238	10	all	all	PRON
ejpam-1851	238	11	(	(	PUNCT
ejpam-1851	238	12	p	p	X
ejpam-1851	238	13	,	,	PUNCT
ejpam-1851	238	14	π	π	NOUN
ejpam-1851	238	15	)	)	PUNCT
ejpam-1851	238	16	∈	∈	PROPN
ejpam-1851	238	17	γ	γ	X
ejpam-1851	238	18	.	.	PROPN
ejpam-1851	238	19	(	(	PUNCT
ejpam-1851	238	20	ii	ii	NOUN
ejpam-1851	238	21	)	)	PUNCT
ejpam-1851	238	22	if	if	SCONJ
ejpam-1851	238	23	p0	p0	NOUN
ejpam-1851	238	24	,	,	PUNCT
ejpam-1851	238	25	p	p	PROPN
ejpam-1851	238	26	∈	∈	PROPN
ejpam-1851	238	27	l	l	NOUN
ejpam-1851	238	28	are	be	AUX
ejpam-1851	238	29	two	two	NUM
ejpam-1851	238	30	different	different	ADJ
ejpam-1851	238	31	points	point	NOUN
ejpam-1851	238	32	such	such	ADJ
ejpam-1851	238	33	that	that	SCONJ
ejpam-1851	238	34	p0	p0	NOUN
ejpam-1851	238	35	=	=	NOUN
ejpam-1851	238	36	p	p	X
ejpam-1851	238	37	(	(	PUNCT
ejpam-1851	238	38	ℓ0	ℓ0	ADV
ejpam-1851	238	39	)	)	PUNCT
ejpam-1851	238	40	and	and	CCONJ
ejpam-1851	238	41	p	p	X
ejpam-1851	238	42	=	=	ADJ
ejpam-1851	238	43	p	p	X
ejpam-1851	238	44	(	(	PUNCT
ejpam-1851	238	45	ℓ	ℓ	NOUN
ejpam-1851	238	46	)	)	PUNCT
ejpam-1851	238	47	,	,	PUNCT
ejpam-1851	238	48	then	then	ADV
ejpam-1851	238	49	ℓ0∩ℓ=	ℓ0∩ℓ=	PROPN
ejpam-1851	238	50	{	{	PUNCT
ejpam-1851	238	51	p	p	NOUN
ejpam-1851	238	52	}	}	PUNCT
ejpam-1851	238	53	and	and	CCONJ
ejpam-1851	238	54	they	they	PRON
ejpam-1851	238	55	determine	determine	VERB
ejpam-1851	238	56	the	the	DET
ejpam-1851	238	57	plane	plane	NOUN
ejpam-1851	238	58	π	π	NOUN
ejpam-1851	238	59	=	=	PUNCT
ejpam-1851	238	60	ℓ0,ℓ	ℓ0,ℓ	PROPN
ejpam-1851	238	61	�	�	PROPN
ejpam-1851	238	62	.	.	PUNCT
ejpam-1851	239	1	in	in	ADP
ejpam-1851	239	2	fact	fact	NOUN
ejpam-1851	239	3	,	,	PUNCT
ejpam-1851	239	4	for	for	ADP
ejpam-1851	239	5	every	every	DET
ejpam-1851	239	6	point	point	NOUN
ejpam-1851	239	7	q	q	NOUN
ejpam-1851	239	8	=	=	X
ejpam-1851	239	9	p	p	X
ejpam-1851	239	10	(	(	PUNCT
ejpam-1851	239	11	m	m	NOUN
ejpam-1851	239	12	)	)	PUNCT
ejpam-1851	239	13	∈	∈	PROPN
ejpam-1851	239	14	l	l	NOUN
ejpam-1851	239	15	holds	hold	VERB
ejpam-1851	239	16	that	that	SCONJ
ejpam-1851	239	17	m	m	VERB
ejpam-1851	239	18	∈	∈	NOUN
ejpam-1851	239	19	ωp(π	ωp(π	NUM
ejpam-1851	239	20	)	)	PUNCT
ejpam-1851	239	21	.	.	PUNCT
ejpam-1851	240	1	planes	plane	NOUN
ejpam-1851	240	2	in	in	ADP
ejpam-1851	240	3	the	the	DET
ejpam-1851	240	4	plücker	plücker	NOUN
ejpam-1851	240	5	’s	’s	PART
ejpam-1851	240	6	quadric	quadric	ADJ
ejpam-1851	240	7	q.	q.	PROPN
ejpam-1851	240	8	there	there	PRON
ejpam-1851	240	9	are	be	VERB
ejpam-1851	240	10	two	two	NUM
ejpam-1851	240	11	families	family	NOUN
ejpam-1851	240	12	of	of	ADP
ejpam-1851	240	13	planes	plane	NOUN
ejpam-1851	240	14	in	in	ADP
ejpam-1851	240	15	the	the	DET
ejpam-1851	240	16	plücker	plücker	NOUN
ejpam-1851	240	17	’s	’s	PART
ejpam-1851	240	18	quadric	quadric	ADJ
ejpam-1851	241	1	q	q	NOUN
ejpam-1851	242	1	parametrized	parametrize	VERB
ejpam-1851	242	2	by	by	ADP
ejpam-1851	242	3	p3	p3	PROPN
ejpam-1851	242	4	and	and	CCONJ
ejpam-1851	242	5	its	its	PRON
ejpam-1851	242	6	dual	dual	ADJ
ejpam-1851	242	7	∨	∨	NOUN
ejpam-1851	242	8	p	p	PROPN
ejpam-1851	242	9	3=	3=	NUM
ejpam-1851	242	10	g2(p	g2(p	PROPN
ejpam-1851	242	11	3	3	NUM
ejpam-1851	242	12	)	)	PUNCT
ejpam-1851	242	13	,	,	PUNCT
ejpam-1851	242	14	respectively	respectively	ADV
ejpam-1851	242	15	.	.	PUNCT
ejpam-1851	243	1	in	in	ADP
ejpam-1851	243	2	fact	fact	NOUN
ejpam-1851	243	3	,	,	PUNCT
ejpam-1851	243	4	let	let	VERB
ejpam-1851	243	5	p	p	PROPN
ejpam-1851	243	6	∈	∈	PROPN
ejpam-1851	243	7	p3	p3	NOUN
ejpam-1851	243	8	and	and	CCONJ
ejpam-1851	243	9	π	π	PROPN
ejpam-1851	243	10	⊂	⊂	PROPN
ejpam-1851	243	11	p3	p3	PROPN
ejpam-1851	243	12	be	be	AUX
ejpam-1851	243	13	a	a	DET
ejpam-1851	243	14	plane	plane	NOUN
ejpam-1851	243	15	.	.	PUNCT
ejpam-1851	244	1	set	set	VERB
ejpam-1851	244	2	ωp	ωp	ADP
ejpam-1851	244	3	=	=	PUNCT
ejpam-1851	244	4	¦	¦	PROPN
ejpam-1851	244	5	ℓ	ℓ	PROPN
ejpam-1851	244	6	∈	∈	PROPN
ejpam-1851	244	7	g1(p	g1(p	NOUN
ejpam-1851	244	8	3	3	X
ejpam-1851	244	9	)	)	PUNCT
ejpam-1851	244	10	|	|	ADV
ejpam-1851	244	11	p	p	X
ejpam-1851	244	12	∈	∈	PROPN
ejpam-1851	244	13	ℓ	ℓ	NOUN
ejpam-1851	244	14	©	©	PROPN
ejpam-1851	244	15	and	and	CCONJ
ejpam-1851	244	16	ω(π	ω(π	NOUN
ejpam-1851	244	17	)	)	PUNCT
ejpam-1851	245	1	=	=	SYM
ejpam-1851	245	2	¦	¦	PROPN
ejpam-1851	245	3	ℓ	ℓ	PROPN
ejpam-1851	245	4	∈	∈	PROPN
ejpam-1851	245	5	g1(p	g1(p	NOUN
ejpam-1851	245	6	3	3	X
ejpam-1851	245	7	)	)	PUNCT
ejpam-1851	245	8	|	|	NOUN
ejpam-1851	245	9	ℓ	ℓ	PROPN
ejpam-1851	245	10	⊂	⊂	PROPN
ejpam-1851	246	1	π	π	X
ejpam-1851	246	2	©	©	PROPN
ejpam-1851	246	3	.	.	PUNCT
ejpam-1851	247	1	proposition	proposition	NOUN
ejpam-1851	247	2	6	6	NUM
ejpam-1851	247	3	.	.	PUNCT
ejpam-1851	248	1	let	let	VERB
ejpam-1851	248	2	p	p	NOUN
ejpam-1851	248	3	:	:	PUNCT
ejpam-1851	248	4	g1(p	g1(p	ADJ
ejpam-1851	248	5	3	3	X
ejpam-1851	248	6	)	)	PUNCT
ejpam-1851	248	7	→	→	PUNCT
ejpam-1851	248	8	p5	p5	ADJ
ejpam-1851	248	9	be	be	AUX
ejpam-1851	248	10	the	the	DET
ejpam-1851	248	11	plücker	plücker	NOUN
ejpam-1851	248	12	embedding	embed	VERB
ejpam-1851	248	13	in	in	ADP
ejpam-1851	248	14	(	(	PUNCT
ejpam-1851	248	15	1	1	NUM
ejpam-1851	248	16	)	)	PUNCT
ejpam-1851	248	17	.	.	PUNCT
ejpam-1851	249	1	if	if	SCONJ
ejpam-1851	249	2	λ	λ	PROPN
ejpam-1851	249	3	is	be	AUX
ejpam-1851	249	4	a	a	DET
ejpam-1851	249	5	plane	plane	NOUN
ejpam-1851	249	6	in	in	ADP
ejpam-1851	249	7	p5	p5	NOUN
ejpam-1851	249	8	contained	contain	VERB
ejpam-1851	249	9	in	in	ADP
ejpam-1851	249	10	q	q	NOUN
ejpam-1851	249	11	,	,	PUNCT
ejpam-1851	249	12	then	then	ADV
ejpam-1851	249	13	we	we	PRON
ejpam-1851	249	14	have	have	VERB
ejpam-1851	249	15	:	:	PUNCT
ejpam-1851	249	16	(	(	PUNCT
ejpam-1851	249	17	i	i	NOUN
ejpam-1851	249	18	)	)	PUNCT
ejpam-1851	249	19	p	p	X
ejpam-1851	249	20	(	(	PUNCT
ejpam-1851	249	21	ωp	ωp	NOUN
ejpam-1851	249	22	)	)	PUNCT
ejpam-1851	249	23	and	and	CCONJ
ejpam-1851	249	24	p	p	X
ejpam-1851	249	25	(	(	PUNCT
ejpam-1851	249	26	ω(π	ω(π	NOUN
ejpam-1851	249	27	)	)	PUNCT
ejpam-1851	249	28	)	)	PUNCT
ejpam-1851	249	29	are	be	AUX
ejpam-1851	249	30	planes	plane	NOUN
ejpam-1851	249	31	in	in	ADP
ejpam-1851	249	32	p5	p5	ADJ
ejpam-1851	249	33	contained	contain	VERB
ejpam-1851	249	34	in	in	ADP
ejpam-1851	249	35	q.	q.	PROPN
ejpam-1851	249	36	(	(	PUNCT
ejpam-1851	249	37	ii	ii	PROPN
ejpam-1851	249	38	)	)	PUNCT
ejpam-1851	249	39	the	the	DET
ejpam-1851	249	40	pre	pre	NOUN
ejpam-1851	249	41	-	-	NOUN
ejpam-1851	249	42	image	image	NOUN
ejpam-1851	249	43	of	of	ADP
ejpam-1851	249	44	λ	λ	PROPN
ejpam-1851	249	45	under	under	ADP
ejpam-1851	249	46	the	the	DET
ejpam-1851	249	47	plücker	plücker	NOUN
ejpam-1851	249	48	embedding	embed	VERB
ejpam-1851	249	49	p	p	NOUN
ejpam-1851	249	50	is	be	AUX
ejpam-1851	249	51	either	either	ADV
ejpam-1851	249	52	:	:	PUNCT
ejpam-1851	249	53	ωp	ωp	NOUN
ejpam-1851	249	54	for	for	ADP
ejpam-1851	249	55	some	some	DET
ejpam-1851	249	56	p	p	PROPN
ejpam-1851	249	57	∈	∈	PROPN
ejpam-1851	249	58	p3	p3	NOUN
ejpam-1851	249	59	or	or	CCONJ
ejpam-1851	249	60	ω(π	ω(π	NOUN
ejpam-1851	249	61	)	)	PUNCT
ejpam-1851	249	62	for	for	ADP
ejpam-1851	249	63	some	some	DET
ejpam-1851	249	64	π	π	PROPN
ejpam-1851	249	65	∈	∈	PROPN
ejpam-1851	249	66	g2(p	g2(p	PROPN
ejpam-1851	249	67	3	3	NUM
ejpam-1851	249	68	)	)	PUNCT
ejpam-1851	249	69	.	.	PUNCT
ejpam-1851	250	1	in	in	ADP
ejpam-1851	250	2	fact	fact	NOUN
ejpam-1851	250	3	,	,	PUNCT
ejpam-1851	250	4	if	if	SCONJ
ejpam-1851	250	5	pi	pi	NOUN
ejpam-1851	250	6	=	=	NOUN
ejpam-1851	250	7	p	p	X
ejpam-1851	250	8	(	(	PUNCT
ejpam-1851	250	9	ℓi	ℓi	PROPN
ejpam-1851	250	10	)	)	PUNCT
ejpam-1851	250	11	,	,	PUNCT
ejpam-1851	251	1	i	i	PRON
ejpam-1851	251	2	=	=	NOUN
ejpam-1851	252	1	0,1,2	0,1,2	NUM
ejpam-1851	252	2	are	be	AUX
ejpam-1851	252	3	three	three	NUM
ejpam-1851	252	4	points	point	NOUN
ejpam-1851	252	5	in	in	ADP
ejpam-1851	252	6	q	q	NOUN
ejpam-1851	252	7	whose	whose	DET
ejpam-1851	252	8	linear	linear	ADJ
ejpam-1851	252	9	span	span	NOUN
ejpam-1851	252	10	it	it	PRON
ejpam-1851	252	11	is	be	AUX
ejpam-1851	252	12	equal	equal	ADJ
ejpam-1851	252	13	to	to	ADP
ejpam-1851	252	14	λ	λ	PROPN
ejpam-1851	252	15	,	,	PUNCT
ejpam-1851	252	16	then	then	ADV
ejpam-1851	252	17	λ	λ	X
ejpam-1851	252	18	=	=	VERB
ejpam-1851	252	19	ωp	ωp	PRON
ejpam-1851	252	20	if	if	SCONJ
ejpam-1851	252	21	p	p	PROPN
ejpam-1851	252	22	∈	∈	PROPN
ejpam-1851	252	23	∩2	∩2	ADV
ejpam-1851	253	1	i=0ℓi	i=0ℓi	PROPN
ejpam-1851	253	2	(	(	PUNCT
ejpam-1851	253	3	and	and	CCONJ
ejpam-1851	253	4	these	these	DET
ejpam-1851	253	5	lines	line	NOUN
ejpam-1851	253	6	are	be	AUX
ejpam-1851	253	7	non	non	ADJ
ejpam-1851	253	8	-	-	ADJ
ejpam-1851	253	9	coplanar	coplanar	ADJ
ejpam-1851	253	10	)	)	PUNCT
ejpam-1851	253	11	else	else	ADV
ejpam-1851	253	12	λ	λ	X
ejpam-1851	253	13	=	=	PUNCT
ejpam-1851	253	14	ω(π	ω(π	NOUN
ejpam-1851	253	15	)	)	PUNCT
ejpam-1851	253	16	with	with	ADP
ejpam-1851	253	17	π	π	PROPN
ejpam-1851	253	18	=	=	PUNCT
ejpam-1851	253	19	〈	〈	PROPN
ejpam-1851	253	20	ℓ0,ℓ1,ℓ2	ℓ0,ℓ1,ℓ2	X
ejpam-1851	253	21	〉	〉	NOUN
ejpam-1851	253	22	.	.	PUNCT
ejpam-1851	254	1	4	4	NUM
ejpam-1851	254	2	.	.	X
ejpam-1851	254	3	description	description	NOUN
ejpam-1851	254	4	of	of	ADP
ejpam-1851	254	5	λ∩q	λ∩q	PROPN
ejpam-1851	254	6	according	accord	VERB
ejpam-1851	254	7	to	to	ADP
ejpam-1851	254	8	the	the	DET
ejpam-1851	254	9	relative	relative	ADJ
ejpam-1851	254	10	position	position	NOUN
ejpam-1851	254	11	of	of	ADP
ejpam-1851	254	12	the	the	DET
ejpam-1851	254	13	four	four	NUM
ejpam-1851	254	14	given	give	VERB
ejpam-1851	254	15	lines	line	NOUN
ejpam-1851	254	16	in	in	ADP
ejpam-1851	254	17	p3	p3	PROPN
ejpam-1851	254	18	we	we	PRON
ejpam-1851	254	19	begin	begin	VERB
ejpam-1851	254	20	this	this	DET
ejpam-1851	254	21	section	section	NOUN
ejpam-1851	254	22	by	by	ADP
ejpam-1851	254	23	introducing	introduce	VERB
ejpam-1851	254	24	some	some	DET
ejpam-1851	254	25	more	more	ADJ
ejpam-1851	254	26	notation	notation	NOUN
ejpam-1851	254	27	.	.	PUNCT
ejpam-1851	255	1	as	as	SCONJ
ejpam-1851	255	2	we	we	PRON
ejpam-1851	255	3	did	do	VERB
ejpam-1851	255	4	for	for	ADP
ejpam-1851	255	5	lines	line	NOUN
ejpam-1851	255	6	we	we	PRON
ejpam-1851	255	7	denote	denote	VERB
ejpam-1851	255	8	by	by	ADP
ejpam-1851	255	9	〈	〈	PROPN
ejpam-1851	255	10	p1	p1	NOUN
ejpam-1851	255	11	,	,	PUNCT
ejpam-1851	255	12	.	.	PUNCT
ejpam-1851	256	1	.	.	PUNCT
ejpam-1851	257	1	.	.	PUNCT
ejpam-1851	258	1	,	,	PUNCT
ejpam-1851	258	2	pk	pk	NOUN
ejpam-1851	258	3	〉	〉	NOUN
ejpam-1851	258	4	the	the	DET
ejpam-1851	258	5	linear	linear	ADJ
ejpam-1851	258	6	span	span	NOUN
ejpam-1851	258	7	of	of	ADP
ejpam-1851	258	8	p1	p1	NOUN
ejpam-1851	258	9	,	,	PUNCT
ejpam-1851	258	10	.	.	PUNCT
ejpam-1851	258	11	.	.	PUNCT
ejpam-1851	258	12	.	.	PUNCT
ejpam-1851	259	1	,	,	PUNCT
ejpam-1851	259	2	pk	pk	NOUN
ejpam-1851	259	3	∈	∈	PROPN
ejpam-1851	259	4	p	p	PROPN
ejpam-1851	259	5	n	n	PROPN
ejpam-1851	259	6	(	(	PUNCT
ejpam-1851	259	7	i.e.	i.e.	X
ejpam-1851	259	8	the	the	DET
ejpam-1851	259	9	smallest	small	ADJ
ejpam-1851	259	10	linear	linear	ADJ
ejpam-1851	259	11	subspace	subspace	NOUN
ejpam-1851	259	12	of	of	ADP
ejpam-1851	259	13	pn	pn	PROPN
ejpam-1851	259	14	containing	contain	VERB
ejpam-1851	259	15	these	these	DET
ejpam-1851	259	16	points	point	NOUN
ejpam-1851	259	17	)	)	PUNCT
ejpam-1851	259	18	.	.	PUNCT
ejpam-1851	260	1	moreover	moreover	ADV
ejpam-1851	260	2	,	,	PUNCT
ejpam-1851	260	3	if	if	SCONJ
ejpam-1851	260	4	p	p	NOUN
ejpam-1851	260	5	is	be	AUX
ejpam-1851	260	6	a	a	DET
ejpam-1851	260	7	point	point	NOUN
ejpam-1851	260	8	and	and	CCONJ
ejpam-1851	260	9	ℓ	ℓ	PROPN
ejpam-1851	260	10	is	be	AUX
ejpam-1851	260	11	a	a	DET
ejpam-1851	260	12	line	line	NOUN
ejpam-1851	260	13	in	in	ADP
ejpam-1851	260	14	pn	pn	PROPN
ejpam-1851	260	15	,	,	PUNCT
ejpam-1851	260	16	then	then	ADV
ejpam-1851	260	17	〈	〈	NOUN
ejpam-1851	260	18	p,ℓ	p,ℓ	VERB
ejpam-1851	260	19	〉	〉	NOUN
ejpam-1851	260	20	also	also	ADV
ejpam-1851	260	21	denote	denote	VERB
ejpam-1851	260	22	the	the	DET
ejpam-1851	260	23	linear	linear	ADJ
ejpam-1851	260	24	span	span	NOUN
ejpam-1851	260	25	of	of	ADP
ejpam-1851	260	26	p	p	PROPN
ejpam-1851	260	27	and	and	CCONJ
ejpam-1851	260	28	ℓ	ℓ	PROPN
ejpam-1851	260	29	in	in	ADP
ejpam-1851	260	30	pn	pn	PROPN
ejpam-1851	260	31	.	.	PUNCT
ejpam-1851	261	1	so	so	ADV
ejpam-1851	261	2	〈	〈	ADP
ejpam-1851	261	3	p,ℓ〉=	p,ℓ〉=	PROPN
ejpam-1851	261	4	ℓ	ℓ	PROPN
ejpam-1851	261	5	,	,	PUNCT
ejpam-1851	261	6	if	if	SCONJ
ejpam-1851	261	7	p	p	PROPN
ejpam-1851	261	8	∈	∈	PROPN
ejpam-1851	261	9	ℓ	ℓ	PROPN
ejpam-1851	261	10	else	else	NOUN
ejpam-1851	261	11	〈	〈	PROPN
ejpam-1851	261	12	p,ℓ	p,ℓ	ADJ
ejpam-1851	261	13	〉	〉	NOUN
ejpam-1851	261	14	is	be	AUX
ejpam-1851	261	15	a	a	DET
ejpam-1851	261	16	3	3	NUM
ejpam-1851	261	17	-	-	PUNCT
ejpam-1851	261	18	plane	plane	NOUN
ejpam-1851	261	19	.	.	PUNCT
ejpam-1851	262	1	let	let	VERB
ejpam-1851	262	2	ℓ1,ℓ2,ℓ3	ℓ1,ℓ2,ℓ3	PROPN
ejpam-1851	262	3	and	and	CCONJ
ejpam-1851	262	4	ℓ4	ℓ4	PROPN
ejpam-1851	262	5	be	be	AUX
ejpam-1851	262	6	four	four	NUM
ejpam-1851	262	7	distinct	distinct	ADJ
ejpam-1851	262	8	lines	line	NOUN
ejpam-1851	262	9	in	in	ADP
ejpam-1851	262	10	p3	p3	PROPN
ejpam-1851	262	11	.	.	PUNCT
ejpam-1851	263	1	let	let	VERB
ejpam-1851	263	2	pi	pi	PROPN
ejpam-1851	263	3	∈	∈	PROPN
ejpam-1851	263	4	q	q	X
ejpam-1851	264	1	(	(	PUNCT
ejpam-1851	264	2	i	i	NOUN
ejpam-1851	264	3	=	=	NOUN
ejpam-1851	264	4	1	1	NUM
ejpam-1851	264	5	,	,	PUNCT
ejpam-1851	264	6	.	.	PUNCT
ejpam-1851	264	7	.	.	PUNCT
ejpam-1851	264	8	.	.	PUNCT
ejpam-1851	265	1	,	,	PUNCT
ejpam-1851	265	2	4	4	X
ejpam-1851	265	3	)	)	PUNCT
ejpam-1851	265	4	be	be	VERB
ejpam-1851	265	5	the	the	DET
ejpam-1851	265	6	image	image	NOUN
ejpam-1851	265	7	of	of	ADP
ejpam-1851	265	8	ℓi	ℓi	NOUN
ejpam-1851	265	9	under	under	ADP
ejpam-1851	265	10	the	the	DET
ejpam-1851	265	11	plücker	plücker	NOUN
ejpam-1851	265	12	embedding	embed	VERB
ejpam-1851	265	13	p	p	X
ejpam-1851	265	14	(	(	PUNCT
ejpam-1851	265	15	in	in	ADP
ejpam-1851	265	16	(	(	PUNCT
ejpam-1851	265	17	1	1	NUM
ejpam-1851	265	18	)	)	PUNCT
ejpam-1851	265	19	)	)	PUNCT
ejpam-1851	266	1	and	and	CCONJ
ejpam-1851	266	2	λ	λ	X
ejpam-1851	266	3	=	=	SYM
ejpam-1851	266	4	〈	〈	PROPN
ejpam-1851	266	5	p1	p1	PROPN
ejpam-1851	266	6	,	,	PUNCT
ejpam-1851	266	7	p2	p2	NOUN
ejpam-1851	266	8	,	,	PUNCT
ejpam-1851	266	9	p3	p3	NOUN
ejpam-1851	266	10	,	,	PUNCT
ejpam-1851	266	11	p4	p4	ADJ
ejpam-1851	266	12	〉	〉	NOUN
ejpam-1851	266	13	⊂	⊂	X
ejpam-1851	266	14	p	p	X
ejpam-1851	266	15	5	5	X
ejpam-1851	266	16	.	.	PUNCT
ejpam-1851	267	1	set	set	NOUN
ejpam-1851	267	2	s	s	VERB
ejpam-1851	267	3	be	be	AUX
ejpam-1851	267	4	the	the	DET
ejpam-1851	267	5	set	set	NOUN
ejpam-1851	267	6	of	of	ADP
ejpam-1851	267	7	solutions	solution	NOUN
ejpam-1851	267	8	of	of	ADP
ejpam-1851	267	9	the	the	DET
ejpam-1851	267	10	4	4	NUM
ejpam-1851	267	11	-	-	PUNCT
ejpam-1851	267	12	lines	line	NOUN
ejpam-1851	267	13	problem	problem	NOUN
ejpam-1851	267	14	in	in	ADP
ejpam-1851	267	15	schubert	schubert	PROPN
ejpam-1851	267	16	calculus	calculus	PROPN
ejpam-1851	267	17	,	,	PUNCT
ejpam-1851	267	18	i.e.	i.e.	X
ejpam-1851	267	19	s	s	X
ejpam-1851	267	20	=	=	SYM
ejpam-1851	267	21	�	�	PROPN
ejpam-1851	267	22	ℓ	ℓ	PROPN
ejpam-1851	267	23	∈	∈	PROPN
ejpam-1851	267	24	g1(p	g1(p	NOUN
ejpam-1851	267	25	3	3	X
ejpam-1851	267	26	)	)	PUNCT
ejpam-1851	268	1	|	|	ADV
ejpam-1851	268	2	ℓ∩	ℓ∩	PROPN
ejpam-1851	268	3	ℓi	ℓi	PROPN
ejpam-1851	268	4	6=	6=	PROPN
ejpam-1851	268	5	;	;	PUNCT
ejpam-1851	268	6	for	for	ADP
ejpam-1851	268	7	1≤	1≤	NUM
ejpam-1851	268	8	i	i	PRON
ejpam-1851	268	9	≤	≤	ADV
ejpam-1851	268	10	4	4	NUM
ejpam-1851	268	11	.	.	PUNCT
ejpam-1851	269	1	we	we	PRON
ejpam-1851	269	2	have	have	VERB
ejpam-1851	269	3	three	three	NUM
ejpam-1851	269	4	major	major	ADJ
ejpam-1851	269	5	cases	case	NOUN
ejpam-1851	269	6	to	to	PART
ejpam-1851	269	7	be	be	AUX
ejpam-1851	269	8	considered	consider	VERB
ejpam-1851	269	9	(	(	PUNCT
ejpam-1851	269	10	cf	cf	NOUN
ejpam-1851	269	11	.	.	PUNCT
ejpam-1851	270	1	subsections	subsection	NOUN
ejpam-1851	270	2	4.1	4.1	NUM
ejpam-1851	270	3	,	,	PUNCT
ejpam-1851	270	4	4.2	4.2	NUM
ejpam-1851	270	5	and	and	CCONJ
ejpam-1851	270	6	4.3	4.3	NUM
ejpam-1851	270	7	)	)	PUNCT
ejpam-1851	270	8	.	.	PUNCT
ejpam-1851	271	1	in	in	ADP
ejpam-1851	271	2	each	each	PRON
ejpam-1851	271	3	of	of	ADP
ejpam-1851	271	4	these	these	DET
ejpam-1851	271	5	cases	case	NOUN
ejpam-1851	271	6	,	,	PUNCT
ejpam-1851	271	7	the	the	DET
ejpam-1851	271	8	linear	linear	ADJ
ejpam-1851	271	9	space	space	NOUN
ejpam-1851	271	10	λ	λ	PROPN
ejpam-1851	271	11	and	and	CCONJ
ejpam-1851	271	12	λ	λ	PROPN
ejpam-1851	271	13	∩q	∩q	PROPN
ejpam-1851	271	14	are	be	AUX
ejpam-1851	271	15	reported	report	VERB
ejpam-1851	271	16	in	in	ADP
ejpam-1851	271	17	tables	table	NOUN
ejpam-1851	271	18	,	,	PUNCT
ejpam-1851	271	19	whose	whose	DET
ejpam-1851	271	20	rows	row	NOUN
ejpam-1851	271	21	are	be	AUX
ejpam-1851	271	22	labeled	label	VERB
ejpam-1851	271	23	according	accord	VERB
ejpam-1851	271	24	to	to	ADP
ejpam-1851	271	25	the	the	DET
ejpam-1851	271	26	position	position	NOUN
ejpam-1851	271	27	that	that	SCONJ
ejpam-1851	271	28	the	the	DET
ejpam-1851	271	29	lines	line	NOUN
ejpam-1851	271	30	can	can	AUX
ejpam-1851	271	31	have	have	VERB
ejpam-1851	271	32	in	in	ADP
ejpam-1851	271	33	p3	p3	PROPN
ejpam-1851	271	34	.	.	PUNCT
ejpam-1851	272	1	next	next	ADV
ejpam-1851	272	2	,	,	PUNCT
ejpam-1851	272	3	we	we	PRON
ejpam-1851	272	4	determine	determine	VERB
ejpam-1851	272	5	λ	λ	PROPN
ejpam-1851	272	6	and	and	CCONJ
ejpam-1851	272	7	λ	λ	X
ejpam-1851	272	8	∩q	∩q	PROPN
ejpam-1851	272	9	in	in	ADP
ejpam-1851	272	10	some	some	PRON
ejpam-1851	272	11	of	of	ADP
ejpam-1851	272	12	these	these	DET
ejpam-1851	272	13	cases	case	NOUN
ejpam-1851	272	14	,	,	PUNCT
ejpam-1851	272	15	which	which	PRON
ejpam-1851	272	16	we	we	PRON
ejpam-1851	272	17	believe	believe	VERB
ejpam-1851	272	18	are	be	AUX
ejpam-1851	272	19	sufficient	sufficient	ADJ
ejpam-1851	272	20	to	to	PART
ejpam-1851	272	21	illustrate	illustrate	VERB
ejpam-1851	272	22	the	the	DET
ejpam-1851	272	23	technique	technique	NOUN
ejpam-1851	272	24	used	use	VERB
ejpam-1851	272	25	in	in	ADP
ejpam-1851	272	26	this	this	DET
ejpam-1851	272	27	computations	computation	NOUN
ejpam-1851	272	28	,	,	PUNCT
ejpam-1851	272	29	we	we	PRON
ejpam-1851	272	30	left	leave	VERB
ejpam-1851	272	31	to	to	ADP
ejpam-1851	272	32	the	the	DET
ejpam-1851	272	33	reader	reader	NOUN
ejpam-1851	272	34	the	the	DET
ejpam-1851	272	35	other	other	ADJ
ejpam-1851	272	36	cases	case	NOUN
ejpam-1851	272	37	in	in	ADP
ejpam-1851	272	38	what	what	PRON
ejpam-1851	272	39	follows	follow	VERB
ejpam-1851	272	40	,	,	PUNCT
ejpam-1851	272	41	for	for	ADP
ejpam-1851	272	42	simplicity	simplicity	NOUN
ejpam-1851	272	43	,	,	PUNCT
ejpam-1851	272	44	we	we	PRON
ejpam-1851	272	45	will	will	AUX
ejpam-1851	272	46	use	use	VERB
ejpam-1851	272	47	the	the	DET
ejpam-1851	272	48	notation	notation	NOUN
ejpam-1851	272	49	ωp(π	ωp(π	NUM
ejpam-1851	272	50	)	)	PUNCT
ejpam-1851	272	51	,	,	PUNCT
ejpam-1851	272	52	ωp	ωp	NOUN
ejpam-1851	272	53	and	and	CCONJ
ejpam-1851	272	54	ω(π	ω(π	NOUN
ejpam-1851	272	55	)	)	PUNCT
ejpam-1851	272	56	for	for	ADP
ejpam-1851	272	57	lines	line	NOUN
ejpam-1851	272	58	and	and	CCONJ
ejpam-1851	272	59	planes	plane	NOUN
ejpam-1851	272	60	in	in	ADP
ejpam-1851	272	61	q	q	NOUN
ejpam-1851	272	62	,	,	PUNCT
ejpam-1851	272	63	according	accord	VERB
ejpam-1851	272	64	to	to	ADP
ejpam-1851	272	65	the	the	DET
ejpam-1851	272	66	identification	identification	NOUN
ejpam-1851	272	67	given	give	VERB
ejpam-1851	272	68	by	by	ADP
ejpam-1851	272	69	the	the	DET
ejpam-1851	272	70	plücker	plücker	NOUN
ejpam-1851	272	71	embedding	embed	VERB
ejpam-1851	272	72	in	in	ADP
ejpam-1851	272	73	propositions	proposition	NOUN
ejpam-1851	272	74	5	5	NUM
ejpam-1851	272	75	and	and	CCONJ
ejpam-1851	272	76	6	6	NUM
ejpam-1851	272	77	,	,	PUNCT
ejpam-1851	272	78	respectively	respectively	ADV
ejpam-1851	272	79	.	.	PUNCT
ejpam-1851	273	1	j.	j.	PROPN
ejpam-1851	273	2	rojas	rojas	PROPN
ejpam-1851	273	3	,	,	PUNCT
ejpam-1851	273	4	r.	r.	PROPN
ejpam-1851	273	5	mendoza	mendoza	PROPN
ejpam-1851	273	6	/	/	SYM
ejpam-1851	273	7	eur	eur	PROPN
ejpam-1851	273	8	.	.	PUNCT
ejpam-1851	274	1	j.	j.	PROPN
ejpam-1851	274	2	pure	pure	PROPN
ejpam-1851	274	3	appl	appl	PROPN
ejpam-1851	274	4	.	.	PROPN
ejpam-1851	274	5	math	math	PROPN
ejpam-1851	274	6	,	,	PUNCT
ejpam-1851	274	7	7	7	NUM
ejpam-1851	274	8	(	(	PUNCT
ejpam-1851	274	9	2014	2014	NUM
ejpam-1851	274	10	)	)	PUNCT
ejpam-1851	274	11	,	,	PUNCT
ejpam-1851	274	12	472	472	NUM
ejpam-1851	274	13	-	-	SYM
ejpam-1851	274	14	485	485	NUM
ejpam-1851	274	15	479	479	NUM
ejpam-1851	274	16	4.1	4.1	NUM
ejpam-1851	274	17	.	.	PUNCT
ejpam-1851	275	1	the	the	DET
ejpam-1851	275	2	four	four	NUM
ejpam-1851	275	3	lines	line	NOUN
ejpam-1851	275	4	are	be	AUX
ejpam-1851	275	5	concurrent	concurrent	ADJ
ejpam-1851	275	6	,	,	PUNCT
ejpam-1851	275	7	say	say	VERB
ejpam-1851	275	8	p	p	PROPN
ejpam-1851	275	9	∈	∈	PROPN
ejpam-1851	275	10	∩4	∩4	NOUN
ejpam-1851	275	11	i=1ℓi	i=1ℓi	VERB
ejpam-1851	275	12	.	.	PUNCT
ejpam-1851	276	1	table	table	NOUN
ejpam-1851	276	2	1	1	NUM
ejpam-1851	276	3	:	:	PUNCT
ejpam-1851	276	4	four	four	NUM
ejpam-1851	276	5	lines	line	NOUN
ejpam-1851	276	6	are	be	AUX
ejpam-1851	276	7	concurrent	concurrent	ADJ
ejpam-1851	276	8	.	.	PUNCT
ejpam-1851	277	1	subcase	subcase	PROPN
ejpam-1851	277	2	position	position	NOUN
ejpam-1851	277	3	of	of	ADP
ejpam-1851	277	4	the	the	DET
ejpam-1851	277	5	lines	line	NOUN
ejpam-1851	277	6	s	s	PART
ejpam-1851	277	7	λ	λ	X
ejpam-1851	277	8	λ∩q	λ∩q	VERB
ejpam-1851	277	9	1.1	1.1	NUM
ejpam-1851	277	10	ℓ1,ℓ2,ℓ3	ℓ1,ℓ2,ℓ3	ADJ
ejpam-1851	277	11	and	and	CCONJ
ejpam-1851	277	12	ℓ4	ℓ4	PROPN
ejpam-1851	277	13	are	be	AUX
ejpam-1851	277	14	contained	contain	VERB
ejpam-1851	277	15	in	in	ADP
ejpam-1851	277	16	the	the	DET
ejpam-1851	277	17	plane	plane	NOUN
ejpam-1851	277	18	π	π	PROPN
ejpam-1851	277	19	ωp	ωp	PRON
ejpam-1851	277	20	∪ω(π	∪ω(π	NOUN
ejpam-1851	277	21	)	)	PUNCT
ejpam-1851	277	22	line	line	NOUN
ejpam-1851	277	23	ωp(π	ωp(π	NUM
ejpam-1851	277	24	)	)	PUNCT
ejpam-1851	277	25	1.2	1.2	NUM
ejpam-1851	277	26	ℓ1,ℓ2,ℓ3	ℓ1,ℓ2,ℓ3	NOUN
ejpam-1851	277	27	and	and	CCONJ
ejpam-1851	277	28	ℓ4	ℓ4	PROPN
ejpam-1851	277	29	are	be	AUX
ejpam-1851	277	30	non	non	ADJ
ejpam-1851	277	31	coplanar	coplanar	ADJ
ejpam-1851	277	32	lines	line	NOUN
ejpam-1851	277	33	ωp	ωp	PRON
ejpam-1851	277	34	plane	plane	NOUN
ejpam-1851	277	35	ωp	ωp	PRON
ejpam-1851	277	36	1.1	1.1	NUM
ejpam-1851	277	37	naturally	naturally	ADV
ejpam-1851	277	38	any	any	DET
ejpam-1851	277	39	line	line	NOUN
ejpam-1851	277	40	passing	pass	VERB
ejpam-1851	277	41	trough	trough	NOUN
ejpam-1851	277	42	p	p	NOUN
ejpam-1851	277	43	is	be	AUX
ejpam-1851	277	44	a	a	DET
ejpam-1851	277	45	solution	solution	NOUN
ejpam-1851	277	46	.	.	PUNCT
ejpam-1851	278	1	on	on	ADP
ejpam-1851	278	2	the	the	DET
ejpam-1851	278	3	other	other	ADJ
ejpam-1851	278	4	hand	hand	NOUN
ejpam-1851	278	5	,	,	PUNCT
ejpam-1851	278	6	if	if	SCONJ
ejpam-1851	278	7	ℓ	ℓ	PROPN
ejpam-1851	278	8	∈	∈	PROPN
ejpam-1851	278	9	s	s	VERB
ejpam-1851	278	10	is	be	AUX
ejpam-1851	278	11	a	a	DET
ejpam-1851	278	12	solution	solution	NOUN
ejpam-1851	278	13	such	such	ADJ
ejpam-1851	278	14	that	that	SCONJ
ejpam-1851	278	15	p	p	PROPN
ejpam-1851	278	16	6∈	6∈	PROPN
ejpam-1851	278	17	ℓ	ℓ	PROPN
ejpam-1851	278	18	,	,	PUNCT
ejpam-1851	278	19	it	it	PRON
ejpam-1851	278	20	certainly	certainly	ADV
ejpam-1851	278	21	meets	meet	VERB
ejpam-1851	278	22	the	the	DET
ejpam-1851	278	23	lines	line	NOUN
ejpam-1851	278	24	ℓ1	ℓ1	VERB
ejpam-1851	278	25	and	and	CCONJ
ejpam-1851	278	26	ℓ2	ℓ2	NOUN
ejpam-1851	278	27	at	at	ADP
ejpam-1851	278	28	two	two	NUM
ejpam-1851	278	29	different	different	ADJ
ejpam-1851	278	30	points	point	NOUN
ejpam-1851	278	31	,	,	PUNCT
ejpam-1851	278	32	which	which	PRON
ejpam-1851	278	33	implies	imply	VERB
ejpam-1851	278	34	that	that	SCONJ
ejpam-1851	278	35	ℓ	ℓ	PROPN
ejpam-1851	278	36	⊂	⊂	PROPN
ejpam-1851	278	37	π	π	PROPN
ejpam-1851	278	38	.	.	PUNCT
ejpam-1851	279	1	therefore	therefore	ADV
ejpam-1851	279	2	s	s	VERB
ejpam-1851	279	3	=	=	PUNCT
ejpam-1851	279	4	ωp	ωp	NOUN
ejpam-1851	279	5	∪	∪	ADJ
ejpam-1851	279	6	ω(π	ω(π	NOUN
ejpam-1851	279	7	)	)	PUNCT
ejpam-1851	279	8	.	.	PUNCT
ejpam-1851	280	1	on	on	ADP
ejpam-1851	280	2	the	the	DET
ejpam-1851	280	3	other	other	ADJ
ejpam-1851	280	4	hand	hand	NOUN
ejpam-1851	280	5	,	,	PUNCT
ejpam-1851	280	6	since	since	SCONJ
ejpam-1851	280	7	pi	pi	PROPN
ejpam-1851	280	8	∈	∈	PROPN
ejpam-1851	280	9	ωp(π	ωp(π	NUM
ejpam-1851	280	10	)	)	PUNCT
ejpam-1851	280	11	for	for	ADP
ejpam-1851	280	12	i	i	PROPN
ejpam-1851	280	13	=	=	NOUN
ejpam-1851	280	14	1	1	NUM
ejpam-1851	280	15	,	,	PUNCT
ejpam-1851	280	16	.	.	PUNCT
ejpam-1851	280	17	.	.	PUNCT
ejpam-1851	280	18	.	.	PUNCT
ejpam-1851	281	1	,	,	PUNCT
ejpam-1851	281	2	4	4	NUM
ejpam-1851	281	3	,	,	PUNCT
ejpam-1851	281	4	then	then	ADV
ejpam-1851	281	5	λ	λ	PROPN
ejpam-1851	281	6	≡	≡	PROPN
ejpam-1851	281	7	ωp(π	ωp(π	NUM
ejpam-1851	281	8	)	)	PUNCT
ejpam-1851	281	9	is	be	AUX
ejpam-1851	281	10	a	a	DET
ejpam-1851	281	11	line	line	NOUN
ejpam-1851	281	12	contained	contain	VERB
ejpam-1851	281	13	in	in	ADP
ejpam-1851	281	14	q.	q.	PROPN
ejpam-1851	281	15	figure	figure	NOUN
ejpam-1851	281	16	1	1	NUM
ejpam-1851	281	17	:	:	PUNCT
ejpam-1851	281	18	case	case	NOUN
ejpam-1851	281	19	1.1	1.1	NUM
ejpam-1851	281	20	4.2	4.2	NUM
ejpam-1851	281	21	.	.	PUNCT
ejpam-1851	282	1	the	the	DET
ejpam-1851	282	2	four	four	NUM
ejpam-1851	282	3	lines	line	NOUN
ejpam-1851	282	4	have	have	VERB
ejpam-1851	282	5	no	no	DET
ejpam-1851	282	6	common	common	ADJ
ejpam-1851	282	7	point	point	NOUN
ejpam-1851	282	8	and	and	CCONJ
ejpam-1851	282	9	at	at	ADV
ejpam-1851	282	10	least	least	ADJ
ejpam-1851	282	11	two	two	NUM
ejpam-1851	282	12	are	be	AUX
ejpam-1851	282	13	coplanar	coplanar	ADJ
ejpam-1851	282	14	.	.	PUNCT
ejpam-1851	283	1	of	of	ADV
ejpam-1851	283	2	course	course	ADV
ejpam-1851	283	3	in	in	ADP
ejpam-1851	283	4	all	all	DET
ejpam-1851	283	5	the	the	DET
ejpam-1851	283	6	subcases	subcase	NOUN
ejpam-1851	283	7	listed	list	VERB
ejpam-1851	283	8	below	below	ADP
ejpam-1851	283	9	we	we	PRON
ejpam-1851	283	10	consider	consider	VERB
ejpam-1851	283	11	an	an	DET
ejpam-1851	283	12	index	index	NOUN
ejpam-1851	283	13	reordering	reordering	NOUN
ejpam-1851	283	14	if	if	SCONJ
ejpam-1851	283	15	necessary	necessary	ADJ
ejpam-1851	283	16	.	.	PUNCT
ejpam-1851	284	1	table	table	NOUN
ejpam-1851	284	2	2	2	NUM
ejpam-1851	284	3	:	:	PUNCT
ejpam-1851	284	4	four	four	NUM
ejpam-1851	284	5	lines	line	NOUN
ejpam-1851	284	6	have	have	VERB
ejpam-1851	284	7	no	no	DET
ejpam-1851	284	8	common	common	ADJ
ejpam-1851	284	9	point	point	NOUN
ejpam-1851	284	10	and	and	CCONJ
ejpam-1851	284	11	at	at	ADV
ejpam-1851	284	12	least	least	ADJ
ejpam-1851	284	13	two	two	NUM
ejpam-1851	284	14	are	be	AUX
ejpam-1851	284	15	coplanar	coplanar	ADJ
ejpam-1851	284	16	.	.	PUNCT
ejpam-1851	285	1	subcase	subcase	PROPN
ejpam-1851	285	2	position	position	NOUN
ejpam-1851	285	3	of	of	ADP
ejpam-1851	285	4	the	the	DET
ejpam-1851	285	5	lines	line	NOUN
ejpam-1851	285	6	s	s	PART
ejpam-1851	285	7	λ	λ	X
ejpam-1851	285	8	λ∩q	λ∩q	VERB
ejpam-1851	285	9	2.1	2.1	NUM
ejpam-1851	285	10	ℓ1,ℓ2,ℓ3	ℓ1,ℓ2,ℓ3	NOUN
ejpam-1851	285	11	and	and	CCONJ
ejpam-1851	285	12	ℓ4	ℓ4	PROPN
ejpam-1851	285	13	are	be	AUX
ejpam-1851	285	14	contained	contain	VERB
ejpam-1851	285	15	in	in	ADP
ejpam-1851	285	16	the	the	DET
ejpam-1851	285	17	plane	plane	NOUN
ejpam-1851	285	18	π	π	PROPN
ejpam-1851	285	19	ω(π	ω(π	NOUN
ejpam-1851	285	20	)	)	PUNCT
ejpam-1851	285	21	plane	plane	NOUN
ejpam-1851	285	22	ω(π	ω(π	NOUN
ejpam-1851	285	23	)	)	PUNCT
ejpam-1851	285	24	2.1	2.1	NUM
ejpam-1851	285	25	of	of	ADP
ejpam-1851	285	26	course	course	NOUN
ejpam-1851	285	27	any	any	DET
ejpam-1851	285	28	line	line	NOUN
ejpam-1851	285	29	contained	contain	VERB
ejpam-1851	285	30	in	in	ADP
ejpam-1851	285	31	the	the	DET
ejpam-1851	285	32	plane	plane	NOUN
ejpam-1851	285	33	π	π	NOUN
ejpam-1851	285	34	is	be	AUX
ejpam-1851	285	35	a	a	DET
ejpam-1851	285	36	solution	solution	NOUN
ejpam-1851	285	37	.	.	PUNCT
ejpam-1851	286	1	now	now	ADV
ejpam-1851	286	2	,	,	PUNCT
ejpam-1851	286	3	since	since	SCONJ
ejpam-1851	286	4	the	the	DET
ejpam-1851	286	5	four	four	NUM
ejpam-1851	286	6	given	give	VERB
ejpam-1851	286	7	lines	line	NOUN
ejpam-1851	286	8	have	have	VERB
ejpam-1851	286	9	not	not	PART
ejpam-1851	286	10	common	common	ADJ
ejpam-1851	286	11	point	point	NOUN
ejpam-1851	286	12	,	,	PUNCT
ejpam-1851	286	13	then	then	ADV
ejpam-1851	286	14	any	any	DET
ejpam-1851	286	15	solution	solution	NOUN
ejpam-1851	286	16	will	will	AUX
ejpam-1851	286	17	meets	meet	VERB
ejpam-1851	286	18	at	at	ADV
ejpam-1851	286	19	least	least	ADJ
ejpam-1851	286	20	two	two	NUM
ejpam-1851	286	21	of	of	ADP
ejpam-1851	286	22	these	these	PRON
ejpam-1851	286	23	at	at	ADP
ejpam-1851	286	24	different	different	ADJ
ejpam-1851	286	25	points	point	NOUN
ejpam-1851	286	26	.	.	PUNCT
ejpam-1851	287	1	therefore	therefore	ADV
ejpam-1851	287	2	,	,	PUNCT
ejpam-1851	287	3	s	s	NOUN
ejpam-1851	287	4	=	=	ADJ
ejpam-1851	287	5	ω(π	ω(π	NOUN
ejpam-1851	287	6	)	)	PUNCT
ejpam-1851	287	7	.	.	PUNCT
ejpam-1851	288	1	now	now	ADV
ejpam-1851	288	2	,	,	PUNCT
ejpam-1851	288	3	since	since	SCONJ
ejpam-1851	288	4	the	the	DET
ejpam-1851	288	5	four	four	NUM
ejpam-1851	288	6	lines	line	NOUN
ejpam-1851	288	7	are	be	AUX
ejpam-1851	288	8	not	not	PART
ejpam-1851	288	9	concurrent	concurrent	ADJ
ejpam-1851	288	10	,	,	PUNCT
ejpam-1851	288	11	then	then	ADV
ejpam-1851	288	12	three	three	NUM
ejpam-1851	288	13	of	of	ADP
ejpam-1851	288	14	these	these	DET
ejpam-1851	288	15	four	four	NUM
ejpam-1851	288	16	points	point	NOUN
ejpam-1851	288	17	(	(	PUNCT
ejpam-1851	288	18	the	the	DET
ejpam-1851	288	19	pi	pi	NOUN
ejpam-1851	288	20	’s	’s	PART
ejpam-1851	288	21	)	)	PUNCT
ejpam-1851	288	22	determine	determine	VERB
ejpam-1851	288	23	the	the	DET
ejpam-1851	288	24	plane	plane	NOUN
ejpam-1851	288	25	ω(π	ω(π	NOUN
ejpam-1851	288	26	)	)	PUNCT
ejpam-1851	288	27	.	.	PUNCT
ejpam-1851	289	1	in	in	ADP
ejpam-1851	289	2	fact	fact	NOUN
ejpam-1851	289	3	,	,	PUNCT
ejpam-1851	289	4	λ	λ	PROPN
ejpam-1851	289	5	≡	≡	PROPN
ejpam-1851	289	6	ω(π	ω(π	NOUN
ejpam-1851	289	7	)	)	PUNCT
ejpam-1851	289	8	,	,	PUNCT
ejpam-1851	289	9	so	so	CCONJ
ejpam-1851	289	10	it	it	PRON
ejpam-1851	289	11	is	be	AUX
ejpam-1851	289	12	a	a	DET
ejpam-1851	289	13	plane	plane	NOUN
ejpam-1851	289	14	contained	contain	VERB
ejpam-1851	289	15	in	in	ADP
ejpam-1851	289	16	q.	q.	PROPN
ejpam-1851	289	17	j.	j.	PROPN
ejpam-1851	289	18	rojas	rojas	PROPN
ejpam-1851	289	19	,	,	PUNCT
ejpam-1851	289	20	r.	r.	PROPN
ejpam-1851	289	21	mendoza	mendoza	PROPN
ejpam-1851	289	22	/	/	SYM
ejpam-1851	289	23	eur	eur	PROPN
ejpam-1851	289	24	.	.	PUNCT
ejpam-1851	290	1	j.	j.	PROPN
ejpam-1851	290	2	pure	pure	PROPN
ejpam-1851	290	3	appl	appl	PROPN
ejpam-1851	290	4	.	.	PROPN
ejpam-1851	290	5	math	math	PROPN
ejpam-1851	290	6	,	,	PUNCT
ejpam-1851	290	7	7	7	NUM
ejpam-1851	290	8	(	(	PUNCT
ejpam-1851	290	9	2014	2014	NUM
ejpam-1851	290	10	)	)	PUNCT
ejpam-1851	290	11	,	,	PUNCT
ejpam-1851	290	12	472	472	NUM
ejpam-1851	290	13	-	-	SYM
ejpam-1851	290	14	485	485	NUM
ejpam-1851	290	15	480	480	NUM
ejpam-1851	290	16	table	table	NOUN
ejpam-1851	290	17	3	3	NUM
ejpam-1851	290	18	:	:	SYM
ejpam-1851	290	19	2.2	2.2	NUM
ejpam-1851	290	20	exactly	exactly	ADV
ejpam-1851	290	21	three	three	NUM
ejpam-1851	290	22	lines	line	NOUN
ejpam-1851	290	23	are	be	AUX
ejpam-1851	290	24	coplanar	coplanar	ADJ
ejpam-1851	290	25	.	.	PUNCT
ejpam-1851	291	1	assume	assume	VERB
ejpam-1851	291	2	that	that	SCONJ
ejpam-1851	291	3	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	291	4	and	and	CCONJ
ejpam-1851	291	5	ℓ3	ℓ3	PROPN
ejpam-1851	291	6	are	be	AUX
ejpam-1851	291	7	contained	contain	VERB
ejpam-1851	291	8	in	in	ADP
ejpam-1851	291	9	the	the	DET
ejpam-1851	291	10	plane	plane	NOUN
ejpam-1851	291	11	π	π	NOUN
ejpam-1851	291	12	and	and	CCONJ
ejpam-1851	291	13	π∩	π∩	PROPN
ejpam-1851	291	14	ℓ4	ℓ4	PROPN
ejpam-1851	291	15	=	=	PUNCT
ejpam-1851	291	16	{	{	PUNCT
ejpam-1851	291	17	q	q	X
ejpam-1851	291	18	}	}	PUNCT
ejpam-1851	291	19	.	.	PUNCT
ejpam-1851	292	1	subcase	subcase	PROPN
ejpam-1851	292	2	position	position	NOUN
ejpam-1851	292	3	of	of	ADP
ejpam-1851	292	4	the	the	DET
ejpam-1851	292	5	lines	line	NOUN
ejpam-1851	292	6	s	s	PART
ejpam-1851	292	7	λ	λ	X
ejpam-1851	292	8	λ∩q	λ∩q	NOUN
ejpam-1851	292	9	2.2.1	2.2.1	NUM
ejpam-1851	292	10	⋂3	⋂3	PROPN
ejpam-1851	292	11	i=1	i=1	PROPN
ejpam-1851	292	12	ℓi	ℓi	NOUN
ejpam-1851	293	1	=	=	PUNCT
ejpam-1851	293	2	{	{	PUNCT
ejpam-1851	293	3	p	p	X
ejpam-1851	293	4	}	}	PUNCT
ejpam-1851	293	5	ωp(〈p,ℓ4〉)∪ωq(π	ωp(〈p,ℓ4〉)∪ωq(π	ADJ
ejpam-1851	293	6	)	)	PUNCT
ejpam-1851	293	7	plane	plane	NOUN
ejpam-1851	293	8	union	union	NOUN
ejpam-1851	293	9	of	of	ADP
ejpam-1851	293	10	two	two	NUM
ejpam-1851	293	11	lines	line	NOUN
ejpam-1851	293	12	2.2.2	2.2.2	NUM
ejpam-1851	293	13	⋂3	⋂3	PROPN
ejpam-1851	293	14	i=1	i=1	PROPN
ejpam-1851	293	15	ℓi	ℓi	PROPN
ejpam-1851	293	16	=	=	X
ejpam-1851	293	17	;	;	PUNCT
ejpam-1851	293	18	ωq(π	ωq(π	NOUN
ejpam-1851	293	19	)	)	PUNCT
ejpam-1851	293	20	3	3	NUM
ejpam-1851	293	21	-	-	PUNCT
ejpam-1851	293	22	plane	plane	NOUN
ejpam-1851	293	23	union	union	NOUN
ejpam-1851	293	24	of	of	ADP
ejpam-1851	293	25	two	two	NUM
ejpam-1851	293	26	planes	plane	NOUN
ejpam-1851	293	27	2.2.1	2.2.1	NUM
ejpam-1851	293	28	let	let	VERB
ejpam-1851	293	29	ℓ	ℓ	PROPN
ejpam-1851	293	30	∈	∈	PROPN
ejpam-1851	293	31	s	s	AUX
ejpam-1851	293	32	be	be	AUX
ejpam-1851	293	33	a	a	DET
ejpam-1851	293	34	solution	solution	NOUN
ejpam-1851	293	35	which	which	PRON
ejpam-1851	293	36	meets	meet	VERB
ejpam-1851	293	37	the	the	DET
ejpam-1851	293	38	lines	line	NOUN
ejpam-1851	293	39	ℓ1	ℓ1	VERB
ejpam-1851	293	40	and	and	CCONJ
ejpam-1851	293	41	ℓ2	ℓ2	NOUN
ejpam-1851	293	42	at	at	ADP
ejpam-1851	293	43	two	two	NUM
ejpam-1851	293	44	different	different	ADJ
ejpam-1851	293	45	points	point	NOUN
ejpam-1851	293	46	,	,	PUNCT
ejpam-1851	293	47	then	then	ADV
ejpam-1851	293	48	ℓ	ℓ	PROPN
ejpam-1851	293	49	∈	∈	PROPN
ejpam-1851	293	50	ωq(π	ωq(π	NOUN
ejpam-1851	293	51	)	)	PUNCT
ejpam-1851	293	52	(	(	PUNCT
ejpam-1851	293	53	since	since	SCONJ
ejpam-1851	293	54	π	π	PROPN
ejpam-1851	293	55	∩	∩	ADJ
ejpam-1851	293	56	ℓ4	ℓ4	NOUN
ejpam-1851	293	57	=	=	PUNCT
ejpam-1851	293	58	{	{	PUNCT
ejpam-1851	293	59	q	q	NOUN
ejpam-1851	293	60	}	}	PUNCT
ejpam-1851	293	61	)	)	PUNCT
ejpam-1851	293	62	.	.	PUNCT
ejpam-1851	294	1	else	else	ADV
ejpam-1851	294	2	ℓ	ℓ	PROPN
ejpam-1851	294	3	meets	meet	VERB
ejpam-1851	294	4	ℓ1	ℓ1	NOUN
ejpam-1851	294	5	and	and	CCONJ
ejpam-1851	294	6	ℓ2	ℓ2	NOUN
ejpam-1851	294	7	at	at	ADP
ejpam-1851	294	8	p	p	NOUN
ejpam-1851	294	9	and	and	CCONJ
ejpam-1851	294	10	ℓ4	ℓ4	NOUN
ejpam-1851	294	11	at	at	ADP
ejpam-1851	294	12	some	some	DET
ejpam-1851	294	13	point	point	NOUN
ejpam-1851	294	14	different	different	ADJ
ejpam-1851	294	15	of	of	ADP
ejpam-1851	294	16	p	p	NOUN
ejpam-1851	294	17	,	,	PUNCT
ejpam-1851	294	18	then	then	ADV
ejpam-1851	294	19	ℓ	ℓ	PROPN
ejpam-1851	294	20	⊂	⊂	PROPN
ejpam-1851	295	1	〈	〈	PROPN
ejpam-1851	295	2	p,ℓ4	p,ℓ4	PROPN
ejpam-1851	295	3	〉	〉	NOUN
ejpam-1851	295	4	.	.	PUNCT
ejpam-1851	296	1	so	so	ADV
ejpam-1851	296	2	ℓ	ℓ	PROPN
ejpam-1851	296	3	∈	∈	PROPN
ejpam-1851	296	4	ωp(〈p,ℓ4	ωp(〈p,ℓ4	NOUN
ejpam-1851	296	5	〉	〉	NOUN
ejpam-1851	296	6	)	)	PUNCT
ejpam-1851	296	7	.	.	PUNCT
ejpam-1851	297	1	therefore	therefore	ADV
ejpam-1851	297	2	,	,	PUNCT
ejpam-1851	297	3	s	s	PART
ejpam-1851	297	4	=	=	SYM
ejpam-1851	297	5	ωq(π	ωq(π	NUM
ejpam-1851	297	6	)	)	PUNCT
ejpam-1851	297	7	∪ωp(〈p,ℓ4	∪ωp(〈p,ℓ4	NUM
ejpam-1851	297	8	〉	〉	NOUN
ejpam-1851	297	9	)	)	PUNCT
ejpam-1851	297	10	and	and	CCONJ
ejpam-1851	297	11	it	it	PRON
ejpam-1851	297	12	can	can	AUX
ejpam-1851	297	13	be	be	AUX
ejpam-1851	297	14	identified	identify	VERB
ejpam-1851	297	15	with	with	ADP
ejpam-1851	297	16	two	two	NUM
ejpam-1851	297	17	projective	projective	ADJ
ejpam-1851	297	18	lines	line	NOUN
ejpam-1851	297	19	having	have	VERB
ejpam-1851	297	20	a	a	DET
ejpam-1851	297	21	common	common	ADJ
ejpam-1851	297	22	point	point	NOUN
ejpam-1851	297	23	(	(	PUNCT
ejpam-1851	297	24	just	just	ADV
ejpam-1851	297	25	a	a	DET
ejpam-1851	297	26	cross	cross	NOUN
ejpam-1851	297	27	!	!	PUNCT
ejpam-1851	297	28	)	)	PUNCT
ejpam-1851	297	29	.	.	PUNCT
ejpam-1851	298	1	on	on	ADP
ejpam-1851	298	2	the	the	DET
ejpam-1851	298	3	other	other	ADJ
ejpam-1851	298	4	hand	hand	NOUN
ejpam-1851	298	5	,	,	PUNCT
ejpam-1851	298	6	since	since	SCONJ
ejpam-1851	298	7	p1	p1	NOUN
ejpam-1851	298	8	,	,	PUNCT
ejpam-1851	298	9	p2	p2	PROPN
ejpam-1851	298	10	and	and	CCONJ
ejpam-1851	298	11	pi	pi	NOUN
ejpam-1851	298	12	lies	lie	NOUN
ejpam-1851	298	13	on	on	ADP
ejpam-1851	298	14	the	the	DET
ejpam-1851	298	15	line	line	NOUN
ejpam-1851	298	16	ωp(π	ωp(π	NUM
ejpam-1851	298	17	)	)	PUNCT
ejpam-1851	298	18	and	and	CCONJ
ejpam-1851	298	19	p4	p4	PROPN
ejpam-1851	298	20	6∈	6∈	PROPN
ejpam-1851	298	21	ωp(π	ωp(π	NUM
ejpam-1851	298	22	)	)	PUNCT
ejpam-1851	298	23	(	(	PUNCT
ejpam-1851	298	24	p	p	PROPN
ejpam-1851	298	25	6∈	6∈	PROPN
ejpam-1851	298	26	ℓ4	ℓ4	PROPN
ejpam-1851	298	27	)	)	PUNCT
ejpam-1851	298	28	we	we	PRON
ejpam-1851	298	29	conclude	conclude	VERB
ejpam-1851	298	30	that	that	SCONJ
ejpam-1851	298	31	λ	λ	PROPN
ejpam-1851	298	32	is	be	AUX
ejpam-1851	298	33	a	a	DET
ejpam-1851	298	34	plane	plane	NOUN
ejpam-1851	298	35	in	in	ADP
ejpam-1851	298	36	p5	p5	ADJ
ejpam-1851	298	37	not	not	PART
ejpam-1851	298	38	contained	contain	VERB
ejpam-1851	298	39	in	in	ADP
ejpam-1851	298	40	q.	q.	PROPN
ejpam-1851	298	41	so	so	ADV
ejpam-1851	298	42	λ∩q	λ∩q	PROPN
ejpam-1851	298	43	is	be	AUX
ejpam-1851	298	44	a	a	DET
ejpam-1851	298	45	conic	conic	ADJ
ejpam-1851	298	46	.	.	PUNCT
ejpam-1851	299	1	note	note	VERB
ejpam-1851	299	2	that	that	SCONJ
ejpam-1851	299	3	the	the	DET
ejpam-1851	299	4	line	line	NOUN
ejpam-1851	299	5	l1,2	l1,2	PROPN
ejpam-1851	299	6	=	=	PUNCT
ejpam-1851	299	7	〈	〈	NOUN
ejpam-1851	299	8	p1	p1	NOUN
ejpam-1851	299	9	,	,	PUNCT
ejpam-1851	299	10	p2	p2	X
ejpam-1851	299	11	〉	〉	NOUN
ejpam-1851	299	12	⊂	⊂	X
ejpam-1851	299	13	λ∩q	λ∩q	PROPN
ejpam-1851	299	14	and	and	CCONJ
ejpam-1851	299	15	p4	p4	PROPN
ejpam-1851	299	16	6∈	6∈	PROPN
ejpam-1851	299	17	l1,2	l1,2	PROPN
ejpam-1851	299	18	.	.	PUNCT
ejpam-1851	299	19	therefore	therefore	ADV
ejpam-1851	299	20	,	,	PUNCT
ejpam-1851	299	21	λ∩q	λ∩q	PROPN
ejpam-1851	299	22	is	be	AUX
ejpam-1851	299	23	the	the	DET
ejpam-1851	299	24	union	union	NOUN
ejpam-1851	299	25	two	two	NUM
ejpam-1851	299	26	lines	line	NOUN
ejpam-1851	299	27	.	.	PUNCT
ejpam-1851	300	1	in	in	ADP
ejpam-1851	300	2	fact	fact	NOUN
ejpam-1851	300	3	,	,	PUNCT
ejpam-1851	300	4	λ∩q	λ∩q	PROPN
ejpam-1851	300	5	=	=	PUNCT
ejpam-1851	300	6	l1,2∪	l1,2∪	PROPN
ejpam-1851	300	7	l	l	NOUN
ejpam-1851	300	8	where	where	SCONJ
ejpam-1851	300	9	l	l	NOUN
ejpam-1851	300	10	=	=	PUNCT
ejpam-1851	300	11	〈	〈	PROPN
ejpam-1851	300	12	p4	p4	ADJ
ejpam-1851	300	13	,	,	PUNCT
ejpam-1851	300	14	m	m	VERB
ejpam-1851	300	15	〉	〉	NOUN
ejpam-1851	300	16	with	with	ADP
ejpam-1851	300	17	m	m	PROPN
ejpam-1851	300	18	=	=	NOUN
ejpam-1851	300	19	p	p	X
ejpam-1851	300	20	(	(	PUNCT
ejpam-1851	300	21	〈	〈	PROPN
ejpam-1851	300	22	p	p	NOUN
ejpam-1851	300	23	,	,	PUNCT
ejpam-1851	300	24	q	q	NOUN
ejpam-1851	300	25	〉	〉	NOUN
ejpam-1851	300	26	)	)	PUNCT
ejpam-1851	300	27	.	.	PUNCT
ejpam-1851	301	1	table	table	NOUN
ejpam-1851	301	2	4	4	NUM
ejpam-1851	301	3	:	:	SYM
ejpam-1851	301	4	2.3	2.3	NUM
ejpam-1851	301	5	any	any	DET
ejpam-1851	301	6	three	three	NUM
ejpam-1851	301	7	lines	line	NOUN
ejpam-1851	301	8	are	be	AUX
ejpam-1851	301	9	non	non	ADJ
ejpam-1851	301	10	coplanar	coplanar	ADJ
ejpam-1851	301	11	and	and	CCONJ
ejpam-1851	301	12	at	at	ADV
ejpam-1851	301	13	least	least	ADV
ejpam-1851	301	14	two	two	NUM
ejpam-1851	301	15	pair	pair	NOUN
ejpam-1851	301	16	of	of	ADP
ejpam-1851	301	17	lines	line	NOUN
ejpam-1851	301	18	are	be	AUX
ejpam-1851	301	19	coplanar	coplanar	ADJ
ejpam-1851	301	20	with	with	ADP
ejpam-1851	301	21	ℓi	ℓi	PROPN
ejpam-1851	301	22	6=	6=	PROPN
ejpam-1851	301	23	π1	π1	PROPN
ejpam-1851	301	24	∩π2∀	∩π2∀	NOUN
ejpam-1851	301	25	i.	i.	NOUN
ejpam-1851	301	26	assume	assume	VERB
ejpam-1851	301	27	that	that	SCONJ
ejpam-1851	301	28	ℓ1	ℓ1	ADJ
ejpam-1851	301	29	∩	∩	ADJ
ejpam-1851	301	30	ℓ2	ℓ2	NOUN
ejpam-1851	301	31	=	=	SYM
ejpam-1851	301	32	{	{	PUNCT
ejpam-1851	301	33	p	p	NOUN
ejpam-1851	301	34	}	}	PUNCT
ejpam-1851	301	35	and	and	CCONJ
ejpam-1851	301	36	ℓ3	ℓ3	PROPN
ejpam-1851	302	1	∩	∩	ADJ
ejpam-1851	302	2	ℓ4	ℓ4	NOUN
ejpam-1851	302	3	=	=	PUNCT
ejpam-1851	302	4	{	{	PUNCT
ejpam-1851	302	5	q	q	X
ejpam-1851	302	6	}	}	PUNCT
ejpam-1851	302	7	.	.	PUNCT
ejpam-1851	303	1	set	set	VERB
ejpam-1851	303	2	π1	π1	NOUN
ejpam-1851	303	3	=	=	SYM
ejpam-1851	303	4	〈	〈	PROPN
ejpam-1851	303	5	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	303	6	〉	〉	NOUN
ejpam-1851	303	7	and	and	CCONJ
ejpam-1851	303	8	π2	π2	NOUN
ejpam-1851	303	9	=	=	PUNCT
ejpam-1851	303	10	〈	〈	PROPN
ejpam-1851	303	11	ℓ3,ℓ4	ℓ3,ℓ4	PROPN
ejpam-1851	303	12	〉	〉	NOUN
ejpam-1851	303	13	.	.	PUNCT
ejpam-1851	304	1	subcase	subcase	PROPN
ejpam-1851	304	2	position	position	NOUN
ejpam-1851	304	3	of	of	ADP
ejpam-1851	304	4	the	the	DET
ejpam-1851	304	5	lines	line	NOUN
ejpam-1851	304	6	s	s	PART
ejpam-1851	304	7	λ	λ	X
ejpam-1851	304	8	λ∩q	λ∩q	NOUN
ejpam-1851	304	9	2.3.1	2.3.1	NUM
ejpam-1851	304	10	p	p	NOUN
ejpam-1851	304	11	,	,	PUNCT
ejpam-1851	304	12	q	q	NOUN
ejpam-1851	304	13	∈	∈	PROPN
ejpam-1851	304	14	π1	π1	NOUN
ejpam-1851	304	15	∩π2	∩π2	NOUN
ejpam-1851	304	16	ωp(π2)∪ωq(π1	ωp(π2)∪ωq(π1	NUM
ejpam-1851	304	17	)	)	PUNCT
ejpam-1851	304	18	plane	plane	NOUN
ejpam-1851	304	19	union	union	NOUN
ejpam-1851	304	20	of	of	ADP
ejpam-1851	304	21	two	two	NUM
ejpam-1851	304	22	lines	line	NOUN
ejpam-1851	304	23	2.3.2	2.3.2	NUM
ejpam-1851	304	24	p	p	NOUN
ejpam-1851	304	25	∈	∈	PROPN
ejpam-1851	304	26	π1	π1	NOUN
ejpam-1851	304	27	∩π2	∩π2	PROPN
ejpam-1851	304	28	6∋	6∋	NUM
ejpam-1851	304	29	q	q	NOUN
ejpam-1851	304	30	ωp(π2	ωp(π2	NUM
ejpam-1851	304	31	)	)	PUNCT
ejpam-1851	304	32	3	3	NUM
ejpam-1851	304	33	-	-	PUNCT
ejpam-1851	304	34	plane	plane	NOUN
ejpam-1851	304	35	union	union	NOUN
ejpam-1851	304	36	of	of	ADP
ejpam-1851	304	37	two	two	NUM
ejpam-1851	304	38	planes	plane	NOUN
ejpam-1851	304	39	2.3.3	2.3.3	NUM
ejpam-1851	304	40	p	p	NOUN
ejpam-1851	304	41	6∈	6∈	NOUN
ejpam-1851	304	42	π1	π1	NOUN
ejpam-1851	304	43	∩π2	∩π2	PROPN
ejpam-1851	304	44	6∋	6∋	NUM
ejpam-1851	304	45	q	q	PROPN
ejpam-1851	304	46	¦	¦	PROPN
ejpam-1851	304	47	π1	π1	PROPN
ejpam-1851	304	48	∩π2,ℓp	∩π2,ℓp	PROPN
ejpam-1851	304	49	,	,	PUNCT
ejpam-1851	304	50	q	q	PROPN
ejpam-1851	304	51	©	©	PROPN
ejpam-1851	304	52	3	3	NUM
ejpam-1851	304	53	-	-	PUNCT
ejpam-1851	304	54	plane	plane	NOUN
ejpam-1851	304	55	nonsingular	nonsingular	ADJ
ejpam-1851	304	56	quadric	quadric	ADJ
ejpam-1851	304	57	figure	figure	NOUN
ejpam-1851	304	58	2	2	NUM
ejpam-1851	304	59	:	:	PUNCT
ejpam-1851	304	60	case	case	NOUN
ejpam-1851	304	61	2.3.3	2.3.3	NUM
ejpam-1851	304	62	2.3.3	2.3.3	NUM
ejpam-1851	304	63	let	let	VERB
ejpam-1851	304	64	ℓ	ℓ	NOUN
ejpam-1851	304	65	be	be	AUX
ejpam-1851	304	66	a	a	DET
ejpam-1851	304	67	solution	solution	NOUN
ejpam-1851	304	68	.	.	PUNCT
ejpam-1851	305	1	next	next	ADV
ejpam-1851	305	2	we	we	PRON
ejpam-1851	305	3	consider	consider	VERB
ejpam-1851	305	4	the	the	DET
ejpam-1851	305	5	following	follow	VERB
ejpam-1851	305	6	two	two	NUM
ejpam-1851	305	7	cases	case	NOUN
ejpam-1851	305	8	.	.	PUNCT
ejpam-1851	306	1	(	(	PUNCT
ejpam-1851	306	2	i	i	NOUN
ejpam-1851	306	3	)	)	PUNCT
ejpam-1851	306	4	assume	assume	VERB
ejpam-1851	306	5	that	that	SCONJ
ejpam-1851	306	6	p	p	PROPN
ejpam-1851	306	7	∈	∈	PROPN
ejpam-1851	306	8	ℓ.	ℓ.	NOUN
ejpam-1851	306	9	suppose	suppose	VERB
ejpam-1851	306	10	that	that	SCONJ
ejpam-1851	306	11	ℓ	ℓ	PROPN
ejpam-1851	306	12	meets	meet	VERB
ejpam-1851	306	13	the	the	DET
ejpam-1851	306	14	lines	line	NOUN
ejpam-1851	306	15	ℓ3	ℓ3	PROPN
ejpam-1851	306	16	and	and	CCONJ
ejpam-1851	306	17	ℓ4	ℓ4	PROPN
ejpam-1851	306	18	at	at	ADP
ejpam-1851	306	19	two	two	NUM
ejpam-1851	306	20	different	different	ADJ
ejpam-1851	306	21	points	point	NOUN
ejpam-1851	306	22	,	,	PUNCT
ejpam-1851	306	23	then	then	ADV
ejpam-1851	306	24	ℓ	ℓ	PROPN
ejpam-1851	306	25	⊂	⊂	PROPN
ejpam-1851	306	26	π2	π2	X
ejpam-1851	306	27	which	which	PRON
ejpam-1851	306	28	implies	imply	VERB
ejpam-1851	306	29	that	that	SCONJ
ejpam-1851	306	30	p	p	PROPN
ejpam-1851	306	31	∈	∈	PROPN
ejpam-1851	306	32	π1	π1	NOUN
ejpam-1851	306	33	∩π2	∩π2	PROPN
ejpam-1851	306	34	.	.	PUNCT
ejpam-1851	307	1	but	but	CCONJ
ejpam-1851	307	2	this	this	PRON
ejpam-1851	307	3	is	be	AUX
ejpam-1851	307	4	impossible	impossible	ADJ
ejpam-1851	307	5	,	,	PUNCT
ejpam-1851	307	6	so	so	ADV
ejpam-1851	307	7	q	q	PROPN
ejpam-1851	307	8	∈	∈	PROPN
ejpam-1851	307	9	ℓ.	ℓ.	NOUN
ejpam-1851	307	10	therefore	therefore	ADV
ejpam-1851	307	11	ℓ=	ℓ=	PROPN
ejpam-1851	307	12	ℓp	ℓp	NOUN
ejpam-1851	307	13	,	,	PUNCT
ejpam-1851	307	14	q.	q.	NOUN
ejpam-1851	307	15	similarly	similarly	ADV
ejpam-1851	307	16	,	,	PUNCT
ejpam-1851	307	17	we	we	PRON
ejpam-1851	307	18	conclude	conclude	VERB
ejpam-1851	307	19	that	that	DET
ejpam-1851	307	20	ℓ=	ℓ=	PROPN
ejpam-1851	307	21	ℓp	ℓp	NOUN
ejpam-1851	307	22	,	,	PUNCT
ejpam-1851	307	23	q	q	NOUN
ejpam-1851	307	24	,	,	PUNCT
ejpam-1851	307	25	if	if	SCONJ
ejpam-1851	307	26	q	q	X
ejpam-1851	307	27	∈	∈	PROPN
ejpam-1851	307	28	ℓ.	ℓ.	NOUN
ejpam-1851	307	29	j.	j.	PROPN
ejpam-1851	307	30	rojas	rojas	PROPN
ejpam-1851	307	31	,	,	PUNCT
ejpam-1851	307	32	r.	r.	PROPN
ejpam-1851	307	33	mendoza	mendoza	PROPN
ejpam-1851	307	34	/	/	SYM
ejpam-1851	307	35	eur	eur	PROPN
ejpam-1851	307	36	.	.	PUNCT
ejpam-1851	308	1	j.	j.	PROPN
ejpam-1851	308	2	pure	pure	PROPN
ejpam-1851	308	3	appl	appl	PROPN
ejpam-1851	308	4	.	.	PROPN
ejpam-1851	308	5	math	math	PROPN
ejpam-1851	308	6	,	,	PUNCT
ejpam-1851	308	7	7	7	NUM
ejpam-1851	308	8	(	(	PUNCT
ejpam-1851	308	9	2014	2014	NUM
ejpam-1851	308	10	)	)	PUNCT
ejpam-1851	308	11	,	,	PUNCT
ejpam-1851	308	12	472	472	NUM
ejpam-1851	308	13	-	-	SYM
ejpam-1851	308	14	485	485	NUM
ejpam-1851	308	15	481	481	NUM
ejpam-1851	308	16	(	(	PUNCT
ejpam-1851	308	17	ii	ii	NOUN
ejpam-1851	308	18	)	)	PUNCT
ejpam-1851	308	19	assume	assume	VERB
ejpam-1851	308	20	that	that	SCONJ
ejpam-1851	308	21	p	p	PROPN
ejpam-1851	308	22	6∈	6∈	PROPN
ejpam-1851	308	23	ℓ	ℓ	PROPN
ejpam-1851	308	24	and	and	CCONJ
ejpam-1851	308	25	q	q	PROPN
ejpam-1851	308	26	6∈	6∈	PROPN
ejpam-1851	308	27	ℓ.	ℓ.	NOUN
ejpam-1851	308	28	then	then	ADV
ejpam-1851	308	29	ℓ	ℓ	PROPN
ejpam-1851	308	30	meets	meet	VERB
ejpam-1851	308	31	the	the	DET
ejpam-1851	308	32	lines	line	NOUN
ejpam-1851	308	33	ℓ1	ℓ1	ADJ
ejpam-1851	308	34	and	and	CCONJ
ejpam-1851	308	35	ℓ2	ℓ2	NOUN
ejpam-1851	308	36	(	(	PUNCT
ejpam-1851	308	37	respectively	respectively	ADV
ejpam-1851	308	38	,	,	PUNCT
ejpam-1851	308	39	ℓ3	ℓ3	PROPN
ejpam-1851	308	40	and	and	CCONJ
ejpam-1851	308	41	ℓ4	ℓ4	PROPN
ejpam-1851	308	42	)	)	PUNCT
ejpam-1851	308	43	at	at	ADP
ejpam-1851	308	44	two	two	NUM
ejpam-1851	308	45	different	different	ADJ
ejpam-1851	308	46	points	point	NOUN
ejpam-1851	308	47	which	which	PRON
ejpam-1851	308	48	implies	imply	VERB
ejpam-1851	308	49	that	that	SCONJ
ejpam-1851	308	50	ℓ	ℓ	PROPN
ejpam-1851	308	51	⊂	⊂	PROPN
ejpam-1851	308	52	π1	π1	PROPN
ejpam-1851	308	53	(	(	PUNCT
ejpam-1851	308	54	respectively	respectively	ADV
ejpam-1851	308	55	,	,	PUNCT
ejpam-1851	308	56	ℓ	ℓ	PROPN
ejpam-1851	308	57	⊂	⊂	PROPN
ejpam-1851	308	58	π2	π2	PROPN
ejpam-1851	308	59	)	)	PUNCT
ejpam-1851	308	60	.	.	PUNCT
ejpam-1851	309	1	therefore	therefore	ADV
ejpam-1851	309	2	,	,	PUNCT
ejpam-1851	309	3	ℓ=	ℓ=	PROPN
ejpam-1851	309	4	π1	π1	PROPN
ejpam-1851	309	5	∩π2	∩π2	PROPN
ejpam-1851	309	6	.	.	PUNCT
ejpam-1851	309	7	now	now	ADV
ejpam-1851	309	8	,	,	PUNCT
ejpam-1851	309	9	note	note	VERB
ejpam-1851	309	10	that	that	SCONJ
ejpam-1851	309	11	the	the	DET
ejpam-1851	309	12	line	line	NOUN
ejpam-1851	309	13	l1,2	l1,2	PROPN
ejpam-1851	309	14	=	=	PUNCT
ejpam-1851	309	15	〈	〈	NOUN
ejpam-1851	309	16	p1	p1	NOUN
ejpam-1851	309	17	,	,	PUNCT
ejpam-1851	309	18	p2	p2	X
ejpam-1851	309	19	〉	〉	NOUN
ejpam-1851	309	20	⊂	⊂	PROPN
ejpam-1851	309	21	λ	λ	PROPN
ejpam-1851	309	22	∩	∩	ADJ
ejpam-1851	309	23	q	q	X
ejpam-1851	309	24	and	and	CCONJ
ejpam-1851	309	25	pi	pi	PROPN
ejpam-1851	309	26	6∈	6∈	PROPN
ejpam-1851	309	27	l1,2	l1,2	PROPN
ejpam-1851	309	28	,	,	PUNCT
ejpam-1851	309	29	i	i	NOUN
ejpam-1851	309	30	=	=	PUNCT
ejpam-1851	309	31	3,4	3,4	NUM
ejpam-1851	309	32	.	.	PUNCT
ejpam-1851	310	1	so	so	ADV
ejpam-1851	310	2	〈	〈	PROPN
ejpam-1851	310	3	p1	p1	NOUN
ejpam-1851	310	4	,	,	PUNCT
ejpam-1851	310	5	p2	p2	NOUN
ejpam-1851	310	6	,	,	PUNCT
ejpam-1851	310	7	p3	p3	NOUN
ejpam-1851	310	8	〉	〉	NOUN
ejpam-1851	310	9	is	be	AUX
ejpam-1851	310	10	a	a	DET
ejpam-1851	310	11	plane	plane	NOUN
ejpam-1851	310	12	not	not	PART
ejpam-1851	310	13	contained	contain	VERB
ejpam-1851	310	14	in	in	ADP
ejpam-1851	310	15	q	q	PROPN
ejpam-1851	310	16	(	(	PUNCT
ejpam-1851	310	17	since	since	SCONJ
ejpam-1851	310	18	p	p	PROPN
ejpam-1851	310	19	6∈	6∈	PROPN
ejpam-1851	310	20	ℓi	ℓi	PROPN
ejpam-1851	310	21	and	and	CCONJ
ejpam-1851	310	22	ℓi	ℓi	PROPN
ejpam-1851	310	23	6⊂	6⊂	NUM
ejpam-1851	310	24	π1	π1	NOUN
ejpam-1851	310	25	for	for	ADP
ejpam-1851	310	26	i	i	X
ejpam-1851	310	27	=	=	NOUN
ejpam-1851	310	28	3,4	3,4	NUM
ejpam-1851	310	29	)	)	PUNCT
ejpam-1851	310	30	.	.	PUNCT
ejpam-1851	311	1	on	on	ADP
ejpam-1851	311	2	the	the	DET
ejpam-1851	311	3	other	other	ADJ
ejpam-1851	311	4	hand	hand	NOUN
ejpam-1851	311	5	,	,	PUNCT
ejpam-1851	311	6	the	the	DET
ejpam-1851	311	7	line	line	NOUN
ejpam-1851	311	8	l3,4	l3,4	PROPN
ejpam-1851	311	9	=	=	SYM
ejpam-1851	311	10	〈	〈	PROPN
ejpam-1851	311	11	p3	p3	PROPN
ejpam-1851	311	12	,	,	PUNCT
ejpam-1851	311	13	p4	p4	ADJ
ejpam-1851	311	14	〉	〉	NOUN
ejpam-1851	311	15	is	be	AUX
ejpam-1851	311	16	also	also	ADV
ejpam-1851	311	17	contained	contain	VERB
ejpam-1851	311	18	in	in	ADP
ejpam-1851	311	19	λ∩q	λ∩q	PROPN
ejpam-1851	311	20	and	and	CCONJ
ejpam-1851	311	21	we	we	PRON
ejpam-1851	311	22	observe	observe	VERB
ejpam-1851	311	23	that	that	SCONJ
ejpam-1851	311	24	l1,2	l1,2	ADJ
ejpam-1851	311	25	and	and	CCONJ
ejpam-1851	311	26	l3,4	l3,4	ADJ
ejpam-1851	311	27	are	be	AUX
ejpam-1851	311	28	disjoint	disjoint	NOUN
ejpam-1851	311	29	(	(	PUNCT
ejpam-1851	311	30	since	since	SCONJ
ejpam-1851	311	31	l1,2	l1,2	PROPN
ejpam-1851	311	32	=	=	PUNCT
ejpam-1851	311	33	ωp(π1	ωp(π1	X
ejpam-1851	311	34	)	)	PUNCT
ejpam-1851	311	35	and	and	CCONJ
ejpam-1851	311	36	l3,4	l3,4	PROPN
ejpam-1851	311	37	=	=	SYM
ejpam-1851	311	38	ωq(π2	ωq(π2	NUM
ejpam-1851	311	39	)	)	PUNCT
ejpam-1851	311	40	)	)	PUNCT
ejpam-1851	311	41	,	,	PUNCT
ejpam-1851	311	42	so	so	ADV
ejpam-1851	311	43	p4	p4	ADJ
ejpam-1851	311	44	does	do	AUX
ejpam-1851	311	45	not	not	PART
ejpam-1851	311	46	belong	belong	VERB
ejpam-1851	311	47	to	to	ADP
ejpam-1851	311	48	the	the	DET
ejpam-1851	311	49	plane	plane	NOUN
ejpam-1851	311	50	〈	〈	PROPN
ejpam-1851	311	51	p1	p1	PROPN
ejpam-1851	311	52	,	,	PUNCT
ejpam-1851	311	53	p2	p2	NOUN
ejpam-1851	311	54	,	,	PUNCT
ejpam-1851	311	55	p3	p3	PROPN
ejpam-1851	311	56	〉	〉	NOUN
ejpam-1851	311	57	.	.	PUNCT
ejpam-1851	312	1	therefore	therefore	ADV
ejpam-1851	312	2	,	,	PUNCT
ejpam-1851	312	3	λ	λ	PROPN
ejpam-1851	312	4	is	be	AUX
ejpam-1851	312	5	a	a	DET
ejpam-1851	312	6	3	3	NUM
ejpam-1851	312	7	-	-	PUNCT
ejpam-1851	312	8	plane	plane	NOUN
ejpam-1851	312	9	.	.	PUNCT
ejpam-1851	313	1	keeping	keep	VERB
ejpam-1851	313	2	in	in	ADP
ejpam-1851	313	3	mind	mind	NOUN
ejpam-1851	313	4	that	that	SCONJ
ejpam-1851	313	5	λ∩q	λ∩q	PROPN
ejpam-1851	313	6	is	be	AUX
ejpam-1851	313	7	a	a	DET
ejpam-1851	313	8	quadric	quadric	ADJ
ejpam-1851	313	9	surface	surface	NOUN
ejpam-1851	313	10	containing	contain	VERB
ejpam-1851	313	11	four	four	NUM
ejpam-1851	313	12	non	non	ADJ
ejpam-1851	313	13	coplanar	coplanar	ADJ
ejpam-1851	313	14	points	point	NOUN
ejpam-1851	313	15	and	and	CCONJ
ejpam-1851	313	16	two	two	NUM
ejpam-1851	313	17	disjoint	disjoint	ADJ
ejpam-1851	313	18	lines	line	NOUN
ejpam-1851	313	19	,	,	PUNCT
ejpam-1851	313	20	we	we	PRON
ejpam-1851	313	21	conclude	conclude	VERB
ejpam-1851	313	22	that	that	SCONJ
ejpam-1851	313	23	λ∩q	λ∩q	PROPN
ejpam-1851	313	24	is	be	AUX
ejpam-1851	313	25	a	a	DET
ejpam-1851	313	26	union	union	NOUN
ejpam-1851	313	27	of	of	ADP
ejpam-1851	313	28	two	two	NUM
ejpam-1851	313	29	planes	plane	NOUN
ejpam-1851	313	30	or	or	CCONJ
ejpam-1851	313	31	a	a	DET
ejpam-1851	313	32	nonsingular	nonsingular	ADJ
ejpam-1851	313	33	quadric	quadric	NOUN
ejpam-1851	313	34	.	.	PUNCT
ejpam-1851	314	1	suppose	suppose	VERB
ejpam-1851	314	2	that	that	SCONJ
ejpam-1851	314	3	λ	λ	PROPN
ejpam-1851	314	4	∩q	∩q	PROPN
ejpam-1851	314	5	is	be	AUX
ejpam-1851	314	6	a	a	DET
ejpam-1851	314	7	union	union	NOUN
ejpam-1851	314	8	of	of	ADP
ejpam-1851	314	9	two	two	NUM
ejpam-1851	314	10	planes	plane	NOUN
ejpam-1851	314	11	,	,	PUNCT
ejpam-1851	314	12	say	say	VERB
ejpam-1851	314	13	λ	λ	INTJ
ejpam-1851	314	14	∩q	∩q	PROPN
ejpam-1851	314	15	=	=	PUNCT
ejpam-1851	315	1	λ1	λ1	ADJ
ejpam-1851	315	2	∪	∪	PROPN
ejpam-1851	315	3	λ2	λ2	PROPN
ejpam-1851	315	4	.	.	PUNCT
ejpam-1851	316	1	thus	thus	ADV
ejpam-1851	316	2	we	we	PRON
ejpam-1851	316	3	can	can	AUX
ejpam-1851	316	4	assume	assume	VERB
ejpam-1851	316	5	that	that	SCONJ
ejpam-1851	316	6	l1,2	l1,2	PROPN
ejpam-1851	316	7	⊂	⊂	PROPN
ejpam-1851	316	8	λ1	λ1	PROPN
ejpam-1851	316	9	and	and	CCONJ
ejpam-1851	316	10	l3,4	l3,4	PROPN
ejpam-1851	316	11	⊂	⊂	PROPN
ejpam-1851	316	12	λ2	λ2	PROPN
ejpam-1851	316	13	.	.	PUNCT
ejpam-1851	317	1	then	then	ADV
ejpam-1851	317	2	necessarily	necessarily	ADV
ejpam-1851	317	3	λ1	λ1	ADJ
ejpam-1851	317	4	it	it	PRON
ejpam-1851	317	5	is	be	AUX
ejpam-1851	317	6	either	either	PRON
ejpam-1851	317	7	ωp	ωp	NOUN
ejpam-1851	317	8	or	or	CCONJ
ejpam-1851	317	9	ω(π1	ω(π1	NOUN
ejpam-1851	317	10	)	)	PUNCT
ejpam-1851	317	11	,	,	PUNCT
ejpam-1851	317	12	and	and	CCONJ
ejpam-1851	317	13	in	in	ADP
ejpam-1851	317	14	the	the	DET
ejpam-1851	317	15	same	same	ADJ
ejpam-1851	317	16	form	form	NOUN
ejpam-1851	317	17	λ2	λ2	NOUN
ejpam-1851	317	18	it	it	PRON
ejpam-1851	317	19	is	be	AUX
ejpam-1851	317	20	either	either	PRON
ejpam-1851	317	21	ωq	ωq	ADP
ejpam-1851	317	22	or	or	CCONJ
ejpam-1851	317	23	ω(π2	ω(π2	NUM
ejpam-1851	317	24	)	)	PUNCT
ejpam-1851	317	25	.	.	PUNCT
ejpam-1851	318	1	but	but	CCONJ
ejpam-1851	318	2	in	in	ADP
ejpam-1851	318	3	any	any	DET
ejpam-1851	318	4	case	case	NOUN
ejpam-1851	318	5	,	,	PUNCT
ejpam-1851	318	6	λ1	λ1	PROPN
ejpam-1851	318	7	∩λ2	∩λ2	PROPN
ejpam-1851	318	8	will	will	AUX
ejpam-1851	318	9	be	be	AUX
ejpam-1851	318	10	empty	empty	ADJ
ejpam-1851	318	11	or	or	CCONJ
ejpam-1851	318	12	a	a	DET
ejpam-1851	318	13	point	point	NOUN
ejpam-1851	318	14	.	.	PUNCT
ejpam-1851	319	1	therefore	therefore	ADV
ejpam-1851	319	2	,	,	PUNCT
ejpam-1851	319	3	λ∩q	λ∩q	PROPN
ejpam-1851	319	4	is	be	AUX
ejpam-1851	319	5	a	a	DET
ejpam-1851	319	6	nonsingular	nonsingular	ADJ
ejpam-1851	319	7	quadric	quadric	ADJ
ejpam-1851	319	8	surface	surface	NOUN
ejpam-1851	319	9	.	.	PUNCT
ejpam-1851	320	1	table	table	NOUN
ejpam-1851	320	2	5	5	NUM
ejpam-1851	320	3	:	:	SYM
ejpam-1851	320	4	2.4	2.4	NUM
ejpam-1851	320	5	any	any	DET
ejpam-1851	320	6	three	three	NUM
ejpam-1851	320	7	lines	line	NOUN
ejpam-1851	320	8	are	be	AUX
ejpam-1851	320	9	non	non	ADJ
ejpam-1851	320	10	coplanar	coplanar	ADJ
ejpam-1851	320	11	and	and	CCONJ
ejpam-1851	320	12	at	at	ADV
ejpam-1851	320	13	least	least	ADV
ejpam-1851	320	14	two	two	NUM
ejpam-1851	320	15	pair	pair	NOUN
ejpam-1851	320	16	of	of	ADP
ejpam-1851	320	17	lines	line	NOUN
ejpam-1851	320	18	are	be	AUX
ejpam-1851	320	19	coplanar	coplanar	ADJ
ejpam-1851	320	20	with	with	ADP
ejpam-1851	320	21	π1∩π2	π1∩π2	NOUN
ejpam-1851	320	22	=	=	SYM
ejpam-1851	320	23	ℓ1	ℓ1	NOUN
ejpam-1851	320	24	.	.	PUNCT
ejpam-1851	321	1	assume	assume	VERB
ejpam-1851	321	2	that	that	SCONJ
ejpam-1851	321	3	ℓ1	ℓ1	ADJ
ejpam-1851	321	4	∩	∩	ADJ
ejpam-1851	321	5	ℓ2	ℓ2	NOUN
ejpam-1851	321	6	=	=	SYM
ejpam-1851	321	7	{	{	PUNCT
ejpam-1851	321	8	p	p	NOUN
ejpam-1851	321	9	}	}	PUNCT
ejpam-1851	321	10	and	and	CCONJ
ejpam-1851	321	11	ℓ1	ℓ1	ADJ
ejpam-1851	321	12	∩	∩	ADJ
ejpam-1851	321	13	ℓ3	ℓ3	NOUN
ejpam-1851	321	14	=	=	SYM
ejpam-1851	321	15	{	{	PUNCT
ejpam-1851	321	16	q	q	X
ejpam-1851	321	17	}	}	PUNCT
ejpam-1851	321	18	.	.	PUNCT
ejpam-1851	322	1	set	set	VERB
ejpam-1851	322	2	π1	π1	NOUN
ejpam-1851	322	3	=	=	SYM
ejpam-1851	322	4	〈	〈	PROPN
ejpam-1851	322	5	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	322	6	〉	〉	NOUN
ejpam-1851	322	7	and	and	CCONJ
ejpam-1851	322	8	π2	π2	NOUN
ejpam-1851	322	9	=	=	PUNCT
ejpam-1851	322	10	〈	〈	PROPN
ejpam-1851	322	11	ℓ1,ℓ3	ℓ1,ℓ3	PROPN
ejpam-1851	322	12	〉	〉	NOUN
ejpam-1851	322	13	.	.	PUNCT
ejpam-1851	323	1	let	let	VERB
ejpam-1851	323	2	{	{	PUNCT
ejpam-1851	323	3	ri}=	ri}=	PROPN
ejpam-1851	323	4	πi	πi	ADP
ejpam-1851	323	5	∩	∩	ADJ
ejpam-1851	323	6	ℓ4	ℓ4	NOUN
ejpam-1851	323	7	for	for	ADP
ejpam-1851	323	8	i	i	PROPN
ejpam-1851	323	9	=	=	NOUN
ejpam-1851	323	10	1	1	NUM
ejpam-1851	323	11	,	,	PUNCT
ejpam-1851	323	12	2	2	NUM
ejpam-1851	323	13	.	.	NOUN
ejpam-1851	323	14	subcase	subcase	NOUN
ejpam-1851	323	15	position	position	NOUN
ejpam-1851	323	16	of	of	ADP
ejpam-1851	323	17	the	the	DET
ejpam-1851	323	18	lines	line	NOUN
ejpam-1851	323	19	s	s	PART
ejpam-1851	323	20	λ	λ	X
ejpam-1851	323	21	λ∩q	λ∩q	VERB
ejpam-1851	323	22	2.4.1	2.4.1	NUM
ejpam-1851	323	23	p	p	NOUN
ejpam-1851	323	24	=	=	PUNCT
ejpam-1851	323	25	q	q	NOUN
ejpam-1851	323	26	ωp(〈p,ℓ4	ωp(〈p,ℓ4	NOUN
ejpam-1851	323	27	〉	〉	NOUN
ejpam-1851	323	28	)	)	PUNCT
ejpam-1851	323	29	3	3	NUM
ejpam-1851	323	30	-	-	PUNCT
ejpam-1851	323	31	plane	plane	NOUN
ejpam-1851	323	32	union	union	NOUN
ejpam-1851	323	33	of	of	ADP
ejpam-1851	323	34	two	two	NUM
ejpam-1851	323	35	planes	plane	NOUN
ejpam-1851	323	36	2.4.2	2.4.2	NUM
ejpam-1851	323	37	p	p	NOUN
ejpam-1851	323	38	6=	6=	ADP
ejpam-1851	323	39	q	q	PROPN
ejpam-1851	324	1	and	and	CCONJ
ejpam-1851	324	2	p	p	PROPN
ejpam-1851	324	3	∈	∈	PROPN
ejpam-1851	324	4	ℓ4	ℓ4	NOUN
ejpam-1851	324	5	(	(	PUNCT
ejpam-1851	324	6	or	or	CCONJ
ejpam-1851	324	7	p	p	X
ejpam-1851	324	8	6=	6=	PROPN
ejpam-1851	324	9	q	q	PROPN
ejpam-1851	324	10	and	and	CCONJ
ejpam-1851	324	11	q	q	PROPN
ejpam-1851	324	12	∈	∈	PROPN
ejpam-1851	324	13	ℓ4	ℓ4	NOUN
ejpam-1851	324	14	)	)	PUNCT
ejpam-1851	324	15	ωp(〈p,ℓ3	ωp(〈p,ℓ3	ADJ
ejpam-1851	324	16	〉	〉	NOUN
ejpam-1851	324	17	)	)	PUNCT
ejpam-1851	324	18	(	(	PUNCT
ejpam-1851	324	19	or	or	CCONJ
ejpam-1851	324	20	ωq(〈q,ℓ2	ωq(〈q,ℓ2	NOUN
ejpam-1851	324	21	〉	〉	NOUN
ejpam-1851	324	22	)	)	PUNCT
ejpam-1851	324	23	)	)	PUNCT
ejpam-1851	324	24	3	3	NUM
ejpam-1851	324	25	-	-	PUNCT
ejpam-1851	324	26	plane	plane	NOUN
ejpam-1851	324	27	union	union	NOUN
ejpam-1851	324	28	of	of	ADP
ejpam-1851	324	29	two	two	NUM
ejpam-1851	324	30	planes	plane	NOUN
ejpam-1851	324	31	2.4.3	2.4.3	NUM
ejpam-1851	324	32	r1	r1	NOUN
ejpam-1851	324	33	=	=	SYM
ejpam-1851	324	34	r2	r2	PROPN
ejpam-1851	324	35	and	and	CCONJ
ejpam-1851	324	36	#	#	SYM
ejpam-1851	324	37	{	{	PUNCT
ejpam-1851	324	38	p	p	X
ejpam-1851	324	39	,	,	PUNCT
ejpam-1851	324	40	q	q	ADJ
ejpam-1851	324	41	,	,	PUNCT
ejpam-1851	324	42	r1}=	r1}=	NUM
ejpam-1851	324	43	3	3	NUM
ejpam-1851	324	44	¦	¦	NOUN
ejpam-1851	324	45	ℓ1	ℓ1	NOUN
ejpam-1851	324	46	©	©	PROPN
ejpam-1851	324	47	3	3	NUM
ejpam-1851	324	48	-	-	PUNCT
ejpam-1851	324	49	plane	plane	NOUN
ejpam-1851	324	50	quadric	quadric	ADJ
ejpam-1851	324	51	cone	cone	NOUN
ejpam-1851	324	52	2.4.4	2.4.4	NUM
ejpam-1851	324	53	p	p	NOUN
ejpam-1851	324	54	6=	6=	ADP
ejpam-1851	324	55	q	q	NOUN
ejpam-1851	324	56	and	and	CCONJ
ejpam-1851	324	57	r1	r1	PROPN
ejpam-1851	324	58	6=	6=	ADP
ejpam-1851	324	59	r2	r2	PROPN
ejpam-1851	324	60	¦	¦	PROPN
ejpam-1851	324	61	ℓq	ℓq	PROPN
ejpam-1851	324	62	,	,	PUNCT
ejpam-1851	324	63	r1	r1	NOUN
ejpam-1851	324	64	,	,	PUNCT
ejpam-1851	324	65	ℓp	ℓp	NOUN
ejpam-1851	324	66	,	,	PUNCT
ejpam-1851	324	67	r2	r2	NOUN
ejpam-1851	324	68	©	©	PROPN
ejpam-1851	324	69	3	3	NUM
ejpam-1851	324	70	-	-	PUNCT
ejpam-1851	324	71	plane	plane	NOUN
ejpam-1851	324	72	nonsingular	nonsingular	ADJ
ejpam-1851	324	73	quadric	quadric	PROPN
ejpam-1851	324	74	2.4.4	2.4.4	NUM
ejpam-1851	324	75	let	let	VERB
ejpam-1851	324	76	ℓ	ℓ	PROPN
ejpam-1851	324	77	∈	∈	PROPN
ejpam-1851	324	78	s	s	PART
ejpam-1851	324	79	.	.	PUNCT
ejpam-1851	325	1	here	here	ADV
ejpam-1851	325	2	we	we	PRON
ejpam-1851	325	3	have	have	VERB
ejpam-1851	325	4	two	two	NUM
ejpam-1851	325	5	possibilities	possibility	NOUN
ejpam-1851	325	6	:	:	PUNCT
ejpam-1851	325	7	(	(	PUNCT
ejpam-1851	325	8	i	i	NOUN
ejpam-1851	325	9	)	)	PUNCT
ejpam-1851	325	10	p	p	PROPN
ejpam-1851	325	11	6∈	6∈	PROPN
ejpam-1851	325	12	ℓ.	ℓ.	NOUN
ejpam-1851	325	13	then	then	ADV
ejpam-1851	325	14	ℓ	ℓ	NOUN
ejpam-1851	325	15	∩	∩	ADJ
ejpam-1851	325	16	ℓ1	ℓ1	PROPN
ejpam-1851	325	17	6=	6=	NUM
ejpam-1851	325	18	ℓ	ℓ	NOUN
ejpam-1851	325	19	∩	∩	ADJ
ejpam-1851	325	20	ℓ2	ℓ2	NOUN
ejpam-1851	325	21	and	and	CCONJ
ejpam-1851	325	22	we	we	PRON
ejpam-1851	325	23	have	have	VERB
ejpam-1851	325	24	that	that	DET
ejpam-1851	325	25	ℓ	ℓ	PROPN
ejpam-1851	325	26	⊂	⊂	PROPN
ejpam-1851	325	27	π1	π1	PROPN
ejpam-1851	325	28	.	.	PUNCT
ejpam-1851	326	1	thus	thus	ADV
ejpam-1851	326	2	ℓ	ℓ	PROPN
ejpam-1851	326	3	∩	∩	PROPN
ejpam-1851	326	4	ℓ3	ℓ3	PROPN
ejpam-1851	326	5	⊂	⊂	PROPN
ejpam-1851	326	6	π1	π1	PROPN
ejpam-1851	326	7	∩	∩	ADJ
ejpam-1851	326	8	ℓ3	ℓ3	NOUN
ejpam-1851	326	9	=	=	SYM
ejpam-1851	326	10	{	{	PUNCT
ejpam-1851	326	11	q	q	X
ejpam-1851	326	12	}	}	PUNCT
ejpam-1851	326	13	and	and	CCONJ
ejpam-1851	326	14	ℓ∩	ℓ∩	PROPN
ejpam-1851	326	15	ℓ4	ℓ4	PROPN
ejpam-1851	326	16	⊂	⊂	PROPN
ejpam-1851	326	17	π1	π1	PROPN
ejpam-1851	326	18	∩	∩	ADJ
ejpam-1851	326	19	ℓ4	ℓ4	NOUN
ejpam-1851	326	20	=	=	SYM
ejpam-1851	326	21	{	{	PUNCT
ejpam-1851	326	22	r1	r1	PROPN
ejpam-1851	326	23	}	}	PUNCT
ejpam-1851	326	24	.	.	PUNCT
ejpam-1851	327	1	therefore	therefore	ADV
ejpam-1851	327	2	,	,	PUNCT
ejpam-1851	327	3	ℓ=	ℓ=	PROPN
ejpam-1851	327	4	ℓq	ℓq	PROPN
ejpam-1851	327	5	,	,	PUNCT
ejpam-1851	327	6	r1	r1	PROPN
ejpam-1851	327	7	(	(	PUNCT
ejpam-1851	327	8	q	q	PROPN
ejpam-1851	327	9	6=	6=	NUM
ejpam-1851	327	10	r1	r1	PROPN
ejpam-1851	327	11	since	since	SCONJ
ejpam-1851	327	12	ℓ4	ℓ4	PROPN
ejpam-1851	327	13	6⊂	6⊂	NUM
ejpam-1851	327	14	π1	π1	NOUN
ejpam-1851	327	15	)	)	PUNCT
ejpam-1851	327	16	.	.	PUNCT
ejpam-1851	328	1	(	(	PUNCT
ejpam-1851	328	2	ii	ii	NOUN
ejpam-1851	328	3	)	)	PUNCT
ejpam-1851	328	4	p	p	PROPN
ejpam-1851	328	5	∈	∈	PROPN
ejpam-1851	328	6	ℓ.	ℓ.	NOUN
ejpam-1851	328	7	note	note	NOUN
ejpam-1851	328	8	that	that	SCONJ
ejpam-1851	328	9	ℓ	ℓ	PROPN
ejpam-1851	328	10	meets	meet	VERB
ejpam-1851	328	11	the	the	DET
ejpam-1851	328	12	line	line	NOUN
ejpam-1851	328	13	ℓ3	ℓ3	NOUN
ejpam-1851	328	14	at	at	ADP
ejpam-1851	328	15	a	a	DET
ejpam-1851	328	16	point	point	NOUN
ejpam-1851	328	17	p1	p1	NOUN
ejpam-1851	328	18	(	(	PUNCT
ejpam-1851	328	19	p1	p1	PROPN
ejpam-1851	328	20	∈	∈	PROPN
ejpam-1851	328	21	π2	π2	PROPN
ejpam-1851	328	22	)	)	PUNCT
ejpam-1851	328	23	different	different	ADJ
ejpam-1851	328	24	from	from	ADP
ejpam-1851	328	25	p.	p.	NOUN
ejpam-1851	328	26	thus	thus	ADV
ejpam-1851	328	27	ℓ	ℓ	PROPN
ejpam-1851	328	28	⊂	⊂	PROPN
ejpam-1851	328	29	π2	π2	PROPN
ejpam-1851	328	30	.	.	PUNCT
ejpam-1851	329	1	on	on	ADP
ejpam-1851	329	2	the	the	DET
ejpam-1851	329	3	other	other	ADJ
ejpam-1851	329	4	hand	hand	NOUN
ejpam-1851	329	5	ℓ∩	ℓ∩	PROPN
ejpam-1851	329	6	ℓ4	ℓ4	PROPN
ejpam-1851	329	7	⊂	⊂	PROPN
ejpam-1851	329	8	π2	π2	PROPN
ejpam-1851	329	9	∩	∩	ADJ
ejpam-1851	329	10	ℓ4	ℓ4	NOUN
ejpam-1851	329	11	=	=	SYM
ejpam-1851	329	12	{	{	PUNCT
ejpam-1851	329	13	r2	r2	PROPN
ejpam-1851	329	14	}	}	PUNCT
ejpam-1851	329	15	.	.	PUNCT
ejpam-1851	330	1	therefore	therefore	ADV
ejpam-1851	330	2	,	,	PUNCT
ejpam-1851	330	3	ℓ=	ℓ=	NOUN
ejpam-1851	330	4	ℓp	ℓp	NOUN
ejpam-1851	330	5	,	,	PUNCT
ejpam-1851	330	6	r2	r2	PROPN
ejpam-1851	330	7	.	.	PUNCT
ejpam-1851	331	1	as	as	ADP
ejpam-1851	331	2	in	in	ADP
ejpam-1851	331	3	case	case	NOUN
ejpam-1851	331	4	2.3.3	2.3.3	NUM
ejpam-1851	331	5	we	we	PRON
ejpam-1851	331	6	have	have	VERB
ejpam-1851	331	7	that	that	PRON
ejpam-1851	331	8	〈	〈	PROPN
ejpam-1851	331	9	p1	p1	PROPN
ejpam-1851	331	10	,	,	PUNCT
ejpam-1851	331	11	p2	p2	NOUN
ejpam-1851	331	12	,	,	PUNCT
ejpam-1851	331	13	p3	p3	NOUN
ejpam-1851	331	14	〉	〉	NOUN
ejpam-1851	331	15	is	be	AUX
ejpam-1851	331	16	a	a	DET
ejpam-1851	331	17	plane	plane	NOUN
ejpam-1851	331	18	not	not	PART
ejpam-1851	331	19	contained	contain	VERB
ejpam-1851	331	20	in	in	ADP
ejpam-1851	331	21	q.	q.	NOUN
ejpam-1851	331	22	since	since	SCONJ
ejpam-1851	331	23	the	the	DET
ejpam-1851	331	24	lines	line	NOUN
ejpam-1851	331	25	l1,2	l1,2	PROPN
ejpam-1851	331	26	=	=	SYM
ejpam-1851	331	27	〈	〈	NOUN
ejpam-1851	331	28	p1	p1	NOUN
ejpam-1851	331	29	,	,	PUNCT
ejpam-1851	332	1	p2	p2	X
ejpam-1851	332	2	〉	〉	NOUN
ejpam-1851	332	3	and	and	CCONJ
ejpam-1851	332	4	l1,3	l1,3	ADJ
ejpam-1851	332	5	=	=	SYM
ejpam-1851	332	6	〈	〈	PROPN
ejpam-1851	332	7	p1	p1	PROPN
ejpam-1851	332	8	,	,	PUNCT
ejpam-1851	332	9	p3	p3	NOUN
ejpam-1851	332	10	〉	〉	NOUN
ejpam-1851	332	11	are	be	AUX
ejpam-1851	332	12	contained	contain	VERB
ejpam-1851	332	13	in	in	ADP
ejpam-1851	332	14	〈	〈	PROPN
ejpam-1851	332	15	p1	p1	NOUN
ejpam-1851	332	16	,	,	PUNCT
ejpam-1851	332	17	p2	p2	NOUN
ejpam-1851	332	18	,	,	PUNCT
ejpam-1851	332	19	p3	p3	NOUN
ejpam-1851	332	20	〉	〉	NOUN
ejpam-1851	332	21	∩q	∩q	X
ejpam-1851	332	22	we	we	PRON
ejpam-1851	332	23	conclude	conclude	VERB
ejpam-1851	332	24	that	that	SCONJ
ejpam-1851	332	25	〈	〈	PROPN
ejpam-1851	332	26	p1	p1	PROPN
ejpam-1851	332	27	,	,	PUNCT
ejpam-1851	332	28	p2	p2	NOUN
ejpam-1851	332	29	,	,	PUNCT
ejpam-1851	332	30	p3	p3	NOUN
ejpam-1851	332	31	〉	〉	NOUN
ejpam-1851	332	32	∩q	∩q	X
ejpam-1851	332	33	=	=	PUNCT
ejpam-1851	333	1	l1,2	l1,2	ADJ
ejpam-1851	333	2	∪	∪	ADJ
ejpam-1851	333	3	l1,3	l1,3	NOUN
ejpam-1851	333	4	.	.	PUNCT
ejpam-1851	334	1	so	so	ADV
ejpam-1851	334	2	p4	p4	ADJ
ejpam-1851	334	3	6∈	6∈	PROPN
ejpam-1851	334	4	〈	〈	PROPN
ejpam-1851	334	5	p1	p1	PROPN
ejpam-1851	334	6	,	,	PUNCT
ejpam-1851	334	7	p2	p2	NOUN
ejpam-1851	334	8	,	,	PUNCT
ejpam-1851	334	9	p3	p3	PROPN
ejpam-1851	334	10	〉	〉	NOUN
ejpam-1851	334	11	.	.	PUNCT
ejpam-1851	335	1	therefore	therefore	ADV
ejpam-1851	335	2	,	,	PUNCT
ejpam-1851	335	3	λ	λ	PROPN
ejpam-1851	335	4	is	be	AUX
ejpam-1851	335	5	a	a	DET
ejpam-1851	335	6	3	3	NUM
ejpam-1851	335	7	-	-	PUNCT
ejpam-1851	335	8	plane	plane	NOUN
ejpam-1851	335	9	.	.	PUNCT
ejpam-1851	336	1	keeping	keep	VERB
ejpam-1851	336	2	in	in	ADP
ejpam-1851	336	3	mind	mind	NOUN
ejpam-1851	336	4	that	that	SCONJ
ejpam-1851	336	5	λ	λ	PROPN
ejpam-1851	336	6	∩	∩	NOUN
ejpam-1851	336	7	q	q	X
ejpam-1851	336	8	is	be	AUX
ejpam-1851	336	9	a	a	DET
ejpam-1851	336	10	quadric	quadric	ADJ
ejpam-1851	336	11	surface	surface	NOUN
ejpam-1851	336	12	containing	contain	VERB
ejpam-1851	336	13	four	four	NUM
ejpam-1851	336	14	non	non	ADJ
ejpam-1851	336	15	coplanar	coplanar	ADJ
ejpam-1851	336	16	points	point	NOUN
ejpam-1851	336	17	,	,	PUNCT
ejpam-1851	336	18	{	{	PUNCT
ejpam-1851	336	19	p1	p1	NOUN
ejpam-1851	336	20	}	}	PUNCT
ejpam-1851	336	21	=	=	SYM
ejpam-1851	336	22	l1,2	l1,2	ADJ
ejpam-1851	336	23	∩	∩	ADJ
ejpam-1851	336	24	l1,3	l1,3	ADJ
ejpam-1851	336	25	and	and	CCONJ
ejpam-1851	336	26	l1,4	l1,4	PROPN
ejpam-1851	336	27	=	=	SYM
ejpam-1851	336	28	〈	〈	PROPN
ejpam-1851	336	29	p1	p1	NOUN
ejpam-1851	336	30	,	,	PUNCT
ejpam-1851	336	31	p4	p4	ADJ
ejpam-1851	336	32	〉	〉	NOUN
ejpam-1851	336	33	6⊂	6⊂	NUM
ejpam-1851	336	34	λ∩q	λ∩q	X
ejpam-1851	336	35	(	(	PUNCT
ejpam-1851	336	36	since	since	SCONJ
ejpam-1851	336	37	l1,4	l1,4	PROPN
ejpam-1851	336	38	6⊂	6⊂	NUM
ejpam-1851	336	39	q	q	NOUN
ejpam-1851	336	40	)	)	PUNCT
ejpam-1851	336	41	,	,	PUNCT
ejpam-1851	336	42	we	we	PRON
ejpam-1851	336	43	conclude	conclude	VERB
ejpam-1851	336	44	that	that	SCONJ
ejpam-1851	336	45	λ∩q	λ∩q	PROPN
ejpam-1851	336	46	is	be	AUX
ejpam-1851	336	47	a	a	DET
ejpam-1851	336	48	union	union	NOUN
ejpam-1851	336	49	of	of	ADP
ejpam-1851	336	50	two	two	NUM
ejpam-1851	336	51	planes	plane	NOUN
ejpam-1851	336	52	or	or	CCONJ
ejpam-1851	336	53	a	a	DET
ejpam-1851	336	54	nonsingular	nonsingular	ADJ
ejpam-1851	336	55	quadric	quadric	NOUN
ejpam-1851	336	56	.	.	PUNCT
ejpam-1851	337	1	suppose	suppose	VERB
ejpam-1851	337	2	that	that	SCONJ
ejpam-1851	337	3	λ∩q	λ∩q	PROPN
ejpam-1851	337	4	is	be	AUX
ejpam-1851	337	5	a	a	DET
ejpam-1851	337	6	union	union	NOUN
ejpam-1851	337	7	of	of	ADP
ejpam-1851	337	8	two	two	NUM
ejpam-1851	337	9	planes	plane	NOUN
ejpam-1851	337	10	,	,	PUNCT
ejpam-1851	337	11	say	say	VERB
ejpam-1851	337	12	λ∩q	λ∩q	ADJ
ejpam-1851	337	13	=	=	SYM
ejpam-1851	337	14	λ1∪λ2	λ1∪λ2	NOUN
ejpam-1851	337	15	.	.	PUNCT
ejpam-1851	338	1	thus	thus	ADV
ejpam-1851	338	2	we	we	PRON
ejpam-1851	338	3	can	can	AUX
ejpam-1851	338	4	assume	assume	VERB
ejpam-1851	338	5	that	that	SCONJ
ejpam-1851	338	6	l1,2	l1,2	PROPN
ejpam-1851	338	7	⊂	⊂	PROPN
ejpam-1851	338	8	λ1	λ1	PROPN
ejpam-1851	338	9	.	.	PUNCT
ejpam-1851	339	1	now	now	ADV
ejpam-1851	339	2	,	,	PUNCT
ejpam-1851	339	3	note	note	VERB
ejpam-1851	339	4	that	that	SCONJ
ejpam-1851	339	5	l2,3	l2,3	VERB
ejpam-1851	339	6	6⊂	6⊂	NUM
ejpam-1851	339	7	λ	λ	X
ejpam-1851	339	8	∩q	∩q	PROPN
ejpam-1851	339	9	(	(	PUNCT
ejpam-1851	339	10	since	since	SCONJ
ejpam-1851	339	11	l2,3	l2,3	PROPN
ejpam-1851	339	12	6⊂	6⊂	NUM
ejpam-1851	339	13	q	q	NOUN
ejpam-1851	339	14	)	)	PUNCT
ejpam-1851	339	15	.	.	PUNCT
ejpam-1851	340	1	so	so	ADV
ejpam-1851	340	2	p3	p3	PROPN
ejpam-1851	340	3	6∈	6∈	PROPN
ejpam-1851	340	4	λ1	λ1	PROPN
ejpam-1851	340	5	,	,	PUNCT
ejpam-1851	340	6	which	which	PRON
ejpam-1851	340	7	implies	imply	VERB
ejpam-1851	340	8	l1,3	l1,3	PROPN
ejpam-1851	340	9	⊂	⊂	PROPN
ejpam-1851	340	10	λ2	λ2	PROPN
ejpam-1851	340	11	.	.	PUNCT
ejpam-1851	341	1	finally	finally	ADV
ejpam-1851	341	2	,	,	PUNCT
ejpam-1851	341	3	observe	observe	VERB
ejpam-1851	341	4	that	that	SCONJ
ejpam-1851	341	5	l2,4	l2,4	PROPN
ejpam-1851	341	6	and	and	CCONJ
ejpam-1851	341	7	l3,4	l3,4	PROPN
ejpam-1851	341	8	are	be	AUX
ejpam-1851	341	9	not	not	PART
ejpam-1851	341	10	contained	contain	VERB
ejpam-1851	341	11	in	in	ADP
ejpam-1851	341	12	λ∩q	λ∩q	PROPN
ejpam-1851	341	13	.	.	PUNCT
ejpam-1851	342	1	thus	thus	ADV
ejpam-1851	342	2	p4	p4	ADJ
ejpam-1851	342	3	6∈	6∈	NOUN
ejpam-1851	342	4	λ1	λ1	ADJ
ejpam-1851	342	5	∪λ2	∪λ2	NOUN
ejpam-1851	342	6	.	.	PUNCT
ejpam-1851	343	1	therefore	therefore	ADV
ejpam-1851	343	2	,	,	PUNCT
ejpam-1851	343	3	λ∩q	λ∩q	PROPN
ejpam-1851	343	4	is	be	AUX
ejpam-1851	343	5	a	a	DET
ejpam-1851	343	6	nonsingular	nonsingular	ADJ
ejpam-1851	343	7	quadric	quadric	NOUN
ejpam-1851	343	8	.	.	PUNCT
ejpam-1851	344	1	j.	j.	PROPN
ejpam-1851	344	2	rojas	rojas	PROPN
ejpam-1851	344	3	,	,	PUNCT
ejpam-1851	344	4	r.	r.	PROPN
ejpam-1851	344	5	mendoza	mendoza	PROPN
ejpam-1851	344	6	/	/	SYM
ejpam-1851	344	7	eur	eur	PROPN
ejpam-1851	344	8	.	.	PUNCT
ejpam-1851	345	1	j.	j.	PROPN
ejpam-1851	345	2	pure	pure	PROPN
ejpam-1851	345	3	appl	appl	PROPN
ejpam-1851	345	4	.	.	PROPN
ejpam-1851	345	5	math	math	PROPN
ejpam-1851	345	6	,	,	PUNCT
ejpam-1851	345	7	7	7	NUM
ejpam-1851	345	8	(	(	PUNCT
ejpam-1851	345	9	2014	2014	NUM
ejpam-1851	345	10	)	)	PUNCT
ejpam-1851	345	11	,	,	PUNCT
ejpam-1851	345	12	472	472	NUM
ejpam-1851	345	13	-	-	SYM
ejpam-1851	345	14	485	485	NUM
ejpam-1851	345	15	482	482	NUM
ejpam-1851	345	16	figure	figure	NOUN
ejpam-1851	345	17	3	3	NUM
ejpam-1851	345	18	:	:	PUNCT
ejpam-1851	345	19	case	case	NOUN
ejpam-1851	345	20	2.4.4	2.4.4	NUM
ejpam-1851	345	21	table	table	NOUN
ejpam-1851	345	22	6	6	NUM
ejpam-1851	345	23	:	:	SYM
ejpam-1851	345	24	2.5	2.5	NUM
ejpam-1851	345	25	exactly	exactly	ADV
ejpam-1851	345	26	two	two	NUM
ejpam-1851	345	27	lines	line	NOUN
ejpam-1851	345	28	are	be	AUX
ejpam-1851	345	29	coplanar	coplanar	ADJ
ejpam-1851	345	30	.	.	PUNCT
ejpam-1851	346	1	assume	assume	VERB
ejpam-1851	346	2	that	that	SCONJ
ejpam-1851	346	3	ℓ1	ℓ1	NOUN
ejpam-1851	346	4	and	and	CCONJ
ejpam-1851	346	5	ℓ2	ℓ2	NOUN
ejpam-1851	346	6	are	be	AUX
ejpam-1851	346	7	contained	contain	VERB
ejpam-1851	346	8	in	in	ADP
ejpam-1851	346	9	the	the	DET
ejpam-1851	346	10	plane	plane	NOUN
ejpam-1851	346	11	π	π	NOUN
ejpam-1851	346	12	,	,	PUNCT
ejpam-1851	346	13	ℓ1	ℓ1	ADJ
ejpam-1851	346	14	∩	∩	ADJ
ejpam-1851	346	15	ℓ2	ℓ2	NOUN
ejpam-1851	346	16	=	=	SYM
ejpam-1851	346	17	{	{	PUNCT
ejpam-1851	346	18	p	p	X
ejpam-1851	346	19	}	}	PUNCT
ejpam-1851	346	20	,	,	PUNCT
ejpam-1851	346	21	π∩	π∩	PROPN
ejpam-1851	346	22	ℓi	ℓi	PROPN
ejpam-1851	346	23	=	=	PUNCT
ejpam-1851	346	24	{	{	PUNCT
ejpam-1851	346	25	pi	pi	NOUN
ejpam-1851	346	26	}	}	PUNCT
ejpam-1851	346	27	for	for	ADP
ejpam-1851	346	28	i	i	PROPN
ejpam-1851	346	29	=	=	SYM
ejpam-1851	346	30	3	3	NUM
ejpam-1851	346	31	,	,	PUNCT
ejpam-1851	346	32	4	4	NUM
ejpam-1851	346	33	and	and	CCONJ
ejpam-1851	346	34	p3	p3	PROPN
ejpam-1851	346	35	6=	6=	SYM
ejpam-1851	346	36	p4	p4	ADJ
ejpam-1851	346	37	.	.	PUNCT
ejpam-1851	347	1	let	let	VERB
ejpam-1851	347	2	c	c	NOUN
ejpam-1851	347	3	=	=	PUNCT
ejpam-1851	347	4	{	{	PUNCT
ejpam-1851	347	5	p	p	X
ejpam-1851	347	6	,	,	PUNCT
ejpam-1851	347	7	p3	p3	PROPN
ejpam-1851	347	8	,	,	PUNCT
ejpam-1851	347	9	p4	p4	ADJ
ejpam-1851	347	10	}	}	PUNCT
ejpam-1851	347	11	.	.	PUNCT
ejpam-1851	348	1	subcase	subcase	PROPN
ejpam-1851	348	2	position	position	NOUN
ejpam-1851	348	3	of	of	ADP
ejpam-1851	348	4	the	the	DET
ejpam-1851	348	5	lines	line	NOUN
ejpam-1851	348	6	s	s	PART
ejpam-1851	348	7	λ	λ	X
ejpam-1851	348	8	λ∩q	λ∩q	VERB
ejpam-1851	348	9	2.5.1	2.5.1	NUM
ejpam-1851	348	10	p	p	NOUN
ejpam-1851	348	11	=	=	NOUN
ejpam-1851	348	12	p3	p3	PROPN
ejpam-1851	348	13	(	(	PUNCT
ejpam-1851	348	14	or	or	CCONJ
ejpam-1851	348	15	p	p	NOUN
ejpam-1851	348	16	=	=	ADJ
ejpam-1851	348	17	p4	p4	ADJ
ejpam-1851	348	18	)	)	PUNCT
ejpam-1851	348	19	ωp(〈p,ℓ4	ωp(〈p,ℓ4	NOUN
ejpam-1851	348	20	〉	〉	NOUN
ejpam-1851	348	21	)	)	PUNCT
ejpam-1851	348	22	(	(	PUNCT
ejpam-1851	348	23	or	or	CCONJ
ejpam-1851	348	24	ωp(〈p,ℓ3	ωp(〈p,ℓ3	PRON
ejpam-1851	348	25	〉	〉	NOUN
ejpam-1851	348	26	)	)	PUNCT
ejpam-1851	348	27	)	)	PUNCT
ejpam-1851	348	28	3	3	NUM
ejpam-1851	348	29	-	-	PUNCT
ejpam-1851	348	30	plane	plane	NOUN
ejpam-1851	348	31	union	union	NOUN
ejpam-1851	348	32	of	of	ADP
ejpam-1851	348	33	two	two	NUM
ejpam-1851	348	34	planes	plane	NOUN
ejpam-1851	348	35	2.5.2	2.5.2	NUM
ejpam-1851	348	36	#	#	SYM
ejpam-1851	348	37	c	c	NOUN
ejpam-1851	348	38	=	=	SYM
ejpam-1851	348	39	3	3	NUM
ejpam-1851	348	40	and	and	CCONJ
ejpam-1851	348	41	p	p	NOUN
ejpam-1851	348	42	∈	∈	PROPN
ejpam-1851	348	43	ℓp3,p4	ℓp3,p4	PROPN
ejpam-1851	348	44	¦	¦	PROPN
ejpam-1851	348	45	ℓp3,p4	ℓp3,p4	PROPN
ejpam-1851	349	1	©	©	PROPN
ejpam-1851	349	2	3	3	NUM
ejpam-1851	349	3	-	-	PUNCT
ejpam-1851	349	4	plane	plane	NOUN
ejpam-1851	349	5	quadric	quadric	ADJ
ejpam-1851	349	6	cone	cone	NOUN
ejpam-1851	349	7	2.5.3	2.5.3	NUM
ejpam-1851	349	8	p	p	PROPN
ejpam-1851	349	9	6∈	6∈	PROPN
ejpam-1851	349	10	ℓp3,p4	ℓp3,p4	NOUN
ejpam-1851	349	11	(	(	PUNCT
ejpam-1851	349	12	so	so	ADV
ejpam-1851	349	13	#	#	SYM
ejpam-1851	349	14	c	c	NOUN
ejpam-1851	349	15	=	=	SYM
ejpam-1851	349	16	3	3	X
ejpam-1851	349	17	)	)	PUNCT
ejpam-1851	349	18	¦	¦	PROPN
ejpam-1851	349	19	〈	〈	PROPN
ejpam-1851	349	20	p,ℓ3	p,ℓ3	PROPN
ejpam-1851	349	21	〉	〉	NOUN
ejpam-1851	349	22	∩	∩	NOUN
ejpam-1851	349	23	〈	〈	NOUN
ejpam-1851	349	24	p,ℓ4〉,ℓp3,p4	p,ℓ4〉,ℓp3,p4	NUM
ejpam-1851	349	25	©	©	ADJ
ejpam-1851	349	26	3	3	NUM
ejpam-1851	349	27	-	-	PUNCT
ejpam-1851	349	28	plane	plane	NOUN
ejpam-1851	349	29	nonsingular	nonsingular	ADJ
ejpam-1851	349	30	quadric	quadric	PROPN
ejpam-1851	349	31	2.5.3	2.5.3	PROPN
ejpam-1851	349	32	let	let	VERB
ejpam-1851	349	33	ℓ	ℓ	NOUN
ejpam-1851	349	34	be	be	AUX
ejpam-1851	349	35	a	a	DET
ejpam-1851	349	36	solution	solution	NOUN
ejpam-1851	349	37	and	and	CCONJ
ejpam-1851	349	38	consider	consider	VERB
ejpam-1851	349	39	the	the	DET
ejpam-1851	349	40	following	follow	VERB
ejpam-1851	349	41	two	two	NUM
ejpam-1851	349	42	possibilities	possibility	NOUN
ejpam-1851	349	43	.	.	PUNCT
ejpam-1851	350	1	(	(	PUNCT
ejpam-1851	350	2	i	i	NOUN
ejpam-1851	350	3	)	)	PUNCT
ejpam-1851	350	4	ℓ	ℓ	PROPN
ejpam-1851	350	5	⊂	⊂	PROPN
ejpam-1851	351	1	π	π	PROPN
ejpam-1851	351	2	.	.	PUNCT
ejpam-1851	352	1	since	since	SCONJ
ejpam-1851	352	2	ℓ∩	ℓ∩	PROPN
ejpam-1851	352	3	ℓi	ℓi	PROPN
ejpam-1851	352	4	⊂	⊂	PROPN
ejpam-1851	352	5	π∩	π∩	PROPN
ejpam-1851	352	6	ℓi	ℓi	PROPN
ejpam-1851	352	7	=	=	PUNCT
ejpam-1851	352	8	{	{	PUNCT
ejpam-1851	352	9	pi	pi	NOUN
ejpam-1851	352	10	}	}	PUNCT
ejpam-1851	352	11	then	then	ADV
ejpam-1851	352	12	pi	pi	PROPN
ejpam-1851	352	13	∈	∈	PROPN
ejpam-1851	352	14	ℓ	ℓ	PROPN
ejpam-1851	352	15	for	for	ADP
ejpam-1851	352	16	i	i	PRON
ejpam-1851	352	17	=	=	NOUN
ejpam-1851	352	18	3,4	3,4	NUM
ejpam-1851	352	19	.	.	PUNCT
ejpam-1851	352	20	therefore	therefore	ADV
ejpam-1851	352	21	ℓ=	ℓ=	PROPN
ejpam-1851	352	22	ℓp3,p4	ℓp3,p4	PROPN
ejpam-1851	352	23	.	.	PUNCT
ejpam-1851	353	1	(	(	PUNCT
ejpam-1851	353	2	ii	ii	NOUN
ejpam-1851	353	3	)	)	PUNCT
ejpam-1851	353	4	ℓ	ℓ	PROPN
ejpam-1851	353	5	6⊂	6⊂	NUM
ejpam-1851	353	6	π	π	X
ejpam-1851	353	7	.	.	PUNCT
ejpam-1851	354	1	now	now	ADV
ejpam-1851	354	2	having	have	VERB
ejpam-1851	354	3	in	in	ADP
ejpam-1851	354	4	mind	mind	NOUN
ejpam-1851	354	5	that	that	SCONJ
ejpam-1851	354	6	ℓ∩ℓi	ℓ∩ℓi	NOUN
ejpam-1851	354	7	6=	6=	NUM
ejpam-1851	354	8	;	;	PUNCT
ejpam-1851	354	9	for	for	ADP
ejpam-1851	354	10	i	i	PROPN
ejpam-1851	354	11	=	=	SYM
ejpam-1851	354	12	1,2	1,2	NUM
ejpam-1851	354	13	and	and	CCONJ
ejpam-1851	354	14	ℓ	ℓ	NOUN
ejpam-1851	354	15	6⊂	6⊂	NUM
ejpam-1851	354	16	πwe	πwe	NOUN
ejpam-1851	354	17	concluded	conclude	VERB
ejpam-1851	354	18	that	that	SCONJ
ejpam-1851	354	19	p	p	PROPN
ejpam-1851	354	20	∈	∈	PROPN
ejpam-1851	354	21	ℓ.	ℓ.	NOUN
ejpam-1851	354	22	on	on	ADP
ejpam-1851	354	23	the	the	DET
ejpam-1851	354	24	other	other	ADJ
ejpam-1851	354	25	hand	hand	NOUN
ejpam-1851	354	26	ℓ∩	ℓ∩	PROPN
ejpam-1851	354	27	ℓi	ℓi	PROPN
ejpam-1851	354	28	=	=	SYM
ejpam-1851	354	29	{	{	PUNCT
ejpam-1851	354	30	qi	qi	NOUN
ejpam-1851	354	31	}	}	PUNCT
ejpam-1851	354	32	⊂	⊂	PROPN
ejpam-1851	354	33	ℓi	ℓi	PROPN
ejpam-1851	354	34	⊂	⊂	PROPN
ejpam-1851	354	35	〈	〈	PROPN
ejpam-1851	354	36	p,ℓi	p,ℓi	NUM
ejpam-1851	354	37	〉	〉	NOUN
ejpam-1851	354	38	for	for	ADP
ejpam-1851	354	39	i	i	X
ejpam-1851	354	40	=	=	NOUN
ejpam-1851	354	41	3,4	3,4	NUM
ejpam-1851	354	42	.	.	PUNCT
ejpam-1851	354	43	note	note	VERB
ejpam-1851	354	44	that	that	SCONJ
ejpam-1851	354	45	qi	qi	PROPN
ejpam-1851	354	46	6=	6=	PROPN
ejpam-1851	354	47	p	p	X
ejpam-1851	354	48	(	(	PUNCT
ejpam-1851	354	49	since	since	SCONJ
ejpam-1851	354	50	p	p	PROPN
ejpam-1851	354	51	6∈	6∈	PROPN
ejpam-1851	354	52	ℓi	ℓi	PROPN
ejpam-1851	354	53	for	for	ADP
ejpam-1851	354	54	i	i	X
ejpam-1851	354	55	=	=	NOUN
ejpam-1851	354	56	3,4	3,4	NUM
ejpam-1851	354	57	)	)	PUNCT
ejpam-1851	354	58	which	which	PRON
ejpam-1851	354	59	implies	imply	VERB
ejpam-1851	354	60	ℓ=	ℓ=	NOUN
ejpam-1851	354	61	ℓp	ℓp	PROPN
ejpam-1851	354	62	,	,	PUNCT
ejpam-1851	354	63	qi	qi	PROPN
ejpam-1851	354	64	⊂	⊂	PROPN
ejpam-1851	354	65	〈	〈	PROPN
ejpam-1851	354	66	p,ℓi	p,ℓi	NUM
ejpam-1851	354	67	〉	〉	NOUN
ejpam-1851	354	68	for	for	ADP
ejpam-1851	354	69	i	i	NOUN
ejpam-1851	354	70	=	=	NOUN
ejpam-1851	354	71	3,4	3,4	NUM
ejpam-1851	354	72	.	.	PUNCT
ejpam-1851	355	1	therefore	therefore	ADV
ejpam-1851	355	2	,	,	PUNCT
ejpam-1851	355	3	ℓ=	ℓ=	PROPN
ejpam-1851	355	4	〈	〈	PROPN
ejpam-1851	355	5	p,ℓ3	p,ℓ3	PROPN
ejpam-1851	355	6	〉	〉	NOUN
ejpam-1851	355	7	∩	∩	NOUN
ejpam-1851	355	8	〈	〈	NOUN
ejpam-1851	355	9	p,ℓ4	p,ℓ4	PROPN
ejpam-1851	355	10	〉	〉	NOUN
ejpam-1851	355	11	.	.	PUNCT
ejpam-1851	355	12	note	note	VERB
ejpam-1851	355	13	that	that	PRON
ejpam-1851	355	14	pi	pi	PROPN
ejpam-1851	355	15	6∈	6∈	PROPN
ejpam-1851	355	16	l1,2	l1,2	PROPN
ejpam-1851	355	17	=	=	PUNCT
ejpam-1851	355	18	〈	〈	NOUN
ejpam-1851	355	19	p1	p1	NOUN
ejpam-1851	355	20	,	,	PUNCT
ejpam-1851	355	21	p2	p2	X
ejpam-1851	355	22	〉	〉	NOUN
ejpam-1851	355	23	⊂	⊂	ADJ
ejpam-1851	355	24	q	q	X
ejpam-1851	355	25	for	for	ADP
ejpam-1851	355	26	i	i	PRON
ejpam-1851	355	27	=	=	NOUN
ejpam-1851	355	28	3,4	3,4	NUM
ejpam-1851	355	29	.	.	PUNCT
ejpam-1851	356	1	thus	thus	ADV
ejpam-1851	356	2	〈	〈	PROPN
ejpam-1851	356	3	p1	p1	NOUN
ejpam-1851	356	4	,	,	PUNCT
ejpam-1851	356	5	p2	p2	NOUN
ejpam-1851	356	6	,	,	PUNCT
ejpam-1851	356	7	p3	p3	NOUN
ejpam-1851	356	8	〉	〉	NOUN
ejpam-1851	356	9	is	be	AUX
ejpam-1851	356	10	a	a	DET
ejpam-1851	356	11	plane	plane	NOUN
ejpam-1851	356	12	not	not	PART
ejpam-1851	356	13	contained	contain	VERB
ejpam-1851	356	14	in	in	ADP
ejpam-1851	356	15	q	q	NOUN
ejpam-1851	356	16	since	since	SCONJ
ejpam-1851	356	17	p	p	PROPN
ejpam-1851	356	18	6∈	6∈	PROPN
ejpam-1851	356	19	ℓ3	ℓ3	PROPN
ejpam-1851	356	20	6⊂	6⊂	NUM
ejpam-1851	356	21	π	π	NOUN
ejpam-1851	356	22	.	.	PUNCT
ejpam-1851	357	1	since	since	SCONJ
ejpam-1851	357	2	l1,2	l1,2	ADJ
ejpam-1851	357	3	∪	∪	X
ejpam-1851	357	4	{	{	PUNCT
ejpam-1851	357	5	p3	p3	PROPN
ejpam-1851	357	6	}	}	PUNCT
ejpam-1851	357	7	⊂	⊂	PROPN
ejpam-1851	357	8	〈	〈	PROPN
ejpam-1851	357	9	p1	p1	PROPN
ejpam-1851	357	10	,	,	PUNCT
ejpam-1851	357	11	p2	p2	NOUN
ejpam-1851	357	12	,	,	PUNCT
ejpam-1851	357	13	p3	p3	NOUN
ejpam-1851	357	14	〉	〉	PUNCT
ejpam-1851	357	15	∩q	∩q	PROPN
ejpam-1851	357	16	,	,	PUNCT
ejpam-1851	357	17	if	if	SCONJ
ejpam-1851	357	18	p4	p4	PROPN
ejpam-1851	357	19	∈	∈	PROPN
ejpam-1851	357	20	〈	〈	NOUN
ejpam-1851	357	21	p1	p1	NOUN
ejpam-1851	357	22	,	,	PUNCT
ejpam-1851	357	23	p2	p2	NOUN
ejpam-1851	357	24	,	,	PUNCT
ejpam-1851	357	25	p3	p3	NOUN
ejpam-1851	357	26	〉	〉	NOUN
ejpam-1851	357	27	then	then	ADV
ejpam-1851	357	28	p4	p4	PROPN
ejpam-1851	357	29	∈	∈	PROPN
ejpam-1851	358	1	l1,2	l1,2	ADJ
ejpam-1851	358	2	or	or	CCONJ
ejpam-1851	358	3	p4	p4	ADJ
ejpam-1851	358	4	belong	belong	VERB
ejpam-1851	358	5	to	to	ADP
ejpam-1851	358	6	the	the	DET
ejpam-1851	358	7	component	component	NOUN
ejpam-1851	358	8	l	l	NOUN
ejpam-1851	358	9	passing	pass	VERB
ejpam-1851	358	10	through	through	ADP
ejpam-1851	358	11	p3	p3	PROPN
ejpam-1851	358	12	in	in	ADP
ejpam-1851	358	13	the	the	DET
ejpam-1851	358	14	conic	conic	ADJ
ejpam-1851	358	15	〈	〈	NOUN
ejpam-1851	358	16	p1	p1	NOUN
ejpam-1851	358	17	,	,	PUNCT
ejpam-1851	358	18	p2	p2	NOUN
ejpam-1851	358	19	,	,	PUNCT
ejpam-1851	358	20	p3	p3	NOUN
ejpam-1851	358	21	〉	〉	NOUN
ejpam-1851	358	22	∩q	∩q	X
ejpam-1851	358	23	=	=	PUNCT
ejpam-1851	359	1	l1,2	l1,2	ADJ
ejpam-1851	359	2	∪	∪	ADJ
ejpam-1851	359	3	l.	l.	NOUN
ejpam-1851	359	4	but	but	CCONJ
ejpam-1851	359	5	p4	p4	PROPN
ejpam-1851	359	6	6∈	6∈	NOUN
ejpam-1851	359	7	l1,2	l1,2	PROPN
ejpam-1851	359	8	because	because	SCONJ
ejpam-1851	359	9	ℓ4	ℓ4	PROPN
ejpam-1851	359	10	6⊂	6⊂	NUM
ejpam-1851	359	11	π	π	PROPN
ejpam-1851	359	12	.	.	PUNCT
ejpam-1851	360	1	moreover	moreover	ADV
ejpam-1851	360	2	,	,	PUNCT
ejpam-1851	360	3	p4	p4	ADJ
ejpam-1851	360	4	6∈	6∈	PROPN
ejpam-1851	360	5	l	l	NOUN
ejpam-1851	360	6	since	since	SCONJ
ejpam-1851	360	7	ℓ4	ℓ4	PROPN
ejpam-1851	360	8	∩	∩	ADJ
ejpam-1851	360	9	ℓ3	ℓ3	PROPN
ejpam-1851	360	10	=	=	PUNCT
ejpam-1851	360	11	;	;	PUNCT
ejpam-1851	360	12	.	.	PUNCT
ejpam-1851	361	1	thus	thus	ADV
ejpam-1851	361	2	λ	λ	PRON
ejpam-1851	361	3	is	be	AUX
ejpam-1851	361	4	a	a	DET
ejpam-1851	361	5	3	3	NUM
ejpam-1851	361	6	-	-	PUNCT
ejpam-1851	361	7	plane	plane	NOUN
ejpam-1851	361	8	.	.	PUNCT
ejpam-1851	362	1	keeping	keep	VERB
ejpam-1851	362	2	in	in	ADP
ejpam-1851	362	3	mind	mind	NOUN
ejpam-1851	362	4	that	that	SCONJ
ejpam-1851	362	5	λ	λ	PROPN
ejpam-1851	362	6	∩q	∩q	PROPN
ejpam-1851	362	7	is	be	AUX
ejpam-1851	362	8	a	a	DET
ejpam-1851	362	9	quadric	quadric	ADJ
ejpam-1851	362	10	surface	surface	NOUN
ejpam-1851	362	11	containing	contain	VERB
ejpam-1851	362	12	four	four	NUM
ejpam-1851	362	13	non	non	ADJ
ejpam-1851	362	14	coplanar	coplanar	ADJ
ejpam-1851	362	15	points	point	NOUN
ejpam-1851	362	16	and	and	CCONJ
ejpam-1851	362	17	l1,3	l1,3	NOUN
ejpam-1851	362	18	is	be	AUX
ejpam-1851	362	19	not	not	PART
ejpam-1851	362	20	contained	contain	VERB
ejpam-1851	362	21	in	in	ADP
ejpam-1851	362	22	λ∩q	λ∩q	PROPN
ejpam-1851	362	23	,	,	PUNCT
ejpam-1851	362	24	we	we	PRON
ejpam-1851	362	25	conclude	conclude	VERB
ejpam-1851	362	26	that	that	SCONJ
ejpam-1851	362	27	λ∩q	λ∩q	PROPN
ejpam-1851	362	28	is	be	AUX
ejpam-1851	362	29	a	a	DET
ejpam-1851	362	30	union	union	NOUN
ejpam-1851	362	31	of	of	ADP
ejpam-1851	362	32	two	two	NUM
ejpam-1851	362	33	planes	plane	NOUN
ejpam-1851	362	34	or	or	CCONJ
ejpam-1851	362	35	a	a	DET
ejpam-1851	362	36	nonsingular	nonsingular	ADJ
ejpam-1851	362	37	quadric	quadric	NOUN
ejpam-1851	362	38	.	.	PUNCT
ejpam-1851	363	1	suppose	suppose	VERB
ejpam-1851	363	2	that	that	SCONJ
ejpam-1851	363	3	λ∩q	λ∩q	PROPN
ejpam-1851	363	4	is	be	AUX
ejpam-1851	363	5	a	a	DET
ejpam-1851	363	6	union	union	NOUN
ejpam-1851	363	7	of	of	ADP
ejpam-1851	363	8	two	two	NUM
ejpam-1851	363	9	planes	plane	NOUN
ejpam-1851	363	10	,	,	PUNCT
ejpam-1851	363	11	say	say	VERB
ejpam-1851	363	12	λ∩q	λ∩q	ADJ
ejpam-1851	363	13	=	=	NOUN
ejpam-1851	363	14	λ1	λ1	ADJ
ejpam-1851	363	15	∪λ2	∪λ2	NOUN
ejpam-1851	363	16	.	.	PUNCT
ejpam-1851	364	1	thus	thus	ADV
ejpam-1851	364	2	we	we	PRON
ejpam-1851	364	3	can	can	AUX
ejpam-1851	364	4	assume	assume	VERB
ejpam-1851	364	5	that	that	SCONJ
ejpam-1851	364	6	l1,2	l1,2	PROPN
ejpam-1851	364	7	⊂	⊂	PROPN
ejpam-1851	364	8	λ1	λ1	PROPN
ejpam-1851	364	9	.	.	PUNCT
ejpam-1851	365	1	next	next	ADV
ejpam-1851	365	2	,	,	PUNCT
ejpam-1851	365	3	note	note	VERB
ejpam-1851	365	4	that	that	SCONJ
ejpam-1851	365	5	l2,3	l2,3	VERB
ejpam-1851	365	6	6⊂	6⊂	NUM
ejpam-1851	365	7	λ∩q	λ∩q	X
ejpam-1851	365	8	(	(	PUNCT
ejpam-1851	365	9	since	since	SCONJ
ejpam-1851	365	10	l2,3	l2,3	NOUN
ejpam-1851	365	11	6⊂	6⊂	NUM
ejpam-1851	365	12	q	q	NOUN
ejpam-1851	365	13	)	)	PUNCT
ejpam-1851	365	14	.	.	PUNCT
ejpam-1851	366	1	so	so	ADV
ejpam-1851	366	2	p3	p3	PROPN
ejpam-1851	366	3	6∈	6∈	PROPN
ejpam-1851	366	4	λ1	λ1	PROPN
ejpam-1851	366	5	,	,	PUNCT
ejpam-1851	366	6	which	which	PRON
ejpam-1851	366	7	implies	imply	VERB
ejpam-1851	366	8	l1,3	l1,3	PROPN
ejpam-1851	366	9	⊂	⊂	PROPN
ejpam-1851	366	10	λ2	λ2	NOUN
ejpam-1851	366	11	.	.	PUNCT
ejpam-1851	367	1	now	now	ADV
ejpam-1851	367	2	,	,	PUNCT
ejpam-1851	367	3	since	since	SCONJ
ejpam-1851	367	4	p4	p4	ADJ
ejpam-1851	367	5	∈	∈	NOUN
ejpam-1851	367	6	λ1	λ1	PROPN
ejpam-1851	367	7	∪	∪	NOUN
ejpam-1851	367	8	λ2	λ2	NOUN
ejpam-1851	367	9	we	we	PRON
ejpam-1851	367	10	conclude	conclude	VERB
ejpam-1851	367	11	that	that	SCONJ
ejpam-1851	367	12	l2,4	l2,4	PROPN
ejpam-1851	367	13	and	and	CCONJ
ejpam-1851	367	14	l3,4	l3,4	PROPN
ejpam-1851	367	15	are	be	AUX
ejpam-1851	367	16	contained	contain	VERB
ejpam-1851	367	17	in	in	ADP
ejpam-1851	367	18	λ	λ	PROPN
ejpam-1851	367	19	∩	∩	NOUN
ejpam-1851	367	20	q	q	X
ejpam-1851	367	21	which	which	PRON
ejpam-1851	367	22	it	it	PRON
ejpam-1851	367	23	is	be	AUX
ejpam-1851	367	24	impossible	impossible	ADJ
ejpam-1851	367	25	since	since	SCONJ
ejpam-1851	367	26	ℓ4	ℓ4	PROPN
ejpam-1851	367	27	∩	∩	PROPN
ejpam-1851	367	28	ℓi	ℓi	PROPN
ejpam-1851	367	29	=	=	PUNCT
ejpam-1851	367	30	;	;	PUNCT
ejpam-1851	367	31	,	,	PUNCT
ejpam-1851	367	32	i	i	PRON
ejpam-1851	367	33	=	=	NOUN
ejpam-1851	367	34	2,3	2,3	NUM
ejpam-1851	367	35	.	.	PUNCT
ejpam-1851	368	1	therefore	therefore	ADV
ejpam-1851	368	2	,	,	PUNCT
ejpam-1851	368	3	λ∩q	λ∩q	PROPN
ejpam-1851	368	4	is	be	AUX
ejpam-1851	368	5	a	a	DET
ejpam-1851	368	6	nonsingular	nonsingular	ADJ
ejpam-1851	368	7	quadric	quadric	ADJ
ejpam-1851	368	8	surface	surface	NOUN
ejpam-1851	368	9	.	.	PUNCT
ejpam-1851	369	1	j.	j.	PROPN
ejpam-1851	369	2	rojas	rojas	PROPN
ejpam-1851	369	3	,	,	PUNCT
ejpam-1851	369	4	r.	r.	PROPN
ejpam-1851	369	5	mendoza	mendoza	PROPN
ejpam-1851	369	6	/	/	SYM
ejpam-1851	369	7	eur	eur	PROPN
ejpam-1851	369	8	.	.	PUNCT
ejpam-1851	370	1	j.	j.	PROPN
ejpam-1851	370	2	pure	pure	PROPN
ejpam-1851	370	3	appl	appl	PROPN
ejpam-1851	370	4	.	.	PROPN
ejpam-1851	370	5	math	math	PROPN
ejpam-1851	370	6	,	,	PUNCT
ejpam-1851	370	7	7	7	NUM
ejpam-1851	370	8	(	(	PUNCT
ejpam-1851	370	9	2014	2014	NUM
ejpam-1851	370	10	)	)	PUNCT
ejpam-1851	370	11	,	,	PUNCT
ejpam-1851	370	12	472	472	NUM
ejpam-1851	370	13	-	-	SYM
ejpam-1851	370	14	485	485	NUM
ejpam-1851	370	15	483	483	NUM
ejpam-1851	370	16	figure	figure	NOUN
ejpam-1851	370	17	4	4	NUM
ejpam-1851	370	18	:	:	PUNCT
ejpam-1851	370	19	case	case	NOUN
ejpam-1851	370	20	2.5.3	2.5.3	NUM
ejpam-1851	370	21	4.3	4.3	NUM
ejpam-1851	370	22	.	.	PUNCT
ejpam-1851	371	1	no	no	DET
ejpam-1851	371	2	pair	pair	NOUN
ejpam-1851	371	3	of	of	ADP
ejpam-1851	371	4	lines	line	NOUN
ejpam-1851	371	5	is	be	AUX
ejpam-1851	371	6	coplanar	coplanar	ADJ
ejpam-1851	371	7	let	let	VERB
ejpam-1851	371	8	q	q	PUNCT
ejpam-1851	371	9	be	be	AUX
ejpam-1851	371	10	a	a	DET
ejpam-1851	371	11	nonsingular	nonsingular	ADJ
ejpam-1851	371	12	quadric	quadric	ADJ
ejpam-1851	371	13	surface	surface	NOUN
ejpam-1851	371	14	in	in	ADP
ejpam-1851	371	15	p3	p3	NOUN
ejpam-1851	371	16	containing	contain	VERB
ejpam-1851	371	17	ℓ1	ℓ1	NOUN
ejpam-1851	371	18	,	,	PUNCT
ejpam-1851	371	19	ℓ2	ℓ2	NOUN
ejpam-1851	371	20	and	and	CCONJ
ejpam-1851	371	21	ℓ3	ℓ3	PROPN
ejpam-1851	371	22	(	(	PUNCT
ejpam-1851	371	23	as	as	SCONJ
ejpam-1851	371	24	stated	state	VERB
ejpam-1851	371	25	in	in	ADP
ejpam-1851	371	26	lemma	lemma	PROPN
ejpam-1851	371	27	2	2	NUM
ejpam-1851	371	28	)	)	PUNCT
ejpam-1851	371	29	.	.	PUNCT
ejpam-1851	372	1	since	since	SCONJ
ejpam-1851	372	2	ℓi	ℓi	PROPN
ejpam-1851	372	3	∩	∩	PROPN
ejpam-1851	372	4	ℓ	ℓ	PROPN
ejpam-1851	372	5	j	j	PROPN
ejpam-1851	373	1	=	=	X
ejpam-1851	373	2	;	;	PUNCT
ejpam-1851	373	3	for	for	ADP
ejpam-1851	373	4	1	1	NUM
ejpam-1851	373	5	≤	≤	NUM
ejpam-1851	374	1	i	i	PRON
ejpam-1851	374	2	<	<	X
ejpam-1851	374	3	j	j	PROPN
ejpam-1851	374	4	≤	≤	ADV
ejpam-1851	374	5	3	3	NUM
ejpam-1851	374	6	then	then	ADV
ejpam-1851	374	7	they	they	PRON
ejpam-1851	374	8	belong	belong	VERB
ejpam-1851	374	9	to	to	ADP
ejpam-1851	374	10	the	the	DET
ejpam-1851	374	11	same	same	ADJ
ejpam-1851	374	12	family	family	NOUN
ejpam-1851	374	13	of	of	ADP
ejpam-1851	374	14	lines	line	NOUN
ejpam-1851	374	15	in	in	ADP
ejpam-1851	374	16	q	q	PROPN
ejpam-1851	374	17	(	(	PUNCT
ejpam-1851	374	18	cf	cf	NOUN
ejpam-1851	374	19	.	.	PUNCT
ejpam-1851	375	1	lemma	lemma	PROPN
ejpam-1851	375	2	1	1	NUM
ejpam-1851	375	3	)	)	PUNCT
ejpam-1851	375	4	.	.	PUNCT
ejpam-1851	376	1	so	so	ADV
ejpam-1851	376	2	we	we	PRON
ejpam-1851	376	3	will	will	AUX
ejpam-1851	376	4	assume	assume	VERB
ejpam-1851	376	5	that	that	SCONJ
ejpam-1851	376	6	ℓ1	ℓ1	NOUN
ejpam-1851	376	7	,	,	PUNCT
ejpam-1851	376	8	ℓ2	ℓ2	NOUN
ejpam-1851	376	9	and	and	CCONJ
ejpam-1851	376	10	ℓ3	ℓ3	PROPN
ejpam-1851	376	11	belong	belong	VERB
ejpam-1851	376	12	to	to	ADP
ejpam-1851	376	13	the	the	DET
ejpam-1851	376	14	family	family	NOUN
ejpam-1851	376	15	l	l	NOUN
ejpam-1851	376	16	.	.	PUNCT
ejpam-1851	377	1	note	note	VERB
ejpam-1851	377	2	that	that	SCONJ
ejpam-1851	377	3	any	any	DET
ejpam-1851	377	4	solution	solution	NOUN
ejpam-1851	377	5	ℓ	ℓ	PROPN
ejpam-1851	377	6	∈	∈	NOUN
ejpam-1851	377	7	s	s	PART
ejpam-1851	377	8	meets	meet	VERB
ejpam-1851	377	9	the	the	DET
ejpam-1851	377	10	line	line	NOUN
ejpam-1851	377	11	ℓi	ℓi	PROPN
ejpam-1851	377	12	at	at	ADP
ejpam-1851	377	13	a	a	DET
ejpam-1851	377	14	point	point	NOUN
ejpam-1851	377	15	pi	pi	NOUN
ejpam-1851	377	16	∈	∈	PROPN
ejpam-1851	377	17	ℓi	ℓi	PROPN
ejpam-1851	377	18	⊂	⊂	PROPN
ejpam-1851	377	19	q	q	PROPN
ejpam-1851	377	20	for	for	ADP
ejpam-1851	377	21	i	i	PRON
ejpam-1851	377	22	=	=	NOUN
ejpam-1851	378	1	1,2,3	1,2,3	X
ejpam-1851	378	2	.	.	PUNCT
ejpam-1851	379	1	since	since	SCONJ
ejpam-1851	379	2	this	this	DET
ejpam-1851	379	3	three	three	NUM
ejpam-1851	379	4	points	point	NOUN
ejpam-1851	379	5	p1	p1	NOUN
ejpam-1851	379	6	,	,	PUNCT
ejpam-1851	379	7	p2	p2	PROPN
ejpam-1851	379	8	and	and	CCONJ
ejpam-1851	379	9	p3	p3	PROPN
ejpam-1851	379	10	are	be	AUX
ejpam-1851	379	11	different	different	ADJ
ejpam-1851	379	12	and	and	CCONJ
ejpam-1851	379	13	belong	belong	VERB
ejpam-1851	379	14	to	to	ADP
ejpam-1851	379	15	ℓ	ℓ	X
ejpam-1851	379	16	∩q	∩q	PROPN
ejpam-1851	379	17	.	.	PUNCT
ejpam-1851	380	1	then	then	ADV
ejpam-1851	380	2	follows	follow	VERB
ejpam-1851	380	3	from	from	ADP
ejpam-1851	380	4	proposition	proposition	NOUN
ejpam-1851	380	5	2	2	NUM
ejpam-1851	380	6	that	that	PRON
ejpam-1851	380	7	ℓ	ℓ	X
ejpam-1851	380	8	⊂q	⊂q	PROPN
ejpam-1851	380	9	,	,	PUNCT
ejpam-1851	380	10	which	which	PRON
ejpam-1851	380	11	implies	imply	VERB
ejpam-1851	380	12	that	that	SCONJ
ejpam-1851	380	13	ℓ	ℓ	NOUN
ejpam-1851	380	14	∈m	∈m	NOUN
ejpam-1851	380	15	.	.	PUNCT
ejpam-1851	381	1	therefore	therefore	ADV
ejpam-1851	381	2	any	any	DET
ejpam-1851	381	3	solution	solution	NOUN
ejpam-1851	381	4	belong	belong	VERB
ejpam-1851	381	5	to	to	ADP
ejpam-1851	381	6	the	the	DET
ejpam-1851	381	7	familym	familym	NOUN
ejpam-1851	381	8	.	.	PUNCT
ejpam-1851	382	1	in	in	ADP
ejpam-1851	382	2	order	order	NOUN
ejpam-1851	382	3	to	to	PART
ejpam-1851	382	4	determine	determine	VERB
ejpam-1851	382	5	the	the	DET
ejpam-1851	382	6	set	set	NOUN
ejpam-1851	382	7	of	of	ADP
ejpam-1851	382	8	solutions	solution	NOUN
ejpam-1851	382	9	s	s	PART
ejpam-1851	382	10	and	and	CCONJ
ejpam-1851	382	11	the	the	DET
ejpam-1851	382	12	linear	linear	ADJ
ejpam-1851	382	13	space	space	NOUN
ejpam-1851	382	14	λ	λ	NOUN
ejpam-1851	382	15	,	,	PUNCT
ejpam-1851	382	16	we	we	PRON
ejpam-1851	382	17	will	will	AUX
ejpam-1851	382	18	consider	consider	VERB
ejpam-1851	382	19	the	the	DET
ejpam-1851	382	20	following	follow	VERB
ejpam-1851	382	21	two	two	NUM
ejpam-1851	382	22	subcases	subcase	NOUN
ejpam-1851	382	23	.	.	PUNCT
ejpam-1851	383	1	3.1	3.1	NUM
ejpam-1851	383	2	ℓ4	ℓ4	NOUN
ejpam-1851	383	3	⊂q	⊂q	ADV
ejpam-1851	383	4	.	.	PUNCT
ejpam-1851	384	1	in	in	ADP
ejpam-1851	384	2	this	this	DET
ejpam-1851	384	3	case	case	NOUN
ejpam-1851	384	4	ℓ1,ℓ2,ℓ3	ℓ1,ℓ2,ℓ3	PROPN
ejpam-1851	384	5	and	and	CCONJ
ejpam-1851	384	6	ℓ4	ℓ4	PROPN
ejpam-1851	384	7	belong	belong	VERB
ejpam-1851	384	8	to	to	ADP
ejpam-1851	384	9	the	the	DET
ejpam-1851	384	10	same	same	ADJ
ejpam-1851	384	11	family	family	NOUN
ejpam-1851	384	12	l	l	NOUN
ejpam-1851	384	13	(	(	PUNCT
ejpam-1851	384	14	since	since	SCONJ
ejpam-1851	384	15	ℓ4	ℓ4	PROPN
ejpam-1851	384	16	if	if	SCONJ
ejpam-1851	384	17	disjoint	disjoint	NOUN
ejpam-1851	384	18	to	to	ADP
ejpam-1851	384	19	the	the	DET
ejpam-1851	384	20	others	other	NOUN
ejpam-1851	384	21	three	three	NUM
ejpam-1851	384	22	)	)	PUNCT
ejpam-1851	384	23	.	.	PUNCT
ejpam-1851	385	1	therefore	therefore	ADV
ejpam-1851	385	2	s	s	VERB
ejpam-1851	385	3	=	=	NOUN
ejpam-1851	385	4	m	m	NOUN
ejpam-1851	385	5	.	.	PUNCT
ejpam-1851	386	1	now	now	ADV
ejpam-1851	386	2	,	,	PUNCT
ejpam-1851	386	3	since	since	SCONJ
ejpam-1851	386	4	the	the	DET
ejpam-1851	386	5	line	line	NOUN
ejpam-1851	386	6	li	li	PROPN
ejpam-1851	386	7	,	,	PUNCT
ejpam-1851	386	8	j	j	PROPN
ejpam-1851	387	1	=	=	PUNCT
ejpam-1851	387	2	pi	pi	PROPN
ejpam-1851	387	3	,	,	PUNCT
ejpam-1851	387	4	pj	pj	PROPN
ejpam-1851	387	5	�	�	PROPN
ejpam-1851	387	6	6⊂	6⊂	NUM
ejpam-1851	387	7	q	q	NOUN
ejpam-1851	387	8	for	for	ADP
ejpam-1851	387	9	1	1	NUM
ejpam-1851	387	10	≤	≤	NUM
ejpam-1851	388	1	i	i	PRON
ejpam-1851	388	2	<	<	X
ejpam-1851	388	3	j	j	PROPN
ejpam-1851	388	4	≤	≤	ADV
ejpam-1851	388	5	3	3	NUM
ejpam-1851	388	6	,	,	PUNCT
ejpam-1851	388	7	we	we	PRON
ejpam-1851	388	8	conclude	conclude	VERB
ejpam-1851	388	9	from	from	ADP
ejpam-1851	388	10	proposition	proposition	NOUN
ejpam-1851	388	11	6	6	NUM
ejpam-1851	388	12	that	that	PRON
ejpam-1851	388	13	p1	p1	NOUN
ejpam-1851	388	14	,	,	PUNCT
ejpam-1851	388	15	p2	p2	NOUN
ejpam-1851	388	16	,	,	PUNCT
ejpam-1851	388	17	p3	p3	PROPN
ejpam-1851	388	18	�	�	PROPN
ejpam-1851	388	19	is	be	AUX
ejpam-1851	388	20	a	a	DET
ejpam-1851	388	21	plane	plane	NOUN
ejpam-1851	388	22	not	not	PART
ejpam-1851	388	23	contained	contain	VERB
ejpam-1851	388	24	in	in	ADP
ejpam-1851	388	25	q.	q.	PROPN
ejpam-1851	388	26	but	but	CCONJ
ejpam-1851	388	27	,	,	PUNCT
ejpam-1851	388	28	from	from	ADP
ejpam-1851	388	29	remark	remark	NOUN
ejpam-1851	388	30	1	1	NUM
ejpam-1851	388	31	we	we	PRON
ejpam-1851	388	32	conclude	conclude	VERB
ejpam-1851	388	33	that	that	SCONJ
ejpam-1851	388	34	p1	p1	NOUN
ejpam-1851	388	35	,	,	PUNCT
ejpam-1851	388	36	p2	p2	NOUN
ejpam-1851	388	37	,	,	PUNCT
ejpam-1851	388	38	p3	p3	PROPN
ejpam-1851	388	39	�	�	PROPN
ejpam-1851	388	40	∩q	∩q	PROPN
ejpam-1851	389	1	=	=	PUNCT
ejpam-1851	390	1	p	p	X
ejpam-1851	390	2	(	(	PUNCT
ejpam-1851	390	3	l	l	NOUN
ejpam-1851	390	4	)	)	PUNCT
ejpam-1851	390	5	.	.	PUNCT
ejpam-1851	391	1	consequently	consequently	ADV
ejpam-1851	391	2	,	,	PUNCT
ejpam-1851	391	3	λ	λ	PROPN
ejpam-1851	391	4	=	=	SYM
ejpam-1851	391	5	p1	p1	PROPN
ejpam-1851	391	6	,	,	PUNCT
ejpam-1851	391	7	p2	p2	NOUN
ejpam-1851	391	8	,	,	PUNCT
ejpam-1851	391	9	p3	p3	PROPN
ejpam-1851	391	10	�	�	PROPN
ejpam-1851	391	11	(	(	PUNCT
ejpam-1851	391	12	keep	keep	VERB
ejpam-1851	391	13	in	in	ADP
ejpam-1851	391	14	mind	mind	NOUN
ejpam-1851	391	15	that	that	SCONJ
ejpam-1851	391	16	p4	p4	PROPN
ejpam-1851	391	17	∈	∈	PROPN
ejpam-1851	391	18	p	p	X
ejpam-1851	391	19	(	(	PUNCT
ejpam-1851	391	20	l	l	NOUN
ejpam-1851	391	21	)	)	PUNCT
ejpam-1851	391	22	too	too	ADV
ejpam-1851	391	23	)	)	PUNCT
ejpam-1851	391	24	and	and	CCONJ
ejpam-1851	391	25	λ∩q	λ∩q	PROPN
ejpam-1851	391	26	is	be	AUX
ejpam-1851	391	27	a	a	DET
ejpam-1851	391	28	nonsingular	nonsingular	ADJ
ejpam-1851	391	29	conic	conic	NOUN
ejpam-1851	391	30	.	.	PUNCT
ejpam-1851	392	1	3.2	3.2	NUM
ejpam-1851	392	2	ℓ4	ℓ4	PROPN
ejpam-1851	392	3	6⊂q	6⊂q	PROPN
ejpam-1851	392	4	.	.	PUNCT
ejpam-1851	393	1	in	in	ADP
ejpam-1851	393	2	this	this	DET
ejpam-1851	393	3	case	case	NOUN
ejpam-1851	393	4	follows	follow	VERB
ejpam-1851	393	5	from	from	ADP
ejpam-1851	393	6	proposition	proposition	NOUN
ejpam-1851	393	7	2	2	NUM
ejpam-1851	393	8	that	that	PRON
ejpam-1851	393	9	ℓ4	ℓ4	PROPN
ejpam-1851	393	10	∩q	∩q	PROPN
ejpam-1851	393	11	is	be	AUX
ejpam-1851	393	12	non	non	X
ejpam-1851	393	13	empty	empty	ADJ
ejpam-1851	393	14	and	and	CCONJ
ejpam-1851	393	15	ℓ4	ℓ4	X
ejpam-1851	393	16	∩q	∩q	PROPN
ejpam-1851	393	17	=	=	PUNCT
ejpam-1851	394	1	{	{	PUNCT
ejpam-1851	394	2	x	x	INTJ
ejpam-1851	394	3	,	,	PUNCT
ejpam-1851	394	4	y	y	PROPN
ejpam-1851	394	5	}	}	PUNCT
ejpam-1851	394	6	(	(	PUNCT
ejpam-1851	394	7	with	with	SCONJ
ejpam-1851	394	8	x	x	PUNCT
ejpam-1851	394	9	and	and	CCONJ
ejpam-1851	394	10	y	y	PROPN
ejpam-1851	394	11	do	do	AUX
ejpam-1851	394	12	not	not	PART
ejpam-1851	394	13	necessarily	necessarily	ADV
ejpam-1851	394	14	distinct	distinct	ADJ
ejpam-1851	394	15	)	)	PUNCT
ejpam-1851	394	16	.	.	PUNCT
ejpam-1851	395	1	let	let	VERB
ejpam-1851	395	2	mx	mx	PROPN
ejpam-1851	395	3	and	and	CCONJ
ejpam-1851	395	4	my	my	PRON
ejpam-1851	395	5	be	be	AUX
ejpam-1851	395	6	the	the	DET
ejpam-1851	395	7	unique	unique	ADJ
ejpam-1851	395	8	lines	line	NOUN
ejpam-1851	395	9	in	in	ADP
ejpam-1851	395	10	the	the	DET
ejpam-1851	395	11	familym	familym	NOUN
ejpam-1851	395	12	passing	pass	VERB
ejpam-1851	395	13	through	through	ADP
ejpam-1851	395	14	x	x	PUNCT
ejpam-1851	395	15	and	and	CCONJ
ejpam-1851	395	16	y	y	PROPN
ejpam-1851	395	17	,	,	PUNCT
ejpam-1851	395	18	respectively	respectively	ADV
ejpam-1851	395	19	(	(	PUNCT
ejpam-1851	395	20	cf	cf	NOUN
ejpam-1851	395	21	.	.	PUNCT
ejpam-1851	396	1	(	(	PUNCT
ejpam-1851	396	2	4	4	NUM
ejpam-1851	396	3	)	)	PUNCT
ejpam-1851	396	4	in	in	ADP
ejpam-1851	396	5	lemma	lemma	PROPN
ejpam-1851	396	6	1	1	NUM
ejpam-1851	396	7	)	)	PUNCT
ejpam-1851	396	8	.	.	PUNCT
ejpam-1851	397	1	indeed	indeed	ADV
ejpam-1851	397	2	mx	mx	PROPN
ejpam-1851	397	3	and	and	CCONJ
ejpam-1851	397	4	my	my	PRON
ejpam-1851	397	5	are	be	AUX
ejpam-1851	397	6	solutions	solution	NOUN
ejpam-1851	397	7	(	(	PUNCT
ejpam-1851	397	8	in	in	ADP
ejpam-1851	397	9	fact	fact	NOUN
ejpam-1851	397	10	,	,	PUNCT
ejpam-1851	397	11	if	if	SCONJ
ejpam-1851	397	12	p	p	X
ejpam-1851	397	13	∈	∈	PROPN
ejpam-1851	397	14	{	{	PUNCT
ejpam-1851	397	15	x	x	PROPN
ejpam-1851	397	16	,	,	PUNCT
ejpam-1851	397	17	y	y	PROPN
ejpam-1851	397	18	}	}	PUNCT
ejpam-1851	397	19	then	then	ADV
ejpam-1851	397	20	mp	mp	PROPN
ejpam-1851	397	21	∩	∩	PROPN
ejpam-1851	397	22	ℓi	ℓi	PROPN
ejpam-1851	397	23	is	be	AUX
ejpam-1851	397	24	non	non	X
ejpam-1851	397	25	empty	empty	ADJ
ejpam-1851	397	26	for	for	ADP
ejpam-1851	397	27	i	i	X
ejpam-1851	397	28	=	=	NOUN
ejpam-1851	397	29	1,2,3	1,2,3	NUM
ejpam-1851	397	30	because	because	SCONJ
ejpam-1851	397	31	ℓi	ℓi	PROPN
ejpam-1851	397	32	∈	∈	PROPN
ejpam-1851	397	33	l	l	PROPN
ejpam-1851	397	34	and	and	CCONJ
ejpam-1851	397	35	mp	mp	PROPN
ejpam-1851	397	36	∩	∩	ADJ
ejpam-1851	397	37	ℓ4	ℓ4	PROPN
ejpam-1851	397	38	=	=	PUNCT
ejpam-1851	397	39	{	{	PUNCT
ejpam-1851	397	40	p	p	NOUN
ejpam-1851	397	41	}	}	PUNCT
ejpam-1851	397	42	)	)	PUNCT
ejpam-1851	397	43	.	.	PUNCT
ejpam-1851	398	1	now	now	ADV
ejpam-1851	398	2	,	,	PUNCT
ejpam-1851	398	3	we	we	PRON
ejpam-1851	398	4	will	will	AUX
ejpam-1851	398	5	show	show	VERB
ejpam-1851	398	6	that	that	SCONJ
ejpam-1851	398	7	mx	mx	PROPN
ejpam-1851	398	8	and	and	CCONJ
ejpam-1851	398	9	my	my	PRON
ejpam-1851	398	10	are	be	AUX
ejpam-1851	398	11	the	the	DET
ejpam-1851	398	12	unique	unique	ADJ
ejpam-1851	398	13	solutions	solution	NOUN
ejpam-1851	398	14	.	.	PUNCT
ejpam-1851	399	1	let	let	VERB
ejpam-1851	399	2	ℓ	ℓ	NOUN
ejpam-1851	399	3	∈m	∈m	NOUN
ejpam-1851	399	4	be	be	AUX
ejpam-1851	399	5	a	a	DET
ejpam-1851	399	6	solution	solution	NOUN
ejpam-1851	399	7	,	,	PUNCT
ejpam-1851	399	8	then	then	ADV
ejpam-1851	399	9	ℓ∩	ℓ∩	PROPN
ejpam-1851	399	10	ℓ4	ℓ4	PROPN
ejpam-1851	399	11	⊂q∩	⊂q∩	NOUN
ejpam-1851	399	12	ℓ4	ℓ4	PROPN
ejpam-1851	399	13	=	=	PUNCT
ejpam-1851	399	14	{	{	PUNCT
ejpam-1851	399	15	x	x	PROPN
ejpam-1851	399	16	,	,	PUNCT
ejpam-1851	399	17	y	y	PROPN
ejpam-1851	399	18	}	}	PUNCT
ejpam-1851	399	19	which	which	PRON
ejpam-1851	399	20	implies	imply	VERB
ejpam-1851	399	21	that	that	SCONJ
ejpam-1851	399	22	x	x	PUNCT
ejpam-1851	399	23	∈	∈	PROPN
ejpam-1851	399	24	ℓ	ℓ	PROPN
ejpam-1851	399	25	or	or	CCONJ
ejpam-1851	399	26	y	y	PROPN
ejpam-1851	399	27	∈	∈	PROPN
ejpam-1851	399	28	ℓ.	ℓ.	NOUN
ejpam-1851	399	29	therefore	therefore	ADV
ejpam-1851	399	30	follows	follow	VERB
ejpam-1851	399	31	from	from	ADP
ejpam-1851	399	32	(	(	PUNCT
ejpam-1851	399	33	4	4	NUM
ejpam-1851	399	34	)	)	PUNCT
ejpam-1851	399	35	in	in	ADP
ejpam-1851	399	36	the	the	DET
ejpam-1851	399	37	lemma	lemma	PROPN
ejpam-1851	399	38	1	1	NUM
ejpam-1851	399	39	that	that	PRON
ejpam-1851	399	40	ℓ=	ℓ=	PROPN
ejpam-1851	399	41	mx	mx	PROPN
ejpam-1851	399	42	or	or	CCONJ
ejpam-1851	399	43	ℓ=	ℓ=	VERB
ejpam-1851	399	44	my	my	PRON
ejpam-1851	399	45	.	.	PUNCT
ejpam-1851	400	1	again	again	ADV
ejpam-1851	400	2	,	,	PUNCT
ejpam-1851	400	3	since	since	SCONJ
ejpam-1851	400	4	the	the	DET
ejpam-1851	400	5	line	line	NOUN
ejpam-1851	400	6	li	li	PROPN
ejpam-1851	400	7	,	,	PUNCT
ejpam-1851	400	8	j	j	PROPN
ejpam-1851	400	9	=	=	SYM
ejpam-1851	400	10	pi	pi	PROPN
ejpam-1851	400	11	,	,	PUNCT
ejpam-1851	400	12	pj	pj	PROPN
ejpam-1851	400	13	�	�	PROPN
ejpam-1851	400	14	6⊂	6⊂	NUM
ejpam-1851	400	15	q	q	NOUN
ejpam-1851	400	16	for	for	ADP
ejpam-1851	400	17	1	1	NUM
ejpam-1851	400	18	≤	≤	NUM
ejpam-1851	401	1	i	i	PRON
ejpam-1851	401	2	<	<	X
ejpam-1851	401	3	j	j	PROPN
ejpam-1851	401	4	≤	≤	ADV
ejpam-1851	401	5	3	3	NUM
ejpam-1851	401	6	,	,	PUNCT
ejpam-1851	401	7	we	we	PRON
ejpam-1851	401	8	conclude	conclude	VERB
ejpam-1851	401	9	that	that	SCONJ
ejpam-1851	401	10	p1	p1	NOUN
ejpam-1851	401	11	,	,	PUNCT
ejpam-1851	401	12	p2	p2	NOUN
ejpam-1851	401	13	,	,	PUNCT
ejpam-1851	401	14	p3	p3	PROPN
ejpam-1851	401	15	�	�	PROPN
ejpam-1851	401	16	is	be	AUX
ejpam-1851	401	17	a	a	DET
ejpam-1851	401	18	plane	plane	NOUN
ejpam-1851	401	19	not	not	PART
ejpam-1851	401	20	contained	contain	VERB
ejpam-1851	401	21	in	in	ADP
ejpam-1851	401	22	q	q	PROPN
ejpam-1851	401	23	and	and	CCONJ
ejpam-1851	401	24	p1	p1	PROPN
ejpam-1851	401	25	,	,	PUNCT
ejpam-1851	401	26	p2	p2	NOUN
ejpam-1851	401	27	,	,	PUNCT
ejpam-1851	402	1	p3	p3	PROPN
ejpam-1851	402	2	�	�	PROPN
ejpam-1851	402	3	∩q	∩q	PROPN
ejpam-1851	402	4	is	be	AUX
ejpam-1851	402	5	a	a	DET
ejpam-1851	402	6	nonsingular	nonsingular	ADJ
ejpam-1851	402	7	conic	conic	NOUN
ejpam-1851	402	8	.	.	PUNCT
ejpam-1851	403	1	on	on	ADP
ejpam-1851	403	2	the	the	DET
ejpam-1851	403	3	other	other	ADJ
ejpam-1851	403	4	hand	hand	NOUN
ejpam-1851	403	5	,	,	PUNCT
ejpam-1851	403	6	the	the	DET
ejpam-1851	403	7	condition	condition	NOUN
ejpam-1851	403	8	ℓ4	ℓ4	PROPN
ejpam-1851	403	9	6⊂	6⊂	NUM
ejpam-1851	403	10	q	q	PROPN
ejpam-1851	403	11	implies	imply	VERB
ejpam-1851	403	12	that	that	SCONJ
ejpam-1851	403	13	ℓ4	ℓ4	VERB
ejpam-1851	403	14	6∈	6∈	NUM
ejpam-1851	403	15	l	l	NOUN
ejpam-1851	403	16	.	.	PUNCT
ejpam-1851	404	1	so	so	ADV
ejpam-1851	404	2	,	,	PUNCT
ejpam-1851	404	3	we	we	PRON
ejpam-1851	404	4	are	be	AUX
ejpam-1851	404	5	left	leave	VERB
ejpam-1851	404	6	to	to	PART
ejpam-1851	404	7	conclude	conclude	VERB
ejpam-1851	404	8	that	that	DET
ejpam-1851	404	9	p4	p4	PROPN
ejpam-1851	404	10	6∈	6∈	PROPN
ejpam-1851	404	11	p1	p1	NOUN
ejpam-1851	404	12	,	,	PUNCT
ejpam-1851	404	13	p2	p2	NOUN
ejpam-1851	404	14	,	,	PUNCT
ejpam-1851	404	15	p3	p3	PROPN
ejpam-1851	404	16	�	�	PROPN
ejpam-1851	404	17	.	.	PUNCT
ejpam-1851	405	1	thus	thus	ADV
ejpam-1851	405	2	λ	λ	PRON
ejpam-1851	405	3	is	be	AUX
ejpam-1851	405	4	a	a	DET
ejpam-1851	405	5	3	3	NUM
ejpam-1851	405	6	-	-	PUNCT
ejpam-1851	405	7	plane	plane	NOUN
ejpam-1851	405	8	in	in	ADP
ejpam-1851	405	9	p5	p5	PROPN
ejpam-1851	405	10	.	.	PUNCT
ejpam-1851	406	1	one	one	NUM
ejpam-1851	406	2	more	more	ADJ
ejpam-1851	406	3	time	time	NOUN
ejpam-1851	406	4	,	,	PUNCT
ejpam-1851	406	5	since	since	SCONJ
ejpam-1851	406	6	λ	λ	PROPN
ejpam-1851	406	7	∩q	∩q	PROPN
ejpam-1851	406	8	is	be	AUX
ejpam-1851	406	9	a	a	DET
ejpam-1851	406	10	quadric	quadric	ADJ
ejpam-1851	406	11	surface	surface	NOUN
ejpam-1851	406	12	containing	contain	VERB
ejpam-1851	406	13	four	four	NUM
ejpam-1851	406	14	non	non	ADJ
ejpam-1851	406	15	coplanar	coplanar	ADJ
ejpam-1851	406	16	points	point	NOUN
ejpam-1851	406	17	and	and	CCONJ
ejpam-1851	406	18	,	,	PUNCT
ejpam-1851	406	19	the	the	DET
ejpam-1851	406	20	line	line	NOUN
ejpam-1851	406	21	li	li	PROPN
ejpam-1851	406	22	,	,	PUNCT
ejpam-1851	406	23	j	j	PROPN
ejpam-1851	406	24	6⊂	6⊂	NUM
ejpam-1851	406	25	q	q	NOUN
ejpam-1851	406	26	for	for	ADP
ejpam-1851	406	27	1	1	NUM
ejpam-1851	406	28	≤	≤	NUM
ejpam-1851	407	1	i	i	PRON
ejpam-1851	407	2	<	<	X
ejpam-1851	407	3	j	j	PROPN
ejpam-1851	407	4	≤	≤	ADV
ejpam-1851	407	5	3	3	NUM
ejpam-1851	407	6	,	,	PUNCT
ejpam-1851	407	7	we	we	PRON
ejpam-1851	407	8	conclude	conclude	VERB
ejpam-1851	407	9	that	that	SCONJ
ejpam-1851	407	10	λ	λ	PROPN
ejpam-1851	407	11	∩	∩	NOUN
ejpam-1851	407	12	q	q	X
ejpam-1851	407	13	is	be	AUX
ejpam-1851	407	14	a	a	DET
ejpam-1851	407	15	quadric	quadric	ADJ
ejpam-1851	407	16	cone	cone	NOUN
ejpam-1851	407	17	or	or	CCONJ
ejpam-1851	407	18	a	a	DET
ejpam-1851	407	19	nonsingular	nonsingular	ADJ
ejpam-1851	407	20	quadric	quadric	ADJ
ejpam-1851	407	21	surface	surface	NOUN
ejpam-1851	407	22	.	.	PUNCT
ejpam-1851	408	1	in	in	ADP
ejpam-1851	408	2	fact	fact	NOUN
ejpam-1851	408	3	,	,	PUNCT
ejpam-1851	408	4	since	since	SCONJ
ejpam-1851	408	5	ℓ4	ℓ4	PROPN
ejpam-1851	408	6	6⊂q	6⊂q	PROPN
ejpam-1851	408	7	,	,	PUNCT
ejpam-1851	408	8	it	it	PRON
ejpam-1851	408	9	must	must	AUX
ejpam-1851	408	10	be	be	AUX
ejpam-1851	408	11	either	either	ADV
ejpam-1851	408	12	:	:	PUNCT
ejpam-1851	408	13	tangent	tangent	NOUN
ejpam-1851	408	14	or	or	CCONJ
ejpam-1851	408	15	secant	secant	ADJ
ejpam-1851	408	16	to	to	ADP
ejpam-1851	408	17	the	the	DET
ejpam-1851	408	18	surface	surface	NOUN
ejpam-1851	408	19	q.	q.	PROPN
ejpam-1851	408	20	j.	j.	PROPN
ejpam-1851	408	21	rojas	rojas	PROPN
ejpam-1851	408	22	,	,	PUNCT
ejpam-1851	408	23	r.	r.	PROPN
ejpam-1851	408	24	mendoza	mendoza	PROPN
ejpam-1851	408	25	/	/	SYM
ejpam-1851	408	26	eur	eur	PROPN
ejpam-1851	408	27	.	.	PUNCT
ejpam-1851	409	1	j.	j.	PROPN
ejpam-1851	409	2	pure	pure	PROPN
ejpam-1851	409	3	appl	appl	PROPN
ejpam-1851	409	4	.	.	PROPN
ejpam-1851	409	5	math	math	PROPN
ejpam-1851	409	6	,	,	PUNCT
ejpam-1851	409	7	7	7	NUM
ejpam-1851	409	8	(	(	PUNCT
ejpam-1851	409	9	2014	2014	NUM
ejpam-1851	409	10	)	)	PUNCT
ejpam-1851	409	11	,	,	PUNCT
ejpam-1851	409	12	472	472	NUM
ejpam-1851	409	13	-	-	SYM
ejpam-1851	409	14	485	485	NUM
ejpam-1851	409	15	484	484	NUM
ejpam-1851	409	16	assume	assume	VERB
ejpam-1851	409	17	that	that	SCONJ
ejpam-1851	409	18	ℓ4	ℓ4	NOUN
ejpam-1851	409	19	is	be	AUX
ejpam-1851	409	20	tangent	tangent	ADJ
ejpam-1851	409	21	to	to	ADP
ejpam-1851	409	22	q.	q.	PROPN
ejpam-1851	409	23	let	let	VERB
ejpam-1851	409	24	x	x	SYM
ejpam-1851	409	25	∈	∈	PROPN
ejpam-1851	409	26	ℓ4	ℓ4	NOUN
ejpam-1851	409	27	∩q	∩q	PROPN
ejpam-1851	409	28	be	be	VERB
ejpam-1851	409	29	the	the	DET
ejpam-1851	409	30	tangent	tangent	NOUN
ejpam-1851	409	31	point	point	NOUN
ejpam-1851	409	32	.	.	PUNCT
ejpam-1851	410	1	let	let	VERB
ejpam-1851	410	2	mx	mx	PROPN
ejpam-1851	410	3	∈	∈	PROPN
ejpam-1851	410	4	m	m	PROPN
ejpam-1851	410	5	and	and	CCONJ
ejpam-1851	410	6	lx	lx	ADP
ejpam-1851	410	7	∈	∈	PROPN
ejpam-1851	410	8	l	l	NOUN
ejpam-1851	410	9	be	be	AUX
ejpam-1851	410	10	the	the	DET
ejpam-1851	410	11	(	(	PUNCT
ejpam-1851	410	12	only	only	ADJ
ejpam-1851	410	13	)	)	PUNCT
ejpam-1851	410	14	lines	line	NOUN
ejpam-1851	410	15	in	in	ADP
ejpam-1851	410	16	q	q	NOUN
ejpam-1851	410	17	passing	pass	VERB
ejpam-1851	410	18	through	through	ADP
ejpam-1851	410	19	x	x	X
ejpam-1851	410	20	.	.	PUNCT
ejpam-1851	411	1	it	it	PRON
ejpam-1851	411	2	is	be	AUX
ejpam-1851	411	3	verified	verify	VERB
ejpam-1851	411	4	that	that	SCONJ
ejpam-1851	411	5	ℓ4	ℓ4	PROPN
ejpam-1851	411	6	⊂	⊂	PROPN
ejpam-1851	411	7	txq	txq	PROPN
ejpam-1851	411	8	=	=	PUNCT
ejpam-1851	411	9	〈	〈	PROPN
ejpam-1851	411	10	mx	mx	PROPN
ejpam-1851	411	11	,	,	PUNCT
ejpam-1851	411	12	lx	lx	NOUN
ejpam-1851	411	13	〉	〉	NOUN
ejpam-1851	411	14	.	.	PUNCT
ejpam-1851	412	1	therefore	therefore	ADV
ejpam-1851	412	2	,	,	PUNCT
ejpam-1851	412	3	ℓ4	ℓ4	PROPN
ejpam-1851	412	4	∈	∈	PROPN
ejpam-1851	412	5	ωx(txq	ωx(txq	PROPN
ejpam-1851	412	6	)	)	PUNCT
ejpam-1851	412	7	.	.	PUNCT
ejpam-1851	413	1	set	set	VERB
ejpam-1851	413	2	m	m	PROPN
ejpam-1851	413	3	=	=	SYM
ejpam-1851	413	4	p	p	X
ejpam-1851	413	5	(	(	PUNCT
ejpam-1851	413	6	mx	mx	NOUN
ejpam-1851	413	7	)	)	PUNCT
ejpam-1851	413	8	,	,	PUNCT
ejpam-1851	413	9	l	l	X
ejpam-1851	414	1	=	=	PUNCT
ejpam-1851	414	2	p	p	X
ejpam-1851	414	3	(	(	PUNCT
ejpam-1851	414	4	lx	lx	NOUN
ejpam-1851	414	5	)	)	PUNCT
ejpam-1851	414	6	∈	∈	PROPN
ejpam-1851	414	7	q.	q.	NOUN
ejpam-1851	414	8	thus	thus	ADV
ejpam-1851	414	9	l	l	NOUN
ejpam-1851	414	10	belong	belong	VERB
ejpam-1851	414	11	to	to	ADP
ejpam-1851	414	12	the	the	DET
ejpam-1851	414	13	conic	conic	ADJ
ejpam-1851	414	14	p1	p1	NOUN
ejpam-1851	414	15	,	,	PUNCT
ejpam-1851	414	16	p2	p2	NOUN
ejpam-1851	414	17	,	,	PUNCT
ejpam-1851	414	18	p3	p3	PROPN
ejpam-1851	414	19	�	�	PROPN
ejpam-1851	414	20	∩q	∩q	PROPN
ejpam-1851	415	1	=	=	PUNCT
ejpam-1851	416	1	p	p	X
ejpam-1851	416	2	(	(	PUNCT
ejpam-1851	416	3	l	l	NOUN
ejpam-1851	416	4	)	)	PUNCT
ejpam-1851	416	5	.	.	PUNCT
ejpam-1851	417	1	moreover	moreover	ADV
ejpam-1851	417	2	m	m	VERB
ejpam-1851	417	3	∈	∈	PROPN
ejpam-1851	417	4	〈	〈	PROPN
ejpam-1851	417	5	l	l	NOUN
ejpam-1851	417	6	,	,	PUNCT
ejpam-1851	417	7	p4	p4	ADJ
ejpam-1851	417	8	〉	〉	NOUN
ejpam-1851	417	9	⊂	⊂	X
ejpam-1851	417	10	λ	λ	X
ejpam-1851	417	11	∩q	∩q	PROPN
ejpam-1851	417	12	(	(	PUNCT
ejpam-1851	417	13	since	since	SCONJ
ejpam-1851	417	14	,	,	PUNCT
ejpam-1851	417	15	x	x	PUNCT
ejpam-1851	417	16	∈	∈	PROPN
ejpam-1851	417	17	ℓ4	ℓ4	PROPN
ejpam-1851	417	18	⊂	⊂	PROPN
ejpam-1851	417	19	〈	〈	PROPN
ejpam-1851	417	20	mx	mx	PROPN
ejpam-1851	417	21	,	,	PUNCT
ejpam-1851	417	22	lx	lx	NOUN
ejpam-1851	417	23	〉	〉	NOUN
ejpam-1851	417	24	)	)	PUNCT
ejpam-1851	417	25	.	.	PUNCT
ejpam-1851	418	1	on	on	ADP
ejpam-1851	418	2	the	the	DET
ejpam-1851	418	3	other	other	ADJ
ejpam-1851	418	4	hand	hand	NOUN
ejpam-1851	418	5	the	the	DET
ejpam-1851	418	6	lines	line	NOUN
ejpam-1851	418	7	li	li	PROPN
ejpam-1851	419	1	=	=	PUNCT
ejpam-1851	419	2	〈	〈	PROPN
ejpam-1851	419	3	m	m	PROPN
ejpam-1851	419	4	,	,	PUNCT
ejpam-1851	419	5	pi	pi	NOUN
ejpam-1851	419	6	〉	〉	NOUN
ejpam-1851	419	7	⊂	⊂	X
ejpam-1851	419	8	λ∩q	λ∩q	PROPN
ejpam-1851	419	9	,	,	PUNCT
ejpam-1851	419	10	i	i	PRON
ejpam-1851	419	11	=	=	NOUN
ejpam-1851	419	12	1	1	NUM
ejpam-1851	419	13	,	,	PUNCT
ejpam-1851	419	14	.	.	PUNCT
ejpam-1851	419	15	.	.	PUNCT
ejpam-1851	420	1	.	.	PUNCT
ejpam-1851	421	1	,	,	PUNCT
ejpam-1851	421	2	4	4	X
ejpam-1851	421	3	.	.	X
ejpam-1851	421	4	therefore	therefore	ADV
ejpam-1851	421	5	,	,	PUNCT
ejpam-1851	421	6	λ∩q	λ∩q	PROPN
ejpam-1851	421	7	is	be	AUX
ejpam-1851	421	8	a	a	DET
ejpam-1851	421	9	quadric	quadric	ADJ
ejpam-1851	421	10	cone	cone	NOUN
ejpam-1851	421	11	.	.	PUNCT
ejpam-1851	422	1	thus	thus	ADV
ejpam-1851	422	2	we	we	PRON
ejpam-1851	422	3	have	have	AUX
ejpam-1851	422	4	proved	prove	VERB
ejpam-1851	422	5	the	the	DET
ejpam-1851	422	6	following	follow	VERB
ejpam-1851	422	7	theorem	theorem	NOUN
ejpam-1851	422	8	.	.	PUNCT
ejpam-1851	422	9	theorem	theorem	NOUN
ejpam-1851	422	10	1	1	NUM
ejpam-1851	422	11	.	.	PUNCT
ejpam-1851	423	1	given	give	VERB
ejpam-1851	423	2	four	four	NUM
ejpam-1851	423	3	lines	line	NOUN
ejpam-1851	423	4	ℓ1	ℓ1	NOUN
ejpam-1851	423	5	,	,	PUNCT
ejpam-1851	423	6	ℓ2	ℓ2	NOUN
ejpam-1851	423	7	,	,	PUNCT
ejpam-1851	423	8	ℓ3	ℓ3	PROPN
ejpam-1851	423	9	,	,	PUNCT
ejpam-1851	423	10	ℓ4	ℓ4	NOUN
ejpam-1851	423	11	in	in	ADP
ejpam-1851	423	12	p3	p3	PROPN
ejpam-1851	423	13	.	.	PUNCT
ejpam-1851	424	1	set	set	VERB
ejpam-1851	424	2	s	s	PART
ejpam-1851	424	3	=	=	SYM
ejpam-1851	424	4	�	�	PROPN
ejpam-1851	424	5	ℓ	ℓ	PROPN
ejpam-1851	424	6	∈	∈	PROPN
ejpam-1851	425	1	g1(p	g1(p	NOUN
ejpam-1851	425	2	3	3	X
ejpam-1851	425	3	)	)	PUNCT
ejpam-1851	425	4	|	|	ADV
ejpam-1851	425	5	ℓ∩	ℓ∩	PROPN
ejpam-1851	425	6	ℓi	ℓi	PROPN
ejpam-1851	425	7	6=	6=	PROPN
ejpam-1851	425	8	;	;	PUNCT
ejpam-1851	425	9	for	for	ADP
ejpam-1851	425	10	1≤	1≤	NUM
ejpam-1851	425	11	i	i	PRON
ejpam-1851	425	12	≤	≤	ADV
ejpam-1851	425	13	4	4	NUM
ejpam-1851	425	14	.	.	PUNCT
ejpam-1851	426	1	let	let	VERB
ejpam-1851	426	2	pi	pi	NOUN
ejpam-1851	426	3	(	(	PUNCT
ejpam-1851	426	4	i	i	NOUN
ejpam-1851	426	5	=	=	NOUN
ejpam-1851	426	6	1	1	NUM
ejpam-1851	426	7	,	,	PUNCT
ejpam-1851	426	8	.	.	PUNCT
ejpam-1851	426	9	.	.	PUNCT
ejpam-1851	427	1	.	.	PUNCT
ejpam-1851	428	1	,	,	PUNCT
ejpam-1851	428	2	4	4	X
ejpam-1851	428	3	)	)	PUNCT
ejpam-1851	428	4	be	be	VERB
ejpam-1851	428	5	the	the	DET
ejpam-1851	428	6	image	image	NOUN
ejpam-1851	428	7	of	of	ADP
ejpam-1851	428	8	ℓi	ℓi	PROPN
ejpam-1851	428	9	in	in	ADP
ejpam-1851	428	10	the	the	DET
ejpam-1851	428	11	plücker	plücker	NOUN
ejpam-1851	428	12	’s	’s	PART
ejpam-1851	428	13	quadric	quadric	ADJ
ejpam-1851	428	14	q	q	X
ejpam-1851	428	15	⊂	⊂	PUNCT
ejpam-1851	428	16	p5	p5	ADJ
ejpam-1851	428	17	under	under	ADP
ejpam-1851	428	18	the	the	DET
ejpam-1851	428	19	plücker	plücker	NOUN
ejpam-1851	428	20	embedding	embed	VERB
ejpam-1851	428	21	p	p	X
ejpam-1851	428	22	(	(	PUNCT
ejpam-1851	428	23	in	in	ADP
ejpam-1851	428	24	(	(	PUNCT
ejpam-1851	428	25	1	1	NUM
ejpam-1851	428	26	)	)	PUNCT
ejpam-1851	428	27	)	)	PUNCT
ejpam-1851	428	28	.	.	PUNCT
ejpam-1851	429	1	set	set	VERB
ejpam-1851	429	2	λ	λ	PROPN
ejpam-1851	429	3	=	=	SYM
ejpam-1851	429	4	p1	p1	PROPN
ejpam-1851	429	5	,	,	PUNCT
ejpam-1851	429	6	.	.	PUNCT
ejpam-1851	429	7	.	.	PUNCT
ejpam-1851	430	1	.	.	PUNCT
ejpam-1851	431	1	,	,	PUNCT
ejpam-1851	431	2	p4	p4	PROPN
ejpam-1851	431	3	�	�	PROPN
ejpam-1851	431	4	be	be	AUX
ejpam-1851	431	5	the	the	DET
ejpam-1851	431	6	linear	linear	ADJ
ejpam-1851	431	7	span	span	NOUN
ejpam-1851	431	8	of	of	ADP
ejpam-1851	431	9	those	those	DET
ejpam-1851	431	10	four	four	NUM
ejpam-1851	431	11	points	point	NOUN
ejpam-1851	431	12	in	in	ADP
ejpam-1851	431	13	p5	p5	PROPN
ejpam-1851	431	14	.	.	PUNCT
ejpam-1851	432	1	then	then	ADV
ejpam-1851	432	2	the	the	DET
ejpam-1851	432	3	cardinal	cardinal	NOUN
ejpam-1851	432	4	of	of	ADP
ejpam-1851	432	5	the	the	DET
ejpam-1851	432	6	set	set	NOUN
ejpam-1851	432	7	s	s	PART
ejpam-1851	432	8	is	be	AUX
ejpam-1851	432	9	either	either	PRON
ejpam-1851	432	10	1	1	NUM
ejpam-1851	432	11	,	,	PUNCT
ejpam-1851	432	12	2	2	NUM
ejpam-1851	432	13	or	or	CCONJ
ejpam-1851	432	14	infinite	infinite	NOUN
ejpam-1851	432	15	.	.	PUNCT
ejpam-1851	433	1	moreover	moreover	ADV
ejpam-1851	433	2	we	we	PRON
ejpam-1851	433	3	have	have	VERB
ejpam-1851	433	4	.	.	PUNCT
ejpam-1851	434	1	(	(	PUNCT
ejpam-1851	434	2	i	i	NOUN
ejpam-1851	434	3	)	)	PUNCT
ejpam-1851	435	1	#	#	SYM
ejpam-1851	435	2	s	s	NOUN
ejpam-1851	435	3	=	=	SYM
ejpam-1851	435	4	2	2	NUM
ejpam-1851	435	5	if	if	SCONJ
ejpam-1851	435	6	and	and	CCONJ
ejpam-1851	435	7	only	only	ADV
ejpam-1851	435	8	if	if	SCONJ
ejpam-1851	435	9	λ	λ	NOUN
ejpam-1851	435	10	is	be	AUX
ejpam-1851	435	11	a	a	DET
ejpam-1851	435	12	3	3	NUM
ejpam-1851	435	13	-	-	PUNCT
ejpam-1851	435	14	plane	plane	NOUN
ejpam-1851	435	15	and	and	CCONJ
ejpam-1851	435	16	λ∩q	λ∩q	PROPN
ejpam-1851	435	17	is	be	AUX
ejpam-1851	435	18	a	a	DET
ejpam-1851	435	19	smooth	smooth	ADJ
ejpam-1851	435	20	quadric	quadric	NOUN
ejpam-1851	435	21	.	.	PUNCT
ejpam-1851	436	1	(	(	PUNCT
ejpam-1851	436	2	ii	ii	NOUN
ejpam-1851	436	3	)	)	PUNCT
ejpam-1851	437	1	#	#	SYM
ejpam-1851	437	2	s	s	NOUN
ejpam-1851	437	3	=	=	SYM
ejpam-1851	437	4	1	1	NUM
ejpam-1851	437	5	if	if	SCONJ
ejpam-1851	437	6	and	and	CCONJ
ejpam-1851	437	7	only	only	ADV
ejpam-1851	437	8	if	if	SCONJ
ejpam-1851	437	9	λ	λ	NOUN
ejpam-1851	437	10	is	be	AUX
ejpam-1851	437	11	a	a	DET
ejpam-1851	437	12	3	3	NUM
ejpam-1851	437	13	-	-	PUNCT
ejpam-1851	437	14	plane	plane	NOUN
ejpam-1851	437	15	and	and	CCONJ
ejpam-1851	437	16	λ∩q	λ∩q	PROPN
ejpam-1851	437	17	is	be	AUX
ejpam-1851	437	18	a	a	DET
ejpam-1851	437	19	quadric	quadric	ADJ
ejpam-1851	437	20	cone	cone	NOUN
ejpam-1851	437	21	.	.	PUNCT
ejpam-1851	438	1	(	(	PUNCT
ejpam-1851	438	2	iii	iii	X
ejpam-1851	438	3	)	)	PUNCT
ejpam-1851	438	4	if	if	SCONJ
ejpam-1851	438	5	#	#	SYM
ejpam-1851	438	6	s	s	NOUN
ejpam-1851	438	7	=	=	NOUN
ejpam-1851	438	8	∞	∞	NUM
ejpam-1851	438	9	then	then	ADV
ejpam-1851	438	10	(	(	PUNCT
ejpam-1851	438	11	a	a	X
ejpam-1851	438	12	)	)	PUNCT
ejpam-1851	438	13	s	s	VERB
ejpam-1851	438	14	is	be	AUX
ejpam-1851	438	15	a	a	DET
ejpam-1851	438	16	line	line	NOUN
ejpam-1851	438	17	if	if	SCONJ
ejpam-1851	438	18	and	and	CCONJ
ejpam-1851	438	19	only	only	ADV
ejpam-1851	438	20	if	if	SCONJ
ejpam-1851	438	21	λ	λ	NOUN
ejpam-1851	438	22	is	be	AUX
ejpam-1851	438	23	a	a	DET
ejpam-1851	438	24	3	3	NUM
ejpam-1851	438	25	-	-	PUNCT
ejpam-1851	438	26	plane	plane	NOUN
ejpam-1851	438	27	and	and	CCONJ
ejpam-1851	438	28	λ∩q	λ∩q	PROPN
ejpam-1851	438	29	is	be	AUX
ejpam-1851	438	30	a	a	DET
ejpam-1851	438	31	union	union	NOUN
ejpam-1851	438	32	of	of	ADP
ejpam-1851	438	33	two	two	NUM
ejpam-1851	438	34	planes	plane	NOUN
ejpam-1851	438	35	.	.	PUNCT
ejpam-1851	439	1	(	(	PUNCT
ejpam-1851	439	2	b	b	X
ejpam-1851	439	3	)	)	PUNCT
ejpam-1851	439	4	s	s	VERB
ejpam-1851	439	5	is	be	AUX
ejpam-1851	439	6	a	a	DET
ejpam-1851	439	7	conic	conic	ADJ
ejpam-1851	439	8	if	if	NOUN
ejpam-1851	439	9	and	and	CCONJ
ejpam-1851	439	10	only	only	ADV
ejpam-1851	439	11	if	if	SCONJ
ejpam-1851	439	12	λ	λ	NOUN
ejpam-1851	439	13	is	be	AUX
ejpam-1851	439	14	a	a	DET
ejpam-1851	439	15	plane	plane	NOUN
ejpam-1851	439	16	and	and	CCONJ
ejpam-1851	439	17	λ∩q	λ∩q	PROPN
ejpam-1851	439	18	is	be	AUX
ejpam-1851	439	19	a	a	DET
ejpam-1851	439	20	nonsingular	nonsingular	ADJ
ejpam-1851	439	21	conic	conic	NOUN
ejpam-1851	439	22	.	.	PUNCT
ejpam-1851	440	1	(	(	PUNCT
ejpam-1851	440	2	c	c	X
ejpam-1851	440	3	)	)	PUNCT
ejpam-1851	440	4	s	s	VERB
ejpam-1851	440	5	is	be	AUX
ejpam-1851	440	6	a	a	DET
ejpam-1851	440	7	reduced	reduced	ADJ
ejpam-1851	440	8	and	and	CCONJ
ejpam-1851	440	9	reducible	reducible	ADJ
ejpam-1851	440	10	conic	conic	ADJ
ejpam-1851	440	11	if	if	SCONJ
ejpam-1851	440	12	and	and	CCONJ
ejpam-1851	440	13	only	only	ADV
ejpam-1851	440	14	if	if	SCONJ
ejpam-1851	440	15	λ	λ	NOUN
ejpam-1851	440	16	is	be	AUX
ejpam-1851	440	17	a	a	DET
ejpam-1851	440	18	plane	plane	NOUN
ejpam-1851	440	19	and	and	CCONJ
ejpam-1851	440	20	λ	λ	PROPN
ejpam-1851	440	21	∩q	∩q	PROPN
ejpam-1851	440	22	is	be	AUX
ejpam-1851	440	23	a	a	DET
ejpam-1851	440	24	reduced	reduced	ADJ
ejpam-1851	440	25	and	and	CCONJ
ejpam-1851	440	26	reducible	reducible	ADJ
ejpam-1851	440	27	conic	conic	ADJ
ejpam-1851	440	28	.	.	PUNCT
ejpam-1851	441	1	(	(	PUNCT
ejpam-1851	441	2	d	d	X
ejpam-1851	441	3	)	)	PUNCT
ejpam-1851	441	4	s	s	VERB
ejpam-1851	441	5	is	be	AUX
ejpam-1851	441	6	a	a	DET
ejpam-1851	441	7	plane	plane	NOUN
ejpam-1851	441	8	if	if	SCONJ
ejpam-1851	441	9	and	and	CCONJ
ejpam-1851	441	10	only	only	ADV
ejpam-1851	441	11	if	if	SCONJ
ejpam-1851	441	12	λ	λ	NOUN
ejpam-1851	441	13	is	be	AUX
ejpam-1851	441	14	a	a	DET
ejpam-1851	441	15	plane	plane	NOUN
ejpam-1851	441	16	in	in	ADP
ejpam-1851	441	17	q.	q.	PROPN
ejpam-1851	441	18	(	(	PUNCT
ejpam-1851	441	19	e	e	NOUN
ejpam-1851	441	20	)	)	PUNCT
ejpam-1851	442	1	s	s	AUX
ejpam-1851	442	2	is	be	AUX
ejpam-1851	442	3	a	a	DET
ejpam-1851	442	4	union	union	NOUN
ejpam-1851	442	5	of	of	ADP
ejpam-1851	442	6	two	two	NUM
ejpam-1851	442	7	distinct	distinct	ADJ
ejpam-1851	442	8	planes	plane	NOUN
ejpam-1851	442	9	if	if	SCONJ
ejpam-1851	442	10	and	and	CCONJ
ejpam-1851	442	11	only	only	ADV
ejpam-1851	442	12	if	if	SCONJ
ejpam-1851	442	13	λ	λ	NOUN
ejpam-1851	442	14	is	be	AUX
ejpam-1851	442	15	a	a	DET
ejpam-1851	442	16	line	line	NOUN
ejpam-1851	442	17	in	in	ADP
ejpam-1851	442	18	q.	q.	NOUN
ejpam-1851	442	19	in	in	ADP
ejpam-1851	442	20	terms	term	NOUN
ejpam-1851	442	21	of	of	ADP
ejpam-1851	442	22	the	the	DET
ejpam-1851	442	23	lines	line	NOUN
ejpam-1851	442	24	position	position	NOUN
ejpam-1851	442	25	theorem	theorem	VERB
ejpam-1851	442	26	1	1	NUM
ejpam-1851	442	27	tell	tell	VERB
ejpam-1851	442	28	us	we	PRON
ejpam-1851	442	29	:	:	PUNCT
ejpam-1851	442	30	#	#	SYM
ejpam-1851	442	31	s	s	NOUN
ejpam-1851	442	32	=	=	SYM
ejpam-1851	442	33	2	2	NUM
ejpam-1851	442	34	⇐	⇐	ADJ
ejpam-1851	442	35	⇒	⇒	NOUN
ejpam-1851	442	36			PRON
ejpam-1851	442	37			ADP
ejpam-1851	442	38			PROPN
ejpam-1851	442	39			PROPN
ejpam-1851	442	40			PROPN
ejpam-1851	442	41			PROPN
ejpam-1851	442	42			PROPN
ejpam-1851	442	43			PROPN
ejpam-1851	442	44			PROPN
ejpam-1851	442	45			PROPN
ejpam-1851	442	46			PROPN
ejpam-1851	442	47			PROPN
ejpam-1851	442	48			PROPN
ejpam-1851	442	49			PROPN
ejpam-1851	442	50			PROPN
ejpam-1851	442	51			PROPN
ejpam-1851	442	52			PROPN
ejpam-1851	442	53			NOUN
ejpam-1851	442	54			PROPN
ejpam-1851	442	55			PROPN
ejpam-1851	442	56			PROPN
ejpam-1851	442	57			PROPN
ejpam-1851	442	58			PROPN
ejpam-1851	442	59			PROPN
ejpam-1851	442	60			PROPN
ejpam-1851	442	61			PROPN
ejpam-1851	442	62			PROPN
ejpam-1851	442	63			PROPN
ejpam-1851	442	64			PROPN
ejpam-1851	442	65			PROPN
ejpam-1851	442	66			PROPN
ejpam-1851	442	67			PROPN
ejpam-1851	442	68			PROPN
ejpam-1851	442	69			PROPN
ejpam-1851	442	70			NOUN
ejpam-1851	442	71	•	•	NUM
ejpam-1851	442	72	∩4	∩4	NOUN
ejpam-1851	442	73	i=1ℓi	i=1ℓi	NUM
ejpam-1851	442	74	=	=	PUNCT
ejpam-1851	442	75	;	;	PUNCT
ejpam-1851	442	76	and	and	CCONJ
ejpam-1851	442	77	any	any	DET
ejpam-1851	442	78	three	three	NUM
ejpam-1851	442	79	lines	line	NOUN
ejpam-1851	442	80	are	be	AUX
ejpam-1851	442	81	non	non	X
ejpam-1851	442	82	coplanar	coplanar	ADJ
ejpam-1851	442	83	.	.	PUNCT
ejpam-1851	443	1	(	(	PUNCT
ejpam-1851	443	2	i	i	NOUN
ejpam-1851	443	3	)	)	PUNCT
ejpam-1851	443	4	ℓ1	ℓ1	ADJ
ejpam-1851	443	5	∩	∩	ADJ
ejpam-1851	443	6	ℓ2	ℓ2	NOUN
ejpam-1851	443	7	=	=	SYM
ejpam-1851	443	8	{	{	PUNCT
ejpam-1851	443	9	p	p	NOUN
ejpam-1851	443	10	}	}	PUNCT
ejpam-1851	443	11	,	,	PUNCT
ejpam-1851	443	12	ℓ3	ℓ3	NOUN
ejpam-1851	443	13	∩	∩	NOUN
ejpam-1851	443	14	ℓ4	ℓ4	NOUN
ejpam-1851	443	15	=	=	PUNCT
ejpam-1851	443	16	{	{	PUNCT
ejpam-1851	443	17	q	q	X
ejpam-1851	443	18	}	}	PUNCT
ejpam-1851	443	19	with	with	ADP
ejpam-1851	443	20	p	p	PROPN
ejpam-1851	443	21	6=	6=	ADP
ejpam-1851	443	22	q	q	PROPN
ejpam-1851	444	1	and	and	CCONJ
ejpam-1851	444	2	p	p	PROPN
ejpam-1851	444	3	6∈	6∈	PROPN
ejpam-1851	444	4	π1	π1	NOUN
ejpam-1851	444	5	∩π2	∩π2	PROPN
ejpam-1851	445	1	6∋	6∋	NUM
ejpam-1851	445	2	q	q	ADJ
ejpam-1851	445	3	,	,	PUNCT
ejpam-1851	445	4	if	if	SCONJ
ejpam-1851	445	5	π1	π1	ADJ
ejpam-1851	445	6	=	=	SYM
ejpam-1851	445	7	〈	〈	PROPN
ejpam-1851	445	8	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	445	9	〉	〉	NOUN
ejpam-1851	445	10	,	,	PUNCT
ejpam-1851	445	11	π2	π2	NOUN
ejpam-1851	445	12	=	=	PUNCT
ejpam-1851	445	13	〈	〈	PROPN
ejpam-1851	445	14	ℓ3,ℓ4	ℓ3,ℓ4	PROPN
ejpam-1851	445	15	〉	〉	NOUN
ejpam-1851	445	16	.	.	PUNCT
ejpam-1851	445	17	(	(	PUNCT
ejpam-1851	445	18	ii	ii	NOUN
ejpam-1851	445	19	)	)	PUNCT
ejpam-1851	445	20	ℓ1	ℓ1	ADJ
ejpam-1851	445	21	∩	∩	ADJ
ejpam-1851	445	22	ℓ2	ℓ2	NOUN
ejpam-1851	445	23	=	=	SYM
ejpam-1851	445	24	{	{	PUNCT
ejpam-1851	445	25	p	p	X
ejpam-1851	445	26	}	}	PUNCT
ejpam-1851	445	27	,	,	PUNCT
ejpam-1851	445	28	ℓ1	ℓ1	NOUN
ejpam-1851	445	29	∩	∩	ADJ
ejpam-1851	445	30	ℓ3	ℓ3	NOUN
ejpam-1851	445	31	=	=	SYM
ejpam-1851	445	32	{	{	PUNCT
ejpam-1851	445	33	q	q	X
ejpam-1851	445	34	}	}	PUNCT
ejpam-1851	445	35	with	with	ADP
ejpam-1851	445	36	p	p	PROPN
ejpam-1851	445	37	6=	6=	ADP
ejpam-1851	445	38	q.	q.	NOUN
ejpam-1851	445	39	if	if	SCONJ
ejpam-1851	445	40	πi	πi	ADV
ejpam-1851	445	41	=	=	PUNCT
ejpam-1851	445	42	〈	〈	NOUN
ejpam-1851	445	43	ℓ1,ℓi+1	ℓ1,ℓi+1	NOUN
ejpam-1851	445	44	〉	〉	NOUN
ejpam-1851	445	45	for	for	ADP
ejpam-1851	445	46	i	i	X
ejpam-1851	445	47	=	=	SYM
ejpam-1851	445	48	1,2	1,2	NUM
ejpam-1851	445	49	,	,	PUNCT
ejpam-1851	445	50	then	then	ADV
ejpam-1851	445	51	it	it	PRON
ejpam-1851	445	52	is	be	AUX
ejpam-1851	445	53	verified	verify	VERB
ejpam-1851	445	54	that	that	SCONJ
ejpam-1851	445	55	πi	πi	ADP
ejpam-1851	445	56	∩	∩	ADJ
ejpam-1851	445	57	ℓ4	ℓ4	NOUN
ejpam-1851	445	58	=	=	SYM
ejpam-1851	445	59	{	{	PUNCT
ejpam-1851	445	60	ri	ri	NOUN
ejpam-1851	445	61	}	}	PUNCT
ejpam-1851	445	62	i	i	PROPN
ejpam-1851	445	63	=	=	SYM
ejpam-1851	445	64	1,2	1,2	NUM
ejpam-1851	445	65	and	and	CCONJ
ejpam-1851	445	66	r1	r1	NOUN
ejpam-1851	445	67	6=	6=	ADP
ejpam-1851	445	68	r2	r2	PROPN
ejpam-1851	445	69	.	.	PUNCT
ejpam-1851	446	1	(	(	PUNCT
ejpam-1851	446	2	iii	iii	X
ejpam-1851	446	3	)	)	PUNCT
ejpam-1851	446	4	ℓ1	ℓ1	NOUN
ejpam-1851	446	5	∩	∩	ADJ
ejpam-1851	446	6	ℓ2	ℓ2	NOUN
ejpam-1851	446	7	=	=	SYM
ejpam-1851	446	8	{	{	PUNCT
ejpam-1851	446	9	p	p	X
ejpam-1851	446	10	}	}	PUNCT
ejpam-1851	446	11	.	.	PUNCT
ejpam-1851	447	1	if	if	SCONJ
ejpam-1851	447	2	π	π	PROPN
ejpam-1851	447	3	=	=	SYM
ejpam-1851	447	4	〈	〈	PROPN
ejpam-1851	447	5	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	447	6	〉	〉	NOUN
ejpam-1851	447	7	then	then	ADV
ejpam-1851	447	8	it	it	PRON
ejpam-1851	447	9	is	be	AUX
ejpam-1851	447	10	verified	verify	VERB
ejpam-1851	447	11	for	for	ADP
ejpam-1851	447	12	i	i	X
ejpam-1851	447	13	=	=	NOUN
ejpam-1851	447	14	3,4	3,4	NUM
ejpam-1851	447	15	that	that	PRON
ejpam-1851	447	16	π∩	π∩	PROPN
ejpam-1851	447	17	ℓi	ℓi	PROPN
ejpam-1851	447	18	=	=	SYM
ejpam-1851	447	19	{	{	PUNCT
ejpam-1851	447	20	ri	ri	NOUN
ejpam-1851	447	21	}	}	PUNCT
ejpam-1851	447	22	,	,	PUNCT
ejpam-1851	447	23	r3	r3	PROPN
ejpam-1851	447	24	6=	6=	SYM
ejpam-1851	447	25	r4	r4	PROPN
ejpam-1851	447	26	and	and	CCONJ
ejpam-1851	447	27	p	p	PROPN
ejpam-1851	447	28	6∈	6∈	PROPN
ejpam-1851	447	29	〈	〈	PROPN
ejpam-1851	447	30	r3	r3	PROPN
ejpam-1851	447	31	,	,	PUNCT
ejpam-1851	447	32	r4	r4	NOUN
ejpam-1851	447	33	〉	〉	NOUN
ejpam-1851	447	34	.	.	NOUN
ejpam-1851	447	35	•	•	NUM
ejpam-1851	448	1	no	no	DET
ejpam-1851	448	2	pair	pair	NOUN
ejpam-1851	448	3	of	of	ADP
ejpam-1851	448	4	lines	line	NOUN
ejpam-1851	448	5	is	be	AUX
ejpam-1851	448	6	coplanar	coplanar	ADJ
ejpam-1851	448	7	.	.	PUNCT
ejpam-1851	449	1	(	(	PUNCT
ejpam-1851	449	2	iv	iv	X
ejpam-1851	449	3	)	)	PUNCT
ejpam-1851	449	4	the	the	DET
ejpam-1851	449	5	non	non	ADJ
ejpam-1851	449	6	singular	singular	PROPN
ejpam-1851	449	7	quadric	quadric	PROPN
ejpam-1851	449	8	surface	surface	PROPN
ejpam-1851	449	9	q	q	PROPN
ejpam-1851	449	10	⊂	⊂	PROPN
ejpam-1851	449	11	p3	p3	PROPN
ejpam-1851	449	12	containing	contain	VERB
ejpam-1851	449	13	ℓ1	ℓ1	NOUN
ejpam-1851	449	14	,	,	PUNCT
ejpam-1851	449	15	ℓ2	ℓ2	NOUN
ejpam-1851	449	16	and	and	CCONJ
ejpam-1851	449	17	ℓ3	ℓ3	PROPN
ejpam-1851	449	18	,	,	PUNCT
ejpam-1851	449	19	meets	meet	VERB
ejpam-1851	449	20	ℓ4	ℓ4	NOUN
ejpam-1851	449	21	exactly	exactly	ADV
ejpam-1851	449	22	in	in	ADP
ejpam-1851	449	23	two	two	NUM
ejpam-1851	449	24	distinct	distinct	ADJ
ejpam-1851	449	25	points	point	NOUN
ejpam-1851	449	26	.	.	PUNCT
ejpam-1851	450	1	references	reference	NOUN
ejpam-1851	450	2	485	485	NUM
ejpam-1851	450	3	note	note	NOUN
ejpam-1851	450	4	that	that	SCONJ
ejpam-1851	450	5	(	(	PUNCT
ejpam-1851	450	6	i	i	NOUN
ejpam-1851	450	7	)	)	PUNCT
ejpam-1851	450	8	,	,	PUNCT
ejpam-1851	450	9	(	(	PUNCT
ejpam-1851	450	10	ii	ii	NOUN
ejpam-1851	450	11	)	)	PUNCT
ejpam-1851	450	12	and	and	CCONJ
ejpam-1851	450	13	(	(	PUNCT
ejpam-1851	450	14	iii	iii	NOUN
ejpam-1851	450	15	)	)	PUNCT
ejpam-1851	450	16	above	above	ADP
ejpam-1851	450	17	correspond	correspond	VERB
ejpam-1851	450	18	to	to	ADP
ejpam-1851	450	19	subcases	subcase	NOUN
ejpam-1851	450	20	2.3.3	2.3.3	NUM
ejpam-1851	450	21	,	,	PUNCT
ejpam-1851	450	22	2.4.4	2.4.4	NUM
ejpam-1851	450	23	and	and	CCONJ
ejpam-1851	450	24	2.5.3	2.5.3	NUM
ejpam-1851	450	25	in	in	ADP
ejpam-1851	450	26	subsection	subsection	NOUN
ejpam-1851	450	27	4.2	4.2	NUM
ejpam-1851	450	28	.	.	PUNCT
ejpam-1851	451	1	already	already	ADV
ejpam-1851	451	2	(	(	PUNCT
ejpam-1851	451	3	iv	iv	X
ejpam-1851	451	4	)	)	PUNCT
ejpam-1851	451	5	corresponds	correspond	NOUN
ejpam-1851	451	6	to	to	PART
ejpam-1851	451	7	subcase	subcase	VERB
ejpam-1851	451	8	3.2	3.2	NUM
ejpam-1851	451	9	in	in	ADP
ejpam-1851	451	10	subsection	subsection	NOUN
ejpam-1851	451	11	4.3	4.3	NUM
ejpam-1851	451	12	.	.	PUNCT
ejpam-1851	452	1	#	#	SYM
ejpam-1851	452	2	s	s	NOUN
ejpam-1851	452	3	=	=	SYM
ejpam-1851	452	4	1	1	NUM
ejpam-1851	452	5	⇐	⇐	ADJ
ejpam-1851	452	6	⇒	⇒	NOUN
ejpam-1851	452	7			PRON
ejpam-1851	452	8			ADP
ejpam-1851	452	9			PROPN
ejpam-1851	452	10			PROPN
ejpam-1851	452	11			PROPN
ejpam-1851	452	12			PROPN
ejpam-1851	452	13			PROPN
ejpam-1851	452	14			PROPN
ejpam-1851	452	15			PROPN
ejpam-1851	452	16			PROPN
ejpam-1851	452	17			PROPN
ejpam-1851	452	18			PROPN
ejpam-1851	452	19			PROPN
ejpam-1851	452	20			PROPN
ejpam-1851	452	21			NOUN
ejpam-1851	452	22			PROPN
ejpam-1851	452	23			PROPN
ejpam-1851	452	24			PROPN
ejpam-1851	452	25			PROPN
ejpam-1851	452	26			PROPN
ejpam-1851	452	27			PROPN
ejpam-1851	452	28			PROPN
ejpam-1851	452	29			PROPN
ejpam-1851	452	30			PROPN
ejpam-1851	452	31			PROPN
ejpam-1851	452	32			PROPN
ejpam-1851	452	33			PROPN
ejpam-1851	452	34			PROPN
ejpam-1851	452	35			NOUN
ejpam-1851	452	36	•	•	NUM
ejpam-1851	452	37	∩4	∩4	NOUN
ejpam-1851	452	38	i=1ℓi	i=1ℓi	NUM
ejpam-1851	452	39	=	=	PUNCT
ejpam-1851	452	40	;	;	PUNCT
ejpam-1851	452	41	and	and	CCONJ
ejpam-1851	452	42	any	any	DET
ejpam-1851	452	43	three	three	NUM
ejpam-1851	452	44	lines	line	NOUN
ejpam-1851	452	45	are	be	AUX
ejpam-1851	452	46	non	non	X
ejpam-1851	452	47	coplanar	coplanar	ADJ
ejpam-1851	452	48	.	.	PUNCT
ejpam-1851	453	1	(	(	PUNCT
ejpam-1851	453	2	α	α	NOUN
ejpam-1851	453	3	)	)	PUNCT
ejpam-1851	453	4	ℓ1	ℓ1	ADJ
ejpam-1851	453	5	∩	∩	ADJ
ejpam-1851	453	6	ℓ2	ℓ2	NOUN
ejpam-1851	453	7	=	=	SYM
ejpam-1851	453	8	{	{	PUNCT
ejpam-1851	453	9	p	p	X
ejpam-1851	453	10	}	}	PUNCT
ejpam-1851	453	11	,	,	PUNCT
ejpam-1851	453	12	ℓ1	ℓ1	NOUN
ejpam-1851	453	13	∩	∩	ADJ
ejpam-1851	453	14	ℓ3	ℓ3	NOUN
ejpam-1851	453	15	=	=	SYM
ejpam-1851	453	16	{	{	PUNCT
ejpam-1851	453	17	q	q	X
ejpam-1851	453	18	}	}	PUNCT
ejpam-1851	453	19	with	with	ADP
ejpam-1851	453	20	p	p	PROPN
ejpam-1851	453	21	6=	6=	ADP
ejpam-1851	453	22	q.	q.	NOUN
ejpam-1851	453	23	if	if	SCONJ
ejpam-1851	453	24	πi	πi	ADV
ejpam-1851	453	25	=	=	PUNCT
ejpam-1851	453	26	〈	〈	NOUN
ejpam-1851	453	27	ℓ1,ℓi+1	ℓ1,ℓi+1	NOUN
ejpam-1851	453	28	〉	〉	NOUN
ejpam-1851	453	29	for	for	ADP
ejpam-1851	453	30	i	i	X
ejpam-1851	453	31	=	=	SYM
ejpam-1851	453	32	1,2	1,2	NUM
ejpam-1851	453	33	,	,	PUNCT
ejpam-1851	453	34	then	then	ADV
ejpam-1851	453	35	it	it	PRON
ejpam-1851	453	36	is	be	AUX
ejpam-1851	453	37	verified	verify	VERB
ejpam-1851	453	38	that	that	SCONJ
ejpam-1851	453	39	πi	πi	ADP
ejpam-1851	453	40	∩	∩	ADJ
ejpam-1851	453	41	ℓ4	ℓ4	NOUN
ejpam-1851	453	42	=	=	SYM
ejpam-1851	453	43	{	{	PUNCT
ejpam-1851	453	44	ri	ri	NOUN
ejpam-1851	453	45	}	}	PUNCT
ejpam-1851	453	46	i	i	NOUN
ejpam-1851	453	47	=	=	NOUN
ejpam-1851	453	48	1,2	1,2	NUM
ejpam-1851	453	49	.	.	PUNCT
ejpam-1851	453	50	with	with	ADP
ejpam-1851	453	51	r1	r1	PROPN
ejpam-1851	453	52	=	=	SYM
ejpam-1851	453	53	r2	r2	PROPN
ejpam-1851	453	54	and	and	CCONJ
ejpam-1851	453	55	#	#	SYM
ejpam-1851	453	56	{	{	PUNCT
ejpam-1851	453	57	p	p	X
ejpam-1851	453	58	,	,	PUNCT
ejpam-1851	453	59	q	q	ADJ
ejpam-1851	453	60	,	,	PUNCT
ejpam-1851	453	61	r1}=	r1}=	NUM
ejpam-1851	453	62	3	3	NUM
ejpam-1851	453	63	.	.	PUNCT
ejpam-1851	454	1	(	(	PUNCT
ejpam-1851	454	2	β	β	NOUN
ejpam-1851	454	3	)	)	PUNCT
ejpam-1851	454	4	ℓ1	ℓ1	ADJ
ejpam-1851	454	5	∩	∩	ADJ
ejpam-1851	454	6	ℓ2	ℓ2	NOUN
ejpam-1851	454	7	=	=	SYM
ejpam-1851	454	8	{	{	PUNCT
ejpam-1851	454	9	p	p	X
ejpam-1851	454	10	}	}	PUNCT
ejpam-1851	454	11	.	.	PUNCT
ejpam-1851	455	1	if	if	SCONJ
ejpam-1851	455	2	π	π	PROPN
ejpam-1851	455	3	=	=	SYM
ejpam-1851	455	4	〈	〈	PROPN
ejpam-1851	455	5	ℓ1,ℓ2	ℓ1,ℓ2	PROPN
ejpam-1851	455	6	〉	〉	NOUN
ejpam-1851	455	7	then	then	ADV
ejpam-1851	455	8	it	it	PRON
ejpam-1851	455	9	is	be	AUX
ejpam-1851	455	10	verified	verify	VERB
ejpam-1851	455	11	for	for	ADP
ejpam-1851	455	12	i	i	X
ejpam-1851	455	13	=	=	NOUN
ejpam-1851	455	14	3,4	3,4	NUM
ejpam-1851	455	15	that	that	PRON
ejpam-1851	455	16	π∩	π∩	PROPN
ejpam-1851	455	17	ℓi	ℓi	PROPN
ejpam-1851	455	18	=	=	PUNCT
ejpam-1851	455	19	{	{	PUNCT
ejpam-1851	455	20	ri	ri	PROPN
ejpam-1851	455	21	}	}	PUNCT
ejpam-1851	455	22	,	,	PUNCT
ejpam-1851	455	23	r3	r3	PROPN
ejpam-1851	455	24	6=	6=	SYM
ejpam-1851	455	25	r4	r4	PROPN
ejpam-1851	455	26	,	,	PUNCT
ejpam-1851	455	27	p	p	PROPN
ejpam-1851	455	28	∈	∈	PROPN
ejpam-1851	455	29	〈	〈	PROPN
ejpam-1851	455	30	r3	r3	PROPN
ejpam-1851	455	31	,	,	PUNCT
ejpam-1851	455	32	r4	r4	VERB
ejpam-1851	455	33	〉	〉	NOUN
ejpam-1851	455	34	and	and	CCONJ
ejpam-1851	455	35	p	p	NOUN
ejpam-1851	455	36	6=	6=	PROPN
ejpam-1851	455	37	ri	ri	PROPN
ejpam-1851	455	38	.	.	PUNCT
ejpam-1851	456	1	•	•	NUM
ejpam-1851	456	2	no	no	DET
ejpam-1851	456	3	pair	pair	NOUN
ejpam-1851	456	4	of	of	ADP
ejpam-1851	456	5	lines	line	NOUN
ejpam-1851	456	6	is	be	AUX
ejpam-1851	456	7	coplanar	coplanar	ADJ
ejpam-1851	456	8	.	.	PUNCT
ejpam-1851	457	1	(	(	PUNCT
ejpam-1851	457	2	γ	γ	X
ejpam-1851	457	3	)	)	PUNCT
ejpam-1851	457	4	the	the	DET
ejpam-1851	457	5	non	non	ADJ
ejpam-1851	457	6	singular	singular	PROPN
ejpam-1851	457	7	quadric	quadric	PROPN
ejpam-1851	457	8	surface	surface	PROPN
ejpam-1851	457	9	q	q	PROPN
ejpam-1851	457	10	⊂	⊂	PROPN
ejpam-1851	457	11	p3	p3	PROPN
ejpam-1851	457	12	containing	contain	VERB
ejpam-1851	457	13	ℓ1	ℓ1	NOUN
ejpam-1851	457	14	,	,	PUNCT
ejpam-1851	457	15	ℓ2	ℓ2	NOUN
ejpam-1851	457	16	and	and	CCONJ
ejpam-1851	457	17	ℓ3	ℓ3	PROPN
ejpam-1851	457	18	,	,	PUNCT
ejpam-1851	457	19	meets	meet	VERB
ejpam-1851	457	20	ℓ4	ℓ4	NOUN
ejpam-1851	457	21	exactly	exactly	ADV
ejpam-1851	457	22	in	in	ADP
ejpam-1851	457	23	one	one	NUM
ejpam-1851	457	24	point	point	NOUN
ejpam-1851	457	25	.	.	PUNCT
ejpam-1851	458	1	note	note	VERB
ejpam-1851	458	2	that	that	SCONJ
ejpam-1851	458	3	(	(	PUNCT
ejpam-1851	458	4	α	α	NOUN
ejpam-1851	458	5	)	)	PUNCT
ejpam-1851	458	6	and	and	CCONJ
ejpam-1851	458	7	(	(	PUNCT
ejpam-1851	458	8	β	β	NOUN
ejpam-1851	458	9	)	)	PUNCT
ejpam-1851	458	10	above	above	ADP
ejpam-1851	458	11	correspond	correspond	VERB
ejpam-1851	458	12	to	to	ADP
ejpam-1851	458	13	subcases	subcase	NOUN
ejpam-1851	458	14	2.4.3	2.4.3	NUM
ejpam-1851	458	15	and	and	CCONJ
ejpam-1851	458	16	2.5.2	2.5.2	NUM
ejpam-1851	458	17	in	in	ADP
ejpam-1851	458	18	subsection	subsection	NOUN
ejpam-1851	458	19	4.2	4.2	NUM
ejpam-1851	458	20	.	.	PUNCT
ejpam-1851	459	1	already	already	ADV
ejpam-1851	459	2	(	(	PUNCT
ejpam-1851	459	3	γ	γ	X
ejpam-1851	459	4	)	)	PUNCT
ejpam-1851	459	5	correspond	correspond	VERB
ejpam-1851	459	6	to	to	PART
ejpam-1851	459	7	subcase	subcase	VERB
ejpam-1851	459	8	3.2	3.2	NUM
ejpam-1851	459	9	in	in	ADP
ejpam-1851	459	10	subsection	subsection	NOUN
ejpam-1851	459	11	4.3	4.3	NUM
ejpam-1851	459	12	.	.	PUNCT
ejpam-1851	460	1	acknowledgements	acknowledgement	NOUN
ejpam-1851	460	2	the	the	DET
ejpam-1851	460	3	first	first	ADJ
ejpam-1851	460	4	author	author	NOUN
ejpam-1851	460	5	wishes	wish	VERB
ejpam-1851	460	6	to	to	PART
ejpam-1851	460	7	express	express	VERB
ejpam-1851	460	8	her	her	PRON
ejpam-1851	460	9	gratitude	gratitude	NOUN
ejpam-1851	460	10	to	to	ADP
ejpam-1851	460	11	i.	i.	PROPN
ejpam-1851	460	12	vainsencher	vainsencher	PROPN
ejpam-1851	460	13	(	(	PUNCT
ejpam-1851	460	14	dm	dm	NOUN
ejpam-1851	460	15	-	-	PUNCT
ejpam-1851	460	16	ufmg	ufmg	NOUN
ejpam-1851	460	17	)	)	PUNCT
ejpam-1851	460	18	for	for	ADP
ejpam-1851	460	19	a	a	DET
ejpam-1851	460	20	helpful	helpful	ADJ
ejpam-1851	460	21	conversation	conversation	NOUN
ejpam-1851	460	22	about	about	ADP
ejpam-1851	460	23	the	the	DET
ejpam-1851	460	24	subject	subject	NOUN
ejpam-1851	460	25	of	of	ADP
ejpam-1851	460	26	this	this	DET
ejpam-1851	460	27	paper	paper	NOUN
ejpam-1851	460	28	.	.	PUNCT
ejpam-1851	461	1	references	reference	NOUN
ejpam-1851	461	2	[	[	X
ejpam-1851	461	3	1	1	NUM
ejpam-1851	461	4	]	]	PUNCT
ejpam-1851	461	5	d.	d.	PROPN
ejpam-1851	461	6	avritzer	avritzer	PROPN
ejpam-1851	461	7	.	.	PUNCT
ejpam-1851	462	1	introdução	introdução	PROPN
ejpam-1851	462	2	à	à	PROPN
ejpam-1851	462	3	geometria	geometria	PROPN
ejpam-1851	462	4	enumerativa	enumerativa	PROPN
ejpam-1851	462	5	via	via	ADP
ejpam-1851	462	6	teoria	teoria	PROPN
ejpam-1851	462	7	de	de	X
ejpam-1851	462	8	deformações	deformações	NOUN
ejpam-1851	462	9	,	,	PUNCT
ejpam-1851	462	10	2a	2a	NUM
ejpam-1851	462	11	bienal	bienal	PROPN
ejpam-1851	462	12	da	da	PROPN
ejpam-1851	462	13	sociedade	sociedade	PROPN
ejpam-1851	462	14	brasileira	brasileira	PROPN
ejpam-1851	462	15	de	de	PROPN
ejpam-1851	462	16	matemática	matemática	PROPN
ejpam-1851	462	17	,	,	PUNCT
ejpam-1851	462	18	mini	mini	ADJ
ejpam-1851	462	19	course	course	NOUN
ejpam-1851	462	20	,	,	PUNCT
ejpam-1851	462	21	salvador	salvador	PROPN
ejpam-1851	462	22	,	,	PUNCT
ejpam-1851	462	23	universidade	universidade	PROPN
ejpam-1851	462	24	federal	federal	PROPN
ejpam-1851	462	25	da	da	PROPN
ejpam-1851	462	26	bahia	bahia	PROPN
ejpam-1851	462	27	.	.	PUNCT
ejpam-1851	462	28	2004	2004	NUM
ejpam-1851	462	29	.	.	PUNCT
ejpam-1851	463	1	[	[	X
ejpam-1851	463	2	2	2	X
ejpam-1851	463	3	]	]	X
ejpam-1851	463	4	d.	d.	PROPN
ejpam-1851	463	5	cox	cox	PROPN
ejpam-1851	463	6	,	,	PUNCT
ejpam-1851	463	7	j.	j.	PROPN
ejpam-1851	463	8	little	little	PROPN
ejpam-1851	463	9	,	,	PUNCT
ejpam-1851	463	10	and	and	CCONJ
ejpam-1851	463	11	d.	d.	PROPN
ejpam-1851	463	12	o’shea	o’shea	PROPN
ejpam-1851	463	13	.	.	PUNCT
ejpam-1851	464	1	ideals	ideal	NOUN
ejpam-1851	464	2	,	,	PUNCT
ejpam-1851	464	3	varieties	variety	NOUN
ejpam-1851	464	4	,	,	PUNCT
ejpam-1851	464	5	and	and	CCONJ
ejpam-1851	464	6	algorithms	algorithm	NOUN
ejpam-1851	464	7	,	,	PUNCT
ejpam-1851	464	8	new	new	PROPN
ejpam-1851	464	9	york	york	PROPN
ejpam-1851	464	10	:	:	PUNCT
ejpam-1851	464	11	springerverlag	springerverlag	PROPN
ejpam-1851	464	12	.	.	PUNCT
ejpam-1851	464	13	1996	1996	NUM
ejpam-1851	464	14	.	.	PUNCT
ejpam-1851	465	1	[	[	X
ejpam-1851	465	2	3	3	X
ejpam-1851	465	3	]	]	X
ejpam-1851	465	4	w.	w.	PROPN
ejpam-1851	465	5	fulton	fulton	PROPN
ejpam-1851	465	6	.	.	PUNCT
ejpam-1851	466	1	intersection	intersection	NOUN
ejpam-1851	466	2	theory	theory	NOUN
ejpam-1851	466	3	.	.	PUNCT
ejpam-1851	467	1	graduate	graduate	ADJ
ejpam-1851	467	2	texts	text	NOUN
ejpam-1851	467	3	in	in	ADP
ejpam-1851	467	4	math	math	NOUN
ejpam-1851	467	5	,	,	PUNCT
ejpam-1851	467	6	springer	springer	NOUN
ejpam-1851	467	7	-	-	PUNCT
ejpam-1851	467	8	verlag	verlag	PROPN
ejpam-1851	467	9	.	.	PUNCT
ejpam-1851	468	1	1977	1977	NUM
ejpam-1851	468	2	.	.	PUNCT
ejpam-1851	469	1	[	[	X
ejpam-1851	469	2	4	4	X
ejpam-1851	469	3	]	]	X
ejpam-1851	469	4	p.	p.	PROPN
ejpam-1851	469	5	griffiths	griffiths	PROPN
ejpam-1851	469	6	and	and	CCONJ
ejpam-1851	469	7	j.	j.	PROPN
ejpam-1851	469	8	harris	harris	PROPN
ejpam-1851	469	9	.	.	PUNCT
ejpam-1851	470	1	principles	principle	NOUN
ejpam-1851	470	2	of	of	ADP
ejpam-1851	470	3	algebraic	algebraic	ADJ
ejpam-1851	470	4	geometry	geometry	NOUN
ejpam-1851	470	5	.	.	PUNCT
ejpam-1851	471	1	wiley	wiley	NOUN
ejpam-1851	471	2	-	-	PUNCT
ejpam-1851	471	3	interscience	interscience	NOUN
ejpam-1851	471	4	.	.	PUNCT
ejpam-1851	472	1	1994	1994	NUM
ejpam-1851	472	2	.	.	PUNCT
ejpam-1851	473	1	[	[	X
ejpam-1851	473	2	5	5	NUM
ejpam-1851	473	3	]	]	PUNCT
ejpam-1851	473	4	r.	r.	PROPN
ejpam-1851	473	5	hartshorne	hartshorne	PROPN
ejpam-1851	473	6	.	.	PUNCT
ejpam-1851	474	1	algebraic	algebraic	ADJ
ejpam-1851	474	2	geometry	geometry	NOUN
ejpam-1851	474	3	.	.	PUNCT
ejpam-1851	475	1	graduate	graduate	NOUN
ejpam-1851	475	2	texts	text	NOUN
ejpam-1851	475	3	in	in	ADP
ejpam-1851	475	4	math	math	NOUN
ejpam-1851	475	5	.	.	PUNCT
ejpam-1851	476	1	52	52	NUM
ejpam-1851	476	2	,	,	PUNCT
ejpam-1851	476	3	springer	springer	NOUN
ejpam-1851	476	4	,	,	PUNCT
ejpam-1851	476	5	new	new	PROPN
ejpam-1851	476	6	york	york	PROPN
ejpam-1851	476	7	.	.	PUNCT
ejpam-1851	476	8	1977	1977	NUM
ejpam-1851	476	9	.	.	PUNCT
ejpam-1851	477	1	[	[	X
ejpam-1851	477	2	6	6	X
ejpam-1851	477	3	]	]	PUNCT
ejpam-1851	477	4	j.	j.	PROPN
ejpam-1851	477	5	harris	harris	PROPN
ejpam-1851	477	6	and	and	CCONJ
ejpam-1851	477	7	d.	d.	PROPN
ejpam-1851	477	8	eisenbud	eisenbud	PROPN
ejpam-1851	477	9	.	.	PUNCT
ejpam-1851	478	1	the	the	DET
ejpam-1851	478	2	geometry	geometry	NOUN
ejpam-1851	478	3	of	of	ADP
ejpam-1851	478	4	schemes	scheme	NOUN
ejpam-1851	478	5	.	.	PUNCT
ejpam-1851	479	1	graduate	graduate	NOUN
ejpam-1851	479	2	texts	text	NOUN
ejpam-1851	479	3	in	in	ADP
ejpam-1851	479	4	math	math	NOUN
ejpam-1851	479	5	.	.	PUNCT
ejpam-1851	480	1	197	197	NUM
ejpam-1851	480	2	,	,	PUNCT
ejpam-1851	480	3	springer	springer	NOUN
ejpam-1851	480	4	,	,	PUNCT
ejpam-1851	480	5	new	new	PROPN
ejpam-1851	480	6	york	york	PROPN
ejpam-1851	480	7	.	.	PUNCT
ejpam-1851	480	8	1999	1999	NUM
ejpam-1851	480	9	.	.	PUNCT
ejpam-1851	481	1	[	[	X
ejpam-1851	481	2	7	7	X
ejpam-1851	481	3	]	]	PUNCT
ejpam-1851	481	4	m.	m.	NOUN
ejpam-1851	481	5	hohmeyer	hohmeyer	NOUN
ejpam-1851	481	6	and	and	CCONJ
ejpam-1851	481	7	s.	s.	PROPN
ejpam-1851	481	8	teller	teller	NOUN
ejpam-1851	481	9	.	.	PUNCT
ejpam-1851	482	1	determining	determine	VERB
ejpam-1851	482	2	the	the	DET
ejpam-1851	482	3	lines	line	NOUN
ejpam-1851	482	4	through	through	ADP
ejpam-1851	482	5	four	four	NUM
ejpam-1851	482	6	lines	line	NOUN
ejpam-1851	482	7	.	.	PUNCT
ejpam-1851	483	1	journal	journal	PROPN
ejpam-1851	483	2	of	of	ADP
ejpam-1851	483	3	graphics	graphic	NOUN
ejpam-1851	483	4	tools	tool	NOUN
ejpam-1851	483	5	,	,	PUNCT
ejpam-1851	483	6	4(3):11–22	4(3):11–22	NUM
ejpam-1851	483	7	.	.	PUNCT
ejpam-1851	483	8	1999	1999	NUM
ejpam-1851	483	9	.	.	PUNCT
ejpam-1851	484	1	[	[	X
ejpam-1851	484	2	8	8	NUM
ejpam-1851	484	3	]	]	X
ejpam-1851	484	4	r.	r.	PROPN
ejpam-1851	484	5	mendoza	mendoza	PROPN
ejpam-1851	484	6	and	and	CCONJ
ejpam-1851	484	7	j.	j.	PROPN
ejpam-1851	484	8	rojas	rojas	PROPN
ejpam-1851	484	9	.	.	PUNCT
ejpam-1851	485	1	álgebra	álgebra	NOUN
ejpam-1851	485	2	linear	linear	NOUN
ejpam-1851	485	3	e	e	X
ejpam-1851	485	4	o	o	PROPN
ejpam-1851	485	5	problema	problema	NOUN
ejpam-1851	485	6	das	das	PROPN
ejpam-1851	485	7	quatro	quatro	PROPN
ejpam-1851	485	8	retas	reta	NOUN
ejpam-1851	485	9	do	do	VERB
ejpam-1851	485	10	cálculo	cálculo	PROPN
ejpam-1851	485	11	de	de	PROPN
ejpam-1851	485	12	schubert	schubert	PROPN
ejpam-1851	485	13	,	,	PUNCT
ejpam-1851	485	14	revista	revista	PROPN
ejpam-1851	485	15	matemática	matemática	PROPN
ejpam-1851	485	16	universitária	universitária	PROPN
ejpam-1851	485	17	,	,	PUNCT
ejpam-1851	485	18	45:55–69	45:55–69	PROPN
ejpam-1851	485	19	.	.	PUNCT
ejpam-1851	485	20	2009	2009	NUM
ejpam-1851	485	21	.	.	PUNCT