id	sid	tid	token	lemma	pos
ejpam-3495	1	1	european	european	PROPN
ejpam-3495	1	2	journal	journal	PROPN
ejpam-3495	1	3	of	of	ADP
ejpam-3495	1	4	pure	pure	ADJ
ejpam-3495	1	5	and	and	CCONJ
ejpam-3495	1	6	applied	apply	VERB
ejpam-3495	1	7	mathematics	mathematic	NOUN
ejpam-3495	1	8	vol	vol	NOUN
ejpam-3495	1	9	.	.	PROPN
ejpam-3495	2	1	12	12	NUM
ejpam-3495	2	2	,	,	PUNCT
ejpam-3495	2	3	no	no	INTJ
ejpam-3495	2	4	.	.	NOUN
ejpam-3495	2	5	3	3	NUM
ejpam-3495	2	6	,	,	PUNCT
ejpam-3495	2	7	2019	2019	NUM
ejpam-3495	2	8	,	,	PUNCT
ejpam-3495	2	9	1248	1248	NUM
ejpam-3495	2	10	-	-	SYM
ejpam-3495	2	11	1259	1259	NUM
ejpam-3495	2	12	issn	issn	PROPN
ejpam-3495	2	13	1307	1307	NUM
ejpam-3495	2	14	-	-	SYM
ejpam-3495	2	15	5543	5543	NUM
ejpam-3495	2	16	–	–	PUNCT
ejpam-3495	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3495	2	18	published	publish	VERB
ejpam-3495	2	19	by	by	ADP
ejpam-3495	2	20	new	new	PROPN
ejpam-3495	2	21	york	york	PROPN
ejpam-3495	2	22	business	business	PROPN
ejpam-3495	2	23	global	global	PROPN
ejpam-3495	2	24	on	on	ADP
ejpam-3495	2	25	companion	companion	PROPN
ejpam-3495	2	26	b	b	PROPN
ejpam-3495	2	27	-	-	PUNCT
ejpam-3495	2	28	algebras	algebras	PROPN
ejpam-3495	2	29	lynnel	lynnel	PROPN
ejpam-3495	2	30	d.	d.	PROPN
ejpam-3495	2	31	naingue1,∗	naingue1,∗	PROPN
ejpam-3495	2	32	,	,	PUNCT
ejpam-3495	2	33	jocelyn	jocelyn	PROPN
ejpam-3495	2	34	p.	p.	PROPN
ejpam-3495	2	35	vilela1	vilela1	NOUN
ejpam-3495	3	1	1	1	NUM
ejpam-3495	3	2	department	department	NOUN
ejpam-3495	3	3	of	of	ADP
ejpam-3495	3	4	mathematics	mathematic	NOUN
ejpam-3495	3	5	and	and	CCONJ
ejpam-3495	3	6	statistics	statistic	NOUN
ejpam-3495	3	7	,	,	PUNCT
ejpam-3495	3	8	college	college	NOUN
ejpam-3495	3	9	of	of	ADP
ejpam-3495	3	10	science	science	NOUN
ejpam-3495	3	11	and	and	CCONJ
ejpam-3495	3	12	mathematics	mathematic	NOUN
ejpam-3495	3	13	,	,	PUNCT
ejpam-3495	3	14	mindanao	mindanao	PROPN
ejpam-3495	3	15	state	state	PROPN
ejpam-3495	3	16	universityiligan	universityiligan	PROPN
ejpam-3495	3	17	institute	institute	PROPN
ejpam-3495	3	18	of	of	ADP
ejpam-3495	3	19	technology	technology	PROPN
ejpam-3495	3	20	,	,	PUNCT
ejpam-3495	3	21	9200	9200	NUM
ejpam-3495	3	22	iligan	iligan	ADJ
ejpam-3495	3	23	city	city	NOUN
ejpam-3495	3	24	,	,	PUNCT
ejpam-3495	3	25	philippines	philippine	NOUN
ejpam-3495	3	26	abstract	abstract	ADJ
ejpam-3495	3	27	.	.	PUNCT
ejpam-3495	4	1	this	this	DET
ejpam-3495	4	2	study	study	NOUN
ejpam-3495	4	3	introduces	introduce	VERB
ejpam-3495	4	4	the	the	DET
ejpam-3495	4	5	concept	concept	NOUN
ejpam-3495	4	6	of	of	ADP
ejpam-3495	4	7	companion	companion	NOUN
ejpam-3495	4	8	b	b	PROPN
ejpam-3495	4	9	-algebra	-algebra	NOUN
ejpam-3495	4	10	and	and	CCONJ
ejpam-3495	4	11	establishes	establish	VERB
ejpam-3495	4	12	some	some	PRON
ejpam-3495	4	13	of	of	ADP
ejpam-3495	4	14	its	its	PRON
ejpam-3495	4	15	properties	property	NOUN
ejpam-3495	4	16	.	.	PUNCT
ejpam-3495	5	1	also	also	ADV
ejpam-3495	5	2	,	,	PUNCT
ejpam-3495	5	3	this	this	DET
ejpam-3495	5	4	paper	paper	NOUN
ejpam-3495	5	5	introduces	introduce	VERB
ejpam-3495	5	6	the	the	DET
ejpam-3495	5	7	notions	notion	NOUN
ejpam-3495	5	8	of	of	ADP
ejpam-3495	5	9	�	�	NOUN
ejpam-3495	5	10	-subalgebra	-subalgebra	NOUN
ejpam-3495	5	11	and	and	CCONJ
ejpam-3495	5	12	�	�	NOUN
ejpam-3495	5	13	-ideal	-ideal	NOUN
ejpam-3495	5	14	of	of	ADP
ejpam-3495	5	15	a	a	DET
ejpam-3495	5	16	companion	companion	NOUN
ejpam-3495	5	17	b	b	NOUN
ejpam-3495	5	18	algebra	algebra	NOUN
ejpam-3495	5	19	and	and	CCONJ
ejpam-3495	5	20	investigates	investigate	VERB
ejpam-3495	5	21	their	their	PRON
ejpam-3495	5	22	relationship	relationship	NOUN
ejpam-3495	5	23	.	.	PUNCT
ejpam-3495	6	1	furthermore	furthermore	ADV
ejpam-3495	6	2	,	,	PUNCT
ejpam-3495	6	3	this	this	DET
ejpam-3495	6	4	study	study	NOUN
ejpam-3495	6	5	establishes	establish	VERB
ejpam-3495	6	6	some	some	DET
ejpam-3495	6	7	homomorphic	homomorphic	ADJ
ejpam-3495	6	8	properties	property	NOUN
ejpam-3495	6	9	of	of	ADP
ejpam-3495	6	10	�	�	NOUN
ejpam-3495	6	11	-subalgebra	-subalgebra	NOUN
ejpam-3495	6	12	and	and	CCONJ
ejpam-3495	6	13	�	�	NOUN
ejpam-3495	6	14	-ideal	-ideal	NOUN
ejpam-3495	6	15	.	.	PUNCT
ejpam-3495	7	1	2010	2010	NUM
ejpam-3495	7	2	mathematics	mathematic	NOUN
ejpam-3495	7	3	subject	subject	NOUN
ejpam-3495	7	4	classifications	classification	NOUN
ejpam-3495	7	5	:	:	PUNCT
ejpam-3495	7	6	08c99	08c99	NUM
ejpam-3495	7	7	,	,	PUNCT
ejpam-3495	7	8	08a05	08a05	NUM
ejpam-3495	7	9	,	,	PUNCT
ejpam-3495	7	10	08a30	08a30	VERB
ejpam-3495	7	11	key	key	ADJ
ejpam-3495	7	12	words	word	NOUN
ejpam-3495	7	13	and	and	CCONJ
ejpam-3495	7	14	phrases	phrase	NOUN
ejpam-3495	7	15	:	:	PUNCT
ejpam-3495	7	16	companion	companion	PROPN
ejpam-3495	7	17	b	b	PROPN
ejpam-3495	7	18	-algebra	-algebra	PROPN
ejpam-3495	7	19	,	,	PUNCT
ejpam-3495	7	20	�	�	NOUN
ejpam-3495	7	21	-subalgebra	-subalgebra	PROPN
ejpam-3495	7	22	,	,	PUNCT
ejpam-3495	7	23	�	�	NOUN
ejpam-3495	7	24	-ideal	-ideal	NOUN
ejpam-3495	7	25	,	,	PUNCT
ejpam-3495	7	26	companion	companion	NOUN
ejpam-3495	7	27	b	b	PROPN
ejpam-3495	7	28	homomorphism	homomorphism	NOUN
ejpam-3495	7	29	1	1	X
ejpam-3495	7	30	.	.	PUNCT
ejpam-3495	7	31	introduction	introduction	NOUN
ejpam-3495	7	32	y.	y.	PROPN
ejpam-3495	7	33	imai	imai	PROPN
ejpam-3495	7	34	and	and	CCONJ
ejpam-3495	7	35	k.	k.	PROPN
ejpam-3495	7	36	iséki	iséki	PROPN
ejpam-3495	8	1	[	[	X
ejpam-3495	8	2	7	7	X
ejpam-3495	8	3	]	]	PUNCT
ejpam-3495	8	4	first	first	ADV
ejpam-3495	8	5	initiated	initiate	VERB
ejpam-3495	8	6	the	the	DET
ejpam-3495	8	7	study	study	NOUN
ejpam-3495	8	8	of	of	ADP
ejpam-3495	8	9	bck	bck	PROPN
ejpam-3495	8	10	-algebras	-algebras	PROPN
ejpam-3495	8	11	in	in	ADP
ejpam-3495	8	12	1966	1966	NUM
ejpam-3495	8	13	.	.	PUNCT
ejpam-3495	9	1	in	in	ADP
ejpam-3495	9	2	the	the	DET
ejpam-3495	9	3	same	same	ADJ
ejpam-3495	9	4	year	year	NOUN
ejpam-3495	9	5	,	,	PUNCT
ejpam-3495	9	6	k.	k.	PROPN
ejpam-3495	9	7	iséki	iséki	PUNCT
ejpam-3495	10	1	[	[	X
ejpam-3495	10	2	6	6	NUM
ejpam-3495	10	3	]	]	PUNCT
ejpam-3495	10	4	introduced	introduce	VERB
ejpam-3495	10	5	another	another	DET
ejpam-3495	10	6	class	class	NOUN
ejpam-3495	10	7	of	of	ADP
ejpam-3495	10	8	algebras	algebra	NOUN
ejpam-3495	10	9	,	,	PUNCT
ejpam-3495	10	10	called	call	VERB
ejpam-3495	10	11	bci	bci	PROPN
ejpam-3495	10	12	-algebras	-algebra	NOUN
ejpam-3495	10	13	,	,	PUNCT
ejpam-3495	10	14	which	which	PRON
ejpam-3495	10	15	are	be	AUX
ejpam-3495	10	16	generalizations	generalization	NOUN
ejpam-3495	10	17	of	of	ADP
ejpam-3495	10	18	bck	bck	PROPN
ejpam-3495	10	19	-algebras	-algebras	PROPN
ejpam-3495	10	20	.	.	PUNCT
ejpam-3495	11	1	in	in	ADP
ejpam-3495	11	2	1999	1999	NUM
ejpam-3495	11	3	,	,	PUNCT
ejpam-3495	11	4	j.	j.	PROPN
ejpam-3495	11	5	neggers	neggers	PROPN
ejpam-3495	11	6	and	and	CCONJ
ejpam-3495	11	7	h.	h.	PROPN
ejpam-3495	11	8	s.	s.	PROPN
ejpam-3495	11	9	kim	kim	PROPN
ejpam-3495	12	1	[	[	X
ejpam-3495	12	2	9	9	NUM
ejpam-3495	12	3	]	]	PUNCT
ejpam-3495	12	4	,	,	PUNCT
ejpam-3495	12	5	introduced	introduce	VERB
ejpam-3495	12	6	the	the	DET
ejpam-3495	12	7	notion	notion	NOUN
ejpam-3495	12	8	of	of	ADP
ejpam-3495	12	9	d	d	PROPN
ejpam-3495	12	10	-algebra	-algebra	NOUN
ejpam-3495	12	11	which	which	PRON
ejpam-3495	12	12	is	be	AUX
ejpam-3495	12	13	another	another	DET
ejpam-3495	12	14	generalization	generalization	NOUN
ejpam-3495	12	15	of	of	ADP
ejpam-3495	12	16	bck	bck	PROPN
ejpam-3495	12	17	-algebra	-algebra	PROPN
ejpam-3495	12	18	.	.	PUNCT
ejpam-3495	13	1	in	in	ADP
ejpam-3495	13	2	2007	2007	NUM
ejpam-3495	13	3	,	,	PUNCT
ejpam-3495	13	4	p.	p.	PROPN
ejpam-3495	13	5	j.	j.	PROPN
ejpam-3495	13	6	allen	allen	PROPN
ejpam-3495	13	7	,	,	PUNCT
ejpam-3495	13	8	h.	h.	PROPN
ejpam-3495	13	9	s.	s.	PROPN
ejpam-3495	13	10	kim	kim	PROPN
ejpam-3495	13	11	and	and	CCONJ
ejpam-3495	13	12	j.	j.	PROPN
ejpam-3495	13	13	neggers	neggers	PROPN
ejpam-3495	14	1	[	[	X
ejpam-3495	14	2	3	3	NUM
ejpam-3495	14	3	]	]	PUNCT
ejpam-3495	14	4	developed	develop	VERB
ejpam-3495	14	5	the	the	DET
ejpam-3495	14	6	concept	concept	NOUN
ejpam-3495	14	7	of	of	ADP
ejpam-3495	14	8	companion	companion	NOUN
ejpam-3495	14	9	d	d	PROPN
ejpam-3495	14	10	-algebra	-algebra	NOUN
ejpam-3495	14	11	to	to	PART
ejpam-3495	14	12	demonstrate	demonstrate	VERB
ejpam-3495	14	13	considerable	considerable	ADJ
ejpam-3495	14	14	parallelism	parallelism	NOUN
ejpam-3495	14	15	with	with	ADP
ejpam-3495	14	16	the	the	DET
ejpam-3495	14	17	theory	theory	NOUN
ejpam-3495	14	18	of	of	ADP
ejpam-3495	14	19	bck	bck	PROPN
ejpam-3495	14	20	-algebras	-algebras	PROPN
ejpam-3495	14	21	.	.	PUNCT
ejpam-3495	15	1	in	in	ADP
ejpam-3495	15	2	2002	2002	NUM
ejpam-3495	15	3	,	,	PUNCT
ejpam-3495	15	4	j.	j.	PROPN
ejpam-3495	15	5	neggers	neggers	PROPN
ejpam-3495	15	6	and	and	CCONJ
ejpam-3495	15	7	h.	h.	PROPN
ejpam-3495	15	8	s.	s.	PROPN
ejpam-3495	15	9	kim	kim	PROPN
ejpam-3495	16	1	[	[	X
ejpam-3495	16	2	11	11	NUM
ejpam-3495	16	3	]	]	PUNCT
ejpam-3495	16	4	introduced	introduce	VERB
ejpam-3495	16	5	and	and	CCONJ
ejpam-3495	16	6	investigated	investigate	VERB
ejpam-3495	16	7	another	another	DET
ejpam-3495	16	8	class	class	NOUN
ejpam-3495	16	9	of	of	ADP
ejpam-3495	16	10	algebras	algebras	PROPN
ejpam-3495	16	11	called	call	VERB
ejpam-3495	16	12	b	b	PROPN
ejpam-3495	16	13	-algebras	-algebras	PROPN
ejpam-3495	16	14	and	and	CCONJ
ejpam-3495	16	15	described	describe	VERB
ejpam-3495	16	16	it	it	PRON
ejpam-3495	16	17	to	to	PART
ejpam-3495	16	18	have	have	VERB
ejpam-3495	16	19	nice	nice	ADJ
ejpam-3495	16	20	properties	property	NOUN
ejpam-3495	16	21	without	without	ADP
ejpam-3495	16	22	being	be	AUX
ejpam-3495	16	23	complicated	complicate	VERB
ejpam-3495	16	24	.	.	PUNCT
ejpam-3495	17	1	p.	p.	NOUN
ejpam-3495	17	2	j.	j.	PROPN
ejpam-3495	17	3	allen	allen	PROPN
ejpam-3495	17	4	,	,	PUNCT
ejpam-3495	17	5	j.	j.	PROPN
ejpam-3495	17	6	neggers	neggers	PROPN
ejpam-3495	17	7	and	and	CCONJ
ejpam-3495	17	8	h.	h.	PROPN
ejpam-3495	17	9	s.	s.	PROPN
ejpam-3495	17	10	kim	kim	PROPN
ejpam-3495	18	1	[	[	X
ejpam-3495	18	2	2	2	NUM
ejpam-3495	18	3	]	]	PUNCT
ejpam-3495	18	4	proved	prove	VERB
ejpam-3495	18	5	that	that	SCONJ
ejpam-3495	18	6	every	every	DET
ejpam-3495	18	7	group	group	NOUN
ejpam-3495	18	8	,	,	PUNCT
ejpam-3495	18	9	under	under	ADP
ejpam-3495	18	10	some	some	DET
ejpam-3495	18	11	conditions	condition	NOUN
ejpam-3495	18	12	,	,	PUNCT
ejpam-3495	18	13	determines	determine	VERB
ejpam-3495	18	14	a	a	DET
ejpam-3495	18	15	b	b	NOUN
ejpam-3495	18	16	-algebra	-algebra	NOUN
ejpam-3495	18	17	.	.	PUNCT
ejpam-3495	19	1	also	also	ADV
ejpam-3495	19	2	,	,	PUNCT
ejpam-3495	19	3	m.	m.	NOUN
ejpam-3495	19	4	kondo	kondo	PROPN
ejpam-3495	19	5	and	and	CCONJ
ejpam-3495	19	6	y.	y.	PROPN
ejpam-3495	19	7	b.	b.	PROPN
ejpam-3495	19	8	jun	jun	PROPN
ejpam-3495	19	9	[	[	X
ejpam-3495	19	10	8	8	NUM
ejpam-3495	19	11	]	]	PUNCT
ejpam-3495	19	12	proved	prove	VERB
ejpam-3495	19	13	the	the	DET
ejpam-3495	19	14	converse	converse	NOUN
ejpam-3495	19	15	.	.	PUNCT
ejpam-3495	20	1	this	this	DET
ejpam-3495	20	2	paper	paper	NOUN
ejpam-3495	20	3	extends	extend	VERB
ejpam-3495	20	4	the	the	DET
ejpam-3495	20	5	study	study	NOUN
ejpam-3495	20	6	of	of	ADP
ejpam-3495	20	7	b	b	NOUN
ejpam-3495	20	8	-algebras	-algebra	NOUN
ejpam-3495	20	9	by	by	ADP
ejpam-3495	20	10	defining	define	VERB
ejpam-3495	20	11	the	the	DET
ejpam-3495	20	12	concept	concept	NOUN
ejpam-3495	20	13	of	of	ADP
ejpam-3495	20	14	companion	companion	NOUN
ejpam-3495	20	15	operation	operation	NOUN
ejpam-3495	20	16	and	and	CCONJ
ejpam-3495	20	17	companion	companion	PROPN
ejpam-3495	20	18	b	b	PROPN
ejpam-3495	20	19	-algebras	-algebras	PROPN
ejpam-3495	20	20	and	and	CCONJ
ejpam-3495	20	21	establishing	establish	VERB
ejpam-3495	20	22	some	some	PRON
ejpam-3495	20	23	of	of	ADP
ejpam-3495	20	24	its	its	PRON
ejpam-3495	20	25	properties	property	NOUN
ejpam-3495	20	26	.	.	PUNCT
ejpam-3495	21	1	this	this	DET
ejpam-3495	21	2	study	study	NOUN
ejpam-3495	21	3	also	also	ADV
ejpam-3495	21	4	introduces	introduce	VERB
ejpam-3495	21	5	the	the	DET
ejpam-3495	21	6	concepts	concept	NOUN
ejpam-3495	21	7	of	of	ADP
ejpam-3495	21	8	subalgebra	subalgebra	NOUN
ejpam-3495	21	9	and	and	CCONJ
ejpam-3495	21	10	ideal	ideal	NOUN
ejpam-3495	21	11	of	of	ADP
ejpam-3495	21	12	a	a	DET
ejpam-3495	21	13	companion	companion	NOUN
ejpam-3495	21	14	b	b	NOUN
ejpam-3495	21	15	-algebra	-algebra	NOUN
ejpam-3495	21	16	and	and	CCONJ
ejpam-3495	21	17	determines	determine	VERB
ejpam-3495	21	18	some	some	PRON
ejpam-3495	21	19	of	of	ADP
ejpam-3495	21	20	its	its	PRON
ejpam-3495	21	21	homomorphic	homomorphic	ADJ
ejpam-3495	21	22	properties	property	NOUN
ejpam-3495	21	23	.	.	PUNCT
ejpam-3495	22	1	∗corresponding	∗corresponde	VERB
ejpam-3495	22	2	author	author	NOUN
ejpam-3495	22	3	.	.	PUNCT
ejpam-3495	23	1	doi	doi	NOUN
ejpam-3495	23	2	:	:	PUNCT
ejpam-3495	23	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3495	https://doi.org/10.29020/nybg.ejpam.v12i3.3495	NUM
ejpam-3495	23	4	email	email	NOUN
ejpam-3495	23	5	addresses	address	NOUN
ejpam-3495	23	6	:	:	PUNCT
ejpam-3495	23	7	lynneldnaingue13@gmail.com	lynneldnaingue13@gmail.com	X
ejpam-3495	23	8	(	(	PUNCT
ejpam-3495	23	9	l.d	l.d	PROPN
ejpam-3495	23	10	.	.	PROPN
ejpam-3495	23	11	naingue	naingue	PROPN
ejpam-3495	23	12	)	)	PUNCT
ejpam-3495	23	13	,	,	PUNCT
ejpam-3495	23	14	jocelyn.vilela@g.msuiit.edu.ph	jocelyn.vilela@g.msuiit.edu.ph	PROPN
ejpam-3495	23	15	(	(	PUNCT
ejpam-3495	23	16	j.p	j.p	PROPN
ejpam-3495	23	17	.	.	PROPN
ejpam-3495	23	18	vilela	vilela	PROPN
ejpam-3495	23	19	)	)	PUNCT
ejpam-3495	23	20	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3495	24	1	1248	1248	NUM
ejpam-3495	24	2	c	c	X
ejpam-3495	24	3	©	©	PROPN
ejpam-3495	24	4	2019	2019	NUM
ejpam-3495	24	5	ejpam	ejpam	NOUN
ejpam-3495	24	6	all	all	DET
ejpam-3495	24	7	rights	right	NOUN
ejpam-3495	24	8	reserved	reserve	VERB
ejpam-3495	24	9	.	.	PUNCT
ejpam-3495	25	1	l.d	l.d	PROPN
ejpam-3495	25	2	.	.	PROPN
ejpam-3495	25	3	naingue	naingue	PROPN
ejpam-3495	25	4	,	,	PUNCT
ejpam-3495	25	5	j.p	j.p	PROPN
ejpam-3495	25	6	.	.	PROPN
ejpam-3495	25	7	vilela	vilela	PROPN
ejpam-3495	25	8	/	/	SYM
ejpam-3495	25	9	eur	eur	PROPN
ejpam-3495	25	10	.	.	PUNCT
ejpam-3495	26	1	j.	j.	PROPN
ejpam-3495	26	2	pure	pure	PROPN
ejpam-3495	26	3	appl	appl	PROPN
ejpam-3495	26	4	.	.	PROPN
ejpam-3495	26	5	math	math	PROPN
ejpam-3495	26	6	,	,	PUNCT
ejpam-3495	26	7	12	12	NUM
ejpam-3495	26	8	(	(	PUNCT
ejpam-3495	26	9	3	3	NUM
ejpam-3495	26	10	)	)	PUNCT
ejpam-3495	26	11	(	(	PUNCT
ejpam-3495	26	12	2019	2019	NUM
ejpam-3495	26	13	)	)	PUNCT
ejpam-3495	26	14	,	,	PUNCT
ejpam-3495	26	15	1248	1248	NUM
ejpam-3495	26	16	-	-	SYM
ejpam-3495	26	17	1259	1259	NUM
ejpam-3495	26	18	1249	1249	NUM
ejpam-3495	26	19	2	2	NUM
ejpam-3495	26	20	.	.	PUNCT
ejpam-3495	26	21	preliminaries	preliminary	NOUN
ejpam-3495	26	22	definition	definition	NOUN
ejpam-3495	26	23	2.1	2.1	NUM
ejpam-3495	26	24	.	.	PUNCT
ejpam-3495	27	1	[	[	X
ejpam-3495	27	2	11	11	NUM
ejpam-3495	27	3	]	]	PUNCT
ejpam-3495	27	4	a	a	DET
ejpam-3495	27	5	b	b	X
ejpam-3495	27	6	-	-	PUNCT
ejpam-3495	27	7	algebra	algebra	NOUN
ejpam-3495	27	8	(	(	PUNCT
ejpam-3495	27	9	x	x	X
ejpam-3495	27	10	,	,	PUNCT
ejpam-3495	27	11	∗	∗	NOUN
ejpam-3495	27	12	,	,	PUNCT
ejpam-3495	27	13	0	0	NUM
ejpam-3495	27	14	)	)	PUNCT
ejpam-3495	27	15	is	be	AUX
ejpam-3495	27	16	a	a	DET
ejpam-3495	27	17	nonempty	nonempty	ADV
ejpam-3495	27	18	set	set	VERB
ejpam-3495	27	19	x	x	PUNCT
ejpam-3495	27	20	with	with	ADP
ejpam-3495	27	21	a	a	DET
ejpam-3495	27	22	constant	constant	ADJ
ejpam-3495	27	23	0	0	NUM
ejpam-3495	27	24	and	and	CCONJ
ejpam-3495	27	25	a	a	DET
ejpam-3495	27	26	binary	binary	ADJ
ejpam-3495	27	27	operation	operation	NOUN
ejpam-3495	27	28	“	"	PUNCT
ejpam-3495	27	29	∗	∗	NOUN
ejpam-3495	27	30	”	"	PUNCT
ejpam-3495	27	31	satisfying	satisfy	VERB
ejpam-3495	27	32	the	the	DET
ejpam-3495	27	33	following	follow	VERB
ejpam-3495	27	34	axioms	axiom	NOUN
ejpam-3495	27	35	:	:	PUNCT
ejpam-3495	27	36	for	for	ADP
ejpam-3495	27	37	all	all	DET
ejpam-3495	27	38	x	x	NOUN
ejpam-3495	27	39	,	,	PUNCT
ejpam-3495	27	40	y	y	PROPN
ejpam-3495	27	41	,	,	PUNCT
ejpam-3495	27	42	z	z	NOUN
ejpam-3495	27	43	in	in	ADP
ejpam-3495	27	44	x	x	PRON
ejpam-3495	27	45	,	,	PUNCT
ejpam-3495	27	46	(	(	PUNCT
ejpam-3495	27	47	i	i	NOUN
ejpam-3495	27	48	)	)	PUNCT
ejpam-3495	27	49	x	x	SYM
ejpam-3495	27	50	∗	∗	NOUN
ejpam-3495	27	51	x	x	SYM
ejpam-3495	27	52	=	=	SYM
ejpam-3495	27	53	0	0	NUM
ejpam-3495	27	54	,	,	PUNCT
ejpam-3495	27	55	(	(	PUNCT
ejpam-3495	27	56	ii	ii	NOUN
ejpam-3495	27	57	)	)	PUNCT
ejpam-3495	27	58	x	x	SYM
ejpam-3495	27	59	∗	∗	NOUN
ejpam-3495	27	60	0	0	NUM
ejpam-3495	28	1	=	=	SYM
ejpam-3495	28	2	x	x	NOUN
ejpam-3495	28	3	,	,	PUNCT
ejpam-3495	28	4	(	(	PUNCT
ejpam-3495	28	5	iii	iii	NOUN
ejpam-3495	28	6	)	)	PUNCT
ejpam-3495	28	7	(	(	PUNCT
ejpam-3495	28	8	x	x	SYM
ejpam-3495	28	9	∗	∗	PROPN
ejpam-3495	28	10	y	y	NOUN
ejpam-3495	28	11	)	)	PUNCT
ejpam-3495	28	12	∗	∗	NOUN
ejpam-3495	28	13	z	z	NOUN
ejpam-3495	29	1	=	=	SYM
ejpam-3495	29	2	x	x	X
ejpam-3495	29	3	∗	∗	NOUN
ejpam-3495	29	4	(	(	PUNCT
ejpam-3495	29	5	z	z	NOUN
ejpam-3495	29	6	∗	∗	NOUN
ejpam-3495	29	7	(	(	PUNCT
ejpam-3495	29	8	0	0	NUM
ejpam-3495	29	9	∗	∗	PROPN
ejpam-3495	29	10	y	y	PROPN
ejpam-3495	29	11	)	)	PUNCT
ejpam-3495	29	12	)	)	PUNCT
ejpam-3495	29	13	.	.	PUNCT
ejpam-3495	30	1	example	example	NOUN
ejpam-3495	30	2	2.2	2.2	NUM
ejpam-3495	30	3	.	.	PUNCT
ejpam-3495	31	1	the	the	DET
ejpam-3495	31	2	set	set	NOUN
ejpam-3495	31	3	of	of	ADP
ejpam-3495	31	4	integers	integer	NOUN
ejpam-3495	31	5	together	together	ADV
ejpam-3495	31	6	with	with	ADP
ejpam-3495	31	7	the	the	DET
ejpam-3495	31	8	usual	usual	ADJ
ejpam-3495	31	9	subtraction	subtraction	NOUN
ejpam-3495	31	10	and	and	CCONJ
ejpam-3495	31	11	the	the	DET
ejpam-3495	31	12	constant	constant	ADJ
ejpam-3495	31	13	0	0	NUM
ejpam-3495	31	14	is	be	AUX
ejpam-3495	31	15	a	a	DET
ejpam-3495	31	16	b	b	NOUN
ejpam-3495	31	17	-algebra	-algebra	NOUN
ejpam-3495	31	18	.	.	PUNCT
ejpam-3495	32	1	theorem	theorem	VERB
ejpam-3495	32	2	2.3	2.3	NUM
ejpam-3495	32	3	.	.	PUNCT
ejpam-3495	33	1	[	[	X
ejpam-3495	33	2	11	11	NUM
ejpam-3495	33	3	]	]	X
ejpam-3495	33	4	if	if	SCONJ
ejpam-3495	33	5	(	(	PUNCT
ejpam-3495	33	6	x	x	NOUN
ejpam-3495	33	7	,	,	PUNCT
ejpam-3495	33	8	∗	∗	NOUN
ejpam-3495	33	9	,	,	PUNCT
ejpam-3495	33	10	0	0	NUM
ejpam-3495	33	11	)	)	PUNCT
ejpam-3495	33	12	is	be	AUX
ejpam-3495	33	13	a	a	DET
ejpam-3495	33	14	b	b	NOUN
ejpam-3495	33	15	-	-	PUNCT
ejpam-3495	33	16	algebra	algebra	NOUN
ejpam-3495	33	17	,	,	PUNCT
ejpam-3495	33	18	then	then	ADV
ejpam-3495	33	19	the	the	DET
ejpam-3495	33	20	following	follow	VERB
ejpam-3495	33	21	hold	hold	NOUN
ejpam-3495	33	22	:	:	PUNCT
ejpam-3495	33	23	for	for	ADP
ejpam-3495	33	24	any	any	DET
ejpam-3495	33	25	x	x	NOUN
ejpam-3495	33	26	,	,	PUNCT
ejpam-3495	33	27	y	y	PROPN
ejpam-3495	33	28	,	,	PUNCT
ejpam-3495	33	29	z	z	PROPN
ejpam-3495	33	30	∈	∈	PROPN
ejpam-3495	33	31	x	x	X
ejpam-3495	33	32	,	,	PUNCT
ejpam-3495	33	33	(	(	PUNCT
ejpam-3495	33	34	a	a	X
ejpam-3495	33	35	)	)	PUNCT
ejpam-3495	33	36	(	(	PUNCT
ejpam-3495	33	37	x	x	X
ejpam-3495	33	38	∗	∗	PROPN
ejpam-3495	33	39	y	y	NOUN
ejpam-3495	33	40	)	)	PUNCT
ejpam-3495	33	41	∗	∗	NOUN
ejpam-3495	33	42	(	(	PUNCT
ejpam-3495	33	43	0	0	NUM
ejpam-3495	33	44	∗	∗	NUM
ejpam-3495	33	45	y	y	NOUN
ejpam-3495	33	46	)	)	PUNCT
ejpam-3495	34	1	=	=	PUNCT
ejpam-3495	34	2	x	x	X
ejpam-3495	34	3	(	(	PUNCT
ejpam-3495	34	4	b	b	X
ejpam-3495	34	5	)	)	PUNCT
ejpam-3495	34	6	y	y	PROPN
ejpam-3495	34	7	∗	∗	NOUN
ejpam-3495	34	8	z	z	PROPN
ejpam-3495	35	1	=	=	SYM
ejpam-3495	35	2	y	y	PROPN
ejpam-3495	35	3	∗	∗	NOUN
ejpam-3495	35	4	(	(	PUNCT
ejpam-3495	35	5	0	0	NUM
ejpam-3495	35	6	∗	∗	NOUN
ejpam-3495	35	7	(	(	PUNCT
ejpam-3495	35	8	0	0	NUM
ejpam-3495	35	9	∗	∗	NOUN
ejpam-3495	35	10	z	z	NOUN
ejpam-3495	35	11	)	)	PUNCT
ejpam-3495	35	12	)	)	PUNCT
ejpam-3495	36	1	(	(	PUNCT
ejpam-3495	36	2	c	c	X
ejpam-3495	36	3	)	)	PUNCT
ejpam-3495	36	4	x	x	SYM
ejpam-3495	36	5	∗	∗	NOUN
ejpam-3495	36	6	(	(	PUNCT
ejpam-3495	36	7	y	y	PROPN
ejpam-3495	36	8	∗	∗	PROPN
ejpam-3495	36	9	z	z	NOUN
ejpam-3495	36	10	)	)	PUNCT
ejpam-3495	36	11	=	=	SYM
ejpam-3495	36	12	(	(	PUNCT
ejpam-3495	36	13	x	x	SYM
ejpam-3495	36	14	∗	∗	NOUN
ejpam-3495	36	15	(	(	PUNCT
ejpam-3495	36	16	0	0	NUM
ejpam-3495	36	17	∗	∗	NOUN
ejpam-3495	36	18	z	z	NOUN
ejpam-3495	36	19	)	)	PUNCT
ejpam-3495	36	20	)	)	PUNCT
ejpam-3495	36	21	∗	∗	NOUN
ejpam-3495	37	1	y	y	PROPN
ejpam-3495	37	2	(	(	PUNCT
ejpam-3495	37	3	d	d	NOUN
ejpam-3495	37	4	)	)	PUNCT
ejpam-3495	37	5	x	x	SYM
ejpam-3495	37	6	∗	∗	NOUN
ejpam-3495	37	7	y	y	NOUN
ejpam-3495	37	8	=	=	SYM
ejpam-3495	37	9	0	0	NUM
ejpam-3495	37	10	implies	imply	VERB
ejpam-3495	37	11	x	x	PUNCT
ejpam-3495	37	12	=	=	SYM
ejpam-3495	37	13	y	y	PROPN
ejpam-3495	37	14	(	(	PUNCT
ejpam-3495	37	15	e	e	NOUN
ejpam-3495	37	16	)	)	PUNCT
ejpam-3495	37	17	0	0	NUM
ejpam-3495	37	18	∗	∗	NOUN
ejpam-3495	37	19	x	x	X
ejpam-3495	37	20	=	=	SYM
ejpam-3495	37	21	0	0	NUM
ejpam-3495	37	22	∗	∗	NOUN
ejpam-3495	37	23	y	y	PROPN
ejpam-3495	37	24	implies	imply	VERB
ejpam-3495	37	25	x	x	PUNCT
ejpam-3495	37	26	=	=	SYM
ejpam-3495	37	27	y	y	PROPN
ejpam-3495	37	28	(	(	PUNCT
ejpam-3495	37	29	f	f	NOUN
ejpam-3495	37	30	)	)	PUNCT
ejpam-3495	37	31	0	0	NUM
ejpam-3495	37	32	∗	∗	NOUN
ejpam-3495	37	33	(	(	PUNCT
ejpam-3495	37	34	0	0	NUM
ejpam-3495	37	35	∗	∗	NOUN
ejpam-3495	37	36	x	x	NOUN
ejpam-3495	37	37	)	)	PUNCT
ejpam-3495	37	38	=	=	SYM
ejpam-3495	37	39	x.	x.	NOUN
ejpam-3495	37	40	theorem	theorem	VERB
ejpam-3495	37	41	2.4	2.4	NUM
ejpam-3495	37	42	.	.	PUNCT
ejpam-3495	38	1	[	[	X
ejpam-3495	38	2	13	13	NUM
ejpam-3495	38	3	]	]	X
ejpam-3495	38	4	if	if	SCONJ
ejpam-3495	38	5	(	(	PUNCT
ejpam-3495	38	6	x	x	NOUN
ejpam-3495	38	7	,	,	PUNCT
ejpam-3495	38	8	∗	∗	NOUN
ejpam-3495	38	9	,	,	PUNCT
ejpam-3495	38	10	0	0	NUM
ejpam-3495	38	11	)	)	PUNCT
ejpam-3495	38	12	is	be	AUX
ejpam-3495	38	13	a	a	DET
ejpam-3495	38	14	b	b	NOUN
ejpam-3495	38	15	-	-	PUNCT
ejpam-3495	38	16	algebra	algebra	NOUN
ejpam-3495	38	17	,	,	PUNCT
ejpam-3495	38	18	then	then	ADV
ejpam-3495	38	19	the	the	DET
ejpam-3495	38	20	following	follow	VERB
ejpam-3495	38	21	hold	hold	NOUN
ejpam-3495	38	22	:	:	PUNCT
ejpam-3495	38	23	for	for	ADP
ejpam-3495	38	24	any	any	DET
ejpam-3495	38	25	x	x	NOUN
ejpam-3495	38	26	,	,	PUNCT
ejpam-3495	38	27	y	y	PROPN
ejpam-3495	38	28	,	,	PUNCT
ejpam-3495	38	29	z	z	PROPN
ejpam-3495	38	30	∈	∈	PROPN
ejpam-3495	38	31	x	x	SYM
ejpam-3495	38	32	,	,	PUNCT
ejpam-3495	38	33	0	0	NUM
ejpam-3495	38	34	∗	∗	NOUN
ejpam-3495	38	35	(	(	PUNCT
ejpam-3495	38	36	x	x	X
ejpam-3495	38	37	∗	∗	NOUN
ejpam-3495	38	38	y	y	NOUN
ejpam-3495	38	39	)	)	PUNCT
ejpam-3495	39	1	=	=	SYM
ejpam-3495	39	2	y	y	PROPN
ejpam-3495	39	3	∗	∗	NOUN
ejpam-3495	39	4	x.	x.	NOUN
ejpam-3495	39	5	definition	definition	NOUN
ejpam-3495	39	6	2.5	2.5	NUM
ejpam-3495	39	7	.	.	PUNCT
ejpam-3495	40	1	[	[	X
ejpam-3495	40	2	11	11	NUM
ejpam-3495	40	3	]	]	PUNCT
ejpam-3495	40	4	a	a	DET
ejpam-3495	40	5	b	b	PROPN
ejpam-3495	40	6	-algebra	-algebra	NOUN
ejpam-3495	40	7	(	(	PUNCT
ejpam-3495	40	8	x	x	X
ejpam-3495	40	9	,	,	PUNCT
ejpam-3495	40	10	∗	∗	NOUN
ejpam-3495	40	11	,	,	PUNCT
ejpam-3495	40	12	0	0	NUM
ejpam-3495	40	13	)	)	PUNCT
ejpam-3495	40	14	is	be	AUX
ejpam-3495	40	15	commutative	commutative	ADJ
ejpam-3495	40	16	if	if	SCONJ
ejpam-3495	40	17	for	for	ADP
ejpam-3495	40	18	any	any	DET
ejpam-3495	40	19	x	x	NOUN
ejpam-3495	40	20	,	,	PUNCT
ejpam-3495	40	21	y	y	PROPN
ejpam-3495	40	22	∈	∈	PROPN
ejpam-3495	40	23	x	x	NOUN
ejpam-3495	40	24	,	,	PUNCT
ejpam-3495	40	25	x∗	x∗	X
ejpam-3495	40	26	(	(	PUNCT
ejpam-3495	40	27	0∗y	0∗y	NOUN
ejpam-3495	40	28	)	)	PUNCT
ejpam-3495	40	29	=	=	SYM
ejpam-3495	40	30	y	y	PROPN
ejpam-3495	40	31	∗	∗	NOUN
ejpam-3495	40	32	(	(	PUNCT
ejpam-3495	40	33	0	0	NUM
ejpam-3495	40	34	∗	∗	NOUN
ejpam-3495	40	35	x	x	NOUN
ejpam-3495	40	36	)	)	PUNCT
ejpam-3495	40	37	.	.	PUNCT
ejpam-3495	41	1	theorem	theorem	VERB
ejpam-3495	41	2	2.6	2.6	NUM
ejpam-3495	41	3	.	.	PUNCT
ejpam-3495	42	1	[	[	X
ejpam-3495	42	2	2	2	NUM
ejpam-3495	42	3	]	]	X
ejpam-3495	42	4	let	let	VERB
ejpam-3495	42	5	(	(	PUNCT
ejpam-3495	42	6	x	x	NOUN
ejpam-3495	42	7	,	,	PUNCT
ejpam-3495	42	8	∗	∗	NOUN
ejpam-3495	42	9	,	,	PUNCT
ejpam-3495	42	10	0	0	NUM
ejpam-3495	42	11	)	)	PUNCT
ejpam-3495	42	12	be	be	AUX
ejpam-3495	42	13	a	a	DET
ejpam-3495	42	14	b	b	NOUN
ejpam-3495	42	15	-	-	PUNCT
ejpam-3495	42	16	algebra	algebra	NOUN
ejpam-3495	42	17	.	.	PUNCT
ejpam-3495	43	1	if	if	SCONJ
ejpam-3495	43	2	x	x	PART
ejpam-3495	43	3	◦	◦	NOUN
ejpam-3495	43	4	y	y	NOUN
ejpam-3495	43	5	=	=	PUNCT
ejpam-3495	43	6	x	x	SYM
ejpam-3495	43	7	∗	∗	NOUN
ejpam-3495	43	8	(	(	PUNCT
ejpam-3495	43	9	0	0	NUM
ejpam-3495	43	10	∗	∗	NUM
ejpam-3495	43	11	y	y	PROPN
ejpam-3495	43	12	)	)	PUNCT
ejpam-3495	43	13	for	for	ADP
ejpam-3495	43	14	all	all	DET
ejpam-3495	43	15	x	x	NOUN
ejpam-3495	43	16	,	,	PUNCT
ejpam-3495	43	17	y	y	PROPN
ejpam-3495	43	18	∈	∈	PROPN
ejpam-3495	43	19	x	x	X
ejpam-3495	43	20	,	,	PUNCT
ejpam-3495	43	21	then	then	ADV
ejpam-3495	43	22	(	(	PUNCT
ejpam-3495	43	23	x	x	NOUN
ejpam-3495	43	24	,	,	PUNCT
ejpam-3495	43	25	◦	◦	NOUN
ejpam-3495	43	26	)	)	PUNCT
ejpam-3495	43	27	is	be	AUX
ejpam-3495	43	28	a	a	DET
ejpam-3495	43	29	group	group	NOUN
ejpam-3495	43	30	.	.	PUNCT
ejpam-3495	44	1	theorem	theorem	ADJ
ejpam-3495	44	2	2.7	2.7	NUM
ejpam-3495	44	3	.	.	PUNCT
ejpam-3495	45	1	[	[	X
ejpam-3495	45	2	11	11	NUM
ejpam-3495	45	3	]	]	X
ejpam-3495	45	4	let	let	VERB
ejpam-3495	45	5	(	(	PUNCT
ejpam-3495	45	6	g	g	NOUN
ejpam-3495	45	7	,	,	PUNCT
ejpam-3495	45	8	◦	◦	NOUN
ejpam-3495	45	9	)	)	PUNCT
ejpam-3495	45	10	be	be	VERB
ejpam-3495	45	11	a	a	DET
ejpam-3495	45	12	group	group	NOUN
ejpam-3495	45	13	with	with	ADP
ejpam-3495	45	14	identity	identity	NOUN
ejpam-3495	45	15	e.	e.	PROPN
ejpam-3495	45	16	if	if	SCONJ
ejpam-3495	45	17	we	we	PRON
ejpam-3495	45	18	define	define	VERB
ejpam-3495	45	19	x	x	X
ejpam-3495	45	20	∗	∗	NOUN
ejpam-3495	45	21	y	y	NOUN
ejpam-3495	45	22	=	=	PUNCT
ejpam-3495	45	23	x	x	PUNCT
ejpam-3495	45	24	◦	◦	NOUN
ejpam-3495	45	25	y−1	y−1	PROPN
ejpam-3495	45	26	,	,	PUNCT
ejpam-3495	45	27	then	then	ADV
ejpam-3495	45	28	(	(	PUNCT
ejpam-3495	45	29	g	g	NOUN
ejpam-3495	45	30	,	,	PUNCT
ejpam-3495	45	31	∗	∗	NOUN
ejpam-3495	45	32	,	,	PUNCT
ejpam-3495	45	33	e	e	NOUN
ejpam-3495	45	34	)	)	PUNCT
ejpam-3495	45	35	is	be	AUX
ejpam-3495	45	36	a	a	DET
ejpam-3495	45	37	b	b	NOUN
ejpam-3495	45	38	-	-	PUNCT
ejpam-3495	45	39	algebra	algebra	NOUN
ejpam-3495	45	40	.	.	PUNCT
ejpam-3495	46	1	definition	definition	NOUN
ejpam-3495	46	2	2.8	2.8	NUM
ejpam-3495	46	3	.	.	PUNCT
ejpam-3495	47	1	[	[	X
ejpam-3495	47	2	12	12	NUM
ejpam-3495	47	3	]	]	X
ejpam-3495	47	4	let	let	VERB
ejpam-3495	47	5	(	(	PUNCT
ejpam-3495	47	6	x	x	X
ejpam-3495	47	7	,	,	PUNCT
ejpam-3495	47	8	∗	∗	NOUN
ejpam-3495	47	9	,	,	PUNCT
ejpam-3495	47	10	0	0	NUM
ejpam-3495	47	11	)	)	PUNCT
ejpam-3495	47	12	be	be	AUX
ejpam-3495	47	13	a	a	DET
ejpam-3495	47	14	b	b	NOUN
ejpam-3495	47	15	-algebra	-algebra	NOUN
ejpam-3495	47	16	.	.	PUNCT
ejpam-3495	48	1	a	a	DET
ejpam-3495	48	2	nonempty	nonempty	ADV
ejpam-3495	48	3	subset	subset	VERB
ejpam-3495	48	4	h	h	NOUN
ejpam-3495	48	5	of	of	ADP
ejpam-3495	48	6	x	x	PROPN
ejpam-3495	48	7	is	be	AUX
ejpam-3495	48	8	called	call	VERB
ejpam-3495	48	9	a	a	DET
ejpam-3495	48	10	b	b	NOUN
ejpam-3495	48	11	-	-	PUNCT
ejpam-3495	48	12	subalgebra	subalgebra	NOUN
ejpam-3495	48	13	of	of	ADP
ejpam-3495	48	14	x	x	SYM
ejpam-3495	48	15	if	if	SCONJ
ejpam-3495	48	16	x	x	PROPN
ejpam-3495	48	17	∗	∗	VERB
ejpam-3495	48	18	y	y	PROPN
ejpam-3495	48	19	∈	∈	PROPN
ejpam-3495	48	20	h	h	NOUN
ejpam-3495	48	21	for	for	ADP
ejpam-3495	48	22	any	any	DET
ejpam-3495	48	23	x	x	NOUN
ejpam-3495	48	24	,	,	PUNCT
ejpam-3495	48	25	y	y	PROPN
ejpam-3495	48	26	∈	∈	PROPN
ejpam-3495	48	27	h.	h.	PROPN
ejpam-3495	48	28	definition	definition	NOUN
ejpam-3495	48	29	2.9	2.9	NUM
ejpam-3495	48	30	.	.	PUNCT
ejpam-3495	49	1	[	[	X
ejpam-3495	49	2	5	5	NUM
ejpam-3495	49	3	]	]	X
ejpam-3495	49	4	let	let	VERB
ejpam-3495	49	5	(	(	PUNCT
ejpam-3495	49	6	x	x	NOUN
ejpam-3495	49	7	,	,	PUNCT
ejpam-3495	49	8	∗	∗	NOUN
ejpam-3495	49	9	,	,	PUNCT
ejpam-3495	49	10	0	0	NUM
ejpam-3495	49	11	)	)	PUNCT
ejpam-3495	49	12	be	be	AUX
ejpam-3495	49	13	a	a	DET
ejpam-3495	49	14	b	b	NOUN
ejpam-3495	49	15	-algebra	-algebra	NOUN
ejpam-3495	49	16	.	.	PUNCT
ejpam-3495	50	1	a	a	DET
ejpam-3495	50	2	nonempty	nonempty	NOUN
ejpam-3495	50	3	subset	subset	VERB
ejpam-3495	50	4	i	i	PRON
ejpam-3495	50	5	of	of	ADP
ejpam-3495	50	6	x	x	PRON
ejpam-3495	50	7	is	be	AUX
ejpam-3495	50	8	called	call	VERB
ejpam-3495	50	9	a	a	DET
ejpam-3495	50	10	b	b	NOUN
ejpam-3495	50	11	-	-	PUNCT
ejpam-3495	50	12	ideal	ideal	NOUN
ejpam-3495	50	13	of	of	ADP
ejpam-3495	50	14	x	x	SYM
ejpam-3495	50	15	if	if	SCONJ
ejpam-3495	50	16	0	0	NUM
ejpam-3495	50	17	∈	∈	VERB
ejpam-3495	51	1	i	i	PRON
ejpam-3495	51	2	and	and	CCONJ
ejpam-3495	51	3	x	x	PROPN
ejpam-3495	51	4	∗	∗	NOUN
ejpam-3495	51	5	y	y	NOUN
ejpam-3495	51	6	∈	∈	PROPN
ejpam-3495	52	1	i	i	PRON
ejpam-3495	52	2	and	and	CCONJ
ejpam-3495	52	3	y	y	PROPN
ejpam-3495	52	4	∈	∈	PROPN
ejpam-3495	53	1	i	i	PRON
ejpam-3495	53	2	imply	imply	VERB
ejpam-3495	53	3	x	x	X
ejpam-3495	53	4	∈	∈	PROPN
ejpam-3495	53	5	i.	i.	NOUN
ejpam-3495	53	6	theorem	theorem	VERB
ejpam-3495	53	7	2.10	2.10	NUM
ejpam-3495	53	8	.	.	PUNCT
ejpam-3495	54	1	[	[	X
ejpam-3495	54	2	1	1	X
ejpam-3495	54	3	]	]	PUNCT
ejpam-3495	54	4	every	every	DET
ejpam-3495	54	5	subalgebra	subalgebra	NOUN
ejpam-3495	54	6	of	of	ADP
ejpam-3495	54	7	a	a	DET
ejpam-3495	54	8	b	b	NOUN
ejpam-3495	54	9	-	-	PUNCT
ejpam-3495	54	10	algebra	algebra	NOUN
ejpam-3495	54	11	x	x	PUNCT
ejpam-3495	54	12	is	be	AUX
ejpam-3495	54	13	an	an	DET
ejpam-3495	54	14	ideal	ideal	NOUN
ejpam-3495	54	15	.	.	PUNCT
ejpam-3495	55	1	definition	definition	NOUN
ejpam-3495	55	2	2.11	2.11	NUM
ejpam-3495	55	3	.	.	PUNCT
ejpam-3495	56	1	[	[	X
ejpam-3495	56	2	10	10	NUM
ejpam-3495	56	3	]	]	X
ejpam-3495	56	4	let	let	VERB
ejpam-3495	56	5	(	(	PUNCT
ejpam-3495	56	6	a	a	PRON
ejpam-3495	56	7	,	,	PUNCT
ejpam-3495	56	8	∗a	∗a	PROPN
ejpam-3495	56	9	,	,	PUNCT
ejpam-3495	56	10	0a	0a	NUM
ejpam-3495	56	11	)	)	PUNCT
ejpam-3495	56	12	and	and	CCONJ
ejpam-3495	56	13	(	(	PUNCT
ejpam-3495	56	14	b	b	X
ejpam-3495	56	15	,	,	PUNCT
ejpam-3495	56	16	∗b	∗b	PROPN
ejpam-3495	56	17	,	,	PUNCT
ejpam-3495	56	18	0b	0b	NUM
ejpam-3495	56	19	)	)	PUNCT
ejpam-3495	56	20	be	be	AUX
ejpam-3495	56	21	b	b	NOUN
ejpam-3495	56	22	-algebras	-algebra	NOUN
ejpam-3495	56	23	.	.	PUNCT
ejpam-3495	57	1	the	the	DET
ejpam-3495	57	2	mapping	mapping	NOUN
ejpam-3495	57	3	φ	φ	PROPN
ejpam-3495	57	4	:	:	PUNCT
ejpam-3495	57	5	a	a	DET
ejpam-3495	57	6	→	→	SYM
ejpam-3495	57	7	b	b	PROPN
ejpam-3495	57	8	is	be	AUX
ejpam-3495	57	9	called	call	VERB
ejpam-3495	57	10	a	a	DET
ejpam-3495	57	11	b	b	NOUN
ejpam-3495	57	12	-	-	PUNCT
ejpam-3495	57	13	homomorphism	homomorphism	NOUN
ejpam-3495	57	14	if	if	SCONJ
ejpam-3495	57	15	φ(x	φ(x	PROPN
ejpam-3495	57	16	∗a	∗a	PROPN
ejpam-3495	57	17	y	y	PROPN
ejpam-3495	57	18	)	)	PUNCT
ejpam-3495	57	19	=	=	SYM
ejpam-3495	57	20	φ(x	φ(x	X
ejpam-3495	57	21	)	)	PUNCT
ejpam-3495	57	22	∗b	∗b	PROPN
ejpam-3495	57	23	φ(y	φ(y	NOUN
ejpam-3495	57	24	)	)	PUNCT
ejpam-3495	57	25	for	for	ADP
ejpam-3495	57	26	any	any	DET
ejpam-3495	57	27	x	x	NOUN
ejpam-3495	57	28	,	,	PUNCT
ejpam-3495	57	29	y	y	PROPN
ejpam-3495	57	30	∈	∈	PROPN
ejpam-3495	57	31	a.	a.	NOUN
ejpam-3495	57	32	the	the	DET
ejpam-3495	57	33	kernel	kernel	NOUN
ejpam-3495	57	34	of	of	ADP
ejpam-3495	57	35	f	f	PROPN
ejpam-3495	57	36	is	be	AUX
ejpam-3495	57	37	defined	define	VERB
ejpam-3495	57	38	as	as	ADP
ejpam-3495	57	39	kerf	kerf	NOUN
ejpam-3495	57	40	=	=	SYM
ejpam-3495	57	41	{	{	PUNCT
ejpam-3495	57	42	x	x	PROPN
ejpam-3495	57	43	∈	∈	PROPN
ejpam-3495	57	44	a	a	DET
ejpam-3495	57	45	:	:	PUNCT
ejpam-3495	57	46	φ(x	φ(x	NOUN
ejpam-3495	57	47	)	)	PUNCT
ejpam-3495	57	48	=	=	SYM
ejpam-3495	57	49	0b	0b	NOUN
ejpam-3495	57	50	}	}	PUNCT
ejpam-3495	57	51	.	.	PUNCT
ejpam-3495	58	1	l.d	l.d	PROPN
ejpam-3495	58	2	.	.	PROPN
ejpam-3495	58	3	naingue	naingue	PROPN
ejpam-3495	58	4	,	,	PUNCT
ejpam-3495	58	5	j.p	j.p	PROPN
ejpam-3495	58	6	.	.	PROPN
ejpam-3495	58	7	vilela	vilela	PROPN
ejpam-3495	58	8	/	/	SYM
ejpam-3495	58	9	eur	eur	PROPN
ejpam-3495	58	10	.	.	PUNCT
ejpam-3495	59	1	j.	j.	PROPN
ejpam-3495	59	2	pure	pure	PROPN
ejpam-3495	59	3	appl	appl	PROPN
ejpam-3495	59	4	.	.	PROPN
ejpam-3495	59	5	math	math	PROPN
ejpam-3495	59	6	,	,	PUNCT
ejpam-3495	59	7	12	12	NUM
ejpam-3495	59	8	(	(	PUNCT
ejpam-3495	59	9	3	3	NUM
ejpam-3495	59	10	)	)	PUNCT
ejpam-3495	59	11	(	(	PUNCT
ejpam-3495	59	12	2019	2019	NUM
ejpam-3495	59	13	)	)	PUNCT
ejpam-3495	59	14	,	,	PUNCT
ejpam-3495	59	15	1248	1248	NUM
ejpam-3495	59	16	-	-	SYM
ejpam-3495	59	17	1259	1259	NUM
ejpam-3495	59	18	1250	1250	NUM
ejpam-3495	59	19	3	3	NUM
ejpam-3495	59	20	.	.	PUNCT
ejpam-3495	59	21	basic	basic	ADJ
ejpam-3495	59	22	properties	property	NOUN
ejpam-3495	59	23	of	of	ADP
ejpam-3495	59	24	companion	companion	NOUN
ejpam-3495	59	25	b	b	NOUN
ejpam-3495	59	26	-	-	PUNCT
ejpam-3495	59	27	algebra	algebra	NOUN
ejpam-3495	59	28	definition	definition	NOUN
ejpam-3495	59	29	3.1	3.1	NUM
ejpam-3495	59	30	.	.	PUNCT
ejpam-3495	60	1	let	let	AUX
ejpam-3495	60	2	(	(	PUNCT
ejpam-3495	60	3	x	x	X
ejpam-3495	60	4	,	,	PUNCT
ejpam-3495	60	5	∗	∗	NOUN
ejpam-3495	60	6	,	,	PUNCT
ejpam-3495	60	7	0	0	NUM
ejpam-3495	60	8	)	)	PUNCT
ejpam-3495	60	9	be	be	AUX
ejpam-3495	60	10	a	a	DET
ejpam-3495	60	11	b	b	NOUN
ejpam-3495	60	12	-algebra	-algebra	NOUN
ejpam-3495	60	13	.	.	PUNCT
ejpam-3495	61	1	a	a	DET
ejpam-3495	61	2	binary	binary	PROPN
ejpam-3495	61	3	operation	operation	NOUN
ejpam-3495	61	4	�	�	PROPN
ejpam-3495	61	5	on	on	ADP
ejpam-3495	61	6	x	x	PROPN
ejpam-3495	61	7	is	be	AUX
ejpam-3495	61	8	called	call	VERB
ejpam-3495	61	9	a	a	DET
ejpam-3495	61	10	subcompanion	subcompanion	NOUN
ejpam-3495	61	11	operation	operation	NOUN
ejpam-3495	61	12	of	of	ADP
ejpam-3495	61	13	x	x	PRON
ejpam-3495	61	14	if	if	SCONJ
ejpam-3495	61	15	it	it	PRON
ejpam-3495	61	16	satisfies	satisfy	VERB
ejpam-3495	61	17	for	for	ADP
ejpam-3495	61	18	any	any	DET
ejpam-3495	61	19	x	x	NOUN
ejpam-3495	61	20	,	,	PUNCT
ejpam-3495	61	21	y	y	PROPN
ejpam-3495	61	22	∈	∈	PROPN
ejpam-3495	61	23	x	x	X
ejpam-3495	61	24	,	,	PUNCT
ejpam-3495	61	25	(	(	PUNCT
ejpam-3495	61	26	(	(	PUNCT
ejpam-3495	61	27	x	x	X
ejpam-3495	61	28	�	�	PROPN
ejpam-3495	61	29	y	y	PROPN
ejpam-3495	61	30	)	)	PUNCT
ejpam-3495	61	31	∗	∗	NOUN
ejpam-3495	61	32	x	x	NOUN
ejpam-3495	61	33	)	)	PUNCT
ejpam-3495	61	34	∗	∗	NOUN
ejpam-3495	61	35	y	y	NOUN
ejpam-3495	61	36	=	=	SYM
ejpam-3495	61	37	0	0	PUNCT
ejpam-3495	61	38	(	(	PUNCT
ejpam-3495	61	39	sc	sc	PROPN
ejpam-3495	61	40	)	)	PUNCT
ejpam-3495	61	41	a	a	DET
ejpam-3495	61	42	subcompanion	subcompanion	NOUN
ejpam-3495	61	43	operation	operation	NOUN
ejpam-3495	61	44	�	�	PROPN
ejpam-3495	61	45	is	be	AUX
ejpam-3495	61	46	a	a	DET
ejpam-3495	61	47	companion	companion	NOUN
ejpam-3495	61	48	operation	operation	NOUN
ejpam-3495	61	49	of	of	ADP
ejpam-3495	61	50	x	x	PRON
ejpam-3495	61	51	if	if	SCONJ
ejpam-3495	61	52	for	for	ADP
ejpam-3495	61	53	any	any	DET
ejpam-3495	61	54	x	x	NOUN
ejpam-3495	61	55	,	,	PUNCT
ejpam-3495	61	56	y	y	PROPN
ejpam-3495	61	57	,	,	PUNCT
ejpam-3495	61	58	z	z	PROPN
ejpam-3495	61	59	∈	∈	PROPN
ejpam-3495	61	60	x	x	X
ejpam-3495	61	61	,	,	PUNCT
ejpam-3495	61	62	(	(	PUNCT
ejpam-3495	61	63	z	z	NOUN
ejpam-3495	61	64	∗	∗	X
ejpam-3495	61	65	x	x	NOUN
ejpam-3495	61	66	)	)	PUNCT
ejpam-3495	61	67	∗	∗	NOUN
ejpam-3495	61	68	y	y	NOUN
ejpam-3495	61	69	=	=	SYM
ejpam-3495	61	70	0	0	NUM
ejpam-3495	61	71	implies	imply	VERB
ejpam-3495	61	72	z	z	NOUN
ejpam-3495	61	73	∗	∗	NOUN
ejpam-3495	61	74	(	(	PUNCT
ejpam-3495	61	75	x	x	X
ejpam-3495	61	76	�	�	PROPN
ejpam-3495	61	77	y	y	NOUN
ejpam-3495	61	78	)	)	PUNCT
ejpam-3495	61	79	=	=	SYM
ejpam-3495	62	1	0	0	X
ejpam-3495	62	2	.	.	PUNCT
ejpam-3495	63	1	(	(	PUNCT
ejpam-3495	63	2	c	c	X
ejpam-3495	63	3	)	)	PUNCT
ejpam-3495	63	4	a	a	DET
ejpam-3495	63	5	companion	companion	NOUN
ejpam-3495	63	6	b	b	NOUN
ejpam-3495	63	7	-	-	PUNCT
ejpam-3495	63	8	algebra	algebra	NOUN
ejpam-3495	63	9	(	(	PUNCT
ejpam-3495	63	10	x	x	X
ejpam-3495	63	11	,	,	PUNCT
ejpam-3495	63	12	∗	∗	NOUN
ejpam-3495	63	13	,	,	PUNCT
ejpam-3495	63	14	�	�	PROPN
ejpam-3495	63	15	,	,	PUNCT
ejpam-3495	63	16	0	0	NUM
ejpam-3495	63	17	)	)	PUNCT
ejpam-3495	63	18	is	be	AUX
ejpam-3495	63	19	a	a	DET
ejpam-3495	63	20	b	b	NOUN
ejpam-3495	63	21	-algebra	-algebra	NOUN
ejpam-3495	63	22	(	(	PUNCT
ejpam-3495	63	23	x	x	X
ejpam-3495	63	24	,	,	PUNCT
ejpam-3495	63	25	∗	∗	NOUN
ejpam-3495	63	26	,	,	PUNCT
ejpam-3495	63	27	0	0	NUM
ejpam-3495	63	28	)	)	PUNCT
ejpam-3495	63	29	with	with	ADP
ejpam-3495	63	30	companion	companion	PROPN
ejpam-3495	63	31	operation	operation	PROPN
ejpam-3495	63	32	�	�	PROPN
ejpam-3495	63	33	.	.	PROPN
ejpam-3495	63	34	example	example	NOUN
ejpam-3495	63	35	3.2	3.2	NUM
ejpam-3495	63	36	.	.	PUNCT
ejpam-3495	64	1	consider	consider	VERB
ejpam-3495	64	2	the	the	DET
ejpam-3495	64	3	b	b	PROPN
ejpam-3495	64	4	-algebra	-algebra	PROPN
ejpam-3495	64	5	(	(	PUNCT
ejpam-3495	64	6	x	x	X
ejpam-3495	64	7	,	,	PUNCT
ejpam-3495	64	8	∗	∗	NOUN
ejpam-3495	64	9	,	,	PUNCT
ejpam-3495	64	10	0	0	NUM
ejpam-3495	64	11	)	)	PUNCT
ejpam-3495	64	12	with	with	ADP
ejpam-3495	64	13	∗	∗	NOUN
ejpam-3495	64	14	defined	define	VERB
ejpam-3495	64	15	below	below	ADP
ejpam-3495	64	16	[	[	X
ejpam-3495	64	17	11	11	NUM
ejpam-3495	64	18	]	]	PUNCT
ejpam-3495	64	19	.	.	PUNCT
ejpam-3495	65	1	define	define	VERB
ejpam-3495	65	2	an	an	DET
ejpam-3495	65	3	operation	operation	NOUN
ejpam-3495	65	4	�	�	NOUN
ejpam-3495	65	5	on	on	ADP
ejpam-3495	65	6	x	x	PUNCT
ejpam-3495	65	7	as	as	SCONJ
ejpam-3495	65	8	follows	follow	VERB
ejpam-3495	65	9	:	:	PUNCT
ejpam-3495	65	10	∗	∗	NOUN
ejpam-3495	65	11	0	0	NUM
ejpam-3495	66	1	1	1	NUM
ejpam-3495	66	2	2	2	NUM
ejpam-3495	66	3	3	3	NUM
ejpam-3495	66	4	4	4	NUM
ejpam-3495	66	5	5	5	NUM
ejpam-3495	66	6	0	0	NUM
ejpam-3495	66	7	0	0	NUM
ejpam-3495	66	8	2	2	NUM
ejpam-3495	66	9	1	1	NUM
ejpam-3495	66	10	3	3	NUM
ejpam-3495	66	11	4	4	NUM
ejpam-3495	66	12	5	5	NUM
ejpam-3495	66	13	1	1	NUM
ejpam-3495	66	14	1	1	NUM
ejpam-3495	66	15	0	0	NUM
ejpam-3495	66	16	2	2	NUM
ejpam-3495	66	17	4	4	NUM
ejpam-3495	66	18	5	5	NUM
ejpam-3495	66	19	3	3	NUM
ejpam-3495	66	20	2	2	NUM
ejpam-3495	66	21	2	2	NUM
ejpam-3495	66	22	1	1	NUM
ejpam-3495	66	23	0	0	NUM
ejpam-3495	66	24	5	5	NUM
ejpam-3495	66	25	3	3	NUM
ejpam-3495	66	26	4	4	NUM
ejpam-3495	66	27	3	3	NUM
ejpam-3495	66	28	3	3	NUM
ejpam-3495	66	29	4	4	NUM
ejpam-3495	66	30	5	5	NUM
ejpam-3495	66	31	0	0	NUM
ejpam-3495	66	32	2	2	NUM
ejpam-3495	66	33	1	1	NUM
ejpam-3495	66	34	4	4	NUM
ejpam-3495	66	35	4	4	NUM
ejpam-3495	66	36	5	5	NUM
ejpam-3495	66	37	3	3	NUM
ejpam-3495	66	38	1	1	NUM
ejpam-3495	66	39	0	0	NUM
ejpam-3495	66	40	2	2	NUM
ejpam-3495	66	41	5	5	NUM
ejpam-3495	66	42	5	5	NUM
ejpam-3495	66	43	3	3	NUM
ejpam-3495	66	44	4	4	NUM
ejpam-3495	66	45	2	2	NUM
ejpam-3495	66	46	1	1	NUM
ejpam-3495	66	47	0	0	NUM
ejpam-3495	66	48	�	�	NOUN
ejpam-3495	66	49	0	0	NUM
ejpam-3495	66	50	1	1	NUM
ejpam-3495	66	51	2	2	NUM
ejpam-3495	66	52	3	3	NUM
ejpam-3495	66	53	4	4	NUM
ejpam-3495	66	54	5	5	NUM
ejpam-3495	66	55	0	0	NUM
ejpam-3495	66	56	0	0	NUM
ejpam-3495	66	57	1	1	NUM
ejpam-3495	66	58	2	2	NUM
ejpam-3495	66	59	3	3	NUM
ejpam-3495	66	60	4	4	NUM
ejpam-3495	66	61	5	5	NUM
ejpam-3495	66	62	1	1	NUM
ejpam-3495	66	63	1	1	NUM
ejpam-3495	66	64	2	2	NUM
ejpam-3495	66	65	0	0	NUM
ejpam-3495	66	66	5	5	NUM
ejpam-3495	66	67	3	3	NUM
ejpam-3495	66	68	4	4	NUM
ejpam-3495	66	69	2	2	NUM
ejpam-3495	66	70	2	2	NUM
ejpam-3495	66	71	0	0	NUM
ejpam-3495	66	72	1	1	NUM
ejpam-3495	66	73	4	4	NUM
ejpam-3495	66	74	5	5	NUM
ejpam-3495	66	75	3	3	NUM
ejpam-3495	66	76	3	3	NUM
ejpam-3495	66	77	3	3	NUM
ejpam-3495	66	78	4	4	NUM
ejpam-3495	66	79	5	5	NUM
ejpam-3495	66	80	0	0	NUM
ejpam-3495	66	81	1	1	NUM
ejpam-3495	66	82	2	2	NUM
ejpam-3495	66	83	4	4	NUM
ejpam-3495	66	84	4	4	NUM
ejpam-3495	66	85	5	5	NUM
ejpam-3495	66	86	3	3	NUM
ejpam-3495	66	87	2	2	NUM
ejpam-3495	66	88	0	0	NUM
ejpam-3495	66	89	1	1	NUM
ejpam-3495	66	90	5	5	NUM
ejpam-3495	66	91	5	5	NUM
ejpam-3495	66	92	3	3	NUM
ejpam-3495	66	93	4	4	NUM
ejpam-3495	66	94	1	1	NUM
ejpam-3495	66	95	2	2	NUM
ejpam-3495	66	96	0	0	NUM
ejpam-3495	66	97	by	by	ADP
ejpam-3495	66	98	routine	routine	ADJ
ejpam-3495	66	99	calculations	calculation	NOUN
ejpam-3495	66	100	,	,	PUNCT
ejpam-3495	66	101	(	(	PUNCT
ejpam-3495	66	102	x	x	X
ejpam-3495	66	103	,	,	PUNCT
ejpam-3495	66	104	∗	∗	NOUN
ejpam-3495	66	105	,	,	PUNCT
ejpam-3495	66	106	�	�	PROPN
ejpam-3495	66	107	,	,	PUNCT
ejpam-3495	66	108	0	0	NUM
ejpam-3495	66	109	)	)	PUNCT
ejpam-3495	66	110	is	be	AUX
ejpam-3495	66	111	a	a	DET
ejpam-3495	66	112	companion	companion	NOUN
ejpam-3495	66	113	b	b	PROPN
ejpam-3495	66	114	-algebra	-algebra	PROPN
ejpam-3495	66	115	.	.	PUNCT
ejpam-3495	66	116	example	example	NOUN
ejpam-3495	66	117	3.3	3.3	NUM
ejpam-3495	66	118	.	.	PUNCT
ejpam-3495	67	1	consider	consider	VERB
ejpam-3495	67	2	the	the	DET
ejpam-3495	67	3	b	b	NOUN
ejpam-3495	67	4	-algebra	-algebra	NOUN
ejpam-3495	67	5	x	x	X
ejpam-3495	67	6	=	=	SYM
ejpam-3495	67	7	(	(	PUNCT
ejpam-3495	67	8	z,−	z,−	PROPN
ejpam-3495	67	9	,	,	PUNCT
ejpam-3495	67	10	0	0	NUM
ejpam-3495	67	11	)	)	PUNCT
ejpam-3495	67	12	.	.	PUNCT
ejpam-3495	68	1	then	then	ADV
ejpam-3495	68	2	for	for	SCONJ
ejpam-3495	68	3	all	all	DET
ejpam-3495	68	4	x	x	NOUN
ejpam-3495	68	5	,	,	PUNCT
ejpam-3495	68	6	y	y	PROPN
ejpam-3495	68	7	,	,	PUNCT
ejpam-3495	68	8	z	z	PROPN
ejpam-3495	68	9	∈	∈	PROPN
ejpam-3495	68	10	z	z	PROPN
ejpam-3495	68	11	,	,	PUNCT
ejpam-3495	68	12	(	(	PUNCT
ejpam-3495	68	13	(	(	PUNCT
ejpam-3495	68	14	x+	x+	X
ejpam-3495	68	15	y)−	y)−	PROPN
ejpam-3495	68	16	x)−	x)−	PROPN
ejpam-3495	68	17	y	y	PROPN
ejpam-3495	68	18	=	=	PUNCT
ejpam-3495	68	19	0	0	PUNCT
ejpam-3495	69	1	and	and	CCONJ
ejpam-3495	69	2	if	if	SCONJ
ejpam-3495	69	3	(	(	PUNCT
ejpam-3495	69	4	z	z	NOUN
ejpam-3495	69	5	−	−	PROPN
ejpam-3495	69	6	x)−	x)−	PROPN
ejpam-3495	69	7	y	y	PROPN
ejpam-3495	69	8	=	=	SYM
ejpam-3495	69	9	0	0	PROPN
ejpam-3495	69	10	,	,	PUNCT
ejpam-3495	69	11	then	then	ADV
ejpam-3495	69	12	z	z	PROPN
ejpam-3495	69	13	−	−	PROPN
ejpam-3495	69	14	(	(	PUNCT
ejpam-3495	69	15	x+	x+	PROPN
ejpam-3495	69	16	y	y	NOUN
ejpam-3495	69	17	)	)	PUNCT
ejpam-3495	69	18	=	=	PUNCT
ejpam-3495	70	1	(	(	PUNCT
ejpam-3495	70	2	z	z	NOUN
ejpam-3495	70	3	−	−	PROPN
ejpam-3495	70	4	x)−	x)−	PROPN
ejpam-3495	70	5	y	y	PROPN
ejpam-3495	70	6	=	=	PUNCT
ejpam-3495	70	7	0	0	PROPN
ejpam-3495	70	8	.	.	PUNCT
ejpam-3495	71	1	hence	hence	ADV
ejpam-3495	71	2	,	,	PUNCT
ejpam-3495	71	3	the	the	DET
ejpam-3495	71	4	binary	binary	PROPN
ejpam-3495	71	5	operation	operation	NOUN
ejpam-3495	71	6	“	"	PUNCT
ejpam-3495	71	7	+	+	PROPN
ejpam-3495	71	8	”	"	PUNCT
ejpam-3495	71	9	is	be	AUX
ejpam-3495	71	10	a	a	DET
ejpam-3495	71	11	companion	companion	NOUN
ejpam-3495	71	12	operation	operation	NOUN
ejpam-3495	71	13	of	of	ADP
ejpam-3495	71	14	z.	z.	PROPN
ejpam-3495	71	15	therefore	therefore	ADV
ejpam-3495	71	16	,	,	PUNCT
ejpam-3495	71	17	(	(	PUNCT
ejpam-3495	71	18	z,−,+	z,−,+	NOUN
ejpam-3495	71	19	,	,	PUNCT
ejpam-3495	71	20	0	0	NUM
ejpam-3495	71	21	)	)	PUNCT
ejpam-3495	71	22	is	be	AUX
ejpam-3495	71	23	a	a	DET
ejpam-3495	71	24	companion	companion	NOUN
ejpam-3495	71	25	b	b	PROPN
ejpam-3495	71	26	-algebra	-algebra	PROPN
ejpam-3495	71	27	.	.	PUNCT
ejpam-3495	72	1	theorem	theorem	VERB
ejpam-3495	72	2	3.4	3.4	NUM
ejpam-3495	72	3	.	.	PUNCT
ejpam-3495	73	1	let	let	AUX
ejpam-3495	73	2	(	(	PUNCT
ejpam-3495	73	3	x	x	X
ejpam-3495	73	4	,	,	PUNCT
ejpam-3495	73	5	∗	∗	NOUN
ejpam-3495	73	6	,	,	PUNCT
ejpam-3495	73	7	0	0	NUM
ejpam-3495	73	8	)	)	PUNCT
ejpam-3495	73	9	be	be	AUX
ejpam-3495	73	10	a	a	DET
ejpam-3495	73	11	b	b	NOUN
ejpam-3495	73	12	-	-	PUNCT
ejpam-3495	73	13	algebra	algebra	NOUN
ejpam-3495	73	14	.	.	PUNCT
ejpam-3495	74	1	if	if	SCONJ
ejpam-3495	74	2	x	x	PRON
ejpam-3495	74	3	has	have	VERB
ejpam-3495	74	4	a	a	DET
ejpam-3495	74	5	companion	companion	NOUN
ejpam-3495	74	6	operation	operation	NOUN
ejpam-3495	74	7	�	�	PROPN
ejpam-3495	74	8	,	,	PUNCT
ejpam-3495	74	9	then	then	ADV
ejpam-3495	74	10	it	it	PRON
ejpam-3495	74	11	is	be	AUX
ejpam-3495	74	12	unique	unique	ADJ
ejpam-3495	74	13	.	.	PUNCT
ejpam-3495	75	1	proof	proof	NOUN
ejpam-3495	75	2	:	:	PUNCT
ejpam-3495	75	3	assume	assume	VERB
ejpam-3495	75	4	that	that	SCONJ
ejpam-3495	75	5	the	the	DET
ejpam-3495	75	6	binary	binary	PROPN
ejpam-3495	75	7	operations	operation	NOUN
ejpam-3495	75	8	�	�	PROPN
ejpam-3495	75	9	1	1	NUM
ejpam-3495	75	10	and	and	CCONJ
ejpam-3495	75	11	�	�	NOUN
ejpam-3495	75	12	2	2	NUM
ejpam-3495	75	13	are	be	AUX
ejpam-3495	75	14	companion	companion	NOUN
ejpam-3495	75	15	operations	operation	NOUN
ejpam-3495	75	16	on	on	ADP
ejpam-3495	75	17	x.	x.	NOUN
ejpam-3495	75	18	then	then	ADV
ejpam-3495	75	19	by	by	ADP
ejpam-3495	75	20	(	(	PUNCT
ejpam-3495	75	21	sc	sc	PROPN
ejpam-3495	75	22	)	)	PUNCT
ejpam-3495	75	23	applied	apply	VERB
ejpam-3495	75	24	on	on	ADP
ejpam-3495	75	25	�	�	PROPN
ejpam-3495	75	26	1	1	NUM
ejpam-3495	75	27	,	,	PUNCT
ejpam-3495	75	28	for	for	ADP
ejpam-3495	75	29	any	any	DET
ejpam-3495	75	30	x	x	NOUN
ejpam-3495	75	31	,	,	PUNCT
ejpam-3495	75	32	y	y	PROPN
ejpam-3495	75	33	∈	∈	PROPN
ejpam-3495	75	34	x	x	X
ejpam-3495	75	35	,	,	PUNCT
ejpam-3495	75	36	(	(	PUNCT
ejpam-3495	75	37	(	(	PUNCT
ejpam-3495	75	38	x	x	X
ejpam-3495	75	39	�	�	PROPN
ejpam-3495	75	40	1	1	NUM
ejpam-3495	75	41	y	y	NOUN
ejpam-3495	75	42	)	)	PUNCT
ejpam-3495	75	43	∗	∗	NOUN
ejpam-3495	75	44	x	x	NOUN
ejpam-3495	75	45	)	)	PUNCT
ejpam-3495	75	46	∗	∗	NOUN
ejpam-3495	75	47	y	y	NOUN
ejpam-3495	76	1	=	=	SYM
ejpam-3495	76	2	0	0	X
ejpam-3495	76	3	.	.	PUNCT
ejpam-3495	77	1	by	by	ADP
ejpam-3495	77	2	(	(	PUNCT
ejpam-3495	77	3	c	c	NOUN
ejpam-3495	77	4	)	)	PUNCT
ejpam-3495	77	5	applied	apply	VERB
ejpam-3495	77	6	on	on	ADP
ejpam-3495	77	7	�	�	PROPN
ejpam-3495	77	8	2	2	NUM
ejpam-3495	77	9	,	,	PUNCT
ejpam-3495	77	10	(	(	PUNCT
ejpam-3495	77	11	x	x	X
ejpam-3495	77	12	�	�	PROPN
ejpam-3495	77	13	1	1	NUM
ejpam-3495	77	14	y	y	NOUN
ejpam-3495	77	15	)	)	PUNCT
ejpam-3495	77	16	∗	∗	NOUN
ejpam-3495	77	17	(	(	PUNCT
ejpam-3495	77	18	x	x	X
ejpam-3495	77	19	�	�	PROPN
ejpam-3495	77	20	2	2	NUM
ejpam-3495	77	21	y	y	NOUN
ejpam-3495	77	22	)	)	PUNCT
ejpam-3495	77	23	=	=	SYM
ejpam-3495	78	1	0	0	X
ejpam-3495	78	2	.	.	PUNCT
ejpam-3495	78	3	then	then	ADV
ejpam-3495	78	4	by	by	ADP
ejpam-3495	78	5	theorem	theorem	NOUN
ejpam-3495	78	6	2.3(d	2.3(d	NUM
ejpam-3495	78	7	)	)	PUNCT
ejpam-3495	78	8	,	,	PUNCT
ejpam-3495	78	9	x	x	X
ejpam-3495	78	10	�	�	PROPN
ejpam-3495	78	11	1	1	NUM
ejpam-3495	78	12	y	y	NOUN
ejpam-3495	78	13	=	=	SYM
ejpam-3495	78	14	x	x	SYM
ejpam-3495	78	15	�	�	PROPN
ejpam-3495	78	16	2	2	NUM
ejpam-3495	78	17	y.	y.	NOUN
ejpam-3495	78	18	thus	thus	ADV
ejpam-3495	78	19	,	,	PUNCT
ejpam-3495	78	20	�	�	PROPN
ejpam-3495	78	21	1	1	NUM
ejpam-3495	78	22	=	=	SYM
ejpam-3495	78	23	�	�	PROPN
ejpam-3495	78	24	2	2	NUM
ejpam-3495	78	25	and	and	CCONJ
ejpam-3495	78	26	the	the	DET
ejpam-3495	78	27	companion	companion	NOUN
ejpam-3495	78	28	operation	operation	NOUN
ejpam-3495	78	29	is	be	AUX
ejpam-3495	78	30	unique	unique	ADJ
ejpam-3495	78	31	.	.	PUNCT
ejpam-3495	79	1	�	�	PROPN
ejpam-3495	79	2	theorem	theorem	VERB
ejpam-3495	79	3	3.5	3.5	NUM
ejpam-3495	79	4	.	.	PUNCT
ejpam-3495	80	1	let	let	VERB
ejpam-3495	80	2	(	(	PUNCT
ejpam-3495	80	3	x	x	X
ejpam-3495	80	4	,	,	PUNCT
ejpam-3495	80	5	∗	∗	NOUN
ejpam-3495	80	6	,	,	PUNCT
ejpam-3495	80	7	�	�	PROPN
ejpam-3495	80	8	,	,	PUNCT
ejpam-3495	80	9	0	0	NUM
ejpam-3495	80	10	)	)	PUNCT
ejpam-3495	80	11	be	be	AUX
ejpam-3495	80	12	a	a	DET
ejpam-3495	80	13	companion	companion	NOUN
ejpam-3495	80	14	b	b	NOUN
ejpam-3495	80	15	-	-	PUNCT
ejpam-3495	80	16	algebra	algebra	NOUN
ejpam-3495	80	17	.	.	PUNCT
ejpam-3495	81	1	let	let	VERB
ejpam-3495	81	2	?	?	PUNCT
ejpam-3495	82	1	be	be	AUX
ejpam-3495	82	2	a	a	DET
ejpam-3495	82	3	binary	binary	ADJ
ejpam-3495	82	4	operation	operation	NOUN
ejpam-3495	82	5	on	on	ADP
ejpam-3495	82	6	x	x	SYM
ejpam-3495	82	7	such	such	ADJ
ejpam-3495	82	8	that	that	PRON
ejpam-3495	82	9	for	for	ADP
ejpam-3495	82	10	all	all	DET
ejpam-3495	82	11	x	x	NOUN
ejpam-3495	82	12	,	,	PUNCT
ejpam-3495	82	13	y	y	PROPN
ejpam-3495	82	14	,	,	PUNCT
ejpam-3495	82	15	z	z	PROPN
ejpam-3495	82	16	∈	∈	PROPN
ejpam-3495	82	17	x	x	X
ejpam-3495	82	18	,	,	PUNCT
ejpam-3495	82	19	(	(	PUNCT
ejpam-3495	82	20	x	x	X
ejpam-3495	82	21	∗	∗	PROPN
ejpam-3495	82	22	y	y	NOUN
ejpam-3495	82	23	)	)	PUNCT
ejpam-3495	82	24	∗	∗	NOUN
ejpam-3495	82	25	z	z	NOUN
ejpam-3495	83	1	=	=	SYM
ejpam-3495	83	2	x	x	X
ejpam-3495	83	3	∗	∗	NOUN
ejpam-3495	83	4	(	(	PUNCT
ejpam-3495	83	5	y	y	PROPN
ejpam-3495	83	6	?	?	PUNCT
ejpam-3495	84	1	z	z	X
ejpam-3495	84	2	)	)	PUNCT
ejpam-3495	84	3	.	.	PUNCT
ejpam-3495	85	1	then	then	ADV
ejpam-3495	85	2	(	(	PUNCT
ejpam-3495	85	3	x	x	X
ejpam-3495	85	4	,	,	PUNCT
ejpam-3495	85	5	∗	∗	NOUN
ejpam-3495	85	6	,	,	PUNCT
ejpam-3495	85	7	?	?	PUNCT
ejpam-3495	85	8	,	,	PUNCT
ejpam-3495	85	9	0	0	X
ejpam-3495	85	10	)	)	PUNCT
ejpam-3495	85	11	is	be	AUX
ejpam-3495	85	12	a	a	DET
ejpam-3495	85	13	companion	companion	NOUN
ejpam-3495	85	14	b	b	NOUN
ejpam-3495	85	15	-	-	PUNCT
ejpam-3495	85	16	algebra	algebra	NOUN
ejpam-3495	85	17	and	and	CCONJ
ejpam-3495	85	18	?	?	PUNCT
ejpam-3495	86	1	is	be	AUX
ejpam-3495	86	2	exactly	exactly	ADV
ejpam-3495	86	3	the	the	DET
ejpam-3495	86	4	operation	operation	NOUN
ejpam-3495	86	5	�	�	PROPN
ejpam-3495	86	6	.	.	PUNCT
ejpam-3495	87	1	proof	proof	NOUN
ejpam-3495	87	2	:	:	PUNCT
ejpam-3495	87	3	suppose	suppose	VERB
ejpam-3495	87	4	x	x	PRON
ejpam-3495	87	5	,	,	PUNCT
ejpam-3495	87	6	y	y	PROPN
ejpam-3495	87	7	,	,	PUNCT
ejpam-3495	87	8	z	z	NOUN
ejpam-3495	87	9	∈	∈	PROPN
ejpam-3495	87	10	x.	x.	NOUN
ejpam-3495	87	11	by	by	ADP
ejpam-3495	87	12	hypothesis	hypothesis	NOUN
ejpam-3495	87	13	and	and	CCONJ
ejpam-3495	87	14	definition	definition	NOUN
ejpam-3495	87	15	2.1(i	2.1(i	NUM
ejpam-3495	87	16	)	)	PUNCT
ejpam-3495	87	17	,	,	PUNCT
ejpam-3495	87	18	(	(	PUNCT
ejpam-3495	87	19	(	(	PUNCT
ejpam-3495	87	20	x	x	X
ejpam-3495	87	21	?	?	PUNCT
ejpam-3495	88	1	y	y	X
ejpam-3495	88	2	)	)	PUNCT
ejpam-3495	88	3	∗	∗	NOUN
ejpam-3495	88	4	x	x	NOUN
ejpam-3495	88	5	)	)	PUNCT
ejpam-3495	88	6	∗	∗	NOUN
ejpam-3495	88	7	y	y	NOUN
ejpam-3495	88	8	=	=	PUNCT
ejpam-3495	88	9	(	(	PUNCT
ejpam-3495	88	10	x	x	X
ejpam-3495	88	11	?	?	PUNCT
ejpam-3495	88	12	y	y	X
ejpam-3495	88	13	)	)	PUNCT
ejpam-3495	88	14	∗	∗	NOUN
ejpam-3495	88	15	(	(	PUNCT
ejpam-3495	88	16	x	x	X
ejpam-3495	88	17	?	?	PUNCT
ejpam-3495	89	1	y	y	X
ejpam-3495	89	2	)	)	PUNCT
ejpam-3495	90	1	=	=	SYM
ejpam-3495	90	2	0	0	X
ejpam-3495	90	3	.	.	PUNCT
ejpam-3495	91	1	hence	hence	ADV
ejpam-3495	91	2	,	,	PUNCT
ejpam-3495	91	3	?	?	PUNCT
ejpam-3495	91	4	is	be	AUX
ejpam-3495	91	5	a	a	DET
ejpam-3495	91	6	subcompanion	subcompanion	NOUN
ejpam-3495	91	7	operation	operation	NOUN
ejpam-3495	91	8	.	.	PUNCT
ejpam-3495	92	1	now	now	ADV
ejpam-3495	92	2	,	,	PUNCT
ejpam-3495	92	3	let	let	VERB
ejpam-3495	92	4	(	(	PUNCT
ejpam-3495	92	5	z	z	NOUN
ejpam-3495	92	6	∗	∗	NOUN
ejpam-3495	92	7	x	x	NOUN
ejpam-3495	92	8	)	)	PUNCT
ejpam-3495	92	9	∗	∗	NOUN
ejpam-3495	92	10	y	y	NOUN
ejpam-3495	92	11	=	=	SYM
ejpam-3495	93	1	0	0	PROPN
ejpam-3495	93	2	.	.	PUNCT
ejpam-3495	94	1	then	then	ADV
ejpam-3495	94	2	by	by	ADP
ejpam-3495	94	3	hypothesis	hypothesis	NOUN
ejpam-3495	94	4	,	,	PUNCT
ejpam-3495	94	5	z	z	NOUN
ejpam-3495	94	6	∗	∗	NOUN
ejpam-3495	94	7	(	(	PUNCT
ejpam-3495	94	8	x	x	X
ejpam-3495	94	9	?	?	PUNCT
ejpam-3495	95	1	y	y	X
ejpam-3495	95	2	)	)	PUNCT
ejpam-3495	96	1	=	=	PRON
ejpam-3495	96	2	(	(	PUNCT
ejpam-3495	96	3	z	z	NOUN
ejpam-3495	96	4	∗	∗	X
ejpam-3495	96	5	x	x	NOUN
ejpam-3495	96	6	)	)	PUNCT
ejpam-3495	96	7	∗	∗	NOUN
ejpam-3495	96	8	y	y	NOUN
ejpam-3495	96	9	=	=	SYM
ejpam-3495	96	10	0	0	PROPN
ejpam-3495	96	11	.	.	PUNCT
ejpam-3495	97	1	thus	thus	ADV
ejpam-3495	97	2	,	,	PUNCT
ejpam-3495	97	3	?	?	PUNCT
ejpam-3495	97	4	is	be	AUX
ejpam-3495	97	5	a	a	DET
ejpam-3495	97	6	companion	companion	NOUN
ejpam-3495	97	7	operation	operation	NOUN
ejpam-3495	97	8	,	,	PUNCT
ejpam-3495	97	9	which	which	PRON
ejpam-3495	97	10	is	be	AUX
ejpam-3495	97	11	unique	unique	ADJ
ejpam-3495	97	12	by	by	ADP
ejpam-3495	97	13	theorem	theorem	ADJ
ejpam-3495	97	14	3.4	3.4	NUM
ejpam-3495	97	15	.	.	PUNCT
ejpam-3495	98	1	therefore	therefore	ADV
ejpam-3495	98	2	,	,	PUNCT
ejpam-3495	98	3	(	(	PUNCT
ejpam-3495	98	4	x	x	X
ejpam-3495	98	5	,	,	PUNCT
ejpam-3495	98	6	∗	∗	NOUN
ejpam-3495	98	7	,	,	PUNCT
ejpam-3495	98	8	?	?	PUNCT
ejpam-3495	98	9	,	,	PUNCT
ejpam-3495	98	10	0	0	X
ejpam-3495	98	11	)	)	PUNCT
ejpam-3495	98	12	is	be	AUX
ejpam-3495	98	13	a	a	DET
ejpam-3495	98	14	companion	companion	NOUN
ejpam-3495	98	15	b	b	NOUN
ejpam-3495	98	16	-	-	PUNCT
ejpam-3495	98	17	algebra	algebra	NOUN
ejpam-3495	98	18	.	.	PUNCT
ejpam-3495	99	1	�	�	PROPN
ejpam-3495	99	2	example	example	NOUN
ejpam-3495	99	3	3.6	3.6	NUM
ejpam-3495	99	4	.	.	PUNCT
ejpam-3495	100	1	let	let	VERB
ejpam-3495	100	2	x	x	PUNCT
ejpam-3495	100	3	=	=	PUNCT
ejpam-3495	100	4	{	{	PUNCT
ejpam-3495	100	5	0	0	NUM
ejpam-3495	100	6	,	,	PUNCT
ejpam-3495	100	7	1	1	NUM
ejpam-3495	100	8	,	,	PUNCT
ejpam-3495	100	9	2	2	NUM
ejpam-3495	100	10	,	,	PUNCT
ejpam-3495	100	11	3	3	NUM
ejpam-3495	100	12	}	}	PUNCT
ejpam-3495	100	13	be	be	AUX
ejpam-3495	100	14	a	a	DET
ejpam-3495	100	15	set	set	NOUN
ejpam-3495	100	16	with	with	ADP
ejpam-3495	100	17	the	the	DET
ejpam-3495	100	18	following	follow	VERB
ejpam-3495	100	19	table	table	NOUN
ejpam-3495	100	20	of	of	ADP
ejpam-3495	100	21	operations	operation	NOUN
ejpam-3495	100	22	:	:	PUNCT
ejpam-3495	100	23	l.d	l.d	PROPN
ejpam-3495	100	24	.	.	PROPN
ejpam-3495	100	25	naingue	naingue	PROPN
ejpam-3495	100	26	,	,	PUNCT
ejpam-3495	100	27	j.p	j.p	PROPN
ejpam-3495	100	28	.	.	PROPN
ejpam-3495	100	29	vilela	vilela	PROPN
ejpam-3495	100	30	/	/	SYM
ejpam-3495	100	31	eur	eur	PROPN
ejpam-3495	100	32	.	.	PUNCT
ejpam-3495	101	1	j.	j.	PROPN
ejpam-3495	101	2	pure	pure	PROPN
ejpam-3495	101	3	appl	appl	PROPN
ejpam-3495	101	4	.	.	PROPN
ejpam-3495	101	5	math	math	PROPN
ejpam-3495	101	6	,	,	PUNCT
ejpam-3495	101	7	12	12	NUM
ejpam-3495	101	8	(	(	PUNCT
ejpam-3495	101	9	3	3	NUM
ejpam-3495	101	10	)	)	PUNCT
ejpam-3495	101	11	(	(	PUNCT
ejpam-3495	101	12	2019	2019	NUM
ejpam-3495	101	13	)	)	PUNCT
ejpam-3495	101	14	,	,	PUNCT
ejpam-3495	101	15	1248	1248	NUM
ejpam-3495	101	16	-	-	SYM
ejpam-3495	101	17	1259	1259	NUM
ejpam-3495	101	18	1251	1251	NUM
ejpam-3495	101	19	∗	∗	NOUN
ejpam-3495	101	20	0	0	NUM
ejpam-3495	101	21	1	1	NUM
ejpam-3495	101	22	2	2	NUM
ejpam-3495	101	23	3	3	NUM
ejpam-3495	101	24	0	0	NUM
ejpam-3495	101	25	0	0	NUM
ejpam-3495	101	26	3	3	NUM
ejpam-3495	101	27	2	2	NUM
ejpam-3495	101	28	1	1	NUM
ejpam-3495	101	29	1	1	NUM
ejpam-3495	101	30	1	1	NUM
ejpam-3495	101	31	0	0	NUM
ejpam-3495	101	32	3	3	NUM
ejpam-3495	101	33	2	2	NUM
ejpam-3495	101	34	2	2	NUM
ejpam-3495	101	35	2	2	NUM
ejpam-3495	101	36	1	1	NUM
ejpam-3495	101	37	0	0	NUM
ejpam-3495	101	38	3	3	NUM
ejpam-3495	101	39	3	3	NUM
ejpam-3495	101	40	3	3	NUM
ejpam-3495	101	41	2	2	NUM
ejpam-3495	101	42	1	1	NUM
ejpam-3495	101	43	0	0	NUM
ejpam-3495	101	44	�	�	NOUN
ejpam-3495	101	45	0	0	NUM
ejpam-3495	101	46	1	1	NUM
ejpam-3495	101	47	2	2	NUM
ejpam-3495	101	48	3	3	NUM
ejpam-3495	101	49	0	0	NUM
ejpam-3495	101	50	0	0	NUM
ejpam-3495	101	51	1	1	NUM
ejpam-3495	101	52	2	2	NUM
ejpam-3495	101	53	3	3	NUM
ejpam-3495	101	54	1	1	NUM
ejpam-3495	101	55	1	1	NUM
ejpam-3495	101	56	2	2	NUM
ejpam-3495	101	57	3	3	NUM
ejpam-3495	101	58	0	0	NUM
ejpam-3495	101	59	2	2	NUM
ejpam-3495	101	60	2	2	NUM
ejpam-3495	101	61	3	3	NUM
ejpam-3495	101	62	0	0	NUM
ejpam-3495	101	63	1	1	NUM
ejpam-3495	101	64	3	3	NUM
ejpam-3495	101	65	3	3	NUM
ejpam-3495	101	66	0	0	NUM
ejpam-3495	101	67	1	1	NUM
ejpam-3495	101	68	2	2	NUM
ejpam-3495	101	69	then	then	ADV
ejpam-3495	101	70	(	(	PUNCT
ejpam-3495	101	71	x	x	X
ejpam-3495	101	72	,	,	PUNCT
ejpam-3495	101	73	∗	∗	NOUN
ejpam-3495	101	74	,	,	PUNCT
ejpam-3495	101	75	0	0	NUM
ejpam-3495	101	76	)	)	PUNCT
ejpam-3495	101	77	is	be	AUX
ejpam-3495	101	78	a	a	DET
ejpam-3495	101	79	b	b	NOUN
ejpam-3495	101	80	-algebra	-algebra	NOUN
ejpam-3495	101	81	[	[	X
ejpam-3495	101	82	2	2	NUM
ejpam-3495	101	83	]	]	PUNCT
ejpam-3495	101	84	and	and	CCONJ
ejpam-3495	101	85	by	by	ADP
ejpam-3495	101	86	routine	routine	ADJ
ejpam-3495	101	87	calculations	calculation	NOUN
ejpam-3495	101	88	,	,	PUNCT
ejpam-3495	101	89	(	(	PUNCT
ejpam-3495	101	90	x	x	X
ejpam-3495	101	91	,	,	PUNCT
ejpam-3495	101	92	∗	∗	NOUN
ejpam-3495	101	93	,	,	PUNCT
ejpam-3495	101	94	�	�	PROPN
ejpam-3495	101	95	,	,	PUNCT
ejpam-3495	101	96	0	0	NUM
ejpam-3495	101	97	)	)	PUNCT
ejpam-3495	101	98	is	be	AUX
ejpam-3495	101	99	a	a	DET
ejpam-3495	101	100	companion	companion	NOUN
ejpam-3495	101	101	b	b	PROPN
ejpam-3495	101	102	-algebra	-algebra	NOUN
ejpam-3495	101	103	.	.	PUNCT
ejpam-3495	102	1	if	if	SCONJ
ejpam-3495	102	2	x	x	SYM
ejpam-3495	102	3	=	=	SYM
ejpam-3495	102	4	1	1	NUM
ejpam-3495	102	5	and	and	CCONJ
ejpam-3495	102	6	y	y	NOUN
ejpam-3495	102	7	=	=	SYM
ejpam-3495	102	8	3	3	NUM
ejpam-3495	102	9	,	,	PUNCT
ejpam-3495	102	10	then	then	ADV
ejpam-3495	102	11	(	(	PUNCT
ejpam-3495	102	12	(	(	PUNCT
ejpam-3495	102	13	1∗3)∗1)∗3	1∗3)∗1)∗3	NUM
ejpam-3495	102	14	=	=	SYM
ejpam-3495	102	15	2	2	NUM
ejpam-3495	102	16	6=	6=	NUM
ejpam-3495	102	17	0	0	NUM
ejpam-3495	102	18	.	.	PUNCT
ejpam-3495	103	1	hence	hence	ADV
ejpam-3495	103	2	,	,	PUNCT
ejpam-3495	103	3	∗	∗	NOUN
ejpam-3495	103	4	is	be	AUX
ejpam-3495	103	5	not	not	PART
ejpam-3495	103	6	a	a	DET
ejpam-3495	103	7	subcompanion	subcompanion	NOUN
ejpam-3495	103	8	operation	operation	NOUN
ejpam-3495	103	9	and	and	CCONJ
ejpam-3495	103	10	so	so	ADV
ejpam-3495	103	11	not	not	PART
ejpam-3495	103	12	a	a	DET
ejpam-3495	103	13	companion	companion	NOUN
ejpam-3495	103	14	operation	operation	NOUN
ejpam-3495	103	15	.	.	PUNCT
ejpam-3495	104	1	remark	remark	VERB
ejpam-3495	104	2	3.7	3.7	NUM
ejpam-3495	104	3	.	.	PUNCT
ejpam-3495	105	1	if	if	SCONJ
ejpam-3495	105	2	(	(	PUNCT
ejpam-3495	105	3	x	x	X
ejpam-3495	105	4	,	,	PUNCT
ejpam-3495	105	5	∗	∗	NOUN
ejpam-3495	105	6	,	,	PUNCT
ejpam-3495	105	7	0	0	NUM
ejpam-3495	105	8	)	)	PUNCT
ejpam-3495	105	9	is	be	AUX
ejpam-3495	105	10	a	a	DET
ejpam-3495	105	11	b	b	NOUN
ejpam-3495	105	12	-	-	PUNCT
ejpam-3495	105	13	algebra	algebra	NOUN
ejpam-3495	105	14	,	,	PUNCT
ejpam-3495	105	15	then	then	ADV
ejpam-3495	105	16	(	(	PUNCT
ejpam-3495	105	17	x	x	X
ejpam-3495	105	18	,	,	PUNCT
ejpam-3495	105	19	∗	∗	NOUN
ejpam-3495	105	20	,	,	PUNCT
ejpam-3495	105	21	∗	∗	NOUN
ejpam-3495	105	22	,	,	PUNCT
ejpam-3495	105	23	0	0	NUM
ejpam-3495	105	24	)	)	PUNCT
ejpam-3495	105	25	is	be	AUX
ejpam-3495	105	26	not	not	PART
ejpam-3495	105	27	always	always	ADV
ejpam-3495	105	28	a	a	DET
ejpam-3495	105	29	companion	companion	NOUN
ejpam-3495	105	30	balgebra	balgebra	NOUN
ejpam-3495	105	31	.	.	PUNCT
ejpam-3495	106	1	in	in	ADP
ejpam-3495	106	2	example	example	NOUN
ejpam-3495	106	3	3.6	3.6	NUM
ejpam-3495	106	4	,	,	PUNCT
ejpam-3495	106	5	the	the	DET
ejpam-3495	106	6	condition	condition	NOUN
ejpam-3495	106	7	x	x	X
ejpam-3495	106	8	∗	∗	NOUN
ejpam-3495	106	9	y	y	NOUN
ejpam-3495	106	10	=	=	SYM
ejpam-3495	106	11	y	y	PROPN
ejpam-3495	106	12	∗	∗	NOUN
ejpam-3495	106	13	(	(	PUNCT
ejpam-3495	106	14	0	0	NUM
ejpam-3495	106	15	∗	∗	NOUN
ejpam-3495	106	16	x	x	NOUN
ejpam-3495	106	17	)	)	PUNCT
ejpam-3495	106	18	does	do	AUX
ejpam-3495	106	19	not	not	PART
ejpam-3495	106	20	hold	hold	VERB
ejpam-3495	106	21	.	.	PUNCT
ejpam-3495	107	1	example	example	NOUN
ejpam-3495	107	2	3.8	3.8	NUM
ejpam-3495	107	3	.	.	PUNCT
ejpam-3495	108	1	consider	consider	VERB
ejpam-3495	108	2	the	the	DET
ejpam-3495	108	3	klein	klein	PROPN
ejpam-3495	108	4	b	b	PROPN
ejpam-3495	108	5	-algebra	-algebra	PROPN
ejpam-3495	108	6	k4	k4	NOUN
ejpam-3495	108	7	=	=	SYM
ejpam-3495	108	8	{	{	PUNCT
ejpam-3495	108	9	0	0	NUM
ejpam-3495	108	10	,	,	PUNCT
ejpam-3495	108	11	1	1	NUM
ejpam-3495	108	12	,	,	PUNCT
ejpam-3495	108	13	2	2	NUM
ejpam-3495	108	14	,	,	PUNCT
ejpam-3495	108	15	3	3	NUM
ejpam-3495	108	16	}	}	PUNCT
ejpam-3495	108	17	with	with	ADP
ejpam-3495	108	18	the	the	DET
ejpam-3495	108	19	following	follow	VERB
ejpam-3495	108	20	table	table	NOUN
ejpam-3495	108	21	of	of	ADP
ejpam-3495	108	22	operation	operation	NOUN
ejpam-3495	108	23	[	[	X
ejpam-3495	108	24	4	4	NUM
ejpam-3495	108	25	]	]	SYM
ejpam-3495	108	26	:	:	PUNCT
ejpam-3495	108	27	∗	∗	NOUN
ejpam-3495	108	28	0	0	NUM
ejpam-3495	108	29	1	1	NUM
ejpam-3495	108	30	2	2	NUM
ejpam-3495	108	31	3	3	NUM
ejpam-3495	108	32	0	0	NUM
ejpam-3495	108	33	0	0	NUM
ejpam-3495	108	34	1	1	NUM
ejpam-3495	108	35	2	2	NUM
ejpam-3495	108	36	3	3	NUM
ejpam-3495	108	37	1	1	NUM
ejpam-3495	108	38	1	1	NUM
ejpam-3495	108	39	0	0	NUM
ejpam-3495	108	40	3	3	NUM
ejpam-3495	108	41	2	2	NUM
ejpam-3495	108	42	2	2	NUM
ejpam-3495	108	43	2	2	NUM
ejpam-3495	108	44	3	3	NUM
ejpam-3495	108	45	0	0	NUM
ejpam-3495	108	46	1	1	NUM
ejpam-3495	108	47	3	3	NUM
ejpam-3495	108	48	3	3	NUM
ejpam-3495	108	49	2	2	NUM
ejpam-3495	108	50	1	1	NUM
ejpam-3495	108	51	0	0	NUM
ejpam-3495	108	52	then	then	ADV
ejpam-3495	108	53	x	x	X
ejpam-3495	108	54	∗	∗	NOUN
ejpam-3495	108	55	y	y	NOUN
ejpam-3495	108	56	=	=	SYM
ejpam-3495	108	57	y	y	PROPN
ejpam-3495	108	58	∗	∗	NOUN
ejpam-3495	108	59	(	(	PUNCT
ejpam-3495	108	60	0	0	NUM
ejpam-3495	108	61	∗	∗	NOUN
ejpam-3495	108	62	x	x	NOUN
ejpam-3495	108	63	)	)	PUNCT
ejpam-3495	108	64	for	for	ADP
ejpam-3495	108	65	any	any	DET
ejpam-3495	108	66	x	x	NOUN
ejpam-3495	108	67	,	,	PUNCT
ejpam-3495	108	68	y	y	PROPN
ejpam-3495	108	69	∈	∈	PROPN
ejpam-3495	108	70	k4	k4	PROPN
ejpam-3495	108	71	and	and	CCONJ
ejpam-3495	108	72	(	(	PUNCT
ejpam-3495	108	73	k4	k4	PROPN
ejpam-3495	108	74	,	,	PUNCT
ejpam-3495	108	75	∗	∗	NOUN
ejpam-3495	108	76	,	,	PUNCT
ejpam-3495	108	77	∗	∗	NOUN
ejpam-3495	108	78	,	,	PUNCT
ejpam-3495	108	79	0	0	NUM
ejpam-3495	108	80	)	)	PUNCT
ejpam-3495	108	81	is	be	AUX
ejpam-3495	108	82	a	a	DET
ejpam-3495	108	83	companion	companion	NOUN
ejpam-3495	108	84	b	b	PROPN
ejpam-3495	108	85	-algebra	-algebra	PROPN
ejpam-3495	108	86	.	.	PUNCT
ejpam-3495	109	1	the	the	DET
ejpam-3495	109	2	observation	observation	NOUN
ejpam-3495	109	3	in	in	ADP
ejpam-3495	109	4	example	example	NOUN
ejpam-3495	109	5	3.8	3.8	NUM
ejpam-3495	109	6	is	be	AUX
ejpam-3495	109	7	generalized	generalize	VERB
ejpam-3495	109	8	in	in	ADP
ejpam-3495	109	9	the	the	DET
ejpam-3495	109	10	next	next	ADJ
ejpam-3495	109	11	theorem	theorem	PROPN
ejpam-3495	109	12	.	.	PUNCT
ejpam-3495	110	1	theorem	theorem	VERB
ejpam-3495	110	2	3.9	3.9	NUM
ejpam-3495	110	3	.	.	PUNCT
ejpam-3495	111	1	let	let	AUX
ejpam-3495	111	2	(	(	PUNCT
ejpam-3495	111	3	x	x	X
ejpam-3495	111	4	,	,	PUNCT
ejpam-3495	111	5	∗	∗	NOUN
ejpam-3495	111	6	,	,	PUNCT
ejpam-3495	111	7	0	0	NUM
ejpam-3495	111	8	)	)	PUNCT
ejpam-3495	111	9	be	be	AUX
ejpam-3495	111	10	a	a	DET
ejpam-3495	111	11	b	b	NOUN
ejpam-3495	111	12	-	-	PUNCT
ejpam-3495	111	13	algebra	algebra	NOUN
ejpam-3495	111	14	.	.	PUNCT
ejpam-3495	112	1	x	x	X
ejpam-3495	112	2	satisfies	satisfy	VERB
ejpam-3495	112	3	x∗y	x∗y	X
ejpam-3495	112	4	=	=	SYM
ejpam-3495	112	5	y	y	PROPN
ejpam-3495	112	6	∗	∗	NOUN
ejpam-3495	112	7	(	(	PUNCT
ejpam-3495	112	8	0∗x	0∗x	NOUN
ejpam-3495	112	9	)	)	PUNCT
ejpam-3495	112	10	for	for	ADP
ejpam-3495	112	11	any	any	DET
ejpam-3495	112	12	x	x	NOUN
ejpam-3495	112	13	,	,	PUNCT
ejpam-3495	112	14	y	y	PROPN
ejpam-3495	112	15	∈	∈	PROPN
ejpam-3495	112	16	x	x	INTJ
ejpam-3495	112	17	if	if	SCONJ
ejpam-3495	112	18	and	and	CCONJ
ejpam-3495	112	19	only	only	ADV
ejpam-3495	112	20	if	if	SCONJ
ejpam-3495	112	21	(	(	PUNCT
ejpam-3495	112	22	x	x	NOUN
ejpam-3495	112	23	,	,	PUNCT
ejpam-3495	112	24	∗	∗	NOUN
ejpam-3495	112	25	,	,	PUNCT
ejpam-3495	112	26	∗	∗	NOUN
ejpam-3495	112	27	,	,	PUNCT
ejpam-3495	112	28	0	0	NUM
ejpam-3495	112	29	)	)	PUNCT
ejpam-3495	112	30	is	be	AUX
ejpam-3495	112	31	a	a	DET
ejpam-3495	112	32	companion	companion	NOUN
ejpam-3495	112	33	b	b	NOUN
ejpam-3495	112	34	-	-	PUNCT
ejpam-3495	112	35	algebra	algebra	NOUN
ejpam-3495	112	36	.	.	PUNCT
ejpam-3495	113	1	proof	proof	NOUN
ejpam-3495	113	2	:	:	PUNCT
ejpam-3495	113	3	suppose	suppose	VERB
ejpam-3495	113	4	x∗y	x∗y	PUNCT
ejpam-3495	113	5	=	=	SYM
ejpam-3495	113	6	y	y	PROPN
ejpam-3495	113	7	∗	∗	NOUN
ejpam-3495	113	8	(	(	PUNCT
ejpam-3495	113	9	0∗x	0∗x	NOUN
ejpam-3495	113	10	)	)	PUNCT
ejpam-3495	113	11	.	.	PUNCT
ejpam-3495	114	1	by	by	ADP
ejpam-3495	114	2	definition	definition	NOUN
ejpam-3495	114	3	2.1(iii	2.1(iii	NUM
ejpam-3495	114	4	)	)	PUNCT
ejpam-3495	114	5	,	,	PUNCT
ejpam-3495	114	6	assumption	assumption	NOUN
ejpam-3495	114	7	and	and	CCONJ
ejpam-3495	114	8	definition	definition	NOUN
ejpam-3495	114	9	2.1(i	2.1(i	NUM
ejpam-3495	114	10	)	)	PUNCT
ejpam-3495	114	11	,	,	PUNCT
ejpam-3495	114	12	(	(	PUNCT
ejpam-3495	114	13	(	(	PUNCT
ejpam-3495	114	14	x	x	SYM
ejpam-3495	114	15	∗	∗	PROPN
ejpam-3495	114	16	y	y	NOUN
ejpam-3495	114	17	)	)	PUNCT
ejpam-3495	114	18	∗	∗	NOUN
ejpam-3495	114	19	x	x	NOUN
ejpam-3495	114	20	)	)	PUNCT
ejpam-3495	114	21	∗	∗	NOUN
ejpam-3495	114	22	y	y	NOUN
ejpam-3495	114	23	=	=	SYM
ejpam-3495	114	24	(	(	PUNCT
ejpam-3495	114	25	x	x	X
ejpam-3495	114	26	∗	∗	PROPN
ejpam-3495	114	27	y	y	NOUN
ejpam-3495	114	28	)	)	PUNCT
ejpam-3495	114	29	∗	∗	NOUN
ejpam-3495	114	30	(	(	PUNCT
ejpam-3495	114	31	y	y	PROPN
ejpam-3495	114	32	∗	∗	NOUN
ejpam-3495	114	33	(	(	PUNCT
ejpam-3495	114	34	0	0	NUM
ejpam-3495	114	35	∗	∗	NOUN
ejpam-3495	114	36	x	x	NOUN
ejpam-3495	114	37	)	)	PUNCT
ejpam-3495	114	38	)	)	PUNCT
ejpam-3495	115	1	=	=	PRON
ejpam-3495	115	2	(	(	PUNCT
ejpam-3495	115	3	x	x	X
ejpam-3495	115	4	∗	∗	PROPN
ejpam-3495	115	5	y	y	NOUN
ejpam-3495	115	6	)	)	PUNCT
ejpam-3495	115	7	∗	∗	NOUN
ejpam-3495	115	8	(	(	PUNCT
ejpam-3495	115	9	x	x	X
ejpam-3495	115	10	∗	∗	NOUN
ejpam-3495	115	11	y	y	NOUN
ejpam-3495	115	12	)	)	PUNCT
ejpam-3495	115	13	=	=	SYM
ejpam-3495	116	1	0	0	X
ejpam-3495	116	2	.	.	PUNCT
ejpam-3495	116	3	suppose	suppose	VERB
ejpam-3495	116	4	(	(	PUNCT
ejpam-3495	116	5	z	z	NOUN
ejpam-3495	116	6	∗	∗	X
ejpam-3495	116	7	x	x	NOUN
ejpam-3495	116	8	)	)	PUNCT
ejpam-3495	116	9	∗	∗	NOUN
ejpam-3495	116	10	y	y	NOUN
ejpam-3495	116	11	=	=	SYM
ejpam-3495	116	12	0	0	X
ejpam-3495	116	13	.	.	PUNCT
ejpam-3495	117	1	by	by	ADP
ejpam-3495	117	2	definition	definition	NOUN
ejpam-3495	117	3	2.1(iii	2.1(iii	NUM
ejpam-3495	117	4	)	)	PUNCT
ejpam-3495	117	5	,	,	PUNCT
ejpam-3495	117	6	z	z	NOUN
ejpam-3495	117	7	∗	∗	NOUN
ejpam-3495	117	8	(	(	PUNCT
ejpam-3495	117	9	y	y	PROPN
ejpam-3495	117	10	∗	∗	NOUN
ejpam-3495	117	11	(	(	PUNCT
ejpam-3495	117	12	0	0	NUM
ejpam-3495	117	13	∗	∗	NOUN
ejpam-3495	117	14	x	x	NOUN
ejpam-3495	117	15	)	)	PUNCT
ejpam-3495	117	16	)	)	PUNCT
ejpam-3495	118	1	=	=	SYM
ejpam-3495	118	2	0	0	NUM
ejpam-3495	118	3	and	and	CCONJ
ejpam-3495	118	4	by	by	ADP
ejpam-3495	118	5	assumption	assumption	NOUN
ejpam-3495	118	6	,	,	PUNCT
ejpam-3495	118	7	z	z	NOUN
ejpam-3495	118	8	∗	∗	NOUN
ejpam-3495	118	9	(	(	PUNCT
ejpam-3495	118	10	x	x	X
ejpam-3495	118	11	∗	∗	NOUN
ejpam-3495	118	12	y	y	NOUN
ejpam-3495	118	13	)	)	PUNCT
ejpam-3495	119	1	=	=	SYM
ejpam-3495	119	2	0	0	X
ejpam-3495	119	3	.	.	PUNCT
ejpam-3495	120	1	therefore	therefore	ADV
ejpam-3495	120	2	,	,	PUNCT
ejpam-3495	120	3	(	(	PUNCT
ejpam-3495	120	4	x	x	X
ejpam-3495	120	5	,	,	PUNCT
ejpam-3495	120	6	∗	∗	NOUN
ejpam-3495	120	7	,	,	PUNCT
ejpam-3495	120	8	∗	∗	NOUN
ejpam-3495	120	9	,	,	PUNCT
ejpam-3495	120	10	0	0	NUM
ejpam-3495	120	11	)	)	PUNCT
ejpam-3495	120	12	is	be	AUX
ejpam-3495	120	13	a	a	DET
ejpam-3495	120	14	companion	companion	NOUN
ejpam-3495	120	15	b	b	NOUN
ejpam-3495	120	16	-	-	PUNCT
ejpam-3495	120	17	algebra	algebra	NOUN
ejpam-3495	120	18	.	.	PUNCT
ejpam-3495	121	1	conversely	conversely	ADV
ejpam-3495	121	2	,	,	PUNCT
ejpam-3495	121	3	suppose	suppose	VERB
ejpam-3495	121	4	(	(	PUNCT
ejpam-3495	121	5	x	x	X
ejpam-3495	121	6	,	,	PUNCT
ejpam-3495	121	7	∗	∗	NOUN
ejpam-3495	121	8	,	,	PUNCT
ejpam-3495	121	9	∗	∗	NOUN
ejpam-3495	121	10	,	,	PUNCT
ejpam-3495	121	11	0	0	NUM
ejpam-3495	121	12	)	)	PUNCT
ejpam-3495	121	13	is	be	AUX
ejpam-3495	121	14	a	a	DET
ejpam-3495	121	15	companion	companion	NOUN
ejpam-3495	121	16	b	b	PROPN
ejpam-3495	121	17	-algebra	-algebra	PROPN
ejpam-3495	121	18	.	.	PUNCT
ejpam-3495	122	1	by	by	ADP
ejpam-3495	122	2	definition	definition	NOUN
ejpam-3495	122	3	3.1	3.1	NUM
ejpam-3495	122	4	,	,	PUNCT
ejpam-3495	122	5	(	(	PUNCT
ejpam-3495	122	6	x	x	X
ejpam-3495	122	7	,	,	PUNCT
ejpam-3495	122	8	∗	∗	NOUN
ejpam-3495	122	9	,	,	PUNCT
ejpam-3495	122	10	0	0	NUM
ejpam-3495	122	11	)	)	PUNCT
ejpam-3495	122	12	is	be	AUX
ejpam-3495	122	13	a	a	DET
ejpam-3495	122	14	b	b	NOUN
ejpam-3495	122	15	-algebra	-algebra	NOUN
ejpam-3495	122	16	.	.	PUNCT
ejpam-3495	123	1	let	let	VERB
ejpam-3495	123	2	x	x	PRON
ejpam-3495	123	3	,	,	PUNCT
ejpam-3495	123	4	y	y	PROPN
ejpam-3495	123	5	∈	∈	PROPN
ejpam-3495	123	6	x.	x.	NOUN
ejpam-3495	123	7	then	then	ADV
ejpam-3495	123	8	by	by	ADP
ejpam-3495	123	9	(	(	PUNCT
ejpam-3495	123	10	sc	sc	PROPN
ejpam-3495	123	11	)	)	PUNCT
ejpam-3495	123	12	,	,	PUNCT
ejpam-3495	123	13	(	(	PUNCT
ejpam-3495	123	14	(	(	PUNCT
ejpam-3495	123	15	x	x	SYM
ejpam-3495	123	16	∗	∗	PROPN
ejpam-3495	123	17	y	y	NOUN
ejpam-3495	123	18	)	)	PUNCT
ejpam-3495	123	19	∗	∗	NOUN
ejpam-3495	123	20	x	x	NOUN
ejpam-3495	123	21	)	)	PUNCT
ejpam-3495	123	22	∗	∗	NOUN
ejpam-3495	123	23	y	y	NOUN
ejpam-3495	124	1	=	=	SYM
ejpam-3495	125	1	0	0	X
ejpam-3495	125	2	.	.	PUNCT
ejpam-3495	126	1	by	by	ADP
ejpam-3495	126	2	definition	definition	NOUN
ejpam-3495	126	3	2.1(iii	2.1(iii	NUM
ejpam-3495	126	4	)	)	PUNCT
ejpam-3495	126	5	,	,	PUNCT
ejpam-3495	126	6	(	(	PUNCT
ejpam-3495	126	7	x	x	X
ejpam-3495	126	8	∗	∗	PROPN
ejpam-3495	126	9	y	y	NOUN
ejpam-3495	126	10	)	)	PUNCT
ejpam-3495	126	11	∗	∗	NOUN
ejpam-3495	126	12	(	(	PUNCT
ejpam-3495	126	13	y	y	PROPN
ejpam-3495	126	14	∗	∗	NOUN
ejpam-3495	126	15	(	(	PUNCT
ejpam-3495	126	16	0	0	NUM
ejpam-3495	126	17	∗	∗	NOUN
ejpam-3495	126	18	x	x	NOUN
ejpam-3495	126	19	)	)	PUNCT
ejpam-3495	126	20	)	)	PUNCT
ejpam-3495	126	21	=	=	PUNCT
ejpam-3495	127	1	0	0	X
ejpam-3495	127	2	.	.	PUNCT
ejpam-3495	128	1	so	so	ADV
ejpam-3495	128	2	,	,	PUNCT
ejpam-3495	128	3	x	x	X
ejpam-3495	128	4	∗	∗	NOUN
ejpam-3495	128	5	y	y	NOUN
ejpam-3495	128	6	=	=	SYM
ejpam-3495	128	7	y	y	PROPN
ejpam-3495	128	8	∗	∗	NOUN
ejpam-3495	128	9	(	(	PUNCT
ejpam-3495	128	10	0	0	NUM
ejpam-3495	128	11	∗	∗	NOUN
ejpam-3495	128	12	x	x	NOUN
ejpam-3495	128	13	)	)	PUNCT
ejpam-3495	128	14	by	by	ADP
ejpam-3495	128	15	theorem	theorem	NOUN
ejpam-3495	128	16	2.3(d	2.3(d	NUM
ejpam-3495	128	17	)	)	PUNCT
ejpam-3495	128	18	.	.	PUNCT
ejpam-3495	129	1	�	�	PROPN
ejpam-3495	129	2	lemma	lemma	PROPN
ejpam-3495	129	3	3.10	3.10	NUM
ejpam-3495	129	4	.	.	PUNCT
ejpam-3495	130	1	let	let	VERB
ejpam-3495	130	2	(	(	PUNCT
ejpam-3495	130	3	x	x	X
ejpam-3495	130	4	,	,	PUNCT
ejpam-3495	130	5	∗	∗	NOUN
ejpam-3495	130	6	,	,	PUNCT
ejpam-3495	130	7	�	�	PROPN
ejpam-3495	130	8	,	,	PUNCT
ejpam-3495	130	9	0	0	NUM
ejpam-3495	130	10	)	)	PUNCT
ejpam-3495	130	11	be	be	AUX
ejpam-3495	130	12	a	a	DET
ejpam-3495	130	13	companion	companion	NOUN
ejpam-3495	130	14	b	b	NOUN
ejpam-3495	130	15	-	-	PUNCT
ejpam-3495	130	16	algebra	algebra	NOUN
ejpam-3495	130	17	.	.	PUNCT
ejpam-3495	131	1	then	then	ADV
ejpam-3495	131	2	for	for	ADP
ejpam-3495	131	3	any	any	DET
ejpam-3495	131	4	x	x	NOUN
ejpam-3495	131	5	,	,	PUNCT
ejpam-3495	131	6	y	y	PROPN
ejpam-3495	131	7	,	,	PUNCT
ejpam-3495	131	8	z	z	PROPN
ejpam-3495	131	9	∈	∈	PROPN
ejpam-3495	131	10	x	x	SYM
ejpam-3495	131	11	,	,	PUNCT
ejpam-3495	131	12	the	the	DET
ejpam-3495	131	13	following	follow	VERB
ejpam-3495	131	14	hold	hold	NOUN
ejpam-3495	131	15	:	:	PUNCT
ejpam-3495	131	16	(	(	PUNCT
ejpam-3495	131	17	a	a	X
ejpam-3495	131	18	)	)	PUNCT
ejpam-3495	131	19	0	0	NUM
ejpam-3495	131	20	�	�	PROPN
ejpam-3495	131	21	y	y	NOUN
ejpam-3495	131	22	=	=	SYM
ejpam-3495	131	23	y	y	PROPN
ejpam-3495	131	24	and	and	CCONJ
ejpam-3495	131	25	y	y	PROPN
ejpam-3495	131	26	�	�	PROPN
ejpam-3495	131	27	0	0	NUM
ejpam-3495	131	28	=	=	SYM
ejpam-3495	131	29	y	y	PROPN
ejpam-3495	131	30	;	;	PUNCT
ejpam-3495	131	31	(	(	PUNCT
ejpam-3495	131	32	b	b	X
ejpam-3495	131	33	)	)	PUNCT
ejpam-3495	131	34	x	x	NOUN
ejpam-3495	131	35	�	�	PROPN
ejpam-3495	131	36	y	y	PROPN
ejpam-3495	131	37	=	=	SYM
ejpam-3495	131	38	y	y	PROPN
ejpam-3495	131	39	∗	∗	NOUN
ejpam-3495	131	40	(	(	PUNCT
ejpam-3495	131	41	0	0	NUM
ejpam-3495	131	42	∗	∗	NOUN
ejpam-3495	131	43	x	x	NOUN
ejpam-3495	131	44	)	)	PUNCT
ejpam-3495	131	45	;	;	PUNCT
ejpam-3495	131	46	(	(	PUNCT
ejpam-3495	131	47	c	c	X
ejpam-3495	131	48	)	)	PUNCT
ejpam-3495	131	49	if	if	SCONJ
ejpam-3495	131	50	x	x	X
ejpam-3495	131	51	∗	∗	NOUN
ejpam-3495	131	52	z	z	NOUN
ejpam-3495	131	53	=	=	SYM
ejpam-3495	131	54	y	y	PROPN
ejpam-3495	131	55	,	,	PUNCT
ejpam-3495	131	56	then	then	ADV
ejpam-3495	131	57	x	x	X
ejpam-3495	131	58	=	=	SYM
ejpam-3495	131	59	z	z	PROPN
ejpam-3495	131	60	�	�	PROPN
ejpam-3495	131	61	y	y	PROPN
ejpam-3495	131	62	;	;	PUNCT
ejpam-3495	131	63	(	(	PUNCT
ejpam-3495	131	64	d	d	X
ejpam-3495	131	65	)	)	PUNCT
ejpam-3495	131	66	�	�	PROPN
ejpam-3495	131	67	is	be	AUX
ejpam-3495	131	68	associative	associative	ADJ
ejpam-3495	131	69	in	in	ADP
ejpam-3495	131	70	x	x	PRON
ejpam-3495	131	71	;	;	PUNCT
ejpam-3495	131	72	(	(	PUNCT
ejpam-3495	131	73	e	e	NOUN
ejpam-3495	131	74	)	)	PUNCT
ejpam-3495	131	75	x	x	X
ejpam-3495	132	1	=	=	SYM
ejpam-3495	132	2	(	(	PUNCT
ejpam-3495	132	3	x	x	X
ejpam-3495	132	4	�	�	PROPN
ejpam-3495	132	5	y	y	PROPN
ejpam-3495	132	6	)	)	PUNCT
ejpam-3495	132	7	�	�	PROPN
ejpam-3495	132	8	(	(	PUNCT
ejpam-3495	132	9	0	0	NUM
ejpam-3495	132	10	∗	∗	PROPN
ejpam-3495	132	11	y	y	PROPN
ejpam-3495	132	12	)	)	PUNCT
ejpam-3495	132	13	;	;	PUNCT
ejpam-3495	132	14	(	(	PUNCT
ejpam-3495	132	15	f	f	X
ejpam-3495	132	16	)	)	PUNCT
ejpam-3495	132	17	if	if	SCONJ
ejpam-3495	132	18	(	(	PUNCT
ejpam-3495	132	19	x	x	NOUN
ejpam-3495	132	20	,	,	PUNCT
ejpam-3495	132	21	∗	∗	NOUN
ejpam-3495	132	22	,	,	PUNCT
ejpam-3495	132	23	0	0	NUM
ejpam-3495	132	24	)	)	PUNCT
ejpam-3495	132	25	is	be	AUX
ejpam-3495	132	26	commutative	commutative	ADJ
ejpam-3495	132	27	,	,	PUNCT
ejpam-3495	132	28	then	then	ADV
ejpam-3495	132	29	x	x	X
ejpam-3495	132	30	�	�	PROPN
ejpam-3495	132	31	y	y	NOUN
ejpam-3495	132	32	=	=	PUNCT
ejpam-3495	132	33	x	x	SYM
ejpam-3495	132	34	∗	∗	NOUN
ejpam-3495	132	35	(	(	PUNCT
ejpam-3495	132	36	0	0	NUM
ejpam-3495	132	37	∗	∗	PROPN
ejpam-3495	132	38	y	y	PROPN
ejpam-3495	132	39	)	)	PUNCT
ejpam-3495	132	40	.	.	PUNCT
ejpam-3495	133	1	l.d	l.d	PROPN
ejpam-3495	133	2	.	.	PROPN
ejpam-3495	133	3	naingue	naingue	PROPN
ejpam-3495	133	4	,	,	PUNCT
ejpam-3495	133	5	j.p	j.p	PROPN
ejpam-3495	133	6	.	.	PROPN
ejpam-3495	133	7	vilela	vilela	PROPN
ejpam-3495	133	8	/	/	SYM
ejpam-3495	133	9	eur	eur	PROPN
ejpam-3495	133	10	.	.	PUNCT
ejpam-3495	134	1	j.	j.	PROPN
ejpam-3495	134	2	pure	pure	PROPN
ejpam-3495	134	3	appl	appl	PROPN
ejpam-3495	134	4	.	.	PROPN
ejpam-3495	134	5	math	math	PROPN
ejpam-3495	134	6	,	,	PUNCT
ejpam-3495	134	7	12	12	NUM
ejpam-3495	134	8	(	(	PUNCT
ejpam-3495	134	9	3	3	NUM
ejpam-3495	134	10	)	)	PUNCT
ejpam-3495	134	11	(	(	PUNCT
ejpam-3495	134	12	2019	2019	NUM
ejpam-3495	134	13	)	)	PUNCT
ejpam-3495	134	14	,	,	PUNCT
ejpam-3495	134	15	1248	1248	NUM
ejpam-3495	134	16	-	-	SYM
ejpam-3495	134	17	1259	1259	NUM
ejpam-3495	134	18	1252	1252	NUM
ejpam-3495	134	19	proof	proof	NOUN
ejpam-3495	134	20	:	:	PUNCT
ejpam-3495	134	21	let	let	VERB
ejpam-3495	134	22	(	(	PUNCT
ejpam-3495	134	23	x	x	NOUN
ejpam-3495	134	24	,	,	PUNCT
ejpam-3495	134	25	∗	∗	NOUN
ejpam-3495	134	26	,	,	PUNCT
ejpam-3495	134	27	�	�	PROPN
ejpam-3495	134	28	,	,	PUNCT
ejpam-3495	134	29	0	0	NUM
ejpam-3495	134	30	)	)	PUNCT
ejpam-3495	134	31	be	be	AUX
ejpam-3495	134	32	a	a	DET
ejpam-3495	134	33	companion	companion	NOUN
ejpam-3495	134	34	b	b	NOUN
ejpam-3495	134	35	-algebra	-algebra	PROPN
ejpam-3495	134	36	and	and	CCONJ
ejpam-3495	134	37	x	x	NOUN
ejpam-3495	134	38	,	,	PUNCT
ejpam-3495	134	39	y	y	PROPN
ejpam-3495	134	40	,	,	PUNCT
ejpam-3495	134	41	z	z	PROPN
ejpam-3495	134	42	∈	∈	PROPN
ejpam-3495	134	43	x.	x.	NOUN
ejpam-3495	134	44	(	(	PUNCT
ejpam-3495	134	45	a	a	X
ejpam-3495	134	46	)	)	PUNCT
ejpam-3495	134	47	in	in	ADP
ejpam-3495	134	48	(	(	PUNCT
ejpam-3495	134	49	sc	sc	PROPN
ejpam-3495	134	50	)	)	PUNCT
ejpam-3495	134	51	,	,	PUNCT
ejpam-3495	134	52	take	take	VERB
ejpam-3495	134	53	x	x	NOUN
ejpam-3495	134	54	=	=	SYM
ejpam-3495	134	55	0	0	NUM
ejpam-3495	134	56	,	,	PUNCT
ejpam-3495	134	57	that	that	ADV
ejpam-3495	134	58	is	is	ADV
ejpam-3495	134	59	,	,	PUNCT
ejpam-3495	134	60	0	0	PUNCT
ejpam-3495	134	61	=	=	SYM
ejpam-3495	134	62	(	(	PUNCT
ejpam-3495	134	63	(	(	PUNCT
ejpam-3495	134	64	0	0	NUM
ejpam-3495	134	65	�	�	PROPN
ejpam-3495	134	66	y	y	PROPN
ejpam-3495	134	67	)	)	PUNCT
ejpam-3495	134	68	∗	∗	NOUN
ejpam-3495	134	69	0	0	NUM
ejpam-3495	134	70	)	)	PUNCT
ejpam-3495	134	71	∗	∗	NOUN
ejpam-3495	134	72	y	y	NOUN
ejpam-3495	134	73	=	=	SYM
ejpam-3495	134	74	(	(	PUNCT
ejpam-3495	134	75	0	0	NUM
ejpam-3495	134	76	�	�	PROPN
ejpam-3495	134	77	y	y	NOUN
ejpam-3495	134	78	)	)	PUNCT
ejpam-3495	134	79	∗	∗	NOUN
ejpam-3495	134	80	y.	y.	NOUN
ejpam-3495	134	81	by	by	ADP
ejpam-3495	134	82	theorem	theorem	NOUN
ejpam-3495	134	83	2.3(d	2.3(d	NUM
ejpam-3495	134	84	)	)	PUNCT
ejpam-3495	134	85	,	,	PUNCT
ejpam-3495	134	86	0	0	NUM
ejpam-3495	134	87	�	�	NOUN
ejpam-3495	134	88	y	y	NOUN
ejpam-3495	134	89	=	=	SYM
ejpam-3495	134	90	y.	y.	PROPN
ejpam-3495	134	91	now	now	ADV
ejpam-3495	134	92	,	,	PUNCT
ejpam-3495	134	93	take	take	VERB
ejpam-3495	134	94	x	x	X
ejpam-3495	134	95	=	=	SYM
ejpam-3495	134	96	y	y	PROPN
ejpam-3495	134	97	and	and	CCONJ
ejpam-3495	134	98	y	y	PROPN
ejpam-3495	134	99	=	=	NOUN
ejpam-3495	134	100	0	0	NUM
ejpam-3495	135	1	in	in	ADP
ejpam-3495	135	2	(	(	PUNCT
ejpam-3495	135	3	sc	sc	PROPN
ejpam-3495	135	4	)	)	PUNCT
ejpam-3495	135	5	.	.	PUNCT
ejpam-3495	136	1	then	then	ADV
ejpam-3495	136	2	,	,	PUNCT
ejpam-3495	136	3	0	0	PUNCT
ejpam-3495	136	4	=	=	SYM
ejpam-3495	136	5	(	(	PUNCT
ejpam-3495	136	6	(	(	PUNCT
ejpam-3495	136	7	y	y	NOUN
ejpam-3495	136	8	�	�	NOUN
ejpam-3495	136	9	0)∗y)∗0	0)∗y)∗0	NOUN
ejpam-3495	136	10	=	=	SYM
ejpam-3495	136	11	(	(	PUNCT
ejpam-3495	136	12	y	y	PROPN
ejpam-3495	136	13	�	�	PROPN
ejpam-3495	136	14	0)∗y	0)∗y	NOUN
ejpam-3495	136	15	.	.	PUNCT
ejpam-3495	137	1	hence	hence	ADV
ejpam-3495	137	2	,	,	PUNCT
ejpam-3495	137	3	by	by	ADP
ejpam-3495	137	4	theorem	theorem	NOUN
ejpam-3495	137	5	2.3(d	2.3(d	NUM
ejpam-3495	137	6	)	)	PUNCT
ejpam-3495	137	7	,	,	PUNCT
ejpam-3495	137	8	y	y	PROPN
ejpam-3495	137	9	�	�	PROPN
ejpam-3495	137	10	0	0	NUM
ejpam-3495	138	1	=	=	PUNCT
ejpam-3495	138	2	y.	y.	NOUN
ejpam-3495	138	3	(	(	PUNCT
ejpam-3495	138	4	b	b	NOUN
ejpam-3495	138	5	)	)	PUNCT
ejpam-3495	138	6	by	by	ADP
ejpam-3495	138	7	(	(	PUNCT
ejpam-3495	138	8	sc	sc	PROPN
ejpam-3495	138	9	)	)	PUNCT
ejpam-3495	138	10	,	,	PUNCT
ejpam-3495	138	11	(	(	PUNCT
ejpam-3495	138	12	(	(	PUNCT
ejpam-3495	138	13	x	x	X
ejpam-3495	138	14	�	�	PROPN
ejpam-3495	138	15	y	y	PROPN
ejpam-3495	138	16	)	)	PUNCT
ejpam-3495	138	17	∗	∗	NOUN
ejpam-3495	138	18	x	x	NOUN
ejpam-3495	138	19	)	)	PUNCT
ejpam-3495	138	20	∗	∗	NOUN
ejpam-3495	138	21	y	y	NOUN
ejpam-3495	139	1	=	=	SYM
ejpam-3495	139	2	0	0	PROPN
ejpam-3495	139	3	.	.	PUNCT
ejpam-3495	140	1	so	so	ADV
ejpam-3495	140	2	,	,	PUNCT
ejpam-3495	140	3	by	by	ADP
ejpam-3495	140	4	definition	definition	NOUN
ejpam-3495	140	5	2.1(iii	2.1(iii	NUM
ejpam-3495	140	6	)	)	PUNCT
ejpam-3495	140	7	,	,	PUNCT
ejpam-3495	140	8	(	(	PUNCT
ejpam-3495	140	9	x	x	X
ejpam-3495	140	10	�	�	PROPN
ejpam-3495	140	11	y	y	NOUN
ejpam-3495	140	12	)	)	PUNCT
ejpam-3495	140	13	∗	∗	NOUN
ejpam-3495	140	14	(	(	PUNCT
ejpam-3495	140	15	y	y	PROPN
ejpam-3495	140	16	∗	∗	NOUN
ejpam-3495	140	17	(	(	PUNCT
ejpam-3495	140	18	0	0	NUM
ejpam-3495	140	19	∗	∗	NOUN
ejpam-3495	140	20	x	x	NOUN
ejpam-3495	140	21	)	)	PUNCT
ejpam-3495	140	22	)	)	PUNCT
ejpam-3495	141	1	=	=	PUNCT
ejpam-3495	141	2	0	0	X
ejpam-3495	141	3	.	.	PUNCT
ejpam-3495	141	4	thus	thus	ADV
ejpam-3495	141	5	,	,	PUNCT
ejpam-3495	141	6	by	by	ADP
ejpam-3495	141	7	theorem	theorem	NOUN
ejpam-3495	141	8	2.3(d	2.3(d	NUM
ejpam-3495	141	9	)	)	PUNCT
ejpam-3495	141	10	,	,	PUNCT
ejpam-3495	141	11	x	x	X
ejpam-3495	141	12	�	�	PROPN
ejpam-3495	141	13	y	y	PROPN
ejpam-3495	141	14	=	=	SYM
ejpam-3495	141	15	y	y	PROPN
ejpam-3495	141	16	∗	∗	NOUN
ejpam-3495	141	17	(	(	PUNCT
ejpam-3495	141	18	0	0	NUM
ejpam-3495	141	19	∗	∗	NOUN
ejpam-3495	141	20	x	x	NOUN
ejpam-3495	141	21	)	)	PUNCT
ejpam-3495	141	22	.	.	PUNCT
ejpam-3495	142	1	(	(	PUNCT
ejpam-3495	142	2	c	c	X
ejpam-3495	142	3	)	)	PUNCT
ejpam-3495	142	4	if	if	SCONJ
ejpam-3495	142	5	x	x	X
ejpam-3495	142	6	∗	∗	NOUN
ejpam-3495	142	7	z	z	NOUN
ejpam-3495	142	8	=	=	SYM
ejpam-3495	142	9	y	y	PROPN
ejpam-3495	142	10	,	,	PUNCT
ejpam-3495	142	11	then	then	ADV
ejpam-3495	142	12	(	(	PUNCT
ejpam-3495	142	13	x	x	X
ejpam-3495	142	14	∗	∗	PROPN
ejpam-3495	142	15	z	z	NOUN
ejpam-3495	142	16	)	)	PUNCT
ejpam-3495	142	17	∗	∗	NOUN
ejpam-3495	142	18	y	y	NOUN
ejpam-3495	142	19	=	=	SYM
ejpam-3495	142	20	y	y	PROPN
ejpam-3495	142	21	∗	∗	NOUN
ejpam-3495	142	22	y	y	PROPN
ejpam-3495	142	23	=	=	SYM
ejpam-3495	142	24	0	0	X
ejpam-3495	142	25	.	.	PUNCT
ejpam-3495	143	1	by	by	ADP
ejpam-3495	143	2	(	(	PUNCT
ejpam-3495	143	3	c	c	NOUN
ejpam-3495	143	4	)	)	PUNCT
ejpam-3495	143	5	,	,	PUNCT
ejpam-3495	143	6	x	x	X
ejpam-3495	143	7	∗	∗	NOUN
ejpam-3495	143	8	(	(	PUNCT
ejpam-3495	143	9	z	z	PROPN
ejpam-3495	143	10	�	�	PROPN
ejpam-3495	143	11	y	y	NOUN
ejpam-3495	143	12	)	)	PUNCT
ejpam-3495	143	13	=	=	SYM
ejpam-3495	144	1	0	0	X
ejpam-3495	144	2	.	.	PUNCT
ejpam-3495	145	1	hence	hence	ADV
ejpam-3495	145	2	,	,	PUNCT
ejpam-3495	145	3	by	by	ADP
ejpam-3495	145	4	theorem	theorem	NOUN
ejpam-3495	145	5	2.3(d	2.3(d	NUM
ejpam-3495	145	6	)	)	PUNCT
ejpam-3495	145	7	,	,	PUNCT
ejpam-3495	145	8	x	x	X
ejpam-3495	145	9	=	=	SYM
ejpam-3495	145	10	z	z	PROPN
ejpam-3495	145	11	�	�	PROPN
ejpam-3495	145	12	y.	y.	NOUN
ejpam-3495	145	13	(	(	PUNCT
ejpam-3495	145	14	d	d	PROPN
ejpam-3495	145	15	)	)	PUNCT
ejpam-3495	145	16	by	by	ADP
ejpam-3495	145	17	lemma	lemma	PROPN
ejpam-3495	145	18	3.10(b	3.10(b	PROPN
ejpam-3495	145	19	)	)	PUNCT
ejpam-3495	145	20	,	,	PUNCT
ejpam-3495	145	21	definition	definition	NOUN
ejpam-3495	145	22	2.1(iii	2.1(iii	NUM
ejpam-3495	145	23	)	)	PUNCT
ejpam-3495	145	24	and	and	CCONJ
ejpam-3495	145	25	theorem	theorem	VERB
ejpam-3495	145	26	2.3(c	2.3(c	NUM
ejpam-3495	145	27	)	)	PUNCT
ejpam-3495	145	28	,	,	PUNCT
ejpam-3495	145	29	we	we	PRON
ejpam-3495	145	30	have	have	VERB
ejpam-3495	145	31	(	(	PUNCT
ejpam-3495	145	32	x	x	X
ejpam-3495	145	33	�	�	PROPN
ejpam-3495	145	34	y	y	PROPN
ejpam-3495	145	35	)	)	PUNCT
ejpam-3495	145	36	�	�	PROPN
ejpam-3495	145	37	z	z	NOUN
ejpam-3495	145	38	=	=	SYM
ejpam-3495	145	39	z	z	NOUN
ejpam-3495	145	40	∗	∗	NOUN
ejpam-3495	145	41	(	(	PUNCT
ejpam-3495	145	42	0	0	NUM
ejpam-3495	145	43	∗	∗	NOUN
ejpam-3495	145	44	(	(	PUNCT
ejpam-3495	145	45	x	x	X
ejpam-3495	145	46	�	�	PROPN
ejpam-3495	145	47	y	y	PROPN
ejpam-3495	145	48	)	)	PUNCT
ejpam-3495	145	49	)	)	PUNCT
ejpam-3495	146	1	=	=	PUNCT
ejpam-3495	146	2	z	z	NOUN
ejpam-3495	146	3	∗	∗	NOUN
ejpam-3495	146	4	(	(	PUNCT
ejpam-3495	146	5	0	0	NUM
ejpam-3495	146	6	∗	∗	NOUN
ejpam-3495	146	7	(	(	PUNCT
ejpam-3495	146	8	y	y	PROPN
ejpam-3495	146	9	∗	∗	NOUN
ejpam-3495	146	10	(	(	PUNCT
ejpam-3495	146	11	0	0	NUM
ejpam-3495	146	12	∗	∗	NOUN
ejpam-3495	146	13	x	x	NOUN
ejpam-3495	146	14	)	)	PUNCT
ejpam-3495	146	15	)	)	PUNCT
ejpam-3495	146	16	)	)	PUNCT
ejpam-3495	147	1	=	=	PUNCT
ejpam-3495	147	2	z	z	NOUN
ejpam-3495	147	3	∗	∗	NOUN
ejpam-3495	147	4	(	(	PUNCT
ejpam-3495	147	5	(	(	PUNCT
ejpam-3495	147	6	0	0	NUM
ejpam-3495	147	7	∗	∗	NOUN
ejpam-3495	147	8	x	x	NOUN
ejpam-3495	147	9	)	)	PUNCT
ejpam-3495	147	10	∗	∗	PROPN
ejpam-3495	147	11	y	y	NOUN
ejpam-3495	147	12	)	)	PUNCT
ejpam-3495	147	13	=	=	PUNCT
ejpam-3495	147	14	(	(	PUNCT
ejpam-3495	147	15	z	z	NOUN
ejpam-3495	147	16	∗	∗	NOUN
ejpam-3495	147	17	(	(	PUNCT
ejpam-3495	147	18	0	0	NUM
ejpam-3495	147	19	∗	∗	PROPN
ejpam-3495	147	20	y	y	PROPN
ejpam-3495	147	21	)	)	PUNCT
ejpam-3495	147	22	)	)	PUNCT
ejpam-3495	147	23	∗	∗	NOUN
ejpam-3495	147	24	(	(	PUNCT
ejpam-3495	147	25	0	0	NUM
ejpam-3495	147	26	∗	∗	NOUN
ejpam-3495	147	27	x	x	NOUN
ejpam-3495	147	28	)	)	PUNCT
ejpam-3495	147	29	=	=	SYM
ejpam-3495	147	30	(	(	PUNCT
ejpam-3495	147	31	y	y	PROPN
ejpam-3495	147	32	�	�	PROPN
ejpam-3495	147	33	z	z	PROPN
ejpam-3495	147	34	)	)	PUNCT
ejpam-3495	147	35	∗	∗	NOUN
ejpam-3495	147	36	(	(	PUNCT
ejpam-3495	147	37	0	0	NUM
ejpam-3495	147	38	∗	∗	NOUN
ejpam-3495	147	39	x	x	NOUN
ejpam-3495	147	40	)	)	PUNCT
ejpam-3495	147	41	=	=	SYM
ejpam-3495	147	42	x	x	SYM
ejpam-3495	147	43	�	�	PROPN
ejpam-3495	147	44	(	(	PUNCT
ejpam-3495	147	45	y	y	PROPN
ejpam-3495	147	46	�	�	PROPN
ejpam-3495	147	47	z	z	PROPN
ejpam-3495	147	48	)	)	PUNCT
ejpam-3495	147	49	.	.	PUNCT
ejpam-3495	148	1	thus	thus	ADV
ejpam-3495	148	2	,	,	PUNCT
ejpam-3495	148	3	the	the	DET
ejpam-3495	148	4	companion	companion	NOUN
ejpam-3495	148	5	operation	operation	NOUN
ejpam-3495	148	6	�	�	PROPN
ejpam-3495	148	7	is	be	AUX
ejpam-3495	148	8	associative	associative	ADJ
ejpam-3495	148	9	.	.	PUNCT
ejpam-3495	149	1	(	(	PUNCT
ejpam-3495	149	2	e	e	NOUN
ejpam-3495	149	3	)	)	PUNCT
ejpam-3495	149	4	note	note	VERB
ejpam-3495	149	5	that	that	SCONJ
ejpam-3495	149	6	by	by	ADP
ejpam-3495	149	7	theorem	theorem	NOUN
ejpam-3495	149	8	2.3(f	2.3(f	NUM
ejpam-3495	149	9	)	)	PUNCT
ejpam-3495	149	10	,	,	PUNCT
ejpam-3495	149	11	definitions	definition	NOUN
ejpam-3495	149	12	2.1(i	2.1(i	NUM
ejpam-3495	149	13	)	)	PUNCT
ejpam-3495	149	14	,	,	PUNCT
ejpam-3495	149	15	2.1(iii	2.1(iii	NUM
ejpam-3495	149	16	)	)	PUNCT
ejpam-3495	149	17	,	,	PUNCT
ejpam-3495	149	18	theorems	theorem	NOUN
ejpam-3495	149	19	2.3(b	2.3(b	NUM
ejpam-3495	149	20	)	)	PUNCT
ejpam-3495	149	21	and	and	CCONJ
ejpam-3495	149	22	2.4	2.4	NUM
ejpam-3495	149	23	,	,	PUNCT
ejpam-3495	149	24	and	and	CCONJ
ejpam-3495	149	25	lemma	lemma	PROPN
ejpam-3495	149	26	3.10(b	3.10(b	PROPN
ejpam-3495	149	27	)	)	PUNCT
ejpam-3495	149	28	,	,	PUNCT
ejpam-3495	149	29	x	x	PUNCT
ejpam-3495	150	1	=	=	SYM
ejpam-3495	150	2	0	0	NUM
ejpam-3495	150	3	∗	∗	NOUN
ejpam-3495	150	4	(	(	PUNCT
ejpam-3495	150	5	0	0	NUM
ejpam-3495	150	6	∗	∗	NOUN
ejpam-3495	150	7	x	x	NOUN
ejpam-3495	150	8	)	)	PUNCT
ejpam-3495	150	9	=	=	SYM
ejpam-3495	150	10	(	(	PUNCT
ejpam-3495	150	11	(	(	PUNCT
ejpam-3495	150	12	0	0	NUM
ejpam-3495	150	13	∗	∗	PROPN
ejpam-3495	150	14	y	y	NOUN
ejpam-3495	150	15	)	)	PUNCT
ejpam-3495	150	16	∗	∗	NOUN
ejpam-3495	150	17	(	(	PUNCT
ejpam-3495	150	18	0	0	NUM
ejpam-3495	150	19	∗	∗	PROPN
ejpam-3495	150	20	y	y	PROPN
ejpam-3495	150	21	)	)	PUNCT
ejpam-3495	150	22	)	)	PUNCT
ejpam-3495	150	23	∗	∗	NOUN
ejpam-3495	150	24	(	(	PUNCT
ejpam-3495	150	25	0	0	NUM
ejpam-3495	150	26	∗	∗	NOUN
ejpam-3495	150	27	x	x	NOUN
ejpam-3495	150	28	)	)	PUNCT
ejpam-3495	150	29	=	=	SYM
ejpam-3495	150	30	(	(	PUNCT
ejpam-3495	150	31	0	0	NUM
ejpam-3495	150	32	∗	∗	PROPN
ejpam-3495	150	33	y	y	NOUN
ejpam-3495	150	34	)	)	PUNCT
ejpam-3495	150	35	∗	∗	NOUN
ejpam-3495	150	36	(	(	PUNCT
ejpam-3495	150	37	(	(	PUNCT
ejpam-3495	150	38	0	0	NUM
ejpam-3495	150	39	∗	∗	NOUN
ejpam-3495	150	40	x	x	NOUN
ejpam-3495	150	41	)	)	PUNCT
ejpam-3495	150	42	∗	∗	NOUN
ejpam-3495	150	43	(	(	PUNCT
ejpam-3495	150	44	0	0	NUM
ejpam-3495	150	45	∗	∗	NOUN
ejpam-3495	150	46	(	(	PUNCT
ejpam-3495	150	47	0	0	NUM
ejpam-3495	150	48	∗	∗	PROPN
ejpam-3495	150	49	y	y	PROPN
ejpam-3495	150	50	)	)	PUNCT
ejpam-3495	150	51	)	)	PUNCT
ejpam-3495	150	52	)	)	PUNCT
ejpam-3495	151	1	=	=	PUNCT
ejpam-3495	151	2	(	(	PUNCT
ejpam-3495	151	3	0	0	NUM
ejpam-3495	151	4	∗	∗	PROPN
ejpam-3495	151	5	y	y	NOUN
ejpam-3495	151	6	)	)	PUNCT
ejpam-3495	151	7	∗	∗	NOUN
ejpam-3495	151	8	(	(	PUNCT
ejpam-3495	151	9	(	(	PUNCT
ejpam-3495	151	10	0	0	NUM
ejpam-3495	151	11	∗	∗	NOUN
ejpam-3495	151	12	x	x	NOUN
ejpam-3495	151	13	)	)	PUNCT
ejpam-3495	151	14	∗	∗	PROPN
ejpam-3495	151	15	y	y	NOUN
ejpam-3495	151	16	)	)	PUNCT
ejpam-3495	151	17	=	=	PUNCT
ejpam-3495	151	18	(	(	PUNCT
ejpam-3495	151	19	0	0	NUM
ejpam-3495	151	20	∗	∗	PROPN
ejpam-3495	151	21	y	y	NOUN
ejpam-3495	151	22	)	)	PUNCT
ejpam-3495	151	23	∗	∗	NOUN
ejpam-3495	151	24	(	(	PUNCT
ejpam-3495	151	25	0	0	NUM
ejpam-3495	151	26	∗	∗	NOUN
ejpam-3495	151	27	(	(	PUNCT
ejpam-3495	151	28	y	y	PROPN
ejpam-3495	151	29	∗	∗	NOUN
ejpam-3495	151	30	(	(	PUNCT
ejpam-3495	151	31	0	0	NUM
ejpam-3495	151	32	∗	∗	NOUN
ejpam-3495	151	33	x	x	NOUN
ejpam-3495	151	34	)	)	PUNCT
ejpam-3495	151	35	)	)	PUNCT
ejpam-3495	151	36	)	)	PUNCT
ejpam-3495	152	1	=	=	PUNCT
ejpam-3495	152	2	(	(	PUNCT
ejpam-3495	152	3	0	0	NUM
ejpam-3495	152	4	∗	∗	PROPN
ejpam-3495	152	5	y	y	NOUN
ejpam-3495	152	6	)	)	PUNCT
ejpam-3495	152	7	∗	∗	NOUN
ejpam-3495	152	8	(	(	PUNCT
ejpam-3495	152	9	0	0	NUM
ejpam-3495	152	10	∗	∗	NOUN
ejpam-3495	152	11	(	(	PUNCT
ejpam-3495	152	12	x	x	X
ejpam-3495	152	13	�	�	PROPN
ejpam-3495	152	14	y	y	PROPN
ejpam-3495	152	15	)	)	PUNCT
ejpam-3495	152	16	)	)	PUNCT
ejpam-3495	153	1	=	=	PRON
ejpam-3495	154	1	(	(	PUNCT
ejpam-3495	154	2	x	x	X
ejpam-3495	154	3	�	�	PROPN
ejpam-3495	154	4	y	y	PROPN
ejpam-3495	154	5	)	)	PUNCT
ejpam-3495	154	6	�	�	PROPN
ejpam-3495	154	7	(	(	PUNCT
ejpam-3495	154	8	0	0	NUM
ejpam-3495	154	9	∗	∗	PROPN
ejpam-3495	154	10	y	y	PROPN
ejpam-3495	154	11	)	)	PUNCT
ejpam-3495	154	12	.	.	PUNCT
ejpam-3495	155	1	(	(	PUNCT
ejpam-3495	155	2	f	f	X
ejpam-3495	155	3	)	)	PUNCT
ejpam-3495	155	4	suppose	suppose	VERB
ejpam-3495	155	5	(	(	PUNCT
ejpam-3495	155	6	x	x	X
ejpam-3495	155	7	,	,	PUNCT
ejpam-3495	155	8	∗	∗	NOUN
ejpam-3495	155	9	,	,	PUNCT
ejpam-3495	155	10	0	0	NUM
ejpam-3495	155	11	)	)	PUNCT
ejpam-3495	155	12	is	be	AUX
ejpam-3495	155	13	commutative	commutative	ADJ
ejpam-3495	155	14	.	.	PUNCT
ejpam-3495	156	1	by	by	ADP
ejpam-3495	156	2	lemma	lemma	PROPN
ejpam-3495	156	3	3.10(b	3.10(b	PROPN
ejpam-3495	156	4	)	)	PUNCT
ejpam-3495	156	5	and	and	CCONJ
ejpam-3495	156	6	definition	definition	NOUN
ejpam-3495	156	7	2.5	2.5	NUM
ejpam-3495	156	8	,	,	PUNCT
ejpam-3495	156	9	x	x	PUNCT
ejpam-3495	156	10	�	�	PROPN
ejpam-3495	156	11	y	y	PROPN
ejpam-3495	156	12	=	=	SYM
ejpam-3495	156	13	y	y	PROPN
ejpam-3495	156	14	∗	∗	NOUN
ejpam-3495	156	15	(	(	PUNCT
ejpam-3495	156	16	0	0	NUM
ejpam-3495	156	17	∗	∗	NOUN
ejpam-3495	156	18	x	x	NOUN
ejpam-3495	156	19	)	)	PUNCT
ejpam-3495	156	20	=	=	SYM
ejpam-3495	156	21	x	x	SYM
ejpam-3495	156	22	∗	∗	NOUN
ejpam-3495	156	23	(	(	PUNCT
ejpam-3495	156	24	0	0	NUM
ejpam-3495	156	25	∗	∗	PROPN
ejpam-3495	156	26	y	y	PROPN
ejpam-3495	156	27	)	)	PUNCT
ejpam-3495	156	28	.	.	PUNCT
ejpam-3495	157	1	�	�	PROPN
ejpam-3495	157	2	notice	notice	VERB
ejpam-3495	157	3	that	that	SCONJ
ejpam-3495	157	4	in	in	ADP
ejpam-3495	157	5	example	example	NOUN
ejpam-3495	157	6	3.6	3.6	NUM
ejpam-3495	157	7	,	,	PUNCT
ejpam-3495	157	8	x	x	PRON
ejpam-3495	157	9	is	be	AUX
ejpam-3495	157	10	commutative	commutative	ADJ
ejpam-3495	157	11	and	and	CCONJ
ejpam-3495	157	12	1	1	NUM
ejpam-3495	157	13	�	�	PROPN
ejpam-3495	157	14	1	1	NUM
ejpam-3495	157	15	=	=	SYM
ejpam-3495	157	16	2	2	NUM
ejpam-3495	157	17	6=	6=	NUM
ejpam-3495	157	18	0	0	NUM
ejpam-3495	157	19	.	.	PUNCT
ejpam-3495	158	1	hence	hence	ADV
ejpam-3495	158	2	,	,	PUNCT
ejpam-3495	158	3	we	we	PRON
ejpam-3495	158	4	have	have	AUX
ejpam-3495	158	5	found	find	VERB
ejpam-3495	158	6	x	x	X
ejpam-3495	158	7	=	=	SYM
ejpam-3495	158	8	1	1	NUM
ejpam-3495	158	9	∈	∈	NOUN
ejpam-3495	158	10	x	x	PUNCT
ejpam-3495	158	11	such	such	ADJ
ejpam-3495	158	12	that	that	SCONJ
ejpam-3495	158	13	x	x	SYM
ejpam-3495	158	14	�	�	PROPN
ejpam-3495	158	15	x	x	SYM
ejpam-3495	158	16	6=	6=	ADP
ejpam-3495	158	17	0	0	NUM
ejpam-3495	158	18	.	.	PUNCT
ejpam-3495	159	1	also	also	ADV
ejpam-3495	159	2	,	,	PUNCT
ejpam-3495	159	3	1	1	NUM
ejpam-3495	159	4	6=	6=	SYM
ejpam-3495	159	5	3	3	NUM
ejpam-3495	159	6	=	=	SYM
ejpam-3495	159	7	0	0	NUM
ejpam-3495	159	8	∗	∗	NOUN
ejpam-3495	159	9	1	1	NUM
ejpam-3495	159	10	.	.	PUNCT
ejpam-3495	160	1	thus	thus	ADV
ejpam-3495	160	2	,	,	PUNCT
ejpam-3495	160	3	we	we	PRON
ejpam-3495	160	4	have	have	VERB
ejpam-3495	160	5	the	the	DET
ejpam-3495	160	6	following	follow	VERB
ejpam-3495	160	7	remark	remark	NOUN
ejpam-3495	160	8	.	.	PUNCT
ejpam-3495	161	1	remark	remark	PROPN
ejpam-3495	161	2	3.11	3.11	NUM
ejpam-3495	161	3	.	.	PUNCT
ejpam-3495	162	1	if	if	SCONJ
ejpam-3495	162	2	(	(	PUNCT
ejpam-3495	162	3	x	x	NOUN
ejpam-3495	162	4	,	,	PUNCT
ejpam-3495	162	5	∗	∗	NOUN
ejpam-3495	162	6	,	,	PUNCT
ejpam-3495	162	7	�	�	PROPN
ejpam-3495	162	8	,	,	PUNCT
ejpam-3495	162	9	0	0	NUM
ejpam-3495	162	10	)	)	PUNCT
ejpam-3495	162	11	is	be	AUX
ejpam-3495	162	12	a	a	DET
ejpam-3495	162	13	companion	companion	NOUN
ejpam-3495	162	14	b	b	NOUN
ejpam-3495	162	15	-	-	PUNCT
ejpam-3495	162	16	algebra	algebra	NOUN
ejpam-3495	162	17	,	,	PUNCT
ejpam-3495	162	18	then	then	ADV
ejpam-3495	162	19	(	(	PUNCT
ejpam-3495	162	20	x	x	X
ejpam-3495	162	21	,	,	PUNCT
ejpam-3495	162	22	�	�	PROPN
ejpam-3495	162	23	,	,	PUNCT
ejpam-3495	162	24	0	0	NUM
ejpam-3495	162	25	)	)	PUNCT
ejpam-3495	162	26	is	be	AUX
ejpam-3495	162	27	not	not	PART
ejpam-3495	162	28	necessarily	necessarily	ADV
ejpam-3495	162	29	a	a	DET
ejpam-3495	162	30	b	b	NOUN
ejpam-3495	162	31	-	-	PUNCT
ejpam-3495	162	32	algebra	algebra	NOUN
ejpam-3495	162	33	.	.	PUNCT
ejpam-3495	163	1	l.d	l.d	PROPN
ejpam-3495	163	2	.	.	PROPN
ejpam-3495	163	3	naingue	naingue	PROPN
ejpam-3495	163	4	,	,	PUNCT
ejpam-3495	163	5	j.p	j.p	PROPN
ejpam-3495	163	6	.	.	PROPN
ejpam-3495	163	7	vilela	vilela	PROPN
ejpam-3495	163	8	/	/	SYM
ejpam-3495	163	9	eur	eur	PROPN
ejpam-3495	163	10	.	.	PUNCT
ejpam-3495	164	1	j.	j.	PROPN
ejpam-3495	164	2	pure	pure	PROPN
ejpam-3495	164	3	appl	appl	PROPN
ejpam-3495	164	4	.	.	PROPN
ejpam-3495	164	5	math	math	PROPN
ejpam-3495	164	6	,	,	PUNCT
ejpam-3495	164	7	12	12	NUM
ejpam-3495	164	8	(	(	PUNCT
ejpam-3495	164	9	3	3	NUM
ejpam-3495	164	10	)	)	PUNCT
ejpam-3495	164	11	(	(	PUNCT
ejpam-3495	164	12	2019	2019	NUM
ejpam-3495	164	13	)	)	PUNCT
ejpam-3495	164	14	,	,	PUNCT
ejpam-3495	164	15	1248	1248	NUM
ejpam-3495	164	16	-	-	SYM
ejpam-3495	164	17	1259	1259	NUM
ejpam-3495	164	18	1253	1253	NUM
ejpam-3495	164	19	proposition	proposition	NOUN
ejpam-3495	164	20	3.12	3.12	NUM
ejpam-3495	164	21	.	.	PUNCT
ejpam-3495	165	1	suppose	suppose	VERB
ejpam-3495	165	2	(	(	PUNCT
ejpam-3495	165	3	x	x	X
ejpam-3495	165	4	,	,	PUNCT
ejpam-3495	165	5	∗	∗	NOUN
ejpam-3495	165	6	,	,	PUNCT
ejpam-3495	165	7	�	�	PROPN
ejpam-3495	165	8	,	,	PUNCT
ejpam-3495	165	9	0	0	NUM
ejpam-3495	165	10	)	)	PUNCT
ejpam-3495	165	11	is	be	AUX
ejpam-3495	165	12	a	a	DET
ejpam-3495	165	13	companion	companion	NOUN
ejpam-3495	165	14	b	b	NOUN
ejpam-3495	165	15	-	-	PUNCT
ejpam-3495	165	16	algebra	algebra	NOUN
ejpam-3495	165	17	.	.	PUNCT
ejpam-3495	166	1	if	if	SCONJ
ejpam-3495	166	2	(	(	PUNCT
ejpam-3495	166	3	x	x	X
ejpam-3495	166	4	,	,	PUNCT
ejpam-3495	166	5	∗	∗	NOUN
ejpam-3495	166	6	,	,	PUNCT
ejpam-3495	166	7	0	0	NUM
ejpam-3495	166	8	)	)	PUNCT
ejpam-3495	166	9	is	be	AUX
ejpam-3495	166	10	a	a	DET
ejpam-3495	166	11	commutative	commutative	ADJ
ejpam-3495	166	12	b	b	NOUN
ejpam-3495	166	13	-	-	PUNCT
ejpam-3495	166	14	algebra	algebra	NOUN
ejpam-3495	166	15	and	and	CCONJ
ejpam-3495	166	16	x	x	SYM
ejpam-3495	166	17	=	=	SYM
ejpam-3495	166	18	0	0	NUM
ejpam-3495	166	19	∗	∗	NOUN
ejpam-3495	166	20	x	x	PUNCT
ejpam-3495	166	21	for	for	ADP
ejpam-3495	166	22	any	any	DET
ejpam-3495	166	23	x	x	SYM
ejpam-3495	166	24	∈	∈	PROPN
ejpam-3495	166	25	x	x	NOUN
ejpam-3495	166	26	,	,	PUNCT
ejpam-3495	166	27	then	then	ADV
ejpam-3495	166	28	(	(	PUNCT
ejpam-3495	166	29	x	x	NOUN
ejpam-3495	166	30	,	,	PUNCT
ejpam-3495	166	31	�	�	PROPN
ejpam-3495	166	32	,	,	PUNCT
ejpam-3495	166	33	0	0	NUM
ejpam-3495	166	34	)	)	PUNCT
ejpam-3495	166	35	is	be	AUX
ejpam-3495	166	36	a	a	DET
ejpam-3495	166	37	b	b	NOUN
ejpam-3495	166	38	-	-	PUNCT
ejpam-3495	166	39	algebra	algebra	NOUN
ejpam-3495	166	40	.	.	PUNCT
ejpam-3495	167	1	proof	proof	NOUN
ejpam-3495	167	2	:	:	PUNCT
ejpam-3495	167	3	suppose	suppose	VERB
ejpam-3495	167	4	(	(	PUNCT
ejpam-3495	167	5	x	x	X
ejpam-3495	167	6	,	,	PUNCT
ejpam-3495	167	7	∗	∗	NOUN
ejpam-3495	167	8	,	,	PUNCT
ejpam-3495	167	9	0	0	NUM
ejpam-3495	167	10	)	)	PUNCT
ejpam-3495	167	11	is	be	AUX
ejpam-3495	167	12	a	a	DET
ejpam-3495	167	13	commutative	commutative	ADJ
ejpam-3495	167	14	b	b	NOUN
ejpam-3495	167	15	-algebra	-algebra	NOUN
ejpam-3495	167	16	and	and	CCONJ
ejpam-3495	167	17	x	x	NOUN
ejpam-3495	167	18	,	,	PUNCT
ejpam-3495	167	19	y	y	PROPN
ejpam-3495	167	20	∈	∈	PROPN
ejpam-3495	167	21	x.	x.	NOUN
ejpam-3495	167	22	by	by	ADP
ejpam-3495	167	23	lemma	lemma	PROPN
ejpam-3495	167	24	3.10(b	3.10(b	PROPN
ejpam-3495	167	25	)	)	PUNCT
ejpam-3495	167	26	,	,	PUNCT
ejpam-3495	167	27	definition	definition	NOUN
ejpam-3495	167	28	2.5	2.5	NUM
ejpam-3495	167	29	and	and	CCONJ
ejpam-3495	167	30	by	by	ADP
ejpam-3495	167	31	assumption	assumption	NOUN
ejpam-3495	167	32	,	,	PUNCT
ejpam-3495	167	33	x	x	X
ejpam-3495	167	34	�	�	PROPN
ejpam-3495	167	35	y	y	PROPN
ejpam-3495	167	36	=	=	SYM
ejpam-3495	167	37	y	y	PROPN
ejpam-3495	167	38	∗	∗	NOUN
ejpam-3495	167	39	(	(	PUNCT
ejpam-3495	167	40	0	0	NUM
ejpam-3495	167	41	∗	∗	NOUN
ejpam-3495	167	42	x	x	NOUN
ejpam-3495	167	43	)	)	PUNCT
ejpam-3495	167	44	=	=	SYM
ejpam-3495	167	45	x	x	SYM
ejpam-3495	167	46	∗	∗	NOUN
ejpam-3495	167	47	(	(	PUNCT
ejpam-3495	167	48	0	0	NUM
ejpam-3495	167	49	∗	∗	NUM
ejpam-3495	167	50	y	y	NOUN
ejpam-3495	167	51	)	)	PUNCT
ejpam-3495	167	52	=	=	PUNCT
ejpam-3495	168	1	x	x	X
ejpam-3495	168	2	∗	∗	NOUN
ejpam-3495	168	3	y.	y.	PROPN
ejpam-3495	168	4	hence	hence	ADV
ejpam-3495	168	5	,	,	PUNCT
ejpam-3495	168	6	(	(	PUNCT
ejpam-3495	168	7	x	x	X
ejpam-3495	168	8	,	,	PUNCT
ejpam-3495	168	9	�	�	PROPN
ejpam-3495	168	10	,	,	PUNCT
ejpam-3495	168	11	0	0	NUM
ejpam-3495	168	12	)	)	PUNCT
ejpam-3495	168	13	=	=	SYM
ejpam-3495	168	14	(	(	PUNCT
ejpam-3495	168	15	x	x	X
ejpam-3495	168	16	,	,	PUNCT
ejpam-3495	168	17	∗	∗	NOUN
ejpam-3495	168	18	,	,	PUNCT
ejpam-3495	168	19	0	0	NUM
ejpam-3495	168	20	)	)	PUNCT
ejpam-3495	168	21	is	be	AUX
ejpam-3495	168	22	a	a	DET
ejpam-3495	168	23	b	b	NOUN
ejpam-3495	168	24	-algebra	-algebra	NOUN
ejpam-3495	168	25	.	.	PUNCT
ejpam-3495	169	1	�	�	PROPN
ejpam-3495	169	2	example	example	NOUN
ejpam-3495	169	3	3.13	3.13	NUM
ejpam-3495	169	4	.	.	PUNCT
ejpam-3495	170	1	consider	consider	VERB
ejpam-3495	170	2	the	the	DET
ejpam-3495	170	3	companion	companion	NOUN
ejpam-3495	170	4	b	b	PROPN
ejpam-3495	170	5	-algebra	-algebra	PROPN
ejpam-3495	170	6	in	in	ADP
ejpam-3495	170	7	example	example	NOUN
ejpam-3495	170	8	3.2	3.2	NUM
ejpam-3495	170	9	and	and	CCONJ
ejpam-3495	170	10	consider	consider	VERB
ejpam-3495	170	11	the	the	DET
ejpam-3495	170	12	following	follow	VERB
ejpam-3495	170	13	table	table	NOUN
ejpam-3495	170	14	of	of	ADP
ejpam-3495	170	15	operation	operation	NOUN
ejpam-3495	170	16	:	:	PUNCT
ejpam-3495	171	1	⊗	⊗	NOUN
ejpam-3495	171	2	0	0	NUM
ejpam-3495	171	3	1	1	NUM
ejpam-3495	171	4	2	2	NUM
ejpam-3495	171	5	3	3	NUM
ejpam-3495	171	6	4	4	NUM
ejpam-3495	171	7	5	5	NUM
ejpam-3495	171	8	0	0	NUM
ejpam-3495	171	9	0	0	NUM
ejpam-3495	171	10	1	1	NUM
ejpam-3495	171	11	2	2	NUM
ejpam-3495	171	12	3	3	NUM
ejpam-3495	171	13	4	4	NUM
ejpam-3495	171	14	5	5	NUM
ejpam-3495	171	15	1	1	NUM
ejpam-3495	171	16	1	1	NUM
ejpam-3495	171	17	2	2	NUM
ejpam-3495	171	18	0	0	NUM
ejpam-3495	171	19	4	4	NUM
ejpam-3495	171	20	5	5	NUM
ejpam-3495	171	21	3	3	NUM
ejpam-3495	171	22	2	2	NUM
ejpam-3495	171	23	2	2	NUM
ejpam-3495	171	24	0	0	NUM
ejpam-3495	171	25	1	1	NUM
ejpam-3495	171	26	5	5	NUM
ejpam-3495	171	27	3	3	NUM
ejpam-3495	171	28	4	4	NUM
ejpam-3495	171	29	3	3	NUM
ejpam-3495	171	30	3	3	NUM
ejpam-3495	171	31	4	4	NUM
ejpam-3495	171	32	4	4	NUM
ejpam-3495	171	33	0	0	NUM
ejpam-3495	171	34	2	2	NUM
ejpam-3495	171	35	1	1	NUM
ejpam-3495	171	36	4	4	NUM
ejpam-3495	171	37	4	4	NUM
ejpam-3495	171	38	3	3	NUM
ejpam-3495	171	39	5	5	NUM
ejpam-3495	171	40	1	1	NUM
ejpam-3495	171	41	0	0	NUM
ejpam-3495	171	42	2	2	NUM
ejpam-3495	171	43	5	5	NUM
ejpam-3495	171	44	5	5	NUM
ejpam-3495	171	45	4	4	NUM
ejpam-3495	171	46	3	3	NUM
ejpam-3495	171	47	2	2	NUM
ejpam-3495	171	48	1	1	NUM
ejpam-3495	171	49	0	0	NUM
ejpam-3495	171	50	applying	apply	VERB
ejpam-3495	171	51	theorem	theorem	NOUN
ejpam-3495	171	52	2.6	2.6	NUM
ejpam-3495	171	53	,	,	PUNCT
ejpam-3495	171	54	we	we	PRON
ejpam-3495	171	55	conclude	conclude	VERB
ejpam-3495	171	56	that	that	PRON
ejpam-3495	171	57	(	(	PUNCT
ejpam-3495	171	58	x,⊗	x,⊗	NOUN
ejpam-3495	171	59	,	,	PUNCT
ejpam-3495	171	60	0	0	NUM
ejpam-3495	171	61	)	)	PUNCT
ejpam-3495	171	62	is	be	AUX
ejpam-3495	171	63	the	the	DET
ejpam-3495	171	64	group	group	NOUN
ejpam-3495	171	65	where	where	SCONJ
ejpam-3495	171	66	x⊗y	x⊗y	PUNCT
ejpam-3495	171	67	=	=	SYM
ejpam-3495	171	68	x∗(0∗y	x∗(0∗y	PROPN
ejpam-3495	171	69	)	)	PUNCT
ejpam-3495	171	70	.	.	PUNCT
ejpam-3495	172	1	note	note	VERB
ejpam-3495	172	2	that	that	SCONJ
ejpam-3495	172	3	�	�	PROPN
ejpam-3495	172	4	6=	6=	PROPN
ejpam-3495	172	5	⊗	⊗	PROPN
ejpam-3495	172	6	since	since	SCONJ
ejpam-3495	172	7	1	1	NUM
ejpam-3495	172	8	�	�	PROPN
ejpam-3495	172	9	5	5	NUM
ejpam-3495	172	10	=	=	SYM
ejpam-3495	172	11	4	4	NUM
ejpam-3495	172	12	6=	6=	SYM
ejpam-3495	172	13	3	3	NUM
ejpam-3495	172	14	=	=	SYM
ejpam-3495	172	15	1⊗	1⊗	NUM
ejpam-3495	172	16	5	5	NUM
ejpam-3495	172	17	.	.	PUNCT
ejpam-3495	173	1	thus	thus	ADV
ejpam-3495	173	2	,	,	PUNCT
ejpam-3495	173	3	by	by	ADP
ejpam-3495	173	4	definition	definition	NOUN
ejpam-3495	173	5	of	of	ADP
ejpam-3495	173	6	⊗	⊗	PROPN
ejpam-3495	173	7	,	,	PUNCT
ejpam-3495	173	8	x	x	PROPN
ejpam-3495	173	9	�	�	PROPN
ejpam-3495	173	10	y	y	PROPN
ejpam-3495	173	11	6=	6=	PROPN
ejpam-3495	173	12	x⊗	x⊗	PROPN
ejpam-3495	173	13	y	y	PROPN
ejpam-3495	173	14	=	=	PUNCT
ejpam-3495	173	15	x	x	SYM
ejpam-3495	173	16	∗	∗	NOUN
ejpam-3495	173	17	(	(	PUNCT
ejpam-3495	173	18	0	0	NUM
ejpam-3495	173	19	∗	∗	PROPN
ejpam-3495	173	20	y	y	PROPN
ejpam-3495	173	21	)	)	PUNCT
ejpam-3495	173	22	.	.	PUNCT
ejpam-3495	174	1	hence	hence	ADV
ejpam-3495	174	2	,	,	PUNCT
ejpam-3495	174	3	we	we	PRON
ejpam-3495	174	4	can	can	AUX
ejpam-3495	174	5	not	not	PART
ejpam-3495	174	6	apply	apply	VERB
ejpam-3495	174	7	theorem	theorem	VERB
ejpam-3495	174	8	2.6	2.6	NUM
ejpam-3495	174	9	to	to	PART
ejpam-3495	174	10	immediately	immediately	ADV
ejpam-3495	174	11	conclude	conclude	VERB
ejpam-3495	174	12	that	that	PRON
ejpam-3495	174	13	(	(	PUNCT
ejpam-3495	174	14	x	x	X
ejpam-3495	174	15	,	,	PUNCT
ejpam-3495	174	16	�	�	PROPN
ejpam-3495	174	17	,	,	PUNCT
ejpam-3495	174	18	0	0	NUM
ejpam-3495	174	19	)	)	PUNCT
ejpam-3495	174	20	is	be	AUX
ejpam-3495	174	21	a	a	DET
ejpam-3495	174	22	group	group	NOUN
ejpam-3495	174	23	.	.	PUNCT
ejpam-3495	175	1	however	however	ADV
ejpam-3495	175	2	,	,	PUNCT
ejpam-3495	175	3	the	the	DET
ejpam-3495	175	4	following	follow	VERB
ejpam-3495	175	5	theorem	theorem	NOUN
ejpam-3495	175	6	says	say	VERB
ejpam-3495	175	7	so	so	ADV
ejpam-3495	175	8	.	.	PUNCT
ejpam-3495	176	1	theorem	theorem	PROPN
ejpam-3495	176	2	3.14	3.14	NUM
ejpam-3495	176	3	.	.	PUNCT
ejpam-3495	177	1	let	let	VERB
ejpam-3495	177	2	(	(	PUNCT
ejpam-3495	177	3	x	x	X
ejpam-3495	177	4	,	,	PUNCT
ejpam-3495	177	5	∗	∗	NOUN
ejpam-3495	177	6	,	,	PUNCT
ejpam-3495	177	7	�	�	PROPN
ejpam-3495	177	8	,	,	PUNCT
ejpam-3495	177	9	0	0	NUM
ejpam-3495	177	10	)	)	PUNCT
ejpam-3495	177	11	be	be	AUX
ejpam-3495	177	12	a	a	DET
ejpam-3495	177	13	companion	companion	NOUN
ejpam-3495	177	14	b	b	NOUN
ejpam-3495	177	15	-	-	PUNCT
ejpam-3495	177	16	algebra	algebra	NOUN
ejpam-3495	177	17	.	.	PUNCT
ejpam-3495	178	1	then	then	ADV
ejpam-3495	178	2	(	(	PUNCT
ejpam-3495	178	3	x	x	X
ejpam-3495	178	4	,	,	PUNCT
ejpam-3495	178	5	�	�	PROPN
ejpam-3495	178	6	,	,	PUNCT
ejpam-3495	178	7	0	0	NUM
ejpam-3495	178	8	)	)	PUNCT
ejpam-3495	178	9	is	be	AUX
ejpam-3495	178	10	a	a	DET
ejpam-3495	178	11	group	group	NOUN
ejpam-3495	178	12	.	.	PUNCT
ejpam-3495	179	1	proof	proof	NOUN
ejpam-3495	179	2	:	:	PUNCT
ejpam-3495	179	3	note	note	VERB
ejpam-3495	179	4	that	that	SCONJ
ejpam-3495	179	5	x	x	SYM
ejpam-3495	179	6	6=	6=	ADP
ejpam-3495	179	7	∅	∅	NOUN
ejpam-3495	179	8	since	since	SCONJ
ejpam-3495	179	9	0	0	NUM
ejpam-3495	179	10	∈	∈	NOUN
ejpam-3495	179	11	x.	x.	NOUN
ejpam-3495	179	12	by	by	ADP
ejpam-3495	179	13	lemma	lemma	PROPN
ejpam-3495	179	14	3.10(d	3.10(d	PROPN
ejpam-3495	179	15	)	)	PUNCT
ejpam-3495	179	16	the	the	DET
ejpam-3495	179	17	companion	companion	PROPN
ejpam-3495	179	18	operation	operation	NOUN
ejpam-3495	179	19	�	�	PROPN
ejpam-3495	179	20	is	be	AUX
ejpam-3495	179	21	associative	associative	ADJ
ejpam-3495	179	22	.	.	PUNCT
ejpam-3495	180	1	note	note	VERB
ejpam-3495	180	2	that	that	SCONJ
ejpam-3495	180	3	by	by	ADP
ejpam-3495	180	4	lemma	lemma	PROPN
ejpam-3495	180	5	3.10(a	3.10(a	PROPN
ejpam-3495	180	6	)	)	PUNCT
ejpam-3495	180	7	,	,	PUNCT
ejpam-3495	180	8	0	0	NUM
ejpam-3495	180	9	acts	act	VERB
ejpam-3495	180	10	as	as	ADP
ejpam-3495	180	11	the	the	DET
ejpam-3495	180	12	�	�	PROPN
ejpam-3495	180	13	-identity	-identity	PROPN
ejpam-3495	180	14	element	element	NOUN
ejpam-3495	180	15	in	in	ADP
ejpam-3495	180	16	(	(	PUNCT
ejpam-3495	180	17	x	x	NOUN
ejpam-3495	180	18	,	,	PUNCT
ejpam-3495	180	19	�	�	PROPN
ejpam-3495	180	20	,	,	PUNCT
ejpam-3495	180	21	0	0	NUM
ejpam-3495	180	22	)	)	PUNCT
ejpam-3495	180	23	.	.	PUNCT
ejpam-3495	181	1	find	find	VERB
ejpam-3495	181	2	y	y	PRON
ejpam-3495	181	3	such	such	ADJ
ejpam-3495	181	4	that	that	SCONJ
ejpam-3495	181	5	x	x	X
ejpam-3495	181	6	�	�	PROPN
ejpam-3495	181	7	y	y	NOUN
ejpam-3495	181	8	=	=	SYM
ejpam-3495	181	9	0	0	PROPN
ejpam-3495	181	10	and	and	CCONJ
ejpam-3495	181	11	y	y	PROPN
ejpam-3495	181	12	�	�	PROPN
ejpam-3495	181	13	x	x	PUNCT
ejpam-3495	182	1	=	=	PUNCT
ejpam-3495	182	2	0	0	X
ejpam-3495	182	3	.	.	PUNCT
ejpam-3495	182	4	suppose	suppose	VERB
ejpam-3495	182	5	x	x	X
ejpam-3495	182	6	�	�	PROPN
ejpam-3495	182	7	y	y	PROPN
ejpam-3495	182	8	=	=	NOUN
ejpam-3495	182	9	0	0	PROPN
ejpam-3495	182	10	.	.	PUNCT
ejpam-3495	183	1	then	then	ADV
ejpam-3495	183	2	by	by	ADP
ejpam-3495	183	3	lemma	lemma	PROPN
ejpam-3495	183	4	3.10(b	3.10(b	PROPN
ejpam-3495	183	5	)	)	PUNCT
ejpam-3495	183	6	,	,	PUNCT
ejpam-3495	183	7	y	y	PROPN
ejpam-3495	183	8	∗	∗	X
ejpam-3495	183	9	(	(	PUNCT
ejpam-3495	183	10	0	0	NUM
ejpam-3495	183	11	∗	∗	NOUN
ejpam-3495	183	12	x	x	NOUN
ejpam-3495	183	13	)	)	PUNCT
ejpam-3495	183	14	=	=	SYM
ejpam-3495	183	15	0	0	X
ejpam-3495	183	16	.	.	PUNCT
ejpam-3495	184	1	so	so	ADV
ejpam-3495	184	2	,	,	PUNCT
ejpam-3495	184	3	by	by	ADP
ejpam-3495	184	4	theorem	theorem	NOUN
ejpam-3495	184	5	2.3(d	2.3(d	NUM
ejpam-3495	184	6	)	)	PUNCT
ejpam-3495	184	7	,	,	PUNCT
ejpam-3495	184	8	y	y	PROPN
ejpam-3495	184	9	=	=	SYM
ejpam-3495	184	10	0	0	NUM
ejpam-3495	184	11	∗	∗	NOUN
ejpam-3495	184	12	x.	x.	NOUN
ejpam-3495	184	13	also	also	ADV
ejpam-3495	184	14	,	,	PUNCT
ejpam-3495	184	15	suppose	suppose	VERB
ejpam-3495	184	16	y	y	PROPN
ejpam-3495	184	17	�	�	PROPN
ejpam-3495	184	18	x	x	PUNCT
ejpam-3495	185	1	=	=	PUNCT
ejpam-3495	185	2	0	0	X
ejpam-3495	185	3	.	.	PUNCT
ejpam-3495	185	4	by	by	ADP
ejpam-3495	185	5	lemma	lemma	PROPN
ejpam-3495	185	6	3.10(b	3.10(b	PROPN
ejpam-3495	185	7	)	)	PUNCT
ejpam-3495	185	8	,	,	PUNCT
ejpam-3495	185	9	x	x	X
ejpam-3495	185	10	∗	∗	NOUN
ejpam-3495	185	11	(	(	PUNCT
ejpam-3495	185	12	0	0	NUM
ejpam-3495	185	13	∗	∗	NUM
ejpam-3495	185	14	y	y	NOUN
ejpam-3495	185	15	)	)	PUNCT
ejpam-3495	185	16	=	=	SYM
ejpam-3495	185	17	0	0	X
ejpam-3495	185	18	.	.	PUNCT
ejpam-3495	185	19	then	then	ADV
ejpam-3495	185	20	by	by	ADP
ejpam-3495	185	21	theorem	theorem	NOUN
ejpam-3495	185	22	2.3(f	2.3(f	NUM
ejpam-3495	185	23	)	)	PUNCT
ejpam-3495	185	24	,	,	PUNCT
ejpam-3495	185	25	(	(	PUNCT
ejpam-3495	185	26	0	0	NUM
ejpam-3495	185	27	∗	∗	NOUN
ejpam-3495	185	28	(	(	PUNCT
ejpam-3495	185	29	0	0	NUM
ejpam-3495	185	30	∗x	∗x	NOUN
ejpam-3495	185	31	)	)	PUNCT
ejpam-3495	185	32	)	)	PUNCT
ejpam-3495	185	33	∗	∗	NOUN
ejpam-3495	185	34	(	(	PUNCT
ejpam-3495	185	35	0	0	NUM
ejpam-3495	185	36	∗	∗	NUM
ejpam-3495	185	37	y	y	NOUN
ejpam-3495	185	38	)	)	PUNCT
ejpam-3495	185	39	=	=	SYM
ejpam-3495	185	40	0	0	NUM
ejpam-3495	185	41	and	and	CCONJ
ejpam-3495	185	42	by	by	ADP
ejpam-3495	185	43	theorem	theorem	NOUN
ejpam-3495	185	44	2.3(d	2.3(d	NUM
ejpam-3495	185	45	)	)	PUNCT
ejpam-3495	185	46	,	,	PUNCT
ejpam-3495	185	47	0	0	NUM
ejpam-3495	185	48	∗	∗	NOUN
ejpam-3495	185	49	(	(	PUNCT
ejpam-3495	185	50	0	0	NUM
ejpam-3495	185	51	∗	∗	NOUN
ejpam-3495	185	52	x	x	NOUN
ejpam-3495	185	53	)	)	PUNCT
ejpam-3495	185	54	=	=	SYM
ejpam-3495	185	55	0	0	NUM
ejpam-3495	185	56	∗	∗	NOUN
ejpam-3495	185	57	y.	y.	PROPN
ejpam-3495	185	58	hence	hence	ADV
ejpam-3495	185	59	,	,	PUNCT
ejpam-3495	185	60	by	by	ADP
ejpam-3495	185	61	theorem	theorem	NOUN
ejpam-3495	185	62	2.3(e	2.3(e	NUM
ejpam-3495	185	63	)	)	PUNCT
ejpam-3495	185	64	,	,	PUNCT
ejpam-3495	185	65	y	y	PROPN
ejpam-3495	185	66	=	=	SYM
ejpam-3495	185	67	0	0	NUM
ejpam-3495	185	68	∗	∗	NOUN
ejpam-3495	185	69	x.	x.	PUNCT
ejpam-3495	186	1	thus	thus	ADV
ejpam-3495	186	2	,	,	PUNCT
ejpam-3495	186	3	we	we	PRON
ejpam-3495	186	4	have	have	AUX
ejpam-3495	186	5	found	find	VERB
ejpam-3495	186	6	x−1	x−1	X
ejpam-3495	186	7	=	=	PUNCT
ejpam-3495	186	8	y	y	PROPN
ejpam-3495	186	9	=	=	SYM
ejpam-3495	186	10	0	0	NUM
ejpam-3495	186	11	∗	∗	NOUN
ejpam-3495	186	12	x	x	PUNCT
ejpam-3495	186	13	in	in	ADP
ejpam-3495	186	14	(	(	PUNCT
ejpam-3495	186	15	x	x	NOUN
ejpam-3495	186	16	,	,	PUNCT
ejpam-3495	186	17	�	�	PROPN
ejpam-3495	186	18	,	,	PUNCT
ejpam-3495	186	19	0	0	NUM
ejpam-3495	186	20	)	)	PUNCT
ejpam-3495	186	21	.	.	PUNCT
ejpam-3495	187	1	therefore	therefore	ADV
ejpam-3495	187	2	,	,	PUNCT
ejpam-3495	187	3	(	(	PUNCT
ejpam-3495	187	4	x	x	X
ejpam-3495	187	5	,	,	PUNCT
ejpam-3495	187	6	�	�	PROPN
ejpam-3495	187	7	,	,	PUNCT
ejpam-3495	187	8	0	0	NUM
ejpam-3495	187	9	)	)	PUNCT
ejpam-3495	187	10	is	be	AUX
ejpam-3495	187	11	a	a	DET
ejpam-3495	187	12	group	group	NOUN
ejpam-3495	187	13	.	.	PUNCT
ejpam-3495	188	1	�	�	PROPN
ejpam-3495	188	2	remark	remark	VERB
ejpam-3495	188	3	3.15	3.15	NUM
ejpam-3495	188	4	.	.	PUNCT
ejpam-3495	189	1	for	for	ADP
ejpam-3495	189	2	any	any	DET
ejpam-3495	189	3	x	x	SYM
ejpam-3495	189	4	∈	∈	PROPN
ejpam-3495	189	5	x	x	X
ejpam-3495	189	6	,	,	PUNCT
ejpam-3495	189	7	x−1	x−1	PROPN
ejpam-3495	189	8	=	=	NOUN
ejpam-3495	189	9	0∗x	0∗x	PRON
ejpam-3495	189	10	is	be	AUX
ejpam-3495	189	11	called	call	VERB
ejpam-3495	189	12	the	the	DET
ejpam-3495	189	13	inverse	inverse	NOUN
ejpam-3495	189	14	of	of	ADP
ejpam-3495	189	15	x	x	PUNCT
ejpam-3495	189	16	in	in	ADP
ejpam-3495	189	17	the	the	DET
ejpam-3495	189	18	group	group	NOUN
ejpam-3495	189	19	(	(	PUNCT
ejpam-3495	189	20	x	x	NOUN
ejpam-3495	189	21	,	,	PUNCT
ejpam-3495	189	22	�	�	PROPN
ejpam-3495	189	23	,	,	PUNCT
ejpam-3495	189	24	0	0	NUM
ejpam-3495	189	25	)	)	PUNCT
ejpam-3495	189	26	.	.	PUNCT
ejpam-3495	190	1	theorem	theorem	VERB
ejpam-3495	190	2	3.16	3.16	NUM
ejpam-3495	190	3	.	.	PUNCT
ejpam-3495	191	1	let	let	AUX
ejpam-3495	191	2	(	(	PUNCT
ejpam-3495	191	3	g	g	NOUN
ejpam-3495	191	4	,	,	PUNCT
ejpam-3495	191	5	◦	◦	NOUN
ejpam-3495	191	6	)	)	PUNCT
ejpam-3495	191	7	be	be	VERB
ejpam-3495	191	8	a	a	DET
ejpam-3495	191	9	group	group	NOUN
ejpam-3495	191	10	with	with	ADP
ejpam-3495	191	11	identity	identity	NOUN
ejpam-3495	191	12	e.	e.	PROPN
ejpam-3495	191	13	then	then	ADV
ejpam-3495	191	14	g	g	PROPN
ejpam-3495	191	15	determines	determine	VERB
ejpam-3495	191	16	a	a	DET
ejpam-3495	191	17	companion	companion	NOUN
ejpam-3495	191	18	b	b	NOUN
ejpam-3495	191	19	-	-	PUNCT
ejpam-3495	191	20	algebra	algebra	NOUN
ejpam-3495	191	21	(	(	PUNCT
ejpam-3495	191	22	g	g	PROPN
ejpam-3495	191	23	,	,	PUNCT
ejpam-3495	191	24	∗,⊗	∗,⊗	PROPN
ejpam-3495	191	25	,	,	PUNCT
ejpam-3495	191	26	e	e	NOUN
ejpam-3495	191	27	)	)	PUNCT
ejpam-3495	191	28	where	where	SCONJ
ejpam-3495	191	29	x	x	PUNCT
ejpam-3495	191	30	∗	∗	VERB
ejpam-3495	191	31	y	y	NOUN
ejpam-3495	191	32	=	=	PUNCT
ejpam-3495	191	33	x	x	PUNCT
ejpam-3495	191	34	◦	◦	NOUN
ejpam-3495	191	35	y−1	y−1	PROPN
ejpam-3495	191	36	and	and	CCONJ
ejpam-3495	191	37	x⊗	x⊗	PROPN
ejpam-3495	191	38	y	y	PROPN
ejpam-3495	191	39	=	=	SYM
ejpam-3495	191	40	y	y	PROPN
ejpam-3495	191	41	∗	∗	PROPN
ejpam-3495	191	42	x−1	x−1	PROPN
ejpam-3495	191	43	.	.	PUNCT
ejpam-3495	192	1	proof	proof	NOUN
ejpam-3495	192	2	:	:	PUNCT
ejpam-3495	192	3	let	let	VERB
ejpam-3495	192	4	(	(	PUNCT
ejpam-3495	192	5	g	g	NOUN
ejpam-3495	192	6	,	,	PUNCT
ejpam-3495	192	7	◦	◦	NOUN
ejpam-3495	192	8	)	)	PUNCT
ejpam-3495	192	9	be	be	VERB
ejpam-3495	192	10	a	a	DET
ejpam-3495	192	11	group	group	NOUN
ejpam-3495	192	12	with	with	ADP
ejpam-3495	192	13	identity	identity	NOUN
ejpam-3495	192	14	e	e	NOUN
ejpam-3495	192	15	and	and	CCONJ
ejpam-3495	192	16	x	x	NOUN
ejpam-3495	192	17	,	,	PUNCT
ejpam-3495	192	18	y	y	PROPN
ejpam-3495	192	19	∈	∈	PROPN
ejpam-3495	192	20	g.	g.	NOUN
ejpam-3495	192	21	define	define	VERB
ejpam-3495	192	22	two	two	NUM
ejpam-3495	192	23	binary	binary	ADJ
ejpam-3495	192	24	operations	operation	NOUN
ejpam-3495	192	25	∗	∗	NOUN
ejpam-3495	192	26	and	and	CCONJ
ejpam-3495	192	27	⊗	⊗	NUM
ejpam-3495	192	28	by	by	ADP
ejpam-3495	192	29	x	x	PROPN
ejpam-3495	192	30	∗	∗	NOUN
ejpam-3495	192	31	y	y	NOUN
ejpam-3495	192	32	=	=	PUNCT
ejpam-3495	192	33	x	x	PUNCT
ejpam-3495	192	34	◦	◦	NOUN
ejpam-3495	192	35	y−1	y−1	PROPN
ejpam-3495	192	36	and	and	CCONJ
ejpam-3495	192	37	x⊗	x⊗	PROPN
ejpam-3495	192	38	y	y	PROPN
ejpam-3495	192	39	=	=	SYM
ejpam-3495	192	40	y	y	PROPN
ejpam-3495	192	41	∗	∗	PROPN
ejpam-3495	192	42	x−1	x−1	PROPN
ejpam-3495	192	43	.	.	PUNCT
ejpam-3495	193	1	by	by	ADP
ejpam-3495	193	2	theorem	theorem	NOUN
ejpam-3495	193	3	2.7	2.7	NUM
ejpam-3495	193	4	,	,	PUNCT
ejpam-3495	193	5	(	(	PUNCT
ejpam-3495	193	6	g	g	NOUN
ejpam-3495	193	7	,	,	PUNCT
ejpam-3495	193	8	∗	∗	NOUN
ejpam-3495	193	9	,	,	PUNCT
ejpam-3495	193	10	e	e	NOUN
ejpam-3495	193	11	)	)	PUNCT
ejpam-3495	193	12	is	be	AUX
ejpam-3495	193	13	a	a	DET
ejpam-3495	193	14	b	b	NOUN
ejpam-3495	193	15	-algebra	-algebra	NOUN
ejpam-3495	193	16	.	.	PUNCT
ejpam-3495	194	1	observe	observe	VERB
ejpam-3495	194	2	that	that	SCONJ
ejpam-3495	194	3	(	(	PUNCT
ejpam-3495	194	4	(	(	PUNCT
ejpam-3495	194	5	x⊗	x⊗	PROPN
ejpam-3495	194	6	y	y	PROPN
ejpam-3495	194	7	)	)	PUNCT
ejpam-3495	194	8	∗	∗	NOUN
ejpam-3495	194	9	x	x	NOUN
ejpam-3495	194	10	)	)	PUNCT
ejpam-3495	194	11	∗	∗	NOUN
ejpam-3495	194	12	y	y	NOUN
ejpam-3495	194	13	=	=	SYM
ejpam-3495	194	14	(	(	PUNCT
ejpam-3495	194	15	(	(	PUNCT
ejpam-3495	194	16	x⊗	x⊗	PROPN
ejpam-3495	194	17	y	y	PROPN
ejpam-3495	194	18	)	)	PUNCT
ejpam-3495	194	19	◦	◦	NOUN
ejpam-3495	194	20	x−1	x−1	PROPN
ejpam-3495	194	21	)	)	PUNCT
ejpam-3495	194	22	∗	∗	NOUN
ejpam-3495	194	23	y	y	NOUN
ejpam-3495	194	24	=	=	SYM
ejpam-3495	194	25	(	(	PUNCT
ejpam-3495	194	26	(	(	PUNCT
ejpam-3495	194	27	x⊗	x⊗	PROPN
ejpam-3495	194	28	y	y	PROPN
ejpam-3495	194	29	)	)	PUNCT
ejpam-3495	194	30	◦	◦	NOUN
ejpam-3495	194	31	x−1	x−1	PROPN
ejpam-3495	194	32	)	)	PUNCT
ejpam-3495	194	33	◦	◦	NOUN
ejpam-3495	194	34	y−1	y−1	PROPN
ejpam-3495	195	1	=	=	PUNCT
ejpam-3495	195	2	(	(	PUNCT
ejpam-3495	195	3	x⊗	x⊗	PROPN
ejpam-3495	195	4	y	y	X
ejpam-3495	195	5	)	)	PUNCT
ejpam-3495	195	6	◦	◦	NOUN
ejpam-3495	195	7	(	(	PUNCT
ejpam-3495	195	8	x−1	x−1	PROPN
ejpam-3495	195	9	◦	◦	NOUN
ejpam-3495	195	10	y−1	y−1	PROPN
ejpam-3495	195	11	)	)	PUNCT
ejpam-3495	195	12	=	=	PRON
ejpam-3495	195	13	(	(	PUNCT
ejpam-3495	195	14	x⊗	x⊗	PROPN
ejpam-3495	195	15	y	y	X
ejpam-3495	195	16	)	)	PUNCT
ejpam-3495	195	17	◦	◦	NOUN
ejpam-3495	195	18	(	(	PUNCT
ejpam-3495	195	19	y	y	PROPN
ejpam-3495	195	20	◦	◦	NOUN
ejpam-3495	195	21	x)−1	x)−1	PROPN
ejpam-3495	195	22	l.d	l.d	PROPN
ejpam-3495	195	23	.	.	PROPN
ejpam-3495	195	24	naingue	naingue	PROPN
ejpam-3495	195	25	,	,	PUNCT
ejpam-3495	195	26	j.p	j.p	PROPN
ejpam-3495	195	27	.	.	PROPN
ejpam-3495	195	28	vilela	vilela	PROPN
ejpam-3495	195	29	/	/	SYM
ejpam-3495	195	30	eur	eur	PROPN
ejpam-3495	195	31	.	.	PUNCT
ejpam-3495	196	1	j.	j.	PROPN
ejpam-3495	196	2	pure	pure	PROPN
ejpam-3495	196	3	appl	appl	PROPN
ejpam-3495	196	4	.	.	PROPN
ejpam-3495	196	5	math	math	PROPN
ejpam-3495	196	6	,	,	PUNCT
ejpam-3495	196	7	12	12	NUM
ejpam-3495	196	8	(	(	PUNCT
ejpam-3495	196	9	3	3	NUM
ejpam-3495	196	10	)	)	PUNCT
ejpam-3495	196	11	(	(	PUNCT
ejpam-3495	196	12	2019	2019	NUM
ejpam-3495	196	13	)	)	PUNCT
ejpam-3495	196	14	,	,	PUNCT
ejpam-3495	196	15	1248	1248	NUM
ejpam-3495	196	16	-	-	SYM
ejpam-3495	196	17	1259	1259	NUM
ejpam-3495	196	18	1254	1254	NUM
ejpam-3495	196	19	=	=	SYM
ejpam-3495	196	20	(	(	PUNCT
ejpam-3495	196	21	y	y	PROPN
ejpam-3495	196	22	∗	∗	PROPN
ejpam-3495	196	23	x−1	x−1	PROPN
ejpam-3495	196	24	)	)	PUNCT
ejpam-3495	197	1	◦	◦	NOUN
ejpam-3495	197	2	(	(	PUNCT
ejpam-3495	197	3	y	y	PROPN
ejpam-3495	197	4	◦	◦	NOUN
ejpam-3495	197	5	x)−1	x)−1	X
ejpam-3495	198	1	=	=	SYM
ejpam-3495	198	2	(	(	PUNCT
ejpam-3495	198	3	y	y	PROPN
ejpam-3495	198	4	◦	◦	NOUN
ejpam-3495	198	5	(	(	PUNCT
ejpam-3495	198	6	x−1)−1	x−1)−1	NOUN
ejpam-3495	198	7	)	)	PUNCT
ejpam-3495	198	8	◦	◦	NOUN
ejpam-3495	198	9	(	(	PUNCT
ejpam-3495	198	10	y	y	PROPN
ejpam-3495	198	11	◦	◦	NOUN
ejpam-3495	198	12	x)−1	x)−1	X
ejpam-3495	199	1	=	=	SYM
ejpam-3495	199	2	(	(	PUNCT
ejpam-3495	199	3	y	y	PROPN
ejpam-3495	199	4	◦	◦	NOUN
ejpam-3495	199	5	x	x	NOUN
ejpam-3495	199	6	)	)	PUNCT
ejpam-3495	199	7	◦	◦	NOUN
ejpam-3495	199	8	(	(	PUNCT
ejpam-3495	199	9	y	y	PROPN
ejpam-3495	199	10	◦	◦	NOUN
ejpam-3495	199	11	x)−1	x)−1	X
ejpam-3495	199	12	=	=	SYM
ejpam-3495	199	13	e.	e.	PROPN
ejpam-3495	199	14	hence	hence	ADV
ejpam-3495	199	15	,	,	PUNCT
ejpam-3495	199	16	⊗	⊗	PROPN
ejpam-3495	199	17	is	be	AUX
ejpam-3495	199	18	a	a	DET
ejpam-3495	199	19	subcompanion	subcompanion	NOUN
ejpam-3495	199	20	operation	operation	NOUN
ejpam-3495	199	21	on	on	ADP
ejpam-3495	199	22	g.	g.	PROPN
ejpam-3495	199	23	suppose	suppose	VERB
ejpam-3495	199	24	(	(	PUNCT
ejpam-3495	199	25	z	z	NOUN
ejpam-3495	199	26	∗	∗	X
ejpam-3495	199	27	x	x	NOUN
ejpam-3495	199	28	)	)	PUNCT
ejpam-3495	199	29	∗	∗	NOUN
ejpam-3495	199	30	y	y	PROPN
ejpam-3495	199	31	=	=	PUNCT
ejpam-3495	199	32	e.	e.	PROPN
ejpam-3495	200	1	then	then	ADV
ejpam-3495	200	2	z	z	PROPN
ejpam-3495	200	3	◦	◦	NOUN
ejpam-3495	200	4	(	(	PUNCT
ejpam-3495	200	5	y	y	PROPN
ejpam-3495	200	6	◦	◦	NOUN
ejpam-3495	200	7	x)−1	x)−1	X
ejpam-3495	201	1	=	=	SYM
ejpam-3495	201	2	z	z	PART
ejpam-3495	201	3	◦	◦	NOUN
ejpam-3495	201	4	(	(	PUNCT
ejpam-3495	201	5	x−1	x−1	PROPN
ejpam-3495	201	6	◦	◦	NOUN
ejpam-3495	201	7	y−1	y−1	PROPN
ejpam-3495	201	8	)	)	PUNCT
ejpam-3495	202	1	=	=	PRON
ejpam-3495	203	1	(	(	PUNCT
ejpam-3495	203	2	z	z	NOUN
ejpam-3495	203	3	◦	◦	NOUN
ejpam-3495	203	4	x−1	x−1	PROPN
ejpam-3495	203	5	)	)	PUNCT
ejpam-3495	203	6	◦	◦	NOUN
ejpam-3495	203	7	y−1	y−1	PROPN
ejpam-3495	204	1	=	=	PUNCT
ejpam-3495	204	2	(	(	PUNCT
ejpam-3495	204	3	z	z	NOUN
ejpam-3495	204	4	∗	∗	X
ejpam-3495	204	5	x	x	NOUN
ejpam-3495	204	6	)	)	PUNCT
ejpam-3495	204	7	◦	◦	NOUN
ejpam-3495	204	8	y−1	y−1	NOUN
ejpam-3495	204	9	=	=	PUNCT
ejpam-3495	204	10	(	(	PUNCT
ejpam-3495	204	11	z	z	NOUN
ejpam-3495	204	12	∗	∗	X
ejpam-3495	204	13	x	x	NOUN
ejpam-3495	204	14	)	)	PUNCT
ejpam-3495	204	15	∗	∗	NOUN
ejpam-3495	204	16	y	y	PROPN
ejpam-3495	204	17	=	=	SYM
ejpam-3495	204	18	e.	e.	PROPN
ejpam-3495	204	19	observe	observe	VERB
ejpam-3495	204	20	that	that	SCONJ
ejpam-3495	204	21	z	z	NOUN
ejpam-3495	204	22	∗	∗	NOUN
ejpam-3495	204	23	(	(	PUNCT
ejpam-3495	204	24	x⊗	x⊗	PROPN
ejpam-3495	204	25	y	y	PROPN
ejpam-3495	204	26	)	)	PUNCT
ejpam-3495	205	1	=	=	SYM
ejpam-3495	205	2	z	z	NOUN
ejpam-3495	205	3	∗	∗	NOUN
ejpam-3495	205	4	(	(	PUNCT
ejpam-3495	205	5	y	y	PROPN
ejpam-3495	205	6	∗	∗	PROPN
ejpam-3495	205	7	x−1	x−1	PROPN
ejpam-3495	205	8	)	)	PUNCT
ejpam-3495	205	9	=	=	SYM
ejpam-3495	205	10	z	z	NOUN
ejpam-3495	205	11	∗	∗	NOUN
ejpam-3495	205	12	(	(	PUNCT
ejpam-3495	205	13	y	y	PROPN
ejpam-3495	205	14	◦	◦	NOUN
ejpam-3495	205	15	(	(	PUNCT
ejpam-3495	205	16	x−1)−1	x−1)−1	X
ejpam-3495	205	17	)	)	PUNCT
ejpam-3495	205	18	=	=	SYM
ejpam-3495	205	19	z	z	NOUN
ejpam-3495	205	20	∗	∗	NOUN
ejpam-3495	205	21	(	(	PUNCT
ejpam-3495	205	22	y	y	PROPN
ejpam-3495	205	23	◦	◦	NOUN
ejpam-3495	205	24	x	x	X
ejpam-3495	205	25	)	)	PUNCT
ejpam-3495	205	26	=	=	SYM
ejpam-3495	205	27	z	z	PART
ejpam-3495	205	28	◦	◦	NOUN
ejpam-3495	205	29	(	(	PUNCT
ejpam-3495	205	30	y	y	PROPN
ejpam-3495	205	31	◦	◦	NOUN
ejpam-3495	205	32	x)−1	x)−1	X
ejpam-3495	206	1	=	=	SYM
ejpam-3495	206	2	e.	e.	PROPN
ejpam-3495	206	3	hence	hence	ADV
ejpam-3495	206	4	,	,	PUNCT
ejpam-3495	206	5	⊗	⊗	PROPN
ejpam-3495	206	6	is	be	AUX
ejpam-3495	206	7	a	a	DET
ejpam-3495	206	8	companion	companion	NOUN
ejpam-3495	206	9	operation	operation	NOUN
ejpam-3495	206	10	on	on	ADP
ejpam-3495	206	11	g.	g.	PROPN
ejpam-3495	206	12	thus	thus	ADV
ejpam-3495	206	13	,	,	PUNCT
ejpam-3495	206	14	(	(	PUNCT
ejpam-3495	206	15	g	g	NOUN
ejpam-3495	206	16	,	,	PUNCT
ejpam-3495	206	17	∗,⊗	∗,⊗	PROPN
ejpam-3495	206	18	,	,	PUNCT
ejpam-3495	206	19	e	e	X
ejpam-3495	206	20	)	)	PUNCT
ejpam-3495	206	21	is	be	AUX
ejpam-3495	206	22	a	a	DET
ejpam-3495	206	23	companion	companion	NOUN
ejpam-3495	206	24	b	b	PROPN
ejpam-3495	206	25	-algebra	-algebra	PROPN
ejpam-3495	206	26	.	.	PUNCT
ejpam-3495	207	1	�	�	PROPN
ejpam-3495	207	2	consider	consider	VERB
ejpam-3495	207	3	the	the	DET
ejpam-3495	207	4	b	b	NOUN
ejpam-3495	207	5	-algebra	-algebra	NOUN
ejpam-3495	207	6	given	give	VERB
ejpam-3495	207	7	in	in	ADP
ejpam-3495	207	8	example	example	NOUN
ejpam-3495	207	9	3.2	3.2	NUM
ejpam-3495	207	10	.	.	PUNCT
ejpam-3495	208	1	note	note	VERB
ejpam-3495	208	2	that	that	SCONJ
ejpam-3495	208	3	x	x	PRON
ejpam-3495	208	4	is	be	AUX
ejpam-3495	208	5	not	not	PART
ejpam-3495	208	6	commutative	commutative	ADJ
ejpam-3495	208	7	since	since	SCONJ
ejpam-3495	208	8	there	there	PRON
ejpam-3495	208	9	exist	exist	VERB
ejpam-3495	208	10	x	x	X
ejpam-3495	208	11	=	=	SYM
ejpam-3495	208	12	3	3	NUM
ejpam-3495	208	13	and	and	CCONJ
ejpam-3495	208	14	y	y	NOUN
ejpam-3495	208	15	=	=	PROPN
ejpam-3495	208	16	4	4	NUM
ejpam-3495	208	17	such	such	ADJ
ejpam-3495	208	18	that	that	DET
ejpam-3495	208	19	3∗(0∗4	3∗(0∗4	NUM
ejpam-3495	208	20	)	)	PUNCT
ejpam-3495	208	21	=	=	SYM
ejpam-3495	209	1	2	2	NUM
ejpam-3495	209	2	6=	6=	SYM
ejpam-3495	209	3	1	1	NUM
ejpam-3495	209	4	=	=	SYM
ejpam-3495	209	5	4∗(0∗3	4∗(0∗3	NOUN
ejpam-3495	209	6	)	)	PUNCT
ejpam-3495	209	7	.	.	PUNCT
ejpam-3495	210	1	define	define	VERB
ejpam-3495	210	2	x	x	SYM
ejpam-3495	210	3	◦	◦	NOUN
ejpam-3495	210	4	y	y	NOUN
ejpam-3495	210	5	=	=	PUNCT
ejpam-3495	210	6	x∗(0∗y	x∗(0∗y	PROPN
ejpam-3495	210	7	)	)	PUNCT
ejpam-3495	210	8	.	.	PUNCT
ejpam-3495	211	1	if	if	SCONJ
ejpam-3495	211	2	x	x	SYM
ejpam-3495	211	3	=	=	SYM
ejpam-3495	211	4	3	3	NUM
ejpam-3495	211	5	and	and	CCONJ
ejpam-3495	211	6	y	y	NOUN
ejpam-3495	211	7	=	=	SYM
ejpam-3495	211	8	2	2	NUM
ejpam-3495	211	9	,	,	PUNCT
ejpam-3495	211	10	then	then	ADV
ejpam-3495	211	11	(	(	PUNCT
ejpam-3495	211	12	(	(	PUNCT
ejpam-3495	211	13	x	x	X
ejpam-3495	211	14	◦	◦	NOUN
ejpam-3495	211	15	y)∗x)∗y	y)∗x)∗y	NOUN
ejpam-3495	211	16	=	=	SYM
ejpam-3495	211	17	2	2	NUM
ejpam-3495	211	18	6=	6=	NUM
ejpam-3495	211	19	0	0	NUM
ejpam-3495	211	20	.	.	PUNCT
ejpam-3495	212	1	hence	hence	ADV
ejpam-3495	212	2	,	,	PUNCT
ejpam-3495	212	3	◦	◦	NOUN
ejpam-3495	212	4	is	be	AUX
ejpam-3495	212	5	not	not	PART
ejpam-3495	212	6	a	a	DET
ejpam-3495	212	7	subcompanion	subcompanion	NOUN
ejpam-3495	212	8	operation	operation	NOUN
ejpam-3495	212	9	.	.	PUNCT
ejpam-3495	213	1	remark	remark	PROPN
ejpam-3495	213	2	3.17	3.17	NUM
ejpam-3495	213	3	.	.	PUNCT
ejpam-3495	214	1	if	if	SCONJ
ejpam-3495	214	2	(	(	PUNCT
ejpam-3495	214	3	x	x	X
ejpam-3495	214	4	,	,	PUNCT
ejpam-3495	214	5	∗	∗	NOUN
ejpam-3495	214	6	,	,	PUNCT
ejpam-3495	214	7	0	0	NUM
ejpam-3495	214	8	)	)	PUNCT
ejpam-3495	214	9	is	be	AUX
ejpam-3495	214	10	a	a	DET
ejpam-3495	214	11	b	b	NOUN
ejpam-3495	214	12	-	-	PUNCT
ejpam-3495	214	13	algebra	algebra	NOUN
ejpam-3495	214	14	,	,	PUNCT
ejpam-3495	214	15	then	then	ADV
ejpam-3495	214	16	(	(	PUNCT
ejpam-3495	214	17	x	x	X
ejpam-3495	214	18	,	,	PUNCT
ejpam-3495	214	19	∗	∗	NOUN
ejpam-3495	214	20	,	,	PUNCT
ejpam-3495	214	21	◦	◦	NOUN
ejpam-3495	214	22	,	,	PUNCT
ejpam-3495	214	23	0	0	NUM
ejpam-3495	214	24	)	)	PUNCT
ejpam-3495	214	25	is	be	AUX
ejpam-3495	214	26	not	not	PART
ejpam-3495	214	27	necessarily	necessarily	ADV
ejpam-3495	214	28	a	a	DET
ejpam-3495	214	29	companion	companion	NOUN
ejpam-3495	214	30	b	b	X
ejpam-3495	214	31	-	-	PUNCT
ejpam-3495	214	32	algebra	algebra	NOUN
ejpam-3495	214	33	where	where	SCONJ
ejpam-3495	214	34	the	the	DET
ejpam-3495	214	35	operation	operation	NOUN
ejpam-3495	214	36	◦	◦	NOUN
ejpam-3495	214	37	is	be	AUX
ejpam-3495	214	38	defined	define	VERB
ejpam-3495	214	39	by	by	ADP
ejpam-3495	214	40	x	x	SYM
ejpam-3495	214	41	◦	◦	NOUN
ejpam-3495	214	42	y	y	NOUN
ejpam-3495	214	43	=	=	PUNCT
ejpam-3495	214	44	x	x	SYM
ejpam-3495	214	45	∗	∗	NOUN
ejpam-3495	214	46	(	(	PUNCT
ejpam-3495	214	47	0	0	NUM
ejpam-3495	214	48	∗	∗	NOUN
ejpam-3495	214	49	y	y	PROPN
ejpam-3495	214	50	)	)	PUNCT
ejpam-3495	214	51	.	.	PUNCT
ejpam-3495	215	1	example	example	NOUN
ejpam-3495	215	2	3.18	3.18	NUM
ejpam-3495	215	3	.	.	PUNCT
ejpam-3495	216	1	let	let	VERB
ejpam-3495	216	2	x	x	PUNCT
ejpam-3495	216	3	=	=	PUNCT
ejpam-3495	216	4	{	{	PUNCT
ejpam-3495	216	5	0	0	NUM
ejpam-3495	216	6	,	,	PUNCT
ejpam-3495	216	7	1	1	NUM
ejpam-3495	216	8	,	,	PUNCT
ejpam-3495	216	9	2	2	NUM
ejpam-3495	216	10	}	}	PUNCT
ejpam-3495	216	11	be	be	AUX
ejpam-3495	216	12	a	a	DET
ejpam-3495	216	13	set	set	NOUN
ejpam-3495	216	14	with	with	ADP
ejpam-3495	216	15	the	the	DET
ejpam-3495	216	16	following	follow	VERB
ejpam-3495	216	17	table	table	NOUN
ejpam-3495	216	18	of	of	ADP
ejpam-3495	216	19	operations	operation	NOUN
ejpam-3495	216	20	,	,	PUNCT
ejpam-3495	216	21	where	where	SCONJ
ejpam-3495	216	22	x	x	X
ejpam-3495	216	23	◦	◦	NOUN
ejpam-3495	216	24	y	y	NOUN
ejpam-3495	216	25	=	=	PUNCT
ejpam-3495	216	26	x	x	SYM
ejpam-3495	216	27	∗	∗	NOUN
ejpam-3495	216	28	(	(	PUNCT
ejpam-3495	216	29	0	0	NUM
ejpam-3495	216	30	∗	∗	NUM
ejpam-3495	216	31	y	y	PROPN
ejpam-3495	216	32	):	):	PUNCT
ejpam-3495	216	33	∗	∗	NOUN
ejpam-3495	216	34	0	0	NUM
ejpam-3495	216	35	1	1	NUM
ejpam-3495	216	36	2	2	NUM
ejpam-3495	216	37	0	0	NUM
ejpam-3495	216	38	0	0	NUM
ejpam-3495	216	39	2	2	NUM
ejpam-3495	216	40	1	1	NUM
ejpam-3495	216	41	1	1	NUM
ejpam-3495	216	42	1	1	NUM
ejpam-3495	216	43	0	0	NUM
ejpam-3495	216	44	2	2	NUM
ejpam-3495	216	45	2	2	NUM
ejpam-3495	216	46	2	2	NUM
ejpam-3495	216	47	1	1	NUM
ejpam-3495	216	48	0	0	NUM
ejpam-3495	217	1	◦	◦	NOUN
ejpam-3495	217	2	0	0	NUM
ejpam-3495	217	3	1	1	NUM
ejpam-3495	217	4	2	2	NUM
ejpam-3495	217	5	0	0	NUM
ejpam-3495	217	6	0	0	NUM
ejpam-3495	217	7	1	1	NUM
ejpam-3495	217	8	2	2	NUM
ejpam-3495	217	9	1	1	NUM
ejpam-3495	217	10	1	1	NUM
ejpam-3495	217	11	2	2	NUM
ejpam-3495	217	12	0	0	NUM
ejpam-3495	217	13	2	2	NUM
ejpam-3495	217	14	2	2	NUM
ejpam-3495	217	15	0	0	NUM
ejpam-3495	217	16	1	1	NUM
ejpam-3495	217	17	by	by	ADP
ejpam-3495	217	18	routine	routine	ADJ
ejpam-3495	217	19	calculations	calculation	NOUN
ejpam-3495	217	20	,	,	PUNCT
ejpam-3495	217	21	(	(	PUNCT
ejpam-3495	217	22	x	x	X
ejpam-3495	217	23	,	,	PUNCT
ejpam-3495	217	24	∗	∗	NOUN
ejpam-3495	217	25	,	,	PUNCT
ejpam-3495	217	26	0	0	NUM
ejpam-3495	217	27	)	)	PUNCT
ejpam-3495	217	28	is	be	AUX
ejpam-3495	217	29	a	a	DET
ejpam-3495	217	30	commutative	commutative	ADJ
ejpam-3495	217	31	b	b	NOUN
ejpam-3495	217	32	-algebra	-algebra	NOUN
ejpam-3495	217	33	and	and	CCONJ
ejpam-3495	217	34	(	(	PUNCT
ejpam-3495	217	35	x	x	X
ejpam-3495	217	36	,	,	PUNCT
ejpam-3495	217	37	∗	∗	NOUN
ejpam-3495	217	38	,	,	PUNCT
ejpam-3495	217	39	◦	◦	NOUN
ejpam-3495	217	40	,	,	PUNCT
ejpam-3495	217	41	0	0	NUM
ejpam-3495	217	42	)	)	PUNCT
ejpam-3495	217	43	is	be	AUX
ejpam-3495	217	44	a	a	DET
ejpam-3495	217	45	companion	companion	NOUN
ejpam-3495	217	46	b	b	PROPN
ejpam-3495	217	47	-algebra	-algebra	PROPN
ejpam-3495	217	48	.	.	PUNCT
ejpam-3495	218	1	theorem	theorem	VERB
ejpam-3495	218	2	3.19	3.19	NUM
ejpam-3495	218	3	.	.	PUNCT
ejpam-3495	219	1	if	if	SCONJ
ejpam-3495	219	2	(	(	PUNCT
ejpam-3495	219	3	x	x	X
ejpam-3495	219	4	,	,	PUNCT
ejpam-3495	219	5	∗	∗	NOUN
ejpam-3495	219	6	,	,	PUNCT
ejpam-3495	219	7	0	0	NUM
ejpam-3495	219	8	)	)	PUNCT
ejpam-3495	219	9	is	be	AUX
ejpam-3495	219	10	a	a	DET
ejpam-3495	219	11	commutative	commutative	ADJ
ejpam-3495	219	12	b	b	NOUN
ejpam-3495	219	13	-	-	PUNCT
ejpam-3495	219	14	algebra	algebra	NOUN
ejpam-3495	219	15	,	,	PUNCT
ejpam-3495	219	16	then	then	ADV
ejpam-3495	219	17	(	(	PUNCT
ejpam-3495	219	18	x	x	X
ejpam-3495	219	19	,	,	PUNCT
ejpam-3495	219	20	∗	∗	NOUN
ejpam-3495	219	21	,	,	PUNCT
ejpam-3495	219	22	◦	◦	NOUN
ejpam-3495	219	23	,	,	PUNCT
ejpam-3495	219	24	0	0	NUM
ejpam-3495	219	25	)	)	PUNCT
ejpam-3495	219	26	is	be	AUX
ejpam-3495	219	27	a	a	DET
ejpam-3495	219	28	companion	companion	NOUN
ejpam-3495	219	29	b	b	NOUN
ejpam-3495	219	30	-	-	PUNCT
ejpam-3495	219	31	algebra	algebra	NOUN
ejpam-3495	219	32	where	where	SCONJ
ejpam-3495	219	33	x	x	PART
ejpam-3495	219	34	◦	◦	NOUN
ejpam-3495	219	35	y	y	NOUN
ejpam-3495	219	36	=	=	PUNCT
ejpam-3495	219	37	x	x	SYM
ejpam-3495	219	38	∗	∗	NOUN
ejpam-3495	219	39	(	(	PUNCT
ejpam-3495	219	40	0	0	NUM
ejpam-3495	219	41	∗	∗	PROPN
ejpam-3495	219	42	y	y	PROPN
ejpam-3495	219	43	)	)	PUNCT
ejpam-3495	219	44	.	.	PUNCT
ejpam-3495	220	1	proof	proof	NOUN
ejpam-3495	220	2	:	:	PUNCT
ejpam-3495	220	3	let	let	VERB
ejpam-3495	220	4	(	(	PUNCT
ejpam-3495	220	5	x	x	NOUN
ejpam-3495	220	6	,	,	PUNCT
ejpam-3495	220	7	∗	∗	NOUN
ejpam-3495	220	8	,	,	PUNCT
ejpam-3495	220	9	0	0	NUM
ejpam-3495	220	10	)	)	PUNCT
ejpam-3495	220	11	be	be	AUX
ejpam-3495	220	12	a	a	DET
ejpam-3495	220	13	commutative	commutative	ADJ
ejpam-3495	220	14	b	b	NOUN
ejpam-3495	220	15	-algebra	-algebra	NOUN
ejpam-3495	220	16	and	and	CCONJ
ejpam-3495	220	17	x	x	NOUN
ejpam-3495	220	18	,	,	PUNCT
ejpam-3495	220	19	y	y	PROPN
ejpam-3495	220	20	,	,	PUNCT
ejpam-3495	220	21	z	z	PROPN
ejpam-3495	220	22	∈	∈	PROPN
ejpam-3495	220	23	x.	x.	NOUN
ejpam-3495	220	24	define	define	VERB
ejpam-3495	220	25	the	the	DET
ejpam-3495	220	26	operation	operation	NOUN
ejpam-3495	220	27	◦	◦	NOUN
ejpam-3495	220	28	by	by	ADP
ejpam-3495	220	29	x	x	SYM
ejpam-3495	220	30	◦	◦	NOUN
ejpam-3495	220	31	y	y	NOUN
ejpam-3495	220	32	=	=	PUNCT
ejpam-3495	220	33	x	x	SYM
ejpam-3495	220	34	∗	∗	NOUN
ejpam-3495	220	35	(	(	PUNCT
ejpam-3495	220	36	0	0	NUM
ejpam-3495	220	37	∗	∗	PROPN
ejpam-3495	220	38	y	y	PROPN
ejpam-3495	220	39	)	)	PUNCT
ejpam-3495	220	40	.	.	PUNCT
ejpam-3495	221	1	note	note	VERB
ejpam-3495	221	2	that	that	SCONJ
ejpam-3495	221	3	by	by	ADP
ejpam-3495	221	4	definition	definition	NOUN
ejpam-3495	221	5	2.1(iii	2.1(iii	NUM
ejpam-3495	221	6	)	)	PUNCT
ejpam-3495	221	7	,	,	PUNCT
ejpam-3495	221	8	the	the	DET
ejpam-3495	221	9	definition	definition	NOUN
ejpam-3495	221	10	of	of	ADP
ejpam-3495	221	11	◦	◦	NOUN
ejpam-3495	221	12	,	,	PUNCT
ejpam-3495	221	13	definitions	definition	VERB
ejpam-3495	221	14	2.5	2.5	NUM
ejpam-3495	221	15	and	and	CCONJ
ejpam-3495	221	16	2.1(i	2.1(i	NUM
ejpam-3495	221	17	)	)	PUNCT
ejpam-3495	221	18	,	,	PUNCT
ejpam-3495	221	19	we	we	PRON
ejpam-3495	221	20	have	have	VERB
ejpam-3495	221	21	(	(	PUNCT
ejpam-3495	221	22	(	(	PUNCT
ejpam-3495	221	23	x	x	SYM
ejpam-3495	221	24	◦	◦	VERB
ejpam-3495	221	25	y	y	NOUN
ejpam-3495	221	26	)	)	PUNCT
ejpam-3495	221	27	∗	∗	NOUN
ejpam-3495	221	28	x	x	NOUN
ejpam-3495	221	29	)	)	PUNCT
ejpam-3495	221	30	∗	∗	NOUN
ejpam-3495	221	31	y	y	NOUN
ejpam-3495	221	32	=	=	SYM
ejpam-3495	222	1	(	(	PUNCT
ejpam-3495	222	2	x	x	SYM
ejpam-3495	222	3	◦	◦	VERB
ejpam-3495	222	4	y	y	NOUN
ejpam-3495	222	5	)	)	PUNCT
ejpam-3495	222	6	∗	∗	NOUN
ejpam-3495	222	7	(	(	PUNCT
ejpam-3495	222	8	y	y	PROPN
ejpam-3495	222	9	∗	∗	NOUN
ejpam-3495	222	10	(	(	PUNCT
ejpam-3495	222	11	0	0	NUM
ejpam-3495	222	12	∗	∗	NOUN
ejpam-3495	222	13	x	x	NOUN
ejpam-3495	222	14	)	)	PUNCT
ejpam-3495	222	15	)	)	PUNCT
ejpam-3495	223	1	=	=	PRON
ejpam-3495	224	1	(	(	PUNCT
ejpam-3495	224	2	x	x	SYM
ejpam-3495	224	3	∗	∗	NOUN
ejpam-3495	224	4	(	(	PUNCT
ejpam-3495	224	5	0	0	NUM
ejpam-3495	224	6	∗	∗	PROPN
ejpam-3495	224	7	y	y	PROPN
ejpam-3495	224	8	)	)	PUNCT
ejpam-3495	224	9	)	)	PUNCT
ejpam-3495	224	10	∗	∗	NOUN
ejpam-3495	224	11	(	(	PUNCT
ejpam-3495	224	12	y	y	PROPN
ejpam-3495	224	13	∗	∗	NOUN
ejpam-3495	224	14	(	(	PUNCT
ejpam-3495	224	15	0	0	NUM
ejpam-3495	224	16	∗	∗	NOUN
ejpam-3495	224	17	x	x	NOUN
ejpam-3495	224	18	)	)	PUNCT
ejpam-3495	224	19	)	)	PUNCT
ejpam-3495	225	1	=	=	SYM
ejpam-3495	225	2	(	(	PUNCT
ejpam-3495	225	3	y	y	PROPN
ejpam-3495	225	4	∗	∗	X
ejpam-3495	225	5	(	(	PUNCT
ejpam-3495	225	6	0	0	NUM
ejpam-3495	225	7	∗	∗	NOUN
ejpam-3495	225	8	x	x	NOUN
ejpam-3495	225	9	)	)	PUNCT
ejpam-3495	225	10	)	)	PUNCT
ejpam-3495	225	11	∗	∗	NOUN
ejpam-3495	225	12	(	(	PUNCT
ejpam-3495	225	13	y	y	PROPN
ejpam-3495	225	14	∗	∗	NOUN
ejpam-3495	225	15	(	(	PUNCT
ejpam-3495	225	16	0	0	NUM
ejpam-3495	225	17	∗	∗	NOUN
ejpam-3495	225	18	x	x	NOUN
ejpam-3495	225	19	)	)	PUNCT
ejpam-3495	225	20	)	)	PUNCT
ejpam-3495	226	1	=	=	PUNCT
ejpam-3495	226	2	0	0	X
ejpam-3495	226	3	.	.	PUNCT
ejpam-3495	227	1	now	now	ADV
ejpam-3495	227	2	,	,	PUNCT
ejpam-3495	227	3	suppose	suppose	VERB
ejpam-3495	227	4	(	(	PUNCT
ejpam-3495	227	5	z	z	NOUN
ejpam-3495	227	6	∗	∗	X
ejpam-3495	227	7	x	x	NOUN
ejpam-3495	227	8	)	)	PUNCT
ejpam-3495	227	9	∗	∗	NOUN
ejpam-3495	227	10	y	y	NOUN
ejpam-3495	227	11	=	=	SYM
ejpam-3495	227	12	0	0	PROPN
ejpam-3495	227	13	.	.	PUNCT
ejpam-3495	228	1	then	then	ADV
ejpam-3495	228	2	by	by	ADP
ejpam-3495	228	3	the	the	DET
ejpam-3495	228	4	definition	definition	NOUN
ejpam-3495	228	5	of	of	ADP
ejpam-3495	228	6	◦	◦	NOUN
ejpam-3495	228	7	,	,	PUNCT
ejpam-3495	228	8	definition	definition	NOUN
ejpam-3495	228	9	2.5	2.5	NUM
ejpam-3495	228	10	and	and	CCONJ
ejpam-3495	228	11	definition	definition	NOUN
ejpam-3495	228	12	2.1(iii	2.1(iii	NUM
ejpam-3495	228	13	)	)	PUNCT
ejpam-3495	228	14	,	,	PUNCT
ejpam-3495	228	15	z	z	NOUN
ejpam-3495	228	16	∗	∗	NOUN
ejpam-3495	228	17	(	(	PUNCT
ejpam-3495	228	18	x	x	SYM
ejpam-3495	228	19	◦	◦	VERB
ejpam-3495	228	20	y	y	NOUN
ejpam-3495	228	21	)	)	PUNCT
ejpam-3495	229	1	=	=	SYM
ejpam-3495	229	2	z	z	NOUN
ejpam-3495	229	3	∗	∗	NOUN
ejpam-3495	229	4	(	(	PUNCT
ejpam-3495	229	5	x	x	X
ejpam-3495	229	6	∗	∗	NOUN
ejpam-3495	229	7	(	(	PUNCT
ejpam-3495	229	8	0	0	NUM
ejpam-3495	229	9	∗	∗	PROPN
ejpam-3495	229	10	y	y	NOUN
ejpam-3495	229	11	)	)	PUNCT
ejpam-3495	229	12	)	)	PUNCT
ejpam-3495	230	1	=	=	PUNCT
ejpam-3495	230	2	z	z	NOUN
ejpam-3495	230	3	∗	∗	NOUN
ejpam-3495	230	4	(	(	PUNCT
ejpam-3495	230	5	y	y	PROPN
ejpam-3495	230	6	∗	∗	NOUN
ejpam-3495	230	7	(	(	PUNCT
ejpam-3495	230	8	0	0	NUM
ejpam-3495	230	9	∗	∗	NOUN
ejpam-3495	230	10	x	x	NOUN
ejpam-3495	230	11	)	)	PUNCT
ejpam-3495	230	12	)	)	PUNCT
ejpam-3495	231	1	=	=	PUNCT
ejpam-3495	231	2	(	(	PUNCT
ejpam-3495	231	3	z	z	NOUN
ejpam-3495	231	4	∗	∗	X
ejpam-3495	231	5	x	x	NOUN
ejpam-3495	231	6	)	)	PUNCT
ejpam-3495	231	7	∗	∗	NOUN
ejpam-3495	231	8	y	y	NOUN
ejpam-3495	232	1	=	=	SYM
ejpam-3495	232	2	0	0	PROPN
ejpam-3495	232	3	.	.	PUNCT
ejpam-3495	233	1	hence	hence	ADV
ejpam-3495	233	2	,	,	PUNCT
ejpam-3495	233	3	◦	◦	NOUN
ejpam-3495	233	4	is	be	AUX
ejpam-3495	233	5	a	a	DET
ejpam-3495	233	6	companion	companion	NOUN
ejpam-3495	233	7	operation	operation	NOUN
ejpam-3495	233	8	.	.	PUNCT
ejpam-3495	234	1	therefore	therefore	ADV
ejpam-3495	234	2	,	,	PUNCT
ejpam-3495	234	3	(	(	PUNCT
ejpam-3495	234	4	x	x	X
ejpam-3495	234	5	,	,	PUNCT
ejpam-3495	234	6	∗	∗	NOUN
ejpam-3495	234	7	,	,	PUNCT
ejpam-3495	234	8	◦	◦	NOUN
ejpam-3495	234	9	,	,	PUNCT
ejpam-3495	234	10	0	0	NUM
ejpam-3495	234	11	)	)	PUNCT
ejpam-3495	234	12	is	be	AUX
ejpam-3495	234	13	a	a	DET
ejpam-3495	234	14	companion	companion	NOUN
ejpam-3495	234	15	b	b	PROPN
ejpam-3495	234	16	-algebra	-algebra	PROPN
ejpam-3495	234	17	.	.	PUNCT
ejpam-3495	235	1	�	�	PROPN
ejpam-3495	235	2	l.d	l.d	PROPN
ejpam-3495	235	3	.	.	PROPN
ejpam-3495	235	4	naingue	naingue	PROPN
ejpam-3495	235	5	,	,	PUNCT
ejpam-3495	235	6	j.p	j.p	PROPN
ejpam-3495	235	7	.	.	PROPN
ejpam-3495	235	8	vilela	vilela	PROPN
ejpam-3495	235	9	/	/	SYM
ejpam-3495	235	10	eur	eur	PROPN
ejpam-3495	235	11	.	.	PUNCT
ejpam-3495	236	1	j.	j.	PROPN
ejpam-3495	236	2	pure	pure	PROPN
ejpam-3495	236	3	appl	appl	PROPN
ejpam-3495	236	4	.	.	PROPN
ejpam-3495	236	5	math	math	PROPN
ejpam-3495	236	6	,	,	PUNCT
ejpam-3495	236	7	12	12	NUM
ejpam-3495	236	8	(	(	PUNCT
ejpam-3495	236	9	3	3	NUM
ejpam-3495	236	10	)	)	PUNCT
ejpam-3495	236	11	(	(	PUNCT
ejpam-3495	236	12	2019	2019	NUM
ejpam-3495	236	13	)	)	PUNCT
ejpam-3495	236	14	,	,	PUNCT
ejpam-3495	236	15	1248	1248	NUM
ejpam-3495	236	16	-	-	SYM
ejpam-3495	236	17	1259	1259	NUM
ejpam-3495	236	18	1255	1255	NUM
ejpam-3495	236	19	4	4	NUM
ejpam-3495	236	20	.	.	PUNCT
ejpam-3495	237	1	on	on	ADP
ejpam-3495	237	2	�	�	NOUN
ejpam-3495	237	3	-subalgebras	-subalgebra	NOUN
ejpam-3495	237	4	definition	definition	NOUN
ejpam-3495	237	5	4.1	4.1	NUM
ejpam-3495	237	6	.	.	PUNCT
ejpam-3495	238	1	let	let	VERB
ejpam-3495	238	2	(	(	PUNCT
ejpam-3495	238	3	x	x	X
ejpam-3495	238	4	,	,	PUNCT
ejpam-3495	238	5	∗	∗	NOUN
ejpam-3495	238	6	,	,	PUNCT
ejpam-3495	238	7	�	�	PROPN
ejpam-3495	238	8	,	,	PUNCT
ejpam-3495	238	9	0	0	NUM
ejpam-3495	238	10	)	)	PUNCT
ejpam-3495	238	11	be	be	AUX
ejpam-3495	238	12	a	a	DET
ejpam-3495	238	13	companion	companion	NOUN
ejpam-3495	238	14	b	b	NOUN
ejpam-3495	238	15	-algebra	-algebra	NOUN
ejpam-3495	239	1	and	and	CCONJ
ejpam-3495	239	2	i	i	PRON
ejpam-3495	239	3	be	be	VERB
ejpam-3495	239	4	a	a	DET
ejpam-3495	239	5	nonempty	nonempty	ADJ
ejpam-3495	239	6	subset	subset	NOUN
ejpam-3495	239	7	of	of	ADP
ejpam-3495	239	8	x.	x.	NOUN
ejpam-3495	240	1	then	then	ADV
ejpam-3495	240	2	i	i	PRON
ejpam-3495	240	3	is	be	AUX
ejpam-3495	240	4	called	call	VERB
ejpam-3495	240	5	a	a	DET
ejpam-3495	240	6	�	�	NOUN
ejpam-3495	240	7	-subalgebra	-subalgebra	NOUN
ejpam-3495	240	8	if	if	SCONJ
ejpam-3495	240	9	x	x	X
ejpam-3495	240	10	�	�	PROPN
ejpam-3495	240	11	y	y	PROPN
ejpam-3495	240	12	∈	∈	PROPN
ejpam-3495	240	13	i	i	PRON
ejpam-3495	240	14	for	for	ADP
ejpam-3495	240	15	any	any	DET
ejpam-3495	240	16	x	x	NOUN
ejpam-3495	240	17	,	,	PUNCT
ejpam-3495	240	18	y,∈	y,∈	PROPN
ejpam-3495	240	19	i.	i.	NOUN
ejpam-3495	240	20	example	example	NOUN
ejpam-3495	240	21	4.2	4.2	NUM
ejpam-3495	240	22	.	.	PUNCT
ejpam-3495	241	1	in	in	ADP
ejpam-3495	241	2	example	example	NOUN
ejpam-3495	241	3	3.2	3.2	NUM
ejpam-3495	241	4	,	,	PUNCT
ejpam-3495	241	5	the	the	DET
ejpam-3495	241	6	set	set	NOUN
ejpam-3495	241	7	i1	i1	PROPN
ejpam-3495	241	8	=	=	PUNCT
ejpam-3495	241	9	{	{	PUNCT
ejpam-3495	241	10	0	0	NUM
ejpam-3495	241	11	,	,	PUNCT
ejpam-3495	241	12	1	1	NUM
ejpam-3495	241	13	,	,	PUNCT
ejpam-3495	241	14	2	2	NUM
ejpam-3495	241	15	}	}	PUNCT
ejpam-3495	241	16	is	be	AUX
ejpam-3495	241	17	a	a	DET
ejpam-3495	241	18	�	�	NOUN
ejpam-3495	241	19	-subalgebra	-subalgebra	NOUN
ejpam-3495	241	20	of	of	ADP
ejpam-3495	241	21	x	x	PRON
ejpam-3495	241	22	,	,	PUNCT
ejpam-3495	241	23	while	while	SCONJ
ejpam-3495	241	24	i2	i2	PROPN
ejpam-3495	241	25	=	=	PUNCT
ejpam-3495	241	26	{	{	PUNCT
ejpam-3495	241	27	3	3	NUM
ejpam-3495	241	28	,	,	PUNCT
ejpam-3495	241	29	4	4	NUM
ejpam-3495	241	30	,	,	PUNCT
ejpam-3495	241	31	5	5	NUM
ejpam-3495	241	32	}	}	PUNCT
ejpam-3495	241	33	is	be	AUX
ejpam-3495	241	34	not	not	PART
ejpam-3495	241	35	a	a	DET
ejpam-3495	241	36	�	�	NOUN
ejpam-3495	241	37	-subalgebra	-subalgebra	NOUN
ejpam-3495	241	38	since	since	SCONJ
ejpam-3495	241	39	3	3	NUM
ejpam-3495	241	40	�	�	PROPN
ejpam-3495	241	41	4	4	NUM
ejpam-3495	241	42	=	=	SYM
ejpam-3495	241	43	1	1	NUM
ejpam-3495	241	44	/∈	/∈	NOUN
ejpam-3495	241	45	i2	i2	PROPN
ejpam-3495	241	46	.	.	PUNCT
ejpam-3495	242	1	theorem	theorem	VERB
ejpam-3495	242	2	4.3	4.3	NUM
ejpam-3495	242	3	.	.	PUNCT
ejpam-3495	243	1	let	let	VERB
ejpam-3495	243	2	(	(	PUNCT
ejpam-3495	243	3	x	x	X
ejpam-3495	243	4	,	,	PUNCT
ejpam-3495	243	5	∗	∗	NOUN
ejpam-3495	243	6	,	,	PUNCT
ejpam-3495	243	7	�	�	PROPN
ejpam-3495	243	8	,	,	PUNCT
ejpam-3495	243	9	0	0	NUM
ejpam-3495	243	10	)	)	PUNCT
ejpam-3495	243	11	be	be	AUX
ejpam-3495	243	12	a	a	DET
ejpam-3495	243	13	companion	companion	NOUN
ejpam-3495	243	14	b	b	NOUN
ejpam-3495	243	15	-	-	PUNCT
ejpam-3495	243	16	algebra	algebra	NOUN
ejpam-3495	243	17	.	.	PUNCT
ejpam-3495	244	1	if	if	SCONJ
ejpam-3495	244	2	i	i	PRON
ejpam-3495	244	3	is	be	AUX
ejpam-3495	244	4	a	a	DET
ejpam-3495	244	5	b	b	NOUN
ejpam-3495	244	6	-	-	PUNCT
ejpam-3495	244	7	ideal	ideal	NOUN
ejpam-3495	244	8	of	of	ADP
ejpam-3495	244	9	x	x	PRON
ejpam-3495	244	10	,	,	PUNCT
ejpam-3495	244	11	then	then	ADV
ejpam-3495	244	12	i	i	PRON
ejpam-3495	244	13	is	be	AUX
ejpam-3495	244	14	a	a	DET
ejpam-3495	244	15	�	�	NOUN
ejpam-3495	244	16	-subalgebra	-subalgebra	NOUN
ejpam-3495	244	17	of	of	ADP
ejpam-3495	244	18	x.	x.	NOUN
ejpam-3495	244	19	proof	proof	NOUN
ejpam-3495	244	20	:	:	PUNCT
ejpam-3495	244	21	let	let	VERB
ejpam-3495	244	22	(	(	PUNCT
ejpam-3495	244	23	x	x	NOUN
ejpam-3495	244	24	,	,	PUNCT
ejpam-3495	244	25	∗	∗	NOUN
ejpam-3495	244	26	,	,	PUNCT
ejpam-3495	244	27	�	�	PROPN
ejpam-3495	244	28	,	,	PUNCT
ejpam-3495	244	29	0	0	NUM
ejpam-3495	244	30	)	)	PUNCT
ejpam-3495	244	31	be	be	AUX
ejpam-3495	244	32	a	a	DET
ejpam-3495	244	33	companion	companion	NOUN
ejpam-3495	244	34	b	b	NOUN
ejpam-3495	244	35	-algebra	-algebra	NOUN
ejpam-3495	244	36	and	and	CCONJ
ejpam-3495	244	37	i	i	PRON
ejpam-3495	244	38	be	be	VERB
ejpam-3495	244	39	a	a	DET
ejpam-3495	244	40	b	b	NOUN
ejpam-3495	244	41	-ideal	-ideal	NOUN
ejpam-3495	244	42	of	of	ADP
ejpam-3495	244	43	x.	x.	NOUN
ejpam-3495	244	44	then	then	ADV
ejpam-3495	244	45	i	i	PRON
ejpam-3495	244	46	6=	6=	VERB
ejpam-3495	244	47	∅.	∅.	AUX
ejpam-3495	244	48	let	let	VERB
ejpam-3495	244	49	x	x	PRON
ejpam-3495	244	50	,	,	PUNCT
ejpam-3495	244	51	y	y	PROPN
ejpam-3495	244	52	∈	∈	PROPN
ejpam-3495	244	53	i.	i.	NOUN
ejpam-3495	244	54	by	by	ADP
ejpam-3495	244	55	(	(	PUNCT
ejpam-3495	244	56	sc	sc	PROPN
ejpam-3495	244	57	)	)	PUNCT
ejpam-3495	244	58	,	,	PUNCT
ejpam-3495	244	59	(	(	PUNCT
ejpam-3495	244	60	(	(	PUNCT
ejpam-3495	244	61	x	x	PROPN
ejpam-3495	244	62	�	�	PROPN
ejpam-3495	244	63	y	y	PROPN
ejpam-3495	244	64	)	)	PUNCT
ejpam-3495	244	65	∗	∗	NOUN
ejpam-3495	244	66	x	x	NOUN
ejpam-3495	244	67	)	)	PUNCT
ejpam-3495	244	68	∗	∗	NOUN
ejpam-3495	244	69	y	y	NOUN
ejpam-3495	244	70	=	=	SYM
ejpam-3495	244	71	0	0	NUM
ejpam-3495	244	72	∈	∈	PROPN
ejpam-3495	244	73	i.	i.	NOUN
ejpam-3495	244	74	since	since	SCONJ
ejpam-3495	244	75	i	i	PRON
ejpam-3495	244	76	is	be	AUX
ejpam-3495	244	77	a	a	DET
ejpam-3495	244	78	b	b	NOUN
ejpam-3495	244	79	-ideal	-ideal	NOUN
ejpam-3495	244	80	of	of	ADP
ejpam-3495	244	81	x	x	PUNCT
ejpam-3495	244	82	and	and	CCONJ
ejpam-3495	244	83	y	y	PROPN
ejpam-3495	244	84	∈	∈	PROPN
ejpam-3495	244	85	i	i	PRON
ejpam-3495	244	86	,	,	PUNCT
ejpam-3495	244	87	(	(	PUNCT
ejpam-3495	244	88	x	x	PROPN
ejpam-3495	244	89	�	�	PROPN
ejpam-3495	244	90	y	y	PROPN
ejpam-3495	244	91	)	)	PUNCT
ejpam-3495	244	92	∗	∗	NOUN
ejpam-3495	244	93	x	x	X
ejpam-3495	245	1	∈	∈	NOUN
ejpam-3495	245	2	i	i	PRON
ejpam-3495	245	3	by	by	ADP
ejpam-3495	245	4	definition	definition	NOUN
ejpam-3495	245	5	2.9	2.9	NUM
ejpam-3495	245	6	.	.	PUNCT
ejpam-3495	246	1	furthermore	furthermore	ADV
ejpam-3495	246	2	,	,	PUNCT
ejpam-3495	246	3	since	since	SCONJ
ejpam-3495	246	4	x	x	PROPN
ejpam-3495	246	5	∈	∈	PROPN
ejpam-3495	246	6	i	i	PRON
ejpam-3495	246	7	,	,	PUNCT
ejpam-3495	246	8	x	x	PROPN
ejpam-3495	246	9	�	�	PROPN
ejpam-3495	246	10	y	y	PROPN
ejpam-3495	246	11	∈	∈	PROPN
ejpam-3495	246	12	i.	i.	NOUN
ejpam-3495	246	13	therefore	therefore	ADV
ejpam-3495	246	14	,	,	PUNCT
ejpam-3495	246	15	i	i	PRON
ejpam-3495	246	16	is	be	AUX
ejpam-3495	246	17	a	a	DET
ejpam-3495	246	18	�	�	NOUN
ejpam-3495	246	19	-subalgebra	-subalgebra	NOUN
ejpam-3495	246	20	of	of	ADP
ejpam-3495	246	21	x.	x.	PROPN
ejpam-3495	246	22	�	�	PROPN
ejpam-3495	246	23	the	the	DET
ejpam-3495	246	24	converse	converse	NOUN
ejpam-3495	246	25	of	of	ADP
ejpam-3495	246	26	theorem	theorem	NOUN
ejpam-3495	246	27	4.3	4.3	NUM
ejpam-3495	246	28	need	need	AUX
ejpam-3495	246	29	not	not	PART
ejpam-3495	246	30	be	be	AUX
ejpam-3495	246	31	true	true	ADJ
ejpam-3495	246	32	in	in	ADP
ejpam-3495	246	33	general	general	ADJ
ejpam-3495	246	34	.	.	PUNCT
ejpam-3495	247	1	in	in	ADP
ejpam-3495	247	2	the	the	DET
ejpam-3495	247	3	companion	companion	PROPN
ejpam-3495	247	4	b	b	PROPN
ejpam-3495	247	5	-algebra	-algebra	PROPN
ejpam-3495	247	6	(	(	PUNCT
ejpam-3495	247	7	z,−,+	z,−,+	NUM
ejpam-3495	247	8	,	,	PUNCT
ejpam-3495	247	9	0	0	NUM
ejpam-3495	247	10	)	)	PUNCT
ejpam-3495	247	11	in	in	ADP
ejpam-3495	247	12	example	example	NOUN
ejpam-3495	247	13	3.3	3.3	NUM
ejpam-3495	248	1	,	,	PUNCT
ejpam-3495	248	2	i	i	PRON
ejpam-3495	248	3	=	=	NOUN
ejpam-3495	248	4	z+	z+	NUM
ejpam-3495	248	5	is	be	AUX
ejpam-3495	248	6	a	a	DET
ejpam-3495	248	7	�	�	NOUN
ejpam-3495	248	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	248	9	since	since	SCONJ
ejpam-3495	248	10	for	for	ADP
ejpam-3495	248	11	all	all	DET
ejpam-3495	248	12	x	x	NOUN
ejpam-3495	248	13	,	,	PUNCT
ejpam-3495	248	14	y	y	PROPN
ejpam-3495	248	15	∈	∈	PROPN
ejpam-3495	249	1	i	i	PRON
ejpam-3495	249	2	,	,	PUNCT
ejpam-3495	249	3	x	x	PROPN
ejpam-3495	250	1	+	+	CCONJ
ejpam-3495	250	2	y	y	PROPN
ejpam-3495	250	3	∈	∈	PROPN
ejpam-3495	250	4	i.	i.	NOUN
ejpam-3495	250	5	however	however	ADV
ejpam-3495	250	6	,	,	PUNCT
ejpam-3495	250	7	0	0	NUM
ejpam-3495	250	8	/∈	/∈	PUNCT
ejpam-3495	251	1	i	i	PRON
ejpam-3495	251	2	,	,	PUNCT
ejpam-3495	251	3	thus	thus	ADV
ejpam-3495	251	4	,	,	PUNCT
ejpam-3495	251	5	i	i	PRON
ejpam-3495	251	6	is	be	AUX
ejpam-3495	251	7	not	not	PART
ejpam-3495	251	8	a	a	DET
ejpam-3495	251	9	b	b	NOUN
ejpam-3495	251	10	-ideal	-ideal	NOUN
ejpam-3495	251	11	.	.	PUNCT
ejpam-3495	252	1	hence	hence	ADV
ejpam-3495	252	2	,	,	PUNCT
ejpam-3495	252	3	we	we	PRON
ejpam-3495	252	4	have	have	VERB
ejpam-3495	252	5	the	the	DET
ejpam-3495	252	6	following	follow	VERB
ejpam-3495	252	7	remark	remark	NOUN
ejpam-3495	252	8	.	.	PUNCT
ejpam-3495	253	1	remark	remark	PROPN
ejpam-3495	253	2	4.4	4.4	NUM
ejpam-3495	253	3	.	.	PUNCT
ejpam-3495	254	1	if	if	SCONJ
ejpam-3495	254	2	i	i	PRON
ejpam-3495	254	3	is	be	AUX
ejpam-3495	254	4	a	a	DET
ejpam-3495	254	5	�	�	NOUN
ejpam-3495	254	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	254	7	of	of	ADP
ejpam-3495	254	8	a	a	DET
ejpam-3495	254	9	companion	companion	NOUN
ejpam-3495	254	10	b	b	NOUN
ejpam-3495	254	11	-	-	PUNCT
ejpam-3495	254	12	algebra	algebra	NOUN
ejpam-3495	254	13	(	(	PUNCT
ejpam-3495	254	14	x	x	X
ejpam-3495	254	15	,	,	PUNCT
ejpam-3495	254	16	∗	∗	NOUN
ejpam-3495	254	17	,	,	PUNCT
ejpam-3495	254	18	�	�	PROPN
ejpam-3495	254	19	,	,	PUNCT
ejpam-3495	254	20	0	0	NUM
ejpam-3495	254	21	)	)	PUNCT
ejpam-3495	254	22	,	,	PUNCT
ejpam-3495	254	23	then	then	ADV
ejpam-3495	254	24	i	i	PRON
ejpam-3495	254	25	is	be	AUX
ejpam-3495	254	26	not	not	PART
ejpam-3495	254	27	necessarily	necessarily	ADV
ejpam-3495	254	28	a	a	DET
ejpam-3495	254	29	b	b	NOUN
ejpam-3495	254	30	-	-	PUNCT
ejpam-3495	254	31	ideal	ideal	NOUN
ejpam-3495	254	32	.	.	PUNCT
ejpam-3495	255	1	let	let	AUX
ejpam-3495	255	2	(	(	PUNCT
ejpam-3495	255	3	z,−,+	z,−,+	VERB
ejpam-3495	255	4	,	,	PUNCT
ejpam-3495	255	5	0	0	NUM
ejpam-3495	255	6	)	)	PUNCT
ejpam-3495	255	7	be	be	VERB
ejpam-3495	255	8	the	the	DET
ejpam-3495	255	9	companion	companion	PROPN
ejpam-3495	255	10	b	b	PROPN
ejpam-3495	255	11	-algebra	-algebra	NOUN
ejpam-3495	255	12	given	give	VERB
ejpam-3495	255	13	in	in	ADP
ejpam-3495	255	14	example	example	NOUN
ejpam-3495	255	15	3.3	3.3	NUM
ejpam-3495	255	16	.	.	PUNCT
ejpam-3495	256	1	then	then	ADV
ejpam-3495	256	2	i	i	PRON
ejpam-3495	256	3	=	=	PUNCT
ejpam-3495	256	4	z+	z+	NUM
ejpam-3495	256	5	is	be	AUX
ejpam-3495	256	6	a	a	DET
ejpam-3495	256	7	+	+	ADJ
ejpam-3495	256	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	256	9	.	.	PUNCT
ejpam-3495	257	1	note	note	VERB
ejpam-3495	257	2	that	that	SCONJ
ejpam-3495	257	3	i1	i1	PROPN
ejpam-3495	257	4	=	=	PUNCT
ejpam-3495	257	5	z+	z+	PROPN
ejpam-3495	257	6	∪	∪	X
ejpam-3495	257	7	{	{	PUNCT
ejpam-3495	257	8	0	0	NUM
ejpam-3495	257	9	}	}	PUNCT
ejpam-3495	257	10	is	be	AUX
ejpam-3495	257	11	a	a	DET
ejpam-3495	257	12	b	b	NOUN
ejpam-3495	257	13	-ideal	-ideal	NOUN
ejpam-3495	257	14	since	since	SCONJ
ejpam-3495	257	15	0	0	NUM
ejpam-3495	257	16	∈	∈	PROPN
ejpam-3495	257	17	i1	i1	PROPN
ejpam-3495	257	18	.	.	PUNCT
ejpam-3495	258	1	now	now	ADV
ejpam-3495	258	2	,	,	PUNCT
ejpam-3495	258	3	let	let	VERB
ejpam-3495	258	4	x−	x−	PROPN
ejpam-3495	258	5	y	y	PROPN
ejpam-3495	258	6	∈	∈	PROPN
ejpam-3495	258	7	i1	i1	PROPN
ejpam-3495	258	8	and	and	CCONJ
ejpam-3495	258	9	y	y	PROPN
ejpam-3495	258	10	∈	∈	PROPN
ejpam-3495	258	11	i1	i1	PROPN
ejpam-3495	258	12	.	.	PUNCT
ejpam-3495	259	1	then	then	ADV
ejpam-3495	259	2	x−	x−	PROPN
ejpam-3495	259	3	y	y	PROPN
ejpam-3495	259	4	≥	≥	NUM
ejpam-3495	259	5	0	0	NUM
ejpam-3495	259	6	and	and	CCONJ
ejpam-3495	259	7	y	y	PROPN
ejpam-3495	259	8	≥	≥	PROPN
ejpam-3495	259	9	0	0	NUM
ejpam-3495	259	10	.	.	PUNCT
ejpam-3495	260	1	so	so	ADV
ejpam-3495	260	2	x	x	X
ejpam-3495	260	3	≥	≥	NOUN
ejpam-3495	260	4	0	0	NUM
ejpam-3495	261	1	and	and	CCONJ
ejpam-3495	261	2	x	x	PROPN
ejpam-3495	261	3	∈	∈	PROPN
ejpam-3495	261	4	i1	i1	PROPN
ejpam-3495	261	5	.	.	PUNCT
ejpam-3495	262	1	theorem	theorem	VERB
ejpam-3495	262	2	4.5	4.5	NUM
ejpam-3495	262	3	.	.	PUNCT
ejpam-3495	263	1	let	let	VERB
ejpam-3495	263	2	(	(	PUNCT
ejpam-3495	263	3	x	x	X
ejpam-3495	263	4	,	,	PUNCT
ejpam-3495	263	5	∗	∗	NOUN
ejpam-3495	263	6	,	,	PUNCT
ejpam-3495	263	7	�	�	PROPN
ejpam-3495	263	8	,	,	PUNCT
ejpam-3495	263	9	0	0	NUM
ejpam-3495	263	10	)	)	PUNCT
ejpam-3495	263	11	be	be	AUX
ejpam-3495	263	12	a	a	DET
ejpam-3495	263	13	companion	companion	NOUN
ejpam-3495	263	14	b	b	NOUN
ejpam-3495	263	15	-	-	PUNCT
ejpam-3495	263	16	algebra	algebra	NOUN
ejpam-3495	263	17	.	.	PUNCT
ejpam-3495	264	1	suppose	suppose	VERB
ejpam-3495	264	2	i	i	PRON
ejpam-3495	264	3	is	be	AUX
ejpam-3495	264	4	a	a	DET
ejpam-3495	264	5	�	�	NOUN
ejpam-3495	264	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	264	7	and	and	CCONJ
ejpam-3495	264	8	0	0	NUM
ejpam-3495	264	9	∈	∈	PROPN
ejpam-3495	264	10	i.	i.	NOUN
ejpam-3495	264	11	then	then	ADV
ejpam-3495	264	12	i	i	PRON
ejpam-3495	264	13	is	be	AUX
ejpam-3495	264	14	a	a	DET
ejpam-3495	264	15	b	b	NOUN
ejpam-3495	264	16	-	-	PUNCT
ejpam-3495	264	17	ideal	ideal	NOUN
ejpam-3495	264	18	.	.	PUNCT
ejpam-3495	265	1	proof	proof	NOUN
ejpam-3495	265	2	:	:	PUNCT
ejpam-3495	265	3	suppose	suppose	VERB
ejpam-3495	265	4	i	i	PRON
ejpam-3495	265	5	is	be	AUX
ejpam-3495	265	6	a	a	DET
ejpam-3495	265	7	�	�	NOUN
ejpam-3495	265	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	265	9	of	of	ADP
ejpam-3495	265	10	x	x	X
ejpam-3495	265	11	and	and	CCONJ
ejpam-3495	265	12	0	0	NUM
ejpam-3495	265	13	∈	∈	PROPN
ejpam-3495	265	14	i.	i.	NOUN
ejpam-3495	265	15	let	let	VERB
ejpam-3495	265	16	u	u	PRON
ejpam-3495	265	17	∗	∗	NOUN
ejpam-3495	265	18	v	v	NOUN
ejpam-3495	265	19	∈	∈	NOUN
ejpam-3495	266	1	i	i	PRON
ejpam-3495	266	2	and	and	CCONJ
ejpam-3495	266	3	v	v	ADP
ejpam-3495	266	4	∈	∈	PROPN
ejpam-3495	266	5	i.	i.	NOUN
ejpam-3495	266	6	then	then	ADV
ejpam-3495	266	7	by	by	ADP
ejpam-3495	266	8	theorem	theorem	ADJ
ejpam-3495	266	9	2.3(a	2.3(a	NUM
ejpam-3495	266	10	)	)	PUNCT
ejpam-3495	266	11	and	and	CCONJ
ejpam-3495	266	12	lemma	lemma	PROPN
ejpam-3495	266	13	3.10(b	3.10(b	PROPN
ejpam-3495	266	14	)	)	PUNCT
ejpam-3495	266	15	,	,	PUNCT
ejpam-3495	266	16	u	u	NOUN
ejpam-3495	266	17	=	=	PUNCT
ejpam-3495	266	18	(	(	PUNCT
ejpam-3495	266	19	u	u	NOUN
ejpam-3495	266	20	∗	∗	PROPN
ejpam-3495	266	21	v	v	NOUN
ejpam-3495	266	22	)	)	PUNCT
ejpam-3495	266	23	∗	∗	NOUN
ejpam-3495	266	24	(	(	PUNCT
ejpam-3495	266	25	0	0	NUM
ejpam-3495	266	26	∗	∗	NUM
ejpam-3495	266	27	v	v	NOUN
ejpam-3495	266	28	)	)	PUNCT
ejpam-3495	266	29	=	=	SYM
ejpam-3495	266	30	v	v	PROPN
ejpam-3495	266	31	�	�	PROPN
ejpam-3495	266	32	(	(	PUNCT
ejpam-3495	266	33	u	u	NOUN
ejpam-3495	266	34	∗	∗	PROPN
ejpam-3495	266	35	v	v	NOUN
ejpam-3495	266	36	)	)	PUNCT
ejpam-3495	266	37	∈	∈	PROPN
ejpam-3495	266	38	i.	i.	NOUN
ejpam-3495	266	39	therefore	therefore	ADV
ejpam-3495	266	40	,	,	PUNCT
ejpam-3495	266	41	i	i	PRON
ejpam-3495	266	42	is	be	AUX
ejpam-3495	266	43	a	a	DET
ejpam-3495	266	44	b	b	NOUN
ejpam-3495	266	45	-ideal	-ideal	NOUN
ejpam-3495	266	46	.	.	PUNCT
ejpam-3495	267	1	�	�	PROPN
ejpam-3495	267	2	the	the	DET
ejpam-3495	267	3	following	following	ADJ
ejpam-3495	267	4	result	result	NOUN
ejpam-3495	267	5	follows	follow	VERB
ejpam-3495	267	6	from	from	ADP
ejpam-3495	267	7	theorem	theorem	ADJ
ejpam-3495	267	8	4.3	4.3	NUM
ejpam-3495	267	9	and	and	CCONJ
ejpam-3495	267	10	theorem	theorem	VERB
ejpam-3495	267	11	2.10	2.10	NUM
ejpam-3495	267	12	.	.	PUNCT
ejpam-3495	268	1	corollary	corollary	ADJ
ejpam-3495	268	2	4.6	4.6	NUM
ejpam-3495	268	3	.	.	PUNCT
ejpam-3495	269	1	let	let	VERB
ejpam-3495	269	2	(	(	PUNCT
ejpam-3495	269	3	x	x	X
ejpam-3495	269	4	,	,	PUNCT
ejpam-3495	269	5	∗	∗	NOUN
ejpam-3495	269	6	,	,	PUNCT
ejpam-3495	269	7	�	�	PROPN
ejpam-3495	269	8	,	,	PUNCT
ejpam-3495	269	9	0	0	NUM
ejpam-3495	269	10	)	)	PUNCT
ejpam-3495	269	11	be	be	AUX
ejpam-3495	269	12	a	a	DET
ejpam-3495	269	13	companion	companion	NOUN
ejpam-3495	269	14	b	b	NOUN
ejpam-3495	269	15	-	-	PUNCT
ejpam-3495	269	16	algebra	algebra	NOUN
ejpam-3495	269	17	.	.	PUNCT
ejpam-3495	270	1	if	if	SCONJ
ejpam-3495	270	2	s	s	PROPN
ejpam-3495	270	3	is	be	AUX
ejpam-3495	270	4	a	a	DET
ejpam-3495	270	5	b	b	NOUN
ejpam-3495	270	6	-	-	PUNCT
ejpam-3495	270	7	subalgebra	subalgebra	NOUN
ejpam-3495	270	8	of	of	ADP
ejpam-3495	270	9	x	x	PRON
ejpam-3495	270	10	,	,	PUNCT
ejpam-3495	270	11	then	then	ADV
ejpam-3495	270	12	s	s	VERB
ejpam-3495	270	13	is	be	AUX
ejpam-3495	270	14	a	a	DET
ejpam-3495	270	15	�	�	NOUN
ejpam-3495	270	16	-subalgebra	-subalgebra	NOUN
ejpam-3495	270	17	of	of	ADP
ejpam-3495	270	18	x.	x.	NOUN
ejpam-3495	270	19	consider	consider	VERB
ejpam-3495	270	20	again	again	ADV
ejpam-3495	270	21	the	the	DET
ejpam-3495	270	22	companion	companion	NOUN
ejpam-3495	270	23	b	b	PROPN
ejpam-3495	270	24	-algebra	-algebra	PROPN
ejpam-3495	270	25	(	(	PUNCT
ejpam-3495	270	26	z,−,+	z,−,+	NUM
ejpam-3495	270	27	,	,	PUNCT
ejpam-3495	270	28	0	0	NUM
ejpam-3495	270	29	)	)	PUNCT
ejpam-3495	270	30	and	and	CCONJ
ejpam-3495	270	31	+	+	SYM
ejpam-3495	270	32	-subalgebra	-subalgebra	NOUN
ejpam-3495	270	33	i	i	X
ejpam-3495	270	34	=	=	SYM
ejpam-3495	270	35	z+	z+	X
ejpam-3495	270	36	.	.	PUNCT
ejpam-3495	270	37	notice	notice	VERB
ejpam-3495	270	38	that	that	SCONJ
ejpam-3495	270	39	3−5	3−5	NUM
ejpam-3495	270	40	=	=	SYM
ejpam-3495	270	41	−2	−2	PROPN
ejpam-3495	270	42	/∈	/∈	PUNCT
ejpam-3495	270	43	i.	i.	PROPN
ejpam-3495	270	44	hence	hence	ADV
ejpam-3495	270	45	,	,	PUNCT
ejpam-3495	270	46	i	i	PRON
ejpam-3495	270	47	is	be	AUX
ejpam-3495	270	48	not	not	PART
ejpam-3495	270	49	a	a	DET
ejpam-3495	270	50	b	b	NOUN
ejpam-3495	270	51	-subalgebra	-subalgebra	NOUN
ejpam-3495	270	52	.	.	PUNCT
ejpam-3495	271	1	thus	thus	ADV
ejpam-3495	271	2	,	,	PUNCT
ejpam-3495	271	3	we	we	PRON
ejpam-3495	271	4	have	have	VERB
ejpam-3495	271	5	the	the	DET
ejpam-3495	271	6	following	follow	VERB
ejpam-3495	271	7	remark	remark	NOUN
ejpam-3495	271	8	.	.	PUNCT
ejpam-3495	272	1	remark	remark	PROPN
ejpam-3495	272	2	4.7	4.7	NUM
ejpam-3495	272	3	.	.	PUNCT
ejpam-3495	273	1	a	a	DET
ejpam-3495	273	2	�	�	PROPN
ejpam-3495	273	3	-subalgebra	-subalgebra	NOUN
ejpam-3495	273	4	of	of	ADP
ejpam-3495	273	5	x	x	SYM
ejpam-3495	273	6	is	be	AUX
ejpam-3495	273	7	not	not	PART
ejpam-3495	273	8	necessarily	necessarily	ADV
ejpam-3495	273	9	a	a	DET
ejpam-3495	273	10	b	b	NOUN
ejpam-3495	273	11	-	-	PUNCT
ejpam-3495	273	12	subalgebra	subalgebra	NOUN
ejpam-3495	273	13	.	.	PUNCT
ejpam-3495	274	1	example	example	NOUN
ejpam-3495	274	2	4.8	4.8	NUM
ejpam-3495	274	3	.	.	PUNCT
ejpam-3495	275	1	consider	consider	VERB
ejpam-3495	275	2	example	example	NOUN
ejpam-3495	275	3	3.2	3.2	NUM
ejpam-3495	275	4	and	and	CCONJ
ejpam-3495	275	5	�	�	NOUN
ejpam-3495	275	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	275	7	i	i	NOUN
ejpam-3495	275	8	=	=	SYM
ejpam-3495	275	9	{	{	PUNCT
ejpam-3495	275	10	0	0	NUM
ejpam-3495	275	11	,	,	PUNCT
ejpam-3495	275	12	1	1	NUM
ejpam-3495	275	13	,	,	PUNCT
ejpam-3495	275	14	2	2	NUM
ejpam-3495	275	15	}	}	PUNCT
ejpam-3495	275	16	.	.	PUNCT
ejpam-3495	276	1	it	it	PRON
ejpam-3495	276	2	is	be	AUX
ejpam-3495	276	3	easy	easy	ADJ
ejpam-3495	276	4	to	to	PART
ejpam-3495	276	5	see	see	VERB
ejpam-3495	276	6	that	that	SCONJ
ejpam-3495	276	7	i	i	PRON
ejpam-3495	276	8	is	be	AUX
ejpam-3495	276	9	a	a	DET
ejpam-3495	276	10	b	b	NOUN
ejpam-3495	276	11	-subalgebra	-subalgebra	NOUN
ejpam-3495	276	12	and	and	CCONJ
ejpam-3495	276	13	0	0	NUM
ejpam-3495	276	14	∗	∗	NOUN
ejpam-3495	276	15	a	a	DET
ejpam-3495	276	16	∈	∈	NOUN
ejpam-3495	277	1	i	i	PRON
ejpam-3495	277	2	,	,	PUNCT
ejpam-3495	277	3	for	for	ADP
ejpam-3495	277	4	any	any	DET
ejpam-3495	277	5	a	a	DET
ejpam-3495	277	6	∈	∈	PROPN
ejpam-3495	277	7	i.	i.	NOUN
ejpam-3495	277	8	theorem	theorem	VERB
ejpam-3495	277	9	4.9	4.9	NUM
ejpam-3495	277	10	.	.	PUNCT
ejpam-3495	278	1	let	let	VERB
ejpam-3495	278	2	(	(	PUNCT
ejpam-3495	278	3	x	x	X
ejpam-3495	278	4	,	,	PUNCT
ejpam-3495	278	5	∗	∗	NOUN
ejpam-3495	278	6	,	,	PUNCT
ejpam-3495	278	7	�	�	PROPN
ejpam-3495	278	8	,	,	PUNCT
ejpam-3495	278	9	0	0	NUM
ejpam-3495	278	10	)	)	PUNCT
ejpam-3495	278	11	be	be	AUX
ejpam-3495	278	12	a	a	DET
ejpam-3495	278	13	companion	companion	NOUN
ejpam-3495	278	14	b	b	NOUN
ejpam-3495	278	15	-	-	PUNCT
ejpam-3495	278	16	algebra	algebra	NOUN
ejpam-3495	278	17	.	.	PUNCT
ejpam-3495	279	1	suppose	suppose	VERB
ejpam-3495	279	2	i	i	PRON
ejpam-3495	279	3	is	be	AUX
ejpam-3495	279	4	a	a	DET
ejpam-3495	279	5	�	�	NOUN
ejpam-3495	279	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	279	7	and	and	CCONJ
ejpam-3495	279	8	0	0	NUM
ejpam-3495	279	9	∗	∗	NOUN
ejpam-3495	279	10	a	a	DET
ejpam-3495	279	11	∈	∈	NOUN
ejpam-3495	280	1	i	i	PRON
ejpam-3495	280	2	,	,	PUNCT
ejpam-3495	280	3	for	for	ADP
ejpam-3495	280	4	any	any	DET
ejpam-3495	280	5	a	a	DET
ejpam-3495	280	6	∈	∈	PROPN
ejpam-3495	280	7	i.	i.	NOUN
ejpam-3495	280	8	then	then	ADV
ejpam-3495	280	9	i	i	PRON
ejpam-3495	280	10	is	be	AUX
ejpam-3495	280	11	a	a	DET
ejpam-3495	280	12	b	b	NOUN
ejpam-3495	280	13	-	-	PUNCT
ejpam-3495	280	14	subalgebra	subalgebra	NOUN
ejpam-3495	280	15	.	.	PUNCT
ejpam-3495	281	1	l.d	l.d	PROPN
ejpam-3495	281	2	.	.	PROPN
ejpam-3495	281	3	naingue	naingue	PROPN
ejpam-3495	281	4	,	,	PUNCT
ejpam-3495	281	5	j.p	j.p	PROPN
ejpam-3495	281	6	.	.	PROPN
ejpam-3495	281	7	vilela	vilela	PROPN
ejpam-3495	281	8	/	/	SYM
ejpam-3495	281	9	eur	eur	PROPN
ejpam-3495	281	10	.	.	PUNCT
ejpam-3495	282	1	j.	j.	PROPN
ejpam-3495	282	2	pure	pure	PROPN
ejpam-3495	282	3	appl	appl	PROPN
ejpam-3495	282	4	.	.	PROPN
ejpam-3495	282	5	math	math	PROPN
ejpam-3495	282	6	,	,	PUNCT
ejpam-3495	282	7	12	12	NUM
ejpam-3495	282	8	(	(	PUNCT
ejpam-3495	282	9	3	3	NUM
ejpam-3495	282	10	)	)	PUNCT
ejpam-3495	282	11	(	(	PUNCT
ejpam-3495	282	12	2019	2019	NUM
ejpam-3495	282	13	)	)	PUNCT
ejpam-3495	282	14	,	,	PUNCT
ejpam-3495	282	15	1248	1248	NUM
ejpam-3495	282	16	-	-	SYM
ejpam-3495	282	17	1259	1259	NUM
ejpam-3495	282	18	1256	1256	NUM
ejpam-3495	282	19	proof	proof	NOUN
ejpam-3495	282	20	:	:	PUNCT
ejpam-3495	282	21	suppose	suppose	VERB
ejpam-3495	282	22	i	i	PRON
ejpam-3495	282	23	is	be	AUX
ejpam-3495	282	24	a	a	DET
ejpam-3495	282	25	�	�	NOUN
ejpam-3495	282	26	-subalgebra	-subalgebra	NOUN
ejpam-3495	282	27	and	and	CCONJ
ejpam-3495	282	28	0	0	NUM
ejpam-3495	282	29	∗	∗	NOUN
ejpam-3495	282	30	a	a	DET
ejpam-3495	282	31	∈	∈	NOUN
ejpam-3495	283	1	i	i	PRON
ejpam-3495	283	2	,	,	PUNCT
ejpam-3495	283	3	for	for	ADP
ejpam-3495	283	4	any	any	DET
ejpam-3495	283	5	a	a	DET
ejpam-3495	283	6	∈	∈	PROPN
ejpam-3495	283	7	i.	i.	NOUN
ejpam-3495	283	8	let	let	VERB
ejpam-3495	283	9	x	x	PRON
ejpam-3495	283	10	,	,	PUNCT
ejpam-3495	283	11	y	y	PROPN
ejpam-3495	283	12	∈	∈	PROPN
ejpam-3495	283	13	i.	i.	NOUN
ejpam-3495	283	14	then	then	ADV
ejpam-3495	283	15	0	0	NUM
ejpam-3495	283	16	∗	∗	NOUN
ejpam-3495	283	17	y	y	PROPN
ejpam-3495	283	18	∈	∈	PROPN
ejpam-3495	283	19	i.	i.	NOUN
ejpam-3495	283	20	by	by	ADP
ejpam-3495	283	21	theorem	theorem	PROPN
ejpam-3495	283	22	2.3(b	2.3(b	NUM
ejpam-3495	283	23	)	)	PUNCT
ejpam-3495	283	24	and	and	CCONJ
ejpam-3495	283	25	lemma	lemma	PROPN
ejpam-3495	283	26	3.10(b	3.10(b	PROPN
ejpam-3495	283	27	)	)	PUNCT
ejpam-3495	283	28	,	,	PUNCT
ejpam-3495	284	1	x	x	X
ejpam-3495	284	2	∗	∗	NOUN
ejpam-3495	284	3	y	y	NOUN
ejpam-3495	284	4	=	=	PUNCT
ejpam-3495	284	5	x	x	SYM
ejpam-3495	284	6	∗	∗	NOUN
ejpam-3495	284	7	(	(	PUNCT
ejpam-3495	284	8	0	0	NUM
ejpam-3495	284	9	∗	∗	NOUN
ejpam-3495	284	10	(	(	PUNCT
ejpam-3495	284	11	0	0	NUM
ejpam-3495	284	12	∗	∗	PROPN
ejpam-3495	284	13	y	y	NOUN
ejpam-3495	284	14	)	)	PUNCT
ejpam-3495	284	15	)	)	PUNCT
ejpam-3495	285	1	=	=	PUNCT
ejpam-3495	285	2	(	(	PUNCT
ejpam-3495	285	3	0	0	NUM
ejpam-3495	285	4	∗	∗	NOUN
ejpam-3495	285	5	y)	y)	NOUN
ejpam-3495	285	6	�	�	PROPN
ejpam-3495	285	7	x	x	SYM
ejpam-3495	285	8	∈	∈	PROPN
ejpam-3495	285	9	i.	i.	NOUN
ejpam-3495	285	10	thus	thus	ADV
ejpam-3495	285	11	,	,	PUNCT
ejpam-3495	285	12	i	i	PRON
ejpam-3495	285	13	is	be	AUX
ejpam-3495	285	14	a	a	DET
ejpam-3495	285	15	b	b	PROPN
ejpam-3495	285	16	-subalgebra	-subalgebra	NOUN
ejpam-3495	285	17	.	.	PUNCT
ejpam-3495	286	1	�	�	PROPN
ejpam-3495	286	2	consider	consider	VERB
ejpam-3495	286	3	again	again	ADV
ejpam-3495	286	4	the	the	DET
ejpam-3495	286	5	companion	companion	NOUN
ejpam-3495	286	6	b	b	PROPN
ejpam-3495	286	7	-algebra	-algebra	PROPN
ejpam-3495	286	8	(	(	PUNCT
ejpam-3495	286	9	z,−,+	z,−,+	NUM
ejpam-3495	286	10	,	,	PUNCT
ejpam-3495	286	11	0	0	NUM
ejpam-3495	286	12	)	)	PUNCT
ejpam-3495	286	13	and	and	CCONJ
ejpam-3495	286	14	+	+	SYM
ejpam-3495	286	15	-subalgebra	-subalgebra	NOUN
ejpam-3495	286	16	i	i	X
ejpam-3495	286	17	=	=	SYM
ejpam-3495	286	18	z+	z+	X
ejpam-3495	286	19	.	.	X
ejpam-3495	287	1	take	take	VERB
ejpam-3495	287	2	a	a	DET
ejpam-3495	287	3	=	=	SYM
ejpam-3495	287	4	2	2	NUM
ejpam-3495	287	5	and	and	CCONJ
ejpam-3495	287	6	b	b	NOUN
ejpam-3495	287	7	=	=	SYM
ejpam-3495	287	8	3	3	NUM
ejpam-3495	287	9	∈	∈	PROPN
ejpam-3495	287	10	i.	i.	NOUN
ejpam-3495	287	11	then	then	ADV
ejpam-3495	287	12	b−1	b−1	PROPN
ejpam-3495	287	13	=	=	PUNCT
ejpam-3495	287	14	0	0	NUM
ejpam-3495	288	1	−	−	PROPN
ejpam-3495	288	2	b	b	X
ejpam-3495	288	3	=	=	X
ejpam-3495	288	4	−3	−3	PROPN
ejpam-3495	288	5	and	and	CCONJ
ejpam-3495	288	6	a	a	DET
ejpam-3495	288	7	+	+	ADJ
ejpam-3495	288	8	b−1	b−1	PROPN
ejpam-3495	288	9	=	=	SYM
ejpam-3495	288	10	−1	−1	NOUN
ejpam-3495	288	11	/∈	/∈	PUNCT
ejpam-3495	288	12	i.	i.	PROPN
ejpam-3495	289	1	hence	hence	ADV
ejpam-3495	289	2	,	,	PUNCT
ejpam-3495	289	3	i	i	PRON
ejpam-3495	289	4	is	be	AUX
ejpam-3495	289	5	not	not	PART
ejpam-3495	289	6	a	a	DET
ejpam-3495	289	7	subgroup	subgroup	NOUN
ejpam-3495	289	8	of	of	ADP
ejpam-3495	289	9	the	the	DET
ejpam-3495	289	10	group	group	NOUN
ejpam-3495	289	11	(	(	PUNCT
ejpam-3495	289	12	z,+	z,+	NUM
ejpam-3495	289	13	,	,	PUNCT
ejpam-3495	289	14	0	0	NUM
ejpam-3495	289	15	)	)	PUNCT
ejpam-3495	289	16	.	.	PUNCT
ejpam-3495	290	1	so	so	ADV
ejpam-3495	290	2	,	,	PUNCT
ejpam-3495	290	3	we	we	PRON
ejpam-3495	290	4	have	have	VERB
ejpam-3495	290	5	the	the	DET
ejpam-3495	290	6	following	follow	VERB
ejpam-3495	290	7	remark	remark	NOUN
ejpam-3495	290	8	.	.	PUNCT
ejpam-3495	291	1	remark	remark	VERB
ejpam-3495	291	2	4.10	4.10	NUM
ejpam-3495	291	3	.	.	PUNCT
ejpam-3495	292	1	if	if	SCONJ
ejpam-3495	292	2	i	i	PRON
ejpam-3495	292	3	is	be	AUX
ejpam-3495	292	4	a	a	DET
ejpam-3495	292	5	�	�	PROPN
ejpam-3495	292	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	292	7	,	,	PUNCT
ejpam-3495	292	8	then	then	ADV
ejpam-3495	292	9	i	i	PRON
ejpam-3495	292	10	is	be	AUX
ejpam-3495	292	11	not	not	PART
ejpam-3495	292	12	necessarily	necessarily	ADV
ejpam-3495	292	13	a	a	DET
ejpam-3495	292	14	subgroup	subgroup	NOUN
ejpam-3495	292	15	.	.	PUNCT
ejpam-3495	293	1	consider	consider	VERB
ejpam-3495	293	2	the	the	DET
ejpam-3495	293	3	companion	companion	NOUN
ejpam-3495	293	4	b	b	PROPN
ejpam-3495	293	5	-algebra	-algebra	PROPN
ejpam-3495	293	6	(	(	PUNCT
ejpam-3495	293	7	z,−,+	z,−,+	NOUN
ejpam-3495	293	8	,	,	PUNCT
ejpam-3495	293	9	0	0	NUM
ejpam-3495	293	10	)	)	PUNCT
ejpam-3495	293	11	,	,	PUNCT
ejpam-3495	293	12	h1	h1	NOUN
ejpam-3495	293	13	=	=	SYM
ejpam-3495	293	14	z+	z+	NUM
ejpam-3495	293	15	and	and	CCONJ
ejpam-3495	293	16	h2	h2	NOUN
ejpam-3495	293	17	=	=	SYM
ejpam-3495	294	1	z−.	z−.	NOUN
ejpam-3495	294	2	then	then	ADV
ejpam-3495	294	3	h1	h1	VERB
ejpam-3495	294	4	and	and	CCONJ
ejpam-3495	294	5	h2	h2	NOUN
ejpam-3495	294	6	are	be	AUX
ejpam-3495	294	7	+	+	X
ejpam-3495	294	8	-subalgebras	-subalgebra	NOUN
ejpam-3495	294	9	.	.	PUNCT
ejpam-3495	295	1	however	however	ADV
ejpam-3495	295	2	,	,	PUNCT
ejpam-3495	295	3	h1	h1	VERB
ejpam-3495	295	4	∩h2	∩h2	PROPN
ejpam-3495	295	5	=	=	PUNCT
ejpam-3495	295	6	∅	∅	NOUN
ejpam-3495	295	7	and	and	CCONJ
ejpam-3495	295	8	hence	hence	ADV
ejpam-3495	295	9	,	,	PUNCT
ejpam-3495	295	10	not	not	PART
ejpam-3495	295	11	a	a	DET
ejpam-3495	295	12	+	+	NOUN
ejpam-3495	295	13	-subalgebra	-subalgebra	NOUN
ejpam-3495	295	14	.	.	PUNCT
ejpam-3495	296	1	thus	thus	ADV
ejpam-3495	296	2	,	,	PUNCT
ejpam-3495	296	3	we	we	PRON
ejpam-3495	296	4	have	have	VERB
ejpam-3495	296	5	the	the	DET
ejpam-3495	296	6	following	follow	VERB
ejpam-3495	296	7	remark	remark	NOUN
ejpam-3495	296	8	.	.	PUNCT
ejpam-3495	297	1	remark	remark	PROPN
ejpam-3495	297	2	4.11	4.11	NUM
ejpam-3495	297	3	.	.	PUNCT
ejpam-3495	298	1	the	the	DET
ejpam-3495	298	2	intersection	intersection	NOUN
ejpam-3495	298	3	of	of	ADP
ejpam-3495	298	4	�	�	NOUN
ejpam-3495	298	5	-subalgebras	-subalgebra	NOUN
ejpam-3495	298	6	need	need	AUX
ejpam-3495	298	7	not	not	PART
ejpam-3495	298	8	be	be	AUX
ejpam-3495	298	9	a	a	DET
ejpam-3495	298	10	�	�	NOUN
ejpam-3495	298	11	-subalgebra	-subalgebra	NOUN
ejpam-3495	298	12	.	.	PUNCT
ejpam-3495	299	1	the	the	DET
ejpam-3495	299	2	proof	proof	NOUN
ejpam-3495	299	3	of	of	ADP
ejpam-3495	299	4	the	the	DET
ejpam-3495	299	5	following	follow	VERB
ejpam-3495	299	6	theorem	theorem	NOUN
ejpam-3495	299	7	is	be	AUX
ejpam-3495	299	8	straightforward	straightforward	ADJ
ejpam-3495	299	9	.	.	PUNCT
ejpam-3495	300	1	theorem	theorem	VERB
ejpam-3495	300	2	4.12	4.12	NUM
ejpam-3495	300	3	.	.	PUNCT
ejpam-3495	301	1	let	let	VERB
ejpam-3495	301	2	{	{	PUNCT
ejpam-3495	301	3	ik	ik	PROPN
ejpam-3495	301	4	:	:	PUNCT
ejpam-3495	301	5	k	k	PROPN
ejpam-3495	301	6	∈	∈	PROPN
ejpam-3495	301	7	k	k	AUX
ejpam-3495	301	8	}	}	PUNCT
ejpam-3495	301	9	be	be	AUX
ejpam-3495	301	10	a	a	DET
ejpam-3495	301	11	nonempty	nonempty	ADJ
ejpam-3495	301	12	collection	collection	NOUN
ejpam-3495	301	13	of	of	ADP
ejpam-3495	301	14	�	�	NOUN
ejpam-3495	301	15	-subalgebras	-subalgebra	NOUN
ejpam-3495	301	16	of	of	ADP
ejpam-3495	301	17	a	a	DET
ejpam-3495	301	18	companion	companion	NOUN
ejpam-3495	301	19	b	b	NOUN
ejpam-3495	301	20	-	-	PUNCT
ejpam-3495	301	21	algebra	algebra	NOUN
ejpam-3495	301	22	.	.	PUNCT
ejpam-3495	302	1	if	if	SCONJ
ejpam-3495	302	2	i	i	PRON
ejpam-3495	302	3	=	=	SYM
ejpam-3495	302	4	⋂	⋂	PROPN
ejpam-3495	302	5	k∈k	k∈k	NOUN
ejpam-3495	302	6	ik	ik	PROPN
ejpam-3495	302	7	6=	6=	PROPN
ejpam-3495	302	8	∅	∅	NOUN
ejpam-3495	302	9	,	,	PUNCT
ejpam-3495	302	10	then	then	ADV
ejpam-3495	302	11	i	i	PRON
ejpam-3495	302	12	is	be	AUX
ejpam-3495	302	13	a	a	DET
ejpam-3495	302	14	�	�	NOUN
ejpam-3495	302	15	-subalgebra	-subalgebra	NOUN
ejpam-3495	302	16	.	.	PUNCT
ejpam-3495	303	1	consider	consider	VERB
ejpam-3495	303	2	example	example	NOUN
ejpam-3495	303	3	3.2	3.2	NUM
ejpam-3495	303	4	.	.	PUNCT
ejpam-3495	304	1	take	take	VERB
ejpam-3495	304	2	a	a	PRON
ejpam-3495	304	3	=	=	PUNCT
ejpam-3495	304	4	{	{	PUNCT
ejpam-3495	304	5	0	0	NUM
ejpam-3495	304	6	,	,	PUNCT
ejpam-3495	304	7	3	3	NUM
ejpam-3495	304	8	}	}	PUNCT
ejpam-3495	304	9	and	and	CCONJ
ejpam-3495	304	10	b	b	X
ejpam-3495	304	11	=	=	SYM
ejpam-3495	304	12	{	{	PUNCT
ejpam-3495	304	13	0	0	NUM
ejpam-3495	304	14	,	,	PUNCT
ejpam-3495	304	15	4	4	NUM
ejpam-3495	304	16	}	}	PUNCT
ejpam-3495	304	17	.	.	PUNCT
ejpam-3495	305	1	then	then	ADV
ejpam-3495	305	2	a	a	PRON
ejpam-3495	305	3	and	and	CCONJ
ejpam-3495	305	4	b	b	NOUN
ejpam-3495	305	5	are	be	AUX
ejpam-3495	305	6	�	�	NOUN
ejpam-3495	305	7	subalgebras	subalgebras	PROPN
ejpam-3495	305	8	.	.	PUNCT
ejpam-3495	306	1	however	however	ADV
ejpam-3495	306	2	,	,	PUNCT
ejpam-3495	306	3	a	a	DET
ejpam-3495	306	4	∪	∪	X
ejpam-3495	306	5	b	b	NOUN
ejpam-3495	306	6	=	=	SYM
ejpam-3495	306	7	{	{	PUNCT
ejpam-3495	306	8	0	0	NUM
ejpam-3495	306	9	,	,	PUNCT
ejpam-3495	306	10	3	3	NUM
ejpam-3495	306	11	,	,	PUNCT
ejpam-3495	306	12	4	4	NUM
ejpam-3495	306	13	}	}	PUNCT
ejpam-3495	306	14	is	be	AUX
ejpam-3495	306	15	not	not	PART
ejpam-3495	306	16	a	a	DET
ejpam-3495	306	17	�	�	NOUN
ejpam-3495	306	18	-subalgebra	-subalgebra	NOUN
ejpam-3495	306	19	since	since	SCONJ
ejpam-3495	306	20	3	3	NUM
ejpam-3495	306	21	�	�	PROPN
ejpam-3495	306	22	4	4	NUM
ejpam-3495	306	23	=	=	SYM
ejpam-3495	306	24	1	1	NUM
ejpam-3495	306	25	/∈	/∈	PUNCT
ejpam-3495	306	26	a	a	DET
ejpam-3495	306	27	∪	∪	NOUN
ejpam-3495	306	28	b.	b.	NOUN
ejpam-3495	306	29	hence	hence	ADV
ejpam-3495	306	30	,	,	PUNCT
ejpam-3495	306	31	we	we	PRON
ejpam-3495	306	32	have	have	VERB
ejpam-3495	306	33	the	the	DET
ejpam-3495	306	34	following	follow	VERB
ejpam-3495	306	35	remark	remark	NOUN
ejpam-3495	306	36	.	.	PUNCT
ejpam-3495	307	1	remark	remark	PROPN
ejpam-3495	307	2	4.13	4.13	NUM
ejpam-3495	307	3	.	.	PUNCT
ejpam-3495	308	1	the	the	DET
ejpam-3495	308	2	union	union	PROPN
ejpam-3495	308	3	of	of	ADP
ejpam-3495	308	4	�	�	PROPN
ejpam-3495	308	5	-subalgebras	-subalgebra	NOUN
ejpam-3495	308	6	need	need	AUX
ejpam-3495	308	7	not	not	PART
ejpam-3495	308	8	be	be	AUX
ejpam-3495	308	9	a	a	DET
ejpam-3495	308	10	�	�	NOUN
ejpam-3495	308	11	-subalgebra	-subalgebra	NOUN
ejpam-3495	308	12	.	.	PUNCT
ejpam-3495	309	1	5	5	NUM
ejpam-3495	309	2	.	.	X
ejpam-3495	309	3	on	on	ADP
ejpam-3495	309	4	�	�	NOUN
ejpam-3495	309	5	-ideals	-ideal	NOUN
ejpam-3495	309	6	definition	definition	NOUN
ejpam-3495	309	7	5.1	5.1	NUM
ejpam-3495	309	8	.	.	PUNCT
ejpam-3495	310	1	let	let	VERB
ejpam-3495	310	2	(	(	PUNCT
ejpam-3495	310	3	x	x	X
ejpam-3495	310	4	,	,	PUNCT
ejpam-3495	310	5	∗	∗	NOUN
ejpam-3495	310	6	,	,	PUNCT
ejpam-3495	310	7	�	�	PROPN
ejpam-3495	310	8	,	,	PUNCT
ejpam-3495	310	9	0	0	NUM
ejpam-3495	310	10	)	)	PUNCT
ejpam-3495	310	11	be	be	AUX
ejpam-3495	310	12	a	a	DET
ejpam-3495	310	13	companion	companion	NOUN
ejpam-3495	310	14	b	b	PROPN
ejpam-3495	310	15	-algebra	-algebra	PROPN
ejpam-3495	310	16	.	.	PUNCT
ejpam-3495	311	1	a	a	DET
ejpam-3495	311	2	nonempty	nonempty	NOUN
ejpam-3495	311	3	subset	subset	VERB
ejpam-3495	311	4	i	i	PRON
ejpam-3495	311	5	of	of	ADP
ejpam-3495	311	6	x	x	PRON
ejpam-3495	311	7	is	be	AUX
ejpam-3495	311	8	called	call	VERB
ejpam-3495	311	9	a	a	DET
ejpam-3495	311	10	�	�	NOUN
ejpam-3495	311	11	-ideal	-ideal	NOUN
ejpam-3495	311	12	if	if	SCONJ
ejpam-3495	311	13	it	it	PRON
ejpam-3495	311	14	satisfies	satisfy	VERB
ejpam-3495	311	15	:	:	PUNCT
ejpam-3495	311	16	for	for	ADP
ejpam-3495	311	17	any	any	DET
ejpam-3495	311	18	x	x	NOUN
ejpam-3495	311	19	,	,	PUNCT
ejpam-3495	311	20	y	y	PROPN
ejpam-3495	311	21	∈	∈	PROPN
ejpam-3495	311	22	x	x	X
ejpam-3495	311	23	,	,	PUNCT
ejpam-3495	311	24	(	(	PUNCT
ejpam-3495	311	25	i	i	NOUN
ejpam-3495	311	26	)	)	PUNCT
ejpam-3495	311	27	0	0	PUNCT
ejpam-3495	312	1	∈	∈	PROPN
ejpam-3495	313	1	i	i	PRON
ejpam-3495	313	2	and	and	CCONJ
ejpam-3495	313	3	(	(	PUNCT
ejpam-3495	313	4	ii	ii	NOUN
ejpam-3495	313	5	)	)	PUNCT
ejpam-3495	313	6	x	x	NOUN
ejpam-3495	313	7	�	�	PROPN
ejpam-3495	313	8	y	y	PROPN
ejpam-3495	313	9	∈	∈	PROPN
ejpam-3495	314	1	i	i	PRON
ejpam-3495	314	2	and	and	CCONJ
ejpam-3495	314	3	y	y	PROPN
ejpam-3495	314	4	∈	∈	PROPN
ejpam-3495	315	1	i	i	PRON
ejpam-3495	315	2	imply	imply	VERB
ejpam-3495	315	3	x	x	X
ejpam-3495	315	4	∈	∈	PROPN
ejpam-3495	315	5	i.	i.	NOUN
ejpam-3495	315	6	example	example	NOUN
ejpam-3495	315	7	5.2	5.2	NUM
ejpam-3495	315	8	.	.	PUNCT
ejpam-3495	316	1	in	in	ADP
ejpam-3495	316	2	example	example	NOUN
ejpam-3495	316	3	3.2	3.2	NUM
ejpam-3495	316	4	,	,	PUNCT
ejpam-3495	316	5	{	{	PUNCT
ejpam-3495	316	6	0	0	NUM
ejpam-3495	316	7	,	,	PUNCT
ejpam-3495	316	8	3	3	NUM
ejpam-3495	316	9	}	}	PUNCT
ejpam-3495	316	10	is	be	AUX
ejpam-3495	316	11	a	a	DET
ejpam-3495	316	12	�	�	NOUN
ejpam-3495	316	13	-ideal	-ideal	NOUN
ejpam-3495	316	14	of	of	ADP
ejpam-3495	316	15	x.	x.	NOUN
ejpam-3495	316	16	but	but	CCONJ
ejpam-3495	316	17	,	,	PUNCT
ejpam-3495	316	18	i	i	PRON
ejpam-3495	316	19	=	=	PUNCT
ejpam-3495	316	20	{	{	PUNCT
ejpam-3495	316	21	0	0	NUM
ejpam-3495	316	22	,	,	PUNCT
ejpam-3495	316	23	1	1	NUM
ejpam-3495	316	24	}	}	PUNCT
ejpam-3495	316	25	is	be	AUX
ejpam-3495	316	26	not	not	PART
ejpam-3495	316	27	a	a	DET
ejpam-3495	316	28	�	�	NOUN
ejpam-3495	316	29	-ideal	-ideal	NOUN
ejpam-3495	316	30	since	since	SCONJ
ejpam-3495	316	31	2	2	NUM
ejpam-3495	316	32	�	�	PROPN
ejpam-3495	316	33	1	1	NUM
ejpam-3495	316	34	=	=	SYM
ejpam-3495	316	35	0	0	PUNCT
ejpam-3495	316	36	∈	∈	PROPN
ejpam-3495	316	37	i	i	PRON
ejpam-3495	316	38	and	and	CCONJ
ejpam-3495	316	39	1	1	NUM
ejpam-3495	316	40	∈	∈	NOUN
ejpam-3495	316	41	i	i	PRON
ejpam-3495	316	42	but	but	CCONJ
ejpam-3495	316	43	2	2	NUM
ejpam-3495	316	44	/∈	/∈	NOUN
ejpam-3495	316	45	i.	i.	PROPN
ejpam-3495	316	46	lemma	lemma	PROPN
ejpam-3495	316	47	5.3	5.3	NUM
ejpam-3495	316	48	.	.	PUNCT
ejpam-3495	317	1	let	let	VERB
ejpam-3495	317	2	(	(	PUNCT
ejpam-3495	317	3	x	x	X
ejpam-3495	317	4	,	,	PUNCT
ejpam-3495	317	5	∗	∗	NOUN
ejpam-3495	317	6	,	,	PUNCT
ejpam-3495	317	7	�	�	PROPN
ejpam-3495	317	8	,	,	PUNCT
ejpam-3495	317	9	0	0	NUM
ejpam-3495	317	10	)	)	PUNCT
ejpam-3495	317	11	be	be	AUX
ejpam-3495	317	12	a	a	DET
ejpam-3495	317	13	companion	companion	NOUN
ejpam-3495	317	14	b	b	NOUN
ejpam-3495	317	15	-	-	PUNCT
ejpam-3495	317	16	algebra	algebra	NOUN
ejpam-3495	317	17	and	and	CCONJ
ejpam-3495	317	18	let	let	VERB
ejpam-3495	317	19	i	i	PRON
ejpam-3495	317	20	be	be	AUX
ejpam-3495	317	21	a	a	DET
ejpam-3495	317	22	�	�	NOUN
ejpam-3495	317	23	-ideal	-ideal	NOUN
ejpam-3495	317	24	.	.	PUNCT
ejpam-3495	318	1	if	if	SCONJ
ejpam-3495	318	2	x	x	SYM
ejpam-3495	318	3	∈	∈	PROPN
ejpam-3495	318	4	i	i	PRON
ejpam-3495	318	5	,	,	PUNCT
ejpam-3495	318	6	then	then	ADV
ejpam-3495	318	7	x−1	x−1	PUNCT
ejpam-3495	318	8	=	=	NOUN
ejpam-3495	318	9	0	0	NUM
ejpam-3495	318	10	∗	∗	NOUN
ejpam-3495	318	11	x	x	SYM
ejpam-3495	318	12	∈	∈	NOUN
ejpam-3495	318	13	i.	i.	NOUN
ejpam-3495	318	14	proof	proof	NOUN
ejpam-3495	318	15	:	:	PUNCT
ejpam-3495	318	16	by	by	ADP
ejpam-3495	318	17	remark	remark	NOUN
ejpam-3495	318	18	3.15	3.15	NUM
ejpam-3495	318	19	,	,	PUNCT
ejpam-3495	318	20	x−1	x−1	PUNCT
ejpam-3495	319	1	=	=	SYM
ejpam-3495	319	2	0	0	NUM
ejpam-3495	319	3	∗	∗	NOUN
ejpam-3495	319	4	x	x	VERB
ejpam-3495	319	5	is	be	AUX
ejpam-3495	319	6	the	the	DET
ejpam-3495	319	7	inverse	inverse	NOUN
ejpam-3495	319	8	of	of	ADP
ejpam-3495	319	9	x.	x.	NOUN
ejpam-3495	319	10	thus	thus	ADV
ejpam-3495	319	11	,	,	PUNCT
ejpam-3495	319	12	(	(	PUNCT
ejpam-3495	319	13	0	0	NUM
ejpam-3495	319	14	∗	∗	NOUN
ejpam-3495	319	15	x	x	NOUN
ejpam-3495	319	16	)	)	PUNCT
ejpam-3495	319	17	�	�	PROPN
ejpam-3495	319	18	x	x	PUNCT
ejpam-3495	319	19	=	=	SYM
ejpam-3495	319	20	0	0	NUM
ejpam-3495	319	21	∈	∈	PROPN
ejpam-3495	319	22	i.	i.	NOUN
ejpam-3495	319	23	since	since	SCONJ
ejpam-3495	319	24	x	x	PROPN
ejpam-3495	319	25	∈	∈	PROPN
ejpam-3495	319	26	i	i	PRON
ejpam-3495	319	27	and	and	CCONJ
ejpam-3495	319	28	i	i	PRON
ejpam-3495	319	29	is	be	AUX
ejpam-3495	319	30	a	a	DET
ejpam-3495	319	31	�	�	NOUN
ejpam-3495	319	32	-ideal	-ideal	NOUN
ejpam-3495	319	33	,	,	PUNCT
ejpam-3495	319	34	then	then	ADV
ejpam-3495	319	35	0	0	NUM
ejpam-3495	319	36	∗	∗	NOUN
ejpam-3495	319	37	x	x	SYM
ejpam-3495	319	38	∈	∈	PROPN
ejpam-3495	319	39	i.	i.	PROPN
ejpam-3495	319	40	�	�	PROPN
ejpam-3495	319	41	theorem	theorem	VERB
ejpam-3495	319	42	5.4	5.4	NUM
ejpam-3495	319	43	.	.	PUNCT
ejpam-3495	320	1	let	let	VERB
ejpam-3495	320	2	(	(	PUNCT
ejpam-3495	320	3	x	x	X
ejpam-3495	320	4	,	,	PUNCT
ejpam-3495	320	5	∗	∗	NOUN
ejpam-3495	320	6	,	,	PUNCT
ejpam-3495	320	7	�	�	PROPN
ejpam-3495	320	8	,	,	PUNCT
ejpam-3495	320	9	0	0	NUM
ejpam-3495	320	10	)	)	PUNCT
ejpam-3495	320	11	be	be	AUX
ejpam-3495	320	12	a	a	DET
ejpam-3495	320	13	companion	companion	NOUN
ejpam-3495	320	14	b	b	NOUN
ejpam-3495	320	15	-	-	PUNCT
ejpam-3495	320	16	algebra	algebra	NOUN
ejpam-3495	320	17	.	.	PUNCT
ejpam-3495	321	1	if	if	SCONJ
ejpam-3495	321	2	i	i	PRON
ejpam-3495	321	3	is	be	AUX
ejpam-3495	321	4	a	a	DET
ejpam-3495	321	5	�	�	NOUN
ejpam-3495	321	6	-ideal	-ideal	NOUN
ejpam-3495	321	7	of	of	ADP
ejpam-3495	321	8	x	x	NOUN
ejpam-3495	321	9	,	,	PUNCT
ejpam-3495	321	10	then	then	ADV
ejpam-3495	321	11	i	i	PRON
ejpam-3495	321	12	is	be	AUX
ejpam-3495	321	13	a	a	DET
ejpam-3495	321	14	�	�	NOUN
ejpam-3495	321	15	-subalgebra	-subalgebra	NOUN
ejpam-3495	321	16	.	.	PUNCT
ejpam-3495	322	1	l.d	l.d	PROPN
ejpam-3495	322	2	.	.	PROPN
ejpam-3495	322	3	naingue	naingue	PROPN
ejpam-3495	322	4	,	,	PUNCT
ejpam-3495	322	5	j.p	j.p	PROPN
ejpam-3495	322	6	.	.	PROPN
ejpam-3495	322	7	vilela	vilela	PROPN
ejpam-3495	322	8	/	/	SYM
ejpam-3495	322	9	eur	eur	PROPN
ejpam-3495	322	10	.	.	PUNCT
ejpam-3495	323	1	j.	j.	PROPN
ejpam-3495	323	2	pure	pure	PROPN
ejpam-3495	323	3	appl	appl	PROPN
ejpam-3495	323	4	.	.	PROPN
ejpam-3495	323	5	math	math	PROPN
ejpam-3495	323	6	,	,	PUNCT
ejpam-3495	323	7	12	12	NUM
ejpam-3495	323	8	(	(	PUNCT
ejpam-3495	323	9	3	3	NUM
ejpam-3495	323	10	)	)	PUNCT
ejpam-3495	323	11	(	(	PUNCT
ejpam-3495	323	12	2019	2019	NUM
ejpam-3495	323	13	)	)	PUNCT
ejpam-3495	323	14	,	,	PUNCT
ejpam-3495	323	15	1248	1248	NUM
ejpam-3495	323	16	-	-	SYM
ejpam-3495	323	17	1259	1259	NUM
ejpam-3495	323	18	1257	1257	NUM
ejpam-3495	323	19	proof	proof	NOUN
ejpam-3495	323	20	:	:	PUNCT
ejpam-3495	323	21	let	let	VERB
ejpam-3495	323	22	x	x	PRON
ejpam-3495	323	23	,	,	PUNCT
ejpam-3495	323	24	y	y	PROPN
ejpam-3495	323	25	∈	∈	PROPN
ejpam-3495	323	26	i.	i.	NOUN
ejpam-3495	323	27	note	note	VERB
ejpam-3495	323	28	that	that	SCONJ
ejpam-3495	323	29	by	by	ADP
ejpam-3495	323	30	lemma	lemma	PROPN
ejpam-3495	323	31	5.3	5.3	NUM
ejpam-3495	323	32	,	,	PUNCT
ejpam-3495	323	33	0	0	NUM
ejpam-3495	323	34	∗	∗	NOUN
ejpam-3495	323	35	y	y	PROPN
ejpam-3495	323	36	∈	∈	PROPN
ejpam-3495	323	37	i.	i.	NOUN
ejpam-3495	323	38	observe	observe	VERB
ejpam-3495	323	39	that	that	SCONJ
ejpam-3495	323	40	by	by	ADP
ejpam-3495	323	41	lemma	lemma	PROPN
ejpam-3495	323	42	3.10(b	3.10(b	PROPN
ejpam-3495	323	43	)	)	PUNCT
ejpam-3495	323	44	,	,	PUNCT
ejpam-3495	323	45	theorems	theorem	VERB
ejpam-3495	323	46	2.4	2.4	NUM
ejpam-3495	323	47	,	,	PUNCT
ejpam-3495	323	48	2.3(c	2.3(c	NUM
ejpam-3495	323	49	)	)	PUNCT
ejpam-3495	323	50	,	,	PUNCT
ejpam-3495	323	51	definition	definition	NOUN
ejpam-3495	323	52	2.1(i	2.1(i	NUM
ejpam-3495	323	53	)	)	PUNCT
ejpam-3495	323	54	and	and	CCONJ
ejpam-3495	323	55	theorem	theorem	VERB
ejpam-3495	323	56	2.3(f	2.3(f	NUM
ejpam-3495	323	57	)	)	PUNCT
ejpam-3495	323	58	,	,	PUNCT
ejpam-3495	323	59	(	(	PUNCT
ejpam-3495	323	60	x	x	X
ejpam-3495	323	61	�	�	PROPN
ejpam-3495	323	62	y	y	PROPN
ejpam-3495	323	63	)	)	PUNCT
ejpam-3495	323	64	�	�	PROPN
ejpam-3495	323	65	(	(	PUNCT
ejpam-3495	323	66	0	0	NUM
ejpam-3495	323	67	∗	∗	NUM
ejpam-3495	323	68	y	y	NOUN
ejpam-3495	323	69	)	)	PUNCT
ejpam-3495	323	70	=	=	PRON
ejpam-3495	324	1	(	(	PUNCT
ejpam-3495	324	2	y	y	PROPN
ejpam-3495	324	3	∗	∗	X
ejpam-3495	324	4	(	(	PUNCT
ejpam-3495	324	5	0	0	NUM
ejpam-3495	324	6	∗	∗	NOUN
ejpam-3495	324	7	x	x	NOUN
ejpam-3495	324	8	)	)	PUNCT
ejpam-3495	324	9	)	)	PUNCT
ejpam-3495	324	10	�	�	PROPN
ejpam-3495	324	11	(	(	PUNCT
ejpam-3495	324	12	0	0	NUM
ejpam-3495	324	13	∗	∗	NUM
ejpam-3495	324	14	y	y	NOUN
ejpam-3495	324	15	)	)	PUNCT
ejpam-3495	324	16	=	=	SYM
ejpam-3495	324	17	(	(	PUNCT
ejpam-3495	324	18	0	0	NUM
ejpam-3495	324	19	∗	∗	PROPN
ejpam-3495	324	20	y	y	NOUN
ejpam-3495	324	21	)	)	PUNCT
ejpam-3495	324	22	∗	∗	NOUN
ejpam-3495	324	23	(	(	PUNCT
ejpam-3495	324	24	0	0	NUM
ejpam-3495	324	25	∗	∗	NOUN
ejpam-3495	324	26	(	(	PUNCT
ejpam-3495	324	27	y	y	PROPN
ejpam-3495	324	28	∗	∗	NOUN
ejpam-3495	324	29	(	(	PUNCT
ejpam-3495	324	30	0	0	NUM
ejpam-3495	324	31	∗	∗	NOUN
ejpam-3495	324	32	x	x	NOUN
ejpam-3495	324	33	)	)	PUNCT
ejpam-3495	324	34	)	)	PUNCT
ejpam-3495	324	35	)	)	PUNCT
ejpam-3495	325	1	=	=	PUNCT
ejpam-3495	325	2	(	(	PUNCT
ejpam-3495	325	3	0	0	NUM
ejpam-3495	325	4	∗	∗	PROPN
ejpam-3495	325	5	y	y	NOUN
ejpam-3495	325	6	)	)	PUNCT
ejpam-3495	325	7	∗	∗	NOUN
ejpam-3495	325	8	(	(	PUNCT
ejpam-3495	325	9	(	(	PUNCT
ejpam-3495	325	10	0	0	NUM
ejpam-3495	325	11	∗	∗	NOUN
ejpam-3495	325	12	x	x	NOUN
ejpam-3495	325	13	)	)	PUNCT
ejpam-3495	325	14	∗	∗	PROPN
ejpam-3495	325	15	y	y	NOUN
ejpam-3495	325	16	)	)	PUNCT
ejpam-3495	325	17	=	=	SYM
ejpam-3495	325	18	(	(	PUNCT
ejpam-3495	325	19	(	(	PUNCT
ejpam-3495	325	20	0	0	NUM
ejpam-3495	325	21	∗	∗	PROPN
ejpam-3495	325	22	y	y	NOUN
ejpam-3495	325	23	)	)	PUNCT
ejpam-3495	325	24	∗	∗	NOUN
ejpam-3495	325	25	(	(	PUNCT
ejpam-3495	325	26	0	0	NUM
ejpam-3495	325	27	∗	∗	PROPN
ejpam-3495	325	28	y	y	PROPN
ejpam-3495	325	29	)	)	PUNCT
ejpam-3495	325	30	)	)	PUNCT
ejpam-3495	325	31	∗	∗	NOUN
ejpam-3495	325	32	(	(	PUNCT
ejpam-3495	325	33	0	0	NUM
ejpam-3495	325	34	∗	∗	NOUN
ejpam-3495	325	35	x	x	NOUN
ejpam-3495	325	36	)	)	PUNCT
ejpam-3495	325	37	=	=	SYM
ejpam-3495	325	38	0	0	NUM
ejpam-3495	325	39	∗	∗	NOUN
ejpam-3495	325	40	(	(	PUNCT
ejpam-3495	325	41	0	0	NUM
ejpam-3495	325	42	∗	∗	NOUN
ejpam-3495	325	43	x	x	NOUN
ejpam-3495	325	44	)	)	PUNCT
ejpam-3495	325	45	=	=	SYM
ejpam-3495	326	1	x.	x.	NOUN
ejpam-3495	326	2	since	since	SCONJ
ejpam-3495	326	3	x	x	PROPN
ejpam-3495	326	4	∈	∈	PROPN
ejpam-3495	326	5	i	i	PRON
ejpam-3495	326	6	,	,	PUNCT
ejpam-3495	326	7	0	0	NUM
ejpam-3495	326	8	∗	∗	NOUN
ejpam-3495	326	9	y	y	NOUN
ejpam-3495	326	10	∈	∈	PROPN
ejpam-3495	327	1	i	i	PRON
ejpam-3495	327	2	and	and	CCONJ
ejpam-3495	327	3	i	i	PRON
ejpam-3495	327	4	is	be	AUX
ejpam-3495	327	5	a	a	DET
ejpam-3495	327	6	�	�	NOUN
ejpam-3495	327	7	-ideal	-ideal	NOUN
ejpam-3495	327	8	,	,	PUNCT
ejpam-3495	327	9	x	x	X
ejpam-3495	327	10	�	�	PROPN
ejpam-3495	327	11	y	y	PROPN
ejpam-3495	327	12	∈	∈	PROPN
ejpam-3495	327	13	i.	i.	NOUN
ejpam-3495	327	14	therefore	therefore	ADV
ejpam-3495	327	15	,	,	PUNCT
ejpam-3495	327	16	i	i	PRON
ejpam-3495	327	17	is	be	AUX
ejpam-3495	327	18	a	a	DET
ejpam-3495	327	19	�	�	NOUN
ejpam-3495	327	20	-subalgebra	-subalgebra	NOUN
ejpam-3495	327	21	.	.	PUNCT
ejpam-3495	328	1	�	�	PROPN
ejpam-3495	328	2	the	the	DET
ejpam-3495	328	3	converse	converse	NOUN
ejpam-3495	328	4	of	of	ADP
ejpam-3495	328	5	theorem	theorem	NOUN
ejpam-3495	328	6	5.4	5.4	NUM
ejpam-3495	328	7	need	need	AUX
ejpam-3495	328	8	not	not	PART
ejpam-3495	328	9	be	be	AUX
ejpam-3495	328	10	true	true	ADJ
ejpam-3495	328	11	in	in	ADP
ejpam-3495	328	12	general	general	ADJ
ejpam-3495	328	13	.	.	PUNCT
ejpam-3495	329	1	note	note	VERB
ejpam-3495	329	2	that	that	SCONJ
ejpam-3495	330	1	i	i	PRON
ejpam-3495	330	2	=	=	PUNCT
ejpam-3495	330	3	z+	z+	NUM
ejpam-3495	330	4	is	be	AUX
ejpam-3495	330	5	a	a	DET
ejpam-3495	330	6	�	�	NOUN
ejpam-3495	330	7	-subalgebra	-subalgebra	NOUN
ejpam-3495	330	8	of	of	ADP
ejpam-3495	330	9	(	(	PUNCT
ejpam-3495	330	10	z,−,+	z,−,+	PROPN
ejpam-3495	330	11	,	,	PUNCT
ejpam-3495	330	12	0	0	NUM
ejpam-3495	330	13	)	)	PUNCT
ejpam-3495	330	14	since	since	SCONJ
ejpam-3495	330	15	for	for	ADP
ejpam-3495	330	16	all	all	DET
ejpam-3495	330	17	x	x	NOUN
ejpam-3495	330	18	,	,	PUNCT
ejpam-3495	330	19	y	y	PROPN
ejpam-3495	330	20	∈	∈	PROPN
ejpam-3495	331	1	i	i	PRON
ejpam-3495	331	2	,	,	PUNCT
ejpam-3495	331	3	x	x	PROPN
ejpam-3495	332	1	+	+	CCONJ
ejpam-3495	332	2	y	y	PROPN
ejpam-3495	332	3	∈	∈	PROPN
ejpam-3495	332	4	i.	i.	NOUN
ejpam-3495	332	5	however	however	ADV
ejpam-3495	332	6	,	,	PUNCT
ejpam-3495	332	7	0	0	NUM
ejpam-3495	332	8	/∈	/∈	NOUN
ejpam-3495	332	9	i.	i.	NOUN
ejpam-3495	332	10	hence	hence	ADV
ejpam-3495	332	11	,	,	PUNCT
ejpam-3495	332	12	i	i	PRON
ejpam-3495	332	13	is	be	AUX
ejpam-3495	332	14	not	not	PART
ejpam-3495	332	15	a	a	DET
ejpam-3495	332	16	�	�	NOUN
ejpam-3495	332	17	-ideal	-ideal	NOUN
ejpam-3495	332	18	.	.	PUNCT
ejpam-3495	333	1	thus	thus	ADV
ejpam-3495	333	2	,	,	PUNCT
ejpam-3495	333	3	we	we	PRON
ejpam-3495	333	4	have	have	VERB
ejpam-3495	333	5	the	the	DET
ejpam-3495	333	6	following	follow	VERB
ejpam-3495	333	7	remark	remark	NOUN
ejpam-3495	333	8	.	.	PUNCT
ejpam-3495	334	1	remark	remark	VERB
ejpam-3495	334	2	5.5	5.5	NUM
ejpam-3495	334	3	.	.	PUNCT
ejpam-3495	335	1	if	if	SCONJ
ejpam-3495	335	2	i	i	PRON
ejpam-3495	335	3	is	be	AUX
ejpam-3495	335	4	a	a	DET
ejpam-3495	335	5	�	�	PROPN
ejpam-3495	335	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	335	7	,	,	PUNCT
ejpam-3495	335	8	then	then	ADV
ejpam-3495	335	9	i	i	PRON
ejpam-3495	335	10	is	be	AUX
ejpam-3495	335	11	not	not	PART
ejpam-3495	335	12	necessarily	necessarily	ADV
ejpam-3495	335	13	a	a	DET
ejpam-3495	335	14	�	�	NOUN
ejpam-3495	335	15	-ideal	-ideal	NOUN
ejpam-3495	335	16	.	.	PUNCT
ejpam-3495	335	17	example	example	NOUN
ejpam-3495	336	1	5.6	5.6	NUM
ejpam-3495	336	2	.	.	PUNCT
ejpam-3495	336	3	consider	consider	VERB
ejpam-3495	336	4	example	example	NOUN
ejpam-3495	336	5	3.6	3.6	NUM
ejpam-3495	336	6	and	and	CCONJ
ejpam-3495	336	7	�	�	NOUN
ejpam-3495	336	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	336	9	i	i	NOUN
ejpam-3495	336	10	=	=	SYM
ejpam-3495	336	11	{	{	PUNCT
ejpam-3495	336	12	0	0	NUM
ejpam-3495	336	13	,	,	PUNCT
ejpam-3495	336	14	2	2	NUM
ejpam-3495	336	15	}	}	PUNCT
ejpam-3495	336	16	.	.	PUNCT
ejpam-3495	337	1	observe	observe	VERB
ejpam-3495	337	2	that	that	SCONJ
ejpam-3495	337	3	0	0	NUM
ejpam-3495	337	4	∗	∗	NOUN
ejpam-3495	337	5	0	0	NUM
ejpam-3495	338	1	=	=	SYM
ejpam-3495	338	2	0	0	PUNCT
ejpam-3495	338	3	∈	∈	PROPN
ejpam-3495	338	4	i	i	PRON
ejpam-3495	338	5	and	and	CCONJ
ejpam-3495	338	6	0	0	NUM
ejpam-3495	338	7	∗	∗	NOUN
ejpam-3495	338	8	2	2	NUM
ejpam-3495	338	9	=	=	SYM
ejpam-3495	338	10	2	2	NUM
ejpam-3495	338	11	∈	∈	NOUN
ejpam-3495	338	12	i	i	PRON
ejpam-3495	338	13	,	,	PUNCT
ejpam-3495	338	14	so	so	ADV
ejpam-3495	338	15	,	,	PUNCT
ejpam-3495	338	16	0	0	NUM
ejpam-3495	338	17	∗	∗	NOUN
ejpam-3495	338	18	a	a	DET
ejpam-3495	338	19	∈	∈	NOUN
ejpam-3495	339	1	i	i	PRON
ejpam-3495	339	2	,	,	PUNCT
ejpam-3495	339	3	for	for	ADP
ejpam-3495	339	4	any	any	DET
ejpam-3495	339	5	a	a	DET
ejpam-3495	339	6	∈	∈	PROPN
ejpam-3495	339	7	i.	i.	NOUN
ejpam-3495	339	8	it	it	PRON
ejpam-3495	339	9	is	be	AUX
ejpam-3495	339	10	clear	clear	ADJ
ejpam-3495	339	11	that	that	SCONJ
ejpam-3495	339	12	i	i	PRON
ejpam-3495	339	13	is	be	AUX
ejpam-3495	339	14	also	also	ADV
ejpam-3495	339	15	a	a	DET
ejpam-3495	339	16	�	�	NOUN
ejpam-3495	339	17	-ideal	-ideal	NOUN
ejpam-3495	339	18	.	.	PUNCT
ejpam-3495	340	1	theorem	theorem	VERB
ejpam-3495	340	2	5.7	5.7	NUM
ejpam-3495	340	3	.	.	PUNCT
ejpam-3495	341	1	let	let	VERB
ejpam-3495	341	2	(	(	PUNCT
ejpam-3495	341	3	x	x	X
ejpam-3495	341	4	,	,	PUNCT
ejpam-3495	341	5	∗	∗	NOUN
ejpam-3495	341	6	,	,	PUNCT
ejpam-3495	341	7	�	�	PROPN
ejpam-3495	341	8	,	,	PUNCT
ejpam-3495	341	9	0	0	NUM
ejpam-3495	341	10	)	)	PUNCT
ejpam-3495	341	11	be	be	AUX
ejpam-3495	341	12	a	a	DET
ejpam-3495	341	13	companion	companion	NOUN
ejpam-3495	341	14	b	b	NOUN
ejpam-3495	341	15	-	-	PUNCT
ejpam-3495	341	16	algebra	algebra	NOUN
ejpam-3495	341	17	.	.	PUNCT
ejpam-3495	342	1	suppose	suppose	VERB
ejpam-3495	342	2	i	i	PRON
ejpam-3495	342	3	is	be	AUX
ejpam-3495	342	4	a	a	DET
ejpam-3495	342	5	�	�	NOUN
ejpam-3495	342	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	342	7	of	of	ADP
ejpam-3495	342	8	x	x	X
ejpam-3495	342	9	and	and	CCONJ
ejpam-3495	342	10	0	0	NUM
ejpam-3495	342	11	∗	∗	NOUN
ejpam-3495	342	12	a	a	DET
ejpam-3495	342	13	∈	∈	NOUN
ejpam-3495	343	1	i	i	PRON
ejpam-3495	343	2	for	for	ADP
ejpam-3495	343	3	any	any	DET
ejpam-3495	343	4	a	a	DET
ejpam-3495	343	5	∈	∈	PROPN
ejpam-3495	343	6	i.	i.	NOUN
ejpam-3495	343	7	then	then	ADV
ejpam-3495	343	8	i	i	PRON
ejpam-3495	343	9	is	be	AUX
ejpam-3495	343	10	a	a	DET
ejpam-3495	343	11	�	�	NOUN
ejpam-3495	343	12	-ideal	-ideal	NOUN
ejpam-3495	343	13	.	.	PUNCT
ejpam-3495	344	1	proof	proof	NOUN
ejpam-3495	344	2	:	:	PUNCT
ejpam-3495	344	3	suppose	suppose	VERB
ejpam-3495	344	4	i	i	PRON
ejpam-3495	344	5	is	be	AUX
ejpam-3495	344	6	a	a	DET
ejpam-3495	344	7	�	�	NOUN
ejpam-3495	344	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	344	9	and	and	CCONJ
ejpam-3495	344	10	0∗a	0∗a	NUM
ejpam-3495	344	11	∈	∈	PROPN
ejpam-3495	344	12	i	i	PRON
ejpam-3495	344	13	for	for	ADP
ejpam-3495	344	14	any	any	DET
ejpam-3495	344	15	a	a	DET
ejpam-3495	344	16	∈	∈	PROPN
ejpam-3495	344	17	i.	i.	NOUN
ejpam-3495	344	18	let	let	VERB
ejpam-3495	344	19	x	x	X
ejpam-3495	344	20	∈	∈	PROPN
ejpam-3495	344	21	i.	i.	NOUN
ejpam-3495	344	22	then	then	ADV
ejpam-3495	344	23	0∗x	0∗x	PROPN
ejpam-3495	344	24	∈	∈	PROPN
ejpam-3495	344	25	i.	i.	NOUN
ejpam-3495	344	26	since	since	SCONJ
ejpam-3495	344	27	i	i	PRON
ejpam-3495	344	28	is	be	AUX
ejpam-3495	344	29	�	�	PROPN
ejpam-3495	344	30	-subalgebra	-subalgebra	NOUN
ejpam-3495	344	31	,	,	PUNCT
ejpam-3495	344	32	0	0	NUM
ejpam-3495	344	33	=	=	SYM
ejpam-3495	344	34	x	x	SYM
ejpam-3495	344	35	�	�	PROPN
ejpam-3495	344	36	(	(	PUNCT
ejpam-3495	344	37	0	0	NUM
ejpam-3495	344	38	∗	∗	NOUN
ejpam-3495	344	39	x	x	NOUN
ejpam-3495	344	40	)	)	PUNCT
ejpam-3495	344	41	∈	∈	PROPN
ejpam-3495	344	42	i.	i.	NOUN
ejpam-3495	344	43	now	now	ADV
ejpam-3495	344	44	,	,	PUNCT
ejpam-3495	344	45	suppose	suppose	VERB
ejpam-3495	344	46	u	u	PRON
ejpam-3495	344	47	�	�	PROPN
ejpam-3495	344	48	v	v	ADP
ejpam-3495	344	49	∈	∈	PROPN
ejpam-3495	344	50	i	i	PRON
ejpam-3495	344	51	and	and	CCONJ
ejpam-3495	344	52	v	v	ADP
ejpam-3495	344	53	∈	∈	PROPN
ejpam-3495	344	54	i.	i.	NOUN
ejpam-3495	344	55	then	then	ADV
ejpam-3495	344	56	0	0	NUM
ejpam-3495	344	57	∗	∗	NOUN
ejpam-3495	344	58	v	v	NOUN
ejpam-3495	344	59	∈	∈	PROPN
ejpam-3495	344	60	i.	i.	NOUN
ejpam-3495	344	61	by	by	ADP
ejpam-3495	344	62	lemma	lemma	PROPN
ejpam-3495	344	63	3.10(e	3.10(e	PROPN
ejpam-3495	344	64	)	)	PUNCT
ejpam-3495	344	65	,	,	PUNCT
ejpam-3495	344	66	u	u	NOUN
ejpam-3495	344	67	=	=	PUNCT
ejpam-3495	344	68	(	(	PUNCT
ejpam-3495	344	69	u	u	PROPN
ejpam-3495	344	70	�	�	PROPN
ejpam-3495	344	71	v	v	NOUN
ejpam-3495	344	72	)	)	PUNCT
ejpam-3495	344	73	�	�	PROPN
ejpam-3495	344	74	(	(	PUNCT
ejpam-3495	344	75	0	0	NUM
ejpam-3495	344	76	∗	∗	NUM
ejpam-3495	344	77	v	v	NOUN
ejpam-3495	344	78	)	)	PUNCT
ejpam-3495	344	79	.	.	PUNCT
ejpam-3495	345	1	since	since	SCONJ
ejpam-3495	345	2	i	i	PRON
ejpam-3495	345	3	is	be	AUX
ejpam-3495	345	4	a	a	DET
ejpam-3495	345	5	�	�	PROPN
ejpam-3495	345	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	345	7	,	,	PUNCT
ejpam-3495	345	8	u	u	PROPN
ejpam-3495	345	9	∈	∈	PROPN
ejpam-3495	345	10	i.	i.	NOUN
ejpam-3495	345	11	therefore	therefore	ADV
ejpam-3495	345	12	,	,	PUNCT
ejpam-3495	345	13	i	i	PRON
ejpam-3495	345	14	is	be	AUX
ejpam-3495	345	15	a	a	DET
ejpam-3495	345	16	�	�	NOUN
ejpam-3495	345	17	-ideal	-ideal	NOUN
ejpam-3495	345	18	.	.	PUNCT
ejpam-3495	346	1	�	�	PROPN
ejpam-3495	346	2	theorem	theorem	VERB
ejpam-3495	346	3	5.8	5.8	NUM
ejpam-3495	346	4	.	.	PUNCT
ejpam-3495	347	1	let	let	AUX
ejpam-3495	347	2	(	(	PUNCT
ejpam-3495	347	3	g	g	NOUN
ejpam-3495	347	4	,	,	PUNCT
ejpam-3495	347	5	∗	∗	NOUN
ejpam-3495	347	6	,	,	PUNCT
ejpam-3495	347	7	�	�	PROPN
ejpam-3495	347	8	,	,	PUNCT
ejpam-3495	347	9	0	0	NUM
ejpam-3495	347	10	)	)	PUNCT
ejpam-3495	347	11	be	be	AUX
ejpam-3495	347	12	a	a	DET
ejpam-3495	347	13	companion	companion	NOUN
ejpam-3495	347	14	b	b	NOUN
ejpam-3495	347	15	-	-	PUNCT
ejpam-3495	347	16	algebra	algebra	NOUN
ejpam-3495	347	17	.	.	PUNCT
ejpam-3495	348	1	a	a	DET
ejpam-3495	348	2	nonempty	nonempty	NOUN
ejpam-3495	348	3	subset	subset	VERB
ejpam-3495	348	4	i	i	PRON
ejpam-3495	348	5	of	of	ADP
ejpam-3495	348	6	g	g	PROPN
ejpam-3495	348	7	is	be	AUX
ejpam-3495	348	8	a	a	DET
ejpam-3495	348	9	�	�	NOUN
ejpam-3495	348	10	-ideal	-ideal	NOUN
ejpam-3495	348	11	of	of	ADP
ejpam-3495	348	12	g	g	NOUN
ejpam-3495	348	13	if	if	SCONJ
ejpam-3495	349	1	and	and	CCONJ
ejpam-3495	349	2	only	only	ADV
ejpam-3495	349	3	if	if	SCONJ
ejpam-3495	349	4	i	i	PRON
ejpam-3495	349	5	is	be	AUX
ejpam-3495	349	6	a	a	DET
ejpam-3495	349	7	subgroup	subgroup	NOUN
ejpam-3495	349	8	of	of	ADP
ejpam-3495	349	9	the	the	DET
ejpam-3495	349	10	group	group	NOUN
ejpam-3495	349	11	(	(	PUNCT
ejpam-3495	349	12	g	g	PROPN
ejpam-3495	349	13	,	,	PUNCT
ejpam-3495	349	14	�	�	PROPN
ejpam-3495	349	15	,	,	PUNCT
ejpam-3495	349	16	0	0	NUM
ejpam-3495	349	17	)	)	PUNCT
ejpam-3495	349	18	.	.	PUNCT
ejpam-3495	350	1	proof	proof	NOUN
ejpam-3495	350	2	:	:	PUNCT
ejpam-3495	350	3	let	let	VERB
ejpam-3495	350	4	i	i	PRON
ejpam-3495	350	5	be	be	AUX
ejpam-3495	350	6	a	a	DET
ejpam-3495	350	7	�	�	NOUN
ejpam-3495	350	8	-ideal	-ideal	NOUN
ejpam-3495	350	9	and	and	CCONJ
ejpam-3495	350	10	a	a	DET
ejpam-3495	350	11	,	,	PUNCT
ejpam-3495	350	12	b	b	PROPN
ejpam-3495	350	13	∈	∈	PROPN
ejpam-3495	350	14	i.	i.	NOUN
ejpam-3495	350	15	by	by	ADP
ejpam-3495	350	16	lemma	lemma	PROPN
ejpam-3495	350	17	5.3	5.3	NUM
ejpam-3495	350	18	,	,	PUNCT
ejpam-3495	350	19	b−1	b−1	PROPN
ejpam-3495	350	20	=	=	SYM
ejpam-3495	350	21	0	0	NUM
ejpam-3495	350	22	∗	∗	NOUN
ejpam-3495	350	23	b	b	PROPN
ejpam-3495	350	24	∈	∈	PROPN
ejpam-3495	350	25	i.	i.	NOUN
ejpam-3495	350	26	because	because	SCONJ
ejpam-3495	350	27	i	i	PRON
ejpam-3495	350	28	is	be	AUX
ejpam-3495	350	29	also	also	ADV
ejpam-3495	350	30	a	a	DET
ejpam-3495	350	31	�	�	NOUN
ejpam-3495	350	32	-subalgebra	-subalgebra	NOUN
ejpam-3495	350	33	by	by	ADP
ejpam-3495	350	34	theorem	theorem	NOUN
ejpam-3495	350	35	5.4	5.4	NUM
ejpam-3495	350	36	,	,	PUNCT
ejpam-3495	350	37	a	a	DET
ejpam-3495	350	38	�	�	PROPN
ejpam-3495	350	39	b−1	b−1	PROPN
ejpam-3495	350	40	∈	∈	PROPN
ejpam-3495	350	41	i.	i.	NOUN
ejpam-3495	350	42	hence	hence	ADV
ejpam-3495	350	43	,	,	PUNCT
ejpam-3495	350	44	i	i	PRON
ejpam-3495	350	45	is	be	AUX
ejpam-3495	350	46	a	a	DET
ejpam-3495	350	47	subgroup	subgroup	NOUN
ejpam-3495	350	48	.	.	PUNCT
ejpam-3495	351	1	conversely	conversely	ADV
ejpam-3495	351	2	,	,	PUNCT
ejpam-3495	351	3	suppose	suppose	VERB
ejpam-3495	351	4	i	i	PRON
ejpam-3495	351	5	is	be	AUX
ejpam-3495	351	6	a	a	DET
ejpam-3495	351	7	subgroup	subgroup	NOUN
ejpam-3495	351	8	of	of	ADP
ejpam-3495	351	9	the	the	DET
ejpam-3495	351	10	group	group	NOUN
ejpam-3495	351	11	(	(	PUNCT
ejpam-3495	351	12	g	g	PROPN
ejpam-3495	351	13	,	,	PUNCT
ejpam-3495	351	14	�	�	PROPN
ejpam-3495	351	15	,	,	PUNCT
ejpam-3495	351	16	0	0	NUM
ejpam-3495	351	17	)	)	PUNCT
ejpam-3495	351	18	and	and	CCONJ
ejpam-3495	351	19	a	a	DET
ejpam-3495	351	20	,	,	PUNCT
ejpam-3495	351	21	b	b	PROPN
ejpam-3495	351	22	∈	∈	PROPN
ejpam-3495	351	23	i.	i.	NOUN
ejpam-3495	351	24	then	then	ADV
ejpam-3495	351	25	a	a	DET
ejpam-3495	351	26	�	�	PROPN
ejpam-3495	351	27	b−1	b−1	PROPN
ejpam-3495	351	28	∈	∈	PROPN
ejpam-3495	351	29	i.	i.	NOUN
ejpam-3495	351	30	note	note	VERB
ejpam-3495	351	31	that	that	SCONJ
ejpam-3495	351	32	a	a	DET
ejpam-3495	351	33	�	�	PROPN
ejpam-3495	351	34	a−1	a−1	PROPN
ejpam-3495	351	35	=	=	PUNCT
ejpam-3495	351	36	0	0	PROPN
ejpam-3495	351	37	.	.	PUNCT
ejpam-3495	352	1	so	so	ADV
ejpam-3495	352	2	,	,	PUNCT
ejpam-3495	352	3	0	0	NUM
ejpam-3495	352	4	∈	∈	PROPN
ejpam-3495	352	5	i.	i.	NOUN
ejpam-3495	352	6	suppose	suppose	VERB
ejpam-3495	352	7	x	x	X
ejpam-3495	352	8	�	�	PROPN
ejpam-3495	352	9	y	y	PROPN
ejpam-3495	352	10	∈	∈	PROPN
ejpam-3495	353	1	i	i	PRON
ejpam-3495	353	2	and	and	CCONJ
ejpam-3495	353	3	y	y	PROPN
ejpam-3495	353	4	∈	∈	PROPN
ejpam-3495	353	5	i.	i.	NOUN
ejpam-3495	353	6	then	then	ADV
ejpam-3495	353	7	by	by	ADP
ejpam-3495	353	8	lemma	lemma	PROPN
ejpam-3495	353	9	3.10(e	3.10(e	PROPN
ejpam-3495	353	10	)	)	PUNCT
ejpam-3495	353	11	,	,	PUNCT
ejpam-3495	353	12	x	x	X
ejpam-3495	353	13	=	=	PRON
ejpam-3495	353	14	(	(	PUNCT
ejpam-3495	353	15	x	x	X
ejpam-3495	353	16	�	�	PROPN
ejpam-3495	353	17	y	y	PROPN
ejpam-3495	353	18	)	)	PUNCT
ejpam-3495	353	19	�	�	PROPN
ejpam-3495	353	20	(	(	PUNCT
ejpam-3495	353	21	0∗	0∗	NOUN
ejpam-3495	353	22	y	y	X
ejpam-3495	353	23	)	)	PUNCT
ejpam-3495	353	24	=	=	SYM
ejpam-3495	353	25	(	(	PUNCT
ejpam-3495	353	26	x	x	X
ejpam-3495	353	27	�	�	PROPN
ejpam-3495	353	28	y	y	PROPN
ejpam-3495	353	29	)	)	PUNCT
ejpam-3495	353	30	�	�	PROPN
ejpam-3495	353	31	y−1	y−1	PROPN
ejpam-3495	353	32	∈	∈	PROPN
ejpam-3495	353	33	i.	i.	NOUN
ejpam-3495	353	34	thus	thus	ADV
ejpam-3495	353	35	,	,	PUNCT
ejpam-3495	353	36	i	i	PRON
ejpam-3495	353	37	is	be	AUX
ejpam-3495	353	38	a	a	DET
ejpam-3495	353	39	�	�	NOUN
ejpam-3495	353	40	-ideal	-ideal	NOUN
ejpam-3495	353	41	.	.	PUNCT
ejpam-3495	354	1	�	�	PROPN
ejpam-3495	354	2	the	the	DET
ejpam-3495	354	3	following	follow	VERB
ejpam-3495	354	4	corollary	corollary	NOUN
ejpam-3495	354	5	follows	follow	VERB
ejpam-3495	354	6	from	from	ADP
ejpam-3495	354	7	theorem	theorem	ADJ
ejpam-3495	354	8	5.8	5.8	NUM
ejpam-3495	354	9	and	and	CCONJ
ejpam-3495	354	10	5.4	5.4	NUM
ejpam-3495	354	11	.	.	PUNCT
ejpam-3495	355	1	corollary	corollary	ADJ
ejpam-3495	355	2	5.9	5.9	NUM
ejpam-3495	355	3	.	.	PUNCT
ejpam-3495	356	1	let	let	AUX
ejpam-3495	356	2	(	(	PUNCT
ejpam-3495	356	3	g	g	NOUN
ejpam-3495	356	4	,	,	PUNCT
ejpam-3495	356	5	∗	∗	NOUN
ejpam-3495	356	6	,	,	PUNCT
ejpam-3495	356	7	�	�	PROPN
ejpam-3495	356	8	,	,	PUNCT
ejpam-3495	356	9	0	0	NUM
ejpam-3495	356	10	)	)	PUNCT
ejpam-3495	356	11	be	be	AUX
ejpam-3495	356	12	a	a	DET
ejpam-3495	356	13	companion	companion	NOUN
ejpam-3495	356	14	b	b	NOUN
ejpam-3495	356	15	-	-	PUNCT
ejpam-3495	356	16	algebra	algebra	NOUN
ejpam-3495	356	17	.	.	PUNCT
ejpam-3495	357	1	if	if	SCONJ
ejpam-3495	357	2	i	i	PRON
ejpam-3495	357	3	is	be	AUX
ejpam-3495	357	4	a	a	DET
ejpam-3495	357	5	subgroup	subgroup	NOUN
ejpam-3495	357	6	of	of	ADP
ejpam-3495	357	7	the	the	DET
ejpam-3495	357	8	group	group	NOUN
ejpam-3495	357	9	(	(	PUNCT
ejpam-3495	357	10	g	g	PROPN
ejpam-3495	357	11	,	,	PUNCT
ejpam-3495	357	12	�	�	PROPN
ejpam-3495	357	13	,	,	PUNCT
ejpam-3495	357	14	0	0	NUM
ejpam-3495	357	15	)	)	PUNCT
ejpam-3495	357	16	,	,	PUNCT
ejpam-3495	357	17	then	then	ADV
ejpam-3495	357	18	i	i	PRON
ejpam-3495	357	19	is	be	AUX
ejpam-3495	357	20	a	a	DET
ejpam-3495	357	21	�	�	NOUN
ejpam-3495	357	22	-subalgebra	-subalgebra	NOUN
ejpam-3495	357	23	.	.	PUNCT
ejpam-3495	358	1	the	the	DET
ejpam-3495	358	2	following	follow	VERB
ejpam-3495	358	3	corollary	corollary	NOUN
ejpam-3495	358	4	follows	follow	VERB
ejpam-3495	358	5	from	from	ADP
ejpam-3495	358	6	theorem	theorem	ADJ
ejpam-3495	358	7	5.8	5.8	NUM
ejpam-3495	358	8	.	.	PUNCT
ejpam-3495	359	1	corollary	corollary	ADJ
ejpam-3495	359	2	5.10	5.10	NUM
ejpam-3495	359	3	.	.	PUNCT
ejpam-3495	360	1	let	let	VERB
ejpam-3495	360	2	{	{	PUNCT
ejpam-3495	360	3	ik	ik	PROPN
ejpam-3495	360	4	:	:	PUNCT
ejpam-3495	360	5	k	k	PROPN
ejpam-3495	360	6	∈	∈	PROPN
ejpam-3495	360	7	k	k	AUX
ejpam-3495	360	8	}	}	PUNCT
ejpam-3495	360	9	be	be	AUX
ejpam-3495	360	10	a	a	DET
ejpam-3495	360	11	nonempty	nonempty	ADJ
ejpam-3495	360	12	collection	collection	NOUN
ejpam-3495	360	13	of	of	ADP
ejpam-3495	360	14	�	�	NOUN
ejpam-3495	360	15	-ideals	-ideal	NOUN
ejpam-3495	360	16	of	of	ADP
ejpam-3495	360	17	a	a	DET
ejpam-3495	360	18	companion	companion	NOUN
ejpam-3495	360	19	b	b	NOUN
ejpam-3495	360	20	-	-	PUNCT
ejpam-3495	360	21	algebra	algebra	NOUN
ejpam-3495	360	22	.	.	PUNCT
ejpam-3495	361	1	if	if	SCONJ
ejpam-3495	361	2	i	i	PRON
ejpam-3495	361	3	=	=	SYM
ejpam-3495	361	4	⋂	⋂	PROPN
ejpam-3495	361	5	k∈k	k∈k	NOUN
ejpam-3495	361	6	ik	ik	PROPN
ejpam-3495	361	7	6=	6=	PROPN
ejpam-3495	361	8	∅	∅	NOUN
ejpam-3495	361	9	,	,	PUNCT
ejpam-3495	361	10	then	then	ADV
ejpam-3495	361	11	i	i	PRON
ejpam-3495	361	12	is	be	AUX
ejpam-3495	361	13	a	a	DET
ejpam-3495	361	14	�	�	NOUN
ejpam-3495	361	15	-ideal	-ideal	NOUN
ejpam-3495	361	16	.	.	PUNCT
ejpam-3495	362	1	l.d	l.d	PROPN
ejpam-3495	362	2	.	.	PROPN
ejpam-3495	362	3	naingue	naingue	PROPN
ejpam-3495	362	4	,	,	PUNCT
ejpam-3495	362	5	j.p	j.p	PROPN
ejpam-3495	362	6	.	.	PROPN
ejpam-3495	362	7	vilela	vilela	PROPN
ejpam-3495	362	8	/	/	SYM
ejpam-3495	362	9	eur	eur	PROPN
ejpam-3495	362	10	.	.	PUNCT
ejpam-3495	363	1	j.	j.	PROPN
ejpam-3495	363	2	pure	pure	PROPN
ejpam-3495	363	3	appl	appl	PROPN
ejpam-3495	363	4	.	.	PROPN
ejpam-3495	363	5	math	math	PROPN
ejpam-3495	363	6	,	,	PUNCT
ejpam-3495	363	7	12	12	NUM
ejpam-3495	363	8	(	(	PUNCT
ejpam-3495	363	9	3	3	NUM
ejpam-3495	363	10	)	)	PUNCT
ejpam-3495	363	11	(	(	PUNCT
ejpam-3495	363	12	2019	2019	NUM
ejpam-3495	363	13	)	)	PUNCT
ejpam-3495	363	14	,	,	PUNCT
ejpam-3495	363	15	1248	1248	NUM
ejpam-3495	363	16	-	-	SYM
ejpam-3495	363	17	1259	1259	NUM
ejpam-3495	363	18	1258	1258	NUM
ejpam-3495	363	19	observe	observe	VERB
ejpam-3495	363	20	that	that	SCONJ
ejpam-3495	363	21	in	in	ADP
ejpam-3495	363	22	example	example	NOUN
ejpam-3495	363	23	3.2	3.2	NUM
ejpam-3495	363	24	,	,	PUNCT
ejpam-3495	363	25	i1	i1	PROPN
ejpam-3495	363	26	=	=	PUNCT
ejpam-3495	363	27	{	{	PUNCT
ejpam-3495	363	28	0	0	NUM
ejpam-3495	363	29	,	,	PUNCT
ejpam-3495	363	30	3	3	NUM
ejpam-3495	363	31	}	}	PUNCT
ejpam-3495	363	32	and	and	CCONJ
ejpam-3495	363	33	i2	i2	PROPN
ejpam-3495	363	34	=	=	PUNCT
ejpam-3495	363	35	{	{	PUNCT
ejpam-3495	363	36	0	0	NUM
ejpam-3495	363	37	,	,	PUNCT
ejpam-3495	363	38	4	4	NUM
ejpam-3495	363	39	}	}	PUNCT
ejpam-3495	363	40	are	be	AUX
ejpam-3495	363	41	�	�	NOUN
ejpam-3495	363	42	-ideals	-ideal	NOUN
ejpam-3495	363	43	.	.	PUNCT
ejpam-3495	364	1	but	but	CCONJ
ejpam-3495	364	2	their	their	PRON
ejpam-3495	364	3	union	union	NOUN
ejpam-3495	364	4	,	,	PUNCT
ejpam-3495	364	5	i	i	PROPN
ejpam-3495	364	6	=	=	PROPN
ejpam-3495	364	7	i1	i1	PROPN
ejpam-3495	364	8	∪	∪	PROPN
ejpam-3495	364	9	i2	i2	PROPN
ejpam-3495	364	10	=	=	PUNCT
ejpam-3495	364	11	{	{	PUNCT
ejpam-3495	364	12	0	0	NUM
ejpam-3495	364	13	,	,	PUNCT
ejpam-3495	364	14	3	3	NUM
ejpam-3495	364	15	,	,	PUNCT
ejpam-3495	364	16	4	4	NUM
ejpam-3495	364	17	}	}	PUNCT
ejpam-3495	364	18	is	be	AUX
ejpam-3495	364	19	not	not	PART
ejpam-3495	364	20	a	a	DET
ejpam-3495	364	21	�	�	NOUN
ejpam-3495	364	22	-ideal	-ideal	NOUN
ejpam-3495	365	1	because	because	SCONJ
ejpam-3495	365	2	1	1	NUM
ejpam-3495	365	3	�	�	PROPN
ejpam-3495	365	4	4	4	NUM
ejpam-3495	365	5	=	=	SYM
ejpam-3495	365	6	3	3	NUM
ejpam-3495	365	7	∈	∈	NOUN
ejpam-3495	365	8	i	i	PRON
ejpam-3495	365	9	and	and	CCONJ
ejpam-3495	365	10	4	4	NUM
ejpam-3495	365	11	∈	∈	NOUN
ejpam-3495	365	12	i	i	PRON
ejpam-3495	365	13	but	but	CCONJ
ejpam-3495	365	14	1	1	NUM
ejpam-3495	365	15	/∈	/∈	NOUN
ejpam-3495	365	16	i.	i.	NOUN
ejpam-3495	365	17	thus	thus	ADV
ejpam-3495	365	18	,	,	PUNCT
ejpam-3495	365	19	we	we	PRON
ejpam-3495	365	20	have	have	VERB
ejpam-3495	365	21	the	the	DET
ejpam-3495	365	22	following	follow	VERB
ejpam-3495	365	23	remark	remark	NOUN
ejpam-3495	365	24	.	.	PUNCT
ejpam-3495	366	1	remark	remark	PROPN
ejpam-3495	366	2	5.11	5.11	NUM
ejpam-3495	366	3	.	.	PUNCT
ejpam-3495	367	1	the	the	DET
ejpam-3495	367	2	union	union	PROPN
ejpam-3495	367	3	of	of	ADP
ejpam-3495	367	4	�	�	PROPN
ejpam-3495	367	5	-ideals	-ideal	NOUN
ejpam-3495	367	6	need	need	AUX
ejpam-3495	367	7	not	not	PART
ejpam-3495	367	8	be	be	AUX
ejpam-3495	367	9	a	a	DET
ejpam-3495	367	10	�	�	NOUN
ejpam-3495	367	11	-ideal	-ideal	NOUN
ejpam-3495	367	12	.	.	PUNCT
ejpam-3495	368	1	6	6	NUM
ejpam-3495	368	2	.	.	X
ejpam-3495	368	3	on	on	ADP
ejpam-3495	368	4	companion	companion	NOUN
ejpam-3495	368	5	-	-	PUNCT
ejpam-3495	368	6	b	b	NOUN
ejpam-3495	368	7	-	-	PUNCT
ejpam-3495	368	8	homomorphisms	homomorphisms	ADJ
ejpam-3495	368	9	definition	definition	NOUN
ejpam-3495	368	10	6.1	6.1	NUM
ejpam-3495	368	11	.	.	PUNCT
ejpam-3495	369	1	let	let	VERB
ejpam-3495	369	2	(	(	PUNCT
ejpam-3495	369	3	x	x	NOUN
ejpam-3495	369	4	,	,	PUNCT
ejpam-3495	369	5	∗x	∗x	NOUN
ejpam-3495	369	6	,	,	PUNCT
ejpam-3495	369	7	�	�	PROPN
ejpam-3495	369	8	x	x	SYM
ejpam-3495	369	9	,	,	PUNCT
ejpam-3495	369	10	0x	0x	NOUN
ejpam-3495	369	11	)	)	PUNCT
ejpam-3495	369	12	and	and	CCONJ
ejpam-3495	369	13	(	(	PUNCT
ejpam-3495	369	14	y	y	PROPN
ejpam-3495	369	15	,	,	PUNCT
ejpam-3495	369	16	∗y	∗y	PROPN
ejpam-3495	369	17	,	,	PUNCT
ejpam-3495	369	18	�	�	PROPN
ejpam-3495	369	19	y	y	PROPN
ejpam-3495	369	20	,	,	PUNCT
ejpam-3495	369	21	0y	0y	NUM
ejpam-3495	369	22	)	)	PUNCT
ejpam-3495	369	23	be	be	AUX
ejpam-3495	369	24	companion	companion	NOUN
ejpam-3495	369	25	b	b	PROPN
ejpam-3495	369	26	-algebras	-algebras	PROPN
ejpam-3495	369	27	.	.	PUNCT
ejpam-3495	370	1	a	a	DET
ejpam-3495	370	2	map	map	NOUN
ejpam-3495	370	3	f	f	X
ejpam-3495	370	4	:	:	PUNCT
ejpam-3495	370	5	x	x	X
ejpam-3495	370	6	→	→	SYM
ejpam-3495	370	7	y	y	PROPN
ejpam-3495	370	8	is	be	AUX
ejpam-3495	370	9	called	call	VERB
ejpam-3495	370	10	a	a	DET
ejpam-3495	370	11	companion	companion	NOUN
ejpam-3495	370	12	-	-	PUNCT
ejpam-3495	370	13	b	b	NOUN
ejpam-3495	370	14	-	-	PUNCT
ejpam-3495	370	15	homomorphism	homomorphism	NOUN
ejpam-3495	370	16	if	if	SCONJ
ejpam-3495	370	17	for	for	ADP
ejpam-3495	370	18	any	any	DET
ejpam-3495	370	19	a	a	NOUN
ejpam-3495	370	20	,	,	PUNCT
ejpam-3495	370	21	b	b	PROPN
ejpam-3495	370	22	∈	∈	PROPN
ejpam-3495	370	23	x	x	NOUN
ejpam-3495	370	24	,	,	PUNCT
ejpam-3495	370	25	f(a	f(a	PROPN
ejpam-3495	370	26	∗x	∗x	PROPN
ejpam-3495	370	27	b	b	X
ejpam-3495	370	28	)	)	PUNCT
ejpam-3495	370	29	=	=	SYM
ejpam-3495	370	30	f(a	f(a	NOUN
ejpam-3495	370	31	)	)	PUNCT
ejpam-3495	370	32	∗y	∗y	PROPN
ejpam-3495	370	33	f(b	f(b	PROPN
ejpam-3495	370	34	)	)	PUNCT
ejpam-3495	370	35	and	and	CCONJ
ejpam-3495	370	36	f(a	f(a	PROPN
ejpam-3495	370	37	�	�	PROPN
ejpam-3495	370	38	x	x	SYM
ejpam-3495	370	39	b	b	NOUN
ejpam-3495	370	40	)	)	PUNCT
ejpam-3495	370	41	=	=	SYM
ejpam-3495	370	42	f(a)	f(a)	PROPN
ejpam-3495	370	43	�	�	PROPN
ejpam-3495	370	44	y	y	PROPN
ejpam-3495	370	45	f(b	f(b	PROPN
ejpam-3495	370	46	)	)	PUNCT
ejpam-3495	370	47	.	.	PUNCT
ejpam-3495	370	48	example	example	NOUN
ejpam-3495	371	1	6.2	6.2	NUM
ejpam-3495	371	2	.	.	PUNCT
ejpam-3495	372	1	let	let	VERB
ejpam-3495	372	2	m	m	PRON
ejpam-3495	372	3	∈	∈	PROPN
ejpam-3495	372	4	z	z	AUX
ejpam-3495	372	5	be	be	AUX
ejpam-3495	372	6	fixed	fix	VERB
ejpam-3495	372	7	.	.	PUNCT
ejpam-3495	373	1	the	the	DET
ejpam-3495	373	2	function	function	NOUN
ejpam-3495	373	3	f	f	X
ejpam-3495	373	4	:	:	PUNCT
ejpam-3495	373	5	z→	z→	PROPN
ejpam-3495	373	6	z	z	NOUN
ejpam-3495	373	7	defined	define	VERB
ejpam-3495	373	8	by	by	ADP
ejpam-3495	373	9	f(x	f(x	PROPN
ejpam-3495	373	10	)	)	PUNCT
ejpam-3495	373	11	=	=	SYM
ejpam-3495	373	12	mx	mx	PROPN
ejpam-3495	373	13	,	,	PUNCT
ejpam-3495	373	14	x	x	SYM
ejpam-3495	373	15	∈	∈	PROPN
ejpam-3495	373	16	z	z	PROPN
ejpam-3495	373	17	,	,	PUNCT
ejpam-3495	373	18	is	be	AUX
ejpam-3495	373	19	a	a	DET
ejpam-3495	373	20	companion	companion	NOUN
ejpam-3495	373	21	-	-	PUNCT
ejpam-3495	373	22	b	b	NOUN
ejpam-3495	373	23	-homomorphism	-homomorphism	PROPN
ejpam-3495	373	24	.	.	PUNCT
ejpam-3495	374	1	remark	remark	NOUN
ejpam-3495	374	2	6.3	6.3	NUM
ejpam-3495	374	3	.	.	PUNCT
ejpam-3495	375	1	a	a	DET
ejpam-3495	375	2	companion	companion	NOUN
ejpam-3495	375	3	b	b	X
ejpam-3495	375	4	-	-	PUNCT
ejpam-3495	375	5	homomorphism	homomorphism	NOUN
ejpam-3495	375	6	is	be	AUX
ejpam-3495	375	7	a	a	DET
ejpam-3495	375	8	b	b	NOUN
ejpam-3495	375	9	-	-	PUNCT
ejpam-3495	375	10	homomorphism	homomorphism	NOUN
ejpam-3495	375	11	and	and	CCONJ
ejpam-3495	375	12	a	a	DET
ejpam-3495	375	13	group	group	NOUN
ejpam-3495	375	14	homomorphism	homomorphism	NOUN
ejpam-3495	375	15	.	.	PUNCT
ejpam-3495	376	1	example	example	NOUN
ejpam-3495	376	2	6.4	6.4	NUM
ejpam-3495	376	3	.	.	PUNCT
ejpam-3495	377	1	consider	consider	VERB
ejpam-3495	377	2	the	the	DET
ejpam-3495	377	3	companion	companion	NOUN
ejpam-3495	377	4	b	b	PROPN
ejpam-3495	377	5	-algebra	-algebra	PROPN
ejpam-3495	377	6	(	(	PUNCT
ejpam-3495	377	7	x	x	NOUN
ejpam-3495	377	8	,	,	PUNCT
ejpam-3495	377	9	∗1,	∗1,	PROPN
ejpam-3495	377	10	�	�	NOUN
ejpam-3495	377	11	1	1	NUM
ejpam-3495	377	12	,	,	PUNCT
ejpam-3495	377	13	0	0	NUM
ejpam-3495	377	14	)	)	PUNCT
ejpam-3495	377	15	in	in	ADP
ejpam-3495	377	16	example	example	NOUN
ejpam-3495	377	17	3.6	3.6	NUM
ejpam-3495	377	18	and	and	CCONJ
ejpam-3495	377	19	(	(	PUNCT
ejpam-3495	377	20	y	y	PROPN
ejpam-3495	377	21	,	,	PUNCT
ejpam-3495	377	22	∗2,	∗2,	PROPN
ejpam-3495	377	23	�	�	PROPN
ejpam-3495	377	24	2	2	NUM
ejpam-3495	377	25	,	,	PUNCT
ejpam-3495	377	26	0	0	NUM
ejpam-3495	377	27	)	)	PUNCT
ejpam-3495	377	28	in	in	ADP
ejpam-3495	377	29	example	example	NOUN
ejpam-3495	377	30	3.8	3.8	NUM
ejpam-3495	377	31	where	where	SCONJ
ejpam-3495	377	32	�	�	NOUN
ejpam-3495	377	33	2	2	NUM
ejpam-3495	377	34	=	=	SYM
ejpam-3495	377	35	∗2	∗2	NOUN
ejpam-3495	377	36	.	.	PUNCT
ejpam-3495	378	1	let	let	VERB
ejpam-3495	378	2	f	f	NOUN
ejpam-3495	378	3	:	:	PUNCT
ejpam-3495	378	4	x	x	X
ejpam-3495	378	5	→	→	SYM
ejpam-3495	378	6	y	y	PROPN
ejpam-3495	378	7	and	and	CCONJ
ejpam-3495	378	8	f(x	f(x	PROPN
ejpam-3495	378	9	)	)	PUNCT
ejpam-3495	379	1	=	=	PRON
ejpam-3495	379	2	{	{	PUNCT
ejpam-3495	379	3	0	0	NUM
ejpam-3495	379	4	,	,	PUNCT
ejpam-3495	379	5	if	if	SCONJ
ejpam-3495	379	6	x	x	ADP
ejpam-3495	379	7	=	=	SYM
ejpam-3495	379	8	0	0	NUM
ejpam-3495	379	9	,	,	PUNCT
ejpam-3495	379	10	2	2	NUM
ejpam-3495	379	11	,	,	PUNCT
ejpam-3495	379	12	3	3	NUM
ejpam-3495	379	13	,	,	PUNCT
ejpam-3495	379	14	if	if	SCONJ
ejpam-3495	379	15	x	x	ADP
ejpam-3495	379	16	=	=	SYM
ejpam-3495	379	17	1	1	NUM
ejpam-3495	379	18	,	,	PUNCT
ejpam-3495	379	19	3	3	NUM
ejpam-3495	379	20	.	.	PUNCT
ejpam-3495	380	1	then	then	ADV
ejpam-3495	380	2	f	f	PROPN
ejpam-3495	380	3	is	be	AUX
ejpam-3495	380	4	a	a	DET
ejpam-3495	380	5	companion	companion	NOUN
ejpam-3495	380	6	-	-	PUNCT
ejpam-3495	380	7	b	b	NOUN
ejpam-3495	380	8	-homomorphism	-homomorphism	PROPN
ejpam-3495	380	9	.	.	PUNCT
ejpam-3495	381	1	theorem	theorem	NOUN
ejpam-3495	381	2	6.5	6.5	NUM
ejpam-3495	381	3	.	.	PUNCT
ejpam-3495	382	1	suppose	suppose	VERB
ejpam-3495	382	2	f	f	X
ejpam-3495	382	3	:	:	PUNCT
ejpam-3495	382	4	x	x	X
ejpam-3495	382	5	→	→	SYM
ejpam-3495	382	6	y	y	PROPN
ejpam-3495	382	7	is	be	AUX
ejpam-3495	382	8	a	a	DET
ejpam-3495	382	9	companion	companion	NOUN
ejpam-3495	382	10	b	b	NOUN
ejpam-3495	382	11	-	-	PUNCT
ejpam-3495	382	12	homomorphism	homomorphism	NOUN
ejpam-3495	382	13	.	.	PUNCT
ejpam-3495	383	1	then	then	ADV
ejpam-3495	383	2	kerf	kerf	NOUN
ejpam-3495	383	3	is	be	AUX
ejpam-3495	383	4	a	a	DET
ejpam-3495	383	5	�	�	NOUN
ejpam-3495	383	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	383	7	of	of	ADP
ejpam-3495	383	8	x.	x.	NOUN
ejpam-3495	383	9	proof	proof	NOUN
ejpam-3495	383	10	:	:	PUNCT
ejpam-3495	383	11	note	note	VERB
ejpam-3495	383	12	that	that	SCONJ
ejpam-3495	383	13	by	by	ADP
ejpam-3495	383	14	remark	remark	NOUN
ejpam-3495	383	15	6.3	6.3	NUM
ejpam-3495	383	16	,	,	PUNCT
ejpam-3495	383	17	kerf	kerf	NOUN
ejpam-3495	383	18	is	be	AUX
ejpam-3495	383	19	a	a	DET
ejpam-3495	383	20	subgroup	subgroup	NOUN
ejpam-3495	383	21	of	of	ADP
ejpam-3495	383	22	x.	x.	NOUN
ejpam-3495	383	23	thus	thus	ADV
ejpam-3495	383	24	,	,	PUNCT
ejpam-3495	383	25	by	by	SCONJ
ejpam-3495	383	26	corollary	corollary	ADJ
ejpam-3495	383	27	5.9	5.9	NUM
ejpam-3495	383	28	,	,	PUNCT
ejpam-3495	383	29	kerf	kerf	NOUN
ejpam-3495	383	30	is	be	AUX
ejpam-3495	383	31	also	also	ADV
ejpam-3495	383	32	a	a	DET
ejpam-3495	383	33	�	�	PROPN
ejpam-3495	383	34	-subalgebra	-subalgebra	NOUN
ejpam-3495	383	35	.	.	PUNCT
ejpam-3495	384	1	�	�	PROPN
ejpam-3495	384	2	the	the	DET
ejpam-3495	384	3	proof	proof	NOUN
ejpam-3495	384	4	of	of	ADP
ejpam-3495	384	5	the	the	DET
ejpam-3495	384	6	following	follow	VERB
ejpam-3495	384	7	theorem	theorem	NOUN
ejpam-3495	384	8	is	be	AUX
ejpam-3495	384	9	straightforward	straightforward	ADJ
ejpam-3495	384	10	.	.	PUNCT
ejpam-3495	385	1	theorem	theorem	VERB
ejpam-3495	385	2	6.6	6.6	NUM
ejpam-3495	385	3	.	.	PUNCT
ejpam-3495	386	1	suppose	suppose	VERB
ejpam-3495	386	2	f	f	X
ejpam-3495	386	3	:	:	PUNCT
ejpam-3495	386	4	x	x	X
ejpam-3495	386	5	→	→	SYM
ejpam-3495	386	6	y	y	PROPN
ejpam-3495	386	7	is	be	AUX
ejpam-3495	386	8	a	a	DET
ejpam-3495	386	9	companion	companion	NOUN
ejpam-3495	386	10	b	b	NOUN
ejpam-3495	386	11	-	-	PUNCT
ejpam-3495	386	12	homomorphism	homomorphism	NOUN
ejpam-3495	386	13	.	.	PUNCT
ejpam-3495	387	1	if	if	SCONJ
ejpam-3495	387	2	i	i	PRON
ejpam-3495	387	3	is	be	AUX
ejpam-3495	387	4	a	a	DET
ejpam-3495	387	5	�	�	NOUN
ejpam-3495	387	6	subalgebra	subalgebra	NOUN
ejpam-3495	387	7	of	of	ADP
ejpam-3495	387	8	x	x	PRON
ejpam-3495	387	9	,	,	PUNCT
ejpam-3495	387	10	then	then	ADV
ejpam-3495	387	11	f(i	f(i	NUM
ejpam-3495	387	12	)	)	PUNCT
ejpam-3495	387	13	is	be	AUX
ejpam-3495	387	14	a	a	DET
ejpam-3495	387	15	�	�	NOUN
ejpam-3495	387	16	-subalgebra	-subalgebra	NOUN
ejpam-3495	387	17	of	of	ADP
ejpam-3495	387	18	y	y	PROPN
ejpam-3495	387	19	.	.	PUNCT
ejpam-3495	388	1	theorem	theorem	VERB
ejpam-3495	388	2	6.7	6.7	NUM
ejpam-3495	388	3	.	.	PUNCT
ejpam-3495	389	1	suppose	suppose	VERB
ejpam-3495	389	2	f	f	X
ejpam-3495	389	3	:	:	PUNCT
ejpam-3495	389	4	x	x	X
ejpam-3495	389	5	→	→	SYM
ejpam-3495	389	6	y	y	PROPN
ejpam-3495	389	7	is	be	AUX
ejpam-3495	389	8	a	a	DET
ejpam-3495	389	9	companion	companion	NOUN
ejpam-3495	389	10	b	b	NOUN
ejpam-3495	389	11	-	-	PUNCT
ejpam-3495	389	12	epimorphism	epimorphism	NOUN
ejpam-3495	389	13	and	and	CCONJ
ejpam-3495	389	14	b	b	PROPN
ejpam-3495	389	15	is	be	AUX
ejpam-3495	389	16	a	a	DET
ejpam-3495	389	17	�	�	NOUN
ejpam-3495	389	18	subalgebra	subalgebra	NOUN
ejpam-3495	389	19	of	of	ADP
ejpam-3495	389	20	y	y	PROPN
ejpam-3495	389	21	.	.	PUNCT
ejpam-3495	390	1	then	then	ADV
ejpam-3495	390	2	f−1(b	f−1(b	PROPN
ejpam-3495	390	3	)	)	PUNCT
ejpam-3495	390	4	is	be	AUX
ejpam-3495	390	5	a	a	DET
ejpam-3495	390	6	�	�	NOUN
ejpam-3495	390	7	-subalgebra	-subalgebra	NOUN
ejpam-3495	390	8	of	of	ADP
ejpam-3495	390	9	x.	x.	NOUN
ejpam-3495	390	10	proof	proof	NOUN
ejpam-3495	390	11	:	:	PUNCT
ejpam-3495	390	12	let	let	VERB
ejpam-3495	390	13	b	b	PRON
ejpam-3495	390	14	⊆	⊆	NUM
ejpam-3495	390	15	y	y	NOUN
ejpam-3495	390	16	be	be	AUX
ejpam-3495	390	17	a	a	DET
ejpam-3495	390	18	�	�	NOUN
ejpam-3495	390	19	-subalgebra	-subalgebra	NOUN
ejpam-3495	390	20	of	of	ADP
ejpam-3495	390	21	y	y	PROPN
ejpam-3495	390	22	.	.	PUNCT
ejpam-3495	391	1	since	since	SCONJ
ejpam-3495	391	2	b	b	PROPN
ejpam-3495	391	3	6=	6=	NUM
ejpam-3495	391	4	∅	∅	NOUN
ejpam-3495	391	5	and	and	CCONJ
ejpam-3495	391	6	f	f	PROPN
ejpam-3495	391	7	is	be	AUX
ejpam-3495	391	8	onto	onto	ADP
ejpam-3495	391	9	,	,	PUNCT
ejpam-3495	391	10	there	there	PRON
ejpam-3495	391	11	exist	exist	VERB
ejpam-3495	391	12	a	a	DET
ejpam-3495	391	13	∈	∈	PROPN
ejpam-3495	391	14	b	b	NOUN
ejpam-3495	391	15	and	and	CCONJ
ejpam-3495	391	16	x	x	SYM
ejpam-3495	391	17	∈	∈	PROPN
ejpam-3495	391	18	x	x	PUNCT
ejpam-3495	391	19	such	such	ADJ
ejpam-3495	391	20	that	that	SCONJ
ejpam-3495	391	21	f(x	f(x	NOUN
ejpam-3495	391	22	)	)	PUNCT
ejpam-3495	392	1	=	=	SYM
ejpam-3495	392	2	a.	a.	NOUN
ejpam-3495	392	3	hence	hence	ADV
ejpam-3495	392	4	,	,	PUNCT
ejpam-3495	392	5	x	x	X
ejpam-3495	392	6	∈	∈	PROPN
ejpam-3495	392	7	f−1(b	f−1(b	PROPN
ejpam-3495	392	8	)	)	PUNCT
ejpam-3495	392	9	.	.	PUNCT
ejpam-3495	393	1	so	so	ADV
ejpam-3495	393	2	,	,	PUNCT
ejpam-3495	393	3	f−1(b	f−1(b	PROPN
ejpam-3495	393	4	)	)	PUNCT
ejpam-3495	394	1	6=	6=	ADP
ejpam-3495	394	2	∅.	∅.	VERB
ejpam-3495	394	3	note	note	VERB
ejpam-3495	394	4	that	that	DET
ejpam-3495	394	5	f−1(b	f−1(b	PROPN
ejpam-3495	394	6	)	)	PUNCT
ejpam-3495	395	1	=	=	PRON
ejpam-3495	395	2	{	{	PUNCT
ejpam-3495	395	3	a	a	DET
ejpam-3495	395	4	∈	∈	NOUN
ejpam-3495	395	5	x	x	X
ejpam-3495	395	6	:	:	PUNCT
ejpam-3495	395	7	f(a	f(a	X
ejpam-3495	395	8	)	)	PUNCT
ejpam-3495	395	9	∈	∈	PROPN
ejpam-3495	395	10	b	b	SYM
ejpam-3495	395	11	}	}	PUNCT
ejpam-3495	395	12	⊆	⊆	NUM
ejpam-3495	395	13	x.	x.	NOUN
ejpam-3495	395	14	now	now	ADV
ejpam-3495	395	15	,	,	PUNCT
ejpam-3495	395	16	let	let	VERB
ejpam-3495	395	17	x	x	PRON
ejpam-3495	395	18	,	,	PUNCT
ejpam-3495	395	19	y	y	PROPN
ejpam-3495	395	20	∈	∈	PROPN
ejpam-3495	395	21	f−1(b	f−1(b	PROPN
ejpam-3495	395	22	)	)	PUNCT
ejpam-3495	395	23	.	.	PUNCT
ejpam-3495	396	1	then	then	ADV
ejpam-3495	396	2	f(x	f(x	PROPN
ejpam-3495	396	3	)	)	PUNCT
ejpam-3495	396	4	,	,	PUNCT
ejpam-3495	396	5	f(y	f(y	NOUN
ejpam-3495	396	6	)	)	PUNCT
ejpam-3495	396	7	∈	∈	PROPN
ejpam-3495	396	8	b.	b.	PROPN
ejpam-3495	397	1	because	because	SCONJ
ejpam-3495	397	2	b	b	PROPN
ejpam-3495	397	3	is	be	AUX
ejpam-3495	397	4	a	a	DET
ejpam-3495	397	5	�	�	PROPN
ejpam-3495	397	6	-subalgebra	-subalgebra	NOUN
ejpam-3495	397	7	,	,	PUNCT
ejpam-3495	397	8	f(x	f(x	PROPN
ejpam-3495	397	9	�	�	PROPN
ejpam-3495	397	10	y	y	PROPN
ejpam-3495	397	11	)	)	PUNCT
ejpam-3495	397	12	=	=	SYM
ejpam-3495	397	13	f(x	f(x	PROPN
ejpam-3495	397	14	)	)	PUNCT
ejpam-3495	397	15	�	�	PROPN
ejpam-3495	397	16	f(y	f(y	PROPN
ejpam-3495	397	17	)	)	PUNCT
ejpam-3495	397	18	∈	∈	PROPN
ejpam-3495	397	19	b.	b.	PROPN
ejpam-3495	397	20	hence	hence	ADV
ejpam-3495	397	21	,	,	PUNCT
ejpam-3495	397	22	x	x	X
ejpam-3495	397	23	�	�	PROPN
ejpam-3495	397	24	y	y	PROPN
ejpam-3495	397	25	∈	∈	PROPN
ejpam-3495	397	26	f−1(b	f−1(b	PROPN
ejpam-3495	397	27	)	)	PUNCT
ejpam-3495	397	28	.	.	PUNCT
ejpam-3495	398	1	therefore	therefore	ADV
ejpam-3495	398	2	,	,	PUNCT
ejpam-3495	398	3	f−1(b	f−1(b	PROPN
ejpam-3495	398	4	)	)	PUNCT
ejpam-3495	398	5	is	be	AUX
ejpam-3495	398	6	a	a	DET
ejpam-3495	398	7	�	�	NOUN
ejpam-3495	398	8	-subalgebra	-subalgebra	NOUN
ejpam-3495	398	9	of	of	ADP
ejpam-3495	398	10	x.	x.	PROPN
ejpam-3495	398	11	�	�	PROPN
ejpam-3495	398	12	by	by	ADP
ejpam-3495	398	13	theorem	theorem	NOUN
ejpam-3495	398	14	5.8	5.8	NUM
ejpam-3495	398	15	,	,	PUNCT
ejpam-3495	398	16	a	a	DET
ejpam-3495	398	17	�	�	NOUN
ejpam-3495	398	18	-ideal	-ideal	NOUN
ejpam-3495	398	19	is	be	AUX
ejpam-3495	398	20	equivalent	equivalent	ADJ
ejpam-3495	398	21	to	to	ADP
ejpam-3495	398	22	a	a	DET
ejpam-3495	398	23	subgroup	subgroup	NOUN
ejpam-3495	398	24	of	of	ADP
ejpam-3495	398	25	(	(	PUNCT
ejpam-3495	398	26	x	x	NOUN
ejpam-3495	398	27	,	,	PUNCT
ejpam-3495	398	28	�	�	PROPN
ejpam-3495	398	29	)	)	PUNCT
ejpam-3495	398	30	.	.	PUNCT
ejpam-3495	399	1	thus	thus	ADV
ejpam-3495	399	2	,	,	PUNCT
ejpam-3495	399	3	the	the	DET
ejpam-3495	399	4	following	follow	VERB
ejpam-3495	399	5	corollary	corollary	ADJ
ejpam-3495	399	6	holds	hold	NOUN
ejpam-3495	399	7	:	:	PUNCT
ejpam-3495	399	8	corollary	corollary	ADJ
ejpam-3495	399	9	6.8	6.8	NUM
ejpam-3495	399	10	.	.	PUNCT
ejpam-3495	399	11	suppose	suppose	VERB
ejpam-3495	399	12	f	f	X
ejpam-3495	399	13	:	:	PUNCT
ejpam-3495	399	14	x	x	X
ejpam-3495	399	15	→	→	SYM
ejpam-3495	399	16	y	y	PROPN
ejpam-3495	399	17	is	be	AUX
ejpam-3495	399	18	a	a	DET
ejpam-3495	399	19	companion	companion	NOUN
ejpam-3495	399	20	b	b	NOUN
ejpam-3495	399	21	-	-	PUNCT
ejpam-3495	399	22	homomorphism	homomorphism	NOUN
ejpam-3495	399	23	.	.	PUNCT
ejpam-3495	400	1	references	reference	NOUN
ejpam-3495	400	2	1259	1259	NUM
ejpam-3495	400	3	(	(	PUNCT
ejpam-3495	400	4	i	i	NOUN
ejpam-3495	400	5	)	)	PUNCT
ejpam-3495	400	6	if	if	SCONJ
ejpam-3495	400	7	i	i	PRON
ejpam-3495	400	8	is	be	AUX
ejpam-3495	400	9	a	a	DET
ejpam-3495	400	10	�	�	NOUN
ejpam-3495	400	11	-ideal	-ideal	NOUN
ejpam-3495	400	12	of	of	ADP
ejpam-3495	400	13	x	x	NOUN
ejpam-3495	400	14	,	,	PUNCT
ejpam-3495	400	15	then	then	ADV
ejpam-3495	400	16	f(i	f(i	NUM
ejpam-3495	400	17	)	)	PUNCT
ejpam-3495	400	18	is	be	AUX
ejpam-3495	400	19	a	a	DET
ejpam-3495	400	20	�	�	NOUN
ejpam-3495	400	21	-ideal	-ideal	NOUN
ejpam-3495	400	22	of	of	ADP
ejpam-3495	400	23	y	y	PROPN
ejpam-3495	400	24	.	.	PUNCT
ejpam-3495	401	1	(	(	PUNCT
ejpam-3495	401	2	i	i	NOUN
ejpam-3495	401	3	)	)	PUNCT
ejpam-3495	401	4	if	if	SCONJ
ejpam-3495	401	5	b	b	PROPN
ejpam-3495	401	6	⊆	⊆	NUM
ejpam-3495	401	7	y	y	PROPN
ejpam-3495	401	8	is	be	AUX
ejpam-3495	401	9	a	a	DET
ejpam-3495	401	10	�	�	NOUN
ejpam-3495	401	11	-ideal	-ideal	NOUN
ejpam-3495	401	12	of	of	ADP
ejpam-3495	401	13	y	y	PROPN
ejpam-3495	401	14	,	,	PUNCT
ejpam-3495	401	15	then	then	ADV
ejpam-3495	401	16	f−1(b	f−1(b	PROPN
ejpam-3495	401	17	)	)	PUNCT
ejpam-3495	401	18	is	be	AUX
ejpam-3495	401	19	a	a	DET
ejpam-3495	401	20	�	�	NOUN
ejpam-3495	401	21	-ideal	-ideal	NOUN
ejpam-3495	401	22	of	of	ADP
ejpam-3495	401	23	x.	x.	NOUN
ejpam-3495	401	24	(	(	PUNCT
ejpam-3495	401	25	iii	iii	NOUN
ejpam-3495	401	26	)	)	PUNCT
ejpam-3495	401	27	kerf	kerf	NOUN
ejpam-3495	401	28	is	be	AUX
ejpam-3495	401	29	a	a	DET
ejpam-3495	401	30	�	�	NOUN
ejpam-3495	401	31	-ideal	-ideal	NOUN
ejpam-3495	401	32	of	of	ADP
ejpam-3495	401	33	x.	x.	NOUN
ejpam-3495	401	34	references	reference	NOUN
ejpam-3495	401	35	[	[	X
ejpam-3495	401	36	1	1	NUM
ejpam-3495	401	37	]	]	X
ejpam-3495	401	38	h.k	h.k	PROPN
ejpam-3495	401	39	.	.	PROPN
ejpam-3495	401	40	abdullah	abdullah	PROPN
ejpam-3495	401	41	and	and	CCONJ
ejpam-3495	401	42	a.a	a.a	PROPN
ejpam-3495	401	43	.	.	PROPN
ejpam-3495	401	44	atshan	atshan	PROPN
ejpam-3495	401	45	.	.	PUNCT
ejpam-3495	402	1	complete	complete	ADJ
ejpam-3495	402	2	ideal	ideal	NOUN
ejpam-3495	402	3	and	and	CCONJ
ejpam-3495	402	4	n	n	CCONJ
ejpam-3495	402	5	-	-	PUNCT
ejpam-3495	402	6	ideal	ideal	NOUN
ejpam-3495	402	7	of	of	ADP
ejpam-3495	402	8	b	b	PROPN
ejpam-3495	402	9	-algebra	-algebra	PROPN
ejpam-3495	402	10	.	.	PUNCT
ejpam-3495	403	1	applied	apply	VERB
ejpam-3495	403	2	mathematical	mathematical	ADJ
ejpam-3495	403	3	sciences	science	NOUN
ejpam-3495	403	4	,	,	PUNCT
ejpam-3495	403	5	11(35):1705–1713	11(35):1705–1713	NUM
ejpam-3495	403	6	,	,	PUNCT
ejpam-3495	403	7	2017	2017	NUM
ejpam-3495	403	8	.	.	PUNCT
ejpam-3495	404	1	[	[	X
ejpam-3495	404	2	2	2	NUM
ejpam-3495	404	3	]	]	X
ejpam-3495	404	4	p.j	p.j	PROPN
ejpam-3495	404	5	.	.	PROPN
ejpam-3495	404	6	allen	allen	PROPN
ejpam-3495	404	7	,	,	PUNCT
ejpam-3495	404	8	h.s	h.s	PROPN
ejpam-3495	404	9	.	.	PROPN
ejpam-3495	404	10	kim	kim	PROPN
ejpam-3495	404	11	,	,	PUNCT
ejpam-3495	404	12	and	and	CCONJ
ejpam-3495	404	13	j.	j.	PROPN
ejpam-3495	404	14	neggers	neggers	PROPN
ejpam-3495	404	15	.	.	PUNCT
ejpam-3495	405	1	b	b	X
ejpam-3495	405	2	-algebras	-algebra	NOUN
ejpam-3495	405	3	and	and	CCONJ
ejpam-3495	405	4	groups	group	NOUN
ejpam-3495	405	5	.	.	PUNCT
ejpam-3495	406	1	scientiae	scientiae	PROPN
ejpam-3495	406	2	mathematicae	mathematicae	VERB
ejpam-3495	406	3	japonicae	japonicae	PROPN
ejpam-3495	406	4	online	online	NOUN
ejpam-3495	406	5	,	,	PUNCT
ejpam-3495	406	6	9:159–165	9:159–165	NUM
ejpam-3495	406	7	,	,	PUNCT
ejpam-3495	406	8	2003	2003	NUM
ejpam-3495	406	9	.	.	PUNCT
ejpam-3495	407	1	[	[	X
ejpam-3495	407	2	3	3	NUM
ejpam-3495	407	3	]	]	X
ejpam-3495	407	4	p.j	p.j	PROPN
ejpam-3495	407	5	.	.	PROPN
ejpam-3495	407	6	allen	allen	PROPN
ejpam-3495	407	7	,	,	PUNCT
ejpam-3495	407	8	h.s	h.s	PROPN
ejpam-3495	407	9	.	.	PROPN
ejpam-3495	407	10	kim	kim	PROPN
ejpam-3495	407	11	,	,	PUNCT
ejpam-3495	407	12	and	and	CCONJ
ejpam-3495	407	13	j.	j.	PROPN
ejpam-3495	407	14	neggers	neggers	PROPN
ejpam-3495	407	15	.	.	PUNCT
ejpam-3495	408	1	companion	companion	NOUN
ejpam-3495	408	2	d	d	X
ejpam-3495	408	3	-algebras	-algebras	PROPN
ejpam-3495	408	4	.	.	PUNCT
ejpam-3495	409	1	mathematica	mathematica	PROPN
ejpam-3495	409	2	slovaca	slovaca	PROPN
ejpam-3495	409	3	,	,	PUNCT
ejpam-3495	409	4	57(2):93–106	57(2):93–106	NUM
ejpam-3495	409	5	,	,	PUNCT
ejpam-3495	409	6	2007	2007	NUM
ejpam-3495	409	7	.	.	PUNCT
ejpam-3495	410	1	[	[	X
ejpam-3495	410	2	4	4	NUM
ejpam-3495	410	3	]	]	X
ejpam-3495	410	4	j.c	j.c	PROPN
ejpam-3495	410	5	.	.	PROPN
ejpam-3495	410	6	endam	endam	PROPN
ejpam-3495	410	7	and	and	CCONJ
ejpam-3495	410	8	r.c	r.c	PROPN
ejpam-3495	410	9	.	.	PROPN
ejpam-3495	410	10	teves	teves	PROPN
ejpam-3495	410	11	.	.	PUNCT
ejpam-3495	411	1	some	some	DET
ejpam-3495	411	2	properties	property	NOUN
ejpam-3495	411	3	of	of	ADP
ejpam-3495	411	4	cyclic	cyclic	ADJ
ejpam-3495	411	5	b	b	PROPN
ejpam-3495	411	6	-algebras	-algebras	PROPN
ejpam-3495	411	7	.	.	PUNCT
ejpam-3495	412	1	international	international	PROPN
ejpam-3495	412	2	mathematical	mathematical	PROPN
ejpam-3495	412	3	forum	forum	PROPN
ejpam-3495	412	4	,	,	PUNCT
ejpam-3495	412	5	11(8):387–394	11(8):387–394	NUM
ejpam-3495	412	6	,	,	PUNCT
ejpam-3495	412	7	2016	2016	NUM
ejpam-3495	412	8	.	.	PUNCT
ejpam-3495	413	1	[	[	X
ejpam-3495	413	2	5	5	X
ejpam-3495	413	3	]	]	PUNCT
ejpam-3495	413	4	e.	e.	PROPN
ejpam-3495	413	5	fitria	fitria	PROPN
ejpam-3495	413	6	,	,	PUNCT
ejpam-3495	413	7	s.	s.	PROPN
ejpam-3495	413	8	gemawati	gemawati	PROPN
ejpam-3495	413	9	,	,	PUNCT
ejpam-3495	413	10	and	and	CCONJ
ejpam-3495	413	11	kartini	kartini	NOUN
ejpam-3495	413	12	.	.	PUNCT
ejpam-3495	414	1	prime	prime	ADJ
ejpam-3495	414	2	ideals	ideal	NOUN
ejpam-3495	414	3	in	in	ADP
ejpam-3495	414	4	b	b	NOUN
ejpam-3495	414	5	-algebras	-algebras	PROPN
ejpam-3495	414	6	.	.	PUNCT
ejpam-3495	415	1	international	international	ADJ
ejpam-3495	415	2	journal	journal	NOUN
ejpam-3495	415	3	of	of	ADP
ejpam-3495	415	4	algebra	algebra	PROPN
ejpam-3495	415	5	,	,	PUNCT
ejpam-3495	415	6	11(7):301–309	11(7):301–309	PROPN
ejpam-3495	415	7	,	,	PUNCT
ejpam-3495	415	8	2017	2017	NUM
ejpam-3495	415	9	.	.	PUNCT
ejpam-3495	416	1	[	[	X
ejpam-3495	416	2	6	6	NUM
ejpam-3495	416	3	]	]	PUNCT
ejpam-3495	416	4	k.	k.	PROPN
ejpam-3495	416	5	iséki	iséki	PROPN
ejpam-3495	416	6	.	.	PROPN
ejpam-3495	416	7	on	on	ADP
ejpam-3495	416	8	bci	bci	PROPN
ejpam-3495	416	9	-algebras	-algebras	PROPN
ejpam-3495	416	10	.	.	PUNCT
ejpam-3495	416	11	math	math	NOUN
ejpam-3495	416	12	.	.	PUNCT
ejpam-3495	417	1	seminar	seminar	NOUN
ejpam-3495	417	2	notes	note	NOUN
ejpam-3495	417	3	,	,	PUNCT
ejpam-3495	417	4	8:125–130	8:125–130	NOUN
ejpam-3495	417	5	,	,	PUNCT
ejpam-3495	417	6	1980	1980	NUM
ejpam-3495	417	7	.	.	PUNCT
ejpam-3495	418	1	[	[	X
ejpam-3495	418	2	7	7	X
ejpam-3495	418	3	]	]	X
ejpam-3495	418	4	k.	k.	PROPN
ejpam-3495	418	5	iséki	iséki	PROPN
ejpam-3495	418	6	and	and	CCONJ
ejpam-3495	418	7	tanaka	tanaka	PROPN
ejpam-3495	418	8	.	.	PUNCT
ejpam-3495	419	1	an	an	DET
ejpam-3495	419	2	introduction	introduction	NOUN
ejpam-3495	419	3	to	to	ADP
ejpam-3495	419	4	theory	theory	NOUN
ejpam-3495	419	5	of	of	ADP
ejpam-3495	419	6	bck	bck	PROPN
ejpam-3495	419	7	-algebras	-algebras	PROPN
ejpam-3495	419	8	.	.	PROPN
ejpam-3495	419	9	math	math	NOUN
ejpam-3495	419	10	.	.	PUNCT
ejpam-3495	420	1	japonica	japonica	PROPN
ejpam-3495	420	2	,	,	PUNCT
ejpam-3495	420	3	23:1–26	23:1–26	NUM
ejpam-3495	420	4	,	,	PUNCT
ejpam-3495	420	5	1978	1978	NUM
ejpam-3495	420	6	.	.	PUNCT
ejpam-3495	421	1	[	[	X
ejpam-3495	421	2	8	8	NUM
ejpam-3495	421	3	]	]	X
ejpam-3495	421	4	m.	m.	NOUN
ejpam-3495	421	5	kondo	kondo	PROPN
ejpam-3495	421	6	and	and	CCONJ
ejpam-3495	421	7	y.b	y.b	PROPN
ejpam-3495	421	8	.	.	PROPN
ejpam-3495	421	9	jun	jun	PROPN
ejpam-3495	421	10	.	.	PUNCT
ejpam-3495	422	1	the	the	DET
ejpam-3495	422	2	class	class	NOUN
ejpam-3495	422	3	of	of	ADP
ejpam-3495	422	4	b	b	NOUN
ejpam-3495	422	5	-algebras	-algebras	PROPN
ejpam-3495	422	6	coincides	coincide	VERB
ejpam-3495	422	7	with	with	ADP
ejpam-3495	422	8	the	the	DET
ejpam-3495	422	9	class	class	NOUN
ejpam-3495	422	10	of	of	ADP
ejpam-3495	422	11	groups	group	NOUN
ejpam-3495	422	12	.	.	PUNCT
ejpam-3495	423	1	scientiae	scientiae	PROPN
ejpam-3495	423	2	mathematicae	mathematicae	VERB
ejpam-3495	423	3	japonicae	japonicae	PROPN
ejpam-3495	423	4	online	online	NOUN
ejpam-3495	423	5	,	,	PUNCT
ejpam-3495	423	6	2:175–177	2:175–177	NUM
ejpam-3495	423	7	,	,	PUNCT
ejpam-3495	423	8	2002	2002	NUM
ejpam-3495	423	9	.	.	PUNCT
ejpam-3495	424	1	[	[	X
ejpam-3495	424	2	9	9	NUM
ejpam-3495	424	3	]	]	X
ejpam-3495	424	4	j.	j.	PROPN
ejpam-3495	424	5	neggers	neggers	PROPN
ejpam-3495	424	6	and	and	CCONJ
ejpam-3495	424	7	h.s	h.s	PROPN
ejpam-3495	424	8	.	.	PROPN
ejpam-3495	424	9	kim	kim	PROPN
ejpam-3495	424	10	.	.	PUNCT
ejpam-3495	425	1	on	on	ADP
ejpam-3495	425	2	d	d	PROPN
ejpam-3495	425	3	-algebras	-algebras	PROPN
ejpam-3495	425	4	.	.	PUNCT
ejpam-3495	425	5	mathematica	mathematica	PROPN
ejpam-3495	425	6	slovaca	slovaca	PROPN
ejpam-3495	425	7	,	,	PUNCT
ejpam-3495	425	8	49(1):19–26	49(1):19–26	NUM
ejpam-3495	425	9	,	,	PUNCT
ejpam-3495	425	10	1999	1999	NUM
ejpam-3495	425	11	.	.	PUNCT
ejpam-3495	426	1	[	[	X
ejpam-3495	426	2	10	10	NUM
ejpam-3495	426	3	]	]	X
ejpam-3495	426	4	j.	j.	PROPN
ejpam-3495	426	5	neggers	neggers	PROPN
ejpam-3495	426	6	and	and	CCONJ
ejpam-3495	426	7	h.s	h.s	PROPN
ejpam-3495	426	8	.	.	PROPN
ejpam-3495	426	9	kim	kim	PROPN
ejpam-3495	426	10	.	.	PUNCT
ejpam-3495	427	1	a	a	DET
ejpam-3495	427	2	fundalmental	fundalmental	ADJ
ejpam-3495	427	3	theorem	theorem	NOUN
ejpam-3495	427	4	of	of	ADP
ejpam-3495	427	5	b	b	PROPN
ejpam-3495	427	6	-homomorphism	-homomorphism	PROPN
ejpam-3495	427	7	for	for	ADP
ejpam-3495	427	8	b	b	PROPN
ejpam-3495	427	9	algebras	algebras	X
ejpam-3495	427	10	.	.	PUNCT
ejpam-3495	428	1	int.math.j	int.math.j	PROPN
ejpam-3495	428	2	.	.	PROPN
ejpam-3495	428	3	,	,	PUNCT
ejpam-3495	428	4	2(3):207–214	2(3):207–214	NUM
ejpam-3495	428	5	,	,	PUNCT
ejpam-3495	428	6	2002	2002	NUM
ejpam-3495	428	7	.	.	PUNCT
ejpam-3495	429	1	[	[	X
ejpam-3495	429	2	11	11	NUM
ejpam-3495	429	3	]	]	PUNCT
ejpam-3495	429	4	j.	j.	PROPN
ejpam-3495	429	5	neggers	neggers	PROPN
ejpam-3495	429	6	and	and	CCONJ
ejpam-3495	429	7	h.s	h.s	PROPN
ejpam-3495	429	8	.	.	PROPN
ejpam-3495	429	9	kim	kim	PROPN
ejpam-3495	429	10	.	.	PUNCT
ejpam-3495	430	1	on	on	ADP
ejpam-3495	430	2	b	b	PROPN
ejpam-3495	430	3	-algebras	-algebras	PROPN
ejpam-3495	430	4	.	.	PROPN
ejpam-3495	430	5	math	math	PROPN
ejpam-3495	430	6	.	.	PUNCT
ejpam-3495	431	1	vesnik	vesnik	PROPN
ejpam-3495	431	2	,	,	PUNCT
ejpam-3495	431	3	54:21–29	54:21–29	NUM
ejpam-3495	431	4	,	,	PUNCT
ejpam-3495	431	5	2002	2002	NUM
ejpam-3495	431	6	.	.	PUNCT
ejpam-3495	432	1	[	[	X
ejpam-3495	432	2	12	12	NUM
ejpam-3495	432	3	]	]	PUNCT
ejpam-3495	432	4	a.	a.	NOUN
ejpam-3495	432	5	walendziak	walendziak	PROPN
ejpam-3495	432	6	.	.	PUNCT
ejpam-3495	433	1	a	a	DET
ejpam-3495	433	2	note	note	NOUN
ejpam-3495	433	3	on	on	ADP
ejpam-3495	433	4	normal	normal	ADJ
ejpam-3495	433	5	subalgebras	subalgebra	NOUN
ejpam-3495	433	6	in	in	ADP
ejpam-3495	433	7	b	b	PROPN
ejpam-3495	433	8	-algebras	-algebras	PROPN
ejpam-3495	433	9	.	.	PUNCT
ejpam-3495	434	1	scientiae	scientiae	PROPN
ejpam-3495	434	2	mathematicae	mathematicae	VERB
ejpam-3495	434	3	japonicae	japonicae	PROPN
ejpam-3495	434	4	online	online	ADV
ejpam-3495	434	5	,	,	PUNCT
ejpam-3495	434	6	pages	page	NOUN
ejpam-3495	434	7	49–53	49–53	NUM
ejpam-3495	434	8	,	,	PUNCT
ejpam-3495	434	9	2005	2005	NUM
ejpam-3495	434	10	.	.	PUNCT
ejpam-3495	435	1	[	[	X
ejpam-3495	435	2	13	13	NUM
ejpam-3495	435	3	]	]	PUNCT
ejpam-3495	435	4	a.	a.	NOUN
ejpam-3495	435	5	walendziak	walendziak	PROPN
ejpam-3495	435	6	.	.	PUNCT
ejpam-3495	436	1	some	some	DET
ejpam-3495	436	2	axiomizations	axiomization	NOUN
ejpam-3495	436	3	of	of	ADP
ejpam-3495	436	4	b	b	PROPN
ejpam-3495	436	5	-algebras	-algebras	PROPN
ejpam-3495	436	6	.	.	PUNCT
ejpam-3495	437	1	mathematica	mathematica	PROPN
ejpam-3495	437	2	slovaca	slovaca	PROPN
ejpam-3495	437	3	,	,	PUNCT
ejpam-3495	437	4	56(3):301	56(3):301	PROPN
ejpam-3495	437	5	–	–	PUNCT
ejpam-3495	437	6	306	306	NUM
ejpam-3495	437	7	,	,	PUNCT
ejpam-3495	437	8	2006	2006	NUM
ejpam-3495	437	9	.	.	PUNCT