id	sid	tid	token	lemma	pos
ejpam-5153	1	1	european	european	PROPN
ejpam-5153	1	2	journal	journal	PROPN
ejpam-5153	1	3	of	of	ADP
ejpam-5153	1	4	pure	pure	ADJ
ejpam-5153	1	5	and	and	CCONJ
ejpam-5153	1	6	applied	apply	VERB
ejpam-5153	1	7	mathematics	mathematic	NOUN
ejpam-5153	1	8	vol	vol	NOUN
ejpam-5153	1	9	.	.	PROPN
ejpam-5153	2	1	17	17	NUM
ejpam-5153	2	2	,	,	PUNCT
ejpam-5153	2	3	no	no	INTJ
ejpam-5153	2	4	.	.	NOUN
ejpam-5153	2	5	3	3	NUM
ejpam-5153	2	6	,	,	PUNCT
ejpam-5153	2	7	2024	2024	NUM
ejpam-5153	2	8	,	,	PUNCT
ejpam-5153	2	9	2276	2276	NUM
ejpam-5153	2	10	-	-	SYM
ejpam-5153	2	11	2287	2287	NUM
ejpam-5153	2	12	issn	issn	PROPN
ejpam-5153	2	13	1307	1307	NUM
ejpam-5153	2	14	-	-	SYM
ejpam-5153	2	15	5543	5543	NUM
ejpam-5153	2	16	–	–	PUNCT
ejpam-5153	2	17	ejpam.com	ejpam.com	X
ejpam-5153	2	18	published	publish	VERB
ejpam-5153	2	19	by	by	ADP
ejpam-5153	2	20	new	new	PROPN
ejpam-5153	2	21	york	york	PROPN
ejpam-5153	2	22	business	business	PROPN
ejpam-5153	2	23	global	global	ADJ
ejpam-5153	2	24	real	real	ADJ
ejpam-5153	2	25	division	division	NOUN
ejpam-5153	2	26	algebras	algebra	VERB
ejpam-5153	2	27	with	with	ADP
ejpam-5153	2	28	a	a	DET
ejpam-5153	2	29	left	left	ADJ
ejpam-5153	2	30	unit	unit	NOUN
ejpam-5153	2	31	element	element	NOUN
ejpam-5153	2	32	that	that	PRON
ejpam-5153	2	33	satisfy	satisfy	VERB
ejpam-5153	2	34	certain	certain	ADJ
ejpam-5153	2	35	identities	identity	NOUN
ejpam-5153	3	1	andré	andré	VERB
ejpam-5153	3	2	souleye	souleye	PROPN
ejpam-5153	3	3	diabang1,∗	diabang1,∗	PROPN
ejpam-5153	3	4	,	,	PUNCT
ejpam-5153	3	5	ama	ama	NOUN
ejpam-5153	3	6	sékou	sékou	PROPN
ejpam-5153	3	7	mballo1	mballo1	PROPN
ejpam-5153	3	8	,	,	PUNCT
ejpam-5153	3	9	papa	papa	VERB
ejpam-5153	3	10	cheikhou	cheikhou	PROPN
ejpam-5153	3	11	diop1	diop1	PROPN
ejpam-5153	3	12	1	1	NUM
ejpam-5153	3	13	département	département	X
ejpam-5153	3	14	de	de	X
ejpam-5153	3	15	mathématiques	mathématique	NOUN
ejpam-5153	3	16	,	,	PUNCT
ejpam-5153	3	17	ufr	ufr	PROPN
ejpam-5153	3	18	science	science	NOUN
ejpam-5153	3	19	et	et	PROPN
ejpam-5153	3	20	technologies	technology	NOUN
ejpam-5153	3	21	,	,	PUNCT
ejpam-5153	3	22	université	université	PROPN
ejpam-5153	3	23	iba	iba	PROPN
ejpam-5153	3	24	der	der	PROPN
ejpam-5153	3	25	thiam	thiam	PROPN
ejpam-5153	3	26	de	de	PROPN
ejpam-5153	3	27	thiès	thiès	PROPN
ejpam-5153	3	28	(	(	PUNCT
ejpam-5153	3	29	uidt	uidt	PROPN
ejpam-5153	3	30	)	)	PUNCT
ejpam-5153	3	31	,	,	PUNCT
ejpam-5153	3	32	sénégal	sénégal	PROPN
ejpam-5153	3	33	.	.	PUNCT
ejpam-5153	4	1	abstract	abstract	ADJ
ejpam-5153	4	2	.	.	PUNCT
ejpam-5153	5	1	we	we	PRON
ejpam-5153	5	2	study	study	VERB
ejpam-5153	5	3	a	a	DET
ejpam-5153	5	4	,	,	PUNCT
ejpam-5153	5	5	finite	finite	ADJ
ejpam-5153	5	6	dimensional	dimensional	ADJ
ejpam-5153	5	7	real	real	ADJ
ejpam-5153	5	8	division	division	NOUN
ejpam-5153	5	9	algebra	algebra	NOUN
ejpam-5153	5	10	with	with	ADP
ejpam-5153	5	11	left	left	ADJ
ejpam-5153	5	12	unit	unit	NOUN
ejpam-5153	5	13	e	e	NOUN
ejpam-5153	5	14	,	,	PUNCT
ejpam-5153	5	15	satisfying	satisfy	VERB
ejpam-5153	5	16	:	:	PUNCT
ejpam-5153	5	17	for	for	ADP
ejpam-5153	5	18	all	all	PRON
ejpam-5153	5	19	x	x	SYM
ejpam-5153	5	20	∈	∈	PROPN
ejpam-5153	5	21	a	a	DET
ejpam-5153	5	22	,	,	PUNCT
ejpam-5153	5	23	(	(	PUNCT
ejpam-5153	5	24	e1	e1	PROPN
ejpam-5153	5	25	)	)	PUNCT
ejpam-5153	5	26	(	(	PUNCT
ejpam-5153	5	27	x	x	X
ejpam-5153	5	28	,	,	PUNCT
ejpam-5153	5	29	x	x	X
ejpam-5153	5	30	,	,	PUNCT
ejpam-5153	5	31	x	x	X
ejpam-5153	5	32	)	)	PUNCT
ejpam-5153	5	33	=	=	SYM
ejpam-5153	5	34	0	0	NUM
ejpam-5153	5	35	,	,	PUNCT
ejpam-5153	5	36	(	(	PUNCT
ejpam-5153	5	37	e2	e2	PROPN
ejpam-5153	5	38	)	)	PUNCT
ejpam-5153	5	39	(	(	PUNCT
ejpam-5153	5	40	x2	x2	PROPN
ejpam-5153	5	41	,	,	PUNCT
ejpam-5153	5	42	x2	x2	PROPN
ejpam-5153	5	43	,	,	PUNCT
ejpam-5153	5	44	x2	x2	PROPN
ejpam-5153	5	45	)	)	PUNCT
ejpam-5153	5	46	=	=	SYM
ejpam-5153	5	47	0	0	NUM
ejpam-5153	5	48	,	,	PUNCT
ejpam-5153	5	49	(	(	PUNCT
ejpam-5153	5	50	e3	e3	NOUN
ejpam-5153	5	51	)	)	PUNCT
ejpam-5153	5	52	x2e	x2e	NOUN
ejpam-5153	6	1	=	=	SYM
ejpam-5153	6	2	x2	x2	PROPN
ejpam-5153	6	3	and	and	CCONJ
ejpam-5153	6	4	(	(	PUNCT
ejpam-5153	6	5	e4	e4	PROPN
ejpam-5153	6	6	)	)	PUNCT
ejpam-5153	6	7	(	(	PUNCT
ejpam-5153	6	8	xe)e	xe)e	PROPN
ejpam-5153	6	9	=	=	PUNCT
ejpam-5153	6	10	x.	x.	NOUN
ejpam-5153	7	1	we	we	PRON
ejpam-5153	7	2	show	show	VERB
ejpam-5153	7	3	that	that	SCONJ
ejpam-5153	7	4	:	:	PUNCT
ejpam-5153	7	5	•	•	X
ejpam-5153	7	6	if	if	SCONJ
ejpam-5153	7	7	a	a	DET
ejpam-5153	7	8	satisfies	satisfie	NOUN
ejpam-5153	7	9	to	to	ADP
ejpam-5153	7	10	(	(	PUNCT
ejpam-5153	7	11	e1	e1	PROPN
ejpam-5153	7	12	)	)	PUNCT
ejpam-5153	7	13	,	,	PUNCT
ejpam-5153	7	14	then	then	ADV
ejpam-5153	7	15	e	e	PROPN
ejpam-5153	7	16	is	be	AUX
ejpam-5153	7	17	the	the	DET
ejpam-5153	7	18	unit	unit	NOUN
ejpam-5153	7	19	element	element	NOUN
ejpam-5153	7	20	of	of	ADP
ejpam-5153	7	21	a.	a.	NOUN
ejpam-5153	7	22	•	•	PROPN
ejpam-5153	7	23	(	(	PUNCT
ejpam-5153	7	24	e1	e1	NOUN
ejpam-5153	7	25	)	)	PUNCT
ejpam-5153	8	1	=	=	NOUN
ejpam-5153	8	2	⇒	⇒	NOUN
ejpam-5153	8	3	(	(	PUNCT
ejpam-5153	8	4	e2	e2	PROPN
ejpam-5153	8	5	)	)	PUNCT
ejpam-5153	9	1	=	=	VERB
ejpam-5153	9	2	⇒	⇒	NOUN
ejpam-5153	9	3	(	(	PUNCT
ejpam-5153	9	4	e3	e3	NOUN
ejpam-5153	9	5	)	)	PUNCT
ejpam-5153	10	1	=	=	NOUN
ejpam-5153	10	2	⇒	⇒	NOUN
ejpam-5153	10	3	(	(	PUNCT
ejpam-5153	10	4	e4	e4	PROPN
ejpam-5153	10	5	)	)	PUNCT
ejpam-5153	10	6	.	.	PUNCT
ejpam-5153	11	1	in	in	ADP
ejpam-5153	11	2	two	two	NUM
ejpam-5153	11	3	-	-	PUNCT
ejpam-5153	11	4	dimensional	dimensional	ADJ
ejpam-5153	11	5	,	,	PUNCT
ejpam-5153	11	6	we	we	PRON
ejpam-5153	11	7	determine	determine	VERB
ejpam-5153	11	8	a	a	DET
ejpam-5153	11	9	satisfying	satisfying	NOUN
ejpam-5153	11	10	(	(	PUNCT
ejpam-5153	11	11	ei)i∈{1,2,3,4	ei)i∈{1,2,3,4	ADJ
ejpam-5153	11	12	}	}	PUNCT
ejpam-5153	11	13	.	.	PUNCT
ejpam-5153	12	1	we	we	PRON
ejpam-5153	12	2	have	have	VERB
ejpam-5153	12	3	a	a	DET
ejpam-5153	12	4	satisfies	satisfie	NOUN
ejpam-5153	12	5	to	to	ADP
ejpam-5153	12	6	(	(	PUNCT
ejpam-5153	12	7	e1	e1	PROPN
ejpam-5153	12	8	)	)	PUNCT
ejpam-5153	12	9	(	(	PUNCT
ejpam-5153	12	10	e2	e2	PROPN
ejpam-5153	12	11	)	)	PUNCT
ejpam-5153	12	12	(	(	PUNCT
ejpam-5153	12	13	e3	e3	NOUN
ejpam-5153	12	14	)	)	PUNCT
ejpam-5153	12	15	(	(	PUNCT
ejpam-5153	12	16	e4	e4	PROPN
ejpam-5153	12	17	)	)	PUNCT
ejpam-5153	12	18	a	a	DET
ejpam-5153	12	19	isomorphic	isomorphic	ADJ
ejpam-5153	12	20	to	to	ADP
ejpam-5153	12	21	r	r	NOUN
ejpam-5153	12	22	;	;	PUNCT
ejpam-5153	12	23	c	c	NOUN
ejpam-5153	12	24	r	r	NOUN
ejpam-5153	12	25	;	;	PUNCT
ejpam-5153	12	26	c	c	X
ejpam-5153	12	27	;	;	PUNCT
ejpam-5153	12	28	⋆c	⋆c	PROPN
ejpam-5153	12	29	r	r	X
ejpam-5153	12	30	;	;	PUNCT
ejpam-5153	12	31	c	c	X
ejpam-5153	12	32	;	;	PUNCT
ejpam-5153	12	33	⋆c	⋆c	PROPN
ejpam-5153	12	34	r	r	X
ejpam-5153	12	35	;	;	PUNCT
ejpam-5153	12	36	c	c	X
ejpam-5153	12	37	;	;	PUNCT
ejpam-5153	12	38	⋆c	⋆c	NOUN
ejpam-5153	12	39	;	;	PUNCT
ejpam-5153	12	40	l(1	l(1	PROPN
ejpam-5153	12	41	,	,	PUNCT
ejpam-5153	12	42	−1	−1	NOUN
ejpam-5153	12	43	,	,	PUNCT
ejpam-5153	12	44	γ	γ	X
ejpam-5153	12	45	,	,	PUNCT
ejpam-5153	12	46	1	1	NUM
ejpam-5153	12	47	)	)	PUNCT
ejpam-5153	12	48	we	we	PRON
ejpam-5153	12	49	show	show	VERB
ejpam-5153	12	50	as	as	ADV
ejpam-5153	12	51	well	well	ADV
ejpam-5153	12	52	as	as	ADP
ejpam-5153	12	53	(	(	PUNCT
ejpam-5153	12	54	e1	e1	NOUN
ejpam-5153	12	55	)	)	PUNCT
ejpam-5153	13	1	=	=	NOUN
ejpam-5153	13	2	⇒	⇒	NOUN
ejpam-5153	13	3	(	(	PUNCT
ejpam-5153	13	4	e2	e2	PROPN
ejpam-5153	13	5	)	)	PUNCT
ejpam-5153	13	6	⇐	⇐	ADJ
ejpam-5153	13	7	⇒	⇒	NOUN
ejpam-5153	13	8	(	(	PUNCT
ejpam-5153	13	9	e3	e3	NOUN
ejpam-5153	13	10	)	)	PUNCT
ejpam-5153	14	1	=	=	NOUN
ejpam-5153	14	2	⇒	⇒	NOUN
ejpam-5153	14	3	(	(	PUNCT
ejpam-5153	14	4	e4	e4	PROPN
ejpam-5153	14	5	)	)	PUNCT
ejpam-5153	14	6	.	.	PUNCT
ejpam-5153	15	1	we	we	PRON
ejpam-5153	15	2	finally	finally	ADV
ejpam-5153	15	3	study	study	VERB
ejpam-5153	15	4	the	the	DET
ejpam-5153	15	5	fused	fuse	VERB
ejpam-5153	15	6	four	four	NUM
ejpam-5153	15	7	-	-	PUNCT
ejpam-5153	15	8	dimensional	dimensional	ADJ
ejpam-5153	15	9	real	real	ADJ
ejpam-5153	15	10	division	division	NOUN
ejpam-5153	15	11	algebras	algebra	VERB
ejpam-5153	15	12	satisfying	satisfy	VERB
ejpam-5153	15	13	(	(	PUNCT
ejpam-5153	15	14	ei)i∈{1,2	ei)i∈{1,2	ADJ
ejpam-5153	15	15	}	}	PUNCT
ejpam-5153	15	16	.	.	PUNCT
ejpam-5153	16	1	we	we	PRON
ejpam-5153	16	2	have	have	AUX
ejpam-5153	16	3	shown	show	VERB
ejpam-5153	16	4	that	that	SCONJ
ejpam-5153	16	5	those	those	PRON
ejpam-5153	16	6	which	which	PRON
ejpam-5153	16	7	verify	verify	VERB
ejpam-5153	16	8	(	(	PUNCT
ejpam-5153	16	9	e2	e2	PROPN
ejpam-5153	16	10	)	)	PUNCT
ejpam-5153	16	11	are	be	AUX
ejpam-5153	16	12	h	h	NOUN
ejpam-5153	16	13	,	,	PUNCT
ejpam-5153	16	14	⋆h	⋆h	PROPN
ejpam-5153	16	15	and	and	CCONJ
ejpam-5153	16	16	c	c	PROPN
ejpam-5153	16	17	⊕	⊕	PROPN
ejpam-5153	16	18	b.	b.	PROPN
ejpam-5153	16	19	and	and	CCONJ
ejpam-5153	16	20	that	that	SCONJ
ejpam-5153	16	21	h	h	NOUN
ejpam-5153	16	22	is	be	AUX
ejpam-5153	16	23	the	the	DET
ejpam-5153	16	24	only	only	ADJ
ejpam-5153	16	25	fused	fuse	VERB
ejpam-5153	16	26	algebra	algebra	NOUN
ejpam-5153	16	27	division	division	NOUN
ejpam-5153	16	28	with	with	ADP
ejpam-5153	16	29	left	left	ADJ
ejpam-5153	16	30	unit	unit	NOUN
ejpam-5153	16	31	satisfies	satisfie	NOUN
ejpam-5153	16	32	to	to	ADP
ejpam-5153	16	33	(	(	PUNCT
ejpam-5153	16	34	e1	e1	PROPN
ejpam-5153	16	35	)	)	PUNCT
ejpam-5153	16	36	.	.	PUNCT
ejpam-5153	17	1	2020	2020	NUM
ejpam-5153	17	2	mathematics	mathematic	NOUN
ejpam-5153	17	3	subject	subject	NOUN
ejpam-5153	17	4	classifications	classification	NOUN
ejpam-5153	17	5	:	:	PUNCT
ejpam-5153	17	6	17a30	17a30	NUM
ejpam-5153	17	7	,	,	PUNCT
ejpam-5153	17	8	17a35	17a35	NUM
ejpam-5153	17	9	,	,	PUNCT
ejpam-5153	17	10	17a36	17a36	NUM
ejpam-5153	17	11	key	key	ADJ
ejpam-5153	17	12	words	word	NOUN
ejpam-5153	17	13	and	and	CCONJ
ejpam-5153	17	14	phrases	phrase	NOUN
ejpam-5153	17	15	:	:	PUNCT
ejpam-5153	17	16	division	division	NOUN
ejpam-5153	17	17	algebra	algebra	NOUN
ejpam-5153	17	18	,	,	PUNCT
ejpam-5153	17	19	left	left	ADJ
ejpam-5153	17	20	unit	unit	NOUN
ejpam-5153	17	21	,	,	PUNCT
ejpam-5153	17	22	fused	fuse	VERB
ejpam-5153	17	23	algebras	algebra	NOUN
ejpam-5153	17	24	and	and	CCONJ
ejpam-5153	17	25	isomorphism	isomorphism	NOUN
ejpam-5153	17	26	of	of	ADP
ejpam-5153	17	27	algebras	algebra	NOUN
ejpam-5153	17	28	.	.	PUNCT
ejpam-5153	18	1	1	1	X
ejpam-5153	18	2	.	.	X
ejpam-5153	18	3	introduction	introduction	NOUN
ejpam-5153	18	4	the	the	DET
ejpam-5153	18	5	discovery	discovery	NOUN
ejpam-5153	18	6	of	of	ADP
ejpam-5153	18	7	the	the	DET
ejpam-5153	18	8	real	real	ADJ
ejpam-5153	18	9	algebra	algebra	NOUN
ejpam-5153	18	10	of	of	ADP
ejpam-5153	18	11	quaternions	quaternion	NOUN
ejpam-5153	18	12	h	h	NOUN
ejpam-5153	18	13	by	by	ADP
ejpam-5153	18	14	hamilton	hamilton	PROPN
ejpam-5153	18	15	in	in	ADP
ejpam-5153	18	16	1843	1843	NUM
ejpam-5153	18	17	,	,	PUNCT
ejpam-5153	18	18	caused	cause	VERB
ejpam-5153	18	19	:	:	PUNCT
ejpam-5153	18	20	on	on	ADP
ejpam-5153	18	21	the	the	DET
ejpam-5153	18	22	one	one	NUM
ejpam-5153	18	23	hand	hand	NOUN
ejpam-5153	18	24	to	to	ADP
ejpam-5153	18	25	the	the	DET
ejpam-5153	18	26	study	study	NOUN
ejpam-5153	18	27	of	of	ADP
ejpam-5153	18	28	real	real	ADJ
ejpam-5153	18	29	division	division	NOUN
ejpam-5153	18	30	algebras	algebra	NOUN
ejpam-5153	18	31	by	by	ADP
ejpam-5153	18	32	great	great	ADJ
ejpam-5153	18	33	researchers	researcher	NOUN
ejpam-5153	18	34	mathematians	mathematian	NOUN
ejpam-5153	18	35	and	and	CCONJ
ejpam-5153	18	36	physicists	physicist	NOUN
ejpam-5153	18	37	.	.	PUNCT
ejpam-5153	19	1	one	one	NUM
ejpam-5153	19	2	of	of	ADP
ejpam-5153	19	3	the	the	DET
ejpam-5153	19	4	fundamental	fundamental	ADJ
ejpam-5153	19	5	results	result	NOUN
ejpam-5153	19	6	is	be	AUX
ejpam-5153	19	7	kervair	kervair	PROPN
ejpam-5153	19	8	milnor	milnor	PROPN
ejpam-5153	19	9	bolt	bolt	PROPN
ejpam-5153	19	10	’s	’s	PART
ejpam-5153	19	11	theorem	theorem	NOUN
ejpam-5153	19	12	which	which	PRON
ejpam-5153	19	13	states	state	VERB
ejpam-5153	19	14	that	that	SCONJ
ejpam-5153	19	15	the	the	DET
ejpam-5153	19	16	dimension	dimension	NOUN
ejpam-5153	19	17	of	of	ADP
ejpam-5153	19	18	real	real	ADJ
ejpam-5153	19	19	finite	finite	ADJ
ejpam-5153	19	20	-	-	ADJ
ejpam-5153	19	21	dimensional	dimensional	ADJ
ejpam-5153	19	22	division	division	NOUN
ejpam-5153	19	23	algebra	algebra	NOUN
ejpam-5153	19	24	is	be	AUX
ejpam-5153	19	25	1	1	NUM
ejpam-5153	19	26	,	,	PUNCT
ejpam-5153	19	27	2	2	NUM
ejpam-5153	19	28	,	,	PUNCT
ejpam-5153	19	29	4	4	NUM
ejpam-5153	19	30	or	or	CCONJ
ejpam-5153	19	31	8	8	NUM
ejpam-5153	19	32	(	(	PUNCT
ejpam-5153	19	33	[	[	X
ejpam-5153	19	34	4	4	NUM
ejpam-5153	19	35	]	]	PUNCT
ejpam-5153	19	36	,	,	PUNCT
ejpam-5153	19	37	[	[	X
ejpam-5153	19	38	12	12	NUM
ejpam-5153	19	39	]	]	PUNCT
ejpam-5153	19	40	)	)	PUNCT
ejpam-5153	19	41	.	.	PUNCT
ejpam-5153	20	1	r	r	NOUN
ejpam-5153	20	2	is	be	AUX
ejpam-5153	20	3	the	the	DET
ejpam-5153	20	4	only	only	ADJ
ejpam-5153	20	5	one	one	NUM
ejpam-5153	20	6	-	-	PUNCT
ejpam-5153	20	7	dimensional	dimensional	ADJ
ejpam-5153	20	8	real	real	ADJ
ejpam-5153	20	9	division	division	NOUN
ejpam-5153	20	10	algebra	algebra	NOUN
ejpam-5153	20	11	.	.	PUNCT
ejpam-5153	21	1	in	in	ADP
ejpam-5153	21	2	two	two	NUM
ejpam-5153	21	3	-	-	PUNCT
ejpam-5153	21	4	dimensional	dimensional	ADJ
ejpam-5153	21	5	,	,	PUNCT
ejpam-5153	21	6	these	these	DET
ejpam-5153	21	7	algebras	algebra	NOUN
ejpam-5153	21	8	have	have	AUX
ejpam-5153	21	9	been	be	AUX
ejpam-5153	21	10	classified	classify	VERB
ejpam-5153	21	11	by	by	ADP
ejpam-5153	21	12	steven	steven	PROPN
ejpam-5153	21	13	c.	c.	PROPN
ejpam-5153	21	14	althoen	althoen	PROPN
ejpam-5153	21	15	and	and	CCONJ
ejpam-5153	21	16	lawrence	lawrence	PROPN
ejpam-5153	21	17	d.	d.	PROPN
ejpam-5153	21	18	kugler	kugler	PROPN
ejpam-5153	22	1	[	[	X
ejpam-5153	22	2	3	3	NUM
ejpam-5153	22	3	]	]	PUNCT
ejpam-5153	22	4	,	,	PUNCT
ejpam-5153	22	5	reviewed	review	VERB
ejpam-5153	22	6	∗corresponding	∗corresponde	VERB
ejpam-5153	22	7	author	author	NOUN
ejpam-5153	22	8	.	.	PUNCT
ejpam-5153	23	1	doi	doi	NOUN
ejpam-5153	23	2	:	:	PUNCT
ejpam-5153	23	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5153	https://doi.org/10.29020/nybg.ejpam.v17i3.5153	ADJ
ejpam-5153	23	4	email	email	NOUN
ejpam-5153	23	5	addresses	address	NOUN
ejpam-5153	23	6	:	:	PUNCT
ejpam-5153	23	7	andre.diabang@univ-thies.sn	andre.diabang@univ-thies.sn	PROPN
ejpam-5153	23	8	(	(	PUNCT
ejpam-5153	23	9	a.	a.	PROPN
ejpam-5153	23	10	s.	s.	PROPN
ejpam-5153	23	11	diabang	diabang	PROPN
ejpam-5153	23	12	)	)	PUNCT
ejpam-5153	23	13	,	,	PUNCT
ejpam-5153	23	14	amasekou.mballo@univ-thies.sn	amasekou.mballo@univ-thies.sn	PROPN
ejpam-5153	23	15	(	(	PUNCT
ejpam-5153	23	16	a.	a.	PROPN
ejpam-5153	23	17	s.	s.	PROPN
ejpam-5153	23	18	mballo	mballo	PROPN
ejpam-5153	23	19	)	)	PUNCT
ejpam-5153	23	20	,	,	PUNCT
ejpam-5153	23	21	cheikh.diop@univ-thies.sn	cheikh.diop@univ-thies.sn	PROPN
ejpam-5153	23	22	(	(	PUNCT
ejpam-5153	23	23	p.	p.	NOUN
ejpam-5153	23	24	c.	c.	PROPN
ejpam-5153	23	25	diop	diop	PROPN
ejpam-5153	23	26	)	)	PUNCT
ejpam-5153	23	27	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5153	23	28	2276	2276	NUM
ejpam-5153	23	29	©	©	ADP
ejpam-5153	23	30	2024	2024	NUM
ejpam-5153	23	31	ejpam	ejpam	NOUN
ejpam-5153	23	32	all	all	DET
ejpam-5153	23	33	rights	right	NOUN
ejpam-5153	23	34	reserved	reserve	VERB
ejpam-5153	23	35	.	.	PUNCT
ejpam-5153	24	1	a.	a.	PROPN
ejpam-5153	24	2	s.	s.	PROPN
ejpam-5153	24	3	diabang	diabang	PROPN
ejpam-5153	24	4	,	,	PUNCT
ejpam-5153	24	5	a.	a.	PROPN
ejpam-5153	24	6	s.	s.	PROPN
ejpam-5153	24	7	mballo	mballo	PROPN
ejpam-5153	24	8	,	,	PUNCT
ejpam-5153	24	9	p.	p.	PROPN
ejpam-5153	24	10	c.	c.	PROPN
ejpam-5153	24	11	diop	diop	PROPN
ejpam-5153	24	12	/	/	SYM
ejpam-5153	24	13	eur	eur	PROPN
ejpam-5153	24	14	.	.	PUNCT
ejpam-5153	25	1	j.	j.	PROPN
ejpam-5153	25	2	pure	pure	PROPN
ejpam-5153	25	3	appl	appl	PROPN
ejpam-5153	25	4	.	.	PROPN
ejpam-5153	25	5	math	math	PROPN
ejpam-5153	25	6	,	,	PUNCT
ejpam-5153	25	7	17	17	NUM
ejpam-5153	25	8	(	(	PUNCT
ejpam-5153	25	9	3	3	NUM
ejpam-5153	25	10	)	)	PUNCT
ejpam-5153	25	11	(	(	PUNCT
ejpam-5153	25	12	2024	2024	NUM
ejpam-5153	25	13	)	)	PUNCT
ejpam-5153	25	14	,	,	PUNCT
ejpam-5153	25	15	2276	2276	NUM
ejpam-5153	25	16	-	-	SYM
ejpam-5153	25	17	2287	2287	NUM
ejpam-5153	25	18	2277	2277	NUM
ejpam-5153	25	19	by	by	ADP
ejpam-5153	25	20	marion	marion	NOUN
ejpam-5153	25	21	hübner	hübner	NOUN
ejpam-5153	25	22	and	and	CCONJ
ejpam-5153	25	23	holger	holger	NOUN
ejpam-5153	25	24	p.	p.	PROPN
ejpam-5153	25	25	petersson	petersson	PROPN
ejpam-5153	26	1	[	[	X
ejpam-5153	26	2	11	11	NUM
ejpam-5153	26	3	]	]	PUNCT
ejpam-5153	26	4	.	.	PUNCT
ejpam-5153	27	1	ana	ana	PROPN
ejpam-5153	27	2	lucia	lucia	PROPN
ejpam-5153	27	3	cali	cali	PROPN
ejpam-5153	27	4	and	and	CCONJ
ejpam-5153	27	5	michael	michael	PROPN
ejpam-5153	27	6	josephy	josephy	PROPN
ejpam-5153	27	7	classified	classify	VERB
ejpam-5153	27	8	the	the	DET
ejpam-5153	27	9	two	two	NUM
ejpam-5153	27	10	-	-	PUNCT
ejpam-5153	27	11	dimensional	dimensional	ADJ
ejpam-5153	27	12	real	real	ADJ
ejpam-5153	27	13	division	division	NOUN
ejpam-5153	27	14	algebras	algebra	VERB
ejpam-5153	27	15	with	with	ADP
ejpam-5153	27	16	left	left	ADJ
ejpam-5153	27	17	unit	unit	NOUN
ejpam-5153	27	18	[	[	X
ejpam-5153	27	19	7	7	NUM
ejpam-5153	27	20	]	]	PUNCT
ejpam-5153	27	21	.	.	PUNCT
ejpam-5153	28	1	the	the	DET
ejpam-5153	28	2	classification	classification	NOUN
ejpam-5153	28	3	problem	problem	NOUN
ejpam-5153	28	4	of	of	ADP
ejpam-5153	28	5	finite	finite	ADJ
ejpam-5153	28	6	-	-	ADJ
ejpam-5153	28	7	dimensional	dimensional	ADJ
ejpam-5153	28	8	real	real	ADJ
ejpam-5153	28	9	division	division	NOUN
ejpam-5153	28	10	algebras	algebra	NOUN
ejpam-5153	28	11	opened	open	VERB
ejpam-5153	28	12	in	in	ADP
ejpam-5153	28	13	dimension	dimension	NOUN
ejpam-5153	28	14	four	four	NUM
ejpam-5153	28	15	and	and	CCONJ
ejpam-5153	28	16	eight	eight	NUM
ejpam-5153	28	17	.	.	PUNCT
ejpam-5153	29	1	[	[	X
ejpam-5153	29	2	1	1	X
ejpam-5153	29	3	]	]	PUNCT
ejpam-5153	29	4	have	have	AUX
ejpam-5153	29	5	studied	study	VERB
ejpam-5153	29	6	real	real	ADJ
ejpam-5153	29	7	division	division	NOUN
ejpam-5153	29	8	algebras	algebra	NOUN
ejpam-5153	29	9	satisfying	satisfy	VERB
ejpam-5153	29	10	some	some	DET
ejpam-5153	29	11	identities	identity	NOUN
ejpam-5153	29	12	.	.	PUNCT
ejpam-5153	30	1	[	[	X
ejpam-5153	30	2	13	13	NUM
ejpam-5153	30	3	]	]	PUNCT
ejpam-5153	30	4	have	have	AUX
ejpam-5153	30	5	studied	study	VERB
ejpam-5153	30	6	the	the	DET
ejpam-5153	30	7	commuting	commuting	NOUN
ejpam-5153	30	8	maps	map	NOUN
ejpam-5153	30	9	and	and	CCONJ
ejpam-5153	30	10	identities	identity	NOUN
ejpam-5153	30	11	with	with	ADP
ejpam-5153	30	12	inverses	inverse	NOUN
ejpam-5153	30	13	on	on	ADP
ejpam-5153	30	14	alternative	alternative	ADJ
ejpam-5153	30	15	division	division	NOUN
ejpam-5153	30	16	rings	ring	NOUN
ejpam-5153	30	17	and	and	CCONJ
ejpam-5153	30	18	[	[	X
ejpam-5153	30	19	5	5	NUM
ejpam-5153	30	20	]	]	PUNCT
ejpam-5153	30	21	studied	study	VERB
ejpam-5153	30	22	some	some	DET
ejpam-5153	30	23	results	result	NOUN
ejpam-5153	30	24	in	in	ADP
ejpam-5153	30	25	the	the	DET
ejpam-5153	30	26	theory	theory	NOUN
ejpam-5153	30	27	of	of	ADP
ejpam-5153	30	28	linear	linear	PROPN
ejpam-5153	30	29	non	non	ADJ
ejpam-5153	30	30	-	-	ADJ
ejpam-5153	30	31	associative	associative	ADJ
ejpam-5153	30	32	algebras	algebra	NOUN
ejpam-5153	30	33	.	.	PUNCT
ejpam-5153	31	1	on	on	ADP
ejpam-5153	31	2	the	the	DET
ejpam-5153	31	3	other	other	ADJ
ejpam-5153	31	4	hand	hand	NOUN
ejpam-5153	31	5	,	,	PUNCT
ejpam-5153	31	6	finite	finite	ADJ
ejpam-5153	31	7	-	-	ADJ
ejpam-5153	31	8	dimensional	dimensional	ADJ
ejpam-5153	31	9	absolute	absolute	ADJ
ejpam-5153	31	10	valued	value	VERB
ejpam-5153	31	11	algebras	algebra	NOUN
ejpam-5153	31	12	were	be	AUX
ejpam-5153	31	13	classified	classify	VERB
ejpam-5153	31	14	by	by	ADP
ejpam-5153	31	15	a.	a.	PROPN
ejpam-5153	31	16	calderon	calderon	PROPN
ejpam-5153	31	17	,	,	PUNCT
ejpam-5153	31	18	a.	a.	NOUN
ejpam-5153	31	19	kaidi	kaidi	PROPN
ejpam-5153	31	20	,	,	PUNCT
ejpam-5153	31	21	c.	c.	PROPN
ejpam-5153	31	22	martin	martin	PROPN
ejpam-5153	31	23	,	,	PUNCT
ejpam-5153	31	24	a.	a.	NOUN
ejpam-5153	31	25	morales	morales	PROPN
ejpam-5153	31	26	,	,	PUNCT
ejpam-5153	31	27	m.	m.	NOUN
ejpam-5153	31	28	ramirez	ramirez	PROPN
ejpam-5153	31	29	,	,	PUNCT
ejpam-5153	31	30	and	and	CCONJ
ejpam-5153	31	31	a.	a.	NOUN
ejpam-5153	31	32	rochdi	rochdi	NOUN
ejpam-5153	32	1	[	[	X
ejpam-5153	32	2	6	6	NUM
ejpam-5153	32	3	]	]	PUNCT
ejpam-5153	32	4	.	.	PUNCT
ejpam-5153	33	1	a.	a.	NOUN
ejpam-5153	33	2	chandid	chandid	PROPN
ejpam-5153	33	3	and	and	CCONJ
ejpam-5153	33	4	a.	a.	NOUN
ejpam-5153	33	5	rochdi	rochdi	NOUN
ejpam-5153	33	6	have	have	AUX
ejpam-5153	33	7	studied	study	VERB
ejpam-5153	33	8	absolute	absolute	ADJ
ejpam-5153	33	9	valued	value	VERB
ejpam-5153	33	10	algebras	algebra	NOUN
ejpam-5153	33	11	satisfies	satisfie	NOUN
ejpam-5153	33	12	(	(	PUNCT
ejpam-5153	33	13	xi	xi	PROPN
ejpam-5153	33	14	,	,	PUNCT
ejpam-5153	33	15	xj	xj	PROPN
ejpam-5153	33	16	,	,	PUNCT
ejpam-5153	33	17	xk	xk	PROPN
ejpam-5153	33	18	)	)	PUNCT
ejpam-5153	33	19	=	=	SYM
ejpam-5153	33	20	0	0	PUNCT
ejpam-5153	34	1	[	[	X
ejpam-5153	34	2	8	8	NUM
ejpam-5153	34	3	]	]	PUNCT
ejpam-5153	34	4	.	.	PUNCT
ejpam-5153	34	5	o.	o.	PROPN
ejpam-5153	34	6	diankha	diankha	PROPN
ejpam-5153	34	7	,	,	PUNCT
ejpam-5153	34	8	a.	a.	NOUN
ejpam-5153	34	9	diouf	diouf	PROPN
ejpam-5153	34	10	,	,	PUNCT
ejpam-5153	34	11	m.i	m.i	PROPN
ejpam-5153	34	12	.	.	PROPN
ejpam-5153	34	13	ramirez	ramirez	PROPN
ejpam-5153	34	14	and	and	CCONJ
ejpam-5153	34	15	a.	a.	NOUN
ejpam-5153	34	16	rochdi	rochdi	NOUN
ejpam-5153	34	17	have	have	AUX
ejpam-5153	34	18	studied	study	VERB
ejpam-5153	34	19	the	the	DET
ejpam-5153	34	20	absolute	absolute	ADJ
ejpam-5153	34	21	valued	value	VERB
ejpam-5153	34	22	algebras	algebra	NOUN
ejpam-5153	34	23	with	with	ADP
ejpam-5153	34	24	one	one	NUM
ejpam-5153	34	25	sided	sided	ADJ
ejpam-5153	34	26	unit	unit	NOUN
ejpam-5153	34	27	satisfying	satisfy	VERB
ejpam-5153	34	28	(	(	PUNCT
ejpam-5153	34	29	x2	x2	PROPN
ejpam-5153	34	30	,	,	PUNCT
ejpam-5153	34	31	x2	x2	PROPN
ejpam-5153	34	32	,	,	PUNCT
ejpam-5153	34	33	x2	x2	PROPN
ejpam-5153	34	34	)	)	PUNCT
ejpam-5153	34	35	=	=	SYM
ejpam-5153	34	36	0	0	PUNCT
ejpam-5153	35	1	[	[	X
ejpam-5153	35	2	9	9	NUM
ejpam-5153	35	3	]	]	PUNCT
ejpam-5153	35	4	.	.	PUNCT
ejpam-5153	36	1	the	the	DET
ejpam-5153	36	2	latter	latter	ADJ
ejpam-5153	36	3	have	have	AUX
ejpam-5153	36	4	shown	show	VERB
ejpam-5153	36	5	in	in	ADP
ejpam-5153	36	6	absolute	absolute	ADJ
ejpam-5153	36	7	valued	value	VERB
ejpam-5153	36	8	algebras	algebra	NOUN
ejpam-5153	36	9	having	have	VERB
ejpam-5153	36	10	a	a	DET
ejpam-5153	36	11	left	left	ADJ
ejpam-5153	36	12	unit	unit	NOUN
ejpam-5153	36	13	element	element	NOUN
ejpam-5153	36	14	of	of	ADP
ejpam-5153	36	15	finite	finite	PROPN
ejpam-5153	36	16	dimensional	dimensional	ADJ
ejpam-5153	36	17	that	that	SCONJ
ejpam-5153	36	18	(	(	PUNCT
ejpam-5153	36	19	e1	e1	NOUN
ejpam-5153	36	20	)	)	PUNCT
ejpam-5153	36	21	=	=	NOUN
ejpam-5153	36	22	⇒	⇒	NOUN
ejpam-5153	36	23	(	(	PUNCT
ejpam-5153	36	24	e2	e2	PROPN
ejpam-5153	36	25	)	)	PUNCT
ejpam-5153	36	26	⇐	⇐	ADJ
ejpam-5153	36	27	⇒	⇒	NOUN
ejpam-5153	36	28	(	(	PUNCT
ejpam-5153	36	29	e3	e3	NOUN
ejpam-5153	36	30	)	)	PUNCT
ejpam-5153	36	31	=	=	NOUN
ejpam-5153	36	32	⇒	⇒	NOUN
ejpam-5153	36	33	(	(	PUNCT
ejpam-5153	36	34	e4	e4	PROPN
ejpam-5153	36	35	)	)	PUNCT
ejpam-5153	36	36	.	.	PUNCT
ejpam-5153	37	1	it	it	PRON
ejpam-5153	37	2	is	be	AUX
ejpam-5153	37	3	with	with	ADP
ejpam-5153	37	4	this	this	PRON
ejpam-5153	37	5	in	in	ADP
ejpam-5153	37	6	mind	mind	NOUN
ejpam-5153	37	7	that	that	SCONJ
ejpam-5153	37	8	we	we	PRON
ejpam-5153	37	9	study	study	VERB
ejpam-5153	37	10	,	,	PUNCT
ejpam-5153	37	11	in	in	ADP
ejpam-5153	37	12	this	this	DET
ejpam-5153	37	13	paper	paper	NOUN
ejpam-5153	37	14	,	,	PUNCT
ejpam-5153	37	15	these	these	DET
ejpam-5153	37	16	identities	identity	NOUN
ejpam-5153	37	17	in	in	ADP
ejpam-5153	37	18	the	the	DET
ejpam-5153	37	19	case	case	NOUN
ejpam-5153	37	20	of	of	ADP
ejpam-5153	37	21	real	real	ADJ
ejpam-5153	37	22	division	division	NOUN
ejpam-5153	37	23	algebras	algebra	VERB
ejpam-5153	37	24	with	with	ADP
ejpam-5153	37	25	left	left	ADJ
ejpam-5153	37	26	unit	unit	NOUN
ejpam-5153	37	27	of	of	ADP
ejpam-5153	37	28	finite	finite	PROPN
ejpam-5153	37	29	dimensional	dimensional	ADJ
ejpam-5153	37	30	n.	n.	NOUN
ejpam-5153	37	31	we	we	PRON
ejpam-5153	37	32	obtained	obtain	VERB
ejpam-5153	37	33	the	the	DET
ejpam-5153	37	34	same	same	ADJ
ejpam-5153	37	35	result	result	NOUN
ejpam-5153	37	36	if	if	SCONJ
ejpam-5153	37	37	n	n	ADV
ejpam-5153	37	38	∈	∈	PROPN
ejpam-5153	37	39	{	{	PUNCT
ejpam-5153	37	40	1	1	NUM
ejpam-5153	37	41	,	,	PUNCT
ejpam-5153	37	42	2	2	NUM
ejpam-5153	37	43	}	}	PUNCT
ejpam-5153	37	44	.	.	PUNCT
ejpam-5153	38	1	but	but	CCONJ
ejpam-5153	38	2	if	if	SCONJ
ejpam-5153	38	3	n	n	X
ejpam-5153	38	4	∈	∈	NOUN
ejpam-5153	38	5	{	{	PUNCT
ejpam-5153	38	6	4	4	NUM
ejpam-5153	38	7	,	,	PUNCT
ejpam-5153	38	8	8	8	NUM
ejpam-5153	38	9	}	}	PUNCT
ejpam-5153	38	10	we	we	PRON
ejpam-5153	38	11	have	have	VERB
ejpam-5153	38	12	(	(	PUNCT
ejpam-5153	38	13	e1	e1	NOUN
ejpam-5153	38	14	)	)	PUNCT
ejpam-5153	39	1	=	=	NOUN
ejpam-5153	39	2	⇒	⇒	NOUN
ejpam-5153	39	3	(	(	PUNCT
ejpam-5153	39	4	e2	e2	PROPN
ejpam-5153	39	5	)	)	PUNCT
ejpam-5153	40	1	=	=	VERB
ejpam-5153	40	2	⇒	⇒	NOUN
ejpam-5153	40	3	(	(	PUNCT
ejpam-5153	40	4	e3	e3	NOUN
ejpam-5153	40	5	)	)	PUNCT
ejpam-5153	41	1	=	=	NOUN
ejpam-5153	41	2	⇒	⇒	NOUN
ejpam-5153	41	3	(	(	PUNCT
ejpam-5153	41	4	e4	e4	PROPN
ejpam-5153	41	5	)	)	PUNCT
ejpam-5153	41	6	.	.	PUNCT
ejpam-5153	42	1	we	we	PRON
ejpam-5153	42	2	study	study	VERB
ejpam-5153	42	3	fused	fuse	VERB
ejpam-5153	42	4	-	-	PUNCT
ejpam-5153	42	5	algebras	algebras	ADJ
ejpam-5153	42	6	division	division	NOUN
ejpam-5153	42	7	with	with	ADP
ejpam-5153	42	8	left	left	ADJ
ejpam-5153	42	9	unit	unit	NOUN
ejpam-5153	42	10	satisfying	satisfy	VERB
ejpam-5153	42	11	to	to	ADP
ejpam-5153	42	12	(	(	PUNCT
ejpam-5153	42	13	ei)i∈{1,2	ei)i∈{1,2	ADJ
ejpam-5153	42	14	}	}	PUNCT
ejpam-5153	42	15	.	.	PUNCT
ejpam-5153	43	1	we	we	PRON
ejpam-5153	43	2	show	show	VERB
ejpam-5153	43	3	that	that	SCONJ
ejpam-5153	43	4	those	those	PRON
ejpam-5153	43	5	which	which	PRON
ejpam-5153	43	6	satisfy	satisfy	VERB
ejpam-5153	43	7	to	to	ADP
ejpam-5153	43	8	(	(	PUNCT
ejpam-5153	43	9	e2	e2	PROPN
ejpam-5153	43	10	)	)	PUNCT
ejpam-5153	43	11	are	be	AUX
ejpam-5153	43	12	h	h	NOUN
ejpam-5153	43	13	,	,	PUNCT
ejpam-5153	43	14	⋆h	⋆h	PUNCT
ejpam-5153	43	15	and	and	CCONJ
ejpam-5153	43	16	c⊕b	c⊕b	NOUN
ejpam-5153	43	17	with	with	ADP
ejpam-5153	43	18	b	b	NOUN
ejpam-5153	43	19	=	=	SYM
ejpam-5153	43	20	(	(	PUNCT
ejpam-5153	43	21	r2	r2	PROPN
ejpam-5153	43	22	,	,	PUNCT
ejpam-5153	43	23	•	•	NUM
ejpam-5153	43	24	)	)	PUNCT
ejpam-5153	43	25	is	be	AUX
ejpam-5153	43	26	the	the	DET
ejpam-5153	43	27	real	real	ADJ
ejpam-5153	43	28	algebra	algebra	NOUN
ejpam-5153	43	29	whose	whose	DET
ejpam-5153	43	30	multiplication	multiplication	NOUN
ejpam-5153	43	31	table	table	NOUN
ejpam-5153	43	32	in	in	ADP
ejpam-5153	43	33	the	the	DET
ejpam-5153	43	34	basis	basis	NOUN
ejpam-5153	43	35	β	β	X
ejpam-5153	43	36	=	=	PUNCT
ejpam-5153	43	37	{	{	PUNCT
ejpam-5153	43	38	u	u	NOUN
ejpam-5153	43	39	,	,	PUNCT
ejpam-5153	43	40	v	v	NOUN
ejpam-5153	43	41	}	}	PUNCT
ejpam-5153	43	42	is	be	AUX
ejpam-5153	43	43	given	give	VERB
ejpam-5153	43	44	by	by	ADP
ejpam-5153	43	45	(	(	PUNCT
ejpam-5153	43	46	b	b	NOUN
ejpam-5153	43	47	)	)	PUNCT
ejpam-5153	43	48	•	•	ADP
ejpam-5153	43	49	u	u	NOUN
ejpam-5153	43	50	v	v	ADP
ejpam-5153	43	51	u	u	X
ejpam-5153	43	52	u	u	NOUN
ejpam-5153	43	53	−v	−v	NOUN
ejpam-5153	43	54	v	v	ADP
ejpam-5153	43	55	c21u	c21u	NOUN
ejpam-5153	43	56	−	−	PROPN
ejpam-5153	43	57	v	v	NOUN
ejpam-5153	43	58	c22u	c22u	PUNCT
ejpam-5153	43	59	where	where	SCONJ
ejpam-5153	43	60	c21	c21	NOUN
ejpam-5153	43	61	,	,	PUNCT
ejpam-5153	43	62	c22	c22	PROPN
ejpam-5153	43	63	are	be	AUX
ejpam-5153	43	64	real	real	ADJ
ejpam-5153	43	65	nombers	nomber	NOUN
ejpam-5153	43	66	thus	thus	ADV
ejpam-5153	43	67	that	that	PRON
ejpam-5153	43	68	c2	c2	PROPN
ejpam-5153	43	69	12	12	NUM
ejpam-5153	43	70	<	<	X
ejpam-5153	43	71	−4c22	−4c22	PROPN
ejpam-5153	43	72	.	.	PUNCT
ejpam-5153	44	1	we	we	PRON
ejpam-5153	44	2	prove	prove	VERB
ejpam-5153	44	3	that	that	SCONJ
ejpam-5153	44	4	h	h	NOUN
ejpam-5153	44	5	is	be	AUX
ejpam-5153	44	6	the	the	DET
ejpam-5153	44	7	unique	unique	ADJ
ejpam-5153	44	8	fused	fuse	VERB
ejpam-5153	44	9	algebra	algebra	NOUN
ejpam-5153	44	10	division	division	NOUN
ejpam-5153	44	11	with	with	ADP
ejpam-5153	44	12	left	left	ADJ
ejpam-5153	44	13	unit	unit	NOUN
ejpam-5153	44	14	satisfies	satisfie	NOUN
ejpam-5153	44	15	to	to	ADP
ejpam-5153	44	16	(	(	PUNCT
ejpam-5153	44	17	e1	e1	PROPN
ejpam-5153	44	18	)	)	PUNCT
ejpam-5153	44	19	.	.	PUNCT
ejpam-5153	45	1	2	2	X
ejpam-5153	45	2	.	.	X
ejpam-5153	45	3	notes	note	NOUN
ejpam-5153	45	4	and	and	CCONJ
ejpam-5153	45	5	preliminary	preliminary	ADJ
ejpam-5153	45	6	results	result	NOUN
ejpam-5153	45	7	let	let	VERB
ejpam-5153	45	8	a	a	PRON
ejpam-5153	45	9	be	be	AUX
ejpam-5153	45	10	an	an	DET
ejpam-5153	45	11	arbitrary	arbitrary	ADJ
ejpam-5153	45	12	real	real	ADJ
ejpam-5153	45	13	algebra	algebra	NOUN
ejpam-5153	45	14	.	.	PUNCT
ejpam-5153	46	1	let	let	VERB
ejpam-5153	46	2	x	x	PRON
ejpam-5153	46	3	,	,	PUNCT
ejpam-5153	46	4	y	y	PROPN
ejpam-5153	46	5	e	e	PROPN
ejpam-5153	46	6	∈	∈	PROPN
ejpam-5153	46	7	a	a	X
ejpam-5153	46	8	,	,	PUNCT
ejpam-5153	46	9	we	we	PRON
ejpam-5153	46	10	define	define	VERB
ejpam-5153	46	11	:	:	PUNCT
ejpam-5153	47	1	[	[	X
ejpam-5153	47	2	x	x	X
ejpam-5153	47	3	,	,	PUNCT
ejpam-5153	47	4	y	y	PROPN
ejpam-5153	47	5	]	]	X
ejpam-5153	47	6	=	=	PUNCT
ejpam-5153	47	7	xy	xy	PROPN
ejpam-5153	47	8	−	−	PROPN
ejpam-5153	47	9	yx	yx	PROPN
ejpam-5153	47	10	,	,	PUNCT
ejpam-5153	47	11	(	(	PUNCT
ejpam-5153	47	12	x	x	X
ejpam-5153	47	13	,	,	PUNCT
ejpam-5153	47	14	y	y	PROPN
ejpam-5153	47	15	,	,	PUNCT
ejpam-5153	47	16	z	z	NOUN
ejpam-5153	47	17	)	)	PUNCT
ejpam-5153	47	18	=	=	SYM
ejpam-5153	47	19	(	(	PUNCT
ejpam-5153	47	20	xy)z	xy)z	PROPN
ejpam-5153	47	21	−	−	NOUN
ejpam-5153	47	22	x(yz	x(yz	PROPN
ejpam-5153	47	23	)	)	PUNCT
ejpam-5153	47	24	,	,	PUNCT
ejpam-5153	47	25	and	and	CCONJ
ejpam-5153	47	26	ida	ida	PROPN
ejpam-5153	47	27	:	:	PUNCT
ejpam-5153	47	28	a	a	DET
ejpam-5153	47	29	−→	−→	NOUN
ejpam-5153	47	30	a	a	DET
ejpam-5153	47	31	the	the	DET
ejpam-5153	47	32	identity	identity	NOUN
ejpam-5153	47	33	application	application	NOUN
ejpam-5153	47	34	of	of	ADP
ejpam-5153	47	35	a.	a.	NOUN
ejpam-5153	47	36	we	we	PRON
ejpam-5153	47	37	define	define	VERB
ejpam-5153	47	38	:	:	PUNCT
ejpam-5153	47	39	⋆a	⋆a	PROPN
ejpam-5153	47	40	:	:	PUNCT
ejpam-5153	47	41	=	=	SYM
ejpam-5153	47	42	(	(	PUNCT
ejpam-5153	47	43	a	a	PRON
ejpam-5153	47	44	,	,	PUNCT
ejpam-5153	47	45	⊙	⊙	PROPN
ejpam-5153	47	46	)	)	PUNCT
ejpam-5153	47	47	the	the	DET
ejpam-5153	47	48	real	real	ADJ
ejpam-5153	47	49	algebra	algebra	NOUN
ejpam-5153	47	50	whose	whose	DET
ejpam-5153	47	51	vector	vector	NOUN
ejpam-5153	47	52	space	space	NOUN
ejpam-5153	47	53	is	be	AUX
ejpam-5153	47	54	the	the	DET
ejpam-5153	47	55	set	set	NOUN
ejpam-5153	47	56	a	a	PRON
ejpam-5153	47	57	and	and	CCONJ
ejpam-5153	47	58	the	the	DET
ejpam-5153	47	59	product	product	NOUN
ejpam-5153	47	60	⊙	⊙	NOUN
ejpam-5153	47	61	is	be	AUX
ejpam-5153	47	62	defined	define	VERB
ejpam-5153	47	63	by	by	ADP
ejpam-5153	47	64	x	x	PROPN
ejpam-5153	47	65	⊙	⊙	PROPN
ejpam-5153	47	66	y	y	PROPN
ejpam-5153	47	67	=	=	PUNCT
ejpam-5153	47	68	xy	xy	PROPN
ejpam-5153	47	69	∀x	∀x	X
ejpam-5153	47	70	,	,	PUNCT
ejpam-5153	47	71	y	y	PROPN
ejpam-5153	47	72	∈	∈	PROPN
ejpam-5153	47	73	a	a	PRON
ejpam-5153	47	74	with	with	ADP
ejpam-5153	47	75	a	a	DET
ejpam-5153	47	76	equal	equal	ADJ
ejpam-5153	47	77	to	to	ADP
ejpam-5153	47	78	either	either	CCONJ
ejpam-5153	47	79	c	c	PROPN
ejpam-5153	47	80	or	or	CCONJ
ejpam-5153	47	81	h	h	NOUN
ejpam-5153	47	82	,	,	PUNCT
ejpam-5153	47	83	and	and	CCONJ
ejpam-5153	47	84	x	x	SYM
ejpam-5153	47	85	7→	7→	NUM
ejpam-5153	47	86	x	x	PUNCT
ejpam-5153	47	87	mains	main	VERB
ejpam-5153	47	88	standard	standard	ADJ
ejpam-5153	47	89	involution	involution	NOUN
ejpam-5153	47	90	.	.	PUNCT
ejpam-5153	48	1	a(x	a(x	NOUN
ejpam-5153	48	2	)	)	PUNCT
ejpam-5153	48	3	is	be	AUX
ejpam-5153	48	4	a	a	DET
ejpam-5153	48	5	subalgebra	subalgebra	NOUN
ejpam-5153	48	6	of	of	ADP
ejpam-5153	48	7	a	a	DET
ejpam-5153	48	8	generated	generate	VERB
ejpam-5153	48	9	by	by	ADP
ejpam-5153	48	10	x.	x.	PROPN
ejpam-5153	48	11	a	a	PRON
ejpam-5153	48	12	is	be	AUX
ejpam-5153	48	13	said	say	VERB
ejpam-5153	48	14	:	:	PUNCT
ejpam-5153	48	15	•	•	NUM
ejpam-5153	48	16	division	division	NOUN
ejpam-5153	48	17	if	if	SCONJ
ejpam-5153	48	18	the	the	DET
ejpam-5153	48	19	operateros	operatero	NOUN
ejpam-5153	48	20	lx	lx	NOUN
ejpam-5153	48	21	:	:	PUNCT
ejpam-5153	48	22	a	a	DET
ejpam-5153	48	23	→	→	SYM
ejpam-5153	48	24	a	a	X
ejpam-5153	48	25	,	,	PUNCT
ejpam-5153	48	26	y	y	PROPN
ejpam-5153	48	27	7→	7→	PROPN
ejpam-5153	48	28	xy	xy	PROPN
ejpam-5153	48	29	and	and	CCONJ
ejpam-5153	48	30	rx	rx	VERB
ejpam-5153	48	31	:	:	PUNCT
ejpam-5153	48	32	a	a	DET
ejpam-5153	48	33	→	→	SYM
ejpam-5153	48	34	a	a	NOUN
ejpam-5153	48	35	,	,	PUNCT
ejpam-5153	48	36	y	y	PROPN
ejpam-5153	48	37	7→	7→	PROPN
ejpam-5153	48	38	yx	yx	NOUN
ejpam-5153	48	39	are	be	AUX
ejpam-5153	48	40	bijective	bijective	ADJ
ejpam-5153	48	41	,	,	PUNCT
ejpam-5153	48	42	for	for	ADP
ejpam-5153	48	43	all	all	DET
ejpam-5153	48	44	x	x	SYM
ejpam-5153	48	45	∈	∈	PROPN
ejpam-5153	48	46	a	a	PRON
ejpam-5153	48	47	,	,	PUNCT
ejpam-5153	48	48	x	x	SYM
ejpam-5153	48	49	̸=	̸=	PROPN
ejpam-5153	48	50	0	0	NUM
ejpam-5153	48	51	.	.	NOUN
ejpam-5153	48	52	•	•	NUM
ejpam-5153	48	53	at	at	ADP
ejpam-5153	48	54	third	third	ADJ
ejpam-5153	48	55	power	power	NOUN
ejpam-5153	48	56	-	-	PUNCT
ejpam-5153	48	57	associative	associative	NOUN
ejpam-5153	48	58	if	if	SCONJ
ejpam-5153	48	59	(	(	PUNCT
ejpam-5153	48	60	x	x	NOUN
ejpam-5153	48	61	,	,	PUNCT
ejpam-5153	48	62	x	x	X
ejpam-5153	48	63	,	,	PUNCT
ejpam-5153	48	64	x	x	X
ejpam-5153	48	65	)	)	PUNCT
ejpam-5153	48	66	=	=	SYM
ejpam-5153	48	67	0	0	NUM
ejpam-5153	48	68	for	for	ADP
ejpam-5153	48	69	all	all	DET
ejpam-5153	48	70	x	x	SYM
ejpam-5153	48	71	∈	∈	PROPN
ejpam-5153	48	72	a	a	PRON
ejpam-5153	48	73	,	,	PUNCT
ejpam-5153	48	74	.	.	PUNCT
ejpam-5153	49	1	•	•	X
ejpam-5153	49	2	at	at	ADP
ejpam-5153	49	3	(	(	PUNCT
ejpam-5153	49	4	222	222	NUM
ejpam-5153	49	5	)	)	PUNCT
ejpam-5153	49	6	power	power	NOUN
ejpam-5153	49	7	-	-	PUNCT
ejpam-5153	49	8	associative	associative	NOUN
ejpam-5153	49	9	if	if	SCONJ
ejpam-5153	49	10	(	(	PUNCT
ejpam-5153	49	11	x2	x2	PROPN
ejpam-5153	49	12	,	,	PUNCT
ejpam-5153	49	13	x2	x2	PROPN
ejpam-5153	49	14	,	,	PUNCT
ejpam-5153	49	15	x2	x2	PROPN
ejpam-5153	49	16	)	)	PUNCT
ejpam-5153	49	17	=	=	SYM
ejpam-5153	49	18	0	0	NUM
ejpam-5153	49	19	for	for	ADP
ejpam-5153	49	20	all	all	DET
ejpam-5153	49	21	x	x	SYM
ejpam-5153	49	22	∈	∈	NOUN
ejpam-5153	49	23	a.	a.	NOUN
ejpam-5153	49	24	•	•	NOUN
ejpam-5153	49	25	at	at	ADP
ejpam-5153	49	26	(	(	PUNCT
ejpam-5153	49	27	121	121	NUM
ejpam-5153	49	28	)	)	PUNCT
ejpam-5153	49	29	power	power	NOUN
ejpam-5153	49	30	-	-	PUNCT
ejpam-5153	49	31	associative	associative	NOUN
ejpam-5153	49	32	if	if	SCONJ
ejpam-5153	49	33	(	(	PUNCT
ejpam-5153	49	34	x	x	NOUN
ejpam-5153	49	35	,	,	PUNCT
ejpam-5153	49	36	x2	x2	PROPN
ejpam-5153	49	37	,	,	PUNCT
ejpam-5153	49	38	x	x	X
ejpam-5153	49	39	)	)	PUNCT
ejpam-5153	49	40	=	=	SYM
ejpam-5153	49	41	0	0	NUM
ejpam-5153	49	42	for	for	ADP
ejpam-5153	49	43	all	all	DET
ejpam-5153	49	44	x	x	SYM
ejpam-5153	49	45	∈	∈	NOUN
ejpam-5153	49	46	a.	a.	NOUN
ejpam-5153	49	47	•	•	NOUN
ejpam-5153	49	48	at	at	ADP
ejpam-5153	49	49	power	power	NOUN
ejpam-5153	49	50	-	-	PUNCT
ejpam-5153	49	51	commutative	commutative	ADJ
ejpam-5153	49	52	if	if	SCONJ
ejpam-5153	49	53	any	any	DET
ejpam-5153	49	54	subalgebra	subalgebra	NOUN
ejpam-5153	49	55	generated	generate	VERB
ejpam-5153	49	56	by	by	ADP
ejpam-5153	49	57	a	a	DET
ejpam-5153	49	58	single	single	ADJ
ejpam-5153	49	59	element	element	NOUN
ejpam-5153	49	60	is	be	AUX
ejpam-5153	49	61	commutative	commutative	ADJ
ejpam-5153	49	62	.	.	PUNCT
ejpam-5153	50	1	we	we	PRON
ejpam-5153	50	2	define	define	VERB
ejpam-5153	50	3	the	the	DET
ejpam-5153	50	4	identities	identity	NOUN
ejpam-5153	50	5	:	:	PUNCT
ejpam-5153	50	6	a.	a.	PROPN
ejpam-5153	50	7	s.	s.	PROPN
ejpam-5153	50	8	diabang	diabang	PROPN
ejpam-5153	50	9	,	,	PUNCT
ejpam-5153	50	10	a.	a.	PROPN
ejpam-5153	50	11	s.	s.	PROPN
ejpam-5153	50	12	mballo	mballo	PROPN
ejpam-5153	50	13	,	,	PUNCT
ejpam-5153	50	14	p.	p.	PROPN
ejpam-5153	50	15	c.	c.	PROPN
ejpam-5153	50	16	diop	diop	PROPN
ejpam-5153	50	17	/	/	SYM
ejpam-5153	50	18	eur	eur	PROPN
ejpam-5153	50	19	.	.	PUNCT
ejpam-5153	51	1	j.	j.	PROPN
ejpam-5153	51	2	pure	pure	PROPN
ejpam-5153	51	3	appl	appl	PROPN
ejpam-5153	51	4	.	.	PROPN
ejpam-5153	51	5	math	math	PROPN
ejpam-5153	51	6	,	,	PUNCT
ejpam-5153	51	7	17	17	NUM
ejpam-5153	51	8	(	(	PUNCT
ejpam-5153	51	9	3	3	NUM
ejpam-5153	51	10	)	)	PUNCT
ejpam-5153	51	11	(	(	PUNCT
ejpam-5153	51	12	2024	2024	NUM
ejpam-5153	51	13	)	)	PUNCT
ejpam-5153	51	14	,	,	PUNCT
ejpam-5153	51	15	2276	2276	NUM
ejpam-5153	51	16	-	-	SYM
ejpam-5153	51	17	2287	2287	NUM
ejpam-5153	51	18	2278	2278	NUM
ejpam-5153	51	19	(	(	PUNCT
ejpam-5153	51	20	e1	e1	PROPN
ejpam-5153	51	21	)	)	PUNCT
ejpam-5153	51	22	(	(	PUNCT
ejpam-5153	51	23	x	x	X
ejpam-5153	51	24	,	,	PUNCT
ejpam-5153	51	25	x	x	X
ejpam-5153	51	26	,	,	PUNCT
ejpam-5153	51	27	x	x	X
ejpam-5153	51	28	)	)	PUNCT
ejpam-5153	51	29	=	=	SYM
ejpam-5153	51	30	0	0	NUM
ejpam-5153	51	31	,	,	PUNCT
ejpam-5153	51	32	(	(	PUNCT
ejpam-5153	51	33	e2	e2	PROPN
ejpam-5153	51	34	)	)	PUNCT
ejpam-5153	51	35	(	(	PUNCT
ejpam-5153	51	36	x2	x2	PROPN
ejpam-5153	51	37	,	,	PUNCT
ejpam-5153	51	38	x2	x2	PROPN
ejpam-5153	51	39	,	,	PUNCT
ejpam-5153	51	40	x2	x2	PROPN
ejpam-5153	51	41	)	)	PUNCT
ejpam-5153	51	42	=	=	SYM
ejpam-5153	51	43	0	0	NUM
ejpam-5153	51	44	,	,	PUNCT
ejpam-5153	51	45	(	(	PUNCT
ejpam-5153	51	46	e3	e3	NOUN
ejpam-5153	51	47	)	)	PUNCT
ejpam-5153	51	48	x2e	x2e	NOUN
ejpam-5153	52	1	=	=	SYM
ejpam-5153	52	2	x2	x2	PROPN
ejpam-5153	52	3	and	and	CCONJ
ejpam-5153	52	4	(	(	PUNCT
ejpam-5153	52	5	e4	e4	PROPN
ejpam-5153	52	6	)	)	PUNCT
ejpam-5153	52	7	(	(	PUNCT
ejpam-5153	52	8	xe)e	xe)e	PROPN
ejpam-5153	52	9	=	=	PUNCT
ejpam-5153	52	10	x.	x.	NOUN
ejpam-5153	52	11	a	a	DET
ejpam-5153	52	12	∼=	∼=	PROPN
ejpam-5153	52	13	b	b	NOUN
ejpam-5153	52	14	if	if	SCONJ
ejpam-5153	52	15	only	only	ADV
ejpam-5153	52	16	if	if	SCONJ
ejpam-5153	52	17	a	a	PRON
ejpam-5153	52	18	and	and	CCONJ
ejpam-5153	52	19	b	b	NOUN
ejpam-5153	52	20	are	be	AUX
ejpam-5153	52	21	isomorphic	isomorphic	ADJ
ejpam-5153	52	22	.	.	PUNCT
ejpam-5153	53	1	we	we	PRON
ejpam-5153	53	2	state	state	VERB
ejpam-5153	53	3	now	now	ADV
ejpam-5153	53	4	some	some	DET
ejpam-5153	53	5	preliminary	preliminary	ADJ
ejpam-5153	53	6	results	result	NOUN
ejpam-5153	53	7	:	:	PUNCT
ejpam-5153	53	8	lemma	lemma	PROPN
ejpam-5153	53	9	1	1	NUM
ejpam-5153	53	10	.	.	PUNCT
ejpam-5153	53	11	.	.	PUNCT
ejpam-5153	54	1	let	let	VERB
ejpam-5153	54	2	a	a	PRON
ejpam-5153	54	3	be	be	AUX
ejpam-5153	54	4	a	a	DET
ejpam-5153	54	5	finite	finite	ADJ
ejpam-5153	54	6	-	-	ADJ
ejpam-5153	54	7	dimensional	dimensional	ADJ
ejpam-5153	54	8	real	real	ADJ
ejpam-5153	54	9	division	division	NOUN
ejpam-5153	54	10	algebra	algebra	NOUN
ejpam-5153	54	11	with	with	ADP
ejpam-5153	54	12	left	left	ADJ
ejpam-5153	54	13	unit	unit	NOUN
ejpam-5153	54	14	e.	e.	PROPN
ejpam-5153	54	15	we	we	PRON
ejpam-5153	54	16	have	have	VERB
ejpam-5153	54	17	1	1	NUM
ejpam-5153	54	18	)	)	PUNCT
ejpam-5153	54	19	it	it	PRON
ejpam-5153	54	20	is	be	AUX
ejpam-5153	54	21	obvious	obvious	ADJ
ejpam-5153	54	22	(	(	PUNCT
ejpam-5153	54	23	e1	e1	NOUN
ejpam-5153	54	24	)	)	PUNCT
ejpam-5153	55	1	=	=	NOUN
ejpam-5153	55	2	⇒	⇒	NOUN
ejpam-5153	55	3	(	(	PUNCT
ejpam-5153	55	4	e2	e2	PROPN
ejpam-5153	55	5	)	)	PUNCT
ejpam-5153	55	6	2	2	NUM
ejpam-5153	55	7	)	)	PUNCT
ejpam-5153	56	1	[	[	X
ejpam-5153	56	2	8	8	NUM
ejpam-5153	56	3	]	]	PUNCT
ejpam-5153	56	4	have	have	AUX
ejpam-5153	56	5	shown	show	VERB
ejpam-5153	56	6	that	that	SCONJ
ejpam-5153	56	7	(	(	PUNCT
ejpam-5153	56	8	e3	e3	NOUN
ejpam-5153	56	9	)	)	PUNCT
ejpam-5153	57	1	=	=	NOUN
ejpam-5153	57	2	⇒	⇒	NOUN
ejpam-5153	57	3	(	(	PUNCT
ejpam-5153	57	4	e4	e4	PROPN
ejpam-5153	57	5	)	)	PUNCT
ejpam-5153	57	6	3	3	NUM
ejpam-5153	57	7	)	)	PUNCT
ejpam-5153	58	1	[	[	X
ejpam-5153	58	2	9	9	NUM
ejpam-5153	58	3	]	]	PUNCT
ejpam-5153	58	4	have	have	AUX
ejpam-5153	58	5	shown	show	VERB
ejpam-5153	58	6	that	that	SCONJ
ejpam-5153	58	7	(	(	PUNCT
ejpam-5153	58	8	e2	e2	PROPN
ejpam-5153	58	9	)	)	PUNCT
ejpam-5153	59	1	=	=	NOUN
ejpam-5153	59	2	⇒	⇒	NOUN
ejpam-5153	59	3	(	(	PUNCT
ejpam-5153	59	4	e4	e4	PROPN
ejpam-5153	59	5	)	)	PUNCT
ejpam-5153	59	6	lemma	lemma	PROPN
ejpam-5153	59	7	2	2	NUM
ejpam-5153	59	8	.	.	PUNCT
ejpam-5153	59	9	let	let	VERB
ejpam-5153	59	10	a	a	PRON
ejpam-5153	59	11	be	be	AUX
ejpam-5153	59	12	a	a	DET
ejpam-5153	59	13	real	real	ADJ
ejpam-5153	59	14	division	division	NOUN
ejpam-5153	59	15	algebra	algebra	NOUN
ejpam-5153	59	16	of	of	ADP
ejpam-5153	59	17	finite	finite	ADJ
ejpam-5153	59	18	dimension	dimension	NOUN
ejpam-5153	59	19	n	n	PRON
ejpam-5153	59	20	≥	≥	NOUN
ejpam-5153	59	21	2	2	NUM
ejpam-5153	59	22	with	with	ADP
ejpam-5153	59	23	left	left	ADJ
ejpam-5153	59	24	unit	unit	NOUN
ejpam-5153	59	25	element	element	PROPN
ejpam-5153	59	26	e.	e.	PROPN
ejpam-5153	59	27	let	let	VERB
ejpam-5153	59	28	x	x	SYM
ejpam-5153	59	29	∈	∈	VERB
ejpam-5153	59	30	a	a	DET
ejpam-5153	59	31	−	−	PROPN
ejpam-5153	59	32	{	{	PUNCT
ejpam-5153	59	33	0	0	NUM
ejpam-5153	59	34	}	}	PUNCT
ejpam-5153	59	35	.	.	PUNCT
ejpam-5153	60	1	the	the	DET
ejpam-5153	60	2	following	follow	VERB
ejpam-5153	60	3	propositions	proposition	NOUN
ejpam-5153	60	4	are	be	AUX
ejpam-5153	60	5	equivalent	equivalent	ADJ
ejpam-5153	60	6	:	:	PUNCT
ejpam-5153	60	7	(	(	PUNCT
ejpam-5153	60	8	1	1	X
ejpam-5153	60	9	)	)	PUNCT
ejpam-5153	60	10	xe	xe	PROPN
ejpam-5153	60	11	∈	∈	PROPN
ejpam-5153	60	12	re	re	PROPN
ejpam-5153	60	13	,	,	PUNCT
ejpam-5153	60	14	(	(	PUNCT
ejpam-5153	60	15	2	2	NUM
ejpam-5153	60	16	)	)	PUNCT
ejpam-5153	60	17	x	x	SYM
ejpam-5153	60	18	∈	∈	PROPN
ejpam-5153	60	19	re	re	NOUN
ejpam-5153	60	20	,	,	PUNCT
ejpam-5153	60	21	(	(	PUNCT
ejpam-5153	60	22	3	3	X
ejpam-5153	60	23	)	)	PUNCT
ejpam-5153	60	24	x2	x2	PROPN
ejpam-5153	60	25	∈	∈	PROPN
ejpam-5153	60	26	rx	rx	VERB
ejpam-5153	60	27	.	.	PUNCT
ejpam-5153	60	28	proof	proof	NOUN
ejpam-5153	60	29	.	.	PUNCT
ejpam-5153	61	1	(	(	PUNCT
ejpam-5153	61	2	1	1	X
ejpam-5153	61	3	)	)	PUNCT
ejpam-5153	61	4	=	=	NOUN
ejpam-5153	61	5	⇒	⇒	NOUN
ejpam-5153	61	6	(	(	PUNCT
ejpam-5153	61	7	2	2	X
ejpam-5153	61	8	)	)	PUNCT
ejpam-5153	61	9	suppose	suppose	VERB
ejpam-5153	61	10	that	that	SCONJ
ejpam-5153	61	11	xe	xe	PROPN
ejpam-5153	61	12	∈	∈	PROPN
ejpam-5153	61	13	re	re	X
ejpam-5153	61	14	and	and	CCONJ
ejpam-5153	61	15	x	x	PROPN
ejpam-5153	61	16	/∈	/∈	PUNCT
ejpam-5153	62	1	re	re	VERB
ejpam-5153	62	2	.	.	PUNCT
ejpam-5153	63	1	as	as	ADP
ejpam-5153	63	2	xe	xe	PROPN
ejpam-5153	63	3	∈	∈	PROPN
ejpam-5153	63	4	re	re	VERB
ejpam-5153	63	5	so	so	ADV
ejpam-5153	63	6	there	there	PRON
ejpam-5153	63	7	exists	exist	VERB
ejpam-5153	63	8	α	α	PROPN
ejpam-5153	63	9	∈	∈	PROPN
ejpam-5153	63	10	r	r	NOUN
ejpam-5153	63	11	−	−	NOUN
ejpam-5153	63	12	{	{	PUNCT
ejpam-5153	63	13	0	0	NUM
ejpam-5153	63	14	}	}	PUNCT
ejpam-5153	63	15	such	such	ADJ
ejpam-5153	63	16	as	as	ADP
ejpam-5153	63	17	xe	xe	PROPN
ejpam-5153	63	18	=	=	PROPN
ejpam-5153	63	19	αe	αe	PROPN
ejpam-5153	63	20	.	.	PROPN
ejpam-5153	64	1	then	then	ADV
ejpam-5153	64	2	(	(	PUNCT
ejpam-5153	64	3	x	x	X
ejpam-5153	64	4	−	−	PROPN
ejpam-5153	64	5	αe)(xe	αe)(xe	NOUN
ejpam-5153	64	6	)	)	PUNCT
ejpam-5153	64	7	=	=	SYM
ejpam-5153	64	8	0	0	NUM
ejpam-5153	65	1	and	and	CCONJ
ejpam-5153	65	2	as	as	ADP
ejpam-5153	65	3	a	a	DET
ejpam-5153	65	4	is	is	NOUN
ejpam-5153	65	5	of	of	ADP
ejpam-5153	65	6	division	division	NOUN
ejpam-5153	65	7	therefore	therefore	ADV
ejpam-5153	65	8	x	x	PUNCT
ejpam-5153	65	9	=	=	NOUN
ejpam-5153	65	10	αe	αe	NUM
ejpam-5153	65	11	absurd	absurd	ADJ
ejpam-5153	65	12	,	,	PUNCT
ejpam-5153	65	13	hence	hence	ADV
ejpam-5153	65	14	the	the	DET
ejpam-5153	65	15	result	result	NOUN
ejpam-5153	65	16	.	.	PUNCT
ejpam-5153	66	1	(	(	PUNCT
ejpam-5153	66	2	2	2	X
ejpam-5153	66	3	)	)	PUNCT
ejpam-5153	66	4	=	=	NOUN
ejpam-5153	66	5	⇒	⇒	NOUN
ejpam-5153	66	6	(	(	PUNCT
ejpam-5153	66	7	3	3	X
ejpam-5153	66	8	)	)	PUNCT
ejpam-5153	66	9	suppose	suppose	VERB
ejpam-5153	66	10	that	that	SCONJ
ejpam-5153	66	11	x	x	SYM
ejpam-5153	66	12	∈	∈	PROPN
ejpam-5153	66	13	re	re	VERB
ejpam-5153	66	14	,	,	PUNCT
ejpam-5153	66	15	so	so	SCONJ
ejpam-5153	66	16	there	there	PRON
ejpam-5153	66	17	exists	exist	VERB
ejpam-5153	66	18	α	α	PRON
ejpam-5153	66	19	∈	∈	NOUN
ejpam-5153	66	20	r	r	NOUN
ejpam-5153	66	21	such	such	ADJ
ejpam-5153	66	22	as	as	ADP
ejpam-5153	66	23	x	x	X
ejpam-5153	66	24	=	=	SYM
ejpam-5153	67	1	αe	αe	NOUN
ejpam-5153	67	2	=	=	NOUN
ejpam-5153	67	3	⇒	⇒	X
ejpam-5153	67	4	x2	x2	NOUN
ejpam-5153	68	1	=	=	PUNCT
ejpam-5153	68	2	α2e2	α2e2	X
ejpam-5153	68	3	=	=	SYM
ejpam-5153	68	4	α2e	α2e	NOUN
ejpam-5153	68	5	=	=	SYM
ejpam-5153	68	6	α(αe	α(αe	X
ejpam-5153	68	7	)	)	PUNCT
ejpam-5153	68	8	=	=	PUNCT
ejpam-5153	69	1	αx	αx	NOUN
ejpam-5153	69	2	as	as	ADP
ejpam-5153	69	3	a	a	DET
ejpam-5153	69	4	result	result	NOUN
ejpam-5153	69	5	x2	x2	PROPN
ejpam-5153	69	6	∈	∈	PROPN
ejpam-5153	69	7	rx	rx	AUX
ejpam-5153	69	8	.	.	PUNCT
ejpam-5153	70	1	(	(	PUNCT
ejpam-5153	70	2	3	3	X
ejpam-5153	70	3	)	)	PUNCT
ejpam-5153	70	4	=	=	NOUN
ejpam-5153	70	5	⇒	⇒	NOUN
ejpam-5153	70	6	(	(	PUNCT
ejpam-5153	70	7	1	1	X
ejpam-5153	70	8	)	)	PUNCT
ejpam-5153	70	9	suppose	suppose	VERB
ejpam-5153	70	10	that	that	SCONJ
ejpam-5153	70	11	x2	x2	PROPN
ejpam-5153	70	12	∈	∈	PROPN
ejpam-5153	70	13	rx	rx	VERB
ejpam-5153	70	14	,	,	PUNCT
ejpam-5153	70	15	so	so	SCONJ
ejpam-5153	70	16	there	there	PRON
ejpam-5153	70	17	exists	exist	VERB
ejpam-5153	70	18	β	β	X
ejpam-5153	70	19	∈	∈	PROPN
ejpam-5153	70	20	r	r	NOUN
ejpam-5153	70	21	−	−	NOUN
ejpam-5153	70	22	{	{	PUNCT
ejpam-5153	70	23	0	0	NUM
ejpam-5153	70	24	}	}	PUNCT
ejpam-5153	70	25	such	such	ADJ
ejpam-5153	70	26	as	as	ADP
ejpam-5153	70	27	x2	x2	PROPN
ejpam-5153	70	28	=	=	SYM
ejpam-5153	70	29	βx	βx	PROPN
ejpam-5153	70	30	as	as	ADP
ejpam-5153	70	31	a	a	DET
ejpam-5153	70	32	result	result	NOUN
ejpam-5153	70	33	(	(	PUNCT
ejpam-5153	70	34	x	x	X
ejpam-5153	70	35	−	−	PROPN
ejpam-5153	70	36	βe)x	βe)x	NOUN
ejpam-5153	70	37	=	=	SYM
ejpam-5153	70	38	0	0	NUM
ejpam-5153	70	39	and	and	CCONJ
ejpam-5153	70	40	dividing	divide	VERB
ejpam-5153	70	41	a	a	PRON
ejpam-5153	70	42	gives	give	VERB
ejpam-5153	70	43	us	we	PRON
ejpam-5153	70	44	x	x	PUNCT
ejpam-5153	71	1	=	=	PUNCT
ejpam-5153	71	2	βe	βe	PROPN
ejpam-5153	72	1	=	=	AUX
ejpam-5153	72	2	⇒	⇒	X
ejpam-5153	72	3	xe	xe	PROPN
ejpam-5153	72	4	=	=	PUNCT
ejpam-5153	72	5	βe2	βe2	X
ejpam-5153	73	1	=	=	SYM
ejpam-5153	73	2	βe	βe	PROPN
ejpam-5153	73	3	,	,	PUNCT
ejpam-5153	73	4	then	then	ADV
ejpam-5153	73	5	xe	xe	PROPN
ejpam-5153	73	6	∈	∈	PROPN
ejpam-5153	73	7	re	re	NOUN
ejpam-5153	73	8	.	.	PROPN
ejpam-5153	74	1	3	3	NUM
ejpam-5153	74	2	.	.	X
ejpam-5153	74	3	two	two	NUM
ejpam-5153	74	4	-	-	PUNCT
ejpam-5153	74	5	dimensional	dimensional	ADJ
ejpam-5153	74	6	real	real	ADJ
ejpam-5153	74	7	division	division	NOUN
ejpam-5153	74	8	algebras	algebra	VERB
ejpam-5153	74	9	with	with	ADP
ejpam-5153	74	10	left	left	ADJ
ejpam-5153	74	11	unit	unit	NOUN
ejpam-5153	74	12	let	let	VERB
ejpam-5153	74	13	a	a	PRON
ejpam-5153	74	14	be	be	AUX
ejpam-5153	74	15	a	a	DET
ejpam-5153	74	16	two	two	NUM
ejpam-5153	74	17	-	-	PUNCT
ejpam-5153	74	18	dimensional	dimensional	ADJ
ejpam-5153	74	19	real	real	ADJ
ejpam-5153	74	20	algebra	algebra	NOUN
ejpam-5153	74	21	having	have	VERB
ejpam-5153	74	22	a	a	DET
ejpam-5153	74	23	basis	basis	NOUN
ejpam-5153	74	24	b	b	NOUN
ejpam-5153	74	25	=	=	SYM
ejpam-5153	74	26	{	{	PUNCT
ejpam-5153	74	27	e	e	NOUN
ejpam-5153	74	28	;	;	PUNCT
ejpam-5153	74	29	u	u	NOUN
ejpam-5153	74	30	}	}	PUNCT
ejpam-5153	74	31	such	such	ADJ
ejpam-5153	74	32	that	that	SCONJ
ejpam-5153	74	33	the	the	DET
ejpam-5153	74	34	products	product	NOUN
ejpam-5153	74	35	in	in	ADP
ejpam-5153	74	36	the	the	DET
ejpam-5153	74	37	base	base	NOUN
ejpam-5153	74	38	are	be	AUX
ejpam-5153	74	39	given	give	VERB
ejpam-5153	74	40	by	by	ADP
ejpam-5153	74	41	the	the	DET
ejpam-5153	74	42	multiplications	multiplication	NOUN
ejpam-5153	74	43	table1	table1	PROPN
ejpam-5153	74	44	and	and	CCONJ
ejpam-5153	74	45	table2	table2	PROPN
ejpam-5153	74	46	.	.	PUNCT
ejpam-5153	75	1	e	e	X
ejpam-5153	75	2	u	u	X
ejpam-5153	75	3	e	e	X
ejpam-5153	75	4	e	e	X
ejpam-5153	75	5	u	u	X
ejpam-5153	75	6	u	u	NOUN
ejpam-5153	75	7	αe	αe	PROPN
ejpam-5153	75	8	+	+	X
ejpam-5153	75	9	βu	βu	SYM
ejpam-5153	75	10	γe	γe	NOUN
ejpam-5153	75	11	+	+	ADJ
ejpam-5153	75	12	λu	λu	X
ejpam-5153	75	13	.	.	PUNCT
ejpam-5153	76	1	e	e	X
ejpam-5153	76	2	u	u	X
ejpam-5153	76	3	e	e	X
ejpam-5153	76	4	e	e	X
ejpam-5153	76	5	αe	αe	PROPN
ejpam-5153	76	6	+	+	NOUN
ejpam-5153	76	7	βu	βu	PUNCT
ejpam-5153	76	8	u	u	NOUN
ejpam-5153	76	9	u	u	NOUN
ejpam-5153	76	10	γe	γe	INTJ
ejpam-5153	76	11	+	+	CCONJ
ejpam-5153	76	12	λu	λu	X
ejpam-5153	76	13	.	.	PUNCT
ejpam-5153	77	1	table1	table1	PROPN
ejpam-5153	77	2	table2	table2	PROPN
ejpam-5153	77	3	with	with	ADP
ejpam-5153	77	4	α	α	PROPN
ejpam-5153	77	5	,	,	PUNCT
ejpam-5153	77	6	β	β	X
ejpam-5153	77	7	,	,	PUNCT
ejpam-5153	77	8	γ	γ	X
ejpam-5153	77	9	,	,	PUNCT
ejpam-5153	77	10	λ	λ	PROPN
ejpam-5153	77	11	∈	∈	PROPN
ejpam-5153	77	12	r.	r.	NOUN
ejpam-5153	77	13	we	we	PRON
ejpam-5153	77	14	will	will	AUX
ejpam-5153	77	15	note	note	VERB
ejpam-5153	77	16	respectively	respectively	ADV
ejpam-5153	77	17	l(α	l(α	PROPN
ejpam-5153	77	18	,	,	PUNCT
ejpam-5153	77	19	β	β	X
ejpam-5153	77	20	,	,	PUNCT
ejpam-5153	77	21	γ	γ	X
ejpam-5153	77	22	,	,	PUNCT
ejpam-5153	77	23	λ	λ	NOUN
ejpam-5153	77	24	)	)	PUNCT
ejpam-5153	77	25	and	and	CCONJ
ejpam-5153	77	26	r(α	r(α	PROPN
ejpam-5153	77	27	,	,	PUNCT
ejpam-5153	77	28	β	β	X
ejpam-5153	77	29	,	,	PUNCT
ejpam-5153	77	30	γ	γ	X
ejpam-5153	77	31	,	,	PUNCT
ejpam-5153	77	32	λ	λ	NOUN
ejpam-5153	77	33	)	)	PUNCT
ejpam-5153	77	34	algebras	algebra	NOUN
ejpam-5153	77	35	of	of	ADP
ejpam-5153	77	36	to	to	PART
ejpam-5153	77	37	dimension	dimension	VERB
ejpam-5153	77	38	2	2	NUM
ejpam-5153	77	39	whose	whose	DET
ejpam-5153	77	40	product	product	NOUN
ejpam-5153	77	41	in	in	ADP
ejpam-5153	77	42	the	the	DET
ejpam-5153	77	43	base	base	NOUN
ejpam-5153	77	44	{	{	PUNCT
ejpam-5153	77	45	e	e	NOUN
ejpam-5153	77	46	,	,	PUNCT
ejpam-5153	77	47	u	u	NOUN
ejpam-5153	77	48	}	}	PUNCT
ejpam-5153	77	49	is	be	AUX
ejpam-5153	77	50	given	give	VERB
ejpam-5153	77	51	respectively	respectively	ADV
ejpam-5153	77	52	by	by	ADP
ejpam-5153	77	53	table1	table1	PROPN
ejpam-5153	77	54	and	and	CCONJ
ejpam-5153	77	55	table2	table2	PROPN
ejpam-5153	77	56	.	.	PUNCT
ejpam-5153	78	1	[	[	X
ejpam-5153	78	2	7	7	X
ejpam-5153	78	3	]	]	PUNCT
ejpam-5153	78	4	gives	give	VERB
ejpam-5153	78	5	the	the	DET
ejpam-5153	78	6	following	follow	VERB
ejpam-5153	78	7	results	result	NOUN
ejpam-5153	78	8	:	:	PUNCT
ejpam-5153	78	9	theorem	theorem	NOUN
ejpam-5153	78	10	1	1	NUM
ejpam-5153	78	11	and	and	CCONJ
ejpam-5153	78	12	proposition	proposition	NOUN
ejpam-5153	78	13	1	1	NUM
ejpam-5153	78	14	.	.	PUNCT
ejpam-5153	78	15	theorem	theorem	NOUN
ejpam-5153	78	16	1	1	NUM
ejpam-5153	78	17	.	.	PUNCT
ejpam-5153	79	1	let	let	VERB
ejpam-5153	79	2	a	a	PRON
ejpam-5153	79	3	be	be	AUX
ejpam-5153	79	4	a	a	DET
ejpam-5153	79	5	real	real	ADJ
ejpam-5153	79	6	division	division	NOUN
ejpam-5153	79	7	algebra	algebra	NOUN
ejpam-5153	79	8	of	of	ADP
ejpam-5153	79	9	dimension	dimension	NOUN
ejpam-5153	79	10	two	two	NUM
ejpam-5153	79	11	with	with	ADP
ejpam-5153	79	12	left	left	ADJ
ejpam-5153	79	13	identity	identity	NOUN
ejpam-5153	79	14	.	.	PUNCT
ejpam-5153	80	1	then	then	ADV
ejpam-5153	80	2	a	a	PRON
ejpam-5153	80	3	is	be	AUX
ejpam-5153	80	4	isomorphic	isomorphic	ADJ
ejpam-5153	80	5	to	to	ADP
ejpam-5153	80	6	precisely	precisely	ADV
ejpam-5153	80	7	one	one	NUM
ejpam-5153	80	8	of	of	ADP
ejpam-5153	80	9	the	the	DET
ejpam-5153	80	10	following	following	ADJ
ejpam-5153	80	11	algebras	algebra	NOUN
ejpam-5153	80	12	:	:	PUNCT
ejpam-5153	80	13	(	(	PUNCT
ejpam-5153	80	14	class	class	NOUN
ejpam-5153	80	15	i	i	NOUN
ejpam-5153	80	16	)	)	PUNCT
ejpam-5153	80	17	l(0	l(0	PROPN
ejpam-5153	80	18	,	,	PUNCT
ejpam-5153	80	19	β	β	X
ejpam-5153	80	20	,	,	PUNCT
ejpam-5153	80	21	−1	−1	NOUN
ejpam-5153	80	22	,	,	PUNCT
ejpam-5153	80	23	0	0	NUM
ejpam-5153	80	24	)	)	PUNCT
ejpam-5153	80	25	for	for	ADP
ejpam-5153	80	26	some	some	DET
ejpam-5153	80	27	β	β	X
ejpam-5153	80	28	>	>	X
ejpam-5153	80	29	0	0	NUM
ejpam-5153	80	30	;	;	PUNCT
ejpam-5153	80	31	(	(	PUNCT
ejpam-5153	80	32	class	class	PROPN
ejpam-5153	80	33	ii	ii	NOUN
ejpam-5153	80	34	)	)	PUNCT
ejpam-5153	80	35	l(0	l(0	PROPN
ejpam-5153	80	36	,	,	PUNCT
ejpam-5153	80	37	β	β	NOUN
ejpam-5153	80	38	,	,	PUNCT
ejpam-5153	80	39	1	1	NUM
ejpam-5153	80	40	,	,	PUNCT
ejpam-5153	80	41	0	0	NUM
ejpam-5153	80	42	)	)	PUNCT
ejpam-5153	80	43	for	for	ADP
ejpam-5153	80	44	some	some	DET
ejpam-5153	80	45	β	β	NOUN
ejpam-5153	80	46	<	<	X
ejpam-5153	80	47	0	0	NUM
ejpam-5153	80	48	;	;	PUNCT
ejpam-5153	80	49	(	(	PUNCT
ejpam-5153	80	50	class	class	NOUN
ejpam-5153	80	51	iii	iii	NOUN
ejpam-5153	80	52	)	)	PUNCT
ejpam-5153	80	53	l(1	l(1	PROPN
ejpam-5153	80	54	,	,	PUNCT
ejpam-5153	80	55	β	β	X
ejpam-5153	80	56	,	,	PUNCT
ejpam-5153	80	57	γ	γ	PROPN
ejpam-5153	80	58	,	,	PUNCT
ejpam-5153	80	59	0	0	NUM
ejpam-5153	80	60	)	)	PUNCT
ejpam-5153	80	61	for	for	ADP
ejpam-5153	80	62	some	some	DET
ejpam-5153	80	63	β	β	X
ejpam-5153	80	64	̸=	̸=	PROPN
ejpam-5153	80	65	−1	−1	NOUN
ejpam-5153	80	66	and	and	CCONJ
ejpam-5153	80	67	γ	γ	NOUN
ejpam-5153	80	68	with	with	ADP
ejpam-5153	80	69	4βγ	4βγ	ADJ
ejpam-5153	80	70	<	<	X
ejpam-5153	80	71	−1	−1	NOUN
ejpam-5153	80	72	;	;	PUNCT
ejpam-5153	80	73	(	(	PUNCT
ejpam-5153	80	74	class	class	NOUN
ejpam-5153	80	75	iv	iv	NOUN
ejpam-5153	80	76	)	)	PUNCT
ejpam-5153	80	77	l(1	l(1	PROPN
ejpam-5153	80	78	,	,	PUNCT
ejpam-5153	80	79	−1	−1	NOUN
ejpam-5153	80	80	,	,	PUNCT
ejpam-5153	80	81	γ	γ	X
ejpam-5153	80	82	,	,	PUNCT
ejpam-5153	80	83	1	1	NUM
ejpam-5153	80	84	)	)	PUNCT
ejpam-5153	80	85	for	for	ADP
ejpam-5153	80	86	some	some	DET
ejpam-5153	80	87	γ	γ	NOUN
ejpam-5153	80	88	>	>	X
ejpam-5153	80	89	0	0	NUM
ejpam-5153	80	90	;	;	PUNCT
ejpam-5153	80	91	a.	a.	PROPN
ejpam-5153	80	92	s.	s.	PROPN
ejpam-5153	80	93	diabang	diabang	PROPN
ejpam-5153	80	94	,	,	PUNCT
ejpam-5153	80	95	a.	a.	PROPN
ejpam-5153	80	96	s.	s.	PROPN
ejpam-5153	80	97	mballo	mballo	PROPN
ejpam-5153	80	98	,	,	PUNCT
ejpam-5153	80	99	p.	p.	PROPN
ejpam-5153	80	100	c.	c.	PROPN
ejpam-5153	80	101	diop	diop	PROPN
ejpam-5153	80	102	/	/	SYM
ejpam-5153	80	103	eur	eur	PROPN
ejpam-5153	80	104	.	.	PUNCT
ejpam-5153	81	1	j.	j.	PROPN
ejpam-5153	81	2	pure	pure	PROPN
ejpam-5153	81	3	appl	appl	PROPN
ejpam-5153	81	4	.	.	PROPN
ejpam-5153	81	5	math	math	PROPN
ejpam-5153	81	6	,	,	PUNCT
ejpam-5153	81	7	17	17	NUM
ejpam-5153	81	8	(	(	PUNCT
ejpam-5153	81	9	3	3	NUM
ejpam-5153	81	10	)	)	PUNCT
ejpam-5153	81	11	(	(	PUNCT
ejpam-5153	81	12	2024	2024	NUM
ejpam-5153	81	13	)	)	PUNCT
ejpam-5153	81	14	,	,	PUNCT
ejpam-5153	81	15	2276	2276	NUM
ejpam-5153	81	16	-	-	SYM
ejpam-5153	81	17	2287	2287	NUM
ejpam-5153	81	18	2279	2279	NUM
ejpam-5153	81	19	proposition	proposition	NOUN
ejpam-5153	81	20	1	1	NUM
ejpam-5153	81	21	.	.	PUNCT
ejpam-5153	82	1	β	β	NOUN
ejpam-5153	82	2	and	and	CCONJ
ejpam-5153	82	3	sgnγ	sgnγ	NOUN
ejpam-5153	82	4	are	be	AUX
ejpam-5153	82	5	invariants	invariant	NOUN
ejpam-5153	82	6	among	among	ADP
ejpam-5153	82	7	division	division	NOUN
ejpam-5153	82	8	algebras	algebra	NOUN
ejpam-5153	82	9	,	,	PUNCT
ejpam-5153	82	10	that	that	ADV
ejpam-5153	82	11	is	is	ADV
ejpam-5153	82	12	,	,	PUNCT
ejpam-5153	82	13	if	if	SCONJ
ejpam-5153	82	14	a	a	PRON
ejpam-5153	82	15	=	=	X
ejpam-5153	82	16	l(α	l(α	PROPN
ejpam-5153	82	17	,	,	PUNCT
ejpam-5153	82	18	β	β	X
ejpam-5153	82	19	,	,	PUNCT
ejpam-5153	82	20	γ	γ	PROPN
ejpam-5153	82	21	,	,	PUNCT
ejpam-5153	82	22	δ	δ	PROPN
ejpam-5153	82	23	)	)	PUNCT
ejpam-5153	82	24	and	and	CCONJ
ejpam-5153	82	25	a′	a′	PROPN
ejpam-5153	82	26	=	=	SYM
ejpam-5153	82	27	l(α′	l(α′	PROPN
ejpam-5153	82	28	,	,	PUNCT
ejpam-5153	82	29	β′	β′	NUM
ejpam-5153	82	30	,	,	PUNCT
ejpam-5153	82	31	γ′	γ′	PROPN
ejpam-5153	82	32	,	,	PUNCT
ejpam-5153	82	33	δ′	δ′	PROPN
ejpam-5153	82	34	)	)	PUNCT
ejpam-5153	82	35	are	be	AUX
ejpam-5153	82	36	isomorphic	isomorphic	ADJ
ejpam-5153	82	37	division	division	NOUN
ejpam-5153	82	38	algebras	algebra	NOUN
ejpam-5153	82	39	,	,	PUNCT
ejpam-5153	82	40	then	then	ADV
ejpam-5153	82	41	β	β	X
ejpam-5153	82	42	=	=	PUNCT
ejpam-5153	82	43	β′	β′	NUM
ejpam-5153	82	44	and	and	CCONJ
ejpam-5153	82	45	sgnγ	sgnγ	NOUN
ejpam-5153	82	46	=	=	SYM
ejpam-5153	82	47	sgnγ′	sgnγ′	NOUN
ejpam-5153	82	48	remark	remark	NOUN
ejpam-5153	82	49	1	1	NUM
ejpam-5153	82	50	.	.	PUNCT
ejpam-5153	83	1	a	a	PRON
ejpam-5153	83	2	be	be	AUX
ejpam-5153	83	3	a	a	DET
ejpam-5153	83	4	finite	finite	ADJ
ejpam-5153	83	5	-	-	ADJ
ejpam-5153	83	6	dimensional	dimensional	ADJ
ejpam-5153	83	7	real	real	ADJ
ejpam-5153	83	8	division	division	NOUN
ejpam-5153	83	9	algebra	algebra	NOUN
ejpam-5153	83	10	with	with	ADP
ejpam-5153	83	11	left	left	ADJ
ejpam-5153	83	12	unit	unit	NOUN
ejpam-5153	83	13	e	e	NOUN
ejpam-5153	83	14	equals	equal	VERB
ejpam-5153	83	15	aopp	aopp	PROPN
ejpam-5153	83	16	=	=	SYM
ejpam-5153	83	17	(	(	PUNCT
ejpam-5153	83	18	a	a	PRON
ejpam-5153	83	19	,	,	PUNCT
ejpam-5153	83	20	⊙	⊙	PROPN
ejpam-5153	83	21	)	)	PUNCT
ejpam-5153	83	22	with	with	ADP
ejpam-5153	83	23	x	x	PROPN
ejpam-5153	83	24	⊙	⊙	PROPN
ejpam-5153	83	25	y	y	PROPN
ejpam-5153	83	26	=	=	PROPN
ejpam-5153	83	27	yx	yx	PROPN
ejpam-5153	83	28	be	be	AUX
ejpam-5153	83	29	a	a	DET
ejpam-5153	83	30	finite	finite	ADJ
ejpam-5153	83	31	-	-	ADJ
ejpam-5153	83	32	dimensional	dimensional	ADJ
ejpam-5153	83	33	real	real	ADJ
ejpam-5153	83	34	division	division	NOUN
ejpam-5153	83	35	algebra	algebra	NOUN
ejpam-5153	83	36	with	with	ADP
ejpam-5153	83	37	right	right	ADJ
ejpam-5153	83	38	unit	unit	NOUN
ejpam-5153	83	39	e.	e.	PROPN
ejpam-5153	83	40	therefore	therefore	ADV
ejpam-5153	83	41	the	the	DET
ejpam-5153	83	42	results	result	NOUN
ejpam-5153	83	43	on	on	ADP
ejpam-5153	83	44	algebras	algebra	NOUN
ejpam-5153	83	45	with	with	ADP
ejpam-5153	83	46	right	right	ADJ
ejpam-5153	83	47	unit	unit	NOUN
ejpam-5153	83	48	are	be	AUX
ejpam-5153	83	49	obtained	obtain	VERB
ejpam-5153	83	50	by	by	ADP
ejpam-5153	83	51	analogy	analogy	NOUN
ejpam-5153	83	52	the	the	DET
ejpam-5153	83	53	results	result	NOUN
ejpam-5153	83	54	of	of	ADP
ejpam-5153	83	55	algebras	algebra	NOUN
ejpam-5153	83	56	with	with	ADP
ejpam-5153	83	57	left	left	ADJ
ejpam-5153	83	58	unit	unit	NOUN
ejpam-5153	83	59	.	.	PUNCT
ejpam-5153	84	1	lemma	lemma	PROPN
ejpam-5153	84	2	3	3	X
ejpam-5153	84	3	.	.	PUNCT
ejpam-5153	85	1	let	let	VERB
ejpam-5153	85	2	a	a	PRON
ejpam-5153	85	3	be	be	AUX
ejpam-5153	85	4	a	a	DET
ejpam-5153	85	5	two	two	NUM
ejpam-5153	85	6	-	-	PUNCT
ejpam-5153	85	7	dimensional	dimensional	ADJ
ejpam-5153	85	8	real	real	ADJ
ejpam-5153	85	9	division	division	NOUN
ejpam-5153	85	10	algebra	algebra	NOUN
ejpam-5153	85	11	with	with	ADP
ejpam-5153	85	12	left	left	ADJ
ejpam-5153	85	13	unit	unit	NOUN
ejpam-5153	85	14	e.	e.	PROPN
ejpam-5153	86	1	the	the	DET
ejpam-5153	86	2	following	follow	VERB
ejpam-5153	86	3	propositions	proposition	NOUN
ejpam-5153	86	4	are	be	AUX
ejpam-5153	86	5	equivalent	equivalent	ADJ
ejpam-5153	86	6	:	:	PUNCT
ejpam-5153	86	7	(	(	PUNCT
ejpam-5153	86	8	1	1	X
ejpam-5153	86	9	)	)	PUNCT
ejpam-5153	86	10	a	a	DET
ejpam-5153	86	11	satisfies	satisfie	NOUN
ejpam-5153	86	12	to	to	ADP
ejpam-5153	86	13	(	(	PUNCT
ejpam-5153	86	14	e4	e4	PROPN
ejpam-5153	86	15	)	)	PUNCT
ejpam-5153	86	16	.	.	PUNCT
ejpam-5153	87	1	(	(	PUNCT
ejpam-5153	87	2	2	2	X
ejpam-5153	87	3	)	)	PUNCT
ejpam-5153	87	4	a	a	PRON
ejpam-5153	87	5	is	be	AUX
ejpam-5153	87	6	isomorphic	isomorphic	ADJ
ejpam-5153	87	7	to	to	ADP
ejpam-5153	87	8	either	either	CCONJ
ejpam-5153	87	9	c	c	PROPN
ejpam-5153	87	10	,	,	PUNCT
ejpam-5153	87	11	⋆c	⋆c	PROPN
ejpam-5153	87	12	,	,	PUNCT
ejpam-5153	87	13	l(1	l(1	PROPN
ejpam-5153	87	14	,	,	PUNCT
ejpam-5153	87	15	−1	−1	NOUN
ejpam-5153	87	16	,	,	PUNCT
ejpam-5153	87	17	γ	γ	X
ejpam-5153	87	18	,	,	PUNCT
ejpam-5153	87	19	1	1	NUM
ejpam-5153	87	20	)	)	PUNCT
ejpam-5153	87	21	with	with	ADP
ejpam-5153	87	22	γ	γ	PROPN
ejpam-5153	87	23	>	>	X
ejpam-5153	87	24	0	0	PROPN
ejpam-5153	87	25	.	.	PUNCT
ejpam-5153	87	26	proof	proof	NOUN
ejpam-5153	87	27	.	.	PUNCT
ejpam-5153	88	1	(	(	PUNCT
ejpam-5153	88	2	1	1	X
ejpam-5153	88	3	)	)	PUNCT
ejpam-5153	88	4	⇒	⇒	NOUN
ejpam-5153	88	5	(	(	PUNCT
ejpam-5153	88	6	2	2	NUM
ejpam-5153	88	7	)	)	PUNCT
ejpam-5153	88	8	according	accord	VERB
ejpam-5153	88	9	to	to	ADP
ejpam-5153	88	10	the	the	DET
ejpam-5153	88	11	theorem	theorem	NOUN
ejpam-5153	88	12	1	1	NUM
ejpam-5153	88	13	,	,	PUNCT
ejpam-5153	88	14	we	we	PRON
ejpam-5153	88	15	have	have	VERB
ejpam-5153	88	16	a	a	PRON
ejpam-5153	88	17	is	be	AUX
ejpam-5153	88	18	isomorphic	isomorphic	ADJ
ejpam-5153	88	19	to	to	ADP
ejpam-5153	88	20	l(α	l(α	PROPN
ejpam-5153	88	21	,	,	PUNCT
ejpam-5153	88	22	β	β	X
ejpam-5153	88	23	,	,	PUNCT
ejpam-5153	88	24	γ	γ	X
ejpam-5153	88	25	,	,	PUNCT
ejpam-5153	88	26	λ	λ	NOUN
ejpam-5153	88	27	)	)	PUNCT
ejpam-5153	88	28	.	.	PUNCT
ejpam-5153	89	1	we	we	PRON
ejpam-5153	89	2	have	have	VERB
ejpam-5153	89	3	(	(	PUNCT
ejpam-5153	89	4	ue)e	ue)e	PROPN
ejpam-5153	89	5	=	=	SYM
ejpam-5153	89	6	u	u	NOUN
ejpam-5153	89	7	⇐	⇐	ADJ
ejpam-5153	89	8	⇒	⇒	NOUN
ejpam-5153	89	9	α(1	α(1	PROPN
ejpam-5153	89	10	+	+	PUNCT
ejpam-5153	89	11	β)e	β)e	PUNCT
ejpam-5153	90	1	+	+	CCONJ
ejpam-5153	90	2	(	(	PUNCT
ejpam-5153	90	3	β2	β2	NOUN
ejpam-5153	90	4	−	−	PROPN
ejpam-5153	90	5	1)u	1)u	NUM
ejpam-5153	90	6	=	=	NOUN
ejpam-5153	90	7	0	0	NUM
ejpam-5153	90	8	⇐	⇐	ADJ
ejpam-5153	90	9	⇒	⇒	NOUN
ejpam-5153	90	10	{	{	PUNCT
ejpam-5153	90	11	α(1	α(1	PROPN
ejpam-5153	90	12	+	+	NUM
ejpam-5153	90	13	β	β	X
ejpam-5153	90	14	)	)	PUNCT
ejpam-5153	90	15	=	=	SYM
ejpam-5153	90	16	0	0	NUM
ejpam-5153	90	17	β2	β2	NOUN
ejpam-5153	90	18	=	=	NOUN
ejpam-5153	90	19	1	1	NUM
ejpam-5153	90	20	•	•	NOUN
ejpam-5153	90	21	if	if	SCONJ
ejpam-5153	90	22	β	β	X
ejpam-5153	90	23	=	=	SYM
ejpam-5153	90	24	1	1	NUM
ejpam-5153	90	25	,	,	PUNCT
ejpam-5153	90	26	then	then	ADV
ejpam-5153	90	27	α	α	NOUN
ejpam-5153	90	28	=	=	SYM
ejpam-5153	90	29	0	0	PROPN
ejpam-5153	90	30	,	,	PUNCT
ejpam-5153	90	31	the	the	DET
ejpam-5153	90	32	theorem	theorem	ADJ
ejpam-5153	90	33	1	1	NUM
ejpam-5153	90	34	shows	show	VERB
ejpam-5153	90	35	that	that	SCONJ
ejpam-5153	90	36	a	a	PRON
ejpam-5153	90	37	is	be	AUX
ejpam-5153	90	38	isomorphic	isomorphic	ADJ
ejpam-5153	90	39	to	to	ADP
ejpam-5153	90	40	l(0	l(0	PROPN
ejpam-5153	90	41	,	,	PUNCT
ejpam-5153	90	42	1	1	NUM
ejpam-5153	90	43	,	,	PUNCT
ejpam-5153	90	44	−1	−1	NOUN
ejpam-5153	90	45	,	,	PUNCT
ejpam-5153	90	46	0	0	NUM
ejpam-5153	90	47	)	)	PUNCT
ejpam-5153	90	48	∼=	∼=	PROPN
ejpam-5153	90	49	c.	c.	NOUN
ejpam-5153	90	50	•	•	ADP
ejpam-5153	90	51	if	if	SCONJ
ejpam-5153	90	52	β	β	X
ejpam-5153	90	53	=	=	SYM
ejpam-5153	90	54	−1	−1	NOUN
ejpam-5153	90	55	,	,	PUNCT
ejpam-5153	90	56	the	the	DET
ejpam-5153	90	57	theorem	theorem	ADJ
ejpam-5153	90	58	1	1	NUM
ejpam-5153	90	59	shows	show	VERB
ejpam-5153	90	60	that	that	SCONJ
ejpam-5153	90	61	a	a	DET
ejpam-5153	90	62	isomorphic	isomorphic	NOUN
ejpam-5153	90	63	to	to	ADP
ejpam-5153	90	64	l(0	l(0	PROPN
ejpam-5153	90	65	,	,	PUNCT
ejpam-5153	90	66	−1	−1	NOUN
ejpam-5153	90	67	,	,	PUNCT
ejpam-5153	90	68	1	1	NUM
ejpam-5153	90	69	,	,	PUNCT
ejpam-5153	90	70	0	0	NUM
ejpam-5153	90	71	)	)	PUNCT
ejpam-5153	90	72	∼=	∼=	PROPN
ejpam-5153	90	73	⋆c	⋆c	PROPN
ejpam-5153	90	74	.	.	NOUN
ejpam-5153	90	75	or	or	CCONJ
ejpam-5153	90	76	l(1	l(1	PROPN
ejpam-5153	90	77	,	,	PUNCT
ejpam-5153	90	78	−1	−1	NOUN
ejpam-5153	90	79	,	,	PUNCT
ejpam-5153	90	80	γ	γ	X
ejpam-5153	90	81	,	,	PUNCT
ejpam-5153	90	82	1	1	NUM
ejpam-5153	90	83	)	)	PUNCT
ejpam-5153	90	84	with	with	ADP
ejpam-5153	90	85	γ	γ	X
ejpam-5153	90	86	>	>	X
ejpam-5153	90	87	0	0	PUNCT
ejpam-5153	90	88	(	(	PUNCT
ejpam-5153	90	89	2	2	NUM
ejpam-5153	90	90	)	)	PUNCT
ejpam-5153	90	91	⇒	⇒	NOUN
ejpam-5153	90	92	(	(	PUNCT
ejpam-5153	90	93	1	1	NUM
ejpam-5153	90	94	)	)	PUNCT
ejpam-5153	90	95	obvious	obvious	ADJ
ejpam-5153	90	96	.	.	PUNCT
ejpam-5153	91	1	lemma	lemma	PROPN
ejpam-5153	91	2	4	4	X
ejpam-5153	91	3	.	.	PUNCT
ejpam-5153	91	4	let	let	VERB
ejpam-5153	91	5	a	a	PRON
ejpam-5153	91	6	be	be	AUX
ejpam-5153	91	7	a	a	DET
ejpam-5153	91	8	two	two	NUM
ejpam-5153	91	9	-	-	PUNCT
ejpam-5153	91	10	dimensional	dimensional	ADJ
ejpam-5153	91	11	real	real	ADJ
ejpam-5153	91	12	division	division	NOUN
ejpam-5153	91	13	algebra	algebra	NOUN
ejpam-5153	91	14	with	with	ADP
ejpam-5153	91	15	left	left	ADJ
ejpam-5153	91	16	unit	unit	NOUN
ejpam-5153	91	17	e.	e.	PROPN
ejpam-5153	92	1	the	the	DET
ejpam-5153	92	2	following	follow	VERB
ejpam-5153	92	3	propositions	proposition	NOUN
ejpam-5153	92	4	are	be	AUX
ejpam-5153	92	5	equivalent	equivalent	ADJ
ejpam-5153	92	6	:	:	PUNCT
ejpam-5153	92	7	(	(	PUNCT
ejpam-5153	92	8	1	1	X
ejpam-5153	92	9	)	)	PUNCT
ejpam-5153	92	10	a	a	DET
ejpam-5153	92	11	satisfies	satisfie	NOUN
ejpam-5153	92	12	to	to	ADP
ejpam-5153	92	13	(	(	PUNCT
ejpam-5153	92	14	e3	e3	NOUN
ejpam-5153	92	15	)	)	PUNCT
ejpam-5153	92	16	(	(	PUNCT
ejpam-5153	92	17	2	2	X
ejpam-5153	92	18	)	)	PUNCT
ejpam-5153	92	19	a	a	PRON
ejpam-5153	92	20	is	be	AUX
ejpam-5153	92	21	isomorphic	isomorphic	ADJ
ejpam-5153	92	22	to	to	ADP
ejpam-5153	92	23	either	either	CCONJ
ejpam-5153	92	24	c	c	PROPN
ejpam-5153	92	25	,	,	PUNCT
ejpam-5153	92	26	⋆c	⋆c	PROPN
ejpam-5153	92	27	.	.	PUNCT
ejpam-5153	93	1	proof	proof	NOUN
ejpam-5153	93	2	.	.	PUNCT
ejpam-5153	94	1	(	(	PUNCT
ejpam-5153	94	2	1	1	X
ejpam-5153	94	3	)	)	PUNCT
ejpam-5153	94	4	⇒	⇒	NOUN
ejpam-5153	94	5	(	(	PUNCT
ejpam-5153	94	6	2	2	NUM
ejpam-5153	94	7	)	)	PUNCT
ejpam-5153	94	8	according	accord	VERB
ejpam-5153	94	9	to	to	ADP
ejpam-5153	94	10	the	the	DET
ejpam-5153	94	11	theorem	theorem	NOUN
ejpam-5153	94	12	1	1	NUM
ejpam-5153	94	13	,	,	PUNCT
ejpam-5153	94	14	we	we	PRON
ejpam-5153	94	15	have	have	VERB
ejpam-5153	94	16	a	a	PRON
ejpam-5153	94	17	is	be	AUX
ejpam-5153	94	18	isomorphic	isomorphic	ADJ
ejpam-5153	94	19	to	to	ADP
ejpam-5153	94	20	l(α	l(α	PROPN
ejpam-5153	94	21	,	,	PUNCT
ejpam-5153	94	22	β	β	X
ejpam-5153	94	23	,	,	PUNCT
ejpam-5153	94	24	γ	γ	X
ejpam-5153	94	25	,	,	PUNCT
ejpam-5153	94	26	λ	λ	NOUN
ejpam-5153	94	27	)	)	PUNCT
ejpam-5153	94	28	•	•	NOUN
ejpam-5153	94	29	if	if	SCONJ
ejpam-5153	94	30	a	a	DET
ejpam-5153	94	31	∼=	∼=	PROPN
ejpam-5153	94	32	l(0	l(0	PROPN
ejpam-5153	94	33	,	,	PUNCT
ejpam-5153	94	34	β	β	X
ejpam-5153	94	35	,	,	PUNCT
ejpam-5153	94	36	−1	−1	NOUN
ejpam-5153	94	37	,	,	PUNCT
ejpam-5153	94	38	0	0	NUM
ejpam-5153	94	39	)	)	PUNCT
ejpam-5153	94	40	,	,	PUNCT
ejpam-5153	94	41	we	we	PRON
ejpam-5153	94	42	have	have	VERB
ejpam-5153	94	43	(	(	PUNCT
ejpam-5153	94	44	e	e	NOUN
ejpam-5153	94	45	+	+	NOUN
ejpam-5153	94	46	u)2e	u)2e	PUNCT
ejpam-5153	94	47	=	=	SYM
ejpam-5153	94	48	(	(	PUNCT
ejpam-5153	94	49	e	e	X
ejpam-5153	94	50	+	+	NOUN
ejpam-5153	94	51	u)2	u)2	ADJ
ejpam-5153	94	52	⇒	⇒	NOUN
ejpam-5153	94	53	β	β	X
ejpam-5153	94	54	=	=	NOUN
ejpam-5153	94	55	1	1	NUM
ejpam-5153	94	56	thus	thus	ADV
ejpam-5153	94	57	a	a	DET
ejpam-5153	94	58	∼=	∼=	PROPN
ejpam-5153	94	59	l(0	l(0	PROPN
ejpam-5153	94	60	,	,	PUNCT
ejpam-5153	94	61	1	1	NUM
ejpam-5153	94	62	,	,	PUNCT
ejpam-5153	94	63	−1	−1	NOUN
ejpam-5153	94	64	,	,	PUNCT
ejpam-5153	94	65	0	0	NUM
ejpam-5153	94	66	)	)	PUNCT
ejpam-5153	94	67	∼=	∼=	PROPN
ejpam-5153	94	68	c.	c.	NOUN
ejpam-5153	94	69	•	•	NOUN
ejpam-5153	94	70	if	if	SCONJ
ejpam-5153	94	71	a	a	DET
ejpam-5153	94	72	∼=	∼=	PROPN
ejpam-5153	94	73	l(0	l(0	PROPN
ejpam-5153	94	74	,	,	PUNCT
ejpam-5153	94	75	β	β	NOUN
ejpam-5153	94	76	,	,	PUNCT
ejpam-5153	94	77	1	1	NUM
ejpam-5153	94	78	,	,	PUNCT
ejpam-5153	94	79	0	0	NUM
ejpam-5153	94	80	)	)	PUNCT
ejpam-5153	94	81	,	,	PUNCT
ejpam-5153	94	82	we	we	PRON
ejpam-5153	94	83	have	have	VERB
ejpam-5153	94	84	(	(	PUNCT
ejpam-5153	94	85	e+u)2e	e+u)2e	X
ejpam-5153	94	86	=	=	PUNCT
ejpam-5153	94	87	(	(	PUNCT
ejpam-5153	94	88	e+u)2	e+u)2	ADV
ejpam-5153	94	89	⇒	⇒	NOUN
ejpam-5153	94	90	β	β	X
ejpam-5153	94	91	=	=	SYM
ejpam-5153	94	92	−1	−1	NOUN
ejpam-5153	94	93	thus	thus	ADV
ejpam-5153	94	94	a	a	DET
ejpam-5153	94	95	∼=	∼=	PROPN
ejpam-5153	94	96	l(0	l(0	PROPN
ejpam-5153	94	97	,	,	PUNCT
ejpam-5153	94	98	−1	−1	NOUN
ejpam-5153	94	99	,	,	PUNCT
ejpam-5153	94	100	1	1	NUM
ejpam-5153	94	101	,	,	PUNCT
ejpam-5153	94	102	0	0	NUM
ejpam-5153	94	103	)	)	PUNCT
ejpam-5153	94	104	∼=⋆	∼=⋆	PROPN
ejpam-5153	94	105	c.	c.	NOUN
ejpam-5153	94	106	•	•	INTJ
ejpam-5153	94	107	if	if	SCONJ
ejpam-5153	94	108	a	a	DET
ejpam-5153	94	109	∼=	∼=	PROPN
ejpam-5153	94	110	l(1	l(1	PROPN
ejpam-5153	94	111	,	,	PUNCT
ejpam-5153	94	112	β	β	X
ejpam-5153	94	113	,	,	PUNCT
ejpam-5153	94	114	γ	γ	PROPN
ejpam-5153	94	115	,	,	PUNCT
ejpam-5153	94	116	0	0	NUM
ejpam-5153	94	117	)	)	PUNCT
ejpam-5153	94	118	or	or	CCONJ
ejpam-5153	94	119	l(1	l(1	PROPN
ejpam-5153	94	120	,	,	PUNCT
ejpam-5153	94	121	−1	−1	NOUN
ejpam-5153	94	122	,	,	PUNCT
ejpam-5153	94	123	γ	γ	X
ejpam-5153	94	124	,	,	PUNCT
ejpam-5153	94	125	1	1	NUM
ejpam-5153	94	126	)	)	PUNCT
ejpam-5153	94	127	,	,	PUNCT
ejpam-5153	94	128	then	then	ADV
ejpam-5153	94	129	a	a	PRON
ejpam-5153	94	130	does	do	AUX
ejpam-5153	94	131	not	not	PART
ejpam-5153	94	132	satisfy	satisfy	VERB
ejpam-5153	94	133	(	(	PUNCT
ejpam-5153	94	134	e3	e3	NOUN
ejpam-5153	94	135	)	)	PUNCT
ejpam-5153	94	136	.	.	PUNCT
ejpam-5153	95	1	indeed	indeed	ADV
ejpam-5153	95	2	(	(	PUNCT
ejpam-5153	95	3	e	e	NOUN
ejpam-5153	95	4	+	+	PUNCT
ejpam-5153	95	5	u)2e	u)2e	ADP
ejpam-5153	95	6	̸=	̸=	PROPN
ejpam-5153	95	7	(	(	PUNCT
ejpam-5153	95	8	e	e	NOUN
ejpam-5153	95	9	+	+	CCONJ
ejpam-5153	95	10	u)2	u)2	ADJ
ejpam-5153	95	11	.	.	PUNCT
ejpam-5153	96	1	(	(	PUNCT
ejpam-5153	96	2	2	2	X
ejpam-5153	96	3	)	)	PUNCT
ejpam-5153	96	4	⇒	⇒	NOUN
ejpam-5153	96	5	(	(	PUNCT
ejpam-5153	96	6	1	1	NUM
ejpam-5153	96	7	)	)	PUNCT
ejpam-5153	96	8	obvious	obvious	ADJ
ejpam-5153	96	9	.	.	PUNCT
ejpam-5153	97	1	proposition	proposition	NOUN
ejpam-5153	97	2	2	2	NUM
ejpam-5153	97	3	.	.	PUNCT
ejpam-5153	97	4	let	let	VERB
ejpam-5153	97	5	a	a	PRON
ejpam-5153	97	6	be	be	AUX
ejpam-5153	97	7	a	a	DET
ejpam-5153	97	8	two	two	NUM
ejpam-5153	97	9	-	-	PUNCT
ejpam-5153	97	10	dimensional	dimensional	ADJ
ejpam-5153	97	11	real	real	ADJ
ejpam-5153	97	12	division	division	NOUN
ejpam-5153	97	13	algebra	algebra	NOUN
ejpam-5153	97	14	with	with	ADP
ejpam-5153	97	15	left	left	ADJ
ejpam-5153	97	16	unit	unit	NOUN
ejpam-5153	97	17	e.	e.	PROPN
ejpam-5153	98	1	the	the	DET
ejpam-5153	98	2	following	follow	VERB
ejpam-5153	98	3	propositions	proposition	NOUN
ejpam-5153	98	4	are	be	AUX
ejpam-5153	98	5	equivalent	equivalent	ADJ
ejpam-5153	98	6	:	:	PUNCT
ejpam-5153	98	7	(	(	PUNCT
ejpam-5153	98	8	1	1	X
ejpam-5153	98	9	)	)	PUNCT
ejpam-5153	98	10	a	a	DET
ejpam-5153	98	11	satisfies	satisfie	NOUN
ejpam-5153	98	12	to	to	ADP
ejpam-5153	98	13	(	(	PUNCT
ejpam-5153	98	14	e2	e2	PROPN
ejpam-5153	98	15	)	)	PUNCT
ejpam-5153	98	16	,	,	PUNCT
ejpam-5153	98	17	(	(	PUNCT
ejpam-5153	98	18	2	2	X
ejpam-5153	98	19	)	)	PUNCT
ejpam-5153	98	20	a	a	DET
ejpam-5153	98	21	satisfies	satisfie	NOUN
ejpam-5153	98	22	to	to	ADP
ejpam-5153	98	23	(	(	PUNCT
ejpam-5153	98	24	e3	e3	NOUN
ejpam-5153	98	25	)	)	PUNCT
ejpam-5153	98	26	,	,	PUNCT
ejpam-5153	98	27	(	(	PUNCT
ejpam-5153	98	28	3	3	X
ejpam-5153	98	29	)	)	PUNCT
ejpam-5153	98	30	a	a	PRON
ejpam-5153	98	31	is	be	AUX
ejpam-5153	98	32	isomorphic	isomorphic	ADJ
ejpam-5153	98	33	to	to	ADP
ejpam-5153	98	34	either	either	CCONJ
ejpam-5153	98	35	c	c	PROPN
ejpam-5153	98	36	,	,	PUNCT
ejpam-5153	98	37	⋆c	⋆c	PROPN
ejpam-5153	98	38	.	.	PUNCT
ejpam-5153	99	1	proof	proof	NOUN
ejpam-5153	99	2	.	.	PUNCT
ejpam-5153	100	1	(	(	PUNCT
ejpam-5153	100	2	1	1	X
ejpam-5153	100	3	)	)	PUNCT
ejpam-5153	100	4	=	=	NOUN
ejpam-5153	100	5	⇒	⇒	NOUN
ejpam-5153	100	6	(	(	PUNCT
ejpam-5153	100	7	2	2	X
ejpam-5153	100	8	)	)	PUNCT
ejpam-5153	100	9	a	a	DET
ejpam-5153	100	10	satisfies	satisfie	NOUN
ejpam-5153	100	11	to	to	ADP
ejpam-5153	100	12	(	(	PUNCT
ejpam-5153	100	13	e2	e2	PROPN
ejpam-5153	100	14	)	)	PUNCT
ejpam-5153	100	15	,	,	PUNCT
ejpam-5153	100	16	then	then	ADV
ejpam-5153	100	17	a	a	DET
ejpam-5153	100	18	satisfies	satisfie	NOUN
ejpam-5153	100	19	to	to	ADP
ejpam-5153	100	20	(	(	PUNCT
ejpam-5153	100	21	e4	e4	PROPN
ejpam-5153	100	22	)	)	PUNCT
ejpam-5153	100	23	,	,	PUNCT
ejpam-5153	100	24	the	the	DET
ejpam-5153	100	25	lemma	lemma	PROPN
ejpam-5153	100	26	3	3	NUM
ejpam-5153	100	27	shows	show	VERB
ejpam-5153	100	28	that	that	SCONJ
ejpam-5153	100	29	a	a	DET
ejpam-5153	100	30	isomorphic	isomorphic	NOUN
ejpam-5153	100	31	to	to	ADP
ejpam-5153	100	32	either	either	CCONJ
ejpam-5153	100	33	c	c	PROPN
ejpam-5153	100	34	,	,	PUNCT
ejpam-5153	100	35	⋆c	⋆c	PROPN
ejpam-5153	100	36	,	,	PUNCT
ejpam-5153	100	37	l(1	l(1	PROPN
ejpam-5153	100	38	,	,	PUNCT
ejpam-5153	100	39	−1	−1	NOUN
ejpam-5153	100	40	,	,	PUNCT
ejpam-5153	100	41	γ	γ	X
ejpam-5153	100	42	,	,	PUNCT
ejpam-5153	100	43	1	1	NUM
ejpam-5153	100	44	)	)	PUNCT
ejpam-5153	100	45	.	.	PUNCT
ejpam-5153	101	1	therefore	therefore	ADV
ejpam-5153	101	2	a	a	DET
ejpam-5153	101	3	isomorphic	isomorphic	NOUN
ejpam-5153	101	4	to	to	ADP
ejpam-5153	101	5	either	either	CCONJ
ejpam-5153	101	6	c	c	PROPN
ejpam-5153	101	7	,	,	PUNCT
ejpam-5153	101	8	⋆c	⋆c	PROPN
ejpam-5153	101	9	.	.	PUNCT
ejpam-5153	101	10	because	because	SCONJ
ejpam-5153	101	11	a.	a.	PROPN
ejpam-5153	101	12	s.	s.	PROPN
ejpam-5153	101	13	diabang	diabang	PROPN
ejpam-5153	101	14	,	,	PUNCT
ejpam-5153	101	15	a.	a.	PROPN
ejpam-5153	101	16	s.	s.	PROPN
ejpam-5153	101	17	mballo	mballo	PROPN
ejpam-5153	101	18	,	,	PUNCT
ejpam-5153	101	19	p.	p.	PROPN
ejpam-5153	101	20	c.	c.	PROPN
ejpam-5153	101	21	diop	diop	PROPN
ejpam-5153	101	22	/	/	SYM
ejpam-5153	101	23	eur	eur	PROPN
ejpam-5153	101	24	.	.	PUNCT
ejpam-5153	102	1	j.	j.	PROPN
ejpam-5153	102	2	pure	pure	PROPN
ejpam-5153	102	3	appl	appl	PROPN
ejpam-5153	102	4	.	.	PROPN
ejpam-5153	102	5	math	math	PROPN
ejpam-5153	102	6	,	,	PUNCT
ejpam-5153	102	7	17	17	NUM
ejpam-5153	102	8	(	(	PUNCT
ejpam-5153	102	9	3	3	NUM
ejpam-5153	102	10	)	)	PUNCT
ejpam-5153	102	11	(	(	PUNCT
ejpam-5153	102	12	2024	2024	NUM
ejpam-5153	102	13	)	)	PUNCT
ejpam-5153	102	14	,	,	PUNCT
ejpam-5153	102	15	2276	2276	NUM
ejpam-5153	102	16	-	-	SYM
ejpam-5153	102	17	2287	2287	NUM
ejpam-5153	102	18	2280	2280	NUM
ejpam-5153	102	19	l(1	l(1	NOUN
ejpam-5153	102	20	,	,	PUNCT
ejpam-5153	102	21	−1	−1	NOUN
ejpam-5153	102	22	,	,	PUNCT
ejpam-5153	102	23	γ	γ	X
ejpam-5153	102	24	,	,	PUNCT
ejpam-5153	102	25	1	1	NUM
ejpam-5153	102	26	)	)	PUNCT
ejpam-5153	102	27	does	do	AUX
ejpam-5153	102	28	not	not	PART
ejpam-5153	102	29	satisfy	satisfy	VERB
ejpam-5153	102	30	to	to	ADP
ejpam-5153	102	31	(	(	PUNCT
ejpam-5153	102	32	e2	e2	PROPN
ejpam-5153	102	33	)	)	PUNCT
ejpam-5153	102	34	,	,	PUNCT
ejpam-5153	102	35	indeed	indeed	ADV
ejpam-5153	102	36	for	for	ADP
ejpam-5153	102	37	this	this	DET
ejpam-5153	102	38	algebra	algebra	NOUN
ejpam-5153	102	39	(	(	PUNCT
ejpam-5153	102	40	u2	u2	PROPN
ejpam-5153	102	41	,	,	PUNCT
ejpam-5153	102	42	u2	u2	NOUN
ejpam-5153	102	43	,	,	PUNCT
ejpam-5153	102	44	u2	u2	NOUN
ejpam-5153	102	45	)	)	PUNCT
ejpam-5153	102	46	̸=	̸=	PROPN
ejpam-5153	102	47	0	0	NUM
ejpam-5153	102	48	.	.	PUNCT
ejpam-5153	103	1	as	as	ADP
ejpam-5153	103	2	a	a	DET
ejpam-5153	103	3	result	result	NOUN
ejpam-5153	103	4	a	a	DET
ejpam-5153	103	5	satisfies	satisfie	NOUN
ejpam-5153	103	6	to	to	ADP
ejpam-5153	103	7	(	(	PUNCT
ejpam-5153	103	8	e3	e3	NOUN
ejpam-5153	103	9	)	)	PUNCT
ejpam-5153	103	10	;	;	PUNCT
ejpam-5153	103	11	(	(	PUNCT
ejpam-5153	103	12	2	2	X
ejpam-5153	103	13	)	)	PUNCT
ejpam-5153	103	14	=	=	NOUN
ejpam-5153	103	15	⇒	⇒	NOUN
ejpam-5153	103	16	(	(	PUNCT
ejpam-5153	103	17	3	3	X
ejpam-5153	103	18	)	)	PUNCT
ejpam-5153	103	19	lemma	lemma	PROPN
ejpam-5153	103	20	4	4	NUM
ejpam-5153	103	21	gives	give	VERB
ejpam-5153	103	22	the	the	DET
ejpam-5153	103	23	result	result	NOUN
ejpam-5153	103	24	;	;	PUNCT
ejpam-5153	103	25	(	(	PUNCT
ejpam-5153	103	26	3	3	X
ejpam-5153	103	27	)	)	PUNCT
ejpam-5153	103	28	=	=	NOUN
ejpam-5153	103	29	⇒	⇒	NOUN
ejpam-5153	103	30	(	(	PUNCT
ejpam-5153	103	31	1	1	NUM
ejpam-5153	103	32	)	)	PUNCT
ejpam-5153	103	33	obvious	obvious	ADJ
ejpam-5153	103	34	.	.	PUNCT
ejpam-5153	104	1	proposition	proposition	NOUN
ejpam-5153	104	2	3	3	X
ejpam-5153	104	3	.	.	PUNCT
ejpam-5153	104	4	let	let	VERB
ejpam-5153	104	5	a	a	PRON
ejpam-5153	104	6	be	be	AUX
ejpam-5153	104	7	a	a	DET
ejpam-5153	104	8	two	two	NUM
ejpam-5153	104	9	-	-	PUNCT
ejpam-5153	104	10	dimensional	dimensional	ADJ
ejpam-5153	104	11	real	real	ADJ
ejpam-5153	104	12	division	division	NOUN
ejpam-5153	104	13	algebra	algebra	NOUN
ejpam-5153	104	14	with	with	ADP
ejpam-5153	104	15	left	left	ADJ
ejpam-5153	104	16	unit	unit	NOUN
ejpam-5153	104	17	e.	e.	PROPN
ejpam-5153	105	1	the	the	DET
ejpam-5153	105	2	following	follow	VERB
ejpam-5153	105	3	propositions	proposition	NOUN
ejpam-5153	105	4	are	be	AUX
ejpam-5153	105	5	equivalent	equivalent	ADJ
ejpam-5153	105	6	:	:	PUNCT
ejpam-5153	105	7	(	(	PUNCT
ejpam-5153	105	8	1	1	X
ejpam-5153	105	9	)	)	PUNCT
ejpam-5153	105	10	a	a	PRON
ejpam-5153	105	11	is	be	AUX
ejpam-5153	105	12	commutative	commutative	ADJ
ejpam-5153	105	13	,	,	PUNCT
ejpam-5153	105	14	(	(	PUNCT
ejpam-5153	105	15	2	2	X
ejpam-5153	105	16	)	)	PUNCT
ejpam-5153	105	17	a	a	PRON
ejpam-5153	105	18	is	be	AUX
ejpam-5153	105	19	power	power	NOUN
ejpam-5153	105	20	-	-	PUNCT
ejpam-5153	105	21	commutative	commutative	ADJ
ejpam-5153	105	22	;	;	PUNCT
ejpam-5153	105	23	(	(	PUNCT
ejpam-5153	105	24	3	3	X
ejpam-5153	105	25	)	)	PUNCT
ejpam-5153	105	26	a	a	DET
ejpam-5153	105	27	satisfies	satisfie	NOUN
ejpam-5153	105	28	to	to	ADP
ejpam-5153	105	29	(	(	PUNCT
ejpam-5153	105	30	e1	e1	PROPN
ejpam-5153	105	31	)	)	PUNCT
ejpam-5153	105	32	;	;	PUNCT
ejpam-5153	105	33	(	(	PUNCT
ejpam-5153	105	34	4	4	X
ejpam-5153	105	35	)	)	PUNCT
ejpam-5153	105	36	a	a	PRON
ejpam-5153	105	37	is	be	AUX
ejpam-5153	105	38	isomorphic	isomorphic	ADJ
ejpam-5153	105	39	to	to	ADP
ejpam-5153	105	40	c.	c.	PROPN
ejpam-5153	105	41	(	(	PUNCT
ejpam-5153	105	42	5	5	NUM
ejpam-5153	105	43	)	)	PUNCT
ejpam-5153	105	44	a	a	PRON
ejpam-5153	105	45	is	be	AUX
ejpam-5153	105	46	(	(	PUNCT
ejpam-5153	105	47	121	121	NUM
ejpam-5153	105	48	)	)	PUNCT
ejpam-5153	105	49	power	power	NOUN
ejpam-5153	105	50	-	-	PUNCT
ejpam-5153	105	51	associative	associative	NOUN
ejpam-5153	105	52	.	.	PUNCT
ejpam-5153	106	1	proof	proof	NOUN
ejpam-5153	106	2	.	.	PUNCT
ejpam-5153	107	1	in	in	ADP
ejpam-5153	107	2	[	[	X
ejpam-5153	107	3	10	10	NUM
ejpam-5153	107	4	]	]	PUNCT
ejpam-5153	107	5	,	,	PUNCT
ejpam-5153	107	6	they	they	PRON
ejpam-5153	107	7	proved	prove	VERB
ejpam-5153	107	8	in	in	ADP
ejpam-5153	107	9	theorem	theorem	NOUN
ejpam-5153	107	10	1	1	NUM
ejpam-5153	107	11	that	that	SCONJ
ejpam-5153	107	12	(	(	PUNCT
ejpam-5153	107	13	1	1	X
ejpam-5153	107	14	)	)	PUNCT
ejpam-5153	107	15	⇔	⇔	X
ejpam-5153	107	16	(	(	PUNCT
ejpam-5153	107	17	2	2	NUM
ejpam-5153	107	18	)	)	PUNCT
ejpam-5153	107	19	⇔	⇔	X
ejpam-5153	107	20	(	(	PUNCT
ejpam-5153	107	21	3	3	NUM
ejpam-5153	107	22	)	)	PUNCT
ejpam-5153	107	23	⇔	⇔	X
ejpam-5153	107	24	(	(	PUNCT
ejpam-5153	107	25	5	5	NUM
ejpam-5153	107	26	)	)	PUNCT
ejpam-5153	107	27	.	.	PUNCT
ejpam-5153	108	1	show	show	VERB
ejpam-5153	108	2	that	that	SCONJ
ejpam-5153	108	3	(	(	PUNCT
ejpam-5153	108	4	3	3	X
ejpam-5153	108	5	)	)	PUNCT
ejpam-5153	108	6	⇒	⇒	NOUN
ejpam-5153	108	7	(	(	PUNCT
ejpam-5153	108	8	4	4	X
ejpam-5153	108	9	)	)	PUNCT
ejpam-5153	108	10	a	a	DET
ejpam-5153	108	11	satisfies	satisfie	NOUN
ejpam-5153	108	12	to	to	ADP
ejpam-5153	108	13	(	(	PUNCT
ejpam-5153	108	14	e1	e1	PROPN
ejpam-5153	108	15	)	)	PUNCT
ejpam-5153	108	16	,	,	PUNCT
ejpam-5153	108	17	then	then	ADV
ejpam-5153	108	18	satisfies	satisfy	VERB
ejpam-5153	108	19	to	to	ADP
ejpam-5153	108	20	(	(	PUNCT
ejpam-5153	108	21	e2	e2	PROPN
ejpam-5153	108	22	)	)	PUNCT
ejpam-5153	108	23	.	.	PUNCT
ejpam-5153	109	1	the	the	DET
ejpam-5153	109	2	proposition	proposition	NOUN
ejpam-5153	109	3	2	2	NUM
ejpam-5153	109	4	shows	show	VERB
ejpam-5153	109	5	that	that	SCONJ
ejpam-5153	109	6	a	a	PRON
ejpam-5153	109	7	is	be	AUX
ejpam-5153	109	8	isomorphic	isomorphic	ADJ
ejpam-5153	109	9	to	to	ADP
ejpam-5153	109	10	either	either	CCONJ
ejpam-5153	109	11	c	c	PROPN
ejpam-5153	109	12	,	,	PUNCT
ejpam-5153	109	13	⋆c	⋆c	PROPN
ejpam-5153	109	14	.	.	PUNCT
ejpam-5153	110	1	consequently	consequently	ADV
ejpam-5153	110	2	a	a	PRON
ejpam-5153	110	3	is	be	AUX
ejpam-5153	110	4	isomorphic	isomorphic	ADJ
ejpam-5153	110	5	to	to	ADP
ejpam-5153	110	6	c.	c.	NOUN
ejpam-5153	110	7	(	(	PUNCT
ejpam-5153	110	8	4	4	NUM
ejpam-5153	110	9	)	)	PUNCT
ejpam-5153	110	10	⇒	⇒	NOUN
ejpam-5153	110	11	(	(	PUNCT
ejpam-5153	110	12	5	5	NUM
ejpam-5153	110	13	)	)	PUNCT
ejpam-5153	110	14	obvious	obvious	ADJ
ejpam-5153	110	15	,	,	PUNCT
ejpam-5153	110	16	corollary	corollary	ADJ
ejpam-5153	110	17	1	1	NUM
ejpam-5153	110	18	.	.	PUNCT
ejpam-5153	111	1	let	let	VERB
ejpam-5153	111	2	a	a	DET
ejpam-5153	111	3	be	be	AUX
ejpam-5153	111	4	a	a	DET
ejpam-5153	111	5	n	n	ADV
ejpam-5153	111	6	-	-	PUNCT
ejpam-5153	111	7	dimensional	dimensional	ADJ
ejpam-5153	111	8	real	real	ADJ
ejpam-5153	111	9	division	division	NOUN
ejpam-5153	111	10	algebra	algebra	NOUN
ejpam-5153	111	11	with	with	ADP
ejpam-5153	111	12	left	left	ADJ
ejpam-5153	111	13	unit	unit	NOUN
ejpam-5153	111	14	and	and	CCONJ
ejpam-5153	111	15	n	n	PRON
ejpam-5153	111	16	≤	≤	NOUN
ejpam-5153	111	17	2	2	NUM
ejpam-5153	111	18	.	.	PUNCT
ejpam-5153	112	1	let	let	VERB
ejpam-5153	112	2	γ	γ	X
ejpam-5153	112	3	∈	∈	VERB
ejpam-5153	112	4	r	r	NOUN
ejpam-5153	112	5	with	with	ADP
ejpam-5153	112	6	γ	γ	X
ejpam-5153	112	7	>	>	X
ejpam-5153	112	8	0	0	NUM
ejpam-5153	112	9	.	.	PUNCT
ejpam-5153	113	1	we	we	PRON
ejpam-5153	113	2	have	have	VERB
ejpam-5153	113	3	a	a	DET
ejpam-5153	113	4	satisfies	satisfie	NOUN
ejpam-5153	113	5	to	to	ADP
ejpam-5153	113	6	(	(	PUNCT
ejpam-5153	113	7	e1	e1	PROPN
ejpam-5153	113	8	)	)	PUNCT
ejpam-5153	113	9	(	(	PUNCT
ejpam-5153	113	10	e2	e2	PROPN
ejpam-5153	113	11	)	)	PUNCT
ejpam-5153	113	12	(	(	PUNCT
ejpam-5153	113	13	e3	e3	NOUN
ejpam-5153	113	14	)	)	PUNCT
ejpam-5153	113	15	(	(	PUNCT
ejpam-5153	113	16	e4	e4	PROPN
ejpam-5153	113	17	)	)	PUNCT
ejpam-5153	113	18	a	a	DET
ejpam-5153	113	19	isomorphic	isomorphic	ADJ
ejpam-5153	113	20	to	to	ADP
ejpam-5153	113	21	r	r	NOUN
ejpam-5153	113	22	;	;	PUNCT
ejpam-5153	113	23	c	c	NOUN
ejpam-5153	113	24	r	r	NOUN
ejpam-5153	113	25	;	;	PUNCT
ejpam-5153	113	26	c	c	X
ejpam-5153	113	27	;	;	PUNCT
ejpam-5153	113	28	⋆c	⋆c	PROPN
ejpam-5153	113	29	r	r	X
ejpam-5153	113	30	;	;	PUNCT
ejpam-5153	113	31	c	c	X
ejpam-5153	113	32	;	;	PUNCT
ejpam-5153	113	33	⋆c	⋆c	PROPN
ejpam-5153	113	34	r	r	X
ejpam-5153	113	35	;	;	PUNCT
ejpam-5153	113	36	c	c	X
ejpam-5153	113	37	;	;	PUNCT
ejpam-5153	113	38	⋆c	⋆c	NOUN
ejpam-5153	113	39	;	;	PUNCT
ejpam-5153	113	40	l(1	l(1	PROPN
ejpam-5153	113	41	,	,	PUNCT
ejpam-5153	113	42	−1	−1	NOUN
ejpam-5153	113	43	,	,	PUNCT
ejpam-5153	113	44	γ	γ	X
ejpam-5153	113	45	,	,	PUNCT
ejpam-5153	113	46	1	1	NUM
ejpam-5153	113	47	)	)	PUNCT
ejpam-5153	113	48	and	and	CCONJ
ejpam-5153	113	49	we	we	PRON
ejpam-5153	113	50	have	have	VERB
ejpam-5153	113	51	the	the	DET
ejpam-5153	113	52	result	result	NOUN
ejpam-5153	113	53	(	(	PUNCT
ejpam-5153	113	54	e1	e1	NOUN
ejpam-5153	113	55	)	)	PUNCT
ejpam-5153	113	56	⇒	⇒	NOUN
ejpam-5153	113	57	(	(	PUNCT
ejpam-5153	113	58	e2	e2	PROPN
ejpam-5153	113	59	)	)	PUNCT
ejpam-5153	113	60	⇔	⇔	X
ejpam-5153	113	61	(	(	PUNCT
ejpam-5153	113	62	e3	e3	NOUN
ejpam-5153	113	63	)	)	PUNCT
ejpam-5153	113	64	⇒	⇒	NOUN
ejpam-5153	113	65	(	(	PUNCT
ejpam-5153	113	66	e4	e4	PROPN
ejpam-5153	113	67	)	)	PUNCT
ejpam-5153	113	68	.	.	PUNCT
ejpam-5153	114	1	4	4	X
ejpam-5153	114	2	.	.	X
ejpam-5153	114	3	on	on	ADP
ejpam-5153	114	4	finite	finite	ADJ
ejpam-5153	114	5	-	-	ADJ
ejpam-5153	114	6	dimensional	dimensional	ADJ
ejpam-5153	114	7	real	real	ADJ
ejpam-5153	114	8	division	division	NOUN
ejpam-5153	114	9	algebras	algebra	VERB
ejpam-5153	114	10	with	with	ADP
ejpam-5153	114	11	left	left	ADJ
ejpam-5153	114	12	unit	unit	NOUN
ejpam-5153	114	13	.	.	PUNCT
ejpam-5153	115	1	we	we	PRON
ejpam-5153	115	2	denote	denote	VERB
ejpam-5153	115	3	ae	ae	PROPN
ejpam-5153	115	4	=	=	PUNCT
ejpam-5153	115	5	{	{	PUNCT
ejpam-5153	115	6	x	x	PUNCT
ejpam-5153	115	7	∈	∈	PROPN
ejpam-5153	115	8	a	a	X
ejpam-5153	115	9	,	,	PUNCT
ejpam-5153	115	10	xe	xe	PROPN
ejpam-5153	115	11	=	=	SYM
ejpam-5153	115	12	x	x	PROPN
ejpam-5153	115	13	}	}	PUNCT
ejpam-5153	115	14	for	for	ADP
ejpam-5153	115	15	all	all	DET
ejpam-5153	115	16	x	x	SYM
ejpam-5153	115	17	∈	∈	PROPN
ejpam-5153	115	18	a.	a.	NOUN
ejpam-5153	115	19	lemma	lemma	PROPN
ejpam-5153	115	20	5	5	X
ejpam-5153	115	21	.	.	PUNCT
ejpam-5153	115	22	let	let	VERB
ejpam-5153	115	23	a	a	PRON
ejpam-5153	115	24	be	be	AUX
ejpam-5153	115	25	a	a	DET
ejpam-5153	115	26	finite	finite	ADJ
ejpam-5153	115	27	-	-	ADJ
ejpam-5153	115	28	dimensional	dimensional	ADJ
ejpam-5153	115	29	real	real	ADJ
ejpam-5153	115	30	division	division	NOUN
ejpam-5153	115	31	algebra	algebra	NOUN
ejpam-5153	115	32	with	with	ADP
ejpam-5153	115	33	left	left	ADJ
ejpam-5153	115	34	unit	unit	NOUN
ejpam-5153	115	35	e	e	NOUN
ejpam-5153	115	36	,	,	PUNCT
ejpam-5153	115	37	satisfies	satisfie	NOUN
ejpam-5153	115	38	to	to	ADP
ejpam-5153	115	39	(	(	PUNCT
ejpam-5153	115	40	e3	e3	NOUN
ejpam-5153	115	41	)	)	PUNCT
ejpam-5153	115	42	.	.	PUNCT
ejpam-5153	116	1	then	then	ADV
ejpam-5153	116	2	for	for	ADP
ejpam-5153	116	3	all	all	PRON
ejpam-5153	116	4	x	x	SYM
ejpam-5153	116	5	and	and	CCONJ
ejpam-5153	116	6	y	y	PROPN
ejpam-5153	116	7	∈	∈	PROPN
ejpam-5153	116	8	a	a	PRON
ejpam-5153	116	9	;	;	PUNCT
ejpam-5153	116	10	xy	xy	PROPN
ejpam-5153	116	11	+	+	SYM
ejpam-5153	116	12	yx	yx	PROPN
ejpam-5153	116	13	∈	∈	PROPN
ejpam-5153	116	14	ae	ae	PROPN
ejpam-5153	116	15	.	.	PUNCT
ejpam-5153	117	1	proof	proof	NOUN
ejpam-5153	117	2	.	.	PUNCT
ejpam-5153	118	1	immediate	immediate	ADJ
ejpam-5153	118	2	consequence	consequence	NOUN
ejpam-5153	118	3	of	of	ADP
ejpam-5153	118	4	equality	equality	NOUN
ejpam-5153	118	5	(	(	PUNCT
ejpam-5153	118	6	x	x	SYM
ejpam-5153	118	7	+	+	NUM
ejpam-5153	118	8	y)2e	y)2e	NOUN
ejpam-5153	118	9	=	=	SYM
ejpam-5153	118	10	(	(	PUNCT
ejpam-5153	118	11	x	x	X
ejpam-5153	118	12	+	+	PUNCT
ejpam-5153	118	13	y)2	y)2	NOUN
ejpam-5153	118	14	.	.	PUNCT
ejpam-5153	119	1	proposition	proposition	NOUN
ejpam-5153	119	2	4	4	NUM
ejpam-5153	119	3	.	.	PUNCT
ejpam-5153	120	1	let	let	VERB
ejpam-5153	120	2	a	a	PRON
ejpam-5153	120	3	be	be	AUX
ejpam-5153	120	4	a	a	DET
ejpam-5153	120	5	finite	finite	ADJ
ejpam-5153	120	6	-	-	ADJ
ejpam-5153	120	7	dimensional	dimensional	ADJ
ejpam-5153	120	8	real	real	ADJ
ejpam-5153	120	9	division	division	NOUN
ejpam-5153	120	10	algebra	algebra	NOUN
ejpam-5153	120	11	with	with	ADP
ejpam-5153	120	12	left	left	ADJ
ejpam-5153	120	13	unit	unit	NOUN
ejpam-5153	120	14	e	e	NOUN
ejpam-5153	120	15	,	,	PUNCT
ejpam-5153	120	16	satisfies	satisfie	NOUN
ejpam-5153	120	17	to	to	ADP
ejpam-5153	120	18	(	(	PUNCT
ejpam-5153	120	19	e1	e1	PROPN
ejpam-5153	120	20	)	)	PUNCT
ejpam-5153	120	21	.	.	PUNCT
ejpam-5153	121	1	the	the	DET
ejpam-5153	121	2	following	follow	VERB
ejpam-5153	121	3	propositions	proposition	NOUN
ejpam-5153	121	4	are	be	AUX
ejpam-5153	121	5	equivalent	equivalent	ADJ
ejpam-5153	121	6	:	:	PUNCT
ejpam-5153	121	7	(	(	PUNCT
ejpam-5153	121	8	1	1	X
ejpam-5153	121	9	)	)	PUNCT
ejpam-5153	121	10	a	a	DET
ejpam-5153	121	11	satisfies	satisfie	NOUN
ejpam-5153	121	12	to	to	ADP
ejpam-5153	121	13	(	(	PUNCT
ejpam-5153	121	14	e4	e4	PROPN
ejpam-5153	121	15	)	)	PUNCT
ejpam-5153	121	16	,	,	PUNCT
ejpam-5153	121	17	(	(	PUNCT
ejpam-5153	121	18	2	2	X
ejpam-5153	121	19	)	)	PUNCT
ejpam-5153	121	20	e	e	NOUN
ejpam-5153	121	21	is	be	AUX
ejpam-5153	121	22	the	the	DET
ejpam-5153	121	23	unit	unit	NOUN
ejpam-5153	121	24	element	element	NOUN
ejpam-5153	121	25	.	.	PUNCT
ejpam-5153	122	1	proof	proof	NOUN
ejpam-5153	122	2	.	.	PUNCT
ejpam-5153	123	1	(	(	PUNCT
ejpam-5153	123	2	1	1	X
ejpam-5153	123	3	)	)	PUNCT
ejpam-5153	123	4	=	=	NOUN
ejpam-5153	123	5	⇒	⇒	NOUN
ejpam-5153	123	6	(	(	PUNCT
ejpam-5153	123	7	2	2	X
ejpam-5153	123	8	)	)	PUNCT
ejpam-5153	123	9	let	let	VERB
ejpam-5153	123	10	x	x	PUNCT
ejpam-5153	123	11	∈	∈	PROPN
ejpam-5153	123	12	a	a	PRON
ejpam-5153	123	13	,	,	PUNCT
ejpam-5153	123	14	as	as	ADP
ejpam-5153	123	15	a	a	DET
ejpam-5153	123	16	satisfies	satisfie	NOUN
ejpam-5153	123	17	to	to	ADP
ejpam-5153	123	18	(	(	PUNCT
ejpam-5153	123	19	e1	e1	PROPN
ejpam-5153	123	20	)	)	PUNCT
ejpam-5153	123	21	,	,	PUNCT
ejpam-5153	123	22	then	then	ADV
ejpam-5153	123	23	a	a	DET
ejpam-5153	123	24	checks	check	NOUN
ejpam-5153	123	25	the	the	DET
ejpam-5153	123	26	equation	equation	NOUN
ejpam-5153	123	27	(	(	PUNCT
ejpam-5153	123	28	2.1	2.1	NUM
ejpam-5153	123	29	)	)	PUNCT
ejpam-5153	123	30	of	of	ADP
ejpam-5153	123	31	[	[	X
ejpam-5153	123	32	10	10	NUM
ejpam-5153	123	33	]	]	PUNCT
ejpam-5153	123	34	.	.	PUNCT
ejpam-5153	124	1	we	we	PRON
ejpam-5153	124	2	have	have	VERB
ejpam-5153	124	3	[	[	X
ejpam-5153	124	4	e2	e2	NOUN
ejpam-5153	124	5	,	,	PUNCT
ejpam-5153	124	6	x	x	X
ejpam-5153	124	7	]	]	X
ejpam-5153	125	1	+	+	CCONJ
ejpam-5153	126	1	[	[	X
ejpam-5153	126	2	ex	ex	X
ejpam-5153	126	3	+	+	ADJ
ejpam-5153	126	4	xe	xe	PROPN
ejpam-5153	126	5	,	,	PUNCT
ejpam-5153	126	6	e	e	X
ejpam-5153	126	7	]	]	X
ejpam-5153	126	8	=	=	SYM
ejpam-5153	126	9	0	0	X
ejpam-5153	126	10	.	.	PUNCT
ejpam-5153	127	1	the	the	DET
ejpam-5153	127	2	fact	fact	NOUN
ejpam-5153	127	3	that	that	SCONJ
ejpam-5153	127	4	a	a	DET
ejpam-5153	127	5	satisfies	satisfie	NOUN
ejpam-5153	127	6	to	to	ADP
ejpam-5153	127	7	(	(	PUNCT
ejpam-5153	127	8	e4	e4	PROPN
ejpam-5153	127	9	)	)	PUNCT
ejpam-5153	127	10	we	we	PRON
ejpam-5153	127	11	have	have	AUX
ejpam-5153	127	12	x	x	X
ejpam-5153	127	13	=	=	SYM
ejpam-5153	127	14	xe	xe	PROPN
ejpam-5153	127	15	,	,	PUNCT
ejpam-5153	127	16	as	as	ADP
ejpam-5153	127	17	a	a	DET
ejpam-5153	127	18	result	result	NOUN
ejpam-5153	127	19	e	e	NOUN
ejpam-5153	127	20	is	be	AUX
ejpam-5153	127	21	the	the	DET
ejpam-5153	127	22	unit	unit	NOUN
ejpam-5153	127	23	element	element	NOUN
ejpam-5153	127	24	.	.	PUNCT
ejpam-5153	128	1	(	(	PUNCT
ejpam-5153	128	2	2	2	X
ejpam-5153	128	3	)	)	PUNCT
ejpam-5153	128	4	=	=	NOUN
ejpam-5153	128	5	⇒	⇒	NOUN
ejpam-5153	128	6	(	(	PUNCT
ejpam-5153	128	7	1	1	NUM
ejpam-5153	128	8	)	)	PUNCT
ejpam-5153	128	9	obvious	obvious	ADJ
ejpam-5153	128	10	.	.	PUNCT
ejpam-5153	129	1	a.	a.	PROPN
ejpam-5153	129	2	s.	s.	PROPN
ejpam-5153	129	3	diabang	diabang	PROPN
ejpam-5153	129	4	,	,	PUNCT
ejpam-5153	129	5	a.	a.	PROPN
ejpam-5153	129	6	s.	s.	PROPN
ejpam-5153	129	7	mballo	mballo	PROPN
ejpam-5153	129	8	,	,	PUNCT
ejpam-5153	129	9	p.	p.	PROPN
ejpam-5153	129	10	c.	c.	PROPN
ejpam-5153	129	11	diop	diop	PROPN
ejpam-5153	129	12	/	/	SYM
ejpam-5153	129	13	eur	eur	PROPN
ejpam-5153	129	14	.	.	PUNCT
ejpam-5153	130	1	j.	j.	PROPN
ejpam-5153	130	2	pure	pure	PROPN
ejpam-5153	130	3	appl	appl	PROPN
ejpam-5153	130	4	.	.	PROPN
ejpam-5153	130	5	math	math	PROPN
ejpam-5153	130	6	,	,	PUNCT
ejpam-5153	130	7	17	17	NUM
ejpam-5153	130	8	(	(	PUNCT
ejpam-5153	130	9	3	3	NUM
ejpam-5153	130	10	)	)	PUNCT
ejpam-5153	130	11	(	(	PUNCT
ejpam-5153	130	12	2024	2024	NUM
ejpam-5153	130	13	)	)	PUNCT
ejpam-5153	130	14	,	,	PUNCT
ejpam-5153	130	15	2276	2276	NUM
ejpam-5153	130	16	-	-	SYM
ejpam-5153	130	17	2287	2287	NUM
ejpam-5153	130	18	2281	2281	NUM
ejpam-5153	130	19	corollary	corollary	NOUN
ejpam-5153	130	20	2	2	NUM
ejpam-5153	130	21	.	.	PUNCT
ejpam-5153	131	1	let	let	VERB
ejpam-5153	131	2	a	a	PRON
ejpam-5153	131	3	be	be	AUX
ejpam-5153	131	4	a	a	DET
ejpam-5153	131	5	finite	finite	ADJ
ejpam-5153	131	6	-	-	ADJ
ejpam-5153	131	7	dimensional	dimensional	ADJ
ejpam-5153	131	8	real	real	ADJ
ejpam-5153	131	9	division	division	NOUN
ejpam-5153	131	10	algebra	algebra	NOUN
ejpam-5153	131	11	with	with	ADP
ejpam-5153	131	12	left	left	ADJ
ejpam-5153	131	13	unit	unit	NOUN
ejpam-5153	131	14	e	e	NOUN
ejpam-5153	131	15	,	,	PUNCT
ejpam-5153	131	16	satisfies	satisfie	NOUN
ejpam-5153	131	17	to	to	ADP
ejpam-5153	131	18	(	(	PUNCT
ejpam-5153	131	19	e1	e1	PROPN
ejpam-5153	131	20	)	)	PUNCT
ejpam-5153	131	21	,	,	PUNCT
ejpam-5153	131	22	then	then	ADV
ejpam-5153	131	23	e	e	PROPN
ejpam-5153	131	24	is	be	AUX
ejpam-5153	131	25	the	the	DET
ejpam-5153	131	26	unit	unit	NOUN
ejpam-5153	131	27	element	element	NOUN
ejpam-5153	131	28	.	.	PUNCT
ejpam-5153	132	1	proof	proof	NOUN
ejpam-5153	132	2	.	.	PUNCT
ejpam-5153	133	1	a	a	DET
ejpam-5153	133	2	satisfies	satisfie	NOUN
ejpam-5153	133	3	to	to	ADP
ejpam-5153	133	4	(	(	PUNCT
ejpam-5153	133	5	e1	e1	PROPN
ejpam-5153	133	6	)	)	PUNCT
ejpam-5153	133	7	,	,	PUNCT
ejpam-5153	133	8	then	then	ADV
ejpam-5153	133	9	satisfies	satisfy	VERB
ejpam-5153	133	10	to	to	ADP
ejpam-5153	133	11	(	(	PUNCT
ejpam-5153	133	12	e2	e2	PROPN
ejpam-5153	133	13	)	)	PUNCT
ejpam-5153	133	14	,	,	PUNCT
ejpam-5153	133	15	as	as	ADP
ejpam-5153	133	16	a	a	DET
ejpam-5153	133	17	result	result	NOUN
ejpam-5153	133	18	a	a	DET
ejpam-5153	133	19	satisfies	satisfie	NOUN
ejpam-5153	133	20	to	to	ADP
ejpam-5153	133	21	(	(	PUNCT
ejpam-5153	133	22	e4	e4	PROPN
ejpam-5153	133	23	)	)	PUNCT
ejpam-5153	133	24	.	.	PUNCT
ejpam-5153	134	1	the	the	DET
ejpam-5153	134	2	proposition	proposition	NOUN
ejpam-5153	134	3	4	4	NUM
ejpam-5153	134	4	shows	show	VERB
ejpam-5153	134	5	that	that	SCONJ
ejpam-5153	134	6	e	e	NOUN
ejpam-5153	134	7	is	be	AUX
ejpam-5153	134	8	the	the	DET
ejpam-5153	134	9	unit	unit	NOUN
ejpam-5153	134	10	element	element	NOUN
ejpam-5153	134	11	.	.	PUNCT
ejpam-5153	135	1	proposition	proposition	NOUN
ejpam-5153	135	2	5	5	NUM
ejpam-5153	135	3	.	.	PUNCT
ejpam-5153	135	4	let	let	VERB
ejpam-5153	135	5	a	a	PRON
ejpam-5153	135	6	be	be	AUX
ejpam-5153	135	7	a	a	DET
ejpam-5153	135	8	four	four	NUM
ejpam-5153	135	9	-	-	PUNCT
ejpam-5153	135	10	dimensional	dimensional	ADJ
ejpam-5153	135	11	real	real	ADJ
ejpam-5153	135	12	division	division	NOUN
ejpam-5153	135	13	algebra	algebra	NOUN
ejpam-5153	135	14	with	with	ADP
ejpam-5153	135	15	left	left	ADJ
ejpam-5153	135	16	unit	unit	NOUN
ejpam-5153	135	17	e	e	NOUN
ejpam-5153	135	18	satisfies	satisfie	NOUN
ejpam-5153	135	19	to	to	ADP
ejpam-5153	135	20	(	(	PUNCT
ejpam-5153	135	21	e2	e2	PROPN
ejpam-5153	135	22	)	)	PUNCT
ejpam-5153	135	23	.	.	PUNCT
ejpam-5153	136	1	the	the	DET
ejpam-5153	136	2	following	follow	VERB
ejpam-5153	136	3	propositions	proposition	NOUN
ejpam-5153	136	4	are	be	AUX
ejpam-5153	136	5	equivalent	equivalent	ADJ
ejpam-5153	136	6	:	:	PUNCT
ejpam-5153	136	7	(	(	PUNCT
ejpam-5153	136	8	1	1	X
ejpam-5153	136	9	)	)	PUNCT
ejpam-5153	136	10	a	a	PRON
ejpam-5153	136	11	contains	contain	VERB
ejpam-5153	136	12	a	a	DET
ejpam-5153	136	13	central	central	ADJ
ejpam-5153	136	14	element	element	NOUN
ejpam-5153	136	15	(	(	PUNCT
ejpam-5153	136	16	2	2	NUM
ejpam-5153	136	17	)	)	PUNCT
ejpam-5153	136	18	a	a	PRON
ejpam-5153	136	19	is	be	AUX
ejpam-5153	136	20	power	power	NOUN
ejpam-5153	136	21	-	-	PUNCT
ejpam-5153	136	22	commutative	commutative	ADJ
ejpam-5153	136	23	.	.	PUNCT
ejpam-5153	137	1	proof	proof	NOUN
ejpam-5153	137	2	.	.	PUNCT
ejpam-5153	138	1	(	(	PUNCT
ejpam-5153	138	2	1	1	X
ejpam-5153	138	3	)	)	PUNCT
ejpam-5153	138	4	=	=	NOUN
ejpam-5153	138	5	⇒	⇒	NOUN
ejpam-5153	138	6	(	(	PUNCT
ejpam-5153	138	7	2	2	NUM
ejpam-5153	138	8	)	)	PUNCT
ejpam-5153	139	1	[	[	X
ejpam-5153	139	2	1	1	NUM
ejpam-5153	139	3	,	,	PUNCT
ejpam-5153	139	4	theorem	theorem	VERB
ejpam-5153	139	5	3	3	NUM
ejpam-5153	139	6	]	]	PUNCT
ejpam-5153	139	7	shows	show	VERB
ejpam-5153	139	8	that	that	SCONJ
ejpam-5153	139	9	a	a	PRON
ejpam-5153	139	10	is	be	AUX
ejpam-5153	139	11	power	power	NOUN
ejpam-5153	139	12	-	-	PUNCT
ejpam-5153	139	13	commutative	commutative	ADJ
ejpam-5153	139	14	.	.	PUNCT
ejpam-5153	140	1	(	(	PUNCT
ejpam-5153	140	2	2	2	X
ejpam-5153	140	3	)	)	PUNCT
ejpam-5153	140	4	=	=	NOUN
ejpam-5153	140	5	⇒	⇒	NOUN
ejpam-5153	140	6	(	(	PUNCT
ejpam-5153	140	7	1	1	X
ejpam-5153	140	8	)	)	PUNCT
ejpam-5153	140	9	a	a	PRON
ejpam-5153	140	10	is	be	AUX
ejpam-5153	140	11	power	power	NOUN
ejpam-5153	140	12	-	-	PUNCT
ejpam-5153	140	13	commutative	commutative	ADJ
ejpam-5153	140	14	,	,	PUNCT
ejpam-5153	140	15	then	then	ADV
ejpam-5153	140	16	satisfies	satisfie	NOUN
ejpam-5153	140	17	to	to	ADP
ejpam-5153	140	18	(	(	PUNCT
ejpam-5153	140	19	e1	e1	PROPN
ejpam-5153	140	20	)	)	PUNCT
ejpam-5153	140	21	.	.	PUNCT
ejpam-5153	141	1	therefore	therefore	ADV
ejpam-5153	141	2	satisfies	satisfy	VERB
ejpam-5153	141	3	to	to	ADP
ejpam-5153	141	4	(	(	PUNCT
ejpam-5153	141	5	e2	e2	PROPN
ejpam-5153	141	6	)	)	PUNCT
ejpam-5153	141	7	and	and	CCONJ
ejpam-5153	141	8	to	to	PART
ejpam-5153	141	9	(	(	PUNCT
ejpam-5153	141	10	e4	e4	PROPN
ejpam-5153	141	11	)	)	PUNCT
ejpam-5153	141	12	.	.	PUNCT
ejpam-5153	142	1	the	the	DET
ejpam-5153	142	2	proposition	proposition	NOUN
ejpam-5153	142	3	4	4	NUM
ejpam-5153	142	4	,	,	PUNCT
ejpam-5153	142	5	shows	show	VERB
ejpam-5153	142	6	that	that	SCONJ
ejpam-5153	142	7	e	e	NOUN
ejpam-5153	142	8	is	be	AUX
ejpam-5153	142	9	the	the	DET
ejpam-5153	142	10	unit	unit	NOUN
ejpam-5153	142	11	element	element	NOUN
ejpam-5153	142	12	.	.	PUNCT
ejpam-5153	143	1	therefore	therefore	ADV
ejpam-5153	143	2	e	e	PROPN
ejpam-5153	143	3	is	be	AUX
ejpam-5153	143	4	a	a	DET
ejpam-5153	143	5	central	central	ADJ
ejpam-5153	143	6	element	element	NOUN
ejpam-5153	143	7	of	of	ADP
ejpam-5153	143	8	a.	a.	PROPN
ejpam-5153	143	9	lemma	lemma	PROPN
ejpam-5153	143	10	6	6	NUM
ejpam-5153	143	11	.	.	PUNCT
ejpam-5153	144	1	let	let	VERB
ejpam-5153	144	2	a	a	PRON
ejpam-5153	144	3	be	be	AUX
ejpam-5153	144	4	a	a	DET
ejpam-5153	144	5	finite	finite	ADJ
ejpam-5153	144	6	-	-	ADJ
ejpam-5153	144	7	dimensional	dimensional	ADJ
ejpam-5153	144	8	real	real	ADJ
ejpam-5153	144	9	division	division	NOUN
ejpam-5153	144	10	algebra	algebra	NOUN
ejpam-5153	144	11	with	with	ADP
ejpam-5153	144	12	left	left	ADJ
ejpam-5153	144	13	unit	unit	NOUN
ejpam-5153	144	14	e	e	NOUN
ejpam-5153	144	15	satisfies	satisfie	NOUN
ejpam-5153	144	16	to	to	ADP
ejpam-5153	144	17	(	(	PUNCT
ejpam-5153	144	18	e2	e2	PROPN
ejpam-5153	144	19	)	)	PUNCT
ejpam-5153	144	20	.	.	PUNCT
ejpam-5153	145	1	if	if	SCONJ
ejpam-5153	145	2	x	x	SYM
ejpam-5153	145	3	∈	∈	PROPN
ejpam-5153	145	4	ae	ae	PROPN
ejpam-5153	145	5	,	,	PUNCT
ejpam-5153	145	6	then	then	ADV
ejpam-5153	145	7	x2	x2	PROPN
ejpam-5153	145	8	∈	∈	PROPN
ejpam-5153	145	9	ae	ae	PROPN
ejpam-5153	145	10	.	.	PUNCT
ejpam-5153	146	1	proof	proof	NOUN
ejpam-5153	146	2	.	.	PUNCT
ejpam-5153	147	1	let	let	VERB
ejpam-5153	147	2	x	x	SYM
ejpam-5153	147	3	∈	∈	PROPN
ejpam-5153	147	4	ae	ae	PROPN
ejpam-5153	147	5	,	,	PUNCT
ejpam-5153	147	6	equality	equality	NOUN
ejpam-5153	147	7	(	(	PUNCT
ejpam-5153	147	8	2	2	NUM
ejpam-5153	147	9	,	,	PUNCT
ejpam-5153	147	10	2	2	NUM
ejpam-5153	147	11	)	)	PUNCT
ejpam-5153	147	12	in	in	ADP
ejpam-5153	147	13	[	[	X
ejpam-5153	147	14	9	9	NUM
ejpam-5153	147	15	]	]	PUNCT
ejpam-5153	147	16	is	be	AUX
ejpam-5153	147	17	verified	verify	VERB
ejpam-5153	147	18	,	,	PUNCT
ejpam-5153	147	19	then	then	ADV
ejpam-5153	147	20	(	(	PUNCT
ejpam-5153	147	21	e2	e2	PROPN
ejpam-5153	147	22	,	,	PUNCT
ejpam-5153	147	23	e2	e2	PROPN
ejpam-5153	147	24	,	,	PUNCT
ejpam-5153	147	25	x2	x2	PROPN
ejpam-5153	147	26	)	)	PUNCT
ejpam-5153	148	1	+	+	CCONJ
ejpam-5153	148	2	(	(	PUNCT
ejpam-5153	148	3	e2	e2	PROPN
ejpam-5153	148	4	,	,	PUNCT
ejpam-5153	148	5	x2	x2	PROPN
ejpam-5153	148	6	,	,	PUNCT
ejpam-5153	148	7	e2	e2	PROPN
ejpam-5153	148	8	)	)	PUNCT
ejpam-5153	148	9	+	+	CCONJ
ejpam-5153	148	10	(	(	PUNCT
ejpam-5153	148	11	x2	x2	PROPN
ejpam-5153	148	12	,	,	PUNCT
ejpam-5153	148	13	e2	e2	PROPN
ejpam-5153	148	14	,	,	PUNCT
ejpam-5153	148	15	e2	e2	PROPN
ejpam-5153	148	16	)	)	PUNCT
ejpam-5153	148	17	+	+	CCONJ
ejpam-5153	148	18	(	(	PUNCT
ejpam-5153	148	19	e2	e2	PROPN
ejpam-5153	148	20	,	,	PUNCT
ejpam-5153	148	21	y	y	PROPN
ejpam-5153	148	22	,	,	PUNCT
ejpam-5153	148	23	y	y	PROPN
ejpam-5153	148	24	)	)	PUNCT
ejpam-5153	149	1	+	+	CCONJ
ejpam-5153	149	2	(	(	PUNCT
ejpam-5153	149	3	y	y	PROPN
ejpam-5153	149	4	,	,	PUNCT
ejpam-5153	149	5	e2	e2	PROPN
ejpam-5153	149	6	,	,	PUNCT
ejpam-5153	149	7	y	y	PROPN
ejpam-5153	149	8	)	)	PUNCT
ejpam-5153	150	1	+	+	CCONJ
ejpam-5153	150	2	(	(	PUNCT
ejpam-5153	150	3	y	y	PROPN
ejpam-5153	150	4	,	,	PUNCT
ejpam-5153	150	5	y	y	PROPN
ejpam-5153	150	6	,	,	PUNCT
ejpam-5153	150	7	e2	e2	PROPN
ejpam-5153	150	8	)	)	PUNCT
ejpam-5153	151	1	=	=	SYM
ejpam-5153	151	2	0	0	PUNCT
ejpam-5153	152	1	(	(	PUNCT
ejpam-5153	152	2	a	a	NOUN
ejpam-5153	152	3	)	)	PUNCT
ejpam-5153	152	4	with	with	ADP
ejpam-5153	152	5	y	y	PROPN
ejpam-5153	152	6	=	=	PUNCT
ejpam-5153	152	7	x	x	PROPN
ejpam-5153	152	8	+	+	NUM
ejpam-5153	152	9	xe	xe	PROPN
ejpam-5153	152	10	=	=	SYM
ejpam-5153	152	11	2x	2x	NUM
ejpam-5153	152	12	.	.	PUNCT
ejpam-5153	153	1	a	a	DET
ejpam-5153	153	2	also	also	ADV
ejpam-5153	153	3	satisfies	satisfie	NOUN
ejpam-5153	153	4	to	to	ADP
ejpam-5153	153	5	(	(	PUNCT
ejpam-5153	153	6	e4	e4	PROPN
ejpam-5153	153	7	)	)	PUNCT
ejpam-5153	153	8	,	,	PUNCT
ejpam-5153	153	9	thus	thus	ADV
ejpam-5153	153	10	(	(	PUNCT
ejpam-5153	153	11	a	a	X
ejpam-5153	153	12	)	)	PUNCT
ejpam-5153	153	13	⇒	⇒	NOUN
ejpam-5153	153	14	(	(	PUNCT
ejpam-5153	153	15	x2	x2	PROPN
ejpam-5153	153	16	,	,	PUNCT
ejpam-5153	153	17	e	e	NOUN
ejpam-5153	153	18	,	,	PUNCT
ejpam-5153	153	19	e	e	NOUN
ejpam-5153	153	20	)	)	PUNCT
ejpam-5153	153	21	+	+	CCONJ
ejpam-5153	153	22	(	(	PUNCT
ejpam-5153	153	23	2x	2x	NUM
ejpam-5153	153	24	,	,	PUNCT
ejpam-5153	153	25	2x	2x	NUM
ejpam-5153	153	26	,	,	PUNCT
ejpam-5153	153	27	e	e	NOUN
ejpam-5153	153	28	)	)	PUNCT
ejpam-5153	153	29	=	=	SYM
ejpam-5153	153	30	0	0	NUM
ejpam-5153	153	31	⇒	⇒	NOUN
ejpam-5153	153	32	x2e	x2e	PUNCT
ejpam-5153	154	1	=	=	SYM
ejpam-5153	154	2	x2	x2	PROPN
ejpam-5153	154	3	.	.	PUNCT
ejpam-5153	155	1	so	so	ADV
ejpam-5153	155	2	x2	x2	PROPN
ejpam-5153	155	3	∈	∈	PROPN
ejpam-5153	155	4	ae	ae	PROPN
ejpam-5153	155	5	.	.	PUNCT
ejpam-5153	156	1	proposition	proposition	NOUN
ejpam-5153	156	2	6	6	NUM
ejpam-5153	156	3	.	.	PUNCT
ejpam-5153	157	1	let	let	VERB
ejpam-5153	157	2	a	a	PRON
ejpam-5153	157	3	be	be	AUX
ejpam-5153	157	4	a	a	DET
ejpam-5153	157	5	finite	finite	ADJ
ejpam-5153	157	6	-	-	ADJ
ejpam-5153	157	7	dimensional	dimensional	ADJ
ejpam-5153	157	8	real	real	ADJ
ejpam-5153	157	9	division	division	NOUN
ejpam-5153	157	10	algebra	algebra	NOUN
ejpam-5153	157	11	with	with	ADP
ejpam-5153	157	12	left	left	ADJ
ejpam-5153	157	13	unit	unit	NOUN
ejpam-5153	158	1	e.	e.	PROPN
ejpam-5153	159	1	we	we	PRON
ejpam-5153	159	2	have	have	VERB
ejpam-5153	159	3	the	the	DET
ejpam-5153	159	4	following	follow	VERB
ejpam-5153	159	5	result	result	NOUN
ejpam-5153	159	6	(	(	PUNCT
ejpam-5153	159	7	e2	e2	NOUN
ejpam-5153	159	8	)	)	PUNCT
ejpam-5153	160	1	=	=	VERB
ejpam-5153	160	2	⇒	⇒	NOUN
ejpam-5153	160	3	(	(	PUNCT
ejpam-5153	160	4	e3	e3	NOUN
ejpam-5153	160	5	)	)	PUNCT
ejpam-5153	160	6	.	.	PUNCT
ejpam-5153	161	1	proof	proof	NOUN
ejpam-5153	161	2	.	.	PUNCT
ejpam-5153	162	1	let	let	VERB
ejpam-5153	162	2	x	x	PUNCT
ejpam-5153	162	3	∈	∈	PROPN
ejpam-5153	162	4	a	a	X
ejpam-5153	162	5	,	,	PUNCT
ejpam-5153	162	6	a	a	DET
ejpam-5153	162	7	satisfies	satisfie	NOUN
ejpam-5153	162	8	to	to	ADP
ejpam-5153	162	9	(	(	PUNCT
ejpam-5153	162	10	e2	e2	PROPN
ejpam-5153	162	11	)	)	PUNCT
ejpam-5153	162	12	,	,	PUNCT
ejpam-5153	162	13	equality	equality	NOUN
ejpam-5153	162	14	(	(	PUNCT
ejpam-5153	162	15	2	2	NUM
ejpam-5153	162	16	,	,	PUNCT
ejpam-5153	162	17	2	2	NUM
ejpam-5153	162	18	)	)	PUNCT
ejpam-5153	162	19	in	in	ADP
ejpam-5153	162	20	[	[	X
ejpam-5153	162	21	9	9	NUM
ejpam-5153	162	22	]	]	PUNCT
ejpam-5153	162	23	is	be	AUX
ejpam-5153	162	24	verified	verify	VERB
ejpam-5153	162	25	,	,	PUNCT
ejpam-5153	162	26	then	then	ADV
ejpam-5153	162	27	(	(	PUNCT
ejpam-5153	162	28	e2	e2	PROPN
ejpam-5153	162	29	,	,	PUNCT
ejpam-5153	162	30	e2	e2	PROPN
ejpam-5153	162	31	,	,	PUNCT
ejpam-5153	162	32	x2	x2	PROPN
ejpam-5153	162	33	)	)	PUNCT
ejpam-5153	163	1	+	+	CCONJ
ejpam-5153	163	2	(	(	PUNCT
ejpam-5153	163	3	e2	e2	PROPN
ejpam-5153	163	4	,	,	PUNCT
ejpam-5153	163	5	x2	x2	PROPN
ejpam-5153	163	6	,	,	PUNCT
ejpam-5153	163	7	e2	e2	PROPN
ejpam-5153	163	8	)	)	PUNCT
ejpam-5153	163	9	+	+	CCONJ
ejpam-5153	163	10	(	(	PUNCT
ejpam-5153	163	11	x2	x2	PROPN
ejpam-5153	163	12	,	,	PUNCT
ejpam-5153	163	13	e2	e2	PROPN
ejpam-5153	163	14	,	,	PUNCT
ejpam-5153	163	15	e2	e2	PROPN
ejpam-5153	163	16	)	)	PUNCT
ejpam-5153	163	17	+	+	CCONJ
ejpam-5153	163	18	(	(	PUNCT
ejpam-5153	163	19	e2	e2	PROPN
ejpam-5153	163	20	,	,	PUNCT
ejpam-5153	163	21	y	y	PROPN
ejpam-5153	163	22	,	,	PUNCT
ejpam-5153	163	23	y	y	PROPN
ejpam-5153	163	24	)	)	PUNCT
ejpam-5153	164	1	+	+	CCONJ
ejpam-5153	164	2	(	(	PUNCT
ejpam-5153	164	3	y	y	PROPN
ejpam-5153	164	4	,	,	PUNCT
ejpam-5153	164	5	e2	e2	PROPN
ejpam-5153	164	6	,	,	PUNCT
ejpam-5153	164	7	y	y	PROPN
ejpam-5153	164	8	)	)	PUNCT
ejpam-5153	165	1	+	+	CCONJ
ejpam-5153	165	2	(	(	PUNCT
ejpam-5153	165	3	y	y	PROPN
ejpam-5153	165	4	,	,	PUNCT
ejpam-5153	165	5	y	y	PROPN
ejpam-5153	165	6	,	,	PUNCT
ejpam-5153	165	7	e2	e2	PROPN
ejpam-5153	165	8	)	)	PUNCT
ejpam-5153	165	9	=	=	SYM
ejpam-5153	165	10	0	0	PUNCT
ejpam-5153	166	1	(	(	PUNCT
ejpam-5153	166	2	b	b	NOUN
ejpam-5153	166	3	)	)	PUNCT
ejpam-5153	166	4	with	with	ADP
ejpam-5153	166	5	y	y	PROPN
ejpam-5153	166	6	=	=	PUNCT
ejpam-5153	166	7	x+xe	x+xe	PROPN
ejpam-5153	166	8	,	,	PUNCT
ejpam-5153	166	9	thus	thus	ADV
ejpam-5153	166	10	(	(	PUNCT
ejpam-5153	166	11	b	b	X
ejpam-5153	166	12	)	)	PUNCT
ejpam-5153	166	13	⇒	⇒	NOUN
ejpam-5153	167	1	x2	x2	INTJ
ejpam-5153	167	2	−x2e+y2e−y2	−x2e+y2e−y2	PROPN
ejpam-5153	167	3	=	=	SYM
ejpam-5153	167	4	0	0	PROPN
ejpam-5153	167	5	,	,	PUNCT
ejpam-5153	167	6	as	as	SCONJ
ejpam-5153	167	7	y	y	PROPN
ejpam-5153	167	8	∈	∈	PROPN
ejpam-5153	167	9	ae	ae	PROPN
ejpam-5153	167	10	⇒	⇒	VERB
ejpam-5153	167	11	y2	y2	PROPN
ejpam-5153	167	12	∈	∈	PROPN
ejpam-5153	167	13	ae	ae	PROPN
ejpam-5153	167	14	⇒	⇒	VERB
ejpam-5153	167	15	y2e−y2	y2e−y2	PROPN
ejpam-5153	167	16	=	=	SYM
ejpam-5153	167	17	0	0	PROPN
ejpam-5153	167	18	,	,	PUNCT
ejpam-5153	167	19	therefore	therefore	ADV
ejpam-5153	167	20	we	we	PRON
ejpam-5153	167	21	have	have	VERB
ejpam-5153	167	22	x2	x2	INTJ
ejpam-5153	167	23	−	−	PROPN
ejpam-5153	167	24	x2e	x2e	NOUN
ejpam-5153	167	25	=	=	SYM
ejpam-5153	167	26	0	0	NUM
ejpam-5153	167	27	⇒	⇒	NOUN
ejpam-5153	167	28	x2e	x2e	PUNCT
ejpam-5153	168	1	=	=	SYM
ejpam-5153	168	2	x2	x2	PROPN
ejpam-5153	168	3	.	.	PUNCT
ejpam-5153	169	1	lemma	lemma	PROPN
ejpam-5153	169	2	7	7	X
ejpam-5153	169	3	.	.	PUNCT
ejpam-5153	170	1	let	let	VERB
ejpam-5153	170	2	a	a	PRON
ejpam-5153	170	3	be	be	AUX
ejpam-5153	170	4	a	a	DET
ejpam-5153	170	5	four	four	NUM
ejpam-5153	170	6	-	-	PUNCT
ejpam-5153	170	7	dimensional	dimensional	ADJ
ejpam-5153	170	8	real	real	ADJ
ejpam-5153	170	9	division	division	NOUN
ejpam-5153	170	10	algebra	algebra	NOUN
ejpam-5153	170	11	with	with	ADP
ejpam-5153	170	12	left	left	ADJ
ejpam-5153	170	13	unit	unit	NOUN
ejpam-5153	170	14	e.	e.	PROPN
ejpam-5153	171	1	if	if	SCONJ
ejpam-5153	171	2	there	there	PRON
ejpam-5153	171	3	exists	exist	VERB
ejpam-5153	171	4	u	u	PROPN
ejpam-5153	171	5	∈	∈	PROPN
ejpam-5153	171	6	a	a	DET
ejpam-5153	171	7	such	such	ADJ
ejpam-5153	171	8	that	that	SCONJ
ejpam-5153	171	9	the	the	DET
ejpam-5153	171	10	subalgebra	subalgebra	NOUN
ejpam-5153	171	11	of	of	ADP
ejpam-5153	171	12	a	a	DET
ejpam-5153	171	13	generated	generate	VERB
ejpam-5153	171	14	by	by	ADP
ejpam-5153	171	15	u	u	NOUN
ejpam-5153	171	16	,	,	PUNCT
ejpam-5153	171	17	a(u	a(u	PROPN
ejpam-5153	171	18	)	)	PUNCT
ejpam-5153	171	19	:	:	PUNCT
ejpam-5153	172	1	=	=	SYM
ejpam-5153	172	2	b	b	X
ejpam-5153	172	3	,	,	PUNCT
ejpam-5153	172	4	is	be	AUX
ejpam-5153	172	5	of	of	ADP
ejpam-5153	172	6	two	two	NUM
ejpam-5153	172	7	-	-	PUNCT
ejpam-5153	172	8	dimensional	dimensional	ADJ
ejpam-5153	172	9	.	.	PUNCT
ejpam-5153	173	1	then	then	ADV
ejpam-5153	173	2	for	for	ADP
ejpam-5153	173	3	all	all	DET
ejpam-5153	173	4	v	v	ADP
ejpam-5153	173	5	∈	∈	PRON
ejpam-5153	173	6	a	a	DET
ejpam-5153	173	7	−	−	PROPN
ejpam-5153	173	8	b	b	NOUN
ejpam-5153	173	9	,	,	PUNCT
ejpam-5153	173	10	{	{	PUNCT
ejpam-5153	173	11	e	e	NOUN
ejpam-5153	173	12	,	,	PUNCT
ejpam-5153	173	13	u	u	NOUN
ejpam-5153	173	14	,	,	PUNCT
ejpam-5153	173	15	v	v	NOUN
ejpam-5153	173	16	,	,	PUNCT
ejpam-5153	173	17	uv	uv	NOUN
ejpam-5153	173	18	}	}	PUNCT
ejpam-5153	173	19	is	be	AUX
ejpam-5153	173	20	a	a	DET
ejpam-5153	173	21	basis	basis	NOUN
ejpam-5153	173	22	of	of	ADP
ejpam-5153	173	23	a.	a.	NOUN
ejpam-5153	173	24	proof	proof	NOUN
ejpam-5153	173	25	.	.	PUNCT
ejpam-5153	174	1	let	let	VERB
ejpam-5153	174	2	v	v	X
ejpam-5153	174	3	∈	∈	VERB
ejpam-5153	174	4	a	a	DET
ejpam-5153	174	5	−	−	PROPN
ejpam-5153	174	6	b	b	NOUN
ejpam-5153	174	7	,	,	PUNCT
ejpam-5153	174	8	we	we	PRON
ejpam-5153	174	9	have	have	VERB
ejpam-5153	174	10	e	e	NOUN
ejpam-5153	174	11	,	,	PUNCT
ejpam-5153	174	12	u	u	NOUN
ejpam-5153	174	13	,	,	PUNCT
ejpam-5153	174	14	,	,	PUNCT
ejpam-5153	174	15	v	v	NOUN
ejpam-5153	174	16	and	and	CCONJ
ejpam-5153	174	17	uv	uv	NOUN
ejpam-5153	174	18	are	be	AUX
ejpam-5153	174	19	linearly	linearly	ADV
ejpam-5153	174	20	independent	independent	ADJ
ejpam-5153	174	21	.	.	PUNCT
ejpam-5153	175	1	suppose	suppose	VERB
ejpam-5153	175	2	that	that	SCONJ
ejpam-5153	175	3	uv	uv	NOUN
ejpam-5153	175	4	=	=	PUNCT
ejpam-5153	175	5	αe	αe	PROPN
ejpam-5153	175	6	+	+	NOUN
ejpam-5153	175	7	βu	βu	PUNCT
ejpam-5153	176	1	+	+	NUM
ejpam-5153	176	2	γv	γv	NOUN
ejpam-5153	176	3	with	with	ADP
ejpam-5153	176	4	α	α	PROPN
ejpam-5153	176	5	,	,	PUNCT
ejpam-5153	176	6	β	β	NOUN
ejpam-5153	176	7	,	,	PUNCT
ejpam-5153	176	8	and	and	CCONJ
ejpam-5153	176	9	γ	γ	PROPN
ejpam-5153	176	10	∈	∈	PROPN
ejpam-5153	176	11	r.	r.	PROPN
ejpam-5153	176	12	uv	uv	PROPN
ejpam-5153	177	1	=	=	PUNCT
ejpam-5153	177	2	αe	αe	PROPN
ejpam-5153	177	3	+	+	X
ejpam-5153	177	4	βu	βu	PUNCT
ejpam-5153	178	1	+	+	NUM
ejpam-5153	178	2	γv	γv	X
ejpam-5153	178	3	=	=	VERB
ejpam-5153	178	4	⇒	⇒	NOUN
ejpam-5153	178	5	uv	uv	NOUN
ejpam-5153	178	6	−	−	NOUN
ejpam-5153	178	7	γv	γv	NOUN
ejpam-5153	179	1	=	=	PUNCT
ejpam-5153	179	2	αe	αe	PROPN
ejpam-5153	179	3	+	+	NUM
ejpam-5153	179	4	βu	βu	PUNCT
ejpam-5153	179	5	=	=	NOUN
ejpam-5153	179	6	⇒	⇒	NOUN
ejpam-5153	179	7	(	(	PUNCT
ejpam-5153	179	8	u	u	NOUN
ejpam-5153	179	9	−	−	PROPN
ejpam-5153	179	10	γe)v	γe)v	PROPN
ejpam-5153	179	11	=	=	SYM
ejpam-5153	179	12	αe	αe	NOUN
ejpam-5153	179	13	+	+	NOUN
ejpam-5153	179	14	βu	βu	VERB
ejpam-5153	179	15	a.	a.	PROPN
ejpam-5153	179	16	s.	s.	PROPN
ejpam-5153	179	17	diabang	diabang	PROPN
ejpam-5153	179	18	,	,	PUNCT
ejpam-5153	179	19	a.	a.	PROPN
ejpam-5153	179	20	s.	s.	PROPN
ejpam-5153	179	21	mballo	mballo	PROPN
ejpam-5153	179	22	,	,	PUNCT
ejpam-5153	179	23	p.	p.	PROPN
ejpam-5153	179	24	c.	c.	PROPN
ejpam-5153	179	25	diop	diop	PROPN
ejpam-5153	179	26	/	/	SYM
ejpam-5153	179	27	eur	eur	PROPN
ejpam-5153	179	28	.	.	PUNCT
ejpam-5153	180	1	j.	j.	PROPN
ejpam-5153	180	2	pure	pure	PROPN
ejpam-5153	180	3	appl	appl	PROPN
ejpam-5153	180	4	.	.	PROPN
ejpam-5153	180	5	math	math	PROPN
ejpam-5153	180	6	,	,	PUNCT
ejpam-5153	180	7	17	17	NUM
ejpam-5153	180	8	(	(	PUNCT
ejpam-5153	180	9	3	3	NUM
ejpam-5153	180	10	)	)	PUNCT
ejpam-5153	180	11	(	(	PUNCT
ejpam-5153	180	12	2024	2024	NUM
ejpam-5153	180	13	)	)	PUNCT
ejpam-5153	180	14	,	,	PUNCT
ejpam-5153	180	15	2276	2276	NUM
ejpam-5153	180	16	-	-	SYM
ejpam-5153	180	17	2287	2287	NUM
ejpam-5153	180	18	2282	2282	NUM
ejpam-5153	180	19	=	=	NOUN
ejpam-5153	180	20	⇒	⇒	NOUN
ejpam-5153	180	21	(	(	PUNCT
ejpam-5153	180	22	u	u	NOUN
ejpam-5153	180	23	−	−	PROPN
ejpam-5153	180	24	γe)v	γe)v	PROPN
ejpam-5153	180	25	∈	∈	PROPN
ejpam-5153	180	26	b	b	NOUN
ejpam-5153	180	27	as	as	SCONJ
ejpam-5153	180	28	b	b	PROPN
ejpam-5153	180	29	is	be	AUX
ejpam-5153	180	30	a	a	DET
ejpam-5153	180	31	subalgebra	subalgebra	NOUN
ejpam-5153	180	32	of	of	ADP
ejpam-5153	180	33	a	a	PRON
ejpam-5153	180	34	,	,	PUNCT
ejpam-5153	180	35	it	it	PRON
ejpam-5153	180	36	exists	exist	VERB
ejpam-5153	180	37	v′	v′	NOUN
ejpam-5153	180	38	∈	∈	PROPN
ejpam-5153	180	39	b	b	PROPN
ejpam-5153	181	1	such	such	ADJ
ejpam-5153	181	2	that	that	PRON
ejpam-5153	181	3	(	(	PUNCT
ejpam-5153	181	4	u	u	NOUN
ejpam-5153	181	5	−	−	PROPN
ejpam-5153	181	6	γe)v′	γe)v′	X
ejpam-5153	181	7	=	=	SYM
ejpam-5153	181	8	αe	αe	PROPN
ejpam-5153	181	9	+	+	X
ejpam-5153	181	10	βu	βu	X
ejpam-5153	181	11	.	.	PUNCT
ejpam-5153	181	12	thus	thus	ADV
ejpam-5153	181	13	(	(	PUNCT
ejpam-5153	181	14	u	u	NOUN
ejpam-5153	181	15	−	−	PROPN
ejpam-5153	181	16	γe)v	γe)v	PROPN
ejpam-5153	181	17	=	=	PUNCT
ejpam-5153	181	18	(	(	PUNCT
ejpam-5153	181	19	u	u	NOUN
ejpam-5153	181	20	−	−	NOUN
ejpam-5153	181	21	γe)v′	γe)v′	X
ejpam-5153	182	1	=	=	NOUN
ejpam-5153	182	2	⇒	⇒	X
ejpam-5153	182	3	v	v	X
ejpam-5153	182	4	=	=	SYM
ejpam-5153	182	5	v′	v′	NOUN
ejpam-5153	182	6	∈	∈	PROPN
ejpam-5153	182	7	b	b	X
ejpam-5153	182	8	absurd	absurd	ADJ
ejpam-5153	182	9	.	.	PUNCT
ejpam-5153	183	1	therefore	therefore	ADV
ejpam-5153	183	2	{	{	PUNCT
ejpam-5153	183	3	e	e	NOUN
ejpam-5153	183	4	,	,	PUNCT
ejpam-5153	183	5	u	u	NOUN
ejpam-5153	183	6	,	,	PUNCT
ejpam-5153	183	7	v	v	NOUN
ejpam-5153	183	8	,	,	PUNCT
ejpam-5153	183	9	uv	uv	NOUN
ejpam-5153	183	10	}	}	PUNCT
ejpam-5153	183	11	is	be	AUX
ejpam-5153	183	12	a	a	DET
ejpam-5153	183	13	basis	basis	NOUN
ejpam-5153	183	14	of	of	ADP
ejpam-5153	183	15	a.	a.	NOUN
ejpam-5153	183	16	example	example	NOUN
ejpam-5153	183	17	1	1	X
ejpam-5153	183	18	.	.	PUNCT
ejpam-5153	184	1	the	the	DET
ejpam-5153	184	2	real	real	ADJ
ejpam-5153	184	3	division	division	NOUN
ejpam-5153	184	4	algebra	algebra	VERB
ejpam-5153	184	5	a	a	DET
ejpam-5153	184	6	whose	whose	DET
ejpam-5153	184	7	product	product	NOUN
ejpam-5153	184	8	in	in	ADP
ejpam-5153	184	9	the	the	DET
ejpam-5153	184	10	basis	basis	NOUN
ejpam-5153	184	11	b	b	NOUN
ejpam-5153	184	12	=	=	SYM
ejpam-5153	184	13	{	{	PUNCT
ejpam-5153	184	14	e	e	NOUN
ejpam-5153	184	15	,	,	PUNCT
ejpam-5153	184	16	u	u	NOUN
ejpam-5153	184	17	,	,	PUNCT
ejpam-5153	184	18	v	v	NOUN
ejpam-5153	184	19	,	,	PUNCT
ejpam-5153	184	20	uv	uv	NOUN
ejpam-5153	184	21	}	}	PUNCT
ejpam-5153	184	22	is	be	AUX
ejpam-5153	184	23	given	give	VERB
ejpam-5153	184	24	by	by	ADP
ejpam-5153	184	25	:	:	PUNCT
ejpam-5153	184	26	.	.	PUNCT
ejpam-5153	185	1	e	e	X
ejpam-5153	185	2	u	u	NOUN
ejpam-5153	185	3	v	v	INTJ
ejpam-5153	185	4	uv	uv	NOUN
ejpam-5153	185	5	e	e	NOUN
ejpam-5153	185	6	e	e	X
ejpam-5153	185	7	u	u	NOUN
ejpam-5153	185	8	v	v	INTJ
ejpam-5153	185	9	uv	uv	NOUN
ejpam-5153	185	10	u	u	NOUN
ejpam-5153	185	11	u	u	NOUN
ejpam-5153	185	12	−e	−e	ADJ
ejpam-5153	185	13	uv	uv	NOUN
ejpam-5153	185	14	−v	−v	NOUN
ejpam-5153	185	15	v	v	ADP
ejpam-5153	185	16	v	v	NOUN
ejpam-5153	185	17	v	v	NOUN
ejpam-5153	185	18	−	−	NOUN
ejpam-5153	186	1	uv	uv	NOUN
ejpam-5153	186	2	−e	−e	NOUN
ejpam-5153	186	3	−e	−e	NOUN
ejpam-5153	186	4	+	+	CCONJ
ejpam-5153	186	5	u	u	NOUN
ejpam-5153	186	6	uv	uv	NOUN
ejpam-5153	186	7	uv	uv	NOUN
ejpam-5153	186	8	v	v	NOUN
ejpam-5153	186	9	+	+	CCONJ
ejpam-5153	186	10	2uv	2uv	ADJ
ejpam-5153	186	11	u	u	NOUN
ejpam-5153	186	12	−e	−e	NOUN
ejpam-5153	186	13	−	−	PROPN
ejpam-5153	186	14	2u	2u	NOUN
ejpam-5153	186	15	satisfies	satisfie	NOUN
ejpam-5153	186	16	to	to	ADP
ejpam-5153	186	17	(	(	PUNCT
ejpam-5153	186	18	e3	e3	NOUN
ejpam-5153	186	19	)	)	PUNCT
ejpam-5153	186	20	and	and	CCONJ
ejpam-5153	186	21	not	not	PART
ejpam-5153	186	22	to	to	PART
ejpam-5153	186	23	(	(	PUNCT
ejpam-5153	186	24	e2	e2	PROPN
ejpam-5153	186	25	)	)	PUNCT
ejpam-5153	186	26	.	.	PUNCT
ejpam-5153	187	1	because	because	SCONJ
ejpam-5153	187	2	(	(	PUNCT
ejpam-5153	187	3	(	(	PUNCT
ejpam-5153	187	4	u	u	NOUN
ejpam-5153	187	5	+	+	X
ejpam-5153	187	6	uv)2	uv)2	PROPN
ejpam-5153	187	7	,	,	PUNCT
ejpam-5153	187	8	(	(	PUNCT
ejpam-5153	187	9	u	u	NOUN
ejpam-5153	187	10	+	+	X
ejpam-5153	187	11	uv)2	uv)2	PROPN
ejpam-5153	187	12	,	,	PUNCT
ejpam-5153	187	13	(	(	PUNCT
ejpam-5153	187	14	u	u	NOUN
ejpam-5153	187	15	+	+	X
ejpam-5153	187	16	uv)2	uv)2	PROPN
ejpam-5153	187	17	)	)	PUNCT
ejpam-5153	187	18	̸=	̸=	PROPN
ejpam-5153	187	19	0	0	NUM
ejpam-5153	187	20	.	.	PUNCT
ejpam-5153	188	1	consequently	consequently	ADV
ejpam-5153	188	2	,	,	PUNCT
ejpam-5153	188	3	lemma	lemma	PROPN
ejpam-5153	188	4	1	1	NUM
ejpam-5153	188	5	and	and	CCONJ
ejpam-5153	188	6	proposition	proposition	NOUN
ejpam-5153	188	7	6	6	NUM
ejpam-5153	188	8	,	,	PUNCT
ejpam-5153	188	9	shows	show	VERB
ejpam-5153	188	10	that	that	SCONJ
ejpam-5153	188	11	(	(	PUNCT
ejpam-5153	188	12	e1	e1	NOUN
ejpam-5153	188	13	)	)	PUNCT
ejpam-5153	188	14	⇒	⇒	NOUN
ejpam-5153	188	15	(	(	PUNCT
ejpam-5153	188	16	e2	e2	PROPN
ejpam-5153	188	17	)	)	PUNCT
ejpam-5153	188	18	⇒	⇒	NOUN
ejpam-5153	188	19	(	(	PUNCT
ejpam-5153	188	20	e3	e3	NOUN
ejpam-5153	188	21	)	)	PUNCT
ejpam-5153	188	22	⇒	⇒	NOUN
ejpam-5153	188	23	(	(	PUNCT
ejpam-5153	188	24	e4	e4	PROPN
ejpam-5153	188	25	)	)	PUNCT
ejpam-5153	188	26	.	.	PUNCT
ejpam-5153	189	1	proposition	proposition	NOUN
ejpam-5153	189	2	7	7	NUM
ejpam-5153	189	3	.	.	PUNCT
ejpam-5153	190	1	let	let	VERB
ejpam-5153	190	2	a	a	PRON
ejpam-5153	190	3	be	be	AUX
ejpam-5153	190	4	a	a	DET
ejpam-5153	190	5	real	real	ADJ
ejpam-5153	190	6	division	division	NOUN
ejpam-5153	190	7	algebra	algebra	NOUN
ejpam-5153	190	8	with	with	ADP
ejpam-5153	190	9	left	left	ADJ
ejpam-5153	190	10	unit	unit	NOUN
ejpam-5153	190	11	e	e	PROPN
ejpam-5153	190	12	of	of	ADP
ejpam-5153	190	13	finite	finite	ADJ
ejpam-5153	190	14	-	-	ADJ
ejpam-5153	190	15	dimensional	dimensional	ADJ
ejpam-5153	190	16	n	n	CCONJ
ejpam-5153	190	17	∈	∈	NOUN
ejpam-5153	190	18	{	{	PUNCT
ejpam-5153	190	19	2	2	NUM
ejpam-5153	190	20	,	,	PUNCT
ejpam-5153	190	21	4	4	NUM
ejpam-5153	190	22	,	,	PUNCT
ejpam-5153	190	23	8	8	NUM
ejpam-5153	190	24	}	}	PUNCT
ejpam-5153	190	25	,	,	PUNCT
ejpam-5153	190	26	satisfies	satisfie	NOUN
ejpam-5153	190	27	to	to	ADP
ejpam-5153	190	28	(	(	PUNCT
ejpam-5153	190	29	e2	e2	PROPN
ejpam-5153	190	30	)	)	PUNCT
ejpam-5153	190	31	.	.	PUNCT
ejpam-5153	191	1	then	then	ADV
ejpam-5153	191	2	there	there	PRON
ejpam-5153	191	3	exists	exist	VERB
ejpam-5153	191	4	u	u	PROPN
ejpam-5153	191	5	∈	∈	PROPN
ejpam-5153	191	6	a	a	DET
ejpam-5153	191	7	−	−	NOUN
ejpam-5153	191	8	re	re	NOUN
ejpam-5153	191	9	such	such	ADJ
ejpam-5153	191	10	as	as	ADP
ejpam-5153	191	11	ue	ue	PROPN
ejpam-5153	191	12	/∈	/∈	PUNCT
ejpam-5153	191	13	re	re	PROPN
ejpam-5153	191	14	and	and	CCONJ
ejpam-5153	191	15	ue.u	ue.u	NOUN
ejpam-5153	191	16	=	=	SYM
ejpam-5153	191	17	−e	−e	NOUN
ejpam-5153	191	18	.	.	PUNCT
ejpam-5153	192	1	we	we	PRON
ejpam-5153	192	2	note	note	VERB
ejpam-5153	192	3	that	that	SCONJ
ejpam-5153	192	4	,	,	PUNCT
ejpam-5153	192	5	if	if	SCONJ
ejpam-5153	192	6	ue	ue	PROPN
ejpam-5153	192	7	∈	∈	PROPN
ejpam-5153	192	8	re	re	PROPN
ejpam-5153	192	9	+	+	PROPN
ejpam-5153	192	10	ru	ru	PROPN
ejpam-5153	192	11	,	,	PUNCT
ejpam-5153	192	12	then	then	ADV
ejpam-5153	192	13	a(u	a(u	NOUN
ejpam-5153	192	14	)	)	PUNCT
ejpam-5153	192	15	is	be	AUX
ejpam-5153	192	16	isomorphic	isomorphic	ADJ
ejpam-5153	192	17	to	to	ADP
ejpam-5153	192	18	either	either	CCONJ
ejpam-5153	192	19	c	c	PROPN
ejpam-5153	192	20	,	,	PUNCT
ejpam-5153	192	21	⋆c	⋆c	PROPN
ejpam-5153	192	22	;	;	PUNCT
ejpam-5153	192	23	otherwise	otherwise	ADV
ejpam-5153	192	24	the	the	DET
ejpam-5153	192	25	dimension	dimension	NOUN
ejpam-5153	192	26	of	of	ADP
ejpam-5153	192	27	a(u	a(u	PROPN
ejpam-5153	192	28	)	)	PUNCT
ejpam-5153	192	29	is	be	AUX
ejpam-5153	192	30	greater	great	ADJ
ejpam-5153	192	31	than	than	ADP
ejpam-5153	192	32	four	four	NUM
ejpam-5153	192	33	.	.	PUNCT
ejpam-5153	193	1	proof	proof	NOUN
ejpam-5153	193	2	.	.	PUNCT
ejpam-5153	194	1	a	a	DET
ejpam-5153	194	2	satisfies	satisfie	NOUN
ejpam-5153	194	3	to	to	ADP
ejpam-5153	194	4	(	(	PUNCT
ejpam-5153	194	5	e2	e2	PROPN
ejpam-5153	194	6	)	)	PUNCT
ejpam-5153	194	7	,	,	PUNCT
ejpam-5153	194	8	then	then	ADV
ejpam-5153	194	9	satisfies	satisfie	NOUN
ejpam-5153	194	10	to	to	ADP
ejpam-5153	194	11	(	(	PUNCT
ejpam-5153	194	12	e4	e4	PROPN
ejpam-5153	194	13	)	)	PUNCT
ejpam-5153	194	14	.	.	PUNCT
ejpam-5153	195	1	the	the	DET
ejpam-5153	195	2	real	real	ADJ
ejpam-5153	195	3	division	division	NOUN
ejpam-5153	195	4	algebra	algebra	VERB
ejpam-5153	195	5	a′	a′	PROPN
ejpam-5153	195	6	=	=	SYM
ejpam-5153	195	7	(	(	PUNCT
ejpam-5153	195	8	a	a	PRON
ejpam-5153	195	9	,	,	PUNCT
ejpam-5153	195	10	⊙	⊙	PROPN
ejpam-5153	195	11	)	)	PUNCT
ejpam-5153	195	12	with	with	ADP
ejpam-5153	195	13	x	x	PROPN
ejpam-5153	195	14	⊙	⊙	PROPN
ejpam-5153	195	15	y	y	PROPN
ejpam-5153	195	16	=	=	PRON
ejpam-5153	196	1	(	(	PUNCT
ejpam-5153	196	2	xe)y	xe)y	PROPN
ejpam-5153	196	3	contains	contain	VERB
ejpam-5153	196	4	e	e	NOUN
ejpam-5153	196	5	as	as	ADP
ejpam-5153	196	6	unit	unit	NOUN
ejpam-5153	196	7	element	element	NOUN
ejpam-5153	196	8	.	.	PUNCT
ejpam-5153	197	1	[	[	X
ejpam-5153	197	2	15	15	NUM
ejpam-5153	197	3	]	]	PUNCT
ejpam-5153	197	4	shows	show	VERB
ejpam-5153	197	5	that	that	SCONJ
ejpam-5153	197	6	there	there	PRON
ejpam-5153	197	7	exists	exist	VERB
ejpam-5153	197	8	u	u	PROPN
ejpam-5153	197	9	∈	∈	PROPN
ejpam-5153	197	10	a	a	DET
ejpam-5153	197	11	−	−	NOUN
ejpam-5153	197	12	re	re	PROPN
ejpam-5153	197	13	,	,	PUNCT
ejpam-5153	197	14	shus	shu	VERB
ejpam-5153	197	15	that	that	PRON
ejpam-5153	197	16	u	u	PROPN
ejpam-5153	197	17	⊙	⊙	NOUN
ejpam-5153	197	18	u	u	NOUN
ejpam-5153	197	19	=	=	NOUN
ejpam-5153	197	20	−e	−e	NOUN
ejpam-5153	197	21	⇐	⇐	ADJ
ejpam-5153	197	22	⇒	⇒	PROPN
ejpam-5153	197	23	ue.u	ue.u	PROPN
ejpam-5153	197	24	=	=	PUNCT
ejpam-5153	197	25	−e	−e	NOUN
ejpam-5153	197	26	.	.	PUNCT
ejpam-5153	197	27	suppose	suppose	VERB
ejpam-5153	197	28	that	that	SCONJ
ejpam-5153	197	29	ue	ue	PROPN
ejpam-5153	197	30	∈	∈	PROPN
ejpam-5153	197	31	re	re	PROPN
ejpam-5153	197	32	,	,	PUNCT
ejpam-5153	197	33	lemma	lemma	PROPN
ejpam-5153	197	34	2	2	NUM
ejpam-5153	197	35	shows	show	VERB
ejpam-5153	197	36	u	u	PROPN
ejpam-5153	197	37	∈	∈	PROPN
ejpam-5153	197	38	re	re	ADP
ejpam-5153	197	39	absurd	absurd	ADJ
ejpam-5153	197	40	,	,	PUNCT
ejpam-5153	197	41	so	so	ADV
ejpam-5153	197	42	ue	ue	PROPN
ejpam-5153	197	43	/∈	/∈	PUNCT
ejpam-5153	197	44	re	re	PROPN
ejpam-5153	197	45	.	.	PROPN
ejpam-5153	197	46	•	•	INTJ
ejpam-5153	197	47	if	if	SCONJ
ejpam-5153	197	48	ue	ue	PROPN
ejpam-5153	197	49	∈	∈	PROPN
ejpam-5153	197	50	re	re	PROPN
ejpam-5153	197	51	+	+	PROPN
ejpam-5153	197	52	ru	ru	PROPN
ejpam-5153	197	53	,	,	PUNCT
ejpam-5153	197	54	then	then	ADV
ejpam-5153	197	55	they	they	PRON
ejpam-5153	197	56	exist	exist	VERB
ejpam-5153	197	57	α	α	PRON
ejpam-5153	197	58	,	,	PUNCT
ejpam-5153	197	59	β	β	X
ejpam-5153	197	60	∈	∈	PROPN
ejpam-5153	197	61	r	r	NOUN
ejpam-5153	197	62	shus	shus	NOUN
ejpam-5153	197	63	that	that	PRON
ejpam-5153	197	64	ue	ue	INTJ
ejpam-5153	198	1	=	=	PUNCT
ejpam-5153	198	2	αe	αe	PROPN
ejpam-5153	198	3	+	+	NOUN
ejpam-5153	198	4	βu	βu	X
ejpam-5153	198	5	.	.	NOUN
ejpam-5153	199	1	we	we	PRON
ejpam-5153	199	2	have	have	VERB
ejpam-5153	199	3	(	(	PUNCT
ejpam-5153	199	4	ue)e	ue)e	PROPN
ejpam-5153	199	5	=	=	SYM
ejpam-5153	199	6	u	u	NOUN
ejpam-5153	199	7	=	=	NOUN
ejpam-5153	199	8	⇒	⇒	VERB
ejpam-5153	199	9	α(1	α(1	PROPN
ejpam-5153	199	10	+	+	PUNCT
ejpam-5153	199	11	β)e	β)e	PUNCT
ejpam-5153	200	1	+	+	CCONJ
ejpam-5153	200	2	(	(	PUNCT
ejpam-5153	200	3	β2	β2	NOUN
ejpam-5153	200	4	−	−	PROPN
ejpam-5153	200	5	1)u	1)u	NUM
ejpam-5153	200	6	=	=	NOUN
ejpam-5153	200	7	0	0	PUNCT
ejpam-5153	201	1	=	=	NOUN
ejpam-5153	201	2	⇒	⇒	NOUN
ejpam-5153	201	3	(	(	PUNCT
ejpam-5153	201	4	s1	s1	NOUN
ejpam-5153	201	5	)	)	PUNCT
ejpam-5153	201	6	{	{	PUNCT
ejpam-5153	201	7	α(1	α(1	PROPN
ejpam-5153	201	8	+	+	NUM
ejpam-5153	201	9	β	β	X
ejpam-5153	201	10	)	)	PUNCT
ejpam-5153	201	11	=	=	SYM
ejpam-5153	201	12	0	0	NUM
ejpam-5153	201	13	β2	β2	NOUN
ejpam-5153	201	14	=	=	NOUN
ejpam-5153	201	15	1	1	NUM
ejpam-5153	202	1	thus	thus	ADV
ejpam-5153	202	2	if	if	SCONJ
ejpam-5153	202	3	β	β	X
ejpam-5153	202	4	=	=	SYM
ejpam-5153	202	5	1	1	NUM
ejpam-5153	202	6	⇒	⇒	NOUN
ejpam-5153	202	7	α	α	NOUN
ejpam-5153	202	8	=	=	SYM
ejpam-5153	202	9	0	0	PUNCT
ejpam-5153	202	10	then	then	ADV
ejpam-5153	202	11	ue.u	ue.u	PROPN
ejpam-5153	202	12	=	=	PUNCT
ejpam-5153	202	13	−e	−e	NOUN
ejpam-5153	202	14	=	=	AUX
ejpam-5153	202	15	⇒	⇒	VERB
ejpam-5153	202	16	u2	u2	NOUN
ejpam-5153	202	17	=	=	PUNCT
ejpam-5153	202	18	−e	−e	VERB
ejpam-5153	202	19	therefore	therefore	ADV
ejpam-5153	202	20	a(u	a(u	NOUN
ejpam-5153	202	21	)	)	PUNCT
ejpam-5153	202	22	is	be	AUX
ejpam-5153	202	23	isomorphic	isomorphic	ADJ
ejpam-5153	202	24	to	to	ADP
ejpam-5153	202	25	c.	c.	NOUN
ejpam-5153	202	26	if	if	SCONJ
ejpam-5153	202	27	β	β	X
ejpam-5153	202	28	=	=	SYM
ejpam-5153	202	29	−1	−1	NOUN
ejpam-5153	202	30	,	,	PUNCT
ejpam-5153	202	31	then	then	ADV
ejpam-5153	202	32	(	(	PUNCT
ejpam-5153	202	33	u(ue	u(ue	PROPN
ejpam-5153	202	34	)	)	PUNCT
ejpam-5153	202	35	)	)	PUNCT
ejpam-5153	203	1	e	e	X
ejpam-5153	203	2	=	=	SYM
ejpam-5153	203	3	u(ue	u(ue	VERB
ejpam-5153	203	4	)	)	PUNCT
ejpam-5153	204	1	=	=	NOUN
ejpam-5153	204	2	⇒	⇒	VERB
ejpam-5153	204	3	α	α	NOUN
ejpam-5153	204	4	=	=	SYM
ejpam-5153	204	5	0	0	NUM
ejpam-5153	204	6	,	,	PUNCT
ejpam-5153	204	7	thus	thus	ADV
ejpam-5153	204	8	ue	ue	ADJ
ejpam-5153	204	9	=	=	PUNCT
ejpam-5153	204	10	−u	−u	PROPN
ejpam-5153	204	11	and	and	CCONJ
ejpam-5153	204	12	ue.u	ue.u	NOUN
ejpam-5153	204	13	=	=	NOUN
ejpam-5153	204	14	−e	−e	NOUN
ejpam-5153	204	15	=	=	AUX
ejpam-5153	204	16	⇒	⇒	VERB
ejpam-5153	204	17	u2	u2	NOUN
ejpam-5153	204	18	=	=	SYM
ejpam-5153	204	19	e	e	X
ejpam-5153	204	20	therefore	therefore	ADV
ejpam-5153	204	21	a(u	a(u	X
ejpam-5153	204	22	)	)	PUNCT
ejpam-5153	204	23	is	be	AUX
ejpam-5153	204	24	isomorphic	isomorphic	ADJ
ejpam-5153	204	25	to	to	ADP
ejpam-5153	204	26	⋆c	⋆c	PROPN
ejpam-5153	204	27	.	.	PUNCT
ejpam-5153	205	1	•	•	INTJ
ejpam-5153	205	2	if	if	SCONJ
ejpam-5153	205	3	ue	ue	PROPN
ejpam-5153	205	4	/∈	/∈	PUNCT
ejpam-5153	206	1	re+ru	re+ru	NOUN
ejpam-5153	206	2	,	,	PUNCT
ejpam-5153	206	3	then	then	ADV
ejpam-5153	206	4	the	the	DET
ejpam-5153	206	5	elements	element	NOUN
ejpam-5153	206	6	e	e	NOUN
ejpam-5153	206	7	,	,	PUNCT
ejpam-5153	206	8	u	u	NOUN
ejpam-5153	206	9	and	and	CCONJ
ejpam-5153	206	10	ue	ue	PROPN
ejpam-5153	206	11	are	be	AUX
ejpam-5153	206	12	linearly	linearly	ADV
ejpam-5153	206	13	independent	independent	ADJ
ejpam-5153	206	14	and	and	CCONJ
ejpam-5153	206	15	they	they	PRON
ejpam-5153	206	16	belong	belong	VERB
ejpam-5153	206	17	a(u	a(u	PROPN
ejpam-5153	206	18	)	)	PUNCT
ejpam-5153	206	19	therefore	therefore	ADV
ejpam-5153	206	20	dim(a(u	dim(a(u	PROPN
ejpam-5153	206	21	)	)	PUNCT
ejpam-5153	206	22	)	)	PUNCT
ejpam-5153	206	23	≥	≥	NOUN
ejpam-5153	207	1	4	4	X
ejpam-5153	207	2	.	.	X
ejpam-5153	207	3	proposition	proposition	NOUN
ejpam-5153	207	4	8	8	NUM
ejpam-5153	207	5	.	.	PUNCT
ejpam-5153	208	1	let	let	VERB
ejpam-5153	208	2	a	a	PRON
ejpam-5153	208	3	be	be	AUX
ejpam-5153	208	4	a	a	DET
ejpam-5153	208	5	four	four	NUM
ejpam-5153	208	6	-	-	PUNCT
ejpam-5153	208	7	dimensional	dimensional	ADJ
ejpam-5153	208	8	real	real	ADJ
ejpam-5153	208	9	division	division	NOUN
ejpam-5153	208	10	algebra	algebra	NOUN
ejpam-5153	208	11	with	with	ADP
ejpam-5153	208	12	left	left	ADJ
ejpam-5153	208	13	unit	unit	NOUN
ejpam-5153	208	14	e.	e.	PROPN
ejpam-5153	209	1	the	the	DET
ejpam-5153	209	2	following	follow	VERB
ejpam-5153	209	3	propositions	proposition	NOUN
ejpam-5153	209	4	are	be	AUX
ejpam-5153	209	5	equivalent	equivalent	ADJ
ejpam-5153	209	6	.	.	PUNCT
ejpam-5153	210	1	(	(	PUNCT
ejpam-5153	210	2	1	1	X
ejpam-5153	210	3	)	)	PUNCT
ejpam-5153	210	4	a	a	DET
ejpam-5153	210	5	satisfies	satisfie	NOUN
ejpam-5153	210	6	to	to	ADP
ejpam-5153	210	7	(	(	PUNCT
ejpam-5153	210	8	e1	e1	PROPN
ejpam-5153	210	9	)	)	PUNCT
ejpam-5153	210	10	(	(	PUNCT
ejpam-5153	210	11	2	2	X
ejpam-5153	210	12	)	)	PUNCT
ejpam-5153	210	13	a	a	PRON
ejpam-5153	210	14	contains	contain	VERB
ejpam-5153	210	15	a	a	DET
ejpam-5153	210	16	central	central	ADJ
ejpam-5153	210	17	element	element	NOUN
ejpam-5153	210	18	(	(	PUNCT
ejpam-5153	210	19	3	3	X
ejpam-5153	210	20	)	)	PUNCT
ejpam-5153	210	21	a	a	PRON
ejpam-5153	210	22	is	be	AUX
ejpam-5153	210	23	is	be	AUX
ejpam-5153	210	24	power	power	NOUN
ejpam-5153	210	25	-	-	PUNCT
ejpam-5153	210	26	commutative	commutative	ADJ
ejpam-5153	210	27	proof	proof	NOUN
ejpam-5153	210	28	.	.	PUNCT
ejpam-5153	211	1	(	(	PUNCT
ejpam-5153	211	2	1	1	X
ejpam-5153	211	3	)	)	PUNCT
ejpam-5153	211	4	=	=	NOUN
ejpam-5153	211	5	⇒	⇒	NOUN
ejpam-5153	211	6	(	(	PUNCT
ejpam-5153	211	7	2	2	NUM
ejpam-5153	211	8	)	)	PUNCT
ejpam-5153	211	9	,	,	PUNCT
ejpam-5153	211	10	corollary	corollary	ADJ
ejpam-5153	211	11	2	2	NUM
ejpam-5153	211	12	gives	give	VERB
ejpam-5153	211	13	the	the	DET
ejpam-5153	211	14	reeults	reeult	NOUN
ejpam-5153	211	15	,	,	PUNCT
ejpam-5153	211	16	(	(	PUNCT
ejpam-5153	211	17	2	2	X
ejpam-5153	211	18	)	)	PUNCT
ejpam-5153	211	19	⇐	⇐	ADJ
ejpam-5153	211	20	⇒	⇒	NOUN
ejpam-5153	211	21	(	(	PUNCT
ejpam-5153	211	22	3	3	X
ejpam-5153	211	23	)	)	PUNCT
ejpam-5153	211	24	theorem	theorem	NOUN
ejpam-5153	211	25	3	3	NUM
ejpam-5153	211	26	in	in	ADP
ejpam-5153	211	27	[	[	X
ejpam-5153	211	28	10	10	NUM
ejpam-5153	211	29	]	]	PUNCT
ejpam-5153	211	30	,	,	PUNCT
ejpam-5153	211	31	(	(	PUNCT
ejpam-5153	211	32	3	3	X
ejpam-5153	211	33	)	)	PUNCT
ejpam-5153	211	34	⇐	⇐	ADJ
ejpam-5153	211	35	⇒	⇒	NOUN
ejpam-5153	211	36	(	(	PUNCT
ejpam-5153	211	37	1	1	NUM
ejpam-5153	211	38	)	)	PUNCT
ejpam-5153	211	39	obvious	obvious	ADJ
ejpam-5153	211	40	.	.	PUNCT
ejpam-5153	212	1	a.	a.	PROPN
ejpam-5153	212	2	s.	s.	PROPN
ejpam-5153	212	3	diabang	diabang	PROPN
ejpam-5153	212	4	,	,	PUNCT
ejpam-5153	212	5	a.	a.	PROPN
ejpam-5153	212	6	s.	s.	PROPN
ejpam-5153	212	7	mballo	mballo	PROPN
ejpam-5153	212	8	,	,	PUNCT
ejpam-5153	212	9	p.	p.	PROPN
ejpam-5153	212	10	c.	c.	PROPN
ejpam-5153	212	11	diop	diop	PROPN
ejpam-5153	212	12	/	/	SYM
ejpam-5153	212	13	eur	eur	PROPN
ejpam-5153	212	14	.	.	PUNCT
ejpam-5153	213	1	j.	j.	PROPN
ejpam-5153	213	2	pure	pure	PROPN
ejpam-5153	213	3	appl	appl	PROPN
ejpam-5153	213	4	.	.	PROPN
ejpam-5153	213	5	math	math	PROPN
ejpam-5153	213	6	,	,	PUNCT
ejpam-5153	213	7	17	17	NUM
ejpam-5153	213	8	(	(	PUNCT
ejpam-5153	213	9	3	3	NUM
ejpam-5153	213	10	)	)	PUNCT
ejpam-5153	213	11	(	(	PUNCT
ejpam-5153	213	12	2024	2024	NUM
ejpam-5153	213	13	)	)	PUNCT
ejpam-5153	213	14	,	,	PUNCT
ejpam-5153	213	15	2276	2276	NUM
ejpam-5153	213	16	-	-	SYM
ejpam-5153	213	17	2287	2287	NUM
ejpam-5153	213	18	2283	2283	NUM
ejpam-5153	213	19	5	5	NUM
ejpam-5153	213	20	.	.	PUNCT
ejpam-5153	213	21	fused	fuse	VERB
ejpam-5153	213	22	algebras	algebras	PROPN
ejpam-5153	213	23	division	division	NOUN
ejpam-5153	213	24	with	with	ADP
ejpam-5153	213	25	left	left	ADJ
ejpam-5153	213	26	unit	unit	NOUN
ejpam-5153	213	27	satisfies	satisfie	NOUN
ejpam-5153	213	28	to	to	ADP
ejpam-5153	213	29	(	(	PUNCT
ejpam-5153	213	30	ei)i∈{1,2	ei)i∈{1,2	ADJ
ejpam-5153	213	31	}	}	PUNCT
ejpam-5153	213	32	.	.	PUNCT
ejpam-5153	214	1	[	[	X
ejpam-5153	214	2	2	2	X
ejpam-5153	214	3	]	]	PUNCT
ejpam-5153	214	4	gives	give	VERB
ejpam-5153	214	5	the	the	DET
ejpam-5153	214	6	definition	definition	NOUN
ejpam-5153	214	7	1	1	NUM
ejpam-5153	214	8	and	and	CCONJ
ejpam-5153	214	9	the	the	DET
ejpam-5153	214	10	theorem	theorem	ADJ
ejpam-5153	214	11	2	2	NUM
ejpam-5153	214	12	definition	definition	NOUN
ejpam-5153	214	13	1	1	NUM
ejpam-5153	214	14	.	.	PUNCT
ejpam-5153	215	1	let	let	VERB
ejpam-5153	215	2	a	a	DET
ejpam-5153	215	3	=	=	SYM
ejpam-5153	215	4	(	(	PUNCT
ejpam-5153	215	5	r2	r2	PROPN
ejpam-5153	215	6	,	,	PUNCT
ejpam-5153	215	7	◦	◦	NOUN
ejpam-5153	215	8	)	)	PUNCT
ejpam-5153	215	9	and	and	CCONJ
ejpam-5153	215	10	b	b	X
ejpam-5153	215	11	=	=	SYM
ejpam-5153	215	12	(	(	PUNCT
ejpam-5153	215	13	r2	r2	PROPN
ejpam-5153	215	14	,	,	PUNCT
ejpam-5153	215	15	•	•	NUM
ejpam-5153	215	16	)	)	PUNCT
ejpam-5153	215	17	the	the	DET
ejpam-5153	215	18	two	two	NUM
ejpam-5153	215	19	-	-	PUNCT
ejpam-5153	215	20	dimensional	dimensional	ADJ
ejpam-5153	215	21	real	real	ADJ
ejpam-5153	215	22	algebras	algebra	NOUN
ejpam-5153	215	23	with	with	ADP
ejpam-5153	215	24	the	the	DET
ejpam-5153	215	25	following	follow	VERB
ejpam-5153	215	26	multiplication	multiplication	NOUN
ejpam-5153	215	27	tables	table	NOUN
ejpam-5153	215	28	with	with	ADP
ejpam-5153	215	29	respect	respect	NOUN
ejpam-5153	215	30	to	to	ADP
ejpam-5153	215	31	a	a	DET
ejpam-5153	215	32	basis	basis	NOUN
ejpam-5153	215	33	b	b	NOUN
ejpam-5153	215	34	=	=	SYM
ejpam-5153	215	35	{	{	PUNCT
ejpam-5153	215	36	u	u	NOUN
ejpam-5153	215	37	,	,	PUNCT
ejpam-5153	215	38	v	v	NOUN
ejpam-5153	215	39	}	}	PUNCT
ejpam-5153	215	40	of	of	ADP
ejpam-5153	215	41	r2	r2	PROPN
ejpam-5153	215	42	:	:	PUNCT
ejpam-5153	215	43	(	(	PUNCT
ejpam-5153	215	44	a	a	X
ejpam-5153	215	45	)	)	PUNCT
ejpam-5153	215	46	◦	◦	NOUN
ejpam-5153	215	47	u	u	NOUN
ejpam-5153	215	48	v	v	NOUN
ejpam-5153	215	49	u	u	NOUN
ejpam-5153	215	50	a11u	a11u	ADP
ejpam-5153	215	51	+	+	CCONJ
ejpam-5153	215	52	b11v	b11v	X
ejpam-5153	215	53	a12u	a12u	PRON
ejpam-5153	216	1	+	+	CCONJ
ejpam-5153	217	1	b12v	b12v	NOUN
ejpam-5153	217	2	v	v	NOUN
ejpam-5153	217	3	a21u	a21u	NOUN
ejpam-5153	217	4	+	+	NUM
ejpam-5153	217	5	b21v	b21v	NOUN
ejpam-5153	217	6	a22u	a22u	PROPN
ejpam-5153	218	1	+	+	CCONJ
ejpam-5153	218	2	b22v	b22v	PROPN
ejpam-5153	218	3	(	(	PUNCT
ejpam-5153	218	4	b	b	NOUN
ejpam-5153	218	5	)	)	PUNCT
ejpam-5153	218	6	•	•	ADP
ejpam-5153	218	7	u	u	PROPN
ejpam-5153	218	8	v	v	NUM
ejpam-5153	218	9	u	u	NOUN
ejpam-5153	218	10	c11u	c11u	NOUN
ejpam-5153	218	11	+	+	CCONJ
ejpam-5153	218	12	d11v	d11v	PROPN
ejpam-5153	218	13	c12u	c12u	NOUN
ejpam-5153	218	14	+	+	CCONJ
ejpam-5153	218	15	d12v	d12v	PROPN
ejpam-5153	218	16	v	v	ADJ
ejpam-5153	218	17	c21u	c21u	NOUN
ejpam-5153	218	18	+	+	CCONJ
ejpam-5153	218	19	d21v	d21v	NOUN
ejpam-5153	218	20	c22u	c22u	PUNCT
ejpam-5153	219	1	+	+	SYM
ejpam-5153	219	2	d22v	d22v	VERB
ejpam-5153	219	3	we	we	PRON
ejpam-5153	219	4	define	define	VERB
ejpam-5153	219	5	multiplication	multiplication	NOUN
ejpam-5153	219	6	on	on	ADP
ejpam-5153	219	7	the	the	DET
ejpam-5153	219	8	direct	direct	ADJ
ejpam-5153	219	9	sum	sum	NOUN
ejpam-5153	219	10	a	a	DET
ejpam-5153	219	11	⊕	⊕	PROPN
ejpam-5153	219	12	b	b	PROPN
ejpam-5153	219	13	by	by	ADP
ejpam-5153	219	14	:	:	PUNCT
ejpam-5153	219	15	(	(	PUNCT
ejpam-5153	219	16	a	a	PRON
ejpam-5153	219	17	,	,	PUNCT
ejpam-5153	219	18	b).(c	b).(c	PROPN
ejpam-5153	219	19	,	,	PUNCT
ejpam-5153	219	20	d	d	NOUN
ejpam-5153	219	21	)	)	PUNCT
ejpam-5153	220	1	=	=	SYM
ejpam-5153	220	2	(	(	PUNCT
ejpam-5153	220	3	a	a	DET
ejpam-5153	220	4	◦	◦	NOUN
ejpam-5153	220	5	c	c	NOUN
ejpam-5153	220	6	−	−	PROPN
ejpam-5153	220	7	b	b	PROPN
ejpam-5153	220	8	•	•	NUM
ejpam-5153	220	9	d	d	PROPN
ejpam-5153	220	10	,	,	PUNCT
ejpam-5153	220	11	a	a	DET
ejpam-5153	220	12	◦	◦	NOUN
ejpam-5153	220	13	d	d	NOUN
ejpam-5153	221	1	+	+	NOUN
ejpam-5153	221	2	b	b	NUM
ejpam-5153	221	3	•	•	NUM
ejpam-5153	221	4	c	c	NOUN
ejpam-5153	221	5	)	)	PUNCT
ejpam-5153	221	6	setting	set	VERB
ejpam-5153	221	7	e1	e1	NOUN
ejpam-5153	221	8	=	=	SYM
ejpam-5153	221	9	(	(	PUNCT
ejpam-5153	221	10	u	u	NOUN
ejpam-5153	221	11	,	,	PUNCT
ejpam-5153	221	12	0	0	NUM
ejpam-5153	221	13	)	)	PUNCT
ejpam-5153	221	14	,	,	PUNCT
ejpam-5153	221	15	e2	e2	PROPN
ejpam-5153	221	16	=	=	SYM
ejpam-5153	221	17	(	(	PUNCT
ejpam-5153	221	18	v	v	NOUN
ejpam-5153	221	19	,	,	PUNCT
ejpam-5153	221	20	0	0	NUM
ejpam-5153	221	21	)	)	PUNCT
ejpam-5153	221	22	,	,	PUNCT
ejpam-5153	221	23	e3	e3	NOUN
ejpam-5153	221	24	=	=	SYM
ejpam-5153	221	25	(	(	PUNCT
ejpam-5153	221	26	0	0	NUM
ejpam-5153	221	27	,	,	PUNCT
ejpam-5153	221	28	u	u	NOUN
ejpam-5153	221	29	)	)	PUNCT
ejpam-5153	221	30	and	and	CCONJ
ejpam-5153	221	31	e4	e4	PROPN
ejpam-5153	221	32	=	=	SYM
ejpam-5153	221	33	(	(	PUNCT
ejpam-5153	221	34	0	0	NUM
ejpam-5153	221	35	,	,	PUNCT
ejpam-5153	221	36	v	v	NOUN
ejpam-5153	221	37	)	)	PUNCT
ejpam-5153	221	38	.	.	PUNCT
ejpam-5153	222	1	it	it	PRON
ejpam-5153	222	2	is	be	AUX
ejpam-5153	222	3	easy	easy	ADJ
ejpam-5153	222	4	to	to	PART
ejpam-5153	222	5	verfy	verfy	VERB
ejpam-5153	222	6	that	that	SCONJ
ejpam-5153	222	7	the	the	DET
ejpam-5153	222	8	algebra	algebra	NOUN
ejpam-5153	222	9	a	a	DET
ejpam-5153	222	10	⊕	⊕	PROPN
ejpam-5153	222	11	b	b	PROPN
ejpam-5153	222	12	has	have	VERB
ejpam-5153	222	13	the	the	DET
ejpam-5153	222	14	multiplication	multiplication	NOUN
ejpam-5153	222	15	table	table	NOUN
ejpam-5153	222	16	(	(	PUNCT
ejpam-5153	222	17	a	a	DET
ejpam-5153	222	18	⊕	⊕	PROPN
ejpam-5153	222	19	b	b	NOUN
ejpam-5153	222	20	)	)	PUNCT
ejpam-5153	222	21	.	.	PUNCT
ejpam-5153	223	1	e1	e1	PROPN
ejpam-5153	223	2	e2	e2	PROPN
ejpam-5153	223	3	e3	e3	NOUN
ejpam-5153	223	4	e4	e4	PROPN
ejpam-5153	223	5	e1	e1	NOUN
ejpam-5153	223	6	a11e1	a11e1	NOUN
ejpam-5153	223	7	+	+	CCONJ
ejpam-5153	223	8	b11e2	b11e2	NOUN
ejpam-5153	223	9	a12e1	a12e1	VERB
ejpam-5153	223	10	+	+	CCONJ
ejpam-5153	223	11	b12e2	b12e2	PROPN
ejpam-5153	223	12	a11e3	a11e3	NOUN
ejpam-5153	223	13	+	+	CCONJ
ejpam-5153	223	14	b11e4	b11e4	NOUN
ejpam-5153	223	15	a12e3	a12e3	NOUN
ejpam-5153	223	16	+	+	CCONJ
ejpam-5153	223	17	b12e4	b12e4	PROPN
ejpam-5153	223	18	e2	e2	NOUN
ejpam-5153	223	19	a21e1	a21e1	NOUN
ejpam-5153	223	20	+	+	CCONJ
ejpam-5153	223	21	b21e2	b21e2	NOUN
ejpam-5153	223	22	a22e1	a22e1	X
ejpam-5153	223	23	+	+	CCONJ
ejpam-5153	223	24	b22e2	b22e2	PROPN
ejpam-5153	223	25	a21e3	a21e3	NOUN
ejpam-5153	223	26	+	+	CCONJ
ejpam-5153	223	27	b21e4	b21e4	NOUN
ejpam-5153	223	28	a22e3	a22e3	NOUN
ejpam-5153	223	29	+	+	CCONJ
ejpam-5153	223	30	b22e4	b22e4	VERB
ejpam-5153	223	31	e3	e3	NOUN
ejpam-5153	223	32	c11e3	c11e3	NOUN
ejpam-5153	223	33	+	+	CCONJ
ejpam-5153	223	34	d11e4	d11e4	NOUN
ejpam-5153	223	35	c12e3	c12e3	NOUN
ejpam-5153	223	36	+	+	CCONJ
ejpam-5153	223	37	d12e4	d12e4	PROPN
ejpam-5153	223	38	−c11e1	−c11e1	NOUN
ejpam-5153	223	39	−	−	NOUN
ejpam-5153	223	40	d11e2	d11e2	VERB
ejpam-5153	223	41	−c12e1	−c12e1	NOUN
ejpam-5153	223	42	−	−	PROPN
ejpam-5153	223	43	d12e2	d12e2	PROPN
ejpam-5153	223	44	e4	e4	PROPN
ejpam-5153	223	45	c21e3	c21e3	PROPN
ejpam-5153	224	1	+	+	CCONJ
ejpam-5153	224	2	d21e4	d21e4	NOUN
ejpam-5153	224	3	c22e3	c22e3	NOUN
ejpam-5153	225	1	+	+	CCONJ
ejpam-5153	225	2	d22e4	d22e4	VERB
ejpam-5153	225	3	−c21e1	−c21e1	PRON
ejpam-5153	225	4	−	−	NOUN
ejpam-5153	226	1	d21e2	d21e2	VERB
ejpam-5153	226	2	−c22e1	−c22e1	NOUN
ejpam-5153	226	3	−	−	X
ejpam-5153	226	4	d22e2	d22e2	NOUN
ejpam-5153	226	5	we	we	PRON
ejpam-5153	226	6	call	call	VERB
ejpam-5153	226	7	this	this	DET
ejpam-5153	226	8	table	table	NOUN
ejpam-5153	226	9	a	a	DET
ejpam-5153	226	10	standard	standard	ADJ
ejpam-5153	226	11	table	table	NOUN
ejpam-5153	226	12	for	for	ADP
ejpam-5153	226	13	the	the	DET
ejpam-5153	226	14	a	a	ADV
ejpam-5153	226	15	-	-	PUNCT
ejpam-5153	226	16	based	base	VERB
ejpam-5153	226	17	fused	fuse	VERB
ejpam-5153	226	18	algebras	algebra	NOUN
ejpam-5153	226	19	a	a	DET
ejpam-5153	226	20	⊕	⊕	PROPN
ejpam-5153	226	21	b.	b.	PROPN
ejpam-5153	227	1	a	a	PRON
ejpam-5153	227	2	is	be	AUX
ejpam-5153	227	3	isomorphic	isomorphic	ADJ
ejpam-5153	227	4	to	to	ADP
ejpam-5153	227	5	the	the	DET
ejpam-5153	227	6	subalgebra	subalgebra	NOUN
ejpam-5153	227	7	of	of	ADP
ejpam-5153	227	8	pairs	pair	NOUN
ejpam-5153	227	9	(	(	PUNCT
ejpam-5153	227	10	a,0	a,0	NOUN
ejpam-5153	227	11	)	)	PUNCT
ejpam-5153	227	12	note	note	NOUN
ejpam-5153	227	13	.	.	PUNCT
ejpam-5153	228	1	in	in	ADP
ejpam-5153	228	2	this	this	DET
ejpam-5153	228	3	part	part	NOUN
ejpam-5153	228	4	we	we	PRON
ejpam-5153	228	5	work	work	VERB
ejpam-5153	228	6	with	with	ADP
ejpam-5153	228	7	a	a	DET
ejpam-5153	228	8	,	,	PUNCT
ejpam-5153	228	9	b	b	NOUN
ejpam-5153	228	10	and	and	CCONJ
ejpam-5153	228	11	a	a	DET
ejpam-5153	228	12	⊕	⊕	PROPN
ejpam-5153	228	13	b	b	PROPN
ejpam-5153	228	14	the	the	DET
ejpam-5153	228	15	algebras	algebra	NOUN
ejpam-5153	228	16	of	of	ADP
ejpam-5153	228	17	definition	definition	NOUN
ejpam-5153	228	18	1	1	NUM
ejpam-5153	228	19	.	.	PUNCT
ejpam-5153	228	20	theorem	theorem	NOUN
ejpam-5153	228	21	2	2	NUM
ejpam-5153	228	22	.	.	PUNCT
ejpam-5153	228	23	a	a	DET
ejpam-5153	228	24	fused	fuse	VERB
ejpam-5153	228	25	algebra	algebra	NOUN
ejpam-5153	228	26	a	a	DET
ejpam-5153	228	27	⊕	⊕	PROPN
ejpam-5153	228	28	b	b	PROPN
ejpam-5153	228	29	is	be	AUX
ejpam-5153	228	30	a	a	DET
ejpam-5153	228	31	division	division	NOUN
ejpam-5153	228	32	algebra	algebra	NOUN
ejpam-5153	228	33	if	if	SCONJ
ejpam-5153	228	34	and	and	CCONJ
ejpam-5153	228	35	only	only	ADV
ejpam-5153	228	36	if	if	SCONJ
ejpam-5153	228	37	a	a	PRON
ejpam-5153	228	38	and	and	CCONJ
ejpam-5153	228	39	b	b	NOUN
ejpam-5153	228	40	are	be	AUX
ejpam-5153	228	41	division	division	NOUN
ejpam-5153	228	42	algebras	algebra	NOUN
ejpam-5153	228	43	and	and	CCONJ
ejpam-5153	228	44	in	in	ADP
ejpam-5153	228	45	any	any	DET
ejpam-5153	228	46	standard	standard	ADJ
ejpam-5153	228	47	table	table	NOUN
ejpam-5153	228	48	for	for	ADP
ejpam-5153	228	49	a	a	DET
ejpam-5153	228	50	⊕	⊕	PROPN
ejpam-5153	228	51	b	b	PROPN
ejpam-5153	228	52	(	(	PUNCT
ejpam-5153	228	53	a11b12	a11b12	NOUN
ejpam-5153	228	54	−	−	PROPN
ejpam-5153	228	55	b11a12)(c11d12	b11a12)(c11d12	PROPN
ejpam-5153	229	1	−	−	PROPN
ejpam-5153	229	2	d11c12	d11c12	NOUN
ejpam-5153	229	3	)	)	PUNCT
ejpam-5153	229	4	<	<	X
ejpam-5153	229	5	0	0	NUM
ejpam-5153	229	6	proposition	proposition	NOUN
ejpam-5153	229	7	9	9	NUM
ejpam-5153	229	8	.	.	PUNCT
ejpam-5153	230	1	if	if	SCONJ
ejpam-5153	230	2	a	a	DET
ejpam-5153	230	3	fused	fuse	VERB
ejpam-5153	230	4	algebra	algebra	NOUN
ejpam-5153	230	5	a	a	DET
ejpam-5153	230	6	⊕	⊕	PROPN
ejpam-5153	230	7	b	b	PROPN
ejpam-5153	230	8	is	be	AUX
ejpam-5153	230	9	a	a	DET
ejpam-5153	230	10	division	division	NOUN
ejpam-5153	230	11	with	with	ADP
ejpam-5153	230	12	left	left	ADJ
ejpam-5153	230	13	unit	unit	NOUN
ejpam-5153	230	14	e1	e1	NOUN
ejpam-5153	230	15	satisfies	satisfie	NOUN
ejpam-5153	230	16	to	to	ADP
ejpam-5153	230	17	(	(	PUNCT
ejpam-5153	230	18	e2	e2	PROPN
ejpam-5153	230	19	)	)	PUNCT
ejpam-5153	230	20	,	,	PUNCT
ejpam-5153	230	21	then	then	ADV
ejpam-5153	230	22	a	a	PRON
ejpam-5153	230	23	is	be	AUX
ejpam-5153	230	24	isomorphic	isomorphic	ADJ
ejpam-5153	230	25	to	to	ADP
ejpam-5153	230	26	either	either	CCONJ
ejpam-5153	230	27	c	c	PROPN
ejpam-5153	230	28	,	,	PUNCT
ejpam-5153	230	29	⋆c	⋆c	PROPN
ejpam-5153	230	30	proof	proof	NOUN
ejpam-5153	230	31	.	.	PUNCT
ejpam-5153	231	1	a	a	PRON
ejpam-5153	231	2	is	be	AUX
ejpam-5153	231	3	an	an	DET
ejpam-5153	231	4	two	two	NUM
ejpam-5153	231	5	-	-	PUNCT
ejpam-5153	231	6	dimensional	dimensional	ADJ
ejpam-5153	231	7	real	real	ADJ
ejpam-5153	231	8	division	division	NOUN
ejpam-5153	231	9	algebra	algebra	NOUN
ejpam-5153	231	10	with	with	ADP
ejpam-5153	231	11	left	left	ADJ
ejpam-5153	231	12	unit	unit	NOUN
ejpam-5153	231	13	u	u	NOUN
ejpam-5153	231	14	satisfies	satisfie	NOUN
ejpam-5153	231	15	to	to	ADP
ejpam-5153	231	16	(	(	PUNCT
ejpam-5153	231	17	e2	e2	PROPN
ejpam-5153	231	18	)	)	PUNCT
ejpam-5153	231	19	,	,	PUNCT
ejpam-5153	231	20	the	the	DET
ejpam-5153	231	21	proposition	proposition	NOUN
ejpam-5153	231	22	2	2	NUM
ejpam-5153	231	23	shows	show	VERB
ejpam-5153	231	24	that	that	SCONJ
ejpam-5153	231	25	a	a	PRON
ejpam-5153	231	26	is	be	AUX
ejpam-5153	231	27	isomorphic	isomorphic	ADJ
ejpam-5153	231	28	to	to	ADP
ejpam-5153	231	29	either	either	CCONJ
ejpam-5153	231	30	c	c	PROPN
ejpam-5153	231	31	,	,	PUNCT
ejpam-5153	231	32	⋆c	⋆c	PROPN
ejpam-5153	231	33	.	.	PUNCT
ejpam-5153	232	1	lemma	lemma	PROPN
ejpam-5153	232	2	8	8	NUM
ejpam-5153	232	3	.	.	PUNCT
ejpam-5153	233	1	let	let	VERB
ejpam-5153	233	2	a	a	DET
ejpam-5153	233	3	=	=	SYM
ejpam-5153	233	4	(	(	PUNCT
ejpam-5153	233	5	r2	r2	PROPN
ejpam-5153	233	6	,	,	PUNCT
ejpam-5153	233	7	◦	◦	NOUN
ejpam-5153	233	8	)	)	PUNCT
ejpam-5153	233	9	isomorphic	isomorphic	ADJ
ejpam-5153	233	10	to	to	ADP
ejpam-5153	233	11	c	c	PROPN
ejpam-5153	233	12	and	and	CCONJ
ejpam-5153	233	13	b	b	X
ejpam-5153	233	14	=	=	SYM
ejpam-5153	233	15	(	(	PUNCT
ejpam-5153	233	16	r2	r2	PROPN
ejpam-5153	233	17	,	,	PUNCT
ejpam-5153	233	18	•	•	NUM
ejpam-5153	233	19	)	)	PUNCT
ejpam-5153	233	20	the	the	DET
ejpam-5153	233	21	algebra	algebra	NOUN
ejpam-5153	233	22	whose	whose	DET
ejpam-5153	233	23	multiplication	multiplication	NOUN
ejpam-5153	233	24	table	table	NOUN
ejpam-5153	233	25	in	in	ADP
ejpam-5153	233	26	the	the	DET
ejpam-5153	233	27	basis	basis	NOUN
ejpam-5153	233	28	β	β	X
ejpam-5153	233	29	=	=	PUNCT
ejpam-5153	233	30	{	{	PUNCT
ejpam-5153	233	31	u	u	NOUN
ejpam-5153	233	32	,	,	PUNCT
ejpam-5153	233	33	v	v	NOUN
ejpam-5153	233	34	}	}	PUNCT
ejpam-5153	233	35	(	(	PUNCT
ejpam-5153	233	36	b	b	NOUN
ejpam-5153	233	37	)	)	PUNCT
ejpam-5153	233	38	•	•	ADP
ejpam-5153	233	39	u	u	NOUN
ejpam-5153	233	40	v	v	ADP
ejpam-5153	233	41	u	u	X
ejpam-5153	233	42	u	u	NOUN
ejpam-5153	233	43	−v	−v	NOUN
ejpam-5153	233	44	v	v	ADP
ejpam-5153	233	45	c21u	c21u	NOUN
ejpam-5153	233	46	−	−	PROPN
ejpam-5153	233	47	v	v	NOUN
ejpam-5153	233	48	c22u	c22u	NOUN
ejpam-5153	233	49	with	with	ADP
ejpam-5153	233	50	c21	c21	NOUN
ejpam-5153	233	51	,	,	PUNCT
ejpam-5153	233	52	c22	c22	NOUN
ejpam-5153	233	53	∈	∈	PROPN
ejpam-5153	233	54	r	r	NOUN
ejpam-5153	233	55	thus	thus	ADV
ejpam-5153	233	56	that	that	PRON
ejpam-5153	233	57	c2	c2	PROPN
ejpam-5153	233	58	12	12	NUM
ejpam-5153	233	59	<	<	X
ejpam-5153	233	60	−4c22	−4c22	PROPN
ejpam-5153	233	61	.	.	PUNCT
ejpam-5153	234	1	then	then	ADV
ejpam-5153	234	2	a	a	DET
ejpam-5153	234	3	⊕b	⊕b	NOUN
ejpam-5153	234	4	is	be	AUX
ejpam-5153	234	5	division	division	NOUN
ejpam-5153	234	6	algebra	algebra	NOUN
ejpam-5153	234	7	satisfies	satisfie	NOUN
ejpam-5153	234	8	to	to	ADP
ejpam-5153	234	9	(	(	PUNCT
ejpam-5153	234	10	e2	e2	PROPN
ejpam-5153	234	11	)	)	PUNCT
ejpam-5153	234	12	,	,	PUNCT
ejpam-5153	234	13	not	not	PART
ejpam-5153	234	14	satisfies	satisfie	NOUN
ejpam-5153	234	15	to	to	ADP
ejpam-5153	234	16	(	(	PUNCT
ejpam-5153	234	17	e3	e3	NOUN
ejpam-5153	234	18	)	)	PUNCT
ejpam-5153	234	19	.	.	PUNCT
ejpam-5153	235	1	and	and	CCONJ
ejpam-5153	235	2	not	not	PART
ejpam-5153	235	3	isomorphic	isomorphic	ADJ
ejpam-5153	235	4	to	to	ADP
ejpam-5153	235	5	h	h	PROPN
ejpam-5153	235	6	a.	a.	PROPN
ejpam-5153	235	7	s.	s.	PROPN
ejpam-5153	235	8	diabang	diabang	PROPN
ejpam-5153	235	9	,	,	PUNCT
ejpam-5153	235	10	a.	a.	PROPN
ejpam-5153	235	11	s.	s.	PROPN
ejpam-5153	235	12	mballo	mballo	PROPN
ejpam-5153	235	13	,	,	PUNCT
ejpam-5153	235	14	p.	p.	PROPN
ejpam-5153	235	15	c.	c.	PROPN
ejpam-5153	235	16	diop	diop	PROPN
ejpam-5153	235	17	/	/	SYM
ejpam-5153	235	18	eur	eur	PROPN
ejpam-5153	235	19	.	.	PUNCT
ejpam-5153	236	1	j.	j.	PROPN
ejpam-5153	236	2	pure	pure	PROPN
ejpam-5153	236	3	appl	appl	PROPN
ejpam-5153	236	4	.	.	PROPN
ejpam-5153	236	5	math	math	PROPN
ejpam-5153	236	6	,	,	PUNCT
ejpam-5153	236	7	17	17	NUM
ejpam-5153	236	8	(	(	PUNCT
ejpam-5153	236	9	3	3	NUM
ejpam-5153	236	10	)	)	PUNCT
ejpam-5153	236	11	(	(	PUNCT
ejpam-5153	236	12	2024	2024	NUM
ejpam-5153	236	13	)	)	PUNCT
ejpam-5153	236	14	,	,	PUNCT
ejpam-5153	236	15	2276	2276	NUM
ejpam-5153	236	16	-	-	SYM
ejpam-5153	236	17	2287	2287	NUM
ejpam-5153	236	18	2284	2284	NUM
ejpam-5153	236	19	proof	proof	NOUN
ejpam-5153	236	20	.	.	PUNCT
ejpam-5153	237	1	the	the	DET
ejpam-5153	237	2	theorem	theorem	NOUN
ejpam-5153	237	3	2	2	NUM
ejpam-5153	237	4	,	,	PUNCT
ejpam-5153	237	5	shows	show	VERB
ejpam-5153	237	6	that	that	SCONJ
ejpam-5153	237	7	a	a	DET
ejpam-5153	237	8	⊕	⊕	PROPN
ejpam-5153	237	9	b	b	PROPN
ejpam-5153	237	10	is	be	AUX
ejpam-5153	237	11	division	division	NOUN
ejpam-5153	237	12	algebra	algebra	NOUN
ejpam-5153	237	13	.	.	PUNCT
ejpam-5153	238	1	now	now	ADV
ejpam-5153	238	2	let	let	VERB
ejpam-5153	238	3	x	x	X
ejpam-5153	238	4	=	=	PUNCT
ejpam-5153	238	5	x1e1	x1e1	PROPN
ejpam-5153	239	1	+	+	CCONJ
ejpam-5153	239	2	x2e2	x2e2	PROPN
ejpam-5153	240	1	+	+	CCONJ
ejpam-5153	240	2	x3e3	x3e3	PUNCT
ejpam-5153	240	3	+	+	CCONJ
ejpam-5153	240	4	x4e4	x4e4	X
ejpam-5153	240	5	∈	∈	PROPN
ejpam-5153	240	6	a	a	DET
ejpam-5153	240	7	⊕	⊕	PROPN
ejpam-5153	240	8	b	b	NOUN
ejpam-5153	240	9	we	we	PRON
ejpam-5153	240	10	have	have	VERB
ejpam-5153	240	11	(	(	PUNCT
ejpam-5153	240	12	x2	x2	PROPN
ejpam-5153	240	13	,	,	PUNCT
ejpam-5153	240	14	x2	x2	PROPN
ejpam-5153	240	15	,	,	PUNCT
ejpam-5153	240	16	x2	x2	PROPN
ejpam-5153	240	17	)	)	PUNCT
ejpam-5153	241	1	=	=	SYM
ejpam-5153	241	2	0	0	X
ejpam-5153	241	3	.	.	PUNCT
ejpam-5153	242	1	so	so	ADV
ejpam-5153	242	2	a	a	DET
ejpam-5153	242	3	⊕	⊕	PROPN
ejpam-5153	242	4	b	b	PROPN
ejpam-5153	242	5	satisfies	satisfie	NOUN
ejpam-5153	242	6	to	to	ADP
ejpam-5153	242	7	(	(	PUNCT
ejpam-5153	242	8	e2	e2	PROPN
ejpam-5153	242	9	)	)	PUNCT
ejpam-5153	242	10	.	.	PUNCT
ejpam-5153	243	1	we	we	PRON
ejpam-5153	243	2	have	have	VERB
ejpam-5153	243	3	for	for	ADP
ejpam-5153	243	4	a	a	DET
ejpam-5153	243	5	=	=	NOUN
ejpam-5153	243	6	e1	e1	PROPN
ejpam-5153	243	7	+	+	CCONJ
ejpam-5153	243	8	e2	e2	NOUN
ejpam-5153	243	9	+	+	CCONJ
ejpam-5153	243	10	e3	e3	NOUN
ejpam-5153	243	11	+	+	X
ejpam-5153	243	12	e4	e4	PROPN
ejpam-5153	243	13	,	,	PUNCT
ejpam-5153	243	14	(	(	PUNCT
ejpam-5153	243	15	a	a	PRON
ejpam-5153	243	16	,	,	PUNCT
ejpam-5153	243	17	a	a	PRON
ejpam-5153	243	18	,	,	PUNCT
ejpam-5153	243	19	a	a	X
ejpam-5153	243	20	)	)	PUNCT
ejpam-5153	243	21	̸=	̸=	PROPN
ejpam-5153	243	22	0	0	NUM
ejpam-5153	243	23	and	and	CCONJ
ejpam-5153	243	24	(	(	PUNCT
ejpam-5153	243	25	e2	e2	PROPN
ejpam-5153	243	26	,	,	PUNCT
ejpam-5153	243	27	e3	e3	NOUN
ejpam-5153	243	28	,	,	PUNCT
ejpam-5153	243	29	e4	e4	PROPN
ejpam-5153	243	30	)	)	PUNCT
ejpam-5153	243	31	̸=	̸=	PROPN
ejpam-5153	243	32	0	0	NUM
ejpam-5153	243	33	.	.	PUNCT
ejpam-5153	244	1	so	so	ADV
ejpam-5153	244	2	a	a	DET
ejpam-5153	244	3	⊕	⊕	PROPN
ejpam-5153	244	4	b	b	PROPN
ejpam-5153	244	5	not	not	PART
ejpam-5153	244	6	satisfies	satisfy	VERB
ejpam-5153	244	7	to	to	ADP
ejpam-5153	244	8	(	(	PUNCT
ejpam-5153	244	9	e1	e1	PROPN
ejpam-5153	244	10	)	)	PUNCT
ejpam-5153	244	11	and	and	CCONJ
ejpam-5153	244	12	not	not	PART
ejpam-5153	244	13	isomorphic	isomorphic	ADJ
ejpam-5153	244	14	to	to	ADP
ejpam-5153	244	15	h.	h.	PROPN
ejpam-5153	244	16	5.1	5.1	NUM
ejpam-5153	244	17	.	.	PUNCT
ejpam-5153	245	1	study	study	NOUN
ejpam-5153	245	2	of	of	ADP
ejpam-5153	245	3	fused	fuse	VERB
ejpam-5153	245	4	algebras	algebra	NOUN
ejpam-5153	245	5	with	with	ADP
ejpam-5153	245	6	a	a	PRON
ejpam-5153	245	7	is	be	AUX
ejpam-5153	245	8	isomorphic	isomorphic	ADJ
ejpam-5153	245	9	to	to	ADP
ejpam-5153	245	10	c.	c.	PROPN
ejpam-5153	245	11	lemma	lemma	PROPN
ejpam-5153	246	1	9	9	X
ejpam-5153	246	2	.	.	PUNCT
ejpam-5153	247	1	we	we	PRON
ejpam-5153	247	2	have	have	VERB
ejpam-5153	247	3	(	(	PUNCT
ejpam-5153	247	4	1	1	X
ejpam-5153	247	5	)	)	PUNCT
ejpam-5153	247	6	c	c	PROPN
ejpam-5153	247	7	⊕	⊕	PROPN
ejpam-5153	247	8	r(0	r(0	PROPN
ejpam-5153	247	9	,	,	PUNCT
ejpam-5153	247	10	d12	d12	PROPN
ejpam-5153	247	11	,	,	PUNCT
ejpam-5153	247	12	1	1	NUM
ejpam-5153	247	13	,	,	PUNCT
ejpam-5153	247	14	0	0	NUM
ejpam-5153	247	15	)	)	PUNCT
ejpam-5153	247	16	satisfies	satisfie	NOUN
ejpam-5153	247	17	to	to	ADP
ejpam-5153	247	18	(	(	PUNCT
ejpam-5153	247	19	e2	e2	PROPN
ejpam-5153	247	20	)	)	PUNCT
ejpam-5153	248	1	if	if	SCONJ
ejpam-5153	248	2	and	and	CCONJ
ejpam-5153	248	3	only	only	ADV
ejpam-5153	248	4	if	if	SCONJ
ejpam-5153	248	5	d12	d12	NUM
ejpam-5153	248	6	=	=	SYM
ejpam-5153	248	7	−1	−1	NOUN
ejpam-5153	248	8	(	(	PUNCT
ejpam-5153	248	9	2	2	NUM
ejpam-5153	248	10	)	)	PUNCT
ejpam-5153	249	1	if	if	SCONJ
ejpam-5153	249	2	d12	d12	ADJ
ejpam-5153	249	3	̸=	̸=	PROPN
ejpam-5153	249	4	−1	−1	NOUN
ejpam-5153	249	5	and	and	CCONJ
ejpam-5153	249	6	4d12c22	4d12c22	NOUN
ejpam-5153	249	7	<	<	X
ejpam-5153	249	8	−1	−1	NOUN
ejpam-5153	249	9	,	,	PUNCT
ejpam-5153	249	10	then	then	ADV
ejpam-5153	249	11	c	c	PROPN
ejpam-5153	249	12	⊕	⊕	PROPN
ejpam-5153	249	13	r(1	r(1	PROPN
ejpam-5153	249	14	,	,	PUNCT
ejpam-5153	249	15	d12	d12	PROPN
ejpam-5153	249	16	,	,	PUNCT
ejpam-5153	249	17	c22	c22	NOUN
ejpam-5153	249	18	,	,	PUNCT
ejpam-5153	249	19	0	0	NUM
ejpam-5153	249	20	)	)	PUNCT
ejpam-5153	249	21	not	not	PART
ejpam-5153	249	22	satisfies	satisfie	NOUN
ejpam-5153	249	23	to	to	ADP
ejpam-5153	249	24	(	(	PUNCT
ejpam-5153	249	25	e2	e2	PROPN
ejpam-5153	249	26	)	)	PUNCT
ejpam-5153	249	27	.	.	PUNCT
ejpam-5153	250	1	(	(	PUNCT
ejpam-5153	250	2	3	3	X
ejpam-5153	250	3	)	)	PUNCT
ejpam-5153	250	4	if	if	SCONJ
ejpam-5153	250	5	c22	c22	PROPN
ejpam-5153	250	6	>	>	X
ejpam-5153	250	7	0	0	PROPN
ejpam-5153	250	8	,	,	PUNCT
ejpam-5153	250	9	then	then	ADV
ejpam-5153	250	10	c	c	PROPN
ejpam-5153	250	11	⊕	⊕	PROPN
ejpam-5153	250	12	r(1	r(1	PROPN
ejpam-5153	250	13	,	,	PUNCT
ejpam-5153	250	14	−1	−1	NOUN
ejpam-5153	250	15	,	,	PUNCT
ejpam-5153	250	16	c22	c22	NOUN
ejpam-5153	250	17	,	,	PUNCT
ejpam-5153	250	18	1	1	NUM
ejpam-5153	250	19	)	)	PUNCT
ejpam-5153	250	20	not	not	PART
ejpam-5153	250	21	satisfies	satisfie	NOUN
ejpam-5153	250	22	to	to	ADP
ejpam-5153	250	23	(	(	PUNCT
ejpam-5153	250	24	e2	e2	PROPN
ejpam-5153	250	25	)	)	PUNCT
ejpam-5153	250	26	.	.	PUNCT
ejpam-5153	251	1	proof	proof	NOUN
ejpam-5153	251	2	.	.	PUNCT
ejpam-5153	252	1	taking	take	VERB
ejpam-5153	252	2	u	u	NOUN
ejpam-5153	252	3	=	=	NOUN
ejpam-5153	252	4	1	1	NUM
ejpam-5153	252	5	and	and	CCONJ
ejpam-5153	252	6	v	v	NOUN
ejpam-5153	252	7	=	=	SYM
ejpam-5153	252	8	i.	i.	NOUN
ejpam-5153	252	9	(	(	PUNCT
ejpam-5153	252	10	1	1	X
ejpam-5153	252	11	)	)	PUNCT
ejpam-5153	252	12	suppose	suppose	VERB
ejpam-5153	252	13	that	that	SCONJ
ejpam-5153	252	14	c	c	PROPN
ejpam-5153	252	15	⊕	⊕	PROPN
ejpam-5153	252	16	r(0	r(0	PROPN
ejpam-5153	252	17	,	,	PUNCT
ejpam-5153	252	18	d12	d12	PROPN
ejpam-5153	252	19	,	,	PUNCT
ejpam-5153	252	20	1	1	NUM
ejpam-5153	252	21	,	,	PUNCT
ejpam-5153	252	22	0	0	NUM
ejpam-5153	252	23	)	)	PUNCT
ejpam-5153	252	24	satisfies	satisfie	NOUN
ejpam-5153	252	25	to	to	ADP
ejpam-5153	252	26	(	(	PUNCT
ejpam-5153	252	27	e2	e2	PROPN
ejpam-5153	252	28	)	)	PUNCT
ejpam-5153	252	29	.	.	PUNCT
ejpam-5153	253	1	let	let	VERB
ejpam-5153	253	2	a	a	DET
ejpam-5153	253	3	=	=	NOUN
ejpam-5153	253	4	e1	e1	PROPN
ejpam-5153	253	5	+	+	CCONJ
ejpam-5153	253	6	e2	e2	NOUN
ejpam-5153	253	7	+	+	CCONJ
ejpam-5153	253	8	e3	e3	NOUN
ejpam-5153	253	9	+	+	CCONJ
ejpam-5153	253	10	e4	e4	PROPN
ejpam-5153	253	11	∈	∈	PROPN
ejpam-5153	253	12	c	c	PROPN
ejpam-5153	253	13	⊕	⊕	PROPN
ejpam-5153	253	14	r(0	r(0	PROPN
ejpam-5153	253	15	,	,	PUNCT
ejpam-5153	253	16	d12	d12	PROPN
ejpam-5153	253	17	,	,	PUNCT
ejpam-5153	253	18	1	1	NUM
ejpam-5153	253	19	,	,	PUNCT
ejpam-5153	253	20	0	0	NUM
ejpam-5153	253	21	)	)	PUNCT
ejpam-5153	253	22	,	,	PUNCT
ejpam-5153	253	23	we	we	PRON
ejpam-5153	253	24	have	have	VERB
ejpam-5153	253	25	(	(	PUNCT
ejpam-5153	253	26	a2	a2	PROPN
ejpam-5153	253	27	,	,	PUNCT
ejpam-5153	253	28	a2	a2	PROPN
ejpam-5153	253	29	,	,	PUNCT
ejpam-5153	253	30	a2	a2	PROPN
ejpam-5153	253	31	)	)	PUNCT
ejpam-5153	253	32	=	=	PUNCT
ejpam-5153	253	33	0	0	PUNCT
ejpam-5153	254	1	=	=	NOUN
ejpam-5153	254	2	⇒	⇒	X
ejpam-5153	254	3	d12	d12	NOUN
ejpam-5153	254	4	=	=	SYM
ejpam-5153	254	5	−1	−1	NOUN
ejpam-5153	254	6	.	.	PUNCT
ejpam-5153	255	1	now	now	ADV
ejpam-5153	255	2	if	if	SCONJ
ejpam-5153	255	3	d12	d12	NUM
ejpam-5153	255	4	=	=	SYM
ejpam-5153	255	5	−1	−1	NOUN
ejpam-5153	255	6	,	,	PUNCT
ejpam-5153	255	7	we	we	PRON
ejpam-5153	255	8	have	have	VERB
ejpam-5153	255	9	c	c	PROPN
ejpam-5153	255	10	⊕	⊕	PROPN
ejpam-5153	255	11	r(0	r(0	PROPN
ejpam-5153	255	12	,	,	PUNCT
ejpam-5153	255	13	d12	d12	PROPN
ejpam-5153	255	14	,	,	PUNCT
ejpam-5153	255	15	1	1	NUM
ejpam-5153	255	16	,	,	PUNCT
ejpam-5153	255	17	0	0	NUM
ejpam-5153	255	18	)	)	PUNCT
ejpam-5153	255	19	is	be	AUX
ejpam-5153	255	20	isomorphic	isomorphic	ADJ
ejpam-5153	255	21	to	to	ADP
ejpam-5153	255	22	h	h	NOUN
ejpam-5153	255	23	,	,	PUNCT
ejpam-5153	255	24	then	then	ADV
ejpam-5153	255	25	satisfies	satisfie	NOUN
ejpam-5153	255	26	to	to	ADP
ejpam-5153	255	27	(	(	PUNCT
ejpam-5153	255	28	e2	e2	PROPN
ejpam-5153	255	29	)	)	PUNCT
ejpam-5153	255	30	.	.	PUNCT
ejpam-5153	256	1	(	(	PUNCT
ejpam-5153	256	2	2	2	X
ejpam-5153	256	3	)	)	PUNCT
ejpam-5153	256	4	suppose	suppose	VERB
ejpam-5153	256	5	that	that	SCONJ
ejpam-5153	256	6	d12	d12	PROPN
ejpam-5153	256	7	̸=	̸=	PROPN
ejpam-5153	256	8	−1	−1	NOUN
ejpam-5153	256	9	,	,	PUNCT
ejpam-5153	256	10	4d12c22	4d12c22	NOUN
ejpam-5153	256	11	<	<	X
ejpam-5153	256	12	−1	−1	NOUN
ejpam-5153	256	13	and	and	CCONJ
ejpam-5153	256	14	c	c	PROPN
ejpam-5153	256	15	⊕	⊕	PROPN
ejpam-5153	256	16	r(1	r(1	PROPN
ejpam-5153	256	17	,	,	PUNCT
ejpam-5153	256	18	d12	d12	PROPN
ejpam-5153	256	19	,	,	PUNCT
ejpam-5153	256	20	c22	c22	NOUN
ejpam-5153	256	21	,	,	PUNCT
ejpam-5153	256	22	0	0	NUM
ejpam-5153	256	23	)	)	PUNCT
ejpam-5153	256	24	satisfies	satisfie	NOUN
ejpam-5153	256	25	to	to	ADP
ejpam-5153	256	26	(	(	PUNCT
ejpam-5153	256	27	e2	e2	PROPN
ejpam-5153	256	28	)	)	PUNCT
ejpam-5153	256	29	.	.	PUNCT
ejpam-5153	257	1	let	let	VERB
ejpam-5153	257	2	b	b	NOUN
ejpam-5153	257	3	=	=	PROPN
ejpam-5153	257	4	e2	e2	PROPN
ejpam-5153	257	5	+	+	CCONJ
ejpam-5153	257	6	e3	e3	VERB
ejpam-5153	257	7	∈	∈	PROPN
ejpam-5153	257	8	c	c	PROPN
ejpam-5153	257	9	⊕	⊕	PROPN
ejpam-5153	257	10	r(1	r(1	PROPN
ejpam-5153	257	11	,	,	PUNCT
ejpam-5153	257	12	d12	d12	PROPN
ejpam-5153	257	13	,	,	PUNCT
ejpam-5153	257	14	c22	c22	NOUN
ejpam-5153	257	15	,	,	PUNCT
ejpam-5153	257	16	0	0	NUM
ejpam-5153	257	17	)	)	PUNCT
ejpam-5153	257	18	,	,	PUNCT
ejpam-5153	257	19	then	then	ADV
ejpam-5153	257	20	(	(	PUNCT
ejpam-5153	257	21	b2	b2	NOUN
ejpam-5153	257	22	,	,	PUNCT
ejpam-5153	257	23	b2	b2	NOUN
ejpam-5153	257	24	,	,	PUNCT
ejpam-5153	257	25	b2	b2	NOUN
ejpam-5153	257	26	)	)	PUNCT
ejpam-5153	257	27	̸=	̸=	PROPN
ejpam-5153	257	28	0	0	NUM
ejpam-5153	257	29	absurd	absurd	ADJ
ejpam-5153	257	30	.	.	PUNCT
ejpam-5153	258	1	(	(	PUNCT
ejpam-5153	258	2	3	3	X
ejpam-5153	258	3	)	)	PUNCT
ejpam-5153	258	4	suppose	suppose	VERB
ejpam-5153	258	5	that	that	SCONJ
ejpam-5153	258	6	c22	c22	PROPN
ejpam-5153	258	7	>	>	X
ejpam-5153	258	8	0	0	PROPN
ejpam-5153	258	9	,	,	PUNCT
ejpam-5153	258	10	and	and	CCONJ
ejpam-5153	258	11	c	c	PROPN
ejpam-5153	258	12	⊕	⊕	PROPN
ejpam-5153	258	13	r(1	r(1	PROPN
ejpam-5153	258	14	,	,	PUNCT
ejpam-5153	258	15	−1	−1	NOUN
ejpam-5153	258	16	,	,	PUNCT
ejpam-5153	258	17	c22	c22	NOUN
ejpam-5153	258	18	,	,	PUNCT
ejpam-5153	258	19	1	1	NUM
ejpam-5153	258	20	)	)	PUNCT
ejpam-5153	258	21	satisfies	satisfie	NOUN
ejpam-5153	258	22	to	to	ADP
ejpam-5153	258	23	(	(	PUNCT
ejpam-5153	258	24	e2	e2	PROPN
ejpam-5153	258	25	)	)	PUNCT
ejpam-5153	258	26	.	.	PUNCT
ejpam-5153	259	1	let	let	VERB
ejpam-5153	259	2	c	c	NOUN
ejpam-5153	259	3	=	=	SYM
ejpam-5153	259	4	e1	e1	PROPN
ejpam-5153	259	5	+	+	CCONJ
ejpam-5153	259	6	e4	e4	PROPN
ejpam-5153	259	7	∈	∈	PROPN
ejpam-5153	259	8	c	c	PROPN
ejpam-5153	259	9	⊕	⊕	PROPN
ejpam-5153	259	10	r(1	r(1	PROPN
ejpam-5153	259	11	,	,	PUNCT
ejpam-5153	259	12	−1	−1	NOUN
ejpam-5153	259	13	,	,	PUNCT
ejpam-5153	259	14	c22	c22	NOUN
ejpam-5153	259	15	,	,	PUNCT
ejpam-5153	259	16	1	1	NUM
ejpam-5153	259	17	)	)	PUNCT
ejpam-5153	259	18	,	,	PUNCT
ejpam-5153	259	19	then	then	ADV
ejpam-5153	259	20	(	(	PUNCT
ejpam-5153	259	21	c2	c2	PROPN
ejpam-5153	259	22	,	,	PUNCT
ejpam-5153	259	23	c2	c2	PROPN
ejpam-5153	259	24	,	,	PUNCT
ejpam-5153	259	25	c2	c2	PROPN
ejpam-5153	259	26	)	)	PUNCT
ejpam-5153	259	27	̸=	̸=	PROPN
ejpam-5153	259	28	0	0	NUM
ejpam-5153	259	29	absurd	absurd	ADJ
ejpam-5153	259	30	.	.	PUNCT
ejpam-5153	260	1	lemma	lemma	PROPN
ejpam-5153	260	2	10	10	NUM
ejpam-5153	260	3	.	.	PUNCT
ejpam-5153	261	1	let	let	VERB
ejpam-5153	261	2	a	a	DET
ejpam-5153	261	3	be	be	AUX
ejpam-5153	261	4	an	an	DET
ejpam-5153	261	5	algebra	algebra	NOUN
ejpam-5153	261	6	isomorphic	isomorphic	ADJ
ejpam-5153	261	7	to	to	ADP
ejpam-5153	261	8	c	c	PROPN
ejpam-5153	261	9	and	and	CCONJ
ejpam-5153	261	10	b	b	X
ejpam-5153	261	11	be	be	AUX
ejpam-5153	261	12	a	a	DET
ejpam-5153	261	13	two	two	NUM
ejpam-5153	261	14	-	-	PUNCT
ejpam-5153	261	15	dimensional	dimensional	ADJ
ejpam-5153	261	16	real	real	ADJ
ejpam-5153	261	17	algebras	algebra	NOUN
ejpam-5153	261	18	with	with	ADP
ejpam-5153	261	19	right	right	ADJ
ejpam-5153	261	20	unit	unit	NOUN
ejpam-5153	261	21	.	.	PUNCT
ejpam-5153	262	1	the	the	DET
ejpam-5153	262	2	following	follow	VERB
ejpam-5153	262	3	propositions	proposition	NOUN
ejpam-5153	262	4	are	be	AUX
ejpam-5153	262	5	equivalent	equivalent	ADJ
ejpam-5153	262	6	(	(	PUNCT
ejpam-5153	262	7	1	1	NUM
ejpam-5153	262	8	)	)	PUNCT
ejpam-5153	262	9	a	a	DET
ejpam-5153	262	10	⊕	⊕	PROPN
ejpam-5153	262	11	b	b	PROPN
ejpam-5153	262	12	is	be	AUX
ejpam-5153	262	13	division	division	NOUN
ejpam-5153	262	14	algebra	algebra	NOUN
ejpam-5153	262	15	satisfies	satisfie	NOUN
ejpam-5153	262	16	to	to	ADP
ejpam-5153	262	17	(	(	PUNCT
ejpam-5153	262	18	e2	e2	PROPN
ejpam-5153	262	19	)	)	PUNCT
ejpam-5153	262	20	,	,	PUNCT
ejpam-5153	262	21	(	(	PUNCT
ejpam-5153	262	22	2	2	X
ejpam-5153	262	23	)	)	PUNCT
ejpam-5153	262	24	a	a	DET
ejpam-5153	262	25	⊕	⊕	PROPN
ejpam-5153	262	26	b	b	PROPN
ejpam-5153	262	27	is	be	AUX
ejpam-5153	262	28	isomorphic	isomorphic	ADJ
ejpam-5153	262	29	to	to	ADP
ejpam-5153	262	30	h.	h.	NOUN
ejpam-5153	262	31	proof	proof	NOUN
ejpam-5153	262	32	.	.	PUNCT
ejpam-5153	263	1	(	(	PUNCT
ejpam-5153	263	2	1	1	X
ejpam-5153	263	3	)	)	PUNCT
ejpam-5153	263	4	⇒	⇒	NOUN
ejpam-5153	263	5	(	(	PUNCT
ejpam-5153	263	6	2	2	X
ejpam-5153	263	7	)	)	PUNCT
ejpam-5153	263	8	we	we	PRON
ejpam-5153	263	9	can	can	AUX
ejpam-5153	263	10	take	take	VERB
ejpam-5153	263	11	u	u	PRON
ejpam-5153	263	12	the	the	DET
ejpam-5153	263	13	unit	unit	NOUN
ejpam-5153	263	14	element	element	NOUN
ejpam-5153	263	15	of	of	ADP
ejpam-5153	263	16	a	a	PRON
ejpam-5153	263	17	and	and	CCONJ
ejpam-5153	263	18	also	also	ADV
ejpam-5153	263	19	the	the	DET
ejpam-5153	263	20	right	right	ADJ
ejpam-5153	263	21	unit	unit	NOUN
ejpam-5153	263	22	of	of	ADP
ejpam-5153	263	23	b.	b.	PROPN
ejpam-5153	263	24	the	the	DET
ejpam-5153	263	25	theorem	theorem	ADJ
ejpam-5153	263	26	2	2	NUM
ejpam-5153	263	27	shows	show	VERB
ejpam-5153	263	28	that	that	SCONJ
ejpam-5153	263	29	b	b	NOUN
ejpam-5153	263	30	be	be	AUX
ejpam-5153	263	31	a	a	DET
ejpam-5153	263	32	two	two	NUM
ejpam-5153	263	33	-	-	PUNCT
ejpam-5153	263	34	dimensional	dimensional	ADJ
ejpam-5153	263	35	real	real	ADJ
ejpam-5153	263	36	division	division	NOUN
ejpam-5153	263	37	algebra	algebra	NOUN
ejpam-5153	263	38	with	with	ADP
ejpam-5153	263	39	right	right	ADJ
ejpam-5153	263	40	unit	unit	NOUN
ejpam-5153	263	41	and	and	CCONJ
ejpam-5153	263	42	d12	d12	NOUN
ejpam-5153	263	43	<	<	X
ejpam-5153	263	44	0	0	NUM
ejpam-5153	263	45	.	.	PUNCT
ejpam-5153	263	46	by	by	ADP
ejpam-5153	263	47	analogy	analogy	NOUN
ejpam-5153	263	48	of	of	ADP
ejpam-5153	263	49	the	the	DET
ejpam-5153	263	50	theorem	theorem	NOUN
ejpam-5153	263	51	1	1	NUM
ejpam-5153	263	52	.	.	X
ejpam-5153	263	53	b	b	NOUN
ejpam-5153	263	54	is	be	AUX
ejpam-5153	263	55	isomorphic	isomorphic	ADJ
ejpam-5153	263	56	to	to	ADP
ejpam-5153	263	57	either	either	CCONJ
ejpam-5153	263	58	r(0	r(0	PROPN
ejpam-5153	263	59	,	,	PUNCT
ejpam-5153	263	60	d12	d12	NOUN
ejpam-5153	263	61	,	,	PUNCT
ejpam-5153	263	62	1	1	NUM
ejpam-5153	263	63	,	,	PUNCT
ejpam-5153	263	64	0	0	NUM
ejpam-5153	263	65	)	)	PUNCT
ejpam-5153	263	66	with	with	ADP
ejpam-5153	263	67	d12	d12	X
ejpam-5153	263	68	<	<	X
ejpam-5153	263	69	0	0	NUM
ejpam-5153	263	70	.	.	PUNCT
ejpam-5153	264	1	r(1	r(1	PROPN
ejpam-5153	264	2	,	,	PUNCT
ejpam-5153	264	3	d12	d12	PROPN
ejpam-5153	264	4	,	,	PUNCT
ejpam-5153	264	5	c22	c22	NOUN
ejpam-5153	264	6	,	,	PUNCT
ejpam-5153	264	7	0	0	NUM
ejpam-5153	264	8	)	)	PUNCT
ejpam-5153	264	9	with	with	ADP
ejpam-5153	264	10	d12	d12	ADJ
ejpam-5153	264	11	̸=	̸=	PROPN
ejpam-5153	264	12	−1	−1	NOUN
ejpam-5153	264	13	and	and	CCONJ
ejpam-5153	264	14	4d12c22	4d12c22	NOUN
ejpam-5153	264	15	<	<	X
ejpam-5153	264	16	−1	−1	NOUN
ejpam-5153	264	17	,	,	PUNCT
ejpam-5153	264	18	r(1	r(1	PROPN
ejpam-5153	264	19	,	,	PUNCT
ejpam-5153	264	20	−1	−1	NOUN
ejpam-5153	264	21	,	,	PUNCT
ejpam-5153	264	22	c22	c22	NOUN
ejpam-5153	264	23	,	,	PUNCT
ejpam-5153	264	24	1	1	NUM
ejpam-5153	264	25	)	)	PUNCT
ejpam-5153	264	26	with	with	ADP
ejpam-5153	264	27	c22	c22	PROPN
ejpam-5153	264	28	>	>	X
ejpam-5153	264	29	0	0	PROPN
ejpam-5153	264	30	.	.	PUNCT
ejpam-5153	265	1	the	the	DET
ejpam-5153	265	2	lemma	lemma	PROPN
ejpam-5153	265	3	9	9	NUM
ejpam-5153	265	4	,	,	PUNCT
ejpam-5153	265	5	shows	show	VERB
ejpam-5153	265	6	that	that	SCONJ
ejpam-5153	265	7	b	b	X
ejpam-5153	265	8	∼=	∼=	PROPN
ejpam-5153	265	9	r(0	r(0	PROPN
ejpam-5153	265	10	,	,	PUNCT
ejpam-5153	265	11	−1	−1	NOUN
ejpam-5153	265	12	,	,	PUNCT
ejpam-5153	265	13	1	1	NUM
ejpam-5153	265	14	,	,	PUNCT
ejpam-5153	265	15	0	0	NUM
ejpam-5153	265	16	)	)	PUNCT
ejpam-5153	265	17	,	,	PUNCT
ejpam-5153	265	18	therefore	therefore	ADV
ejpam-5153	265	19	a	a	DET
ejpam-5153	265	20	⊕	⊕	PROPN
ejpam-5153	265	21	b	b	NOUN
ejpam-5153	265	22	∼=	∼=	PROPN
ejpam-5153	265	23	c	c	PROPN
ejpam-5153	265	24	⊕	⊕	PROPN
ejpam-5153	265	25	r(0	r(0	PROPN
ejpam-5153	265	26	,	,	PUNCT
ejpam-5153	265	27	−1	−1	NOUN
ejpam-5153	265	28	,	,	PUNCT
ejpam-5153	265	29	1	1	NUM
ejpam-5153	265	30	,	,	PUNCT
ejpam-5153	265	31	0	0	NUM
ejpam-5153	265	32	)	)	PUNCT
ejpam-5153	265	33	thus	thus	ADV
ejpam-5153	265	34	a	a	DET
ejpam-5153	265	35	⊕	⊕	PROPN
ejpam-5153	265	36	b	b	PROPN
ejpam-5153	265	37	is	be	AUX
ejpam-5153	265	38	isomorphic	isomorphic	ADJ
ejpam-5153	265	39	to	to	ADP
ejpam-5153	265	40	h.	h.	PROPN
ejpam-5153	265	41	(	(	PUNCT
ejpam-5153	265	42	2	2	NUM
ejpam-5153	265	43	)	)	PUNCT
ejpam-5153	265	44	⇒	⇒	NOUN
ejpam-5153	265	45	(	(	PUNCT
ejpam-5153	265	46	1	1	NUM
ejpam-5153	265	47	)	)	PUNCT
ejpam-5153	265	48	obvious	obvious	ADJ
ejpam-5153	265	49	.	.	PUNCT
ejpam-5153	266	1	theorem	theorem	NOUN
ejpam-5153	266	2	3	3	NUM
ejpam-5153	266	3	.	.	PUNCT
ejpam-5153	266	4	a	a	DET
ejpam-5153	266	5	⊕	⊕	PROPN
ejpam-5153	266	6	b	b	PROPN
ejpam-5153	266	7	be	be	AUX
ejpam-5153	266	8	a	a	DET
ejpam-5153	266	9	division	division	NOUN
ejpam-5153	266	10	algebra	algebra	NOUN
ejpam-5153	266	11	with	with	ADP
ejpam-5153	266	12	left	left	ADJ
ejpam-5153	266	13	unit	unit	NOUN
ejpam-5153	266	14	e1	e1	NOUN
ejpam-5153	266	15	satisfies	satisfie	NOUN
ejpam-5153	266	16	to	to	ADP
ejpam-5153	266	17	(	(	PUNCT
ejpam-5153	266	18	e2	e2	PROPN
ejpam-5153	266	19	)	)	PUNCT
ejpam-5153	266	20	.	.	PUNCT
ejpam-5153	267	1	the	the	DET
ejpam-5153	267	2	following	follow	VERB
ejpam-5153	267	3	propositions	proposition	NOUN
ejpam-5153	267	4	are	be	AUX
ejpam-5153	267	5	equivalent	equivalent	ADJ
ejpam-5153	267	6	:	:	PUNCT
ejpam-5153	267	7	(	(	PUNCT
ejpam-5153	267	8	1	1	X
ejpam-5153	267	9	)	)	PUNCT
ejpam-5153	267	10	a	a	PRON
ejpam-5153	267	11	is	be	AUX
ejpam-5153	267	12	ismorphic	ismorphic	ADJ
ejpam-5153	267	13	to	to	ADP
ejpam-5153	267	14	c	c	PROPN
ejpam-5153	267	15	,	,	PUNCT
ejpam-5153	267	16	(	(	PUNCT
ejpam-5153	267	17	2	2	X
ejpam-5153	267	18	)	)	PUNCT
ejpam-5153	267	19	a	a	DET
ejpam-5153	267	20	⊕	⊕	PROPN
ejpam-5153	267	21	b	b	PROPN
ejpam-5153	267	22	is	be	AUX
ejpam-5153	267	23	isomorphic	isomorphic	ADJ
ejpam-5153	267	24	to	to	ADP
ejpam-5153	267	25	either	either	DET
ejpam-5153	267	26	h	h	NOUN
ejpam-5153	267	27	,	,	PUNCT
ejpam-5153	268	1	c	c	PROPN
ejpam-5153	268	2	⊕	⊕	PROPN
ejpam-5153	268	3	b.	b.	PROPN
ejpam-5153	268	4	a.	a.	PROPN
ejpam-5153	268	5	s.	s.	PROPN
ejpam-5153	268	6	diabang	diabang	PROPN
ejpam-5153	268	7	,	,	PUNCT
ejpam-5153	268	8	a.	a.	PROPN
ejpam-5153	268	9	s.	s.	PROPN
ejpam-5153	268	10	mballo	mballo	PROPN
ejpam-5153	268	11	,	,	PUNCT
ejpam-5153	268	12	p.	p.	PROPN
ejpam-5153	268	13	c.	c.	PROPN
ejpam-5153	268	14	diop	diop	PROPN
ejpam-5153	268	15	/	/	SYM
ejpam-5153	268	16	eur	eur	PROPN
ejpam-5153	268	17	.	.	PUNCT
ejpam-5153	269	1	j.	j.	PROPN
ejpam-5153	269	2	pure	pure	PROPN
ejpam-5153	269	3	appl	appl	PROPN
ejpam-5153	269	4	.	.	PROPN
ejpam-5153	269	5	math	math	PROPN
ejpam-5153	269	6	,	,	PUNCT
ejpam-5153	269	7	17	17	NUM
ejpam-5153	269	8	(	(	PUNCT
ejpam-5153	269	9	3	3	NUM
ejpam-5153	269	10	)	)	PUNCT
ejpam-5153	269	11	(	(	PUNCT
ejpam-5153	269	12	2024	2024	NUM
ejpam-5153	269	13	)	)	PUNCT
ejpam-5153	269	14	,	,	PUNCT
ejpam-5153	269	15	2276	2276	NUM
ejpam-5153	269	16	-	-	SYM
ejpam-5153	269	17	2287	2287	NUM
ejpam-5153	269	18	2285	2285	NUM
ejpam-5153	269	19	proof	proof	NOUN
ejpam-5153	269	20	.	.	PUNCT
ejpam-5153	270	1	(	(	PUNCT
ejpam-5153	270	2	1	1	X
ejpam-5153	270	3	)	)	PUNCT
ejpam-5153	270	4	⇒	⇒	NOUN
ejpam-5153	270	5	(	(	PUNCT
ejpam-5153	270	6	2	2	X
ejpam-5153	270	7	)	)	PUNCT
ejpam-5153	270	8	we	we	PRON
ejpam-5153	270	9	have	have	VERB
ejpam-5153	270	10	b	b	NOUN
ejpam-5153	270	11	be	be	AUX
ejpam-5153	270	12	a	a	DET
ejpam-5153	270	13	two	two	NUM
ejpam-5153	270	14	-	-	PUNCT
ejpam-5153	270	15	dimensional	dimensional	ADJ
ejpam-5153	270	16	real	real	ADJ
ejpam-5153	270	17	division	division	NOUN
ejpam-5153	270	18	algebra	algebra	NOUN
ejpam-5153	270	19	,	,	PUNCT
ejpam-5153	270	20	[	[	X
ejpam-5153	270	21	14	14	NUM
ejpam-5153	270	22	]	]	PUNCT
ejpam-5153	270	23	shows	show	VERB
ejpam-5153	270	24	that	that	SCONJ
ejpam-5153	270	25	b	b	PROPN
ejpam-5153	270	26	contains	contain	VERB
ejpam-5153	270	27	a	a	DET
ejpam-5153	270	28	non	non	ADJ
ejpam-5153	270	29	-	-	ADJ
ejpam-5153	270	30	zero	zero	ADJ
ejpam-5153	270	31	idempotent	idempotent	NOUN
ejpam-5153	270	32	.	.	PUNCT
ejpam-5153	271	1	thus	thus	ADV
ejpam-5153	271	2	we	we	PRON
ejpam-5153	271	3	can	can	AUX
ejpam-5153	271	4	take	take	VERB
ejpam-5153	271	5	u	u	PRON
ejpam-5153	271	6	the	the	DET
ejpam-5153	271	7	unit	unit	NOUN
ejpam-5153	271	8	element	element	NOUN
ejpam-5153	271	9	of	of	ADP
ejpam-5153	271	10	a	a	PRON
ejpam-5153	271	11	and	and	CCONJ
ejpam-5153	271	12	the	the	DET
ejpam-5153	271	13	non	non	ADJ
ejpam-5153	271	14	-	-	ADJ
ejpam-5153	271	15	zero	zero	ADJ
ejpam-5153	271	16	idempotent	idempotent	NOUN
ejpam-5153	271	17	of	of	ADP
ejpam-5153	271	18	b.	b.	PROPN
ejpam-5153	271	19	the	the	DET
ejpam-5153	271	20	lemma	lemma	PROPN
ejpam-5153	271	21	1	1	NUM
ejpam-5153	271	22	(	(	PUNCT
ejpam-5153	271	23	3	3	NUM
ejpam-5153	271	24	)	)	PUNCT
ejpam-5153	271	25	shows	show	VERB
ejpam-5153	271	26	that	that	SCONJ
ejpam-5153	271	27	a	a	DET
ejpam-5153	271	28	⊕	⊕	PROPN
ejpam-5153	271	29	b	b	PROPN
ejpam-5153	271	30	satisfies	satisfie	NOUN
ejpam-5153	271	31	to	to	ADP
ejpam-5153	271	32	(	(	PUNCT
ejpam-5153	271	33	e4	e4	PROPN
ejpam-5153	271	34	)	)	PUNCT
ejpam-5153	271	35	then	then	ADV
ejpam-5153	271	36	(	(	PUNCT
ejpam-5153	271	37	e4.e1).e1	e4.e1).e1	NOUN
ejpam-5153	271	38	=	=	SYM
ejpam-5153	271	39	e4	e4	PROPN
ejpam-5153	271	40	=	=	AUX
ejpam-5153	271	41	⇒	⇒	PROPN
ejpam-5153	271	42	c21(1	c21(1	PROPN
ejpam-5153	271	43	+	+	CCONJ
ejpam-5153	271	44	d21	d21	NOUN
ejpam-5153	271	45	)	)	PUNCT
ejpam-5153	271	46	=	=	SYM
ejpam-5153	271	47	0	0	NUM
ejpam-5153	271	48	and	and	CCONJ
ejpam-5153	271	49	d2	d2	PROPN
ejpam-5153	271	50	21	21	NUM
ejpam-5153	271	51	=	=	SYM
ejpam-5153	271	52	1	1	NUM
ejpam-5153	271	53	.	.	NOUN
ejpam-5153	271	54	•	•	NOUN
ejpam-5153	271	55	if	if	SCONJ
ejpam-5153	271	56	d21	d21	NOUN
ejpam-5153	271	57	=	=	NOUN
ejpam-5153	271	58	1	1	NUM
ejpam-5153	271	59	,	,	PUNCT
ejpam-5153	271	60	we	we	PRON
ejpam-5153	271	61	have	have	VERB
ejpam-5153	271	62	c21	c21	NOUN
ejpam-5153	271	63	=	=	SYM
ejpam-5153	271	64	0	0	PROPN
ejpam-5153	271	65	,	,	PUNCT
ejpam-5153	271	66	thus	thus	ADV
ejpam-5153	271	67	b	b	AUX
ejpam-5153	271	68	be	be	AUX
ejpam-5153	271	69	an	an	DET
ejpam-5153	271	70	two	two	NUM
ejpam-5153	271	71	-	-	PUNCT
ejpam-5153	271	72	dimensional	dimensional	ADJ
ejpam-5153	271	73	real	real	ADJ
ejpam-5153	271	74	division	division	NOUN
ejpam-5153	271	75	with	with	ADP
ejpam-5153	271	76	right	right	ADJ
ejpam-5153	271	77	unit.the	unit.the	DET
ejpam-5153	271	78	lemma	lemma	PROPN
ejpam-5153	271	79	10	10	NUM
ejpam-5153	271	80	shows	show	VERB
ejpam-5153	271	81	that	that	SCONJ
ejpam-5153	271	82	a	a	DET
ejpam-5153	271	83	⊕	⊕	PROPN
ejpam-5153	271	84	b	b	PROPN
ejpam-5153	271	85	is	be	AUX
ejpam-5153	271	86	isomorphic	isomorphic	ADJ
ejpam-5153	271	87	to	to	ADP
ejpam-5153	271	88	h.	h.	PROPN
ejpam-5153	271	89	•	•	PROPN
ejpam-5153	271	90	if	if	SCONJ
ejpam-5153	271	91	d21	d21	PROPN
ejpam-5153	271	92	=	=	SYM
ejpam-5153	271	93	−1	−1	NOUN
ejpam-5153	271	94	,	,	PUNCT
ejpam-5153	271	95	the	the	DET
ejpam-5153	271	96	a	a	PRON
ejpam-5153	271	97	⊕	⊕	PROPN
ejpam-5153	271	98	b	b	PROPN
ejpam-5153	271	99	satisfies	satisfie	NOUN
ejpam-5153	271	100	to	to	ADP
ejpam-5153	271	101	(	(	PUNCT
ejpam-5153	271	102	e3	e3	NOUN
ejpam-5153	271	103	)	)	PUNCT
ejpam-5153	271	104	,	,	PUNCT
ejpam-5153	271	105	the	the	DET
ejpam-5153	271	106	lemme	lemme	NOUN
ejpam-5153	271	107	5	5	NUM
ejpam-5153	271	108	gives	give	VERB
ejpam-5153	271	109	.	.	PUNCT
ejpam-5153	272	1	(	(	PUNCT
ejpam-5153	272	2	e2e3	e2e3	X
ejpam-5153	272	3	+	+	X
ejpam-5153	272	4	e3e2)e1	e3e2)e1	ADJ
ejpam-5153	272	5	=	=	SYM
ejpam-5153	272	6	e2e3	e2e3	NOUN
ejpam-5153	272	7	+	+	CCONJ
ejpam-5153	272	8	e3e2	e3e2	NOUN
ejpam-5153	272	9	=	=	NOUN
ejpam-5153	272	10	⇒	⇒	NOUN
ejpam-5153	272	11	d12	d12	NOUN
ejpam-5153	272	12	=	=	SYM
ejpam-5153	272	13	−1	−1	NOUN
ejpam-5153	272	14	.	.	PUNCT
ejpam-5153	273	1	(	(	PUNCT
ejpam-5153	273	2	e2e4	e2e4	X
ejpam-5153	273	3	+	+	X
ejpam-5153	273	4	e4e2)e1	e4e2)e1	NOUN
ejpam-5153	273	5	=	=	PUNCT
ejpam-5153	273	6	e2e4	e2e4	X
ejpam-5153	273	7	+	+	X
ejpam-5153	273	8	e4e2	e4e2	X
ejpam-5153	273	9	=	=	NOUN
ejpam-5153	273	10	⇒	⇒	NOUN
ejpam-5153	273	11	d22	d22	NOUN
ejpam-5153	273	12	=	=	SYM
ejpam-5153	273	13	0	0	NUM
ejpam-5153	274	1	the	the	DET
ejpam-5153	274	2	division	division	NOUN
ejpam-5153	274	3	of	of	ADP
ejpam-5153	274	4	a	a	DET
ejpam-5153	274	5	⊕	⊕	PROPN
ejpam-5153	274	6	b	b	PROPN
ejpam-5153	274	7	gives	give	VERB
ejpam-5153	274	8	c2	c2	PROPN
ejpam-5153	274	9	12	12	NUM
ejpam-5153	274	10	<	<	X
ejpam-5153	274	11	−4c22	−4c22	NOUN
ejpam-5153	274	12	let	let	VERB
ejpam-5153	274	13	a	a	DET
ejpam-5153	274	14	=	=	ADJ
ejpam-5153	274	15	c12e1	c12e1	PROPN
ejpam-5153	274	16	+	+	PROPN
ejpam-5153	274	17	e2	e2	PROPN
ejpam-5153	274	18	+	+	CCONJ
ejpam-5153	274	19	e3	e3	PROPN
ejpam-5153	274	20	,	,	PUNCT
ejpam-5153	274	21	(	(	PUNCT
ejpam-5153	274	22	a2	a2	PROPN
ejpam-5153	274	23	,	,	PUNCT
ejpam-5153	274	24	a2	a2	PROPN
ejpam-5153	274	25	,	,	PUNCT
ejpam-5153	274	26	a2	a2	PROPN
ejpam-5153	274	27	)	)	PUNCT
ejpam-5153	274	28	=	=	PUNCT
ejpam-5153	274	29	0	0	PUNCT
ejpam-5153	275	1	=	=	AUX
ejpam-5153	275	2	⇒	⇒	X
ejpam-5153	275	3	c12	c12	PROPN
ejpam-5153	275	4	=	=	SYM
ejpam-5153	275	5	0	0	PROPN
ejpam-5153	275	6	.	.	PUNCT
ejpam-5153	276	1	so	so	ADV
ejpam-5153	276	2	a	a	DET
ejpam-5153	276	3	⊕	⊕	PROPN
ejpam-5153	276	4	b	b	PROPN
ejpam-5153	276	5	is	be	AUX
ejpam-5153	276	6	isomorphic	isomorphic	ADJ
ejpam-5153	276	7	to	to	ADP
ejpam-5153	276	8	c	c	PROPN
ejpam-5153	276	9	⊕	⊕	PROPN
ejpam-5153	276	10	b.	b.	PROPN
ejpam-5153	276	11	(	(	PUNCT
ejpam-5153	276	12	2	2	NUM
ejpam-5153	276	13	)	)	PUNCT
ejpam-5153	276	14	⇒	⇒	NOUN
ejpam-5153	276	15	(	(	PUNCT
ejpam-5153	276	16	1	1	NUM
ejpam-5153	276	17	)	)	PUNCT
ejpam-5153	276	18	obvious	obvious	ADJ
ejpam-5153	276	19	..	..	PUNCT
ejpam-5153	277	1	5.2	5.2	NUM
ejpam-5153	277	2	.	.	PUNCT
ejpam-5153	278	1	study	study	NOUN
ejpam-5153	278	2	of	of	ADP
ejpam-5153	278	3	fused	fuse	VERB
ejpam-5153	278	4	algebras	algebra	NOUN
ejpam-5153	278	5	with	with	ADP
ejpam-5153	278	6	a	a	PRON
ejpam-5153	278	7	is	be	AUX
ejpam-5153	278	8	isomorphic	isomorphic	ADJ
ejpam-5153	278	9	to	to	ADP
ejpam-5153	278	10	⋆c	⋆c	PROPN
ejpam-5153	278	11	.	.	PUNCT
ejpam-5153	279	1	lemma	lemma	PROPN
ejpam-5153	279	2	11	11	NUM
ejpam-5153	279	3	.	.	PUNCT
ejpam-5153	280	1	let	let	VERB
ejpam-5153	280	2	a	a	DET
ejpam-5153	280	3	be	be	AUX
ejpam-5153	280	4	an	an	DET
ejpam-5153	280	5	algebra	algebra	NOUN
ejpam-5153	280	6	isomorphic	isomorphic	ADJ
ejpam-5153	280	7	to	to	ADP
ejpam-5153	280	8	⋆c	⋆c	PROPN
ejpam-5153	280	9	and	and	CCONJ
ejpam-5153	280	10	b	b	X
ejpam-5153	280	11	be	be	AUX
ejpam-5153	280	12	a	a	DET
ejpam-5153	280	13	two	two	NUM
ejpam-5153	280	14	-	-	PUNCT
ejpam-5153	280	15	dimensional	dimensional	ADJ
ejpam-5153	280	16	real	real	ADJ
ejpam-5153	280	17	algebras	algebra	NOUN
ejpam-5153	280	18	with	with	ADP
ejpam-5153	280	19	right	right	ADJ
ejpam-5153	280	20	unit	unit	NOUN
ejpam-5153	280	21	.	.	PUNCT
ejpam-5153	281	1	the	the	DET
ejpam-5153	281	2	following	follow	VERB
ejpam-5153	281	3	propositions	proposition	NOUN
ejpam-5153	281	4	are	be	AUX
ejpam-5153	281	5	equivalent	equivalent	ADJ
ejpam-5153	281	6	(	(	PUNCT
ejpam-5153	281	7	1	1	NUM
ejpam-5153	281	8	)	)	PUNCT
ejpam-5153	281	9	a	a	DET
ejpam-5153	281	10	⊕	⊕	PROPN
ejpam-5153	281	11	b	b	PROPN
ejpam-5153	281	12	is	be	AUX
ejpam-5153	281	13	division	division	NOUN
ejpam-5153	281	14	algebra	algebra	NOUN
ejpam-5153	281	15	satisfies	satisfie	NOUN
ejpam-5153	281	16	to	to	ADP
ejpam-5153	281	17	(	(	PUNCT
ejpam-5153	281	18	e2	e2	PROPN
ejpam-5153	281	19	)	)	PUNCT
ejpam-5153	281	20	,	,	PUNCT
ejpam-5153	281	21	(	(	PUNCT
ejpam-5153	281	22	2	2	X
ejpam-5153	281	23	)	)	PUNCT
ejpam-5153	281	24	a	a	DET
ejpam-5153	281	25	⊕	⊕	PROPN
ejpam-5153	281	26	b	b	PROPN
ejpam-5153	281	27	is	be	AUX
ejpam-5153	281	28	isomorphic	isomorphic	ADJ
ejpam-5153	281	29	to	to	ADP
ejpam-5153	281	30	⋆h	⋆h	PRON
ejpam-5153	281	31	.	.	PUNCT
ejpam-5153	282	1	proof	proof	NOUN
ejpam-5153	282	2	.	.	PUNCT
ejpam-5153	283	1	(	(	PUNCT
ejpam-5153	283	2	1	1	X
ejpam-5153	283	3	)	)	PUNCT
ejpam-5153	283	4	⇒	⇒	NOUN
ejpam-5153	283	5	(	(	PUNCT
ejpam-5153	283	6	2	2	X
ejpam-5153	283	7	)	)	PUNCT
ejpam-5153	283	8	we	we	PRON
ejpam-5153	283	9	can	can	AUX
ejpam-5153	283	10	take	take	VERB
ejpam-5153	283	11	u	u	PRON
ejpam-5153	283	12	the	the	DET
ejpam-5153	283	13	left	left	ADJ
ejpam-5153	283	14	unit	unit	NOUN
ejpam-5153	283	15	element	element	NOUN
ejpam-5153	283	16	of	of	ADP
ejpam-5153	283	17	a	a	DET
ejpam-5153	283	18	and	and	CCONJ
ejpam-5153	283	19	also	also	ADV
ejpam-5153	283	20	the	the	DET
ejpam-5153	283	21	right	right	ADJ
ejpam-5153	283	22	unit	unit	NOUN
ejpam-5153	283	23	element	element	NOUN
ejpam-5153	283	24	of	of	ADP
ejpam-5153	283	25	b.	b.	PROPN
ejpam-5153	283	26	the	the	DET
ejpam-5153	283	27	theorem	theorem	ADJ
ejpam-5153	283	28	2	2	NUM
ejpam-5153	283	29	shows	show	VERB
ejpam-5153	283	30	that	that	SCONJ
ejpam-5153	283	31	b	b	NOUN
ejpam-5153	283	32	be	be	AUX
ejpam-5153	283	33	a	a	DET
ejpam-5153	283	34	two	two	NUM
ejpam-5153	283	35	-	-	PUNCT
ejpam-5153	283	36	dimensional	dimensional	ADJ
ejpam-5153	283	37	real	real	ADJ
ejpam-5153	283	38	division	division	NOUN
ejpam-5153	283	39	algebra	algebra	NOUN
ejpam-5153	283	40	with	with	ADP
ejpam-5153	283	41	right	right	ADJ
ejpam-5153	283	42	unit	unit	NOUN
ejpam-5153	283	43	and	and	CCONJ
ejpam-5153	283	44	d12	d12	NOUN
ejpam-5153	283	45	<	<	X
ejpam-5153	283	46	0	0	NUM
ejpam-5153	283	47	.	.	PUNCT
ejpam-5153	284	1	we	we	PRON
ejpam-5153	284	2	have	have	AUX
ejpam-5153	284	3	(	(	PUNCT
ejpam-5153	284	4	e4e3	e4e3	X
ejpam-5153	284	5	+	+	PUNCT
ejpam-5153	284	6	e3e4)e1	e3e4)e1	ADJ
ejpam-5153	284	7	=	=	PUNCT
ejpam-5153	284	8	e4e3	e4e3	X
ejpam-5153	284	9	+	+	CCONJ
ejpam-5153	284	10	e3e4	e3e4	X
ejpam-5153	284	11	=	=	NOUN
ejpam-5153	284	12	⇒	⇒	X
ejpam-5153	284	13	d12	d12	NOUN
ejpam-5153	284	14	=	=	SYM
ejpam-5153	284	15	−1	−1	NOUN
ejpam-5153	284	16	and	and	CCONJ
ejpam-5153	284	17	e2	e2	PROPN
ejpam-5153	284	18	4e1	4e1	NUM
ejpam-5153	285	1	=	=	NOUN
ejpam-5153	285	2	e4	e4	PROPN
ejpam-5153	285	3	=	=	AUX
ejpam-5153	285	4	⇒	⇒	VERB
ejpam-5153	285	5	d22	d22	NOUN
ejpam-5153	285	6	=	=	SYM
ejpam-5153	285	7	0	0	NUM
ejpam-5153	285	8	by	by	ADP
ejpam-5153	285	9	analogy	analogy	NOUN
ejpam-5153	285	10	of	of	ADP
ejpam-5153	285	11	the	the	DET
ejpam-5153	285	12	theorem	theorem	NOUN
ejpam-5153	285	13	1	1	NUM
ejpam-5153	285	14	.	.	X
ejpam-5153	286	1	b	b	NOUN
ejpam-5153	286	2	is	be	AUX
ejpam-5153	286	3	isomorphic	isomorphic	ADJ
ejpam-5153	286	4	to	to	ADP
ejpam-5153	286	5	r(0	r(0	PROPN
ejpam-5153	286	6	,	,	PUNCT
ejpam-5153	286	7	−1	−1	NOUN
ejpam-5153	286	8	,	,	PUNCT
ejpam-5153	286	9	1	1	NUM
ejpam-5153	286	10	,	,	PUNCT
ejpam-5153	286	11	0	0	NUM
ejpam-5153	286	12	)	)	PUNCT
ejpam-5153	286	13	.	.	PUNCT
ejpam-5153	287	1	therefore	therefore	ADV
ejpam-5153	287	2	a	a	DET
ejpam-5153	287	3	⊕	⊕	PROPN
ejpam-5153	287	4	b	b	PROPN
ejpam-5153	287	5	≃⋆	≃⋆	PROPN
ejpam-5153	287	6	c	c	PROPN
ejpam-5153	287	7	⊕	⊕	PROPN
ejpam-5153	287	8	r(0	r(0	PROPN
ejpam-5153	287	9	,	,	PUNCT
ejpam-5153	287	10	−1	−1	NOUN
ejpam-5153	287	11	,	,	PUNCT
ejpam-5153	287	12	1	1	NUM
ejpam-5153	287	13	,	,	PUNCT
ejpam-5153	287	14	0	0	NUM
ejpam-5153	287	15	)	)	PUNCT
ejpam-5153	287	16	isomorphic	isomorphic	ADJ
ejpam-5153	287	17	to	to	ADP
ejpam-5153	287	18	⋆h	⋆h	PROPN
ejpam-5153	287	19	.	.	PUNCT
ejpam-5153	288	1	(	(	PUNCT
ejpam-5153	288	2	2	2	X
ejpam-5153	288	3	)	)	PUNCT
ejpam-5153	288	4	⇒	⇒	NOUN
ejpam-5153	288	5	(	(	PUNCT
ejpam-5153	288	6	1	1	NUM
ejpam-5153	288	7	)	)	PUNCT
ejpam-5153	288	8	obvious	obvious	ADJ
ejpam-5153	288	9	.	.	PUNCT
ejpam-5153	289	1	theorem	theorem	VERB
ejpam-5153	289	2	4	4	NUM
ejpam-5153	289	3	.	.	PUNCT
ejpam-5153	290	1	a	a	DET
ejpam-5153	290	2	⊕	⊕	PROPN
ejpam-5153	290	3	b	b	PROPN
ejpam-5153	290	4	be	be	AUX
ejpam-5153	290	5	a	a	DET
ejpam-5153	290	6	division	division	NOUN
ejpam-5153	290	7	algebra	algebra	NOUN
ejpam-5153	290	8	with	with	ADP
ejpam-5153	290	9	left	left	ADJ
ejpam-5153	290	10	unit	unit	NOUN
ejpam-5153	290	11	e1	e1	NOUN
ejpam-5153	290	12	satisfies	satisfie	NOUN
ejpam-5153	290	13	to	to	ADP
ejpam-5153	290	14	(	(	PUNCT
ejpam-5153	290	15	e2	e2	PROPN
ejpam-5153	290	16	)	)	PUNCT
ejpam-5153	290	17	.	.	PUNCT
ejpam-5153	291	1	the	the	DET
ejpam-5153	291	2	following	follow	VERB
ejpam-5153	291	3	propositions	proposition	NOUN
ejpam-5153	291	4	are	be	AUX
ejpam-5153	291	5	equivalent	equivalent	ADJ
ejpam-5153	291	6	:	:	PUNCT
ejpam-5153	291	7	(	(	PUNCT
ejpam-5153	291	8	1	1	X
ejpam-5153	291	9	)	)	PUNCT
ejpam-5153	291	10	a	a	PRON
ejpam-5153	291	11	is	be	AUX
ejpam-5153	291	12	isomorphic	isomorphic	ADJ
ejpam-5153	291	13	to	to	ADP
ejpam-5153	291	14	⋆c	⋆c	PROPN
ejpam-5153	291	15	,	,	PUNCT
ejpam-5153	291	16	(	(	PUNCT
ejpam-5153	291	17	2	2	X
ejpam-5153	291	18	)	)	PUNCT
ejpam-5153	291	19	a	a	DET
ejpam-5153	291	20	⊕	⊕	PROPN
ejpam-5153	291	21	b	b	PROPN
ejpam-5153	291	22	is	be	AUX
ejpam-5153	291	23	isomorphic	isomorphic	ADJ
ejpam-5153	291	24	to	to	ADP
ejpam-5153	291	25	⋆h	⋆h	PRON
ejpam-5153	291	26	.	.	PUNCT
ejpam-5153	292	1	proof	proof	NOUN
ejpam-5153	292	2	.	.	PUNCT
ejpam-5153	293	1	(	(	PUNCT
ejpam-5153	293	2	1	1	X
ejpam-5153	293	3	)	)	PUNCT
ejpam-5153	293	4	⇒	⇒	NOUN
ejpam-5153	293	5	(	(	PUNCT
ejpam-5153	293	6	2	2	NUM
ejpam-5153	293	7	)	)	PUNCT
ejpam-5153	293	8	b	b	NOUN
ejpam-5153	293	9	be	be	AUX
ejpam-5153	293	10	a	a	DET
ejpam-5153	293	11	two	two	NUM
ejpam-5153	293	12	-	-	PUNCT
ejpam-5153	293	13	dimensional	dimensional	ADJ
ejpam-5153	293	14	real	real	ADJ
ejpam-5153	293	15	division	division	NOUN
ejpam-5153	293	16	alegbra	alegbra	NOUN
ejpam-5153	293	17	and	and	CCONJ
ejpam-5153	293	18	d12	d12	NOUN
ejpam-5153	293	19	<	<	X
ejpam-5153	293	20	0	0	NUM
ejpam-5153	293	21	.	.	PUNCT
ejpam-5153	294	1	[	[	X
ejpam-5153	294	2	14	14	NUM
ejpam-5153	294	3	]	]	PUNCT
ejpam-5153	294	4	shows	show	VERB
ejpam-5153	294	5	that	that	SCONJ
ejpam-5153	294	6	b	b	PROPN
ejpam-5153	294	7	contains	contain	VERB
ejpam-5153	294	8	a	a	DET
ejpam-5153	294	9	non	non	ADJ
ejpam-5153	294	10	-	-	ADJ
ejpam-5153	294	11	zero	zero	ADJ
ejpam-5153	294	12	idempotent	idempotent	NOUN
ejpam-5153	294	13	.	.	PUNCT
ejpam-5153	295	1	thus	thus	ADV
ejpam-5153	295	2	we	we	PRON
ejpam-5153	295	3	can	can	AUX
ejpam-5153	295	4	take	take	VERB
ejpam-5153	295	5	u	u	PRON
ejpam-5153	295	6	the	the	DET
ejpam-5153	295	7	left	left	ADJ
ejpam-5153	295	8	unit	unit	NOUN
ejpam-5153	295	9	element	element	NOUN
ejpam-5153	295	10	of	of	ADP
ejpam-5153	295	11	a	a	PRON
ejpam-5153	295	12	and	and	CCONJ
ejpam-5153	295	13	the	the	DET
ejpam-5153	295	14	non	non	ADJ
ejpam-5153	295	15	-	-	ADJ
ejpam-5153	295	16	zero	zero	ADJ
ejpam-5153	295	17	idempotent	idempotent	NOUN
ejpam-5153	295	18	of	of	ADP
ejpam-5153	295	19	b.	b.	PROPN
ejpam-5153	295	20	a	a	DET
ejpam-5153	295	21	⊕	⊕	PROPN
ejpam-5153	295	22	b	b	PROPN
ejpam-5153	295	23	satisfies	satisfie	NOUN
ejpam-5153	295	24	to	to	ADP
ejpam-5153	295	25	(	(	PUNCT
ejpam-5153	295	26	e4	e4	PROPN
ejpam-5153	295	27	)	)	PUNCT
ejpam-5153	295	28	,	,	PUNCT
ejpam-5153	295	29	we	we	PRON
ejpam-5153	295	30	have	have	VERB
ejpam-5153	295	31	(	(	PUNCT
ejpam-5153	295	32	e4.e1).e1	e4.e1).e1	NOUN
ejpam-5153	295	33	=	=	SYM
ejpam-5153	295	34	e4	e4	PROPN
ejpam-5153	295	35	=	=	AUX
ejpam-5153	295	36	⇒	⇒	PROPN
ejpam-5153	295	37	c21(1	c21(1	PROPN
ejpam-5153	295	38	+	+	CCONJ
ejpam-5153	295	39	d21	d21	NOUN
ejpam-5153	295	40	)	)	PUNCT
ejpam-5153	295	41	=	=	SYM
ejpam-5153	296	1	0	0	PUNCT
ejpam-5153	296	2	et	et	NOUN
ejpam-5153	296	3	d2	d2	PROPN
ejpam-5153	296	4	21	21	NUM
ejpam-5153	296	5	=	=	SYM
ejpam-5153	296	6	1	1	NUM
ejpam-5153	296	7	.	.	NOUN
ejpam-5153	296	8	•	•	NOUN
ejpam-5153	296	9	if	if	SCONJ
ejpam-5153	296	10	d21	d21	NOUN
ejpam-5153	296	11	=	=	NOUN
ejpam-5153	296	12	1	1	NUM
ejpam-5153	296	13	,	,	PUNCT
ejpam-5153	296	14	then	then	ADV
ejpam-5153	296	15	c21	c21	PROPN
ejpam-5153	296	16	=	=	SYM
ejpam-5153	296	17	0	0	PROPN
ejpam-5153	296	18	,	,	PUNCT
ejpam-5153	296	19	thus	thus	ADV
ejpam-5153	296	20	b	b	AUX
ejpam-5153	296	21	be	be	AUX
ejpam-5153	296	22	a	a	DET
ejpam-5153	296	23	two	two	NUM
ejpam-5153	296	24	-	-	PUNCT
ejpam-5153	296	25	dimensional	dimensional	ADJ
ejpam-5153	296	26	real	real	ADJ
ejpam-5153	296	27	division	division	NOUN
ejpam-5153	296	28	alegbra	alegbra	NOUN
ejpam-5153	296	29	with	with	ADP
ejpam-5153	296	30	right	right	ADJ
ejpam-5153	296	31	unit	unit	NOUN
ejpam-5153	296	32	.	.	PUNCT
ejpam-5153	297	1	the	the	DET
ejpam-5153	297	2	lemma	lemma	PROPN
ejpam-5153	297	3	11	11	NUM
ejpam-5153	297	4	,	,	PUNCT
ejpam-5153	297	5	shows	show	VERB
ejpam-5153	297	6	that	that	SCONJ
ejpam-5153	297	7	a	a	DET
ejpam-5153	297	8	⊕	⊕	PROPN
ejpam-5153	297	9	b	b	PROPN
ejpam-5153	297	10	is	be	AUX
ejpam-5153	297	11	isomorphic	isomorphic	ADJ
ejpam-5153	297	12	to	to	ADP
ejpam-5153	297	13	⋆h	⋆h	PROPN
ejpam-5153	297	14	.	.	PUNCT
ejpam-5153	298	1	•	•	INTJ
ejpam-5153	298	2	if	if	SCONJ
ejpam-5153	298	3	d21	d21	PROPN
ejpam-5153	298	4	=	=	SYM
ejpam-5153	298	5	−1	−1	NOUN
ejpam-5153	298	6	,	,	PUNCT
ejpam-5153	298	7	we	we	PRON
ejpam-5153	298	8	have	have	VERB
ejpam-5153	298	9	(	(	PUNCT
ejpam-5153	298	10	e4e3	e4e3	X
ejpam-5153	298	11	+	+	PUNCT
ejpam-5153	298	12	e3e4)e1	e3e4)e1	ADJ
ejpam-5153	298	13	=	=	PUNCT
ejpam-5153	298	14	e4e3	e4e3	X
ejpam-5153	299	1	+	+	CCONJ
ejpam-5153	300	1	e3e4	e3e4	X
ejpam-5153	300	2	=	=	NOUN
ejpam-5153	300	3	⇒	⇒	X
ejpam-5153	300	4	d12	d12	NOUN
ejpam-5153	300	5	=	=	SYM
ejpam-5153	300	6	1	1	NUM
ejpam-5153	300	7	,	,	PUNCT
ejpam-5153	300	8	absurd	absurd	ADJ
ejpam-5153	300	9	because	because	SCONJ
ejpam-5153	300	10	d12	d12	ADJ
ejpam-5153	300	11	<	<	X
ejpam-5153	300	12	0	0	NUM
ejpam-5153	300	13	.	.	PUNCT
ejpam-5153	300	14	(	(	PUNCT
ejpam-5153	300	15	2	2	X
ejpam-5153	300	16	)	)	PUNCT
ejpam-5153	300	17	⇒	⇒	NOUN
ejpam-5153	300	18	(	(	PUNCT
ejpam-5153	300	19	1	1	NUM
ejpam-5153	300	20	)	)	PUNCT
ejpam-5153	300	21	obvious	obvious	ADJ
ejpam-5153	300	22	.	.	PUNCT
ejpam-5153	301	1	references	reference	NOUN
ejpam-5153	301	2	2286	2286	NUM
ejpam-5153	301	3	5.3	5.3	NUM
ejpam-5153	301	4	.	.	PUNCT
ejpam-5153	301	5	fused	fuse	VERB
ejpam-5153	301	6	algebras	algebras	PROPN
ejpam-5153	301	7	division	division	NOUN
ejpam-5153	301	8	with	with	ADP
ejpam-5153	301	9	left	left	ADJ
ejpam-5153	301	10	unit	unit	NOUN
ejpam-5153	301	11	satisfies	satisfie	NOUN
ejpam-5153	301	12	to	to	ADP
ejpam-5153	301	13	(	(	PUNCT
ejpam-5153	301	14	e1	e1	PROPN
ejpam-5153	301	15	)	)	PUNCT
ejpam-5153	301	16	and	and	CCONJ
ejpam-5153	301	17	to	to	ADP
ejpam-5153	301	18	(	(	PUNCT
ejpam-5153	301	19	e2	e2	PROPN
ejpam-5153	301	20	)	)	PUNCT
ejpam-5153	301	21	.	.	PUNCT
ejpam-5153	302	1	theorem	theorem	NOUN
ejpam-5153	302	2	5	5	NUM
ejpam-5153	302	3	.	.	PUNCT
ejpam-5153	302	4	a	a	DET
ejpam-5153	302	5	⊕	⊕	PROPN
ejpam-5153	302	6	b	b	PROPN
ejpam-5153	302	7	be	be	AUX
ejpam-5153	302	8	a	a	DET
ejpam-5153	302	9	division	division	NOUN
ejpam-5153	302	10	algebra	algebra	NOUN
ejpam-5153	302	11	with	with	ADP
ejpam-5153	302	12	left	left	ADJ
ejpam-5153	302	13	unit	unit	NOUN
ejpam-5153	302	14	e1	e1	PROPN
ejpam-5153	302	15	.	.	PUNCT
ejpam-5153	303	1	the	the	DET
ejpam-5153	303	2	following	follow	VERB
ejpam-5153	303	3	propositions	proposition	NOUN
ejpam-5153	303	4	are	be	AUX
ejpam-5153	303	5	equivalent	equivalent	ADJ
ejpam-5153	303	6	:	:	PUNCT
ejpam-5153	303	7	(	(	PUNCT
ejpam-5153	303	8	1	1	X
ejpam-5153	303	9	)	)	PUNCT
ejpam-5153	303	10	a	a	DET
ejpam-5153	303	11	⊕	⊕	PROPN
ejpam-5153	303	12	b	b	PROPN
ejpam-5153	303	13	satisfies	satisfie	NOUN
ejpam-5153	303	14	to	to	ADP
ejpam-5153	303	15	(	(	PUNCT
ejpam-5153	303	16	e2	e2	PROPN
ejpam-5153	303	17	)	)	PUNCT
ejpam-5153	303	18	,	,	PUNCT
ejpam-5153	303	19	(	(	PUNCT
ejpam-5153	303	20	2	2	X
ejpam-5153	303	21	)	)	PUNCT
ejpam-5153	303	22	a	a	DET
ejpam-5153	303	23	⊕	⊕	PROPN
ejpam-5153	303	24	b	b	PROPN
ejpam-5153	303	25	is	be	AUX
ejpam-5153	303	26	isomorphic	isomorphic	ADJ
ejpam-5153	303	27	to	to	ADP
ejpam-5153	303	28	either	either	DET
ejpam-5153	303	29	h	h	NOUN
ejpam-5153	303	30	,	,	PUNCT
ejpam-5153	303	31	⋆h	⋆h	PROPN
ejpam-5153	303	32	,	,	PUNCT
ejpam-5153	303	33	c	c	PROPN
ejpam-5153	303	34	⊕	⊕	PROPN
ejpam-5153	303	35	b.	b.	PROPN
ejpam-5153	303	36	proof	proof	NOUN
ejpam-5153	303	37	.	.	PUNCT
ejpam-5153	304	1	(	(	PUNCT
ejpam-5153	304	2	1	1	X
ejpam-5153	304	3	)	)	PUNCT
ejpam-5153	304	4	=	=	NOUN
ejpam-5153	304	5	⇒	⇒	NOUN
ejpam-5153	304	6	(	(	PUNCT
ejpam-5153	304	7	2	2	X
ejpam-5153	304	8	)	)	PUNCT
ejpam-5153	304	9	a	a	DET
ejpam-5153	304	10	⊕	⊕	PROPN
ejpam-5153	304	11	b	b	PROPN
ejpam-5153	304	12	satisfies	satisfie	NOUN
ejpam-5153	304	13	to	to	ADP
ejpam-5153	304	14	(	(	PUNCT
ejpam-5153	304	15	e2	e2	PROPN
ejpam-5153	304	16	)	)	PUNCT
ejpam-5153	304	17	,	,	PUNCT
ejpam-5153	304	18	the	the	DET
ejpam-5153	304	19	proposition	proposition	NOUN
ejpam-5153	304	20	9	9	NUM
ejpam-5153	304	21	shows	show	VERB
ejpam-5153	304	22	that	that	SCONJ
ejpam-5153	304	23	a	a	PRON
ejpam-5153	304	24	is	be	AUX
ejpam-5153	304	25	isomorphic	isomorphic	ADJ
ejpam-5153	304	26	to	to	PART
ejpam-5153	304	27	eihter	eihter	VERB
ejpam-5153	304	28	c	c	PROPN
ejpam-5153	304	29	,	,	PUNCT
ejpam-5153	304	30	⋆c	⋆c	PROPN
ejpam-5153	304	31	.	.	NOUN
ejpam-5153	304	32	•	•	NOUN
ejpam-5153	304	33	if	if	SCONJ
ejpam-5153	304	34	a	a	DET
ejpam-5153	304	35	∼=	∼=	PROPN
ejpam-5153	304	36	c	c	NOUN
ejpam-5153	304	37	,	,	PUNCT
ejpam-5153	304	38	the	the	DET
ejpam-5153	304	39	theorem	theorem	NOUN
ejpam-5153	304	40	3	3	NUM
ejpam-5153	304	41	,	,	PUNCT
ejpam-5153	304	42	shows	show	VERB
ejpam-5153	304	43	that	that	SCONJ
ejpam-5153	304	44	a	a	DET
ejpam-5153	304	45	⊕	⊕	PROPN
ejpam-5153	304	46	b	b	PROPN
ejpam-5153	304	47	is	be	AUX
ejpam-5153	304	48	isomorphic	isomorphic	ADJ
ejpam-5153	304	49	to	to	ADP
ejpam-5153	304	50	either	either	DET
ejpam-5153	304	51	h	h	NOUN
ejpam-5153	304	52	,	,	PUNCT
ejpam-5153	304	53	c	c	PROPN
ejpam-5153	304	54	⊕	⊕	PROPN
ejpam-5153	304	55	b.	b.	PROPN
ejpam-5153	305	1	•	•	INTJ
ejpam-5153	305	2	if	if	SCONJ
ejpam-5153	305	3	a	a	DET
ejpam-5153	305	4	∼=⋆	∼=⋆	PROPN
ejpam-5153	305	5	c	c	PROPN
ejpam-5153	305	6	,	,	PUNCT
ejpam-5153	305	7	the	the	DET
ejpam-5153	305	8	theorem	theorem	ADJ
ejpam-5153	305	9	4	4	NUM
ejpam-5153	305	10	shows	show	VERB
ejpam-5153	305	11	that	that	SCONJ
ejpam-5153	305	12	a	a	DET
ejpam-5153	305	13	⊕	⊕	PROPN
ejpam-5153	305	14	b	b	PROPN
ejpam-5153	305	15	is	be	AUX
ejpam-5153	305	16	isomorphic	isomorphic	ADJ
ejpam-5153	305	17	to	to	ADP
ejpam-5153	305	18	⋆h	⋆h	PROPN
ejpam-5153	305	19	.	.	PUNCT
ejpam-5153	306	1	therefore	therefore	ADV
ejpam-5153	306	2	a	a	DET
ejpam-5153	306	3	⊕	⊕	PROPN
ejpam-5153	306	4	b	b	PROPN
ejpam-5153	306	5	is	be	AUX
ejpam-5153	306	6	isomorphic	isomorphic	ADJ
ejpam-5153	306	7	to	to	ADP
ejpam-5153	306	8	either	either	DET
ejpam-5153	306	9	h	h	NOUN
ejpam-5153	306	10	,	,	PUNCT
ejpam-5153	306	11	⋆h	⋆h	PROPN
ejpam-5153	306	12	,	,	PUNCT
ejpam-5153	306	13	c	c	PROPN
ejpam-5153	306	14	⊕	⊕	PROPN
ejpam-5153	306	15	b.	b.	PROPN
ejpam-5153	307	1	(	(	PUNCT
ejpam-5153	307	2	2	2	NUM
ejpam-5153	307	3	)	)	PUNCT
ejpam-5153	307	4	=	=	NOUN
ejpam-5153	307	5	⇒	⇒	NOUN
ejpam-5153	307	6	(	(	PUNCT
ejpam-5153	307	7	1	1	NUM
ejpam-5153	307	8	)	)	PUNCT
ejpam-5153	307	9	obvious	obvious	ADJ
ejpam-5153	307	10	.	.	PUNCT
ejpam-5153	308	1	corollary	corollary	ADJ
ejpam-5153	308	2	3	3	NUM
ejpam-5153	308	3	.	.	PUNCT
ejpam-5153	309	1	a	a	DET
ejpam-5153	309	2	⊕	⊕	PROPN
ejpam-5153	309	3	b	b	PROPN
ejpam-5153	309	4	be	be	AUX
ejpam-5153	309	5	a	a	DET
ejpam-5153	309	6	division	division	NOUN
ejpam-5153	309	7	algebra	algebra	NOUN
ejpam-5153	309	8	with	with	ADP
ejpam-5153	309	9	left	left	ADJ
ejpam-5153	309	10	unit	unit	NOUN
ejpam-5153	309	11	e1	e1	PROPN
ejpam-5153	309	12	.	.	PUNCT
ejpam-5153	310	1	we	we	PRON
ejpam-5153	310	2	have	have	VERB
ejpam-5153	310	3	a	a	DET
ejpam-5153	310	4	⊕	⊕	PROPN
ejpam-5153	310	5	b	b	PROPN
ejpam-5153	310	6	satisfies	satisfie	NOUN
ejpam-5153	310	7	to	to	ADP
ejpam-5153	310	8	(	(	PUNCT
ejpam-5153	310	9	e1	e1	PROPN
ejpam-5153	310	10	)	)	PUNCT
ejpam-5153	310	11	(	(	PUNCT
ejpam-5153	310	12	e2	e2	PROPN
ejpam-5153	310	13	)	)	PUNCT
ejpam-5153	310	14	a	a	DET
ejpam-5153	310	15	⊕	⊕	PROPN
ejpam-5153	310	16	b	b	PROPN
ejpam-5153	310	17	isomorphic	isomorphic	ADJ
ejpam-5153	310	18	to	to	ADP
ejpam-5153	310	19	h	h	PROPN
ejpam-5153	310	20	h	h	PROPN
ejpam-5153	310	21	,	,	PUNCT
ejpam-5153	310	22	⋆h	⋆h	PROPN
ejpam-5153	310	23	,	,	PUNCT
ejpam-5153	310	24	c	c	PROPN
ejpam-5153	310	25	⊕	⊕	PROPN
ejpam-5153	310	26	b	b	PROPN
ejpam-5153	310	27	references	reference	NOUN
ejpam-5153	310	28	[	[	X
ejpam-5153	310	29	1	1	NUM
ejpam-5153	310	30	]	]	SYM
ejpam-5153	310	31	b	b	X
ejpam-5153	310	32	aharmim	aharmim	NOUN
ejpam-5153	310	33	,	,	PUNCT
ejpam-5153	310	34	o	o	PROPN
ejpam-5153	310	35	fayz	fayz	NOUN
ejpam-5153	310	36	,	,	PUNCT
ejpam-5153	310	37	e	e	NOUN
ejpam-5153	310	38	idnarour	idnarour	PROPN
ejpam-5153	310	39	,	,	PUNCT
ejpam-5153	310	40	and	and	CCONJ
ejpam-5153	310	41	a	a	DET
ejpam-5153	310	42	rochdi	rochdi	NOUN
ejpam-5153	310	43	.	.	PUNCT
ejpam-5153	311	1	real	real	ADJ
ejpam-5153	311	2	division	division	NOUN
ejpam-5153	311	3	algebras	algebra	VERB
ejpam-5153	311	4	satisfying	satisfy	VERB
ejpam-5153	311	5	some	some	DET
ejpam-5153	311	6	identities	identity	NOUN
ejpam-5153	311	7	.	.	PUNCT
ejpam-5153	312	1	ija	ija	NOUN
ejpam-5153	312	2	,	,	PUNCT
ejpam-5153	312	3	pages	page	NOUN
ejpam-5153	312	4	123–128	123–128	NUM
ejpam-5153	312	5	,	,	PUNCT
ejpam-5153	312	6	2021	2021	NUM
ejpam-5153	312	7	.	.	PUNCT
ejpam-5153	313	1	[	[	X
ejpam-5153	313	2	2	2	NUM
ejpam-5153	313	3	]	]	PUNCT
ejpam-5153	313	4	s.c	s.c	PROPN
ejpam-5153	313	5	althoen	althoen	PROPN
ejpam-5153	313	6	,	,	PUNCT
ejpam-5153	313	7	k.	k.	PROPN
ejpam-5153	313	8	d	d	PROPN
ejpam-5153	313	9	hansen	hansen	PROPN
ejpam-5153	313	10	,	,	PUNCT
ejpam-5153	313	11	and	and	CCONJ
ejpam-5153	313	12	l.	l.	PROPN
ejpam-5153	313	13	d	d	PROPN
ejpam-5153	313	14	kugler	kugler	PROPN
ejpam-5153	313	15	.	.	PUNCT
ejpam-5153	313	16	fused	fuse	VERB
ejpam-5153	313	17	four	four	NUM
ejpam-5153	313	18	-	-	PUNCT
ejpam-5153	313	19	dimensional	dimensional	ADJ
ejpam-5153	313	20	real	real	ADJ
ejpam-5153	313	21	division	division	NOUN
ejpam-5153	313	22	algebras	algebra	NOUN
ejpam-5153	313	23	.	.	PUNCT
ejpam-5153	314	1	j.	j.	PROPN
ejpam-5153	314	2	algebra	algebra	PROPN
ejpam-5153	314	3	,	,	PUNCT
ejpam-5153	314	4	pages	page	NOUN
ejpam-5153	314	5	649–660	649–660	NUM
ejpam-5153	314	6	,	,	PUNCT
ejpam-5153	314	7	1994	1994	NUM
ejpam-5153	314	8	.	.	PUNCT
ejpam-5153	315	1	[	[	X
ejpam-5153	315	2	3	3	NUM
ejpam-5153	315	3	]	]	X
ejpam-5153	315	4	steven	steven	PROPN
ejpam-5153	315	5	c	c	PROPN
ejpam-5153	315	6	althoen	althoen	PROPN
ejpam-5153	315	7	and	and	CCONJ
ejpam-5153	315	8	lawrence	lawrence	PROPN
ejpam-5153	315	9	d	d	PROPN
ejpam-5153	315	10	kugler	kugler	PROPN
ejpam-5153	315	11	.	.	PUNCT
ejpam-5153	316	1	when	when	SCONJ
ejpam-5153	316	2	r2	r2	PROPN
ejpam-5153	316	3	is	be	AUX
ejpam-5153	316	4	a	a	DET
ejpam-5153	316	5	division	division	NOUN
ejpam-5153	316	6	algebra	algebra	NOUN
ejpam-5153	316	7	?	?	PUNCT
ejpam-5153	317	1	the	the	DET
ejpam-5153	317	2	american	american	PROPN
ejpam-5153	317	3	mathematical	mathematical	PROPN
ejpam-5153	317	4	monthly	monthly	ADV
ejpam-5153	317	5	,	,	PUNCT
ejpam-5153	317	6	pages	page	NOUN
ejpam-5153	317	7	625–635	625–635	NUM
ejpam-5153	317	8	,	,	PUNCT
ejpam-5153	317	9	1983	1983	NUM
ejpam-5153	317	10	.	.	PUNCT
ejpam-5153	318	1	[	[	X
ejpam-5153	318	2	4	4	NUM
ejpam-5153	318	3	]	]	X
ejpam-5153	318	4	r	r	NOUN
ejpam-5153	318	5	bott	bott	PROPN
ejpam-5153	318	6	and	and	CCONJ
ejpam-5153	318	7	j	j	PROPN
ejpam-5153	318	8	milnor	milnor	PROPN
ejpam-5153	318	9	.	.	PUNCT
ejpam-5153	319	1	on	on	ADP
ejpam-5153	319	2	the	the	DET
ejpam-5153	319	3	parallelizability	parallelizability	NOUN
ejpam-5153	319	4	of	of	ADP
ejpam-5153	319	5	the	the	DET
ejpam-5153	319	6	spheres	sphere	NOUN
ejpam-5153	319	7	.	.	PUNCT
ejpam-5153	320	1	bulletin	bulletin	NOUN
ejpam-5153	320	2	(	(	PUNCT
ejpam-5153	320	3	new	new	ADJ
ejpam-5153	320	4	series	series	NOUN
ejpam-5153	320	5	)	)	PUNCT
ejpam-5153	320	6	of	of	ADP
ejpam-5153	320	7	the	the	DET
ejpam-5153	320	8	american	american	PROPN
ejpam-5153	320	9	mathematical	mathematical	PROPN
ejpam-5153	320	10	society	society	NOUN
ejpam-5153	320	11	/	/	SYM
ejpam-5153	320	12	bulletin	bulletin	NOUN
ejpam-5153	320	13	,	,	PUNCT
ejpam-5153	320	14	new	new	ADJ
ejpam-5153	320	15	series	series	NOUN
ejpam-5153	320	16	,	,	PUNCT
ejpam-5153	320	17	of	of	ADP
ejpam-5153	320	18	the	the	DET
ejpam-5153	320	19	american	american	PROPN
ejpam-5153	320	20	mathematical	mathematical	PROPN
ejpam-5153	320	21	society	society	NOUN
ejpam-5153	320	22	,	,	PUNCT
ejpam-5153	320	23	pages	page	VERB
ejpam-5153	320	24	87–89	87–89	NUM
ejpam-5153	320	25	,	,	PUNCT
ejpam-5153	320	26	1958	1958	NUM
ejpam-5153	320	27	.	.	PUNCT
ejpam-5153	321	1	[	[	X
ejpam-5153	321	2	5	5	X
ejpam-5153	321	3	]	]	X
ejpam-5153	321	4	r.h	r.h	NOUN
ejpam-5153	321	5	bruck	bruck	NOUN
ejpam-5153	321	6	.	.	PUNCT
ejpam-5153	322	1	some	some	DET
ejpam-5153	322	2	results	result	NOUN
ejpam-5153	322	3	in	in	ADP
ejpam-5153	322	4	the	the	DET
ejpam-5153	322	5	theory	theory	NOUN
ejpam-5153	322	6	of	of	ADP
ejpam-5153	322	7	linear	linear	PROPN
ejpam-5153	322	8	non	non	ADJ
ejpam-5153	322	9	-	-	ADJ
ejpam-5153	322	10	associative	associative	ADJ
ejpam-5153	322	11	algebras	algebra	NOUN
ejpam-5153	322	12	.	.	PUNCT
ejpam-5153	323	1	university	university	PROPN
ejpam-5153	323	2	of	of	ADP
ejpam-5153	323	3	wisconsin	wisconsin	PROPN
ejpam-5153	323	4	,	,	PUNCT
ejpam-5153	323	5	madison	madison	PROPN
ejpam-5153	323	6	,	,	PUNCT
ejpam-5153	323	7	wis	wis	PROPN
ejpam-5153	323	8	,	,	PUNCT
ejpam-5153	323	9	,	,	PUNCT
ejpam-5153	323	10	pages	page	NOUN
ejpam-5153	323	11	141–199	141–199	NUM
ejpam-5153	323	12	,	,	PUNCT
ejpam-5153	323	13	1994	1994	NUM
ejpam-5153	323	14	.	.	PUNCT
ejpam-5153	324	1	[	[	X
ejpam-5153	324	2	6	6	NUM
ejpam-5153	324	3	]	]	PUNCT
ejpam-5153	324	4	a	a	DET
ejpam-5153	324	5	calderon	calderon	NOUN
ejpam-5153	324	6	,	,	PUNCT
ejpam-5153	324	7	a	a	DET
ejpam-5153	324	8	kaidi	kaidi	NOUN
ejpam-5153	324	9	,	,	PUNCT
ejpam-5153	324	10	c	c	PROPN
ejpam-5153	324	11	martin	martin	PROPN
ejpam-5153	324	12	,	,	PUNCT
ejpam-5153	324	13	a	a	DET
ejpam-5153	324	14	morales	morale	NOUN
ejpam-5153	324	15	,	,	PUNCT
ejpam-5153	324	16	m	m	PROPN
ejpam-5153	324	17	ramirez	ramirez	NOUN
ejpam-5153	324	18	,	,	PUNCT
ejpam-5153	324	19	and	and	CCONJ
ejpam-5153	324	20	a	a	DET
ejpam-5153	324	21	rochdi	rochdi	NOUN
ejpam-5153	324	22	.	.	PUNCT
ejpam-5153	325	1	finitedimensional	finitedimensional	ADJ
ejpam-5153	325	2	absolute	absolute	ADJ
ejpam-5153	325	3	valued	value	VERB
ejpam-5153	325	4	algebras	algebra	NOUN
ejpam-5153	325	5	.	.	PUNCT
ejpam-5153	326	1	israel	israel	PROPN
ejpam-5153	326	2	j.	j.	PROPN
ejpam-5153	326	3	mathematics	mathematics	PROPN
ejpam-5153	326	4	,	,	PUNCT
ejpam-5153	326	5	pages	page	NOUN
ejpam-5153	326	6	193–220	193–220	NUM
ejpam-5153	326	7	,	,	PUNCT
ejpam-5153	326	8	2011	2011	NUM
ejpam-5153	326	9	.	.	PUNCT
ejpam-5153	327	1	[	[	X
ejpam-5153	327	2	7	7	X
ejpam-5153	327	3	]	]	PUNCT
ejpam-5153	327	4	ana	ana	PROPN
ejpam-5153	327	5	lucía	lucía	ADJ
ejpam-5153	327	6	calí	calí	NOUN
ejpam-5153	327	7	and	and	CCONJ
ejpam-5153	327	8	michael	michael	PROPN
ejpam-5153	327	9	josephy	josephy	PROPN
ejpam-5153	327	10	.	.	PUNCT
ejpam-5153	328	1	two	two	NUM
ejpam-5153	328	2	-	-	PUNCT
ejpam-5153	328	3	dimensional	dimensional	ADJ
ejpam-5153	328	4	real	real	ADJ
ejpam-5153	328	5	division	division	NOUN
ejpam-5153	328	6	algebras	algebra	NOUN
ejpam-5153	328	7	.	.	PUNCT
ejpam-5153	329	1	semanticscholar	semanticscholar	NOUN
ejpam-5153	329	2	,	,	PUNCT
ejpam-5153	329	3	pages	page	NOUN
ejpam-5153	329	4	58–63	58–63	NUM
ejpam-5153	329	5	,	,	PUNCT
ejpam-5153	329	6	1985	1985	NUM
ejpam-5153	329	7	.	.	PUNCT
ejpam-5153	330	1	[	[	X
ejpam-5153	330	2	8	8	NUM
ejpam-5153	330	3	]	]	X
ejpam-5153	330	4	a	a	DET
ejpam-5153	330	5	chandid	chandid	NOUN
ejpam-5153	330	6	and	and	CCONJ
ejpam-5153	330	7	a	a	DET
ejpam-5153	330	8	rochdi	rochdi	NOUN
ejpam-5153	330	9	.	.	PUNCT
ejpam-5153	331	1	a	a	DET
ejpam-5153	331	2	survey	survey	NOUN
ejpam-5153	331	3	on	on	ADP
ejpam-5153	331	4	absolute	absolute	ADJ
ejpam-5153	331	5	valued	value	VERB
ejpam-5153	331	6	algebras	algebra	NOUN
ejpam-5153	331	7	satisfying	satisfy	VERB
ejpam-5153	331	8	(	(	PUNCT
ejpam-5153	331	9	xi	xi	PROPN
ejpam-5153	331	10	,	,	PUNCT
ejpam-5153	331	11	xj	xj	PROPN
ejpam-5153	331	12	,	,	PUNCT
ejpam-5153	331	13	xk	xk	PROPN
ejpam-5153	331	14	)	)	PUNCT
ejpam-5153	331	15	=	=	SYM
ejpam-5153	331	16	0	0	X
ejpam-5153	331	17	.	.	PUNCT
ejpam-5153	331	18	international	international	ADJ
ejpam-5153	331	19	journal	journal	PROPN
ejpam-5153	331	20	of	of	ADP
ejpam-5153	331	21	algebra	algebra	PROPN
ejpam-5153	331	22	,	,	PUNCT
ejpam-5153	331	23	2:837–852	2:837–852	NUM
ejpam-5153	331	24	,	,	PUNCT
ejpam-5153	331	25	2008	2008	NUM
ejpam-5153	331	26	.	.	PUNCT
ejpam-5153	332	1	references	reference	NOUN
ejpam-5153	332	2	2287	2287	NUM
ejpam-5153	333	1	[	[	X
ejpam-5153	333	2	9	9	NUM
ejpam-5153	333	3	]	]	SYM
ejpam-5153	333	4	o	o	NOUN
ejpam-5153	333	5	diankha	diankha	NOUN
ejpam-5153	333	6	,	,	PUNCT
ejpam-5153	333	7	a	a	DET
ejpam-5153	333	8	diouf	diouf	PROPN
ejpam-5153	333	9	,	,	PUNCT
ejpam-5153	333	10	m.	m.	NOUN
ejpam-5153	333	11	i	i	PROPN
ejpam-5153	333	12	ramirez	ramirez	PROPN
ejpam-5153	333	13	,	,	PUNCT
ejpam-5153	333	14	and	and	CCONJ
ejpam-5153	333	15	a	a	DET
ejpam-5153	333	16	rochdi	rochdi	NOUN
ejpam-5153	333	17	.	.	PUNCT
ejpam-5153	334	1	absolute	absolute	ADJ
ejpam-5153	334	2	valued	value	VERB
ejpam-5153	334	3	algebras	algebra	NOUN
ejpam-5153	334	4	with	with	ADP
ejpam-5153	334	5	one	one	NUM
ejpam-5153	334	6	sided	sided	ADJ
ejpam-5153	334	7	unit	unit	NOUN
ejpam-5153	334	8	satisfying	satisfy	VERB
ejpam-5153	334	9	(	(	PUNCT
ejpam-5153	334	10	x2	x2	PROPN
ejpam-5153	334	11	,	,	PUNCT
ejpam-5153	334	12	x2	x2	PROPN
ejpam-5153	334	13	,	,	PUNCT
ejpam-5153	334	14	x2	x2	PROPN
ejpam-5153	334	15	)	)	PUNCT
ejpam-5153	334	16	=	=	SYM
ejpam-5153	335	1	0	0	X
ejpam-5153	335	2	.	.	PUNCT
ejpam-5153	336	1	international	international	ADJ
ejpam-5153	336	2	journal	journal	PROPN
ejpam-5153	336	3	of	of	ADP
ejpam-5153	336	4	algebra	algebra	PROPN
ejpam-5153	336	5	,	,	PUNCT
ejpam-5153	336	6	7:932	7:932	NUM
ejpam-5153	336	7	–	–	PUNCT
ejpam-5153	336	8	958	958	NUM
ejpam-5153	336	9	,	,	PUNCT
ejpam-5153	336	10	2013	2013	NUM
ejpam-5153	336	11	.	.	PUNCT
ejpam-5153	337	1	[	[	X
ejpam-5153	337	2	10	10	NUM
ejpam-5153	337	3	]	]	X
ejpam-5153	337	4	o	o	X
ejpam-5153	337	5	diankha	diankha	NOUN
ejpam-5153	337	6	,	,	PUNCT
ejpam-5153	337	7	m	m	NOUN
ejpam-5153	337	8	traoré	traoré	NOUN
ejpam-5153	337	9	,	,	PUNCT
ejpam-5153	337	10	m.	m.	NOUN
ejpam-5153	337	11	i	i	PROPN
ejpam-5153	337	12	ramirez	ramirez	PROPN
ejpam-5153	337	13	,	,	PUNCT
ejpam-5153	337	14	and	and	CCONJ
ejpam-5153	337	15	a	a	DET
ejpam-5153	337	16	rochdi	rochdi	NOUN
ejpam-5153	337	17	.	.	PUNCT
ejpam-5153	338	1	four	four	NUM
ejpam-5153	338	2	-	-	PUNCT
ejpam-5153	338	3	dimensional	dimensional	ADJ
ejpam-5153	338	4	real	real	ADJ
ejpam-5153	338	5	thirdpower	thirdpower	NOUN
ejpam-5153	338	6	associative	associative	NOUN
ejpam-5153	338	7	division	division	NOUN
ejpam-5153	338	8	algebras	algebra	NOUN
ejpam-5153	338	9	.	.	PUNCT
ejpam-5153	339	1	comm	comm	NOUN
ejpam-5153	339	2	.	.	PUNCT
ejpam-5153	340	1	alg	alg	PROPN
ejpam-5153	340	2	,	,	PUNCT
ejpam-5153	340	3	pages	page	NOUN
ejpam-5153	340	4	3397–3406	3397–3406	NUM
ejpam-5153	340	5	,	,	PUNCT
ejpam-5153	340	6	2016	2016	NUM
ejpam-5153	340	7	.	.	PUNCT
ejpam-5153	341	1	[	[	X
ejpam-5153	341	2	11	11	NUM
ejpam-5153	341	3	]	]	X
ejpam-5153	341	4	issai	issai	VERB
ejpam-5153	341	5	kantor	kantor	PROPN
ejpam-5153	341	6	,	,	PUNCT
ejpam-5153	341	7	marion	marion	PROPN
ejpam-5153	341	8	hubner	hubner	NOUN
ejpam-5153	341	9	,	,	PUNCT
ejpam-5153	341	10	and	and	CCONJ
ejpam-5153	341	11	holger	holger	NOUN
ejpam-5153	341	12	p	p	PROPN
ejpam-5153	341	13	petersson	petersson	PROPN
ejpam-5153	341	14	.	.	PUNCT
ejpam-5153	342	1	two	two	NUM
ejpam-5153	342	2	-	-	PUNCT
ejpam-5153	342	3	dimensional	dimensional	ADJ
ejpam-5153	342	4	real	real	ADJ
ejpam-5153	342	5	division	division	NOUN
ejpam-5153	342	6	algebras	algebra	NOUN
ejpam-5153	342	7	revisited	revisit	VERB
ejpam-5153	342	8	.	.	PUNCT
ejpam-5153	343	1	beiträge	beiträge	PROPN
ejpam-5153	343	2	zur	zur	PROPN
ejpam-5153	343	3	algebra	algebra	NOUN
ejpam-5153	343	4	geometrie	geometrie	NOUN
ejpam-5153	343	5	,	,	PUNCT
ejpam-5153	343	6	pages	page	NOUN
ejpam-5153	343	7	29–36	29–36	NUM
ejpam-5153	343	8	,	,	PUNCT
ejpam-5153	343	9	2004	2004	NUM
ejpam-5153	343	10	.	.	PUNCT
ejpam-5153	344	1	[	[	X
ejpam-5153	344	2	12	12	NUM
ejpam-5153	344	3	]	]	PUNCT
ejpam-5153	344	4	michel	michel	PROPN
ejpam-5153	344	5	kervaire	kervaire	PROPN
ejpam-5153	344	6	.	.	PUNCT
ejpam-5153	345	1	non	non	ADJ
ejpam-5153	345	2	-	-	NOUN
ejpam-5153	345	3	parallelizability	parallelizability	NOUN
ejpam-5153	345	4	of	of	ADP
ejpam-5153	345	5	the	the	DET
ejpam-5153	345	6	n	n	NOUN
ejpam-5153	345	7	-	-	PUNCT
ejpam-5153	345	8	sphere	sphere	NOUN
ejpam-5153	345	9	for	for	ADP
ejpam-5153	345	10	n	n	X
ejpam-5153	345	11	>	>	X
ejpam-5153	345	12	7	7	NUM
ejpam-5153	345	13	.	.	PUNCT
ejpam-5153	345	14	proceedings	proceeding	NOUN
ejpam-5153	345	15	of	of	ADP
ejpam-5153	345	16	the	the	DET
ejpam-5153	345	17	national	national	PROPN
ejpam-5153	345	18	academy	academy	PROPN
ejpam-5153	345	19	of	of	ADP
ejpam-5153	345	20	sciences	sciences	PROPN
ejpam-5153	345	21	of	of	ADP
ejpam-5153	345	22	the	the	DET
ejpam-5153	345	23	united	united	PROPN
ejpam-5153	345	24	states	states	PROPN
ejpam-5153	345	25	of	of	ADP
ejpam-5153	345	26	america	america	PROPN
ejpam-5153	345	27	,	,	PUNCT
ejpam-5153	345	28	pages	page	NOUN
ejpam-5153	345	29	280–283	280–283	NUM
ejpam-5153	345	30	,	,	PUNCT
ejpam-5153	345	31	1958	1958	NUM
ejpam-5153	345	32	.	.	PUNCT
ejpam-5153	346	1	[	[	X
ejpam-5153	346	2	13	13	NUM
ejpam-5153	346	3	]	]	SYM
ejpam-5153	346	4	leonardo	leonardo	PROPN
ejpam-5153	346	5	,	,	PUNCT
ejpam-5153	346	6	bruno	bruno	PROPN
ejpam-5153	346	7	bruno	bruno	PROPN
ejpam-5153	346	8	,	,	PUNCT
ejpam-5153	346	9	julius	julius	PROPN
ejpam-5153	346	10	,	,	PUNCT
ejpam-5153	346	11	hayden	hayden	PROPN
ejpam-5153	346	12	,	,	PUNCT
ejpam-5153	346	13	smigly	smigly	ADV
ejpam-5153	346	14	,	,	PUNCT
ejpam-5153	346	15	and	and	CCONJ
ejpam-5153	346	16	douglas	douglas	PROPN
ejpam-5153	346	17	.	.	PUNCT
ejpam-5153	347	1	commuting	commute	VERB
ejpam-5153	347	2	maps	map	NOUN
ejpam-5153	347	3	and	and	CCONJ
ejpam-5153	347	4	identities	identity	NOUN
ejpam-5153	347	5	with	with	ADP
ejpam-5153	347	6	inverses	inverse	NOUN
ejpam-5153	347	7	on	on	ADP
ejpam-5153	347	8	alternative	alternative	ADJ
ejpam-5153	347	9	division	division	NOUN
ejpam-5153	347	10	rings	ring	NOUN
ejpam-5153	347	11	.	.	PUNCT
ejpam-5153	348	1	journal	journal	PROPN
ejpam-5153	348	2	of	of	ADP
ejpam-5153	348	3	algebra	algebra	PROPN
ejpam-5153	348	4	,	,	PUNCT
ejpam-5153	348	5	pages	page	NOUN
ejpam-5153	348	6	488–505	488–505	NUM
ejpam-5153	348	7	,	,	PUNCT
ejpam-5153	348	8	2024	2024	NUM
ejpam-5153	348	9	.	.	PUNCT
ejpam-5153	349	1	[	[	X
ejpam-5153	349	2	14	14	NUM
ejpam-5153	349	3	]	]	X
ejpam-5153	349	4	b	b	NOUN
ejpam-5153	349	5	segre	segre	NOUN
ejpam-5153	349	6	.	.	PUNCT
ejpam-5153	350	1	la	la	PROPN
ejpam-5153	350	2	teoria	teoria	PROPN
ejpam-5153	350	3	delle	delle	NOUN
ejpam-5153	350	4	algebre	algebre	NOUN
ejpam-5153	350	5	ed	ed	PROPN
ejpam-5153	350	6	alcune	alcune	PROPN
ejpam-5153	350	7	questione	questione	PROPN
ejpam-5153	350	8	di	di	PROPN
ejpam-5153	350	9	realta	realta	PROPN
ejpam-5153	350	10	.	.	PUNCT
ejpam-5153	351	1	pages	page	NOUN
ejpam-5153	351	2	157–188	157–188	NUM
ejpam-5153	351	3	,	,	PUNCT
ejpam-5153	351	4	1954	1954	NUM
ejpam-5153	351	5	.	.	PUNCT
ejpam-5153	352	1	[	[	X
ejpam-5153	352	2	15	15	NUM
ejpam-5153	352	3	]	]	X
ejpam-5153	352	4	t	t	PROPN
ejpam-5153	352	5	yang	yang	PROPN
ejpam-5153	352	6	.	.	PUNCT
ejpam-5153	353	1	division	division	NOUN
ejpam-5153	353	2	algebras	algebra	NOUN
ejpam-5153	353	3	and	and	CCONJ
ejpam-5153	353	4	fibrations	fibration	NOUN
ejpam-5153	353	5	of	of	ADP
ejpam-5153	353	6	spheres	sphere	NOUN
ejpam-5153	353	7	by	by	ADP
ejpam-5153	353	8	great	great	ADJ
ejpam-5153	353	9	spheres	sphere	NOUN
ejpam-5153	353	10	.	.	PUNCT
ejpam-5153	354	1	j.	j.	PROPN
ejpam-5153	354	2	differential	differential	PROPN
ejpam-5153	354	3	geometry	geometry	NOUN
ejpam-5153	354	4	,	,	PUNCT
ejpam-5153	354	5	pages	page	NOUN
ejpam-5153	354	6	577–593	577–593	NUM
ejpam-5153	354	7	,	,	PUNCT
ejpam-5153	354	8	1981	1981	NUM
ejpam-5153	354	9	.	.	PUNCT