id	sid	tid	token	lemma	pos
ejpam-3873	1	1	european	european	PROPN
ejpam-3873	1	2	journal	journal	PROPN
ejpam-3873	1	3	of	of	ADP
ejpam-3873	1	4	pure	pure	ADJ
ejpam-3873	1	5	and	and	CCONJ
ejpam-3873	1	6	applied	apply	VERB
ejpam-3873	1	7	mathematics	mathematic	NOUN
ejpam-3873	1	8	vol	vol	NOUN
ejpam-3873	1	9	.	.	PUNCT
ejpam-3873	2	1	14	14	NUM
ejpam-3873	2	2	,	,	PUNCT
ejpam-3873	2	3	no	no	INTJ
ejpam-3873	2	4	.	.	NOUN
ejpam-3873	2	5	1	1	NUM
ejpam-3873	2	6	,	,	PUNCT
ejpam-3873	2	7	2021	2021	NUM
ejpam-3873	2	8	,	,	PUNCT
ejpam-3873	2	9	314	314	NUM
ejpam-3873	2	10	-	-	SYM
ejpam-3873	2	11	326	326	NUM
ejpam-3873	2	12	issn	issn	PROPN
ejpam-3873	2	13	1307	1307	NUM
ejpam-3873	2	14	-	-	SYM
ejpam-3873	2	15	5543	5543	NUM
ejpam-3873	2	16	–	–	PUNCT
ejpam-3873	3	1	ejpam.com	ejpam.com	X
ejpam-3873	3	2	published	publish	VERB
ejpam-3873	3	3	by	by	ADP
ejpam-3873	3	4	new	new	PROPN
ejpam-3873	3	5	york	york	PROPN
ejpam-3873	3	6	business	business	PROPN
ejpam-3873	3	7	global	global	PROPN
ejpam-3873	3	8	on	on	ADP
ejpam-3873	3	9	γ	γ	NOUN
ejpam-3873	3	10	-	-	PUNCT
ejpam-3873	3	11	sets	set	NOUN
ejpam-3873	3	12	in	in	ADP
ejpam-3873	3	13	rings	ring	NOUN
ejpam-3873	3	14	eva	eva	PROPN
ejpam-3873	3	15	jenny	jenny	PROPN
ejpam-3873	3	16	c.	c.	PROPN
ejpam-3873	3	17	sigasig1	sigasig1	PROPN
ejpam-3873	3	18	,	,	PUNCT
ejpam-3873	3	19	cristoper	cristoper	NOUN
ejpam-3873	3	20	john	john	PROPN
ejpam-3873	3	21	s.	s.	PROPN
ejpam-3873	3	22	rosero2	rosero2	PROPN
ejpam-3873	3	23	,	,	PUNCT
ejpam-3873	3	24	michael	michael	PROPN
ejpam-3873	3	25	p.	p.	PROPN
ejpam-3873	3	26	baldado	baldado	NOUN
ejpam-3873	3	27	jr.3,∗	jr.3,∗	PROPN
ejpam-3873	3	28	1	1	NUM
ejpam-3873	3	29	lourdes	lourdes	PROPN
ejpam-3873	3	30	ledesma	ledesma	PROPN
ejpam-3873	3	31	del	del	PROPN
ejpam-3873	3	32	prado	prado	PROPN
ejpam-3873	3	33	memorial	memorial	PROPN
ejpam-3873	3	34	national	national	ADJ
ejpam-3873	3	35	high	high	ADJ
ejpam-3873	3	36	school	school	NOUN
ejpam-3873	3	37	,	,	PUNCT
ejpam-3873	3	38	tanjay	tanjay	NOUN
ejpam-3873	3	39	city	city	NOUN
ejpam-3873	3	40	,	,	PUNCT
ejpam-3873	3	41	philippines	philippine	NOUN
ejpam-3873	3	42	2	2	NUM
ejpam-3873	3	43	mathematics	mathematic	NOUN
ejpam-3873	3	44	and	and	CCONJ
ejpam-3873	3	45	ict	ict	PROPN
ejpam-3873	3	46	department	department	PROPN
ejpam-3873	3	47	,	,	PUNCT
ejpam-3873	3	48	cebu	cebu	NOUN
ejpam-3873	3	49	normal	normal	ADJ
ejpam-3873	3	50	university	university	NOUN
ejpam-3873	3	51	,	,	PUNCT
ejpam-3873	3	52	cebu	cebu	NOUN
ejpam-3873	3	53	city	city	NOUN
ejpam-3873	3	54	,	,	PUNCT
ejpam-3873	3	55	philippines	philippine	NOUN
ejpam-3873	3	56	3	3	NUM
ejpam-3873	3	57	mathematics	mathematics	PROPN
ejpam-3873	3	58	department	department	NOUN
ejpam-3873	3	59	,	,	PUNCT
ejpam-3873	3	60	negros	negros	PROPN
ejpam-3873	3	61	oriental	oriental	ADJ
ejpam-3873	3	62	state	state	PROPN
ejpam-3873	3	63	university	university	PROPN
ejpam-3873	3	64	,	,	PUNCT
ejpam-3873	3	65	dumaguete	dumaguete	PROPN
ejpam-3873	3	66	city	city	PROPN
ejpam-3873	3	67	,	,	PUNCT
ejpam-3873	3	68	philippines	philippine	NOUN
ejpam-3873	3	69	abstract	abstract	ADJ
ejpam-3873	3	70	.	.	PUNCT
ejpam-3873	4	1	let	let	VERB
ejpam-3873	4	2	r	r	PRON
ejpam-3873	4	3	be	be	AUX
ejpam-3873	4	4	a	a	DET
ejpam-3873	4	5	ring	ring	NOUN
ejpam-3873	4	6	with	with	ADP
ejpam-3873	4	7	identity	identity	NOUN
ejpam-3873	4	8	1r	1r	NUM
ejpam-3873	4	9	.	.	PUNCT
ejpam-3873	5	1	a	a	DET
ejpam-3873	5	2	subset	subset	NOUN
ejpam-3873	5	3	j	j	PROPN
ejpam-3873	5	4	of	of	ADP
ejpam-3873	5	5	r	r	NOUN
ejpam-3873	5	6	is	be	AUX
ejpam-3873	5	7	called	call	VERB
ejpam-3873	5	8	a	a	DET
ejpam-3873	5	9	γ	γ	NOUN
ejpam-3873	5	10	-	-	PUNCT
ejpam-3873	5	11	set	set	ADJ
ejpam-3873	5	12	if	if	SCONJ
ejpam-3873	5	13	for	for	SCONJ
ejpam-3873	5	14	every	every	DET
ejpam-3873	5	15	a	a	DET
ejpam-3873	5	16	∈	∈	NOUN
ejpam-3873	5	17	r\j	r\j	ADV
ejpam-3873	5	18	,	,	PUNCT
ejpam-3873	5	19	there	there	PRON
ejpam-3873	5	20	exist	exist	VERB
ejpam-3873	5	21	b	b	NOUN
ejpam-3873	5	22	,	,	PUNCT
ejpam-3873	5	23	c	c	PROPN
ejpam-3873	5	24	∈	∈	PROPN
ejpam-3873	5	25	j	j	PROPN
ejpam-3873	5	26	such	such	ADJ
ejpam-3873	5	27	that	that	SCONJ
ejpam-3873	5	28	a+	a+	PRON
ejpam-3873	5	29	b	b	X
ejpam-3873	5	30	=	=	SYM
ejpam-3873	5	31	0	0	PROPN
ejpam-3873	5	32	and	and	CCONJ
ejpam-3873	5	33	ac	ac	ADJ
ejpam-3873	5	34	=	=	NOUN
ejpam-3873	5	35	1r	1r	NUM
ejpam-3873	5	36	=	=	PUNCT
ejpam-3873	5	37	ca	ca	NOUN
ejpam-3873	5	38	.	.	PUNCT
ejpam-3873	6	1	a	a	DET
ejpam-3873	6	2	γ	γ	NOUN
ejpam-3873	6	3	-	-	PUNCT
ejpam-3873	6	4	set	set	NOUN
ejpam-3873	6	5	of	of	ADP
ejpam-3873	6	6	minimum	minimum	ADJ
ejpam-3873	6	7	cardinality	cardinality	NOUN
ejpam-3873	6	8	is	be	AUX
ejpam-3873	6	9	called	call	VERB
ejpam-3873	6	10	a	a	DET
ejpam-3873	6	11	minimum	minimum	ADJ
ejpam-3873	6	12	γ	γ	X
ejpam-3873	6	13	-	-	PUNCT
ejpam-3873	6	14	set	set	NOUN
ejpam-3873	6	15	.	.	PUNCT
ejpam-3873	7	1	in	in	ADP
ejpam-3873	7	2	this	this	DET
ejpam-3873	7	3	study	study	NOUN
ejpam-3873	7	4	,	,	PUNCT
ejpam-3873	7	5	we	we	PRON
ejpam-3873	7	6	identified	identify	VERB
ejpam-3873	7	7	some	some	DET
ejpam-3873	7	8	elements	element	NOUN
ejpam-3873	7	9	of	of	ADP
ejpam-3873	7	10	r	r	NOUN
ejpam-3873	7	11	that	that	PRON
ejpam-3873	7	12	are	be	AUX
ejpam-3873	7	13	necessarily	necessarily	ADV
ejpam-3873	7	14	in	in	ADP
ejpam-3873	7	15	a	a	DET
ejpam-3873	7	16	γ	γ	NOUN
ejpam-3873	7	17	-	-	PUNCT
ejpam-3873	7	18	sets	set	NOUN
ejpam-3873	7	19	,	,	PUNCT
ejpam-3873	7	20	and	and	CCONJ
ejpam-3873	7	21	we	we	PRON
ejpam-3873	7	22	presented	present	VERB
ejpam-3873	7	23	a	a	DET
ejpam-3873	7	24	method	method	NOUN
ejpam-3873	7	25	of	of	ADP
ejpam-3873	7	26	constructing	construct	VERB
ejpam-3873	7	27	a	a	DET
ejpam-3873	7	28	new	new	ADJ
ejpam-3873	7	29	γ	γ	X
ejpam-3873	7	30	-	-	PUNCT
ejpam-3873	7	31	set	set	NOUN
ejpam-3873	7	32	.	.	PUNCT
ejpam-3873	8	1	moreover	moreover	ADV
ejpam-3873	8	2	,	,	PUNCT
ejpam-3873	8	3	we	we	PRON
ejpam-3873	8	4	gave	give	VERB
ejpam-3873	8	5	:	:	PUNCT
ejpam-3873	8	6	necessary	necessary	ADJ
ejpam-3873	8	7	and	and	CCONJ
ejpam-3873	8	8	sufficient	sufficient	ADJ
ejpam-3873	8	9	conditions	condition	NOUN
ejpam-3873	8	10	for	for	ADP
ejpam-3873	8	11	rings	ring	NOUN
ejpam-3873	8	12	to	to	PART
ejpam-3873	8	13	have	have	VERB
ejpam-3873	8	14	a	a	DET
ejpam-3873	8	15	unique	unique	ADJ
ejpam-3873	8	16	γ	γ	NOUN
ejpam-3873	8	17	-	-	PUNCT
ejpam-3873	8	18	set	set	NOUN
ejpam-3873	8	19	;	;	PUNCT
ejpam-3873	8	20	an	an	DET
ejpam-3873	8	21	upper	upper	ADJ
ejpam-3873	8	22	bound	bind	VERB
ejpam-3873	8	23	for	for	ADP
ejpam-3873	8	24	the	the	DET
ejpam-3873	8	25	total	total	ADJ
ejpam-3873	8	26	number	number	NOUN
ejpam-3873	8	27	of	of	ADP
ejpam-3873	8	28	minimum	minimum	ADJ
ejpam-3873	8	29	γ	γ	NOUN
ejpam-3873	8	30	-	-	PUNCT
ejpam-3873	8	31	sets	set	NOUN
ejpam-3873	8	32	in	in	ADP
ejpam-3873	8	33	a	a	DET
ejpam-3873	8	34	division	division	NOUN
ejpam-3873	8	35	ring	ring	NOUN
ejpam-3873	8	36	;	;	PUNCT
ejpam-3873	8	37	a	a	DET
ejpam-3873	8	38	lower	lower	ADV
ejpam-3873	8	39	bound	bind	VERB
ejpam-3873	8	40	for	for	ADP
ejpam-3873	8	41	the	the	DET
ejpam-3873	8	42	total	total	ADJ
ejpam-3873	8	43	number	number	NOUN
ejpam-3873	8	44	of	of	ADP
ejpam-3873	8	45	minimum	minimum	ADJ
ejpam-3873	8	46	γ	γ	NOUN
ejpam-3873	8	47	-	-	PUNCT
ejpam-3873	8	48	sets	set	NOUN
ejpam-3873	8	49	in	in	ADP
ejpam-3873	8	50	a	a	DET
ejpam-3873	8	51	division	division	NOUN
ejpam-3873	8	52	ring	ring	NOUN
ejpam-3873	8	53	;	;	PUNCT
ejpam-3873	8	54	necessary	necessary	ADJ
ejpam-3873	8	55	and	and	CCONJ
ejpam-3873	8	56	sufficient	sufficient	ADJ
ejpam-3873	8	57	conditions	condition	NOUN
ejpam-3873	8	58	for	for	ADP
ejpam-3873	8	59	t	t	PROPN
ejpam-3873	8	60	(	(	PUNCT
ejpam-3873	8	61	x	x	NOUN
ejpam-3873	8	62	)	)	PUNCT
ejpam-3873	8	63	and	and	CCONJ
ejpam-3873	8	64	t	t	X
ejpam-3873	8	65	to	to	PART
ejpam-3873	8	66	be	be	AUX
ejpam-3873	8	67	equal	equal	ADJ
ejpam-3873	8	68	;	;	PUNCT
ejpam-3873	8	69	necessary	necessary	ADJ
ejpam-3873	8	70	and	and	CCONJ
ejpam-3873	8	71	sufficient	sufficient	ADJ
ejpam-3873	8	72	conditions	condition	NOUN
ejpam-3873	8	73	for	for	ADP
ejpam-3873	8	74	a	a	DET
ejpam-3873	8	75	ring	ring	NOUN
ejpam-3873	8	76	to	to	PART
ejpam-3873	8	77	have	have	AUX
ejpam-3873	8	78	a	a	DET
ejpam-3873	8	79	trivial	trivial	ADJ
ejpam-3873	8	80	γ	γ	NOUN
ejpam-3873	8	81	-	-	PUNCT
ejpam-3873	8	82	set	set	ADJ
ejpam-3873	8	83	;	;	PUNCT
ejpam-3873	8	84	necessary	necessary	ADJ
ejpam-3873	8	85	and	and	CCONJ
ejpam-3873	8	86	sufficient	sufficient	ADJ
ejpam-3873	8	87	conditions	condition	NOUN
ejpam-3873	8	88	for	for	ADP
ejpam-3873	8	89	an	an	DET
ejpam-3873	8	90	image	image	NOUN
ejpam-3873	8	91	of	of	ADP
ejpam-3873	8	92	a	a	DET
ejpam-3873	8	93	γ	γ	NOUN
ejpam-3873	8	94	-	-	PUNCT
ejpam-3873	8	95	set	set	NOUN
ejpam-3873	8	96	to	to	PART
ejpam-3873	8	97	be	be	AUX
ejpam-3873	8	98	a	a	DET
ejpam-3873	8	99	γ	γ	NOUN
ejpam-3873	8	100	-	-	PUNCT
ejpam-3873	8	101	set	set	NOUN
ejpam-3873	8	102	also	also	ADV
ejpam-3873	8	103	;	;	PUNCT
ejpam-3873	8	104	necessary	necessary	ADJ
ejpam-3873	8	105	and	and	CCONJ
ejpam-3873	8	106	sufficient	sufficient	ADJ
ejpam-3873	8	107	conditions	condition	NOUN
ejpam-3873	8	108	for	for	ADP
ejpam-3873	8	109	a	a	DET
ejpam-3873	8	110	ring	ring	NOUN
ejpam-3873	8	111	to	to	PART
ejpam-3873	8	112	have	have	AUX
ejpam-3873	8	113	a	a	DET
ejpam-3873	8	114	trivial	trivial	ADJ
ejpam-3873	8	115	γ	γ	NOUN
ejpam-3873	8	116	-	-	PUNCT
ejpam-3873	8	117	set	set	NOUN
ejpam-3873	8	118	;	;	PUNCT
ejpam-3873	8	119	and	and	CCONJ
ejpam-3873	8	120	,	,	PUNCT
ejpam-3873	8	121	necessary	necessary	ADJ
ejpam-3873	8	122	and	and	CCONJ
ejpam-3873	8	123	sufficient	sufficient	ADJ
ejpam-3873	8	124	conditions	condition	NOUN
ejpam-3873	8	125	for	for	ADP
ejpam-3873	8	126	the	the	DET
ejpam-3873	8	127	families	family	NOUN
ejpam-3873	8	128	of	of	ADP
ejpam-3873	8	129	γ	γ	NOUN
ejpam-3873	8	130	-	-	PUNCT
ejpam-3873	8	131	sets	set	NOUN
ejpam-3873	8	132	of	of	ADP
ejpam-3873	8	133	two	two	NUM
ejpam-3873	8	134	division	division	NOUN
ejpam-3873	8	135	rings	ring	NOUN
ejpam-3873	8	136	to	to	PART
ejpam-3873	8	137	be	be	AUX
ejpam-3873	8	138	isomorphic	isomorphic	ADJ
ejpam-3873	8	139	.	.	PUNCT
ejpam-3873	9	1	2020	2020	NUM
ejpam-3873	9	2	mathematics	mathematic	NOUN
ejpam-3873	9	3	subject	subject	NOUN
ejpam-3873	9	4	classifications	classification	NOUN
ejpam-3873	9	5	:	:	PUNCT
ejpam-3873	9	6	16	16	NUM
ejpam-3873	9	7	key	key	ADJ
ejpam-3873	9	8	words	word	NOUN
ejpam-3873	9	9	and	and	CCONJ
ejpam-3873	9	10	phrases	phrase	NOUN
ejpam-3873	9	11	:	:	PUNCT
ejpam-3873	9	12	d	d	X
ejpam-3873	9	13	-	-	PUNCT
ejpam-3873	9	14	set	set	ADJ
ejpam-3873	9	15	,	,	PUNCT
ejpam-3873	9	16	γ	γ	NOUN
ejpam-3873	9	17	-	-	PUNCT
ejpam-3873	9	18	set	set	ADJ
ejpam-3873	9	19	,	,	PUNCT
ejpam-3873	9	20	minimum	minimum	ADJ
ejpam-3873	9	21	γ	γ	X
ejpam-3873	9	22	-	-	PUNCT
ejpam-3873	9	23	set	set	NOUN
ejpam-3873	9	24	,	,	PUNCT
ejpam-3873	9	25	separating	separate	VERB
ejpam-3873	9	26	γ	γ	NOUN
ejpam-3873	9	27	-	-	PUNCT
ejpam-3873	9	28	set	set	ADJ
ejpam-3873	9	29	,	,	PUNCT
ejpam-3873	9	30	ring	ring	NOUN
ejpam-3873	9	31	1	1	NUM
ejpam-3873	9	32	.	.	PUNCT
ejpam-3873	10	1	introduction	introduction	NOUN
ejpam-3873	10	2	let	let	VERB
ejpam-3873	10	3	g	g	NOUN
ejpam-3873	10	4	be	be	AUX
ejpam-3873	10	5	a	a	DET
ejpam-3873	10	6	group	group	NOUN
ejpam-3873	10	7	with	with	ADP
ejpam-3873	10	8	identity	identity	NOUN
ejpam-3873	10	9	e.	e.	PROPN
ejpam-3873	11	1	a	a	DET
ejpam-3873	11	2	subset	subset	NOUN
ejpam-3873	11	3	d	d	NOUN
ejpam-3873	11	4	of	of	ADP
ejpam-3873	11	5	g	g	PROPN
ejpam-3873	11	6	is	be	AUX
ejpam-3873	11	7	called	call	VERB
ejpam-3873	11	8	a	a	DET
ejpam-3873	11	9	d	d	NOUN
ejpam-3873	11	10	-	-	PUNCT
ejpam-3873	11	11	set	set	NOUN
ejpam-3873	11	12	of	of	ADP
ejpam-3873	11	13	g	g	PROPN
ejpam-3873	11	14	if	if	SCONJ
ejpam-3873	11	15	for	for	ADP
ejpam-3873	11	16	every	every	DET
ejpam-3873	11	17	x	x	NOUN
ejpam-3873	11	18	in	in	ADP
ejpam-3873	11	19	g\d	g\d	NOUN
ejpam-3873	11	20	,	,	PUNCT
ejpam-3873	11	21	there	there	PRON
ejpam-3873	11	22	exists	exist	VERB
ejpam-3873	11	23	y	y	PROPN
ejpam-3873	11	24	∈	∈	PROPN
ejpam-3873	12	1	d	d	ADP
ejpam-3873	12	2	such	such	ADJ
ejpam-3873	12	3	that	that	PRON
ejpam-3873	12	4	xy	xy	PROPN
ejpam-3873	12	5	=	=	PUNCT
ejpam-3873	12	6	e	e	PROPN
ejpam-3873	12	7	=	=	SYM
ejpam-3873	12	8	yx	yx	PROPN
ejpam-3873	12	9	.	.	PROPN
ejpam-3873	13	1	in	in	ADP
ejpam-3873	13	2	other	other	ADJ
ejpam-3873	13	3	words	word	NOUN
ejpam-3873	13	4	,	,	PUNCT
ejpam-3873	13	5	a	a	DET
ejpam-3873	13	6	subset	subset	NOUN
ejpam-3873	13	7	of	of	ADP
ejpam-3873	13	8	a	a	DET
ejpam-3873	13	9	group	group	NOUN
ejpam-3873	13	10	g	g	NOUN
ejpam-3873	13	11	is	be	AUX
ejpam-3873	13	12	a	a	DET
ejpam-3873	13	13	d	d	NOUN
ejpam-3873	13	14	-	-	PUNCT
ejpam-3873	13	15	set	set	ADJ
ejpam-3873	13	16	only	only	ADV
ejpam-3873	13	17	if	if	SCONJ
ejpam-3873	13	18	every	every	DET
ejpam-3873	13	19	element	element	NOUN
ejpam-3873	13	20	not	not	PART
ejpam-3873	13	21	in	in	ADP
ejpam-3873	13	22	d	d	PROPN
ejpam-3873	13	23	has	have	VERB
ejpam-3873	13	24	its	its	PRON
ejpam-3873	13	25	inverse	inverse	NOUN
ejpam-3873	13	26	in	in	ADP
ejpam-3873	13	27	d.	d.	PROPN
ejpam-3873	13	28	a	a	DET
ejpam-3873	13	29	smallest	small	ADJ
ejpam-3873	13	30	d	d	NOUN
ejpam-3873	13	31	-	-	PUNCT
ejpam-3873	13	32	set	set	NOUN
ejpam-3873	13	33	of	of	ADP
ejpam-3873	13	34	g	g	PROPN
ejpam-3873	13	35	is	be	AUX
ejpam-3873	13	36	called	call	VERB
ejpam-3873	13	37	a	a	DET
ejpam-3873	13	38	minimum	minimum	NOUN
ejpam-3873	13	39	d	d	NOUN
ejpam-3873	13	40	-	-	PUNCT
ejpam-3873	13	41	set	set	NOUN
ejpam-3873	13	42	of	of	ADP
ejpam-3873	13	43	g.	g.	PROPN
ejpam-3873	13	44	the	the	DET
ejpam-3873	13	45	number	number	NOUN
ejpam-3873	13	46	of	of	ADP
ejpam-3873	13	47	minimum	minimum	NOUN
ejpam-3873	13	48	d	d	NOUN
ejpam-3873	13	49	-	-	PUNCT
ejpam-3873	13	50	set	set	NOUN
ejpam-3873	13	51	of	of	ADP
ejpam-3873	13	52	g	g	PROPN
ejpam-3873	13	53	is	be	AUX
ejpam-3873	13	54	called	call	VERB
ejpam-3873	13	55	the	the	DET
ejpam-3873	13	56	index	index	NOUN
ejpam-3873	13	57	minimum	minimum	NOUN
ejpam-3873	13	58	.	.	PUNCT
ejpam-3873	14	1	if	if	SCONJ
ejpam-3873	14	2	g	g	PROPN
ejpam-3873	14	3	is	be	AUX
ejpam-3873	14	4	a	a	DET
ejpam-3873	14	5	finite	finite	ADJ
ejpam-3873	14	6	group	group	NOUN
ejpam-3873	14	7	and	and	CCONJ
ejpam-3873	14	8	s	s	NOUN
ejpam-3873	14	9	=	=	X
ejpam-3873	14	10	{	{	PUNCT
ejpam-3873	14	11	s	s	NOUN
ejpam-3873	14	12	∈	∈	PROPN
ejpam-3873	14	13	g	g	NOUN
ejpam-3873	14	14	:	:	PUNCT
ejpam-3873	14	15	s2	s2	NOUN
ejpam-3873	14	16	=	=	SYM
ejpam-3873	14	17	e	e	X
ejpam-3873	14	18	}	}	PUNCT
ejpam-3873	14	19	(	(	PUNCT
ejpam-3873	14	20	the	the	DET
ejpam-3873	14	21	elements	element	NOUN
ejpam-3873	14	22	of	of	ADP
ejpam-3873	14	23	s	s	PRON
ejpam-3873	14	24	will	will	AUX
ejpam-3873	14	25	be	be	AUX
ejpam-3873	14	26	called	call	VERB
ejpam-3873	14	27	involutions	involution	NOUN
ejpam-3873	14	28	)	)	PUNCT
ejpam-3873	14	29	,	,	PUNCT
ejpam-3873	14	30	then	then	ADV
ejpam-3873	14	31	the	the	DET
ejpam-3873	14	32	c	c	NOUN
ejpam-3873	14	33	-	-	PUNCT
ejpam-3873	14	34	number	number	NOUN
ejpam-3873	14	35	of	of	ADP
ejpam-3873	14	36	g	g	NOUN
ejpam-3873	14	37	is	be	AUX
ejpam-3873	14	38	given	give	VERB
ejpam-3873	14	39	by	by	ADP
ejpam-3873	14	40	|(g\s)|	|(g\s)|	PROPN
ejpam-3873	14	41	/2	/2	PROPN
ejpam-3873	14	42	.	.	PUNCT
ejpam-3873	15	1	let	let	VERB
ejpam-3873	15	2	r	r	PRON
ejpam-3873	15	3	be	be	AUX
ejpam-3873	15	4	a	a	DET
ejpam-3873	15	5	ring	ring	NOUN
ejpam-3873	15	6	with	with	ADP
ejpam-3873	15	7	identity	identity	NOUN
ejpam-3873	15	8	1r	1r	NUM
ejpam-3873	15	9	.	.	PUNCT
ejpam-3873	16	1	a	a	DET
ejpam-3873	16	2	subset	subset	NOUN
ejpam-3873	16	3	j	j	PROPN
ejpam-3873	16	4	of	of	ADP
ejpam-3873	16	5	r	r	NOUN
ejpam-3873	16	6	is	be	AUX
ejpam-3873	16	7	called	call	VERB
ejpam-3873	16	8	a	a	DET
ejpam-3873	16	9	γ	γ	NOUN
ejpam-3873	16	10	-	-	PUNCT
ejpam-3873	16	11	set	set	NOUN
ejpam-3873	16	12	of	of	ADP
ejpam-3873	16	13	r	r	NOUN
ejpam-3873	16	14	if	if	SCONJ
ejpam-3873	16	15	for	for	SCONJ
ejpam-3873	16	16	every	every	DET
ejpam-3873	16	17	a	a	DET
ejpam-3873	16	18	∈	∈	NOUN
ejpam-3873	16	19	r\j	r\j	ADV
ejpam-3873	16	20	,	,	PUNCT
ejpam-3873	16	21	there	there	PRON
ejpam-3873	16	22	exist	exist	VERB
ejpam-3873	16	23	b	b	NOUN
ejpam-3873	16	24	,	,	PUNCT
ejpam-3873	16	25	c	c	PROPN
ejpam-3873	16	26	∈	∈	PROPN
ejpam-3873	16	27	j	j	PROPN
ejpam-3873	16	28	such	such	ADJ
ejpam-3873	16	29	that	that	SCONJ
ejpam-3873	16	30	a+	a+	PRON
ejpam-3873	16	31	b	b	X
ejpam-3873	16	32	=	=	SYM
ejpam-3873	16	33	0	0	PROPN
ejpam-3873	16	34	and	and	CCONJ
ejpam-3873	16	35	ac	ac	ADJ
ejpam-3873	16	36	=	=	NOUN
ejpam-3873	16	37	1r	1r	NUM
ejpam-3873	16	38	=	=	SYM
ejpam-3873	16	39	ca	ca	NOUN
ejpam-3873	16	40	.	.	PUNCT
ejpam-3873	17	1	for	for	ADP
ejpam-3873	17	2	example	example	NOUN
ejpam-3873	17	3	,	,	PUNCT
ejpam-3873	17	4	consider	consider	VERB
ejpam-3873	17	5	the	the	DET
ejpam-3873	17	6	field	field	NOUN
ejpam-3873	17	7	z5	z5	PROPN
ejpam-3873	17	8	.	.	PUNCT
ejpam-3873	18	1	then	then	ADV
ejpam-3873	18	2	the	the	DET
ejpam-3873	18	3	γ	γ	NOUN
ejpam-3873	18	4	-	-	PUNCT
ejpam-3873	18	5	sets	set	NOUN
ejpam-3873	18	6	of	of	ADP
ejpam-3873	18	7	z5	z5	NOUN
ejpam-3873	18	8	are	be	AUX
ejpam-3873	18	9	{	{	PUNCT
ejpam-3873	18	10	0	0	NUM
ejpam-3873	18	11	,	,	PUNCT
ejpam-3873	18	12	1	1	NUM
ejpam-3873	18	13	,	,	PUNCT
ejpam-3873	18	14	4	4	NUM
ejpam-3873	18	15	,	,	PUNCT
ejpam-3873	18	16	2	2	NUM
ejpam-3873	18	17	}	}	PUNCT
ejpam-3873	18	18	,	,	PUNCT
ejpam-3873	18	19	{	{	PUNCT
ejpam-3873	18	20	0	0	NUM
ejpam-3873	18	21	,	,	PUNCT
ejpam-3873	18	22	1	1	NUM
ejpam-3873	18	23	,	,	PUNCT
ejpam-3873	18	24	4	4	NUM
ejpam-3873	18	25	,	,	PUNCT
ejpam-3873	18	26	3	3	NUM
ejpam-3873	18	27	}	}	PUNCT
ejpam-3873	18	28	,	,	PUNCT
ejpam-3873	18	29	and	and	CCONJ
ejpam-3873	18	30	z5	z5	PROPN
ejpam-3873	18	31	.	.	PUNCT
ejpam-3873	19	1	a	a	DET
ejpam-3873	19	2	γ	γ	NOUN
ejpam-3873	19	3	-	-	PUNCT
ejpam-3873	19	4	set	set	NOUN
ejpam-3873	19	5	of	of	ADP
ejpam-3873	19	6	a	a	DET
ejpam-3873	19	7	finite	finite	ADJ
ejpam-3873	19	8	∗corresponding	∗corresponde	VERB
ejpam-3873	19	9	author	author	NOUN
ejpam-3873	19	10	.	.	PUNCT
ejpam-3873	20	1	doi	doi	NOUN
ejpam-3873	20	2	:	:	PUNCT
ejpam-3873	20	3	https://doi.org/10.29020/nybg.ejpam.v14i1.3873	https://doi.org/10.29020/nybg.ejpam.v14i1.3873	ADJ
ejpam-3873	20	4	email	email	NOUN
ejpam-3873	20	5	addresses	address	VERB
ejpam-3873	20	6	:	:	PUNCT
ejpam-3873	20	7	evajenny@yahoo.com	evajenny@yahoo.com	X
ejpam-3873	20	8	(	(	PUNCT
ejpam-3873	20	9	e.j	e.j	PROPN
ejpam-3873	20	10	.	.	PROPN
ejpam-3873	20	11	sigasig	sigasig	PROPN
ejpam-3873	20	12	)	)	PUNCT
ejpam-3873	20	13	,	,	PUNCT
ejpam-3873	20	14	crisrose	crisrose	VERB
ejpam-3873	20	15	18@yahoo.com	18@yahoo.com	NUM
ejpam-3873	20	16	(	(	PUNCT
ejpam-3873	20	17	c.j	c.j	PROPN
ejpam-3873	20	18	.	.	NOUN
ejpam-3873	20	19	rosero	rosero	PROPN
ejpam-3873	20	20	)	)	PUNCT
ejpam-3873	20	21	,	,	PUNCT
ejpam-3873	20	22	michaelpbaldadojr@yahoo.com	michaelpbaldadojr@yahoo.com	X
ejpam-3873	21	1	(	(	PUNCT
ejpam-3873	21	2	m.	m.	PROPN
ejpam-3873	21	3	baldado	baldado	PROPN
ejpam-3873	21	4	jr	jr	PROPN
ejpam-3873	21	5	.	.	PUNCT
ejpam-3873	21	6	)	)	PUNCT
ejpam-3873	22	1	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3873	23	1	314	314	NUM
ejpam-3873	23	2	c	c	X
ejpam-3873	23	3	©	©	PROPN
ejpam-3873	23	4	2021	2021	NUM
ejpam-3873	23	5	ejpam	ejpam	VERB
ejpam-3873	23	6	all	all	DET
ejpam-3873	23	7	rights	right	NOUN
ejpam-3873	23	8	reserved	reserve	VERB
ejpam-3873	23	9	.	.	PUNCT
ejpam-3873	24	1	e.j	e.j	PROPN
ejpam-3873	24	2	.	.	PROPN
ejpam-3873	24	3	sigasig	sigasig	PROPN
ejpam-3873	24	4	,	,	PUNCT
ejpam-3873	24	5	c.j	c.j	PROPN
ejpam-3873	24	6	.	.	PROPN
ejpam-3873	24	7	rosero	rosero	PROPN
ejpam-3873	24	8	,	,	PUNCT
ejpam-3873	24	9	m.	m.	NOUN
ejpam-3873	24	10	baldado	baldado	PROPN
ejpam-3873	24	11	jr	jr	PROPN
ejpam-3873	24	12	.	.	PROPN
ejpam-3873	24	13	/	/	SYM
ejpam-3873	24	14	eur	eur	PROPN
ejpam-3873	24	15	.	.	PUNCT
ejpam-3873	25	1	j.	j.	PROPN
ejpam-3873	25	2	pure	pure	PROPN
ejpam-3873	25	3	appl	appl	PROPN
ejpam-3873	25	4	.	.	PROPN
ejpam-3873	25	5	math	math	PROPN
ejpam-3873	25	6	,	,	PUNCT
ejpam-3873	25	7	14	14	NUM
ejpam-3873	25	8	(	(	PUNCT
ejpam-3873	25	9	1	1	NUM
ejpam-3873	25	10	)	)	PUNCT
ejpam-3873	25	11	(	(	PUNCT
ejpam-3873	25	12	2021	2021	NUM
ejpam-3873	25	13	)	)	PUNCT
ejpam-3873	25	14	,	,	PUNCT
ejpam-3873	25	15	314	314	NUM
ejpam-3873	25	16	-	-	SYM
ejpam-3873	25	17	326	326	NUM
ejpam-3873	25	18	315	315	NUM
ejpam-3873	25	19	ring	ring	NOUN
ejpam-3873	25	20	having	have	VERB
ejpam-3873	25	21	minimum	minimum	ADJ
ejpam-3873	25	22	cardinality	cardinality	NOUN
ejpam-3873	25	23	is	be	AUX
ejpam-3873	25	24	called	call	VERB
ejpam-3873	25	25	a	a	DET
ejpam-3873	25	26	minimum	minimum	ADJ
ejpam-3873	25	27	γ	γ	X
ejpam-3873	25	28	-	-	PUNCT
ejpam-3873	25	29	set	set	NOUN
ejpam-3873	25	30	.	.	PUNCT
ejpam-3873	26	1	for	for	ADP
ejpam-3873	26	2	example	example	NOUN
ejpam-3873	26	3	,	,	PUNCT
ejpam-3873	26	4	{	{	PUNCT
ejpam-3873	26	5	0	0	NUM
ejpam-3873	26	6	,	,	PUNCT
ejpam-3873	26	7	1	1	NUM
ejpam-3873	26	8	,	,	PUNCT
ejpam-3873	26	9	4	4	NUM
ejpam-3873	26	10	,	,	PUNCT
ejpam-3873	26	11	2	2	NUM
ejpam-3873	26	12	}	}	PUNCT
ejpam-3873	26	13	and	and	CCONJ
ejpam-3873	26	14	{	{	PUNCT
ejpam-3873	26	15	0	0	NUM
ejpam-3873	26	16	,	,	PUNCT
ejpam-3873	26	17	1	1	NUM
ejpam-3873	26	18	,	,	PUNCT
ejpam-3873	26	19	4	4	NUM
ejpam-3873	26	20	,	,	PUNCT
ejpam-3873	26	21	3	3	NUM
ejpam-3873	26	22	}	}	PUNCT
ejpam-3873	26	23	are	be	AUX
ejpam-3873	26	24	minimum	minimum	ADJ
ejpam-3873	26	25	γ	γ	NOUN
ejpam-3873	26	26	-	-	PUNCT
ejpam-3873	26	27	sets	set	NOUN
ejpam-3873	26	28	z5	z5	NOUN
ejpam-3873	26	29	.	.	PUNCT
ejpam-3873	27	1	here	here	ADV
ejpam-3873	27	2	after	after	SCONJ
ejpam-3873	27	3	please	please	INTJ
ejpam-3873	27	4	refer	refer	VERB
ejpam-3873	27	5	to	to	ADP
ejpam-3873	27	6	[	[	X
ejpam-3873	27	7	4	4	NUM
ejpam-3873	27	8	]	]	PUNCT
ejpam-3873	27	9	,	,	PUNCT
ejpam-3873	27	10	[	[	X
ejpam-3873	27	11	5	5	NUM
ejpam-3873	27	12	]	]	PUNCT
ejpam-3873	27	13	,	,	PUNCT
ejpam-3873	27	14	[	[	X
ejpam-3873	27	15	6	6	NUM
ejpam-3873	27	16	]	]	PUNCT
ejpam-3873	27	17	,	,	PUNCT
ejpam-3873	27	18	[	[	X
ejpam-3873	27	19	7	7	NUM
ejpam-3873	27	20	]	]	PUNCT
ejpam-3873	27	21	,	,	PUNCT
ejpam-3873	27	22	[	[	X
ejpam-3873	27	23	8	8	NUM
ejpam-3873	27	24	]	]	PUNCT
ejpam-3873	27	25	,	,	PUNCT
ejpam-3873	27	26	[	[	X
ejpam-3873	27	27	9	9	NUM
ejpam-3873	27	28	]	]	PUNCT
ejpam-3873	27	29	,	,	PUNCT
ejpam-3873	27	30	[	[	X
ejpam-3873	27	31	10	10	NUM
ejpam-3873	27	32	]	]	PUNCT
ejpam-3873	27	33	for	for	ADP
ejpam-3873	27	34	the	the	DET
ejpam-3873	27	35	other	other	ADJ
ejpam-3873	27	36	concepts	concept	NOUN
ejpam-3873	27	37	.	.	PUNCT
ejpam-3873	28	1	motivated	motivate	VERB
ejpam-3873	28	2	by	by	ADP
ejpam-3873	28	3	the	the	DET
ejpam-3873	28	4	concept	concept	NOUN
ejpam-3873	28	5	dominating	dominating	NOUN
ejpam-3873	28	6	sets	set	NOUN
ejpam-3873	28	7	in	in	ADP
ejpam-3873	28	8	graphs	graph	NOUN
ejpam-3873	28	9	,	,	PUNCT
ejpam-3873	28	10	buloron	buloron	PROPN
ejpam-3873	28	11	et	et	PROPN
ejpam-3873	28	12	al	al	PROPN
ejpam-3873	28	13	.	.	PUNCT
ejpam-3873	29	1	[	[	X
ejpam-3873	29	2	3	3	X
ejpam-3873	29	3	]	]	PUNCT
ejpam-3873	29	4	introduced	introduce	VERB
ejpam-3873	29	5	the	the	DET
ejpam-3873	29	6	concept	concept	NOUN
ejpam-3873	29	7	d	d	NOUN
ejpam-3873	29	8	-	-	PUNCT
ejpam-3873	29	9	set	set	VERB
ejpam-3873	29	10	in	in	ADP
ejpam-3873	29	11	a	a	DET
ejpam-3873	29	12	group	group	NOUN
ejpam-3873	29	13	.	.	PUNCT
ejpam-3873	30	1	the	the	DET
ejpam-3873	30	2	concept	concept	NOUN
ejpam-3873	30	3	d	d	NOUN
ejpam-3873	30	4	-	-	PUNCT
ejpam-3873	30	5	set	set	NOUN
ejpam-3873	30	6	uses	use	VERB
ejpam-3873	30	7	the	the	DET
ejpam-3873	30	8	idea	idea	NOUN
ejpam-3873	30	9	of	of	ADP
ejpam-3873	30	10	dominating	dominating	NOUN
ejpam-3873	30	11	sets	set	NOUN
ejpam-3873	30	12	in	in	ADP
ejpam-3873	30	13	some	some	DET
ejpam-3873	30	14	sense	sense	NOUN
ejpam-3873	30	15	.	.	PUNCT
ejpam-3873	31	1	for	for	ADP
ejpam-3873	31	2	example	example	NOUN
ejpam-3873	31	3	,	,	PUNCT
ejpam-3873	31	4	a	a	DET
ejpam-3873	31	5	d	d	NOUN
ejpam-3873	31	6	-	-	PUNCT
ejpam-3873	31	7	set	set	ADJ
ejpam-3873	31	8	e	e	NOUN
ejpam-3873	31	9	in	in	ADP
ejpam-3873	31	10	a	a	DET
ejpam-3873	31	11	group	group	NOUN
ejpam-3873	31	12	requires	require	VERB
ejpam-3873	31	13	that	that	SCONJ
ejpam-3873	31	14	every	every	DET
ejpam-3873	31	15	element	element	NOUN
ejpam-3873	31	16	not	not	PART
ejpam-3873	31	17	in	in	ADP
ejpam-3873	31	18	e	e	NOUN
ejpam-3873	31	19	must	must	AUX
ejpam-3873	31	20	have	have	VERB
ejpam-3873	31	21	its	its	PRON
ejpam-3873	31	22	inverse	inverse	NOUN
ejpam-3873	31	23	in	in	ADP
ejpam-3873	31	24	e	e	NOUN
ejpam-3873	31	25	in	in	ADP
ejpam-3873	31	26	the	the	DET
ejpam-3873	31	27	same	same	ADJ
ejpam-3873	31	28	way	way	NOUN
ejpam-3873	31	29	that	that	PRON
ejpam-3873	31	30	a	a	DET
ejpam-3873	31	31	dominating	dominating	NOUN
ejpam-3873	31	32	set	set	NOUN
ejpam-3873	31	33	d	d	NOUN
ejpam-3873	31	34	in	in	ADP
ejpam-3873	31	35	a	a	DET
ejpam-3873	31	36	graph	graph	NOUN
ejpam-3873	31	37	requires	require	VERB
ejpam-3873	31	38	every	every	DET
ejpam-3873	31	39	element	element	NOUN
ejpam-3873	31	40	not	not	PART
ejpam-3873	31	41	in	in	ADP
ejpam-3873	31	42	d	d	PROPN
ejpam-3873	31	43	must	must	AUX
ejpam-3873	31	44	be	be	AUX
ejpam-3873	31	45	a	a	DET
ejpam-3873	31	46	neighbor	neighbor	NOUN
ejpam-3873	31	47	of	of	ADP
ejpam-3873	31	48	some	some	DET
ejpam-3873	31	49	element	element	NOUN
ejpam-3873	31	50	in	in	ADP
ejpam-3873	31	51	d.	d.	PROPN
ejpam-3873	31	52	buloron	buloron	PROPN
ejpam-3873	31	53	et	et	PROPN
ejpam-3873	31	54	al	al	PROPN
ejpam-3873	31	55	.	.	PUNCT
ejpam-3873	32	1	[	[	X
ejpam-3873	32	2	3	3	X
ejpam-3873	32	3	]	]	PUNCT
ejpam-3873	32	4	gave	give	VERB
ejpam-3873	32	5	some	some	DET
ejpam-3873	32	6	fundamental	fundamental	ADJ
ejpam-3873	32	7	properties	property	NOUN
ejpam-3873	32	8	of	of	ADP
ejpam-3873	32	9	d	d	NOUN
ejpam-3873	32	10	-	-	PUNCT
ejpam-3873	32	11	sets	set	NOUN
ejpam-3873	32	12	and	and	CCONJ
ejpam-3873	32	13	some	some	DET
ejpam-3873	32	14	characterizations	characterization	NOUN
ejpam-3873	32	15	.	.	PUNCT
ejpam-3873	33	1	ontolan	ontolan	NOUN
ejpam-3873	33	2	et	et	PROPN
ejpam-3873	33	3	al	al	PROPN
ejpam-3873	33	4	.	.	PUNCT
ejpam-3873	34	1	[	[	X
ejpam-3873	34	2	12	12	NUM
ejpam-3873	34	3	]	]	PUNCT
ejpam-3873	34	4	gave	give	VERB
ejpam-3873	34	5	the	the	DET
ejpam-3873	34	6	number	number	NOUN
ejpam-3873	34	7	of	of	ADP
ejpam-3873	34	8	minimum	minimum	ADJ
ejpam-3873	34	9	d	d	NOUN
ejpam-3873	34	10	-	-	PUNCT
ejpam-3873	34	11	sets	set	NOUN
ejpam-3873	34	12	in	in	ADP
ejpam-3873	34	13	a	a	DET
ejpam-3873	34	14	group	group	NOUN
ejpam-3873	34	15	.	.	PUNCT
ejpam-3873	35	1	corcino	corcino	NOUN
ejpam-3873	35	2	et	et	PROPN
ejpam-3873	35	3	al	al	PROPN
ejpam-3873	35	4	.	.	PUNCT
ejpam-3873	36	1	[	[	X
ejpam-3873	36	2	2	2	X
ejpam-3873	36	3	]	]	PUNCT
ejpam-3873	36	4	presented	present	VERB
ejpam-3873	36	5	some	some	DET
ejpam-3873	36	6	isomorphism	isomorphism	NOUN
ejpam-3873	36	7	results	result	NOUN
ejpam-3873	36	8	for	for	ADP
ejpam-3873	36	9	some	some	DET
ejpam-3873	36	10	families	family	NOUN
ejpam-3873	36	11	of	of	ADP
ejpam-3873	36	12	d	d	NOUN
ejpam-3873	36	13	-	-	PUNCT
ejpam-3873	36	14	sets	set	NOUN
ejpam-3873	36	15	.	.	PUNCT
ejpam-3873	37	1	rosero	rosero	VERB
ejpam-3873	37	2	and	and	CCONJ
ejpam-3873	37	3	baldado	baldado	NOUN
ejpam-3873	38	1	[	[	X
ejpam-3873	38	2	1	1	X
ejpam-3873	38	3	]	]	PUNCT
ejpam-3873	38	4	continued	continue	VERB
ejpam-3873	38	5	the	the	DET
ejpam-3873	38	6	study	study	NOUN
ejpam-3873	38	7	of	of	ADP
ejpam-3873	38	8	d	d	NOUN
ejpam-3873	38	9	-	-	PUNCT
ejpam-3873	38	10	sets	set	NOUN
ejpam-3873	38	11	by	by	ADP
ejpam-3873	38	12	investigating	investigate	VERB
ejpam-3873	38	13	the	the	DET
ejpam-3873	38	14	d	d	NOUN
ejpam-3873	38	15	-	-	PUNCT
ejpam-3873	38	16	sets	set	NOUN
ejpam-3873	38	17	that	that	PRON
ejpam-3873	38	18	are	be	AUX
ejpam-3873	38	19	generated	generate	VERB
ejpam-3873	38	20	by	by	ADP
ejpam-3873	38	21	a	a	DET
ejpam-3873	38	22	set	set	NOUN
ejpam-3873	38	23	.	.	PUNCT
ejpam-3873	39	1	moreover	moreover	ADV
ejpam-3873	39	2	,	,	PUNCT
ejpam-3873	39	3	they	they	PRON
ejpam-3873	39	4	introduced	introduce	VERB
ejpam-3873	39	5	and	and	CCONJ
ejpam-3873	39	6	investigated	investigate	VERB
ejpam-3873	39	7	a	a	DET
ejpam-3873	39	8	parallel	parallel	ADJ
ejpam-3873	39	9	concept	concept	NOUN
ejpam-3873	39	10	for	for	ADP
ejpam-3873	39	11	rings	ring	NOUN
ejpam-3873	39	12	,	,	PUNCT
ejpam-3873	39	13	called	call	VERB
ejpam-3873	39	14	γ	γ	NOUN
ejpam-3873	39	15	-	-	PUNCT
ejpam-3873	39	16	sets	set	NOUN
ejpam-3873	39	17	[	[	X
ejpam-3873	39	18	11	11	NUM
ejpam-3873	39	19	]	]	PUNCT
ejpam-3873	39	20	.	.	PUNCT
ejpam-3873	40	1	in	in	ADP
ejpam-3873	40	2	this	this	DET
ejpam-3873	40	3	study	study	NOUN
ejpam-3873	40	4	,	,	PUNCT
ejpam-3873	40	5	we	we	PRON
ejpam-3873	40	6	continued	continue	VERB
ejpam-3873	40	7	the	the	DET
ejpam-3873	40	8	investigation	investigation	NOUN
ejpam-3873	40	9	of	of	ADP
ejpam-3873	40	10	γ	γ	NOUN
ejpam-3873	40	11	-	-	PUNCT
ejpam-3873	40	12	sets	set	NOUN
ejpam-3873	40	13	.	.	PUNCT
ejpam-3873	41	1	2	2	X
ejpam-3873	41	2	.	.	X
ejpam-3873	41	3	preliminary	preliminary	ADJ
ejpam-3873	41	4	results	result	NOUN
ejpam-3873	41	5	this	this	DET
ejpam-3873	41	6	section	section	NOUN
ejpam-3873	41	7	presents	present	VERB
ejpam-3873	41	8	some	some	DET
ejpam-3873	41	9	elementary	elementary	ADJ
ejpam-3873	41	10	properties	property	NOUN
ejpam-3873	41	11	of	of	ADP
ejpam-3873	41	12	a	a	DET
ejpam-3873	41	13	γ	γ	NOUN
ejpam-3873	41	14	-	-	PUNCT
ejpam-3873	41	15	set	set	NOUN
ejpam-3873	41	16	.	.	PUNCT
ejpam-3873	42	1	we	we	PRON
ejpam-3873	42	2	denote	denote	VERB
ejpam-3873	42	3	by	by	ADP
ejpam-3873	42	4	tr	tr	PRON
ejpam-3873	42	5	the	the	DET
ejpam-3873	42	6	set	set	NOUN
ejpam-3873	42	7	of	of	ADP
ejpam-3873	42	8	all	all	DET
ejpam-3873	42	9	γ	γ	NOUN
ejpam-3873	42	10	-	-	NOUN
ejpam-3873	42	11	sets	set	NOUN
ejpam-3873	42	12	of	of	ADP
ejpam-3873	42	13	r.	r.	PROPN
ejpam-3873	42	14	note	note	VERB
ejpam-3873	42	15	that	that	SCONJ
ejpam-3873	42	16	tr	tr	PUNCT
ejpam-3873	42	17	6=	6=	NOUN
ejpam-3873	42	18	∅	∅	NOUN
ejpam-3873	42	19	since	since	SCONJ
ejpam-3873	42	20	r	r	NOUN
ejpam-3873	42	21	is	be	AUX
ejpam-3873	42	22	a	a	DET
ejpam-3873	42	23	γ	γ	NOUN
ejpam-3873	42	24	-	-	PUNCT
ejpam-3873	42	25	set	set	NOUN
ejpam-3873	42	26	.	.	PUNCT
ejpam-3873	43	1	the	the	DET
ejpam-3873	43	2	next	next	ADJ
ejpam-3873	43	3	theorem	theorem	NOUN
ejpam-3873	43	4	,	,	PUNCT
ejpam-3873	43	5	theorem	theorem	ADJ
ejpam-3873	43	6	1	1	NUM
ejpam-3873	43	7	,	,	PUNCT
ejpam-3873	43	8	is	be	AUX
ejpam-3873	43	9	taken	take	VERB
ejpam-3873	43	10	from	from	ADP
ejpam-3873	43	11	[	[	X
ejpam-3873	43	12	3	3	NUM
ejpam-3873	43	13	]	]	PUNCT
ejpam-3873	43	14	.	.	PUNCT
ejpam-3873	44	1	it	it	PRON
ejpam-3873	44	2	shows	show	VERB
ejpam-3873	44	3	that	that	SCONJ
ejpam-3873	44	4	the	the	DET
ejpam-3873	44	5	set	set	NOUN
ejpam-3873	44	6	of	of	ADP
ejpam-3873	44	7	all	all	DET
ejpam-3873	44	8	γ	γ	NOUN
ejpam-3873	44	9	-	-	NOUN
ejpam-3873	44	10	sets	set	NOUN
ejpam-3873	44	11	in	in	ADP
ejpam-3873	44	12	a	a	DET
ejpam-3873	44	13	ring	ring	NOUN
ejpam-3873	44	14	is	be	AUX
ejpam-3873	44	15	a	a	DET
ejpam-3873	44	16	semi	semi	NOUN
ejpam-3873	44	17	-	-	NOUN
ejpam-3873	44	18	group	group	NOUN
ejpam-3873	44	19	under	under	ADP
ejpam-3873	44	20	the	the	DET
ejpam-3873	44	21	set	set	NOUN
ejpam-3873	44	22	operation	operation	NOUN
ejpam-3873	44	23	union	union	NOUN
ejpam-3873	44	24	,	,	PUNCT
ejpam-3873	44	25	and	and	CCONJ
ejpam-3873	44	26	the	the	DET
ejpam-3873	44	27	set	set	NOUN
ejpam-3873	44	28	tc	tc	NOUN
ejpam-3873	44	29	r	r	NOUN
ejpam-3873	44	30	=	=	PUNCT
ejpam-3873	44	31	{	{	PUNCT
ejpam-3873	44	32	jc	jc	PROPN
ejpam-3873	44	33	:	:	PUNCT
ejpam-3873	44	34	j	j	PROPN
ejpam-3873	44	35	is	be	AUX
ejpam-3873	44	36	a	a	DET
ejpam-3873	44	37	γ	γ	NOUN
ejpam-3873	44	38	-	-	PUNCT
ejpam-3873	44	39	set	set	ADJ
ejpam-3873	44	40	}	}	PUNCT
ejpam-3873	44	41	is	be	AUX
ejpam-3873	44	42	a	a	DET
ejpam-3873	44	43	semi	semi	NOUN
ejpam-3873	44	44	-	-	NOUN
ejpam-3873	44	45	group	group	NOUN
ejpam-3873	44	46	under	under	ADP
ejpam-3873	44	47	the	the	DET
ejpam-3873	44	48	set	set	ADJ
ejpam-3873	44	49	operation	operation	NOUN
ejpam-3873	44	50	intersection	intersection	NOUN
ejpam-3873	44	51	.	.	PUNCT
ejpam-3873	45	1	theorem	theorem	NOUN
ejpam-3873	45	2	1	1	NUM
ejpam-3873	45	3	.	.	PUNCT
ejpam-3873	46	1	[	[	X
ejpam-3873	46	2	3	3	X
ejpam-3873	46	3	]	]	PUNCT
ejpam-3873	46	4	let	let	VERB
ejpam-3873	46	5	r	r	PRON
ejpam-3873	46	6	be	be	AUX
ejpam-3873	46	7	a	a	DET
ejpam-3873	46	8	ring	ring	NOUN
ejpam-3873	46	9	with	with	ADP
ejpam-3873	46	10	identity	identity	NOUN
ejpam-3873	46	11	1r	1r	NOUN
ejpam-3873	46	12	.	.	PUNCT
ejpam-3873	47	1	let	let	VERB
ejpam-3873	47	2	tr	tr	PRON
ejpam-3873	47	3	be	be	AUX
ejpam-3873	47	4	the	the	DET
ejpam-3873	47	5	set	set	NOUN
ejpam-3873	47	6	of	of	ADP
ejpam-3873	47	7	all	all	DET
ejpam-3873	47	8	γ	γ	NOUN
ejpam-3873	47	9	-	-	NOUN
ejpam-3873	47	10	sets	set	NOUN
ejpam-3873	47	11	of	of	ADP
ejpam-3873	47	12	r	r	NOUN
ejpam-3873	47	13	and	and	CCONJ
ejpam-3873	47	14	tc	tc	NOUN
ejpam-3873	47	15	r	r	NOUN
ejpam-3873	47	16	=	=	PUNCT
ejpam-3873	47	17	{	{	PUNCT
ejpam-3873	47	18	jc	jc	PROPN
ejpam-3873	47	19	:	:	PUNCT
ejpam-3873	47	20	j	j	PROPN
ejpam-3873	47	21	is	be	AUX
ejpam-3873	47	22	a	a	DET
ejpam-3873	47	23	γ	γ	NOUN
ejpam-3873	47	24	-	-	PUNCT
ejpam-3873	47	25	set	set	NOUN
ejpam-3873	47	26	}	}	PUNCT
ejpam-3873	47	27	.	.	PUNCT
ejpam-3873	48	1	then	then	ADV
ejpam-3873	48	2	a.	a.	PROPN
ejpam-3873	48	3	)	)	PUNCT
ejpam-3873	49	1	the	the	DET
ejpam-3873	49	2	set	set	NOUN
ejpam-3873	49	3	tr	tr	VERB
ejpam-3873	49	4	is	be	AUX
ejpam-3873	49	5	a	a	DET
ejpam-3873	49	6	semi	semi	NOUN
ejpam-3873	49	7	-	-	NOUN
ejpam-3873	49	8	group	group	NOUN
ejpam-3873	49	9	under	under	ADP
ejpam-3873	49	10	the	the	DET
ejpam-3873	49	11	set	set	NOUN
ejpam-3873	49	12	operation	operation	NOUN
ejpam-3873	49	13	union	union	NOUN
ejpam-3873	49	14	;	;	PUNCT
ejpam-3873	49	15	b.	b.	PROPN
ejpam-3873	49	16	)	)	PUNCT
ejpam-3873	50	1	the	the	DET
ejpam-3873	50	2	set	set	NOUN
ejpam-3873	50	3	tc	tc	NOUN
ejpam-3873	50	4	r	r	NOUN
ejpam-3873	50	5	is	be	AUX
ejpam-3873	50	6	a	a	DET
ejpam-3873	50	7	semi	semi	NOUN
ejpam-3873	50	8	-	-	NOUN
ejpam-3873	50	9	group	group	NOUN
ejpam-3873	50	10	under	under	ADP
ejpam-3873	50	11	the	the	DET
ejpam-3873	50	12	set	set	ADJ
ejpam-3873	50	13	operation	operation	NOUN
ejpam-3873	50	14	intersection	intersection	NOUN
ejpam-3873	50	15	.	.	PUNCT
ejpam-3873	51	1	remark	remark	NOUN
ejpam-3873	51	2	1	1	NUM
ejpam-3873	51	3	(	(	PUNCT
ejpam-3873	51	4	b	b	NOUN
ejpam-3873	51	5	)	)	PUNCT
ejpam-3873	51	6	is	be	AUX
ejpam-3873	51	7	found	find	VERB
ejpam-3873	51	8	in	in	ADP
ejpam-3873	51	9	[	[	X
ejpam-3873	51	10	4	4	NUM
ejpam-3873	51	11	]	]	PUNCT
ejpam-3873	51	12	,	,	PUNCT
ejpam-3873	51	13	while	while	SCONJ
ejpam-3873	51	14	(	(	PUNCT
ejpam-3873	51	15	c	c	X
ejpam-3873	51	16	)	)	PUNCT
ejpam-3873	51	17	is	be	AUX
ejpam-3873	51	18	an	an	DET
ejpam-3873	51	19	exercise	exercise	NOUN
ejpam-3873	51	20	in	in	ADP
ejpam-3873	51	21	[	[	X
ejpam-3873	51	22	6	6	NUM
ejpam-3873	51	23	]	]	PUNCT
ejpam-3873	51	24	(	(	PUNCT
ejpam-3873	51	25	prob	prob	NOUN
ejpam-3873	51	26	24e	24e	PROPN
ejpam-3873	51	27	,	,	PUNCT
ejpam-3873	51	28	chapter	chapter	NOUN
ejpam-3873	51	29	5.1	5.1	NUM
ejpam-3873	51	30	)	)	PUNCT
ejpam-3873	51	31	.	.	PUNCT
ejpam-3873	52	1	remark	remark	NOUN
ejpam-3873	52	2	1	1	NUM
ejpam-3873	52	3	(	(	PUNCT
ejpam-3873	52	4	d	d	NOUN
ejpam-3873	52	5	)	)	PUNCT
ejpam-3873	52	6	is	be	AUX
ejpam-3873	52	7	a	a	DET
ejpam-3873	52	8	contrapositive	contrapositive	NOUN
ejpam-3873	52	9	of	of	ADP
ejpam-3873	52	10	(	(	PUNCT
ejpam-3873	52	11	c	c	NOUN
ejpam-3873	52	12	)	)	PUNCT
ejpam-3873	52	13	.	.	PUNCT
ejpam-3873	53	1	remark	remark	PROPN
ejpam-3873	53	2	1	1	NUM
ejpam-3873	53	3	.	.	PUNCT
ejpam-3873	54	1	let	let	VERB
ejpam-3873	54	2	r	r	PRON
ejpam-3873	54	3	be	be	AUX
ejpam-3873	54	4	a	a	DET
ejpam-3873	54	5	ring	ring	NOUN
ejpam-3873	54	6	with	with	ADP
ejpam-3873	54	7	identity	identity	NOUN
ejpam-3873	54	8	1r	1r	NOUN
ejpam-3873	54	9	and	and	CCONJ
ejpam-3873	54	10	a	a	DET
ejpam-3873	54	11	∈	∈	PROPN
ejpam-3873	54	12	r.	r.	PROPN
ejpam-3873	54	13	a.	a.	PROPN
ejpam-3873	54	14	)	)	PUNCT
ejpam-3873	54	15	if	if	SCONJ
ejpam-3873	54	16	a	a	PRON
ejpam-3873	54	17	is	be	AUX
ejpam-3873	54	18	a	a	DET
ejpam-3873	54	19	unit	unit	NOUN
ejpam-3873	54	20	,	,	PUNCT
ejpam-3873	54	21	then	then	ADV
ejpam-3873	54	22	−(a−1	−(a−1	PROPN
ejpam-3873	54	23	)	)	PUNCT
ejpam-3873	55	1	=	=	PRON
ejpam-3873	55	2	(	(	PUNCT
ejpam-3873	55	3	−a)−1	−a)−1	NOUN
ejpam-3873	55	4	.	.	PUNCT
ejpam-3873	55	5	b.	b.	PROPN
ejpam-3873	55	6	)	)	PUNCT
ejpam-3873	56	1	if	if	SCONJ
ejpam-3873	56	2	a	a	PRON
ejpam-3873	56	3	is	be	AUX
ejpam-3873	56	4	a	a	DET
ejpam-3873	56	5	unit	unit	NOUN
ejpam-3873	56	6	,	,	PUNCT
ejpam-3873	56	7	then	then	ADV
ejpam-3873	56	8	so	so	ADV
ejpam-3873	56	9	is	be	AUX
ejpam-3873	56	10	−a	−a	ADJ
ejpam-3873	56	11	.	.	PUNCT
ejpam-3873	57	1	c.	c.	PROPN
ejpam-3873	57	2	)	)	PUNCT
ejpam-3873	58	1	if	if	SCONJ
ejpam-3873	58	2	a	a	PRON
ejpam-3873	58	3	is	be	AUX
ejpam-3873	58	4	a	a	DET
ejpam-3873	58	5	unit	unit	NOUN
ejpam-3873	58	6	,	,	PUNCT
ejpam-3873	58	7	then	then	ADV
ejpam-3873	58	8	a	a	PRON
ejpam-3873	58	9	is	be	AUX
ejpam-3873	58	10	not	not	PART
ejpam-3873	58	11	a	a	DET
ejpam-3873	58	12	zero	zero	NUM
ejpam-3873	58	13	divisor	divisor	NOUN
ejpam-3873	58	14	.	.	PUNCT
ejpam-3873	59	1	d.	d.	PROPN
ejpam-3873	59	2	)	)	PUNCT
ejpam-3873	60	1	if	if	SCONJ
ejpam-3873	60	2	a	a	PRON
ejpam-3873	60	3	is	be	AUX
ejpam-3873	60	4	a	a	DET
ejpam-3873	60	5	zero	zero	NUM
ejpam-3873	60	6	divisor	divisor	NOUN
ejpam-3873	60	7	,	,	PUNCT
ejpam-3873	60	8	then	then	ADV
ejpam-3873	60	9	a	a	PRON
ejpam-3873	60	10	is	be	AUX
ejpam-3873	60	11	not	not	PART
ejpam-3873	60	12	a	a	DET
ejpam-3873	60	13	unit	unit	NOUN
ejpam-3873	60	14	.	.	PUNCT
ejpam-3873	61	1	remark	remark	PROPN
ejpam-3873	61	2	2	2	NUM
ejpam-3873	61	3	is	be	AUX
ejpam-3873	61	4	clear	clear	ADJ
ejpam-3873	61	5	,	,	PUNCT
ejpam-3873	61	6	and	and	CCONJ
ejpam-3873	61	7	sometimes	sometimes	ADV
ejpam-3873	61	8	are	be	AUX
ejpam-3873	61	9	given	give	VERB
ejpam-3873	61	10	in	in	ADP
ejpam-3873	61	11	the	the	DET
ejpam-3873	61	12	exercises	exercise	NOUN
ejpam-3873	61	13	of	of	ADP
ejpam-3873	61	14	some	some	DET
ejpam-3873	61	15	books	book	NOUN
ejpam-3873	61	16	.	.	PUNCT
ejpam-3873	62	1	e.j	e.j	PROPN
ejpam-3873	62	2	.	.	PROPN
ejpam-3873	62	3	sigasig	sigasig	PROPN
ejpam-3873	62	4	,	,	PUNCT
ejpam-3873	62	5	c.j	c.j	PROPN
ejpam-3873	62	6	.	.	PROPN
ejpam-3873	62	7	rosero	rosero	PROPN
ejpam-3873	62	8	,	,	PUNCT
ejpam-3873	62	9	m.	m.	NOUN
ejpam-3873	62	10	baldado	baldado	PROPN
ejpam-3873	62	11	jr	jr	PROPN
ejpam-3873	62	12	.	.	PROPN
ejpam-3873	62	13	/	/	SYM
ejpam-3873	62	14	eur	eur	PROPN
ejpam-3873	62	15	.	.	PUNCT
ejpam-3873	63	1	j.	j.	PROPN
ejpam-3873	63	2	pure	pure	PROPN
ejpam-3873	63	3	appl	appl	PROPN
ejpam-3873	63	4	.	.	PROPN
ejpam-3873	63	5	math	math	PROPN
ejpam-3873	63	6	,	,	PUNCT
ejpam-3873	63	7	14	14	NUM
ejpam-3873	63	8	(	(	PUNCT
ejpam-3873	63	9	1	1	NUM
ejpam-3873	63	10	)	)	PUNCT
ejpam-3873	63	11	(	(	PUNCT
ejpam-3873	63	12	2021	2021	NUM
ejpam-3873	63	13	)	)	PUNCT
ejpam-3873	63	14	,	,	PUNCT
ejpam-3873	63	15	314	314	NUM
ejpam-3873	63	16	-	-	SYM
ejpam-3873	63	17	326	326	NUM
ejpam-3873	63	18	316	316	NUM
ejpam-3873	63	19	remark	remark	NOUN
ejpam-3873	63	20	2	2	NUM
ejpam-3873	63	21	.	.	PUNCT
ejpam-3873	64	1	let	let	VERB
ejpam-3873	64	2	r	r	PRON
ejpam-3873	64	3	be	be	AUX
ejpam-3873	64	4	a	a	DET
ejpam-3873	64	5	ring	ring	NOUN
ejpam-3873	64	6	with	with	ADP
ejpam-3873	64	7	identity	identity	NOUN
ejpam-3873	64	8	1r	1r	NUM
ejpam-3873	64	9	6=	6=	PRON
ejpam-3873	64	10	0	0	NUM
ejpam-3873	64	11	and	and	CCONJ
ejpam-3873	64	12	a	a	PRON
ejpam-3873	64	13	be	be	AUX
ejpam-3873	64	14	a	a	DET
ejpam-3873	64	15	unit	unit	NOUN
ejpam-3873	64	16	of	of	ADP
ejpam-3873	64	17	r.	r.	PROPN
ejpam-3873	64	18	a.	a.	PROPN
ejpam-3873	64	19	)	)	PUNCT
ejpam-3873	65	1	−a	−a	NOUN
ejpam-3873	65	2	=	=	PUNCT
ejpam-3873	65	3	a−1	a−1	PROPN
ejpam-3873	66	1	if	if	SCONJ
ejpam-3873	66	2	and	and	CCONJ
ejpam-3873	66	3	only	only	ADV
ejpam-3873	66	4	if	if	SCONJ
ejpam-3873	66	5	a	a	DET
ejpam-3873	66	6	=	=	X
ejpam-3873	66	7	(	(	PUNCT
ejpam-3873	66	8	−a)−1	−a)−1	NOUN
ejpam-3873	66	9	.	.	PUNCT
ejpam-3873	66	10	b.	b.	PROPN
ejpam-3873	66	11	)	)	PUNCT
ejpam-3873	66	12	−a	−a	NOUN
ejpam-3873	66	13	6=	6=	NUM
ejpam-3873	67	1	a−1	a−1	PROPN
ejpam-3873	67	2	if	if	SCONJ
ejpam-3873	67	3	and	and	CCONJ
ejpam-3873	67	4	only	only	ADV
ejpam-3873	67	5	if	if	SCONJ
ejpam-3873	67	6	a	a	PRON
ejpam-3873	67	7	6=	6=	NUM
ejpam-3873	67	8	(	(	PUNCT
ejpam-3873	67	9	−a)−1	−a)−1	PROPN
ejpam-3873	67	10	c.	c.	PROPN
ejpam-3873	67	11	)	)	PUNCT
ejpam-3873	67	12	a2	a2	PROPN
ejpam-3873	67	13	=	=	PUNCT
ejpam-3873	67	14	1r	1r	PROPN
ejpam-3873	67	15	if	if	SCONJ
ejpam-3873	67	16	and	and	CCONJ
ejpam-3873	67	17	only	only	ADV
ejpam-3873	67	18	if	if	SCONJ
ejpam-3873	67	19	−a	−a	ADJ
ejpam-3873	67	20	=	=	SYM
ejpam-3873	67	21	(	(	PUNCT
ejpam-3873	67	22	−a)−1	−a)−1	NOUN
ejpam-3873	67	23	.	.	PUNCT
ejpam-3873	67	24	d.	d.	PROPN
ejpam-3873	67	25	)	)	PUNCT
ejpam-3873	67	26	a2	a2	PROPN
ejpam-3873	67	27	6=	6=	PRON
ejpam-3873	67	28	1r	1r	NOUN
ejpam-3873	67	29	if	if	SCONJ
ejpam-3873	67	30	and	and	CCONJ
ejpam-3873	67	31	only	only	ADV
ejpam-3873	67	32	if	if	SCONJ
ejpam-3873	67	33	−a	−a	ADJ
ejpam-3873	67	34	6=	6=	NOUN
ejpam-3873	67	35	(	(	PUNCT
ejpam-3873	67	36	−a)−1	−a)−1	NOUN
ejpam-3873	67	37	.	.	PUNCT
ejpam-3873	68	1	e.	e.	PROPN
ejpam-3873	68	2	)	)	PUNCT
ejpam-3873	68	3	2a	2a	NUM
ejpam-3873	69	1	=	=	SYM
ejpam-3873	69	2	0	0	PUNCT
ejpam-3873	70	1	if	if	SCONJ
ejpam-3873	70	2	and	and	CCONJ
ejpam-3873	70	3	only	only	ADV
ejpam-3873	70	4	if	if	SCONJ
ejpam-3873	70	5	(	(	PUNCT
ejpam-3873	70	6	a)−1	a)−1	NOUN
ejpam-3873	70	7	=	=	SYM
ejpam-3873	70	8	(	(	PUNCT
ejpam-3873	70	9	−a)−1	−a)−1	NOUN
ejpam-3873	70	10	.	.	PUNCT
ejpam-3873	70	11	f.	f.	PROPN
ejpam-3873	70	12	)	)	PUNCT
ejpam-3873	70	13	2a	2a	NUM
ejpam-3873	70	14	6=	6=	ADP
ejpam-3873	70	15	0	0	PUNCT
ejpam-3873	71	1	if	if	SCONJ
ejpam-3873	71	2	and	and	CCONJ
ejpam-3873	71	3	only	only	ADV
ejpam-3873	71	4	if	if	SCONJ
ejpam-3873	71	5	(	(	PUNCT
ejpam-3873	71	6	a)−1	a)−1	NOUN
ejpam-3873	71	7	6=	6=	PROPN
ejpam-3873	71	8	(	(	PUNCT
ejpam-3873	71	9	−a)−1	−a)−1	NOUN
ejpam-3873	71	10	.	.	PUNCT
ejpam-3873	71	11	theorem	theorem	NOUN
ejpam-3873	71	12	2	2	NUM
ejpam-3873	71	13	,	,	PUNCT
ejpam-3873	71	14	identified	identify	VERB
ejpam-3873	71	15	the	the	DET
ejpam-3873	71	16	elements	element	NOUN
ejpam-3873	71	17	of	of	ADP
ejpam-3873	71	18	a	a	DET
ejpam-3873	71	19	ring	ring	NOUN
ejpam-3873	71	20	that	that	PRON
ejpam-3873	71	21	are	be	AUX
ejpam-3873	71	22	necessarily	necessarily	ADV
ejpam-3873	71	23	in	in	ADP
ejpam-3873	71	24	a	a	DET
ejpam-3873	71	25	γ	γ	NOUN
ejpam-3873	71	26	-	-	PUNCT
ejpam-3873	71	27	sets	set	NOUN
ejpam-3873	71	28	.	.	PUNCT
ejpam-3873	72	1	theorem	theorem	NOUN
ejpam-3873	72	2	2	2	NUM
ejpam-3873	72	3	.	.	PUNCT
ejpam-3873	73	1	let	let	VERB
ejpam-3873	73	2	r	r	PRON
ejpam-3873	73	3	be	be	AUX
ejpam-3873	73	4	a	a	DET
ejpam-3873	73	5	ring	ring	NOUN
ejpam-3873	73	6	with	with	ADP
ejpam-3873	73	7	identity	identity	NOUN
ejpam-3873	73	8	1r	1r	NUM
ejpam-3873	73	9	6=	6=	PRON
ejpam-3873	73	10	0	0	NUM
ejpam-3873	73	11	and	and	CCONJ
ejpam-3873	73	12	j	j	PROPN
ejpam-3873	73	13	be	be	AUX
ejpam-3873	73	14	a	a	DET
ejpam-3873	73	15	γ	γ	NOUN
ejpam-3873	73	16	-	-	PUNCT
ejpam-3873	73	17	set	set	NOUN
ejpam-3873	73	18	of	of	ADP
ejpam-3873	73	19	r.	r.	PROPN
ejpam-3873	73	20	a.	a.	PROPN
ejpam-3873	73	21	)	)	PUNCT
ejpam-3873	74	1	if	if	SCONJ
ejpam-3873	74	2	2a	2a	NUM
ejpam-3873	74	3	=	=	SYM
ejpam-3873	74	4	0	0	NUM
ejpam-3873	74	5	,	,	PUNCT
ejpam-3873	74	6	then	then	ADV
ejpam-3873	74	7	a	a	DET
ejpam-3873	74	8	∈	∈	PROPN
ejpam-3873	74	9	j	j	PROPN
ejpam-3873	74	10	.	.	PUNCT
ejpam-3873	74	11	b.	b.	PROPN
ejpam-3873	74	12	)	)	PUNCT
ejpam-3873	75	1	if	if	SCONJ
ejpam-3873	75	2	a2	a2	PROPN
ejpam-3873	75	3	=	=	SYM
ejpam-3873	75	4	1r	1r	NUM
ejpam-3873	75	5	,	,	PUNCT
ejpam-3873	75	6	then	then	ADV
ejpam-3873	75	7	a	a	DET
ejpam-3873	75	8	∈	∈	PROPN
ejpam-3873	75	9	j	j	PROPN
ejpam-3873	75	10	.	.	PUNCT
ejpam-3873	75	11	c.	c.	PROPN
ejpam-3873	75	12	)	)	PUNCT
ejpam-3873	76	1	if	if	SCONJ
ejpam-3873	76	2	a	a	PRON
ejpam-3873	76	3	is	be	AUX
ejpam-3873	76	4	not	not	PART
ejpam-3873	76	5	a	a	DET
ejpam-3873	76	6	unit	unit	NOUN
ejpam-3873	76	7	,	,	PUNCT
ejpam-3873	76	8	then	then	ADV
ejpam-3873	76	9	a	a	DET
ejpam-3873	76	10	∈	∈	PROPN
ejpam-3873	76	11	j	j	PROPN
ejpam-3873	76	12	.	.	PUNCT
ejpam-3873	77	1	d.	d.	PROPN
ejpam-3873	77	2	)	)	PUNCT
ejpam-3873	78	1	if	if	SCONJ
ejpam-3873	78	2	a	a	PRON
ejpam-3873	78	3	is	be	AUX
ejpam-3873	78	4	a	a	DET
ejpam-3873	78	5	zero	zero	NUM
ejpam-3873	78	6	-	-	PUNCT
ejpam-3873	78	7	divisor	divisor	NOUN
ejpam-3873	78	8	,	,	PUNCT
ejpam-3873	78	9	then	then	ADV
ejpam-3873	78	10	a	a	DET
ejpam-3873	78	11	∈	∈	PROPN
ejpam-3873	78	12	j	j	PROPN
ejpam-3873	78	13	.	.	PUNCT
ejpam-3873	79	1	proof	proof	NOUN
ejpam-3873	79	2	.	.	PUNCT
ejpam-3873	80	1	let	let	VERB
ejpam-3873	80	2	r	r	PRON
ejpam-3873	80	3	be	be	AUX
ejpam-3873	80	4	a	a	DET
ejpam-3873	80	5	ring	ring	NOUN
ejpam-3873	80	6	with	with	ADP
ejpam-3873	80	7	identity	identity	NOUN
ejpam-3873	80	8	1r	1r	NUM
ejpam-3873	80	9	6=	6=	PRON
ejpam-3873	80	10	0	0	NUM
ejpam-3873	80	11	and	and	CCONJ
ejpam-3873	80	12	j	j	PROPN
ejpam-3873	80	13	be	be	AUX
ejpam-3873	80	14	a	a	DET
ejpam-3873	80	15	γ	γ	NOUN
ejpam-3873	80	16	-	-	PUNCT
ejpam-3873	80	17	set	set	NOUN
ejpam-3873	80	18	of	of	ADP
ejpam-3873	80	19	r.	r.	PROPN
ejpam-3873	80	20	(	(	PUNCT
ejpam-3873	80	21	a	a	X
ejpam-3873	80	22	)	)	PUNCT
ejpam-3873	80	23	assume	assume	VERB
ejpam-3873	80	24	that	that	SCONJ
ejpam-3873	80	25	2a	2a	NUM
ejpam-3873	80	26	=	=	SYM
ejpam-3873	80	27	0	0	NUM
ejpam-3873	80	28	and	and	CCONJ
ejpam-3873	80	29	a	a	DET
ejpam-3873	80	30	/∈	/∈	INTJ
ejpam-3873	80	31	j	j	PROPN
ejpam-3873	80	32	.	.	PUNCT
ejpam-3873	81	1	since	since	SCONJ
ejpam-3873	81	2	j	j	PROPN
ejpam-3873	81	3	is	be	AUX
ejpam-3873	81	4	a	a	DET
ejpam-3873	81	5	γ	γ	NOUN
ejpam-3873	81	6	-	-	PUNCT
ejpam-3873	81	7	set	set	NOUN
ejpam-3873	81	8	,	,	PUNCT
ejpam-3873	81	9	there	there	PRON
ejpam-3873	81	10	exists	exist	VERB
ejpam-3873	81	11	b	b	PROPN
ejpam-3873	81	12	∈	∈	PROPN
ejpam-3873	81	13	j	j	NOUN
ejpam-3873	81	14	such	such	ADJ
ejpam-3873	81	15	that	that	SCONJ
ejpam-3873	81	16	a	a	DET
ejpam-3873	81	17	+	+	NOUN
ejpam-3873	81	18	b	b	NOUN
ejpam-3873	81	19	=	=	SYM
ejpam-3873	81	20	0	0	NUM
ejpam-3873	81	21	.	.	PUNCT
ejpam-3873	82	1	hence	hence	ADV
ejpam-3873	82	2	,	,	PUNCT
ejpam-3873	82	3	a	a	PRON
ejpam-3873	82	4	=	=	X
ejpam-3873	82	5	a	a	DET
ejpam-3873	82	6	+	+	NOUN
ejpam-3873	82	7	0	0	NUM
ejpam-3873	82	8	=	=	SYM
ejpam-3873	82	9	a	a	PRON
ejpam-3873	82	10	+	+	X
ejpam-3873	82	11	(	(	PUNCT
ejpam-3873	82	12	a	a	DET
ejpam-3873	82	13	+	+	NOUN
ejpam-3873	82	14	b	b	NOUN
ejpam-3873	82	15	)	)	PUNCT
ejpam-3873	82	16	=	=	SYM
ejpam-3873	82	17	(	(	PUNCT
ejpam-3873	82	18	a	a	DET
ejpam-3873	82	19	+	+	NOUN
ejpam-3873	82	20	a	a	X
ejpam-3873	82	21	)	)	PUNCT
ejpam-3873	82	22	+	+	NUM
ejpam-3873	82	23	b	b	X
ejpam-3873	82	24	=	=	SYM
ejpam-3873	82	25	2a	2a	NUM
ejpam-3873	83	1	+	+	CCONJ
ejpam-3873	83	2	b	b	X
ejpam-3873	83	3	=	=	SYM
ejpam-3873	83	4	0	0	PUNCT
ejpam-3873	83	5	+	+	NUM
ejpam-3873	83	6	b	b	X
ejpam-3873	83	7	=	=	SYM
ejpam-3873	83	8	b	b	PROPN
ejpam-3873	83	9	,	,	PUNCT
ejpam-3873	83	10	that	that	PRON
ejpam-3873	83	11	is	be	AUX
ejpam-3873	83	12	a	a	DET
ejpam-3873	83	13	=	=	X
ejpam-3873	83	14	b.	b.	NOUN
ejpam-3873	83	15	this	this	PRON
ejpam-3873	83	16	is	be	AUX
ejpam-3873	83	17	a	a	DET
ejpam-3873	83	18	contradiction	contradiction	NOUN
ejpam-3873	83	19	.	.	PUNCT
ejpam-3873	84	1	(	(	PUNCT
ejpam-3873	84	2	b	b	X
ejpam-3873	84	3	)	)	PUNCT
ejpam-3873	84	4	assume	assume	VERB
ejpam-3873	84	5	that	that	SCONJ
ejpam-3873	84	6	a2	a2	PROPN
ejpam-3873	84	7	=	=	PUNCT
ejpam-3873	84	8	1r	1r	NUM
ejpam-3873	84	9	and	and	CCONJ
ejpam-3873	84	10	a	a	DET
ejpam-3873	84	11	/∈	/∈	INTJ
ejpam-3873	84	12	j	j	PROPN
ejpam-3873	84	13	.	.	PUNCT
ejpam-3873	85	1	since	since	SCONJ
ejpam-3873	85	2	j	j	PROPN
ejpam-3873	85	3	is	be	AUX
ejpam-3873	85	4	a	a	DET
ejpam-3873	85	5	γ	γ	NOUN
ejpam-3873	85	6	-	-	PUNCT
ejpam-3873	85	7	set	set	NOUN
ejpam-3873	85	8	,	,	PUNCT
ejpam-3873	85	9	there	there	PRON
ejpam-3873	85	10	exists	exist	VERB
ejpam-3873	85	11	c	c	PROPN
ejpam-3873	85	12	∈	∈	PROPN
ejpam-3873	85	13	j	j	PROPN
ejpam-3873	85	14	such	such	ADJ
ejpam-3873	85	15	that	that	SCONJ
ejpam-3873	85	16	ac	ac	PROPN
ejpam-3873	86	1	=	=	SYM
ejpam-3873	86	2	1r	1r	NUM
ejpam-3873	86	3	=	=	SYM
ejpam-3873	86	4	ca	ca	NOUN
ejpam-3873	86	5	.	.	PUNCT
ejpam-3873	87	1	hence	hence	ADV
ejpam-3873	87	2	,	,	PUNCT
ejpam-3873	87	3	a	a	DET
ejpam-3873	87	4	=	=	X
ejpam-3873	87	5	a1r	a1r	PROPN
ejpam-3873	87	6	=	=	SYM
ejpam-3873	87	7	a(ac	a(ac	PROPN
ejpam-3873	87	8	)	)	PUNCT
ejpam-3873	87	9	=	=	PUNCT
ejpam-3873	88	1	(	(	PUNCT
ejpam-3873	88	2	aa)c	aa)c	NOUN
ejpam-3873	88	3	=	=	SYM
ejpam-3873	88	4	1rc	1rc	NOUN
ejpam-3873	89	1	=	=	PUNCT
ejpam-3873	89	2	c	c	X
ejpam-3873	89	3	,	,	PUNCT
ejpam-3873	89	4	that	that	PRON
ejpam-3873	89	5	is	be	AUX
ejpam-3873	89	6	a	a	DET
ejpam-3873	89	7	=	=	ADJ
ejpam-3873	89	8	c.	c.	NOUN
ejpam-3873	89	9	this	this	PRON
ejpam-3873	89	10	is	be	AUX
ejpam-3873	89	11	a	a	DET
ejpam-3873	89	12	contradiction	contradiction	NOUN
ejpam-3873	89	13	.	.	PUNCT
ejpam-3873	90	1	(	(	PUNCT
ejpam-3873	90	2	c	c	X
ejpam-3873	90	3	)	)	PUNCT
ejpam-3873	90	4	if	if	SCONJ
ejpam-3873	90	5	a	a	PRON
ejpam-3873	90	6	is	be	AUX
ejpam-3873	90	7	not	not	PART
ejpam-3873	90	8	a	a	DET
ejpam-3873	90	9	unit	unit	NOUN
ejpam-3873	90	10	,	,	PUNCT
ejpam-3873	90	11	then	then	ADV
ejpam-3873	90	12	a	a	PRON
ejpam-3873	90	13	has	have	VERB
ejpam-3873	90	14	no	no	DET
ejpam-3873	90	15	multiplicative	multiplicative	ADJ
ejpam-3873	90	16	inverse	inverse	NOUN
ejpam-3873	90	17	.	.	PUNCT
ejpam-3873	91	1	clearly	clearly	ADV
ejpam-3873	91	2	,	,	PUNCT
ejpam-3873	91	3	a	a	PRON
ejpam-3873	91	4	is	be	AUX
ejpam-3873	91	5	necessarily	necessarily	ADV
ejpam-3873	91	6	in	in	ADP
ejpam-3873	91	7	j	j	PROPN
ejpam-3873	91	8	.	.	PUNCT
ejpam-3873	92	1	(	(	PUNCT
ejpam-3873	92	2	d	d	X
ejpam-3873	92	3	)	)	PUNCT
ejpam-3873	92	4	if	if	SCONJ
ejpam-3873	92	5	a	a	PRON
ejpam-3873	92	6	is	be	AUX
ejpam-3873	92	7	a	a	DET
ejpam-3873	92	8	zero	zero	NUM
ejpam-3873	92	9	-	-	PUNCT
ejpam-3873	92	10	divisor	divisor	NOUN
ejpam-3873	92	11	,	,	PUNCT
ejpam-3873	92	12	then	then	ADV
ejpam-3873	92	13	by	by	ADP
ejpam-3873	92	14	remark	remark	NOUN
ejpam-3873	92	15	2	2	NUM
ejpam-3873	92	16	(	(	PUNCT
ejpam-3873	92	17	b	b	NOUN
ejpam-3873	92	18	)	)	PUNCT
ejpam-3873	92	19	,	,	PUNCT
ejpam-3873	92	20	a	a	PRON
ejpam-3873	92	21	is	be	AUX
ejpam-3873	92	22	not	not	PART
ejpam-3873	92	23	a	a	DET
ejpam-3873	92	24	unit	unit	NOUN
ejpam-3873	92	25	.	.	PUNCT
ejpam-3873	93	1	hence	hence	ADV
ejpam-3873	93	2	,	,	PUNCT
ejpam-3873	93	3	by	by	ADP
ejpam-3873	93	4	(	(	PUNCT
ejpam-3873	93	5	c	c	X
ejpam-3873	93	6	)	)	PUNCT
ejpam-3873	93	7	a	a	PRON
ejpam-3873	93	8	must	must	AUX
ejpam-3873	93	9	be	be	AUX
ejpam-3873	93	10	in	in	ADP
ejpam-3873	93	11	j	j	PROPN
ejpam-3873	93	12	.	.	PUNCT
ejpam-3873	94	1	3	3	X
ejpam-3873	94	2	.	.	X
ejpam-3873	94	3	constructing	construct	VERB
ejpam-3873	94	4	a	a	DET
ejpam-3873	94	5	γ	γ	X
ejpam-3873	94	6	-	-	PUNCT
ejpam-3873	94	7	set	set	NOUN
ejpam-3873	94	8	in	in	ADP
ejpam-3873	94	9	this	this	DET
ejpam-3873	94	10	section	section	NOUN
ejpam-3873	94	11	,	,	PUNCT
ejpam-3873	94	12	we	we	PRON
ejpam-3873	94	13	presented	present	VERB
ejpam-3873	94	14	a	a	DET
ejpam-3873	94	15	method	method	NOUN
ejpam-3873	94	16	of	of	ADP
ejpam-3873	94	17	constructing	construct	VERB
ejpam-3873	94	18	a	a	DET
ejpam-3873	94	19	γ	γ	X
ejpam-3873	94	20	-	-	NOUN
ejpam-3873	94	21	set	set	VERB
ejpam-3873	94	22	from	from	ADP
ejpam-3873	94	23	a	a	DET
ejpam-3873	94	24	γ	γ	NOUN
ejpam-3873	94	25	-	-	PUNCT
ejpam-3873	94	26	set	set	NOUN
ejpam-3873	94	27	.	.	PUNCT
ejpam-3873	95	1	the	the	DET
ejpam-3873	95	2	next	next	ADJ
ejpam-3873	95	3	theorem	theorem	NOUN
ejpam-3873	95	4	,	,	PUNCT
ejpam-3873	95	5	theorem	theorem	VERB
ejpam-3873	95	6	3	3	NUM
ejpam-3873	95	7	,	,	PUNCT
ejpam-3873	95	8	says	say	VERB
ejpam-3873	95	9	that	that	SCONJ
ejpam-3873	95	10	a	a	DET
ejpam-3873	95	11	unit	unit	NOUN
ejpam-3873	95	12	a	a	PRON
ejpam-3873	95	13	with	with	ADP
ejpam-3873	95	14	a2	a2	PROPN
ejpam-3873	95	15	6=	6=	ADP
ejpam-3873	95	16	1r	1r	NUM
ejpam-3873	95	17	and	and	CCONJ
ejpam-3873	95	18	2a	2a	NUM
ejpam-3873	95	19	6=	6=	NUM
ejpam-3873	95	20	0	0	NUM
ejpam-3873	95	21	determines	determine	VERB
ejpam-3873	95	22	a	a	DET
ejpam-3873	95	23	γ	γ	X
ejpam-3873	95	24	-	-	PUNCT
ejpam-3873	95	25	set	set	NOUN
ejpam-3873	95	26	.	.	PUNCT
ejpam-3873	96	1	theorem	theorem	NOUN
ejpam-3873	96	2	3	3	X
ejpam-3873	96	3	.	.	PUNCT
ejpam-3873	97	1	let	let	VERB
ejpam-3873	97	2	r	r	PRON
ejpam-3873	97	3	be	be	AUX
ejpam-3873	97	4	a	a	DET
ejpam-3873	97	5	ring	ring	NOUN
ejpam-3873	97	6	with	with	ADP
ejpam-3873	97	7	identity	identity	NOUN
ejpam-3873	97	8	1r	1r	NUM
ejpam-3873	97	9	6=	6=	ADP
ejpam-3873	97	10	0	0	NUM
ejpam-3873	97	11	,	,	PUNCT
ejpam-3873	97	12	and	and	CCONJ
ejpam-3873	97	13	j	j	PROPN
ejpam-3873	97	14	is	be	AUX
ejpam-3873	97	15	a	a	DET
ejpam-3873	97	16	γ	γ	NOUN
ejpam-3873	97	17	-	-	PUNCT
ejpam-3873	97	18	set	set	NOUN
ejpam-3873	97	19	of	of	ADP
ejpam-3873	97	20	r.	r.	PROPN
ejpam-3873	97	21	if	if	SCONJ
ejpam-3873	97	22	a	a	PRON
ejpam-3873	97	23	is	be	AUX
ejpam-3873	97	24	a	a	DET
ejpam-3873	97	25	unit	unit	NOUN
ejpam-3873	97	26	with	with	ADP
ejpam-3873	97	27	a2	a2	PROPN
ejpam-3873	97	28	6=	6=	ADP
ejpam-3873	97	29	1r	1r	NUM
ejpam-3873	97	30	and	and	CCONJ
ejpam-3873	97	31	2a	2a	NUM
ejpam-3873	97	32	6=	6=	NUM
ejpam-3873	97	33	0	0	NUM
ejpam-3873	97	34	,	,	PUNCT
ejpam-3873	97	35	then	then	ADV
ejpam-3873	97	36	(	(	PUNCT
ejpam-3873	97	37	j\{a	j\{a	X
ejpam-3873	97	38	,	,	PUNCT
ejpam-3873	97	39	(	(	PUNCT
ejpam-3873	97	40	−a)−1	−a)−1	NOUN
ejpam-3873	97	41	}	}	PUNCT
ejpam-3873	97	42	)	)	PUNCT
ejpam-3873	97	43	∪	∪	ADP
ejpam-3873	97	44	{	{	PUNCT
ejpam-3873	97	45	a−1,−a	a−1,−a	PROPN
ejpam-3873	97	46	}	}	PUNCT
ejpam-3873	97	47	and	and	CCONJ
ejpam-3873	97	48	(	(	PUNCT
ejpam-3873	97	49	j\{a−1,−a	j\{a−1,−a	ADV
ejpam-3873	97	50	}	}	PUNCT
ejpam-3873	97	51	)	)	PUNCT
ejpam-3873	97	52	∪	∪	ADP
ejpam-3873	97	53	{	{	PUNCT
ejpam-3873	97	54	a	a	PRON
ejpam-3873	97	55	,	,	PUNCT
ejpam-3873	97	56	(	(	PUNCT
ejpam-3873	97	57	−a)−1	−a)−1	NOUN
ejpam-3873	97	58	}	}	PUNCT
ejpam-3873	97	59	are	be	AUX
ejpam-3873	97	60	γ	γ	NOUN
ejpam-3873	97	61	-	-	PUNCT
ejpam-3873	97	62	sets	set	NOUN
ejpam-3873	97	63	of	of	ADP
ejpam-3873	97	64	r.	r.	PROPN
ejpam-3873	97	65	e.j	e.j	PROPN
ejpam-3873	97	66	.	.	PROPN
ejpam-3873	97	67	sigasig	sigasig	PROPN
ejpam-3873	97	68	,	,	PUNCT
ejpam-3873	97	69	c.j	c.j	PROPN
ejpam-3873	97	70	.	.	PROPN
ejpam-3873	97	71	rosero	rosero	PROPN
ejpam-3873	97	72	,	,	PUNCT
ejpam-3873	97	73	m.	m.	NOUN
ejpam-3873	97	74	baldado	baldado	PROPN
ejpam-3873	97	75	jr	jr	PROPN
ejpam-3873	97	76	.	.	PROPN
ejpam-3873	97	77	/	/	SYM
ejpam-3873	97	78	eur	eur	PROPN
ejpam-3873	97	79	.	.	PUNCT
ejpam-3873	98	1	j.	j.	PROPN
ejpam-3873	98	2	pure	pure	PROPN
ejpam-3873	98	3	appl	appl	PROPN
ejpam-3873	98	4	.	.	PROPN
ejpam-3873	98	5	math	math	PROPN
ejpam-3873	98	6	,	,	PUNCT
ejpam-3873	98	7	14	14	NUM
ejpam-3873	98	8	(	(	PUNCT
ejpam-3873	98	9	1	1	NUM
ejpam-3873	98	10	)	)	PUNCT
ejpam-3873	98	11	(	(	PUNCT
ejpam-3873	98	12	2021	2021	NUM
ejpam-3873	98	13	)	)	PUNCT
ejpam-3873	98	14	,	,	PUNCT
ejpam-3873	98	15	314	314	NUM
ejpam-3873	98	16	-	-	SYM
ejpam-3873	98	17	326	326	NUM
ejpam-3873	98	18	317	317	NUM
ejpam-3873	98	19	proof	proof	NOUN
ejpam-3873	98	20	.	.	PUNCT
ejpam-3873	99	1	let	let	VERB
ejpam-3873	99	2	r	r	PRON
ejpam-3873	99	3	be	be	AUX
ejpam-3873	99	4	a	a	DET
ejpam-3873	99	5	ring	ring	NOUN
ejpam-3873	99	6	with	with	ADP
ejpam-3873	99	7	identity	identity	NOUN
ejpam-3873	99	8	1r	1r	NUM
ejpam-3873	99	9	6=	6=	ADP
ejpam-3873	99	10	0	0	NUM
ejpam-3873	99	11	,	,	PUNCT
ejpam-3873	99	12	and	and	CCONJ
ejpam-3873	99	13	j	j	PROPN
ejpam-3873	99	14	is	be	AUX
ejpam-3873	99	15	a	a	DET
ejpam-3873	99	16	γ	γ	NOUN
ejpam-3873	99	17	-	-	PUNCT
ejpam-3873	99	18	set	set	NOUN
ejpam-3873	99	19	of	of	ADP
ejpam-3873	99	20	r.	r.	PROPN
ejpam-3873	99	21	let	let	VERB
ejpam-3873	99	22	a	a	PRON
ejpam-3873	99	23	be	be	AUX
ejpam-3873	99	24	a	a	DET
ejpam-3873	99	25	unit	unit	NOUN
ejpam-3873	99	26	of	of	ADP
ejpam-3873	99	27	r	r	NOUN
ejpam-3873	99	28	with	with	ADP
ejpam-3873	99	29	a2	a2	PROPN
ejpam-3873	99	30	6=	6=	ADP
ejpam-3873	99	31	1r	1r	NUM
ejpam-3873	99	32	and	and	CCONJ
ejpam-3873	99	33	2a	2a	NUM
ejpam-3873	99	34	6=	6=	ADP
ejpam-3873	99	35	0	0	X
ejpam-3873	99	36	.	.	PUNCT
ejpam-3873	100	1	then	then	ADV
ejpam-3873	100	2	by	by	ADP
ejpam-3873	100	3	remark	remark	NOUN
ejpam-3873	100	4	2	2	NUM
ejpam-3873	100	5	(	(	PUNCT
ejpam-3873	100	6	d	d	NOUN
ejpam-3873	100	7	)	)	PUNCT
ejpam-3873	100	8	and	and	CCONJ
ejpam-3873	100	9	remark	remark	NOUN
ejpam-3873	100	10	2	2	NUM
ejpam-3873	100	11	(	(	PUNCT
ejpam-3873	100	12	f	f	NOUN
ejpam-3873	100	13	)	)	PUNCT
ejpam-3873	100	14	,	,	PUNCT
ejpam-3873	100	15	−a	−a	VERB
ejpam-3873	100	16	6=	6=	NUM
ejpam-3873	100	17	(	(	PUNCT
ejpam-3873	100	18	−a)−1	−a)−1	NOUN
ejpam-3873	100	19	and	and	CCONJ
ejpam-3873	100	20	a−1	a−1	PROPN
ejpam-3873	100	21	6=	6=	PROPN
ejpam-3873	100	22	(	(	PUNCT
ejpam-3873	100	23	−a)−1	−a)−1	NOUN
ejpam-3873	100	24	.	.	PUNCT
ejpam-3873	101	1	consider	consider	VERB
ejpam-3873	101	2	j1	j1	NOUN
ejpam-3873	101	3	=	=	SYM
ejpam-3873	101	4	(	(	PUNCT
ejpam-3873	101	5	j\{a	j\{a	NOUN
ejpam-3873	101	6	,	,	PUNCT
ejpam-3873	101	7	(	(	PUNCT
ejpam-3873	101	8	−a)−1	−a)−1	NOUN
ejpam-3873	101	9	}	}	PUNCT
ejpam-3873	101	10	)	)	PUNCT
ejpam-3873	101	11	∪	∪	ADP
ejpam-3873	101	12	{	{	PUNCT
ejpam-3873	101	13	a−1,−a	a−1,−a	PROPN
ejpam-3873	101	14	}	}	PUNCT
ejpam-3873	101	15	and	and	CCONJ
ejpam-3873	101	16	j2	j2	PROPN
ejpam-3873	101	17	=	=	SYM
ejpam-3873	101	18	(	(	PUNCT
ejpam-3873	101	19	j\{a−1,−a	j\{a−1,−a	ADV
ejpam-3873	101	20	}	}	PUNCT
ejpam-3873	101	21	)	)	PUNCT
ejpam-3873	101	22	∪	∪	ADP
ejpam-3873	101	23	{	{	PUNCT
ejpam-3873	101	24	a	a	PRON
ejpam-3873	101	25	,	,	PUNCT
ejpam-3873	101	26	(	(	PUNCT
ejpam-3873	101	27	−a)−1	−a)−1	NOUN
ejpam-3873	101	28	}	}	PUNCT
ejpam-3873	101	29	.	.	PUNCT
ejpam-3873	102	1	claim	claim	NOUN
ejpam-3873	102	2	1	1	NUM
ejpam-3873	102	3	.	.	PUNCT
ejpam-3873	103	1	j1	j1	PROPN
ejpam-3873	103	2	=	=	PUNCT
ejpam-3873	103	3	(	(	PUNCT
ejpam-3873	103	4	j\{a	j\{a	NOUN
ejpam-3873	103	5	,	,	PUNCT
ejpam-3873	103	6	(	(	PUNCT
ejpam-3873	103	7	−a)−1	−a)−1	NOUN
ejpam-3873	103	8	}	}	PUNCT
ejpam-3873	103	9	)	)	PUNCT
ejpam-3873	103	10	∪	∪	ADP
ejpam-3873	103	11	{	{	PUNCT
ejpam-3873	103	12	a−1,−a	a−1,−a	PROPN
ejpam-3873	103	13	}	}	PUNCT
ejpam-3873	103	14	is	be	AUX
ejpam-3873	103	15	a	a	DET
ejpam-3873	103	16	γ	γ	NOUN
ejpam-3873	103	17	-	-	PUNCT
ejpam-3873	103	18	set	set	NOUN
ejpam-3873	103	19	to	to	PART
ejpam-3873	103	20	show	show	VERB
ejpam-3873	103	21	claim	claim	NOUN
ejpam-3873	103	22	1	1	NUM
ejpam-3873	103	23	consider	consider	VERB
ejpam-3873	103	24	the	the	DET
ejpam-3873	103	25	following	follow	VERB
ejpam-3873	103	26	cases	case	NOUN
ejpam-3873	103	27	:	:	PUNCT
ejpam-3873	103	28	case	case	NOUN
ejpam-3873	103	29	1	1	NUM
ejpam-3873	103	30	.	.	PUNCT
ejpam-3873	104	1	a	a	DET
ejpam-3873	104	2	/∈	/∈	NOUN
ejpam-3873	104	3	j	j	NOUN
ejpam-3873	104	4	if	if	SCONJ
ejpam-3873	104	5	a	a	PRON
ejpam-3873	104	6	/∈	/∈	NOUN
ejpam-3873	104	7	j	j	NOUN
ejpam-3873	104	8	,	,	PUNCT
ejpam-3873	104	9	then	then	ADV
ejpam-3873	104	10	j	j	PROPN
ejpam-3873	104	11	=	=	PROPN
ejpam-3873	104	12	j1	j1	PROPN
ejpam-3873	104	13	.	.	PUNCT
ejpam-3873	105	1	hence	hence	ADV
ejpam-3873	105	2	j1	j1	PROPN
ejpam-3873	105	3	is	be	AUX
ejpam-3873	105	4	a	a	DET
ejpam-3873	105	5	γ	γ	NOUN
ejpam-3873	105	6	-	-	PUNCT
ejpam-3873	105	7	set	set	VERB
ejpam-3873	105	8	.	.	PUNCT
ejpam-3873	106	1	case	case	NOUN
ejpam-3873	106	2	2	2	NUM
ejpam-3873	106	3	.	.	PUNCT
ejpam-3873	106	4	a	a	DET
ejpam-3873	106	5	∈	∈	PROPN
ejpam-3873	106	6	j	j	NOUN
ejpam-3873	106	7	if	if	SCONJ
ejpam-3873	106	8	a	a	DET
ejpam-3873	106	9	∈	∈	PROPN
ejpam-3873	106	10	j	j	NOUN
ejpam-3873	106	11	,	,	PUNCT
ejpam-3873	106	12	then	then	ADV
ejpam-3873	106	13	let	let	VERB
ejpam-3873	106	14	b	b	X
ejpam-3873	106	15	∈	∈	PROPN
ejpam-3873	106	16	r\j1	r\j1	NOUN
ejpam-3873	106	17	and	and	CCONJ
ejpam-3873	106	18	consider	consider	VERB
ejpam-3873	106	19	the	the	DET
ejpam-3873	106	20	following	follow	VERB
ejpam-3873	106	21	subcases	subcase	NOUN
ejpam-3873	106	22	:	:	PUNCT
ejpam-3873	106	23	subcase	subcase	NOUN
ejpam-3873	106	24	1	1	NUM
ejpam-3873	106	25	.	.	PUNCT
ejpam-3873	107	1	b	b	X
ejpam-3873	107	2	6=	6=	ADP
ejpam-3873	107	3	a	a	PRON
ejpam-3873	107	4	and	and	CCONJ
ejpam-3873	107	5	b	b	NOUN
ejpam-3873	107	6	6=	6=	PROPN
ejpam-3873	107	7	(	(	PUNCT
ejpam-3873	107	8	−a)−1	−a)−1	NOUN
ejpam-3873	107	9	if	if	SCONJ
ejpam-3873	107	10	b	b	PROPN
ejpam-3873	107	11	6=	6=	ADP
ejpam-3873	107	12	a	a	PRON
ejpam-3873	107	13	and	and	CCONJ
ejpam-3873	107	14	b	b	NOUN
ejpam-3873	107	15	6=	6=	PROPN
ejpam-3873	107	16	(	(	PUNCT
ejpam-3873	107	17	−a)−1	−a)−1	NOUN
ejpam-3873	107	18	,	,	PUNCT
ejpam-3873	107	19	then	then	ADV
ejpam-3873	107	20	b	b	PROPN
ejpam-3873	107	21	∈	∈	PROPN
ejpam-3873	107	22	r\j	r\j	VERB
ejpam-3873	107	23	∪{a−1,−a	∪{a−1,−a	ADP
ejpam-3873	107	24	}	}	PUNCT
ejpam-3873	107	25	.	.	PUNCT
ejpam-3873	108	1	since	since	SCONJ
ejpam-3873	108	2	j	j	PROPN
ejpam-3873	108	3	is	be	AUX
ejpam-3873	108	4	a	a	DET
ejpam-3873	108	5	γ	γ	NOUN
ejpam-3873	108	6	-	-	PUNCT
ejpam-3873	108	7	set	set	NOUN
ejpam-3873	108	8	,	,	PUNCT
ejpam-3873	108	9	there	there	PRON
ejpam-3873	108	10	exist	exist	VERB
ejpam-3873	108	11	c	c	NOUN
ejpam-3873	108	12	,	,	PUNCT
ejpam-3873	108	13	d	d	PROPN
ejpam-3873	108	14	∈	∈	PROPN
ejpam-3873	108	15	j1	j1	NOUN
ejpam-3873	108	16	such	such	ADJ
ejpam-3873	108	17	that	that	PRON
ejpam-3873	108	18	b+	b+	ADP
ejpam-3873	108	19	c	c	NOUN
ejpam-3873	108	20	=	=	SYM
ejpam-3873	108	21	0	0	PUNCT
ejpam-3873	109	1	=	=	PUNCT
ejpam-3873	109	2	c+	c+	PROPN
ejpam-3873	109	3	b	b	NOUN
ejpam-3873	109	4	and	and	CCONJ
ejpam-3873	109	5	bd	bd	PROPN
ejpam-3873	109	6	=	=	NOUN
ejpam-3873	109	7	1r	1r	NUM
ejpam-3873	109	8	=	=	SYM
ejpam-3873	109	9	db	db	PROPN
ejpam-3873	109	10	.	.	PROPN
ejpam-3873	109	11	subcase	subcase	PROPN
ejpam-3873	109	12	2	2	NUM
ejpam-3873	109	13	.	.	X
ejpam-3873	110	1	b	b	X
ejpam-3873	110	2	=	=	PUNCT
ejpam-3873	111	1	a	a	DET
ejpam-3873	111	2	if	if	SCONJ
ejpam-3873	111	3	b	b	X
ejpam-3873	111	4	=	=	SYM
ejpam-3873	111	5	a	a	NOUN
ejpam-3873	111	6	,	,	PUNCT
ejpam-3873	111	7	then	then	ADV
ejpam-3873	111	8	a+	a+	PUNCT
ejpam-3873	111	9	(	(	PUNCT
ejpam-3873	111	10	−a	−a	ADJ
ejpam-3873	111	11	)	)	PUNCT
ejpam-3873	111	12	=	=	SYM
ejpam-3873	111	13	0	0	PUNCT
ejpam-3873	112	1	=	=	SYM
ejpam-3873	112	2	(	(	PUNCT
ejpam-3873	112	3	−a	−a	ADV
ejpam-3873	112	4	)	)	PUNCT
ejpam-3873	113	1	+	+	CCONJ
ejpam-3873	113	2	a	a	PRON
ejpam-3873	113	3	and	and	CCONJ
ejpam-3873	113	4	aa−1	aa−1	NOUN
ejpam-3873	113	5	=	=	SYM
ejpam-3873	113	6	1r	1r	NUM
ejpam-3873	114	1	=	=	SYM
ejpam-3873	114	2	a−1a	a−1a	PROPN
ejpam-3873	114	3	.	.	PUNCT
ejpam-3873	114	4	subcase	subcase	PROPN
ejpam-3873	114	5	3	3	NUM
ejpam-3873	114	6	.	.	PUNCT
ejpam-3873	115	1	b	b	X
ejpam-3873	115	2	=	=	PRON
ejpam-3873	115	3	(	(	PUNCT
ejpam-3873	115	4	−a)−1	−a)−1	NOUN
ejpam-3873	115	5	if	if	SCONJ
ejpam-3873	115	6	b	b	PROPN
ejpam-3873	115	7	=	=	PUNCT
ejpam-3873	115	8	(	(	PUNCT
ejpam-3873	115	9	−a)−1	−a)−1	NOUN
ejpam-3873	115	10	,	,	PUNCT
ejpam-3873	115	11	then	then	ADV
ejpam-3873	115	12	by	by	ADP
ejpam-3873	115	13	remark	remark	NOUN
ejpam-3873	115	14	1	1	NUM
ejpam-3873	115	15	(	(	PUNCT
ejpam-3873	115	16	a	a	NOUN
ejpam-3873	115	17	)	)	PUNCT
ejpam-3873	115	18	(	(	PUNCT
ejpam-3873	115	19	−a)−1	−a)−1	NOUN
ejpam-3873	115	20	+	+	CCONJ
ejpam-3873	115	21	a−1	a−1	NOUN
ejpam-3873	115	22	=	=	PUNCT
ejpam-3873	115	23	−a−1	−a−1	PROPN
ejpam-3873	115	24	+	+	CCONJ
ejpam-3873	115	25	a−1	a−1	NOUN
ejpam-3873	115	26	=	=	SYM
ejpam-3873	115	27	0	0	PUNCT
ejpam-3873	116	1	=	=	SYM
ejpam-3873	116	2	a−1	a−1	NOUN
ejpam-3873	116	3	+	+	CCONJ
ejpam-3873	116	4	−a−1	−a−1	X
ejpam-3873	116	5	=	=	PUNCT
ejpam-3873	116	6	a−1	a−1	PROPN
ejpam-3873	116	7	+	+	CCONJ
ejpam-3873	116	8	(	(	PUNCT
ejpam-3873	116	9	−a)−1	−a)−1	NOUN
ejpam-3873	116	10	and	and	CCONJ
ejpam-3873	116	11	(	(	PUNCT
ejpam-3873	116	12	−a)−1(−a	−a)−1(−a	ADJ
ejpam-3873	116	13	)	)	PUNCT
ejpam-3873	116	14	=	=	SYM
ejpam-3873	116	15	1r	1r	NUM
ejpam-3873	116	16	=	=	SYM
ejpam-3873	116	17	(	(	PUNCT
ejpam-3873	116	18	−a)(−a)−1	−a)(−a)−1	NOUN
ejpam-3873	116	19	.	.	PUNCT
ejpam-3873	117	1	this	this	PRON
ejpam-3873	117	2	shows	show	VERB
ejpam-3873	117	3	the	the	DET
ejpam-3873	117	4	claim	claim	NOUN
ejpam-3873	117	5	.	.	PUNCT
ejpam-3873	118	1	claim	claim	NOUN
ejpam-3873	118	2	2	2	NUM
ejpam-3873	118	3	.	.	X
ejpam-3873	118	4	j2	j2	PROPN
ejpam-3873	118	5	=	=	SYM
ejpam-3873	118	6	(	(	PUNCT
ejpam-3873	118	7	j\{a−1,−a	j\{a−1,−a	ADV
ejpam-3873	118	8	}	}	PUNCT
ejpam-3873	118	9	)	)	PUNCT
ejpam-3873	118	10	∪	∪	ADP
ejpam-3873	118	11	{	{	PUNCT
ejpam-3873	118	12	a	a	PRON
ejpam-3873	118	13	,	,	PUNCT
ejpam-3873	118	14	(	(	PUNCT
ejpam-3873	118	15	−a)−1	−a)−1	NOUN
ejpam-3873	118	16	}	}	PUNCT
ejpam-3873	118	17	is	be	AUX
ejpam-3873	118	18	a	a	DET
ejpam-3873	118	19	γ	γ	NOUN
ejpam-3873	118	20	-	-	PUNCT
ejpam-3873	118	21	set	set	NOUN
ejpam-3873	118	22	proved	prove	VERB
ejpam-3873	118	23	similarly	similarly	ADV
ejpam-3873	118	24	.	.	PUNCT
ejpam-3873	119	1	let	let	VERB
ejpam-3873	119	2	r	r	PRON
ejpam-3873	119	3	be	be	AUX
ejpam-3873	119	4	a	a	DET
ejpam-3873	119	5	ring	ring	NOUN
ejpam-3873	119	6	with	with	ADP
ejpam-3873	119	7	identity	identity	NOUN
ejpam-3873	119	8	1r	1r	NUM
ejpam-3873	119	9	6=	6=	ADP
ejpam-3873	119	10	0	0	NUM
ejpam-3873	119	11	,	,	PUNCT
ejpam-3873	119	12	and	and	CCONJ
ejpam-3873	119	13	j	j	PROPN
ejpam-3873	119	14	is	be	AUX
ejpam-3873	119	15	a	a	DET
ejpam-3873	119	16	γ	γ	NOUN
ejpam-3873	119	17	-	-	PUNCT
ejpam-3873	119	18	set	set	NOUN
ejpam-3873	119	19	of	of	ADP
ejpam-3873	119	20	r.	r.	PROPN
ejpam-3873	119	21	a	a	DET
ejpam-3873	119	22	unit	unit	NOUN
ejpam-3873	119	23	a	a	DET
ejpam-3873	119	24	with	with	ADP
ejpam-3873	119	25	a2	a2	PROPN
ejpam-3873	119	26	6=	6=	ADP
ejpam-3873	119	27	1r	1r	NUM
ejpam-3873	119	28	and	and	CCONJ
ejpam-3873	119	29	2a	2a	NUM
ejpam-3873	119	30	6=	6=	NUM
ejpam-3873	119	31	0	0	NUM
ejpam-3873	119	32	is	be	AUX
ejpam-3873	119	33	called	call	VERB
ejpam-3873	119	34	a	a	DET
ejpam-3873	119	35	super	super	NOUN
ejpam-3873	119	36	-	-	NOUN
ejpam-3873	119	37	couple	couple	ADJ
ejpam-3873	119	38	.	.	PUNCT
ejpam-3873	120	1	theorem	theorem	ADJ
ejpam-3873	120	2	5	5	NUM
ejpam-3873	120	3	suggests	suggest	VERB
ejpam-3873	120	4	that	that	SCONJ
ejpam-3873	120	5	every	every	DET
ejpam-3873	120	6	super	super	ADJ
ejpam-3873	120	7	-	-	ADJ
ejpam-3873	120	8	couple	couple	ADJ
ejpam-3873	120	9	determines	determine	VERB
ejpam-3873	120	10	a	a	DET
ejpam-3873	120	11	minimum	minimum	ADJ
ejpam-3873	120	12	γ	γ	X
ejpam-3873	120	13	-	-	PUNCT
ejpam-3873	120	14	set	set	NOUN
ejpam-3873	120	15	,	,	PUNCT
ejpam-3873	120	16	in	in	ADP
ejpam-3873	120	17	the	the	DET
ejpam-3873	120	18	same	same	ADJ
ejpam-3873	120	19	way	way	NOUN
ejpam-3873	120	20	as	as	ADP
ejpam-3873	120	21	in	in	ADP
ejpam-3873	120	22	[	[	X
ejpam-3873	120	23	3	3	X
ejpam-3873	120	24	]	]	PUNCT
ejpam-3873	120	25	that	that	SCONJ
ejpam-3873	120	26	every	every	DET
ejpam-3873	120	27	non	non	ADJ
ejpam-3873	120	28	-	-	ADJ
ejpam-3873	120	29	involution	involution	ADJ
ejpam-3873	120	30	determines	determine	VERB
ejpam-3873	120	31	a	a	DET
ejpam-3873	120	32	d	d	NOUN
ejpam-3873	120	33	-	-	PUNCT
ejpam-3873	120	34	set	set	ADJ
ejpam-3873	120	35	.	.	PUNCT
ejpam-3873	121	1	theorem	theorem	NOUN
ejpam-3873	121	2	4	4	NUM
ejpam-3873	121	3	give	give	VERB
ejpam-3873	121	4	some	some	PRON
ejpam-3873	121	5	of	of	ADP
ejpam-3873	121	6	the	the	DET
ejpam-3873	121	7	conditions	condition	NOUN
ejpam-3873	121	8	wherein	wherein	SCONJ
ejpam-3873	121	9	a	a	DET
ejpam-3873	121	10	ring	ring	NOUN
ejpam-3873	121	11	r	r	NOUN
ejpam-3873	121	12	has	have	VERB
ejpam-3873	121	13	a	a	DET
ejpam-3873	121	14	unique	unique	ADJ
ejpam-3873	121	15	γ	γ	NOUN
ejpam-3873	121	16	-	-	NOUN
ejpam-3873	121	17	set	set	NOUN
ejpam-3873	121	18	,	,	PUNCT
ejpam-3873	121	19	that	that	ADV
ejpam-3873	121	20	is	is	ADV
ejpam-3873	121	21	,	,	PUNCT
ejpam-3873	121	22	|tr|	|tr|	PROPN
ejpam-3873	121	23	=	=	SYM
ejpam-3873	121	24	1	1	X
ejpam-3873	121	25	.	.	PUNCT
ejpam-3873	121	26	theorem	theorem	NOUN
ejpam-3873	121	27	4	4	NUM
ejpam-3873	121	28	.	.	PUNCT
ejpam-3873	122	1	let	let	VERB
ejpam-3873	122	2	r	r	PRON
ejpam-3873	122	3	be	be	AUX
ejpam-3873	122	4	a	a	DET
ejpam-3873	122	5	ring	ring	NOUN
ejpam-3873	122	6	with	with	ADP
ejpam-3873	122	7	identity	identity	NOUN
ejpam-3873	122	8	1r	1r	NUM
ejpam-3873	122	9	6=	6=	PRON
ejpam-3873	122	10	0	0	NUM
ejpam-3873	122	11	and	and	CCONJ
ejpam-3873	122	12	j	j	PROPN
ejpam-3873	122	13	be	be	AUX
ejpam-3873	122	14	a	a	DET
ejpam-3873	122	15	γ	γ	NOUN
ejpam-3873	122	16	-	-	PUNCT
ejpam-3873	122	17	set	set	NOUN
ejpam-3873	122	18	of	of	ADP
ejpam-3873	122	19	r.	r.	PROPN
ejpam-3873	122	20	then	then	ADV
ejpam-3873	122	21	|tr|	|tr|	PROPN
ejpam-3873	122	22	>	>	X
ejpam-3873	122	23	1	1	NUM
ejpam-3873	122	24	if	if	SCONJ
ejpam-3873	122	25	and	and	CCONJ
ejpam-3873	122	26	only	only	ADV
ejpam-3873	122	27	if	if	SCONJ
ejpam-3873	122	28	there	there	PRON
ejpam-3873	122	29	exists	exist	VERB
ejpam-3873	122	30	a	a	DET
ejpam-3873	122	31	unit	unit	NOUN
ejpam-3873	122	32	u	u	NOUN
ejpam-3873	122	33	∈	∈	PROPN
ejpam-3873	122	34	r	r	NOUN
ejpam-3873	122	35	such	such	ADJ
ejpam-3873	122	36	that	that	DET
ejpam-3873	122	37	u2	u2	PROPN
ejpam-3873	122	38	6=	6=	PROPN
ejpam-3873	122	39	1r	1r	NUM
ejpam-3873	122	40	and	and	CCONJ
ejpam-3873	122	41	2u	2u	ADJ
ejpam-3873	122	42	6=	6=	PROPN
ejpam-3873	122	43	0	0	NUM
ejpam-3873	122	44	.	.	PUNCT
ejpam-3873	123	1	proof	proof	NOUN
ejpam-3873	123	2	.	.	PUNCT
ejpam-3873	124	1	let	let	VERB
ejpam-3873	124	2	r	r	PRON
ejpam-3873	124	3	be	be	AUX
ejpam-3873	124	4	a	a	DET
ejpam-3873	124	5	ring	ring	NOUN
ejpam-3873	124	6	with	with	ADP
ejpam-3873	124	7	identity	identity	NOUN
ejpam-3873	124	8	1r	1r	NUM
ejpam-3873	124	9	6=	6=	PRON
ejpam-3873	124	10	0	0	NUM
ejpam-3873	124	11	and	and	CCONJ
ejpam-3873	124	12	j	j	PROPN
ejpam-3873	124	13	be	be	AUX
ejpam-3873	124	14	a	a	DET
ejpam-3873	124	15	γ	γ	NOUN
ejpam-3873	124	16	-	-	PUNCT
ejpam-3873	124	17	set	set	NOUN
ejpam-3873	124	18	of	of	ADP
ejpam-3873	124	19	r.	r.	PROPN
ejpam-3873	124	20	suppose	suppose	VERB
ejpam-3873	124	21	that	that	SCONJ
ejpam-3873	124	22	|tr|	|tr|	PROPN
ejpam-3873	124	23	>	>	X
ejpam-3873	124	24	1	1	NUM
ejpam-3873	124	25	.	.	PUNCT
ejpam-3873	125	1	then	then	ADV
ejpam-3873	125	2	there	there	PRON
ejpam-3873	125	3	exists	exist	VERB
ejpam-3873	125	4	a	a	DET
ejpam-3873	125	5	γ	γ	X
ejpam-3873	125	6	-	-	PUNCT
ejpam-3873	125	7	set	set	VERB
ejpam-3873	125	8	j	j	NOUN
ejpam-3873	125	9	in	in	ADP
ejpam-3873	125	10	r	r	NOUN
ejpam-3873	125	11	with	with	ADP
ejpam-3873	125	12	j	j	PROPN
ejpam-3873	125	13	6=	6=	PROPN
ejpam-3873	125	14	r.	r.	PROPN
ejpam-3873	125	15	let	let	VERB
ejpam-3873	125	16	x	x	SYM
ejpam-3873	125	17	∈	∈	PROPN
ejpam-3873	125	18	r\j	r\j	ADV
ejpam-3873	125	19	.	.	PUNCT
ejpam-3873	126	1	since	since	SCONJ
ejpam-3873	126	2	j	j	PROPN
ejpam-3873	126	3	is	be	AUX
ejpam-3873	126	4	a	a	DET
ejpam-3873	126	5	γ	γ	NOUN
ejpam-3873	126	6	-	-	PUNCT
ejpam-3873	126	7	set	set	NOUN
ejpam-3873	126	8	,	,	PUNCT
ejpam-3873	126	9	there	there	PRON
ejpam-3873	126	10	exists	exist	VERB
ejpam-3873	126	11	y	y	PROPN
ejpam-3873	126	12	,	,	PUNCT
ejpam-3873	126	13	z	z	PROPN
ejpam-3873	126	14	∈	∈	PROPN
ejpam-3873	126	15	j	j	NOUN
ejpam-3873	126	16	such	such	ADJ
ejpam-3873	126	17	that	that	SCONJ
ejpam-3873	126	18	x	x	X
ejpam-3873	127	1	+	+	NUM
ejpam-3873	127	2	y	y	NOUN
ejpam-3873	127	3	=	=	SYM
ejpam-3873	127	4	0	0	PUNCT
ejpam-3873	127	5	=	=	SYM
ejpam-3873	127	6	y	y	PROPN
ejpam-3873	127	7	+	+	NOUN
ejpam-3873	127	8	x	x	SYM
ejpam-3873	127	9	and	and	CCONJ
ejpam-3873	127	10	xz	xz	NOUN
ejpam-3873	127	11	=	=	SYM
ejpam-3873	127	12	1r	1r	NUM
ejpam-3873	127	13	=	=	SYM
ejpam-3873	127	14	zx	zx	NUM
ejpam-3873	127	15	.	.	PUNCT
ejpam-3873	128	1	thus	thus	ADV
ejpam-3873	128	2	,	,	PUNCT
ejpam-3873	128	3	x	x	PRON
ejpam-3873	128	4	is	be	AUX
ejpam-3873	128	5	a	a	DET
ejpam-3873	128	6	unit	unit	NOUN
ejpam-3873	128	7	.	.	PUNCT
ejpam-3873	129	1	moreover	moreover	ADV
ejpam-3873	129	2	,	,	PUNCT
ejpam-3873	129	3	since	since	SCONJ
ejpam-3873	129	4	y	y	PROPN
ejpam-3873	129	5	,	,	PUNCT
ejpam-3873	129	6	z	z	PROPN
ejpam-3873	129	7	∈	∈	PROPN
ejpam-3873	129	8	j	j	PROPN
ejpam-3873	129	9	and	and	CCONJ
ejpam-3873	129	10	x	x	PUNCT
ejpam-3873	129	11	∈	∈	PROPN
ejpam-3873	129	12	r\j	r\j	VERB
ejpam-3873	129	13	,	,	PUNCT
ejpam-3873	129	14	x	x	SYM
ejpam-3873	129	15	6=	6=	ADP
ejpam-3873	129	16	y	y	PROPN
ejpam-3873	129	17	and	and	CCONJ
ejpam-3873	129	18	x	x	PROPN
ejpam-3873	129	19	6=	6=	PROPN
ejpam-3873	129	20	z.	z.	PROPN
ejpam-3873	129	21	hence	hence	ADV
ejpam-3873	129	22	,	,	PUNCT
ejpam-3873	129	23	by	by	ADP
ejpam-3873	129	24	remark	remark	NOUN
ejpam-3873	129	25	2	2	NUM
ejpam-3873	129	26	(	(	PUNCT
ejpam-3873	129	27	d	d	NOUN
ejpam-3873	129	28	)	)	PUNCT
ejpam-3873	129	29	and	and	CCONJ
ejpam-3873	129	30	remark	remark	NOUN
ejpam-3873	129	31	2	2	NUM
ejpam-3873	129	32	(	(	PUNCT
ejpam-3873	129	33	f	f	X
ejpam-3873	129	34	)	)	PUNCT
ejpam-3873	129	35	,	,	PUNCT
ejpam-3873	130	1	x2	x2	PROPN
ejpam-3873	130	2	6=	6=	NUM
ejpam-3873	130	3	1r	1r	NUM
ejpam-3873	130	4	and	and	CCONJ
ejpam-3873	130	5	2x	2x	NUM
ejpam-3873	130	6	6=	6=	NUM
ejpam-3873	130	7	0	0	NUM
ejpam-3873	130	8	,	,	PUNCT
ejpam-3873	130	9	respectively	respectively	ADV
ejpam-3873	130	10	.	.	PUNCT
ejpam-3873	131	1	conversely	conversely	ADV
ejpam-3873	131	2	,	,	PUNCT
ejpam-3873	131	3	assume	assume	VERB
ejpam-3873	131	4	that	that	SCONJ
ejpam-3873	131	5	there	there	PRON
ejpam-3873	131	6	exists	exist	VERB
ejpam-3873	131	7	a	a	DET
ejpam-3873	131	8	unit	unit	NOUN
ejpam-3873	131	9	x	x	SYM
ejpam-3873	131	10	∈	∈	NOUN
ejpam-3873	131	11	r	r	NOUN
ejpam-3873	131	12	such	such	ADJ
ejpam-3873	131	13	that	that	DET
ejpam-3873	131	14	x2	x2	PROPN
ejpam-3873	131	15	6=	6=	NUM
ejpam-3873	131	16	1r	1r	NUM
ejpam-3873	131	17	and	and	CCONJ
ejpam-3873	131	18	2x	2x	NUM
ejpam-3873	131	19	6=	6=	X
ejpam-3873	131	20	0	0	NUM
ejpam-3873	131	21	.	.	PUNCT
ejpam-3873	132	1	then	then	ADV
ejpam-3873	132	2	by	by	ADP
ejpam-3873	132	3	theorem	theorem	NOUN
ejpam-3873	132	4	5	5	NUM
ejpam-3873	132	5	,	,	PUNCT
ejpam-3873	132	6	(	(	PUNCT
ejpam-3873	132	7	r\{x−1,−x	r\{x−1,−x	NOUN
ejpam-3873	132	8	}	}	PUNCT
ejpam-3873	132	9	)	)	PUNCT
ejpam-3873	132	10	∪	∪	ADP
ejpam-3873	132	11	{	{	PUNCT
ejpam-3873	132	12	x	x	NOUN
ejpam-3873	132	13	,	,	PUNCT
ejpam-3873	132	14	(	(	PUNCT
ejpam-3873	132	15	−x)−1	−x)−1	NOUN
ejpam-3873	132	16	}	}	PUNCT
ejpam-3873	132	17	is	be	AUX
ejpam-3873	132	18	a	a	DET
ejpam-3873	132	19	nontrivial	nontrivial	ADJ
ejpam-3873	132	20	γ	γ	NOUN
ejpam-3873	132	21	-	-	PUNCT
ejpam-3873	132	22	sets	set	NOUN
ejpam-3873	132	23	of	of	ADP
ejpam-3873	132	24	r.	r.	PROPN
ejpam-3873	132	25	therefore	therefore	ADV
ejpam-3873	132	26	,	,	PUNCT
ejpam-3873	132	27	|tr|	|tr|	PROPN
ejpam-3873	132	28	>	>	X
ejpam-3873	132	29	1	1	X
ejpam-3873	132	30	.	.	PUNCT
ejpam-3873	132	31	theorem	theorem	NOUN
ejpam-3873	132	32	5	5	NUM
ejpam-3873	132	33	.	.	PUNCT
ejpam-3873	133	1	let	let	VERB
ejpam-3873	133	2	r	r	PRON
ejpam-3873	133	3	be	be	AUX
ejpam-3873	133	4	a	a	DET
ejpam-3873	133	5	ring	ring	NOUN
ejpam-3873	133	6	with	with	ADP
ejpam-3873	133	7	identity	identity	NOUN
ejpam-3873	133	8	1r	1r	NUM
ejpam-3873	133	9	6=	6=	PRON
ejpam-3873	133	10	0	0	NUM
ejpam-3873	133	11	and	and	CCONJ
ejpam-3873	133	12	j	j	PROPN
ejpam-3873	133	13	be	be	AUX
ejpam-3873	133	14	a	a	DET
ejpam-3873	133	15	γ	γ	NOUN
ejpam-3873	133	16	-	-	PUNCT
ejpam-3873	133	17	set	set	NOUN
ejpam-3873	133	18	of	of	ADP
ejpam-3873	133	19	r.	r.	PROPN
ejpam-3873	133	20	then	then	ADV
ejpam-3873	133	21	|tr|	|tr|	PROPN
ejpam-3873	133	22	=	=	SYM
ejpam-3873	133	23	1	1	NUM
ejpam-3873	133	24	if	if	SCONJ
ejpam-3873	133	25	and	and	CCONJ
ejpam-3873	133	26	only	only	ADV
ejpam-3873	133	27	if	if	SCONJ
ejpam-3873	133	28	for	for	ADP
ejpam-3873	133	29	all	all	DET
ejpam-3873	133	30	a	a	DET
ejpam-3873	133	31	∈	∈	NOUN
ejpam-3873	133	32	r	r	NOUN
ejpam-3873	133	33	either	either	CCONJ
ejpam-3873	133	34	a2	a2	PROPN
ejpam-3873	133	35	=	=	SYM
ejpam-3873	133	36	1r	1r	NUM
ejpam-3873	133	37	or	or	CCONJ
ejpam-3873	133	38	2a	2a	NUM
ejpam-3873	133	39	=	=	SYM
ejpam-3873	133	40	0	0	NUM
ejpam-3873	133	41	or	or	CCONJ
ejpam-3873	133	42	a	a	PRON
ejpam-3873	133	43	is	be	AUX
ejpam-3873	133	44	a	a	DET
ejpam-3873	133	45	zero	zero	NUM
ejpam-3873	133	46	-	-	PUNCT
ejpam-3873	133	47	divisor	divisor	NOUN
ejpam-3873	133	48	.	.	PUNCT
ejpam-3873	134	1	proof	proof	NOUN
ejpam-3873	134	2	.	.	PUNCT
ejpam-3873	135	1	proved	prove	VERB
ejpam-3873	135	2	similarly	similarly	ADV
ejpam-3873	135	3	.	.	PUNCT
ejpam-3873	136	1	e.j	e.j	PROPN
ejpam-3873	136	2	.	.	PROPN
ejpam-3873	136	3	sigasig	sigasig	PROPN
ejpam-3873	136	4	,	,	PUNCT
ejpam-3873	136	5	c.j	c.j	PROPN
ejpam-3873	136	6	.	.	PROPN
ejpam-3873	136	7	rosero	rosero	PROPN
ejpam-3873	136	8	,	,	PUNCT
ejpam-3873	136	9	m.	m.	NOUN
ejpam-3873	136	10	baldado	baldado	PROPN
ejpam-3873	136	11	jr	jr	PROPN
ejpam-3873	136	12	.	.	PROPN
ejpam-3873	136	13	/	/	SYM
ejpam-3873	136	14	eur	eur	PROPN
ejpam-3873	136	15	.	.	PUNCT
ejpam-3873	137	1	j.	j.	PROPN
ejpam-3873	137	2	pure	pure	PROPN
ejpam-3873	137	3	appl	appl	PROPN
ejpam-3873	137	4	.	.	PROPN
ejpam-3873	137	5	math	math	PROPN
ejpam-3873	137	6	,	,	PUNCT
ejpam-3873	137	7	14	14	NUM
ejpam-3873	137	8	(	(	PUNCT
ejpam-3873	137	9	1	1	NUM
ejpam-3873	137	10	)	)	PUNCT
ejpam-3873	137	11	(	(	PUNCT
ejpam-3873	137	12	2021	2021	NUM
ejpam-3873	137	13	)	)	PUNCT
ejpam-3873	137	14	,	,	PUNCT
ejpam-3873	137	15	314	314	NUM
ejpam-3873	137	16	-	-	SYM
ejpam-3873	137	17	326	326	NUM
ejpam-3873	137	18	318	318	NUM
ejpam-3873	137	19	4	4	NUM
ejpam-3873	137	20	.	.	PUNCT
ejpam-3873	138	1	an	an	DET
ejpam-3873	138	2	equivalence	equivalence	NOUN
ejpam-3873	138	3	relation	relation	NOUN
ejpam-3873	138	4	in	in	ADP
ejpam-3873	138	5	r\s	r\s	NOUN
ejpam-3873	138	6	in	in	ADP
ejpam-3873	138	7	this	this	DET
ejpam-3873	138	8	section	section	NOUN
ejpam-3873	138	9	,	,	PUNCT
ejpam-3873	138	10	we	we	PRON
ejpam-3873	138	11	presented	present	VERB
ejpam-3873	138	12	an	an	DET
ejpam-3873	138	13	equivalence	equivalence	NOUN
ejpam-3873	138	14	relation	relation	NOUN
ejpam-3873	138	15	in	in	ADP
ejpam-3873	138	16	r\s	r\s	NOUN
ejpam-3873	138	17	which	which	PRON
ejpam-3873	138	18	will	will	AUX
ejpam-3873	138	19	be	be	AUX
ejpam-3873	138	20	useful	useful	ADJ
ejpam-3873	138	21	in	in	ADP
ejpam-3873	138	22	the	the	DET
ejpam-3873	138	23	next	next	ADJ
ejpam-3873	138	24	section	section	NOUN
ejpam-3873	138	25	.	.	PUNCT
ejpam-3873	139	1	lemma	lemma	PROPN
ejpam-3873	139	2	1	1	X
ejpam-3873	139	3	.	.	PUNCT
ejpam-3873	140	1	let	let	VERB
ejpam-3873	140	2	r	r	PRON
ejpam-3873	140	3	be	be	AUX
ejpam-3873	140	4	a	a	DET
ejpam-3873	140	5	division	division	NOUN
ejpam-3873	140	6	ring	ring	NOUN
ejpam-3873	140	7	and	and	CCONJ
ejpam-3873	140	8	s	s	NOUN
ejpam-3873	140	9	=	=	PUNCT
ejpam-3873	140	10	{	{	PUNCT
ejpam-3873	140	11	x	x	SYM
ejpam-3873	140	12	∈	∈	PROPN
ejpam-3873	140	13	r	r	NOUN
ejpam-3873	140	14	:	:	PUNCT
ejpam-3873	141	1	x2	x2	NOUN
ejpam-3873	141	2	=	=	PUNCT
ejpam-3873	141	3	1r	1r	NUM
ejpam-3873	141	4	or	or	CCONJ
ejpam-3873	141	5	2x	2x	NUM
ejpam-3873	141	6	=	=	SYM
ejpam-3873	141	7	0	0	NUM
ejpam-3873	141	8	}	}	PUNCT
ejpam-3873	141	9	.	.	PUNCT
ejpam-3873	142	1	the	the	DET
ejpam-3873	142	2	relation	relation	NOUN
ejpam-3873	142	3	∼	∼	NOUN
ejpam-3873	142	4	on	on	ADP
ejpam-3873	142	5	r\s	r\s	NOUN
ejpam-3873	142	6	given	give	VERB
ejpam-3873	142	7	by	by	ADP
ejpam-3873	142	8	x	x	PUNCT
ejpam-3873	142	9	∼	∼	NOUN
ejpam-3873	142	10	y	y	NOUN
ejpam-3873	142	11	if	if	SCONJ
ejpam-3873	143	1	and	and	CCONJ
ejpam-3873	143	2	only	only	ADV
ejpam-3873	143	3	if	if	SCONJ
ejpam-3873	143	4	x	x	NOUN
ejpam-3873	143	5	=	=	VERB
ejpam-3873	143	6	y	y	PROPN
ejpam-3873	143	7	or	or	CCONJ
ejpam-3873	143	8	x	x	X
ejpam-3873	143	9	=	=	SYM
ejpam-3873	143	10	y−1	y−1	PROPN
ejpam-3873	143	11	or	or	CCONJ
ejpam-3873	143	12	x	x	X
ejpam-3873	143	13	=	=	SYM
ejpam-3873	143	14	y	y	PROPN
ejpam-3873	143	15	or	or	CCONJ
ejpam-3873	143	16	x	x	X
ejpam-3873	143	17	=	=	PRON
ejpam-3873	143	18	(	(	PUNCT
ejpam-3873	143	19	−y)−1	−y)−1	NOUN
ejpam-3873	143	20	is	be	AUX
ejpam-3873	143	21	an	an	DET
ejpam-3873	143	22	equivalence	equivalence	NOUN
ejpam-3873	143	23	relation	relation	NOUN
ejpam-3873	143	24	.	.	PUNCT
ejpam-3873	144	1	proof	proof	NOUN
ejpam-3873	144	2	.	.	PUNCT
ejpam-3873	145	1	let	let	VERB
ejpam-3873	145	2	r	r	PRON
ejpam-3873	145	3	be	be	AUX
ejpam-3873	145	4	a	a	DET
ejpam-3873	145	5	division	division	NOUN
ejpam-3873	145	6	ring	ring	NOUN
ejpam-3873	145	7	and	and	CCONJ
ejpam-3873	145	8	s	s	NOUN
ejpam-3873	145	9	=	=	PUNCT
ejpam-3873	145	10	{	{	PUNCT
ejpam-3873	145	11	x	x	SYM
ejpam-3873	145	12	∈	∈	PROPN
ejpam-3873	145	13	r	r	NOUN
ejpam-3873	145	14	:	:	PUNCT
ejpam-3873	146	1	x2	x2	NOUN
ejpam-3873	146	2	=	=	PUNCT
ejpam-3873	146	3	1r	1r	NUM
ejpam-3873	146	4	or	or	CCONJ
ejpam-3873	146	5	2x	2x	NUM
ejpam-3873	146	6	=	=	SYM
ejpam-3873	146	7	0	0	NUM
ejpam-3873	146	8	}	}	PUNCT
ejpam-3873	146	9	.	.	PUNCT
ejpam-3873	147	1	define	define	VERB
ejpam-3873	147	2	a	a	DET
ejpam-3873	147	3	relation	relation	NOUN
ejpam-3873	147	4	∼	∼	NOUN
ejpam-3873	147	5	on	on	ADP
ejpam-3873	147	6	r\s	r\s	NOUN
ejpam-3873	147	7	as	as	SCONJ
ejpam-3873	147	8	follows	follow	VERB
ejpam-3873	147	9	:	:	PUNCT
ejpam-3873	147	10	x	x	PUNCT
ejpam-3873	147	11	∼	∼	NOUN
ejpam-3873	147	12	y	y	NOUN
ejpam-3873	147	13	if	if	SCONJ
ejpam-3873	147	14	and	and	CCONJ
ejpam-3873	147	15	only	only	ADV
ejpam-3873	147	16	if	if	SCONJ
ejpam-3873	147	17	x	x	NOUN
ejpam-3873	147	18	=	=	VERB
ejpam-3873	147	19	y	y	PROPN
ejpam-3873	147	20	or	or	CCONJ
ejpam-3873	147	21	x	x	X
ejpam-3873	147	22	=	=	SYM
ejpam-3873	147	23	y−1	y−1	PROPN
ejpam-3873	147	24	or	or	CCONJ
ejpam-3873	147	25	x	x	X
ejpam-3873	147	26	=	=	PRON
ejpam-3873	147	27	(	(	PUNCT
ejpam-3873	147	28	−y)−1	−y)−1	NOUN
ejpam-3873	147	29	.	.	PUNCT
ejpam-3873	148	1	since	since	SCONJ
ejpam-3873	148	2	x	x	X
ejpam-3873	148	3	=	=	PUNCT
ejpam-3873	148	4	x	x	PROPN
ejpam-3873	148	5	for	for	ADP
ejpam-3873	148	6	all	all	DET
ejpam-3873	148	7	x	x	SYM
ejpam-3873	148	8	∈	∈	PROPN
ejpam-3873	148	9	r	r	NOUN
ejpam-3873	148	10	,	,	PUNCT
ejpam-3873	148	11	we	we	PRON
ejpam-3873	148	12	have	have	VERB
ejpam-3873	148	13	x	x	X
ejpam-3873	148	14	∼	∼	NOUN
ejpam-3873	148	15	x	x	PUNCT
ejpam-3873	148	16	for	for	ADP
ejpam-3873	148	17	all	all	DET
ejpam-3873	148	18	x	x	SYM
ejpam-3873	148	19	∈	∈	PROPN
ejpam-3873	148	20	r\s	r\s	NOUN
ejpam-3873	148	21	.	.	PUNCT
ejpam-3873	149	1	hence	hence	ADV
ejpam-3873	149	2	,	,	PUNCT
ejpam-3873	149	3	∼	∼	NOUN
ejpam-3873	149	4	is	be	AUX
ejpam-3873	149	5	reflexive	reflexive	ADJ
ejpam-3873	149	6	.	.	PUNCT
ejpam-3873	150	1	it	it	PRON
ejpam-3873	150	2	can	can	AUX
ejpam-3873	150	3	easily	easily	ADV
ejpam-3873	150	4	be	be	AUX
ejpam-3873	150	5	shown	show	VERB
ejpam-3873	150	6	that	that	SCONJ
ejpam-3873	150	7	∼	∼	NOUN
ejpam-3873	150	8	is	be	AUX
ejpam-3873	150	9	symmetric	symmetric	ADJ
ejpam-3873	150	10	and	and	CCONJ
ejpam-3873	150	11	transitive	transitive	ADJ
ejpam-3873	150	12	.	.	PUNCT
ejpam-3873	151	1	thus	thus	ADV
ejpam-3873	151	2	,	,	PUNCT
ejpam-3873	151	3	∼	∼	NOUN
ejpam-3873	151	4	is	be	AUX
ejpam-3873	151	5	an	an	DET
ejpam-3873	151	6	equivalence	equivalence	NOUN
ejpam-3873	151	7	relation	relation	NOUN
ejpam-3873	151	8	.	.	PUNCT
ejpam-3873	152	1	remark	remark	PROPN
ejpam-3873	152	2	3	3	NUM
ejpam-3873	152	3	.	.	PUNCT
ejpam-3873	153	1	let	let	VERB
ejpam-3873	153	2	r	r	PRON
ejpam-3873	153	3	be	be	AUX
ejpam-3873	153	4	a	a	DET
ejpam-3873	153	5	division	division	NOUN
ejpam-3873	153	6	ring	ring	NOUN
ejpam-3873	153	7	,	,	PUNCT
ejpam-3873	153	8	and	and	CCONJ
ejpam-3873	153	9	let	let	VERB
ejpam-3873	153	10	s	s	AUX
ejpam-3873	153	11	=	=	PUNCT
ejpam-3873	153	12	{	{	PUNCT
ejpam-3873	153	13	x	x	SYM
ejpam-3873	153	14	∈	∈	PROPN
ejpam-3873	153	15	r	r	NOUN
ejpam-3873	153	16	:	:	PUNCT
ejpam-3873	153	17	x2	x2	NOUN
ejpam-3873	153	18	=	=	PUNCT
ejpam-3873	153	19	1r	1r	NUM
ejpam-3873	153	20	or	or	CCONJ
ejpam-3873	153	21	2x	2x	NUM
ejpam-3873	153	22	=	=	SYM
ejpam-3873	153	23	0	0	NUM
ejpam-3873	153	24	}	}	PUNCT
ejpam-3873	153	25	.	.	PUNCT
ejpam-3873	154	1	the	the	DET
ejpam-3873	154	2	equivalence	equivalence	NOUN
ejpam-3873	154	3	relation	relation	NOUN
ejpam-3873	154	4	∼	∼	NOUN
ejpam-3873	154	5	in	in	ADP
ejpam-3873	154	6	r\s	r\s	NOUN
ejpam-3873	154	7	of	of	ADP
ejpam-3873	154	8	lemma	lemma	PROPN
ejpam-3873	154	9	1	1	NUM
ejpam-3873	154	10	partitions	partition	NOUN
ejpam-3873	154	11	r\s	r\s	NOUN
ejpam-3873	154	12	into	into	ADP
ejpam-3873	154	13	equivalence	equivalence	NOUN
ejpam-3873	154	14	classes	class	NOUN
ejpam-3873	154	15	[	[	X
ejpam-3873	154	16	a	a	X
ejpam-3873	154	17	]	]	X
ejpam-3873	154	18	=	=	SYM
ejpam-3873	154	19	{	{	PUNCT
ejpam-3873	154	20	x	x	SYM
ejpam-3873	154	21	∈	∈	PROPN
ejpam-3873	154	22	r\s	r\s	NOUN
ejpam-3873	154	23	:	:	PUNCT
ejpam-3873	154	24	x	x	PUNCT
ejpam-3873	154	25	∼	∼	VERB
ejpam-3873	154	26	a	a	PRON
ejpam-3873	154	27	}	}	PUNCT
ejpam-3873	154	28	=	=	SYM
ejpam-3873	154	29	{	{	PUNCT
ejpam-3873	154	30	x	x	SYM
ejpam-3873	154	31	∈	∈	PROPN
ejpam-3873	154	32	r\s	r\s	NOUN
ejpam-3873	154	33	:	:	PUNCT
ejpam-3873	154	34	x	x	X
ejpam-3873	154	35	=	=	PUNCT
ejpam-3873	154	36	a	a	NOUN
ejpam-3873	154	37	,	,	PUNCT
ejpam-3873	154	38	or	or	CCONJ
ejpam-3873	154	39	x	x	X
ejpam-3873	154	40	=	=	SYM
ejpam-3873	154	41	a−1	a−1	PROPN
ejpam-3873	154	42	,	,	PUNCT
ejpam-3873	154	43	or	or	CCONJ
ejpam-3873	154	44	x	x	X
ejpam-3873	154	45	=	=	SYM
ejpam-3873	154	46	−a	−a	NOUN
ejpam-3873	154	47	,	,	PUNCT
ejpam-3873	154	48	or	or	CCONJ
ejpam-3873	154	49	x	x	X
ejpam-3873	154	50	=	=	SYM
ejpam-3873	154	51	(	(	PUNCT
ejpam-3873	154	52	−a)−1	−a)−1	NOUN
ejpam-3873	154	53	}	}	PUNCT
ejpam-3873	154	54	.	.	PUNCT
ejpam-3873	155	1	5	5	X
ejpam-3873	155	2	.	.	X
ejpam-3873	156	1	some	some	DET
ejpam-3873	156	2	bounds	bound	NOUN
ejpam-3873	156	3	on	on	ADP
ejpam-3873	156	4	the	the	DET
ejpam-3873	156	5	number	number	NOUN
ejpam-3873	156	6	of	of	ADP
ejpam-3873	156	7	minimum	minimum	ADJ
ejpam-3873	156	8	γ	γ	X
ejpam-3873	156	9	-	-	NOUN
ejpam-3873	156	10	set	set	NOUN
ejpam-3873	156	11	in	in	ADP
ejpam-3873	156	12	this	this	DET
ejpam-3873	156	13	section	section	NOUN
ejpam-3873	156	14	,	,	PUNCT
ejpam-3873	156	15	we	we	PRON
ejpam-3873	156	16	established	establish	VERB
ejpam-3873	156	17	a	a	DET
ejpam-3873	156	18	sharp	sharp	ADJ
ejpam-3873	156	19	upperbound	upperbound	NOUN
ejpam-3873	156	20	and	and	CCONJ
ejpam-3873	156	21	a	a	DET
ejpam-3873	156	22	sharp	sharp	ADJ
ejpam-3873	156	23	lowerbound	lowerbound	NOUN
ejpam-3873	156	24	for	for	ADP
ejpam-3873	156	25	the	the	DET
ejpam-3873	156	26	number	number	NOUN
ejpam-3873	156	27	of	of	ADP
ejpam-3873	156	28	minimum	minimum	ADJ
ejpam-3873	156	29	γ	γ	X
ejpam-3873	156	30	-	-	PUNCT
ejpam-3873	156	31	set	set	VERB
ejpam-3873	156	32	in	in	ADP
ejpam-3873	156	33	a	a	DET
ejpam-3873	156	34	finite	finite	ADJ
ejpam-3873	156	35	division	division	NOUN
ejpam-3873	156	36	ring	ring	NOUN
ejpam-3873	156	37	.	.	PUNCT
ejpam-3873	157	1	if	if	SCONJ
ejpam-3873	157	2	r	r	NOUN
ejpam-3873	157	3	is	be	AUX
ejpam-3873	157	4	a	a	DET
ejpam-3873	157	5	finite	finite	ADJ
ejpam-3873	157	6	division	division	NOUN
ejpam-3873	157	7	ring	ring	NOUN
ejpam-3873	157	8	,	,	PUNCT
ejpam-3873	157	9	then	then	ADV
ejpam-3873	157	10	we	we	PRON
ejpam-3873	157	11	denote	denote	VERB
ejpam-3873	157	12	the	the	DET
ejpam-3873	157	13	partition	partition	NOUN
ejpam-3873	157	14	of	of	ADP
ejpam-3873	157	15	r\s	r\s	NOUN
ejpam-3873	157	16	in	in	ADP
ejpam-3873	157	17	remark	remark	NOUN
ejpam-3873	157	18	3	3	NUM
ejpam-3873	157	19	by	by	ADP
ejpam-3873	157	20	c	c	NOUN
ejpam-3873	157	21	=	=	SYM
ejpam-3873	157	22	{	{	PUNCT
ejpam-3873	157	23	[	[	X
ejpam-3873	157	24	a1	a1	NOUN
ejpam-3873	157	25	]	]	PUNCT
ejpam-3873	157	26	,	,	PUNCT
ejpam-3873	157	27	[	[	X
ejpam-3873	157	28	a2	a2	X
ejpam-3873	157	29	]	]	PUNCT
ejpam-3873	157	30	,	,	PUNCT
ejpam-3873	157	31	.	.	PUNCT
ejpam-3873	157	32	.	.	PUNCT
ejpam-3873	157	33	.	.	PUNCT
ejpam-3873	158	1	,	,	PUNCT
ejpam-3873	158	2	[	[	X
ejpam-3873	158	3	ac	ac	X
ejpam-3873	158	4	]	]	X
ejpam-3873	158	5	}	}	PUNCT
ejpam-3873	158	6	.	.	PUNCT
ejpam-3873	159	1	in	in	ADP
ejpam-3873	159	2	this	this	DET
ejpam-3873	159	3	case	case	NOUN
ejpam-3873	159	4	,	,	PUNCT
ejpam-3873	159	5	we	we	PRON
ejpam-3873	159	6	call	call	VERB
ejpam-3873	159	7	c	c	VERB
ejpam-3873	159	8	the	the	DET
ejpam-3873	159	9	c	c	NOUN
ejpam-3873	159	10	-	-	PUNCT
ejpam-3873	159	11	number	number	NOUN
ejpam-3873	159	12	of	of	ADP
ejpam-3873	159	13	r.	r.	PROPN
ejpam-3873	159	14	lemma	lemma	PROPN
ejpam-3873	160	1	2	2	X
ejpam-3873	160	2	.	.	PUNCT
ejpam-3873	160	3	let	let	VERB
ejpam-3873	160	4	r	r	PRON
ejpam-3873	160	5	be	be	AUX
ejpam-3873	160	6	a	a	DET
ejpam-3873	160	7	finite	finite	ADJ
ejpam-3873	160	8	division	division	NOUN
ejpam-3873	160	9	ring	ring	NOUN
ejpam-3873	160	10	,	,	PUNCT
ejpam-3873	160	11	and	and	CCONJ
ejpam-3873	160	12	let	let	VERB
ejpam-3873	160	13	s	s	AUX
ejpam-3873	160	14	=	=	PUNCT
ejpam-3873	160	15	{	{	PUNCT
ejpam-3873	160	16	x	x	SYM
ejpam-3873	160	17	∈	∈	PROPN
ejpam-3873	160	18	r	r	NOUN
ejpam-3873	160	19	:	:	PUNCT
ejpam-3873	160	20	x2	x2	NOUN
ejpam-3873	160	21	=	=	PUNCT
ejpam-3873	160	22	1r	1r	NUM
ejpam-3873	160	23	or	or	CCONJ
ejpam-3873	160	24	2x	2x	NUM
ejpam-3873	160	25	=	=	SYM
ejpam-3873	160	26	0	0	NUM
ejpam-3873	160	27	}	}	PUNCT
ejpam-3873	160	28	.	.	PUNCT
ejpam-3873	161	1	if	if	SCONJ
ejpam-3873	161	2	c	c	AUX
ejpam-3873	161	3	=	=	PRON
ejpam-3873	161	4	{	{	PUNCT
ejpam-3873	161	5	[	[	X
ejpam-3873	161	6	a1	a1	NOUN
ejpam-3873	161	7	]	]	PUNCT
ejpam-3873	161	8	,	,	PUNCT
ejpam-3873	162	1	[	[	X
ejpam-3873	162	2	a2	a2	X
ejpam-3873	162	3	]	]	PUNCT
ejpam-3873	162	4	,	,	PUNCT
ejpam-3873	162	5	.	.	PUNCT
ejpam-3873	162	6	.	.	PUNCT
ejpam-3873	162	7	.	.	PUNCT
ejpam-3873	163	1	,	,	PUNCT
ejpam-3873	163	2	[	[	X
ejpam-3873	163	3	ac	ac	ADP
ejpam-3873	163	4	]	]	X
ejpam-3873	163	5	}	}	PUNCT
ejpam-3873	163	6	is	be	AUX
ejpam-3873	163	7	the	the	DET
ejpam-3873	163	8	partition	partition	NOUN
ejpam-3873	163	9	of	of	ADP
ejpam-3873	163	10	r\s	r\s	NOUN
ejpam-3873	163	11	in	in	ADP
ejpam-3873	163	12	the	the	DET
ejpam-3873	163	13	sense	sense	NOUN
ejpam-3873	163	14	of	of	ADP
ejpam-3873	163	15	remark	remark	NOUN
ejpam-3873	163	16	3	3	NUM
ejpam-3873	163	17	,	,	PUNCT
ejpam-3873	163	18	then	then	ADV
ejpam-3873	163	19	2	2	NUM
ejpam-3873	163	20	≤	≤	NOUN
ejpam-3873	163	21	|[ai]|	|[ai]|	NOUN
ejpam-3873	163	22	≤	≤	ADV
ejpam-3873	163	23	4	4	NUM
ejpam-3873	163	24	.	.	PUNCT
ejpam-3873	164	1	proof	proof	NOUN
ejpam-3873	164	2	.	.	PUNCT
ejpam-3873	165	1	let	let	VERB
ejpam-3873	165	2	r	r	PRON
ejpam-3873	165	3	be	be	AUX
ejpam-3873	165	4	a	a	DET
ejpam-3873	165	5	finite	finite	ADJ
ejpam-3873	165	6	division	division	NOUN
ejpam-3873	165	7	ring	ring	NOUN
ejpam-3873	165	8	and	and	CCONJ
ejpam-3873	165	9	let	let	VERB
ejpam-3873	165	10	s	s	AUX
ejpam-3873	165	11	=	=	PUNCT
ejpam-3873	165	12	{	{	PUNCT
ejpam-3873	165	13	x	x	SYM
ejpam-3873	165	14	∈	∈	PROPN
ejpam-3873	165	15	r	r	NOUN
ejpam-3873	165	16	:	:	PUNCT
ejpam-3873	165	17	x2	x2	NOUN
ejpam-3873	165	18	=	=	PUNCT
ejpam-3873	165	19	1r	1r	NUM
ejpam-3873	165	20	or	or	CCONJ
ejpam-3873	165	21	2x	2x	NUM
ejpam-3873	165	22	=	=	SYM
ejpam-3873	165	23	0	0	NUM
ejpam-3873	165	24	}	}	PUNCT
ejpam-3873	165	25	.	.	PUNCT
ejpam-3873	166	1	if	if	SCONJ
ejpam-3873	166	2	c	c	AUX
ejpam-3873	166	3	=	=	PRON
ejpam-3873	166	4	{	{	PUNCT
ejpam-3873	166	5	[	[	X
ejpam-3873	166	6	a1	a1	NOUN
ejpam-3873	166	7	]	]	PUNCT
ejpam-3873	166	8	,	,	PUNCT
ejpam-3873	167	1	[	[	X
ejpam-3873	167	2	a2	a2	X
ejpam-3873	167	3	]	]	PUNCT
ejpam-3873	167	4	,	,	PUNCT
ejpam-3873	167	5	.	.	PUNCT
ejpam-3873	167	6	.	.	PUNCT
ejpam-3873	167	7	.	.	PUNCT
ejpam-3873	168	1	,	,	PUNCT
ejpam-3873	168	2	[	[	X
ejpam-3873	168	3	ac	ac	ADP
ejpam-3873	168	4	]	]	X
ejpam-3873	168	5	}	}	PUNCT
ejpam-3873	168	6	is	be	AUX
ejpam-3873	168	7	the	the	DET
ejpam-3873	168	8	partition	partition	NOUN
ejpam-3873	168	9	of	of	ADP
ejpam-3873	168	10	r\s	r\s	NOUN
ejpam-3873	168	11	in	in	ADP
ejpam-3873	168	12	the	the	DET
ejpam-3873	168	13	sense	sense	NOUN
ejpam-3873	168	14	of	of	ADP
ejpam-3873	168	15	remark	remark	NOUN
ejpam-3873	168	16	3	3	NUM
ejpam-3873	168	17	,	,	PUNCT
ejpam-3873	168	18	then	then	ADV
ejpam-3873	168	19	[	[	X
ejpam-3873	168	20	ai	ai	NOUN
ejpam-3873	168	21	]	]	X
ejpam-3873	168	22	=	=	PRON
ejpam-3873	168	23	{	{	PUNCT
ejpam-3873	168	24	ai,−ai	ai,−ai	NOUN
ejpam-3873	168	25	,	,	PUNCT
ejpam-3873	168	26	a−1i	a−1i	NOUN
ejpam-3873	168	27	,	,	PUNCT
ejpam-3873	168	28	−a−1i	−a−1i	NOUN
ejpam-3873	168	29	,	,	PUNCT
ejpam-3873	168	30	}	}	PUNCT
ejpam-3873	168	31	for	for	ADP
ejpam-3873	168	32	all	all	DET
ejpam-3873	168	33	i	i	PRON
ejpam-3873	168	34	=	=	NOUN
ejpam-3873	168	35	1	1	NUM
ejpam-3873	168	36	,	,	PUNCT
ejpam-3873	168	37	2	2	NUM
ejpam-3873	168	38	,	,	PUNCT
ejpam-3873	168	39	.	.	PUNCT
ejpam-3873	168	40	.	.	PUNCT
ejpam-3873	169	1	.	.	PUNCT
ejpam-3873	170	1	,	,	PUNCT
ejpam-3873	170	2	c.	c.	PROPN
ejpam-3873	170	3	since	since	SCONJ
ejpam-3873	170	4	a2i	a2i	PROPN
ejpam-3873	170	5	6=	6=	SYM
ejpam-3873	170	6	1r	1r	NUM
ejpam-3873	170	7	and	and	CCONJ
ejpam-3873	170	8	2ai	2ai	ADJ
ejpam-3873	170	9	6=	6=	ADP
ejpam-3873	170	10	0	0	NUM
ejpam-3873	170	11	for	for	ADP
ejpam-3873	170	12	all	all	DET
ejpam-3873	170	13	i	i	PROPN
ejpam-3873	170	14	,	,	PUNCT
ejpam-3873	170	15	remark	remark	VERB
ejpam-3873	170	16	2	2	NUM
ejpam-3873	170	17	implies	imply	VERB
ejpam-3873	170	18	that	that	SCONJ
ejpam-3873	170	19	{	{	PUNCT
ejpam-3873	170	20	ai,−a−1i	ai,−a−1i	NOUN
ejpam-3873	170	21	}	}	PUNCT
ejpam-3873	170	22	∩	∩	ADJ
ejpam-3873	170	23	{	{	PUNCT
ejpam-3873	170	24	−ai	−ai	NOUN
ejpam-3873	170	25	,	,	PUNCT
ejpam-3873	170	26	a	a	DET
ejpam-3873	170	27	−1	−1	NOUN
ejpam-3873	170	28	i	i	NOUN
ejpam-3873	170	29	}	}	PUNCT
ejpam-3873	170	30	=	=	PUNCT
ejpam-3873	170	31	∅	∅	NOUN
ejpam-3873	170	32	for	for	ADP
ejpam-3873	170	33	all	all	DET
ejpam-3873	170	34	i.	i.	NOUN
ejpam-3873	170	35	if	if	SCONJ
ejpam-3873	170	36	ai	ai	VERB
ejpam-3873	170	37	=	=	ADJ
ejpam-3873	170	38	−a−1i	−a−1i	PROPN
ejpam-3873	170	39	,	,	PUNCT
ejpam-3873	170	40	then	then	ADV
ejpam-3873	170	41	by	by	ADP
ejpam-3873	170	42	remark	remark	NOUN
ejpam-3873	170	43	2	2	NUM
ejpam-3873	170	44	,	,	PUNCT
ejpam-3873	170	45	−ai	−ai	NOUN
ejpam-3873	170	46	=	=	NOUN
ejpam-3873	170	47	a−1i	a−1i	NOUN
ejpam-3873	170	48	.	.	PUNCT
ejpam-3873	171	1	hence	hence	ADV
ejpam-3873	171	2	,	,	PUNCT
ejpam-3873	171	3	in	in	ADP
ejpam-3873	171	4	this	this	DET
ejpam-3873	171	5	case	case	NOUN
ejpam-3873	171	6	|[ai]|	|[ai]|	NOUN
ejpam-3873	171	7	=	=	PUNCT
ejpam-3873	171	8	2	2	X
ejpam-3873	171	9	.	.	PUNCT
ejpam-3873	171	10	on	on	ADP
ejpam-3873	171	11	the	the	DET
ejpam-3873	171	12	hand	hand	NOUN
ejpam-3873	171	13	,	,	PUNCT
ejpam-3873	171	14	if	if	SCONJ
ejpam-3873	171	15	ai	ai	VERB
ejpam-3873	171	16	6=	6=	NUM
ejpam-3873	171	17	−a−1i	−a−1i	NOUN
ejpam-3873	171	18	,	,	PUNCT
ejpam-3873	171	19	then	then	ADV
ejpam-3873	171	20	by	by	ADP
ejpam-3873	171	21	remark	remark	NOUN
ejpam-3873	171	22	2	2	NUM
ejpam-3873	171	23	,	,	PUNCT
ejpam-3873	171	24	−ai	−ai	NOUN
ejpam-3873	171	25	6=	6=	NOUN
ejpam-3873	171	26	a−1i	a−1i	NOUN
ejpam-3873	171	27	.	.	PUNCT
ejpam-3873	172	1	hence	hence	ADV
ejpam-3873	172	2	,	,	PUNCT
ejpam-3873	172	3	in	in	ADP
ejpam-3873	172	4	this	this	DET
ejpam-3873	172	5	case	case	NOUN
ejpam-3873	172	6	|[ai]|	|[ai]|	NOUN
ejpam-3873	172	7	=	=	PUNCT
ejpam-3873	172	8	4	4	X
ejpam-3873	172	9	.	.	PUNCT
ejpam-3873	172	10	accordingly	accordingly	ADV
ejpam-3873	172	11	,	,	PUNCT
ejpam-3873	172	12	2	2	NUM
ejpam-3873	172	13	≤	≤	NUM
ejpam-3873	172	14	|[ai]|	|[ai]|	NOUN
ejpam-3873	172	15	≤	≤	ADJ
ejpam-3873	172	16	4	4	NUM
ejpam-3873	172	17	.	.	PUNCT
ejpam-3873	172	18	by	by	ADP
ejpam-3873	172	19	theorem	theorem	NOUN
ejpam-3873	172	20	3	3	NUM
ejpam-3873	172	21	,	,	PUNCT
ejpam-3873	172	22	each	each	DET
ejpam-3873	172	23	equivalence	equivalence	NOUN
ejpam-3873	172	24	class	class	NOUN
ejpam-3873	172	25	[	[	X
ejpam-3873	172	26	ai	ai	NOUN
ejpam-3873	172	27	]	]	X
ejpam-3873	172	28	determines	determine	VERB
ejpam-3873	172	29	two	two	NUM
ejpam-3873	172	30	minimum	minimum	ADJ
ejpam-3873	172	31	γ	γ	NOUN
ejpam-3873	172	32	-	-	PUNCT
ejpam-3873	172	33	sets	set	NOUN
ejpam-3873	172	34	.	.	PUNCT
ejpam-3873	173	1	lemma	lemma	PROPN
ejpam-3873	173	2	3	3	X
ejpam-3873	173	3	.	.	PUNCT
ejpam-3873	174	1	let	let	VERB
ejpam-3873	174	2	r	r	PRON
ejpam-3873	174	3	be	be	AUX
ejpam-3873	174	4	a	a	DET
ejpam-3873	174	5	finite	finite	ADJ
ejpam-3873	174	6	division	division	NOUN
ejpam-3873	174	7	ring	ring	NOUN
ejpam-3873	174	8	,	,	PUNCT
ejpam-3873	174	9	and	and	CCONJ
ejpam-3873	174	10	let	let	VERB
ejpam-3873	174	11	s	s	AUX
ejpam-3873	174	12	=	=	PUNCT
ejpam-3873	174	13	{	{	PUNCT
ejpam-3873	174	14	x	x	SYM
ejpam-3873	174	15	∈	∈	PROPN
ejpam-3873	174	16	r	r	NOUN
ejpam-3873	174	17	:	:	PUNCT
ejpam-3873	174	18	x2	x2	NOUN
ejpam-3873	174	19	=	=	PUNCT
ejpam-3873	174	20	1r	1r	NUM
ejpam-3873	174	21	or	or	CCONJ
ejpam-3873	174	22	2x	2x	NUM
ejpam-3873	174	23	=	=	SYM
ejpam-3873	174	24	0	0	NUM
ejpam-3873	174	25	}	}	PUNCT
ejpam-3873	174	26	.	.	PUNCT
ejpam-3873	175	1	then	then	ADV
ejpam-3873	175	2	,	,	PUNCT
ejpam-3873	175	3	c	c	PROPN
ejpam-3873	175	4	≥	≥	X
ejpam-3873	175	5	(	(	PUNCT
ejpam-3873	175	6	|r|	|r|	NOUN
ejpam-3873	175	7	−	−	NOUN
ejpam-3873	175	8	|s|)/4	|s|)/4	ADJ
ejpam-3873	175	9	.	.	PUNCT
ejpam-3873	176	1	proof	proof	NOUN
ejpam-3873	176	2	.	.	PUNCT
ejpam-3873	177	1	let	let	VERB
ejpam-3873	177	2	r	r	PRON
ejpam-3873	177	3	be	be	AUX
ejpam-3873	177	4	a	a	DET
ejpam-3873	177	5	finite	finite	ADJ
ejpam-3873	177	6	division	division	NOUN
ejpam-3873	177	7	ring	ring	NOUN
ejpam-3873	177	8	and	and	CCONJ
ejpam-3873	177	9	let	let	VERB
ejpam-3873	177	10	s	s	AUX
ejpam-3873	177	11	=	=	PUNCT
ejpam-3873	177	12	{	{	PUNCT
ejpam-3873	177	13	x	x	SYM
ejpam-3873	177	14	∈	∈	PROPN
ejpam-3873	177	15	r	r	NOUN
ejpam-3873	177	16	:	:	PUNCT
ejpam-3873	177	17	x2	x2	NOUN
ejpam-3873	177	18	=	=	PUNCT
ejpam-3873	177	19	1r	1r	NUM
ejpam-3873	177	20	or	or	CCONJ
ejpam-3873	177	21	2x	2x	NUM
ejpam-3873	177	22	=	=	SYM
ejpam-3873	177	23	0	0	NUM
ejpam-3873	177	24	}	}	PUNCT
ejpam-3873	177	25	.	.	PUNCT
ejpam-3873	178	1	by	by	ADP
ejpam-3873	178	2	lemma	lemma	PROPN
ejpam-3873	178	3	2	2	NUM
ejpam-3873	178	4	,	,	PUNCT
ejpam-3873	178	5	|[ai]|	|[ai]|	VERB
ejpam-3873	178	6	≤	≤	NUM
ejpam-3873	178	7	4	4	NUM
ejpam-3873	178	8	.	.	PUNCT
ejpam-3873	179	1	therefore	therefore	ADV
ejpam-3873	179	2	,	,	PUNCT
ejpam-3873	179	3	c	c	PROPN
ejpam-3873	179	4	≥	≥	NUM
ejpam-3873	179	5	(	(	PUNCT
ejpam-3873	179	6	|r|	|r|	NOUN
ejpam-3873	179	7	−	−	NOUN
ejpam-3873	179	8	|s|)/4	|s|)/4	PROPN
ejpam-3873	179	9	.	.	PUNCT
ejpam-3873	180	1	e.j	e.j	PROPN
ejpam-3873	180	2	.	.	PROPN
ejpam-3873	180	3	sigasig	sigasig	PROPN
ejpam-3873	180	4	,	,	PUNCT
ejpam-3873	180	5	c.j	c.j	PROPN
ejpam-3873	180	6	.	.	PROPN
ejpam-3873	180	7	rosero	rosero	PROPN
ejpam-3873	180	8	,	,	PUNCT
ejpam-3873	180	9	m.	m.	NOUN
ejpam-3873	180	10	baldado	baldado	PROPN
ejpam-3873	180	11	jr	jr	PROPN
ejpam-3873	180	12	.	.	PROPN
ejpam-3873	180	13	/	/	SYM
ejpam-3873	180	14	eur	eur	PROPN
ejpam-3873	180	15	.	.	PUNCT
ejpam-3873	181	1	j.	j.	PROPN
ejpam-3873	181	2	pure	pure	PROPN
ejpam-3873	181	3	appl	appl	PROPN
ejpam-3873	181	4	.	.	PROPN
ejpam-3873	181	5	math	math	PROPN
ejpam-3873	181	6	,	,	PUNCT
ejpam-3873	181	7	14	14	NUM
ejpam-3873	181	8	(	(	PUNCT
ejpam-3873	181	9	1	1	NUM
ejpam-3873	181	10	)	)	PUNCT
ejpam-3873	181	11	(	(	PUNCT
ejpam-3873	181	12	2021	2021	NUM
ejpam-3873	181	13	)	)	PUNCT
ejpam-3873	181	14	,	,	PUNCT
ejpam-3873	181	15	314	314	NUM
ejpam-3873	181	16	-	-	SYM
ejpam-3873	181	17	326	326	NUM
ejpam-3873	181	18	319	319	NUM
ejpam-3873	181	19	lemma	lemma	PROPN
ejpam-3873	181	20	4	4	X
ejpam-3873	181	21	.	.	PUNCT
ejpam-3873	182	1	let	let	VERB
ejpam-3873	182	2	r	r	PRON
ejpam-3873	182	3	be	be	AUX
ejpam-3873	182	4	a	a	DET
ejpam-3873	182	5	finite	finite	ADJ
ejpam-3873	182	6	division	division	NOUN
ejpam-3873	182	7	ring	ring	NOUN
ejpam-3873	182	8	,	,	PUNCT
ejpam-3873	182	9	and	and	CCONJ
ejpam-3873	182	10	let	let	VERB
ejpam-3873	182	11	s	s	AUX
ejpam-3873	182	12	=	=	PUNCT
ejpam-3873	182	13	{	{	PUNCT
ejpam-3873	182	14	x	x	SYM
ejpam-3873	182	15	∈	∈	PROPN
ejpam-3873	182	16	r	r	NOUN
ejpam-3873	182	17	:	:	PUNCT
ejpam-3873	182	18	x2	x2	NOUN
ejpam-3873	182	19	=	=	PUNCT
ejpam-3873	182	20	1r	1r	NUM
ejpam-3873	182	21	or	or	CCONJ
ejpam-3873	182	22	2x	2x	NUM
ejpam-3873	182	23	=	=	SYM
ejpam-3873	182	24	0	0	NUM
ejpam-3873	182	25	}	}	PUNCT
ejpam-3873	182	26	.	.	PUNCT
ejpam-3873	183	1	then	then	ADV
ejpam-3873	183	2	,	,	PUNCT
ejpam-3873	183	3	c	c	PROPN
ejpam-3873	183	4	≤	≤	NUM
ejpam-3873	183	5	(	(	PUNCT
ejpam-3873	183	6	|r|	|r|	NOUN
ejpam-3873	183	7	−	−	NOUN
ejpam-3873	183	8	|s|)/2	|s|)/2	NOUN
ejpam-3873	183	9	.	.	PUNCT
ejpam-3873	184	1	proof	proof	NOUN
ejpam-3873	184	2	.	.	PUNCT
ejpam-3873	185	1	let	let	VERB
ejpam-3873	185	2	r	r	PRON
ejpam-3873	185	3	be	be	AUX
ejpam-3873	185	4	a	a	DET
ejpam-3873	185	5	finite	finite	ADJ
ejpam-3873	185	6	division	division	NOUN
ejpam-3873	185	7	ring	ring	NOUN
ejpam-3873	185	8	and	and	CCONJ
ejpam-3873	185	9	let	let	VERB
ejpam-3873	185	10	s	s	AUX
ejpam-3873	185	11	=	=	PUNCT
ejpam-3873	185	12	{	{	PUNCT
ejpam-3873	185	13	x	x	SYM
ejpam-3873	185	14	∈	∈	PROPN
ejpam-3873	185	15	r	r	NOUN
ejpam-3873	185	16	:	:	PUNCT
ejpam-3873	185	17	x2	x2	NOUN
ejpam-3873	185	18	=	=	PUNCT
ejpam-3873	185	19	1r	1r	NUM
ejpam-3873	185	20	or	or	CCONJ
ejpam-3873	185	21	2x	2x	NUM
ejpam-3873	185	22	=	=	SYM
ejpam-3873	185	23	0	0	NUM
ejpam-3873	185	24	}	}	PUNCT
ejpam-3873	185	25	.	.	PUNCT
ejpam-3873	186	1	by	by	ADP
ejpam-3873	186	2	lemma	lemma	PROPN
ejpam-3873	186	3	2	2	NUM
ejpam-3873	186	4	,	,	PUNCT
ejpam-3873	186	5	2	2	NUM
ejpam-3873	186	6	≤	≤	NUM
ejpam-3873	186	7	|[ai]|	|[ai]|	NOUN
ejpam-3873	186	8	.	.	PUNCT
ejpam-3873	187	1	therefore	therefore	ADV
ejpam-3873	187	2	,	,	PUNCT
ejpam-3873	187	3	c	c	PROPN
ejpam-3873	187	4	≤	≤	NUM
ejpam-3873	187	5	(	(	PUNCT
ejpam-3873	187	6	|r|	|r|	NOUN
ejpam-3873	187	7	−	−	NOUN
ejpam-3873	187	8	|s|)/2	|s|)/2	PROPN
ejpam-3873	187	9	.	.	PUNCT
ejpam-3873	188	1	theorem	theorem	VERB
ejpam-3873	188	2	6	6	NUM
ejpam-3873	188	3	give	give	VERB
ejpam-3873	188	4	a	a	DET
ejpam-3873	188	5	necessary	necessary	ADJ
ejpam-3873	188	6	and	and	CCONJ
ejpam-3873	188	7	sufficient	sufficient	ADJ
ejpam-3873	188	8	condition	condition	NOUN
ejpam-3873	188	9	for	for	ADP
ejpam-3873	188	10	a	a	DET
ejpam-3873	188	11	γ	γ	X
ejpam-3873	188	12	-	-	PUNCT
ejpam-3873	188	13	set	set	NOUN
ejpam-3873	188	14	to	to	PART
ejpam-3873	188	15	be	be	AUX
ejpam-3873	188	16	minimum	minimum	ADJ
ejpam-3873	188	17	.	.	PUNCT
ejpam-3873	189	1	theorem	theorem	VERB
ejpam-3873	189	2	6	6	NUM
ejpam-3873	189	3	.	.	PUNCT
ejpam-3873	190	1	let	let	VERB
ejpam-3873	190	2	r	r	PRON
ejpam-3873	190	3	be	be	AUX
ejpam-3873	190	4	a	a	DET
ejpam-3873	190	5	finite	finite	ADJ
ejpam-3873	190	6	division	division	NOUN
ejpam-3873	190	7	ring	ring	NOUN
ejpam-3873	190	8	.	.	PUNCT
ejpam-3873	191	1	then	then	ADV
ejpam-3873	191	2	,	,	PUNCT
ejpam-3873	191	3	e	e	X
ejpam-3873	191	4	is	be	AUX
ejpam-3873	191	5	a	a	DET
ejpam-3873	191	6	minimum	minimum	ADJ
ejpam-3873	191	7	γ	γ	X
ejpam-3873	191	8	-	-	NOUN
ejpam-3873	191	9	set	set	NOUN
ejpam-3873	191	10	of	of	ADP
ejpam-3873	191	11	r	r	NOUN
ejpam-3873	191	12	if	if	SCONJ
ejpam-3873	192	1	and	and	CCONJ
ejpam-3873	192	2	only	only	ADV
ejpam-3873	192	3	if	if	SCONJ
ejpam-3873	192	4	e	e	X
ejpam-3873	192	5	=	=	SYM
ejpam-3873	192	6	s	s	PART
ejpam-3873	192	7	∪	∪	X
ejpam-3873	192	8	{	{	PUNCT
ejpam-3873	192	9	−x1	−x1	NOUN
ejpam-3873	192	10	,	,	PUNCT
ejpam-3873	192	11	x−11	x−11	X
ejpam-3873	192	12	,	,	PUNCT
ejpam-3873	192	13	−x2	−x2	PROPN
ejpam-3873	192	14	,	,	PUNCT
ejpam-3873	192	15	x−12	x−12	PROPN
ejpam-3873	192	16	,	,	PUNCT
ejpam-3873	192	17	.	.	PUNCT
ejpam-3873	192	18	.	.	PUNCT
ejpam-3873	192	19	.	.	PUNCT
ejpam-3873	193	1	,	,	PUNCT
ejpam-3873	193	2	−xc	−xc	NUM
ejpam-3873	193	3	,	,	PUNCT
ejpam-3873	193	4	x−1c	x−1c	NOUN
ejpam-3873	193	5	}	}	PUNCT
ejpam-3873	193	6	where	where	SCONJ
ejpam-3873	193	7	xi	xi	X
ejpam-3873	193	8	∈	∈	PROPN
ejpam-3873	193	9	[	[	X
ejpam-3873	193	10	ai	ai	X
ejpam-3873	193	11	]	]	X
ejpam-3873	193	12	for	for	ADP
ejpam-3873	193	13	i	i	PROPN
ejpam-3873	193	14	=	=	NOUN
ejpam-3873	193	15	1	1	NUM
ejpam-3873	193	16	,	,	PUNCT
ejpam-3873	193	17	2	2	NUM
ejpam-3873	193	18	,	,	PUNCT
ejpam-3873	193	19	.	.	PUNCT
ejpam-3873	193	20	.	.	PUNCT
ejpam-3873	194	1	.	.	PUNCT
ejpam-3873	195	1	,	,	PUNCT
ejpam-3873	195	2	c	c	X
ejpam-3873	195	3	,	,	PUNCT
ejpam-3873	195	4	and	and	CCONJ
ejpam-3873	195	5	{	{	PUNCT
ejpam-3873	196	1	[	[	X
ejpam-3873	196	2	a1	a1	NOUN
ejpam-3873	196	3	]	]	PUNCT
ejpam-3873	196	4	,	,	PUNCT
ejpam-3873	197	1	[	[	X
ejpam-3873	197	2	a2	a2	X
ejpam-3873	197	3	]	]	PUNCT
ejpam-3873	197	4	,	,	PUNCT
ejpam-3873	197	5	.	.	PUNCT
ejpam-3873	197	6	.	.	PUNCT
ejpam-3873	197	7	.	.	PUNCT
ejpam-3873	198	1	,	,	PUNCT
ejpam-3873	198	2	[	[	X
ejpam-3873	198	3	ac	ac	ADP
ejpam-3873	198	4	]	]	X
ejpam-3873	198	5	}	}	PUNCT
ejpam-3873	198	6	is	be	AUX
ejpam-3873	198	7	the	the	DET
ejpam-3873	198	8	partition	partition	NOUN
ejpam-3873	198	9	of	of	ADP
ejpam-3873	198	10	r\s	r\s	NOUN
ejpam-3873	198	11	in	in	ADP
ejpam-3873	198	12	the	the	DET
ejpam-3873	198	13	sense	sense	NOUN
ejpam-3873	198	14	of	of	ADP
ejpam-3873	198	15	remark	remark	NOUN
ejpam-3873	198	16	3	3	NUM
ejpam-3873	198	17	.	.	PUNCT
ejpam-3873	199	1	proof	proof	NOUN
ejpam-3873	199	2	.	.	PUNCT
ejpam-3873	200	1	suppose	suppose	VERB
ejpam-3873	200	2	that	that	SCONJ
ejpam-3873	200	3	e	e	PROPN
ejpam-3873	200	4	is	be	AUX
ejpam-3873	200	5	a	a	DET
ejpam-3873	200	6	minimum	minimum	ADJ
ejpam-3873	200	7	γ	γ	X
ejpam-3873	200	8	-	-	PUNCT
ejpam-3873	200	9	set	set	NOUN
ejpam-3873	200	10	of	of	ADP
ejpam-3873	200	11	r	r	NOUN
ejpam-3873	200	12	and	and	CCONJ
ejpam-3873	200	13	e	e	NOUN
ejpam-3873	200	14	is	be	AUX
ejpam-3873	200	15	not	not	PART
ejpam-3873	200	16	of	of	ADP
ejpam-3873	200	17	the	the	DET
ejpam-3873	200	18	form	form	NOUN
ejpam-3873	200	19	s	s	PART
ejpam-3873	200	20	∪	∪	X
ejpam-3873	200	21	{	{	PUNCT
ejpam-3873	200	22	−x1	−x1	NOUN
ejpam-3873	200	23	,	,	PUNCT
ejpam-3873	200	24	x−11	x−11	X
ejpam-3873	200	25	,	,	PUNCT
ejpam-3873	200	26	−x2	−x2	PROPN
ejpam-3873	200	27	,	,	PUNCT
ejpam-3873	200	28	x−12	x−12	PROPN
ejpam-3873	200	29	,	,	PUNCT
ejpam-3873	200	30	.	.	PUNCT
ejpam-3873	200	31	.	.	PUNCT
ejpam-3873	201	1	.	.	PUNCT
ejpam-3873	202	1	,	,	PUNCT
ejpam-3873	202	2	−xc	−xc	NUM
ejpam-3873	202	3	,	,	PUNCT
ejpam-3873	202	4	x−1c	x−1c	PUNCT
ejpam-3873	202	5	}	}	PUNCT
ejpam-3873	202	6	.	.	PUNCT
ejpam-3873	203	1	if	if	SCONJ
ejpam-3873	203	2	e	e	NOUN
ejpam-3873	203	3	is	be	AUX
ejpam-3873	203	4	not	not	PART
ejpam-3873	203	5	of	of	ADP
ejpam-3873	203	6	the	the	DET
ejpam-3873	203	7	form	form	NOUN
ejpam-3873	203	8	s	s	PART
ejpam-3873	203	9	∪	∪	X
ejpam-3873	203	10	{	{	PUNCT
ejpam-3873	203	11	−x1	−x1	NOUN
ejpam-3873	203	12	,	,	PUNCT
ejpam-3873	203	13	x−11	x−11	X
ejpam-3873	203	14	,	,	PUNCT
ejpam-3873	203	15	−x2	−x2	PROPN
ejpam-3873	203	16	,	,	PUNCT
ejpam-3873	203	17	x−12	x−12	PROPN
ejpam-3873	203	18	,	,	PUNCT
ejpam-3873	203	19	.	.	PUNCT
ejpam-3873	203	20	.	.	PUNCT
ejpam-3873	204	1	.	.	PUNCT
ejpam-3873	205	1	,	,	PUNCT
ejpam-3873	205	2	−xc	−xc	NUM
ejpam-3873	205	3	,	,	PUNCT
ejpam-3873	205	4	x−1c	x−1c	NUM
ejpam-3873	205	5	}	}	PUNCT
ejpam-3873	205	6	,	,	PUNCT
ejpam-3873	205	7	then	then	ADV
ejpam-3873	205	8	there	there	PRON
ejpam-3873	205	9	exists	exist	VERB
ejpam-3873	205	10	i	i	PRON
ejpam-3873	205	11	∈	∈	PROPN
ejpam-3873	205	12	{	{	PUNCT
ejpam-3873	205	13	1	1	NUM
ejpam-3873	205	14	,	,	PUNCT
ejpam-3873	205	15	2	2	NUM
ejpam-3873	205	16	,	,	PUNCT
ejpam-3873	205	17	.	.	PUNCT
ejpam-3873	205	18	.	.	PUNCT
ejpam-3873	206	1	.	.	PUNCT
ejpam-3873	207	1	,	,	PUNCT
ejpam-3873	207	2	j	j	NOUN
ejpam-3873	207	3	}	}	PUNCT
ejpam-3873	208	1	such	such	ADJ
ejpam-3873	208	2	that	that	SCONJ
ejpam-3873	208	3	−xi	−xi	PROPN
ejpam-3873	208	4	,	,	PUNCT
ejpam-3873	208	5	x−1i	x−1i	PROPN
ejpam-3873	208	6	,	,	PUNCT
ejpam-3873	208	7	(	(	PUNCT
ejpam-3873	208	8	−xi)−1	−xi)−1	NOUN
ejpam-3873	208	9	∈	∈	NOUN
ejpam-3873	208	10	e	e	NOUN
ejpam-3873	208	11	or	or	CCONJ
ejpam-3873	208	12	xi,−xi	xi,−xi	PROPN
ejpam-3873	208	13	,	,	PUNCT
ejpam-3873	208	14	x−1i	x−1i	PROPN
ejpam-3873	208	15	,	,	PUNCT
ejpam-3873	208	16	(	(	PUNCT
ejpam-3873	208	17	−xi)−1	−xi)−1	NOUN
ejpam-3873	208	18	∈	∈	PROPN
ejpam-3873	208	19	e.	e.	PROPN
ejpam-3873	208	20	thus	thus	ADV
ejpam-3873	208	21	,	,	PUNCT
ejpam-3873	208	22	s	s	VERB
ejpam-3873	208	23	∪	∪	X
ejpam-3873	208	24	{	{	PUNCT
ejpam-3873	208	25	−x1	−x1	NOUN
ejpam-3873	208	26	,	,	PUNCT
ejpam-3873	208	27	x−11	x−11	X
ejpam-3873	208	28	,	,	PUNCT
ejpam-3873	208	29	−x2	−x2	PROPN
ejpam-3873	208	30	,	,	PUNCT
ejpam-3873	208	31	x−12	x−12	PROPN
ejpam-3873	208	32	,	,	PUNCT
ejpam-3873	208	33	.	.	PUNCT
ejpam-3873	208	34	.	.	PUNCT
ejpam-3873	209	1	.	.	PUNCT
ejpam-3873	210	1	,	,	PUNCT
ejpam-3873	210	2	−xc	−xc	NUM
ejpam-3873	210	3	,	,	PUNCT
ejpam-3873	210	4	x−1c	x−1c	X
ejpam-3873	210	5	}	}	PUNCT
ejpam-3873	210	6	is	be	AUX
ejpam-3873	210	7	a	a	DET
ejpam-3873	210	8	γ	γ	NOUN
ejpam-3873	210	9	-	-	PUNCT
ejpam-3873	210	10	set	set	VERB
ejpam-3873	210	11	smaller	small	ADJ
ejpam-3873	210	12	than	than	SCONJ
ejpam-3873	210	13	e.	e.	PROPN
ejpam-3873	210	14	this	this	PRON
ejpam-3873	210	15	is	be	AUX
ejpam-3873	210	16	a	a	DET
ejpam-3873	210	17	contradiction	contradiction	NOUN
ejpam-3873	210	18	.	.	PUNCT
ejpam-3873	211	1	conversely	conversely	ADV
ejpam-3873	211	2	,	,	PUNCT
ejpam-3873	211	3	suppose	suppose	VERB
ejpam-3873	211	4	that	that	SCONJ
ejpam-3873	211	5	e	e	PROPN
ejpam-3873	211	6	is	be	AUX
ejpam-3873	211	7	of	of	ADP
ejpam-3873	211	8	the	the	DET
ejpam-3873	211	9	form	form	NOUN
ejpam-3873	211	10	s	s	PART
ejpam-3873	211	11	∪	∪	X
ejpam-3873	211	12	{	{	PUNCT
ejpam-3873	211	13	−x1	−x1	NOUN
ejpam-3873	211	14	,	,	PUNCT
ejpam-3873	211	15	x−11	x−11	X
ejpam-3873	211	16	,	,	PUNCT
ejpam-3873	211	17	−x2	−x2	PROPN
ejpam-3873	211	18	,	,	PUNCT
ejpam-3873	211	19	x−12	x−12	PROPN
ejpam-3873	211	20	,	,	PUNCT
ejpam-3873	211	21	.	.	PUNCT
ejpam-3873	211	22	.	.	PUNCT
ejpam-3873	211	23	.	.	PUNCT
ejpam-3873	212	1	,	,	PUNCT
ejpam-3873	212	2	−xc	−xc	NUM
ejpam-3873	212	3	,	,	PUNCT
ejpam-3873	212	4	x−1c	x−1c	PUNCT
ejpam-3873	212	5	}	}	PUNCT
ejpam-3873	212	6	and	and	CCONJ
ejpam-3873	212	7	e	e	NOUN
ejpam-3873	212	8	is	be	AUX
ejpam-3873	212	9	not	not	PART
ejpam-3873	212	10	a	a	DET
ejpam-3873	212	11	minimum	minimum	ADJ
ejpam-3873	212	12	γ	γ	X
ejpam-3873	212	13	-	-	PUNCT
ejpam-3873	212	14	set	set	NOUN
ejpam-3873	212	15	of	of	ADP
ejpam-3873	212	16	r.	r.	PROPN
ejpam-3873	212	17	if	if	SCONJ
ejpam-3873	212	18	e	e	PROPN
ejpam-3873	212	19	is	be	AUX
ejpam-3873	212	20	not	not	PART
ejpam-3873	212	21	a	a	DET
ejpam-3873	212	22	minimum	minimum	ADJ
ejpam-3873	212	23	γ	γ	X
ejpam-3873	212	24	-	-	PUNCT
ejpam-3873	212	25	set	set	NOUN
ejpam-3873	212	26	of	of	ADP
ejpam-3873	212	27	r	r	NOUN
ejpam-3873	212	28	,	,	PUNCT
ejpam-3873	212	29	then	then	ADV
ejpam-3873	212	30	then	then	ADV
ejpam-3873	212	31	there	there	PRON
ejpam-3873	212	32	exists	exist	VERB
ejpam-3873	212	33	i	i	PRON
ejpam-3873	212	34	∈	∈	PROPN
ejpam-3873	212	35	{	{	PUNCT
ejpam-3873	212	36	1	1	NUM
ejpam-3873	212	37	,	,	PUNCT
ejpam-3873	212	38	2	2	NUM
ejpam-3873	212	39	,	,	PUNCT
ejpam-3873	212	40	.	.	PUNCT
ejpam-3873	212	41	.	.	PUNCT
ejpam-3873	213	1	.	.	PUNCT
ejpam-3873	214	1	,	,	PUNCT
ejpam-3873	214	2	j	j	NOUN
ejpam-3873	214	3	}	}	PUNCT
ejpam-3873	214	4	such	such	ADJ
ejpam-3873	214	5	that	that	SCONJ
ejpam-3873	214	6	xi,−xi	xi,−xi	PROPN
ejpam-3873	214	7	,	,	PUNCT
ejpam-3873	214	8	x−1i	x−1i	PROPN
ejpam-3873	214	9	/∈	/∈	PROPN
ejpam-3873	215	1	e.	e.	PROPN
ejpam-3873	215	2	since	since	SCONJ
ejpam-3873	215	3	e	e	PROPN
ejpam-3873	215	4	is	be	AUX
ejpam-3873	215	5	a	a	DET
ejpam-3873	215	6	γ	γ	NOUN
ejpam-3873	215	7	-	-	PUNCT
ejpam-3873	215	8	set	set	VERB
ejpam-3873	215	9	and	and	CCONJ
ejpam-3873	215	10	x−1i	x−1i	NOUN
ejpam-3873	215	11	,	,	PUNCT
ejpam-3873	215	12	xi	xi	X
ejpam-3873	215	13	=	=	SYM
ejpam-3873	215	14	(	(	PUNCT
ejpam-3873	215	15	x−1i	x−1i	NUM
ejpam-3873	215	16	)	)	PUNCT
ejpam-3873	215	17	−1	−1	NOUN
ejpam-3873	216	1	∈	∈	PROPN
ejpam-3873	216	2	e.	e.	PROPN
ejpam-3873	216	3	this	this	PRON
ejpam-3873	216	4	is	be	AUX
ejpam-3873	216	5	a	a	DET
ejpam-3873	216	6	contradiction	contradiction	NOUN
ejpam-3873	216	7	.	.	PUNCT
ejpam-3873	217	1	the	the	DET
ejpam-3873	217	2	index	index	NOUN
ejpam-3873	217	3	minimum	minimum	NOUN
ejpam-3873	217	4	of	of	ADP
ejpam-3873	217	5	a	a	DET
ejpam-3873	217	6	finite	finite	NOUN
ejpam-3873	217	7	ring	ring	NOUN
ejpam-3873	217	8	r	r	NOUN
ejpam-3873	217	9	is	be	AUX
ejpam-3873	217	10	the	the	DET
ejpam-3873	217	11	number	number	NOUN
ejpam-3873	217	12	of	of	ADP
ejpam-3873	217	13	minimum	minimum	ADJ
ejpam-3873	217	14	γ	γ	NOUN
ejpam-3873	217	15	-	-	PUNCT
ejpam-3873	217	16	sets	set	NOUN
ejpam-3873	217	17	of	of	ADP
ejpam-3873	217	18	r	r	NOUN
ejpam-3873	217	19	and	and	CCONJ
ejpam-3873	217	20	is	be	AUX
ejpam-3873	217	21	denoted	denote	VERB
ejpam-3873	217	22	by	by	ADP
ejpam-3873	217	23	ind(r	ind(r	NOUN
ejpam-3873	217	24	)	)	PUNCT
ejpam-3873	217	25	.	.	PUNCT
ejpam-3873	218	1	corollary	corollary	ADJ
ejpam-3873	218	2	1	1	NUM
ejpam-3873	218	3	gives	give	VERB
ejpam-3873	218	4	an	an	DET
ejpam-3873	218	5	upper	upper	ADJ
ejpam-3873	218	6	bound	bind	VERB
ejpam-3873	218	7	on	on	ADP
ejpam-3873	218	8	the	the	DET
ejpam-3873	218	9	number	number	NOUN
ejpam-3873	218	10	of	of	ADP
ejpam-3873	218	11	minimum	minimum	ADJ
ejpam-3873	218	12	γ	γ	X
ejpam-3873	218	13	-	-	PUNCT
ejpam-3873	218	14	set	set	NOUN
ejpam-3873	218	15	of	of	ADP
ejpam-3873	218	16	a	a	DET
ejpam-3873	218	17	finite	finite	ADJ
ejpam-3873	218	18	division	division	NOUN
ejpam-3873	218	19	ring	ring	NOUN
ejpam-3873	218	20	,	,	PUNCT
ejpam-3873	218	21	while	while	SCONJ
ejpam-3873	218	22	corollary	corollary	ADJ
ejpam-3873	218	23	2	2	NUM
ejpam-3873	218	24	gives	give	VERB
ejpam-3873	218	25	a	a	DET
ejpam-3873	218	26	lower	low	ADJ
ejpam-3873	218	27	bound	bind	VERB
ejpam-3873	218	28	on	on	ADP
ejpam-3873	218	29	the	the	DET
ejpam-3873	218	30	number	number	NOUN
ejpam-3873	218	31	of	of	ADP
ejpam-3873	218	32	minimum	minimum	ADJ
ejpam-3873	218	33	γ	γ	X
ejpam-3873	218	34	-	-	PUNCT
ejpam-3873	218	35	set	set	NOUN
ejpam-3873	218	36	of	of	ADP
ejpam-3873	218	37	a	a	DET
ejpam-3873	218	38	finite	finite	ADJ
ejpam-3873	218	39	division	division	NOUN
ejpam-3873	218	40	ring	ring	NOUN
ejpam-3873	218	41	.	.	PUNCT
ejpam-3873	219	1	corollary	corollary	ADJ
ejpam-3873	219	2	1	1	NUM
ejpam-3873	219	3	.	.	PUNCT
ejpam-3873	220	1	let	let	VERB
ejpam-3873	220	2	r	r	PRON
ejpam-3873	220	3	be	be	AUX
ejpam-3873	220	4	a	a	DET
ejpam-3873	220	5	finite	finite	ADJ
ejpam-3873	220	6	division	division	NOUN
ejpam-3873	220	7	ring	ring	NOUN
ejpam-3873	220	8	.	.	PUNCT
ejpam-3873	221	1	then	then	ADV
ejpam-3873	221	2	ind(r	ind(r	PROPN
ejpam-3873	221	3	)	)	PUNCT
ejpam-3873	221	4	≤	≤	NOUN
ejpam-3873	221	5	2(|r|−|s|)/2	2(|r|−|s|)/2	NUM
ejpam-3873	221	6	.	.	PUNCT
ejpam-3873	222	1	proof	proof	NOUN
ejpam-3873	222	2	.	.	PUNCT
ejpam-3873	223	1	letr	letr	PROPN
ejpam-3873	223	2	be	be	AUX
ejpam-3873	223	3	a	a	DET
ejpam-3873	223	4	finite	finite	ADJ
ejpam-3873	223	5	division	division	NOUN
ejpam-3873	223	6	ring	ring	NOUN
ejpam-3873	223	7	and	and	CCONJ
ejpam-3873	223	8	let	let	VERB
ejpam-3873	223	9	s	s	AUX
ejpam-3873	223	10	=	=	PUNCT
ejpam-3873	223	11	{	{	PUNCT
ejpam-3873	223	12	x	x	SYM
ejpam-3873	223	13	∈	∈	PROPN
ejpam-3873	223	14	r	r	NOUN
ejpam-3873	223	15	:	:	PUNCT
ejpam-3873	223	16	x2	x2	NOUN
ejpam-3873	223	17	=	=	PUNCT
ejpam-3873	223	18	1r	1r	NUM
ejpam-3873	223	19	or	or	CCONJ
ejpam-3873	223	20	2x	2x	NUM
ejpam-3873	223	21	=	=	SYM
ejpam-3873	223	22	0	0	NUM
ejpam-3873	223	23	}	}	PUNCT
ejpam-3873	223	24	.	.	PUNCT
ejpam-3873	224	1	then	then	ADV
ejpam-3873	224	2	by	by	ADP
ejpam-3873	224	3	lemma	lemma	PROPN
ejpam-3873	224	4	4	4	NUM
ejpam-3873	224	5	,	,	PUNCT
ejpam-3873	224	6	c	c	NOUN
ejpam-3873	224	7	≤	≤	NUM
ejpam-3873	224	8	(	(	PUNCT
ejpam-3873	224	9	|r|−|s|)/2	|r|−|s|)/2	NOUN
ejpam-3873	224	10	.	.	PUNCT
ejpam-3873	225	1	moreover	moreover	ADV
ejpam-3873	225	2	,	,	PUNCT
ejpam-3873	225	3	by	by	ADP
ejpam-3873	225	4	theorem	theorem	NOUN
ejpam-3873	225	5	3	3	NUM
ejpam-3873	225	6	,	,	PUNCT
ejpam-3873	225	7	each	each	DET
ejpam-3873	225	8	equivalence	equivalence	NOUN
ejpam-3873	225	9	class	class	NOUN
ejpam-3873	225	10	[	[	X
ejpam-3873	225	11	ai	ai	NOUN
ejpam-3873	225	12	]	]	X
ejpam-3873	225	13	determines	determine	VERB
ejpam-3873	225	14	two	two	NUM
ejpam-3873	225	15	minimum	minimum	ADJ
ejpam-3873	225	16	γ	γ	NOUN
ejpam-3873	225	17	-	-	PUNCT
ejpam-3873	225	18	sets	set	NOUN
ejpam-3873	225	19	.	.	PUNCT
ejpam-3873	226	1	therefore	therefore	ADV
ejpam-3873	226	2	,	,	PUNCT
ejpam-3873	226	3	by	by	ADP
ejpam-3873	226	4	the	the	DET
ejpam-3873	226	5	multiplication	multiplication	NOUN
ejpam-3873	226	6	principle	principle	NOUN
ejpam-3873	226	7	,	,	PUNCT
ejpam-3873	226	8	ind(r	ind(r	NOUN
ejpam-3873	226	9	)	)	PUNCT
ejpam-3873	226	10	≤	≤	NOUN
ejpam-3873	226	11	2(|r|−|s|)/2	2(|r|−|s|)/2	NUM
ejpam-3873	226	12	.	.	PUNCT
ejpam-3873	227	1	the	the	DET
ejpam-3873	227	2	bound	bind	VERB
ejpam-3873	227	3	in	in	ADP
ejpam-3873	227	4	corollary	corollary	ADJ
ejpam-3873	227	5	1	1	NUM
ejpam-3873	227	6	is	be	AUX
ejpam-3873	227	7	sharp	sharp	ADJ
ejpam-3873	227	8	.	.	PUNCT
ejpam-3873	228	1	equality	equality	NOUN
ejpam-3873	228	2	holds	hold	VERB
ejpam-3873	228	3	for	for	ADP
ejpam-3873	228	4	some	some	DET
ejpam-3873	228	5	fields	field	NOUN
ejpam-3873	228	6	.	.	PUNCT
ejpam-3873	229	1	for	for	ADP
ejpam-3873	229	2	example	example	NOUN
ejpam-3873	229	3	,	,	PUNCT
ejpam-3873	229	4	if	if	SCONJ
ejpam-3873	229	5	r	r	NOUN
ejpam-3873	229	6	is	be	AUX
ejpam-3873	229	7	z5	z5	PROPN
ejpam-3873	229	8	,	,	PUNCT
ejpam-3873	229	9	then	then	ADV
ejpam-3873	229	10	the	the	DET
ejpam-3873	229	11	equality	equality	NOUN
ejpam-3873	229	12	holds	hold	VERB
ejpam-3873	229	13	.	.	PUNCT
ejpam-3873	230	1	to	to	PART
ejpam-3873	230	2	see	see	VERB
ejpam-3873	230	3	this	this	PRON
ejpam-3873	230	4	,	,	PUNCT
ejpam-3873	230	5	we	we	PRON
ejpam-3873	230	6	note	note	VERB
ejpam-3873	230	7	that	that	SCONJ
ejpam-3873	230	8	the	the	DET
ejpam-3873	230	9	minimum	minimum	ADJ
ejpam-3873	230	10	γ	γ	NOUN
ejpam-3873	230	11	-	-	PUNCT
ejpam-3873	230	12	sets	set	NOUN
ejpam-3873	230	13	of	of	ADP
ejpam-3873	230	14	z5	z5	NOUN
ejpam-3873	230	15	are	be	AUX
ejpam-3873	230	16	j1	j1	NOUN
ejpam-3873	230	17	=	=	PUNCT
ejpam-3873	230	18	{	{	PUNCT
ejpam-3873	230	19	0	0	NUM
ejpam-3873	230	20	,	,	PUNCT
ejpam-3873	230	21	1	1	NUM
ejpam-3873	230	22	,	,	PUNCT
ejpam-3873	230	23	4	4	NUM
ejpam-3873	230	24	,	,	PUNCT
ejpam-3873	230	25	3	3	NUM
ejpam-3873	230	26	}	}	PUNCT
ejpam-3873	230	27	and	and	CCONJ
ejpam-3873	230	28	j1	j1	PROPN
ejpam-3873	230	29	=	=	PUNCT
ejpam-3873	230	30	{	{	PUNCT
ejpam-3873	230	31	0	0	NUM
ejpam-3873	230	32	,	,	PUNCT
ejpam-3873	230	33	1	1	NUM
ejpam-3873	230	34	,	,	PUNCT
ejpam-3873	230	35	4	4	NUM
ejpam-3873	230	36	,	,	PUNCT
ejpam-3873	230	37	3	3	NUM
ejpam-3873	230	38	}	}	PUNCT
ejpam-3873	230	39	.	.	PUNCT
ejpam-3873	231	1	hence	hence	ADV
ejpam-3873	231	2	,	,	PUNCT
ejpam-3873	231	3	ind(z5	ind(z5	PROPN
ejpam-3873	231	4	)	)	PUNCT
ejpam-3873	231	5	=	=	SYM
ejpam-3873	231	6	2	2	X
ejpam-3873	231	7	.	.	PUNCT
ejpam-3873	232	1	this	this	PRON
ejpam-3873	232	2	is	be	AUX
ejpam-3873	232	3	equal	equal	ADJ
ejpam-3873	232	4	to	to	ADP
ejpam-3873	232	5	2(|r|−|s|)/2	2(|r|−|s|)/2	NUM
ejpam-3873	232	6	=	=	SYM
ejpam-3873	232	7	2(|z5|−|{0,1,4}|)/2	2(|z5|−|{0,1,4}|)/2	NUM
ejpam-3873	232	8	=	=	SYM
ejpam-3873	232	9	2(5−3)/2	2(5−3)/2	NUM
ejpam-3873	232	10	=	=	SYM
ejpam-3873	232	11	2	2	X
ejpam-3873	232	12	.	.	PUNCT
ejpam-3873	232	13	corollary	corollary	ADJ
ejpam-3873	232	14	2	2	NUM
ejpam-3873	232	15	.	.	PUNCT
ejpam-3873	233	1	let	let	VERB
ejpam-3873	233	2	r	r	PRON
ejpam-3873	233	3	be	be	AUX
ejpam-3873	233	4	a	a	DET
ejpam-3873	233	5	finite	finite	ADJ
ejpam-3873	233	6	division	division	NOUN
ejpam-3873	233	7	ring	ring	NOUN
ejpam-3873	233	8	.	.	PUNCT
ejpam-3873	234	1	then	then	ADV
ejpam-3873	234	2	ind(r	ind(r	PROPN
ejpam-3873	234	3	)	)	PUNCT
ejpam-3873	234	4	≥	≥	NOUN
ejpam-3873	234	5	2(|r|−|s|)/4	2(|r|−|s|)/4	NUM
ejpam-3873	234	6	.	.	PUNCT
ejpam-3873	235	1	proof	proof	NOUN
ejpam-3873	235	2	.	.	PUNCT
ejpam-3873	236	1	letr	letr	PROPN
ejpam-3873	236	2	be	be	AUX
ejpam-3873	236	3	a	a	DET
ejpam-3873	236	4	finite	finite	ADJ
ejpam-3873	236	5	division	division	NOUN
ejpam-3873	236	6	ring	ring	NOUN
ejpam-3873	236	7	and	and	CCONJ
ejpam-3873	236	8	let	let	VERB
ejpam-3873	236	9	s	s	AUX
ejpam-3873	236	10	=	=	PUNCT
ejpam-3873	236	11	{	{	PUNCT
ejpam-3873	236	12	x	x	SYM
ejpam-3873	236	13	∈	∈	PROPN
ejpam-3873	236	14	r	r	NOUN
ejpam-3873	236	15	:	:	PUNCT
ejpam-3873	236	16	x2	x2	NOUN
ejpam-3873	236	17	=	=	PUNCT
ejpam-3873	236	18	1r	1r	NUM
ejpam-3873	236	19	or	or	CCONJ
ejpam-3873	236	20	2x	2x	NUM
ejpam-3873	236	21	=	=	SYM
ejpam-3873	236	22	0	0	NUM
ejpam-3873	236	23	}	}	PUNCT
ejpam-3873	236	24	.	.	PUNCT
ejpam-3873	237	1	then	then	ADV
ejpam-3873	237	2	by	by	ADP
ejpam-3873	237	3	lemma	lemma	PROPN
ejpam-3873	237	4	3	3	NUM
ejpam-3873	237	5	,	,	PUNCT
ejpam-3873	237	6	c	c	X
ejpam-3873	237	7	≥	≥	X
ejpam-3873	237	8	(	(	PUNCT
ejpam-3873	237	9	|r|−|s|)/4	|r|−|s|)/4	NOUN
ejpam-3873	237	10	.	.	PUNCT
ejpam-3873	238	1	moreover	moreover	ADV
ejpam-3873	238	2	,	,	PUNCT
ejpam-3873	238	3	by	by	ADP
ejpam-3873	238	4	theorem	theorem	NOUN
ejpam-3873	238	5	3	3	NUM
ejpam-3873	238	6	,	,	PUNCT
ejpam-3873	238	7	each	each	DET
ejpam-3873	238	8	equivalence	equivalence	NOUN
ejpam-3873	238	9	class	class	NOUN
ejpam-3873	238	10	[	[	X
ejpam-3873	238	11	ai	ai	NOUN
ejpam-3873	238	12	]	]	X
ejpam-3873	238	13	determines	determine	VERB
ejpam-3873	238	14	two	two	NUM
ejpam-3873	238	15	minimum	minimum	ADJ
ejpam-3873	238	16	γ	γ	NOUN
ejpam-3873	238	17	-	-	PUNCT
ejpam-3873	238	18	sets	set	NOUN
ejpam-3873	238	19	.	.	PUNCT
ejpam-3873	239	1	therefore	therefore	ADV
ejpam-3873	239	2	,	,	PUNCT
ejpam-3873	239	3	by	by	ADP
ejpam-3873	239	4	multiplication	multiplication	NOUN
ejpam-3873	239	5	principle	principle	NOUN
ejpam-3873	239	6	,	,	PUNCT
ejpam-3873	239	7	ind(r	ind(r	NOUN
ejpam-3873	239	8	)	)	PUNCT
ejpam-3873	239	9	≤	≤	NOUN
ejpam-3873	239	10	2(|r|−|s|)/2	2(|r|−|s|)/2	NUM
ejpam-3873	239	11	.	.	PUNCT
ejpam-3873	240	1	the	the	DET
ejpam-3873	240	2	bound	bind	VERB
ejpam-3873	240	3	in	in	ADP
ejpam-3873	240	4	corollary	corollary	ADJ
ejpam-3873	240	5	2	2	NUM
ejpam-3873	240	6	is	be	AUX
ejpam-3873	240	7	also	also	ADV
ejpam-3873	240	8	sharp	sharp	ADJ
ejpam-3873	240	9	.	.	PUNCT
ejpam-3873	241	1	equality	equality	NOUN
ejpam-3873	241	2	holds	hold	VERB
ejpam-3873	241	3	for	for	ADP
ejpam-3873	241	4	some	some	DET
ejpam-3873	241	5	fields	field	NOUN
ejpam-3873	241	6	.	.	PUNCT
ejpam-3873	242	1	for	for	ADP
ejpam-3873	242	2	example	example	NOUN
ejpam-3873	242	3	,	,	PUNCT
ejpam-3873	242	4	if	if	SCONJ
ejpam-3873	242	5	r	r	NOUN
ejpam-3873	242	6	is	be	AUX
ejpam-3873	242	7	z7	z7	PROPN
ejpam-3873	242	8	,	,	PUNCT
ejpam-3873	242	9	then	then	ADV
ejpam-3873	242	10	the	the	DET
ejpam-3873	242	11	equality	equality	NOUN
ejpam-3873	242	12	holds	hold	VERB
ejpam-3873	242	13	.	.	PUNCT
ejpam-3873	243	1	to	to	PART
ejpam-3873	243	2	see	see	VERB
ejpam-3873	243	3	this	this	PRON
ejpam-3873	243	4	,	,	PUNCT
ejpam-3873	243	5	we	we	PRON
ejpam-3873	243	6	note	note	VERB
ejpam-3873	243	7	that	that	SCONJ
ejpam-3873	243	8	the	the	DET
ejpam-3873	243	9	minimum	minimum	ADJ
ejpam-3873	243	10	γ	γ	NOUN
ejpam-3873	243	11	-	-	PUNCT
ejpam-3873	243	12	sets	set	NOUN
ejpam-3873	243	13	of	of	ADP
ejpam-3873	243	14	z7	z7	PROPN
ejpam-3873	243	15	are	be	AUX
ejpam-3873	243	16	j1	j1	PROPN
ejpam-3873	243	17	=	=	PUNCT
ejpam-3873	243	18	{	{	PUNCT
ejpam-3873	243	19	0	0	NUM
ejpam-3873	243	20	,	,	PUNCT
ejpam-3873	243	21	1	1	NUM
ejpam-3873	243	22	,	,	PUNCT
ejpam-3873	243	23	2	2	NUM
ejpam-3873	243	24	,	,	PUNCT
ejpam-3873	243	25	3	3	NUM
ejpam-3873	243	26	,	,	PUNCT
ejpam-3873	243	27	6	6	NUM
ejpam-3873	243	28	}	}	PUNCT
ejpam-3873	243	29	and	and	CCONJ
ejpam-3873	243	30	j2	j2	PROPN
ejpam-3873	243	31	=	=	SYM
ejpam-3873	243	32	{	{	PUNCT
ejpam-3873	243	33	0	0	NUM
ejpam-3873	243	34	,	,	PUNCT
ejpam-3873	243	35	1	1	NUM
ejpam-3873	243	36	,	,	PUNCT
ejpam-3873	243	37	4	4	NUM
ejpam-3873	243	38	,	,	PUNCT
ejpam-3873	243	39	5	5	NUM
ejpam-3873	243	40	,	,	PUNCT
ejpam-3873	243	41	6	6	NUM
ejpam-3873	243	42	}	}	PUNCT
ejpam-3873	243	43	.	.	PUNCT
ejpam-3873	244	1	hence	hence	ADV
ejpam-3873	244	2	,	,	PUNCT
ejpam-3873	244	3	ind(z7	ind(z7	NOUN
ejpam-3873	244	4	)	)	PUNCT
ejpam-3873	244	5	=	=	SYM
ejpam-3873	244	6	2	2	X
ejpam-3873	244	7	.	.	PUNCT
ejpam-3873	245	1	this	this	PRON
ejpam-3873	245	2	is	be	AUX
ejpam-3873	245	3	equal	equal	ADJ
ejpam-3873	245	4	to	to	ADP
ejpam-3873	245	5	2(|r|−|s|)/2	2(|r|−|s|)/2	PROPN
ejpam-3873	245	6	=	=	SYM
ejpam-3873	245	7	2(|z7|−|{0,1,6}|)/2	2(|z7|−|{0,1,6}|)/2	NUM
ejpam-3873	246	1	=	=	SYM
ejpam-3873	246	2	2(7−3)/4	2(7−3)/4	NUM
ejpam-3873	246	3	=	=	SYM
ejpam-3873	246	4	2	2	X
ejpam-3873	246	5	.	.	X
ejpam-3873	246	6	e.j	e.j	PROPN
ejpam-3873	246	7	.	.	PROPN
ejpam-3873	246	8	sigasig	sigasig	PROPN
ejpam-3873	246	9	,	,	PUNCT
ejpam-3873	246	10	c.j	c.j	PROPN
ejpam-3873	246	11	.	.	PROPN
ejpam-3873	246	12	rosero	rosero	PROPN
ejpam-3873	246	13	,	,	PUNCT
ejpam-3873	246	14	m.	m.	NOUN
ejpam-3873	246	15	baldado	baldado	PROPN
ejpam-3873	246	16	jr	jr	PROPN
ejpam-3873	246	17	.	.	PROPN
ejpam-3873	246	18	/	/	SYM
ejpam-3873	246	19	eur	eur	PROPN
ejpam-3873	246	20	.	.	PUNCT
ejpam-3873	247	1	j.	j.	PROPN
ejpam-3873	247	2	pure	pure	PROPN
ejpam-3873	247	3	appl	appl	PROPN
ejpam-3873	247	4	.	.	PROPN
ejpam-3873	247	5	math	math	PROPN
ejpam-3873	247	6	,	,	PUNCT
ejpam-3873	247	7	14	14	NUM
ejpam-3873	247	8	(	(	PUNCT
ejpam-3873	247	9	1	1	NUM
ejpam-3873	247	10	)	)	PUNCT
ejpam-3873	247	11	(	(	PUNCT
ejpam-3873	247	12	2021	2021	NUM
ejpam-3873	247	13	)	)	PUNCT
ejpam-3873	247	14	,	,	PUNCT
ejpam-3873	247	15	314	314	NUM
ejpam-3873	247	16	-	-	SYM
ejpam-3873	247	17	326	326	NUM
ejpam-3873	247	18	320	320	NUM
ejpam-3873	247	19	6	6	NUM
ejpam-3873	247	20	.	.	PUNCT
ejpam-3873	248	1	γ	γ	NOUN
ejpam-3873	248	2	-	-	PUNCT
ejpam-3873	248	3	sets	set	NOUN
ejpam-3873	248	4	and	and	CCONJ
ejpam-3873	248	5	homomorphism	homomorphism	NOUN
ejpam-3873	248	6	of	of	ADP
ejpam-3873	248	7	rings	ring	NOUN
ejpam-3873	248	8	in	in	ADP
ejpam-3873	248	9	this	this	DET
ejpam-3873	248	10	section	section	NOUN
ejpam-3873	248	11	,	,	PUNCT
ejpam-3873	248	12	we	we	PRON
ejpam-3873	248	13	gave	give	VERB
ejpam-3873	248	14	some	some	DET
ejpam-3873	248	15	properties	property	NOUN
ejpam-3873	248	16	of	of	ADP
ejpam-3873	248	17	γ	γ	NOUN
ejpam-3873	248	18	-	-	PUNCT
ejpam-3873	248	19	sets	set	NOUN
ejpam-3873	248	20	in	in	ADP
ejpam-3873	248	21	relation	relation	NOUN
ejpam-3873	248	22	to	to	ADP
ejpam-3873	248	23	its	its	PRON
ejpam-3873	248	24	homomorphic	homomorphic	ADJ
ejpam-3873	248	25	image	image	NOUN
ejpam-3873	248	26	.	.	PUNCT
ejpam-3873	249	1	we	we	PRON
ejpam-3873	249	2	say	say	VERB
ejpam-3873	249	3	that	that	SCONJ
ejpam-3873	249	4	a	a	DET
ejpam-3873	249	5	set	set	NOUN
ejpam-3873	249	6	a	a	DET
ejpam-3873	249	7	precedes	precede	NOUN
ejpam-3873	249	8	a	a	DET
ejpam-3873	249	9	set	set	NOUN
ejpam-3873	249	10	b	b	NOUN
ejpam-3873	249	11	if	if	SCONJ
ejpam-3873	249	12	there	there	PRON
ejpam-3873	249	13	exists	exist	VERB
ejpam-3873	249	14	an	an	DET
ejpam-3873	249	15	injective	injective	ADJ
ejpam-3873	249	16	map	map	NOUN
ejpam-3873	249	17	from	from	ADP
ejpam-3873	249	18	a	a	PRON
ejpam-3873	249	19	to	to	ADP
ejpam-3873	249	20	b.	b.	PROPN
ejpam-3873	249	21	in	in	ADP
ejpam-3873	249	22	this	this	DET
ejpam-3873	249	23	case	case	NOUN
ejpam-3873	249	24	,	,	PUNCT
ejpam-3873	249	25	we	we	PRON
ejpam-3873	249	26	write	write	VERB
ejpam-3873	249	27	a≺b	a≺b	PROPN
ejpam-3873	249	28	.	.	PUNCT
ejpam-3873	249	29	theorem	theorem	VERB
ejpam-3873	249	30	7	7	NUM
ejpam-3873	249	31	.	.	PUNCT
ejpam-3873	250	1	let	let	VERB
ejpam-3873	250	2	j	j	PROPN
ejpam-3873	250	3	be	be	AUX
ejpam-3873	250	4	a	a	DET
ejpam-3873	250	5	γ	γ	NOUN
ejpam-3873	250	6	-	-	PUNCT
ejpam-3873	250	7	set	set	NOUN
ejpam-3873	250	8	of	of	ADP
ejpam-3873	250	9	a	a	DET
ejpam-3873	250	10	ring	ring	NOUN
ejpam-3873	250	11	.	.	PUNCT
ejpam-3873	251	1	then	then	ADV
ejpam-3873	251	2	r\j≺j	r\j≺j	X
ejpam-3873	251	3	.	.	PUNCT
ejpam-3873	252	1	proof	proof	NOUN
ejpam-3873	252	2	.	.	PUNCT
ejpam-3873	253	1	let	let	VERB
ejpam-3873	253	2	f	f	NOUN
ejpam-3873	253	3	:	:	PUNCT
ejpam-3873	253	4	r\j	r\j	ADV
ejpam-3873	253	5	→	→	PUNCT
ejpam-3873	253	6	j	j	PROPN
ejpam-3873	253	7	be	be	AUX
ejpam-3873	253	8	a	a	DET
ejpam-3873	253	9	given	give	VERB
ejpam-3873	253	10	by	by	ADP
ejpam-3873	253	11	f(x	f(x	PROPN
ejpam-3873	253	12	)	)	PUNCT
ejpam-3873	253	13	=	=	SYM
ejpam-3873	254	1	x−1	x−1	PROPN
ejpam-3873	254	2	,	,	PUNCT
ejpam-3873	254	3	and	and	CCONJ
ejpam-3873	254	4	let	let	VERB
ejpam-3873	254	5	x	x	PRON
ejpam-3873	254	6	,	,	PUNCT
ejpam-3873	254	7	y	y	PROPN
ejpam-3873	254	8	∈	∈	PROPN
ejpam-3873	254	9	j	j	PROPN
ejpam-3873	254	10	with	with	ADP
ejpam-3873	254	11	x	x	PROPN
ejpam-3873	254	12	=	=	PUNCT
ejpam-3873	254	13	y.	y.	NOUN
ejpam-3873	254	14	since	since	SCONJ
ejpam-3873	254	15	j	j	PROPN
ejpam-3873	254	16	is	be	AUX
ejpam-3873	254	17	a	a	DET
ejpam-3873	254	18	γ	γ	NOUN
ejpam-3873	254	19	-	-	PUNCT
ejpam-3873	254	20	set	set	VERB
ejpam-3873	254	21	and	and	CCONJ
ejpam-3873	254	22	each	each	DET
ejpam-3873	254	23	unit	unit	NOUN
ejpam-3873	254	24	of	of	ADP
ejpam-3873	254	25	a	a	DET
ejpam-3873	254	26	ring	ring	NOUN
ejpam-3873	254	27	has	have	VERB
ejpam-3873	254	28	a	a	DET
ejpam-3873	254	29	unique	unique	ADJ
ejpam-3873	254	30	multiplicative	multiplicative	ADJ
ejpam-3873	254	31	inverse	inverse	NOUN
ejpam-3873	254	32	,	,	PUNCT
ejpam-3873	254	33	x	x	SYM
ejpam-3873	254	34	=	=	SYM
ejpam-3873	254	35	y	y	PROPN
ejpam-3873	254	36	implies	imply	VERB
ejpam-3873	254	37	that	that	DET
ejpam-3873	254	38	f−1(x	f−1(x	NOUN
ejpam-3873	254	39	)	)	PUNCT
ejpam-3873	254	40	=	=	SYM
ejpam-3873	254	41	f−1(y	f−1(y	PROPN
ejpam-3873	254	42	)	)	PUNCT
ejpam-3873	254	43	.	.	PUNCT
ejpam-3873	255	1	this	this	PRON
ejpam-3873	255	2	means	mean	VERB
ejpam-3873	255	3	that	that	SCONJ
ejpam-3873	255	4	f	f	PROPN
ejpam-3873	255	5	is	be	AUX
ejpam-3873	255	6	injective	injective	ADJ
ejpam-3873	255	7	.	.	PUNCT
ejpam-3873	256	1	hence	hence	ADV
ejpam-3873	256	2	,	,	PUNCT
ejpam-3873	256	3	r\j≺j	r\j≺j	X
ejpam-3873	256	4	.	.	PUNCT
ejpam-3873	257	1	theorem	theorem	VERB
ejpam-3873	257	2	7	7	NUM
ejpam-3873	257	3	says	say	VERB
ejpam-3873	257	4	that	that	SCONJ
ejpam-3873	257	5	in	in	ADP
ejpam-3873	257	6	a	a	DET
ejpam-3873	257	7	finite	finite	ADJ
ejpam-3873	257	8	ring	ring	NOUN
ejpam-3873	257	9	,	,	PUNCT
ejpam-3873	257	10	a	a	DET
ejpam-3873	257	11	γ	γ	NOUN
ejpam-3873	257	12	-	-	PUNCT
ejpam-3873	257	13	set	set	NOUN
ejpam-3873	257	14	has	have	VERB
ejpam-3873	257	15	more	more	ADJ
ejpam-3873	257	16	elements	element	NOUN
ejpam-3873	257	17	than	than	ADP
ejpam-3873	257	18	its	its	PRON
ejpam-3873	257	19	complement	complement	NOUN
ejpam-3873	257	20	.	.	PUNCT
ejpam-3873	258	1	lemma	lemma	PROPN
ejpam-3873	258	2	5	5	X
ejpam-3873	258	3	.	.	PUNCT
ejpam-3873	259	1	let	let	VERB
ejpam-3873	259	2	r	r	PRON
ejpam-3873	259	3	be	be	AUX
ejpam-3873	259	4	a	a	DET
ejpam-3873	259	5	ring	ring	NOUN
ejpam-3873	259	6	,	,	PUNCT
ejpam-3873	259	7	and	and	CCONJ
ejpam-3873	259	8	x	x	X
ejpam-3873	259	9	∈	∈	PROPN
ejpam-3873	259	10	r.	r.	PROPN
ejpam-3873	259	11	then	then	ADV
ejpam-3873	259	12	t	t	PROPN
ejpam-3873	259	13	(	(	PUNCT
ejpam-3873	259	14	x	x	X
ejpam-3873	259	15	)	)	PUNCT
ejpam-3873	259	16	=	=	PRON
ejpam-3873	259	17	{	{	PUNCT
ejpam-3873	259	18	j	j	NOUN
ejpam-3873	259	19	⊆	⊆	NUM
ejpam-3873	259	20	r	r	NOUN
ejpam-3873	259	21	:	:	PUNCT
ejpam-3873	259	22	j	j	PROPN
ejpam-3873	259	23	is	be	AUX
ejpam-3873	259	24	a	a	DET
ejpam-3873	259	25	γ	γ	NOUN
ejpam-3873	259	26	-	-	PUNCT
ejpam-3873	259	27	set	set	VERB
ejpam-3873	259	28	and	and	CCONJ
ejpam-3873	259	29	x	x	SYM
ejpam-3873	259	30	∈	∈	PROPN
ejpam-3873	259	31	j	j	PROPN
ejpam-3873	259	32	}	}	PUNCT
ejpam-3873	259	33	is	be	AUX
ejpam-3873	259	34	a	a	DET
ejpam-3873	259	35	semigroup	semigroup	NOUN
ejpam-3873	259	36	under	under	ADP
ejpam-3873	259	37	the	the	DET
ejpam-3873	259	38	operation	operation	NOUN
ejpam-3873	259	39	union	union	NOUN
ejpam-3873	259	40	.	.	PUNCT
ejpam-3873	260	1	proof	proof	NOUN
ejpam-3873	260	2	.	.	PUNCT
ejpam-3873	261	1	it	it	PRON
ejpam-3873	261	2	suffices	suffice	VERB
ejpam-3873	261	3	to	to	PART
ejpam-3873	261	4	show	show	VERB
ejpam-3873	261	5	that	that	SCONJ
ejpam-3873	261	6	t	t	NOUN
ejpam-3873	261	7	(	(	PUNCT
ejpam-3873	261	8	x	x	X
ejpam-3873	261	9	)	)	PUNCT
ejpam-3873	261	10	is	be	AUX
ejpam-3873	261	11	closed	close	VERB
ejpam-3873	261	12	under	under	ADP
ejpam-3873	261	13	the	the	DET
ejpam-3873	261	14	operation	operation	NOUN
ejpam-3873	261	15	union	union	NOUN
ejpam-3873	261	16	.	.	PUNCT
ejpam-3873	262	1	let	let	VERB
ejpam-3873	262	2	j1	j1	PROPN
ejpam-3873	262	3	and	and	CCONJ
ejpam-3873	262	4	j2	j2	PROPN
ejpam-3873	262	5	be	be	AUX
ejpam-3873	262	6	element	element	NOUN
ejpam-3873	262	7	of	of	ADP
ejpam-3873	262	8	t	t	PROPN
ejpam-3873	262	9	(	(	PUNCT
ejpam-3873	262	10	x	x	NOUN
ejpam-3873	262	11	)	)	PUNCT
ejpam-3873	262	12	.	.	PUNCT
ejpam-3873	263	1	then	then	ADV
ejpam-3873	263	2	by	by	ADP
ejpam-3873	263	3	theorem	theorem	NOUN
ejpam-3873	263	4	1	1	NUM
ejpam-3873	263	5	,	,	PUNCT
ejpam-3873	263	6	j1	j1	PROPN
ejpam-3873	263	7	∪	∪	ADJ
ejpam-3873	263	8	j2	j2	PROPN
ejpam-3873	263	9	is	be	AUX
ejpam-3873	263	10	a	a	DET
ejpam-3873	263	11	γ	γ	NOUN
ejpam-3873	263	12	-	-	PUNCT
ejpam-3873	263	13	set	set	NOUN
ejpam-3873	263	14	.	.	PUNCT
ejpam-3873	264	1	since	since	SCONJ
ejpam-3873	264	2	clearly	clearly	ADV
ejpam-3873	264	3	x	x	SYM
ejpam-3873	264	4	∈	∈	PROPN
ejpam-3873	264	5	j1	j1	PROPN
ejpam-3873	264	6	∪	∪	PROPN
ejpam-3873	264	7	j2	j2	PROPN
ejpam-3873	264	8	,	,	PUNCT
ejpam-3873	264	9	j1	j1	PROPN
ejpam-3873	264	10	∪	∪	PROPN
ejpam-3873	264	11	j2	j2	PROPN
ejpam-3873	264	12	∈	∈	PROPN
ejpam-3873	264	13	t	t	PROPN
ejpam-3873	264	14	(	(	PUNCT
ejpam-3873	264	15	x	x	NOUN
ejpam-3873	264	16	)	)	PUNCT
ejpam-3873	264	17	.	.	PUNCT
ejpam-3873	265	1	an	an	DET
ejpam-3873	265	2	element	element	NOUN
ejpam-3873	265	3	x	x	PUNCT
ejpam-3873	265	4	of	of	ADP
ejpam-3873	265	5	a	a	DET
ejpam-3873	265	6	ring	ring	NOUN
ejpam-3873	265	7	r	r	NOUN
ejpam-3873	265	8	is	be	AUX
ejpam-3873	265	9	called	call	VERB
ejpam-3873	265	10	an	an	DET
ejpam-3873	265	11	involution	involution	NOUN
ejpam-3873	265	12	if	if	SCONJ
ejpam-3873	265	13	x2	x2	PROPN
ejpam-3873	265	14	=	=	NOUN
ejpam-3873	265	15	1r	1r	NUM
ejpam-3873	265	16	,	,	PUNCT
ejpam-3873	265	17	or	or	CCONJ
ejpam-3873	265	18	2x	2x	NUM
ejpam-3873	265	19	=	=	SYM
ejpam-3873	265	20	0	0	NUM
ejpam-3873	265	21	,	,	PUNCT
ejpam-3873	265	22	or	or	CCONJ
ejpam-3873	265	23	x	x	SYM
ejpam-3873	265	24	6=	6=	ADP
ejpam-3873	265	25	−x−1	−x−1	NUM
ejpam-3873	265	26	.	.	PUNCT
ejpam-3873	266	1	theorem	theorem	ADJ
ejpam-3873	266	2	8	8	NUM
ejpam-3873	266	3	.	.	PUNCT
ejpam-3873	267	1	let	let	VERB
ejpam-3873	267	2	x	x	PRON
ejpam-3873	267	3	be	be	AUX
ejpam-3873	267	4	a	a	DET
ejpam-3873	267	5	non	non	ADJ
ejpam-3873	267	6	-	-	ADJ
ejpam-3873	267	7	identity	identity	ADJ
ejpam-3873	267	8	element	element	NOUN
ejpam-3873	267	9	of	of	ADP
ejpam-3873	267	10	a	a	DET
ejpam-3873	267	11	ring	ring	NOUN
ejpam-3873	267	12	.	.	PUNCT
ejpam-3873	268	1	then	then	ADV
ejpam-3873	268	2	x	x	PRON
ejpam-3873	268	3	is	be	AUX
ejpam-3873	268	4	an	an	DET
ejpam-3873	268	5	involution	involution	NOUN
ejpam-3873	268	6	if	if	SCONJ
ejpam-3873	268	7	and	and	CCONJ
ejpam-3873	268	8	only	only	ADV
ejpam-3873	268	9	if	if	SCONJ
ejpam-3873	268	10	t	t	PROPN
ejpam-3873	268	11	(	(	PUNCT
ejpam-3873	268	12	x	x	NOUN
ejpam-3873	268	13	)	)	PUNCT
ejpam-3873	268	14	=	=	SYM
ejpam-3873	268	15	t	t	NOUN
ejpam-3873	268	16	.	.	PUNCT
ejpam-3873	269	1	proof	proof	NOUN
ejpam-3873	269	2	.	.	PUNCT
ejpam-3873	270	1	let	let	VERB
ejpam-3873	270	2	r	r	PRON
ejpam-3873	270	3	be	be	AUX
ejpam-3873	270	4	a	a	DET
ejpam-3873	270	5	ring	ring	NOUN
ejpam-3873	270	6	,	,	PUNCT
ejpam-3873	270	7	and	and	CCONJ
ejpam-3873	270	8	suppose	suppose	VERB
ejpam-3873	270	9	that	that	SCONJ
ejpam-3873	270	10	x	x	PRON
ejpam-3873	270	11	is	be	AUX
ejpam-3873	270	12	an	an	DET
ejpam-3873	270	13	involution	involution	NOUN
ejpam-3873	270	14	of	of	ADP
ejpam-3873	270	15	r.	r.	PROPN
ejpam-3873	270	16	if	if	SCONJ
ejpam-3873	270	17	x	x	PRON
ejpam-3873	270	18	is	be	AUX
ejpam-3873	270	19	an	an	DET
ejpam-3873	270	20	involution	involution	NOUN
ejpam-3873	270	21	,	,	PUNCT
ejpam-3873	270	22	then	then	ADV
ejpam-3873	270	23	by	by	ADP
ejpam-3873	270	24	theorem	theorem	NOUN
ejpam-3873	270	25	2	2	NUM
ejpam-3873	270	26	x	x	VERB
ejpam-3873	270	27	is	be	AUX
ejpam-3873	270	28	contained	contain	VERB
ejpam-3873	270	29	in	in	ADP
ejpam-3873	270	30	every	every	DET
ejpam-3873	270	31	γ	γ	NOUN
ejpam-3873	270	32	-	-	PUNCT
ejpam-3873	270	33	set	set	NOUN
ejpam-3873	270	34	of	of	ADP
ejpam-3873	270	35	r.	r.	PROPN
ejpam-3873	270	36	thus	thus	ADV
ejpam-3873	270	37	,	,	PUNCT
ejpam-3873	270	38	if	if	SCONJ
ejpam-3873	270	39	j	j	PROPN
ejpam-3873	270	40	is	be	AUX
ejpam-3873	270	41	a	a	DET
ejpam-3873	270	42	γ	γ	NOUN
ejpam-3873	270	43	-	-	PUNCT
ejpam-3873	270	44	set	set	NOUN
ejpam-3873	270	45	,	,	PUNCT
ejpam-3873	270	46	then	then	ADV
ejpam-3873	270	47	j	j	PROPN
ejpam-3873	270	48	∈	∈	PROPN
ejpam-3873	270	49	t	t	PROPN
ejpam-3873	270	50	(	(	PUNCT
ejpam-3873	270	51	x	x	NOUN
ejpam-3873	270	52	)	)	PUNCT
ejpam-3873	270	53	,	,	PUNCT
ejpam-3873	270	54	that	that	PRON
ejpam-3873	270	55	is	is	ADV
ejpam-3873	270	56	t	t	PROPN
ejpam-3873	270	57	⊆	⊆	NUM
ejpam-3873	270	58	t	t	PROPN
ejpam-3873	270	59	(	(	PUNCT
ejpam-3873	270	60	x	x	NOUN
ejpam-3873	270	61	)	)	PUNCT
ejpam-3873	270	62	.	.	PUNCT
ejpam-3873	271	1	since	since	SCONJ
ejpam-3873	271	2	clearly	clearly	ADV
ejpam-3873	271	3	t	t	PROPN
ejpam-3873	271	4	⊇	⊇	PROPN
ejpam-3873	271	5	t	t	PROPN
ejpam-3873	271	6	(	(	PUNCT
ejpam-3873	271	7	x	x	NOUN
ejpam-3873	271	8	)	)	PUNCT
ejpam-3873	271	9	,	,	PUNCT
ejpam-3873	271	10	we	we	PRON
ejpam-3873	271	11	must	must	AUX
ejpam-3873	271	12	have	have	VERB
ejpam-3873	271	13	t	t	PROPN
ejpam-3873	271	14	=	=	SYM
ejpam-3873	271	15	t	t	PROPN
ejpam-3873	271	16	(	(	PUNCT
ejpam-3873	271	17	x	x	NOUN
ejpam-3873	271	18	)	)	PUNCT
ejpam-3873	271	19	.	.	PUNCT
ejpam-3873	272	1	conversely	conversely	ADV
ejpam-3873	272	2	,	,	PUNCT
ejpam-3873	272	3	assume	assume	VERB
ejpam-3873	272	4	that	that	SCONJ
ejpam-3873	272	5	t	t	PROPN
ejpam-3873	272	6	(	(	PUNCT
ejpam-3873	272	7	x	x	X
ejpam-3873	272	8	)	)	PUNCT
ejpam-3873	272	9	=	=	SYM
ejpam-3873	272	10	t	t	PROPN
ejpam-3873	272	11	and	and	CCONJ
ejpam-3873	272	12	x	x	X
ejpam-3873	272	13	is	be	AUX
ejpam-3873	272	14	not	not	PART
ejpam-3873	272	15	an	an	DET
ejpam-3873	272	16	involution	involution	NOUN
ejpam-3873	272	17	.	.	PUNCT
ejpam-3873	273	1	if	if	SCONJ
ejpam-3873	273	2	x	x	PRON
ejpam-3873	273	3	is	be	AUX
ejpam-3873	273	4	not	not	PART
ejpam-3873	273	5	an	an	DET
ejpam-3873	273	6	involution	involution	NOUN
ejpam-3873	273	7	and	and	CCONJ
ejpam-3873	273	8	x	x	SYM
ejpam-3873	273	9	6=	6=	NUM
ejpam-3873	273	10	1r	1r	NUM
ejpam-3873	273	11	,	,	PUNCT
ejpam-3873	273	12	then	then	ADV
ejpam-3873	273	13	x	x	X
ejpam-3873	273	14	6=	6=	PROPN
ejpam-3873	273	15	x−1	x−1	PROPN
ejpam-3873	273	16	.	.	PUNCT
ejpam-3873	274	1	by	by	ADP
ejpam-3873	274	2	theorem	theorem	NOUN
ejpam-3873	274	3	3	3	NUM
ejpam-3873	274	4	,	,	PUNCT
ejpam-3873	274	5	j	j	NOUN
ejpam-3873	274	6	′	′	NUM
ejpam-3873	275	1	=	=	SYM
ejpam-3873	275	2	(	(	PUNCT
ejpam-3873	275	3	j\{x	j\{x	X
ejpam-3873	275	4	,	,	PUNCT
ejpam-3873	275	5	(	(	PUNCT
ejpam-3873	275	6	−x)−1	−x)−1	NOUN
ejpam-3873	275	7	}	}	PUNCT
ejpam-3873	275	8	)	)	PUNCT
ejpam-3873	275	9	∪	∪	ADP
ejpam-3873	275	10	{	{	PUNCT
ejpam-3873	275	11	x−1,−x	x−1,−x	PROPN
ejpam-3873	275	12	}	}	PUNCT
ejpam-3873	275	13	is	be	AUX
ejpam-3873	275	14	also	also	ADV
ejpam-3873	275	15	a	a	DET
ejpam-3873	275	16	γ	γ	NOUN
ejpam-3873	275	17	-	-	PUNCT
ejpam-3873	275	18	set	set	NOUN
ejpam-3873	275	19	.	.	PUNCT
ejpam-3873	276	1	note	note	VERB
ejpam-3873	276	2	that	that	SCONJ
ejpam-3873	276	3	j	j	PROPN
ejpam-3873	276	4	′	′	NUM
ejpam-3873	276	5	∈	∈	PROPN
ejpam-3873	276	6	t	t	PROPN
ejpam-3873	276	7	,	,	PUNCT
ejpam-3873	276	8	but	but	CCONJ
ejpam-3873	277	1	j	j	PROPN
ejpam-3873	278	1	′	′	NUM
ejpam-3873	278	2	/∈	/∈	PUNCT
ejpam-3873	279	1	t	t	PROPN
ejpam-3873	279	2	(	(	PUNCT
ejpam-3873	279	3	x	x	NOUN
ejpam-3873	279	4	)	)	PUNCT
ejpam-3873	279	5	,	,	PUNCT
ejpam-3873	279	6	that	that	PRON
ejpam-3873	279	7	is	is	ADV
ejpam-3873	279	8	t	t	PROPN
ejpam-3873	279	9	(	(	PUNCT
ejpam-3873	279	10	x	x	X
ejpam-3873	279	11	)	)	PUNCT
ejpam-3873	279	12	6=	6=	ADP
ejpam-3873	279	13	t	t	PROPN
ejpam-3873	279	14	.	.	PUNCT
ejpam-3873	280	1	this	this	PRON
ejpam-3873	280	2	is	be	AUX
ejpam-3873	280	3	a	a	DET
ejpam-3873	280	4	contradiction	contradiction	NOUN
ejpam-3873	280	5	.	.	PUNCT
ejpam-3873	281	1	theorem	theorem	NOUN
ejpam-3873	281	2	9	9	NUM
ejpam-3873	281	3	.	.	PUNCT
ejpam-3873	282	1	let	let	VERB
ejpam-3873	282	2	r	r	PRON
ejpam-3873	282	3	be	be	AUX
ejpam-3873	282	4	a	a	DET
ejpam-3873	282	5	ring	ring	NOUN
ejpam-3873	282	6	with	with	ADP
ejpam-3873	282	7	identity	identity	NOUN
ejpam-3873	282	8	and	and	CCONJ
ejpam-3873	282	9	x	x	PART
ejpam-3873	282	10	be	be	AUX
ejpam-3873	282	11	a	a	DET
ejpam-3873	282	12	non	non	ADJ
ejpam-3873	282	13	-	-	ADJ
ejpam-3873	282	14	zero	zero	NUM
ejpam-3873	282	15	element	element	NOUN
ejpam-3873	282	16	of	of	ADP
ejpam-3873	282	17	r	r	NOUN
ejpam-3873	282	18	with	with	ADP
ejpam-3873	282	19	2x	2x	NUM
ejpam-3873	282	20	=	=	SYM
ejpam-3873	282	21	0	0	X
ejpam-3873	282	22	.	.	PUNCT
ejpam-3873	283	1	then	then	ADV
ejpam-3873	283	2	every	every	DET
ejpam-3873	283	3	γ	γ	PROPN
ejpam-3873	283	4	-	-	PUNCT
ejpam-3873	283	5	set	set	NOUN
ejpam-3873	283	6	of	of	ADP
ejpam-3873	283	7	r	r	NOUN
ejpam-3873	283	8	contains	contain	VERB
ejpam-3873	283	9	a	a	DET
ejpam-3873	283	10	non	non	ADJ
ejpam-3873	283	11	-	-	ADJ
ejpam-3873	283	12	trivial	trivial	ADJ
ejpam-3873	283	13	subring	subring	NOUN
ejpam-3873	283	14	.	.	PUNCT
ejpam-3873	284	1	proof	proof	NOUN
ejpam-3873	284	2	.	.	PUNCT
ejpam-3873	285	1	let	let	VERB
ejpam-3873	285	2	j	j	PROPN
ejpam-3873	285	3	be	be	AUX
ejpam-3873	285	4	a	a	DET
ejpam-3873	285	5	γ	γ	NOUN
ejpam-3873	285	6	-	-	PUNCT
ejpam-3873	285	7	set	set	NOUN
ejpam-3873	285	8	of	of	ADP
ejpam-3873	285	9	a	a	DET
ejpam-3873	285	10	ring	ring	NOUN
ejpam-3873	285	11	r	r	NOUN
ejpam-3873	285	12	and	and	CCONJ
ejpam-3873	285	13	x	x	ADJ
ejpam-3873	285	14	be	be	AUX
ejpam-3873	285	15	a	a	DET
ejpam-3873	285	16	non	non	ADJ
ejpam-3873	285	17	-	-	ADJ
ejpam-3873	285	18	zero	zero	NUM
ejpam-3873	285	19	element	element	NOUN
ejpam-3873	285	20	of	of	ADP
ejpam-3873	285	21	r	r	NOUN
ejpam-3873	285	22	with	with	ADP
ejpam-3873	285	23	2x	2x	NUM
ejpam-3873	285	24	=	=	SYM
ejpam-3873	285	25	0	0	X
ejpam-3873	285	26	.	.	PUNCT
ejpam-3873	285	27	theorem	theorem	ADJ
ejpam-3873	285	28	2	2	NUM
ejpam-3873	285	29	implies	imply	VERB
ejpam-3873	285	30	that	that	SCONJ
ejpam-3873	285	31	0	0	PUNCT
ejpam-3873	286	1	and	and	CCONJ
ejpam-3873	286	2	x	x	PRON
ejpam-3873	286	3	are	be	AUX
ejpam-3873	286	4	elements	element	NOUN
ejpam-3873	286	5	of	of	ADP
ejpam-3873	286	6	j	j	PROPN
ejpam-3873	286	7	.	.	PUNCT
ejpam-3873	287	1	thus	thus	ADV
ejpam-3873	287	2	,	,	PUNCT
ejpam-3873	287	3	{	{	PUNCT
ejpam-3873	287	4	0	0	NUM
ejpam-3873	287	5	,	,	PUNCT
ejpam-3873	287	6	x	x	PRON
ejpam-3873	287	7	}	}	PUNCT
ejpam-3873	287	8	is	be	AUX
ejpam-3873	287	9	a	a	DET
ejpam-3873	287	10	subring	subring	NOUN
ejpam-3873	287	11	of	of	ADP
ejpam-3873	287	12	r	r	NOUN
ejpam-3873	287	13	contained	contain	VERB
ejpam-3873	287	14	in	in	ADP
ejpam-3873	287	15	j	j	PROPN
ejpam-3873	287	16	.	.	PUNCT
ejpam-3873	288	1	theorem	theorem	PROPN
ejpam-3873	288	2	10	10	NUM
ejpam-3873	288	3	.	.	PUNCT
ejpam-3873	289	1	let	let	VERB
ejpam-3873	289	2	r	r	PRON
ejpam-3873	289	3	be	be	AUX
ejpam-3873	289	4	a	a	DET
ejpam-3873	289	5	ring	ring	NOUN
ejpam-3873	289	6	with	with	ADP
ejpam-3873	289	7	identity	identity	NOUN
ejpam-3873	289	8	.	.	PUNCT
ejpam-3873	290	1	r	r	NOUN
ejpam-3873	290	2	has	have	VERB
ejpam-3873	290	3	a	a	DET
ejpam-3873	290	4	trivial	trivial	ADJ
ejpam-3873	290	5	γ	γ	NOUN
ejpam-3873	290	6	-	-	PUNCT
ejpam-3873	290	7	set	set	VERB
ejpam-3873	290	8	if	if	SCONJ
ejpam-3873	290	9	and	and	CCONJ
ejpam-3873	290	10	only	only	ADV
ejpam-3873	290	11	if	if	SCONJ
ejpam-3873	290	12	r	r	NOUN
ejpam-3873	290	13	is	be	AUX
ejpam-3873	290	14	trivial	trivial	ADJ
ejpam-3873	290	15	.	.	PUNCT
ejpam-3873	291	1	proof	proof	NOUN
ejpam-3873	291	2	.	.	PUNCT
ejpam-3873	292	1	let	let	VERB
ejpam-3873	292	2	r	r	NOUN
ejpam-3873	292	3	=	=	SYM
ejpam-3873	292	4	{	{	PUNCT
ejpam-3873	292	5	0	0	NUM
ejpam-3873	292	6	,	,	PUNCT
ejpam-3873	292	7	1	1	NUM
ejpam-3873	292	8	}	}	PUNCT
ejpam-3873	292	9	.	.	PUNCT
ejpam-3873	293	1	then	then	ADV
ejpam-3873	293	2	clearly	clearly	ADV
ejpam-3873	293	3	{	{	PUNCT
ejpam-3873	293	4	0	0	NUM
ejpam-3873	293	5	,	,	PUNCT
ejpam-3873	293	6	1	1	NUM
ejpam-3873	293	7	}	}	PUNCT
ejpam-3873	293	8	is	be	AUX
ejpam-3873	293	9	a	a	DET
ejpam-3873	293	10	γ	γ	NOUN
ejpam-3873	293	11	-	-	PUNCT
ejpam-3873	293	12	set	set	NOUN
ejpam-3873	293	13	of	of	ADP
ejpam-3873	293	14	r.	r.	PROPN
ejpam-3873	293	15	conversely	conversely	ADV
ejpam-3873	293	16	,	,	PUNCT
ejpam-3873	293	17	suppose	suppose	VERB
ejpam-3873	293	18	that	that	SCONJ
ejpam-3873	293	19	j	j	PROPN
ejpam-3873	293	20	=	=	PRON
ejpam-3873	293	21	{	{	PUNCT
ejpam-3873	293	22	0	0	NUM
ejpam-3873	293	23	,	,	PUNCT
ejpam-3873	293	24	1	1	NUM
ejpam-3873	293	25	}	}	PUNCT
ejpam-3873	293	26	is	be	AUX
ejpam-3873	293	27	a	a	DET
ejpam-3873	293	28	γ	γ	NOUN
ejpam-3873	293	29	-	-	PUNCT
ejpam-3873	293	30	set	set	NOUN
ejpam-3873	293	31	of	of	ADP
ejpam-3873	293	32	r	r	NOUN
ejpam-3873	293	33	and	and	CCONJ
ejpam-3873	293	34	r	r	NOUN
ejpam-3873	293	35	is	be	AUX
ejpam-3873	293	36	non	non	ADJ
ejpam-3873	293	37	-	-	ADJ
ejpam-3873	293	38	trivial	trivial	ADJ
ejpam-3873	293	39	.	.	PUNCT
ejpam-3873	294	1	let	let	VERB
ejpam-3873	294	2	x	x	PUNCT
ejpam-3873	294	3	∈	∈	VERB
ejpam-3873	294	4	r	r	NOUN
ejpam-3873	294	5	with	with	ADP
ejpam-3873	294	6	x	x	SYM
ejpam-3873	294	7	6=	6=	ADP
ejpam-3873	294	8	0	0	NUM
ejpam-3873	294	9	and	and	CCONJ
ejpam-3873	294	10	x	x	SYM
ejpam-3873	294	11	6=	6=	NUM
ejpam-3873	294	12	1r	1r	NUM
ejpam-3873	294	13	.	.	PUNCT
ejpam-3873	295	1	since	since	SCONJ
ejpam-3873	295	2	j	j	PROPN
ejpam-3873	295	3	is	be	AUX
ejpam-3873	295	4	a	a	DET
ejpam-3873	295	5	γ	γ	NOUN
ejpam-3873	295	6	-	-	PUNCT
ejpam-3873	295	7	set	set	NOUN
ejpam-3873	295	8	of	of	ADP
ejpam-3873	295	9	r	r	NOUN
ejpam-3873	295	10	,	,	PUNCT
ejpam-3873	295	11	there	there	PRON
ejpam-3873	295	12	exists	exist	VERB
ejpam-3873	295	13	y	y	PROPN
ejpam-3873	295	14	,	,	PUNCT
ejpam-3873	295	15	z	z	PROPN
ejpam-3873	295	16	∈	∈	PROPN
ejpam-3873	295	17	j	j	NOUN
ejpam-3873	295	18	such	such	ADJ
ejpam-3873	295	19	that	that	SCONJ
ejpam-3873	295	20	x+	x+	ADJ
ejpam-3873	295	21	y	y	PROPN
ejpam-3873	295	22	=	=	SYM
ejpam-3873	295	23	0	0	PROPN
ejpam-3873	295	24	and	and	CCONJ
ejpam-3873	295	25	xz	xz	NOUN
ejpam-3873	295	26	=	=	PUNCT
ejpam-3873	295	27	1r	1r	NUM
ejpam-3873	295	28	.	.	PUNCT
ejpam-3873	296	1	since	since	SCONJ
ejpam-3873	296	2	the	the	DET
ejpam-3873	296	3	elements	element	NOUN
ejpam-3873	296	4	of	of	ADP
ejpam-3873	296	5	j	j	PROPN
ejpam-3873	296	6	are	be	AUX
ejpam-3873	296	7	0	0	NUM
ejpam-3873	296	8	and	and	CCONJ
ejpam-3873	296	9	1	1	NUM
ejpam-3873	296	10	only	only	ADV
ejpam-3873	296	11	,	,	PUNCT
ejpam-3873	296	12	this	this	PRON
ejpam-3873	296	13	implies	imply	VERB
ejpam-3873	296	14	that	that	SCONJ
ejpam-3873	296	15	x	x	X
ejpam-3873	297	1	+	+	NOUN
ejpam-3873	297	2	1	1	NUM
ejpam-3873	297	3	=	=	SYM
ejpam-3873	297	4	0	0	NUM
ejpam-3873	297	5	,	,	PUNCT
ejpam-3873	297	6	that	that	PRON
ejpam-3873	297	7	is	be	AUX
ejpam-3873	297	8	x	x	NOUN
ejpam-3873	297	9	=	=	SYM
ejpam-3873	297	10	−1	−1	NOUN
ejpam-3873	297	11	.	.	PUNCT
ejpam-3873	298	1	hence	hence	ADV
ejpam-3873	298	2	,	,	PUNCT
ejpam-3873	298	3	x2	x2	PROPN
ejpam-3873	298	4	=	=	PRON
ejpam-3873	298	5	(	(	PUNCT
ejpam-3873	298	6	−1)2	−1)2	X
ejpam-3873	298	7	=	=	SYM
ejpam-3873	298	8	1r	1r	NUM
ejpam-3873	298	9	.	.	PUNCT
ejpam-3873	299	1	by	by	ADP
ejpam-3873	299	2	theorem	theorem	NOUN
ejpam-3873	299	3	2	2	NUM
ejpam-3873	299	4	,	,	PUNCT
ejpam-3873	299	5	x	x	SYM
ejpam-3873	299	6	∈	∈	PROPN
ejpam-3873	299	7	j	j	PROPN
ejpam-3873	299	8	.	.	PUNCT
ejpam-3873	300	1	this	this	PRON
ejpam-3873	300	2	is	be	AUX
ejpam-3873	300	3	a	a	DET
ejpam-3873	300	4	contradiction	contradiction	NOUN
ejpam-3873	300	5	.	.	PUNCT
ejpam-3873	301	1	e.j	e.j	PROPN
ejpam-3873	301	2	.	.	PROPN
ejpam-3873	301	3	sigasig	sigasig	PROPN
ejpam-3873	301	4	,	,	PUNCT
ejpam-3873	301	5	c.j	c.j	PROPN
ejpam-3873	301	6	.	.	PROPN
ejpam-3873	301	7	rosero	rosero	PROPN
ejpam-3873	301	8	,	,	PUNCT
ejpam-3873	301	9	m.	m.	NOUN
ejpam-3873	301	10	baldado	baldado	PROPN
ejpam-3873	301	11	jr	jr	PROPN
ejpam-3873	301	12	.	.	PROPN
ejpam-3873	301	13	/	/	SYM
ejpam-3873	301	14	eur	eur	PROPN
ejpam-3873	301	15	.	.	PUNCT
ejpam-3873	302	1	j.	j.	PROPN
ejpam-3873	302	2	pure	pure	PROPN
ejpam-3873	302	3	appl	appl	PROPN
ejpam-3873	302	4	.	.	PROPN
ejpam-3873	302	5	math	math	PROPN
ejpam-3873	302	6	,	,	PUNCT
ejpam-3873	302	7	14	14	NUM
ejpam-3873	302	8	(	(	PUNCT
ejpam-3873	302	9	1	1	NUM
ejpam-3873	302	10	)	)	PUNCT
ejpam-3873	302	11	(	(	PUNCT
ejpam-3873	302	12	2021	2021	NUM
ejpam-3873	302	13	)	)	PUNCT
ejpam-3873	302	14	,	,	PUNCT
ejpam-3873	302	15	314	314	NUM
ejpam-3873	302	16	-	-	SYM
ejpam-3873	302	17	326	326	NUM
ejpam-3873	302	18	321	321	NUM
ejpam-3873	302	19	theorem	theorem	NOUN
ejpam-3873	302	20	11	11	NUM
ejpam-3873	302	21	.	.	PUNCT
ejpam-3873	303	1	let	let	VERB
ejpam-3873	303	2	t	t	PROPN
ejpam-3873	303	3	be	be	AUX
ejpam-3873	303	4	the	the	DET
ejpam-3873	303	5	set	set	NOUN
ejpam-3873	303	6	of	of	ADP
ejpam-3873	303	7	all	all	DET
ejpam-3873	303	8	γ	γ	NOUN
ejpam-3873	303	9	-	-	NOUN
ejpam-3873	303	10	sets	set	NOUN
ejpam-3873	303	11	of	of	ADP
ejpam-3873	303	12	a	a	DET
ejpam-3873	303	13	division	division	NOUN
ejpam-3873	303	14	ring	ring	NOUN
ejpam-3873	303	15	r	r	NOUN
ejpam-3873	303	16	,	,	PUNCT
ejpam-3873	303	17	and	and	CCONJ
ejpam-3873	304	1	s	s	VERB
ejpam-3873	304	2	=	=	PUNCT
ejpam-3873	304	3	{	{	PUNCT
ejpam-3873	304	4	x	x	SYM
ejpam-3873	304	5	∈	∈	PROPN
ejpam-3873	304	6	r	r	NOUN
ejpam-3873	304	7	:	:	PUNCT
ejpam-3873	304	8	x2	x2	NOUN
ejpam-3873	304	9	=	=	NOUN
ejpam-3873	304	10	1r	1r	NUM
ejpam-3873	304	11	}	}	PUNCT
ejpam-3873	304	12	∪	∪	X
ejpam-3873	304	13	{	{	PUNCT
ejpam-3873	304	14	0	0	NUM
ejpam-3873	304	15	}	}	PUNCT
ejpam-3873	304	16	.	.	PUNCT
ejpam-3873	305	1	then	then	ADV
ejpam-3873	305	2	|t	|t	VERB
ejpam-3873	306	1	|	|	ADV
ejpam-3873	306	2	=	=	SYM
ejpam-3873	306	3	1	1	NUM
ejpam-3873	306	4	if	if	SCONJ
ejpam-3873	306	5	and	and	CCONJ
ejpam-3873	306	6	only	only	ADV
ejpam-3873	306	7	if	if	SCONJ
ejpam-3873	306	8	r	r	NOUN
ejpam-3873	306	9	=	=	PUNCT
ejpam-3873	306	10	s.	s.	PROPN
ejpam-3873	306	11	proof	proof	PROPN
ejpam-3873	306	12	.	.	PUNCT
ejpam-3873	307	1	assume	assume	VERB
ejpam-3873	307	2	that	that	SCONJ
ejpam-3873	307	3	|t	|t	VERB
ejpam-3873	307	4	|	|	ADV
ejpam-3873	307	5	=	=	NOUN
ejpam-3873	307	6	1	1	NUM
ejpam-3873	307	7	,	,	PUNCT
ejpam-3873	307	8	and	and	CCONJ
ejpam-3873	307	9	r	r	NOUN
ejpam-3873	307	10	6=	6=	NUM
ejpam-3873	307	11	s.	s.	PROPN
ejpam-3873	307	12	if	if	SCONJ
ejpam-3873	307	13	r	r	PROPN
ejpam-3873	307	14	6=	6=	PROPN
ejpam-3873	307	15	s	s	PART
ejpam-3873	307	16	,	,	PUNCT
ejpam-3873	307	17	then	then	ADV
ejpam-3873	307	18	there	there	PRON
ejpam-3873	307	19	exists	exist	VERB
ejpam-3873	307	20	x	x	X
ejpam-3873	307	21	∈	∈	PROPN
ejpam-3873	307	22	r\s	r\s	NOUN
ejpam-3873	307	23	such	such	ADJ
ejpam-3873	307	24	that	that	SCONJ
ejpam-3873	307	25	x2	x2	PROPN
ejpam-3873	307	26	=	=	NOUN
ejpam-3873	307	27	1r	1r	NUM
ejpam-3873	307	28	.	.	PUNCT
ejpam-3873	308	1	let	let	VERB
ejpam-3873	308	2	j	j	PROPN
ejpam-3873	308	3	∈	∈	PROPN
ejpam-3873	308	4	t	t	PROPN
ejpam-3873	308	5	and	and	CCONJ
ejpam-3873	308	6	consider	consider	VERB
ejpam-3873	308	7	the	the	DET
ejpam-3873	308	8	following	follow	VERB
ejpam-3873	308	9	cases	case	NOUN
ejpam-3873	308	10	:	:	PUNCT
ejpam-3873	308	11	case	case	NOUN
ejpam-3873	308	12	1	1	NUM
ejpam-3873	308	13	.	.	PUNCT
ejpam-3873	309	1	x	x	X
ejpam-3873	309	2	/∈	/∈	PROPN
ejpam-3873	310	1	j	j	PROPN
ejpam-3873	311	1	if	if	SCONJ
ejpam-3873	311	2	x	x	PROPN
ejpam-3873	311	3	∈	∈	PROPN
ejpam-3873	311	4	j	j	PROPN
ejpam-3873	311	5	,	,	PUNCT
ejpam-3873	311	6	then	then	ADV
ejpam-3873	311	7	by	by	ADP
ejpam-3873	311	8	theorem	theorem	NOUN
ejpam-3873	311	9	3	3	NUM
ejpam-3873	311	10	,	,	PUNCT
ejpam-3873	311	11	j	j	NOUN
ejpam-3873	311	12	′	′	NUM
ejpam-3873	312	1	=	=	SYM
ejpam-3873	312	2	(	(	PUNCT
ejpam-3873	312	3	j\{x−1,−x	j\{x−1,−x	PROPN
ejpam-3873	312	4	}	}	PUNCT
ejpam-3873	312	5	)	)	PUNCT
ejpam-3873	312	6	∪{x	∪{x	NOUN
ejpam-3873	312	7	,	,	PUNCT
ejpam-3873	312	8	(	(	PUNCT
ejpam-3873	312	9	−x)−1	−x)−1	NOUN
ejpam-3873	312	10	}	}	PUNCT
ejpam-3873	312	11	is	be	AUX
ejpam-3873	312	12	another	another	DET
ejpam-3873	312	13	γ	γ	NOUN
ejpam-3873	312	14	-	-	PUNCT
ejpam-3873	312	15	set	set	NOUN
ejpam-3873	312	16	.	.	PUNCT
ejpam-3873	313	1	this	this	PRON
ejpam-3873	313	2	is	be	AUX
ejpam-3873	313	3	a	a	DET
ejpam-3873	313	4	contradiction	contradiction	NOUN
ejpam-3873	313	5	.	.	PUNCT
ejpam-3873	314	1	case	case	NOUN
ejpam-3873	314	2	2	2	NUM
ejpam-3873	314	3	.	.	PUNCT
ejpam-3873	314	4	x	x	SYM
ejpam-3873	315	1	∈	∈	PROPN
ejpam-3873	315	2	j	j	NOUN
ejpam-3873	315	3	if	if	SCONJ
ejpam-3873	315	4	x	x	PROPN
ejpam-3873	315	5	∈	∈	PROPN
ejpam-3873	315	6	j	j	PROPN
ejpam-3873	315	7	,	,	PUNCT
ejpam-3873	315	8	then	then	ADV
ejpam-3873	315	9	by	by	ADP
ejpam-3873	315	10	theorem	theorem	NOUN
ejpam-3873	315	11	3	3	NUM
ejpam-3873	315	12	,	,	PUNCT
ejpam-3873	315	13	j	j	NOUN
ejpam-3873	315	14	′	′	NUM
ejpam-3873	316	1	=	=	SYM
ejpam-3873	316	2	(	(	PUNCT
ejpam-3873	316	3	j\{x	j\{x	X
ejpam-3873	316	4	,	,	PUNCT
ejpam-3873	316	5	(	(	PUNCT
ejpam-3873	316	6	−x)−1	−x)−1	NOUN
ejpam-3873	316	7	}	}	PUNCT
ejpam-3873	316	8	)	)	PUNCT
ejpam-3873	317	1	∪{x−1,−x	∪{x−1,−x	NOUN
ejpam-3873	317	2	}	}	PUNCT
ejpam-3873	317	3	is	be	AUX
ejpam-3873	317	4	another	another	DET
ejpam-3873	317	5	γ	γ	NOUN
ejpam-3873	317	6	-	-	PUNCT
ejpam-3873	317	7	set	set	NOUN
ejpam-3873	317	8	.	.	PUNCT
ejpam-3873	318	1	this	this	PRON
ejpam-3873	318	2	is	be	AUX
ejpam-3873	318	3	a	a	DET
ejpam-3873	318	4	contradiction	contradiction	NOUN
ejpam-3873	318	5	.	.	PUNCT
ejpam-3873	319	1	conversely	conversely	ADV
ejpam-3873	319	2	,	,	PUNCT
ejpam-3873	319	3	suppose	suppose	VERB
ejpam-3873	319	4	that	that	SCONJ
ejpam-3873	319	5	r	r	NOUN
ejpam-3873	319	6	=	=	PUNCT
ejpam-3873	319	7	s.	s.	PROPN
ejpam-3873	319	8	since	since	SCONJ
ejpam-3873	319	9	r	r	NOUN
ejpam-3873	319	10	is	be	AUX
ejpam-3873	319	11	a	a	DET
ejpam-3873	319	12	division	division	NOUN
ejpam-3873	319	13	ring	ring	NOUN
ejpam-3873	319	14	,	,	PUNCT
ejpam-3873	319	15	every	every	DET
ejpam-3873	319	16	non	non	ADJ
ejpam-3873	319	17	-	-	ADJ
ejpam-3873	319	18	zero	zero	NUM
ejpam-3873	319	19	element	element	NOUN
ejpam-3873	319	20	is	be	AUX
ejpam-3873	319	21	an	an	DET
ejpam-3873	319	22	involution	involution	NOUN
ejpam-3873	319	23	.	.	PUNCT
ejpam-3873	320	1	hence	hence	ADV
ejpam-3873	320	2	,	,	PUNCT
ejpam-3873	320	3	by	by	ADP
ejpam-3873	320	4	theorem	theorem	NOUN
ejpam-3873	320	5	10	10	NUM
ejpam-3873	320	6	if	if	SCONJ
ejpam-3873	320	7	j	j	PROPN
ejpam-3873	320	8	is	be	AUX
ejpam-3873	320	9	a	a	DET
ejpam-3873	320	10	γ	γ	NOUN
ejpam-3873	320	11	-	-	PUNCT
ejpam-3873	320	12	set	set	NOUN
ejpam-3873	320	13	of	of	ADP
ejpam-3873	320	14	r	r	NOUN
ejpam-3873	320	15	,	,	PUNCT
ejpam-3873	320	16	we	we	PRON
ejpam-3873	320	17	must	must	AUX
ejpam-3873	320	18	have	have	VERB
ejpam-3873	320	19	j	j	NOUN
ejpam-3873	320	20	=	=	SYM
ejpam-3873	320	21	r	r	NOUN
ejpam-3873	320	22	,	,	PUNCT
ejpam-3873	320	23	that	that	PRON
ejpam-3873	320	24	is	is	ADV
ejpam-3873	320	25	r	r	NOUN
ejpam-3873	320	26	is	be	AUX
ejpam-3873	320	27	the	the	DET
ejpam-3873	320	28	only	only	ADV
ejpam-3873	320	29	γ	γ	PROPN
ejpam-3873	320	30	-	-	PUNCT
ejpam-3873	320	31	set	set	VERB
ejpam-3873	320	32	r.	r.	PROPN
ejpam-3873	320	33	thus	thus	ADV
ejpam-3873	320	34	,	,	PUNCT
ejpam-3873	320	35	|t	|t	VERB
ejpam-3873	320	36	|	|	ADV
ejpam-3873	320	37	=	=	SYM
ejpam-3873	320	38	1	1	X
ejpam-3873	320	39	.	.	PUNCT
ejpam-3873	320	40	theorem	theorem	NOUN
ejpam-3873	320	41	12	12	NUM
ejpam-3873	320	42	.	.	PUNCT
ejpam-3873	321	1	let	let	VERB
ejpam-3873	321	2	q	q	PRON
ejpam-3873	321	3	be	be	AUX
ejpam-3873	321	4	a	a	DET
ejpam-3873	321	5	subring	subring	NOUN
ejpam-3873	321	6	of	of	ADP
ejpam-3873	321	7	r	r	NOUN
ejpam-3873	321	8	,	,	PUNCT
ejpam-3873	321	9	and	and	CCONJ
ejpam-3873	321	10	j	j	PROPN
ejpam-3873	321	11	be	be	VERB
ejpam-3873	321	12	a	a	DET
ejpam-3873	321	13	γ	γ	NOUN
ejpam-3873	321	14	-	-	PUNCT
ejpam-3873	321	15	set	set	NOUN
ejpam-3873	321	16	of	of	ADP
ejpam-3873	321	17	r.	r.	PROPN
ejpam-3873	321	18	then	then	ADV
ejpam-3873	321	19	j	j	PROPN
ejpam-3873	321	20	is	be	AUX
ejpam-3873	321	21	a	a	DET
ejpam-3873	321	22	γ	γ	NOUN
ejpam-3873	321	23	-	-	PUNCT
ejpam-3873	321	24	set	set	NOUN
ejpam-3873	321	25	of	of	ADP
ejpam-3873	321	26	q	q	NOUN
ejpam-3873	321	27	if	if	SCONJ
ejpam-3873	321	28	and	and	CCONJ
ejpam-3873	321	29	only	only	ADV
ejpam-3873	321	30	if	if	SCONJ
ejpam-3873	321	31	q	q	PROPN
ejpam-3873	321	32	=	=	SYM
ejpam-3873	321	33	r.	r.	NOUN
ejpam-3873	321	34	proof	proof	NOUN
ejpam-3873	321	35	.	.	PUNCT
ejpam-3873	322	1	assume	assume	VERB
ejpam-3873	322	2	that	that	SCONJ
ejpam-3873	322	3	j	j	PROPN
ejpam-3873	322	4	is	be	AUX
ejpam-3873	322	5	a	a	DET
ejpam-3873	322	6	γ	γ	NOUN
ejpam-3873	322	7	-	-	PUNCT
ejpam-3873	322	8	set	set	NOUN
ejpam-3873	322	9	of	of	ADP
ejpam-3873	322	10	q	q	NOUN
ejpam-3873	322	11	,	,	PUNCT
ejpam-3873	322	12	and	and	CCONJ
ejpam-3873	322	13	q	q	PROPN
ejpam-3873	322	14	6=	6=	NOUN
ejpam-3873	322	15	r.	r.	PROPN
ejpam-3873	322	16	if	if	SCONJ
ejpam-3873	322	17	q	q	PROPN
ejpam-3873	322	18	6=	6=	NUM
ejpam-3873	322	19	r	r	NOUN
ejpam-3873	322	20	,	,	PUNCT
ejpam-3873	322	21	then	then	ADV
ejpam-3873	322	22	there	there	PRON
ejpam-3873	322	23	exists	exist	VERB
ejpam-3873	322	24	x	x	X
ejpam-3873	322	25	∈	∈	PROPN
ejpam-3873	322	26	r\q	r\q	PROPN
ejpam-3873	322	27	.	.	PUNCT
ejpam-3873	323	1	since	since	SCONJ
ejpam-3873	323	2	x	x	PROPN
ejpam-3873	323	3	/∈	/∈	PUNCT
ejpam-3873	323	4	q	q	NOUN
ejpam-3873	323	5	,	,	PUNCT
ejpam-3873	323	6	x	x	PROPN
ejpam-3873	323	7	/∈	/∈	PROPN
ejpam-3873	323	8	j	j	PROPN
ejpam-3873	323	9	.	.	PUNCT
ejpam-3873	324	1	since	since	SCONJ
ejpam-3873	324	2	j	j	PROPN
ejpam-3873	324	3	,	,	PUNCT
ejpam-3873	324	4	x−1	x−1	PROPN
ejpam-3873	324	5	∈	∈	PROPN
ejpam-3873	324	6	j	j	PROPN
ejpam-3873	324	7	.	.	PUNCT
ejpam-3873	325	1	since	since	SCONJ
ejpam-3873	325	2	j	j	PROPN
ejpam-3873	325	3	is	be	AUX
ejpam-3873	325	4	also	also	ADV
ejpam-3873	325	5	a	a	DET
ejpam-3873	325	6	γ	γ	X
ejpam-3873	325	7	-	-	PUNCT
ejpam-3873	325	8	set	set	NOUN
ejpam-3873	325	9	of	of	ADP
ejpam-3873	325	10	q	q	NOUN
ejpam-3873	325	11	,	,	PUNCT
ejpam-3873	325	12	x	x	X
ejpam-3873	325	13	=	=	SYM
ejpam-3873	325	14	xx−1	xx−1	PROPN
ejpam-3873	325	15	∈	∈	PROPN
ejpam-3873	325	16	t	t	PROPN
ejpam-3873	325	17	.	.	PUNCT
ejpam-3873	326	1	this	this	PRON
ejpam-3873	326	2	is	be	AUX
ejpam-3873	326	3	a	a	DET
ejpam-3873	326	4	contradiction	contradiction	NOUN
ejpam-3873	326	5	.	.	PUNCT
ejpam-3873	327	1	the	the	DET
ejpam-3873	327	2	converse	converse	NOUN
ejpam-3873	327	3	is	be	AUX
ejpam-3873	327	4	clear	clear	ADJ
ejpam-3873	327	5	.	.	PUNCT
ejpam-3873	328	1	theorem	theorem	ADJ
ejpam-3873	328	2	13	13	NUM
ejpam-3873	328	3	.	.	PUNCT
ejpam-3873	329	1	let	let	VERB
ejpam-3873	329	2	r1	r1	PROPN
ejpam-3873	329	3	and	and	CCONJ
ejpam-3873	329	4	r2	r2	PROPN
ejpam-3873	329	5	be	be	VERB
ejpam-3873	329	6	rings	ring	NOUN
ejpam-3873	329	7	,	,	PUNCT
ejpam-3873	329	8	and	and	CCONJ
ejpam-3873	329	9	φ	φ	NUM
ejpam-3873	329	10	:	:	PUNCT
ejpam-3873	330	1	r1	r1	PROPN
ejpam-3873	330	2	→	→	SYM
ejpam-3873	330	3	r2	r2	PROPN
ejpam-3873	330	4	be	be	AUX
ejpam-3873	330	5	an	an	DET
ejpam-3873	330	6	epimorphism	epimorphism	NOUN
ejpam-3873	330	7	of	of	ADP
ejpam-3873	330	8	rings	ring	NOUN
ejpam-3873	330	9	.	.	PUNCT
ejpam-3873	331	1	if	if	SCONJ
ejpam-3873	331	2	j	j	PROPN
ejpam-3873	331	3	is	be	AUX
ejpam-3873	331	4	a	a	DET
ejpam-3873	331	5	γ	γ	NOUN
ejpam-3873	331	6	-	-	PUNCT
ejpam-3873	331	7	set	set	NOUN
ejpam-3873	331	8	of	of	ADP
ejpam-3873	331	9	r1	r1	NOUN
ejpam-3873	331	10	,	,	PUNCT
ejpam-3873	331	11	then	then	ADV
ejpam-3873	331	12	φ(j	φ(j	PROPN
ejpam-3873	331	13	)	)	PUNCT
ejpam-3873	331	14	is	be	AUX
ejpam-3873	331	15	a	a	DET
ejpam-3873	331	16	γ	γ	NOUN
ejpam-3873	331	17	-	-	PUNCT
ejpam-3873	331	18	set	set	NOUN
ejpam-3873	331	19	of	of	ADP
ejpam-3873	331	20	r2	r2	NOUN
ejpam-3873	331	21	.	.	PUNCT
ejpam-3873	332	1	proof	proof	NOUN
ejpam-3873	332	2	.	.	PUNCT
ejpam-3873	333	1	let	let	VERB
ejpam-3873	333	2	j	j	PROPN
ejpam-3873	333	3	be	be	AUX
ejpam-3873	333	4	a	a	DET
ejpam-3873	333	5	γ	γ	NOUN
ejpam-3873	333	6	-	-	PUNCT
ejpam-3873	333	7	set	set	NOUN
ejpam-3873	333	8	of	of	ADP
ejpam-3873	333	9	r1	r1	NOUN
ejpam-3873	333	10	,	,	PUNCT
ejpam-3873	333	11	and	and	CCONJ
ejpam-3873	333	12	y	y	PROPN
ejpam-3873	333	13	∈	∈	PROPN
ejpam-3873	333	14	r2\φ(j	r2\φ(j	NOUN
ejpam-3873	333	15	)	)	PUNCT
ejpam-3873	333	16	.	.	PUNCT
ejpam-3873	334	1	if	if	SCONJ
ejpam-3873	334	2	y	y	PROPN
ejpam-3873	334	3	∈	∈	PROPN
ejpam-3873	334	4	r2\φ(j	r2\φ(j	NOUN
ejpam-3873	334	5	)	)	PUNCT
ejpam-3873	334	6	and	and	CCONJ
ejpam-3873	334	7	φ	φ	PROPN
ejpam-3873	334	8	is	be	AUX
ejpam-3873	334	9	an	an	DET
ejpam-3873	334	10	epimorphism	epimorphism	NOUN
ejpam-3873	334	11	,	,	PUNCT
ejpam-3873	334	12	then	then	ADV
ejpam-3873	334	13	there	there	PRON
ejpam-3873	334	14	exists	exist	VERB
ejpam-3873	334	15	x	x	X
ejpam-3873	334	16	∈	∈	PROPN
ejpam-3873	334	17	r1	r1	NOUN
ejpam-3873	334	18	such	such	ADJ
ejpam-3873	334	19	that	that	SCONJ
ejpam-3873	334	20	φ(x	φ(x	NOUN
ejpam-3873	334	21	)	)	PUNCT
ejpam-3873	334	22	=	=	SYM
ejpam-3873	334	23	y.	y.	NOUN
ejpam-3873	334	24	note	note	VERB
ejpam-3873	334	25	that	that	SCONJ
ejpam-3873	334	26	x	x	SYM
ejpam-3873	334	27	∈	∈	NOUN
ejpam-3873	334	28	r1\j	r1\j	NOUN
ejpam-3873	334	29	,	,	PUNCT
ejpam-3873	334	30	otherwise	otherwise	ADV
ejpam-3873	334	31	y	y	PROPN
ejpam-3873	334	32	=	=	SYM
ejpam-3873	334	33	φ(x	φ(x	X
ejpam-3873	334	34	)	)	PUNCT
ejpam-3873	334	35	∈	∈	NOUN
ejpam-3873	334	36	φ(j	φ(j	PROPN
ejpam-3873	334	37	)	)	PUNCT
ejpam-3873	334	38	.	.	PUNCT
ejpam-3873	335	1	since	since	SCONJ
ejpam-3873	335	2	j	j	PROPN
ejpam-3873	335	3	is	be	AUX
ejpam-3873	335	4	a	a	DET
ejpam-3873	335	5	γ	γ	NOUN
ejpam-3873	335	6	-	-	PUNCT
ejpam-3873	335	7	set	set	NOUN
ejpam-3873	335	8	,	,	PUNCT
ejpam-3873	335	9	there	there	PRON
ejpam-3873	335	10	exists	exist	VERB
ejpam-3873	335	11	u	u	NOUN
ejpam-3873	335	12	,	,	PUNCT
ejpam-3873	335	13	v	v	PROPN
ejpam-3873	335	14	∈	∈	PROPN
ejpam-3873	335	15	j	j	NOUN
ejpam-3873	335	16	such	such	ADJ
ejpam-3873	335	17	that	that	SCONJ
ejpam-3873	335	18	xu	xu	PROPN
ejpam-3873	336	1	=	=	SYM
ejpam-3873	336	2	1r1	1r1	NUM
ejpam-3873	336	3	and	and	CCONJ
ejpam-3873	336	4	x+	x+	NUM
ejpam-3873	336	5	v	v	NOUN
ejpam-3873	336	6	=	=	NOUN
ejpam-3873	336	7	0r1	0r1	NUM
ejpam-3873	336	8	.	.	PUNCT
ejpam-3873	337	1	clearly	clearly	ADV
ejpam-3873	337	2	,	,	PUNCT
ejpam-3873	337	3	φ(u	φ(u	PROPN
ejpam-3873	337	4	)	)	PUNCT
ejpam-3873	337	5	,	,	PUNCT
ejpam-3873	337	6	φ(v	φ(v	X
ejpam-3873	337	7	)	)	PUNCT
ejpam-3873	337	8	∈	∈	PROPN
ejpam-3873	337	9	φ(j	φ(j	PROPN
ejpam-3873	337	10	)	)	PUNCT
ejpam-3873	337	11	.	.	PUNCT
ejpam-3873	338	1	since	since	SCONJ
ejpam-3873	338	2	φ	φ	PROPN
ejpam-3873	338	3	is	be	AUX
ejpam-3873	338	4	a	a	DET
ejpam-3873	338	5	homomorphism	homomorphism	NOUN
ejpam-3873	338	6	,	,	PUNCT
ejpam-3873	338	7	φ(x)φ(u	φ(x)φ(u	NOUN
ejpam-3873	338	8	)	)	PUNCT
ejpam-3873	338	9	=	=	SYM
ejpam-3873	338	10	φ(xu	φ(xu	NOUN
ejpam-3873	338	11	)	)	PUNCT
ejpam-3873	338	12	=	=	SYM
ejpam-3873	338	13	φ(1r1	φ(1r1	NUM
ejpam-3873	338	14	)	)	PUNCT
ejpam-3873	338	15	=	=	PUNCT
ejpam-3873	338	16	1r2	1r2	NUM
ejpam-3873	338	17	and	and	CCONJ
ejpam-3873	338	18	φ(x	φ(x	NOUN
ejpam-3873	338	19	)	)	PUNCT
ejpam-3873	339	1	+	+	SYM
ejpam-3873	339	2	φ(v	φ(v	X
ejpam-3873	339	3	)	)	PUNCT
ejpam-3873	339	4	=	=	PUNCT
ejpam-3873	340	1	φ(x	φ(x	PROPN
ejpam-3873	340	2	+	+	CCONJ
ejpam-3873	340	3	v	v	NOUN
ejpam-3873	340	4	)	)	PUNCT
ejpam-3873	340	5	=	=	SYM
ejpam-3873	340	6	φ(0r1	φ(0r1	X
ejpam-3873	340	7	)	)	PUNCT
ejpam-3873	340	8	=	=	NOUN
ejpam-3873	340	9	0r2	0r2	NOUN
ejpam-3873	340	10	.	.	PUNCT
ejpam-3873	341	1	hence	hence	ADV
ejpam-3873	341	2	,	,	PUNCT
ejpam-3873	341	3	there	there	PRON
ejpam-3873	341	4	exists	exist	VERB
ejpam-3873	341	5	φ(u	φ(u	NOUN
ejpam-3873	341	6	)	)	PUNCT
ejpam-3873	341	7	,	,	PUNCT
ejpam-3873	341	8	φ(v	φ(v	X
ejpam-3873	341	9	)	)	PUNCT
ejpam-3873	341	10	∈	∈	PROPN
ejpam-3873	341	11	φ(j	φ(j	PROPN
ejpam-3873	341	12	)	)	PUNCT
ejpam-3873	341	13	such	such	ADJ
ejpam-3873	341	14	that	that	SCONJ
ejpam-3873	341	15	φ(x)φ(u	φ(x)φ(u	NOUN
ejpam-3873	341	16	)	)	PUNCT
ejpam-3873	341	17	=	=	PUNCT
ejpam-3873	341	18	1r2	1r2	NUM
ejpam-3873	341	19	and	and	CCONJ
ejpam-3873	341	20	φ(x	φ(x	NOUN
ejpam-3873	341	21	)	)	PUNCT
ejpam-3873	342	1	+	+	SYM
ejpam-3873	342	2	φ(v	φ(v	X
ejpam-3873	342	3	)	)	PUNCT
ejpam-3873	342	4	=	=	PUNCT
ejpam-3873	342	5	0r2	0r2	NOUN
ejpam-3873	342	6	.	.	PUNCT
ejpam-3873	343	1	this	this	PRON
ejpam-3873	343	2	shows	show	VERB
ejpam-3873	343	3	that	that	SCONJ
ejpam-3873	343	4	φ(j	φ(j	PROPN
ejpam-3873	343	5	)	)	PUNCT
ejpam-3873	343	6	is	be	AUX
ejpam-3873	343	7	a	a	DET
ejpam-3873	343	8	γ	γ	NOUN
ejpam-3873	343	9	-	-	PUNCT
ejpam-3873	343	10	set	set	NOUN
ejpam-3873	343	11	of	of	ADP
ejpam-3873	343	12	r2	r2	PROPN
ejpam-3873	343	13	.	.	PUNCT
ejpam-3873	344	1	theorem	theorem	VERB
ejpam-3873	344	2	14	14	NUM
ejpam-3873	344	3	.	.	PUNCT
ejpam-3873	345	1	let	let	VERB
ejpam-3873	345	2	r1	r1	PROPN
ejpam-3873	345	3	and	and	CCONJ
ejpam-3873	345	4	r2	r2	PROPN
ejpam-3873	345	5	be	be	VERB
ejpam-3873	345	6	rings	ring	NOUN
ejpam-3873	345	7	,	,	PUNCT
ejpam-3873	345	8	and	and	CCONJ
ejpam-3873	345	9	φ	φ	NUM
ejpam-3873	345	10	:	:	PUNCT
ejpam-3873	346	1	r1	r1	PROPN
ejpam-3873	346	2	→	→	SYM
ejpam-3873	346	3	r2	r2	PROPN
ejpam-3873	346	4	be	be	AUX
ejpam-3873	346	5	an	an	DET
ejpam-3873	346	6	isomorphism	isomorphism	NOUN
ejpam-3873	346	7	of	of	ADP
ejpam-3873	346	8	rings	ring	NOUN
ejpam-3873	346	9	.	.	PUNCT
ejpam-3873	347	1	then	then	ADV
ejpam-3873	347	2	,	,	PUNCT
ejpam-3873	347	3	j	j	PROPN
ejpam-3873	347	4	is	be	AUX
ejpam-3873	347	5	a	a	DET
ejpam-3873	347	6	γ	γ	NOUN
ejpam-3873	347	7	-	-	PUNCT
ejpam-3873	347	8	set	set	NOUN
ejpam-3873	347	9	of	of	ADP
ejpam-3873	347	10	r1	r1	PROPN
ejpam-3873	347	11	if	if	SCONJ
ejpam-3873	347	12	and	and	CCONJ
ejpam-3873	347	13	only	only	ADV
ejpam-3873	347	14	if	if	SCONJ
ejpam-3873	347	15	φ(j	φ(j	PROPN
ejpam-3873	347	16	)	)	PUNCT
ejpam-3873	347	17	is	be	AUX
ejpam-3873	347	18	a	a	DET
ejpam-3873	347	19	γ	γ	NOUN
ejpam-3873	347	20	-	-	PUNCT
ejpam-3873	347	21	set	set	NOUN
ejpam-3873	347	22	of	of	ADP
ejpam-3873	347	23	r2	r2	NOUN
ejpam-3873	347	24	.	.	PUNCT
ejpam-3873	348	1	proof	proof	NOUN
ejpam-3873	348	2	.	.	PUNCT
ejpam-3873	349	1	let	let	VERB
ejpam-3873	349	2	j	j	PROPN
ejpam-3873	349	3	be	be	AUX
ejpam-3873	349	4	a	a	DET
ejpam-3873	349	5	γ	γ	NOUN
ejpam-3873	349	6	-	-	PUNCT
ejpam-3873	349	7	set	set	NOUN
ejpam-3873	349	8	of	of	ADP
ejpam-3873	349	9	r1	r1	PROPN
ejpam-3873	349	10	.	.	PUNCT
ejpam-3873	350	1	then	then	ADV
ejpam-3873	350	2	by	by	ADP
ejpam-3873	350	3	theorem	theorem	ADJ
ejpam-3873	350	4	7	7	NUM
ejpam-3873	350	5	,	,	PUNCT
ejpam-3873	350	6	φ(j	φ(j	PROPN
ejpam-3873	350	7	)	)	PUNCT
ejpam-3873	350	8	is	be	AUX
ejpam-3873	350	9	a	a	DET
ejpam-3873	350	10	γ	γ	NOUN
ejpam-3873	350	11	-	-	PUNCT
ejpam-3873	350	12	set	set	NOUN
ejpam-3873	350	13	of	of	ADP
ejpam-3873	350	14	r2	r2	PROPN
ejpam-3873	350	15	.	.	PUNCT
ejpam-3873	351	1	conversely	conversely	ADV
ejpam-3873	351	2	,	,	PUNCT
ejpam-3873	351	3	let	let	VERB
ejpam-3873	351	4	j	j	PROPN
ejpam-3873	351	5	be	be	AUX
ejpam-3873	351	6	a	a	DET
ejpam-3873	351	7	subset	subset	NOUN
ejpam-3873	351	8	of	of	ADP
ejpam-3873	351	9	r1	r1	NOUN
ejpam-3873	351	10	,	,	PUNCT
ejpam-3873	351	11	and	and	CCONJ
ejpam-3873	351	12	suppose	suppose	VERB
ejpam-3873	351	13	that	that	SCONJ
ejpam-3873	351	14	φ(j	φ(j	PROPN
ejpam-3873	351	15	)	)	PUNCT
ejpam-3873	351	16	is	be	AUX
ejpam-3873	351	17	a	a	DET
ejpam-3873	351	18	γ	γ	NOUN
ejpam-3873	351	19	-	-	PUNCT
ejpam-3873	351	20	set	set	NOUN
ejpam-3873	351	21	of	of	ADP
ejpam-3873	351	22	r2	r2	PROPN
ejpam-3873	351	23	.	.	PUNCT
ejpam-3873	352	1	let	let	VERB
ejpam-3873	352	2	x	x	PUNCT
ejpam-3873	352	3	∈	∈	PROPN
ejpam-3873	352	4	r1\j	r1\j	NOUN
ejpam-3873	352	5	.	.	PUNCT
ejpam-3873	353	1	then	then	ADV
ejpam-3873	353	2	φ(x	φ(x	NOUN
ejpam-3873	353	3	)	)	PUNCT
ejpam-3873	353	4	∈	∈	PROPN
ejpam-3873	353	5	r2\φ(j	r2\φ(j	NOUN
ejpam-3873	353	6	)	)	PUNCT
ejpam-3873	353	7	,	,	PUNCT
ejpam-3873	353	8	otherwise	otherwise	ADV
ejpam-3873	353	9	x	x	X
ejpam-3873	353	10	=	=	SYM
ejpam-3873	353	11	φ−1φ(x	φ−1φ(x	PROPN
ejpam-3873	353	12	)	)	PUNCT
ejpam-3873	353	13	∈	∈	PROPN
ejpam-3873	353	14	j	j	PROPN
ejpam-3873	353	15	.	.	PUNCT
ejpam-3873	354	1	since	since	SCONJ
ejpam-3873	354	2	φ(j	φ(j	PROPN
ejpam-3873	354	3	)	)	PUNCT
ejpam-3873	354	4	is	be	AUX
ejpam-3873	354	5	a	a	DET
ejpam-3873	354	6	γ	γ	NOUN
ejpam-3873	354	7	-	-	PUNCT
ejpam-3873	354	8	set	set	NOUN
ejpam-3873	354	9	of	of	ADP
ejpam-3873	354	10	r2	r2	NOUN
ejpam-3873	354	11	,	,	PUNCT
ejpam-3873	354	12	there	there	PRON
ejpam-3873	354	13	exists	exist	VERB
ejpam-3873	354	14	u	u	NOUN
ejpam-3873	354	15	,	,	PUNCT
ejpam-3873	354	16	v	v	NOUN
ejpam-3873	354	17	∈	∈	NOUN
ejpam-3873	354	18	φ(j	φ(j	PROPN
ejpam-3873	354	19	)	)	PUNCT
ejpam-3873	354	20	such	such	ADJ
ejpam-3873	354	21	that	that	DET
ejpam-3873	354	22	φ(x)u	φ(x)u	NOUN
ejpam-3873	354	23	=	=	NUM
ejpam-3873	354	24	1r2	1r2	NUM
ejpam-3873	354	25	and	and	CCONJ
ejpam-3873	354	26	φ(x	φ(x	NOUN
ejpam-3873	354	27	)	)	PUNCT
ejpam-3873	355	1	+	+	CCONJ
ejpam-3873	355	2	v	v	X
ejpam-3873	355	3	=	=	NOUN
ejpam-3873	355	4	0r2	0r2	NOUN
ejpam-3873	355	5	.	.	PUNCT
ejpam-3873	356	1	note	note	VERB
ejpam-3873	356	2	that	that	SCONJ
ejpam-3873	356	3	φ−1(u	φ−1(u	PROPN
ejpam-3873	356	4	)	)	PUNCT
ejpam-3873	356	5	,	,	PUNCT
ejpam-3873	356	6	φ−1(v	φ−1(v	PROPN
ejpam-3873	356	7	)	)	PUNCT
ejpam-3873	357	1	∈	∈	PROPN
ejpam-3873	357	2	j	j	PROPN
ejpam-3873	357	3	,	,	PUNCT
ejpam-3873	357	4	otherwise	otherwise	ADV
ejpam-3873	357	5	u	u	X
ejpam-3873	357	6	=	=	PUNCT
ejpam-3873	357	7	φ(φ−1(u	φ(φ−1(u	PROPN
ejpam-3873	357	8	)	)	PUNCT
ejpam-3873	357	9	)	)	PUNCT
ejpam-3873	358	1	∈	∈	PROPN
ejpam-3873	358	2	r2\φ(j	r2\φ(j	NOUN
ejpam-3873	358	3	)	)	PUNCT
ejpam-3873	358	4	and	and	CCONJ
ejpam-3873	358	5	v	v	NOUN
ejpam-3873	358	6	=	=	SYM
ejpam-3873	358	7	φ(φ−1(v	φ(φ−1(v	PROPN
ejpam-3873	358	8	)	)	PUNCT
ejpam-3873	358	9	)	)	PUNCT
ejpam-3873	359	1	∈	∈	PROPN
ejpam-3873	359	2	r2\φ(j	r2\φ(j	NOUN
ejpam-3873	359	3	)	)	PUNCT
ejpam-3873	359	4	.	.	PUNCT
ejpam-3873	360	1	since	since	SCONJ
ejpam-3873	360	2	φ	φ	PROPN
ejpam-3873	360	3	is	be	AUX
ejpam-3873	360	4	an	an	DET
ejpam-3873	360	5	isomorphism	isomorphism	NOUN
ejpam-3873	360	6	,	,	PUNCT
ejpam-3873	360	7	xφ−1(u	xφ−1(u	PROPN
ejpam-3873	360	8	)	)	PUNCT
ejpam-3873	361	1	=	=	SYM
ejpam-3873	361	2	φ−1(φ(x))φ−1(u	φ−1(φ(x))φ−1(u	NOUN
ejpam-3873	361	3	)	)	PUNCT
ejpam-3873	361	4	=	=	SYM
ejpam-3873	361	5	φ−1(φ(x)u	φ−1(φ(x)u	NOUN
ejpam-3873	361	6	)	)	PUNCT
ejpam-3873	361	7	=	=	SYM
ejpam-3873	361	8	φ−1(1r2	φ−1(1r2	X
ejpam-3873	361	9	)	)	PUNCT
ejpam-3873	361	10	=	=	SYM
ejpam-3873	361	11	1r1	1r1	NUM
ejpam-3873	361	12	and	and	CCONJ
ejpam-3873	361	13	x	x	PROPN
ejpam-3873	361	14	+	+	ADJ
ejpam-3873	361	15	φ−1(v	φ−1(v	PROPN
ejpam-3873	361	16	)	)	PUNCT
ejpam-3873	361	17	=	=	SYM
ejpam-3873	361	18	φ−1(φ(x	φ−1(φ(x	NOUN
ejpam-3873	361	19	)	)	PUNCT
ejpam-3873	361	20	)	)	PUNCT
ejpam-3873	362	1	+	+	CCONJ
ejpam-3873	362	2	φ−1(v	φ−1(v	PROPN
ejpam-3873	362	3	)	)	PUNCT
ejpam-3873	362	4	=	=	SYM
ejpam-3873	362	5	φ−1(φ(x	φ−1(φ(x	NOUN
ejpam-3873	362	6	)	)	PUNCT
ejpam-3873	363	1	+	+	SYM
ejpam-3873	363	2	v	v	NOUN
ejpam-3873	363	3	)	)	PUNCT
ejpam-3873	363	4	=	=	SYM
ejpam-3873	364	1	φ−1(0r2	φ−1(0r2	X
ejpam-3873	364	2	)	)	PUNCT
ejpam-3873	364	3	=	=	NOUN
ejpam-3873	364	4	0r1	0r1	NUM
ejpam-3873	364	5	.	.	PUNCT
ejpam-3873	365	1	hence	hence	ADV
ejpam-3873	365	2	,	,	PUNCT
ejpam-3873	365	3	there	there	PRON
ejpam-3873	365	4	exists	exist	VERB
ejpam-3873	365	5	φ−1(u	φ−1(u	PROPN
ejpam-3873	365	6	)	)	PUNCT
ejpam-3873	365	7	,	,	PUNCT
ejpam-3873	365	8	φ−1(v	φ−1(v	PROPN
ejpam-3873	365	9	)	)	PUNCT
ejpam-3873	366	1	∈	∈	PROPN
ejpam-3873	366	2	j	j	PROPN
ejpam-3873	366	3	such	such	ADJ
ejpam-3873	366	4	that	that	SCONJ
ejpam-3873	366	5	xφ−1(u	xφ−1(u	PROPN
ejpam-3873	366	6	)	)	PUNCT
ejpam-3873	367	1	=	=	SYM
ejpam-3873	367	2	1r1	1r1	NUM
ejpam-3873	367	3	and	and	CCONJ
ejpam-3873	367	4	x	x	PROPN
ejpam-3873	367	5	+	+	ADJ
ejpam-3873	367	6	φ−1(v	φ−1(v	PROPN
ejpam-3873	367	7	)	)	PUNCT
ejpam-3873	368	1	=	=	NOUN
ejpam-3873	368	2	0r1	0r1	NUM
ejpam-3873	368	3	.	.	PUNCT
ejpam-3873	369	1	this	this	PRON
ejpam-3873	369	2	shows	show	VERB
ejpam-3873	369	3	that	that	SCONJ
ejpam-3873	369	4	j	j	PROPN
ejpam-3873	369	5	is	be	AUX
ejpam-3873	369	6	a	a	DET
ejpam-3873	369	7	γ	γ	NOUN
ejpam-3873	369	8	-	-	PUNCT
ejpam-3873	369	9	set	set	NOUN
ejpam-3873	369	10	of	of	ADP
ejpam-3873	369	11	r1	r1	PROPN
ejpam-3873	369	12	.	.	PUNCT
ejpam-3873	370	1	e.j	e.j	PROPN
ejpam-3873	370	2	.	.	PROPN
ejpam-3873	370	3	sigasig	sigasig	PROPN
ejpam-3873	370	4	,	,	PUNCT
ejpam-3873	370	5	c.j	c.j	PROPN
ejpam-3873	370	6	.	.	PROPN
ejpam-3873	370	7	rosero	rosero	PROPN
ejpam-3873	370	8	,	,	PUNCT
ejpam-3873	370	9	m.	m.	NOUN
ejpam-3873	370	10	baldado	baldado	PROPN
ejpam-3873	370	11	jr	jr	PROPN
ejpam-3873	370	12	.	.	PROPN
ejpam-3873	370	13	/	/	SYM
ejpam-3873	370	14	eur	eur	PROPN
ejpam-3873	370	15	.	.	PUNCT
ejpam-3873	371	1	j.	j.	PROPN
ejpam-3873	371	2	pure	pure	PROPN
ejpam-3873	371	3	appl	appl	PROPN
ejpam-3873	371	4	.	.	PROPN
ejpam-3873	371	5	math	math	PROPN
ejpam-3873	371	6	,	,	PUNCT
ejpam-3873	371	7	14	14	NUM
ejpam-3873	371	8	(	(	PUNCT
ejpam-3873	371	9	1	1	NUM
ejpam-3873	371	10	)	)	PUNCT
ejpam-3873	371	11	(	(	PUNCT
ejpam-3873	371	12	2021	2021	NUM
ejpam-3873	371	13	)	)	PUNCT
ejpam-3873	371	14	,	,	PUNCT
ejpam-3873	371	15	314	314	NUM
ejpam-3873	371	16	-	-	SYM
ejpam-3873	371	17	326	326	NUM
ejpam-3873	371	18	322	322	NUM
ejpam-3873	371	19	theorem	theorem	NOUN
ejpam-3873	371	20	15	15	NUM
ejpam-3873	371	21	.	.	PUNCT
ejpam-3873	372	1	let	let	VERB
ejpam-3873	372	2	r	r	PRON
ejpam-3873	372	3	be	be	AUX
ejpam-3873	372	4	a	a	DET
ejpam-3873	372	5	ring	ring	NOUN
ejpam-3873	372	6	,	,	PUNCT
ejpam-3873	372	7	t	t	NOUN
ejpam-3873	372	8	=	=	SYM
ejpam-3873	372	9	{	{	PUNCT
ejpam-3873	372	10	j	j	NOUN
ejpam-3873	372	11	⊆	⊆	NUM
ejpam-3873	372	12	r	r	NOUN
ejpam-3873	372	13	:	:	PUNCT
ejpam-3873	372	14	j	j	PROPN
ejpam-3873	372	15	is	be	AUX
ejpam-3873	372	16	a	a	DET
ejpam-3873	372	17	γ	γ	NOUN
ejpam-3873	372	18	-	-	PUNCT
ejpam-3873	372	19	set	set	NOUN
ejpam-3873	372	20	of	of	ADP
ejpam-3873	372	21	r	r	NOUN
ejpam-3873	372	22	}	}	PUNCT
ejpam-3873	372	23	,	,	PUNCT
ejpam-3873	372	24	and	and	CCONJ
ejpam-3873	372	25	t	t	NOUN
ejpam-3873	372	26	′	′	NUM
ejpam-3873	373	1	=	=	PUNCT
ejpam-3873	373	2	{	{	PUNCT
ejpam-3873	373	3	j	j	NOUN
ejpam-3873	373	4	′	′	NUM
ejpam-3873	373	5	:	:	PUNCT
ejpam-3873	374	1	j	j	PROPN
ejpam-3873	374	2	∈	∈	PROPN
ejpam-3873	374	3	t	t	PROPN
ejpam-3873	374	4	}	}	PUNCT
ejpam-3873	374	5	where	where	SCONJ
ejpam-3873	374	6	j	j	PROPN
ejpam-3873	374	7	′	′	PROPN
ejpam-3873	374	8	is	be	AUX
ejpam-3873	374	9	the	the	DET
ejpam-3873	374	10	complement	complement	NOUN
ejpam-3873	374	11	of	of	ADP
ejpam-3873	374	12	j	j	PROPN
ejpam-3873	374	13	.	.	PUNCT
ejpam-3873	375	1	then	then	ADV
ejpam-3873	375	2	,	,	PUNCT
ejpam-3873	375	3	t	t	PROPN
ejpam-3873	375	4	is	be	AUX
ejpam-3873	375	5	isomorphic	isomorphic	ADJ
ejpam-3873	375	6	to	to	ADP
ejpam-3873	375	7	t	t	PROPN
ejpam-3873	375	8	′.	′.	NOUN
ejpam-3873	375	9	proof	proof	NOUN
ejpam-3873	375	10	.	.	PUNCT
ejpam-3873	376	1	let	let	VERB
ejpam-3873	376	2	r	r	PRON
ejpam-3873	376	3	be	be	AUX
ejpam-3873	376	4	a	a	DET
ejpam-3873	376	5	ring	ring	NOUN
ejpam-3873	376	6	,	,	PUNCT
ejpam-3873	376	7	t	t	NOUN
ejpam-3873	376	8	=	=	SYM
ejpam-3873	376	9	{	{	PUNCT
ejpam-3873	376	10	j	j	NOUN
ejpam-3873	376	11	⊆	⊆	NUM
ejpam-3873	376	12	r	r	NOUN
ejpam-3873	376	13	:	:	PUNCT
ejpam-3873	376	14	j	j	PROPN
ejpam-3873	376	15	is	be	AUX
ejpam-3873	376	16	a	a	DET
ejpam-3873	376	17	γ	γ	NOUN
ejpam-3873	376	18	-	-	PUNCT
ejpam-3873	376	19	set	set	NOUN
ejpam-3873	376	20	of	of	ADP
ejpam-3873	376	21	r	r	NOUN
ejpam-3873	376	22	}	}	PUNCT
ejpam-3873	376	23	,	,	PUNCT
ejpam-3873	376	24	and	and	CCONJ
ejpam-3873	376	25	t	t	NOUN
ejpam-3873	376	26	′	′	NUM
ejpam-3873	377	1	=	=	PUNCT
ejpam-3873	377	2	{	{	PUNCT
ejpam-3873	377	3	j	j	NOUN
ejpam-3873	377	4	′	′	NUM
ejpam-3873	377	5	:	:	PUNCT
ejpam-3873	378	1	j	j	PROPN
ejpam-3873	378	2	∈	∈	PROPN
ejpam-3873	378	3	t	t	PROPN
ejpam-3873	378	4	}	}	PUNCT
ejpam-3873	378	5	.	.	PUNCT
ejpam-3873	379	1	define	define	VERB
ejpam-3873	379	2	φ	φ	NOUN
ejpam-3873	379	3	:	:	PUNCT
ejpam-3873	379	4	t	t	PROPN
ejpam-3873	379	5	→	→	SYM
ejpam-3873	379	6	t	t	PROPN
ejpam-3873	379	7	′	′	NUM
ejpam-3873	379	8	by	by	ADP
ejpam-3873	379	9	j	j	PROPN
ejpam-3873	379	10	7→	7→	PROPN
ejpam-3873	379	11	j	j	NOUN
ejpam-3873	380	1	′	′	NUM
ejpam-3873	380	2	where	where	SCONJ
ejpam-3873	380	3	j	j	PROPN
ejpam-3873	380	4	′	′	PROPN
ejpam-3873	380	5	is	be	AUX
ejpam-3873	380	6	the	the	DET
ejpam-3873	380	7	complement	complement	NOUN
ejpam-3873	380	8	of	of	ADP
ejpam-3873	380	9	j	j	PROPN
ejpam-3873	380	10	.	.	PUNCT
ejpam-3873	381	1	then	then	ADV
ejpam-3873	381	2	clearly	clearly	ADV
ejpam-3873	381	3	φ	φ	PROPN
ejpam-3873	381	4	is	be	AUX
ejpam-3873	381	5	bijective	bijective	ADJ
ejpam-3873	381	6	.	.	PUNCT
ejpam-3873	382	1	now	now	ADV
ejpam-3873	382	2	,	,	PUNCT
ejpam-3873	382	3	let	let	VERB
ejpam-3873	382	4	j1	j1	PROPN
ejpam-3873	382	5	,	,	PUNCT
ejpam-3873	382	6	j2	j2	PROPN
ejpam-3873	382	7	∈	∈	PROPN
ejpam-3873	382	8	t	t	PROPN
ejpam-3873	382	9	.	.	PUNCT
ejpam-3873	383	1	then	then	ADV
ejpam-3873	383	2	φ(j1	φ(j1	NOUN
ejpam-3873	383	3	∪	∪	PROPN
ejpam-3873	383	4	j2	j2	PROPN
ejpam-3873	383	5	)	)	PUNCT
ejpam-3873	383	6	=	=	PUNCT
ejpam-3873	383	7	(	(	PUNCT
ejpam-3873	383	8	j1	j1	PROPN
ejpam-3873	383	9	∪	∪	NOUN
ejpam-3873	383	10	j2)′	j2)′	NOUN
ejpam-3873	383	11	=	=	SYM
ejpam-3873	383	12	j	j	PROPN
ejpam-3873	383	13	′1	′1	X
ejpam-3873	383	14	∩	∩	NOUN
ejpam-3873	383	15	j	j	X
ejpam-3873	383	16	′2	′2	X
ejpam-3873	383	17	=	=	PUNCT
ejpam-3873	383	18	φ(j1	φ(j1	X
ejpam-3873	383	19	)	)	PUNCT
ejpam-3873	383	20	∩	∩	NOUN
ejpam-3873	383	21	φ(j2	φ(j2	NUM
ejpam-3873	383	22	)	)	PUNCT
ejpam-3873	383	23	.	.	PUNCT
ejpam-3873	384	1	this	this	PRON
ejpam-3873	384	2	shows	show	VERB
ejpam-3873	384	3	that	that	SCONJ
ejpam-3873	384	4	φ	φ	PROPN
ejpam-3873	384	5	is	be	AUX
ejpam-3873	384	6	an	an	DET
ejpam-3873	384	7	isomorphism	isomorphism	NOUN
ejpam-3873	384	8	,	,	PUNCT
ejpam-3873	384	9	that	that	PRON
ejpam-3873	384	10	is	is	ADV
ejpam-3873	384	11	t	t	NOUN
ejpam-3873	384	12	is	be	AUX
ejpam-3873	384	13	isomorphic	isomorphic	ADJ
ejpam-3873	384	14	to	to	ADP
ejpam-3873	384	15	t	t	PROPN
ejpam-3873	384	16	′	′	NUM
ejpam-3873	384	17	as	as	ADP
ejpam-3873	384	18	a	a	DET
ejpam-3873	384	19	semigroup	semigroup	NOUN
ejpam-3873	384	20	.	.	PUNCT
ejpam-3873	385	1	7	7	X
ejpam-3873	385	2	.	.	X
ejpam-3873	385	3	separating	separate	VERB
ejpam-3873	385	4	γ	γ	NOUN
ejpam-3873	385	5	-	-	PUNCT
ejpam-3873	385	6	sets	set	VERB
ejpam-3873	385	7	our	our	PRON
ejpam-3873	385	8	objective	objective	NOUN
ejpam-3873	385	9	in	in	ADP
ejpam-3873	385	10	this	this	DET
ejpam-3873	385	11	section	section	NOUN
ejpam-3873	385	12	is	be	AUX
ejpam-3873	385	13	to	to	PART
ejpam-3873	385	14	show	show	VERB
ejpam-3873	385	15	the	the	DET
ejpam-3873	385	16	statement	statement	NOUN
ejpam-3873	385	17	:	:	PUNCT
ejpam-3873	385	18	let	let	VERB
ejpam-3873	385	19	r	r	NOUN
ejpam-3873	385	20	and	and	CCONJ
ejpam-3873	385	21	s	s	AUX
ejpam-3873	385	22	be	be	AUX
ejpam-3873	385	23	rings	ring	NOUN
ejpam-3873	385	24	.	.	PUNCT
ejpam-3873	386	1	then	then	ADV
ejpam-3873	386	2	tr	tr	NOUN
ejpam-3873	386	3	is	be	AUX
ejpam-3873	386	4	isomorphic	isomorphic	ADJ
ejpam-3873	386	5	to	to	ADP
ejpam-3873	386	6	ts	ts	ADP
ejpam-3873	386	7	if	if	SCONJ
ejpam-3873	386	8	and	and	CCONJ
ejpam-3873	386	9	only	only	ADV
ejpam-3873	386	10	if	if	SCONJ
ejpam-3873	386	11	|g\sr|	|g\sr|	PROPN
ejpam-3873	386	12	=	=	SYM
ejpam-3873	386	13	|h\ss	|h\ss	PROPN
ejpam-3873	386	14	|	|	NOUN
ejpam-3873	386	15	.	.	PUNCT
ejpam-3873	387	1	we	we	PRON
ejpam-3873	387	2	borrowed	borrow	VERB
ejpam-3873	387	3	here	here	ADV
ejpam-3873	387	4	some	some	DET
ejpam-3873	387	5	ideas	idea	NOUN
ejpam-3873	387	6	presented	present	VERB
ejpam-3873	387	7	by	by	ADP
ejpam-3873	387	8	joris	joris	PROPN
ejpam-3873	387	9	n.	n.	PROPN
ejpam-3873	387	10	buloron	buloron	PROPN
ejpam-3873	387	11	in	in	ADP
ejpam-3873	387	12	[	[	X
ejpam-3873	387	13	2	2	NUM
ejpam-3873	387	14	]	]	PUNCT
ejpam-3873	387	15	to	to	PART
ejpam-3873	387	16	show	show	VERB
ejpam-3873	387	17	the	the	DET
ejpam-3873	387	18	results	result	NOUN
ejpam-3873	387	19	.	.	PUNCT
ejpam-3873	388	1	we	we	PRON
ejpam-3873	388	2	denote	denote	VERB
ejpam-3873	388	3	the	the	DET
ejpam-3873	388	4	set	set	NOUN
ejpam-3873	388	5	of	of	ADP
ejpam-3873	388	6	all	all	DET
ejpam-3873	388	7	involutions	involution	NOUN
ejpam-3873	388	8	of	of	ADP
ejpam-3873	388	9	a	a	DET
ejpam-3873	388	10	division	division	NOUN
ejpam-3873	388	11	ring	ring	NOUN
ejpam-3873	388	12	d	d	NOUN
ejpam-3873	388	13	by	by	ADP
ejpam-3873	388	14	sd	sd	NOUN
ejpam-3873	388	15	,	,	PUNCT
ejpam-3873	388	16	that	that	ADV
ejpam-3873	388	17	is	is	ADV
ejpam-3873	388	18	,	,	PUNCT
ejpam-3873	388	19	sd	sd	ADP
ejpam-3873	388	20	=	=	SYM
ejpam-3873	388	21	{	{	PUNCT
ejpam-3873	388	22	x	x	PUNCT
ejpam-3873	388	23	∈	∈	PROPN
ejpam-3873	388	24	r	r	NOUN
ejpam-3873	388	25	:	:	PUNCT
ejpam-3873	388	26	x2	x2	PROPN
ejpam-3873	388	27	=	=	SYM
ejpam-3873	388	28	1d	1d	NUM
ejpam-3873	388	29	,	,	PUNCT
ejpam-3873	388	30	or	or	CCONJ
ejpam-3873	388	31	2x	2x	NUM
ejpam-3873	388	32	=	=	SYM
ejpam-3873	388	33	0	0	NUM
ejpam-3873	388	34	,	,	PUNCT
ejpam-3873	388	35	or	or	CCONJ
ejpam-3873	388	36	a	a	DET
ejpam-3873	388	37	6=	6=	ADP
ejpam-3873	388	38	−a−1	−a−1	NUM
ejpam-3873	388	39	}	}	PUNCT
ejpam-3873	388	40	.	.	PUNCT
ejpam-3873	389	1	let	let	VERB
ejpam-3873	389	2	j	j	PROPN
ejpam-3873	389	3	be	be	AUX
ejpam-3873	389	4	a	a	DET
ejpam-3873	389	5	γ	γ	NOUN
ejpam-3873	389	6	-	-	PUNCT
ejpam-3873	389	7	set	set	NOUN
ejpam-3873	389	8	of	of	ADP
ejpam-3873	389	9	a	a	DET
ejpam-3873	389	10	division	division	NOUN
ejpam-3873	389	11	ring	ring	NOUN
ejpam-3873	389	12	d.	d.	PROPN
ejpam-3873	389	13	then	then	ADV
ejpam-3873	389	14	j	j	PROPN
ejpam-3873	389	15	is	be	AUX
ejpam-3873	389	16	called	call	VERB
ejpam-3873	389	17	a	a	DET
ejpam-3873	389	18	separating	separate	VERB
ejpam-3873	389	19	γ	γ	NOUN
ejpam-3873	389	20	-	-	PUNCT
ejpam-3873	389	21	set	set	NOUN
ejpam-3873	389	22	of	of	ADP
ejpam-3873	389	23	d	d	NOUN
ejpam-3873	389	24	if	if	SCONJ
ejpam-3873	389	25	for	for	ADP
ejpam-3873	389	26	every	every	DET
ejpam-3873	389	27	x	x	PROPN
ejpam-3873	389	28	∈	∈	PROPN
ejpam-3873	389	29	j\sd	j\sd	PROPN
ejpam-3873	389	30	,	,	PUNCT
ejpam-3873	389	31	x−1	x−1	PROPN
ejpam-3873	389	32	/∈	/∈	PROPN
ejpam-3873	390	1	d.	d.	PROPN
ejpam-3873	390	2	note	note	VERB
ejpam-3873	390	3	that	that	SCONJ
ejpam-3873	390	4	for	for	ADP
ejpam-3873	390	5	a	a	DET
ejpam-3873	390	6	finite	finite	ADJ
ejpam-3873	390	7	division	division	NOUN
ejpam-3873	390	8	ring	ring	NOUN
ejpam-3873	390	9	d	d	PROPN
ejpam-3873	390	10	,	,	PUNCT
ejpam-3873	390	11	the	the	DET
ejpam-3873	390	12	separating	separate	VERB
ejpam-3873	390	13	γ	γ	NOUN
ejpam-3873	390	14	-	-	PUNCT
ejpam-3873	390	15	sets	set	NOUN
ejpam-3873	390	16	are	be	AUX
ejpam-3873	390	17	just	just	ADV
ejpam-3873	390	18	the	the	DET
ejpam-3873	390	19	minimum	minimum	ADJ
ejpam-3873	390	20	γ	γ	NOUN
ejpam-3873	390	21	-	-	PUNCT
ejpam-3873	390	22	sets	set	NOUN
ejpam-3873	390	23	.	.	PUNCT
ejpam-3873	391	1	also	also	ADV
ejpam-3873	391	2	note	note	VERB
ejpam-3873	391	3	that	that	SCONJ
ejpam-3873	391	4	if	if	SCONJ
ejpam-3873	391	5	j	j	PROPN
ejpam-3873	391	6	is	be	AUX
ejpam-3873	391	7	not	not	PART
ejpam-3873	391	8	a	a	DET
ejpam-3873	391	9	separating	separate	VERB
ejpam-3873	391	10	γ	γ	NOUN
ejpam-3873	391	11	-	-	PUNCT
ejpam-3873	391	12	set	set	NOUN
ejpam-3873	391	13	,	,	PUNCT
ejpam-3873	391	14	then	then	ADV
ejpam-3873	391	15	there	there	PRON
ejpam-3873	391	16	exists	exist	VERB
ejpam-3873	391	17	x	x	X
ejpam-3873	391	18	∈	∈	PROPN
ejpam-3873	391	19	d\sd	d\sd	PROPN
ejpam-3873	392	1	such	such	DET
ejpam-3873	392	2	that	that	SCONJ
ejpam-3873	392	3	x	x	NOUN
ejpam-3873	392	4	,	,	PUNCT
ejpam-3873	392	5	x−1	x−1	PROPN
ejpam-3873	392	6	∈	∈	PROPN
ejpam-3873	392	7	j	j	PROPN
ejpam-3873	392	8	.	.	PUNCT
ejpam-3873	393	1	lemma	lemma	PROPN
ejpam-3873	393	2	6	6	NUM
ejpam-3873	393	3	.	.	PUNCT
ejpam-3873	394	1	let	let	VERB
ejpam-3873	394	2	d	d	PRON
ejpam-3873	394	3	be	be	AUX
ejpam-3873	394	4	a	a	DET
ejpam-3873	394	5	division	division	NOUN
ejpam-3873	394	6	ring	ring	NOUN
ejpam-3873	394	7	and	and	CCONJ
ejpam-3873	394	8	j	j	PROPN
ejpam-3873	394	9	be	be	AUX
ejpam-3873	394	10	a	a	DET
ejpam-3873	394	11	γ	γ	NOUN
ejpam-3873	394	12	-	-	PUNCT
ejpam-3873	394	13	set	set	NOUN
ejpam-3873	394	14	.	.	PUNCT
ejpam-3873	395	1	if	if	SCONJ
ejpam-3873	395	2	j	j	PROPN
ejpam-3873	395	3	is	be	AUX
ejpam-3873	395	4	not	not	PART
ejpam-3873	395	5	a	a	DET
ejpam-3873	395	6	separating	separate	VERB
ejpam-3873	395	7	γ	γ	NOUN
ejpam-3873	395	8	-	-	PUNCT
ejpam-3873	395	9	set	set	NOUN
ejpam-3873	395	10	,	,	PUNCT
ejpam-3873	395	11	then	then	ADV
ejpam-3873	395	12	j	j	PROPN
ejpam-3873	395	13	can	can	AUX
ejpam-3873	395	14	be	be	AUX
ejpam-3873	395	15	expressed	express	VERB
ejpam-3873	395	16	as	as	ADP
ejpam-3873	395	17	a	a	DET
ejpam-3873	395	18	union	union	NOUN
ejpam-3873	395	19	of	of	ADP
ejpam-3873	395	20	two	two	NUM
ejpam-3873	395	21	distinct	distinct	ADJ
ejpam-3873	395	22	separating	separate	VERB
ejpam-3873	395	23	γ	γ	NOUN
ejpam-3873	395	24	-	-	PUNCT
ejpam-3873	395	25	sets	set	NOUN
ejpam-3873	395	26	.	.	PUNCT
ejpam-3873	396	1	proof	proof	NOUN
ejpam-3873	396	2	.	.	PUNCT
ejpam-3873	397	1	let	let	VERB
ejpam-3873	397	2	j	j	PROPN
ejpam-3873	397	3	be	be	AUX
ejpam-3873	397	4	a	a	DET
ejpam-3873	397	5	γ	γ	NOUN
ejpam-3873	397	6	-	-	PUNCT
ejpam-3873	397	7	set	set	NOUN
ejpam-3873	397	8	that	that	PRON
ejpam-3873	397	9	is	be	AUX
ejpam-3873	397	10	not	not	PART
ejpam-3873	397	11	separating	separate	VERB
ejpam-3873	397	12	.	.	PUNCT
ejpam-3873	398	1	define	define	VERB
ejpam-3873	398	2	a	a	DET
ejpam-3873	398	3	relation	relation	NOUN
ejpam-3873	398	4	∼	∼	NOUN
ejpam-3873	398	5	on	on	ADP
ejpam-3873	398	6	j\sd	j\sd	PROPN
ejpam-3873	398	7	as	as	SCONJ
ejpam-3873	398	8	follows	follow	VERB
ejpam-3873	398	9	:	:	PUNCT
ejpam-3873	398	10	x	x	PUNCT
ejpam-3873	398	11	∼	∼	NOUN
ejpam-3873	398	12	y	y	NOUN
ejpam-3873	398	13	if	if	SCONJ
ejpam-3873	398	14	and	and	CCONJ
ejpam-3873	398	15	only	only	ADV
ejpam-3873	398	16	if	if	SCONJ
ejpam-3873	398	17	x	x	NOUN
ejpam-3873	398	18	=	=	SYM
ejpam-3873	398	19	y	y	PROPN
ejpam-3873	398	20	or	or	CCONJ
ejpam-3873	398	21	y	y	PROPN
ejpam-3873	398	22	=	=	SYM
ejpam-3873	398	23	x−1	x−1	PROPN
ejpam-3873	398	24	.	.	PUNCT
ejpam-3873	399	1	then	then	ADV
ejpam-3873	399	2	∼	∼	NOUN
ejpam-3873	399	3	is	be	AUX
ejpam-3873	399	4	an	an	DET
ejpam-3873	399	5	equivalence	equivalence	NOUN
ejpam-3873	399	6	relation	relation	NOUN
ejpam-3873	399	7	,	,	PUNCT
ejpam-3873	399	8	that	that	ADV
ejpam-3873	399	9	is	is	ADV
ejpam-3873	399	10	,	,	PUNCT
ejpam-3873	399	11	∼	∼	NOUN
ejpam-3873	399	12	partitions	partition	NOUN
ejpam-3873	399	13	j\sd	j\sd	PROPN
ejpam-3873	399	14	into	into	ADP
ejpam-3873	399	15	equivalence	equivalence	NOUN
ejpam-3873	399	16	classes	class	NOUN
ejpam-3873	399	17	.	.	PUNCT
ejpam-3873	400	1	for	for	ADP
ejpam-3873	400	2	each	each	DET
ejpam-3873	400	3	x	x	SYM
ejpam-3873	400	4	∈	∈	PROPN
ejpam-3873	400	5	j\sd	j\sd	PROPN
ejpam-3873	400	6	,	,	PUNCT
ejpam-3873	400	7	the	the	DET
ejpam-3873	400	8	equivalence	equivalence	NOUN
ejpam-3873	400	9	class	class	NOUN
ejpam-3873	400	10	containing	contain	VERB
ejpam-3873	400	11	x	x	SYM
ejpam-3873	400	12	is	be	AUX
ejpam-3873	400	13	x̄	x̄	NOUN
ejpam-3873	400	14	=	=	SYM
ejpam-3873	400	15	{	{	PUNCT
ejpam-3873	400	16	x	x	NOUN
ejpam-3873	400	17	,	,	PUNCT
ejpam-3873	400	18	x−1	x−1	PROPN
ejpam-3873	400	19	}	}	PUNCT
ejpam-3873	400	20	.	.	PUNCT
ejpam-3873	401	1	by	by	ADP
ejpam-3873	401	2	the	the	DET
ejpam-3873	401	3	axiom	axiom	NOUN
ejpam-3873	401	4	of	of	ADP
ejpam-3873	401	5	choice	choice	NOUN
ejpam-3873	401	6	,	,	PUNCT
ejpam-3873	401	7	there	there	PRON
ejpam-3873	401	8	exists	exist	VERB
ejpam-3873	401	9	a	a	DET
ejpam-3873	401	10	set	set	NOUN
ejpam-3873	401	11	∆	∆	PROPN
ejpam-3873	401	12	such	such	ADJ
ejpam-3873	401	13	that	that	SCONJ
ejpam-3873	401	14	∆	∆	PROPN
ejpam-3873	401	15	∩	∩	NOUN
ejpam-3873	401	16	x̄	x̄	PRON
ejpam-3873	401	17	is	be	AUX
ejpam-3873	401	18	a	a	DET
ejpam-3873	401	19	singleton	singleton	NOUN
ejpam-3873	401	20	set	set	NOUN
ejpam-3873	401	21	for	for	ADP
ejpam-3873	401	22	all	all	DET
ejpam-3873	401	23	x	x	SYM
ejpam-3873	401	24	∈	∈	PROPN
ejpam-3873	401	25	j\sd	j\sd	NOUN
ejpam-3873	401	26	.	.	PUNCT
ejpam-3873	402	1	it	it	PRON
ejpam-3873	402	2	is	be	AUX
ejpam-3873	402	3	easy	easy	ADJ
ejpam-3873	402	4	to	to	PART
ejpam-3873	402	5	see	see	VERB
ejpam-3873	402	6	that	that	SCONJ
ejpam-3873	402	7	∆	∆	PROPN
ejpam-3873	402	8	∪	∪	NOUN
ejpam-3873	402	9	sd	sd	NOUN
ejpam-3873	402	10	and	and	CCONJ
ejpam-3873	402	11	j\∆	j\∆	PROPN
ejpam-3873	402	12	is	be	AUX
ejpam-3873	402	13	a	a	DET
ejpam-3873	402	14	separating	separate	VERB
ejpam-3873	402	15	γ	γ	NOUN
ejpam-3873	402	16	-	-	PUNCT
ejpam-3873	402	17	set	set	NOUN
ejpam-3873	402	18	,	,	PUNCT
ejpam-3873	402	19	and	and	CCONJ
ejpam-3873	402	20	j	j	PROPN
ejpam-3873	402	21	=	=	PRON
ejpam-3873	402	22	(	(	PUNCT
ejpam-3873	402	23	∆	∆	PROPN
ejpam-3873	402	24	∪	∪	ADP
ejpam-3873	402	25	sd	sd	NOUN
ejpam-3873	402	26	)	)	PUNCT
ejpam-3873	402	27	∪	∪	NOUN
ejpam-3873	402	28	(	(	PUNCT
ejpam-3873	402	29	j\∆	j\∆	NOUN
ejpam-3873	402	30	)	)	PUNCT
ejpam-3873	402	31	.	.	PUNCT
ejpam-3873	403	1	a	a	DET
ejpam-3873	403	2	careful	careful	ADJ
ejpam-3873	403	3	observation	observation	NOUN
ejpam-3873	403	4	would	would	AUX
ejpam-3873	403	5	suggest	suggest	VERB
ejpam-3873	403	6	that	that	SCONJ
ejpam-3873	403	7	a	a	DET
ejpam-3873	403	8	separating	separate	VERB
ejpam-3873	403	9	γ	γ	NOUN
ejpam-3873	403	10	-	-	PUNCT
ejpam-3873	403	11	set	set	ADJ
ejpam-3873	403	12	can	can	AUX
ejpam-3873	403	13	not	not	PART
ejpam-3873	403	14	be	be	AUX
ejpam-3873	403	15	expressed	express	VERB
ejpam-3873	403	16	as	as	ADP
ejpam-3873	403	17	a	a	DET
ejpam-3873	403	18	union	union	NOUN
ejpam-3873	403	19	of	of	ADP
ejpam-3873	403	20	two	two	NUM
ejpam-3873	403	21	distinct	distinct	ADJ
ejpam-3873	403	22	γ	γ	NOUN
ejpam-3873	403	23	-	-	PUNCT
ejpam-3873	403	24	sets	set	NOUN
ejpam-3873	403	25	.	.	PUNCT
ejpam-3873	404	1	the	the	DET
ejpam-3873	404	2	next	next	ADJ
ejpam-3873	404	3	lemma	lemma	PROPN
ejpam-3873	404	4	is	be	AUX
ejpam-3873	404	5	anchored	anchor	VERB
ejpam-3873	404	6	on	on	ADP
ejpam-3873	404	7	this	this	DET
ejpam-3873	404	8	idea	idea	NOUN
ejpam-3873	404	9	.	.	PUNCT
ejpam-3873	405	1	lemma	lemma	PROPN
ejpam-3873	405	2	7	7	X
ejpam-3873	405	3	.	.	PUNCT
ejpam-3873	406	1	let	let	VERB
ejpam-3873	406	2	d	d	PRON
ejpam-3873	406	3	be	be	AUX
ejpam-3873	406	4	a	a	DET
ejpam-3873	406	5	division	division	NOUN
ejpam-3873	406	6	ring	ring	NOUN
ejpam-3873	406	7	and	and	CCONJ
ejpam-3873	406	8	j	j	PROPN
ejpam-3873	406	9	be	be	AUX
ejpam-3873	406	10	a	a	DET
ejpam-3873	406	11	γ	γ	NOUN
ejpam-3873	406	12	-	-	PUNCT
ejpam-3873	406	13	set	set	NOUN
ejpam-3873	406	14	of	of	ADP
ejpam-3873	406	15	d.	d.	PROPN
ejpam-3873	406	16	j	j	PROPN
ejpam-3873	406	17	is	be	AUX
ejpam-3873	406	18	not	not	PART
ejpam-3873	406	19	a	a	DET
ejpam-3873	406	20	separating	separate	VERB
ejpam-3873	406	21	d	d	NOUN
ejpam-3873	406	22	-	-	PUNCT
ejpam-3873	406	23	set	set	VERB
ejpam-3873	406	24	if	if	SCONJ
ejpam-3873	406	25	and	and	CCONJ
ejpam-3873	406	26	only	only	ADV
ejpam-3873	406	27	if	if	SCONJ
ejpam-3873	406	28	it	it	PRON
ejpam-3873	406	29	is	be	AUX
ejpam-3873	406	30	a	a	DET
ejpam-3873	406	31	union	union	NOUN
ejpam-3873	406	32	of	of	ADP
ejpam-3873	406	33	two	two	NUM
ejpam-3873	406	34	or	or	CCONJ
ejpam-3873	406	35	more	more	ADV
ejpam-3873	406	36	distinct	distinct	ADJ
ejpam-3873	406	37	γ	γ	NOUN
ejpam-3873	406	38	-	-	PUNCT
ejpam-3873	406	39	sets	set	NOUN
ejpam-3873	406	40	.	.	PUNCT
ejpam-3873	407	1	e.j	e.j	PROPN
ejpam-3873	407	2	.	.	PROPN
ejpam-3873	407	3	sigasig	sigasig	PROPN
ejpam-3873	407	4	,	,	PUNCT
ejpam-3873	407	5	c.j	c.j	PROPN
ejpam-3873	407	6	.	.	PROPN
ejpam-3873	407	7	rosero	rosero	PROPN
ejpam-3873	407	8	,	,	PUNCT
ejpam-3873	407	9	m.	m.	NOUN
ejpam-3873	407	10	baldado	baldado	PROPN
ejpam-3873	407	11	jr	jr	PROPN
ejpam-3873	407	12	.	.	PROPN
ejpam-3873	407	13	/	/	SYM
ejpam-3873	407	14	eur	eur	PROPN
ejpam-3873	407	15	.	.	PUNCT
ejpam-3873	408	1	j.	j.	PROPN
ejpam-3873	408	2	pure	pure	PROPN
ejpam-3873	408	3	appl	appl	PROPN
ejpam-3873	408	4	.	.	PROPN
ejpam-3873	408	5	math	math	PROPN
ejpam-3873	408	6	,	,	PUNCT
ejpam-3873	408	7	14	14	NUM
ejpam-3873	408	8	(	(	PUNCT
ejpam-3873	408	9	1	1	NUM
ejpam-3873	408	10	)	)	PUNCT
ejpam-3873	408	11	(	(	PUNCT
ejpam-3873	408	12	2021	2021	NUM
ejpam-3873	408	13	)	)	PUNCT
ejpam-3873	408	14	,	,	PUNCT
ejpam-3873	408	15	314	314	NUM
ejpam-3873	408	16	-	-	SYM
ejpam-3873	408	17	326	326	NUM
ejpam-3873	408	18	323	323	NUM
ejpam-3873	408	19	proof	proof	NOUN
ejpam-3873	408	20	.	.	PUNCT
ejpam-3873	409	1	let	let	VERB
ejpam-3873	409	2	j	j	PROPN
ejpam-3873	409	3	be	be	AUX
ejpam-3873	409	4	a	a	DET
ejpam-3873	409	5	γ	γ	NOUN
ejpam-3873	409	6	-	-	PUNCT
ejpam-3873	409	7	set	set	NOUN
ejpam-3873	409	8	of	of	ADP
ejpam-3873	409	9	d	d	PROPN
ejpam-3873	409	10	and	and	CCONJ
ejpam-3873	409	11	assume	assume	VERB
ejpam-3873	409	12	that	that	SCONJ
ejpam-3873	409	13	j	j	PROPN
ejpam-3873	409	14	is	be	AUX
ejpam-3873	409	15	not	not	PART
ejpam-3873	409	16	a	a	DET
ejpam-3873	409	17	separating	separate	VERB
ejpam-3873	409	18	γ	γ	NOUN
ejpam-3873	409	19	-	-	PUNCT
ejpam-3873	409	20	set	set	NOUN
ejpam-3873	409	21	.	.	PUNCT
ejpam-3873	410	1	then	then	ADV
ejpam-3873	410	2	by	by	ADP
ejpam-3873	410	3	lemma	lemma	PROPN
ejpam-3873	410	4	6	6	NUM
ejpam-3873	410	5	,	,	PUNCT
ejpam-3873	410	6	j	j	X
ejpam-3873	410	7	=	=	SYM
ejpam-3873	410	8	e	e	PROPN
ejpam-3873	410	9	∪	∪	VERB
ejpam-3873	410	10	f	f	PROPN
ejpam-3873	410	11	for	for	ADP
ejpam-3873	410	12	some	some	PRON
ejpam-3873	410	13	separating	separate	VERB
ejpam-3873	410	14	γ	γ	NOUN
ejpam-3873	410	15	-	-	PUNCT
ejpam-3873	410	16	sets	set	NOUN
ejpam-3873	410	17	e	e	NOUN
ejpam-3873	410	18	and	and	CCONJ
ejpam-3873	410	19	f	f	PROPN
ejpam-3873	410	20	.	.	PUNCT
ejpam-3873	411	1	note	note	VERB
ejpam-3873	411	2	that	that	SCONJ
ejpam-3873	411	3	e	e	PROPN
ejpam-3873	411	4	and	and	CCONJ
ejpam-3873	411	5	f	f	PROPN
ejpam-3873	411	6	must	must	AUX
ejpam-3873	411	7	be	be	AUX
ejpam-3873	411	8	distinct	distinct	ADJ
ejpam-3873	411	9	,	,	PUNCT
ejpam-3873	411	10	otherwise	otherwise	ADV
ejpam-3873	411	11	,	,	PUNCT
ejpam-3873	411	12	j	j	PROPN
ejpam-3873	411	13	=	=	SYM
ejpam-3873	411	14	e	e	X
ejpam-3873	411	15	∪f	∪f	X
ejpam-3873	411	16	=	=	SYM
ejpam-3873	411	17	e	e	X
ejpam-3873	411	18	(	(	PUNCT
ejpam-3873	411	19	which	which	PRON
ejpam-3873	411	20	is	be	AUX
ejpam-3873	411	21	a	a	DET
ejpam-3873	411	22	contradiction	contradiction	NOUN
ejpam-3873	411	23	since	since	SCONJ
ejpam-3873	411	24	e	e	NOUN
ejpam-3873	411	25	is	be	AUX
ejpam-3873	411	26	a	a	DET
ejpam-3873	411	27	separating	separate	VERB
ejpam-3873	411	28	γ	γ	NOUN
ejpam-3873	411	29	-	-	PUNCT
ejpam-3873	411	30	set	set	VERB
ejpam-3873	411	31	while	while	SCONJ
ejpam-3873	411	32	j	j	PROPN
ejpam-3873	411	33	is	be	AUX
ejpam-3873	411	34	not	not	PART
ejpam-3873	411	35	)	)	PUNCT
ejpam-3873	411	36	.	.	PUNCT
ejpam-3873	412	1	conversely	conversely	ADV
ejpam-3873	412	2	,	,	PUNCT
ejpam-3873	412	3	assume	assume	VERB
ejpam-3873	412	4	that	that	SCONJ
ejpam-3873	412	5	j	j	PROPN
ejpam-3873	412	6	=	=	SYM
ejpam-3873	412	7	e	e	X
ejpam-3873	412	8	∪f	∪f	PROPN
ejpam-3873	412	9	for	for	ADP
ejpam-3873	412	10	some	some	DET
ejpam-3873	412	11	γ	γ	NOUN
ejpam-3873	412	12	-	-	PUNCT
ejpam-3873	412	13	sets	set	NOUN
ejpam-3873	412	14	e	e	NOUN
ejpam-3873	412	15	and	and	CCONJ
ejpam-3873	412	16	f	f	PROPN
ejpam-3873	412	17	,	,	PUNCT
ejpam-3873	412	18	with	with	ADP
ejpam-3873	412	19	e	e	PROPN
ejpam-3873	412	20	6=	6=	PROPN
ejpam-3873	412	21	f	f	PROPN
ejpam-3873	412	22	.	.	PUNCT
ejpam-3873	413	1	if	if	SCONJ
ejpam-3873	413	2	one	one	NUM
ejpam-3873	413	3	of	of	ADP
ejpam-3873	413	4	e	e	PROPN
ejpam-3873	413	5	and	and	CCONJ
ejpam-3873	413	6	f	f	PROPN
ejpam-3873	413	7	is	be	AUX
ejpam-3873	413	8	not	not	PART
ejpam-3873	413	9	a	a	DET
ejpam-3873	413	10	separating	separate	VERB
ejpam-3873	413	11	γ	γ	NOUN
ejpam-3873	413	12	-	-	PUNCT
ejpam-3873	413	13	set	set	NOUN
ejpam-3873	413	14	,	,	PUNCT
ejpam-3873	413	15	then	then	ADV
ejpam-3873	413	16	clearly	clearly	ADV
ejpam-3873	413	17	,	,	PUNCT
ejpam-3873	413	18	j	j	PROPN
ejpam-3873	413	19	=	=	SYM
ejpam-3873	413	20	e	e	PROPN
ejpam-3873	413	21	∪	∪	NOUN
ejpam-3873	413	22	f	f	PROPN
ejpam-3873	413	23	is	be	AUX
ejpam-3873	413	24	not	not	PART
ejpam-3873	413	25	a	a	DET
ejpam-3873	413	26	separating	separate	VERB
ejpam-3873	413	27	γ	γ	NOUN
ejpam-3873	413	28	-	-	PUNCT
ejpam-3873	413	29	set	set	NOUN
ejpam-3873	413	30	.	.	PUNCT
ejpam-3873	414	1	so	so	ADV
ejpam-3873	414	2	we	we	PRON
ejpam-3873	414	3	assume	assume	VERB
ejpam-3873	414	4	that	that	SCONJ
ejpam-3873	414	5	e	e	PROPN
ejpam-3873	414	6	and	and	CCONJ
ejpam-3873	414	7	f	f	PROPN
ejpam-3873	414	8	are	be	AUX
ejpam-3873	414	9	both	both	PRON
ejpam-3873	414	10	separating	separate	VERB
ejpam-3873	414	11	γ	γ	NOUN
ejpam-3873	414	12	-	-	PUNCT
ejpam-3873	414	13	sets	set	NOUN
ejpam-3873	414	14	.	.	PUNCT
ejpam-3873	415	1	since	since	SCONJ
ejpam-3873	415	2	e	e	PROPN
ejpam-3873	415	3	6=	6=	PROPN
ejpam-3873	415	4	f	f	PROPN
ejpam-3873	415	5	,	,	PUNCT
ejpam-3873	415	6	e\f	e\f	PROPN
ejpam-3873	415	7	6=	6=	ADP
ejpam-3873	415	8	∅.	∅.	ADV
ejpam-3873	415	9	let	let	VERB
ejpam-3873	415	10	x	x	X
ejpam-3873	415	11	∈	∈	PROPN
ejpam-3873	415	12	e\f	e\f	NOUN
ejpam-3873	415	13	.	.	PUNCT
ejpam-3873	416	1	since	since	SCONJ
ejpam-3873	416	2	f	f	PROPN
ejpam-3873	416	3	is	be	AUX
ejpam-3873	416	4	a	a	DET
ejpam-3873	416	5	γ	γ	NOUN
ejpam-3873	416	6	-	-	PUNCT
ejpam-3873	416	7	set	set	NOUN
ejpam-3873	416	8	,	,	PUNCT
ejpam-3873	416	9	x−1	x−1	PROPN
ejpam-3873	416	10	∈	∈	PROPN
ejpam-3873	416	11	f	f	PROPN
ejpam-3873	416	12	.	.	PUNCT
ejpam-3873	417	1	hence	hence	ADV
ejpam-3873	417	2	,	,	PUNCT
ejpam-3873	417	3	x	x	X
ejpam-3873	417	4	,	,	PUNCT
ejpam-3873	417	5	x−1	x−1	PROPN
ejpam-3873	417	6	∈	∈	PROPN
ejpam-3873	417	7	d.	d.	PROPN
ejpam-3873	417	8	this	this	PRON
ejpam-3873	417	9	implies	imply	VERB
ejpam-3873	417	10	that	that	SCONJ
ejpam-3873	417	11	d	d	NOUN
ejpam-3873	417	12	is	be	AUX
ejpam-3873	417	13	not	not	PART
ejpam-3873	417	14	a	a	DET
ejpam-3873	417	15	separating	separate	VERB
ejpam-3873	417	16	γ	γ	NOUN
ejpam-3873	417	17	-	-	PUNCT
ejpam-3873	417	18	set	set	NOUN
ejpam-3873	417	19	.	.	PUNCT
ejpam-3873	418	1	at	at	ADP
ejpam-3873	418	2	this	this	DET
ejpam-3873	418	3	point	point	NOUN
ejpam-3873	418	4	,	,	PUNCT
ejpam-3873	418	5	we	we	PRON
ejpam-3873	418	6	will	will	AUX
ejpam-3873	418	7	now	now	ADV
ejpam-3873	418	8	state	state	VERB
ejpam-3873	418	9	some	some	DET
ejpam-3873	418	10	consequence	consequence	NOUN
ejpam-3873	418	11	of	of	ADP
ejpam-3873	418	12	the	the	DET
ejpam-3873	418	13	above	above	ADJ
ejpam-3873	418	14	lemma	lemma	PROPN
ejpam-3873	418	15	.	.	PUNCT
ejpam-3873	419	1	the	the	DET
ejpam-3873	419	2	following	follow	VERB
ejpam-3873	419	3	definitions	definition	NOUN
ejpam-3873	419	4	are	be	AUX
ejpam-3873	419	5	helpful	helpful	ADJ
ejpam-3873	419	6	in	in	ADP
ejpam-3873	419	7	the	the	DET
ejpam-3873	419	8	succeeding	succeed	VERB
ejpam-3873	419	9	statements	statement	NOUN
ejpam-3873	419	10	.	.	PUNCT
ejpam-3873	420	1	let	let	VERB
ejpam-3873	420	2	x	x	PRON
ejpam-3873	420	3	be	be	AUX
ejpam-3873	420	4	an	an	DET
ejpam-3873	420	5	element	element	NOUN
ejpam-3873	420	6	of	of	ADP
ejpam-3873	420	7	a	a	DET
ejpam-3873	420	8	division	division	NOUN
ejpam-3873	420	9	ring	ring	NOUN
ejpam-3873	420	10	d.	d.	PROPN
ejpam-3873	420	11	we	we	PRON
ejpam-3873	420	12	denote	denote	VERB
ejpam-3873	420	13	by	by	ADP
ejpam-3873	420	14	td(x	td(x	NOUN
ejpam-3873	420	15	)	)	PUNCT
ejpam-3873	420	16	the	the	DET
ejpam-3873	420	17	family	family	NOUN
ejpam-3873	420	18	of	of	ADP
ejpam-3873	420	19	all	all	DET
ejpam-3873	420	20	γ	γ	NOUN
ejpam-3873	420	21	-	-	NOUN
ejpam-3873	420	22	sets	set	NOUN
ejpam-3873	420	23	containing	contain	VERB
ejpam-3873	420	24	x	x	X
ejpam-3873	420	25	,	,	PUNCT
ejpam-3873	420	26	that	that	ADV
ejpam-3873	420	27	is	is	ADV
ejpam-3873	420	28	,	,	PUNCT
ejpam-3873	420	29	td(x	td(x	NOUN
ejpam-3873	420	30	)	)	PUNCT
ejpam-3873	420	31	=	=	SYM
ejpam-3873	421	1	{	{	PUNCT
ejpam-3873	421	2	d	d	X
ejpam-3873	421	3	∈	∈	PROPN
ejpam-3873	421	4	td	td	NOUN
ejpam-3873	421	5	:	:	PUNCT
ejpam-3873	421	6	x	x	X
ejpam-3873	421	7	∈	∈	PROPN
ejpam-3873	421	8	d	d	NOUN
ejpam-3873	421	9	}	}	PUNCT
ejpam-3873	421	10	.	.	PUNCT
ejpam-3873	422	1	similarly	similarly	ADV
ejpam-3873	422	2	,	,	PUNCT
ejpam-3873	422	3	we	we	PRON
ejpam-3873	422	4	denote	denote	VERB
ejpam-3873	422	5	by	by	ADP
ejpam-3873	422	6	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	422	7	)	)	PUNCT
ejpam-3873	422	8	the	the	DET
ejpam-3873	422	9	family	family	NOUN
ejpam-3873	422	10	of	of	ADP
ejpam-3873	422	11	all	all	PRON
ejpam-3873	422	12	separating	separate	VERB
ejpam-3873	422	13	γ	γ	NOUN
ejpam-3873	422	14	-	-	PUNCT
ejpam-3873	422	15	sets	set	NOUN
ejpam-3873	422	16	containing	contain	VERB
ejpam-3873	422	17	x	x	X
ejpam-3873	422	18	,	,	PUNCT
ejpam-3873	422	19	that	that	ADV
ejpam-3873	422	20	is	is	ADV
ejpam-3873	422	21	,	,	PUNCT
ejpam-3873	422	22	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	422	23	)	)	PUNCT
ejpam-3873	422	24	=	=	PRON
ejpam-3873	423	1	{	{	PUNCT
ejpam-3873	423	2	d	d	PROPN
ejpam-3873	423	3	∈	∈	PROPN
ejpam-3873	423	4	tsep(d	tsep(d	PROPN
ejpam-3873	423	5	)	)	PUNCT
ejpam-3873	423	6	:	:	PUNCT
ejpam-3873	424	1	x	x	X
ejpam-3873	424	2	∈	∈	PROPN
ejpam-3873	424	3	d	d	NOUN
ejpam-3873	424	4	}	}	PUNCT
ejpam-3873	424	5	.	.	PUNCT
ejpam-3873	425	1	lemma	lemma	PROPN
ejpam-3873	425	2	8	8	NUM
ejpam-3873	425	3	.	.	PUNCT
ejpam-3873	426	1	let	let	VERB
ejpam-3873	426	2	d	d	PRON
ejpam-3873	426	3	be	be	AUX
ejpam-3873	426	4	a	a	DET
ejpam-3873	426	5	non	non	ADJ
ejpam-3873	426	6	-	-	ADJ
ejpam-3873	426	7	trivial	trivial	ADJ
ejpam-3873	426	8	division	division	NOUN
ejpam-3873	426	9	ring	ring	NOUN
ejpam-3873	426	10	and	and	CCONJ
ejpam-3873	426	11	x	x	ADJ
ejpam-3873	426	12	be	be	AUX
ejpam-3873	426	13	an	an	DET
ejpam-3873	426	14	element	element	NOUN
ejpam-3873	426	15	of	of	ADP
ejpam-3873	426	16	d.	d.	PROPN
ejpam-3873	426	17	then	then	ADV
ejpam-3873	426	18	the	the	DET
ejpam-3873	426	19	following	follow	VERB
ejpam-3873	426	20	statements	statement	NOUN
ejpam-3873	426	21	are	be	AUX
ejpam-3873	426	22	equivalent	equivalent	ADJ
ejpam-3873	426	23	.	.	PUNCT
ejpam-3873	427	1	(	(	PUNCT
ejpam-3873	427	2	i	i	NOUN
ejpam-3873	427	3	)	)	PUNCT
ejpam-3873	427	4	x	x	X
ejpam-3873	427	5	is	be	AUX
ejpam-3873	427	6	an	an	DET
ejpam-3873	427	7	involution	involution	NOUN
ejpam-3873	427	8	.	.	PUNCT
ejpam-3873	428	1	(	(	PUNCT
ejpam-3873	428	2	ii	ii	NOUN
ejpam-3873	428	3	)	)	PUNCT
ejpam-3873	428	4	td(x	td(x	PUNCT
ejpam-3873	428	5	)	)	PUNCT
ejpam-3873	429	1	=	=	SYM
ejpam-3873	429	2	td	td	PROPN
ejpam-3873	429	3	.	.	PUNCT
ejpam-3873	429	4	(	(	PUNCT
ejpam-3873	429	5	iii	iii	X
ejpam-3873	429	6	)	)	PUNCT
ejpam-3873	429	7	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	429	8	)	)	PUNCT
ejpam-3873	429	9	=	=	SYM
ejpam-3873	429	10	tsep(d	tsep(d	NOUN
ejpam-3873	429	11	)	)	PUNCT
ejpam-3873	429	12	.	.	PUNCT
ejpam-3873	430	1	proof	proof	NOUN
ejpam-3873	430	2	.	.	PUNCT
ejpam-3873	431	1	(	(	PUNCT
ejpam-3873	431	2	1	1	X
ejpam-3873	431	3	)	)	PUNCT
ejpam-3873	431	4	⇒	⇒	NOUN
ejpam-3873	431	5	(	(	PUNCT
ejpam-3873	431	6	2	2	X
ejpam-3873	431	7	)	)	PUNCT
ejpam-3873	431	8	suppose	suppose	VERB
ejpam-3873	431	9	that	that	SCONJ
ejpam-3873	431	10	x	x	PRON
ejpam-3873	431	11	is	be	AUX
ejpam-3873	431	12	an	an	DET
ejpam-3873	431	13	involution	involution	NOUN
ejpam-3873	431	14	and	and	CCONJ
ejpam-3873	431	15	td(x	td(x	NOUN
ejpam-3873	431	16	)	)	PUNCT
ejpam-3873	431	17	6=	6=	ADP
ejpam-3873	432	1	td	td	NOUN
ejpam-3873	432	2	.	.	PUNCT
ejpam-3873	433	1	if	if	SCONJ
ejpam-3873	433	2	td(x	td(x	VERB
ejpam-3873	433	3	)	)	PUNCT
ejpam-3873	434	1	6=	6=	ADP
ejpam-3873	434	2	td	td	NOUN
ejpam-3873	434	3	,	,	PUNCT
ejpam-3873	434	4	then	then	ADV
ejpam-3873	434	5	there	there	PRON
ejpam-3873	434	6	exists	exist	VERB
ejpam-3873	434	7	a	a	DET
ejpam-3873	434	8	γ	γ	NOUN
ejpam-3873	434	9	-	-	PUNCT
ejpam-3873	434	10	set	set	VERB
ejpam-3873	434	11	b	b	NOUN
ejpam-3873	434	12	such	such	ADJ
ejpam-3873	434	13	that	that	SCONJ
ejpam-3873	434	14	x	x	PROPN
ejpam-3873	434	15	/∈	/∈	PROPN
ejpam-3873	434	16	b.	b.	PROPN
ejpam-3873	435	1	since	since	SCONJ
ejpam-3873	435	2	x	x	PRON
ejpam-3873	435	3	is	be	AUX
ejpam-3873	435	4	an	an	DET
ejpam-3873	435	5	involution	involution	NOUN
ejpam-3873	435	6	and	and	CCONJ
ejpam-3873	435	7	b	b	NOUN
ejpam-3873	435	8	is	be	AUX
ejpam-3873	435	9	a	a	DET
ejpam-3873	435	10	γ	γ	NOUN
ejpam-3873	435	11	-	-	PUNCT
ejpam-3873	435	12	set	set	NOUN
ejpam-3873	435	13	,	,	PUNCT
ejpam-3873	435	14	x	x	PUNCT
ejpam-3873	435	15	=	=	PUNCT
ejpam-3873	435	16	x−1	x−1	PROPN
ejpam-3873	435	17	∈	∈	PROPN
ejpam-3873	435	18	b.	b.	PROPN
ejpam-3873	436	1	this	this	PRON
ejpam-3873	436	2	is	be	AUX
ejpam-3873	436	3	a	a	DET
ejpam-3873	436	4	contradiction	contradiction	NOUN
ejpam-3873	436	5	.	.	PUNCT
ejpam-3873	437	1	hence	hence	ADV
ejpam-3873	437	2	,	,	PUNCT
ejpam-3873	437	3	td(x	td(x	NOUN
ejpam-3873	437	4	)	)	PUNCT
ejpam-3873	437	5	=	=	SYM
ejpam-3873	437	6	td	td	NOUN
ejpam-3873	437	7	.	.	PUNCT
ejpam-3873	438	1	(	(	PUNCT
ejpam-3873	438	2	2	2	X
ejpam-3873	438	3	)	)	PUNCT
ejpam-3873	438	4	⇒	⇒	NOUN
ejpam-3873	438	5	(	(	PUNCT
ejpam-3873	438	6	1	1	X
ejpam-3873	438	7	)	)	PUNCT
ejpam-3873	438	8	suppose	suppose	VERB
ejpam-3873	438	9	that	that	SCONJ
ejpam-3873	438	10	td(x	td(x	VERB
ejpam-3873	438	11	)	)	PUNCT
ejpam-3873	438	12	=	=	SYM
ejpam-3873	438	13	td	td	NOUN
ejpam-3873	438	14	and	and	CCONJ
ejpam-3873	438	15	x	x	NOUN
ejpam-3873	438	16	is	be	AUX
ejpam-3873	438	17	not	not	PART
ejpam-3873	438	18	an	an	DET
ejpam-3873	438	19	involution	involution	NOUN
ejpam-3873	438	20	.	.	PUNCT
ejpam-3873	439	1	let	let	VERB
ejpam-3873	439	2	j	j	PROPN
ejpam-3873	439	3	be	be	AUX
ejpam-3873	439	4	a	a	DET
ejpam-3873	439	5	γ	γ	NOUN
ejpam-3873	439	6	-	-	PUNCT
ejpam-3873	439	7	set	set	NOUN
ejpam-3873	439	8	of	of	ADP
ejpam-3873	439	9	d	d	PROPN
ejpam-3873	439	10	and	and	CCONJ
ejpam-3873	439	11	x	x	ADJ
ejpam-3873	439	12	be	be	AUX
ejpam-3873	439	13	a	a	DET
ejpam-3873	439	14	non	non	ADJ
ejpam-3873	439	15	-	-	NOUN
ejpam-3873	439	16	involution	involution	NOUN
ejpam-3873	439	17	.	.	PUNCT
ejpam-3873	440	1	consider	consider	VERB
ejpam-3873	440	2	the	the	DET
ejpam-3873	440	3	following	follow	VERB
ejpam-3873	440	4	cases	case	NOUN
ejpam-3873	440	5	:	:	PUNCT
ejpam-3873	440	6	case	case	NOUN
ejpam-3873	440	7	1	1	NUM
ejpam-3873	440	8	.	.	X
ejpam-3873	441	1	x2	x2	PROPN
ejpam-3873	442	1	6=	6=	NUM
ejpam-3873	442	2	1d	1d	NUM
ejpam-3873	442	3	if	if	SCONJ
ejpam-3873	442	4	x2	x2	PROPN
ejpam-3873	442	5	6=	6=	NUM
ejpam-3873	442	6	1d	1d	NUM
ejpam-3873	442	7	,	,	PUNCT
ejpam-3873	442	8	that	that	ADV
ejpam-3873	442	9	is	is	ADV
ejpam-3873	442	10	,	,	PUNCT
ejpam-3873	442	11	x	x	PROPN
ejpam-3873	442	12	6=	6=	PROPN
ejpam-3873	442	13	x−1	x−1	PROPN
ejpam-3873	442	14	,	,	PUNCT
ejpam-3873	442	15	then	then	ADV
ejpam-3873	442	16	we	we	PRON
ejpam-3873	442	17	note	note	VERB
ejpam-3873	442	18	that	that	SCONJ
ejpam-3873	442	19	j\{x	j\{x	NOUN
ejpam-3873	442	20	}	}	PUNCT
ejpam-3873	442	21	is	be	AUX
ejpam-3873	442	22	a	a	DET
ejpam-3873	442	23	γ	γ	NOUN
ejpam-3873	442	24	-	-	PUNCT
ejpam-3873	442	25	set	set	NOUN
ejpam-3873	442	26	that	that	PRON
ejpam-3873	442	27	do	do	AUX
ejpam-3873	442	28	not	not	PART
ejpam-3873	442	29	contain	contain	VERB
ejpam-3873	442	30	x.	x.	NOUN
ejpam-3873	443	1	this	this	PRON
ejpam-3873	443	2	is	be	AUX
ejpam-3873	443	3	a	a	DET
ejpam-3873	443	4	contradiction	contradiction	NOUN
ejpam-3873	443	5	since	since	SCONJ
ejpam-3873	443	6	td(x	td(x	NOUN
ejpam-3873	443	7	)	)	PUNCT
ejpam-3873	444	1	=	=	SYM
ejpam-3873	444	2	td	td	PROPN
ejpam-3873	444	3	.	.	PUNCT
ejpam-3873	445	1	therefore	therefore	ADV
ejpam-3873	445	2	,	,	PUNCT
ejpam-3873	445	3	x	x	PRON
ejpam-3873	445	4	must	must	AUX
ejpam-3873	445	5	be	be	AUX
ejpam-3873	445	6	an	an	DET
ejpam-3873	445	7	involution	involution	NOUN
ejpam-3873	445	8	.	.	PUNCT
ejpam-3873	446	1	case	case	NOUN
ejpam-3873	446	2	2	2	NUM
ejpam-3873	446	3	.	.	NUM
ejpam-3873	446	4	2x	2x	NUM
ejpam-3873	446	5	6=	6=	SYM
ejpam-3873	446	6	0	0	PUNCT
ejpam-3873	447	1	if	if	SCONJ
ejpam-3873	447	2	2x	2x	NUM
ejpam-3873	447	3	6=	6=	ADP
ejpam-3873	447	4	0	0	NUM
ejpam-3873	447	5	,	,	PUNCT
ejpam-3873	447	6	that	that	ADV
ejpam-3873	447	7	is	is	ADV
ejpam-3873	447	8	,	,	PUNCT
ejpam-3873	447	9	x	x	X
ejpam-3873	447	10	6=	6=	NUM
ejpam-3873	447	11	−x	−x	NOUN
ejpam-3873	447	12	,	,	PUNCT
ejpam-3873	447	13	then	then	ADV
ejpam-3873	447	14	we	we	PRON
ejpam-3873	447	15	note	note	VERB
ejpam-3873	447	16	that	that	SCONJ
ejpam-3873	447	17	j\{x	j\{x	NOUN
ejpam-3873	447	18	}	}	PUNCT
ejpam-3873	447	19	is	be	AUX
ejpam-3873	447	20	a	a	DET
ejpam-3873	447	21	γ	γ	NOUN
ejpam-3873	447	22	-	-	PUNCT
ejpam-3873	447	23	set	set	NOUN
ejpam-3873	447	24	that	that	PRON
ejpam-3873	447	25	do	do	AUX
ejpam-3873	447	26	not	not	PART
ejpam-3873	447	27	contain	contain	VERB
ejpam-3873	447	28	x.	x.	NOUN
ejpam-3873	448	1	this	this	PRON
ejpam-3873	448	2	is	be	AUX
ejpam-3873	448	3	a	a	DET
ejpam-3873	448	4	contradiction	contradiction	NOUN
ejpam-3873	448	5	since	since	SCONJ
ejpam-3873	448	6	td(x	td(x	NOUN
ejpam-3873	448	7	)	)	PUNCT
ejpam-3873	449	1	=	=	SYM
ejpam-3873	449	2	td	td	PROPN
ejpam-3873	449	3	.	.	PUNCT
ejpam-3873	450	1	therefore	therefore	ADV
ejpam-3873	450	2	,	,	PUNCT
ejpam-3873	450	3	x	x	PRON
ejpam-3873	450	4	must	must	AUX
ejpam-3873	450	5	be	be	AUX
ejpam-3873	450	6	an	an	DET
ejpam-3873	450	7	involution	involution	NOUN
ejpam-3873	450	8	.	.	PUNCT
ejpam-3873	451	1	case	case	NOUN
ejpam-3873	451	2	3	3	NUM
ejpam-3873	451	3	.	.	PUNCT
ejpam-3873	451	4	x	x	SYM
ejpam-3873	452	1	=	=	PUNCT
ejpam-3873	452	2	−x−1	−x−1	NUM
ejpam-3873	452	3	if	if	SCONJ
ejpam-3873	452	4	x	x	PROPN
ejpam-3873	452	5	=	=	SYM
ejpam-3873	452	6	−x−1	−x−1	NUM
ejpam-3873	452	7	,	,	PUNCT
ejpam-3873	452	8	then	then	ADV
ejpam-3873	452	9	we	we	PRON
ejpam-3873	452	10	note	note	VERB
ejpam-3873	452	11	that	that	SCONJ
ejpam-3873	452	12	j\{x	j\{x	NOUN
ejpam-3873	452	13	}	}	PUNCT
ejpam-3873	452	14	is	be	AUX
ejpam-3873	452	15	a	a	DET
ejpam-3873	452	16	γ	γ	NOUN
ejpam-3873	452	17	-	-	PUNCT
ejpam-3873	452	18	set	set	NOUN
ejpam-3873	452	19	that	that	PRON
ejpam-3873	452	20	do	do	AUX
ejpam-3873	452	21	not	not	PART
ejpam-3873	452	22	contain	contain	VERB
ejpam-3873	452	23	x.	x.	NOUN
ejpam-3873	453	1	this	this	PRON
ejpam-3873	453	2	is	be	AUX
ejpam-3873	453	3	a	a	DET
ejpam-3873	453	4	contradiction	contradiction	NOUN
ejpam-3873	453	5	since	since	SCONJ
ejpam-3873	453	6	td(x	td(x	NOUN
ejpam-3873	453	7	)	)	PUNCT
ejpam-3873	454	1	=	=	SYM
ejpam-3873	454	2	td	td	PROPN
ejpam-3873	454	3	.	.	PUNCT
ejpam-3873	455	1	therefore	therefore	ADV
ejpam-3873	455	2	,	,	PUNCT
ejpam-3873	455	3	x	x	PRON
ejpam-3873	455	4	must	must	AUX
ejpam-3873	455	5	be	be	AUX
ejpam-3873	455	6	an	an	DET
ejpam-3873	455	7	involution	involution	NOUN
ejpam-3873	455	8	.	.	PUNCT
ejpam-3873	456	1	(	(	PUNCT
ejpam-3873	456	2	1	1	X
ejpam-3873	456	3	)	)	PUNCT
ejpam-3873	456	4	⇒	⇒	NOUN
ejpam-3873	456	5	(	(	PUNCT
ejpam-3873	456	6	3	3	X
ejpam-3873	456	7	)	)	PUNCT
ejpam-3873	456	8	suppose	suppose	VERB
ejpam-3873	456	9	that	that	SCONJ
ejpam-3873	456	10	x	x	PRON
ejpam-3873	456	11	is	be	AUX
ejpam-3873	456	12	an	an	DET
ejpam-3873	456	13	involution	involution	NOUN
ejpam-3873	456	14	and	and	CCONJ
ejpam-3873	456	15	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	456	16	)	)	PUNCT
ejpam-3873	456	17	6=	6=	X
ejpam-3873	457	1	tsep(d	tsep(d	NOUN
ejpam-3873	457	2	)	)	PUNCT
ejpam-3873	457	3	.	.	PUNCT
ejpam-3873	458	1	if	if	SCONJ
ejpam-3873	458	2	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	458	3	)	)	PUNCT
ejpam-3873	458	4	6=	6=	X
ejpam-3873	458	5	tsep(d	tsep(d	NOUN
ejpam-3873	458	6	)	)	PUNCT
ejpam-3873	458	7	,	,	PUNCT
ejpam-3873	458	8	then	then	ADV
ejpam-3873	458	9	there	there	PRON
ejpam-3873	458	10	exists	exist	VERB
ejpam-3873	458	11	a	a	DET
ejpam-3873	458	12	separating	separate	VERB
ejpam-3873	458	13	γ	γ	NOUN
ejpam-3873	458	14	-	-	PUNCT
ejpam-3873	458	15	set	set	VERB
ejpam-3873	458	16	b	b	NOUN
ejpam-3873	458	17	such	such	ADJ
ejpam-3873	458	18	that	that	SCONJ
ejpam-3873	458	19	x	x	PROPN
ejpam-3873	458	20	/∈	/∈	PROPN
ejpam-3873	458	21	b.	b.	PROPN
ejpam-3873	459	1	since	since	SCONJ
ejpam-3873	459	2	x	x	PRON
ejpam-3873	459	3	is	be	AUX
ejpam-3873	459	4	an	an	DET
ejpam-3873	459	5	involution	involution	NOUN
ejpam-3873	459	6	and	and	CCONJ
ejpam-3873	459	7	b	b	NOUN
ejpam-3873	459	8	is	be	AUX
ejpam-3873	459	9	a	a	DET
ejpam-3873	459	10	γ	γ	NOUN
ejpam-3873	459	11	-	-	PUNCT
ejpam-3873	459	12	set	set	NOUN
ejpam-3873	459	13	,	,	PUNCT
ejpam-3873	459	14	x	x	PUNCT
ejpam-3873	459	15	=	=	PUNCT
ejpam-3873	459	16	x−1	x−1	PROPN
ejpam-3873	459	17	∈	∈	PROPN
ejpam-3873	459	18	b.	b.	PROPN
ejpam-3873	460	1	this	this	PRON
ejpam-3873	460	2	is	be	AUX
ejpam-3873	460	3	a	a	DET
ejpam-3873	460	4	contradiction	contradiction	NOUN
ejpam-3873	460	5	.	.	PUNCT
ejpam-3873	461	1	hence	hence	ADV
ejpam-3873	461	2	,	,	PUNCT
ejpam-3873	461	3	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	461	4	)	)	PUNCT
ejpam-3873	461	5	=	=	SYM
ejpam-3873	461	6	tsep(d	tsep(d	PROPN
ejpam-3873	461	7	)	)	PUNCT
ejpam-3873	461	8	.	.	PUNCT
ejpam-3873	462	1	(	(	PUNCT
ejpam-3873	462	2	3	3	X
ejpam-3873	462	3	)	)	PUNCT
ejpam-3873	462	4	⇒	⇒	NOUN
ejpam-3873	462	5	(	(	PUNCT
ejpam-3873	462	6	1	1	X
ejpam-3873	462	7	)	)	PUNCT
ejpam-3873	462	8	suppose	suppose	VERB
ejpam-3873	462	9	that	that	SCONJ
ejpam-3873	462	10	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	462	11	)	)	PUNCT
ejpam-3873	462	12	=	=	SYM
ejpam-3873	462	13	tsep(d	tsep(d	NOUN
ejpam-3873	462	14	)	)	PUNCT
ejpam-3873	462	15	and	and	CCONJ
ejpam-3873	462	16	x	x	X
ejpam-3873	462	17	is	be	AUX
ejpam-3873	462	18	not	not	PART
ejpam-3873	462	19	an	an	DET
ejpam-3873	462	20	involution	involution	NOUN
ejpam-3873	462	21	.	.	PUNCT
ejpam-3873	463	1	if	if	SCONJ
ejpam-3873	463	2	x	x	PRON
ejpam-3873	463	3	is	be	AUX
ejpam-3873	463	4	not	not	PART
ejpam-3873	463	5	an	an	DET
ejpam-3873	463	6	involution	involution	NOUN
ejpam-3873	463	7	,	,	PUNCT
ejpam-3873	463	8	then	then	ADV
ejpam-3873	463	9	consider	consider	VERB
ejpam-3873	463	10	the	the	DET
ejpam-3873	463	11	following	follow	VERB
ejpam-3873	463	12	cases	case	NOUN
ejpam-3873	463	13	:	:	PUNCT
ejpam-3873	463	14	case	case	NOUN
ejpam-3873	463	15	1	1	NUM
ejpam-3873	463	16	.	.	X
ejpam-3873	464	1	x2	x2	PROPN
ejpam-3873	465	1	6=	6=	NUM
ejpam-3873	465	2	1d	1d	NUM
ejpam-3873	465	3	e.j	e.j	PROPN
ejpam-3873	465	4	.	.	PROPN
ejpam-3873	465	5	sigasig	sigasig	PROPN
ejpam-3873	465	6	,	,	PUNCT
ejpam-3873	465	7	c.j	c.j	PROPN
ejpam-3873	465	8	.	.	PROPN
ejpam-3873	465	9	rosero	rosero	PROPN
ejpam-3873	465	10	,	,	PUNCT
ejpam-3873	465	11	m.	m.	NOUN
ejpam-3873	465	12	baldado	baldado	PROPN
ejpam-3873	465	13	jr	jr	PROPN
ejpam-3873	465	14	.	.	PROPN
ejpam-3873	465	15	/	/	SYM
ejpam-3873	465	16	eur	eur	PROPN
ejpam-3873	465	17	.	.	PUNCT
ejpam-3873	466	1	j.	j.	PROPN
ejpam-3873	466	2	pure	pure	PROPN
ejpam-3873	466	3	appl	appl	PROPN
ejpam-3873	466	4	.	.	PROPN
ejpam-3873	466	5	math	math	PROPN
ejpam-3873	466	6	,	,	PUNCT
ejpam-3873	466	7	14	14	NUM
ejpam-3873	466	8	(	(	PUNCT
ejpam-3873	466	9	1	1	NUM
ejpam-3873	466	10	)	)	PUNCT
ejpam-3873	466	11	(	(	PUNCT
ejpam-3873	466	12	2021	2021	NUM
ejpam-3873	466	13	)	)	PUNCT
ejpam-3873	466	14	,	,	PUNCT
ejpam-3873	466	15	314	314	NUM
ejpam-3873	466	16	-	-	SYM
ejpam-3873	466	17	326	326	NUM
ejpam-3873	466	18	324	324	NUM
ejpam-3873	466	19	if	if	SCONJ
ejpam-3873	466	20	x2	x2	PROPN
ejpam-3873	466	21	6=	6=	NUM
ejpam-3873	466	22	1d	1d	NUM
ejpam-3873	466	23	,	,	PUNCT
ejpam-3873	466	24	then	then	ADV
ejpam-3873	466	25	x	x	X
ejpam-3873	466	26	6=	6=	PROPN
ejpam-3873	467	1	x−1	x−1	PROPN
ejpam-3873	467	2	.	.	PUNCT
ejpam-3873	468	1	let	let	VERB
ejpam-3873	468	2	h	h	PRON
ejpam-3873	468	3	be	be	AUX
ejpam-3873	468	4	a	a	DET
ejpam-3873	468	5	separating	separate	VERB
ejpam-3873	468	6	γ	γ	NOUN
ejpam-3873	468	7	-	-	PUNCT
ejpam-3873	468	8	set	set	NOUN
ejpam-3873	468	9	.	.	PUNCT
ejpam-3873	469	1	then	then	ADV
ejpam-3873	469	2	(	(	PUNCT
ejpam-3873	469	3	h\{x	h\{x	NOUN
ejpam-3873	469	4	}	}	PUNCT
ejpam-3873	469	5	)	)	PUNCT
ejpam-3873	469	6	∪	∪	ADP
ejpam-3873	469	7	{	{	PUNCT
ejpam-3873	469	8	x−1	x−1	NOUN
ejpam-3873	469	9	}	}	PUNCT
ejpam-3873	469	10	is	be	AUX
ejpam-3873	469	11	a	a	DET
ejpam-3873	469	12	separating	separate	VERB
ejpam-3873	469	13	γ	γ	NOUN
ejpam-3873	469	14	-	-	PUNCT
ejpam-3873	469	15	set	set	NOUN
ejpam-3873	469	16	that	that	PRON
ejpam-3873	469	17	do	do	AUX
ejpam-3873	469	18	not	not	PART
ejpam-3873	469	19	contain	contain	VERB
ejpam-3873	469	20	x.	x.	NOUN
ejpam-3873	470	1	this	this	PRON
ejpam-3873	470	2	is	be	AUX
ejpam-3873	470	3	a	a	DET
ejpam-3873	470	4	contradiction	contradiction	NOUN
ejpam-3873	470	5	since	since	SCONJ
ejpam-3873	470	6	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	470	7	)	)	PUNCT
ejpam-3873	470	8	=	=	SYM
ejpam-3873	470	9	tsep(d	tsep(d	PROPN
ejpam-3873	470	10	)	)	PUNCT
ejpam-3873	470	11	.	.	PUNCT
ejpam-3873	471	1	therefore	therefore	ADV
ejpam-3873	471	2	,	,	PUNCT
ejpam-3873	471	3	x	x	PRON
ejpam-3873	471	4	must	must	AUX
ejpam-3873	471	5	be	be	AUX
ejpam-3873	471	6	an	an	DET
ejpam-3873	471	7	involution	involution	NOUN
ejpam-3873	471	8	.	.	PUNCT
ejpam-3873	472	1	case	case	NOUN
ejpam-3873	472	2	2	2	NUM
ejpam-3873	472	3	.	.	NUM
ejpam-3873	472	4	2x	2x	NUM
ejpam-3873	472	5	6=	6=	SYM
ejpam-3873	472	6	0	0	PUNCT
ejpam-3873	473	1	if	if	SCONJ
ejpam-3873	473	2	2x	2x	NUM
ejpam-3873	473	3	6=	6=	NUM
ejpam-3873	473	4	0	0	NUM
ejpam-3873	473	5	,	,	PUNCT
ejpam-3873	473	6	then	then	ADV
ejpam-3873	473	7	x	x	X
ejpam-3873	473	8	6=	6=	ADP
ejpam-3873	473	9	−x	−x	NOUN
ejpam-3873	473	10	.	.	PUNCT
ejpam-3873	474	1	let	let	VERB
ejpam-3873	474	2	h	h	PRON
ejpam-3873	474	3	be	be	AUX
ejpam-3873	474	4	a	a	DET
ejpam-3873	474	5	separating	separate	VERB
ejpam-3873	474	6	γ	γ	NOUN
ejpam-3873	474	7	-	-	PUNCT
ejpam-3873	474	8	set	set	NOUN
ejpam-3873	474	9	.	.	PUNCT
ejpam-3873	475	1	then	then	ADV
ejpam-3873	475	2	(	(	PUNCT
ejpam-3873	475	3	h\{x})∪{−x	h\{x})∪{−x	NOUN
ejpam-3873	475	4	}	}	PUNCT
ejpam-3873	475	5	is	be	AUX
ejpam-3873	475	6	a	a	DET
ejpam-3873	475	7	separating	separate	VERB
ejpam-3873	475	8	γ	γ	NOUN
ejpam-3873	475	9	-	-	PUNCT
ejpam-3873	475	10	set	set	NOUN
ejpam-3873	475	11	that	that	PRON
ejpam-3873	475	12	do	do	AUX
ejpam-3873	475	13	not	not	PART
ejpam-3873	475	14	contain	contain	VERB
ejpam-3873	475	15	x.	x.	NOUN
ejpam-3873	476	1	this	this	PRON
ejpam-3873	476	2	is	be	AUX
ejpam-3873	476	3	a	a	DET
ejpam-3873	476	4	contradiction	contradiction	NOUN
ejpam-3873	476	5	since	since	SCONJ
ejpam-3873	476	6	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	476	7	)	)	PUNCT
ejpam-3873	476	8	=	=	SYM
ejpam-3873	476	9	tsep(d	tsep(d	PROPN
ejpam-3873	476	10	)	)	PUNCT
ejpam-3873	476	11	.	.	PUNCT
ejpam-3873	477	1	therefore	therefore	ADV
ejpam-3873	477	2	,	,	PUNCT
ejpam-3873	477	3	x	x	PRON
ejpam-3873	477	4	must	must	AUX
ejpam-3873	477	5	be	be	AUX
ejpam-3873	477	6	an	an	DET
ejpam-3873	477	7	involution	involution	NOUN
ejpam-3873	477	8	.	.	PUNCT
ejpam-3873	478	1	case	case	NOUN
ejpam-3873	478	2	3	3	NUM
ejpam-3873	478	3	.	.	PUNCT
ejpam-3873	478	4	x	x	SYM
ejpam-3873	479	1	=	=	PUNCT
ejpam-3873	479	2	−x−1	−x−1	NUM
ejpam-3873	479	3	if	if	SCONJ
ejpam-3873	479	4	x	x	PROPN
ejpam-3873	479	5	=	=	SYM
ejpam-3873	479	6	−x−1	−x−1	NUM
ejpam-3873	479	7	,	,	PUNCT
ejpam-3873	479	8	then	then	ADV
ejpam-3873	479	9	x	x	X
ejpam-3873	479	10	=	=	PRON
ejpam-3873	479	11	(	(	PUNCT
ejpam-3873	479	12	−x)−1	−x)−1	NOUN
ejpam-3873	479	13	.	.	PUNCT
ejpam-3873	480	1	let	let	VERB
ejpam-3873	480	2	h	h	PRON
ejpam-3873	480	3	be	be	AUX
ejpam-3873	480	4	a	a	DET
ejpam-3873	480	5	separating	separate	VERB
ejpam-3873	480	6	γ	γ	NOUN
ejpam-3873	480	7	-	-	PUNCT
ejpam-3873	480	8	set	set	NOUN
ejpam-3873	480	9	.	.	PUNCT
ejpam-3873	481	1	then	then	ADV
ejpam-3873	481	2	(	(	PUNCT
ejpam-3873	481	3	h\{x	h\{x	NOUN
ejpam-3873	481	4	}	}	PUNCT
ejpam-3873	481	5	)	)	PUNCT
ejpam-3873	481	6	∪	∪	ADP
ejpam-3873	481	7	{	{	PUNCT
ejpam-3873	481	8	−x	−x	NOUN
ejpam-3873	481	9	}	}	PUNCT
ejpam-3873	481	10	is	be	AUX
ejpam-3873	481	11	a	a	DET
ejpam-3873	481	12	separating	separate	VERB
ejpam-3873	481	13	γ	γ	NOUN
ejpam-3873	481	14	-	-	PUNCT
ejpam-3873	481	15	set	set	NOUN
ejpam-3873	481	16	that	that	PRON
ejpam-3873	481	17	do	do	AUX
ejpam-3873	481	18	not	not	PART
ejpam-3873	481	19	contain	contain	VERB
ejpam-3873	481	20	x.	x.	NOUN
ejpam-3873	482	1	this	this	PRON
ejpam-3873	482	2	is	be	AUX
ejpam-3873	482	3	a	a	DET
ejpam-3873	482	4	contradiction	contradiction	NOUN
ejpam-3873	482	5	since	since	SCONJ
ejpam-3873	482	6	tsep(d)(x	tsep(d)(x	PROPN
ejpam-3873	482	7	)	)	PUNCT
ejpam-3873	482	8	=	=	SYM
ejpam-3873	482	9	tsep(d	tsep(d	PROPN
ejpam-3873	482	10	)	)	PUNCT
ejpam-3873	482	11	.	.	PUNCT
ejpam-3873	483	1	therefore	therefore	ADV
ejpam-3873	483	2	,	,	PUNCT
ejpam-3873	483	3	x	x	PRON
ejpam-3873	483	4	must	must	AUX
ejpam-3873	483	5	be	be	AUX
ejpam-3873	483	6	an	an	DET
ejpam-3873	483	7	involution	involution	NOUN
ejpam-3873	483	8	.	.	PUNCT
ejpam-3873	484	1	the	the	DET
ejpam-3873	484	2	next	next	ADJ
ejpam-3873	484	3	proposition	proposition	NOUN
ejpam-3873	484	4	shows	show	VERB
ejpam-3873	484	5	that	that	SCONJ
ejpam-3873	484	6	an	an	DET
ejpam-3873	484	7	isomorphism	isomorphism	NOUN
ejpam-3873	484	8	preserves	preserve	VERB
ejpam-3873	484	9	the	the	DET
ejpam-3873	484	10	state	state	NOUN
ejpam-3873	484	11	of	of	ADP
ejpam-3873	484	12	being	be	AUX
ejpam-3873	484	13	separating	separate	VERB
ejpam-3873	484	14	in	in	ADP
ejpam-3873	484	15	the	the	DET
ejpam-3873	484	16	same	same	ADJ
ejpam-3873	484	17	way	way	NOUN
ejpam-3873	484	18	as	as	SCONJ
ejpam-3873	484	19	it	it	PRON
ejpam-3873	484	20	preserves	preserve	VERB
ejpam-3873	484	21	other	other	ADJ
ejpam-3873	484	22	properties	property	NOUN
ejpam-3873	484	23	.	.	PUNCT
ejpam-3873	485	1	lemma	lemma	PROPN
ejpam-3873	485	2	9	9	NUM
ejpam-3873	485	3	.	.	PUNCT
ejpam-3873	486	1	let	let	VERB
ejpam-3873	486	2	d1	d1	PROPN
ejpam-3873	486	3	and	and	CCONJ
ejpam-3873	486	4	d2	d2	PROPN
ejpam-3873	486	5	be	be	PROPN
ejpam-3873	486	6	division	division	NOUN
ejpam-3873	486	7	rings	ring	NOUN
ejpam-3873	486	8	,	,	PUNCT
ejpam-3873	486	9	and	and	CCONJ
ejpam-3873	486	10	ϕ	ϕ	X
ejpam-3873	486	11	:	:	PUNCT
ejpam-3873	486	12	td1	td1	PROPN
ejpam-3873	486	13	→	→	SYM
ejpam-3873	486	14	td2	td2	AUX
ejpam-3873	486	15	be	be	AUX
ejpam-3873	486	16	an	an	DET
ejpam-3873	486	17	isomorphism	isomorphism	NOUN
ejpam-3873	486	18	.	.	PUNCT
ejpam-3873	487	1	then	then	ADV
ejpam-3873	487	2	j	j	PROPN
ejpam-3873	487	3	is	be	AUX
ejpam-3873	487	4	a	a	DET
ejpam-3873	487	5	separating	separate	VERB
ejpam-3873	487	6	γ	γ	NOUN
ejpam-3873	487	7	-	-	PUNCT
ejpam-3873	487	8	set	set	NOUN
ejpam-3873	487	9	of	of	ADP
ejpam-3873	487	10	d1	d1	PROPN
ejpam-3873	487	11	if	if	SCONJ
ejpam-3873	487	12	and	and	CCONJ
ejpam-3873	487	13	only	only	ADV
ejpam-3873	487	14	if	if	SCONJ
ejpam-3873	487	15	ϕ(j	ϕ(j	PROPN
ejpam-3873	487	16	)	)	PUNCT
ejpam-3873	487	17	is	be	AUX
ejpam-3873	487	18	a	a	DET
ejpam-3873	487	19	separating	separate	VERB
ejpam-3873	487	20	γ	γ	NOUN
ejpam-3873	487	21	-	-	PUNCT
ejpam-3873	487	22	set	set	NOUN
ejpam-3873	487	23	of	of	ADP
ejpam-3873	487	24	d2	d2	PROPN
ejpam-3873	487	25	.	.	PUNCT
ejpam-3873	488	1	proof	proof	NOUN
ejpam-3873	488	2	.	.	PUNCT
ejpam-3873	489	1	let	let	VERB
ejpam-3873	489	2	d1	d1	PROPN
ejpam-3873	489	3	and	and	CCONJ
ejpam-3873	489	4	d2	d2	PROPN
ejpam-3873	489	5	be	be	PROPN
ejpam-3873	489	6	division	division	NOUN
ejpam-3873	489	7	rings	ring	NOUN
ejpam-3873	489	8	,	,	PUNCT
ejpam-3873	489	9	and	and	CCONJ
ejpam-3873	489	10	ϕ	ϕ	X
ejpam-3873	489	11	:	:	PUNCT
ejpam-3873	489	12	td1	td1	PROPN
ejpam-3873	489	13	→	→	SYM
ejpam-3873	489	14	td2	td2	AUX
ejpam-3873	489	15	be	be	AUX
ejpam-3873	489	16	an	an	DET
ejpam-3873	489	17	isomorphism	isomorphism	NOUN
ejpam-3873	489	18	.	.	PUNCT
ejpam-3873	490	1	suppose	suppose	VERB
ejpam-3873	490	2	that	that	SCONJ
ejpam-3873	490	3	j	j	PROPN
ejpam-3873	490	4	is	be	AUX
ejpam-3873	490	5	a	a	DET
ejpam-3873	490	6	separating	separate	VERB
ejpam-3873	490	7	γ	γ	NOUN
ejpam-3873	490	8	-	-	PUNCT
ejpam-3873	490	9	set	set	NOUN
ejpam-3873	490	10	of	of	ADP
ejpam-3873	490	11	d1	d1	PROPN
ejpam-3873	490	12	and	and	CCONJ
ejpam-3873	490	13	ϕ(j	ϕ(j	PROPN
ejpam-3873	490	14	)	)	PUNCT
ejpam-3873	490	15	is	be	AUX
ejpam-3873	490	16	not	not	PART
ejpam-3873	490	17	a	a	DET
ejpam-3873	490	18	separating	separate	VERB
ejpam-3873	490	19	γ	γ	NOUN
ejpam-3873	490	20	-	-	PUNCT
ejpam-3873	490	21	set	set	NOUN
ejpam-3873	490	22	of	of	ADP
ejpam-3873	490	23	d2	d2	PROPN
ejpam-3873	490	24	.	.	PUNCT
ejpam-3873	491	1	if	if	SCONJ
ejpam-3873	491	2	ϕ(j	ϕ(j	PROPN
ejpam-3873	491	3	)	)	PUNCT
ejpam-3873	491	4	is	be	AUX
ejpam-3873	491	5	not	not	PART
ejpam-3873	491	6	a	a	DET
ejpam-3873	491	7	separating	separate	VERB
ejpam-3873	491	8	γ	γ	NOUN
ejpam-3873	491	9	-	-	PUNCT
ejpam-3873	491	10	set	set	NOUN
ejpam-3873	491	11	of	of	ADP
ejpam-3873	491	12	d2	d2	NOUN
ejpam-3873	491	13	,	,	PUNCT
ejpam-3873	491	14	then	then	ADV
ejpam-3873	491	15	by	by	ADP
ejpam-3873	491	16	lemma	lemma	PROPN
ejpam-3873	491	17	6	6	NUM
ejpam-3873	491	18	,	,	PUNCT
ejpam-3873	491	19	ϕ(j	ϕ(j	PROPN
ejpam-3873	491	20	)	)	PUNCT
ejpam-3873	492	1	=	=	PUNCT
ejpam-3873	492	2	e	e	NOUN
ejpam-3873	492	3	∪	∪	VERB
ejpam-3873	492	4	f	f	PROPN
ejpam-3873	492	5	for	for	ADP
ejpam-3873	492	6	some	some	DET
ejpam-3873	492	7	distinct	distinct	NOUN
ejpam-3873	492	8	separating	separate	VERB
ejpam-3873	492	9	γ	γ	NOUN
ejpam-3873	492	10	-	-	PUNCT
ejpam-3873	492	11	sets	set	NOUN
ejpam-3873	492	12	e	e	NOUN
ejpam-3873	492	13	and	and	CCONJ
ejpam-3873	492	14	f	f	PROPN
ejpam-3873	492	15	in	in	ADP
ejpam-3873	492	16	d2	d2	PROPN
ejpam-3873	492	17	.	.	PUNCT
ejpam-3873	493	1	it	it	PRON
ejpam-3873	493	2	is	be	AUX
ejpam-3873	493	3	easy	easy	ADJ
ejpam-3873	493	4	to	to	PART
ejpam-3873	493	5	see	see	VERB
ejpam-3873	493	6	that	that	SCONJ
ejpam-3873	493	7	there	there	PRON
ejpam-3873	493	8	exist	exist	VERB
ejpam-3873	493	9	distinct	distinct	ADJ
ejpam-3873	493	10	γ	γ	NOUN
ejpam-3873	493	11	-	-	PUNCT
ejpam-3873	493	12	sets	set	NOUN
ejpam-3873	493	13	e′	e′	X
ejpam-3873	493	14	and	and	CCONJ
ejpam-3873	493	15	f	f	PROPN
ejpam-3873	493	16	′	′	NUM
ejpam-3873	494	1	such	such	ADJ
ejpam-3873	494	2	that	that	DET
ejpam-3873	494	3	ϕ(e′	ϕ(e′	PROPN
ejpam-3873	494	4	)	)	PUNCT
ejpam-3873	494	5	=	=	SYM
ejpam-3873	494	6	e	e	PROPN
ejpam-3873	494	7	and	and	CCONJ
ejpam-3873	494	8	ϕ(f	ϕ(f	PROPN
ejpam-3873	494	9	′	′	NOUN
ejpam-3873	494	10	)	)	PUNCT
ejpam-3873	495	1	=	=	SYM
ejpam-3873	495	2	f	f	PROPN
ejpam-3873	495	3	.	.	PUNCT
ejpam-3873	496	1	thus	thus	ADV
ejpam-3873	496	2	,	,	PUNCT
ejpam-3873	496	3	ϕ(j	ϕ(j	PROPN
ejpam-3873	496	4	)	)	PUNCT
ejpam-3873	497	1	=	=	PUNCT
ejpam-3873	497	2	e	e	X
ejpam-3873	497	3	∪	∪	VERB
ejpam-3873	497	4	f	f	PROPN
ejpam-3873	497	5	=	=	SYM
ejpam-3873	497	6	ϕ(e′)∪	ϕ(e′)∪	PROPN
ejpam-3873	497	7	ϕ(f	ϕ(f	NOUN
ejpam-3873	497	8	′	′	NUM
ejpam-3873	497	9	)	)	PUNCT
ejpam-3873	498	1	=	=	PRON
ejpam-3873	498	2	ϕ(e′	ϕ(e′	PROPN
ejpam-3873	498	3	∪	∪	PROPN
ejpam-3873	498	4	f	f	PROPN
ejpam-3873	498	5	′	′	NUM
ejpam-3873	498	6	)	)	PUNCT
ejpam-3873	498	7	.	.	PUNCT
ejpam-3873	499	1	since	since	SCONJ
ejpam-3873	499	2	ϕ	ϕ	PROPN
ejpam-3873	499	3	is	be	AUX
ejpam-3873	499	4	injective	injective	ADJ
ejpam-3873	499	5	,	,	PUNCT
ejpam-3873	499	6	we	we	PRON
ejpam-3873	499	7	have	have	VERB
ejpam-3873	499	8	j	j	PROPN
ejpam-3873	499	9	=	=	SYM
ejpam-3873	499	10	e′∪f	e′∪f	PROPN
ejpam-3873	499	11	′.	′.	NOUN
ejpam-3873	499	12	this	this	PRON
ejpam-3873	499	13	is	be	AUX
ejpam-3873	499	14	a	a	DET
ejpam-3873	499	15	contradiction	contradiction	NOUN
ejpam-3873	499	16	(	(	PUNCT
ejpam-3873	499	17	by	by	ADP
ejpam-3873	499	18	lemma	lemma	PROPN
ejpam-3873	499	19	6	6	NUM
ejpam-3873	499	20	)	)	PUNCT
ejpam-3873	499	21	.	.	PUNCT
ejpam-3873	500	1	therefore	therefore	ADV
ejpam-3873	500	2	,	,	PUNCT
ejpam-3873	500	3	ϕ(j	ϕ(j	PROPN
ejpam-3873	500	4	)	)	PUNCT
ejpam-3873	500	5	must	must	AUX
ejpam-3873	500	6	be	be	AUX
ejpam-3873	500	7	a	a	DET
ejpam-3873	500	8	separating	separate	VERB
ejpam-3873	500	9	γ	γ	NOUN
ejpam-3873	500	10	-	-	PUNCT
ejpam-3873	500	11	set	set	NOUN
ejpam-3873	500	12	of	of	ADP
ejpam-3873	500	13	d2	d2	PROPN
ejpam-3873	500	14	.	.	PUNCT
ejpam-3873	501	1	conversely	conversely	ADV
ejpam-3873	501	2	,	,	PUNCT
ejpam-3873	501	3	assume	assume	VERB
ejpam-3873	501	4	that	that	SCONJ
ejpam-3873	501	5	ϕ(j	ϕ(j	PROPN
ejpam-3873	501	6	)	)	PUNCT
ejpam-3873	501	7	is	be	AUX
ejpam-3873	501	8	a	a	DET
ejpam-3873	501	9	separating	separate	VERB
ejpam-3873	501	10	γ	γ	NOUN
ejpam-3873	501	11	-	-	PUNCT
ejpam-3873	501	12	set	set	NOUN
ejpam-3873	501	13	of	of	ADP
ejpam-3873	501	14	h	h	NOUN
ejpam-3873	501	15	and	and	CCONJ
ejpam-3873	501	16	j	j	PROPN
ejpam-3873	501	17	is	be	AUX
ejpam-3873	501	18	not	not	PART
ejpam-3873	501	19	a	a	DET
ejpam-3873	501	20	separating	separate	VERB
ejpam-3873	501	21	γ	γ	NOUN
ejpam-3873	501	22	-	-	PUNCT
ejpam-3873	501	23	set	set	NOUN
ejpam-3873	501	24	of	of	ADP
ejpam-3873	501	25	d1	d1	PROPN
ejpam-3873	501	26	.	.	PUNCT
ejpam-3873	502	1	if	if	SCONJ
ejpam-3873	502	2	j	j	PROPN
ejpam-3873	502	3	is	be	AUX
ejpam-3873	502	4	not	not	PART
ejpam-3873	502	5	a	a	DET
ejpam-3873	502	6	separating	separate	VERB
ejpam-3873	502	7	γ	γ	NOUN
ejpam-3873	502	8	-	-	PUNCT
ejpam-3873	502	9	set	set	NOUN
ejpam-3873	502	10	of	of	ADP
ejpam-3873	502	11	d1	d1	NOUN
ejpam-3873	502	12	,	,	PUNCT
ejpam-3873	502	13	then	then	ADV
ejpam-3873	502	14	by	by	ADP
ejpam-3873	502	15	lemma	lemma	PROPN
ejpam-3873	502	16	6	6	NUM
ejpam-3873	502	17	,	,	PUNCT
ejpam-3873	502	18	j	j	X
ejpam-3873	502	19	=	=	SYM
ejpam-3873	502	20	e	e	PROPN
ejpam-3873	502	21	∪	∪	VERB
ejpam-3873	502	22	f	f	PROPN
ejpam-3873	502	23	for	for	ADP
ejpam-3873	502	24	some	some	DET
ejpam-3873	502	25	γ	γ	NOUN
ejpam-3873	502	26	-	-	PUNCT
ejpam-3873	502	27	sets	set	NOUN
ejpam-3873	502	28	e	e	NOUN
ejpam-3873	502	29	and	and	CCONJ
ejpam-3873	502	30	f	f	PROPN
ejpam-3873	502	31	with	with	ADP
ejpam-3873	502	32	e	e	PROPN
ejpam-3873	502	33	6=	6=	PROPN
ejpam-3873	502	34	f	f	PROPN
ejpam-3873	502	35	.	.	PUNCT
ejpam-3873	503	1	thus	thus	ADV
ejpam-3873	503	2	,	,	PUNCT
ejpam-3873	503	3	ϕ(j	ϕ(j	PROPN
ejpam-3873	503	4	)	)	PUNCT
ejpam-3873	504	1	=	=	PUNCT
ejpam-3873	504	2	ϕ(e	ϕ(e	NOUN
ejpam-3873	504	3	∪f	∪f	X
ejpam-3873	504	4	)	)	PUNCT
ejpam-3873	504	5	=	=	SYM
ejpam-3873	504	6	ϕ(e)∪ϕ(f	ϕ(e)∪ϕ(f	PROPN
ejpam-3873	504	7	)	)	PUNCT
ejpam-3873	504	8	.	.	PUNCT
ejpam-3873	505	1	since	since	SCONJ
ejpam-3873	505	2	ϕ	ϕ	PROPN
ejpam-3873	505	3	is	be	AUX
ejpam-3873	505	4	injective	injective	ADJ
ejpam-3873	505	5	,	,	PUNCT
ejpam-3873	505	6	ϕ(e	ϕ(e	PROPN
ejpam-3873	505	7	)	)	PUNCT
ejpam-3873	506	1	6=	6=	ADP
ejpam-3873	506	2	ϕ(f	ϕ(f	PROPN
ejpam-3873	506	3	)	)	PUNCT
ejpam-3873	506	4	.	.	PUNCT
ejpam-3873	507	1	this	this	PRON
ejpam-3873	507	2	is	be	AUX
ejpam-3873	507	3	a	a	DET
ejpam-3873	507	4	contradiction	contradiction	NOUN
ejpam-3873	507	5	(	(	PUNCT
ejpam-3873	507	6	by	by	ADP
ejpam-3873	507	7	lemma	lemma	PROPN
ejpam-3873	507	8	6	6	NUM
ejpam-3873	507	9	)	)	PUNCT
ejpam-3873	507	10	.	.	PUNCT
ejpam-3873	508	1	therefore	therefore	ADV
ejpam-3873	508	2	,	,	PUNCT
ejpam-3873	508	3	j	j	PROPN
ejpam-3873	508	4	must	must	AUX
ejpam-3873	508	5	be	be	AUX
ejpam-3873	508	6	a	a	DET
ejpam-3873	508	7	separating	separate	VERB
ejpam-3873	508	8	γ	γ	NOUN
ejpam-3873	508	9	-	-	PUNCT
ejpam-3873	508	10	set	set	NOUN
ejpam-3873	508	11	of	of	ADP
ejpam-3873	508	12	d1	d1	PROPN
ejpam-3873	508	13	.	.	PUNCT
ejpam-3873	509	1	lemma	lemma	PROPN
ejpam-3873	509	2	10	10	NUM
ejpam-3873	509	3	.	.	PUNCT
ejpam-3873	510	1	let	let	VERB
ejpam-3873	510	2	d1	d1	PROPN
ejpam-3873	510	3	and	and	CCONJ
ejpam-3873	510	4	d2	d2	PROPN
ejpam-3873	510	5	be	be	PROPN
ejpam-3873	510	6	division	division	NOUN
ejpam-3873	510	7	rings	ring	NOUN
ejpam-3873	510	8	and	and	CCONJ
ejpam-3873	510	9	ϕ	ϕ	NOUN
ejpam-3873	510	10	:	:	PUNCT
ejpam-3873	510	11	td1	td1	PROPN
ejpam-3873	510	12	→	→	SYM
ejpam-3873	510	13	td2	td2	AUX
ejpam-3873	510	14	be	be	AUX
ejpam-3873	510	15	an	an	DET
ejpam-3873	510	16	isomorphism	isomorphism	NOUN
ejpam-3873	510	17	.	.	PUNCT
ejpam-3873	511	1	let	let	VERB
ejpam-3873	511	2	j	j	PROPN
ejpam-3873	511	3	be	be	AUX
ejpam-3873	511	4	a	a	DET
ejpam-3873	511	5	separating	separate	VERB
ejpam-3873	511	6	γ	γ	NOUN
ejpam-3873	511	7	-	-	PUNCT
ejpam-3873	511	8	set	set	NOUN
ejpam-3873	511	9	of	of	ADP
ejpam-3873	511	10	d1	d1	PROPN
ejpam-3873	511	11	and	and	CCONJ
ejpam-3873	511	12	x	x	PUNCT
ejpam-3873	511	13	∈	∈	NOUN
ejpam-3873	511	14	d1\j	d1\j	NOUN
ejpam-3873	511	15	.	.	PUNCT
ejpam-3873	512	1	then	then	ADV
ejpam-3873	512	2	there	there	PRON
ejpam-3873	512	3	exists	exist	VERB
ejpam-3873	512	4	a	a	DET
ejpam-3873	512	5	unique	unique	ADJ
ejpam-3873	512	6	y	y	PROPN
ejpam-3873	512	7	∈	∈	PROPN
ejpam-3873	512	8	d2\ϕ(j	d2\ϕ(j	NOUN
ejpam-3873	512	9	)	)	PUNCT
ejpam-3873	512	10	such	such	ADJ
ejpam-3873	512	11	that	that	SCONJ
ejpam-3873	512	12	ϕ(j	ϕ(j	PROPN
ejpam-3873	512	13	∪	∪	ADP
ejpam-3873	512	14	{	{	PUNCT
ejpam-3873	512	15	x	x	NOUN
ejpam-3873	512	16	}	}	PUNCT
ejpam-3873	512	17	)	)	PUNCT
ejpam-3873	512	18	=	=	SYM
ejpam-3873	512	19	ϕ(j	ϕ(j	PROPN
ejpam-3873	512	20	)	)	PUNCT
ejpam-3873	512	21	∪	∪	ADP
ejpam-3873	512	22	{	{	PUNCT
ejpam-3873	512	23	y	y	NOUN
ejpam-3873	512	24	}	}	PUNCT
ejpam-3873	512	25	.	.	PUNCT
ejpam-3873	513	1	proof	proof	NOUN
ejpam-3873	513	2	.	.	PUNCT
ejpam-3873	514	1	let	let	VERB
ejpam-3873	514	2	d1	d1	PROPN
ejpam-3873	514	3	and	and	CCONJ
ejpam-3873	514	4	d2	d2	PROPN
ejpam-3873	514	5	be	be	PROPN
ejpam-3873	514	6	division	division	NOUN
ejpam-3873	514	7	rings	ring	NOUN
ejpam-3873	514	8	and	and	CCONJ
ejpam-3873	514	9	ϕ	ϕ	NOUN
ejpam-3873	514	10	:	:	PUNCT
ejpam-3873	514	11	td1	td1	PROPN
ejpam-3873	514	12	→	→	SYM
ejpam-3873	514	13	td2	td2	AUX
ejpam-3873	514	14	be	be	AUX
ejpam-3873	514	15	an	an	DET
ejpam-3873	514	16	isomorphism	isomorphism	NOUN
ejpam-3873	514	17	.	.	PUNCT
ejpam-3873	515	1	let	let	VERB
ejpam-3873	515	2	j	j	PROPN
ejpam-3873	515	3	be	be	AUX
ejpam-3873	515	4	a	a	DET
ejpam-3873	515	5	separating	separate	VERB
ejpam-3873	515	6	γ	γ	NOUN
ejpam-3873	515	7	-	-	PUNCT
ejpam-3873	515	8	set	set	NOUN
ejpam-3873	515	9	of	of	ADP
ejpam-3873	515	10	d1	d1	PROPN
ejpam-3873	515	11	and	and	CCONJ
ejpam-3873	515	12	x	x	PUNCT
ejpam-3873	515	13	∈	∈	NOUN
ejpam-3873	515	14	d1\j	d1\j	NOUN
ejpam-3873	515	15	.	.	PUNCT
ejpam-3873	516	1	if	if	SCONJ
ejpam-3873	516	2	x	x	PUNCT
ejpam-3873	516	3	∈	∈	PROPN
ejpam-3873	516	4	d1\j	d1\j	VERB
ejpam-3873	516	5	,	,	PUNCT
ejpam-3873	516	6	then	then	ADV
ejpam-3873	516	7	x	x	X
ejpam-3873	516	8	/∈	/∈	PROPN
ejpam-3873	517	1	j	j	PROPN
ejpam-3873	517	2	.	.	PUNCT
ejpam-3873	518	1	note	note	VERB
ejpam-3873	518	2	that	that	SCONJ
ejpam-3873	518	3	ϕ(j	ϕ(j	PROPN
ejpam-3873	518	4	)	)	PUNCT
ejpam-3873	518	5	∪	∪	ADP
ejpam-3873	518	6	{	{	PUNCT
ejpam-3873	518	7	x	x	NOUN
ejpam-3873	518	8	}	}	PUNCT
ejpam-3873	518	9	6=	6=	PUNCT
ejpam-3873	518	10	ϕ(j	ϕ(j	PROPN
ejpam-3873	518	11	)	)	PUNCT
ejpam-3873	518	12	since	since	SCONJ
ejpam-3873	518	13	j	j	PROPN
ejpam-3873	518	14	is	be	AUX
ejpam-3873	518	15	a	a	DET
ejpam-3873	518	16	separating	separate	VERB
ejpam-3873	518	17	γ	γ	NOUN
ejpam-3873	518	18	-	-	PUNCT
ejpam-3873	518	19	set	set	VERB
ejpam-3873	518	20	and	and	CCONJ
ejpam-3873	518	21	ϕ(j	ϕ(j	NUM
ejpam-3873	518	22	)	)	PUNCT
ejpam-3873	518	23	∪	∪	ADP
ejpam-3873	518	24	{	{	PUNCT
ejpam-3873	518	25	x	x	NOUN
ejpam-3873	518	26	}	}	PUNCT
ejpam-3873	518	27	is	be	AUX
ejpam-3873	518	28	not	not	PART
ejpam-3873	518	29	(	(	PUNCT
ejpam-3873	518	30	by	by	ADP
ejpam-3873	518	31	lemma	lemma	PROPN
ejpam-3873	518	32	9	9	NUM
ejpam-3873	518	33	)	)	PUNCT
ejpam-3873	518	34	.	.	PUNCT
ejpam-3873	519	1	hence	hence	ADV
ejpam-3873	519	2	,	,	PUNCT
ejpam-3873	519	3	(	(	PUNCT
ejpam-3873	519	4	ϕ(j	ϕ(j	PROPN
ejpam-3873	519	5	)	)	PUNCT
ejpam-3873	519	6	∪	∪	NOUN
ejpam-3873	519	7	{	{	PUNCT
ejpam-3873	519	8	x})\ϕ(j	x})\ϕ(j	PROPN
ejpam-3873	519	9	)	)	PUNCT
ejpam-3873	519	10	6=	6=	ADP
ejpam-3873	519	11	∅.	∅.	ADP
ejpam-3873	519	12	now	now	ADV
ejpam-3873	519	13	,	,	PUNCT
ejpam-3873	519	14	we	we	PRON
ejpam-3873	519	15	claim	claim	VERB
ejpam-3873	519	16	that	that	SCONJ
ejpam-3873	519	17	(	(	PUNCT
ejpam-3873	519	18	ϕ(j	ϕ(j	PROPN
ejpam-3873	519	19	)	)	PUNCT
ejpam-3873	519	20	∪	∪	NOUN
ejpam-3873	519	21	{	{	PUNCT
ejpam-3873	519	22	x})\ϕ(j	x})\ϕ(j	PROPN
ejpam-3873	519	23	)	)	PUNCT
ejpam-3873	519	24	is	be	AUX
ejpam-3873	519	25	singleton	singleton	NOUN
ejpam-3873	519	26	.	.	PUNCT
ejpam-3873	520	1	suppose	suppose	VERB
ejpam-3873	520	2	it	it	PRON
ejpam-3873	520	3	is	be	AUX
ejpam-3873	520	4	not	not	PART
ejpam-3873	520	5	.	.	PUNCT
ejpam-3873	521	1	without	without	ADP
ejpam-3873	521	2	loss	loss	NOUN
ejpam-3873	521	3	of	of	ADP
ejpam-3873	521	4	generality	generality	NOUN
ejpam-3873	521	5	,	,	PUNCT
ejpam-3873	521	6	assume	assume	VERB
ejpam-3873	521	7	that	that	SCONJ
ejpam-3873	521	8	{	{	PUNCT
ejpam-3873	521	9	u	u	NOUN
ejpam-3873	521	10	,	,	PUNCT
ejpam-3873	521	11	v	v	NOUN
ejpam-3873	521	12	}	}	PUNCT
ejpam-3873	521	13	=	=	SYM
ejpam-3873	521	14	(	(	PUNCT
ejpam-3873	521	15	ϕ(j	ϕ(j	PROPN
ejpam-3873	521	16	)	)	PUNCT
ejpam-3873	521	17	∪	∪	NOUN
ejpam-3873	521	18	{	{	PUNCT
ejpam-3873	521	19	x})\ϕ(j	x})\ϕ(j	PROPN
ejpam-3873	521	20	)	)	PUNCT
ejpam-3873	521	21	.	.	PUNCT
ejpam-3873	522	1	if	if	SCONJ
ejpam-3873	522	2	{	{	PUNCT
ejpam-3873	522	3	u	u	NOUN
ejpam-3873	522	4	,	,	PUNCT
ejpam-3873	522	5	v	v	NOUN
ejpam-3873	522	6	}	}	PUNCT
ejpam-3873	522	7	=	=	SYM
ejpam-3873	522	8	(	(	PUNCT
ejpam-3873	522	9	ϕ(j)∪{x})\ϕ(j	ϕ(j)∪{x})\ϕ(j	NOUN
ejpam-3873	522	10	)	)	PUNCT
ejpam-3873	522	11	,	,	PUNCT
ejpam-3873	522	12	then	then	ADV
ejpam-3873	522	13	u	u	NOUN
ejpam-3873	522	14	,	,	PUNCT
ejpam-3873	522	15	v	v	NOUN
ejpam-3873	522	16	/∈	/∈	PUNCT
ejpam-3873	522	17	ϕ(j	ϕ(j	PROPN
ejpam-3873	522	18	)	)	PUNCT
ejpam-3873	522	19	.	.	PUNCT
ejpam-3873	523	1	since	since	SCONJ
ejpam-3873	523	2	ϕ(j	ϕ(j	PROPN
ejpam-3873	523	3	)	)	PUNCT
ejpam-3873	523	4	is	be	AUX
ejpam-3873	523	5	a	a	DET
ejpam-3873	523	6	γ	γ	NOUN
ejpam-3873	523	7	-	-	PUNCT
ejpam-3873	523	8	set	set	ADJ
ejpam-3873	523	9	u−1	u−1	PROPN
ejpam-3873	523	10	,	,	PUNCT
ejpam-3873	523	11	u−1	u−1	PROPN
ejpam-3873	523	12	∈	∈	PROPN
ejpam-3873	523	13	ϕ(j	ϕ(j	PROPN
ejpam-3873	523	14	)	)	PUNCT
ejpam-3873	523	15	.	.	PUNCT
ejpam-3873	524	1	thus	thus	ADV
ejpam-3873	524	2	,	,	PUNCT
ejpam-3873	524	3	a	a	DET
ejpam-3873	524	4	=	=	X
ejpam-3873	524	5	ϕ(j	ϕ(j	PROPN
ejpam-3873	524	6	)	)	PUNCT
ejpam-3873	524	7	,	,	PUNCT
ejpam-3873	524	8	b	b	X
ejpam-3873	524	9	=	=	SYM
ejpam-3873	524	10	(	(	PUNCT
ejpam-3873	524	11	ϕ(j)\{u−1})∪	ϕ(j)\{u−1})∪	PROPN
ejpam-3873	524	12	{	{	PUNCT
ejpam-3873	524	13	u	u	NOUN
ejpam-3873	524	14	}	}	PUNCT
ejpam-3873	524	15	,	,	PUNCT
ejpam-3873	524	16	and	and	CCONJ
ejpam-3873	524	17	c	c	X
ejpam-3873	524	18	=	=	SYM
ejpam-3873	524	19	(	(	PUNCT
ejpam-3873	524	20	ϕ(j)\{v−1})∪	ϕ(j)\{v−1})∪	PROPN
ejpam-3873	524	21	{	{	PUNCT
ejpam-3873	524	22	v	v	NOUN
ejpam-3873	524	23	}	}	PUNCT
ejpam-3873	524	24	are	be	AUX
ejpam-3873	524	25	three	three	NUM
ejpam-3873	524	26	distinct	distinct	ADJ
ejpam-3873	524	27	separating	separate	VERB
ejpam-3873	524	28	γ	γ	NOUN
ejpam-3873	524	29	-	-	PUNCT
ejpam-3873	524	30	sets	set	NOUN
ejpam-3873	524	31	.	.	PUNCT
ejpam-3873	525	1	note	note	VERB
ejpam-3873	525	2	that	that	SCONJ
ejpam-3873	525	3	ϕ(j	ϕ(j	PROPN
ejpam-3873	525	4	∪	∪	ADP
ejpam-3873	525	5	{	{	PUNCT
ejpam-3873	525	6	x	x	NOUN
ejpam-3873	525	7	}	}	PUNCT
ejpam-3873	525	8	)	)	PUNCT
ejpam-3873	525	9	=	=	SYM
ejpam-3873	526	1	a∪b	a∪b	NOUN
ejpam-3873	526	2	∪c	∪c	PROPN
ejpam-3873	526	3	.	.	PUNCT
ejpam-3873	527	1	hence	hence	ADV
ejpam-3873	527	2	,	,	PUNCT
ejpam-3873	527	3	j	j	PROPN
ejpam-3873	527	4	∪	∪	X
ejpam-3873	527	5	{	{	PUNCT
ejpam-3873	527	6	x	x	NOUN
ejpam-3873	527	7	}	}	PUNCT
ejpam-3873	527	8	=	=	SYM
ejpam-3873	527	9	j	j	PROPN
ejpam-3873	527	10	∪ϕ−1(b)∪ϕ−1(c	∪ϕ−1(b)∪ϕ−1(c	PROPN
ejpam-3873	527	11	)	)	PUNCT
ejpam-3873	527	12	where	where	SCONJ
ejpam-3873	527	13	j	j	PROPN
ejpam-3873	527	14	,	,	PUNCT
ejpam-3873	527	15	ϕ−1(b	ϕ−1(b	PROPN
ejpam-3873	527	16	)	)	PUNCT
ejpam-3873	527	17	,	,	PUNCT
ejpam-3873	527	18	ϕ−1(c	ϕ−1(c	PROPN
ejpam-3873	527	19	)	)	PUNCT
ejpam-3873	527	20	are	be	AUX
ejpam-3873	527	21	three	three	NUM
ejpam-3873	527	22	distinct	distinct	ADJ
ejpam-3873	527	23	separating	separate	VERB
ejpam-3873	527	24	γ	γ	NOUN
ejpam-3873	527	25	-	-	PUNCT
ejpam-3873	527	26	sets	set	NOUN
ejpam-3873	527	27	.	.	PUNCT
ejpam-3873	528	1	this	this	PRON
ejpam-3873	528	2	is	be	AUX
ejpam-3873	528	3	a	a	DET
ejpam-3873	528	4	contradiction	contradiction	NOUN
ejpam-3873	528	5	.	.	PUNCT
ejpam-3873	529	1	e.j	e.j	PROPN
ejpam-3873	529	2	.	.	PROPN
ejpam-3873	529	3	sigasig	sigasig	PROPN
ejpam-3873	529	4	,	,	PUNCT
ejpam-3873	529	5	c.j	c.j	PROPN
ejpam-3873	529	6	.	.	PROPN
ejpam-3873	529	7	rosero	rosero	PROPN
ejpam-3873	529	8	,	,	PUNCT
ejpam-3873	529	9	m.	m.	NOUN
ejpam-3873	529	10	baldado	baldado	PROPN
ejpam-3873	529	11	jr	jr	PROPN
ejpam-3873	529	12	.	.	PROPN
ejpam-3873	529	13	/	/	SYM
ejpam-3873	529	14	eur	eur	PROPN
ejpam-3873	529	15	.	.	PUNCT
ejpam-3873	530	1	j.	j.	PROPN
ejpam-3873	530	2	pure	pure	PROPN
ejpam-3873	530	3	appl	appl	PROPN
ejpam-3873	530	4	.	.	PROPN
ejpam-3873	530	5	math	math	PROPN
ejpam-3873	530	6	,	,	PUNCT
ejpam-3873	530	7	14	14	NUM
ejpam-3873	530	8	(	(	PUNCT
ejpam-3873	530	9	1	1	NUM
ejpam-3873	530	10	)	)	PUNCT
ejpam-3873	530	11	(	(	PUNCT
ejpam-3873	530	12	2021	2021	NUM
ejpam-3873	530	13	)	)	PUNCT
ejpam-3873	530	14	,	,	PUNCT
ejpam-3873	530	15	314	314	NUM
ejpam-3873	530	16	-	-	SYM
ejpam-3873	530	17	326	326	NUM
ejpam-3873	530	18	325	325	NUM
ejpam-3873	530	19	therefore	therefore	ADV
ejpam-3873	530	20	,	,	PUNCT
ejpam-3873	530	21	(	(	PUNCT
ejpam-3873	530	22	ϕ(j	ϕ(j	PROPN
ejpam-3873	530	23	)	)	PUNCT
ejpam-3873	530	24	∪	∪	NOUN
ejpam-3873	530	25	{	{	PUNCT
ejpam-3873	530	26	x})\ϕ(j	x})\ϕ(j	PROPN
ejpam-3873	530	27	)	)	PUNCT
ejpam-3873	530	28	must	must	AUX
ejpam-3873	530	29	be	be	AUX
ejpam-3873	530	30	singleton	singleton	NOUN
ejpam-3873	530	31	.	.	PUNCT
ejpam-3873	531	1	let	let	VERB
ejpam-3873	531	2	y	y	PROPN
ejpam-3873	531	3	∈	∈	PROPN
ejpam-3873	531	4	(	(	PUNCT
ejpam-3873	531	5	ϕ(j	ϕ(j	PROPN
ejpam-3873	531	6	)	)	PUNCT
ejpam-3873	531	7	∪	∪	NOUN
ejpam-3873	531	8	{	{	PUNCT
ejpam-3873	531	9	x})\ϕ(j	x})\ϕ(j	PROPN
ejpam-3873	531	10	)	)	PUNCT
ejpam-3873	531	11	.	.	PUNCT
ejpam-3873	532	1	then	then	ADV
ejpam-3873	532	2	there	there	PRON
ejpam-3873	532	3	exists	exist	VERB
ejpam-3873	532	4	y	y	PROPN
ejpam-3873	532	5	∈	∈	PROPN
ejpam-3873	532	6	d2\ϕ(j	d2\ϕ(j	NOUN
ejpam-3873	532	7	)	)	PUNCT
ejpam-3873	532	8	such	such	ADJ
ejpam-3873	532	9	that	that	SCONJ
ejpam-3873	532	10	ϕ(j	ϕ(j	PROPN
ejpam-3873	532	11	∪	∪	ADP
ejpam-3873	532	12	{	{	PUNCT
ejpam-3873	532	13	x	x	NOUN
ejpam-3873	532	14	}	}	PUNCT
ejpam-3873	532	15	)	)	PUNCT
ejpam-3873	532	16	=	=	SYM
ejpam-3873	532	17	ϕ(j	ϕ(j	PROPN
ejpam-3873	532	18	)	)	PUNCT
ejpam-3873	532	19	∪	∪	ADP
ejpam-3873	532	20	{	{	PUNCT
ejpam-3873	532	21	y	y	NOUN
ejpam-3873	532	22	}	}	PUNCT
ejpam-3873	532	23	.	.	PUNCT
ejpam-3873	533	1	the	the	DET
ejpam-3873	533	2	next	next	ADJ
ejpam-3873	533	3	result	result	NOUN
ejpam-3873	533	4	give	give	VERB
ejpam-3873	533	5	necessary	necessary	ADJ
ejpam-3873	533	6	and	and	CCONJ
ejpam-3873	533	7	sufficient	sufficient	ADJ
ejpam-3873	533	8	conditions	condition	NOUN
ejpam-3873	533	9	for	for	ADP
ejpam-3873	533	10	two	two	NUM
ejpam-3873	533	11	division	division	NOUN
ejpam-3873	533	12	rings	ring	NOUN
ejpam-3873	533	13	to	to	PART
ejpam-3873	533	14	have	have	VERB
ejpam-3873	533	15	isomorphic	isomorphic	ADJ
ejpam-3873	533	16	families	family	NOUN
ejpam-3873	533	17	of	of	ADP
ejpam-3873	533	18	γ	γ	PROPN
ejpam-3873	533	19	-	-	PUNCT
ejpam-3873	533	20	set	set	NOUN
ejpam-3873	533	21	.	.	PUNCT
ejpam-3873	534	1	theorem	theorem	VERB
ejpam-3873	534	2	16	16	NUM
ejpam-3873	534	3	.	.	PUNCT
ejpam-3873	535	1	let	let	VERB
ejpam-3873	535	2	d1	d1	PROPN
ejpam-3873	535	3	and	and	CCONJ
ejpam-3873	535	4	d2	d2	PROPN
ejpam-3873	535	5	be	be	PROPN
ejpam-3873	535	6	division	division	NOUN
ejpam-3873	535	7	rings	ring	NOUN
ejpam-3873	535	8	.	.	PUNCT
ejpam-3873	536	1	then	then	ADV
ejpam-3873	536	2	td1	td1	PROPN
ejpam-3873	536	3	is	be	AUX
ejpam-3873	536	4	isomorphic	isomorphic	ADJ
ejpam-3873	536	5	to	to	PART
ejpam-3873	536	6	td2	td2	VERB
ejpam-3873	536	7	if	if	SCONJ
ejpam-3873	536	8	and	and	CCONJ
ejpam-3873	536	9	only	only	ADV
ejpam-3873	536	10	if	if	SCONJ
ejpam-3873	536	11	there	there	PRON
ejpam-3873	536	12	exists	exist	VERB
ejpam-3873	536	13	a	a	DET
ejpam-3873	536	14	bijection	bijection	NOUN
ejpam-3873	536	15	σ	σ	NOUN
ejpam-3873	536	16	:	:	PUNCT
ejpam-3873	536	17	d1\sd1	d1\sd1	PROPN
ejpam-3873	536	18	→	→	SYM
ejpam-3873	536	19	d2\sd2	d2\sd2	NOUN
ejpam-3873	536	20	.	.	PUNCT
ejpam-3873	537	1	proof	proof	NOUN
ejpam-3873	537	2	.	.	PUNCT
ejpam-3873	538	1	let	let	VERB
ejpam-3873	538	2	ϕ	ϕ	NOUN
ejpam-3873	538	3	:	:	PUNCT
ejpam-3873	538	4	td1	td1	PROPN
ejpam-3873	538	5	→	→	SYM
ejpam-3873	538	6	td2	td2	AUX
ejpam-3873	538	7	be	be	AUX
ejpam-3873	538	8	an	an	DET
ejpam-3873	538	9	isomorphism	isomorphism	NOUN
ejpam-3873	538	10	.	.	PUNCT
ejpam-3873	539	1	define	define	VERB
ejpam-3873	539	2	σ	σ	NOUN
ejpam-3873	539	3	:	:	PUNCT
ejpam-3873	539	4	d1\sd1	d1\sd1	PROPN
ejpam-3873	539	5	→	→	PUNCT
ejpam-3873	539	6	d2\sd2	d2\sd2	NOUN
ejpam-3873	539	7	as	as	SCONJ
ejpam-3873	539	8	follows	follow	VERB
ejpam-3873	539	9	.	.	PUNCT
ejpam-3873	540	1	let	let	VERB
ejpam-3873	540	2	j	j	PROPN
ejpam-3873	540	3	be	be	AUX
ejpam-3873	540	4	a	a	DET
ejpam-3873	540	5	separating	separate	VERB
ejpam-3873	540	6	γ	γ	NOUN
ejpam-3873	540	7	-	-	PUNCT
ejpam-3873	540	8	set	set	NOUN
ejpam-3873	540	9	of	of	ADP
ejpam-3873	540	10	d1	d1	PROPN
ejpam-3873	540	11	and	and	CCONJ
ejpam-3873	540	12	x	x	PUNCT
ejpam-3873	540	13	∈	∈	NOUN
ejpam-3873	540	14	d1\sd1	d1\sd1	PROPN
ejpam-3873	540	15	.	.	PUNCT
ejpam-3873	541	1	without	without	ADP
ejpam-3873	541	2	loss	loss	NOUN
ejpam-3873	541	3	of	of	ADP
ejpam-3873	541	4	generality	generality	NOUN
ejpam-3873	541	5	,	,	PUNCT
ejpam-3873	541	6	choose	choose	VERB
ejpam-3873	541	7	x	x	X
ejpam-3873	541	8	/∈	/∈	PROPN
ejpam-3873	541	9	j	j	PROPN
ejpam-3873	541	10	.	.	PUNCT
ejpam-3873	542	1	if	if	SCONJ
ejpam-3873	542	2	x	x	PROPN
ejpam-3873	542	3	/∈	/∈	PROPN
ejpam-3873	543	1	j	j	PROPN
ejpam-3873	543	2	,	,	PUNCT
ejpam-3873	543	3	then	then	ADV
ejpam-3873	543	4	x	x	SYM
ejpam-3873	543	5	∈	∈	PROPN
ejpam-3873	543	6	d1\j	d1\j	ADV
ejpam-3873	543	7	.	.	PUNCT
ejpam-3873	544	1	by	by	ADP
ejpam-3873	544	2	lemma	lemma	PROPN
ejpam-3873	544	3	10	10	NUM
ejpam-3873	544	4	,	,	PUNCT
ejpam-3873	544	5	there	there	PRON
ejpam-3873	544	6	exists	exist	VERB
ejpam-3873	544	7	y	y	PROPN
ejpam-3873	544	8	∈	∈	PROPN
ejpam-3873	544	9	d2\ϕ(j	d2\ϕ(j	NOUN
ejpam-3873	544	10	)	)	PUNCT
ejpam-3873	544	11	with	with	ADP
ejpam-3873	544	12	ϕ(j	ϕ(j	PROPN
ejpam-3873	544	13	∪	∪	X
ejpam-3873	544	14	{	{	PUNCT
ejpam-3873	544	15	x	x	NOUN
ejpam-3873	544	16	}	}	PUNCT
ejpam-3873	544	17	)	)	PUNCT
ejpam-3873	544	18	=	=	SYM
ejpam-3873	544	19	ϕ(j	ϕ(j	PROPN
ejpam-3873	544	20	)	)	PUNCT
ejpam-3873	544	21	∪	∪	ADP
ejpam-3873	544	22	{	{	PUNCT
ejpam-3873	544	23	y	y	NOUN
ejpam-3873	544	24	}	}	PUNCT
ejpam-3873	544	25	.	.	PUNCT
ejpam-3873	545	1	now	now	ADV
ejpam-3873	545	2	,	,	PUNCT
ejpam-3873	545	3	we	we	PRON
ejpam-3873	545	4	define	define	VERB
ejpam-3873	545	5	σ(x	σ(x	NOUN
ejpam-3873	545	6	)	)	PUNCT
ejpam-3873	545	7	=	=	SYM
ejpam-3873	545	8	y	y	PROPN
ejpam-3873	545	9	and	and	CCONJ
ejpam-3873	545	10	σ(x−1	σ(x−1	NOUN
ejpam-3873	545	11	)	)	PUNCT
ejpam-3873	546	1	=	=	SYM
ejpam-3873	546	2	y−1	y−1	PROPN
ejpam-3873	546	3	.	.	PUNCT
ejpam-3873	547	1	we	we	PRON
ejpam-3873	547	2	first	first	ADV
ejpam-3873	547	3	show	show	VERB
ejpam-3873	547	4	that	that	SCONJ
ejpam-3873	547	5	σ	σ	PROPN
ejpam-3873	547	6	is	be	AUX
ejpam-3873	547	7	injective	injective	ADJ
ejpam-3873	547	8	.	.	PUNCT
ejpam-3873	548	1	let	let	VERB
ejpam-3873	548	2	a	a	DET
ejpam-3873	548	3	,	,	PUNCT
ejpam-3873	548	4	b	b	NOUN
ejpam-3873	548	5	∈	∈	NOUN
ejpam-3873	548	6	d1\sd1	d1\sd1	NUM
ejpam-3873	548	7	with	with	ADP
ejpam-3873	548	8	a	a	DET
ejpam-3873	548	9	6=	6=	PROPN
ejpam-3873	548	10	b.	b.	PROPN
ejpam-3873	548	11	let	let	VERB
ejpam-3873	548	12	jj	jj	PROPN
ejpam-3873	548	13	=	=	PRON
ejpam-3873	548	14	(	(	PUNCT
ejpam-3873	548	15	j\{a	j\{a	X
ejpam-3873	548	16	}	}	PUNCT
ejpam-3873	548	17	)	)	PUNCT
ejpam-3873	548	18	∪	∪	ADP
ejpam-3873	548	19	{	{	PUNCT
ejpam-3873	548	20	a−1	a−1	PROPN
ejpam-3873	548	21	}	}	PUNCT
ejpam-3873	548	22	and	and	CCONJ
ejpam-3873	548	23	jk	jk	PROPN
ejpam-3873	548	24	=	=	PRON
ejpam-3873	548	25	(	(	PUNCT
ejpam-3873	548	26	j\{b	j\{b	X
ejpam-3873	548	27	}	}	PUNCT
ejpam-3873	548	28	)	)	PUNCT
ejpam-3873	548	29	∪	∪	ADP
ejpam-3873	548	30	{	{	PUNCT
ejpam-3873	548	31	b−1	b−1	NOUN
ejpam-3873	548	32	}	}	PUNCT
ejpam-3873	548	33	.	.	PUNCT
ejpam-3873	549	1	then	then	ADV
ejpam-3873	549	2	a	a	DET
ejpam-3873	549	3	∈	∈	NOUN
ejpam-3873	549	4	d1\jj	d1\jj	PROPN
ejpam-3873	549	5	and	and	CCONJ
ejpam-3873	549	6	b	b	X
ejpam-3873	549	7	∈	∈	PROPN
ejpam-3873	549	8	d1\jk	d1\jk	PROPN
ejpam-3873	549	9	.	.	PUNCT
ejpam-3873	550	1	by	by	ADP
ejpam-3873	550	2	lemma	lemma	PROPN
ejpam-3873	550	3	10	10	NUM
ejpam-3873	550	4	,	,	PUNCT
ejpam-3873	550	5	there	there	PRON
ejpam-3873	550	6	exist	exist	VERB
ejpam-3873	550	7	u	u	NOUN
ejpam-3873	550	8	∈	∈	NOUN
ejpam-3873	550	9	d2\ϕ(jj	d2\ϕ(jj	PROPN
ejpam-3873	550	10	)	)	PUNCT
ejpam-3873	550	11	,	,	PUNCT
ejpam-3873	550	12	and	and	CCONJ
ejpam-3873	550	13	v	v	ADP
ejpam-3873	550	14	∈	∈	NOUN
ejpam-3873	550	15	d2\ϕ(jk	d2\ϕ(jk	PROPN
ejpam-3873	550	16	)	)	PUNCT
ejpam-3873	550	17	such	such	ADJ
ejpam-3873	550	18	that	that	SCONJ
ejpam-3873	550	19	ϕ(jj	ϕ(jj	PROPN
ejpam-3873	550	20	∪	∪	X
ejpam-3873	550	21	{	{	PUNCT
ejpam-3873	550	22	a	a	PRON
ejpam-3873	550	23	}	}	PUNCT
ejpam-3873	550	24	)	)	PUNCT
ejpam-3873	550	25	=	=	SYM
ejpam-3873	550	26	ϕ(jj	ϕ(jj	X
ejpam-3873	550	27	)	)	PUNCT
ejpam-3873	550	28	∪	∪	ADP
ejpam-3873	550	29	{	{	PUNCT
ejpam-3873	550	30	u	u	NOUN
ejpam-3873	550	31	}	}	PUNCT
ejpam-3873	550	32	and	and	CCONJ
ejpam-3873	550	33	ϕ(jk	ϕ(jk	PROPN
ejpam-3873	550	34	∪	∪	VERB
ejpam-3873	550	35	{	{	PUNCT
ejpam-3873	550	36	b	b	NOUN
ejpam-3873	550	37	}	}	PUNCT
ejpam-3873	550	38	)	)	PUNCT
ejpam-3873	551	1	=	=	SYM
ejpam-3873	551	2	ϕ(jk	ϕ(jk	PROPN
ejpam-3873	551	3	)	)	PUNCT
ejpam-3873	551	4	∪	∪	ADP
ejpam-3873	551	5	{	{	PUNCT
ejpam-3873	551	6	v	v	NOUN
ejpam-3873	551	7	}	}	PUNCT
ejpam-3873	551	8	.	.	PUNCT
ejpam-3873	552	1	without	without	ADP
ejpam-3873	552	2	loss	loss	NOUN
ejpam-3873	552	3	of	of	ADP
ejpam-3873	552	4	generality	generality	NOUN
ejpam-3873	552	5	,	,	PUNCT
ejpam-3873	552	6	assume	assume	VERB
ejpam-3873	552	7	that	that	SCONJ
ejpam-3873	552	8	a	a	DET
ejpam-3873	552	9	/∈	/∈	SYM
ejpam-3873	552	10	j	j	PROPN
ejpam-3873	552	11	and	and	CCONJ
ejpam-3873	552	12	b	b	PROPN
ejpam-3873	552	13	/∈	/∈	PROPN
ejpam-3873	552	14	j	j	PROPN
ejpam-3873	552	15	.	.	PUNCT
ejpam-3873	553	1	if	if	SCONJ
ejpam-3873	553	2	a	a	DET
ejpam-3873	553	3	/∈	/∈	SYM
ejpam-3873	553	4	j	j	PROPN
ejpam-3873	553	5	and	and	CCONJ
ejpam-3873	553	6	b	b	PROPN
ejpam-3873	553	7	/∈	/∈	PROPN
ejpam-3873	553	8	j	j	PROPN
ejpam-3873	553	9	,	,	PUNCT
ejpam-3873	553	10	then	then	ADV
ejpam-3873	553	11	σ(a	σ(a	PROPN
ejpam-3873	553	12	)	)	PUNCT
ejpam-3873	554	1	=	=	SYM
ejpam-3873	554	2	u	u	NOUN
ejpam-3873	554	3	and	and	CCONJ
ejpam-3873	554	4	σ(b	σ(b	PROPN
ejpam-3873	554	5	)	)	PUNCT
ejpam-3873	555	1	=	=	PUNCT
ejpam-3873	556	1	v.	v.	CCONJ
ejpam-3873	556	2	in	in	ADP
ejpam-3873	556	3	the	the	DET
ejpam-3873	556	4	sense	sense	NOUN
ejpam-3873	556	5	of	of	ADP
ejpam-3873	556	6	the	the	DET
ejpam-3873	556	7	proof	proof	NOUN
ejpam-3873	556	8	of	of	ADP
ejpam-3873	556	9	lemma	lemma	PROPN
ejpam-3873	556	10	10	10	NUM
ejpam-3873	556	11	,	,	PUNCT
ejpam-3873	556	12	ϕ(j)\ϕ(jj	ϕ(j)\ϕ(jj	PUNCT
ejpam-3873	556	13	)	)	PUNCT
ejpam-3873	556	14	and	and	CCONJ
ejpam-3873	556	15	ϕ(j)\ϕ(jj	ϕ(j)\ϕ(jj	PROPN
ejpam-3873	556	16	)	)	PUNCT
ejpam-3873	556	17	are	be	AUX
ejpam-3873	556	18	singleton	singleton	NOUN
ejpam-3873	556	19	sets	set	NOUN
ejpam-3873	556	20	.	.	PUNCT
ejpam-3873	557	1	thus	thus	ADV
ejpam-3873	557	2	,	,	PUNCT
ejpam-3873	557	3	if	if	SCONJ
ejpam-3873	557	4	u	u	PROPN
ejpam-3873	557	5	=	=	PROPN
ejpam-3873	557	6	v	v	NOUN
ejpam-3873	557	7	,	,	PUNCT
ejpam-3873	557	8	then	then	ADV
ejpam-3873	557	9	ϕ(j∪{a	ϕ(j∪{a	NUM
ejpam-3873	557	10	}	}	PUNCT
ejpam-3873	557	11	)	)	PUNCT
ejpam-3873	557	12	=	=	PUNCT
ejpam-3873	557	13	ϕ(j∪{b	ϕ(j∪{b	NOUN
ejpam-3873	557	14	}	}	PUNCT
ejpam-3873	557	15	)	)	PUNCT
ejpam-3873	557	16	.	.	PUNCT
ejpam-3873	558	1	since	since	SCONJ
ejpam-3873	558	2	ϕ	ϕ	PROPN
ejpam-3873	558	3	is	be	AUX
ejpam-3873	558	4	an	an	DET
ejpam-3873	558	5	isomorphism	isomorphism	NOUN
ejpam-3873	558	6	,	,	PUNCT
ejpam-3873	558	7	j	j	PROPN
ejpam-3873	558	8	∪	∪	X
ejpam-3873	558	9	{	{	PUNCT
ejpam-3873	558	10	a	a	PRON
ejpam-3873	558	11	}	}	PUNCT
ejpam-3873	558	12	=	=	SYM
ejpam-3873	558	13	j	j	PROPN
ejpam-3873	558	14	∪	∪	X
ejpam-3873	558	15	{	{	PUNCT
ejpam-3873	558	16	b	b	NOUN
ejpam-3873	558	17	}	}	PUNCT
ejpam-3873	558	18	.	.	PUNCT
ejpam-3873	559	1	thus	thus	ADV
ejpam-3873	559	2	,	,	PUNCT
ejpam-3873	559	3	if	if	SCONJ
ejpam-3873	559	4	a	a	PRON
ejpam-3873	559	5	,	,	PUNCT
ejpam-3873	559	6	b	b	PROPN
ejpam-3873	559	7	/∈	/∈	PROPN
ejpam-3873	559	8	j	j	PROPN
ejpam-3873	559	9	,	,	PUNCT
ejpam-3873	559	10	then	then	ADV
ejpam-3873	559	11	a	a	DET
ejpam-3873	559	12	=	=	X
ejpam-3873	559	13	b.	b.	PROPN
ejpam-3873	560	1	this	this	PRON
ejpam-3873	560	2	is	be	AUX
ejpam-3873	560	3	a	a	DET
ejpam-3873	560	4	contradiction.this	contradiction.this	NOUN
ejpam-3873	560	5	shows	show	NOUN
ejpam-3873	560	6	that	that	SCONJ
ejpam-3873	560	7	σ	σ	PROPN
ejpam-3873	560	8	is	be	AUX
ejpam-3873	560	9	injective	injective	ADJ
ejpam-3873	560	10	.	.	PUNCT
ejpam-3873	561	1	next	next	ADV
ejpam-3873	561	2	,	,	PUNCT
ejpam-3873	561	3	we	we	PRON
ejpam-3873	561	4	show	show	VERB
ejpam-3873	561	5	that	that	SCONJ
ejpam-3873	561	6	σ	σ	PROPN
ejpam-3873	561	7	is	be	AUX
ejpam-3873	561	8	surjective	surjective	ADJ
ejpam-3873	561	9	.	.	PUNCT
ejpam-3873	562	1	let	let	VERB
ejpam-3873	562	2	y	y	PROPN
ejpam-3873	562	3	∈	∈	PROPN
ejpam-3873	562	4	d2\sd2	d2\sd2	PROPN
ejpam-3873	562	5	and	and	CCONJ
ejpam-3873	562	6	j	j	PROPN
ejpam-3873	562	7	be	be	AUX
ejpam-3873	562	8	a	a	DET
ejpam-3873	562	9	separating	separate	VERB
ejpam-3873	562	10	γ	γ	NOUN
ejpam-3873	562	11	-	-	PUNCT
ejpam-3873	562	12	set	set	NOUN
ejpam-3873	562	13	of	of	ADP
ejpam-3873	562	14	d1	d1	PROPN
ejpam-3873	562	15	.	.	PUNCT
ejpam-3873	563	1	without	without	ADP
ejpam-3873	563	2	loss	loss	NOUN
ejpam-3873	563	3	of	of	ADP
ejpam-3873	563	4	generality	generality	NOUN
ejpam-3873	563	5	,	,	PUNCT
ejpam-3873	563	6	assume	assume	VERB
ejpam-3873	563	7	that	that	SCONJ
ejpam-3873	563	8	y	y	PROPN
ejpam-3873	563	9	/∈	/∈	PROPN
ejpam-3873	563	10	j	j	PROPN
ejpam-3873	563	11	.	.	PUNCT
ejpam-3873	564	1	if	if	SCONJ
ejpam-3873	564	2	y	y	PROPN
ejpam-3873	564	3	/∈	/∈	PROPN
ejpam-3873	565	1	j	j	PROPN
ejpam-3873	565	2	,	,	PUNCT
ejpam-3873	565	3	then	then	ADV
ejpam-3873	565	4	y	y	PROPN
ejpam-3873	565	5	∈	∈	PROPN
ejpam-3873	565	6	d2\ϕ(j	d2\ϕ(j	NOUN
ejpam-3873	565	7	)	)	PUNCT
ejpam-3873	565	8	.	.	PUNCT
ejpam-3873	566	1	since	since	SCONJ
ejpam-3873	566	2	ϕ−1	ϕ−1	PROPN
ejpam-3873	566	3	is	be	AUX
ejpam-3873	566	4	also	also	ADV
ejpam-3873	566	5	an	an	DET
ejpam-3873	566	6	isomorphism	isomorphism	NOUN
ejpam-3873	566	7	,	,	PUNCT
ejpam-3873	566	8	by	by	ADP
ejpam-3873	566	9	lemma	lemma	PROPN
ejpam-3873	566	10	10	10	NUM
ejpam-3873	566	11	,	,	PUNCT
ejpam-3873	566	12	there	there	PRON
ejpam-3873	566	13	exists	exist	VERB
ejpam-3873	566	14	x	x	X
ejpam-3873	566	15	∈	∈	PROPN
ejpam-3873	566	16	d1\j	d1\j	VERB
ejpam-3873	566	17	such	such	DET
ejpam-3873	566	18	that	that	DET
ejpam-3873	566	19	ϕ−1(ϕ(j	ϕ−1(ϕ(j	NOUN
ejpam-3873	566	20	)	)	PUNCT
ejpam-3873	566	21	∪	∪	ADP
ejpam-3873	566	22	{	{	PUNCT
ejpam-3873	566	23	y	y	NOUN
ejpam-3873	566	24	}	}	PUNCT
ejpam-3873	566	25	)	)	PUNCT
ejpam-3873	567	1	=	=	SYM
ejpam-3873	567	2	j	j	PROPN
ejpam-3873	567	3	∪	∪	X
ejpam-3873	567	4	{	{	PUNCT
ejpam-3873	567	5	x	x	NOUN
ejpam-3873	567	6	}	}	PUNCT
ejpam-3873	567	7	,	,	PUNCT
ejpam-3873	567	8	that	that	PRON
ejpam-3873	567	9	is	is	ADV
ejpam-3873	567	10	ϕ(j)∪	ϕ(j)∪	PROPN
ejpam-3873	567	11	{	{	PUNCT
ejpam-3873	567	12	y	y	NOUN
ejpam-3873	567	13	}	}	PUNCT
ejpam-3873	567	14	=	=	SYM
ejpam-3873	567	15	ϕ(j	ϕ(j	ADP
ejpam-3873	567	16	∪	∪	X
ejpam-3873	567	17	{	{	PUNCT
ejpam-3873	567	18	x	x	NOUN
ejpam-3873	567	19	}	}	PUNCT
ejpam-3873	567	20	)	)	PUNCT
ejpam-3873	567	21	.	.	PUNCT
ejpam-3873	568	1	this	this	PRON
ejpam-3873	568	2	implies	imply	VERB
ejpam-3873	568	3	that	that	SCONJ
ejpam-3873	568	4	there	there	PRON
ejpam-3873	568	5	exists	exist	VERB
ejpam-3873	568	6	x	x	X
ejpam-3873	568	7	∈	∈	NOUN
ejpam-3873	568	8	d1\sd1	d1\sd1	NUM
ejpam-3873	568	9	such	such	ADJ
ejpam-3873	568	10	that	that	PRON
ejpam-3873	568	11	σ(x	σ(x	NOUN
ejpam-3873	568	12	)	)	PUNCT
ejpam-3873	568	13	=	=	VERB
ejpam-3873	569	1	y.	y.	NOUN
ejpam-3873	569	2	this	this	PRON
ejpam-3873	569	3	shows	show	VERB
ejpam-3873	569	4	that	that	SCONJ
ejpam-3873	569	5	σ	σ	PROPN
ejpam-3873	569	6	is	be	AUX
ejpam-3873	569	7	surjective	surjective	ADJ
ejpam-3873	569	8	.	.	PUNCT
ejpam-3873	570	1	accordingly	accordingly	ADV
ejpam-3873	570	2	,	,	PUNCT
ejpam-3873	570	3	σ	σ	PROPN
ejpam-3873	570	4	is	be	AUX
ejpam-3873	570	5	bijective	bijective	ADJ
ejpam-3873	570	6	.	.	PUNCT
ejpam-3873	571	1	for	for	ADP
ejpam-3873	571	2	the	the	DET
ejpam-3873	571	3	converse	converse	NOUN
ejpam-3873	571	4	,	,	PUNCT
ejpam-3873	571	5	consider	consider	VERB
ejpam-3873	571	6	the	the	DET
ejpam-3873	571	7	bijective	bijective	ADJ
ejpam-3873	571	8	function	function	NOUN
ejpam-3873	571	9	σ	σ	NOUN
ejpam-3873	571	10	:	:	PUNCT
ejpam-3873	571	11	d1\sd1	d1\sd1	PROPN
ejpam-3873	571	12	→	→	PUNCT
ejpam-3873	571	13	d2\sd2	d2\sd2	NOUN
ejpam-3873	571	14	given	give	VERB
ejpam-3873	571	15	by	by	ADP
ejpam-3873	571	16	σ(x	σ(x	NOUN
ejpam-3873	571	17	)	)	PUNCT
ejpam-3873	571	18	=	=	SYM
ejpam-3873	571	19	y	y	PROPN
ejpam-3873	571	20	and	and	CCONJ
ejpam-3873	571	21	σ(x−1	σ(x−1	NOUN
ejpam-3873	571	22	)	)	PUNCT
ejpam-3873	572	1	=	=	SYM
ejpam-3873	572	2	y−1	y−1	PROPN
ejpam-3873	572	3	where	where	SCONJ
ejpam-3873	572	4	x	x	X
ejpam-3873	572	5	/∈	/∈	PROPN
ejpam-3873	572	6	j	j	PROPN
ejpam-3873	572	7	,	,	PUNCT
ejpam-3873	572	8	and	and	CCONJ
ejpam-3873	572	9	y	y	PROPN
ejpam-3873	572	10	and	and	CCONJ
ejpam-3873	572	11	j	j	PROPN
ejpam-3873	572	12	are	be	AUX
ejpam-3873	572	13	in	in	ADP
ejpam-3873	572	14	the	the	DET
ejpam-3873	572	15	same	same	ADJ
ejpam-3873	572	16	sense	sense	NOUN
ejpam-3873	572	17	as	as	ADP
ejpam-3873	572	18	in	in	ADP
ejpam-3873	572	19	the	the	DET
ejpam-3873	572	20	above	above	ADJ
ejpam-3873	572	21	arguments	argument	NOUN
ejpam-3873	572	22	.	.	PUNCT
ejpam-3873	573	1	define	define	VERB
ejpam-3873	573	2	ϕ	ϕ	NOUN
ejpam-3873	573	3	:	:	PUNCT
ejpam-3873	573	4	td1	td1	PROPN
ejpam-3873	573	5	→	→	SYM
ejpam-3873	573	6	td2	td2	PROPN
ejpam-3873	573	7	as	as	SCONJ
ejpam-3873	573	8	follows	follow	VERB
ejpam-3873	573	9	.	.	PUNCT
ejpam-3873	574	1	let	let	VERB
ejpam-3873	574	2	j	j	PROPN
ejpam-3873	574	3	be	be	AUX
ejpam-3873	574	4	in	in	ADP
ejpam-3873	574	5	td1	td1	PROPN
ejpam-3873	574	6	,	,	PUNCT
ejpam-3873	574	7	then	then	ADV
ejpam-3873	574	8	j	j	PROPN
ejpam-3873	574	9	=	=	PROPN
ejpam-3873	574	10	sd1	sd1	PROPN
ejpam-3873	574	11	∪	∪	VERB
ejpam-3873	574	12	a	a	PRON
ejpam-3873	574	13	for	for	ADP
ejpam-3873	574	14	some	some	PRON
ejpam-3873	574	15	subset	subset	NOUN
ejpam-3873	574	16	a	a	PRON
ejpam-3873	574	17	of	of	ADP
ejpam-3873	574	18	d1\sd1	d1\sd1	PROPN
ejpam-3873	574	19	.	.	PUNCT
ejpam-3873	575	1	let	let	VERB
ejpam-3873	575	2	ϕ(j	ϕ(j	PUNCT
ejpam-3873	575	3	)	)	PUNCT
ejpam-3873	576	1	=	=	PUNCT
ejpam-3873	576	2	sd2	sd2	PROPN
ejpam-3873	576	3	∪	∪	PROPN
ejpam-3873	576	4	σ(a	σ(a	PROPN
ejpam-3873	576	5	)	)	PUNCT
ejpam-3873	576	6	.	.	PUNCT
ejpam-3873	577	1	then	then	ADV
ejpam-3873	577	2	it	it	PRON
ejpam-3873	577	3	is	be	AUX
ejpam-3873	577	4	easy	easy	ADJ
ejpam-3873	577	5	to	to	PART
ejpam-3873	577	6	show	show	VERB
ejpam-3873	577	7	that	that	SCONJ
ejpam-3873	577	8	ϕ	ϕ	NOUN
ejpam-3873	577	9	is	be	AUX
ejpam-3873	577	10	an	an	DET
ejpam-3873	577	11	isomorphism	isomorphism	NOUN
ejpam-3873	577	12	.	.	PUNCT
ejpam-3873	578	1	corollary	corollary	ADJ
ejpam-3873	578	2	3	3	NUM
ejpam-3873	578	3	.	.	PUNCT
ejpam-3873	579	1	let	let	VERB
ejpam-3873	579	2	d1	d1	PROPN
ejpam-3873	579	3	and	and	CCONJ
ejpam-3873	579	4	d2	d2	PROPN
ejpam-3873	579	5	be	be	PROPN
ejpam-3873	579	6	division	division	NOUN
ejpam-3873	579	7	rings	ring	NOUN
ejpam-3873	579	8	.	.	PUNCT
ejpam-3873	580	1	then	then	ADV
ejpam-3873	580	2	,	,	PUNCT
ejpam-3873	580	3	td1	td1	PROPN
ejpam-3873	580	4	is	be	AUX
ejpam-3873	580	5	isomorphic	isomorphic	ADJ
ejpam-3873	580	6	to	to	PART
ejpam-3873	580	7	td2	td2	VERB
ejpam-3873	580	8	if	if	SCONJ
ejpam-3873	580	9	and	and	CCONJ
ejpam-3873	581	1	only	only	ADV
ejpam-3873	581	2	if	if	SCONJ
ejpam-3873	581	3	|d1\sd1	|d1\sd1	PROPN
ejpam-3873	581	4	|	|	NOUN
ejpam-3873	581	5	=	=	SYM
ejpam-3873	581	6	|d2\sd2	|d2\sd2	PROPN
ejpam-3873	581	7	|	|	ADV
ejpam-3873	581	8	.	.	PUNCT
ejpam-3873	582	1	proof	proof	NOUN
ejpam-3873	582	2	.	.	PUNCT
ejpam-3873	583	1	the	the	DET
ejpam-3873	583	2	given	give	VERB
ejpam-3873	583	3	statement	statement	NOUN
ejpam-3873	583	4	follows	follow	VERB
ejpam-3873	583	5	from	from	ADP
ejpam-3873	583	6	theorem	theorem	ADJ
ejpam-3873	583	7	16	16	NUM
ejpam-3873	583	8	.	.	NOUN
ejpam-3873	583	9	8	8	NUM
ejpam-3873	583	10	.	.	PUNCT
ejpam-3873	584	1	acknowledgements	acknowledgement	NOUN
ejpam-3873	584	2	the	the	DET
ejpam-3873	584	3	authors	author	NOUN
ejpam-3873	584	4	would	would	AUX
ejpam-3873	584	5	like	like	VERB
ejpam-3873	584	6	to	to	PART
ejpam-3873	584	7	thank	thank	VERB
ejpam-3873	584	8	rural	rural	ADJ
ejpam-3873	584	9	engineering	engineering	NOUN
ejpam-3873	584	10	and	and	CCONJ
ejpam-3873	584	11	technology	technology	NOUN
ejpam-3873	584	12	center	center	NOUN
ejpam-3873	584	13	of	of	ADP
ejpam-3873	584	14	negros	negros	PROPN
ejpam-3873	584	15	oriental	oriental	ADJ
ejpam-3873	584	16	state	state	PROPN
ejpam-3873	584	17	university	university	PROPN
ejpam-3873	584	18	for	for	ADP
ejpam-3873	584	19	partially	partially	ADV
ejpam-3873	584	20	supporting	support	VERB
ejpam-3873	584	21	this	this	DET
ejpam-3873	584	22	research	research	NOUN
ejpam-3873	584	23	.	.	PUNCT
ejpam-3873	585	1	references	reference	NOUN
ejpam-3873	585	2	326	326	NUM
ejpam-3873	585	3	references	reference	NOUN
ejpam-3873	585	4	[	[	X
ejpam-3873	585	5	1	1	NUM
ejpam-3873	585	6	]	]	PUNCT
ejpam-3873	585	7	michael	michael	PROPN
ejpam-3873	585	8	patula	patula	PROPN
ejpam-3873	585	9	baldado	baldado	PROPN
ejpam-3873	585	10	jr	jr	PROPN
ejpam-3873	585	11	and	and	CCONJ
ejpam-3873	585	12	cristopher	cristopher	PROPN
ejpam-3873	585	13	john	john	PROPN
ejpam-3873	585	14	salvador	salvador	PROPN
ejpam-3873	585	15	rosero	rosero	VERB
ejpam-3873	585	16	.	.	PUNCT
ejpam-3873	586	1	d	d	X
ejpam-3873	586	2	-	-	PUNCT
ejpam-3873	586	3	sets	set	NOUN
ejpam-3873	586	4	generated	generate	VERB
ejpam-3873	586	5	by	by	ADP
ejpam-3873	586	6	a	a	DET
ejpam-3873	586	7	subset	subset	NOUN
ejpam-3873	586	8	of	of	ADP
ejpam-3873	586	9	a	a	DET
ejpam-3873	586	10	group	group	NOUN
ejpam-3873	586	11	.	.	PUNCT
ejpam-3873	587	1	european	european	PROPN
ejpam-3873	587	2	journal	journal	PROPN
ejpam-3873	587	3	of	of	ADP
ejpam-3873	587	4	pure	pure	ADJ
ejpam-3873	587	5	and	and	CCONJ
ejpam-3873	587	6	applied	applied	ADJ
ejpam-3873	587	7	mathematics	mathematic	NOUN
ejpam-3873	587	8	,	,	PUNCT
ejpam-3873	587	9	9(1):34–38	9(1):34–38	NUM
ejpam-3873	587	10	,	,	PUNCT
ejpam-3873	587	11	2016	2016	NUM
ejpam-3873	587	12	.	.	PUNCT
ejpam-3873	588	1	[	[	X
ejpam-3873	588	2	2	2	NUM
ejpam-3873	588	3	]	]	X
ejpam-3873	588	4	joris	joris	PROPN
ejpam-3873	588	5	n	n	PROPN
ejpam-3873	588	6	buloron	buloron	NOUN
ejpam-3873	588	7	,	,	PUNCT
ejpam-3873	588	8	roberto	roberto	PROPN
ejpam-3873	588	9	b	b	PROPN
ejpam-3873	588	10	corcino	corcino	PROPN
ejpam-3873	588	11	,	,	PUNCT
ejpam-3873	588	12	lorna	lorna	PROPN
ejpam-3873	588	13	s	s	PART
ejpam-3873	588	14	almocera	almocera	NOUN
ejpam-3873	588	15	,	,	PUNCT
ejpam-3873	588	16	and	and	CCONJ
ejpam-3873	588	17	michael	michael	PROPN
ejpam-3873	588	18	p	p	PROPN
ejpam-3873	588	19	baldado	baldado	PROPN
ejpam-3873	588	20	jr	jr	PROPN
ejpam-3873	588	21	.	.	PUNCT
ejpam-3873	589	1	d	d	X
ejpam-3873	589	2	-	-	PUNCT
ejpam-3873	589	3	sets	set	NOUN
ejpam-3873	589	4	and	and	CCONJ
ejpam-3873	589	5	structure	structure	NOUN
ejpam-3873	589	6	-	-	PUNCT
ejpam-3873	589	7	preserving	preserve	VERB
ejpam-3873	589	8	maps	map	NOUN
ejpam-3873	589	9	.	.	PUNCT
ejpam-3873	590	1	turkish	turkish	ADJ
ejpam-3873	590	2	journal	journal	NOUN
ejpam-3873	590	3	of	of	ADP
ejpam-3873	590	4	analysis	analysis	NOUN
ejpam-3873	590	5	and	and	CCONJ
ejpam-3873	590	6	number	number	NOUN
ejpam-3873	590	7	theory	theory	NOUN
ejpam-3873	590	8	,	,	PUNCT
ejpam-3873	590	9	3(6):160–164	3(6):160–164	NUM
ejpam-3873	590	10	,	,	PUNCT
ejpam-3873	590	11	2015	2015	NUM
ejpam-3873	590	12	.	.	PUNCT
ejpam-3873	591	1	[	[	X
ejpam-3873	591	2	3	3	NUM
ejpam-3873	591	3	]	]	X
ejpam-3873	591	4	joris	joris	PROPN
ejpam-3873	591	5	n	n	PROPN
ejpam-3873	591	6	buloron	buloron	PROPN
ejpam-3873	591	7	,	,	PUNCT
ejpam-3873	591	8	cristopher	cristopher	PROPN
ejpam-3873	591	9	john	john	PROPN
ejpam-3873	591	10	s	s	PROPN
ejpam-3873	591	11	rosero	rosero	PROPN
ejpam-3873	591	12	,	,	PUNCT
ejpam-3873	591	13	jay	jay	PROPN
ejpam-3873	591	14	m	m	VERB
ejpam-3873	591	15	ontolan	ontolan	ADJ
ejpam-3873	591	16	,	,	PUNCT
ejpam-3873	591	17	and	and	CCONJ
ejpam-3873	591	18	mp	mp	PROPN
ejpam-3873	591	19	baldado	baldado	PROPN
ejpam-3873	591	20	jr	jr	PROPN
ejpam-3873	591	21	.	.	PUNCT
ejpam-3873	592	1	some	some	DET
ejpam-3873	592	2	properties	property	NOUN
ejpam-3873	592	3	of	of	ADP
ejpam-3873	592	4	d	d	NOUN
ejpam-3873	592	5	-	-	PUNCT
ejpam-3873	592	6	sets	set	NOUN
ejpam-3873	592	7	of	of	ADP
ejpam-3873	592	8	a	a	DET
ejpam-3873	592	9	group1	group1	PROPN
ejpam-3873	592	10	.	.	PUNCT
ejpam-3873	593	1	in	in	ADP
ejpam-3873	593	2	international	international	PROPN
ejpam-3873	593	3	mathematical	mathematical	ADJ
ejpam-3873	593	4	forum	forum	PROPN
ejpam-3873	593	5	,	,	PUNCT
ejpam-3873	593	6	volume	volume	NOUN
ejpam-3873	593	7	9	9	NUM
ejpam-3873	593	8	,	,	PUNCT
ejpam-3873	593	9	pages	page	NOUN
ejpam-3873	593	10	1035–1040	1035–1040	NUM
ejpam-3873	593	11	,	,	PUNCT
ejpam-3873	593	12	2014	2014	NUM
ejpam-3873	593	13	.	.	PUNCT
ejpam-3873	594	1	[	[	X
ejpam-3873	594	2	4	4	X
ejpam-3873	594	3	]	]	X
ejpam-3873	594	4	john	john	PROPN
ejpam-3873	594	5	b	b	PROPN
ejpam-3873	594	6	fraleigh	fraleigh	PROPN
ejpam-3873	594	7	.	.	PUNCT
ejpam-3873	595	1	a	a	DET
ejpam-3873	595	2	first	first	ADJ
ejpam-3873	595	3	course	course	NOUN
ejpam-3873	595	4	in	in	ADP
ejpam-3873	595	5	abstract	abstract	ADJ
ejpam-3873	595	6	algebra	algebra	NOUN
ejpam-3873	595	7	.	.	PUNCT
ejpam-3873	596	1	pearson	pearson	PROPN
ejpam-3873	596	2	education	education	PROPN
ejpam-3873	596	3	india	india	PROPN
ejpam-3873	596	4	,	,	PUNCT
ejpam-3873	596	5	2003	2003	NUM
ejpam-3873	596	6	.	.	PUNCT
ejpam-3873	597	1	[	[	X
ejpam-3873	597	2	5	5	X
ejpam-3873	597	3	]	]	X
ejpam-3873	597	4	joseph	joseph	PROPN
ejpam-3873	597	5	gallian	gallian	PROPN
ejpam-3873	597	6	.	.	PUNCT
ejpam-3873	598	1	contemporary	contemporary	ADJ
ejpam-3873	598	2	abstract	abstract	ADJ
ejpam-3873	598	3	algebra	algebra	PROPN
ejpam-3873	598	4	.	.	PUNCT
ejpam-3873	599	1	nelson	nelson	PROPN
ejpam-3873	599	2	education	education	PROPN
ejpam-3873	599	3	,	,	PUNCT
ejpam-3873	599	4	2012	2012	NUM
ejpam-3873	599	5	.	.	PUNCT
ejpam-3873	600	1	[	[	X
ejpam-3873	600	2	6	6	NUM
ejpam-3873	600	3	]	]	X
ejpam-3873	600	4	linda	linda	PROPN
ejpam-3873	600	5	gilbert	gilbert	PROPN
ejpam-3873	600	6	.	.	PUNCT
ejpam-3873	601	1	elements	element	NOUN
ejpam-3873	601	2	of	of	ADP
ejpam-3873	601	3	modern	modern	ADJ
ejpam-3873	601	4	algebra	algebra	NOUN
ejpam-3873	601	5	.	.	PUNCT
ejpam-3873	602	1	nelson	nelson	PROPN
ejpam-3873	602	2	education	education	PROPN
ejpam-3873	602	3	,	,	PUNCT
ejpam-3873	602	4	2014	2014	NUM
ejpam-3873	602	5	.	.	PUNCT
ejpam-3873	603	1	[	[	X
ejpam-3873	603	2	7	7	X
ejpam-3873	603	3	]	]	X
ejpam-3873	603	4	israel	israel	PROPN
ejpam-3873	603	5	n	n	PROPN
ejpam-3873	603	6	herstein	herstein	NOUN
ejpam-3873	603	7	.	.	PUNCT
ejpam-3873	604	1	abstract	abstract	ADJ
ejpam-3873	604	2	algebra	algebra	PROPN
ejpam-3873	604	3	.	.	PUNCT
ejpam-3873	605	1	prentice	prentice	PROPN
ejpam-3873	605	2	hall	hall	PROPN
ejpam-3873	605	3	,	,	PUNCT
ejpam-3873	605	4	1996	1996	NUM
ejpam-3873	605	5	.	.	PUNCT
ejpam-3873	606	1	[	[	X
ejpam-3873	606	2	8	8	NUM
ejpam-3873	606	3	]	]	X
ejpam-3873	606	4	thomas	thomas	PROPN
ejpam-3873	606	5	w	w	PROPN
ejpam-3873	606	6	hungerford	hungerford	PROPN
ejpam-3873	606	7	.	.	PUNCT
ejpam-3873	607	1	algebra	algebra	PROPN
ejpam-3873	607	2	,	,	PUNCT
ejpam-3873	607	3	volume	volume	NOUN
ejpam-3873	607	4	73	73	NUM
ejpam-3873	607	5	of	of	ADP
ejpam-3873	607	6	.	.	PUNCT
ejpam-3873	608	1	graduate	graduate	NOUN
ejpam-3873	608	2	texts	text	NOUN
ejpam-3873	608	3	in	in	ADP
ejpam-3873	608	4	mathematics	mathematic	NOUN
ejpam-3873	608	5	,	,	PUNCT
ejpam-3873	608	6	pages	page	NOUN
ejpam-3873	608	7	20–31	20–31	PROPN
ejpam-3873	608	8	,	,	PUNCT
ejpam-3873	608	9	1980	1980	NUM
ejpam-3873	608	10	.	.	PUNCT
ejpam-3873	609	1	[	[	X
ejpam-3873	609	2	9	9	NUM
ejpam-3873	609	3	]	]	X
ejpam-3873	609	4	david	david	PROPN
ejpam-3873	609	5	c	c	PROPN
ejpam-3873	609	6	kurtz	kurtz	PROPN
ejpam-3873	609	7	.	.	PUNCT
ejpam-3873	610	1	foundations	foundation	NOUN
ejpam-3873	610	2	of	of	ADP
ejpam-3873	610	3	abstract	abstract	ADJ
ejpam-3873	610	4	mathematics	mathematic	NOUN
ejpam-3873	610	5	.	.	PUNCT
ejpam-3873	610	6	1992	1992	NUM
ejpam-3873	610	7	.	.	PUNCT
ejpam-3873	611	1	[	[	X
ejpam-3873	611	2	10	10	NUM
ejpam-3873	611	3	]	]	X
ejpam-3873	611	4	davender	davender	PROPN
ejpam-3873	611	5	s	s	PROPN
ejpam-3873	611	6	malik	malik	PROPN
ejpam-3873	611	7	,	,	PUNCT
ejpam-3873	611	8	john	john	PROPN
ejpam-3873	611	9	m	m	PROPN
ejpam-3873	611	10	mordeson	mordeson	PROPN
ejpam-3873	611	11	,	,	PUNCT
ejpam-3873	611	12	and	and	CCONJ
ejpam-3873	611	13	mk	mk	PROPN
ejpam-3873	611	14	sen	sen	PROPN
ejpam-3873	611	15	.	.	PROPN
ejpam-3873	611	16	fundamentals	fundamental	NOUN
ejpam-3873	611	17	of	of	ADP
ejpam-3873	611	18	abstract	abstract	ADJ
ejpam-3873	611	19	algebra	algebra	NOUN
ejpam-3873	611	20	.	.	PUNCT
ejpam-3873	612	1	mcgraw	mcgraw	PROPN
ejpam-3873	612	2	-	-	PUNCT
ejpam-3873	612	3	hill	hill	PROPN
ejpam-3873	612	4	,	,	PUNCT
ejpam-3873	612	5	1997	1997	NUM
ejpam-3873	612	6	.	.	PUNCT
ejpam-3873	613	1	[	[	X
ejpam-3873	613	2	11	11	NUM
ejpam-3873	613	3	]	]	PUNCT
ejpam-3873	613	4	cristopher	cristopher	PROPN
ejpam-3873	613	5	john	john	PROPN
ejpam-3873	613	6	s	s	PROPN
ejpam-3873	613	7	rosero	rosero	PROPN
ejpam-3873	613	8	and	and	CCONJ
ejpam-3873	613	9	michael	michael	PROPN
ejpam-3873	613	10	p	p	PROPN
ejpam-3873	613	11	baldado	baldado	PROPN
ejpam-3873	613	12	jr	jr	PROPN
ejpam-3873	613	13	.	.	PUNCT
ejpam-3873	614	1	some	some	DET
ejpam-3873	614	2	properties	property	NOUN
ejpam-3873	614	3	of	of	ADP
ejpam-3873	614	4	γ	γ	NOUN
ejpam-3873	614	5	-	-	NOUN
ejpam-3873	614	6	sets	set	NOUN
ejpam-3873	614	7	in	in	ADP
ejpam-3873	614	8	a	a	DET
ejpam-3873	614	9	ring	ring	NOUN
ejpam-3873	614	10	.	.	PUNCT
ejpam-3873	615	1	international	international	ADJ
ejpam-3873	615	2	journal	journal	NOUN
ejpam-3873	615	3	of	of	ADP
ejpam-3873	615	4	algebra	algebra	PROPN
ejpam-3873	615	5	,	,	PUNCT
ejpam-3873	615	6	8(18):883–888	8(18):883–888	NUM
ejpam-3873	615	7	,	,	PUNCT
ejpam-3873	615	8	2014	2014	NUM
ejpam-3873	615	9	.	.	PUNCT
ejpam-3873	616	1	[	[	X
ejpam-3873	616	2	12	12	NUM
ejpam-3873	616	3	]	]	PUNCT
ejpam-3873	616	4	cristopher	cristopher	PROPN
ejpam-3873	616	5	john	john	PROPN
ejpam-3873	616	6	s	s	PROPN
ejpam-3873	616	7	rosero	rosero	PROPN
ejpam-3873	616	8	,	,	PUNCT
ejpam-3873	616	9	joris	joris	PROPN
ejpam-3873	616	10	n	n	PART
ejpam-3873	616	11	buloron	buloron	NOUN
ejpam-3873	616	12	,	,	PUNCT
ejpam-3873	616	13	jay	jay	PROPN
ejpam-3873	616	14	m	m	VERB
ejpam-3873	616	15	ontolan	ontolan	ADJ
ejpam-3873	616	16	,	,	PUNCT
ejpam-3873	616	17	and	and	CCONJ
ejpam-3873	616	18	michael	michael	PROPN
ejpam-3873	616	19	p	p	PROPN
ejpam-3873	616	20	baldado	baldado	PROPN
ejpam-3873	616	21	jr	jr	PROPN
ejpam-3873	616	22	.	.	PUNCT
ejpam-3873	617	1	d	d	X
ejpam-3873	617	2	-	-	PUNCT
ejpam-3873	617	3	sets	set	NOUN
ejpam-3873	617	4	of	of	ADP
ejpam-3873	617	5	finite	finite	ADJ
ejpam-3873	617	6	groups	group	NOUN
ejpam-3873	617	7	.	.	PUNCT
ejpam-3873	618	1	international	international	ADJ
ejpam-3873	618	2	journal	journal	PROPN
ejpam-3873	618	3	of	of	ADP
ejpam-3873	618	4	algebra	algebra	PROPN
ejpam-3873	618	5	,	,	PUNCT
ejpam-3873	618	6	8(13):623–628	8(13):623–628	NUM
ejpam-3873	618	7	,	,	PUNCT
ejpam-3873	618	8	2014	2014	NUM
ejpam-3873	618	9	.	.	PUNCT