id	sid	tid	token	lemma	pos
ejpam-6172	1	1	european	european	PROPN
ejpam-6172	1	2	journal	journal	PROPN
ejpam-6172	1	3	of	of	ADP
ejpam-6172	1	4	pure	pure	ADJ
ejpam-6172	1	5	and	and	CCONJ
ejpam-6172	1	6	applied	applied	ADJ
ejpam-6172	1	7	mathematics	mathematic	NOUN
ejpam-6172	1	8	2025	2025	NUM
ejpam-6172	1	9	,	,	PUNCT
ejpam-6172	1	10	vol	vol	NOUN
ejpam-6172	1	11	.	.	PROPN
ejpam-6172	1	12	18	18	NUM
ejpam-6172	1	13	,	,	PUNCT
ejpam-6172	1	14	issue	issue	NOUN
ejpam-6172	1	15	3	3	NUM
ejpam-6172	1	16	,	,	PUNCT
ejpam-6172	1	17	article	article	NOUN
ejpam-6172	1	18	number	number	NOUN
ejpam-6172	1	19	6172	6172	NUM
ejpam-6172	1	20	issn	issn	VERB
ejpam-6172	1	21	1307	1307	NUM
ejpam-6172	1	22	-	-	SYM
ejpam-6172	1	23	5543	5543	NUM
ejpam-6172	1	24	–	–	PUNCT
ejpam-6172	1	25	ejpam.com	ejpam.com	X
ejpam-6172	1	26	published	publish	VERB
ejpam-6172	1	27	by	by	ADP
ejpam-6172	1	28	new	new	PROPN
ejpam-6172	1	29	york	york	PROPN
ejpam-6172	1	30	business	business	PROPN
ejpam-6172	1	31	global	global	PROPN
ejpam-6172	1	32	finite	finite	PROPN
ejpam-6172	1	33	γ	γ	PROPN
ejpam-6172	1	34	-	-	PUNCT
ejpam-6172	1	35	ag	ag	NOUN
ejpam-6172	1	36	-	-	PUNCT
ejpam-6172	1	37	groupoids	groupoid	NOUN
ejpam-6172	1	38	with	with	ADP
ejpam-6172	1	39	left	left	ADJ
ejpam-6172	1	40	identities	identity	NOUN
ejpam-6172	1	41	and	and	CCONJ
ejpam-6172	1	42	left	leave	VERB
ejpam-6172	1	43	zeros	zero	NOUN
ejpam-6172	1	44	cholathorn	cholathorn	VERB
ejpam-6172	1	45	chanoi1	chanoi1	PROPN
ejpam-6172	1	46	,	,	PUNCT
ejpam-6172	1	47	apichaya	apichaya	NOUN
ejpam-6172	1	48	kauppamung1	kauppamung1	NOUN
ejpam-6172	1	49	,	,	PUNCT
ejpam-6172	1	50	panuwat	panuwat	VERB
ejpam-6172	1	51	luangchaisri1	luangchaisri1	NOUN
ejpam-6172	1	52	,	,	PUNCT
ejpam-6172	1	53	thawhat	thawhat	PROPN
ejpam-6172	1	54	changphas1,∗	changphas1,∗	NOUN
ejpam-6172	1	55	1department	1department	NUM
ejpam-6172	1	56	of	of	ADP
ejpam-6172	1	57	mathematics	mathematic	NOUN
ejpam-6172	1	58	,	,	PUNCT
ejpam-6172	1	59	faculty	faculty	NOUN
ejpam-6172	1	60	of	of	ADP
ejpam-6172	1	61	science	science	NOUN
ejpam-6172	1	62	,	,	PUNCT
ejpam-6172	1	63	khon	khon	PROPN
ejpam-6172	1	64	kaen	kaen	PROPN
ejpam-6172	1	65	university	university	PROPN
ejpam-6172	1	66	,	,	PUNCT
ejpam-6172	1	67	khon	khon	PROPN
ejpam-6172	1	68	kaen	kaen	PROPN
ejpam-6172	1	69	40002	40002	NUM
ejpam-6172	1	70	,	,	PUNCT
ejpam-6172	1	71	thailand	thailand	PROPN
ejpam-6172	1	72	abstract	abstract	PROPN
ejpam-6172	1	73	.	.	PUNCT
ejpam-6172	2	1	let	let	VERB
ejpam-6172	2	2	γ	γ	X
ejpam-6172	2	3	be	be	AUX
ejpam-6172	2	4	a	a	DET
ejpam-6172	2	5	nonempty	nonempty	ADV
ejpam-6172	2	6	set	set	VERB
ejpam-6172	2	7	.	.	PUNCT
ejpam-6172	3	1	a	a	DET
ejpam-6172	3	2	nonempty	nonempty	ADV
ejpam-6172	3	3	set	set	VERB
ejpam-6172	3	4	a	a	PRON
ejpam-6172	3	5	is	be	AUX
ejpam-6172	3	6	called	call	VERB
ejpam-6172	3	7	a	a	DET
ejpam-6172	3	8	γ	γ	PROPN
ejpam-6172	3	9	-	-	PUNCT
ejpam-6172	3	10	ag	ag	ADJ
ejpam-6172	3	11	-	-	PUNCT
ejpam-6172	3	12	groupoid	groupoid	NOUN
ejpam-6172	3	13	if	if	SCONJ
ejpam-6172	3	14	there	there	PRON
ejpam-6172	3	15	is	be	VERB
ejpam-6172	3	16	a	a	DET
ejpam-6172	3	17	function	function	NOUN
ejpam-6172	3	18	f	f	NOUN
ejpam-6172	3	19	from	from	ADP
ejpam-6172	3	20	a	a	DET
ejpam-6172	3	21	×	×	NOUN
ejpam-6172	3	22	γ	γ	X
ejpam-6172	3	23	×	×	NOUN
ejpam-6172	3	24	a	a	PRON
ejpam-6172	3	25	into	into	ADP
ejpam-6172	3	26	a	a	DET
ejpam-6172	3	27	,	,	PUNCT
ejpam-6172	3	28	customary	customary	ADJ
ejpam-6172	3	29	denoted	denote	VERB
ejpam-6172	3	30	aγb	aγb	NOUN
ejpam-6172	3	31	for	for	ADP
ejpam-6172	3	32	f(a	f(a	PROPN
ejpam-6172	3	33	,	,	PUNCT
ejpam-6172	3	34	γ	γ	X
ejpam-6172	3	35	,	,	PUNCT
ejpam-6172	3	36	b	b	NOUN
ejpam-6172	3	37	)	)	PUNCT
ejpam-6172	3	38	,	,	PUNCT
ejpam-6172	3	39	satisfying	satisfy	VERB
ejpam-6172	3	40	the	the	DET
ejpam-6172	3	41	identity	identity	NOUN
ejpam-6172	3	42	(	(	PUNCT
ejpam-6172	3	43	aγb)βc	aγb)βc	NOUN
ejpam-6172	3	44	=	=	PUNCT
ejpam-6172	3	45	(	(	PUNCT
ejpam-6172	3	46	cγb)βa	cγb)βa	VERB
ejpam-6172	3	47	for	for	ADP
ejpam-6172	3	48	any	any	DET
ejpam-6172	3	49	a	a	DET
ejpam-6172	3	50	,	,	PUNCT
ejpam-6172	3	51	b	b	NOUN
ejpam-6172	3	52	,	,	PUNCT
ejpam-6172	3	53	c	c	PROPN
ejpam-6172	3	54	∈	∈	PROPN
ejpam-6172	3	55	a	a	PRON
ejpam-6172	3	56	and	and	CCONJ
ejpam-6172	3	57	γ	γ	NOUN
ejpam-6172	3	58	,	,	PUNCT
ejpam-6172	3	59	β	β	PROPN
ejpam-6172	3	60	∈	∈	PROPN
ejpam-6172	3	61	γ	γ	X
ejpam-6172	3	62	.	.	PROPN
ejpam-6172	3	63	for	for	ADP
ejpam-6172	3	64	each	each	DET
ejpam-6172	3	65	γ	γ	PROPN
ejpam-6172	3	66	∈	∈	PROPN
ejpam-6172	3	67	γ	γ	X
ejpam-6172	3	68	,	,	PUNCT
ejpam-6172	3	69	an	an	DET
ejpam-6172	3	70	operation	operation	NOUN
ejpam-6172	3	71	on	on	ADP
ejpam-6172	3	72	a	a	DET
ejpam-6172	3	73	associated	associate	VERB
ejpam-6172	3	74	to	to	ADP
ejpam-6172	3	75	γ	γ	PROPN
ejpam-6172	3	76	is	be	AUX
ejpam-6172	3	77	given	give	VERB
ejpam-6172	3	78	by	by	ADP
ejpam-6172	3	79	ab	ab	PROPN
ejpam-6172	3	80	=	=	PUNCT
ejpam-6172	3	81	aγb	aγb	NOUN
ejpam-6172	3	82	.	.	PUNCT
ejpam-6172	3	83	suppose	suppose	VERB
ejpam-6172	3	84	further	far	ADV
ejpam-6172	3	85	that	that	SCONJ
ejpam-6172	3	86	a	a	PRON
ejpam-6172	3	87	is	be	AUX
ejpam-6172	3	88	finite	finite	ADJ
ejpam-6172	3	89	,	,	PUNCT
ejpam-6172	3	90	contains	contain	VERB
ejpam-6172	3	91	a	a	DET
ejpam-6172	3	92	left	left	ADJ
ejpam-6172	3	93	identity	identity	NOUN
ejpam-6172	3	94	and	and	CCONJ
ejpam-6172	3	95	a	a	DET
ejpam-6172	3	96	left	left	ADJ
ejpam-6172	3	97	zero	zero	NUM
ejpam-6172	3	98	a0	a0	NOUN
ejpam-6172	3	99	.	.	PUNCT
ejpam-6172	4	1	the	the	DET
ejpam-6172	4	2	objective	objective	NOUN
ejpam-6172	4	3	of	of	ADP
ejpam-6172	4	4	this	this	DET
ejpam-6172	4	5	paper	paper	NOUN
ejpam-6172	4	6	is	be	AUX
ejpam-6172	4	7	to	to	PART
ejpam-6172	4	8	provide	provide	VERB
ejpam-6172	4	9	sufficient	sufficient	ADJ
ejpam-6172	4	10	conditions	condition	NOUN
ejpam-6172	4	11	under	under	ADP
ejpam-6172	4	12	which	which	PRON
ejpam-6172	4	13	the	the	DET
ejpam-6172	4	14	set	set	NOUN
ejpam-6172	4	15	a	a	DET
ejpam-6172	4	16	\	\	PROPN
ejpam-6172	4	17	{	{	PUNCT
ejpam-6172	4	18	a0	a0	PROPN
ejpam-6172	4	19	}	}	PUNCT
ejpam-6172	4	20	is	be	AUX
ejpam-6172	4	21	a	a	DET
ejpam-6172	4	22	commutative	commutative	ADJ
ejpam-6172	4	23	group	group	NOUN
ejpam-6172	4	24	under	under	ADP
ejpam-6172	4	25	the	the	DET
ejpam-6172	4	26	operation	operation	NOUN
ejpam-6172	4	27	on	on	ADP
ejpam-6172	4	28	a	a	DET
ejpam-6172	4	29	determined	determine	VERB
ejpam-6172	4	30	by	by	ADP
ejpam-6172	4	31	γ	γ	NOUN
ejpam-6172	4	32	for	for	ADP
ejpam-6172	4	33	all	all	DET
ejpam-6172	4	34	γ	γ	PROPN
ejpam-6172	4	35	∈	∈	PROPN
ejpam-6172	4	36	γ	γ	X
ejpam-6172	4	37	.	.	PROPN
ejpam-6172	4	38	2020	2020	NUM
ejpam-6172	4	39	mathematics	mathematic	NOUN
ejpam-6172	4	40	subject	subject	NOUN
ejpam-6172	4	41	classifications	classification	NOUN
ejpam-6172	4	42	:	:	PUNCT
ejpam-6172	4	43	20n02	20n02	NUM
ejpam-6172	4	44	key	key	ADJ
ejpam-6172	4	45	words	word	NOUN
ejpam-6172	4	46	and	and	CCONJ
ejpam-6172	4	47	phrases	phrase	NOUN
ejpam-6172	4	48	:	:	PUNCT
ejpam-6172	4	49	ag	ag	PROPN
ejpam-6172	4	50	-	-	PROPN
ejpam-6172	4	51	groupoid	groupoid	PROPN
ejpam-6172	4	52	,	,	PUNCT
ejpam-6172	4	53	γ	γ	PROPN
ejpam-6172	4	54	-	-	PUNCT
ejpam-6172	4	55	ag	ag	ADJ
ejpam-6172	4	56	-	-	PUNCT
ejpam-6172	4	57	groupoid	groupoid	PROPN
ejpam-6172	4	58	,	,	PUNCT
ejpam-6172	4	59	semigroup	semigroup	PROPN
ejpam-6172	4	60	,	,	PUNCT
ejpam-6172	4	61	γ	γ	PROPN
ejpam-6172	4	62	-	-	PUNCT
ejpam-6172	4	63	semigroup	semigroup	NOUN
ejpam-6172	4	64	,	,	PUNCT
ejpam-6172	4	65	group	group	NOUN
ejpam-6172	4	66	.	.	PUNCT
ejpam-6172	5	1	1	1	X
ejpam-6172	5	2	.	.	X
ejpam-6172	5	3	introduction	introduction	NOUN
ejpam-6172	5	4	an	an	DET
ejpam-6172	5	5	abel	abel	PROPN
ejpam-6172	5	6	-	-	PUNCT
ejpam-6172	5	7	grassmann	grassmann	PROPN
ejpam-6172	5	8	’s	’s	PART
ejpam-6172	5	9	groupoid	groupoid	NOUN
ejpam-6172	5	10	,	,	PUNCT
ejpam-6172	5	11	abreviated	abreviate	VERB
ejpam-6172	5	12	by	by	ADP
ejpam-6172	5	13	ag	ag	PROPN
ejpam-6172	5	14	-	-	PROPN
ejpam-6172	5	15	groupoid	groupoid	PROPN
ejpam-6172	5	16	,	,	PUNCT
ejpam-6172	5	17	is	be	AUX
ejpam-6172	5	18	a	a	DET
ejpam-6172	5	19	groupoid	groupoid	NOUN
ejpam-6172	5	20	a	a	PRON
ejpam-6172	5	21	such	such	ADJ
ejpam-6172	5	22	that	that	SCONJ
ejpam-6172	5	23	the	the	DET
ejpam-6172	5	24	identity	identity	NOUN
ejpam-6172	5	25	(	(	PUNCT
ejpam-6172	5	26	ab)c	ab)c	PROPN
ejpam-6172	5	27	=	=	SYM
ejpam-6172	5	28	(	(	PUNCT
ejpam-6172	5	29	cb)a	cb)a	PROPN
ejpam-6172	5	30	holds	hold	VERB
ejpam-6172	5	31	for	for	ADP
ejpam-6172	5	32	any	any	DET
ejpam-6172	5	33	a	a	DET
ejpam-6172	5	34	,	,	PUNCT
ejpam-6172	5	35	b	b	NOUN
ejpam-6172	5	36	,	,	PUNCT
ejpam-6172	5	37	c	c	PROPN
ejpam-6172	5	38	∈	∈	PROPN
ejpam-6172	5	39	a.	a.	NOUN
ejpam-6172	5	40	an	an	DET
ejpam-6172	5	41	ag	ag	PROPN
ejpam-6172	5	42	-	-	PUNCT
ejpam-6172	5	43	groupoid	groupoid	PROPN
ejpam-6172	5	44	is	be	AUX
ejpam-6172	5	45	also	also	ADV
ejpam-6172	5	46	called	call	VERB
ejpam-6172	5	47	a	a	DET
ejpam-6172	5	48	left	left	NOUN
ejpam-6172	5	49	almost	almost	ADV
ejpam-6172	5	50	semigroup	semigroup	ADJ
ejpam-6172	5	51	(	(	PUNCT
ejpam-6172	5	52	it	it	PRON
ejpam-6172	5	53	is	be	AUX
ejpam-6172	5	54	abreviated	abreviate	VERB
ejpam-6172	5	55	by	by	ADP
ejpam-6172	5	56	la	la	PROPN
ejpam-6172	5	57	-	-	PUNCT
ejpam-6172	5	58	semigroup	semigroup	NOUN
ejpam-6172	5	59	)	)	PUNCT
ejpam-6172	5	60	,	,	PUNCT
ejpam-6172	5	61	a	a	DET
ejpam-6172	5	62	left	left	ADJ
ejpam-6172	5	63	invertive	invertive	ADJ
ejpam-6172	5	64	groupoid	groupoid	NOUN
ejpam-6172	5	65	,	,	PUNCT
ejpam-6172	5	66	or	or	CCONJ
ejpam-6172	5	67	a	a	DET
ejpam-6172	5	68	right	right	ADJ
ejpam-6172	5	69	modular	modular	ADJ
ejpam-6172	5	70	groupoid	groupoid	NOUN
ejpam-6172	5	71	(	(	PUNCT
ejpam-6172	5	72	cf	cf	NOUN
ejpam-6172	5	73	.	.	PUNCT
ejpam-6172	6	1	[	[	X
ejpam-6172	6	2	1	1	NUM
ejpam-6172	6	3	]	]	PUNCT
ejpam-6172	6	4	,	,	PUNCT
ejpam-6172	6	5	[	[	X
ejpam-6172	6	6	2	2	NUM
ejpam-6172	6	7	]	]	PUNCT
ejpam-6172	6	8	,	,	PUNCT
ejpam-6172	6	9	[	[	X
ejpam-6172	6	10	3	3	NUM
ejpam-6172	6	11	]	]	PUNCT
ejpam-6172	6	12	,	,	PUNCT
ejpam-6172	6	13	[	[	X
ejpam-6172	6	14	4	4	NUM
ejpam-6172	6	15	]	]	NUM
ejpam-6172	6	16	)	)	PUNCT
ejpam-6172	6	17	.	.	PUNCT
ejpam-6172	7	1	an	an	DET
ejpam-6172	7	2	ag	ag	PROPN
ejpam-6172	7	3	-	-	PUNCT
ejpam-6172	7	4	groupoid	groupoid	PROPN
ejpam-6172	7	5	is	be	AUX
ejpam-6172	7	6	closely	closely	ADV
ejpam-6172	7	7	related	relate	VERB
ejpam-6172	7	8	to	to	ADP
ejpam-6172	7	9	a	a	DET
ejpam-6172	7	10	commutative	commutative	ADJ
ejpam-6172	7	11	semigroup	semigroup	NOUN
ejpam-6172	7	12	,	,	PUNCT
ejpam-6172	7	13	because	because	SCONJ
ejpam-6172	7	14	if	if	SCONJ
ejpam-6172	7	15	an	an	DET
ejpam-6172	7	16	aggroupoid	aggroupoid	NOUN
ejpam-6172	7	17	contains	contain	VERB
ejpam-6172	7	18	a	a	DET
ejpam-6172	7	19	right	right	ADJ
ejpam-6172	7	20	identity	identity	NOUN
ejpam-6172	7	21	,	,	PUNCT
ejpam-6172	7	22	then	then	ADV
ejpam-6172	7	23	it	it	PRON
ejpam-6172	7	24	becomes	become	VERB
ejpam-6172	7	25	a	a	DET
ejpam-6172	7	26	commutative	commutative	ADJ
ejpam-6172	7	27	monoid	monoid	NOUN
ejpam-6172	7	28	.	.	PUNCT
ejpam-6172	8	1	moreover	moreover	ADV
ejpam-6172	8	2	,	,	PUNCT
ejpam-6172	8	3	if	if	SCONJ
ejpam-6172	8	4	an	an	DET
ejpam-6172	8	5	ag	ag	PROPN
ejpam-6172	8	6	-	-	PROPN
ejpam-6172	8	7	groupoid	groupoid	PROPN
ejpam-6172	8	8	a	a	PRON
ejpam-6172	8	9	with	with	ADP
ejpam-6172	8	10	a	a	DET
ejpam-6172	8	11	left	left	ADJ
ejpam-6172	8	12	identity	identity	NOUN
ejpam-6172	8	13	and	and	CCONJ
ejpam-6172	8	14	a	a	DET
ejpam-6172	8	15	left	left	ADJ
ejpam-6172	8	16	zero	zero	NUM
ejpam-6172	8	17	a0	a0	PROPN
ejpam-6172	8	18	is	be	AUX
ejpam-6172	8	19	finite	finite	ADJ
ejpam-6172	8	20	,	,	PUNCT
ejpam-6172	8	21	then	then	ADV
ejpam-6172	8	22	(	(	PUNCT
ejpam-6172	8	23	under	under	ADP
ejpam-6172	8	24	certain	certain	ADJ
ejpam-6172	8	25	conditions	condition	NOUN
ejpam-6172	8	26	)	)	PUNCT
ejpam-6172	8	27	a	a	DET
ejpam-6172	8	28	\	\	PROPN
ejpam-6172	8	29	{	{	PUNCT
ejpam-6172	8	30	a0	a0	PROPN
ejpam-6172	8	31	}	}	PUNCT
ejpam-6172	8	32	is	be	AUX
ejpam-6172	8	33	a	a	DET
ejpam-6172	8	34	commutative	commutative	ADJ
ejpam-6172	8	35	group	group	NOUN
ejpam-6172	8	36	(	(	PUNCT
ejpam-6172	8	37	cf	cf	NOUN
ejpam-6172	8	38	.	.	PUNCT
ejpam-6172	9	1	[	[	X
ejpam-6172	9	2	5	5	NUM
ejpam-6172	9	3	]	]	PUNCT
ejpam-6172	9	4	theorem	theorem	VERB
ejpam-6172	9	5	2.2	2.2	NUM
ejpam-6172	9	6	)	)	PUNCT
ejpam-6172	9	7	.	.	PUNCT
ejpam-6172	10	1	the	the	DET
ejpam-6172	10	2	purpose	purpose	NOUN
ejpam-6172	10	3	of	of	ADP
ejpam-6172	10	4	this	this	DET
ejpam-6172	10	5	paper	paper	NOUN
ejpam-6172	10	6	is	be	AUX
ejpam-6172	10	7	to	to	PART
ejpam-6172	10	8	extend	extend	VERB
ejpam-6172	10	9	this	this	DET
ejpam-6172	10	10	result	result	NOUN
ejpam-6172	10	11	to	to	ADP
ejpam-6172	10	12	γ	γ	PROPN
ejpam-6172	10	13	-	-	PUNCT
ejpam-6172	10	14	ag	ag	ADJ
ejpam-6172	10	15	-	-	PUNCT
ejpam-6172	10	16	groupoids	groupoid	NOUN
ejpam-6172	10	17	.	.	PUNCT
ejpam-6172	11	1	∗corresponding	∗corresponde	VERB
ejpam-6172	11	2	author	author	NOUN
ejpam-6172	11	3	.	.	PUNCT
ejpam-6172	12	1	doi	doi	NOUN
ejpam-6172	12	2	:	:	PUNCT
ejpam-6172	12	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6172	https://doi.org/10.29020/nybg.ejpam.v18i3.6172	NOUN
ejpam-6172	12	4	email	email	NOUN
ejpam-6172	12	5	addresses	address	NOUN
ejpam-6172	12	6	:	:	PUNCT
ejpam-6172	12	7	thacha@kku.ac.th	thacha@kku.ac.th	NOUN
ejpam-6172	12	8	(	(	PUNCT
ejpam-6172	12	9	t.	t.	NOUN
ejpam-6172	12	10	changphas	changphas	PROPN
ejpam-6172	12	11	)	)	PUNCT
ejpam-6172	12	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6172	13	1	1	1	NUM
ejpam-6172	13	2	copyright	copyright	NOUN
ejpam-6172	13	3	:	:	PUNCT
ejpam-6172	13	4	©	©	PROPN
ejpam-6172	13	5	2025	2025	NUM
ejpam-6172	13	6	the	the	DET
ejpam-6172	13	7	author(s	author(s	NOUN
ejpam-6172	13	8	)	)	PUNCT
ejpam-6172	13	9	.	.	PUNCT
ejpam-6172	14	1	(	(	PUNCT
ejpam-6172	14	2	cc	cc	NOUN
ejpam-6172	14	3	by	by	ADP
ejpam-6172	14	4	-	-	PUNCT
ejpam-6172	14	5	nc	nc	PROPN
ejpam-6172	14	6	4.0	4.0	NUM
ejpam-6172	14	7	)	)	PUNCT
ejpam-6172	14	8	c.	c.	NOUN
ejpam-6172	14	9	chanoi	chanoi	PROPN
ejpam-6172	14	10	et	et	PROPN
ejpam-6172	14	11	al	al	PROPN
ejpam-6172	14	12	.	.	PUNCT
ejpam-6172	14	13	/	/	SYM
ejpam-6172	14	14	eur	eur	PROPN
ejpam-6172	14	15	.	.	PUNCT
ejpam-6172	15	1	j.	j.	PROPN
ejpam-6172	15	2	pure	pure	PROPN
ejpam-6172	15	3	appl	appl	PROPN
ejpam-6172	15	4	.	.	PROPN
ejpam-6172	15	5	math	math	PROPN
ejpam-6172	15	6	,	,	PUNCT
ejpam-6172	15	7	18	18	NUM
ejpam-6172	15	8	(	(	PUNCT
ejpam-6172	15	9	3	3	NUM
ejpam-6172	15	10	)	)	PUNCT
ejpam-6172	15	11	(	(	PUNCT
ejpam-6172	15	12	2025	2025	NUM
ejpam-6172	15	13	)	)	PUNCT
ejpam-6172	15	14	,	,	PUNCT
ejpam-6172	15	15	6172	6172	NUM
ejpam-6172	15	16	2	2	NUM
ejpam-6172	15	17	of	of	ADP
ejpam-6172	15	18	9	9	NUM
ejpam-6172	15	19	2	2	NUM
ejpam-6172	15	20	.	.	PUNCT
ejpam-6172	16	1	preliminaries	preliminary	NOUN
ejpam-6172	16	2	let	let	VERB
ejpam-6172	16	3	γ	γ	NOUN
ejpam-6172	16	4	be	be	AUX
ejpam-6172	16	5	a	a	DET
ejpam-6172	16	6	nonempty	nonempty	ADV
ejpam-6172	16	7	set	set	VERB
ejpam-6172	16	8	.	.	PUNCT
ejpam-6172	17	1	a	a	DET
ejpam-6172	17	2	nonempty	nonempty	ADV
ejpam-6172	17	3	set	set	VERB
ejpam-6172	17	4	a	a	PRON
ejpam-6172	17	5	is	be	AUX
ejpam-6172	17	6	called	call	VERB
ejpam-6172	17	7	a	a	DET
ejpam-6172	17	8	γ	γ	NOUN
ejpam-6172	17	9	-	-	ADJ
ejpam-6172	17	10	groupoid	groupoid	NOUN
ejpam-6172	17	11	if	if	SCONJ
ejpam-6172	17	12	there	there	PRON
ejpam-6172	17	13	is	be	VERB
ejpam-6172	17	14	a	a	DET
ejpam-6172	17	15	function	function	NOUN
ejpam-6172	17	16	f	f	NOUN
ejpam-6172	17	17	of	of	ADP
ejpam-6172	17	18	a×	a×	PROPN
ejpam-6172	17	19	γ×	γ×	PROPN
ejpam-6172	17	20	a	a	PRON
ejpam-6172	17	21	into	into	ADP
ejpam-6172	17	22	a	a	PRON
ejpam-6172	17	23	,	,	PUNCT
ejpam-6172	17	24	we	we	PRON
ejpam-6172	17	25	often	often	ADV
ejpam-6172	17	26	write	write	VERB
ejpam-6172	17	27	aγb	aγb	NOUN
ejpam-6172	17	28	for	for	ADP
ejpam-6172	17	29	f(a	f(a	PROPN
ejpam-6172	17	30	,	,	PUNCT
ejpam-6172	17	31	γ	γ	X
ejpam-6172	17	32	,	,	PUNCT
ejpam-6172	17	33	b	b	NOUN
ejpam-6172	17	34	)	)	PUNCT
ejpam-6172	17	35	.	.	PUNCT
ejpam-6172	18	1	suppose	suppose	VERB
ejpam-6172	18	2	a	a	PRON
ejpam-6172	18	3	is	be	AUX
ejpam-6172	18	4	a	a	DET
ejpam-6172	18	5	γ	γ	NOUN
ejpam-6172	18	6	-	-	NOUN
ejpam-6172	18	7	groupoid	groupoid	NOUN
ejpam-6172	18	8	.	.	PUNCT
ejpam-6172	19	1	for	for	ADP
ejpam-6172	19	2	each	each	DET
ejpam-6172	19	3	γ	γ	PROPN
ejpam-6172	19	4	∈	∈	PROPN
ejpam-6172	19	5	γ	γ	X
ejpam-6172	19	6	,	,	PUNCT
ejpam-6172	19	7	an	an	DET
ejpam-6172	19	8	operation	operation	NOUN
ejpam-6172	19	9	on	on	ADP
ejpam-6172	19	10	a	a	DET
ejpam-6172	19	11	determined	determine	VERB
ejpam-6172	19	12	by	by	ADP
ejpam-6172	19	13	γ	γ	NOUN
ejpam-6172	19	14	is	be	AUX
ejpam-6172	19	15	defined	define	VERB
ejpam-6172	19	16	by	by	ADP
ejpam-6172	19	17	for	for	ADP
ejpam-6172	19	18	any	any	DET
ejpam-6172	19	19	a	a	NOUN
ejpam-6172	19	20	,	,	PUNCT
ejpam-6172	19	21	b	b	PROPN
ejpam-6172	19	22	∈	∈	PROPN
ejpam-6172	19	23	a	a	X
ejpam-6172	20	1	,	,	PUNCT
ejpam-6172	20	2	ab	ab	PROPN
ejpam-6172	20	3	=	=	PUNCT
ejpam-6172	20	4	aγb	aγb	NOUN
ejpam-6172	20	5	.	.	PUNCT
ejpam-6172	21	1	a	a	DET
ejpam-6172	21	2	γ	γ	X
ejpam-6172	21	3	-	-	PUNCT
ejpam-6172	21	4	groupoid	groupoid	NOUN
ejpam-6172	21	5	a	a	PRON
ejpam-6172	21	6	is	be	AUX
ejpam-6172	21	7	called	call	VERB
ejpam-6172	21	8	a	a	DET
ejpam-6172	21	9	γ	γ	NOUN
ejpam-6172	21	10	-	-	PUNCT
ejpam-6172	21	11	semigroup	semigroup	NOUN
ejpam-6172	21	12	if	if	SCONJ
ejpam-6172	21	13	the	the	DET
ejpam-6172	21	14	identity	identity	NOUN
ejpam-6172	21	15	(	(	PUNCT
ejpam-6172	21	16	aγb)βc	aγb)βc	NOUN
ejpam-6172	21	17	=	=	SYM
ejpam-6172	21	18	aγ(bβc	aγ(bβc	PROPN
ejpam-6172	21	19	)	)	PUNCT
ejpam-6172	21	20	holds	hold	VERB
ejpam-6172	21	21	for	for	ADP
ejpam-6172	21	22	all	all	DET
ejpam-6172	21	23	a	a	DET
ejpam-6172	21	24	,	,	PUNCT
ejpam-6172	21	25	b	b	NOUN
ejpam-6172	21	26	,	,	PUNCT
ejpam-6172	21	27	c	c	PROPN
ejpam-6172	21	28	∈	∈	PROPN
ejpam-6172	21	29	a	a	PRON
ejpam-6172	21	30	and	and	CCONJ
ejpam-6172	21	31	γ	γ	NOUN
ejpam-6172	21	32	,	,	PUNCT
ejpam-6172	21	33	β	β	PROPN
ejpam-6172	21	34	∈	∈	PROPN
ejpam-6172	21	35	γ	γ	X
ejpam-6172	21	36	.	.	PUNCT
ejpam-6172	22	1	a	a	DET
ejpam-6172	22	2	γ	γ	PROPN
ejpam-6172	22	3	-	-	PUNCT
ejpam-6172	22	4	semigroup	semigroup	NOUN
ejpam-6172	22	5	a	a	PRON
ejpam-6172	22	6	is	be	AUX
ejpam-6172	22	7	said	say	VERB
ejpam-6172	22	8	to	to	PART
ejpam-6172	22	9	be	be	AUX
ejpam-6172	22	10	commutative	commutative	ADJ
ejpam-6172	22	11	if	if	SCONJ
ejpam-6172	22	12	aγb	aγb	ADV
ejpam-6172	22	13	=	=	PUNCT
ejpam-6172	22	14	bγa	bγa	ADJ
ejpam-6172	22	15	for	for	ADP
ejpam-6172	22	16	all	all	DET
ejpam-6172	22	17	a	a	PRON
ejpam-6172	22	18	,	,	PUNCT
ejpam-6172	22	19	b	b	X
ejpam-6172	22	20	∈	∈	PROPN
ejpam-6172	22	21	a	a	PRON
ejpam-6172	22	22	and	and	CCONJ
ejpam-6172	22	23	γ	γ	PROPN
ejpam-6172	22	24	∈	∈	PROPN
ejpam-6172	22	25	γ	γ	X
ejpam-6172	22	26	.	.	PUNCT
ejpam-6172	23	1	the	the	DET
ejpam-6172	23	2	notion	notion	NOUN
ejpam-6172	23	3	of	of	ADP
ejpam-6172	23	4	γ	γ	PROPN
ejpam-6172	23	5	-	-	PUNCT
ejpam-6172	23	6	semigroup	semigroup	PROPN
ejpam-6172	23	7	was	be	AUX
ejpam-6172	23	8	introduced	introduce	VERB
ejpam-6172	23	9	and	and	CCONJ
ejpam-6172	23	10	studied	study	VERB
ejpam-6172	23	11	by	by	ADP
ejpam-6172	23	12	m.	m.	PROPN
ejpam-6172	23	13	k.	k.	PROPN
ejpam-6172	23	14	sen	sen	PROPN
ejpam-6172	23	15	(	(	PUNCT
ejpam-6172	23	16	cf	cf	NOUN
ejpam-6172	23	17	.	.	PUNCT
ejpam-6172	24	1	[	[	X
ejpam-6172	24	2	6	6	NUM
ejpam-6172	24	3	]	]	PUNCT
ejpam-6172	24	4	,	,	PUNCT
ejpam-6172	24	5	[	[	X
ejpam-6172	24	6	7	7	NUM
ejpam-6172	24	7	]	]	PUNCT
ejpam-6172	24	8	)	)	PUNCT
ejpam-6172	24	9	.	.	PUNCT
ejpam-6172	25	1	suppose	suppose	VERB
ejpam-6172	25	2	a	a	PRON
ejpam-6172	25	3	is	be	AUX
ejpam-6172	25	4	a	a	DET
ejpam-6172	25	5	semigroup	semigroup	NOUN
ejpam-6172	25	6	.	.	PUNCT
ejpam-6172	26	1	for	for	ADP
ejpam-6172	26	2	a	a	DET
ejpam-6172	26	3	nonempty	nonempty	ADJ
ejpam-6172	26	4	set	set	VERB
ejpam-6172	26	5	γ	γ	NOUN
ejpam-6172	26	6	,	,	PUNCT
ejpam-6172	26	7	define	define	VERB
ejpam-6172	26	8	aγb	aγb	NOUN
ejpam-6172	26	9	=	=	SYM
ejpam-6172	26	10	ab	ab	PROPN
ejpam-6172	26	11	for	for	ADP
ejpam-6172	26	12	any	any	DET
ejpam-6172	26	13	a	a	PRON
ejpam-6172	26	14	,	,	PUNCT
ejpam-6172	26	15	b	b	X
ejpam-6172	26	16	∈	∈	PROPN
ejpam-6172	26	17	a	a	PRON
ejpam-6172	26	18	and	and	CCONJ
ejpam-6172	26	19	γ	γ	PROPN
ejpam-6172	26	20	∈	∈	PROPN
ejpam-6172	26	21	γ	γ	X
ejpam-6172	26	22	;	;	PUNCT
ejpam-6172	26	23	then	then	ADV
ejpam-6172	26	24	a	a	PRON
ejpam-6172	26	25	is	be	AUX
ejpam-6172	26	26	a	a	DET
ejpam-6172	26	27	γ	γ	NOUN
ejpam-6172	26	28	-	-	PUNCT
ejpam-6172	26	29	semigroup	semigroup	NOUN
ejpam-6172	26	30	.	.	PUNCT
ejpam-6172	27	1	if	if	SCONJ
ejpam-6172	27	2	a	a	PRON
ejpam-6172	27	3	is	be	AUX
ejpam-6172	27	4	a	a	DET
ejpam-6172	27	5	γ	γ	NOUN
ejpam-6172	27	6	-	-	PUNCT
ejpam-6172	27	7	semigroup	semigroup	NOUN
ejpam-6172	27	8	,	,	PUNCT
ejpam-6172	27	9	then	then	ADV
ejpam-6172	27	10	for	for	ADP
ejpam-6172	27	11	any	any	DET
ejpam-6172	27	12	γ	γ	PROPN
ejpam-6172	27	13	∈	∈	PROPN
ejpam-6172	27	14	γ	γ	X
ejpam-6172	27	15	,	,	PUNCT
ejpam-6172	27	16	a	a	PRON
ejpam-6172	27	17	is	be	AUX
ejpam-6172	27	18	a	a	DET
ejpam-6172	27	19	semigroup	semigroup	NOUN
ejpam-6172	27	20	under	under	ADP
ejpam-6172	27	21	the	the	DET
ejpam-6172	27	22	operation	operation	NOUN
ejpam-6172	27	23	determined	determine	VERB
ejpam-6172	27	24	by	by	ADP
ejpam-6172	27	25	γ	γ	PROPN
ejpam-6172	27	26	.	.	PROPN
ejpam-6172	27	27	example	example	NOUN
ejpam-6172	28	1	1	1	NUM
ejpam-6172	28	2	.	.	PUNCT
ejpam-6172	29	1	let	let	VERB
ejpam-6172	29	2	a	a	PRON
ejpam-6172	29	3	and	and	CCONJ
ejpam-6172	29	4	γ	γ	NOUN
ejpam-6172	29	5	be	be	AUX
ejpam-6172	29	6	the	the	DET
ejpam-6172	29	7	set	set	NOUN
ejpam-6172	29	8	of	of	ADP
ejpam-6172	29	9	all	all	DET
ejpam-6172	29	10	nonpositive	nonpositive	ADJ
ejpam-6172	29	11	integers	integer	NOUN
ejpam-6172	29	12	and	and	CCONJ
ejpam-6172	29	13	the	the	DET
ejpam-6172	29	14	set	set	NOUN
ejpam-6172	29	15	of	of	ADP
ejpam-6172	29	16	all	all	DET
ejpam-6172	29	17	nonpositive	nonpositive	ADJ
ejpam-6172	29	18	even	even	ADV
ejpam-6172	29	19	integers	integer	NOUN
ejpam-6172	29	20	,	,	PUNCT
ejpam-6172	29	21	respectively	respectively	ADV
ejpam-6172	29	22	.	.	PUNCT
ejpam-6172	30	1	for	for	ADP
ejpam-6172	30	2	a	a	DET
ejpam-6172	30	3	,	,	PUNCT
ejpam-6172	30	4	b	b	PROPN
ejpam-6172	30	5	∈	∈	PROPN
ejpam-6172	30	6	a	a	PRON
ejpam-6172	30	7	and	and	CCONJ
ejpam-6172	30	8	γ	γ	PROPN
ejpam-6172	30	9	∈	∈	PROPN
ejpam-6172	30	10	γ	γ	X
ejpam-6172	30	11	,	,	PUNCT
ejpam-6172	30	12	define	define	VERB
ejpam-6172	30	13	aγb	aγb	NOUN
ejpam-6172	30	14	to	to	PART
ejpam-6172	30	15	be	be	AUX
ejpam-6172	30	16	the	the	DET
ejpam-6172	30	17	usual	usual	ADJ
ejpam-6172	30	18	multiplication	multiplication	NOUN
ejpam-6172	30	19	of	of	ADP
ejpam-6172	30	20	integers	integer	NOUN
ejpam-6172	30	21	;	;	PUNCT
ejpam-6172	30	22	then	then	ADV
ejpam-6172	30	23	a	a	PRON
ejpam-6172	30	24	is	be	AUX
ejpam-6172	30	25	a	a	DET
ejpam-6172	30	26	γ	γ	NOUN
ejpam-6172	30	27	-	-	PUNCT
ejpam-6172	30	28	semigroup	semigroup	NOUN
ejpam-6172	30	29	.	.	PUNCT
ejpam-6172	30	30	example	example	NOUN
ejpam-6172	31	1	2	2	NUM
ejpam-6172	31	2	.	.	PUNCT
ejpam-6172	31	3	let	let	AUX
ejpam-6172	31	4	mat2×3(q	mat2×3(q	NOUN
ejpam-6172	31	5	)	)	PUNCT
ejpam-6172	31	6	denote	denote	VERB
ejpam-6172	31	7	the	the	DET
ejpam-6172	31	8	set	set	NOUN
ejpam-6172	31	9	of	of	ADP
ejpam-6172	31	10	all	all	DET
ejpam-6172	31	11	2×3	2×3	NOUN
ejpam-6172	31	12	matrices	matrix	NOUN
ejpam-6172	31	13	over	over	ADP
ejpam-6172	31	14	q	q	NOUN
ejpam-6172	31	15	,	,	PUNCT
ejpam-6172	31	16	the	the	DET
ejpam-6172	31	17	set	set	NOUN
ejpam-6172	31	18	of	of	ADP
ejpam-6172	31	19	rational	rational	ADJ
ejpam-6172	31	20	numbers	number	NOUN
ejpam-6172	31	21	.	.	PUNCT
ejpam-6172	32	1	and	and	CCONJ
ejpam-6172	32	2	,	,	PUNCT
ejpam-6172	32	3	let	let	VERB
ejpam-6172	32	4	γ	γ	PRON
ejpam-6172	32	5	denote	denote	VERB
ejpam-6172	32	6	the	the	DET
ejpam-6172	32	7	set	set	NOUN
ejpam-6172	32	8	of	of	ADP
ejpam-6172	32	9	all	all	DET
ejpam-6172	32	10	3×	3×	NUM
ejpam-6172	32	11	2	2	NUM
ejpam-6172	32	12	matrices	matrix	NOUN
ejpam-6172	32	13	over	over	ADP
ejpam-6172	32	14	q.	q.	NOUN
ejpam-6172	32	15	for	for	ADP
ejpam-6172	32	16	a	a	DET
ejpam-6172	32	17	,	,	PUNCT
ejpam-6172	32	18	b	b	NOUN
ejpam-6172	32	19	∈	∈	PROPN
ejpam-6172	32	20	mat2×3(q	mat2×3(q	PROPN
ejpam-6172	32	21	)	)	PUNCT
ejpam-6172	32	22	and	and	CCONJ
ejpam-6172	32	23	γ	γ	PROPN
ejpam-6172	32	24	∈	∈	PROPN
ejpam-6172	32	25	γ	γ	X
ejpam-6172	32	26	,	,	PUNCT
ejpam-6172	32	27	define	define	VERB
ejpam-6172	32	28	aγb	aγb	NOUN
ejpam-6172	32	29	to	to	PART
ejpam-6172	32	30	be	be	AUX
ejpam-6172	32	31	the	the	DET
ejpam-6172	32	32	usual	usual	ADJ
ejpam-6172	32	33	matrix	matrix	NOUN
ejpam-6172	32	34	product	product	NOUN
ejpam-6172	32	35	.	.	PUNCT
ejpam-6172	33	1	then	then	ADV
ejpam-6172	33	2	mat2×3(q	mat2×3(q	ADJ
ejpam-6172	33	3	)	)	PUNCT
ejpam-6172	33	4	is	be	AUX
ejpam-6172	33	5	a	a	DET
ejpam-6172	33	6	γ	γ	NOUN
ejpam-6172	33	7	-	-	PUNCT
ejpam-6172	33	8	semigroup	semigroup	NOUN
ejpam-6172	33	9	.	.	PUNCT
ejpam-6172	34	1	note	note	VERB
ejpam-6172	34	2	that	that	SCONJ
ejpam-6172	34	3	mat2×3(q	mat2×3(q	NOUN
ejpam-6172	34	4	)	)	PUNCT
ejpam-6172	34	5	is	be	AUX
ejpam-6172	34	6	not	not	PART
ejpam-6172	34	7	a	a	DET
ejpam-6172	34	8	semigroup	semigroup	NOUN
ejpam-6172	34	9	under	under	ADP
ejpam-6172	34	10	the	the	DET
ejpam-6172	34	11	usual	usual	ADJ
ejpam-6172	34	12	product	product	NOUN
ejpam-6172	34	13	of	of	ADP
ejpam-6172	34	14	matrices	matrix	NOUN
ejpam-6172	34	15	.	.	PUNCT
ejpam-6172	35	1	we	we	PRON
ejpam-6172	35	2	need	need	VERB
ejpam-6172	35	3	the	the	DET
ejpam-6172	35	4	following	follow	VERB
ejpam-6172	35	5	theorem	theorem	NOUN
ejpam-6172	35	6	proved	prove	VERB
ejpam-6172	35	7	in	in	ADP
ejpam-6172	35	8	[	[	X
ejpam-6172	35	9	8	8	NUM
ejpam-6172	35	10	]	]	PUNCT
ejpam-6172	35	11	(	(	PUNCT
ejpam-6172	35	12	also	also	ADV
ejpam-6172	35	13	,	,	PUNCT
ejpam-6172	35	14	in	in	ADP
ejpam-6172	35	15	[	[	PUNCT
ejpam-6172	35	16	9	9	NUM
ejpam-6172	35	17	]	]	SYM
ejpam-6172	35	18	)	)	PUNCT
ejpam-6172	35	19	.	.	PUNCT
ejpam-6172	36	1	theorem	theorem	NOUN
ejpam-6172	36	2	1	1	X
ejpam-6172	36	3	.	.	PUNCT
ejpam-6172	36	4	suppose	suppose	VERB
ejpam-6172	36	5	a	a	PRON
ejpam-6172	36	6	is	be	AUX
ejpam-6172	36	7	a	a	DET
ejpam-6172	36	8	γ	γ	NOUN
ejpam-6172	36	9	-	-	PUNCT
ejpam-6172	36	10	semigroup	semigroup	NOUN
ejpam-6172	36	11	.	.	PUNCT
ejpam-6172	37	1	if	if	SCONJ
ejpam-6172	37	2	a	a	PRON
ejpam-6172	37	3	is	be	AUX
ejpam-6172	37	4	a	a	DET
ejpam-6172	37	5	group	group	NOUN
ejpam-6172	37	6	under	under	ADP
ejpam-6172	37	7	the	the	DET
ejpam-6172	37	8	operation	operation	NOUN
ejpam-6172	37	9	defined	define	VERB
ejpam-6172	37	10	by	by	ADP
ejpam-6172	37	11	γ	γ	NOUN
ejpam-6172	37	12	for	for	ADP
ejpam-6172	37	13	some	some	DET
ejpam-6172	37	14	γ	γ	PROPN
ejpam-6172	37	15	∈	∈	PROPN
ejpam-6172	37	16	γ	γ	X
ejpam-6172	37	17	,	,	PUNCT
ejpam-6172	37	18	then	then	ADV
ejpam-6172	37	19	a	a	PRON
ejpam-6172	37	20	is	be	AUX
ejpam-6172	37	21	a	a	DET
ejpam-6172	37	22	group	group	NOUN
ejpam-6172	37	23	under	under	ADP
ejpam-6172	37	24	the	the	DET
ejpam-6172	37	25	operation	operation	NOUN
ejpam-6172	37	26	determined	determine	VERB
ejpam-6172	37	27	by	by	ADP
ejpam-6172	37	28	γ	γ	NOUN
ejpam-6172	37	29	for	for	ADP
ejpam-6172	37	30	all	all	DET
ejpam-6172	37	31	γ	γ	PROPN
ejpam-6172	37	32	∈	∈	PROPN
ejpam-6172	37	33	γ	γ	X
ejpam-6172	37	34	.	.	PUNCT
ejpam-6172	37	35	let	let	VERB
ejpam-6172	37	36	γ	γ	NOUN
ejpam-6172	37	37	be	be	AUX
ejpam-6172	37	38	a	a	DET
ejpam-6172	37	39	nonempty	nonempty	ADV
ejpam-6172	37	40	set	set	VERB
ejpam-6172	37	41	.	.	PUNCT
ejpam-6172	38	1	a	a	DET
ejpam-6172	38	2	nonempty	nonempty	ADV
ejpam-6172	38	3	set	set	VERB
ejpam-6172	38	4	a	a	PRON
ejpam-6172	38	5	is	be	AUX
ejpam-6172	38	6	called	call	VERB
ejpam-6172	38	7	a	a	DET
ejpam-6172	38	8	γ	γ	PROPN
ejpam-6172	38	9	-	-	PUNCT
ejpam-6172	38	10	ag	ag	ADJ
ejpam-6172	38	11	-	-	PUNCT
ejpam-6172	38	12	groupoid	groupoid	NOUN
ejpam-6172	38	13	if	if	SCONJ
ejpam-6172	38	14	there	there	PRON
ejpam-6172	38	15	is	be	VERB
ejpam-6172	38	16	a	a	DET
ejpam-6172	38	17	function	function	NOUN
ejpam-6172	38	18	f	f	NOUN
ejpam-6172	38	19	from	from	ADP
ejpam-6172	38	20	a×	a×	PROPN
ejpam-6172	38	21	γ×a	γ×a	PROPN
ejpam-6172	38	22	into	into	ADP
ejpam-6172	38	23	a	a	PRON
ejpam-6172	38	24	,	,	PUNCT
ejpam-6172	38	25	it	it	PRON
ejpam-6172	38	26	is	be	AUX
ejpam-6172	38	27	customary	customary	ADJ
ejpam-6172	38	28	to	to	PART
ejpam-6172	38	29	write	write	VERB
ejpam-6172	38	30	aγb	aγb	NOUN
ejpam-6172	38	31	for	for	ADP
ejpam-6172	38	32	f(a	f(a	PROPN
ejpam-6172	38	33	,	,	PUNCT
ejpam-6172	38	34	γ	γ	X
ejpam-6172	38	35	,	,	PUNCT
ejpam-6172	38	36	b	b	NOUN
ejpam-6172	38	37	)	)	PUNCT
ejpam-6172	38	38	,	,	PUNCT
ejpam-6172	39	1	such	such	ADJ
ejpam-6172	39	2	that	that	SCONJ
ejpam-6172	39	3	(	(	PUNCT
ejpam-6172	39	4	aγb)βc	aγb)βc	NOUN
ejpam-6172	39	5	=	=	PUNCT
ejpam-6172	39	6	(	(	PUNCT
ejpam-6172	39	7	cγb)βa	cγb)βa	VERB
ejpam-6172	39	8	for	for	ADP
ejpam-6172	39	9	all	all	DET
ejpam-6172	39	10	a	a	DET
ejpam-6172	39	11	,	,	PUNCT
ejpam-6172	39	12	b	b	NOUN
ejpam-6172	39	13	,	,	PUNCT
ejpam-6172	39	14	c	c	PROPN
ejpam-6172	39	15	∈	∈	PROPN
ejpam-6172	39	16	a	a	PRON
ejpam-6172	39	17	and	and	CCONJ
ejpam-6172	39	18	γ	γ	NOUN
ejpam-6172	39	19	,	,	PUNCT
ejpam-6172	39	20	β	β	PROPN
ejpam-6172	39	21	∈	∈	PROPN
ejpam-6172	39	22	γ	γ	X
ejpam-6172	39	23	.	.	PROPN
ejpam-6172	39	24	suppose	suppose	VERB
ejpam-6172	39	25	a	a	PRON
ejpam-6172	39	26	is	be	AUX
ejpam-6172	39	27	an	an	DET
ejpam-6172	39	28	ag	ag	PROPN
ejpam-6172	39	29	-	-	PUNCT
ejpam-6172	39	30	groupoid	groupoid	PROPN
ejpam-6172	39	31	and	and	CCONJ
ejpam-6172	39	32	γ	γ	PROPN
ejpam-6172	39	33	is	be	AUX
ejpam-6172	39	34	a	a	DET
ejpam-6172	39	35	nonempty	nonempty	ADV
ejpam-6172	39	36	set	set	VERB
ejpam-6172	39	37	.	.	PUNCT
ejpam-6172	40	1	then	then	ADV
ejpam-6172	40	2	a	a	PRON
ejpam-6172	40	3	is	be	AUX
ejpam-6172	40	4	a	a	DET
ejpam-6172	40	5	γ	γ	PROPN
ejpam-6172	40	6	-	-	PUNCT
ejpam-6172	40	7	ag	ag	ADJ
ejpam-6172	40	8	-	-	NOUN
ejpam-6172	40	9	groupoid	groupoid	NOUN
ejpam-6172	40	10	under	under	ADP
ejpam-6172	40	11	the	the	DET
ejpam-6172	40	12	function	function	NOUN
ejpam-6172	40	13	defined	define	VERB
ejpam-6172	40	14	by	by	ADP
ejpam-6172	40	15	aγb	aγb	NOUN
ejpam-6172	40	16	=	=	SYM
ejpam-6172	40	17	ab	ab	PROPN
ejpam-6172	40	18	for	for	ADP
ejpam-6172	40	19	all	all	DET
ejpam-6172	40	20	a	a	DET
ejpam-6172	40	21	,	,	PUNCT
ejpam-6172	40	22	b	b	X
ejpam-6172	40	23	∈	∈	PROPN
ejpam-6172	40	24	a	a	PRON
ejpam-6172	40	25	and	and	CCONJ
ejpam-6172	40	26	γ	γ	PROPN
ejpam-6172	40	27	∈	∈	PROPN
ejpam-6172	40	28	γ	γ	X
ejpam-6172	40	29	.	.	PUNCT
ejpam-6172	41	1	if	if	SCONJ
ejpam-6172	41	2	a	a	PRON
ejpam-6172	41	3	is	be	AUX
ejpam-6172	41	4	a	a	DET
ejpam-6172	41	5	γ	γ	PROPN
ejpam-6172	41	6	-	-	PUNCT
ejpam-6172	41	7	ag	ag	ADJ
ejpam-6172	41	8	-	-	PUNCT
ejpam-6172	41	9	groupoid	groupoid	PROPN
ejpam-6172	41	10	,	,	PUNCT
ejpam-6172	41	11	then	then	ADV
ejpam-6172	41	12	for	for	ADP
ejpam-6172	41	13	any	any	DET
ejpam-6172	41	14	γ	γ	PROPN
ejpam-6172	41	15	∈	∈	PROPN
ejpam-6172	41	16	γ	γ	X
ejpam-6172	41	17	,	,	PUNCT
ejpam-6172	41	18	a	a	PRON
ejpam-6172	41	19	is	be	AUX
ejpam-6172	41	20	an	an	DET
ejpam-6172	41	21	ag	ag	PROPN
ejpam-6172	41	22	-	-	NOUN
ejpam-6172	41	23	groupoid	groupoid	PROPN
ejpam-6172	41	24	under	under	ADP
ejpam-6172	41	25	the	the	DET
ejpam-6172	41	26	operation	operation	NOUN
ejpam-6172	41	27	determined	determine	VERB
ejpam-6172	41	28	by	by	ADP
ejpam-6172	41	29	γ	γ	PROPN
ejpam-6172	41	30	.	.	PROPN
ejpam-6172	41	31	example	example	NOUN
ejpam-6172	41	32	3	3	X
ejpam-6172	41	33	.	.	PUNCT
ejpam-6172	42	1	let	let	VERB
ejpam-6172	42	2	γ	γ	X
ejpam-6172	42	3	=	=	SYM
ejpam-6172	42	4	{	{	PUNCT
ejpam-6172	42	5	1	1	NUM
ejpam-6172	42	6	,	,	PUNCT
ejpam-6172	42	7	2	2	NUM
ejpam-6172	42	8	,	,	PUNCT
ejpam-6172	42	9	.	.	PUNCT
ejpam-6172	42	10	.	.	PUNCT
ejpam-6172	43	1	.	.	PUNCT
ejpam-6172	43	2	,	,	PUNCT
ejpam-6172	43	3	n	n	CCONJ
ejpam-6172	43	4	}	}	PUNCT
ejpam-6172	43	5	.	.	PUNCT
ejpam-6172	44	1	define	define	VERB
ejpam-6172	44	2	a	a	DET
ejpam-6172	44	3	function	function	NOUN
ejpam-6172	44	4	from	from	ADP
ejpam-6172	44	5	z	z	PROPN
ejpam-6172	44	6	×	×	PROPN
ejpam-6172	44	7	γ	γ	X
ejpam-6172	44	8	×	×	NOUN
ejpam-6172	44	9	z	z	NOUN
ejpam-6172	44	10	into	into	ADP
ejpam-6172	44	11	z	z	PROPN
ejpam-6172	44	12	,	,	PUNCT
ejpam-6172	44	13	the	the	DET
ejpam-6172	44	14	set	set	NOUN
ejpam-6172	44	15	of	of	ADP
ejpam-6172	44	16	all	all	DET
ejpam-6172	44	17	integers	integer	NOUN
ejpam-6172	44	18	,	,	PUNCT
ejpam-6172	44	19	by	by	ADP
ejpam-6172	44	20	aγb	aγb	NOUN
ejpam-6172	44	21	=	=	SYM
ejpam-6172	44	22	b−	b−	PROPN
ejpam-6172	44	23	γ	γ	NOUN
ejpam-6172	44	24	−	−	PROPN
ejpam-6172	44	25	a	a	PRON
ejpam-6172	44	26	for	for	ADP
ejpam-6172	44	27	all	all	DET
ejpam-6172	44	28	a	a	DET
ejpam-6172	44	29	,	,	PUNCT
ejpam-6172	44	30	b	b	X
ejpam-6172	44	31	∈	∈	PROPN
ejpam-6172	44	32	z	z	NOUN
ejpam-6172	44	33	and	and	CCONJ
ejpam-6172	44	34	γ	γ	PROPN
ejpam-6172	44	35	∈	∈	PROPN
ejpam-6172	44	36	γ	γ	X
ejpam-6172	44	37	,	,	PUNCT
ejpam-6172	44	38	where	where	SCONJ
ejpam-6172	44	39	−	−	PROPN
ejpam-6172	44	40	is	be	AUX
ejpam-6172	44	41	a	a	DET
ejpam-6172	44	42	usual	usual	ADJ
ejpam-6172	44	43	subtraction	subtraction	NOUN
ejpam-6172	44	44	of	of	ADP
ejpam-6172	44	45	integers	integer	NOUN
ejpam-6172	44	46	.	.	PUNCT
ejpam-6172	45	1	then	then	ADV
ejpam-6172	45	2	z	z	PROPN
ejpam-6172	45	3	is	be	AUX
ejpam-6172	45	4	a	a	DET
ejpam-6172	45	5	γ	γ	PROPN
ejpam-6172	45	6	-	-	PUNCT
ejpam-6172	45	7	ag	ag	ADJ
ejpam-6172	45	8	-	-	PUNCT
ejpam-6172	45	9	groupoid	groupoid	PROPN
ejpam-6172	45	10	.	.	PUNCT
ejpam-6172	46	1	c.	c.	PROPN
ejpam-6172	46	2	chanoi	chanoi	PROPN
ejpam-6172	46	3	et	et	PROPN
ejpam-6172	46	4	al	al	PROPN
ejpam-6172	46	5	.	.	PUNCT
ejpam-6172	46	6	/	/	SYM
ejpam-6172	46	7	eur	eur	PROPN
ejpam-6172	46	8	.	.	PUNCT
ejpam-6172	47	1	j.	j.	PROPN
ejpam-6172	47	2	pure	pure	PROPN
ejpam-6172	47	3	appl	appl	PROPN
ejpam-6172	47	4	.	.	PROPN
ejpam-6172	47	5	math	math	PROPN
ejpam-6172	47	6	,	,	PUNCT
ejpam-6172	47	7	18	18	NUM
ejpam-6172	47	8	(	(	PUNCT
ejpam-6172	47	9	3	3	NUM
ejpam-6172	47	10	)	)	PUNCT
ejpam-6172	47	11	(	(	PUNCT
ejpam-6172	47	12	2025	2025	NUM
ejpam-6172	47	13	)	)	PUNCT
ejpam-6172	47	14	,	,	PUNCT
ejpam-6172	47	15	6172	6172	NUM
ejpam-6172	47	16	3	3	NUM
ejpam-6172	47	17	of	of	ADP
ejpam-6172	47	18	9	9	NUM
ejpam-6172	47	19	example	example	NOUN
ejpam-6172	47	20	4	4	NUM
ejpam-6172	47	21	.	.	PUNCT
ejpam-6172	48	1	let	let	VERB
ejpam-6172	48	2	a	a	DET
ejpam-6172	48	3	=	=	SYM
ejpam-6172	48	4	γ	γ	X
ejpam-6172	48	5	=	=	SYM
ejpam-6172	48	6	{	{	PUNCT
ejpam-6172	48	7	0	0	NUM
ejpam-6172	48	8	,	,	PUNCT
ejpam-6172	48	9	i,−i	i,−i	NOUN
ejpam-6172	48	10	}	}	PUNCT
ejpam-6172	48	11	.	.	PUNCT
ejpam-6172	49	1	define	define	VERB
ejpam-6172	49	2	a	a	DET
ejpam-6172	49	3	function	function	NOUN
ejpam-6172	49	4	from	from	ADP
ejpam-6172	49	5	a×γ×a	a×γ×a	NOUN
ejpam-6172	49	6	into	into	ADP
ejpam-6172	49	7	a	a	DET
ejpam-6172	49	8	by	by	ADP
ejpam-6172	49	9	aγb	aγb	NOUN
ejpam-6172	49	10	for	for	ADP
ejpam-6172	49	11	all	all	DET
ejpam-6172	49	12	(	(	PUNCT
ejpam-6172	49	13	a	a	PRON
ejpam-6172	49	14	,	,	PUNCT
ejpam-6172	49	15	γ	γ	PROPN
ejpam-6172	49	16	,	,	PUNCT
ejpam-6172	49	17	b	b	NOUN
ejpam-6172	49	18	)	)	PUNCT
ejpam-6172	49	19	∈	∈	PROPN
ejpam-6172	49	20	a	a	DET
ejpam-6172	49	21	×	×	NOUN
ejpam-6172	49	22	γ	γ	X
ejpam-6172	49	23	×	×	NOUN
ejpam-6172	49	24	a	a	X
ejpam-6172	49	25	;	;	PUNCT
ejpam-6172	49	26	here	here	ADV
ejpam-6172	49	27	aγb	aγb	VERB
ejpam-6172	49	28	is	be	VERB
ejpam-6172	49	29	the	the	DET
ejpam-6172	49	30	usual	usual	ADJ
ejpam-6172	49	31	multiplication	multiplication	NOUN
ejpam-6172	49	32	of	of	ADP
ejpam-6172	49	33	complex	complex	ADJ
ejpam-6172	49	34	numbers	number	NOUN
ejpam-6172	49	35	.	.	PUNCT
ejpam-6172	50	1	then	then	ADV
ejpam-6172	50	2	a	a	PRON
ejpam-6172	50	3	is	be	AUX
ejpam-6172	50	4	a	a	DET
ejpam-6172	50	5	γ	γ	PROPN
ejpam-6172	50	6	-	-	PUNCT
ejpam-6172	50	7	ag	ag	ADJ
ejpam-6172	50	8	-	-	PUNCT
ejpam-6172	50	9	groupoid	groupoid	PROPN
ejpam-6172	50	10	,	,	PUNCT
ejpam-6172	50	11	where	where	SCONJ
ejpam-6172	50	12	as	as	ADP
ejpam-6172	50	13	a	a	PRON
ejpam-6172	50	14	is	be	AUX
ejpam-6172	50	15	not	not	PART
ejpam-6172	50	16	an	an	DET
ejpam-6172	50	17	ag	ag	PROPN
ejpam-6172	50	18	-	-	NOUN
ejpam-6172	50	19	groupoid	groupoid	PROPN
ejpam-6172	50	20	.	.	PUNCT
ejpam-6172	51	1	the	the	DET
ejpam-6172	51	2	following	follow	VERB
ejpam-6172	51	3	theorem	theorem	NOUN
ejpam-6172	51	4	is	be	AUX
ejpam-6172	51	5	in	in	ADP
ejpam-6172	51	6	[	[	X
ejpam-6172	51	7	10	10	NUM
ejpam-6172	51	8	]	]	PUNCT
ejpam-6172	51	9	(	(	PUNCT
ejpam-6172	51	10	see	see	VERB
ejpam-6172	51	11	also	also	ADV
ejpam-6172	51	12	in	in	ADP
ejpam-6172	51	13	[	[	PUNCT
ejpam-6172	51	14	11	11	NUM
ejpam-6172	51	15	]	]	NUM
ejpam-6172	51	16	)	)	PUNCT
ejpam-6172	51	17	.	.	PUNCT
ejpam-6172	52	1	theorem	theorem	NOUN
ejpam-6172	52	2	2	2	NUM
ejpam-6172	52	3	.	.	PUNCT
ejpam-6172	53	1	any	any	DET
ejpam-6172	53	2	γ	γ	PROPN
ejpam-6172	53	3	-	-	PUNCT
ejpam-6172	53	4	ag	ag	ADJ
ejpam-6172	53	5	-	-	PUNCT
ejpam-6172	53	6	groupoid	groupoid	PROPN
ejpam-6172	53	7	satisfies	satisfy	VERB
ejpam-6172	53	8	the	the	DET
ejpam-6172	53	9	medial	medial	ADJ
ejpam-6172	53	10	law	law	NOUN
ejpam-6172	53	11	.	.	PUNCT
ejpam-6172	54	1	that	that	PRON
ejpam-6172	54	2	is	be	AUX
ejpam-6172	54	3	,	,	PUNCT
ejpam-6172	54	4	if	if	SCONJ
ejpam-6172	54	5	a	a	PRON
ejpam-6172	54	6	is	be	AUX
ejpam-6172	54	7	a	a	DET
ejpam-6172	54	8	γ	γ	NOUN
ejpam-6172	54	9	-	-	ADJ
ejpam-6172	54	10	aggroupoid	aggroupoid	ADJ
ejpam-6172	54	11	,	,	PUNCT
ejpam-6172	54	12	then	then	ADV
ejpam-6172	54	13	(	(	PUNCT
ejpam-6172	54	14	aγb)β(cαd	aγb)β(cαd	X
ejpam-6172	54	15	)	)	PUNCT
ejpam-6172	54	16	=	=	SYM
ejpam-6172	54	17	(	(	PUNCT
ejpam-6172	54	18	aγc)β(bαd	aγc)β(bαd	X
ejpam-6172	54	19	)	)	PUNCT
ejpam-6172	54	20	for	for	ADP
ejpam-6172	54	21	any	any	DET
ejpam-6172	54	22	a	a	DET
ejpam-6172	54	23	,	,	PUNCT
ejpam-6172	54	24	b	b	NOUN
ejpam-6172	54	25	,	,	PUNCT
ejpam-6172	54	26	c	c	NOUN
ejpam-6172	54	27	,	,	PUNCT
ejpam-6172	54	28	d	d	PROPN
ejpam-6172	54	29	∈	∈	PROPN
ejpam-6172	54	30	a	a	PRON
ejpam-6172	54	31	and	and	CCONJ
ejpam-6172	54	32	γ	γ	NOUN
ejpam-6172	54	33	,	,	PUNCT
ejpam-6172	54	34	β	β	X
ejpam-6172	54	35	,	,	PUNCT
ejpam-6172	54	36	α	α	PROPN
ejpam-6172	54	37	∈	∈	PROPN
ejpam-6172	54	38	γ	γ	X
ejpam-6172	54	39	.	.	PUNCT
ejpam-6172	55	1	an	an	DET
ejpam-6172	55	2	element	element	NOUN
ejpam-6172	55	3	e	e	NOUN
ejpam-6172	55	4	of	of	ADP
ejpam-6172	55	5	a	a	DET
ejpam-6172	55	6	γ	γ	PROPN
ejpam-6172	55	7	-	-	PUNCT
ejpam-6172	55	8	ag	ag	NOUN
ejpam-6172	55	9	-	-	PROPN
ejpam-6172	55	10	groupoid	groupoid	PROPN
ejpam-6172	55	11	a	a	PRON
ejpam-6172	55	12	is	be	AUX
ejpam-6172	55	13	said	say	VERB
ejpam-6172	55	14	to	to	PART
ejpam-6172	55	15	be	be	AUX
ejpam-6172	55	16	a	a	DET
ejpam-6172	55	17	left	left	ADJ
ejpam-6172	55	18	identity	identity	NOUN
ejpam-6172	55	19	if	if	SCONJ
ejpam-6172	55	20	for	for	ADP
ejpam-6172	55	21	all	all	DET
ejpam-6172	55	22	a	a	DET
ejpam-6172	55	23	∈	∈	PROPN
ejpam-6172	55	24	a	a	PRON
ejpam-6172	55	25	and	and	CCONJ
ejpam-6172	55	26	γ	γ	PROPN
ejpam-6172	55	27	∈	∈	PROPN
ejpam-6172	55	28	γ	γ	X
ejpam-6172	55	29	,	,	PUNCT
ejpam-6172	55	30	eγa	eγa	NOUN
ejpam-6172	55	31	=	=	NOUN
ejpam-6172	55	32	a.	a.	NOUN
ejpam-6172	55	33	an	an	DET
ejpam-6172	55	34	element	element	NOUN
ejpam-6172	55	35	a0	a0	NOUN
ejpam-6172	55	36	of	of	ADP
ejpam-6172	55	37	a	a	DET
ejpam-6172	55	38	γ	γ	PROPN
ejpam-6172	55	39	-	-	PUNCT
ejpam-6172	55	40	ag	ag	NOUN
ejpam-6172	55	41	-	-	PROPN
ejpam-6172	55	42	groupoid	groupoid	PROPN
ejpam-6172	55	43	a	a	PRON
ejpam-6172	55	44	is	be	AUX
ejpam-6172	55	45	said	say	VERB
ejpam-6172	55	46	to	to	PART
ejpam-6172	55	47	be	be	AUX
ejpam-6172	55	48	a	a	DET
ejpam-6172	55	49	left	left	ADJ
ejpam-6172	55	50	zero	zero	NUM
ejpam-6172	55	51	if	if	SCONJ
ejpam-6172	55	52	for	for	ADP
ejpam-6172	55	53	all	all	DET
ejpam-6172	55	54	a	a	DET
ejpam-6172	55	55	∈	∈	PROPN
ejpam-6172	55	56	a	a	PRON
ejpam-6172	55	57	and	and	CCONJ
ejpam-6172	55	58	γ	γ	PROPN
ejpam-6172	55	59	∈	∈	PROPN
ejpam-6172	55	60	γ	γ	X
ejpam-6172	55	61	,	,	PUNCT
ejpam-6172	55	62	a0γa	a0γa	PUNCT
ejpam-6172	55	63	=	=	SYM
ejpam-6172	55	64	a0	a0	PROPN
ejpam-6172	55	65	.	.	PUNCT
ejpam-6172	56	1	a	a	DET
ejpam-6172	56	2	γ	γ	PROPN
ejpam-6172	56	3	-	-	PUNCT
ejpam-6172	56	4	ag	ag	NOUN
ejpam-6172	56	5	-	-	PROPN
ejpam-6172	56	6	groupoid	groupoid	PROPN
ejpam-6172	56	7	a	a	PRON
ejpam-6172	56	8	is	be	AUX
ejpam-6172	56	9	said	say	VERB
ejpam-6172	56	10	to	to	PART
ejpam-6172	56	11	be	be	AUX
ejpam-6172	56	12	cancellative	cancellative	ADJ
ejpam-6172	56	13	if	if	SCONJ
ejpam-6172	56	14	for	for	ADP
ejpam-6172	56	15	all	all	DET
ejpam-6172	56	16	a	a	DET
ejpam-6172	56	17	,	,	PUNCT
ejpam-6172	56	18	b	b	NOUN
ejpam-6172	56	19	,	,	PUNCT
ejpam-6172	56	20	c	c	PROPN
ejpam-6172	56	21	∈	∈	PROPN
ejpam-6172	56	22	a	a	PRON
ejpam-6172	56	23	and	and	CCONJ
ejpam-6172	56	24	γ	γ	PROPN
ejpam-6172	56	25	∈	∈	PROPN
ejpam-6172	56	26	γ	γ	X
ejpam-6172	56	27	,	,	PUNCT
ejpam-6172	56	28	(	(	PUNCT
ejpam-6172	56	29	aγc	aγc	NOUN
ejpam-6172	56	30	=	=	PUNCT
ejpam-6172	56	31	bγc	bγc	NOUN
ejpam-6172	56	32	or	or	CCONJ
ejpam-6172	56	33	cγa	cγa	PROPN
ejpam-6172	56	34	=	=	SYM
ejpam-6172	56	35	cγb	cγb	NOUN
ejpam-6172	56	36	)	)	PUNCT
ejpam-6172	56	37	imply	imply	VERB
ejpam-6172	56	38	a	a	DET
ejpam-6172	56	39	=	=	X
ejpam-6172	56	40	b.	b.	PROPN
ejpam-6172	56	41	example	example	NOUN
ejpam-6172	56	42	5	5	X
ejpam-6172	56	43	.	.	X
ejpam-6172	56	44	consider	consider	VERB
ejpam-6172	56	45	an	an	DET
ejpam-6172	56	46	ag	ag	PROPN
ejpam-6172	56	47	-	-	NOUN
ejpam-6172	56	48	groupoid	groupoid	PROPN
ejpam-6172	56	49	a	a	X
ejpam-6172	56	50	=	=	PUNCT
ejpam-6172	56	51	{	{	PUNCT
ejpam-6172	56	52	1	1	NUM
ejpam-6172	56	53	,	,	PUNCT
ejpam-6172	56	54	2	2	NUM
ejpam-6172	56	55	,	,	PUNCT
ejpam-6172	56	56	3	3	NUM
ejpam-6172	56	57	,	,	PUNCT
ejpam-6172	56	58	4	4	NUM
ejpam-6172	56	59	}	}	PUNCT
ejpam-6172	56	60	with	with	ADP
ejpam-6172	56	61	the	the	DET
ejpam-6172	56	62	operation	operation	NOUN
ejpam-6172	56	63	defined	define	VERB
ejpam-6172	56	64	as	as	SCONJ
ejpam-6172	56	65	follows	follow	VERB
ejpam-6172	56	66	:	:	PUNCT
ejpam-6172	56	67	·	·	PUNCT
ejpam-6172	56	68	1	1	NUM
ejpam-6172	56	69	2	2	NUM
ejpam-6172	56	70	3	3	NUM
ejpam-6172	56	71	5	5	NUM
ejpam-6172	56	72	1	1	NUM
ejpam-6172	56	73	1	1	NUM
ejpam-6172	56	74	2	2	NUM
ejpam-6172	56	75	3	3	NUM
ejpam-6172	56	76	5	5	NUM
ejpam-6172	56	77	2	2	NUM
ejpam-6172	56	78	3	3	NUM
ejpam-6172	56	79	3	3	NUM
ejpam-6172	56	80	3	3	NUM
ejpam-6172	56	81	3	3	NUM
ejpam-6172	56	82	3	3	NUM
ejpam-6172	56	83	3	3	NUM
ejpam-6172	56	84	3	3	NUM
ejpam-6172	56	85	3	3	NUM
ejpam-6172	56	86	3	3	NUM
ejpam-6172	56	87	5	5	NUM
ejpam-6172	56	88	2	2	NUM
ejpam-6172	56	89	3	3	NUM
ejpam-6172	56	90	3	3	NUM
ejpam-6172	56	91	3	3	NUM
ejpam-6172	56	92	let	let	VERB
ejpam-6172	56	93	γ	γ	NOUN
ejpam-6172	56	94	be	be	AUX
ejpam-6172	56	95	a	a	DET
ejpam-6172	56	96	nonempty	nonempty	ADV
ejpam-6172	56	97	set	set	VERB
ejpam-6172	56	98	.	.	PUNCT
ejpam-6172	57	1	for	for	ADP
ejpam-6172	57	2	a	a	DET
ejpam-6172	57	3	,	,	PUNCT
ejpam-6172	57	4	b	b	PROPN
ejpam-6172	57	5	∈	∈	PROPN
ejpam-6172	57	6	a	a	PRON
ejpam-6172	57	7	and	and	CCONJ
ejpam-6172	57	8	γ	γ	PROPN
ejpam-6172	57	9	∈	∈	PROPN
ejpam-6172	57	10	γ	γ	X
ejpam-6172	57	11	,	,	PUNCT
ejpam-6172	57	12	define	define	VERB
ejpam-6172	57	13	aγb	aγb	NOUN
ejpam-6172	57	14	=	=	PUNCT
ejpam-6172	57	15	a	a	DET
ejpam-6172	57	16	·	·	PUNCT
ejpam-6172	57	17	b.	b.	NOUN
ejpam-6172	58	1	we	we	PRON
ejpam-6172	58	2	have	have	VERB
ejpam-6172	58	3	that	that	SCONJ
ejpam-6172	58	4	a	a	PRON
ejpam-6172	58	5	is	be	AUX
ejpam-6172	58	6	a	a	DET
ejpam-6172	58	7	finite	finite	NOUN
ejpam-6172	58	8	γ	γ	PROPN
ejpam-6172	58	9	-	-	PUNCT
ejpam-6172	58	10	ag	ag	NOUN
ejpam-6172	58	11	-	-	NOUN
ejpam-6172	58	12	groupoid	groupoid	NOUN
ejpam-6172	58	13	with	with	ADP
ejpam-6172	58	14	left	left	ADJ
ejpam-6172	58	15	identity	identity	NOUN
ejpam-6172	58	16	1	1	NUM
ejpam-6172	58	17	,	,	PUNCT
ejpam-6172	58	18	and	and	CCONJ
ejpam-6172	58	19	left	leave	VERB
ejpam-6172	58	20	zero	zero	NUM
ejpam-6172	58	21	3	3	NUM
ejpam-6172	58	22	.	.	NOUN
ejpam-6172	59	1	3	3	NUM
ejpam-6172	59	2	.	.	X
ejpam-6172	59	3	results	result	NOUN
ejpam-6172	59	4	we	we	PRON
ejpam-6172	59	5	begin	begin	VERB
ejpam-6172	59	6	this	this	DET
ejpam-6172	59	7	section	section	NOUN
ejpam-6172	59	8	with	with	ADP
ejpam-6172	59	9	the	the	DET
ejpam-6172	59	10	following	follow	VERB
ejpam-6172	59	11	theorem	theorem	NOUN
ejpam-6172	59	12	.	.	PUNCT
ejpam-6172	59	13	theorem	theorem	NOUN
ejpam-6172	59	14	3	3	NUM
ejpam-6172	59	15	.	.	PUNCT
ejpam-6172	60	1	if	if	SCONJ
ejpam-6172	60	2	a	a	PRON
ejpam-6172	60	3	is	be	AUX
ejpam-6172	60	4	a	a	DET
ejpam-6172	60	5	γ	γ	PROPN
ejpam-6172	60	6	-	-	PUNCT
ejpam-6172	60	7	ag	ag	ADJ
ejpam-6172	60	8	-	-	PUNCT
ejpam-6172	60	9	groupoid	groupoid	NOUN
ejpam-6172	60	10	satisfying	satisfy	VERB
ejpam-6172	60	11	the	the	DET
ejpam-6172	60	12	identity	identity	NOUN
ejpam-6172	60	13	aγ(bβc	aγ(bβc	ADV
ejpam-6172	60	14	)	)	PUNCT
ejpam-6172	60	15	=	=	PUNCT
ejpam-6172	60	16	(	(	PUNCT
ejpam-6172	60	17	cγb)βa	cγb)βa	VERB
ejpam-6172	60	18	for	for	ADP
ejpam-6172	60	19	all	all	DET
ejpam-6172	60	20	a	a	DET
ejpam-6172	60	21	,	,	PUNCT
ejpam-6172	60	22	b	b	NOUN
ejpam-6172	60	23	,	,	PUNCT
ejpam-6172	60	24	c	c	PROPN
ejpam-6172	60	25	∈	∈	PROPN
ejpam-6172	60	26	a	a	PRON
ejpam-6172	60	27	and	and	CCONJ
ejpam-6172	60	28	γ	γ	NOUN
ejpam-6172	60	29	,	,	PUNCT
ejpam-6172	60	30	β	β	PROPN
ejpam-6172	60	31	∈	∈	PROPN
ejpam-6172	60	32	γ	γ	X
ejpam-6172	60	33	,	,	PUNCT
ejpam-6172	60	34	then	then	ADV
ejpam-6172	60	35	a	a	PRON
ejpam-6172	60	36	is	be	AUX
ejpam-6172	60	37	a	a	DET
ejpam-6172	60	38	γ	γ	NOUN
ejpam-6172	60	39	-	-	PUNCT
ejpam-6172	60	40	semigroup	semigroup	NOUN
ejpam-6172	60	41	.	.	PUNCT
ejpam-6172	61	1	proof	proof	NOUN
ejpam-6172	61	2	.	.	PUNCT
ejpam-6172	62	1	assume	assume	VERB
ejpam-6172	62	2	the	the	DET
ejpam-6172	62	3	condition	condition	NOUN
ejpam-6172	62	4	holds	hold	VERB
ejpam-6172	62	5	.	.	PUNCT
ejpam-6172	63	1	then	then	ADV
ejpam-6172	63	2	,	,	PUNCT
ejpam-6172	63	3	for	for	ADP
ejpam-6172	63	4	a	a	DET
ejpam-6172	63	5	,	,	PUNCT
ejpam-6172	63	6	b	b	NOUN
ejpam-6172	63	7	,	,	PUNCT
ejpam-6172	63	8	c	c	PROPN
ejpam-6172	63	9	∈	∈	PROPN
ejpam-6172	63	10	a	a	PRON
ejpam-6172	63	11	and	and	CCONJ
ejpam-6172	63	12	γ	γ	NOUN
ejpam-6172	63	13	,	,	PUNCT
ejpam-6172	63	14	β	β	PROPN
ejpam-6172	63	15	∈	∈	PROPN
ejpam-6172	63	16	γ	γ	X
ejpam-6172	63	17	,	,	PUNCT
ejpam-6172	63	18	we	we	PRON
ejpam-6172	63	19	have	have	VERB
ejpam-6172	63	20	(	(	PUNCT
ejpam-6172	63	21	aγb)βc	aγb)βc	VERB
ejpam-6172	63	22	=	=	SYM
ejpam-6172	63	23	(	(	PUNCT
ejpam-6172	63	24	cγb)βa	cγb)βa	NOUN
ejpam-6172	63	25	=	=	PUNCT
ejpam-6172	63	26	aγ(bβc	aγ(bβc	PROPN
ejpam-6172	63	27	)	)	PUNCT
ejpam-6172	63	28	.	.	PUNCT
ejpam-6172	64	1	thus	thus	ADV
ejpam-6172	64	2	a	a	PRON
ejpam-6172	64	3	is	be	AUX
ejpam-6172	64	4	a	a	DET
ejpam-6172	64	5	γ	γ	NOUN
ejpam-6172	64	6	-	-	PUNCT
ejpam-6172	64	7	semigroup	semigroup	NOUN
ejpam-6172	64	8	.	.	PUNCT
ejpam-6172	65	1	c.	c.	PROPN
ejpam-6172	65	2	chanoi	chanoi	PROPN
ejpam-6172	65	3	et	et	PROPN
ejpam-6172	65	4	al	al	PROPN
ejpam-6172	65	5	.	.	PUNCT
ejpam-6172	65	6	/	/	SYM
ejpam-6172	65	7	eur	eur	PROPN
ejpam-6172	65	8	.	.	PUNCT
ejpam-6172	66	1	j.	j.	PROPN
ejpam-6172	66	2	pure	pure	PROPN
ejpam-6172	66	3	appl	appl	PROPN
ejpam-6172	66	4	.	.	PROPN
ejpam-6172	66	5	math	math	PROPN
ejpam-6172	66	6	,	,	PUNCT
ejpam-6172	66	7	18	18	NUM
ejpam-6172	66	8	(	(	PUNCT
ejpam-6172	66	9	3	3	NUM
ejpam-6172	66	10	)	)	PUNCT
ejpam-6172	66	11	(	(	PUNCT
ejpam-6172	66	12	2025	2025	NUM
ejpam-6172	66	13	)	)	PUNCT
ejpam-6172	66	14	,	,	PUNCT
ejpam-6172	66	15	6172	6172	NUM
ejpam-6172	66	16	4	4	NUM
ejpam-6172	66	17	of	of	ADP
ejpam-6172	66	18	9	9	NUM
ejpam-6172	66	19	theorem	theorem	NOUN
ejpam-6172	66	20	4	4	NUM
ejpam-6172	66	21	.	.	PUNCT
ejpam-6172	67	1	if	if	SCONJ
ejpam-6172	67	2	a	a	PRON
ejpam-6172	67	3	is	be	AUX
ejpam-6172	67	4	a	a	DET
ejpam-6172	67	5	cancellative	cancellative	ADJ
ejpam-6172	67	6	γ	γ	PROPN
ejpam-6172	67	7	-	-	PUNCT
ejpam-6172	67	8	ag	ag	ADJ
ejpam-6172	67	9	-	-	PUNCT
ejpam-6172	67	10	groupoid	groupoid	NOUN
ejpam-6172	67	11	satisfying	satisfy	VERB
ejpam-6172	67	12	the	the	DET
ejpam-6172	67	13	identity	identity	NOUN
ejpam-6172	67	14	aγ(bβc	aγ(bβc	ADV
ejpam-6172	67	15	)	)	PUNCT
ejpam-6172	67	16	=	=	PUNCT
ejpam-6172	67	17	(	(	PUNCT
ejpam-6172	67	18	cγb)βa	cγb)βa	VERB
ejpam-6172	67	19	for	for	ADP
ejpam-6172	67	20	all	all	DET
ejpam-6172	67	21	a	a	DET
ejpam-6172	67	22	,	,	PUNCT
ejpam-6172	67	23	b	b	NOUN
ejpam-6172	67	24	,	,	PUNCT
ejpam-6172	67	25	c	c	PROPN
ejpam-6172	67	26	∈	∈	PROPN
ejpam-6172	67	27	a	a	PRON
ejpam-6172	67	28	and	and	CCONJ
ejpam-6172	67	29	β	β	NOUN
ejpam-6172	67	30	,	,	PUNCT
ejpam-6172	67	31	γ	γ	PROPN
ejpam-6172	67	32	∈	∈	PROPN
ejpam-6172	67	33	γ	γ	X
ejpam-6172	67	34	,	,	PUNCT
ejpam-6172	67	35	then	then	ADV
ejpam-6172	67	36	for	for	ADP
ejpam-6172	67	37	any	any	DET
ejpam-6172	67	38	γ	γ	PROPN
ejpam-6172	67	39	∈	∈	PROPN
ejpam-6172	67	40	γ	γ	X
ejpam-6172	67	41	,	,	PUNCT
ejpam-6172	67	42	a	a	PRON
ejpam-6172	67	43	is	be	AUX
ejpam-6172	67	44	a	a	DET
ejpam-6172	67	45	commutative	commutative	ADJ
ejpam-6172	67	46	semigroup	semigroup	NOUN
ejpam-6172	67	47	under	under	ADP
ejpam-6172	67	48	the	the	DET
ejpam-6172	67	49	operation	operation	NOUN
ejpam-6172	67	50	determined	determine	VERB
ejpam-6172	67	51	by	by	ADP
ejpam-6172	67	52	γ	γ	PROPN
ejpam-6172	67	53	.	.	PROPN
ejpam-6172	67	54	proof	proof	NOUN
ejpam-6172	67	55	.	.	PUNCT
ejpam-6172	68	1	assume	assume	VERB
ejpam-6172	68	2	the	the	DET
ejpam-6172	68	3	condition	condition	NOUN
ejpam-6172	68	4	holds	hold	VERB
ejpam-6172	68	5	.	.	PUNCT
ejpam-6172	69	1	by	by	ADP
ejpam-6172	69	2	theorem	theorem	NOUN
ejpam-6172	69	3	3	3	NUM
ejpam-6172	69	4	,	,	PUNCT
ejpam-6172	69	5	for	for	ADP
ejpam-6172	69	6	any	any	DET
ejpam-6172	69	7	γ	γ	PROPN
ejpam-6172	69	8	∈	∈	PROPN
ejpam-6172	69	9	γ	γ	X
ejpam-6172	69	10	,	,	PUNCT
ejpam-6172	69	11	a	a	PRON
ejpam-6172	69	12	is	be	AUX
ejpam-6172	69	13	a	a	DET
ejpam-6172	69	14	semigroup	semigroup	NOUN
ejpam-6172	69	15	under	under	ADP
ejpam-6172	69	16	the	the	DET
ejpam-6172	69	17	operation	operation	NOUN
ejpam-6172	69	18	determined	determine	VERB
ejpam-6172	69	19	by	by	ADP
ejpam-6172	69	20	γ	γ	PROPN
ejpam-6172	69	21	.	.	PROPN
ejpam-6172	69	22	let	let	VERB
ejpam-6172	69	23	a	a	DET
ejpam-6172	69	24	,	,	PUNCT
ejpam-6172	69	25	b	b	X
ejpam-6172	69	26	∈	∈	PROPN
ejpam-6172	69	27	a	a	PRON
ejpam-6172	69	28	and	and	CCONJ
ejpam-6172	69	29	γ	γ	PROPN
ejpam-6172	69	30	∈	∈	PROPN
ejpam-6172	69	31	γ	γ	X
ejpam-6172	69	32	.	.	PUNCT
ejpam-6172	70	1	consider	consider	VERB
ejpam-6172	70	2	:	:	PUNCT
ejpam-6172	70	3	(	(	PUNCT
ejpam-6172	70	4	aγ(aγb))γa	aγ(aγb))γa	ADP
ejpam-6172	70	5	=	=	PUNCT
ejpam-6172	70	6	(	(	PUNCT
ejpam-6172	70	7	(	(	PUNCT
ejpam-6172	70	8	aγa)γb)γa	aγa)γb)γa	NOUN
ejpam-6172	70	9	=	=	SYM
ejpam-6172	70	10	(	(	PUNCT
ejpam-6172	70	11	aγa)γ(bγa	aγa)γ(bγa	PROPN
ejpam-6172	70	12	)	)	PUNCT
ejpam-6172	70	13	=	=	SYM
ejpam-6172	70	14	(	(	PUNCT
ejpam-6172	70	15	aγb)γ(aγa	aγb)γ(aγa	PROPN
ejpam-6172	70	16	)	)	PUNCT
ejpam-6172	70	17	=	=	SYM
ejpam-6172	71	1	(	(	PUNCT
ejpam-6172	71	2	(	(	PUNCT
ejpam-6172	71	3	aγb)γa)γa	aγb)γa)γa	NOUN
ejpam-6172	71	4	.	.	PUNCT
ejpam-6172	72	1	so	so	ADV
ejpam-6172	72	2	(	(	PUNCT
ejpam-6172	72	3	aγ(aγb))γa	aγ(aγb))γa	PROPN
ejpam-6172	72	4	=	=	PUNCT
ejpam-6172	72	5	(	(	PUNCT
ejpam-6172	72	6	(	(	PUNCT
ejpam-6172	72	7	aγb)γa)γa	aγb)γa)γa	NOUN
ejpam-6172	72	8	.	.	PUNCT
ejpam-6172	73	1	by	by	ADP
ejpam-6172	73	2	cancellative	cancellative	ADJ
ejpam-6172	73	3	law	law	NOUN
ejpam-6172	73	4	,	,	PUNCT
ejpam-6172	73	5	aγ(aγb	aγ(aγb	NOUN
ejpam-6172	73	6	)	)	PUNCT
ejpam-6172	73	7	=	=	SYM
ejpam-6172	73	8	(	(	PUNCT
ejpam-6172	73	9	aγb)γa	aγb)γa	NOUN
ejpam-6172	73	10	.	.	PUNCT
ejpam-6172	74	1	by	by	ADP
ejpam-6172	74	2	assumption	assumption	NOUN
ejpam-6172	74	3	,	,	PUNCT
ejpam-6172	74	4	aγ(aγb	aγ(aγb	NOUN
ejpam-6172	74	5	)	)	PUNCT
ejpam-6172	74	6	=	=	SYM
ejpam-6172	74	7	aγ(bγa	aγ(bγa	PROPN
ejpam-6172	74	8	)	)	PUNCT
ejpam-6172	74	9	.	.	PUNCT
ejpam-6172	75	1	using	use	VERB
ejpam-6172	75	2	cancellative	cancellative	ADJ
ejpam-6172	75	3	law	law	NOUN
ejpam-6172	75	4	,	,	PUNCT
ejpam-6172	75	5	aγb	aγb	NOUN
ejpam-6172	75	6	=	=	NOUN
ejpam-6172	75	7	bγa	bγa	PROPN
ejpam-6172	75	8	.	.	PUNCT
ejpam-6172	76	1	hence	hence	ADV
ejpam-6172	76	2	a	a	PRON
ejpam-6172	76	3	is	be	AUX
ejpam-6172	76	4	a	a	DET
ejpam-6172	76	5	commutative	commutative	ADJ
ejpam-6172	76	6	semigroup	semigroup	NOUN
ejpam-6172	76	7	under	under	ADP
ejpam-6172	76	8	the	the	DET
ejpam-6172	76	9	operation	operation	NOUN
ejpam-6172	76	10	determined	determine	VERB
ejpam-6172	76	11	by	by	ADP
ejpam-6172	76	12	γ	γ	PROPN
ejpam-6172	76	13	.	.	PROPN
ejpam-6172	76	14	an	an	DET
ejpam-6172	76	15	ag	ag	PROPN
ejpam-6172	76	16	-	-	PROPN
ejpam-6172	76	17	groupoid	groupoid	PROPN
ejpam-6172	76	18	a	a	PRON
ejpam-6172	76	19	is	be	AUX
ejpam-6172	76	20	said	say	VERB
ejpam-6172	76	21	to	to	PART
ejpam-6172	76	22	be	be	AUX
ejpam-6172	76	23	cancellative	cancellative	ADJ
ejpam-6172	76	24	if	if	SCONJ
ejpam-6172	76	25	for	for	ADP
ejpam-6172	76	26	all	all	DET
ejpam-6172	76	27	a	a	DET
ejpam-6172	76	28	,	,	PUNCT
ejpam-6172	76	29	b	b	NOUN
ejpam-6172	76	30	,	,	PUNCT
ejpam-6172	76	31	c	c	PROPN
ejpam-6172	76	32	∈	∈	PROPN
ejpam-6172	76	33	a	a	PRON
ejpam-6172	76	34	,	,	PUNCT
ejpam-6172	76	35	ac	ac	PROPN
ejpam-6172	76	36	=	=	PUNCT
ejpam-6172	76	37	bc	bc	PROPN
ejpam-6172	76	38	or	or	CCONJ
ejpam-6172	76	39	ca	ca	NOUN
ejpam-6172	76	40	=	=	SYM
ejpam-6172	76	41	cb	cb	NOUN
ejpam-6172	76	42	imply	imply	VERB
ejpam-6172	76	43	a	a	DET
ejpam-6172	76	44	=	=	X
ejpam-6172	76	45	b.	b.	NOUN
ejpam-6172	76	46	we	we	PRON
ejpam-6172	76	47	specifically	specifically	ADV
ejpam-6172	76	48	have	have	VERB
ejpam-6172	76	49	the	the	DET
ejpam-6172	76	50	following	follow	VERB
ejpam-6172	76	51	corollary	corollary	ADJ
ejpam-6172	76	52	:	:	PUNCT
ejpam-6172	76	53	corollary	corollary	ADJ
ejpam-6172	76	54	1	1	NUM
ejpam-6172	76	55	.	.	PUNCT
ejpam-6172	77	1	if	if	SCONJ
ejpam-6172	77	2	a	a	PRON
ejpam-6172	77	3	is	be	AUX
ejpam-6172	77	4	a	a	DET
ejpam-6172	77	5	cancellative	cancellative	ADJ
ejpam-6172	77	6	ag	ag	PROPN
ejpam-6172	77	7	-	-	NOUN
ejpam-6172	77	8	groupoid	groupoid	NOUN
ejpam-6172	77	9	satisfying	satisfy	VERB
ejpam-6172	77	10	the	the	DET
ejpam-6172	77	11	identity	identity	NOUN
ejpam-6172	77	12	a(bc	a(bc	NOUN
ejpam-6172	77	13	)	)	PUNCT
ejpam-6172	77	14	=	=	SYM
ejpam-6172	77	15	(	(	PUNCT
ejpam-6172	77	16	cb)a	cb)a	PROPN
ejpam-6172	77	17	for	for	ADP
ejpam-6172	77	18	all	all	DET
ejpam-6172	77	19	a	a	DET
ejpam-6172	77	20	,	,	PUNCT
ejpam-6172	77	21	b	b	NOUN
ejpam-6172	77	22	,	,	PUNCT
ejpam-6172	77	23	c	c	PROPN
ejpam-6172	77	24	∈	∈	PROPN
ejpam-6172	77	25	a	a	PRON
ejpam-6172	77	26	,	,	PUNCT
ejpam-6172	77	27	then	then	ADV
ejpam-6172	77	28	a	a	PRON
ejpam-6172	77	29	is	be	AUX
ejpam-6172	77	30	a	a	DET
ejpam-6172	77	31	commutative	commutative	ADJ
ejpam-6172	77	32	semigroup	semigroup	NOUN
ejpam-6172	77	33	.	.	PUNCT
ejpam-6172	78	1	now	now	ADV
ejpam-6172	78	2	,	,	PUNCT
ejpam-6172	78	3	we	we	PRON
ejpam-6172	78	4	present	present	VERB
ejpam-6172	78	5	the	the	DET
ejpam-6172	78	6	main	main	ADJ
ejpam-6172	78	7	result	result	NOUN
ejpam-6172	78	8	.	.	PUNCT
ejpam-6172	79	1	theorem	theorem	NOUN
ejpam-6172	79	2	5	5	NUM
ejpam-6172	79	3	.	.	PUNCT
ejpam-6172	80	1	let	let	VERB
ejpam-6172	80	2	a	a	PRON
ejpam-6172	80	3	be	be	AUX
ejpam-6172	80	4	a	a	DET
ejpam-6172	80	5	finite	finite	NOUN
ejpam-6172	80	6	γ	γ	PROPN
ejpam-6172	80	7	-	-	PUNCT
ejpam-6172	80	8	ag	ag	ADJ
ejpam-6172	80	9	-	-	PUNCT
ejpam-6172	80	10	groupoid	groupoid	NOUN
ejpam-6172	80	11	containing	contain	VERB
ejpam-6172	80	12	at	at	ADV
ejpam-6172	80	13	least	least	ADV
ejpam-6172	80	14	two	two	NUM
ejpam-6172	80	15	elements	element	NOUN
ejpam-6172	80	16	(	(	PUNCT
ejpam-6172	80	17	|a|	|a|	NOUN
ejpam-6172	80	18	>	>	X
ejpam-6172	80	19	1	1	NUM
ejpam-6172	80	20	)	)	PUNCT
ejpam-6172	80	21	.	.	PUNCT
ejpam-6172	81	1	suppose	suppose	VERB
ejpam-6172	81	2	a	a	PRON
ejpam-6172	81	3	contains	contain	VERB
ejpam-6172	81	4	a	a	DET
ejpam-6172	81	5	left	left	ADJ
ejpam-6172	81	6	identity	identity	NOUN
ejpam-6172	81	7	e	e	NOUN
ejpam-6172	81	8	and	and	CCONJ
ejpam-6172	81	9	a	a	DET
ejpam-6172	81	10	left	left	ADJ
ejpam-6172	81	11	zero	zero	NUM
ejpam-6172	81	12	a0	a0	NOUN
ejpam-6172	81	13	,	,	PUNCT
ejpam-6172	81	14	and	and	CCONJ
ejpam-6172	81	15	a	a	DET
ejpam-6172	81	16	satisfies	satisfie	NOUN
ejpam-6172	81	17	the	the	DET
ejpam-6172	81	18	identity	identity	NOUN
ejpam-6172	81	19	aγ(bβc	aγ(bβc	ADV
ejpam-6172	81	20	)	)	PUNCT
ejpam-6172	81	21	=	=	PUNCT
ejpam-6172	81	22	(	(	PUNCT
ejpam-6172	81	23	cγb)βa	cγb)βa	VERB
ejpam-6172	81	24	for	for	ADP
ejpam-6172	81	25	all	all	DET
ejpam-6172	81	26	a	a	DET
ejpam-6172	81	27	,	,	PUNCT
ejpam-6172	81	28	b	b	NOUN
ejpam-6172	81	29	,	,	PUNCT
ejpam-6172	81	30	c	c	PROPN
ejpam-6172	81	31	∈	∈	PROPN
ejpam-6172	81	32	a	a	PRON
ejpam-6172	81	33	and	and	CCONJ
ejpam-6172	81	34	γ	γ	NOUN
ejpam-6172	81	35	,	,	PUNCT
ejpam-6172	81	36	β	β	PROPN
ejpam-6172	81	37	∈	∈	PROPN
ejpam-6172	81	38	γ	γ	PROPN
ejpam-6172	81	39	.	.	PROPN
ejpam-6172	81	40	suppose	suppose	VERB
ejpam-6172	81	41	further	far	ADV
ejpam-6172	81	42	that	that	SCONJ
ejpam-6172	81	43	there	there	PRON
ejpam-6172	81	44	exist	exist	VERB
ejpam-6172	81	45	γ0	γ0	NOUN
ejpam-6172	81	46	∈	∈	PROPN
ejpam-6172	81	47	γ	γ	NOUN
ejpam-6172	81	48	and	and	CCONJ
ejpam-6172	81	49	an	an	DET
ejpam-6172	81	50	operation	operation	NOUN
ejpam-6172	81	51	∗	∗	NOUN
ejpam-6172	81	52	of	of	ADP
ejpam-6172	81	53	a×a	a×a	PROPN
ejpam-6172	81	54	into	into	ADP
ejpam-6172	81	55	a	a	PRON
ejpam-6172	81	56	,	,	PUNCT
ejpam-6172	81	57	write	write	VERB
ejpam-6172	81	58	a	a	DET
ejpam-6172	81	59	∗	∗	NOUN
ejpam-6172	81	60	b	b	NOUN
ejpam-6172	81	61	for	for	ADP
ejpam-6172	81	62	∗(a	∗(a	NOUN
ejpam-6172	81	63	,	,	PUNCT
ejpam-6172	81	64	b	b	NOUN
ejpam-6172	81	65	)	)	PUNCT
ejpam-6172	81	66	,	,	PUNCT
ejpam-6172	81	67	such	such	ADJ
ejpam-6172	81	68	that	that	SCONJ
ejpam-6172	81	69	(	(	PUNCT
ejpam-6172	81	70	i)-(v	i)-(v	PROPN
ejpam-6172	81	71	)	)	PUNCT
ejpam-6172	81	72	hold	hold	VERB
ejpam-6172	81	73	:	:	PUNCT
ejpam-6172	81	74	(	(	PUNCT
ejpam-6172	81	75	i	i	NOUN
ejpam-6172	81	76	)	)	PUNCT
ejpam-6172	81	77	a	a	PRON
ejpam-6172	81	78	is	be	AUX
ejpam-6172	81	79	an	an	DET
ejpam-6172	81	80	ag	ag	PROPN
ejpam-6172	81	81	-	-	PUNCT
ejpam-6172	81	82	groupoid	groupoid	PROPN
ejpam-6172	81	83	under	under	ADP
ejpam-6172	81	84	∗.	∗.	PROPN
ejpam-6172	81	85	(	(	PUNCT
ejpam-6172	81	86	ii	ii	NOUN
ejpam-6172	81	87	)	)	PUNCT
ejpam-6172	81	88	for	for	ADP
ejpam-6172	81	89	any	any	DET
ejpam-6172	81	90	a	a	DET
ejpam-6172	81	91	∈	∈	PROPN
ejpam-6172	81	92	a	a	DET
ejpam-6172	81	93	there	there	PRON
ejpam-6172	81	94	exists	exist	VERB
ejpam-6172	81	95	b	b	PROPN
ejpam-6172	81	96	∈	∈	PROPN
ejpam-6172	81	97	a	a	DET
ejpam-6172	81	98	such	such	ADJ
ejpam-6172	81	99	that	that	DET
ejpam-6172	81	100	b	b	NOUN
ejpam-6172	81	101	∗	∗	NOUN
ejpam-6172	81	102	a	a	DET
ejpam-6172	81	103	=	=	SYM
ejpam-6172	81	104	a0	a0	PROPN
ejpam-6172	81	105	.	.	PUNCT
ejpam-6172	82	1	(	(	PUNCT
ejpam-6172	82	2	iii	iii	X
ejpam-6172	82	3	)	)	PUNCT
ejpam-6172	82	4	a0	a0	PROPN
ejpam-6172	82	5	∗	∗	NOUN
ejpam-6172	82	6	a	a	DET
ejpam-6172	82	7	=	=	NOUN
ejpam-6172	82	8	a	a	PRON
ejpam-6172	82	9	for	for	ADP
ejpam-6172	82	10	all	all	DET
ejpam-6172	82	11	a	a	DET
ejpam-6172	82	12	∈	∈	PROPN
ejpam-6172	82	13	a.	a.	NOUN
ejpam-6172	82	14	(	(	PUNCT
ejpam-6172	82	15	iv	iv	X
ejpam-6172	82	16	)	)	PUNCT
ejpam-6172	82	17	(	(	PUNCT
ejpam-6172	82	18	a	a	DET
ejpam-6172	82	19	∗	∗	NOUN
ejpam-6172	82	20	b)γ0c	b)γ0c	NOUN
ejpam-6172	82	21	=	=	SYM
ejpam-6172	82	22	(	(	PUNCT
ejpam-6172	82	23	aγ0c	aγ0c	NOUN
ejpam-6172	82	24	)	)	PUNCT
ejpam-6172	82	25	∗	∗	NOUN
ejpam-6172	82	26	(	(	PUNCT
ejpam-6172	82	27	bγ0c	bγ0c	PROPN
ejpam-6172	82	28	)	)	PUNCT
ejpam-6172	82	29	for	for	ADP
ejpam-6172	82	30	all	all	DET
ejpam-6172	82	31	a	a	DET
ejpam-6172	82	32	,	,	PUNCT
ejpam-6172	82	33	b	b	NOUN
ejpam-6172	82	34	,	,	PUNCT
ejpam-6172	82	35	c	c	PROPN
ejpam-6172	82	36	∈	∈	PROPN
ejpam-6172	82	37	a.	a.	NOUN
ejpam-6172	82	38	c.	c.	PROPN
ejpam-6172	82	39	chanoi	chanoi	PROPN
ejpam-6172	82	40	et	et	PROPN
ejpam-6172	82	41	al	al	PROPN
ejpam-6172	82	42	.	.	PUNCT
ejpam-6172	82	43	/	/	SYM
ejpam-6172	82	44	eur	eur	PROPN
ejpam-6172	82	45	.	.	PUNCT
ejpam-6172	83	1	j.	j.	PROPN
ejpam-6172	83	2	pure	pure	PROPN
ejpam-6172	83	3	appl	appl	PROPN
ejpam-6172	83	4	.	.	PROPN
ejpam-6172	83	5	math	math	PROPN
ejpam-6172	83	6	,	,	PUNCT
ejpam-6172	83	7	18	18	NUM
ejpam-6172	83	8	(	(	PUNCT
ejpam-6172	83	9	3	3	NUM
ejpam-6172	83	10	)	)	PUNCT
ejpam-6172	83	11	(	(	PUNCT
ejpam-6172	83	12	2025	2025	NUM
ejpam-6172	83	13	)	)	PUNCT
ejpam-6172	83	14	,	,	PUNCT
ejpam-6172	83	15	6172	6172	NUM
ejpam-6172	83	16	5	5	NUM
ejpam-6172	83	17	of	of	ADP
ejpam-6172	83	18	9	9	NUM
ejpam-6172	83	19	(	(	PUNCT
ejpam-6172	83	20	v	v	NOUN
ejpam-6172	83	21	)	)	PUNCT
ejpam-6172	83	22	for	for	ADP
ejpam-6172	83	23	any	any	DET
ejpam-6172	83	24	a	a	PRON
ejpam-6172	83	25	,	,	PUNCT
ejpam-6172	83	26	b	b	PROPN
ejpam-6172	83	27	∈	∈	PROPN
ejpam-6172	83	28	a	a	PRON
ejpam-6172	83	29	,	,	PUNCT
ejpam-6172	83	30	if	if	SCONJ
ejpam-6172	83	31	aγ0b	aγ0b	PROPN
ejpam-6172	83	32	=	=	SYM
ejpam-6172	83	33	a0	a0	PROPN
ejpam-6172	83	34	then	then	ADV
ejpam-6172	83	35	a	a	DET
ejpam-6172	83	36	=	=	X
ejpam-6172	83	37	a0	a0	PROPN
ejpam-6172	83	38	or	or	CCONJ
ejpam-6172	83	39	b	b	PROPN
ejpam-6172	83	40	=	=	PROPN
ejpam-6172	83	41	a0	a0	PROPN
ejpam-6172	83	42	.	.	PUNCT
ejpam-6172	84	1	then	then	ADV
ejpam-6172	84	2	a	a	DET
ejpam-6172	84	3	\	\	PROPN
ejpam-6172	84	4	{	{	PUNCT
ejpam-6172	84	5	a0	a0	PROPN
ejpam-6172	84	6	}	}	PUNCT
ejpam-6172	84	7	is	be	AUX
ejpam-6172	84	8	a	a	DET
ejpam-6172	84	9	commutative	commutative	ADJ
ejpam-6172	84	10	group	group	NOUN
ejpam-6172	84	11	under	under	ADP
ejpam-6172	84	12	the	the	DET
ejpam-6172	84	13	operation	operation	NOUN
ejpam-6172	84	14	determined	determine	VERB
ejpam-6172	84	15	by	by	ADP
ejpam-6172	84	16	γ	γ	NOUN
ejpam-6172	84	17	for	for	ADP
ejpam-6172	84	18	all	all	DET
ejpam-6172	84	19	γ	γ	PROPN
ejpam-6172	84	20	∈	∈	PROPN
ejpam-6172	84	21	γ	γ	X
ejpam-6172	84	22	.	.	PUNCT
ejpam-6172	84	23	proof	proof	NOUN
ejpam-6172	84	24	.	.	PUNCT
ejpam-6172	85	1	let	let	VERB
ejpam-6172	85	2	a	a	DET
ejpam-6172	85	3	=	=	SYM
ejpam-6172	85	4	{	{	PUNCT
ejpam-6172	85	5	a0	a0	PROPN
ejpam-6172	85	6	,	,	PUNCT
ejpam-6172	85	7	a1	a1	NOUN
ejpam-6172	85	8	,	,	PUNCT
ejpam-6172	85	9	.	.	PUNCT
ejpam-6172	85	10	.	.	PUNCT
ejpam-6172	86	1	.	.	PUNCT
ejpam-6172	87	1	,	,	PUNCT
ejpam-6172	87	2	am	be	AUX
ejpam-6172	87	3	}	}	PUNCT
ejpam-6172	87	4	,	,	PUNCT
ejpam-6172	87	5	where	where	SCONJ
ejpam-6172	87	6	m	m	PROPN
ejpam-6172	87	7	≥	≥	NOUN
ejpam-6172	87	8	1	1	NUM
ejpam-6172	87	9	.	.	PUNCT
ejpam-6172	87	10	claim	claim	VERB
ejpam-6172	87	11	1	1	NUM
ejpam-6172	87	12	:	:	PUNCT
ejpam-6172	87	13	a	a	DET
ejpam-6172	87	14	\	\	PROPN
ejpam-6172	87	15	{	{	PUNCT
ejpam-6172	87	16	a0	a0	PROPN
ejpam-6172	87	17	}	}	PUNCT
ejpam-6172	87	18	is	be	AUX
ejpam-6172	87	19	an	an	DET
ejpam-6172	87	20	ag	ag	PROPN
ejpam-6172	87	21	-	-	NOUN
ejpam-6172	87	22	groupoid	groupoid	PROPN
ejpam-6172	87	23	under	under	ADP
ejpam-6172	87	24	the	the	DET
ejpam-6172	87	25	operation	operation	NOUN
ejpam-6172	87	26	determined	determine	VERB
ejpam-6172	87	27	by	by	ADP
ejpam-6172	87	28	γ0	γ0	NOUN
ejpam-6172	87	29	.	.	PUNCT
ejpam-6172	88	1	since	since	SCONJ
ejpam-6172	88	2	m	m	PROPN
ejpam-6172	88	3	≥	≥	NUM
ejpam-6172	88	4	1	1	NUM
ejpam-6172	88	5	,	,	PUNCT
ejpam-6172	88	6	a	a	DET
ejpam-6172	88	7	\	\	PROPN
ejpam-6172	88	8	{	{	PUNCT
ejpam-6172	88	9	a0	a0	NOUN
ejpam-6172	88	10	}	}	PUNCT
ejpam-6172	88	11	=	=	NOUN
ejpam-6172	88	12	̸	̸	ADV
ejpam-6172	88	13	∅.	∅.	ADV
ejpam-6172	88	14	suppose	suppose	VERB
ejpam-6172	88	15	aiγ0aj	aiγ0aj	PROPN
ejpam-6172	88	16	=	=	SYM
ejpam-6172	88	17	a0	a0	PROPN
ejpam-6172	88	18	for	for	ADP
ejpam-6172	88	19	some	some	DET
ejpam-6172	88	20	ai	ai	NOUN
ejpam-6172	88	21	,	,	PUNCT
ejpam-6172	88	22	aj	aj	PROPN
ejpam-6172	88	23	∈	∈	PROPN
ejpam-6172	88	24	a	a	DET
ejpam-6172	88	25	\	\	PROPN
ejpam-6172	88	26	{	{	PUNCT
ejpam-6172	88	27	a0	a0	NOUN
ejpam-6172	88	28	}	}	PUNCT
ejpam-6172	88	29	.	.	PUNCT
ejpam-6172	89	1	by	by	ADP
ejpam-6172	89	2	(	(	PUNCT
ejpam-6172	89	3	v	v	NOUN
ejpam-6172	89	4	)	)	PUNCT
ejpam-6172	89	5	,	,	PUNCT
ejpam-6172	89	6	ai	ai	VERB
ejpam-6172	89	7	=	=	SYM
ejpam-6172	89	8	a0	a0	PROPN
ejpam-6172	89	9	or	or	CCONJ
ejpam-6172	89	10	aj	aj	PROPN
ejpam-6172	89	11	=	=	PROPN
ejpam-6172	89	12	a0	a0	PROPN
ejpam-6172	89	13	.	.	PUNCT
ejpam-6172	90	1	this	this	PRON
ejpam-6172	90	2	is	be	AUX
ejpam-6172	90	3	a	a	DET
ejpam-6172	90	4	contradiction	contradiction	NOUN
ejpam-6172	90	5	.	.	PUNCT
ejpam-6172	91	1	thus	thus	ADV
ejpam-6172	91	2	aiγ0aj	aiγ0aj	ADP
ejpam-6172	91	3	∈	∈	PROPN
ejpam-6172	91	4	a	a	DET
ejpam-6172	91	5	\	\	PROPN
ejpam-6172	91	6	{	{	PUNCT
ejpam-6172	91	7	a0	a0	PROPN
ejpam-6172	91	8	}	}	PUNCT
ejpam-6172	91	9	for	for	ADP
ejpam-6172	91	10	all	all	PRON
ejpam-6172	91	11	ai	ai	VERB
ejpam-6172	91	12	,	,	PUNCT
ejpam-6172	91	13	aj	aj	PROPN
ejpam-6172	91	14	∈	∈	PROPN
ejpam-6172	91	15	a	a	DET
ejpam-6172	91	16	\	\	PROPN
ejpam-6172	91	17	{	{	PUNCT
ejpam-6172	91	18	a0	a0	PROPN
ejpam-6172	91	19	}	}	PUNCT
ejpam-6172	91	20	;	;	PUNCT
ejpam-6172	91	21	so	so	ADV
ejpam-6172	91	22	a	a	DET
ejpam-6172	91	23	\	\	PROPN
ejpam-6172	91	24	{	{	PUNCT
ejpam-6172	91	25	a0	a0	PROPN
ejpam-6172	91	26	}	}	PUNCT
ejpam-6172	91	27	is	be	AUX
ejpam-6172	91	28	a	a	DET
ejpam-6172	91	29	groupoid	groupoid	NOUN
ejpam-6172	91	30	under	under	ADP
ejpam-6172	91	31	the	the	DET
ejpam-6172	91	32	operation	operation	NOUN
ejpam-6172	91	33	determined	determine	VERB
ejpam-6172	91	34	by	by	ADP
ejpam-6172	91	35	γ0	γ0	PROPN
ejpam-6172	91	36	.	.	PUNCT
ejpam-6172	92	1	from	from	ADP
ejpam-6172	92	2	a	a	DET
ejpam-6172	92	3	\	\	PROPN
ejpam-6172	92	4	{	{	PUNCT
ejpam-6172	92	5	a0	a0	PROPN
ejpam-6172	92	6	}	}	PUNCT
ejpam-6172	92	7	⊆	⊆	PROPN
ejpam-6172	92	8	a	a	PRON
ejpam-6172	92	9	,	,	PUNCT
ejpam-6172	92	10	it	it	PRON
ejpam-6172	92	11	follows	follow	VERB
ejpam-6172	92	12	that	that	SCONJ
ejpam-6172	92	13	(	(	PUNCT
ejpam-6172	92	14	aiγ0aj)γ0ak	aiγ0aj)γ0ak	NOUN
ejpam-6172	92	15	=	=	X
ejpam-6172	92	16	(	(	PUNCT
ejpam-6172	92	17	akγ0aj)γ0ai	akγ0aj)γ0ai	VERB
ejpam-6172	92	18	for	for	ADP
ejpam-6172	92	19	all	all	DET
ejpam-6172	92	20	ai	ai	VERB
ejpam-6172	92	21	,	,	PUNCT
ejpam-6172	92	22	aj	aj	PROPN
ejpam-6172	92	23	,	,	PUNCT
ejpam-6172	92	24	ak	ak	PROPN
ejpam-6172	92	25	∈	∈	PROPN
ejpam-6172	92	26	a	a	DET
ejpam-6172	92	27	\	\	PROPN
ejpam-6172	92	28	{	{	PUNCT
ejpam-6172	92	29	a0	a0	PROPN
ejpam-6172	92	30	}	}	PUNCT
ejpam-6172	92	31	.	.	PUNCT
ejpam-6172	93	1	claim	claim	NOUN
ejpam-6172	93	2	2	2	NUM
ejpam-6172	93	3	:	:	PUNCT
ejpam-6172	93	4	e	e	PROPN
ejpam-6172	93	5	̸=	̸=	PROPN
ejpam-6172	93	6	a0	a0	PROPN
ejpam-6172	93	7	.	.	PUNCT
ejpam-6172	93	8	suppose	suppose	VERB
ejpam-6172	93	9	not	not	PART
ejpam-6172	93	10	.	.	PUNCT
ejpam-6172	94	1	if	if	SCONJ
ejpam-6172	94	2	ai	ai	VERB
ejpam-6172	94	3	∈	∈	PROPN
ejpam-6172	94	4	a	a	DET
ejpam-6172	94	5	then	then	ADV
ejpam-6172	94	6	ai	ai	VERB
ejpam-6172	94	7	=	=	NOUN
ejpam-6172	94	8	eγ0ai	eγ0ai	PROPN
ejpam-6172	94	9	=	=	SYM
ejpam-6172	94	10	a0γ0ai	a0γ0ai	PROPN
ejpam-6172	94	11	=	=	SYM
ejpam-6172	94	12	a0	a0	PROPN
ejpam-6172	94	13	.	.	PUNCT
ejpam-6172	95	1	thus	thus	ADV
ejpam-6172	95	2	a	a	PRON
ejpam-6172	95	3	=	=	SYM
ejpam-6172	95	4	{	{	PUNCT
ejpam-6172	95	5	a0	a0	PROPN
ejpam-6172	95	6	}	}	PUNCT
ejpam-6172	95	7	,	,	PUNCT
ejpam-6172	95	8	this	this	PRON
ejpam-6172	95	9	is	be	AUX
ejpam-6172	95	10	a	a	DET
ejpam-6172	95	11	contradiction	contradiction	NOUN
ejpam-6172	95	12	.	.	PUNCT
ejpam-6172	96	1	so	so	ADV
ejpam-6172	96	2	e	e	PROPN
ejpam-6172	96	3	̸=	̸=	PROPN
ejpam-6172	96	4	a0	a0	PROPN
ejpam-6172	96	5	.	.	PUNCT
ejpam-6172	97	1	claim	claim	VERB
ejpam-6172	97	2	3	3	NUM
ejpam-6172	97	3	:	:	PUNCT
ejpam-6172	97	4	a0γ0ai	a0γ0ai	PROPN
ejpam-6172	97	5	=	=	PUNCT
ejpam-6172	97	6	aiγ0a0	aiγ0a0	PROPN
ejpam-6172	97	7	for	for	ADP
ejpam-6172	97	8	all	all	PRON
ejpam-6172	97	9	ai	ai	AUX
ejpam-6172	97	10	∈	∈	NOUN
ejpam-6172	97	11	a.	a.	NOUN
ejpam-6172	97	12	let	let	VERB
ejpam-6172	97	13	ai	ai	AUX
ejpam-6172	97	14	∈	∈	PROPN
ejpam-6172	97	15	a.	a.	NOUN
ejpam-6172	97	16	consider	consider	NOUN
ejpam-6172	97	17	:	:	PUNCT
ejpam-6172	97	18	(	(	PUNCT
ejpam-6172	97	19	aiγ0a0)γ0e	aiγ0a0)γ0e	X
ejpam-6172	97	20	=	=	PUNCT
ejpam-6172	97	21	(	(	PUNCT
ejpam-6172	97	22	eγ0a0)γ0ai	eγ0a0)γ0ai	NOUN
ejpam-6172	97	23	=	=	SYM
ejpam-6172	97	24	a0γ0ai	a0γ0ai	PROPN
ejpam-6172	97	25	=	=	SYM
ejpam-6172	97	26	a0	a0	PROPN
ejpam-6172	97	27	.	.	PUNCT
ejpam-6172	98	1	by	by	ADP
ejpam-6172	98	2	(	(	PUNCT
ejpam-6172	98	3	v	v	NOUN
ejpam-6172	98	4	)	)	PUNCT
ejpam-6172	98	5	,	,	PUNCT
ejpam-6172	98	6	aiγ0a0	aiγ0a0	PROPN
ejpam-6172	98	7	=	=	SYM
ejpam-6172	98	8	a0	a0	PROPN
ejpam-6172	98	9	or	or	CCONJ
ejpam-6172	98	10	e	e	NOUN
ejpam-6172	98	11	=	=	PROPN
ejpam-6172	98	12	a0	a0	PROPN
ejpam-6172	98	13	.	.	PUNCT
ejpam-6172	99	1	by	by	ADP
ejpam-6172	99	2	claim	claim	NOUN
ejpam-6172	99	3	2	2	NUM
ejpam-6172	99	4	,	,	PUNCT
ejpam-6172	99	5	aiγ0a0	aiγ0a0	PROPN
ejpam-6172	99	6	=	=	SYM
ejpam-6172	99	7	a0	a0	PROPN
ejpam-6172	99	8	.	.	PUNCT
ejpam-6172	100	1	hence	hence	ADV
ejpam-6172	100	2	a0γ0a	a0γ0a	PUNCT
ejpam-6172	100	3	=	=	SYM
ejpam-6172	100	4	a0	a0	NOUN
ejpam-6172	100	5	=	=	SYM
ejpam-6172	100	6	aiγ0a0	aiγ0a0	PROPN
ejpam-6172	100	7	.	.	PUNCT
ejpam-6172	101	1	claim	claim	VERB
ejpam-6172	101	2	4	4	NUM
ejpam-6172	101	3	:	:	PUNCT
ejpam-6172	101	4	for	for	ADP
ejpam-6172	101	5	each	each	DET
ejpam-6172	101	6	ak	ak	PROPN
ejpam-6172	101	7	∈	∈	PROPN
ejpam-6172	101	8	a	a	DET
ejpam-6172	101	9	\	\	PROPN
ejpam-6172	101	10	{	{	PUNCT
ejpam-6172	101	11	a0	a0	PROPN
ejpam-6172	101	12	}	}	PUNCT
ejpam-6172	101	13	there	there	PRON
ejpam-6172	101	14	exists	exist	VERB
ejpam-6172	101	15	a−1	a−1	PROPN
ejpam-6172	101	16	k	k	PROPN
ejpam-6172	101	17	∈	∈	PROPN
ejpam-6172	101	18	a	a	DET
ejpam-6172	101	19	\	\	PROPN
ejpam-6172	101	20	{	{	PUNCT
ejpam-6172	101	21	a0	a0	NOUN
ejpam-6172	101	22	}	}	PUNCT
ejpam-6172	102	1	such	such	ADJ
ejpam-6172	102	2	that	that	SCONJ
ejpam-6172	102	3	akγ0a	akγ0a	PROPN
ejpam-6172	102	4	−1	−1	NOUN
ejpam-6172	102	5	k	k	NOUN
ejpam-6172	102	6	=	=	PUNCT
ejpam-6172	102	7	e	e	X
ejpam-6172	102	8	=	=	SYM
ejpam-6172	102	9	a−1	a−1	PROPN
ejpam-6172	102	10	k	k	PROPN
ejpam-6172	102	11	γ0ak	γ0ak	PROPN
ejpam-6172	102	12	.	.	PUNCT
ejpam-6172	103	1	let	let	VERB
ejpam-6172	103	2	ak	ak	PROPN
ejpam-6172	103	3	∈	∈	PROPN
ejpam-6172	103	4	a	a	DET
ejpam-6172	103	5	\	\	PROPN
ejpam-6172	103	6	{	{	PUNCT
ejpam-6172	103	7	a0	a0	NOUN
ejpam-6172	103	8	}	}	PUNCT
ejpam-6172	103	9	.	.	PUNCT
ejpam-6172	104	1	consider	consider	VERB
ejpam-6172	104	2	:	:	PUNCT
ejpam-6172	104	3	hk	hk	PROPN
ejpam-6172	104	4	,	,	PUNCT
ejpam-6172	104	5	γ0	γ0	NOUN
ejpam-6172	104	6	=	=	SYM
ejpam-6172	104	7	{	{	PUNCT
ejpam-6172	104	8	akγ0a1	akγ0a1	PROPN
ejpam-6172	104	9	,	,	PUNCT
ejpam-6172	104	10	akγ0a2	akγ0a2	NOUN
ejpam-6172	104	11	,	,	PUNCT
ejpam-6172	104	12	.	.	PUNCT
ejpam-6172	104	13	.	.	PUNCT
ejpam-6172	104	14	.	.	PUNCT
ejpam-6172	105	1	,	,	PUNCT
ejpam-6172	105	2	akγ0am	akγ0am	PROPN
ejpam-6172	105	3	}	}	PUNCT
ejpam-6172	105	4	.	.	PUNCT
ejpam-6172	106	1	to	to	PART
ejpam-6172	106	2	show	show	VERB
ejpam-6172	106	3	that	that	SCONJ
ejpam-6172	106	4	|hk	|hk	NOUN
ejpam-6172	106	5	,	,	PUNCT
ejpam-6172	106	6	γ0	γ0	NOUN
ejpam-6172	106	7	|	|	NOUN
ejpam-6172	106	8	=	=	SYM
ejpam-6172	106	9	m	m	PROPN
ejpam-6172	106	10	,	,	PUNCT
ejpam-6172	106	11	suppose	suppose	VERB
ejpam-6172	106	12	akγ0ar	akγ0ar	PROPN
ejpam-6172	106	13	=	=	PUNCT
ejpam-6172	106	14	akγ0as	akγ0as	PROPN
ejpam-6172	106	15	for	for	ADP
ejpam-6172	106	16	some	some	DET
ejpam-6172	106	17	r	r	NOUN
ejpam-6172	106	18	̸=	̸=	PROPN
ejpam-6172	106	19	s.	s.	PROPN
ejpam-6172	106	20	consider	consider	VERB
ejpam-6172	106	21	:	:	PUNCT
ejpam-6172	106	22	arγ0ak	arγ0ak	PROPN
ejpam-6172	106	23	=	=	PUNCT
ejpam-6172	106	24	(	(	PUNCT
ejpam-6172	106	25	eγ0ar)γ0ak	eγ0ar)γ0ak	X
ejpam-6172	106	26	=	=	X
ejpam-6172	106	27	(	(	PUNCT
ejpam-6172	106	28	akγ0ar)γ0e	akγ0ar)γ0e	X
ejpam-6172	106	29	=	=	SYM
ejpam-6172	106	30	(	(	PUNCT
ejpam-6172	106	31	akγ0as)γ0e	akγ0as)γ0e	NUM
ejpam-6172	106	32	=	=	SYM
ejpam-6172	106	33	(	(	PUNCT
ejpam-6172	106	34	eγ0as)γ0ak	eγ0as)γ0ak	X
ejpam-6172	106	35	=	=	SYM
ejpam-6172	106	36	asγ0ak	asγ0ak	X
ejpam-6172	106	37	.	.	PUNCT
ejpam-6172	107	1	by	by	ADP
ejpam-6172	107	2	(	(	PUNCT
ejpam-6172	107	3	ii	ii	NOUN
ejpam-6172	107	4	)	)	PUNCT
ejpam-6172	107	5	,	,	PUNCT
ejpam-6172	107	6	there	there	PRON
ejpam-6172	107	7	exists	exist	VERB
ejpam-6172	107	8	a−1	a−1	PROPN
ejpam-6172	107	9	r	r	NOUN
ejpam-6172	107	10	∈	∈	PROPN
ejpam-6172	107	11	a	a	DET
ejpam-6172	107	12	such	such	ADJ
ejpam-6172	107	13	that	that	SCONJ
ejpam-6172	107	14	a−1	a−1	PROPN
ejpam-6172	107	15	r	r	PROPN
ejpam-6172	107	16	∗	∗	X
ejpam-6172	107	17	ar	ar	NOUN
ejpam-6172	107	18	=	=	PROPN
ejpam-6172	107	19	a0	a0	PROPN
ejpam-6172	107	20	.	.	PUNCT
ejpam-6172	108	1	consider	consider	VERB
ejpam-6172	108	2	(	(	PUNCT
ejpam-6172	108	3	using	use	VERB
ejpam-6172	108	4	(	(	PUNCT
ejpam-6172	108	5	i	i	NOUN
ejpam-6172	108	6	)	)	PUNCT
ejpam-6172	108	7	,	,	PUNCT
ejpam-6172	108	8	(	(	PUNCT
ejpam-6172	108	9	iii	iii	NOUN
ejpam-6172	108	10	)	)	PUNCT
ejpam-6172	108	11	,	,	PUNCT
ejpam-6172	108	12	(	(	PUNCT
ejpam-6172	108	13	iv	iv	X
ejpam-6172	108	14	)	)	PUNCT
ejpam-6172	108	15	):	):	PUNCT
ejpam-6172	108	16	(	(	PUNCT
ejpam-6172	108	17	as	as	SCONJ
ejpam-6172	108	18	∗	∗	NOUN
ejpam-6172	108	19	a−1	a−1	PROPN
ejpam-6172	108	20	r	r	NOUN
ejpam-6172	108	21	)	)	PUNCT
ejpam-6172	108	22	γ0ak	γ0ak	X
ejpam-6172	109	1	=	=	SYM
ejpam-6172	109	2	(	(	PUNCT
ejpam-6172	109	3	asγ0ak	asγ0ak	X
ejpam-6172	109	4	)	)	PUNCT
ejpam-6172	109	5	∗	∗	NOUN
ejpam-6172	109	6	(	(	PUNCT
ejpam-6172	109	7	a−1	a−1	PROPN
ejpam-6172	109	8	r	r	NOUN
ejpam-6172	109	9	γ0ak	γ0ak	PUNCT
ejpam-6172	109	10	)	)	PUNCT
ejpam-6172	109	11	=	=	SYM
ejpam-6172	109	12	(	(	PUNCT
ejpam-6172	109	13	arγ0ak	arγ0ak	PROPN
ejpam-6172	109	14	)	)	PUNCT
ejpam-6172	109	15	∗	∗	NOUN
ejpam-6172	109	16	(	(	PUNCT
ejpam-6172	109	17	a−1	a−1	PROPN
ejpam-6172	109	18	r	r	NOUN
ejpam-6172	109	19	γ0ak	γ0ak	PUNCT
ejpam-6172	109	20	)	)	PUNCT
ejpam-6172	109	21	=	=	SYM
ejpam-6172	109	22	(	(	PUNCT
ejpam-6172	109	23	ar	ar	NOUN
ejpam-6172	109	24	∗	∗	NOUN
ejpam-6172	109	25	a−1	a−1	PROPN
ejpam-6172	109	26	r	r	NOUN
ejpam-6172	109	27	)	)	PUNCT
ejpam-6172	109	28	γ0ak	γ0ak	X
ejpam-6172	110	1	=	=	SYM
ejpam-6172	110	2	(	(	PUNCT
ejpam-6172	110	3	a0	a0	PROPN
ejpam-6172	110	4	∗	∗	PROPN
ejpam-6172	110	5	(	(	PUNCT
ejpam-6172	110	6	ar	ar	NOUN
ejpam-6172	110	7	∗	∗	NOUN
ejpam-6172	110	8	a−1	a−1	PROPN
ejpam-6172	110	9	r	r	NOUN
ejpam-6172	110	10	)	)	PUNCT
ejpam-6172	110	11	)	)	PUNCT
ejpam-6172	110	12	γ0ak	γ0ak	PUNCT
ejpam-6172	110	13	c.	c.	PROPN
ejpam-6172	110	14	chanoi	chanoi	PROPN
ejpam-6172	110	15	et	et	PROPN
ejpam-6172	110	16	al	al	PROPN
ejpam-6172	110	17	.	.	PUNCT
ejpam-6172	110	18	/	/	SYM
ejpam-6172	110	19	eur	eur	PROPN
ejpam-6172	110	20	.	.	PUNCT
ejpam-6172	111	1	j.	j.	PROPN
ejpam-6172	111	2	pure	pure	PROPN
ejpam-6172	111	3	appl	appl	PROPN
ejpam-6172	111	4	.	.	PROPN
ejpam-6172	111	5	math	math	PROPN
ejpam-6172	111	6	,	,	PUNCT
ejpam-6172	111	7	18	18	NUM
ejpam-6172	111	8	(	(	PUNCT
ejpam-6172	111	9	3	3	NUM
ejpam-6172	111	10	)	)	PUNCT
ejpam-6172	111	11	(	(	PUNCT
ejpam-6172	111	12	2025	2025	NUM
ejpam-6172	111	13	)	)	PUNCT
ejpam-6172	111	14	,	,	PUNCT
ejpam-6172	111	15	6172	6172	NUM
ejpam-6172	111	16	6	6	NUM
ejpam-6172	111	17	of	of	ADP
ejpam-6172	111	18	9	9	NUM
ejpam-6172	111	19	=	=	SYM
ejpam-6172	111	20	(	(	PUNCT
ejpam-6172	111	21	(	(	PUNCT
ejpam-6172	111	22	a0	a0	PROPN
ejpam-6172	111	23	∗	∗	PROPN
ejpam-6172	111	24	a0	a0	PROPN
ejpam-6172	111	25	)	)	PUNCT
ejpam-6172	111	26	∗	∗	NOUN
ejpam-6172	111	27	(	(	PUNCT
ejpam-6172	111	28	ar	ar	NOUN
ejpam-6172	111	29	∗	∗	NOUN
ejpam-6172	111	30	a−1	a−1	PROPN
ejpam-6172	111	31	r	r	NOUN
ejpam-6172	111	32	)	)	PUNCT
ejpam-6172	111	33	)	)	PUNCT
ejpam-6172	111	34	γ0ak	γ0ak	X
ejpam-6172	112	1	=	=	SYM
ejpam-6172	112	2	(	(	PUNCT
ejpam-6172	112	3	(	(	PUNCT
ejpam-6172	112	4	(	(	PUNCT
ejpam-6172	112	5	ar	ar	NOUN
ejpam-6172	112	6	∗	∗	NOUN
ejpam-6172	112	7	a−1	a−1	PROPN
ejpam-6172	112	8	r	r	NOUN
ejpam-6172	112	9	)	)	PUNCT
ejpam-6172	112	10	∗	∗	NOUN
ejpam-6172	112	11	a0	a0	NOUN
ejpam-6172	112	12	)	)	PUNCT
ejpam-6172	112	13	∗	∗	NOUN
ejpam-6172	112	14	a0)γ0ak	a0)γ0ak	NOUN
ejpam-6172	113	1	=	=	SYM
ejpam-6172	113	2	(	(	PUNCT
ejpam-6172	113	3	(	(	PUNCT
ejpam-6172	113	4	(	(	PUNCT
ejpam-6172	113	5	a0	a0	PROPN
ejpam-6172	113	6	∗	∗	NOUN
ejpam-6172	113	7	a−1	a−1	PROPN
ejpam-6172	113	8	r	r	NOUN
ejpam-6172	113	9	)	)	PUNCT
ejpam-6172	113	10	∗	∗	PROPN
ejpam-6172	113	11	ar	ar	PROPN
ejpam-6172	113	12	)	)	PUNCT
ejpam-6172	113	13	∗	∗	NOUN
ejpam-6172	113	14	a0)γ0ak	a0)γ0ak	NOUN
ejpam-6172	113	15	=	=	SYM
ejpam-6172	113	16	(	(	PUNCT
ejpam-6172	113	17	(	(	PUNCT
ejpam-6172	113	18	a−1	a−1	PROPN
ejpam-6172	113	19	r	r	PROPN
ejpam-6172	113	20	∗	∗	X
ejpam-6172	113	21	ar	ar	PROPN
ejpam-6172	113	22	)	)	PUNCT
ejpam-6172	113	23	∗	∗	NOUN
ejpam-6172	113	24	a0)γ0ak	a0)γ0ak	NOUN
ejpam-6172	114	1	=	=	SYM
ejpam-6172	114	2	(	(	PUNCT
ejpam-6172	114	3	a0	a0	PROPN
ejpam-6172	114	4	∗	∗	X
ejpam-6172	114	5	a0)γ0ak	a0)γ0ak	PROPN
ejpam-6172	114	6	=	=	SYM
ejpam-6172	114	7	a0γ0ak	a0γ0ak	X
ejpam-6172	114	8	=	=	SYM
ejpam-6172	114	9	a0	a0	PROPN
ejpam-6172	114	10	.	.	PUNCT
ejpam-6172	115	1	by	by	ADP
ejpam-6172	115	2	(	(	PUNCT
ejpam-6172	115	3	v	v	NOUN
ejpam-6172	115	4	)	)	PUNCT
ejpam-6172	115	5	,	,	PUNCT
ejpam-6172	115	6	as	as	ADP
ejpam-6172	115	7	∗	∗	NOUN
ejpam-6172	115	8	a−1	a−1	PROPN
ejpam-6172	115	9	r	r	NOUN
ejpam-6172	115	10	=	=	SYM
ejpam-6172	115	11	a0	a0	PROPN
ejpam-6172	115	12	or	or	CCONJ
ejpam-6172	115	13	ak	ak	PROPN
ejpam-6172	115	14	=	=	PROPN
ejpam-6172	115	15	a0	a0	PROPN
ejpam-6172	115	16	.	.	PUNCT
ejpam-6172	116	1	since	since	SCONJ
ejpam-6172	116	2	ak	ak	PROPN
ejpam-6172	116	3	̸=	̸=	PROPN
ejpam-6172	116	4	a0	a0	NOUN
ejpam-6172	116	5	,	,	PUNCT
ejpam-6172	116	6	as	as	ADP
ejpam-6172	116	7	∗	∗	NOUN
ejpam-6172	116	8	a−1	a−1	PROPN
ejpam-6172	116	9	r	r	NOUN
ejpam-6172	116	10	=	=	SYM
ejpam-6172	116	11	a0	a0	PROPN
ejpam-6172	116	12	.	.	PUNCT
ejpam-6172	116	13	consider	consider	VERB
ejpam-6172	116	14	:	:	PUNCT
ejpam-6172	116	15	ar	ar	PROPN
ejpam-6172	116	16	=	=	PROPN
ejpam-6172	116	17	a0	a0	PROPN
ejpam-6172	116	18	∗	∗	PROPN
ejpam-6172	116	19	ar	ar	PROPN
ejpam-6172	116	20	=	=	PUNCT
ejpam-6172	116	21	(	(	PUNCT
ejpam-6172	116	22	as	as	ADP
ejpam-6172	116	23	∗	∗	NOUN
ejpam-6172	116	24	a−1	a−1	PROPN
ejpam-6172	116	25	r	r	NOUN
ejpam-6172	116	26	)	)	PUNCT
ejpam-6172	116	27	∗	∗	NOUN
ejpam-6172	116	28	ar	ar	NOUN
ejpam-6172	116	29	=	=	PUNCT
ejpam-6172	116	30	(	(	PUNCT
ejpam-6172	116	31	ar	ar	PROPN
ejpam-6172	116	32	∗	∗	NOUN
ejpam-6172	116	33	a−1	a−1	PROPN
ejpam-6172	116	34	r	r	NOUN
ejpam-6172	116	35	)	)	PUNCT
ejpam-6172	116	36	∗	∗	NOUN
ejpam-6172	116	37	as	as	ADP
ejpam-6172	116	38	=	=	SYM
ejpam-6172	116	39	(	(	PUNCT
ejpam-6172	116	40	a0	a0	PROPN
ejpam-6172	116	41	∗	∗	PROPN
ejpam-6172	116	42	(	(	PUNCT
ejpam-6172	116	43	ar	ar	NOUN
ejpam-6172	116	44	∗	∗	NOUN
ejpam-6172	116	45	a−1	a−1	PROPN
ejpam-6172	116	46	r	r	NOUN
ejpam-6172	116	47	)	)	PUNCT
ejpam-6172	116	48	)	)	PUNCT
ejpam-6172	116	49	∗	∗	PROPN
ejpam-6172	116	50	ak	ak	PROPN
ejpam-6172	116	51	=	=	SYM
ejpam-6172	116	52	(	(	PUNCT
ejpam-6172	116	53	(	(	PUNCT
ejpam-6172	116	54	a0	a0	PROPN
ejpam-6172	116	55	∗	∗	PROPN
ejpam-6172	116	56	a0	a0	PROPN
ejpam-6172	116	57	)	)	PUNCT
ejpam-6172	116	58	∗	∗	NOUN
ejpam-6172	116	59	(	(	PUNCT
ejpam-6172	116	60	ar	ar	NOUN
ejpam-6172	116	61	∗	∗	NOUN
ejpam-6172	116	62	a−1	a−1	PROPN
ejpam-6172	116	63	r	r	NOUN
ejpam-6172	116	64	)	)	PUNCT
ejpam-6172	116	65	)	)	PUNCT
ejpam-6172	116	66	∗	∗	PROPN
ejpam-6172	116	67	ak	ak	PROPN
ejpam-6172	116	68	=	=	PUNCT
ejpam-6172	116	69	(	(	PUNCT
ejpam-6172	116	70	(	(	PUNCT
ejpam-6172	116	71	(	(	PUNCT
ejpam-6172	116	72	ar	ar	NOUN
ejpam-6172	116	73	∗	∗	NOUN
ejpam-6172	116	74	a−1	a−1	PROPN
ejpam-6172	116	75	r	r	NOUN
ejpam-6172	116	76	)	)	PUNCT
ejpam-6172	116	77	∗	∗	NOUN
ejpam-6172	116	78	a0	a0	NOUN
ejpam-6172	116	79	)	)	PUNCT
ejpam-6172	116	80	∗	∗	NOUN
ejpam-6172	116	81	a0	a0	PROPN
ejpam-6172	116	82	)	)	PUNCT
ejpam-6172	116	83	∗	∗	NOUN
ejpam-6172	116	84	ak	ak	PROPN
ejpam-6172	116	85	=	=	PUNCT
ejpam-6172	116	86	(	(	PUNCT
ejpam-6172	116	87	(	(	PUNCT
ejpam-6172	116	88	(	(	PUNCT
ejpam-6172	116	89	a0	a0	PROPN
ejpam-6172	116	90	∗	∗	NOUN
ejpam-6172	116	91	a−1	a−1	PROPN
ejpam-6172	116	92	r	r	NOUN
ejpam-6172	116	93	)	)	PUNCT
ejpam-6172	116	94	∗	∗	PROPN
ejpam-6172	116	95	ar	ar	PROPN
ejpam-6172	116	96	)	)	PUNCT
ejpam-6172	116	97	∗	∗	NOUN
ejpam-6172	116	98	a0	a0	PROPN
ejpam-6172	116	99	)	)	PUNCT
ejpam-6172	116	100	∗	∗	NOUN
ejpam-6172	116	101	ak	ak	PROPN
ejpam-6172	116	102	=	=	PUNCT
ejpam-6172	116	103	(	(	PUNCT
ejpam-6172	116	104	(	(	PUNCT
ejpam-6172	116	105	a−1	a−1	PROPN
ejpam-6172	116	106	r	r	PROPN
ejpam-6172	116	107	∗	∗	X
ejpam-6172	116	108	ar	ar	PROPN
ejpam-6172	116	109	)	)	PUNCT
ejpam-6172	116	110	∗	∗	NOUN
ejpam-6172	116	111	a0	a0	PROPN
ejpam-6172	116	112	)	)	PUNCT
ejpam-6172	116	113	∗	∗	NOUN
ejpam-6172	116	114	ak	ak	PROPN
ejpam-6172	116	115	=	=	PUNCT
ejpam-6172	116	116	(	(	PUNCT
ejpam-6172	116	117	a0	a0	PROPN
ejpam-6172	116	118	∗	∗	PROPN
ejpam-6172	116	119	a0	a0	PROPN
ejpam-6172	116	120	)	)	PUNCT
ejpam-6172	116	121	∗	∗	PROPN
ejpam-6172	116	122	ak	ak	PROPN
ejpam-6172	116	123	=	=	PROPN
ejpam-6172	116	124	a0	a0	PROPN
ejpam-6172	116	125	∗	∗	NOUN
ejpam-6172	116	126	as	as	ADP
ejpam-6172	116	127	=	=	PUNCT
ejpam-6172	116	128	as	as	ADP
ejpam-6172	116	129	.	.	PUNCT
ejpam-6172	117	1	then	then	ADV
ejpam-6172	117	2	ar	ar	PROPN
ejpam-6172	117	3	=	=	PROPN
ejpam-6172	117	4	as	as	ADP
ejpam-6172	117	5	;	;	PUNCT
ejpam-6172	117	6	this	this	PRON
ejpam-6172	117	7	is	be	AUX
ejpam-6172	117	8	a	a	DET
ejpam-6172	117	9	contradiction	contradiction	NOUN
ejpam-6172	117	10	.	.	PUNCT
ejpam-6172	118	1	hence	hence	ADV
ejpam-6172	118	2	|hk	|hk	NOUN
ejpam-6172	118	3	,	,	PUNCT
ejpam-6172	118	4	γ0	γ0	NOUN
ejpam-6172	118	5	|	|	NOUN
ejpam-6172	118	6	=	=	SYM
ejpam-6172	118	7	m.	m.	NOUN
ejpam-6172	118	8	let	let	VERB
ejpam-6172	118	9	akγ0aj	akγ0aj	PROPN
ejpam-6172	118	10	∈	∈	PROPN
ejpam-6172	118	11	hk	hk	PROPN
ejpam-6172	118	12	,	,	PUNCT
ejpam-6172	118	13	γ0	γ0	PROPN
ejpam-6172	118	14	.	.	PUNCT
ejpam-6172	119	1	suppose	suppose	VERB
ejpam-6172	119	2	akγ0aj	akγ0aj	PROPN
ejpam-6172	119	3	=	=	SYM
ejpam-6172	119	4	a0	a0	PROPN
ejpam-6172	119	5	.	.	PUNCT
ejpam-6172	120	1	by	by	ADP
ejpam-6172	120	2	(	(	PUNCT
ejpam-6172	120	3	v	v	NOUN
ejpam-6172	120	4	)	)	PUNCT
ejpam-6172	120	5	,	,	PUNCT
ejpam-6172	120	6	ak	ak	PROPN
ejpam-6172	120	7	=	=	PROPN
ejpam-6172	120	8	a0	a0	PROPN
ejpam-6172	120	9	or	or	CCONJ
ejpam-6172	120	10	aj	aj	PROPN
ejpam-6172	120	11	=	=	PROPN
ejpam-6172	120	12	a0	a0	PROPN
ejpam-6172	120	13	.	.	PUNCT
ejpam-6172	121	1	this	this	PRON
ejpam-6172	121	2	is	be	AUX
ejpam-6172	121	3	a	a	DET
ejpam-6172	121	4	contradiction	contradiction	NOUN
ejpam-6172	121	5	.	.	PUNCT
ejpam-6172	122	1	then	then	ADV
ejpam-6172	122	2	akγ0aj	akγ0aj	PROPN
ejpam-6172	122	3	̸=	̸=	PROPN
ejpam-6172	122	4	a0	a0	NOUN
ejpam-6172	122	5	,	,	PUNCT
ejpam-6172	122	6	and	and	CCONJ
ejpam-6172	122	7	akγ0aj	akγ0aj	PROPN
ejpam-6172	122	8	∈	∈	PROPN
ejpam-6172	122	9	a	a	DET
ejpam-6172	122	10	\	\	PROPN
ejpam-6172	122	11	{	{	PUNCT
ejpam-6172	122	12	a0	a0	NOUN
ejpam-6172	122	13	}	}	PUNCT
ejpam-6172	122	14	.	.	PUNCT
ejpam-6172	123	1	so	so	ADV
ejpam-6172	123	2	hk	hk	PROPN
ejpam-6172	123	3	,	,	PUNCT
ejpam-6172	123	4	γ0	γ0	VERB
ejpam-6172	123	5	⊆	⊆	NUM
ejpam-6172	123	6	a	a	DET
ejpam-6172	123	7	\	\	PROPN
ejpam-6172	123	8	{	{	PUNCT
ejpam-6172	123	9	a0	a0	PROPN
ejpam-6172	123	10	}	}	PUNCT
ejpam-6172	123	11	.	.	PUNCT
ejpam-6172	124	1	from	from	ADP
ejpam-6172	124	2	|hk	|hk	NOUN
ejpam-6172	124	3	,	,	PUNCT
ejpam-6172	124	4	γ0	γ0	NOUN
ejpam-6172	124	5	|	|	NOUN
ejpam-6172	124	6	=	=	NOUN
ejpam-6172	124	7	m	m	VERB
ejpam-6172	124	8	=	=	PUNCT
ejpam-6172	124	9	|a	|a	VERB
ejpam-6172	124	10	\	\	PROPN
ejpam-6172	124	11	{	{	PUNCT
ejpam-6172	124	12	a0}|	a0}|	PROPN
ejpam-6172	124	13	,	,	PUNCT
ejpam-6172	124	14	it	it	PRON
ejpam-6172	124	15	follows	follow	VERB
ejpam-6172	124	16	that	that	SCONJ
ejpam-6172	124	17	hk	hk	PROPN
ejpam-6172	124	18	,	,	PUNCT
ejpam-6172	124	19	γ0	γ0	NOUN
ejpam-6172	124	20	=	=	PUNCT
ejpam-6172	124	21	a	a	DET
ejpam-6172	124	22	\	\	PROPN
ejpam-6172	124	23	{	{	PUNCT
ejpam-6172	124	24	a0	a0	NOUN
ejpam-6172	124	25	}	}	PUNCT
ejpam-6172	124	26	.	.	PUNCT
ejpam-6172	125	1	since	since	SCONJ
ejpam-6172	125	2	e	e	PROPN
ejpam-6172	125	3	∈	∈	PROPN
ejpam-6172	125	4	hk	hk	PROPN
ejpam-6172	125	5	,	,	PUNCT
ejpam-6172	125	6	γ0	γ0	NOUN
ejpam-6172	125	7	,	,	PUNCT
ejpam-6172	125	8	e	e	PROPN
ejpam-6172	125	9	=	=	PROPN
ejpam-6172	125	10	akγ0ai	akγ0ai	PROPN
ejpam-6172	125	11	for	for	ADP
ejpam-6172	125	12	some	some	DET
ejpam-6172	125	13	ai	ai	VERB
ejpam-6172	125	14	∈	∈	PROPN
ejpam-6172	125	15	a	a	DET
ejpam-6172	125	16	\	\	PROPN
ejpam-6172	125	17	{	{	PUNCT
ejpam-6172	125	18	a0	a0	PROPN
ejpam-6172	125	19	}	}	PUNCT
ejpam-6172	125	20	.	.	PUNCT
ejpam-6172	126	1	moreover	moreover	ADV
ejpam-6172	126	2	,	,	PUNCT
ejpam-6172	126	3	aiγ0ak	aiγ0ak	X
ejpam-6172	126	4	=	=	SYM
ejpam-6172	126	5	eγ0(aiγ0ak	eγ0(aiγ0ak	PROPN
ejpam-6172	126	6	)	)	PUNCT
ejpam-6172	126	7	=	=	SYM
ejpam-6172	126	8	(	(	PUNCT
ejpam-6172	126	9	akγ0ai)γ0e	akγ0ai)γ0e	NOUN
ejpam-6172	126	10	=	=	SYM
ejpam-6172	126	11	eγ0e	eγ0e	NOUN
ejpam-6172	126	12	=	=	SYM
ejpam-6172	126	13	e.	e.	NOUN
ejpam-6172	126	14	setting	set	VERB
ejpam-6172	126	15	a−1	a−1	PROPN
ejpam-6172	126	16	k	k	PROPN
ejpam-6172	127	1	=	=	PUNCT
ejpam-6172	127	2	ai	ai	PROPN
ejpam-6172	127	3	,	,	PUNCT
ejpam-6172	127	4	we	we	PRON
ejpam-6172	127	5	then	then	ADV
ejpam-6172	127	6	have	have	VERB
ejpam-6172	127	7	akγ0a	akγ0a	PROPN
ejpam-6172	127	8	−1	−1	NOUN
ejpam-6172	127	9	k	k	NOUN
ejpam-6172	127	10	=	=	PUNCT
ejpam-6172	127	11	e	e	X
ejpam-6172	127	12	=	=	SYM
ejpam-6172	127	13	a−1	a−1	PROPN
ejpam-6172	127	14	k	k	PROPN
ejpam-6172	127	15	γ0ak	γ0ak	PROPN
ejpam-6172	127	16	.	.	PUNCT
ejpam-6172	128	1	by	by	ADP
ejpam-6172	128	2	claim	claim	NOUN
ejpam-6172	128	3	4	4	NUM
ejpam-6172	128	4	,	,	PUNCT
ejpam-6172	128	5	a	a	DET
ejpam-6172	128	6	\	\	PROPN
ejpam-6172	128	7	{	{	PUNCT
ejpam-6172	128	8	a0	a0	PROPN
ejpam-6172	128	9	}	}	PUNCT
ejpam-6172	128	10	is	be	AUX
ejpam-6172	128	11	a	a	DET
ejpam-6172	128	12	group	group	NOUN
ejpam-6172	128	13	under	under	ADP
ejpam-6172	128	14	the	the	DET
ejpam-6172	128	15	operation	operation	NOUN
ejpam-6172	128	16	determined	determine	VERB
ejpam-6172	128	17	by	by	ADP
ejpam-6172	128	18	γ0	γ0	NOUN
ejpam-6172	128	19	.	.	PUNCT
ejpam-6172	129	1	and	and	CCONJ
ejpam-6172	129	2	,	,	PUNCT
ejpam-6172	129	3	by	by	ADP
ejpam-6172	129	4	theorem	theorem	NOUN
ejpam-6172	129	5	1	1	NUM
ejpam-6172	129	6	,	,	PUNCT
ejpam-6172	129	7	a	a	DET
ejpam-6172	129	8	\	\	PROPN
ejpam-6172	129	9	{	{	PUNCT
ejpam-6172	129	10	a0	a0	PROPN
ejpam-6172	129	11	}	}	PUNCT
ejpam-6172	129	12	is	be	AUX
ejpam-6172	129	13	a	a	DET
ejpam-6172	129	14	group	group	NOUN
ejpam-6172	129	15	under	under	ADP
ejpam-6172	129	16	the	the	DET
ejpam-6172	129	17	operation	operation	NOUN
ejpam-6172	129	18	determined	determine	VERB
ejpam-6172	129	19	by	by	ADP
ejpam-6172	129	20	γ	γ	NOUN
ejpam-6172	129	21	for	for	ADP
ejpam-6172	129	22	all	all	DET
ejpam-6172	129	23	γ	γ	PROPN
ejpam-6172	129	24	∈	∈	PROPN
ejpam-6172	129	25	γ	γ	X
ejpam-6172	129	26	.	.	PUNCT
ejpam-6172	129	27	finally	finally	ADV
ejpam-6172	129	28	,	,	PUNCT
ejpam-6172	129	29	by	by	ADP
ejpam-6172	129	30	theorem	theorem	NOUN
ejpam-6172	129	31	4	4	NUM
ejpam-6172	129	32	,	,	PUNCT
ejpam-6172	129	33	we	we	PRON
ejpam-6172	129	34	conclude	conclude	VERB
ejpam-6172	129	35	that	that	SCONJ
ejpam-6172	129	36	a	a	DET
ejpam-6172	129	37	\	\	PROPN
ejpam-6172	129	38	{	{	PUNCT
ejpam-6172	129	39	a0	a0	PROPN
ejpam-6172	129	40	}	}	PUNCT
ejpam-6172	129	41	is	be	AUX
ejpam-6172	129	42	a	a	DET
ejpam-6172	129	43	commutative	commutative	ADJ
ejpam-6172	129	44	group	group	NOUN
ejpam-6172	129	45	under	under	ADP
ejpam-6172	129	46	the	the	DET
ejpam-6172	129	47	operation	operation	NOUN
ejpam-6172	129	48	determined	determine	VERB
ejpam-6172	129	49	by	by	ADP
ejpam-6172	129	50	γ	γ	NOUN
ejpam-6172	129	51	for	for	ADP
ejpam-6172	129	52	all	all	DET
ejpam-6172	129	53	γ	γ	PROPN
ejpam-6172	129	54	∈	∈	PROPN
ejpam-6172	129	55	γ	γ	X
ejpam-6172	129	56	.	.	PUNCT
ejpam-6172	130	1	this	this	PRON
ejpam-6172	130	2	completes	complete	VERB
ejpam-6172	130	3	the	the	DET
ejpam-6172	130	4	proof	proof	NOUN
ejpam-6172	130	5	.	.	PUNCT
ejpam-6172	131	1	an	an	DET
ejpam-6172	131	2	element	element	NOUN
ejpam-6172	131	3	e	e	NOUN
ejpam-6172	131	4	of	of	ADP
ejpam-6172	131	5	an	an	DET
ejpam-6172	131	6	ag	ag	PROPN
ejpam-6172	131	7	-	-	PROPN
ejpam-6172	131	8	groupoid	groupoid	PROPN
ejpam-6172	131	9	a	a	PRON
ejpam-6172	131	10	is	be	AUX
ejpam-6172	131	11	said	say	VERB
ejpam-6172	131	12	to	to	PART
ejpam-6172	131	13	be	be	AUX
ejpam-6172	131	14	a	a	DET
ejpam-6172	131	15	left	left	ADJ
ejpam-6172	131	16	identity	identity	NOUN
ejpam-6172	131	17	if	if	SCONJ
ejpam-6172	131	18	for	for	ADP
ejpam-6172	131	19	all	all	DET
ejpam-6172	131	20	a	a	DET
ejpam-6172	131	21	∈	∈	PROPN
ejpam-6172	131	22	a	a	PRON
ejpam-6172	131	23	,	,	PUNCT
ejpam-6172	131	24	ea	ea	X
ejpam-6172	131	25	=	=	PUNCT
ejpam-6172	131	26	a.	a.	NOUN
ejpam-6172	131	27	an	an	DET
ejpam-6172	131	28	element	element	NOUN
ejpam-6172	131	29	a0	a0	NOUN
ejpam-6172	131	30	of	of	ADP
ejpam-6172	131	31	a	a	PRON
ejpam-6172	131	32	is	be	AUX
ejpam-6172	131	33	said	say	VERB
ejpam-6172	131	34	to	to	PART
ejpam-6172	131	35	be	be	AUX
ejpam-6172	131	36	a	a	DET
ejpam-6172	131	37	left	left	ADJ
ejpam-6172	131	38	zero	zero	NUM
ejpam-6172	131	39	if	if	SCONJ
ejpam-6172	131	40	for	for	ADP
ejpam-6172	131	41	all	all	DET
ejpam-6172	131	42	a	a	DET
ejpam-6172	131	43	∈	∈	PROPN
ejpam-6172	131	44	a	a	DET
ejpam-6172	131	45	,	,	PUNCT
ejpam-6172	131	46	a0a	a0a	PROPN
ejpam-6172	131	47	=	=	SYM
ejpam-6172	131	48	a0	a0	PROPN
ejpam-6172	131	49	.	.	PUNCT
ejpam-6172	132	1	the	the	DET
ejpam-6172	132	2	following	follow	VERB
ejpam-6172	132	3	corollary	corollary	NOUN
ejpam-6172	132	4	is	be	AUX
ejpam-6172	132	5	particularly	particularly	ADV
ejpam-6172	132	6	true	true	ADJ
ejpam-6172	132	7	.	.	PUNCT
ejpam-6172	133	1	c.	c.	PROPN
ejpam-6172	133	2	chanoi	chanoi	PROPN
ejpam-6172	133	3	et	et	PROPN
ejpam-6172	133	4	al	al	PROPN
ejpam-6172	133	5	.	.	PUNCT
ejpam-6172	133	6	/	/	SYM
ejpam-6172	133	7	eur	eur	PROPN
ejpam-6172	133	8	.	.	PUNCT
ejpam-6172	134	1	j.	j.	PROPN
ejpam-6172	134	2	pure	pure	PROPN
ejpam-6172	134	3	appl	appl	PROPN
ejpam-6172	134	4	.	.	PROPN
ejpam-6172	134	5	math	math	PROPN
ejpam-6172	134	6	,	,	PUNCT
ejpam-6172	134	7	18	18	NUM
ejpam-6172	134	8	(	(	PUNCT
ejpam-6172	134	9	3	3	NUM
ejpam-6172	134	10	)	)	PUNCT
ejpam-6172	134	11	(	(	PUNCT
ejpam-6172	134	12	2025	2025	NUM
ejpam-6172	134	13	)	)	PUNCT
ejpam-6172	134	14	,	,	PUNCT
ejpam-6172	134	15	6172	6172	NUM
ejpam-6172	134	16	7	7	NUM
ejpam-6172	134	17	of	of	ADP
ejpam-6172	134	18	9	9	NUM
ejpam-6172	134	19	corollary	corollary	ADJ
ejpam-6172	134	20	2	2	NUM
ejpam-6172	134	21	.	.	PUNCT
ejpam-6172	135	1	let	let	VERB
ejpam-6172	135	2	(	(	PUNCT
ejpam-6172	135	3	a	a	PRON
ejpam-6172	135	4	,	,	PUNCT
ejpam-6172	135	5	·	·	PUNCT
ejpam-6172	135	6	)	)	PUNCT
ejpam-6172	135	7	be	be	AUX
ejpam-6172	135	8	a	a	DET
ejpam-6172	135	9	finite	finite	ADJ
ejpam-6172	135	10	ag	ag	PROPN
ejpam-6172	135	11	-	-	NOUN
ejpam-6172	135	12	groupoid	groupoid	PROPN
ejpam-6172	135	13	with	with	ADP
ejpam-6172	135	14	|a|	|a|	PROPN
ejpam-6172	135	15	>	>	SYM
ejpam-6172	135	16	1	1	NUM
ejpam-6172	135	17	,	,	PUNCT
ejpam-6172	135	18	a	a	DET
ejpam-6172	135	19	left	left	ADJ
ejpam-6172	135	20	identity	identity	NOUN
ejpam-6172	135	21	e	e	NOUN
ejpam-6172	135	22	,	,	PUNCT
ejpam-6172	135	23	a	a	DET
ejpam-6172	135	24	left	left	ADJ
ejpam-6172	135	25	zero	zero	NUM
ejpam-6172	135	26	a0	a0	NOUN
ejpam-6172	135	27	,	,	PUNCT
ejpam-6172	135	28	and	and	CCONJ
ejpam-6172	135	29	a	a	DET
ejpam-6172	135	30	·	·	PUNCT
ejpam-6172	135	31	(	(	PUNCT
ejpam-6172	135	32	b	b	X
ejpam-6172	135	33	·	·	PUNCT
ejpam-6172	135	34	c	c	X
ejpam-6172	135	35	)	)	PUNCT
ejpam-6172	135	36	=	=	SYM
ejpam-6172	136	1	(	(	PUNCT
ejpam-6172	136	2	c	c	X
ejpam-6172	136	3	·	·	PUNCT
ejpam-6172	136	4	b	b	X
ejpam-6172	136	5	)	)	PUNCT
ejpam-6172	136	6	·	·	PUNCT
ejpam-6172	137	1	a	a	PRON
ejpam-6172	137	2	for	for	ADP
ejpam-6172	137	3	all	all	DET
ejpam-6172	137	4	a	a	DET
ejpam-6172	137	5	,	,	PUNCT
ejpam-6172	137	6	b	b	NOUN
ejpam-6172	137	7	,	,	PUNCT
ejpam-6172	137	8	c	c	PROPN
ejpam-6172	137	9	∈	∈	PROPN
ejpam-6172	137	10	a.	a.	NOUN
ejpam-6172	137	11	suppose	suppose	VERB
ejpam-6172	137	12	there	there	PRON
ejpam-6172	137	13	exists	exist	VERB
ejpam-6172	137	14	an	an	DET
ejpam-6172	137	15	operation	operation	NOUN
ejpam-6172	137	16	∗	∗	NOUN
ejpam-6172	137	17	on	on	ADP
ejpam-6172	137	18	a	a	DET
ejpam-6172	137	19	such	such	ADJ
ejpam-6172	137	20	that	that	PRON
ejpam-6172	137	21	(	(	PUNCT
ejpam-6172	137	22	i)-(v	i)-(v	PROPN
ejpam-6172	137	23	)	)	PUNCT
ejpam-6172	137	24	hold	hold	VERB
ejpam-6172	137	25	:	:	PUNCT
ejpam-6172	137	26	(	(	PUNCT
ejpam-6172	137	27	i	i	NOUN
ejpam-6172	137	28	)	)	PUNCT
ejpam-6172	137	29	(	(	PUNCT
ejpam-6172	137	30	a	a	DET
ejpam-6172	137	31	,	,	PUNCT
ejpam-6172	137	32	∗	∗	NOUN
ejpam-6172	137	33	)	)	PUNCT
ejpam-6172	137	34	is	be	AUX
ejpam-6172	137	35	an	an	DET
ejpam-6172	137	36	ag	ag	PROPN
ejpam-6172	137	37	-	-	NOUN
ejpam-6172	137	38	groupoid	groupoid	PROPN
ejpam-6172	137	39	.	.	PUNCT
ejpam-6172	138	1	(	(	PUNCT
ejpam-6172	138	2	ii	ii	NOUN
ejpam-6172	138	3	)	)	PUNCT
ejpam-6172	138	4	for	for	ADP
ejpam-6172	138	5	any	any	DET
ejpam-6172	138	6	a	a	DET
ejpam-6172	138	7	∈	∈	PROPN
ejpam-6172	138	8	a	a	DET
ejpam-6172	138	9	there	there	PRON
ejpam-6172	138	10	exists	exist	VERB
ejpam-6172	138	11	b	b	PROPN
ejpam-6172	138	12	∈	∈	PROPN
ejpam-6172	138	13	a	a	DET
ejpam-6172	138	14	such	such	ADJ
ejpam-6172	138	15	that	that	DET
ejpam-6172	138	16	b	b	NOUN
ejpam-6172	138	17	∗	∗	NOUN
ejpam-6172	138	18	a	a	DET
ejpam-6172	138	19	=	=	SYM
ejpam-6172	138	20	a0	a0	PROPN
ejpam-6172	138	21	.	.	PUNCT
ejpam-6172	139	1	(	(	PUNCT
ejpam-6172	139	2	iii	iii	X
ejpam-6172	139	3	)	)	PUNCT
ejpam-6172	139	4	a0	a0	PROPN
ejpam-6172	139	5	∗	∗	NOUN
ejpam-6172	139	6	a	a	DET
ejpam-6172	139	7	=	=	NOUN
ejpam-6172	139	8	a	a	PRON
ejpam-6172	139	9	for	for	ADP
ejpam-6172	139	10	all	all	DET
ejpam-6172	139	11	a	a	DET
ejpam-6172	139	12	∈	∈	PROPN
ejpam-6172	139	13	a.	a.	NOUN
ejpam-6172	139	14	(	(	PUNCT
ejpam-6172	139	15	iv	iv	X
ejpam-6172	139	16	)	)	PUNCT
ejpam-6172	139	17	(	(	PUNCT
ejpam-6172	139	18	a	a	DET
ejpam-6172	139	19	∗	∗	NOUN
ejpam-6172	139	20	b	b	NOUN
ejpam-6172	139	21	)	)	PUNCT
ejpam-6172	139	22	·	·	PUNCT
ejpam-6172	140	1	c	c	X
ejpam-6172	140	2	=	=	SYM
ejpam-6172	140	3	(	(	PUNCT
ejpam-6172	140	4	a	a	PRON
ejpam-6172	140	5	·	·	PUNCT
ejpam-6172	140	6	c	c	X
ejpam-6172	140	7	)	)	PUNCT
ejpam-6172	140	8	∗	∗	NOUN
ejpam-6172	140	9	(	(	PUNCT
ejpam-6172	140	10	b	b	X
ejpam-6172	140	11	·	·	PUNCT
ejpam-6172	140	12	c	c	X
ejpam-6172	140	13	)	)	PUNCT
ejpam-6172	140	14	for	for	ADP
ejpam-6172	140	15	all	all	DET
ejpam-6172	140	16	a	a	DET
ejpam-6172	140	17	,	,	PUNCT
ejpam-6172	140	18	b	b	NOUN
ejpam-6172	140	19	,	,	PUNCT
ejpam-6172	140	20	c	c	PROPN
ejpam-6172	140	21	∈	∈	PROPN
ejpam-6172	140	22	a.	a.	NOUN
ejpam-6172	140	23	(	(	PUNCT
ejpam-6172	140	24	v	v	NOUN
ejpam-6172	140	25	)	)	PUNCT
ejpam-6172	140	26	for	for	ADP
ejpam-6172	140	27	any	any	DET
ejpam-6172	140	28	a	a	PRON
ejpam-6172	140	29	,	,	PUNCT
ejpam-6172	140	30	b	b	PROPN
ejpam-6172	140	31	∈	∈	PROPN
ejpam-6172	140	32	a	a	X
ejpam-6172	140	33	,	,	PUNCT
ejpam-6172	140	34	a	a	DET
ejpam-6172	140	35	·	·	PUNCT
ejpam-6172	140	36	b	b	X
ejpam-6172	140	37	=	=	SYM
ejpam-6172	140	38	a0	a0	PROPN
ejpam-6172	140	39	implies	imply	VERB
ejpam-6172	140	40	a	a	DET
ejpam-6172	140	41	=	=	SYM
ejpam-6172	140	42	a0	a0	PROPN
ejpam-6172	140	43	or	or	CCONJ
ejpam-6172	140	44	b	b	PROPN
ejpam-6172	140	45	=	=	PROPN
ejpam-6172	140	46	a0	a0	PROPN
ejpam-6172	140	47	.	.	PUNCT
ejpam-6172	141	1	then	then	ADV
ejpam-6172	141	2	(	(	PUNCT
ejpam-6172	141	3	a	a	DET
ejpam-6172	141	4	\	\	PROPN
ejpam-6172	141	5	{	{	PUNCT
ejpam-6172	141	6	a0	a0	NOUN
ejpam-6172	141	7	}	}	PUNCT
ejpam-6172	141	8	,	,	PUNCT
ejpam-6172	141	9	·	·	PUNCT
ejpam-6172	141	10	)	)	PUNCT
ejpam-6172	141	11	is	be	AUX
ejpam-6172	141	12	a	a	DET
ejpam-6172	141	13	commutative	commutative	ADJ
ejpam-6172	141	14	group	group	NOUN
ejpam-6172	141	15	.	.	PUNCT
ejpam-6172	142	1	the	the	DET
ejpam-6172	142	2	following	follow	VERB
ejpam-6172	142	3	proposition	proposition	NOUN
ejpam-6172	142	4	demonstrates	demonstrate	VERB
ejpam-6172	142	5	the	the	DET
ejpam-6172	142	6	necessity	necessity	NOUN
ejpam-6172	142	7	of	of	ADP
ejpam-6172	142	8	the	the	DET
ejpam-6172	142	9	identity	identity	NOUN
ejpam-6172	142	10	aγ(bβc	aγ(bβc	ADV
ejpam-6172	142	11	)	)	PUNCT
ejpam-6172	142	12	=	=	PUNCT
ejpam-6172	142	13	(	(	PUNCT
ejpam-6172	142	14	cγb)βa	cγb)βa	VERB
ejpam-6172	142	15	for	for	ADP
ejpam-6172	142	16	all	all	DET
ejpam-6172	142	17	a	a	DET
ejpam-6172	142	18	,	,	PUNCT
ejpam-6172	142	19	b	b	NOUN
ejpam-6172	142	20	,	,	PUNCT
ejpam-6172	142	21	c	c	PROPN
ejpam-6172	142	22	∈	∈	PROPN
ejpam-6172	142	23	a	a	PRON
ejpam-6172	142	24	and	and	CCONJ
ejpam-6172	142	25	γ	γ	NOUN
ejpam-6172	142	26	,	,	PUNCT
ejpam-6172	142	27	β	β	PROPN
ejpam-6172	142	28	∈	∈	NOUN
ejpam-6172	142	29	γ	γ	NOUN
ejpam-6172	142	30	as	as	SCONJ
ejpam-6172	142	31	stated	state	VERB
ejpam-6172	142	32	in	in	ADP
ejpam-6172	142	33	theorem	theorem	NOUN
ejpam-6172	142	34	5	5	NUM
ejpam-6172	142	35	.	.	PUNCT
ejpam-6172	142	36	proposition	proposition	NOUN
ejpam-6172	142	37	1	1	NUM
ejpam-6172	142	38	.	.	PUNCT
ejpam-6172	143	1	let	let	VERB
ejpam-6172	143	2	a	a	PRON
ejpam-6172	143	3	be	be	AUX
ejpam-6172	143	4	a	a	DET
ejpam-6172	143	5	finite	finite	NOUN
ejpam-6172	143	6	γ	γ	PROPN
ejpam-6172	143	7	-	-	PUNCT
ejpam-6172	143	8	ag	ag	ADJ
ejpam-6172	143	9	-	-	PUNCT
ejpam-6172	143	10	groupoid	groupoid	NOUN
ejpam-6172	143	11	containing	contain	VERB
ejpam-6172	143	12	at	at	ADV
ejpam-6172	143	13	least	least	ADV
ejpam-6172	143	14	two	two	NUM
ejpam-6172	143	15	elements	element	NOUN
ejpam-6172	143	16	(	(	PUNCT
ejpam-6172	143	17	|a|	|a|	NOUN
ejpam-6172	143	18	>	>	X
ejpam-6172	143	19	1	1	NUM
ejpam-6172	143	20	)	)	PUNCT
ejpam-6172	143	21	.	.	PUNCT
ejpam-6172	144	1	suppose	suppose	VERB
ejpam-6172	144	2	a	a	PRON
ejpam-6172	144	3	contains	contain	VERB
ejpam-6172	144	4	a	a	DET
ejpam-6172	144	5	left	left	ADJ
ejpam-6172	144	6	identity	identity	NOUN
ejpam-6172	144	7	e	e	NOUN
ejpam-6172	144	8	and	and	CCONJ
ejpam-6172	144	9	a	a	DET
ejpam-6172	144	10	left	left	ADJ
ejpam-6172	144	11	zero	zero	NUM
ejpam-6172	144	12	a0	a0	PROPN
ejpam-6172	144	13	.	.	PUNCT
ejpam-6172	144	14	suppose	suppose	VERB
ejpam-6172	144	15	further	far	ADV
ejpam-6172	144	16	that	that	SCONJ
ejpam-6172	144	17	there	there	PRON
ejpam-6172	144	18	exist	exist	VERB
ejpam-6172	144	19	γ0	γ0	NOUN
ejpam-6172	144	20	∈	∈	PROPN
ejpam-6172	144	21	γ	γ	NOUN
ejpam-6172	144	22	and	and	CCONJ
ejpam-6172	144	23	an	an	DET
ejpam-6172	144	24	operation	operation	NOUN
ejpam-6172	144	25	∗	∗	NOUN
ejpam-6172	144	26	of	of	ADP
ejpam-6172	144	27	a×a	a×a	PROPN
ejpam-6172	144	28	into	into	ADP
ejpam-6172	144	29	a	a	PRON
ejpam-6172	144	30	,	,	PUNCT
ejpam-6172	144	31	write	write	VERB
ejpam-6172	144	32	a	a	DET
ejpam-6172	144	33	∗	∗	NOUN
ejpam-6172	144	34	b	b	NOUN
ejpam-6172	144	35	for	for	ADP
ejpam-6172	144	36	∗(a	∗(a	NOUN
ejpam-6172	144	37	,	,	PUNCT
ejpam-6172	144	38	b	b	NOUN
ejpam-6172	144	39	)	)	PUNCT
ejpam-6172	144	40	,	,	PUNCT
ejpam-6172	144	41	such	such	ADJ
ejpam-6172	144	42	that	that	SCONJ
ejpam-6172	144	43	(	(	PUNCT
ejpam-6172	144	44	i)-(v	i)-(v	PROPN
ejpam-6172	144	45	)	)	PUNCT
ejpam-6172	144	46	hold	hold	VERB
ejpam-6172	144	47	:	:	PUNCT
ejpam-6172	144	48	(	(	PUNCT
ejpam-6172	144	49	i	i	NOUN
ejpam-6172	144	50	)	)	PUNCT
ejpam-6172	144	51	a	a	PRON
ejpam-6172	144	52	is	be	AUX
ejpam-6172	144	53	an	an	DET
ejpam-6172	144	54	ag	ag	PROPN
ejpam-6172	144	55	-	-	PUNCT
ejpam-6172	144	56	groupoid	groupoid	PROPN
ejpam-6172	144	57	under	under	ADP
ejpam-6172	144	58	∗.	∗.	PROPN
ejpam-6172	144	59	(	(	PUNCT
ejpam-6172	144	60	ii	ii	NOUN
ejpam-6172	144	61	)	)	PUNCT
ejpam-6172	144	62	for	for	ADP
ejpam-6172	144	63	any	any	DET
ejpam-6172	144	64	a	a	DET
ejpam-6172	144	65	∈	∈	PROPN
ejpam-6172	145	1	a	a	DET
ejpam-6172	145	2	there	there	PRON
ejpam-6172	145	3	exists	exist	VERB
ejpam-6172	145	4	b	b	PROPN
ejpam-6172	145	5	∈	∈	PROPN
ejpam-6172	145	6	a	a	DET
ejpam-6172	145	7	such	such	ADJ
ejpam-6172	145	8	that	that	DET
ejpam-6172	145	9	b	b	NOUN
ejpam-6172	145	10	∗	∗	NOUN
ejpam-6172	145	11	a	a	DET
ejpam-6172	145	12	=	=	SYM
ejpam-6172	145	13	a0	a0	PROPN
ejpam-6172	145	14	.	.	PUNCT
ejpam-6172	146	1	(	(	PUNCT
ejpam-6172	146	2	iii	iii	X
ejpam-6172	146	3	)	)	PUNCT
ejpam-6172	146	4	a0	a0	PROPN
ejpam-6172	146	5	∗	∗	NOUN
ejpam-6172	146	6	a	a	DET
ejpam-6172	146	7	=	=	NOUN
ejpam-6172	146	8	a	a	PRON
ejpam-6172	146	9	for	for	ADP
ejpam-6172	146	10	all	all	DET
ejpam-6172	146	11	a	a	DET
ejpam-6172	146	12	∈	∈	PROPN
ejpam-6172	146	13	a.	a.	NOUN
ejpam-6172	146	14	(	(	PUNCT
ejpam-6172	146	15	iv	iv	X
ejpam-6172	146	16	)	)	PUNCT
ejpam-6172	146	17	(	(	PUNCT
ejpam-6172	146	18	a	a	DET
ejpam-6172	146	19	∗	∗	NOUN
ejpam-6172	146	20	b)γ0c	b)γ0c	NOUN
ejpam-6172	146	21	=	=	SYM
ejpam-6172	146	22	(	(	PUNCT
ejpam-6172	146	23	aγ0c	aγ0c	NOUN
ejpam-6172	146	24	)	)	PUNCT
ejpam-6172	146	25	∗	∗	NOUN
ejpam-6172	146	26	(	(	PUNCT
ejpam-6172	146	27	bγ0c	bγ0c	PROPN
ejpam-6172	146	28	)	)	PUNCT
ejpam-6172	146	29	for	for	ADP
ejpam-6172	146	30	all	all	DET
ejpam-6172	146	31	a	a	DET
ejpam-6172	146	32	,	,	PUNCT
ejpam-6172	146	33	b	b	NOUN
ejpam-6172	146	34	,	,	PUNCT
ejpam-6172	146	35	c	c	PROPN
ejpam-6172	146	36	∈	∈	PROPN
ejpam-6172	146	37	a.	a.	NOUN
ejpam-6172	146	38	(	(	PUNCT
ejpam-6172	146	39	v	v	NOUN
ejpam-6172	146	40	)	)	PUNCT
ejpam-6172	146	41	for	for	ADP
ejpam-6172	146	42	any	any	DET
ejpam-6172	146	43	a	a	PRON
ejpam-6172	146	44	,	,	PUNCT
ejpam-6172	146	45	b	b	PROPN
ejpam-6172	146	46	∈	∈	PROPN
ejpam-6172	146	47	a	a	PRON
ejpam-6172	146	48	,	,	PUNCT
ejpam-6172	146	49	if	if	SCONJ
ejpam-6172	146	50	aγ0b	aγ0b	PROPN
ejpam-6172	146	51	=	=	SYM
ejpam-6172	146	52	a0	a0	PROPN
ejpam-6172	146	53	then	then	ADV
ejpam-6172	146	54	a	a	DET
ejpam-6172	146	55	=	=	X
ejpam-6172	146	56	a0	a0	PROPN
ejpam-6172	146	57	or	or	CCONJ
ejpam-6172	146	58	b	b	PROPN
ejpam-6172	146	59	=	=	PROPN
ejpam-6172	146	60	a0	a0	PROPN
ejpam-6172	146	61	.	.	PUNCT
ejpam-6172	147	1	then	then	ADV
ejpam-6172	147	2	,	,	PUNCT
ejpam-6172	147	3	under	under	ADP
ejpam-6172	147	4	the	the	DET
ejpam-6172	147	5	operation	operation	NOUN
ejpam-6172	147	6	determined	determine	VERB
ejpam-6172	147	7	by	by	ADP
ejpam-6172	147	8	γ0	γ0	NOUN
ejpam-6172	147	9	,	,	PUNCT
ejpam-6172	147	10	a	a	DET
ejpam-6172	147	11	\	\	PROPN
ejpam-6172	147	12	{	{	PUNCT
ejpam-6172	147	13	a0	a0	PROPN
ejpam-6172	147	14	}	}	PUNCT
ejpam-6172	147	15	is	be	AUX
ejpam-6172	147	16	a	a	DET
ejpam-6172	147	17	cancellative	cancellative	ADJ
ejpam-6172	147	18	ag	ag	PROPN
ejpam-6172	147	19	-	-	NOUN
ejpam-6172	147	20	groupoid	groupoid	PROPN
ejpam-6172	147	21	with	with	ADP
ejpam-6172	147	22	left	left	ADJ
ejpam-6172	147	23	identity	identity	NOUN
ejpam-6172	147	24	and	and	CCONJ
ejpam-6172	147	25	inverses	inverse	NOUN
ejpam-6172	147	26	(	(	PUNCT
ejpam-6172	147	27	i.e.	i.e.	X
ejpam-6172	147	28	,	,	PUNCT
ejpam-6172	147	29	for	for	ADP
ejpam-6172	147	30	each	each	DET
ejpam-6172	147	31	ak	ak	PROPN
ejpam-6172	147	32	∈	∈	PROPN
ejpam-6172	147	33	a\{a0	a\{a0	PROPN
ejpam-6172	147	34	}	}	PUNCT
ejpam-6172	147	35	there	there	PRON
ejpam-6172	147	36	exists	exist	VERB
ejpam-6172	147	37	a−1	a−1	PROPN
ejpam-6172	147	38	k	k	PROPN
ejpam-6172	147	39	∈	∈	PROPN
ejpam-6172	147	40	a\{a0	a\{a0	PROPN
ejpam-6172	147	41	}	}	PUNCT
ejpam-6172	147	42	such	such	ADJ
ejpam-6172	147	43	that	that	SCONJ
ejpam-6172	147	44	akγ0a	akγ0a	PROPN
ejpam-6172	147	45	−1	−1	NOUN
ejpam-6172	147	46	k	k	NOUN
ejpam-6172	147	47	=	=	PUNCT
ejpam-6172	147	48	e	e	X
ejpam-6172	147	49	=	=	SYM
ejpam-6172	147	50	a−1	a−1	PROPN
ejpam-6172	147	51	k	k	PROPN
ejpam-6172	147	52	γ0ak	γ0ak	PROPN
ejpam-6172	147	53	)	)	PUNCT
ejpam-6172	147	54	.	.	PUNCT
ejpam-6172	148	1	proof	proof	NOUN
ejpam-6172	148	2	.	.	PUNCT
ejpam-6172	149	1	as	as	ADP
ejpam-6172	149	2	the	the	DET
ejpam-6172	149	3	proof	proof	NOUN
ejpam-6172	149	4	of	of	ADP
ejpam-6172	149	5	theorem	theorem	NOUN
ejpam-6172	149	6	5	5	NUM
ejpam-6172	149	7	,	,	PUNCT
ejpam-6172	149	8	under	under	ADP
ejpam-6172	149	9	the	the	DET
ejpam-6172	149	10	operation	operation	NOUN
ejpam-6172	149	11	determined	determine	VERB
ejpam-6172	149	12	by	by	ADP
ejpam-6172	149	13	γ0	γ0	NOUN
ejpam-6172	149	14	,	,	PUNCT
ejpam-6172	149	15	we	we	PRON
ejpam-6172	149	16	have	have	VERB
ejpam-6172	149	17	a	a	DET
ejpam-6172	149	18	\	\	PROPN
ejpam-6172	149	19	{	{	PUNCT
ejpam-6172	149	20	a0	a0	PROPN
ejpam-6172	149	21	}	}	PUNCT
ejpam-6172	149	22	is	be	AUX
ejpam-6172	149	23	an	an	DET
ejpam-6172	149	24	ag	ag	PROPN
ejpam-6172	149	25	-	-	PUNCT
ejpam-6172	149	26	groupoid	groupoid	PROPN
ejpam-6172	149	27	with	with	ADP
ejpam-6172	149	28	left	left	ADJ
ejpam-6172	149	29	identity	identity	NOUN
ejpam-6172	149	30	,	,	PUNCT
ejpam-6172	149	31	and	and	CCONJ
ejpam-6172	149	32	for	for	ADP
ejpam-6172	149	33	any	any	DET
ejpam-6172	149	34	ak	ak	PROPN
ejpam-6172	149	35	∈	∈	PROPN
ejpam-6172	149	36	a	a	DET
ejpam-6172	149	37	\	\	PROPN
ejpam-6172	149	38	{	{	PUNCT
ejpam-6172	149	39	a0	a0	NOUN
ejpam-6172	149	40	}	}	PUNCT
ejpam-6172	149	41	,	,	PUNCT
ejpam-6172	149	42	e	e	X
ejpam-6172	149	43	=	=	PUNCT
ejpam-6172	149	44	akγ0ai	akγ0ai	PROPN
ejpam-6172	149	45	for	for	ADP
ejpam-6172	149	46	some	some	DET
ejpam-6172	149	47	ai	ai	VERB
ejpam-6172	149	48	∈	∈	PROPN
ejpam-6172	149	49	a	a	DET
ejpam-6172	149	50	\	\	PROPN
ejpam-6172	149	51	{	{	PUNCT
ejpam-6172	149	52	a0	a0	PROPN
ejpam-6172	149	53	}	}	PUNCT
ejpam-6172	149	54	.	.	PUNCT
ejpam-6172	150	1	consider	consider	VERB
ejpam-6172	150	2	(	(	PUNCT
ejpam-6172	150	3	using	use	VERB
ejpam-6172	150	4	theorem	theorem	NOUN
ejpam-6172	150	5	2	2	NUM
ejpam-6172	150	6	):	):	PUNCT
ejpam-6172	150	7	aiγ0ak	aiγ0ak	X
ejpam-6172	150	8	=	=	SYM
ejpam-6172	150	9	eγ0(aiγ0ak	eγ0(aiγ0ak	PROPN
ejpam-6172	150	10	)	)	PUNCT
ejpam-6172	150	11	=	=	SYM
ejpam-6172	150	12	(	(	PUNCT
ejpam-6172	150	13	eγ0e)γ0(aiγ0ak	eγ0e)γ0(aiγ0ak	PROPN
ejpam-6172	150	14	)	)	PUNCT
ejpam-6172	150	15	=	=	PUNCT
ejpam-6172	150	16	(	(	PUNCT
ejpam-6172	150	17	eγ0ai)γ0(eγ0ak	eγ0ai)γ0(eγ0ak	PROPN
ejpam-6172	150	18	)	)	PUNCT
ejpam-6172	151	1	=	=	SYM
ejpam-6172	151	2	(	(	PUNCT
ejpam-6172	151	3	(	(	PUNCT
ejpam-6172	151	4	eγ0ak)γ0ai)γ0e	eγ0ak)γ0ai)γ0e	NOUN
ejpam-6172	151	5	=	=	SYM
ejpam-6172	151	6	(	(	PUNCT
ejpam-6172	151	7	akγ0ai)γ0e	akγ0ai)γ0e	PROPN
ejpam-6172	151	8	c.	c.	PROPN
ejpam-6172	151	9	chanoi	chanoi	PROPN
ejpam-6172	151	10	et	et	PROPN
ejpam-6172	151	11	al	al	PROPN
ejpam-6172	151	12	.	.	PUNCT
ejpam-6172	151	13	/	/	SYM
ejpam-6172	151	14	eur	eur	PROPN
ejpam-6172	151	15	.	.	PUNCT
ejpam-6172	152	1	j.	j.	PROPN
ejpam-6172	152	2	pure	pure	PROPN
ejpam-6172	152	3	appl	appl	PROPN
ejpam-6172	152	4	.	.	PROPN
ejpam-6172	152	5	math	math	PROPN
ejpam-6172	152	6	,	,	PUNCT
ejpam-6172	152	7	18	18	NUM
ejpam-6172	152	8	(	(	PUNCT
ejpam-6172	152	9	3	3	NUM
ejpam-6172	152	10	)	)	PUNCT
ejpam-6172	152	11	(	(	PUNCT
ejpam-6172	152	12	2025	2025	NUM
ejpam-6172	152	13	)	)	PUNCT
ejpam-6172	152	14	,	,	PUNCT
ejpam-6172	152	15	6172	6172	NUM
ejpam-6172	152	16	8	8	NUM
ejpam-6172	152	17	of	of	ADP
ejpam-6172	152	18	9	9	NUM
ejpam-6172	152	19	=	=	SYM
ejpam-6172	152	20	eγ0e	eγ0e	NOUN
ejpam-6172	152	21	=	=	SYM
ejpam-6172	152	22	e.	e.	PROPN
ejpam-6172	152	23	then	then	ADV
ejpam-6172	152	24	aiγ0ak	aiγ0ak	PROPN
ejpam-6172	152	25	=	=	PUNCT
ejpam-6172	152	26	e.	e.	PROPN
ejpam-6172	152	27	finally	finally	ADV
ejpam-6172	152	28	,	,	PUNCT
ejpam-6172	152	29	let	let	VERB
ejpam-6172	152	30	ai	ai	VERB
ejpam-6172	152	31	,	,	PUNCT
ejpam-6172	152	32	aj	aj	PROPN
ejpam-6172	152	33	,	,	PUNCT
ejpam-6172	152	34	ak	ak	PROPN
ejpam-6172	152	35	∈	∈	PROPN
ejpam-6172	152	36	a	a	DET
ejpam-6172	152	37	\	\	PROPN
ejpam-6172	152	38	{	{	PUNCT
ejpam-6172	152	39	a0	a0	PROPN
ejpam-6172	152	40	}	}	PUNCT
ejpam-6172	152	41	be	be	VERB
ejpam-6172	152	42	such	such	ADJ
ejpam-6172	152	43	that	that	SCONJ
ejpam-6172	152	44	aiγ0ak	aiγ0ak	PROPN
ejpam-6172	152	45	=	=	SYM
ejpam-6172	152	46	ajγ0ak	ajγ0ak	PROPN
ejpam-6172	152	47	.	.	PUNCT
ejpam-6172	153	1	moreover	moreover	ADV
ejpam-6172	153	2	,	,	PUNCT
ejpam-6172	153	3	as	as	ADP
ejpam-6172	153	4	the	the	DET
ejpam-6172	153	5	proof	proof	NOUN
ejpam-6172	153	6	of	of	ADP
ejpam-6172	153	7	theorem	theorem	NOUN
ejpam-6172	153	8	5	5	NUM
ejpam-6172	153	9	,	,	PUNCT
ejpam-6172	153	10	there	there	PRON
ejpam-6172	153	11	exists	exist	VERB
ejpam-6172	153	12	a−1	a−1	PROPN
ejpam-6172	153	13	k	k	PROPN
ejpam-6172	153	14	∈	∈	PROPN
ejpam-6172	153	15	a	a	DET
ejpam-6172	153	16	\	\	PROPN
ejpam-6172	153	17	{	{	PUNCT
ejpam-6172	153	18	a0	a0	NOUN
ejpam-6172	153	19	}	}	PUNCT
ejpam-6172	153	20	such	such	ADJ
ejpam-6172	153	21	that	that	SCONJ
ejpam-6172	153	22	a−1	a−1	PROPN
ejpam-6172	153	23	k	k	PROPN
ejpam-6172	153	24	γ0ak	γ0ak	PUNCT
ejpam-6172	153	25	=	=	PUNCT
ejpam-6172	153	26	e	e	X
ejpam-6172	153	27	=	=	PUNCT
ejpam-6172	153	28	akγ0a	akγ0a	PROPN
ejpam-6172	153	29	−1	−1	NOUN
ejpam-6172	153	30	k	k	X
ejpam-6172	153	31	.	.	PUNCT
ejpam-6172	154	1	consider	consider	VERB
ejpam-6172	154	2	:	:	PUNCT
ejpam-6172	154	3	ai	ai	VERB
ejpam-6172	154	4	=	=	NOUN
ejpam-6172	154	5	eγ0ai	eγ0ai	PROPN
ejpam-6172	154	6	=	=	PUNCT
ejpam-6172	154	7	(	(	PUNCT
ejpam-6172	154	8	a−1	a−1	PROPN
ejpam-6172	154	9	k	k	PROPN
ejpam-6172	154	10	γ0ak)γ0ai	γ0ak)γ0ai	PROPN
ejpam-6172	154	11	=	=	SYM
ejpam-6172	154	12	(	(	PUNCT
ejpam-6172	154	13	aiγ0ak)γ0a	aiγ0ak)γ0a	ADJ
ejpam-6172	154	14	−1	−1	NOUN
ejpam-6172	154	15	k	k	NOUN
ejpam-6172	155	1	=	=	PUNCT
ejpam-6172	156	1	(	(	PUNCT
ejpam-6172	156	2	ajγ0ak)γ0a	ajγ0ak)γ0a	PROPN
ejpam-6172	156	3	−1	−1	NOUN
ejpam-6172	156	4	k	k	PROPN
ejpam-6172	157	1	=	=	PUNCT
ejpam-6172	157	2	(	(	PUNCT
ejpam-6172	157	3	a−1	a−1	PROPN
ejpam-6172	157	4	k	k	PROPN
ejpam-6172	157	5	γ0ak)γ0aj	γ0ak)γ0aj	PROPN
ejpam-6172	157	6	=	=	SYM
ejpam-6172	157	7	eγ0aj	eγ0aj	PROPN
ejpam-6172	157	8	=	=	SYM
ejpam-6172	157	9	aj	aj	PROPN
ejpam-6172	157	10	.	.	PUNCT
ejpam-6172	158	1	similarly	similarly	ADV
ejpam-6172	158	2	,	,	PUNCT
ejpam-6172	158	3	if	if	SCONJ
ejpam-6172	158	4	ai	ai	VERB
ejpam-6172	158	5	,	,	PUNCT
ejpam-6172	158	6	aj	aj	PROPN
ejpam-6172	158	7	,	,	PUNCT
ejpam-6172	158	8	ak	ak	PROPN
ejpam-6172	158	9	∈	∈	PROPN
ejpam-6172	158	10	a	a	DET
ejpam-6172	158	11	\	\	PROPN
ejpam-6172	158	12	{	{	PUNCT
ejpam-6172	158	13	a0	a0	NOUN
ejpam-6172	158	14	}	}	PUNCT
ejpam-6172	158	15	such	such	ADJ
ejpam-6172	158	16	that	that	DET
ejpam-6172	158	17	akγ0ai	akγ0ai	PROPN
ejpam-6172	158	18	=	=	PRON
ejpam-6172	158	19	akγ0aj	akγ0aj	PROPN
ejpam-6172	158	20	then	then	ADV
ejpam-6172	158	21	ai	ai	VERB
ejpam-6172	158	22	=	=	PROPN
ejpam-6172	158	23	aj	aj	PROPN
ejpam-6172	158	24	.	.	PUNCT
ejpam-6172	159	1	hence	hence	ADV
ejpam-6172	159	2	the	the	DET
ejpam-6172	159	3	proof	proof	NOUN
ejpam-6172	159	4	is	be	AUX
ejpam-6172	159	5	complete	complete	ADJ
ejpam-6172	159	6	.	.	PUNCT
ejpam-6172	160	1	acknowledgements	acknowledgement	VERB
ejpam-6172	160	2	the	the	DET
ejpam-6172	160	3	research	research	NOUN
ejpam-6172	160	4	on	on	ADP
ejpam-6172	160	5	”	"	PUNCT
ejpam-6172	160	6	finite	finite	PROPN
ejpam-6172	160	7	γ	γ	PROPN
ejpam-6172	160	8	-	-	PUNCT
ejpam-6172	160	9	ag	ag	NOUN
ejpam-6172	160	10	-	-	PUNCT
ejpam-6172	160	11	groupoids	groupoid	NOUN
ejpam-6172	160	12	with	with	ADP
ejpam-6172	160	13	left	left	ADJ
ejpam-6172	160	14	identities	identity	NOUN
ejpam-6172	160	15	and	and	CCONJ
ejpam-6172	160	16	left	leave	VERB
ejpam-6172	160	17	zeros	zero	NOUN
ejpam-6172	160	18	”	"	PUNCT
ejpam-6172	160	19	is	be	AUX
ejpam-6172	160	20	supported	support	VERB
ejpam-6172	160	21	by	by	ADP
ejpam-6172	160	22	research	research	NOUN
ejpam-6172	160	23	,	,	PUNCT
ejpam-6172	160	24	innovation	innovation	NOUN
ejpam-6172	160	25	and	and	CCONJ
ejpam-6172	160	26	academic	academic	ADJ
ejpam-6172	160	27	services	service	NOUN
ejpam-6172	160	28	fund	fund	NOUN
ejpam-6172	160	29	,	,	PUNCT
ejpam-6172	160	30	faculty	faculty	NOUN
ejpam-6172	160	31	of	of	ADP
ejpam-6172	160	32	science	science	NOUN
ejpam-6172	160	33	,	,	PUNCT
ejpam-6172	160	34	khon	khon	PROPN
ejpam-6172	160	35	kaen	kaen	PROPN
ejpam-6172	160	36	university	university	PROPN
ejpam-6172	160	37	.	.	PUNCT
ejpam-6172	161	1	references	reference	NOUN
ejpam-6172	161	2	[	[	X
ejpam-6172	161	3	1	1	NUM
ejpam-6172	161	4	]	]	PUNCT
ejpam-6172	161	5	w.	w.	PROPN
ejpam-6172	161	6	a.	a.	PROPN
ejpam-6172	161	7	dudek	dudek	PROPN
ejpam-6172	161	8	and	and	CCONJ
ejpam-6172	161	9	r.	r.	PROPN
ejpam-6172	161	10	s.	s.	PROPN
ejpam-6172	162	1	gigoń.	gigoń.	PROPN
ejpam-6172	162	2	completely	completely	ADV
ejpam-6172	162	3	inverse	inverse	ADJ
ejpam-6172	162	4	ag**-groupoids	ag**-groupoid	NOUN
ejpam-6172	162	5	.	.	PUNCT
ejpam-6172	163	1	semigroup	semigroup	PROPN
ejpam-6172	163	2	forum	forum	PROPN
ejpam-6172	163	3	,	,	PUNCT
ejpam-6172	163	4	87(1):201–229	87(1):201–229	PROPN
ejpam-6172	163	5	,	,	PUNCT
ejpam-6172	163	6	2013	2013	NUM
ejpam-6172	163	7	.	.	PUNCT
ejpam-6172	164	1	[	[	X
ejpam-6172	164	2	2	2	NUM
ejpam-6172	164	3	]	]	PUNCT
ejpam-6172	164	4	m.	m.	NOUN
ejpam-6172	164	5	a.	a.	PROPN
ejpam-6172	164	6	kazim	kazim	PROPN
ejpam-6172	164	7	and	and	CCONJ
ejpam-6172	164	8	m.	m.	PROPN
ejpam-6172	164	9	naseeruddin	naseeruddin	PROPN
ejpam-6172	164	10	.	.	PUNCT
ejpam-6172	165	1	on	on	ADP
ejpam-6172	165	2	almost	almost	ADV
ejpam-6172	165	3	semigroups	semigroup	NOUN
ejpam-6172	165	4	.	.	PUNCT
ejpam-6172	166	1	portugaliae	portugaliae	PROPN
ejpam-6172	166	2	mathematica	mathematica	PROPN
ejpam-6172	166	3	,	,	PUNCT
ejpam-6172	166	4	36(1):41–47	36(1):41–47	NUM
ejpam-6172	166	5	,	,	PUNCT
ejpam-6172	166	6	1977	1977	NUM
ejpam-6172	166	7	.	.	PUNCT
ejpam-6172	167	1	[	[	X
ejpam-6172	167	2	3	3	NUM
ejpam-6172	167	3	]	]	PUNCT
ejpam-6172	167	4	a.	a.	NOUN
ejpam-6172	167	5	d.	d.	PROPN
ejpam-6172	167	6	keedwell	keedwell	PROPN
ejpam-6172	167	7	and	and	CCONJ
ejpam-6172	167	8	j.	j.	PROPN
ejpam-6172	167	9	dénes	dénes	PROPN
ejpam-6172	167	10	.	.	PUNCT
ejpam-6172	168	1	latin	latin	ADJ
ejpam-6172	168	2	squares	square	NOUN
ejpam-6172	168	3	and	and	CCONJ
ejpam-6172	168	4	their	their	PRON
ejpam-6172	168	5	applications	application	NOUN
ejpam-6172	168	6	.	.	PUNCT
ejpam-6172	169	1	1974	1974	NUM
ejpam-6172	169	2	.	.	PUNCT
ejpam-6172	170	1	[	[	X
ejpam-6172	170	2	4	4	X
ejpam-6172	170	3	]	]	X
ejpam-6172	170	4	p.	p.	NOUN
ejpam-6172	170	5	v.	v.	ADP
ejpam-6172	170	6	protic	protic	PROPN
ejpam-6172	170	7	and	and	CCONJ
ejpam-6172	170	8	n.	n.	NOUN
ejpam-6172	170	9	stevanovic	stevanovic	PROPN
ejpam-6172	170	10	.	.	PUNCT
ejpam-6172	171	1	on	on	ADP
ejpam-6172	171	2	abel	abel	PROPN
ejpam-6172	171	3	-	-	PUNCT
ejpam-6172	171	4	grassmann	grassmann	PROPN
ejpam-6172	171	5	’s	’s	PART
ejpam-6172	171	6	groupoids	groupoid	NOUN
ejpam-6172	171	7	.	.	PUNCT
ejpam-6172	172	1	proc	proc	PROPN
ejpam-6172	172	2	.	.	PUNCT
ejpam-6172	173	1	math	math	NOUN
ejpam-6172	173	2	.	.	PUNCT
ejpam-6172	173	3	conf	conf	PROPN
ejpam-6172	173	4	.	.	PUNCT
ejpam-6172	174	1	priötina	priötina	NOUN
ejpam-6172	174	2	,	,	PUNCT
ejpam-6172	174	3	page	page	NOUN
ejpam-6172	174	4	31–38	31–38	NUM
ejpam-6172	174	5	,	,	PUNCT
ejpam-6172	174	6	1994	1994	NUM
ejpam-6172	174	7	.	.	PUNCT
ejpam-6172	175	1	[	[	X
ejpam-6172	175	2	5	5	X
ejpam-6172	175	3	]	]	PUNCT
ejpam-6172	175	4	q.	q.	PROPN
ejpam-6172	175	5	mushtaq	mushtaq	PROPN
ejpam-6172	175	6	and	and	CCONJ
ejpam-6172	175	7	m.	m.	PROPN
ejpam-6172	175	8	s.	s.	PROPN
ejpam-6172	175	9	kamran	kamran	PROPN
ejpam-6172	175	10	.	.	PUNCT
ejpam-6172	176	1	finite	finite	PROPN
ejpam-6172	176	2	ag	ag	PROPN
ejpam-6172	176	3	-	-	PROPN
ejpam-6172	176	4	groupoid	groupoid	PROPN
ejpam-6172	176	5	with	with	ADP
ejpam-6172	176	6	left	left	ADJ
ejpam-6172	176	7	identity	identity	NOUN
ejpam-6172	176	8	and	and	CCONJ
ejpam-6172	176	9	left	leave	VERB
ejpam-6172	176	10	zero	zero	NUM
ejpam-6172	176	11	.	.	PUNCT
ejpam-6172	177	1	international	international	ADJ
ejpam-6172	177	2	journal	journal	NOUN
ejpam-6172	177	3	of	of	ADP
ejpam-6172	177	4	mathematics	mathematics	PROPN
ejpam-6172	177	5	and	and	CCONJ
ejpam-6172	177	6	mathematical	mathematical	ADJ
ejpam-6172	177	7	sciences	science	NOUN
ejpam-6172	177	8	,	,	PUNCT
ejpam-6172	177	9	27(6):3873–389	27(6):3873–389	NUM
ejpam-6172	177	10	,	,	PUNCT
ejpam-6172	177	11	2000	2000	NUM
ejpam-6172	177	12	.	.	PUNCT
ejpam-6172	178	1	[	[	X
ejpam-6172	178	2	6	6	NUM
ejpam-6172	178	3	]	]	PUNCT
ejpam-6172	178	4	m.	m.	NOUN
ejpam-6172	178	5	k.	k.	PROPN
ejpam-6172	178	6	sen	sen	PROPN
ejpam-6172	178	7	.	.	PROPN
ejpam-6172	179	1	on	on	ADP
ejpam-6172	179	2	γ	γ	NOUN
ejpam-6172	179	3	-	-	PUNCT
ejpam-6172	179	4	semigroups	semigroup	NOUN
ejpam-6172	179	5	,	,	PUNCT
ejpam-6172	179	6	algebra	algebra	NOUN
ejpam-6172	179	7	and	and	CCONJ
ejpam-6172	179	8	its	its	PRON
ejpam-6172	179	9	applications	application	NOUN
ejpam-6172	179	10	.	.	PUNCT
ejpam-6172	180	1	algebra	algebra	NOUN
ejpam-6172	180	2	and	and	CCONJ
ejpam-6172	180	3	its	its	PRON
ejpam-6172	180	4	applications	application	NOUN
ejpam-6172	180	5	,	,	PUNCT
ejpam-6172	180	6	page	page	NOUN
ejpam-6172	180	7	301–308	301–308	NUM
ejpam-6172	180	8	,	,	PUNCT
ejpam-6172	180	9	1981	1981	NUM
ejpam-6172	180	10	.	.	PUNCT
ejpam-6172	181	1	c.	c.	PROPN
ejpam-6172	181	2	chanoi	chanoi	PROPN
ejpam-6172	181	3	et	et	PROPN
ejpam-6172	181	4	al	al	PROPN
ejpam-6172	181	5	.	.	PUNCT
ejpam-6172	181	6	/	/	SYM
ejpam-6172	181	7	eur	eur	PROPN
ejpam-6172	181	8	.	.	PUNCT
ejpam-6172	182	1	j.	j.	PROPN
ejpam-6172	182	2	pure	pure	PROPN
ejpam-6172	182	3	appl	appl	PROPN
ejpam-6172	182	4	.	.	PROPN
ejpam-6172	182	5	math	math	PROPN
ejpam-6172	182	6	,	,	PUNCT
ejpam-6172	182	7	18	18	NUM
ejpam-6172	182	8	(	(	PUNCT
ejpam-6172	182	9	3	3	NUM
ejpam-6172	182	10	)	)	PUNCT
ejpam-6172	182	11	(	(	PUNCT
ejpam-6172	182	12	2025	2025	NUM
ejpam-6172	182	13	)	)	PUNCT
ejpam-6172	182	14	,	,	PUNCT
ejpam-6172	182	15	6172	6172	NUM
ejpam-6172	182	16	9	9	NUM
ejpam-6172	182	17	of	of	ADP
ejpam-6172	182	18	9	9	NUM
ejpam-6172	182	19	[	[	SYM
ejpam-6172	182	20	7	7	NUM
ejpam-6172	182	21	]	]	PUNCT
ejpam-6172	182	22	m.	m.	NOUN
ejpam-6172	182	23	k.	k.	PROPN
ejpam-6172	182	24	sen	sen	PROPN
ejpam-6172	182	25	and	and	CCONJ
ejpam-6172	182	26	n.	n.	PROPN
ejpam-6172	182	27	saha	saha	PROPN
ejpam-6172	182	28	.	.	PUNCT
ejpam-6172	183	1	k	k	X
ejpam-6172	183	2	:	:	PUNCT
ejpam-6172	183	3	on	on	ADP
ejpam-6172	183	4	γ	γ	PROPN
ejpam-6172	183	5	-	-	PUNCT
ejpam-6172	183	6	semigroup	semigroup	PROPN
ejpam-6172	183	7	i.	i.	NOUN
ejpam-6172	183	8	bulletin	bulletin	NOUN
ejpam-6172	183	9	of	of	ADP
ejpam-6172	183	10	calcutta	calcutta	PROPN
ejpam-6172	183	11	mathematical	mathematical	ADJ
ejpam-6172	183	12	society	society	NOUN
ejpam-6172	183	13	,	,	PUNCT
ejpam-6172	183	14	78:181–186	78:181–186	PROPN
ejpam-6172	183	15	,	,	PUNCT
ejpam-6172	183	16	1986	1986	NUM
ejpam-6172	183	17	.	.	PUNCT
ejpam-6172	184	1	[	[	X
ejpam-6172	184	2	8	8	NUM
ejpam-6172	184	3	]	]	PUNCT
ejpam-6172	184	4	m.	m.	NOUN
ejpam-6172	184	5	k.	k.	PROPN
ejpam-6172	184	6	sen	sen	PROPN
ejpam-6172	184	7	and	and	CCONJ
ejpam-6172	184	8	s.	s.	PROPN
ejpam-6172	184	9	chattopadhyay	chattopadhyay	PROPN
ejpam-6172	184	10	.	.	PUNCT
ejpam-6172	185	1	wreath	wreath	NOUN
ejpam-6172	185	2	product	product	NOUN
ejpam-6172	185	3	of	of	ADP
ejpam-6172	185	4	a	a	DET
ejpam-6172	185	5	semigroup	semigroup	NOUN
ejpam-6172	185	6	and	and	CCONJ
ejpam-6172	185	7	a	a	DET
ejpam-6172	185	8	γ	γ	NOUN
ejpam-6172	185	9	-	-	PUNCT
ejpam-6172	185	10	semigroup	semigroup	NOUN
ejpam-6172	185	11	.	.	PUNCT
ejpam-6172	186	1	discussiones	discussione	NOUN
ejpam-6172	186	2	mathematicae	mathematicae	VERB
ejpam-6172	186	3	general	general	ADJ
ejpam-6172	186	4	algebra	algebra	PROPN
ejpam-6172	186	5	and	and	CCONJ
ejpam-6172	186	6	applications	application	NOUN
ejpam-6172	186	7	,	,	PUNCT
ejpam-6172	186	8	28:161–178	28:161–178	NUM
ejpam-6172	186	9	,	,	PUNCT
ejpam-6172	186	10	2008	2008	NUM
ejpam-6172	186	11	.	.	PUNCT
ejpam-6172	187	1	[	[	X
ejpam-6172	187	2	9	9	NUM
ejpam-6172	187	3	]	]	PUNCT
ejpam-6172	187	4	k.	k.	NOUN
ejpam-6172	187	5	wattanatripop	wattanatripop	PROPN
ejpam-6172	187	6	and	and	CCONJ
ejpam-6172	187	7	t.	t.	PROPN
ejpam-6172	187	8	changphas	changphas	PROPN
ejpam-6172	187	9	.	.	PUNCT
ejpam-6172	188	1	on	on	ADP
ejpam-6172	188	2	left	left	ADJ
ejpam-6172	188	3	and	and	CCONJ
ejpam-6172	188	4	right	right	ADJ
ejpam-6172	188	5	a	a	DET
ejpam-6172	188	6	-	-	PUNCT
ejpam-6172	188	7	ideals	ideal	NOUN
ejpam-6172	188	8	of	of	ADP
ejpam-6172	188	9	a	a	DET
ejpam-6172	188	10	γ	γ	NOUN
ejpam-6172	188	11	-	-	PUNCT
ejpam-6172	188	12	semigroup	semigroup	NOUN
ejpam-6172	188	13	.	.	PUNCT
ejpam-6172	188	14	thai	thai	PROPN
ejpam-6172	188	15	journal	journal	PROPN
ejpam-6172	188	16	of	of	ADP
ejpam-6172	188	17	mathematics	mathematic	NOUN
ejpam-6172	188	18	,	,	PUNCT
ejpam-6172	188	19	page	page	NOUN
ejpam-6172	188	20	87–96	87–96	NUM
ejpam-6172	188	21	,	,	PUNCT
ejpam-6172	188	22	2018	2018	NUM
ejpam-6172	188	23	.	.	PUNCT
ejpam-6172	189	1	[	[	X
ejpam-6172	189	2	10	10	NUM
ejpam-6172	189	3	]	]	PUNCT
ejpam-6172	189	4	t.	t.	NOUN
ejpam-6172	189	5	shah	shah	PROPN
ejpam-6172	189	6	and	and	CCONJ
ejpam-6172	189	7	i.	i.	PROPN
ejpam-6172	189	8	rehman	rehman	PROPN
ejpam-6172	189	9	.	.	PUNCT
ejpam-6172	190	1	on	on	ADP
ejpam-6172	190	2	γ	γ	NOUN
ejpam-6172	190	3	-	-	PUNCT
ejpam-6172	190	4	ideals	ideal	NOUN
ejpam-6172	190	5	and	and	CCONJ
ejpam-6172	190	6	γ	γ	NOUN
ejpam-6172	190	7	-	-	ADJ
ejpam-6172	190	8	bi	bi	NOUN
ejpam-6172	190	9	-	-	NOUN
ejpam-6172	190	10	ideals	ideal	NOUN
ejpam-6172	190	11	in	in	ADP
ejpam-6172	190	12	γ	γ	PROPN
ejpam-6172	190	13	-	-	PUNCT
ejpam-6172	190	14	ag	ag	ADJ
ejpam-6172	190	15	-	-	PUNCT
ejpam-6172	190	16	groupoids	groupoid	NOUN
ejpam-6172	190	17	.	.	PUNCT
ejpam-6172	191	1	international	international	ADJ
ejpam-6172	191	2	journal	journal	PROPN
ejpam-6172	191	3	of	of	ADP
ejpam-6172	191	4	algebra	algebra	PROPN
ejpam-6172	191	5	,	,	PUNCT
ejpam-6172	191	6	4:267–276	4:267–276	NUM
ejpam-6172	191	7	,	,	PUNCT
ejpam-6172	191	8	2010	2010	NUM
ejpam-6172	191	9	.	.	PUNCT
ejpam-6172	192	1	[	[	X
ejpam-6172	192	2	11	11	NUM
ejpam-6172	192	3	]	]	PUNCT
ejpam-6172	192	4	m.	m.	PROPN
ejpam-6172	192	5	khan	khan	PROPN
ejpam-6172	192	6	,	,	PUNCT
ejpam-6172	192	7	s.	s.	PROPN
ejpam-6172	192	8	anis	anis	PROPN
ejpam-6172	192	9	,	,	PUNCT
ejpam-6172	192	10	and	and	CCONJ
ejpam-6172	192	11	f.	f.	PROPN
ejpam-6172	192	12	faisal	faisal	PROPN
ejpam-6172	192	13	.	.	PUNCT
ejpam-6172	193	1	on	on	ADP
ejpam-6172	193	2	fuzzy	fuzzy	ADJ
ejpam-6172	193	3	-	-	PUNCT
ejpam-6172	193	4	γ	γ	NOUN
ejpam-6172	193	5	-	-	PUNCT
ejpam-6172	193	6	ideals	ideal	NOUN
ejpam-6172	193	7	of	of	ADP
ejpam-6172	193	8	γ	γ	PROPN
ejpam-6172	193	9	-	-	PUNCT
ejpam-6172	193	10	abel	abel	NOUN
ejpam-6172	193	11	-	-	PUNCT
ejpam-6172	193	12	grassmann	grassmann	PROPN
ejpam-6172	193	13	’s	’s	PART
ejpam-6172	193	14	groupoids	groupoid	NOUN
ejpam-6172	193	15	.	.	PUNCT
ejpam-6172	194	1	research	research	NOUN
ejpam-6172	194	2	journal	journal	PROPN
ejpam-6172	194	3	of	of	ADP
ejpam-6172	194	4	applied	apply	VERB
ejpam-6172	194	5	sciences	science	NOUN
ejpam-6172	194	6	,	,	PUNCT
ejpam-6172	194	7	engineering	engineering	NOUN
ejpam-6172	194	8	and	and	CCONJ
ejpam-6172	194	9	technology	technology	NOUN
ejpam-6172	194	10	,	,	PUNCT
ejpam-6172	194	11	6:1326–1334	6:1326–1334	NUM
ejpam-6172	194	12	,	,	PUNCT
ejpam-6172	194	13	2013	2013	NUM
ejpam-6172	194	14	.	.	PUNCT