id	sid	tid	token	lemma	pos
iajs-2723	1	1	108	108	NUM
iajs-2723	1	2	this	this	DET
iajs-2723	1	3	work	work	NOUN
iajs-2723	1	4	is	be	AUX
iajs-2723	1	5	licensed	license	VERB
iajs-2723	1	6	under	under	ADP
iajs-2723	1	7	a	a	DET
iajs-2723	1	8	creative	creative	ADJ
iajs-2723	1	9	commons	common	NOUN
iajs-2723	1	10	attribution	attribution	NOUN
iajs-2723	1	11	4.0	4.0	NUM
iajs-2723	1	12	international	international	ADJ
iajs-2723	1	13	license	license	NOUN
iajs-2723	1	14	.	.	PUNCT
iajs-2723	2	1	(	(	PUNCT
iajs-2723	2	2	𝜽𝟏,𝜽𝟐)-derivation	𝜽𝟏,𝜽𝟐)-derivation	NOUN
iajs-2723	2	3	pair	pair	NOUN
iajs-2723	2	4	on	on	ADP
iajs-2723	2	5	rings	ring	NOUN
iajs-2723	2	6	mohammed	mohammed	PROPN
iajs-2723	2	7	khalid	khalid	PROPN
iajs-2723	2	8	shahoodh	shahoodh	PROPN
iajs-2723	2	9	ministry	ministry	PROPN
iajs-2723	2	10	of	of	ADP
iajs-2723	2	11	education	education	PROPN
iajs-2723	2	12	,	,	PUNCT
iajs-2723	2	13	ramadi	ramadi	NOUN
iajs-2723	2	14	directorate	directorate	NOUN
iajs-2723	2	15	of	of	ADP
iajs-2723	2	16	education	education	NOUN
iajs-2723	2	17	,	,	PUNCT
iajs-2723	2	18	anbar	anbar	NOUN
iajs-2723	2	19	-	-	PUNCT
iajs-2723	2	20	iraq	iraq	PROPN
iajs-2723	2	21	moha861122@yahoo.com	moha861122@yahoo.com	NOUN
iajs-2723	3	1	abstract	abstract	ADJ
iajs-2723	3	2	ring	ring	NOUN
iajs-2723	3	3	theory	theory	NOUN
iajs-2723	3	4	is	be	AUX
iajs-2723	3	5	one	one	NUM
iajs-2723	3	6	of	of	ADP
iajs-2723	3	7	the	the	DET
iajs-2723	3	8	influential	influential	ADJ
iajs-2723	3	9	branches	branch	NOUN
iajs-2723	3	10	of	of	ADP
iajs-2723	3	11	abstract	abstract	ADJ
iajs-2723	3	12	algebra	algebra	NOUN
iajs-2723	3	13	.	.	PUNCT
iajs-2723	4	1	in	in	ADP
iajs-2723	4	2	this	this	DET
iajs-2723	4	3	field	field	NOUN
iajs-2723	4	4	,	,	PUNCT
iajs-2723	4	5	many	many	ADJ
iajs-2723	4	6	algebraic	algebraic	ADJ
iajs-2723	4	7	problems	problem	NOUN
iajs-2723	4	8	have	have	AUX
iajs-2723	4	9	been	be	AUX
iajs-2723	4	10	considered	consider	VERB
iajs-2723	4	11	by	by	ADP
iajs-2723	4	12	mathematical	mathematical	ADJ
iajs-2723	4	13	researchers	researcher	NOUN
iajs-2723	4	14	who	who	PRON
iajs-2723	4	15	are	be	AUX
iajs-2723	4	16	working	work	VERB
iajs-2723	4	17	in	in	ADP
iajs-2723	4	18	this	this	DET
iajs-2723	4	19	field	field	NOUN
iajs-2723	4	20	.	.	PUNCT
iajs-2723	5	1	however	however	ADV
iajs-2723	5	2	,	,	PUNCT
iajs-2723	5	3	some	some	DET
iajs-2723	5	4	new	new	ADJ
iajs-2723	5	5	concepts	concept	NOUN
iajs-2723	5	6	have	have	AUX
iajs-2723	5	7	been	be	AUX
iajs-2723	5	8	created	create	VERB
iajs-2723	5	9	and	and	CCONJ
iajs-2723	5	10	developed	develop	VERB
iajs-2723	5	11	to	to	PART
iajs-2723	5	12	present	present	VERB
iajs-2723	5	13	some	some	DET
iajs-2723	5	14	algebraic	algebraic	ADJ
iajs-2723	5	15	structures	structure	NOUN
iajs-2723	5	16	with	with	ADP
iajs-2723	5	17	their	their	PRON
iajs-2723	5	18	properties	property	NOUN
iajs-2723	5	19	.	.	PUNCT
iajs-2723	6	1	rings	ring	NOUN
iajs-2723	6	2	with	with	ADP
iajs-2723	6	3	derivations	derivation	NOUN
iajs-2723	6	4	have	have	AUX
iajs-2723	6	5	been	be	AUX
iajs-2723	6	6	studied	study	VERB
iajs-2723	6	7	fifty	fifty	NUM
iajs-2723	6	8	years	year	NOUN
iajs-2723	6	9	ago	ago	ADV
iajs-2723	6	10	,	,	PUNCT
iajs-2723	6	11	especially	especially	ADV
iajs-2723	6	12	the	the	DET
iajs-2723	6	13	relationships	relationship	NOUN
iajs-2723	6	14	between	between	ADP
iajs-2723	6	15	the	the	DET
iajs-2723	6	16	derivations	derivation	NOUN
iajs-2723	6	17	and	and	CCONJ
iajs-2723	6	18	the	the	DET
iajs-2723	6	19	structure	structure	NOUN
iajs-2723	6	20	of	of	ADP
iajs-2723	6	21	a	a	DET
iajs-2723	6	22	ring	ring	NOUN
iajs-2723	6	23	.	.	PUNCT
iajs-2723	7	1	by	by	ADP
iajs-2723	7	2	using	use	VERB
iajs-2723	7	3	the	the	DET
iajs-2723	7	4	notatin	notatin	NOUN
iajs-2723	7	5	of	of	ADP
iajs-2723	7	6	derivation	derivation	NOUN
iajs-2723	7	7	,	,	PUNCT
iajs-2723	7	8	many	many	ADJ
iajs-2723	7	9	results	result	NOUN
iajs-2723	7	10	have	have	AUX
iajs-2723	7	11	been	be	AUX
iajs-2723	7	12	obtained	obtain	VERB
iajs-2723	7	13	in	in	ADP
iajs-2723	7	14	the	the	DET
iajs-2723	7	15	literature	literature	NOUN
iajs-2723	7	16	with	with	ADP
iajs-2723	7	17	different	different	ADJ
iajs-2723	7	18	types	type	NOUN
iajs-2723	7	19	of	of	ADP
iajs-2723	7	20	derivations	derivation	NOUN
iajs-2723	7	21	.	.	PUNCT
iajs-2723	8	1	in	in	ADP
iajs-2723	8	2	this	this	DET
iajs-2723	8	3	paper	paper	NOUN
iajs-2723	8	4	,	,	PUNCT
iajs-2723	8	5	the	the	DET
iajs-2723	8	6	concept	concept	NOUN
iajs-2723	8	7	of	of	ADP
iajs-2723	8	8	the	the	DET
iajs-2723	8	9	derivation	derivation	NOUN
iajs-2723	8	10	theory	theory	NOUN
iajs-2723	8	11	of	of	ADP
iajs-2723	8	12	a	a	DET
iajs-2723	8	13	ring	ring	NOUN
iajs-2723	8	14	has	have	AUX
iajs-2723	8	15	been	be	AUX
iajs-2723	8	16	considered	consider	VERB
iajs-2723	8	17	.	.	PUNCT
iajs-2723	9	1	this	this	DET
iajs-2723	9	2	study	study	NOUN
iajs-2723	9	3	presented	present	VERB
iajs-2723	9	4	the	the	DET
iajs-2723	9	5	definition	definition	NOUN
iajs-2723	9	6	of	of	ADP
iajs-2723	9	7	(	(	PUNCT
iajs-2723	9	8	𝜃1,𝜃2)-derivation	𝜃1,𝜃2)-derivation	NOUN
iajs-2723	9	9	pair	pair	NOUN
iajs-2723	9	10	and	and	CCONJ
iajs-2723	9	11	jordan	jordan	PROPN
iajs-2723	9	12	(	(	PUNCT
iajs-2723	9	13	𝜃1	𝜃1	PROPN
iajs-2723	9	14	,	,	PUNCT
iajs-2723	9	15	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	9	16	pair	pair	NOUN
iajs-2723	9	17	on	on	ADP
iajs-2723	9	18	an	an	DET
iajs-2723	9	19	associative	associative	ADJ
iajs-2723	9	20	ring	ring	NOUN
iajs-2723	9	21	γ	γ	NOUN
iajs-2723	9	22	,	,	PUNCT
iajs-2723	9	23	and	and	CCONJ
iajs-2723	9	24	the	the	DET
iajs-2723	9	25	relation	relation	NOUN
iajs-2723	9	26	between	between	ADP
iajs-2723	9	27	them	they	PRON
iajs-2723	9	28	.	.	PUNCT
iajs-2723	10	1	furthermore	furthermore	ADV
iajs-2723	10	2	,	,	PUNCT
iajs-2723	10	3	we	we	PRON
iajs-2723	10	4	study	study	VERB
iajs-2723	10	5	the	the	DET
iajs-2723	10	6	concept	concept	NOUN
iajs-2723	10	7	of	of	ADP
iajs-2723	10	8	prime	prime	ADJ
iajs-2723	10	9	rings	ring	NOUN
iajs-2723	10	10	under	under	ADP
iajs-2723	10	11	this	this	DET
iajs-2723	10	12	notion	notion	NOUN
iajs-2723	10	13	by	by	ADP
iajs-2723	10	14	introducing	introduce	VERB
iajs-2723	10	15	some	some	PRON
iajs-2723	10	16	of	of	ADP
iajs-2723	10	17	its	its	PRON
iajs-2723	10	18	properties	property	NOUN
iajs-2723	10	19	where	where	SCONJ
iajs-2723	10	20	𝜃1	𝜃1	VERB
iajs-2723	10	21	and	and	CCONJ
iajs-2723	10	22	𝜃2	𝜃2	NOUN
iajs-2723	10	23	are	be	AUX
iajs-2723	10	24	two	two	NUM
iajs-2723	10	25	mappings	mapping	NOUN
iajs-2723	10	26	of	of	ADP
iajs-2723	10	27	γ	γ	NOUN
iajs-2723	10	28	into	into	ADP
iajs-2723	10	29	itself	itself	PRON
iajs-2723	10	30	.	.	PUNCT
iajs-2723	11	1	keywords	keyword	NOUN
iajs-2723	11	2	:	:	PUNCT
iajs-2723	11	3	ring	ring	NOUN
iajs-2723	11	4	theory	theory	NOUN
iajs-2723	11	5	,	,	PUNCT
iajs-2723	11	6	derivation	derivation	NOUN
iajs-2723	11	7	theory	theory	NOUN
iajs-2723	11	8	,	,	PUNCT
iajs-2723	11	9	prime	prime	ADJ
iajs-2723	11	10	ring	ring	NOUN
iajs-2723	11	11	,	,	PUNCT
iajs-2723	11	12	derivation	derivation	NOUN
iajs-2723	11	13	pair	pair	NOUN
iajs-2723	11	14	,	,	PUNCT
iajs-2723	11	15	semiprime	semiprime	NOUN
iajs-2723	11	16	ring	ring	NOUN
iajs-2723	11	17	.	.	PUNCT
iajs-2723	12	1	1	1	X
iajs-2723	12	2	.	.	X
iajs-2723	12	3	introduction	introduction	NOUN
iajs-2723	12	4	the	the	DET
iajs-2723	12	5	study	study	NOUN
iajs-2723	12	6	of	of	ADP
iajs-2723	12	7	derivation	derivation	NOUN
iajs-2723	12	8	has	have	AUX
iajs-2723	12	9	been	be	AUX
iajs-2723	12	10	initiated	initiate	VERB
iajs-2723	12	11	from	from	ADP
iajs-2723	12	12	the	the	DET
iajs-2723	12	13	development	development	NOUN
iajs-2723	12	14	of	of	ADP
iajs-2723	12	15	galois	galois	PROPN
iajs-2723	12	16	theory	theory	NOUN
iajs-2723	12	17	and	and	CCONJ
iajs-2723	12	18	the	the	DET
iajs-2723	12	19	theory	theory	NOUN
iajs-2723	12	20	of	of	ADP
iajs-2723	12	21	invariants	invariant	NOUN
iajs-2723	12	22	.	.	PUNCT
iajs-2723	13	1	this	this	DET
iajs-2723	13	2	theory	theory	NOUN
iajs-2723	13	3	has	have	AUX
iajs-2723	13	4	been	be	AUX
iajs-2723	13	5	studied	study	VERB
iajs-2723	13	6	very	very	ADV
iajs-2723	13	7	widely	widely	ADV
iajs-2723	13	8	by	by	ADP
iajs-2723	13	9	many	many	ADJ
iajs-2723	13	10	researchers	researcher	NOUN
iajs-2723	13	11	on	on	ADP
iajs-2723	13	12	various	various	ADJ
iajs-2723	13	13	algebraic	algebraic	ADJ
iajs-2723	13	14	structures	structure	NOUN
iajs-2723	13	15	.	.	PUNCT
iajs-2723	14	1	the	the	DET
iajs-2723	14	2	author	author	NOUN
iajs-2723	14	3	in	in	ADP
iajs-2723	14	4	[	[	X
iajs-2723	14	5	1	1	NUM
iajs-2723	14	6	]	]	PUNCT
iajs-2723	14	7	studied	study	VERB
iajs-2723	14	8	this	this	DET
iajs-2723	14	9	topic	topic	NOUN
iajs-2723	14	10	on	on	ADP
iajs-2723	14	11	𝐻∗-algebra	𝐻∗-algebra	PROPN
iajs-2723	14	12	by	by	ADP
iajs-2723	14	13	introducing	introduce	VERB
iajs-2723	14	14	jordan	jordan	PROPN
iajs-2723	14	15	∗-derivation	∗-derivation	NOUN
iajs-2723	14	16	pair	pair	NOUN
iajs-2723	14	17	.	.	PUNCT
iajs-2723	15	1	while	while	SCONJ
iajs-2723	15	2	the	the	DET
iajs-2723	15	3	authors	author	NOUN
iajs-2723	15	4	in	in	ADP
iajs-2723	15	5	[	[	X
iajs-2723	15	6	2	2	NUM
iajs-2723	15	7	]	]	PUNCT
iajs-2723	15	8	considered	consider	VERB
iajs-2723	15	9	the	the	DET
iajs-2723	15	10	topic	topic	NOUN
iajs-2723	15	11	of	of	ADP
iajs-2723	15	12	bci	bci	NOUN
iajs-2723	15	13	-	-	PUNCT
iajs-2723	15	14	algebras	algebra	NOUN
iajs-2723	15	15	,	,	PUNCT
iajs-2723	15	16	and	and	CCONJ
iajs-2723	15	17	the	the	DET
iajs-2723	15	18	same	same	ADJ
iajs-2723	15	19	topic	topic	NOUN
iajs-2723	15	20	has	have	AUX
iajs-2723	15	21	been	be	AUX
iajs-2723	15	22	investigated	investigate	VERB
iajs-2723	15	23	on	on	ADP
iajs-2723	15	24	bcc	bcc	PROPN
iajs-2723	15	25	-	-	PUNCT
iajs-2723	15	26	algebras	algebras	PROPN
iajs-2723	15	27	by	by	ADP
iajs-2723	15	28	the	the	DET
iajs-2723	15	29	authors	author	NOUN
iajs-2723	15	30	in	in	ADP
iajs-2723	15	31	[	[	X
iajs-2723	15	32	3	3	NUM
iajs-2723	15	33	]	]	PUNCT
iajs-2723	15	34	.	.	PUNCT
iajs-2723	16	1	moreover	moreover	ADV
iajs-2723	16	2	,	,	PUNCT
iajs-2723	16	3	some	some	DET
iajs-2723	16	4	other	other	ADJ
iajs-2723	16	5	works	work	VERB
iajs-2723	16	6	with	with	ADP
iajs-2723	16	7	different	different	ADJ
iajs-2723	16	8	algebraic	algebraic	ADJ
iajs-2723	16	9	structures	structure	NOUN
iajs-2723	16	10	can	can	AUX
iajs-2723	16	11	be	be	AUX
iajs-2723	16	12	found	find	VERB
iajs-2723	16	13	in	in	ADP
iajs-2723	16	14	[	[	PUNCT
iajs-2723	16	15	4	4	NUM
iajs-2723	16	16	-	-	SYM
iajs-2723	16	17	5	5	NUM
iajs-2723	16	18	]	]	PUNCT
iajs-2723	16	19	.	.	PUNCT
iajs-2723	17	1	on	on	ADP
iajs-2723	17	2	the	the	DET
iajs-2723	17	3	other	other	ADJ
iajs-2723	17	4	hand	hand	NOUN
iajs-2723	17	5	,	,	PUNCT
iajs-2723	17	6	some	some	DET
iajs-2723	17	7	other	other	ADJ
iajs-2723	17	8	studies	study	NOUN
iajs-2723	17	9	have	have	AUX
iajs-2723	17	10	studied	study	VERB
iajs-2723	17	11	this	this	DET
iajs-2723	17	12	topic	topic	NOUN
iajs-2723	17	13	with	with	ADP
iajs-2723	17	14	some	some	DET
iajs-2723	17	15	types	type	NOUN
iajs-2723	17	16	of	of	ADP
iajs-2723	17	17	rings	ring	NOUN
iajs-2723	17	18	such	such	ADJ
iajs-2723	17	19	as	as	ADP
iajs-2723	17	20	prime	prime	ADJ
iajs-2723	17	21	and	and	CCONJ
iajs-2723	17	22	semiprime	semiprime	NOUN
iajs-2723	17	23	rings	ring	NOUN
iajs-2723	17	24	,	,	PUNCT
iajs-2723	17	25	see	see	VERB
iajs-2723	17	26	[	[	X
iajs-2723	17	27	6	6	NUM
iajs-2723	17	28	-	-	SYM
iajs-2723	17	29	8	8	NUM
iajs-2723	17	30	]	]	PUNCT
iajs-2723	17	31	.	.	PUNCT
iajs-2723	18	1	[	[	X
iajs-2723	18	2	12	12	NUM
iajs-2723	18	3	]	]	PUNCT
iajs-2723	18	4	presented	present	VERB
iajs-2723	18	5	a	a	DET
iajs-2723	18	6	new	new	ADJ
iajs-2723	18	7	definition	definition	NOUN
iajs-2723	18	8	of	of	ADP
iajs-2723	18	9	derivation	derivation	NOUN
iajs-2723	18	10	pair	pair	NOUN
iajs-2723	18	11	instead	instead	ADV
iajs-2723	18	12	of	of	ADP
iajs-2723	18	13	jordan	jordan	PROPN
iajs-2723	18	14	∗-derivation	∗-derivation	PROPN
iajs-2723	18	15	pair	pair	NOUN
iajs-2723	18	16	which	which	PRON
iajs-2723	18	17	was	be	AUX
iajs-2723	18	18	provided	provide	VERB
iajs-2723	18	19	by	by	ADP
iajs-2723	18	20	[	[	X
iajs-2723	18	21	1	1	NUM
iajs-2723	18	22	]	]	PUNCT
iajs-2723	18	23	.	.	PUNCT
iajs-2723	19	1	in	in	ADP
iajs-2723	19	2	this	this	DET
iajs-2723	19	3	paper	paper	NOUN
iajs-2723	19	4	,	,	PUNCT
iajs-2723	19	5	we	we	PRON
iajs-2723	19	6	extended	extend	VERB
iajs-2723	19	7	the	the	DET
iajs-2723	19	8	results	result	NOUN
iajs-2723	19	9	of	of	ADP
iajs-2723	19	10	[	[	X
iajs-2723	19	11	12	12	NUM
iajs-2723	19	12	]	]	PUNCT
iajs-2723	19	13	by	by	ADP
iajs-2723	19	14	introducing	introduce	VERB
iajs-2723	19	15	the	the	DET
iajs-2723	19	16	notion	notion	NOUN
iajs-2723	19	17	of	of	ADP
iajs-2723	19	18	(	(	PUNCT
iajs-2723	19	19	𝜃1,𝜃2)-derivation	𝜃1,𝜃2)-derivation	NOUN
iajs-2723	19	20	pair	pair	NOUN
iajs-2723	19	21	and	and	CCONJ
iajs-2723	19	22	studied	study	VERB
iajs-2723	19	23	some	some	PRON
iajs-2723	19	24	of	of	ADP
iajs-2723	19	25	its	its	PRON
iajs-2723	19	26	properties	property	NOUN
iajs-2723	19	27	.	.	PUNCT
iajs-2723	20	1	2	2	X
iajs-2723	20	2	.	.	NUM
iajs-2723	20	3	basic	basic	ADJ
iajs-2723	20	4	concepts	concept	NOUN
iajs-2723	20	5	this	this	DET
iajs-2723	20	6	section	section	NOUN
iajs-2723	20	7	contains	contain	VERB
iajs-2723	20	8	some	some	PRON
iajs-2723	20	9	of	of	ADP
iajs-2723	20	10	the	the	DET
iajs-2723	20	11	previous	previous	ADJ
iajs-2723	20	12	results	result	NOUN
iajs-2723	20	13	that	that	PRON
iajs-2723	20	14	are	be	AUX
iajs-2723	20	15	needed	need	VERB
iajs-2723	20	16	in	in	ADP
iajs-2723	20	17	this	this	DET
iajs-2723	20	18	study	study	NOUN
iajs-2723	20	19	which	which	PRON
iajs-2723	20	20	are	be	AUX
iajs-2723	20	21	as	as	SCONJ
iajs-2723	20	22	follows	follow	VERB
iajs-2723	20	23	:	:	PUNCT
iajs-2723	20	24	definition	definition	NOUN
iajs-2723	20	25	2.1[9	2.1[9	NUM
iajs-2723	20	26	]	]	X
iajs-2723	21	1	ibn	ibn	PROPN
iajs-2723	21	2	al	al	PROPN
iajs-2723	21	3	haitham	haitham	PROPN
iajs-2723	21	4	journal	journal	PROPN
iajs-2723	21	5	for	for	ADP
iajs-2723	21	6	pure	pure	ADJ
iajs-2723	21	7	and	and	CCONJ
iajs-2723	21	8	applied	applied	ADJ
iajs-2723	21	9	sciences	sciences	PROPN
iajs-2723	21	10	journal	journal	PROPN
iajs-2723	21	11	homepage	homepage	NOUN
iajs-2723	21	12	:	:	PUNCT
iajs-2723	21	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2723	21	14	doi	doi	NOUN
iajs-2723	21	15	:	:	PUNCT
iajs-2723	21	16	10.30526/35.2.2723	10.30526/35.2.2723	PROPN
iajs-2723	21	17	article	article	NOUN
iajs-2723	21	18	history	history	NOUN
iajs-2723	21	19	:	:	PUNCT
iajs-2723	21	20	received	receive	VERB
iajs-2723	21	21	21	21	NUM
iajs-2723	21	22	november	november	PROPN
iajs-2723	21	23	,	,	PUNCT
iajs-2723	21	24	2021	2021	NUM
iajs-2723	21	25	,	,	PUNCT
iajs-2723	21	26	accepted,25	accepted,25	PROPN
iajs-2723	21	27	,	,	PUNCT
iajs-2723	21	28	january	january	PROPN
iajs-2723	21	29	,	,	PUNCT
iajs-2723	21	30	2022	2022	NUM
iajs-2723	21	31	,	,	PUNCT
iajs-2723	21	32	published	publish	VERB
iajs-2723	21	33	in	in	ADP
iajs-2723	21	34	april	april	PROPN
iajs-2723	21	35	2022	2022	NUM
iajs-2723	21	36	.	.	PUNCT
iajs-2723	22	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2723	22	2	mailto:moha861122@yahoo.com	mailto:moha861122@yahoo.com	PROPN
iajs-2723	22	3	ibn	ibn	PROPN
iajs-2723	22	4	al	al	PROPN
iajs-2723	22	5	-	-	PUNCT
iajs-2723	22	6	haitham	haitham	PROPN
iajs-2723	22	7	jour	jour	X
iajs-2723	22	8	.	.	PROPN
iajs-2723	22	9	for	for	ADP
iajs-2723	22	10	pure	pure	ADJ
iajs-2723	22	11	&	&	CCONJ
iajs-2723	22	12	appl	appl	PROPN
iajs-2723	22	13	.	.	PUNCT
iajs-2723	23	1	sci	sci	PROPN
iajs-2723	23	2	.	.	PROPN
iajs-2723	24	1	53	53	NUM
iajs-2723	24	2	(	(	PUNCT
iajs-2723	24	3	2)2022	2)2022	VERB
iajs-2723	24	4	109	109	NUM
iajs-2723	24	5	a	a	DET
iajs-2723	24	6	non	non	ADJ
iajs-2723	24	7	-	-	ADJ
iajs-2723	24	8	empty	empty	ADJ
iajs-2723	24	9	set	set	NOUN
iajs-2723	24	10	γ	γ	PROPN
iajs-2723	24	11	is	be	AUX
iajs-2723	24	12	said	say	VERB
iajs-2723	24	13	to	to	PART
iajs-2723	24	14	be	be	AUX
iajs-2723	24	15	an	an	DET
iajs-2723	24	16	associative	associative	ADJ
iajs-2723	24	17	ring	ring	NOUN
iajs-2723	24	18	,	,	PUNCT
iajs-2723	24	19	if	if	SCONJ
iajs-2723	24	20	for	for	ADP
iajs-2723	24	21	all	all	DET
iajs-2723	24	22	𝑐1	𝑐1	NOUN
iajs-2723	24	23	,	,	PUNCT
iajs-2723	24	24	𝑐2	𝑐2	NOUN
iajs-2723	24	25	,	,	PUNCT
iajs-2723	24	26	𝑐3	𝑐3	AUX
iajs-2723	24	27	∈	∈	PROPN
iajs-2723	24	28	γ	γ	NOUN
iajs-2723	24	29	there	there	PRON
iajs-2723	24	30	exist	exist	VERB
iajs-2723	24	31	two	two	NUM
iajs-2723	24	32	binary	binary	ADJ
iajs-2723	24	33	operations	operation	NOUN
iajs-2723	24	34	defined	define	VERB
iajs-2723	24	35	on	on	ADP
iajs-2723	24	36	γ	γ	NOUN
iajs-2723	24	37	and	and	CCONJ
iajs-2723	24	38	denoted	denote	VERB
iajs-2723	24	39	by	by	ADP
iajs-2723	24	40	+	+	SYM
iajs-2723	24	41	and	and	CCONJ
iajs-2723	24	42	⋅	⋅	PROPN
iajs-2723	24	43	respectively	respectively	ADV
iajs-2723	24	44	,	,	PUNCT
iajs-2723	25	1	such	such	ADJ
iajs-2723	25	2	that	that	DET
iajs-2723	25	3	i.	i.	NOUN
iajs-2723	25	4	𝑐1	𝑐1	NOUN
iajs-2723	25	5	+	+	CCONJ
iajs-2723	25	6	𝑐2	𝑐2	NOUN
iajs-2723	25	7	=	=	SYM
iajs-2723	25	8	𝑐2	𝑐2	NOUN
iajs-2723	25	9	+	+	CCONJ
iajs-2723	25	10	𝑐1	𝑐1	NOUN
iajs-2723	25	11	ii	ii	NOUN
iajs-2723	25	12	.	.	PUNCT
iajs-2723	26	1	(	(	PUNCT
iajs-2723	26	2	𝑐1	𝑐1	NOUN
iajs-2723	26	3	+	+	NOUN
iajs-2723	26	4	𝑐2	𝑐2	NOUN
iajs-2723	26	5	)	)	PUNCT
iajs-2723	27	1	+	+	NUM
iajs-2723	27	2	𝑐3	𝑐3	NOUN
iajs-2723	27	3	=	=	NOUN
iajs-2723	27	4	𝑐1	𝑐1	NOUN
iajs-2723	27	5	+	+	CCONJ
iajs-2723	27	6	(	(	PUNCT
iajs-2723	27	7	𝑐2	𝑐2	NOUN
iajs-2723	27	8	+	+	NUM
iajs-2723	27	9	𝑐3	𝑐3	NOUN
iajs-2723	27	10	)	)	PUNCT
iajs-2723	27	11	iii	iii	PROPN
iajs-2723	27	12	.	.	PUNCT
iajs-2723	28	1	∀𝑐1	∀𝑐1	PROPN
iajs-2723	28	2	∈	∈	PROPN
iajs-2723	28	3	γ	γ	X
iajs-2723	28	4	∃	∃	PROPN
iajs-2723	28	5	0	0	NUM
iajs-2723	28	6	∈	∈	PROPN
iajs-2723	28	7	γ	γ	NOUN
iajs-2723	28	8	such	such	ADJ
iajs-2723	28	9	that	that	SCONJ
iajs-2723	28	10	𝑐1	𝑐1	NOUN
iajs-2723	28	11	+	+	X
iajs-2723	28	12	0	0	NUM
iajs-2723	28	13	=	=	SYM
iajs-2723	28	14	0	0	PUNCT
iajs-2723	29	1	+	+	CCONJ
iajs-2723	29	2	𝑐1	𝑐1	NOUN
iajs-2723	29	3	=	=	SYM
iajs-2723	29	4	𝑐1	𝑐1	NOUN
iajs-2723	29	5	iv	iv	X
iajs-2723	29	6	.	.	PUNCT
iajs-2723	30	1	∀𝑐1	∀𝑐1	PROPN
iajs-2723	30	2	∈	∈	PROPN
iajs-2723	30	3	γ	γ	PROPN
iajs-2723	30	4	∃	∃	PROPN
iajs-2723	30	5	−	−	PROPN
iajs-2723	30	6	𝑐1	𝑐1	NOUN
iajs-2723	30	7	∈	∈	PROPN
iajs-2723	30	8	γ	γ	NOUN
iajs-2723	30	9	such	such	ADJ
iajs-2723	30	10	that	that	SCONJ
iajs-2723	30	11	−𝑐1	−𝑐1	PROPN
iajs-2723	30	12	+	+	CCONJ
iajs-2723	30	13	𝑐1	𝑐1	NOUN
iajs-2723	30	14	=	=	PUNCT
iajs-2723	31	1	𝑐1	𝑐1	NOUN
iajs-2723	31	2	+	+	X
iajs-2723	31	3	(	(	PUNCT
iajs-2723	31	4	−𝑐1	−𝑐1	PROPN
iajs-2723	31	5	)	)	PUNCT
iajs-2723	31	6	=	=	SYM
iajs-2723	31	7	0	0	NUM
iajs-2723	31	8	v.	v.	CCONJ
iajs-2723	31	9	𝑐1	𝑐1	NOUN
iajs-2723	31	10	⋅	⋅	PROPN
iajs-2723	31	11	𝑐2	𝑐2	NOUN
iajs-2723	31	12	∈	∈	PROPN
iajs-2723	31	13	γ	γ	X
iajs-2723	31	14	vi	vi	PROPN
iajs-2723	31	15	.	.	PUNCT
iajs-2723	32	1	(	(	PUNCT
iajs-2723	32	2	𝑐1	𝑐1	NOUN
iajs-2723	32	3	⋅	⋅	PROPN
iajs-2723	32	4	𝑐2	𝑐2	NOUN
iajs-2723	32	5	)	)	PUNCT
iajs-2723	32	6	⋅	⋅	PROPN
iajs-2723	32	7	𝑐3	𝑐3	NOUN
iajs-2723	32	8	=	=	PUNCT
iajs-2723	32	9	𝑐1	𝑐1	NOUN
iajs-2723	32	10	⋅	⋅	X
iajs-2723	32	11	(	(	PUNCT
iajs-2723	32	12	𝑐2	𝑐2	PROPN
iajs-2723	32	13	⋅	⋅	PROPN
iajs-2723	32	14	𝑐3	𝑐3	PROPN
iajs-2723	32	15	)	)	PUNCT
iajs-2723	32	16	vii	vii	PROPN
iajs-2723	32	17	.	.	PUNCT
iajs-2723	33	1	𝑐1	𝑐1	NOUN
iajs-2723	33	2	⋅	⋅	X
iajs-2723	33	3	(	(	PUNCT
iajs-2723	33	4	𝑐2	𝑐2	NOUN
iajs-2723	33	5	+	+	NUM
iajs-2723	33	6	𝑐3	𝑐3	NOUN
iajs-2723	33	7	)	)	PUNCT
iajs-2723	33	8	=	=	SYM
iajs-2723	33	9	𝑐1	𝑐1	NOUN
iajs-2723	33	10	⋅	⋅	PROPN
iajs-2723	33	11	𝑐2	𝑐2	NOUN
iajs-2723	33	12	+	+	CCONJ
iajs-2723	33	13	𝑐1	𝑐1	NOUN
iajs-2723	33	14	⋅	⋅	PROPN
iajs-2723	33	15	𝑐3	𝑐3	NOUN
iajs-2723	33	16	viii	viii	VERB
iajs-2723	33	17	.	.	PUNCT
iajs-2723	34	1	(	(	PUNCT
iajs-2723	34	2	𝑐1	𝑐1	NOUN
iajs-2723	34	3	+	+	NOUN
iajs-2723	34	4	𝑐2	𝑐2	NOUN
iajs-2723	34	5	)	)	PUNCT
iajs-2723	34	6	⋅	⋅	PROPN
iajs-2723	34	7	𝑐3	𝑐3	NOUN
iajs-2723	34	8	=	=	PUNCT
iajs-2723	34	9	𝑐1	𝑐1	NOUN
iajs-2723	34	10	⋅	⋅	PROPN
iajs-2723	34	11	𝑐3	𝑐3	NOUN
iajs-2723	34	12	+	+	CCONJ
iajs-2723	34	13	𝑐2	𝑐2	NOUN
iajs-2723	34	14	⋅	⋅	PROPN
iajs-2723	34	15	𝑐3	𝑐3	NOUN
iajs-2723	34	16	.	.	PUNCT
iajs-2723	35	1	definition	definition	NOUN
iajs-2723	35	2	2.2	2.2	NUM
iajs-2723	35	3	[	[	X
iajs-2723	35	4	9	9	NUM
iajs-2723	35	5	]	]	X
iajs-2723	35	6	a	a	DET
iajs-2723	35	7	ring	ring	NOUN
iajs-2723	35	8	γ	γ	PROPN
iajs-2723	35	9	is	be	AUX
iajs-2723	35	10	said	say	VERB
iajs-2723	35	11	to	to	PART
iajs-2723	35	12	be	be	AUX
iajs-2723	35	13	a	a	DET
iajs-2723	35	14	prime	prime	ADJ
iajs-2723	35	15	ring	ring	NOUN
iajs-2723	35	16	if	if	SCONJ
iajs-2723	35	17	for	for	ADP
iajs-2723	35	18	each	each	DET
iajs-2723	35	19	𝑐1	𝑐1	NOUN
iajs-2723	35	20	,	,	PUNCT
iajs-2723	35	21	𝑐2	𝑐2	NOUN
iajs-2723	35	22	∈	∈	PROPN
iajs-2723	35	23	γ	γ	X
iajs-2723	35	24	,	,	PUNCT
iajs-2723	35	25	𝑐1γ𝑐2	𝑐1γ𝑐2	PROPN
iajs-2723	35	26	=	=	SYM
iajs-2723	35	27	0	0	NUM
iajs-2723	35	28	implies	imply	VERB
iajs-2723	35	29	that	that	SCONJ
iajs-2723	35	30	𝑐1	𝑐1	NOUN
iajs-2723	35	31	=	=	NOUN
iajs-2723	35	32	0	0	NUM
iajs-2723	35	33	or	or	CCONJ
iajs-2723	35	34	𝑐2	𝑐2	NOUN
iajs-2723	35	35	=	=	SYM
iajs-2723	36	1	0	0	X
iajs-2723	36	2	.	.	PUNCT
iajs-2723	37	1	definition	definition	NOUN
iajs-2723	37	2	2.3	2.3	NUM
iajs-2723	38	1	[	[	X
iajs-2723	38	2	9	9	NUM
iajs-2723	38	3	]	]	X
iajs-2723	38	4	a	a	DET
iajs-2723	38	5	ring	ring	NOUN
iajs-2723	38	6	γ	γ	PROPN
iajs-2723	38	7	is	be	AUX
iajs-2723	38	8	said	say	VERB
iajs-2723	38	9	to	to	PART
iajs-2723	38	10	be	be	AUX
iajs-2723	38	11	𝑘-torsion	𝑘-torsion	NOUN
iajs-2723	38	12	-	-	PUNCT
iajs-2723	38	13	free	free	ADJ
iajs-2723	38	14	if	if	SCONJ
iajs-2723	38	15	whenever	whenever	SCONJ
iajs-2723	38	16	𝑘𝑐	𝑘𝑐	PRON
iajs-2723	38	17	=	=	SYM
iajs-2723	38	18	0	0	PROPN
iajs-2723	38	19	implies	imply	VERB
iajs-2723	38	20	that	that	SCONJ
iajs-2723	38	21	𝑐	𝑐	PROPN
iajs-2723	38	22	=	=	SYM
iajs-2723	38	23	0	0	PROPN
iajs-2723	38	24	,	,	PUNCT
iajs-2723	38	25	where	where	SCONJ
iajs-2723	38	26	𝑐	𝑐	PROPN
iajs-2723	38	27	∈	∈	PROPN
iajs-2723	38	28	γ	γ	NOUN
iajs-2723	38	29	and	and	CCONJ
iajs-2723	38	30	𝑘	𝑘	DET
iajs-2723	38	31	≠	≠	PROPN
iajs-2723	38	32	0	0	NUM
iajs-2723	38	33	.	.	PUNCT
iajs-2723	39	1	definition	definition	NOUN
iajs-2723	39	2	2.4	2.4	NUM
iajs-2723	39	3	[	[	SYM
iajs-2723	39	4	10	10	NUM
iajs-2723	39	5	]	]	PUNCT
iajs-2723	39	6	let	let	VERB
iajs-2723	39	7	γ	γ	X
iajs-2723	39	8	be	be	AUX
iajs-2723	39	9	a	a	DET
iajs-2723	39	10	ring	ring	NOUN
iajs-2723	39	11	,	,	PUNCT
iajs-2723	39	12	then	then	ADV
iajs-2723	39	13	[	[	X
iajs-2723	39	14	𝑐1	𝑐1	NOUN
iajs-2723	39	15	,	,	PUNCT
iajs-2723	39	16	𝑐2	𝑐2	SYM
iajs-2723	39	17	]	]	PUNCT
iajs-2723	39	18	is	be	AUX
iajs-2723	39	19	said	say	VERB
iajs-2723	39	20	to	to	PART
iajs-2723	39	21	be	be	AUX
iajs-2723	39	22	lie	lie	NOUN
iajs-2723	39	23	product	product	NOUN
iajs-2723	39	24	and	and	CCONJ
iajs-2723	39	25	given	give	VERB
iajs-2723	39	26	as	as	ADP
iajs-2723	39	27	[	[	NOUN
iajs-2723	39	28	𝑐1	𝑐1	NOUN
iajs-2723	39	29	,	,	PUNCT
iajs-2723	39	30	𝑐2	𝑐2	SYM
iajs-2723	39	31	]	]	X
iajs-2723	39	32	=	=	PUNCT
iajs-2723	40	1	𝑐1𝑐2	𝑐1𝑐2	NOUN
iajs-2723	40	2	−	−	X
iajs-2723	41	1	𝑐2𝑐1	𝑐2𝑐1	X
iajs-2723	41	2	and	and	CCONJ
iajs-2723	41	3	𝑐1	𝑐1	NOUN
iajs-2723	41	4	∘	∘	NOUN
iajs-2723	41	5	𝑐2	𝑐2	NOUN
iajs-2723	41	6	is	be	AUX
iajs-2723	41	7	said	say	VERB
iajs-2723	41	8	to	to	PART
iajs-2723	41	9	be	be	AUX
iajs-2723	41	10	jordan	jordan	PROPN
iajs-2723	41	11	product	product	NOUN
iajs-2723	41	12	and	and	CCONJ
iajs-2723	41	13	given	give	VERB
iajs-2723	41	14	as	as	ADP
iajs-2723	41	15	𝑐1	𝑐1	NOUN
iajs-2723	41	16	∘	∘	NOUN
iajs-2723	41	17	𝑐2	𝑐2	NOUN
iajs-2723	42	1	=	=	SYM
iajs-2723	43	1	𝑐1𝑐2	𝑐1𝑐2	PUNCT
iajs-2723	43	2	+	+	CCONJ
iajs-2723	43	3	𝑐2𝑐1	𝑐2𝑐1	ADJ
iajs-2723	43	4	.	.	PUNCT
iajs-2723	44	1	definition	definition	NOUN
iajs-2723	44	2	2.5	2.5	NUM
iajs-2723	45	1	[	[	X
iajs-2723	45	2	11	11	NUM
iajs-2723	45	3	]	]	PUNCT
iajs-2723	45	4	the	the	DET
iajs-2723	45	5	characteristic	characteristic	NOUN
iajs-2723	45	6	of	of	ADP
iajs-2723	45	7	a	a	DET
iajs-2723	45	8	ring	ring	NOUN
iajs-2723	45	9	γ	γ	X
iajs-2723	45	10	(	(	PUNCT
iajs-2723	45	11	for	for	ADP
iajs-2723	45	12	short	short	ADJ
iajs-2723	45	13	char(γ	char(γ	NOUN
iajs-2723	45	14	)	)	PUNCT
iajs-2723	45	15	)	)	PUNCT
iajs-2723	45	16	is	be	AUX
iajs-2723	45	17	the	the	DET
iajs-2723	45	18	smallest	small	ADJ
iajs-2723	45	19	positive	positive	ADJ
iajs-2723	45	20	integer	integer	NOUN
iajs-2723	45	21	𝑧	𝑧	ADP
iajs-2723	45	22	such	such	ADJ
iajs-2723	45	23	that	that	DET
iajs-2723	45	24	𝑧𝑟	𝑧𝑟	NOUN
iajs-2723	45	25	=	=	NOUN
iajs-2723	45	26	0	0	NUM
iajs-2723	45	27	with	with	ADP
iajs-2723	45	28	𝑟	𝑟	DET
iajs-2723	45	29	∈	∈	PROPN
iajs-2723	45	30	γ	γ	X
iajs-2723	45	31	.	.	PROPN
iajs-2723	45	32	otherwise	otherwise	ADV
iajs-2723	45	33	,	,	PUNCT
iajs-2723	45	34	char(γ	char(γ	NOUN
iajs-2723	45	35	)	)	PUNCT
iajs-2723	45	36	=	=	SYM
iajs-2723	45	37	0	0	X
iajs-2723	45	38	.	.	PUNCT
iajs-2723	46	1	definition	definition	NOUN
iajs-2723	46	2	2.6	2.6	NUM
iajs-2723	46	3	[	[	X
iajs-2723	46	4	12	12	NUM
iajs-2723	46	5	]	]	PUNCT
iajs-2723	46	6	let	let	VERB
iajs-2723	46	7	γ	γ	X
iajs-2723	46	8	be	be	AUX
iajs-2723	46	9	a	a	DET
iajs-2723	46	10	ring	ring	NOUN
iajs-2723	46	11	and	and	CCONJ
iajs-2723	46	12	let	let	VERB
iajs-2723	46	13	𝜇	𝜇	ADP
iajs-2723	46	14	,	,	PUNCT
iajs-2723	46	15	𝜎	𝜎	PROPN
iajs-2723	46	16	:	:	PUNCT
iajs-2723	46	17	γ	γ	PROPN
iajs-2723	46	18	⟶	⟶	NOUN
iajs-2723	46	19	γ	γ	X
iajs-2723	46	20	be	be	AUX
iajs-2723	46	21	two	two	NUM
iajs-2723	46	22	additive	additive	ADJ
iajs-2723	46	23	mappings	mapping	NOUN
iajs-2723	46	24	,	,	PUNCT
iajs-2723	46	25	then	then	ADV
iajs-2723	46	26	𝜇	𝜇	X
iajs-2723	46	27	,	,	PUNCT
iajs-2723	46	28	𝜎	𝜎	PROPN
iajs-2723	46	29	are	be	AUX
iajs-2723	46	30	said	say	VERB
iajs-2723	46	31	to	to	PART
iajs-2723	46	32	be	be	AUX
iajs-2723	46	33	derivation	derivation	NOUN
iajs-2723	46	34	pair	pair	NOUN
iajs-2723	46	35	(	(	PUNCT
iajs-2723	46	36	𝜇	𝜇	X
iajs-2723	46	37	,	,	PUNCT
iajs-2723	46	38	𝜎	𝜎	NOUN
iajs-2723	46	39	)	)	PUNCT
iajs-2723	46	40	if	if	SCONJ
iajs-2723	46	41	the	the	DET
iajs-2723	46	42	following	follow	VERB
iajs-2723	46	43	equations	equation	NOUN
iajs-2723	46	44	are	be	AUX
iajs-2723	46	45	holds	hold	NOUN
iajs-2723	46	46	:	:	PUNCT
iajs-2723	46	47	𝜇(𝑢𝑣𝑢	𝜇(𝑢𝑣𝑢	ADJ
iajs-2723	46	48	)	)	PUNCT
iajs-2723	46	49	=	=	PRON
iajs-2723	46	50	𝜇(𝑢)𝑣𝑢	𝜇(𝑢)𝑣𝑢	VERB
iajs-2723	46	51	+	+	X
iajs-2723	46	52	𝑢𝜎(𝑣)𝑢	𝑢𝜎(𝑣)𝑢	X
iajs-2723	46	53	+	+	SYM
iajs-2723	46	54	𝑢𝑣𝜇(𝑢	𝑢𝑣𝜇(𝑢	NOUN
iajs-2723	46	55	)	)	PUNCT
iajs-2723	46	56	,	,	PUNCT
iajs-2723	46	57	for	for	ADP
iajs-2723	46	58	each	each	DET
iajs-2723	46	59	𝑢	𝑢	NOUN
iajs-2723	46	60	,	,	PUNCT
iajs-2723	46	61	𝑣	𝑣	PROPN
iajs-2723	46	62	∈	∈	PROPN
iajs-2723	46	63	γ	γ	NOUN
iajs-2723	46	64	𝜎(𝑢𝑣𝑢	𝜎(𝑢𝑣𝑢	PROPN
iajs-2723	46	65	)	)	PUNCT
iajs-2723	46	66	=	=	SYM
iajs-2723	46	67	𝜎(𝑢)𝑣𝑢	𝜎(𝑢)𝑣𝑢	NUM
iajs-2723	46	68	+	+	NUM
iajs-2723	46	69	𝑢𝜇(𝑣)𝑢	𝑢𝜇(𝑣)𝑢	PROPN
iajs-2723	46	70	+	+	NUM
iajs-2723	46	71	𝑢𝑣𝜎(𝑢	𝑢𝑣𝜎(𝑢	X
iajs-2723	46	72	)	)	PUNCT
iajs-2723	46	73	,	,	PUNCT
iajs-2723	46	74	for	for	ADP
iajs-2723	46	75	each	each	DET
iajs-2723	46	76	𝑢	𝑢	NOUN
iajs-2723	46	77	,	,	PUNCT
iajs-2723	46	78	𝑣	𝑣	PROPN
iajs-2723	46	79	∈	∈	NOUN
iajs-2723	46	80	γ	γ	NOUN
iajs-2723	46	81	and	and	CCONJ
iajs-2723	46	82	are	be	AUX
iajs-2723	46	83	called	call	VERB
iajs-2723	46	84	jordan	jordan	PROPN
iajs-2723	46	85	derivation	derivation	PROPN
iajs-2723	46	86	pair	pair	NOUN
iajs-2723	47	1	if	if	SCONJ
iajs-2723	47	2	:	:	PUNCT
iajs-2723	47	3	𝜇(𝑢3	𝜇(𝑢3	NOUN
iajs-2723	47	4	)	)	PUNCT
iajs-2723	47	5	=	=	PUNCT
iajs-2723	47	6	𝜇(𝑢)𝑢2	𝜇(𝑢)𝑢2	NOUN
iajs-2723	47	7	+	+	CCONJ
iajs-2723	47	8	𝑢𝜎(𝑢)𝑢	𝑢𝜎(𝑢)𝑢	X
iajs-2723	47	9	+	+	CCONJ
iajs-2723	47	10	𝑢2𝜇(𝑢	𝑢2𝜇(𝑢	PROPN
iajs-2723	47	11	)	)	PUNCT
iajs-2723	47	12	,	,	PUNCT
iajs-2723	47	13	for	for	ADP
iajs-2723	47	14	each	each	DET
iajs-2723	47	15	𝑢	𝑢	PROPN
iajs-2723	47	16	∈	∈	PROPN
iajs-2723	47	17	γ	γ	X
iajs-2723	47	18	𝜎(𝑢3	𝜎(𝑢3	NOUN
iajs-2723	47	19	)	)	PUNCT
iajs-2723	47	20	=	=	PUNCT
iajs-2723	48	1	𝜎(𝑢)𝑢2	𝜎(𝑢)𝑢2	PROPN
iajs-2723	48	2	+	+	NOUN
iajs-2723	48	3	𝑢𝜇(𝑢)𝑢	𝑢𝜇(𝑢)𝑢	PROPN
iajs-2723	48	4	+	+	CCONJ
iajs-2723	48	5	𝑢2𝜎(𝑢	𝑢2𝜎(𝑢	PROPN
iajs-2723	48	6	)	)	PUNCT
iajs-2723	48	7	,	,	PUNCT
iajs-2723	48	8	for	for	ADP
iajs-2723	48	9	each	each	DET
iajs-2723	48	10	𝑢	𝑢	PROPN
iajs-2723	48	11	∈	∈	PROPN
iajs-2723	48	12	γ	γ	X
iajs-2723	48	13	.	.	PROPN
iajs-2723	48	14	3	3	NUM
iajs-2723	48	15	.	.	X
iajs-2723	48	16	main	main	ADJ
iajs-2723	48	17	results	result	NOUN
iajs-2723	48	18	in	in	ADP
iajs-2723	48	19	this	this	DET
iajs-2723	48	20	section	section	NOUN
iajs-2723	48	21	,	,	PUNCT
iajs-2723	48	22	we	we	PRON
iajs-2723	48	23	presented	present	VERB
iajs-2723	48	24	the	the	DET
iajs-2723	48	25	notion	notion	NOUN
iajs-2723	48	26	of	of	ADP
iajs-2723	48	27	(	(	PUNCT
iajs-2723	48	28	𝜃1	𝜃1	VERB
iajs-2723	48	29	,	,	PUNCT
iajs-2723	48	30	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	48	31	pair	pair	NOUN
iajs-2723	48	32	on	on	ADP
iajs-2723	48	33	the	the	DET
iajs-2723	48	34	ring	ring	NOUN
iajs-2723	48	35	γ	γ	PROPN
iajs-2723	48	36	where	where	SCONJ
iajs-2723	48	37	𝜃1	𝜃1	VERB
iajs-2723	48	38	and	and	CCONJ
iajs-2723	48	39	𝜃2	𝜃2	NOUN
iajs-2723	48	40	are	be	AUX
iajs-2723	48	41	two	two	NUM
iajs-2723	48	42	mappings	mapping	NOUN
iajs-2723	48	43	from	from	ADP
iajs-2723	48	44	the	the	DET
iajs-2723	48	45	ring	ring	NOUN
iajs-2723	48	46	γ	γ	NOUN
iajs-2723	48	47	into	into	ADP
iajs-2723	48	48	itself	itself	PRON
iajs-2723	48	49	.	.	PUNCT
iajs-2723	49	1	moreover	moreover	ADV
iajs-2723	49	2	,	,	PUNCT
iajs-2723	49	3	some	some	DET
iajs-2723	49	4	properties	property	NOUN
iajs-2723	49	5	of	of	ADP
iajs-2723	49	6	this	this	DET
iajs-2723	49	7	concept	concept	NOUN
iajs-2723	49	8	have	have	AUX
iajs-2723	49	9	been	be	AUX
iajs-2723	49	10	proved	prove	VERB
iajs-2723	49	11	.	.	PUNCT
iajs-2723	50	1	definition	definition	NOUN
iajs-2723	50	2	3.1	3.1	NUM
iajs-2723	50	3	let	let	VERB
iajs-2723	50	4	γ	γ	NOUN
iajs-2723	50	5	be	be	AUX
iajs-2723	50	6	a	a	DET
iajs-2723	50	7	ring	ring	NOUN
iajs-2723	50	8	.	.	PUNCT
iajs-2723	51	1	let	let	VERB
iajs-2723	51	2	𝛿1	𝛿1	NOUN
iajs-2723	51	3	,	,	PUNCT
iajs-2723	51	4	𝛿2	𝛿2	NOUN
iajs-2723	51	5	:	:	PUNCT
iajs-2723	51	6	γ	γ	PROPN
iajs-2723	51	7	⟶	⟶	NOUN
iajs-2723	51	8	γ	γ	X
iajs-2723	51	9	be	be	AUX
iajs-2723	51	10	additive	additive	ADJ
iajs-2723	51	11	mappings	mapping	NOUN
iajs-2723	51	12	,	,	PUNCT
iajs-2723	51	13	then	then	ADV
iajs-2723	51	14	(	(	PUNCT
iajs-2723	51	15	𝛿1	𝛿1	PROPN
iajs-2723	51	16	,	,	PUNCT
iajs-2723	51	17	𝛿2	𝛿2	PROPN
iajs-2723	51	18	)	)	PUNCT
iajs-2723	51	19	is	be	AUX
iajs-2723	51	20	said	say	VERB
iajs-2723	51	21	to	to	PART
iajs-2723	51	22	be	be	AUX
iajs-2723	51	23	(	(	PUNCT
iajs-2723	51	24	𝜃1	𝜃1	VERB
iajs-2723	51	25	,	,	PUNCT
iajs-2723	51	26	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	51	27	pair	pair	NOUN
iajs-2723	51	28	,	,	PUNCT
iajs-2723	51	29	if	if	SCONJ
iajs-2723	51	30	the	the	DET
iajs-2723	51	31	following	following	NOUN
iajs-2723	51	32	are	be	AUX
iajs-2723	51	33	holds	hold	VERB
iajs-2723	51	34	:	:	PUNCT
iajs-2723	51	35	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NUM
iajs-2723	51	36	)	)	PUNCT
iajs-2723	51	37	=	=	SYM
iajs-2723	51	38	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	51	39	)	)	PUNCT
iajs-2723	51	40	+	+	PUNCT
iajs-2723	52	1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	52	2	)	)	PUNCT
iajs-2723	52	3	+	+	ADJ
iajs-2723	52	4	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	52	5	)	)	PUNCT
iajs-2723	52	6	,	,	PUNCT
iajs-2723	52	7	for	for	ADP
iajs-2723	52	8	each	each	DET
iajs-2723	52	9	𝑢	𝑢	NOUN
iajs-2723	52	10	,	,	PUNCT
iajs-2723	52	11	𝑣	𝑣	PROPN
iajs-2723	52	12	∈	∈	PROPN
iajs-2723	52	13	γ	γ	X
iajs-2723	52	14	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	X
iajs-2723	52	15	)	)	PUNCT
iajs-2723	52	16	=	=	SYM
iajs-2723	52	17	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	52	18	)	)	PUNCT
iajs-2723	52	19	+	+	PUNCT
iajs-2723	52	20	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	52	21	)	)	PUNCT
iajs-2723	52	22	+	+	CCONJ
iajs-2723	52	23	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	NOUN
iajs-2723	52	24	)	)	PUNCT
iajs-2723	52	25	,	,	PUNCT
iajs-2723	52	26	for	for	ADP
iajs-2723	52	27	each	each	DET
iajs-2723	52	28	𝑢	𝑢	NOUN
iajs-2723	52	29	,	,	PUNCT
iajs-2723	52	30	𝑣	𝑣	PRON
iajs-2723	52	31	∈	∈	PROPN
iajs-2723	52	32	γ	γ	X
iajs-2723	52	33	.	.	PROPN
iajs-2723	52	34	and	and	CCONJ
iajs-2723	52	35	are	be	AUX
iajs-2723	52	36	said	say	VERB
iajs-2723	52	37	to	to	PART
iajs-2723	52	38	be	be	AUX
iajs-2723	52	39	jordan	jordan	PROPN
iajs-2723	52	40	(	(	PUNCT
iajs-2723	52	41	𝜃1	𝜃1	PROPN
iajs-2723	52	42	,	,	PUNCT
iajs-2723	52	43	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	52	44	pair	pair	NOUN
iajs-2723	52	45	,	,	PUNCT
iajs-2723	52	46	if	if	SCONJ
iajs-2723	52	47	the	the	DET
iajs-2723	52	48	following	following	NOUN
iajs-2723	52	49	are	be	AUX
iajs-2723	52	50	holds	hold	VERB
iajs-2723	52	51	:	:	PUNCT
iajs-2723	52	52	𝛿1(𝑢3	𝛿1(𝑢3	NUM
iajs-2723	52	53	)	)	PUNCT
iajs-2723	52	54	=	=	SYM
iajs-2723	52	55	𝛿1(𝑢)𝜃1(𝑢2	𝛿1(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	52	56	)	)	PUNCT
iajs-2723	52	57	+	+	X
iajs-2723	52	58	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	NUM
iajs-2723	52	59	)	)	PUNCT
iajs-2723	52	60	+	+	CCONJ
iajs-2723	52	61	𝜃2(𝑢2)𝛿1(𝑢	𝜃2(𝑢2)𝛿1(𝑢	NUM
iajs-2723	52	62	)	)	PUNCT
iajs-2723	52	63	for	for	ADP
iajs-2723	52	64	all	all	DET
iajs-2723	52	65	𝑢	𝑢	PRON
iajs-2723	52	66	∈	∈	PROPN
iajs-2723	52	67	γ	γ	PROPN
iajs-2723	52	68	ibn	ibn	PROPN
iajs-2723	52	69	al	al	PROPN
iajs-2723	52	70	-	-	PUNCT
iajs-2723	52	71	haitham	haitham	PROPN
iajs-2723	52	72	jour	jour	X
iajs-2723	52	73	.	.	PROPN
iajs-2723	53	1	for	for	ADP
iajs-2723	53	2	pure	pure	ADJ
iajs-2723	53	3	&	&	CCONJ
iajs-2723	53	4	appl	appl	PROPN
iajs-2723	53	5	.	.	PUNCT
iajs-2723	54	1	sci	sci	PROPN
iajs-2723	54	2	.	.	PROPN
iajs-2723	55	1	53	53	NUM
iajs-2723	55	2	(	(	PUNCT
iajs-2723	55	3	2)2022	2)2022	VERB
iajs-2723	55	4	110	110	NUM
iajs-2723	55	5	𝛿2(𝑢3	𝛿2(𝑢3	NOUN
iajs-2723	55	6	)	)	PUNCT
iajs-2723	55	7	=	=	SYM
iajs-2723	55	8	𝛿2(𝑢)𝜃1(𝑢2	𝛿2(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	55	9	)	)	PUNCT
iajs-2723	55	10	+	+	CCONJ
iajs-2723	55	11	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	PROPN
iajs-2723	55	12	)	)	PUNCT
iajs-2723	55	13	+	+	NUM
iajs-2723	55	14	𝜃2(𝑢2)𝛿2(𝑢	𝜃2(𝑢2)𝛿2(𝑢	NOUN
iajs-2723	55	15	)	)	PUNCT
iajs-2723	55	16	for	for	ADP
iajs-2723	55	17	all	all	DET
iajs-2723	55	18	𝑢	𝑢	PROPN
iajs-2723	55	19	∈	∈	PROPN
iajs-2723	55	20	γ	γ	PROPN
iajs-2723	55	21	.	.	PROPN
iajs-2723	55	22	example	example	NOUN
iajs-2723	55	23	3.1	3.1	NUM
iajs-2723	55	24	let	let	VERB
iajs-2723	55	25	γ	γ	NOUN
iajs-2723	55	26	be	be	AUX
iajs-2723	55	27	a	a	DET
iajs-2723	55	28	non	non	ADJ
iajs-2723	55	29	-	-	ADJ
iajs-2723	55	30	commutative	commutative	ADJ
iajs-2723	55	31	ring	ring	NOUN
iajs-2723	55	32	,	,	PUNCT
iajs-2723	55	33	let	let	VERB
iajs-2723	55	34	𝑐1	𝑐1	NOUN
iajs-2723	55	35	,	,	PUNCT
iajs-2723	55	36	𝑐2	𝑐2	NOUN
iajs-2723	55	37	∈	∈	PROPN
iajs-2723	55	38	γ	γ	NOUN
iajs-2723	55	39	such	such	ADJ
iajs-2723	55	40	that	that	SCONJ
iajs-2723	55	41	𝜃2(𝑢)𝑐1	𝜃2(𝑢)𝑐1	PUNCT
iajs-2723	55	42	=	=	SYM
iajs-2723	55	43	𝜃2(𝑢)𝑐2	𝜃2(𝑢)𝑐2	NUM
iajs-2723	55	44	=	=	SYM
iajs-2723	55	45	0	0	NUM
iajs-2723	55	46	(	(	PUNCT
iajs-2723	55	47	resp	resp	NOUN
iajs-2723	55	48	.	.	PUNCT
iajs-2723	55	49	𝜃2(𝑣)𝑐1	𝜃2(𝑣)𝑐1	VERB
iajs-2723	56	1	=	=	PUNCT
iajs-2723	56	2	𝜃2(𝑣)𝑐2	𝜃2(𝑣)𝑐2	X
iajs-2723	56	3	=	=	SYM
iajs-2723	56	4	0	0	NUM
iajs-2723	56	5	)	)	PUNCT
iajs-2723	56	6	for	for	SCONJ
iajs-2723	56	7	all	all	DET
iajs-2723	56	8	𝑢	𝑢	NOUN
iajs-2723	56	9	,	,	PUNCT
iajs-2723	56	10	𝑣	𝑣	PROPN
iajs-2723	56	11	∈	∈	PROPN
iajs-2723	56	12	γ	γ	X
iajs-2723	56	13	.define	.define	NOUN
iajs-2723	56	14	𝛿1	𝛿1	PROPN
iajs-2723	56	15	,	,	PUNCT
iajs-2723	56	16	𝛿2	𝛿2	NOUN
iajs-2723	56	17	:	:	PUNCT
iajs-2723	56	18	γ	γ	PROPN
iajs-2723	56	19	⟶	⟶	PROPN
iajs-2723	56	20	γ	γ	NOUN
iajs-2723	56	21	as	as	SCONJ
iajs-2723	56	22	follows	follow	VERB
iajs-2723	56	23	:	:	PUNCT
iajs-2723	56	24	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	56	25	)	)	PUNCT
iajs-2723	56	26	=	=	SYM
iajs-2723	56	27	𝑐1𝜃1(𝑢	𝑐1𝜃1(𝑢	NOUN
iajs-2723	56	28	)	)	PUNCT
iajs-2723	56	29	and	and	CCONJ
iajs-2723	56	30	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	56	31	)	)	PUNCT
iajs-2723	56	32	=	=	SYM
iajs-2723	56	33	𝑐2𝜃1(𝑢	𝑐2𝜃1(𝑢	NOUN
iajs-2723	56	34	)	)	PUNCT
iajs-2723	56	35	,	,	PUNCT
iajs-2723	56	36	∀𝑢	∀𝑢	DET
iajs-2723	56	37	∈	∈	PROPN
iajs-2723	56	38	γ	γ	PROPN
iajs-2723	56	39	,	,	PUNCT
iajs-2723	56	40	where	where	SCONJ
iajs-2723	56	41	𝜃1	𝜃1	NOUN
iajs-2723	56	42	,	,	PUNCT
iajs-2723	56	43	𝜃2	𝜃2	PROPN
iajs-2723	56	44	:	:	PUNCT
iajs-2723	56	45	γ	γ	PROPN
iajs-2723	56	46	⟶	⟶	NOUN
iajs-2723	56	47	γ	γ	NOUN
iajs-2723	56	48	are	be	AUX
iajs-2723	56	49	two	two	NUM
iajs-2723	56	50	endomorphism	endomorphism	NOUN
iajs-2723	56	51	mappings	mapping	NOUN
iajs-2723	56	52	.	.	PUNCT
iajs-2723	57	1	then	then	ADV
iajs-2723	57	2	(	(	PUNCT
iajs-2723	57	3	𝛿1	𝛿1	PROPN
iajs-2723	57	4	,	,	PUNCT
iajs-2723	57	5	𝛿2	𝛿2	PROPN
iajs-2723	57	6	)	)	PUNCT
iajs-2723	57	7	is	be	AUX
iajs-2723	57	8	a	a	DET
iajs-2723	57	9	(	(	PUNCT
iajs-2723	57	10	𝜃1	𝜃1	NOUN
iajs-2723	57	11	,	,	PUNCT
iajs-2723	57	12	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	57	13	pair	pair	NOUN
iajs-2723	57	14	of	of	ADP
iajs-2723	57	15	γ	γ	PROPN
iajs-2723	57	16	.	.	PUNCT
iajs-2723	58	1	let	let	VERB
iajs-2723	58	2	𝑢	𝑢	NOUN
iajs-2723	58	3	,	,	PUNCT
iajs-2723	58	4	𝑣	𝑣	PROPN
iajs-2723	58	5	∈	∈	PROPN
iajs-2723	58	6	γ	γ	X
iajs-2723	58	7	,	,	PUNCT
iajs-2723	58	8	then	then	ADV
iajs-2723	58	9	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NOUN
iajs-2723	58	10	)	)	PUNCT
iajs-2723	58	11	=	=	SYM
iajs-2723	59	1	𝑐1𝜃1(𝑢𝑣𝑢	𝑐1𝜃1(𝑢𝑣𝑢	X
iajs-2723	59	2	)	)	PUNCT
iajs-2723	59	3	=	=	SYM
iajs-2723	59	4	𝑐1𝜃1(𝑢(𝑣𝑢	𝑐1𝜃1(𝑢(𝑣𝑢	PROPN
iajs-2723	59	5	)	)	PUNCT
iajs-2723	59	6	)	)	PUNCT
iajs-2723	60	1	=	=	SYM
iajs-2723	60	2	𝑐1𝜃1(𝑢)𝜃1(𝑣𝑢	𝑐1𝜃1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	60	3	)	)	PUNCT
iajs-2723	60	4	=	=	SYM
iajs-2723	60	5	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	60	6	)	)	PUNCT
iajs-2723	60	7	+	+	PUNCT
iajs-2723	60	8	𝜃2(𝑢)𝑐2𝜃1(𝑣𝑢	𝜃2(𝑢)𝑐2𝜃1(𝑣𝑢	NOUN
iajs-2723	60	9	)	)	PUNCT
iajs-2723	60	10	=	=	SYM
iajs-2723	60	11	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	60	12	)	)	PUNCT
iajs-2723	60	13	+	+	CCONJ
iajs-2723	60	14	𝜃2(𝑢)𝑐2𝜃1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝑐2𝜃1(𝑣)𝜃1(𝑢	X
iajs-2723	60	15	)	)	PUNCT
iajs-2723	60	16	=	=	SYM
iajs-2723	60	17	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	60	18	)	)	PUNCT
iajs-2723	60	19	+	+	PUNCT
iajs-2723	60	20	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	NOUN
iajs-2723	60	21	)	)	PUNCT
iajs-2723	61	1	+	+	CCONJ
iajs-2723	61	2	𝜃2(𝑢)𝜃2(𝑣)𝑐1𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑣)𝑐1𝜃1(𝑢	X
iajs-2723	61	3	)	)	PUNCT
iajs-2723	61	4	=	=	SYM
iajs-2723	61	5	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	61	6	)	)	PUNCT
iajs-2723	61	7	+	+	PUNCT
iajs-2723	61	8	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	61	9	)	)	PUNCT
iajs-2723	62	1	+	+	PUNCT
iajs-2723	62	2	𝜃2(𝑢)𝜃2(𝑣)𝛿1(𝑢	𝜃2(𝑢)𝜃2(𝑣)𝛿1(𝑢	NOUN
iajs-2723	62	3	)	)	PUNCT
iajs-2723	62	4	=	=	SYM
iajs-2723	62	5	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	62	6	)	)	PUNCT
iajs-2723	62	7	+	+	PUNCT
iajs-2723	62	8	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	62	9	)	)	PUNCT
iajs-2723	62	10	+	+	ADJ
iajs-2723	62	11	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	62	12	)	)	PUNCT
iajs-2723	62	13	also	also	ADV
iajs-2723	62	14	,	,	PUNCT
iajs-2723	62	15	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	ADJ
iajs-2723	62	16	)	)	PUNCT
iajs-2723	62	17	=	=	SYM
iajs-2723	62	18	𝑐2𝜃1(𝑢𝑣𝑢	𝑐2𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	62	19	)	)	PUNCT
iajs-2723	62	20	=	=	SYM
iajs-2723	63	1	𝑐2𝜃1(𝑢(𝑣𝑢	𝑐2𝜃1(𝑢(𝑣𝑢	NUM
iajs-2723	63	2	)	)	PUNCT
iajs-2723	63	3	)	)	PUNCT
iajs-2723	64	1	=	=	SYM
iajs-2723	64	2	𝑐2𝜃1(𝑢)𝜃1(𝑣𝑢	𝑐2𝜃1(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	64	3	)	)	PUNCT
iajs-2723	64	4	=	=	SYM
iajs-2723	64	5	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	64	6	)	)	PUNCT
iajs-2723	64	7	+	+	SYM
iajs-2723	64	8	𝜃2(𝑢)𝑐1𝜃1(𝑣𝑢	𝜃2(𝑢)𝑐1𝜃1(𝑣𝑢	X
iajs-2723	64	9	)	)	PUNCT
iajs-2723	64	10	=	=	SYM
iajs-2723	64	11	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	64	12	)	)	PUNCT
iajs-2723	64	13	+	+	NUM
iajs-2723	64	14	𝜃2(𝑢)𝑐1𝜃1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝑐1𝜃1(𝑣)𝜃1(𝑢	NUM
iajs-2723	64	15	)	)	PUNCT
iajs-2723	64	16	=	=	SYM
iajs-2723	64	17	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	64	18	)	)	PUNCT
iajs-2723	64	19	+	+	PUNCT
iajs-2723	64	20	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	64	21	)	)	PUNCT
iajs-2723	65	1	+	+	CCONJ
iajs-2723	65	2	𝜃2(𝑢)𝜃2(𝑣)𝑐2𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑣)𝑐2𝜃1(𝑢	X
iajs-2723	65	3	)	)	PUNCT
iajs-2723	65	4	=	=	SYM
iajs-2723	65	5	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	65	6	)	)	PUNCT
iajs-2723	65	7	+	+	PUNCT
iajs-2723	65	8	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	65	9	)	)	PUNCT
iajs-2723	65	10	+	+	PUNCT
iajs-2723	65	11	𝜃2(𝑢)𝜃2(𝑣)𝛿2(𝑢	𝜃2(𝑢)𝜃2(𝑣)𝛿2(𝑢	PROPN
iajs-2723	65	12	)	)	PUNCT
iajs-2723	65	13	=	=	SYM
iajs-2723	65	14	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	65	15	)	)	PUNCT
iajs-2723	65	16	+	+	PUNCT
iajs-2723	65	17	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	65	18	)	)	PUNCT
iajs-2723	65	19	+	+	NUM
iajs-2723	65	20	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	SYM
iajs-2723	65	21	)	)	PUNCT
iajs-2723	65	22	thus	thus	ADV
iajs-2723	65	23	,	,	PUNCT
iajs-2723	65	24	(	(	PUNCT
iajs-2723	65	25	𝛿1	𝛿1	NOUN
iajs-2723	65	26	,	,	PUNCT
iajs-2723	65	27	𝛿2	𝛿2	PROPN
iajs-2723	65	28	)	)	PUNCT
iajs-2723	65	29	is	be	AUX
iajs-2723	65	30	a	a	DET
iajs-2723	65	31	(	(	PUNCT
iajs-2723	65	32	𝜃1	𝜃1	NOUN
iajs-2723	65	33	,	,	PUNCT
iajs-2723	65	34	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	65	35	pair	pair	NOUN
iajs-2723	65	36	of	of	ADP
iajs-2723	65	37	γ	γ	PROPN
iajs-2723	65	38	.	.	PROPN
iajs-2723	65	39	remark	remark	PROPN
iajs-2723	65	40	3.1	3.1	NUM
iajs-2723	65	41	:	:	PUNCT
iajs-2723	65	42	every	every	PRON
iajs-2723	65	43	(	(	PUNCT
iajs-2723	65	44	𝜃1	𝜃1	NOUN
iajs-2723	65	45	,	,	PUNCT
iajs-2723	65	46	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	65	47	pair	pair	NOUN
iajs-2723	65	48	is	be	AUX
iajs-2723	65	49	a	a	DET
iajs-2723	65	50	jordan	jordan	PROPN
iajs-2723	65	51	(	(	PUNCT
iajs-2723	65	52	𝜃1	𝜃1	PROPN
iajs-2723	65	53	,	,	PUNCT
iajs-2723	65	54	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	65	55	pair	pair	NOUN
iajs-2723	65	56	,	,	PUNCT
iajs-2723	65	57	but	but	CCONJ
iajs-2723	65	58	the	the	DET
iajs-2723	65	59	converse	converse	NOUN
iajs-2723	65	60	is	be	AUX
iajs-2723	65	61	not	not	PART
iajs-2723	65	62	true	true	ADJ
iajs-2723	65	63	in	in	ADP
iajs-2723	65	64	general	general	ADJ
iajs-2723	65	65	.	.	PUNCT
iajs-2723	65	66	example	example	NOUN
iajs-2723	65	67	3.2	3.2	NUM
iajs-2723	65	68	:	:	PUNCT
iajs-2723	65	69	let	let	VERB
iajs-2723	65	70	γ	γ	PART
iajs-2723	65	71	be	be	AUX
iajs-2723	65	72	a	a	DET
iajs-2723	65	73	2	2	NUM
iajs-2723	65	74	-	-	PUNCT
iajs-2723	65	75	torsion	torsion	NOUN
iajs-2723	65	76	free	free	ADJ
iajs-2723	65	77	non	non	ADJ
iajs-2723	65	78	-	-	ADJ
iajs-2723	65	79	commutative	commutative	ADJ
iajs-2723	65	80	ring	ring	NOUN
iajs-2723	65	81	,	,	PUNCT
iajs-2723	65	82	let	let	VERB
iajs-2723	65	83	𝑐	𝑐	PROPN
iajs-2723	65	84	∈	∈	VERB
iajs-2723	65	85	γ	γ	NOUN
iajs-2723	65	86	such	such	ADJ
iajs-2723	65	87	that	that	PRON
iajs-2723	65	88	𝜃2(𝑢)𝑐𝜃1(𝑢	𝜃2(𝑢)𝑐𝜃1(𝑢	PRON
iajs-2723	65	89	)	)	PUNCT
iajs-2723	65	90	=	=	SYM
iajs-2723	65	91	0	0	NUM
iajs-2723	65	92	,	,	PUNCT
iajs-2723	65	93	∀𝑢	∀𝑢	PRON
iajs-2723	65	94	∈	∈	PROPN
iajs-2723	65	95	γ	γ	X
iajs-2723	65	96	,	,	PUNCT
iajs-2723	65	97	but	but	CCONJ
iajs-2723	65	98	𝜃2(𝑢)𝑐𝜃1(𝑣	𝜃2(𝑢)𝑐𝜃1(𝑣	NOUN
iajs-2723	65	99	)	)	PUNCT
iajs-2723	65	100	≠	≠	PROPN
iajs-2723	65	101	0	0	NUM
iajs-2723	65	102	for	for	SCONJ
iajs-2723	65	103	some	some	DET
iajs-2723	65	104	𝑢	𝑢	NOUN
iajs-2723	65	105	≠	≠	NOUN
iajs-2723	65	106	𝑣	𝑣	ADP
iajs-2723	65	107	∈	∈	PROPN
iajs-2723	65	108	γ	γ	PROPN
iajs-2723	65	109	.	.	PROPN
iajs-2723	65	110	define	define	VERB
iajs-2723	65	111	𝛿1	𝛿1	PROPN
iajs-2723	65	112	,	,	PUNCT
iajs-2723	65	113	𝛿2	𝛿2	NOUN
iajs-2723	65	114	:	:	PUNCT
iajs-2723	65	115	γ	γ	PROPN
iajs-2723	65	116	⟶	⟶	PROPN
iajs-2723	65	117	γ	γ	NOUN
iajs-2723	65	118	as	as	SCONJ
iajs-2723	65	119	follows	follow	VERB
iajs-2723	65	120	:	:	PUNCT
iajs-2723	65	121	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	65	122	)	)	PUNCT
iajs-2723	65	123	=	=	SYM
iajs-2723	65	124	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	65	125	+	+	CCONJ
iajs-2723	65	126	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	65	127	)	)	PUNCT
iajs-2723	65	128	and	and	CCONJ
iajs-2723	65	129	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	65	130	)	)	PUNCT
iajs-2723	66	1	=	=	SYM
iajs-2723	66	2	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	66	3	−	−	PROPN
iajs-2723	66	4	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	66	5	)	)	PUNCT
iajs-2723	66	6	,	,	PUNCT
iajs-2723	66	7	∀𝑢	∀𝑢	PROPN
iajs-2723	66	8	∈	∈	PROPN
iajs-2723	66	9	γ	γ	PROPN
iajs-2723	66	10	,	,	PUNCT
iajs-2723	66	11	where	where	SCONJ
iajs-2723	66	12	𝜃1	𝜃1	NOUN
iajs-2723	66	13	,	,	PUNCT
iajs-2723	66	14	𝜃2	𝜃2	PROPN
iajs-2723	66	15	:	:	PUNCT
iajs-2723	66	16	γ	γ	PROPN
iajs-2723	66	17	⟶	⟶	NOUN
iajs-2723	66	18	γ	γ	NOUN
iajs-2723	66	19	are	be	AUX
iajs-2723	66	20	two	two	NUM
iajs-2723	66	21	endomorphisms	endomorphism	NOUN
iajs-2723	66	22	.	.	PUNCT
iajs-2723	67	1	then	then	ADV
iajs-2723	67	2	(	(	PUNCT
iajs-2723	67	3	𝛿1	𝛿1	PROPN
iajs-2723	67	4	,	,	PUNCT
iajs-2723	67	5	𝛿2	𝛿2	PROPN
iajs-2723	67	6	)	)	PUNCT
iajs-2723	67	7	is	be	AUX
iajs-2723	67	8	a	a	DET
iajs-2723	67	9	jordan	jordan	PROPN
iajs-2723	67	10	(	(	PUNCT
iajs-2723	67	11	𝜃1	𝜃1	PROPN
iajs-2723	67	12	,	,	PUNCT
iajs-2723	67	13	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	67	14	but	but	CCONJ
iajs-2723	67	15	not	not	PART
iajs-2723	67	16	(	(	PUNCT
iajs-2723	67	17	𝜃1	𝜃1	VERB
iajs-2723	67	18	,	,	PUNCT
iajs-2723	67	19	𝜃2)-derivation	𝜃2)-derivation	PROPN
iajs-2723	67	20	.	.	PUNCT
iajs-2723	68	1	let	let	VERB
iajs-2723	68	2	𝑢	𝑢	NOUN
iajs-2723	68	3	,	,	PUNCT
iajs-2723	68	4	𝑣	𝑣	PROPN
iajs-2723	68	5	∈	∈	PROPN
iajs-2723	68	6	γ	γ	NOUN
iajs-2723	68	7	,	,	PUNCT
iajs-2723	68	8	then	then	ADV
iajs-2723	68	9	𝛿1(𝑢3	𝛿1(𝑢3	NOUN
iajs-2723	68	10	)	)	PUNCT
iajs-2723	68	11	=	=	PUNCT
iajs-2723	68	12	𝜃2(𝑢3)𝑐	𝜃2(𝑢3)𝑐	X
iajs-2723	68	13	+	+	X
iajs-2723	68	14	𝑐𝜃1(𝑢3	𝑐𝜃1(𝑢3	NOUN
iajs-2723	68	15	)	)	PUNCT
iajs-2723	68	16	and	and	CCONJ
iajs-2723	68	17	𝛿1(𝑢3	𝛿1(𝑢3	NUM
iajs-2723	68	18	)	)	PUNCT
iajs-2723	68	19	=	=	SYM
iajs-2723	68	20	𝛿1(𝑢)𝜃1(𝑢2	𝛿1(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	68	21	)	)	PUNCT
iajs-2723	69	1	+	+	X
iajs-2723	70	1	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	NUM
iajs-2723	70	2	)	)	PUNCT
iajs-2723	71	1	+	+	CCONJ
iajs-2723	72	1	𝜃2(𝑢2)𝛿1(𝑢	𝜃2(𝑢2)𝛿1(𝑢	NUM
iajs-2723	72	2	)	)	PUNCT
iajs-2723	72	3	.	.	PUNCT
iajs-2723	73	1	thus	thus	ADV
iajs-2723	73	2	,	,	PUNCT
iajs-2723	73	3	𝛿1(𝑢)𝜃1(𝑢2	𝛿1(𝑢)𝜃1(𝑢2	NOUN
iajs-2723	73	4	)	)	PUNCT
iajs-2723	73	5	+	+	X
iajs-2723	73	6	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	NUM
iajs-2723	73	7	)	)	PUNCT
iajs-2723	74	1	+	+	CCONJ
iajs-2723	74	2	𝜃2(𝑢2)𝛿1(𝑢	𝜃2(𝑢2)𝛿1(𝑢	NUM
iajs-2723	74	3	)	)	PUNCT
iajs-2723	74	4	=	=	PUNCT
iajs-2723	74	5	(	(	PUNCT
iajs-2723	74	6	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	74	7	+	+	NUM
iajs-2723	74	8	𝑐𝜃1(𝑢))𝜃1(𝑢2	𝑐𝜃1(𝑢))𝜃1(𝑢2	NOUN
iajs-2723	74	9	)	)	PUNCT
iajs-2723	74	10	+	+	CCONJ
iajs-2723	74	11	𝜃2(𝑢)(𝜃2(𝑢)𝑐	𝜃2(𝑢)(𝜃2(𝑢)𝑐	PUNCT
iajs-2723	74	12	−	−	NOUN
iajs-2723	74	13	𝑐𝜃1(𝑢))𝜃1(𝑢	𝑐𝜃1(𝑢))𝜃1(𝑢	ADJ
iajs-2723	74	14	)	)	PUNCT
iajs-2723	75	1	+	+	CCONJ
iajs-2723	75	2	𝜃2(𝑢2)(𝜃2(𝑢)𝑐	𝜃2(𝑢2)(𝜃2(𝑢)𝑐	PROPN
iajs-2723	75	3	+	+	CCONJ
iajs-2723	75	4	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	75	5	)	)	PUNCT
iajs-2723	75	6	)	)	PUNCT
iajs-2723	76	1	=	=	SYM
iajs-2723	76	2	(	(	PUNCT
iajs-2723	76	3	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	X
iajs-2723	76	4	)	)	PUNCT
iajs-2723	77	1	+	+	CCONJ
iajs-2723	77	2	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	NUM
iajs-2723	77	3	)	)	PUNCT
iajs-2723	77	4	)	)	PUNCT
iajs-2723	78	1	+	+	CCONJ
iajs-2723	78	2	(	(	PUNCT
iajs-2723	78	3	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	PROPN
iajs-2723	78	4	)	)	PUNCT
iajs-2723	78	5	−	−	ADP
iajs-2723	78	6	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	NUM
iajs-2723	78	7	)	)	PUNCT
iajs-2723	78	8	)	)	PUNCT
iajs-2723	79	1	+	+	CCONJ
iajs-2723	79	2	(	(	PUNCT
iajs-2723	79	3	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	NUM
iajs-2723	79	4	+	+	CCONJ
iajs-2723	79	5	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	PROPN
iajs-2723	79	6	)	)	PUNCT
iajs-2723	79	7	)	)	PUNCT
iajs-2723	80	1	=	=	SYM
iajs-2723	80	2	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	NUM
iajs-2723	80	3	)	)	PUNCT
iajs-2723	81	1	+	+	CCONJ
iajs-2723	81	2	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	NUM
iajs-2723	81	3	=	=	PUNCT
iajs-2723	81	4	𝜃2(𝑢3)𝑐	𝜃2(𝑢3)𝑐	X
iajs-2723	81	5	+	+	X
iajs-2723	81	6	𝑐𝜃1(𝑢3	𝑐𝜃1(𝑢3	NOUN
iajs-2723	81	7	)	)	PUNCT
iajs-2723	81	8	.	.	PUNCT
iajs-2723	82	1	also	also	ADV
iajs-2723	82	2	,	,	PUNCT
iajs-2723	82	3	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	82	4	)	)	PUNCT
iajs-2723	82	5	=	=	SYM
iajs-2723	82	6	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	82	7	−	−	PROPN
iajs-2723	82	8	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	82	9	)	)	PUNCT
iajs-2723	82	10	and	and	CCONJ
iajs-2723	82	11	𝛿2(𝑢3	𝛿2(𝑢3	NOUN
iajs-2723	82	12	)	)	PUNCT
iajs-2723	82	13	=	=	SYM
iajs-2723	82	14	𝛿2(𝑢)𝜃1(𝑢2	𝛿2(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	82	15	)	)	PUNCT
iajs-2723	82	16	+	+	CCONJ
iajs-2723	82	17	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	PROPN
iajs-2723	82	18	)	)	PUNCT
iajs-2723	82	19	+	+	NUM
iajs-2723	82	20	𝜃2(𝑢2)𝛿2(𝑢	𝜃2(𝑢2)𝛿2(𝑢	NOUN
iajs-2723	82	21	)	)	PUNCT
iajs-2723	82	22	.	.	PUNCT
iajs-2723	83	1	thus	thus	ADV
iajs-2723	83	2	,	,	PUNCT
iajs-2723	83	3	𝛿2(𝑢)𝜃1(𝑢2	𝛿2(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	83	4	)	)	PUNCT
iajs-2723	84	1	+	+	CCONJ
iajs-2723	85	1	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	PROPN
iajs-2723	85	2	)	)	PUNCT
iajs-2723	85	3	+	+	NUM
iajs-2723	85	4	𝜃2(𝑢2)𝛿2(𝑢	𝜃2(𝑢2)𝛿2(𝑢	NOUN
iajs-2723	85	5	)	)	PUNCT
iajs-2723	85	6	=	=	SYM
iajs-2723	85	7	(	(	PUNCT
iajs-2723	85	8	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	85	9	−	−	PROPN
iajs-2723	85	10	𝑐𝜃1(𝑢))𝜃1(𝑢2	𝑐𝜃1(𝑢))𝜃1(𝑢2	PROPN
iajs-2723	85	11	)	)	PUNCT
iajs-2723	85	12	+	+	CCONJ
iajs-2723	85	13	𝜃2(𝑢)(𝜃2(𝑢)𝑐	𝜃2(𝑢)(𝜃2(𝑢)𝑐	PUNCT
iajs-2723	85	14	+	+	PUNCT
iajs-2723	85	15	𝑐𝜃1(𝑢))𝜃1(𝑢	𝑐𝜃1(𝑢))𝜃1(𝑢	ADV
iajs-2723	85	16	)	)	PUNCT
iajs-2723	86	1	+	+	CCONJ
iajs-2723	86	2	𝜃2(𝑢2)(𝜃2(𝑢)𝑐	𝜃2(𝑢2)(𝜃2(𝑢)𝑐	PROPN
iajs-2723	86	3	−	−	PROPN
iajs-2723	86	4	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	86	5	)	)	PUNCT
iajs-2723	86	6	)	)	PUNCT
iajs-2723	87	1	=	=	SYM
iajs-2723	87	2	(	(	PUNCT
iajs-2723	87	3	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	X
iajs-2723	87	4	)	)	PUNCT
iajs-2723	87	5	−	−	PROPN
iajs-2723	87	6	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	NUM
iajs-2723	87	7	)	)	PUNCT
iajs-2723	87	8	)	)	PUNCT
iajs-2723	88	1	+	+	CCONJ
iajs-2723	88	2	(	(	PUNCT
iajs-2723	88	3	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	PROPN
iajs-2723	88	4	)	)	PUNCT
iajs-2723	89	1	+	+	CCONJ
iajs-2723	89	2	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝑐𝜃1(𝑢)𝜃1(𝑢	NUM
iajs-2723	89	3	)	)	PUNCT
iajs-2723	89	4	)	)	PUNCT
iajs-2723	90	1	+	+	PROPN
iajs-2723	90	2	(	(	PUNCT
iajs-2723	90	3	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	PROPN
iajs-2723	90	4	−	−	ADP
iajs-2723	90	5	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	𝜃2(𝑢)𝜃2(𝑢)𝑐𝜃1(𝑢	PROPN
iajs-2723	90	6	)	)	PUNCT
iajs-2723	90	7	)	)	PUNCT
iajs-2723	91	1	=	=	PUNCT
iajs-2723	91	2	−𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	−𝑐𝜃1(𝑢)𝜃1(𝑢)𝜃1(𝑢	PROPN
iajs-2723	91	3	)	)	PUNCT
iajs-2723	92	1	+	+	CCONJ
iajs-2723	92	2	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	𝜃2(𝑢)𝜃2(𝑢)𝜃2(𝑢)𝑐	NUM
iajs-2723	92	3	=	=	PUNCT
iajs-2723	92	4	𝜃2(𝑢3)𝑐	𝜃2(𝑢3)𝑐	VERB
iajs-2723	92	5	−	−	NOUN
iajs-2723	92	6	𝑐𝜃1(𝑢3	𝑐𝜃1(𝑢3	NOUN
iajs-2723	92	7	)	)	PUNCT
iajs-2723	92	8	.	.	PUNCT
iajs-2723	93	1	therefore	therefore	ADV
iajs-2723	93	2	,	,	PUNCT
iajs-2723	93	3	(	(	PUNCT
iajs-2723	93	4	𝛿1	𝛿1	PROPN
iajs-2723	93	5	,	,	PUNCT
iajs-2723	93	6	𝛿2	𝛿2	PROPN
iajs-2723	93	7	)	)	PUNCT
iajs-2723	93	8	is	be	AUX
iajs-2723	93	9	a	a	DET
iajs-2723	93	10	jordan(𝜃1	jordan(𝜃1	NOUN
iajs-2723	93	11	,	,	PUNCT
iajs-2723	93	12	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	93	13	pair	pair	NOUN
iajs-2723	93	14	.	.	PUNCT
iajs-2723	94	1	ibn	ibn	PROPN
iajs-2723	94	2	al	al	PROPN
iajs-2723	94	3	-	-	PUNCT
iajs-2723	94	4	haitham	haitham	PROPN
iajs-2723	94	5	jour	jour	X
iajs-2723	94	6	.	.	PROPN
iajs-2723	94	7	for	for	ADP
iajs-2723	94	8	pure	pure	ADJ
iajs-2723	94	9	&	&	CCONJ
iajs-2723	94	10	appl	appl	PROPN
iajs-2723	94	11	.	.	PUNCT
iajs-2723	95	1	sci	sci	PROPN
iajs-2723	95	2	.	.	PROPN
iajs-2723	96	1	53	53	NUM
iajs-2723	96	2	(	(	PUNCT
iajs-2723	96	3	2)2022	2)2022	NOUN
iajs-2723	96	4	111	111	NUM
iajs-2723	96	5	now	now	ADV
iajs-2723	96	6	,	,	PUNCT
iajs-2723	96	7	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NOUN
iajs-2723	96	8	)	)	PUNCT
iajs-2723	96	9	=	=	SYM
iajs-2723	97	1	𝜃2(𝑢𝑣𝑢)𝑐	𝜃2(𝑢𝑣𝑢)𝑐	NOUN
iajs-2723	97	2	+	+	NUM
iajs-2723	97	3	𝑐𝜃1(𝑢𝑣𝑢	𝑐𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	97	4	)	)	PUNCT
iajs-2723	97	5	and	and	CCONJ
iajs-2723	97	6	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	97	7	)	)	PUNCT
iajs-2723	98	1	+	+	PUNCT
iajs-2723	98	2	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	98	3	)	)	PUNCT
iajs-2723	99	1	+	+	ADJ
iajs-2723	99	2	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	99	3	)	)	PUNCT
iajs-2723	100	1	=	=	SYM
iajs-2723	100	2	(	(	PUNCT
iajs-2723	100	3	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	100	4	+	+	CCONJ
iajs-2723	100	5	𝑐𝜃1(𝑢))𝜃1(𝑣𝑢	𝑐𝜃1(𝑢))𝜃1(𝑣𝑢	PROPN
iajs-2723	100	6	)	)	PUNCT
iajs-2723	101	1	+	+	CCONJ
iajs-2723	101	2	𝜃2(𝑢)(𝜃2(𝑣)𝑐	𝜃2(𝑢)(𝜃2(𝑣)𝑐	PUNCT
iajs-2723	101	3	−	−	NOUN
iajs-2723	101	4	𝑐𝜃1(𝑣))𝜃1(𝑢	𝑐𝜃1(𝑣))𝜃1(𝑢	NUM
iajs-2723	101	5	)	)	PUNCT
iajs-2723	102	1	+	+	CCONJ
iajs-2723	102	2	𝜃2(𝑢𝑣)(𝜃2(𝑢)𝑐	𝜃2(𝑢𝑣)(𝜃2(𝑢)𝑐	NOUN
iajs-2723	102	3	+	+	CCONJ
iajs-2723	102	4	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	102	5	)	)	PUNCT
iajs-2723	102	6	)	)	PUNCT
iajs-2723	103	1	=	=	SYM
iajs-2723	103	2	𝑐𝜃1(𝑢𝑣𝑢	𝑐𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	103	3	)	)	PUNCT
iajs-2723	103	4	+	+	NUM
iajs-2723	103	5	𝜃2(𝑢𝑣𝑢)𝑐	𝜃2(𝑢𝑣𝑢)𝑐	NOUN
iajs-2723	103	6	+	+	CCONJ
iajs-2723	103	7	2𝜃2(𝑢𝑣)𝑐𝜃1(𝑢	2𝜃2(𝑢𝑣)𝑐𝜃1(𝑢	NUM
iajs-2723	103	8	)	)	PUNCT
iajs-2723	104	1	=	=	SYM
iajs-2723	104	2	𝜃2(𝑢𝑣𝑢)𝑐	𝜃2(𝑢𝑣𝑢)𝑐	NOUN
iajs-2723	104	3	+	+	NUM
iajs-2723	104	4	𝑐𝜃1(𝑢𝑣𝑢	𝑐𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	104	5	)	)	PUNCT
iajs-2723	104	6	.	.	PUNCT
iajs-2723	105	1	also	also	ADV
iajs-2723	105	2	,	,	PUNCT
iajs-2723	105	3	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	ADJ
iajs-2723	105	4	)	)	PUNCT
iajs-2723	105	5	=	=	SYM
iajs-2723	105	6	𝜃2(𝑢𝑣𝑢)𝑐	𝜃2(𝑢𝑣𝑢)𝑐	NOUN
iajs-2723	105	7	−	−	PROPN
iajs-2723	105	8	𝑐𝜃1(𝑢𝑣𝑢	𝑐𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	105	9	)	)	PUNCT
iajs-2723	105	10	and	and	CCONJ
iajs-2723	105	11	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	105	12	)	)	PUNCT
iajs-2723	106	1	+	+	PUNCT
iajs-2723	106	2	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	106	3	)	)	PUNCT
iajs-2723	106	4	+	+	NUM
iajs-2723	106	5	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	SYM
iajs-2723	106	6	)	)	PUNCT
iajs-2723	106	7	=	=	PUNCT
iajs-2723	107	1	(	(	PUNCT
iajs-2723	107	2	𝜃2(𝑢)𝑐	𝜃2(𝑢)𝑐	NOUN
iajs-2723	107	3	−	−	PROPN
iajs-2723	107	4	𝑐𝜃1(𝑢))𝜃1(𝑣𝑢	𝑐𝜃1(𝑢))𝜃1(𝑣𝑢	PROPN
iajs-2723	107	5	)	)	PUNCT
iajs-2723	107	6	+	+	CCONJ
iajs-2723	107	7	𝜃2(𝑢)(𝜃2(𝑣)𝑐	𝜃2(𝑢)(𝜃2(𝑣)𝑐	PUNCT
iajs-2723	107	8	+	+	CCONJ
iajs-2723	107	9	𝑐𝜃1(𝑣))𝜃1(𝑢	𝑐𝜃1(𝑣))𝜃1(𝑢	NUM
iajs-2723	107	10	)	)	PUNCT
iajs-2723	108	1	+	+	CCONJ
iajs-2723	108	2	𝜃2(𝑢𝑣)(𝜃2(𝑢)𝑐	𝜃2(𝑢𝑣)(𝜃2(𝑢)𝑐	NOUN
iajs-2723	108	3	−	−	PROPN
iajs-2723	108	4	𝑐𝜃1(𝑢	𝑐𝜃1(𝑢	NUM
iajs-2723	108	5	)	)	PUNCT
iajs-2723	108	6	)	)	PUNCT
iajs-2723	109	1	=	=	SYM
iajs-2723	109	2	−𝑐𝜃1(𝑢𝑣𝑢	−𝑐𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	109	3	)	)	PUNCT
iajs-2723	110	1	+	+	NUM
iajs-2723	110	2	𝜃2(𝑢𝑣𝑢)𝑐	𝜃2(𝑢𝑣𝑢)𝑐	NOUN
iajs-2723	110	3	+	+	CCONJ
iajs-2723	110	4	2𝜃2(𝑢)𝑐𝜃1(𝑣𝑢	2𝜃2(𝑢)𝑐𝜃1(𝑣𝑢	NOUN
iajs-2723	110	5	)	)	PUNCT
iajs-2723	110	6	.	.	PUNCT
iajs-2723	111	1	since	since	SCONJ
iajs-2723	111	2	𝜃2(𝑢)𝑐𝜃1(𝑣	𝜃2(𝑢)𝑐𝜃1(𝑣	NOUN
iajs-2723	111	3	)	)	PUNCT
iajs-2723	111	4	≠	≠	PROPN
iajs-2723	111	5	0	0	NUM
iajs-2723	111	6	for	for	ADP
iajs-2723	111	7	some	some	DET
iajs-2723	111	8	𝑢	𝑢	NOUN
iajs-2723	111	9	≠	≠	NOUN
iajs-2723	111	10	𝑣	𝑣	ADP
iajs-2723	111	11	∈	∈	PROPN
iajs-2723	111	12	γ	γ	NOUN
iajs-2723	111	13	,	,	PUNCT
iajs-2723	111	14	this	this	PRON
iajs-2723	111	15	means	mean	VERB
iajs-2723	111	16	that	that	SCONJ
iajs-2723	111	17	(	(	PUNCT
iajs-2723	111	18	𝛿1	𝛿1	NOUN
iajs-2723	111	19	,	,	PUNCT
iajs-2723	111	20	𝛿2	𝛿2	PROPN
iajs-2723	111	21	)	)	PUNCT
iajs-2723	111	22	is	be	AUX
iajs-2723	111	23	not	not	PART
iajs-2723	111	24	(	(	PUNCT
iajs-2723	111	25	𝜃1	𝜃1	VERB
iajs-2723	111	26	,	,	PUNCT
iajs-2723	111	27	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	111	28	pair	pair	NOUN
iajs-2723	111	29	.	.	PUNCT
iajs-2723	112	1	theorem	theorem	VERB
iajs-2723	112	2	3.1	3.1	NUM
iajs-2723	112	3	let	let	VERB
iajs-2723	112	4	γ	γ	NOUN
iajs-2723	112	5	be	be	AUX
iajs-2723	112	6	a	a	DET
iajs-2723	112	7	prime	prime	ADJ
iajs-2723	112	8	ring	ring	NOUN
iajs-2723	112	9	.	.	PUNCT
iajs-2723	113	1	let	let	VERB
iajs-2723	113	2	𝜃1	𝜃1	VERB
iajs-2723	113	3	and	and	CCONJ
iajs-2723	113	4	𝜃2	𝜃2	NOUN
iajs-2723	113	5	be	be	VERB
iajs-2723	113	6	two	two	NUM
iajs-2723	113	7	automorphisms	automorphism	NOUN
iajs-2723	113	8	of	of	ADP
iajs-2723	113	9	γ	γ	PROPN
iajs-2723	113	10	.	.	PROPN
iajs-2723	114	1	if	if	SCONJ
iajs-2723	114	2	γ	γ	PROPN
iajs-2723	114	3	is	be	AUX
iajs-2723	114	4	a	a	DET
iajs-2723	114	5	(	(	PUNCT
iajs-2723	114	6	𝛿1	𝛿1	NOUN
iajs-2723	114	7	,	,	PUNCT
iajs-2723	114	8	𝛿2)derivation	𝛿2)derivation	NOUN
iajs-2723	114	9	pair	pair	NOUN
iajs-2723	114	10	such	such	ADJ
iajs-2723	114	11	that	that	SCONJ
iajs-2723	114	12	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	114	13	)	)	PUNCT
iajs-2723	114	14	=	=	VERB
iajs-2723	114	15	∓	∓	PROPN
iajs-2723	114	16	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	114	17	)	)	PUNCT
iajs-2723	114	18	for	for	ADP
iajs-2723	114	19	each	each	DET
iajs-2723	114	20	𝑢	𝑢	PROPN
iajs-2723	114	21	∈	∈	PROPN
iajs-2723	114	22	γ	γ	X
iajs-2723	114	23	,	,	PUNCT
iajs-2723	114	24	then	then	ADV
iajs-2723	114	25	𝛿2(𝑢	𝛿2(𝑢	X
iajs-2723	114	26	)	)	PUNCT
iajs-2723	114	27	=	=	SYM
iajs-2723	114	28	0	0	X
iajs-2723	114	29	.	.	PUNCT
iajs-2723	115	1	proof	proof	NOUN
iajs-2723	115	2	:	:	PUNCT
iajs-2723	115	3	let	let	VERB
iajs-2723	115	4	𝑢	𝑢	PRON
iajs-2723	115	5	∈	∈	PROPN
iajs-2723	115	6	γ	γ	X
iajs-2723	115	7	.	.	PUNCT
iajs-2723	116	1	if	if	SCONJ
iajs-2723	116	2	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	116	3	)	)	PUNCT
iajs-2723	116	4	=	=	SYM
iajs-2723	117	1	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	117	2	)	)	PUNCT
iajs-2723	117	3	for	for	ADP
iajs-2723	117	4	each	each	DET
iajs-2723	117	5	𝑢	𝑢	PROPN
iajs-2723	117	6	∈	∈	PROPN
iajs-2723	117	7	γ	γ	X
iajs-2723	117	8	(	(	PUNCT
iajs-2723	117	9	1	1	NUM
iajs-2723	117	10	)	)	PUNCT
iajs-2723	117	11	replacing	replace	VERB
iajs-2723	117	12	𝑢	𝑢	PRON
iajs-2723	117	13	by	by	ADP
iajs-2723	117	14	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	117	15	in	in	ADP
iajs-2723	117	16	(	(	PUNCT
iajs-2723	117	17	1	1	NUM
iajs-2723	117	18	)	)	PUNCT
iajs-2723	117	19	,	,	PUNCT
iajs-2723	117	20	we	we	PRON
iajs-2723	117	21	get	get	VERB
iajs-2723	117	22	:	:	PUNCT
iajs-2723	117	23	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NOUN
iajs-2723	117	24	)	)	PUNCT
iajs-2723	118	1	=	=	SYM
iajs-2723	118	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	118	3	)	)	PUNCT
iajs-2723	118	4	for	for	ADP
iajs-2723	118	5	each	each	DET
iajs-2723	118	6	𝑢	𝑢	NOUN
iajs-2723	118	7	,	,	PUNCT
iajs-2723	118	8	𝑣	𝑣	PROPN
iajs-2723	118	9	∈	∈	PROPN
iajs-2723	118	10	γ	γ	X
iajs-2723	118	11	(	(	PUNCT
iajs-2723	118	12	2	2	NUM
iajs-2723	118	13	)	)	PUNCT
iajs-2723	118	14	that	that	PRON
iajs-2723	118	15	is	be	AUX
iajs-2723	118	16	:	:	PUNCT
iajs-2723	118	17	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	118	18	)	)	PUNCT
iajs-2723	118	19	+	+	PUNCT
iajs-2723	118	20	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	118	21	)	)	PUNCT
iajs-2723	118	22	+	+	ADJ
iajs-2723	118	23	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	118	24	)	)	PUNCT
iajs-2723	118	25	=	=	SYM
iajs-2723	118	26	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	118	27	)	)	PUNCT
iajs-2723	118	28	for	for	ADP
iajs-2723	118	29	each	each	DET
iajs-2723	118	30	𝑢	𝑢	NOUN
iajs-2723	118	31	,	,	PUNCT
iajs-2723	118	32	𝑣	𝑣	PROPN
iajs-2723	118	33	∈	∈	PROPN
iajs-2723	118	34	γ	γ	X
iajs-2723	118	35	(	(	PUNCT
iajs-2723	118	36	3	3	NUM
iajs-2723	118	37	)	)	PUNCT
iajs-2723	118	38	by	by	ADP
iajs-2723	118	39	using	use	VERB
iajs-2723	118	40	(	(	PUNCT
iajs-2723	118	41	1	1	X
iajs-2723	118	42	)	)	PUNCT
iajs-2723	118	43	we	we	PRON
iajs-2723	118	44	have	have	VERB
iajs-2723	118	45	:	:	PUNCT
iajs-2723	118	46	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	ADJ
iajs-2723	118	47	)	)	PUNCT
iajs-2723	118	48	+	+	CCONJ
iajs-2723	118	49	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	118	50	)	)	PUNCT
iajs-2723	118	51	+	+	PUNCT
iajs-2723	118	52	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	118	53	)	)	PUNCT
iajs-2723	118	54	−	−	NOUN
iajs-2723	118	55	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	118	56	)	)	PUNCT
iajs-2723	118	57	=	=	SYM
iajs-2723	118	58	0	0	NUM
iajs-2723	118	59	for	for	ADP
iajs-2723	118	60	each	each	DET
iajs-2723	118	61	𝑢	𝑢	NOUN
iajs-2723	118	62	,	,	PUNCT
iajs-2723	118	63	𝑣	𝑣	PROPN
iajs-2723	118	64	∈	∈	PROPN
iajs-2723	118	65	γ	γ	X
iajs-2723	118	66	(	(	PUNCT
iajs-2723	118	67	4	4	NUM
iajs-2723	118	68	)	)	PUNCT
iajs-2723	118	69	that	that	PRON
iajs-2723	118	70	is	be	AUX
iajs-2723	118	71	:	:	PUNCT
iajs-2723	118	72	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	118	73	)	)	PUNCT
iajs-2723	118	74	+	+	PUNCT
iajs-2723	118	75	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	118	76	)	)	PUNCT
iajs-2723	119	1	=	=	SYM
iajs-2723	119	2	0	0	PUNCT
iajs-2723	119	3	for	for	ADP
iajs-2723	119	4	each	each	DET
iajs-2723	119	5	𝑢	𝑢	NOUN
iajs-2723	119	6	,	,	PUNCT
iajs-2723	119	7	𝑣	𝑣	PROPN
iajs-2723	119	8	∈	∈	PROPN
iajs-2723	119	9	γ	γ	X
iajs-2723	119	10	(	(	PUNCT
iajs-2723	119	11	5	5	NUM
iajs-2723	119	12	)	)	PUNCT
iajs-2723	119	13	that	that	PRON
iajs-2723	119	14	is	be	AUX
iajs-2723	119	15	:	:	PUNCT
iajs-2723	119	16	𝜃2(𝑢)(𝛿2(𝑣	𝜃2(𝑢)(𝛿2(𝑣	NOUN
iajs-2723	119	17	)	)	PUNCT
iajs-2723	119	18	+	+	ADJ
iajs-2723	119	19	𝜃2(𝑣))𝜃1(𝑢	𝜃2(𝑣))𝜃1(𝑢	SYM
iajs-2723	119	20	)	)	PUNCT
iajs-2723	119	21	=	=	SYM
iajs-2723	119	22	0	0	NUM
iajs-2723	119	23	for	for	ADP
iajs-2723	119	24	each	each	DET
iajs-2723	119	25	𝑢	𝑢	NOUN
iajs-2723	119	26	,	,	PUNCT
iajs-2723	119	27	𝑣	𝑣	PROPN
iajs-2723	119	28	∈	∈	PROPN
iajs-2723	119	29	γ	γ	X
iajs-2723	119	30	(	(	PUNCT
iajs-2723	119	31	6	6	NUM
iajs-2723	119	32	)	)	PUNCT
iajs-2723	119	33	replacing	replace	VERB
iajs-2723	119	34	𝛿2(𝑣	𝛿2(𝑣	NOUN
iajs-2723	119	35	)	)	PUNCT
iajs-2723	119	36	+	+	PUNCT
iajs-2723	119	37	𝜃2(𝑣	𝜃2(𝑣	X
iajs-2723	119	38	)	)	PUNCT
iajs-2723	119	39	by	by	ADP
iajs-2723	119	40	𝜃2(𝑣	𝜃2(𝑣	PRON
iajs-2723	119	41	)	)	PUNCT
iajs-2723	119	42	in	in	ADP
iajs-2723	119	43	(	(	PUNCT
iajs-2723	119	44	6	6	NUM
iajs-2723	119	45	)	)	PUNCT
iajs-2723	119	46	,	,	PUNCT
iajs-2723	119	47	and	and	CCONJ
iajs-2723	119	48	using	use	VERB
iajs-2723	119	49	(	(	PUNCT
iajs-2723	119	50	1	1	NUM
iajs-2723	119	51	)	)	PUNCT
iajs-2723	119	52	we	we	PRON
iajs-2723	119	53	get	get	VERB
iajs-2723	119	54	:	:	PUNCT
iajs-2723	119	55	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	119	56	)	)	PUNCT
iajs-2723	119	57	=	=	SYM
iajs-2723	119	58	0	0	NUM
iajs-2723	119	59	for	for	ADP
iajs-2723	119	60	each	each	DET
iajs-2723	119	61	𝑢	𝑢	NOUN
iajs-2723	119	62	,	,	PUNCT
iajs-2723	119	63	𝑣	𝑣	PROPN
iajs-2723	119	64	∈	∈	PROPN
iajs-2723	119	65	γ	γ	X
iajs-2723	119	66	(	(	PUNCT
iajs-2723	119	67	7	7	NUM
iajs-2723	119	68	)	)	PUNCT
iajs-2723	119	69	left	leave	VERB
iajs-2723	119	70	multiplying	multiplying	NOUN
iajs-2723	119	71	of	of	ADP
iajs-2723	119	72	(	(	PUNCT
iajs-2723	119	73	7	7	NUM
iajs-2723	119	74	)	)	PUNCT
iajs-2723	119	75	by	by	ADP
iajs-2723	119	76	𝛿2(𝑢	𝛿2(𝑢	NOUN
iajs-2723	119	77	)	)	PUNCT
iajs-2723	119	78	we	we	PRON
iajs-2723	119	79	have	have	VERB
iajs-2723	119	80	:	:	PUNCT
iajs-2723	119	81	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	119	82	)	)	PUNCT
iajs-2723	120	1	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	120	2	)	)	PUNCT
iajs-2723	121	1	=	=	SYM
iajs-2723	121	2	0	0	NUM
iajs-2723	121	3	for	for	ADP
iajs-2723	121	4	each	each	DET
iajs-2723	121	5	𝑢	𝑢	NOUN
iajs-2723	121	6	,	,	PUNCT
iajs-2723	121	7	𝑣	𝑣	PROPN
iajs-2723	121	8	∈	∈	PROPN
iajs-2723	121	9	γ	γ	X
iajs-2723	121	10	(	(	PUNCT
iajs-2723	121	11	8)	8)	NUM
iajs-2723	121	12	since	since	SCONJ
iajs-2723	121	13	γ	γ	X
iajs-2723	121	14	is	be	AUX
iajs-2723	121	15	a	a	DET
iajs-2723	121	16	prime	prime	ADJ
iajs-2723	121	17	ring	ring	NOUN
iajs-2723	121	18	,	,	PUNCT
iajs-2723	121	19	(	(	PUNCT
iajs-2723	121	20	8)	8)	NUM
iajs-2723	121	21	gives	give	VERB
iajs-2723	121	22	:	:	PUNCT
iajs-2723	121	23	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	121	24	)	)	PUNCT
iajs-2723	121	25	=	=	SYM
iajs-2723	121	26	0	0	NUM
iajs-2723	121	27	for	for	ADP
iajs-2723	121	28	each	each	DET
iajs-2723	121	29	𝑢	𝑢	PROPN
iajs-2723	121	30	∈	∈	PROPN
iajs-2723	121	31	γ	γ	X
iajs-2723	121	32	.	.	PUNCT
iajs-2723	122	1	now	now	ADV
iajs-2723	122	2	,	,	PUNCT
iajs-2723	122	3	if	if	SCONJ
iajs-2723	122	4	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	122	5	)	)	PUNCT
iajs-2723	122	6	=	=	PUNCT
iajs-2723	123	1	−	−	PROPN
iajs-2723	124	1	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	124	2	)	)	PUNCT
iajs-2723	124	3	for	for	ADP
iajs-2723	124	4	each	each	DET
iajs-2723	124	5	𝑢	𝑢	PROPN
iajs-2723	124	6	∈	∈	PROPN
iajs-2723	124	7	γ	γ	X
iajs-2723	124	8	(	(	PUNCT
iajs-2723	124	9	9	9	NUM
iajs-2723	124	10	)	)	PUNCT
iajs-2723	124	11	replacing	replace	VERB
iajs-2723	124	12	𝑢	𝑢	PRON
iajs-2723	124	13	by	by	ADP
iajs-2723	124	14	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	124	15	in	in	ADP
iajs-2723	124	16	(	(	PUNCT
iajs-2723	124	17	9	9	NUM
iajs-2723	124	18	)	)	PUNCT
iajs-2723	124	19	,	,	PUNCT
iajs-2723	124	20	we	we	PRON
iajs-2723	124	21	get	get	VERB
iajs-2723	124	22	:	:	PUNCT
iajs-2723	124	23	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NUM
iajs-2723	124	24	)	)	PUNCT
iajs-2723	124	25	=	=	SYM
iajs-2723	125	1	−	−	NOUN
iajs-2723	125	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	125	3	)	)	PUNCT
iajs-2723	125	4	for	for	ADP
iajs-2723	125	5	each	each	DET
iajs-2723	125	6	𝑢	𝑢	NOUN
iajs-2723	125	7	,	,	PUNCT
iajs-2723	125	8	𝑣	𝑣	PROPN
iajs-2723	125	9	∈	∈	PROPN
iajs-2723	125	10	γ	γ	X
iajs-2723	125	11	(	(	PUNCT
iajs-2723	125	12	10	10	NUM
iajs-2723	125	13	)	)	PUNCT
iajs-2723	125	14	that	that	PRON
iajs-2723	125	15	is	be	AUX
iajs-2723	125	16	:	:	PUNCT
iajs-2723	125	17	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	125	18	)	)	PUNCT
iajs-2723	125	19	+	+	PUNCT
iajs-2723	126	1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	126	2	)	)	PUNCT
iajs-2723	126	3	+	+	ADJ
iajs-2723	126	4	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	126	5	)	)	PUNCT
iajs-2723	126	6	=	=	SYM
iajs-2723	127	1	−	−	NOUN
iajs-2723	127	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	127	3	)	)	PUNCT
iajs-2723	127	4	for	for	ADP
iajs-2723	127	5	each	each	DET
iajs-2723	127	6	𝑢	𝑢	NOUN
iajs-2723	127	7	,	,	PUNCT
iajs-2723	127	8	𝑣	𝑣	PROPN
iajs-2723	127	9	∈	∈	PROPN
iajs-2723	127	10	γ	γ	X
iajs-2723	127	11	(	(	PUNCT
iajs-2723	127	12	11	11	NUM
iajs-2723	127	13	)	)	PUNCT
iajs-2723	127	14	by	by	ADP
iajs-2723	127	15	using	use	VERB
iajs-2723	127	16	(	(	PUNCT
iajs-2723	127	17	9	9	NUM
iajs-2723	127	18	)	)	PUNCT
iajs-2723	127	19	we	we	PRON
iajs-2723	127	20	have	have	VERB
iajs-2723	127	21	:	:	PUNCT
iajs-2723	127	22	−𝜃1(𝑢𝑣𝑢	−𝜃1(𝑢𝑣𝑢	X
iajs-2723	127	23	)	)	PUNCT
iajs-2723	127	24	+	+	PUNCT
iajs-2723	128	1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	128	2	)	)	PUNCT
iajs-2723	128	3	−	−	ADP
iajs-2723	129	1	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	129	2	)	)	PUNCT
iajs-2723	130	1	+	+	CCONJ
iajs-2723	130	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	X
iajs-2723	130	3	)	)	PUNCT
iajs-2723	130	4	=	=	SYM
iajs-2723	130	5	0	0	NUM
iajs-2723	130	6	for	for	ADP
iajs-2723	130	7	each	each	DET
iajs-2723	130	8	𝑢	𝑢	NOUN
iajs-2723	130	9	,	,	PUNCT
iajs-2723	130	10	𝑣	𝑣	PROPN
iajs-2723	130	11	∈	∈	PROPN
iajs-2723	130	12	γ	γ	X
iajs-2723	130	13	(	(	PUNCT
iajs-2723	130	14	12	12	NUM
iajs-2723	130	15	)	)	PUNCT
iajs-2723	130	16	that	that	PRON
iajs-2723	130	17	is	be	AUX
iajs-2723	130	18	:	:	PUNCT
iajs-2723	130	19	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	NUM
iajs-2723	130	20	)	)	PUNCT
iajs-2723	130	21	−	−	ADP
iajs-2723	130	22	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	130	23	)	)	PUNCT
iajs-2723	131	1	=	=	SYM
iajs-2723	131	2	0	0	NUM
iajs-2723	131	3	for	for	ADP
iajs-2723	131	4	each	each	DET
iajs-2723	131	5	𝑢	𝑢	NOUN
iajs-2723	131	6	,	,	PUNCT
iajs-2723	131	7	𝑣	𝑣	PROPN
iajs-2723	131	8	∈	∈	PROPN
iajs-2723	131	9	γ	γ	X
iajs-2723	131	10	(	(	PUNCT
iajs-2723	131	11	13	13	NUM
iajs-2723	131	12	)	)	PUNCT
iajs-2723	131	13	that	that	PRON
iajs-2723	131	14	is	be	AUX
iajs-2723	131	15	:	:	PUNCT
iajs-2723	131	16	𝜃2(𝑢)(𝜃2(𝑣	𝜃2(𝑢)(𝜃2(𝑣	ADP
iajs-2723	131	17	)	)	PUNCT
iajs-2723	131	18	−	−	PROPN
iajs-2723	131	19	𝛿2(𝑣))(−𝜃1(𝑢	𝛿2(𝑣))(−𝜃1(𝑢	PROPN
iajs-2723	131	20	)	)	PUNCT
iajs-2723	131	21	)	)	PUNCT
iajs-2723	132	1	=	=	SYM
iajs-2723	132	2	0	0	PUNCT
iajs-2723	133	1	for	for	ADP
iajs-2723	133	2	each	each	DET
iajs-2723	133	3	𝑢	𝑢	NOUN
iajs-2723	133	4	,	,	PUNCT
iajs-2723	133	5	𝑣	𝑣	PROPN
iajs-2723	133	6	∈	∈	PROPN
iajs-2723	133	7	γ	γ	X
iajs-2723	133	8	(	(	PUNCT
iajs-2723	133	9	14	14	NUM
iajs-2723	133	10	)	)	PUNCT
iajs-2723	133	11	by	by	ADP
iajs-2723	133	12	using	use	VERB
iajs-2723	133	13	(	(	PUNCT
iajs-2723	133	14	9	9	NUM
iajs-2723	133	15	)	)	PUNCT
iajs-2723	133	16	we	we	PRON
iajs-2723	133	17	have	have	VERB
iajs-2723	133	18	:	:	PUNCT
iajs-2723	133	19	𝜃2(𝑢)(𝜃2(𝑣	𝜃2(𝑢)(𝜃2(𝑣	NUM
iajs-2723	133	20	)	)	PUNCT
iajs-2723	133	21	−	−	PROPN
iajs-2723	133	22	𝛿2(𝑣))𝛿1(𝑢	𝛿2(𝑣))𝛿1(𝑢	NUM
iajs-2723	133	23	)	)	PUNCT
iajs-2723	133	24	=	=	SYM
iajs-2723	133	25	0	0	NUM
iajs-2723	133	26	for	for	ADP
iajs-2723	133	27	each	each	DET
iajs-2723	133	28	𝑢	𝑢	NOUN
iajs-2723	133	29	,	,	PUNCT
iajs-2723	133	30	𝑣	𝑣	PROPN
iajs-2723	133	31	∈	∈	PROPN
iajs-2723	133	32	γ	γ	X
iajs-2723	133	33	(	(	PUNCT
iajs-2723	133	34	15	15	NUM
iajs-2723	133	35	)	)	PUNCT
iajs-2723	133	36	ibn	ibn	PROPN
iajs-2723	133	37	al	al	PROPN
iajs-2723	133	38	-	-	PUNCT
iajs-2723	133	39	haitham	haitham	PROPN
iajs-2723	133	40	jour	jour	X
iajs-2723	133	41	.	.	PROPN
iajs-2723	134	1	for	for	ADP
iajs-2723	134	2	pure	pure	ADJ
iajs-2723	134	3	&	&	CCONJ
iajs-2723	134	4	appl	appl	PROPN
iajs-2723	134	5	.	.	PUNCT
iajs-2723	135	1	sci	sci	PROPN
iajs-2723	135	2	.	.	PROPN
iajs-2723	136	1	53	53	NUM
iajs-2723	136	2	(	(	PUNCT
iajs-2723	136	3	2)2022	2)2022	VERB
iajs-2723	136	4	112	112	NUM
iajs-2723	136	5	replacing	replace	VERB
iajs-2723	136	6	𝜃2(𝑣	𝜃2(𝑣	NUM
iajs-2723	136	7	)	)	PUNCT
iajs-2723	136	8	−	−	PROPN
iajs-2723	136	9	𝛿2(𝑣	𝛿2(𝑣	NOUN
iajs-2723	136	10	)	)	PUNCT
iajs-2723	136	11	by	by	ADP
iajs-2723	136	12	𝜃2(𝑣	𝜃2(𝑣	PRON
iajs-2723	136	13	)	)	PUNCT
iajs-2723	136	14	in	in	ADP
iajs-2723	136	15	(	(	PUNCT
iajs-2723	136	16	15	15	NUM
iajs-2723	136	17	)	)	PUNCT
iajs-2723	136	18	,	,	PUNCT
iajs-2723	136	19	we	we	PRON
iajs-2723	136	20	get	get	VERB
iajs-2723	136	21	:	:	PUNCT
iajs-2723	136	22	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	136	23	)	)	PUNCT
iajs-2723	136	24	=	=	SYM
iajs-2723	136	25	0	0	NUM
iajs-2723	136	26	for	for	ADP
iajs-2723	136	27	each	each	DET
iajs-2723	136	28	𝑢	𝑢	NOUN
iajs-2723	136	29	,	,	PUNCT
iajs-2723	136	30	𝑣	𝑣	PROPN
iajs-2723	136	31	∈	∈	PROPN
iajs-2723	136	32	γ	γ	X
iajs-2723	136	33	(	(	PUNCT
iajs-2723	136	34	16	16	NUM
iajs-2723	136	35	)	)	PUNCT
iajs-2723	136	36	left	leave	VERB
iajs-2723	136	37	multiplying	multiplying	NOUN
iajs-2723	136	38	of	of	ADP
iajs-2723	136	39	(	(	PUNCT
iajs-2723	136	40	16	16	NUM
iajs-2723	136	41	)	)	PUNCT
iajs-2723	136	42	by	by	ADP
iajs-2723	136	43	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	136	44	)	)	PUNCT
iajs-2723	136	45	we	we	PRON
iajs-2723	136	46	have	have	VERB
iajs-2723	136	47	:	:	PUNCT
iajs-2723	136	48	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	136	49	)	)	PUNCT
iajs-2723	136	50	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	136	51	)	)	PUNCT
iajs-2723	137	1	=	=	SYM
iajs-2723	137	2	0	0	NUM
iajs-2723	137	3	for	for	ADP
iajs-2723	137	4	each	each	DET
iajs-2723	137	5	𝑢	𝑢	NOUN
iajs-2723	137	6	,	,	PUNCT
iajs-2723	137	7	𝑣	𝑣	PROPN
iajs-2723	137	8	∈	∈	PROPN
iajs-2723	137	9	γ	γ	X
iajs-2723	137	10	(	(	PUNCT
iajs-2723	137	11	17	17	NUM
iajs-2723	137	12	)	)	PUNCT
iajs-2723	137	13	since	since	SCONJ
iajs-2723	137	14	γ	γ	X
iajs-2723	137	15	is	be	AUX
iajs-2723	137	16	a	a	DET
iajs-2723	137	17	prime	prime	ADJ
iajs-2723	137	18	ring	ring	NOUN
iajs-2723	137	19	,	,	PUNCT
iajs-2723	137	20	(	(	PUNCT
iajs-2723	137	21	17	17	NUM
iajs-2723	137	22	)	)	PUNCT
iajs-2723	137	23	gives	give	VERB
iajs-2723	137	24	:	:	PUNCT
iajs-2723	137	25	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	137	26	)	)	PUNCT
iajs-2723	137	27	=	=	SYM
iajs-2723	137	28	0	0	NUM
iajs-2723	137	29	for	for	SCONJ
iajs-2723	137	30	each	each	DET
iajs-2723	137	31	𝑢	𝑢	PROPN
iajs-2723	137	32	∈	∈	PROPN
iajs-2723	137	33	γ	γ	X
iajs-2723	137	34	.	.	PUNCT
iajs-2723	137	35	∎	∎	PROPN
iajs-2723	137	36	theorem	theorem	VERB
iajs-2723	137	37	3.2	3.2	NUM
iajs-2723	137	38	let	let	VERB
iajs-2723	137	39	γ	γ	NOUN
iajs-2723	137	40	be	be	AUX
iajs-2723	137	41	a	a	DET
iajs-2723	137	42	prime	prime	ADJ
iajs-2723	137	43	ring	ring	NOUN
iajs-2723	137	44	.	.	PUNCT
iajs-2723	138	1	let	let	VERB
iajs-2723	138	2	𝜃1	𝜃1	VERB
iajs-2723	138	3	and	and	CCONJ
iajs-2723	138	4	𝜃2	𝜃2	NOUN
iajs-2723	138	5	be	be	VERB
iajs-2723	138	6	two	two	NUM
iajs-2723	138	7	automorphisms	automorphism	NOUN
iajs-2723	138	8	of	of	ADP
iajs-2723	138	9	γ	γ	PROPN
iajs-2723	138	10	.	.	PROPN
iajs-2723	139	1	if	if	SCONJ
iajs-2723	139	2	γ	γ	PROPN
iajs-2723	139	3	is	be	AUX
iajs-2723	139	4	a	a	DET
iajs-2723	139	5	(	(	PUNCT
iajs-2723	139	6	𝛿1	𝛿1	NOUN
iajs-2723	139	7	,	,	PUNCT
iajs-2723	139	8	𝛿2)derivation	𝛿2)derivation	NOUN
iajs-2723	139	9	pair	pair	NOUN
iajs-2723	139	10	such	such	ADJ
iajs-2723	139	11	that	that	SCONJ
iajs-2723	139	12	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	139	13	)	)	PUNCT
iajs-2723	139	14	=	=	VERB
iajs-2723	139	15	∓	∓	PROPN
iajs-2723	139	16	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	139	17	)	)	PUNCT
iajs-2723	139	18	for	for	ADP
iajs-2723	139	19	each	each	DET
iajs-2723	139	20	𝑢	𝑢	PROPN
iajs-2723	139	21	∈	∈	PROPN
iajs-2723	139	22	γ	γ	X
iajs-2723	139	23	,	,	PUNCT
iajs-2723	139	24	then	then	ADV
iajs-2723	139	25	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	139	26	)	)	PUNCT
iajs-2723	139	27	=	=	SYM
iajs-2723	139	28	0	0	X
iajs-2723	139	29	.	.	PUNCT
iajs-2723	140	1	proof	proof	NOUN
iajs-2723	140	2	:	:	PUNCT
iajs-2723	140	3	let	let	VERB
iajs-2723	140	4	𝑢	𝑢	PRON
iajs-2723	140	5	∈	∈	PROPN
iajs-2723	140	6	γ	γ	X
iajs-2723	140	7	.	.	PUNCT
iajs-2723	141	1	if	if	SCONJ
iajs-2723	141	2	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	141	3	)	)	PUNCT
iajs-2723	142	1	=	=	SYM
iajs-2723	142	2	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	142	3	)	)	PUNCT
iajs-2723	142	4	for	for	ADP
iajs-2723	142	5	each	each	DET
iajs-2723	142	6	𝑢	𝑢	PROPN
iajs-2723	142	7	∈	∈	PROPN
iajs-2723	142	8	γ	γ	X
iajs-2723	142	9	(	(	PUNCT
iajs-2723	142	10	18	18	NUM
iajs-2723	142	11	)	)	PUNCT
iajs-2723	142	12	replacing	replace	VERB
iajs-2723	142	13	𝑢	𝑢	PRON
iajs-2723	142	14	by	by	ADP
iajs-2723	142	15	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	142	16	in	in	ADP
iajs-2723	142	17	(	(	PUNCT
iajs-2723	142	18	18	18	NUM
iajs-2723	142	19	)	)	PUNCT
iajs-2723	142	20	,	,	PUNCT
iajs-2723	142	21	we	we	PRON
iajs-2723	142	22	get	get	VERB
iajs-2723	142	23	:	:	PUNCT
iajs-2723	142	24	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	X
iajs-2723	142	25	)	)	PUNCT
iajs-2723	142	26	=	=	SYM
iajs-2723	142	27	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	142	28	)	)	PUNCT
iajs-2723	142	29	for	for	ADP
iajs-2723	142	30	each	each	DET
iajs-2723	142	31	𝑢	𝑢	NOUN
iajs-2723	142	32	,	,	PUNCT
iajs-2723	142	33	𝑣	𝑣	PROPN
iajs-2723	142	34	∈	∈	PROPN
iajs-2723	142	35	γ	γ	X
iajs-2723	142	36	(	(	PUNCT
iajs-2723	142	37	19	19	NUM
iajs-2723	142	38	)	)	PUNCT
iajs-2723	142	39	that	that	PRON
iajs-2723	142	40	is	be	AUX
iajs-2723	142	41	:	:	PUNCT
iajs-2723	142	42	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	142	43	)	)	PUNCT
iajs-2723	142	44	+	+	SYM
iajs-2723	142	45	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	142	46	)	)	PUNCT
iajs-2723	143	1	+	+	NUM
iajs-2723	143	2	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	SYM
iajs-2723	143	3	)	)	PUNCT
iajs-2723	143	4	=	=	SYM
iajs-2723	143	5	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	PROPN
iajs-2723	143	6	)	)	PUNCT
iajs-2723	143	7	for	for	ADP
iajs-2723	143	8	each	each	DET
iajs-2723	143	9	𝑢	𝑢	NOUN
iajs-2723	143	10	,	,	PUNCT
iajs-2723	143	11	𝑣	𝑣	PROPN
iajs-2723	143	12	∈	∈	PROPN
iajs-2723	143	13	γ	γ	X
iajs-2723	143	14	(	(	PUNCT
iajs-2723	143	15	20	20	NUM
iajs-2723	143	16	)	)	PUNCT
iajs-2723	143	17	by	by	ADP
iajs-2723	143	18	using	use	VERB
iajs-2723	143	19	(	(	PUNCT
iajs-2723	143	20	18	18	NUM
iajs-2723	143	21	)	)	PUNCT
iajs-2723	143	22	we	we	PRON
iajs-2723	143	23	have	have	VERB
iajs-2723	143	24	:	:	PUNCT
iajs-2723	143	25	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	ADJ
iajs-2723	143	26	)	)	PUNCT
iajs-2723	143	27	+	+	CCONJ
iajs-2723	143	28	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	143	29	)	)	PUNCT
iajs-2723	144	1	+	+	PUNCT
iajs-2723	144	2	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	144	3	)	)	PUNCT
iajs-2723	144	4	−	−	NOUN
iajs-2723	144	5	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	144	6	)	)	PUNCT
iajs-2723	144	7	=	=	SYM
iajs-2723	144	8	0	0	NUM
iajs-2723	144	9	for	for	ADP
iajs-2723	144	10	each	each	DET
iajs-2723	144	11	𝑢	𝑢	NOUN
iajs-2723	144	12	,	,	PUNCT
iajs-2723	144	13	𝑣	𝑣	PROPN
iajs-2723	144	14	∈	∈	PROPN
iajs-2723	144	15	γ	γ	X
iajs-2723	144	16	(	(	PUNCT
iajs-2723	144	17	21	21	NUM
iajs-2723	144	18	)	)	PUNCT
iajs-2723	144	19	that	that	PRON
iajs-2723	144	20	is	be	AUX
iajs-2723	144	21	:	:	PUNCT
iajs-2723	144	22	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	144	23	)	)	PUNCT
iajs-2723	144	24	+	+	PUNCT
iajs-2723	144	25	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	144	26	)	)	PUNCT
iajs-2723	144	27	=	=	SYM
iajs-2723	144	28	0	0	PUNCT
iajs-2723	144	29	for	for	ADP
iajs-2723	144	30	each	each	DET
iajs-2723	144	31	𝑢	𝑢	NOUN
iajs-2723	144	32	,	,	PUNCT
iajs-2723	144	33	𝑣	𝑣	PROPN
iajs-2723	144	34	∈	∈	PROPN
iajs-2723	144	35	γ	γ	X
iajs-2723	144	36	(	(	PUNCT
iajs-2723	144	37	22	22	NUM
iajs-2723	144	38	)	)	PUNCT
iajs-2723	144	39	that	that	PRON
iajs-2723	144	40	is	be	AUX
iajs-2723	144	41	:	:	PUNCT
iajs-2723	144	42	𝜃2(𝑢)(𝛿1(𝑣	𝜃2(𝑢)(𝛿1(𝑣	PRON
iajs-2723	144	43	)	)	PUNCT
iajs-2723	144	44	+	+	PUNCT
iajs-2723	144	45	𝜃2(𝑣))𝜃1(𝑢	𝜃2(𝑣))𝜃1(𝑢	SYM
iajs-2723	144	46	)	)	PUNCT
iajs-2723	144	47	=	=	SYM
iajs-2723	144	48	0	0	NUM
iajs-2723	144	49	for	for	ADP
iajs-2723	144	50	each	each	DET
iajs-2723	144	51	𝑢	𝑢	NOUN
iajs-2723	144	52	,	,	PUNCT
iajs-2723	144	53	𝑣	𝑣	PROPN
iajs-2723	144	54	∈	∈	PROPN
iajs-2723	144	55	γ	γ	X
iajs-2723	144	56	(	(	PUNCT
iajs-2723	144	57	23	23	NUM
iajs-2723	144	58	)	)	PUNCT
iajs-2723	144	59	replacing	replace	VERB
iajs-2723	144	60	𝛿1(𝑣	𝛿1(𝑣	PROPN
iajs-2723	144	61	)	)	PUNCT
iajs-2723	144	62	+	+	PUNCT
iajs-2723	145	1	𝜃2(𝑣	𝜃2(𝑣	X
iajs-2723	145	2	)	)	PUNCT
iajs-2723	145	3	by	by	ADP
iajs-2723	145	4	𝜃2(𝑣	𝜃2(𝑣	PRON
iajs-2723	145	5	)	)	PUNCT
iajs-2723	145	6	in	in	ADP
iajs-2723	145	7	(	(	PUNCT
iajs-2723	145	8	23	23	NUM
iajs-2723	145	9	)	)	PUNCT
iajs-2723	145	10	,	,	PUNCT
iajs-2723	145	11	and	and	CCONJ
iajs-2723	145	12	using	use	VERB
iajs-2723	145	13	(	(	PUNCT
iajs-2723	145	14	18	18	NUM
iajs-2723	145	15	)	)	PUNCT
iajs-2723	145	16	we	we	PRON
iajs-2723	145	17	get	get	VERB
iajs-2723	145	18	:	:	PUNCT
iajs-2723	145	19	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	145	20	)	)	PUNCT
iajs-2723	145	21	=	=	SYM
iajs-2723	145	22	0	0	NUM
iajs-2723	145	23	for	for	ADP
iajs-2723	145	24	each	each	DET
iajs-2723	145	25	𝑢	𝑢	NOUN
iajs-2723	145	26	,	,	PUNCT
iajs-2723	145	27	𝑣	𝑣	PROPN
iajs-2723	145	28	∈	∈	PROPN
iajs-2723	145	29	γ	γ	X
iajs-2723	145	30	(	(	PUNCT
iajs-2723	145	31	24	24	NUM
iajs-2723	145	32	)	)	PUNCT
iajs-2723	145	33	left	leave	VERB
iajs-2723	145	34	multiplying	multiplying	NOUN
iajs-2723	145	35	of	of	ADP
iajs-2723	145	36	(	(	PUNCT
iajs-2723	145	37	24	24	NUM
iajs-2723	145	38	)	)	PUNCT
iajs-2723	145	39	by	by	ADP
iajs-2723	145	40	𝛿1(𝑢	𝛿1(𝑢	NOUN
iajs-2723	145	41	)	)	PUNCT
iajs-2723	145	42	we	we	PRON
iajs-2723	145	43	have	have	VERB
iajs-2723	145	44	:	:	PUNCT
iajs-2723	145	45	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	145	46	)	)	PUNCT
iajs-2723	145	47	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	NOUN
iajs-2723	145	48	)	)	PUNCT
iajs-2723	145	49	=	=	SYM
iajs-2723	145	50	0	0	NUM
iajs-2723	145	51	for	for	ADP
iajs-2723	145	52	each	each	DET
iajs-2723	145	53	𝑢	𝑢	NOUN
iajs-2723	145	54	,	,	PUNCT
iajs-2723	145	55	𝑣	𝑣	PROPN
iajs-2723	145	56	∈	∈	PROPN
iajs-2723	145	57	γ	γ	X
iajs-2723	145	58	(	(	PUNCT
iajs-2723	145	59	25	25	NUM
iajs-2723	145	60	)	)	PUNCT
iajs-2723	145	61	since	since	SCONJ
iajs-2723	145	62	γ	γ	X
iajs-2723	145	63	is	be	AUX
iajs-2723	145	64	a	a	DET
iajs-2723	145	65	prime	prime	ADJ
iajs-2723	145	66	ring	ring	NOUN
iajs-2723	145	67	,	,	PUNCT
iajs-2723	145	68	(	(	PUNCT
iajs-2723	145	69	25	25	NUM
iajs-2723	145	70	)	)	PUNCT
iajs-2723	145	71	gives	give	VERB
iajs-2723	145	72	:	:	PUNCT
iajs-2723	145	73	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	145	74	)	)	PUNCT
iajs-2723	145	75	=	=	SYM
iajs-2723	145	76	0	0	NUM
iajs-2723	145	77	for	for	ADP
iajs-2723	145	78	each	each	DET
iajs-2723	145	79	𝑢	𝑢	PROPN
iajs-2723	145	80	∈	∈	PROPN
iajs-2723	145	81	γ	γ	X
iajs-2723	145	82	.	.	PUNCT
iajs-2723	146	1	now	now	ADV
iajs-2723	146	2	,	,	PUNCT
iajs-2723	146	3	if	if	SCONJ
iajs-2723	146	4	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	146	5	)	)	PUNCT
iajs-2723	146	6	=	=	SYM
iajs-2723	147	1	−	−	PROPN
iajs-2723	148	1	𝜃1(𝑢	𝜃1(𝑢	NOUN
iajs-2723	148	2	)	)	PUNCT
iajs-2723	148	3	for	for	ADP
iajs-2723	148	4	each	each	DET
iajs-2723	148	5	𝑢	𝑢	PROPN
iajs-2723	148	6	∈	∈	PROPN
iajs-2723	148	7	γ	γ	X
iajs-2723	148	8	(	(	PUNCT
iajs-2723	148	9	26	26	NUM
iajs-2723	148	10	)	)	PUNCT
iajs-2723	148	11	replacing	replace	VERB
iajs-2723	148	12	𝑢	𝑢	PRON
iajs-2723	148	13	by	by	ADP
iajs-2723	148	14	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	148	15	in	in	ADP
iajs-2723	148	16	(	(	PUNCT
iajs-2723	148	17	26	26	NUM
iajs-2723	148	18	)	)	PUNCT
iajs-2723	148	19	,	,	PUNCT
iajs-2723	148	20	we	we	PRON
iajs-2723	148	21	get	get	VERB
iajs-2723	148	22	:	:	PUNCT
iajs-2723	148	23	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	X
iajs-2723	148	24	)	)	PUNCT
iajs-2723	148	25	=	=	SYM
iajs-2723	149	1	−	−	NOUN
iajs-2723	149	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	149	3	)	)	PUNCT
iajs-2723	149	4	for	for	ADP
iajs-2723	149	5	each	each	DET
iajs-2723	149	6	𝑢	𝑢	NOUN
iajs-2723	149	7	,	,	PUNCT
iajs-2723	149	8	𝑣	𝑣	PROPN
iajs-2723	149	9	∈	∈	PROPN
iajs-2723	149	10	γ	γ	X
iajs-2723	149	11	(	(	PUNCT
iajs-2723	149	12	27	27	NUM
iajs-2723	149	13	)	)	PUNCT
iajs-2723	149	14	that	that	PRON
iajs-2723	149	15	is	be	AUX
iajs-2723	149	16	:	:	PUNCT
iajs-2723	149	17	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	149	18	)	)	PUNCT
iajs-2723	149	19	+	+	SYM
iajs-2723	149	20	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	149	21	)	)	PUNCT
iajs-2723	149	22	+	+	NUM
iajs-2723	149	23	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	SYM
iajs-2723	149	24	)	)	PUNCT
iajs-2723	149	25	=	=	SYM
iajs-2723	150	1	−	−	NOUN
iajs-2723	150	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	NOUN
iajs-2723	150	3	)	)	PUNCT
iajs-2723	150	4	for	for	ADP
iajs-2723	150	5	each	each	DET
iajs-2723	150	6	𝑢	𝑢	NOUN
iajs-2723	150	7	,	,	PUNCT
iajs-2723	150	8	𝑣	𝑣	PROPN
iajs-2723	150	9	∈	∈	PROPN
iajs-2723	150	10	γ	γ	X
iajs-2723	150	11	(	(	PUNCT
iajs-2723	150	12	28	28	NUM
iajs-2723	150	13	)	)	PUNCT
iajs-2723	150	14	by	by	ADP
iajs-2723	150	15	using	use	VERB
iajs-2723	150	16	(	(	PUNCT
iajs-2723	150	17	26	26	NUM
iajs-2723	150	18	)	)	PUNCT
iajs-2723	150	19	we	we	PRON
iajs-2723	150	20	have	have	VERB
iajs-2723	150	21	:	:	PUNCT
iajs-2723	150	22	−𝜃1(𝑢𝑣𝑢	−𝜃1(𝑢𝑣𝑢	X
iajs-2723	150	23	)	)	PUNCT
iajs-2723	150	24	+	+	PUNCT
iajs-2723	150	25	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	150	26	)	)	PUNCT
iajs-2723	150	27	−	−	ADP
iajs-2723	151	1	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	151	2	)	)	PUNCT
iajs-2723	152	1	+	+	CCONJ
iajs-2723	152	2	𝜃1(𝑢𝑣𝑢	𝜃1(𝑢𝑣𝑢	X
iajs-2723	152	3	)	)	PUNCT
iajs-2723	152	4	=	=	SYM
iajs-2723	152	5	0	0	NUM
iajs-2723	152	6	for	for	ADP
iajs-2723	152	7	each	each	DET
iajs-2723	152	8	𝑢	𝑢	NOUN
iajs-2723	152	9	,	,	PUNCT
iajs-2723	152	10	𝑣	𝑣	PROPN
iajs-2723	152	11	∈	∈	PROPN
iajs-2723	152	12	γ	γ	X
iajs-2723	152	13	(	(	PUNCT
iajs-2723	152	14	29	29	NUM
iajs-2723	152	15	)	)	PUNCT
iajs-2723	152	16	that	that	PRON
iajs-2723	152	17	is	be	AUX
iajs-2723	152	18	:	:	PUNCT
iajs-2723	152	19	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	152	20	)	)	PUNCT
iajs-2723	152	21	−	−	ADP
iajs-2723	152	22	𝜃2(𝑢𝑣)𝜃1(𝑢	𝜃2(𝑢𝑣)𝜃1(𝑢	NUM
iajs-2723	152	23	)	)	PUNCT
iajs-2723	153	1	=	=	SYM
iajs-2723	153	2	0	0	NUM
iajs-2723	153	3	for	for	ADP
iajs-2723	153	4	each	each	DET
iajs-2723	153	5	𝑢	𝑢	NOUN
iajs-2723	153	6	,	,	PUNCT
iajs-2723	153	7	𝑣	𝑣	PROPN
iajs-2723	153	8	∈	∈	PROPN
iajs-2723	153	9	γ	γ	X
iajs-2723	153	10	(	(	PUNCT
iajs-2723	153	11	30	30	NUM
iajs-2723	153	12	)	)	PUNCT
iajs-2723	153	13	that	that	PRON
iajs-2723	153	14	is	be	AUX
iajs-2723	153	15	:	:	PUNCT
iajs-2723	153	16	𝜃2(𝑢)(𝜃2(𝑣	𝜃2(𝑢)(𝜃2(𝑣	ADP
iajs-2723	153	17	)	)	PUNCT
iajs-2723	153	18	−	−	PROPN
iajs-2723	153	19	𝛿1(𝑣))(−𝜃1(𝑢	𝛿1(𝑣))(−𝜃1(𝑢	PROPN
iajs-2723	153	20	)	)	PUNCT
iajs-2723	153	21	)	)	PUNCT
iajs-2723	153	22	=	=	SYM
iajs-2723	153	23	0	0	PUNCT
iajs-2723	153	24	for	for	ADP
iajs-2723	153	25	each	each	DET
iajs-2723	153	26	𝑢	𝑢	NOUN
iajs-2723	153	27	,	,	PUNCT
iajs-2723	153	28	𝑣	𝑣	PROPN
iajs-2723	153	29	∈	∈	PROPN
iajs-2723	153	30	γ	γ	X
iajs-2723	153	31	(	(	PUNCT
iajs-2723	153	32	31	31	NUM
iajs-2723	153	33	)	)	PUNCT
iajs-2723	153	34	by	by	ADP
iajs-2723	153	35	using	use	VERB
iajs-2723	153	36	(	(	PUNCT
iajs-2723	153	37	26	26	NUM
iajs-2723	153	38	)	)	PUNCT
iajs-2723	153	39	we	we	PRON
iajs-2723	153	40	have	have	VERB
iajs-2723	153	41	:	:	PUNCT
iajs-2723	153	42	𝜃2(𝑢)(𝜃2(𝑣	𝜃2(𝑢)(𝜃2(𝑣	NUM
iajs-2723	153	43	)	)	PUNCT
iajs-2723	153	44	−	−	PRON
iajs-2723	153	45	𝛿1(𝑣))𝛿2(𝑢	𝛿1(𝑣))𝛿2(𝑢	NOUN
iajs-2723	153	46	)	)	PUNCT
iajs-2723	153	47	=	=	SYM
iajs-2723	153	48	0	0	NUM
iajs-2723	153	49	for	for	ADP
iajs-2723	153	50	each	each	DET
iajs-2723	153	51	𝑢	𝑢	NOUN
iajs-2723	153	52	,	,	PUNCT
iajs-2723	153	53	𝑣	𝑣	PROPN
iajs-2723	153	54	∈	∈	PROPN
iajs-2723	153	55	γ	γ	X
iajs-2723	153	56	(	(	PUNCT
iajs-2723	153	57	32	32	NUM
iajs-2723	153	58	)	)	PUNCT
iajs-2723	153	59	replacing	replace	VERB
iajs-2723	153	60	𝜃2(𝑣	𝜃2(𝑣	NUM
iajs-2723	153	61	)	)	PUNCT
iajs-2723	153	62	−	−	PROPN
iajs-2723	153	63	𝛿1(𝑣	𝛿1(𝑣	PROPN
iajs-2723	153	64	)	)	PUNCT
iajs-2723	153	65	by	by	ADP
iajs-2723	153	66	𝜃2(𝑣	𝜃2(𝑣	PRON
iajs-2723	153	67	)	)	PUNCT
iajs-2723	153	68	in	in	ADP
iajs-2723	153	69	(	(	PUNCT
iajs-2723	153	70	32	32	NUM
iajs-2723	153	71	)	)	PUNCT
iajs-2723	153	72	,	,	PUNCT
iajs-2723	153	73	we	we	PRON
iajs-2723	153	74	get	get	VERB
iajs-2723	153	75	:	:	PUNCT
iajs-2723	153	76	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	153	77	)	)	PUNCT
iajs-2723	153	78	=	=	SYM
iajs-2723	153	79	0	0	NUM
iajs-2723	153	80	for	for	ADP
iajs-2723	153	81	each	each	DET
iajs-2723	153	82	𝑢	𝑢	NOUN
iajs-2723	153	83	,	,	PUNCT
iajs-2723	153	84	𝑣	𝑣	PROPN
iajs-2723	153	85	∈	∈	PROPN
iajs-2723	153	86	γ	γ	X
iajs-2723	153	87	(	(	PUNCT
iajs-2723	153	88	33	33	NUM
iajs-2723	153	89	)	)	PUNCT
iajs-2723	153	90	left	leave	VERB
iajs-2723	153	91	multiplying	multiplying	NOUN
iajs-2723	153	92	of	of	ADP
iajs-2723	153	93	(	(	PUNCT
iajs-2723	153	94	33	33	NUM
iajs-2723	153	95	)	)	PUNCT
iajs-2723	153	96	by	by	ADP
iajs-2723	153	97	δ1(𝑢	δ1(𝑢	NOUN
iajs-2723	153	98	)	)	PUNCT
iajs-2723	154	1	we	we	PRON
iajs-2723	154	2	have	have	VERB
iajs-2723	154	3	:	:	PUNCT
iajs-2723	154	4	ibn	ibn	PROPN
iajs-2723	154	5	al	al	PROPN
iajs-2723	154	6	-	-	PUNCT
iajs-2723	154	7	haitham	haitham	PROPN
iajs-2723	154	8	jour	jour	X
iajs-2723	154	9	.	.	PROPN
iajs-2723	155	1	for	for	ADP
iajs-2723	155	2	pure	pure	ADJ
iajs-2723	155	3	&	&	CCONJ
iajs-2723	155	4	appl	appl	PROPN
iajs-2723	155	5	.	.	PUNCT
iajs-2723	156	1	sci	sci	PROPN
iajs-2723	156	2	.	.	PROPN
iajs-2723	157	1	53	53	NUM
iajs-2723	157	2	(	(	PUNCT
iajs-2723	157	3	2)2022	2)2022	VERB
iajs-2723	157	4	113	113	NUM
iajs-2723	157	5	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	157	6	)	)	PUNCT
iajs-2723	157	7	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	157	8	)	)	PUNCT
iajs-2723	157	9	=	=	SYM
iajs-2723	157	10	0	0	NUM
iajs-2723	157	11	for	for	ADP
iajs-2723	157	12	each	each	DET
iajs-2723	157	13	𝑢	𝑢	NOUN
iajs-2723	157	14	,	,	PUNCT
iajs-2723	157	15	𝑣	𝑣	PROPN
iajs-2723	157	16	∈	∈	PROPN
iajs-2723	157	17	γ	γ	X
iajs-2723	157	18	(	(	PUNCT
iajs-2723	157	19	34	34	NUM
iajs-2723	157	20	)	)	PUNCT
iajs-2723	157	21	since	since	SCONJ
iajs-2723	157	22	γ	γ	X
iajs-2723	157	23	is	be	AUX
iajs-2723	157	24	a	a	DET
iajs-2723	157	25	prime	prime	ADJ
iajs-2723	157	26	ring	ring	NOUN
iajs-2723	157	27	,	,	PUNCT
iajs-2723	157	28	(	(	PUNCT
iajs-2723	157	29	34	34	NUM
iajs-2723	157	30	)	)	PUNCT
iajs-2723	157	31	gives	give	VERB
iajs-2723	157	32	:	:	PUNCT
iajs-2723	157	33	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	157	34	)	)	PUNCT
iajs-2723	157	35	=	=	SYM
iajs-2723	157	36	0	0	NUM
iajs-2723	157	37	for	for	SCONJ
iajs-2723	157	38	each	each	DET
iajs-2723	157	39	𝑢	𝑢	PROPN
iajs-2723	157	40	∈	∈	PROPN
iajs-2723	157	41	γ	γ	X
iajs-2723	157	42	.	.	PUNCT
iajs-2723	157	43	∎	∎	PROPN
iajs-2723	157	44	theorem	theorem	VERB
iajs-2723	157	45	3.3	3.3	NUM
iajs-2723	157	46	let	let	VERB
iajs-2723	157	47	γ	γ	NOUN
iajs-2723	157	48	be	be	AUX
iajs-2723	157	49	a	a	DET
iajs-2723	157	50	prime	prime	ADJ
iajs-2723	157	51	ring	ring	NOUN
iajs-2723	157	52	.	.	PUNCT
iajs-2723	158	1	let	let	VERB
iajs-2723	158	2	𝜃1	𝜃1	VERB
iajs-2723	158	3	and	and	CCONJ
iajs-2723	158	4	𝜃2	𝜃2	NOUN
iajs-2723	158	5	be	be	VERB
iajs-2723	158	6	two	two	NUM
iajs-2723	158	7	automorphisms	automorphism	NOUN
iajs-2723	158	8	of	of	ADP
iajs-2723	158	9	γ	γ	PROPN
iajs-2723	158	10	.	.	PROPN
iajs-2723	159	1	if	if	SCONJ
iajs-2723	159	2	γ	γ	PROPN
iajs-2723	159	3	is	be	AUX
iajs-2723	159	4	a	a	DET
iajs-2723	159	5	(	(	PUNCT
iajs-2723	159	6	𝛿1	𝛿1	NOUN
iajs-2723	159	7	,	,	PUNCT
iajs-2723	159	8	𝛿2	𝛿2	NOUN
iajs-2723	159	9	)	)	PUNCT
iajs-2723	159	10	derivation	derivation	NOUN
iajs-2723	159	11	pair	pair	VERB
iajs-2723	159	12	such	such	ADJ
iajs-2723	159	13	that	that	DET
iajs-2723	159	14	𝛿2(𝑢)𝛿1(𝑣	𝛿2(𝑢)𝛿1(𝑣	NOUN
iajs-2723	159	15	)	)	PUNCT
iajs-2723	159	16	=	=	SYM
iajs-2723	159	17	0	0	NUM
iajs-2723	159	18	(	(	PUNCT
iajs-2723	159	19	resp	resp	NOUN
iajs-2723	159	20	.	.	PUNCT
iajs-2723	160	1	𝛿1(𝑢)𝛿2(𝑣	𝛿1(𝑢)𝛿2(𝑣	NOUN
iajs-2723	160	2	)	)	PUNCT
iajs-2723	160	3	=	=	SYM
iajs-2723	160	4	0	0	NUM
iajs-2723	160	5	)	)	PUNCT
iajs-2723	160	6	for	for	ADP
iajs-2723	160	7	each	each	DET
iajs-2723	160	8	𝑢	𝑢	NOUN
iajs-2723	160	9	,	,	PUNCT
iajs-2723	160	10	𝑣	𝑣	PROPN
iajs-2723	160	11	∈	∈	PROPN
iajs-2723	160	12	γ	γ	PROPN
iajs-2723	160	13	,	,	PUNCT
iajs-2723	160	14	then	then	ADV
iajs-2723	160	15	𝛿2(𝑢	𝛿2(𝑢	X
iajs-2723	160	16	)	)	PUNCT
iajs-2723	160	17	=	=	SYM
iajs-2723	160	18	0	0	NUM
iajs-2723	160	19	(	(	PUNCT
iajs-2723	160	20	resp	resp	NOUN
iajs-2723	160	21	.	.	PUNCT
iajs-2723	161	1	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	161	2	)	)	PUNCT
iajs-2723	161	3	=	=	SYM
iajs-2723	161	4	0	0	NUM
iajs-2723	161	5	)	)	PUNCT
iajs-2723	161	6	.	.	PUNCT
iajs-2723	162	1	proof	proof	NOUN
iajs-2723	162	2	:	:	PUNCT
iajs-2723	162	3	let	let	VERB
iajs-2723	162	4	𝑢	𝑢	NOUN
iajs-2723	162	5	,	,	PUNCT
iajs-2723	162	6	𝑣	𝑣	PRON
iajs-2723	162	7	∈	∈	PROPN
iajs-2723	162	8	γ	γ	X
iajs-2723	162	9	.	.	PROPN
iajs-2723	163	1	if	if	SCONJ
iajs-2723	163	2	𝛿2(𝑢)𝛿1(𝑣	𝛿2(𝑢)𝛿1(𝑣	NOUN
iajs-2723	163	3	)	)	PUNCT
iajs-2723	164	1	=	=	SYM
iajs-2723	164	2	0	0	NUM
iajs-2723	164	3	for	for	ADP
iajs-2723	164	4	each	each	DET
iajs-2723	164	5	𝑢	𝑢	NOUN
iajs-2723	164	6	,	,	PUNCT
iajs-2723	164	7	𝑣	𝑣	PROPN
iajs-2723	164	8	∈	∈	PROPN
iajs-2723	164	9	γ	γ	X
iajs-2723	164	10	(	(	PUNCT
iajs-2723	164	11	35	35	NUM
iajs-2723	164	12	)	)	PUNCT
iajs-2723	164	13	replacing	replace	VERB
iajs-2723	164	14	𝑢	𝑢	PRON
iajs-2723	164	15	by	by	ADP
iajs-2723	164	16	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	164	17	in	in	ADP
iajs-2723	164	18	(	(	PUNCT
iajs-2723	164	19	35	35	NUM
iajs-2723	164	20	)	)	PUNCT
iajs-2723	164	21	,	,	PUNCT
iajs-2723	164	22	we	we	PRON
iajs-2723	164	23	get	get	VERB
iajs-2723	164	24	:	:	PUNCT
iajs-2723	164	25	𝛿2(𝑢𝑣𝑢)𝛿1(𝑣	𝛿2(𝑢𝑣𝑢)𝛿1(𝑣	NOUN
iajs-2723	164	26	)	)	PUNCT
iajs-2723	165	1	=	=	SYM
iajs-2723	165	2	0	0	NUM
iajs-2723	166	1	for	for	ADP
iajs-2723	166	2	each	each	DET
iajs-2723	166	3	𝑢	𝑢	NOUN
iajs-2723	166	4	,	,	PUNCT
iajs-2723	166	5	𝑣	𝑣	PROPN
iajs-2723	166	6	∈	∈	PROPN
iajs-2723	166	7	γ	γ	X
iajs-2723	166	8	(	(	PUNCT
iajs-2723	166	9	36	36	NUM
iajs-2723	166	10	)	)	PUNCT
iajs-2723	166	11	that	that	PRON
iajs-2723	166	12	is	be	AUX
iajs-2723	166	13	:	:	PUNCT
iajs-2723	166	14	(	(	PUNCT
iajs-2723	166	15	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	166	16	)	)	PUNCT
iajs-2723	166	17	+	+	PUNCT
iajs-2723	166	18	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	166	19	)	)	PUNCT
iajs-2723	167	1	+	+	PUNCT
iajs-2723	167	2	𝜃2(𝑢𝑣)𝛿2(𝑢))𝛿1(𝑣	𝜃2(𝑢𝑣)𝛿2(𝑢))𝛿1(𝑣	NUM
iajs-2723	167	3	)	)	PUNCT
iajs-2723	168	1	=	=	SYM
iajs-2723	168	2	0	0	NUM
iajs-2723	169	1	for	for	ADP
iajs-2723	169	2	each	each	DET
iajs-2723	169	3	𝑢	𝑢	NOUN
iajs-2723	169	4	,	,	PUNCT
iajs-2723	169	5	𝑣	𝑣	PROPN
iajs-2723	169	6	∈	∈	PROPN
iajs-2723	169	7	γ	γ	X
iajs-2723	169	8	(	(	PUNCT
iajs-2723	169	9	37	37	NUM
iajs-2723	169	10	)	)	PUNCT
iajs-2723	169	11	that	that	PRON
iajs-2723	169	12	is	be	AUX
iajs-2723	169	13	:	:	PUNCT
iajs-2723	169	14	𝛿2(𝑢)𝜃1(𝑣𝑢)𝛿1(𝑣	𝛿2(𝑢)𝜃1(𝑣𝑢)𝛿1(𝑣	ADJ
iajs-2723	169	15	)	)	PUNCT
iajs-2723	169	16	+	+	SYM
iajs-2723	169	17	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢)𝛿1(𝑣	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢)𝛿1(𝑣	NOUN
iajs-2723	169	18	)	)	PUNCT
iajs-2723	169	19	+	+	NUM
iajs-2723	169	20	𝜃2(𝑢𝑣)𝛿2(𝑢)𝛿1(𝑣	𝜃2(𝑢𝑣)𝛿2(𝑢)𝛿1(𝑣	NOUN
iajs-2723	169	21	)	)	PUNCT
iajs-2723	169	22	=	=	SYM
iajs-2723	169	23	0	0	NUM
iajs-2723	169	24	for	for	ADP
iajs-2723	169	25	each	each	DET
iajs-2723	169	26	𝑢	𝑢	NOUN
iajs-2723	169	27	,	,	PUNCT
iajs-2723	169	28	𝑣	𝑣	PROPN
iajs-2723	169	29	∈	∈	PROPN
iajs-2723	169	30	γ	γ	X
iajs-2723	169	31	(	(	PUNCT
iajs-2723	169	32	38	38	NUM
iajs-2723	169	33	)	)	PUNCT
iajs-2723	169	34	by	by	ADP
iajs-2723	169	35	using	use	VERB
iajs-2723	169	36	(	(	PUNCT
iajs-2723	169	37	35	35	NUM
iajs-2723	169	38	)	)	PUNCT
iajs-2723	169	39	,	,	PUNCT
iajs-2723	169	40	we	we	PRON
iajs-2723	169	41	have	have	AUX
iajs-2723	169	42	:	:	PUNCT
iajs-2723	169	43	𝛿2(𝑢)𝜃1(𝑣𝑢)𝛿1(𝑣	𝛿2(𝑢)𝜃1(𝑣𝑢)𝛿1(𝑣	ADJ
iajs-2723	169	44	)	)	PUNCT
iajs-2723	169	45	+	+	SYM
iajs-2723	169	46	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢)𝛿1(𝑣	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢)𝛿1(𝑣	NOUN
iajs-2723	169	47	)	)	PUNCT
iajs-2723	169	48	=	=	SYM
iajs-2723	169	49	0	0	PUNCT
iajs-2723	169	50	for	for	ADP
iajs-2723	169	51	each	each	DET
iajs-2723	169	52	𝑢	𝑢	NOUN
iajs-2723	169	53	,	,	PUNCT
iajs-2723	169	54	𝑣	𝑣	PROPN
iajs-2723	169	55	∈	∈	PROPN
iajs-2723	169	56	γ	γ	X
iajs-2723	169	57	(	(	PUNCT
iajs-2723	169	58	39	39	NUM
iajs-2723	169	59	)	)	PUNCT
iajs-2723	169	60	that	that	PRON
iajs-2723	169	61	is	be	AUX
iajs-2723	169	62	:	:	PUNCT
iajs-2723	169	63	(	(	PUNCT
iajs-2723	169	64	𝛿2(𝑢)𝜃1(𝑣	𝛿2(𝑢)𝜃1(𝑣	NOUN
iajs-2723	169	65	)	)	PUNCT
iajs-2723	169	66	+	+	PUNCT
iajs-2723	169	67	𝜃2(𝑢)𝛿1(𝑣))𝜃1(𝑢)𝛿1(𝑣	𝜃2(𝑢)𝛿1(𝑣))𝜃1(𝑢)𝛿1(𝑣	NOUN
iajs-2723	169	68	)	)	PUNCT
iajs-2723	169	69	=	=	SYM
iajs-2723	169	70	0	0	NUM
iajs-2723	169	71	for	for	ADP
iajs-2723	169	72	each	each	DET
iajs-2723	169	73	𝑢	𝑢	NOUN
iajs-2723	169	74	,	,	PUNCT
iajs-2723	169	75	𝑣	𝑣	PROPN
iajs-2723	169	76	∈	∈	PROPN
iajs-2723	169	77	γ	γ	X
iajs-2723	169	78	(	(	PUNCT
iajs-2723	169	79	40	40	NUM
iajs-2723	169	80	)	)	PUNCT
iajs-2723	169	81	replacing	replace	VERB
iajs-2723	169	82	𝛿2(𝑢)𝜃1(𝑣	𝛿2(𝑢)𝜃1(𝑣	NOUN
iajs-2723	169	83	)	)	PUNCT
iajs-2723	169	84	+	+	NUM
iajs-2723	169	85	𝜃2(𝑢)𝛿1(𝑣	𝜃2(𝑢)𝛿1(𝑣	X
iajs-2723	169	86	)	)	PUNCT
iajs-2723	169	87	by	by	ADP
iajs-2723	169	88	𝜃1(𝑣	𝜃1(𝑣	NOUN
iajs-2723	169	89	)	)	PUNCT
iajs-2723	169	90	in	in	ADP
iajs-2723	169	91	(	(	PUNCT
iajs-2723	169	92	40	40	NUM
iajs-2723	169	93	)	)	PUNCT
iajs-2723	169	94	,	,	PUNCT
iajs-2723	169	95	we	we	PRON
iajs-2723	169	96	get	get	VERB
iajs-2723	169	97	:	:	PUNCT
iajs-2723	169	98	𝜃1(𝑣𝑢)𝛿1(𝑣	𝜃1(𝑣𝑢)𝛿1(𝑣	ADJ
iajs-2723	169	99	)	)	PUNCT
iajs-2723	170	1	=	=	SYM
iajs-2723	170	2	0	0	NUM
iajs-2723	170	3	for	for	ADP
iajs-2723	170	4	each	each	DET
iajs-2723	170	5	𝑢	𝑢	NOUN
iajs-2723	170	6	,	,	PUNCT
iajs-2723	170	7	𝑣	𝑣	PROPN
iajs-2723	170	8	∈	∈	PROPN
iajs-2723	170	9	γ	γ	X
iajs-2723	170	10	(	(	PUNCT
iajs-2723	170	11	41	41	NUM
iajs-2723	170	12	)	)	PUNCT
iajs-2723	170	13	left	leave	VERB
iajs-2723	170	14	multiplying	multiplying	NOUN
iajs-2723	170	15	(	(	PUNCT
iajs-2723	170	16	41	41	NUM
iajs-2723	170	17	)	)	PUNCT
iajs-2723	170	18	by	by	ADP
iajs-2723	170	19	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	170	20	)	)	PUNCT
iajs-2723	170	21	we	we	PRON
iajs-2723	170	22	have	have	VERB
iajs-2723	170	23	:	:	PUNCT
iajs-2723	170	24	𝛿2(𝑢	𝛿2(𝑢	NOUN
iajs-2723	170	25	)	)	PUNCT
iajs-2723	170	26	𝜃1(𝑣𝑢)𝛿1(𝑣	𝜃1(𝑣𝑢)𝛿1(𝑣	NUM
iajs-2723	170	27	)	)	PUNCT
iajs-2723	171	1	=	=	SYM
iajs-2723	171	2	0	0	NUM
iajs-2723	171	3	for	for	ADP
iajs-2723	171	4	each	each	DET
iajs-2723	171	5	𝑢	𝑢	NOUN
iajs-2723	171	6	,	,	PUNCT
iajs-2723	171	7	𝑣	𝑣	PROPN
iajs-2723	171	8	∈	∈	PROPN
iajs-2723	171	9	γ	γ	X
iajs-2723	171	10	(	(	PUNCT
iajs-2723	171	11	42	42	NUM
iajs-2723	171	12	)	)	PUNCT
iajs-2723	171	13	since	since	SCONJ
iajs-2723	171	14	γ	γ	X
iajs-2723	171	15	is	be	AUX
iajs-2723	171	16	a	a	DET
iajs-2723	171	17	prime	prime	ADJ
iajs-2723	171	18	ring	ring	NOUN
iajs-2723	171	19	,	,	PUNCT
iajs-2723	171	20	(	(	PUNCT
iajs-2723	171	21	42	42	NUM
iajs-2723	171	22	)	)	PUNCT
iajs-2723	171	23	gives	give	VERB
iajs-2723	171	24	:	:	PUNCT
iajs-2723	171	25	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	171	26	)	)	PUNCT
iajs-2723	171	27	=	=	SYM
iajs-2723	171	28	0	0	NUM
iajs-2723	171	29	for	for	ADP
iajs-2723	171	30	each	each	DET
iajs-2723	171	31	𝑢	𝑢	PROPN
iajs-2723	171	32	∈	∈	PROPN
iajs-2723	171	33	γ	γ	X
iajs-2723	171	34	.	.	PUNCT
iajs-2723	172	1	now	now	ADV
iajs-2723	172	2	,	,	PUNCT
iajs-2723	172	3	if	if	SCONJ
iajs-2723	172	4	𝛿1(𝑢)𝛿2(𝑣	𝛿1(𝑢)𝛿2(𝑣	NOUN
iajs-2723	172	5	)	)	PUNCT
iajs-2723	172	6	=	=	SYM
iajs-2723	172	7	0	0	NUM
iajs-2723	172	8	for	for	ADP
iajs-2723	172	9	each	each	DET
iajs-2723	172	10	𝑢	𝑢	NOUN
iajs-2723	172	11	,	,	PUNCT
iajs-2723	172	12	𝑣	𝑣	PROPN
iajs-2723	172	13	∈	∈	PROPN
iajs-2723	172	14	γ	γ	X
iajs-2723	172	15	(	(	PUNCT
iajs-2723	172	16	43	43	NUM
iajs-2723	172	17	)	)	PUNCT
iajs-2723	172	18	replacing	replace	VERB
iajs-2723	172	19	𝑢	𝑢	PRON
iajs-2723	172	20	by	by	ADP
iajs-2723	172	21	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	172	22	in	in	ADP
iajs-2723	172	23	(	(	PUNCT
iajs-2723	172	24	43	43	NUM
iajs-2723	172	25	)	)	PUNCT
iajs-2723	172	26	,	,	PUNCT
iajs-2723	172	27	we	we	PRON
iajs-2723	172	28	get	get	VERB
iajs-2723	172	29	:	:	PUNCT
iajs-2723	172	30	𝛿1(𝑢𝑣𝑢)𝛿2(𝑣	𝛿1(𝑢𝑣𝑢)𝛿2(𝑣	NOUN
iajs-2723	172	31	)	)	PUNCT
iajs-2723	172	32	=	=	SYM
iajs-2723	172	33	0	0	NUM
iajs-2723	173	1	for	for	ADP
iajs-2723	173	2	each	each	DET
iajs-2723	173	3	𝑢	𝑢	NOUN
iajs-2723	173	4	,	,	PUNCT
iajs-2723	173	5	𝑣	𝑣	PROPN
iajs-2723	173	6	∈	∈	PROPN
iajs-2723	173	7	γ	γ	X
iajs-2723	173	8	(	(	PUNCT
iajs-2723	173	9	44	44	NUM
iajs-2723	173	10	)	)	PUNCT
iajs-2723	173	11	that	that	PRON
iajs-2723	173	12	is	be	AUX
iajs-2723	173	13	:	:	PUNCT
iajs-2723	173	14	(	(	PUNCT
iajs-2723	173	15	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	173	16	)	)	PUNCT
iajs-2723	173	17	+	+	PUNCT
iajs-2723	173	18	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	173	19	)	)	PUNCT
iajs-2723	173	20	+	+	NUM
iajs-2723	173	21	𝜃2(𝑢𝑣)𝛿1(𝑢))𝛿2(𝑣	𝜃2(𝑢𝑣)𝛿1(𝑢))𝛿2(𝑣	NOUN
iajs-2723	173	22	)	)	PUNCT
iajs-2723	173	23	=	=	SYM
iajs-2723	173	24	0	0	NUM
iajs-2723	173	25	for	for	ADP
iajs-2723	173	26	each	each	DET
iajs-2723	173	27	𝑢	𝑢	NOUN
iajs-2723	173	28	,	,	PUNCT
iajs-2723	173	29	𝑣	𝑣	PROPN
iajs-2723	173	30	∈	∈	PROPN
iajs-2723	173	31	γ	γ	X
iajs-2723	173	32	(	(	PUNCT
iajs-2723	173	33	45	45	NUM
iajs-2723	173	34	)	)	PUNCT
iajs-2723	173	35	that	that	PRON
iajs-2723	173	36	is	be	AUX
iajs-2723	173	37	:	:	PUNCT
iajs-2723	173	38	𝛿1(𝑢)𝜃1(𝑣𝑢)𝛿2(𝑣	𝛿1(𝑢)𝜃1(𝑣𝑢)𝛿2(𝑣	NOUN
iajs-2723	173	39	)	)	PUNCT
iajs-2723	173	40	+	+	SYM
iajs-2723	173	41	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝛿2(𝑣	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝛿2(𝑣	NOUN
iajs-2723	173	42	)	)	PUNCT
iajs-2723	173	43	+	+	SYM
iajs-2723	173	44	𝜃2(𝑢𝑣)𝛿1(𝑢)𝛿2(𝑣	𝜃2(𝑢𝑣)𝛿1(𝑢)𝛿2(𝑣	NOUN
iajs-2723	173	45	)	)	PUNCT
iajs-2723	173	46	=	=	SYM
iajs-2723	173	47	0	0	NUM
iajs-2723	173	48	for	for	ADP
iajs-2723	173	49	each	each	DET
iajs-2723	173	50	𝑢	𝑢	NOUN
iajs-2723	173	51	,	,	PUNCT
iajs-2723	173	52	𝑣	𝑣	PROPN
iajs-2723	173	53	∈	∈	PROPN
iajs-2723	173	54	γ	γ	X
iajs-2723	173	55	(	(	PUNCT
iajs-2723	173	56	46	46	NUM
iajs-2723	173	57	)	)	PUNCT
iajs-2723	173	58	by	by	ADP
iajs-2723	173	59	using	use	VERB
iajs-2723	173	60	(	(	PUNCT
iajs-2723	173	61	43	43	NUM
iajs-2723	173	62	)	)	PUNCT
iajs-2723	173	63	,	,	PUNCT
iajs-2723	173	64	we	we	PRON
iajs-2723	173	65	have	have	VERB
iajs-2723	173	66	:	:	PUNCT
iajs-2723	173	67	𝛿1(𝑢)𝜃1(𝑣𝑢)𝛿2(𝑣	𝛿1(𝑢)𝜃1(𝑣𝑢)𝛿2(𝑣	NOUN
iajs-2723	173	68	)	)	PUNCT
iajs-2723	173	69	+	+	SYM
iajs-2723	173	70	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝛿2(𝑣	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝛿2(𝑣	X
iajs-2723	173	71	)	)	PUNCT
iajs-2723	173	72	=	=	SYM
iajs-2723	173	73	0	0	NUM
iajs-2723	173	74	for	for	ADP
iajs-2723	173	75	each	each	DET
iajs-2723	173	76	𝑢	𝑢	NOUN
iajs-2723	173	77	,	,	PUNCT
iajs-2723	173	78	𝑣	𝑣	PROPN
iajs-2723	173	79	∈	∈	PROPN
iajs-2723	173	80	γ	γ	X
iajs-2723	173	81	(	(	PUNCT
iajs-2723	173	82	47	47	NUM
iajs-2723	173	83	)	)	PUNCT
iajs-2723	173	84	that	that	PRON
iajs-2723	173	85	is	be	AUX
iajs-2723	173	86	:	:	PUNCT
iajs-2723	173	87	(	(	PUNCT
iajs-2723	173	88	𝛿1(𝑢)𝜃1(𝑣	𝛿1(𝑢)𝜃1(𝑣	NOUN
iajs-2723	173	89	)	)	PUNCT
iajs-2723	173	90	+	+	CCONJ
iajs-2723	173	91	𝜃2(𝑢)𝛿2(𝑣))𝜃1(𝑢)𝛿2(𝑣	𝜃2(𝑢)𝛿2(𝑣))𝜃1(𝑢)𝛿2(𝑣	NOUN
iajs-2723	173	92	)	)	PUNCT
iajs-2723	173	93	=	=	SYM
iajs-2723	173	94	0	0	NUM
iajs-2723	173	95	for	for	ADP
iajs-2723	173	96	each	each	DET
iajs-2723	173	97	𝑢	𝑢	NOUN
iajs-2723	173	98	,	,	PUNCT
iajs-2723	173	99	𝑣	𝑣	PROPN
iajs-2723	173	100	∈	∈	PROPN
iajs-2723	173	101	γ	γ	X
iajs-2723	173	102	(	(	PUNCT
iajs-2723	173	103	48	48	NUM
iajs-2723	173	104	)	)	PUNCT
iajs-2723	173	105	replacing	replace	VERB
iajs-2723	173	106	𝛿1(𝑢)𝜃1(𝑣	𝛿1(𝑢)𝜃1(𝑣	NOUN
iajs-2723	173	107	)	)	PUNCT
iajs-2723	173	108	+	+	CCONJ
iajs-2723	173	109	𝜃2(𝑢)𝛿2(𝑣	𝜃2(𝑢)𝛿2(𝑣	VERB
iajs-2723	173	110	)	)	PUNCT
iajs-2723	173	111	by	by	ADP
iajs-2723	173	112	𝜃1(𝑣	𝜃1(𝑣	NOUN
iajs-2723	173	113	)	)	PUNCT
iajs-2723	173	114	in	in	ADP
iajs-2723	173	115	(	(	PUNCT
iajs-2723	173	116	48	48	NUM
iajs-2723	173	117	)	)	PUNCT
iajs-2723	173	118	,	,	PUNCT
iajs-2723	173	119	we	we	PRON
iajs-2723	173	120	get	get	VERB
iajs-2723	173	121	𝜃1(𝑣𝑢)𝛿2(𝑣	𝜃1(𝑣𝑢)𝛿2(𝑣	NUM
iajs-2723	173	122	)	)	PUNCT
iajs-2723	173	123	=	=	SYM
iajs-2723	173	124	0	0	NUM
iajs-2723	173	125	for	for	ADP
iajs-2723	173	126	each	each	DET
iajs-2723	173	127	𝑢	𝑢	NOUN
iajs-2723	173	128	,	,	PUNCT
iajs-2723	173	129	𝑣	𝑣	PROPN
iajs-2723	173	130	∈	∈	PROPN
iajs-2723	173	131	γ	γ	X
iajs-2723	173	132	(	(	PUNCT
iajs-2723	173	133	49	49	NUM
iajs-2723	173	134	)	)	PUNCT
iajs-2723	173	135	left	leave	VERB
iajs-2723	173	136	multiplying	multiplying	NOUN
iajs-2723	173	137	of	of	ADP
iajs-2723	173	138	(	(	PUNCT
iajs-2723	173	139	49	49	NUM
iajs-2723	173	140	)	)	PUNCT
iajs-2723	173	141	by	by	ADP
iajs-2723	173	142	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	173	143	)	)	PUNCT
iajs-2723	173	144	,	,	PUNCT
iajs-2723	173	145	we	we	PRON
iajs-2723	173	146	have	have	AUX
iajs-2723	173	147	:	:	PUNCT
iajs-2723	173	148	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	173	149	)	)	PUNCT
iajs-2723	173	150	𝜃1(𝑣𝑢)𝛿2(𝑣	𝜃1(𝑣𝑢)𝛿2(𝑣	NUM
iajs-2723	173	151	)	)	PUNCT
iajs-2723	173	152	=	=	SYM
iajs-2723	173	153	0	0	NUM
iajs-2723	173	154	for	for	ADP
iajs-2723	173	155	each	each	DET
iajs-2723	173	156	𝑢	𝑢	NOUN
iajs-2723	173	157	,	,	PUNCT
iajs-2723	173	158	𝑣	𝑣	PROPN
iajs-2723	173	159	∈	∈	PROPN
iajs-2723	173	160	γ	γ	X
iajs-2723	173	161	(	(	PUNCT
iajs-2723	173	162	50	50	NUM
iajs-2723	173	163	)	)	PUNCT
iajs-2723	173	164	since	since	SCONJ
iajs-2723	173	165	γ	γ	X
iajs-2723	173	166	is	be	AUX
iajs-2723	173	167	a	a	DET
iajs-2723	173	168	prime	prime	ADJ
iajs-2723	173	169	ring	ring	NOUN
iajs-2723	173	170	,	,	PUNCT
iajs-2723	173	171	(	(	PUNCT
iajs-2723	173	172	50	50	NUM
iajs-2723	173	173	)	)	PUNCT
iajs-2723	173	174	gives	give	VERB
iajs-2723	173	175	:	:	PUNCT
iajs-2723	173	176	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	173	177	)	)	PUNCT
iajs-2723	173	178	=	=	SYM
iajs-2723	173	179	0	0	NUM
iajs-2723	173	180	for	for	SCONJ
iajs-2723	173	181	each	each	DET
iajs-2723	173	182	𝑢	𝑢	PROPN
iajs-2723	173	183	∈	∈	PROPN
iajs-2723	173	184	γ	γ	X
iajs-2723	173	185	.	.	PUNCT
iajs-2723	173	186	∎	∎	PROPN
iajs-2723	173	187	theorem	theorem	VERB
iajs-2723	173	188	3.4	3.4	NUM
iajs-2723	173	189	let	let	VERB
iajs-2723	173	190	γ	γ	NOUN
iajs-2723	173	191	be	be	AUX
iajs-2723	173	192	a	a	DET
iajs-2723	173	193	prime	prime	ADJ
iajs-2723	173	194	ring	ring	NOUN
iajs-2723	173	195	.	.	PUNCT
iajs-2723	174	1	let	let	VERB
iajs-2723	174	2	𝜃1	𝜃1	VERB
iajs-2723	174	3	and	and	CCONJ
iajs-2723	174	4	𝜃2	𝜃2	NOUN
iajs-2723	174	5	be	be	VERB
iajs-2723	174	6	two	two	NUM
iajs-2723	174	7	automorphisms	automorphism	NOUN
iajs-2723	174	8	of	of	ADP
iajs-2723	174	9	γ	γ	PROPN
iajs-2723	174	10	.	.	PROPN
iajs-2723	175	1	if	if	SCONJ
iajs-2723	175	2	γ	γ	PROPN
iajs-2723	175	3	is	be	AUX
iajs-2723	175	4	a	a	DET
iajs-2723	175	5	(	(	PUNCT
iajs-2723	175	6	𝛿1	𝛿1	NOUN
iajs-2723	175	7	,	,	PUNCT
iajs-2723	175	8	𝛿2	𝛿2	NOUN
iajs-2723	175	9	)	)	PUNCT
iajs-2723	175	10	derivation	derivation	NOUN
iajs-2723	175	11	pair	pair	VERB
iajs-2723	175	12	such	such	ADJ
iajs-2723	175	13	that	that	SCONJ
iajs-2723	175	14	𝑐𝛿1(𝑢	𝑐𝛿1(𝑢	PROPN
iajs-2723	175	15	)	)	PUNCT
iajs-2723	176	1	=	=	SYM
iajs-2723	176	2	0	0	NUM
iajs-2723	176	3	or	or	CCONJ
iajs-2723	176	4	𝛿1(𝑢)𝑐	𝛿1(𝑢)𝑐	PROPN
iajs-2723	176	5	=	=	SYM
iajs-2723	176	6	0	0	NUM
iajs-2723	176	7	(	(	PUNCT
iajs-2723	176	8	resp	resp	NOUN
iajs-2723	176	9	.	.	PUNCT
iajs-2723	177	1	c𝛿2(𝑢	c𝛿2(𝑢	NOUN
iajs-2723	177	2	)	)	PUNCT
iajs-2723	177	3	=	=	SYM
iajs-2723	177	4	0	0	NUM
iajs-2723	177	5	or	or	CCONJ
iajs-2723	177	6	𝛿2(𝑢)𝑐	𝛿2(𝑢)𝑐	NOUN
iajs-2723	177	7	=	=	SYM
iajs-2723	177	8	0	0	NUM
iajs-2723	177	9	)	)	PUNCT
iajs-2723	177	10	for	for	ADP
iajs-2723	177	11	each	each	DET
iajs-2723	177	12	𝑢	𝑢	NOUN
iajs-2723	177	13	,	,	PUNCT
iajs-2723	177	14	𝑐	𝑐	PROPN
iajs-2723	177	15	∈	∈	PROPN
iajs-2723	177	16	γ	γ	X
iajs-2723	177	17	,	,	PUNCT
iajs-2723	177	18	then	then	ADV
iajs-2723	177	19	either	either	CCONJ
iajs-2723	177	20	𝑐	𝑐	PROPN
iajs-2723	177	21	=	=	SYM
iajs-2723	177	22	0	0	NUM
iajs-2723	177	23	or	or	CCONJ
iajs-2723	177	24	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	177	25	)	)	PUNCT
iajs-2723	178	1	=	=	SYM
iajs-2723	178	2	0	0	PUNCT
iajs-2723	178	3	(	(	PUNCT
iajs-2723	178	4	resp.𝑐	resp.𝑐	PROPN
iajs-2723	178	5	=	=	SYM
iajs-2723	178	6	0	0	NUM
iajs-2723	178	7	or	or	CCONJ
iajs-2723	178	8	𝛿2(𝑢	𝛿2(𝑢	ADJ
iajs-2723	178	9	)	)	PUNCT
iajs-2723	178	10	=	=	SYM
iajs-2723	178	11	0	0	NUM
iajs-2723	178	12	)	)	PUNCT
iajs-2723	178	13	.	.	PUNCT
iajs-2723	179	1	ibn	ibn	PROPN
iajs-2723	179	2	al	al	PROPN
iajs-2723	179	3	-	-	PUNCT
iajs-2723	179	4	haitham	haitham	PROPN
iajs-2723	179	5	jour	jour	X
iajs-2723	179	6	.	.	PROPN
iajs-2723	179	7	for	for	ADP
iajs-2723	179	8	pure	pure	ADJ
iajs-2723	179	9	&	&	CCONJ
iajs-2723	179	10	appl	appl	PROPN
iajs-2723	179	11	.	.	PUNCT
iajs-2723	180	1	sci	sci	PROPN
iajs-2723	180	2	.	.	PROPN
iajs-2723	181	1	53	53	NUM
iajs-2723	181	2	(	(	PUNCT
iajs-2723	181	3	2)2022	2)2022	VERB
iajs-2723	181	4	114	114	NUM
iajs-2723	181	5	proof	proof	NOUN
iajs-2723	181	6	:	:	PUNCT
iajs-2723	181	7	let	let	VERB
iajs-2723	181	8	𝑢	𝑢	NOUN
iajs-2723	181	9	,	,	PUNCT
iajs-2723	181	10	𝑐	𝑐	NOUN
iajs-2723	181	11	≠	≠	PROPN
iajs-2723	181	12	0	0	NUM
iajs-2723	181	13	∈	∈	PROPN
iajs-2723	181	14	γ	γ	X
iajs-2723	181	15	.	.	PUNCT
iajs-2723	182	1	if	if	SCONJ
iajs-2723	182	2	𝑐𝛿1(𝑢	𝑐𝛿1(𝑢	PROPN
iajs-2723	182	3	)	)	PUNCT
iajs-2723	182	4	=	=	SYM
iajs-2723	182	5	0	0	NUM
iajs-2723	182	6	for	for	ADP
iajs-2723	182	7	each	each	DET
iajs-2723	182	8	𝑐	𝑐	NOUN
iajs-2723	182	9	,	,	PUNCT
iajs-2723	182	10	𝑢	𝑢	PROPN
iajs-2723	182	11	∈	∈	PROPN
iajs-2723	182	12	γ	γ	X
iajs-2723	182	13	(	(	PUNCT
iajs-2723	182	14	51	51	NUM
iajs-2723	182	15	)	)	PUNCT
iajs-2723	182	16	replacing	replace	VERB
iajs-2723	182	17	𝑢	𝑢	PRON
iajs-2723	182	18	by	by	ADP
iajs-2723	182	19	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	182	20	in	in	ADP
iajs-2723	182	21	(	(	PUNCT
iajs-2723	182	22	51	51	NUM
iajs-2723	182	23	)	)	PUNCT
iajs-2723	182	24	,	,	PUNCT
iajs-2723	182	25	we	we	PRON
iajs-2723	182	26	get	get	VERB
iajs-2723	182	27	:	:	PUNCT
iajs-2723	182	28	𝑐𝛿1(𝑢𝑣𝑢	𝑐𝛿1(𝑢𝑣𝑢	NUM
iajs-2723	182	29	)	)	PUNCT
iajs-2723	183	1	=	=	SYM
iajs-2723	183	2	0	0	NUM
iajs-2723	183	3	for	for	ADP
iajs-2723	183	4	each	each	DET
iajs-2723	183	5	𝑢	𝑢	NOUN
iajs-2723	183	6	,	,	PUNCT
iajs-2723	183	7	𝑣	𝑣	PROPN
iajs-2723	183	8	,	,	PUNCT
iajs-2723	183	9	𝑐	𝑐	PROPN
iajs-2723	183	10	∈	∈	PROPN
iajs-2723	183	11	γ	γ	X
iajs-2723	183	12	(	(	PUNCT
iajs-2723	183	13	52	52	NUM
iajs-2723	183	14	)	)	PUNCT
iajs-2723	183	15	that	that	PRON
iajs-2723	183	16	is	be	AUX
iajs-2723	183	17	𝑐(𝛿1(𝑢)𝜃1(𝑣𝑢	𝑐(𝛿1(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	183	18	)	)	PUNCT
iajs-2723	183	19	+	+	X
iajs-2723	183	20	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	183	21	)	)	PUNCT
iajs-2723	183	22	+	+	ADJ
iajs-2723	183	23	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	183	24	)	)	PUNCT
iajs-2723	183	25	)	)	PUNCT
iajs-2723	184	1	=	=	SYM
iajs-2723	184	2	0	0	PUNCT
iajs-2723	184	3	for	for	ADP
iajs-2723	184	4	each	each	DET
iajs-2723	184	5	𝑢	𝑢	NOUN
iajs-2723	184	6	,	,	PUNCT
iajs-2723	184	7	𝑣	𝑣	PROPN
iajs-2723	184	8	,	,	PUNCT
iajs-2723	184	9	𝑐	𝑐	PROPN
iajs-2723	184	10	∈	∈	PROPN
iajs-2723	184	11	γ	γ	X
iajs-2723	184	12	(	(	PUNCT
iajs-2723	184	13	53	53	NUM
iajs-2723	184	14	)	)	PUNCT
iajs-2723	184	15	that	that	PRON
iajs-2723	184	16	is	be	AUX
iajs-2723	184	17	𝑐𝛿1(𝑢)𝜃1(𝑣𝑢	𝑐𝛿1(𝑢)𝜃1(𝑣𝑢	VERB
iajs-2723	184	18	)	)	PUNCT
iajs-2723	184	19	+	+	CCONJ
iajs-2723	184	20	𝑐𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	NUM
iajs-2723	184	21	)	)	PUNCT
iajs-2723	185	1	+	+	CCONJ
iajs-2723	185	2	𝑐𝜃2(𝑢𝑣)𝛿1(𝑢	𝑐𝜃2(𝑢𝑣)𝛿1(𝑢	X
iajs-2723	185	3	)	)	PUNCT
iajs-2723	185	4	=	=	SYM
iajs-2723	185	5	0	0	NUM
iajs-2723	185	6	for	for	ADP
iajs-2723	185	7	each	each	DET
iajs-2723	185	8	𝑢	𝑢	NOUN
iajs-2723	185	9	,	,	PUNCT
iajs-2723	185	10	𝑣	𝑣	PROPN
iajs-2723	185	11	,	,	PUNCT
iajs-2723	185	12	𝑐	𝑐	PROPN
iajs-2723	185	13	∈	∈	PROPN
iajs-2723	185	14	γ	γ	X
iajs-2723	185	15	(	(	PUNCT
iajs-2723	185	16	54	54	NUM
iajs-2723	185	17	)	)	PUNCT
iajs-2723	185	18	by	by	ADP
iajs-2723	185	19	using	use	VERB
iajs-2723	185	20	(	(	PUNCT
iajs-2723	185	21	51	51	NUM
iajs-2723	185	22	)	)	PUNCT
iajs-2723	185	23	,	,	PUNCT
iajs-2723	185	24	we	we	PRON
iajs-2723	185	25	have	have	AUX
iajs-2723	185	26	:	:	PUNCT
iajs-2723	185	27	𝑐𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	VERB
iajs-2723	185	28	)	)	PUNCT
iajs-2723	186	1	+	+	CCONJ
iajs-2723	186	2	𝑐𝜃2(𝑢𝑣)𝛿1(𝑢	𝑐𝜃2(𝑢𝑣)𝛿1(𝑢	X
iajs-2723	186	3	)	)	PUNCT
iajs-2723	186	4	=	=	SYM
iajs-2723	186	5	0	0	NUM
iajs-2723	186	6	for	for	ADP
iajs-2723	186	7	each	each	DET
iajs-2723	186	8	𝑢	𝑢	NOUN
iajs-2723	186	9	,	,	PUNCT
iajs-2723	186	10	𝑣	𝑣	PROPN
iajs-2723	186	11	,	,	PUNCT
iajs-2723	186	12	𝑐	𝑐	PROPN
iajs-2723	186	13	∈	∈	PROPN
iajs-2723	186	14	γ	γ	X
iajs-2723	186	15	(	(	PUNCT
iajs-2723	186	16	55	55	NUM
iajs-2723	186	17	)	)	PUNCT
iajs-2723	186	18	that	that	PRON
iajs-2723	186	19	is	be	AUX
iajs-2723	186	20	𝑐𝜃2(𝑢)(𝛿2(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)(𝛿2(𝑣)𝜃1(𝑢	X
iajs-2723	186	21	)	)	PUNCT
iajs-2723	186	22	+	+	CCONJ
iajs-2723	186	23	𝜃2(𝑣)𝛿1(𝑢	𝜃2(𝑣)𝛿1(𝑢	NUM
iajs-2723	186	24	)	)	PUNCT
iajs-2723	186	25	)	)	PUNCT
iajs-2723	187	1	=	=	SYM
iajs-2723	187	2	0	0	PUNCT
iajs-2723	187	3	for	for	ADP
iajs-2723	187	4	each	each	DET
iajs-2723	187	5	𝑢	𝑢	NOUN
iajs-2723	187	6	,	,	PUNCT
iajs-2723	187	7	𝑣	𝑣	PROPN
iajs-2723	187	8	,	,	PUNCT
iajs-2723	187	9	𝑐	𝑐	PROPN
iajs-2723	187	10	∈	∈	PROPN
iajs-2723	187	11	γ	γ	X
iajs-2723	187	12	(	(	PUNCT
iajs-2723	187	13	56	56	NUM
iajs-2723	187	14	)	)	PUNCT
iajs-2723	187	15	replacing	replace	VERB
iajs-2723	187	16	𝛿2(𝑣)𝜃1(𝑢	𝛿2(𝑣)𝜃1(𝑢	PUNCT
iajs-2723	187	17	)	)	PUNCT
iajs-2723	187	18	+	+	CCONJ
iajs-2723	187	19	𝜃2(𝑣)𝛿1(𝑢	𝜃2(𝑣)𝛿1(𝑢	PUNCT
iajs-2723	187	20	)	)	PUNCT
iajs-2723	187	21	by	by	ADP
iajs-2723	187	22	𝛿1(𝑢	𝛿1(𝑢	NOUN
iajs-2723	187	23	)	)	PUNCT
iajs-2723	187	24	in	in	ADP
iajs-2723	187	25	(	(	PUNCT
iajs-2723	187	26	56	56	NUM
iajs-2723	187	27	)	)	PUNCT
iajs-2723	187	28	,	,	PUNCT
iajs-2723	187	29	we	we	PRON
iajs-2723	187	30	get	get	VERB
iajs-2723	187	31	:	:	PUNCT
iajs-2723	187	32	𝑐𝜃2(𝑢)𝛿1(𝑢	𝑐𝜃2(𝑢)𝛿1(𝑢	NUM
iajs-2723	187	33	)	)	PUNCT
iajs-2723	187	34	=	=	SYM
iajs-2723	187	35	0	0	PUNCT
iajs-2723	187	36	for	for	ADP
iajs-2723	187	37	each	each	DET
iajs-2723	187	38	𝑢	𝑢	NOUN
iajs-2723	187	39	,	,	PUNCT
iajs-2723	187	40	𝑐	𝑐	PROPN
iajs-2723	187	41	∈	∈	PROPN
iajs-2723	187	42	γ	γ	X
iajs-2723	187	43	(	(	PUNCT
iajs-2723	187	44	57	57	NUM
iajs-2723	187	45	)	)	PUNCT
iajs-2723	187	46	since	since	SCONJ
iajs-2723	187	47	𝑐	𝑐	PROPN
iajs-2723	187	48	≠	≠	PROPN
iajs-2723	187	49	0	0	NUM
iajs-2723	187	50	and	and	CCONJ
iajs-2723	187	51	γ	γ	PROPN
iajs-2723	187	52	is	be	AUX
iajs-2723	187	53	a	a	DET
iajs-2723	187	54	prime	prime	ADJ
iajs-2723	187	55	rings	ring	NOUN
iajs-2723	187	56	,	,	PUNCT
iajs-2723	187	57	then	then	ADV
iajs-2723	187	58	(	(	PUNCT
iajs-2723	187	59	57	57	NUM
iajs-2723	187	60	)	)	PUNCT
iajs-2723	187	61	gives	give	VERB
iajs-2723	187	62	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	187	63	)	)	PUNCT
iajs-2723	187	64	=	=	SYM
iajs-2723	188	1	0	0	X
iajs-2723	188	2	.	.	PUNCT
iajs-2723	189	1	now	now	ADV
iajs-2723	189	2	,	,	PUNCT
iajs-2723	189	3	let	let	VERB
iajs-2723	189	4	𝑢	𝑢	NOUN
iajs-2723	189	5	,	,	PUNCT
iajs-2723	189	6	𝑐	𝑐	NOUN
iajs-2723	189	7	≠	≠	PROPN
iajs-2723	189	8	0	0	NUM
iajs-2723	189	9	∈	∈	PROPN
iajs-2723	189	10	γ	γ	X
iajs-2723	189	11	.	.	PROPN
iajs-2723	190	1	if	if	SCONJ
iajs-2723	190	2	𝑐𝛿2(𝑢	𝑐𝛿2(𝑢	PROPN
iajs-2723	190	3	)	)	PUNCT
iajs-2723	190	4	=	=	SYM
iajs-2723	190	5	0	0	NUM
iajs-2723	190	6	for	for	ADP
iajs-2723	190	7	each	each	DET
iajs-2723	190	8	𝑐	𝑐	NOUN
iajs-2723	190	9	,	,	PUNCT
iajs-2723	190	10	𝑢	𝑢	PROPN
iajs-2723	190	11	∈	∈	PROPN
iajs-2723	190	12	γ	γ	X
iajs-2723	190	13	(	(	PUNCT
iajs-2723	190	14	58	58	NUM
iajs-2723	190	15	)	)	PUNCT
iajs-2723	190	16	replacing	replace	VERB
iajs-2723	190	17	𝑢	𝑢	PRON
iajs-2723	190	18	by	by	ADP
iajs-2723	190	19	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	190	20	in	in	ADP
iajs-2723	190	21	(	(	PUNCT
iajs-2723	190	22	58	58	NUM
iajs-2723	190	23	)	)	PUNCT
iajs-2723	190	24	,	,	PUNCT
iajs-2723	190	25	we	we	PRON
iajs-2723	190	26	get	get	VERB
iajs-2723	190	27	:	:	PUNCT
iajs-2723	190	28	𝑐𝛿2(𝑢𝑣𝑢	𝑐𝛿2(𝑢𝑣𝑢	X
iajs-2723	190	29	)	)	PUNCT
iajs-2723	191	1	=	=	SYM
iajs-2723	191	2	0	0	NUM
iajs-2723	191	3	for	for	ADP
iajs-2723	191	4	each	each	DET
iajs-2723	191	5	𝑢	𝑢	NOUN
iajs-2723	191	6	,	,	PUNCT
iajs-2723	191	7	𝑣	𝑣	PROPN
iajs-2723	191	8	,	,	PUNCT
iajs-2723	191	9	𝑐	𝑐	PROPN
iajs-2723	191	10	∈	∈	PROPN
iajs-2723	191	11	γ	γ	X
iajs-2723	191	12	(	(	PUNCT
iajs-2723	191	13	59	59	NUM
iajs-2723	191	14	)	)	PUNCT
iajs-2723	191	15	that	that	PRON
iajs-2723	191	16	is	be	AUX
iajs-2723	191	17	𝑐(𝛿2(𝑢)𝜃1(𝑣𝑢	𝑐(𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	191	18	)	)	PUNCT
iajs-2723	191	19	+	+	CCONJ
iajs-2723	191	20	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	191	21	)	)	PUNCT
iajs-2723	191	22	+	+	NUM
iajs-2723	191	23	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	NOUN
iajs-2723	191	24	)	)	PUNCT
iajs-2723	191	25	)	)	PUNCT
iajs-2723	192	1	=	=	SYM
iajs-2723	192	2	0	0	PUNCT
iajs-2723	192	3	for	for	ADP
iajs-2723	192	4	each	each	DET
iajs-2723	192	5	𝑢	𝑢	NOUN
iajs-2723	192	6	,	,	PUNCT
iajs-2723	192	7	𝑣	𝑣	PROPN
iajs-2723	192	8	,	,	PUNCT
iajs-2723	192	9	𝑐	𝑐	PROPN
iajs-2723	192	10	∈	∈	PROPN
iajs-2723	192	11	γ	γ	X
iajs-2723	192	12	(	(	PUNCT
iajs-2723	192	13	60	60	NUM
iajs-2723	192	14	)	)	PUNCT
iajs-2723	192	15	that	that	PRON
iajs-2723	192	16	is	be	AUX
iajs-2723	192	17	𝑐𝛿2(𝑢)𝜃1(𝑣𝑢	𝑐𝛿2(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	192	18	)	)	PUNCT
iajs-2723	193	1	+	+	X
iajs-2723	193	2	𝑐𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	X
iajs-2723	193	3	)	)	PUNCT
iajs-2723	193	4	+	+	NUM
iajs-2723	194	1	𝑐𝜃2(𝑢𝑣)𝛿2(𝑢	𝑐𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	194	2	)	)	PUNCT
iajs-2723	194	3	=	=	SYM
iajs-2723	194	4	0	0	NUM
iajs-2723	194	5	for	for	ADP
iajs-2723	194	6	each	each	DET
iajs-2723	194	7	𝑢	𝑢	NOUN
iajs-2723	194	8	,	,	PUNCT
iajs-2723	194	9	𝑣	𝑣	PROPN
iajs-2723	194	10	,	,	PUNCT
iajs-2723	194	11	𝑐	𝑐	PROPN
iajs-2723	194	12	∈	∈	PROPN
iajs-2723	194	13	γ	γ	X
iajs-2723	194	14	(	(	PUNCT
iajs-2723	194	15	61	61	NUM
iajs-2723	194	16	)	)	PUNCT
iajs-2723	194	17	by	by	ADP
iajs-2723	194	18	using	use	VERB
iajs-2723	194	19	(	(	PUNCT
iajs-2723	194	20	58	58	NUM
iajs-2723	194	21	)	)	PUNCT
iajs-2723	194	22	,	,	PUNCT
iajs-2723	194	23	we	we	PRON
iajs-2723	194	24	have	have	VERB
iajs-2723	194	25	:	:	PUNCT
iajs-2723	194	26	𝑐𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	X
iajs-2723	194	27	)	)	PUNCT
iajs-2723	194	28	+	+	CCONJ
iajs-2723	195	1	𝑐𝜃2(𝑢𝑣)𝛿2(𝑢	𝑐𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	195	2	)	)	PUNCT
iajs-2723	195	3	=	=	SYM
iajs-2723	195	4	0	0	NUM
iajs-2723	195	5	for	for	SCONJ
iajs-2723	195	6	each	each	DET
iajs-2723	195	7	𝑢	𝑢	NOUN
iajs-2723	195	8	,	,	PUNCT
iajs-2723	195	9	𝑣	𝑣	PROPN
iajs-2723	195	10	,	,	PUNCT
iajs-2723	195	11	𝑐	𝑐	PROPN
iajs-2723	195	12	∈	∈	PROPN
iajs-2723	195	13	γ	γ	X
iajs-2723	195	14	(	(	PUNCT
iajs-2723	195	15	62	62	NUM
iajs-2723	195	16	)	)	PUNCT
iajs-2723	195	17	that	that	PRON
iajs-2723	195	18	is	be	AUX
iajs-2723	195	19	𝑐𝜃2(𝑢)(𝛿1(𝑣)𝜃1(𝑢	𝑐𝜃2(𝑢)(𝛿1(𝑣)𝜃1(𝑢	VERB
iajs-2723	195	20	)	)	PUNCT
iajs-2723	195	21	+	+	NUM
iajs-2723	195	22	𝜃2(𝑣)𝛿2(𝑢	𝜃2(𝑣)𝛿2(𝑢	NOUN
iajs-2723	195	23	)	)	PUNCT
iajs-2723	195	24	)	)	PUNCT
iajs-2723	196	1	=	=	SYM
iajs-2723	196	2	0	0	PUNCT
iajs-2723	196	3	for	for	ADP
iajs-2723	196	4	each	each	DET
iajs-2723	196	5	𝑢	𝑢	NOUN
iajs-2723	196	6	,	,	PUNCT
iajs-2723	196	7	𝑣	𝑣	PROPN
iajs-2723	196	8	,	,	PUNCT
iajs-2723	196	9	𝑐	𝑐	PROPN
iajs-2723	196	10	∈	∈	PROPN
iajs-2723	196	11	γ	γ	X
iajs-2723	196	12	(	(	PUNCT
iajs-2723	196	13	63	63	NUM
iajs-2723	196	14	)	)	PUNCT
iajs-2723	196	15	replacing	replace	VERB
iajs-2723	196	16	𝛿1(𝑣)𝜃1(𝑢	𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	196	17	)	)	PUNCT
iajs-2723	196	18	+	+	CCONJ
iajs-2723	196	19	𝜃2(𝑣)𝛿2(𝑢	𝜃2(𝑣)𝛿2(𝑢	PRON
iajs-2723	196	20	)	)	PUNCT
iajs-2723	196	21	by	by	ADP
iajs-2723	196	22	𝛿2(𝑢	𝛿2(𝑢	NOUN
iajs-2723	196	23	)	)	PUNCT
iajs-2723	196	24	in	in	ADP
iajs-2723	196	25	(	(	PUNCT
iajs-2723	196	26	63	63	NUM
iajs-2723	196	27	)	)	PUNCT
iajs-2723	196	28	,	,	PUNCT
iajs-2723	196	29	we	we	PRON
iajs-2723	196	30	get	get	VERB
iajs-2723	196	31	:	:	PUNCT
iajs-2723	196	32	𝑐𝜃2(𝑢)𝛿2(𝑢	𝑐𝜃2(𝑢)𝛿2(𝑢	X
iajs-2723	196	33	)	)	PUNCT
iajs-2723	196	34	=	=	SYM
iajs-2723	196	35	0	0	NUM
iajs-2723	196	36	for	for	ADP
iajs-2723	196	37	each	each	DET
iajs-2723	196	38	𝑢	𝑢	NOUN
iajs-2723	196	39	,	,	PUNCT
iajs-2723	196	40	𝑐	𝑐	PROPN
iajs-2723	196	41	∈	∈	PROPN
iajs-2723	196	42	γ	γ	X
iajs-2723	196	43	(	(	PUNCT
iajs-2723	196	44	64	64	NUM
iajs-2723	196	45	)	)	PUNCT
iajs-2723	196	46	since	since	SCONJ
iajs-2723	196	47	𝑐	𝑐	PROPN
iajs-2723	196	48	≠	≠	PROPN
iajs-2723	196	49	0	0	NUM
iajs-2723	196	50	and	and	CCONJ
iajs-2723	196	51	γ	γ	PROPN
iajs-2723	196	52	is	be	AUX
iajs-2723	196	53	a	a	DET
iajs-2723	196	54	prime	prime	ADJ
iajs-2723	196	55	rings	ring	NOUN
iajs-2723	196	56	,	,	PUNCT
iajs-2723	196	57	then	then	ADV
iajs-2723	196	58	(	(	PUNCT
iajs-2723	196	59	64	64	NUM
iajs-2723	196	60	)	)	PUNCT
iajs-2723	196	61	gives	give	VERB
iajs-2723	196	62	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	196	63	)	)	PUNCT
iajs-2723	196	64	=	=	SYM
iajs-2723	196	65	0	0	X
iajs-2723	196	66	.	.	PUNCT
iajs-2723	197	1	∎	∎	PROPN
iajs-2723	197	2	theorem	theorem	VERB
iajs-2723	197	3	3.5	3.5	NUM
iajs-2723	197	4	let	let	VERB
iajs-2723	197	5	γ	γ	NOUN
iajs-2723	197	6	be	be	AUX
iajs-2723	197	7	a	a	DET
iajs-2723	197	8	prime	prime	ADJ
iajs-2723	197	9	ring	ring	NOUN
iajs-2723	197	10	with	with	ADP
iajs-2723	197	11	char(γ	char(γ	NOUN
iajs-2723	197	12	)	)	PUNCT
iajs-2723	197	13	≠	≠	PROPN
iajs-2723	197	14	2	2	X
iajs-2723	197	15	.	.	PUNCT
iajs-2723	198	1	let	let	VERB
iajs-2723	198	2	𝜃1and	𝜃1and	CCONJ
iajs-2723	198	3	𝜃2	𝜃2	NOUN
iajs-2723	198	4	be	be	VERB
iajs-2723	198	5	two	two	NUM
iajs-2723	198	6	endomorphisms	endomorphism	NOUN
iajs-2723	198	7	of	of	ADP
iajs-2723	198	8	γ	γ	PROPN
iajs-2723	198	9	.	.	PUNCT
iajs-2723	199	1	if	if	SCONJ
iajs-2723	199	2	γ	γ	PROPN
iajs-2723	199	3	is	be	AUX
iajs-2723	199	4	a	a	DET
iajs-2723	199	5	(	(	PUNCT
iajs-2723	199	6	𝛿1	𝛿1	NOUN
iajs-2723	199	7	,	,	PUNCT
iajs-2723	199	8	𝛿2)-derivation	𝛿2)-derivation	NOUN
iajs-2723	199	9	pair	pair	NOUN
iajs-2723	199	10	such	such	ADJ
iajs-2723	199	11	that	that	SCONJ
iajs-2723	199	12	𝑐1𝑞𝑐2𝛿1(𝑢	𝑐1𝑞𝑐2𝛿1(𝑢	VERB
iajs-2723	199	13	)	)	PUNCT
iajs-2723	199	14	+	+	CCONJ
iajs-2723	199	15	𝛿1(𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢)𝑐2𝑞𝑐1	PROPN
iajs-2723	199	16	=	=	SYM
iajs-2723	199	17	0	0	NUM
iajs-2723	199	18	(	(	PUNCT
iajs-2723	199	19	resp	resp	NOUN
iajs-2723	199	20	.	.	PUNCT
iajs-2723	199	21	𝑐1𝑞𝑐2𝛿2(𝑢	𝑐1𝑞𝑐2𝛿2(𝑢	X
iajs-2723	199	22	)	)	PUNCT
iajs-2723	200	1	+	+	NUM
iajs-2723	200	2	𝛿2(𝑢)𝑐2𝑞𝑐1	𝛿2(𝑢)𝑐2𝑞𝑐1	PUNCT
iajs-2723	200	3	=	=	SYM
iajs-2723	200	4	0	0	NUM
iajs-2723	200	5	)	)	PUNCT
iajs-2723	200	6	for	for	ADP
iajs-2723	200	7	each	each	DET
iajs-2723	200	8	𝑢	𝑢	NOUN
iajs-2723	200	9	,	,	PUNCT
iajs-2723	200	10	𝑐1	𝑐1	NOUN
iajs-2723	200	11	,	,	PUNCT
iajs-2723	200	12	𝑐2	𝑐2	NOUN
iajs-2723	200	13	,	,	PUNCT
iajs-2723	200	14	𝑞	𝑞	PROPN
iajs-2723	200	15	∈	∈	PROPN
iajs-2723	200	16	γ	γ	X
iajs-2723	200	17	,	,	PUNCT
iajs-2723	200	18	then	then	ADV
iajs-2723	200	19	𝑐1	𝑐1	NOUN
iajs-2723	200	20	=	=	PUNCT
iajs-2723	200	21	0	0	NUM
iajs-2723	200	22	or	or	CCONJ
iajs-2723	200	23	𝑐2	𝑐2	NOUN
iajs-2723	200	24	=	=	SYM
iajs-2723	200	25	0	0	X
iajs-2723	200	26	.	.	X
iajs-2723	201	1	proof	proof	NOUN
iajs-2723	201	2	:	:	PUNCT
iajs-2723	201	3	from	from	ADP
iajs-2723	201	4	the	the	DET
iajs-2723	201	5	assumption	assumption	NOUN
iajs-2723	201	6	we	we	PRON
iajs-2723	201	7	have	have	VERB
iajs-2723	201	8	:	:	PUNCT
iajs-2723	201	9	𝑐1𝑞𝑐2𝛿1(𝑢	𝑐1𝑞𝑐2𝛿1(𝑢	ADJ
iajs-2723	201	10	)	)	PUNCT
iajs-2723	202	1	+	+	CCONJ
iajs-2723	202	2	𝛿1(𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢)𝑐2𝑞𝑐1	PROPN
iajs-2723	202	3	=	=	NOUN
iajs-2723	202	4	0	0	NUM
iajs-2723	202	5	for	for	ADP
iajs-2723	202	6	each	each	DET
iajs-2723	202	7	𝑢	𝑢	NOUN
iajs-2723	202	8	,	,	PUNCT
iajs-2723	202	9	𝑐1	𝑐1	NOUN
iajs-2723	202	10	,	,	PUNCT
iajs-2723	202	11	𝑐2	𝑐2	NOUN
iajs-2723	202	12	,	,	PUNCT
iajs-2723	202	13	𝑞	𝑞	PROPN
iajs-2723	202	14	∈	∈	PROPN
iajs-2723	202	15	γ	γ	X
iajs-2723	202	16	(	(	PUNCT
iajs-2723	202	17	65	65	NUM
iajs-2723	202	18	)	)	PUNCT
iajs-2723	202	19	replacing	replace	VERB
iajs-2723	202	20	𝑢	𝑢	PRON
iajs-2723	202	21	by	by	ADP
iajs-2723	202	22	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	202	23	in	in	ADP
iajs-2723	202	24	(	(	PUNCT
iajs-2723	202	25	65	65	NUM
iajs-2723	202	26	)	)	PUNCT
iajs-2723	202	27	,	,	PUNCT
iajs-2723	202	28	we	we	PRON
iajs-2723	202	29	have	have	VERB
iajs-2723	202	30	:	:	PUNCT
iajs-2723	202	31	𝑐1𝑞𝑐2𝛿1(𝑢𝑣𝑢	𝑐1𝑞𝑐2𝛿1(𝑢𝑣𝑢	PRON
iajs-2723	202	32	)	)	PUNCT
iajs-2723	203	1	+	+	CCONJ
iajs-2723	203	2	𝛿1(𝑢𝑣𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢𝑣𝑢)𝑐2𝑞𝑐1	X
iajs-2723	203	3	=	=	SYM
iajs-2723	203	4	0	0	NUM
iajs-2723	203	5	for	for	ADP
iajs-2723	203	6	each	each	DET
iajs-2723	203	7	𝑢	𝑢	NOUN
iajs-2723	203	8	,	,	PUNCT
iajs-2723	203	9	𝑣	𝑣	NOUN
iajs-2723	203	10	,	,	PUNCT
iajs-2723	203	11	𝑐1	𝑐1	NOUN
iajs-2723	203	12	,	,	PUNCT
iajs-2723	203	13	𝑐2	𝑐2	NOUN
iajs-2723	203	14	,	,	PUNCT
iajs-2723	203	15	𝑞	𝑞	PROPN
iajs-2723	203	16	∈	∈	PROPN
iajs-2723	203	17	γ	γ	X
iajs-2723	203	18	(	(	PUNCT
iajs-2723	203	19	66	66	NUM
iajs-2723	203	20	)	)	PUNCT
iajs-2723	203	21	that	that	PRON
iajs-2723	203	22	is	be	AUX
iajs-2723	203	23	𝑐1𝑞𝑐2(𝛿1(𝑢)𝜃1(𝑣𝑢	𝑐1𝑞𝑐2(𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	203	24	)	)	PUNCT
iajs-2723	203	25	+	+	PUNCT
iajs-2723	203	26	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	203	27	)	)	PUNCT
iajs-2723	203	28	+	+	ADJ
iajs-2723	203	29	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	203	30	)	)	PUNCT
iajs-2723	203	31	)	)	PUNCT
iajs-2723	204	1	+	+	CCONJ
iajs-2723	204	2	(	(	PUNCT
iajs-2723	204	3	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	204	4	)	)	PUNCT
iajs-2723	204	5	+	+	PUNCT
iajs-2723	204	6	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	NOUN
iajs-2723	204	7	)	)	PUNCT
iajs-2723	205	1	+	+	CCONJ
iajs-2723	205	2	𝜃2(𝑢𝑣)𝛿1(𝑢))𝑐2𝑞𝑐1	𝜃2(𝑢𝑣)𝛿1(𝑢))𝑐2𝑞𝑐1	PUNCT
iajs-2723	205	3	=	=	SYM
iajs-2723	205	4	0	0	NUM
iajs-2723	205	5	for	for	ADP
iajs-2723	205	6	each	each	DET
iajs-2723	205	7	𝑢	𝑢	NOUN
iajs-2723	205	8	,	,	PUNCT
iajs-2723	205	9	𝑣	𝑣	NOUN
iajs-2723	205	10	,	,	PUNCT
iajs-2723	205	11	𝑐1	𝑐1	NOUN
iajs-2723	205	12	,	,	PUNCT
iajs-2723	205	13	𝑐2	𝑐2	NOUN
iajs-2723	205	14	,	,	PUNCT
iajs-2723	205	15	𝑞	𝑞	PROPN
iajs-2723	205	16	∈	∈	PROPN
iajs-2723	205	17	γ	γ	X
iajs-2723	205	18	(	(	PUNCT
iajs-2723	205	19	67	67	NUM
iajs-2723	205	20	)	)	PUNCT
iajs-2723	205	21	that	that	PRON
iajs-2723	205	22	is	is	ADV
iajs-2723	205	23	(	(	PUNCT
iajs-2723	205	24	𝑐1𝑞𝑐2𝛿1(𝑢)𝜃1(𝑣𝑢	𝑐1𝑞𝑐2𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	205	25	)	)	PUNCT
iajs-2723	206	1	+	+	CCONJ
iajs-2723	206	2	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	206	3	)	)	PUNCT
iajs-2723	207	1	+	+	CCONJ
iajs-2723	207	2	𝑐1𝑞𝑐2𝜃2(𝑢𝑣)𝛿1(𝑢	𝑐1𝑞𝑐2𝜃2(𝑢𝑣)𝛿1(𝑢	NOUN
iajs-2723	207	3	)	)	PUNCT
iajs-2723	207	4	)	)	PUNCT
iajs-2723	208	1	+	+	CCONJ
iajs-2723	208	2	(	(	PUNCT
iajs-2723	208	3	𝛿1(𝑢)𝜃1(𝑣𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢)𝜃1(𝑣𝑢)𝑐2𝑞𝑐1	NOUN
iajs-2723	208	4	+	+	X
iajs-2723	208	5	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	NOUN
iajs-2723	208	6	+	+	NUM
iajs-2723	208	7	𝜃2(𝑢𝑣)𝛿1(𝑢)𝑐2𝑞𝑐1	𝜃2(𝑢𝑣)𝛿1(𝑢)𝑐2𝑞𝑐1	NOUN
iajs-2723	208	8	)	)	PUNCT
iajs-2723	209	1	=	=	SYM
iajs-2723	209	2	0	0	NUM
iajs-2723	210	1	for	for	ADP
iajs-2723	210	2	each	each	DET
iajs-2723	210	3	𝑢	𝑢	NOUN
iajs-2723	210	4	,	,	PUNCT
iajs-2723	210	5	𝑣	𝑣	NOUN
iajs-2723	210	6	,	,	PUNCT
iajs-2723	210	7	𝑐1	𝑐1	NOUN
iajs-2723	210	8	,	,	PUNCT
iajs-2723	210	9	𝑐2	𝑐2	NOUN
iajs-2723	210	10	,	,	PUNCT
iajs-2723	210	11	𝑞	𝑞	PROPN
iajs-2723	210	12	∈	∈	PROPN
iajs-2723	210	13	γ	γ	X
iajs-2723	210	14	(	(	PUNCT
iajs-2723	210	15	68	68	NUM
iajs-2723	210	16	)	)	PUNCT
iajs-2723	210	17	by	by	ADP
iajs-2723	210	18	setting	set	VERB
iajs-2723	210	19	𝜃1(𝑣𝑢	𝜃1(𝑣𝑢	NOUN
iajs-2723	210	20	)	)	PUNCT
iajs-2723	210	21	=	=	SYM
iajs-2723	210	22	𝜃2(𝑢𝑣	𝜃2(𝑢𝑣	PROPN
iajs-2723	210	23	)	)	PUNCT
iajs-2723	210	24	=	=	SYM
iajs-2723	210	25	1	1	NUM
iajs-2723	210	26	in	in	ADP
iajs-2723	210	27	(	(	PUNCT
iajs-2723	210	28	68	68	NUM
iajs-2723	210	29	)	)	PUNCT
iajs-2723	210	30	,	,	PUNCT
iajs-2723	210	31	we	we	PRON
iajs-2723	210	32	have	have	VERB
iajs-2723	210	33	:	:	PUNCT
iajs-2723	210	34	(	(	PUNCT
iajs-2723	210	35	𝑐1𝑞𝑐2𝛿1(𝑢	𝑐1𝑞𝑐2𝛿1(𝑢	ADJ
iajs-2723	210	36	)	)	PUNCT
iajs-2723	210	37	+	+	CCONJ
iajs-2723	210	38	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	210	39	)	)	PUNCT
iajs-2723	211	1	+	+	CCONJ
iajs-2723	211	2	𝑐1𝑞𝑐2𝛿1(𝑢	𝑐1𝑞𝑐2𝛿1(𝑢	NOUN
iajs-2723	211	3	)	)	PUNCT
iajs-2723	211	4	)	)	PUNCT
iajs-2723	212	1	+	+	CCONJ
iajs-2723	212	2	(	(	PUNCT
iajs-2723	212	3	𝛿1(𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢)𝑐2𝑞𝑐1	PROPN
iajs-2723	212	4	+	+	CCONJ
iajs-2723	212	5	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	PROPN
iajs-2723	212	6	+	+	NUM
iajs-2723	212	7	𝛿1(𝑢)𝑐2𝑞𝑐1	𝛿1(𝑢)𝑐2𝑞𝑐1	PROPN
iajs-2723	212	8	)	)	PUNCT
iajs-2723	213	1	=	=	PUNCT
iajs-2723	213	2	0	0	NUM
iajs-2723	214	1	for	for	ADP
iajs-2723	214	2	each	each	DET
iajs-2723	214	3	𝑢	𝑢	NOUN
iajs-2723	214	4	,	,	PUNCT
iajs-2723	214	5	𝑣	𝑣	NOUN
iajs-2723	214	6	,	,	PUNCT
iajs-2723	214	7	𝑐1	𝑐1	NOUN
iajs-2723	214	8	,	,	PUNCT
iajs-2723	214	9	𝑐2	𝑐2	NOUN
iajs-2723	214	10	,	,	PUNCT
iajs-2723	214	11	𝑞	𝑞	PROPN
iajs-2723	214	12	∈	∈	PROPN
iajs-2723	214	13	γ	γ	X
iajs-2723	214	14	(	(	PUNCT
iajs-2723	214	15	69	69	NUM
iajs-2723	214	16	)	)	PUNCT
iajs-2723	214	17	by	by	ADP
iajs-2723	214	18	using	use	VERB
iajs-2723	214	19	(	(	PUNCT
iajs-2723	214	20	65	65	NUM
iajs-2723	214	21	)	)	PUNCT
iajs-2723	214	22	,	,	PUNCT
iajs-2723	214	23	we	we	PRON
iajs-2723	214	24	get	get	VERB
iajs-2723	214	25	:	:	PUNCT
iajs-2723	214	26	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝑐1𝑞𝑐2𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	214	27	)	)	PUNCT
iajs-2723	215	1	+	+	CCONJ
iajs-2723	215	2	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢)𝑐2𝑞𝑐1	NOUN
iajs-2723	215	3	=	=	SYM
iajs-2723	215	4	0	0	NUM
iajs-2723	215	5	for	for	ADP
iajs-2723	215	6	each	each	DET
iajs-2723	215	7	𝑢	𝑢	NOUN
iajs-2723	215	8	,	,	PUNCT
iajs-2723	215	9	𝑣	𝑣	NOUN
iajs-2723	215	10	,	,	PUNCT
iajs-2723	215	11	𝑐1	𝑐1	NOUN
iajs-2723	215	12	,	,	PUNCT
iajs-2723	215	13	𝑐2	𝑐2	NOUN
iajs-2723	215	14	,	,	PUNCT
iajs-2723	215	15	𝑞	𝑞	PROPN
iajs-2723	215	16	∈	∈	PROPN
iajs-2723	215	17	γ	γ	X
iajs-2723	215	18	(	(	PUNCT
iajs-2723	215	19	70	70	NUM
iajs-2723	215	20	)	)	PUNCT
iajs-2723	215	21	ibn	ibn	PROPN
iajs-2723	215	22	al	al	PROPN
iajs-2723	215	23	-	-	PUNCT
iajs-2723	215	24	haitham	haitham	PROPN
iajs-2723	215	25	jour	jour	X
iajs-2723	215	26	.	.	PROPN
iajs-2723	216	1	for	for	ADP
iajs-2723	216	2	pure	pure	ADJ
iajs-2723	216	3	&	&	CCONJ
iajs-2723	216	4	appl	appl	PROPN
iajs-2723	216	5	.	.	PUNCT
iajs-2723	217	1	sci	sci	PROPN
iajs-2723	217	2	.	.	PROPN
iajs-2723	218	1	53	53	NUM
iajs-2723	218	2	(	(	PUNCT
iajs-2723	218	3	2)2022	2)2022	NOUN
iajs-2723	218	4	115	115	NUM
iajs-2723	218	5	by	by	ADP
iajs-2723	218	6	setting	set	VERB
iajs-2723	218	7	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	NOUN
iajs-2723	218	8	)	)	PUNCT
iajs-2723	218	9	=	=	SYM
iajs-2723	218	10	1	1	NUM
iajs-2723	218	11	in	in	ADP
iajs-2723	218	12	(	(	PUNCT
iajs-2723	218	13	70	70	NUM
iajs-2723	218	14	)	)	PUNCT
iajs-2723	218	15	,	,	PUNCT
iajs-2723	218	16	we	we	PRON
iajs-2723	218	17	have	have	AUX
iajs-2723	218	18	:	:	PUNCT
iajs-2723	218	19	𝑐1𝑞𝑐2	𝑐1𝑞𝑐2	VERB
iajs-2723	218	20	+	+	PROPN
iajs-2723	218	21	𝑐2𝑞𝑐1	𝑐2𝑞𝑐1	NOUN
iajs-2723	218	22	=	=	NOUN
iajs-2723	218	23	0	0	NUM
iajs-2723	218	24	for	for	ADP
iajs-2723	218	25	each	each	DET
iajs-2723	218	26	𝑢	𝑢	NOUN
iajs-2723	218	27	,	,	PUNCT
iajs-2723	218	28	𝑣	𝑣	NOUN
iajs-2723	218	29	,	,	PUNCT
iajs-2723	218	30	𝑐1	𝑐1	NOUN
iajs-2723	218	31	,	,	PUNCT
iajs-2723	218	32	𝑐2	𝑐2	NOUN
iajs-2723	218	33	,	,	PUNCT
iajs-2723	218	34	𝑞	𝑞	PROPN
iajs-2723	218	35	∈	∈	PROPN
iajs-2723	218	36	γ	γ	X
iajs-2723	218	37	(	(	PUNCT
iajs-2723	218	38	71	71	NUM
iajs-2723	218	39	)	)	PUNCT
iajs-2723	218	40	replacing	replace	VERB
iajs-2723	218	41	𝑞	𝑞	PRON
iajs-2723	218	42	by	by	ADP
iajs-2723	218	43	𝑥𝑐1𝑦	𝑥𝑐1𝑦	PROPN
iajs-2723	218	44	in	in	ADP
iajs-2723	218	45	(	(	PUNCT
iajs-2723	218	46	71	71	NUM
iajs-2723	218	47	)	)	PUNCT
iajs-2723	218	48	,	,	PUNCT
iajs-2723	218	49	we	we	PRON
iajs-2723	218	50	get	get	VERB
iajs-2723	218	51	:	:	PUNCT
iajs-2723	218	52	𝑐1𝑥𝑐1𝑦𝑐2	𝑐1𝑥𝑐1𝑦𝑐2	X
iajs-2723	218	53	+	+	CCONJ
iajs-2723	218	54	𝑐2𝑥𝑐1𝑦𝑐1	𝑐2𝑥𝑐1𝑦𝑐1	NOUN
iajs-2723	218	55	=	=	SYM
iajs-2723	218	56	0	0	NUM
iajs-2723	218	57	for	for	ADP
iajs-2723	218	58	each	each	DET
iajs-2723	218	59	𝑥	𝑥	PROPN
iajs-2723	218	60	,	,	PUNCT
iajs-2723	218	61	𝑦	𝑦	NOUN
iajs-2723	218	62	,	,	PUNCT
iajs-2723	218	63	𝑐1	𝑐1	NOUN
iajs-2723	218	64	,	,	PUNCT
iajs-2723	218	65	𝑐2	𝑐2	NOUN
iajs-2723	218	66	∈	∈	PROPN
iajs-2723	218	67	γ	γ	X
iajs-2723	218	68	(	(	PUNCT
iajs-2723	218	69	72	72	NUM
iajs-2723	218	70	)	)	PUNCT
iajs-2723	218	71	that	that	PRON
iajs-2723	218	72	is	be	AUX
iajs-2723	218	73	𝑐1𝑦𝑐2	𝑐1𝑦𝑐2	NOUN
iajs-2723	218	74	=	=	SYM
iajs-2723	218	75	−𝑐2𝑦𝑐1	−𝑐2𝑦𝑐1	NOUN
iajs-2723	218	76	and	and	CCONJ
iajs-2723	218	77	𝑐2𝑥𝑐1	𝑐2𝑥𝑐1	NOUN
iajs-2723	218	78	=	=	SYM
iajs-2723	219	1	−𝑐1𝑥𝑐2	−𝑐1𝑥𝑐2	PROPN
iajs-2723	219	2	(	(	PUNCT
iajs-2723	219	3	73	73	NUM
iajs-2723	219	4	)	)	PUNCT
iajs-2723	219	5	substituting	substitute	VERB
iajs-2723	219	6	(	(	PUNCT
iajs-2723	219	7	73	73	NUM
iajs-2723	219	8	)	)	PUNCT
iajs-2723	219	9	in	in	ADP
iajs-2723	219	10	(	(	PUNCT
iajs-2723	219	11	72	72	X
iajs-2723	219	12	)	)	PUNCT
iajs-2723	219	13	we	we	PRON
iajs-2723	219	14	have	have	VERB
iajs-2723	219	15	:	:	PUNCT
iajs-2723	219	16	−𝑐1𝑥𝑐2𝑦𝑐1	−𝑐1𝑥𝑐2𝑦𝑐1	NOUN
iajs-2723	219	17	−	−	PROPN
iajs-2723	219	18	𝑐1𝑥𝑐2𝑦𝑐1	𝑐1𝑥𝑐2𝑦𝑐1	SYM
iajs-2723	219	19	=	=	SYM
iajs-2723	219	20	0	0	PUNCT
iajs-2723	219	21	(	(	PUNCT
iajs-2723	219	22	74	74	NUM
iajs-2723	219	23	)	)	PUNCT
iajs-2723	219	24	that	that	PRON
iajs-2723	219	25	is	be	AUX
iajs-2723	219	26	2𝑐1γ𝑐2γ𝑐1	2𝑐1γ𝑐2γ𝑐1	NUM
iajs-2723	219	27	=	=	SYM
iajs-2723	219	28	(	(	PUNCT
iajs-2723	219	29	0	0	NUM
iajs-2723	219	30	)	)	PUNCT
iajs-2723	219	31	(	(	PUNCT
iajs-2723	219	32	75	75	NUM
iajs-2723	219	33	)	)	PUNCT
iajs-2723	219	34	since	since	SCONJ
iajs-2723	219	35	char(γ	char(γ	PROPN
iajs-2723	219	36	)	)	PUNCT
iajs-2723	219	37	≠	≠	PROPN
iajs-2723	219	38	2	2	NUM
iajs-2723	219	39	and	and	CCONJ
iajs-2723	219	40	γ	γ	NOUN
iajs-2723	219	41	is	be	AUX
iajs-2723	219	42	a	a	DET
iajs-2723	219	43	prime	prime	NOUN
iajs-2723	219	44	,	,	PUNCT
iajs-2723	219	45	then	then	ADV
iajs-2723	219	46	(	(	PUNCT
iajs-2723	219	47	75	75	NUM
iajs-2723	219	48	)	)	PUNCT
iajs-2723	219	49	gives	give	VERB
iajs-2723	219	50	𝑐1	𝑐1	NOUN
iajs-2723	219	51	=	=	SYM
iajs-2723	219	52	0	0	NUM
iajs-2723	219	53	or	or	CCONJ
iajs-2723	219	54	𝑐2	𝑐2	NOUN
iajs-2723	219	55	=	=	SYM
iajs-2723	219	56	0	0	X
iajs-2723	219	57	.	.	PUNCT
iajs-2723	220	1	∎	∎	PROPN
iajs-2723	220	2	theorem	theorem	VERB
iajs-2723	220	3	3.6	3.6	NUM
iajs-2723	220	4	let	let	VERB
iajs-2723	220	5	γ	γ	NOUN
iajs-2723	220	6	be	be	AUX
iajs-2723	220	7	a	a	DET
iajs-2723	220	8	2	2	NUM
iajs-2723	220	9	-	-	PUNCT
iajs-2723	220	10	torsion	torsion	NOUN
iajs-2723	220	11	free	free	ADJ
iajs-2723	220	12	ring	ring	NOUN
iajs-2723	220	13	with	with	ADP
iajs-2723	220	14	an	an	DET
iajs-2723	220	15	identity	identity	NOUN
iajs-2723	220	16	element	element	NOUN
iajs-2723	220	17	.	.	PUNCT
iajs-2723	221	1	furthermore	furthermore	ADV
iajs-2723	221	2	,	,	PUNCT
iajs-2723	221	3	let	let	VERB
iajs-2723	221	4	(	(	PUNCT
iajs-2723	221	5	𝛿1	𝛿1	ADJ
iajs-2723	221	6	,	,	PUNCT
iajs-2723	221	7	𝛿2	𝛿2	PROPN
iajs-2723	221	8	)	)	PUNCT
iajs-2723	221	9	be	be	AUX
iajs-2723	221	10	a	a	DET
iajs-2723	221	11	jordan	jordan	PROPN
iajs-2723	221	12	(	(	PUNCT
iajs-2723	221	13	𝜃1	𝜃1	PROPN
iajs-2723	221	14	,	,	PUNCT
iajs-2723	221	15	𝜃2)-derivation	𝜃2)-derivation	PRON
iajs-2723	221	16	pair	pair	VERB
iajs-2723	221	17	such	such	ADJ
iajs-2723	221	18	that	that	DET
iajs-2723	221	19	𝛿1(1	𝛿1(1	ADJ
iajs-2723	221	20	)	)	PUNCT
iajs-2723	221	21	=	=	SYM
iajs-2723	221	22	𝛿2(1	𝛿2(1	PROPN
iajs-2723	221	23	)	)	PUNCT
iajs-2723	221	24	.	.	PUNCT
iajs-2723	222	1	then	then	ADV
iajs-2723	222	2	𝛿1(𝑢	𝛿1(𝑢	NUM
iajs-2723	222	3	)	)	PUNCT
iajs-2723	222	4	=	=	SYM
iajs-2723	222	5	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	222	6	)	)	PUNCT
iajs-2723	222	7	,	,	PUNCT
iajs-2723	222	8	∀𝑢	∀𝑢	DET
iajs-2723	222	9	∈	∈	PROPN
iajs-2723	222	10	γ	γ	X
iajs-2723	222	11	where	where	SCONJ
iajs-2723	222	12	𝜃1	𝜃1	VERB
iajs-2723	222	13	and	and	CCONJ
iajs-2723	222	14	𝜃2	𝜃2	NOUN
iajs-2723	222	15	are	be	AUX
iajs-2723	222	16	two	two	NUM
iajs-2723	222	17	mappings	mapping	NOUN
iajs-2723	222	18	of	of	ADP
iajs-2723	222	19	γ	γ	PROPN
iajs-2723	223	1	.	.	PUNCT
iajs-2723	223	2	proof	proof	NOUN
iajs-2723	223	3	:	:	PUNCT
iajs-2723	223	4	let	let	VERB
iajs-2723	223	5	𝜑	𝜑	PRON
iajs-2723	223	6	:	:	PUNCT
iajs-2723	223	7	γ	γ	X
iajs-2723	223	8	→	→	SYM
iajs-2723	223	9	γ	γ	X
iajs-2723	223	10	be	be	AUX
iajs-2723	223	11	a	a	DET
iajs-2723	223	12	mapping	mapping	NOUN
iajs-2723	223	13	given	give	VERB
iajs-2723	223	14	by	by	ADP
iajs-2723	223	15	𝜑(𝑢	𝜑(𝑢	PRON
iajs-2723	223	16	)	)	PUNCT
iajs-2723	223	17	=	=	SYM
iajs-2723	224	1	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	224	2	)	)	PUNCT
iajs-2723	224	3	−	−	PROPN
iajs-2723	224	4	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	224	5	)	)	PUNCT
iajs-2723	224	6	,	,	PUNCT
iajs-2723	224	7	∀𝑢	∀𝑢	DET
iajs-2723	224	8	∈	∈	PROPN
iajs-2723	224	9	γ	γ	X
iajs-2723	224	10	.	.	PUNCT
iajs-2723	224	11	by	by	ADP
iajs-2723	224	12	definition	definition	NOUN
iajs-2723	224	13	3.1	3.1	NUM
iajs-2723	224	14	,	,	PUNCT
iajs-2723	224	15	we	we	PRON
iajs-2723	224	16	have	have	VERB
iajs-2723	224	17	:	:	PUNCT
iajs-2723	224	18	𝛿1(𝑢3	𝛿1(𝑢3	NUM
iajs-2723	224	19	)	)	PUNCT
iajs-2723	224	20	=	=	SYM
iajs-2723	224	21	𝛿1(𝑢)𝜃1(𝑢2	𝛿1(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	224	22	)	)	PUNCT
iajs-2723	224	23	+	+	X
iajs-2723	224	24	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑢)𝜃1(𝑢	NUM
iajs-2723	224	25	)	)	PUNCT
iajs-2723	224	26	+	+	CCONJ
iajs-2723	224	27	𝜃2(𝑢2)𝛿1(𝑢	𝜃2(𝑢2)𝛿1(𝑢	NUM
iajs-2723	224	28	)	)	PUNCT
iajs-2723	224	29	for	for	ADP
iajs-2723	224	30	all	all	DET
iajs-2723	224	31	𝑢	𝑢	PRON
iajs-2723	224	32	∈	∈	PROPN
iajs-2723	224	33	γ	γ	X
iajs-2723	224	34	(	(	PUNCT
iajs-2723	224	35	76	76	NUM
iajs-2723	224	36	)	)	PUNCT
iajs-2723	224	37	𝛿2(𝑢3	𝛿2(𝑢3	NOUN
iajs-2723	224	38	)	)	PUNCT
iajs-2723	224	39	=	=	SYM
iajs-2723	224	40	𝛿2(𝑢)𝜃1(𝑢2	𝛿2(𝑢)𝜃1(𝑢2	PROPN
iajs-2723	224	41	)	)	PUNCT
iajs-2723	224	42	+	+	CCONJ
iajs-2723	224	43	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑢)𝜃1(𝑢	PROPN
iajs-2723	224	44	)	)	PUNCT
iajs-2723	224	45	+	+	NUM
iajs-2723	224	46	𝜃2(𝑢2)𝛿2(𝑢	𝜃2(𝑢2)𝛿2(𝑢	NOUN
iajs-2723	224	47	)	)	PUNCT
iajs-2723	224	48	for	for	ADP
iajs-2723	224	49	all	all	PRON
iajs-2723	224	50	𝑢	𝑢	PRON
iajs-2723	224	51	∈	∈	PROPN
iajs-2723	224	52	γ	γ	X
iajs-2723	224	53	(	(	PUNCT
iajs-2723	224	54	77	77	NUM
iajs-2723	224	55	)	)	PUNCT
iajs-2723	224	56	subtracting	subtract	VERB
iajs-2723	224	57	(	(	PUNCT
iajs-2723	224	58	77	77	NUM
iajs-2723	224	59	)	)	PUNCT
iajs-2723	224	60	from	from	ADP
iajs-2723	224	61	(	(	PUNCT
iajs-2723	224	62	76	76	NUM
iajs-2723	224	63	)	)	PUNCT
iajs-2723	224	64	,	,	PUNCT
iajs-2723	224	65	we	we	PRON
iajs-2723	224	66	get	get	VERB
iajs-2723	224	67	:	:	PUNCT
iajs-2723	224	68	𝜑(𝑢3	𝜑(𝑢3	NOUN
iajs-2723	224	69	)	)	PUNCT
iajs-2723	224	70	=	=	SYM
iajs-2723	224	71	𝜑(𝑢)𝜃1(𝑢2	𝜑(𝑢)𝜃1(𝑢2	NOUN
iajs-2723	224	72	)	)	PUNCT
iajs-2723	224	73	−	−	NOUN
iajs-2723	225	1	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑢	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑢	NOUN
iajs-2723	225	2	)	)	PUNCT
iajs-2723	225	3	+	+	NUM
iajs-2723	225	4	𝜃2(𝑢2)𝜑(𝑢	𝜃2(𝑢2)𝜑(𝑢	PROPN
iajs-2723	225	5	)	)	PUNCT
iajs-2723	225	6	for	for	ADP
iajs-2723	225	7	all	all	PRON
iajs-2723	225	8	𝑢	𝑢	PRON
iajs-2723	225	9	∈	∈	PROPN
iajs-2723	225	10	γ	γ	X
iajs-2723	225	11	(	(	PUNCT
iajs-2723	225	12	78	78	NUM
iajs-2723	225	13	)	)	PUNCT
iajs-2723	225	14	linearizing	linearize	VERB
iajs-2723	225	15	(	(	PUNCT
iajs-2723	225	16	78	78	NUM
iajs-2723	225	17	)	)	PUNCT
iajs-2723	225	18	,	,	PUNCT
iajs-2723	225	19	we	we	PRON
iajs-2723	225	20	have	have	VERB
iajs-2723	225	21	:	:	PUNCT
iajs-2723	225	22	𝜑(𝑢2𝑣	𝜑(𝑢2𝑣	X
iajs-2723	225	23	+	+	X
iajs-2723	225	24	𝑣𝑢2	𝑣𝑢2	X
iajs-2723	225	25	+	+	CCONJ
iajs-2723	225	26	𝑢𝑣2	𝑢𝑣2	NOUN
iajs-2723	225	27	+	+	CCONJ
iajs-2723	225	28	𝑣2𝑢	𝑣2𝑢	ADJ
iajs-2723	225	29	+	+	NUM
iajs-2723	225	30	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	225	31	+	+	CCONJ
iajs-2723	225	32	𝑣𝑢𝑣	𝑣𝑢𝑣	NOUN
iajs-2723	225	33	)	)	PUNCT
iajs-2723	225	34	=	=	SYM
iajs-2723	225	35	𝜑(𝑢)𝜃1(𝑢𝑣	𝜑(𝑢)𝜃1(𝑢𝑣	NOUN
iajs-2723	225	36	)	)	PUNCT
iajs-2723	225	37	+	+	NUM
iajs-2723	225	38	𝜑(𝑢)𝜃1(𝑣𝑢	𝜑(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	225	39	)	)	PUNCT
iajs-2723	225	40	+	+	CCONJ
iajs-2723	225	41	𝜑(𝑢)𝜃1(𝑣2	𝜑(𝑢)𝜃1(𝑣2	VERB
iajs-2723	225	42	)	)	PUNCT
iajs-2723	226	1	+	+	NUM
iajs-2723	226	2	𝜑(𝑣)𝜃1(𝑢𝑣	𝜑(𝑣)𝜃1(𝑢𝑣	NOUN
iajs-2723	226	3	)	)	PUNCT
iajs-2723	226	4	+	+	NUM
iajs-2723	226	5	𝜑(𝑣)𝜃1(𝑣𝑢	𝜑(𝑣)𝜃1(𝑣𝑢	NOUN
iajs-2723	226	6	)	)	PUNCT
iajs-2723	226	7	+	+	SYM
iajs-2723	226	8	𝜑(𝑣)𝜃1(𝑢2	𝜑(𝑣)𝜃1(𝑢2	NOUN
iajs-2723	226	9	)	)	PUNCT
iajs-2723	226	10	−	−	ADP
iajs-2723	226	11	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑣	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑣	NOUN
iajs-2723	226	12	)	)	PUNCT
iajs-2723	227	1	−	−	NOUN
iajs-2723	227	2	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑢	NOUN
iajs-2723	227	3	)	)	PUNCT
iajs-2723	227	4	−	−	ADP
iajs-2723	227	5	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑢	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑢	NOUN
iajs-2723	227	6	)	)	PUNCT
iajs-2723	227	7	−	−	ADP
iajs-2723	227	8	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	NOUN
iajs-2723	227	9	)	)	PUNCT
iajs-2723	227	10	−	−	ADP
iajs-2723	227	11	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	X
iajs-2723	227	12	)	)	PUNCT
iajs-2723	227	13	−	−	NOUN
iajs-2723	227	14	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	ADV
iajs-2723	227	15	)	)	PUNCT
iajs-2723	228	1	+	+	CCONJ
iajs-2723	228	2	𝜃2(𝑢𝑣)𝜑(𝑢	𝜃2(𝑢𝑣)𝜑(𝑢	PRON
iajs-2723	228	3	)	)	PUNCT
iajs-2723	229	1	+	+	PUNCT
iajs-2723	229	2	𝜃2(𝑣𝑢)𝜑(𝑢	𝜃2(𝑣𝑢)𝜑(𝑢	X
iajs-2723	229	3	)	)	PUNCT
iajs-2723	229	4	+	+	CCONJ
iajs-2723	229	5	𝜃2(𝑣2)𝜑(𝑢	𝜃2(𝑣2)𝜑(𝑢	PROPN
iajs-2723	229	6	)	)	PUNCT
iajs-2723	229	7	+	+	PUNCT
iajs-2723	229	8	𝜃2(𝑢𝑣)𝜑(𝑣	𝜃2(𝑢𝑣)𝜑(𝑣	SYM
iajs-2723	229	9	)	)	PUNCT
iajs-2723	229	10	+	+	SYM
iajs-2723	229	11	𝜃2(𝑣𝑢)𝜑(𝑣)+𝜃2(𝑢2)𝜑(𝑣	𝜃2(𝑣𝑢)𝜑(𝑣)+𝜃2(𝑢2)𝜑(𝑣	NOUN
iajs-2723	229	12	)	)	PUNCT
iajs-2723	229	13	for	for	ADP
iajs-2723	229	14	all	all	DET
iajs-2723	229	15	𝑢	𝑢	NOUN
iajs-2723	229	16	,	,	PUNCT
iajs-2723	229	17	𝑣	𝑣	PROPN
iajs-2723	229	18	∈	∈	PROPN
iajs-2723	229	19	γ	γ	X
iajs-2723	229	20	(	(	PUNCT
iajs-2723	229	21	79	79	NUM
iajs-2723	229	22	)	)	PUNCT
iajs-2723	229	23	replacing	replace	VERB
iajs-2723	229	24	𝑢	𝑢	PRON
iajs-2723	229	25	by	by	ADP
iajs-2723	229	26	−𝑢	−𝑢	NOUN
iajs-2723	229	27	in	in	ADP
iajs-2723	229	28	(	(	PUNCT
iajs-2723	229	29	79	79	NUM
iajs-2723	229	30	)	)	PUNCT
iajs-2723	229	31	,	,	PUNCT
iajs-2723	229	32	we	we	PRON
iajs-2723	229	33	get	get	VERB
iajs-2723	229	34	:	:	PUNCT
iajs-2723	229	35	𝜑(𝑢2𝑣	𝜑(𝑢2𝑣	PUNCT
iajs-2723	229	36	+	+	CCONJ
iajs-2723	229	37	𝑣𝑢2	𝑣𝑢2	ADP
iajs-2723	229	38	−	−	NOUN
iajs-2723	229	39	𝑢𝑣2	𝑢𝑣2	NOUN
iajs-2723	229	40	−	−	NOUN
iajs-2723	229	41	𝑣2𝑢	𝑣2𝑢	NOUN
iajs-2723	229	42	+	+	NUM
iajs-2723	229	43	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	229	44	−	−	DET
iajs-2723	229	45	𝑣𝑢𝑣	𝑣𝑢𝑣	NOUN
iajs-2723	229	46	)	)	PUNCT
iajs-2723	229	47	=	=	SYM
iajs-2723	229	48	𝜑(𝑢)𝜃1(𝑢𝑣	𝜑(𝑢)𝜃1(𝑢𝑣	NOUN
iajs-2723	229	49	)	)	PUNCT
iajs-2723	229	50	+	+	NUM
iajs-2723	229	51	𝜑(𝑢)𝜃1(𝑣𝑢	𝜑(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	229	52	)	)	PUNCT
iajs-2723	229	53	−	−	ADP
iajs-2723	229	54	𝜑(𝑢)𝜃1(𝑣2	𝜑(𝑢)𝜃1(𝑣2	NOUN
iajs-2723	229	55	)	)	PUNCT
iajs-2723	229	56	−	−	PROPN
iajs-2723	229	57	𝜑(𝑣)𝜃1(𝑢𝑣	𝜑(𝑣)𝜃1(𝑢𝑣	NOUN
iajs-2723	229	58	)	)	PUNCT
iajs-2723	229	59	−	−	PROPN
iajs-2723	229	60	𝜑(𝑣)𝜃1(𝑣𝑢	𝜑(𝑣)𝜃1(𝑣𝑢	PROPN
iajs-2723	229	61	)	)	PUNCT
iajs-2723	229	62	+	+	NUM
iajs-2723	229	63	𝜑(𝑣)𝜃1(𝑢2	𝜑(𝑣)𝜃1(𝑢2	NOUN
iajs-2723	229	64	)	)	PUNCT
iajs-2723	229	65	+	+	PUNCT
iajs-2723	229	66	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑣	𝜃2(𝑢)𝜑(𝑢)𝜃1(𝑣	NUM
iajs-2723	229	67	)	)	PUNCT
iajs-2723	229	68	+	+	NUM
iajs-2723	229	69	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑢	NOUN
iajs-2723	229	70	)	)	PUNCT
iajs-2723	230	1	+	+	CCONJ
iajs-2723	230	2	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑢	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑢	NOUN
iajs-2723	230	3	)	)	PUNCT
iajs-2723	230	4	−	−	NOUN
iajs-2723	230	5	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	NOUN
iajs-2723	230	6	)	)	PUNCT
iajs-2723	230	7	−	−	ADP
iajs-2723	230	8	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	X
iajs-2723	230	9	)	)	PUNCT
iajs-2723	230	10	−	−	NOUN
iajs-2723	230	11	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	ADV
iajs-2723	230	12	)	)	PUNCT
iajs-2723	230	13	+	+	CCONJ
iajs-2723	230	14	𝜃2(𝑢𝑣)𝜑(𝑢	𝜃2(𝑢𝑣)𝜑(𝑢	PRON
iajs-2723	230	15	)	)	PUNCT
iajs-2723	231	1	+	+	PUNCT
iajs-2723	231	2	𝜃2(𝑣𝑢)𝜑(𝑢	𝜃2(𝑣𝑢)𝜑(𝑢	X
iajs-2723	231	3	)	)	PUNCT
iajs-2723	231	4	−	−	PROPN
iajs-2723	231	5	𝜃2(𝑣2)𝜑(𝑢	𝜃2(𝑣2)𝜑(𝑢	PROPN
iajs-2723	231	6	)	)	PUNCT
iajs-2723	231	7	−	−	PART
iajs-2723	231	8	𝜃2(𝑢𝑣)𝜑(𝑣	𝜃2(𝑢𝑣)𝜑(𝑣	SYM
iajs-2723	231	9	)	)	PUNCT
iajs-2723	231	10	−	−	PRON
iajs-2723	231	11	𝜃2(𝑣𝑢)𝜑(𝑣	𝜃2(𝑣𝑢)𝜑(𝑣	NOUN
iajs-2723	231	12	)	)	PUNCT
iajs-2723	231	13	+	+	CCONJ
iajs-2723	231	14	𝜃2(𝑢2)𝜑(𝑣	𝜃2(𝑢2)𝜑(𝑣	NOUN
iajs-2723	231	15	)	)	PUNCT
iajs-2723	231	16	for	for	ADP
iajs-2723	231	17	all	all	DET
iajs-2723	231	18	𝑢	𝑢	NOUN
iajs-2723	231	19	,	,	PUNCT
iajs-2723	231	20	𝑣	𝑣	PROPN
iajs-2723	231	21	∈	∈	PROPN
iajs-2723	231	22	γ	γ	X
iajs-2723	231	23	(	(	PUNCT
iajs-2723	231	24	80	80	NUM
iajs-2723	231	25	)	)	PUNCT
iajs-2723	231	26	according	accord	VERB
iajs-2723	231	27	to	to	ADP
iajs-2723	231	28	(	(	PUNCT
iajs-2723	231	29	79	79	NUM
iajs-2723	231	30	)	)	PUNCT
iajs-2723	231	31	and	and	CCONJ
iajs-2723	231	32	(	(	PUNCT
iajs-2723	231	33	80	80	NUM
iajs-2723	231	34	)	)	PUNCT
iajs-2723	231	35	,	,	PUNCT
iajs-2723	231	36	we	we	PRON
iajs-2723	231	37	have	have	VERB
iajs-2723	231	38	:	:	PUNCT
iajs-2723	231	39	𝜑(𝑢2𝑣	𝜑(𝑢2𝑣	X
iajs-2723	231	40	+	+	CCONJ
iajs-2723	231	41	𝑣𝑢2	𝑣𝑢2	PROPN
iajs-2723	231	42	+	+	CCONJ
iajs-2723	231	43	𝑢𝑣𝑢	𝑢𝑣𝑢	NOUN
iajs-2723	231	44	)	)	PUNCT
iajs-2723	231	45	=	=	SYM
iajs-2723	231	46	𝜑(𝑢)𝜃1(𝑢𝑣	𝜑(𝑢)𝜃1(𝑢𝑣	NOUN
iajs-2723	231	47	)	)	PUNCT
iajs-2723	231	48	+	+	NUM
iajs-2723	231	49	𝜑(𝑢)𝜃1(𝑣𝑢	𝜑(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	231	50	)	)	PUNCT
iajs-2723	231	51	+	+	NUM
iajs-2723	231	52	𝜑(𝑣)𝜃1(𝑢2	𝜑(𝑣)𝜃1(𝑢2	NOUN
iajs-2723	231	53	)	)	PUNCT
iajs-2723	231	54	−	−	NOUN
iajs-2723	231	55	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	𝜃2(𝑣)𝜑(𝑣)𝜃1(𝑢	NOUN
iajs-2723	231	56	)	)	PUNCT
iajs-2723	231	57	−	−	ADP
iajs-2723	231	58	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	𝜃2(𝑢)𝜑(𝑣)𝜃1(𝑣	X
iajs-2723	231	59	)	)	PUNCT
iajs-2723	231	60	−	−	NOUN
iajs-2723	231	61	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	𝜃2(𝑣)𝜑(𝑢)𝜃1(𝑣	ADV
iajs-2723	231	62	)	)	PUNCT
iajs-2723	231	63	+	+	CCONJ
iajs-2723	231	64	𝜃2(𝑢𝑣)𝜑(𝑢	𝜃2(𝑢𝑣)𝜑(𝑢	PRON
iajs-2723	231	65	)	)	PUNCT
iajs-2723	232	1	+	+	CCONJ
iajs-2723	232	2	𝜃2(𝑣𝑢)𝜑(𝑢	𝜃2(𝑣𝑢)𝜑(𝑢	PROPN
iajs-2723	232	3	)	)	PUNCT
iajs-2723	232	4	+	+	NUM
iajs-2723	232	5	𝜃2(𝑢2)𝜑(𝑣	𝜃2(𝑢2)𝜑(𝑣	NOUN
iajs-2723	232	6	)	)	PUNCT
iajs-2723	232	7	for	for	ADP
iajs-2723	232	8	all	all	DET
iajs-2723	232	9	𝑢	𝑢	NOUN
iajs-2723	232	10	,	,	PUNCT
iajs-2723	232	11	𝑣	𝑣	PROPN
iajs-2723	232	12	∈	∈	PROPN
iajs-2723	232	13	γ	γ	X
iajs-2723	232	14	(	(	PUNCT
iajs-2723	232	15	81	81	NUM
iajs-2723	232	16	)	)	PUNCT
iajs-2723	232	17	replacing	replace	VERB
iajs-2723	232	18	𝑢	𝑢	PRON
iajs-2723	232	19	by	by	ADP
iajs-2723	232	20	1	1	NUM
iajs-2723	232	21	in	in	ADP
iajs-2723	232	22	(	(	PUNCT
iajs-2723	232	23	81	81	NUM
iajs-2723	232	24	)	)	PUNCT
iajs-2723	232	25	,	,	PUNCT
iajs-2723	232	26	we	we	PRON
iajs-2723	232	27	get	get	VERB
iajs-2723	232	28	:	:	PUNCT
iajs-2723	232	29	2𝜑(𝑣	2𝜑(𝑣	NUM
iajs-2723	232	30	)	)	PUNCT
iajs-2723	232	31	=	=	PUNCT
iajs-2723	233	1	𝜑(𝑣)𝜃1(1	𝜑(𝑣)𝜃1(1	ADJ
iajs-2723	233	2	)	)	PUNCT
iajs-2723	233	3	−	−	NOUN
iajs-2723	233	4	𝜃2(𝑣)𝜑(𝑣)𝜃1(1	𝜃2(𝑣)𝜑(𝑣)𝜃1(1	ADJ
iajs-2723	233	5	)	)	PUNCT
iajs-2723	233	6	−	−	ADP
iajs-2723	233	7	𝜃2(1)𝜑(𝑣)𝜃1(𝑣	𝜃2(1)𝜑(𝑣)𝜃1(𝑣	NOUN
iajs-2723	233	8	)	)	PUNCT
iajs-2723	234	1	+	+	CCONJ
iajs-2723	234	2	𝜃2(1)𝜑(𝑣)for	𝜃2(1)𝜑(𝑣)for	VERB
iajs-2723	234	3	all	all	DET
iajs-2723	234	4	𝑣	𝑣	ADP
iajs-2723	234	5	∈	∈	PROPN
iajs-2723	234	6	γ	γ	X
iajs-2723	234	7	(	(	PUNCT
iajs-2723	234	8	82	82	NUM
iajs-2723	234	9	)	)	PUNCT
iajs-2723	234	10	by	by	ADP
iajs-2723	234	11	setting	set	VERB
iajs-2723	234	12	𝜃1(1	𝜃1(1	ADJ
iajs-2723	234	13	)	)	PUNCT
iajs-2723	234	14	=	=	SYM
iajs-2723	234	15	𝜃2(1	𝜃2(1	NOUN
iajs-2723	234	16	)	)	PUNCT
iajs-2723	234	17	=	=	SYM
iajs-2723	234	18	0	0	NUM
iajs-2723	234	19	in	in	ADP
iajs-2723	234	20	(	(	PUNCT
iajs-2723	234	21	82	82	NUM
iajs-2723	234	22	)	)	PUNCT
iajs-2723	234	23	,	,	PUNCT
iajs-2723	234	24	then	then	ADV
iajs-2723	234	25	we	we	PRON
iajs-2723	234	26	have	have	VERB
iajs-2723	234	27	:	:	PUNCT
iajs-2723	234	28	2𝜑(𝑣	2𝜑(𝑣	NUM
iajs-2723	234	29	)	)	PUNCT
iajs-2723	235	1	=	=	SYM
iajs-2723	235	2	0	0	NUM
iajs-2723	235	3	for	for	ADP
iajs-2723	235	4	all	all	PRON
iajs-2723	235	5	𝑣	𝑣	DET
iajs-2723	235	6	∈	∈	PROPN
iajs-2723	235	7	γ	γ	X
iajs-2723	235	8	(	(	PUNCT
iajs-2723	235	9	83	83	NUM
iajs-2723	235	10	)	)	PUNCT
iajs-2723	235	11	since	since	SCONJ
iajs-2723	235	12	γ	γ	X
iajs-2723	235	13	is	be	AUX
iajs-2723	235	14	a	a	DET
iajs-2723	235	15	2	2	NUM
iajs-2723	235	16	-	-	PUNCT
iajs-2723	235	17	torsion	torsion	NOUN
iajs-2723	235	18	free	free	ADJ
iajs-2723	235	19	ring	ring	NOUN
iajs-2723	235	20	,	,	PUNCT
iajs-2723	235	21	then	then	ADV
iajs-2723	235	22	𝜑(𝑣	𝜑(𝑣	NUM
iajs-2723	235	23	)	)	PUNCT
iajs-2723	235	24	=	=	SYM
iajs-2723	235	25	0	0	NUM
iajs-2723	235	26	for	for	ADP
iajs-2723	235	27	all	all	PRON
iajs-2723	235	28	𝑣	𝑣	DET
iajs-2723	235	29	∈	∈	PROPN
iajs-2723	235	30	γ	γ	X
iajs-2723	235	31	(	(	PUNCT
iajs-2723	235	32	84	84	NUM
iajs-2723	235	33	)	)	PUNCT
iajs-2723	235	34	therefore	therefore	ADV
iajs-2723	235	35	,	,	PUNCT
iajs-2723	235	36	(	(	PUNCT
iajs-2723	235	37	84	84	NUM
iajs-2723	235	38	)	)	PUNCT
iajs-2723	235	39	gives	give	VERB
iajs-2723	235	40	𝛿1(𝑢	𝛿1(𝑢	PROPN
iajs-2723	235	41	)	)	PUNCT
iajs-2723	235	42	=	=	SYM
iajs-2723	235	43	𝛿2(𝑢	𝛿2(𝑢	PROPN
iajs-2723	235	44	)	)	PUNCT
iajs-2723	235	45	for	for	ADP
iajs-2723	235	46	all	all	DET
iajs-2723	235	47	𝑢	𝑢	PROPN
iajs-2723	235	48	∈	∈	PROPN
iajs-2723	235	49	γ	γ	X
iajs-2723	235	50	.	.	PUNCT
iajs-2723	236	1	∎	∎	PROPN
iajs-2723	236	2	proposition	proposition	NOUN
iajs-2723	236	3	3.1	3.1	NUM
iajs-2723	236	4	let	let	VERB
iajs-2723	236	5	γ	γ	NOUN
iajs-2723	236	6	be	be	AUX
iajs-2723	236	7	a	a	DET
iajs-2723	236	8	ring	ring	NOUN
iajs-2723	236	9	,	,	PUNCT
iajs-2723	236	10	and	and	CCONJ
iajs-2723	236	11	𝜃1	𝜃1	VERB
iajs-2723	236	12	,	,	PUNCT
iajs-2723	236	13	𝜃2	𝜃2	NOUN
iajs-2723	236	14	be	be	VERB
iajs-2723	236	15	two	two	NUM
iajs-2723	236	16	mappings	mapping	NOUN
iajs-2723	236	17	of	of	ADP
iajs-2723	236	18	γ	γ	PROPN
iajs-2723	236	19	.	.	PROPN
iajs-2723	236	20	then	then	ADV
iajs-2723	236	21	1if	1if	PROPN
iajs-2723	236	22	(	(	PUNCT
iajs-2723	236	23	𝛿1	𝛿1	PROPN
iajs-2723	236	24	,	,	PUNCT
iajs-2723	236	25	𝛿2	𝛿2	PROPN
iajs-2723	236	26	)	)	PUNCT
iajs-2723	236	27	is	be	AUX
iajs-2723	236	28	a	a	DET
iajs-2723	236	29	(	(	PUNCT
iajs-2723	236	30	𝜃1	𝜃1	NOUN
iajs-2723	236	31	,	,	PUNCT
iajs-2723	236	32	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	236	33	pair	pair	NOUN
iajs-2723	236	34	on	on	ADP
iajs-2723	236	35	γ	γ	NOUN
iajs-2723	236	36	,	,	PUNCT
iajs-2723	236	37	then	then	ADV
iajs-2723	236	38	𝛿1	𝛿1	NOUN
iajs-2723	236	39	+	+	CCONJ
iajs-2723	236	40	𝛿2	𝛿2	NOUN
iajs-2723	236	41	is	be	AUX
iajs-2723	236	42	a	a	DET
iajs-2723	236	43	(	(	PUNCT
iajs-2723	236	44	𝜃1	𝜃1	NOUN
iajs-2723	236	45	,	,	PUNCT
iajs-2723	236	46	𝜃2)-derivation	𝜃2)-derivation	PROPN
iajs-2723	236	47	.	.	PUNCT
iajs-2723	237	1	ibn	ibn	PROPN
iajs-2723	237	2	al	al	PROPN
iajs-2723	237	3	-	-	PUNCT
iajs-2723	237	4	haitham	haitham	PROPN
iajs-2723	237	5	jour	jour	X
iajs-2723	237	6	.	.	PROPN
iajs-2723	237	7	for	for	ADP
iajs-2723	237	8	pure	pure	ADJ
iajs-2723	237	9	&	&	CCONJ
iajs-2723	237	10	appl	appl	PROPN
iajs-2723	237	11	.	.	PUNCT
iajs-2723	238	1	sci	sci	PROPN
iajs-2723	238	2	.	.	PROPN
iajs-2723	239	1	53	53	NUM
iajs-2723	239	2	(	(	PUNCT
iajs-2723	239	3	2)2022	2)2022	NOUN
iajs-2723	239	4	116	116	NUM
iajs-2723	239	5	2if	2if	NOUN
iajs-2723	239	6	(	(	PUNCT
iajs-2723	239	7	𝛿1	𝛿1	NOUN
iajs-2723	239	8	,	,	PUNCT
iajs-2723	239	9	𝛿2	𝛿2	PROPN
iajs-2723	239	10	)	)	PUNCT
iajs-2723	239	11	is	be	AUX
iajs-2723	239	12	a	a	DET
iajs-2723	239	13	jordan	jordan	PROPN
iajs-2723	239	14	(	(	PUNCT
iajs-2723	239	15	𝜃1	𝜃1	PROPN
iajs-2723	239	16	,	,	PUNCT
iajs-2723	239	17	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	239	18	pair	pair	NOUN
iajs-2723	239	19	on	on	ADP
iajs-2723	239	20	γ	γ	NOUN
iajs-2723	239	21	,	,	PUNCT
iajs-2723	239	22	then	then	ADV
iajs-2723	239	23	𝛿1	𝛿1	NOUN
iajs-2723	239	24	+	+	CCONJ
iajs-2723	239	25	𝛿2	𝛿2	NOUN
iajs-2723	239	26	is	be	AUX
iajs-2723	239	27	a	a	DET
iajs-2723	239	28	jordan	jordan	PROPN
iajs-2723	239	29	(	(	PUNCT
iajs-2723	239	30	𝜃1	𝜃1	PROPN
iajs-2723	239	31	,	,	PUNCT
iajs-2723	239	32	𝜃2)derivation	𝜃2)derivation	NOUN
iajs-2723	239	33	.	.	PUNCT
iajs-2723	240	1	proof	proof	NOUN
iajs-2723	240	2	:	:	PUNCT
iajs-2723	240	3	(	(	PUNCT
iajs-2723	240	4	1	1	X
iajs-2723	240	5	)	)	PUNCT
iajs-2723	240	6	since	since	SCONJ
iajs-2723	240	7	𝛿1	𝛿1	NOUN
iajs-2723	240	8	and	and	CCONJ
iajs-2723	240	9	𝛿2	𝛿2	NOUN
iajs-2723	240	10	is	be	AUX
iajs-2723	240	11	a	a	DET
iajs-2723	240	12	(	(	PUNCT
iajs-2723	240	13	𝜃1	𝜃1	NOUN
iajs-2723	240	14	,	,	PUNCT
iajs-2723	240	15	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	240	16	pair	pair	NOUN
iajs-2723	240	17	,	,	PUNCT
iajs-2723	240	18	then	then	ADV
iajs-2723	240	19	by	by	ADP
iajs-2723	240	20	definition	definition	NOUN
iajs-2723	240	21	3.1	3.1	NUM
iajs-2723	240	22	,	,	PUNCT
iajs-2723	240	23	we	we	PRON
iajs-2723	240	24	have	have	VERB
iajs-2723	240	25	:	:	PUNCT
iajs-2723	240	26	𝛿1(𝑢𝑣𝑢	𝛿1(𝑢𝑣𝑢	NUM
iajs-2723	240	27	)	)	PUNCT
iajs-2723	240	28	=	=	SYM
iajs-2723	240	29	𝛿1(𝑢)𝜃1(𝑣𝑢	𝛿1(𝑢)𝜃1(𝑣𝑢	NOUN
iajs-2723	240	30	)	)	PUNCT
iajs-2723	240	31	+	+	PUNCT
iajs-2723	240	32	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿2(𝑣)𝜃1(𝑢	PROPN
iajs-2723	240	33	)	)	PUNCT
iajs-2723	240	34	+	+	ADJ
iajs-2723	240	35	𝜃2(𝑢𝑣)𝛿1(𝑢	𝜃2(𝑢𝑣)𝛿1(𝑢	NUM
iajs-2723	240	36	)	)	PUNCT
iajs-2723	240	37	for	for	ADP
iajs-2723	240	38	all	all	DET
iajs-2723	240	39	𝑢	𝑢	NOUN
iajs-2723	240	40	,	,	PUNCT
iajs-2723	240	41	𝑣	𝑣	PROPN
iajs-2723	240	42	∈	∈	PROPN
iajs-2723	240	43	γ	γ	X
iajs-2723	240	44	(	(	PUNCT
iajs-2723	240	45	85	85	NUM
iajs-2723	240	46	)	)	PUNCT
iajs-2723	240	47	𝛿2(𝑢𝑣𝑢	𝛿2(𝑢𝑣𝑢	NOUN
iajs-2723	240	48	)	)	PUNCT
iajs-2723	240	49	=	=	PUNCT
iajs-2723	240	50	𝛿2(𝑢)𝜃1(𝑣𝑢	𝛿2(𝑢)𝜃1(𝑣𝑢	PROPN
iajs-2723	240	51	)	)	PUNCT
iajs-2723	240	52	+	+	PUNCT
iajs-2723	240	53	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	𝜃2(𝑢)𝛿1(𝑣)𝜃1(𝑢	PROPN
iajs-2723	240	54	)	)	PUNCT
iajs-2723	240	55	+	+	CCONJ
iajs-2723	240	56	𝜃2(𝑢𝑣)𝛿2(𝑢	𝜃2(𝑢𝑣)𝛿2(𝑢	X
iajs-2723	240	57	)	)	PUNCT
iajs-2723	240	58	for	for	ADP
iajs-2723	240	59	all	all	DET
iajs-2723	240	60	𝑢	𝑢	NOUN
iajs-2723	240	61	,	,	PUNCT
iajs-2723	240	62	𝑣	𝑣	PROPN
iajs-2723	240	63	∈	∈	PROPN
iajs-2723	240	64	γ	γ	X
iajs-2723	240	65	(	(	PUNCT
iajs-2723	240	66	86	86	NUM
iajs-2723	240	67	)	)	PUNCT
iajs-2723	240	68	by	by	ADP
iajs-2723	240	69	adding	add	VERB
iajs-2723	240	70	(	(	PUNCT
iajs-2723	240	71	85	85	NUM
iajs-2723	240	72	)	)	PUNCT
iajs-2723	240	73	and	and	CCONJ
iajs-2723	240	74	(	(	PUNCT
iajs-2723	240	75	86	86	NUM
iajs-2723	240	76	)	)	PUNCT
iajs-2723	240	77	,	,	PUNCT
iajs-2723	240	78	we	we	PRON
iajs-2723	240	79	have	have	VERB
iajs-2723	240	80	:	:	PUNCT
iajs-2723	240	81	(	(	PUNCT
iajs-2723	240	82	𝛿1	𝛿1	NOUN
iajs-2723	240	83	+	+	CCONJ
iajs-2723	240	84	𝛿2)(𝑢𝑣𝑢	𝛿2)(𝑢𝑣𝑢	ADJ
iajs-2723	240	85	)	)	PUNCT
iajs-2723	240	86	=	=	PUNCT
iajs-2723	240	87	(	(	PUNCT
iajs-2723	240	88	𝛿1	𝛿1	NOUN
iajs-2723	240	89	+	+	CCONJ
iajs-2723	240	90	𝛿2)(𝑢)𝜃1(𝑣𝑢	𝛿2)(𝑢)𝜃1(𝑣𝑢	NUM
iajs-2723	240	91	)	)	PUNCT
iajs-2723	241	1	+	+	CCONJ
iajs-2723	241	2	𝜃2(𝑢)(𝛿1	𝜃2(𝑢)(𝛿1	SYM
iajs-2723	242	1	+	+	NUM
iajs-2723	242	2	𝛿2)(𝑣)𝜃1(𝑢	𝛿2)(𝑣)𝜃1(𝑢	NUM
iajs-2723	242	3	)	)	PUNCT
iajs-2723	243	1	+	+	CCONJ
iajs-2723	243	2	𝜃2(𝑢𝑣)(𝛿1	𝜃2(𝑢𝑣)(𝛿1	NOUN
iajs-2723	243	3	+	+	CCONJ
iajs-2723	243	4	𝛿2)(𝑢	𝛿2)(𝑢	NOUN
iajs-2723	243	5	)	)	PUNCT
iajs-2723	243	6	thus	thus	ADV
iajs-2723	243	7	,	,	PUNCT
iajs-2723	243	8	𝛿1	𝛿1	NOUN
iajs-2723	243	9	+	+	CCONJ
iajs-2723	243	10	𝛿2	𝛿2	NOUN
iajs-2723	243	11	is	be	AUX
iajs-2723	243	12	a	a	DET
iajs-2723	243	13	(	(	PUNCT
iajs-2723	243	14	𝜃1	𝜃1	NOUN
iajs-2723	243	15	,	,	PUNCT
iajs-2723	243	16	𝜃2)-derivation	𝜃2)-derivation	NUM
iajs-2723	243	17	.	.	PUNCT
iajs-2723	244	1	by	by	ADP
iajs-2723	244	2	a	a	DET
iajs-2723	244	3	similar	similar	ADJ
iajs-2723	244	4	way	way	NOUN
iajs-2723	244	5	to	to	PART
iajs-2723	244	6	prove	prove	VERB
iajs-2723	244	7	(	(	PUNCT
iajs-2723	244	8	2	2	NUM
iajs-2723	244	9	)	)	PUNCT
iajs-2723	244	10	.	.	PUNCT
iajs-2723	245	1	∎	∎	PROPN
iajs-2723	245	2	4	4	NUM
iajs-2723	245	3	.	.	X
iajs-2723	245	4	conclusion	conclusion	NOUN
iajs-2723	245	5	as	as	ADP
iajs-2723	245	6	a	a	DET
iajs-2723	245	7	conclusion	conclusion	NOUN
iajs-2723	245	8	,	,	PUNCT
iajs-2723	245	9	this	this	DET
iajs-2723	245	10	article	article	NOUN
iajs-2723	245	11	presented	present	VERB
iajs-2723	245	12	the	the	DET
iajs-2723	245	13	notion	notion	NOUN
iajs-2723	245	14	of	of	ADP
iajs-2723	245	15	(	(	PUNCT
iajs-2723	245	16	𝜃1	𝜃1	VERB
iajs-2723	245	17	,	,	PUNCT
iajs-2723	245	18	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	245	19	pair	pair	NOUN
iajs-2723	245	20	with	with	ADP
iajs-2723	245	21	some	some	PRON
iajs-2723	245	22	of	of	ADP
iajs-2723	245	23	its	its	PRON
iajs-2723	245	24	properties	property	NOUN
iajs-2723	245	25	.	.	PUNCT
iajs-2723	246	1	this	this	DET
iajs-2723	246	2	study	study	NOUN
iajs-2723	246	3	displayed	display	VERB
iajs-2723	246	4	that	that	SCONJ
iajs-2723	246	5	the	the	DET
iajs-2723	246	6	sum	sum	NOUN
iajs-2723	246	7	of	of	ADP
iajs-2723	246	8	two	two	NUM
iajs-2723	246	9	(	(	PUNCT
iajs-2723	246	10	𝜃1	𝜃1	NOUN
iajs-2723	246	11	,	,	PUNCT
iajs-2723	246	12	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	246	13	pair	pair	NOUN
iajs-2723	246	14	is	be	AUX
iajs-2723	246	15	a	a	DET
iajs-2723	246	16	(	(	PUNCT
iajs-2723	246	17	𝜃1	𝜃1	NOUN
iajs-2723	246	18	,	,	PUNCT
iajs-2723	246	19	𝜃2)derivation	𝜃2)derivation	NOUN
iajs-2723	246	20	and	and	CCONJ
iajs-2723	246	21	the	the	DET
iajs-2723	246	22	sum	sum	NOUN
iajs-2723	246	23	of	of	ADP
iajs-2723	246	24	two	two	NUM
iajs-2723	246	25	jordan	jordan	PROPN
iajs-2723	246	26	(	(	PUNCT
iajs-2723	246	27	𝜃1	𝜃1	PROPN
iajs-2723	246	28	,	,	PUNCT
iajs-2723	246	29	𝜃2)-derivation	𝜃2)-derivation	NOUN
iajs-2723	246	30	pair	pair	NOUN
iajs-2723	246	31	is	be	AUX
iajs-2723	246	32	a	a	DET
iajs-2723	246	33	jordan	jordan	PROPN
iajs-2723	246	34	(	(	PUNCT
iajs-2723	246	35	𝜃1	𝜃1	PROPN
iajs-2723	246	36	,	,	PUNCT
iajs-2723	246	37	𝜃2)-derivation	𝜃2)-derivation	NUM
iajs-2723	246	38	.	.	PUNCT
iajs-2723	247	1	references	reference	NOUN
iajs-2723	247	2	1	1	NUM
iajs-2723	247	3	.	.	PUNCT
iajs-2723	248	1	zalar	zalar	PROPN
iajs-2723	248	2	,	,	PUNCT
iajs-2723	248	3	b.	b.	PROPN
iajs-2723	248	4	jordan	jordan	PROPN
iajs-2723	248	5	-	-	PUNCT
iajs-2723	248	6	von	von	PROPN
iajs-2723	248	7	neumann	neumann	PROPN
iajs-2723	248	8	theorem	theorem	PROPN
iajs-2723	248	9	for	for	ADP
iajs-2723	248	10	saworotnow	saworotnow	PROPN
iajs-2723	248	11	's	's	PART
iajs-2723	248	12	generalized	generalized	ADJ
iajs-2723	248	13	hilbert	hilbert	NOUN
iajs-2723	248	14	space	space	NOUN
iajs-2723	248	15	,	,	PUNCT
iajs-2723	248	16	acta	acta	PROPN
iajs-2723	248	17	math	math	PROPN
iajs-2723	248	18	.	.	PUNCT
iajs-2723	249	1	hungar	hungar	NOUN
iajs-2723	249	2	.	.	PUNCT
iajs-2723	250	1	1995,69	1995,69	NUM
iajs-2723	250	2	:	:	PUNCT
iajs-2723	250	3	301	301	NUM
iajs-2723	250	4	-	-	SYM
iajs-2723	250	5	325	325	NUM
iajs-2723	250	6	.	.	NOUN
iajs-2723	251	1	2	2	NUM
iajs-2723	251	2	.	.	X
iajs-2723	251	3	abujabal	abujabal	PROPN
iajs-2723	251	4	,	,	PUNCT
iajs-2723	251	5	h.	h.	PROPN
iajs-2723	251	6	a.	a.	PROPN
iajs-2723	251	7	;	;	PUNCT
iajs-2723	251	8	al	al	PROPN
iajs-2723	251	9	-	-	PUNCT
iajs-2723	251	10	shehri	shehri	PROPN
iajs-2723	251	11	n.	n.	PROPN
iajs-2723	251	12	some	some	DET
iajs-2723	251	13	results	result	NOUN
iajs-2723	251	14	on	on	ADP
iajs-2723	251	15	derivations	derivation	NOUN
iajs-2723	251	16	of	of	ADP
iajs-2723	251	17	bci	bci	PROPN
iajs-2723	251	18	-	-	PUNCT
iajs-2723	251	19	algebras	algebra	NOUN
iajs-2723	251	20	,	,	PUNCT
iajs-2723	251	21	journal	journal	NOUN
iajs-2723	251	22	of	of	ADP
iajs-2723	251	23	natural	natural	ADJ
iajs-2723	251	24	sciences	science	NOUN
iajs-2723	251	25	and	and	CCONJ
iajs-2723	251	26	mathematics	mathematic	NOUN
iajs-2723	251	27	,	,	PUNCT
iajs-2723	251	28	2006	2006	NUM
iajs-2723	251	29	,	,	PUNCT
iajs-2723	251	30	46(1	46(1	PROPN
iajs-2723	251	31	-	-	SYM
iajs-2723	251	32	2):13	2):13	NUM
iajs-2723	251	33	-	-	SYM
iajs-2723	251	34	19	19	NUM
iajs-2723	251	35	.	.	NOUN
iajs-2723	252	1	3	3	NUM
iajs-2723	252	2	.	.	X
iajs-2723	252	3	prabpayak	prabpayak	NOUN
iajs-2723	252	4	,	,	PUNCT
iajs-2723	252	5	c.	c.	PROPN
iajs-2723	252	6	;	;	PUNCT
iajs-2723	252	7	leerawat	leerawat	NOUN
iajs-2723	252	8	,	,	PUNCT
iajs-2723	252	9	u.	u.	NOUN
iajs-2723	252	10	on	on	ADP
iajs-2723	252	11	derivations	derivation	NOUN
iajs-2723	252	12	of	of	ADP
iajs-2723	252	13	bcc	bcc	PROPN
iajs-2723	252	14	-	-	PUNCT
iajs-2723	252	15	algebras	algebras	PROPN
iajs-2723	252	16	,	,	PUNCT
iajs-2723	252	17	kasetsart	kasetsart	PROPN
iajs-2723	252	18	journal	journal	PROPN
iajs-2723	252	19	,	,	PUNCT
iajs-2723	252	20	2009	2009	NUM
iajs-2723	252	21	,	,	PUNCT
iajs-2723	252	22	43(2	43(2	NUM
iajs-2723	252	23	)	)	PUNCT
iajs-2723	252	24	:	:	PUNCT
iajs-2723	252	25	398	398	NUM
iajs-2723	252	26	-	-	SYM
iajs-2723	252	27	401	401	NUM
iajs-2723	252	28	.	.	PUNCT
iajs-2723	253	1	4	4	NUM
iajs-2723	253	2	.	.	X
iajs-2723	254	1	al	al	PROPN
iajs-2723	254	2	-	-	PUNCT
iajs-2723	254	3	shehrie	shehrie	PROPN
iajs-2723	254	4	,	,	PUNCT
iajs-2723	254	5	n.	n.	NOUN
iajs-2723	254	6	derivations	derivation	NOUN
iajs-2723	254	7	of	of	ADP
iajs-2723	254	8	b	b	NOUN
iajs-2723	254	9	-	-	PUNCT
iajs-2723	254	10	algebras	algebras	PROPN
iajs-2723	254	11	,	,	PUNCT
iajs-2723	254	12	journal	journal	NOUN
iajs-2723	254	13	of	of	ADP
iajs-2723	254	14	king	king	PROPN
iajs-2723	254	15	abdulaziz	abdulaziz	PROPN
iajs-2723	254	16	university	university	PROPN
iajs-2723	254	17	-	-	PUNCT
iajs-2723	254	18	science	science	NOUN
iajs-2723	254	19	,	,	PUNCT
iajs-2723	254	20	2010	2010	NUM
iajs-2723	254	21	,	,	PUNCT
iajs-2723	254	22	22(1):71	22(1):71	PROPN
iajs-2723	254	23	-	-	PUNCT
iajs-2723	254	24	83	83	NUM
iajs-2723	254	25	.	.	PUNCT
iajs-2723	255	1	5	5	NUM
iajs-2723	255	2	.	.	X
iajs-2723	255	3	al	al	PROPN
iajs-2723	255	4	-	-	PUNCT
iajs-2723	255	5	kadi	kadi	PROPN
iajs-2723	255	6	,	,	PUNCT
iajs-2723	255	7	d.	d.	PROPN
iajs-2723	255	8	𝑓𝑞-derivations	𝑓𝑞-derivation	NOUN
iajs-2723	255	9	of	of	ADP
iajs-2723	255	10	𝐺-algebra	𝐺-algebra	PROPN
iajs-2723	255	11	,	,	PUNCT
iajs-2723	255	12	international	international	ADJ
iajs-2723	255	13	journal	journal	NOUN
iajs-2723	255	14	of	of	ADP
iajs-2723	255	15	mathematics	mathematics	PROPN
iajs-2723	255	16	and	and	CCONJ
iajs-2723	255	17	mathematical	mathematical	ADJ
iajs-2723	255	18	sciences	science	NOUN
iajs-2723	255	19	,	,	PUNCT
iajs-2723	255	20	2016	2016	NUM
iajs-2723	255	21	:1	:1	PUNCT
iajs-2723	255	22	-	-	PUNCT
iajs-2723	255	23	5	5	NUM
iajs-2723	255	24	.	.	NOUN
iajs-2723	255	25	6	6	NUM
iajs-2723	255	26	.	.	X
iajs-2723	255	27	yass	yass	PROPN
iajs-2723	255	28	,	,	PUNCT
iajs-2723	255	29	s.	s.	PROPN
iajs-2723	255	30	strongly	strongly	ADV
iajs-2723	255	31	derivation	derivation	NOUN
iajs-2723	255	32	pairs	pair	NOUN
iajs-2723	255	33	on	on	ADP
iajs-2723	255	34	prime	prime	ADJ
iajs-2723	255	35	and	and	CCONJ
iajs-2723	255	36	seniprime	seniprime	NOUN
iajs-2723	255	37	rings	ring	NOUN
iajs-2723	255	38	.	.	PUNCT
iajs-2723	256	1	msc	msc	PROPN
iajs-2723	256	2	.	.	PROPN
iajs-2723	257	1	thesis	thesis	PROPN
iajs-2723	257	2	,	,	PUNCT
iajs-2723	257	3	university	university	NOUN
iajs-2723	257	4	of	of	ADP
iajs-2723	257	5	baghdad	baghdad	PROPN
iajs-2723	257	6	,	,	PUNCT
iajs-2723	257	7	2010	2010	NUM
iajs-2723	257	8	.	.	PUNCT
iajs-2723	258	1	7	7	X
iajs-2723	258	2	.	.	X
iajs-2723	258	3	kumar	kumar	PROPN
iajs-2723	258	4	,	,	PUNCT
iajs-2723	258	5	d.	d.	PROPN
iajs-2723	258	6	;	;	PUNCT
iajs-2723	258	7	sandhu	sandhu	PROPN
iajs-2723	258	8	,	,	PUNCT
iajs-2723	258	9	g.	g.	PROPN
iajs-2723	258	10	on	on	ADP
iajs-2723	258	11	multiplicative	multiplicative	PROPN
iajs-2723	258	12	(	(	PUNCT
iajs-2723	258	13	generalized)-derivations	generalized)-derivation	NOUN
iajs-2723	258	14	in	in	ADP
iajs-2723	258	15	semiprime	semiprime	NOUN
iajs-2723	258	16	rings	ring	NOUN
iajs-2723	258	17	,	,	PUNCT
iajs-2723	258	18	,	,	PUNCT
iajs-2723	258	19	international	international	ADJ
iajs-2723	258	20	journal	journal	NOUN
iajs-2723	258	21	of	of	ADP
iajs-2723	258	22	pure	pure	ADJ
iajs-2723	258	23	and	and	CCONJ
iajs-2723	258	24	applied	applied	ADJ
iajs-2723	258	25	mathematics	mathematic	NOUN
iajs-2723	258	26	,	,	PUNCT
iajs-2723	258	27	2016	2016	NUM
iajs-2723	258	28	,	,	PUNCT
iajs-2723	258	29	106(1):249	106(1):249	NUM
iajs-2723	258	30	-	-	SYM
iajs-2723	258	31	257	257	NUM
iajs-2723	258	32	.	.	NOUN
iajs-2723	258	33	8	8	NUM
iajs-2723	258	34	.	.	X
iajs-2723	259	1	samman	samman	PROPN
iajs-2723	259	2	,	,	PUNCT
iajs-2723	259	3	m.	m.	NOUN
iajs-2723	259	4	;	;	PUNCT
iajs-2723	259	5	alyamani	alyamani	PROPN
iajs-2723	259	6	,	,	PUNCT
iajs-2723	259	7	n.	n.	NOUN
iajs-2723	259	8	derivations	derivation	NOUN
iajs-2723	259	9	and	and	CCONJ
iajs-2723	259	10	reverse	reverse	ADJ
iajs-2723	259	11	derivations	derivation	NOUN
iajs-2723	259	12	in	in	ADP
iajs-2723	259	13	semiprime	semiprime	NOUN
iajs-2723	259	14	rings	ring	NOUN
iajs-2723	259	15	,	,	PUNCT
iajs-2723	259	16	international	international	PROPN
iajs-2723	259	17	mathematical	mathematical	ADJ
iajs-2723	259	18	forum	forum	PROPN
iajs-2723	259	19	,	,	PUNCT
iajs-2723	259	20	2007	2007	NUM
iajs-2723	259	21	,	,	PUNCT
iajs-2723	259	22	2(39):1895	2(39):1895	NUM
iajs-2723	259	23	-	-	SYM
iajs-2723	259	24	1902	1902	NUM
iajs-2723	259	25	.	.	PUNCT
iajs-2723	260	1	9	9	X
iajs-2723	260	2	.	.	X
iajs-2723	260	3	herstein	herstein	NOUN
iajs-2723	260	4	,	,	PUNCT
iajs-2723	260	5	i.	i.	NOUN
iajs-2723	260	6	topics	topic	NOUN
iajs-2723	260	7	in	in	ADP
iajs-2723	260	8	ring	ring	NOUN
iajs-2723	260	9	theory	theory	NOUN
iajs-2723	260	10	,	,	PUNCT
iajs-2723	260	11	university	university	PROPN
iajs-2723	260	12	of	of	ADP
iajs-2723	260	13	chicago	chicago	PROPN
iajs-2723	260	14	press	press	PROPN
iajs-2723	260	15	,	,	PUNCT
iajs-2723	260	16	chicago	chicago	PROPN
iajs-2723	260	17	,	,	PUNCT
iajs-2723	260	18	1969	1969	NUM
iajs-2723	260	19	.	.	PUNCT
iajs-2723	261	1	10	10	NUM
iajs-2723	261	2	.	.	X
iajs-2723	261	3	ashraf	ashraf	PROPN
iajs-2723	261	4	,	,	PUNCT
iajs-2723	261	5	m	m	PROPN
iajs-2723	261	6	,	,	PUNCT
iajs-2723	261	7	;	;	PUNCT
iajs-2723	261	8	ali	ali	PROPN
iajs-2723	261	9	.	.	PROPN
iajs-2723	261	10	s	s	PART
iajs-2723	261	11	,	,	PUNCT
iajs-2723	261	12	;	;	PUNCT
iajs-2723	261	13	haetinger	haetinger	X
iajs-2723	261	14	,	,	PUNCT
iajs-2723	261	15	c.	c.	NOUN
iajs-2723	261	16	on	on	ADP
iajs-2723	261	17	derivations	derivation	NOUN
iajs-2723	261	18	in	in	ADP
iajs-2723	261	19	rings	ring	NOUN
iajs-2723	261	20	and	and	CCONJ
iajs-2723	261	21	their	their	PRON
iajs-2723	261	22	applications	application	NOUN
iajs-2723	261	23	,	,	PUNCT
iajs-2723	261	24	aligarh	aligarh	NOUN
iajs-2723	261	25	bull	bull	PROPN
iajs-2723	261	26	.	.	PUNCT
iajs-2723	262	1	of	of	ADP
iajs-2723	262	2	mathematics	mathematic	NOUN
iajs-2723	262	3	,	,	PUNCT
iajs-2723	262	4	2006	2006	NUM
iajs-2723	262	5	,	,	PUNCT
iajs-2723	262	6	25(2):79	25(2):79	NUM
iajs-2723	262	7	-	-	SYM
iajs-2723	262	8	107	107	NUM
iajs-2723	262	9	.	.	PUNCT
iajs-2723	263	1	11	11	NUM
iajs-2723	263	2	.	.	X
iajs-2723	264	1	jackson	jackson	PROPN
iajs-2723	264	2	,	,	PUNCT
iajs-2723	264	3	n.	n.	VERB
iajs-2723	264	4	a	a	DET
iajs-2723	264	5	first	first	ADJ
iajs-2723	264	6	course	course	NOUN
iajs-2723	264	7	in	in	ADP
iajs-2723	264	8	abstract	abstract	ADJ
iajs-2723	264	9	algebra	algebra	NOUN
iajs-2723	264	10	,	,	PUNCT
iajs-2723	264	11	2016	2016	NUM
iajs-2723	264	12	.	.	PUNCT
iajs-2723	265	1	12	12	NUM
iajs-2723	265	2	.	.	PUNCT
iajs-2723	265	3	majeed	majeed	PROPN
iajs-2723	265	4	,	,	PUNCT
iajs-2723	265	5	a.	a.	PROPN
iajs-2723	265	6	h.	h.	PROPN
iajs-2723	265	7	;	;	PUNCT
iajs-2723	265	8	altay	altay	NOUN
iajs-2723	265	9	,	,	PUNCT
iajs-2723	265	10	a.	a.	NOUN
iajs-2723	265	11	a.	a.	NOUN
iajs-2723	265	12	on	on	ADP
iajs-2723	265	13	jordan	jordan	PROPN
iajs-2723	265	14	derivation	derivation	NOUN
iajs-2723	265	15	pairs	pair	NOUN
iajs-2723	265	16	in	in	ADP
iajs-2723	265	17	rings	ring	NOUN
iajs-2723	265	18	,	,	PUNCT
iajs-2723	265	19	iraqi	iraqi	ADJ
iajs-2723	265	20	j.	j.	PROPN
iajs-2723	265	21	of	of	ADP
iajs-2723	265	22	science.(to	science.(to	NOUN
iajs-2723	265	23	appear	appear	VERB
iajs-2723	265	24	)	)	PUNCT
iajs-2723	265	25	.	.	PUNCT