id	sid	tid	token	lemma	pos
iajs-2920	1	1	ihjpas	ihjpas	PROPN
iajs-2920	1	2	.	.	PUNCT
iajs-2920	2	1	36(1)2023	36(1)2023	NUM
iajs-2920	2	2	367	367	NUM
iajs-2920	2	3	this	this	DET
iajs-2920	2	4	work	work	NOUN
iajs-2920	2	5	is	be	AUX
iajs-2920	2	6	licensed	license	VERB
iajs-2920	2	7	under	under	ADP
iajs-2920	2	8	a	a	DET
iajs-2920	2	9	creative	creative	ADJ
iajs-2920	2	10	commons	common	NOUN
iajs-2920	2	11	attribution	attribution	NOUN
iajs-2920	2	12	4.0	4.0	NUM
iajs-2920	2	13	international	international	ADJ
iajs-2920	2	14	license	license	NOUN
iajs-2920	2	15	cubic	cubic	ADJ
iajs-2920	2	16	ideals	ideal	NOUN
iajs-2920	2	17	of	of	ADP
iajs-2920	2	18	tm	tm	NOUN
iajs-2920	2	19	-	-	PUNCT
iajs-2920	2	20	algebras	algebras	ADJ
iajs-2920	2	21	abstract	abstract	NOUN
iajs-2920	2	22	for	for	ADP
iajs-2920	2	23	the	the	DET
iajs-2920	2	24	generality	generality	NOUN
iajs-2920	2	25	of	of	ADP
iajs-2920	2	26	fuzzy	fuzzy	ADJ
iajs-2920	2	27	ideals	ideal	NOUN
iajs-2920	2	28	in	in	ADP
iajs-2920	2	29	tm	tm	NOUN
iajs-2920	2	30	-	-	NOUN
iajs-2920	2	31	algebra	algebra	NOUN
iajs-2920	2	32	,	,	PUNCT
iajs-2920	2	33	a	a	DET
iajs-2920	2	34	cubic	cubic	ADJ
iajs-2920	2	35	ideal	ideal	NOUN
iajs-2920	2	36	in	in	ADP
iajs-2920	2	37	this	this	DET
iajs-2920	2	38	algebra	algebra	NOUN
iajs-2920	2	39	has	have	AUX
iajs-2920	2	40	been	be	AUX
iajs-2920	2	41	studied	study	VERB
iajs-2920	2	42	,	,	PUNCT
iajs-2920	2	43	such	such	ADJ
iajs-2920	2	44	as	as	ADP
iajs-2920	2	45	cubic	cubic	ADJ
iajs-2920	2	46	ideals	ideal	NOUN
iajs-2920	2	47	and	and	CCONJ
iajs-2920	2	48	cubic	cubic	ADJ
iajs-2920	2	49	t	t	PROPN
iajs-2920	2	50	-	-	PUNCT
iajs-2920	2	51	ideals	ideal	NOUN
iajs-2920	2	52	.	.	PUNCT
iajs-2920	3	1	some	some	DET
iajs-2920	3	2	properties	property	NOUN
iajs-2920	3	3	of	of	ADP
iajs-2920	3	4	these	these	DET
iajs-2920	3	5	ideals	ideal	NOUN
iajs-2920	3	6	are	be	AUX
iajs-2920	3	7	investigated	investigate	VERB
iajs-2920	3	8	.	.	PUNCT
iajs-2920	4	1	also	also	ADV
iajs-2920	4	2	,	,	PUNCT
iajs-2920	4	3	we	we	PRON
iajs-2920	4	4	show	show	VERB
iajs-2920	4	5	that	that	SCONJ
iajs-2920	4	6	the	the	DET
iajs-2920	4	7	cubic	cubic	ADJ
iajs-2920	4	8	t	t	PROPN
iajs-2920	4	9	-	-	PUNCT
iajs-2920	4	10	ideal	ideal	NOUN
iajs-2920	4	11	is	be	AUX
iajs-2920	4	12	a	a	DET
iajs-2920	4	13	cubic	cubic	ADJ
iajs-2920	4	14	ideal	ideal	NOUN
iajs-2920	4	15	,	,	PUNCT
iajs-2920	4	16	but	but	CCONJ
iajs-2920	4	17	the	the	DET
iajs-2920	4	18	converse	converse	NOUN
iajs-2920	4	19	is	be	AUX
iajs-2920	4	20	not	not	PART
iajs-2920	4	21	generally	generally	ADV
iajs-2920	4	22	valid	valid	ADJ
iajs-2920	4	23	.	.	PUNCT
iajs-2920	5	1	in	in	ADP
iajs-2920	5	2	addition	addition	NOUN
iajs-2920	5	3	,	,	PUNCT
iajs-2920	5	4	a	a	DET
iajs-2920	5	5	cubic	cubic	ADJ
iajs-2920	5	6	sub	sub	NOUN
iajs-2920	5	7	-	-	NOUN
iajs-2920	5	8	algebra	algebra	NOUN
iajs-2920	5	9	is	be	AUX
iajs-2920	5	10	defined	define	VERB
iajs-2920	5	11	,	,	PUNCT
iajs-2920	5	12	and	and	CCONJ
iajs-2920	5	13	new	new	ADJ
iajs-2920	5	14	relations	relation	NOUN
iajs-2920	5	15	between	between	ADP
iajs-2920	5	16	the	the	DET
iajs-2920	5	17	level	level	NOUN
iajs-2920	5	18	subset	subset	NOUN
iajs-2920	5	19	and	and	CCONJ
iajs-2920	5	20	a	a	DET
iajs-2920	5	21	cubic	cubic	ADJ
iajs-2920	5	22	sub	sub	NOUN
iajs-2920	5	23	-	-	NOUN
iajs-2920	5	24	algebra	algebra	NOUN
iajs-2920	5	25	are	be	AUX
iajs-2920	5	26	discussed	discuss	VERB
iajs-2920	5	27	.	.	PUNCT
iajs-2920	6	1	after	after	ADP
iajs-2920	6	2	that	that	PRON
iajs-2920	6	3	,	,	PUNCT
iajs-2920	6	4	cubic	cubic	ADJ
iajs-2920	6	5	ideals	ideal	NOUN
iajs-2920	6	6	and	and	CCONJ
iajs-2920	6	7	cubic	cubic	ADJ
iajs-2920	6	8	t	t	PROPN
iajs-2920	6	9	-	-	PUNCT
iajs-2920	6	10	ideals	ideal	NOUN
iajs-2920	6	11	under	under	ADP
iajs-2920	6	12	homomorphism	homomorphism	NOUN
iajs-2920	6	13	are	be	AUX
iajs-2920	6	14	studied	study	VERB
iajs-2920	6	15	,	,	PUNCT
iajs-2920	6	16	and	and	CCONJ
iajs-2920	6	17	the	the	DET
iajs-2920	6	18	image	image	NOUN
iajs-2920	6	19	(	(	PUNCT
iajs-2920	6	20	pre	pre	NOUN
iajs-2920	6	21	-	-	NOUN
iajs-2920	6	22	image	image	NOUN
iajs-2920	6	23	)	)	PUNCT
iajs-2920	6	24	of	of	ADP
iajs-2920	6	25	cubic	cubic	ADJ
iajs-2920	6	26	t	t	PROPN
iajs-2920	6	27	-	-	PUNCT
iajs-2920	6	28	ideals	ideal	NOUN
iajs-2920	6	29	is	be	AUX
iajs-2920	6	30	discussed	discuss	VERB
iajs-2920	6	31	.	.	PUNCT
iajs-2920	7	1	finally	finally	ADV
iajs-2920	7	2	,	,	PUNCT
iajs-2920	7	3	the	the	DET
iajs-2920	7	4	cartesian	cartesian	ADJ
iajs-2920	7	5	product	product	NOUN
iajs-2920	7	6	of	of	ADP
iajs-2920	7	7	cubic	cubic	ADJ
iajs-2920	7	8	ideals	ideal	NOUN
iajs-2920	7	9	in	in	ADP
iajs-2920	7	10	cartesian	cartesian	ADJ
iajs-2920	7	11	product	product	NOUN
iajs-2920	7	12	tm	tm	NOUN
iajs-2920	7	13	-	-	PUNCT
iajs-2920	7	14	algebras	algebras	PROPN
iajs-2920	7	15	is	be	AUX
iajs-2920	7	16	given	give	VERB
iajs-2920	7	17	.	.	PUNCT
iajs-2920	8	1	we	we	PRON
iajs-2920	8	2	proved	prove	VERB
iajs-2920	8	3	that	that	SCONJ
iajs-2920	8	4	the	the	DET
iajs-2920	8	5	product	product	NOUN
iajs-2920	8	6	of	of	ADP
iajs-2920	8	7	two	two	NUM
iajs-2920	8	8	cubic	cubic	ADJ
iajs-2920	8	9	ideals	ideal	NOUN
iajs-2920	8	10	of	of	ADP
iajs-2920	8	11	the	the	DET
iajs-2920	8	12	cartesian	cartesian	ADJ
iajs-2920	8	13	product	product	NOUN
iajs-2920	8	14	of	of	ADP
iajs-2920	8	15	two	two	NUM
iajs-2920	8	16	tm	tm	NOUN
iajs-2920	8	17	-	-	PUNCT
iajs-2920	8	18	algebras	algebras	PROPN
iajs-2920	8	19	is	be	AUX
iajs-2920	8	20	also	also	ADV
iajs-2920	8	21	a	a	DET
iajs-2920	8	22	cubic	cubic	ADJ
iajs-2920	8	23	ideal	ideal	NOUN
iajs-2920	8	24	.	.	PUNCT
iajs-2920	9	1	key	key	ADJ
iajs-2920	9	2	words	word	NOUN
iajs-2920	9	3	:	:	PUNCT
iajs-2920	9	4	tm	tm	NOUN
iajs-2920	9	5	-	-	NOUN
iajs-2920	9	6	algebra	algebra	PROPN
iajs-2920	9	7	,	,	PUNCT
iajs-2920	9	8	cubic	cubic	ADJ
iajs-2920	9	9	t	t	PROPN
iajs-2920	9	10	-	-	PUNCT
iajs-2920	9	11	ideal	ideal	ADJ
iajs-2920	9	12	,	,	PUNCT
iajs-2920	9	13	cubic	cubic	ADJ
iajs-2920	9	14	ideal	ideal	ADJ
iajs-2920	9	15	,	,	PUNCT
iajs-2920	9	16	fuzzy	fuzzy	ADJ
iajs-2920	9	17	ideals	ideal	NOUN
iajs-2920	9	18	.	.	PUNCT
iajs-2920	10	1	1.introduction	1.introduction	NUM
iajs-2920	10	2	in	in	ADP
iajs-2920	10	3	2010	2010	NUM
iajs-2920	10	4	the	the	DET
iajs-2920	10	5	notion	notion	NOUN
iajs-2920	10	6	of	of	ADP
iajs-2920	10	7	tm	tm	PROPN
iajs-2920	10	8	-	-	PUNCT
iajs-2920	10	9	algebras	algebras	PROPN
iajs-2920	10	10	was	be	AUX
iajs-2920	10	11	introduced	introduce	VERB
iajs-2920	10	12	by	by	ADP
iajs-2920	10	13	[	[	X
iajs-2920	10	14	1	1	NUM
iajs-2920	10	15	]	]	PUNCT
iajs-2920	10	16	as	as	ADP
iajs-2920	10	17	a	a	DET
iajs-2920	10	18	generalization	generalization	NOUN
iajs-2920	10	19	of	of	ADP
iajs-2920	10	20	bck	bck	PROPN
iajs-2920	10	21	and	and	CCONJ
iajs-2920	10	22	bci	bci	PROPN
iajs-2920	10	23	algebras	algebra	NOUN
iajs-2920	10	24	.	.	PUNCT
iajs-2920	11	1	after	after	ADP
iajs-2920	11	2	that	that	PRON
iajs-2920	11	3	,	,	PUNCT
iajs-2920	11	4	many	many	ADJ
iajs-2920	11	5	authors	author	NOUN
iajs-2920	11	6	studied	study	VERB
iajs-2920	11	7	this	this	DET
iajs-2920	11	8	structure	structure	NOUN
iajs-2920	11	9	differently	differently	ADV
iajs-2920	11	10	;	;	PUNCT
iajs-2920	11	11	see	see	VERB
iajs-2920	11	12	[	[	X
iajs-2920	11	13	2	2	NUM
iajs-2920	11	14	-	-	SYM
iajs-2920	11	15	6	6	NUM
iajs-2920	11	16	]	]	PUNCT
iajs-2920	11	17	.	.	PUNCT
iajs-2920	12	1	the	the	DET
iajs-2920	12	2	cubic	cubic	ADJ
iajs-2920	12	3	set	set	NOUN
iajs-2920	12	4	is	be	AUX
iajs-2920	12	5	an	an	DET
iajs-2920	12	6	essential	essential	ADJ
iajs-2920	12	7	concept	concept	NOUN
iajs-2920	12	8	for	for	ADP
iajs-2920	12	9	generalizing	generalize	VERB
iajs-2920	12	10	the	the	DET
iajs-2920	12	11	fuzzy	fuzzy	ADJ
iajs-2920	12	12	set	set	NOUN
iajs-2920	12	13	.	.	PUNCT
iajs-2920	13	1	so	so	ADV
iajs-2920	13	2	,	,	PUNCT
iajs-2920	13	3	jun	jun	PROPN
iajs-2920	13	4	et	et	PROPN
iajs-2920	13	5	al	al	PROPN
iajs-2920	13	6	.	.	PUNCT
iajs-2920	14	1	[	[	X
iajs-2920	14	2	78	78	NUM
iajs-2920	14	3	]	]	PUNCT
iajs-2920	14	4	introduced	introduce	VERB
iajs-2920	14	5	subalgebras	subalgebra	NOUN
iajs-2920	14	6	and	and	CCONJ
iajs-2920	14	7	ideals	ideal	NOUN
iajs-2920	14	8	in	in	ADP
iajs-2920	14	9	bck	bck	PROPN
iajs-2920	14	10	/	/	SYM
iajs-2920	14	11	bci	bci	NOUN
iajs-2920	14	12	-	-	PUNCT
iajs-2920	14	13	algebras	algebras	PROPN
iajs-2920	14	14	and	and	CCONJ
iajs-2920	14	15	discussed	discuss	VERB
iajs-2920	14	16	the	the	DET
iajs-2920	14	17	relationship	relationship	NOUN
iajs-2920	14	18	between	between	ADP
iajs-2920	14	19	a	a	DET
iajs-2920	14	20	cubic	cubic	ADJ
iajs-2920	14	21	subalgebra	subalgebra	NOUN
iajs-2920	14	22	and	and	CCONJ
iajs-2920	14	23	a	a	DET
iajs-2920	14	24	cubic	cubic	ADJ
iajs-2920	14	25	ideal	ideal	NOUN
iajs-2920	14	26	.	.	PUNCT
iajs-2920	15	1	in	in	ADP
iajs-2920	15	2	[	[	X
iajs-2920	15	3	9	9	NUM
iajs-2920	15	4	]	]	PUNCT
iajs-2920	15	5	,	,	PUNCT
iajs-2920	15	6	yaqoob	yaqoob	NOUN
iajs-2920	15	7	et	et	NOUN
iajs-2920	15	8	al	al	PROPN
iajs-2920	15	9	.	.	PROPN
iajs-2920	15	10	introduced	introduce	VERB
iajs-2920	15	11	the	the	DET
iajs-2920	15	12	cubic	cubic	ADJ
iajs-2920	15	13	ku	ku	PROPN
iajs-2920	15	14	-	-	PUNCT
iajs-2920	15	15	algebra	algebra	PROPN
iajs-2920	15	16	,	,	PUNCT
iajs-2920	15	17	a	a	DET
iajs-2920	15	18	generalization	generalization	NOUN
iajs-2920	15	19	of	of	ADP
iajs-2920	15	20	fuzzy	fuzzy	ADJ
iajs-2920	15	21	ku	ku	NOUN
iajs-2920	15	22	-	-	PUNCT
iajs-2920	15	23	ideals	ideal	NOUN
iajs-2920	15	24	of	of	ADP
iajs-2920	15	25	ku	ku	PROPN
iajs-2920	15	26	-	-	PUNCT
iajs-2920	15	27	algebras	algebras	PROPN
iajs-2920	15	28	.	.	PUNCT
iajs-2920	16	1	after	after	ADP
iajs-2920	16	2	that	that	PRON
iajs-2920	16	3	,	,	PUNCT
iajs-2920	16	4	some	some	DET
iajs-2920	16	5	authors	author	NOUN
iajs-2920	16	6	introduced	introduce	VERB
iajs-2920	16	7	a	a	DET
iajs-2920	16	8	cubic	cubic	ADJ
iajs-2920	16	9	set	set	NOUN
iajs-2920	16	10	of	of	ADP
iajs-2920	16	11	different	different	ADJ
iajs-2920	16	12	structures	structure	NOUN
iajs-2920	16	13	.	.	PUNCT
iajs-2920	17	1	see	see	VERB
iajs-2920	17	2	[	[	X
iajs-2920	17	3	10	10	NUM
iajs-2920	17	4	-	-	SYM
iajs-2920	17	5	13	13	NUM
iajs-2920	17	6	]	]	PUNCT
iajs-2920	17	7	.	.	PUNCT
iajs-2920	18	1	this	this	DET
iajs-2920	18	2	paper	paper	NOUN
iajs-2920	18	3	introduces	introduce	VERB
iajs-2920	18	4	the	the	DET
iajs-2920	18	5	concept	concept	NOUN
iajs-2920	18	6	of	of	ADP
iajs-2920	18	7	cubic	cubic	ADJ
iajs-2920	18	8	t	t	PROPN
iajs-2920	18	9	-	-	PUNCT
iajs-2920	18	10	ideals	ideal	NOUN
iajs-2920	18	11	in	in	ADP
iajs-2920	18	12	tm	tm	NOUN
iajs-2920	18	13	-	-	NOUN
iajs-2920	18	14	algebra	algebra	NOUN
iajs-2920	18	15	,	,	PUNCT
iajs-2920	18	16	and	and	CCONJ
iajs-2920	18	17	investigate	investigate	VERB
iajs-2920	18	18	some	some	DET
iajs-2920	18	19	properties	property	NOUN
iajs-2920	18	20	of	of	ADP
iajs-2920	18	21	these	these	DET
iajs-2920	18	22	ideals	ideal	NOUN
iajs-2920	18	23	.	.	PUNCT
iajs-2920	19	1	also	also	ADV
iajs-2920	19	2	,	,	PUNCT
iajs-2920	19	3	a	a	DET
iajs-2920	19	4	few	few	ADJ
iajs-2920	19	5	relations	relation	NOUN
iajs-2920	19	6	between	between	ADP
iajs-2920	19	7	a	a	DET
iajs-2920	19	8	cubic	cubic	ADJ
iajs-2920	19	9	ideal	ideal	NOUN
iajs-2920	19	10	and	and	CCONJ
iajs-2920	19	11	a	a	DET
iajs-2920	19	12	cubic	cubic	ADJ
iajs-2920	19	13	t	t	PROPN
iajs-2920	19	14	-	-	PUNCT
iajs-2920	19	15	ideal	ideal	NOUN
iajs-2920	19	16	are	be	AUX
iajs-2920	19	17	discussed	discuss	VERB
iajs-2920	19	18	.	.	PUNCT
iajs-2920	20	1	the	the	DET
iajs-2920	20	2	cartesian	cartesian	ADJ
iajs-2920	20	3	product	product	NOUN
iajs-2920	20	4	of	of	ADP
iajs-2920	20	5	cubic	cubic	ADJ
iajs-2920	20	6	t	t	PROPN
iajs-2920	20	7	-	-	PUNCT
iajs-2920	20	8	ideals	ideal	NOUN
iajs-2920	20	9	in	in	ADP
iajs-2920	20	10	cartesian	cartesian	ADJ
iajs-2920	20	11	product	product	NOUN
iajs-2920	20	12	tm	tm	NOUN
iajs-2920	20	13	-	-	PUNCT
iajs-2920	20	14	algebras	algebras	PROPN
iajs-2920	20	15	is	be	AUX
iajs-2920	20	16	given	give	VERB
iajs-2920	20	17	.	.	PUNCT
iajs-2920	21	1	doi.org/10.30526/36.1.2920	doi.org/10.30526/36.1.2920	ADJ
iajs-2920	21	2	article	article	NOUN
iajs-2920	21	3	history	history	NOUN
iajs-2920	21	4	:	:	PUNCT
iajs-2920	21	5	received	receive	VERB
iajs-2920	21	6	26	26	NUM
iajs-2920	21	7	june	june	PROPN
iajs-2920	21	8	2022	2022	NUM
iajs-2920	21	9	,	,	PUNCT
iajs-2920	21	10	accepted	accept	VERB
iajs-2920	21	11	23	23	NUM
iajs-2920	21	12	augest	aug	ADJ
iajs-2920	21	13	2022	2022	NUM
iajs-2920	21	14	,	,	PUNCT
iajs-2920	21	15	published	publish	VERB
iajs-2920	21	16	in	in	ADP
iajs-2920	21	17	january	january	PROPN
iajs-2920	21	18	2023	2023	NUM
iajs-2920	21	19	.	.	PUNCT
iajs-2920	22	1	ibn	ibn	PROPN
iajs-2920	22	2	al	al	PROPN
iajs-2920	22	3	-	-	PUNCT
iajs-2920	22	4	haitham	haitham	PROPN
iajs-2920	22	5	journal	journal	PROPN
iajs-2920	22	6	for	for	ADP
iajs-2920	22	7	pure	pure	ADJ
iajs-2920	22	8	and	and	CCONJ
iajs-2920	22	9	applied	applied	ADJ
iajs-2920	22	10	sciences	sciences	PROPN
iajs-2920	22	11	journal	journal	PROPN
iajs-2920	22	12	homepage	homepage	NOUN
iajs-2920	22	13	:	:	PUNCT
iajs-2920	22	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	PROPN
iajs-2920	22	15	fatima	fatima	PROPN
iajs-2920	22	16	m.	m.	PROPN
iajs-2920	22	17	ghlaim	ghlaim	PROPN
iajs-2920	22	18	department	department	PROPN
iajs-2920	22	19	of	of	ADP
iajs-2920	22	20	mathematics	mathematics	PROPN
iajs-2920	22	21	,	,	PUNCT
iajs-2920	22	22	college	college	NOUN
iajs-2920	22	23	of	of	ADP
iajs-2920	22	24	education	education	NOUN
iajs-2920	22	25	for	for	ADP
iajs-2920	22	26	pure	pure	ADJ
iajs-2920	22	27	sciences	science	NOUN
iajs-2920	22	28	,	,	PUNCT
iajs-2920	22	29	ibn	ibn	PROPN
iajs-2920	22	30	al	al	PROPN
iajs-2920	22	31	–	–	PUNCT
iajs-2920	22	32	haitham/	haitham/	NUM
iajs-2920	22	33	university	university	NOUN
iajs-2920	22	34	of	of	ADP
iajs-2920	22	35	baghdadiraq	baghdadiraq	PROPN
iajs-2920	22	36	.	.	PUNCT
iajs-2920	23	1	fa560fa@gmail.com	fa560fa@gmail.com	PROPN
iajs-2920	23	2	fatema	fatema	PROPN
iajs-2920	23	3	f.kareem	f.kareem	PROPN
iajs-2920	23	4	department	department	PROPN
iajs-2920	23	5	of	of	ADP
iajs-2920	23	6	mathematics	mathematics	PROPN
iajs-2920	23	7	,	,	PUNCT
iajs-2920	23	8	college	college	NOUN
iajs-2920	23	9	of	of	ADP
iajs-2920	23	10	education	education	NOUN
iajs-2920	23	11	for	for	ADP
iajs-2920	23	12	pure	pure	ADJ
iajs-2920	23	13	sciences	science	NOUN
iajs-2920	23	14	,	,	PUNCT
iajs-2920	23	15	ibn	ibn	PROPN
iajs-2920	23	16	al	al	PROPN
iajs-2920	23	17	–	–	PUNCT
iajs-2920	23	18	haitham/	haitham/	NUM
iajs-2920	23	19	university	university	NOUN
iajs-2920	23	20	of	of	ADP
iajs-2920	23	21	baghdadiraq	baghdadiraq	PROPN
iajs-2920	23	22	.	.	PUNCT
iajs-2920	24	1	fatma.f.k@ihcoedu.uobaghdad.edu.iq	fatma.f.k@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-2920	24	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2920	24	3	mailto:fa560fa@gmail.com	mailto:fa560fa@gmail.com	PROPN
iajs-2920	24	4	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-2920	24	5	ihjpas	ihjpa	NOUN
iajs-2920	24	6	.	.	PUNCT
iajs-2920	25	1	36(1)2023	36(1)2023	NUM
iajs-2920	25	2	368	368	NUM
iajs-2920	25	3	2	2	NUM
iajs-2920	25	4	.	.	PUNCT
iajs-2920	25	5	basic	basic	ADJ
iajs-2920	25	6	concepts	concept	NOUN
iajs-2920	25	7	we	we	PRON
iajs-2920	25	8	will	will	AUX
iajs-2920	25	9	recall	recall	VERB
iajs-2920	25	10	some	some	DET
iajs-2920	25	11	concepts	concept	NOUN
iajs-2920	25	12	related	relate	VERB
iajs-2920	25	13	to	to	ADP
iajs-2920	25	14	tm	tm	PROPN
iajs-2920	25	15	algebra	algebra	PROPN
iajs-2920	25	16	and	and	CCONJ
iajs-2920	25	17	cubic	cubic	ADJ
iajs-2920	25	18	sets	set	NOUN
iajs-2920	25	19	.	.	PUNCT
iajs-2920	26	1	definition	definition	NOUN
iajs-2920	26	2	(	(	PUNCT
iajs-2920	26	3	1)[1	1)[1	NOUN
iajs-2920	26	4	]	]	X
iajs-2920	26	5	.	.	PUNCT
iajs-2920	27	1	a	a	DET
iajs-2920	27	2	tm	tm	NOUN
iajs-2920	27	3	-	-	PUNCT
iajs-2920	27	4	algebra	algebra	NOUN
iajs-2920	27	5	is	be	AUX
iajs-2920	27	6	a	a	DET
iajs-2920	27	7	nonempty	nonempty	NOUN
iajs-2920	27	8	subset	subset	VERB
iajs-2920	27	9	with	with	ADP
iajs-2920	27	10	a	a	DET
iajs-2920	27	11	constant	constant	ADJ
iajs-2920	27	12	“	"	PUNCT
iajs-2920	27	13	0	0	NUM
iajs-2920	27	14	”	"	PUNCT
iajs-2920	27	15	and	and	CCONJ
iajs-2920	27	16	a	a	DET
iajs-2920	27	17	binary	binary	ADJ
iajs-2920	27	18	operation	operation	NOUN
iajs-2920	27	19	“	"	PUNCT
iajs-2920	27	20	*	*	NOUN
iajs-2920	27	21	”	"	PUNCT
iajs-2920	27	22	satisfying	satisfy	VERB
iajs-2920	27	23	the	the	DET
iajs-2920	27	24	following	following	NOUN
iajs-2920	27	25	:	:	PUNCT
iajs-2920	27	26	(	(	PUNCT
iajs-2920	27	27	tm1	tm1	NOUN
iajs-2920	27	28	)	)	PUNCT
iajs-2920	27	29	𝜌	𝜌	ADP
iajs-2920	27	30	∗	∗	X
iajs-2920	27	31	0	0	NUM
iajs-2920	27	32	=	=	SYM
iajs-2920	27	33	𝜌	𝜌	X
iajs-2920	27	34	,	,	PUNCT
iajs-2920	27	35	(	(	PUNCT
iajs-2920	27	36	tm2)(𝜌	tm2)(𝜌	PROPN
iajs-2920	27	37	∗	∗	NOUN
iajs-2920	27	38	𝜏	𝜏	NOUN
iajs-2920	27	39	)	)	PUNCT
iajs-2920	27	40	∗	∗	NOUN
iajs-2920	27	41	(	(	PUNCT
iajs-2920	27	42	𝜌	𝜌	X
iajs-2920	27	43	∗	∗	X
iajs-2920	27	44	휀	휀	NOUN
iajs-2920	27	45	)	)	PUNCT
iajs-2920	27	46	=	=	SYM
iajs-2920	27	47	휀	휀	DET
iajs-2920	27	48	∗	∗	NOUN
iajs-2920	27	49	𝜏,∀	𝜏,∀	PUNCT
iajs-2920	27	50	𝜌	𝜌	X
iajs-2920	27	51	,	,	PUNCT
iajs-2920	27	52	𝜏	𝜏	NOUN
iajs-2920	27	53	,	,	PUNCT
iajs-2920	27	54	휀	휀	DET
iajs-2920	27	55	∈	∈	PROPN
iajs-2920	27	56	ℵ	ℵ	NOUN
iajs-2920	27	57	.	.	PUNCT
iajs-2920	28	1	for	for	ADP
iajs-2920	28	2	ℵ	ℵ	NOUN
iajs-2920	28	3	we	we	PRON
iajs-2920	28	4	can	can	AUX
iajs-2920	28	5	define	define	VERB
iajs-2920	28	6	a	a	DET
iajs-2920	28	7	binary	binary	ADJ
iajs-2920	28	8	operation	operation	NOUN
iajs-2920	28	9	≤	≤	NUM
iajs-2920	28	10	by	by	ADP
iajs-2920	28	11	𝜌	𝜌	ADP
iajs-2920	28	12	≤	≤	NUM
iajs-2920	28	13	𝜏	𝜏	NOUN
iajs-2920	28	14	if	if	SCONJ
iajs-2920	29	1	and	and	CCONJ
iajs-2920	29	2	only	only	ADV
iajs-2920	29	3	if	if	SCONJ
iajs-2920	29	4	𝜌	𝜌	X
iajs-2920	29	5	∗	∗	NOUN
iajs-2920	29	6	𝜏	𝜏	X
iajs-2920	29	7	=	=	SYM
iajs-2920	29	8	0	0	NUM
iajs-2920	29	9	.	.	PUNCT
iajs-2920	30	1	for	for	ADP
iajs-2920	30	2	any	any	DET
iajs-2920	30	3	tm	tm	NOUN
iajs-2920	30	4	-	-	PUNCT
iajs-2920	30	5	algebra(ℵ,∗	algebra(ℵ,∗	PROPN
iajs-2920	30	6	,	,	PUNCT
iajs-2920	30	7	0),the	0),the	DET
iajs-2920	30	8	following	follow	VERB
iajs-2920	30	9	axioms	axiom	NOUN
iajs-2920	30	10	hold.∀	hold.∀	ADV
iajs-2920	30	11	𝜌	𝜌	X
iajs-2920	30	12	,	,	PUNCT
iajs-2920	30	13	𝜏	𝜏	X
iajs-2920	30	14	,	,	PUNCT
iajs-2920	30	15	휀	휀	PROPN
iajs-2920	30	16	∈	∈	PROPN
iajs-2920	30	17	ℵ	ℵ	NOUN
iajs-2920	30	18	a	a	NOUN
iajs-2920	30	19	)	)	PUNCT
iajs-2920	30	20	𝜌	𝜌	ADP
iajs-2920	30	21	∗	∗	NOUN
iajs-2920	30	22	𝜌	𝜌	X
iajs-2920	30	23	=	=	SYM
iajs-2920	30	24	0	0	NUM
iajs-2920	30	25	,	,	PUNCT
iajs-2920	30	26	b	b	NOUN
iajs-2920	30	27	)	)	PUNCT
iajs-2920	30	28	(	(	PUNCT
iajs-2920	30	29	𝜌	𝜌	X
iajs-2920	30	30	∗	∗	X
iajs-2920	30	31	𝜏	𝜏	NOUN
iajs-2920	30	32	)	)	PUNCT
iajs-2920	30	33	∗	∗	NOUN
iajs-2920	30	34	𝜌	𝜌	ADP
iajs-2920	30	35	=	=	SYM
iajs-2920	30	36	0	0	NUM
iajs-2920	30	37	∗	∗	NOUN
iajs-2920	30	38	𝜏	𝜏	NOUN
iajs-2920	30	39	,	,	PUNCT
iajs-2920	30	40	c	c	NOUN
iajs-2920	30	41	)	)	PUNCT
iajs-2920	30	42	𝜌	𝜌	ADP
iajs-2920	30	43	∗	∗	X
iajs-2920	30	44	(	(	PUNCT
iajs-2920	30	45	𝜌	𝜌	X
iajs-2920	30	46	∗	∗	X
iajs-2920	30	47	𝜏	𝜏	NOUN
iajs-2920	30	48	)	)	PUNCT
iajs-2920	30	49	=	=	SYM
iajs-2920	30	50	𝜏	𝜏	NOUN
iajs-2920	30	51	,	,	PUNCT
iajs-2920	30	52	d	d	NOUN
iajs-2920	30	53	)	)	PUNCT
iajs-2920	30	54	(	(	PUNCT
iajs-2920	30	55	𝜌	𝜌	X
iajs-2920	30	56	∗	∗	X
iajs-2920	30	57	휀	휀	NOUN
iajs-2920	30	58	)	)	PUNCT
iajs-2920	30	59	∗	∗	NOUN
iajs-2920	30	60	(	(	PUNCT
iajs-2920	30	61	𝜏	𝜏	NOUN
iajs-2920	30	62	∗	∗	PRON
iajs-2920	30	63	휀	휀	NOUN
iajs-2920	30	64	)	)	PUNCT
iajs-2920	30	65	≤	≤	NOUN
iajs-2920	30	66	𝜌	𝜌	ADP
iajs-2920	30	67	∗	∗	NOUN
iajs-2920	30	68	𝜏	𝜏	NUM
iajs-2920	30	69	,	,	PUNCT
iajs-2920	30	70	e	e	NOUN
iajs-2920	30	71	)	)	PUNCT
iajs-2920	30	72	(	(	PUNCT
iajs-2920	30	73	𝜌	𝜌	X
iajs-2920	30	74	∗	∗	X
iajs-2920	30	75	𝜏	𝜏	NOUN
iajs-2920	30	76	)	)	PUNCT
iajs-2920	30	77	∗	∗	NOUN
iajs-2920	30	78	휀	휀	NOUN
iajs-2920	30	79	=	=	X
iajs-2920	30	80	(	(	PUNCT
iajs-2920	30	81	𝜌	𝜌	X
iajs-2920	30	82	∗	∗	X
iajs-2920	30	83	휀	휀	NOUN
iajs-2920	30	84	)	)	PUNCT
iajs-2920	30	85	∗	∗	NOUN
iajs-2920	30	86	𝜏	𝜏	PROPN
iajs-2920	30	87	,	,	PUNCT
iajs-2920	30	88	f	f	X
iajs-2920	30	89	)	)	PUNCT
iajs-2920	30	90	𝜌	𝜌	ADP
iajs-2920	30	91	∗	∗	NOUN
iajs-2920	30	92	0	0	NUM
iajs-2920	30	93	=	=	SYM
iajs-2920	30	94	0	0	NUM
iajs-2920	30	95	⇒	⇒	NOUN
iajs-2920	30	96	𝜌	𝜌	ADP
iajs-2920	30	97	=	=	SYM
iajs-2920	30	98	0	0	NUM
iajs-2920	30	99	,	,	PUNCT
iajs-2920	30	100	g	g	NOUN
iajs-2920	30	101	)	)	PUNCT
iajs-2920	30	102	𝜌	𝜌	ADP
iajs-2920	30	103	≤	≤	NUM
iajs-2920	30	104	𝜏	𝜏	X
iajs-2920	30	105	⇒	⇒	NOUN
iajs-2920	30	106	𝜌	𝜌	ADP
iajs-2920	30	107	∗	∗	NOUN
iajs-2920	30	108	휀	휀	PRON
iajs-2920	30	109	≤	≤	NUM
iajs-2920	30	110	𝜏	𝜏	NUM
iajs-2920	30	111	∗	∗	NOUN
iajs-2920	30	112	휀	휀	NOUN
iajs-2920	30	113	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	30	114	휀	휀	NOUN
iajs-2920	30	115	∗	∗	NOUN
iajs-2920	30	116	𝜏	𝜏	NOUN
iajs-2920	30	117	≤	≤	NOUN
iajs-2920	30	118	휀	휀	DET
iajs-2920	30	119	∗	∗	NOUN
iajs-2920	30	120	𝜌	𝜌	ADP
iajs-2920	30	121	,	,	PUNCT
iajs-2920	30	122	h	h	NOUN
iajs-2920	30	123	)	)	PUNCT
iajs-2920	30	124	𝜌	𝜌	ADP
iajs-2920	30	125	∗	∗	X
iajs-2920	30	126	(	(	PUNCT
iajs-2920	30	127	𝜌	𝜌	X
iajs-2920	30	128	∗	∗	X
iajs-2920	30	129	(	(	PUNCT
iajs-2920	30	130	𝜌	𝜌	X
iajs-2920	30	131	∗	∗	X
iajs-2920	30	132	𝜏	𝜏	NOUN
iajs-2920	30	133	)	)	PUNCT
iajs-2920	30	134	)	)	PUNCT
iajs-2920	31	1	=	=	PUNCT
iajs-2920	31	2	𝜌	𝜌	ADP
iajs-2920	31	3	∗	∗	X
iajs-2920	31	4	𝜏	𝜏	NUM
iajs-2920	31	5	,	,	PUNCT
iajs-2920	31	6	i	i	NOUN
iajs-2920	31	7	)	)	PUNCT
iajs-2920	31	8	0	0	NUM
iajs-2920	31	9	∗	∗	NOUN
iajs-2920	31	10	(	(	PUNCT
iajs-2920	31	11	𝜌	𝜌	X
iajs-2920	31	12	∗	∗	X
iajs-2920	31	13	𝜏	𝜏	NOUN
iajs-2920	31	14	)	)	PUNCT
iajs-2920	31	15	=	=	SYM
iajs-2920	31	16	𝜏	𝜏	PART
iajs-2920	31	17	∗	∗	NOUN
iajs-2920	31	18	𝜌	𝜌	X
iajs-2920	31	19	=	=	SYM
iajs-2920	31	20	(	(	PUNCT
iajs-2920	31	21	0	0	NUM
iajs-2920	31	22	∗	∗	NOUN
iajs-2920	31	23	𝜌	𝜌	NOUN
iajs-2920	31	24	)	)	PUNCT
iajs-2920	31	25	∗	∗	NOUN
iajs-2920	31	26	(	(	PUNCT
iajs-2920	31	27	0	0	NUM
iajs-2920	31	28	∗	∗	NOUN
iajs-2920	31	29	𝜏	𝜏	NUM
iajs-2920	31	30	)	)	PUNCT
iajs-2920	31	31	,	,	PUNCT
iajs-2920	31	32	j	j	NOUN
iajs-2920	31	33	)	)	PUNCT
iajs-2920	31	34	(	(	PUNCT
iajs-2920	31	35	𝜌	𝜌	X
iajs-2920	31	36	∗	∗	X
iajs-2920	31	37	(	(	PUNCT
iajs-2920	31	38	𝜌	𝜌	X
iajs-2920	31	39	∗	∗	X
iajs-2920	31	40	𝜏	𝜏	NOUN
iajs-2920	31	41	)	)	PUNCT
iajs-2920	31	42	)	)	PUNCT
iajs-2920	31	43	∗	∗	NOUN
iajs-2920	31	44	𝜏	𝜏	NOUN
iajs-2920	31	45	=	=	SYM
iajs-2920	31	46	0	0	NUM
iajs-2920	31	47	,	,	PUNCT
iajs-2920	31	48	k	k	NOUN
iajs-2920	31	49	)	)	PUNCT
iajs-2920	31	50	if	if	SCONJ
iajs-2920	31	51	𝜌	𝜌	X
iajs-2920	31	52	∗	∗	NOUN
iajs-2920	31	53	𝜏	𝜏	NOUN
iajs-2920	31	54	=	=	SYM
iajs-2920	31	55	0	0	NUM
iajs-2920	31	56	and	and	CCONJ
iajs-2920	31	57	𝜏	𝜏	NOUN
iajs-2920	31	58	∗	∗	NOUN
iajs-2920	31	59	𝜌	𝜌	X
iajs-2920	31	60	=	=	SYM
iajs-2920	31	61	0	0	NUM
iajs-2920	31	62	imply	imply	VERB
iajs-2920	31	63	𝜌	𝜌	PUNCT
iajs-2920	31	64	=	=	PUNCT
iajs-2920	31	65	𝜏.	𝜏.	PROPN
iajs-2920	31	66	example	example	NOUN
iajs-2920	31	67	(	(	PUNCT
iajs-2920	31	68	2	2	X
iajs-2920	31	69	)	)	PUNCT
iajs-2920	32	1	[	[	X
iajs-2920	32	2	1	1	NUM
iajs-2920	32	3	]	]	PUNCT
iajs-2920	32	4	.	.	PUNCT
iajs-2920	33	1	letℵ=	letℵ=	NUM
iajs-2920	33	2	{	{	PUNCT
iajs-2920	33	3	0,1,2,3}be	0,1,2,3}be	X
iajs-2920	33	4	a	a	DET
iajs-2920	33	5	set	set	NOUN
iajs-2920	33	6	with	with	ADP
iajs-2920	33	7	the	the	DET
iajs-2920	33	8	following	follow	VERB
iajs-2920	33	9	table	table	NOUN
iajs-2920	33	10	.	.	PUNCT
iajs-2920	34	1	*	*	PUNCT
iajs-2920	34	2	0	0	NUM
iajs-2920	35	1	1	1	NUM
iajs-2920	35	2	2	2	NUM
iajs-2920	35	3	3	3	NUM
iajs-2920	35	4	0	0	NUM
iajs-2920	35	5	0	0	NUM
iajs-2920	35	6	1	1	NUM
iajs-2920	35	7	2	2	NUM
iajs-2920	35	8	3	3	NUM
iajs-2920	35	9	1	1	NUM
iajs-2920	35	10	1	1	NUM
iajs-2920	35	11	0	0	NUM
iajs-2920	35	12	3	3	NUM
iajs-2920	35	13	2	2	NUM
iajs-2920	35	14	2	2	NUM
iajs-2920	35	15	2	2	NUM
iajs-2920	35	16	3	3	NUM
iajs-2920	35	17	0	0	NUM
iajs-2920	35	18	1	1	NUM
iajs-2920	35	19	3	3	NUM
iajs-2920	35	20	3	3	NUM
iajs-2920	35	21	2	2	NUM
iajs-2920	35	22	1	1	NUM
iajs-2920	35	23	0	0	NUM
iajs-2920	35	24	then,(ℵ,∗	then,(ℵ,∗	PRON
iajs-2920	35	25	,	,	PUNCT
iajs-2920	35	26	0	0	NUM
iajs-2920	35	27	)	)	PUNCT
iajs-2920	35	28	is	be	AUX
iajs-2920	35	29	a	a	DET
iajs-2920	35	30	tm	tm	NOUN
iajs-2920	35	31	-	-	NOUN
iajs-2920	35	32	algebra	algebra	NOUN
iajs-2920	35	33	.	.	PUNCT
iajs-2920	36	1	definition	definition	NOUN
iajs-2920	36	2	(	(	PUNCT
iajs-2920	36	3	3	3	X
iajs-2920	36	4	)	)	PUNCT
iajs-2920	36	5	[	[	X
iajs-2920	36	6	2	2	NUM
iajs-2920	36	7	]	]	PUNCT
iajs-2920	36	8	.	.	PUNCT
iajs-2920	37	1	a	a	DET
iajs-2920	37	2	non	non	ADJ
iajs-2920	37	3	-	-	ADJ
iajs-2920	37	4	empty	empty	ADJ
iajs-2920	37	5	subset	subset	ADJ
iajs-2920	37	6	𝑆	𝑆	PROPN
iajs-2920	37	7	of	of	ADP
iajs-2920	37	8	a	a	DET
iajs-2920	37	9	tm	tm	NOUN
iajs-2920	37	10	-	-	NOUN
iajs-2920	37	11	algebra	algebra	PROPN
iajs-2920	37	12	(	(	PUNCT
iajs-2920	37	13	ℵ,∗	ℵ,∗	PROPN
iajs-2920	37	14	,	,	PUNCT
iajs-2920	37	15	0	0	NUM
iajs-2920	37	16	)	)	PUNCT
iajs-2920	37	17	is	be	AUX
iajs-2920	37	18	called	call	VERB
iajs-2920	37	19	a	a	DET
iajs-2920	37	20	tm	tm	NOUN
iajs-2920	37	21	-	-	PUNCT
iajs-2920	37	22	subalgebra	subalgebra	NOUN
iajs-2920	37	23	ofℵif𝜌	ofℵif𝜌	ADP
iajs-2920	37	24	∗	∗	NOUN
iajs-2920	37	25	𝜏	𝜏	PRON
iajs-2920	37	26	∈	∈	PROPN
iajs-2920	37	27	𝑆whenever	𝑆whenever	PROPN
iajs-2920	37	28	𝜌	𝜌	PROPN
iajs-2920	37	29	,	,	PUNCT
iajs-2920	37	30	𝜏	𝜏	PRON
iajs-2920	37	31	∈	∈	NOUN
iajs-2920	37	32	𝑆.	𝑆.	NOUN
iajs-2920	37	33	definition	definition	NOUN
iajs-2920	37	34	(	(	PUNCT
iajs-2920	37	35	4	4	NUM
iajs-2920	37	36	)	)	PUNCT
iajs-2920	38	1	[	[	X
iajs-2920	38	2	2	2	NUM
iajs-2920	38	3	]	]	PUNCT
iajs-2920	38	4	.	.	PUNCT
iajs-2920	39	1	a	a	DET
iajs-2920	39	2	non	non	ADJ
iajs-2920	39	3	-	-	ADJ
iajs-2920	39	4	empty	empty	ADJ
iajs-2920	39	5	subset	subset	NOUN
iajs-2920	39	6	𝜓	𝜓	ADP
iajs-2920	39	7	ofantm	ofantm	NOUN
iajs-2920	39	8	-	-	PROPN
iajs-2920	39	9	algebra	algebra	PROPN
iajs-2920	39	10	(	(	PUNCT
iajs-2920	39	11	ℵ,∗	ℵ,∗	PROPN
iajs-2920	39	12	,	,	PUNCT
iajs-2920	39	13	0	0	NUM
iajs-2920	39	14	)	)	PUNCT
iajs-2920	39	15	is	be	AUX
iajs-2920	39	16	said	say	VERB
iajs-2920	39	17	to	to	PART
iajs-2920	39	18	be	be	AUX
iajs-2920	39	19	an	an	DET
iajs-2920	39	20	ideal	ideal	NOUN
iajs-2920	39	21	of	of	ADP
iajs-2920	39	22	ℵif	ℵif	NOUN
iajs-2920	39	23	it	it	PRON
iajs-2920	39	24	satisfies	satisfy	VERB
iajs-2920	39	25	,	,	PUNCT
iajs-2920	39	26	for	for	ADP
iajs-2920	39	27	any	any	DET
iajs-2920	39	28	𝜌	𝜌	NOUN
iajs-2920	39	29	,	,	PUNCT
iajs-2920	39	30	𝜏	𝜏	PROPN
iajs-2920	39	31	∈	∈	PROPN
iajs-2920	39	32	𝜓	𝜓	PROPN
iajs-2920	39	33	i	i	PROPN
iajs-2920	39	34	)	)	PUNCT
iajs-2920	39	35	0	0	NUM
iajs-2920	39	36	∈	∈	PROPN
iajs-2920	39	37	𝜓	𝜓	PROPN
iajs-2920	39	38	,	,	PUNCT
iajs-2920	39	39	ihjpas	ihjpa	NOUN
iajs-2920	39	40	.	.	PUNCT
iajs-2920	40	1	36(1)2023	36(1)2023	NUM
iajs-2920	40	2	369	369	NUM
iajs-2920	40	3	ii	ii	NOUN
iajs-2920	40	4	)	)	PUNCT
iajs-2920	40	5	𝜌	𝜌	ADP
iajs-2920	40	6	∗	∗	NOUN
iajs-2920	40	7	𝜏	𝜏	PRON
iajs-2920	40	8	∈	∈	PROPN
iajs-2920	40	9	𝜓	𝜓	NOUN
iajs-2920	40	10	and	and	CCONJ
iajs-2920	40	11	𝜏	𝜏	PRON
iajs-2920	40	12	∈	∈	PROPN
iajs-2920	40	13	𝜓	𝜓	PROPN
iajs-2920	40	14	implies	imply	VERB
iajs-2920	40	15	that	that	SCONJ
iajs-2920	40	16	𝜌	𝜌	ADP
iajs-2920	40	17	∈	∈	PROPN
iajs-2920	40	18	𝜓.	𝜓.	PROPN
iajs-2920	40	19	example	example	NOUN
iajs-2920	40	20	(	(	PUNCT
iajs-2920	40	21	5	5	X
iajs-2920	40	22	)	)	PUNCT
iajs-2920	40	23	[	[	X
iajs-2920	40	24	2	2	NUM
iajs-2920	40	25	]	]	PUNCT
iajs-2920	40	26	.	.	PUNCT
iajs-2920	41	1	letℵ	letℵ	VERB
iajs-2920	41	2	=	=	PUNCT
iajs-2920	41	3	{	{	PUNCT
iajs-2920	41	4	0	0	NUM
iajs-2920	41	5	,	,	PUNCT
iajs-2920	41	6	1	1	NUM
iajs-2920	41	7	,	,	PUNCT
iajs-2920	41	8	2	2	NUM
iajs-2920	41	9	,	,	PUNCT
iajs-2920	41	10	3	3	NUM
iajs-2920	41	11	}	}	PUNCT
iajs-2920	41	12	be	be	AUX
iajs-2920	41	13	a	a	DET
iajs-2920	41	14	set	set	NOUN
iajs-2920	41	15	with	with	ADP
iajs-2920	41	16	a	a	DET
iajs-2920	41	17	binary	binary	NOUN
iajs-2920	41	18	operation∗defined	operation∗defined	ADJ
iajs-2920	41	19	inthe	inthe	DET
iajs-2920	41	20	following	follow	VERB
iajs-2920	41	21	table	table	NOUN
iajs-2920	41	22	:	:	PUNCT
iajs-2920	42	1	*	*	SYM
iajs-2920	42	2	0	0	NUM
iajs-2920	43	1	1	1	NUM
iajs-2920	43	2	2	2	NUM
iajs-2920	43	3	3	3	NUM
iajs-2920	43	4	0	0	NUM
iajs-2920	43	5	0	0	NUM
iajs-2920	43	6	0	0	NUM
iajs-2920	43	7	3	3	NUM
iajs-2920	43	8	2	2	NUM
iajs-2920	43	9	1	1	NUM
iajs-2920	43	10	1	1	NUM
iajs-2920	43	11	0	0	NUM
iajs-2920	43	12	3	3	NUM
iajs-2920	43	13	2	2	NUM
iajs-2920	43	14	2	2	NUM
iajs-2920	43	15	2	2	NUM
iajs-2920	43	16	2	2	NUM
iajs-2920	43	17	0	0	NUM
iajs-2920	43	18	3	3	NUM
iajs-2920	43	19	3	3	NUM
iajs-2920	43	20	3	3	NUM
iajs-2920	43	21	3	3	NUM
iajs-2920	43	22	b	b	NOUN
iajs-2920	43	23	0	0	NUM
iajs-2920	43	24	then	then	ADV
iajs-2920	43	25	(	(	PUNCT
iajs-2920	43	26	ℵ,∗	ℵ,∗	PROPN
iajs-2920	43	27	,	,	PUNCT
iajs-2920	43	28	0)is	0)is	PROPN
iajs-2920	43	29	a	a	DET
iajs-2920	43	30	tm	tm	NOUN
iajs-2920	43	31	-	-	NOUN
iajs-2920	43	32	algebra	algebra	NOUN
iajs-2920	43	33	and	and	CCONJ
iajs-2920	43	34	𝜓	𝜓	NOUN
iajs-2920	43	35	=	=	SYM
iajs-2920	43	36	{	{	PUNCT
iajs-2920	43	37	0,1	0,1	NOUN
iajs-2920	43	38	}	}	PUNCT
iajs-2920	43	39	is	be	AUX
iajs-2920	43	40	an	an	DET
iajs-2920	43	41	ideal	ideal	NOUN
iajs-2920	43	42	of	of	ADP
iajs-2920	43	43	ℵ.	ℵ.	PROPN
iajs-2920	43	44	definition	definition	NOUN
iajs-2920	43	45	(	(	PUNCT
iajs-2920	43	46	6	6	NUM
iajs-2920	43	47	)	)	PUNCT
iajs-2920	44	1	[	[	X
iajs-2920	44	2	1	1	NUM
iajs-2920	44	3	]	]	PUNCT
iajs-2920	44	4	.	.	PUNCT
iajs-2920	45	1	a	a	DET
iajs-2920	45	2	non	non	ADJ
iajs-2920	45	3	-	-	ADJ
iajs-2920	45	4	empty	empty	ADJ
iajs-2920	45	5	subset	subset	ADJ
iajs-2920	45	6	𝐸	𝐸	PROPN
iajs-2920	45	7	of	of	ADP
iajs-2920	45	8	a	a	DET
iajs-2920	45	9	tm	tm	NOUN
iajs-2920	45	10	-	-	PUNCT
iajs-2920	45	11	algebra	algebra	NOUN
iajs-2920	45	12	ℵ	ℵ	NOUN
iajs-2920	45	13	is	be	AUX
iajs-2920	45	14	a	a	DET
iajs-2920	45	15	t	t	NOUN
iajs-2920	45	16	-	-	PUNCT
iajs-2920	45	17	ideal	ideal	NOUN
iajs-2920	45	18	,	,	PUNCT
iajs-2920	45	19	if	if	SCONJ
iajs-2920	45	20	i	i	PRON
iajs-2920	45	21	)	)	PUNCT
iajs-2920	45	22	0	0	PUNCT
iajs-2920	46	1	∈	∈	PROPN
iajs-2920	46	2	𝐸	𝐸	PROPN
iajs-2920	46	3	ii	ii	PROPN
iajs-2920	46	4	)	)	PUNCT
iajs-2920	46	5	∀	∀	PUNCT
iajs-2920	47	1	𝜌	𝜌	X
iajs-2920	47	2	,	,	PUNCT
iajs-2920	47	3	𝜏	𝜏	NOUN
iajs-2920	47	4	,	,	PUNCT
iajs-2920	47	5	휀	휀	PRON
iajs-2920	47	6	∈	∈	NOUN
iajs-2920	47	7	ℵ	ℵ	NOUN
iajs-2920	47	8	,	,	PUNCT
iajs-2920	47	9	(	(	PUNCT
iajs-2920	47	10	𝜌	𝜌	X
iajs-2920	47	11	∗	∗	X
iajs-2920	47	12	𝜏	𝜏	NOUN
iajs-2920	47	13	)	)	PUNCT
iajs-2920	47	14	∗	∗	NOUN
iajs-2920	47	15	휀	휀	DET
iajs-2920	47	16	∈	∈	PROPN
iajs-2920	47	17	𝐸	𝐸	PROPN
iajs-2920	47	18	and	and	CCONJ
iajs-2920	47	19	𝜏	𝜏	NOUN
iajs-2920	47	20	∈	∈	NOUN
iajs-2920	47	21	𝐸	𝐸	NOUN
iajs-2920	47	22	imply	imply	VERB
iajs-2920	47	23	(	(	PUNCT
iajs-2920	47	24	𝜌	𝜌	X
iajs-2920	47	25	∗	∗	X
iajs-2920	47	26	휀	휀	NOUN
iajs-2920	47	27	)	)	PUNCT
iajs-2920	47	28	∈	∈	PROPN
iajs-2920	47	29	𝐸.	𝐸.	PROPN
iajs-2920	47	30	definition	definition	NOUN
iajs-2920	47	31	(	(	PUNCT
iajs-2920	47	32	7	7	X
iajs-2920	47	33	)	)	PUNCT
iajs-2920	48	1	[	[	X
iajs-2920	48	2	5	5	NUM
iajs-2920	48	3	]	]	PUNCT
iajs-2920	48	4	.	.	PUNCT
iajs-2920	49	1	let	let	VERB
iajs-2920	49	2	(	(	PUNCT
iajs-2920	49	3	ℵ	ℵ	NOUN
iajs-2920	49	4	,	,	PUNCT
iajs-2920	49	5	∗	∗	NOUN
iajs-2920	49	6	,	,	PUNCT
iajs-2920	49	7	0	0	NUM
iajs-2920	49	8	)	)	PUNCT
iajs-2920	49	9	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
iajs-2920	49	10	(	(	PUNCT
iajs-2920	49	11	ℵ′,∗′	ℵ′,∗′	PROPN
iajs-2920	49	12	,	,	PUNCT
iajs-2920	49	13	0′)be	0′)be	VERB
iajs-2920	49	14	a	a	DET
iajs-2920	49	15	tm	tm	NOUN
iajs-2920	49	16	-	-	PUNCT
iajs-2920	49	17	algebras	algebras	PROPN
iajs-2920	49	18	.	.	PUNCT
iajs-2920	50	1	a	a	DET
iajs-2920	50	2	homomorphism	homomorphism	NOUN
iajs-2920	50	3	is	be	AUX
iajs-2920	50	4	a	a	DET
iajs-2920	50	5	map	map	NOUN
iajs-2920	50	6	𝑓	𝑓	PRON
iajs-2920	50	7	:	:	PUNCT
iajs-2920	50	8	ℵ	ℵ	PROPN
iajs-2920	50	9	→	→	SYM
iajs-2920	50	10	ℵ′	ℵ′	X
iajs-2920	50	11	satisfying	satisfying	NOUN
iajs-2920	50	12	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	50	13	∗	∗	NOUN
iajs-2920	50	14	𝜏	𝜏	NOUN
iajs-2920	50	15	)	)	PUNCT
iajs-2920	50	16	=	=	PUNCT
iajs-2920	50	17	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	50	18	)	)	PUNCT
iajs-2920	50	19	∗′	∗′	ADJ
iajs-2920	50	20	𝑓(𝜏	𝑓(𝜏	NOUN
iajs-2920	50	21	)	)	PUNCT
iajs-2920	50	22	,	,	PUNCT
iajs-2920	50	23	for	for	ADP
iajs-2920	50	24	all	all	DET
iajs-2920	50	25	𝜌	𝜌	NOUN
iajs-2920	50	26	,	,	PUNCT
iajs-2920	50	27	𝜏	𝜏	PROPN
iajs-2920	50	28	∈	∈	NOUN
iajs-2920	50	29	ℵ.	ℵ.	NOUN
iajs-2920	51	1	now	now	ADV
iajs-2920	51	2	,	,	PUNCT
iajs-2920	51	3	we	we	PRON
iajs-2920	51	4	review	review	VERB
iajs-2920	51	5	an	an	DET
iajs-2920	51	6	interval	interval	NOUN
iajs-2920	51	7	-	-	PUNCT
iajs-2920	51	8	valued	value	VERB
iajs-2920	51	9	fuzzy	fuzzy	ADJ
iajs-2920	51	10	set	set	VERB
iajs-2920	51	11	concepts	concept	NOUN
iajs-2920	51	12	.	.	PUNCT
iajs-2920	52	1	definition	definition	NOUN
iajs-2920	52	2	(	(	PUNCT
iajs-2920	52	3	8)	8)	NUM
iajs-2920	52	4	[	[	X
iajs-2920	52	5	5	5	NUM
iajs-2920	52	6	]	]	PUNCT
iajs-2920	52	7	.	.	PUNCT
iajs-2920	53	1	let	let	VERB
iajs-2920	53	2	�	�	PROPN
iajs-2920	53	3	̃	̃	PROPN
iajs-2920	53	4	�	�	NOUN
iajs-2920	53	5	=	=	PUNCT
iajs-2920	54	1	[	[	X
iajs-2920	54	2	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	54	3	,	,	PUNCT
iajs-2920	54	4	𝑎𝑈	𝑎𝑈	PROPN
iajs-2920	54	5	]	]	X
iajs-2920	54	6	be	be	VERB
iajs-2920	54	7	an	an	DET
iajs-2920	54	8	interval	interval	NOUN
iajs-2920	54	9	number	number	NOUN
iajs-2920	54	10	,	,	PUNCT
iajs-2920	54	11	where	where	SCONJ
iajs-2920	54	12	0	0	NUM
iajs-2920	54	13	≤	≤	NUM
iajs-2920	54	14	𝑎𝐿	𝑎𝐿	VERB
iajs-2920	54	15	≤	≤	NUM
iajs-2920	54	16	𝑎𝑈	𝑎𝑈	VERB
iajs-2920	54	17	≤	≤	NUM
iajs-2920	54	18	1	1	NUM
iajs-2920	54	19	and	and	CCONJ
iajs-2920	54	20	let	let	VERB
iajs-2920	54	21	𝐷[0,1]be	𝐷[0,1]be	NOUN
iajs-2920	54	22	denoted	denote	VERB
iajs-2920	54	23	the	the	DET
iajs-2920	54	24	family	family	NOUN
iajs-2920	54	25	of	of	ADP
iajs-2920	54	26	all	all	DET
iajs-2920	54	27	closed	closed	ADJ
iajs-2920	54	28	subinterval	subinterval	NOUN
iajs-2920	54	29	of	of	ADP
iajs-2920	54	30	[	[	X
iajs-2920	54	31	0,1	0,1	NUM
iajs-2920	54	32	]	]	PUNCT
iajs-2920	54	33	,	,	PUNCT
iajs-2920	54	34	that	that	ADV
iajs-2920	54	35	is	is	ADV
iajs-2920	54	36	,	,	PUNCT
iajs-2920	54	37	𝐷[0,1	𝐷[0,1	PRON
iajs-2920	54	38	]	]	X
iajs-2920	54	39	=	=	PRON
iajs-2920	54	40	{	{	PUNCT
iajs-2920	54	41	�	�	PROPN
iajs-2920	54	42	̃	̃	NOUN
iajs-2920	54	43	�	�	NOUN
iajs-2920	54	44	=	=	PUNCT
iajs-2920	55	1	[	[	X
iajs-2920	55	2	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	55	3	,	,	PUNCT
iajs-2920	55	4	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	55	5	]	]	X
iajs-2920	55	6	∶	∶	NOUN
iajs-2920	55	7	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	55	8	≤	≤	NUM
iajs-2920	55	9	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	55	10	,	,	PUNCT
iajs-2920	55	11	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2920	55	12	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	55	13	≤	≤	PROPN
iajs-2920	55	14	𝑎𝑈	𝑎𝑈	VERB
iajs-2920	55	15	∈	∈	PROPN
iajs-2920	56	1	[	[	X
iajs-2920	56	2	0,1	0,1	NUM
iajs-2920	56	3	]	]	PUNCT
iajs-2920	56	4	}	}	PUNCT
iajs-2920	56	5	.	.	PUNCT
iajs-2920	57	1	the	the	DET
iajs-2920	57	2	operations≥	operations≥	NOUN
iajs-2920	57	3	,	,	PUNCT
iajs-2920	57	4	≤	≤	ADJ
iajs-2920	57	5	,	,	PUNCT
iajs-2920	57	6	=	=	SYM
iajs-2920	57	7	,	,	PUNCT
iajs-2920	57	8	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	57	9	,	,	PUNCT
iajs-2920	57	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	57	11	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
iajs-2920	57	12	of	of	ADP
iajs-2920	57	13	two	two	NUM
iajs-2920	57	14	elements	element	NOUN
iajs-2920	57	15	in	in	ADP
iajs-2920	57	16	𝐷[0,1	𝐷[0,1	PROPN
iajs-2920	57	17	]	]	PUNCT
iajs-2920	57	18	is	be	AUX
iajs-2920	57	19	defined	define	VERB
iajs-2920	57	20	as	as	SCONJ
iajs-2920	57	21	follows	follow	VERB
iajs-2920	57	22	:	:	PUNCT
iajs-2920	57	23	let	let	VERB
iajs-2920	57	24	�	�	PROPN
iajs-2920	57	25	̃	̃	PROPN
iajs-2920	57	26	�	�	NOUN
iajs-2920	57	27	=	=	PUNCT
iajs-2920	58	1	[	[	X
iajs-2920	58	2	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	58	3	,	,	PUNCT
iajs-2920	58	4	𝑎𝑈	𝑎𝑈	PROPN
iajs-2920	58	5	]	]	X
iajs-2920	58	6	,	,	PUNCT
iajs-2920	58	7	�	�	PROPN
iajs-2920	58	8	̃	̃	NOUN
iajs-2920	58	9	�	�	NOUN
iajs-2920	58	10	=	=	PUNCT
iajs-2920	59	1	[	[	X
iajs-2920	59	2	𝑏𝐿	𝑏𝐿	PROPN
iajs-2920	59	3	,	,	PUNCT
iajs-2920	59	4	𝑏𝑈	𝑏𝑈	PROPN
iajs-2920	59	5	]	]	X
iajs-2920	59	6	in	in	ADP
iajs-2920	59	7	𝐷[0,1	𝐷[0,1	PROPN
iajs-2920	59	8	]	]	PUNCT
iajs-2920	59	9	,	,	PUNCT
iajs-2920	59	10	then	then	ADV
iajs-2920	59	11	(	(	PUNCT
iajs-2920	59	12	1	1	X
iajs-2920	59	13	)	)	PUNCT
iajs-2920	59	14	�	�	PROPN
iajs-2920	59	15	̃	̃	PROPN
iajs-2920	59	16	�	�	PROPN
iajs-2920	59	17	≥	≥	NUM
iajs-2920	59	18	�	�	PROPN
iajs-2920	59	19	̃	̃	PROPN
iajs-2920	59	20	�	�	PROPN
iajs-2920	59	21	𝑖𝑓	𝑖𝑓	NUM
iajs-2920	59	22	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2920	59	23	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
iajs-2920	59	24	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	60	1	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	60	2	≥	≥	PUNCT
iajs-2920	60	3	𝑏𝐿	𝑏𝐿	ADJ
iajs-2920	60	4	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2920	60	5	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	60	6	≥	≥	NOUN
iajs-2920	60	7	𝑏𝑈	𝑏𝑈	PROPN
iajs-2920	60	8	,	,	PUNCT
iajs-2920	60	9	(	(	PUNCT
iajs-2920	60	10	2	2	X
iajs-2920	60	11	)	)	PUNCT
iajs-2920	60	12	�	�	PROPN
iajs-2920	60	13	̃	̃	PROPN
iajs-2920	60	14	�	�	PROPN
iajs-2920	60	15	≤	≤	NUM
iajs-2920	60	16	�	�	PROPN
iajs-2920	60	17	̃	̃	PROPN
iajs-2920	60	18	�	�	PROPN
iajs-2920	60	19	𝑖𝑓	𝑖𝑓	NUM
iajs-2920	60	20	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2920	60	21	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
iajs-2920	60	22	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	60	23	𝑎𝐿	𝑎𝐿	PROPN
iajs-2920	61	1	≤	≤	ADV
iajs-2920	61	2	𝑏𝐿	𝑏𝐿	VERB
iajs-2920	61	3	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	61	4	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	61	5	≤	≤	ADJ
iajs-2920	61	6	𝑏𝑈	𝑏𝑈	PROPN
iajs-2920	61	7	,	,	PUNCT
iajs-2920	61	8	(	(	PUNCT
iajs-2920	61	9	3	3	X
iajs-2920	61	10	)	)	PUNCT
iajs-2920	61	11	�	�	PROPN
iajs-2920	61	12	̃	̃	PROPN
iajs-2920	61	13	�	�	PROPN
iajs-2920	61	14	=	=	SYM
iajs-2920	61	15	�	�	PROPN
iajs-2920	61	16	̃	̃	PROPN
iajs-2920	61	17	�	�	PROPN
iajs-2920	61	18	𝑖𝑓	𝑖𝑓	NUM
iajs-2920	61	19	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2920	61	20	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
iajs-2920	61	21	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	61	22	𝑎𝐿	𝑎𝐿	NOUN
iajs-2920	62	1	=	=	PUNCT
iajs-2920	62	2	𝑏𝐿	𝑏𝐿	ADJ
iajs-2920	62	3	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	62	4	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	62	5	=	=	SYM
iajs-2920	62	6	𝑎𝑈	𝑎𝑈	NOUN
iajs-2920	62	7	,	,	PUNCT
iajs-2920	62	8	(	(	PUNCT
iajs-2920	62	9	4	4	X
iajs-2920	62	10	)	)	PUNCT
iajs-2920	62	11	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	62	12	�	�	PROPN
iajs-2920	62	13	̃	̃	PROPN
iajs-2920	62	14	�	�	PROPN
iajs-2920	62	15	,	,	PUNCT
iajs-2920	62	16	�	�	PROPN
iajs-2920	62	17	̃	̃	NOUN
iajs-2920	62	18	�	�	NOUN
iajs-2920	62	19	}	}	PUNCT
iajs-2920	62	20	=	=	PUNCT
iajs-2920	63	1	[	[	X
iajs-2920	63	2	𝑚𝑖𝑛{𝑎𝐿	𝑚𝑖𝑛{𝑎𝐿	NOUN
iajs-2920	63	3	,	,	PUNCT
iajs-2920	63	4	𝑏𝐿	𝑏𝐿	PROPN
iajs-2920	63	5	}	}	PUNCT
iajs-2920	63	6	,	,	PUNCT
iajs-2920	63	7	𝑚𝑖𝑛{𝑎𝑈	𝑚𝑖𝑛{𝑎𝑈	ADV
iajs-2920	63	8	,	,	PUNCT
iajs-2920	63	9	𝑏𝑈	𝑏𝑈	PROPN
iajs-2920	63	10	}	}	PUNCT
iajs-2920	63	11	]	]	PUNCT
iajs-2920	63	12	,	,	PUNCT
iajs-2920	63	13	(	(	PUNCT
iajs-2920	63	14	5	5	X
iajs-2920	63	15	)	)	PUNCT
iajs-2920	63	16	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2920	63	17	�	�	PROPN
iajs-2920	63	18	̃	̃	PROPN
iajs-2920	63	19	�	�	PROPN
iajs-2920	63	20	,	,	PUNCT
iajs-2920	63	21	�	�	PROPN
iajs-2920	63	22	̃	̃	NOUN
iajs-2920	63	23	�	�	NOUN
iajs-2920	63	24	}	}	PUNCT
iajs-2920	63	25	=	=	PUNCT
iajs-2920	64	1	[	[	X
iajs-2920	64	2	𝑚𝑎𝑥{𝑎𝐿	𝑚𝑎𝑥{𝑎𝐿	NOUN
iajs-2920	64	3	,	,	PUNCT
iajs-2920	64	4	𝑏𝐿	𝑏𝐿	NOUN
iajs-2920	64	5	}	}	PUNCT
iajs-2920	64	6	,	,	PUNCT
iajs-2920	64	7	𝑚𝑎𝑥{𝑎𝑈	𝑚𝑎𝑥{𝑎𝑈	NOUN
iajs-2920	64	8	,	,	PUNCT
iajs-2920	64	9	𝑏𝑈	𝑏𝑈	PROPN
iajs-2920	64	10	}	}	PUNCT
iajs-2920	64	11	]	]	PUNCT
iajs-2920	64	12	,	,	PUNCT
iajs-2920	64	13	and	and	CCONJ
iajs-2920	64	14	if	if	SCONJ
iajs-2920	64	15	�	�	PROPN
iajs-2920	64	16	̃	̃	NOUN
iajs-2920	64	17	�	�	NOUN
iajs-2920	64	18	𝑖	𝑖	ADP
iajs-2920	64	19	∈	∈	PROPN
iajs-2920	64	20	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	64	21	]	]	X
iajs-2920	64	22	where	where	SCONJ
iajs-2920	64	23	𝑖	𝑖	PUNCT
iajs-2920	64	24	∈	∈	NOUN
iajs-2920	64	25	⋀.	⋀.	NOUN
iajs-2920	64	26	we	we	PRON
iajs-2920	64	27	define	define	VERB
iajs-2920	64	28	𝑟	𝑟	DET
iajs-2920	64	29	inf	inf	PROPN
iajs-2920	64	30	𝑖∈⋀	𝑖∈⋀	PROPN
iajs-2920	64	31	�	�	PROPN
iajs-2920	64	32	̃	̃	PROPN
iajs-2920	64	33	�	�	NOUN
iajs-2920	64	34	𝑖	𝑖	ADP
iajs-2920	64	35	=	=	PUNCT
iajs-2920	65	1	[	[	X
iajs-2920	65	2	inf	inf	ADJ
iajs-2920	65	3	𝑖∈⋀	𝑖∈⋀	PROPN
iajs-2920	65	4	𝑎𝑖	𝑎𝑖	PROPN
iajs-2920	65	5	𝐿	𝐿	PROPN
iajs-2920	65	6	,	,	PUNCT
iajs-2920	65	7	inf	inf	PROPN
iajs-2920	65	8	𝑖∈⋀	𝑖∈⋀	PROPN
iajs-2920	65	9	𝑎𝑖	𝑎𝑖	ADP
iajs-2920	65	10	𝑈	𝑈	PROPN
iajs-2920	65	11	]	]	PUNCT
iajs-2920	65	12	,	,	PUNCT
iajs-2920	65	13	𝑟	𝑟	PRON
iajs-2920	65	14	sup	sup	VERB
iajs-2920	65	15	𝑖∈⋀	𝑖∈⋀	PROPN
iajs-2920	65	16	�	�	PROPN
iajs-2920	65	17	̃	̃	PROPN
iajs-2920	65	18	�	�	NOUN
iajs-2920	65	19	𝑖	𝑖	ADP
iajs-2920	65	20	=	=	PUNCT
iajs-2920	66	1	[	[	X
iajs-2920	66	2	sup	sup	NOUN
iajs-2920	66	3	𝑖∈⋀	𝑖∈⋀	NOUN
iajs-2920	66	4	𝑎𝑖	𝑎𝑖	ADP
iajs-2920	66	5	𝐿	𝐿	PROPN
iajs-2920	66	6	,	,	PUNCT
iajs-2920	66	7	sup	sup	NOUN
iajs-2920	66	8	𝑖∈⋀	𝑖∈⋀	PROPN
iajs-2920	66	9	𝑎𝑖	𝑎𝑖	ADP
iajs-2920	66	10	𝑈	𝑈	PROPN
iajs-2920	66	11	]	]	PUNCT
iajs-2920	66	12	.	.	PUNCT
iajs-2920	67	1	an	an	DET
iajs-2920	67	2	interval	interval	NOUN
iajs-2920	67	3	-	-	PUNCT
iajs-2920	67	4	valued	value	VERB
iajs-2920	67	5	fuzzy	fuzzy	ADJ
iajs-2920	67	6	set	set	VERB
iajs-2920	67	7	𝑉	𝑉	PROPN
iajs-2920	67	8	=	=	PROPN
iajs-2920	67	9	<	<	X
iajs-2920	67	10	𝜌	𝜌	X
iajs-2920	67	11	,	,	PUNCT
iajs-2920	67	12	�	�	PROPN
iajs-2920	67	13	̃	̃	PROPN
iajs-2920	67	14	�	�	NOUN
iajs-2920	67	15	(𝜌	(𝜌	NOUN
iajs-2920	67	16	)	)	PUNCT
iajs-2920	67	17	>	>	X
iajs-2920	67	18	on	on	ADP
iajs-2920	67	19	ℵ	ℵ	PRON
iajs-2920	67	20	is	be	AUX
iajs-2920	67	21	defined	define	VERB
iajs-2920	67	22	as	as	ADP
iajs-2920	67	23	�	�	PROPN
iajs-2920	67	24	̃	̃	PROPN
iajs-2920	67	25	�	�	NOUN
iajs-2920	67	26	(𝜌	(𝜌	NOUN
iajs-2920	67	27	)	)	PUNCT
iajs-2920	67	28	=	=	SYM
iajs-2920	67	29	{	{	PUNCT
iajs-2920	67	30	〈	〈	NOUN
iajs-2920	67	31	𝜌	𝜌	X
iajs-2920	67	32	,	,	PUNCT
iajs-2920	67	33	[	[	X
iajs-2920	67	34	𝜗𝐿(𝜌	𝜗𝐿(𝜌	NOUN
iajs-2920	67	35	)	)	PUNCT
iajs-2920	67	36	,	,	PUNCT
iajs-2920	67	37	𝜗𝐿(𝜌	𝜗𝐿(𝜌	PROPN
iajs-2920	67	38	)	)	PUNCT
iajs-2920	67	39	]	]	PUNCT
iajs-2920	67	40	〉	〉	NOUN
iajs-2920	67	41	∶	∶	NOUN
iajs-2920	67	42	𝜌	𝜌	ADP
iajs-2920	67	43	∈	∈	PROPN
iajs-2920	67	44	ℵ	ℵ	NOUN
iajs-2920	67	45	}	}	PUNCT
iajs-2920	67	46	,	,	PUNCT
iajs-2920	67	47	where	where	SCONJ
iajs-2920	67	48	𝜗𝐿(𝜌	𝜗𝐿(𝜌	ADP
iajs-2920	67	49	)	)	PUNCT
iajs-2920	67	50	≤	≤	NOUN
iajs-2920	67	51	𝜗𝑈(𝜌	𝜗𝑈(𝜌	NUM
iajs-2920	67	52	)	)	PUNCT
iajs-2920	67	53	,	,	PUNCT
iajs-2920	67	54	for	for	ADP
iajs-2920	67	55	all	all	DET
iajs-2920	67	56	𝜌	𝜌	ADP
iajs-2920	67	57	∈	∈	PRON
iajs-2920	67	58	ℵ.	ℵ.	NOUN
iajs-2920	67	59	then,𝜗𝐿(𝜌	then,𝜗𝐿(𝜌	PROPN
iajs-2920	67	60	)	)	PUNCT
iajs-2920	67	61	∶	∶	NOUN
iajs-2920	67	62	ℵ	ℵ	NOUN
iajs-2920	67	63	→	→	X
iajs-2920	67	64	[	[	X
iajs-2920	67	65	0,1	0,1	NUM
iajs-2920	67	66	]	]	PUNCT
iajs-2920	67	67	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
iajs-2920	67	68	𝜗𝑈	𝜗𝑈	PROPN
iajs-2920	67	69	∶	∶	NOUN
iajs-2920	67	70	ℵ	ℵ	X
iajs-2920	67	71	→	→	X
iajs-2920	67	72	[	[	X
iajs-2920	67	73	0,1	0,1	NUM
iajs-2920	67	74	]	]	PUNCT
iajs-2920	67	75	are	be	AUX
iajs-2920	67	76	called	call	VERB
iajs-2920	67	77	a	a	DET
iajs-2920	67	78	lower	low	ADJ
iajs-2920	67	79	fuzzy	fuzzy	ADJ
iajs-2920	67	80	set	set	NOUN
iajs-2920	67	81	and	and	CCONJ
iajs-2920	67	82	an	an	DET
iajs-2920	67	83	upper	upper	ADJ
iajs-2920	67	84	fuzzy	fuzzy	ADJ
iajs-2920	67	85	set	set	NOUN
iajs-2920	67	86	of	of	ADP
iajs-2920	67	87	𝜗,̃	𝜗,̃	PUNCT
iajs-2920	67	88	respectively	respectively	ADV
iajs-2920	67	89	.	.	PUNCT
iajs-2920	68	1	definition	definition	NOUN
iajs-2920	68	2	(	(	PUNCT
iajs-2920	68	3	9	9	NUM
iajs-2920	68	4	)	)	PUNCT
iajs-2920	68	5	[	[	X
iajs-2920	68	6	5	5	NUM
iajs-2920	68	7	]	]	PUNCT
iajs-2920	68	8	.	.	PUNCT
iajs-2920	69	1	let	let	VERB
iajs-2920	69	2	(	(	PUNCT
iajs-2920	69	3	ℵ,∗	ℵ,∗	PROPN
iajs-2920	69	4	,	,	PUNCT
iajs-2920	69	5	0	0	NUM
iajs-2920	69	6	)	)	PUNCT
iajs-2920	69	7	be	be	AUX
iajs-2920	69	8	a	a	DET
iajs-2920	69	9	tm	tm	NOUN
iajs-2920	69	10	-	-	NOUN
iajs-2920	69	11	algebra	algebra	NOUN
iajs-2920	69	12	and	and	CCONJ
iajs-2920	69	13	�	�	PROPN
iajs-2920	69	14	̃	̃	PROPN
iajs-2920	69	15	�	�	PROPN
iajs-2920	69	16	:	:	PUNCT
iajs-2920	69	17	ℵ	ℵ	PROPN
iajs-2920	69	18	→	→	SYM
iajs-2920	69	19	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	69	20	]	]	X
iajs-2920	69	21	.	.	PUNCT
iajs-2920	70	1	then	then	ADV
iajs-2920	70	2	,	,	PUNCT
iajs-2920	70	3	𝑉	𝑉	PROPN
iajs-2920	70	4	=	=	PROPN
iajs-2920	70	5	<	<	X
iajs-2920	70	6	𝜌	𝜌	X
iajs-2920	70	7	,	,	PUNCT
iajs-2920	70	8	�	�	PROPN
iajs-2920	70	9	̃	̃	PROPN
iajs-2920	70	10	�	�	NOUN
iajs-2920	70	11	(𝜌	(𝜌	NOUN
iajs-2920	70	12	)	)	PUNCT
iajs-2920	70	13	>	>	X
iajs-2920	70	14	is	be	AUX
iajs-2920	70	15	called	call	VERB
iajs-2920	70	16	an	an	DET
iajs-2920	70	17	interval	interval	NOUN
iajs-2920	70	18	valued	value	VERB
iajs-2920	70	19	fuzzy	fuzzy	ADJ
iajs-2920	70	20	sub	sub	NOUN
iajs-2920	70	21	tm	tm	NOUN
iajs-2920	70	22	-	-	NOUN
iajs-2920	70	23	algebra	algebra	NOUN
iajs-2920	70	24	ℵ	ℵ	NOUN
iajs-2920	70	25	,	,	PUNCT
iajs-2920	70	26	if	if	SCONJ
iajs-2920	70	27	ihjpas	ihjpa	NOUN
iajs-2920	70	28	.	.	PUNCT
iajs-2920	71	1	36(1)2023	36(1)2023	NUM
iajs-2920	71	2	370	370	NUM
iajs-2920	71	3	�	�	PROPN
iajs-2920	71	4	̃	̃	PROPN
iajs-2920	71	5	�	�	PROPN
iajs-2920	71	6	(𝜌	(𝜌	NOUN
iajs-2920	71	7	∗	∗	NOUN
iajs-2920	71	8	𝛾	𝛾	NOUN
iajs-2920	71	9	)	)	PUNCT
iajs-2920	71	10	≥	≥	PROPN
iajs-2920	71	11	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	71	12	�	�	PROPN
iajs-2920	71	13	̃	̃	PROPN
iajs-2920	71	14	�	�	NOUN
iajs-2920	71	15	(𝜌	(𝜌	NOUN
iajs-2920	71	16	)	)	PUNCT
iajs-2920	71	17	,	,	PUNCT
iajs-2920	71	18	�	�	PROPN
iajs-2920	71	19	̃	̃	PROPN
iajs-2920	71	20	�	�	NOUN
iajs-2920	71	21	(𝛾	(𝛾	NOUN
iajs-2920	71	22	)	)	PUNCT
iajs-2920	71	23	}	}	PUNCT
iajs-2920	71	24	,	,	PUNCT
iajs-2920	71	25	∀𝜌	∀𝜌	PUNCT
iajs-2920	71	26	,	,	PUNCT
iajs-2920	71	27	𝛾	𝛾	PROPN
iajs-2920	71	28	∈	∈	PRON
iajs-2920	71	29	ℵ.	ℵ.	NOUN
iajs-2920	71	30	definition	definition	NOUN
iajs-2920	71	31	(	(	PUNCT
iajs-2920	71	32	10	10	NUM
iajs-2920	71	33	)	)	PUNCT
iajs-2920	72	1	[	[	X
iajs-2920	72	2	5	5	NUM
iajs-2920	72	3	]	]	PUNCT
iajs-2920	72	4	.	.	PUNCT
iajs-2920	73	1	let(ℵ,∗	let(ℵ,∗	PRON
iajs-2920	73	2	,	,	PUNCT
iajs-2920	73	3	0	0	X
iajs-2920	73	4	)	)	PUNCT
iajs-2920	73	5	be	be	AUX
iajs-2920	73	6	a	a	DET
iajs-2920	73	7	tm	tm	NOUN
iajs-2920	73	8	-	-	NOUN
iajs-2920	73	9	algebra	algebra	NOUN
iajs-2920	73	10	and	and	CCONJ
iajs-2920	73	11	�	�	PROPN
iajs-2920	73	12	̃	̃	PROPN
iajs-2920	73	13	�	�	PROPN
iajs-2920	73	14	:	:	PUNCT
iajs-2920	73	15	ℵ	ℵ	PROPN
iajs-2920	73	16	→	→	SYM
iajs-2920	73	17	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	73	18	]	]	PUNCT
iajs-2920	73	19	.	.	PUNCT
iajs-2920	74	1	then	then	ADV
iajs-2920	74	2	𝑉	𝑉	PROPN
iajs-2920	74	3	=	=	PROPN
iajs-2920	74	4	<	<	X
iajs-2920	74	5	𝜌	𝜌	X
iajs-2920	74	6	,	,	PUNCT
iajs-2920	74	7	�	�	PROPN
iajs-2920	74	8	̃	̃	PROPN
iajs-2920	74	9	�	�	NOUN
iajs-2920	74	10	(𝜌	(𝜌	NOUN
iajs-2920	74	11	)	)	PUNCT
iajs-2920	74	12	>	>	X
iajs-2920	74	13	is	be	AUX
iajs-2920	74	14	said	say	VERB
iajs-2920	74	15	to	to	PART
iajs-2920	74	16	be	be	AUX
iajs-2920	74	17	an	an	DET
iajs-2920	74	18	interval	interval	NOUN
iajs-2920	74	19	valued	value	VERB
iajs-2920	74	20	fuzzy	fuzzy	ADJ
iajs-2920	74	21	ideal	ideal	NOUN
iajs-2920	74	22	if	if	SCONJ
iajs-2920	74	23	(	(	PUNCT
iajs-2920	74	24	i1	i1	PROPN
iajs-2920	74	25	)	)	PUNCT
iajs-2920	74	26	�	�	PROPN
iajs-2920	74	27	̃	̃	PROPN
iajs-2920	74	28	�	�	PROPN
iajs-2920	74	29	(0	(0	X
iajs-2920	74	30	)	)	PUNCT
iajs-2920	74	31	≥	≥	PROPN
iajs-2920	74	32	�	�	PROPN
iajs-2920	74	33	̃	̃	PROPN
iajs-2920	74	34	�	�	NOUN
iajs-2920	74	35	(𝜌	(𝜌	NOUN
iajs-2920	74	36	)	)	PUNCT
iajs-2920	74	37	,	,	PUNCT
iajs-2920	74	38	∀𝜌	∀𝜌	PUNCT
iajs-2920	74	39	∈	∈	NOUN
iajs-2920	74	40	ℵ	ℵ	NOUN
iajs-2920	74	41	,	,	PUNCT
iajs-2920	74	42	(	(	PUNCT
iajs-2920	74	43	i2	i2	PROPN
iajs-2920	74	44	)	)	PUNCT
iajs-2920	74	45	for	for	ADP
iajs-2920	74	46	all	all	DET
iajs-2920	74	47	𝜌	𝜌	NOUN
iajs-2920	74	48	,	,	PUNCT
iajs-2920	74	49	𝛾	𝛾	PROPN
iajs-2920	74	50	∈	∈	NOUN
iajs-2920	74	51	ℵ	ℵ	NOUN
iajs-2920	74	52	,	,	PUNCT
iajs-2920	74	53	�	�	PROPN
iajs-2920	74	54	̃	̃	PROPN
iajs-2920	74	55	�	�	NOUN
iajs-2920	74	56	(𝜌	(𝜌	NOUN
iajs-2920	74	57	)	)	PUNCT
iajs-2920	74	58	≥	≥	PROPN
iajs-2920	74	59	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	74	60	�	�	PROPN
iajs-2920	74	61	̃	̃	PROPN
iajs-2920	74	62	�	�	PROPN
iajs-2920	74	63	(𝜌	(𝜌	NOUN
iajs-2920	74	64	∗	∗	NOUN
iajs-2920	74	65	𝛾	𝛾	NOUN
iajs-2920	74	66	)	)	PUNCT
iajs-2920	74	67	,	,	PUNCT
iajs-2920	74	68	�	�	PROPN
iajs-2920	74	69	̃	̃	PROPN
iajs-2920	74	70	�	�	NOUN
iajs-2920	74	71	(𝛾	(𝛾	NOUN
iajs-2920	74	72	)	)	PUNCT
iajs-2920	74	73	}	}	PUNCT
iajs-2920	74	74	.	.	PUNCT
iajs-2920	75	1	definition	definition	NOUN
iajs-2920	75	2	(	(	PUNCT
iajs-2920	75	3	11	11	NUM
iajs-2920	75	4	)	)	PUNCT
iajs-2920	76	1	[	[	X
iajs-2920	76	2	5	5	NUM
iajs-2920	76	3	]	]	PUNCT
iajs-2920	76	4	.	.	PUNCT
iajs-2920	77	1	let(ℵ,∗	let(ℵ,∗	PRON
iajs-2920	77	2	,	,	PUNCT
iajs-2920	77	3	0	0	X
iajs-2920	77	4	)	)	PUNCT
iajs-2920	77	5	be	be	AUX
iajs-2920	77	6	a	a	DET
iajs-2920	77	7	tm	tm	NOUN
iajs-2920	77	8	-	-	NOUN
iajs-2920	77	9	algebra	algebra	NOUN
iajs-2920	77	10	and	and	CCONJ
iajs-2920	77	11	�	�	PROPN
iajs-2920	77	12	̃	̃	PROPN
iajs-2920	77	13	�	�	PROPN
iajs-2920	77	14	:	:	PUNCT
iajs-2920	77	15	ℵ	ℵ	PROPN
iajs-2920	77	16	→	→	SYM
iajs-2920	77	17	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	77	18	]	]	PUNCT
iajs-2920	77	19	.	.	PUNCT
iajs-2920	78	1	then	then	ADV
iajs-2920	78	2	𝑉	𝑉	PROPN
iajs-2920	78	3	=	=	PROPN
iajs-2920	78	4	<	<	X
iajs-2920	78	5	𝜌	𝜌	X
iajs-2920	78	6	,	,	PUNCT
iajs-2920	78	7	�	�	PROPN
iajs-2920	78	8	̃	̃	PROPN
iajs-2920	78	9	�	�	NOUN
iajs-2920	78	10	(𝜌	(𝜌	NOUN
iajs-2920	78	11	)	)	PUNCT
iajs-2920	78	12	>	>	PUNCT
iajs-2920	78	13	is	be	AUX
iajs-2920	78	14	said	say	VERB
iajs-2920	78	15	to	to	ADP
iajs-2920	78	16	bean	bean	NOUN
iajs-2920	78	17	interval	interval	NOUN
iajs-2920	78	18	valued	value	VERB
iajs-2920	78	19	fuzzy	fuzzy	ADJ
iajs-2920	78	20	t	t	PROPN
iajs-2920	78	21	-	-	PUNCT
iajs-2920	78	22	ideal	ideal	NOUN
iajs-2920	78	23	if	if	SCONJ
iajs-2920	78	24	(	(	PUNCT
iajs-2920	78	25	i1	i1	PROPN
iajs-2920	78	26	)	)	PUNCT
iajs-2920	78	27	�	�	PROPN
iajs-2920	78	28	̃	̃	PROPN
iajs-2920	78	29	�	�	PROPN
iajs-2920	78	30	(0	(0	X
iajs-2920	78	31	)	)	PUNCT
iajs-2920	78	32	≥	≥	PROPN
iajs-2920	78	33	�	�	PROPN
iajs-2920	78	34	̃	̃	PROPN
iajs-2920	78	35	�	�	NOUN
iajs-2920	78	36	(𝜌	(𝜌	NOUN
iajs-2920	78	37	)	)	PUNCT
iajs-2920	78	38	,	,	PUNCT
iajs-2920	78	39	∀𝜌	∀𝜌	PUNCT
iajs-2920	78	40	∈	∈	NOUN
iajs-2920	78	41	ℵ	ℵ	NOUN
iajs-2920	78	42	,	,	PUNCT
iajs-2920	78	43	(	(	PUNCT
iajs-2920	78	44	i2	i2	PROPN
iajs-2920	78	45	)	)	PUNCT
iajs-2920	78	46	for	for	ADP
iajs-2920	78	47	all	all	DET
iajs-2920	78	48	𝜌	𝜌	NOUN
iajs-2920	78	49	,	,	PUNCT
iajs-2920	78	50	𝛾	𝛾	PROPN
iajs-2920	78	51	,	,	PUNCT
iajs-2920	78	52	휀	휀	PRON
iajs-2920	78	53	∈	∈	PROPN
iajs-2920	78	54	ℵ	ℵ	NOUN
iajs-2920	78	55	,	,	PUNCT
iajs-2920	78	56	�	�	PROPN
iajs-2920	78	57	̃	̃	PROPN
iajs-2920	78	58	�	�	NOUN
iajs-2920	78	59	(𝜌	(𝜌	NOUN
iajs-2920	78	60	∗	∗	NOUN
iajs-2920	78	61	휀	휀	NOUN
iajs-2920	78	62	)	)	PUNCT
iajs-2920	78	63	≥	≥	NOUN
iajs-2920	78	64	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	78	65	�	�	PROPN
iajs-2920	78	66	̃	̃	PROPN
iajs-2920	78	67	�	�	PROPN
iajs-2920	78	68	((𝜌	((𝜌	PROPN
iajs-2920	78	69	∗	∗	PROPN
iajs-2920	78	70	𝛾	𝛾	PROPN
iajs-2920	78	71	)	)	PUNCT
iajs-2920	78	72	∗	∗	NOUN
iajs-2920	78	73	휀	휀	NOUN
iajs-2920	78	74	)	)	PUNCT
iajs-2920	78	75	,	,	PUNCT
iajs-2920	78	76	�	�	PROPN
iajs-2920	78	77	̃	̃	PROPN
iajs-2920	78	78	�	�	NOUN
iajs-2920	78	79	(𝛾	(𝛾	NOUN
iajs-2920	78	80	)	)	PUNCT
iajs-2920	78	81	}	}	PUNCT
iajs-2920	78	82	.	.	PUNCT
iajs-2920	79	1	3	3	X
iajs-2920	79	2	.	.	X
iajs-2920	79	3	cubic	cubic	PROPN
iajs-2920	79	4	t	t	PROPN
iajs-2920	79	5	-	-	PUNCT
iajs-2920	79	6	ideals	ideal	NOUN
iajs-2920	79	7	of	of	ADP
iajs-2920	79	8	tm	tm	NOUN
iajs-2920	79	9	-	-	NOUN
iajs-2920	79	10	algebra	algebra	NOUN
iajs-2920	79	11	we	we	PRON
iajs-2920	79	12	recall	recall	VERB
iajs-2920	79	13	that	that	SCONJ
iajs-2920	79	14	a	a	DET
iajs-2920	79	15	cubic	cubic	ADJ
iajs-2920	79	16	set	set	NOUN
iajs-2920	79	17	𝛿	𝛿	NOUN
iajs-2920	79	18	in	in	ADP
iajs-2920	79	19	a	a	DET
iajs-2920	79	20	set	set	NOUN
iajs-2920	79	21	ℵ	ℵ	NOUN
iajs-2920	79	22	is	be	AUX
iajs-2920	79	23	the	the	DET
iajs-2920	79	24	structure	structure	NOUN
iajs-2920	79	25	𝛿	𝛿	NOUN
iajs-2920	79	26	=	=	PUNCT
iajs-2920	79	27	{	{	PUNCT
iajs-2920	79	28	〈	〈	NOUN
iajs-2920	79	29	𝜌	𝜌	X
iajs-2920	79	30	,	,	PUNCT
iajs-2920	79	31	�	�	PROPN
iajs-2920	79	32	̃	̃	PROPN
iajs-2920	79	33	�	�	NOUN
iajs-2920	79	34	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	79	35	)	)	PUNCT
iajs-2920	79	36	,	,	PUNCT
iajs-2920	79	37	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	79	38	)	)	PUNCT
iajs-2920	79	39	〉	〉	NOUN
iajs-2920	79	40	∶	∶	NOUN
iajs-2920	79	41	𝜌	𝜌	ADP
iajs-2920	79	42	∈	∈	PROPN
iajs-2920	79	43	ℵ	ℵ	NOUN
iajs-2920	79	44	}	}	PUNCT
iajs-2920	79	45	,	,	PUNCT
iajs-2920	79	46	where	where	SCONJ
iajs-2920	79	47	�	�	PROPN
iajs-2920	79	48	̃	̃	NOUN
iajs-2920	79	49	�	�	PROPN
iajs-2920	79	50	𝛿	𝛿	DET
iajs-2920	79	51	∶	∶	NOUN
iajs-2920	79	52	ℵ	ℵ	X
iajs-2920	79	53	→	→	SYM
iajs-2920	79	54	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	79	55	]	]	X
iajs-2920	79	56	such	such	ADJ
iajs-2920	79	57	that	that	DET
iajs-2920	79	58	�	�	PROPN
iajs-2920	79	59	̃	̃	PROPN
iajs-2920	79	60	�	�	NOUN
iajs-2920	79	61	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	79	62	)	)	PUNCT
iajs-2920	79	63	=	=	PUNCT
iajs-2920	80	1	[	[	X
iajs-2920	80	2	𝜗𝛿	𝜗𝛿	INTJ
iajs-2920	80	3	𝐿(𝜌	𝐿(𝜌	NOUN
iajs-2920	80	4	)	)	PUNCT
iajs-2920	80	5	,	,	PUNCT
iajs-2920	80	6	𝜗𝛿	𝜗𝛿	CCONJ
iajs-2920	80	7	𝑈(𝜌	𝑈(𝜌	NOUN
iajs-2920	80	8	)	)	PUNCT
iajs-2920	80	9	]	]	PUNCT
iajs-2920	80	10	is	be	AUX
iajs-2920	80	11	an	an	DET
iajs-2920	80	12	interval	interval	NOUN
iajs-2920	80	13	valued	value	VERB
iajs-2920	80	14	fuzzy	fuzzy	ADJ
iajs-2920	80	15	set	set	VERB
iajs-2920	80	16	in	in	ADP
iajs-2920	80	17	ℵ	ℵ	NOUN
iajs-2920	80	18	and	and	CCONJ
iajs-2920	80	19	𝛼𝛿	𝛼𝛿	NOUN
iajs-2920	80	20	is	be	AUX
iajs-2920	80	21	a	a	DET
iajs-2920	80	22	fuzzy	fuzzy	ADJ
iajs-2920	80	23	set	set	NOUN
iajs-2920	80	24	in	in	ADP
iajs-2920	80	25	ℵ.we	ℵ.we	PRON
iajs-2920	80	26	write	write	VERB
iajs-2920	80	27	a	a	DET
iajs-2920	80	28	cubic	cubic	ADJ
iajs-2920	80	29	set	set	VERB
iajs-2920	80	30	by	by	ADP
iajs-2920	80	31	as	as	SCONJ
iajs-2920	80	32	follows	follow	VERB
iajs-2920	80	33	.	.	PUNCT
iajs-2920	81	1	𝛿	𝛿	PRON
iajs-2920	81	2	=	=	PUNCT
iajs-2920	81	3	〈	〈	PROPN
iajs-2920	81	4	�	�	PROPN
iajs-2920	81	5	̃	̃	PROPN
iajs-2920	81	6	�	�	PROPN
iajs-2920	81	7	𝛿	𝛿	NOUN
iajs-2920	81	8	,	,	PUNCT
iajs-2920	81	9	𝛼𝛿	𝛼𝛿	PART
iajs-2920	81	10	〉	〉	NOUN
iajs-2920	81	11	and	and	CCONJ
iajs-2920	81	12	we	we	PRON
iajs-2920	81	13	can	can	AUX
iajs-2920	81	14	define	define	VERB
iajs-2920	81	15	the	the	DET
iajs-2920	81	16	level	level	NOUN
iajs-2920	81	17	subset	subset	NOUN
iajs-2920	81	18	of	of	ADP
iajs-2920	81	19	𝛿	𝛿	PROPN
iajs-2920	81	20	=	=	PUNCT
iajs-2920	81	21	〈	〈	PROPN
iajs-2920	81	22	�	�	PROPN
iajs-2920	81	23	̃	̃	PROPN
iajs-2920	81	24	�	�	PROPN
iajs-2920	81	25	𝛿	𝛿	NOUN
iajs-2920	81	26	,	,	PUNCT
iajs-2920	81	27	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	81	28	〉	〉	NOUN
iajs-2920	81	29	which	which	PRON
iajs-2920	81	30	is	be	AUX
iajs-2920	81	31	denoted	denote	VERB
iajs-2920	81	32	by	by	ADP
iajs-2920	81	33	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	81	34	,	,	PUNCT
iajs-2920	81	35	�	�	PROPN
iajs-2920	81	36	̃	̃	PROPN
iajs-2920	81	37	�	�	PROPN
iajs-2920	81	38	,	,	PUNCT
iajs-2920	81	39	𝑠	𝑠	NOUN
iajs-2920	81	40	)	)	PUNCT
iajs-2920	81	41	as	as	ADP
iajs-2920	81	42	follows𝑈(𝛿	follows𝑈(𝛿	PROPN
iajs-2920	81	43	,	,	PUNCT
iajs-2920	81	44	�	�	PROPN
iajs-2920	81	45	̃	̃	PROPN
iajs-2920	81	46	�	�	PROPN
iajs-2920	81	47	,	,	PUNCT
iajs-2920	81	48	𝑠	𝑠	PROPN
iajs-2920	81	49	)	)	PUNCT
iajs-2920	81	50	=	=	SYM
iajs-2920	81	51	{	{	PUNCT
iajs-2920	81	52	𝜌	𝜌	X
iajs-2920	81	53	∈	∈	NOUN
iajs-2920	81	54	ℵ	ℵ	NOUN
iajs-2920	81	55	:	:	PUNCT
iajs-2920	81	56	�	�	PROPN
iajs-2920	81	57	̃	̃	PROPN
iajs-2920	81	58	�	�	PROPN
iajs-2920	81	59	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	81	60	)	)	PUNCT
iajs-2920	81	61	≥	≥	NOUN
iajs-2920	81	62	�	�	PROPN
iajs-2920	81	63	̃	̃	PROPN
iajs-2920	81	64	�	�	PROPN
iajs-2920	81	65	,	,	PUNCT
iajs-2920	81	66	𝛼𝛿	𝛼𝛿	ADP
iajs-2920	81	67	≤	≤	NUM
iajs-2920	81	68	𝑠	𝑠	NUM
iajs-2920	81	69	}	}	PUNCT
iajs-2920	81	70	,	,	PUNCT
iajs-2920	81	71	for	for	SCONJ
iajs-2920	81	72	every	every	DET
iajs-2920	81	73	[	[	NOUN
iajs-2920	81	74	0,0	0,0	NOUN
iajs-2920	81	75	]	]	PUNCT
iajs-2920	81	76	≤	≤	NUM
iajs-2920	81	77	�	�	PROPN
iajs-2920	81	78	̃	̃	PROPN
iajs-2920	81	79	�	�	PROPN
iajs-2920	81	80	≤	≤	NOUN
iajs-2920	82	1	[	[	X
iajs-2920	82	2	1,1	1,1	NUM
iajs-2920	82	3	]	]	PUNCT
iajs-2920	82	4	and	and	CCONJ
iajs-2920	82	5	𝑠	𝑠	PRON
iajs-2920	82	6	∈	∈	PROPN
iajs-2920	82	7	[	[	X
iajs-2920	82	8	0,1	0,1	NUM
iajs-2920	82	9	]	]	PUNCT
iajs-2920	82	10	.	.	PUNCT
iajs-2920	83	1	definition	definition	NOUN
iajs-2920	83	2	(	(	PUNCT
iajs-2920	83	3	12).let	12).let	NUM
iajs-2920	83	4	ℵ	ℵ	NOUN
iajs-2920	83	5	be	be	VERB
iajs-2920	83	6	a	a	DET
iajs-2920	83	7	tm	tm	NOUN
iajs-2920	83	8	-	-	NOUN
iajs-2920	83	9	algebra	algebra	NOUN
iajs-2920	83	10	.	.	PUNCT
iajs-2920	84	1	a	a	DET
iajs-2920	84	2	cubic	cubic	ADJ
iajs-2920	84	3	set	set	VERB
iajs-2920	84	4	𝛿	𝛿	PROPN
iajs-2920	84	5	=	=	PROPN
iajs-2920	84	6	〈	〈	PROPN
iajs-2920	84	7	�	�	PROPN
iajs-2920	84	8	̃	̃	PROPN
iajs-2920	84	9	�	�	PROPN
iajs-2920	84	10	𝛿	𝛿	NOUN
iajs-2920	84	11	,	,	PUNCT
iajs-2920	84	12	𝛼𝛿	𝛼𝛿	PART
iajs-2920	84	13	〉	〉	NOUN
iajs-2920	84	14	in	in	ADP
iajs-2920	84	15	ℵ	ℵ	PROPN
iajs-2920	84	16	is	be	AUX
iajs-2920	84	17	called	call	VERB
iajs-2920	84	18	a	a	DET
iajs-2920	84	19	cubic	cubic	ADJ
iajs-2920	84	20	sub	sub	NOUN
iajs-2920	84	21	-	-	NOUN
iajs-2920	84	22	algebra	algebra	ADJ
iajs-2920	84	23	if	if	SCONJ
iajs-2920	84	24	(	(	PUNCT
iajs-2920	84	25	1	1	NUM
iajs-2920	84	26	)	)	PUNCT
iajs-2920	84	27	�	�	PROPN
iajs-2920	84	28	̃	̃	PROPN
iajs-2920	84	29	�	�	PROPN
iajs-2920	84	30	δ(𝜌	δ(𝜌	NOUN
iajs-2920	84	31	∗	∗	NOUN
iajs-2920	84	32	𝜏	𝜏	NOUN
iajs-2920	84	33	)	)	PUNCT
iajs-2920	84	34	≥	≥	NOUN
iajs-2920	84	35	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	84	36	{	{	PUNCT
iajs-2920	84	37	�	�	PROPN
iajs-2920	84	38	̃	̃	PROPN
iajs-2920	84	39	�	�	NOUN
iajs-2920	84	40	(𝜌	(𝜌	NOUN
iajs-2920	84	41	)	)	PUNCT
iajs-2920	84	42	,	,	PUNCT
iajs-2920	84	43	�	�	PROPN
iajs-2920	84	44	̃	̃	PROPN
iajs-2920	84	45	�	�	PROPN
iajs-2920	84	46	(𝜏	(𝜏	VERB
iajs-2920	84	47	)	)	PUNCT
iajs-2920	84	48	}	}	PUNCT
iajs-2920	84	49	.	.	PUNCT
iajs-2920	85	1	(	(	PUNCT
iajs-2920	85	2	2	2	NUM
iajs-2920	85	3	)	)	PUNCT
iajs-2920	85	4	𝛼δ(𝜌	𝛼δ(𝜌	X
iajs-2920	85	5	∗	∗	NOUN
iajs-2920	85	6	𝜏	𝜏	NOUN
iajs-2920	85	7	)	)	PUNCT
iajs-2920	85	8	≤	≤	NOUN
iajs-2920	85	9	𝑚𝑎𝑥{𝛼δ(𝜌	𝑚𝑎𝑥{𝛼δ(𝜌	X
iajs-2920	85	10	)	)	PUNCT
iajs-2920	85	11	,	,	PUNCT
iajs-2920	85	12	𝛼δ(𝜏)},∀	𝛼δ(𝜏)},∀	PROPN
iajs-2920	85	13	𝜌	𝜌	X
iajs-2920	85	14	,	,	PUNCT
iajs-2920	85	15	𝜏	𝜏	PROPN
iajs-2920	85	16	∈	∈	PROPN
iajs-2920	85	17	ℵ.	ℵ.	NOUN
iajs-2920	85	18	example	example	NOUN
iajs-2920	85	19	(	(	PUNCT
iajs-2920	85	20	13	13	NUM
iajs-2920	85	21	)	)	PUNCT
iajs-2920	85	22	.	.	PUNCT
iajs-2920	86	1	let	let	VERB
iajs-2920	86	2	ℵ	ℵ	NOUN
iajs-2920	86	3	=	=	SYM
iajs-2920	86	4	{	{	PUNCT
iajs-2920	86	5	0	0	NUM
iajs-2920	86	6	,	,	PUNCT
iajs-2920	86	7	a	a	DET
iajs-2920	86	8	,	,	PUNCT
iajs-2920	86	9	b	b	NOUN
iajs-2920	86	10	,	,	PUNCT
iajs-2920	86	11	c	c	AUX
iajs-2920	86	12	}	}	PUNCT
iajs-2920	86	13	be	be	AUX
iajs-2920	86	14	a	a	DET
iajs-2920	86	15	set	set	NOUN
iajs-2920	86	16	with	with	ADP
iajs-2920	86	17	the	the	DET
iajs-2920	86	18	following	follow	VERB
iajs-2920	86	19	table	table	NOUN
iajs-2920	86	20	:	:	PUNCT
iajs-2920	87	1	*	*	PUNCT
iajs-2920	87	2	0	0	PUNCT
iajs-2920	87	3	a	a	DET
iajs-2920	87	4	b	b	X
iajs-2920	87	5	c	c	NOUN
iajs-2920	87	6	0	0	NUM
iajs-2920	87	7	0	0	NUM
iajs-2920	87	8	a	a	DET
iajs-2920	87	9	b	b	NOUN
iajs-2920	87	10	c	c	NOUN
iajs-2920	87	11	a	a	DET
iajs-2920	87	12	a	a	DET
iajs-2920	87	13	0	0	NUM
iajs-2920	87	14	c	c	NOUN
iajs-2920	87	15	b	b	PROPN
iajs-2920	87	16	b	b	PROPN
iajs-2920	87	17	b	b	PROPN
iajs-2920	87	18	c	c	PROPN
iajs-2920	87	19	0	0	NUM
iajs-2920	88	1	a	a	DET
iajs-2920	88	2	c	c	NOUN
iajs-2920	88	3	c	c	NOUN
iajs-2920	88	4	b	b	PROPN
iajs-2920	88	5	a	a	DET
iajs-2920	88	6	0	0	NUM
iajs-2920	88	7	then	then	ADV
iajs-2920	88	8	(	(	PUNCT
iajs-2920	88	9	ℵ,∗	ℵ,∗	PROPN
iajs-2920	88	10	,	,	PUNCT
iajs-2920	88	11	0	0	NUM
iajs-2920	88	12	)	)	PUNCT
iajs-2920	88	13	is	be	AUX
iajs-2920	88	14	a	a	DET
iajs-2920	88	15	tm	tm	NOUN
iajs-2920	88	16	-	-	NOUN
iajs-2920	88	17	algebra	algebra	NOUN
iajs-2920	88	18	.	.	PUNCT
iajs-2920	89	1	define	define	VERB
iajs-2920	89	2	�	�	PROPN
iajs-2920	89	3	̃	̃	PROPN
iajs-2920	89	4	�	�	NOUN
iajs-2920	89	5	δ(𝜌	δ(𝜌	NOUN
iajs-2920	89	6	)	)	PUNCT
iajs-2920	89	7	ˑand	ˑand	CCONJ
iajs-2920	89	8	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	89	9	)	)	PUNCT
iajs-2920	89	10	by	by	ADP
iajs-2920	89	11	�	�	PROPN
iajs-2920	89	12	̃	̃	PROPN
iajs-2920	89	13	�	�	NOUN
iajs-2920	89	14	δ(𝜌	δ(𝜌	NOUN
iajs-2920	89	15	)	)	PUNCT
iajs-2920	90	1	=	=	PRON
iajs-2920	90	2	{	{	PUNCT
iajs-2920	91	1	[	[	X
iajs-2920	91	2	0.2,0.9	0.2,0.9	X
iajs-2920	91	3	]	]	X
iajs-2920	91	4	𝑖𝑓	𝑖𝑓	X
iajs-2920	91	5	𝜌	𝜌	X
iajs-2920	91	6	=	=	SYM
iajs-2920	91	7	{	{	PUNCT
iajs-2920	91	8	0	0	NUM
iajs-2920	91	9	,	,	PUNCT
iajs-2920	91	10	𝑎	𝑎	NOUN
iajs-2920	91	11	,	,	PUNCT
iajs-2920	91	12	𝑏	𝑏	NOUN
iajs-2920	91	13	}	}	PUNCT
iajs-2920	91	14	[	[	X
iajs-2920	91	15	0.1,0.3	0.1,0.3	X
iajs-2920	91	16	]	]	X
iajs-2920	91	17	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	91	18	𝜌	𝜌	X
iajs-2920	91	19	=	=	SYM
iajs-2920	91	20	𝑐	𝑐	PROPN
iajs-2920	91	21	,	,	PUNCT
iajs-2920	91	22	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	91	23	)	)	PUNCT
iajs-2920	91	24	=	=	NOUN
iajs-2920	91	25	{	{	PUNCT
iajs-2920	91	26	0.2	0.2	NUM
iajs-2920	91	27	𝑖𝑓	𝑖𝑓	NUM
iajs-2920	91	28	𝜌	𝜌	X
iajs-2920	91	29	=	=	SYM
iajs-2920	91	30	{	{	PUNCT
iajs-2920	91	31	0	0	NUM
iajs-2920	91	32	,	,	PUNCT
iajs-2920	91	33	𝑎	𝑎	NOUN
iajs-2920	91	34	,	,	PUNCT
iajs-2920	91	35	𝑏	𝑏	NOUN
iajs-2920	91	36	}	}	PUNCT
iajs-2920	91	37	0.4	0.4	NUM
iajs-2920	91	38	𝑖𝑓	𝑖𝑓	NOUN
iajs-2920	91	39	𝜌	𝜌	X
iajs-2920	91	40	=	=	SYM
iajs-2920	91	41	𝑐	𝑐	PROPN
iajs-2920	91	42	,	,	PUNCT
iajs-2920	91	43	by	by	AUX
iajs-2920	91	44	apply	apply	VERB
iajs-2920	91	45	definition(12	definition(12	NOUN
iajs-2920	91	46	)	)	PUNCT
iajs-2920	91	47	,	,	PUNCT
iajs-2920	91	48	we	we	PRON
iajs-2920	91	49	can	can	AUX
iajs-2920	91	50	prove	prove	VERB
iajs-2920	91	51	that𝛿	that𝛿	ADJ
iajs-2920	91	52	=	=	SYM
iajs-2920	91	53	〈	〈	PROPN
iajs-2920	91	54	�	�	PROPN
iajs-2920	91	55	̃	̃	PROPN
iajs-2920	91	56	�	�	PROPN
iajs-2920	91	57	𝛿	𝛿	NOUN
iajs-2920	91	58	,	,	PUNCT
iajs-2920	91	59	𝛼𝛿〉is	𝛼𝛿〉is	PROPN
iajs-2920	91	60	a	a	DET
iajs-2920	91	61	cubic	cubic	ADJ
iajs-2920	91	62	sub	sub	NOUN
iajs-2920	91	63	-	-	ADJ
iajs-2920	91	64	algebra	algebra	ADJ
iajs-2920	91	65	ofℵˑ.	ofℵˑ.	NOUN
iajs-2920	91	66	ihjpas	ihjpas	PROPN
iajs-2920	91	67	.	.	PUNCT
iajs-2920	92	1	36(1)2023	36(1)2023	NUM
iajs-2920	92	2	371	371	NUM
iajs-2920	92	3	proposition	proposition	NOUN
iajs-2920	92	4	(	(	PUNCT
iajs-2920	92	5	14	14	NUM
iajs-2920	92	6	)	)	PUNCT
iajs-2920	92	7	.	.	PUNCT
iajs-2920	93	1	if	if	SCONJ
iajs-2920	93	2	𝛿	𝛿	ADJ
iajs-2920	93	3	=	=	PROPN
iajs-2920	93	4	〈	〈	PROPN
iajs-2920	93	5	�	�	PROPN
iajs-2920	93	6	̃	̃	PROPN
iajs-2920	93	7	�	�	PROPN
iajs-2920	93	8	𝛿	𝛿	NOUN
iajs-2920	93	9	,	,	PUNCT
iajs-2920	93	10	𝛼𝛿〉is	𝛼𝛿〉is	PROPN
iajs-2920	93	11	a	a	DET
iajs-2920	93	12	cubic	cubic	ADJ
iajs-2920	93	13	sub	sub	ADJ
iajs-2920	93	14	-	-	ADJ
iajs-2920	93	15	algebra	algebra	ADJ
iajs-2920	93	16	ofℵˑ	ofℵˑ	NOUN
iajs-2920	93	17	,	,	PUNCT
iajs-2920	93	18	then	then	ADV
iajs-2920	93	19	�	�	PROPN
iajs-2920	93	20	̃	̃	PROPN
iajs-2920	93	21	�	�	NOUN
iajs-2920	93	22	δ(0	δ(0	PROPN
iajs-2920	93	23	)	)	PUNCT
iajs-2920	93	24	≥	≥	NOUN
iajs-2920	93	25	�	�	PROPN
iajs-2920	93	26	̃	̃	PROPN
iajs-2920	93	27	�	�	NOUN
iajs-2920	93	28	δ(𝜌	δ(𝜌	NOUN
iajs-2920	93	29	)	)	PUNCT
iajs-2920	93	30	and	and	CCONJ
iajs-2920	93	31	𝛼δ(0	𝛼δ(0	NOUN
iajs-2920	93	32	)	)	PUNCT
iajs-2920	93	33	≤	≤	NOUN
iajs-2920	93	34	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	93	35	)	)	PUNCT
iajs-2920	93	36	,	,	PUNCT
iajs-2920	93	37	∀𝜌	∀𝜌	PUNCT
iajs-2920	93	38	∈	∈	PROPN
iajs-2920	93	39	ℵ	ℵ	NOUN
iajs-2920	93	40	proof.since	proof.since	NOUN
iajs-2920	93	41	𝜌	𝜌	ADP
iajs-2920	93	42	∗	∗	NOUN
iajs-2920	93	43	𝜌	𝜌	X
iajs-2920	93	44	=	=	SYM
iajs-2920	93	45	0	0	NUM
iajs-2920	93	46	,	,	PUNCT
iajs-2920	93	47	then	then	ADV
iajs-2920	93	48	�	�	PROPN
iajs-2920	93	49	̃	̃	PROPN
iajs-2920	93	50	�	�	NOUN
iajs-2920	93	51	δ(0	δ(0	NOUN
iajs-2920	93	52	)	)	PUNCT
iajs-2920	93	53	=	=	SYM
iajs-2920	93	54	�	�	PROPN
iajs-2920	93	55	̃	̃	PROPN
iajs-2920	93	56	�	�	PROPN
iajs-2920	93	57	δ(𝜌	δ(𝜌	NOUN
iajs-2920	93	58	∗	∗	NOUN
iajs-2920	93	59	𝜌	𝜌	ADP
iajs-2920	93	60	)	)	PUNCT
iajs-2920	93	61	≥	≥	PROPN
iajs-2920	93	62	𝑟𝑚𝑖	𝑟𝑚𝑖	PROPN
iajs-2920	93	63	𝑛{	𝑛{	PROPN
iajs-2920	93	64	�	�	SYM
iajs-2920	93	65	̃	̃	PROPN
iajs-2920	93	66	�	�	PROPN
iajs-2920	93	67	δ(𝜌)ˑ	δ(𝜌)ˑ	PROPN
iajs-2920	93	68	,	,	PUNCT
iajs-2920	93	69	�	�	PROPN
iajs-2920	93	70	̃	̃	PROPN
iajs-2920	93	71	�	�	NOUN
iajs-2920	93	72	δ(𝜌	δ(𝜌	NOUN
iajs-2920	93	73	)	)	PUNCT
iajs-2920	93	74	}	}	PUNCT
iajs-2920	93	75	=	=	SYM
iajs-2920	93	76	�	�	PROPN
iajs-2920	93	77	̃	̃	PROPN
iajs-2920	93	78	�	�	PROPN
iajs-2920	93	79	δ(𝜌)and	δ(𝜌)and	NOUN
iajs-2920	93	80	𝛼δ(0	𝛼δ(0	NOUN
iajs-2920	93	81	)	)	PUNCT
iajs-2920	93	82	=	=	PUNCT
iajs-2920	93	83	𝛼δ(𝜌	𝛼δ(𝜌	X
iajs-2920	93	84	∗	∗	NOUN
iajs-2920	93	85	𝜌	𝜌	ADP
iajs-2920	93	86	)	)	PUNCT
iajs-2920	93	87	≤	≤	NOUN
iajs-2920	93	88	𝑚𝑎	𝑚𝑎	ADP
iajs-2920	93	89	𝑥{𝛼δ(𝜌)ˑ	𝑥{𝛼δ(𝜌)ˑ	NOUN
iajs-2920	93	90	,	,	PUNCT
iajs-2920	93	91	ˑ𝛼δ(𝜌	ˑ𝛼δ(𝜌	ADJ
iajs-2920	93	92	)	)	PUNCT
iajs-2920	93	93	}	}	PUNCT
iajs-2920	93	94	=	=	SYM
iajs-2920	93	95	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	93	96	)	)	PUNCT
iajs-2920	93	97	.	.	PUNCT
iajs-2920	94	1	theorem	theorem	NOUN
iajs-2920	94	2	(	(	PUNCT
iajs-2920	94	3	15	15	NUM
iajs-2920	94	4	)	)	PUNCT
iajs-2920	94	5	.	.	PUNCT
iajs-2920	95	1	let	let	VERB
iajs-2920	95	2	𝛿	𝛿	ADJ
iajs-2920	95	3	=	=	PROPN
iajs-2920	95	4	〈	〈	PROPN
iajs-2920	95	5	�	�	PROPN
iajs-2920	95	6	̃	̃	PROPN
iajs-2920	95	7	�	�	PROPN
iajs-2920	95	8	𝛿	𝛿	NOUN
iajs-2920	95	9	,	,	PUNCT
iajs-2920	95	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	95	11	〉	〉	NOUN
iajs-2920	95	12	be	be	VERB
iajs-2920	95	13	a	a	DET
iajs-2920	95	14	cubic	cubic	ADJ
iajs-2920	95	15	set	set	NOUN
iajs-2920	95	16	in	in	ADP
iajs-2920	95	17	ℵ	ℵ	NOUN
iajs-2920	95	18	,	,	PUNCT
iajs-2920	95	19	then	then	ADV
iajs-2920	95	20	𝛿	𝛿	ADJ
iajs-2920	95	21	=	=	PROPN
iajs-2920	95	22	〈	〈	PROPN
iajs-2920	95	23	�	�	PROPN
iajs-2920	95	24	̃	̃	PROPN
iajs-2920	95	25	�	�	PROPN
iajs-2920	95	26	𝛿	𝛿	NOUN
iajs-2920	95	27	,	,	PUNCT
iajs-2920	95	28	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	95	29	〉	〉	NOUN
iajs-2920	95	30	is	be	AUX
iajs-2920	95	31	a	a	DET
iajs-2920	95	32	cubic	cubic	ADJ
iajs-2920	95	33	sub	sub	NOUN
iajs-2920	95	34	-	-	NOUN
iajs-2920	95	35	algebra	algebra	NOUN
iajs-2920	95	36	of	of	ADP
iajs-2920	95	37	ℵ	ℵ	NOUN
iajs-2920	95	38	if	if	NOUN
iajs-2920	95	39	and	and	CCONJ
iajs-2920	95	40	only	only	ADV
iajs-2920	95	41	if	if	SCONJ
iajs-2920	95	42	for	for	ADP
iajs-2920	95	43	all	all	DET
iajs-2920	95	44	�	�	PROPN
iajs-2920	95	45	̃	̃	PROPN
iajs-2920	95	46	�	�	PROPN
iajs-2920	95	47	∈	∈	PROPN
iajs-2920	95	48	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	95	49	]	]	PUNCT
iajs-2920	95	50	and	and	CCONJ
iajs-2920	95	51	𝑠	𝑠	PRON
iajs-2920	95	52	∈	∈	PROPN
iajs-2920	96	1	[	[	X
iajs-2920	96	2	0,1	0,1	NUM
iajs-2920	96	3	]	]	PUNCT
iajs-2920	96	4	,	,	PUNCT
iajs-2920	96	5	the	the	DET
iajs-2920	96	6	set	set	NOUN
iajs-2920	96	7	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	96	8	;	;	PUNCT
iajs-2920	96	9	�	�	PROPN
iajs-2920	96	10	̃	̃	PROPN
iajs-2920	96	11	�	�	PROPN
iajs-2920	96	12	,	,	PUNCT
iajs-2920	96	13	𝑠	𝑠	NOUN
iajs-2920	96	14	)	)	PUNCT
iajs-2920	96	15	is	be	AUX
iajs-2920	96	16	either	either	CCONJ
iajs-2920	96	17	empty	empty	ADJ
iajs-2920	96	18	or	or	CCONJ
iajs-2920	96	19	a	a	DET
iajs-2920	96	20	sub	sub	NOUN
iajs-2920	96	21	-	-	NOUN
iajs-2920	96	22	algebra	algebra	NOUN
iajs-2920	96	23	of	of	ADP
iajs-2920	96	24	ℵ	ℵ	NOUN
iajs-2920	96	25	.	.	PUNCT
iajs-2920	97	1	proof	proof	NOUN
iajs-2920	97	2	.	.	PUNCT
iajs-2920	98	1	assumethat	assumethat	PROPN
iajs-2920	98	2	𝛿	𝛿	ADJ
iajs-2920	98	3	=	=	PROPN
iajs-2920	98	4	〈	〈	PROPN
iajs-2920	98	5	�	�	PROPN
iajs-2920	98	6	̃	̃	PROPN
iajs-2920	98	7	�	�	PROPN
iajs-2920	98	8	𝛿	𝛿	NOUN
iajs-2920	98	9	,	,	PUNCT
iajs-2920	98	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	98	11	〉	〉	NOUN
iajs-2920	98	12	is	be	AUX
iajs-2920	98	13	a	a	DET
iajs-2920	98	14	cubic	cubic	ADJ
iajs-2920	98	15	sub	sub	NOUN
iajs-2920	98	16	-	-	NOUN
iajs-2920	98	17	algebra	algebra	NOUN
iajs-2920	98	18	of	of	ADP
iajs-2920	98	19	ℵ	ℵ	NOUN
iajs-2920	98	20	,	,	PUNCT
iajs-2920	98	21	let	let	VERB
iajs-2920	98	22	�	�	PROPN
iajs-2920	98	23	̃	̃	PROPN
iajs-2920	98	24	�	�	PROPN
iajs-2920	98	25	∈	∈	PROPN
iajs-2920	98	26	𝐷[0,1	𝐷[0,1	NOUN
iajs-2920	98	27	]	]	PUNCT
iajs-2920	98	28	and	and	CCONJ
iajs-2920	98	29	𝑠	𝑠	PRON
iajs-2920	98	30	∈	∈	PROPN
iajs-2920	99	1	[	[	X
iajs-2920	99	2	0,1	0,1	NUM
iajs-2920	99	3	]	]	PUNCT
iajs-2920	99	4	,	,	PUNCT
iajs-2920	99	5	be	be	AUX
iajs-2920	99	6	such	such	ADJ
iajs-2920	99	7	that	that	SCONJ
iajs-2920	99	8	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	99	9	;	;	PUNCT
iajs-2920	99	10	�	�	PROPN
iajs-2920	99	11	̃	̃	PROPN
iajs-2920	99	12	�	�	PROPN
iajs-2920	99	13	,	,	PUNCT
iajs-2920	99	14	𝑠	𝑠	NOUN
iajs-2920	99	15	)	)	PUNCT
iajs-2920	99	16	≠	≠	PROPN
iajs-2920	99	17	∅.	∅.	VERB
iajs-2920	99	18	then	then	ADV
iajs-2920	99	19	,	,	PUNCT
iajs-2920	99	20	for	for	ADP
iajs-2920	99	21	any	any	DET
iajs-2920	99	22	𝜌	𝜌	NOUN
iajs-2920	99	23	,	,	PUNCT
iajs-2920	99	24	𝜏	𝜏	PROPN
iajs-2920	99	25	∈	∈	PROPN
iajs-2920	99	26	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	99	27	,	,	PUNCT
iajs-2920	99	28	�	�	PROPN
iajs-2920	99	29	̃	̃	PROPN
iajs-2920	99	30	�	�	PROPN
iajs-2920	99	31	,	,	PUNCT
iajs-2920	99	32	𝑠	𝑠	PROPN
iajs-2920	99	33	)	)	PUNCT
iajs-2920	99	34	we	we	PRON
iajs-2920	99	35	have	have	VERB
iajs-2920	99	36	�	�	PROPN
iajs-2920	99	37	̃	̃	PROPN
iajs-2920	99	38	�	�	PROPN
iajs-2920	99	39	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	99	40	)	)	PUNCT
iajs-2920	99	41	≥	≥	NOUN
iajs-2920	99	42	�	�	PROPN
iajs-2920	99	43	̃	̃	PROPN
iajs-2920	99	44	�	�	PROPN
iajs-2920	99	45	,	,	PUNCT
iajs-2920	99	46	�	�	PROPN
iajs-2920	99	47	̃	̃	PROPN
iajs-2920	99	48	�	�	NOUN
iajs-2920	99	49	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	99	50	)	)	PUNCT
iajs-2920	99	51	≥	≥	NOUN
iajs-2920	99	52	�	�	PROPN
iajs-2920	99	53	̃	̃	PROPN
iajs-2920	99	54	�	�	PROPN
iajs-2920	99	55	and	and	CCONJ
iajs-2920	99	56	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	99	57	)	)	PUNCT
iajs-2920	99	58	≤	≤	NOUN
iajs-2920	99	59	𝑠,𝛼𝛿(𝜏	𝑠,𝛼𝛿(𝜏	NOUN
iajs-2920	99	60	)	)	PUNCT
iajs-2920	100	1	≤	≤	NOUN
iajs-2920	100	2	𝑠	𝑠	INTJ
iajs-2920	101	1	and	and	CCONJ
iajs-2920	101	2	since	since	SCONJ
iajs-2920	101	3	𝛿	𝛿	ADJ
iajs-2920	101	4	=	=	PROPN
iajs-2920	101	5	〈	〈	PROPN
iajs-2920	101	6	�	�	PROPN
iajs-2920	101	7	̃	̃	PROPN
iajs-2920	101	8	�	�	PROPN
iajs-2920	101	9	𝛿	𝛿	NOUN
iajs-2920	101	10	,	,	PUNCT
iajs-2920	101	11	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	101	12	〉	〉	NOUN
iajs-2920	101	13	is	be	AUX
iajs-2920	101	14	a	a	DET
iajs-2920	101	15	cubic	cubic	ADJ
iajs-2920	101	16	sub	sub	NOUN
iajs-2920	101	17	-	-	NOUN
iajs-2920	101	18	algebra	algebra	ADJ
iajs-2920	101	19	,	,	PUNCT
iajs-2920	101	20	we	we	PRON
iajs-2920	101	21	have	have	VERB
iajs-2920	101	22	�	�	PROPN
iajs-2920	101	23	̃	̃	PROPN
iajs-2920	101	24	�	�	PROPN
iajs-2920	101	25	δ(𝜌	δ(𝜌	NOUN
iajs-2920	101	26	∗	∗	NOUN
iajs-2920	101	27	𝜏	𝜏	NOUN
iajs-2920	101	28	)	)	PUNCT
iajs-2920	101	29	≥	≥	NOUN
iajs-2920	101	30	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	101	31	�	�	PROPN
iajs-2920	101	32	̃	̃	PROPN
iajs-2920	101	33	�	�	NOUN
iajs-2920	101	34	(𝜌	(𝜌	NOUN
iajs-2920	101	35	)	)	PUNCT
iajs-2920	101	36	,	,	PUNCT
iajs-2920	101	37	�	�	PROPN
iajs-2920	101	38	̃	̃	PROPN
iajs-2920	101	39	�	�	PROPN
iajs-2920	101	40	(𝜏	(𝜏	VERB
iajs-2920	101	41	)	)	PUNCT
iajs-2920	101	42	}	}	PUNCT
iajs-2920	101	43	=	=	SYM
iajs-2920	101	44	�	�	PROPN
iajs-2920	101	45	̃	̃	PROPN
iajs-2920	101	46	�	�	PROPN
iajs-2920	101	47	.	.	PROPN
iajs-2920	101	48	𝛼δ(𝜌	𝛼δ(𝜌	X
iajs-2920	101	49	∗	∗	NOUN
iajs-2920	101	50	𝜏	𝜏	NOUN
iajs-2920	101	51	)	)	PUNCT
iajs-2920	101	52	≤	≤	NOUN
iajs-2920	101	53	𝑚𝑎𝑥{𝛼δ(𝜌	𝑚𝑎𝑥{𝛼δ(𝜌	X
iajs-2920	101	54	)	)	PUNCT
iajs-2920	101	55	,	,	PUNCT
iajs-2920	101	56	𝛼δ(𝜏	𝛼δ(𝜏	NOUN
iajs-2920	101	57	)	)	PUNCT
iajs-2920	101	58	}	}	PUNCT
iajs-2920	101	59	=	=	SYM
iajs-2920	101	60	𝑠	𝑠	PROPN
iajs-2920	101	61	,	,	PUNCT
iajs-2920	101	62	so	so	SCONJ
iajs-2920	101	63	that	that	SCONJ
iajs-2920	101	64	𝜌	𝜌	ADP
iajs-2920	101	65	∗	∗	NOUN
iajs-2920	101	66	𝜏	𝜏	X
iajs-2920	101	67	∈	∈	NOUN
iajs-2920	101	68	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	101	69	;	;	PUNCT
iajs-2920	101	70	�	�	PROPN
iajs-2920	101	71	̃	̃	PROPN
iajs-2920	101	72	�	�	PROPN
iajs-2920	101	73	,	,	PUNCT
iajs-2920	101	74	𝑠	𝑠	NOUN
iajs-2920	101	75	)	)	PUNCT
iajs-2920	101	76	.	.	PUNCT
iajs-2920	102	1	hence	hence	ADV
iajs-2920	102	2	,	,	PUNCT
iajs-2920	102	3	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	102	4	;	;	PUNCT
iajs-2920	102	5	�	�	PROPN
iajs-2920	102	6	̃	̃	PROPN
iajs-2920	102	7	�	�	PROPN
iajs-2920	102	8	,	,	PUNCT
iajs-2920	102	9	𝑠	𝑠	NOUN
iajs-2920	102	10	)	)	PUNCT
iajs-2920	102	11	is	be	AUX
iajs-2920	102	12	a	a	DET
iajs-2920	102	13	sub	sub	NOUN
iajs-2920	102	14	-	-	NOUN
iajs-2920	102	15	algebraof	algebraof	NOUN
iajs-2920	102	16	ℵ.	ℵ.	NOUN
iajs-2920	102	17	conversely	conversely	ADV
iajs-2920	102	18	,	,	PUNCT
iajs-2920	102	19	suppose	suppose	VERB
iajs-2920	102	20	that	that	SCONJ
iajs-2920	102	21	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	102	22	;	;	PUNCT
iajs-2920	102	23	�	�	PROPN
iajs-2920	102	24	̃	̃	PROPN
iajs-2920	102	25	�	�	PROPN
iajs-2920	102	26	,	,	PUNCT
iajs-2920	102	27	𝑠	𝑠	NOUN
iajs-2920	102	28	)	)	PUNCT
iajs-2920	102	29	is	be	AUX
iajs-2920	102	30	a	a	DET
iajs-2920	102	31	sub	sub	ADJ
iajs-2920	102	32	-	-	NOUN
iajs-2920	102	33	algebraof	algebraof	NOUN
iajs-2920	102	34	ℵand	ℵand	NOUN
iajs-2920	102	35	let	let	VERB
iajs-2920	102	36	𝜌	𝜌	X
iajs-2920	102	37	,	,	PUNCT
iajs-2920	102	38	𝜏	𝜏	PROPN
iajs-2920	102	39	∈	∈	NOUN
iajs-2920	102	40	ℵ.	ℵ.	NOUN
iajs-2920	102	41	take	take	VERB
iajs-2920	102	42	�	�	PROPN
iajs-2920	102	43	̃	̃	PROPN
iajs-2920	102	44	�	�	PROPN
iajs-2920	102	45	=	=	SYM
iajs-2920	102	46	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	102	47	�	�	PROPN
iajs-2920	102	48	̃	̃	PROPN
iajs-2920	102	49	�	�	NOUN
iajs-2920	102	50	(𝜌	(𝜌	NOUN
iajs-2920	102	51	)	)	PUNCT
iajs-2920	102	52	,	,	PUNCT
iajs-2920	102	53	�	�	PROPN
iajs-2920	102	54	̃	̃	PROPN
iajs-2920	102	55	�	�	PROPN
iajs-2920	102	56	(𝜏	(𝜏	VERB
iajs-2920	102	57	)	)	PUNCT
iajs-2920	102	58	}	}	PUNCT
iajs-2920	102	59	and	and	CCONJ
iajs-2920	102	60	𝑠	𝑠	PROPN
iajs-2920	102	61	=	=	PUNCT
iajs-2920	102	62	𝑚𝑎𝑥{𝛼δ(𝜌	𝑚𝑎𝑥{𝛼δ(𝜌	PROPN
iajs-2920	102	63	)	)	PUNCT
iajs-2920	102	64	,	,	PUNCT
iajs-2920	102	65	𝛼δ(𝜏	𝛼δ(𝜏	NOUN
iajs-2920	102	66	)	)	PUNCT
iajs-2920	102	67	}	}	PUNCT
iajs-2920	102	68	by	by	ADP
iajs-2920	102	69	assumption	assumption	NOUN
iajs-2920	102	70	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	102	71	;	;	PUNCT
iajs-2920	102	72	�	�	PROPN
iajs-2920	102	73	̃	̃	PROPN
iajs-2920	102	74	�	�	PROPN
iajs-2920	102	75	,	,	PUNCT
iajs-2920	102	76	𝑠	𝑠	NOUN
iajs-2920	102	77	)	)	PUNCT
iajs-2920	102	78	is	be	AUX
iajs-2920	102	79	sub	sub	NOUN
iajs-2920	102	80	algebra	algebra	NOUN
iajs-2920	102	81	of	of	ADP
iajs-2920	102	82	ℵ	ℵ	ADJ
iajs-2920	102	83	implies	implie	NOUN
iajs-2920	102	84	:	:	PUNCT
iajs-2920	102	85	𝜌	𝜌	X
iajs-2920	102	86	∗	∗	X
iajs-2920	102	87	𝜏	𝜏	X
iajs-2920	102	88	∈	∈	NOUN
iajs-2920	103	1	𝑈(𝛿	𝑈(𝛿	PROPN
iajs-2920	103	2	;	;	PUNCT
iajs-2920	103	3	�	�	PROPN
iajs-2920	103	4	̃	̃	PROPN
iajs-2920	103	5	�	�	PROPN
iajs-2920	103	6	,	,	PUNCT
iajs-2920	103	7	𝑠	𝑠	PROPN
iajs-2920	103	8	)	)	PUNCT
iajs-2920	103	9	,	,	PUNCT
iajs-2920	103	10	therefore	therefore	ADV
iajs-2920	103	11	�	�	PROPN
iajs-2920	103	12	̃	̃	PROPN
iajs-2920	103	13	�	�	PROPN
iajs-2920	103	14	δ(𝜌	δ(𝜌	NOUN
iajs-2920	103	15	∗	∗	NOUN
iajs-2920	103	16	𝜏	𝜏	NOUN
iajs-2920	103	17	)	)	PUNCT
iajs-2920	103	18	≥	≥	NOUN
iajs-2920	103	19	�	�	PROPN
iajs-2920	103	20	̃	̃	PROPN
iajs-2920	103	21	�	�	PROPN
iajs-2920	103	22	=	=	SYM
iajs-2920	103	23	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	103	24	�	�	PROPN
iajs-2920	103	25	̃	̃	PROPN
iajs-2920	103	26	�	�	NOUN
iajs-2920	103	27	(𝜌	(𝜌	NOUN
iajs-2920	103	28	)	)	PUNCT
iajs-2920	103	29	,	,	PUNCT
iajs-2920	103	30	�	�	PROPN
iajs-2920	103	31	̃	̃	PROPN
iajs-2920	103	32	�	�	PROPN
iajs-2920	103	33	(𝜏	(𝜏	VERB
iajs-2920	103	34	)	)	PUNCT
iajs-2920	103	35	}	}	PUNCT
iajs-2920	103	36	and	and	CCONJ
iajs-2920	103	37	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	103	38	∗	∗	NOUN
iajs-2920	103	39	𝜏	𝜏	NOUN
iajs-2920	103	40	)	)	PUNCT
iajs-2920	103	41	≤	≤	NUM
iajs-2920	103	42	𝑠	𝑠	PROPN
iajs-2920	103	43	=	=	SYM
iajs-2920	103	44	𝑚𝑎𝑥{𝛼δ(𝜌	𝑚𝑎𝑥{𝛼δ(𝜌	PROPN
iajs-2920	103	45	)	)	PUNCT
iajs-2920	103	46	,	,	PUNCT
iajs-2920	103	47	𝛼δ(𝜏	𝛼δ(𝜏	NOUN
iajs-2920	103	48	)	)	PUNCT
iajs-2920	103	49	}	}	PUNCT
iajs-2920	103	50	.	.	PUNCT
iajs-2920	104	1	hence	hence	ADV
iajs-2920	104	2	𝛿	𝛿	ADJ
iajs-2920	104	3	=	=	PUNCT
iajs-2920	104	4	〈	〈	PROPN
iajs-2920	104	5	�	�	PROPN
iajs-2920	104	6	̃	̃	PROPN
iajs-2920	104	7	�	�	PROPN
iajs-2920	104	8	𝛿	𝛿	NOUN
iajs-2920	104	9	,	,	PUNCT
iajs-2920	104	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	104	11	〉	〉	NOUN
iajs-2920	104	12	is	be	AUX
iajs-2920	104	13	a	a	DET
iajs-2920	104	14	cubic	cubic	ADJ
iajs-2920	104	15	sub	sub	NOUN
iajs-2920	104	16	-	-	NOUN
iajs-2920	104	17	algebra	algebra	NOUN
iajs-2920	104	18	of	of	ADP
iajs-2920	104	19	ℵ	ℵ	NOUN
iajs-2920	104	20	.	.	PUNCT
iajs-2920	104	21	definition(16	definition(16	NOUN
iajs-2920	104	22	)	)	PUNCT
iajs-2920	104	23	.	.	PUNCT
iajs-2920	105	1	let	let	VERB
iajs-2920	105	2	ℵ	ℵ	NOUN
iajs-2920	105	3	be	be	AUX
iajs-2920	105	4	a	a	DET
iajs-2920	105	5	tm	tm	NOUN
iajs-2920	105	6	-	-	NOUN
iajs-2920	105	7	algebra	algebra	NOUN
iajs-2920	105	8	.	.	PUNCT
iajs-2920	106	1	a	a	DET
iajs-2920	106	2	cubic	cubic	ADJ
iajs-2920	106	3	set𝛿	set𝛿	NOUN
iajs-2920	106	4	=	=	SYM
iajs-2920	106	5	〈	〈	PROPN
iajs-2920	106	6	�	�	PROPN
iajs-2920	106	7	̃	̃	PROPN
iajs-2920	106	8	�	�	PROPN
iajs-2920	106	9	𝛿	𝛿	NOUN
iajs-2920	106	10	,	,	PUNCT
iajs-2920	106	11	𝛼𝛿	𝛼𝛿	PART
iajs-2920	106	12	〉	〉	NOUN
iajs-2920	106	13	in	in	ADP
iajs-2920	106	14	ℵ	ℵ	NOUN
iajs-2920	106	15	is	be	AUX
iajs-2920	106	16	said	say	VERB
iajs-2920	106	17	to	to	PART
iajs-2920	106	18	be	be	AUX
iajs-2920	106	19	a	a	DET
iajs-2920	106	20	cubic	cubic	ADJ
iajs-2920	106	21	ideal	ideal	NOUN
iajs-2920	106	22	if	if	SCONJ
iajs-2920	106	23	:	:	PUNCT
iajs-2920	106	24	(	(	PUNCT
iajs-2920	106	25	h1)	h1)	PROPN
iajs-2920	106	26	�	�	PROPN
iajs-2920	106	27	̃	̃	PROPN
iajs-2920	106	28	�	�	NOUN
iajs-2920	106	29	δ(0	δ(0	PROPN
iajs-2920	106	30	)	)	PUNCT
iajs-2920	106	31	≥	≥	NOUN
iajs-2920	106	32	�	�	PROPN
iajs-2920	106	33	̃	̃	PROPN
iajs-2920	106	34	�	�	NOUN
iajs-2920	106	35	δ(𝜌	δ(𝜌	NOUN
iajs-2920	106	36	)	)	PUNCT
iajs-2920	106	37	and	and	CCONJ
iajs-2920	106	38	𝛼δ(0	𝛼δ(0	NOUN
iajs-2920	106	39	)	)	PUNCT
iajs-2920	106	40	≤	≤	NOUN
iajs-2920	106	41	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	106	42	)	)	PUNCT
iajs-2920	106	43	.	.	PUNCT
iajs-2920	107	1	(	(	PUNCT
iajs-2920	107	2	h2)	h2)	PROPN
iajs-2920	107	3	�	�	PROPN
iajs-2920	107	4	̃	̃	PROPN
iajs-2920	107	5	�	�	NOUN
iajs-2920	107	6	δ(𝜌	δ(𝜌	NOUN
iajs-2920	107	7	)	)	PUNCT
iajs-2920	107	8	≥	≥	NOUN
iajs-2920	107	9	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	107	10	�	�	PROPN
iajs-2920	107	11	̃	̃	PROPN
iajs-2920	107	12	�	�	PROPN
iajs-2920	107	13	δ(𝜌	δ(𝜌	NOUN
iajs-2920	107	14	∗	∗	NOUN
iajs-2920	107	15	𝜏	𝜏	NOUN
iajs-2920	107	16	)	)	PUNCT
iajs-2920	107	17	,	,	PUNCT
iajs-2920	107	18	�	�	PROPN
iajs-2920	107	19	̃	̃	PROPN
iajs-2920	107	20	�	�	NOUN
iajs-2920	107	21	δ(𝜏	δ(𝜏	NOUN
iajs-2920	107	22	)	)	PUNCT
iajs-2920	107	23	}	}	PUNCT
iajs-2920	107	24	and	and	CCONJ
iajs-2920	107	25	𝛼δ(𝜌	𝛼δ(𝜌	X
iajs-2920	107	26	)	)	PUNCT
iajs-2920	107	27	≤	≤	NOUN
iajs-2920	107	28	𝑚𝑎𝑥{𝛼δ(𝜌	𝑚𝑎𝑥{𝛼δ(𝜌	X
iajs-2920	107	29	∗	∗	X
iajs-2920	107	30	𝜏	𝜏	NOUN
iajs-2920	107	31	)	)	PUNCT
iajs-2920	107	32	,	,	PUNCT
iajs-2920	107	33	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	107	34	)	)	PUNCT
iajs-2920	107	35	}	}	PUNCT
iajs-2920	108	1	,	,	PUNCT
iajs-2920	108	2	for	for	ADP
iajs-2920	108	3	all	all	DET
iajs-2920	108	4	𝜌	𝜌	NOUN
iajs-2920	108	5	,	,	PUNCT
iajs-2920	108	6	𝜏	𝜏	PROPN
iajs-2920	108	7	∈	∈	NOUN
iajs-2920	108	8	ℵ.	ℵ.	NOUN
iajs-2920	108	9	definition	definition	NOUN
iajs-2920	108	10	(	(	PUNCT
iajs-2920	108	11	17	17	NUM
iajs-2920	108	12	)	)	PUNCT
iajs-2920	108	13	.	.	PUNCT
iajs-2920	109	1	let	let	VERB
iajs-2920	109	2	ℵ	ℵ	NOUN
iajs-2920	109	3	be	be	AUX
iajs-2920	109	4	a	a	DET
iajs-2920	109	5	tm	tm	NOUN
iajs-2920	109	6	-	-	NOUN
iajs-2920	109	7	algebra	algebra	NOUN
iajs-2920	109	8	.	.	PUNCT
iajs-2920	110	1	a	a	DET
iajs-2920	110	2	cubic	cubic	ADJ
iajs-2920	110	3	set	set	VERB
iajs-2920	110	4	𝛿	𝛿	PROPN
iajs-2920	110	5	=	=	PUNCT
iajs-2920	110	6	〈	〈	NOUN
iajs-2920	110	7	�	�	PROPN
iajs-2920	110	8	̌	̌	NUM
iajs-2920	110	9	�	�	NOUN
iajs-2920	110	10	𝛿	𝛿	NOUN
iajs-2920	110	11	,	,	PUNCT
iajs-2920	110	12	𝛼𝛿	𝛼𝛿	PART
iajs-2920	110	13	〉	〉	NOUN
iajs-2920	110	14	in	in	ADP
iajs-2920	110	15	ℵ	ℵ	NOUN
iajs-2920	110	16	is	be	AUX
iajs-2920	110	17	said	say	VERB
iajs-2920	110	18	to	to	PART
iajs-2920	110	19	be	be	AUX
iajs-2920	110	20	a	a	DET
iajs-2920	110	21	cubic	cubic	ADJ
iajs-2920	110	22	tideal	tideal	NOUN
iajs-2920	110	23	if	if	SCONJ
iajs-2920	110	24	:	:	PUNCT
iajs-2920	110	25	(	(	PUNCT
iajs-2920	110	26	𝐵1)	𝐵1)	PROPN
iajs-2920	110	27	�	�	PROPN
iajs-2920	110	28	̃	̃	PROPN
iajs-2920	110	29	�	�	PROPN
iajs-2920	110	30	𝛿(0	𝛿(0	PROPN
iajs-2920	110	31	)	)	PUNCT
iajs-2920	110	32	≥	≥	NOUN
iajs-2920	110	33	�	�	PROPN
iajs-2920	110	34	̃	̃	PROPN
iajs-2920	110	35	�	�	PROPN
iajs-2920	110	36	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	110	37	)	)	PUNCT
iajs-2920	110	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	110	39	𝛼𝛿(0	𝛼𝛿(0	PROPN
iajs-2920	110	40	)	)	PUNCT
iajs-2920	110	41	≤	≤	NOUN
iajs-2920	110	42	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	110	43	)	)	PUNCT
iajs-2920	110	44	,	,	PUNCT
iajs-2920	110	45	(	(	PUNCT
iajs-2920	110	46	𝐵2)	𝐵2)	PROPN
iajs-2920	110	47	�	�	PROPN
iajs-2920	110	48	̃	̃	PROPN
iajs-2920	110	49	�	�	PROPN
iajs-2920	110	50	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	110	51	∗	∗	NOUN
iajs-2920	110	52	휀	휀	NOUN
iajs-2920	110	53	)	)	PUNCT
iajs-2920	110	54	≥	≥	NOUN
iajs-2920	110	55	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	110	56	�	�	PROPN
iajs-2920	110	57	̃	̃	PROPN
iajs-2920	110	58	�	�	PROPN
iajs-2920	110	59	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	110	60	∗	∗	PROPN
iajs-2920	110	61	𝜏	𝜏	NOUN
iajs-2920	110	62	)	)	PUNCT
iajs-2920	110	63	∗	∗	NOUN
iajs-2920	110	64	휀	휀	NOUN
iajs-2920	110	65	)	)	PUNCT
iajs-2920	110	66	,	,	PUNCT
iajs-2920	110	67	�	�	PROPN
iajs-2920	110	68	̃	̃	PROPN
iajs-2920	110	69	�	�	NOUN
iajs-2920	110	70	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	110	71	)	)	PUNCT
iajs-2920	110	72	}	}	PUNCT
iajs-2920	110	73	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
iajs-2920	110	74	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	110	75	∗	∗	NOUN
iajs-2920	110	76	휀	휀	NOUN
iajs-2920	110	77	)	)	PUNCT
iajs-2920	110	78	≤	≤	NOUN
iajs-2920	110	79	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	110	80	∗	∗	NOUN
iajs-2920	110	81	𝜏	𝜏	NOUN
iajs-2920	110	82	)	)	PUNCT
iajs-2920	110	83	∗	∗	NOUN
iajs-2920	110	84	휀	휀	NOUN
iajs-2920	110	85	)	)	PUNCT
iajs-2920	110	86	,	,	PUNCT
iajs-2920	110	87	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	110	88	)	)	PUNCT
iajs-2920	110	89	}	}	PUNCT
iajs-2920	110	90	.	.	PUNCT
iajs-2920	111	1	example	example	NOUN
iajs-2920	111	2	(	(	PUNCT
iajs-2920	111	3	18	18	NUM
iajs-2920	111	4	)	)	PUNCT
iajs-2920	111	5	.	.	PUNCT
iajs-2920	112	1	let	let	VERB
iajs-2920	112	2	ℵ	ℵ	NOUN
iajs-2920	112	3	=	=	SYM
iajs-2920	112	4	{	{	PUNCT
iajs-2920	112	5	0	0	NUM
iajs-2920	112	6	,	,	PUNCT
iajs-2920	112	7	𝑎	𝑎	NOUN
iajs-2920	112	8	,	,	PUNCT
iajs-2920	112	9	𝑏	𝑏	NOUN
iajs-2920	112	10	,	,	PUNCT
iajs-2920	112	11	𝑐	𝑐	NOUN
iajs-2920	112	12	}	}	PUNCT
iajs-2920	112	13	in	in	ADP
iajs-2920	112	14	example(13	example(13	NOUN
iajs-2920	112	15	)	)	PUNCT
iajs-2920	112	16	.	.	PUNCT
iajs-2920	113	1	define	define	VERB
iajs-2920	113	2	a	a	DET
iajs-2920	113	3	cubic	cubic	ADJ
iajs-2920	113	4	set	set	NOUN
iajs-2920	113	5	𝛿	𝛿	PROPN
iajs-2920	113	6	=	=	PROPN
iajs-2920	113	7	〈	〈	PROPN
iajs-2920	113	8	�	�	PROPN
iajs-2920	113	9	̃	̃	PROPN
iajs-2920	113	10	�	�	PROPN
iajs-2920	113	11	𝛿	𝛿	NOUN
iajs-2920	113	12	,	,	PUNCT
iajs-2920	113	13	𝛼𝛿	𝛼𝛿	PART
iajs-2920	113	14	〉	〉	NOUN
iajs-2920	113	15	in	in	ADP
iajs-2920	113	16	ℵ	ℵ	NOUN
iajs-2920	113	17	as	as	SCONJ
iajs-2920	113	18	follows	follow	VERB
iajs-2920	113	19	:	:	PUNCT
iajs-2920	113	20	ihjpas	ihjpas	PROPN
iajs-2920	113	21	.	.	PUNCT
iajs-2920	114	1	36(1)2023	36(1)2023	NUM
iajs-2920	114	2	372	372	NUM
iajs-2920	114	3	�	�	PROPN
iajs-2920	114	4	̃	̃	PROPN
iajs-2920	114	5	�	�	NOUN
iajs-2920	114	6	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	114	7	)	)	PUNCT
iajs-2920	115	1	=	=	PRON
iajs-2920	115	2	{	{	PUNCT
iajs-2920	116	1	[	[	X
iajs-2920	116	2	0.1,0.7	0.1,0.7	NOUN
iajs-2920	116	3	]	]	PUNCT
iajs-2920	116	4	,	,	PUNCT
iajs-2920	116	5	if	if	SCONJ
iajs-2920	116	6	𝜌	𝜌	ADP
iajs-2920	116	7	=	=	SYM
iajs-2920	116	8	0	0	NUM
iajs-2920	116	9	,	,	PUNCT
iajs-2920	116	10	[	[	X
iajs-2920	116	11	0.4,0.5	0.4,0.5	X
iajs-2920	116	12	]	]	X
iajs-2920	116	13	,	,	PUNCT
iajs-2920	116	14	if	if	SCONJ
iajs-2920	116	15	𝜌	𝜌	ADP
iajs-2920	116	16	∈	∈	PROPN
iajs-2920	116	17	{	{	PUNCT
iajs-2920	116	18	𝑎	𝑎	NOUN
iajs-2920	116	19	,	,	PUNCT
iajs-2920	116	20	𝑏	𝑏	NOUN
iajs-2920	116	21	}	}	PUNCT
iajs-2920	116	22	[	[	X
iajs-2920	116	23	0.1,0.3	0.1,0.3	PROPN
iajs-2920	116	24	]	]	X
iajs-2920	116	25	,	,	PUNCT
iajs-2920	116	26	if	if	SCONJ
iajs-2920	116	27	𝜌	𝜌	ADP
iajs-2920	116	28	∈	∈	PROPN
iajs-2920	116	29	𝑐	𝑐	PROPN
iajs-2920	116	30	,	,	PUNCT
iajs-2920	116	31	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	116	32	)	)	PUNCT
iajs-2920	116	33	=	=	NOUN
iajs-2920	116	34	{	{	PUNCT
iajs-2920	116	35	0.1	0.1	NUM
iajs-2920	116	36	,	,	PUNCT
iajs-2920	116	37	if	if	SCONJ
iajs-2920	116	38	𝜌	𝜌	ADP
iajs-2920	116	39	=	=	SYM
iajs-2920	116	40	0	0	NUM
iajs-2920	116	41	,	,	PUNCT
iajs-2920	116	42	0.3	0.3	NUM
iajs-2920	116	43	,	,	PUNCT
iajs-2920	116	44	if	if	SCONJ
iajs-2920	116	45	𝜌	𝜌	X
iajs-2920	116	46	∈	∈	X
iajs-2920	116	47	{	{	PUNCT
iajs-2920	116	48	𝑎	𝑎	NOUN
iajs-2920	116	49	,	,	PUNCT
iajs-2920	116	50	𝑏	𝑏	NOUN
iajs-2920	116	51	}	}	PUNCT
iajs-2920	116	52	,	,	PUNCT
iajs-2920	116	53	0.6	0.6	NUM
iajs-2920	116	54	,	,	PUNCT
iajs-2920	116	55	if	if	SCONJ
iajs-2920	116	56	𝜌	𝜌	ADP
iajs-2920	116	57	∈	∈	NOUN
iajs-2920	116	58	𝑐	𝑐	NOUN
iajs-2920	116	59	then	then	ADV
iajs-2920	116	60	,	,	PUNCT
iajs-2920	116	61	we	we	PRON
iajs-2920	116	62	can	can	AUX
iajs-2920	116	63	easy	easy	ADV
iajs-2920	116	64	show	show	VERB
iajs-2920	116	65	that	that	SCONJ
iajs-2920	116	66	a	a	DET
iajs-2920	116	67	cubic	cubic	ADJ
iajs-2920	116	68	set	set	VERB
iajs-2920	116	69	𝛿	𝛿	PROPN
iajs-2920	116	70	=	=	PROPN
iajs-2920	116	71	〈	〈	PROPN
iajs-2920	116	72	�	�	PROPN
iajs-2920	116	73	̃	̃	PROPN
iajs-2920	116	74	�	�	PROPN
iajs-2920	116	75	𝛿	𝛿	NOUN
iajs-2920	116	76	,	,	PUNCT
iajs-2920	116	77	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	116	78	〉	〉	NOUN
iajs-2920	116	79	is	be	AUX
iajs-2920	116	80	a	a	DET
iajs-2920	116	81	cubic	cubic	ADJ
iajs-2920	116	82	t	t	NOUN
iajs-2920	116	83	-	-	PUNCT
iajs-2920	116	84	ideal	ideal	NOUN
iajs-2920	116	85	of	of	ADP
iajs-2920	116	86	ℵ	ℵ	NOUN
iajs-2920	116	87	.	.	PUNCT
iajs-2920	117	1	proposition	proposition	NOUN
iajs-2920	117	2	(	(	PUNCT
iajs-2920	117	3	19	19	NUM
iajs-2920	117	4	)	)	PUNCT
iajs-2920	117	5	.	.	PUNCT
iajs-2920	118	1	if	if	SCONJ
iajs-2920	118	2	𝛿	𝛿	ADJ
iajs-2920	118	3	=	=	PROPN
iajs-2920	118	4	〈	〈	PROPN
iajs-2920	118	5	�	�	PROPN
iajs-2920	118	6	̃	̃	PROPN
iajs-2920	118	7	�	�	PROPN
iajs-2920	118	8	𝛿	𝛿	NOUN
iajs-2920	118	9	,	,	PUNCT
iajs-2920	118	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	118	11	〉	〉	NOUN
iajs-2920	118	12	is	be	AUX
iajs-2920	118	13	a	a	DET
iajs-2920	118	14	cubic	cubic	ADJ
iajs-2920	118	15	t	t	NOUN
iajs-2920	118	16	-	-	PUNCT
iajs-2920	118	17	ideal	ideal	NOUN
iajs-2920	118	18	of	of	ADP
iajs-2920	118	19	tm	tm	NOUN
iajs-2920	118	20	-	-	NOUN
iajs-2920	118	21	algebra	algebra	NOUN
iajs-2920	118	22	ℵ	ℵ	NOUN
iajs-2920	118	23	,	,	PUNCT
iajs-2920	118	24	then	then	ADV
iajs-2920	118	25	�	�	PROPN
iajs-2920	118	26	̃	̃	PROPN
iajs-2920	118	27	�	�	PROPN
iajs-2920	118	28	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	118	29	∗	∗	NOUN
iajs-2920	118	30	(	(	PUNCT
iajs-2920	118	31	𝜌	𝜌	X
iajs-2920	118	32	∗	∗	X
iajs-2920	118	33	𝜏	𝜏	NOUN
iajs-2920	118	34	)	)	PUNCT
iajs-2920	118	35	)	)	PUNCT
iajs-2920	118	36	≥	≥	PROPN
iajs-2920	118	37	�	�	PROPN
iajs-2920	118	38	̃	̃	PROPN
iajs-2920	118	39	�	�	NOUN
iajs-2920	118	40	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	118	41	)	)	PUNCT
iajs-2920	118	42	,	,	PUNCT
iajs-2920	118	43	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	118	44	∗	∗	NOUN
iajs-2920	118	45	(	(	PUNCT
iajs-2920	118	46	𝜌	𝜌	X
iajs-2920	118	47	∗	∗	X
iajs-2920	118	48	𝜏	𝜏	NOUN
iajs-2920	118	49	)	)	PUNCT
iajs-2920	118	50	)	)	PUNCT
iajs-2920	118	51	≤	≤	NOUN
iajs-2920	118	52	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	118	53	)	)	PUNCT
iajs-2920	118	54	.	.	PUNCT
iajs-2920	119	1	proof	proof	NOUN
iajs-2920	119	2	.	.	PUNCT
iajs-2920	120	1	taking	take	VERB
iajs-2920	120	2	휀	휀	PRON
iajs-2920	120	3	=	=	X
iajs-2920	120	4	𝜌	𝜌	PART
iajs-2920	120	5	∗	∗	NOUN
iajs-2920	120	6	𝜏	𝜏	NOUN
iajs-2920	120	7	in	in	ADP
iajs-2920	120	8	definition	definition	NOUN
iajs-2920	120	9	3.6	3.6	NUM
iajs-2920	120	10	we	we	PRON
iajs-2920	120	11	get	get	VERB
iajs-2920	120	12	�	�	PROPN
iajs-2920	120	13	̃	̃	PROPN
iajs-2920	120	14	�	�	NOUN
iajs-2920	120	15	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	120	16	∗	∗	NOUN
iajs-2920	120	17	휀	휀	NOUN
iajs-2920	120	18	)	)	PUNCT
iajs-2920	120	19	≥	≥	NOUN
iajs-2920	120	20	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	120	21	�	�	PROPN
iajs-2920	120	22	̃	̃	PROPN
iajs-2920	120	23	�	�	PROPN
iajs-2920	120	24	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	120	25	∗	∗	PROPN
iajs-2920	120	26	𝜏	𝜏	NOUN
iajs-2920	120	27	)	)	PUNCT
iajs-2920	120	28	∗	∗	NOUN
iajs-2920	120	29	휀	휀	NOUN
iajs-2920	120	30	)	)	PUNCT
iajs-2920	120	31	,	,	PUNCT
iajs-2920	120	32	�	�	PROPN
iajs-2920	120	33	̃	̃	PROPN
iajs-2920	120	34	�	�	NOUN
iajs-2920	120	35	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	120	36	)	)	PUNCT
iajs-2920	120	37	}	}	PUNCT
iajs-2920	120	38	�	�	PROPN
iajs-2920	120	39	̃	̃	PROPN
iajs-2920	120	40	�	�	PROPN
iajs-2920	120	41	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	120	42	∗	∗	NOUN
iajs-2920	120	43	(	(	PUNCT
iajs-2920	120	44	𝜌	𝜌	X
iajs-2920	120	45	∗	∗	X
iajs-2920	120	46	𝜏	𝜏	NOUN
iajs-2920	120	47	)	)	PUNCT
iajs-2920	120	48	)	)	PUNCT
iajs-2920	120	49	≥	≥	PROPN
iajs-2920	120	50	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	120	51	�	�	PROPN
iajs-2920	120	52	̃	̃	PROPN
iajs-2920	120	53	�	�	PROPN
iajs-2920	120	54	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	120	55	∗	∗	NOUN
iajs-2920	120	56	𝜏	𝜏	NOUN
iajs-2920	120	57	)	)	PUNCT
iajs-2920	120	58	∗	∗	NOUN
iajs-2920	120	59	(	(	PUNCT
iajs-2920	120	60	𝜌	𝜌	X
iajs-2920	120	61	∗	∗	X
iajs-2920	120	62	𝜏	𝜏	NOUN
iajs-2920	120	63	)	)	PUNCT
iajs-2920	120	64	)	)	PUNCT
iajs-2920	120	65	,	,	PUNCT
iajs-2920	120	66	�	�	PROPN
iajs-2920	120	67	̃	̃	PROPN
iajs-2920	120	68	�	�	NOUN
iajs-2920	120	69	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	120	70	)	)	PUNCT
iajs-2920	120	71	}	}	PUNCT
iajs-2920	120	72	=	=	SYM
iajs-2920	120	73	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	120	74	�	�	PROPN
iajs-2920	120	75	̃	̃	PROPN
iajs-2920	120	76	�	�	PROPN
iajs-2920	120	77	𝛿(0	𝛿(0	PROPN
iajs-2920	120	78	)	)	PUNCT
iajs-2920	120	79	,	,	PUNCT
iajs-2920	120	80	�	�	PROPN
iajs-2920	120	81	̃	̃	PROPN
iajs-2920	120	82	�	�	NOUN
iajs-2920	120	83	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	120	84	)	)	PUNCT
iajs-2920	120	85	}	}	PUNCT
iajs-2920	120	86	=	=	SYM
iajs-2920	120	87	�	�	PROPN
iajs-2920	120	88	̃	̃	PROPN
iajs-2920	120	89	�	�	NOUN
iajs-2920	120	90	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	120	91	)	)	PUNCT
iajs-2920	120	92	and	and	CCONJ
iajs-2920	120	93	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	120	94	∗	∗	PROPN
iajs-2920	120	95	휀	휀	NOUN
iajs-2920	120	96	)	)	PUNCT
iajs-2920	120	97	≤	≤	NOUN
iajs-2920	120	98	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	120	99	∗	∗	NOUN
iajs-2920	120	100	𝜏	𝜏	NOUN
iajs-2920	120	101	)	)	PUNCT
iajs-2920	120	102	∗	∗	NOUN
iajs-2920	120	103	휀	휀	NOUN
iajs-2920	120	104	)	)	PUNCT
iajs-2920	120	105	,	,	PUNCT
iajs-2920	120	106	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	120	107	)	)	PUNCT
iajs-2920	120	108	}	}	PUNCT
iajs-2920	121	1	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	121	2	∗	∗	NOUN
iajs-2920	121	3	(	(	PUNCT
iajs-2920	121	4	𝜌	𝜌	X
iajs-2920	121	5	∗	∗	X
iajs-2920	121	6	𝜏	𝜏	NOUN
iajs-2920	121	7	)	)	PUNCT
iajs-2920	121	8	)	)	PUNCT
iajs-2920	121	9	≤	≤	NOUN
iajs-2920	121	10	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	121	11	∗	∗	NOUN
iajs-2920	121	12	𝜏	𝜏	NOUN
iajs-2920	121	13	)	)	PUNCT
iajs-2920	121	14	∗	∗	NOUN
iajs-2920	121	15	(	(	PUNCT
iajs-2920	121	16	𝜌	𝜌	X
iajs-2920	121	17	∗	∗	X
iajs-2920	121	18	𝜏	𝜏	NOUN
iajs-2920	121	19	)	)	PUNCT
iajs-2920	121	20	)	)	PUNCT
iajs-2920	121	21	,	,	PUNCT
iajs-2920	121	22	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	121	23	)	)	PUNCT
iajs-2920	121	24	}	}	PUNCT
iajs-2920	121	25	=	=	SYM
iajs-2920	121	26	𝑚𝑎𝑥{𝛼𝛿(0	𝑚𝑎𝑥{𝛼𝛿(0	PROPN
iajs-2920	121	27	)	)	PUNCT
iajs-2920	121	28	,	,	PUNCT
iajs-2920	121	29	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	121	30	)	)	PUNCT
iajs-2920	121	31	}	}	PUNCT
iajs-2920	121	32	=	=	SYM
iajs-2920	121	33	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	121	34	)	)	PUNCT
iajs-2920	121	35	.	.	PUNCT
iajs-2920	122	1	proposition	proposition	NOUN
iajs-2920	122	2	(	(	PUNCT
iajs-2920	122	3	20	20	NUM
iajs-2920	122	4	)	)	PUNCT
iajs-2920	122	5	.	.	PUNCT
iajs-2920	123	1	let𝛿	let𝛿	PROPN
iajs-2920	123	2	=	=	PUNCT
iajs-2920	123	3	〈	〈	PROPN
iajs-2920	123	4	�	�	PROPN
iajs-2920	123	5	̃	̃	PROPN
iajs-2920	123	6	�	�	PROPN
iajs-2920	123	7	𝛿	𝛿	NOUN
iajs-2920	123	8	,	,	PUNCT
iajs-2920	123	9	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	123	10	〉	〉	NOUN
iajs-2920	123	11	be	be	VERB
iajs-2920	123	12	a	a	DET
iajs-2920	123	13	cubic	cubic	ADJ
iajs-2920	123	14	t	t	NOUN
iajs-2920	123	15	-	-	PUNCT
iajs-2920	123	16	ideal	ideal	NOUN
iajs-2920	123	17	of	of	ADP
iajs-2920	123	18	tm	tm	NOUN
iajs-2920	123	19	-	-	NOUN
iajs-2920	123	20	algebraℵ	algebraℵ	NOUN
iajs-2920	123	21	.	.	PUNCT
iajs-2920	124	1	if	if	SCONJ
iajs-2920	124	2	the	the	DET
iajs-2920	124	3	inequality	inequality	NOUN
iajs-2920	124	4	𝜌	𝜌	ADP
iajs-2920	124	5	∗	∗	NOUN
iajs-2920	124	6	𝜏	𝜏	NOUN
iajs-2920	124	7	≤	≤	NOUN
iajs-2920	124	8	휀	휀	PRON
iajs-2920	124	9	holds	hold	NOUN
iajs-2920	124	10	in	in	ADP
iajs-2920	124	11	ℵ	ℵ	NOUN
iajs-2920	124	12	,	,	PUNCT
iajs-2920	124	13	then	then	ADV
iajs-2920	124	14	�	�	PROPN
iajs-2920	124	15	̃	̃	PROPN
iajs-2920	124	16	�	�	PROPN
iajs-2920	124	17	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	124	18	)	)	PUNCT
iajs-2920	124	19	≥	≥	NOUN
iajs-2920	124	20	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	124	21	�	�	PROPN
iajs-2920	124	22	̃	̃	PROPN
iajs-2920	124	23	�	�	NOUN
iajs-2920	124	24	𝛿(휀	𝛿(휀	NOUN
iajs-2920	124	25	)	)	PUNCT
iajs-2920	124	26	,	,	PUNCT
iajs-2920	124	27	�	�	PROPN
iajs-2920	124	28	̃	̃	PROPN
iajs-2920	124	29	�	�	NOUN
iajs-2920	124	30	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	124	31	)	)	PUNCT
iajs-2920	124	32	}	}	PUNCT
iajs-2920	124	33	and	and	CCONJ
iajs-2920	124	34	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	124	35	)	)	PUNCT
iajs-2920	124	36	≤	≤	NUM
iajs-2920	124	37	𝑚𝑎𝑥{𝛼𝛿(휀	𝑚𝑎𝑥{𝛼𝛿(휀	X
iajs-2920	124	38	)	)	PUNCT
iajs-2920	124	39	,	,	PUNCT
iajs-2920	124	40	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	124	41	)	)	PUNCT
iajs-2920	124	42	}	}	PUNCT
iajs-2920	124	43	.	.	PUNCT
iajs-2920	125	1	proof	proof	NOUN
iajs-2920	125	2	.	.	PUNCT
iajs-2920	126	1	assume	assume	VERB
iajs-2920	126	2	that	that	SCONJ
iajs-2920	126	3	the	the	DET
iajs-2920	126	4	inequality	inequality	NOUN
iajs-2920	126	5	𝜌	𝜌	ADP
iajs-2920	126	6	∗	∗	NOUN
iajs-2920	126	7	𝜏	𝜏	NOUN
iajs-2920	126	8	≤	≤	NOUN
iajs-2920	126	9	휀	휀	PRON
iajs-2920	126	10	holds	hold	NOUN
iajs-2920	126	11	inℵ	inℵ	PROPN
iajs-2920	126	12	,	,	PUNCT
iajs-2920	126	13	then	then	ADV
iajs-2920	126	14	(	(	PUNCT
iajs-2920	126	15	𝜌	𝜌	X
iajs-2920	126	16	∗	∗	X
iajs-2920	126	17	𝜏	𝜏	NOUN
iajs-2920	126	18	)	)	PUNCT
iajs-2920	126	19	∗	∗	NOUN
iajs-2920	126	20	휀	휀	NOUN
iajs-2920	126	21	=	=	NOUN
iajs-2920	126	22	0	0	NUM
iajs-2920	126	23	and	and	CCONJ
iajs-2920	126	24	by	by	ADP
iajs-2920	126	25	�	�	PROPN
iajs-2920	126	26	̃	̃	PROPN
iajs-2920	126	27	�	�	PROPN
iajs-2920	126	28	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	29	∗	∗	NOUN
iajs-2920	126	30	휀	휀	NOUN
iajs-2920	126	31	)	)	PUNCT
iajs-2920	126	32	≥	≥	NOUN
iajs-2920	126	33	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	34	�	�	PROPN
iajs-2920	126	35	̃	̃	PROPN
iajs-2920	126	36	�	�	PROPN
iajs-2920	126	37	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	126	38	∗	∗	PROPN
iajs-2920	126	39	𝜏	𝜏	NOUN
iajs-2920	126	40	)	)	PUNCT
iajs-2920	126	41	∗	∗	NOUN
iajs-2920	126	42	휀	휀	NOUN
iajs-2920	126	43	)	)	PUNCT
iajs-2920	126	44	,	,	PUNCT
iajs-2920	126	45	�	�	PROPN
iajs-2920	126	46	̃	̃	PROPN
iajs-2920	126	47	�	�	NOUN
iajs-2920	126	48	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	126	49	)	)	PUNCT
iajs-2920	126	50	}	}	PUNCT
iajs-2920	126	51	,	,	PUNCT
iajs-2920	126	52	if	if	SCONJ
iajs-2920	126	53	we	we	PRON
iajs-2920	126	54	put	put	VERB
iajs-2920	126	55	휀	휀	PRON
iajs-2920	126	56	=	=	NOUN
iajs-2920	126	57	0	0	NUM
iajs-2920	126	58	then,	then,	PROPN
iajs-2920	126	59	�	�	PROPN
iajs-2920	126	60	̃	̃	PROPN
iajs-2920	126	61	�	�	PROPN
iajs-2920	126	62	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	63	∗	∗	NOUN
iajs-2920	126	64	0	0	NUM
iajs-2920	126	65	)	)	PUNCT
iajs-2920	126	66	≥	≥	NOUN
iajs-2920	126	67	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	68	�	�	PROPN
iajs-2920	126	69	̃	̃	PROPN
iajs-2920	126	70	�	�	PROPN
iajs-2920	126	71	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	126	72	∗	∗	PROPN
iajs-2920	126	73	𝜏	𝜏	NOUN
iajs-2920	126	74	)	)	PUNCT
iajs-2920	126	75	∗	∗	NOUN
iajs-2920	126	76	0	0	NUM
iajs-2920	126	77	)	)	PUNCT
iajs-2920	126	78	,	,	PUNCT
iajs-2920	126	79	�	�	PROPN
iajs-2920	126	80	̃	̃	PROPN
iajs-2920	126	81	�	�	NOUN
iajs-2920	126	82	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	126	83	)	)	PUNCT
iajs-2920	126	84	}	}	PUNCT
iajs-2920	126	85	�	�	PROPN
iajs-2920	126	86	̃	̃	PROPN
iajs-2920	126	87	�	�	PROPN
iajs-2920	126	88	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	89	)	)	PUNCT
iajs-2920	126	90	≥	≥	NOUN
iajs-2920	126	91	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	92	�	�	PROPN
iajs-2920	126	93	̃	̃	PROPN
iajs-2920	126	94	�	�	PROPN
iajs-2920	126	95	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	96	∗	∗	NOUN
iajs-2920	126	97	𝜏	𝜏	NOUN
iajs-2920	126	98	)	)	PUNCT
iajs-2920	126	99	,	,	PUNCT
iajs-2920	126	100	�	�	PROPN
iajs-2920	126	101	̃	̃	PROPN
iajs-2920	126	102	�	�	NOUN
iajs-2920	126	103	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	126	104	)	)	PUNCT
iajs-2920	126	105	}	}	PUNCT
iajs-2920	126	106	…	…	PUNCT
iajs-2920	126	107	…	…	PUNCT
iajs-2920	126	108	…	…	PUNCT
iajs-2920	126	109	(	(	PUNCT
iajs-2920	126	110	𝑖	𝑖	X
iajs-2920	126	111	)	)	PUNCT
iajs-2920	126	112	but	but	CCONJ
iajs-2920	126	113	�	�	PROPN
iajs-2920	126	114	̃	̃	PROPN
iajs-2920	126	115	�	�	PROPN
iajs-2920	126	116	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	117	∗	∗	NOUN
iajs-2920	126	118	𝜏	𝜏	NOUN
iajs-2920	126	119	)	)	PUNCT
iajs-2920	126	120	≥	≥	NOUN
iajs-2920	126	121	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	122	�	�	PROPN
iajs-2920	126	123	̃	̃	PROPN
iajs-2920	126	124	�	�	PROPN
iajs-2920	126	125	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	126	126	∗	∗	VERB
iajs-2920	126	127	휀	휀	NOUN
iajs-2920	126	128	)	)	PUNCT
iajs-2920	126	129	∗	∗	NOUN
iajs-2920	126	130	𝜏	𝜏	NOUN
iajs-2920	126	131	)	)	PUNCT
iajs-2920	126	132	,	,	PUNCT
iajs-2920	126	133	�	�	PROPN
iajs-2920	126	134	̃	̃	PROPN
iajs-2920	126	135	�	�	NOUN
iajs-2920	126	136	𝛿(휀	𝛿(휀	NOUN
iajs-2920	126	137	)	)	PUNCT
iajs-2920	126	138	}	}	PUNCT
iajs-2920	126	139	�	�	PROPN
iajs-2920	126	140	̃	̃	PROPN
iajs-2920	126	141	�	�	PROPN
iajs-2920	126	142	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	143	∗	∗	NOUN
iajs-2920	126	144	𝜏	𝜏	NOUN
iajs-2920	126	145	)	)	PUNCT
iajs-2920	126	146	≥	≥	NOUN
iajs-2920	126	147	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	148	�	�	PROPN
iajs-2920	126	149	̃	̃	PROPN
iajs-2920	126	150	�	�	PROPN
iajs-2920	126	151	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	126	152	∗	∗	PROPN
iajs-2920	126	153	𝜏	𝜏	NOUN
iajs-2920	126	154	)	)	PUNCT
iajs-2920	126	155	∗	∗	NOUN
iajs-2920	126	156	휀	휀	NOUN
iajs-2920	126	157	)	)	PUNCT
iajs-2920	126	158	,	,	PUNCT
iajs-2920	126	159	�	�	PROPN
iajs-2920	126	160	̃	̃	PROPN
iajs-2920	126	161	�	�	NOUN
iajs-2920	126	162	𝛿(휀	𝛿(휀	NOUN
iajs-2920	126	163	)	)	PUNCT
iajs-2920	126	164	}	}	PUNCT
iajs-2920	126	165	=	=	SYM
iajs-2920	126	166	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	167	�	�	PROPN
iajs-2920	126	168	̃	̃	PROPN
iajs-2920	126	169	�	�	PROPN
iajs-2920	126	170	𝛿(0	𝛿(0	PROPN
iajs-2920	126	171	)	)	PUNCT
iajs-2920	126	172	,	,	PUNCT
iajs-2920	126	173	�	�	PROPN
iajs-2920	126	174	̃	̃	PROPN
iajs-2920	126	175	�	�	NOUN
iajs-2920	126	176	𝛿(휀	𝛿(휀	NOUN
iajs-2920	126	177	)	)	PUNCT
iajs-2920	126	178	}	}	PUNCT
iajs-2920	126	179	=	=	SYM
iajs-2920	126	180	�	�	PROPN
iajs-2920	126	181	̃	̃	PROPN
iajs-2920	126	182	�	�	NOUN
iajs-2920	126	183	𝛿(휀	𝛿(휀	NOUN
iajs-2920	126	184	)	)	PUNCT
iajs-2920	126	185	…	…	PUNCT
iajs-2920	126	186	…	…	PUNCT
iajs-2920	126	187	(	(	PUNCT
iajs-2920	126	188	𝑖𝑖	𝑖𝑖	NOUN
iajs-2920	126	189	)	)	PUNCT
iajs-2920	126	190	from	from	ADP
iajs-2920	126	191	(	(	PUNCT
iajs-2920	126	192	𝑖	𝑖	X
iajs-2920	126	193	)	)	PUNCT
iajs-2920	126	194	and	and	CCONJ
iajs-2920	126	195	(	(	PUNCT
iajs-2920	126	196	𝑖𝑖	𝑖𝑖	NOUN
iajs-2920	126	197	)	)	PUNCT
iajs-2920	126	198	,	,	PUNCT
iajs-2920	126	199	we	we	PRON
iajs-2920	126	200	get	get	VERB
iajs-2920	126	201	�	�	PROPN
iajs-2920	126	202	̃	̃	PROPN
iajs-2920	126	203	�	�	NOUN
iajs-2920	126	204	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	126	205	)	)	PUNCT
iajs-2920	126	206	≥	≥	NOUN
iajs-2920	126	207	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	126	208	�	�	PROPN
iajs-2920	126	209	̃	̃	PROPN
iajs-2920	126	210	�	�	NOUN
iajs-2920	126	211	𝛿(휀	𝛿(휀	NOUN
iajs-2920	126	212	)	)	PUNCT
iajs-2920	126	213	,	,	PUNCT
iajs-2920	126	214	�	�	PROPN
iajs-2920	126	215	̃	̃	PROPN
iajs-2920	126	216	�	�	NOUN
iajs-2920	126	217	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	126	218	)	)	PUNCT
iajs-2920	126	219	}	}	PUNCT
iajs-2920	126	220	.	.	PUNCT
iajs-2920	127	1	similarly	similarly	ADV
iajs-2920	127	2	,	,	PUNCT
iajs-2920	127	3	we	we	PRON
iajs-2920	127	4	can	can	AUX
iajs-2920	127	5	show	show	VERB
iajs-2920	127	6	that	that	SCONJ
iajs-2920	127	7	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	127	8	)	)	PUNCT
iajs-2920	127	9	≤	≤	NUM
iajs-2920	127	10	𝑚𝑎𝑥{𝛼𝛿(휀	𝑚𝑎𝑥{𝛼𝛿(휀	X
iajs-2920	127	11	)	)	PUNCT
iajs-2920	127	12	,	,	PUNCT
iajs-2920	127	13	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	127	14	)	)	PUNCT
iajs-2920	127	15	}	}	PUNCT
iajs-2920	127	16	.	.	PUNCT
iajs-2920	128	1	proposition	proposition	NOUN
iajs-2920	128	2	(	(	PUNCT
iajs-2920	128	3	21	21	NUM
iajs-2920	128	4	)	)	PUNCT
iajs-2920	128	5	.	.	PUNCT
iajs-2920	129	1	if	if	SCONJ
iajs-2920	129	2	𝛿	𝛿	ADJ
iajs-2920	129	3	=	=	PROPN
iajs-2920	129	4	〈	〈	PROPN
iajs-2920	129	5	�	�	PROPN
iajs-2920	129	6	̃	̃	PROPN
iajs-2920	129	7	�	�	PROPN
iajs-2920	129	8	𝛿	𝛿	NOUN
iajs-2920	129	9	,	,	PUNCT
iajs-2920	129	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	129	11	〉	〉	NOUN
iajs-2920	129	12	is	be	AUX
iajs-2920	129	13	a	a	DET
iajs-2920	129	14	cubic	cubic	ADJ
iajs-2920	129	15	t	t	NOUN
iajs-2920	129	16	-	-	PUNCT
iajs-2920	129	17	ideal	ideal	NOUN
iajs-2920	129	18	of	of	ADP
iajs-2920	129	19	tm	tm	NOUN
iajs-2920	129	20	-	-	NOUN
iajs-2920	129	21	algebra	algebra	NOUN
iajs-2920	129	22	ℵand	ℵand	NOUN
iajs-2920	129	23	𝜌	𝜌	ADP
iajs-2920	129	24	≤	≤	NUM
iajs-2920	129	25	𝜏	𝜏	PRON
iajs-2920	129	26	then	then	ADV
iajs-2920	129	27	�	�	PROPN
iajs-2920	129	28	̃	̃	PROPN
iajs-2920	129	29	�	�	PROPN
iajs-2920	129	30	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	129	31	)	)	PUNCT
iajs-2920	129	32	≥	≥	NOUN
iajs-2920	129	33	�	�	PROPN
iajs-2920	129	34	̃	̃	PROPN
iajs-2920	129	35	�	�	NOUN
iajs-2920	129	36	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	129	37	)	)	PUNCT
iajs-2920	129	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	129	39	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	129	40	)	)	PUNCT
iajs-2920	129	41	≤	≤	NOUN
iajs-2920	129	42	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	129	43	)	)	PUNCT
iajs-2920	129	44	.	.	PUNCT
iajs-2920	130	1	proof	proof	NOUN
iajs-2920	130	2	.	.	PUNCT
iajs-2920	131	1	if	if	SCONJ
iajs-2920	131	2	𝜌	𝜌	ADP
iajs-2920	131	3	≤	≤	NUM
iajs-2920	131	4	𝜏then	𝜏then	ADV
iajs-2920	131	5	𝜌	𝜌	ADP
iajs-2920	131	6	∗	∗	NOUN
iajs-2920	131	7	𝜏	𝜏	X
iajs-2920	131	8	=	=	SYM
iajs-2920	131	9	0	0	NUM
iajs-2920	131	10	.	.	PUNCT
iajs-2920	132	1	this	this	PRON
iajs-2920	132	2	is	be	AUX
iajs-2920	132	3	together	together	ADV
iajs-2920	132	4	with	with	ADP
iajs-2920	132	5	𝜌	𝜌	ADP
iajs-2920	132	6	∗	∗	NOUN
iajs-2920	132	7	0	0	NUM
iajs-2920	132	8	=	=	SYM
iajs-2920	132	9	𝜌	𝜌	X
iajs-2920	132	10	and	and	CCONJ
iajs-2920	132	11	�	�	PROPN
iajs-2920	132	12	̃	̃	PROPN
iajs-2920	132	13	�	�	PROPN
iajs-2920	132	14	𝛿(0	𝛿(0	PROPN
iajs-2920	132	15	)	)	PUNCT
iajs-2920	132	16	≥	≥	NOUN
iajs-2920	132	17	�	�	PROPN
iajs-2920	132	18	̃	̃	PROPN
iajs-2920	132	19	�	�	NOUN
iajs-2920	132	20	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	132	21	)	)	PUNCT
iajs-2920	132	22	also	also	ADV
iajs-2920	132	23	𝛼𝛿(0	𝛼𝛿(0	NOUN
iajs-2920	132	24	)	)	PUNCT
iajs-2920	132	25	≤	≤	NOUN
iajs-2920	132	26	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	132	27	)	)	PUNCT
iajs-2920	132	28	,	,	PUNCT
iajs-2920	132	29	we	we	PRON
iajs-2920	132	30	get	get	VERB
iajs-2920	132	31	ihjpas	ihjpa	NOUN
iajs-2920	132	32	.	.	PUNCT
iajs-2920	133	1	36(1)2023	36(1)2023	NUM
iajs-2920	133	2	373	373	NUM
iajs-2920	133	3	�	�	SYM
iajs-2920	133	4	̃	̃	PROPN
iajs-2920	133	5	�	�	PROPN
iajs-2920	133	6	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	133	7	∗	∗	NOUN
iajs-2920	133	8	0	0	NUM
iajs-2920	133	9	)	)	PUNCT
iajs-2920	133	10	=	=	SYM
iajs-2920	133	11	�	�	PROPN
iajs-2920	133	12	̃	̃	PROPN
iajs-2920	133	13	�	�	PROPN
iajs-2920	133	14	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	133	15	)	)	PUNCT
iajs-2920	133	16	≥	≥	NOUN
iajs-2920	133	17	𝑟𝑚𝑖𝑛{((𝜌	𝑟𝑚𝑖𝑛{((𝜌	PROPN
iajs-2920	133	18	∗	∗	PROPN
iajs-2920	133	19	𝜏	𝜏	NOUN
iajs-2920	133	20	)	)	PUNCT
iajs-2920	133	21	∗	∗	NOUN
iajs-2920	133	22	0	0	NUM
iajs-2920	133	23	)	)	PUNCT
iajs-2920	133	24	,	,	PUNCT
iajs-2920	133	25	�	�	PROPN
iajs-2920	133	26	̃	̃	PROPN
iajs-2920	133	27	�	�	NOUN
iajs-2920	133	28	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	133	29	)	)	PUNCT
iajs-2920	133	30	}	}	PUNCT
iajs-2920	134	1	=	=	SYM
iajs-2920	134	2	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	134	3	�	�	PROPN
iajs-2920	134	4	̃	̃	PROPN
iajs-2920	134	5	�	�	PROPN
iajs-2920	134	6	𝛿(0	𝛿(0	NOUN
iajs-2920	134	7	∗	∗	NOUN
iajs-2920	134	8	0	0	NUM
iajs-2920	134	9	)	)	PUNCT
iajs-2920	134	10	,	,	PUNCT
iajs-2920	134	11	�	�	PROPN
iajs-2920	134	12	̃	̃	PROPN
iajs-2920	134	13	�	�	NOUN
iajs-2920	134	14	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	134	15	)	)	PUNCT
iajs-2920	134	16	}	}	PUNCT
iajs-2920	134	17	=	=	SYM
iajs-2920	134	18	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	134	19	�	�	PROPN
iajs-2920	134	20	̃	̃	PROPN
iajs-2920	134	21	�	�	PROPN
iajs-2920	134	22	𝛿(0	𝛿(0	PROPN
iajs-2920	134	23	)	)	PUNCT
iajs-2920	134	24	,	,	PUNCT
iajs-2920	134	25	�	�	PROPN
iajs-2920	134	26	̃	̃	PROPN
iajs-2920	134	27	�	�	NOUN
iajs-2920	134	28	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	134	29	)	)	PUNCT
iajs-2920	134	30	}	}	PUNCT
iajs-2920	134	31	=	=	SYM
iajs-2920	134	32	�	�	PROPN
iajs-2920	134	33	̃	̃	PROPN
iajs-2920	134	34	�	�	NOUN
iajs-2920	134	35	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	134	36	)	)	PUNCT
iajs-2920	134	37	,	,	PUNCT
iajs-2920	134	38	also	also	ADV
iajs-2920	134	39	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	134	40	∗	∗	NOUN
iajs-2920	134	41	0	0	NUM
iajs-2920	134	42	)	)	PUNCT
iajs-2920	134	43	=	=	SYM
iajs-2920	134	44	𝛼𝛿(𝜌	𝛼𝛿(𝜌	X
iajs-2920	134	45	)	)	PUNCT
iajs-2920	134	46	≤	≤	NOUN
iajs-2920	134	47	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	134	48	∗	∗	NOUN
iajs-2920	134	49	𝜏	𝜏	NOUN
iajs-2920	134	50	)	)	PUNCT
iajs-2920	134	51	∗	∗	NOUN
iajs-2920	134	52	0	0	NUM
iajs-2920	134	53	)	)	PUNCT
iajs-2920	134	54	,	,	PUNCT
iajs-2920	134	55	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	134	56	)	)	PUNCT
iajs-2920	134	57	}	}	PUNCT
iajs-2920	135	1	=	=	SYM
iajs-2920	135	2	𝑚𝑎𝑥{𝛼𝛿(0	𝑚𝑎𝑥{𝛼𝛿(0	PROPN
iajs-2920	135	3	∗	∗	NOUN
iajs-2920	135	4	0	0	NUM
iajs-2920	135	5	)	)	PUNCT
iajs-2920	135	6	,	,	PUNCT
iajs-2920	135	7	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	135	8	)	)	PUNCT
iajs-2920	135	9	}	}	PUNCT
iajs-2920	135	10	=	=	SYM
iajs-2920	135	11	𝑚𝑎𝑥{𝛼𝛿(0	𝑚𝑎𝑥{𝛼𝛿(0	PROPN
iajs-2920	135	12	)	)	PUNCT
iajs-2920	135	13	,	,	PUNCT
iajs-2920	135	14	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	135	15	)	)	PUNCT
iajs-2920	135	16	}	}	PUNCT
iajs-2920	135	17	=	=	SYM
iajs-2920	135	18	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	135	19	)	)	PUNCT
iajs-2920	135	20	.	.	PUNCT
iajs-2920	136	1	theorem	theorem	NOUN
iajs-2920	136	2	(	(	PUNCT
iajs-2920	136	3	22	22	NUM
iajs-2920	136	4	)	)	PUNCT
iajs-2920	136	5	.	.	PUNCT
iajs-2920	137	1	let	let	VERB
iajs-2920	137	2	ℵ	ℵ	NOUN
iajs-2920	137	3	be	be	AUX
iajs-2920	137	4	a	a	DET
iajs-2920	137	5	tm	tm	NOUN
iajs-2920	137	6	-	-	NOUN
iajs-2920	137	7	algebra	algebra	NOUN
iajs-2920	137	8	,	,	PUNCT
iajs-2920	137	9	a	a	DET
iajs-2920	137	10	cubic	cubic	ADJ
iajs-2920	137	11	set𝛿	set𝛿	NOUN
iajs-2920	137	12	=	=	SYM
iajs-2920	137	13	〈	〈	PROPN
iajs-2920	137	14	�	�	PROPN
iajs-2920	137	15	̃	̃	PROPN
iajs-2920	137	16	�	�	PROPN
iajs-2920	137	17	𝛿	𝛿	NOUN
iajs-2920	137	18	,	,	PUNCT
iajs-2920	137	19	𝛼𝛿〉of	𝛼𝛿〉of	NUM
iajs-2920	137	20	ℵ	ℵ	NOUN
iajs-2920	137	21	is	be	AUX
iajs-2920	137	22	a	a	DET
iajs-2920	137	23	cubic	cubic	ADJ
iajs-2920	137	24	t	t	NOUN
iajs-2920	137	25	-	-	PUNCT
iajs-2920	137	26	ideal	ideal	NOUN
iajs-2920	137	27	if	if	SCONJ
iajs-2920	137	28	𝛿	𝛿	ADJ
iajs-2920	137	29	is	be	AUX
iajs-2920	137	30	a	a	DET
iajs-2920	137	31	cubic	cubic	ADJ
iajs-2920	137	32	ideal	ideal	NOUN
iajs-2920	137	33	of	of	ADP
iajs-2920	137	34	ℵ.	ℵ.	PROPN
iajs-2920	137	35	proof	proof	NOUN
iajs-2920	137	36	.	.	PUNCT
iajs-2920	138	1	if	if	SCONJ
iajs-2920	138	2	we	we	PRON
iajs-2920	138	3	put	put	VERB
iajs-2920	138	4	휀	휀	NOUN
iajs-2920	138	5	=	=	NOUN
iajs-2920	138	6	0	0	NUM
iajs-2920	138	7	in	in	ADP
iajs-2920	138	8	(	(	PUNCT
iajs-2920	138	9	𝐵2	𝐵2	NOUN
iajs-2920	138	10	)	)	PUNCT
iajs-2920	138	11	,	,	PUNCT
iajs-2920	138	12	then	then	ADV
iajs-2920	138	13	�	�	PROPN
iajs-2920	138	14	̃	̃	PROPN
iajs-2920	138	15	�	�	PROPN
iajs-2920	138	16	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	138	17	)	)	PUNCT
iajs-2920	138	18	≥	≥	NOUN
iajs-2920	138	19	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	138	20	�	�	PROPN
iajs-2920	138	21	̃	̃	PROPN
iajs-2920	138	22	�	�	PROPN
iajs-2920	138	23	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	138	24	∗	∗	NOUN
iajs-2920	138	25	𝜏	𝜏	NOUN
iajs-2920	138	26	)	)	PUNCT
iajs-2920	138	27	)	)	PUNCT
iajs-2920	138	28	,	,	PUNCT
iajs-2920	138	29	�	�	PROPN
iajs-2920	138	30	̃	̃	PROPN
iajs-2920	138	31	�	�	NOUN
iajs-2920	138	32	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	138	33	)	)	PUNCT
iajs-2920	138	34	}	}	PUNCT
iajs-2920	138	35	𝑎𝑛𝑑𝛼𝛿(𝜌	𝑎𝑛𝑑𝛼𝛿(𝜌	PROPN
iajs-2920	138	36	)	)	PUNCT
iajs-2920	138	37	≤	≤	NOUN
iajs-2920	138	38	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	138	39	∗	∗	NOUN
iajs-2920	138	40	𝜏	𝜏	NOUN
iajs-2920	138	41	)	)	PUNCT
iajs-2920	138	42	)	)	PUNCT
iajs-2920	138	43	,	,	PUNCT
iajs-2920	138	44	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	138	45	)	)	PUNCT
iajs-2920	138	46	}	}	PUNCT
iajs-2920	138	47	.	.	PUNCT
iajs-2920	139	1	hence	hence	ADV
iajs-2920	139	2	𝛿	𝛿	ADJ
iajs-2920	139	3	=	=	PUNCT
iajs-2920	139	4	〈	〈	PROPN
iajs-2920	139	5	�	�	PROPN
iajs-2920	139	6	̃	̃	PROPN
iajs-2920	139	7	�	�	PROPN
iajs-2920	139	8	𝛿	𝛿	NOUN
iajs-2920	139	9	,	,	PUNCT
iajs-2920	139	10	𝛼𝛿〉is	𝛼𝛿〉is	PROPN
iajs-2920	139	11	a	a	DET
iajs-2920	139	12	cubic	cubic	ADJ
iajs-2920	139	13	ideal	ideal	NOUN
iajs-2920	139	14	of	of	ADP
iajs-2920	139	15	ℵ.	ℵ.	PROPN
iajs-2920	139	16	remark	remark	PROPN
iajs-2920	139	17	(	(	PUNCT
iajs-2920	139	18	23	23	NUM
iajs-2920	139	19	)	)	PUNCT
iajs-2920	139	20	.	.	PUNCT
iajs-2920	140	1	the	the	DET
iajs-2920	140	2	converse	converse	NOUN
iajs-2920	140	3	of	of	ADP
iajs-2920	140	4	theorem	theorem	NOUN
iajs-2920	140	5	(	(	PUNCT
iajs-2920	140	6	22	22	NUM
iajs-2920	140	7	)	)	PUNCT
iajs-2920	140	8	is	be	AUX
iajs-2920	140	9	not	not	PART
iajs-2920	140	10	true	true	ADJ
iajs-2920	140	11	.	.	PUNCT
iajs-2920	141	1	the	the	DET
iajs-2920	141	2	following	follow	VERB
iajs-2920	141	3	example	example	NOUN
iajs-2920	141	4	shows	show	VERB
iajs-2920	141	5	the	the	DET
iajs-2920	141	6	reverse	reverse	ADJ
iajs-2920	141	7	direction	direction	NOUN
iajs-2920	141	8	of	of	ADP
iajs-2920	141	9	theorem	theorem	NOUN
iajs-2920	141	10	(	(	PUNCT
iajs-2920	141	11	22	22	NUM
iajs-2920	141	12	)	)	PUNCT
iajs-2920	141	13	.	.	PUNCT
iajs-2920	142	1	example	example	NOUN
iajs-2920	142	2	(	(	PUNCT
iajs-2920	142	3	24	24	NUM
iajs-2920	142	4	)	)	PUNCT
iajs-2920	142	5	.	.	PUNCT
iajs-2920	143	1	let	let	VERB
iajs-2920	143	2	ℵ	ℵ	NOUN
iajs-2920	143	3	=	=	SYM
iajs-2920	143	4	{	{	PUNCT
iajs-2920	143	5	0	0	NUM
iajs-2920	143	6	,	,	PUNCT
iajs-2920	143	7	a	a	DET
iajs-2920	143	8	,	,	PUNCT
iajs-2920	143	9	b	b	NOUN
iajs-2920	143	10	,	,	PUNCT
iajs-2920	143	11	c	c	AUX
iajs-2920	143	12	}	}	PUNCT
iajs-2920	143	13	be	be	AUX
iajs-2920	143	14	a	a	DET
iajs-2920	143	15	set	set	NOUN
iajs-2920	143	16	with	with	ADP
iajs-2920	143	17	the	the	DET
iajs-2920	143	18	following	follow	VERB
iajs-2920	143	19	table	table	NOUN
iajs-2920	143	20	:	:	PUNCT
iajs-2920	144	1	*	*	PUNCT
iajs-2920	144	2	0	0	PUNCT
iajs-2920	144	3	a	a	DET
iajs-2920	144	4	b	b	X
iajs-2920	144	5	c	c	NOUN
iajs-2920	144	6	0	0	NUM
iajs-2920	144	7	0	0	NUM
iajs-2920	144	8	a	a	DET
iajs-2920	144	9	b	b	NOUN
iajs-2920	144	10	c	c	NOUN
iajs-2920	144	11	a	a	DET
iajs-2920	144	12	a	a	DET
iajs-2920	144	13	0	0	NUM
iajs-2920	144	14	c	c	NOUN
iajs-2920	144	15	b	b	PROPN
iajs-2920	144	16	b	b	PROPN
iajs-2920	144	17	b	b	PROPN
iajs-2920	144	18	c	c	PROPN
iajs-2920	144	19	0	0	NUM
iajs-2920	145	1	a	a	DET
iajs-2920	145	2	c	c	NOUN
iajs-2920	145	3	c	c	NOUN
iajs-2920	145	4	b	b	PROPN
iajs-2920	145	5	a	a	DET
iajs-2920	145	6	0	0	NUM
iajs-2920	145	7	then	then	ADV
iajs-2920	145	8	,	,	PUNCT
iajs-2920	145	9	(	(	PUNCT
iajs-2920	145	10	ℵ,∗	ℵ,∗	PROPN
iajs-2920	145	11	,	,	PUNCT
iajs-2920	145	12	0	0	NUM
iajs-2920	145	13	)	)	PUNCT
iajs-2920	145	14	is	be	AUX
iajs-2920	145	15	a	a	DET
iajs-2920	145	16	tm	tm	NOUN
iajs-2920	145	17	-	-	NOUN
iajs-2920	145	18	algebra	algebra	NOUN
iajs-2920	145	19	.	.	PUNCT
iajs-2920	146	1	define	define	VERB
iajs-2920	146	2	�	�	PROPN
iajs-2920	146	3	̃	̃	PROPN
iajs-2920	146	4	�	�	NOUN
iajs-2920	146	5	δ(𝜌	δ(𝜌	NOUN
iajs-2920	146	6	)	)	PUNCT
iajs-2920	146	7	ˑand	ˑand	CCONJ
iajs-2920	146	8	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	146	9	)	)	PUNCT
iajs-2920	146	10	by	by	ADP
iajs-2920	146	11	�	�	PROPN
iajs-2920	146	12	̃	̃	PROPN
iajs-2920	146	13	�	�	NOUN
iajs-2920	146	14	δ(𝜌	δ(𝜌	NOUN
iajs-2920	146	15	)	)	PUNCT
iajs-2920	147	1	=	=	PRON
iajs-2920	147	2	{	{	PUNCT
iajs-2920	148	1	[	[	X
iajs-2920	148	2	0.1,0.8	0.1,0.8	X
iajs-2920	148	3	]	]	X
iajs-2920	148	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	148	5	𝜌	𝜌	X
iajs-2920	148	6	=	=	SYM
iajs-2920	148	7	{	{	PUNCT
iajs-2920	148	8	0	0	NUM
iajs-2920	148	9	,	,	PUNCT
iajs-2920	148	10	𝑎	𝑎	NOUN
iajs-2920	148	11	,	,	PUNCT
iajs-2920	148	12	𝑏	𝑏	NOUN
iajs-2920	148	13	}	}	PUNCT
iajs-2920	148	14	[	[	X
iajs-2920	148	15	0.1,0.3	0.1,0.3	X
iajs-2920	148	16	]	]	X
iajs-2920	148	17	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	148	18	𝜌	𝜌	X
iajs-2920	148	19	=	=	SYM
iajs-2920	148	20	𝑐	𝑐	PROPN
iajs-2920	148	21	,	,	PUNCT
iajs-2920	148	22	𝛼δ(𝜌	𝛼δ(𝜌	NUM
iajs-2920	148	23	)	)	PUNCT
iajs-2920	149	1	=	=	NOUN
iajs-2920	149	2	{	{	PUNCT
iajs-2920	149	3	0.1	0.1	NUM
iajs-2920	149	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2920	149	5	𝜌	𝜌	X
iajs-2920	149	6	=	=	SYM
iajs-2920	149	7	{	{	PUNCT
iajs-2920	149	8	0	0	NUM
iajs-2920	149	9	,	,	PUNCT
iajs-2920	149	10	𝑎	𝑎	NOUN
iajs-2920	149	11	,	,	PUNCT
iajs-2920	149	12	𝑏	𝑏	NOUN
iajs-2920	149	13	}	}	PUNCT
iajs-2920	149	14	0.8	0.8	NUM
iajs-2920	149	15	𝑖𝑓	𝑖𝑓	NUM
iajs-2920	149	16	𝜌	𝜌	X
iajs-2920	149	17	=	=	SYM
iajs-2920	149	18	𝑐	𝑐	PROPN
iajs-2920	149	19	,	,	PUNCT
iajs-2920	149	20	then	then	ADV
iajs-2920	149	21	,	,	PUNCT
iajs-2920	149	22	it	it	PRON
iajs-2920	149	23	is	be	AUX
iajs-2920	149	24	easy	easy	ADJ
iajs-2920	149	25	to	to	PART
iajs-2920	149	26	show	show	VERB
iajs-2920	149	27	that	that	SCONJ
iajs-2920	149	28	𝛿	𝛿	ADJ
iajs-2920	149	29	=	=	PROPN
iajs-2920	149	30	〈	〈	PROPN
iajs-2920	149	31	�	�	PROPN
iajs-2920	149	32	̃	̃	PROPN
iajs-2920	149	33	�	�	PROPN
iajs-2920	149	34	𝛿	𝛿	NOUN
iajs-2920	149	35	,	,	PUNCT
iajs-2920	149	36	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	149	37	〉	〉	NOUN
iajs-2920	149	38	is	be	AUX
iajs-2920	149	39	a	a	DET
iajs-2920	149	40	cubic	cubic	ADJ
iajs-2920	149	41	ideal	ideal	NOUN
iajs-2920	149	42	of	of	ADP
iajs-2920	149	43	ℵ.	ℵ.	PROPN
iajs-2920	149	44	but	but	CCONJ
iajs-2920	149	45	not	not	PART
iajs-2920	149	46	a	a	DET
iajs-2920	149	47	cubic	cubic	ADJ
iajs-2920	149	48	t	t	NOUN
iajs-2920	149	49	-	-	PUNCT
iajs-2920	149	50	ideal	ideal	NOUN
iajs-2920	149	51	since	since	SCONJ
iajs-2920	149	52	�	�	PROPN
iajs-2920	149	53	̃	̃	PROPN
iajs-2920	149	54	�	�	PROPN
iajs-2920	149	55	𝛿(𝑎	𝛿(𝑎	PROPN
iajs-2920	149	56	∗	∗	VERB
iajs-2920	149	57	𝑏	𝑏	NOUN
iajs-2920	149	58	)	)	PUNCT
iajs-2920	149	59	≤	≤	PROPN
iajs-2920	149	60	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	149	61	�	�	PROPN
iajs-2920	149	62	̃	̃	PROPN
iajs-2920	149	63	�	�	NOUN
iajs-2920	149	64	𝛿((𝑎	𝛿((𝑎	NOUN
iajs-2920	149	65	∗	∗	NOUN
iajs-2920	149	66	𝑐	𝑐	NOUN
iajs-2920	149	67	)	)	PUNCT
iajs-2920	149	68	∗	∗	NOUN
iajs-2920	149	69	𝑏	𝑏	NOUN
iajs-2920	149	70	)	)	PUNCT
iajs-2920	149	71	,	,	PUNCT
iajs-2920	149	72	�	�	PROPN
iajs-2920	149	73	̃	̃	PROPN
iajs-2920	149	74	�	�	NOUN
iajs-2920	149	75	𝛿(𝑐	𝛿(𝑐	NOUN
iajs-2920	149	76	)	)	PUNCT
iajs-2920	149	77	}	}	PUNCT
iajs-2920	149	78	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	149	79	𝛼𝛿(𝑎	𝛼𝛿(𝑎	NUM
iajs-2920	149	80	∗	∗	X
iajs-2920	149	81	𝑏	𝑏	NOUN
iajs-2920	149	82	)	)	PUNCT
iajs-2920	149	83	≥	≥	NOUN
iajs-2920	149	84	𝑚𝑎𝑥{𝛼𝛿((𝑎	𝑚𝑎𝑥{𝛼𝛿((𝑎	NOUN
iajs-2920	149	85	∗	∗	NOUN
iajs-2920	149	86	𝑐	𝑐	NOUN
iajs-2920	149	87	)	)	PUNCT
iajs-2920	149	88	∗	∗	NOUN
iajs-2920	149	89	𝑏	𝑏	NOUN
iajs-2920	149	90	)	)	PUNCT
iajs-2920	149	91	,	,	PUNCT
iajs-2920	149	92	𝛼𝛿(𝑐	𝛼𝛿(𝑐	NUM
iajs-2920	149	93	)	)	PUNCT
iajs-2920	149	94	}	}	PUNCT
iajs-2920	149	95	.	.	PUNCT
iajs-2920	150	1	4	4	X
iajs-2920	150	2	.	.	X
iajs-2920	150	3	image	image	NOUN
iajs-2920	150	4	and	and	CCONJ
iajs-2920	150	5	pre	pre	NOUN
iajs-2920	150	6	-	-	NOUN
iajs-2920	150	7	image	image	NOUN
iajs-2920	150	8	of	of	ADP
iajs-2920	150	9	cubic	cubic	ADJ
iajs-2920	150	10	t	t	PROPN
iajs-2920	150	11	-	-	PUNCT
iajs-2920	150	12	ideals	ideal	NOUN
iajs-2920	150	13	definition	definition	NOUN
iajs-2920	150	14	(	(	PUNCT
iajs-2920	150	15	25	25	NUM
iajs-2920	150	16	)	)	PUNCT
iajs-2920	150	17	.	.	PUNCT
iajs-2920	151	1	let	let	VERB
iajs-2920	151	2	𝑓	𝑓	PRON
iajs-2920	151	3	:	:	PUNCT
iajs-2920	151	4	ℵ	ℵ	PROPN
iajs-2920	151	5	→	→	SYM
iajs-2920	151	6	𝑌	𝑌	PROPN
iajs-2920	151	7	be	be	VERB
iajs-2920	151	8	a	a	DET
iajs-2920	151	9	mapping	mapping	NOUN
iajs-2920	151	10	.	.	PUNCT
iajs-2920	152	1	if	if	SCONJ
iajs-2920	152	2	𝛿	𝛿	ADJ
iajs-2920	152	3	=	=	PROPN
iajs-2920	152	4	〈	〈	PROPN
iajs-2920	152	5	�	�	PROPN
iajs-2920	152	6	̃	̃	PROPN
iajs-2920	152	7	�	�	PROPN
iajs-2920	152	8	𝛿	𝛿	NOUN
iajs-2920	152	9	,	,	PUNCT
iajs-2920	152	10	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	152	11	〉	〉	NOUN
iajs-2920	152	12	is	be	AUX
iajs-2920	152	13	a	a	DET
iajs-2920	152	14	cubic	cubic	ADJ
iajs-2920	152	15	set	set	NOUN
iajs-2920	152	16	of	of	ADP
iajs-2920	152	17	ℵ	ℵ	NOUN
iajs-2920	152	18	,	,	PUNCT
iajs-2920	152	19	then	then	ADV
iajs-2920	152	20	the	the	DET
iajs-2920	152	21	cubic	cubic	ADJ
iajs-2920	152	22	set	set	NOUN
iajs-2920	152	23	𝜔	𝜔	PROPN
iajs-2920	152	24	=	=	PUNCT
iajs-2920	152	25	〈	〈	PROPN
iajs-2920	152	26	�	�	PROPN
iajs-2920	152	27	̃	̃	PROPN
iajs-2920	152	28	�	�	NOUN
iajs-2920	152	29	𝜔	𝜔	NOUN
iajs-2920	152	30	,	,	PUNCT
iajs-2920	152	31	𝛼𝜔	𝛼𝜔	NOUN
iajs-2920	152	32	〉	〉	NOUN
iajs-2920	152	33	of	of	ADP
iajs-2920	152	34	𝑌	𝑌	PROPN
iajs-2920	152	35	is	be	AUX
iajs-2920	152	36	define	define	VERB
iajs-2920	152	37	by	by	ADP
iajs-2920	152	38	ihjpas	ihjpa	NOUN
iajs-2920	152	39	.	.	PUNCT
iajs-2920	153	1	36(1)2023	36(1)2023	NUM
iajs-2920	153	2	374	374	NUM
iajs-2920	153	3	𝑓(	𝑓(	NUM
iajs-2920	153	4	�	�	PROPN
iajs-2920	153	5	̃	̃	PROPN
iajs-2920	153	6	�	�	NOUN
iajs-2920	153	7	𝛿)(𝜏	𝛿)(𝜏	NOUN
iajs-2920	153	8	)	)	PUNCT
iajs-2920	154	1	=	=	SYM
iajs-2920	154	2	�	�	PROPN
iajs-2920	154	3	̃	̃	PROPN
iajs-2920	154	4	�	�	NOUN
iajs-2920	154	5	𝜔(𝜏	𝜔(𝜏	NOUN
iajs-2920	154	6	)	)	PUNCT
iajs-2920	154	7	=	=	PRON
iajs-2920	154	8	{	{	PUNCT
iajs-2920	154	9	𝑟𝑠𝑢𝑝	𝑟𝑠𝑢𝑝	ADV
iajs-2920	154	10	𝜌∈𝑓−1(𝜏	𝜌∈𝑓−1(𝜏	NOUN
iajs-2920	154	11	)	)	PUNCT
iajs-2920	154	12	�	�	PROPN
iajs-2920	154	13	̃	̃	PROPN
iajs-2920	154	14	�	�	PROPN
iajs-2920	154	15	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	154	16	)	)	PUNCT
iajs-2920	154	17	,	,	PUNCT
iajs-2920	154	18	𝑖𝑓	𝑖𝑓	X
iajs-2920	154	19	𝑓−1(𝜏	𝑓−1(𝜏	NOUN
iajs-2920	154	20	)	)	PUNCT
iajs-2920	155	1	=	=	PRON
iajs-2920	155	2	{	{	PUNCT
iajs-2920	155	3	𝜌	𝜌	X
iajs-2920	155	4	∈	∈	PROPN
iajs-2920	155	5	ℵ	ℵ	NOUN
iajs-2920	155	6	,	,	PUNCT
iajs-2920	155	7	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	155	8	)	)	PUNCT
iajs-2920	155	9	=	=	SYM
iajs-2920	155	10	𝜏	𝜏	X
iajs-2920	155	11	}	}	PUNCT
iajs-2920	155	12	≠	≠	PROPN
iajs-2920	155	13	∅	∅	NOUN
iajs-2920	155	14	0	0	NUM
iajs-2920	155	15	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2920	155	16	𝑓(𝛼𝛿)(𝜏	𝑓(𝛼𝛿)(𝜏	PROPN
iajs-2920	155	17	)	)	PUNCT
iajs-2920	155	18	=	=	SYM
iajs-2920	155	19	𝛼𝜔(𝜏	𝛼𝜔(𝜏	NOUN
iajs-2920	155	20	)	)	PUNCT
iajs-2920	155	21	=	=	PRON
iajs-2920	155	22	{	{	PUNCT
iajs-2920	155	23	inf	inf	NOUN
iajs-2920	155	24	𝜌∈𝑓−1(𝜏	𝜌∈𝑓−1(𝜏	NOUN
iajs-2920	155	25	)	)	PUNCT
iajs-2920	155	26	𝛼𝛿	𝛼𝛿	NOUN
iajs-2920	155	27	(	(	PUNCT
iajs-2920	155	28	𝜌	𝜌	NOUN
iajs-2920	155	29	)	)	PUNCT
iajs-2920	156	1	𝑖𝑓	𝑖𝑓	X
iajs-2920	156	2	𝑓−1(𝜏	𝑓−1(𝜏	PROPN
iajs-2920	156	3	)	)	PUNCT
iajs-2920	157	1	=	=	PRON
iajs-2920	157	2	{	{	PUNCT
iajs-2920	157	3	𝜌	𝜌	X
iajs-2920	157	4	∈	∈	PROPN
iajs-2920	157	5	ℵ	ℵ	NOUN
iajs-2920	157	6	,	,	PUNCT
iajs-2920	157	7	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	157	8	)	)	PUNCT
iajs-2920	157	9	=	=	SYM
iajs-2920	157	10	𝜏	𝜏	X
iajs-2920	157	11	}	}	PUNCT
iajs-2920	157	12	≠	≠	PROPN
iajs-2920	157	13	∅	∅	NOUN
iajs-2920	157	14	1	1	NUM
iajs-2920	157	15	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2920	157	16	it	it	PRON
iajs-2920	157	17	is	be	AUX
iajs-2920	157	18	called	call	VERB
iajs-2920	157	19	the	the	DET
iajs-2920	157	20	image	image	NOUN
iajs-2920	157	21	of	of	ADP
iajs-2920	157	22	𝛿	𝛿	ADJ
iajs-2920	157	23	=	=	PUNCT
iajs-2920	157	24	〈	〈	PROPN
iajs-2920	157	25	�	�	PROPN
iajs-2920	157	26	̃	̃	PROPN
iajs-2920	157	27	�	�	PROPN
iajs-2920	157	28	𝛿	𝛿	NOUN
iajs-2920	157	29	,	,	PUNCT
iajs-2920	157	30	𝛼𝛿	𝛼𝛿	PART
iajs-2920	157	31	〉	〉	NOUN
iajs-2920	157	32	under	under	ADP
iajs-2920	157	33	𝑓.	𝑓.	NOUN
iajs-2920	157	34	similarly	similarly	ADV
iajs-2920	157	35	,	,	PUNCT
iajs-2920	157	36	if	if	SCONJ
iajs-2920	157	37	𝜔	𝜔	PRON
iajs-2920	157	38	=	=	NUM
iajs-2920	157	39	〈	〈	PROPN
iajs-2920	157	40	�	�	PROPN
iajs-2920	157	41	̃	̃	PROPN
iajs-2920	157	42	�	�	NOUN
iajs-2920	157	43	𝜔	𝜔	NOUN
iajs-2920	157	44	,	,	PUNCT
iajs-2920	157	45	𝛼𝜔	𝛼𝜔	NOUN
iajs-2920	157	46	〉	〉	NOUN
iajs-2920	157	47	is	be	AUX
iajs-2920	157	48	a	a	DET
iajs-2920	157	49	cubic	cubic	ADJ
iajs-2920	157	50	subset	subset	NOUN
iajs-2920	157	51	of	of	ADP
iajs-2920	157	52	𝑌	𝑌	PROPN
iajs-2920	157	53	,	,	PUNCT
iajs-2920	157	54	then	then	ADV
iajs-2920	157	55	the	the	DET
iajs-2920	157	56	cubic	cubic	ADJ
iajs-2920	157	57	subset	subset	NOUN
iajs-2920	157	58	defined	define	VERB
iajs-2920	157	59	by	by	ADP
iajs-2920	157	60	�	�	PROPN
iajs-2920	157	61	̃	̃	PROPN
iajs-2920	157	62	�	�	NOUN
iajs-2920	157	63	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	157	64	)	)	PUNCT
iajs-2920	157	65	=	=	SYM
iajs-2920	157	66	�	�	PROPN
iajs-2920	157	67	̃	̃	PROPN
iajs-2920	157	68	�	�	PROPN
iajs-2920	157	69	𝜔(𝑓(𝜌	𝜔(𝑓(𝜌	PROPN
iajs-2920	157	70	)	)	PUNCT
iajs-2920	157	71	)	)	PUNCT
iajs-2920	157	72	and	and	CCONJ
iajs-2920	157	73	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	157	74	)	)	PUNCT
iajs-2920	157	75	=	=	PUNCT
iajs-2920	158	1	𝛼𝜔(𝑓(𝜌)),for	𝛼𝜔(𝑓(𝜌)),for	ADP
iajs-2920	158	2	any	any	DET
iajs-2920	158	3	𝜌	𝜌	X
iajs-2920	158	4	∈	∈	PROPN
iajs-2920	158	5	ℵis	ℵis	NOUN
iajs-2920	158	6	said	say	VERB
iajs-2920	158	7	to	to	PART
iajs-2920	158	8	be	be	AUX
iajs-2920	158	9	the	the	DET
iajs-2920	158	10	pre	pre	NOUN
iajs-2920	158	11	-	-	NOUN
iajs-2920	158	12	image	image	NOUN
iajs-2920	158	13	of	of	ADP
iajs-2920	158	14	𝜔under	𝜔under	NOUN
iajs-2920	158	15	𝑓.	𝑓.	NOUN
iajs-2920	158	16	theorem	theorem	NOUN
iajs-2920	158	17	(	(	PUNCT
iajs-2920	158	18	26	26	NUM
iajs-2920	158	19	)	)	PUNCT
iajs-2920	158	20	.	.	PUNCT
iajs-2920	159	1	an	an	DET
iajs-2920	159	2	epimorphism	epimorphism	NOUN
iajs-2920	159	3	pre	pre	NOUN
iajs-2920	159	4	-	-	NOUN
iajs-2920	159	5	image	image	NOUN
iajs-2920	159	6	of	of	ADP
iajs-2920	159	7	a	a	DET
iajs-2920	159	8	cubic	cubic	ADJ
iajs-2920	159	9	t	t	PROPN
iajs-2920	159	10	-	-	PUNCT
iajs-2920	159	11	ideal	ideal	NOUN
iajs-2920	159	12	is	be	AUX
iajs-2920	159	13	also	also	ADV
iajs-2920	159	14	a	a	DET
iajs-2920	159	15	cubic	cubic	ADJ
iajs-2920	159	16	t	t	NOUN
iajs-2920	159	17	-	-	PUNCT
iajs-2920	159	18	ideal	ideal	NOUN
iajs-2920	159	19	.	.	PUNCT
iajs-2920	160	1	proof	proof	NOUN
iajs-2920	160	2	.	.	PUNCT
iajs-2920	161	1	let	let	VERB
iajs-2920	161	2	𝑓	𝑓	DET
iajs-2920	161	3	∶	∶	NOUN
iajs-2920	161	4	ℵ	ℵ	X
iajs-2920	161	5	→	→	SYM
iajs-2920	161	6	ℵ′	ℵ′	PUNCT
iajs-2920	161	7	be	be	AUX
iajs-2920	161	8	an	an	DET
iajs-2920	161	9	epimorphism	epimorphism	NOUN
iajs-2920	161	10	mapping	mapping	NOUN
iajs-2920	161	11	of	of	ADP
iajs-2920	161	12	tm	tm	NOUN
iajs-2920	161	13	-	-	NOUN
iajs-2920	161	14	algebra	algebra	NOUN
iajs-2920	161	15	,	,	PUNCT
iajs-2920	161	16	𝜔	𝜔	PROPN
iajs-2920	161	17	=	=	PUNCT
iajs-2920	161	18	〈	〈	PROPN
iajs-2920	161	19	�	�	PROPN
iajs-2920	161	20	̃	̃	PROPN
iajs-2920	161	21	�	�	NOUN
iajs-2920	161	22	𝜔	𝜔	PART
iajs-2920	161	23	,	,	PUNCT
iajs-2920	161	24	𝛼𝜔〉be	𝛼𝜔〉be	PROPN
iajs-2920	161	25	a	a	DET
iajs-2920	161	26	cubic	cubic	ADJ
iajs-2920	161	27	tideal	tideal	NOUN
iajs-2920	161	28	of	of	ADP
iajs-2920	161	29	ℵ′	ℵ′	NUM
iajs-2920	161	30	and	and	CCONJ
iajs-2920	161	31	𝛿	𝛿	ADJ
iajs-2920	161	32	=	=	ADJ
iajs-2920	161	33	〈	〈	PROPN
iajs-2920	161	34	�	�	PROPN
iajs-2920	161	35	̃	̃	PROPN
iajs-2920	161	36	�	�	NOUN
iajs-2920	161	37	𝛿	𝛿	NOUN
iajs-2920	161	38	,	,	PUNCT
iajs-2920	161	39	𝛼𝛿〉be	𝛼𝛿〉be	PROPN
iajs-2920	161	40	the	the	DET
iajs-2920	161	41	pre	pre	NOUN
iajs-2920	161	42	-	-	NOUN
iajs-2920	161	43	image	image	NOUN
iajs-2920	161	44	of	of	ADP
iajs-2920	161	45	𝜔under	𝜔under	NOUN
iajs-2920	161	46	𝑓	𝑓	PRON
iajs-2920	161	47	,	,	PUNCT
iajs-2920	161	48	then	then	ADV
iajs-2920	161	49	�	�	PROPN
iajs-2920	161	50	̃	̃	PROPN
iajs-2920	161	51	�	�	NOUN
iajs-2920	161	52	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	161	53	)	)	PUNCT
iajs-2920	161	54	=	=	SYM
iajs-2920	161	55	�	�	PROPN
iajs-2920	161	56	̃	̃	PROPN
iajs-2920	161	57	�	�	PROPN
iajs-2920	161	58	𝜔(𝑓(𝜌	𝜔(𝑓(𝜌	PROPN
iajs-2920	161	59	)	)	PUNCT
iajs-2920	161	60	)	)	PUNCT
iajs-2920	161	61	and	and	CCONJ
iajs-2920	161	62	𝛼𝛼(𝜌	𝛼𝛼(𝜌	NUM
iajs-2920	161	63	)	)	PUNCT
iajs-2920	161	64	=	=	SYM
iajs-2920	161	65	𝛼𝜔(𝑓(𝜌	𝛼𝜔(𝑓(𝜌	NUM
iajs-2920	161	66	)	)	PUNCT
iajs-2920	161	67	)	)	PUNCT
iajs-2920	161	68	for	for	ADP
iajs-2920	161	69	any	any	DET
iajs-2920	161	70	𝜌	𝜌	X
iajs-2920	161	71	∈	∈	PROPN
iajs-2920	161	72	ℵ	ℵ	NOUN
iajs-2920	161	73	,	,	PUNCT
iajs-2920	161	74	then	then	ADV
iajs-2920	161	75	�	�	PROPN
iajs-2920	161	76	̃	̃	PROPN
iajs-2920	161	77	�	�	PROPN
iajs-2920	161	78	𝛿(0	𝛿(0	PROPN
iajs-2920	161	79	)	)	PUNCT
iajs-2920	161	80	=	=	SYM
iajs-2920	161	81	�	�	PROPN
iajs-2920	161	82	̃	̃	PROPN
iajs-2920	161	83	�	�	PROPN
iajs-2920	161	84	𝜔(𝑓(0	𝜔(𝑓(0	NOUN
iajs-2920	161	85	)	)	PUNCT
iajs-2920	161	86	)	)	PUNCT
iajs-2920	161	87	≥	≥	PROPN
iajs-2920	161	88	�	�	PROPN
iajs-2920	161	89	̃	̃	PROPN
iajs-2920	161	90	�	�	PROPN
iajs-2920	161	91	𝜔(𝑓(𝜌	𝜔(𝑓(𝜌	PROPN
iajs-2920	161	92	)	)	PUNCT
iajs-2920	161	93	)	)	PUNCT
iajs-2920	162	1	=	=	PUNCT
iajs-2920	162	2	�	�	PROPN
iajs-2920	162	3	̃	̃	PROPN
iajs-2920	162	4	�	�	PROPN
iajs-2920	162	5	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	162	6	)	)	PUNCT
iajs-2920	162	7	,	,	PUNCT
iajs-2920	162	8	𝛼𝛿(0	𝛼𝛿(0	NOUN
iajs-2920	162	9	)	)	PUNCT
iajs-2920	162	10	=	=	PUNCT
iajs-2920	162	11	𝛼𝜔(𝑓(0	𝛼𝜔(𝑓(0	NUM
iajs-2920	162	12	)	)	PUNCT
iajs-2920	162	13	)	)	PUNCT
iajs-2920	163	1	≤	≤	PROPN
iajs-2920	163	2	𝛼𝜔(𝑓(𝜌	𝛼𝜔(𝑓(𝜌	PROPN
iajs-2920	163	3	)	)	PUNCT
iajs-2920	163	4	)	)	PUNCT
iajs-2920	163	5	=	=	SYM
iajs-2920	164	1	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	164	2	)	)	PUNCT
iajs-2920	164	3	.	.	PUNCT
iajs-2920	165	1	now	now	ADV
iajs-2920	165	2	,	,	PUNCT
iajs-2920	165	3	let	let	VERB
iajs-2920	165	4	𝜌	𝜌	PART
iajs-2920	165	5	,	,	PUNCT
iajs-2920	165	6	𝜏	𝜏	NOUN
iajs-2920	165	7	,	,	PUNCT
iajs-2920	165	8	휀	휀	DET
iajs-2920	165	9	∈	∈	PROPN
iajs-2920	165	10	ℵ	ℵ	NOUN
iajs-2920	165	11	,	,	PUNCT
iajs-2920	165	12	then	then	ADV
iajs-2920	165	13	�	�	PROPN
iajs-2920	165	14	̃	̃	PROPN
iajs-2920	165	15	�	�	PROPN
iajs-2920	165	16	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	165	17	∗	∗	NOUN
iajs-2920	165	18	휀	휀	NOUN
iajs-2920	165	19	)	)	PUNCT
iajs-2920	165	20	=	=	SYM
iajs-2920	165	21	�	�	PROPN
iajs-2920	165	22	̃	̃	PROPN
iajs-2920	165	23	�	�	NOUN
iajs-2920	165	24	𝜔(𝑓(𝜌	𝜔(𝑓(𝜌	PROPN
iajs-2920	165	25	∗	∗	NOUN
iajs-2920	165	26	휀	휀	NOUN
iajs-2920	165	27	)	)	PUNCT
iajs-2920	165	28	)	)	PUNCT
iajs-2920	166	1	=	=	SYM
iajs-2920	166	2	�	�	PROPN
iajs-2920	166	3	̃	̃	PROPN
iajs-2920	166	4	�	�	NOUN
iajs-2920	166	5	𝜔(𝑓(𝜌	𝜔(𝑓(𝜌	PROPN
iajs-2920	166	6	)	)	PUNCT
iajs-2920	166	7	∗′	∗′	PROPN
iajs-2920	166	8	𝑓(휀	𝑓(휀	PROPN
iajs-2920	166	9	)	)	PUNCT
iajs-2920	166	10	)	)	PUNCT
iajs-2920	166	11	≥	≥	PROPN
iajs-2920	166	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	166	13	{	{	PUNCT
iajs-2920	166	14	�	�	PROPN
iajs-2920	166	15	̃	̃	PROPN
iajs-2920	166	16	�	�	NOUN
iajs-2920	166	17	𝜔	𝜔	PART
iajs-2920	166	18	(	(	PUNCT
iajs-2920	166	19	(	(	PUNCT
iajs-2920	166	20	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	166	21	)	)	PUNCT
iajs-2920	166	22	∗′	∗′	ADJ
iajs-2920	166	23	𝑓(𝜏	𝑓(𝜏	NOUN
iajs-2920	166	24	)	)	PUNCT
iajs-2920	166	25	)	)	PUNCT
iajs-2920	167	1	∗′	∗′	PROPN
iajs-2920	167	2	𝑓(휀	𝑓(휀	PROPN
iajs-2920	167	3	)	)	PUNCT
iajs-2920	167	4	)	)	PUNCT
iajs-2920	167	5	,	,	PUNCT
iajs-2920	167	6	�	�	PROPN
iajs-2920	167	7	̃	̃	PROPN
iajs-2920	167	8	�	�	NOUN
iajs-2920	167	9	𝜔(𝑓(𝜏	𝜔(𝑓(𝜏	NOUN
iajs-2920	167	10	)	)	PUNCT
iajs-2920	167	11	)	)	PUNCT
iajs-2920	167	12	}	}	PUNCT
iajs-2920	168	1	=	=	SYM
iajs-2920	168	2	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	168	3	{	{	PUNCT
iajs-2920	168	4	�	�	PROPN
iajs-2920	168	5	̃	̃	PROPN
iajs-2920	168	6	�	�	NOUN
iajs-2920	168	7	𝜔	𝜔	NOUN
iajs-2920	168	8	(	(	PUNCT
iajs-2920	168	9	𝑓((𝜌	𝑓((𝜌	PROPN
iajs-2920	168	10	∗	∗	PROPN
iajs-2920	168	11	𝜏	𝜏	NOUN
iajs-2920	168	12	)	)	PUNCT
iajs-2920	168	13	∗	∗	NOUN
iajs-2920	168	14	휀	휀	NOUN
iajs-2920	168	15	)	)	PUNCT
iajs-2920	168	16	)	)	PUNCT
iajs-2920	168	17	,	,	PUNCT
iajs-2920	168	18	�	�	PROPN
iajs-2920	168	19	̃	̃	PROPN
iajs-2920	168	20	�	�	NOUN
iajs-2920	168	21	𝜔(𝑓(𝜏	𝜔(𝑓(𝜏	NOUN
iajs-2920	168	22	)	)	PUNCT
iajs-2920	168	23	)	)	PUNCT
iajs-2920	168	24	}	}	PUNCT
iajs-2920	168	25	=	=	SYM
iajs-2920	168	26	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	168	27	�	�	PROPN
iajs-2920	168	28	̃	̃	PROPN
iajs-2920	168	29	�	�	PROPN
iajs-2920	168	30	𝛿((𝜌	𝛿((𝜌	PROPN
iajs-2920	168	31	∗	∗	PROPN
iajs-2920	168	32	𝜏	𝜏	NOUN
iajs-2920	168	33	)	)	PUNCT
iajs-2920	168	34	∗	∗	NOUN
iajs-2920	168	35	휀	휀	NOUN
iajs-2920	168	36	)	)	PUNCT
iajs-2920	168	37	,	,	PUNCT
iajs-2920	168	38	�	�	PROPN
iajs-2920	168	39	̃	̃	PROPN
iajs-2920	168	40	�	�	NOUN
iajs-2920	168	41	𝛿(𝜏	𝛿(𝜏	NOUN
iajs-2920	168	42	)	)	PUNCT
iajs-2920	168	43	}	}	PUNCT
iajs-2920	168	44	,	,	PUNCT
iajs-2920	168	45	𝛼𝛿(𝜌	𝛼𝛿(𝜌	PUNCT
iajs-2920	168	46	∗	∗	NOUN
iajs-2920	168	47	휀	휀	NOUN
iajs-2920	168	48	)	)	PUNCT
iajs-2920	168	49	=	=	SYM
iajs-2920	168	50	𝛼𝜔(𝑓(𝜌	𝛼𝜔(𝑓(𝜌	PROPN
iajs-2920	168	51	∗	∗	NOUN
iajs-2920	168	52	휀	휀	NOUN
iajs-2920	168	53	)	)	PUNCT
iajs-2920	168	54	)	)	PUNCT
iajs-2920	168	55	=	=	SYM
iajs-2920	168	56	𝛼𝜔(𝑓(𝜌	𝛼𝜔(𝑓(𝜌	VERB
iajs-2920	168	57	)	)	PUNCT
iajs-2920	168	58	∗′	∗′	PROPN
iajs-2920	168	59	𝑓(휀	𝑓(휀	NOUN
iajs-2920	168	60	)	)	PUNCT
iajs-2920	168	61	)	)	PUNCT
iajs-2920	168	62	≤	≤	NUM
iajs-2920	169	1	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2920	169	2	{	{	PUNCT
iajs-2920	169	3	𝛼𝜔	𝛼𝜔	PROPN
iajs-2920	169	4	(	(	PUNCT
iajs-2920	169	5	(	(	PUNCT
iajs-2920	169	6	𝑓(𝜌	𝑓(𝜌	PROPN
iajs-2920	169	7	)	)	PUNCT
iajs-2920	169	8	∗′	∗′	ADJ
iajs-2920	169	9	𝑓(𝜏	𝑓(𝜏	NOUN
iajs-2920	169	10	)	)	PUNCT
iajs-2920	169	11	)	)	PUNCT
iajs-2920	170	1	∗′	∗′	PROPN
iajs-2920	170	2	𝑓(휀	𝑓(휀	PROPN
iajs-2920	170	3	)	)	PUNCT
iajs-2920	170	4	)	)	PUNCT
iajs-2920	170	5	,	,	PUNCT
iajs-2920	170	6	𝛼𝜔(𝑓(𝜏	𝛼𝜔(𝑓(𝜏	NUM
iajs-2920	170	7	)	)	PUNCT
iajs-2920	170	8	)	)	PUNCT
iajs-2920	170	9	}	}	PUNCT
iajs-2920	171	1	=	=	PUNCT
iajs-2920	171	2	𝑚𝑎𝑥	𝑚𝑎𝑥	X
iajs-2920	171	3	{	{	PUNCT
iajs-2920	171	4	𝛼𝜔	𝛼𝜔	PROPN
iajs-2920	171	5	(	(	PUNCT
iajs-2920	171	6	𝑓((𝜌	𝑓((𝜌	PROPN
iajs-2920	171	7	∗	∗	PROPN
iajs-2920	171	8	𝜏	𝜏	NOUN
iajs-2920	171	9	)	)	PUNCT
iajs-2920	171	10	∗	∗	NOUN
iajs-2920	171	11	휀	휀	NOUN
iajs-2920	171	12	)	)	PUNCT
iajs-2920	171	13	)	)	PUNCT
iajs-2920	171	14	,	,	PUNCT
iajs-2920	171	15	𝛼𝜔(𝑓(𝜏	𝛼𝜔(𝑓(𝜏	NUM
iajs-2920	171	16	)	)	PUNCT
iajs-2920	171	17	)	)	PUNCT
iajs-2920	171	18	}	}	PUNCT
iajs-2920	171	19	=	=	SYM
iajs-2920	171	20	𝑚𝑎𝑥{𝛼𝛿((𝜌	𝑚𝑎𝑥{𝛼𝛿((𝜌	X
iajs-2920	171	21	∗	∗	X
iajs-2920	171	22	𝜏	𝜏	NOUN
iajs-2920	171	23	)	)	PUNCT
iajs-2920	171	24	∗	∗	NOUN
iajs-2920	171	25	휀	휀	NOUN
iajs-2920	171	26	)	)	PUNCT
iajs-2920	171	27	,	,	PUNCT
iajs-2920	171	28	𝛼𝛿(𝜏	𝛼𝛿(𝜏	NOUN
iajs-2920	171	29	)	)	PUNCT
iajs-2920	171	30	}	}	PUNCT
iajs-2920	171	31	.	.	PUNCT
iajs-2920	172	1	definition	definition	NOUN
iajs-2920	172	2	(	(	PUNCT
iajs-2920	172	3	27	27	NUM
iajs-2920	172	4	)	)	PUNCT
iajs-2920	172	5	.	.	PUNCT
iajs-2920	173	1	a	a	DET
iajs-2920	173	2	cubic	cubic	ADJ
iajs-2920	173	3	subset	subset	VERB
iajs-2920	173	4	𝛿	𝛿	PROPN
iajs-2920	173	5	=	=	PROPN
iajs-2920	173	6	〈	〈	PROPN
iajs-2920	173	7	�	�	PROPN
iajs-2920	173	8	̃	̃	PROPN
iajs-2920	173	9	�	�	PROPN
iajs-2920	173	10	𝛿	𝛿	NOUN
iajs-2920	173	11	,	,	PUNCT
iajs-2920	173	12	𝛼𝛿	𝛼𝛿	PART
iajs-2920	173	13	〉	〉	NOUN
iajs-2920	173	14	of	of	ADP
iajs-2920	173	15	ℵhas	ℵha	NOUN
iajs-2920	173	16	sup	sup	NOUN
iajs-2920	173	17	and	and	CCONJ
iajs-2920	173	18	inf	inf	NOUN
iajs-2920	173	19	properties	property	NOUN
iajs-2920	173	20	if	if	SCONJ
iajs-2920	173	21	for	for	ADP
iajs-2920	173	22	any	any	DET
iajs-2920	173	23	subset	subset	NOUN
iajs-2920	173	24	𝑇	𝑇	PROPN
iajs-2920	173	25	𝑜𝑓	𝑜𝑓	ADP
iajs-2920	173	26	ℵ	ℵ	NOUN
iajs-2920	173	27	,	,	PUNCT
iajs-2920	173	28	there	there	PRON
iajs-2920	173	29	exist	exist	VERB
iajs-2920	173	30	𝑡	𝑡	NOUN
iajs-2920	173	31	,	,	PUNCT
iajs-2920	173	32	𝑠	𝑠	PROPN
iajs-2920	173	33	∈	∈	PROPN
iajs-2920	173	34	𝑇	𝑇	PROPN
iajs-2920	173	35	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-2920	173	36	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-2920	173	37	�	�	PROPN
iajs-2920	173	38	̃	̃	PROPN
iajs-2920	173	39	�	�	NOUN
iajs-2920	173	40	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	173	41	)	)	PUNCT
iajs-2920	174	1	=	=	PUNCT
iajs-2920	174	2	𝑟𝑠𝑢𝑝𝑡∈𝑇	𝑟𝑠𝑢𝑝𝑡∈𝑇	PROPN
iajs-2920	174	3	�	�	PROPN
iajs-2920	174	4	̃	̃	PROPN
iajs-2920	174	5	�	�	NOUN
iajs-2920	174	6	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	174	7	)	)	PUNCT
iajs-2920	174	8	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	174	9	𝛼𝛿(𝑠	𝛼𝛿(𝑠	PUNCT
iajs-2920	174	10	)	)	PUNCT
iajs-2920	174	11	=	=	SYM
iajs-2920	174	12	𝑖𝑛𝑓𝑡∈𝑇𝛼𝛿(𝑠	𝑖𝑛𝑓𝑡∈𝑇𝛼𝛿(𝑠	X
iajs-2920	174	13	)	)	PUNCT
iajs-2920	174	14	.	.	PUNCT
iajs-2920	175	1	theorem	theorem	NOUN
iajs-2920	175	2	(	(	PUNCT
iajs-2920	175	3	28	28	NUM
iajs-2920	175	4	)	)	PUNCT
iajs-2920	175	5	.	.	PUNCT
iajs-2920	176	1	let	let	VERB
iajs-2920	176	2	𝑓	𝑓	PRON
iajs-2920	176	3	:	:	PUNCT
iajs-2920	176	4	ℵ	ℵ	PROPN
iajs-2920	176	5	→	→	SYM
iajs-2920	176	6	𝑌	𝑌	PROPN
iajs-2920	176	7	be	be	VERB
iajs-2920	176	8	anepimorphism	anepimorphism	NOUN
iajs-2920	176	9	between	between	ADP
iajs-2920	176	10	tm	tm	NOUN
iajs-2920	176	11	-	-	NOUN
iajs-2920	176	12	algebra	algebra	ADJ
iajs-2920	176	13	ℵ	ℵ	ADJ
iajs-2920	176	14	ihjpas	ihjpa	NOUN
iajs-2920	176	15	.	.	PUNCT
iajs-2920	177	1	36(1)2023	36(1)2023	NUM
iajs-2920	177	2	375	375	NUM
iajs-2920	177	3	and	and	CCONJ
iajs-2920	177	4	𝑌.	𝑌.	PROPN
iajs-2920	177	5	for	for	ADP
iajs-2920	177	6	every	every	DET
iajs-2920	177	7	cubic	cubic	ADJ
iajs-2920	177	8	t	t	PROPN
iajs-2920	177	9	-	-	PUNCT
iajs-2920	177	10	ideal	ideal	NOUN
iajs-2920	177	11	𝛿	𝛿	ADJ
iajs-2920	177	12	=	=	PUNCT
iajs-2920	177	13	〈	〈	PROPN
iajs-2920	177	14	�	�	PROPN
iajs-2920	177	15	̃	̃	PROPN
iajs-2920	177	16	�	�	PROPN
iajs-2920	177	17	𝛿	𝛿	NOUN
iajs-2920	177	18	,	,	PUNCT
iajs-2920	177	19	𝛼𝛿	𝛼𝛿	X
iajs-2920	177	20	〉	〉	NOUN
iajs-2920	177	21	𝑖𝑛	𝑖𝑛	NOUN
iajs-2920	177	22	ℵ	ℵ	NOUN
iajs-2920	177	23	,	,	PUNCT
iajs-2920	177	24	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	NOUN
iajs-2920	177	25	𝑓(𝛿)is	𝑓(𝛿)is	ADJ
iajs-2920	177	26	cubic	cubic	PROPN
iajs-2920	177	27	t	t	PROPN
iajs-2920	177	28	-	-	PUNCT
iajs-2920	177	29	ideal	ideal	NOUN
iajs-2920	177	30	of	of	ADP
iajs-2920	177	31	𝑌.	𝑌.	PROPN
iajs-2920	177	32	proof	proof	NOUN
iajs-2920	177	33	.	.	PUNCT
iajs-2920	178	1	by	by	ADP
iajs-2920	178	2	definition25	definition25	PROPN
iajs-2920	178	3	�	�	PROPN
iajs-2920	178	4	̃	̃	PROPN
iajs-2920	178	5	�	�	NOUN
iajs-2920	178	6	𝜔(𝜏′	𝜔(𝜏′	NUM
iajs-2920	178	7	)	)	PUNCT
iajs-2920	178	8	=	=	SYM
iajs-2920	178	9	𝑓(	𝑓(	PROPN
iajs-2920	178	10	�	�	PROPN
iajs-2920	178	11	̃	̃	PROPN
iajs-2920	178	12	�	�	NOUN
iajs-2920	178	13	𝛿)(𝜏′	𝛿)(𝜏′	NUM
iajs-2920	178	14	)	)	PUNCT
iajs-2920	178	15	=	=	SYM
iajs-2920	178	16	𝑟𝑠𝑢𝑝𝜌∈𝑓−1(𝜏′)	𝑟𝑠𝑢𝑝𝜌∈𝑓−1(𝜏′)	NOUN
iajs-2920	178	17	�	�	PROPN
iajs-2920	178	18	̃	̃	PROPN
iajs-2920	178	19	�	�	NOUN
iajs-2920	178	20	𝛿(𝜌	𝛿(𝜌	NOUN
iajs-2920	178	21	)	)	PUNCT
iajs-2920	178	22	and	and	CCONJ
iajs-2920	178	23	𝛼𝜔(𝜏′	𝛼𝜔(𝜏′	X
iajs-2920	178	24	)	)	PUNCT
iajs-2920	178	25	=	=	PUNCT
iajs-2920	178	26	𝑓(𝛼𝛿)(𝜏′	𝑓(𝛼𝛿)(𝜏′	PROPN
iajs-2920	178	27	)	)	PUNCT
iajs-2920	178	28	=	=	PUNCT
iajs-2920	178	29	𝑖𝑛𝑓𝜌∈𝑓−1(𝜏′)𝛼𝛿(𝜌	𝑖𝑛𝑓𝜌∈𝑓−1(𝜏′)𝛼𝛿(𝜌	X
iajs-2920	178	30	)	)	PUNCT
iajs-2920	178	31	for	for	ADP
iajs-2920	178	32	any	any	DET
iajs-2920	178	33	𝜏′	𝜏′	PROPN
iajs-2920	178	34	∈	∈	PROPN
iajs-2920	178	35	𝑌	𝑌	PROPN
iajs-2920	178	36	and	and	CCONJ
iajs-2920	178	37	𝑟𝑠𝑢𝑝	𝑟𝑠𝑢𝑝	ADJ
iajs-2920	178	38	∅	∅	NOUN
iajs-2920	178	39	=	=	PUNCT
iajs-2920	179	1	[	[	X
iajs-2920	179	2	0,0	0,0	NOUN
iajs-2920	179	3	]	]	X
iajs-2920	179	4	=	=	SYM
iajs-2920	179	5	0	0	X
iajs-2920	179	6	.	.	PUNCT
iajs-2920	180	1	we	we	PRON
iajs-2920	180	2	must	must	AUX
iajs-2920	180	3	prove	prove	VERB
iajs-2920	180	4	that	that	SCONJ
iajs-2920	180	5	�	�	PROPN
iajs-2920	180	6	̃	̃	PROPN
iajs-2920	180	7	�	�	PROPN
iajs-2920	180	8	𝜔(𝜌′	𝜔(𝜌′	PROPN
iajs-2920	180	9	∗	∗	NOUN
iajs-2920	180	10	휀′	휀′	NUM
iajs-2920	180	11	)	)	PUNCT
iajs-2920	180	12	≥	≥	PROPN
iajs-2920	180	13	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	180	14	�	�	PROPN
iajs-2920	180	15	̃	̃	PROPN
iajs-2920	180	16	�	�	PROPN
iajs-2920	180	17	𝜔((𝜌′	𝜔((𝜌′	PROPN
iajs-2920	180	18	∗	∗	PROPN
iajs-2920	180	19	𝜏′	𝜏′	PROPN
iajs-2920	180	20	)	)	PUNCT
iajs-2920	180	21	∗	∗	NOUN
iajs-2920	180	22	휀′	휀′	NUM
iajs-2920	180	23	)	)	PUNCT
iajs-2920	180	24	,	,	PUNCT
iajs-2920	180	25	�	�	PROPN
iajs-2920	180	26	̃	̃	PROPN
iajs-2920	180	27	�	�	NOUN
iajs-2920	180	28	𝜔(𝜏′	𝜔(𝜏′	NUM
iajs-2920	180	29	)	)	PUNCT
iajs-2920	180	30	}	}	PUNCT
iajs-2920	180	31	and	and	CCONJ
iajs-2920	180	32	𝛼𝜔(𝜌′	𝛼𝜔(𝜌′	NUM
iajs-2920	180	33	∗	∗	NOUN
iajs-2920	180	34	휀′	휀′	NUM
iajs-2920	180	35	)	)	PUNCT
iajs-2920	180	36	≤	≤	NOUN
iajs-2920	181	1	𝑚𝑎𝑥{𝛼𝜔((𝜌′	𝑚𝑎𝑥{𝛼𝜔((𝜌′	NUM
iajs-2920	181	2	∗	∗	NOUN
iajs-2920	181	3	𝜏′	𝜏′	PROPN
iajs-2920	181	4	)	)	PUNCT
iajs-2920	181	5	∗	∗	NOUN
iajs-2920	181	6	휀′	휀′	NUM
iajs-2920	181	7	)	)	PUNCT
iajs-2920	182	1	,	,	PUNCT
iajs-2920	182	2	𝛼𝜔(𝜏′	𝛼𝜔(𝜏′	NOUN
iajs-2920	182	3	)	)	PUNCT
iajs-2920	182	4	}	}	PUNCT
iajs-2920	182	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-2920	182	6	𝑎𝑛𝑦	𝑎𝑛𝑦	VERB
iajs-2920	182	7	𝜌′	𝜌′	PROPN
iajs-2920	182	8	,	,	PUNCT
iajs-2920	182	9	𝜏′	𝜏′	PROPN
iajs-2920	182	10	,	,	PUNCT
iajs-2920	182	11	휀′	휀′	NUM
iajs-2920	182	12	∈	∈	NOUN
iajs-2920	182	13	𝑌.	𝑌.	PROPN
iajs-2920	182	14	let	let	VERB
iajs-2920	182	15	𝑓	𝑓	PRON
iajs-2920	182	16	:	:	PUNCT
iajs-2920	182	17	ℵ	ℵ	PROPN
iajs-2920	182	18	→	→	SYM
iajs-2920	182	19	𝑌	𝑌	PROPN
iajs-2920	182	20	be	be	VERB
iajs-2920	182	21	an	an	DET
iajs-2920	182	22	epimorphism	epimorphism	NOUN
iajs-2920	182	23	mappingof	mappingof	PROPN
iajs-2920	182	24	ℵ	ℵ	NOUN
iajs-2920	182	25	,	,	PUNCT
iajs-2920	182	26	𝛿	𝛿	ADJ
iajs-2920	182	27	=	=	PROPN
iajs-2920	182	28	〈	〈	PROPN
iajs-2920	182	29	�	�	PROPN
iajs-2920	182	30	̃	̃	PROPN
iajs-2920	182	31	�	�	NOUN
iajs-2920	182	32	𝛿	𝛿	NOUN
iajs-2920	182	33	,	,	PUNCT
iajs-2920	182	34	𝛼𝛿〉be	𝛼𝛿〉be	PROPN
iajs-2920	182	35	a	a	DET
iajs-2920	182	36	cubic	cubic	ADJ
iajs-2920	182	37	t	t	PROPN
iajs-2920	182	38	-	-	PUNCT
iajs-2920	182	39	ideal	ideal	NOUN
iajs-2920	182	40	of	of	ADP
iajs-2920	182	41	ℵwith	ℵwith	ADJ
iajs-2920	182	42	sup	sup	NOUN
iajs-2920	182	43	and	and	CCONJ
iajs-2920	182	44	inf	inf	NOUN
iajs-2920	182	45	properties	property	NOUN
iajs-2920	182	46	and	and	CCONJ
iajs-2920	182	47	ω	ω	NUM
iajs-2920	182	48	=	=	SYM
iajs-2920	182	49	〈	〈	PROPN
iajs-2920	182	50	�	�	PROPN
iajs-2920	182	51	̃	̃	PROPN
iajs-2920	182	52	�	�	NOUN
iajs-2920	182	53	𝜔	𝜔	NOUN
iajs-2920	182	54	,	,	PUNCT
iajs-2920	182	55	𝛼𝜔〉be	𝛼𝜔〉be	PROPN
iajs-2920	182	56	the	the	DET
iajs-2920	182	57	image	image	NOUN
iajs-2920	182	58	of	of	ADP
iajs-2920	182	59	𝛿	𝛿	ADJ
iajs-2920	182	60	=	=	PUNCT
iajs-2920	182	61	〈	〈	PROPN
iajs-2920	182	62	�	�	PROPN
iajs-2920	182	63	̃	̃	PROPN
iajs-2920	182	64	�	�	PROPN
iajs-2920	182	65	𝛿	𝛿	NOUN
iajs-2920	182	66	,	,	PUNCT
iajs-2920	182	67	𝛼𝛿	𝛼𝛿	PART
iajs-2920	182	68	〉	〉	NOUN
iajs-2920	182	69	under	under	ADP
iajs-2920	182	70	𝑓.	𝑓.	NOUN
iajs-2920	182	71	since	since	SCONJ
iajs-2920	182	72	𝛿	𝛿	PROPN
iajs-2920	182	73	=	=	PROPN
iajs-2920	182	74	〈	〈	PROPN
iajs-2920	182	75	�	�	PROPN
iajs-2920	182	76	̃	̃	PROPN
iajs-2920	182	77	�	�	PROPN
iajs-2920	182	78	𝛿	𝛿	NOUN
iajs-2920	182	79	,	,	PUNCT
iajs-2920	182	80	𝛼𝛿	𝛼𝛿	NUM
iajs-2920	182	81	〉	〉	NOUN
iajs-2920	182	82	is	be	AUX
iajs-2920	182	83	a	a	DET
iajs-2920	182	84	cubict	cubict	NOUN
iajs-2920	182	85	-	-	PUNCT
iajs-2920	182	86	ideal	ideal	NOUN
iajs-2920	182	87	of	of	ADP
iajs-2920	182	88	ℵ	ℵ	NOUN
iajs-2920	182	89	,	,	PUNCT
iajs-2920	182	90	we	we	PRON
iajs-2920	182	91	have	have	VERB
iajs-2920	182	92	�	�	PROPN
iajs-2920	182	93	̃	̃	PROPN
iajs-2920	182	94	�	�	PROPN
iajs-2920	182	95	𝛿(0	𝛿(0	PROPN
iajs-2920	182	96	)	)	PUNCT
iajs-2920	182	97	≥	≥	NOUN
iajs-2920	182	98	�	�	PROPN
iajs-2920	182	99	̃	̃	PROPN
iajs-2920	182	100	�	�	PROPN
iajs-2920	182	101	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	182	102	)	)	PUNCT
iajs-2920	182	103	,	,	PUNCT
iajs-2920	182	104	𝛼𝛿(0	𝛼𝛿(0	NOUN
iajs-2920	182	105	)	)	PUNCT
iajs-2920	182	106	≤	≤	NOUN
iajs-2920	182	107	𝛼𝛿(𝜌	𝛼𝛿(𝜌	NOUN
iajs-2920	182	108	)	)	PUNCT
iajs-2920	182	109	∀𝜌	∀𝜌	PUNCT
iajs-2920	182	110	∈	∈	PROPN
iajs-2920	182	111	ℵ	ℵ	NOUN
iajs-2920	182	112	.	.	PUNCT
iajs-2920	183	1	note	note	VERB
iajs-2920	183	2	that	that	SCONJ
iajs-2920	183	3	0	0	NUM
iajs-2920	183	4	∈	∈	PROPN
iajs-2920	183	5	𝑓−1(0′	𝑓−1(0′	PROPN
iajs-2920	183	6	)	)	PUNCT
iajs-2920	183	7	where	where	SCONJ
iajs-2920	183	8	0	0	NUM
iajs-2920	183	9	,	,	PUNCT
iajs-2920	183	10	0′	0′	NUM
iajs-2920	183	11	are	be	AUX
iajs-2920	183	12	the	the	DET
iajs-2920	183	13	zero	zero	NUM
iajs-2920	183	14	of	of	ADP
iajs-2920	183	15	ℵ	ℵ	DET
iajs-2920	183	16	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	183	17	𝑌	𝑌	PROPN
iajs-2920	183	18	,	,	PUNCT
iajs-2920	183	19	respectively	respectively	ADV
iajs-2920	183	20	.	.	PUNCT
iajs-2920	184	1	thus	thus	ADV
iajs-2920	184	2	,	,	PUNCT
iajs-2920	184	3	�	�	PROPN
iajs-2920	184	4	̃	̃	PROPN
iajs-2920	184	5	�	�	NOUN
iajs-2920	184	6	𝛿(0′	𝛿(0′	VERB
iajs-2920	184	7	)	)	PUNCT
iajs-2920	185	1	=	=	SYM
iajs-2920	185	2	rsup	rsup	ADJ
iajs-2920	185	3	𝑡∈𝑓−1(0	𝑡∈𝑓−1(0	SYM
iajs-2920	185	4	)	)	PUNCT
iajs-2920	185	5	�	�	PROPN
iajs-2920	185	6	̃	̃	PROPN
iajs-2920	185	7	�	�	NOUN
iajs-2920	185	8	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	185	9	)	)	PUNCT
iajs-2920	185	10	=	=	SYM
iajs-2920	185	11	�	�	PROPN
iajs-2920	185	12	̃	̃	PROPN
iajs-2920	185	13	�	�	PROPN
iajs-2920	185	14	𝛿(0	𝛿(0	PROPN
iajs-2920	185	15	)	)	PUNCT
iajs-2920	185	16	≥	≥	NOUN
iajs-2920	185	17	�	�	PROPN
iajs-2920	185	18	̃	̃	PROPN
iajs-2920	185	19	�	�	PROPN
iajs-2920	185	20	𝛿(𝜌	𝛿(𝜌	PROPN
iajs-2920	185	21	)	)	PUNCT
iajs-2920	185	22	∀𝜌	∀𝜌	PUNCT
iajs-2920	185	23	∈	∈	PROPN
iajs-2920	185	24	ℵ	ℵ	NOUN
iajs-2920	185	25	,	,	PUNCT
iajs-2920	185	26	𝛼𝜔(0′	𝛼𝜔(0′	PROPN
iajs-2920	185	27	)	)	PUNCT
iajs-2920	185	28	≤	≤	NUM
iajs-2920	185	29	inf	inf	PROPN
iajs-2920	185	30	𝑡∈𝑓−1(0′	𝑡∈𝑓−1(0′	NOUN
iajs-2920	185	31	)	)	PUNCT
iajs-2920	185	32	𝛼𝜔(𝑡	𝛼𝜔(𝑡	PUNCT
iajs-2920	185	33	)	)	PUNCT
iajs-2920	185	34	=	=	SYM
iajs-2920	185	35	𝛼𝜔(0	𝛼𝜔(0	NUM
iajs-2920	185	36	)	)	PUNCT
iajs-2920	185	37	≤	≤	NOUN
iajs-2920	185	38	𝛼𝜔(𝜌	𝛼𝜔(𝜌	PUNCT
iajs-2920	185	39	)	)	PUNCT
iajs-2920	185	40	∀𝜌	∀𝜌	PUNCT
iajs-2920	185	41	∈	∈	PROPN
iajs-2920	185	42	ℵ	ℵ	NOUN
iajs-2920	185	43	,	,	PUNCT
iajs-2920	185	44	which	which	PRON
iajs-2920	185	45	implies	imply	VERB
iajs-2920	185	46	that	that	SCONJ
iajs-2920	185	47	�	�	PROPN
iajs-2920	185	48	̃	̃	PROPN
iajs-2920	185	49	�	�	PROPN
iajs-2920	185	50	𝜔(0′	𝜔(0′	PROPN
iajs-2920	185	51	)	)	PUNCT
iajs-2920	185	52	≥	≥	NOUN
iajs-2920	185	53	rsup	rsup	PROPN
iajs-2920	185	54	𝑡∈𝑓−1(𝜌′	𝑡∈𝑓−1(𝜌′	PROPN
iajs-2920	185	55	)	)	PUNCT
iajs-2920	185	56	�	�	PROPN
iajs-2920	185	57	̃	̃	PROPN
iajs-2920	185	58	�	�	NOUN
iajs-2920	185	59	𝜔(𝜌′	𝜔(𝜌′	NUM
iajs-2920	185	60	)	)	PUNCT
iajs-2920	185	61	and	and	CCONJ
iajs-2920	185	62	𝛼𝜔(0′	𝛼𝜔(0′	NUM
iajs-2920	185	63	)	)	PUNCT
iajs-2920	185	64	≤	≤	NUM
iajs-2920	185	65	inf	inf	PROPN
iajs-2920	185	66	𝑡∈𝑓−1(𝜌′	𝑡∈𝑓−1(𝜌′	PROPN
iajs-2920	185	67	)	)	PUNCT
iajs-2920	185	68	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NUM
iajs-2920	185	69	)	)	PUNCT
iajs-2920	185	70	=	=	SYM
iajs-2920	186	1	𝛼𝜔(𝜌′)for	𝛼𝜔(𝜌′)for	PROPN
iajs-2920	186	2	any𝜌′	any𝜌′	NOUN
iajs-2920	186	3	∈	∈	PROPN
iajs-2920	186	4	𝑌.	𝑌.	PROPN
iajs-2920	186	5	for	for	ADP
iajs-2920	186	6	any	any	DET
iajs-2920	186	7	𝜌′	𝜌′	NOUN
iajs-2920	186	8	,	,	PUNCT
iajs-2920	186	9	𝜏′	𝜏′	NUM
iajs-2920	186	10	,	,	PUNCT
iajs-2920	186	11	휀′	휀′	NUM
iajs-2920	186	12	∈	∈	PROPN
iajs-2920	186	13	𝑌	𝑌	PROPN
iajs-2920	186	14	,	,	PUNCT
iajs-2920	186	15	let	let	VERB
iajs-2920	186	16	𝜌0	𝜌0	PROPN
iajs-2920	186	17	∈	∈	PROPN
iajs-2920	186	18	𝑓−1(𝜌′	𝑓−1(𝜌′	NOUN
iajs-2920	186	19	)	)	PUNCT
iajs-2920	186	20	,	,	PUNCT
iajs-2920	186	21	𝜏0	𝜏0	PROPN
iajs-2920	186	22	∈	∈	PROPN
iajs-2920	186	23	𝑓−1(𝜏′	𝑓−1(𝜏′	PRON
iajs-2920	186	24	)	)	PUNCT
iajs-2920	186	25	,	,	PUNCT
iajs-2920	186	26	and휀0	and휀0	PROPN
iajs-2920	186	27	∈	∈	PROPN
iajs-2920	186	28	𝑓−1(휀′	𝑓−1(휀′	NOUN
iajs-2920	186	29	)	)	PUNCT
iajs-2920	186	30	be	be	VERB
iajs-2920	186	31	such	such	ADJ
iajs-2920	186	32	that	that	DET
iajs-2920	186	33	�	�	PROPN
iajs-2920	186	34	̃	̃	PROPN
iajs-2920	186	35	�	�	PROPN
iajs-2920	186	36	𝛿(𝜌0	𝛿(𝜌0	NOUN
iajs-2920	186	37	∗	∗	NOUN
iajs-2920	186	38	휀0	휀0	NOUN
iajs-2920	186	39	)	)	PUNCT
iajs-2920	186	40	=	=	SYM
iajs-2920	186	41	rsup	rsup	ADJ
iajs-2920	186	42	𝑡∈𝑓−1(𝜌′∗	𝑡∈𝑓−1(𝜌′∗	PROPN
iajs-2920	186	43	′	′	NUM
iajs-2920	186	44	)	)	PUNCT
iajs-2920	186	45	�	�	PROPN
iajs-2920	186	46	̃	̃	PROPN
iajs-2920	186	47	�	�	NOUN
iajs-2920	186	48	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	186	49	)	)	PUNCT
iajs-2920	186	50	,	,	PUNCT
iajs-2920	186	51	�	�	PROPN
iajs-2920	186	52	̃	̃	PROPN
iajs-2920	186	53	�	�	NOUN
iajs-2920	186	54	𝛿(𝜏0	𝛿(𝜏0	NOUN
iajs-2920	186	55	)	)	PUNCT
iajs-2920	186	56	=	=	SYM
iajs-2920	186	57	rsup	rsup	ADJ
iajs-2920	186	58	𝑡∈𝑓−1(𝜏′	𝑡∈𝑓−1(𝜏′	PROPN
iajs-2920	186	59	)	)	PUNCT
iajs-2920	186	60	�	�	PROPN
iajs-2920	186	61	̃	̃	PROPN
iajs-2920	186	62	�	�	NOUN
iajs-2920	186	63	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	186	64	)	)	PUNCT
iajs-2920	186	65	,	,	PUNCT
iajs-2920	186	66	�	�	PROPN
iajs-2920	186	67	̃	̃	PROPN
iajs-2920	186	68	�	�	NOUN
iajs-2920	186	69	𝛿((𝜌0	𝛿((𝜌0	NOUN
iajs-2920	186	70	∗	∗	NOUN
iajs-2920	186	71	𝜏0	𝜏0	PROPN
iajs-2920	186	72	)	)	PUNCT
iajs-2920	186	73	∗	∗	NOUN
iajs-2920	186	74	휀0	휀0	NOUN
iajs-2920	186	75	)	)	PUNCT
iajs-2920	186	76	=	=	SYM
iajs-2920	186	77	�	�	PROPN
iajs-2920	186	78	̃	̃	PROPN
iajs-2920	186	79	�	�	NOUN
iajs-2920	186	80	𝜔{𝑓((𝜌0	𝜔{𝑓((𝜌0	PART
iajs-2920	186	81	∗	∗	NUM
iajs-2920	186	82	𝜏0	𝜏0	PROPN
iajs-2920	186	83	)	)	PUNCT
iajs-2920	186	84	∗	∗	NOUN
iajs-2920	186	85	휀0	휀0	NOUN
iajs-2920	186	86	)	)	PUNCT
iajs-2920	186	87	}	}	PUNCT
iajs-2920	186	88	=	=	SYM
iajs-2920	186	89	�	�	PROPN
iajs-2920	186	90	̃	̃	PROPN
iajs-2920	186	91	�	�	PROPN
iajs-2920	186	92	𝜔((𝜌′	𝜔((𝜌′	PROPN
iajs-2920	186	93	∗	∗	PROPN
iajs-2920	186	94	𝜏′	𝜏′	PROPN
iajs-2920	186	95	)	)	PUNCT
iajs-2920	186	96	∗	∗	NOUN
iajs-2920	186	97	휀′	휀′	NUM
iajs-2920	186	98	)	)	PUNCT
iajs-2920	187	1	=	=	PRON
iajs-2920	187	2	rsup	rsup	NOUN
iajs-2920	187	3	(	(	PUNCT
iajs-2920	187	4	(	(	PUNCT
iajs-2920	187	5	𝜌0∗𝜏0)∗	𝜌0∗𝜏0)∗	PROPN
iajs-2920	187	6	0)∈𝑓−1((𝜌′∗𝜏′)∗	0)∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	187	7	′	′	NOUN
iajs-2920	187	8	)	)	PUNCT
iajs-2920	187	9	�	�	PROPN
iajs-2920	187	10	̃	̃	PROPN
iajs-2920	187	11	�	�	PROPN
iajs-2920	187	12	𝛿	𝛿	ADJ
iajs-2920	187	13	(	(	PUNCT
iajs-2920	187	14	(	(	PUNCT
iajs-2920	187	15	𝜌0	𝜌0	ADJ
iajs-2920	187	16	∗	∗	NOUN
iajs-2920	187	17	𝜏0	𝜏0	PROPN
iajs-2920	187	18	)	)	PUNCT
iajs-2920	187	19	∗	∗	NOUN
iajs-2920	187	20	휀0	휀0	NOUN
iajs-2920	187	21	)	)	PUNCT
iajs-2920	187	22	=	=	SYM
iajs-2920	187	23	rsup	rsup	ADJ
iajs-2920	187	24	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	187	25	′	′	NUM
iajs-2920	187	26	)	)	PUNCT
iajs-2920	187	27	�	�	PROPN
iajs-2920	187	28	̃	̃	PROPN
iajs-2920	187	29	�	�	NOUN
iajs-2920	187	30	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	187	31	)	)	PUNCT
iajs-2920	187	32	.also	.also	PUNCT
iajs-2920	187	33	𝛼𝛿(𝜌0	𝛼𝛿(𝜌0	PROPN
iajs-2920	187	34	∗	∗	NOUN
iajs-2920	187	35	휀0	휀0	NOUN
iajs-2920	187	36	)	)	PUNCT
iajs-2920	187	37	=	=	SYM
iajs-2920	187	38	inf	inf	NOUN
iajs-2920	187	39	𝑡∈𝑓−1(𝜌′∗	𝑡∈𝑓−1(𝜌′∗	PROPN
iajs-2920	187	40	′	′	NUM
iajs-2920	187	41	)	)	PUNCT
iajs-2920	187	42	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NUM
iajs-2920	187	43	)	)	PUNCT
iajs-2920	187	44	.	.	PUNCT
iajs-2920	188	1	𝛼𝛿(𝜏0	𝛼𝛿(𝜏0	NOUN
iajs-2920	188	2	)	)	PUNCT
iajs-2920	188	3	=	=	SYM
iajs-2920	188	4	inf	inf	PROPN
iajs-2920	188	5	𝑡∈𝑓−1(𝜏′	𝑡∈𝑓−1(𝜏′	PROPN
iajs-2920	188	6	)	)	PUNCT
iajs-2920	188	7	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NOUN
iajs-2920	188	8	)	)	PUNCT
iajs-2920	188	9	,	,	PUNCT
iajs-2920	188	10	𝛼𝛿((𝜌0	𝛼𝛿((𝜌0	PROPN
iajs-2920	188	11	∗	∗	NOUN
iajs-2920	188	12	𝜏0	𝜏0	PROPN
iajs-2920	188	13	)	)	PUNCT
iajs-2920	188	14	∗	∗	NOUN
iajs-2920	188	15	휀0	휀0	NOUN
iajs-2920	188	16	)	)	PUNCT
iajs-2920	188	17	=	=	PUNCT
iajs-2920	188	18	𝛼𝜔{𝑓((𝜌0	𝛼𝜔{𝑓((𝜌0	NUM
iajs-2920	188	19	∗	∗	NUM
iajs-2920	188	20	𝜏0	𝜏0	PROPN
iajs-2920	188	21	)	)	PUNCT
iajs-2920	188	22	∗	∗	NOUN
iajs-2920	188	23	휀0	휀0	NOUN
iajs-2920	188	24	)	)	PUNCT
iajs-2920	188	25	}	}	PUNCT
iajs-2920	188	26	ihjpas	ihjpa	VERB
iajs-2920	188	27	.	.	PUNCT
iajs-2920	189	1	36(1)2023	36(1)2023	NUM
iajs-2920	189	2	376	376	NUM
iajs-2920	189	3	=	=	SYM
iajs-2920	189	4	𝛼𝜔((𝜌′	𝛼𝜔((𝜌′	PROPN
iajs-2920	189	5	∗	∗	NOUN
iajs-2920	189	6	𝜏′	𝜏′	PROPN
iajs-2920	189	7	)	)	PUNCT
iajs-2920	189	8	∗	∗	NOUN
iajs-2920	189	9	휀′	휀′	NUM
iajs-2920	189	10	)	)	PUNCT
iajs-2920	190	1	=	=	SYM
iajs-2920	190	2	inf	inf	NOUN
iajs-2920	190	3	(	(	PUNCT
iajs-2920	190	4	(	(	PUNCT
iajs-2920	190	5	𝜌0∗𝜏0)∗	𝜌0∗𝜏0)∗	PROPN
iajs-2920	190	6	0)∈𝑓−1((𝜌′∗𝜏′)∗	0)∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	190	7	′	′	NUM
iajs-2920	190	8	)	)	PUNCT
iajs-2920	190	9	𝛼𝛿((𝜌0	𝛼𝛿((𝜌0	NOUN
iajs-2920	190	10	∗	∗	NOUN
iajs-2920	190	11	𝜏0	𝜏0	PROPN
iajs-2920	190	12	)	)	PUNCT
iajs-2920	190	13	∗	∗	NOUN
iajs-2920	190	14	휀0	휀0	NOUN
iajs-2920	190	15	)	)	PUNCT
iajs-2920	190	16	=	=	SYM
iajs-2920	190	17	inf	inf	NOUN
iajs-2920	190	18	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	190	19	′	′	NUM
iajs-2920	190	20	)	)	PUNCT
iajs-2920	190	21	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NUM
iajs-2920	190	22	)	)	PUNCT
iajs-2920	190	23	.	.	PUNCT
iajs-2920	191	1	then	then	ADV
iajs-2920	191	2	�	�	PROPN
iajs-2920	191	3	̃	̃	PROPN
iajs-2920	191	4	�	�	PROPN
iajs-2920	191	5	𝜔(𝜌′	𝜔(𝜌′	PROPN
iajs-2920	191	6	∗	∗	NOUN
iajs-2920	191	7	𝜏′	𝜏′	NUM
iajs-2920	191	8	)	)	PUNCT
iajs-2920	192	1	=	=	SYM
iajs-2920	192	2	rsup	rsup	ADJ
iajs-2920	192	3	𝑡∈𝑓−1(𝜌′∗𝜏′	𝑡∈𝑓−1(𝜌′∗𝜏′	PROPN
iajs-2920	192	4	)	)	PUNCT
iajs-2920	192	5	�	�	PROPN
iajs-2920	192	6	̃	̃	PROPN
iajs-2920	192	7	�	�	NOUN
iajs-2920	192	8	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	192	9	)	)	PUNCT
iajs-2920	192	10	=	=	SYM
iajs-2920	192	11	�	�	PROPN
iajs-2920	192	12	̃	̃	PROPN
iajs-2920	192	13	�	�	NOUN
iajs-2920	192	14	𝛿(𝜌0	𝛿(𝜌0	NOUN
iajs-2920	192	15	∗	∗	NOUN
iajs-2920	192	16	𝜏0	𝜏0	PROPN
iajs-2920	192	17	)	)	PUNCT
iajs-2920	192	18	≥	≥	PROPN
iajs-2920	192	19	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	192	20	�	�	PROPN
iajs-2920	192	21	̃	̃	PROPN
iajs-2920	192	22	�	�	NOUN
iajs-2920	192	23	𝛿((𝜌0	𝛿((𝜌0	NOUN
iajs-2920	192	24	∗	∗	NOUN
iajs-2920	192	25	𝜏0	𝜏0	PROPN
iajs-2920	192	26	)	)	PUNCT
iajs-2920	192	27	∗	∗	NOUN
iajs-2920	192	28	휀0	휀0	NOUN
iajs-2920	192	29	)	)	PUNCT
iajs-2920	192	30	,	,	PUNCT
iajs-2920	192	31	�	�	PROPN
iajs-2920	192	32	̃	̃	PROPN
iajs-2920	192	33	�	�	NOUN
iajs-2920	192	34	𝛿(𝜏0	𝛿(𝜏0	NOUN
iajs-2920	192	35	)	)	PUNCT
iajs-2920	192	36	}	}	PUNCT
iajs-2920	192	37	=	=	SYM
iajs-2920	192	38	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	192	39	{	{	PUNCT
iajs-2920	192	40	rsup	rsup	ADJ
iajs-2920	192	41	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	192	42	′	′	NOUN
iajs-2920	192	43	)	)	PUNCT
iajs-2920	192	44	�	�	PROPN
iajs-2920	192	45	̃	̃	PROPN
iajs-2920	192	46	�	�	NOUN
iajs-2920	192	47	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	192	48	)	)	PUNCT
iajs-2920	192	49	,	,	PUNCT
iajs-2920	192	50	rsup	rsup	ADJ
iajs-2920	192	51	𝑡∈𝑓−1(𝜏′	𝑡∈𝑓−1(𝜏′	PROPN
iajs-2920	192	52	)	)	PUNCT
iajs-2920	192	53	�	�	PROPN
iajs-2920	192	54	̃	̃	PROPN
iajs-2920	192	55	�	�	NOUN
iajs-2920	192	56	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-2920	192	57	)	)	PUNCT
iajs-2920	192	58	}	}	PUNCT
iajs-2920	192	59	=	=	SYM
iajs-2920	192	60	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	192	61	�	�	PROPN
iajs-2920	192	62	̃	̃	PROPN
iajs-2920	192	63	�	�	PROPN
iajs-2920	192	64	𝜔((𝜌′	𝜔((𝜌′	PROPN
iajs-2920	192	65	∗	∗	PROPN
iajs-2920	192	66	𝜏′	𝜏′	PROPN
iajs-2920	192	67	)	)	PUNCT
iajs-2920	192	68	∗	∗	NOUN
iajs-2920	192	69	휀′	휀′	NUM
iajs-2920	192	70	)	)	PUNCT
iajs-2920	192	71	,	,	PUNCT
iajs-2920	192	72	�	�	PROPN
iajs-2920	192	73	̃	̃	PROPN
iajs-2920	192	74	�	�	NOUN
iajs-2920	192	75	𝜔(𝜏′	𝜔(𝜏′	NUM
iajs-2920	192	76	)	)	PUNCT
iajs-2920	192	77	}	}	PUNCT
iajs-2920	192	78	,	,	PUNCT
iajs-2920	192	79	𝛼𝜔(𝜌′	𝛼𝜔(𝜌′	PROPN
iajs-2920	192	80	∗	∗	NOUN
iajs-2920	192	81	휀′	휀′	NUM
iajs-2920	192	82	)	)	PUNCT
iajs-2920	193	1	=	=	SYM
iajs-2920	193	2	inf	inf	NOUN
iajs-2920	193	3	𝑡∈𝑓−1(𝜌′∗	𝑡∈𝑓−1(𝜌′∗	PROPN
iajs-2920	193	4	′	′	NUM
iajs-2920	193	5	)	)	PUNCT
iajs-2920	193	6	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NUM
iajs-2920	193	7	)	)	PUNCT
iajs-2920	193	8	=	=	SYM
iajs-2920	193	9	𝛼𝛿	𝛼𝛿	X
iajs-2920	193	10	(	(	PUNCT
iajs-2920	193	11	𝜌0	𝜌0	ADJ
iajs-2920	193	12	∗	∗	NOUN
iajs-2920	193	13	𝜏0	𝜏0	PROPN
iajs-2920	193	14	)	)	PUNCT
iajs-2920	193	15	≤	≤	PUNCT
iajs-2920	193	16	𝑚𝑎𝑥{𝛼𝛿((𝜌0	𝑚𝑎𝑥{𝛼𝛿((𝜌0	PROPN
iajs-2920	193	17	∗	∗	NOUN
iajs-2920	193	18	𝜏0	𝜏0	PROPN
iajs-2920	193	19	)	)	PUNCT
iajs-2920	193	20	∗	∗	NOUN
iajs-2920	193	21	휀0	휀0	NOUN
iajs-2920	193	22	)	)	PUNCT
iajs-2920	193	23	,	,	PUNCT
iajs-2920	193	24	𝛼𝛿(𝜏0	𝛼𝛿(𝜏0	NOUN
iajs-2920	193	25	)	)	PUNCT
iajs-2920	193	26	}	}	PUNCT
iajs-2920	193	27	=	=	PUNCT
iajs-2920	193	28	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2920	193	29	{	{	PUNCT
iajs-2920	193	30	inf	inf	NOUN
iajs-2920	193	31	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	𝑡∈𝑓−1((𝜌′∗𝜏′)∗	PROPN
iajs-2920	193	32	′	′	NUM
iajs-2920	193	33	)	)	PUNCT
iajs-2920	193	34	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NUM
iajs-2920	193	35	)	)	PUNCT
iajs-2920	193	36	,	,	PUNCT
iajs-2920	193	37	inf	inf	PROPN
iajs-2920	193	38	𝑡∈𝑓−1(𝜏′	𝑡∈𝑓−1(𝜏′	PROPN
iajs-2920	193	39	)	)	PUNCT
iajs-2920	193	40	𝛼𝛿(𝑡	𝛼𝛿(𝑡	NOUN
iajs-2920	193	41	)	)	PUNCT
iajs-2920	193	42	}	}	PUNCT
iajs-2920	193	43	=	=	SYM
iajs-2920	193	44	𝑚𝑎𝑥{𝛼𝜔((𝜌′	𝑚𝑎𝑥{𝛼𝜔((𝜌′	PROPN
iajs-2920	193	45	∗	∗	PROPN
iajs-2920	193	46	𝜏′	𝜏′	NOUN
iajs-2920	193	47	)	)	PUNCT
iajs-2920	193	48	∗	∗	NOUN
iajs-2920	193	49	휀′	휀′	NUM
iajs-2920	193	50	)	)	PUNCT
iajs-2920	193	51	,	,	PUNCT
iajs-2920	193	52	𝛼𝜔(𝜏′	𝛼𝜔(𝜏′	NOUN
iajs-2920	193	53	)	)	PUNCT
iajs-2920	193	54	}	}	PUNCT
iajs-2920	193	55	.	.	PUNCT
iajs-2920	194	1	hence	hence	ADV
iajs-2920	194	2	,	,	PUNCT
iajs-2920	194	3	𝜔is	𝜔is	PROPN
iajs-2920	194	4	a	a	DET
iajs-2920	194	5	cubic	cubic	ADJ
iajs-2920	194	6	t	t	PROPN
iajs-2920	194	7	-	-	PUNCT
iajs-2920	194	8	ideal	ideal	NOUN
iajs-2920	194	9	of	of	ADP
iajs-2920	194	10	𝑌.	𝑌.	PROPN
iajs-2920	194	11	5	5	NUM
iajs-2920	194	12	.	.	PUNCT
iajs-2920	195	1	cartesian	cartesian	ADJ
iajs-2920	195	2	product	product	NOUN
iajs-2920	195	3	of	of	ADP
iajs-2920	195	4	cubic	cubic	ADJ
iajs-2920	195	5	t	t	PROPN
iajs-2920	195	6	-	-	PUNCT
iajs-2920	195	7	ideals	ideal	NOUN
iajs-2920	195	8	in	in	ADP
iajs-2920	195	9	this	this	DET
iajs-2920	195	10	section	section	NOUN
iajs-2920	195	11	,	,	PUNCT
iajs-2920	195	12	we	we	PRON
iajs-2920	195	13	provide	provide	VERB
iajs-2920	195	14	some	some	DET
iajs-2920	195	15	definitions	definition	NOUN
iajs-2920	195	16	of	of	ADP
iajs-2920	195	17	the	the	DET
iajs-2920	195	18	cartesian	cartesian	ADJ
iajs-2920	195	19	product	product	NOUN
iajs-2920	195	20	of	of	ADP
iajs-2920	195	21	cubic	cubic	ADJ
iajs-2920	195	22	t	t	PROPN
iajs-2920	195	23	-	-	PUNCT
iajs-2920	195	24	ideals	ideal	NOUN
iajs-2920	195	25	in	in	ADP
iajs-2920	195	26	tmalgebras	tmalgebras	ADJ
iajs-2920	195	27	.	.	PUNCT
iajs-2920	196	1	definition	definition	NOUN
iajs-2920	196	2	(	(	PUNCT
iajs-2920	196	3	29	29	NUM
iajs-2920	196	4	)	)	PUNCT
iajs-2920	196	5	.	.	PUNCT
iajs-2920	197	1	let𝛿1	let𝛿1	NOUN
iajs-2920	198	1	=	=	PUNCT
iajs-2920	198	2	〈	〈	PROPN
iajs-2920	198	3	�	�	PROPN
iajs-2920	198	4	̃	̃	PROPN
iajs-2920	198	5	�	�	NOUN
iajs-2920	198	6	𝛿1	𝛿1	NOUN
iajs-2920	198	7	,	,	PUNCT
iajs-2920	198	8	𝛼𝛿1	𝛼𝛿1	PROPN
iajs-2920	198	9	〉	〉	NOUN
iajs-2920	198	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	198	11	𝛿2	𝛿2	PROPN
iajs-2920	198	12	=	=	SYM
iajs-2920	198	13	〈	〈	PROPN
iajs-2920	198	14	�	�	PROPN
iajs-2920	198	15	̃	̃	PROPN
iajs-2920	198	16	�	�	NOUN
iajs-2920	198	17	𝛿2	𝛿2	NOUN
iajs-2920	198	18	,	,	PUNCT
iajs-2920	198	19	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	198	20	〉	〉	NOUN
iajs-2920	198	21	be	be	AUX
iajs-2920	198	22	two	two	NUM
iajs-2920	198	23	cubic	cubic	ADJ
iajs-2920	198	24	subsets	subset	NOUN
iajs-2920	198	25	of	of	ADP
iajs-2920	198	26	tm	tm	NOUN
iajs-2920	198	27	-	-	PUNCT
iajs-2920	198	28	algebras	algebras	PROPN
iajs-2920	198	29	ℵ1	ℵ1	PROPN
iajs-2920	198	30	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	198	31	ℵ2	ℵ2	NOUN
iajs-2920	198	32	,	,	PUNCT
iajs-2920	198	33	respectively	respectively	ADV
iajs-2920	198	34	.	.	PUNCT
iajs-2920	199	1	we	we	PRON
iajs-2920	199	2	definethe	definethe	VERB
iajs-2920	199	3	cartesian	cartesian	ADJ
iajs-2920	199	4	product	product	NOUN
iajs-2920	199	5	of	of	ADP
iajs-2920	199	6	two	two	NUM
iajs-2920	199	7	cubic	cubic	ADJ
iajs-2920	199	8	sets	set	NOUN
iajs-2920	199	9	𝛿1	𝛿1	PROPN
iajs-2920	199	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	199	11	𝛿2	𝛿2	NOUN
iajs-2920	199	12	by	by	ADP
iajs-2920	199	13	𝛿1	𝛿1	PROPN
iajs-2920	199	14	×	×	PROPN
iajs-2920	199	15	𝛿2	𝛿2	NOUN
iajs-2920	199	16	=	=	PUNCT
iajs-2920	199	17	〈	〈	PROPN
iajs-2920	199	18	�	�	PROPN
iajs-2920	199	19	̃	̃	PROPN
iajs-2920	199	20	�	�	NOUN
iajs-2920	199	21	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	199	22	,	,	PUNCT
iajs-2920	199	23	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	199	24	〉	〉	NOUN
iajs-2920	199	25	and	and	CCONJ
iajs-2920	199	26	�	�	PROPN
iajs-2920	199	27	̃	̃	PROPN
iajs-2920	199	28	�	�	PROPN
iajs-2920	199	29	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	199	30	(	(	PUNCT
iajs-2920	199	31	𝜌	𝜌	X
iajs-2920	199	32	,	,	PUNCT
iajs-2920	199	33	𝜏	𝜏	NOUN
iajs-2920	199	34	)	)	PUNCT
iajs-2920	199	35	=	=	SYM
iajs-2920	199	36	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	199	37	�	�	PROPN
iajs-2920	199	38	̃	̃	PROPN
iajs-2920	199	39	�	�	NOUN
iajs-2920	199	40	𝛿1	𝛿1	NOUN
iajs-2920	199	41	(	(	PUNCT
iajs-2920	199	42	𝜌	𝜌	NOUN
iajs-2920	199	43	)	)	PUNCT
iajs-2920	199	44	,	,	PUNCT
iajs-2920	199	45	�	�	PROPN
iajs-2920	199	46	̃	̃	PROPN
iajs-2920	199	47	�	�	NOUN
iajs-2920	199	48	𝛿2	𝛿2	NOUN
iajs-2920	199	49	(	(	PUNCT
iajs-2920	199	50	𝜏	𝜏	NOUN
iajs-2920	199	51	)	)	PUNCT
iajs-2920	199	52	}	}	PUNCT
iajs-2920	199	53	,	,	PUNCT
iajs-2920	199	54	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	199	55	(	(	PUNCT
iajs-2920	199	56	𝜌	𝜌	X
iajs-2920	199	57	,	,	PUNCT
iajs-2920	199	58	𝜏	𝜏	NOUN
iajs-2920	199	59	)	)	PUNCT
iajs-2920	199	60	=	=	SYM
iajs-2920	199	61	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	PROPN
iajs-2920	199	62	(	(	PUNCT
iajs-2920	199	63	𝜌	𝜌	NOUN
iajs-2920	199	64	)	)	PUNCT
iajs-2920	199	65	,	,	PUNCT
iajs-2920	199	66	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	199	67	(	(	PUNCT
iajs-2920	199	68	𝜏	𝜏	NOUN
iajs-2920	199	69	)	)	PUNCT
iajs-2920	199	70	}	}	PUNCT
iajs-2920	199	71	,	,	PUNCT
iajs-2920	199	72	for	for	ADP
iajs-2920	199	73	any	any	DET
iajs-2920	199	74	(	(	PUNCT
iajs-2920	199	75	𝜌	𝜌	X
iajs-2920	199	76	,	,	PUNCT
iajs-2920	199	77	𝜏	𝜏	NOUN
iajs-2920	199	78	)	)	PUNCT
iajs-2920	199	79	∈	∈	NOUN
iajs-2920	199	80	ℵ1	ℵ1	NOUN
iajs-2920	199	81	×	×	NOUN
iajs-2920	199	82	ℵ2	ℵ2	PROPN
iajs-2920	199	83	.	.	PUNCT
iajs-2920	200	1	remark	remark	PROPN
iajs-2920	200	2	(	(	PUNCT
iajs-2920	200	3	30	30	NUM
iajs-2920	200	4	)	)	PUNCT
iajs-2920	200	5	.	.	PUNCT
iajs-2920	201	1	let	let	VERB
iajs-2920	201	2	ℵ	ℵ	NOUN
iajs-2920	201	3	and	and	CCONJ
iajs-2920	201	4	𝑌	𝑌	PROPN
iajs-2920	201	5	be	be	VERB
iajs-2920	201	6	tm	tm	NOUN
iajs-2920	201	7	-	-	PUNCT
iajs-2920	201	8	algebras	algebras	X
iajs-2920	201	9	.	.	PUNCT
iajs-2920	202	1	we	we	PRON
iajs-2920	202	2	define	define	VERB
iajs-2920	202	3	∗	∗	NOUN
iajs-2920	202	4	𝑜𝑛	𝑜𝑛	PROPN
iajs-2920	202	5	ℵ	ℵ	NOUN
iajs-2920	202	6	×	×	NOUN
iajs-2920	202	7	𝑌	𝑌	PROPN
iajs-2920	202	8	by	by	ADP
iajs-2920	202	9	(	(	PUNCT
iajs-2920	202	10	𝜌	𝜌	X
iajs-2920	202	11	,	,	PUNCT
iajs-2920	202	12	𝜏	𝜏	NOUN
iajs-2920	202	13	)	)	PUNCT
iajs-2920	202	14	∗	∗	NOUN
iajs-2920	202	15	(	(	PUNCT
iajs-2920	202	16	𝑢	𝑢	X
iajs-2920	202	17	,	,	PUNCT
iajs-2920	202	18	𝑣	𝑣	NOUN
iajs-2920	202	19	)	)	PUNCT
iajs-2920	202	20	=	=	SYM
iajs-2920	202	21	(	(	PUNCT
iajs-2920	202	22	𝜌	𝜌	X
iajs-2920	202	23	∗	∗	X
iajs-2920	202	24	𝑢	𝑢	NOUN
iajs-2920	202	25	,	,	PUNCT
iajs-2920	202	26	𝜏	𝜏	PRON
iajs-2920	202	27	∗	∗	NOUN
iajs-2920	202	28	𝑣	𝑣	NOUN
iajs-2920	202	29	)	)	PUNCT
iajs-2920	202	30	for	for	ADP
iajs-2920	202	31	every	every	DET
iajs-2920	202	32	(	(	PUNCT
iajs-2920	202	33	𝜌	𝜌	X
iajs-2920	202	34	,	,	PUNCT
iajs-2920	202	35	𝜏	𝜏	NOUN
iajs-2920	202	36	)	)	PUNCT
iajs-2920	202	37	,	,	PUNCT
iajs-2920	202	38	(	(	PUNCT
iajs-2920	202	39	𝑢	𝑢	X
iajs-2920	202	40	,	,	PUNCT
iajs-2920	202	41	𝑣	𝑣	NOUN
iajs-2920	202	42	)	)	PUNCT
iajs-2920	202	43	belong	belong	VERB
iajs-2920	202	44	to	to	ADP
iajs-2920	202	45	ℵ	ℵ	PRON
iajs-2920	202	46	×	×	PROPN
iajs-2920	202	47	𝑌	𝑌	PROPN
iajs-2920	202	48	,	,	PUNCT
iajs-2920	202	49	then	then	ADV
iajs-2920	202	50	,	,	PUNCT
iajs-2920	202	51	clearly	clearly	ADV
iajs-2920	202	52	(	(	PUNCT
iajs-2920	202	53	ℵ	ℵ	PROPN
iajs-2920	202	54	×	×	PROPN
iajs-2920	202	55	𝑌,∗	𝑌,∗	NOUN
iajs-2920	202	56	,	,	PUNCT
iajs-2920	202	57	(	(	PUNCT
iajs-2920	202	58	0,0	0,0	NOUN
iajs-2920	202	59	)	)	PUNCT
iajs-2920	202	60	)	)	PUNCT
iajs-2920	202	61	is	be	AUX
iajs-2920	202	62	a	a	DET
iajs-2920	202	63	tm	tm	NOUN
iajs-2920	202	64	-	-	NOUN
iajs-2920	202	65	algebra	algebra	NOUN
iajs-2920	202	66	.	.	PUNCT
iajs-2920	203	1	definition	definition	NOUN
iajs-2920	203	2	(	(	PUNCT
iajs-2920	203	3	31	31	NUM
iajs-2920	203	4	)	)	PUNCT
iajs-2920	203	5	.	.	PUNCT
iajs-2920	204	1	a	a	DET
iajs-2920	204	2	cubic	cubic	ADJ
iajs-2920	204	3	subset	subset	VERB
iajs-2920	204	4	𝛿1	𝛿1	PROPN
iajs-2920	204	5	×	×	PROPN
iajs-2920	204	6	𝛿2	𝛿2	NOUN
iajs-2920	204	7	=	=	PUNCT
iajs-2920	204	8	〈	〈	PROPN
iajs-2920	204	9	�	�	PROPN
iajs-2920	204	10	̃	̃	PROPN
iajs-2920	204	11	�	�	NOUN
iajs-2920	204	12	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	204	13	,	,	PUNCT
iajs-2920	204	14	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	204	15	〉	〉	NOUN
iajs-2920	204	16	of	of	ADP
iajs-2920	204	17	ℵ1	ℵ1	PROPN
iajs-2920	204	18	×	×	PROPN
iajs-2920	204	19	ℵ2	ℵ2	PROPN
iajs-2920	204	20	is	be	AUX
iajs-2920	204	21	called	call	VERB
iajs-2920	204	22	a	a	DET
iajs-2920	204	23	cubic	cubic	ADJ
iajs-2920	204	24	ideal	ideal	NOUN
iajs-2920	204	25	of	of	ADP
iajs-2920	204	26	ℵ1	ℵ1	PROPN
iajs-2920	204	27	×	×	NOUN
iajs-2920	204	28	ℵ2	ℵ2	ADJ
iajs-2920	204	29	if	if	SCONJ
iajs-2920	204	30	(	(	PUNCT
iajs-2920	204	31	𝐂𝐏𝟏)	𝐂𝐏𝟏)	PROPN
iajs-2920	204	32	�	�	PROPN
iajs-2920	204	33	̃	̃	PROPN
iajs-2920	204	34	�	�	PROPN
iajs-2920	204	35	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	204	36	(	(	PUNCT
iajs-2920	204	37	0,0	0,0	NUM
iajs-2920	204	38	)	)	PUNCT
iajs-2920	204	39	≥	≥	NUM
iajs-2920	204	40	�	�	PROPN
iajs-2920	204	41	̃	̃	PROPN
iajs-2920	204	42	�	�	PROPN
iajs-2920	204	43	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	204	44	(	(	PUNCT
iajs-2920	204	45	𝜌	𝜌	X
iajs-2920	204	46	,	,	PUNCT
iajs-2920	204	47	𝜏	𝜏	NOUN
iajs-2920	204	48	)	)	PUNCT
iajs-2920	204	49	and	and	CCONJ
iajs-2920	204	50	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	204	51	(	(	PUNCT
iajs-2920	204	52	0,0	0,0	NUM
iajs-2920	204	53	)	)	PUNCT
iajs-2920	204	54	≤	≤	NOUN
iajs-2920	205	1	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	205	2	(	(	PUNCT
iajs-2920	205	3	𝜌	𝜌	X
iajs-2920	205	4	,	,	PUNCT
iajs-2920	205	5	𝜏	𝜏	NOUN
iajs-2920	205	6	)	)	PUNCT
iajs-2920	205	7	,	,	PUNCT
iajs-2920	205	8	ihjpas	ihjpa	VERB
iajs-2920	205	9	.	.	PUNCT
iajs-2920	206	1	36(1)2023	36(1)2023	NUM
iajs-2920	206	2	377	377	NUM
iajs-2920	206	3	(	(	PUNCT
iajs-2920	206	4	𝐂𝐏𝟐)	𝐂𝐏𝟐)	PROPN
iajs-2920	206	5	�	�	PROPN
iajs-2920	206	6	̃	̃	PROPN
iajs-2920	206	7	�	�	PROPN
iajs-2920	206	8	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	206	9	(	(	PUNCT
iajs-2920	206	10	𝜌1	𝜌1	PROPN
iajs-2920	206	11	,	,	PUNCT
iajs-2920	206	12	𝜏1	𝜏1	NOUN
iajs-2920	206	13	)	)	PUNCT
iajs-2920	206	14	≥	≥	NOUN
iajs-2920	206	15	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	206	16	�	�	PROPN
iajs-2920	206	17	̃	̃	PROPN
iajs-2920	206	18	�	�	PROPN
iajs-2920	206	19	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	206	20	(	(	PUNCT
iajs-2920	206	21	(	(	PUNCT
iajs-2920	206	22	𝜌1	𝜌1	NOUN
iajs-2920	206	23	,	,	PUNCT
iajs-2920	206	24	𝜏1	𝜏1	NOUN
iajs-2920	206	25	)	)	PUNCT
iajs-2920	206	26	∗	∗	NOUN
iajs-2920	206	27	(	(	PUNCT
iajs-2920	206	28	𝜌2	𝜌2	ADJ
iajs-2920	206	29	,	,	PUNCT
iajs-2920	206	30	𝜏2	𝜏2	PROPN
iajs-2920	206	31	)	)	PUNCT
iajs-2920	206	32	)	)	PUNCT
iajs-2920	206	33	,	,	PUNCT
iajs-2920	206	34	�	�	PROPN
iajs-2920	206	35	̃	̃	PROPN
iajs-2920	206	36	�	�	NOUN
iajs-2920	206	37	𝛿1×𝛿1	𝛿1×𝛿1	VERB
iajs-2920	206	38	(	(	PUNCT
iajs-2920	206	39	𝜌2	𝜌2	ADJ
iajs-2920	206	40	,	,	PUNCT
iajs-2920	206	41	𝜏2	𝜏2	PROPN
iajs-2920	206	42	)	)	PUNCT
iajs-2920	206	43	}	}	PUNCT
iajs-2920	206	44	(	(	PUNCT
iajs-2920	206	45	𝐂𝐏𝟑)𝛼𝛿1×𝛿2	𝐂𝐏𝟑)𝛼𝛿1×𝛿2	NOUN
iajs-2920	206	46	(	(	PUNCT
iajs-2920	206	47	𝜌1	𝜌1	NOUN
iajs-2920	206	48	,	,	PUNCT
iajs-2920	206	49	𝜏1	𝜏1	NOUN
iajs-2920	206	50	)	)	PUNCT
iajs-2920	206	51	≤	≤	NOUN
iajs-2920	206	52	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	ADP
iajs-2920	206	53	(	(	PUNCT
iajs-2920	206	54	(	(	PUNCT
iajs-2920	206	55	𝜌1	𝜌1	NOUN
iajs-2920	206	56	,	,	PUNCT
iajs-2920	206	57	𝜏1	𝜏1	NOUN
iajs-2920	206	58	)	)	PUNCT
iajs-2920	206	59	∗	∗	NOUN
iajs-2920	206	60	(	(	PUNCT
iajs-2920	206	61	𝜌2	𝜌2	ADJ
iajs-2920	206	62	,	,	PUNCT
iajs-2920	206	63	𝜏2	𝜏2	PROPN
iajs-2920	206	64	)	)	PUNCT
iajs-2920	206	65	)	)	PUNCT
iajs-2920	206	66	,	,	PUNCT
iajs-2920	206	67	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	206	68	(	(	PUNCT
iajs-2920	206	69	𝜌2	𝜌2	ADJ
iajs-2920	206	70	,	,	PUNCT
iajs-2920	206	71	𝜏2	𝜏2	PROPN
iajs-2920	206	72	)	)	PUNCT
iajs-2920	206	73	}	}	PUNCT
iajs-2920	206	74	,	,	PUNCT
iajs-2920	206	75	for	for	ADP
iajs-2920	206	76	any	any	DET
iajs-2920	206	77	(	(	PUNCT
iajs-2920	206	78	𝜌1	𝜌1	NOUN
iajs-2920	206	79	,	,	PUNCT
iajs-2920	206	80	𝜏1	𝜏1	NOUN
iajs-2920	206	81	)	)	PUNCT
iajs-2920	206	82	,	,	PUNCT
iajs-2920	206	83	(	(	PUNCT
iajs-2920	206	84	𝜌2	𝜌2	ADJ
iajs-2920	206	85	,	,	PUNCT
iajs-2920	206	86	𝜏2	𝜏2	ADJ
iajs-2920	206	87	)	)	PUNCT
iajs-2920	206	88	∈	∈	PROPN
iajs-2920	206	89	ℵ1	ℵ1	NOUN
iajs-2920	206	90	×	×	NOUN
iajs-2920	206	91	ℵ2	ℵ2	ADJ
iajs-2920	206	92	.	.	PUNCT
iajs-2920	207	1	definition	definition	NOUN
iajs-2920	207	2	(	(	PUNCT
iajs-2920	207	3	32	32	NUM
iajs-2920	207	4	)	)	PUNCT
iajs-2920	207	5	.	.	PUNCT
iajs-2920	208	1	a	a	DET
iajs-2920	208	2	cubic	cubic	ADJ
iajs-2920	208	3	subset	subset	VERB
iajs-2920	208	4	𝛿1	𝛿1	PROPN
iajs-2920	208	5	×	×	PROPN
iajs-2920	208	6	𝛿2	𝛿2	NOUN
iajs-2920	208	7	=	=	PUNCT
iajs-2920	208	8	〈	〈	PROPN
iajs-2920	208	9	�	�	PROPN
iajs-2920	208	10	̃	̃	PROPN
iajs-2920	208	11	�	�	NOUN
iajs-2920	208	12	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	208	13	,	,	PUNCT
iajs-2920	208	14	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	208	15	〉	〉	NOUN
iajs-2920	208	16	of	of	ADP
iajs-2920	208	17	ℵ1	ℵ1	PROPN
iajs-2920	208	18	×	×	PROPN
iajs-2920	208	19	ℵ2	ℵ2	PROPN
iajs-2920	208	20	is	be	AUX
iajs-2920	208	21	called	call	VERB
iajs-2920	208	22	a	a	DET
iajs-2920	208	23	cubic	cubic	ADJ
iajs-2920	208	24	t	t	PROPN
iajs-2920	208	25	-	-	PUNCT
iajs-2920	208	26	ideal	ideal	NOUN
iajs-2920	208	27	of	of	ADP
iajs-2920	208	28	ℵ1	ℵ1	PROPN
iajs-2920	208	29	×	×	NOUN
iajs-2920	208	30	ℵ2	ℵ2	ADJ
iajs-2920	208	31	if	if	SCONJ
iajs-2920	208	32	(	(	PUNCT
iajs-2920	208	33	𝐂𝐏𝟏)	𝐂𝐏𝟏)	PROPN
iajs-2920	208	34	�	�	PROPN
iajs-2920	208	35	̃	̃	PROPN
iajs-2920	208	36	�	�	PROPN
iajs-2920	208	37	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	208	38	(	(	PUNCT
iajs-2920	208	39	0,0	0,0	NUM
iajs-2920	208	40	)	)	PUNCT
iajs-2920	208	41	≥	≥	NUM
iajs-2920	208	42	�	�	PROPN
iajs-2920	208	43	̃	̃	PROPN
iajs-2920	208	44	�	�	PROPN
iajs-2920	208	45	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	208	46	(	(	PUNCT
iajs-2920	208	47	𝜌	𝜌	X
iajs-2920	208	48	,	,	PUNCT
iajs-2920	208	49	𝜏	𝜏	NOUN
iajs-2920	208	50	)	)	PUNCT
iajs-2920	208	51	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	208	52	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	208	53	(	(	PUNCT
iajs-2920	208	54	0,0	0,0	NOUN
iajs-2920	208	55	)	)	PUNCT
iajs-2920	208	56	≤	≤	NOUN
iajs-2920	209	1	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	209	2	(	(	PUNCT
iajs-2920	209	3	𝜌	𝜌	X
iajs-2920	209	4	,	,	PUNCT
iajs-2920	209	5	𝜏	𝜏	NOUN
iajs-2920	209	6	)	)	PUNCT
iajs-2920	209	7	,	,	PUNCT
iajs-2920	209	8	(	(	PUNCT
iajs-2920	209	9	𝐂𝐏𝟐)	𝐂𝐏𝟐)	PROPN
iajs-2920	209	10	�	�	PROPN
iajs-2920	209	11	̃	̃	PROPN
iajs-2920	209	12	�	�	NOUN
iajs-2920	209	13	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	209	14	(	(	PUNCT
iajs-2920	209	15	(	(	PUNCT
iajs-2920	209	16	𝜌1	𝜌1	NOUN
iajs-2920	209	17	,	,	PUNCT
iajs-2920	209	18	𝜏1	𝜏1	NOUN
iajs-2920	209	19	)	)	PUNCT
iajs-2920	209	20	∗	∗	NOUN
iajs-2920	209	21	(	(	PUNCT
iajs-2920	209	22	𝜌3	𝜌3	PROPN
iajs-2920	209	23	,	,	PUNCT
iajs-2920	209	24	𝜏3	𝜏3	NOUN
iajs-2920	209	25	)	)	PUNCT
iajs-2920	209	26	)	)	PUNCT
iajs-2920	209	27	≥	≥	PROPN
iajs-2920	209	28	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	209	29	{	{	PUNCT
iajs-2920	209	30	�	�	PROPN
iajs-2920	209	31	̃	̃	PROPN
iajs-2920	209	32	�	�	PROPN
iajs-2920	209	33	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	209	34	(	(	PUNCT
iajs-2920	209	35	(	(	PUNCT
iajs-2920	209	36	(	(	PUNCT
iajs-2920	209	37	𝜌1	𝜌1	NOUN
iajs-2920	209	38	,	,	PUNCT
iajs-2920	209	39	𝜏1	𝜏1	NOUN
iajs-2920	209	40	)	)	PUNCT
iajs-2920	209	41	∗	∗	NOUN
iajs-2920	209	42	(	(	PUNCT
iajs-2920	209	43	𝜌2	𝜌2	ADJ
iajs-2920	209	44	,	,	PUNCT
iajs-2920	209	45	𝜏2	𝜏2	PROPN
iajs-2920	209	46	)	)	PUNCT
iajs-2920	209	47	)	)	PUNCT
iajs-2920	209	48	∗	∗	NOUN
iajs-2920	209	49	(	(	PUNCT
iajs-2920	209	50	𝜌3	𝜌3	PROPN
iajs-2920	209	51	,	,	PUNCT
iajs-2920	209	52	𝜏3	𝜏3	NOUN
iajs-2920	209	53	)	)	PUNCT
iajs-2920	209	54	)	)	PUNCT
iajs-2920	209	55	,	,	PUNCT
iajs-2920	209	56	�	�	PROPN
iajs-2920	209	57	̃	̃	PROPN
iajs-2920	209	58	�	�	PROPN
iajs-2920	209	59	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	209	60	(	(	PUNCT
iajs-2920	209	61	𝜌2	𝜌2	ADJ
iajs-2920	209	62	,	,	PUNCT
iajs-2920	209	63	𝜏2	𝜏2	PROPN
iajs-2920	209	64	)	)	PUNCT
iajs-2920	209	65	}	}	PUNCT
iajs-2920	209	66	(	(	PUNCT
iajs-2920	209	67	𝐂𝐏𝟑)𝛼𝛿1×𝛿2	𝐂𝐏𝟑)𝛼𝛿1×𝛿2	NOUN
iajs-2920	209	68	(	(	PUNCT
iajs-2920	209	69	(	(	PUNCT
iajs-2920	209	70	𝜌1	𝜌1	NOUN
iajs-2920	209	71	,	,	PUNCT
iajs-2920	209	72	𝜏1	𝜏1	NOUN
iajs-2920	209	73	)	)	PUNCT
iajs-2920	209	74	∗	∗	NOUN
iajs-2920	209	75	(	(	PUNCT
iajs-2920	209	76	𝜌3	𝜌3	PROPN
iajs-2920	209	77	,	,	PUNCT
iajs-2920	209	78	𝜏3	𝜏3	NOUN
iajs-2920	209	79	)	)	PUNCT
iajs-2920	209	80	)	)	PUNCT
iajs-2920	209	81	≤	≤	NUM
iajs-2920	209	82	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2920	209	83	{	{	PUNCT
iajs-2920	209	84	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	209	85	(	(	PUNCT
iajs-2920	209	86	(	(	PUNCT
iajs-2920	209	87	(	(	PUNCT
iajs-2920	209	88	𝜌1	𝜌1	NOUN
iajs-2920	209	89	,	,	PUNCT
iajs-2920	209	90	𝜏1	𝜏1	NOUN
iajs-2920	209	91	)	)	PUNCT
iajs-2920	209	92	∗	∗	NOUN
iajs-2920	209	93	(	(	PUNCT
iajs-2920	209	94	𝜌2	𝜌2	ADJ
iajs-2920	209	95	,	,	PUNCT
iajs-2920	209	96	𝜏2	𝜏2	PROPN
iajs-2920	209	97	)	)	PUNCT
iajs-2920	209	98	)	)	PUNCT
iajs-2920	209	99	∗	∗	NOUN
iajs-2920	209	100	(	(	PUNCT
iajs-2920	209	101	𝜌3	𝜌3	PROPN
iajs-2920	209	102	,	,	PUNCT
iajs-2920	209	103	𝜏3	𝜏3	NOUN
iajs-2920	209	104	)	)	PUNCT
iajs-2920	209	105	)	)	PUNCT
iajs-2920	209	106	,	,	PUNCT
iajs-2920	209	107	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	209	108	(	(	PUNCT
iajs-2920	209	109	𝜌2	𝜌2	ADJ
iajs-2920	209	110	,	,	PUNCT
iajs-2920	209	111	𝜏2	𝜏2	PROPN
iajs-2920	209	112	)	)	PUNCT
iajs-2920	209	113	}	}	PUNCT
iajs-2920	209	114	,	,	PUNCT
iajs-2920	209	115	for	for	ADP
iajs-2920	209	116	any	any	DET
iajs-2920	209	117	(	(	PUNCT
iajs-2920	209	118	𝜌1	𝜌1	NOUN
iajs-2920	209	119	,	,	PUNCT
iajs-2920	209	120	𝜏1	𝜏1	NOUN
iajs-2920	209	121	)	)	PUNCT
iajs-2920	209	122	,	,	PUNCT
iajs-2920	209	123	(	(	PUNCT
iajs-2920	209	124	𝜌2	𝜌2	ADJ
iajs-2920	209	125	,	,	PUNCT
iajs-2920	209	126	𝜏2	𝜏2	PROPN
iajs-2920	209	127	)	)	PUNCT
iajs-2920	209	128	,	,	PUNCT
iajs-2920	209	129	(	(	PUNCT
iajs-2920	209	130	𝜌3	𝜌3	PROPN
iajs-2920	209	131	,	,	PUNCT
iajs-2920	209	132	𝜏3	𝜏3	PROPN
iajs-2920	209	133	)	)	PUNCT
iajs-2920	209	134	∈	∈	PROPN
iajs-2920	209	135	ℵ1	ℵ1	NOUN
iajs-2920	209	136	×	×	NOUN
iajs-2920	209	137	ℵ2	ℵ2	ADJ
iajs-2920	209	138	.	.	PUNCT
iajs-2920	210	1	proposition	proposition	NOUN
iajs-2920	210	2	(	(	PUNCT
iajs-2920	210	3	33	33	NUM
iajs-2920	210	4	)	)	PUNCT
iajs-2920	210	5	.	.	PUNCT
iajs-2920	211	1	if	if	SCONJ
iajs-2920	211	2	𝛿1	𝛿1	NOUN
iajs-2920	211	3	×	×	PROPN
iajs-2920	211	4	𝛿2	𝛿2	NOUN
iajs-2920	211	5	=	=	PUNCT
iajs-2920	211	6	〈	〈	PROPN
iajs-2920	211	7	�	�	PROPN
iajs-2920	211	8	̃	̃	PROPN
iajs-2920	211	9	�	�	NOUN
iajs-2920	211	10	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	211	11	,	,	PUNCT
iajs-2920	211	12	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	211	13	〉	〉	NOUN
iajs-2920	211	14	is	be	AUX
iajs-2920	211	15	a	a	DET
iajs-2920	211	16	cubic	cubic	ADJ
iajs-2920	211	17	t	t	NOUN
iajs-2920	211	18	-	-	PUNCT
iajs-2920	211	19	ideal	ideal	NOUN
iajs-2920	211	20	of	of	ADP
iajs-2920	211	21	tm	tm	NOUN
iajs-2920	211	22	-	-	NOUN
iajs-2920	211	23	algebra	algebra	NOUN
iajs-2920	211	24	ℵ1	ℵ1	NOUN
iajs-2920	211	25	×	×	NOUN
iajs-2920	211	26	ℵ2	ℵ2	ADJ
iajs-2920	211	27	and	and	CCONJ
iajs-2920	211	28	if	if	SCONJ
iajs-2920	211	29	(	(	PUNCT
iajs-2920	211	30	𝜌1	𝜌1	NOUN
iajs-2920	211	31	,	,	PUNCT
iajs-2920	211	32	𝜏1	𝜏1	NOUN
iajs-2920	211	33	)	)	PUNCT
iajs-2920	211	34	≤	≤	NOUN
iajs-2920	211	35	(	(	PUNCT
iajs-2920	211	36	𝜌2	𝜌2	ADJ
iajs-2920	211	37	,	,	PUNCT
iajs-2920	211	38	𝜏2	𝜏2	PROPN
iajs-2920	211	39	)	)	PUNCT
iajs-2920	211	40	,	,	PUNCT
iajs-2920	211	41	we	we	PRON
iajs-2920	211	42	have	have	VERB
iajs-2920	211	43	�	�	PROPN
iajs-2920	211	44	̃	̃	PROPN
iajs-2920	211	45	�	�	NOUN
iajs-2920	211	46	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	211	47	(	(	PUNCT
iajs-2920	211	48	𝜌2	𝜌2	ADJ
iajs-2920	211	49	,	,	PUNCT
iajs-2920	211	50	𝜏2	𝜏2	PROPN
iajs-2920	211	51	)	)	PUNCT
iajs-2920	211	52	≤	≤	NUM
iajs-2920	211	53	�	�	PROPN
iajs-2920	211	54	̃	̃	PROPN
iajs-2920	211	55	�	�	PROPN
iajs-2920	211	56	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	211	57	(	(	PUNCT
iajs-2920	211	58	𝜌1	𝜌1	PROPN
iajs-2920	211	59	,	,	PUNCT
iajs-2920	211	60	𝜏1	𝜏1	NOUN
iajs-2920	211	61	)	)	PUNCT
iajs-2920	211	62	and	and	CCONJ
iajs-2920	212	1	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	212	2	(	(	PUNCT
iajs-2920	212	3	𝜌2	𝜌2	ADJ
iajs-2920	212	4	,	,	PUNCT
iajs-2920	212	5	𝜏2	𝜏2	PROPN
iajs-2920	212	6	)	)	PUNCT
iajs-2920	212	7	≥	≥	NOUN
iajs-2920	212	8	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	212	9	(	(	PUNCT
iajs-2920	212	10	𝜌1	𝜌1	PROPN
iajs-2920	212	11	,	,	PUNCT
iajs-2920	212	12	𝜏1	𝜏1	NOUN
iajs-2920	212	13	)	)	PUNCT
iajs-2920	212	14	.	.	PUNCT
iajs-2920	213	1	for	for	ADP
iajs-2920	213	2	all	all	PRON
iajs-2920	213	3	(	(	PUNCT
iajs-2920	213	4	𝜌1	𝜌1	NOUN
iajs-2920	213	5	,	,	PUNCT
iajs-2920	213	6	𝜏1	𝜏1	NOUN
iajs-2920	213	7	)	)	PUNCT
iajs-2920	213	8	,	,	PUNCT
iajs-2920	213	9	(	(	PUNCT
iajs-2920	213	10	𝜌2	𝜌2	ADJ
iajs-2920	213	11	,	,	PUNCT
iajs-2920	213	12	𝜏2	𝜏2	ADJ
iajs-2920	213	13	)	)	PUNCT
iajs-2920	213	14	∈	∈	PROPN
iajs-2920	213	15	ℵ1	ℵ1	NOUN
iajs-2920	213	16	×	×	NOUN
iajs-2920	213	17	ℵ2	ℵ2	ADJ
iajs-2920	213	18	.	.	PUNCT
iajs-2920	214	1	proof	proof	NOUN
iajs-2920	214	2	.	.	PUNCT
iajs-2920	215	1	let	let	VERB
iajs-2920	215	2	(	(	PUNCT
iajs-2920	215	3	𝜌1	𝜌1	NOUN
iajs-2920	215	4	,	,	PUNCT
iajs-2920	215	5	𝜏1	𝜏1	NOUN
iajs-2920	215	6	)	)	PUNCT
iajs-2920	215	7	,	,	PUNCT
iajs-2920	215	8	(	(	PUNCT
iajs-2920	215	9	𝜌2	𝜌2	ADJ
iajs-2920	215	10	,	,	PUNCT
iajs-2920	215	11	𝜏2	𝜏2	ADJ
iajs-2920	215	12	)	)	PUNCT
iajs-2920	215	13	∈	∈	PROPN
iajs-2920	215	14	ℵ1	ℵ1	NOUN
iajs-2920	215	15	×	×	NOUN
iajs-2920	215	16	ℵ2	ℵ2	ADJ
iajs-2920	215	17	,	,	PUNCT
iajs-2920	215	18	such	such	ADJ
iajs-2920	215	19	that	that	SCONJ
iajs-2920	215	20	(	(	PUNCT
iajs-2920	215	21	𝜌1	𝜌1	NOUN
iajs-2920	215	22	,	,	PUNCT
iajs-2920	215	23	𝜏1	𝜏1	NOUN
iajs-2920	215	24	)	)	PUNCT
iajs-2920	215	25	≤	≤	NOUN
iajs-2920	215	26	(	(	PUNCT
iajs-2920	215	27	𝜌2	𝜌2	ADJ
iajs-2920	215	28	,	,	PUNCT
iajs-2920	215	29	𝜏2	𝜏2	ADJ
iajs-2920	215	30	)	)	PUNCT
iajs-2920	215	31	⇒	⇒	NOUN
iajs-2920	215	32	(	(	PUNCT
iajs-2920	215	33	𝜌2	𝜌2	ADJ
iajs-2920	215	34	,	,	PUNCT
iajs-2920	215	35	𝜏2	𝜏2	PROPN
iajs-2920	215	36	)	)	PUNCT
iajs-2920	215	37	∗	∗	NOUN
iajs-2920	215	38	(	(	PUNCT
iajs-2920	215	39	𝜌1	𝜌1	NOUN
iajs-2920	215	40	,	,	PUNCT
iajs-2920	215	41	𝜏1	𝜏1	NOUN
iajs-2920	215	42	)	)	PUNCT
iajs-2920	215	43	=	=	PUNCT
iajs-2920	216	1	(	(	PUNCT
iajs-2920	216	2	0,0).this	0,0).this	PROPN
iajs-2920	216	3	together	together	ADV
iajs-2920	216	4	with	with	ADP
iajs-2920	216	5	(	(	PUNCT
iajs-2920	216	6	0,0	0,0	NOUN
iajs-2920	216	7	)	)	PUNCT
iajs-2920	216	8	∗	∗	NOUN
iajs-2920	216	9	(	(	PUNCT
iajs-2920	216	10	𝜌1	𝜌1	NOUN
iajs-2920	216	11	,	,	PUNCT
iajs-2920	216	12	𝜏1	𝜏1	NOUN
iajs-2920	216	13	)	)	PUNCT
iajs-2920	216	14	=	=	PUNCT
iajs-2920	216	15	(	(	PUNCT
iajs-2920	216	16	𝜌1	𝜌1	NOUN
iajs-2920	216	17	,	,	PUNCT
iajs-2920	216	18	𝜏1)and	𝜏1)and	PROPN
iajs-2920	216	19	�	�	PROPN
iajs-2920	216	20	̃	̃	PROPN
iajs-2920	216	21	�	�	PROPN
iajs-2920	216	22	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	216	23	(	(	PUNCT
iajs-2920	216	24	𝜌2	𝜌2	ADJ
iajs-2920	216	25	,	,	PUNCT
iajs-2920	216	26	𝜏2	𝜏2	PROPN
iajs-2920	216	27	)	)	PUNCT
iajs-2920	216	28	≤	≤	NUM
iajs-2920	216	29	�	�	PROPN
iajs-2920	216	30	̃	̃	PROPN
iajs-2920	216	31	�	�	PROPN
iajs-2920	216	32	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	216	33	(	(	PUNCT
iajs-2920	216	34	0,0	0,0	NOUN
iajs-2920	216	35	)	)	PUNCT
iajs-2920	216	36	also	also	ADV
iajs-2920	216	37	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	216	38	(	(	PUNCT
iajs-2920	216	39	𝜌2	𝜌2	ADJ
iajs-2920	216	40	,	,	PUNCT
iajs-2920	216	41	𝜏2	𝜏2	PROPN
iajs-2920	216	42	)	)	PUNCT
iajs-2920	216	43	≥	≥	NOUN
iajs-2920	217	1	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	217	2	(	(	PUNCT
iajs-2920	217	3	0,0	0,0	NOUN
iajs-2920	217	4	)	)	PUNCT
iajs-2920	217	5	.	.	PUNCT
iajs-2920	218	1	consider	consider	VERB
iajs-2920	218	2	𝜗𝛿1×𝛿2	𝜗𝛿1×𝛿2	ADP
iajs-2920	218	3	(	(	PUNCT
iajs-2920	218	4	(	(	PUNCT
iajs-2920	218	5	0,0	0,0	NOUN
iajs-2920	218	6	)	)	PUNCT
iajs-2920	218	7	∗	∗	NOUN
iajs-2920	218	8	(	(	PUNCT
iajs-2920	218	9	𝜌1	𝜌1	NOUN
iajs-2920	218	10	,	,	PUNCT
iajs-2920	218	11	𝜏1	𝜏1	NOUN
iajs-2920	218	12	)	)	PUNCT
iajs-2920	218	13	)	)	PUNCT
iajs-2920	219	1	=	=	PUNCT
iajs-2920	219	2	�	�	PROPN
iajs-2920	219	3	̃	̃	PROPN
iajs-2920	219	4	�	�	PROPN
iajs-2920	219	5	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	6	(	(	PUNCT
iajs-2920	219	7	𝜌1	𝜌1	PROPN
iajs-2920	219	8	,	,	PUNCT
iajs-2920	219	9	𝜏1	𝜏1	NOUN
iajs-2920	219	10	)	)	PUNCT
iajs-2920	219	11	≥	≥	PROPN
iajs-2920	219	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	219	13	{	{	PUNCT
iajs-2920	219	14	�	�	PROPN
iajs-2920	219	15	̃	̃	PROPN
iajs-2920	219	16	�	�	PROPN
iajs-2920	219	17	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	18	(	(	PUNCT
iajs-2920	219	19	(	(	PUNCT
iajs-2920	219	20	(	(	PUNCT
iajs-2920	219	21	0,0	0,0	NOUN
iajs-2920	219	22	)	)	PUNCT
iajs-2920	219	23	∗	∗	NOUN
iajs-2920	219	24	(	(	PUNCT
iajs-2920	219	25	𝜌2	𝜌2	ADJ
iajs-2920	219	26	,	,	PUNCT
iajs-2920	219	27	𝜏2	𝜏2	PROPN
iajs-2920	219	28	)	)	PUNCT
iajs-2920	219	29	)	)	PUNCT
iajs-2920	219	30	∗	∗	NOUN
iajs-2920	219	31	(	(	PUNCT
iajs-2920	219	32	𝜌1	𝜌1	NOUN
iajs-2920	219	33	,	,	PUNCT
iajs-2920	219	34	𝜏1	𝜏1	NOUN
iajs-2920	219	35	)	)	PUNCT
iajs-2920	219	36	)	)	PUNCT
iajs-2920	219	37	,	,	PUNCT
iajs-2920	219	38	�	�	PROPN
iajs-2920	219	39	̃	̃	PROPN
iajs-2920	219	40	�	�	PROPN
iajs-2920	219	41	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	42	(	(	PUNCT
iajs-2920	219	43	𝜌2	𝜌2	ADJ
iajs-2920	219	44	,	,	PUNCT
iajs-2920	219	45	𝜏2	𝜏2	PROPN
iajs-2920	219	46	)	)	PUNCT
iajs-2920	219	47	}	}	PUNCT
iajs-2920	219	48	=	=	SYM
iajs-2920	219	49	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	219	50	�	�	PROPN
iajs-2920	219	51	̃	̃	PROPN
iajs-2920	219	52	�	�	PROPN
iajs-2920	219	53	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	54	(	(	PUNCT
iajs-2920	219	55	(	(	PUNCT
iajs-2920	219	56	0,0	0,0	NOUN
iajs-2920	219	57	)	)	PUNCT
iajs-2920	219	58	∗	∗	NOUN
iajs-2920	219	59	(	(	PUNCT
iajs-2920	219	60	0,0	0,0	NOUN
iajs-2920	219	61	)	)	PUNCT
iajs-2920	219	62	)	)	PUNCT
iajs-2920	219	63	,	,	PUNCT
iajs-2920	219	64	�	�	PROPN
iajs-2920	219	65	̃	̃	PROPN
iajs-2920	219	66	�	�	PROPN
iajs-2920	219	67	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	68	(	(	PUNCT
iajs-2920	219	69	𝜌2	𝜌2	ADJ
iajs-2920	219	70	,	,	PUNCT
iajs-2920	219	71	𝜏2	𝜏2	PROPN
iajs-2920	219	72	)	)	PUNCT
iajs-2920	219	73	}	}	PUNCT
iajs-2920	219	74	=	=	SYM
iajs-2920	219	75	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	219	76	�	�	PROPN
iajs-2920	219	77	̃	̃	PROPN
iajs-2920	219	78	�	�	PROPN
iajs-2920	219	79	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	80	(	(	PUNCT
iajs-2920	219	81	0,0	0,0	NOUN
iajs-2920	219	82	)	)	PUNCT
iajs-2920	219	83	,	,	PUNCT
iajs-2920	219	84	�	�	PROPN
iajs-2920	219	85	̃	̃	PROPN
iajs-2920	219	86	�	�	PROPN
iajs-2920	219	87	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	88	(	(	PUNCT
iajs-2920	219	89	𝜌2	𝜌2	ADJ
iajs-2920	219	90	,	,	PUNCT
iajs-2920	219	91	𝜏2	𝜏2	PROPN
iajs-2920	219	92	)	)	PUNCT
iajs-2920	219	93	}	}	PUNCT
iajs-2920	219	94	=	=	SYM
iajs-2920	219	95	�	�	PROPN
iajs-2920	219	96	̃	̃	PROPN
iajs-2920	219	97	�	�	PROPN
iajs-2920	219	98	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	219	99	(	(	PUNCT
iajs-2920	219	100	𝜌2	𝜌2	ADJ
iajs-2920	219	101	,	,	PUNCT
iajs-2920	219	102	𝜏2	𝜏2	PROPN
iajs-2920	219	103	)	)	PUNCT
iajs-2920	219	104	,	,	PUNCT
iajs-2920	219	105	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	219	106	(	(	PUNCT
iajs-2920	219	107	(	(	PUNCT
iajs-2920	219	108	0,0	0,0	NOUN
iajs-2920	219	109	)	)	PUNCT
iajs-2920	219	110	∗	∗	NOUN
iajs-2920	219	111	(	(	PUNCT
iajs-2920	219	112	𝜌1	𝜌1	NOUN
iajs-2920	219	113	,	,	PUNCT
iajs-2920	219	114	𝜏1	𝜏1	NOUN
iajs-2920	219	115	)	)	PUNCT
iajs-2920	219	116	)	)	PUNCT
iajs-2920	220	1	=	=	SYM
iajs-2920	220	2	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	220	3	(	(	PUNCT
iajs-2920	220	4	𝜌1	𝜌1	PROPN
iajs-2920	220	5	,	,	PUNCT
iajs-2920	220	6	𝜏1	𝜏1	NOUN
iajs-2920	220	7	)	)	PUNCT
iajs-2920	220	8	≤	≤	NUM
iajs-2920	221	1	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2920	222	1	{	{	PUNCT
iajs-2920	223	1	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	223	2	(	(	PUNCT
iajs-2920	223	3	(	(	PUNCT
iajs-2920	223	4	(	(	PUNCT
iajs-2920	223	5	0,0	0,0	NOUN
iajs-2920	223	6	)	)	PUNCT
iajs-2920	223	7	∗	∗	NOUN
iajs-2920	223	8	(	(	PUNCT
iajs-2920	223	9	𝜌2	𝜌2	ADJ
iajs-2920	223	10	,	,	PUNCT
iajs-2920	223	11	𝜏2	𝜏2	PROPN
iajs-2920	223	12	)	)	PUNCT
iajs-2920	223	13	)	)	PUNCT
iajs-2920	223	14	∗	∗	NOUN
iajs-2920	223	15	(	(	PUNCT
iajs-2920	223	16	𝜌1	𝜌1	NOUN
iajs-2920	223	17	,	,	PUNCT
iajs-2920	223	18	𝜏1	𝜏1	NOUN
iajs-2920	223	19	)	)	PUNCT
iajs-2920	223	20	)	)	PUNCT
iajs-2920	223	21	,	,	PUNCT
iajs-2920	223	22	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	223	23	(	(	PUNCT
iajs-2920	223	24	𝜌2	𝜌2	ADJ
iajs-2920	223	25	,	,	PUNCT
iajs-2920	223	26	𝜏2	𝜏2	PROPN
iajs-2920	223	27	)	)	PUNCT
iajs-2920	223	28	}	}	PUNCT
iajs-2920	223	29	=	=	SYM
iajs-2920	223	30	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	ADP
iajs-2920	223	31	(	(	PUNCT
iajs-2920	223	32	(	(	PUNCT
iajs-2920	223	33	0,0	0,0	NOUN
iajs-2920	223	34	)	)	PUNCT
iajs-2920	223	35	∗	∗	NOUN
iajs-2920	223	36	(	(	PUNCT
iajs-2920	223	37	0,0	0,0	NOUN
iajs-2920	223	38	)	)	PUNCT
iajs-2920	223	39	)	)	PUNCT
iajs-2920	223	40	,	,	PUNCT
iajs-2920	223	41	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	223	42	(	(	PUNCT
iajs-2920	223	43	𝜌2	𝜌2	ADJ
iajs-2920	223	44	,	,	PUNCT
iajs-2920	223	45	𝜏2	𝜏2	PROPN
iajs-2920	223	46	)	)	PUNCT
iajs-2920	223	47	}	}	PUNCT
iajs-2920	223	48	=	=	SYM
iajs-2920	223	49	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	ADP
iajs-2920	223	50	(	(	PUNCT
iajs-2920	223	51	0,0	0,0	NOUN
iajs-2920	223	52	)	)	PUNCT
iajs-2920	223	53	,	,	PUNCT
iajs-2920	223	54	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	223	55	(	(	PUNCT
iajs-2920	223	56	𝜌2	𝜌2	ADJ
iajs-2920	223	57	,	,	PUNCT
iajs-2920	223	58	𝜏2	𝜏2	PROPN
iajs-2920	223	59	)	)	PUNCT
iajs-2920	223	60	}	}	PUNCT
iajs-2920	223	61	ihjpas	ihjpa	VERB
iajs-2920	223	62	.	.	PUNCT
iajs-2920	224	1	36(1)2023	36(1)2023	NUM
iajs-2920	224	2	378	378	NUM
iajs-2920	224	3	=	=	SYM
iajs-2920	224	4	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	224	5	(	(	PUNCT
iajs-2920	224	6	𝜌2	𝜌2	ADJ
iajs-2920	224	7	,	,	PUNCT
iajs-2920	224	8	𝜏2	𝜏2	PROPN
iajs-2920	224	9	)	)	PUNCT
iajs-2920	224	10	this	this	PRON
iajs-2920	224	11	shows	show	VERB
iajs-2920	224	12	that	that	SCONJ
iajs-2920	224	13	�	�	PROPN
iajs-2920	224	14	̃	̃	PROPN
iajs-2920	224	15	�	�	PROPN
iajs-2920	224	16	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	224	17	(	(	PUNCT
iajs-2920	224	18	𝜌2	𝜌2	ADJ
iajs-2920	224	19	,	,	PUNCT
iajs-2920	224	20	𝜏2	𝜏2	PROPN
iajs-2920	224	21	)	)	PUNCT
iajs-2920	224	22	≤	≤	NUM
iajs-2920	224	23	�	�	PROPN
iajs-2920	224	24	̃	̃	PROPN
iajs-2920	224	25	�	�	PROPN
iajs-2920	224	26	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	224	27	(	(	PUNCT
iajs-2920	224	28	𝜌1	𝜌1	PROPN
iajs-2920	224	29	,	,	PUNCT
iajs-2920	224	30	𝜏1	𝜏1	NOUN
iajs-2920	224	31	)	)	PUNCT
iajs-2920	224	32	and	and	CCONJ
iajs-2920	224	33	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	224	34	(	(	PUNCT
iajs-2920	224	35	𝜌2	𝜌2	ADJ
iajs-2920	224	36	,	,	PUNCT
iajs-2920	224	37	𝜏2	𝜏2	PROPN
iajs-2920	224	38	)	)	PUNCT
iajs-2920	224	39	≥	≥	NOUN
iajs-2920	224	40	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	224	41	(	(	PUNCT
iajs-2920	224	42	𝜌1	𝜌1	PROPN
iajs-2920	224	43	,	,	PUNCT
iajs-2920	224	44	𝜏2	𝜏2	PROPN
iajs-2920	224	45	)	)	PUNCT
iajs-2920	224	46	,	,	PUNCT
iajs-2920	224	47	for	for	ADP
iajs-2920	224	48	all	all	DET
iajs-2920	224	49	(	(	PUNCT
iajs-2920	224	50	𝜌1	𝜌1	NOUN
iajs-2920	224	51	,	,	PUNCT
iajs-2920	224	52	𝜏1	𝜏1	NOUN
iajs-2920	224	53	)	)	PUNCT
iajs-2920	224	54	,	,	PUNCT
iajs-2920	224	55	(	(	PUNCT
iajs-2920	224	56	𝜌2	𝜌2	ADJ
iajs-2920	224	57	,	,	PUNCT
iajs-2920	224	58	𝜏2	𝜏2	ADJ
iajs-2920	224	59	)	)	PUNCT
iajs-2920	224	60	∈	∈	PROPN
iajs-2920	224	61	ℵ1	ℵ1	NOUN
iajs-2920	224	62	×	×	NOUN
iajs-2920	224	63	ℵ2	ℵ2	ADJ
iajs-2920	224	64	.	.	PUNCT
iajs-2920	225	1	theorem	theorem	NOUN
iajs-2920	225	2	(	(	PUNCT
iajs-2920	225	3	34)	34)	NUM
iajs-2920	225	4	..	..	PUNCT
iajs-2920	225	5	let	let	VERB
iajs-2920	225	6	𝛿1	𝛿1	NOUN
iajs-2920	225	7	=	=	SYM
iajs-2920	225	8	〈	〈	PROPN
iajs-2920	225	9	�	�	PROPN
iajs-2920	225	10	̃	̃	PROPN
iajs-2920	225	11	�	�	NOUN
iajs-2920	225	12	𝛿1	𝛿1	NOUN
iajs-2920	225	13	,	,	PUNCT
iajs-2920	225	14	𝛼𝛿1	𝛼𝛿1	PROPN
iajs-2920	225	15	〉	〉	NOUN
iajs-2920	225	16	and	and	CCONJ
iajs-2920	225	17	𝛿2	𝛿2	PROPN
iajs-2920	225	18	=	=	SYM
iajs-2920	225	19	〈	〈	PROPN
iajs-2920	225	20	�	�	PROPN
iajs-2920	225	21	̃	̃	PROPN
iajs-2920	225	22	�	�	NOUN
iajs-2920	225	23	𝛿2	𝛿2	NOUN
iajs-2920	225	24	,	,	PUNCT
iajs-2920	225	25	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	225	26	〉	〉	NOUN
iajs-2920	225	27	be	be	AUX
iajs-2920	225	28	two	two	NUM
iajs-2920	225	29	cubic	cubic	ADJ
iajs-2920	225	30	ideal	ideal	NOUN
iajs-2920	225	31	of	of	ADP
iajs-2920	225	32	tm	tm	NOUN
iajs-2920	225	33	-	-	NOUN
iajs-2920	225	34	algebra	algebra	NOUN
iajs-2920	225	35	ℵ1	ℵ1	PROPN
iajs-2920	225	36	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2920	225	37	ℵ2	ℵ2	NOUN
iajs-2920	225	38	,	,	PUNCT
iajs-2920	225	39	respectively	respectively	ADV
iajs-2920	225	40	.	.	PUNCT
iajs-2920	226	1	then	then	ADV
iajs-2920	226	2	𝛿1	𝛿1	VERB
iajs-2920	226	3	×	×	PROPN
iajs-2920	226	4	𝛿2	𝛿2	NOUN
iajs-2920	226	5	=	=	PUNCT
iajs-2920	226	6	〈	〈	PROPN
iajs-2920	226	7	�	�	PROPN
iajs-2920	226	8	̃	̃	PROPN
iajs-2920	226	9	�	�	NOUN
iajs-2920	226	10	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	226	11	,	,	PUNCT
iajs-2920	226	12	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	226	13	〉	〉	NOUN
iajs-2920	226	14	is	be	AUX
iajs-2920	226	15	a	a	DET
iajs-2920	226	16	cubic	cubic	ADJ
iajs-2920	226	17	ideal	ideal	NOUN
iajs-2920	226	18	of	of	ADP
iajs-2920	226	19	ℵ1	ℵ1	PROPN
iajs-2920	226	20	×	×	NOUN
iajs-2920	226	21	ℵ2	ℵ2	ADJ
iajs-2920	226	22	.	.	PUNCT
iajs-2920	227	1	proof	proof	NOUN
iajs-2920	227	2	.	.	PUNCT
iajs-2920	228	1	for	for	ADP
iajs-2920	228	2	any	any	DET
iajs-2920	228	3	(	(	PUNCT
iajs-2920	228	4	𝜌	𝜌	X
iajs-2920	228	5	,	,	PUNCT
iajs-2920	228	6	𝜏	𝜏	NOUN
iajs-2920	228	7	)	)	PUNCT
iajs-2920	228	8	∈	∈	NOUN
iajs-2920	229	1	ℵ1	ℵ1	NOUN
iajs-2920	229	2	×	×	NOUN
iajs-2920	229	3	ℵ2	ℵ2	ADJ
iajs-2920	229	4	,	,	PUNCT
iajs-2920	229	5	�	�	PROPN
iajs-2920	229	6	̃	̃	PROPN
iajs-2920	229	7	�	�	NOUN
iajs-2920	230	1	𝜹𝟏×𝜹𝟏	𝜹𝟏×𝜹𝟏	PROPN
iajs-2920	230	2	(	(	PUNCT
iajs-2920	230	3	0,0	0,0	NUM
iajs-2920	230	4	)	)	PUNCT
iajs-2920	230	5	=	=	SYM
iajs-2920	230	6	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	230	7	�	�	PROPN
iajs-2920	230	8	̃	̃	PROPN
iajs-2920	230	9	�	�	NOUN
iajs-2920	230	10	𝛿1	𝛿1	NOUN
iajs-2920	230	11	(	(	PUNCT
iajs-2920	230	12	0	0	NUM
iajs-2920	230	13	)	)	PUNCT
iajs-2920	230	14	,	,	PUNCT
iajs-2920	230	15	�	�	PROPN
iajs-2920	230	16	̃	̃	PROPN
iajs-2920	230	17	�	�	NOUN
iajs-2920	230	18	𝛿2	𝛿2	NOUN
iajs-2920	230	19	(	(	PUNCT
iajs-2920	230	20	0	0	NUM
iajs-2920	230	21	)	)	PUNCT
iajs-2920	230	22	}	}	PUNCT
iajs-2920	230	23	≥	≥	PROPN
iajs-2920	230	24	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	230	25	�	�	PROPN
iajs-2920	230	26	̃	̃	PROPN
iajs-2920	230	27	�	�	NOUN
iajs-2920	230	28	𝛿1	𝛿1	NOUN
iajs-2920	230	29	(	(	PUNCT
iajs-2920	230	30	𝜌	𝜌	NOUN
iajs-2920	230	31	)	)	PUNCT
iajs-2920	230	32	,	,	PUNCT
iajs-2920	230	33	�	�	PROPN
iajs-2920	230	34	̃	̃	PROPN
iajs-2920	230	35	�	�	NOUN
iajs-2920	230	36	𝛿2	𝛿2	NOUN
iajs-2920	230	37	(	(	PUNCT
iajs-2920	230	38	𝜏	𝜏	NOUN
iajs-2920	230	39	)	)	PUNCT
iajs-2920	230	40	}	}	PUNCT
iajs-2920	230	41	=	=	SYM
iajs-2920	230	42	�	�	PROPN
iajs-2920	230	43	̃	̃	PROPN
iajs-2920	230	44	�	�	PROPN
iajs-2920	230	45	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	230	46	(	(	PUNCT
iajs-2920	230	47	𝜌	𝜌	X
iajs-2920	230	48	,	,	PUNCT
iajs-2920	230	49	𝜏	𝜏	NOUN
iajs-2920	230	50	)	)	PUNCT
iajs-2920	230	51	,	,	PUNCT
iajs-2920	230	52	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	230	53	(	(	PUNCT
iajs-2920	230	54	0,0	0,0	NUM
iajs-2920	230	55	)	)	PUNCT
iajs-2920	230	56	=	=	SYM
iajs-2920	230	57	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	X
iajs-2920	230	58	(	(	PUNCT
iajs-2920	230	59	0	0	NUM
iajs-2920	230	60	)	)	PUNCT
iajs-2920	230	61	,	,	PUNCT
iajs-2920	230	62	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	230	63	(	(	PUNCT
iajs-2920	230	64	0	0	NUM
iajs-2920	230	65	)	)	PUNCT
iajs-2920	230	66	}	}	PUNCT
iajs-2920	230	67	≤	≤	ADJ
iajs-2920	230	68	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	NOUN
iajs-2920	230	69	(	(	PUNCT
iajs-2920	230	70	𝜌	𝜌	NOUN
iajs-2920	230	71	)	)	PUNCT
iajs-2920	230	72	,	,	PUNCT
iajs-2920	230	73	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	230	74	(	(	PUNCT
iajs-2920	230	75	𝜏	𝜏	NOUN
iajs-2920	230	76	)	)	PUNCT
iajs-2920	230	77	}	}	PUNCT
iajs-2920	230	78	=	=	SYM
iajs-2920	230	79	𝛼𝛿1×𝛿𝟐	𝛼𝛿1×𝛿𝟐	PROPN
iajs-2920	230	80	(	(	PUNCT
iajs-2920	230	81	𝜌	𝜌	X
iajs-2920	230	82	,	,	PUNCT
iajs-2920	230	83	𝜏	𝜏	NOUN
iajs-2920	230	84	)	)	PUNCT
iajs-2920	230	85	.	.	PUNCT
iajs-2920	231	1	for	for	ADP
iajs-2920	231	2	any	any	DET
iajs-2920	231	3	(	(	PUNCT
iajs-2920	231	4	𝜌1	𝜌1	NOUN
iajs-2920	231	5	,	,	PUNCT
iajs-2920	231	6	𝜏1	𝜏1	NOUN
iajs-2920	231	7	)	)	PUNCT
iajs-2920	231	8	,	,	PUNCT
iajs-2920	231	9	(	(	PUNCT
iajs-2920	231	10	𝜌2	𝜌2	ADJ
iajs-2920	231	11	,	,	PUNCT
iajs-2920	231	12	𝜏2	𝜏2	ADJ
iajs-2920	231	13	)	)	PUNCT
iajs-2920	231	14	∈	∈	PROPN
iajs-2920	231	15	ℵ1	ℵ1	NOUN
iajs-2920	231	16	×	×	NOUN
iajs-2920	231	17	ℵ2	ℵ2	ADJ
iajs-2920	231	18	.	.	PUNCT
iajs-2920	232	1	then	then	ADV
iajs-2920	232	2	�	�	PROPN
iajs-2920	232	3	̃	̃	PROPN
iajs-2920	232	4	�	�	PROPN
iajs-2920	232	5	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	232	6	(	(	PUNCT
iajs-2920	232	7	𝜌1	𝜌1	PROPN
iajs-2920	232	8	,	,	PUNCT
iajs-2920	232	9	𝜏1	𝜏1	NOUN
iajs-2920	232	10	)	)	PUNCT
iajs-2920	232	11	=	=	SYM
iajs-2920	232	12	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	232	13	�	�	PROPN
iajs-2920	232	14	̃	̃	PROPN
iajs-2920	232	15	�	�	PROPN
iajs-2920	232	16	𝛿1	𝛿1	NOUN
iajs-2920	232	17	(	(	PUNCT
iajs-2920	232	18	𝜌1	𝜌1	NOUN
iajs-2920	232	19	)	)	PUNCT
iajs-2920	232	20	,	,	PUNCT
iajs-2920	232	21	�	�	PROPN
iajs-2920	232	22	̃	̃	PROPN
iajs-2920	232	23	�	�	NOUN
iajs-2920	232	24	𝛿2	𝛿2	NOUN
iajs-2920	232	25	(	(	PUNCT
iajs-2920	232	26	𝜏1	𝜏1	NOUN
iajs-2920	232	27	)	)	PUNCT
iajs-2920	232	28	}	}	PUNCT
iajs-2920	232	29	≥	≥	PROPN
iajs-2920	232	30	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2920	232	31	{	{	PUNCT
iajs-2920	232	32	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	232	33	�	�	PROPN
iajs-2920	232	34	̃	̃	PROPN
iajs-2920	232	35	�	�	PROPN
iajs-2920	232	36	𝛿1	𝛿1	NOUN
iajs-2920	232	37	(	(	PUNCT
iajs-2920	232	38	𝜌1	𝜌1	PROPN
iajs-2920	232	39	∗	∗	PROPN
iajs-2920	232	40	𝜌2	𝜌2	ADJ
iajs-2920	232	41	)	)	PUNCT
iajs-2920	232	42	,	,	PUNCT
iajs-2920	232	43	�	�	PROPN
iajs-2920	232	44	̃	̃	PROPN
iajs-2920	232	45	�	�	NOUN
iajs-2920	233	1	𝛿1	𝛿1	NOUN
iajs-2920	233	2	(	(	PUNCT
iajs-2920	233	3	𝜌2	𝜌2	ADJ
iajs-2920	233	4	)	)	PUNCT
iajs-2920	233	5	}	}	PUNCT
iajs-2920	233	6	,	,	PUNCT
iajs-2920	233	7	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	233	8	�	�	PROPN
iajs-2920	233	9	̃	̃	PROPN
iajs-2920	233	10	�	�	NOUN
iajs-2920	233	11	𝛿2	𝛿2	NOUN
iajs-2920	233	12	(	(	PUNCT
iajs-2920	233	13	𝜏1	𝜏1	NOUN
iajs-2920	233	14	∗	∗	NOUN
iajs-2920	233	15	𝜏2	𝜏2	PROPN
iajs-2920	233	16	)	)	PUNCT
iajs-2920	233	17	,	,	PUNCT
iajs-2920	233	18	�	�	PROPN
iajs-2920	233	19	̃	̃	PROPN
iajs-2920	233	20	�	�	NOUN
iajs-2920	233	21	𝛿2	𝛿2	NOUN
iajs-2920	233	22	(	(	PUNCT
iajs-2920	233	23	𝜏2	𝜏2	PROPN
iajs-2920	233	24	)	)	PUNCT
iajs-2920	233	25	}	}	PUNCT
iajs-2920	233	26	}	}	PUNCT
iajs-2920	233	27	=	=	SYM
iajs-2920	233	28	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2920	233	29	{	{	PUNCT
iajs-2920	233	30	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	233	31	�	�	PROPN
iajs-2920	233	32	̃	̃	PROPN
iajs-2920	233	33	�	�	PROPN
iajs-2920	233	34	𝛿1	𝛿1	NOUN
iajs-2920	233	35	(	(	PUNCT
iajs-2920	233	36	𝜌1	𝜌1	PROPN
iajs-2920	233	37	∗	∗	PROPN
iajs-2920	233	38	𝜌2	𝜌2	ADJ
iajs-2920	233	39	)	)	PUNCT
iajs-2920	233	40	,	,	PUNCT
iajs-2920	233	41	�	�	PROPN
iajs-2920	233	42	̃	̃	PROPN
iajs-2920	233	43	�	�	NOUN
iajs-2920	233	44	𝛿2	𝛿2	NOUN
iajs-2920	233	45	(	(	PUNCT
iajs-2920	233	46	𝜏1	𝜏1	NOUN
iajs-2920	233	47	∗	∗	NOUN
iajs-2920	233	48	𝜏2	𝜏2	PROPN
iajs-2920	233	49	)	)	PUNCT
iajs-2920	233	50	}	}	PUNCT
iajs-2920	233	51	,	,	PUNCT
iajs-2920	233	52	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	233	53	�	�	PROPN
iajs-2920	233	54	̃	̃	PROPN
iajs-2920	233	55	�	�	PROPN
iajs-2920	233	56	𝛿1	𝛿1	NOUN
iajs-2920	233	57	(	(	PUNCT
iajs-2920	233	58	𝜌2	𝜌2	ADJ
iajs-2920	233	59	)	)	PUNCT
iajs-2920	233	60	,	,	PUNCT
iajs-2920	233	61	�	�	PROPN
iajs-2920	233	62	̃	̃	PROPN
iajs-2920	233	63	�	�	NOUN
iajs-2920	233	64	𝛿2	𝛿2	NOUN
iajs-2920	233	65	(	(	PUNCT
iajs-2920	233	66	𝜏2	𝜏2	PROPN
iajs-2920	233	67	)	)	PUNCT
iajs-2920	233	68	}	}	PUNCT
iajs-2920	233	69	}	}	PUNCT
iajs-2920	233	70	=	=	SYM
iajs-2920	233	71	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	233	72	�	�	PROPN
iajs-2920	233	73	̃	̃	PROPN
iajs-2920	233	74	�	�	PROPN
iajs-2920	233	75	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	233	76	(	(	PUNCT
iajs-2920	233	77	𝜌1	𝜌1	PROPN
iajs-2920	233	78	∗	∗	PROPN
iajs-2920	233	79	𝜌2	𝜌2	ADJ
iajs-2920	233	80	,	,	PUNCT
iajs-2920	233	81	𝜏1	𝜏1	NOUN
iajs-2920	233	82	∗	∗	NOUN
iajs-2920	233	83	𝜏2	𝜏2	PROPN
iajs-2920	233	84	)	)	PUNCT
iajs-2920	233	85	,	,	PUNCT
iajs-2920	233	86	�	�	PROPN
iajs-2920	233	87	̃	̃	PROPN
iajs-2920	233	88	�	�	PROPN
iajs-2920	233	89	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	233	90	(	(	PUNCT
iajs-2920	233	91	𝜌2	𝜌2	ADJ
iajs-2920	233	92	,	,	PUNCT
iajs-2920	233	93	𝜏2	𝜏2	PROPN
iajs-2920	233	94	)	)	PUNCT
iajs-2920	233	95	}	}	PUNCT
iajs-2920	233	96	≥	≥	PROPN
iajs-2920	233	97	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2920	233	98	�	�	PROPN
iajs-2920	233	99	̃	̃	PROPN
iajs-2920	233	100	�	�	PROPN
iajs-2920	233	101	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	233	102	(	(	PUNCT
iajs-2920	233	103	(	(	PUNCT
iajs-2920	233	104	𝜌1	𝜌1	NOUN
iajs-2920	233	105	∗	∗	NOUN
iajs-2920	233	106	𝜏1)(𝜌2	𝜏1)(𝜌2	PROPN
iajs-2920	233	107	∗	∗	NOUN
iajs-2920	233	108	𝜏2	𝜏2	PROPN
iajs-2920	233	109	)	)	PUNCT
iajs-2920	233	110	)	)	PUNCT
iajs-2920	233	111	,	,	PUNCT
iajs-2920	233	112	�	�	PROPN
iajs-2920	233	113	̃	̃	PROPN
iajs-2920	233	114	�	�	PROPN
iajs-2920	233	115	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	233	116	(	(	PUNCT
iajs-2920	233	117	𝜌2	𝜌2	ADJ
iajs-2920	233	118	,	,	PUNCT
iajs-2920	233	119	𝜏2	𝜏2	PROPN
iajs-2920	233	120	)	)	PUNCT
iajs-2920	233	121	}	}	PUNCT
iajs-2920	233	122	,	,	PUNCT
iajs-2920	233	123	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	233	124	(	(	PUNCT
iajs-2920	233	125	𝜌1	𝜌1	PROPN
iajs-2920	233	126	,	,	PUNCT
iajs-2920	233	127	𝜏1	𝜏1	NOUN
iajs-2920	233	128	)	)	PUNCT
iajs-2920	233	129	=	=	SYM
iajs-2920	233	130	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	PROPN
iajs-2920	233	131	(	(	PUNCT
iajs-2920	233	132	𝜌1	𝜌1	NOUN
iajs-2920	233	133	)	)	PUNCT
iajs-2920	233	134	,	,	PUNCT
iajs-2920	233	135	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	233	136	(	(	PUNCT
iajs-2920	233	137	𝜏1	𝜏1	NOUN
iajs-2920	233	138	)	)	PUNCT
iajs-2920	233	139	}	}	PUNCT
iajs-2920	233	140	≤	≤	NUM
iajs-2920	233	141	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2920	233	142	{	{	PUNCT
iajs-2920	233	143	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	PROPN
iajs-2920	233	144	(	(	PUNCT
iajs-2920	233	145	𝜌1	𝜌1	PROPN
iajs-2920	233	146	∗	∗	PROPN
iajs-2920	233	147	𝜌2	𝜌2	ADJ
iajs-2920	233	148	)	)	PUNCT
iajs-2920	233	149	,	,	PUNCT
iajs-2920	233	150	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	233	151	(	(	PUNCT
iajs-2920	233	152	𝜏2	𝜏2	PROPN
iajs-2920	233	153	)	)	PUNCT
iajs-2920	233	154	}	}	PUNCT
iajs-2920	233	155	,	,	PUNCT
iajs-2920	233	156	𝑚𝑎𝑥{𝛼𝛿2	𝑚𝑎𝑥{𝛼𝛿2	PROPN
iajs-2920	233	157	(	(	PUNCT
iajs-2920	233	158	𝜏1	𝜏1	NOUN
iajs-2920	233	159	∗	∗	NOUN
iajs-2920	233	160	𝜏2	𝜏2	PROPN
iajs-2920	233	161	)	)	PUNCT
iajs-2920	233	162	,	,	PUNCT
iajs-2920	233	163	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	233	164	(	(	PUNCT
iajs-2920	233	165	𝜏2	𝜏2	PROPN
iajs-2920	233	166	)	)	PUNCT
iajs-2920	233	167	}	}	PUNCT
iajs-2920	233	168	}	}	PUNCT
iajs-2920	233	169	=	=	PUNCT
iajs-2920	233	170	𝑚𝑎𝑥	𝑚𝑎𝑥	X
iajs-2920	233	171	{	{	PUNCT
iajs-2920	233	172	𝑚𝑎𝑥{𝛼𝛿1	𝑚𝑎𝑥{𝛼𝛿1	PROPN
iajs-2920	233	173	(	(	PUNCT
iajs-2920	233	174	𝜌1	𝜌1	PROPN
iajs-2920	233	175	∗	∗	PROPN
iajs-2920	233	176	𝜌2	𝜌2	ADJ
iajs-2920	233	177	)	)	PUNCT
iajs-2920	233	178	,	,	PUNCT
iajs-2920	233	179	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	233	180	(	(	PUNCT
iajs-2920	233	181	𝜏1	𝜏1	NOUN
iajs-2920	233	182	∗	∗	NOUN
iajs-2920	233	183	𝜏2	𝜏2	PROPN
iajs-2920	233	184	)	)	PUNCT
iajs-2920	233	185	}	}	PUNCT
iajs-2920	233	186	,	,	PUNCT
iajs-2920	233	187	𝑚𝑎𝑥{𝛼𝛿2	𝑚𝑎𝑥{𝛼𝛿2	PROPN
iajs-2920	233	188	(	(	PUNCT
iajs-2920	233	189	𝜌2	𝜌2	ADJ
iajs-2920	233	190	)	)	PUNCT
iajs-2920	233	191	,	,	PUNCT
iajs-2920	233	192	𝛼𝛿2	𝛼𝛿2	PROPN
iajs-2920	233	193	(	(	PUNCT
iajs-2920	233	194	𝜏2	𝜏2	PROPN
iajs-2920	233	195	)	)	PUNCT
iajs-2920	233	196	}	}	PUNCT
iajs-2920	233	197	}	}	PUNCT
iajs-2920	233	198	=	=	SYM
iajs-2920	233	199	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	𝑚𝑎𝑥{𝛼𝛿1×𝛿2	ADP
iajs-2920	233	200	(	(	PUNCT
iajs-2920	233	201	𝜌1	𝜌1	PROPN
iajs-2920	233	202	∗	∗	PROPN
iajs-2920	233	203	𝜌2	𝜌2	ADJ
iajs-2920	233	204	,	,	PUNCT
iajs-2920	233	205	𝜏1	𝜏1	NOUN
iajs-2920	233	206	∗	∗	NOUN
iajs-2920	233	207	𝜏2	𝜏2	PROPN
iajs-2920	233	208	)	)	PUNCT
iajs-2920	233	209	,	,	PUNCT
iajs-2920	233	210	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	233	211	(	(	PUNCT
iajs-2920	233	212	𝜌2	𝜌2	ADJ
iajs-2920	233	213	,	,	PUNCT
iajs-2920	233	214	𝜏2	𝜏2	PROPN
iajs-2920	233	215	)	)	PUNCT
iajs-2920	233	216	}	}	PUNCT
iajs-2920	233	217	hence	hence	ADV
iajs-2920	233	218	,	,	PUNCT
iajs-2920	233	219	for	for	ADP
iajs-2920	233	220	all	all	DET
iajs-2920	233	221	(	(	PUNCT
iajs-2920	233	222	𝜌1	𝜌1	NOUN
iajs-2920	233	223	,	,	PUNCT
iajs-2920	233	224	𝜏1	𝜏1	NOUN
iajs-2920	233	225	)	)	PUNCT
iajs-2920	233	226	,	,	PUNCT
iajs-2920	233	227	(	(	PUNCT
iajs-2920	233	228	𝜌2	𝜌2	ADJ
iajs-2920	233	229	,	,	PUNCT
iajs-2920	233	230	𝜏2	𝜏2	ADJ
iajs-2920	233	231	)	)	PUNCT
iajs-2920	233	232	∈	∈	PROPN
iajs-2920	233	233	ℵ1	ℵ1	NOUN
iajs-2920	233	234	×	×	PROPN
iajs-2920	233	235	ℵ1	ℵ1	PROPN
iajs-2920	233	236	,	,	PUNCT
iajs-2920	233	237	𝛿1	𝛿1	NOUN
iajs-2920	233	238	×	×	NOUN
iajs-2920	233	239	𝛿2	𝛿2	NOUN
iajs-2920	233	240	=	=	PUNCT
iajs-2920	233	241	〈	〈	PROPN
iajs-2920	233	242	�	�	PROPN
iajs-2920	233	243	̃	̃	PROPN
iajs-2920	233	244	�	�	NOUN
iajs-2920	233	245	𝛿1×𝛿2	𝛿1×𝛿2	PROPN
iajs-2920	233	246	,	,	PUNCT
iajs-2920	233	247	𝛼𝛿1×𝛿2	𝛼𝛿1×𝛿2	PROPN
iajs-2920	233	248	〉	〉	NOUN
iajs-2920	233	249	is	be	AUX
iajs-2920	233	250	a	a	DET
iajs-2920	233	251	cubic	cubic	ADJ
iajs-2920	233	252	idealof	idealof	NOUN
iajs-2920	233	253	tmalgebra	tmalgebra	NOUN
iajs-2920	233	254	ℵ1	ℵ1	PROPN
iajs-2920	233	255	×	×	NOUN
iajs-2920	233	256	ℵ2	ℵ2	ADJ
iajs-2920	233	257	.	.	PUNCT
iajs-2920	234	1	6.conclusion	6.conclusion	NUM
iajs-2920	234	2	the	the	DET
iajs-2920	234	3	goal	goal	NOUN
iajs-2920	234	4	of	of	ADP
iajs-2920	234	5	this	this	DET
iajs-2920	234	6	paper	paper	NOUN
iajs-2920	234	7	is	be	AUX
iajs-2920	234	8	to	to	PART
iajs-2920	234	9	introduce	introduce	VERB
iajs-2920	234	10	the	the	DET
iajs-2920	234	11	definition	definition	NOUN
iajs-2920	234	12	of	of	ADP
iajs-2920	234	13	a	a	DET
iajs-2920	234	14	cubic	cubic	ADJ
iajs-2920	234	15	ideal	ideal	NOUN
iajs-2920	234	16	and	and	CCONJ
iajs-2920	234	17	a	a	DET
iajs-2920	234	18	cubic	cubic	ADJ
iajs-2920	234	19	t	t	NOUN
iajs-2920	234	20	-	-	PUNCT
iajs-2920	234	21	ideal	ideal	NOUN
iajs-2920	234	22	.	.	PUNCT
iajs-2920	235	1	the	the	DET
iajs-2920	235	2	homomorphism	homomorphism	NOUN
iajs-2920	235	3	of	of	ADP
iajs-2920	235	4	these	these	DET
iajs-2920	235	5	ideals	ideal	NOUN
iajs-2920	235	6	is	be	AUX
iajs-2920	235	7	defined	define	VERB
iajs-2920	235	8	,	,	PUNCT
iajs-2920	235	9	and	and	CCONJ
iajs-2920	235	10	the	the	DET
iajs-2920	235	11	cartesian	cartesian	ADJ
iajs-2920	235	12	product	product	NOUN
iajs-2920	235	13	of	of	ADP
iajs-2920	235	14	cubic	cubic	ADJ
iajs-2920	235	15	ideals	ideal	NOUN
iajs-2920	235	16	in	in	ADP
iajs-2920	235	17	cartesian	cartesian	ADJ
iajs-2920	235	18	product	product	NOUN
iajs-2920	235	19	tm	tm	NOUN
iajs-2920	235	20	-	-	PUNCT
iajs-2920	235	21	algebras	algebras	PROPN
iajs-2920	235	22	is	be	AUX
iajs-2920	235	23	given	give	VERB
iajs-2920	235	24	.	.	PUNCT
iajs-2920	236	1	cubic	cubic	ADJ
iajs-2920	236	2	ideals	ideal	NOUN
iajs-2920	236	3	are	be	AUX
iajs-2920	236	4	presented	present	VERB
iajs-2920	236	5	and	and	CCONJ
iajs-2920	236	6	studied	study	VERB
iajs-2920	236	7	by	by	ADP
iajs-2920	236	8	more	more	ADJ
iajs-2920	236	9	than	than	ADP
iajs-2920	236	10	one	one	NUM
iajs-2920	236	11	author	author	NOUN
iajs-2920	236	12	on	on	ADP
iajs-2920	236	13	ihjpas	ihjpas	PROPN
iajs-2920	236	14	.	.	PUNCT
iajs-2920	237	1	36(1)2023	36(1)2023	NUM
iajs-2920	237	2	379	379	NUM
iajs-2920	237	3	different	different	ADJ
iajs-2920	237	4	algebraic	algebraic	ADJ
iajs-2920	237	5	structures	structure	NOUN
iajs-2920	237	6	.	.	PUNCT
iajs-2920	238	1	also	also	ADV
iajs-2920	238	2	,	,	PUNCT
iajs-2920	238	3	new	new	ADJ
iajs-2920	238	4	relations	relation	NOUN
iajs-2920	238	5	between	between	ADP
iajs-2920	238	6	a	a	DET
iajs-2920	238	7	cubic	cubic	ADJ
iajs-2920	238	8	ideal	ideal	NOUN
iajs-2920	238	9	and	and	CCONJ
iajs-2920	238	10	a	a	DET
iajs-2920	238	11	cubic	cubic	ADJ
iajs-2920	238	12	t	t	PROPN
iajs-2920	238	13	-	-	PUNCT
iajs-2920	238	14	ideal	ideal	NOUN
iajs-2920	238	15	are	be	AUX
iajs-2920	238	16	discussed	discuss	VERB
iajs-2920	238	17	.	.	PUNCT
iajs-2920	239	1	references	reference	NOUN
iajs-2920	239	2	1	1	NUM
iajs-2920	239	3	.	.	X
iajs-2920	240	1	megalai	megalai	PROPN
iajs-2920	240	2	,	,	PUNCT
iajs-2920	240	3	k.	k.	PROPN
iajs-2920	240	4	;	;	PUNCT
iajs-2920	240	5	tamilarasi	tamilarasi	NOUN
iajs-2920	240	6	,	,	PUNCT
iajs-2920	240	7	a.	a.	NOUN
iajs-2920	240	8	classification	classification	NOUN
iajs-2920	240	9	of	of	ADP
iajs-2920	240	10	tm	tm	NOUN
iajs-2920	240	11	-	-	NOUN
iajs-2920	240	12	algebra	algebra	NOUN
iajs-2920	240	13	,	,	PUNCT
iajs-2920	240	14	computer	computer	NOUN
iajs-2920	240	15	aided	aid	VERB
iajs-2920	240	16	soft	soft	ADJ
iajs-2920	240	17	computing	computing	NOUN
iajs-2920	240	18	techniques	technique	NOUN
iajs-2920	240	19	for	for	ADP
iajs-2920	240	20	imaging	imaging	NOUN
iajs-2920	240	21	and	and	CCONJ
iajs-2920	240	22	biomedical	biomedical	ADJ
iajs-2920	240	23	applications	application	NOUN
iajs-2920	240	24	.	.	PUNCT
iajs-2920	241	1	special	special	ADJ
iajs-2920	241	2	issue	issue	NOUN
iajs-2920	241	3	2010	2010	NUM
iajs-2920	241	4	.	.	PUNCT
iajs-2920	242	1	2	2	X
iajs-2920	242	2	.	.	X
iajs-2920	242	3	ganeshkumar	ganeshkumar	PROPN
iajs-2920	242	4	,	,	PUNCT
iajs-2920	242	5	t.	t.	PROPN
iajs-2920	242	6	;	;	PUNCT
iajs-2920	242	7	chandramouleeswaran	chandramouleeswaran	ADJ
iajs-2920	242	8	,	,	PUNCT
iajs-2920	242	9	m.	m.	NOUN
iajs-2920	242	10	t−derivations	t−derivation	NOUN
iajs-2920	242	11	on	on	ADP
iajs-2920	242	12	tm	tm	NOUN
iajs-2920	242	13	-	-	PUNCT
iajs-2920	242	14	algebras	algebras	PROPN
iajs-2920	242	15	.	.	PUNCT
iajs-2920	243	1	international	international	ADJ
iajs-2920	243	2	journal	journal	PROPN
iajs-2920	243	3	of	of	ADP
iajs-2920	243	4	pure	pure	ADJ
iajs-2920	243	5	and	and	CCONJ
iajs-2920	243	6	applied	applied	ADJ
iajs-2920	243	7	mathematics	mathematic	NOUN
iajs-2920	243	8	.	.	PUNCT
iajs-2920	244	1	2013,85,1	2013,85,1	NUM
iajs-2920	244	2	,	,	PUNCT
iajs-2920	244	3	95	95	NUM
iajs-2920	244	4	-	-	SYM
iajs-2920	244	5	107	107	NUM
iajs-2920	244	6	.	.	PUNCT
iajs-2920	245	1	3	3	X
iajs-2920	245	2	.	.	X
iajs-2920	245	3	megalai	megalai	PROPN
iajs-2920	245	4	,	,	PUNCT
iajs-2920	245	5	k.	k.	PROPN
iajs-2920	245	6	;	;	PUNCT
iajs-2920	246	1	tamilarasi	tamilarasi	NOUN
iajs-2920	246	2	,	,	PUNCT
iajs-2920	246	3	a.	a.	NOUN
iajs-2920	246	4	fuzzy	fuzzy	ADJ
iajs-2920	246	5	subalgebras	subalgebras	PROPN
iajs-2920	246	6	and	and	CCONJ
iajs-2920	246	7	fuzzy	fuzzy	ADJ
iajs-2920	246	8	t	t	NOUN
iajs-2920	246	9	-	-	PUNCT
iajs-2920	246	10	ideals	ideal	NOUN
iajs-2920	246	11	in	in	ADP
iajs-2920	246	12	tm	tm	NOUN
iajs-2920	246	13	-	-	PUNCT
iajs-2920	246	14	algebras	algebras	PROPN
iajs-2920	246	15	.	.	PUNCT
iajs-2920	246	16	journal	journal	PROPN
iajs-2920	246	17	of	of	ADP
iajs-2920	246	18	mathematics	mathematic	NOUN
iajs-2920	246	19	and	and	CCONJ
iajs-2920	246	20	statistics	statistic	NOUN
iajs-2920	246	21	.	.	PUNCT
iajs-2920	247	1	2011	2011	NUM
iajs-2920	247	2	,	,	PUNCT
iajs-2920	247	3	7,2	7,2	NUM
iajs-2920	247	4	,	,	PUNCT
iajs-2920	247	5	107	107	NUM
iajs-2920	247	6	-	-	SYM
iajs-2920	247	7	111	111	NUM
iajs-2920	247	8	.	.	PUNCT
iajs-2920	248	1	4	4	NUM
iajs-2920	248	2	.	.	X
iajs-2920	248	3	thomas	thomas	PROPN
iajs-2920	248	4	,	,	PUNCT
iajs-2920	248	5	j.	j.	PROPN
iajs-2920	248	6	;	;	PUNCT
iajs-2920	248	7	indhira	indhira	PROPN
iajs-2920	248	8	,	,	PUNCT
iajs-2920	248	9	k.	k.	PROPN
iajs-2920	248	10	;	;	PUNCT
iajs-2920	248	11	chandrasekaran	chandrasekaran	VERB
iajs-2920	248	12	,	,	PUNCT
iajs-2920	248	13	v.	v.	ADP
iajs-2920	248	14	m.	m.	PROPN
iajs-2920	248	15	t	t	PROPN
iajs-2920	248	16	-	-	PUNCT
iajs-2920	248	17	normed	norme	VERB
iajs-2920	248	18	fuzzy	fuzzy	ADJ
iajs-2920	248	19	tm	tm	NOUN
iajs-2920	248	20	-	-	NOUN
iajs-2920	248	21	subalgebra	subalgebra	NOUN
iajs-2920	248	22	of	of	ADP
iajs-2920	248	23	tmalgebras	tmalgebras	ADJ
iajs-2920	248	24	.	.	PUNCT
iajs-2920	249	1	international	international	ADJ
iajs-2920	249	2	journal	journal	PROPN
iajs-2920	249	3	of	of	ADP
iajs-2920	249	4	computational	computational	ADJ
iajs-2920	249	5	intelligence	intelligence	NOUN
iajs-2920	249	6	systems	system	NOUN
iajs-2920	249	7	.	.	PUNCT
iajs-2920	250	1	2019,12,2	2019,12,2	NUM
iajs-2920	250	2	,	,	PUNCT
iajs-2920	250	3	706–712	706–712	NUM
iajs-2920	250	4	.	.	PUNCT
iajs-2920	251	1	5	5	NUM
iajs-2920	251	2	.	.	X
iajs-2920	251	3	ghlaim	ghlaim	PROPN
iajs-2920	251	4	,	,	PUNCT
iajs-2920	251	5	f.	f.	PROPN
iajs-2920	251	6	m.	m.	PROPN
iajs-2920	251	7	;	;	PUNCT
iajs-2920	251	8	kareem	kareem	PROPN
iajs-2920	251	9	,	,	PUNCT
iajs-2920	251	10	f.	f.	PROPN
iajs-2920	251	11	f.	f.	PROPN
iajs-2920	251	12	interval	interval	PROPN
iajs-2920	251	13	valued	value	VERB
iajs-2920	251	14	fuzzy	fuzzy	ADJ
iajs-2920	251	15	ideals	ideal	NOUN
iajs-2920	251	16	of	of	ADP
iajs-2920	251	17	tm	tm	NOUN
iajs-2920	251	18	-	-	NOUN
iajs-2920	251	19	algebra	algebra	NOUN
iajs-2920	251	20	.	.	PUNCT
iajs-2920	252	1	accepted	accept	VERB
iajs-2920	252	2	in	in	ADP
iajs-2920	252	3	journal	journal	NOUN
iajs-2920	252	4	of	of	ADP
iajs-2920	252	5	interdisciplinary	interdisciplinary	ADJ
iajs-2920	252	6	mathematics	mathematic	NOUN
iajs-2920	252	7	.	.	PUNCT
iajs-2920	253	1	6	6	NUM
iajs-2920	253	2	.	.	X
iajs-2920	253	3	ganeshkumar	ganeshkumar	PROPN
iajs-2920	253	4	,	,	PUNCT
iajs-2920	253	5	t.	t.	NOUN
iajs-2920	253	6	generalized	generalized	ADJ
iajs-2920	253	7	derivation	derivation	NOUN
iajs-2920	253	8	on	on	ADP
iajs-2920	253	9	tm	tm	PROPN
iajs-2920	253	10	–	–	PUNCT
iajs-2920	253	11	algebras	algebras	PROPN
iajs-2920	253	12	.	.	PUNCT
iajs-2920	254	1	international	international	ADJ
iajs-2920	254	2	journal	journal	PROPN
iajs-2920	254	3	of	of	ADP
iajs-2920	254	4	algebra	algebra	PROPN
iajs-2920	254	5	.	.	PUNCT
iajs-2920	255	1	2013	2013	NUM
iajs-2920	255	2	,	,	PUNCT
iajs-2920	255	3	7	7	NUM
iajs-2920	255	4	,	,	PUNCT
iajs-2920	255	5	6	6	NUM
iajs-2920	255	6	,	,	PUNCT
iajs-2920	255	7	251	251	NUM
iajs-2920	255	8	–	–	SYM
iajs-2920	255	9	258	258	NUM
iajs-2920	255	10	.	.	X
iajs-2920	256	1	7	7	NUM
iajs-2920	256	2	.	.	X
iajs-2920	256	3	jun	jun	PROPN
iajs-2920	256	4	,	,	PUNCT
iajs-2920	256	5	y.	y.	PROPN
iajs-2920	256	6	b.	b.	PROPN
iajs-2920	256	7	;	;	PUNCT
iajs-2920	256	8	kim	kim	PROPN
iajs-2920	256	9	,	,	PUNCT
iajs-2920	256	10	c.	c.	PROPN
iajs-2920	256	11	s.	s.	PROPN
iajs-2920	256	12	;	;	PUNCT
iajs-2920	256	13	kang	kang	PROPN
iajs-2920	256	14	,	,	PUNCT
iajs-2920	256	15	m.	m.	PROPN
iajs-2920	256	16	s.	s.	PROPN
iajs-2920	256	17	cubic	cubic	PROPN
iajs-2920	256	18	subalgebras	subalgebras	PROPN
iajs-2920	256	19	and	and	CCONJ
iajs-2920	256	20	ideals	ideal	NOUN
iajs-2920	256	21	of	of	ADP
iajs-2920	256	22	bck	bck	PROPN
iajs-2920	256	23	/	/	SYM
iajs-2920	256	24	bci	bci	NOUN
iajs-2920	256	25	-	-	PUNCT
iajs-2920	256	26	algebras	algebra	NOUN
iajs-2920	256	27	.	.	PUNCT
iajs-2920	257	1	far	far	PROPN
iajs-2920	257	2	east	east	PROPN
iajs-2920	257	3	journal	journal	PROPN
iajs-2920	257	4	of	of	ADP
iajs-2920	257	5	mathematical	mathematical	ADJ
iajs-2920	257	6	sciences	science	NOUN
iajs-2920	257	7	.	.	PUNCT
iajs-2920	258	1	2010	2010	NUM
iajs-2920	258	2	,	,	PUNCT
iajs-2920	258	3	2	2	NUM
iajs-2920	258	4	,	,	PUNCT
iajs-2920	258	5	44	44	NUM
iajs-2920	258	6	,	,	PUNCT
iajs-2920	258	7	239–250	239–250	NUM
iajs-2920	258	8	.	.	PUNCT
iajs-2920	259	1	8	8	NUM
iajs-2920	259	2	.	.	X
iajs-2920	260	1	jun	jun	PROPN
iajs-2920	260	2	,	,	PUNCT
iajs-2920	260	3	y.	y.	PROPN
iajs-2920	260	4	b.	b.	PROPN
iajs-2920	260	5	;	;	PUNCT
iajs-2920	260	6	kim	kim	PROPN
iajs-2920	260	7	,	,	PUNCT
iajs-2920	260	8	c.	c.	PROPN
iajs-2920	260	9	s.	s.	PROPN
iajs-2920	260	10	;	;	PUNCT
iajs-2920	260	11	yang	yang	PROPN
iajs-2920	260	12	,	,	PUNCT
iajs-2920	260	13	k.o	k.o	PROPN
iajs-2920	260	14	.	.	PROPN
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iajs-2920	260	16	sets	set	NOUN
iajs-2920	260	17	.	.	PUNCT
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iajs-2920	261	3	fuzzy	fuzzy	ADJ
iajs-2920	261	4	mathematics	mathematic	NOUN
iajs-2920	261	5	and	and	CCONJ
iajs-2920	261	6	informatics	informatic	NOUN
iajs-2920	261	7	.	.	PUNCT
iajs-2920	262	1	2012	2012	NUM
iajs-2920	262	2	,	,	PUNCT
iajs-2920	262	3	1,4	1,4	NUM
iajs-2920	262	4	,	,	PUNCT
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iajs-2920	263	1	9	9	X
iajs-2920	263	2	.	.	X
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iajs-2920	263	4	,	,	PUNCT
iajs-2920	263	5	n.	n.	PROPN
iajs-2920	263	6	;	;	PUNCT
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iajs-2920	263	8	,	,	PUNCT
iajs-2920	263	9	s.m	s.m	PROPN
iajs-2920	263	10	.	.	PROPN
iajs-2920	263	11	;	;	PUNCT
iajs-2920	263	12	ansari	ansari	PROPN
iajs-2920	263	13	,	,	PUNCT
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iajs-2920	263	18	ku	ku	PROPN
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iajs-2920	263	24	.	.	PUNCT
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iajs-2920	264	2	.	.	PUNCT
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iajs-2920	265	6	.	.	PUNCT
iajs-2920	266	1	10	10	NUM
iajs-2920	266	2	.	.	X
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iajs-2920	266	5	f.	f.	PROPN
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iajs-2920	267	4	a.	a.	PROPN
iajs-2920	267	5	cubic	cubic	PROPN
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iajs-2920	267	9	in	in	ADP
iajs-2920	267	10	ku	ku	PROPN
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iajs-2920	267	13	.	.	PUNCT
iajs-2920	268	1	j.	j.	PROPN
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iajs-2920	268	3	.	.	PUNCT
iajs-2920	268	4	:	:	PUNCT
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iajs-2920	269	2	.	.	PUNCT
iajs-2920	269	3	ser	ser	PROPN
iajs-2920	269	4	.	.	PROPN
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iajs-2920	269	6	.	.	PUNCT
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iajs-2920	270	2	,	,	PUNCT
iajs-2920	270	3	012018	012018	NUM
iajs-2920	270	4	.	.	PUNCT
iajs-2920	271	1	11	11	NUM
iajs-2920	271	2	.	.	X
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iajs-2920	272	2	,	,	PUNCT
iajs-2920	272	3	m.	m.	NOUN
iajs-2920	272	4	;	;	PUNCT
iajs-2920	272	5	yaqoob	yaqoob	NOUN
iajs-2920	272	6	,	,	PUNCT
iajs-2920	272	7	n.	n.	PROPN
iajs-2920	272	8	;	;	PUNCT
iajs-2920	272	9	gulistan	gulistan	PROPN
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iajs-2920	272	11	m.	m.	NOUN
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iajs-2920	272	13	ku	ku	PROPN
iajs-2920	272	14	-	-	PUNCT
iajs-2920	272	15	subalgebras	subalgebras	PROPN
iajs-2920	272	16	.	.	PUNCT
iajs-2920	273	1	int	int	NOUN
iajs-2920	273	2	.	.	PUNCT
iajs-2920	274	1	j.	j.	PROPN
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iajs-2920	274	4	.	.	PUNCT
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iajs-2920	274	6	.	.	PUNCT
iajs-2920	275	1	2013,89	2013,89	ADJ
iajs-2920	275	2	,	,	PUNCT
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iajs-2920	275	4	,	,	PUNCT
iajs-2920	275	5	659665	659665	NUM
iajs-2920	275	6	.	.	PUNCT
iajs-2920	276	1	12	12	NUM
iajs-2920	276	2	.	.	PUNCT
iajs-2920	277	1	jun	jun	PROPN
iajs-2920	277	2	,	,	PUNCT
iajs-2920	277	3	y.	y.	PROPN
iajs-2920	277	4	b.	b.	PROPN
iajs-2920	277	5	;	;	PUNCT
iajs-2920	277	6	kim	kim	PROPN
iajs-2920	277	7	,	,	PUNCT
iajs-2920	277	8	c.	c.	PROPN
iajs-2920	277	9	s.	s.	PROPN
iajs-2920	277	10	;	;	PUNCT
iajs-2920	277	11	kang	kang	PROPN
iajs-2920	277	12	;	;	PUNCT
iajs-2920	277	13	kang	kang	PROPN
iajs-2920	277	14	,	,	PUNCT
iajs-2920	277	15	j.	j.	PROPN
iajs-2920	277	16	g.	g.	PROPN
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iajs-2920	278	1	q	q	PROPN
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iajs-2920	278	5	-	-	PUNCT
iajs-2920	278	6	algebras	algebra	NOUN
iajs-2920	278	7	.	.	PUNCT
iajs-2920	279	1	annals	annal	NOUN
iajs-2920	279	2	of	of	ADP
iajs-2920	279	3	fuzzy	fuzzy	ADJ
iajs-2920	279	4	mathematics	mathematic	NOUN
iajs-2920	279	5	and	and	CCONJ
iajs-2920	279	6	informatics	informatic	NOUN
iajs-2920	279	7	.	.	PUNCT
iajs-2920	280	1	2011	2011	NUM
iajs-2920	280	2	,	,	PUNCT
iajs-2920	280	3	1	1	NUM
iajs-2920	280	4	,	,	PUNCT
iajs-2920	280	5	1	1	NUM
iajs-2920	280	6	,	,	PUNCT
iajs-2920	280	7	2534	2534	NUM
iajs-2920	280	8	.	.	PUNCT
iajs-2920	281	1	13	13	NUM
iajs-2920	281	2	.	.	PUNCT
iajs-2920	282	1	janaa	janaa	PROPN
iajs-2920	282	2	,	,	PUNCT
iajs-2920	282	3	c.	c.	PROPN
iajs-2920	282	4	;	;	PUNCT
iajs-2920	282	5	senapati	senapati	PROPN
iajs-2920	282	6	,	,	PUNCT
iajs-2920	282	7	t.	t.	PROPN
iajs-2920	282	8	cubic	cubic	PROPN
iajs-2920	282	9	g	g	PROPN
iajs-2920	282	10	-	-	PUNCT
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iajs-2920	282	12	of	of	ADP
iajs-2920	282	13	g	g	NOUN
iajs-2920	282	14	-	-	PUNCT
iajs-2920	282	15	algebras	algebras	NOUN
iajs-2920	282	16	.	.	PUNCT
iajs-2920	283	1	annals	annal	NOUN
iajs-2920	283	2	of	of	ADP
iajs-2920	283	3	pure	pure	ADJ
iajs-2920	283	4	and	and	CCONJ
iajs-2920	283	5	applied	applied	ADJ
iajs-2920	283	6	mathematics	mathematic	NOUN
iajs-2920	283	7	.	.	PUNCT
iajs-2920	284	1	2015	2015	NUM
iajs-2920	284	2	,	,	PUNCT
iajs-2920	284	3	10,1	10,1	NUM
iajs-2920	284	4	,	,	PUNCT
iajs-2920	284	5	105	105	NUM
iajs-2920	284	6	-	-	SYM
iajs-2920	284	7	115	115	NUM
iajs-2920	284	8	.	.	PUNCT