id	sid	tid	token	lemma	pos
cana-3815	1	1	communications	communication	NOUN
cana-3815	1	2	on	on	ADP
cana-3815	1	3	applied	apply	VERB
cana-3815	1	4	nonlinear	nonlinear	ADJ
cana-3815	1	5	analysis	analysis	NOUN
cana-3815	1	6	issn	issn	NOUN
cana-3815	1	7	:	:	PUNCT
cana-3815	1	8	1074	1074	NUM
cana-3815	1	9	-	-	PUNCT
cana-3815	1	10	133x	133x	NUM
cana-3815	1	11	vol	vol	NOUN
cana-3815	1	12	32	32	NUM
cana-3815	1	13	no	no	NOUN
cana-3815	1	14	.	.	NOUN
cana-3815	1	15	3	3	NUM
cana-3815	1	16	(	(	PUNCT
cana-3815	1	17	2025	2025	NUM
cana-3815	1	18	)	)	PUNCT
cana-3815	1	19	extension	extension	NOUN
cana-3815	1	20	of	of	ADP
cana-3815	1	21	primary	primary	ADJ
cana-3815	1	22	and	and	CCONJ
cana-3815	1	23	semiprimary	semiprimary	ADJ
cana-3815	1	24	bi	bi	NOUN
cana-3815	1	25	-	-	NOUN
cana-3815	1	26	ideals	ideal	NOUN
cana-3815	1	27	of	of	ADP
cana-3815	1	28	semirings	semiring	NOUN
cana-3815	1	29	g.	g.	PROPN
cana-3815	1	30	shanmugam1	shanmugam1	PROPN
cana-3815	1	31	,	,	PUNCT
cana-3815	1	32	p.	p.	NOUN
cana-3815	1	33	lakshmi	lakshmi	PROPN
cana-3815	1	34	pallavi2	pallavi2	PROPN
cana-3815	1	35	,	,	PUNCT
cana-3815	1	36	k.arulmozhi3	k.arulmozhi3	PROPN
cana-3815	1	37	,	,	PUNCT
cana-3815	1	38	aiyared	aiyare	VERB
cana-3815	1	39	iampan4,∗	iampan4,∗	ADJ
cana-3815	1	40	1department	1department	NUM
cana-3815	1	41	of	of	ADP
cana-3815	1	42	mathematics	mathematic	NOUN
cana-3815	1	43	,	,	PUNCT
cana-3815	1	44	saveetha	saveetha	PROPN
cana-3815	1	45	school	school	PROPN
cana-3815	1	46	of	of	ADP
cana-3815	1	47	engineering	engineering	PROPN
cana-3815	1	48	,	,	PUNCT
cana-3815	1	49	saveetha	saveetha	PROPN
cana-3815	1	50	institute	institute	PROPN
cana-3815	1	51	of	of	ADP
cana-3815	1	52	medical	medical	ADJ
cana-3815	1	53	and	and	CCONJ
cana-3815	1	54	technical	technical	ADJ
cana-3815	1	55	sciences	science	NOUN
cana-3815	1	56	,	,	PUNCT
cana-3815	1	57	chennai-602105	chennai-602105	ADJ
cana-3815	1	58	,	,	PUNCT
cana-3815	1	59	india	india	PROPN
cana-3815	1	60	.	.	PUNCT
cana-3815	1	61	2b	2b	PROPN
cana-3815	1	62	v	v	X
cana-3815	1	63	raju	raju	PROPN
cana-3815	1	64	institute	institute	PROPN
cana-3815	1	65	of	of	ADP
cana-3815	1	66	technology	technology	PROPN
cana-3815	1	67	,	,	PUNCT
cana-3815	1	68	narsapur	narsapur	NOUN
cana-3815	1	69	medak	medak	PROPN
cana-3815	1	70	dist	dist	PROPN
cana-3815	1	71	,	,	PUNCT
cana-3815	1	72	telangana	telangana	PROPN
cana-3815	1	73	state-502313	state-502313	PROPN
cana-3815	1	74	,	,	PUNCT
cana-3815	1	75	india	india	PROPN
cana-3815	1	76	.	.	PUNCT
cana-3815	2	1	3department	3department	NUM
cana-3815	2	2	of	of	ADP
cana-3815	2	3	mathematics	mathematic	NOUN
cana-3815	2	4	,	,	PUNCT
cana-3815	2	5	bharath	bharath	PROPN
cana-3815	2	6	institute	institute	PROPN
cana-3815	2	7	of	of	ADP
cana-3815	2	8	higher	high	ADJ
cana-3815	2	9	education	education	NOUN
cana-3815	2	10	and	and	CCONJ
cana-3815	2	11	research	research	NOUN
cana-3815	2	12	,	,	PUNCT
cana-3815	2	13	tamil	tamil	PROPN
cana-3815	2	14	nadu	nadu	PROPN
cana-3815	2	15	,	,	PUNCT
cana-3815	2	16	chennai	chennai	PROPN
cana-3815	2	17	600073	600073	NUM
cana-3815	2	18	,	,	PUNCT
cana-3815	2	19	india	india	PROPN
cana-3815	2	20	.	.	PUNCT
cana-3815	3	1	4department	4department	NUM
cana-3815	3	2	of	of	ADP
cana-3815	3	3	mathematics	mathematic	NOUN
cana-3815	3	4	,	,	PUNCT
cana-3815	3	5	school	school	NOUN
cana-3815	3	6	of	of	ADP
cana-3815	3	7	science	science	NOUN
cana-3815	3	8	,	,	PUNCT
cana-3815	3	9	university	university	NOUN
cana-3815	3	10	of	of	ADP
cana-3815	3	11	phayao	phayao	NOUN
cana-3815	3	12	,	,	PUNCT
cana-3815	3	13	19	19	NUM
cana-3815	3	14	moo	moo	NOUN
cana-3815	3	15	2	2	NUM
cana-3815	3	16	,	,	PUNCT
cana-3815	3	17	tambon	tambon	PROPN
cana-3815	3	18	mae	mae	PROPN
cana-3815	3	19	ka	ka	PROPN
cana-3815	3	20	,	,	PUNCT
cana-3815	3	21	amphur	amphur	PROPN
cana-3815	3	22	mueang	mueang	NOUN
cana-3815	3	23	,	,	PUNCT
cana-3815	3	24	phayao	phayao	NOUN
cana-3815	3	25	56000	56000	NUM
cana-3815	3	26	,	,	PUNCT
cana-3815	3	27	thailand	thailand	PROPN
cana-3815	3	28	.	.	PUNCT
cana-3815	4	1	e-mails:1gsm.maths@gmail.com	e-mails:1gsm.maths@gmail.com	PROPN
cana-3815	4	2	,	,	PUNCT
cana-3815	4	3	2lakshmipallavi.p@bvrit.ac.in	2lakshmipallavi.p@bvrit.ac.in	NUM
cana-3815	4	4	,	,	PUNCT
cana-3815	4	5	3arulmozhiems@gmail.com	3arulmozhiems@gmail.com	NUM
cana-3815	4	6	,	,	PUNCT
cana-3815	4	7	4aiyared.ia@up.ac.th	4aiyared.ia@up.ac.th	NUM
cana-3815	4	8	,	,	PUNCT
cana-3815	4	9	∗corresponding	∗corresponde	VERB
cana-3815	4	10	author	author	NOUN
cana-3815	4	11	:	:	PUNCT
cana-3815	4	12	aiyared	aiyared	PROPN
cana-3815	4	13	iampan	iampan	PROPN
cana-3815	4	14	.	.	PUNCT
cana-3815	5	1	received	receive	VERB
cana-3815	5	2	:	:	PUNCT
cana-3815	5	3	04	04	NUM
cana-3815	5	4	-	-	SYM
cana-3815	5	5	11	11	NUM
cana-3815	5	6	-	-	PUNCT
cana-3815	5	7	2024	2024	NUM
cana-3815	5	8	revised	revise	VERB
cana-3815	5	9	:	:	PUNCT
cana-3815	5	10	12	12	NUM
cana-3815	5	11	-	-	SYM
cana-3815	5	12	12	12	NUM
cana-3815	5	13	-	-	PUNCT
cana-3815	5	14	2024	2024	NUM
cana-3815	5	15	accepted	accept	VERB
cana-3815	5	16	:	:	PUNCT
cana-3815	5	17	01	01	NUM
cana-3815	5	18	-	-	SYM
cana-3815	5	19	01	01	NUM
cana-3815	5	20	-	-	PUNCT
cana-3815	5	21	2025	2025	NUM
cana-3815	5	22	.	.	PUNCT
cana-3815	6	1	abstract	abstract	ADP
cana-3815	6	2	the	the	DET
cana-3815	6	3	1pbid	1pbid	NUM
cana-3815	6	4	,	,	PUNCT
cana-3815	6	5	2pbid	2pbid	NUM
cana-3815	6	6	,	,	PUNCT
cana-3815	6	7	and	and	CCONJ
cana-3815	6	8	3pbid	3pbid	NUM
cana-3815	6	9	of	of	ADP
cana-3815	6	10	semirings	semiring	NOUN
cana-3815	6	11	are	be	AUX
cana-3815	6	12	introduced	introduce	VERB
cana-3815	6	13	.	.	PUNCT
cana-3815	7	1	additionally	additionally	ADV
cana-3815	7	2	,	,	PUNCT
cana-3815	7	3	we	we	PRON
cana-3815	7	4	communicate	communicate	VERB
cana-3815	7	5	with	with	ADP
cana-3815	7	6	a	a	DET
cana-3815	7	7	number	number	NOUN
cana-3815	7	8	of	of	ADP
cana-3815	7	9	the	the	DET
cana-3815	7	10	various	various	ADJ
cana-3815	7	11	spbis	spbis	ADJ
cana-3815	7	12	’	'	PUNCT
cana-3815	7	13	attributes	attribute	NOUN
cana-3815	7	14	.	.	PUNCT
cana-3815	8	1	2pbid	2pbid	NUM
cana-3815	8	2	and	and	CCONJ
cana-3815	8	3	3pbid	3pbid	NUM
cana-3815	8	4	are	be	AUX
cana-3815	8	5	generalizations	generalization	NOUN
cana-3815	8	6	of	of	ADP
cana-3815	8	7	1pbid	1pbid	NUM
cana-3815	8	8	and	and	CCONJ
cana-3815	8	9	2pbid	2pbid	NUM
cana-3815	8	10	,	,	PUNCT
cana-3815	8	11	respectively	respectively	ADV
cana-3815	8	12	,	,	PUNCT
cana-3815	8	13	which	which	PRON
cana-3815	8	14	we	we	PRON
cana-3815	8	15	explain	explain	VERB
cana-3815	8	16	.	.	PUNCT
cana-3815	9	1	we	we	PRON
cana-3815	9	2	go	go	VERB
cana-3815	9	3	over	over	ADP
cana-3815	9	4	the	the	DET
cana-3815	9	5	m1	m1	NOUN
cana-3815	9	6	,	,	PUNCT
cana-3815	9	7	m2	m2	PROPN
cana-3815	9	8	and	and	CCONJ
cana-3815	9	9	m3	m3	PROPN
cana-3815	9	10	-	-	PUNCT
cana-3815	9	11	systems	system	NOUN
cana-3815	9	12	and	and	CCONJ
cana-3815	9	13	bi	bi	PROPN
cana-3815	9	14	generator	generator	NOUN
cana-3815	9	15	.	.	PUNCT
cana-3815	10	1	it	it	PRON
cana-3815	10	2	is	be	AUX
cana-3815	10	3	possible	possible	ADJ
cana-3815	10	4	to	to	PART
cana-3815	10	5	generalize	generalize	VERB
cana-3815	10	6	the	the	DET
cana-3815	10	7	m1	m1	NOUN
cana-3815	10	8	-	-	PUNCT
cana-3815	10	9	system	system	NOUN
cana-3815	10	10	to	to	ADP
cana-3815	10	11	the	the	DET
cana-3815	10	12	m2	m2	NOUN
cana-3815	10	13	-	-	PUNCT
cana-3815	10	14	system	system	NOUN
cana-3815	10	15	and	and	CCONJ
cana-3815	10	16	the	the	DET
cana-3815	10	17	m2	m2	NOUN
cana-3815	10	18	-	-	PUNCT
cana-3815	10	19	system	system	NOUN
cana-3815	10	20	to	to	ADP
cana-3815	10	21	the	the	DET
cana-3815	10	22	m3	m3	NOUN
cana-3815	10	23	-	-	PUNCT
cana-3815	10	24	system	system	NOUN
cana-3815	10	25	.	.	PUNCT
cana-3815	11	1	keywords	keyword	NOUN
cana-3815	11	2	:	:	PUNCT
cana-3815	11	3	1pbid	1pbid	NUM
cana-3815	11	4	;	;	PUNCT
cana-3815	11	5	2pbid	2pbid	NUM
cana-3815	11	6	;	;	PUNCT
cana-3815	11	7	3pbid	3pbid	NUM
cana-3815	11	8	;	;	PUNCT
cana-3815	11	9	m1	m1	NOUN
cana-3815	11	10	-	-	PUNCT
cana-3815	11	11	system	system	NOUN
cana-3815	11	12	,	,	PUNCT
cana-3815	11	13	m2	m2	NOUN
cana-3815	11	14	-	-	PUNCT
cana-3815	11	15	system	system	NOUN
cana-3815	11	16	and	and	CCONJ
cana-3815	11	17	m3	m3	NOUN
cana-3815	11	18	-	-	PUNCT
cana-3815	11	19	system	system	NOUN
cana-3815	11	20	.	.	PUNCT
cana-3815	12	1	1	1	NUM
cana-3815	12	2	introduction	introduction	NOUN
cana-3815	12	3	numerous	numerous	ADJ
cana-3815	12	4	studies	study	NOUN
cana-3815	12	5	have	have	AUX
cana-3815	12	6	looked	look	VERB
cana-3815	12	7	at	at	ADP
cana-3815	12	8	different	different	ADJ
cana-3815	12	9	kinds	kind	NOUN
cana-3815	12	10	of	of	ADP
cana-3815	12	11	ideals	ideal	NOUN
cana-3815	12	12	in	in	ADP
cana-3815	12	13	mathematical	mathematical	ADJ
cana-3815	12	14	structures	structure	NOUN
cana-3815	12	15	,	,	PUNCT
cana-3815	12	16	such	such	ADJ
cana-3815	12	17	as	as	ADP
cana-3815	12	18	rings	ring	NOUN
cana-3815	12	19	and	and	CCONJ
cana-3815	12	20	semirings,3	semirings,3	NOUN
cana-3815	12	21	,	,	PUNCT
cana-3815	12	22	4	4	NUM
cana-3815	12	23	respectively	respectively	ADV
cana-3815	12	24	.	.	PUNCT
cana-3815	13	1	associative	associative	PROPN
cana-3815	13	2	rings	ring	NOUN
cana-3815	13	3	made	make	VERB
cana-3815	13	4	up	up	ADP
cana-3815	13	5	the	the	DET
cana-3815	13	6	notion	notion	NOUN
cana-3815	13	7	of	of	ADP
cana-3815	13	8	ideals	ideal	NOUN
cana-3815	13	9	that	that	PRON
cana-3815	13	10	dedekind	dedekind	NOUN
cana-3815	13	11	introduced	introduce	VERB
cana-3815	13	12	into	into	ADP
cana-3815	13	13	the	the	DET
cana-3815	13	14	theory	theory	NOUN
cana-3815	13	15	of	of	ADP
cana-3815	13	16	algebraic	algebraic	ADJ
cana-3815	13	17	numbers	number	NOUN
cana-3815	13	18	.	.	PUNCT
cana-3815	14	1	in	in	ADP
cana-3815	14	2	this	this	DET
cana-3815	14	3	approach	approach	NOUN
cana-3815	14	4	the	the	DET
cana-3815	14	5	notion	notion	NOUN
cana-3815	14	6	was	be	AUX
cana-3815	14	7	expanded	expand	VERB
cana-3815	14	8	to	to	PART
cana-3815	14	9	include	include	VERB
cana-3815	14	10	algebraic	algebraic	ADJ
cana-3815	14	11	numbers	number	NOUN
cana-3815	14	12	.	.	PUNCT
cana-3815	15	1	moreover	moreover	ADV
cana-3815	15	2	,	,	PUNCT
cana-3815	15	3	it	it	PRON
cana-3815	15	4	is	be	AUX
cana-3815	15	5	a	a	DET
cana-3815	15	6	particular	particular	ADJ
cana-3815	15	7	case	case	NOUN
cana-3815	15	8	of	of	ADP
cana-3815	15	9	lajos	lajos	NOUN
cana-3815	15	10	(	(	PUNCT
cana-3815	15	11	m	m	PROPN
cana-3815	15	12	,	,	PUNCT
cana-3815	15	13	n)-ideal	n)-ideal	NOUN
cana-3815	15	14	.	.	PUNCT
cana-3815	16	1	in	in	ADP
cana-3815	16	2	order	order	NOUN
cana-3815	16	3	to	to	PART
cana-3815	16	4	analyze	analyze	VERB
cana-3815	16	5	regular	regular	ADJ
cana-3815	16	6	and	and	CCONJ
cana-3815	16	7	intra	intra	ADJ
cana-3815	16	8	-	-	ADJ
cana-3815	16	9	regular	regular	ADJ
cana-3815	16	10	semigroups	semigroup	NOUN
cana-3815	16	11	,	,	PUNCT
cana-3815	16	12	lajos	lajos	NOUN
cana-3815	16	13	used	use	VERB
cana-3815	16	14	generalized	generalized	ADJ
cana-3815	16	15	bids	bid	NOUN
cana-3815	16	16	and	and	CCONJ
cana-3815	16	17	quasi	quasi	NOUN
cana-3815	16	18	-	-	NOUN
cana-3815	16	19	ideals	ideal	NOUN
cana-3815	16	20	.	.	PUNCT
cana-3815	17	1	to	to	ADP
cana-3815	17	2	reference5	reference5	NOUN
cana-3815	17	3	while	while	SCONJ
cana-3815	17	4	discussing	discuss	VERB
cana-3815	17	5	different	different	ADJ
cana-3815	17	6	types	type	NOUN
cana-3815	17	7	of	of	ADP
cana-3815	17	8	semigroups	semigroup	NOUN
cana-3815	17	9	is	be	AUX
cana-3815	17	10	a	a	DET
cana-3815	17	11	bid	bid	NOUN
cana-3815	17	12	.	.	PUNCT
cana-3815	18	1	the	the	DET
cana-3815	18	2	associative	associative	ADJ
cana-3815	18	3	rings	ring	NOUN
cana-3815	18	4	are	be	AUX
cana-3815	18	5	somewhat	somewhat	ADV
cana-3815	18	6	arbitrary	arbitrary	ADJ
cana-3815	18	7	,	,	PUNCT
cana-3815	18	8	however	however	ADV
cana-3815	18	9	they	they	PRON
cana-3815	18	10	are	be	AUX
cana-3815	18	11	described	describe	VERB
cana-3815	18	12	in	in	ADP
cana-3815	18	13	terms	term	NOUN
cana-3815	18	14	of	of	ADP
cana-3815	18	15	bids	bid	NOUN
cana-3815	18	16	.	.	PUNCT
cana-3815	19	1	it	it	PRON
cana-3815	19	2	would	would	AUX
cana-3815	19	3	be	be	AUX
cana-3815	19	4	almost	almost	ADV
cana-3815	19	5	perfect	perfect	ADJ
cana-3815	19	6	to	to	PART
cana-3815	19	7	expand	expand	VERB
cana-3815	19	8	lis	li	NOUN
cana-3815	19	9	and	and	CCONJ
cana-3815	19	10	ris	ris	PROPN
cana-3815	19	11	,	,	PUNCT
cana-3815	19	12	which	which	PRON
cana-3815	19	13	are	be	AUX
cana-3815	19	14	specific	specific	ADJ
cana-3815	19	15	types	type	NOUN
cana-3815	19	16	of	of	ADP
cana-3815	19	17	bids	bid	NOUN
cana-3815	19	18	.	.	PUNCT
cana-3815	20	1	semigroups	semigroup	NOUN
cana-3815	20	2	and	and	CCONJ
cana-3815	20	3	rings	ring	NOUN
cana-3815	20	4	now	now	ADV
cana-3815	20	5	referred	refer	VERB
cana-3815	20	6	to	to	ADP
cana-3815	20	7	as	as	SCONJ
cana-3815	20	8	quasi	quasi	ADJ
cana-3815	20	9	ideals	ideal	NOUN
cana-3815	20	10	were	be	AUX
cana-3815	20	11	introduced	introduce	VERB
cana-3815	20	12	by	by	ADP
cana-3815	20	13	otto	otto	PROPN
cana-3815	20	14	steinfeld	steinfeld	PROPN
cana-3815	20	15	.	.	PUNCT
cana-3815	21	1	quote3	quote3	PROPN
cana-3815	21	2	states	state	VERB
cana-3815	21	3	that	that	SCONJ
cana-3815	21	4	semirings	semiring	NOUN
cana-3815	21	5	provide	provide	VERB
cana-3815	21	6	a	a	DET
cana-3815	21	7	range	range	NOUN
cana-3815	21	8	of	of	ADP
cana-3815	21	9	methods	method	NOUN
cana-3815	21	10	for	for	ADP
cana-3815	21	11	elucidating	elucidate	VERB
cana-3815	21	12	prime	prime	ADJ
cana-3815	21	13	ideals	ideal	NOUN
cana-3815	21	14	.	.	PUNCT
cana-3815	22	1	commutative	commutative	ADJ
cana-3815	22	2	ring	ring	NOUN
cana-3815	22	3	theory	theory	NOUN
cana-3815	22	4	has	have	AUX
cana-3815	22	5	been	be	AUX
cana-3815	22	6	extensively	extensively	ADV
cana-3815	22	7	influenced	influence	VERB
cana-3815	22	8	by	by	ADP
cana-3815	22	9	prime	prime	ADJ
cana-3815	22	10	ideal	ideal	PROPN
cana-3815	22	11	theory	theory	NOUN
cana-3815	22	12	.	.	PUNCT
cana-3815	23	1	it	it	PRON
cana-3815	23	2	has	have	AUX
cana-3815	23	3	been	be	AUX
cana-3815	23	4	less	less	ADV
cana-3815	23	5	frequently	frequently	ADV
cana-3815	23	6	used	use	VERB
cana-3815	23	7	for	for	ADP
cana-3815	23	8	non	non	ADJ
cana-3815	23	9	-	-	ADJ
cana-3815	23	10	commutative	commutative	ADJ
cana-3815	23	11	rings	ring	NOUN
cana-3815	23	12	than	than	ADP
cana-3815	23	13	for	for	ADP
cana-3815	23	14	commutative	commutative	ADJ
cana-3815	23	15	rings	ring	NOUN
cana-3815	23	16	.	.	PUNCT
cana-3815	24	1	palanikumar	palanikumar	PROPN
cana-3815	24	2	et	et	NOUN
cana-3815	24	3	al.6	al.6	PROPN
cana-3815	24	4	investigated	investigate	VERB
cana-3815	24	5	different	different	ADJ
cana-3815	24	6	prime	prime	ADJ
cana-3815	24	7	partial	partial	ADJ
cana-3815	24	8	bis	bis	NOUN
cana-3815	24	9	in	in	ADP
cana-3815	24	10	non	non	ADJ
cana-3815	24	11	-	-	ADJ
cana-3815	24	12	commutative	commutative	ADJ
cana-3815	24	13	partial	partial	ADJ
cana-3815	24	14	rings	ring	NOUN
cana-3815	24	15	.	.	PUNCT
cana-3815	25	1	walt9	walt9	PROPN
cana-3815	25	2	studied	study	VERB
cana-3815	25	3	the	the	DET
cana-3815	25	4	prime	prime	ADJ
cana-3815	25	5	and	and	CCONJ
cana-3815	25	6	semiprimate	semiprimate	ADJ
cana-3815	25	7	bis	bis	NOUN
cana-3815	25	8	of	of	ADP
cana-3815	25	9	associative	associative	ADJ
cana-3815	25	10	rings	ring	NOUN
cana-3815	25	11	with	with	ADP
cana-3815	25	12	unity	unity	NOUN
cana-3815	25	13	.	.	PUNCT
cana-3815	26	1	associative	associative	ADJ
cana-3815	26	2	rings	ring	NOUN
cana-3815	26	3	without	without	ADP
cana-3815	26	4	unity	unity	NOUN
cana-3815	26	5	were	be	AUX
cana-3815	26	6	extended	extend	VERB
cana-3815	26	7	to	to	ADP
cana-3815	26	8	prime	prime	ADJ
cana-3815	26	9	and	and	CCONJ
cana-3815	26	10	semiprime	semiprime	NOUN
cana-3815	26	11	bis	bi	VERB
cana-3815	26	12	by	by	ADP
cana-3815	26	13	roux.10	roux.10	NOUN
cana-3815	26	14	some	some	DET
cana-3815	26	15	descriptions	description	NOUN
cana-3815	26	16	of	of	ADP
cana-3815	26	17	bis	bis	NOUN
cana-3815	26	18	in	in	ADP
cana-3815	26	19	basic	basic	ADJ
cana-3815	26	20	semirings	semiring	NOUN
cana-3815	26	21	were	be	AUX
cana-3815	26	22	given	give	VERB
cana-3815	26	23	by	by	ADP
cana-3815	26	24	flaska	flaska	ADJ
cana-3815	26	25	et	et	PROPN
cana-3815	26	26	al.11	al.11	PROPN
cana-3815	26	27	atani12	atani12	NOUN
cana-3815	26	28	also	also	ADV
cana-3815	26	29	provided	provide	VERB
cana-3815	26	30	some	some	DET
cana-3815	26	31	results	result	NOUN
cana-3815	26	32	for	for	ADP
cana-3815	26	33	the	the	DET
cana-3815	26	34	ideal	ideal	ADJ
cana-3815	26	35	theory	theory	NOUN
cana-3815	26	36	of	of	ADP
cana-3815	26	37	commutative	commutative	ADJ
cana-3815	26	38	semirings	semiring	NOUN
cana-3815	26	39	with	with	ADP
cana-3815	26	40	non	non	ADJ
cana-3815	26	41	-	-	ADJ
cana-3815	26	42	zero	zero	NUM
cana-3815	26	43	identities	identity	NOUN
cana-3815	26	44	.	.	PUNCT
cana-3815	27	1	mccoy	mccoy	PROPN
cana-3815	27	2	gives	give	VERB
cana-3815	27	3	some	some	DET
cana-3815	27	4	details	detail	NOUN
cana-3815	27	5	on	on	ADP
cana-3815	27	6	prime	prime	ADJ
cana-3815	27	7	ideals	ideal	NOUN
cana-3815	27	8	in	in	ADP
cana-3815	27	9	general	general	ADJ
cana-3815	27	10	rings.4	rings.4	PROPN
cana-3815	27	11	information	information	NOUN
cana-3815	27	12	on	on	ADP
cana-3815	27	13	the	the	DET
cana-3815	27	14	pid	pid	NOUN
cana-3815	27	15	for	for	ADP
cana-3815	27	16	rings	ring	NOUN
cana-3815	27	17	and	and	CCONJ
cana-3815	27	18	semirings	semiring	NOUN
cana-3815	27	19	was	be	AUX
cana-3815	27	20	given	give	VERB
cana-3815	27	21	in.3	in.3	PROPN
cana-3815	27	22	,	,	PUNCT
cana-3815	27	23	13	13	NUM
cana-3815	27	24	,	,	PUNCT
cana-3815	27	25	14	14	NUM
cana-3815	27	26	van	van	PROPN
cana-3815	27	27	der	der	PROPN
cana-3815	27	28	walt	walt	PROPN
cana-3815	27	29	coined	coin	VERB
cana-3815	27	30	the	the	DET
cana-3815	27	31	words	word	NOUN
cana-3815	27	32	prime	prime	ADJ
cana-3815	27	33	bid	bid	NOUN
cana-3815	27	34	and	and	CCONJ
cana-3815	27	35	semiprime	semiprime	NOUN
cana-3815	27	36	bid.9	bid.9	PROPN
cana-3815	27	37	the	the	DET
cana-3815	27	38	subsets	subset	NOUN
cana-3815	27	39	x1	x1	PROPN
cana-3815	27	40	and	and	CCONJ
cana-3815	27	41	x2	x2	PROPN
cana-3815	27	42	of	of	ADP
cana-3815	27	43	§	§	PROPN
cana-3815	27	44	and	and	CCONJ
cana-3815	27	45	the	the	DET
cana-3815	27	46	product	product	NOUN
cana-3815	27	47	x1	x1	PROPN
cana-3815	27	48	·	·	PUNCT
cana-3815	28	1	x2	x2	PRON
cana-3815	28	2	may	may	AUX
cana-3815	28	3	be	be	AUX
cana-3815	28	4	understood	understand	VERB
cana-3815	28	5	as	as	SCONJ
cana-3815	28	6	follows	follow	VERB
cana-3815	28	7	:	:	PUNCT
cana-3815	28	8	the	the	DET
cana-3815	28	9	subring	subring	NOUN
cana-3815	28	10	of	of	ADP
cana-3815	28	11	§	§	PROPN
cana-3815	28	12	is	be	AUX
cana-3815	28	13	produced	produce	VERB
cana-3815	28	14	by	by	ADP
cana-3815	28	15	the	the	DET
cana-3815	28	16	set	set	NOUN
cana-3815	28	17	of	of	ADP
cana-3815	28	18	all	all	DET
cana-3815	28	19	products	product	NOUN
cana-3815	29	1	x1	x1	PROPN
cana-3815	29	2	·	·	PUNCT
cana-3815	29	3	x2	x2	INTJ
cana-3815	29	4	,	,	PUNCT
cana-3815	29	5	where	where	SCONJ
cana-3815	29	6	x1	x1	PROPN
cana-3815	29	7	∈	∈	PROPN
cana-3815	29	8	x1	x1	PROPN
cana-3815	29	9	,	,	PUNCT
cana-3815	29	10	x2	x2	PROPN
cana-3815	29	11	∈	∈	PROPN
cana-3815	29	12	x2	x2	PROPN
cana-3815	29	13	.	.	PUNCT
cana-3815	30	1	a	a	DET
cana-3815	30	2	bid	bid	NOUN
cana-3815	30	3	a1	a1	NOUN
cana-3815	30	4	of	of	ADP
cana-3815	30	5	a	a	DET
cana-3815	30	6	ring	ring	NOUN
cana-3815	30	7	§	§	NOUN
cana-3815	30	8	is	be	AUX
cana-3815	30	9	defined	define	VERB
cana-3815	30	10	as	as	ADP
cana-3815	30	11	a	a	DET
cana-3815	30	12	subring	subre	VERB
cana-3815	30	13	a1	a1	NOUN
cana-3815	30	14	of	of	ADP
cana-3815	30	15	§	§	PROPN
cana-3815	30	16	that	that	SCONJ
cana-3815	30	17	satisfies	satisfy	VERB
cana-3815	30	18	a1§a1	a1§a1	NOUN
cana-3815	30	19	⊆	⊆	NUM
cana-3815	30	20	a1	a1	NOUN
cana-3815	30	21	.	.	PUNCT
cana-3815	31	1	for	for	ADP
cana-3815	31	2	ideals	ideal	NOUN
cana-3815	31	3	a	a	PRON
cana-3815	31	4	and	and	CCONJ
cana-3815	31	5	a1	a1	NOUN
cana-3815	31	6	of	of	ADP
cana-3815	31	7	§	§	PROPN
cana-3815	31	8	,	,	PUNCT
cana-3815	31	9	then	then	ADV
cana-3815	31	10	a	a	DET
cana-3815	31	11	⊆	⊆	NUM
cana-3815	31	12	ℵ	ℵ	NOUN
cana-3815	31	13	or	or	CCONJ
cana-3815	31	14	a1	a1	VERB
cana-3815	31	15	⊆	⊆	NUM
cana-3815	31	16	ℵ	ℵ	NOUN
cana-3815	31	17	means	mean	NOUN
cana-3815	31	18	that	that	SCONJ
cana-3815	31	19	an	an	DET
cana-3815	31	20	i	i	PROPN
cana-3815	31	21	d	d	NOUN
cana-3815	31	22	ℵ	ℵ	NOUN
cana-3815	31	23	of	of	ADP
cana-3815	31	24	a	a	DET
cana-3815	31	25	ring	ring	NOUN
cana-3815	31	26	§	§	NOUN
cana-3815	31	27	is	be	AUX
cana-3815	31	28	pid	pid	NOUN
cana-3815	32	1	if	if	SCONJ
cana-3815	32	2	and	and	CCONJ
cana-3815	32	3	only	only	ADV
cana-3815	32	4	if	if	SCONJ
cana-3815	32	5	whenever	whenever	SCONJ
cana-3815	32	6	aa1	aa1	PROPN
cana-3815	32	7	⊆	⊆	NUM
cana-3815	32	8	ℵ.4	ℵ.4	ADP
cana-3815	32	9	each	each	PRON
cana-3815	32	10	of	of	ADP
cana-3815	32	11	the	the	DET
cana-3815	32	12	five	five	NUM
cana-3815	32	13	components	component	NOUN
cana-3815	32	14	that	that	PRON
cana-3815	32	15	make	make	VERB
cana-3815	32	16	up	up	ADP
cana-3815	32	17	this	this	DET
cana-3815	32	18	document	document	NOUN
cana-3815	32	19	is	be	AUX
cana-3815	32	20	arranged	arrange	VERB
cana-3815	32	21	differently	differently	ADV
cana-3815	32	22	.	.	PUNCT
cana-3815	33	1	in	in	ADP
cana-3815	33	2	section	section	NOUN
cana-3815	33	3	2	2	NUM
cana-3815	33	4	,	,	PUNCT
cana-3815	33	5	we	we	PRON
cana-3815	33	6	address	address	VERB
cana-3815	33	7	the	the	DET
cana-3815	33	8	various	various	ADJ
cana-3815	33	9	kinds	kind	NOUN
cana-3815	33	10	of	of	ADP
cana-3815	33	11	main	main	ADJ
cana-3815	33	12	bids	bid	NOUN
cana-3815	33	13	and	and	CCONJ
cana-3815	33	14	their	their	PRON
cana-3815	33	15	expansions	expansion	NOUN
cana-3815	33	16	.	.	PUNCT
cana-3815	34	1	section	section	NOUN
cana-3815	34	2	3	3	NUM
cana-3815	34	3	offers	offer	VERB
cana-3815	34	4	a	a	DET
cana-3815	34	5	discussion	discussion	NOUN
cana-3815	34	6	of	of	ADP
cana-3815	34	7	the	the	DET
cana-3815	34	8	semiprimary	semiprimary	ADJ
cana-3815	34	9	bids	bid	NOUN
cana-3815	34	10	.	.	PUNCT
cana-3815	35	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	35	2	755	755	NUM
cana-3815	35	3	communications	communication	NOUN
cana-3815	35	4	on	on	ADP
cana-3815	35	5	applied	apply	VERB
cana-3815	35	6	nonlinear	nonlinear	ADJ
cana-3815	35	7	analysis	analysis	NOUN
cana-3815	35	8	issn	issn	NOUN
cana-3815	35	9	:	:	PUNCT
cana-3815	35	10	1074	1074	NUM
cana-3815	35	11	-	-	PUNCT
cana-3815	35	12	133x	133x	NUM
cana-3815	35	13	vol	vol	NOUN
cana-3815	35	14	32	32	NUM
cana-3815	35	15	no	no	NOUN
cana-3815	35	16	.	.	NOUN
cana-3815	35	17	3	3	NUM
cana-3815	35	18	(	(	PUNCT
cana-3815	35	19	2025	2025	NUM
cana-3815	35	20	)	)	PUNCT
cana-3815	35	21	list	list	NOUN
cana-3815	35	22	of	of	ADP
cana-3815	35	23	abbreviations	abbreviation	NOUN
cana-3815	35	24	rid	rid	VERB
cana-3815	35	25	right	right	ADJ
cana-3815	35	26	ideal	ideal	ADJ
cana-3815	35	27	lid	lid	NOUN
cana-3815	35	28	left	leave	VERB
cana-3815	35	29	ideal	ideal	NOUN
cana-3815	36	1	i	i	PROPN
cana-3815	36	2	d	d	PROPN
cana-3815	36	3	ideal	ideal	PROPN
cana-3815	36	4	primary	primary	ADJ
cana-3815	36	5	bid	bid	NOUN
cana-3815	36	6	primary	primary	ADJ
cana-3815	36	7	bi	bi	ADJ
cana-3815	36	8	-	-	ADJ
cana-3815	36	9	ideal	ideal	ADJ
cana-3815	36	10	primary	primary	NOUN
cana-3815	36	11	i	i	PROPN
cana-3815	36	12	d	d	PROPN
cana-3815	36	13	primary	primary	ADJ
cana-3815	36	14	ideal	ideal	NOUN
cana-3815	36	15	tid	tid	PROPN
cana-3815	36	16	two	two	NUM
cana-3815	36	17	sided	sided	ADJ
cana-3815	36	18	ideal	ideal	NOUN
cana-3815	36	19	semi	semi	ADV
cana-3815	36	20	primary	primary	ADJ
cana-3815	36	21	bid	bid	NOUN
cana-3815	36	22	semi	semi	ADP
cana-3815	36	23	primary	primary	ADJ
cana-3815	36	24	bi	bi	ADJ
cana-3815	36	25	-	-	ADJ
cana-3815	36	26	ideal	ideal	ADJ
cana-3815	36	27	semi	semi	ADJ
cana-3815	36	28	primary	primary	ADJ
cana-3815	36	29	i	i	PROPN
cana-3815	36	30	d	d	PROPN
cana-3815	36	31	semi	semi	ADV
cana-3815	36	32	primary	primary	ADJ
cana-3815	36	33	ideal	ideal	ADJ
cana-3815	36	34	2	2	NUM
cana-3815	36	35	characterization	characterization	NOUN
cana-3815	36	36	of	of	ADP
cana-3815	36	37	pbids	pbid	NOUN
cana-3815	36	38	definition	definition	NOUN
cana-3815	36	39	2.1	2.1	NUM
cana-3815	36	40	.	.	PUNCT
cana-3815	37	1	a	a	DET
cana-3815	37	2	bid	bid	NOUN
cana-3815	37	3	ℵ	ℵ	NOUN
cana-3815	37	4	of	of	ADP
cana-3815	37	5	§	§	PROPN
cana-3815	37	6	is	be	AUX
cana-3815	37	7	said	say	VERB
cana-3815	37	8	to	to	PART
cana-3815	37	9	be	be	AUX
cana-3815	37	10	(	(	PUNCT
cana-3815	37	11	1	1	NUM
cana-3815	37	12	)	)	PUNCT
cana-3815	37	13	1pbid	1pbid	NUM
cana-3815	37	14	if	if	SCONJ
cana-3815	37	15	a1a2	a1a2	ADP
cana-3815	37	16	⊆	⊆	NUM
cana-3815	37	17	ℵ	ℵ	NOUN
cana-3815	37	18	implies	implie	NOUN
cana-3815	37	19	a1	a1	VERB
cana-3815	37	20	⊆	⊆	NUM
cana-3815	37	21	ℵ	ℵ	NOUN
cana-3815	37	22	or	or	CCONJ
cana-3815	37	23	a2	a2	PROPN
cana-3815	37	24	⊆	⊆	NUM
cana-3815	37	25	√	√	NUM
cana-3815	37	26	ℵ	ℵ	NOUN
cana-3815	37	27	for	for	ADP
cana-3815	37	28	any	any	DET
cana-3815	37	29	bids	bid	NOUN
cana-3815	37	30	a1	a1	NOUN
cana-3815	37	31	and	and	CCONJ
cana-3815	37	32	a2	a2	PROPN
cana-3815	37	33	of	of	ADP
cana-3815	37	34	§	§	PROPN
cana-3815	37	35	.	.	PUNCT
cana-3815	38	1	(	(	PUNCT
cana-3815	38	2	ii	ii	NOUN
cana-3815	38	3	)	)	PUNCT
cana-3815	38	4	2pbid	2pbid	NUM
cana-3815	38	5	if	if	SCONJ
cana-3815	38	6	∝	∝	PROPN
cana-3815	38	7	§	§	VERB
cana-3815	38	8	℘	℘	VERB
cana-3815	38	9	⊆	⊆	NUM
cana-3815	38	10	ℵ	ℵ	NOUN
cana-3815	38	11	implies	implie	NOUN
cana-3815	38	12	∝∈	∝∈	PUNCT
cana-3815	38	13	ℵ	ℵ	ADP
cana-3815	38	14	or	or	CCONJ
cana-3815	38	15	℘	℘	PROPN
cana-3815	38	16	∈	∈	NOUN
cana-3815	38	17	√	√	ADJ
cana-3815	38	18	ℵ	ℵ	NOUN
cana-3815	38	19	.	.	PUNCT
cana-3815	39	1	(	(	PUNCT
cana-3815	39	2	iii	iii	X
cana-3815	39	3	)	)	PUNCT
cana-3815	39	4	3pbid	3pbid	NUM
cana-3815	39	5	if	if	SCONJ
cana-3815	39	6	£	£	SYM
cana-3815	39	7	1£2	1£2	NUM
cana-3815	39	8	⊆	⊆	NUM
cana-3815	39	9	ℵ	ℵ	NOUN
cana-3815	39	10	implies	imply	VERB
cana-3815	39	11	£	£	SYM
cana-3815	39	12	1	1	NUM
cana-3815	39	13	⊆	⊆	NUM
cana-3815	39	14	ℵ	ℵ	NOUN
cana-3815	39	15	or	or	CCONJ
cana-3815	39	16	£	£	SYM
cana-3815	39	17	2	2	NUM
cana-3815	39	18	⊆	⊆	NUM
cana-3815	39	19	√	√	NUM
cana-3815	39	20	ℵ	ℵ	NOUN
cana-3815	39	21	for	for	ADP
cana-3815	39	22	any	any	DET
cana-3815	39	23	ids	id	NOUN
cana-3815	39	24	£	£	SYM
cana-3815	39	25	1	1	NUM
cana-3815	39	26	and	and	CCONJ
cana-3815	39	27	£	£	SYM
cana-3815	39	28	2	2	NUM
cana-3815	39	29	of	of	ADP
cana-3815	39	30	§	§	PROPN
cana-3815	39	31	.	.	PUNCT
cana-3815	40	1	theorem	theorem	VERB
cana-3815	40	2	2.2	2.2	NUM
cana-3815	40	3	.	.	PUNCT
cana-3815	41	1	every	every	DET
cana-3815	41	2	1pbid	1pbid	PROPN
cana-3815	41	3	is	be	AUX
cana-3815	41	4	a	a	DET
cana-3815	41	5	2pbid	2pbid	NUM
cana-3815	41	6	.	.	PUNCT
cana-3815	42	1	proof	proof	NOUN
cana-3815	42	2	.	.	PUNCT
cana-3815	43	1	let	let	VERB
cana-3815	43	2	ℵ	ℵ	NOUN
cana-3815	43	3	be	be	AUX
cana-3815	43	4	an	an	DET
cana-3815	43	5	1pbid	1pbid	NUM
cana-3815	43	6	of	of	ADP
cana-3815	43	7	§	§	PROPN
cana-3815	43	8	.	.	PUNCT
cana-3815	44	1	let	let	VERB
cana-3815	44	2	∝	∝	PRON
cana-3815	44	3	,	,	PUNCT
cana-3815	44	4	℘	℘	PROPN
cana-3815	44	5	∈	∈	PROPN
cana-3815	44	6	§	§	PROPN
cana-3815	44	7	and	and	CCONJ
cana-3815	44	8	∝	∝	PROPN
cana-3815	44	9	§	§	PROPN
cana-3815	44	10	℘	℘	PROPN
cana-3815	44	11	⊆	⊆	NUM
cana-3815	44	12	ℵ.	ℵ.	NOUN
cana-3815	44	13	now	now	ADV
cana-3815	44	14	,	,	PUNCT
cana-3815	44	15	(	(	PUNCT
cana-3815	44	16	∝	∝	PROPN
cana-3815	44	17	§	§	PROPN
cana-3815	44	18	)	)	PUNCT
cana-3815	44	19	·	·	PUNCT
cana-3815	45	1	(	(	PUNCT
cana-3815	45	2	§	§	NOUN
cana-3815	45	3	℘	℘	NOUN
cana-3815	45	4	)	)	PUNCT
cana-3815	45	5	⊆∝	⊆∝	ADP
cana-3815	45	6	§	§	PROPN
cana-3815	45	7	℘	℘	NUM
cana-3815	45	8	⊆	⊆	NUM
cana-3815	45	9	ℵ	ℵ	NOUN
cana-3815	45	10	,	,	PUNCT
cana-3815	45	11	since	since	SCONJ
cana-3815	45	12	∝	∝	PROPN
cana-3815	45	13	§	§	PROPN
cana-3815	45	14	and	and	CCONJ
cana-3815	45	15	§	§	PROPN
cana-3815	45	16	℘	℘	PROPN
cana-3815	45	17	are	be	AUX
cana-3815	45	18	bids	bid	NOUN
cana-3815	45	19	.	.	PUNCT
cana-3815	46	1	hence	hence	ADV
cana-3815	46	2	∝	∝	PROPN
cana-3815	46	3	§	§	PROPN
cana-3815	46	4	⊆	⊆	NUM
cana-3815	46	5	ℵ	ℵ	NOUN
cana-3815	46	6	or	or	CCONJ
cana-3815	46	7	§	§	VERB
cana-3815	46	8	℘	℘	NUM
cana-3815	46	9	⊆	⊆	SYM
cana-3815	46	10	√	√	NOUN
cana-3815	46	11	ℵ.	ℵ.	PROPN
cana-3815	46	12	suppose	suppose	VERB
cana-3815	46	13	that	that	SCONJ
cana-3815	46	14	∝	∝	PROPN
cana-3815	46	15	§	§	PROPN
cana-3815	46	16	⊆	⊆	NUM
cana-3815	46	17	ℵ.	ℵ.	NOUN
cana-3815	46	18	consider	consider	VERB
cana-3815	46	19	<	<	PRON
cana-3815	46	20	∝>b	∝>b	NOUN
cana-3815	46	21	·	·	PUNCT
cana-3815	47	1	<	<	X
cana-3815	47	2	∝>b⊆∝	∝>b⊆∝	PROPN
cana-3815	47	3	§	§	PROPN
cana-3815	47	4	⊆	⊆	NUM
cana-3815	47	5	ℵ.	ℵ.	NOUN
cana-3815	47	6	then	then	ADV
cana-3815	47	7	∝∈	∝∈	PUNCT
cana-3815	47	8	ℵ.	ℵ.	PROPN
cana-3815	47	9	similarly	similarly	ADV
cana-3815	47	10	if	if	SCONJ
cana-3815	47	11	§	§	NOUN
cana-3815	47	12	℘	℘	VERB
cana-3815	47	13	⊆	⊆	SYM
cana-3815	47	14	√	√	NUM
cana-3815	47	15	ℵ	ℵ	NOUN
cana-3815	47	16	then	then	ADV
cana-3815	47	17	℘	℘	PROPN
cana-3815	47	18	∈	∈	NOUN
cana-3815	47	19	√	√	NOUN
cana-3815	47	20	ℵ.	ℵ.	PUNCT
cana-3815	48	1	thus	thus	ADV
cana-3815	48	2	ℵ	ℵ	NOUN
cana-3815	48	3	is	be	AUX
cana-3815	48	4	a	a	DET
cana-3815	48	5	2pbid	2pbid	NUM
cana-3815	48	6	of	of	ADP
cana-3815	48	7	§	§	PROPN
cana-3815	48	8	.	.	PUNCT
cana-3815	49	1	converse	converse	NOUN
cana-3815	49	2	is	be	AUX
cana-3815	49	3	does	do	AUX
cana-3815	49	4	not	not	PART
cana-3815	49	5	hold	hold	VERB
cana-3815	49	6	.	.	PUNCT
cana-3815	50	1	example	example	NOUN
cana-3815	50	2	2.3	2.3	NUM
cana-3815	50	3	.	.	PUNCT
cana-3815	51	1	consider	consider	VERB
cana-3815	51	2	the	the	DET
cana-3815	51	3	semiring	semire	VERB
cana-3815	51	4	§	§	PROPN
cana-3815	51	5	=	=	SYM
cana-3815	51	6	d2(z2	d2(z2	PROPN
cana-3815	51	7	)	)	PUNCT
cana-3815	51	8	and	and	CCONJ
cana-3815	51	9	ℵ	ℵ	X
cana-3815	51	10	=	=	SYM
cana-3815	51	11	{	{	PUNCT
cana-3815	51	12	(	(	PUNCT
cana-3815	51	13	0	0	NUM
cana-3815	51	14	0	0	NUM
cana-3815	51	15	0	0	NUM
cana-3815	51	16	0	0	NUM
cana-3815	51	17	)	)	PUNCT
cana-3815	51	18	}	}	PUNCT
cana-3815	51	19	is	be	AUX
cana-3815	51	20	a	a	DET
cana-3815	51	21	2pbid	2pbid	NUM
cana-3815	51	22	,	,	PUNCT
cana-3815	51	23	but	but	CCONJ
cana-3815	51	24	not	not	PART
cana-3815	51	25	1pbid	1pbid	NUM
cana-3815	51	26	.	.	PUNCT
cana-3815	51	27	theorem	theorem	VERB
cana-3815	51	28	2.4	2.4	NUM
cana-3815	51	29	.	.	PUNCT
cana-3815	52	1	every	every	DET
cana-3815	52	2	2pbid	2pbid	PROPN
cana-3815	52	3	is	be	AUX
cana-3815	52	4	a	a	DET
cana-3815	52	5	3pbid	3pbid	NOUN
cana-3815	52	6	.	.	PUNCT
cana-3815	53	1	proof	proof	NOUN
cana-3815	53	2	.	.	PUNCT
cana-3815	54	1	let	let	VERB
cana-3815	54	2	ℵ	ℵ	NOUN
cana-3815	54	3	be	be	AUX
cana-3815	54	4	an	an	DET
cana-3815	54	5	2pbid	2pbid	NUM
cana-3815	54	6	of	of	ADP
cana-3815	54	7	§	§	PROPN
cana-3815	54	8	.	.	PUNCT
cana-3815	55	1	for	for	ADP
cana-3815	55	2	the	the	DET
cana-3815	55	3	ids	id	NOUN
cana-3815	55	4	£	£	SYM
cana-3815	55	5	1	1	NUM
cana-3815	55	6	and	and	CCONJ
cana-3815	55	7	£	£	SYM
cana-3815	55	8	2	2	NUM
cana-3815	55	9	of	of	ADP
cana-3815	55	10	§	§	PROPN
cana-3815	55	11	such	such	ADJ
cana-3815	55	12	that	that	SCONJ
cana-3815	55	13	£	£	SYM
cana-3815	55	14	1	1	NUM
cana-3815	55	15	·	·	SYM
cana-3815	55	16	£	£	SYM
cana-3815	55	17	2	2	NUM
cana-3815	55	18	⊆	⊆	NUM
cana-3815	55	19	ℵ.	ℵ.	NOUN
cana-3815	55	20	if	if	SCONJ
cana-3815	55	21	£	£	SYM
cana-3815	55	22	1	1	NUM
cana-3815	55	23	6⊆	6⊆	NUM
cana-3815	55	24	ℵ	ℵ	NOUN
cana-3815	55	25	,	,	PUNCT
cana-3815	55	26	let	let	VERB
cana-3815	55	27	∝∈	∝∈	PUNCT
cana-3815	55	28	£	£	SYM
cana-3815	55	29	1	1	NUM
cana-3815	55	30	\	\	NOUN
cana-3815	55	31	ℵ.	ℵ.	NOUN
cana-3815	55	32	for	for	ADP
cana-3815	55	33	any	any	DET
cana-3815	55	34	℘	℘	PROPN
cana-3815	55	35	∈	∈	PROPN
cana-3815	55	36	£	£	SYM
cana-3815	55	37	2	2	NUM
cana-3815	55	38	,	,	PUNCT
cana-3815	55	39	∝	∝	PROPN
cana-3815	55	40	§	§	PROPN
cana-3815	55	41	℘	℘	PROPN
cana-3815	55	42	⊆<∝	⊆<∝	NOUN
cana-3815	55	43	>	>	X
cana-3815	55	44	·	·	PUNCT
cana-3815	55	45	<	<	X
cana-3815	55	46	℘	℘	PROPN
cana-3815	55	47	>	>	SYM
cana-3815	55	48	⊆	⊆	NUM
cana-3815	55	49	£	£	SYM
cana-3815	55	50	1	1	NUM
cana-3815	55	51	·	·	SYM
cana-3815	55	52	£	£	SYM
cana-3815	55	53	2	2	NUM
cana-3815	55	54	⊆	⊆	NUM
cana-3815	55	55	ℵ.	ℵ.	NOUN
cana-3815	55	56	hence	hence	ADV
cana-3815	55	57	℘	℘	VERB
cana-3815	55	58	∈	∈	NOUN
cana-3815	55	59	√	√	NOUN
cana-3815	55	60	ℵ.	ℵ.	NOUN
cana-3815	56	1	then	then	ADV
cana-3815	56	2	£	£	SYM
cana-3815	56	3	2	2	NUM
cana-3815	56	4	⊆	⊆	NUM
cana-3815	56	5	√	√	NOUN
cana-3815	56	6	ℵ.	ℵ.	NOUN
cana-3815	57	1	thus	thus	ADV
cana-3815	57	2	ℵ	ℵ	NOUN
cana-3815	57	3	is	be	AUX
cana-3815	57	4	a	a	DET
cana-3815	57	5	3pbid	3pbid	NUM
cana-3815	57	6	of	of	ADP
cana-3815	57	7	§	§	PROPN
cana-3815	57	8	.	.	PUNCT
cana-3815	58	1	definition	definition	NOUN
cana-3815	58	2	2.5	2.5	NUM
cana-3815	58	3	.	.	PUNCT
cana-3815	59	1	a	a	DET
cana-3815	59	2	subset	subset	NOUN
cana-3815	59	3	d	d	NOUN
cana-3815	59	4	of	of	ADP
cana-3815	59	5	§	§	PROPN
cana-3815	59	6	is	be	AUX
cana-3815	59	7	said	say	VERB
cana-3815	59	8	to	to	PART
cana-3815	59	9	be	be	AUX
cana-3815	59	10	(	(	PUNCT
cana-3815	59	11	i	i	NOUN
cana-3815	59	12	)	)	PUNCT
cana-3815	59	13	mp1	mp1	NOUN
cana-3815	59	14	-sys	-sys	PUNCT
cana-3815	59	15	if	if	SCONJ
cana-3815	59	16	for	for	ADP
cana-3815	59	17	any	any	DET
cana-3815	59	18	∝	∝	NOUN
cana-3815	59	19	,	,	PUNCT
cana-3815	59	20	℘	℘	X
cana-3815	59	21	∈	∈	PROPN
cana-3815	59	22	d	d	NOUN
cana-3815	59	23	,	,	PUNCT
cana-3815	59	24	∃	∃	PROPN
cana-3815	59	25	∝1∈<∝>b	∝1∈<∝>b	PROPN
cana-3815	59	26	and	and	CCONJ
cana-3815	59	27	℘1	℘1	VERB
cana-3815	59	28	<	<	X
cana-3815	59	29	℘	℘	PROPN
cana-3815	59	30	>	>	PUNCT
cana-3815	59	31	b	b	NOUN
cana-3815	59	32	such	such	ADJ
cana-3815	59	33	that	that	PRON
cana-3815	59	34	∝1	∝1	ADJ
cana-3815	59	35	℘1	℘1	VERB
cana-3815	59	36	∈	∈	PROPN
cana-3815	59	37	d.	d.	X
cana-3815	59	38	(	(	PUNCT
cana-3815	59	39	ii	ii	PROPN
cana-3815	59	40	)	)	PUNCT
cana-3815	59	41	mp2	mp2	PROPN
cana-3815	59	42	-sys	-sys	PUNCT
cana-3815	59	43	if	if	SCONJ
cana-3815	59	44	for	for	ADP
cana-3815	59	45	any	any	DET
cana-3815	59	46	∝	∝	NOUN
cana-3815	59	47	,	,	PUNCT
cana-3815	59	48	℘	℘	X
cana-3815	59	49	∈	∈	PROPN
cana-3815	59	50	d	d	NOUN
cana-3815	59	51	,	,	PUNCT
cana-3815	59	52	∃	∃	PROPN
cana-3815	59	53	∝1∈<∝>r	∝1∈<∝>r	PROPN
cana-3815	59	54	and	and	CCONJ
cana-3815	59	55	℘1	℘1	PROPN
cana-3815	59	56	∈	∈	PROPN
cana-3815	59	57	<	<	X
cana-3815	59	58	℘	℘	PROPN
cana-3815	59	59	>	>	PUNCT
cana-3815	59	60	l	l	NOUN
cana-3815	59	61	such	such	ADJ
cana-3815	59	62	that	that	SCONJ
cana-3815	59	63	∝1	∝1	ADJ
cana-3815	59	64	℘1	℘1	VERB
cana-3815	59	65	∈	∈	PROPN
cana-3815	59	66	d.	d.	X
cana-3815	59	67	(	(	PUNCT
cana-3815	59	68	iii	iii	NOUN
cana-3815	59	69	)	)	PUNCT
cana-3815	59	70	mp3	mp3	NOUN
cana-3815	59	71	-sys	-sys	PUNCT
cana-3815	59	72	if	if	SCONJ
cana-3815	59	73	for	for	ADP
cana-3815	59	74	any	any	DET
cana-3815	59	75	∝	∝	NOUN
cana-3815	59	76	,	,	PUNCT
cana-3815	59	77	℘	℘	X
cana-3815	59	78	∈	∈	PROPN
cana-3815	59	79	d	d	NOUN
cana-3815	59	80	,	,	PUNCT
cana-3815	59	81	∃	∃	PROPN
cana-3815	59	82	∝1∈<∝	∝1∈<∝	PROPN
cana-3815	59	83	>	>	X
cana-3815	59	84	and	and	CCONJ
cana-3815	59	85	℘1	℘1	ADJ
cana-3815	59	86	∈	∈	PROPN
cana-3815	59	87	<	<	X
cana-3815	59	88	℘	℘	PROPN
cana-3815	59	89	>	>	PUNCT
cana-3815	59	90	such	such	ADJ
cana-3815	59	91	that	that	SCONJ
cana-3815	59	92	∝1	∝1	ADJ
cana-3815	59	93	℘1	℘1	VERB
cana-3815	59	94	∈	∈	PROPN
cana-3815	59	95	d.	d.	PROPN
cana-3815	59	96	theorem	theorem	VERB
cana-3815	59	97	2.6	2.6	NUM
cana-3815	59	98	.	.	PUNCT
cana-3815	60	1	if	if	SCONJ
cana-3815	60	2	ℵ	ℵ	NOUN
cana-3815	60	3	is	be	AUX
cana-3815	60	4	a	a	DET
cana-3815	60	5	bid	bid	NOUN
cana-3815	60	6	of	of	ADP
cana-3815	60	7	§	§	PROPN
cana-3815	60	8	,	,	PUNCT
cana-3815	60	9	then	then	ADV
cana-3815	60	10	ℵ	ℵ	NOUN
cana-3815	60	11	is	be	AUX
cana-3815	60	12	a	a	DET
cana-3815	60	13	1pbid	1pbid	NUM
cana-3815	60	14	(	(	PUNCT
cana-3815	60	15	2pbid,3pbid	2pbid,3pbid	NUM
cana-3815	60	16	)	)	PUNCT
cana-3815	60	17	if	if	SCONJ
cana-3815	60	18	and	and	CCONJ
cana-3815	60	19	only	only	ADV
cana-3815	60	20	if	if	SCONJ
cana-3815	60	21	§	§	PROPN
cana-3815	60	22	\ℵ	\ℵ	NOUN
cana-3815	60	23	is	be	AUX
cana-3815	60	24	anmp1	anmp1	NOUN
cana-3815	60	25	-sys	-sys	PUNCT
cana-3815	60	26	(	(	PUNCT
cana-3815	60	27	mp2	mp2	PROPN
cana-3815	60	28	-sys	-sys	PROPN
cana-3815	60	29	,	,	PUNCT
cana-3815	60	30	mp3	mp3	NOUN
cana-3815	60	31	-sys	-sy	NOUN
cana-3815	60	32	)	)	PUNCT
cana-3815	60	33	of	of	ADP
cana-3815	60	34	§	§	PROPN
cana-3815	60	35	.	.	PUNCT
cana-3815	61	1	proof	proof	NOUN
cana-3815	61	2	.	.	PUNCT
cana-3815	62	1	let	let	VERB
cana-3815	62	2	ς	ς	NOUN
cana-3815	62	3	,	,	PUNCT
cana-3815	62	4	ε	ε	PROPN
cana-3815	62	5	∈	∈	PROPN
cana-3815	62	6	§	§	PROPN
cana-3815	62	7	\	\	PROPN
cana-3815	63	1	ℵ.	ℵ.	PROPN
cana-3815	64	1	hence	hence	ADV
cana-3815	64	2	ς	ς	PROPN
cana-3815	64	3	,	,	PUNCT
cana-3815	64	4	ε	ε	PROPN
cana-3815	64	5	∈	∈	PROPN
cana-3815	64	6	§	§	PROPN
cana-3815	64	7	but	but	CCONJ
cana-3815	64	8	ς	ς	PROPN
cana-3815	64	9	,	,	PUNCT
cana-3815	64	10	ε	ε	PROPN
cana-3815	64	11	/∈	/∈	PUNCT
cana-3815	64	12	ℵ.	ℵ.	PROPN
cana-3815	65	1	so	so	ADV
cana-3815	65	2	<	<	X
cana-3815	65	3	ς	ς	X
cana-3815	65	4	>	>	X
cana-3815	65	5	b	b	PROPN
cana-3815	65	6	·	·	PUNCT
cana-3815	65	7	<	<	X
cana-3815	65	8	ε	ε	PROPN
cana-3815	65	9	>	>	SYM
cana-3815	65	10	b	b	PROPN
cana-3815	65	11	6⊆	6⊆	NUM
cana-3815	65	12	ℵ.	ℵ.	PROPN
cana-3815	65	13	there	there	PRON
cana-3815	65	14	exists	exist	VERB
cana-3815	65	15	ς	ς	PROPN
cana-3815	65	16	′	′	NUM
cana-3815	65	17	∈	∈	PROPN
cana-3815	65	18	<	<	X
cana-3815	65	19	ς	ς	PROPN
cana-3815	65	20	>	>	X
cana-3815	65	21	b	b	PROPN
cana-3815	65	22	and	and	CCONJ
cana-3815	65	23	ε	ε	PROPN
cana-3815	65	24	′	′	NUM
cana-3815	65	25	∈	∈	PROPN
cana-3815	65	26	<	<	X
cana-3815	65	27	ε	ε	PROPN
cana-3815	65	28	>	>	PUNCT
cana-3815	65	29	b	b	PROPN
cana-3815	65	30	such	such	ADJ
cana-3815	65	31	that	that	DET
cana-3815	65	32	ς	ς	PROPN
cana-3815	65	33	′	′	NUM
cana-3815	65	34	·	·	PUNCT
cana-3815	65	35	ε′	ε′	NUM
cana-3815	65	36	/∈	/∈	PUNCT
cana-3815	66	1	ℵ.	ℵ.	PROPN
cana-3815	66	2	hence	hence	ADV
cana-3815	66	3	ς	ς	PROPN
cana-3815	66	4	′	′	NUM
cana-3815	66	5	·	·	PUNCT
cana-3815	66	6	ε′	ε′	X
cana-3815	66	7	∈	∈	PROPN
cana-3815	66	8	§	§	NOUN
cana-3815	66	9	\	\	PROPN
cana-3815	67	1	ℵ.	ℵ.	PROPN
cana-3815	68	1	so	so	ADV
cana-3815	68	2	we	we	PRON
cana-3815	68	3	have	have	AUX
cana-3815	68	4	proved	prove	VERB
cana-3815	68	5	that	that	SCONJ
cana-3815	68	6	for	for	ADP
cana-3815	68	7	ς	ς	PROPN
cana-3815	68	8	,	,	PUNCT
cana-3815	68	9	ε	ε	PROPN
cana-3815	68	10	∈	∈	PROPN
cana-3815	68	11	§	§	PROPN
cana-3815	68	12	\	\	PROPN
cana-3815	68	13	ℵ	ℵ	PROPN
cana-3815	68	14	∃	∃	PROPN
cana-3815	68	15	ς	ς	PROPN
cana-3815	68	16	′	′	NUM
cana-3815	68	17	∈	∈	PROPN
cana-3815	68	18	<	<	X
cana-3815	68	19	ς	ς	PROPN
cana-3815	68	20	>	>	X
cana-3815	68	21	b	b	PROPN
cana-3815	68	22	and	and	CCONJ
cana-3815	68	23	ε	ε	PROPN
cana-3815	68	24	′	′	NUM
cana-3815	68	25	∈	∈	PROPN
cana-3815	68	26	<	<	X
cana-3815	68	27	ε	ε	PROPN
cana-3815	68	28	>	>	PUNCT
cana-3815	68	29	b	b	PROPN
cana-3815	68	30	such	such	ADJ
cana-3815	68	31	that	that	PRON
cana-3815	68	32	ς	ς	PROPN
cana-3815	68	33	′	′	NUM
cana-3815	68	34	·	·	PUNCT
cana-3815	68	35	ε′	ε′	X
cana-3815	68	36	∈	∈	PROPN
cana-3815	68	37	§	§	NOUN
cana-3815	68	38	\	\	PROPN
cana-3815	68	39	ℵ.	ℵ.	PROPN
cana-3815	69	1	so	so	ADV
cana-3815	69	2	§	§	NOUN
cana-3815	69	3	\	\	PROPN
cana-3815	69	4	ℵ	ℵ	NOUN
cana-3815	69	5	is	be	AUX
cana-3815	69	6	an	an	DET
cana-3815	69	7	mp1	mp1	NOUN
cana-3815	69	8	-sys	-sy	NOUN
cana-3815	69	9	.	.	PUNCT
cana-3815	70	1	conversely	conversely	ADV
cana-3815	70	2	,	,	PUNCT
cana-3815	70	3	let	let	VERB
cana-3815	70	4	§	§	NOUN
cana-3815	70	5	\ℵ	\ℵ	NOUN
cana-3815	70	6	is	be	AUX
cana-3815	70	7	anmp1	anmp1	NOUN
cana-3815	70	8	-sys	-sys	PUNCT
cana-3815	70	9	.	.	PUNCT
cana-3815	71	1	let	let	VERB
cana-3815	71	2	us	we	PRON
cana-3815	71	3	shows	show	VERB
cana-3815	71	4	that	that	SCONJ
cana-3815	71	5	a1	a1	NOUN
cana-3815	71	6	⊆	⊆	NUM
cana-3815	71	7	ℵ	ℵ	NOUN
cana-3815	71	8	or	or	CCONJ
cana-3815	71	9	a2	a2	PROPN
cana-3815	71	10	⊆	⊆	NUM
cana-3815	71	11	√	√	PROPN
cana-3815	71	12	ℵ.	ℵ.	PROPN
cana-3815	72	1	let	let	VERB
cana-3815	72	2	us	we	PRON
cana-3815	72	3	arrive	arrive	VERB
cana-3815	72	4	at	at	ADP
cana-3815	72	5	a	a	DET
cana-3815	72	6	contradiction	contradiction	NOUN
cana-3815	72	7	.	.	PUNCT
cana-3815	73	1	if	if	SCONJ
cana-3815	73	2	a1	a1	NOUN
cana-3815	73	3	6⊆	6⊆	NUM
cana-3815	73	4	ℵ	ℵ	NOUN
cana-3815	73	5	and	and	CCONJ
cana-3815	73	6	a2	a2	PROPN
cana-3815	73	7	6⊆	6⊆	NUM
cana-3815	73	8	√	√	NUM
cana-3815	73	9	ℵ	ℵ	NOUN
cana-3815	73	10	,	,	PUNCT
cana-3815	73	11	let	let	VERB
cana-3815	73	12	℘1	℘1	NOUN
cana-3815	73	13	∈	∈	NOUN
cana-3815	73	14	a1	a1	NOUN
cana-3815	73	15	\	\	NOUN
cana-3815	73	16	ℵ	ℵ	NOUN
cana-3815	73	17	and	and	CCONJ
cana-3815	73	18	let	let	VERB
cana-3815	73	19	℘2	℘2	PROPN
cana-3815	73	20	∈	∈	PROPN
cana-3815	73	21	a2	a2	PROPN
cana-3815	73	22	\	\	PROPN
cana-3815	73	23	√	√	PROPN
cana-3815	73	24	ℵ.	ℵ.	PROPN
cana-3815	74	1	since	since	SCONJ
cana-3815	74	2	℘2	℘2	PROPN
cana-3815	74	3	/∈	/∈	NOUN
cana-3815	74	4	√	√	NUM
cana-3815	74	5	ℵ	ℵ	NOUN
cana-3815	74	6	,	,	PUNCT
cana-3815	74	7	so	so	ADV
cana-3815	74	8	∃	∃	PROPN
cana-3815	74	9	an	an	DET
cana-3815	74	10	mp1	mp1	NOUN
cana-3815	74	11	-sys	-sys	PUNCT
cana-3815	74	12	§	§	PROPN
cana-3815	74	13	\	\	PROPN
cana-3815	74	14	ℵ	ℵ	NOUN
cana-3815	74	15	in	in	ADP
cana-3815	74	16	§	§	PROPN
cana-3815	74	17	such	such	ADJ
cana-3815	74	18	that	that	SCONJ
cana-3815	74	19	℘2	℘2	PROPN
cana-3815	74	20	∈	∈	PROPN
cana-3815	74	21	§	§	NOUN
cana-3815	74	22	\	\	PROPN
cana-3815	74	23	ℵ	ℵ	NOUN
cana-3815	74	24	and	and	CCONJ
cana-3815	74	25	(	(	PUNCT
cana-3815	74	26	§	§	PROPN
cana-3815	74	27	\	\	PROPN
cana-3815	74	28	ℵ)∩	ℵ)∩	NOUN
cana-3815	74	29	ℵ	ℵ	X
cana-3815	74	30	=	=	SYM
cana-3815	74	31	φ	φ	PROPN
cana-3815	74	32	.	.	PUNCT
cana-3815	75	1	thus	thus	ADV
cana-3815	75	2	℘1	℘1	NOUN
cana-3815	75	3	,	,	PUNCT
cana-3815	75	4	℘2	℘2	NOUN
cana-3815	75	5	∈	∈	PROPN
cana-3815	75	6	§	§	NOUN
cana-3815	75	7	\	\	PROPN
cana-3815	75	8	ℵ	ℵ	NOUN
cana-3815	75	9	implies	imply	VERB
cana-3815	75	10	<	<	X
cana-3815	75	11	℘1	℘1	PROPN
cana-3815	75	12	>	>	X
cana-3815	75	13	b	b	X
cana-3815	75	14	·	·	PUNCT
cana-3815	75	15	<	<	X
cana-3815	75	16	℘2	℘2	PROPN
cana-3815	75	17	>	>	SYM
cana-3815	75	18	b	b	PROPN
cana-3815	75	19	6⊆	6⊆	NOUN
cana-3815	75	20	ℵ.	ℵ.	PROPN
cana-3815	76	1	thus	thus	ADV
cana-3815	76	2	a1	a1	VERB
cana-3815	76	3	⊆	⊆	NUM
cana-3815	76	4	ℵ	ℵ	NOUN
cana-3815	76	5	or	or	CCONJ
cana-3815	76	6	a2	a2	PROPN
cana-3815	76	7	⊆	⊆	NUM
cana-3815	76	8	√	√	PROPN
cana-3815	76	9	ℵ.	ℵ.	NOUN
cana-3815	76	10	hence	hence	ADV
cana-3815	76	11	ℵ	ℵ	ADV
cana-3815	76	12	is	be	AUX
cana-3815	76	13	a	a	DET
cana-3815	76	14	1pbid	1pbid	NUM
cana-3815	76	15	of	of	ADP
cana-3815	76	16	§	§	PROPN
cana-3815	76	17	.	.	PUNCT
cana-3815	77	1	corollary	corollary	ADJ
cana-3815	77	2	2.7	2.7	NUM
cana-3815	77	3	.	.	PUNCT
cana-3815	78	1	every	every	DET
cana-3815	78	2	mp1	mp1	NOUN
cana-3815	78	3	-sys	-sys	PUNCT
cana-3815	78	4	is	be	AUX
cana-3815	78	5	an	an	DET
cana-3815	78	6	mp2	mp2	PROPN
cana-3815	78	7	-sys	-sys	PUNCT
cana-3815	78	8	.	.	PUNCT
cana-3815	79	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	79	2	756	756	NUM
cana-3815	79	3	communications	communication	NOUN
cana-3815	79	4	on	on	ADP
cana-3815	79	5	applied	apply	VERB
cana-3815	79	6	nonlinear	nonlinear	ADJ
cana-3815	79	7	analysis	analysis	NOUN
cana-3815	79	8	issn	issn	NOUN
cana-3815	79	9	:	:	PUNCT
cana-3815	79	10	1074	1074	NUM
cana-3815	79	11	-	-	PUNCT
cana-3815	79	12	133x	133x	NUM
cana-3815	79	13	vol	vol	NOUN
cana-3815	79	14	32	32	NUM
cana-3815	79	15	no	no	NOUN
cana-3815	79	16	.	.	NOUN
cana-3815	79	17	3	3	NUM
cana-3815	79	18	(	(	PUNCT
cana-3815	79	19	2025	2025	NUM
cana-3815	79	20	)	)	PUNCT
cana-3815	79	21	proof	proof	NOUN
cana-3815	79	22	.	.	PUNCT
cana-3815	80	1	given	give	VERB
cana-3815	80	2	that	that	PRON
cana-3815	80	3	d	d	NOUN
cana-3815	80	4	be	be	AUX
cana-3815	80	5	an	an	DET
cana-3815	80	6	mp1	mp1	NOUN
cana-3815	80	7	-sys	-sys	PUNCT
cana-3815	80	8	of	of	ADP
cana-3815	80	9	§	§	PROPN
cana-3815	80	10	.	.	AUX
cana-3815	81	1	for	for	ADP
cana-3815	81	2	any	any	DET
cana-3815	81	3	∝	∝	NOUN
cana-3815	81	4	,	,	PUNCT
cana-3815	81	5	℘	℘	X
cana-3815	81	6	∈	∈	PROPN
cana-3815	81	7	d	d	NOUN
cana-3815	81	8	,	,	PUNCT
cana-3815	81	9	∃	∃	PROPN
cana-3815	81	10	∝1∈<∝>b	∝1∈<∝>b	PROPN
cana-3815	81	11	and	and	CCONJ
cana-3815	81	12	℘1	℘1	VERB
cana-3815	81	13	∈	∈	PROPN
cana-3815	81	14	<	<	X
cana-3815	81	15	℘	℘	PROPN
cana-3815	81	16	>	>	PUNCT
cana-3815	81	17	b	b	NOUN
cana-3815	81	18	such	such	ADJ
cana-3815	81	19	that	that	PRON
cana-3815	81	20	∝1	∝1	NOUN
cana-3815	81	21	·	·	PUNCT
cana-3815	81	22	℘1	℘1	VERB
cana-3815	81	23	∈	∈	PROPN
cana-3815	81	24	d.	d.	NOUN
cana-3815	81	25	let	let	VERB
cana-3815	81	26	us	we	PRON
cana-3815	81	27	shows	show	VERB
cana-3815	81	28	that	that	SCONJ
cana-3815	81	29	d	d	NOUN
cana-3815	81	30	is	be	AUX
cana-3815	81	31	an	an	DET
cana-3815	81	32	mp2	mp2	PROPN
cana-3815	81	33	-sys	-sys	X
cana-3815	81	34	.	.	PUNCT
cana-3815	82	1	for	for	ADP
cana-3815	82	2	∝	∝	PROPN
cana-3815	82	3	,	,	PUNCT
cana-3815	82	4	℘	℘	X
cana-3815	82	5	∈	∈	PROPN
cana-3815	82	6	d	d	NOUN
cana-3815	82	7	,	,	PUNCT
cana-3815	82	8	∃	∃	PROPN
cana-3815	82	9	∝1∈<∝>r	∝1∈<∝>r	PROPN
cana-3815	82	10	and	and	CCONJ
cana-3815	82	11	℘1	℘1	PROPN
cana-3815	82	12	∈	∈	PROPN
cana-3815	82	13	<	<	X
cana-3815	82	14	℘	℘	PROPN
cana-3815	82	15	>	>	PUNCT
cana-3815	82	16	l.	l.	PROPN
cana-3815	82	17	since	since	SCONJ
cana-3815	82	18	right	right	ADV
cana-3815	82	19	and	and	CCONJ
cana-3815	82	20	lids	lid	NOUN
cana-3815	82	21	are	be	AUX
cana-3815	82	22	bids	bid	NOUN
cana-3815	82	23	also	also	ADV
cana-3815	82	24	,	,	PUNCT
cana-3815	82	25	we	we	PRON
cana-3815	82	26	have	have	VERB
cana-3815	82	27	∝1	∝1	ADJ
cana-3815	82	28	·	·	PUNCT
cana-3815	82	29	℘1	℘1	VERB
cana-3815	82	30	∈	∈	PROPN
cana-3815	82	31	d.	d.	NOUN
cana-3815	82	32	hence	hence	ADV
cana-3815	82	33	d	d	PROPN
cana-3815	82	34	is	be	AUX
cana-3815	82	35	an	an	DET
cana-3815	82	36	mp2	mp2	NOUN
cana-3815	82	37	-sys	-sys	PUNCT
cana-3815	82	38	of	of	ADP
cana-3815	82	39	§	§	PROPN
cana-3815	82	40	.	.	PUNCT
cana-3815	83	1	corollary	corollary	ADJ
cana-3815	83	2	2.8	2.8	NUM
cana-3815	83	3	.	.	PUNCT
cana-3815	84	1	every	every	DET
cana-3815	84	2	mp2	mp2	PROPN
cana-3815	84	3	-sys	-sys	PUNCT
cana-3815	84	4	is	be	AUX
cana-3815	84	5	an	an	DET
cana-3815	84	6	mp3	mp3	NOUN
cana-3815	84	7	-sys	-sy	NOUN
cana-3815	84	8	.	.	PUNCT
cana-3815	85	1	proof	proof	NOUN
cana-3815	85	2	.	.	PUNCT
cana-3815	86	1	given	give	VERB
cana-3815	86	2	that	that	PRON
cana-3815	86	3	d	d	NOUN
cana-3815	86	4	be	be	AUX
cana-3815	86	5	an	an	DET
cana-3815	86	6	mp2	mp2	NOUN
cana-3815	86	7	-sys	-sys	PUNCT
cana-3815	86	8	of	of	ADP
cana-3815	86	9	§	§	PROPN
cana-3815	86	10	.	.	AUX
cana-3815	87	1	for	for	ADP
cana-3815	87	2	any	any	DET
cana-3815	87	3	∝	∝	NOUN
cana-3815	87	4	,	,	PUNCT
cana-3815	87	5	℘	℘	X
cana-3815	87	6	∈	∈	PROPN
cana-3815	87	7	d	d	NOUN
cana-3815	87	8	,	,	PUNCT
cana-3815	87	9	∃	∃	PROPN
cana-3815	87	10	∝1∈<∝>r	∝1∈<∝>r	PROPN
cana-3815	87	11	and	and	CCONJ
cana-3815	87	12	℘1	℘1	PROPN
cana-3815	87	13	∈	∈	PROPN
cana-3815	87	14	<	<	X
cana-3815	87	15	℘	℘	PROPN
cana-3815	87	16	>	>	PUNCT
cana-3815	87	17	l	l	NOUN
cana-3815	87	18	such	such	ADJ
cana-3815	87	19	that	that	SCONJ
cana-3815	87	20	∝1	∝1	ADJ
cana-3815	87	21	℘1	℘1	VERB
cana-3815	87	22	∈	∈	PROPN
cana-3815	87	23	d.	d.	NOUN
cana-3815	87	24	let	let	VERB
cana-3815	87	25	us	we	PRON
cana-3815	87	26	shows	show	VERB
cana-3815	87	27	that	that	SCONJ
cana-3815	87	28	d	d	NOUN
cana-3815	87	29	is	be	AUX
cana-3815	87	30	an	an	DET
cana-3815	87	31	mp3	mp3	NOUN
cana-3815	87	32	-sys	-sy	NOUN
cana-3815	87	33	.	.	PUNCT
cana-3815	88	1	for	for	ADP
cana-3815	88	2	∝	∝	PROPN
cana-3815	88	3	,	,	PUNCT
cana-3815	88	4	℘	℘	X
cana-3815	88	5	∈	∈	PROPN
cana-3815	88	6	d	d	NOUN
cana-3815	88	7	,	,	PUNCT
cana-3815	88	8	∃	∃	PROPN
cana-3815	88	9	∝1∈<∝	∝1∈<∝	PROPN
cana-3815	88	10	>	>	X
cana-3815	88	11	and	and	CCONJ
cana-3815	88	12	℘1	℘1	ADJ
cana-3815	88	13	∈	∈	PROPN
cana-3815	88	14	<	<	X
cana-3815	88	15	℘	℘	PROPN
cana-3815	88	16	>	>	PUNCT
cana-3815	88	17	.	.	PUNCT
cana-3815	89	1	since	since	SCONJ
cana-3815	89	2	ids	id	NOUN
cana-3815	89	3	are	be	AUX
cana-3815	89	4	rid	rid	VERB
cana-3815	89	5	and	and	CCONJ
cana-3815	89	6	lids	lid	NOUN
cana-3815	89	7	also	also	ADV
cana-3815	89	8	,	,	PUNCT
cana-3815	89	9	we	we	PRON
cana-3815	89	10	have	have	VERB
cana-3815	89	11	∝1	∝1	NUM
cana-3815	89	12	℘1	℘1	VERB
cana-3815	89	13	∈	∈	PROPN
cana-3815	89	14	d.	d.	NOUN
cana-3815	90	1	hence	hence	ADV
cana-3815	90	2	d	d	PROPN
cana-3815	90	3	is	be	AUX
cana-3815	90	4	an	an	DET
cana-3815	90	5	mp3	mp3	NOUN
cana-3815	90	6	-sys	-sys	PUNCT
cana-3815	90	7	of	of	ADP
cana-3815	90	8	§	§	PROPN
cana-3815	90	9	.	.	PROPN
cana-3815	90	10	remark	remark	PROPN
cana-3815	90	11	2.9	2.9	NUM
cana-3815	90	12	.	.	PUNCT
cana-3815	91	1	let	let	VERB
cana-3815	91	2	√	√	VERB
cana-3815	91	3	a	a	DET
cana-3815	91	4	be	be	AUX
cana-3815	91	5	any	any	DET
cana-3815	91	6	bid	bid	NOUN
cana-3815	91	7	of	of	ADP
cana-3815	91	8	a	a	DET
cana-3815	91	9	ring	ring	NOUN
cana-3815	91	10	§	§	PROPN
cana-3815	91	11	.	.	PUNCT
cana-3815	92	1	then	then	ADV
cana-3815	92	2	√	√	VERB
cana-3815	92	3	ßa	ßa	NOUN
cana-3815	92	4	=	=	PUNCT
cana-3815	92	5	{	{	PUNCT
cana-3815	92	6	ς	ς	PROPN
cana-3815	92	7	∈	∈	PROPN
cana-3815	92	8	√	√	ADP
cana-3815	92	9	a	a	DET
cana-3815	92	10	|	|	NOUN
cana-3815	92	11	§	§	PROPN
cana-3815	92	12	ς	ς	PROPN
cana-3815	92	13	⊆	⊆	NUM
cana-3815	92	14	√	√	PROPN
cana-3815	92	15	a	a	PRON
cana-3815	92	16	}	}	PUNCT
cana-3815	92	17	and	and	CCONJ
cana-3815	92	18	√	√	ADJ
cana-3815	92	19	`	`	PUNCT
cana-3815	92	20	a	a	PRON
cana-3815	92	21	=	=	X
cana-3815	92	22	{	{	PUNCT
cana-3815	92	23	ε	ε	PROPN
cana-3815	92	24	∈√	∈√	PROPN
cana-3815	92	25	ßa	ßa	PROPN
cana-3815	92	26	|	|	ADV
cana-3815	92	27	ε§	ε§	VERB
cana-3815	92	28	⊆	⊆	X
cana-3815	92	29	√	√	ADP
cana-3815	92	30	ßa	ßa	NOUN
cana-3815	92	31	}	}	PUNCT
cana-3815	92	32	.	.	PUNCT
cana-3815	93	1	corollary	corollary	ADJ
cana-3815	93	2	2.10	2.10	NUM
cana-3815	93	3	.	.	PUNCT
cana-3815	94	1	let	let	VERB
cana-3815	94	2	√	√	VERB
cana-3815	94	3	a	a	DET
cana-3815	94	4	be	be	AUX
cana-3815	94	5	a	a	DET
cana-3815	94	6	bid	bid	NOUN
cana-3815	94	7	of	of	ADP
cana-3815	94	8	§	§	PROPN
cana-3815	94	9	.	.	PUNCT
cana-3815	95	1	then	then	ADV
cana-3815	95	2	√	√	VERB
cana-3815	95	3	ßa	ßa	PROPN
cana-3815	95	4	is	be	AUX
cana-3815	95	5	a	a	DET
cana-3815	95	6	lid	lid	NOUN
cana-3815	95	7	of	of	ADP
cana-3815	95	8	§	§	PROPN
cana-3815	95	9	such	such	ADJ
cana-3815	95	10	that	that	PRON
cana-3815	95	11	√	√	NOUN
cana-3815	95	12	ßa	ßa	NUM
cana-3815	95	13	⊆	⊆	NUM
cana-3815	95	14	√	√	NOUN
cana-3815	95	15	a.	a.	NOUN
cana-3815	95	16	proof	proof	NOUN
cana-3815	95	17	.	.	PUNCT
cana-3815	96	1	let	let	VERB
cana-3815	96	2	ς	ς	NOUN
cana-3815	96	3	,	,	PUNCT
cana-3815	96	4	ε	ε	PROPN
cana-3815	96	5	∈	∈	PROPN
cana-3815	96	6	√	√	PROPN
cana-3815	96	7	ßa	ßa	PROPN
cana-3815	96	8	.	.	PUNCT
cana-3815	97	1	then	then	ADV
cana-3815	97	2	ς	ς	PROPN
cana-3815	97	3	,	,	PUNCT
cana-3815	97	4	ε	ε	PROPN
cana-3815	97	5	∈	∈	PROPN
cana-3815	97	6	√	√	PROPN
cana-3815	97	7	a	a	PRON
cana-3815	97	8	and	and	CCONJ
cana-3815	97	9	§	§	PROPN
cana-3815	97	10	ς	ς	PROPN
cana-3815	97	11	⊆	⊆	NUM
cana-3815	97	12	√	√	PROPN
cana-3815	97	13	a	a	PRON
cana-3815	97	14	and	and	CCONJ
cana-3815	97	15	§	§	PROPN
cana-3815	97	16	ε	ε	PROPN
cana-3815	97	17	⊆	⊆	NUM
cana-3815	97	18	√	√	ADJ
cana-3815	97	19	a.	a.	NOUN
cana-3815	97	20	since	since	SCONJ
cana-3815	97	21	√	√	PROPN
cana-3815	97	22	a	a	PRON
cana-3815	97	23	is	be	AUX
cana-3815	97	24	a	a	DET
cana-3815	97	25	bid	bid	NOUN
cana-3815	97	26	of	of	ADP
cana-3815	97	27	§	§	PROPN
cana-3815	97	28	,	,	PUNCT
cana-3815	97	29	ς	ς	PROPN
cana-3815	97	30	+	+	NOUN
cana-3815	97	31	ε	ε	PROPN
cana-3815	97	32	∈	∈	PROPN
cana-3815	97	33	√	√	PROPN
cana-3815	97	34	a	a	PRON
cana-3815	97	35	and	and	CCONJ
cana-3815	97	36	ςε	ςε	INTJ
cana-3815	97	37	∈	∈	NOUN
cana-3815	97	38	√	√	NOUN
cana-3815	97	39	a.	a.	NOUN
cana-3815	97	40	now	now	ADV
cana-3815	97	41	,	,	PUNCT
cana-3815	97	42	§	§	PROPN
cana-3815	97	43	(	(	PUNCT
cana-3815	97	44	ς	ς	PROPN
cana-3815	97	45	+	+	PROPN
cana-3815	97	46	ε	ε	PROPN
cana-3815	97	47	)	)	PUNCT
cana-3815	97	48	⊆	⊆	NUM
cana-3815	97	49	§	§	PROPN
cana-3815	97	50	ς	ς	PROPN
cana-3815	97	51	+	+	PROPN
cana-3815	97	52	§	§	PROPN
cana-3815	97	53	ε	ε	PROPN
cana-3815	97	54	⊆	⊆	NUM
cana-3815	97	55	√	√	PROPN
cana-3815	97	56	a.	a.	NOUN
cana-3815	97	57	thus	thus	ADV
cana-3815	97	58	,	,	PUNCT
cana-3815	97	59	ς	ς	PROPN
cana-3815	97	60	+	+	CCONJ
cana-3815	97	61	ε	ε	PROPN
cana-3815	97	62	∈	∈	PROPN
cana-3815	97	63	√	√	PROPN
cana-3815	97	64	ßa	ßa	PROPN
cana-3815	97	65	.	.	PUNCT
cana-3815	98	1	now	now	ADV
cana-3815	98	2	,	,	PUNCT
cana-3815	98	3	§	§	PROPN
cana-3815	98	4	(	(	PUNCT
cana-3815	98	5	ςε	ςε	NOUN
cana-3815	98	6	)	)	PUNCT
cana-3815	98	7	⊆	⊆	NUM
cana-3815	98	8	(	(	PUNCT
cana-3815	98	9	§	§	NOUN
cana-3815	98	10	ς)(§ε	ς)(§ε	NOUN
cana-3815	98	11	)	)	PUNCT
cana-3815	98	12	⊆	⊆	NUM
cana-3815	98	13	√	√	NOUN
cana-3815	98	14	a.	a.	NOUN
cana-3815	98	15	thus	thus	ADV
cana-3815	98	16	,	,	PUNCT
cana-3815	98	17	ςε	ςε	PROPN
cana-3815	98	18	∈	∈	PROPN
cana-3815	98	19	√	√	PROPN
cana-3815	98	20	ßa	ßa	PROPN
cana-3815	98	21	.	.	PUNCT
cana-3815	99	1	hence	hence	ADV
cana-3815	99	2	√	√	VERB
cana-3815	99	3	ßa	ßa	NOUN
cana-3815	99	4	is	be	AUX
cana-3815	99	5	a	a	DET
cana-3815	99	6	ssr	ssr	NOUN
cana-3815	99	7	of	of	ADP
cana-3815	99	8	§	§	PROPN
cana-3815	99	9	.	.	PUNCT
cana-3815	100	1	let	let	VERB
cana-3815	100	2	ς	ς	PROPN
cana-3815	100	3	∈	∈	NOUN
cana-3815	100	4	√	√	VERB
cana-3815	100	5	ßa	ßa	PROPN
cana-3815	100	6	and	and	CCONJ
cana-3815	100	7	~	~	PUNCT
cana-3815	100	8	∈	∈	PROPN
cana-3815	100	9	§	§	PROPN
cana-3815	100	10	.	.	PUNCT
cana-3815	101	1	since	since	SCONJ
cana-3815	101	2	~ς	~ς	PUNCT
cana-3815	101	3	∈	∈	PROPN
cana-3815	101	4	§	§	PROPN
cana-3815	101	5	ς	ς	PROPN
cana-3815	101	6	⊆	⊆	NUM
cana-3815	101	7	√	√	ADP
cana-3815	101	8	a	a	PRON
cana-3815	101	9	,	,	PUNCT
cana-3815	101	10	we	we	PRON
cana-3815	101	11	have	have	VERB
cana-3815	101	12	~ς	~ς	X
cana-3815	101	13	∈	∈	NOUN
cana-3815	101	14	√	√	VERB
cana-3815	101	15	a	a	PRON
cana-3815	101	16	and	and	CCONJ
cana-3815	101	17	§	§	PROPN
cana-3815	101	18	~ς	~ς	PRON
cana-3815	101	19	⊆	⊆	NUM
cana-3815	101	20	§	§	PROPN
cana-3815	101	21	§	§	PROPN
cana-3815	101	22	ς	ς	PROPN
cana-3815	101	23	⊆	⊆	NUM
cana-3815	101	24	§	§	PROPN
cana-3815	101	25	ς	ς	PROPN
cana-3815	101	26	⊆	⊆	NUM
cana-3815	101	27	√	√	NOUN
cana-3815	101	28	a.	a.	NOUN
cana-3815	101	29	thus	thus	ADV
cana-3815	101	30	,	,	PUNCT
cana-3815	101	31	~ς	~ς	PUNCT
cana-3815	101	32	∈	∈	PROPN
cana-3815	101	33	√	√	VERB
cana-3815	101	34	ßa	ßa	PROPN
cana-3815	101	35	.	.	PUNCT
cana-3815	102	1	hence	hence	ADV
cana-3815	102	2	√	√	VERB
cana-3815	102	3	ßa	ßa	NOUN
cana-3815	102	4	is	be	AUX
cana-3815	102	5	a	a	DET
cana-3815	102	6	lid	lid	NOUN
cana-3815	102	7	of	of	ADP
cana-3815	102	8	§	§	PROPN
cana-3815	102	9	and	and	CCONJ
cana-3815	102	10	√	√	ADV
cana-3815	102	11	ßa	ßa	NUM
cana-3815	102	12	⊆	⊆	NUM
cana-3815	102	13	√	√	PROPN
cana-3815	102	14	a.	a.	NOUN
cana-3815	102	15	corollary	corollary	NOUN
cana-3815	102	16	2.11	2.11	NUM
cana-3815	102	17	.	.	PUNCT
cana-3815	103	1	let	let	VERB
cana-3815	103	2	√	√	VERB
cana-3815	103	3	a	a	DET
cana-3815	103	4	be	be	AUX
cana-3815	103	5	a	a	DET
cana-3815	103	6	bid	bid	NOUN
cana-3815	103	7	of	of	ADP
cana-3815	103	8	§	§	PROPN
cana-3815	103	9	.	.	PUNCT
cana-3815	104	1	then	then	ADV
cana-3815	104	2	√	√	VERB
cana-3815	104	3	`	`	PUNCT
cana-3815	104	4	a	a	PRON
cana-3815	104	5	is	be	AUX
cana-3815	104	6	a	a	DET
cana-3815	104	7	ssr	ssr	NOUN
cana-3815	104	8	of	of	ADP
cana-3815	104	9	§	§	PROPN
cana-3815	104	10	.	.	PUNCT
cana-3815	105	1	proof	proof	NOUN
cana-3815	105	2	.	.	PUNCT
cana-3815	106	1	let	let	VERB
cana-3815	106	2	ς	ς	NOUN
cana-3815	106	3	,	,	PUNCT
cana-3815	106	4	ε	ε	PROPN
cana-3815	106	5	∈	∈	PROPN
cana-3815	106	6	√	√	NOUN
cana-3815	106	7	`	`	PUNCT
cana-3815	106	8	a.	a.	NOUN
cana-3815	106	9	then	then	ADV
cana-3815	106	10	ς	ς	PROPN
cana-3815	106	11	,	,	PUNCT
cana-3815	106	12	ε	ε	PROPN
cana-3815	106	13	∈	∈	PROPN
cana-3815	106	14	√	√	PROPN
cana-3815	106	15	ßa	ßa	PROPN
cana-3815	106	16	and	and	CCONJ
cana-3815	106	17	ς§	ς§	PROPN
cana-3815	106	18	⊆	⊆	NUM
cana-3815	106	19	√	√	ADP
cana-3815	106	20	ßa	ßa	PROPN
cana-3815	106	21	and	and	CCONJ
cana-3815	106	22	ε§	ε§	PROPN
cana-3815	106	23	⊆	⊆	NUM
cana-3815	106	24	√	√	PROPN
cana-3815	106	25	ßa	ßa	PROPN
cana-3815	106	26	.	.	PUNCT
cana-3815	107	1	since	since	SCONJ
cana-3815	107	2	ς	ς	PROPN
cana-3815	107	3	∈	∈	PROPN
cana-3815	107	4	√	√	PROPN
cana-3815	107	5	ßa	ßa	PROPN
cana-3815	107	6	,	,	PUNCT
cana-3815	107	7	ς	ς	PROPN
cana-3815	107	8	∈	∈	PROPN
cana-3815	107	9	√	√	VERB
cana-3815	107	10	a	a	PRON
cana-3815	107	11	and	and	CCONJ
cana-3815	107	12	§	§	PROPN
cana-3815	107	13	ς	ς	PROPN
cana-3815	107	14	⊆	⊆	NUM
cana-3815	107	15	√	√	NOUN
cana-3815	107	16	a.	a.	NOUN
cana-3815	107	17	since	since	SCONJ
cana-3815	107	18	ε	ε	PROPN
cana-3815	107	19	∈	∈	PROPN
cana-3815	107	20	√	√	PROPN
cana-3815	107	21	ßa	ßa	PROPN
cana-3815	107	22	,	,	PUNCT
cana-3815	107	23	ε	ε	PROPN
cana-3815	107	24	∈	∈	PROPN
cana-3815	107	25	√	√	PROPN
cana-3815	107	26	a	a	PRON
cana-3815	107	27	and	and	CCONJ
cana-3815	107	28	§	§	PROPN
cana-3815	107	29	ε	ε	PROPN
cana-3815	107	30	⊆	⊆	NUM
cana-3815	107	31	√	√	PROPN
cana-3815	107	32	a.	a.	NOUN
cana-3815	107	33	since	since	SCONJ
cana-3815	107	34	ς	ς	PROPN
cana-3815	107	35	,	,	PUNCT
cana-3815	107	36	ε	ε	PROPN
cana-3815	107	37	∈	∈	PROPN
cana-3815	107	38	√	√	PROPN
cana-3815	107	39	a	a	PRON
cana-3815	107	40	and	and	CCONJ
cana-3815	107	41	√	√	DET
cana-3815	107	42	a	a	PRON
cana-3815	107	43	is	be	AUX
cana-3815	107	44	a	a	DET
cana-3815	107	45	ssr	ssr	NOUN
cana-3815	107	46	of	of	ADP
cana-3815	107	47	§	§	PROPN
cana-3815	107	48	.	.	PUNCT
cana-3815	108	1	we	we	PRON
cana-3815	108	2	have	have	VERB
cana-3815	108	3	ς+	ς+	X
cana-3815	108	4	ε	ε	PROPN
cana-3815	108	5	∈	∈	PROPN
cana-3815	108	6	√	√	PROPN
cana-3815	108	7	a	a	PRON
cana-3815	108	8	and	and	CCONJ
cana-3815	108	9	ςε	ςε	INTJ
cana-3815	108	10	∈	∈	NOUN
cana-3815	108	11	√	√	NOUN
cana-3815	108	12	a.	a.	NOUN
cana-3815	108	13	now	now	ADV
cana-3815	108	14	,	,	PUNCT
cana-3815	108	15	§	§	PROPN
cana-3815	108	16	(	(	PUNCT
cana-3815	108	17	ς	ς	PROPN
cana-3815	108	18	+	+	PROPN
cana-3815	108	19	ε	ε	PROPN
cana-3815	108	20	)	)	PUNCT
cana-3815	108	21	⊆	⊆	NUM
cana-3815	108	22	§	§	PROPN
cana-3815	108	23	ς	ς	PROPN
cana-3815	108	24	+	+	PROPN
cana-3815	108	25	§	§	PROPN
cana-3815	108	26	ε	ε	PROPN
cana-3815	108	27	⊆	⊆	NUM
cana-3815	108	28	√	√	ADP
cana-3815	108	29	a	a	DET
cana-3815	108	30	implies	implie	NOUN
cana-3815	108	31	ς	ς	PROPN
cana-3815	108	32	+	+	CCONJ
cana-3815	108	33	ε	ε	PROPN
cana-3815	108	34	∈	∈	PROPN
cana-3815	108	35	√	√	PROPN
cana-3815	108	36	ßa	ßa	PROPN
cana-3815	108	37	.	.	PUNCT
cana-3815	109	1	now	now	ADV
cana-3815	109	2	,	,	PUNCT
cana-3815	109	3	(	(	PUNCT
cana-3815	109	4	ς	ς	PROPN
cana-3815	109	5	+	+	X
cana-3815	109	6	ε)§	ε)§	NOUN
cana-3815	109	7	⊆	⊆	NUM
cana-3815	109	8	ς§	ς§	NOUN
cana-3815	109	9	+	+	CCONJ
cana-3815	109	10	ε§	ε§	PROPN
cana-3815	109	11	⊆	⊆	X
cana-3815	109	12	√	√	PROPN
cana-3815	109	13	ßa	ßa	PROPN
cana-3815	109	14	.	.	PUNCT
cana-3815	110	1	hence	hence	ADV
cana-3815	110	2	ς	ς	PROPN
cana-3815	110	3	+	+	CCONJ
cana-3815	110	4	ε	ε	PROPN
cana-3815	110	5	∈	∈	PROPN
cana-3815	110	6	√	√	PUNCT
cana-3815	110	7	`	`	PUNCT
cana-3815	110	8	a.	a.	NOUN
cana-3815	110	9	now	now	ADV
cana-3815	110	10	,	,	PUNCT
cana-3815	110	11	§	§	PROPN
cana-3815	110	12	(	(	PUNCT
cana-3815	110	13	ςε	ςε	NOUN
cana-3815	110	14	)	)	PUNCT
cana-3815	110	15	⊆	⊆	NUM
cana-3815	110	16	(	(	PUNCT
cana-3815	110	17	§	§	NOUN
cana-3815	110	18	ς)(§ε	ς)(§ε	NOUN
cana-3815	110	19	)	)	PUNCT
cana-3815	110	20	⊆	⊆	NUM
cana-3815	110	21	√	√	ADP
cana-3815	110	22	a	a	DET
cana-3815	110	23	implies	imply	VERB
cana-3815	110	24	ςε	ςε	INTJ
cana-3815	110	25	∈	∈	NOUN
cana-3815	110	26	√	√	PROPN
cana-3815	110	27	ßa	ßa	PROPN
cana-3815	110	28	and	and	CCONJ
cana-3815	110	29	(	(	PUNCT
cana-3815	110	30	ςε)§	ςε)§	NOUN
cana-3815	110	31	⊆	⊆	NUM
cana-3815	110	32	(	(	PUNCT
cana-3815	110	33	ς§)(ε§	ς§)(ε§	NOUN
cana-3815	110	34	)	)	PUNCT
cana-3815	110	35	⊆	⊆	NUM
cana-3815	110	36	√	√	PROPN
cana-3815	110	37	ßa	ßa	NOUN
cana-3815	110	38	.	.	PUNCT
cana-3815	111	1	that	that	PRON
cana-3815	111	2	is	be	AUX
cana-3815	111	3	ςε	ςε	PART
cana-3815	111	4	∈	∈	NOUN
cana-3815	111	5	√	√	NOUN
cana-3815	111	6	`	`	PUNCT
cana-3815	111	7	a.	a.	NOUN
cana-3815	111	8	hence	hence	ADV
cana-3815	111	9	√	√	NOUN
cana-3815	111	10	`	`	PUNCT
cana-3815	111	11	a	a	PRON
cana-3815	111	12	is	be	AUX
cana-3815	111	13	a	a	DET
cana-3815	111	14	ssr	ssr	NOUN
cana-3815	111	15	of	of	ADP
cana-3815	111	16	§	§	PROPN
cana-3815	111	17	.	.	PUNCT
cana-3815	112	1	corollary	corollary	ADJ
cana-3815	112	2	2.12	2.12	NUM
cana-3815	112	3	.	.	PUNCT
cana-3815	113	1	let	let	VERB
cana-3815	113	2	√	√	VERB
cana-3815	113	3	a	a	DET
cana-3815	113	4	be	be	AUX
cana-3815	113	5	a	a	DET
cana-3815	113	6	lid	lid	NOUN
cana-3815	113	7	of	of	ADP
cana-3815	113	8	§	§	PROPN
cana-3815	113	9	.	.	PUNCT
cana-3815	114	1	then	then	ADV
cana-3815	114	2	√	√	VERB
cana-3815	114	3	ßa	ßa	NOUN
cana-3815	114	4	=	=	PUNCT
cana-3815	114	5	√	√	NOUN
cana-3815	114	6	a.	a.	NOUN
cana-3815	114	7	proof	proof	NOUN
cana-3815	114	8	.	.	PUNCT
cana-3815	115	1	clearly	clearly	ADV
cana-3815	115	2	,	,	PUNCT
cana-3815	115	3	√	√	ADV
cana-3815	115	4	ßa	ßa	NUM
cana-3815	115	5	⊆	⊆	NUM
cana-3815	115	6	√	√	PROPN
cana-3815	115	7	a.	a.	NOUN
cana-3815	115	8	let	let	VERB
cana-3815	115	9	ς	ς	PROPN
cana-3815	115	10	∈	∈	PROPN
cana-3815	115	11	√	√	VERB
cana-3815	115	12	a	a	PRON
cana-3815	115	13	,	,	PUNCT
cana-3815	115	14	since	since	SCONJ
cana-3815	115	15	√	√	PROPN
cana-3815	115	16	a	a	PRON
cana-3815	115	17	is	be	AUX
cana-3815	115	18	a	a	DET
cana-3815	115	19	lid	lid	NOUN
cana-3815	115	20	of	of	ADP
cana-3815	115	21	§	§	PROPN
cana-3815	115	22	.	.	PUNCT
cana-3815	116	1	we	we	PRON
cana-3815	116	2	have	have	VERB
cana-3815	116	3	§	§	VERB
cana-3815	116	4	ς	ς	PROPN
cana-3815	116	5	⊆	⊆	NUM
cana-3815	116	6	√	√	ADP
cana-3815	116	7	a	a	DET
cana-3815	116	8	implies	imply	VERB
cana-3815	116	9	ς	ς	PROPN
cana-3815	116	10	∈	∈	PROPN
cana-3815	116	11	√	√	PROPN
cana-3815	116	12	ßa	ßa	NOUN
cana-3815	116	13	.	.	PUNCT
cana-3815	117	1	thus,√	thus,√	NOUN
cana-3815	117	2	a	a	DET
cana-3815	117	3	⊆	⊆	NUM
cana-3815	117	4	√	√	PROPN
cana-3815	117	5	ßa	ßa	PROPN
cana-3815	117	6	.	.	PUNCT
cana-3815	118	1	hence	hence	ADV
cana-3815	118	2	√	√	VERB
cana-3815	118	3	ßa	ßa	NOUN
cana-3815	118	4	=	=	SYM
cana-3815	118	5	√	√	PROPN
cana-3815	118	6	a.	a.	NOUN
cana-3815	118	7	theorem	theorem	NOUN
cana-3815	118	8	2.13	2.13	NUM
cana-3815	118	9	.	.	PUNCT
cana-3815	119	1	let	let	VERB
cana-3815	119	2	√	√	VERB
cana-3815	119	3	a	a	PRON
cana-3815	119	4	is	be	AUX
cana-3815	119	5	a	a	DET
cana-3815	119	6	bid	bid	NOUN
cana-3815	119	7	of	of	ADP
cana-3815	119	8	§	§	PROPN
cana-3815	119	9	.	.	PUNCT
cana-3815	120	1	then	then	ADV
cana-3815	120	2	√	√	VERB
cana-3815	120	3	`	`	PUNCT
cana-3815	120	4	a	a	PRON
cana-3815	120	5	is	be	AUX
cana-3815	120	6	the	the	DET
cana-3815	120	7	unique	unique	ADJ
cana-3815	120	8	largest	large	ADJ
cana-3815	120	9	tid	tid	NOUN
cana-3815	120	10	of	of	ADP
cana-3815	120	11	§	§	PROPN
cana-3815	120	12	contained	contain	VERB
cana-3815	120	13	in	in	ADP
cana-3815	120	14	√	√	NUM
cana-3815	120	15	a.	a.	NOUN
cana-3815	120	16	proof	proof	NOUN
cana-3815	120	17	.	.	PUNCT
cana-3815	121	1	let	let	VERB
cana-3815	121	2	√	√	PROPN
cana-3815	121	3	a	a	PRON
cana-3815	121	4	is	be	AUX
cana-3815	121	5	any	any	DET
cana-3815	121	6	bid	bid	NOUN
cana-3815	121	7	of	of	ADP
cana-3815	121	8	§	§	PROPN
cana-3815	121	9	.	.	PUNCT
cana-3815	121	10	to	to	PART
cana-3815	121	11	prove	prove	VERB
cana-3815	121	12	that	that	SCONJ
cana-3815	121	13	√	√	PROPN
cana-3815	121	14	`	`	PUNCT
cana-3815	121	15	a	a	PRON
cana-3815	121	16	is	be	AUX
cana-3815	121	17	the	the	DET
cana-3815	121	18	tid	tid	NOUN
cana-3815	121	19	of	of	ADP
cana-3815	121	20	§	§	PROPN
cana-3815	121	21	.	.	PUNCT
cana-3815	122	1	since	since	SCONJ
cana-3815	122	2	√	√	PROPN
cana-3815	122	3	ßa	ßa	NUM
cana-3815	122	4	⊆	⊆	NUM
cana-3815	122	5	√	√	ADP
cana-3815	122	6	a	a	PRON
cana-3815	122	7	and	and	CCONJ
cana-3815	122	8	√	√	NUM
cana-3815	122	9	`	`	PUNCT
cana-3815	122	10	a	a	DET
cana-3815	122	11	⊆	⊆	NUM
cana-3815	122	12	√	√	PROPN
cana-3815	122	13	ßa	ßa	PROPN
cana-3815	122	14	.	.	PUNCT
cana-3815	123	1	therefore	therefore	ADV
cana-3815	123	2	√	√	VERB
cana-3815	123	3	`	`	PUNCT
cana-3815	123	4	a	a	DET
cana-3815	123	5	⊆	⊆	NUM
cana-3815	123	6	√	√	ADJ
cana-3815	123	7	ßa	ßa	NOUN
cana-3815	123	8	⊆	⊆	NUM
cana-3815	123	9	√	√	PROPN
cana-3815	123	10	a.	a.	NOUN
cana-3815	123	11	let	let	VERB
cana-3815	123	12	ς	ς	PROPN
cana-3815	123	13	∈	∈	NOUN
cana-3815	123	14	√	√	NOUN
cana-3815	123	15	`	`	PUNCT
cana-3815	123	16	a	a	PRON
cana-3815	123	17	and	and	CCONJ
cana-3815	123	18	1∈	1∈	PROPN
cana-3815	123	19	§	§	PROPN
cana-3815	123	20	.	.	PUNCT
cana-3815	124	1	then	then	ADV
cana-3815	124	2	ς	ς	PROPN
cana-3815	124	3	∈	∈	PROPN
cana-3815	124	4	√	√	NOUN
cana-3815	124	5	`	`	PUNCT
cana-3815	124	6	a	a	DET
cana-3815	124	7	⊆	⊆	NUM
cana-3815	124	8	√	√	ADP
cana-3815	124	9	a	a	DET
cana-3815	124	10	=	=	NOUN
cana-3815	124	11	⇒	⇒	NOUN
cana-3815	124	12	ς	ς	PROPN
cana-3815	124	13	∈	∈	NOUN
cana-3815	124	14	√	√	NOUN
cana-3815	124	15	a.	a.	NOUN
cana-3815	124	16	since	since	SCONJ
cana-3815	124	17	ς	ς	PROPN
cana-3815	124	18	is	be	AUX
cana-3815	124	19	an	an	DET
cana-3815	124	20	element	element	NOUN
cana-3815	124	21	of	of	ADP
cana-3815	124	22	√	√	PROPN
cana-3815	124	23	ßa	ßa	PROPN
cana-3815	124	24	.	.	PUNCT
cana-3815	125	1	we	we	PRON
cana-3815	125	2	have	have	VERB
cana-3815	125	3	§	§	VERB
cana-3815	125	4	ς	ς	PROPN
cana-3815	125	5	⊆	⊆	NUM
cana-3815	125	6	√	√	PROPN
cana-3815	125	7	a	a	PRON
cana-3815	125	8	and	and	CCONJ
cana-3815	125	9	ς§	ς§	PROPN
cana-3815	125	10	⊆	⊆	NUM
cana-3815	125	11	√	√	PROPN
cana-3815	125	12	ßa	ßa	PROPN
cana-3815	125	13	.	.	PUNCT
cana-3815	126	1	then	then	ADV
cana-3815	126	2	1	1	NUM
cana-3815	126	3	ς	ς	PROPN
cana-3815	126	4	∈	∈	PROPN
cana-3815	126	5	§	§	PROPN
cana-3815	126	6	ς	ς	PROPN
cana-3815	126	7	⊆	⊆	NUM
cana-3815	126	8	√	√	ADP
cana-3815	126	9	a	a	PRON
cana-3815	126	10	implies	imply	VERB
cana-3815	126	11	1	1	NUM
cana-3815	126	12	ς	ς	PROPN
cana-3815	126	13	∈	∈	PROPN
cana-3815	126	14	√	√	NOUN
cana-3815	126	15	a	a	PRON
cana-3815	126	16	and	and	CCONJ
cana-3815	126	17	§	§	PROPN
cana-3815	126	18	1	1	NUM
cana-3815	126	19	ς	ς	PROPN
cana-3815	126	20	⊆	⊆	NUM
cana-3815	126	21	§	§	PROPN
cana-3815	126	22	§	§	PROPN
cana-3815	126	23	ς	ς	PROPN
cana-3815	126	24	⊆	⊆	NUM
cana-3815	126	25	§	§	PROPN
cana-3815	126	26	ς	ς	PROPN
cana-3815	126	27	⊆	⊆	NUM
cana-3815	126	28	√	√	ADP
cana-3815	126	29	a	a	DET
cana-3815	126	30	=	=	NOUN
cana-3815	126	31	⇒	⇒	NOUN
cana-3815	126	32	1	1	NUM
cana-3815	126	33	ς	ς	PROPN
cana-3815	126	34	∈	∈	PROPN
cana-3815	126	35	√	√	PROPN
cana-3815	126	36	ßa	ßa	PROPN
cana-3815	126	37	.	.	PUNCT
cana-3815	127	1	now	now	ADV
cana-3815	127	2	,	,	PUNCT
cana-3815	127	3	ς	ς	PROPN
cana-3815	127	4	1∈	1∈	PROPN
cana-3815	127	5	ς§	ς§	PROPN
cana-3815	127	6	⊆	⊆	NUM
cana-3815	127	7	√	√	PROPN
cana-3815	127	8	ßa	ßa	PROPN
cana-3815	127	9	.	.	PUNCT
cana-3815	128	1	hence	hence	ADV
cana-3815	128	2	ς	ς	PROPN
cana-3815	128	3	1∈	1∈	PROPN
cana-3815	128	4	√	√	PROPN
cana-3815	128	5	ßa	ßa	PROPN
cana-3815	128	6	and	and	CCONJ
cana-3815	128	7	1	1	NUM
cana-3815	128	8	ς	ς	PROPN
cana-3815	128	9	∈	∈	PROPN
cana-3815	128	10	√	√	NOUN
cana-3815	128	11	ßa	ßa	PROPN
cana-3815	128	12	.	.	PUNCT
cana-3815	129	1	first	first	ADV
cana-3815	129	2	to	to	PART
cana-3815	129	3	prove	prove	VERB
cana-3815	129	4	that	that	SCONJ
cana-3815	129	5	ς	ς	PROPN
cana-3815	129	6	1∈	1∈	PROPN
cana-3815	129	7	√	√	PROPN
cana-3815	129	8	`	`	PUNCT
cana-3815	129	9	a	a	PRON
cana-3815	129	10	and	and	CCONJ
cana-3815	129	11	1	1	NUM
cana-3815	129	12	ς	ς	PROPN
cana-3815	129	13	∈	∈	NOUN
cana-3815	129	14	√	√	PUNCT
cana-3815	129	15	`	`	PUNCT
cana-3815	129	16	a.	a.	NOUN
cana-3815	129	17	now	now	ADV
cana-3815	129	18	,	,	PUNCT
cana-3815	129	19	ς	ς	PROPN
cana-3815	129	20	1	1	NUM
cana-3815	129	21	§	§	PROPN
cana-3815	129	22	⊆	⊆	NUM
cana-3815	129	23	ς§§	ς§§	PROPN
cana-3815	129	24	⊆	⊆	NUM
cana-3815	129	25	ς§	ς§	NOUN
cana-3815	129	26	⊆	⊆	NUM
cana-3815	129	27	√	√	PROPN
cana-3815	129	28	ßa	ßa	PROPN
cana-3815	129	29	.	.	PUNCT
cana-3815	130	1	hence	hence	ADV
cana-3815	130	2	ς	ς	PROPN
cana-3815	130	3	1	1	NUM
cana-3815	130	4	§	§	PROPN
cana-3815	130	5	⊆	⊆	NUM
cana-3815	130	6	√	√	PROPN
cana-3815	130	7	ßa	ßa	PROPN
cana-3815	130	8	implies	imply	VERB
cana-3815	130	9	ς	ς	PROPN
cana-3815	130	10	1∈	1∈	PROPN
cana-3815	130	11	√	√	PROPN
cana-3815	130	12	`	`	PUNCT
cana-3815	130	13	a.	a.	NOUN
cana-3815	130	14	now	now	ADV
cana-3815	130	15	,	,	PUNCT
cana-3815	130	16	1	1	NUM
cana-3815	130	17	ς§	ς§	NOUN
cana-3815	130	18	⊆	⊆	NUM
cana-3815	130	19	§	§	PROPN
cana-3815	130	20	ς§	ς§	PROPN
cana-3815	130	21	⊆	⊆	NUM
cana-3815	130	22	§	§	PROPN
cana-3815	130	23	√	√	NUM
cana-3815	130	24	ßa	ßa	ADP
cana-3815	130	25	⊆	⊆	NUM
cana-3815	130	26	√	√	PROPN
cana-3815	130	27	ßa	ßa	PROPN
cana-3815	130	28	.	.	PUNCT
cana-3815	131	1	since	since	SCONJ
cana-3815	131	2	√	√	PROPN
cana-3815	131	3	ßa	ßa	PROPN
cana-3815	131	4	is	be	AUX
cana-3815	131	5	a	a	DET
cana-3815	131	6	lid	lid	NOUN
cana-3815	131	7	of	of	ADP
cana-3815	131	8	§	§	PROPN
cana-3815	131	9	,	,	PUNCT
cana-3815	131	10	1	1	NUM
cana-3815	131	11	ς	ς	PROPN
cana-3815	131	12	∈	∈	NOUN
cana-3815	131	13	√	√	PUNCT
cana-3815	131	14	`	`	PUNCT
cana-3815	131	15	a.	a.	NOUN
cana-3815	131	16	hence	hence	ADV
cana-3815	131	17	√	√	NOUN
cana-3815	131	18	`	`	PUNCT
cana-3815	131	19	a	a	PRON
cana-3815	131	20	is	be	AUX
cana-3815	131	21	a	a	DET
cana-3815	131	22	tid	tid	NOUN
cana-3815	131	23	of	of	ADP
cana-3815	131	24	§	§	PROPN
cana-3815	131	25	.	.	PUNCT
cana-3815	132	1	it	it	PRON
cana-3815	132	2	enough	enough	ADV
cana-3815	132	3	to	to	PART
cana-3815	132	4	prove	prove	VERB
cana-3815	132	5	√	√	INTJ
cana-3815	132	6	`	`	PUNCT
cana-3815	132	7	a	a	PRON
cana-3815	132	8	is	be	AUX
cana-3815	132	9	a	a	DET
cana-3815	132	10	largest	large	ADJ
cana-3815	132	11	two	two	NUM
cana-3815	132	12	sided	sided	ADJ
cana-3815	132	13	i	i	PROPN
cana-3815	132	14	d	d	PROPN
cana-3815	132	15	of	of	ADP
cana-3815	132	16	§	§	PROPN
cana-3815	132	17	.	.	PUNCT
cana-3815	133	1	let	let	VERB
cana-3815	133	2	√	√	PROPN
cana-3815	133	3	§	§	PROPN
cana-3815	133	4	be	be	AUX
cana-3815	133	5	any	any	DET
cana-3815	133	6	i	i	PROPN
cana-3815	133	7	d	d	PROPN
cana-3815	133	8	of	of	ADP
cana-3815	133	9	§	§	PROPN
cana-3815	133	10	and	and	CCONJ
cana-3815	133	11	√	√	PROPN
cana-3815	133	12	§	§	PROPN
cana-3815	133	13	⊆	⊆	NUM
cana-3815	133	14	√	√	ADJ
cana-3815	133	15	a.	a.	NOUN
cana-3815	133	16	let	let	VERB
cana-3815	133	17	%	%	NOUN
cana-3815	133	18	∈	∈	PROPN
cana-3815	133	19	√	√	PROPN
cana-3815	133	20	§	§	PROPN
cana-3815	133	21	,	,	PUNCT
cana-3815	133	22	then	then	ADV
cana-3815	133	23	%	%	INTJ
cana-3815	133	24	∈	∈	PROPN
cana-3815	133	25	√	√	PROPN
cana-3815	133	26	a	a	PRON
cana-3815	133	27	and	and	CCONJ
cana-3815	133	28	§	§	ADJ
cana-3815	133	29	%	%	NOUN
cana-3815	133	30	⊆	⊆	NUM
cana-3815	133	31	√	√	ADP
cana-3815	133	32	§	§	PROPN
cana-3815	133	33	⊆	⊆	NUM
cana-3815	133	34	√	√	ADJ
cana-3815	133	35	a.	a.	NOUN
cana-3815	133	36	hence	hence	ADV
cana-3815	133	37	§	§	NOUN
cana-3815	133	38	%	%	NOUN
cana-3815	133	39	⊆	⊆	NUM
cana-3815	133	40	√	√	ADP
cana-3815	133	41	a	a	DET
cana-3815	133	42	=	=	NOUN
cana-3815	133	43	⇒	⇒	X
cana-3815	133	44	%	%	NOUN
cana-3815	133	45	∈	∈	PROPN
cana-3815	133	46	√	√	PROPN
cana-3815	133	47	ßa	ßa	PROPN
cana-3815	133	48	.	.	PUNCT
cana-3815	134	1	hence	hence	ADV
cana-3815	134	2	√	√	PROPN
cana-3815	134	3	§	§	PROPN
cana-3815	134	4	⊆	⊆	NUM
cana-3815	134	5	√	√	PROPN
cana-3815	134	6	ßa	ßa	PROPN
cana-3815	134	7	.	.	PUNCT
cana-3815	135	1	next	next	ADJ
cana-3815	135	2	,	,	PUNCT
cana-3815	135	3	%	%	INTJ
cana-3815	135	4	∈	∈	PROPN
cana-3815	136	1	√	√	VERB
cana-3815	136	2	ßa	ßa	PROPN
cana-3815	137	1	and	and	CCONJ
cana-3815	137	2	%	%	INTJ
cana-3815	137	3	§	§	PROPN
cana-3815	137	4	⊆	⊆	NUM
cana-3815	137	5	√	√	PROPN
cana-3815	137	6	§	§	PROPN
cana-3815	137	7	⊆	⊆	NUM
cana-3815	137	8	√	√	PROPN
cana-3815	137	9	ßa	ßa	PROPN
cana-3815	137	10	.	.	PUNCT
cana-3815	138	1	therefore	therefore	ADV
cana-3815	138	2	%	%	INTJ
cana-3815	138	3	§	§	PROPN
cana-3815	138	4	⊆	⊆	NUM
cana-3815	138	5	√	√	PROPN
cana-3815	138	6	ßa	ßa	PROPN
cana-3815	138	7	.	.	PUNCT
cana-3815	139	1	thus	thus	ADV
cana-3815	139	2	,	,	PUNCT
cana-3815	139	3	%	%	INTJ
cana-3815	139	4	∈	∈	PROPN
cana-3815	139	5	√	√	PUNCT
cana-3815	139	6	`	`	PUNCT
cana-3815	139	7	a.	a.	NOUN
cana-3815	139	8	hence	hence	ADV
cana-3815	139	9	√	√	PROPN
cana-3815	139	10	§	§	PROPN
cana-3815	139	11	⊆	⊆	NUM
cana-3815	139	12	√	√	NUM
cana-3815	139	13	`	`	PUNCT
cana-3815	139	14	a.	a.	NOUN
cana-3815	139	15	theorem	theorem	NOUN
cana-3815	139	16	2.14	2.14	NUM
cana-3815	139	17	.	.	PUNCT
cana-3815	140	1	a	a	DET
cana-3815	140	2	bid	bid	NOUN
cana-3815	140	3	a	a	PRON
cana-3815	140	4	of	of	ADP
cana-3815	140	5	a	a	DET
cana-3815	140	6	semiring	semiring	NOUN
cana-3815	140	7	§	§	NOUN
cana-3815	140	8	is	be	AUX
cana-3815	140	9	2pbid	2pbid	NUM
cana-3815	140	10	if	if	SCONJ
cana-3815	140	11	and	and	CCONJ
cana-3815	140	12	only	only	ADV
cana-3815	140	13	if	if	SCONJ
cana-3815	140	14	£	£	SYM
cana-3815	140	15	1£2	1£2	NUM
cana-3815	140	16	⊆	⊆	NUM
cana-3815	140	17	a	a	PRON
cana-3815	140	18	,	,	PUNCT
cana-3815	140	19	with	with	ADP
cana-3815	140	20	£	£	SYM
cana-3815	140	21	1	1	NUM
cana-3815	140	22	is	be	AUX
cana-3815	140	23	a	a	DET
cana-3815	140	24	rid	rid	NOUN
cana-3815	140	25	of	of	ADP
cana-3815	140	26	§	§	PROPN
cana-3815	140	27	and	and	CCONJ
cana-3815	140	28	£	£	SYM
cana-3815	140	29	2	2	NUM
cana-3815	140	30	is	be	AUX
cana-3815	140	31	a	a	DET
cana-3815	140	32	lid	lid	NOUN
cana-3815	140	33	of	of	ADP
cana-3815	140	34	§	§	PROPN
cana-3815	140	35	implies	imply	VERB
cana-3815	140	36	£	£	SYM
cana-3815	140	37	1	1	NUM
cana-3815	140	38	⊆	⊆	NUM
cana-3815	140	39	a	a	PRON
cana-3815	140	40	or	or	CCONJ
cana-3815	140	41	£	£	SYM
cana-3815	140	42	2	2	NUM
cana-3815	140	43	⊆	⊆	NUM
cana-3815	140	44	√	√	NOUN
cana-3815	140	45	a.	a.	NOUN
cana-3815	140	46	proof	proof	NOUN
cana-3815	140	47	.	.	PUNCT
cana-3815	141	1	let	let	VERB
cana-3815	141	2	a	a	PRON
cana-3815	141	3	be	be	AUX
cana-3815	141	4	a	a	DET
cana-3815	141	5	2pbid	2pbid	NUM
cana-3815	141	6	and	and	CCONJ
cana-3815	141	7	£	£	SYM
cana-3815	141	8	1£2	1£2	NUM
cana-3815	141	9	⊆	⊆	NUM
cana-3815	141	10	a.	a.	NOUN
cana-3815	141	11	suppose	suppose	VERB
cana-3815	141	12	£	£	SYM
cana-3815	141	13	1	1	NUM
cana-3815	141	14	6⊆	6⊆	NUM
cana-3815	141	15	a.	a.	NOUN
cana-3815	141	16	for	for	ADP
cana-3815	141	17	all	all	DET
cana-3815	141	18	℘	℘	PROPN
cana-3815	141	19	∈	∈	PROPN
cana-3815	141	20	£	£	SYM
cana-3815	141	21	2	2	NUM
cana-3815	141	22	and	and	CCONJ
cana-3815	141	23	∝∈	∝∈	X
cana-3815	141	24	£	£	SYM
cana-3815	141	25	1	1	NUM
cana-3815	141	26	\	\	NOUN
cana-3815	141	27	a	a	PRON
cana-3815	141	28	,	,	PUNCT
cana-3815	141	29	we	we	PRON
cana-3815	141	30	have	have	VERB
cana-3815	141	31	∝	∝	PROPN
cana-3815	141	32	§	§	NOUN
cana-3815	141	33	℘	℘	PROPN
cana-3815	141	34	⊆	⊆	NUM
cana-3815	141	35	£	£	SYM
cana-3815	141	36	1£2	1£2	NUM
cana-3815	141	37	⊆	⊆	NUM
cana-3815	141	38	a.	a.	NOUN
cana-3815	141	39	since	since	SCONJ
cana-3815	141	40	a	a	PRON
cana-3815	141	41	is	be	AUX
cana-3815	141	42	primary	primary	ADJ
cana-3815	141	43	and	and	CCONJ
cana-3815	141	44	∝6∈	∝6∈	ADJ
cana-3815	141	45	a	a	PRON
cana-3815	141	46	and	and	CCONJ
cana-3815	141	47	℘	℘	PROPN
cana-3815	141	48	∈	∈	PROPN
cana-3815	141	49	√	√	ADP
cana-3815	141	50	a	a	PRON
cana-3815	141	51	for	for	ADP
cana-3815	141	52	all	all	DET
cana-3815	141	53	℘	℘	PROPN
cana-3815	141	54	∈	∈	PROPN
cana-3815	141	55	£	£	SYM
cana-3815	141	56	2	2	NUM
cana-3815	141	57	.	.	PUNCT
cana-3815	142	1	so	so	ADV
cana-3815	142	2	£	£	SYM
cana-3815	142	3	2	2	NUM
cana-3815	142	4	⊆	⊆	NUM
cana-3815	142	5	√	√	NOUN
cana-3815	142	6	a.	a.	NOUN
cana-3815	142	7	conversely	conversely	ADV
cana-3815	142	8	,	,	PUNCT
cana-3815	142	9	suppose	suppose	VERB
cana-3815	142	10	that	that	SCONJ
cana-3815	142	11	∝	∝	PROPN
cana-3815	142	12	§	§	NOUN
cana-3815	142	13	℘	℘	PROPN
cana-3815	142	14	⊆	⊆	NUM
cana-3815	142	15	a.	a.	NOUN
cana-3815	142	16	now	now	ADV
cana-3815	142	17	,	,	PUNCT
cana-3815	142	18	(	(	PUNCT
cana-3815	142	19	∝	∝	PROPN
cana-3815	142	20	§	§	PROPN
cana-3815	142	21	)	)	PUNCT
cana-3815	142	22	(	(	PUNCT
cana-3815	142	23	§	§	NOUN
cana-3815	142	24	℘	℘	NOUN
cana-3815	142	25	)	)	PUNCT
cana-3815	142	26	⊆∝	⊆∝	ADP
cana-3815	142	27	§	§	PROPN
cana-3815	142	28	℘	℘	PROPN
cana-3815	142	29	implies	imply	VERB
cana-3815	142	30	∝	∝	PROPN
cana-3815	142	31	§	§	PROPN
cana-3815	142	32	⊆	⊆	NUM
cana-3815	142	33	a	a	DET
cana-3815	142	34	or	or	CCONJ
cana-3815	142	35	§	§	VERB
cana-3815	142	36	℘	℘	PROPN
cana-3815	142	37	⊆	⊆	SYM
cana-3815	142	38	√	√	NOUN
cana-3815	142	39	a.	a.	NOUN
cana-3815	142	40	if	if	SCONJ
cana-3815	142	41	∝	∝	PROPN
cana-3815	142	42	§	§	PROPN
cana-3815	142	43	⊆	⊆	SYM
cana-3815	142	44	a	a	PRON
cana-3815	142	45	,	,	PUNCT
cana-3815	142	46	then	then	ADV
cana-3815	142	47	<	<	X
cana-3815	142	48	∝>r	∝>r	ADJ
cana-3815	142	49	<	<	X
cana-3815	142	50	℘	℘	PROPN
cana-3815	142	51	>	>	PUNCT
cana-3815	142	52	l=	l=	ADJ
cana-3815	142	53	{	{	PUNCT
cana-3815	142	54	n	n	CCONJ
cana-3815	142	55	∝	∝	PROPN
cana-3815	142	56	+	+	CCONJ
cana-3815	142	57	∝	∝	PROPN
cana-3815	142	58	§	§	PROPN
cana-3815	142	59	|n	|n	NOUN
cana-3815	142	60	∈	∈	NOUN
cana-3815	142	61	z+	z+	X
cana-3815	142	62	}	}	PUNCT
cana-3815	142	63	·	·	PUNCT
cana-3815	142	64	{	{	PUNCT
cana-3815	142	65	m℘+	m℘+	PROPN
cana-3815	142	66	§	§	NOUN
cana-3815	142	67	℘|m	℘|m	ADJ
cana-3815	142	68	∈	∈	NOUN
cana-3815	142	69	z+	z+	NUM
cana-3815	142	70	}	}	PUNCT
cana-3815	142	71	=	=	SYM
cana-3815	142	72	n	n	CCONJ
cana-3815	142	73	∝	∝	PROPN
cana-3815	142	74	m℘+n	m℘+n	NUM
cana-3815	142	75	∝	∝	PROPN
cana-3815	142	76	§	§	PROPN
cana-3815	142	77	℘+	℘+	ADP
cana-3815	142	78	∝	∝	PROPN
cana-3815	142	79	℘+	℘+	ADP
cana-3815	142	80	∝	∝	PROPN
cana-3815	142	81	§	§	PROPN
cana-3815	142	82	§	§	PROPN
cana-3815	142	83	℘	℘	PROPN
cana-3815	142	84	⊆∝	⊆∝	ADP
cana-3815	142	85	§	§	PROPN
cana-3815	142	86	⊆	⊆	NUM
cana-3815	142	87	a.	a.	NOUN
cana-3815	142	88	thus	thus	ADV
cana-3815	142	89	,	,	PUNCT
cana-3815	142	90	∝∈	∝∈	X
cana-3815	142	91	a	a	PRON
cana-3815	142	92	or	or	CCONJ
cana-3815	142	93	℘	℘	PROPN
cana-3815	142	94	∈	∈	NOUN
cana-3815	142	95	√	√	NOUN
cana-3815	142	96	a.	a.	NOUN
cana-3815	142	97	similarly	similarly	ADV
cana-3815	142	98	,	,	PUNCT
cana-3815	142	99	suppose	suppose	VERB
cana-3815	142	100	that	that	SCONJ
cana-3815	142	101	§	§	NOUN
cana-3815	142	102	℘	℘	NOUN
cana-3815	142	103	⊆	⊆	SYM
cana-3815	142	104	√	√	ADP
cana-3815	142	105	a	a	DET
cana-3815	142	106	=	=	NOUN
cana-3815	142	107	⇒	⇒	NOUN
cana-3815	142	108	<	<	X
cana-3815	142	109	∝>r	∝>r	X
cana-3815	142	110	<	<	X
cana-3815	142	111	℘	℘	PROPN
cana-3815	142	112	>	>	SYM
cana-3815	142	113	l⊆	l⊆	NOUN
cana-3815	142	114	§	§	NOUN
cana-3815	142	115	℘	℘	DET
cana-3815	142	116	⊆	⊆	SYM
cana-3815	142	117	√	√	NOUN
cana-3815	142	118	a.	a.	NOUN
cana-3815	142	119	thus	thus	ADV
cana-3815	142	120	,	,	PUNCT
cana-3815	142	121	∝∈	∝∈	X
cana-3815	142	122	a	a	DET
cana-3815	142	123	or	or	CCONJ
cana-3815	142	124	℘	℘	PROPN
cana-3815	142	125	∈	∈	NOUN
cana-3815	142	126	√	√	NOUN
cana-3815	142	127	a.	a.	NOUN
cana-3815	142	128	theorem	theorem	NOUN
cana-3815	142	129	2.15	2.15	NUM
cana-3815	142	130	.	.	PUNCT
cana-3815	143	1	a	a	DET
cana-3815	143	2	bid	bid	NOUN
cana-3815	143	3	a	a	PRON
cana-3815	143	4	is	be	AUX
cana-3815	143	5	a	a	DET
cana-3815	143	6	3pbid	3pbid	NUM
cana-3815	143	7	of	of	ADP
cana-3815	143	8	§	§	PROPN
cana-3815	143	9	if	if	SCONJ
cana-3815	143	10	and	and	CCONJ
cana-3815	143	11	only	only	ADV
cana-3815	143	12	if	if	SCONJ
cana-3815	143	13	`	`	PUNCT
cana-3815	143	14	a	a	PRON
cana-3815	143	15	is	be	AUX
cana-3815	143	16	a	a	DET
cana-3815	143	17	pid	pid	NOUN
cana-3815	143	18	of	of	ADP
cana-3815	143	19	§	§	PROPN
cana-3815	143	20	.	.	PUNCT
cana-3815	144	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	144	2	757	757	NUM
cana-3815	144	3	communications	communication	NOUN
cana-3815	144	4	on	on	ADP
cana-3815	144	5	applied	apply	VERB
cana-3815	144	6	nonlinear	nonlinear	ADJ
cana-3815	144	7	analysis	analysis	NOUN
cana-3815	144	8	issn	issn	NOUN
cana-3815	144	9	:	:	PUNCT
cana-3815	144	10	1074	1074	NUM
cana-3815	144	11	-	-	PUNCT
cana-3815	144	12	133x	133x	NUM
cana-3815	144	13	vol	vol	NOUN
cana-3815	144	14	32	32	NUM
cana-3815	144	15	no	no	NOUN
cana-3815	144	16	.	.	NOUN
cana-3815	144	17	3	3	NUM
cana-3815	144	18	(	(	PUNCT
cana-3815	144	19	2025	2025	NUM
cana-3815	144	20	)	)	PUNCT
cana-3815	144	21	proof	proof	NOUN
cana-3815	144	22	.	.	PUNCT
cana-3815	145	1	let	let	VERB
cana-3815	145	2	a	a	DET
cana-3815	145	3	be	be	AUX
cana-3815	145	4	an	an	DET
cana-3815	145	5	3pbid	3pbid	NUM
cana-3815	145	6	of	of	ADP
cana-3815	145	7	§	§	PROPN
cana-3815	145	8	.	.	PUNCT
cana-3815	146	1	to	to	PART
cana-3815	146	2	show	show	VERB
cana-3815	146	3	that	that	SCONJ
cana-3815	146	4	`	`	PUNCT
cana-3815	146	5	a	a	PRON
cana-3815	146	6	is	be	AUX
cana-3815	146	7	a	a	DET
cana-3815	146	8	pid	pid	NOUN
cana-3815	146	9	of	of	ADP
cana-3815	146	10	§	§	PROPN
cana-3815	146	11	.	.	PUNCT
cana-3815	147	1	let	let	VERB
cana-3815	147	2	£	£	SYM
cana-3815	147	3	1	1	NUM
cana-3815	147	4	and	and	CCONJ
cana-3815	147	5	£	£	SYM
cana-3815	147	6	2	2	NUM
cana-3815	147	7	be	be	AUX
cana-3815	147	8	the	the	DET
cana-3815	147	9	ids	id	NOUN
cana-3815	147	10	of	of	ADP
cana-3815	147	11	§	§	PROPN
cana-3815	147	12	such	such	ADJ
cana-3815	147	13	that	that	SCONJ
cana-3815	147	14	£	£	SYM
cana-3815	147	15	1	1	NUM
cana-3815	147	16	·	·	SYM
cana-3815	147	17	£	£	SYM
cana-3815	147	18	2	2	NUM
cana-3815	147	19	⊆	⊆	NUM
cana-3815	147	20	`	`	PUNCT
cana-3815	147	21	a.	a.	NOUN
cana-3815	147	22	by	by	ADP
cana-3815	147	23	thm	thm	PROPN
cana-3815	147	24	2.13	2.13	NUM
cana-3815	147	25	,	,	PUNCT
cana-3815	147	26	2.14	2.14	NUM
cana-3815	147	27	and	and	CCONJ
cana-3815	147	28	proposition	proposition	NOUN
cana-3815	147	29	6,10	6,10	NUM
cana-3815	147	30	`	`	PUNCT
cana-3815	147	31	a	a	PRON
cana-3815	147	32	and	and	CCONJ
cana-3815	147	33	√	√	NUM
cana-3815	147	34	`	`	PUNCT
cana-3815	147	35	a	a	PRON
cana-3815	147	36	are	be	AUX
cana-3815	147	37	unique	unique	ADJ
cana-3815	147	38	largest	large	ADJ
cana-3815	147	39	tid	tid	NOUN
cana-3815	147	40	contained	contain	VERB
cana-3815	147	41	in	in	ADP
cana-3815	147	42	a	a	DET
cana-3815	147	43	and√	and√	NOUN
cana-3815	147	44	a	a	PRON
cana-3815	147	45	respectively	respectively	ADV
cana-3815	147	46	.	.	PUNCT
cana-3815	148	1	thus	thus	ADV
cana-3815	148	2	£	£	SYM
cana-3815	148	3	1	1	NUM
cana-3815	148	4	⊆	⊆	NUM
cana-3815	148	5	`	`	PUNCT
cana-3815	148	6	a	a	PRON
cana-3815	148	7	or	or	CCONJ
cana-3815	148	8	£	£	SYM
cana-3815	148	9	2	2	NUM
cana-3815	148	10	⊆	⊆	NUM
cana-3815	148	11	√	√	NUM
cana-3815	148	12	`	`	PUNCT
cana-3815	148	13	a.	a.	NOUN
cana-3815	148	14	conversely	conversely	ADV
cana-3815	148	15	,	,	PUNCT
cana-3815	148	16	suppose	suppose	VERB
cana-3815	148	17	that	that	SCONJ
cana-3815	148	18	£	£	SYM
cana-3815	148	19	1	1	NUM
cana-3815	148	20	and	and	CCONJ
cana-3815	148	21	£	£	SYM
cana-3815	148	22	2	2	NUM
cana-3815	148	23	are	be	AUX
cana-3815	148	24	ids	id	NOUN
cana-3815	148	25	of	of	ADP
cana-3815	148	26	§	§	PROPN
cana-3815	148	27	such	such	ADJ
cana-3815	148	28	that	that	SCONJ
cana-3815	148	29	£	£	SYM
cana-3815	148	30	1	1	NUM
cana-3815	148	31	·	·	SYM
cana-3815	148	32	£	£	SYM
cana-3815	148	33	2	2	NUM
cana-3815	148	34	⊆	⊆	NUM
cana-3815	148	35	a.	a.	NOUN
cana-3815	148	36	then	then	ADV
cana-3815	148	37	£	£	SYM
cana-3815	148	38	1	1	NUM
cana-3815	148	39	·	·	SYM
cana-3815	148	40	£	£	SYM
cana-3815	148	41	2	2	NUM
cana-3815	148	42	⊆	⊆	NUM
cana-3815	148	43	`	`	PUNCT
cana-3815	148	44	a	a	PRON
cana-3815	148	45	,	,	PUNCT
cana-3815	148	46	implies	imply	VERB
cana-3815	148	47	£	£	SYM
cana-3815	148	48	1	1	NUM
cana-3815	148	49	⊆	⊆	NUM
cana-3815	148	50	`	`	PUNCT
cana-3815	148	51	a	a	DET
cana-3815	148	52	⊆	⊆	NUM
cana-3815	148	53	a	a	PRON
cana-3815	148	54	or	or	CCONJ
cana-3815	148	55	£	£	SYM
cana-3815	148	56	2	2	NUM
cana-3815	148	57	⊆	⊆	NUM
cana-3815	148	58	√	√	NUM
cana-3815	148	59	`	`	PUNCT
cana-3815	148	60	a	a	DET
cana-3815	148	61	⊆	⊆	NUM
cana-3815	148	62	√	√	NOUN
cana-3815	148	63	a.	a.	NOUN
cana-3815	148	64	hence	hence	ADV
cana-3815	148	65	a	a	PRON
cana-3815	148	66	is	be	AUX
cana-3815	148	67	a	a	DET
cana-3815	148	68	3pbids	3pbids	NUM
cana-3815	148	69	of	of	ADP
cana-3815	148	70	§	§	PROPN
cana-3815	148	71	.	.	PUNCT
cana-3815	149	1	corollary	corollary	ADJ
cana-3815	149	2	2.16	2.16	NUM
cana-3815	149	3	.	.	PUNCT
cana-3815	150	1	if	if	SCONJ
cana-3815	150	2	a	a	PRON
cana-3815	150	3	is	be	AUX
cana-3815	150	4	a	a	DET
cana-3815	150	5	1pbid	1pbid	NUM
cana-3815	150	6	of	of	ADP
cana-3815	150	7	§	§	PROPN
cana-3815	150	8	,	,	PUNCT
cana-3815	150	9	then	then	ADV
cana-3815	150	10	`	`	PUNCT
cana-3815	150	11	a	a	PRON
cana-3815	150	12	is	be	AUX
cana-3815	150	13	a	a	DET
cana-3815	150	14	pid	pid	NOUN
cana-3815	150	15	of	of	ADP
cana-3815	150	16	§	§	PROPN
cana-3815	150	17	.	.	PUNCT
cana-3815	151	1	proof	proof	NOUN
cana-3815	151	2	.	.	PUNCT
cana-3815	152	1	let	let	VERB
cana-3815	152	2	a	a	PRON
cana-3815	152	3	be	be	AUX
cana-3815	152	4	an	an	DET
cana-3815	152	5	1pbid	1pbid	NUM
cana-3815	152	6	of	of	ADP
cana-3815	152	7	§	§	PROPN
cana-3815	152	8	.	.	PUNCT
cana-3815	153	1	let	let	VERB
cana-3815	153	2	us	we	PRON
cana-3815	153	3	show	show	VERB
cana-3815	153	4	that	that	SCONJ
cana-3815	153	5	`	`	PUNCT
cana-3815	153	6	a	a	PRON
cana-3815	153	7	is	be	AUX
cana-3815	153	8	a	a	DET
cana-3815	153	9	pid	pid	NOUN
cana-3815	153	10	of	of	ADP
cana-3815	153	11	§	§	PROPN
cana-3815	153	12	.	.	PUNCT
cana-3815	154	1	let	let	VERB
cana-3815	154	2	£	£	SYM
cana-3815	154	3	1	1	NUM
cana-3815	154	4	and	and	CCONJ
cana-3815	154	5	£	£	SYM
cana-3815	154	6	2	2	NUM
cana-3815	154	7	be	be	AUX
cana-3815	154	8	an	an	DET
cana-3815	154	9	ids	id	NOUN
cana-3815	154	10	of	of	ADP
cana-3815	154	11	§	§	PROPN
cana-3815	154	12	such	such	ADJ
cana-3815	154	13	that	that	SCONJ
cana-3815	154	14	£	£	SYM
cana-3815	154	15	1£2	1£2	NUM
cana-3815	154	16	⊆	⊆	NUM
cana-3815	154	17	`	`	PUNCT
cana-3815	154	18	a.	a.	NOUN
cana-3815	154	19	to	to	PART
cana-3815	154	20	show	show	VERB
cana-3815	154	21	that	that	SCONJ
cana-3815	154	22	£	£	SYM
cana-3815	154	23	1	1	NUM
cana-3815	154	24	⊆	⊆	NUM
cana-3815	154	25	`	`	PUNCT
cana-3815	154	26	a	a	DET
cana-3815	154	27	or	or	CCONJ
cana-3815	154	28	£	£	SYM
cana-3815	154	29	2	2	NUM
cana-3815	154	30	⊆	⊆	NUM
cana-3815	154	31	√	√	NUM
cana-3815	154	32	`	`	PUNCT
cana-3815	154	33	a.	a.	NOUN
cana-3815	154	34	since	since	SCONJ
cana-3815	154	35	`	`	PUNCT
cana-3815	154	36	a	a	DET
cana-3815	154	37	⊆	⊆	NUM
cana-3815	154	38	a	a	PRON
cana-3815	154	39	and	and	CCONJ
cana-3815	154	40	√	√	ADJ
cana-3815	154	41	`	`	PUNCT
cana-3815	154	42	a	a	DET
cana-3815	154	43	⊆	⊆	NUM
cana-3815	154	44	√	√	ADJ
cana-3815	154	45	a.	a.	NOUN
cana-3815	154	46	hence	hence	ADV
cana-3815	154	47	£	£	SYM
cana-3815	154	48	1£2	1£2	NUM
cana-3815	154	49	⊆	⊆	NUM
cana-3815	154	50	a.	a.	NOUN
cana-3815	154	51	since	since	SCONJ
cana-3815	154	52	£	£	SYM
cana-3815	154	53	1	1	NUM
cana-3815	154	54	and	and	CCONJ
cana-3815	154	55	£	£	SYM
cana-3815	154	56	2	2	NUM
cana-3815	154	57	are	be	AUX
cana-3815	154	58	ids	id	NOUN
cana-3815	154	59	of	of	ADP
cana-3815	154	60	§	§	PROPN
cana-3815	154	61	is	be	AUX
cana-3815	154	62	a	a	DET
cana-3815	154	63	bids	bid	NOUN
cana-3815	154	64	also	also	ADV
cana-3815	154	65	and	and	CCONJ
cana-3815	154	66	a	a	PRON
cana-3815	154	67	is	be	AUX
cana-3815	154	68	an	an	DET
cana-3815	154	69	1pbid	1pbid	NUM
cana-3815	154	70	of	of	ADP
cana-3815	154	71	§	§	PROPN
cana-3815	154	72	.	.	PUNCT
cana-3815	155	1	hence	hence	ADV
cana-3815	155	2	£	£	SYM
cana-3815	155	3	1	1	NUM
cana-3815	155	4	⊆	⊆	NUM
cana-3815	155	5	a	a	PRON
cana-3815	155	6	or	or	CCONJ
cana-3815	155	7	£	£	SYM
cana-3815	155	8	2	2	NUM
cana-3815	155	9	⊆	⊆	NUM
cana-3815	155	10	√	√	NOUN
cana-3815	155	11	a.	a.	NOUN
cana-3815	155	12	by	by	ADP
cana-3815	155	13	proposition	proposition	NOUN
cana-3815	155	14	6,10	6,10	NUM
cana-3815	155	15	`	`	PUNCT
cana-3815	155	16	a	a	PRON
cana-3815	155	17	is	be	AUX
cana-3815	155	18	the	the	DET
cana-3815	155	19	largest	large	ADJ
cana-3815	155	20	i	i	NOUN
cana-3815	155	21	d	d	PROPN
cana-3815	155	22	of	of	ADP
cana-3815	155	23	§	§	PROPN
cana-3815	155	24	such	such	ADJ
cana-3815	155	25	that	that	SCONJ
cana-3815	155	26	`	`	PUNCT
cana-3815	155	27	a	a	DET
cana-3815	155	28	⊆	⊆	NUM
cana-3815	155	29	a	a	PRON
cana-3815	155	30	and	and	CCONJ
cana-3815	155	31	by	by	ADP
cana-3815	155	32	thm	thm	PROPN
cana-3815	155	33	2.13	2.13	NUM
cana-3815	155	34	,	,	PUNCT
cana-3815	155	35	√	√	NUM
cana-3815	155	36	`	`	PUNCT
cana-3815	155	37	a	a	PRON
cana-3815	155	38	is	be	AUX
cana-3815	155	39	the	the	DET
cana-3815	155	40	largest	large	ADJ
cana-3815	155	41	i	i	NOUN
cana-3815	155	42	d	d	PROPN
cana-3815	155	43	of	of	ADP
cana-3815	155	44	§	§	PROPN
cana-3815	155	45	such	such	ADJ
cana-3815	155	46	that	that	PRON
cana-3815	155	47	√	√	ADP
cana-3815	155	48	`	`	PUNCT
cana-3815	155	49	a	a	DET
cana-3815	155	50	⊆	⊆	NUM
cana-3815	155	51	√	√	NOUN
cana-3815	155	52	a.	a.	NOUN
cana-3815	155	53	thus	thus	ADV
cana-3815	155	54	£	£	SYM
cana-3815	155	55	1	1	NUM
cana-3815	155	56	⊆	⊆	NUM
cana-3815	155	57	`	`	PUNCT
cana-3815	155	58	a	a	PRON
cana-3815	155	59	or	or	CCONJ
cana-3815	155	60	£	£	SYM
cana-3815	155	61	2	2	NUM
cana-3815	155	62	⊆	⊆	NUM
cana-3815	155	63	√	√	NUM
cana-3815	155	64	`	`	PUNCT
cana-3815	155	65	a.	a.	NOUN
cana-3815	155	66	hence	hence	ADV
cana-3815	155	67	`	`	PUNCT
cana-3815	155	68	a	a	PRON
cana-3815	155	69	is	be	AUX
cana-3815	155	70	a	a	DET
cana-3815	155	71	pid	pid	NOUN
cana-3815	155	72	of	of	ADP
cana-3815	155	73	§	§	PROPN
cana-3815	155	74	.	.	PUNCT
cana-3815	156	1	corollary	corollary	ADJ
cana-3815	156	2	2.17	2.17	NUM
cana-3815	156	3	.	.	PUNCT
cana-3815	157	1	if	if	SCONJ
cana-3815	157	2	a	a	PRON
cana-3815	157	3	is	be	AUX
cana-3815	157	4	a	a	DET
cana-3815	157	5	2pbid	2pbid	NUM
cana-3815	157	6	of	of	ADP
cana-3815	157	7	§	§	PROPN
cana-3815	157	8	,	,	PUNCT
cana-3815	157	9	then	then	ADV
cana-3815	157	10	`	`	PUNCT
cana-3815	157	11	a	a	PRON
cana-3815	157	12	is	be	AUX
cana-3815	157	13	a	a	DET
cana-3815	157	14	pid	pid	NOUN
cana-3815	157	15	of	of	ADP
cana-3815	157	16	§	§	PROPN
cana-3815	157	17	.	.	PUNCT
cana-3815	158	1	proof	proof	NOUN
cana-3815	158	2	.	.	PUNCT
cana-3815	159	1	let	let	VERB
cana-3815	159	2	a	a	PRON
cana-3815	159	3	be	be	AUX
cana-3815	159	4	an	an	DET
cana-3815	159	5	2pbid	2pbid	NUM
cana-3815	159	6	of	of	ADP
cana-3815	159	7	§	§	PROPN
cana-3815	159	8	.	.	PUNCT
cana-3815	160	1	let	let	VERB
cana-3815	160	2	us	we	PRON
cana-3815	160	3	show	show	VERB
cana-3815	160	4	that	that	SCONJ
cana-3815	160	5	`	`	PUNCT
cana-3815	160	6	a	a	PRON
cana-3815	160	7	is	be	AUX
cana-3815	160	8	a	a	DET
cana-3815	160	9	pid	pid	NOUN
cana-3815	160	10	of	of	ADP
cana-3815	160	11	§	§	PROPN
cana-3815	160	12	.	.	PUNCT
cana-3815	161	1	let	let	VERB
cana-3815	161	2	£	£	SYM
cana-3815	161	3	1	1	NUM
cana-3815	161	4	and	and	CCONJ
cana-3815	161	5	£	£	SYM
cana-3815	161	6	2	2	NUM
cana-3815	161	7	be	be	AUX
cana-3815	161	8	an	an	DET
cana-3815	161	9	ids	id	NOUN
cana-3815	161	10	of	of	ADP
cana-3815	161	11	§	§	PROPN
cana-3815	161	12	such	such	ADJ
cana-3815	161	13	that	that	SCONJ
cana-3815	161	14	£	£	SYM
cana-3815	161	15	1£2	1£2	NUM
cana-3815	161	16	⊆	⊆	NUM
cana-3815	161	17	`	`	PUNCT
cana-3815	161	18	a.	a.	NOUN
cana-3815	161	19	to	to	PART
cana-3815	161	20	show	show	VERB
cana-3815	161	21	that	that	SCONJ
cana-3815	161	22	£	£	SYM
cana-3815	161	23	1	1	NUM
cana-3815	161	24	⊆	⊆	NUM
cana-3815	161	25	`	`	PUNCT
cana-3815	161	26	a	a	DET
cana-3815	161	27	or	or	CCONJ
cana-3815	161	28	£	£	SYM
cana-3815	161	29	2	2	NUM
cana-3815	161	30	⊆	⊆	NUM
cana-3815	161	31	√	√	NUM
cana-3815	161	32	`	`	PUNCT
cana-3815	161	33	a.	a.	NOUN
cana-3815	161	34	since	since	SCONJ
cana-3815	161	35	`	`	PUNCT
cana-3815	161	36	a	a	DET
cana-3815	161	37	⊆	⊆	NUM
cana-3815	161	38	a	a	PRON
cana-3815	161	39	and	and	CCONJ
cana-3815	161	40	√	√	ADJ
cana-3815	161	41	`	`	PUNCT
cana-3815	161	42	a	a	DET
cana-3815	161	43	⊆	⊆	NUM
cana-3815	161	44	√	√	ADJ
cana-3815	161	45	a.	a.	NOUN
cana-3815	161	46	hence	hence	ADV
cana-3815	161	47	£	£	SYM
cana-3815	161	48	1£2	1£2	NUM
cana-3815	161	49	⊆	⊆	NUM
cana-3815	161	50	a.	a.	NOUN
cana-3815	161	51	since	since	SCONJ
cana-3815	161	52	£	£	SYM
cana-3815	161	53	1	1	NUM
cana-3815	161	54	is	be	AUX
cana-3815	161	55	an	an	DET
cana-3815	161	56	i	i	PROPN
cana-3815	161	57	d	d	PROPN
cana-3815	161	58	of	of	ADP
cana-3815	161	59	§	§	PROPN
cana-3815	161	60	is	be	AUX
cana-3815	161	61	an	an	DET
cana-3815	161	62	rid	rid	NOUN
cana-3815	161	63	also	also	ADV
cana-3815	161	64	and	and	CCONJ
cana-3815	161	65	£	£	SYM
cana-3815	161	66	2	2	NUM
cana-3815	161	67	is	be	AUX
cana-3815	161	68	an	an	DET
cana-3815	161	69	i	i	PROPN
cana-3815	161	70	d	d	PROPN
cana-3815	161	71	of	of	ADP
cana-3815	161	72	the	the	DET
cana-3815	161	73	ring	ring	NOUN
cana-3815	161	74	§	§	PROPN
cana-3815	161	75	is	be	AUX
cana-3815	161	76	an	an	DET
cana-3815	161	77	lid	lid	NOUN
cana-3815	161	78	also	also	ADV
cana-3815	161	79	.	.	PUNCT
cana-3815	162	1	since	since	SCONJ
cana-3815	162	2	a	a	PRON
cana-3815	162	3	is	be	AUX
cana-3815	162	4	an	an	DET
cana-3815	162	5	2pbid	2pbid	NUM
cana-3815	162	6	of	of	ADP
cana-3815	162	7	§	§	PROPN
cana-3815	162	8	.	.	PUNCT
cana-3815	163	1	hence	hence	ADV
cana-3815	163	2	£	£	SYM
cana-3815	163	3	1	1	NUM
cana-3815	163	4	⊆	⊆	NUM
cana-3815	163	5	a	a	PRON
cana-3815	163	6	or	or	CCONJ
cana-3815	163	7	£	£	SYM
cana-3815	163	8	2	2	NUM
cana-3815	163	9	⊆	⊆	NUM
cana-3815	163	10	√	√	NOUN
cana-3815	163	11	a.	a.	NOUN
cana-3815	163	12	by	by	ADP
cana-3815	163	13	proposition	proposition	NOUN
cana-3815	163	14	6,10	6,10	NUM
cana-3815	163	15	`	`	PUNCT
cana-3815	163	16	a	a	PRON
cana-3815	163	17	is	be	AUX
cana-3815	163	18	the	the	DET
cana-3815	163	19	largest	large	ADJ
cana-3815	163	20	i	i	NOUN
cana-3815	163	21	d	d	PROPN
cana-3815	163	22	of	of	ADP
cana-3815	163	23	§	§	PROPN
cana-3815	163	24	such	such	ADJ
cana-3815	163	25	that	that	SCONJ
cana-3815	163	26	`	`	PUNCT
cana-3815	163	27	a	a	DET
cana-3815	163	28	⊆	⊆	NUM
cana-3815	163	29	a	a	PRON
cana-3815	163	30	and	and	CCONJ
cana-3815	163	31	by	by	ADP
cana-3815	163	32	thm	thm	PROPN
cana-3815	163	33	2.13	2.13	NUM
cana-3815	163	34	,	,	PUNCT
cana-3815	163	35	√	√	NUM
cana-3815	163	36	`	`	PUNCT
cana-3815	163	37	a	a	PRON
cana-3815	163	38	is	be	AUX
cana-3815	163	39	the	the	DET
cana-3815	163	40	largest	large	ADJ
cana-3815	163	41	i	i	NOUN
cana-3815	163	42	d	d	PROPN
cana-3815	163	43	of	of	ADP
cana-3815	163	44	§	§	PROPN
cana-3815	163	45	such	such	ADJ
cana-3815	163	46	that	that	PRON
cana-3815	163	47	√	√	ADP
cana-3815	163	48	`	`	PUNCT
cana-3815	163	49	a	a	DET
cana-3815	163	50	⊆	⊆	NUM
cana-3815	163	51	√	√	NOUN
cana-3815	163	52	a.	a.	NOUN
cana-3815	163	53	thus	thus	ADV
cana-3815	163	54	£	£	SYM
cana-3815	163	55	1	1	NUM
cana-3815	163	56	⊆	⊆	NUM
cana-3815	163	57	`	`	PUNCT
cana-3815	163	58	a	a	PRON
cana-3815	163	59	or	or	CCONJ
cana-3815	163	60	£	£	SYM
cana-3815	163	61	2	2	NUM
cana-3815	163	62	⊆	⊆	NUM
cana-3815	163	63	√	√	NUM
cana-3815	163	64	`	`	PUNCT
cana-3815	163	65	a.	a.	NOUN
cana-3815	163	66	hence	hence	ADV
cana-3815	163	67	`	`	PUNCT
cana-3815	163	68	a	a	PRON
cana-3815	163	69	is	be	AUX
cana-3815	163	70	a	a	DET
cana-3815	163	71	pid	pid	NOUN
cana-3815	163	72	of	of	ADP
cana-3815	163	73	§	§	PROPN
cana-3815	163	74	.	.	PUNCT
cana-3815	164	1	theorem	theorem	VERB
cana-3815	164	2	2.18	2.18	NUM
cana-3815	164	3	.	.	PUNCT
cana-3815	165	1	let	let	VERB
cana-3815	165	2	d	d	PRON
cana-3815	165	3	be	be	AUX
cana-3815	165	4	a	a	DET
cana-3815	165	5	mp3	mp3	NOUN
cana-3815	165	6	sys	sys	NOUN
cana-3815	165	7	and	and	CCONJ
cana-3815	165	8	a	a	DET
cana-3815	165	9	be	be	AUX
cana-3815	165	10	a	a	DET
cana-3815	165	11	bid	bid	NOUN
cana-3815	165	12	of	of	ADP
cana-3815	165	13	§	§	PROPN
cana-3815	165	14	with	with	ADP
cana-3815	165	15	a	a	DET
cana-3815	165	16	∩d	∩d	NOUN
cana-3815	165	17	=	=	SYM
cana-3815	165	18	φ	φ	PROPN
cana-3815	165	19	.	.	PUNCT
cana-3815	166	1	then	then	ADV
cana-3815	166	2	∃	∃	PROPN
cana-3815	166	3	a	a	DET
cana-3815	166	4	3pbid	3pbid	NUM
cana-3815	166	5	ℵ	ℵ	NOUN
cana-3815	166	6	of	of	ADP
cana-3815	166	7	§	§	PROPN
cana-3815	166	8	containing	contain	VERB
cana-3815	166	9	a	a	PRON
cana-3815	166	10	with	with	ADP
cana-3815	166	11	ℵ	ℵ	NOUN
cana-3815	166	12	∩d	∩d	NOUN
cana-3815	166	13	=	=	X
cana-3815	166	14	φ	φ	PROPN
cana-3815	166	15	.	.	PUNCT
cana-3815	166	16	proof	proof	NOUN
cana-3815	166	17	.	.	PUNCT
cana-3815	167	1	let	let	VERB
cana-3815	167	2	x	x	PUNCT
cana-3815	167	3	=	=	PRON
cana-3815	167	4	{	{	PUNCT
cana-3815	167	5	£	£	SYM
cana-3815	167	6	2|£2	2|£2	NUM
cana-3815	167	7	is	be	AUX
cana-3815	167	8	a	a	DET
cana-3815	167	9	bid	bid	NOUN
cana-3815	167	10	with	with	ADP
cana-3815	167	11	a	a	DET
cana-3815	167	12	⊆	⊆	NUM
cana-3815	167	13	£	£	SYM
cana-3815	167	14	2	2	NUM
cana-3815	167	15	and	and	CCONJ
cana-3815	167	16	£	£	SYM
cana-3815	167	17	2	2	NUM
cana-3815	167	18	∩d	∩d	NOUN
cana-3815	167	19	=	=	X
cana-3815	167	20	φ	φ	PROPN
cana-3815	167	21	}	}	PUNCT
cana-3815	167	22	.	.	PUNCT
cana-3815	168	1	clearly	clearly	ADV
cana-3815	168	2	x	x	PRON
cana-3815	168	3	is	be	AUX
cana-3815	168	4	non	non	ADJ
cana-3815	168	5	-	-	ADJ
cana-3815	168	6	empty	empty	ADJ
cana-3815	168	7	.	.	PUNCT
cana-3815	169	1	by	by	ADP
cana-3815	169	2	zorn	zorn	PROPN
cana-3815	169	3	’s	’s	PART
cana-3815	169	4	lem	lem	PROPN
cana-3815	169	5	,	,	PUNCT
cana-3815	169	6	∃	∃	PROPN
cana-3815	169	7	an	an	DET
cana-3815	169	8	maximal	maximal	ADJ
cana-3815	169	9	element	element	NOUN
cana-3815	169	10	ℵ	ℵ	NOUN
cana-3815	169	11	in	in	ADP
cana-3815	169	12	x	x	X
cana-3815	169	13	.	.	PUNCT
cana-3815	170	1	we	we	PRON
cana-3815	170	2	claim	claim	VERB
cana-3815	170	3	that	that	SCONJ
cana-3815	170	4	ℵ	ℵ	NOUN
cana-3815	170	5	is	be	AUX
cana-3815	170	6	a	a	DET
cana-3815	170	7	3pbid	3pbid	NUM
cana-3815	170	8	of	of	ADP
cana-3815	170	9	§	§	PROPN
cana-3815	170	10	.	.	PUNCT
cana-3815	171	1	it	it	PRON
cana-3815	171	2	is	be	AUX
cana-3815	171	3	enough	enough	ADJ
cana-3815	171	4	if	if	SCONJ
cana-3815	171	5	we	we	PRON
cana-3815	171	6	show	show	VERB
cana-3815	171	7	that	that	SCONJ
cana-3815	171	8	`	`	PUNCT
cana-3815	171	9	ℵ	ℵ	NOUN
cana-3815	171	10	is	be	AUX
cana-3815	171	11	a	a	DET
cana-3815	171	12	pid	pid	NOUN
cana-3815	171	13	in	in	ADP
cana-3815	171	14	§	§	PROPN
cana-3815	171	15	.	.	PUNCT
cana-3815	172	1	since	since	SCONJ
cana-3815	172	2	`	`	PUNCT
cana-3815	172	3	ℵ	ℵ	PRON
cana-3815	172	4	⊆	⊆	NUM
cana-3815	172	5	ℵ	ℵ	NOUN
cana-3815	172	6	and	and	CCONJ
cana-3815	172	7	ℵ	ℵ	NOUN
cana-3815	172	8	∩d	∩d	NOUN
cana-3815	172	9	=	=	PUNCT
cana-3815	172	10	φ	φ	PROPN
cana-3815	172	11	,	,	PUNCT
cana-3815	172	12	this	this	DET
cana-3815	172	13	=	=	AUX
cana-3815	172	14	⇒	⇒	VERB
cana-3815	172	15	`	`	PUNCT
cana-3815	172	16	ℵ	ℵ	NOUN
cana-3815	172	17	∩d	∩d	NOUN
cana-3815	172	18	=	=	PUNCT
cana-3815	172	19	φ	φ	PROPN
cana-3815	172	20	.	.	PUNCT
cana-3815	173	1	then	then	ADV
cana-3815	173	2	`	`	PUNCT
cana-3815	173	3	ℵ	ℵ	NOUN
cana-3815	173	4	is	be	AUX
cana-3815	173	5	a	a	DET
cana-3815	173	6	largest	large	ADJ
cana-3815	173	7	i	i	NOUN
cana-3815	173	8	d	d	PROPN
cana-3815	173	9	in	in	ADP
cana-3815	173	10	§	§	PROPN
cana-3815	174	1	such	such	ADJ
cana-3815	174	2	that	that	SCONJ
cana-3815	174	3	`	`	PUNCT
cana-3815	174	4	ℵ	ℵ	ADJ
cana-3815	174	5	∩d	∩d	NOUN
cana-3815	174	6	=	=	PUNCT
cana-3815	174	7	φ	φ	X
cana-3815	174	8	.	.	PUNCT
cana-3815	175	1	we	we	PRON
cana-3815	175	2	claim	claim	VERB
cana-3815	175	3	that	that	SCONJ
cana-3815	175	4	<	<	X
cana-3815	175	5	∝	∝	X
cana-3815	175	6	>	>	X
cana-3815	175	7	<	<	X
cana-3815	175	8	℘	℘	PROPN
cana-3815	175	9	>	>	SYM
cana-3815	175	10	⊆	⊆	NUM
cana-3815	175	11	`	`	PUNCT
cana-3815	175	12	ℵ.	ℵ.	PROPN
cana-3815	175	13	then	then	ADV
cana-3815	175	14	<	<	X
cana-3815	175	15	∝>⊆	∝>⊆	PROPN
cana-3815	175	16	`	`	PUNCT
cana-3815	175	17	(	(	PUNCT
cana-3815	175	18	q	q	X
cana-3815	175	19	)	)	PUNCT
cana-3815	175	20	or	or	CCONJ
cana-3815	175	21	<	<	X
cana-3815	175	22	℘	℘	PROPN
cana-3815	175	23	>	>	SYM
cana-3815	175	24	⊆	⊆	NUM
cana-3815	175	25	`	`	PUNCT
cana-3815	175	26	(	(	PUNCT
cana-3815	175	27	q	q	NOUN
cana-3815	175	28	)	)	PUNCT
cana-3815	175	29	.	.	PUNCT
cana-3815	176	1	if	if	SCONJ
cana-3815	176	2	<	<	X
cana-3815	176	3	∝>6⊆	∝>6⊆	X
cana-3815	176	4	`	`	PUNCT
cana-3815	176	5	ℵ	ℵ	NOUN
cana-3815	176	6	and	and	CCONJ
cana-3815	176	7	<	<	X
cana-3815	176	8	℘	℘	PROPN
cana-3815	176	9	>	>	PUNCT
cana-3815	176	10	6⊆	6⊆	NUM
cana-3815	176	11	√	√	NUM
cana-3815	176	12	`	`	PUNCT
cana-3815	176	13	ℵ	ℵ	PROPN
cana-3815	176	14	,	,	PUNCT
cana-3815	176	15	then	then	ADV
cana-3815	176	16	ς	ς	PROPN
cana-3815	176	17	∈<∝	∈<∝	PROPN
cana-3815	176	18	>	>	X
cana-3815	176	19	\`ℵ	\`ℵ	NOUN
cana-3815	176	20	and	and	CCONJ
cana-3815	176	21	ε	ε	PROPN
cana-3815	176	22	∈	∈	PROPN
cana-3815	176	23	<	<	X
cana-3815	176	24	℘	℘	PROPN
cana-3815	176	25	>	>	PUNCT
cana-3815	176	26	\	\	NOUN
cana-3815	176	27	√	√	PUNCT
cana-3815	177	1	`	`	PUNCT
cana-3815	177	2	ℵ.	ℵ.	PROPN
cana-3815	177	3	then	then	ADV
cana-3815	177	4	<	<	X
cana-3815	177	5	ς	ς	PROPN
cana-3815	177	6	>	>	X
cana-3815	177	7	⊆<∝	⊆<∝	NOUN
cana-3815	177	8	>	>	X
cana-3815	177	9	and	and	CCONJ
cana-3815	177	10	<	<	X
cana-3815	177	11	ε	ε	PROPN
cana-3815	177	12	>	>	PUNCT
cana-3815	177	13	⊆	⊆	X
cana-3815	177	14	<	<	X
cana-3815	177	15	℘	℘	PROPN
cana-3815	177	16	>	>	PUNCT
cana-3815	177	17	.	.	PUNCT
cana-3815	178	1	if	if	SCONJ
cana-3815	178	2	<	<	X
cana-3815	178	3	∝	∝	X
cana-3815	178	4	>	>	X
cana-3815	178	5	<	<	X
cana-3815	178	6	℘	℘	PROPN
cana-3815	178	7	>	>	SYM
cana-3815	178	8	⊆	⊆	NUM
cana-3815	178	9	`	`	PUNCT
cana-3815	178	10	ℵ	ℵ	NOUN
cana-3815	178	11	then	then	ADV
cana-3815	178	12	<	<	X
cana-3815	178	13	ς	ς	X
cana-3815	178	14	>	>	X
cana-3815	178	15	<	<	X
cana-3815	178	16	ε	ε	PROPN
cana-3815	178	17	>	>	X
cana-3815	178	18	⊆<∝	⊆<∝	PROPN
cana-3815	178	19	>	>	X
cana-3815	178	20	<	<	X
cana-3815	178	21	℘	℘	PROPN
cana-3815	178	22	>	>	PUNCT
cana-3815	178	23	⊆	⊆	NUM
cana-3815	178	24	`	`	PUNCT
cana-3815	178	25	ℵ.	ℵ.	PROPN
cana-3815	178	26	since	since	SCONJ
cana-3815	178	27	<	<	X
cana-3815	178	28	℘	℘	PROPN
cana-3815	178	29	>	>	SYM
cana-3815	178	30	6⊆	6⊆	NUM
cana-3815	178	31	√	√	ADP
cana-3815	178	32	`	`	PUNCT
cana-3815	178	33	ℵ	ℵ	NOUN
cana-3815	178	34	and	and	CCONJ
cana-3815	178	35	hence	hence	ADV
cana-3815	178	36	(	(	PUNCT
cana-3815	178	37	<	<	X
cana-3815	178	38	℘	℘	PROPN
cana-3815	178	39	>	>	SYM
cana-3815	178	40	)	)	PUNCT
cana-3815	178	41	n	n	PROPN
cana-3815	178	42	6⊆	6⊆	NUM
cana-3815	178	43	`	`	PUNCT
cana-3815	178	44	ℵ	ℵ	PROPN
cana-3815	178	45	=	=	NOUN
cana-3815	178	46	⇒	⇒	NOUN
cana-3815	178	47	<	<	X
cana-3815	178	48	℘	℘	PROPN
cana-3815	178	49	>	>	PUNCT
cana-3815	178	50	6⊆	6⊆	NUM
cana-3815	179	1	`	`	PUNCT
cana-3815	179	2	ℵ.	ℵ.	PROPN
cana-3815	179	3	then	then	ADV
cana-3815	179	4	(	(	PUNCT
cana-3815	179	5	`	`	PUNCT
cana-3815	179	6	ℵ+	ℵ+	X
cana-3815	179	7	<	<	X
cana-3815	179	8	ς	ς	PROPN
cana-3815	179	9	>	>	PUNCT
cana-3815	179	10	)	)	PUNCT
cana-3815	179	11	∩d	∩d	VERB
cana-3815	180	1	6=	6=	PROPN
cana-3815	180	2	φ	φ	PROPN
cana-3815	180	3	and	and	CCONJ
cana-3815	180	4	(	(	PUNCT
cana-3815	180	5	`	`	PUNCT
cana-3815	180	6	ℵ+	ℵ+	X
cana-3815	180	7	<	<	X
cana-3815	180	8	ε	ε	PROPN
cana-3815	180	9	>	>	PUNCT
cana-3815	180	10	)	)	PUNCT
cana-3815	180	11	∩d	∩d	VERB
cana-3815	181	1	6=	6=	PROPN
cana-3815	181	2	φ	φ	PROPN
cana-3815	181	3	.	.	PUNCT
cana-3815	182	1	thus	thus	ADV
cana-3815	182	2	(	(	PUNCT
cana-3815	182	3	`	`	PUNCT
cana-3815	182	4	ℵ+	ℵ+	X
cana-3815	182	5	<	<	X
cana-3815	182	6	ς	ς	PROPN
cana-3815	182	7	>	>	PROPN
cana-3815	182	8	)	)	PUNCT
cana-3815	182	9	(	(	PUNCT
cana-3815	182	10	`	`	PUNCT
cana-3815	182	11	ℵ+	ℵ+	X
cana-3815	182	12	<	<	X
cana-3815	182	13	ε	ε	PROPN
cana-3815	182	14	>	>	PUNCT
cana-3815	182	15	)	)	PUNCT
cana-3815	182	16	⊆	⊆	NUM
cana-3815	183	1	`	`	PUNCT
cana-3815	183	2	ℵ.	ℵ.	PROPN
cana-3815	183	3	then	then	ADV
cana-3815	183	4	the	the	DET
cana-3815	183	5	bid	bid	NOUN
cana-3815	183	6	(	(	PUNCT
cana-3815	183	7	`	`	PUNCT
cana-3815	183	8	(	(	PUNCT
cana-3815	183	9	q	q	X
cana-3815	183	10	)	)	PUNCT
cana-3815	183	11	+	+	CCONJ
cana-3815	183	12	ς	ς	X
cana-3815	183	13	)	)	PUNCT
cana-3815	183	14	contains	contain	VERB
cana-3815	183	15	an	an	DET
cana-3815	183	16	element	element	NOUN
cana-3815	183	17	mp1	mp1	NOUN
cana-3815	183	18	of	of	ADP
cana-3815	183	19	d.	d.	PROPN
cana-3815	183	20	then	then	ADV
cana-3815	183	21	∃	∃	PROPN
cana-3815	183	22	$	$	SYM
cana-3815	183	23	1	1	NUM
cana-3815	183	24	∈	∈	NOUN
cana-3815	183	25	(	(	PUNCT
cana-3815	183	26	`	`	PUNCT
cana-3815	183	27	ℵ+	ℵ+	X
cana-3815	183	28	<	<	X
cana-3815	183	29	ς	ς	PROPN
cana-3815	183	30	>	>	PUNCT
cana-3815	183	31	)	)	PUNCT
cana-3815	184	1	∩d	∩d	NOUN
cana-3815	184	2	.	.	PUNCT
cana-3815	185	1	similarly	similarly	ADV
cana-3815	185	2	the	the	DET
cana-3815	185	3	bid	bid	NOUN
cana-3815	185	4	(	(	PUNCT
cana-3815	185	5	`	`	PUNCT
cana-3815	185	6	(	(	PUNCT
cana-3815	185	7	q	q	X
cana-3815	185	8	)	)	PUNCT
cana-3815	185	9	+	+	CCONJ
cana-3815	185	10	ε	ε	PROPN
cana-3815	185	11	)	)	PUNCT
cana-3815	185	12	contains	contain	VERB
cana-3815	185	13	an	an	DET
cana-3815	185	14	element	element	ADJ
cana-3815	185	15	mp2	mp2	PROPN
cana-3815	185	16	of	of	ADP
cana-3815	185	17	d.	d.	PROPN
cana-3815	185	18	then	then	ADV
cana-3815	185	19	∃$2	∃$2	PROPN
cana-3815	185	20	∈	∈	PROPN
cana-3815	185	21	(	(	PUNCT
cana-3815	185	22	`	`	PUNCT
cana-3815	185	23	ℵ+	ℵ+	X
cana-3815	185	24	<	<	X
cana-3815	185	25	ε	ε	PROPN
cana-3815	185	26	>	>	PUNCT
cana-3815	185	27	)	)	PUNCT
cana-3815	185	28	∩d	∩d	NOUN
cana-3815	185	29	.	.	PUNCT
cana-3815	186	1	since	since	SCONJ
cana-3815	186	2	d	d	PROPN
cana-3815	186	3	is	be	AUX
cana-3815	186	4	mp3	mp3	NOUN
cana-3815	186	5	-sys	-sys	PUNCT
cana-3815	186	6	of	of	ADP
cana-3815	186	7	a	a	DET
cana-3815	186	8	,	,	PUNCT
cana-3815	186	9	$	$	SYM
cana-3815	186	10	1	1	NUM
cana-3815	186	11	′	′	NUM
cana-3815	186	12	∈	∈	PROPN
cana-3815	186	13	<	<	X
cana-3815	186	14	$	$	SYM
cana-3815	186	15	1	1	NUM
cana-3815	186	16	>	>	PUNCT
cana-3815	186	17	and	and	CCONJ
cana-3815	186	18	$	$	SYM
cana-3815	186	19	2	2	NUM
cana-3815	186	20	′	′	NUM
cana-3815	186	21	∈	∈	PROPN
cana-3815	186	22	<	<	X
cana-3815	186	23	$	$	SYM
cana-3815	186	24	2	2	NUM
cana-3815	186	25	>	>	SYM
cana-3815	186	26	$	$	SYM
cana-3815	186	27	1	1	NUM
cana-3815	186	28	′	′	NUM
cana-3815	186	29	$	$	SYM
cana-3815	186	30	2	2	NUM
cana-3815	186	31	′	′	NUM
cana-3815	186	32	∈	∈	PROPN
cana-3815	186	33	d	d	NOUN
cana-3815	186	34	for	for	ADP
cana-3815	186	35	some	some	DET
cana-3815	186	36	$	$	SYM
cana-3815	186	37	1	1	NUM
cana-3815	186	38	′	′	NUM
cana-3815	186	39	∈	∈	PROPN
cana-3815	186	40	<	<	X
cana-3815	186	41	$	$	SYM
cana-3815	186	42	1	1	NUM
cana-3815	186	43	>	>	SYM
cana-3815	186	44	⊆	⊆	NUM
cana-3815	186	45	(	(	PUNCT
cana-3815	186	46	`	`	PUNCT
cana-3815	186	47	ℵ+	ℵ+	X
cana-3815	186	48	<	<	X
cana-3815	186	49	ς	ς	PROPN
cana-3815	186	50	>	>	PUNCT
cana-3815	186	51	)	)	PUNCT
cana-3815	186	52	and	and	CCONJ
cana-3815	186	53	$	$	SYM
cana-3815	186	54	2	2	NUM
cana-3815	186	55	′	′	NUM
cana-3815	186	56	∈	∈	PROPN
cana-3815	186	57	<	<	X
cana-3815	186	58	$	$	SYM
cana-3815	186	59	2	2	NUM
cana-3815	186	60	>	>	SYM
cana-3815	186	61	⊆	⊆	NUM
cana-3815	186	62	(	(	PUNCT
cana-3815	186	63	`	`	PUNCT
cana-3815	186	64	ℵ+	ℵ+	X
cana-3815	186	65	<	<	X
cana-3815	186	66	ε	ε	PROPN
cana-3815	186	67	>	>	PUNCT
cana-3815	186	68	)	)	PUNCT
cana-3815	186	69	.	.	PUNCT
cana-3815	187	1	hence	hence	ADV
cana-3815	187	2	$	$	SYM
cana-3815	187	3	1	1	NUM
cana-3815	187	4	′	′	NUM
cana-3815	187	5	$	$	SYM
cana-3815	187	6	2	2	NUM
cana-3815	187	7	′	′	NUM
cana-3815	187	8	∈	∈	NOUN
cana-3815	187	9	(	(	PUNCT
cana-3815	187	10	`	`	PUNCT
cana-3815	187	11	ℵ+	ℵ+	X
cana-3815	187	12	<	<	X
cana-3815	187	13	ς	ς	PROPN
cana-3815	187	14	>	>	PROPN
cana-3815	187	15	)	)	PUNCT
cana-3815	187	16	(	(	PUNCT
cana-3815	187	17	`	`	PUNCT
cana-3815	187	18	ℵ+	ℵ+	X
cana-3815	187	19	<	<	X
cana-3815	187	20	ε	ε	PROPN
cana-3815	187	21	>	>	PUNCT
cana-3815	187	22	)	)	PUNCT
cana-3815	187	23	⊆	⊆	NUM
cana-3815	187	24	`	`	PUNCT
cana-3815	187	25	ℵ.	ℵ.	NOUN
cana-3815	187	26	which	which	PRON
cana-3815	187	27	is	be	AUX
cana-3815	187	28	a	a	DET
cana-3815	187	29	contradiction	contradiction	NOUN
cana-3815	187	30	.	.	PUNCT
cana-3815	188	1	thus	thus	ADV
cana-3815	188	2	<	<	X
cana-3815	188	3	∝	∝	X
cana-3815	188	4	>	>	X
cana-3815	188	5	<	<	X
cana-3815	188	6	℘	℘	PROPN
cana-3815	188	7	>	>	X
cana-3815	188	8	6⊆	6⊆	NUM
cana-3815	188	9	`	`	PUNCT
cana-3815	188	10	ℵ.	ℵ.	PROPN
cana-3815	188	11	hence	hence	ADV
cana-3815	188	12	`	`	PUNCT
cana-3815	188	13	ℵ	ℵ	NOUN
cana-3815	188	14	is	be	AUX
cana-3815	188	15	a	a	DET
cana-3815	188	16	primary	primary	ADJ
cana-3815	188	17	i	i	PROPN
cana-3815	188	18	d	d	PROPN
cana-3815	188	19	of	of	ADP
cana-3815	188	20	§	§	PROPN
cana-3815	188	21	.	.	PUNCT
cana-3815	189	1	by	by	ADP
cana-3815	189	2	thm	thm	PROPN
cana-3815	189	3	2.15	2.15	NUM
cana-3815	189	4	,	,	PUNCT
cana-3815	189	5	then	then	ADV
cana-3815	189	6	there	there	PRON
cana-3815	189	7	is	be	VERB
cana-3815	189	8	an	an	DET
cana-3815	189	9	maximal	maximal	ADJ
cana-3815	189	10	i	i	NOUN
cana-3815	189	11	d	d	NOUN
cana-3815	189	12	ℵ′	ℵ′	PUNCT
cana-3815	189	13	in	in	ADP
cana-3815	189	14	§	§	PROPN
cana-3815	189	15	such	such	ADJ
cana-3815	189	16	that	that	SCONJ
cana-3815	189	17	`	`	PUNCT
cana-3815	189	18	ℵ	ℵ	ADP
cana-3815	189	19	⊆	⊆	NUM
cana-3815	189	20	ℵ′	ℵ′	NUM
cana-3815	189	21	and	and	CCONJ
cana-3815	189	22	ℵ′	ℵ′	ADV
cana-3815	189	23	∩d	∩d	NOUN
cana-3815	190	1	=	=	X
cana-3815	190	2	φ	φ	X
cana-3815	190	3	.	.	PUNCT
cana-3815	191	1	hence	hence	ADV
cana-3815	191	2	ℵ′	ℵ′	ADV
cana-3815	191	3	is	be	AUX
cana-3815	191	4	the	the	DET
cana-3815	191	5	bid	bid	NOUN
cana-3815	191	6	of	of	ADP
cana-3815	191	7	§	§	PROPN
cana-3815	191	8	.	.	PROPN
cana-3815	191	9	3	3	NUM
cana-3815	191	10	characterization	characterization	NOUN
cana-3815	191	11	of	of	ADP
cana-3815	191	12	spbids	spbid	NOUN
cana-3815	191	13	definition	definition	NOUN
cana-3815	191	14	3.1	3.1	NUM
cana-3815	191	15	.	.	PUNCT
cana-3815	192	1	a	a	DET
cana-3815	192	2	bid	bid	NOUN
cana-3815	192	3	ℵ	ℵ	NOUN
cana-3815	192	4	of	of	ADP
cana-3815	192	5	§	§	PROPN
cana-3815	192	6	is	be	AUX
cana-3815	192	7	said	say	VERB
cana-3815	192	8	to	to	PART
cana-3815	192	9	be	be	AUX
cana-3815	192	10	(	(	PUNCT
cana-3815	192	11	i	i	NOUN
cana-3815	192	12	)	)	PUNCT
cana-3815	192	13	1spbid	1spbid	NUM
cana-3815	192	14	if	if	SCONJ
cana-3815	192	15	a2	a2	PROPN
cana-3815	192	16	⊆	⊆	NUM
cana-3815	192	17	ℵ	ℵ	NOUN
cana-3815	192	18	implies	imply	VERB
cana-3815	192	19	a	a	DET
cana-3815	192	20	⊆	⊆	NUM
cana-3815	192	21	ℵ	ℵ	NOUN
cana-3815	192	22	or	or	CCONJ
cana-3815	192	23	a	a	DET
cana-3815	192	24	⊆	⊆	NUM
cana-3815	192	25	√	√	NUM
cana-3815	192	26	ℵ	ℵ	NOUN
cana-3815	192	27	for	for	ADP
cana-3815	192	28	any	any	DET
cana-3815	192	29	bid	bid	NOUN
cana-3815	192	30	a	a	PRON
cana-3815	192	31	of	of	ADP
cana-3815	192	32	§	§	PROPN
cana-3815	192	33	.	.	PUNCT
cana-3815	193	1	(	(	PUNCT
cana-3815	193	2	ii	ii	NOUN
cana-3815	193	3	)	)	PUNCT
cana-3815	193	4	2spbid	2spbid	NUM
cana-3815	193	5	if	if	SCONJ
cana-3815	193	6	∝	∝	PROPN
cana-3815	193	7	§	§	PROPN
cana-3815	193	8	∝⊆	∝⊆	X
cana-3815	193	9	ℵ	ℵ	ADJ
cana-3815	193	10	implies	implie	NOUN
cana-3815	193	11	∝∈	∝∈	PUNCT
cana-3815	193	12	ℵ	ℵ	NOUN
cana-3815	193	13	or	or	CCONJ
cana-3815	193	14	∝∈	∝∈	PUNCT
cana-3815	193	15	√	√	ADJ
cana-3815	193	16	ℵ.	ℵ.	NOUN
cana-3815	193	17	(	(	PUNCT
cana-3815	193	18	iii	iii	X
cana-3815	193	19	)	)	PUNCT
cana-3815	193	20	3spbid	3spbid	PROPN
cana-3815	193	21	if	if	SCONJ
cana-3815	193	22	£	£	SYM
cana-3815	193	23	2	2	NUM
cana-3815	193	24	1	1	NUM
cana-3815	193	25	⊆	⊆	NUM
cana-3815	193	26	ℵ	ℵ	NOUN
cana-3815	193	27	implies	imply	VERB
cana-3815	193	28	£	£	SYM
cana-3815	193	29	1	1	NUM
cana-3815	193	30	⊆	⊆	NUM
cana-3815	193	31	ℵ	ℵ	NOUN
cana-3815	193	32	or	or	CCONJ
cana-3815	193	33	£	£	SYM
cana-3815	193	34	1	1	NUM
cana-3815	193	35	⊆	⊆	NUM
cana-3815	193	36	√	√	NUM
cana-3815	193	37	ℵ	ℵ	NOUN
cana-3815	193	38	for	for	ADP
cana-3815	193	39	any	any	DET
cana-3815	193	40	i	i	PROPN
cana-3815	193	41	d	d	PROPN
cana-3815	193	42	£	£	SYM
cana-3815	193	43	1	1	NUM
cana-3815	193	44	of	of	ADP
cana-3815	193	45	§	§	PROPN
cana-3815	193	46	.	.	PUNCT
cana-3815	193	47	theorem	theorem	ADJ
cana-3815	193	48	3.2	3.2	NUM
cana-3815	193	49	.	.	PUNCT
cana-3815	194	1	every	every	DET
cana-3815	194	2	1spbid	1spbid	PROPN
cana-3815	194	3	is	be	AUX
cana-3815	194	4	a	a	DET
cana-3815	194	5	2spbid	2spbid	NUM
cana-3815	194	6	of	of	ADP
cana-3815	194	7	§	§	PROPN
cana-3815	194	8	.	.	PUNCT
cana-3815	195	1	proof	proof	NOUN
cana-3815	195	2	.	.	PUNCT
cana-3815	196	1	let	let	VERB
cana-3815	196	2	ℵ	ℵ	NOUN
cana-3815	196	3	is	be	AUX
cana-3815	196	4	a	a	DET
cana-3815	196	5	1spbid	1spbid	NUM
cana-3815	196	6	of	of	ADP
cana-3815	196	7	§	§	PROPN
cana-3815	196	8	.	.	PUNCT
cana-3815	197	1	let	let	VERB
cana-3815	197	2	∝∈	∝∈	PUNCT
cana-3815	197	3	§	§	PROPN
cana-3815	197	4	and	and	CCONJ
cana-3815	197	5	∝	∝	PROPN
cana-3815	197	6	§	§	PROPN
cana-3815	197	7	∝⊆	∝⊆	PROPN
cana-3815	197	8	ℵ.	ℵ.	PROPN
cana-3815	197	9	now	now	ADV
cana-3815	197	10	,	,	PUNCT
cana-3815	197	11	(	(	PUNCT
cana-3815	197	12	∝	∝	PROPN
cana-3815	197	13	§	§	PROPN
cana-3815	197	14	)	)	PUNCT
cana-3815	197	15	·	·	PUNCT
cana-3815	198	1	(	(	PUNCT
cana-3815	198	2	§	§	PROPN
cana-3815	198	3	∝	∝	PROPN
cana-3815	198	4	)	)	PUNCT
cana-3815	198	5	⊆∝	⊆∝	ADP
cana-3815	198	6	§	§	PROPN
cana-3815	198	7	∝⊆	∝⊆	NOUN
cana-3815	198	8	ℵ	ℵ	NOUN
cana-3815	198	9	,	,	PUNCT
cana-3815	198	10	since	since	SCONJ
cana-3815	198	11	∝	∝	PROPN
cana-3815	198	12	§	§	PROPN
cana-3815	198	13	and	and	CCONJ
cana-3815	198	14	§	§	PROPN
cana-3815	198	15	∝	∝	PROPN
cana-3815	198	16	are	be	AUX
cana-3815	198	17	bids	bid	NOUN
cana-3815	198	18	.	.	PUNCT
cana-3815	199	1	hence	hence	ADV
cana-3815	199	2	∝	∝	PROPN
cana-3815	199	3	§	§	PROPN
cana-3815	199	4	⊆	⊆	NUM
cana-3815	199	5	ℵ	ℵ	NOUN
cana-3815	199	6	or	or	CCONJ
cana-3815	199	7	§	§	PROPN
cana-3815	199	8	∝⊆	∝⊆	VERB
cana-3815	199	9	√	√	NOUN
cana-3815	199	10	ℵ.	ℵ.	PROPN
cana-3815	199	11	suppose	suppose	VERB
cana-3815	199	12	that	that	SCONJ
cana-3815	199	13	∝	∝	PROPN
cana-3815	199	14	§	§	PROPN
cana-3815	199	15	⊆	⊆	NUM
cana-3815	199	16	ℵ.	ℵ.	NOUN
cana-3815	199	17	consider	consider	VERB
cana-3815	199	18	<	<	PRON
cana-3815	199	19	∝>b	∝>b	NOUN
cana-3815	199	20	·	·	PUNCT
cana-3815	200	1	<	<	X
cana-3815	200	2	∝>b⊆∝	∝>b⊆∝	PROPN
cana-3815	200	3	§	§	PROPN
cana-3815	200	4	⊆	⊆	NUM
cana-3815	200	5	ℵ.	ℵ.	NOUN
cana-3815	200	6	then	then	ADV
cana-3815	200	7	∝∈	∝∈	PUNCT
cana-3815	200	8	ℵ.	ℵ.	PROPN
cana-3815	200	9	similarly	similarly	ADV
cana-3815	200	10	if	if	SCONJ
cana-3815	200	11	§	§	PROPN
cana-3815	200	12	∝⊆	∝⊆	NOUN
cana-3815	200	13	√	√	NOUN
cana-3815	200	14	ℵ	ℵ	NOUN
cana-3815	200	15	then	then	ADV
cana-3815	200	16	∝∈	∝∈	PUNCT
cana-3815	200	17	√	√	ADV
cana-3815	200	18	ℵ.	ℵ.	NOUN
cana-3815	201	1	thus	thus	ADV
cana-3815	201	2	ℵ	ℵ	NOUN
cana-3815	201	3	is	be	AUX
cana-3815	201	4	a	a	DET
cana-3815	201	5	2spbid	2spbid	NUM
cana-3815	201	6	of	of	ADP
cana-3815	201	7	§	§	PROPN
cana-3815	201	8	.	.	PUNCT
cana-3815	202	1	theorem	theorem	VERB
cana-3815	202	2	3.3	3.3	NUM
cana-3815	202	3	.	.	PUNCT
cana-3815	203	1	every	every	DET
cana-3815	203	2	2spbid(2pbid	2spbid(2pbid	NOUN
cana-3815	203	3	)	)	PUNCT
cana-3815	203	4	is	be	AUX
cana-3815	203	5	a	a	DET
cana-3815	203	6	3spbid	3spbid	PROPN
cana-3815	203	7	of	of	ADP
cana-3815	203	8	§	§	PROPN
cana-3815	203	9	.	.	PUNCT
cana-3815	204	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	204	2	758	758	NUM
cana-3815	204	3	communications	communication	NOUN
cana-3815	204	4	on	on	ADP
cana-3815	204	5	applied	apply	VERB
cana-3815	204	6	nonlinear	nonlinear	ADJ
cana-3815	204	7	analysis	analysis	NOUN
cana-3815	204	8	issn	issn	NOUN
cana-3815	204	9	:	:	PUNCT
cana-3815	204	10	1074	1074	NUM
cana-3815	204	11	-	-	PUNCT
cana-3815	204	12	133x	133x	NUM
cana-3815	204	13	vol	vol	NOUN
cana-3815	204	14	32	32	NUM
cana-3815	204	15	no	no	NOUN
cana-3815	204	16	.	.	NOUN
cana-3815	204	17	3	3	NUM
cana-3815	204	18	(	(	PUNCT
cana-3815	204	19	2025	2025	NUM
cana-3815	204	20	)	)	PUNCT
cana-3815	204	21	proof	proof	NOUN
cana-3815	204	22	.	.	PUNCT
cana-3815	205	1	suppose	suppose	VERB
cana-3815	205	2	that	that	SCONJ
cana-3815	205	3	ℵ	ℵ	NOUN
cana-3815	205	4	is	be	AUX
cana-3815	205	5	a	a	DET
cana-3815	205	6	2spbid	2spbid	NUM
cana-3815	205	7	and	and	CCONJ
cana-3815	205	8	£	£	SYM
cana-3815	205	9	2	2	NUM
cana-3815	205	10	⊆	⊆	NUM
cana-3815	205	11	ℵ	ℵ	NOUN
cana-3815	205	12	for	for	ADP
cana-3815	205	13	an	an	DET
cana-3815	205	14	i	i	PROPN
cana-3815	205	15	d	d	PROPN
cana-3815	205	16	£	£	PROPN
cana-3815	205	17	of	of	ADP
cana-3815	205	18	§	§	PROPN
cana-3815	205	19	.	.	PUNCT
cana-3815	205	20	to	to	PART
cana-3815	205	21	show	show	VERB
cana-3815	205	22	that	that	SCONJ
cana-3815	205	23	£	£	SYM
cana-3815	205	24	⊆	⊆	NUM
cana-3815	205	25	ℵ	ℵ	NOUN
cana-3815	205	26	or	or	CCONJ
cana-3815	205	27	£	£	SYM
cana-3815	205	28	⊆	⊆	NUM
cana-3815	205	29	√	√	ADJ
cana-3815	205	30	ℵ.	ℵ.	NOUN
cana-3815	206	1	if	if	SCONJ
cana-3815	206	2	£	£	SYM
cana-3815	206	3	6⊆	6⊆	NUM
cana-3815	206	4	ℵ	ℵ	NOUN
cana-3815	206	5	and	and	CCONJ
cana-3815	206	6	£	£	SYM
cana-3815	206	7	6⊆	6⊆	NOUN
cana-3815	206	8	√	√	NUM
cana-3815	206	9	ℵ.	ℵ.	PROPN
cana-3815	206	10	for	for	ADP
cana-3815	206	11	∝∈	∝∈	X
cana-3815	206	12	£	£	NOUN
cana-3815	206	13	,	,	PUNCT
cana-3815	206	14	but	but	CCONJ
cana-3815	206	15	∝/∈	∝/∈	NUM
cana-3815	206	16	ℵ	ℵ	NOUN
cana-3815	206	17	and	and	CCONJ
cana-3815	206	18	∝/∈	∝/∈	NUM
cana-3815	206	19	√	√	PROPN
cana-3815	206	20	ℵ.	ℵ.	PROPN
cana-3815	207	1	now	now	ADV
cana-3815	207	2	∝	∝	PROPN
cana-3815	207	3	§	§	PROPN
cana-3815	207	4	∝⊆<∝	∝⊆<∝	NOUN
cana-3815	207	5	>	>	X
cana-3815	207	6	·	·	PUNCT
cana-3815	208	1	<	<	X
cana-3815	208	2	∝>⊆	∝>⊆	PRON
cana-3815	208	3	£	£	SYM
cana-3815	208	4	2	2	NUM
cana-3815	208	5	⊆	⊆	NUM
cana-3815	208	6	ℵ.	ℵ.	NOUN
cana-3815	208	7	since	since	SCONJ
cana-3815	208	8	ℵ	ℵ	NOUN
cana-3815	208	9	is	be	AUX
cana-3815	208	10	a	a	DET
cana-3815	208	11	2spbid	2spbid	NUM
cana-3815	208	12	of	of	ADP
cana-3815	208	13	§	§	PROPN
cana-3815	208	14	,	,	PUNCT
cana-3815	208	15	then	then	ADV
cana-3815	208	16	∝∈	∝∈	PUNCT
cana-3815	208	17	ℵ	ℵ	NOUN
cana-3815	208	18	or	or	CCONJ
cana-3815	208	19	∝∈	∝∈	PUNCT
cana-3815	208	20	√	√	ADJ
cana-3815	208	21	ℵ.	ℵ.	NOUN
cana-3815	208	22	which	which	PRON
cana-3815	208	23	is	be	AUX
cana-3815	208	24	contradiction	contradiction	NOUN
cana-3815	208	25	,	,	PUNCT
cana-3815	208	26	hence	hence	ADV
cana-3815	208	27	£	£	SYM
cana-3815	208	28	⊆	⊆	NUM
cana-3815	208	29	ℵ	ℵ	NOUN
cana-3815	208	30	or	or	CCONJ
cana-3815	208	31	£	£	SYM
cana-3815	208	32	⊆	⊆	NUM
cana-3815	208	33	√	√	NUM
cana-3815	208	34	ℵ.	ℵ.	NOUN
cana-3815	209	1	thus	thus	ADV
cana-3815	209	2	ℵ	ℵ	NOUN
cana-3815	209	3	is	be	AUX
cana-3815	209	4	a	a	DET
cana-3815	209	5	3spbid	3spbid	PROPN
cana-3815	209	6	of	of	ADP
cana-3815	209	7	§	§	PROPN
cana-3815	209	8	.	.	PUNCT
cana-3815	210	1	definition	definition	NOUN
cana-3815	210	2	3.4	3.4	NUM
cana-3815	210	3	.	.	PUNCT
cana-3815	211	1	a	a	DET
cana-3815	211	2	subset	subset	NOUN
cana-3815	211	3	n	n	NOUN
cana-3815	211	4	of	of	ADP
cana-3815	211	5	§	§	PROPN
cana-3815	211	6	is	be	AUX
cana-3815	211	7	said	say	VERB
cana-3815	211	8	to	to	PART
cana-3815	211	9	be	be	AUX
cana-3815	211	10	(	(	PUNCT
cana-3815	211	11	i	i	NOUN
cana-3815	211	12	)	)	PUNCT
cana-3815	211	13	np1	np1	PROPN
cana-3815	211	14	-sys	-sys	PUNCT
cana-3815	211	15	if	if	SCONJ
cana-3815	211	16	for	for	ADP
cana-3815	211	17	any	any	DET
cana-3815	211	18	∝∈	∝∈	X
cana-3815	211	19	n	n	NOUN
cana-3815	211	20	,	,	PUNCT
cana-3815	211	21	∃	∃	PROPN
cana-3815	211	22	∝1,∝2∈<∝>b	∝1,∝2∈<∝>b	NUM
cana-3815	211	23	such	such	ADJ
cana-3815	211	24	that	that	SCONJ
cana-3815	211	25	∝1∝2∈	∝1∝2∈	NOUN
cana-3815	211	26	n	n	CCONJ
cana-3815	211	27	.	.	PUNCT
cana-3815	212	1	(	(	PUNCT
cana-3815	212	2	ii	ii	NOUN
cana-3815	212	3	)	)	PUNCT
cana-3815	212	4	np2	np2	NOUN
cana-3815	212	5	-sys	-sys	PUNCT
cana-3815	212	6	if	if	SCONJ
cana-3815	212	7	for	for	ADP
cana-3815	212	8	any	any	DET
cana-3815	212	9	∝∈	∝∈	X
cana-3815	212	10	n	n	NOUN
cana-3815	212	11	,	,	PUNCT
cana-3815	212	12	∃	∃	PROPN
cana-3815	212	13	∝1,∝2∈<∝>r	∝1,∝2∈<∝>r	PROPN
cana-3815	212	14	(	(	PUNCT
cana-3815	212	15	∝1,∝2∈<∝>l	∝1,∝2∈<∝>l	NOUN
cana-3815	212	16	)	)	PUNCT
cana-3815	212	17	such	such	ADJ
cana-3815	212	18	that	that	SCONJ
cana-3815	212	19	∝1∝2∈	∝1∝2∈	NOUN
cana-3815	212	20	n	n	CCONJ
cana-3815	212	21	.	.	PUNCT
cana-3815	213	1	(	(	PUNCT
cana-3815	213	2	iii	iii	X
cana-3815	213	3	)	)	PUNCT
cana-3815	213	4	np3	np3	PROPN
cana-3815	213	5	-sys	-sys	PUNCT
cana-3815	213	6	if	if	SCONJ
cana-3815	213	7	for	for	ADP
cana-3815	213	8	any	any	DET
cana-3815	213	9	∝∈	∝∈	X
cana-3815	213	10	n	n	CCONJ
cana-3815	213	11	,	,	PUNCT
cana-3815	213	12	∃	∃	PROPN
cana-3815	213	13	∝1,∝2∈<∝	∝1,∝2∈<∝	PROPN
cana-3815	213	14	>	>	X
cana-3815	213	15	such	such	ADJ
cana-3815	213	16	that	that	PRON
cana-3815	213	17	∝1∝2∈	∝1∝2∈	NOUN
cana-3815	213	18	n	n	CCONJ
cana-3815	213	19	.	.	PUNCT
cana-3815	213	20	theorem	theorem	VERB
cana-3815	213	21	3.5	3.5	NUM
cana-3815	213	22	.	.	PUNCT
cana-3815	214	1	if	if	SCONJ
cana-3815	214	2	ℵ	ℵ	NOUN
cana-3815	214	3	is	be	AUX
cana-3815	214	4	a	a	DET
cana-3815	214	5	bid	bid	NOUN
cana-3815	214	6	of	of	ADP
cana-3815	214	7	§	§	PROPN
cana-3815	214	8	,	,	PUNCT
cana-3815	214	9	then	then	ADV
cana-3815	214	10	ℵ	ℵ	NOUN
cana-3815	214	11	is	be	AUX
cana-3815	214	12	a	a	DET
cana-3815	214	13	1spbid	1spbid	NUM
cana-3815	214	14	(	(	PUNCT
cana-3815	214	15	2spbid	2spbid	NUM
cana-3815	214	16	,	,	PUNCT
cana-3815	214	17	3spbid	3spbid	NUM
cana-3815	214	18	)	)	PUNCT
cana-3815	214	19	if	if	SCONJ
cana-3815	214	20	and	and	CCONJ
cana-3815	214	21	only	only	ADV
cana-3815	214	22	if	if	SCONJ
cana-3815	214	23	§	§	PROPN
cana-3815	214	24	\	\	PROPN
cana-3815	214	25	ℵ	ℵ	NOUN
cana-3815	214	26	is	be	AUX
cana-3815	214	27	an	an	DET
cana-3815	214	28	np1	np1	NOUN
cana-3815	214	29	-sys	-sys	PUNCT
cana-3815	214	30	(	(	PUNCT
cana-3815	214	31	np2	np2	NOUN
cana-3815	214	32	-sys	-sy	NOUN
cana-3815	214	33	,	,	PUNCT
cana-3815	214	34	np3	np3	PROPN
cana-3815	214	35	-sys	-sy	NOUN
cana-3815	214	36	)	)	PUNCT
cana-3815	214	37	.	.	PUNCT
cana-3815	215	1	proof	proof	NOUN
cana-3815	215	2	.	.	PUNCT
cana-3815	216	1	let	let	VERB
cana-3815	216	2	ℵ	ℵ	NOUN
cana-3815	216	3	be	be	AUX
cana-3815	216	4	a	a	DET
cana-3815	216	5	1spbid	1spbid	NUM
cana-3815	216	6	of	of	ADP
cana-3815	216	7	§	§	PROPN
cana-3815	216	8	.	.	PUNCT
cana-3815	217	1	let	let	VERB
cana-3815	217	2	∝∈	∝∈	PUNCT
cana-3815	217	3	§	§	PROPN
cana-3815	217	4	\	\	PROPN
cana-3815	217	5	ℵ.	ℵ.	PROPN
cana-3815	218	1	hence	hence	ADV
cana-3815	218	2	∝∈	∝∈	PUNCT
cana-3815	218	3	§	§	PROPN
cana-3815	218	4	but	but	CCONJ
cana-3815	218	5	∝/∈	∝/∈	NUM
cana-3815	218	6	ℵ.	ℵ.	PROPN
cana-3815	219	1	so	so	ADV
cana-3815	219	2	<	<	X
cana-3815	219	3	∝>b	∝>b	X
cana-3815	219	4	·	·	PUNCT
cana-3815	219	5	<	<	X
cana-3815	219	6	∝>b	∝>b	PROPN
cana-3815	219	7	6⊆	6⊆	NUM
cana-3815	219	8	ℵ.	ℵ.	PROPN
cana-3815	220	1	there	there	PRON
cana-3815	220	2	exists	exist	VERB
cana-3815	220	3	∝′	∝′	NUM
cana-3815	220	4	,	,	PUNCT
cana-3815	220	5	∝′′∈<∝>b	∝′′∈<∝>b	INTJ
cana-3815	220	6	such	such	ADJ
cana-3815	220	7	that	that	PRON
cana-3815	220	8	∝′	∝′	PROPN
cana-3815	220	9	·	·	PUNCT
cana-3815	220	10	∝′′	∝′′	X
cana-3815	220	11	/∈	/∈	PUNCT
cana-3815	221	1	ℵ.	ℵ.	PROPN
cana-3815	221	2	hence	hence	ADV
cana-3815	221	3	∝′	∝′	PROPN
cana-3815	221	4	·	·	PUNCT
cana-3815	221	5	∝′′∈	∝′′∈	PUNCT
cana-3815	222	1	§	§	PROPN
cana-3815	222	2	\	\	PROPN
cana-3815	222	3	ℵ.	ℵ.	PROPN
cana-3815	223	1	so	so	ADV
cana-3815	223	2	we	we	PRON
cana-3815	223	3	have	have	AUX
cana-3815	223	4	proved	prove	VERB
cana-3815	223	5	that	that	SCONJ
cana-3815	223	6	for	for	ADP
cana-3815	223	7	∝∈	∝∈	PUNCT
cana-3815	223	8	§	§	PROPN
cana-3815	223	9	\	\	PROPN
cana-3815	223	10	ℵ	ℵ	PROPN
cana-3815	223	11	∃	∃	PROPN
cana-3815	223	12	∝′	∝′	PROPN
cana-3815	223	13	,	,	PUNCT
cana-3815	223	14	∝′′∈<∝>b	∝′′∈<∝>b	INTJ
cana-3815	223	15	such	such	ADJ
cana-3815	223	16	that	that	PRON
cana-3815	223	17	∝′	∝′	PROPN
cana-3815	223	18	·	·	PUNCT
cana-3815	223	19	∝′′∈	∝′′∈	PUNCT
cana-3815	224	1	§	§	PROPN
cana-3815	224	2	\	\	PROPN
cana-3815	224	3	ℵ.	ℵ.	PROPN
cana-3815	225	1	so	so	ADV
cana-3815	225	2	§	§	NOUN
cana-3815	225	3	\	\	PROPN
cana-3815	225	4	ℵ	ℵ	NOUN
cana-3815	225	5	is	be	AUX
cana-3815	225	6	an	an	DET
cana-3815	225	7	np1	np1	PROPN
cana-3815	225	8	-sys	-sy	NOUN
cana-3815	225	9	.	.	PUNCT
cana-3815	226	1	conversely	conversely	ADV
cana-3815	226	2	,	,	PUNCT
cana-3815	226	3	let	let	VERB
cana-3815	226	4	§	§	PROPN
cana-3815	226	5	\	\	PROPN
cana-3815	226	6	ℵ	ℵ	NOUN
cana-3815	226	7	is	be	AUX
cana-3815	226	8	an	an	DET
cana-3815	226	9	np1	np1	PROPN
cana-3815	226	10	-sys	-sys	PUNCT
cana-3815	226	11	.	.	PUNCT
cana-3815	227	1	let	let	VERB
cana-3815	227	2	a2	a2	PROPN
cana-3815	227	3	⊆	⊆	NUM
cana-3815	227	4	ℵ	ℵ	NOUN
cana-3815	227	5	for	for	ADP
cana-3815	227	6	the	the	DET
cana-3815	227	7	bid	bid	NOUN
cana-3815	227	8	a	a	PRON
cana-3815	227	9	of	of	ADP
cana-3815	227	10	§	§	PROPN
cana-3815	227	11	.	.	PUNCT
cana-3815	228	1	let	let	VERB
cana-3815	228	2	us	we	PRON
cana-3815	228	3	shows	show	VERB
cana-3815	228	4	that	that	SCONJ
cana-3815	228	5	a	a	DET
cana-3815	228	6	⊆	⊆	NUM
cana-3815	228	7	ℵ	ℵ	NOUN
cana-3815	228	8	or	or	CCONJ
cana-3815	228	9	a	a	DET
cana-3815	228	10	⊆	⊆	NUM
cana-3815	228	11	√	√	NOUN
cana-3815	228	12	ℵ.	ℵ.	PUNCT
cana-3815	229	1	if	if	SCONJ
cana-3815	229	2	a	a	DET
cana-3815	229	3	6⊆	6⊆	NUM
cana-3815	229	4	ℵ	ℵ	NOUN
cana-3815	229	5	and	and	CCONJ
cana-3815	229	6	a	a	DET
cana-3815	229	7	6⊆	6⊆	NUM
cana-3815	229	8	√	√	NUM
cana-3815	229	9	ℵ	ℵ	NOUN
cana-3815	229	10	,	,	PUNCT
cana-3815	229	11	let	let	VERB
cana-3815	229	12	℘1	℘1	PRON
cana-3815	229	13	∈	∈	VERB
cana-3815	229	14	a	a	DET
cana-3815	229	15	\	\	NOUN
cana-3815	229	16	ℵ	ℵ	NOUN
cana-3815	229	17	and	and	CCONJ
cana-3815	229	18	℘1	℘1	VERB
cana-3815	229	19	∈	∈	VERB
cana-3815	229	20	a	a	DET
cana-3815	229	21	\	\	NOUN
cana-3815	229	22	√	√	PROPN
cana-3815	229	23	ℵ.	ℵ.	PROPN
cana-3815	229	24	since	since	SCONJ
cana-3815	229	25	℘1	℘1	NOUN
cana-3815	229	26	/∈	/∈	PUNCT
cana-3815	229	27	√	√	ADP
cana-3815	229	28	ℵ	ℵ	NOUN
cana-3815	229	29	,	,	PUNCT
cana-3815	229	30	so	so	ADV
cana-3815	229	31	∃	∃	PROPN
cana-3815	229	32	an	an	DET
cana-3815	229	33	np1	np1	PROPN
cana-3815	229	34	-sys	-sys	PUNCT
cana-3815	229	35	§	§	PROPN
cana-3815	229	36	\	\	PROPN
cana-3815	229	37	ℵ	ℵ	NOUN
cana-3815	229	38	in	in	ADP
cana-3815	229	39	§	§	PROPN
cana-3815	229	40	such	such	ADJ
cana-3815	229	41	that	that	SCONJ
cana-3815	229	42	℘1	℘1	VERB
cana-3815	229	43	∈	∈	PROPN
cana-3815	229	44	§	§	NOUN
cana-3815	229	45	\	\	PROPN
cana-3815	229	46	ℵ	ℵ	NOUN
cana-3815	229	47	and	and	CCONJ
cana-3815	229	48	(	(	PUNCT
cana-3815	229	49	§	§	PROPN
cana-3815	229	50	\	\	PROPN
cana-3815	229	51	ℵ	ℵ	NOUN
cana-3815	229	52	)	)	PUNCT
cana-3815	229	53	∩	∩	NOUN
cana-3815	229	54	ℵ	ℵ	NOUN
cana-3815	229	55	=	=	SYM
cana-3815	229	56	φ	φ	PROPN
cana-3815	229	57	.	.	PUNCT
cana-3815	230	1	thus	thus	ADV
cana-3815	230	2	℘1	℘1	VERB
cana-3815	230	3	∈	∈	PROPN
cana-3815	230	4	§	§	NOUN
cana-3815	230	5	\	\	PROPN
cana-3815	230	6	ℵ	ℵ	NOUN
cana-3815	230	7	implies	imply	VERB
cana-3815	230	8	<	<	X
cana-3815	230	9	℘1	℘1	PROPN
cana-3815	230	10	>	>	X
cana-3815	230	11	b	b	X
cana-3815	230	12	·	·	PUNCT
cana-3815	230	13	<	<	X
cana-3815	230	14	℘1	℘1	PROPN
cana-3815	230	15	>	>	X
cana-3815	230	16	b	b	NOUN
cana-3815	230	17	6⊆	6⊆	NUM
cana-3815	230	18	ℵ	ℵ	NOUN
cana-3815	230	19	,	,	PUNCT
cana-3815	230	20	which	which	PRON
cana-3815	230	21	is	be	AUX
cana-3815	230	22	a	a	DET
cana-3815	230	23	contradiction	contradiction	NOUN
cana-3815	230	24	.	.	PUNCT
cana-3815	231	1	thus	thus	ADV
cana-3815	231	2	a	a	DET
cana-3815	231	3	⊆	⊆	NUM
cana-3815	231	4	ℵ	ℵ	NOUN
cana-3815	231	5	or	or	CCONJ
cana-3815	231	6	a	a	DET
cana-3815	231	7	⊆	⊆	NUM
cana-3815	231	8	√	√	NOUN
cana-3815	231	9	ℵ.	ℵ.	NOUN
cana-3815	232	1	hence	hence	ADV
cana-3815	232	2	ℵ	ℵ	ADV
cana-3815	232	3	is	be	AUX
cana-3815	232	4	a	a	DET
cana-3815	232	5	1spbid	1spbid	NUM
cana-3815	232	6	of	of	ADP
cana-3815	232	7	§	§	PROPN
cana-3815	232	8	.	.	PUNCT
cana-3815	233	1	corollary	corollary	ADJ
cana-3815	233	2	3.6	3.6	NUM
cana-3815	233	3	.	.	PUNCT
cana-3815	234	1	every	every	DET
cana-3815	234	2	np1	np1	NOUN
cana-3815	234	3	-sys	-sys	PUNCT
cana-3815	234	4	is	be	AUX
cana-3815	234	5	an	an	DET
cana-3815	234	6	np2	np2	NOUN
cana-3815	234	7	-sys	-sys	PUNCT
cana-3815	234	8	.	.	PUNCT
cana-3815	235	1	theorem	theorem	VERB
cana-3815	235	2	3.7	3.7	NUM
cana-3815	235	3	.	.	PUNCT
cana-3815	236	1	let	let	VERB
cana-3815	236	2	a	a	DET
cana-3815	236	3	be	be	AUX
cana-3815	236	4	a	a	DET
cana-3815	236	5	2spbid	2spbid	NUM
cana-3815	236	6	of	of	ADP
cana-3815	236	7	a	a	DET
cana-3815	236	8	ring	ring	NOUN
cana-3815	236	9	§	§	PROPN
cana-3815	236	10	.	.	PUNCT
cana-3815	237	1	then	then	ADV
cana-3815	237	2	£	£	SYM
cana-3815	237	3	2	2	NUM
cana-3815	237	4	⊆	⊆	NUM
cana-3815	237	5	a	a	PRON
cana-3815	237	6	implies	imply	VERB
cana-3815	237	7	£	£	SYM
cana-3815	237	8	⊆	⊆	NUM
cana-3815	237	9	a	a	PRON
cana-3815	237	10	or	or	CCONJ
cana-3815	237	11	£	£	SYM
cana-3815	237	12	⊆	⊆	NUM
cana-3815	237	13	√	√	ADP
cana-3815	237	14	a	a	PRON
cana-3815	237	15	for	for	ADP
cana-3815	237	16	any	any	DET
cana-3815	237	17	lid	lid	NOUN
cana-3815	237	18	(	(	PUNCT
cana-3815	237	19	rid)£	rid)£	NOUN
cana-3815	237	20	of	of	ADP
cana-3815	237	21	§	§	PROPN
cana-3815	237	22	.	.	PUNCT
cana-3815	238	1	theorem	theorem	VERB
cana-3815	238	2	3.8	3.8	NUM
cana-3815	238	3	.	.	PUNCT
cana-3815	239	1	a	a	DET
cana-3815	239	2	bid	bid	NOUN
cana-3815	239	3	a	a	PRON
cana-3815	239	4	is	be	AUX
cana-3815	239	5	a	a	DET
cana-3815	239	6	3spbid	3spbid	PROPN
cana-3815	239	7	of	of	ADP
cana-3815	239	8	§	§	PROPN
cana-3815	239	9	if	if	SCONJ
cana-3815	239	10	and	and	CCONJ
cana-3815	239	11	only	only	ADV
cana-3815	239	12	if	if	SCONJ
cana-3815	239	13	`	`	PUNCT
cana-3815	239	14	a	a	PRON
cana-3815	239	15	is	be	AUX
cana-3815	239	16	a	a	DET
cana-3815	239	17	spid	spid	NOUN
cana-3815	239	18	of	of	ADP
cana-3815	239	19	§	§	PROPN
cana-3815	239	20	.	.	PUNCT
cana-3815	240	1	corollary	corollary	ADJ
cana-3815	240	2	3.9	3.9	NUM
cana-3815	240	3	.	.	PUNCT
cana-3815	241	1	if	if	SCONJ
cana-3815	241	2	a	a	PRON
cana-3815	241	3	is	be	AUX
cana-3815	241	4	a	a	DET
cana-3815	241	5	1spbid	1spbid	NUM
cana-3815	241	6	(	(	PUNCT
cana-3815	241	7	2spbid	2spbid	NUM
cana-3815	241	8	)	)	PUNCT
cana-3815	241	9	of	of	ADP
cana-3815	241	10	§	§	PROPN
cana-3815	241	11	,	,	PUNCT
cana-3815	241	12	then	then	ADV
cana-3815	241	13	`	`	PUNCT
cana-3815	241	14	a	a	PRON
cana-3815	241	15	is	be	AUX
cana-3815	241	16	a	a	DET
cana-3815	241	17	spid	spid	NOUN
cana-3815	241	18	of	of	ADP
cana-3815	241	19	§	§	PROPN
cana-3815	241	20	.	.	PUNCT
cana-3815	242	1	acknowledgment	acknowledgment	NOUN
cana-3815	242	2	.	.	PUNCT
cana-3815	243	1	this	this	DET
cana-3815	243	2	research	research	NOUN
cana-3815	243	3	was	be	AUX
cana-3815	243	4	supported	support	VERB
cana-3815	243	5	by	by	ADP
cana-3815	243	6	university	university	NOUN
cana-3815	243	7	of	of	ADP
cana-3815	243	8	phayao	phayao	NOUN
cana-3815	243	9	and	and	CCONJ
cana-3815	243	10	thailand	thailand	PROPN
cana-3815	243	11	science	science	PROPN
cana-3815	243	12	research	research	PROPN
cana-3815	243	13	and	and	CCONJ
cana-3815	243	14	innovation	innovation	NOUN
cana-3815	243	15	fund	fund	NOUN
cana-3815	243	16	(	(	PUNCT
cana-3815	243	17	fundamental	fundamental	ADJ
cana-3815	243	18	fund	fund	NOUN
cana-3815	243	19	2025	2025	NUM
cana-3815	243	20	,	,	PUNCT
cana-3815	243	21	grant	grant	VERB
cana-3815	243	22	no	no	NOUN
cana-3815	243	23	.	.	PROPN
cana-3815	244	1	5027/2567	5027/2567	NUM
cana-3815	244	2	)	)	PUNCT
cana-3815	244	3	.	.	PUNCT
cana-3815	245	1	conflicts	conflict	NOUN
cana-3815	245	2	of	of	ADP
cana-3815	245	3	interest	interest	NOUN
cana-3815	245	4	the	the	DET
cana-3815	245	5	author(s	author(s	NOUN
cana-3815	245	6	)	)	PUNCT
cana-3815	245	7	declare	declare	VERB
cana-3815	245	8	that	that	SCONJ
cana-3815	245	9	there	there	PRON
cana-3815	245	10	are	be	VERB
cana-3815	245	11	no	no	DET
cana-3815	245	12	conflicts	conflict	NOUN
cana-3815	245	13	of	of	ADP
cana-3815	245	14	interest	interest	NOUN
cana-3815	245	15	regarding	regard	VERB
cana-3815	245	16	the	the	DET
cana-3815	245	17	publication	publication	NOUN
cana-3815	245	18	of	of	ADP
cana-3815	245	19	this	this	DET
cana-3815	245	20	paper	paper	NOUN
cana-3815	245	21	.	.	PUNCT
cana-3815	246	1	references	reference	NOUN
cana-3815	246	2	[	[	X
cana-3815	246	3	1	1	X
cana-3815	246	4	]	]	X
cana-3815	246	5	lam	lam	PROPN
cana-3815	246	6	t.y	t.y	PROPN
cana-3815	246	7	,	,	PUNCT
cana-3815	246	8	a	a	DET
cana-3815	246	9	first	first	ADJ
cana-3815	246	10	course	course	NOUN
cana-3815	246	11	in	in	ADP
cana-3815	246	12	non	non	ADJ
cana-3815	246	13	commutative	commutative	ADJ
cana-3815	246	14	rings	ring	NOUN
cana-3815	246	15	,	,	PUNCT
cana-3815	246	16	graduate	graduate	NOUN
cana-3815	246	17	text	text	NOUN
cana-3815	246	18	in	in	ADP
cana-3815	246	19	mathematics	mathematics	PROPN
cana-3815	246	20	131	131	NUM
cana-3815	246	21	,	,	PUNCT
cana-3815	246	22	springer	springer	NOUN
cana-3815	246	23	-	-	PUNCT
cana-3815	246	24	verlag	verlag	PROPN
cana-3815	246	25	,	,	PUNCT
cana-3815	246	26	new	new	PROPN
cana-3815	246	27	york	york	PROPN
cana-3815	246	28	.	.	PUNCT
cana-3815	246	29	1991	1991	NUM
cana-3815	246	30	.	.	PUNCT
cana-3815	247	1	[	[	X
cana-3815	247	2	2	2	NUM
cana-3815	247	3	]	]	PUNCT
cana-3815	247	4	veldsman	veldsman	NOUN
cana-3815	247	5	.s	.s	NOUN
cana-3815	247	6	,	,	PUNCT
cana-3815	247	7	a	a	DET
cana-3815	247	8	note	note	NOUN
cana-3815	247	9	on	on	ADP
cana-3815	247	10	radicals	radical	NOUN
cana-3815	247	11	of	of	ADP
cana-3815	247	12	idealisations	idealisation	NOUN
cana-3815	247	13	,	,	PUNCT
cana-3815	247	14	southeast	southeast	ADJ
cana-3815	247	15	asian	asian	ADJ
cana-3815	247	16	bull	bull	NOUN
cana-3815	247	17	.	.	PUNCT
cana-3815	248	1	math	math	NOUN
cana-3815	248	2	.	.	PUNCT
cana-3815	249	1	2008	2008	NUM
cana-3815	249	2	,	,	PUNCT
cana-3815	249	3	32	32	NUM
cana-3815	249	4	,	,	PUNCT
cana-3815	249	5	545–551	545–551	NUM
cana-3815	249	6	.	.	PUNCT
cana-3815	250	1	[	[	X
cana-3815	250	2	3	3	NUM
cana-3815	250	3	]	]	X
cana-3815	250	4	golan.s.j	golan.s.j	NOUN
cana-3815	250	5	,	,	PUNCT
cana-3815	250	6	semirings	semiring	NOUN
cana-3815	250	7	and	and	CCONJ
cana-3815	250	8	their	their	PRON
cana-3815	250	9	applications	application	NOUN
cana-3815	250	10	,	,	PUNCT
cana-3815	250	11	kluwer	kluwer	NOUN
cana-3815	250	12	academic	academic	ADJ
cana-3815	250	13	publishers	publisher	NOUN
cana-3815	250	14	,	,	PUNCT
cana-3815	250	15	london	london	PROPN
cana-3815	250	16	.	.	PUNCT
cana-3815	251	1	1999	1999	NUM
cana-3815	251	2	.	.	PUNCT
cana-3815	252	1	[	[	X
cana-3815	252	2	4	4	NUM
cana-3815	252	3	]	]	X
cana-3815	252	4	mccoy.n.h	mccoy.n.h	PROPN
cana-3815	252	5	,	,	PUNCT
cana-3815	252	6	the	the	DET
cana-3815	252	7	theory	theory	NOUN
cana-3815	252	8	of	of	ADP
cana-3815	252	9	rings	ring	NOUN
cana-3815	252	10	,	,	PUNCT
cana-3815	252	11	chelsea	chelsea	PROPN
cana-3815	252	12	publishing	publishing	PROPN
cana-3815	252	13	company	company	NOUN
cana-3815	252	14	,	,	PUNCT
cana-3815	252	15	bronx	bronx	PROPN
cana-3815	252	16	new	new	PROPN
cana-3815	252	17	york	york	PROPN
cana-3815	252	18	.	.	PUNCT
cana-3815	252	19	1973	1973	NUM
cana-3815	252	20	.	.	PUNCT
cana-3815	253	1	[	[	X
cana-3815	253	2	5	5	X
cana-3815	253	3	]	]	SYM
cana-3815	253	4	kemprasit.y	kemprasit.y	PROPN
cana-3815	253	5	,	,	PUNCT
cana-3815	253	6	quasi	quasi	NOUN
cana-3815	253	7	-	-	NOUN
cana-3815	253	8	ideals	ideal	NOUN
cana-3815	253	9	and	and	CCONJ
cana-3815	253	10	bi	bi	NOUN
cana-3815	253	11	-	-	NOUN
cana-3815	253	12	ideals	ideal	NOUN
cana-3815	253	13	in	in	ADP
cana-3815	253	14	semigroups	semigroup	NOUN
cana-3815	253	15	and	and	CCONJ
cana-3815	253	16	rings	ring	NOUN
cana-3815	253	17	,	,	PUNCT
cana-3815	253	18	proceedings	proceeding	NOUN
cana-3815	253	19	of	of	ADP
cana-3815	253	20	the	the	DET
cana-3815	253	21	international	international	ADJ
cana-3815	253	22	conference	conference	NOUN
cana-3815	253	23	on	on	ADP
cana-3815	253	24	algebra	algebra	PROPN
cana-3815	253	25	and	and	CCONJ
cana-3815	253	26	its	its	PRON
cana-3815	253	27	applications	application	NOUN
cana-3815	253	28	,	,	PUNCT
cana-3815	253	29	2002	2002	NUM
cana-3815	253	30	,	,	PUNCT
cana-3815	253	31	30–46	30–46	NUM
cana-3815	253	32	.	.	PUNCT
cana-3815	254	1	[	[	X
cana-3815	254	2	6	6	NUM
cana-3815	254	3	]	]	X
cana-3815	254	4	palanikumar	palanikumar	NOUN
cana-3815	254	5	.	.	PUNCT
cana-3815	255	1	m	m	PROPN
cana-3815	255	2	,	,	PUNCT
cana-3815	255	3	shanqiti	shanqiti	ADV
cana-3815	255	4	,	,	PUNCT
cana-3815	255	5	o.al	o.al	PROPN
cana-3815	255	6	,	,	PUNCT
cana-3815	255	7	jana	jana	PROPN
cana-3815	255	8	,	,	PUNCT
cana-3815	255	9	c.	c.	PROPN
cana-3815	255	10	pal	pal	PROPN
cana-3815	255	11	.	.	PUNCT
cana-3815	256	1	m	m	PROPN
cana-3815	256	2	,	,	PUNCT
cana-3815	256	3	novelty	novelty	NOUN
cana-3815	256	4	for	for	ADP
cana-3815	256	5	different	different	ADJ
cana-3815	256	6	prime	prime	ADJ
cana-3815	256	7	partial	partial	ADJ
cana-3815	256	8	bi	bi	NOUN
cana-3815	256	9	-	-	NOUN
cana-3815	256	10	ideals	ideal	NOUN
cana-3815	256	11	in	in	ADP
cana-3815	256	12	noncommutative	noncommutative	ADJ
cana-3815	256	13	partial	partial	ADJ
cana-3815	256	14	rings	ring	NOUN
cana-3815	256	15	and	and	CCONJ
cana-3815	256	16	its	its	PRON
cana-3815	256	17	extension	extension	NOUN
cana-3815	256	18	,	,	PUNCT
cana-3815	256	19	mathematics	mathematics	NOUN
cana-3815	256	20	2023	2023	NUM
cana-3815	256	21	,	,	PUNCT
cana-3815	256	22	11(6	11(6	NUM
cana-3815	256	23	)	)	PUNCT
cana-3815	256	24	,	,	PUNCT
cana-3815	256	25	1–11	1–11	PROPN
cana-3815	256	26	.	.	PUNCT
cana-3815	257	1	[	[	X
cana-3815	257	2	7	7	NUM
cana-3815	257	3	]	]	X
cana-3815	257	4	palanikumar	palanikumar	NOUN
cana-3815	257	5	,	,	PUNCT
cana-3815	257	6	m.	m.	NOUN
cana-3815	257	7	;	;	PUNCT
cana-3815	257	8	arulmozhi	arulmozhi	PROPN
cana-3815	257	9	,	,	PUNCT
cana-3815	257	10	k.	k.	PROPN
cana-3815	257	11	;	;	PUNCT
cana-3815	257	12	jana	jana	PROPN
cana-3815	257	13	,	,	PUNCT
cana-3815	257	14	c	c	PROPN
cana-3815	257	15	,	,	PUNCT
cana-3815	257	16	;	;	PUNCT
cana-3815	257	17	pal	pal	NOUN
cana-3815	257	18	,	,	PUNCT
cana-3815	257	19	m.	m.	NOUN
cana-3815	257	20	;	;	PUNCT
cana-3815	257	21	shum	shum	X
cana-3815	257	22	,	,	PUNCT
cana-3815	257	23	k.p	k.p	PROPN
cana-3815	257	24	.	.	PROPN
cana-3815	257	25	new	new	ADJ
cana-3815	257	26	approach	approach	NOUN
cana-3815	257	27	towards	towards	ADP
cana-3815	257	28	different	different	ADJ
cana-3815	257	29	bi	bi	NOUN
cana-3815	257	30	-	-	NOUN
cana-3815	257	31	basis	basis	NOUN
cana-3815	257	32	of	of	ADP
cana-3815	257	33	ordered	order	VERB
cana-3815	257	34	b	b	X
cana-3815	257	35	-	-	PUNCT
cana-3815	257	36	semiring	semiring	NOUN
cana-3815	257	37	.	.	PUNCT
cana-3815	258	1	asian	asian	ADJ
cana-3815	258	2	-	-	PUNCT
cana-3815	258	3	european	european	ADJ
cana-3815	258	4	journal	journal	NOUN
cana-3815	258	5	of	of	ADP
cana-3815	258	6	mathematics	mathematic	NOUN
cana-3815	258	7	.	.	PUNCT
cana-3815	259	1	2023	2023	NUM
cana-3815	259	2	,	,	PUNCT
cana-3815	259	3	16(2	16(2	NUM
cana-3815	259	4	)	)	PUNCT
cana-3815	259	5	,	,	PUNCT
cana-3815	259	6	2350020	2350020	NUM
cana-3815	259	7	.	.	PUNCT
cana-3815	260	1	[	[	X
cana-3815	260	2	8	8	NUM
cana-3815	260	3	]	]	X
cana-3815	260	4	palanikumar	palanikumar	NOUN
cana-3815	260	5	,	,	PUNCT
cana-3815	260	6	m.	m.	NOUN
cana-3815	260	7	;	;	PUNCT
cana-3815	260	8	iampan	iampan	PROPN
cana-3815	260	9	,	,	PUNCT
cana-3815	260	10	a.	a.	NOUN
cana-3815	260	11	;	;	PUNCT
cana-3815	260	12	manavalan	manavalan	ADJ
cana-3815	260	13	,	,	PUNCT
cana-3815	260	14	l.j	l.j	PROPN
cana-3815	260	15	.	.	PROPN
cana-3815	260	16	m	m	PROPN
cana-3815	260	17	-	-	PUNCT
cana-3815	260	18	bi	bi	ADJ
cana-3815	260	19	-	-	ADJ
cana-3815	260	20	basis	basis	NOUN
cana-3815	260	21	generator	generator	NOUN
cana-3815	260	22	of	of	ADP
cana-3815	260	23	ordered	order	VERB
cana-3815	260	24	gamma	gamma	NOUN
cana-3815	260	25	-	-	PUNCT
cana-3815	260	26	semigroups	semigroup	NOUN
cana-3815	260	27	.	.	PUNCT
cana-3815	261	1	icic	icic	PROPN
cana-3815	261	2	express	express	VERB
cana-3815	261	3	letters	letter	NOUN
cana-3815	261	4	part	part	NOUN
cana-3815	261	5	b	b	NOUN
cana-3815	261	6	:	:	PUNCT
cana-3815	261	7	applications	application	NOUN
cana-3815	261	8	.	.	PUNCT
cana-3815	262	1	13(8	13(8	NUM
cana-3815	262	2	)	)	PUNCT
cana-3815	262	3	,	,	PUNCT
cana-3815	262	4	2022	2022	NUM
cana-3815	262	5	,	,	PUNCT
cana-3815	262	6	795–802	795–802	NUM
cana-3815	262	7	.	.	PUNCT
cana-3815	263	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	263	2	759	759	NUM
cana-3815	263	3	communications	communication	NOUN
cana-3815	263	4	on	on	ADP
cana-3815	263	5	applied	apply	VERB
cana-3815	263	6	nonlinear	nonlinear	ADJ
cana-3815	263	7	analysis	analysis	NOUN
cana-3815	263	8	issn	issn	NOUN
cana-3815	263	9	:	:	PUNCT
cana-3815	263	10	1074	1074	NUM
cana-3815	263	11	-	-	PUNCT
cana-3815	263	12	133x	133x	NUM
cana-3815	263	13	vol	vol	NOUN
cana-3815	263	14	32	32	NUM
cana-3815	263	15	no	no	NOUN
cana-3815	263	16	.	.	NOUN
cana-3815	263	17	3	3	NUM
cana-3815	263	18	(	(	PUNCT
cana-3815	263	19	2025	2025	NUM
cana-3815	263	20	)	)	PUNCT
cana-3815	264	1	[	[	X
cana-3815	264	2	9	9	NUM
cana-3815	264	3	]	]	X
cana-3815	264	4	van	van	PROPN
cana-3815	264	5	der	der	PROPN
cana-3815	264	6	walt	walt	PROPN
cana-3815	264	7	a.	a.	PROPN
cana-3815	264	8	p.	p.	PROPN
cana-3815	264	9	j.	j.	PROPN
cana-3815	264	10	,	,	PUNCT
cana-3815	264	11	prime	prime	ADJ
cana-3815	264	12	and	and	CCONJ
cana-3815	264	13	semiprime	semiprime	NOUN
cana-3815	264	14	bi	bi	NOUN
cana-3815	264	15	-	-	NOUN
cana-3815	264	16	ideals	ideal	NOUN
cana-3815	264	17	,	,	PUNCT
cana-3815	264	18	quaestiones	quaestione	NOUN
cana-3815	264	19	mathematicae	mathematicae	PROPN
cana-3815	264	20	.	.	PUNCT
cana-3815	264	21	1983	1983	NUM
cana-3815	264	22	,	,	PUNCT
cana-3815	264	23	5	5	NUM
cana-3815	264	24	341–345	341–345	NUM
cana-3815	264	25	.	.	PUNCT
cana-3815	265	1	[	[	X
cana-3815	265	2	10	10	NUM
cana-3815	265	3	]	]	PUNCT
cana-3815	265	4	roux	roux	NOUN
cana-3815	265	5	h.	h.	PROPN
cana-3815	265	6	j.	j.	PROPN
cana-3815	265	7	le	le	PROPN
cana-3815	265	8	.	.	PROPN
cana-3815	265	9	,	,	PUNCT
cana-3815	265	10	a	a	DET
cana-3815	265	11	note	note	NOUN
cana-3815	265	12	on	on	ADP
cana-3815	265	13	prime	prime	ADJ
cana-3815	265	14	and	and	CCONJ
cana-3815	265	15	semiprime	semiprime	NOUN
cana-3815	265	16	bi	bi	NOUN
cana-3815	265	17	-	-	NOUN
cana-3815	265	18	ideals	ideal	NOUN
cana-3815	265	19	,	,	PUNCT
cana-3815	265	20	kyungpook	kyungpook	NOUN
cana-3815	265	21	math	math	NOUN
cana-3815	265	22	.	.	PUNCT
cana-3815	266	1	j.	j.	PROPN
cana-3815	266	2	1995	1995	NUM
cana-3815	266	3	,	,	PUNCT
cana-3815	266	4	35	35	NUM
cana-3815	266	5	,	,	PUNCT
cana-3815	266	6	243–247	243–247	NUM
cana-3815	266	7	.	.	PUNCT
cana-3815	267	1	[	[	X
cana-3815	267	2	11	11	NUM
cana-3815	267	3	]	]	SYM
cana-3815	267	4	flaska.v	flaska.v	NOUN
cana-3815	267	5	,	,	PUNCT
cana-3815	267	6	kepka.t	kepka.t	PROPN
cana-3815	267	7	and	and	CCONJ
cana-3815	267	8	saroch.j	saroch.j	PROPN
cana-3815	267	9	,	,	PUNCT
cana-3815	267	10	bi	bi	ADJ
cana-3815	267	11	-	-	ADJ
cana-3815	267	12	ideal	ideal	ADJ
cana-3815	267	13	-	-	PUNCT
cana-3815	267	14	simple	simple	ADJ
cana-3815	267	15	semirings	semiring	NOUN
cana-3815	267	16	,	,	PUNCT
cana-3815	267	17	commentationes	commentatione	VERB
cana-3815	267	18	mathematicae	mathematicae	VERB
cana-3815	267	19	universitatis	universitatis	PROPN
cana-3815	267	20	carolinae	carolinae	PROPN
cana-3815	267	21	.	.	PUNCT
cana-3815	268	1	2005	2005	NUM
cana-3815	268	2	,	,	PUNCT
cana-3815	268	3	46	46	NUM
cana-3815	268	4	,	,	PUNCT
cana-3815	268	5	391–397	391–397	NUM
cana-3815	268	6	.	.	PUNCT
cana-3815	269	1	[	[	X
cana-3815	269	2	12	12	NUM
cana-3815	269	3	]	]	PUNCT
cana-3815	269	4	atani.r.e	atani.r.e	NOUN
cana-3815	269	5	;	;	PUNCT
cana-3815	269	6	atani.s.e	atani.s.e	NOUN
cana-3815	269	7	,	,	PUNCT
cana-3815	269	8	ideal	ideal	ADJ
cana-3815	269	9	theory	theory	NOUN
cana-3815	269	10	in	in	ADP
cana-3815	269	11	commutative	commutative	ADJ
cana-3815	269	12	semirings	semiring	NOUN
cana-3815	269	13	,	,	PUNCT
cana-3815	269	14	buletinul	buletinul	NOUN
cana-3815	269	15	academiei	academiei	PROPN
cana-3815	269	16	de	de	X
cana-3815	269	17	stiinte	stiinte	VERB
cana-3815	269	18	a	a	DET
cana-3815	269	19	republicii	republicii	PROPN
cana-3815	269	20	moldova	moldova	PROPN
cana-3815	269	21	matematica	matematica	PROPN
cana-3815	269	22	.	.	PROPN
cana-3815	269	23	2008	2008	NUM
cana-3815	269	24	,	,	PUNCT
cana-3815	269	25	57	57	NUM
cana-3815	269	26	,	,	PUNCT
cana-3815	269	27	14–23	14–23	NUM
cana-3815	269	28	.	.	PUNCT
cana-3815	270	1	[	[	X
cana-3815	270	2	13	13	NUM
cana-3815	270	3	]	]	SYM
cana-3815	270	4	dubey.m.k	dubey.m.k	PROPN
cana-3815	270	5	,	,	PUNCT
cana-3815	270	6	prime	prime	ADJ
cana-3815	270	7	and	and	CCONJ
cana-3815	270	8	weakly	weakly	ADJ
cana-3815	270	9	prime	prime	ADJ
cana-3815	270	10	ideals	ideal	NOUN
cana-3815	270	11	in	in	ADP
cana-3815	270	12	semirings	semiring	NOUN
cana-3815	270	13	,	,	PUNCT
cana-3815	270	14	quasigroups	quasigroup	NOUN
cana-3815	270	15	and	and	CCONJ
cana-3815	270	16	related	related	ADJ
cana-3815	270	17	systems	system	NOUN
cana-3815	270	18	.	.	PUNCT
cana-3815	271	1	2012	2012	NUM
cana-3815	271	2	,	,	PUNCT
cana-3815	271	3	20	20	NUM
cana-3815	271	4	,	,	PUNCT
cana-3815	271	5	151–156	151–156	NUM
cana-3815	271	6	.	.	PUNCT
cana-3815	272	1	[	[	X
cana-3815	272	2	14	14	NUM
cana-3815	272	3	]	]	X
cana-3815	272	4	sharp.r.y	sharp.r.y	PROPN
cana-3815	272	5	,	,	PUNCT
cana-3815	272	6	steps	step	NOUN
cana-3815	272	7	in	in	ADP
cana-3815	272	8	commutative	commutative	ADJ
cana-3815	272	9	algebra	algebra	NOUN
cana-3815	272	10	,	,	PUNCT
cana-3815	272	11	second	second	ADJ
cana-3815	272	12	edition	edition	NOUN
cana-3815	272	13	,	,	PUNCT
cana-3815	272	14	cambridge	cambridge	PROPN
cana-3815	272	15	university	university	PROPN
cana-3815	272	16	press	press	PROPN
cana-3815	272	17	,	,	PUNCT
cana-3815	272	18	cambridge	cambridge	PROPN
cana-3815	272	19	.	.	PROPN
cana-3815	272	20	2000	2000	NUM
cana-3815	272	21	.	.	PUNCT
cana-3815	273	1	[	[	X
cana-3815	273	2	15	15	X
cana-3815	273	3	]	]	X
cana-3815	273	4	giri	giri	PROPN
cana-3815	273	5	r.	r.	PROPN
cana-3815	273	6	d.	d.	PROPN
cana-3815	273	7	and	and	CCONJ
cana-3815	273	8	wazalwar	wazalwar	PROPN
cana-3815	273	9	a.	a.	PROPN
cana-3815	273	10	k.	k.	PROPN
cana-3815	273	11	,	,	PUNCT
cana-3815	273	12	prime	prime	ADJ
cana-3815	273	13	ideals	ideal	NOUN
cana-3815	273	14	and	and	CCONJ
cana-3815	273	15	prime	prime	ADJ
cana-3815	273	16	radicals	radical	NOUN
cana-3815	273	17	in	in	ADP
cana-3815	273	18	non	non	ADJ
cana-3815	273	19	-	-	ADJ
cana-3815	273	20	commutative	commutative	ADJ
cana-3815	273	21	semigroups	semigroup	NOUN
cana-3815	273	22	,	,	PUNCT
cana-3815	273	23	kyungpook	kyungpook	ADJ
cana-3815	273	24	mathemematical	mathemematical	ADJ
cana-3815	273	25	journal	journal	NOUN
cana-3815	273	26	.	.	PUNCT
cana-3815	274	1	1993	1993	NUM
cana-3815	274	2	,	,	PUNCT
cana-3815	274	3	33(1	33(1	NUM
cana-3815	274	4	)	)	PUNCT
cana-3815	274	5	,	,	PUNCT
cana-3815	274	6	37–48	37–48	NUM
cana-3815	274	7	.	.	PUNCT
cana-3815	275	1	[	[	X
cana-3815	275	2	16	16	NUM
cana-3815	275	3	]	]	X
cana-3815	275	4	r.p	r.p	PROPN
cana-3815	275	5	.	.	PROPN
cana-3815	275	6	sharma	sharma	PROPN
cana-3815	275	7	and	and	CCONJ
cana-3815	275	8	t.r	t.r	PROPN
cana-3815	275	9	.	.	PROPN
cana-3815	275	10	sharma	sharma	PROPN
cana-3815	275	11	,	,	PUNCT
cana-3815	275	12	primary	primary	ADJ
cana-3815	275	13	ideals	ideal	NOUN
cana-3815	275	14	in	in	ADP
cana-3815	275	15	noncommutative	noncommutative	ADJ
cana-3815	275	16	semirings	semiring	NOUN
cana-3815	275	17	,	,	PUNCT
cana-3815	275	18	southeast	southeast	ADJ
cana-3815	275	19	asian	asian	ADJ
cana-3815	275	20	bulletin	bulletin	NOUN
cana-3815	275	21	of	of	ADP
cana-3815	275	22	mathematics	mathematic	NOUN
cana-3815	275	23	2011	2011	NUM
cana-3815	275	24	,	,	PUNCT
cana-3815	275	25	35	35	NUM
cana-3815	275	26	,	,	PUNCT
cana-3815	275	27	345–360	345–360	NUM
cana-3815	275	28	.	.	PUNCT
cana-3815	276	1	[	[	X
cana-3815	276	2	17	17	NUM
cana-3815	276	3	]	]	PUNCT
cana-3815	276	4	p.	p.	NOUN
cana-3815	276	5	v.	v.	ADP
cana-3815	276	6	srinivasa	srinivasa	PROPN
cana-3815	276	7	rao	rao	PROPN
cana-3815	276	8	and	and	CCONJ
cana-3815	276	9	m.	m.	PROPN
cana-3815	276	10	siva	siva	PROPN
cana-3815	276	11	mala	mala	PROPN
cana-3815	276	12	,	,	PUNCT
cana-3815	276	13	prime	prime	ADJ
cana-3815	276	14	and	and	CCONJ
cana-3815	276	15	semiprime	semiprime	NOUN
cana-3815	276	16	bi	bi	NOUN
cana-3815	276	17	-	-	NOUN
cana-3815	276	18	ideals	ideal	NOUN
cana-3815	276	19	of	of	ADP
cana-3815	276	20	gamma	gamma	NOUN
cana-3815	276	21	so	so	SCONJ
cana-3815	276	22	rings	ring	NOUN
cana-3815	276	23	,	,	PUNCT
cana-3815	276	24	international	international	ADJ
cana-3815	276	25	journal	journal	NOUN
cana-3815	276	26	of	of	ADP
cana-3815	276	27	pure	pure	ADJ
cana-3815	276	28	and	and	CCONJ
cana-3815	276	29	applied	applied	ADJ
cana-3815	276	30	mathematics	mathematic	NOUN
cana-3815	276	31	.	.	PUNCT
cana-3815	277	1	2017	2017	NUM
cana-3815	277	2	,	,	PUNCT
cana-3815	277	3	113(6	113(6	NUM
cana-3815	277	4	)	)	PUNCT
cana-3815	277	5	,	,	PUNCT
cana-3815	277	6	352–361	352–361	NUM
cana-3815	277	7	.	.	PUNCT
cana-3815	278	1	[	[	X
cana-3815	278	2	18	18	NUM
cana-3815	278	3	]	]	PUNCT
cana-3815	278	4	p.	p.	NOUN
cana-3815	278	5	v.	v.	CCONJ
cana-3815	278	6	srinivasa	srinivasa	PROPN
cana-3815	278	7	rao	rao	PROPN
cana-3815	278	8	,	,	PUNCT
cana-3815	278	9	2(1)-semiprime	2(1)-semiprime	NUM
cana-3815	278	10	partial	partial	ADJ
cana-3815	278	11	ideals	ideal	NOUN
cana-3815	278	12	of	of	ADP
cana-3815	278	13	partial	partial	ADJ
cana-3815	278	14	semirings	semiring	NOUN
cana-3815	278	15	,	,	PUNCT
cana-3815	278	16	international	international	ADJ
cana-3815	278	17	journal	journal	NOUN
cana-3815	278	18	of	of	ADP
cana-3815	278	19	mathematics	mathematics	NOUN
cana-3815	278	20	trends	trend	NOUN
cana-3815	278	21	and	and	CCONJ
cana-3815	278	22	technology	technology	NOUN
cana-3815	278	23	.	.	PUNCT
cana-3815	279	1	2015	2015	NUM
cana-3815	279	2	,	,	PUNCT
cana-3815	279	3	19(2	19(2	NUM
cana-3815	279	4	)	)	PUNCT
cana-3815	279	5	,	,	PUNCT
cana-3815	279	6	162–168	162–168	NUM
cana-3815	279	7	.	.	PUNCT
cana-3815	280	1	[	[	X
cana-3815	280	2	19	19	NUM
cana-3815	280	3	]	]	PUNCT
cana-3815	280	4	raed	raed	PROPN
cana-3815	280	5	hatamleh	hatamleh	PROPN
cana-3815	280	6	,	,	PUNCT
cana-3815	280	7	abdallah	abdallah	PROPN
cana-3815	280	8	al	al	PROPN
cana-3815	280	9	-	-	PUNCT
cana-3815	280	10	husban	husban	PROPN
cana-3815	280	11	,	,	PUNCT
cana-3815	280	12	k.	k.	PROPN
cana-3815	280	13	sundareswari	sundareswari	PROPN
cana-3815	280	14	,	,	PUNCT
cana-3815	280	15	g.balaj	g.balaj	NOUN
cana-3815	280	16	,	,	PUNCT
cana-3815	280	17	m.palanikumar	m.palanikumar	ADJ
cana-3815	280	18	,	,	PUNCT
cana-3815	280	19	complex	complex	ADJ
cana-3815	280	20	tangent	tangent	NOUN
cana-3815	280	21	trigonometric	trigonometric	ADJ
cana-3815	280	22	approach	approach	NOUN
cana-3815	280	23	applied	apply	VERB
cana-3815	280	24	to	to	ADP
cana-3815	280	25	(	(	PUNCT
cana-3815	280	26	α	α	X
cana-3815	280	27	,	,	PUNCT
cana-3815	280	28	β)-rung	β)-rung	PUNCT
cana-3815	280	29	fuzzy	fuzzy	ADJ
cana-3815	280	30	set	set	NOUN
cana-3815	280	31	using	use	VERB
cana-3815	280	32	weighted	weight	VERB
cana-3815	280	33	averaging	averaging	NOUN
cana-3815	280	34	,	,	PUNCT
cana-3815	280	35	geometric	geometric	ADJ
cana-3815	280	36	operators	operator	NOUN
cana-3815	280	37	and	and	CCONJ
cana-3815	280	38	its	its	PRON
cana-3815	280	39	extension	extension	NOUN
cana-3815	280	40	.	.	PUNCT
cana-3815	281	1	communications	communication	NOUN
cana-3815	281	2	on	on	ADP
cana-3815	281	3	applied	apply	VERB
cana-3815	281	4	nonlinear	nonlinear	ADJ
cana-3815	281	5	analysis	analysis	NOUN
cana-3815	281	6	,	,	PUNCT
cana-3815	281	7	32	32	NUM
cana-3815	281	8	(	(	PUNCT
cana-3815	281	9	5	5	NUM
cana-3815	281	10	)	)	PUNCT
cana-3815	281	11	,	,	PUNCT
cana-3815	281	12	(	(	PUNCT
cana-3815	281	13	2025	2025	NUM
cana-3815	281	14	)	)	PUNCT
cana-3815	281	15	,	,	PUNCT
cana-3815	281	16	133	133	NUM
cana-3815	281	17	-	-	SYM
cana-3815	281	18	144	144	NUM
cana-3815	281	19	.	.	PUNCT
cana-3815	282	1	[	[	X
cana-3815	282	2	20	20	NUM
cana-3815	282	3	]	]	X
cana-3815	282	4	abdallah	abdallah	PROPN
cana-3815	282	5	shihadeh	shihadeh	PROPN
cana-3815	282	6	,	,	PUNCT
cana-3815	282	7	raed	raed	PROPN
cana-3815	282	8	hatamleh	hatamleh	PROPN
cana-3815	282	9	,	,	PUNCT
cana-3815	282	10	m.palanikumar	m.palanikumar	PROPN
cana-3815	282	11	,	,	PUNCT
cana-3815	282	12	abdallah	abdallah	PROPN
cana-3815	282	13	al	al	PROPN
cana-3815	282	14	-	-	PUNCT
cana-3815	282	15	husban	husban	PROPN
cana-3815	282	16	,	,	PUNCT
cana-3815	282	17	new	new	ADJ
cana-3815	282	18	algebraic	algebraic	ADJ
cana-3815	282	19	structures	structure	NOUN
cana-3815	282	20	towards	towards	ADP
cana-3815	282	21	different	different	ADJ
cana-3815	282	22	(	(	PUNCT
cana-3815	282	23	[	[	X
cana-3815	282	24	,	,	PUNCT
cana-3815	282	25	`	`	PUNCT
cana-3815	282	26	)	)	PUNCT
cana-3815	282	27	intuitionistic	intuitionistic	ADJ
cana-3815	282	28	fuzzy	fuzzy	ADJ
cana-3815	282	29	ideals	ideal	NOUN
cana-3815	282	30	and	and	CCONJ
cana-3815	282	31	it	it	PRON
cana-3815	282	32	characterization	characterization	NOUN
cana-3815	282	33	of	of	ADP
cana-3815	282	34	an	an	DET
cana-3815	282	35	ordered	order	VERB
cana-3815	282	36	ternary	ternary	ADJ
cana-3815	282	37	semigroups	semigroup	NOUN
cana-3815	282	38	.	.	PUNCT
cana-3815	283	1	communications	communication	NOUN
cana-3815	283	2	on	on	ADP
cana-3815	283	3	applied	apply	VERB
cana-3815	283	4	nonlinear	nonlinear	ADJ
cana-3815	283	5	analysis	analysis	NOUN
cana-3815	283	6	,	,	PUNCT
cana-3815	283	7	32	32	NUM
cana-3815	283	8	(	(	PUNCT
cana-3815	283	9	6	6	NUM
cana-3815	283	10	)	)	PUNCT
cana-3815	283	11	,	,	PUNCT
cana-3815	283	12	(	(	PUNCT
cana-3815	283	13	2025	2025	NUM
cana-3815	283	14	)	)	PUNCT
cana-3815	283	15	,	,	PUNCT
cana-3815	283	16	568	568	NUM
cana-3815	283	17	-	-	SYM
cana-3815	283	18	578	578	NUM
cana-3815	283	19	.	.	PUNCT
cana-3815	284	1	[	[	X
cana-3815	284	2	21	21	NUM
cana-3815	284	3	]	]	X
cana-3815	284	4	palanikumar	palanikumar	PROPN
cana-3815	284	5	,	,	PUNCT
cana-3815	284	6	m	m	PROPN
cana-3815	284	7	;	;	PUNCT
cana-3815	284	8	iampan	iampan	PROPN
cana-3815	284	9	,	,	PUNCT
cana-3815	284	10	a	a	DET
cana-3815	284	11	;	;	PUNCT
cana-3815	284	12	manavalan	manavalan	ADJ
cana-3815	284	13	,	,	PUNCT
cana-3815	284	14	l.j	l.j	PROPN
cana-3815	284	15	.	.	PROPN
cana-3815	284	16	m	m	PROPN
cana-3815	284	17	-	-	PUNCT
cana-3815	284	18	bi	bi	ADJ
cana-3815	284	19	-	-	ADJ
cana-3815	284	20	base	base	ADJ
cana-3815	284	21	generator	generator	NOUN
cana-3815	284	22	of	of	ADP
cana-3815	284	23	ordered	order	VERB
cana-3815	284	24	γ	γ	NOUN
cana-3815	284	25	-	-	PUNCT
cana-3815	284	26	semigroups	semigroup	NOUN
cana-3815	284	27	.	.	PUNCT
cana-3815	285	1	icic	icic	PROPN
cana-3815	285	2	express	express	VERB
cana-3815	285	3	letters	letter	NOUN
cana-3815	285	4	part	part	NOUN
cana-3815	285	5	b	b	NOUN
cana-3815	285	6	:	:	PUNCT
cana-3815	285	7	applications	application	NOUN
cana-3815	285	8	.	.	PUNCT
cana-3815	286	1	2022	2022	NUM
cana-3815	286	2	,	,	PUNCT
cana-3815	286	3	13(8	13(8	NUM
cana-3815	286	4	)	)	PUNCT
cana-3815	286	5	,	,	PUNCT
cana-3815	286	6	795	795	NUM
cana-3815	286	7	-	-	SYM
cana-3815	286	8	802	802	NUM
cana-3815	286	9	.	.	PUNCT
cana-3815	287	1	[	[	X
cana-3815	287	2	22	22	NUM
cana-3815	287	3	]	]	PUNCT
cana-3815	287	4	mohanraj	mohanraj	NOUN
cana-3815	287	5	,	,	PUNCT
cana-3815	287	6	g	g	NOUN
cana-3815	287	7	;	;	PUNCT
cana-3815	287	8	palanikumar	palanikumar	NOUN
cana-3815	287	9	,	,	PUNCT
cana-3815	287	10	m.	m.	NOUN
cana-3815	287	11	characterization	characterization	NOUN
cana-3815	287	12	of	of	ADP
cana-3815	287	13	various	various	ADJ
cana-3815	287	14	k	k	NOUN
cana-3815	287	15	-	-	NOUN
cana-3815	287	16	regular	regular	ADJ
cana-3815	287	17	in	in	ADP
cana-3815	287	18	b	b	NOUN
cana-3815	287	19	-	-	PUNCT
cana-3815	287	20	semirings	semiring	NOUN
cana-3815	287	21	,	,	PUNCT
cana-3815	287	22	aip	aip	PROPN
cana-3815	287	23	conference	conference	NOUN
cana-3815	287	24	proceedings	proceeding	NOUN
cana-3815	287	25	,	,	PUNCT
cana-3815	287	26	2019	2019	NUM
cana-3815	287	27	,	,	PUNCT
cana-3815	287	28	2112	2112	NUM
cana-3815	287	29	(	(	PUNCT
cana-3815	287	30	1	1	NUM
cana-3815	287	31	)	)	PUNCT
cana-3815	287	32	,	,	PUNCT
cana-3815	287	33	020021	020021	NUM
cana-3815	287	34	.	.	PUNCT
cana-3815	288	1	[	[	X
cana-3815	288	2	23	23	NUM
cana-3815	288	3	]	]	X
cana-3815	288	4	palanikumar	palanikumar	PROPN
cana-3815	288	5	,	,	PUNCT
cana-3815	288	6	m	m	PROPN
cana-3815	288	7	;	;	PUNCT
cana-3815	288	8	shanqiti	shanqiti	ADV
cana-3815	288	9	,	,	PUNCT
cana-3815	288	10	o.	o.	PROPN
cana-3815	288	11	al	al	PROPN
cana-3815	288	12	;	;	PUNCT
cana-3815	288	13	jana	jana	PROPN
cana-3815	288	14	,	,	PUNCT
cana-3815	288	15	c	c	X
cana-3815	288	16	;	;	PUNCT
cana-3815	288	17	pal	pal	ADJ
cana-3815	288	18	,	,	PUNCT
cana-3815	288	19	m.	m.	NOUN
cana-3815	288	20	novelty	novelty	NOUN
cana-3815	288	21	for	for	ADP
cana-3815	288	22	different	different	ADJ
cana-3815	288	23	prime	prime	ADJ
cana-3815	288	24	partial	partial	ADJ
cana-3815	288	25	bi	bi	NOUN
cana-3815	288	26	-	-	NOUN
cana-3815	288	27	ideals	ideal	NOUN
cana-3815	288	28	in	in	ADP
cana-3815	288	29	noncommutative	noncommutative	ADJ
cana-3815	288	30	partial	partial	ADJ
cana-3815	288	31	rings	ring	NOUN
cana-3815	288	32	and	and	CCONJ
cana-3815	288	33	its	its	PRON
cana-3815	288	34	extension	extension	NOUN
cana-3815	288	35	mathematics	mathematic	NOUN
cana-3815	288	36	,	,	PUNCT
cana-3815	288	37	2023	2023	NUM
cana-3815	288	38	,	,	PUNCT
cana-3815	288	39	11(6	11(6	NUM
cana-3815	288	40	)	)	PUNCT
cana-3815	288	41	,	,	PUNCT
cana-3815	288	42	1309	1309	NUM
cana-3815	288	43	.	.	PUNCT
cana-3815	289	1	[	[	X
cana-3815	289	2	24	24	NUM
cana-3815	289	3	]	]	X
cana-3815	289	4	palanikumar	palanikumar	PROPN
cana-3815	289	5	,	,	PUNCT
cana-3815	289	6	m	m	PROPN
cana-3815	289	7	;	;	PUNCT
cana-3815	289	8	mohanraj	mohanraj	NOUN
cana-3815	289	9	,	,	PUNCT
cana-3815	289	10	g	g	NOUN
cana-3815	289	11	;	;	PUNCT
cana-3815	289	12	iampan	iampan	NOUN
cana-3815	289	13	,	,	PUNCT
cana-3815	289	14	a.	a.	NOUN
cana-3815	289	15	characterization	characterization	NOUN
cana-3815	289	16	of	of	ADP
cana-3815	289	17	different	different	ADJ
cana-3815	289	18	prime	prime	ADJ
cana-3815	289	19	bi	bi	NOUN
cana-3815	289	20	-	-	NOUN
cana-3815	289	21	ideals	ideal	NOUN
cana-3815	289	22	and	and	CCONJ
cana-3815	289	23	its	its	PRON
cana-3815	289	24	generalization	generalization	NOUN
cana-3815	289	25	of	of	ADP
cana-3815	289	26	semirings	semiring	NOUN
cana-3815	289	27	,	,	PUNCT
cana-3815	289	28	international	international	ADJ
cana-3815	289	29	journal	journal	NOUN
cana-3815	289	30	of	of	ADP
cana-3815	289	31	analysis	analysis	NOUN
cana-3815	289	32	and	and	CCONJ
cana-3815	289	33	applications	application	NOUN
cana-3815	289	34	,	,	PUNCT
cana-3815	289	35	2024	2024	NUM
cana-3815	289	36	,	,	PUNCT
cana-3815	289	37	22	22	NUM
cana-3815	289	38	,	,	PUNCT
cana-3815	289	39	112–112	112–112	NUM
cana-3815	289	40	.	.	PUNCT
cana-3815	290	1	[	[	X
cana-3815	290	2	25	25	NUM
cana-3815	290	3	]	]	X
cana-3815	290	4	abdallah	abdallah	PROPN
cana-3815	290	5	al	al	PROPN
cana-3815	290	6	-	-	PROPN
cana-3815	290	7	husban	husban	PROPN
cana-3815	290	8	&	&	CCONJ
cana-3815	290	9	abdul	abdul	PROPN
cana-3815	290	10	razak	razak	PROPN
cana-3815	290	11	salleh	salleh	PROPN
cana-3815	290	12	,	,	PUNCT
cana-3815	290	13	complex	complex	ADJ
cana-3815	290	14	fuzzy	fuzzy	ADJ
cana-3815	290	15	ring	ring	NOUN
cana-3815	290	16	.	.	PUNCT
cana-3815	291	1	proceedings	proceeding	NOUN
cana-3815	291	2	of	of	ADP
cana-3815	291	3	2nd	2nd	ADJ
cana-3815	291	4	international	international	ADJ
cana-3815	291	5	conference	conference	NOUN
cana-3815	291	6	on	on	ADP
cana-3815	291	7	computing	computing	NOUN
cana-3815	291	8	,	,	PUNCT
cana-3815	291	9	mathematics	mathematic	NOUN
cana-3815	291	10	and	and	CCONJ
cana-3815	291	11	statistics	statistic	NOUN
cana-3815	291	12	,	,	PUNCT
cana-3815	291	13	ieee	ieee	NOUN
cana-3815	291	14	,	,	PUNCT
cana-3815	291	15	2015	2015	NUM
cana-3815	291	16	,	,	PUNCT
cana-3815	291	17	241	241	NUM
cana-3815	291	18	-	-	SYM
cana-3815	291	19	245	245	NUM
cana-3815	291	20	.	.	PUNCT
cana-3815	292	1	[	[	X
cana-3815	292	2	26	26	NUM
cana-3815	292	3	]	]	X
cana-3815	292	4	abdallah	abdallah	PROPN
cana-3815	292	5	al	al	PROPN
cana-3815	292	6	-	-	PROPN
cana-3815	292	7	husban	husban	PROPN
cana-3815	292	8	&	&	CCONJ
cana-3815	292	9	abdul	abdul	PROPN
cana-3815	292	10	razak	razak	PROPN
cana-3815	292	11	salleh	salleh	PROPN
cana-3815	292	12	,	,	PUNCT
cana-3815	292	13	complex	complex	ADJ
cana-3815	292	14	fuzzy	fuzzy	ADJ
cana-3815	292	15	hyperring	hyperring	NOUN
cana-3815	292	16	based	base	VERB
cana-3815	292	17	on	on	ADP
cana-3815	292	18	complex	complex	ADJ
cana-3815	292	19	fuzzy	fuzzy	ADJ
cana-3815	292	20	spaces	space	NOUN
cana-3815	292	21	.	.	PUNCT
cana-3815	293	1	proceedings	proceeding	NOUN
cana-3815	293	2	of	of	ADP
cana-3815	293	3	2nd	2nd	ADJ
cana-3815	293	4	innovation	innovation	NOUN
cana-3815	293	5	and	and	CCONJ
cana-3815	293	6	analytics	analytic	NOUN
cana-3815	293	7	conference	conference	NOUN
cana-3815	293	8	&	&	CCONJ
cana-3815	293	9	exhibition	exhibition	PROPN
cana-3815	293	10	(	(	PUNCT
cana-3815	293	11	iace	iace	NOUN
cana-3815	293	12	)	)	PUNCT
cana-3815	293	13	.	.	PUNCT
cana-3815	294	1	vol	vol	NOUN
cana-3815	294	2	.	.	PROPN
cana-3815	294	3	1691	1691	NUM
cana-3815	294	4	.	.	PUNCT
cana-3815	295	1	aip	aip	PROPN
cana-3815	295	2	publishing	publish	VERB
cana-3815	295	3	2015	2015	NUM
cana-3815	295	4	,	,	PUNCT
cana-3815	295	5	040009	040009	NUM
cana-3815	295	6	-	-	SYM
cana-3815	295	7	040017	040017	NUM
cana-3815	295	8	.	.	PUNCT
cana-3815	296	1	[	[	X
cana-3815	296	2	27	27	NUM
cana-3815	296	3	]	]	X
cana-3815	296	4	al	al	PROPN
cana-3815	296	5	-	-	PUNCT
cana-3815	296	6	husban	husban	PROPN
cana-3815	296	7	,	,	PUNCT
cana-3815	296	8	a.	a.	PROPN
cana-3815	296	9	,	,	PUNCT
cana-3815	296	10	&	&	CCONJ
cana-3815	296	11	salleh	salleh	PROPN
cana-3815	296	12	,	,	PUNCT
cana-3815	296	13	a.	a.	PROPN
cana-3815	296	14	r.	r.	PROPN
cana-3815	296	15	complex	complex	PROPN
cana-3815	296	16	fuzzy	fuzzy	ADJ
cana-3815	296	17	hyper	hyper	ADJ
cana-3815	296	18	groups	group	NOUN
cana-3815	296	19	based	base	VERB
cana-3815	296	20	on	on	ADP
cana-3815	296	21	complex	complex	ADJ
cana-3815	296	22	fuzzy	fuzzy	ADJ
cana-3815	296	23	spaces	space	NOUN
cana-3815	296	24	.	.	PUNCT
cana-3815	297	1	international	international	ADJ
cana-3815	297	2	journal	journal	NOUN
cana-3815	297	3	of	of	ADP
cana-3815	297	4	pure	pure	ADJ
cana-3815	297	5	and	and	CCONJ
cana-3815	297	6	applied	applied	ADJ
cana-3815	297	7	mathematics	mathematic	NOUN
cana-3815	297	8	,	,	PUNCT
cana-3815	297	9	107(4	107(4	NUM
cana-3815	297	10	)	)	PUNCT
cana-3815	297	11	,	,	PUNCT
cana-3815	297	12	(	(	PUNCT
cana-3815	297	13	2016	2016	NUM
cana-3815	297	14	)	)	PUNCT
cana-3815	297	15	,	,	PUNCT
cana-3815	297	16	949	949	NUM
cana-3815	297	17	-	-	SYM
cana-3815	297	18	958	958	NUM
cana-3815	297	19	.	.	PUNCT
cana-3815	298	1	[	[	X
cana-3815	298	2	28	28	NUM
cana-3815	298	3	]	]	X
cana-3815	298	4	alsarahead	alsarahead	ADJ
cana-3815	298	5	,	,	PUNCT
cana-3815	298	6	m.	m.	NOUN
cana-3815	298	7	o.	o.	PROPN
cana-3815	298	8	,	,	PUNCT
cana-3815	298	9	&	&	CCONJ
cana-3815	298	10	al	al	PROPN
cana-3815	298	11	-	-	PUNCT
cana-3815	298	12	husban	husban	PROPN
cana-3815	298	13	,	,	PUNCT
cana-3815	298	14	a	a	DET
cana-3815	298	15	,	,	PUNCT
cana-3815	298	16	complex	complex	ADJ
cana-3815	298	17	multi	multi	ADJ
cana-3815	298	18	-	-	ADJ
cana-3815	298	19	fuzzy	fuzzy	ADJ
cana-3815	298	20	subgroups	subgroup	NOUN
cana-3815	298	21	.	.	PUNCT
cana-3815	299	1	journal	journal	NOUN
cana-3815	299	2	of	of	ADP
cana-3815	299	3	discrete	discrete	ADJ
cana-3815	299	4	mathematical	mathematical	ADJ
cana-3815	299	5	sciences	science	NOUN
cana-3815	299	6	and	and	CCONJ
cana-3815	299	7	cryptography	cryptography	NOUN
cana-3815	299	8	,	,	PUNCT
cana-3815	299	9	25(8	25(8	NUM
cana-3815	299	10	)	)	PUNCT
cana-3815	299	11	,	,	PUNCT
cana-3815	299	12	(	(	PUNCT
cana-3815	299	13	2022	2022	NUM
cana-3815	299	14	)	)	PUNCT
cana-3815	299	15	,	,	PUNCT
cana-3815	299	16	2707	2707	NUM
cana-3815	299	17	-	-	SYM
cana-3815	299	18	2716	2716	NUM
cana-3815	299	19	.	.	PUNCT
cana-3815	300	1	[	[	X
cana-3815	300	2	29	29	NUM
cana-3815	300	3	]	]	X
cana-3815	300	4	al	al	PROPN
cana-3815	300	5	-	-	PUNCT
cana-3815	300	6	husban	husban	PROPN
cana-3815	300	7	,	,	PUNCT
cana-3815	300	8	a	a	PRON
cana-3815	300	9	,	,	PUNCT
cana-3815	300	10	multi	multi	ADJ
cana-3815	300	11	-	-	ADJ
cana-3815	300	12	fuzzy	fuzzy	ADJ
cana-3815	300	13	hyper	hyper	ADJ
cana-3815	300	14	groups	group	NOUN
cana-3815	300	15	.	.	PUNCT
cana-3815	301	1	italian	italian	ADJ
cana-3815	301	2	journal	journal	NOUN
cana-3815	301	3	of	of	ADP
cana-3815	301	4	pure	pure	ADJ
cana-3815	301	5	and	and	CCONJ
cana-3815	301	6	applied	applied	ADJ
cana-3815	301	7	mathematics	mathematic	NOUN
cana-3815	301	8	,	,	PUNCT
cana-3815	301	9	46	46	NUM
cana-3815	301	10	,	,	PUNCT
cana-3815	301	11	(	(	PUNCT
cana-3815	301	12	2021	2021	NUM
cana-3815	301	13	)	)	PUNCT
cana-3815	301	14	,	,	PUNCT
cana-3815	301	15	382	382	NUM
cana-3815	301	16	-	-	SYM
cana-3815	301	17	390	390	NUM
cana-3815	301	18	.	.	PUNCT
cana-3815	302	1	[	[	X
cana-3815	302	2	30	30	NUM
cana-3815	302	3	]	]	X
cana-3815	302	4	al	al	PROPN
cana-3815	302	5	-	-	PUNCT
cana-3815	302	6	husban	husban	PROPN
cana-3815	302	7	,	,	PUNCT
cana-3815	302	8	a	a	DET
cana-3815	302	9	,	,	PUNCT
cana-3815	302	10	fuzzy	fuzzy	ADJ
cana-3815	302	11	soft	soft	ADJ
cana-3815	302	12	groups	group	NOUN
cana-3815	302	13	based	base	VERB
cana-3815	302	14	on	on	ADP
cana-3815	302	15	fuzzy	fuzzy	ADJ
cana-3815	302	16	space	space	NOUN
cana-3815	302	17	,	,	PUNCT
cana-3815	302	18	wseas	wseas	VERB
cana-3815	302	19	transactions	transaction	NOUN
cana-3815	302	20	on	on	ADP
cana-3815	302	21	mathematics	mathematic	NOUN
cana-3815	302	22	.	.	PUNCT
cana-3815	303	1	21	21	NUM
cana-3815	303	2	,	,	PUNCT
cana-3815	303	3	(	(	PUNCT
cana-3815	303	4	2021	2021	NUM
cana-3815	303	5	)	)	PUNCT
cana-3815	303	6	,	,	PUNCT
cana-3815	303	7	53	53	NUM
cana-3815	303	8	-	-	SYM
cana-3815	303	9	57	57	NUM
cana-3815	303	10	.	.	PUNCT
cana-3815	304	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3815	304	2	760	760	NUM
cana-3815	304	3	1	1	NUM
cana-3815	304	4	introduction	introduction	NOUN
cana-3815	304	5	2	2	NUM
cana-3815	304	6	characterization	characterization	NOUN
cana-3815	304	7	of	of	ADP
cana-3815	304	8	pbids	pbid	NOUN
cana-3815	304	9	3	3	NUM
cana-3815	304	10	characterization	characterization	NOUN
cana-3815	304	11	of	of	ADP
cana-3815	304	12	spbids	spbid	NOUN