id	sid	tid	token	lemma	pos
cana-540	1	1	communications	communication	NOUN
cana-540	1	2	on	on	ADP
cana-540	1	3	applied	apply	VERB
cana-540	1	4	nonlinear	nonlinear	ADJ
cana-540	1	5	analysis	analysis	NOUN
cana-540	1	6	issn	issn	NOUN
cana-540	1	7	:	:	PUNCT
cana-540	1	8	1074	1074	NUM
cana-540	1	9	-	-	PUNCT
cana-540	1	10	133x	133x	NUM
cana-540	1	11	vol	vol	NOUN
cana-540	1	12	31	31	NUM
cana-540	1	13	no	no	NOUN
cana-540	1	14	.	.	NOUN
cana-540	1	15	2	2	NUM
cana-540	1	16	(	(	PUNCT
cana-540	1	17	2024	2024	NUM
cana-540	1	18	)	)	PUNCT
cana-540	2	1	248	248	NUM
cana-540	2	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	2	3	spgα	spgα	ADJ
cana-540	2	4	-	-	PUNCT
cana-540	2	5	continuous	continuous	ADJ
cana-540	2	6	multifunctions	multifunction	NOUN
cana-540	2	7	m.	m.	NOUN
cana-540	2	8	m.	m.	PROPN
cana-540	2	9	holliyavar1	holliyavar1	PROPN
cana-540	2	10	,	,	PUNCT
cana-540	2	11	t.d	t.d	PROPN
cana-540	2	12	.	.	PUNCT
cana-540	2	13	rayanagoudar2	rayanagoudar2	PROPN
cana-540	2	14	,	,	PUNCT
cana-540	2	15	sarika	sarika	PROPN
cana-540	2	16	m.	m.	PROPN
cana-540	2	17	patil3	patil3	PROPN
cana-540	3	1	1r	1r	NUM
cana-540	3	2	and	and	CCONJ
cana-540	3	3	d	d	PROPN
cana-540	3	4	department	department	NOUN
cana-540	3	5	,	,	PUNCT
cana-540	3	6	bharathiar	bharathiar	PROPN
cana-540	3	7	university	university	PROPN
cana-540	3	8	,	,	PUNCT
cana-540	3	9	coimbatore641046	coimbatore641046	PROPN
cana-540	3	10	,	,	PUNCT
cana-540	3	11	tamilnadu	tamilnadu	NOUN
cana-540	3	12	,	,	PUNCT
cana-540	3	13	india	india	PROPN
cana-540	3	14	,	,	PUNCT
cana-540	3	15	mmholliyavar@yahoo.com	mmholliyavar@yahoo.com	PROPN
cana-540	3	16	2	2	NUM
cana-540	3	17	,	,	PUNCT
cana-540	3	18	3	3	NUM
cana-540	3	19	department	department	NOUN
cana-540	3	20	of	of	ADP
cana-540	3	21	mathematics	mathematic	NOUN
cana-540	3	22	,	,	PUNCT
cana-540	3	23	government	government	NOUN
cana-540	3	24	first	first	ADJ
cana-540	3	25	grade	grade	NOUN
cana-540	3	26	college	college	NOUN
cana-540	3	27	,	,	PUNCT
cana-540	3	28	rajnagar	rajnagar	ADJ
cana-540	3	29	,	,	PUNCT
cana-540	3	30	hubli-580032	hubli-580032	NOUN
cana-540	3	31	,	,	PUNCT
cana-540	3	32	karnataka	karnataka	PROPN
cana-540	3	33	,	,	PUNCT
cana-540	3	34	india	india	PROPN
cana-540	3	35	rgoudar1980@gmail.com	rgoudar1980@gmail.com	PROPN
cana-540	3	36	and	and	CCONJ
cana-540	3	37	sarupatil@rediffmail.com	sarupatil@rediffmail.com	NOUN
cana-540	3	38	article	article	NOUN
cana-540	3	39	history	history	NOUN
cana-540	3	40	:	:	PUNCT
cana-540	3	41	received	receive	VERB
cana-540	3	42	:	:	PUNCT
cana-540	3	43	31	31	NUM
cana-540	3	44	-	-	SYM
cana-540	3	45	01	01	NUM
cana-540	3	46	-	-	PUNCT
cana-540	3	47	2024	2024	NUM
cana-540	3	48	revised	revise	VERB
cana-540	3	49	:	:	PUNCT
cana-540	3	50	16	16	NUM
cana-540	3	51	-	-	PUNCT
cana-540	3	52	04	04	NUM
cana-540	3	53	-	-	PUNCT
cana-540	3	54	2024	2024	NUM
cana-540	3	55	accepted	accept	VERB
cana-540	3	56	:	:	PUNCT
cana-540	3	57	02	02	NUM
cana-540	3	58	-	-	PUNCT
cana-540	3	59	05	05	NUM
cana-540	3	60	-	-	PUNCT
cana-540	3	61	2024	2024	NUM
cana-540	3	62	abstract	abstract	NOUN
cana-540	3	63	:	:	PUNCT
cana-540	3	64	the	the	DET
cana-540	3	65	purpose	purpose	NOUN
cana-540	3	66	of	of	ADP
cana-540	3	67	this	this	DET
cana-540	3	68	paper	paper	NOUN
cana-540	3	69	is	be	AUX
cana-540	3	70	to	to	PART
cana-540	3	71	introduce	introduce	VERB
cana-540	3	72	and	and	CCONJ
cana-540	3	73	study	study	VERB
cana-540	3	74	spgαcontinuous	spgαcontinuous	ADJ
cana-540	3	75	multifunctions	multifunction	NOUN
cana-540	3	76	such	such	ADJ
cana-540	3	77	as	as	ADP
cana-540	3	78	upper	upper	ADJ
cana-540	3	79	(	(	PUNCT
cana-540	3	80	lower	low	ADJ
cana-540	3	81	)	)	PUNCT
cana-540	3	82	spgα	spgα	ADJ
cana-540	3	83	-	-	PUNCT
cana-540	3	84	continuous	continuous	ADJ
cana-540	3	85	multifunctions	multifunction	NOUN
cana-540	3	86	,	,	PUNCT
cana-540	3	87	basic	basic	ADJ
cana-540	3	88	characterizations	characterization	NOUN
cana-540	3	89	and	and	CCONJ
cana-540	3	90	several	several	ADJ
cana-540	3	91	properties	property	NOUN
cana-540	3	92	concerning	concern	VERB
cana-540	3	93	to	to	ADP
cana-540	3	94	upper	upper	ADJ
cana-540	3	95	and	and	CCONJ
cana-540	3	96	lower	low	ADJ
cana-540	3	97	spgα	spgα	ADJ
cana-540	3	98	-	-	PUNCT
cana-540	3	99	continuous	continuous	ADJ
cana-540	3	100	multifunctions	multifunction	NOUN
cana-540	3	101	are	be	AUX
cana-540	3	102	investigated	investigate	VERB
cana-540	3	103	.	.	PUNCT
cana-540	4	1	further	far	ADV
cana-540	4	2	,	,	PUNCT
cana-540	4	3	upper	upper	ADJ
cana-540	4	4	(	(	PUNCT
cana-540	4	5	lower	low	ADJ
cana-540	4	6	)	)	PUNCT
cana-540	4	7	spgα	spgα	ADJ
cana-540	4	8	-	-	PUNCT
cana-540	4	9	irresolute	irresolute	ADJ
cana-540	4	10	multifunctions	multifunction	NOUN
cana-540	4	11	are	be	AUX
cana-540	4	12	also	also	ADV
cana-540	4	13	studied	study	VERB
cana-540	4	14	with	with	ADP
cana-540	4	15	basic	basic	ADJ
cana-540	4	16	properties	property	NOUN
cana-540	4	17	.	.	PUNCT
cana-540	5	1	keywords	keyword	NOUN
cana-540	5	2	:	:	PUNCT
cana-540	5	3	spgα	spgα	ADJ
cana-540	5	4	-	-	PUNCT
cana-540	5	5	closed	close	VERB
cana-540	5	6	set	set	NOUN
cana-540	5	7	,	,	PUNCT
cana-540	5	8	spgα	spgα	ADJ
cana-540	5	9	-	-	PUNCT
cana-540	5	10	open	open	NOUN
cana-540	5	11	set	set	NOUN
cana-540	5	12	,	,	PUNCT
cana-540	5	13	spgα	spgα	ADJ
cana-540	5	14	-	-	PUNCT
cana-540	5	15	ti	ti	NOUN
cana-540	5	16	-	-	NOUN
cana-540	5	17	spaces	space	NOUN
cana-540	5	18	,	,	PUNCT
cana-540	5	19	spgα	spgα	ADJ
cana-540	5	20	continuous	continuous	ADJ
cana-540	5	21	function	function	NOUN
cana-540	5	22	.	.	PUNCT
cana-540	6	1	2010	2010	NUM
cana-540	6	2	ams	am	NOUN
cana-540	6	3	subject	subject	NOUN
cana-540	6	4	classifications	classification	NOUN
cana-540	6	5	:	:	PUNCT
cana-540	6	6	54a05	54a05	NUM
cana-540	6	7	,	,	PUNCT
cana-540	6	8	54c08	54c08	NUM
cana-540	6	9	1	1	NUM
cana-540	6	10	.	.	PUNCT
cana-540	7	1	introduction	introduction	NOUN
cana-540	7	2	many	many	ADJ
cana-540	7	3	authors	author	NOUN
cana-540	7	4	have	have	AUX
cana-540	7	5	continued	continue	VERB
cana-540	7	6	their	their	PRON
cana-540	7	7	research	research	NOUN
cana-540	7	8	and	and	CCONJ
cana-540	7	9	studied	study	VERB
cana-540	7	10	several	several	ADJ
cana-540	7	11	stronger	strong	ADJ
cana-540	7	12	and	and	CCONJ
cana-540	7	13	weaker	weak	ADJ
cana-540	7	14	forms	form	NOUN
cana-540	7	15	of	of	ADP
cana-540	7	16	continuous	continuous	ADJ
cana-540	7	17	functions	function	NOUN
cana-540	7	18	and	and	CCONJ
cana-540	7	19	multifunctions	multifunction	NOUN
cana-540	7	20	in	in	ADP
cana-540	7	21	topological	topological	ADJ
cana-540	7	22	spaces	space	NOUN
cana-540	7	23	.	.	PUNCT
cana-540	8	1	it	it	PRON
cana-540	8	2	is	be	AUX
cana-540	8	3	observed	observe	VERB
cana-540	8	4	that	that	SCONJ
cana-540	8	5	continuity	continuity	NOUN
cana-540	8	6	and	and	CCONJ
cana-540	8	7	multifunctions	multifunction	NOUN
cana-540	8	8	are	be	AUX
cana-540	8	9	basic	basic	ADJ
cana-540	8	10	topics	topic	NOUN
cana-540	8	11	in	in	ADP
cana-540	8	12	general	general	ADJ
cana-540	8	13	topology	topology	NOUN
cana-540	8	14	and	and	CCONJ
cana-540	8	15	in	in	ADP
cana-540	8	16	set	set	NOUN
cana-540	8	17	valued	value	VERB
cana-540	8	18	analysis	analysis	NOUN
cana-540	8	19	in	in	ADP
cana-540	8	20	mathematics	mathematic	NOUN
cana-540	8	21	.	.	PUNCT
cana-540	9	1	continuous	continuous	ADJ
cana-540	9	2	functions	function	NOUN
cana-540	9	3	and	and	CCONJ
cana-540	9	4	continuous	continuous	ADJ
cana-540	9	5	multifunctions	multifunction	NOUN
cana-540	9	6	stand	stand	VERB
cana-540	9	7	among	among	ADP
cana-540	9	8	the	the	DET
cana-540	9	9	most	most	ADV
cana-540	9	10	fundamental	fundamental	ADJ
cana-540	9	11	and	and	CCONJ
cana-540	9	12	most	most	ADV
cana-540	9	13	related	related	ADJ
cana-540	9	14	points	point	NOUN
cana-540	9	15	in	in	ADP
cana-540	9	16	the	the	DET
cana-540	9	17	whole	whole	NOUN
cana-540	9	18	of	of	ADP
cana-540	9	19	mathematical	mathematical	ADJ
cana-540	9	20	sciences	science	NOUN
cana-540	9	21	.	.	PUNCT
cana-540	10	1	in	in	ADP
cana-540	10	2	the	the	DET
cana-540	10	3	literature	literature	NOUN
cana-540	10	4	,	,	PUNCT
cana-540	10	5	the	the	DET
cana-540	10	6	upper	upper	ADJ
cana-540	10	7	and	and	CCONJ
cana-540	10	8	lower	low	ADJ
cana-540	10	9	continuity	continuity	NOUN
cana-540	10	10	for	for	ADP
cana-540	10	11	multifunctions	multifunction	NOUN
cana-540	10	12	were	be	AUX
cana-540	10	13	firstly	firstly	ADV
cana-540	10	14	studied	study	VERB
cana-540	10	15	and	and	CCONJ
cana-540	10	16	introduced	introduce	VERB
cana-540	10	17	by	by	ADP
cana-540	10	18	berge[2	berge[2	NOUN
cana-540	10	19	]	]	PUNCT
cana-540	10	20	.	.	PUNCT
cana-540	11	1	after	after	ADP
cana-540	11	2	the	the	DET
cana-540	11	3	work	work	NOUN
cana-540	11	4	of	of	ADP
cana-540	11	5	him	he	PRON
cana-540	11	6	,	,	PUNCT
cana-540	11	7	many	many	ADJ
cana-540	11	8	authors	author	NOUN
cana-540	11	9	turned	turn	VERB
cana-540	11	10	continued	continue	VERB
cana-540	11	11	their	their	PRON
cana-540	11	12	research	research	NOUN
cana-540	11	13	to	to	PART
cana-540	11	14	investigate	investigate	VERB
cana-540	11	15	several	several	ADJ
cana-540	11	16	weak	weak	ADJ
cana-540	11	17	and	and	CCONJ
cana-540	11	18	strong	strong	ADJ
cana-540	11	19	forms	form	NOUN
cana-540	11	20	of	of	ADP
cana-540	11	21	continuity	continuity	NOUN
cana-540	11	22	.	.	PUNCT
cana-540	12	1	in	in	ADP
cana-540	12	2	1999	1999	NUM
cana-540	12	3	,	,	PUNCT
cana-540	12	4	mahmoad[12	mahmoad[12	PROPN
cana-540	12	5	]	]	PUNCT
cana-540	12	6	introduced	introduce	VERB
cana-540	12	7	the	the	DET
cana-540	12	8	concept	concept	NOUN
cana-540	12	9	of	of	ADP
cana-540	12	10	pre	pre	ADJ
cana-540	12	11	-	-	ADJ
cana-540	12	12	irresolute	irresolute	ADJ
cana-540	12	13	multivalued	multivalued	ADJ
cana-540	12	14	functions	function	NOUN
cana-540	12	15	.	.	PUNCT
cana-540	13	1	neubrunn	neubrunn	NOUN
cana-540	14	1	[	[	X
cana-540	14	2	13	13	NUM
cana-540	14	3	]	]	PUNCT
cana-540	14	4	introduced	introduce	VERB
cana-540	14	5	and	and	CCONJ
cana-540	14	6	studied	study	VERB
cana-540	14	7	the	the	DET
cana-540	14	8	notion	notion	NOUN
cana-540	14	9	of	of	ADP
cana-540	14	10	upper(lower	upper(lower	NOUN
cana-540	14	11	)	)	PUNCT
cana-540	14	12	α	α	NUM
cana-540	14	13	-	-	ADJ
cana-540	14	14	continuous	continuous	ADJ
cana-540	14	15	multifunctions	multifunction	NOUN
cana-540	14	16	.	.	PUNCT
cana-540	15	1	the	the	DET
cana-540	15	2	purpose	purpose	NOUN
cana-540	15	3	of	of	ADP
cana-540	15	4	this	this	DET
cana-540	15	5	paper	paper	NOUN
cana-540	15	6	is	be	AUX
cana-540	15	7	to	to	PART
cana-540	15	8	give	give	VERB
cana-540	15	9	a	a	DET
cana-540	15	10	new	new	ADJ
cana-540	15	11	an	an	DET
cana-540	15	12	idea	idea	NOUN
cana-540	15	13	about	about	ADP
cana-540	15	14	new	new	ADJ
cana-540	15	15	weaker	weak	ADJ
cana-540	15	16	form	form	NOUN
cana-540	15	17	of	of	ADP
cana-540	15	18	continuous	continuous	ADJ
cana-540	15	19	functions	function	NOUN
cana-540	15	20	called	call	VERB
cana-540	15	21	upper	upper	ADJ
cana-540	15	22	spgα	spgα	ADJ
cana-540	15	23	-	-	PUNCT
cana-540	15	24	continuous	continuous	ADJ
cana-540	15	25	multifunctions	multifunction	NOUN
cana-540	15	26	and	and	CCONJ
cana-540	15	27	lower	low	ADJ
cana-540	15	28	spgα	spgα	ADJ
cana-540	15	29	-	-	PUNCT
cana-540	15	30	continuous	continuous	ADJ
cana-540	15	31	multifunctions	multifunction	NOUN
cana-540	15	32	.	.	PUNCT
cana-540	16	1	further	far	ADV
cana-540	16	2	,	,	PUNCT
cana-540	16	3	basic	basic	ADJ
cana-540	16	4	properties	property	NOUN
cana-540	16	5	and	and	CCONJ
cana-540	16	6	preservation	preservation	NOUN
cana-540	16	7	theorems	theorem	NOUN
cana-540	16	8	of	of	ADP
cana-540	16	9	these	these	DET
cana-540	16	10	upper	upper	ADJ
cana-540	16	11	spgαcontinuous	spgαcontinuous	ADJ
cana-540	16	12	multifunctions	multifunction	NOUN
cana-540	16	13	(	(	PUNCT
cana-540	16	14	lower	low	ADJ
cana-540	16	15	spgα	spgα	ADJ
cana-540	16	16	-	-	PUNCT
cana-540	16	17	continuous	continuous	ADJ
cana-540	16	18	multifunctions	multifunction	NOUN
cana-540	16	19	)	)	PUNCT
cana-540	16	20	are	be	AUX
cana-540	16	21	studied	study	VERB
cana-540	16	22	.	.	PUNCT
cana-540	17	1	in	in	ADP
cana-540	17	2	the	the	DET
cana-540	17	3	next	next	ADJ
cana-540	17	4	section	section	NOUN
cana-540	17	5	,	,	PUNCT
cana-540	17	6	the	the	DET
cana-540	17	7	notion	notion	NOUN
cana-540	17	8	of	of	ADP
cana-540	17	9	upper	upper	ADJ
cana-540	17	10	spgα	spgα	ADJ
cana-540	17	11	-	-	PUNCT
cana-540	17	12	irresolute	irresolute	ADJ
cana-540	17	13	multifunctions	multifunction	NOUN
cana-540	17	14	(	(	PUNCT
cana-540	17	15	lower	low	ADJ
cana-540	17	16	spgαcontinuous	spgαcontinuous	ADJ
cana-540	17	17	multifunctions	multifunction	NOUN
cana-540	17	18	)	)	PUNCT
cana-540	17	19	is	be	AUX
cana-540	17	20	introduced	introduce	VERB
cana-540	17	21	and	and	CCONJ
cana-540	17	22	characterizations	characterization	NOUN
cana-540	17	23	and	and	CCONJ
cana-540	17	24	some	some	DET
cana-540	17	25	basic	basic	ADJ
cana-540	17	26	properties	property	NOUN
cana-540	17	27	are	be	AUX
cana-540	17	28	investigated	investigate	VERB
cana-540	17	29	.	.	PUNCT
cana-540	18	1	throughout	throughout	ADP
cana-540	18	2	this	this	DET
cana-540	18	3	paper	paper	NOUN
cana-540	18	4	(	(	PUNCT
cana-540	18	5	r	r	NOUN
cana-540	18	6	,	,	PUNCT
cana-540	18	7			PROPN
cana-540	18	8	)	)	PUNCT
cana-540	18	9	,	,	PUNCT
cana-540	18	10	(	(	PUNCT
cana-540	18	11	s	s	X
cana-540	18	12	,	,	PUNCT
cana-540	18	13			NUM
cana-540	18	14	)	)	PUNCT
cana-540	18	15	and	and	CCONJ
cana-540	18	16	(	(	PUNCT
cana-540	18	17	q	q	ADJ
cana-540	18	18	,	,	PUNCT
cana-540	18	19			NUM
cana-540	18	20	)	)	PUNCT
cana-540	18	21	stands	stand	VERB
cana-540	18	22	for	for	ADP
cana-540	18	23	topological	topological	ADJ
cana-540	18	24	spaces	space	NOUN
cana-540	18	25	with	with	ADP
cana-540	18	26	no	no	DET
cana-540	18	27	separation	separation	NOUN
cana-540	18	28	axioms	axiom	NOUN
cana-540	18	29	are	be	AUX
cana-540	18	30	assumed	assume	VERB
cana-540	18	31	,	,	PUNCT
cana-540	18	32	for	for	ADP
cana-540	18	33	any	any	DET
cana-540	18	34	set	set	NOUN
cana-540	18	35	a	a	PRON
cana-540	18	36	of	of	ADP
cana-540	18	37	a	a	DET
cana-540	18	38	space	space	NOUN
cana-540	18	39	r	r	NOUN
cana-540	18	40	,	,	PUNCT
cana-540	18	41	closure	closure	NOUN
cana-540	18	42	of	of	ADP
cana-540	18	43	a	a	PRON
cana-540	18	44	and	and	CCONJ
cana-540	18	45	interior	interior	ADJ
cana-540	18	46	of	of	ADP
cana-540	18	47	a	a	PRON
cana-540	18	48	is	be	AUX
cana-540	18	49	denoted	denote	VERB
cana-540	18	50	by	by	ADP
cana-540	18	51	cl(a	cl(a	NOUN
cana-540	18	52	)	)	PUNCT
cana-540	18	53	and	and	CCONJ
cana-540	18	54	int(a	int(a	PROPN
cana-540	18	55	)	)	PUNCT
cana-540	18	56	.	.	PUNCT
cana-540	19	1	a	a	DET
cana-540	19	2	multifunction	multifunction	NOUN
cana-540	19	3	p	p	X
cana-540	19	4	:	:	PUNCT
cana-540	19	5	r	r	NOUN
cana-540	19	6	→	→	SYM
cana-540	19	7	s	s	PART
cana-540	19	8	is	be	AUX
cana-540	19	9	a	a	DET
cana-540	19	10	point	point	NOUN
cana-540	19	11	to	to	PART
cana-540	19	12	set	set	VERB
cana-540	19	13	correspondence	correspondence	NOUN
cana-540	19	14	and	and	CCONJ
cana-540	19	15	always	always	ADV
cana-540	19	16	we	we	PRON
cana-540	19	17	assume	assume	VERB
cana-540	19	18	that	that	SCONJ
cana-540	19	19	p(p	p(p	NOUN
cana-540	19	20	)	)	PUNCT
cana-540	19	21			VERB
cana-540	19	22			NOUN
cana-540	19	23	for	for	ADP
cana-540	19	24	every	every	DET
cana-540	19	25	p	p	PROPN
cana-540	19	26			PROPN
cana-540	19	27	r.	r.	PROPN
cana-540	19	28	let	let	VERB
cana-540	19	29	a	a	DET
cana-540	19	30	be	be	AUX
cana-540	19	31	any	any	DET
cana-540	19	32	subset	subset	NOUN
cana-540	19	33	of	of	ADP
cana-540	19	34	r	r	NOUN
cana-540	19	35	and	and	CCONJ
cana-540	19	36	b	b	NOUN
cana-540	19	37	be	be	AUX
cana-540	19	38	any	any	DET
cana-540	19	39	subset	subset	NOUN
cana-540	19	40	of	of	ADP
cana-540	19	41	s.	s.	PROPN
cana-540	19	42	then	then	ADV
cana-540	19	43	p(a	p(a	PROPN
cana-540	19	44	)	)	PUNCT
cana-540	19	45	=	=	SYM
cana-540	19	46			NOUN
cana-540	19	47	{	{	PUNCT
cana-540	19	48	p(p	p(p	NOUN
cana-540	19	49	)	)	PUNCT
cana-540	19	50	:	:	PUNCT
cana-540	19	51	communications	communication	NOUN
cana-540	19	52	on	on	ADP
cana-540	19	53	applied	apply	VERB
cana-540	19	54	nonlinear	nonlinear	ADJ
cana-540	19	55	analysis	analysis	NOUN
cana-540	19	56	issn	issn	NOUN
cana-540	19	57	:	:	PUNCT
cana-540	19	58	1074	1074	NUM
cana-540	19	59	-	-	PUNCT
cana-540	19	60	133x	133x	NUM
cana-540	19	61	vol	vol	NOUN
cana-540	19	62	31	31	NUM
cana-540	19	63	no	no	NOUN
cana-540	19	64	.	.	NOUN
cana-540	19	65	2	2	NUM
cana-540	19	66	(	(	PUNCT
cana-540	19	67	2024	2024	NUM
cana-540	19	68	)	)	PUNCT
cana-540	19	69	249	249	NUM
cana-540	19	70	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	20	1	p	p	PROPN
cana-540	20	2			PROPN
cana-540	20	3	a	a	PRON
cana-540	20	4	}	}	PUNCT
cana-540	20	5	.	.	PUNCT
cana-540	21	1	for	for	ADP
cana-540	21	2	a	a	DET
cana-540	21	3	multifunction	multifunction	NOUN
cana-540	21	4	p	p	NOUN
cana-540	21	5	:	:	PUNCT
cana-540	21	6	r	r	NOUN
cana-540	21	7	→	→	SYM
cana-540	21	8	s	s	NOUN
cana-540	21	9	,	,	PUNCT
cana-540	21	10	following	follow	VERB
cana-540	21	11	[	[	X
cana-540	21	12	8	8	NUM
cana-540	21	13	]	]	PUNCT
cana-540	21	14	we	we	PRON
cana-540	21	15	will	will	AUX
cana-540	21	16	denote	denote	VERB
cana-540	21	17	the	the	DET
cana-540	21	18	upper	upper	ADJ
cana-540	21	19	and	and	CCONJ
cana-540	21	20	lower	low	ADJ
cana-540	21	21	inverse	inverse	NOUN
cana-540	21	22	of	of	ADP
cana-540	21	23	a	a	DET
cana-540	21	24	set	set	NOUN
cana-540	21	25	b	b	PROPN
cana-540	21	26	of	of	ADP
cana-540	21	27	s	s	PRON
cana-540	21	28	by	by	ADP
cana-540	21	29	p+(b	p+(b	NOUN
cana-540	21	30	)	)	PUNCT
cana-540	21	31	and	and	CCONJ
cana-540	21	32	p-(b	p-(b	ADJ
cana-540	21	33	)	)	PUNCT
cana-540	21	34	respectively	respectively	ADV
cana-540	21	35	,	,	PUNCT
cana-540	21	36	that	that	PRON
cana-540	21	37	is	be	AUX
cana-540	21	38	p+(b	p+(b	NOUN
cana-540	21	39	)	)	PUNCT
cana-540	21	40	=	=	PUNCT
cana-540	21	41	{p	{p	VERB
cana-540	21	42			NOUN
cana-540	21	43	r	r	NOUN
cana-540	21	44	:	:	PUNCT
cana-540	21	45	p(p	p(p	NOUN
cana-540	21	46	)	)	PUNCT
cana-540	22	1			PROPN
cana-540	22	2	b	b	AUX
cana-540	22	3	}	}	PUNCT
cana-540	22	4	and	and	CCONJ
cana-540	22	5	p-(b	p-(b	ADJ
cana-540	22	6	)	)	PUNCT
cana-540	22	7	=	=	SYM
cana-540	22	8	{	{	PUNCT
cana-540	22	9	p	p	X
cana-540	22	10			PROPN
cana-540	22	11	r	r	NOUN
cana-540	22	12	:	:	PUNCT
cana-540	22	13	p(p	p(p	NOUN
cana-540	22	14	)	)	PUNCT
cana-540	22	15			PROPN
cana-540	22	16	b	b	PROPN
cana-540	22	17			NOUN
cana-540	22	18	0	0	NUM
cana-540	22	19	}	}	PUNCT
cana-540	22	20	.	.	PUNCT
cana-540	23	1	so	so	ADV
cana-540	23	2	p	p	X
cana-540	23	3	:	:	PUNCT
cana-540	23	4	s	s	X
cana-540	23	5	→	→	SYM
cana-540	23	6	r(p	r(p	NOUN
cana-540	23	7	)	)	PUNCT
cana-540	23	8	and	and	CCONJ
cana-540	23	9	if	if	SCONJ
cana-540	23	10	s	s	PRON
cana-540	23	11			PROPN
cana-540	23	12	s.	s.	PROPN
cana-540	23	13	then	then	ADV
cana-540	23	14	p-(s	p-(s	VERB
cana-540	23	15	)	)	PUNCT
cana-540	23	16	=	=	PRON
cana-540	23	17	{	{	PUNCT
cana-540	23	18	p	p	PROPN
cana-540	23	19			PROPN
cana-540	23	20	s	s	PART
cana-540	23	21	:	:	PUNCT
cana-540	23	22	s	s	VERB
cana-540	23	23			NOUN
cana-540	23	24	p(p	p(p	NOUN
cana-540	23	25	)	)	PUNCT
cana-540	23	26	}	}	PUNCT
cana-540	23	27	where	where	SCONJ
cana-540	23	28	p(a	p(a	PROPN
cana-540	23	29	)	)	PUNCT
cana-540	23	30	be	be	VERB
cana-540	23	31	the	the	DET
cana-540	23	32	collection	collection	NOUN
cana-540	23	33	of	of	ADP
cana-540	23	34	the	the	DET
cana-540	23	35	subsets	subset	NOUN
cana-540	23	36	of	of	ADP
cana-540	23	37	p.	p.	NOUN
cana-540	23	38	thus	thus	ADV
cana-540	23	39	,	,	PUNCT
cana-540	23	40	for	for	ADP
cana-540	23	41	a	a	DET
cana-540	23	42	subset	subset	NOUN
cana-540	23	43	b	b	NOUN
cana-540	23	44	in	in	ADP
cana-540	23	45	s	s	NOUN
cana-540	23	46	,	,	PUNCT
cana-540	23	47	p-(b	p-(b	ADJ
cana-540	23	48	)	)	PUNCT
cana-540	23	49	=	=	SYM
cana-540	23	50	{p-(s	{p-(s	ADJ
cana-540	23	51	)	)	PUNCT
cana-540	23	52	:	:	PUNCT
cana-540	23	53	s	s	VERB
cana-540	23	54			PROPN
cana-540	23	55	b	b	NOUN
cana-540	23	56	}	}	PUNCT
cana-540	23	57	.	.	PUNCT
cana-540	24	1	then	then	ADV
cana-540	24	2	p	p	NOUN
cana-540	24	3	is	be	AUX
cana-540	24	4	said	say	VERB
cana-540	24	5	to	to	PART
cana-540	24	6	be	be	AUX
cana-540	24	7	a	a	DET
cana-540	24	8	surjection	surjection	NOUN
cana-540	24	9	if	if	SCONJ
cana-540	24	10	p(p	p(p	NOUN
cana-540	24	11	)	)	PUNCT
cana-540	24	12	=	=	SYM
cana-540	24	13	s	s	NOUN
cana-540	24	14	or	or	CCONJ
cana-540	24	15	equivalently	equivalently	ADV
cana-540	24	16	,	,	PUNCT
cana-540	24	17	if	if	SCONJ
cana-540	24	18	for	for	ADP
cana-540	24	19	each	each	PRON
cana-540	24	20	s	s	PROPN
cana-540	24	21			PROPN
cana-540	24	22	s	s	PART
cana-540	24	23	,	,	PUNCT
cana-540	24	24	there	there	PRON
cana-540	24	25	exists	exist	VERB
cana-540	24	26	p	p	ADJ
cana-540	24	27			PROPN
cana-540	24	28	r	r	NOUN
cana-540	24	29	such	such	ADJ
cana-540	24	30	that	that	PRON
cana-540	24	31	s	s	VERB
cana-540	24	32			NOUN
cana-540	24	33	p(p	p(p	NOUN
cana-540	24	34	)	)	PUNCT
cana-540	24	35	.	.	PUNCT
cana-540	25	1	for	for	ADP
cana-540	25	2	multifunction	multifunction	NOUN
cana-540	25	3	p	p	NOUN
cana-540	25	4	:	:	PUNCT
cana-540	25	5	r	r	NOUN
cana-540	25	6	→	→	SYM
cana-540	25	7	s	s	X
cana-540	25	8	,	,	PUNCT
cana-540	25	9	the	the	DET
cana-540	25	10	graph	graph	NOUN
cana-540	25	11	multifunction	multifunction	NOUN
cana-540	25	12	gp	gp	NOUN
cana-540	26	1	:	:	PUNCT
cana-540	26	2	r	r	NOUN
cana-540	26	3	→	→	SYM
cana-540	26	4	r	r	NOUN
cana-540	26	5	x	x	SYM
cana-540	26	6	s	s	NOUN
cana-540	26	7	is	be	AUX
cana-540	26	8	defined	define	VERB
cana-540	26	9	as	as	ADP
cana-540	26	10	gp	gp	NOUN
cana-540	26	11	(	(	PUNCT
cana-540	26	12	p	p	X
cana-540	26	13	)	)	PUNCT
cana-540	26	14	=	=	PUNCT
cana-540	27	1	{	{	PUNCT
cana-540	27	2	p	p	X
cana-540	27	3	}	}	PUNCT
cana-540	27	4	x	x	SYM
cana-540	27	5	p(p	p(p	NOUN
cana-540	27	6	)	)	PUNCT
cana-540	27	7	for	for	ADP
cana-540	27	8	each	each	DET
cana-540	27	9	p	p	NOUN
cana-540	27	10			NOUN
cana-540	27	11	r	r	NOUN
cana-540	27	12	and	and	CCONJ
cana-540	27	13	the	the	DET
cana-540	27	14	subset	subset	NOUN
cana-540	27	15	{	{	PUNCT
cana-540	27	16	{	{	PUNCT
cana-540	27	17	p	p	X
cana-540	27	18	}	}	PUNCT
cana-540	27	19	x	x	SYM
cana-540	27	20	p(p	p(p	NOUN
cana-540	27	21	)	)	PUNCT
cana-540	27	22	:	:	PUNCT
cana-540	28	1	p	p	X
cana-540	28	2			PROPN
cana-540	28	3	r	r	NOUN
cana-540	28	4	}	}	PUNCT
cana-540	28	5			PROPN
cana-540	28	6	r	r	NOUN
cana-540	28	7	x	x	PUNCT
cana-540	28	8	s	s	NOUN
cana-540	28	9	is	be	AUX
cana-540	28	10	called	call	VERB
cana-540	28	11	the	the	DET
cana-540	28	12	multifunction	multifunction	NOUN
cana-540	28	13	of	of	ADP
cana-540	28	14	p	p	NOUN
cana-540	28	15	and	and	CCONJ
cana-540	28	16	is	be	AUX
cana-540	28	17	denoted	denote	VERB
cana-540	28	18	by	by	ADP
cana-540	28	19	g(p)[8	g(p)[8	X
cana-540	28	20	]	]	PUNCT
cana-540	28	21	.	.	PUNCT
cana-540	29	1	2	2	X
cana-540	29	2	.	.	X
cana-540	29	3	spgα	spgα	ADJ
cana-540	29	4	-	-	PUNCT
cana-540	29	5	continuous	continuous	ADJ
cana-540	29	6	multifunctions	multifunction	NOUN
cana-540	29	7	this	this	DET
cana-540	29	8	section	section	NOUN
cana-540	29	9	deals	deal	VERB
cana-540	29	10	with	with	ADP
cana-540	29	11	the	the	DET
cana-540	29	12	characterizations	characterization	NOUN
cana-540	29	13	of	of	ADP
cana-540	29	14	upper	upper	ADJ
cana-540	29	15	and	and	CCONJ
cana-540	29	16	lower	low	ADJ
cana-540	29	17	spgα	spgα	ADJ
cana-540	29	18	-	-	PUNCT
cana-540	29	19	continuous	continuous	ADJ
cana-540	29	20	multifunctions	multifunction	NOUN
cana-540	29	21	and	and	CCONJ
cana-540	29	22	introduced	introduce	VERB
cana-540	29	23	basic	basic	ADJ
cana-540	29	24	properties	property	NOUN
cana-540	29	25	related	relate	VERB
cana-540	29	26	to	to	ADP
cana-540	29	27	them	they	PRON
cana-540	29	28	in	in	ADP
cana-540	29	29	topological	topological	ADJ
cana-540	29	30	spaces	space	NOUN
cana-540	29	31	.	.	PUNCT
cana-540	30	1	definition	definition	NOUN
cana-540	30	2	2.1	2.1	NUM
cana-540	30	3	.	.	PUNCT
cana-540	31	1	a	a	DET
cana-540	31	2	multifunction	multifunction	NOUN
cana-540	31	3	p	p	NOUN
cana-540	31	4	:	:	PUNCT
cana-540	31	5	r	r	NOUN
cana-540	31	6	→	→	SYM
cana-540	31	7	s	s	PART
cana-540	31	8	is	be	AUX
cana-540	31	9	called	call	VERB
cana-540	31	10	1	1	NUM
cana-540	31	11	.	.	PUNCT
cana-540	32	1	upper	upper	ADJ
cana-540	32	2	spgα	spgα	NOUN
cana-540	32	3	-	-	PUNCT
cana-540	32	4	continuous	continuous	ADJ
cana-540	32	5	(	(	PUNCT
cana-540	32	6	briefly	briefly	NOUN
cana-540	32	7	u.spgα.c	u.spgα.c	NOUN
cana-540	32	8	)	)	PUNCT
cana-540	32	9	at	at	ADP
cana-540	32	10	a	a	DET
cana-540	32	11	point	point	NOUN
cana-540	32	12	r	r	NOUN
cana-540	32	13			NOUN
cana-540	32	14	r	r	NOUN
cana-540	32	15	,	,	PUNCT
cana-540	32	16	if	if	SCONJ
cana-540	32	17	for	for	ADP
cana-540	32	18	each	each	PRON
cana-540	32	19	v	v	ADJ
cana-540	32	20			PROPN
cana-540	32	21	o(s	o(s	PROPN
cana-540	32	22	)	)	PUNCT
cana-540	32	23	with	with	ADP
cana-540	32	24	p(r	p(r	PROPN
cana-540	32	25	)	)	PUNCT
cana-540	33	1			PROPN
cana-540	33	2	v	v	ADP
cana-540	33	3	,	,	PUNCT
cana-540	33	4	there	there	PRON
cana-540	33	5	exist	exist	VERB
cana-540	33	6	u	u	PRON
cana-540	33	7			PROPN
cana-540	33	8	spgα	spgα	NOUN
cana-540	33	9	-	-	PUNCT
cana-540	33	10	o(r	o(r	PROPN
cana-540	33	11	,	,	PUNCT
cana-540	33	12	r	r	NOUN
cana-540	33	13	)	)	PUNCT
cana-540	33	14	such	such	ADJ
cana-540	33	15	that	that	DET
cana-540	33	16	p(u	p(u	NOUN
cana-540	33	17	)	)	PUNCT
cana-540	34	1			PROPN
cana-540	34	2	v	v	NOUN
cana-540	34	3	.	.	PUNCT
cana-540	35	1	2	2	X
cana-540	35	2	.	.	X
cana-540	35	3	lower	low	ADJ
cana-540	35	4	spgα	spgα	ADJ
cana-540	35	5	-	-	PUNCT
cana-540	35	6	continuous	continuous	ADJ
cana-540	35	7	(	(	PUNCT
cana-540	35	8	briefly	briefly	NOUN
cana-540	35	9	l.spgα.c	l.spgα.c	NOUN
cana-540	35	10	)	)	PUNCT
cana-540	35	11	at	at	ADP
cana-540	35	12	a	a	DET
cana-540	35	13	point	point	NOUN
cana-540	35	14	r	r	NOUN
cana-540	35	15			NOUN
cana-540	35	16	r	r	NOUN
cana-540	35	17	,	,	PUNCT
cana-540	35	18	if	if	SCONJ
cana-540	35	19	for	for	ADP
cana-540	35	20	every	every	DET
cana-540	35	21	v	v	ADJ
cana-540	35	22			PROPN
cana-540	35	23	o(s	o(s	PROPN
cana-540	35	24	)	)	PUNCT
cana-540	35	25	with	with	ADP
cana-540	35	26	p(r	p(r	PROPN
cana-540	35	27	)	)	PUNCT
cana-540	35	28			PROPN
cana-540	35	29	v	v	ADP
cana-540	35	30			NOUN
cana-540	35	31			NOUN
cana-540	35	32	,	,	PUNCT
cana-540	35	33	then	then	ADV
cana-540	35	34	there	there	PRON
cana-540	35	35	exist	exist	VERB
cana-540	35	36	v	v	ADP
cana-540	35	37			PROPN
cana-540	35	38	spgα	spgα	NOUN
cana-540	35	39	-	-	PUNCT
cana-540	35	40	o(r	o(r	PROPN
cana-540	35	41	,	,	PUNCT
cana-540	35	42	z	z	NOUN
cana-540	35	43	)	)	PUNCT
cana-540	35	44	such	such	ADJ
cana-540	35	45	that	that	DET
cana-540	35	46	p(z	p(z	NOUN
cana-540	35	47	)	)	PUNCT
cana-540	35	48			NOUN
cana-540	35	49	v	v	NUM
cana-540	35	50			NOUN
cana-540	35	51			NOUN
cana-540	35	52	holds	hold	VERB
cana-540	35	53	for	for	ADP
cana-540	35	54	each	each	DET
cana-540	35	55	z	z	PROPN
cana-540	35	56			PROPN
cana-540	35	57	r.	r.	PROPN
cana-540	35	58	3	3	NUM
cana-540	35	59	.	.	PUNCT
cana-540	35	60	upper(lower	upper(lower	NOUN
cana-540	35	61	)	)	PUNCT
cana-540	35	62	spgα	spgα	NOUN
cana-540	35	63	-	-	PUNCT
cana-540	35	64	continuous	continuous	ADJ
cana-540	35	65	,	,	PUNCT
cana-540	35	66	if	if	SCONJ
cana-540	35	67	it	it	PRON
cana-540	35	68	is	be	AUX
cana-540	35	69	upper(lower	upper(lower	NOUN
cana-540	35	70	)	)	PUNCT
cana-540	35	71	spgα	spgα	NOUN
cana-540	35	72	-	-	PUNCT
cana-540	35	73	continuous	continuous	ADJ
cana-540	35	74	at	at	ADP
cana-540	35	75	every	every	DET
cana-540	35	76	point	point	NOUN
cana-540	35	77	of	of	ADP
cana-540	35	78	r.	r.	PROPN
cana-540	35	79	example	example	NOUN
cana-540	35	80	2.1	2.1	NUM
cana-540	35	81	.	.	PUNCT
cana-540	36	1	let	let	VERB
cana-540	36	2	r	r	NOUN
cana-540	36	3	=	=	SYM
cana-540	36	4	s	s	PART
cana-540	36	5	=	=	PUNCT
cana-540	36	6	{	{	PUNCT
cana-540	36	7	p1	p1	NOUN
cana-540	36	8	,	,	PUNCT
cana-540	36	9	p2	p2	NOUN
cana-540	36	10	,	,	PUNCT
cana-540	36	11	p3	p3	PROPN
cana-540	36	12	}	}	PUNCT
cana-540	36	13	,	,	PUNCT
cana-540	36	14	=	=	NOUN
cana-540	36	15	{	{	PUNCT
cana-540	36	16	r	r	NOUN
cana-540	36	17	,	,	PUNCT
cana-540	36	18			NOUN
cana-540	36	19	,	,	PUNCT
cana-540	36	20	{	{	PUNCT
cana-540	36	21	p1	p1	NOUN
cana-540	36	22	}	}	PUNCT
cana-540	36	23	}	}	PUNCT
cana-540	36	24	and	and	CCONJ
cana-540	37	1			PROPN
cana-540	37	2	=	=	SYM
cana-540	37	3	{	{	PUNCT
cana-540	37	4	s	s	NOUN
cana-540	37	5	,	,	PUNCT
cana-540	37	6			NOUN
cana-540	37	7	,	,	PUNCT
cana-540	37	8	{	{	PUNCT
cana-540	37	9	p1	p1	NOUN
cana-540	37	10	}	}	PUNCT
cana-540	37	11	}	}	PUNCT
cana-540	37	12	,	,	PUNCT
cana-540	37	13	{	{	PUNCT
cana-540	37	14	p2	p2	X
cana-540	37	15	}	}	PUNCT
cana-540	37	16	}	}	PUNCT
cana-540	37	17	let	let	VERB
cana-540	37	18	p	p	PRON
cana-540	37	19	be	be	AUX
cana-540	37	20	m.f	m.f	NOUN
cana-540	37	21	and	and	CCONJ
cana-540	37	22	p1	p1	PROPN
cana-540	37	23	be	be	VERB
cana-540	37	24	the	the	DET
cana-540	37	25	identity	identity	NOUN
cana-540	37	26	m.f	m.f	NOUN
cana-540	37	27	from	from	ADP
cana-540	37	28	r	r	NOUN
cana-540	37	29	to	to	ADP
cana-540	37	30	s.	s.	PROPN
cana-540	37	31	spgα	spgα	PROPN
cana-540	37	32	-	-	PUNCT
cana-540	37	33	o(r	o(r	PROPN
cana-540	37	34	)	)	PUNCT
cana-540	38	1	=	=	SYM
cana-540	38	2	r	r	NOUN
cana-540	38	3	,	,	PUNCT
cana-540	38	4			NOUN
cana-540	38	5	,	,	PUNCT
cana-540	38	6	{	{	PUNCT
cana-540	38	7	p1	p1	NOUN
cana-540	38	8	}	}	PUNCT
cana-540	38	9	,	,	PUNCT
cana-540	38	10	{	{	PUNCT
cana-540	38	11	p1	p1	NOUN
cana-540	38	12	,	,	PUNCT
cana-540	38	13	p2	p2	PROPN
cana-540	38	14	}	}	PUNCT
cana-540	38	15	,	,	PUNCT
cana-540	38	16	{	{	PUNCT
cana-540	38	17	p1	p1	NOUN
cana-540	38	18	,	,	PUNCT
cana-540	38	19	p3	p3	PROPN
cana-540	38	20	}	}	PUNCT
cana-540	38	21	.	.	PUNCT
cana-540	39	1	spgα	spgα	ADJ
cana-540	39	2	-	-	PUNCT
cana-540	39	3	o(s	o(s	NOUN
cana-540	39	4	)	)	PUNCT
cana-540	40	1	=	=	SYM
cana-540	40	2	s	s	NOUN
cana-540	40	3	,	,	PUNCT
cana-540	40	4			NOUN
cana-540	40	5	,	,	PUNCT
cana-540	40	6	{	{	PUNCT
cana-540	40	7	p1	p1	NOUN
cana-540	40	8	}	}	PUNCT
cana-540	40	9	,	,	PUNCT
cana-540	40	10	{	{	PUNCT
cana-540	40	11	p2	p2	X
cana-540	40	12	}	}	PUNCT
cana-540	40	13	,	,	PUNCT
cana-540	40	14	{	{	PUNCT
cana-540	40	15	p1	p1	NOUN
cana-540	40	16	,	,	PUNCT
cana-540	40	17	p2	p2	PROPN
cana-540	40	18	}	}	PUNCT
cana-540	40	19	,	,	PUNCT
cana-540	40	20	{	{	PUNCT
cana-540	40	21	p1	p1	NOUN
cana-540	40	22	,	,	PUNCT
cana-540	40	23	p3	p3	PROPN
cana-540	40	24	}	}	PUNCT
cana-540	40	25	,	,	PUNCT
cana-540	40	26	{	{	PUNCT
cana-540	40	27	p2	p2	NOUN
cana-540	40	28	,	,	PUNCT
cana-540	40	29	p3	p3	PROPN
cana-540	40	30	}	}	PUNCT
cana-540	40	31	.	.	PUNCT
cana-540	41	1	let	let	VERB
cana-540	41	2	p1	p1	VERB
cana-540	41	3			NOUN
cana-540	41	4	r	r	NOUN
cana-540	41	5	and	and	CCONJ
cana-540	41	6	v	v	NOUN
cana-540	41	7	=	=	SYM
cana-540	41	8	{	{	PUNCT
cana-540	41	9	p1	p1	NOUN
cana-540	41	10	,	,	PUNCT
cana-540	41	11	p2	p2	PROPN
cana-540	41	12	}	}	PUNCT
cana-540	41	13	is	be	AUX
cana-540	41	14	a	a	DET
cana-540	41	15	open	open	ADJ
cana-540	41	16	set	set	NOUN
cana-540	41	17	of	of	ADP
cana-540	41	18	s	s	PRON
cana-540	41	19	where	where	SCONJ
cana-540	41	20	p({p1	p({p1	NOUN
cana-540	41	21	}	}	PUNCT
cana-540	41	22	)	)	PUNCT
cana-540	42	1	=	=	SYM
cana-540	42	2	{	{	PUNCT
cana-540	42	3	p1	p1	PROPN
cana-540	42	4	}	}	PUNCT
cana-540	42	5			PROPN
cana-540	42	6	v.	v.	CCONJ
cana-540	42	7	then	then	ADV
cana-540	42	8	,	,	PUNCT
cana-540	42	9	there	there	PRON
cana-540	42	10	exists	exist	VERB
cana-540	42	11	a	a	DET
cana-540	42	12	spgα	spgα	ADJ
cana-540	42	13	-	-	PUNCT
cana-540	42	14	open	open	ADJ
cana-540	42	15	set	set	NOUN
cana-540	42	16	u	u	NOUN
cana-540	42	17	=	=	SYM
cana-540	42	18	{	{	PUNCT
cana-540	42	19	p1	p1	PROPN
cana-540	42	20	,	,	PUNCT
cana-540	42	21	p2	p2	NOUN
cana-540	42	22	}	}	PUNCT
cana-540	42	23	in	in	ADP
cana-540	42	24	r	r	NOUN
cana-540	42	25	containing	contain	VERB
cana-540	42	26	the	the	DET
cana-540	42	27	point	point	NOUN
cana-540	42	28	p1	p1	NOUN
cana-540	42	29	with	with	ADP
cana-540	42	30	p(u	p(u	NOUN
cana-540	42	31	)	)	PUNCT
cana-540	43	1	=	=	SYM
cana-540	43	2	p({p1	p({p1	ADJ
cana-540	43	3	,	,	PUNCT
cana-540	43	4	p2	p2	NOUN
cana-540	43	5	}	}	PUNCT
cana-540	43	6	)	)	PUNCT
cana-540	43	7	=	=	SYM
cana-540	43	8	{	{	PUNCT
cana-540	43	9	p1	p1	NOUN
cana-540	43	10	,	,	PUNCT
cana-540	43	11	p2	p2	PROPN
cana-540	43	12	}	}	PUNCT
cana-540	43	13			PROPN
cana-540	43	14	v	v	ADP
cana-540	43	15	thus	thus	ADV
cana-540	43	16	p(u	p(u	ADJ
cana-540	43	17	)	)	PUNCT
cana-540	44	1			PROPN
cana-540	44	2	v	v	PROPN
cana-540	44	3	and	and	CCONJ
cana-540	44	4	so	so	ADV
cana-540	44	5	p1	p1	PROPN
cana-540	44	6	is	be	AUX
cana-540	44	7	u.spgα.c	u.spgα.c	PROPN
cana-540	44	8	.	.	PUNCT
cana-540	44	9	example	example	NOUN
cana-540	44	10	2.2	2.2	NUM
cana-540	44	11	.	.	PUNCT
cana-540	45	1	let	let	VERB
cana-540	45	2	r	r	NOUN
cana-540	45	3	=	=	SYM
cana-540	45	4	s	s	PART
cana-540	45	5	=	=	PUNCT
cana-540	45	6	{	{	PUNCT
cana-540	45	7	p1	p1	NOUN
cana-540	45	8	,	,	PUNCT
cana-540	45	9	p2	p2	NOUN
cana-540	45	10	,	,	PUNCT
cana-540	45	11	p3	p3	NOUN
cana-540	45	12	}	}	PUNCT
cana-540	45	13			NOUN
cana-540	45	14	=	=	SYM
cana-540	45	15	{	{	PUNCT
cana-540	45	16	r	r	NOUN
cana-540	45	17	,	,	PUNCT
cana-540	45	18			NOUN
cana-540	45	19	,	,	PUNCT
cana-540	45	20	{	{	PUNCT
cana-540	45	21	p1	p1	NOUN
cana-540	45	22	}	}	PUNCT
cana-540	45	23	,	,	PUNCT
cana-540	45	24	{	{	PUNCT
cana-540	45	25	p1	p1	NOUN
cana-540	45	26	,	,	PUNCT
cana-540	45	27	p2	p2	NOUN
cana-540	45	28	}	}	PUNCT
cana-540	45	29	}	}	PUNCT
cana-540	45	30	.	.	PUNCT
cana-540	46	1	here	here	ADV
cana-540	46	2	spgα	spgα	ADJ
cana-540	46	3	-	-	PUNCT
cana-540	46	4	open	open	ADJ
cana-540	46	5	sets	set	NOUN
cana-540	46	6	are	be	AUX
cana-540	46	7	:	:	PUNCT
cana-540	46	8	r	r	NOUN
cana-540	46	9	,	,	PUNCT
cana-540	46	10			NOUN
cana-540	46	11	,	,	PUNCT
cana-540	46	12	{	{	PUNCT
cana-540	46	13	p1	p1	NOUN
cana-540	46	14	}	}	PUNCT
cana-540	46	15	,	,	PUNCT
cana-540	46	16	{	{	PUNCT
cana-540	46	17	p2	p2	X
cana-540	46	18	}	}	PUNCT
cana-540	46	19	,	,	PUNCT
cana-540	46	20	{	{	PUNCT
cana-540	46	21	p1	p1	NOUN
cana-540	46	22	,	,	PUNCT
cana-540	46	23	p2	p2	PROPN
cana-540	46	24	}	}	PUNCT
cana-540	46	25			X
cana-540	46	26	=	=	SYM
cana-540	46	27	{	{	PUNCT
cana-540	46	28	s	s	NOUN
cana-540	46	29	,	,	PUNCT
cana-540	46	30			NOUN
cana-540	46	31	,	,	PUNCT
cana-540	46	32	{	{	PUNCT
cana-540	46	33	p1	p1	NOUN
cana-540	46	34	}	}	PUNCT
cana-540	46	35	}	}	PUNCT
cana-540	46	36	.	.	PUNCT
cana-540	47	1	here	here	ADV
cana-540	47	2	spgα	spgα	ADJ
cana-540	47	3	-	-	PUNCT
cana-540	47	4	open	open	ADJ
cana-540	47	5	sets	set	NOUN
cana-540	47	6	are	be	AUX
cana-540	47	7	:	:	PUNCT
cana-540	47	8	s	s	X
cana-540	47	9	,	,	PUNCT
cana-540	47	10			NOUN
cana-540	47	11	,	,	PUNCT
cana-540	47	12	{	{	PUNCT
cana-540	47	13	p1	p1	NOUN
cana-540	47	14	}	}	PUNCT
cana-540	47	15	,	,	PUNCT
cana-540	47	16	{	{	PUNCT
cana-540	47	17	p1	p1	NOUN
cana-540	47	18	,	,	PUNCT
cana-540	47	19	p2	p2	PROPN
cana-540	47	20	}	}	PUNCT
cana-540	47	21	,	,	PUNCT
cana-540	47	22	{	{	PUNCT
cana-540	47	23	p1	p1	NOUN
cana-540	47	24	,	,	PUNCT
cana-540	47	25	p3	p3	PROPN
cana-540	47	26	}	}	PUNCT
cana-540	47	27	let	let	VERB
cana-540	47	28	p1	p1	NOUN
cana-540	47	29	:	:	PUNCT
cana-540	47	30	r	r	X
cana-540	47	31	→	→	SYM
cana-540	47	32	s	s	AUX
cana-540	47	33	be	be	AUX
cana-540	47	34	a	a	DET
cana-540	47	35	m.f	m.f	NOUN
cana-540	47	36	and	and	CCONJ
cana-540	47	37	r	r	NOUN
cana-540	47	38	=	=	PUNCT
cana-540	47	39	{	{	PUNCT
cana-540	47	40	p2	p2	PROPN
cana-540	47	41	}	}	PUNCT
cana-540	47	42			NOUN
cana-540	47	43	r	r	NOUN
cana-540	47	44	spgα	spgα	ADJ
cana-540	47	45	-	-	PUNCT
cana-540	47	46	open	open	ADJ
cana-540	47	47	sets	set	NOUN
cana-540	47	48	in	in	ADP
cana-540	47	49	s	s	AUX
cana-540	47	50	containing	contain	VERB
cana-540	47	51	p2	p2	NOUN
cana-540	47	52	are	be	AUX
cana-540	47	53	:	:	PUNCT
cana-540	47	54	s	s	X
cana-540	47	55	,	,	PUNCT
cana-540	47	56	{	{	PUNCT
cana-540	47	57	p1	p1	NOUN
cana-540	47	58	,	,	PUNCT
cana-540	47	59	p2	p2	PROPN
cana-540	47	60	}	}	PUNCT
cana-540	47	61	let	let	VERB
cana-540	47	62	v	v	NOUN
cana-540	47	63	=	=	PUNCT
cana-540	47	64	s.	s.	PROPN
cana-540	48	1	v	v	NOUN
cana-540	48	2	=	=	SYM
cana-540	48	3	s	s	PART
cana-540	48	4	=	=	PUNCT
cana-540	48	5	{	{	PUNCT
cana-540	48	6	p1	p1	NOUN
cana-540	48	7	,	,	PUNCT
cana-540	48	8	p2	p2	NOUN
cana-540	48	9	,	,	PUNCT
cana-540	48	10	p3	p3	PROPN
cana-540	48	11	}	}	PUNCT
cana-540	48	12	→	→	SYM
cana-540	48	13	p({p2	p({p2	NOUN
cana-540	48	14	}	}	PUNCT
cana-540	48	15	)	)	PUNCT
cana-540	49	1			PUNCT
cana-540	49	2	v	v	NOUN
cana-540	49	3	=	=	SYM
cana-540	49	4	{	{	PUNCT
cana-540	49	5	p2	p2	NOUN
cana-540	49	6	}	}	PUNCT
cana-540	49	7			X
cana-540	49	8	{	{	PUNCT
cana-540	49	9	p1	p1	NOUN
cana-540	49	10	,	,	PUNCT
cana-540	49	11	p2	p2	NOUN
cana-540	49	12	,	,	PUNCT
cana-540	49	13	p3	p3	NOUN
cana-540	49	14	}	}	PUNCT
cana-540	49	15	=	=	SYM
cana-540	49	16	{	{	PUNCT
cana-540	49	17	p2	p2	PROPN
cana-540	49	18	}	}	PUNCT
cana-540	49	19			NOUN
cana-540	49	20			NOUN
cana-540	49	21	consider	consider	VERB
cana-540	49	22	v	v	NOUN
cana-540	49	23	=	=	SYM
cana-540	49	24	{	{	PUNCT
cana-540	49	25	p1	p1	NOUN
cana-540	49	26	,	,	PUNCT
cana-540	49	27	p2	p2	PROPN
cana-540	49	28	}	}	PUNCT
cana-540	49	29	v	v	NOUN
cana-540	49	30	=	=	SYM
cana-540	49	31	{	{	PUNCT
cana-540	49	32	p1	p1	NOUN
cana-540	49	33	,	,	PUNCT
cana-540	49	34	p2	p2	X
cana-540	49	35	}	}	PUNCT
cana-540	49	36	→	→	SYM
cana-540	49	37	p({p2	p({p2	NOUN
cana-540	49	38	}	}	PUNCT
cana-540	49	39	)	)	PUNCT
cana-540	49	40			PUNCT
cana-540	49	41	v	v	NOUN
cana-540	49	42	=	=	SYM
cana-540	49	43	{	{	PUNCT
cana-540	49	44	p2	p2	NOUN
cana-540	49	45	}	}	PUNCT
cana-540	49	46			X
cana-540	49	47	{	{	PUNCT
cana-540	49	48	p1	p1	NOUN
cana-540	49	49	,	,	PUNCT
cana-540	49	50	p2	p2	NOUN
cana-540	49	51	}	}	PUNCT
cana-540	49	52	=	=	SYM
cana-540	49	53	{	{	PUNCT
cana-540	49	54	p2	p2	PROPN
cana-540	49	55	}	}	PUNCT
cana-540	49	56			NOUN
cana-540	49	57	.	.	PUNCT
cana-540	49	58	then	then	ADV
cana-540	49	59	there	there	PRON
cana-540	49	60	exists	exist	VERB
cana-540	49	61	spgα	spgα	ADJ
cana-540	49	62	-	-	PUNCT
cana-540	49	63	open	open	ADJ
cana-540	49	64	set	set	NOUN
cana-540	49	65	u	u	NOUN
cana-540	49	66	=	=	SYM
cana-540	49	67	{	{	PUNCT
cana-540	49	68	p1	p1	NOUN
cana-540	49	69	,	,	PUNCT
cana-540	49	70	p2	p2	X
cana-540	49	71	}	}	PUNCT
cana-540	49	72	containing	contain	VERB
cana-540	49	73	{	{	PUNCT
cana-540	49	74	p2	p2	NOUN
cana-540	49	75	}	}	PUNCT
cana-540	49	76	are	be	AUX
cana-540	49	77	:	:	PUNCT
cana-540	49	78	s	s	X
cana-540	49	79	,	,	PUNCT
cana-540	49	80	{	{	PUNCT
cana-540	49	81	p1	p1	NOUN
cana-540	49	82	,	,	PUNCT
cana-540	49	83	p2	p2	NOUN
cana-540	49	84	}	}	PUNCT
cana-540	49	85	communications	communication	NOUN
cana-540	49	86	on	on	ADP
cana-540	49	87	applied	apply	VERB
cana-540	49	88	nonlinear	nonlinear	ADJ
cana-540	49	89	analysis	analysis	NOUN
cana-540	49	90	issn	issn	NOUN
cana-540	49	91	:	:	PUNCT
cana-540	49	92	1074	1074	NUM
cana-540	49	93	-	-	PUNCT
cana-540	49	94	133x	133x	NUM
cana-540	49	95	vol	vol	NOUN
cana-540	49	96	31	31	NUM
cana-540	49	97	no	no	NOUN
cana-540	49	98	.	.	NOUN
cana-540	49	99	2	2	NUM
cana-540	49	100	(	(	PUNCT
cana-540	49	101	2024	2024	NUM
cana-540	49	102	)	)	PUNCT
cana-540	50	1	250	250	NUM
cana-540	50	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	50	3	1	1	NUM
cana-540	50	4	.	.	PUNCT
cana-540	50	5	{	{	PUNCT
cana-540	50	6	p2	p2	NOUN
cana-540	50	7	}	}	PUNCT
cana-540	50	8			X
cana-540	50	9	{	{	PUNCT
cana-540	50	10	p1	p1	NOUN
cana-540	50	11	,	,	PUNCT
cana-540	50	12	p2	p2	NOUN
cana-540	50	13	}	}	PUNCT
cana-540	50	14	=	=	SYM
cana-540	50	15	{	{	PUNCT
cana-540	50	16	p2	p2	PROPN
cana-540	50	17	}	}	PUNCT
cana-540	50	18			NOUN
cana-540	50	19	.	.	X
cana-540	50	20	2	2	NUM
cana-540	50	21	.	.	PUNCT
cana-540	50	22	{	{	PUNCT
cana-540	50	23	p2	p2	PROPN
cana-540	50	24	}	}	PUNCT
cana-540	50	25			NOUN
cana-540	50	26	n	n	NOUN
cana-540	50	27	=	=	SYM
cana-540	50	28	{	{	PUNCT
cana-540	50	29	p2	p2	NOUN
cana-540	50	30	}	}	PUNCT
cana-540	50	31			X
cana-540	50	32	{	{	PUNCT
cana-540	50	33	p1	p1	NOUN
cana-540	50	34	,	,	PUNCT
cana-540	50	35	p2	p2	NOUN
cana-540	50	36	,	,	PUNCT
cana-540	50	37	p3	p3	NOUN
cana-540	50	38	}	}	PUNCT
cana-540	50	39	=	=	SYM
cana-540	50	40	{	{	PUNCT
cana-540	50	41	p2	p2	PROPN
cana-540	50	42	}	}	PUNCT
cana-540	50	43			NOUN
cana-540	50	44	.	.	PUNCT
cana-540	50	45	thus	thus	ADV
cana-540	50	46	p1	p1	NOUN
cana-540	50	47	is	be	AUX
cana-540	50	48	a	a	DET
cana-540	50	49	l.spgα.c	l.spgα.c	PROPN
cana-540	50	50	.	.	PUNCT
cana-540	51	1	theorem	theorem	VERB
cana-540	51	2	2.1	2.1	NUM
cana-540	51	3	.	.	PUNCT
cana-540	52	1	if	if	SCONJ
cana-540	52	2	r	r	NOUN
cana-540	52	3	and	and	CCONJ
cana-540	52	4	s	s	VERB
cana-540	52	5	be	be	AUX
cana-540	52	6	any	any	DET
cana-540	52	7	ts	ts	NOUN
cana-540	52	8	.	.	PUNCT
cana-540	53	1	then	then	ADV
cana-540	53	2	for	for	ADP
cana-540	53	3	a	a	DET
cana-540	53	4	m.f	m.f	NOUN
cana-540	53	5	p	p	X
cana-540	53	6	:	:	PUNCT
cana-540	53	7	r	r	NOUN
cana-540	53	8	→	→	SYM
cana-540	53	9	s	s	VERB
cana-540	53	10	the	the	DET
cana-540	53	11	following	follow	VERB
cana-540	53	12	properties	property	NOUN
cana-540	53	13	are	be	AUX
cana-540	53	14	equivalent	equivalent	ADJ
cana-540	53	15	:	:	PUNCT
cana-540	53	16	1	1	X
cana-540	53	17	.	.	X
cana-540	53	18	r	r	NOUN
cana-540	53	19	is	be	AUX
cana-540	53	20	u.spgα.c	u.spgα.c	PRON
cana-540	53	21	.	.	PUNCT
cana-540	54	1	2	2	X
cana-540	54	2	.	.	X
cana-540	54	3	for	for	ADP
cana-540	54	4	each	each	DET
cana-540	54	5	r	r	NOUN
cana-540	54	6			NOUN
cana-540	54	7	r	r	NOUN
cana-540	54	8	,	,	PUNCT
cana-540	54	9	for	for	ADP
cana-540	54	10	each	each	DET
cana-540	54	11	v	v	ADJ
cana-540	54	12			PROPN
cana-540	54	13	o(s	o(s	PROPN
cana-540	54	14	)	)	PUNCT
cana-540	54	15	such	such	ADJ
cana-540	54	16	that	that	SCONJ
cana-540	54	17	r	r	NOUN
cana-540	54	18			NOUN
cana-540	54	19	p+(v	p+(v	PROPN
cana-540	54	20	)	)	PUNCT
cana-540	54	21	,	,	PUNCT
cana-540	54	22	there	there	PRON
cana-540	54	23	exists	exist	VERB
cana-540	54	24	u	u	NOUN
cana-540	54	25			PROPN
cana-540	54	26	spgα	spgα	NOUN
cana-540	54	27	-	-	PUNCT
cana-540	54	28	o(r	o(r	PROPN
cana-540	54	29	,	,	PUNCT
cana-540	54	30	r	r	NOUN
cana-540	54	31	)	)	PUNCT
cana-540	54	32	with	with	ADP
cana-540	54	33	u	u	NOUN
cana-540	54	34			PROPN
cana-540	54	35	p+(v	p+(v	PROPN
cana-540	54	36	)	)	PUNCT
cana-540	54	37	.	.	PUNCT
cana-540	55	1	3	3	X
cana-540	55	2	.	.	X
cana-540	55	3	for	for	ADP
cana-540	55	4	each	each	DET
cana-540	55	5	r	r	NOUN
cana-540	55	6			NOUN
cana-540	55	7	r	r	NOUN
cana-540	55	8	and	and	CCONJ
cana-540	55	9	each	each	DET
cana-540	55	10	k	k	PROPN
cana-540	55	11			PROPN
cana-540	55	12	c(s	c(	VERB
cana-540	55	13	)	)	PUNCT
cana-540	55	14	such	such	ADJ
cana-540	55	15	that	that	SCONJ
cana-540	55	16	r	r	NOUN
cana-540	55	17			NOUN
cana-540	55	18	p+(s	p+(s	X
cana-540	55	19	k	k	NOUN
cana-540	55	20	)	)	PUNCT
cana-540	55	21	,	,	PUNCT
cana-540	55	22	there	there	PRON
cana-540	55	23	exists	exist	VERB
cana-540	55	24	h	h	PROPN
cana-540	55	25			PROPN
cana-540	55	26	spgα	spgα	NOUN
cana-540	55	27	-	-	PUNCT
cana-540	55	28	c(r	c(r	NOUN
cana-540	55	29	)	)	PUNCT
cana-540	55	30	such	such	ADJ
cana-540	55	31	that	that	SCONJ
cana-540	55	32	r	r	NOUN
cana-540	55	33			NOUN
cana-540	55	34	s	s	PART
cana-540	55	35	h	h	NOUN
cana-540	55	36	and	and	CCONJ
cana-540	55	37	p-(k	p-(k	ADV
cana-540	55	38	)	)	PUNCT
cana-540	56	1			PROPN
cana-540	56	2	h.	h.	PROPN
cana-540	56	3	4	4	NUM
cana-540	56	4	.	.	PUNCT
cana-540	57	1	for	for	ADP
cana-540	57	2	each	each	DET
cana-540	57	3	v	v	ADJ
cana-540	57	4			PROPN
cana-540	57	5	o(s	o(s	PROPN
cana-540	57	6	)	)	PUNCT
cana-540	57	7	,	,	PUNCT
cana-540	57	8	p+(v	p+(v	PROPN
cana-540	57	9	)	)	PUNCT
cana-540	57	10			NOUN
cana-540	57	11	spgα	spgα	NOUN
cana-540	57	12	-	-	PUNCT
cana-540	57	13	o(r	o(r	PROPN
cana-540	57	14	)	)	PUNCT
cana-540	57	15	.	.	PUNCT
cana-540	58	1	5	5	X
cana-540	58	2	.	.	X
cana-540	58	3	for	for	ADP
cana-540	58	4	each	each	DET
cana-540	58	5	k	k	PROPN
cana-540	58	6			PROPN
cana-540	58	7	c(s	c(	VERB
cana-540	58	8	)	)	PUNCT
cana-540	58	9	,	,	PUNCT
cana-540	58	10	p-(v	p-(v	X
cana-540	58	11	)	)	PUNCT
cana-540	58	12			NOUN
cana-540	58	13	spgα	spgα	NOUN
cana-540	58	14	-	-	PUNCT
cana-540	58	15	c(r	c(r	NOUN
cana-540	58	16	)	)	PUNCT
cana-540	58	17	.	.	PUNCT
cana-540	59	1	6	6	X
cana-540	59	2	.	.	X
cana-540	59	3	for	for	ADP
cana-540	59	4	each	each	DET
cana-540	59	5	r	r	NOUN
cana-540	59	6			NOUN
cana-540	59	7	r	r	NOUN
cana-540	59	8	and	and	CCONJ
cana-540	59	9	for	for	ADP
cana-540	59	10	each	each	DET
cana-540	59	11	nbd	nbd	PROPN
cana-540	59	12	.	.	PUNCT
cana-540	60	1	v	v	NOUN
cana-540	60	2	of	of	ADP
cana-540	60	3	p(r	p(r	PROPN
cana-540	60	4	)	)	PUNCT
cana-540	60	5	,	,	PUNCT
cana-540	60	6	p+(v	p+(v	PROPN
cana-540	60	7	)	)	PUNCT
cana-540	60	8	is	be	AUX
cana-540	60	9	spgα	spgα	ADJ
cana-540	60	10	-	-	PUNCT
cana-540	60	11	nbd	nbd	PROPN
cana-540	60	12	.	.	PUNCT
cana-540	60	13	of	of	ADP
cana-540	60	14	r.	r.	PROPN
cana-540	60	15	7	7	NUM
cana-540	60	16	.	.	PUNCT
cana-540	61	1	for	for	ADP
cana-540	61	2	each	each	DET
cana-540	61	3	r	r	NOUN
cana-540	61	4			NOUN
cana-540	61	5	r	r	NOUN
cana-540	61	6	and	and	CCONJ
cana-540	61	7	each	each	DET
cana-540	61	8	nbd	nbd	PROPN
cana-540	61	9	.	.	PUNCT
cana-540	61	10	v	v	NOUN
cana-540	61	11	of	of	ADP
cana-540	61	12	p(r	p(r	PROPN
cana-540	61	13	)	)	PUNCT
cana-540	61	14	,	,	PUNCT
cana-540	61	15	there	there	PRON
cana-540	61	16	exists	exist	VERB
cana-540	61	17	a	a	DET
cana-540	61	18	spgα	spgα	ADJ
cana-540	61	19	-	-	PUNCT
cana-540	61	20	nbd	nbd	PROPN
cana-540	61	21	.	.	PUNCT
cana-540	62	1	u	u	PROPN
cana-540	62	2	of	of	ADP
cana-540	62	3	r	r	NOUN
cana-540	62	4	such	such	ADJ
cana-540	62	5	that	that	DET
cana-540	62	6	p(u	p(u	NOUN
cana-540	62	7	)	)	PUNCT
cana-540	63	1			PROPN
cana-540	63	2	v	v	NOUN
cana-540	63	3	.	.	PUNCT
cana-540	64	1	proof	proof	NOUN
cana-540	64	2	:	:	PUNCT
cana-540	64	3	(	(	PUNCT
cana-540	64	4	1	1	X
cana-540	64	5	)	)	PUNCT
cana-540	64	6	→	→	X
cana-540	64	7	(	(	PUNCT
cana-540	64	8	2	2	NUM
cana-540	64	9	):	):	PUNCT
cana-540	64	10	obvious	obvious	ADJ
cana-540	64	11	from	from	ADP
cana-540	64	12	the	the	DET
cana-540	64	13	definition	definition	NOUN
cana-540	64	14	.	.	PUNCT
cana-540	65	1	(	(	PUNCT
cana-540	65	2	2	2	NUM
cana-540	65	3	)	)	PUNCT
cana-540	65	4	→	→	X
cana-540	65	5	(	(	PUNCT
cana-540	65	6	3	3	NUM
cana-540	65	7	):	):	PUNCT
cana-540	65	8	let	let	VERB
cana-540	65	9	r	r	NOUN
cana-540	65	10			NOUN
cana-540	65	11	r	r	NOUN
cana-540	65	12	and	and	CCONJ
cana-540	65	13	k	k	PROPN
cana-540	65	14			NOUN
cana-540	65	15	c(s	c(	VERB
cana-540	65	16	)	)	PUNCT
cana-540	65	17	with	with	ADP
cana-540	65	18	r	r	NOUN
cana-540	65	19			PROPN
cana-540	65	20	p+(s	p+(s	X
cana-540	65	21	k	k	NOUN
cana-540	65	22	)	)	PUNCT
cana-540	65	23	.	.	PUNCT
cana-540	66	1	from	from	ADP
cana-540	66	2	(	(	PUNCT
cana-540	66	3	2	2	X
cana-540	66	4	)	)	PUNCT
cana-540	66	5	there	there	PRON
cana-540	66	6	exists	exist	VERB
cana-540	66	7	u	u	NOUN
cana-540	66	8	spgα	spgα	PROPN
cana-540	66	9	-	-	PUNCT
cana-540	66	10	o(r	o(r	PROPN
cana-540	66	11	,	,	PUNCT
cana-540	66	12	r	r	NOUN
cana-540	66	13	)	)	PUNCT
cana-540	66	14	such	such	ADJ
cana-540	66	15	that	that	SCONJ
cana-540	66	16	u	u	PROPN
cana-540	66	17			PROPN
cana-540	66	18	p+(s	p+(s	PROPN
cana-540	66	19	k	k	NOUN
cana-540	66	20	)	)	PUNCT
cana-540	66	21	.	.	PUNCT
cana-540	67	1	put	put	VERB
cana-540	67	2	h	h	NOUN
cana-540	68	1	=	=	SYM
cana-540	68	2	s	s	PART
cana-540	68	3	k	k	NOUN
cana-540	68	4	,	,	PUNCT
cana-540	68	5	then	then	ADV
cana-540	68	6	h	h	PROPN
cana-540	68	7			PROPN
cana-540	68	8	spgα	spgα	NOUN
cana-540	68	9	-	-	PUNCT
cana-540	68	10	c(r	c(r	NOUN
cana-540	68	11	)	)	PUNCT
cana-540	68	12	with	with	ADP
cana-540	68	13	r	r	NOUN
cana-540	68	14	=	=	SYM
cana-540	68	15	r	r	NOUN
cana-540	68	16	h.	h.	NOUN
cana-540	68	17	also	also	ADV
cana-540	68	18	u	u	PROPN
cana-540	69	1			PROPN
cana-540	69	2	p+(s	p+(s	PROPN
cana-540	69	3	k	k	PROPN
cana-540	69	4	)	)	PUNCT
cana-540	69	5	=	=	SYM
cana-540	69	6	r	r	NOUN
cana-540	69	7	–	–	PUNCT
cana-540	69	8	p-(k	p-(k	ADV
cana-540	69	9	)	)	PUNCT
cana-540	69	10	,	,	PUNCT
cana-540	69	11	that	that	PRON
cana-540	69	12	is	be	AUX
cana-540	69	13	p-(k	p-(k	ADV
cana-540	69	14	)	)	PUNCT
cana-540	70	1			PROPN
cana-540	70	2	r	r	NOUN
cana-540	70	3	v	v	NOUN
cana-540	70	4	=	=	PUNCT
cana-540	70	5	h.	h.	PROPN
cana-540	70	6	(	(	PUNCT
cana-540	70	7	3	3	NUM
cana-540	70	8	)	)	PUNCT
cana-540	70	9	→	→	X
cana-540	70	10	(	(	PUNCT
cana-540	70	11	2	2	NUM
cana-540	70	12	):	):	PUNCT
cana-540	70	13	let	let	VERB
cana-540	70	14	r	r	NOUN
cana-540	70	15			NOUN
cana-540	70	16	r	r	NOUN
cana-540	70	17	and	and	CCONJ
cana-540	70	18	v	v	ADP
cana-540	70	19			PROPN
cana-540	70	20	o(s	o(s	PROPN
cana-540	70	21	)	)	PUNCT
cana-540	70	22	with	with	ADP
cana-540	70	23	r	r	NOUN
cana-540	70	24			NOUN
cana-540	70	25	p+(v	p+(v	PROPN
cana-540	70	26	)	)	PUNCT
cana-540	70	27	.	.	PUNCT
cana-540	71	1	put	put	VERB
cana-540	71	2	k	k	NOUN
cana-540	72	1	=	=	PUNCT
cana-540	72	2	s	s	PROPN
cana-540	72	3	v	v	NOUN
cana-540	72	4	,	,	PUNCT
cana-540	72	5	where	where	SCONJ
cana-540	72	6	k	k	PROPN
cana-540	72	7			NOUN
cana-540	72	8	c(s	c(	VERB
cana-540	72	9	)	)	PUNCT
cana-540	72	10	with	with	ADP
cana-540	72	11	r	r	NOUN
cana-540	72	12			PROPN
cana-540	72	13	p+(s	p+(s	X
cana-540	72	14	k	k	NOUN
cana-540	72	15	)	)	PUNCT
cana-540	72	16	.	.	PUNCT
cana-540	73	1	from	from	ADP
cana-540	73	2	(	(	PUNCT
cana-540	73	3	3	3	NUM
cana-540	73	4	)	)	PUNCT
cana-540	73	5	,	,	PUNCT
cana-540	73	6	there	there	PRON
cana-540	73	7	exists	exist	VERB
cana-540	73	8	h	h	PROPN
cana-540	73	9			PROPN
cana-540	73	10	spgα	spgα	NOUN
cana-540	73	11	-	-	PUNCT
cana-540	73	12	c(r	c(r	NOUN
cana-540	73	13	)	)	PUNCT
cana-540	73	14	such	such	ADJ
cana-540	73	15	that	that	SCONJ
cana-540	73	16	r	r	NOUN
cana-540	73	17			NOUN
cana-540	73	18	r	r	NOUN
cana-540	73	19	h	h	NOUN
cana-540	73	20	and	and	CCONJ
cana-540	73	21	p+(k	p+(k	PROPN
cana-540	73	22	)	)	PUNCT
cana-540	74	1			PROPN
cana-540	74	2	h.	h.	PROPN
cana-540	74	3	let	let	VERB
cana-540	74	4	u	u	NOUN
cana-540	74	5	=	=	NOUN
cana-540	74	6	r	r	NOUN
cana-540	74	7	h	h	NOUN
cana-540	74	8	then	then	ADV
cana-540	74	9	u	u	PRON
cana-540	74	10			PROPN
cana-540	74	11	spgα	spgα	NOUN
cana-540	74	12	-	-	PUNCT
cana-540	74	13	o(r	o(r	PROPN
cana-540	74	14	,	,	PUNCT
cana-540	74	15	r	r	NOUN
cana-540	74	16	)	)	PUNCT
cana-540	74	17	and	and	CCONJ
cana-540	74	18	also	also	ADV
cana-540	74	19	p-(k	p-(k	ADV
cana-540	74	20	)	)	PUNCT
cana-540	75	1			PROPN
cana-540	75	2	h.	h.	PROPN
cana-540	75	3	thus	thus	ADV
cana-540	75	4	,	,	PUNCT
cana-540	75	5	rp-(s	rp-(s	PUNCT
cana-540	75	6	k	k	X
cana-540	75	7	)	)	PUNCT
cana-540	75	8			PROPN
cana-540	75	9	h	h	NOUN
cana-540	76	1	and	and	CCONJ
cana-540	76	2	so	so	ADV
cana-540	76	3	r	r	NOUN
cana-540	76	4	h	h	NOUN
cana-540	76	5			PROPN
cana-540	76	6	p(s	p(s	PROPN
cana-540	76	7	k	k	NOUN
cana-540	76	8	)	)	PUNCT
cana-540	76	9	.	.	PUNCT
cana-540	77	1	so	so	ADV
cana-540	77	2	u	u	X
cana-540	77	3			PROPN
cana-540	77	4	p+(v	p+(v	PROPN
cana-540	77	5	)	)	PUNCT
cana-540	77	6	.	.	PUNCT
cana-540	78	1	(	(	PUNCT
cana-540	78	2	2	2	X
cana-540	78	3	)	)	PUNCT
cana-540	78	4	→	→	X
cana-540	78	5	(	(	PUNCT
cana-540	78	6	4	4	NUM
cana-540	78	7	):	):	PUNCT
cana-540	78	8	let	let	VERB
cana-540	78	9	v	v	ADP
cana-540	78	10			PROPN
cana-540	78	11	o(s	o(s	PROPN
cana-540	78	12	)	)	PUNCT
cana-540	78	13	and	and	CCONJ
cana-540	78	14	r	r	NOUN
cana-540	78	15			PROPN
cana-540	78	16	p-(v	p-(v	NOUN
cana-540	78	17	)	)	PUNCT
cana-540	78	18	.	.	PUNCT
cana-540	79	1	then	then	ADV
cana-540	79	2	from	from	ADP
cana-540	79	3	(	(	PUNCT
cana-540	79	4	2	2	NUM
cana-540	79	5	)	)	PUNCT
cana-540	79	6	,	,	PUNCT
cana-540	79	7	there	there	PRON
cana-540	79	8	exists	exist	VERB
cana-540	79	9	u	u	NOUN
cana-540	79	10			PROPN
cana-540	79	11	spgα	spgα	NOUN
cana-540	79	12	-	-	PUNCT
cana-540	79	13	o(r	o(r	PROPN
cana-540	79	14	,	,	PUNCT
cana-540	79	15	r	r	NOUN
cana-540	79	16	)	)	PUNCT
cana-540	79	17	with	with	ADP
cana-540	79	18	u	u	NOUN
cana-540	79	19			PROPN
cana-540	79	20	p+(v	p+(v	PROPN
cana-540	79	21	)	)	PUNCT
cana-540	79	22	.	.	PUNCT
cana-540	80	1	so	so	ADV
cana-540	80	2	p+(v	p+(v	PROPN
cana-540	80	3	)	)	PUNCT
cana-540	80	4	=	=	PUNCT
cana-540	81	1	⋃	⋃	NOUN
cana-540	81	2	𝑈𝑈𝑟𝑃+(𝑉	𝑈𝑈𝑟𝑃+(𝑉	NOUN
cana-540	81	3	)	)	PUNCT
cana-540	81	4	.	.	PUNCT
cana-540	82	1	we	we	PRON
cana-540	82	2	know	know	VERB
cana-540	82	3	that	that	SCONJ
cana-540	82	4	,	,	PUNCT
cana-540	82	5	arbitrary	arbitrary	ADJ
cana-540	82	6	union	union	NOUN
cana-540	82	7	of	of	ADP
cana-540	82	8	spgα	spgα	ADJ
cana-540	82	9	-	-	PUNCT
cana-540	82	10	open	open	ADJ
cana-540	82	11	set	set	NOUN
cana-540	82	12	is	be	AUX
cana-540	82	13	again	again	ADV
cana-540	82	14	spgα	spgα	ADJ
cana-540	82	15	-	-	PUNCT
cana-540	82	16	open	open	ADJ
cana-540	82	17	and	and	CCONJ
cana-540	82	18	so	so	ADV
cana-540	82	19	p+(v	p+(v	PROPN
cana-540	82	20	)	)	PUNCT
cana-540	82	21			NOUN
cana-540	82	22	spgα	spgα	NOUN
cana-540	82	23	-	-	PUNCT
cana-540	82	24	o(r	o(r	PROPN
cana-540	82	25	)	)	PUNCT
cana-540	82	26	.	.	PUNCT
cana-540	83	1	(	(	PUNCT
cana-540	83	2	4	4	NUM
cana-540	83	3	)	)	PUNCT
cana-540	83	4	→	→	X
cana-540	83	5	(	(	PUNCT
cana-540	83	6	2	2	NUM
cana-540	83	7	):	):	PUNCT
cana-540	83	8	let	let	VERB
cana-540	83	9	r	r	NOUN
cana-540	83	10			NOUN
cana-540	83	11	r	r	NOUN
cana-540	83	12	and	and	CCONJ
cana-540	83	13	v	v	ADP
cana-540	83	14			PROPN
cana-540	83	15	o(s	o(s	PROPN
cana-540	83	16	)	)	PUNCT
cana-540	83	17	with	with	ADP
cana-540	83	18	r	r	NOUN
cana-540	83	19			NOUN
cana-540	83	20	p+(v	p+(v	PROPN
cana-540	83	21	)	)	PUNCT
cana-540	83	22	.	.	PUNCT
cana-540	84	1	from	from	ADP
cana-540	84	2	(	(	PUNCT
cana-540	84	3	4	4	X
cana-540	84	4	)	)	PUNCT
cana-540	84	5	p+(v	p+(v	PROPN
cana-540	84	6	)	)	PUNCT
cana-540	84	7			NOUN
cana-540	84	8	spgα	spgα	NOUN
cana-540	84	9	-	-	PUNCT
cana-540	84	10	o(r	o(r	PROPN
cana-540	84	11	)	)	PUNCT
cana-540	84	12	.	.	PUNCT
cana-540	85	1	let	let	VERB
cana-540	85	2	u	u	PRON
cana-540	85	3	=	=	NOUN
cana-540	85	4	p+(v	p+(v	PROPN
cana-540	85	5	)	)	PUNCT
cana-540	85	6	,	,	PUNCT
cana-540	85	7	then	then	ADV
cana-540	85	8	u	u	NOUN
cana-540	85	9			PROPN
cana-540	85	10	spgα	spgα	NOUN
cana-540	85	11	-	-	PUNCT
cana-540	85	12	o(r	o(r	PROPN
cana-540	85	13	,	,	PUNCT
cana-540	85	14	r	r	NOUN
cana-540	85	15	)	)	PUNCT
cana-540	85	16	and	and	CCONJ
cana-540	85	17	so	so	ADV
cana-540	85	18	u	u	PRON
cana-540	85	19			NOUN
cana-540	85	20	p+(v	p+(v	PROPN
cana-540	85	21	)	)	PUNCT
cana-540	85	22	.	.	PUNCT
cana-540	86	1	(	(	PUNCT
cana-540	86	2	4	4	NUM
cana-540	86	3	)	)	PUNCT
cana-540	86	4	→	→	X
cana-540	86	5	(	(	PUNCT
cana-540	86	6	5	5	NUM
cana-540	86	7	):	):	PUNCT
cana-540	86	8	u	u	NOUN
cana-540	86	9			NOUN
cana-540	86	10	c(s	c(	VERB
cana-540	86	11	)	)	PUNCT
cana-540	86	12	and	and	CCONJ
cana-540	86	13	so	so	ADV
cana-540	86	14	s	s	VERB
cana-540	86	15	u	u	NOUN
cana-540	86	16	=	=	PROPN
cana-540	86	17	o(s	o(s	PROPN
cana-540	86	18	)	)	PUNCT
cana-540	86	19	.	.	PUNCT
cana-540	87	1	but	but	CCONJ
cana-540	87	2	from	from	ADP
cana-540	87	3	(	(	PUNCT
cana-540	87	4	4	4	NUM
cana-540	87	5	)	)	PUNCT
cana-540	87	6	,	,	PUNCT
cana-540	87	7	p+(s	p+(s	PROPN
cana-540	87	8	u	u	NOUN
cana-540	87	9	)	)	PUNCT
cana-540	87	10			NOUN
cana-540	87	11	spgα	spgα	NOUN
cana-540	87	12	-	-	PUNCT
cana-540	87	13	o(r	o(r	PROPN
cana-540	87	14	)	)	PUNCT
cana-540	87	15	,	,	PUNCT
cana-540	87	16	since	since	SCONJ
cana-540	87	17	p+(s	p+(s	NUM
cana-540	87	18	u	u	NOUN
cana-540	87	19	)	)	PUNCT
cana-540	87	20	=	=	SYM
cana-540	87	21	r	r	NOUN
cana-540	87	22	–	–	PUNCT
cana-540	87	23	p-(u	p-(u	NOUN
cana-540	87	24	)	)	PUNCT
cana-540	87	25	and	and	CCONJ
cana-540	87	26	so	so	ADV
cana-540	87	27	r	r	NOUN
cana-540	87	28	–	–	PUNCT
cana-540	87	29	p-(u	p-(u	NOUN
cana-540	87	30	)	)	PUNCT
cana-540	87	31			NOUN
cana-540	87	32	spgα	spgα	NOUN
cana-540	87	33	-	-	PUNCT
cana-540	87	34	o(r	o(r	PROPN
cana-540	87	35	)	)	PUNCT
cana-540	87	36	.	.	PUNCT
cana-540	88	1	thus	thus	ADV
cana-540	88	2	p-(u	p-(u	X
cana-540	88	3	)	)	PUNCT
cana-540	88	4			NOUN
cana-540	88	5	spgα	spgα	NOUN
cana-540	88	6	-	-	PUNCT
cana-540	88	7	o(r	o(r	PROPN
cana-540	88	8	)	)	PUNCT
cana-540	88	9	.	.	PUNCT
cana-540	89	1	(	(	PUNCT
cana-540	89	2	5	5	NUM
cana-540	89	3	)	)	PUNCT
cana-540	89	4	→	→	X
cana-540	89	5	(	(	PUNCT
cana-540	89	6	6	6	NUM
cana-540	89	7	):	):	PUNCT
cana-540	89	8	v	v	ADP
cana-540	89	9			PROPN
cana-540	89	10	o(s	o(s	PROPN
cana-540	89	11	)	)	PUNCT
cana-540	89	12	and	and	CCONJ
cana-540	89	13	so	so	ADV
cana-540	89	14	s	s	PROPN
cana-540	89	15	v	v	NOUN
cana-540	89	16			NOUN
cana-540	89	17	c(s	c(	VERB
cana-540	89	18	)	)	PUNCT
cana-540	89	19	.	.	PUNCT
cana-540	90	1	but	but	CCONJ
cana-540	90	2	from	from	ADP
cana-540	90	3	(	(	PUNCT
cana-540	90	4	6	6	NUM
cana-540	90	5	)	)	PUNCT
cana-540	90	6	,	,	PUNCT
cana-540	90	7	p(s	p(s	PROPN
cana-540	90	8	v	v	NOUN
cana-540	90	9	)	)	PUNCT
cana-540	90	10			NOUN
cana-540	90	11	spgα	spgα	NOUN
cana-540	90	12	-	-	PUNCT
cana-540	90	13	c(r	c(r	NOUN
cana-540	90	14	)	)	PUNCT
cana-540	90	15	,	,	PUNCT
cana-540	90	16	since	since	SCONJ
cana-540	90	17	p-(s	p-(s	NOUN
cana-540	90	18	v	v	NOUN
cana-540	90	19	)	)	PUNCT
cana-540	90	20	=	=	SYM
cana-540	90	21	r	r	NOUN
cana-540	90	22	p+(v	p+(v	PROPN
cana-540	90	23	)	)	PUNCT
cana-540	90	24	and	and	CCONJ
cana-540	90	25	so	so	ADV
cana-540	90	26	r	r	NOUN
cana-540	90	27	p+(u	p+(u	ADJ
cana-540	90	28	)	)	PUNCT
cana-540	90	29			NOUN
cana-540	90	30	spgα	spgα	NOUN
cana-540	90	31	-	-	PUNCT
cana-540	90	32	c(r	c(r	NOUN
cana-540	90	33	)	)	PUNCT
cana-540	90	34	.	.	PUNCT
cana-540	91	1	thus	thus	ADV
cana-540	91	2	p+(v	p+(v	PROPN
cana-540	91	3	)	)	PUNCT
cana-540	91	4			NOUN
cana-540	91	5	spgα	spgα	NOUN
cana-540	91	6	-	-	PUNCT
cana-540	91	7	o(r	o(r	PROPN
cana-540	91	8	)	)	PUNCT
cana-540	91	9	(	(	PUNCT
cana-540	91	10	4	4	NUM
cana-540	91	11	)	)	PUNCT
cana-540	91	12	→	→	X
cana-540	91	13	(	(	PUNCT
cana-540	91	14	6	6	NUM
cana-540	91	15	):	):	PUNCT
cana-540	91	16	let	let	VERB
cana-540	91	17	r	r	NOUN
cana-540	91	18			NOUN
cana-540	91	19	r	r	NOUN
cana-540	91	20	and	and	CCONJ
cana-540	91	21	v	v	NOUN
cana-540	91	22	be	be	AUX
cana-540	91	23	a	a	DET
cana-540	91	24	nbd	nbd	PROPN
cana-540	91	25	.	.	PROPN
cana-540	91	26	of	of	ADP
cana-540	91	27	p(r	p(r	PROPN
cana-540	91	28	)	)	PUNCT
cana-540	91	29	.	.	PUNCT
cana-540	92	1	then	then	ADV
cana-540	92	2	u	u	PRON
cana-540	92	3			PROPN
cana-540	92	4	o(n	o(n	PROPN
cana-540	92	5	)	)	PUNCT
cana-540	92	6	such	such	ADJ
cana-540	92	7	that	that	SCONJ
cana-540	92	8	p(r	p(r	PROPN
cana-540	92	9	)	)	PUNCT
cana-540	93	1			PROPN
cana-540	93	2	u	u	NOUN
cana-540	93	3			PROPN
cana-540	93	4	v	v	PROPN
cana-540	93	5	,	,	PUNCT
cana-540	93	6	that	that	ADV
cana-540	93	7	is	is	ADV
cana-540	93	8	r	r	NOUN
cana-540	93	9			NOUN
cana-540	93	10	p+(u	p+(u	PUNCT
cana-540	93	11	)	)	PUNCT
cana-540	94	1			PROPN
cana-540	94	2	p+(v	p+(v	PROPN
cana-540	94	3	)	)	PUNCT
cana-540	94	4	.	.	PUNCT
cana-540	95	1	but	but	CCONJ
cana-540	95	2	from	from	ADP
cana-540	95	3	(	(	PUNCT
cana-540	95	4	4	4	NUM
cana-540	95	5	)	)	PUNCT
cana-540	95	6	,	,	PUNCT
cana-540	95	7	p+(u	p+(u	X
cana-540	95	8	)	)	PUNCT
cana-540	95	9			NOUN
cana-540	95	10	spgα	spgα	NOUN
cana-540	95	11	-	-	PUNCT
cana-540	95	12	o(r	o(r	NOUN
cana-540	95	13	)	)	PUNCT
cana-540	95	14	and	and	CCONJ
cana-540	95	15	so	so	ADV
cana-540	95	16	p+(v	p+(v	PROPN
cana-540	95	17	)	)	PUNCT
cana-540	95	18			PROPN
cana-540	95	19	spgα.nbd	spgα.nbd	PROPN
cana-540	95	20	.	.	PROPN
cana-540	95	21	of	of	ADP
cana-540	95	22	r.	r.	PROPN
cana-540	95	23	(	(	PUNCT
cana-540	95	24	6	6	NUM
cana-540	95	25	)	)	PUNCT
cana-540	95	26	→	→	X
cana-540	95	27	(	(	PUNCT
cana-540	95	28	7	7	NUM
cana-540	95	29	):	):	PUNCT
cana-540	95	30	let	let	VERB
cana-540	95	31	r	r	NOUN
cana-540	95	32			NOUN
cana-540	95	33	r	r	NOUN
cana-540	95	34	and	and	CCONJ
cana-540	95	35	v	v	NOUN
cana-540	95	36	be	be	AUX
cana-540	95	37	a	a	DET
cana-540	95	38	nbd	nbd	PROPN
cana-540	95	39	.	.	PROPN
cana-540	95	40	of	of	ADP
cana-540	95	41	p(r	p(r	PROPN
cana-540	95	42	)	)	PUNCT
cana-540	95	43	.	.	PUNCT
cana-540	96	1	from	from	ADP
cana-540	96	2	(	(	PUNCT
cana-540	96	3	6	6	NUM
cana-540	96	4	)	)	PUNCT
cana-540	96	5	,	,	PUNCT
cana-540	96	6	p+(v	p+(v	PROPN
cana-540	96	7	)	)	PUNCT
cana-540	96	8	is	be	AUX
cana-540	96	9	a	a	DET
cana-540	96	10	spgα	spgα	ADJ
cana-540	96	11	-	-	PUNCT
cana-540	96	12	nbd	nbd	PROPN
cana-540	96	13	.	.	PUNCT
cana-540	97	1	of	of	ADP
cana-540	97	2	the	the	DET
cana-540	97	3	point	point	NOUN
cana-540	97	4	r.	r.	PROPN
cana-540	97	5	put	put	VERB
cana-540	97	6	u	u	NOUN
cana-540	97	7	=	=	NOUN
cana-540	97	8	p+(v	p+(v	PROPN
cana-540	97	9	)	)	PUNCT
cana-540	97	10	,	,	PUNCT
cana-540	97	11	so	so	CCONJ
cana-540	97	12	u	u	NOUN
cana-540	97	13	is	be	AUX
cana-540	97	14	a	a	DET
cana-540	97	15	spgα	spgα	ADJ
cana-540	97	16	-	-	PUNCT
cana-540	97	17	nbd	nbd	PROPN
cana-540	97	18	.	.	PUNCT
cana-540	98	1	of	of	ADP
cana-540	98	2	r	r	NOUN
cana-540	98	3	with	with	ADP
cana-540	98	4	p(u	p(u	NOUN
cana-540	98	5	)	)	PUNCT
cana-540	99	1			PROPN
cana-540	99	2	v	v	ADP
cana-540	99	3	.	.	PUNCT
cana-540	100	1	(	(	PUNCT
cana-540	100	2	7	7	NUM
cana-540	100	3	)	)	PUNCT
cana-540	100	4	→	→	X
cana-540	100	5	(	(	PUNCT
cana-540	100	6	1	1	NUM
cana-540	100	7	):	):	PUNCT
cana-540	100	8	let	let	VERB
cana-540	100	9	r	r	NOUN
cana-540	100	10			NOUN
cana-540	100	11	r	r	NOUN
cana-540	100	12	and	and	CCONJ
cana-540	100	13	v	v	ADP
cana-540	100	14			PROPN
cana-540	100	15	o(s	o(s	PROPN
cana-540	100	16	)	)	PUNCT
cana-540	100	17	with	with	ADP
cana-540	100	18	p(r	p(r	PROPN
cana-540	100	19	)	)	PUNCT
cana-540	101	1			PROPN
cana-540	101	2	v	v	NOUN
cana-540	101	3	.	.	PUNCT
cana-540	102	1	then	then	ADV
cana-540	102	2	v	v	NOUN
cana-540	102	3	is	be	AUX
cana-540	102	4	a	a	DET
cana-540	102	5	nbd	nbd	PROPN
cana-540	102	6	.	.	PROPN
cana-540	102	7	of	of	ADP
cana-540	102	8	p(r	p(r	PROPN
cana-540	102	9	)	)	PUNCT
cana-540	102	10	.	.	PUNCT
cana-540	103	1	by	by	ADP
cana-540	103	2	(	(	PUNCT
cana-540	103	3	7	7	NUM
cana-540	103	4	)	)	PUNCT
cana-540	103	5	,	,	PUNCT
cana-540	103	6	there	there	PRON
cana-540	103	7	exists	exist	VERB
cana-540	103	8	spgα	spgα	ADJ
cana-540	103	9	-	-	PUNCT
cana-540	103	10	nbd	nbd	PROPN
cana-540	103	11	.	.	PUNCT
cana-540	104	1	u	u	PROPN
cana-540	104	2	of	of	ADP
cana-540	104	3	r	r	NOUN
cana-540	104	4	such	such	ADJ
cana-540	104	5	that	that	DET
cana-540	104	6	p(u	p(u	NOUN
cana-540	104	7	)	)	PUNCT
cana-540	105	1			PROPN
cana-540	105	2	v	v	X
cana-540	105	3	.	.	PUNCT
cana-540	106	1	so	so	ADV
cana-540	106	2	,	,	PUNCT
cana-540	106	3	there	there	PRON
cana-540	106	4	exists	exist	VERB
cana-540	106	5	a	a	DET
cana-540	106	6	g	g	PROPN
cana-540	106	7			PROPN
cana-540	106	8	spgα	spgα	NOUN
cana-540	106	9	-	-	PUNCT
cana-540	106	10	o(r	o(r	NOUN
cana-540	106	11	)	)	PUNCT
cana-540	106	12	with	with	ADP
cana-540	106	13	r	r	NOUN
cana-540	106	14			NOUN
cana-540	106	15	g	g	PROPN
cana-540	106	16			PROPN
cana-540	106	17	u	u	PROPN
cana-540	106	18	and	and	CCONJ
cana-540	106	19	so	so	ADV
cana-540	106	20	p(r	p(r	PROPN
cana-540	106	21	)	)	PUNCT
cana-540	106	22			PROPN
cana-540	106	23	p(g	p(g	PROPN
cana-540	106	24	)	)	PUNCT
cana-540	107	1			PROPN
cana-540	107	2	p(u	p(u	PROPN
cana-540	107	3	)	)	PUNCT
cana-540	108	1			PROPN
cana-540	108	2	v.	v.	CCONJ
cana-540	108	3	thus	thus	ADV
cana-540	108	4	r	r	NOUN
cana-540	108	5	is	be	AUX
cana-540	108	6	u.spgα.c	u.spgα.c	NOUN
cana-540	108	7	for	for	ADP
cana-540	108	8	each	each	DET
cana-540	108	9	point	point	NOUN
cana-540	108	10	r	r	NOUN
cana-540	108	11			PROPN
cana-540	108	12	r.	r.	PROPN
cana-540	108	13	communications	communication	NOUN
cana-540	108	14	on	on	ADP
cana-540	108	15	applied	apply	VERB
cana-540	108	16	nonlinear	nonlinear	ADJ
cana-540	108	17	analysis	analysis	NOUN
cana-540	108	18	issn	issn	NOUN
cana-540	108	19	:	:	PUNCT
cana-540	108	20	1074	1074	NUM
cana-540	108	21	-	-	PUNCT
cana-540	108	22	133x	133x	NUM
cana-540	108	23	vol	vol	NOUN
cana-540	108	24	31	31	NUM
cana-540	108	25	no	no	NOUN
cana-540	108	26	.	.	NOUN
cana-540	108	27	2	2	NUM
cana-540	108	28	(	(	PUNCT
cana-540	108	29	2024	2024	NUM
cana-540	108	30	)	)	PUNCT
cana-540	108	31	251	251	NUM
cana-540	108	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	108	33	theorem	theorem	VERB
cana-540	108	34	2.2	2.2	NUM
cana-540	108	35	.	.	PUNCT
cana-540	109	1	the	the	DET
cana-540	109	2	following	follow	VERB
cana-540	109	3	properties	property	NOUN
cana-540	109	4	are	be	AUX
cana-540	109	5	equivalent	equivalent	ADJ
cana-540	109	6	for	for	ADP
cana-540	109	7	a	a	DET
cana-540	109	8	m.f	m.f	NOUN
cana-540	109	9	p	p	X
cana-540	109	10	:	:	PUNCT
cana-540	109	11	r	r	NOUN
cana-540	109	12	→	→	SYM
cana-540	109	13	s	s	NOUN
cana-540	109	14	:	:	PUNCT
cana-540	109	15	1	1	NUM
cana-540	109	16	.	.	X
cana-540	110	1	p	p	NOUN
cana-540	110	2	is	be	AUX
cana-540	110	3	l.spgα.c	l.spgα.c	PROPN
cana-540	110	4	.	.	PROPN
cana-540	110	5	2	2	NUM
cana-540	110	6	.	.	X
cana-540	111	1	for	for	ADP
cana-540	111	2	each	each	DET
cana-540	111	3	r	r	NOUN
cana-540	111	4			NOUN
cana-540	111	5	r	r	NOUN
cana-540	111	6	and	and	CCONJ
cana-540	111	7	v	v	ADP
cana-540	111	8			PROPN
cana-540	111	9	o(r	o(r	PROPN
cana-540	111	10	)	)	PUNCT
cana-540	111	11	with	with	ADP
cana-540	111	12	r	r	NOUN
cana-540	111	13			PROPN
cana-540	111	14	p-(v	p-(v	NOUN
cana-540	111	15	)	)	PUNCT
cana-540	111	16	there	there	PRON
cana-540	111	17	exists	exist	VERB
cana-540	111	18	u	u	NOUN
cana-540	111	19			PROPN
cana-540	111	20	spgα	spgα	PROPN
cana-540	111	21	-	-	PUNCT
cana-540	111	22	o(r	o(r	NOUN
cana-540	111	23	)	)	PUNCT
cana-540	111	24	containing	contain	VERB
cana-540	111	25	r	r	NOUN
cana-540	111	26	such	such	ADJ
cana-540	111	27	that	that	DET
cana-540	111	28	u	u	NOUN
cana-540	111	29			PROPN
cana-540	111	30	p-(v	p-(v	PROPN
cana-540	111	31	)	)	PUNCT
cana-540	111	32	.	.	PUNCT
cana-540	112	1	3	3	X
cana-540	112	2	.	.	X
cana-540	112	3	for	for	ADP
cana-540	112	4	each	each	DET
cana-540	112	5	r	r	NOUN
cana-540	112	6			NOUN
cana-540	112	7	r	r	NOUN
cana-540	112	8	and	and	CCONJ
cana-540	112	9	k	k	PROPN
cana-540	112	10			NOUN
cana-540	112	11	c(s	c(	VERB
cana-540	112	12	)	)	PUNCT
cana-540	112	13	with	with	SCONJ
cana-540	112	14	r	r	NOUN
cana-540	112	15			NOUN
cana-540	112	16	p-(s	p-(s	NOUN
cana-540	112	17	k	k	PROPN
cana-540	112	18	)	)	PUNCT
cana-540	112	19	there	there	PRON
cana-540	112	20	exists	exist	VERB
cana-540	112	21	h	h	PROPN
cana-540	112	22			PROPN
cana-540	112	23	spgα	spgα	NOUN
cana-540	112	24	-	-	PUNCT
cana-540	112	25	c(r	c(r	NOUN
cana-540	112	26	)	)	PUNCT
cana-540	112	27	such	such	ADJ
cana-540	112	28	that	that	SCONJ
cana-540	112	29	r	r	NOUN
cana-540	112	30			NOUN
cana-540	112	31	r	r	NOUN
cana-540	112	32	h	h	NOUN
cana-540	112	33	and	and	CCONJ
cana-540	112	34	p+(k	p+(k	NOUN
cana-540	112	35	)	)	PUNCT
cana-540	113	1			PROPN
cana-540	113	2	h.	h.	PROPN
cana-540	113	3	4	4	NUM
cana-540	113	4	.	.	PUNCT
cana-540	114	1	for	for	ADP
cana-540	114	2	each	each	PRON
cana-540	114	3	v	v	ADJ
cana-540	114	4			PROPN
cana-540	114	5	o(s	o(s	PROPN
cana-540	114	6	)	)	PUNCT
cana-540	114	7	,	,	PUNCT
cana-540	114	8	p-(v	p-(v	X
cana-540	114	9	)	)	PUNCT
cana-540	115	1			NOUN
cana-540	115	2	spgα	spgα	NOUN
cana-540	115	3	-	-	PUNCT
cana-540	115	4	o(r	o(r	PROPN
cana-540	115	5	)	)	PUNCT
cana-540	115	6	.	.	PUNCT
cana-540	116	1	5	5	X
cana-540	116	2	.	.	X
cana-540	116	3	for	for	ADP
cana-540	116	4	each	each	DET
cana-540	116	5	k	k	PROPN
cana-540	116	6			PROPN
cana-540	116	7	c(s	c(	VERB
cana-540	116	8	)	)	PUNCT
cana-540	116	9	,	,	PUNCT
cana-540	116	10	p+(k	p+(k	PROPN
cana-540	116	11	)	)	PUNCT
cana-540	116	12			NOUN
cana-540	116	13	spgα	spgα	NOUN
cana-540	116	14	-	-	PUNCT
cana-540	116	15	c(r	c(r	NOUN
cana-540	116	16	)	)	PUNCT
cana-540	116	17	.	.	PUNCT
cana-540	117	1	definition	definition	NOUN
cana-540	117	2	2.2	2.2	NUM
cana-540	117	3	:	:	PUNCT
cana-540	118	1	[	[	X
cana-540	118	2	4	4	X
cana-540	118	3	]	]	X
cana-540	118	4	a	a	DET
cana-540	118	5	space	space	NOUN
cana-540	118	6	r	r	NOUN
cana-540	118	7	is	be	AUX
cana-540	118	8	said	say	VERB
cana-540	118	9	to	to	PART
cana-540	118	10	be	be	AUX
cana-540	118	11	submaximal	submaximal	ADJ
cana-540	118	12	if	if	SCONJ
cana-540	118	13	every	every	DET
cana-540	118	14	dense	dense	ADJ
cana-540	118	15	subset	subset	NOUN
cana-540	118	16	of	of	ADP
cana-540	118	17	r	r	NOUN
cana-540	118	18	is	be	AUX
cana-540	118	19	open	open	ADJ
cana-540	118	20	.	.	PUNCT
cana-540	119	1	theorem	theorem	VERB
cana-540	119	2	2.3	2.3	NUM
cana-540	119	3	.	.	PUNCT
cana-540	120	1	if	if	SCONJ
cana-540	120	2	a	a	DET
cana-540	120	3	m.f	m.f	NOUN
cana-540	120	4	p	p	X
cana-540	120	5	:	:	PUNCT
cana-540	120	6	r	r	NOUN
cana-540	120	7	→	→	SYM
cana-540	120	8	s	s	X
cana-540	120	9	is	be	AUX
cana-540	120	10	u.p.c	u.p.c	NOUN
cana-540	120	11	and	and	CCONJ
cana-540	120	12	s	s	NOUN
cana-540	120	13	is	be	AUX
cana-540	120	14	submaximal	submaximal	ADJ
cana-540	120	15	,	,	PUNCT
cana-540	120	16	then	then	ADV
cana-540	120	17	p	p	PROPN
cana-540	120	18	is	be	AUX
cana-540	120	19	u.spgα.c	u.spgα.c	PROPN
cana-540	120	20	.	.	PUNCT
cana-540	121	1	proof	proof	NOUN
cana-540	121	2	.	.	PUNCT
cana-540	122	1	let	let	VERB
cana-540	122	2	a	a	DET
cana-540	122	3			NOUN
cana-540	122	4	p	p	NOUN
cana-540	122	5	-	-	PUNCT
cana-540	122	6	o(s	o(s	NOUN
cana-540	122	7	)	)	PUNCT
cana-540	122	8	.	.	PUNCT
cana-540	123	1	as	as	SCONJ
cana-540	123	2	s	s	PRON
cana-540	123	3	is	be	AUX
cana-540	123	4	submaximal	submaximal	ADJ
cana-540	123	5	,	,	PUNCT
cana-540	123	6	a	a	DET
cana-540	123	7			PROPN
cana-540	123	8	o(s	o(s	PROPN
cana-540	123	9	)	)	PUNCT
cana-540	123	10	.	.	PUNCT
cana-540	124	1	since	since	SCONJ
cana-540	124	2	p	p	NOUN
cana-540	124	3	is	be	AUX
cana-540	124	4	u.p.c	u.p.c	NOUN
cana-540	124	5	,	,	PUNCT
cana-540	124	6	p+(a	p+(a	PROPN
cana-540	124	7	)	)	PUNCT
cana-540	124	8			NOUN
cana-540	124	9	p	p	PROPN
cana-540	124	10	-	-	PUNCT
cana-540	124	11	o(r	o(r	NOUN
cana-540	124	12	)	)	PUNCT
cana-540	124	13	and	and	CCONJ
cana-540	124	14	hence	hence	ADV
cana-540	124	15	p+(a	p+(a	ADJ
cana-540	124	16	)	)	PUNCT
cana-540	124	17			NOUN
cana-540	124	18	spgα	spgα	NOUN
cana-540	124	19	-	-	PUNCT
cana-540	124	20	o(r	o(r	PROPN
cana-540	124	21	)	)	PUNCT
cana-540	124	22	.	.	PUNCT
cana-540	125	1	thus	thus	ADV
cana-540	125	2	p	p	X
cana-540	125	3	is	be	AUX
cana-540	125	4	u.spgα.c	u.spgα.c	PROPN
cana-540	125	5	.	.	PUNCT
cana-540	126	1	theorem	theorem	VERB
cana-540	126	2	2.4	2.4	NUM
cana-540	126	3	.	.	PUNCT
cana-540	127	1	a	a	DET
cana-540	127	2	m.f	m.f	NOUN
cana-540	127	3	p	p	X
cana-540	127	4	:	:	PUNCT
cana-540	127	5	r	r	NOUN
cana-540	127	6	→	→	SYM
cana-540	127	7	s	s	X
cana-540	127	8	is	be	AUX
cana-540	127	9	u.spgα.c	u.spgα.c	PROPN
cana-540	127	10	if	if	SCONJ
cana-540	127	11	and	and	CCONJ
cana-540	127	12	only	only	ADV
cana-540	127	13	if	if	SCONJ
cana-540	127	14	for	for	ADP
cana-540	127	15	all	all	DET
cana-540	127	16	b	b	PROPN
cana-540	127	17			PROPN
cana-540	127	18	o(s	o(s	PROPN
cana-540	127	19	)	)	PUNCT
cana-540	127	20	,	,	PUNCT
cana-540	127	21	p-(b	p-(b	ADJ
cana-540	127	22	)	)	PUNCT
cana-540	127	23			NOUN
cana-540	127	24	o(r	o(r	PROPN
cana-540	127	25	)	)	PUNCT
cana-540	127	26	.	.	PUNCT
cana-540	128	1	proof	proof	NOUN
cana-540	128	2	.	.	PUNCT
cana-540	129	1	let	let	VERB
cana-540	129	2	b	b	X
cana-540	129	3			PROPN
cana-540	129	4	o(s	o(s	PROPN
cana-540	129	5	)	)	PUNCT
cana-540	129	6	and	and	CCONJ
cana-540	129	7	r	r	NOUN
cana-540	129	8			NOUN
cana-540	129	9	p+(b	p+(b	NOUN
cana-540	129	10	)	)	PUNCT
cana-540	129	11	.	.	PUNCT
cana-540	130	1	then	then	ADV
cana-540	130	2	by	by	ADP
cana-540	130	3	u.spgα.c	u.spgα.c	NOUN
cana-540	130	4	,	,	PUNCT
cana-540	130	5	there	there	PRON
cana-540	130	6	exists	exist	VERB
cana-540	130	7	v	v	ADP
cana-540	130	8			PROPN
cana-540	130	9	spgα	spgα	NOUN
cana-540	130	10	-	-	PUNCT
cana-540	130	11	o(r	o(r	PROPN
cana-540	130	12	)	)	PUNCT
cana-540	130	13	with	with	ADP
cana-540	130	14	p(v	p(v	NOUN
cana-540	130	15	)	)	PUNCT
cana-540	130	16			PROPN
cana-540	130	17	b	b	PROPN
cana-540	130	18	,	,	PUNCT
cana-540	130	19	where	where	SCONJ
cana-540	130	20	p+(b	p+(b	NOUN
cana-540	130	21	)	)	PUNCT
cana-540	130	22			NOUN
cana-540	130	23	o(r	o(r	PROPN
cana-540	130	24	)	)	PUNCT
cana-540	130	25	.	.	PUNCT
cana-540	131	1	let	let	VERB
cana-540	131	2	p+(b	p+(b	NOUN
cana-540	131	3	)	)	PUNCT
cana-540	131	4	is	be	AUX
cana-540	131	5	open	open	ADJ
cana-540	131	6	and	and	CCONJ
cana-540	131	7	r	r	NOUN
cana-540	131	8			NOUN
cana-540	131	9	p-(b	p-(b	NOUN
cana-540	131	10	)	)	PUNCT
cana-540	131	11	.	.	PUNCT
cana-540	132	1	then	then	ADV
cana-540	132	2	p+(b	p+(b	NOUN
cana-540	132	3	)	)	PUNCT
cana-540	132	4	=	=	PRON
cana-540	132	5	{	{	PUNCT
cana-540	132	6	r	r	NOUN
cana-540	132	7			PROPN
cana-540	132	8	b	b	PROPN
cana-540	132	9	:	:	PUNCT
cana-540	132	10	p(r	p(r	PROPN
cana-540	132	11	)	)	PUNCT
cana-540	133	1			PROPN
cana-540	133	2	b	b	PROPN
cana-540	133	3	}	}	PUNCT
cana-540	133	4	.	.	PUNCT
cana-540	134	1	so	so	ADV
cana-540	134	2	p	p	PROPN
cana-540	134	3	is	be	AUX
cana-540	134	4	u.spgα.c	u.spgα.c	PROPN
cana-540	134	5	.	.	PUNCT
cana-540	135	1	theorem	theorem	VERB
cana-540	135	2	2.5	2.5	NUM
cana-540	135	3	.	.	PUNCT
cana-540	136	1	a	a	DET
cana-540	136	2	m.f	m.f	NOUN
cana-540	136	3	p	p	X
cana-540	136	4	:	:	PUNCT
cana-540	136	5	r	r	NOUN
cana-540	136	6	→	→	SYM
cana-540	136	7	s	s	PART
cana-540	136	8	is	be	AUX
cana-540	136	9	l.spgα.c	l.spgα.c	PROPN
cana-540	136	10	if	if	SCONJ
cana-540	136	11	and	and	CCONJ
cana-540	136	12	only	only	ADV
cana-540	136	13	if	if	SCONJ
cana-540	136	14	for	for	ADP
cana-540	136	15	all	all	DET
cana-540	136	16	open	open	ADJ
cana-540	136	17	set	set	ADJ
cana-540	136	18	b	b	PROPN
cana-540	136	19	in	in	ADP
cana-540	136	20	s	s	PROPN
cana-540	136	21	,	,	PUNCT
cana-540	136	22	p-(b	p-(b	ADJ
cana-540	136	23	)	)	PUNCT
cana-540	136	24			NOUN
cana-540	136	25	o(r	o(r	PROPN
cana-540	136	26	)	)	PUNCT
cana-540	136	27	.	.	PUNCT
cana-540	137	1	proof	proof	NOUN
cana-540	137	2	.	.	PUNCT
cana-540	138	1	let	let	VERB
cana-540	138	2	b	b	X
cana-540	138	3			PROPN
cana-540	138	4	o(s	o(s	PROPN
cana-540	138	5	)	)	PUNCT
cana-540	138	6	and	and	CCONJ
cana-540	138	7	r	r	NOUN
cana-540	138	8			NOUN
cana-540	138	9	p+(b	p+(b	NOUN
cana-540	138	10	)	)	PUNCT
cana-540	138	11	.	.	PUNCT
cana-540	139	1	then	then	ADV
cana-540	139	2	by	by	ADP
cana-540	139	3	l.spgα.c	l.spgα.c	PROPN
cana-540	139	4	,	,	PUNCT
cana-540	139	5	there	there	PRON
cana-540	139	6	exists	exist	VERB
cana-540	139	7	v	v	ADP
cana-540	139	8			PROPN
cana-540	139	9	spgα	spgα	NOUN
cana-540	139	10	-	-	PUNCT
cana-540	139	11	o(r	o(r	PROPN
cana-540	139	12	)	)	PUNCT
cana-540	139	13	with	with	ADP
cana-540	139	14	p(v	p(v	NOUN
cana-540	139	15	)	)	PUNCT
cana-540	139	16			PROPN
cana-540	139	17	b	b	PROPN
cana-540	139	18			PROPN
cana-540	139	19	.	.	X
cana-540	139	20	as	as	ADP
cana-540	139	21	v	v	ADP
cana-540	139	22			NOUN
cana-540	139	23	v	v	NOUN
cana-540	139	24	then	then	ADV
cana-540	139	25	p-(b	p-(b	ADJ
cana-540	139	26	)	)	PUNCT
cana-540	139	27			NOUN
cana-540	139	28	o(r	o(r	PROPN
cana-540	139	29	)	)	PUNCT
cana-540	139	30	.	.	PUNCT
cana-540	140	1	suppose	suppose	VERB
cana-540	140	2	p-(b	p-(b	ADJ
cana-540	140	3	)	)	PUNCT
cana-540	140	4			NOUN
cana-540	140	5	o(r	o(r	PROPN
cana-540	140	6	)	)	PUNCT
cana-540	140	7	and	and	CCONJ
cana-540	140	8	r	r	NOUN
cana-540	140	9			NOUN
cana-540	140	10	p-(b	p-(b	NOUN
cana-540	140	11	)	)	PUNCT
cana-540	140	12	,	,	PUNCT
cana-540	140	13	then	then	ADV
cana-540	140	14	p-(b	p-(b	ADJ
cana-540	140	15	)	)	PUNCT
cana-540	140	16	=	=	SYM
cana-540	140	17	{	{	PUNCT
cana-540	140	18	r	r	NOUN
cana-540	140	19			NOUN
cana-540	140	20	r	r	NOUN
cana-540	140	21	:	:	PUNCT
cana-540	140	22	p(r	p(r	PROPN
cana-540	140	23	)	)	PUNCT
cana-540	140	24			PUNCT
cana-540	140	25	b	b	PROPN
cana-540	140	26			NOUN
cana-540	140	27			NOUN
cana-540	140	28	}	}	PUNCT
cana-540	140	29	.	.	PUNCT
cana-540	141	1	so	so	ADV
cana-540	141	2	p	p	PROPN
cana-540	141	3	is	be	AUX
cana-540	141	4	l.spgα.c	l.spgα.c	PROPN
cana-540	141	5	.	.	PUNCT
cana-540	141	6	theorem	theorem	VERB
cana-540	141	7	2.6	2.6	NUM
cana-540	141	8	.	.	PUNCT
cana-540	142	1	the	the	DET
cana-540	142	2	following	follow	VERB
cana-540	142	3	holds	hold	VERB
cana-540	142	4	good	good	ADJ
cana-540	142	5	for	for	ADP
cana-540	142	6	a	a	DET
cana-540	142	7	m.f	m.f	NOUN
cana-540	142	8	p	p	X
cana-540	142	9	:	:	PUNCT
cana-540	142	10	r	r	NOUN
cana-540	142	11	→	→	SYM
cana-540	142	12	s	s	PART
cana-540	142	13	1	1	NUM
cana-540	142	14	.	.	PUNCT
cana-540	143	1	p	p	PROPN
cana-540	143	2	is	be	AUX
cana-540	143	3	u.spgα.c	u.spgα.c	PRON
cana-540	143	4	.	.	PUNCT
cana-540	144	1	2	2	NUM
cana-540	144	2	.	.	NUM
cana-540	144	3	p(spgα	p(spgα	NOUN
cana-540	144	4	-	-	PUNCT
cana-540	144	5	cl(b	cl(b	NOUN
cana-540	144	6	)	)	PUNCT
cana-540	144	7	)	)	PUNCT
cana-540	145	1			PROPN
cana-540	145	2	cl(p(b	cl(p(b	NOUN
cana-540	145	3	)	)	PUNCT
cana-540	145	4	)	)	PUNCT
cana-540	145	5	for	for	ADP
cana-540	145	6	every	every	DET
cana-540	145	7	b	b	PROPN
cana-540	145	8			PROPN
cana-540	145	9	r.	r.	PROPN
cana-540	145	10	3	3	NUM
cana-540	145	11	.	.	PUNCT
cana-540	145	12	spgα	spgα	NOUN
cana-540	145	13	-	-	PUNCT
cana-540	145	14	cl(p+(a	cl(p+(a	NOUN
cana-540	145	15	)	)	PUNCT
cana-540	145	16	)	)	PUNCT
cana-540	146	1			PROPN
cana-540	146	2	p+(cl(a	p+(cl(a	PROPN
cana-540	146	3	)	)	PUNCT
cana-540	146	4	)	)	PUNCT
cana-540	146	5	for	for	ADP
cana-540	146	6	every	every	DET
cana-540	146	7	a	a	DET
cana-540	146	8			PROPN
cana-540	146	9	s.	s.	PROPN
cana-540	146	10	4	4	NUM
cana-540	146	11	.	.	PUNCT
cana-540	147	1	p-(int(a	p-(int(a	NOUN
cana-540	147	2	)	)	PUNCT
cana-540	147	3	)	)	PUNCT
cana-540	148	1			PROPN
cana-540	148	2	spgα	spgα	ADJ
cana-540	148	3	-	-	PUNCT
cana-540	148	4	int(p-(a	int(p-(a	NOUN
cana-540	148	5	)	)	PUNCT
cana-540	148	6	)	)	PUNCT
cana-540	148	7	for	for	ADP
cana-540	148	8	every	every	DET
cana-540	148	9	a	a	DET
cana-540	148	10			PROPN
cana-540	148	11	s.	s.	PROPN
cana-540	148	12	5	5	NUM
cana-540	148	13	.	.	PUNCT
cana-540	148	14	int(p(b	int(p(b	NOUN
cana-540	148	15	)	)	PUNCT
cana-540	148	16	)	)	PUNCT
cana-540	149	1			PROPN
cana-540	149	2	p(spgα	p(spgα	PROPN
cana-540	149	3	-	-	PUNCT
cana-540	149	4	int(b	int(b	NOUN
cana-540	149	5	)	)	PUNCT
cana-540	149	6	)	)	PUNCT
cana-540	149	7	for	for	ADP
cana-540	149	8	every	every	DET
cana-540	149	9	b	b	PROPN
cana-540	149	10			PROPN
cana-540	149	11	r.	r.	PROPN
cana-540	149	12	proof	proof	NOUN
cana-540	149	13	.	.	PUNCT
cana-540	150	1	(	(	PUNCT
cana-540	150	2	1	1	X
cana-540	150	3	)	)	PUNCT
cana-540	150	4	→	→	X
cana-540	150	5	(	(	PUNCT
cana-540	150	6	2	2	NUM
cana-540	150	7	):	):	PUNCT
cana-540	150	8	let	let	VERB
cana-540	150	9	b	b	PROPN
cana-540	150	10			PROPN
cana-540	150	11	r.	r.	PROPN
cana-540	150	12	then	then	ADV
cana-540	150	13	p(b	p(b	PROPN
cana-540	150	14	)	)	PUNCT
cana-540	151	1			PROPN
cana-540	151	2	cl(p(b	cl(p(b	NOUN
cana-540	151	3	)	)	PUNCT
cana-540	151	4	)	)	PUNCT
cana-540	151	5	,	,	PUNCT
cana-540	151	6	where	where	SCONJ
cana-540	151	7	cl(p(b	cl(p(b	NOUN
cana-540	151	8	)	)	PUNCT
cana-540	151	9	)	)	PUNCT
cana-540	151	10			NOUN
cana-540	151	11	c(s	c(	VERB
cana-540	151	12	)	)	PUNCT
cana-540	151	13	.	.	PUNCT
cana-540	152	1	as	as	SCONJ
cana-540	152	2	p	p	PRON
cana-540	152	3	is	be	AUX
cana-540	152	4	u.spgα.c	u.spgα.c	PROPN
cana-540	152	5	,	,	PUNCT
cana-540	152	6	b	b	PROPN
cana-540	152	7			PROPN
cana-540	152	8	p+(cl(p(b	p+(cl(p(b	PROPN
cana-540	152	9	)	)	PUNCT
cana-540	152	10	)	)	PUNCT
cana-540	152	11	)	)	PUNCT
cana-540	152	12	.	.	PUNCT
cana-540	153	1	from	from	ADP
cana-540	153	2	theorem	theorem	ADJ
cana-540	153	3	2.1	2.1	NUM
cana-540	153	4	,	,	PUNCT
cana-540	153	5	p+(cl(p(b	p+(cl(p(b	NOUN
cana-540	153	6	)	)	PUNCT
cana-540	153	7	)	)	PUNCT
cana-540	153	8	)	)	PUNCT
cana-540	153	9			NOUN
cana-540	153	10	spgα	spgα	NOUN
cana-540	153	11	-	-	PUNCT
cana-540	153	12	c(r	c(r	NOUN
cana-540	153	13	)	)	PUNCT
cana-540	153	14	.	.	PUNCT
cana-540	154	1	thus	thus	ADV
cana-540	154	2	spgα	spgα	ADJ
cana-540	154	3	-	-	PUNCT
cana-540	154	4	cl(b	cl(b	NOUN
cana-540	154	5	)	)	PUNCT
cana-540	154	6			PROPN
cana-540	154	7	p+cl(p(b	p+cl(p(b	PROPN
cana-540	154	8	)	)	PUNCT
cana-540	154	9	)	)	PUNCT
cana-540	154	10	)	)	PUNCT
cana-540	154	11	and	and	CCONJ
cana-540	154	12	so	so	ADV
cana-540	154	13	p(spgα	p(spgα	NOUN
cana-540	154	14	-	-	NOUN
cana-540	154	15	cl(b	cl(b	NOUN
cana-540	154	16	)	)	PUNCT
cana-540	154	17	)	)	PUNCT
cana-540	155	1			PROPN
cana-540	155	2	cl(p(b	cl(p(b	NOUN
cana-540	155	3	)	)	PUNCT
cana-540	155	4	)	)	PUNCT
cana-540	155	5	.	.	PUNCT
cana-540	156	1	(	(	PUNCT
cana-540	156	2	2	2	X
cana-540	156	3	)	)	PUNCT
cana-540	156	4	→	→	X
cana-540	156	5	(	(	PUNCT
cana-540	156	6	3	3	NUM
cana-540	156	7	):	):	PUNCT
cana-540	156	8	let	let	VERB
cana-540	156	9	a	a	DET
cana-540	156	10			PROPN
cana-540	156	11	s	s	PROPN
cana-540	156	12	and	and	CCONJ
cana-540	156	13	so	so	ADV
cana-540	156	14	p+(a	p+(a	PROPN
cana-540	156	15	)	)	PUNCT
cana-540	157	1			PROPN
cana-540	157	2	r.	r.	PROPN
cana-540	157	3	from	from	ADP
cana-540	157	4	(	(	PUNCT
cana-540	157	5	2	2	NUM
cana-540	157	6	)	)	PUNCT
cana-540	157	7	,	,	PUNCT
cana-540	157	8	p(spgα	p(spgα	NOUN
cana-540	157	9	-	-	PUNCT
cana-540	157	10	cl(p+(a	cl(p+(a	VERB
cana-540	157	11	)	)	PUNCT
cana-540	157	12	)	)	PUNCT
cana-540	157	13	)	)	PUNCT
cana-540	158	1			PROPN
cana-540	158	2	cl(p(p+(a	cl(p(p+(a	ADJ
cana-540	158	3	)	)	PUNCT
cana-540	158	4	)	)	PUNCT
cana-540	158	5	)	)	PUNCT
cana-540	159	1	=	=	SYM
cana-540	159	2	cl(a	cl(a	X
cana-540	159	3	)	)	PUNCT
cana-540	159	4	.	.	PUNCT
cana-540	160	1	thus	thus	ADV
cana-540	160	2	spgα	spgα	ADJ
cana-540	160	3	-	-	PUNCT
cana-540	160	4	cl(p+(a	cl(p+(a	NOUN
cana-540	160	5	)	)	PUNCT
cana-540	160	6	)	)	PUNCT
cana-540	161	1			PROPN
cana-540	161	2	p+cl(a	p+cl(a	PROPN
cana-540	161	3	)	)	PUNCT
cana-540	161	4	.	.	PUNCT
cana-540	162	1	(	(	PUNCT
cana-540	162	2	3	3	X
cana-540	162	3	)	)	PUNCT
cana-540	162	4	→	→	X
cana-540	162	5	(	(	PUNCT
cana-540	162	6	4	4	NUM
cana-540	162	7	):	):	PUNCT
cana-540	162	8	let	let	VERB
cana-540	162	9	a	a	DET
cana-540	162	10			PROPN
cana-540	162	11	s.	s.	PROPN
cana-540	162	12	apply	apply	VERB
cana-540	162	13	(	(	PUNCT
cana-540	162	14	3	3	NUM
cana-540	162	15	)	)	PUNCT
cana-540	162	16	to	to	ADP
cana-540	162	17	s	s	PRON
cana-540	162	18	a	a	DET
cana-540	162	19	,	,	PUNCT
cana-540	162	20	then	then	ADV
cana-540	162	21	spgα	spgα	ADJ
cana-540	162	22	-	-	PUNCT
cana-540	162	23	cl(p+(s	cl(p+(s	PROPN
cana-540	162	24	a	a	PRON
cana-540	162	25	)	)	PUNCT
cana-540	162	26	)	)	PUNCT
cana-540	163	1			PROPN
cana-540	163	2	p+cl(s	p+cl(	VERB
cana-540	163	3	a	a	PRON
cana-540	163	4	)	)	PUNCT
cana-540	163	5	)	)	PUNCT
cana-540	163	6	,	,	PUNCT
cana-540	163	7	spgαcl(r	spgαcl(r	ADJ
cana-540	163	8	–	–	PUNCT
cana-540	163	9	p-(a	p-(a	NOUN
cana-540	163	10	)	)	PUNCT
cana-540	163	11	)	)	PUNCT
cana-540	164	1			PROPN
cana-540	164	2	p+(s	p+(s	NUM
cana-540	164	3	int(a	int(a	PROPN
cana-540	164	4	)	)	PUNCT
cana-540	164	5	)	)	PUNCT
cana-540	164	6	,	,	PUNCT
cana-540	164	7	r	r	NOUN
cana-540	164	8	spgα	spgα	NOUN
cana-540	164	9	-	-	PUNCT
cana-540	164	10	int(p-(a	int(p-(a	NOUN
cana-540	164	11	)	)	PUNCT
cana-540	164	12	)	)	PUNCT
cana-540	165	1			PROPN
cana-540	165	2	r	r	NOUN
cana-540	165	3	–	–	PUNCT
cana-540	165	4	p-(int(a	p-(int(a	PROPN
cana-540	165	5	)	)	PUNCT
cana-540	165	6	)	)	PUNCT
cana-540	165	7	,	,	PUNCT
cana-540	165	8	p-(int(a	p-(int(a	PROPN
cana-540	165	9	)	)	PUNCT
cana-540	165	10	)	)	PUNCT
cana-540	166	1			PROPN
cana-540	166	2	spgα	spgα	NOUN
cana-540	166	3	-	-	PUNCT
cana-540	166	4	int(p(a	int(p(a	NOUN
cana-540	166	5	)	)	PUNCT
cana-540	166	6	)	)	PUNCT
cana-540	166	7	.	.	PUNCT
cana-540	167	1	communications	communication	NOUN
cana-540	167	2	on	on	ADP
cana-540	167	3	applied	apply	VERB
cana-540	167	4	nonlinear	nonlinear	ADJ
cana-540	167	5	analysis	analysis	NOUN
cana-540	167	6	issn	issn	NOUN
cana-540	167	7	:	:	PUNCT
cana-540	167	8	1074	1074	NUM
cana-540	167	9	-	-	PUNCT
cana-540	167	10	133x	133x	NUM
cana-540	167	11	vol	vol	NOUN
cana-540	167	12	31	31	NUM
cana-540	167	13	no	no	NOUN
cana-540	167	14	.	.	NOUN
cana-540	167	15	2	2	NUM
cana-540	167	16	(	(	PUNCT
cana-540	167	17	2024	2024	NUM
cana-540	167	18	)	)	PUNCT
cana-540	167	19	252	252	NUM
cana-540	167	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	167	21	(	(	PUNCT
cana-540	167	22	4	4	NUM
cana-540	167	23	)	)	PUNCT
cana-540	167	24	→	→	X
cana-540	167	25	(	(	PUNCT
cana-540	167	26	5	5	NUM
cana-540	167	27	):	):	PUNCT
cana-540	167	28	let	let	VERB
cana-540	167	29	b	b	X
cana-540	167	30			PROPN
cana-540	167	31	r	r	PROPN
cana-540	167	32	,	,	PUNCT
cana-540	167	33	so	so	ADV
cana-540	167	34	p(b	p(b	NOUN
cana-540	167	35	)	)	PUNCT
cana-540	168	1			PROPN
cana-540	168	2	s.	s.	PROPN
cana-540	168	3	from	from	ADP
cana-540	168	4	(	(	PUNCT
cana-540	168	5	4	4	NUM
cana-540	168	6	)	)	PUNCT
cana-540	168	7	,	,	PUNCT
cana-540	168	8	p-(int(p(a	p-(int(p(a	NOUN
cana-540	168	9	)	)	PUNCT
cana-540	168	10	)	)	PUNCT
cana-540	169	1			PROPN
cana-540	169	2	spgα	spgα	NOUN
cana-540	169	3	-	-	PUNCT
cana-540	169	4	int(p-(p(a	int(p-(p(a	NOUN
cana-540	169	5	)	)	PUNCT
cana-540	169	6	)	)	PUNCT
cana-540	169	7	)	)	PUNCT
cana-540	170	1	=	=	SYM
cana-540	170	2	spgαint(a	spgαint(a	NOUN
cana-540	170	3	)	)	PUNCT
cana-540	170	4	.	.	PUNCT
cana-540	171	1	thus	thus	ADV
cana-540	171	2	int(p(a	int(p(a	ADJ
cana-540	171	3	)	)	PUNCT
cana-540	171	4	)	)	PUNCT
cana-540	172	1			PRON
cana-540	172	2	p(spgα	p(spgα	NOUN
cana-540	172	3	-	-	PUNCT
cana-540	172	4	int(a	int(a	NOUN
cana-540	172	5	)	)	PUNCT
cana-540	172	6	)	)	PUNCT
cana-540	172	7	.	.	PUNCT
cana-540	173	1	(	(	PUNCT
cana-540	173	2	5	5	NUM
cana-540	173	3	)	)	PUNCT
cana-540	173	4	→	→	X
cana-540	173	5	(	(	PUNCT
cana-540	173	6	1	1	NUM
cana-540	173	7	):	):	PUNCT
cana-540	173	8	let	let	VERB
cana-540	173	9	r	r	NOUN
cana-540	173	10			NOUN
cana-540	173	11	r	r	NOUN
cana-540	173	12	and	and	CCONJ
cana-540	173	13	a	a	DET
cana-540	173	14			PROPN
cana-540	173	15	o(s	o(s	NOUN
cana-540	173	16	,	,	PUNCT
cana-540	173	17	p(r	p(r	PROPN
cana-540	173	18	)	)	PUNCT
cana-540	173	19	)	)	PUNCT
cana-540	173	20	.	.	PUNCT
cana-540	174	1	so	so	ADV
cana-540	174	2	,	,	PUNCT
cana-540	174	3	r	r	NOUN
cana-540	174	4			PROPN
cana-540	174	5	p+(a	p+(a	PROPN
cana-540	174	6	)	)	PUNCT
cana-540	174	7	where	where	SCONJ
cana-540	174	8	p+(a	p+(a	PROPN
cana-540	174	9	)	)	PUNCT
cana-540	175	1			PROPN
cana-540	175	2	r.	r.	PROPN
cana-540	175	3	from	from	ADP
cana-540	175	4	(	(	PUNCT
cana-540	175	5	5	5	NUM
cana-540	175	6	)	)	PUNCT
cana-540	175	7	,	,	PUNCT
cana-540	175	8	we	we	PRON
cana-540	175	9	have	have	VERB
cana-540	175	10	int(p(p+(a	int(p(p+(a	NOUN
cana-540	175	11	)	)	PUNCT
cana-540	175	12	)	)	PUNCT
cana-540	175	13	)	)	PUNCT
cana-540	176	1			PRON
cana-540	176	2	p(spgα	p(spgα	NOUN
cana-540	176	3	-	-	PUNCT
cana-540	176	4	int(p+(a	int(p+(a	ADJ
cana-540	176	5	)	)	PUNCT
cana-540	176	6	)	)	PUNCT
cana-540	176	7	)	)	PUNCT
cana-540	176	8	.	.	PUNCT
cana-540	177	1	then	then	ADV
cana-540	177	2	int(a	int(a	X
cana-540	177	3	)	)	PUNCT
cana-540	178	1			PROPN
cana-540	178	2	p(spgα	p(spgα	NOUN
cana-540	178	3	-	-	PUNCT
cana-540	178	4	int(p+(a	int(p+(a	ADJ
cana-540	178	5	)	)	PUNCT
cana-540	178	6	)	)	PUNCT
cana-540	178	7	)	)	PUNCT
cana-540	178	8	.	.	PUNCT
cana-540	179	1	as	as	ADP
cana-540	179	2	a	a	DET
cana-540	179	3			NOUN
cana-540	179	4	o(r	o(r	PROPN
cana-540	179	5	)	)	PUNCT
cana-540	179	6	,	,	PUNCT
cana-540	179	7	then	then	ADV
cana-540	179	8	a	a	DET
cana-540	179	9			NOUN
cana-540	179	10	p(spgα	p(spgα	NOUN
cana-540	179	11	-	-	PUNCT
cana-540	179	12	int(p+(a	int(p+(a	NOUN
cana-540	179	13	)	)	PUNCT
cana-540	179	14	)	)	PUNCT
cana-540	179	15	)	)	PUNCT
cana-540	179	16	,	,	PUNCT
cana-540	179	17	that	that	PRON
cana-540	179	18	is	is	ADV
cana-540	179	19	p+(a	p+(a	PROPN
cana-540	179	20	)	)	PUNCT
cana-540	180	1			PROPN
cana-540	180	2	spgα	spgα	NOUN
cana-540	180	3	-	-	PUNCT
cana-540	180	4	int(p+(a	int(p+(a	NOUN
cana-540	180	5	)	)	PUNCT
cana-540	180	6	)	)	PUNCT
cana-540	180	7	.	.	PUNCT
cana-540	181	1	thus	thus	ADV
cana-540	181	2	p+(a	p+(a	PROPN
cana-540	181	3	)	)	PUNCT
cana-540	181	4			NOUN
cana-540	181	5	spgα	spgα	NOUN
cana-540	181	6	-	-	PUNCT
cana-540	181	7	o(r	o(r	PROPN
cana-540	181	8	,	,	PUNCT
cana-540	181	9	r	r	NOUN
cana-540	181	10	)	)	PUNCT
cana-540	181	11	and	and	CCONJ
cana-540	181	12	p(p+(a	p(p+(a	VERB
cana-540	181	13	)	)	PUNCT
cana-540	181	14	)	)	PUNCT
cana-540	182	1			PROPN
cana-540	182	2	a.	a.	NOUN
cana-540	182	3	hence	hence	ADV
cana-540	182	4	p	p	PROPN
cana-540	182	5	is	be	AUX
cana-540	182	6	u.spgα.c	u.spgα.c	PROPN
cana-540	182	7	.	.	PUNCT
cana-540	183	1	theorem	theorem	VERB
cana-540	183	2	2.7	2.7	NUM
cana-540	183	3	.	.	PUNCT
cana-540	184	1	let	let	VERB
cana-540	184	2	p	p	PRON
cana-540	184	3	:	:	PUNCT
cana-540	184	4	r	r	NOUN
cana-540	184	5	→	→	SYM
cana-540	184	6	s	s	AUX
cana-540	184	7	be	be	AUX
cana-540	184	8	u.spgα.c	u.spgα.c	PROPN
cana-540	184	9	with	with	ADP
cana-540	184	10	q	q	PROPN
cana-540	184	11			PROPN
cana-540	184	12	s.	s.	PROPN
cana-540	184	13	if	if	SCONJ
cana-540	184	14	q	q	NOUN
cana-540	184	15	is	be	AUX
cana-540	184	16	closed	close	VERB
cana-540	184	17	in	in	ADP
cana-540	184	18	s	s	PROPN
cana-540	184	19	,	,	PUNCT
cana-540	184	20	then	then	ADV
cana-540	184	21	p	p	X
cana-540	184	22	:	:	PUNCT
cana-540	184	23	r	r	NOUN
cana-540	184	24	→	→	X
cana-540	184	25	q	q	X
cana-540	184	26	is	be	AUX
cana-540	184	27	u.spgα.c	u.spgα.c	PRON
cana-540	184	28	.	.	PUNCT
cana-540	185	1	proof	proof	NOUN
cana-540	185	2	.	.	PUNCT
cana-540	186	1	let	let	VERB
cana-540	186	2	q	q	PROPN
cana-540	186	3			PROPN
cana-540	186	4	c(s	c(	NOUN
cana-540	186	5	)	)	PUNCT
cana-540	186	6	,	,	PUNCT
cana-540	186	7	so	so	SCONJ
cana-540	186	8	q	q	PROPN
cana-540	186	9	s	s	PROPN
cana-540	186	10			NOUN
cana-540	186	11	c(s	c(	VERB
cana-540	186	12	)	)	PUNCT
cana-540	186	13	.	.	PUNCT
cana-540	187	1	then	then	ADV
cana-540	187	2	q	q	PROPN
cana-540	187	3	s	s	PROPN
cana-540	187	4			NOUN
cana-540	187	5	spgα	spgα	NOUN
cana-540	187	6	-	-	PUNCT
cana-540	187	7	c(s	c(	NOUN
cana-540	187	8	)	)	PUNCT
cana-540	187	9	.	.	PUNCT
cana-540	188	1	by	by	ADP
cana-540	188	2	u.spgα.c	u.spgα.c	PROPN
cana-540	188	3	,	,	PUNCT
cana-540	188	4	p+(q	p+(q	NOUN
cana-540	188	5	s	s	PART
cana-540	188	6	)	)	PUNCT
cana-540	188	7			NOUN
cana-540	188	8	spgα	spgα	NOUN
cana-540	188	9	-	-	PUNCT
cana-540	188	10	c(r	c(r	NOUN
cana-540	188	11	)	)	PUNCT
cana-540	188	12	with	with	ADP
cana-540	188	13	p(r	p(r	PROPN
cana-540	188	14	)	)	PUNCT
cana-540	188	15			NOUN
cana-540	188	16	s.	s.	PROPN
cana-540	188	17	thus	thus	ADV
cana-540	188	18	p+(q	p+(q	VERB
cana-540	188	19	)	)	PUNCT
cana-540	189	1	=	=	PUNCT
cana-540	189	2	p+(q	p+(q	PUNCT
cana-540	189	3	s	s	X
cana-540	189	4	)	)	PUNCT
cana-540	189	5			NOUN
cana-540	189	6	spgα	spgα	NOUN
cana-540	189	7	-	-	PUNCT
cana-540	189	8	c(r	c(r	NOUN
cana-540	189	9	)	)	PUNCT
cana-540	189	10	.	.	PUNCT
cana-540	190	1	from	from	ADP
cana-540	190	2	theorem	theorem	VERB
cana-540	190	3	2.1	2.1	NUM
cana-540	190	4	p	p	NOUN
cana-540	190	5	:	:	PUNCT
cana-540	190	6	r	r	NOUN
cana-540	190	7	→	→	X
cana-540	190	8	q	q	X
cana-540	190	9	is	be	AUX
cana-540	190	10	u.spgα.c	u.spgα.c	PROPN
cana-540	190	11	.	.	PUNCT
cana-540	191	1	theorem	theorem	VERB
cana-540	191	2	2.8	2.8	NUM
cana-540	191	3	.	.	PUNCT
cana-540	192	1	[	[	X
cana-540	192	2	6	6	NUM
cana-540	192	3	]	]	X
cana-540	192	4	intersection	intersection	NOUN
cana-540	192	5	of	of	ADP
cana-540	192	6	any	any	DET
cana-540	192	7	two	two	NUM
cana-540	192	8	spgα	spgα	ADJ
cana-540	192	9	-	-	PUNCT
cana-540	192	10	closed	close	VERB
cana-540	192	11	set	set	NOUN
cana-540	192	12	is	be	AUX
cana-540	192	13	again	again	ADV
cana-540	192	14	spgα	spgα	ADJ
cana-540	192	15	-	-	PUNCT
cana-540	192	16	closed	closed	ADJ
cana-540	192	17	.	.	PUNCT
cana-540	193	1	theorem	theorem	VERB
cana-540	193	2	2.9	2.9	NUM
cana-540	193	3	.	.	PUNCT
cana-540	194	1	let	let	VERB
cana-540	194	2	p	p	PRON
cana-540	194	3	:	:	PUNCT
cana-540	194	4	r	r	NOUN
cana-540	194	5	→	→	SYM
cana-540	194	6	s	s	PART
cana-540	194	7	be	be	AUX
cana-540	194	8	u.spgα.c	u.spgα.c	NOUN
cana-540	194	9	and	and	CCONJ
cana-540	194	10	a	a	DET
cana-540	194	11			NOUN
cana-540	194	12	spgα	spgα	NOUN
cana-540	194	13	-	-	PUNCT
cana-540	194	14	c(r	c(r	NOUN
cana-540	194	15	)	)	PUNCT
cana-540	194	16	.	.	PUNCT
cana-540	195	1	then	then	ADV
cana-540	195	2	p	p	X
cana-540	195	3	a	a	PRON
cana-540	195	4	:	:	PUNCT
cana-540	195	5	a	a	DET
cana-540	195	6	→	→	SYM
cana-540	195	7	s	s	X
cana-540	195	8	is	be	AUX
cana-540	195	9	u.spgα.c	u.spgα.c	PRON
cana-540	195	10	.	.	PUNCT
cana-540	196	1	proof	proof	NOUN
cana-540	196	2	.	.	PUNCT
cana-540	197	1	let	let	VERB
cana-540	197	2	b	b	X
cana-540	197	3			NOUN
cana-540	197	4	c(s	c(	VERB
cana-540	197	5	)	)	PUNCT
cana-540	197	6	as	as	SCONJ
cana-540	197	7	p	p	PROPN
cana-540	197	8	is	be	AUX
cana-540	197	9	u.spgα.c	u.spgα.c	PROPN
cana-540	197	10	.	.	PUNCT
cana-540	198	1	then	then	ADV
cana-540	198	2	p+(b	p+(b	NOUN
cana-540	198	3	)	)	PUNCT
cana-540	198	4			NOUN
cana-540	198	5	spgα	spgα	NOUN
cana-540	198	6	-	-	PUNCT
cana-540	198	7	c(r	c(r	NOUN
cana-540	198	8	)	)	PUNCT
cana-540	198	9	.	.	PUNCT
cana-540	199	1	as	as	SCONJ
cana-540	199	2	intersection	intersection	NOUN
cana-540	199	3	of	of	ADP
cana-540	199	4	two	two	NUM
cana-540	199	5	spgα	spgα	ADJ
cana-540	199	6	-	-	PUNCT
cana-540	199	7	closed	close	VERB
cana-540	199	8	set	set	NOUN
cana-540	199	9	is	be	AUX
cana-540	199	10	closed	closed	ADJ
cana-540	199	11	,	,	PUNCT
cana-540	199	12	then	then	ADV
cana-540	199	13	p+(b	p+(b	NOUN
cana-540	199	14	)	)	PUNCT
cana-540	199	15	a	a	DET
cana-540	199	16	=	=	NOUN
cana-540	199	17	a1	a1	NOUN
cana-540	199	18	,	,	PUNCT
cana-540	199	19	where	where	SCONJ
cana-540	199	20	a1	a1	NOUN
cana-540	199	21			NOUN
cana-540	199	22	spgα	spgα	NOUN
cana-540	199	23	-	-	PUNCT
cana-540	199	24	c(r	c(r	NOUN
cana-540	199	25	)	)	PUNCT
cana-540	199	26	.	.	PUNCT
cana-540	200	1	then	then	ADV
cana-540	200	2	(	(	PUNCT
cana-540	200	3	p	p	NOUN
cana-540	200	4	a)+(b	a)+(b	NOUN
cana-540	200	5	)	)	PUNCT
cana-540	200	6	=	=	SYM
cana-540	200	7	a1	a1	PROPN
cana-540	200	8			NOUN
cana-540	200	9	spgα	spgα	NOUN
cana-540	200	10	-	-	PUNCT
cana-540	200	11	c(r	c(r	NOUN
cana-540	200	12	)	)	PUNCT
cana-540	200	13	.	.	PUNCT
cana-540	201	1	hence	hence	ADV
cana-540	201	2	p	p	X
cana-540	201	3	a	a	PRON
cana-540	201	4	is	be	AUX
cana-540	201	5	u.spgα.c	u.spgα.c	PROPN
cana-540	201	6	.	.	PUNCT
cana-540	202	1	theorem	theorem	VERB
cana-540	202	2	2.10	2.10	NUM
cana-540	202	3	.	.	PUNCT
cana-540	203	1	if	if	SCONJ
cana-540	203	2	p	p	X
cana-540	203	3	:	:	PUNCT
cana-540	203	4	r	r	NOUN
cana-540	203	5	→	→	SYM
cana-540	203	6	s	s	X
cana-540	203	7	is	be	AUX
cana-540	203	8	u.spgα.c	u.spgα.c	NOUN
cana-540	203	9	injective	injective	ADJ
cana-540	203	10	with	with	ADP
cana-540	203	11	s	s	PROPN
cana-540	203	12	is	be	AUX
cana-540	203	13	t1	t1	NOUN
cana-540	203	14	,	,	PUNCT
cana-540	203	15	then	then	ADV
cana-540	203	16	r	r	NOUN
cana-540	203	17	is	be	AUX
cana-540	203	18	spgα	spgα	ADJ
cana-540	203	19	-	-	PUNCT
cana-540	203	20	t1	t1	NOUN
cana-540	203	21	.	.	PUNCT
cana-540	204	1	proof	proof	NOUN
cana-540	204	2	.	.	PUNCT
cana-540	205	1	let	let	VERB
cana-540	205	2	s	s	PRON
cana-540	205	3	be	be	AUX
cana-540	205	4	t1	t1	NOUN
cana-540	205	5	space	space	NOUN
cana-540	205	6	,	,	PUNCT
cana-540	205	7	so	so	SCONJ
cana-540	205	8	for	for	ADP
cana-540	205	9	each	each	DET
cana-540	205	10	distinct	distinct	ADJ
cana-540	205	11	points	point	NOUN
cana-540	205	12	p1	p1	NOUN
cana-540	205	13	,	,	PUNCT
cana-540	205	14	p2	p2	X
cana-540	205	15			NOUN
cana-540	205	16	r	r	NOUN
cana-540	205	17	,	,	PUNCT
cana-540	205	18	there	there	PRON
cana-540	205	19	exist	exist	VERB
cana-540	205	20	a	a	DET
cana-540	205	21	,	,	PUNCT
cana-540	205	22	b	b	PROPN
cana-540	205	23			PROPN
cana-540	205	24	o(s	o(s	PROPN
cana-540	205	25	)	)	PUNCT
cana-540	205	26	such	such	ADJ
cana-540	205	27	that	that	DET
cana-540	205	28	p(p1	p(p1	NOUN
cana-540	205	29	)	)	PUNCT
cana-540	206	1			NOUN
cana-540	206	2	a	a	PRON
cana-540	206	3	,	,	PUNCT
cana-540	206	4	p(p2	p(p2	ADJ
cana-540	206	5	)	)	PUNCT
cana-540	206	6			NOUN
cana-540	206	7	a	a	PRON
cana-540	206	8	and	and	CCONJ
cana-540	206	9	p(p1	p(p1	NOUN
cana-540	206	10	)	)	PUNCT
cana-540	206	11			NOUN
cana-540	206	12	b	b	NOUN
cana-540	206	13	,	,	PUNCT
cana-540	206	14	p(p2	p(p2	ADJ
cana-540	206	15	)	)	PUNCT
cana-540	206	16			NOUN
cana-540	206	17	a.	a.	NOUN
cana-540	206	18	since	since	SCONJ
cana-540	206	19	p	p	PROPN
cana-540	206	20	is	be	AUX
cana-540	206	21	u.spgα.c	u.spgα.c	PROPN
cana-540	206	22	,	,	PUNCT
cana-540	206	23	there	there	PRON
cana-540	206	24	exist	exist	VERB
cana-540	206	25	u	u	NOUN
cana-540	206	26	,	,	PUNCT
cana-540	206	27	v	v	ADP
cana-540	206	28			PROPN
cana-540	206	29	spgα	spgα	NOUN
cana-540	206	30	-	-	PUNCT
cana-540	206	31	o(r	o(r	PROPN
cana-540	206	32	)	)	PUNCT
cana-540	206	33	with	with	ADP
cana-540	206	34	p1	p1	PROPN
cana-540	206	35			PROPN
cana-540	206	36	u	u	PROPN
cana-540	206	37	,	,	PUNCT
cana-540	206	38	p1	p1	PROPN
cana-540	206	39			NUM
cana-540	206	40	v	v	NOUN
cana-540	206	41	and	and	CCONJ
cana-540	206	42	p2	p2	PROPN
cana-540	206	43			PUNCT
cana-540	206	44	u	u	NOUN
cana-540	206	45	,	,	PUNCT
cana-540	206	46	p2	p2	VERB
cana-540	206	47			PROPN
cana-540	206	48	v	v	NOUN
cana-540	206	49	,	,	PUNCT
cana-540	206	50	that	that	PRON
cana-540	206	51	is	is	AUX
cana-540	206	52	p1	p1	PROPN
cana-540	206	53			PROPN
cana-540	206	54	u	u	NOUN
cana-540	206	55	,	,	PUNCT
cana-540	206	56	p2	p2	VERB
cana-540	206	57			NOUN
cana-540	206	58	v	v	ADP
cana-540	206	59	and	and	CCONJ
cana-540	206	60	p1(u	p1(u	NOUN
cana-540	206	61	)	)	PUNCT
cana-540	207	1			PROPN
cana-540	207	2	a	a	PROPN
cana-540	207	3	,	,	PUNCT
cana-540	207	4	p1(v	p1(v	PROPN
cana-540	207	5	)	)	PUNCT
cana-540	208	1			PROPN
cana-540	208	2	b.	b.	PROPN
cana-540	208	3	hence	hence	ADV
cana-540	208	4	r	r	NOUN
cana-540	208	5	is	be	AUX
cana-540	208	6	u.spgα	u.spgα	PROPN
cana-540	208	7	-	-	PUNCT
cana-540	208	8	t1	t1	NOUN
cana-540	208	9	.	.	PUNCT
cana-540	209	1	theorem	theorem	VERB
cana-540	209	2	2.11	2.11	NUM
cana-540	209	3	.	.	PUNCT
cana-540	210	1	let	let	VERB
cana-540	210	2	p	p	PRON
cana-540	210	3	:	:	PUNCT
cana-540	210	4	r	r	NOUN
cana-540	210	5	→	→	SYM
cana-540	210	6	s	s	X
cana-540	210	7	is	be	AUX
cana-540	210	8	u.spgα.c	u.spgα.c	NOUN
cana-540	210	9	injective	injective	ADJ
cana-540	210	10	and	and	CCONJ
cana-540	210	11	s	s	NOUN
cana-540	210	12	is	be	AUX
cana-540	210	13	t2	t2	NOUN
cana-540	210	14	-space	-space	NOUN
cana-540	210	15	.	.	PUNCT
cana-540	211	1	then	then	ADV
cana-540	211	2	r	r	NOUN
cana-540	211	3	is	be	AUX
cana-540	211	4	spgα	spgα	ADJ
cana-540	211	5	-	-	PUNCT
cana-540	211	6	t2	t2	NOUN
cana-540	211	7	.	.	PUNCT
cana-540	212	1	proof	proof	NOUN
cana-540	212	2	.	.	PUNCT
cana-540	213	1	let	let	VERB
cana-540	213	2	p1	p1	PROPN
cana-540	213	3	,	,	PUNCT
cana-540	213	4	p2	p2	PROPN
cana-540	213	5			NOUN
cana-540	213	6	r	r	NOUN
cana-540	213	7	be	be	VERB
cana-540	213	8	any	any	DET
cana-540	213	9	two	two	NUM
cana-540	213	10	distinct	distinct	ADJ
cana-540	213	11	points	point	NOUN
cana-540	213	12	.	.	PUNCT
cana-540	214	1	then	then	ADV
cana-540	214	2	,	,	PUNCT
cana-540	214	3	g	g	PROPN
cana-540	214	4	,	,	PUNCT
cana-540	214	5	h	h	PROPN
cana-540	214	6			PROPN
cana-540	214	7	o(s	o(s	PROPN
cana-540	214	8	)	)	PUNCT
cana-540	214	9	such	such	ADJ
cana-540	214	10	that	that	DET
cana-540	214	11	p(p1	p(p1	NOUN
cana-540	214	12	)	)	PUNCT
cana-540	214	13			NOUN
cana-540	214	14	g	g	NOUN
cana-540	214	15	,	,	PUNCT
cana-540	214	16	p(p2	p(p2	ADJ
cana-540	214	17	)	)	PUNCT
cana-540	214	18			NOUN
cana-540	214	19	h.	h.	PROPN
cana-540	214	20	as	as	SCONJ
cana-540	214	21	p	p	PROPN
cana-540	214	22	is	be	AUX
cana-540	214	23	u.spgα.c	u.spgα.c	PROPN
cana-540	214	24	,	,	PUNCT
cana-540	214	25	there	there	PRON
cana-540	214	26	exist	exist	VERB
cana-540	214	27	a	a	DET
cana-540	214	28	,	,	PUNCT
cana-540	214	29	b	b	PROPN
cana-540	214	30			PROPN
cana-540	214	31	spgα	spgα	NOUN
cana-540	214	32	-	-	PUNCT
cana-540	214	33	o(r	o(r	NOUN
cana-540	214	34	)	)	PUNCT
cana-540	214	35	such	such	ADJ
cana-540	214	36	that	that	SCONJ
cana-540	214	37	p(g	p(g	NOUN
cana-540	214	38	)	)	PUNCT
cana-540	215	1			PROPN
cana-540	215	2	a	a	PRON
cana-540	215	3	,	,	PUNCT
cana-540	215	4	p(h	p(h	PROPN
cana-540	215	5	)	)	PUNCT
cana-540	216	1			PROPN
cana-540	216	2	b	b	PROPN
cana-540	216	3	with	with	ADP
cana-540	216	4	a	a	DET
cana-540	216	5			NOUN
cana-540	216	6	b	b	NOUN
cana-540	216	7	=	=	SYM
cana-540	216	8			NOUN
cana-540	216	9	and	and	CCONJ
cana-540	216	10	so	so	ADV
cana-540	216	11	g	g	NOUN
cana-540	216	12			PUNCT
cana-540	216	13	h	h	NOUN
cana-540	216	14	=	=	PUNCT
cana-540	216	15	.	.	PUNCT
cana-540	216	16	hence	hence	ADV
cana-540	216	17	r	r	NOUN
cana-540	216	18	is	be	AUX
cana-540	216	19	spgα	spgα	ADJ
cana-540	216	20	-	-	PUNCT
cana-540	216	21	t2	t2	NOUN
cana-540	216	22	.	.	PUNCT
cana-540	217	1	theorem	theorem	VERB
cana-540	217	2	2.12	2.12	NUM
cana-540	217	3	.	.	PUNCT
cana-540	218	1	for	for	ADP
cana-540	218	2	a	a	DET
cana-540	218	3	m.f	m.f	NOUN
cana-540	218	4	p	p	X
cana-540	218	5	:	:	PUNCT
cana-540	218	6	r	r	NOUN
cana-540	218	7	→	→	SYM
cana-540	218	8	s	s	PART
cana-540	218	9	is	be	AUX
cana-540	218	10	u.spgα.c	u.spgα.c	PROPN
cana-540	218	11	,	,	PUNCT
cana-540	218	12	image	image	NOUN
cana-540	218	13	of	of	ADP
cana-540	218	14	spgα	spgα	ADJ
cana-540	218	15	-	-	PUNCT
cana-540	218	16	connected	connect	VERB
cana-540	218	17	space	space	NOUN
cana-540	218	18	is	be	AUX
cana-540	218	19	spgαconnected	spgαconnecte	VERB
cana-540	218	20	.	.	PUNCT
cana-540	219	1	proof	proof	NOUN
cana-540	219	2	.	.	PUNCT
cana-540	220	1	let	let	VERB
cana-540	220	2	p	p	PRON
cana-540	220	3	:	:	PUNCT
cana-540	220	4	r	r	NOUN
cana-540	220	5	→	→	SYM
cana-540	220	6	s	s	PART
cana-540	220	7	is	be	AUX
cana-540	220	8	u.spgα.c	u.spgα.c	PROPN
cana-540	220	9	.	.	PUNCT
cana-540	221	1	suppose	suppose	VERB
cana-540	221	2	s	s	NOUN
cana-540	221	3	is	be	AUX
cana-540	221	4	not	not	PART
cana-540	221	5	connected	connect	VERB
cana-540	221	6	and	and	CCONJ
cana-540	221	7	s	s	VERB
cana-540	221	8	=	=	SYM
cana-540	221	9	a	a	DET
cana-540	221	10			PROPN
cana-540	221	11	b	b	PROPN
cana-540	221	12	with	with	ADP
cana-540	221	13	a	a	DET
cana-540	221	14	partition	partition	NOUN
cana-540	221	15	of	of	ADP
cana-540	221	16	s	s	NOUN
cana-540	221	17	,	,	PUNCT
cana-540	221	18	where	where	SCONJ
cana-540	221	19	a	a	DET
cana-540	221	20			PROPN
cana-540	221	21	o(s	o(s	PROPN
cana-540	221	22	)	)	PUNCT
cana-540	221	23	and	and	CCONJ
cana-540	221	24	b	b	NOUN
cana-540	221	25			NOUN
cana-540	221	26	c(s	c(	VERB
cana-540	221	27	)	)	PUNCT
cana-540	221	28	.	.	PUNCT
cana-540	222	1	since	since	SCONJ
cana-540	222	2	p	p	NOUN
cana-540	222	3	is	be	AUX
cana-540	222	4	u.spgα.c	u.spgα.c	PROPN
cana-540	222	5	,	,	PUNCT
cana-540	222	6	p+(a	p+(a	PROPN
cana-540	222	7	)	)	PUNCT
cana-540	222	8	,	,	PUNCT
cana-540	222	9	p+(b	p+(b	PROPN
cana-540	222	10	)	)	PUNCT
cana-540	222	11			NOUN
cana-540	222	12	spgα	spgα	NOUN
cana-540	222	13	-	-	PUNCT
cana-540	222	14	o(r	o(r	PROPN
cana-540	222	15	)	)	PUNCT
cana-540	223	1	where	where	SCONJ
cana-540	223	2	p+(a	p+(a	NOUN
cana-540	223	3	)	)	PUNCT
cana-540	223	4			NUM
cana-540	223	5	p+(b	p+(b	NOUN
cana-540	223	6	)	)	PUNCT
cana-540	223	7	=	=	NOUN
cana-540	223	8			NOUN
cana-540	223	9	and	and	CCONJ
cana-540	223	10	r	r	NOUN
cana-540	223	11	=	=	SYM
cana-540	223	12	p+(a	p+(a	PROPN
cana-540	223	13	)	)	PUNCT
cana-540	223	14			NOUN
cana-540	223	15	p+(b	p+(b	NOUN
cana-540	223	16	)	)	PUNCT
cana-540	223	17	is	be	AUX
cana-540	223	18	a	a	DET
cana-540	223	19	partition	partition	NOUN
cana-540	223	20	of	of	ADP
cana-540	223	21	r	r	NOUN
cana-540	223	22	,	,	PUNCT
cana-540	223	23	which	which	PRON
cana-540	223	24	contradicts	contradict	VERB
cana-540	223	25	that	that	SCONJ
cana-540	223	26	,	,	PUNCT
cana-540	223	27	r	r	NOUN
cana-540	223	28	is	be	AUX
cana-540	223	29	spgαconnected	spgαconnecte	VERB
cana-540	223	30	.	.	PUNCT
cana-540	224	1	communications	communication	NOUN
cana-540	224	2	on	on	ADP
cana-540	224	3	applied	apply	VERB
cana-540	224	4	nonlinear	nonlinear	ADJ
cana-540	224	5	analysis	analysis	NOUN
cana-540	224	6	issn	issn	NOUN
cana-540	224	7	:	:	PUNCT
cana-540	224	8	1074	1074	NUM
cana-540	224	9	-	-	PUNCT
cana-540	224	10	133x	133x	NUM
cana-540	224	11	vol	vol	NOUN
cana-540	224	12	31	31	NUM
cana-540	224	13	no	no	NOUN
cana-540	224	14	.	.	NOUN
cana-540	224	15	2	2	NUM
cana-540	224	16	(	(	PUNCT
cana-540	224	17	2024	2024	NUM
cana-540	224	18	)	)	PUNCT
cana-540	224	19	253	253	NUM
cana-540	224	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	224	21	definition	definition	NOUN
cana-540	224	22	2.3	2.3	NUM
cana-540	224	23	(	(	PUNCT
cana-540	224	24	7	7	NUM
cana-540	224	25	)	)	PUNCT
cana-540	224	26	.	.	PUNCT
cana-540	225	1	for	for	ADP
cana-540	225	2	a	a	DET
cana-540	225	3	m.f	m.f	NOUN
cana-540	225	4	p	p	X
cana-540	225	5	:	:	PUNCT
cana-540	225	6	r	r	NOUN
cana-540	225	7	→	→	SYM
cana-540	225	8	s	s	X
cana-540	225	9	,	,	PUNCT
cana-540	225	10	the	the	DET
cana-540	225	11	graph	graph	NOUN
cana-540	225	12	m.f	m.f	NOUN
cana-540	225	13	gp	gp	NOUN
cana-540	225	14	:	:	PUNCT
cana-540	225	15	r	r	NOUN
cana-540	225	16	→	→	SYM
cana-540	225	17	r	r	NOUN
cana-540	225	18	x	x	SYM
cana-540	225	19	s	s	NOUN
cana-540	225	20	is	be	AUX
cana-540	225	21	defined	define	VERB
cana-540	225	22	as	as	ADP
cana-540	225	23	gp	gp	NOUN
cana-540	225	24	(	(	PUNCT
cana-540	225	25	p	p	X
cana-540	225	26	)	)	PUNCT
cana-540	225	27	=	=	PUNCT
cana-540	226	1	{	{	PUNCT
cana-540	226	2	p	p	X
cana-540	226	3	}	}	PUNCT
cana-540	226	4	x	x	SYM
cana-540	226	5	p(p	p(p	NOUN
cana-540	226	6	)	)	PUNCT
cana-540	226	7	for	for	ADP
cana-540	226	8	every	every	DET
cana-540	226	9	p	p	PROPN
cana-540	226	10			PROPN
cana-540	226	11	r.	r.	PROPN
cana-540	226	12	lemma	lemma	PROPN
cana-540	226	13	2.1	2.1	NUM
cana-540	226	14	(	(	PUNCT
cana-540	226	15	7	7	NUM
cana-540	226	16	)	)	PUNCT
cana-540	226	17	.	.	PUNCT
cana-540	227	1	for	for	ADP
cana-540	227	2	a	a	DET
cana-540	227	3	m.f	m.f	NOUN
cana-540	227	4	p	p	X
cana-540	227	5	:	:	PUNCT
cana-540	227	6	r	r	NOUN
cana-540	227	7	→	→	SYM
cana-540	227	8	s	s	PROPN
cana-540	227	9	,	,	PUNCT
cana-540	227	10	1	1	NUM
cana-540	227	11	.	.	PUNCT
cana-540	227	12	g+p	g+p	PROPN
cana-540	227	13	(	(	PUNCT
cana-540	227	14	a	a	DET
cana-540	227	15	x	x	NOUN
cana-540	227	16	b	b	NOUN
cana-540	227	17	)	)	PUNCT
cana-540	227	18	=	=	SYM
cana-540	227	19	a	a	DET
cana-540	227	20	x	x	X
cana-540	227	21	p+(b	p+(b	NOUN
cana-540	227	22	)	)	PUNCT
cana-540	227	23	2	2	NUM
cana-540	227	24	.	.	X
cana-540	228	1	g	g	NOUN
cana-540	228	2	-	-	PUNCT
cana-540	228	3	p	p	X
cana-540	228	4	(	(	PUNCT
cana-540	228	5	a	a	DET
cana-540	228	6	x	x	X
cana-540	228	7	b	b	NOUN
cana-540	228	8	)	)	PUNCT
cana-540	228	9	=	=	PUNCT
cana-540	229	1	a	a	DET
cana-540	229	2	x	x	NOUN
cana-540	229	3	p-(b	p-(b	NOUN
cana-540	229	4	)	)	PUNCT
cana-540	229	5	for	for	ADP
cana-540	229	6	a	a	DET
cana-540	229	7			PROPN
cana-540	229	8	r	r	PROPN
cana-540	229	9	and	and	CCONJ
cana-540	229	10	b	b	PROPN
cana-540	230	1			PROPN
cana-540	230	2	s.	s.	PROPN
cana-540	230	3	theorem	theorem	VERB
cana-540	230	4	2.13	2.13	NUM
cana-540	230	5	.	.	PUNCT
cana-540	231	1	let	let	VERB
cana-540	231	2	p	p	PRON
cana-540	231	3	:	:	PUNCT
cana-540	231	4	r	r	NOUN
cana-540	231	5	→	→	SYM
cana-540	231	6	s	s	VERB
cana-540	231	7	be	be	AUX
cana-540	231	8	a	a	DET
cana-540	231	9	m.f	m.f	NOUN
cana-540	231	10	.	.	PUNCT
cana-540	232	1	if	if	SCONJ
cana-540	232	2	the	the	DET
cana-540	232	3	graph	graph	NOUN
cana-540	232	4	m.f	m.f	NOUN
cana-540	232	5	gp	gp	NOUN
cana-540	232	6	is	be	AUX
cana-540	232	7	u.spgα.c	u.spgα.c	PROPN
cana-540	232	8	,	,	PUNCT
cana-540	232	9	then	then	ADV
cana-540	232	10	p	p	PROPN
cana-540	232	11	is	be	AUX
cana-540	232	12	u.spgα.c	u.spgα.c	PROPN
cana-540	232	13	,	,	PUNCT
cana-540	232	14	where	where	SCONJ
cana-540	232	15	gp	gp	NOUN
cana-540	232	16	:	:	PUNCT
cana-540	232	17	r	r	NOUN
cana-540	232	18	→	→	SYM
cana-540	232	19	r	r	NOUN
cana-540	232	20	x	x	SYM
cana-540	232	21	s	s	NOUN
cana-540	232	22	is	be	AUX
cana-540	232	23	defined	define	VERB
cana-540	232	24	as	as	ADP
cana-540	232	25	gp	gp	NOUN
cana-540	232	26	(	(	PUNCT
cana-540	232	27	p	p	X
cana-540	232	28	)	)	PUNCT
cana-540	232	29	=	=	PUNCT
cana-540	232	30	{	{	PUNCT
cana-540	232	31	p	p	X
cana-540	232	32	}	}	PUNCT
cana-540	232	33	x	x	SYM
cana-540	232	34	p(p	p(p	NOUN
cana-540	232	35	)	)	PUNCT
cana-540	232	36	.	.	PUNCT
cana-540	233	1	proof	proof	NOUN
cana-540	233	2	.	.	PUNCT
cana-540	234	1	let	let	VERB
cana-540	234	2	p	p	PRON
cana-540	234	3			PROPN
cana-540	234	4	r	r	NOUN
cana-540	234	5	and	and	CCONJ
cana-540	234	6	v	v	ADP
cana-540	234	7			PROPN
cana-540	234	8	o(s	o(s	PROPN
cana-540	234	9	,	,	PUNCT
cana-540	234	10	p(p	p(p	NOUN
cana-540	234	11	)	)	PUNCT
cana-540	234	12	)	)	PUNCT
cana-540	234	13	.	.	PUNCT
cana-540	235	1	then	then	ADV
cana-540	235	2	r	r	NOUN
cana-540	235	3	x	x	SYM
cana-540	235	4	v	v	ADP
cana-540	235	5			NOUN
cana-540	235	6	o(r	o(r	PROPN
cana-540	235	7	x	x	SYM
cana-540	235	8	s	s	X
cana-540	235	9	)	)	PUNCT
cana-540	235	10	and	and	CCONJ
cana-540	235	11	gp	gp	NOUN
cana-540	235	12	(	(	PUNCT
cana-540	235	13	p	p	X
cana-540	235	14	)	)	PUNCT
cana-540	236	1			PROPN
cana-540	236	2	r	r	NOUN
cana-540	236	3	x	x	SYM
cana-540	236	4	v	v	NOUN
cana-540	236	5	.	.	PUNCT
cana-540	237	1	as	as	SCONJ
cana-540	237	2	gp	gp	NOUN
cana-540	237	3	is	be	AUX
cana-540	237	4	u.spgα.c	u.spgα.c	PROPN
cana-540	237	5	,	,	PUNCT
cana-540	237	6	there	there	PRON
cana-540	237	7	exists	exist	VERB
cana-540	237	8	gp	gp	NOUN
cana-540	237	9	(	(	PUNCT
cana-540	237	10	v	v	NOUN
cana-540	237	11	)	)	PUNCT
cana-540	238	1			PROPN
cana-540	238	2	r	r	NOUN
cana-540	238	3	x	x	SYM
cana-540	238	4	v	v	NOUN
cana-540	238	5	.	.	PUNCT
cana-540	239	1	thus	thus	ADV
cana-540	239	2	u	u	PROPN
cana-540	239	3			PROPN
cana-540	239	4	g+p	g+p	PROPN
cana-540	239	5	(	(	PUNCT
cana-540	239	6	r	r	NOUN
cana-540	239	7	x	x	SYM
cana-540	239	8	v	v	NOUN
cana-540	239	9	)	)	PUNCT
cana-540	239	10	.	.	PUNCT
cana-540	240	1	by	by	ADP
cana-540	240	2	lemma	lemma	PROPN
cana-540	240	3	2.1	2.1	NUM
cana-540	240	4	,	,	PUNCT
cana-540	240	5	g+p	g+p	PROPN
cana-540	240	6	(	(	PUNCT
cana-540	240	7	r	r	NOUN
cana-540	240	8	x	x	SYM
cana-540	240	9	v	v	NOUN
cana-540	240	10	)	)	PUNCT
cana-540	240	11	=	=	SYM
cana-540	240	12	p+(v	p+(v	PROPN
cana-540	240	13	)	)	PUNCT
cana-540	240	14	and	and	CCONJ
cana-540	240	15	so	so	ADV
cana-540	240	16	u	u	X
cana-540	240	17			PROPN
cana-540	240	18	p+(v	p+(v	PROPN
cana-540	240	19	)	)	PUNCT
cana-540	240	20	.	.	PUNCT
cana-540	241	1	thus	thus	ADV
cana-540	241	2	p	p	X
cana-540	241	3	is	be	AUX
cana-540	241	4	u.spgα.c	u.spgα.c	PROPN
cana-540	241	5	.	.	PUNCT
cana-540	242	1	theorem	theorem	VERB
cana-540	242	2	2.14	2.14	NUM
cana-540	242	3	.	.	PUNCT
cana-540	243	1	let	let	VERB
cana-540	243	2	p	p	PRON
cana-540	243	3	:	:	PUNCT
cana-540	243	4	r	r	NOUN
cana-540	243	5	→	→	SYM
cana-540	243	6	s	s	VERB
cana-540	243	7	be	be	AUX
cana-540	243	8	a	a	DET
cana-540	243	9	m.f	m.f	NOUN
cana-540	243	10	.	.	PUNCT
cana-540	244	1	if	if	SCONJ
cana-540	244	2	the	the	DET
cana-540	244	3	graph	graph	NOUN
cana-540	244	4	m.f	m.f	PROPN
cana-540	244	5	gp	gp	NOUN
cana-540	244	6	is	be	AUX
cana-540	244	7	l.spgα.c	l.spgα.c	PROPN
cana-540	244	8	.	.	PUNCT
cana-540	245	1	then	then	ADV
cana-540	245	2	p	p	PROPN
cana-540	245	3	is	be	AUX
cana-540	245	4	l.spgα.c	l.spgα.c	PROPN
cana-540	245	5	.	.	PUNCT
cana-540	245	6	proof	proof	NOUN
cana-540	245	7	.	.	PUNCT
cana-540	246	1	let	let	VERB
cana-540	246	2	p	p	PRON
cana-540	246	3			PROPN
cana-540	246	4	r	r	NOUN
cana-540	246	5	and	and	CCONJ
cana-540	246	6	v	v	ADP
cana-540	246	7			PROPN
cana-540	246	8	o(s	o(s	PROPN
cana-540	246	9	)	)	PUNCT
cana-540	246	10	with	with	ADP
cana-540	246	11	p	p	PROPN
cana-540	246	12			PROPN
cana-540	246	13	p-(v	p-(v	NOUN
cana-540	246	14	)	)	PUNCT
cana-540	246	15	.	.	PUNCT
cana-540	247	1	then	then	ADV
cana-540	247	2	r	r	NOUN
cana-540	247	3	x	x	SYM
cana-540	247	4	v	v	NOUN
cana-540	247	5	is	be	AUX
cana-540	247	6	open	open	ADJ
cana-540	247	7	in	in	ADP
cana-540	247	8	r	r	NOUN
cana-540	247	9	x	x	PUNCT
cana-540	247	10	s.	s.	PROPN
cana-540	247	11	also	also	ADV
cana-540	247	12	,	,	PUNCT
cana-540	247	13	we	we	PRON
cana-540	247	14	have	have	VERB
cana-540	247	15	gp	gp	NOUN
cana-540	247	16	(	(	PUNCT
cana-540	247	17	p	p	NOUN
cana-540	247	18	)	)	PUNCT
cana-540	247	19			NOUN
cana-540	247	20	(	(	PUNCT
cana-540	247	21	r	r	NOUN
cana-540	247	22	x	x	SYM
cana-540	247	23	v	v	NOUN
cana-540	247	24	)	)	PUNCT
cana-540	247	25	=	=	SYM
cana-540	247	26	(	(	PUNCT
cana-540	247	27	{	{	PUNCT
cana-540	247	28	p	p	NOUN
cana-540	247	29	}	}	PUNCT
cana-540	247	30	)	)	PUNCT
cana-540	247	31	x	x	SYM
cana-540	247	32	p(p	p(p	NOUN
cana-540	247	33	)	)	PUNCT
cana-540	247	34	)	)	PUNCT
cana-540	248	1			NOUN
cana-540	248	2	(	(	PUNCT
cana-540	248	3	r	r	NOUN
cana-540	248	4	x	x	SYM
cana-540	248	5	v	v	NOUN
cana-540	248	6	)	)	PUNCT
cana-540	248	7	=	=	SYM
cana-540	248	8	(	(	PUNCT
cana-540	248	9	{	{	PUNCT
cana-540	248	10	p	p	NOUN
cana-540	248	11	}	}	PUNCT
cana-540	248	12	)	)	PUNCT
cana-540	248	13	x	x	SYM
cana-540	248	14	p(p	p(p	NOUN
cana-540	248	15	)	)	PUNCT
cana-540	248	16			PROPN
cana-540	248	17	v	v	NUM
cana-540	248	18			NOUN
cana-540	248	19	.	.	X
cana-540	248	20	as	as	ADP
cana-540	248	21	gp	gp	PROPN
cana-540	248	22	l.spgα.c	l.spgα.c	PROPN
cana-540	248	23	,	,	PUNCT
cana-540	248	24	there	there	PRON
cana-540	248	25	exists	exist	VERB
cana-540	248	26	u	u	NOUN
cana-540	248	27			PROPN
cana-540	248	28	spgα	spgα	NOUN
cana-540	248	29	-	-	PUNCT
cana-540	248	30	o(r	o(r	PROPN
cana-540	248	31	,	,	PUNCT
cana-540	248	32	r	r	NOUN
cana-540	248	33	)	)	PUNCT
cana-540	248	34	such	such	ADJ
cana-540	248	35	that	that	SCONJ
cana-540	248	36	u	u	PROPN
cana-540	248	37			NOUN
cana-540	248	38	g	g	NOUN
cana-540	248	39	-	-	PUNCT
cana-540	248	40	p	p	X
cana-540	248	41	(	(	PUNCT
cana-540	248	42	r	r	NOUN
cana-540	248	43	x	x	SYM
cana-540	248	44	v	v	NOUN
cana-540	248	45	)	)	PUNCT
cana-540	248	46	and	and	CCONJ
cana-540	248	47	from	from	ADP
cana-540	248	48	lemma	lemma	PROPN
cana-540	248	49	2.1	2.1	NUM
cana-540	248	50	g	g	NOUN
cana-540	248	51	-	-	PUNCT
cana-540	248	52	p	p	NOUN
cana-540	248	53	(	(	PUNCT
cana-540	248	54	r	r	NOUN
cana-540	248	55	x	x	SYM
cana-540	248	56	v	v	NOUN
cana-540	248	57	)	)	PUNCT
cana-540	248	58	=	=	SYM
cana-540	248	59	p-(v	p-(v	NOUN
cana-540	248	60	)	)	PUNCT
cana-540	248	61	and	and	CCONJ
cana-540	248	62	so	so	ADV
cana-540	248	63	u	u	X
cana-540	248	64	x	x	NOUN
cana-540	248	65	p_(v	p_(v	NUM
cana-540	248	66	)	)	PUNCT
cana-540	248	67	.	.	PUNCT
cana-540	249	1	thus	thus	ADV
cana-540	249	2	p	p	X
cana-540	249	3	is	be	AUX
cana-540	249	4	l.spgα.c	l.spgα.c	PROPN
cana-540	249	5	.	.	PUNCT
cana-540	249	6	theorem	theorem	PROPN
cana-540	249	7	2.15	2.15	NUM
cana-540	249	8	.	.	PUNCT
cana-540	250	1	let	let	VERB
cana-540	250	2	(	(	PUNCT
cana-540	250	3	r	r	NOUN
cana-540	250	4	,	,	PUNCT
cana-540	250	5			PROPN
cana-540	250	6	)	)	PUNCT
cana-540	250	7	,	,	PUNCT
cana-540	250	8	(	(	PUNCT
cana-540	250	9	s	s	X
cana-540	250	10	,	,	PUNCT
cana-540	250	11			PROPN
cana-540	250	12	)	)	PUNCT
cana-540	250	13	,	,	PUNCT
cana-540	250	14	(	(	PUNCT
cana-540	250	15	q	q	X
cana-540	250	16	,	,	PUNCT
cana-540	250	17	γ	γ	NOUN
cana-540	250	18	)	)	PUNCT
cana-540	250	19	be	be	VERB
cana-540	250	20	ts	ts	ADP
cana-540	250	21	and	and	CCONJ
cana-540	250	22	p1	p1	PROPN
cana-540	250	23	:	:	PUNCT
cana-540	250	24	r	r	NOUN
cana-540	250	25	→	→	SYM
cana-540	250	26	s	s	X
cana-540	250	27	and	and	CCONJ
cana-540	250	28	p2	p2	PROPN
cana-540	250	29	:	:	PUNCT
cana-540	250	30	s	s	AUX
cana-540	250	31	→	→	X
cana-540	250	32	q	q	X
cana-540	250	33	be	be	AUX
cana-540	250	34	m.f	m.f	PROPN
cana-540	250	35	.	.	PUNCT
cana-540	251	1	let	let	VERB
cana-540	251	2	p1	p1	PROPN
cana-540	251	3	x	x	X
cana-540	251	4	p2	p2	PROPN
cana-540	251	5	:	:	PUNCT
cana-540	252	1	r	r	X
cana-540	252	2	→	→	SYM
cana-540	252	3	s	s	NOUN
cana-540	252	4	x	x	X
cana-540	252	5	q	q	X
cana-540	252	6	be	be	AUX
cana-540	252	7	a	a	DET
cana-540	252	8	m.f	m.f	NOUN
cana-540	252	9	defined	define	VERB
cana-540	252	10	by	by	ADP
cana-540	252	11	(	(	PUNCT
cana-540	252	12	p1	p1	PROPN
cana-540	252	13	x	x	SYM
cana-540	252	14	p2)(p	p2)(p	PROPN
cana-540	252	15	)	)	PUNCT
cana-540	252	16	=	=	SYM
cana-540	252	17	p1(p	p1(p	PROPN
cana-540	252	18	)	)	PUNCT
cana-540	252	19	x	x	PUNCT
cana-540	252	20	p2(p	p2(p	NOUN
cana-540	252	21	)	)	PUNCT
cana-540	252	22	for	for	ADP
cana-540	252	23	each	each	DET
cana-540	252	24	p	p	NOUN
cana-540	252	25			NOUN
cana-540	252	26	r	r	NOUN
cana-540	252	27	if	if	SCONJ
cana-540	252	28	(	(	PUNCT
cana-540	252	29	p1	p1	NOUN
cana-540	252	30	x	x	SYM
cana-540	252	31	p2	p2	PROPN
cana-540	252	32	)	)	PUNCT
cana-540	252	33	is	be	AUX
cana-540	252	34	u.spgα.c	u.spgα.c	PROPN
cana-540	252	35	.	.	PUNCT
cana-540	253	1	then	then	ADV
cana-540	253	2	p1	p1	PROPN
cana-540	253	3	and	and	CCONJ
cana-540	253	4	p2	p2	PROPN
cana-540	253	5	are	be	AUX
cana-540	253	6	u.spgα.c	u.spgα.c	PRON
cana-540	253	7	.	.	PUNCT
cana-540	254	1	proof	proof	NOUN
cana-540	254	2	.	.	PUNCT
cana-540	255	1	let	let	VERB
cana-540	255	2	p	p	PRON
cana-540	255	3			PROPN
cana-540	255	4	r	r	NOUN
cana-540	255	5	and	and	CCONJ
cana-540	255	6	v	v	NOUN
cana-540	255	7	x	x	SYM
cana-540	255	8	s	s	NOUN
cana-540	255	9	and	and	CCONJ
cana-540	255	10	w	w	NOUN
cana-540	255	11	x	x	X
cana-540	255	12	q	q	PUNCT
cana-540	255	13	be	be	AUX
cana-540	255	14	open	open	ADJ
cana-540	255	15	sets	set	NOUN
cana-540	255	16	with	with	ADP
cana-540	255	17	p	p	PROPN
cana-540	255	18			PROPN
cana-540	255	19	p+1	p+1	PROPN
cana-540	255	20	(	(	PUNCT
cana-540	255	21	v	v	NOUN
cana-540	255	22	)	)	PUNCT
cana-540	255	23	and	and	CCONJ
cana-540	255	24	p	p	X
cana-540	255	25			NOUN
cana-540	255	26	p+2	p+2	PROPN
cana-540	255	27	(	(	PUNCT
cana-540	255	28	w	w	NOUN
cana-540	255	29	)	)	PUNCT
cana-540	255	30	and	and	CCONJ
cana-540	255	31	so	so	ADV
cana-540	255	32	p1(p	p1(p	PROPN
cana-540	255	33	)	)	PUNCT
cana-540	255	34			PROPN
cana-540	255	35	v	v	NOUN
cana-540	255	36	and	and	CCONJ
cana-540	255	37	p2(p	p2(p	NOUN
cana-540	255	38	)	)	PUNCT
cana-540	255	39			PROPN
cana-540	255	40	w.	w.	PROPN
cana-540	255	41	thus	thus	ADV
cana-540	255	42	(	(	PUNCT
cana-540	255	43	p1	p1	NOUN
cana-540	255	44	x	x	SYM
cana-540	255	45	p2)(p	p2)(p	PROPN
cana-540	255	46	)	)	PUNCT
cana-540	255	47	=	=	SYM
cana-540	255	48	p1(p	p1(p	PROPN
cana-540	255	49	)	)	PUNCT
cana-540	255	50	x	x	PUNCT
cana-540	255	51	p2(p	p2(p	X
cana-540	255	52	)	)	PUNCT
cana-540	255	53			PROPN
cana-540	255	54	v	v	ADP
cana-540	255	55	x	x	SYM
cana-540	255	56	w	w	NOUN
cana-540	256	1	and	and	CCONJ
cana-540	256	2	so	so	ADV
cana-540	256	3	p	p	PRON
cana-540	256	4			NOUN
cana-540	256	5	(	(	PUNCT
cana-540	256	6	p1x	p1x	NOUN
cana-540	256	7	p2)+(v	p2)+(v	PROPN
cana-540	256	8	x	x	SYM
cana-540	256	9	w	w	NOUN
cana-540	256	10	)	)	PUNCT
cana-540	256	11	as	as	ADP
cana-540	256	12	p1	p1	PROPN
cana-540	256	13	x	x	PUNCT
cana-540	256	14	p2	p2	PROPN
cana-540	256	15	is	be	AUX
cana-540	256	16	u.spgα.c	u.spgα.c	PROPN
cana-540	256	17	,	,	PUNCT
cana-540	256	18	there	there	PRON
cana-540	256	19	exists	exist	VERB
cana-540	256	20	spgα	spgα	ADJ
cana-540	256	21	-	-	PUNCT
cana-540	256	22	open	open	ADJ
cana-540	256	23	set	set	NOUN
cana-540	256	24	u	u	NOUN
cana-540	256	25	containing	contain	VERB
cana-540	256	26	p	p	NOUN
cana-540	256	27	such	such	ADJ
cana-540	256	28	that	that	SCONJ
cana-540	256	29	u	u	NOUN
cana-540	256	30			PROPN
cana-540	256	31	(	(	PUNCT
cana-540	256	32	p1	p1	PROPN
cana-540	256	33	x	x	PUNCT
cana-540	256	34	p2)+(v	p2)+(v	PROPN
cana-540	256	35	x	x	SYM
cana-540	256	36	w	w	NOUN
cana-540	256	37	)	)	PUNCT
cana-540	256	38	that	that	PRON
cana-540	256	39	is	be	AUX
cana-540	256	40	u	u	PROPN
cana-540	256	41			PROPN
cana-540	256	42	p+1	p+1	PROPN
cana-540	256	43	(	(	PUNCT
cana-540	256	44	v	v	NOUN
cana-540	256	45	)	)	PUNCT
cana-540	256	46	and	and	CCONJ
cana-540	256	47	u	u	NOUN
cana-540	256	48			PROPN
cana-540	256	49	p+2	p+2	PROPN
cana-540	256	50	(	(	PUNCT
cana-540	256	51	w	w	NOUN
cana-540	256	52	)	)	PUNCT
cana-540	256	53	hence	hence	ADV
cana-540	256	54	p1	p1	NOUN
cana-540	256	55	and	and	CCONJ
cana-540	256	56	p2	p2	PROPN
cana-540	256	57	are	be	AUX
cana-540	256	58	u.spgα.c	u.spgα.c	NOUN
cana-540	256	59	similarlly	similarlly	ADV
cana-540	256	60	we	we	PRON
cana-540	256	61	can	can	AUX
cana-540	256	62	prove	prove	VERB
cana-540	256	63	the	the	DET
cana-540	256	64	results	result	NOUN
cana-540	256	65	for	for	ADP
cana-540	256	66	l.spgα.c	l.spgα.c	PROPN
cana-540	256	67	.	.	PUNCT
cana-540	256	68	theorem	theorem	VERB
cana-540	256	69	2.16	2.16	NUM
cana-540	256	70	.	.	PUNCT
cana-540	257	1	let	let	VERB
cana-540	257	2	p	p	PRON
cana-540	257	3	:	:	PUNCT
cana-540	257	4	r	r	NOUN
cana-540	257	5	→	→	SYM
cana-540	257	6	s	s	AUX
cana-540	257	7	be	be	AUX
cana-540	257	8	compact	compact	ADJ
cana-540	257	9	m.f	m.f	NOUN
cana-540	257	10	.	.	PUNCT
cana-540	258	1	then	then	ADV
cana-540	258	2	gp	gp	PROPN
cana-540	258	3	is	be	AUX
cana-540	258	4	u.spgα.c	u.spgα.c	PROPN
cana-540	258	5	if	if	SCONJ
cana-540	259	1	and	and	CCONJ
cana-540	259	2	only	only	ADV
cana-540	259	3	if	if	SCONJ
cana-540	259	4	p	p	NOUN
cana-540	259	5	is	be	AUX
cana-540	259	6	u.spgα.c	u.spgα.c	PRON
cana-540	259	7	.	.	PUNCT
cana-540	260	1	proof	proof	NOUN
cana-540	260	2	.	.	PUNCT
cana-540	261	1	suppose	suppose	VERB
cana-540	261	2	gp	gp	NOUN
cana-540	261	3	:	:	PUNCT
cana-540	261	4	r	r	X
cana-540	261	5	→	→	SYM
cana-540	261	6	s	s	X
cana-540	261	7	be	be	AUX
cana-540	261	8	u.spgα.c	u.spgα.c	PROPN
cana-540	261	9	.	.	PUNCT
cana-540	262	1	let	let	VERB
cana-540	262	2	p	p	PRON
cana-540	262	3			PROPN
cana-540	262	4	r	r	NOUN
cana-540	262	5	and	and	CCONJ
cana-540	262	6	v	v	ADP
cana-540	262	7			PROPN
cana-540	262	8	o(s	o(s	PROPN
cana-540	262	9	,	,	PUNCT
cana-540	262	10	p(p	p(p	NOUN
cana-540	262	11	)	)	PUNCT
cana-540	262	12	)	)	PUNCT
cana-540	262	13	.	.	PUNCT
cana-540	263	1	since	since	SCONJ
cana-540	263	2	r	r	NOUN
cana-540	263	3	x	x	SYM
cana-540	263	4	v	v	NOUN
cana-540	263	5	is	be	AUX
cana-540	263	6	open	open	ADJ
cana-540	263	7	in	in	ADP
cana-540	263	8	r	r	NOUN
cana-540	263	9	x	x	SYM
cana-540	263	10	s	s	NOUN
cana-540	263	11	and	and	CCONJ
cana-540	263	12	gp	gp	NOUN
cana-540	263	13	(	(	PUNCT
cana-540	263	14	p	p	X
cana-540	263	15	)	)	PUNCT
cana-540	264	1			PROPN
cana-540	264	2	r	r	NOUN
cana-540	264	3	x	x	SYM
cana-540	264	4	s	s	NOUN
cana-540	265	1	and	and	CCONJ
cana-540	265	2	so	so	ADV
cana-540	265	3	there	there	PRON
cana-540	265	4	exists	exist	VERB
cana-540	265	5	u	u	PRON
cana-540	265	6			PROPN
cana-540	265	7	spgα	spgα	NOUN
cana-540	265	8	-	-	PUNCT
cana-540	265	9	o(r	o(r	PROPN
cana-540	265	10	,	,	PUNCT
cana-540	265	11	p	p	NOUN
cana-540	265	12	)	)	PUNCT
cana-540	265	13	such	such	ADJ
cana-540	265	14	that	that	DET
cana-540	265	15	gp	gp	NOUN
cana-540	265	16	(	(	PUNCT
cana-540	265	17	u	u	NOUN
cana-540	265	18	)	)	PUNCT
cana-540	266	1			PROPN
cana-540	266	2	r	r	NOUN
cana-540	266	3	x	x	PUNCT
cana-540	266	4	s.	s.	PROPN
cana-540	266	5	from	from	ADP
cana-540	266	6	lemma	lemma	PROPN
cana-540	266	7	2.1	2.1	NUM
cana-540	266	8	,	,	PUNCT
cana-540	266	9	u	u	NOUN
cana-540	266	10			PROPN
cana-540	266	11	g+p	g+p	PROPN
cana-540	266	12	(	(	PUNCT
cana-540	266	13	r	r	NOUN
cana-540	266	14	x	x	SYM
cana-540	266	15	v	v	NOUN
cana-540	266	16	)	)	PUNCT
cana-540	266	17	=	=	SYM
cana-540	266	18	p+(v	p+(v	PROPN
cana-540	266	19	)	)	PUNCT
cana-540	266	20	and	and	CCONJ
cana-540	266	21	p(u	p(u	NOUN
cana-540	266	22	)	)	PUNCT
cana-540	266	23	x	x	SYM
cana-540	266	24	v	v	NOUN
cana-540	266	25	and	and	CCONJ
cana-540	266	26	hence	hence	ADV
cana-540	266	27	p	p	NOUN
cana-540	266	28	is	be	AUX
cana-540	266	29	u.spgα.c	u.spgα.c	PRON
cana-540	266	30	.	.	PUNCT
cana-540	267	1	conversely	conversely	ADV
cana-540	267	2	,	,	PUNCT
cana-540	267	3	let	let	VERB
cana-540	267	4	p	p	PRON
cana-540	267	5	be	be	AUX
cana-540	267	6	u.spgα.c	u.spgα.c	PROPN
cana-540	267	7	.	.	PUNCT
cana-540	268	1	let	let	VERB
cana-540	268	2	p	p	PRON
cana-540	268	3			PROPN
cana-540	268	4	r	r	NOUN
cana-540	268	5	and	and	CCONJ
cana-540	268	6	w	w	PROPN
cana-540	268	7	be	be	AUX
cana-540	268	8	any	any	DET
cana-540	268	9	open	open	ADJ
cana-540	268	10	set	set	NOUN
cana-540	268	11	in	in	ADP
cana-540	268	12	r	r	NOUN
cana-540	268	13	x	x	X
cana-540	268	14	s	s	AUX
cana-540	268	15	containing	contain	VERB
cana-540	268	16	gp(p	gp(p	NOUN
cana-540	268	17	)	)	PUNCT
cana-540	268	18	.	.	PUNCT
cana-540	269	1	then	then	ADV
cana-540	269	2	for	for	SCONJ
cana-540	269	3	each	each	DET
cana-540	269	4	n	n	ADJ
cana-540	269	5			NOUN
cana-540	269	6	p(p	p(p	NOUN
cana-540	269	7	)	)	PUNCT
cana-540	269	8	,	,	PUNCT
cana-540	269	9	there	there	PRON
cana-540	269	10	exists	exist	VERB
cana-540	269	11	u(n	u(n	PROPN
cana-540	269	12	)	)	PUNCT
cana-540	269	13			NOUN
cana-540	269	14	r	r	NOUN
cana-540	269	15	and	and	CCONJ
cana-540	269	16	v	v	NOUN
cana-540	269	17	(	(	PUNCT
cana-540	269	18	n	n	CCONJ
cana-540	269	19	)	)	PUNCT
cana-540	269	20			NOUN
cana-540	269	21	s	s	VERB
cana-540	270	1	such	such	ADJ
cana-540	270	2	that	that	SCONJ
cana-540	270	3	(	(	PUNCT
cana-540	270	4	p	p	X
cana-540	270	5	,	,	PUNCT
cana-540	270	6	n	n	CCONJ
cana-540	270	7	)	)	PUNCT
cana-540	270	8			PROPN
cana-540	270	9	u(n	u(n	NOUN
cana-540	270	10	)	)	PUNCT
cana-540	270	11	x	x	SYM
cana-540	270	12	v	v	NOUN
cana-540	270	13	(	(	PUNCT
cana-540	270	14	n	n	CCONJ
cana-540	270	15	)	)	PUNCT
cana-540	271	1			PROPN
cana-540	271	2	w.	w.	PROPN
cana-540	271	3	the	the	DET
cana-540	271	4	family	family	NOUN
cana-540	271	5	{	{	PUNCT
cana-540	271	6	v	v	NOUN
cana-540	271	7	(	(	PUNCT
cana-540	271	8	n	n	CCONJ
cana-540	271	9	)	)	PUNCT
cana-540	271	10	:	:	PUNCT
cana-540	272	1	n	n	CCONJ
cana-540	272	2			NOUN
cana-540	272	3	p(p	p(p	NOUN
cana-540	272	4	)	)	PUNCT
cana-540	272	5	}	}	PUNCT
cana-540	272	6	is	be	AUX
cana-540	272	7	an	an	DET
cana-540	272	8	open	open	ADJ
cana-540	272	9	cover	cover	NOUN
cana-540	272	10	of	of	ADP
cana-540	272	11	p(p	p(p	NOUN
cana-540	272	12	)	)	PUNCT
cana-540	272	13	.	.	PUNCT
cana-540	273	1	as	as	ADP
cana-540	273	2	p(p	p(p	NOUN
cana-540	273	3	)	)	PUNCT
cana-540	273	4	is	be	AUX
cana-540	273	5	compact	compact	ADJ
cana-540	273	6	,	,	PUNCT
cana-540	273	7	there	there	PRON
cana-540	273	8	exists	exist	VERB
cana-540	273	9	finite	finite	ADJ
cana-540	273	10	number	number	NOUN
cana-540	273	11	of	of	ADP
cana-540	273	12	points	point	NOUN
cana-540	273	13	say	say	VERB
cana-540	273	14	n1	n1	NOUN
cana-540	273	15	,	,	PUNCT
cana-540	273	16	n2	n2	NOUN
cana-540	273	17	,	,	PUNCT
cana-540	273	18	…	…	PUNCT
cana-540	273	19	.	.	PUNCT
cana-540	274	1	nk	nk	PROPN
cana-540	274	2	in	in	ADP
cana-540	274	3	p(p	p(p	NOUN
cana-540	274	4	)	)	PUNCT
cana-540	275	1	such	such	ADJ
cana-540	275	2	that	that	DET
cana-540	275	3	p(p	p(p	NOUN
cana-540	275	4	)	)	PUNCT
cana-540	276	1			PROPN
cana-540	276	2	{	{	PUNCT
cana-540	276	3	v(ni	v(ni	PROPN
cana-540	276	4	):	):	PUNCT
cana-540	276	5	i	i	PRON
cana-540	276	6	=	=	NOUN
cana-540	276	7	1	1	NUM
cana-540	276	8	;	;	PUNCT
cana-540	276	9	2	2	NUM
cana-540	276	10	;	;	PUNCT
cana-540	276	11	…	…	PUNCT
cana-540	276	12	.k	.k	NOUN
cana-540	276	13	}	}	PUNCT
cana-540	276	14	.	.	PUNCT
cana-540	277	1	put	put	VERB
cana-540	277	2	u	u	NOUN
cana-540	277	3	=	=	NOUN
cana-540	277	4			X
cana-540	277	5	{	{	PUNCT
cana-540	277	6	u(ni	u(ni	PROPN
cana-540	277	7	):	):	PUNCT
cana-540	277	8	i	i	PRON
cana-540	277	9	=	=	PUNCT
cana-540	277	10	i=1,2	i=1,2	ADJ
cana-540	277	11	…	…	X
cana-540	277	12	.k	.k	NOUN
cana-540	277	13	}	}	PUNCT
cana-540	277	14	and	and	CCONJ
cana-540	277	15	v	v	X
cana-540	277	16	=	=	SYM
cana-540	277	17			NOUN
cana-540	277	18	{	{	PUNCT
cana-540	277	19	v	v	NOUN
cana-540	277	20	(	(	PUNCT
cana-540	277	21	ni	ni	PROPN
cana-540	277	22	):	):	PUNCT
cana-540	277	23	i	i	PROPN
cana-540	277	24	=	=	SYM
cana-540	277	25	1,2,	1,2,	NUM
cana-540	277	26	…	…	SYM
cana-540	277	27	k	k	NOUN
cana-540	277	28	}	}	PUNCT
cana-540	277	29	.	.	PUNCT
cana-540	278	1	then	then	ADV
cana-540	278	2	u	u	NOUN
cana-540	278	3	and	and	CCONJ
cana-540	278	4	v	v	NOUN
cana-540	278	5	are	be	AUX
cana-540	278	6	open	open	ADJ
cana-540	278	7	sets	set	NOUN
cana-540	278	8	in	in	ADP
cana-540	278	9	r	r	NOUN
cana-540	278	10	and	and	CCONJ
cana-540	278	11	s	s	NOUN
cana-540	278	12	respectively	respectively	ADV
cana-540	278	13	and	and	CCONJ
cana-540	278	14	{	{	PUNCT
cana-540	278	15	p	p	X
cana-540	278	16	}	}	PUNCT
cana-540	278	17	x	x	SYM
cana-540	278	18	p(p	p(p	NOUN
cana-540	278	19	)	)	PUNCT
cana-540	279	1			PROPN
cana-540	279	2	u	u	NOUN
cana-540	279	3	x	x	X
cana-540	279	4	v	v	ADP
cana-540	279	5			PROPN
cana-540	279	6	w.	w.	PROPN
cana-540	279	7	as	as	SCONJ
cana-540	279	8	p	p	PROPN
cana-540	279	9	is	be	AUX
cana-540	279	10	u.spgα.c	u.spgα.c	NOUN
cana-540	279	11	there	there	ADV
cana-540	279	12	exists	exist	VERB
cana-540	279	13	u	u	NOUN
cana-540	279	14			NOUN
cana-540	279	15	spgαcommunications	spgαcommunication	NOUN
cana-540	279	16	on	on	ADP
cana-540	279	17	applied	apply	VERB
cana-540	279	18	nonlinear	nonlinear	ADJ
cana-540	279	19	analysis	analysis	NOUN
cana-540	279	20	issn	issn	NOUN
cana-540	279	21	:	:	PUNCT
cana-540	279	22	1074	1074	NUM
cana-540	279	23	-	-	PUNCT
cana-540	279	24	133x	133x	NUM
cana-540	279	25	vol	vol	NOUN
cana-540	279	26	31	31	NUM
cana-540	279	27	no	no	NOUN
cana-540	279	28	.	.	NOUN
cana-540	279	29	2	2	NUM
cana-540	279	30	(	(	PUNCT
cana-540	279	31	2024	2024	NUM
cana-540	279	32	)	)	PUNCT
cana-540	279	33	254	254	NUM
cana-540	279	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	279	35	o(r	o(r	PROPN
cana-540	279	36	,	,	PUNCT
cana-540	279	37	p	p	NOUN
cana-540	279	38	)	)	PUNCT
cana-540	279	39	such	such	ADJ
cana-540	279	40	that	that	DET
cana-540	279	41	p(u1	p(u1	NOUN
cana-540	279	42	)	)	PUNCT
cana-540	280	1			PROPN
cana-540	280	2	v	v	NOUN
cana-540	280	3	.	.	PUNCT
cana-540	281	1	b	b	X
cana-540	281	2	lemma	lemma	PROPN
cana-540	281	3	2.1	2.1	NUM
cana-540	281	4	,	,	PUNCT
cana-540	281	5	u	u	NOUN
cana-540	281	6			PUNCT
cana-540	281	7	u1	u1	NOUN
cana-540	281	8			PROPN
cana-540	281	9	u	u	PROPN
cana-540	281	10			X
cana-540	281	11	p+(v	p+(v	PROPN
cana-540	281	12	)	)	PUNCT
cana-540	281	13	=	=	PUNCT
cana-540	281	14	g+p	g+p	X
cana-540	281	15	(	(	PUNCT
cana-540	281	16	u	u	NOUN
cana-540	281	17	x	x	NOUN
cana-540	281	18	v	v	NOUN
cana-540	281	19	)	)	PUNCT
cana-540	281	20			PROPN
cana-540	281	21	g+p	g+p	PROPN
cana-540	281	22	(	(	PUNCT
cana-540	281	23	w	w	NOUN
cana-540	281	24	)	)	PUNCT
cana-540	281	25	.	.	PUNCT
cana-540	282	1	thus	thus	ADV
cana-540	282	2	u	u	PRON
cana-540	282	3			PUNCT
cana-540	282	4	u1	u1	PROPN
cana-540	282	5			PROPN
cana-540	282	6	spgα	spgα	NOUN
cana-540	282	7	-	-	PUNCT
cana-540	282	8	o(r	o(r	PROPN
cana-540	282	9	,	,	PUNCT
cana-540	282	10	p	p	NOUN
cana-540	282	11	)	)	PUNCT
cana-540	282	12	and	and	CCONJ
cana-540	282	13	gp	gp	NOUN
cana-540	282	14	(	(	PUNCT
cana-540	282	15	u	u	NOUN
cana-540	282	16			NOUN
cana-540	282	17	v	v	NOUN
cana-540	282	18	)	)	PUNCT
cana-540	283	1			PROPN
cana-540	283	2	w.	w.	PROPN
cana-540	283	3	hence	hence	ADV
cana-540	283	4	gp	gp	PROPN
cana-540	283	5	is	be	AUX
cana-540	283	6	u.spgα.c	u.spgα.c	PROPN
cana-540	283	7	.	.	PUNCT
cana-540	284	1	theorem	theorem	VERB
cana-540	284	2	2.17	2.17	NUM
cana-540	284	3	.	.	PUNCT
cana-540	285	1	a	a	DET
cana-540	285	2	m.f	m.f	NOUN
cana-540	285	3	p	p	X
cana-540	285	4	:	:	PUNCT
cana-540	285	5	r	r	NOUN
cana-540	285	6	→	→	SYM
cana-540	285	7	s	s	PART
cana-540	285	8	is	be	AUX
cana-540	285	9	if	if	SCONJ
cana-540	285	10	and	and	CCONJ
cana-540	285	11	only	only	ADV
cana-540	285	12	if	if	SCONJ
cana-540	285	13	gp	gp	NOUN
cana-540	285	14	is	be	AUX
cana-540	285	15	l.spgα.c	l.spgα.c	PROPN
cana-540	285	16	.	.	PUNCT
cana-540	285	17	proof	proof	NOUN
cana-540	285	18	.	.	PUNCT
cana-540	286	1	let	let	VERB
cana-540	286	2	p	p	PRON
cana-540	286	3	be	be	AUX
cana-540	286	4	l.spgα.c	l.spgα.c	PROPN
cana-540	286	5	.	.	PUNCT
cana-540	287	1	let	let	VERB
cana-540	287	2	p	p	PRON
cana-540	287	3			PROPN
cana-540	287	4	r	r	NOUN
cana-540	287	5	and	and	CCONJ
cana-540	287	6	w	w	PROPN
cana-540	287	7			PROPN
cana-540	287	8	o(r	o(r	PROPN
cana-540	287	9	x	x	SYM
cana-540	287	10	s	s	X
cana-540	287	11	)	)	PUNCT
cana-540	287	12	with	with	ADP
cana-540	287	13	p	p	PROPN
cana-540	287	14			NOUN
cana-540	287	15	g	g	NOUN
cana-540	287	16	-	-	PUNCT
cana-540	287	17	p	p	X
cana-540	287	18	(	(	PUNCT
cana-540	287	19	w	w	NOUN
cana-540	287	20	)	)	PUNCT
cana-540	287	21	.	.	PUNCT
cana-540	288	1	as	as	ADP
cana-540	288	2	w	w	PROPN
cana-540	288	3			X
cana-540	288	4	(	(	PUNCT
cana-540	288	5	{	{	PUNCT
cana-540	288	6	p	p	X
cana-540	288	7	}	}	PUNCT
cana-540	288	8	x	x	SYM
cana-540	288	9	p(p	p(p	NOUN
cana-540	288	10	)	)	PUNCT
cana-540	288	11	)	)	PUNCT
cana-540	288	12			NOUN
cana-540	288	13			NOUN
cana-540	288	14	,	,	PUNCT
cana-540	288	15	there	there	PRON
cana-540	288	16	exists	exist	VERB
cana-540	288	17	n	n	DET
cana-540	288	18			NOUN
cana-540	288	19	p(p	p(p	NOUN
cana-540	288	20	)	)	PUNCT
cana-540	288	21	with	with	ADP
cana-540	288	22	(	(	PUNCT
cana-540	288	23	p	p	X
cana-540	288	24	,	,	PUNCT
cana-540	288	25	n	n	CCONJ
cana-540	288	26	)	)	PUNCT
cana-540	288	27			NOUN
cana-540	288	28	w	w	NOUN
cana-540	289	1	and	and	CCONJ
cana-540	289	2	so	so	ADV
cana-540	289	3	(	(	PUNCT
cana-540	289	4	p	p	X
cana-540	289	5	,	,	PUNCT
cana-540	289	6	n	n	CCONJ
cana-540	289	7	)	)	PUNCT
cana-540	289	8			NOUN
cana-540	289	9	u	u	NOUN
cana-540	289	10	x	x	SYM
cana-540	289	11	v	v	ADP
cana-540	289	12			PROPN
cana-540	289	13	w	w	PROPN
cana-540	289	14	,	,	PUNCT
cana-540	289	15	where	where	SCONJ
cana-540	289	16	u	u	PRON
cana-540	289	17			PROPN
cana-540	289	18	o(r	o(r	PROPN
cana-540	289	19	)	)	PUNCT
cana-540	289	20	and	and	CCONJ
cana-540	289	21	v	v	ADP
cana-540	289	22			PROPN
cana-540	289	23	o(s	o(s	PROPN
cana-540	289	24	)	)	PUNCT
cana-540	289	25	respectively	respectively	ADV
cana-540	289	26	.	.	PUNCT
cana-540	290	1	since	since	SCONJ
cana-540	290	2	p(p	p(p	NOUN
cana-540	290	3	)	)	PUNCT
cana-540	290	4			NOUN
cana-540	290	5	v	v	ADP
cana-540	290	6			NOUN
cana-540	290	7			NOUN
cana-540	290	8	,	,	PUNCT
cana-540	290	9	there	there	PRON
cana-540	290	10	exists	exist	VERB
cana-540	290	11	g	g	PROPN
cana-540	290	12			PROPN
cana-540	290	13	spgα	spgα	PROPN
cana-540	290	14	-	-	PUNCT
cana-540	290	15	o(r	o(r	PROPN
cana-540	290	16	,	,	PUNCT
cana-540	290	17	p	p	NOUN
cana-540	290	18	)	)	PUNCT
cana-540	290	19	with	with	ADP
cana-540	290	20	g	g	PROPN
cana-540	290	21			PROPN
cana-540	290	22	p(v	p(v	PROPN
cana-540	290	23	)	)	PUNCT
cana-540	290	24	.	.	PUNCT
cana-540	291	1	by	by	ADP
cana-540	291	2	lemma	lemma	PROPN
cana-540	291	3	2.1	2.1	NUM
cana-540	291	4	,	,	PUNCT
cana-540	291	5	u	u	NOUN
cana-540	291	6			PUNCT
cana-540	291	7	g	g	PROPN
cana-540	291	8			PROPN
cana-540	291	9	u	u	PROPN
cana-540	291	10			PUNCT
cana-540	291	11	p-(v	p-(v	NOUN
cana-540	291	12	)	)	PUNCT
cana-540	292	1			PROPN
cana-540	292	2	g	g	PROPN
cana-540	292	3	-	-	PUNCT
cana-540	292	4	p	p	X
cana-540	292	5	(	(	PUNCT
cana-540	292	6	u	u	NOUN
cana-540	292	7	x	x	PROPN
cana-540	292	8	v	v	NOUN
cana-540	292	9	)	)	PUNCT
cana-540	292	10			PROPN
cana-540	292	11	g	g	PROPN
cana-540	292	12	-	-	PUNCT
cana-540	292	13	p	p	X
cana-540	292	14	(	(	PUNCT
cana-540	292	15	w	w	NOUN
cana-540	292	16	)	)	PUNCT
cana-540	292	17	.	.	PUNCT
cana-540	293	1	thus	thus	ADV
cana-540	293	2	p	p	X
cana-540	293	3			PROPN
cana-540	293	4	u	u	NOUN
cana-540	293	5	x	x	PROPN
cana-540	293	6	g	g	PROPN
cana-540	293	7			NOUN
cana-540	293	8	spgαo(r	spgαo(r	NOUN
cana-540	293	9	,	,	PUNCT
cana-540	293	10	p	p	NOUN
cana-540	293	11	)	)	PUNCT
cana-540	293	12	and	and	CCONJ
cana-540	293	13	hence	hence	ADV
cana-540	293	14	gp	gp	NOUN
cana-540	293	15	is	be	AUX
cana-540	293	16	l.spgα.c	l.spgα.c	PROPN
cana-540	293	17	.	.	PUNCT
cana-540	293	18	conversely	conversely	ADV
cana-540	293	19	,	,	PUNCT
cana-540	293	20	let	let	VERB
cana-540	293	21	gp	gp	NOUN
cana-540	293	22	be	be	AUX
cana-540	293	23	l.spgα.c	l.spgα.c	PROPN
cana-540	293	24	,	,	PUNCT
cana-540	293	25	p	p	NOUN
cana-540	293	26			NOUN
cana-540	293	27	r	r	NOUN
cana-540	293	28	and	and	CCONJ
cana-540	293	29	v	v	ADP
cana-540	293	30			PROPN
cana-540	293	31	o(s	o(s	PROPN
cana-540	293	32	)	)	PUNCT
cana-540	293	33	with	with	ADP
cana-540	293	34	p	p	PROPN
cana-540	293	35			PROPN
cana-540	293	36	p-(v	p-(v	NOUN
cana-540	293	37	)	)	PUNCT
cana-540	293	38	.	.	PUNCT
cana-540	294	1	so	so	ADV
cana-540	294	2	r	r	NOUN
cana-540	294	3	x	x	SYM
cana-540	294	4	v	v	X
cana-540	294	5			NOUN
cana-540	294	6	o(r	o(r	PROPN
cana-540	294	7	x	x	SYM
cana-540	294	8	s	s	X
cana-540	294	9	)	)	PUNCT
cana-540	294	10	with	with	ADP
cana-540	294	11	gp	gp	NOUN
cana-540	294	12	(	(	PUNCT
cana-540	294	13	p	p	NOUN
cana-540	294	14	)	)	PUNCT
cana-540	294	15			NOUN
cana-540	294	16	(	(	PUNCT
cana-540	294	17	r	r	NOUN
cana-540	294	18	x	x	SYM
cana-540	294	19	v	v	NOUN
cana-540	294	20	)	)	PUNCT
cana-540	294	21	=	=	SYM
cana-540	294	22	(	(	PUNCT
cana-540	294	23	{	{	PUNCT
cana-540	294	24	p	p	X
cana-540	294	25	}	}	PUNCT
cana-540	294	26	x	x	SYM
cana-540	294	27	p(p	p(p	NOUN
cana-540	294	28	)	)	PUNCT
cana-540	294	29	)	)	PUNCT
cana-540	295	1			NOUN
cana-540	295	2	(	(	PUNCT
cana-540	295	3	r	r	NOUN
cana-540	295	4	x	x	SYM
cana-540	295	5	v	v	NOUN
cana-540	295	6	)	)	PUNCT
cana-540	295	7	=	=	SYM
cana-540	295	8	(	(	PUNCT
cana-540	295	9	{	{	PUNCT
cana-540	295	10	p	p	X
cana-540	295	11	}	}	PUNCT
cana-540	295	12	x	x	X
cana-540	295	13	(	(	PUNCT
cana-540	295	14	p(p	p(p	NOUN
cana-540	295	15	)	)	PUNCT
cana-540	295	16			NUM
cana-540	295	17	v	v	NOUN
cana-540	295	18	)	)	PUNCT
cana-540	295	19			NOUN
cana-540	295	20	.	.	PUNCT
cana-540	295	21	since	since	SCONJ
cana-540	295	22	gp	gp	NOUN
cana-540	295	23	is	be	AUX
cana-540	295	24	l.spgα.c	l.spgα.c	PROPN
cana-540	295	25	,	,	PUNCT
cana-540	295	26	there	there	PRON
cana-540	295	27	exists	exist	VERB
cana-540	295	28	u	u	PRON
cana-540	295	29			PROPN
cana-540	295	30	spgα	spgα	VERB
cana-540	295	31	o(r	o(r	PROPN
cana-540	295	32	,	,	PUNCT
cana-540	295	33	p	p	NOUN
cana-540	295	34	)	)	PUNCT
cana-540	295	35	with	with	ADP
cana-540	295	36	u	u	NOUN
cana-540	295	37			PROPN
cana-540	295	38	(	(	PUNCT
cana-540	295	39	r	r	NOUN
cana-540	295	40	x	x	SYM
cana-540	295	41	v	v	NOUN
cana-540	295	42	)	)	PUNCT
cana-540	295	43	.	.	PUNCT
cana-540	296	1	from	from	ADP
cana-540	296	2	lemma	lemma	PROPN
cana-540	296	3	2.1	2.1	NUM
cana-540	296	4	,	,	PUNCT
cana-540	296	5	u	u	NOUN
cana-540	296	6			PROPN
cana-540	296	7	p-(v	p-(v	PROPN
cana-540	296	8	)	)	PUNCT
cana-540	297	1	and	and	CCONJ
cana-540	297	2	so	so	ADV
cana-540	297	3	p	p	PROPN
cana-540	297	4	is	be	AUX
cana-540	297	5	l.spgα.c	l.spgα.c	PROPN
cana-540	297	6	.	.	PUNCT
cana-540	298	1	3	3	X
cana-540	298	2	.	.	X
cana-540	298	3	upper	upper	ADJ
cana-540	298	4	(	(	PUNCT
cana-540	298	5	lower	low	ADJ
cana-540	298	6	)	)	PUNCT
cana-540	298	7	spgα	spgα	ADJ
cana-540	298	8	-	-	PUNCT
cana-540	298	9	irresolute	irresolute	ADJ
cana-540	298	10	multifunctions	multifunction	NOUN
cana-540	298	11	definition	definition	NOUN
cana-540	298	12	3.1	3.1	NUM
cana-540	298	13	.	.	PUNCT
cana-540	299	1	a	a	DET
cana-540	299	2	m.f	m.f	NOUN
cana-540	299	3	p	p	X
cana-540	299	4	:	:	PUNCT
cana-540	299	5	r	r	NOUN
cana-540	299	6	→	→	SYM
cana-540	299	7	s	s	PART
cana-540	299	8	is	be	AUX
cana-540	299	9	called	call	VERB
cana-540	299	10	1	1	NUM
cana-540	299	11	.	.	PUNCT
cana-540	300	1	upper	upper	ADJ
cana-540	300	2	spgα	spgα	NOUN
cana-540	300	3	-	-	PUNCT
cana-540	300	4	irresolute	irresolute	ADJ
cana-540	300	5	(	(	PUNCT
cana-540	300	6	briefly	briefly	NOUN
cana-540	300	7	u.spgα.i	u.spgα.i	NOUN
cana-540	300	8	)	)	PUNCT
cana-540	300	9	if	if	SCONJ
cana-540	300	10	for	for	ADP
cana-540	300	11	each	each	DET
cana-540	300	12	p	p	NOUN
cana-540	300	13			NOUN
cana-540	300	14	r	r	NOUN
cana-540	300	15	and	and	CCONJ
cana-540	300	16	each	each	PRON
cana-540	300	17	v	v	PROPN
cana-540	300	18			PROPN
cana-540	300	19	spgα	spgα	NOUN
cana-540	300	20	o(s	o(s	PROPN
cana-540	300	21	,	,	PUNCT
cana-540	300	22	p(p	p(p	NOUN
cana-540	300	23	)	)	PUNCT
cana-540	300	24	)	)	PUNCT
cana-540	300	25	,	,	PUNCT
cana-540	300	26	there	there	PRON
cana-540	300	27	exists	exist	VERB
cana-540	300	28	u	u	PRON
cana-540	300	29			PROPN
cana-540	300	30	spgα	spgα	VERB
cana-540	300	31	o(r	o(r	PROPN
cana-540	300	32	,	,	PUNCT
cana-540	300	33	p	p	NOUN
cana-540	300	34	)	)	PUNCT
cana-540	300	35	such	such	ADJ
cana-540	300	36	that	that	DET
cana-540	300	37	p(u	p(u	NOUN
cana-540	300	38	)	)	PUNCT
cana-540	301	1			PROPN
cana-540	301	2	v	v	NOUN
cana-540	301	3	.	.	PUNCT
cana-540	302	1	2	2	X
cana-540	302	2	.	.	X
cana-540	302	3	lower	low	ADJ
cana-540	302	4	spgα	spgα	NOUN
cana-540	302	5	-	-	PUNCT
cana-540	302	6	irresolute	irresolute	ADJ
cana-540	302	7	(	(	PUNCT
cana-540	302	8	briefly	briefly	NOUN
cana-540	302	9	l.spgα.i	l.spgα.i	NOUN
cana-540	302	10	)	)	PUNCT
cana-540	302	11	if	if	SCONJ
cana-540	302	12	for	for	ADP
cana-540	302	13	each	each	DET
cana-540	302	14	p	p	NOUN
cana-540	302	15			NOUN
cana-540	302	16	r	r	NOUN
cana-540	302	17	and	and	CCONJ
cana-540	302	18	each	each	DET
cana-540	302	19	spgα	spgα	ADJ
cana-540	302	20	-	-	PUNCT
cana-540	302	21	open	open	NOUN
cana-540	302	22	set	set	VERB
cana-540	302	23	v	v	NOUN
cana-540	302	24	with	with	ADP
cana-540	302	25	p(p	p(p	NOUN
cana-540	302	26	)	)	PUNCT
cana-540	302	27			NOUN
cana-540	302	28	v	v	ADP
cana-540	302	29			NOUN
cana-540	302	30			NOUN
cana-540	303	1	,	,	PUNCT
cana-540	303	2	there	there	PRON
cana-540	303	3	exists	exist	VERB
cana-540	303	4	u	u	PRON
cana-540	303	5			PROPN
cana-540	303	6	spgα	spgα	VERB
cana-540	303	7	o(r	o(r	PROPN
cana-540	303	8	,	,	PUNCT
cana-540	303	9	p	p	NOUN
cana-540	303	10	)	)	PUNCT
cana-540	303	11	such	such	ADJ
cana-540	303	12	that	that	SCONJ
cana-540	303	13	u	u	NOUN
cana-540	303	14			PROPN
cana-540	303	15	p-(v	p-(v	PROPN
cana-540	303	16	)	)	PUNCT
cana-540	303	17	.	.	PUNCT
cana-540	304	1	theorem	theorem	VERB
cana-540	304	2	3.1	3.1	NUM
cana-540	304	3	.	.	PUNCT
cana-540	305	1	for	for	ADP
cana-540	305	2	a	a	DET
cana-540	305	3	m.f	m.f	NOUN
cana-540	305	4	p	p	X
cana-540	305	5	:	:	PUNCT
cana-540	305	6	r	r	NOUN
cana-540	305	7	→	→	SYM
cana-540	305	8	s	s	VERB
cana-540	305	9	the	the	DET
cana-540	305	10	following	following	ADJ
cana-540	305	11	statements	statement	NOUN
cana-540	305	12	are	be	AUX
cana-540	305	13	equivalent	equivalent	ADJ
cana-540	305	14	:	:	PUNCT
cana-540	306	1	1	1	X
cana-540	306	2	.	.	X
cana-540	306	3	p	p	NOUN
cana-540	306	4	is	be	AUX
cana-540	306	5	u.spgα.i	u.spgα.i	NOUN
cana-540	306	6	.	.	PUNCT
cana-540	307	1	2	2	NUM
cana-540	307	2	.	.	X
cana-540	307	3	for	for	ADP
cana-540	307	4	each	each	DET
cana-540	307	5	p	p	NOUN
cana-540	307	6			PROPN
cana-540	307	7	r	r	NOUN
cana-540	307	8	,	,	PUNCT
cana-540	307	9	for	for	ADP
cana-540	307	10	each	each	DET
cana-540	307	11	spgα	spgα	NOUN
cana-540	307	12	-	-	PUNCT
cana-540	307	13	nbd	nbd	PROPN
cana-540	307	14	.	.	PUNCT
cana-540	308	1	v	v	NOUN
cana-540	308	2	of	of	ADP
cana-540	308	3	p(p	p(p	NOUN
cana-540	308	4	)	)	PUNCT
cana-540	309	1	,	,	PUNCT
cana-540	309	2	we	we	PRON
cana-540	309	3	have	have	AUX
cana-540	309	4	p+(v	p+(v	PROPN
cana-540	309	5	)	)	PUNCT
cana-540	309	6	is	be	AUX
cana-540	309	7	spgαnbd	spgαnbd	PROPN
cana-540	309	8	.	.	PUNCT
cana-540	310	1	of	of	ADP
cana-540	310	2	p.	p.	NOUN
cana-540	310	3	3	3	NUM
cana-540	310	4	.	.	PUNCT
cana-540	311	1	for	for	ADP
cana-540	311	2	each	each	DET
cana-540	311	3	p	p	NOUN
cana-540	311	4			PROPN
cana-540	311	5	r	r	NOUN
cana-540	311	6	,	,	PUNCT
cana-540	311	7	for	for	ADP
cana-540	311	8	each	each	DET
cana-540	311	9	spgα	spgα	NOUN
cana-540	311	10	-	-	PUNCT
cana-540	311	11	nbd	nbd	PROPN
cana-540	311	12	.	.	PUNCT
cana-540	311	13	v	v	NOUN
cana-540	311	14	of	of	ADP
cana-540	311	15	p(p	p(p	NOUN
cana-540	311	16	)	)	PUNCT
cana-540	311	17	,	,	PUNCT
cana-540	311	18	there	there	PRON
cana-540	311	19	exists	exist	VERB
cana-540	311	20	spgα	spgα	ADJ
cana-540	311	21	-	-	PUNCT
cana-540	311	22	nbd	nbd	PROPN
cana-540	311	23	.	.	PUNCT
cana-540	312	1	u	u	PROPN
cana-540	312	2	of	of	ADP
cana-540	312	3	p	p	NOUN
cana-540	312	4	with	with	ADP
cana-540	312	5	p(u	p(u	NOUN
cana-540	312	6	)	)	PUNCT
cana-540	313	1			PROPN
cana-540	313	2	v	v	NOUN
cana-540	313	3	.	.	PUNCT
cana-540	314	1	4	4	X
cana-540	314	2	.	.	X
cana-540	314	3	p+(v	p+(v	PROPN
cana-540	314	4	)	)	PUNCT
cana-540	314	5			NOUN
cana-540	314	6	spgαo(r	spgαo(r	NOUN
cana-540	314	7	)	)	PUNCT
cana-540	314	8	for	for	ADP
cana-540	314	9	each	each	DET
cana-540	314	10	v	v	PROPN
cana-540	314	11			PROPN
cana-540	314	12	spgα	spgα	NOUN
cana-540	314	13	o(s	o(s	PROPN
cana-540	314	14	)	)	PUNCT
cana-540	314	15	.	.	PUNCT
cana-540	315	1	5	5	X
cana-540	315	2	.	.	NUM
cana-540	315	3	p-(v	p-(v	NUM
cana-540	315	4	)	)	PUNCT
cana-540	315	5			NOUN
cana-540	315	6	spgα	spgα	VERB
cana-540	315	7	c(r	c(r	NOUN
cana-540	315	8	)	)	PUNCT
cana-540	315	9	for	for	ADP
cana-540	315	10	each	each	DET
cana-540	315	11	v	v	ADP
cana-540	315	12			NOUN
cana-540	315	13	sgα	sgα	NUM
cana-540	315	14	c(s	c(	NOUN
cana-540	315	15	)	)	PUNCT
cana-540	315	16	.	.	PUNCT
cana-540	316	1	6	6	X
cana-540	316	2	.	.	X
cana-540	316	3	spgα	spgα	PROPN
cana-540	316	4	cl(p-(b	cl(p-(b	NUM
cana-540	316	5	)	)	PUNCT
cana-540	316	6	)	)	PUNCT
cana-540	317	1			PROPN
cana-540	317	2	p-(spgα	p-(spgα	ADJ
cana-540	317	3	cl(b	cl(b	NOUN
cana-540	317	4	)	)	PUNCT
cana-540	317	5	)	)	PUNCT
cana-540	317	6	for	for	ADP
cana-540	317	7	each	each	DET
cana-540	317	8	b	b	NOUN
cana-540	317	9			PROPN
cana-540	317	10	s.	s.	PROPN
cana-540	317	11	proof	proof	PROPN
cana-540	317	12	.	.	PUNCT
cana-540	318	1	(	(	PUNCT
cana-540	318	2	1	1	X
cana-540	318	3	)	)	PUNCT
cana-540	318	4	→	→	X
cana-540	318	5	(	(	PUNCT
cana-540	318	6	2	2	NUM
cana-540	318	7	):	):	PUNCT
cana-540	318	8	let	let	VERB
cana-540	318	9	p	p	PRON
cana-540	318	10			PROPN
cana-540	318	11	r	r	NOUN
cana-540	318	12	and	and	CCONJ
cana-540	318	13	w	w	PROPN
cana-540	318	14	be	be	AUX
cana-540	318	15	spgα	spgα	ADJ
cana-540	318	16	-	-	PUNCT
cana-540	318	17	nbd	nbd	PROPN
cana-540	318	18	.	.	PUNCT
cana-540	318	19	of	of	ADP
cana-540	318	20	p(p	p(p	NOUN
cana-540	318	21	)	)	PUNCT
cana-540	318	22	.	.	PUNCT
cana-540	319	1	then	then	ADV
cana-540	319	2	there	there	PRON
cana-540	319	3	exists	exist	VERB
cana-540	319	4	v	v	ADP
cana-540	319	5			PROPN
cana-540	319	6	spgα	spgα	NOUN
cana-540	319	7	o(s	o(s	PROPN
cana-540	319	8	)	)	PUNCT
cana-540	319	9	with	with	ADP
cana-540	319	10	p(p	p(p	NOUN
cana-540	319	11	)	)	PUNCT
cana-540	320	1			PROPN
cana-540	320	2	v	v	ADP
cana-540	320	3			PROPN
cana-540	320	4	w.	w.	PROPN
cana-540	320	5	as	as	SCONJ
cana-540	320	6	p	p	PROPN
cana-540	320	7	is	be	AUX
cana-540	320	8	u.spgα.i	u.spgα.i	NOUN
cana-540	320	9	,	,	PUNCT
cana-540	320	10	there	there	PRON
cana-540	320	11	exists	exist	VERB
cana-540	320	12	u	u	PRON
cana-540	320	13			PROPN
cana-540	320	14	spgα	spgα	VERB
cana-540	320	15	o(r	o(r	PROPN
cana-540	320	16	,	,	PUNCT
cana-540	320	17	p	p	NOUN
cana-540	320	18	)	)	PUNCT
cana-540	320	19	such	such	ADJ
cana-540	320	20	that	that	DET
cana-540	320	21	p(u	p(u	NOUN
cana-540	320	22	)	)	PUNCT
cana-540	321	1			PROPN
cana-540	321	2	v	v	NOUN
cana-540	321	3	.	.	PUNCT
cana-540	322	1	thus	thus	ADV
cana-540	322	2	p	p	X
cana-540	322	3			PROPN
cana-540	322	4	u	u	NOUN
cana-540	322	5			PROPN
cana-540	322	6	p+(v	p+(v	PROPN
cana-540	322	7	)	)	PUNCT
cana-540	323	1			PROPN
cana-540	323	2	p+(w	p+(w	PROPN
cana-540	323	3	)	)	PUNCT
cana-540	323	4	and	and	CCONJ
cana-540	323	5	so	so	ADV
cana-540	323	6	p+(w	p+(w	NOUN
cana-540	323	7	)	)	PUNCT
cana-540	323	8	is	be	AUX
cana-540	323	9	a	a	DET
cana-540	323	10	spgα	spgα	ADJ
cana-540	323	11	-	-	PUNCT
cana-540	323	12	nbd	nbd	PROPN
cana-540	323	13	.	.	PUNCT
cana-540	324	1	of	of	ADP
cana-540	324	2	p.	p.	NOUN
cana-540	324	3	(	(	PUNCT
cana-540	324	4	2	2	NUM
cana-540	324	5	)	)	PUNCT
cana-540	324	6	→	→	X
cana-540	324	7	(	(	PUNCT
cana-540	324	8	3	3	NUM
cana-540	324	9	):	):	PUNCT
cana-540	324	10	let	let	VERB
cana-540	324	11	p	p	PRON
cana-540	324	12			PROPN
cana-540	324	13	r	r	NOUN
cana-540	324	14	and	and	CCONJ
cana-540	324	15	v	v	NOUN
cana-540	324	16	be	be	AUX
cana-540	324	17	a	a	DET
cana-540	324	18	spgα	spgα	ADJ
cana-540	324	19	-	-	PUNCT
cana-540	324	20	nbd	nbd	PROPN
cana-540	324	21	.	.	PUNCT
cana-540	325	1	of	of	ADP
cana-540	325	2	p(p	p(p	NOUN
cana-540	325	3	)	)	PUNCT
cana-540	325	4	.	.	PUNCT
cana-540	326	1	let	let	VERB
cana-540	326	2	u	u	PRON
cana-540	326	3	=	=	NOUN
cana-540	326	4	p+(v	p+(v	PROPN
cana-540	326	5	)	)	PUNCT
cana-540	326	6	.	.	PUNCT
cana-540	327	1	then	then	ADV
cana-540	327	2	by	by	ADP
cana-540	327	3	(	(	PUNCT
cana-540	327	4	2	2	NUM
cana-540	327	5	)	)	PUNCT
cana-540	327	6	,	,	PUNCT
cana-540	327	7	u	u	PROPN
cana-540	327	8	is	be	AUX
cana-540	327	9	spgαnbd	spgαnbd	NOUN
cana-540	327	10	of	of	ADP
cana-540	327	11	p	p	NOUN
cana-540	327	12	with	with	ADP
cana-540	327	13	p(u	p(u	NOUN
cana-540	327	14	)	)	PUNCT
cana-540	328	1			PROPN
cana-540	328	2	v	v	ADP
cana-540	328	3	.	.	PUNCT
cana-540	329	1	(	(	PUNCT
cana-540	329	2	3	3	NUM
cana-540	329	3	)	)	PUNCT
cana-540	329	4	→	→	X
cana-540	329	5	(	(	PUNCT
cana-540	329	6	4	4	NUM
cana-540	329	7	):	):	PUNCT
cana-540	329	8	let	let	VERB
cana-540	329	9	v	v	ADP
cana-540	329	10			PROPN
cana-540	329	11	spgα	spgα	NOUN
cana-540	329	12	-o(s	-o(s	PROPN
cana-540	329	13	)	)	PUNCT
cana-540	329	14	and	and	CCONJ
cana-540	329	15	p	p	X
cana-540	329	16			PROPN
cana-540	329	17	p+(v	p+(v	PROPN
cana-540	329	18	)	)	PUNCT
cana-540	329	19	.	.	PUNCT
cana-540	330	1	then	then	ADV
cana-540	330	2	there	there	PRON
cana-540	330	3	exists	exist	VERB
cana-540	330	4	a	a	DET
cana-540	330	5	spgα	spgα	ADJ
cana-540	330	6	nbd	nbd	PROPN
cana-540	330	7	g	g	PROPN
cana-540	330	8	of	of	ADP
cana-540	330	9	p	p	NOUN
cana-540	330	10	with	with	ADP
cana-540	330	11	p(g	p(g	NOUN
cana-540	330	12	)	)	PUNCT
cana-540	331	1			PROPN
cana-540	331	2	v	v	NOUN
cana-540	331	3	.	.	PUNCT
cana-540	332	1	thus	thus	ADV
cana-540	332	2	for	for	ADP
cana-540	332	3	some	some	DET
cana-540	332	4	u	u	NOUN
cana-540	332	5			NOUN
cana-540	332	6	spgα	spgα	VERB
cana-540	332	7	o(r	o(r	PROPN
cana-540	332	8	,	,	PUNCT
cana-540	332	9	p	p	NOUN
cana-540	332	10	)	)	PUNCT
cana-540	332	11	with	with	ADP
cana-540	332	12	u	u	NOUN
cana-540	332	13			PROPN
cana-540	332	14	g	g	PROPN
cana-540	332	15	and	and	CCONJ
cana-540	332	16	p(u	p(u	NOUN
cana-540	332	17	)	)	PUNCT
cana-540	333	1			PROPN
cana-540	333	2	v	v	X
cana-540	333	3	.	.	PUNCT
cana-540	334	1	so	so	ADV
cana-540	334	2	p	p	ADP
cana-540	334	3			PROPN
cana-540	334	4	u	u	NOUN
cana-540	334	5			PROPN
cana-540	334	6	p+(v	p+(v	PROPN
cana-540	334	7	)	)	PUNCT
cana-540	334	8	and	and	CCONJ
cana-540	334	9	hence	hence	ADV
cana-540	334	10	p+(v	p+(v	PROPN
cana-540	334	11	)	)	PUNCT
cana-540	334	12			NOUN
cana-540	334	13	spgα	spgα	NOUN
cana-540	334	14	o(s	o(s	PROPN
cana-540	334	15	)	)	PUNCT
cana-540	334	16	.	.	PUNCT
cana-540	335	1	(	(	PUNCT
cana-540	335	2	4	4	NUM
cana-540	335	3	)	)	PUNCT
cana-540	335	4	→	→	X
cana-540	335	5	(	(	PUNCT
cana-540	335	6	5	5	NUM
cana-540	335	7	):	):	PUNCT
cana-540	335	8	let	let	VERB
cana-540	335	9	a	a	DET
cana-540	335	10			NOUN
cana-540	335	11	spgα	spgα	NOUN
cana-540	335	12	c(s	c(	NOUN
cana-540	335	13	)	)	PUNCT
cana-540	335	14	and	and	CCONJ
cana-540	335	15	so	so	ADV
cana-540	335	16	r	r	NOUN
cana-540	335	17	–	–	PUNCT
cana-540	335	18	p-(a	p-(a	NOUN
cana-540	335	19	)	)	PUNCT
cana-540	336	1	=	=	SYM
cana-540	336	2	p+(s	p+(s	NUM
cana-540	336	3	k	k	NOUN
cana-540	336	4	)	)	PUNCT
cana-540	336	5			NOUN
cana-540	336	6	spgα	spgα	VERB
cana-540	336	7	o(r	o(r	PROPN
cana-540	336	8	)	)	PUNCT
cana-540	336	9	.	.	PUNCT
cana-540	337	1	thus	thus	ADV
cana-540	337	2	p-(a	p-(a	NOUN
cana-540	337	3	)	)	PUNCT
cana-540	337	4			NOUN
cana-540	337	5	spgα	spgα	NOUN
cana-540	337	6	-	-	PUNCT
cana-540	337	7	c(s	c(	NOUN
cana-540	337	8	)	)	PUNCT
cana-540	337	9	.	.	PUNCT
cana-540	338	1	communications	communication	NOUN
cana-540	338	2	on	on	ADP
cana-540	338	3	applied	apply	VERB
cana-540	338	4	nonlinear	nonlinear	ADJ
cana-540	338	5	analysis	analysis	NOUN
cana-540	338	6	issn	issn	NOUN
cana-540	338	7	:	:	PUNCT
cana-540	338	8	1074	1074	NUM
cana-540	338	9	-	-	PUNCT
cana-540	338	10	133x	133x	NUM
cana-540	338	11	vol	vol	NOUN
cana-540	338	12	31	31	NUM
cana-540	338	13	no	no	NOUN
cana-540	338	14	.	.	NOUN
cana-540	338	15	2	2	NUM
cana-540	338	16	(	(	PUNCT
cana-540	338	17	2024	2024	NUM
cana-540	338	18	)	)	PUNCT
cana-540	338	19	255	255	NUM
cana-540	338	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	338	21	(	(	PUNCT
cana-540	338	22	5	5	NUM
cana-540	338	23	)	)	PUNCT
cana-540	338	24	→	→	X
cana-540	338	25	(	(	PUNCT
cana-540	338	26	6	6	NUM
cana-540	338	27	):	):	PUNCT
cana-540	338	28	let	let	VERB
cana-540	338	29	b	b	PROPN
cana-540	338	30			PROPN
cana-540	338	31	s.	s.	PROPN
cana-540	338	32	since	since	SCONJ
cana-540	338	33	spgα	spgα	ADJ
cana-540	338	34	cl(b	cl(b	NOUN
cana-540	338	35	)	)	PUNCT
cana-540	338	36	is	be	AUX
cana-540	338	37	spgα	spgα	ADJ
cana-540	338	38	-	-	PUNCT
cana-540	338	39	closed	close	VERB
cana-540	338	40	in	in	ADP
cana-540	338	41	s	s	PROPN
cana-540	338	42	,	,	PUNCT
cana-540	338	43	so	so	ADV
cana-540	338	44	p-(spgα	p-(spgα	ADJ
cana-540	338	45	cl(b	cl(b	NOUN
cana-540	338	46	)	)	PUNCT
cana-540	338	47	)	)	PUNCT
cana-540	339	1			PROPN
cana-540	339	2	spgα	spgα	NOUN
cana-540	339	3	-c(r	-c(r	PROPN
cana-540	339	4	)	)	PUNCT
cana-540	339	5	with	with	ADP
cana-540	339	6	p-(b	p-(b	ADJ
cana-540	339	7	)	)	PUNCT
cana-540	339	8	)	)	PUNCT
cana-540	340	1			PROPN
cana-540	340	2	p-(spgα	p-(spgα	ADJ
cana-540	340	3	cl(b	cl(b	NOUN
cana-540	340	4	)	)	PUNCT
cana-540	340	5	.	.	PUNCT
cana-540	341	1	thus	thus	ADV
cana-540	341	2	spgαcl(p-(b	spgαcl(p-(b	PROPN
cana-540	341	3	)	)	PUNCT
cana-540	341	4	)	)	PUNCT
cana-540	342	1			PROPN
cana-540	342	2	p-(spgαcl(b	p-(spgαcl(b	PROPN
cana-540	342	3	)	)	PUNCT
cana-540	342	4	)	)	PUNCT
cana-540	342	5	.	.	PUNCT
cana-540	343	1	(	(	PUNCT
cana-540	343	2	6	6	NUM
cana-540	343	3	)	)	PUNCT
cana-540	343	4	→	→	X
cana-540	343	5	(	(	PUNCT
cana-540	343	6	1	1	NUM
cana-540	343	7	):	):	PUNCT
cana-540	343	8	let	let	VERB
cana-540	343	9	p	p	PRON
cana-540	343	10			PROPN
cana-540	343	11	r	r	NOUN
cana-540	343	12	and	and	CCONJ
cana-540	343	13	v	v	ADP
cana-540	343	14			NOUN
cana-540	343	15	spgαo(r	spgαo(r	PUNCT
cana-540	343	16	)	)	PUNCT
cana-540	343	17	with	with	ADP
cana-540	343	18	p(p	p(p	NOUN
cana-540	343	19	)	)	PUNCT
cana-540	344	1			PROPN
cana-540	344	2	v	v	NOUN
cana-540	344	3	.	.	PUNCT
cana-540	345	1	so	so	ADV
cana-540	345	2	p(p	p(p	ADV
cana-540	345	3	)	)	PUNCT
cana-540	345	4			NOUN
cana-540	345	5	(	(	PUNCT
cana-540	345	6	s	s	NOUN
cana-540	345	7	v	v	NOUN
cana-540	345	8	)	)	PUNCT
cana-540	345	9	=	=	SYM
cana-540	345	10	.	.	X
cana-540	345	11	hence	hence	ADV
cana-540	345	12	p	p	X
cana-540	345	13			NUM
cana-540	345	14	p-(s	p-(s	NOUN
cana-540	345	15	v	v	NOUN
cana-540	345	16	)	)	PUNCT
cana-540	345	17	.	.	PUNCT
cana-540	346	1	from	from	ADP
cana-540	346	2	(	(	PUNCT
cana-540	346	3	6	6	NUM
cana-540	346	4	)	)	PUNCT
cana-540	346	5	,	,	PUNCT
cana-540	346	6	p	p	NOUN
cana-540	346	7			PROPN
cana-540	346	8	spgα	spgα	NOUN
cana-540	346	9	cl(p-(s	cl(p-(s	NUM
cana-540	346	10	v	v	NOUN
cana-540	346	11	)	)	PUNCT
cana-540	346	12	)	)	PUNCT
cana-540	347	1	and	and	CCONJ
cana-540	347	2	so	so	ADV
cana-540	347	3	there	there	PRON
cana-540	347	4	exists	exist	VERB
cana-540	347	5	u	u	PRON
cana-540	347	6			PROPN
cana-540	347	7	spgα	spgα	NOUN
cana-540	347	8	-	-	PUNCT
cana-540	347	9	o(r	o(r	PROPN
cana-540	347	10	,	,	PUNCT
cana-540	347	11	p	p	NOUN
cana-540	347	12	)	)	PUNCT
cana-540	347	13	such	such	ADJ
cana-540	347	14	that	that	SCONJ
cana-540	347	15	u	u	NOUN
cana-540	347	16			VERB
cana-540	347	17	p-(s	p-(s	NOUN
cana-540	347	18	v	v	NOUN
cana-540	347	19	)	)	PUNCT
cana-540	347	20	=	=	PUNCT
cana-540	347	21	.	.	X
cana-540	347	22	thus	thus	ADV
cana-540	347	23	p(u	p(u	X
cana-540	347	24	)	)	PUNCT
cana-540	348	1			PROPN
cana-540	348	2	v	v	NOUN
cana-540	349	1	and	and	CCONJ
cana-540	349	2	so	so	ADV
cana-540	349	3	p	p	PROPN
cana-540	349	4	is	be	AUX
cana-540	349	5	u.spgα.i	u.spgα.i	PROPN
cana-540	349	6	.	.	PUNCT
cana-540	350	1	theorem	theorem	VERB
cana-540	350	2	3.2	3.2	NUM
cana-540	350	3	.	.	PUNCT
cana-540	351	1	the	the	DET
cana-540	351	2	following	follow	VERB
cana-540	351	3	statements	statement	NOUN
cana-540	351	4	are	be	AUX
cana-540	351	5	equivalent	equivalent	ADJ
cana-540	351	6	for	for	ADP
cana-540	351	7	a	a	DET
cana-540	351	8	m.f	m.f	NOUN
cana-540	351	9	p	p	X
cana-540	351	10	:	:	PUNCT
cana-540	351	11	r	r	NOUN
cana-540	351	12	→	→	SYM
cana-540	351	13	s	s	NOUN
cana-540	351	14	:	:	PUNCT
cana-540	351	15	1	1	NUM
cana-540	351	16	.	.	X
cana-540	352	1	p	p	NOUN
cana-540	352	2	is	be	AUX
cana-540	352	3	l.spgα.i	l.spgα.i	NOUN
cana-540	352	4	.	.	PUNCT
cana-540	353	1	2	2	X
cana-540	353	2	.	.	X
cana-540	353	3	for	for	ADP
cana-540	353	4	each	each	DET
cana-540	353	5	v	v	ADJ
cana-540	353	6			PROPN
cana-540	353	7	spgα	spgα	NOUN
cana-540	353	8	-	-	PUNCT
cana-540	353	9	o(s	o(s	NOUN
cana-540	353	10	)	)	PUNCT
cana-540	353	11	and	and	CCONJ
cana-540	353	12	each	each	DET
cana-540	353	13	p	p	ADJ
cana-540	353	14			PROPN
cana-540	353	15	p-(v	p-(v	PROPN
cana-540	353	16	)	)	PUNCT
cana-540	353	17	,	,	PUNCT
cana-540	353	18	there	there	PRON
cana-540	353	19	exists	exist	VERB
cana-540	353	20	u	u	PRON
cana-540	353	21			PROPN
cana-540	353	22	spgα	spgα	VERB
cana-540	353	23	o(r	o(r	PROPN
cana-540	353	24	,	,	PUNCT
cana-540	353	25	p	p	NOUN
cana-540	353	26	)	)	PUNCT
cana-540	353	27	such	such	ADJ
cana-540	353	28	that	that	SCONJ
cana-540	353	29	u	u	NOUN
cana-540	353	30			PROPN
cana-540	353	31	p-(v	p-(v	PROPN
cana-540	353	32	)	)	PUNCT
cana-540	353	33	.	.	PUNCT
cana-540	354	1	3	3	X
cana-540	354	2	.	.	X
cana-540	354	3	p-(v	p-(v	NUM
cana-540	354	4	)	)	PUNCT
cana-540	354	5			NOUN
cana-540	354	6	spgαo(r	spgαo(r	NOUN
cana-540	354	7	)	)	PUNCT
cana-540	354	8	for	for	ADP
cana-540	354	9	each	each	DET
cana-540	354	10	v	v	ADJ
cana-540	354	11			PROPN
cana-540	354	12	spgα	spgα	NOUN
cana-540	354	13	-	-	PUNCT
cana-540	354	14	o(s	o(s	NOUN
cana-540	354	15	)	)	PUNCT
cana-540	354	16	.	.	PUNCT
cana-540	355	1	4	4	X
cana-540	355	2	.	.	X
cana-540	355	3	p+(k	p+(k	NOUN
cana-540	355	4	)	)	PUNCT
cana-540	355	5			NOUN
cana-540	355	6	spgα	spgα	VERB
cana-540	355	7	c(r	c(r	NOUN
cana-540	355	8	)	)	PUNCT
cana-540	355	9	for	for	ADP
cana-540	355	10	each	each	DET
cana-540	355	11	k	k	PROPN
cana-540	355	12			PROPN
cana-540	355	13	spgα	spgα	NOUN
cana-540	355	14	-	-	PUNCT
cana-540	355	15	c(s	c(	NOUN
cana-540	355	16	)	)	PUNCT
cana-540	355	17	.	.	PUNCT
cana-540	356	1	5	5	X
cana-540	356	2	.	.	X
cana-540	356	3	for	for	ADP
cana-540	356	4	each	each	PRON
cana-540	356	5	a	a	DET
cana-540	356	6			PROPN
cana-540	356	7	r	r	PROPN
cana-540	356	8	,	,	PUNCT
cana-540	356	9	p(spgα	p(spgα	NOUN
cana-540	356	10	-	-	PUNCT
cana-540	356	11	cl(a	cl(a	NUM
cana-540	356	12	)	)	PUNCT
cana-540	356	13	)	)	PUNCT
cana-540	357	1			PROPN
cana-540	357	2	spgα	spgα	NOUN
cana-540	357	3	-	-	PUNCT
cana-540	357	4	cl(p(a	cl(p(a	NOUN
cana-540	357	5	)	)	PUNCT
cana-540	357	6	)	)	PUNCT
cana-540	357	7	.	.	PUNCT
cana-540	358	1	6	6	X
cana-540	358	2	.	.	X
cana-540	358	3	spgα	spgα	ADJ
cana-540	358	4	-	-	PUNCT
cana-540	358	5	cl(p+(b	cl(p+(b	NOUN
cana-540	358	6	)	)	PUNCT
cana-540	358	7	)	)	PUNCT
cana-540	359	1			PROPN
cana-540	359	2	p+(spgα	p+(spgα	NOUN
cana-540	359	3	-	-	PUNCT
cana-540	359	4	cl(b	cl(b	NOUN
cana-540	359	5	)	)	PUNCT
cana-540	359	6	)	)	PUNCT
cana-540	359	7	for	for	ADP
cana-540	359	8	each	each	DET
cana-540	359	9	b	b	NOUN
cana-540	359	10			PROPN
cana-540	359	11	s.	s.	PROPN
cana-540	359	12	proof	proof	PROPN
cana-540	359	13	.	.	PUNCT
cana-540	360	1	(	(	PUNCT
cana-540	360	2	1	1	X
cana-540	360	3	)	)	PUNCT
cana-540	360	4	→	→	X
cana-540	360	5	(	(	PUNCT
cana-540	360	6	2	2	NUM
cana-540	360	7	):	):	PUNCT
cana-540	360	8	follows	follow	VERB
cana-540	360	9	by	by	ADP
cana-540	360	10	the	the	DET
cana-540	360	11	definition	definition	NOUN
cana-540	360	12	.	.	PUNCT
cana-540	361	1	(	(	PUNCT
cana-540	361	2	2	2	NUM
cana-540	361	3	)	)	PUNCT
cana-540	361	4	→	→	X
cana-540	361	5	(	(	PUNCT
cana-540	361	6	3	3	NUM
cana-540	361	7	):	):	PUNCT
cana-540	361	8	let	let	VERB
cana-540	361	9	v	v	ADP
cana-540	361	10			PROPN
cana-540	361	11	spgα	spgα	NOUN
cana-540	361	12	-	-	PUNCT
cana-540	361	13	o(s	o(s	NOUN
cana-540	361	14	)	)	PUNCT
cana-540	361	15	with	with	ADP
cana-540	361	16	p	p	PROPN
cana-540	361	17			PROPN
cana-540	361	18	p-(v	p-(v	NOUN
cana-540	361	19	)	)	PUNCT
cana-540	361	20	.	.	PUNCT
cana-540	362	1	from	from	ADP
cana-540	362	2	(	(	PUNCT
cana-540	362	3	2	2	X
cana-540	362	4	)	)	PUNCT
cana-540	362	5	there	there	PRON
cana-540	362	6	exists	exist	VERB
cana-540	362	7	u	u	NOUN
cana-540	362	8			PROPN
cana-540	362	9	spgα	spgα	NOUN
cana-540	362	10	-	-	PUNCT
cana-540	362	11	o(r	o(r	PROPN
cana-540	362	12	,	,	PUNCT
cana-540	362	13	p	p	NOUN
cana-540	362	14	)	)	PUNCT
cana-540	362	15	such	such	ADJ
cana-540	362	16	that	that	SCONJ
cana-540	362	17	u	u	NOUN
cana-540	362	18			PROPN
cana-540	362	19	p-(v	p-(v	PROPN
cana-540	362	20	)	)	PUNCT
cana-540	362	21	.	.	PUNCT
cana-540	363	1	thus	thus	ADV
cana-540	363	2	,	,	PUNCT
cana-540	363	3	p	p	ADJ
cana-540	363	4			PROPN
cana-540	363	5	u	u	NOUN
cana-540	363	6			PROPN
cana-540	363	7	cl(int(u	cl(int(u	PROPN
cana-540	363	8	)	)	PUNCT
cana-540	363	9	)	)	PUNCT
cana-540	363	10			NOUN
cana-540	363	11	int(cl(u	int(cl(u	PROPN
cana-540	363	12	)	)	PUNCT
cana-540	363	13	)	)	PUNCT
cana-540	364	1			PROPN
cana-540	364	2	cl(int(p-(u	cl(int(p-(u	PROPN
cana-540	364	3	)	)	PUNCT
cana-540	364	4	)	)	PUNCT
cana-540	364	5	)	)	PUNCT
cana-540	364	6			NOUN
cana-540	364	7	int(cl(p-(u	int(cl(p-(u	PROPN
cana-540	364	8	)	)	PUNCT
cana-540	364	9	)	)	PUNCT
cana-540	364	10	)	)	PUNCT
cana-540	364	11	.	.	PUNCT
cana-540	365	1	so	so	ADV
cana-540	365	2	p(v	p(v	NOUN
cana-540	365	3	)	)	PUNCT
cana-540	365	4			NOUN
cana-540	365	5	spgα	spgα	NOUN
cana-540	365	6	-	-	PUNCT
cana-540	365	7	o(r	o(r	PROPN
cana-540	365	8	)	)	PUNCT
cana-540	365	9	.	.	PUNCT
cana-540	366	1	(	(	PUNCT
cana-540	366	2	3	3	X
cana-540	366	3	)	)	PUNCT
cana-540	366	4	→	→	X
cana-540	366	5	(	(	PUNCT
cana-540	366	6	4	4	NUM
cana-540	366	7	):	):	PUNCT
cana-540	366	8	let	let	VERB
cana-540	366	9	k	k	PROPN
cana-540	366	10			PROPN
cana-540	366	11	spgα	spgα	NOUN
cana-540	366	12	-	-	PUNCT
cana-540	366	13	c(s	c(	NOUN
cana-540	366	14	)	)	PUNCT
cana-540	366	15	.	.	PUNCT
cana-540	367	1	then	then	ADV
cana-540	367	2	r	r	PROPN
cana-540	367	3	p+(k	p+(k	PROPN
cana-540	367	4	)	)	PUNCT
cana-540	367	5	=	=	VERB
cana-540	367	6	p-(s	p-(s	ADJ
cana-540	367	7	k	k	PROPN
cana-540	367	8	)	)	PUNCT
cana-540	367	9			NOUN
cana-540	367	10	spgα	spgα	NOUN
cana-540	367	11	-	-	PUNCT
cana-540	367	12	o(r	o(r	NOUN
cana-540	367	13	)	)	PUNCT
cana-540	367	14	and	and	CCONJ
cana-540	367	15	so	so	ADV
cana-540	367	16	p+(k	p+(k	ADJ
cana-540	367	17	)	)	PUNCT
cana-540	367	18			NOUN
cana-540	367	19	spgα	spgα	NOUN
cana-540	367	20	-	-	PUNCT
cana-540	367	21	c(r	c(r	NOUN
cana-540	367	22	)	)	PUNCT
cana-540	367	23	.	.	PUNCT
cana-540	368	1	(	(	PUNCT
cana-540	368	2	4	4	NUM
cana-540	368	3	)	)	PUNCT
cana-540	368	4	→	→	X
cana-540	368	5	(	(	PUNCT
cana-540	368	6	5	5	NUM
cana-540	368	7	)	)	PUNCT
cana-540	368	8	and	and	CCONJ
cana-540	368	9	(	(	PUNCT
cana-540	368	10	5	5	NUM
cana-540	368	11	)	)	PUNCT
cana-540	368	12	→	→	X
cana-540	368	13	(	(	PUNCT
cana-540	368	14	6	6	NUM
cana-540	368	15	)	)	PUNCT
cana-540	368	16	are	be	AUX
cana-540	368	17	straight	straight	ADV
cana-540	368	18	forward	forward	ADV
cana-540	368	19	.	.	PUNCT
cana-540	369	1	(	(	PUNCT
cana-540	369	2	6	6	NUM
cana-540	369	3	)	)	PUNCT
cana-540	369	4	→	→	X
cana-540	369	5	(	(	PUNCT
cana-540	369	6	1	1	NUM
cana-540	369	7	):	):	PUNCT
cana-540	369	8	let	let	VERB
cana-540	369	9	p	p	PRON
cana-540	369	10			PROPN
cana-540	369	11	r	r	NOUN
cana-540	369	12	and	and	CCONJ
cana-540	369	13	v	v	ADP
cana-540	369	14			PROPN
cana-540	369	15	spgα	spgα	NOUN
cana-540	369	16	-	-	PUNCT
cana-540	369	17	o(s	o(s	NOUN
cana-540	369	18	)	)	PUNCT
cana-540	369	19	with	with	ADP
cana-540	369	20	p(p	p(p	NOUN
cana-540	369	21	)	)	PUNCT
cana-540	369	22			PROPN
cana-540	369	23	v	v	NUM
cana-540	369	24			NOUN
cana-540	369	25	.	.	PUNCT
cana-540	369	26	so	so	ADV
cana-540	369	27	p(p	p(p	NOUN
cana-540	369	28	)	)	PUNCT
cana-540	369	29			NOUN
cana-540	369	30	(	(	PUNCT
cana-540	369	31	s	s	NOUN
cana-540	369	32	v	v	NOUN
cana-540	369	33	)	)	PUNCT
cana-540	369	34	=	=	PUNCT
cana-540	369	35	.	.	PUNCT
cana-540	369	36	then	then	ADV
cana-540	369	37	p(p	p(p	NOUN
cana-540	369	38	)	)	PUNCT
cana-540	369	39			PROPN
cana-540	369	40	s	s	NOUN
cana-540	369	41	v	v	NOUN
cana-540	369	42	and	and	CCONJ
cana-540	369	43	p	p	NOUN
cana-540	369	44			NOUN
cana-540	369	45	p+(s	p+(s	PROPN
cana-540	369	46	v	v	NOUN
cana-540	369	47	)	)	PUNCT
cana-540	369	48	.	.	PUNCT
cana-540	370	1	since	since	SCONJ
cana-540	370	2	s	s	PROPN
cana-540	370	3	v	v	PROPN
cana-540	370	4			NOUN
cana-540	370	5	spgα	spgα	NOUN
cana-540	370	6	c(s	c(	NOUN
cana-540	370	7	)	)	PUNCT
cana-540	370	8	and	and	CCONJ
cana-540	370	9	by	by	ADP
cana-540	370	10	(	(	PUNCT
cana-540	370	11	6	6	NUM
cana-540	370	12	)	)	PUNCT
cana-540	370	13	,	,	PUNCT
cana-540	370	14	p	p	NOUN
cana-540	370	15			NOUN
cana-540	370	16	spgα	spgα	ADJ
cana-540	370	17	cl(p+(s	cl(p+(s	PROPN
cana-540	370	18	v	v	NOUN
cana-540	370	19	)	)	PUNCT
cana-540	370	20	)	)	PUNCT
cana-540	371	1	and	and	CCONJ
cana-540	371	2	so	so	ADV
cana-540	371	3	there	there	PRON
cana-540	371	4	exists	exist	VERB
cana-540	371	5	u	u	PRON
cana-540	371	6			PROPN
cana-540	371	7	spgα	spgα	VERB
cana-540	371	8	o(r	o(r	PROPN
cana-540	371	9	,	,	PUNCT
cana-540	371	10	p	p	NOUN
cana-540	371	11	)	)	PUNCT
cana-540	371	12	with	with	ADP
cana-540	371	13	u	u	NOUN
cana-540	371	14			NOUN
cana-540	371	15	p-(s	p-(s	NOUN
cana-540	371	16	v	v	NOUN
cana-540	371	17	)	)	PUNCT
cana-540	371	18	=	=	SYM
cana-540	371	19	u	u	NOUN
cana-540	371	20			PUNCT
cana-540	371	21	(	(	PUNCT
cana-540	371	22	r	r	NOUN
cana-540	371	23	–	–	PUNCT
cana-540	371	24	p-(v	p-(v	NOUN
cana-540	371	25	)	)	PUNCT
cana-540	371	26	)	)	PUNCT
cana-540	372	1	=	=	PUNCT
cana-540	372	2	.	.	X
cana-540	372	3	thus	thus	ADV
cana-540	372	4	u	u	X
cana-540	372	5			PROPN
cana-540	372	6	r	r	PROPN
cana-540	372	7	(	(	PUNCT
cana-540	372	8	r	r	NOUN
cana-540	372	9	–	–	PUNCT
cana-540	372	10	p-(v	p-(v	NOUN
cana-540	372	11	)	)	PUNCT
cana-540	372	12	)	)	PUNCT
cana-540	373	1	=	=	SYM
cana-540	373	2	p-(v	p-(v	NOUN
cana-540	373	3	)	)	PUNCT
cana-540	373	4	,	,	PUNCT
cana-540	373	5	that	that	PRON
cana-540	373	6	is	be	AUX
cana-540	373	7	u	u	NOUN
cana-540	373	8			PROPN
cana-540	373	9	p-(v	p-(v	PROPN
cana-540	373	10	)	)	PUNCT
cana-540	373	11	.	.	PUNCT
cana-540	374	1	so	so	ADV
cana-540	374	2	p	p	PROPN
cana-540	374	3	is	be	AUX
cana-540	374	4	l.spgα.i	l.spgα.i	PROPN
cana-540	374	5	.	.	PUNCT
cana-540	375	1	lemma	lemma	PROPN
cana-540	375	2	3.1	3.1	NUM
cana-540	375	3	.	.	PUNCT
cana-540	376	1	let	let	VERB
cana-540	376	2	p	p	PRON
cana-540	376	3	be	be	AUX
cana-540	376	4	a	a	DET
cana-540	376	5	m.f	m.f	PROPN
cana-540	376	6	.	.	PUNCT
cana-540	377	1	then	then	ADV
cana-540	377	2	(	(	PUNCT
cana-540	377	3	spgα	spgα	ADJ
cana-540	377	4	cl(p))-(v	cl(p))-(v	NOUN
cana-540	377	5	)	)	PUNCT
cana-540	377	6	=	=	SYM
cana-540	377	7	p-(v	p-(v	NOUN
cana-540	377	8	)	)	PUNCT
cana-540	377	9	for	for	ADP
cana-540	377	10	each	each	DET
cana-540	377	11	v	v	NOUN
cana-540	377	12			PROPN
cana-540	377	13	spgα	spgα	VERB
cana-540	377	14	o(r	o(r	PROPN
cana-540	377	15	)	)	PUNCT
cana-540	377	16	.	.	PUNCT
cana-540	378	1	proof	proof	NOUN
cana-540	378	2	.	.	PUNCT
cana-540	379	1	let	let	VERB
cana-540	379	2	v	v	ADP
cana-540	379	3			PROPN
cana-540	379	4	spgα	spgα	NOUN
cana-540	379	5	o(s	o(s	PROPN
cana-540	379	6	)	)	PUNCT
cana-540	379	7	with	with	ADP
cana-540	379	8	p	p	PROPN
cana-540	379	9			NOUN
cana-540	379	10	(	(	PUNCT
cana-540	379	11	spgα	spgα	NOUN
cana-540	379	12	-	-	PUNCT
cana-540	379	13	cl(p))-c(v	cl(p))-c(v	NOUN
cana-540	379	14	)	)	PUNCT
cana-540	379	15	so	so	ADV
cana-540	379	16	v	v	ADP
cana-540	379	17			X
cana-540	379	18	(	(	PUNCT
cana-540	379	19	spgα	spgα	ADJ
cana-540	379	20	cl(p))(p	cl(p))(p	ADJ
cana-540	379	21	)	)	PUNCT
cana-540	379	22			NOUN
cana-540	379	23	.	.	X
cana-540	379	24	as	as	ADP
cana-540	379	25	v	v	ADP
cana-540	379	26			PROPN
cana-540	379	27	spgα	spgα	NOUN
cana-540	379	28	-	-	PUNCT
cana-540	379	29	o(s	o(s	NOUN
cana-540	379	30	)	)	PUNCT
cana-540	379	31	and	and	CCONJ
cana-540	379	32	so	so	ADV
cana-540	379	33	v	v	ADJ
cana-540	379	34			NUM
cana-540	379	35	p(p	p(p	NOUN
cana-540	379	36	)	)	PUNCT
cana-540	379	37			NOUN
cana-540	379	38	.	.	VERB
cana-540	379	39	thus	thus	ADV
cana-540	379	40	p	p	ADJ
cana-540	379	41			PROPN
cana-540	379	42	p-(v	p-(v	NOUN
cana-540	379	43	)	)	PUNCT
cana-540	379	44	.	.	PUNCT
cana-540	380	1	conversely	conversely	ADV
cana-540	380	2	,	,	PUNCT
cana-540	380	3	let	let	VERB
cana-540	380	4	p	p	PRON
cana-540	380	5			NOUN
cana-540	380	6	p-(v	p-(v	PROPN
cana-540	380	7	)	)	PUNCT
cana-540	380	8	.	.	PUNCT
cana-540	381	1	then	then	ADV
cana-540	381	2	v	v	X
cana-540	381	3			PUNCT
cana-540	381	4	p(p	p(p	NOUN
cana-540	381	5	)	)	PUNCT
cana-540	382	1			PROPN
cana-540	382	2	(	(	PUNCT
cana-540	382	3	spgαcl(p))(p	spgαcl(p))(p	NOUN
cana-540	382	4	)	)	PUNCT
cana-540	382	5			PUNCT
cana-540	382	6	v	v	NUM
cana-540	382	7			NOUN
cana-540	382	8			NOUN
cana-540	382	9	and	and	CCONJ
cana-540	382	10	so	so	ADV
cana-540	382	11	p	p	PRON
cana-540	382	12			NOUN
cana-540	382	13	(	(	PUNCT
cana-540	382	14	spgα	spgα	ADJ
cana-540	382	15	cl(p))-(v	cl(p))-(v	NOUN
cana-540	382	16	)	)	PUNCT
cana-540	382	17	.	.	PUNCT
cana-540	383	1	thus	thus	ADV
cana-540	383	2	(	(	PUNCT
cana-540	383	3	spgα	spgα	ADJ
cana-540	383	4	-	-	PUNCT
cana-540	383	5	cl(p))-(v	cl(p))-(v	NOUN
cana-540	383	6	)	)	PUNCT
cana-540	383	7	=	=	SYM
cana-540	384	1	p-(v	p-(v	NOUN
cana-540	384	2	)	)	PUNCT
cana-540	384	3	.	.	PUNCT
cana-540	385	1	lemma	lemma	PROPN
cana-540	385	2	3.2	3.2	NUM
cana-540	385	3	.	.	PUNCT
cana-540	386	1	[	[	X
cana-540	386	2	6	6	NUM
cana-540	386	3	]	]	PUNCT
cana-540	386	4	let	let	VERB
cana-540	386	5	a	a	PRON
cana-540	386	6	,	,	PUNCT
cana-540	386	7	b	b	SYM
cana-540	386	8			PROPN
cana-540	386	9	r.	r.	PROPN
cana-540	386	10	then	then	ADV
cana-540	387	1	1	1	X
cana-540	387	2	.	.	PUNCT
cana-540	388	1	if	if	SCONJ
cana-540	388	2	a	a	DET
cana-540	388	3			PROPN
cana-540	388	4	spgα	spgα	NOUN
cana-540	388	5	-	-	PUNCT
cana-540	388	6	o(r	o(r	NOUN
cana-540	388	7	)	)	PUNCT
cana-540	388	8	and	and	CCONJ
cana-540	388	9	b	b	NOUN
cana-540	388	10			NOUN
cana-540	388	11	r	r	NOUN
cana-540	388	12	then	then	ADV
cana-540	388	13	a	a	DET
cana-540	388	14			PROPN
cana-540	388	15	b	b	PROPN
cana-540	388	16			PROPN
cana-540	388	17	spgα	spgα	NOUN
cana-540	388	18	-	-	PUNCT
cana-540	388	19	o(b	o(b	PROPN
cana-540	388	20	)	)	PUNCT
cana-540	388	21	.	.	PUNCT
cana-540	389	1	2	2	X
cana-540	389	2	.	.	X
cana-540	389	3	if	if	SCONJ
cana-540	389	4	a	a	DET
cana-540	389	5			NOUN
cana-540	389	6	spgα	spgα	NOUN
cana-540	389	7	-	-	PUNCT
cana-540	389	8	o(b	o(b	PROPN
cana-540	389	9	)	)	PUNCT
cana-540	389	10	and	and	CCONJ
cana-540	389	11	b	b	PROPN
cana-540	389	12			PROPN
cana-540	389	13	spgα	spgα	NOUN
cana-540	389	14	-	-	PUNCT
cana-540	389	15	o(r	o(r	NOUN
cana-540	389	16	)	)	PUNCT
cana-540	389	17	then	then	ADV
cana-540	389	18	a	a	DET
cana-540	389	19			PROPN
cana-540	389	20	spgα	spgα	NOUN
cana-540	389	21	-	-	PUNCT
cana-540	389	22	o(r	o(r	PROPN
cana-540	389	23	)	)	PUNCT
cana-540	389	24	.	.	PUNCT
cana-540	390	1	theorem	theorem	VERB
cana-540	390	2	3.3	3.3	NUM
cana-540	390	3	.	.	PUNCT
cana-540	391	1	let	let	VERB
cana-540	391	2	p	p	PRON
cana-540	391	3	be	be	AUX
cana-540	391	4	a	a	DET
cana-540	391	5	m.f	m.f	PROPN
cana-540	391	6	.	.	PUNCT
cana-540	392	1	and	and	CCONJ
cana-540	392	2	u	u	PRON
cana-540	392	3			PROPN
cana-540	392	4	o(r	o(r	PROPN
cana-540	392	5	)	)	PUNCT
cana-540	392	6	.	.	PUNCT
cana-540	393	1	if	if	SCONJ
cana-540	393	2	p	p	NOUN
cana-540	393	3	is	be	AUX
cana-540	393	4	u.spgα.i	u.spgα.i	PROPN
cana-540	393	5	(	(	PUNCT
cana-540	393	6	resp	resp	PROPN
cana-540	393	7	l.spgα.i	l.spgα.i	PROPN
cana-540	393	8	)	)	PUNCT
cana-540	393	9	then	then	ADV
cana-540	393	10	plu	plu	PROPN
cana-540	393	11	:	:	PUNCT
cana-540	393	12	u	u	X
cana-540	393	13	→	→	SYM
cana-540	393	14	s	s	X
cana-540	393	15	is	be	AUX
cana-540	393	16	u.spgα.i	u.spgα.i	PROPN
cana-540	393	17	(	(	PUNCT
cana-540	393	18	resp	resp	PROPN
cana-540	393	19	l.spgα.i	l.spgα.i	PROPN
cana-540	393	20	)	)	PUNCT
cana-540	393	21	.	.	PUNCT
cana-540	394	1	communications	communication	NOUN
cana-540	394	2	on	on	ADP
cana-540	394	3	applied	apply	VERB
cana-540	394	4	nonlinear	nonlinear	ADJ
cana-540	394	5	analysis	analysis	NOUN
cana-540	394	6	issn	issn	NOUN
cana-540	394	7	:	:	PUNCT
cana-540	394	8	1074	1074	NUM
cana-540	394	9	-	-	PUNCT
cana-540	394	10	133x	133x	NUM
cana-540	394	11	vol	vol	NOUN
cana-540	394	12	31	31	NUM
cana-540	394	13	no	no	NOUN
cana-540	394	14	.	.	NOUN
cana-540	394	15	2	2	NUM
cana-540	394	16	(	(	PUNCT
cana-540	394	17	2024	2024	NUM
cana-540	394	18	)	)	PUNCT
cana-540	394	19	256	256	NUM
cana-540	394	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	394	21	proof	proof	NOUN
cana-540	394	22	.	.	PUNCT
cana-540	395	1	let	let	VERB
cana-540	395	2	v	v	ADP
cana-540	395	3			PROPN
cana-540	395	4	spgα	spgα	NOUN
cana-540	395	5	-	-	PUNCT
cana-540	395	6	o(s	o(s	NOUN
cana-540	395	7	)	)	PUNCT
cana-540	395	8	and	and	CCONJ
cana-540	395	9	p	p	X
cana-540	395	10			PROPN
cana-540	395	11	u	u	NOUN
cana-540	395	12	,	,	PUNCT
cana-540	395	13	p	p	NOUN
cana-540	395	14			NOUN
cana-540	395	15	p	p	NOUN
cana-540	395	16	-	-	PUNCT
cana-540	395	17	lu(v	lu(v	NOUN
cana-540	395	18	)	)	PUNCT
cana-540	395	19	.	.	PUNCT
cana-540	396	1	as	as	SCONJ
cana-540	396	2	p	p	PROPN
cana-540	396	3	is	be	AUX
cana-540	396	4	l.spgα.i	l.spgα.i	NOUN
cana-540	396	5	there	there	ADV
cana-540	396	6	exists	exist	VERB
cana-540	396	7	g	g	PROPN
cana-540	396	8			PROPN
cana-540	396	9	spgαo(r	spgαo(r	NOUN
cana-540	396	10	,	,	PUNCT
cana-540	396	11	r	r	NOUN
cana-540	396	12	)	)	PUNCT
cana-540	396	13	with	with	ADP
cana-540	396	14	g	g	PROPN
cana-540	396	15			PROPN
cana-540	396	16	p-(v	p-(v	PROPN
cana-540	396	17	)	)	PUNCT
cana-540	397	1	and	and	CCONJ
cana-540	397	2	so	so	ADV
cana-540	397	3	p	p	PRON
cana-540	397	4			PROPN
cana-540	397	5	g	g	PROPN
cana-540	397	6			PUNCT
cana-540	397	7	u	u	NOUN
cana-540	397	8			NOUN
cana-540	397	9	spgα	spgα	NOUN
cana-540	397	10	-	-	PUNCT
cana-540	397	11	o(u	o(u	ADJ
cana-540	397	12	)	)	PUNCT
cana-540	397	13	with	with	ADP
cana-540	397	14	g	g	PROPN
cana-540	397	15			PUNCT
cana-540	397	16	u	u	NOUN
cana-540	397	17			PROPN
cana-540	397	18	plu	plu	PROPN
cana-540	397	19	(	(	PUNCT
cana-540	397	20	v	v	NOUN
cana-540	397	21	)	)	PUNCT
cana-540	397	22	.	.	PUNCT
cana-540	398	1	thus	thus	ADV
cana-540	398	2	plu	plu	PROPN
cana-540	398	3	is	be	AUX
cana-540	398	4	l.spgα.i	l.spgα.i	PROPN
cana-540	398	5	.	.	PUNCT
cana-540	399	1	similarlly	similarlly	ADV
cana-540	399	2	,	,	PUNCT
cana-540	399	3	we	we	PRON
cana-540	399	4	can	can	AUX
cana-540	399	5	prove	prove	VERB
cana-540	399	6	for	for	ADP
cana-540	399	7	u.spgα.i	u.spgα.i	NOUN
cana-540	399	8	.	.	PUNCT
cana-540	400	1	definition	definition	NOUN
cana-540	400	2	3.2	3.2	NUM
cana-540	400	3	.	.	PUNCT
cana-540	401	1	a	a	DET
cana-540	401	2	subset	subset	NOUN
cana-540	401	3	b	b	NOUN
cana-540	401	4	of	of	ADP
cana-540	401	5	a	a	DET
cana-540	401	6	space	space	NOUN
cana-540	401	7	r	r	NOUN
cana-540	401	8	is	be	AUX
cana-540	401	9	said	say	VERB
cana-540	401	10	to	to	PART
cana-540	401	11	be	be	AUX
cana-540	401	12	1	1	NUM
cana-540	401	13	.	.	PUNCT
cana-540	402	1	spgα	spgα	ADJ
cana-540	402	2	-	-	PUNCT
cana-540	402	3	compact	compact	ADJ
cana-540	402	4	relative	relative	NOUN
cana-540	402	5	to	to	ADP
cana-540	402	6	r	r	NOUN
cana-540	402	7	(	(	PUNCT
cana-540	402	8	resp	resp	NOUN
cana-540	402	9	.	.	PUNCT
cana-540	403	1	spgα	spgα	ADJ
cana-540	403	2	-	-	PUNCT
cana-540	403	3	lindelof	lindelof	NOUN
cana-540	403	4	relative	relative	ADJ
cana-540	403	5	to	to	ADP
cana-540	403	6	r	r	NOUN
cana-540	403	7	)	)	PUNCT
cana-540	403	8	if	if	SCONJ
cana-540	403	9	every	every	DET
cana-540	403	10	cover	cover	NOUN
cana-540	403	11	of	of	ADP
cana-540	403	12	b	b	NOUN
cana-540	403	13	by	by	ADP
cana-540	403	14	spgα	spgα	ADJ
cana-540	403	15	-	-	PUNCT
cana-540	403	16	open	open	ADJ
cana-540	403	17	sets	set	NOUN
cana-540	403	18	of	of	ADP
cana-540	403	19	r	r	NOUN
cana-540	403	20	has	have	VERB
cana-540	403	21	a	a	DET
cana-540	403	22	finite	finite	NOUN
cana-540	403	23	(	(	PUNCT
cana-540	403	24	resp	resp	NOUN
cana-540	403	25	.	.	PUNCT
cana-540	404	1	countable	countable	ADJ
cana-540	404	2	)	)	PUNCT
cana-540	404	3	subcover	subcover	PROPN
cana-540	404	4	.	.	PUNCT
cana-540	405	1	2	2	X
cana-540	405	2	.	.	X
cana-540	405	3	spgα	spgα	ADJ
cana-540	405	4	-	-	PUNCT
cana-540	405	5	compact	compact	ADJ
cana-540	405	6	(	(	PUNCT
cana-540	405	7	spgα	spgα	ADJ
cana-540	405	8	-	-	PUNCT
cana-540	405	9	lindelof	lindelof	NOUN
cana-540	405	10	)	)	PUNCT
cana-540	405	11	if	if	SCONJ
cana-540	405	12	r	r	NOUN
cana-540	405	13	is	be	AUX
cana-540	405	14	spgα	spgα	ADJ
cana-540	405	15	-	-	PUNCT
cana-540	405	16	compact	compact	ADJ
cana-540	405	17	(	(	PUNCT
cana-540	405	18	resp	resp	NOUN
cana-540	405	19	spgα	spgα	ADJ
cana-540	405	20	-	-	PUNCT
cana-540	405	21	lindelof	lindelof	NOUN
cana-540	405	22	)	)	PUNCT
cana-540	405	23	relative	relative	NOUN
cana-540	405	24	to	to	ADP
cana-540	405	25	r.	r.	PROPN
cana-540	405	26	theorem	theorem	VERB
cana-540	405	27	3.4	3.4	NUM
cana-540	405	28	.	.	PUNCT
cana-540	406	1	if	if	SCONJ
cana-540	406	2	p	p	PRON
cana-540	406	3	be	be	VERB
cana-540	406	4	u.spgα.i	u.spgα.i	PROPN
cana-540	406	5	m.f	m.f	PROPN
cana-540	406	6	and	and	CCONJ
cana-540	406	7	p(p	p(p	NOUN
cana-540	406	8	)	)	PUNCT
cana-540	406	9	is	be	AUX
cana-540	406	10	spgα	spgα	ADJ
cana-540	406	11	-	-	PUNCT
cana-540	406	12	compact	compact	ADJ
cana-540	406	13	relative	relative	NOUN
cana-540	406	14	to	to	ADP
cana-540	406	15	s	s	PRON
cana-540	406	16	for	for	ADP
cana-540	406	17	each	each	DET
cana-540	406	18	p	p	PROPN
cana-540	406	19			PROPN
cana-540	406	20	r.	r.	PROPN
cana-540	406	21	if	if	SCONJ
cana-540	406	22	b	b	PROPN
cana-540	406	23	is	be	AUX
cana-540	406	24	spgα	spgα	ADJ
cana-540	406	25	-	-	PUNCT
cana-540	406	26	compact	compact	ADJ
cana-540	406	27	relative	relative	NOUN
cana-540	406	28	to	to	ADP
cana-540	406	29	r	r	NOUN
cana-540	406	30	,	,	PUNCT
cana-540	406	31	then	then	ADV
cana-540	406	32	p(b	p(b	PROPN
cana-540	406	33	)	)	PUNCT
cana-540	406	34	is	be	AUX
cana-540	406	35	spgα	spgα	ADJ
cana-540	406	36	-	-	PUNCT
cana-540	406	37	compact	compact	ADJ
cana-540	406	38	relative	relative	NOUN
cana-540	406	39	to	to	ADP
cana-540	406	40	s.	s.	PROPN
cana-540	406	41	proof	proof	PROPN
cana-540	406	42	.	.	PUNCT
cana-540	407	1	let	let	VERB
cana-540	407	2	{	{	PUNCT
cana-540	407	3	vi	vi	VERB
cana-540	407	4	:	:	PUNCT
cana-540	407	5	i	i	PRON
cana-540	407	6			NOUN
cana-540	407	7			NOUN
cana-540	407	8	}	}	PUNCT
cana-540	407	9	be	be	AUX
cana-540	407	10	a	a	DET
cana-540	407	11	cover	cover	NOUN
cana-540	407	12	of	of	ADP
cana-540	407	13	p(b	p(b	NOUN
cana-540	407	14	)	)	PUNCT
cana-540	407	15	by	by	ADP
cana-540	407	16	spgα	spgα	ADJ
cana-540	407	17	-	-	PUNCT
cana-540	407	18	open	open	ADJ
cana-540	407	19	sets	set	NOUN
cana-540	407	20	in	in	ADP
cana-540	407	21	s.	s.	PROPN
cana-540	407	22	then	then	ADV
cana-540	407	23	for	for	ADP
cana-540	407	24	each	each	DET
cana-540	407	25	p	p	PROPN
cana-540	407	26			PROPN
cana-540	407	27	b	b	NOUN
cana-540	407	28	,	,	PUNCT
cana-540	407	29	there	there	PRON
cana-540	407	30	exists	exist	VERB
cana-540	407	31	a	a	DET
cana-540	407	32	finite	finite	NOUN
cana-540	407	33	subset	subset	VERB
cana-540	407	34	(p	(p	NOUN
cana-540	407	35	)	)	PUNCT
cana-540	407	36			NOUN
cana-540	407	37			NOUN
cana-540	407	38	with	with	ADP
cana-540	407	39	p(p	p(p	NOUN
cana-540	407	40	)	)	PUNCT
cana-540	408	1			PROPN
cana-540	408	2	{vi	{vi	ADV
cana-540	408	3	:	:	PUNCT
cana-540	408	4	i	i	PRON
cana-540	408	5			NOUN
cana-540	408	6	(p	(p	PROPN
cana-540	408	7	)	)	PUNCT
cana-540	408	8	}	}	PUNCT
cana-540	408	9	and	and	CCONJ
cana-540	408	10	so	so	ADV
cana-540	408	11	p(p	p(p	ADV
cana-540	408	12	)	)	PUNCT
cana-540	408	13			PROPN
cana-540	408	14	v(p	v(p	PROPN
cana-540	408	15	)	)	PUNCT
cana-540	408	16			NOUN
cana-540	408	17	spgα	spgα	NOUN
cana-540	408	18	-	-	PUNCT
cana-540	408	19	o(s	o(s	NOUN
cana-540	408	20	)	)	PUNCT
cana-540	408	21	.	.	PUNCT
cana-540	409	1	thus	thus	ADV
cana-540	409	2	there	there	PRON
cana-540	409	3	exists	exist	VERB
cana-540	409	4	a	a	DET
cana-540	409	5	finite	finite	ADJ
cana-540	409	6	number	number	NOUN
cana-540	409	7	of	of	ADP
cana-540	409	8	points	point	NOUN
cana-540	409	9	of	of	ADP
cana-540	409	10	b	b	NOUN
cana-540	409	11	,	,	PUNCT
cana-540	409	12	p1	p1	NOUN
cana-540	409	13	,	,	PUNCT
cana-540	409	14	p2	p2	NOUN
cana-540	409	15	,	,	PUNCT
cana-540	409	16	p3	p3	NOUN
cana-540	409	17	,	,	PUNCT
cana-540	409	18	…	…	PUNCT
cana-540	409	19	pk	pk	NOUN
cana-540	409	20	with	with	ADP
cana-540	409	21	b	b	PROPN
cana-540	409	22			PROPN
cana-540	409	23	{v	{v	PROPN
cana-540	409	24	(	(	PUNCT
cana-540	409	25	pi	pi	NOUN
cana-540	409	26	):	):	PUNCT
cana-540	409	27	i	i	PRON
cana-540	409	28	=	=	NOUN
cana-540	409	29	1	1	NUM
cana-540	409	30	,	,	PUNCT
cana-540	409	31	2	2	NUM
cana-540	409	32	,	,	PUNCT
cana-540	409	33	3	3	NUM
cana-540	409	34	…	…	NUM
cana-540	409	35	.	.	PUNCT
cana-540	410	1	k}.thus	k}.thus	PROPN
cana-540	410	2	p(b	p(b	PROPN
cana-540	410	3	)	)	PUNCT
cana-540	411	1			PROPN
cana-540	411	2	p(⋃	p(⋃	PROPN
cana-540	411	3	{	{	PUNCT
cana-540	411	4	𝑉𝑖(𝑝𝑖	𝑉𝑖(𝑝𝑖	NOUN
cana-540	411	5	)	)	PUNCT
cana-540	411	6	}	}	PUNCT
cana-540	411	7	)	)	PUNCT
cana-540	412	1	𝑘	𝑘	X
cana-540	412	2	𝑖=1	𝑖=1	PUNCT
cana-540	413	1			PROPN
cana-540	413	2	⋃	⋃	ADV
cana-540	413	3	𝑃(𝑉𝑖(𝑃𝑖))𝑘	𝑃(𝑉𝑖(𝑃𝑖))𝑘	X
cana-540	413	4	𝑖=1	𝑖=1	PUNCT
cana-540	414	1			PROPN
cana-540	414	2	⋃	⋃	NOUN
cana-540	414	3	𝑉(𝑃𝑖	𝑉(𝑃𝑖	PROPN
cana-540	414	4	)	)	PUNCT
cana-540	414	5	𝑘	𝑘	X
cana-540	414	6	𝑖=1	𝑖=1	PUNCT
cana-540	415	1			PRON
cana-540	415	2	⋃	⋃	NOUN
cana-540	415	3	⋃	⋃	NOUN
cana-540	415	4	𝑉𝑖𝑖∈(𝑝𝑖	𝑉𝑖𝑖∈(𝑝𝑖	NOUN
cana-540	415	5	)	)	PUNCT
cana-540	415	6	𝑘	𝑘	PROPN
cana-540	415	7	𝑖=1	𝑖=1	PROPN
cana-540	415	8	.	.	PUNCT
cana-540	416	1	thus	thus	ADV
cana-540	416	2	p(b	p(b	PROPN
cana-540	416	3	)	)	PUNCT
cana-540	416	4	is	be	AUX
cana-540	416	5	spgα	spgα	ADJ
cana-540	416	6	-	-	PUNCT
cana-540	416	7	compact	compact	ADJ
cana-540	416	8	relative	relative	NOUN
cana-540	416	9	to	to	ADP
cana-540	416	10	s.	s.	PROPN
cana-540	416	11	corollary	corollary	PROPN
cana-540	416	12	3.1	3.1	NUM
cana-540	416	13	.	.	PUNCT
cana-540	417	1	let	let	VERB
cana-540	417	2	p	p	PRON
cana-540	417	3	be	be	AUX
cana-540	417	4	an	an	DET
cana-540	417	5	u.spgα.i	u.spgα.i	PROPN
cana-540	417	6	surjective	surjective	ADJ
cana-540	417	7	m.f	m.f	NOUN
cana-540	417	8	and	and	CCONJ
cana-540	417	9	p(b	p(b	PROPN
cana-540	417	10	)	)	PUNCT
cana-540	417	11	is	be	AUX
cana-540	417	12	spgα	spgα	ADJ
cana-540	417	13	-	-	PUNCT
cana-540	417	14	compact	compact	ADJ
cana-540	417	15	relative	relative	NOUN
cana-540	417	16	to	to	ADP
cana-540	417	17	s	s	PRON
cana-540	417	18	for	for	ADP
cana-540	417	19	each	each	DET
cana-540	417	20	p	p	PROPN
cana-540	417	21			PROPN
cana-540	417	22	r.	r.	PROPN
cana-540	417	23	if	if	SCONJ
cana-540	417	24	r	r	NOUN
cana-540	417	25	is	be	AUX
cana-540	417	26	spgα	spgα	ADJ
cana-540	417	27	-	-	PUNCT
cana-540	417	28	compact	compact	ADJ
cana-540	417	29	,	,	PUNCT
cana-540	417	30	then	then	ADV
cana-540	417	31	s	s	VERB
cana-540	417	32	is	be	AUX
cana-540	417	33	spgα	spgα	ADJ
cana-540	417	34	-	-	PUNCT
cana-540	417	35	compact	compact	ADJ
cana-540	417	36	.	.	PUNCT
cana-540	418	1	theorem	theorem	VERB
cana-540	418	2	3.5	3.5	NUM
cana-540	418	3	.	.	PUNCT
cana-540	419	1	if	if	SCONJ
cana-540	419	2	p	p	NOUN
cana-540	419	3	is	be	AUX
cana-540	419	4	u.spgα.i	u.spgα.i	PROPN
cana-540	419	5	and	and	CCONJ
cana-540	419	6	p(p	p(p	NOUN
cana-540	419	7	)	)	PUNCT
cana-540	419	8	is	be	AUX
cana-540	419	9	spgα	spgα	ADJ
cana-540	419	10	-	-	PUNCT
cana-540	419	11	lindelof	lindelof	NOUN
cana-540	419	12	relative	relative	ADJ
cana-540	419	13	to	to	ADP
cana-540	419	14	s	s	PRON
cana-540	419	15	for	for	ADP
cana-540	419	16	each	each	DET
cana-540	419	17	p	p	NOUN
cana-540	419	18			NOUN
cana-540	419	19	r	r	NOUN
cana-540	419	20	and	and	CCONJ
cana-540	419	21	if	if	SCONJ
cana-540	419	22	b	b	NOUN
cana-540	419	23	is	be	AUX
cana-540	419	24	spgα	spgα	ADJ
cana-540	419	25	-	-	PUNCT
cana-540	419	26	lindelof	lindelof	NOUN
cana-540	419	27	relative	relative	ADJ
cana-540	419	28	to	to	ADP
cana-540	419	29	r	r	NOUN
cana-540	419	30	,	,	PUNCT
cana-540	419	31	then	then	ADV
cana-540	419	32	p(b	p(b	PROPN
cana-540	419	33	)	)	PUNCT
cana-540	419	34	is	be	AUX
cana-540	419	35	spgα	spgα	ADJ
cana-540	419	36	-	-	PUNCT
cana-540	419	37	lindelof	lindelof	NOUN
cana-540	419	38	relative	relative	ADJ
cana-540	419	39	to	to	ADP
cana-540	419	40	s.	s.	PROPN
cana-540	419	41	proof	proof	PROPN
cana-540	419	42	.	.	PUNCT
cana-540	420	1	similar	similar	ADJ
cana-540	420	2	to	to	ADP
cana-540	420	3	proof	proof	NOUN
cana-540	420	4	of	of	ADP
cana-540	420	5	theorem	theorem	ADJ
cana-540	420	6	3.4	3.4	NUM
cana-540	420	7	.	.	PUNCT
cana-540	420	8	definition	definition	NOUN
cana-540	420	9	3.3	3.3	NUM
cana-540	420	10	.	.	PUNCT
cana-540	421	1	a	a	DET
cana-540	421	2	space	space	NOUN
cana-540	421	3	r	r	NOUN
cana-540	421	4	is	be	AUX
cana-540	421	5	said	say	VERB
cana-540	421	6	to	to	PART
cana-540	421	7	be	be	AUX
cana-540	421	8	spgα	spgα	ADJ
cana-540	421	9	-	-	PUNCT
cana-540	421	10	normal	normal	ADJ
cana-540	421	11	(	(	PUNCT
cana-540	421	12	briefly	briefly	NOUN
cana-540	421	13	spgα.n	spgα.n	NOUN
cana-540	421	14	)	)	PUNCT
cana-540	421	15	if	if	SCONJ
cana-540	421	16	for	for	ADP
cana-540	421	17	any	any	DET
cana-540	421	18	pair	pair	NOUN
cana-540	421	19	of	of	ADP
cana-540	421	20	distinct	distinct	ADJ
cana-540	421	21	spgα	spgα	ADJ
cana-540	421	22	-	-	PUNCT
cana-540	421	23	closed	close	VERB
cana-540	421	24	sets	set	NOUN
cana-540	421	25	a	a	PRON
cana-540	421	26	and	and	CCONJ
cana-540	421	27	b	b	NOUN
cana-540	421	28	in	in	ADP
cana-540	421	29	r	r	NOUN
cana-540	421	30	there	there	PRON
cana-540	421	31	exists	exist	VERB
cana-540	421	32	disjoint	disjoint	NOUN
cana-540	421	33	open	open	ADJ
cana-540	421	34	sets	set	NOUN
cana-540	421	35	u	u	NOUN
cana-540	421	36	and	and	CCONJ
cana-540	421	37	v	v	NOUN
cana-540	421	38	in	in	ADP
cana-540	421	39	r	r	NOUN
cana-540	421	40	such	such	ADJ
cana-540	421	41	that	that	SCONJ
cana-540	421	42	a	a	DET
cana-540	421	43			PROPN
cana-540	421	44	u	u	PROPN
cana-540	421	45	,	,	PUNCT
cana-540	421	46	b	b	PROPN
cana-540	421	47			PROPN
cana-540	421	48	v.	v.	ADP
cana-540	421	49	theorem	theorem	VERB
cana-540	421	50	3.7	3.7	NUM
cana-540	421	51	.	.	PUNCT
cana-540	422	1	the	the	DET
cana-540	422	2	set	set	NOUN
cana-540	422	3	of	of	ADP
cana-540	422	4	a	a	DET
cana-540	422	5	point	point	NOUN
cana-540	422	6	p	p	NOUN
cana-540	422	7	of	of	ADP
cana-540	422	8	r	r	NOUN
cana-540	422	9	at	at	ADP
cana-540	422	10	which	which	PRON
cana-540	422	11	a	a	DET
cana-540	422	12	m.f	m.f	NOUN
cana-540	422	13	p	p	NOUN
cana-540	422	14	is	be	AUX
cana-540	422	15	not	not	PART
cana-540	422	16	u.spgα.i	u.spgα.i	PROPN
cana-540	422	17	(	(	PUNCT
cana-540	422	18	resp	resp	PROPN
cana-540	422	19	l.spgα.i	l.spgα.i	PROPN
cana-540	422	20	)	)	PUNCT
cana-540	422	21	is	be	AUX
cana-540	422	22	identical	identical	ADJ
cana-540	422	23	with	with	ADP
cana-540	422	24	the	the	DET
cana-540	422	25	union	union	NOUN
cana-540	422	26	of	of	ADP
cana-540	422	27	the	the	DET
cana-540	422	28	spgα	spgα	ADJ
cana-540	422	29	-	-	PUNCT
cana-540	422	30	frontiers	frontier	NOUN
cana-540	422	31	of	of	ADP
cana-540	422	32	the	the	DET
cana-540	422	33	upper	upper	ADJ
cana-540	422	34	(	(	PUNCT
cana-540	422	35	lower	low	ADJ
cana-540	422	36	)	)	PUNCT
cana-540	422	37	inverse	inverse	NOUN
cana-540	422	38	image	image	NOUN
cana-540	422	39	of	of	ADP
cana-540	422	40	spgαopen	spgαopen	PROPN
cana-540	422	41	sets	set	NOUN
cana-540	422	42	containing	contain	VERB
cana-540	422	43	(	(	PUNCT
cana-540	422	44	respectively	respectively	ADV
cana-540	422	45	meeting	meeting	NOUN
cana-540	422	46	)	)	PUNCT
cana-540	423	1	p(p	p(p	NOUN
cana-540	423	2	)	)	PUNCT
cana-540	423	3	.	.	PUNCT
cana-540	424	1	proof	proof	NOUN
cana-540	424	2	.	.	PUNCT
cana-540	425	1	let	let	VERB
cana-540	425	2	p	p	PRON
cana-540	425	3			PROPN
cana-540	425	4	r	r	NOUN
cana-540	425	5	at	at	ADP
cana-540	425	6	which	which	PRON
cana-540	425	7	p	p	NOUN
cana-540	425	8	is	be	AUX
cana-540	425	9	not	not	PART
cana-540	425	10	u.spgα.i	u.spgα.i	NOUN
cana-540	425	11	.	.	PUNCT
cana-540	426	1	then	then	ADV
cana-540	426	2	there	there	PRON
cana-540	426	3	exists	exist	VERB
cana-540	426	4	v	v	ADP
cana-540	426	5			PROPN
cana-540	426	6	spgα	spgα	NOUN
cana-540	426	7	o(s	o(s	PROPN
cana-540	426	8	)	)	PUNCT
cana-540	426	9	containing	contain	VERB
cana-540	426	10	p(p	p(p	NOUN
cana-540	426	11	)	)	PUNCT
cana-540	426	12	with	with	ADP
cana-540	426	13	u	u	NOUN
cana-540	426	14			PUNCT
cana-540	426	15	(	(	PUNCT
cana-540	426	16	r	r	NOUN
cana-540	426	17	p+(v	p+(v	PROPN
cana-540	426	18	)	)	PUNCT
cana-540	426	19	)	)	PUNCT
cana-540	427	1			VERB
cana-540	427	2			NOUN
cana-540	427	3	for	for	ADP
cana-540	427	4	each	each	DET
cana-540	427	5	u	u	NOUN
cana-540	427	6			NOUN
cana-540	427	7	spgαo(r	spgαo(r	NOUN
cana-540	427	8	,	,	PUNCT
cana-540	427	9	p	p	NOUN
cana-540	427	10	)	)	PUNCT
cana-540	427	11	.	.	PUNCT
cana-540	428	1	then	then	ADV
cana-540	428	2	p	p	PROPN
cana-540	428	3			PROPN
cana-540	428	4	spgαcl(r	spgαcl(r	PROPN
cana-540	428	5	p+(v	p+(v	PROPN
cana-540	428	6	)	)	PUNCT
cana-540	428	7	)	)	PUNCT
cana-540	428	8	as	as	ADP
cana-540	428	9	p	p	PROPN
cana-540	428	10			PROPN
cana-540	428	11	p+(v	p+(v	PROPN
cana-540	428	12	)	)	PUNCT
cana-540	428	13	.	.	PUNCT
cana-540	429	1	so	so	ADV
cana-540	429	2	p	p	ADP
cana-540	429	3			PROPN
cana-540	429	4	spgα	spgα	NOUN
cana-540	429	5	cl(p+(s	cl(p+(s	PROPN
cana-540	429	6	)	)	PUNCT
cana-540	429	7	)	)	PUNCT
cana-540	429	8	and	and	CCONJ
cana-540	429	9	p	p	X
cana-540	429	10			PROPN
cana-540	429	11	spgα	spgα	NOUN
cana-540	429	12	-	-	PUNCT
cana-540	429	13	fr(p+(b	fr(p+(b	NOUN
cana-540	429	14	)	)	PUNCT
cana-540	429	15	)	)	PUNCT
cana-540	429	16	.	.	PUNCT
cana-540	430	1	on	on	ADP
cana-540	430	2	the	the	DET
cana-540	430	3	other	other	ADJ
cana-540	430	4	hand	hand	NOUN
cana-540	430	5	v	v	PROPN
cana-540	430	6			PROPN
cana-540	430	7	spgαo(s	spgαo(s	NOUN
cana-540	430	8	)	)	PUNCT
cana-540	430	9	containing	contain	VERB
cana-540	430	10	p(p	p(p	NOUN
cana-540	430	11	)	)	PUNCT
cana-540	430	12	and	and	CCONJ
cana-540	430	13	p	p	X
cana-540	430	14			PROPN
cana-540	430	15	spgα	spgα	NOUN
cana-540	430	16	-	-	PUNCT
cana-540	430	17	fr(p+(b	fr(p+(b	NOUN
cana-540	430	18	)	)	PUNCT
cana-540	430	19	)	)	PUNCT
cana-540	430	20	.	.	PUNCT
cana-540	431	1	let	let	VERB
cana-540	431	2	p	p	NOUN
cana-540	431	3	is	be	AUX
cana-540	431	4	u.spgα.i	u.spgα.i	NOUN
cana-540	431	5	,	,	PUNCT
cana-540	431	6	there	there	PRON
cana-540	431	7	exists	exist	VERB
cana-540	431	8	u	u	PRON
cana-540	431	9			NOUN
cana-540	431	10	spgαo(r	spgαo(r	NOUN
cana-540	431	11	,	,	PUNCT
cana-540	431	12	p	p	NOUN
cana-540	431	13	)	)	PUNCT
cana-540	431	14	with	with	ADP
cana-540	431	15	p(u	p(u	NOUN
cana-540	431	16	)	)	PUNCT
cana-540	432	1			PROPN
cana-540	432	2	v.	v.	ADP
cana-540	432	3	thus	thus	ADV
cana-540	432	4	p	p	ADJ
cana-540	432	5			PROPN
cana-540	432	6	u	u	NOUN
cana-540	432	7			PROPN
cana-540	432	8	spgα	spgα	NOUN
cana-540	432	9	-	-	PUNCT
cana-540	432	10	int(p+(v	int(p+(v	NOUN
cana-540	432	11	)	)	PUNCT
cana-540	432	12	)	)	PUNCT
cana-540	432	13	which	which	PRON
cana-540	432	14	contradicts	contradict	VERB
cana-540	432	15	to	to	ADP
cana-540	432	16	the	the	DET
cana-540	432	17	fact	fact	NOUN
cana-540	432	18	that	that	SCONJ
cana-540	432	19	p	p	ADJ
cana-540	432	20			PROPN
cana-540	432	21	spgα	spgα	NOUN
cana-540	432	22	-	-	PUNCT
cana-540	432	23	fr(p+(v	fr(p+(v	NOUN
cana-540	432	24	)	)	PUNCT
cana-540	432	25	)	)	PUNCT
cana-540	432	26	.	.	PUNCT
cana-540	433	1	hence	hence	ADV
cana-540	433	2	p	p	NOUN
cana-540	433	3	is	be	AUX
cana-540	433	4	not	not	PART
cana-540	433	5	u.spgα.i	u.spgα.i	NOUN
cana-540	433	6	.	.	PUNCT
cana-540	434	1	similarlly	similarlly	ADV
cana-540	434	2	we	we	PRON
cana-540	434	3	can	can	AUX
cana-540	434	4	prove	prove	VERB
cana-540	434	5	the	the	DET
cana-540	434	6	theorem	theorem	NOUN
cana-540	434	7	related	relate	VERB
cana-540	434	8	to	to	ADP
cana-540	434	9	l.spgα.i	l.spgα.i	PROPN
cana-540	434	10	.	.	PUNCT
cana-540	435	1	communications	communication	NOUN
cana-540	435	2	on	on	ADP
cana-540	435	3	applied	apply	VERB
cana-540	435	4	nonlinear	nonlinear	ADJ
cana-540	435	5	analysis	analysis	NOUN
cana-540	435	6	issn	issn	NOUN
cana-540	435	7	:	:	PUNCT
cana-540	435	8	1074	1074	NUM
cana-540	435	9	-	-	PUNCT
cana-540	435	10	133x	133x	NUM
cana-540	435	11	vol	vol	NOUN
cana-540	435	12	31	31	NUM
cana-540	435	13	no	no	NOUN
cana-540	435	14	.	.	NOUN
cana-540	435	15	2	2	NUM
cana-540	435	16	(	(	PUNCT
cana-540	435	17	2024	2024	NUM
cana-540	435	18	)	)	PUNCT
cana-540	435	19	257	257	NUM
cana-540	435	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	435	21	theorem	theorem	VERB
cana-540	435	22	3.8	3.8	NUM
cana-540	435	23	.	.	PUNCT
cana-540	436	1	let	let	VERB
cana-540	436	2	p	p	PRON
cana-540	436	3	be	be	AUX
cana-540	436	4	an	an	DET
cana-540	436	5	u.spgα.i	u.spgα.i	PROPN
cana-540	436	6	injective	injective	ADJ
cana-540	436	7	m.f	m.f	NOUN
cana-540	436	8	and	and	CCONJ
cana-540	436	9	point	point	NOUN
cana-540	436	10	closed	close	VERB
cana-540	436	11	from	from	ADP
cana-540	436	12	a	a	DET
cana-540	436	13	ts	ts	ADP
cana-540	436	14	r	r	NOUN
cana-540	436	15	to	to	PART
cana-540	436	16	spgα	spgα	VERB
cana-540	436	17	-	-	PUNCT
cana-540	436	18	n	n	CCONJ
cana-540	436	19	,	,	PUNCT
cana-540	436	20	then	then	ADV
cana-540	436	21	r	r	NOUN
cana-540	436	22	is	be	AUX
cana-540	436	23	spgα	spgα	ADJ
cana-540	436	24	-	-	PUNCT
cana-540	436	25	t2	t2	NOUN
cana-540	436	26	-	-	PUNCT
cana-540	436	27	space	space	NOUN
cana-540	436	28	.	.	PUNCT
cana-540	437	1	proof	proof	NOUN
cana-540	437	2	.	.	PUNCT
cana-540	438	1	let	let	VERB
cana-540	438	2	m	m	PRON
cana-540	438	3	,	,	PUNCT
cana-540	438	4	n	n	CCONJ
cana-540	438	5			NOUN
cana-540	438	6	r	r	NOUN
cana-540	438	7	with	with	ADP
cana-540	438	8	m	m	PROPN
cana-540	438	9			PROPN
cana-540	438	10	n.	n.	PROPN
cana-540	438	11	then	then	ADV
cana-540	438	12	p(m	p(m	NOUN
cana-540	438	13	)	)	PUNCT
cana-540	438	14			PUNCT
cana-540	438	15	p(n	p(n	NOUN
cana-540	438	16	)	)	PUNCT
cana-540	438	17	=	=	NOUN
cana-540	438	18			NOUN
cana-540	438	19	as	as	SCONJ
cana-540	438	20	p	p	NOUN
cana-540	438	21	is	be	AUX
cana-540	438	22	injective	injective	ADJ
cana-540	438	23	.	.	PUNCT
cana-540	439	1	by	by	ADP
cana-540	439	2	spgα	spgα	ADJ
cana-540	439	3	-	-	PUNCT
cana-540	439	4	normality	normality	NOUN
cana-540	439	5	of	of	ADP
cana-540	439	6	s	s	PROPN
cana-540	439	7	,	,	PUNCT
cana-540	439	8	there	there	PRON
cana-540	439	9	exists	exist	VERB
cana-540	439	10	disjoint	disjoint	NOUN
cana-540	439	11	open	open	ADJ
cana-540	439	12	sets	set	NOUN
cana-540	439	13	u	u	NOUN
cana-540	439	14	and	and	CCONJ
cana-540	439	15	v	v	ADP
cana-540	439	16	containing	contain	VERB
cana-540	439	17	p(m	p(m	NOUN
cana-540	439	18	)	)	PUNCT
cana-540	439	19	and	and	CCONJ
cana-540	439	20	p(n	p(n	PROPN
cana-540	439	21	)	)	PUNCT
cana-540	439	22	.	.	PUNCT
cana-540	440	1	thus	thus	ADV
cana-540	440	2	,	,	PUNCT
cana-540	440	3	there	there	PRON
cana-540	440	4	exists	exist	VERB
cana-540	440	5	disjoint	disjoint	ADJ
cana-540	440	6	spgα	spgα	ADJ
cana-540	440	7	-	-	PUNCT
cana-540	440	8	open	open	ADJ
cana-540	440	9	sets	set	NOUN
cana-540	440	10	p+(u	p+(u	NOUN
cana-540	440	11	)	)	PUNCT
cana-540	440	12	and	and	CCONJ
cana-540	440	13	p+(v	p+(v	NOUN
cana-540	440	14	)	)	PUNCT
cana-540	440	15	containing	contain	VERB
cana-540	440	16	m	m	PROPN
cana-540	440	17	and	and	CCONJ
cana-540	440	18	n	n	PRON
cana-540	440	19	respectively	respectively	ADV
cana-540	440	20	such	such	ADJ
cana-540	440	21	that	that	SCONJ
cana-540	440	22	g	g	PROPN
cana-540	440	23			PROPN
cana-540	440	24	p+(u	p+(u	PROPN
cana-540	440	25	)	)	PUNCT
cana-540	440	26	and	and	CCONJ
cana-540	440	27	h	h	NOUN
cana-540	440	28			PROPN
cana-540	440	29	p+(v	p+(v	PROPN
cana-540	440	30	)	)	PUNCT
cana-540	440	31	and	and	CCONJ
cana-540	440	32	so	so	ADV
cana-540	440	33	g	g	NOUN
cana-540	440	34			PUNCT
cana-540	440	35	h	h	NOUN
cana-540	440	36	=	=	PUNCT
cana-540	440	37	.	.	PUNCT
cana-540	440	38	hence	hence	ADV
cana-540	440	39	r	r	NOUN
cana-540	440	40	is	be	AUX
cana-540	440	41	spgα	spgα	ADJ
cana-540	440	42	-	-	PUNCT
cana-540	440	43	t2	t2	NOUN
cana-540	440	44	-	-	PUNCT
cana-540	440	45	space	space	NOUN
cana-540	440	46	.	.	PUNCT
cana-540	441	1	definition	definition	NOUN
cana-540	441	2	3.4	3.4	NUM
cana-540	441	3	.	.	PUNCT
cana-540	442	1	a	a	DET
cana-540	442	2	m.f	m.f	NOUN
cana-540	442	3	p	p	X
cana-540	442	4	:	:	PUNCT
cana-540	442	5	r	r	NOUN
cana-540	442	6	→	→	SYM
cana-540	442	7	s	s	PART
cana-540	442	8	is	be	AUX
cana-540	442	9	said	say	VERB
cana-540	442	10	to	to	PART
cana-540	442	11	have	have	VERB
cana-540	442	12	spgα	spgα	ADJ
cana-540	442	13	-	-	PUNCT
cana-540	442	14	closed	close	VERB
cana-540	442	15	graph	graph	NOUN
cana-540	442	16	if	if	SCONJ
cana-540	442	17	for	for	ADP
cana-540	442	18	each	each	PRON
cana-540	442	19	(	(	PUNCT
cana-540	442	20	m	m	PROPN
cana-540	442	21	,	,	PUNCT
cana-540	442	22	n	n	CCONJ
cana-540	442	23	)	)	PUNCT
cana-540	442	24			VERB
cana-540	442	25	g(p	g(p	PROPN
cana-540	442	26	)	)	PUNCT
cana-540	442	27	there	there	PRON
cana-540	442	28	exists	exist	VERB
cana-540	442	29	u	u	NOUN
cana-540	442	30			PROPN
cana-540	442	31	spgα	spgα	PROPN
cana-540	442	32	-	-	PUNCT
cana-540	442	33	o(r	o(r	PROPN
cana-540	442	34	,	,	PUNCT
cana-540	442	35	m	m	PROPN
cana-540	442	36	)	)	PUNCT
cana-540	442	37	and	and	CCONJ
cana-540	442	38	v	v	ADP
cana-540	442	39			PROPN
cana-540	442	40	spgα	spgα	NOUN
cana-540	442	41	-	-	PUNCT
cana-540	442	42	o(s	o(s	NOUN
cana-540	442	43	,	,	PUNCT
cana-540	442	44	n	n	CCONJ
cana-540	442	45	)	)	PUNCT
cana-540	442	46	with	with	ADP
cana-540	442	47	(	(	PUNCT
cana-540	442	48	u	u	NOUN
cana-540	442	49	x	x	PROPN
cana-540	442	50	v	v	NOUN
cana-540	442	51	)	)	PUNCT
cana-540	442	52			NOUN
cana-540	442	53	g(p	g(p	NOUN
cana-540	442	54	)	)	PUNCT
cana-540	442	55	=	=	PUNCT
cana-540	442	56	.	.	X
cana-540	442	57	theorem	theorem	VERB
cana-540	442	58	3.9	3.9	NUM
cana-540	442	59	.	.	PUNCT
cana-540	443	1	let	let	VERB
cana-540	443	2	p	p	PRON
cana-540	443	3	be	be	AUX
cana-540	443	4	a	a	DET
cana-540	443	5	m.f	m.f	NOUN
cana-540	443	6	from	from	ADP
cana-540	443	7	a	a	DET
cana-540	443	8	space	space	NOUN
cana-540	443	9	r	r	NOUN
cana-540	443	10	into	into	ADP
cana-540	443	11	spgα	spgα	ADJ
cana-540	443	12	-	-	PUNCT
cana-540	443	13	compact	compact	ADJ
cana-540	443	14	space	space	NOUN
cana-540	443	15	s.	s.	PROPN
cana-540	443	16	if	if	SCONJ
cana-540	443	17	g(p	g(p	PROPN
cana-540	443	18	)	)	PUNCT
cana-540	443	19	is	be	AUX
cana-540	443	20	spgαclosed	spgαclose	VERB
cana-540	443	21	then	then	ADV
cana-540	443	22	p	p	NOUN
cana-540	443	23	is	be	AUX
cana-540	443	24	u.spgα.c	u.spgα.c	PROPN
cana-540	443	25	.	.	PUNCT
cana-540	444	1	proof	proof	NOUN
cana-540	444	2	.	.	PUNCT
cana-540	445	1	suppose	suppose	VERB
cana-540	445	2	p	p	NOUN
cana-540	445	3	is	be	AUX
cana-540	445	4	not	not	PART
cana-540	445	5	u.spgα.c	u.spgα.c	PRON
cana-540	445	6	.	.	PUNCT
cana-540	446	1	then	then	ADV
cana-540	446	2	there	there	PRON
cana-540	446	3	exists	exist	VERB
cana-540	446	4	a	a	DET
cana-540	446	5	non	non	X
cana-540	446	6	empty	empty	ADJ
cana-540	446	7	closed	closed	ADJ
cana-540	446	8	subset	subset	NOUN
cana-540	446	9	b	b	NOUN
cana-540	446	10	with	with	ADP
cana-540	446	11	p-(b	p-(b	ADJ
cana-540	446	12	)	)	PUNCT
cana-540	446	13	is	be	AUX
cana-540	446	14	not	not	PART
cana-540	446	15	spgα	spgα	ADJ
cana-540	446	16	-	-	PUNCT
cana-540	446	17	closed	close	VERB
cana-540	446	18	in	in	ADP
cana-540	446	19	r.	r.	PROPN
cana-540	446	20	assume	assume	VERB
cana-540	446	21	that	that	SCONJ
cana-540	446	22	p-(b	p-(b	ADJ
cana-540	446	23	)	)	PUNCT
cana-540	446	24			NOUN
cana-540	446	25			NOUN
cana-540	446	26	,	,	PUNCT
cana-540	446	27	then	then	ADV
cana-540	446	28	there	there	PRON
cana-540	446	29	exists	exist	VERB
cana-540	446	30	a	a	DET
cana-540	446	31	point	point	NOUN
cana-540	446	32	p0	p0	NOUN
cana-540	446	33			NOUN
cana-540	446	34	spgα	spgα	NOUN
cana-540	446	35	-	-	PUNCT
cana-540	446	36	cl(p-(b	cl(p-(b	NUM
cana-540	446	37	)	)	PUNCT
cana-540	446	38	–	–	PUNCT
cana-540	446	39	p-(b	p-(b	ADJ
cana-540	446	40	)	)	PUNCT
cana-540	446	41	)	)	PUNCT
cana-540	446	42	.	.	PUNCT
cana-540	447	1	so	so	ADV
cana-540	447	2	for	for	ADP
cana-540	447	3	each	each	DET
cana-540	447	4	point	point	NOUN
cana-540	447	5	n	n	PRON
cana-540	447	6			NOUN
cana-540	447	7	b	b	PROPN
cana-540	447	8	,	,	PUNCT
cana-540	447	9	(	(	PUNCT
cana-540	447	10	p0	p0	NOUN
cana-540	447	11	,	,	PUNCT
cana-540	447	12	n0	n0	NUM
cana-540	447	13	)	)	PUNCT
cana-540	447	14			PUNCT
cana-540	447	15	g(p	g(p	PROPN
cana-540	447	16	)	)	PUNCT
cana-540	447	17	,	,	PUNCT
cana-540	447	18	as	as	SCONJ
cana-540	447	19	p	p	PROPN
cana-540	447	20	is	be	AUX
cana-540	447	21	spgα	spgα	ADJ
cana-540	447	22	-	-	PUNCT
cana-540	447	23	closed	closed	ADJ
cana-540	447	24	graph	graph	NOUN
cana-540	447	25	.	.	PUNCT
cana-540	448	1	thus	thus	ADV
cana-540	448	2	there	there	PRON
cana-540	448	3	exists	exist	VERB
cana-540	448	4	a	a	DET
cana-540	448	5	spgα	spgα	ADJ
cana-540	448	6	-	-	PUNCT
cana-540	448	7	open	open	ADJ
cana-540	448	8	sets	set	NOUN
cana-540	448	9	u(n	u(n	NOUN
cana-540	448	10	)	)	PUNCT
cana-540	448	11	and	and	CCONJ
cana-540	448	12	v	v	NOUN
cana-540	448	13	(	(	PUNCT
cana-540	448	14	n	n	CCONJ
cana-540	448	15	)	)	PUNCT
cana-540	448	16	containing	contain	VERB
cana-540	448	17	p0	p0	NOUN
cana-540	448	18	and	and	CCONJ
cana-540	448	19	n	n	NOUN
cana-540	448	20	respectively	respectively	ADV
cana-540	448	21	with	with	ADP
cana-540	448	22	(	(	PUNCT
cana-540	448	23	u(n	u(n	PROPN
cana-540	448	24	)	)	PUNCT
cana-540	448	25	x	x	SYM
cana-540	448	26	v	v	NOUN
cana-540	448	27	(	(	PUNCT
cana-540	448	28	n	n	CCONJ
cana-540	448	29	)	)	PUNCT
cana-540	448	30	)	)	PUNCT
cana-540	449	1			NOUN
cana-540	449	2	g(p	g(p	NOUN
cana-540	449	3	)	)	PUNCT
cana-540	449	4	=	=	PUNCT
cana-540	449	5	.	.	PUNCT
cana-540	449	6	then	then	ADV
cana-540	449	7	{	{	PUNCT
cana-540	449	8	s	s	PROPN
cana-540	449	9	–	–	PUNCT
cana-540	449	10	b	b	NOUN
cana-540	449	11	}	}	PUNCT
cana-540	449	12			NOUN
cana-540	449	13	{	{	PUNCT
cana-540	449	14	v(n	v(n	PROPN
cana-540	449	15	)	)	PUNCT
cana-540	449	16	:	:	PUNCT
cana-540	450	1	n	n	CCONJ
cana-540	450	2			NOUN
cana-540	450	3	b	b	X
cana-540	450	4	}	}	PUNCT
cana-540	450	5	is	be	AUX
cana-540	450	6	a	a	DET
cana-540	450	7	spgα	spgα	ADJ
cana-540	450	8	-	-	PUNCT
cana-540	450	9	open	open	ADJ
cana-540	450	10	cover	cover	NOUN
cana-540	450	11	of	of	ADP
cana-540	450	12	sn	sn	PROPN
cana-540	450	13	and	and	CCONJ
cana-540	450	14	so	so	ADV
cana-540	450	15	it	it	PRON
cana-540	450	16	has	have	VERB
cana-540	450	17	a	a	DET
cana-540	450	18	subcover	subcover	PROPN
cana-540	450	19	{	{	PUNCT
cana-540	450	20	s	s	PROPN
cana-540	450	21	–	–	PUNCT
cana-540	450	22	b	b	NOUN
cana-540	450	23	}	}	PUNCT
cana-540	450	24			NOUN
cana-540	450	25	{	{	PUNCT
cana-540	450	26	v(ni	v(ni	PROPN
cana-540	450	27	):	):	PUNCT
cana-540	450	28	ni	ni	PROPN
cana-540	450	29			PROPN
cana-540	450	30	b	b	PROPN
cana-540	450	31	:	:	PUNCT
cana-540	450	32	1	1	NUM
cana-540	450	33	<	<	X
cana-540	451	1	i	i	X
cana-540	451	2	<	<	X
cana-540	451	3	k	k	X
cana-540	451	4	}	}	PUNCT
cana-540	451	5	.	.	PUNCT
cana-540	452	1	put	put	VERB
cana-540	452	2	u	u	NOUN
cana-540	452	3	=	=	NOUN
cana-540	452	4	u(ni	u(ni	PROPN
cana-540	452	5	)	)	PUNCT
cana-540	452	6	and	and	CCONJ
cana-540	452	7	v	v	NOUN
cana-540	452	8	=	=	SYM
cana-540	452	9	v(ni	v(ni	PROPN
cana-540	452	10	)	)	PUNCT
cana-540	452	11	,	,	PUNCT
cana-540	452	12	then	then	ADV
cana-540	452	13	b	b	PROPN
cana-540	452	14			PROPN
cana-540	452	15	v	v	PROPN
cana-540	452	16	and	and	CCONJ
cana-540	452	17	(	(	PUNCT
cana-540	452	18	u	u	NOUN
cana-540	452	19	x	x	PROPN
cana-540	452	20	v	v	NOUN
cana-540	452	21	)	)	PUNCT
cana-540	452	22			NOUN
cana-540	452	23	g(p	g(p	NOUN
cana-540	452	24	)	)	PUNCT
cana-540	453	1	=	=	SYM
cana-540	453	2			NOUN
cana-540	453	3	as	as	SCONJ
cana-540	453	4	u	u	NOUN
cana-540	453	5	is	be	AUX
cana-540	453	6	spgα	spgα	ADJ
cana-540	453	7	-	-	PUNCT
cana-540	453	8	nbd	nbd	PROPN
cana-540	453	9	.	.	PROPN
cana-540	453	10	of	of	ADP
cana-540	453	11	p0	p0	PROPN
cana-540	453	12	u	u	NOUN
cana-540	453	13	–	–	PUNCT
cana-540	453	14	p-(b	p-(b	ADJ
cana-540	453	15	)	)	PUNCT
cana-540	453	16	=	=	NOUN
cana-540	453	17			NOUN
cana-540	453	18	and	and	CCONJ
cana-540	453	19	so	so	ADV
cana-540	453	20			NOUN
cana-540	453	21			NOUN
cana-540	453	22	(	(	PUNCT
cana-540	453	23	u	u	NOUN
cana-540	453	24	x	x	PROPN
cana-540	453	25	b	b	PROPN
cana-540	453	26	)	)	PUNCT
cana-540	453	27			NOUN
cana-540	453	28	g(p	g(p	NOUN
cana-540	453	29	)	)	PUNCT
cana-540	454	1			PROPN
cana-540	454	2	(	(	PUNCT
cana-540	454	3	u	u	NOUN
cana-540	454	4	x	x	PROPN
cana-540	454	5	v	v	NOUN
cana-540	454	6	)	)	PUNCT
cana-540	454	7			NOUN
cana-540	454	8	g(p	g(p	NOUN
cana-540	454	9	)	)	PUNCT
cana-540	454	10	which	which	PRON
cana-540	454	11	is	be	AUX
cana-540	454	12	contradiction	contradiction	NOUN
cana-540	454	13	.	.	PUNCT
cana-540	455	1	thus	thus	ADV
cana-540	455	2	p	p	X
cana-540	455	3	is	be	AUX
cana-540	455	4	u.spgα.c	u.spgα.c	PROPN
cana-540	455	5	.	.	PUNCT
cana-540	456	1	discussion	discussion	NOUN
cana-540	456	2	and	and	CCONJ
cana-540	456	3	conclusion	conclusion	NOUN
cana-540	456	4	topology	topology	NOUN
cana-540	456	5	is	be	AUX
cana-540	456	6	a	a	DET
cana-540	456	7	relatively	relatively	ADV
cana-540	456	8	new	new	ADJ
cana-540	456	9	branch	branch	NOUN
cana-540	456	10	of	of	ADP
cana-540	456	11	mathematics	mathematic	NOUN
cana-540	456	12	,	,	PUNCT
cana-540	456	13	most	most	ADJ
cana-540	456	14	of	of	ADP
cana-540	456	15	the	the	DET
cana-540	456	16	research	research	NOUN
cana-540	456	17	in	in	ADP
cana-540	456	18	topology	topology	NOUN
cana-540	456	19	has	have	AUX
cana-540	456	20	been	be	AUX
cana-540	456	21	done	do	VERB
cana-540	456	22	since	since	SCONJ
cana-540	456	23	1900	1900	NUM
cana-540	456	24	.	.	PUNCT
cana-540	457	1	the	the	DET
cana-540	457	2	topological	topological	ADJ
cana-540	457	3	structures	structure	NOUN
cana-540	457	4	are	be	AUX
cana-540	457	5	modelled	model	VERB
cana-540	457	6	suitably	suitably	ADV
cana-540	457	7	in	in	ADP
cana-540	457	8	the	the	DET
cana-540	457	9	field	field	NOUN
cana-540	457	10	of	of	ADP
cana-540	457	11	computer	computer	NOUN
cana-540	457	12	graphics	graphic	NOUN
cana-540	457	13	,	,	PUNCT
cana-540	457	14	pattern	pattern	NOUN
cana-540	457	15	recognition	recognition	NOUN
cana-540	457	16	,	,	PUNCT
cana-540	457	17	artificial	artificial	ADJ
cana-540	457	18	intelligence	intelligence	NOUN
cana-540	457	19	,	,	PUNCT
cana-540	457	20	data	datum	NOUN
cana-540	457	21	mining	mining	NOUN
cana-540	457	22	,	,	PUNCT
cana-540	457	23	rough	rough	ADJ
cana-540	457	24	set	set	NOUN
cana-540	457	25	theory	theory	NOUN
cana-540	457	26	,	,	PUNCT
cana-540	457	27	information	information	NOUN
cana-540	457	28	systems	system	NOUN
cana-540	457	29	,	,	PUNCT
cana-540	457	30	quantum	quantum	PROPN
cana-540	457	31	physics	physics	NOUN
cana-540	457	32	etc	etc	X
cana-540	457	33	.	.	PUNCT
cana-540	458	1	the	the	DET
cana-540	458	2	investigation	investigation	NOUN
cana-540	458	3	on	on	ADP
cana-540	458	4	generalization	generalization	NOUN
cana-540	458	5	of	of	ADP
cana-540	458	6	open	open	ADJ
cana-540	458	7	set	set	VERB
cana-540	458	8	as	as	ADV
cana-540	458	9	well	well	ADV
cana-540	458	10	as	as	ADP
cana-540	458	11	closed	closed	ADJ
cana-540	458	12	set	set	NOUN
cana-540	458	13	has	have	AUX
cana-540	458	14	led	lead	VERB
cana-540	458	15	to	to	ADP
cana-540	458	16	significant	significant	ADJ
cana-540	458	17	contribution	contribution	NOUN
cana-540	458	18	to	to	ADP
cana-540	458	19	the	the	DET
cana-540	458	20	theory	theory	NOUN
cana-540	458	21	of	of	ADP
cana-540	458	22	generalization	generalization	NOUN
cana-540	458	23	of	of	ADP
cana-540	458	24	continuity	continuity	NOUN
cana-540	458	25	,	,	PUNCT
cana-540	458	26	separation	separation	NOUN
cana-540	458	27	axioms	axiom	NOUN
cana-540	458	28	,	,	PUNCT
cana-540	458	29	covering	cover	VERB
cana-540	458	30	properties	property	NOUN
cana-540	458	31	and	and	CCONJ
cana-540	458	32	compactness	compactness	NOUN
cana-540	458	33	with	with	ADP
cana-540	458	34	the	the	DET
cana-540	458	35	help	help	NOUN
cana-540	458	36	of	of	ADP
cana-540	458	37	open	open	ADJ
cana-540	458	38	sets	set	NOUN
cana-540	458	39	.	.	PUNCT
cana-540	459	1	several	several	ADJ
cana-540	459	2	generalized	generalized	ADJ
cana-540	459	3	form	form	NOUN
cana-540	459	4	of	of	ADP
cana-540	459	5	continuous	continuous	ADJ
cana-540	459	6	functions	function	NOUN
cana-540	459	7	has	have	AUX
cana-540	459	8	been	be	AUX
cana-540	459	9	introduced	introduce	VERB
cana-540	459	10	in	in	ADP
cana-540	459	11	the	the	DET
cana-540	459	12	last	last	ADJ
cana-540	459	13	decades	decade	NOUN
cana-540	459	14	which	which	PRON
cana-540	459	15	helps	help	VERB
cana-540	459	16	us	we	PRON
cana-540	459	17	to	to	PART
cana-540	459	18	understand	understand	VERB
cana-540	459	19	various	various	ADJ
cana-540	459	20	properties	property	NOUN
cana-540	459	21	of	of	ADP
cana-540	459	22	topological	topological	ADJ
cana-540	459	23	spaces	space	NOUN
cana-540	459	24	.	.	PUNCT
cana-540	460	1	in	in	ADP
cana-540	460	2	this	this	DET
cana-540	460	3	way	way	NOUN
cana-540	460	4	,	,	PUNCT
cana-540	460	5	this	this	DET
cana-540	460	6	paper	paper	NOUN
cana-540	460	7	introduce	introduce	VERB
cana-540	460	8	some	some	DET
cana-540	460	9	concepts	concept	NOUN
cana-540	460	10	of	of	ADP
cana-540	460	11	multifunctions	multifunction	NOUN
cana-540	460	12	in	in	ADP
cana-540	460	13	topological	topological	ADJ
cana-540	460	14	spaces	space	NOUN
cana-540	460	15	.	.	PUNCT
cana-540	461	1	we	we	PRON
cana-540	461	2	first	first	ADV
cana-540	461	3	introduced	introduce	VERB
cana-540	461	4	the	the	DET
cana-540	461	5	concept	concept	NOUN
cana-540	461	6	of	of	ADP
cana-540	461	7	upper	upper	ADJ
cana-540	461	8	spgα	spgα	NOUN
cana-540	461	9	-	-	PUNCT
cana-540	461	10	continuous	continuous	ADJ
cana-540	461	11	(	(	PUNCT
cana-540	461	12	resp	resp	NOUN
cana-540	461	13	.	.	PUNCT
cana-540	462	1	lower	low	ADJ
cana-540	462	2	spgα	spgα	ADJ
cana-540	462	3	-	-	PUNCT
cana-540	462	4	continuous	continuous	ADJ
cana-540	462	5	)	)	PUNCT
cana-540	462	6	multifunctions	multifunction	NOUN
cana-540	462	7	and	and	CCONJ
cana-540	462	8	some	some	DET
cana-540	462	9	properties	property	NOUN
cana-540	462	10	and	and	CCONJ
cana-540	462	11	point	point	VERB
cana-540	462	12	out	out	ADP
cana-540	462	13	the	the	DET
cana-540	462	14	relationship	relationship	NOUN
cana-540	462	15	among	among	ADP
cana-540	462	16	them	they	PRON
cana-540	462	17	.	.	PUNCT
cana-540	463	1	further	far	ADV
cana-540	463	2	,	,	PUNCT
cana-540	463	3	we	we	PRON
cana-540	463	4	introduce	introduce	VERB
cana-540	463	5	upper	upper	ADJ
cana-540	463	6	spgα	spgα	NOUN
cana-540	463	7	-	-	PUNCT
cana-540	463	8	irresolute	irresolute	ADJ
cana-540	463	9	(	(	PUNCT
cana-540	463	10	resp	resp	NOUN
cana-540	463	11	.	.	PUNCT
cana-540	464	1	lower	low	ADJ
cana-540	464	2	spgα	spgα	NOUN
cana-540	464	3	-	-	PUNCT
cana-540	464	4	irresolute	irresolute	NOUN
cana-540	464	5	)	)	PUNCT
cana-540	464	6	multifunctions	multifunction	NOUN
cana-540	464	7	and	and	CCONJ
cana-540	464	8	several	several	ADJ
cana-540	464	9	results	result	NOUN
cana-540	464	10	of	of	ADP
cana-540	464	11	these	these	DET
cana-540	464	12	spaces	space	NOUN
cana-540	464	13	in	in	ADP
cana-540	464	14	topological	topological	ADJ
cana-540	464	15	spaces	space	NOUN
cana-540	464	16	.	.	PUNCT
cana-540	465	1	references	reference	NOUN
cana-540	465	2	[	[	X
cana-540	465	3	1	1	NUM
cana-540	465	4	]	]	X
cana-540	465	5	m.e	m.e	PROPN
cana-540	465	6	.	.	PROPN
cana-540	465	7	abd	abd	PROPN
cana-540	465	8	el	el	PROPN
cana-540	465	9	-	-	PROPN
cana-540	465	10	monsef	monsef	ADJ
cana-540	465	11	,	,	PUNCT
cana-540	465	12	a.a	a.a	PROPN
cana-540	465	13	.	.	PROPN
cana-540	465	14	nasef	nasef	PROPN
cana-540	465	15	,	,	PUNCT
cana-540	465	16	on	on	ADP
cana-540	465	17	multifunctions	multifunction	NOUN
cana-540	465	18	,	,	PUNCT
cana-540	465	19	chaos	chaos	NOUN
cana-540	465	20	,	,	PUNCT
cana-540	465	21	solitons	soliton	NOUN
cana-540	465	22	and	and	CCONJ
cana-540	465	23	fractals	fractal	NOUN
cana-540	465	24	12	12	NUM
cana-540	465	25	,	,	PUNCT
cana-540	465	26	(	(	PUNCT
cana-540	465	27	2001	2001	NUM
cana-540	465	28	)	)	PUNCT
cana-540	465	29	,	,	PUNCT
cana-540	465	30	2387	2387	NUM
cana-540	465	31	-	-	SYM
cana-540	465	32	2394	2394	NUM
cana-540	465	33	.	.	PUNCT
cana-540	466	1	[	[	X
cana-540	466	2	2	2	NUM
cana-540	466	3	]	]	PUNCT
cana-540	466	4	c.	c.	PROPN
cana-540	466	5	berge	berge	PROPN
cana-540	466	6	,	,	PUNCT
cana-540	466	7	topological	topological	ADJ
cana-540	466	8	spaces	space	NOUN
cana-540	466	9	,	,	PUNCT
cana-540	466	10	oliver	oliver	ADV
cana-540	466	11	and	and	CCONJ
cana-540	466	12	boyed	boyed	ADJ
cana-540	466	13	,	,	PUNCT
cana-540	466	14	edinburg	edinburg	NOUN
cana-540	466	15	-	-	PUNCT
cana-540	466	16	london	london	PROPN
cana-540	466	17	,	,	PUNCT
cana-540	466	18	1st	1st	ADJ
cana-540	466	19	english	english	PROPN
cana-540	466	20	edition	edition	NOUN
cana-540	466	21	,	,	PUNCT
cana-540	466	22	1963	1963	NUM
cana-540	466	23	.	.	PUNCT
cana-540	467	1	[	[	X
cana-540	467	2	3	3	X
cana-540	467	3	]	]	X
cana-540	467	4	n.	n.	NOUN
cana-540	467	5	bourbaki	bourbaki	PROPN
cana-540	467	6	,	,	PUNCT
cana-540	467	7	general	general	ADJ
cana-540	467	8	topology	topology	NOUN
cana-540	467	9	,	,	PUNCT
cana-540	467	10	addision	addision	NOUN
cana-540	467	11	wesely	wesely	ADV
cana-540	467	12	,	,	PUNCT
cana-540	467	13	mass	mass	PROPN
cana-540	467	14	.	.	PROPN
cana-540	467	15	,	,	PUNCT
cana-540	467	16	1956	1956	NUM
cana-540	467	17	.	.	PUNCT
cana-540	468	1	[	[	X
cana-540	468	2	4	4	X
cana-540	468	3	]	]	X
cana-540	468	4	e.	e.	PROPN
cana-540	468	5	ekici	ekici	PROPN
cana-540	468	6	,	,	PUNCT
cana-540	468	7	j.h	j.h	PROPN
cana-540	468	8	.	.	PROPN
cana-540	468	9	park	park	PROPN
cana-540	468	10	,	,	PUNCT
cana-540	468	11	a	a	DET
cana-540	468	12	weak	weak	ADJ
cana-540	468	13	form	form	NOUN
cana-540	468	14	of	of	ADP
cana-540	468	15	some	some	DET
cana-540	468	16	types	type	NOUN
cana-540	468	17	of	of	ADP
cana-540	468	18	continuous	continuous	ADJ
cana-540	468	19	multifunctions	multifunction	NOUN
cana-540	468	20	,	,	PUNCT
cana-540	468	21	filomat	filomat	NOUN
cana-540	468	22	20(2),(2006	20(2),(2006	NOUN
cana-540	468	23	)	)	PUNCT
cana-540	468	24	,	,	PUNCT
cana-540	468	25	13	13	NUM
cana-540	468	26	-	-	SYM
cana-540	468	27	32	32	NUM
cana-540	468	28	.	.	PUNCT
cana-540	469	1	communications	communication	NOUN
cana-540	469	2	on	on	ADP
cana-540	469	3	applied	apply	VERB
cana-540	469	4	nonlinear	nonlinear	ADJ
cana-540	469	5	analysis	analysis	NOUN
cana-540	469	6	issn	issn	NOUN
cana-540	469	7	:	:	PUNCT
cana-540	469	8	1074	1074	NUM
cana-540	469	9	-	-	PUNCT
cana-540	469	10	133x	133x	NUM
cana-540	469	11	vol	vol	NOUN
cana-540	469	12	31	31	NUM
cana-540	469	13	no	no	NOUN
cana-540	469	14	.	.	NOUN
cana-540	469	15	2	2	NUM
cana-540	469	16	(	(	PUNCT
cana-540	469	17	2024	2024	NUM
cana-540	469	18	)	)	PUNCT
cana-540	469	19	258	258	NUM
cana-540	469	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-540	470	1	[	[	X
cana-540	470	2	5	5	NUM
cana-540	470	3	]	]	X
cana-540	470	4	d.k	d.k	PROPN
cana-540	470	5	.	.	PROPN
cana-540	470	6	ganguly	ganguly	PROPN
cana-540	470	7	,	,	PUNCT
cana-540	470	8	p.	p.	PROPN
cana-540	470	9	mallick	mallick	PROPN
cana-540	470	10	,	,	PUNCT
cana-540	470	11	on	on	ADP
cana-540	470	12	generalized	generalized	ADJ
cana-540	470	13	continuous	continuous	ADJ
cana-540	470	14	multifunctions	multifunction	NOUN
cana-540	470	15	and	and	CCONJ
cana-540	470	16	their	their	PRON
cana-540	470	17	selections	selection	NOUN
cana-540	470	18	,	,	PUNCT
cana-540	470	19	real	real	ADJ
cana-540	470	20	anal	anal	NOUN
cana-540	470	21	.	.	PUNCT
cana-540	471	1	exchange	exchange	NOUN
cana-540	471	2	,	,	PUNCT
cana-540	471	3	33	33	NUM
cana-540	471	4	,	,	PUNCT
cana-540	471	5	(	(	PUNCT
cana-540	471	6	2007	2007	NUM
cana-540	471	7	)	)	PUNCT
cana-540	471	8	,	,	PUNCT
cana-540	471	9	449	449	NUM
cana-540	471	10	-	-	NOUN
cana-540	471	11	456	456	NUM
cana-540	471	12	.	.	PUNCT
cana-540	472	1	[	[	X
cana-540	472	2	6	6	NUM
cana-540	472	3	]	]	PUNCT
cana-540	472	4	m.	m.	NOUN
cana-540	472	5	m.	m.	PROPN
cana-540	472	6	holliyavar	holliyavar	PROPN
cana-540	472	7	,	,	PUNCT
cana-540	472	8	t.	t.	PROPN
cana-540	472	9	d.	d.	PROPN
cana-540	472	10	rayanagoudar	rayanagoudar	PROPN
cana-540	472	11	and	and	CCONJ
cana-540	472	12	sarika	sarika	PROPN
cana-540	472	13	m.	m.	PROPN
cana-540	472	14	patil	patil	PROPN
cana-540	472	15	,	,	PUNCT
cana-540	472	16	on	on	ADP
cana-540	472	17	semi	semi	ADJ
cana-540	472	18	-	-	ADJ
cana-540	472	19	pre	pre	ADJ
cana-540	472	20	generalized	generalized	ADJ
cana-540	472	21	α	α	NOUN
cana-540	472	22	-	-	PUNCT
cana-540	472	23	closed	closed	ADJ
cana-540	472	24	sets	set	NOUN
cana-540	472	25	in	in	ADP
cana-540	472	26	topological	topological	ADJ
cana-540	472	27	spaces	space	NOUN
cana-540	472	28	,	,	PUNCT
cana-540	472	29	global	global	ADJ
cana-540	472	30	jl	jl	PROPN
cana-540	472	31	.	.	PUNCT
cana-540	472	32	of	of	ADP
cana-540	472	33	pure	pure	ADJ
cana-540	472	34	and	and	CCONJ
cana-540	472	35	appl	appl	ADJ
cana-540	472	36	.	.	PUNCT
cana-540	473	1	maths	maths	PROPN
cana-540	473	2	.	.	PUNCT
cana-540	473	3	,	,	PUNCT
cana-540	473	4	vol	vol	NOUN
cana-540	473	5	.	.	PROPN
cana-540	473	6	13	13	NUM
cana-540	473	7	(	(	PUNCT
cana-540	473	8	10	10	NUM
cana-540	473	9	)	)	PUNCT
cana-540	473	10	,	,	PUNCT
cana-540	473	11	(	(	PUNCT
cana-540	473	12	2017	2017	NUM
cana-540	473	13	)	)	PUNCT
cana-540	473	14	,	,	PUNCT
cana-540	473	15	7627	7627	NUM
cana-540	473	16	-	-	SYM
cana-540	473	17	7635	7635	NUM
cana-540	473	18	.	.	PUNCT
cana-540	474	1	[	[	X
cana-540	474	2	7	7	X
cana-540	474	3	]	]	X
cana-540	474	4	sarika	sarika	X
cana-540	474	5	m.	m.	PROPN
cana-540	474	6	patil	patil	PROPN
cana-540	474	7	,	,	PUNCT
cana-540	474	8	t.	t.	PROPN
cana-540	474	9	d.	d.	PROPN
cana-540	474	10	rayanagoudar	rayanagoudar	PROPN
cana-540	474	11	and	and	CCONJ
cana-540	474	12	m.	m.	PROPN
cana-540	474	13	m.	m.	PROPN
cana-540	474	14	holliyavar	holliyavar	PROPN
cana-540	474	15	,	,	PUNCT
cana-540	474	16	spg	spg	PRON
cana-540	474	17	αcontinuous	αcontinuous	ADJ
cana-540	474	18	functions	function	NOUN
cana-540	474	19	in	in	ADP
cana-540	474	20	topological	topological	ADJ
cana-540	474	21	spaces	space	NOUN
cana-540	474	22	,	,	PUNCT
cana-540	474	23	iosr	iosr	ADJ
cana-540	474	24	jl	jl	PROPN
cana-540	474	25	.	.	PROPN
cana-540	474	26	of	of	ADP
cana-540	474	27	mathematics	mathematics	PROPN
cana-540	474	28	,	,	PUNCT
cana-540	474	29	vol	vol	NOUN
cana-540	474	30	.	.	PROPN
cana-540	474	31	19	19	NUM
cana-540	474	32	,	,	PUNCT
cana-540	474	33	issue	issue	NOUN
cana-540	474	34	03	03	NUM
cana-540	474	35	,	,	PUNCT
cana-540	474	36	ser	ser	NOUN
cana-540	474	37	.	.	PROPN
cana-540	474	38	02	02	NUM
cana-540	474	39	,	,	PUNCT
cana-540	474	40	(	(	PUNCT
cana-540	474	41	2023	2023	NUM
cana-540	474	42	)	)	PUNCT
cana-540	474	43	,	,	PUNCT
cana-540	474	44	55	55	NUM
cana-540	474	45	-	-	SYM
cana-540	474	46	57	57	NUM
cana-540	474	47	.	.	PUNCT
cana-540	475	1	[	[	X
cana-540	475	2	8	8	NUM
cana-540	475	3	]	]	PUNCT
cana-540	475	4	t.	t.	PROPN
cana-540	475	5	noiri	noiri	PROPN
cana-540	475	6	,	,	PUNCT
cana-540	475	7	v.	v.	CCONJ
cana-540	475	8	popa	popa	NOUN
cana-540	475	9	,	,	PUNCT
cana-540	475	10	,	,	PUNCT
cana-540	475	11	almost	almost	ADV
cana-540	475	12	weakly	weakly	ADJ
cana-540	475	13	continuous	continuous	ADJ
cana-540	475	14	multifunctions	multifunction	NOUN
cana-540	475	15	,	,	PUNCT
cana-540	475	16	demonstratio	demonstratio	PROPN
cana-540	475	17	math	math	PROPN
cana-540	475	18	.	.	PUNCT
cana-540	476	1	26	26	NUM
cana-540	476	2	,	,	PUNCT
cana-540	476	3	(	(	PUNCT
cana-540	476	4	1993	1993	NUM
cana-540	476	5	)	)	PUNCT
cana-540	476	6	,	,	PUNCT
cana-540	476	7	363	363	NUM
cana-540	476	8	-	-	SYM
cana-540	476	9	380	380	NUM
cana-540	476	10	.	.	PUNCT
cana-540	477	1	[	[	X
cana-540	477	2	9	9	NUM
cana-540	477	3	]	]	X
cana-540	477	4	w.j	w.j	PROPN
cana-540	477	5	.	.	PROPN
cana-540	477	6	perwin	perwin	PROPN
cana-540	477	7	,	,	PUNCT
cana-540	477	8	foundations	foundation	NOUN
cana-540	477	9	of	of	ADP
cana-540	477	10	general	general	ADJ
cana-540	477	11	topology	topology	NOUN
cana-540	477	12	,	,	PUNCT
cana-540	477	13	academic	academic	PROPN
cana-540	477	14	press	press	PROPN
cana-540	477	15	inc	inc	PROPN
cana-540	477	16	.	.	PROPN
cana-540	477	17	,	,	PUNCT
cana-540	477	18	new	new	PROPN
cana-540	477	19	york	york	PROPN
cana-540	477	20	,	,	PUNCT
cana-540	477	21	1965	1965	NUM
cana-540	477	22	.	.	PUNCT
cana-540	478	1	[	[	X
cana-540	478	2	10	10	NUM
cana-540	478	3	]	]	PUNCT
cana-540	478	4	v.	v.	CCONJ
cana-540	478	5	popa	popa	NOUN
cana-540	478	6	,	,	PUNCT
cana-540	478	7	almost	almost	ADV
cana-540	478	8	continuous	continuous	ADJ
cana-540	478	9	multifunctions	multifunction	NOUN
cana-540	478	10	,	,	PUNCT
cana-540	478	11	mat	mat	PROPN
cana-540	478	12	.	.	PROPN
cana-540	478	13	vesnik	vesnik	PROPN
cana-540	478	14	6	6	NUM
cana-540	478	15	(	(	PUNCT
cana-540	478	16	19	19	NUM
cana-540	478	17	)	)	PUNCT
cana-540	478	18	,	,	PUNCT
cana-540	478	19	(	(	PUNCT
cana-540	478	20	1982	1982	NUM
cana-540	478	21	)	)	PUNCT
cana-540	478	22	,	,	PUNCT
cana-540	478	23	7584	7584	NUM
cana-540	478	24	.	.	PUNCT
cana-540	479	1	[	[	X
cana-540	479	2	11	11	NUM
cana-540	479	3	]	]	X
cana-540	479	4	g.t	g.t	PROPN
cana-540	479	5	.	.	PROPN
cana-540	479	6	whyburn	whyburn	NOUN
cana-540	479	7	,	,	PUNCT
cana-540	479	8	continuity	continuity	NOUN
cana-540	479	9	of	of	ADP
cana-540	479	10	multifunctions	multifunction	NOUN
cana-540	479	11	,	,	PUNCT
cana-540	479	12	proc	proc	NOUN
cana-540	479	13	.	.	PUNCT
cana-540	480	1	nat	nat	PROPN
cana-540	480	2	.	.	PUNCT
cana-540	481	1	acad	acad	PROPN
cana-540	481	2	.	.	PUNCT
cana-540	482	1	sci	sci	PROPN
cana-540	482	2	.	.	PUNCT
cana-540	482	3	u.s.a	u.s.a	PROPN
cana-540	482	4	54	54	NUM
cana-540	482	5	,	,	PUNCT
cana-540	482	6	(	(	PUNCT
cana-540	482	7	1964	1964	NUM
cana-540	482	8	)	)	PUNCT
cana-540	482	9	,	,	PUNCT
cana-540	482	10	1494	1494	NUM
cana-540	482	11	-	-	SYM
cana-540	482	12	1501	1501	NUM
cana-540	482	13	.	.	PUNCT
cana-540	483	1	[	[	X
cana-540	483	2	12	12	NUM
cana-540	483	3	]	]	X
cana-540	483	4	r.a	r.a	PROPN
cana-540	483	5	.	.	PROPN
cana-540	483	6	mahmoud	mahmoud	PROPN
cana-540	483	7	,	,	PUNCT
cana-540	483	8	on	on	ADP
cana-540	483	9	preirresolute	preirresolute	ADJ
cana-540	483	10	multivalued	multivalue	VERB
cana-540	483	11	functions	function	NOUN
cana-540	483	12	,	,	PUNCT
cana-540	483	13	demonstratio	demonstratio	PROPN
cana-540	483	14	.	.	PUNCT
cana-540	483	15	math	math	PROPN
cana-540	483	16	.	.	PUNCT
cana-540	483	17	,	,	PUNCT
cana-540	483	18	32(3	32(3	NUM
cana-540	483	19	)	)	PUNCT
cana-540	483	20	,	,	PUNCT
cana-540	483	21	(	(	PUNCT
cana-540	483	22	1999	1999	NUM
cana-540	483	23	)	)	PUNCT
cana-540	483	24	,	,	PUNCT
cana-540	483	25	621	621	NUM
cana-540	483	26	-	-	SYM
cana-540	483	27	628	628	NUM
cana-540	483	28	.	.	PUNCT
cana-540	484	1	[	[	X
cana-540	484	2	13	13	NUM
cana-540	484	3	]	]	PUNCT
cana-540	484	4	t.	t.	NOUN
cana-540	484	5	neubrunn	neubrunn	PROPN
cana-540	484	6	,	,	PUNCT
cana-540	484	7	strongly	strongly	ADV
cana-540	484	8	quasi	quasi	ADJ
cana-540	484	9	-	-	ADJ
cana-540	484	10	continuous	continuous	ADJ
cana-540	484	11	multivalued	multivalued	ADJ
cana-540	484	12	mappings	mapping	NOUN
cana-540	484	13	in	in	ADP
cana-540	484	14	general	general	ADJ
cana-540	484	15	topology	topology	NOUN
cana-540	484	16	and	and	CCONJ
cana-540	484	17	its	its	PRON
cana-540	484	18	relations	relation	NOUN
cana-540	484	19	to	to	ADP
cana-540	484	20	modern	modern	ADJ
cana-540	484	21	analysis	analysis	NOUN
cana-540	484	22	and	and	CCONJ
cana-540	484	23	algebra	algebra	NOUN
cana-540	484	24	,	,	PUNCT
cana-540	484	25	vi	vi	PROPN
cana-540	484	26	(	(	PUNCT
cana-540	484	27	prague	prague	NOUN
cana-540	484	28	1986	1986	NUM
cana-540	484	29	)	)	PUNCT
cana-540	484	30	,	,	PUNCT
cana-540	484	31	berlin	berlin	PROPN
cana-540	484	32	,	,	PUNCT
cana-540	484	33	heldermann	heldermann	NOUN
cana-540	484	34	,	,	PUNCT
cana-540	484	35	1988	1988	NUM
cana-540	484	36	,	,	PUNCT
cana-540	484	37	351	351	NUM
cana-540	484	38	-	-	SYM
cana-540	484	39	359	359	NUM
cana-540	484	40	.	.	PUNCT
cana-540	485	1	[	[	X
cana-540	485	2	14	14	NUM
cana-540	485	3	]	]	X
cana-540	485	4	v.	v.	CCONJ
cana-540	485	5	popa	popa	ADJ
cana-540	485	6	,	,	PUNCT
cana-540	485	7	weakly	weakly	ADJ
cana-540	485	8	continuous	continuous	ADJ
cana-540	485	9	multifunctions	multifunction	NOUN
cana-540	485	10	,	,	PUNCT
cana-540	485	11	boll	boll	NOUN
cana-540	485	12	.	.	PUNCT
cana-540	486	1	un	un	PROPN
cana-540	486	2	.	.	PROPN
cana-540	486	3	mat	mat	PROPN
cana-540	486	4	.	.	PUNCT
cana-540	486	5	ital	ital	PROPN
cana-540	486	6	.	.	PUNCT
cana-540	487	1	(	(	PUNCT
cana-540	487	2	5	5	NUM
cana-540	487	3	)	)	PUNCT
cana-540	487	4	15	15	NUM
cana-540	487	5	-	-	SYM
cana-540	487	6	24	24	NUM
cana-540	487	7	,	,	PUNCT
cana-540	487	8	(	(	PUNCT
cana-540	487	9	1978	1978	NUM
cana-540	487	10	)	)	PUNCT
cana-540	487	11	,	,	PUNCT
cana-540	487	12	379	379	NUM
cana-540	487	13	-	-	SYM
cana-540	487	14	388	388	NUM
cana-540	487	15	.	.	PUNCT