id	sid	tid	token	lemma	pos
cana-5238	1	1	communications	communication	NOUN
cana-5238	1	2	on	on	ADP
cana-5238	1	3	applied	apply	VERB
cana-5238	1	4	nonlinear	nonlinear	ADJ
cana-5238	1	5	analysis	analysis	NOUN
cana-5238	1	6	issn	issn	NOUN
cana-5238	1	7	:	:	PUNCT
cana-5238	1	8	1074	1074	NUM
cana-5238	1	9	-	-	PUNCT
cana-5238	1	10	133x	133x	NUM
cana-5238	1	11	vol	vol	VERB
cana-5238	1	12	32	32	NUM
cana-5238	1	13	no	no	NOUN
cana-5238	1	14	.	.	PUNCT
cana-5238	2	1	10s	10	NOUN
cana-5238	2	2	(	(	PUNCT
cana-5238	2	3	2025	2025	NUM
cana-5238	2	4	)	)	PUNCT
cana-5238	2	5	1358	1358	NUM
cana-5238	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	2	7	decomposition	decomposition	NOUN
cana-5238	2	8	of	of	ADP
cana-5238	2	9	(	(	PUNCT
cana-5238	2	10	gζ	gζ	PROPN
cana-5238	2	11	,	,	PUNCT
cana-5238	2	12	ξ	ξ	NOUN
cana-5238	2	13	)	)	PUNCT
cana-5238	2	14	-continuity	-continuity	PROPN
cana-5238	2	15	r.	r.	PROPN
cana-5238	2	16	ramesh1	ramesh1	PROPN
cana-5238	2	17	,	,	PUNCT
cana-5238	2	18	k.	k.	PROPN
cana-5238	2	19	rajupillai2,∗	rajupillai2,∗	PROPN
cana-5238	2	20	and	and	CCONJ
cana-5238	2	21	r.	r.	PROPN
cana-5238	2	22	uma	uma	PROPN
cana-5238	2	23	3	3	NUM
cana-5238	2	24	1	1	NUM
cana-5238	2	25	department	department	NOUN
cana-5238	2	26	of	of	ADP
cana-5238	2	27	mathematics	mathematics	PROPN
cana-5238	2	28	,	,	PUNCT
cana-5238	2	29	dr	dr	PROPN
cana-5238	2	30	.	.	PROPN
cana-5238	2	31	mahalingam	mahalingam	PROPN
cana-5238	2	32	college	college	PROPN
cana-5238	2	33	of	of	ADP
cana-5238	2	34	engineering	engineering	NOUN
cana-5238	2	35	and	and	CCONJ
cana-5238	2	36	technology	technology	NOUN
cana-5238	2	37	,	,	PUNCT
cana-5238	2	38	pollachi	pollachi	PROPN
cana-5238	2	39	,	,	PUNCT
cana-5238	2	40	tamil	tamil	PROPN
cana-5238	2	41	nadu	nadu	PROPN
cana-5238	2	42	,	,	PUNCT
cana-5238	2	43	india	india	PROPN
cana-5238	2	44	.	.	PUNCT
cana-5238	3	1	2∗	2∗	NUM
cana-5238	3	2	department	department	NOUN
cana-5238	3	3	of	of	ADP
cana-5238	3	4	mathematics	mathematic	NOUN
cana-5238	3	5	,	,	PUNCT
cana-5238	3	6	government	government	NOUN
cana-5238	3	7	college	college	NOUN
cana-5238	3	8	of	of	ADP
cana-5238	3	9	engineering	engineering	PROPN
cana-5238	3	10	,	,	PUNCT
cana-5238	3	11	thanjavur	thanjavur	PROPN
cana-5238	3	12	,	,	PUNCT
cana-5238	3	13	tamil	tamil	PROPN
cana-5238	3	14	nadu	nadu	PROPN
cana-5238	3	15	,	,	PUNCT
cana-5238	3	16	india	india	PROPN
cana-5238	3	17	.	.	PROPN
cana-5238	3	18	3	3	NUM
cana-5238	3	19	department	department	NOUN
cana-5238	3	20	of	of	ADP
cana-5238	3	21	mathematics	mathematics	PROPN
cana-5238	3	22	,	,	PUNCT
cana-5238	3	23	sree	sree	PROPN
cana-5238	3	24	saraswathi	saraswathi	PROPN
cana-5238	3	25	thyagaraja	thyagaraja	PROPN
cana-5238	3	26	college	college	PROPN
cana-5238	3	27	,	,	PUNCT
cana-5238	3	28	tamil	tamil	PROPN
cana-5238	3	29	nadu	nadu	PROPN
cana-5238	3	30	,	,	PUNCT
cana-5238	3	31	india	india	PROPN
cana-5238	3	32	.	.	PUNCT
cana-5238	3	33	∗-correspondance	∗-correspondance	PROPN
cana-5238	3	34	:	:	PUNCT
cana-5238	4	1	rajupillai@gct.ac	rajupillai@gct.ac	ADJ
cana-5238	4	2	.	.	PUNCT
cana-5238	5	1	article	article	NOUN
cana-5238	5	2	history	history	NOUN
cana-5238	5	3	:	:	PUNCT
cana-5238	5	4	received	receive	VERB
cana-5238	5	5	:	:	PUNCT
cana-5238	5	6	12	12	NUM
cana-5238	5	7	-	-	SYM
cana-5238	5	8	01	01	NUM
cana-5238	5	9	-	-	PUNCT
cana-5238	5	10	2025	2025	NUM
cana-5238	5	11	revised	revise	VERB
cana-5238	5	12	:	:	PUNCT
cana-5238	5	13	15	15	NUM
cana-5238	5	14	-	-	NUM
cana-5238	5	15	02	02	NUM
cana-5238	5	16	-	-	PUNCT
cana-5238	5	17	2025	2025	NUM
cana-5238	5	18	accepted	accept	VERB
cana-5238	5	19	:	:	PUNCT
cana-5238	5	20	01	01	NUM
cana-5238	5	21	-	-	SYM
cana-5238	5	22	03	03	NUM
cana-5238	5	23	-	-	PUNCT
cana-5238	5	24	2025	2025	NUM
cana-5238	5	25	abstract	abstract	NOUN
cana-5238	5	26	:	:	PUNCT
cana-5238	5	27	in	in	ADP
cana-5238	5	28	this	this	DET
cana-5238	5	29	work	work	NOUN
cana-5238	5	30	,	,	PUNCT
cana-5238	5	31	we	we	PRON
cana-5238	5	32	introduce	introduce	VERB
cana-5238	5	33	and	and	CCONJ
cana-5238	5	34	investigate	investigate	VERB
cana-5238	5	35	a	a	DET
cana-5238	5	36	new	new	ADJ
cana-5238	5	37	type	type	NOUN
cana-5238	5	38	of	of	ADP
cana-5238	5	39	open	open	ADJ
cana-5238	5	40	sets	set	NOUN
cana-5238	5	41	.	.	PUNCT
cana-5238	6	1	we	we	PRON
cana-5238	6	2	also	also	ADV
cana-5238	6	3	present	present	VERB
cana-5238	6	4	a	a	DET
cana-5238	6	5	decomposition	decomposition	NOUN
cana-5238	6	6	of	of	ADP
cana-5238	6	7	both	both	DET
cana-5238	6	8	(	(	PUNCT
cana-5238	6	9	gζ	gζ	PROPN
cana-5238	6	10	,	,	PUNCT
cana-5238	6	11	ξ	ξ	NOUN
cana-5238	6	12	)	)	PUNCT
cana-5238	6	13	c	c	NOUN
cana-5238	6	14	and	and	CCONJ
cana-5238	6	15	decomposition	decomposition	NOUN
cana-5238	6	16	of	of	ADP
cana-5238	6	17	(	(	PUNCT
cana-5238	6	18	ζ	ζ	NOUN
cana-5238	6	19	,	,	PUNCT
cana-5238	6	20	ξ)-c	ξ)-c	NOUN
cana-5238	6	21	.	.	PUNCT
cana-5238	7	1	keywords	keyword	NOUN
cana-5238	7	2	:	:	PUNCT
cana-5238	7	3	hereditary	hereditary	ADJ
cana-5238	7	4	generalized	generalize	VERB
cana-5238	7	5	topology	topology	NOUN
cana-5238	7	6	,	,	PUNCT
cana-5238	7	7	α	α	PROPN
cana-5238	7	8	-	-	PUNCT
cana-5238	7	9	hg	hg	NOUN
cana-5238	7	10	-	-	PROPN
cana-5238	7	11	o	o	PROPN
cana-5238	7	12	,	,	PUNCT
cana-5238	7	13	σ	σ	PROPN
cana-5238	7	14	-	-	PUNCT
cana-5238	7	15	hg	hg	NOUN
cana-5238	7	16	-	-	NOUN
cana-5238	7	17	o	o	NOUN
cana-5238	7	18	and	and	CCONJ
cana-5238	7	19	π	π	PROPN
cana-5238	7	20	-	-	PUNCT
cana-5238	7	21	hg	hg	NOUN
cana-5238	7	22	-	-	NOUN
cana-5238	7	23	o	o	NOUN
cana-5238	7	24	sets	set	NOUN
cana-5238	7	25	,	,	PUNCT
cana-5238	7	26	β	β	X
cana-5238	7	27	-	-	ADJ
cana-5238	7	28	hg	hg	NOUN
cana-5238	7	29	-	-	NOUN
cana-5238	7	30	o	o	NOUN
cana-5238	7	31	sets	set	NOUN
cana-5238	7	32	.	.	PUNCT
cana-5238	8	1	1	1	X
cana-5238	8	2	.	.	X
cana-5238	8	3	introduction	introduction	NOUN
cana-5238	8	4	in	in	ADP
cana-5238	8	5	2002	2002	NUM
cana-5238	8	6	,	,	PUNCT
cana-5238	8	7	generalized	generalized	ADJ
cana-5238	8	8	topology	topology	NOUN
cana-5238	8	9	and	and	CCONJ
cana-5238	8	10	generalized	generalized	ADJ
cana-5238	8	11	continuity	continuity	NOUN
cana-5238	8	12	introduced	introduce	VERB
cana-5238	8	13	by	by	ADP
cana-5238	8	14	csaszar	csaszar	NOUN
cana-5238	8	15	in	in	ADP
cana-5238	8	16	[	[	X
cana-5238	8	17	1	1	NUM
cana-5238	8	18	]	]	PUNCT
cana-5238	8	19	.	.	PUNCT
cana-5238	9	1	in	in	ADP
cana-5238	9	2	2005	2005	NUM
cana-5238	9	3	,	,	PUNCT
cana-5238	9	4	csaszar	csaszar	VERB
cana-5238	9	5	introduced	introduce	VERB
cana-5238	9	6	and	and	CCONJ
cana-5238	9	7	studied	study	VERB
cana-5238	9	8	generalized	generalized	ADJ
cana-5238	9	9	open	open	ADJ
cana-5238	9	10	sets	set	NOUN
cana-5238	9	11	(	(	PUNCT
cana-5238	9	12	ζ	ζ	NOUN
cana-5238	9	13	-	-	PUNCT
cana-5238	9	14	α	α	NOUN
cana-5238	9	15	-	-	PUNCT
cana-5238	9	16	o	o	NOUN
cana-5238	9	17	,	,	PUNCT
cana-5238	9	18	ζ	ζ	NOUN
cana-5238	9	19	-	-	PUNCT
cana-5238	9	20	σ	σ	NOUN
cana-5238	9	21	-	-	PUNCT
cana-5238	9	22	o	o	NOUN
cana-5238	9	23	,	,	PUNCT
cana-5238	9	24	ζ	ζ	NOUN
cana-5238	9	25	-	-	PUNCT
cana-5238	9	26	π	π	NOUN
cana-5238	9	27	-	-	NOUN
cana-5238	9	28	o	o	NOUN
cana-5238	9	29	,	,	PUNCT
cana-5238	9	30	ζ	ζ	NOUN
cana-5238	9	31	-	-	PUNCT
cana-5238	9	32	β	β	NOUN
cana-5238	9	33	-	-	PUNCT
cana-5238	9	34	o)[2	o)[2	NOUN
cana-5238	9	35	]	]	PUNCT
cana-5238	9	36	.	.	PUNCT
cana-5238	10	1	the	the	DET
cana-5238	10	2	notion	notion	NOUN
cana-5238	10	3	ζ	ζ	PROPN
cana-5238	10	4	-	-	PUNCT
cana-5238	10	5	b	b	NOUN
cana-5238	10	6	-	-	PUNCT
cana-5238	10	7	o	o	NOUN
cana-5238	10	8	introduced	introduce	VERB
cana-5238	10	9	by	by	ADP
cana-5238	10	10	sarsak	sarsak	NOUN
cana-5238	10	11	in	in	ADP
cana-5238	10	12	[	[	X
cana-5238	10	13	11	11	NUM
cana-5238	10	14	]	]	PUNCT
cana-5238	10	15	.	.	PUNCT
cana-5238	11	1	a	a	DET
cana-5238	11	2	space	space	NOUN
cana-5238	11	3	z	z	NOUN
cana-5238	11	4	is	be	AUX
cana-5238	11	5	called	call	VERB
cana-5238	11	6	a	a	DET
cana-5238	11	7	c0	c0	NOUN
cana-5238	11	8	-space	-space	PROPN
cana-5238	12	1	[	[	X
cana-5238	12	2	12	12	NUM
cana-5238	12	3	]	]	PUNCT
cana-5238	12	4	,	,	PUNCT
cana-5238	12	5	if	if	SCONJ
cana-5238	12	6	c0	c0	PROPN
cana-5238	12	7	=	=	PROPN
cana-5238	12	8	z	z	PROPN
cana-5238	12	9	,	,	PUNCT
cana-5238	12	10	where	where	SCONJ
cana-5238	12	11	c0	c0	PROPN
cana-5238	12	12	is	be	AUX
cana-5238	12	13	the	the	DET
cana-5238	12	14	set	set	NOUN
cana-5238	12	15	of	of	ADP
cana-5238	12	16	all	all	DET
cana-5238	12	17	representative	representative	ADJ
cana-5238	12	18	elements	element	NOUN
cana-5238	12	19	of	of	ADP
cana-5238	12	20	sets	set	NOUN
cana-5238	12	21	of	of	ADP
cana-5238	12	22	ζ	ζ	NOUN
cana-5238	12	23	and	and	CCONJ
cana-5238	12	24	x	x	NOUN
cana-5238	12	25	is	be	AUX
cana-5238	12	26	called	call	VERB
cana-5238	12	27	a	a	DET
cana-5238	12	28	represent	represent	ADJ
cana-5238	12	29	element	element	NOUN
cana-5238	12	30	of	of	ADP
cana-5238	12	31	u	u	PROPN
cana-5238	12	32	∈	∈	PROPN
cana-5238	12	33	ζ	ζ	NOUN
cana-5238	12	34	if	if	SCONJ
cana-5238	12	35	u	u	PROPN
cana-5238	12	36	⊂	⊂	X
cana-5238	12	37	v	v	NOUN
cana-5238	12	38	for	for	ADP
cana-5238	12	39	each	each	PRON
cana-5238	12	40	v	v	X
cana-5238	12	41	∈	∈	PROPN
cana-5238	12	42	ζ(x	ζ(x	NOUN
cana-5238	12	43	)	)	PUNCT
cana-5238	12	44	.	.	PUNCT
cana-5238	13	1	a	a	DET
cana-5238	13	2	subset	subset	NOUN
cana-5238	13	3	a	a	PRON
cana-5238	13	4	of	of	ADP
cana-5238	13	5	generalized	generalized	ADJ
cana-5238	13	6	topological	topological	ADJ
cana-5238	13	7	space	space	NOUN
cana-5238	13	8	(	(	PUNCT
cana-5238	13	9	x	x	NOUN
cana-5238	13	10	,	,	PUNCT
cana-5238	13	11	ζ	ζ	NOUN
cana-5238	13	12	)	)	PUNCT
cana-5238	13	13	is	be	AUX
cana-5238	13	14	said	say	VERB
cana-5238	13	15	to	to	PART
cana-5238	13	16	be	be	AUX
cana-5238	13	17	gζ	gζ	ADP
cana-5238	13	18	-closed	-close	VERB
cana-5238	13	19	[	[	X
cana-5238	13	20	4	4	NUM
cana-5238	13	21	]	]	PUNCT
cana-5238	13	22	(	(	PUNCT
cana-5238	13	23	resp	resp	NOUN
cana-5238	13	24	.	.	PUNCT
cana-5238	14	1	ωζ	ωζ	ADP
cana-5238	14	2	-closed	-close	VERB
cana-5238	14	3	[	[	X
cana-5238	14	4	7	7	NUM
cana-5238	14	5	]	]	NUM
cana-5238	14	6	)	)	PUNCT
cana-5238	14	7	,	,	PUNCT
cana-5238	14	8	if	if	SCONJ
cana-5238	14	9	c(a)⊆m	c(a)⊆m	ADP
cana-5238	14	10	whenever	whenever	SCONJ
cana-5238	14	11	a⊆m	a⊆m	ADJ
cana-5238	14	12	and	and	CCONJ
cana-5238	14	13	m	m	NOUN
cana-5238	14	14	is	be	AUX
cana-5238	14	15	ζ	ζ	NOUN
cana-5238	14	16	-	-	PUNCT
cana-5238	14	17	o	o	X
cana-5238	14	18	(	(	PUNCT
cana-5238	14	19	resp	resp	NOUN
cana-5238	14	20	.	.	PUNCT
cana-5238	15	1	ζ	ζ	X
cana-5238	15	2	-	-	PUNCT
cana-5238	15	3	σ	σ	NOUN
cana-5238	15	4	-	-	PUNCT
cana-5238	15	5	o	o	NOUN
cana-5238	15	6	)	)	PUNCT
cana-5238	15	7	in	in	ADP
cana-5238	15	8	x.	x.	NOUN
cana-5238	15	9	the	the	DET
cana-5238	15	10	complement	complement	NOUN
cana-5238	15	11	of	of	ADP
cana-5238	15	12	ωζ	ωζ	PUNCT
cana-5238	15	13	-closed	-closed	PROPN
cana-5238	15	14	(	(	PUNCT
cana-5238	15	15	resp	resp	NOUN
cana-5238	15	16	.	.	PUNCT
cana-5238	16	1	gζ	gζ	PROPN
cana-5238	16	2	-	-	PUNCT
cana-5238	16	3	o	o	PROPN
cana-5238	16	4	)	)	PUNCT
cana-5238	16	5	is	be	AUX
cana-5238	16	6	ωζ	ωζ	NOUN
cana-5238	16	7	-	-	PUNCT
cana-5238	16	8	o[7	o[7	NUM
cana-5238	16	9	]	]	PUNCT
cana-5238	16	10	(	(	PUNCT
cana-5238	16	11	resp	resp	NOUN
cana-5238	16	12	.	.	PUNCT
cana-5238	17	1	gζ	gζ	PROPN
cana-5238	17	2	-	-	PUNCT
cana-5238	17	3	o	o	NOUN
cana-5238	18	1	[	[	X
cana-5238	18	2	4	4	NUM
cana-5238	18	3	]	]	NUM
cana-5238	18	4	)	)	PUNCT
cana-5238	18	5	.	.	PUNCT
cana-5238	19	1	the	the	DET
cana-5238	19	2	gζ	gζ	PROPN
cana-5238	19	3	-	-	PUNCT
cana-5238	19	4	interior	interior	ADJ
cana-5238	19	5	(	(	PUNCT
cana-5238	19	6	resp	resp	NOUN
cana-5238	19	7	.	.	PUNCT
cana-5238	20	1	ωζ	ωζ	PROPN
cana-5238	20	2	-	-	NOUN
cana-5238	20	3	interior	interior	NOUN
cana-5238	20	4	)	)	PUNCT
cana-5238	20	5	is	be	AUX
cana-5238	20	6	the	the	DET
cana-5238	20	7	largest	large	ADJ
cana-5238	20	8	gζ	gζ	NOUN
cana-5238	20	9	-	-	PUNCT
cana-5238	20	10	o	o	PROPN
cana-5238	20	11	(	(	PUNCT
cana-5238	20	12	resp	resp	NOUN
cana-5238	20	13	.	.	PUNCT
cana-5238	21	1	ωζ	ωζ	PROPN
cana-5238	21	2	-	-	PUNCT
cana-5238	21	3	o	o	NOUN
cana-5238	21	4	)	)	PUNCT
cana-5238	21	5	set	set	NOUN
cana-5238	21	6	contained	contain	VERB
cana-5238	21	7	in	in	ADP
cana-5238	21	8	a	a	PRON
cana-5238	21	9	and	and	CCONJ
cana-5238	21	10	is	be	AUX
cana-5238	21	11	denoted	denote	VERB
cana-5238	21	12	by	by	ADP
cana-5238	21	13	ig(a	ig(a	X
cana-5238	21	14	)	)	PUNCT
cana-5238	21	15	(	(	PUNCT
cana-5238	21	16	resp	resp	NOUN
cana-5238	21	17	.	.	PUNCT
cana-5238	21	18	iω(a	iω(a	PROPN
cana-5238	21	19	)	)	PUNCT
cana-5238	21	20	)	)	PUNCT
cana-5238	21	21	.	.	PUNCT
cana-5238	22	1	in	in	ADP
cana-5238	22	2	2005	2005	NUM
cana-5238	22	3	,	,	PUNCT
cana-5238	22	4	csaszar	csaszar	VERB
cana-5238	22	5	introduced	introduce	VERB
cana-5238	22	6	hereditary	hereditary	ADJ
cana-5238	22	7	class	class	NOUN
cana-5238	22	8	in	in	ADP
cana-5238	22	9	[	[	X
cana-5238	22	10	3	3	NUM
cana-5238	22	11	]	]	PUNCT
cana-5238	22	12	.	.	PUNCT
cana-5238	23	1	in	in	ADP
cana-5238	23	2	this	this	DET
cana-5238	23	3	work	work	NOUN
cana-5238	23	4	hereditary	hereditary	ADJ
cana-5238	23	5	generalized	generalized	ADJ
cana-5238	23	6	topological	topological	ADJ
cana-5238	23	7	space	space	NOUN
cana-5238	23	8	(	(	PUNCT
cana-5238	23	9	z	z	NOUN
cana-5238	23	10	,	,	PUNCT
cana-5238	23	11	ζ	ζ	NOUN
cana-5238	23	12	,	,	PUNCT
cana-5238	23	13	h	h	NOUN
cana-5238	23	14	)	)	PUNCT
cana-5238	23	15	is	be	AUX
cana-5238	23	16	denoted	denote	VERB
cana-5238	23	17	by	by	ADP
cana-5238	23	18	hgts	hgts	NOUN
cana-5238	23	19	.	.	PUNCT
cana-5238	24	1	definition	definition	NOUN
cana-5238	24	2	1.1	1.1	NUM
cana-5238	24	3	.	.	PUNCT
cana-5238	25	1	[	[	X
cana-5238	25	2	3	3	X
cana-5238	25	3	]	]	PUNCT
cana-5238	25	4	the	the	DET
cana-5238	25	5	set	set	NOUN
cana-5238	25	6	ψ	ψ	NOUN
cana-5238	25	7	is	be	AUX
cana-5238	25	8	said	say	VERB
cana-5238	25	9	to	to	PART
cana-5238	25	10	be	be	AUX
cana-5238	25	11	α	α	NOUN
cana-5238	25	12	-	-	ADJ
cana-5238	25	13	h	h	NOUN
cana-5238	25	14	-	-	PUNCT
cana-5238	25	15	o	o	X
cana-5238	25	16	(	(	PUNCT
cana-5238	25	17	resp	resp	NOUN
cana-5238	25	18	.	.	PUNCT
cana-5238	26	1	σ	σ	PROPN
cana-5238	26	2	-	-	PUNCT
cana-5238	26	3	h	h	NOUN
cana-5238	26	4	-	-	PUNCT
cana-5238	26	5	o	o	NOUN
cana-5238	26	6	,	,	PUNCT
cana-5238	26	7	π	π	PROPN
cana-5238	26	8	-	-	NOUN
cana-5238	26	9	h	h	NOUN
cana-5238	26	10	-	-	PUNCT
cana-5238	26	11	o	o	NOUN
cana-5238	26	12	,	,	PUNCT
cana-5238	26	13	β	β	PROPN
cana-5238	26	14	-	-	ADJ
cana-5238	26	15	h	h	NOUN
cana-5238	26	16	-	-	PUNCT
cana-5238	26	17	o	o	NOUN
cana-5238	26	18	,	,	PUNCT
cana-5238	26	19	β∗-h	β∗-h	PUNCT
cana-5238	26	20	-	-	PUNCT
cana-5238	26	21	o	o	PROPN
cana-5238	26	22	,	,	PUNCT
cana-5238	26	23	ζ∗	ζ∗	PROPN
cana-5238	26	24	-closed	-close	VERB
cana-5238	26	25	)	)	PUNCT
cana-5238	26	26	,	,	PUNCT
cana-5238	26	27	if	if	SCONJ
cana-5238	26	28	ψ⊆ic∗(ψ	ψ⊆ic∗(ψ	PROPN
cana-5238	26	29	)	)	PUNCT
cana-5238	26	30	(	(	PUNCT
cana-5238	26	31	resp	resp	NOUN
cana-5238	26	32	.	.	PUNCT
cana-5238	27	1	ψ⊆c∗i(ψ	ψ⊆c∗i(ψ	NOUN
cana-5238	27	2	)	)	PUNCT
cana-5238	27	3	,	,	PUNCT
cana-5238	27	4	ψ⊆ic∗(ψ	ψ⊆ic∗(ψ	PROPN
cana-5238	27	5	)	)	PUNCT
cana-5238	27	6	,	,	PUNCT
cana-5238	27	7	ψ⊆cic∗(ψ	ψ⊆cic∗(ψ	PROPN
cana-5238	27	8	)	)	PUNCT
cana-5238	27	9	,	,	PUNCT
cana-5238	27	10	ψ⊆c∗ic∗(ψ	ψ⊆c∗ic∗(ψ	PROPN
cana-5238	27	11	)	)	PUNCT
cana-5238	27	12	,	,	PUNCT
cana-5238	27	13	c∗(ψ)⊂ψ	c∗(ψ)⊂ψ	NOUN
cana-5238	27	14	)	)	PUNCT
cana-5238	27	15	.	.	PUNCT
cana-5238	28	1	definition	definition	NOUN
cana-5238	28	2	1.2	1.2	NUM
cana-5238	28	3	.	.	PUNCT
cana-5238	29	1	a	a	DET
cana-5238	29	2	set	set	NOUN
cana-5238	29	3	ψ	ψ	NOUN
cana-5238	29	4	is	be	AUX
cana-5238	29	5	said	say	VERB
cana-5238	29	6	to	to	PART
cana-5238	29	7	be	be	AUX
cana-5238	29	8	b	b	NUM
cana-5238	29	9	-	-	PUNCT
cana-5238	29	10	h	h	NOUN
cana-5238	29	11	-	-	NOUN
cana-5238	29	12	o	o	NOUN
cana-5238	30	1	[	[	X
cana-5238	30	2	8	8	NUM
cana-5238	30	3	]	]	PUNCT
cana-5238	30	4	,	,	PUNCT
cana-5238	30	5	if	if	SCONJ
cana-5238	30	6	ψ	ψ	ADP
cana-5238	30	7	⊆	⊆	NUM
cana-5238	30	8	ic∗(ψ)∪c∗i	ic∗(ψ)∪c∗i	NOUN
cana-5238	30	9	(	(	PUNCT
cana-5238	30	10	ψ	ψ	NOUN
cana-5238	30	11	)	)	PUNCT
cana-5238	30	12	.	.	PUNCT
cana-5238	31	1	definition	definition	NOUN
cana-5238	31	2	1.3	1.3	NUM
cana-5238	31	3	.	.	PUNCT
cana-5238	32	1	[	[	X
cana-5238	32	2	10	10	NUM
cana-5238	32	3	]	]	X
cana-5238	32	4	a	a	DET
cana-5238	32	5	set	set	NOUN
cana-5238	32	6	ψ	ψ	NOUN
cana-5238	32	7	is	be	AUX
cana-5238	32	8	said	say	VERB
cana-5238	32	9	to	to	PART
cana-5238	32	10	be	be	AUX
cana-5238	32	11	1	1	NUM
cana-5238	32	12	.	.	PUNCT
cana-5238	33	1	α	α	X
cana-5238	33	2	-	-	PUNCT
cana-5238	33	3	hg	hg	NOUN
cana-5238	33	4	-	-	NOUN
cana-5238	33	5	o	o	NOUN
cana-5238	33	6	,	,	PUNCT
cana-5238	33	7	if	if	SCONJ
cana-5238	33	8	ψ	ψ	ADP
cana-5238	33	9	⊆	⊆	NUM
cana-5238	33	10	igc∗ig(ψ	igc∗ig(ψ	NOUN
cana-5238	33	11	)	)	PUNCT
cana-5238	33	12	.	.	PUNCT
cana-5238	34	1	2	2	X
cana-5238	34	2	.	.	X
cana-5238	34	3	σ	σ	PROPN
cana-5238	34	4	-	-	PUNCT
cana-5238	34	5	hg	hg	NOUN
cana-5238	34	6	-	-	NOUN
cana-5238	34	7	o	o	NOUN
cana-5238	34	8	,	,	PUNCT
cana-5238	34	9	if	if	SCONJ
cana-5238	34	10	ψ	ψ	ADP
cana-5238	34	11	⊆	⊆	NUM
cana-5238	34	12	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	34	13	)	)	PUNCT
cana-5238	34	14	.	.	PUNCT
cana-5238	35	1	3	3	X
cana-5238	35	2	.	.	X
cana-5238	35	3	π	π	PROPN
cana-5238	35	4	-	-	PUNCT
cana-5238	35	5	hg	hg	NOUN
cana-5238	35	6	-	-	NOUN
cana-5238	35	7	o	o	NOUN
cana-5238	35	8	,	,	PUNCT
cana-5238	35	9	if	if	SCONJ
cana-5238	35	10	ψ	ψ	ADP
cana-5238	35	11	⊆	⊆	NUM
cana-5238	35	12	igc∗(ψ	igc∗(ψ	NUM
cana-5238	35	13	)	)	PUNCT
cana-5238	35	14	.	.	PUNCT
cana-5238	36	1	4	4	X
cana-5238	36	2	.	.	X
cana-5238	36	3	β	β	X
cana-5238	36	4	-	-	PUNCT
cana-5238	36	5	hg	hg	NOUN
cana-5238	36	6	-	-	NOUN
cana-5238	36	7	o	o	NOUN
cana-5238	36	8	,	,	PUNCT
cana-5238	36	9	if	if	SCONJ
cana-5238	36	10	ψ	ψ	ADP
cana-5238	36	11	⊆	⊆	NUM
cana-5238	36	12	cigc∗(ψ	cigc∗(ψ	NUM
cana-5238	36	13	)	)	PUNCT
cana-5238	36	14	.	.	PUNCT
cana-5238	37	1	5	5	X
cana-5238	37	2	.	.	X
cana-5238	37	3	s	s	X
cana-5238	37	4	-	-	PUNCT
cana-5238	37	5	β	β	NOUN
cana-5238	37	6	-	-	PUNCT
cana-5238	37	7	hgo	hgo	NOUN
cana-5238	37	8	,	,	PUNCT
cana-5238	37	9	if	if	SCONJ
cana-5238	37	10	ψ	ψ	ADP
cana-5238	37	11	⊆	⊆	NUM
cana-5238	37	12	c∗igc∗(ψ	c∗igc∗(ψ	X
cana-5238	37	13	)	)	PUNCT
cana-5238	37	14	.	.	PUNCT
cana-5238	38	1	mailto:-correspondance:%20rajupillai@gct.ac	mailto:-correspondance:%20rajupillai@gct.ac	NOUN
cana-5238	38	2	.	.	PUNCT
cana-5238	39	1	communications	communication	NOUN
cana-5238	39	2	on	on	ADP
cana-5238	39	3	applied	apply	VERB
cana-5238	39	4	nonlinear	nonlinear	ADJ
cana-5238	39	5	analysis	analysis	NOUN
cana-5238	39	6	issn	issn	NOUN
cana-5238	39	7	:	:	PUNCT
cana-5238	39	8	1074	1074	NUM
cana-5238	39	9	-	-	PUNCT
cana-5238	39	10	133x	133x	NUM
cana-5238	39	11	vol	vol	VERB
cana-5238	39	12	32	32	NUM
cana-5238	39	13	no	no	NOUN
cana-5238	39	14	.	.	PUNCT
cana-5238	40	1	10s	10	NOUN
cana-5238	40	2	(	(	PUNCT
cana-5238	40	3	2025	2025	NUM
cana-5238	40	4	)	)	PUNCT
cana-5238	40	5	1359	1359	NUM
cana-5238	40	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	40	7	2	2	NUM
cana-5238	40	8	.	.	X
cana-5238	40	9	b	b	X
cana-5238	40	10	-	-	PUNCT
cana-5238	40	11	hg	hg	NOUN
cana-5238	40	12	-	-	ADJ
cana-5238	40	13	open	open	ADJ
cana-5238	40	14	set	set	VERB
cana-5238	40	15	definition	definition	NOUN
cana-5238	40	16	2.1	2.1	NUM
cana-5238	40	17	.	.	PUNCT
cana-5238	41	1	the	the	DET
cana-5238	41	2	set	set	PROPN
cana-5238	41	3	ψ⊂(z	ψ⊂(z	NOUN
cana-5238	41	4	,	,	PUNCT
cana-5238	41	5	ζ	ζ	NOUN
cana-5238	41	6	,	,	PUNCT
cana-5238	41	7	h	h	NOUN
cana-5238	41	8	)	)	PUNCT
cana-5238	41	9	is	be	AUX
cana-5238	41	10	called	call	VERB
cana-5238	41	11	as	as	ADP
cana-5238	41	12	b	b	NOUN
cana-5238	41	13	-	-	PUNCT
cana-5238	41	14	hg	hg	NOUN
cana-5238	41	15	-	-	ADJ
cana-5238	41	16	open	open	ADJ
cana-5238	41	17	(	(	PUNCT
cana-5238	41	18	b	b	X
cana-5238	41	19	-	-	PUNCT
cana-5238	41	20	hg	hg	NOUN
cana-5238	41	21	-	-	NOUN
cana-5238	41	22	o	o	NOUN
cana-5238	41	23	)	)	PUNCT
cana-5238	41	24	,	,	PUNCT
cana-5238	41	25	if	if	SCONJ
cana-5238	41	26	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	41	27	)	)	PUNCT
cana-5238	41	28	.	.	PUNCT
cana-5238	42	1	proposition	proposition	NOUN
cana-5238	42	2	2.2	2.2	NUM
cana-5238	42	3	.	.	PUNCT
cana-5238	43	1	in	in	ADP
cana-5238	43	2	hgt	hgt	PROPN
cana-5238	43	3	s	s	PROPN
cana-5238	43	4	,	,	PUNCT
cana-5238	43	5	every	every	DET
cana-5238	43	6	ζ	ζ	NOUN
cana-5238	43	7	-	-	PUNCT
cana-5238	43	8	o	o	NOUN
cana-5238	43	9	set	set	NOUN
cana-5238	43	10	is	be	AUX
cana-5238	43	11	b	b	NUM
cana-5238	43	12	-	-	PUNCT
cana-5238	43	13	hg	hg	NOUN
cana-5238	43	14	-	-	ADJ
cana-5238	43	15	o.	o.	ADJ
cana-5238	43	16	proof	proof	NOUN
cana-5238	43	17	.	.	PUNCT
cana-5238	44	1	a	a	DET
cana-5238	44	2	subset	subset	NOUN
cana-5238	44	3	ψ⊂z	ψ⊂z	NOUN
cana-5238	44	4	is	be	AUX
cana-5238	44	5	ζ	ζ	NOUN
cana-5238	44	6	-	-	ADV
cana-5238	44	7	o.	o.	NOUN
cana-5238	44	8	then	then	ADV
cana-5238	44	9	ψ	ψ	X
cana-5238	44	10	=	=	X
cana-5238	44	11	i(ψ	i(ψ	PROPN
cana-5238	44	12	)	)	PUNCT
cana-5238	44	13	.	.	PUNCT
cana-5238	45	1	now	now	ADV
cana-5238	45	2	ψ	ψ	ADP
cana-5238	45	3	⊆	⊆	NUM
cana-5238	45	4	i(ψ	i(ψ	PROPN
cana-5238	45	5	)	)	PUNCT
cana-5238	45	6	⊆	⊆	NUM
cana-5238	45	7	ig(ψ	ig(ψ	NOUN
cana-5238	45	8	)	)	PUNCT
cana-5238	45	9	⊆igc∗(ψ	⊆igc∗(ψ	PROPN
cana-5238	45	10	)	)	PUNCT
cana-5238	45	11	∪	∪	ADP
cana-5238	45	12	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	45	13	)	)	PUNCT
cana-5238	45	14	.	.	PUNCT
cana-5238	46	1	hence	hence	ADV
cana-5238	46	2	ψ	ψ	X
cana-5238	46	3	is	be	AUX
cana-5238	46	4	bhg	bhg	NOUN
cana-5238	46	5	-	-	PUNCT
cana-5238	46	6	o.	o.	NOUN
cana-5238	46	7	remark	remark	NOUN
cana-5238	46	8	2.3	2.3	NUM
cana-5238	46	9	.	.	PUNCT
cana-5238	47	1	the	the	DET
cana-5238	47	2	converse	converse	NOUN
cana-5238	47	3	of	of	ADP
cana-5238	47	4	proposition	proposition	NOUN
cana-5238	47	5	2.2	2.2	NUM
cana-5238	47	6	need	need	AUX
cana-5238	47	7	not	not	PART
cana-5238	47	8	be	be	AUX
cana-5238	47	9	true	true	ADJ
cana-5238	47	10	from	from	ADP
cana-5238	47	11	the	the	DET
cana-5238	47	12	following	follow	VERB
cana-5238	47	13	example	example	NOUN
cana-5238	47	14	.	.	PUNCT
cana-5238	48	1	example	example	NOUN
cana-5238	48	2	2.4	2.4	NUM
cana-5238	48	3	.	.	PUNCT
cana-5238	49	1	assume	assume	VERB
cana-5238	49	2	z	z	NOUN
cana-5238	49	3	=	=	PUNCT
cana-5238	49	4	{	{	PUNCT
cana-5238	49	5	1	1	NUM
cana-5238	49	6	,	,	PUNCT
cana-5238	49	7	2	2	NUM
cana-5238	49	8	,	,	PUNCT
cana-5238	49	9	3	3	NUM
cana-5238	49	10	,	,	PUNCT
cana-5238	49	11	4	4	NUM
cana-5238	49	12	}	}	PUNCT
cana-5238	49	13	,	,	PUNCT
cana-5238	49	14	ζ={∅	ζ={∅	PROPN
cana-5238	49	15	,	,	PUNCT
cana-5238	49	16	{	{	PUNCT
cana-5238	49	17	1	1	NUM
cana-5238	49	18	}	}	PUNCT
cana-5238	49	19	,	,	PUNCT
cana-5238	49	20	{	{	PUNCT
cana-5238	49	21	2	2	NUM
cana-5238	49	22	}	}	PUNCT
cana-5238	49	23	,	,	PUNCT
cana-5238	49	24	{	{	PUNCT
cana-5238	49	25	1	1	NUM
cana-5238	49	26	,	,	PUNCT
cana-5238	49	27	2	2	NUM
cana-5238	49	28	}	}	PUNCT
cana-5238	49	29	,	,	PUNCT
cana-5238	49	30	{	{	PUNCT
cana-5238	49	31	2	2	NUM
cana-5238	49	32	,	,	PUNCT
cana-5238	49	33	3	3	NUM
cana-5238	49	34	,	,	PUNCT
cana-5238	49	35	4	4	NUM
cana-5238	49	36	}	}	PUNCT
cana-5238	49	37	,	,	PUNCT
cana-5238	49	38	z	z	NOUN
cana-5238	49	39	}	}	PUNCT
cana-5238	49	40	,	,	PUNCT
cana-5238	49	41	h	h	NOUN
cana-5238	49	42	=	=	PRON
cana-5238	49	43	{	{	PUNCT
cana-5238	49	44	∅	∅	NOUN
cana-5238	49	45	,	,	PUNCT
cana-5238	49	46	{	{	PUNCT
cana-5238	49	47	1	1	NUM
cana-5238	49	48	}	}	PUNCT
cana-5238	49	49	,	,	PUNCT
cana-5238	49	50	{	{	PUNCT
cana-5238	49	51	3	3	NUM
cana-5238	49	52	}	}	PUNCT
cana-5238	49	53	}	}	PUNCT
cana-5238	49	54	.	.	PUNCT
cana-5238	50	1	then	then	ADV
cana-5238	50	2	ψ	ψ	X
cana-5238	50	3	=	=	PUNCT
cana-5238	50	4	{	{	PUNCT
cana-5238	50	5	1	1	NUM
cana-5238	50	6	,	,	PUNCT
cana-5238	50	7	2	2	NUM
cana-5238	50	8	,	,	PUNCT
cana-5238	50	9	3	3	NUM
cana-5238	50	10	}	}	PUNCT
cana-5238	50	11	is	be	AUX
cana-5238	50	12	b	b	NUM
cana-5238	50	13	-	-	PUNCT
cana-5238	50	14	hg	hg	NOUN
cana-5238	50	15	-	-	NOUN
cana-5238	50	16	o	o	NOUN
cana-5238	50	17	but	but	CCONJ
cana-5238	50	18	not	not	PART
cana-5238	50	19	ζ	ζ	NOUN
cana-5238	50	20	-	-	PUNCT
cana-5238	50	21	o.	o.	ADJ
cana-5238	50	22	proposition	proposition	NOUN
cana-5238	50	23	2.5	2.5	NUM
cana-5238	50	24	.	.	PUNCT
cana-5238	51	1	in	in	ADP
cana-5238	51	2	hgts	hgts	PROPN
cana-5238	51	3	(	(	PUNCT
cana-5238	51	4	z	z	NOUN
cana-5238	51	5	,	,	PUNCT
cana-5238	51	6	ζ	ζ	NOUN
cana-5238	51	7	,	,	PUNCT
cana-5238	51	8	h	h	NOUN
cana-5238	51	9	)	)	PUNCT
cana-5238	51	10	,	,	PUNCT
cana-5238	51	11	every	every	DET
cana-5238	51	12	gζ	gζ	PROPN
cana-5238	51	13	-	-	PUNCT
cana-5238	51	14	o	o	NOUN
cana-5238	51	15	set	set	NOUN
cana-5238	51	16	is	be	AUX
cana-5238	51	17	b	b	NOUN
cana-5238	51	18	-	-	PUNCT
cana-5238	51	19	hg	hg	NOUN
cana-5238	51	20	-	-	NOUN
cana-5238	51	21	o	o	NOUN
cana-5238	51	22	but	but	CCONJ
cana-5238	51	23	not	not	PART
cana-5238	51	24	conversely	conversely	ADV
cana-5238	51	25	.	.	PUNCT
cana-5238	52	1	proof	proof	NOUN
cana-5238	52	2	.	.	PUNCT
cana-5238	53	1	assume	assume	VERB
cana-5238	53	2	a	a	DET
cana-5238	53	3	subset	subset	NOUN
cana-5238	53	4	ψ	ψ	NOUN
cana-5238	53	5	of	of	ADP
cana-5238	53	6	hgts	hgts	NOUN
cana-5238	53	7	(	(	PUNCT
cana-5238	53	8	z	z	NOUN
cana-5238	53	9	,	,	PUNCT
cana-5238	53	10	ζ	ζ	NOUN
cana-5238	53	11	,	,	PUNCT
cana-5238	53	12	h	h	NOUN
cana-5238	53	13	)	)	PUNCT
cana-5238	53	14	is	be	AUX
cana-5238	53	15	gζo	gζo	ADJ
cana-5238	53	16	.	.	PUNCT
cana-5238	54	1	then	then	ADV
cana-5238	54	2	ψ	ψ	X
cana-5238	54	3	=	=	NOUN
cana-5238	54	4	ig(ψ	ig(ψ	NOUN
cana-5238	54	5	)	)	PUNCT
cana-5238	54	6	.	.	PUNCT
cana-5238	55	1	now	now	ADV
cana-5238	55	2	ψ	ψ	ADP
cana-5238	55	3	⊆	⊆	NUM
cana-5238	55	4	ig(ψ	ig(ψ	NOUN
cana-5238	55	5	)	)	PUNCT
cana-5238	55	6	⊆	⊆	NUM
cana-5238	55	7	igc∗(ψ	igc∗(ψ	NUM
cana-5238	55	8	)	)	PUNCT
cana-5238	55	9	∪	∪	ADP
cana-5238	55	10	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	55	11	)	)	PUNCT
cana-5238	55	12	.	.	PUNCT
cana-5238	56	1	hence	hence	ADV
cana-5238	56	2	ψ	ψ	NOUN
cana-5238	56	3	is	be	AUX
cana-5238	56	4	b	b	NUM
cana-5238	56	5	-	-	PUNCT
cana-5238	56	6	hg	hg	NOUN
cana-5238	56	7	-	-	ADJ
cana-5238	56	8	o.	o.	ADJ
cana-5238	56	9	proposition	proposition	NOUN
cana-5238	56	10	2.6	2.6	NUM
cana-5238	56	11	.	.	PUNCT
cana-5238	57	1	in	in	ADP
cana-5238	57	2	hgts	hgts	NOUN
cana-5238	57	3	,	,	PUNCT
cana-5238	57	4	every	every	DET
cana-5238	57	5	ωζ	ωζ	PROPN
cana-5238	57	6	-	-	PUNCT
cana-5238	57	7	o	o	NOUN
cana-5238	57	8	set	set	NOUN
cana-5238	57	9	is	be	AUX
cana-5238	57	10	b	b	NOUN
cana-5238	57	11	-	-	PUNCT
cana-5238	57	12	hg	hg	NOUN
cana-5238	57	13	-	-	NOUN
cana-5238	57	14	o	o	NOUN
cana-5238	57	15	but	but	CCONJ
cana-5238	57	16	not	not	PART
cana-5238	57	17	conversely	conversely	ADV
cana-5238	57	18	.	.	PUNCT
cana-5238	58	1	proof	proof	NOUN
cana-5238	58	2	.	.	PUNCT
cana-5238	59	1	assume	assume	VERB
cana-5238	59	2	a	a	DET
cana-5238	59	3	subset	subset	NOUN
cana-5238	59	4	ψ	ψ	X
cana-5238	59	5	of	of	ADP
cana-5238	59	6	hgt	hgt	PROPN
cana-5238	59	7	s	s	PROPN
cana-5238	59	8	(	(	PUNCT
cana-5238	59	9	z	z	NOUN
cana-5238	59	10	,	,	PUNCT
cana-5238	59	11	ζ	ζ	NOUN
cana-5238	59	12	,	,	PUNCT
cana-5238	59	13	h	h	NOUN
cana-5238	59	14	)	)	PUNCT
cana-5238	59	15	is	be	AUX
cana-5238	59	16	ωζ	ωζ	NOUN
cana-5238	59	17	-	-	NOUN
cana-5238	59	18	o.	o.	NOUN
cana-5238	59	19	then	then	ADV
cana-5238	59	20	ψ	ψ	X
cana-5238	59	21	=	=	X
cana-5238	59	22	iω(ψ	iω(ψ	NOUN
cana-5238	59	23	)	)	PUNCT
cana-5238	59	24	.	.	PUNCT
cana-5238	60	1	now	now	ADV
cana-5238	60	2	ψ⊆iω(ψ	ψ⊆iω(ψ	NOUN
cana-5238	60	3	)	)	PUNCT
cana-5238	60	4	⊆	⊆	NUM
cana-5238	60	5	ig(ψ	ig(ψ	NOUN
cana-5238	60	6	)	)	PUNCT
cana-5238	60	7	⊆	⊆	NUM
cana-5238	60	8	igc∗(ψ	igc∗(ψ	NUM
cana-5238	60	9	)	)	PUNCT
cana-5238	60	10	∪	∪	ADP
cana-5238	60	11	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	60	12	)	)	PUNCT
cana-5238	60	13	.	.	PUNCT
cana-5238	61	1	hence	hence	ADV
cana-5238	61	2	ψ	ψ	NOUN
cana-5238	61	3	is	be	AUX
cana-5238	61	4	b	b	NUM
cana-5238	61	5	-	-	PUNCT
cana-5238	61	6	hg	hg	NOUN
cana-5238	61	7	-	-	PROPN
cana-5238	61	8	o.	o.	ADJ
cana-5238	61	9	example	example	NOUN
cana-5238	61	10	2.7	2.7	NUM
cana-5238	61	11	.	.	PUNCT
cana-5238	62	1	assume	assume	VERB
cana-5238	62	2	z	z	NOUN
cana-5238	62	3	=	=	PUNCT
cana-5238	62	4	{	{	PUNCT
cana-5238	62	5	1	1	NUM
cana-5238	62	6	,	,	PUNCT
cana-5238	62	7	2	2	NUM
cana-5238	62	8	,	,	PUNCT
cana-5238	62	9	3	3	NUM
cana-5238	62	10	,	,	PUNCT
cana-5238	62	11	4	4	NUM
cana-5238	62	12	}	}	PUNCT
cana-5238	62	13	,	,	PUNCT
cana-5238	62	14	ζ={∅	ζ={∅	PROPN
cana-5238	62	15	,	,	PUNCT
cana-5238	62	16	{	{	PUNCT
cana-5238	62	17	1	1	NUM
cana-5238	62	18	}	}	PUNCT
cana-5238	62	19	,	,	PUNCT
cana-5238	62	20	{	{	PUNCT
cana-5238	62	21	2	2	NUM
cana-5238	62	22	}	}	PUNCT
cana-5238	62	23	,	,	PUNCT
cana-5238	62	24	{	{	PUNCT
cana-5238	62	25	1	1	NUM
cana-5238	62	26	,	,	PUNCT
cana-5238	62	27	2	2	NUM
cana-5238	62	28	}	}	PUNCT
cana-5238	62	29	,	,	PUNCT
cana-5238	62	30	{	{	PUNCT
cana-5238	62	31	2	2	NUM
cana-5238	62	32	,	,	PUNCT
cana-5238	62	33	3	3	NUM
cana-5238	62	34	,	,	PUNCT
cana-5238	62	35	4	4	NUM
cana-5238	62	36	}	}	PUNCT
cana-5238	62	37	,	,	PUNCT
cana-5238	62	38	z	z	NOUN
cana-5238	62	39	}	}	PUNCT
cana-5238	62	40	,	,	PUNCT
cana-5238	62	41	h	h	NOUN
cana-5238	62	42	=	=	PRON
cana-5238	62	43	{	{	PUNCT
cana-5238	62	44	∅	∅	NOUN
cana-5238	62	45	,	,	PUNCT
cana-5238	62	46	{	{	PUNCT
cana-5238	62	47	1	1	NUM
cana-5238	62	48	}	}	PUNCT
cana-5238	62	49	,	,	PUNCT
cana-5238	62	50	{	{	PUNCT
cana-5238	62	51	3	3	NUM
cana-5238	62	52	}	}	PUNCT
cana-5238	62	53	}	}	PUNCT
cana-5238	62	54	.	.	PUNCT
cana-5238	63	1	then	then	ADV
cana-5238	63	2	ψ	ψ	X
cana-5238	63	3	=	=	PUNCT
cana-5238	63	4	{	{	PUNCT
cana-5238	63	5	1	1	NUM
cana-5238	63	6	,	,	PUNCT
cana-5238	63	7	2	2	NUM
cana-5238	63	8	,	,	PUNCT
cana-5238	63	9	3	3	NUM
cana-5238	63	10	}	}	PUNCT
cana-5238	63	11	is	be	AUX
cana-5238	63	12	b	b	NUM
cana-5238	63	13	hg	hg	NOUN
cana-5238	63	14	o	o	NOUN
cana-5238	63	15	but	but	CCONJ
cana-5238	63	16	not	not	PART
cana-5238	63	17	gζ	gζ	NOUN
cana-5238	63	18	-	-	PUNCT
cana-5238	63	19	o.	o.	PROPN
cana-5238	63	20	remark	remark	NOUN
cana-5238	63	21	2.8	2.8	NUM
cana-5238	63	22	.	.	PUNCT
cana-5238	64	1	the	the	DET
cana-5238	64	2	notions	notion	NOUN
cana-5238	64	3	of	of	ADP
cana-5238	64	4	b	b	NOUN
cana-5238	64	5	-	-	PUNCT
cana-5238	64	6	hg	hg	NOUN
cana-5238	64	7	-	-	NOUN
cana-5238	64	8	o	o	NOUN
cana-5238	64	9	and	and	CCONJ
cana-5238	64	10	ζ	ζ	NOUN
cana-5238	64	11	-	-	PUNCT
cana-5238	64	12	b	b	NOUN
cana-5238	64	13	-	-	PUNCT
cana-5238	64	14	o	o	NOUN
cana-5238	64	15	are	be	AUX
cana-5238	64	16	independent	independent	ADJ
cana-5238	64	17	.	.	PUNCT
cana-5238	64	18	example	example	NOUN
cana-5238	65	1	2.9	2.9	NUM
cana-5238	65	2	.	.	PUNCT
cana-5238	66	1	assume	assume	VERB
cana-5238	66	2	z	z	NOUN
cana-5238	66	3	=	=	PUNCT
cana-5238	66	4	{	{	PUNCT
cana-5238	66	5	1	1	NUM
cana-5238	66	6	,	,	PUNCT
cana-5238	66	7	2	2	NUM
cana-5238	66	8	,	,	PUNCT
cana-5238	66	9	3	3	NUM
cana-5238	66	10	,	,	PUNCT
cana-5238	66	11	4	4	NUM
cana-5238	66	12	}	}	PUNCT
cana-5238	66	13	,	,	PUNCT
cana-5238	66	14	ζ={∅	ζ={∅	PROPN
cana-5238	66	15	,	,	PUNCT
cana-5238	66	16	{	{	PUNCT
cana-5238	66	17	1	1	NUM
cana-5238	66	18	}	}	PUNCT
cana-5238	66	19	,	,	PUNCT
cana-5238	66	20	{	{	PUNCT
cana-5238	66	21	2	2	NUM
cana-5238	66	22	}	}	PUNCT
cana-5238	66	23	,	,	PUNCT
cana-5238	66	24	{	{	PUNCT
cana-5238	66	25	1	1	NUM
cana-5238	66	26	,	,	PUNCT
cana-5238	66	27	2	2	NUM
cana-5238	66	28	}	}	PUNCT
cana-5238	66	29	,	,	PUNCT
cana-5238	66	30	{	{	PUNCT
cana-5238	66	31	2	2	NUM
cana-5238	66	32	,	,	PUNCT
cana-5238	66	33	3	3	NUM
cana-5238	66	34	,	,	PUNCT
cana-5238	66	35	4	4	NUM
cana-5238	66	36	}	}	PUNCT
cana-5238	66	37	,	,	PUNCT
cana-5238	66	38	z	z	NOUN
cana-5238	66	39	}	}	PUNCT
cana-5238	66	40	,	,	PUNCT
cana-5238	66	41	h	h	NOUN
cana-5238	66	42	=	=	PRON
cana-5238	66	43	{	{	PUNCT
cana-5238	66	44	∅	∅	NOUN
cana-5238	66	45	,	,	PUNCT
cana-5238	66	46	{	{	PUNCT
cana-5238	66	47	1	1	NUM
cana-5238	66	48	}	}	PUNCT
cana-5238	66	49	,	,	PUNCT
cana-5238	66	50	{	{	PUNCT
cana-5238	66	51	3	3	NUM
cana-5238	66	52	}	}	PUNCT
cana-5238	66	53	}	}	PUNCT
cana-5238	66	54	.	.	PUNCT
cana-5238	67	1	then	then	ADV
cana-5238	67	2	ψ	ψ	X
cana-5238	67	3	=	=	PUNCT
cana-5238	67	4	{	{	PUNCT
cana-5238	67	5	4	4	NUM
cana-5238	67	6	}	}	PUNCT
cana-5238	67	7	is	be	AUX
cana-5238	67	8	b	b	NUM
cana-5238	67	9	-	-	PUNCT
cana-5238	67	10	hg	hg	NOUN
cana-5238	67	11	-	-	NOUN
cana-5238	67	12	o	o	NOUN
cana-5238	67	13	but	but	CCONJ
cana-5238	67	14	not	not	PART
cana-5238	67	15	ζ	ζ	NOUN
cana-5238	67	16	-	-	PUNCT
cana-5238	67	17	b	b	NOUN
cana-5238	67	18	-	-	PUNCT
cana-5238	67	19	o.	o.	NOUN
cana-5238	67	20	example	example	NOUN
cana-5238	67	21	2.10	2.10	NUM
cana-5238	67	22	.	.	PUNCT
cana-5238	68	1	assume	assume	VERB
cana-5238	68	2	z	z	NOUN
cana-5238	68	3	=	=	PUNCT
cana-5238	68	4	{	{	PUNCT
cana-5238	68	5	1	1	NUM
cana-5238	68	6	,	,	PUNCT
cana-5238	68	7	2	2	NUM
cana-5238	68	8	,	,	PUNCT
cana-5238	68	9	3	3	NUM
cana-5238	68	10	,	,	PUNCT
cana-5238	68	11	4	4	NUM
cana-5238	68	12	}	}	PUNCT
cana-5238	68	13	,	,	PUNCT
cana-5238	68	14	ζ	ζ	NOUN
cana-5238	68	15	=	=	SYM
cana-5238	68	16	{	{	PUNCT
cana-5238	68	17	∅	∅	NOUN
cana-5238	68	18	,	,	PUNCT
cana-5238	68	19	{	{	PUNCT
cana-5238	68	20	1	1	NUM
cana-5238	68	21	}	}	PUNCT
cana-5238	68	22	,	,	PUNCT
cana-5238	68	23	{	{	PUNCT
cana-5238	68	24	1	1	NUM
cana-5238	68	25	,	,	PUNCT
cana-5238	68	26	2	2	NUM
cana-5238	68	27	,	,	PUNCT
cana-5238	68	28	3	3	NUM
cana-5238	68	29	}	}	PUNCT
cana-5238	68	30	,	,	PUNCT
cana-5238	68	31	{	{	PUNCT
cana-5238	68	32	3	3	NUM
cana-5238	68	33	,	,	PUNCT
cana-5238	68	34	4	4	NUM
cana-5238	68	35	}	}	PUNCT
cana-5238	68	36	,	,	PUNCT
cana-5238	68	37	z	z	NOUN
cana-5238	68	38	}	}	PUNCT
cana-5238	68	39	,	,	PUNCT
cana-5238	68	40	h	h	NOUN
cana-5238	68	41	=	=	NOUN
cana-5238	68	42	{	{	PUNCT
cana-5238	68	43	∅	∅	NOUN
cana-5238	68	44	,	,	PUNCT
cana-5238	68	45	{	{	PUNCT
cana-5238	68	46	1	1	NUM
cana-5238	68	47	}	}	PUNCT
cana-5238	68	48	,	,	PUNCT
cana-5238	68	49	{	{	PUNCT
cana-5238	68	50	3	3	NUM
cana-5238	68	51	}	}	PUNCT
cana-5238	68	52	}	}	PUNCT
cana-5238	68	53	.	.	PUNCT
cana-5238	69	1	then	then	ADV
cana-5238	69	2	m	m	VERB
cana-5238	69	3	=	=	PUNCT
cana-5238	69	4	{	{	PUNCT
cana-5238	69	5	1	1	NUM
cana-5238	69	6	,	,	PUNCT
cana-5238	69	7	4	4	NUM
cana-5238	69	8	}	}	PUNCT
cana-5238	69	9	is	be	AUX
cana-5238	69	10	ζbo	ζbo	ADJ
cana-5238	69	11	but	but	CCONJ
cana-5238	69	12	not	not	PART
cana-5238	69	13	bhg	bhg	NOUN
cana-5238	69	14	-	-	PUNCT
cana-5238	69	15	o.	o.	NOUN
cana-5238	69	16	proposition	proposition	NOUN
cana-5238	69	17	2.11	2.11	NUM
cana-5238	69	18	.	.	PUNCT
cana-5238	70	1	in	in	ADP
cana-5238	70	2	hgts	hgts	PROPN
cana-5238	70	3	(	(	PUNCT
cana-5238	70	4	z	z	NOUN
cana-5238	70	5	,	,	PUNCT
cana-5238	70	6	ζ	ζ	NOUN
cana-5238	70	7	,	,	PUNCT
cana-5238	70	8	h	h	NOUN
cana-5238	70	9	)	)	PUNCT
cana-5238	70	10	every	every	DET
cana-5238	70	11	b	b	X
cana-5238	70	12	-	-	PUNCT
cana-5238	70	13	h	h	NOUN
cana-5238	70	14	-	-	PUNCT
cana-5238	70	15	o	o	NOUN
cana-5238	70	16	is	be	AUX
cana-5238	70	17	b	b	NUM
cana-5238	70	18	-	-	PUNCT
cana-5238	70	19	hg	hg	NOUN
cana-5238	70	20	-	-	ADJ
cana-5238	70	21	o.	o.	ADJ
cana-5238	70	22	proof	proof	NOUN
cana-5238	70	23	.	.	PUNCT
cana-5238	71	1	assume	assume	VERB
cana-5238	71	2	ψ	ψ	PART
cana-5238	71	3	be	be	AUX
cana-5238	71	4	a	a	DET
cana-5238	71	5	b	b	NOUN
cana-5238	71	6	-	-	PUNCT
cana-5238	71	7	h	h	NOUN
cana-5238	71	8	-	-	PUNCT
cana-5238	71	9	o	o	NOUN
cana-5238	71	10	ψ⊆ic∗(ψ)∪c∗i(ψ)⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆ic∗(ψ)∪c∗i(ψ)⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	71	11	)	)	PUNCT
cana-5238	71	12	.	.	PUNCT
cana-5238	72	1	hence	hence	ADV
cana-5238	72	2	ψ	ψ	NOUN
cana-5238	72	3	is	be	AUX
cana-5238	72	4	b-hg-o.remark	b-hg-o.remark	ADJ
cana-5238	72	5	2.12	2.12	NUM
cana-5238	72	6	.	.	PUNCT
cana-5238	73	1	the	the	DET
cana-5238	73	2	converse	converse	NOUN
cana-5238	73	3	of	of	ADP
cana-5238	73	4	proposition	proposition	NOUN
cana-5238	73	5	2.11	2.11	NUM
cana-5238	73	6	need	need	AUX
cana-5238	73	7	not	not	PART
cana-5238	73	8	be	be	AUX
cana-5238	73	9	correct	correct	ADJ
cana-5238	73	10	from	from	ADP
cana-5238	73	11	the	the	DET
cana-5238	73	12	following	following	ADJ
cana-5238	73	13	examples	example	NOUN
cana-5238	73	14	.	.	PUNCT
cana-5238	74	1	example	example	NOUN
cana-5238	74	2	2.13	2.13	NUM
cana-5238	74	3	.	.	PUNCT
cana-5238	75	1	assume	assume	VERB
cana-5238	75	2	z	z	NOUN
cana-5238	75	3	=	=	PUNCT
cana-5238	75	4	{	{	PUNCT
cana-5238	75	5	1	1	NUM
cana-5238	75	6	,	,	PUNCT
cana-5238	75	7	2	2	NUM
cana-5238	75	8	,	,	PUNCT
cana-5238	75	9	3	3	NUM
cana-5238	75	10	,	,	PUNCT
cana-5238	75	11	4	4	NUM
cana-5238	75	12	}	}	PUNCT
cana-5238	75	13	,	,	PUNCT
cana-5238	75	14	ζ	ζ	NOUN
cana-5238	75	15	=	=	SYM
cana-5238	75	16	{	{	PUNCT
cana-5238	75	17	∅	∅	NOUN
cana-5238	75	18	,	,	PUNCT
cana-5238	75	19	{	{	PUNCT
cana-5238	75	20	1	1	NUM
cana-5238	75	21	,	,	PUNCT
cana-5238	75	22	3	3	NUM
cana-5238	75	23	}	}	PUNCT
cana-5238	75	24	,	,	PUNCT
cana-5238	75	25	{	{	PUNCT
cana-5238	75	26	2	2	NUM
cana-5238	75	27	,	,	PUNCT
cana-5238	75	28	3	3	NUM
cana-5238	75	29	}	}	PUNCT
cana-5238	75	30	,	,	PUNCT
cana-5238	75	31	{	{	PUNCT
cana-5238	75	32	1	1	NUM
cana-5238	75	33	,	,	PUNCT
cana-5238	75	34	2	2	NUM
cana-5238	75	35	,	,	PUNCT
cana-5238	75	36	3	3	NUM
cana-5238	75	37	}	}	PUNCT
cana-5238	75	38	,	,	PUNCT
cana-5238	75	39	{	{	PUNCT
cana-5238	75	40	1	1	NUM
cana-5238	75	41	,	,	PUNCT
cana-5238	75	42	4	4	NUM
cana-5238	75	43	}	}	PUNCT
cana-5238	75	44	,	,	PUNCT
cana-5238	75	45	{	{	PUNCT
cana-5238	75	46	1	1	NUM
cana-5238	75	47	,	,	PUNCT
cana-5238	75	48	3	3	NUM
cana-5238	75	49	,	,	PUNCT
cana-5238	75	50	4	4	NUM
cana-5238	75	51	}	}	PUNCT
cana-5238	75	52	,	,	PUNCT
cana-5238	75	53	z	z	NOUN
cana-5238	75	54	}	}	PUNCT
cana-5238	75	55	,	,	PUNCT
cana-5238	75	56	h	h	NOUN
cana-5238	75	57	=	=	PRON
cana-5238	75	58	{	{	PUNCT
cana-5238	75	59	∅	∅	NOUN
cana-5238	75	60	,	,	PUNCT
cana-5238	75	61	{	{	PUNCT
cana-5238	75	62	1	1	NUM
cana-5238	75	63	,	,	PUNCT
cana-5238	75	64	2	2	NUM
cana-5238	75	65	}	}	PUNCT
cana-5238	75	66	}	}	PUNCT
cana-5238	75	67	.	.	PUNCT
cana-5238	76	1	then	then	ADV
cana-5238	76	2	ψ	ψ	X
cana-5238	76	3	=	=	PUNCT
cana-5238	76	4	{	{	PUNCT
cana-5238	76	5	1	1	NUM
cana-5238	76	6	}	}	PUNCT
cana-5238	76	7	is	be	AUX
cana-5238	76	8	b	b	NUM
cana-5238	76	9	-	-	PUNCT
cana-5238	76	10	hg	hg	NOUN
cana-5238	76	11	-	-	NOUN
cana-5238	76	12	o	o	NOUN
cana-5238	76	13	but	but	CCONJ
cana-5238	76	14	not	not	PART
cana-5238	76	15	b	b	PROPN
cana-5238	76	16	-	-	PUNCT
cana-5238	76	17	ho	ho	PROPN
cana-5238	76	18	.	.	PUNCT
cana-5238	77	1	proposition	proposition	NOUN
cana-5238	77	2	2.14	2.14	NUM
cana-5238	77	3	.	.	PUNCT
cana-5238	78	1	every	every	DET
cana-5238	78	2	α	α	PROPN
cana-5238	78	3	-	-	PUNCT
cana-5238	78	4	hg	hg	NOUN
cana-5238	78	5	-	-	NOUN
cana-5238	78	6	o	o	X
cana-5238	78	7	(	(	PUNCT
cana-5238	78	8	resp	resp	NOUN
cana-5238	78	9	.	.	PUNCT
cana-5238	79	1	σ	σ	PROPN
cana-5238	79	2	-	-	PUNCT
cana-5238	79	3	hg	hg	NOUN
cana-5238	79	4	-	-	PROPN
cana-5238	79	5	o	o	NOUN
cana-5238	79	6	,	,	PUNCT
cana-5238	79	7	π	π	PROPN
cana-5238	79	8	-	-	PUNCT
cana-5238	79	9	hg	hg	NOUN
cana-5238	79	10	-	-	NOUN
cana-5238	79	11	o	o	NOUN
cana-5238	79	12	)	)	PUNCT
cana-5238	79	13	is	be	AUX
cana-5238	79	14	b	b	NUM
cana-5238	79	15	-	-	PUNCT
cana-5238	79	16	hg	hg	NOUN
cana-5238	79	17	-	-	NOUN
cana-5238	79	18	o	o	NOUN
cana-5238	79	19	but	but	CCONJ
cana-5238	79	20	not	not	PART
cana-5238	79	21	conversely	conversely	ADV
cana-5238	79	22	.	.	PUNCT
cana-5238	80	1	proof	proof	NOUN
cana-5238	80	2	.	.	PUNCT
cana-5238	81	1	1	1	X
cana-5238	81	2	.	.	X
cana-5238	81	3	assume	assume	VERB
cana-5238	81	4	ψ	ψ	PART
cana-5238	81	5	be	be	AUX
cana-5238	81	6	α	α	X
cana-5238	81	7	-	-	PUNCT
cana-5238	81	8	hg	hg	NOUN
cana-5238	81	9	-	-	NOUN
cana-5238	81	10	o.	o.	PROPN
cana-5238	81	11	then	then	ADV
cana-5238	81	12	ψ⊆	ψ⊆	VERB
cana-5238	81	13	igc∗ig(ψ)⊆c∗ig(ψ)∪igc∗(ψ	igc∗ig(ψ)⊆c∗ig(ψ)∪igc∗(ψ	NOUN
cana-5238	81	14	)	)	PUNCT
cana-5238	81	15	.	.	PUNCT
cana-5238	82	1	which	which	PRON
cana-5238	82	2	implies	imply	VERB
cana-5238	82	3	ψ	ψ	ADP
cana-5238	82	4	is	be	AUX
cana-5238	82	5	b	b	NUM
cana-5238	82	6	-	-	PUNCT
cana-5238	82	7	hg	hg	NOUN
cana-5238	82	8	-	-	NOUN
cana-5238	82	9	o.	o.	ADJ
cana-5238	82	10	2	2	PROPN
cana-5238	82	11	.	.	PUNCT
cana-5238	83	1	assume	assume	VERB
cana-5238	83	2	ψ	ψ	SYM
cana-5238	83	3	is	be	AUX
cana-5238	83	4	σhg	σhg	NOUN
cana-5238	83	5	-	-	PUNCT
cana-5238	83	6	o.	o.	NOUN
cana-5238	83	7	then	then	ADV
cana-5238	83	8	ψ	ψ	VERB
cana-5238	83	9	⊆	⊆	NUM
cana-5238	83	10	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	83	11	)	)	PUNCT
cana-5238	83	12	⊆	⊆	NUM
cana-5238	83	13	c∗ig(ψ)∪igc∗(ψ	c∗ig(ψ)∪igc∗(ψ	NOUN
cana-5238	83	14	)	)	PUNCT
cana-5238	83	15	.	.	PUNCT
cana-5238	84	1	which	which	PRON
cana-5238	84	2	implies	imply	VERB
cana-5238	84	3	ψ	ψ	ADP
cana-5238	84	4	is	be	AUX
cana-5238	84	5	b	b	NUM
cana-5238	84	6	-	-	PUNCT
cana-5238	84	7	hg	hg	NOUN
cana-5238	84	8	-	-	PROPN
cana-5238	84	9	o.	o.	ADJ
cana-5238	84	10	3	3	PROPN
cana-5238	84	11	.	.	PUNCT
cana-5238	85	1	assume	assume	VERB
cana-5238	85	2	ψ	ψ	PART
cana-5238	85	3	be	be	AUX
cana-5238	85	4	π	π	PROPN
cana-5238	85	5	-	-	PUNCT
cana-5238	85	6	hg	hg	NOUN
cana-5238	85	7	-	-	NOUN
cana-5238	85	8	o.	o.	NOUN
cana-5238	85	9	then	then	ADV
cana-5238	85	10	ψ⊆igc∗(ψ	ψ⊆igc∗(ψ	NUM
cana-5238	85	11	)	)	PUNCT
cana-5238	85	12	⊆	⊆	NUM
cana-5238	85	13	c∗ig(ψ)∪	c∗ig(ψ)∪	X
cana-5238	85	14	igc∗(ψ	igc∗(ψ	NUM
cana-5238	85	15	)	)	PUNCT
cana-5238	85	16	.	.	PUNCT
cana-5238	86	1	which	which	PRON
cana-5238	86	2	implies	imply	VERB
cana-5238	86	3	ψ	ψ	NOUN
cana-5238	86	4	is	be	AUX
cana-5238	86	5	bhg	bhg	NOUN
cana-5238	86	6	-	-	PUNCT
cana-5238	86	7	o.	o.	PROPN
cana-5238	86	8	example	example	NOUN
cana-5238	86	9	2.15	2.15	NUM
cana-5238	86	10	.	.	PUNCT
cana-5238	87	1	assume	assume	VERB
cana-5238	87	2	z	z	NOUN
cana-5238	87	3	=	=	PUNCT
cana-5238	87	4	{	{	PUNCT
cana-5238	87	5	1	1	NUM
cana-5238	87	6	,	,	PUNCT
cana-5238	87	7	2	2	NUM
cana-5238	87	8	,	,	PUNCT
cana-5238	87	9	3	3	NUM
cana-5238	87	10	,	,	PUNCT
cana-5238	87	11	4	4	NUM
cana-5238	87	12	}	}	PUNCT
cana-5238	87	13	,	,	PUNCT
cana-5238	87	14	ζ={∅	ζ={∅	PROPN
cana-5238	87	15	,	,	PUNCT
cana-5238	87	16	{	{	PUNCT
cana-5238	87	17	1	1	NUM
cana-5238	87	18	,	,	PUNCT
cana-5238	87	19	3	3	NUM
cana-5238	87	20	}	}	PUNCT
cana-5238	87	21	,	,	PUNCT
cana-5238	87	22	{	{	PUNCT
cana-5238	87	23	2	2	NUM
cana-5238	87	24	,	,	PUNCT
cana-5238	87	25	3	3	NUM
cana-5238	87	26	}	}	PUNCT
cana-5238	87	27	,	,	PUNCT
cana-5238	87	28	{	{	PUNCT
cana-5238	87	29	1	1	NUM
cana-5238	87	30	,	,	PUNCT
cana-5238	87	31	2	2	NUM
cana-5238	87	32	,	,	PUNCT
cana-5238	87	33	3	3	NUM
cana-5238	87	34	}	}	PUNCT
cana-5238	87	35	,	,	PUNCT
cana-5238	87	36	{	{	PUNCT
cana-5238	87	37	1	1	NUM
cana-5238	87	38	,	,	PUNCT
cana-5238	87	39	4	4	NUM
cana-5238	87	40	}	}	PUNCT
cana-5238	87	41	,	,	PUNCT
cana-5238	87	42	{	{	PUNCT
cana-5238	87	43	1	1	NUM
cana-5238	87	44	,	,	PUNCT
cana-5238	87	45	3	3	NUM
cana-5238	87	46	,	,	PUNCT
cana-5238	87	47	4	4	NUM
cana-5238	87	48	}	}	PUNCT
cana-5238	87	49	,	,	PUNCT
cana-5238	87	50	z	z	NOUN
cana-5238	87	51	}	}	PUNCT
cana-5238	87	52	,	,	PUNCT
cana-5238	87	53	h	h	NOUN
cana-5238	87	54	=	=	PRON
cana-5238	87	55	{	{	PUNCT
cana-5238	87	56	∅	∅	NOUN
cana-5238	87	57	,	,	PUNCT
cana-5238	87	58	{	{	PUNCT
cana-5238	87	59	1	1	NUM
cana-5238	87	60	}	}	PUNCT
cana-5238	87	61	,	,	PUNCT
cana-5238	87	62	{	{	PUNCT
cana-5238	87	63	2	2	NUM
cana-5238	87	64	}	}	PUNCT
cana-5238	87	65	}	}	PUNCT
cana-5238	87	66	.	.	PUNCT
cana-5238	88	1	then	then	ADV
cana-5238	88	2	ψ={2	ψ={2	PROPN
cana-5238	88	3	}	}	PUNCT
cana-5238	88	4	is	be	AUX
cana-5238	88	5	b	b	NUM
cana-5238	88	6	-	-	PUNCT
cana-5238	88	7	hg	hg	NOUN
cana-5238	88	8	-	-	NOUN
cana-5238	88	9	o	o	NOUN
cana-5238	88	10	but	but	CCONJ
cana-5238	88	11	not	not	PART
cana-5238	88	12	α	α	PROPN
cana-5238	88	13	-	-	PUNCT
cana-5238	88	14	hg	hg	NOUN
cana-5238	88	15	-	-	NOUN
cana-5238	88	16	o	o	X
cana-5238	88	17	(	(	PUNCT
cana-5238	88	18	resp	resp	NOUN
cana-5238	88	19	.	.	PUNCT
cana-5238	89	1	σ	σ	NOUN
cana-5238	89	2	-	-	PUNCT
cana-5238	89	3	hgo	hgo	PROPN
cana-5238	89	4	,	,	PUNCT
cana-5238	89	5	π	π	PROPN
cana-5238	89	6	-	-	PUNCT
cana-5238	89	7	hg	hg	NOUN
cana-5238	89	8	-	-	NOUN
cana-5238	89	9	o	o	NOUN
cana-5238	89	10	)	)	PUNCT
cana-5238	89	11	.	.	PUNCT
cana-5238	90	1	theorem	theorem	VERB
cana-5238	90	2	2.16	2.16	NUM
cana-5238	90	3	.	.	PUNCT
cana-5238	91	1	if	if	SCONJ
cana-5238	91	2	ψ	ψ	NOUN
cana-5238	91	3	is	be	AUX
cana-5238	91	4	b	b	PROPN
cana-5238	91	5	-	-	PUNCT
cana-5238	91	6	hg	hg	NOUN
cana-5238	91	7	-	-	NOUN
cana-5238	91	8	o	o	NOUN
cana-5238	91	9	and	and	CCONJ
cana-5238	91	10	ζ	ζ	NOUN
cana-5238	91	11	-	-	PUNCT
cana-5238	91	12	σ	σ	NOUN
cana-5238	91	13	-	-	PUNCT
cana-5238	91	14	o	o	NOUN
cana-5238	91	15	,	,	PUNCT
cana-5238	91	16	then	then	ADV
cana-5238	91	17	it	it	PRON
cana-5238	91	18	is	be	AUX
cana-5238	91	19	β	β	NOUN
cana-5238	91	20	-	-	PUNCT
cana-5238	91	21	ho	ho	ADJ
cana-5238	91	22	.	.	PUNCT
cana-5238	91	23	communications	communication	NOUN
cana-5238	91	24	on	on	ADP
cana-5238	91	25	applied	apply	VERB
cana-5238	91	26	nonlinear	nonlinear	ADJ
cana-5238	91	27	analysis	analysis	NOUN
cana-5238	91	28	issn	issn	NOUN
cana-5238	91	29	:	:	PUNCT
cana-5238	91	30	1074	1074	NUM
cana-5238	91	31	-	-	PUNCT
cana-5238	91	32	133x	133x	NUM
cana-5238	91	33	vol	vol	VERB
cana-5238	91	34	32	32	NUM
cana-5238	91	35	no	no	NOUN
cana-5238	91	36	.	.	PUNCT
cana-5238	92	1	10s	10	NOUN
cana-5238	92	2	(	(	PUNCT
cana-5238	92	3	2025	2025	NUM
cana-5238	92	4	)	)	PUNCT
cana-5238	92	5	1360	1360	NUM
cana-5238	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	92	7	ζ	ζ	NOUN
cana-5238	92	8	proof	proof	NOUN
cana-5238	92	9	.	.	PUNCT
cana-5238	93	1	let	let	VERB
cana-5238	93	2	ψ	ψ	NOUN
cana-5238	93	3	is	be	AUX
cana-5238	93	4	b	b	NUM
cana-5238	93	5	-	-	PUNCT
cana-5238	93	6	hg	hg	NOUN
cana-5238	93	7	-	-	NOUN
cana-5238	93	8	o	o	NOUN
cana-5238	93	9	and	and	CCONJ
cana-5238	93	10	ζ	ζ	NOUN
cana-5238	93	11	-	-	PUNCT
cana-5238	93	12	σo	σo	NOUN
cana-5238	93	13	.	.	PUNCT
cana-5238	94	1	then	then	ADV
cana-5238	94	2	ψ	ψ	ADP
cana-5238	94	3	⊆	⊆	NUM
cana-5238	94	4	igc∗(ψ	igc∗(ψ	NUM
cana-5238	94	5	)	)	PUNCT
cana-5238	94	6	∪	∪	ADP
cana-5238	94	7	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	94	8	)	)	PUNCT
cana-5238	94	9	and	and	CCONJ
cana-5238	94	10	ψ	ψ	X
cana-5238	94	11	⊆	⊆	NUM
cana-5238	94	12	ci	ci	NOUN
cana-5238	94	13	(	(	PUNCT
cana-5238	94	14	ψ	ψ	NOUN
cana-5238	94	15	)	)	PUNCT
cana-5238	94	16	.	.	PUNCT
cana-5238	95	1	now	now	ADV
cana-5238	95	2	ψ	ψ	ADP
cana-5238	95	3	⊆	⊆	NUM
cana-5238	95	4	igc∗(ψ)∪c∗ig(ψ)⊆c∗(ψ	igc∗(ψ)∪c∗ig(ψ)⊆c∗(ψ	NUM
cana-5238	95	5	)	)	PUNCT
cana-5238	95	6	,	,	PUNCT
cana-5238	95	7	which	which	PRON
cana-5238	95	8	implies	imply	VERB
cana-5238	95	9	ci	ci	NOUN
cana-5238	95	10	(	(	PUNCT
cana-5238	95	11	ψ	ψ	NOUN
cana-5238	95	12	)	)	PUNCT
cana-5238	95	13	⊆	⊆	NUM
cana-5238	95	14	cic∗(ψ	cic∗(ψ	NUM
cana-5238	95	15	)	)	PUNCT
cana-5238	95	16	.	.	PUNCT
cana-5238	96	1	so	so	ADV
cana-5238	96	2	ψ⊆ci(ψ)⊆ci∗(ψ	ψ⊆ci(ψ)⊆ci∗(ψ	NOUN
cana-5238	96	3	)	)	PUNCT
cana-5238	96	4	.	.	PUNCT
cana-5238	97	1	hence	hence	ADV
cana-5238	97	2	ψ	ψ	NOUN
cana-5238	97	3	is	be	AUX
cana-5238	97	4	β	β	X
cana-5238	97	5	-	-	ADJ
cana-5238	97	6	h	h	ADJ
cana-5238	97	7	-	-	PUNCT
cana-5238	97	8	o.	o.	NOUN
cana-5238	97	9	theorem	theorem	NOUN
cana-5238	97	10	2.17	2.17	NUM
cana-5238	97	11	.	.	PUNCT
cana-5238	98	1	if	if	SCONJ
cana-5238	98	2	ψ	ψ	NOUN
cana-5238	98	3	is	be	AUX
cana-5238	98	4	b	b	NUM
cana-5238	98	5	hg	hg	NOUN
cana-5238	98	6	o	o	NOUN
cana-5238	98	7	and	and	CCONJ
cana-5238	98	8	ζ∗	ζ∗	PROPN
cana-5238	98	9	-closed	-close	VERB
cana-5238	98	10	,	,	PUNCT
cana-5238	98	11	then	then	ADV
cana-5238	98	12	it	it	PRON
cana-5238	98	13	is	be	AUX
cana-5238	98	14	σ	σ	PROPN
cana-5238	98	15	-	-	PUNCT
cana-5238	98	16	hg	hg	NOUN
cana-5238	98	17	-	-	ADJ
cana-5238	98	18	o.	o.	ADJ
cana-5238	98	19	proof	proof	NOUN
cana-5238	98	20	.	.	PUNCT
cana-5238	99	1	let	let	VERB
cana-5238	99	2	ψ	ψ	NOUN
cana-5238	99	3	is	be	AUX
cana-5238	99	4	b	b	NUM
cana-5238	99	5	-	-	PUNCT
cana-5238	99	6	hg	hg	NOUN
cana-5238	99	7	-	-	NOUN
cana-5238	99	8	o	o	NOUN
cana-5238	99	9	and	and	CCONJ
cana-5238	99	10	ζ∗-closed	ζ∗-close	VERB
cana-5238	99	11	.	.	PUNCT
cana-5238	100	1	then	then	ADV
cana-5238	100	2	ψ⊆	ψ⊆	ADJ
cana-5238	100	3	igc∗(ψ)∪c∗ig(ψ	igc∗(ψ)∪c∗ig(ψ	NOUN
cana-5238	100	4	)	)	PUNCT
cana-5238	100	5	and	and	CCONJ
cana-5238	100	6	c∗(ψ	c∗(ψ	NOUN
cana-5238	100	7	)	)	PUNCT
cana-5238	100	8	⊆ψ	⊆ψ	NOUN
cana-5238	100	9	.	.	PUNCT
cana-5238	101	1	now	now	ADV
cana-5238	101	2	ψ	ψ	ADP
cana-5238	101	3	⊆	⊆	NUM
cana-5238	101	4	igc∗(ψ)∪c∗ig(ψ	igc∗(ψ)∪c∗ig(ψ	NOUN
cana-5238	101	5	)	)	PUNCT
cana-5238	101	6	⊆	⊆	NUM
cana-5238	101	7	c∗ig(ψ)∪ig(ψ	c∗ig(ψ)∪ig(ψ	NUM
cana-5238	101	8	)	)	PUNCT
cana-5238	101	9	=	=	SYM
cana-5238	101	10	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	101	11	)	)	PUNCT
cana-5238	101	12	.	.	PUNCT
cana-5238	102	1	hence	hence	ADV
cana-5238	102	2	ψ	ψ	X
cana-5238	102	3	is	be	AUX
cana-5238	102	4	σ	σ	PROPN
cana-5238	102	5	-	-	PUNCT
cana-5238	102	6	hg	hg	NOUN
cana-5238	102	7	-	-	ADJ
cana-5238	102	8	o.	o.	NOUN
cana-5238	102	9	theorem	theorem	PROPN
cana-5238	102	10	2.18	2.18	NUM
cana-5238	102	11	.	.	PUNCT
cana-5238	103	1	if	if	SCONJ
cana-5238	103	2	ψ	ψ	NOUN
cana-5238	103	3	is	be	AUX
cana-5238	103	4	b	b	PROPN
cana-5238	103	5	-	-	PUNCT
cana-5238	103	6	hg	hg	NOUN
cana-5238	103	7	-	-	NOUN
cana-5238	103	8	o	o	NOUN
cana-5238	103	9	and	and	CCONJ
cana-5238	103	10	ζ	ζ	NOUN
cana-5238	103	11	-	-	PUNCT
cana-5238	103	12	closed	closed	ADJ
cana-5238	103	13	,	,	PUNCT
cana-5238	103	14	then	then	ADV
cana-5238	103	15	it	it	PRON
cana-5238	103	16	is	be	AUX
cana-5238	103	17	σ	σ	PROPN
cana-5238	103	18	-	-	PUNCT
cana-5238	103	19	hg	hg	NOUN
cana-5238	103	20	-	-	ADJ
cana-5238	103	21	o.	o.	ADJ
cana-5238	103	22	proof	proof	NOUN
cana-5238	103	23	.	.	PUNCT
cana-5238	104	1	let	let	VERB
cana-5238	104	2	ψ	ψ	NOUN
cana-5238	104	3	is	be	AUX
cana-5238	104	4	b	b	NUM
cana-5238	104	5	-	-	PUNCT
cana-5238	104	6	hg	hg	NOUN
cana-5238	104	7	-	-	NOUN
cana-5238	104	8	o	o	NOUN
cana-5238	104	9	and	and	CCONJ
cana-5238	104	10	ζ	ζ	NOUN
cana-5238	104	11	-	-	PUNCT
cana-5238	104	12	closed	closed	ADJ
cana-5238	104	13	.	.	PUNCT
cana-5238	105	1	then	then	ADV
cana-5238	105	2	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	105	3	)	)	PUNCT
cana-5238	105	4	and	and	CCONJ
cana-5238	105	5	c∗(ψ)⊆ψ	c∗(ψ)⊆ψ	VERB
cana-5238	105	6	by	by	ADP
cana-5238	105	7	proposition	proposition	NOUN
cana-5238	105	8	2.9	2.9	NUM
cana-5238	105	9	of	of	ADP
cana-5238	105	10	[	[	X
cana-5238	105	11	6	6	NUM
cana-5238	105	12	]	]	PUNCT
cana-5238	105	13	.	.	PUNCT
cana-5238	106	1	which	which	PRON
cana-5238	106	2	implies	imply	VERB
cana-5238	106	3	igc∗(ψ	igc∗(ψ	NOUN
cana-5238	106	4	)	)	PUNCT
cana-5238	106	5	⊆	⊆	NUM
cana-5238	106	6	ig(ψ	ig(ψ	NOUN
cana-5238	106	7	)	)	PUNCT
cana-5238	106	8	.	.	PUNCT
cana-5238	107	1	now	now	ADV
cana-5238	107	2	ψ⊆igc∗(ψ)∪c∗ig(ψ)⊆c∗ig(ψ)∪ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ)⊆c∗ig(ψ)∪ig(ψ	PRON
cana-5238	107	3	)	)	PUNCT
cana-5238	107	4	=	=	SYM
cana-5238	107	5	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	107	6	)	)	PUNCT
cana-5238	107	7	.	.	PUNCT
cana-5238	108	1	hence	hence	ADV
cana-5238	108	2	σhg	σhg	NOUN
cana-5238	108	3	-	-	PUNCT
cana-5238	108	4	o.	o.	NOUN
cana-5238	108	5	theorem	theorem	VERB
cana-5238	108	6	2.19	2.19	NUM
cana-5238	108	7	.	.	PUNCT
cana-5238	109	1	if	if	SCONJ
cana-5238	109	2	ψ	ψ	NOUN
cana-5238	109	3	is	be	AUX
cana-5238	109	4	b	b	PROPN
cana-5238	109	5	-	-	PUNCT
cana-5238	109	6	hg	hg	NOUN
cana-5238	109	7	-	-	NOUN
cana-5238	109	8	o	o	NOUN
cana-5238	109	9	such	such	ADJ
cana-5238	109	10	that	that	SCONJ
cana-5238	109	11	ig(ψ	ig(ψ	NOUN
cana-5238	109	12	)	)	PUNCT
cana-5238	109	13	=	=	SYM
cana-5238	109	14	∅	∅	NOUN
cana-5238	109	15	,	,	PUNCT
cana-5238	109	16	then	then	ADV
cana-5238	109	17	it	it	PRON
cana-5238	109	18	is	be	AUX
cana-5238	109	19	π	π	PROPN
cana-5238	109	20	-	-	PUNCT
cana-5238	109	21	hg	hg	ADJ
cana-5238	109	22	-	-	ADJ
cana-5238	109	23	o.	o.	ADJ
cana-5238	109	24	proof	proof	NOUN
cana-5238	109	25	.	.	PUNCT
cana-5238	110	1	let	let	VERB
cana-5238	110	2	ψ	ψ	PART
cana-5238	110	3	be	be	AUX
cana-5238	110	4	a	a	DET
cana-5238	110	5	b	b	PROPN
cana-5238	110	6	-	-	PUNCT
cana-5238	110	7	hg	hg	NOUN
cana-5238	110	8	-	-	NOUN
cana-5238	110	9	o	o	NOUN
cana-5238	110	10	and	and	CCONJ
cana-5238	110	11	ig(ψ)=∅.	ig(ψ)=∅.	NUM
cana-5238	110	12	then	then	ADV
cana-5238	110	13	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	110	14	)	)	PUNCT
cana-5238	110	15	=	=	PUNCT
cana-5238	110	16	igc∗(ψ	igc∗(ψ	NUM
cana-5238	110	17	)	)	PUNCT
cana-5238	110	18	.	.	PUNCT
cana-5238	111	1	hence	hence	ADV
cana-5238	111	2	ψ	ψ	X
cana-5238	111	3	is	be	AUX
cana-5238	111	4	π	π	PROPN
cana-5238	111	5	-	-	PUNCT
cana-5238	111	6	hg	hg	ADJ
cana-5238	111	7	-	-	ADJ
cana-5238	111	8	o.	o.	NOUN
cana-5238	111	9	theorem	theorem	VERB
cana-5238	111	10	2.20	2.20	NUM
cana-5238	111	11	.	.	PUNCT
cana-5238	112	1	if	if	SCONJ
cana-5238	112	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	112	3	is	be	AUX
cana-5238	112	4	b	b	PROPN
cana-5238	112	5	-	-	PUNCT
cana-5238	112	6	hg	hg	NOUN
cana-5238	112	7	-	-	NOUN
cana-5238	112	8	o	o	NOUN
cana-5238	112	9	and	and	CCONJ
cana-5238	112	10	ψ∈h	ψ∈h	NOUN
cana-5238	112	11	,	,	PUNCT
cana-5238	112	12	then	then	ADV
cana-5238	112	13	it	it	PRON
cana-5238	112	14	is	be	AUX
cana-5238	112	15	σ	σ	PROPN
cana-5238	112	16	-	-	PUNCT
cana-5238	112	17	hg	hg	NOUN
cana-5238	112	18	-	-	ADJ
cana-5238	112	19	o.	o.	ADJ
cana-5238	112	20	proof	proof	NOUN
cana-5238	112	21	.	.	PUNCT
cana-5238	113	1	let	let	VERB
cana-5238	113	2	ψ	ψ	NOUN
cana-5238	113	3	is	be	AUX
cana-5238	113	4	b	b	NUM
cana-5238	113	5	-	-	PUNCT
cana-5238	113	6	hg	hg	NOUN
cana-5238	113	7	-	-	NOUN
cana-5238	113	8	o	o	X
cana-5238	113	9	and	and	CCONJ
cana-5238	113	10	ψ∈	ψ∈	VERB
cana-5238	113	11	h.	h.	PROPN
cana-5238	113	12	then	then	ADV
cana-5238	113	13	ψ⊆	ψ⊆	ADJ
cana-5238	113	14	igc∗(ψ)∪c∗ig(ψ	igc∗(ψ)∪c∗ig(ψ	NOUN
cana-5238	113	15	)	)	PUNCT
cana-5238	113	16	and	and	CCONJ
cana-5238	113	17	c∗(ψ	c∗(ψ	NUM
cana-5238	113	18	)	)	PUNCT
cana-5238	114	1	=	=	SYM
cana-5238	114	2	ψ	ψ	NOUN
cana-5238	114	3	by	by	ADP
cana-5238	114	4	remark	remark	NOUN
cana-5238	114	5	2.10	2.10	NUM
cana-5238	114	6	of	of	ADP
cana-5238	114	7	[	[	X
cana-5238	114	8	6	6	NUM
cana-5238	114	9	]	]	PUNCT
cana-5238	114	10	.	.	PUNCT
cana-5238	115	1	now	now	ADV
cana-5238	115	2	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	115	3	)	)	PUNCT
cana-5238	115	4	=	=	PUNCT
cana-5238	116	1	ig(ψ)∪c∗ig(ψ)=c∗ig(ψ	ig(ψ)∪c∗ig(ψ)=c∗ig(ψ	NOUN
cana-5238	116	2	)	)	PUNCT
cana-5238	116	3	.	.	PUNCT
cana-5238	117	1	hence	hence	ADV
cana-5238	117	2	ψ	ψ	X
cana-5238	117	3	is	be	AUX
cana-5238	117	4	σ	σ	PROPN
cana-5238	117	5	-	-	PUNCT
cana-5238	117	6	hg	hg	NOUN
cana-5238	117	7	-	-	ADJ
cana-5238	117	8	o.	o.	NOUN
cana-5238	117	9	theorem	theorem	VERB
cana-5238	117	10	2.21	2.21	NUM
cana-5238	117	11	.	.	PUNCT
cana-5238	118	1	if	if	SCONJ
cana-5238	118	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	118	3	is	be	AUX
cana-5238	118	4	b	b	PROPN
cana-5238	118	5	-	-	PUNCT
cana-5238	118	6	hg	hg	NOUN
cana-5238	118	7	-	-	NOUN
cana-5238	118	8	o	o	NOUN
cana-5238	118	9	and	and	CCONJ
cana-5238	118	10	h	h	NOUN
cana-5238	118	11	=	=	NOUN
cana-5238	118	12	p	p	X
cana-5238	118	13	(	(	PUNCT
cana-5238	118	14	z	z	NOUN
cana-5238	118	15	)	)	PUNCT
cana-5238	118	16	then	then	ADV
cana-5238	118	17	it	it	PRON
cana-5238	118	18	is	be	AUX
cana-5238	118	19	σ	σ	PROPN
cana-5238	118	20	-	-	PUNCT
cana-5238	118	21	hg	hg	NOUN
cana-5238	118	22	-	-	ADJ
cana-5238	118	23	o.	o.	ADJ
cana-5238	118	24	proof	proof	NOUN
cana-5238	118	25	.	.	PUNCT
cana-5238	119	1	let	let	VERB
cana-5238	119	2	ψ	ψ	NOUN
cana-5238	119	3	is	be	AUX
cana-5238	119	4	b	b	NUM
cana-5238	119	5	-	-	PUNCT
cana-5238	119	6	hg	hg	NOUN
cana-5238	119	7	-	-	NOUN
cana-5238	119	8	o	o	NOUN
cana-5238	119	9	and	and	CCONJ
cana-5238	119	10	h	h	NOUN
cana-5238	120	1	=	=	NOUN
cana-5238	120	2	p	p	X
cana-5238	120	3	(	(	PUNCT
cana-5238	120	4	z	z	NOUN
cana-5238	120	5	)	)	PUNCT
cana-5238	120	6	then	then	ADV
cana-5238	120	7	ψ	ψ	ADP
cana-5238	120	8	⊆	⊆	NUM
cana-5238	120	9	igc∗(ψ)∪c∗ig(ψ	igc∗(ψ)∪c∗ig(ψ	NOUN
cana-5238	120	10	)	)	PUNCT
cana-5238	120	11	and	and	CCONJ
cana-5238	120	12	c∗(ψ)=ψ	c∗(ψ)=ψ	VERB
cana-5238	120	13	by	by	ADP
cana-5238	120	14	remark	remark	NOUN
cana-5238	120	15	2.10	2.10	NUM
cana-5238	120	16	of	of	ADP
cana-5238	120	17	[	[	X
cana-5238	120	18	6	6	NUM
cana-5238	120	19	]	]	PUNCT
cana-5238	120	20	.	.	PUNCT
cana-5238	121	1	now	now	ADV
cana-5238	121	2	ψ⊆igc∗(ψ)∪c∗ig(ψ)=ig(ψ)∪c∗ig(ψ)=c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ)=ig(ψ)∪c∗ig(ψ)=c∗ig(ψ	PROPN
cana-5238	121	3	)	)	PUNCT
cana-5238	121	4	.	.	PUNCT
cana-5238	122	1	hence	hence	ADV
cana-5238	122	2	ψ	ψ	X
cana-5238	122	3	is	be	AUX
cana-5238	122	4	σ	σ	PROPN
cana-5238	122	5	-	-	PUNCT
cana-5238	122	6	hg	hg	NOUN
cana-5238	122	7	-	-	ADJ
cana-5238	122	8	o.	o.	ADJ
cana-5238	122	9	definition	definition	NOUN
cana-5238	122	10	2.22	2.22	NUM
cana-5238	122	11	.	.	PUNCT
cana-5238	123	1	for	for	ADP
cana-5238	123	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	123	3	,	,	PUNCT
cana-5238	123	4	ibhg(ψ	ibhg(ψ	PROPN
cana-5238	123	5	)	)	PUNCT
cana-5238	123	6	is	be	AUX
cana-5238	123	7	the	the	DET
cana-5238	123	8	largest	large	ADJ
cana-5238	123	9	b	b	NOUN
cana-5238	123	10	-	-	PUNCT
cana-5238	123	11	hg	hg	NOUN
cana-5238	123	12	-	-	NOUN
cana-5238	123	13	o	o	NOUN
cana-5238	123	14	set	set	NOUN
cana-5238	123	15	contained	contain	VERB
cana-5238	123	16	in	in	ADP
cana-5238	123	17	ψ	ψ	PROPN
cana-5238	123	18	.	.	PUNCT
cana-5238	124	1	definition	definition	NOUN
cana-5238	124	2	2.23	2.23	NUM
cana-5238	124	3	.	.	PUNCT
cana-5238	125	1	a	a	DET
cana-5238	125	2	subset	subset	NOUN
cana-5238	125	3	ψ	ψ	X
cana-5238	125	4	of	of	ADP
cana-5238	125	5	hgt	hgt	PROPN
cana-5238	125	6	s	s	PROPN
cana-5238	125	7	(	(	PUNCT
cana-5238	125	8	z	z	NOUN
cana-5238	125	9	,	,	PUNCT
cana-5238	125	10	ζ	ζ	NOUN
cana-5238	125	11	,	,	PUNCT
cana-5238	125	12	h	h	NOUN
cana-5238	125	13	)	)	PUNCT
cana-5238	125	14	is	be	AUX
cana-5238	125	15	called	call	VERB
cana-5238	125	16	db(c	db(c	PUNCT
cana-5238	125	17	,	,	PUNCT
cana-5238	125	18	hg)-s	hg)-	NOUN
cana-5238	125	19	,	,	PUNCT
cana-5238	125	20	if	if	SCONJ
cana-5238	125	21	ig(ψ)=ibhg	ig(ψ)=ibhg	NOUN
cana-5238	125	22	(	(	PUNCT
cana-5238	125	23	ψ	ψ	NOUN
cana-5238	125	24	)	)	PUNCT
cana-5238	125	25	.	.	PUNCT
cana-5238	126	1	theorem	theorem	VERB
cana-5238	126	2	2.24	2.24	NUM
cana-5238	126	3	.	.	PUNCT
cana-5238	127	1	for	for	ADP
cana-5238	127	2	a	a	DET
cana-5238	127	3	subset	subset	NOUN
cana-5238	127	4	ψ	ψ	X
cana-5238	127	5	of	of	ADP
cana-5238	127	6	hgt	hgt	PROPN
cana-5238	127	7	s	s	PROPN
cana-5238	127	8	(	(	PUNCT
cana-5238	127	9	z	z	NOUN
cana-5238	127	10	,	,	PUNCT
cana-5238	127	11	ζ	ζ	NOUN
cana-5238	127	12	,	,	PUNCT
cana-5238	127	13	h	h	NOUN
cana-5238	127	14	)	)	PUNCT
cana-5238	127	15	,	,	PUNCT
cana-5238	127	16	the	the	DET
cana-5238	127	17	following	follow	VERB
cana-5238	127	18	conditions	condition	NOUN
cana-5238	127	19	are	be	AUX
cana-5238	127	20	equivalent	equivalent	ADJ
cana-5238	127	21	.	.	PUNCT
cana-5238	128	1	1	1	X
cana-5238	128	2	.	.	X
cana-5238	128	3	ψ	ψ	NOUN
cana-5238	128	4	is	be	AUX
cana-5238	128	5	gζ	gζ	NOUN
cana-5238	128	6	-	-	PUNCT
cana-5238	128	7	o	o	NOUN
cana-5238	128	8	,	,	PUNCT
cana-5238	128	9	2	2	NUM
cana-5238	128	10	.	.	PUNCT
cana-5238	128	11	ψ	ψ	NOUN
cana-5238	128	12	is	be	AUX
cana-5238	128	13	b	b	PROPN
cana-5238	128	14	-	-	PUNCT
cana-5238	128	15	hg	hg	NOUN
cana-5238	128	16	-	-	NOUN
cana-5238	128	17	o	o	NOUN
cana-5238	128	18	and	and	CCONJ
cana-5238	128	19	db(c	db(c	NOUN
cana-5238	128	20	,	,	PUNCT
cana-5238	128	21	hg)-s	hg)-s	ADJ
cana-5238	128	22	proof	proof	NOUN
cana-5238	128	23	.	.	PUNCT
cana-5238	129	1	(	(	PUNCT
cana-5238	129	2	1)⇒(2	1)⇒(2	X
cana-5238	129	3	)	)	PUNCT
cana-5238	129	4	let	let	VERB
cana-5238	129	5	ψ	ψ	NOUN
cana-5238	129	6	is	be	AUX
cana-5238	129	7	gζ	gζ	PROPN
cana-5238	129	8	-	-	PUNCT
cana-5238	129	9	o.	o.	PROPN
cana-5238	129	10	then	then	ADV
cana-5238	129	11	ψ	ψ	X
cana-5238	129	12	is	be	AUX
cana-5238	129	13	b	b	NUM
cana-5238	129	14	-	-	PUNCT
cana-5238	129	15	hg	hg	NOUN
cana-5238	129	16	-	-	NOUN
cana-5238	130	1	o.	o.	ADJ
cana-5238	130	2	so	so	CCONJ
cana-5238	130	3	ψ	ψ	X
cana-5238	130	4	=	=	NOUN
cana-5238	130	5	ig(ψ	ig(ψ	NOUN
cana-5238	130	6	)	)	PUNCT
cana-5238	130	7	and	and	CCONJ
cana-5238	130	8	ψ	ψ	X
cana-5238	130	9	=	=	SYM
cana-5238	130	10	ibh(ψ	ibh(ψ	PROPN
cana-5238	130	11	)	)	PUNCT
cana-5238	130	12	.	.	PUNCT
cana-5238	131	1	therefore	therefore	ADV
cana-5238	131	2	ig(ψ	ig(ψ	NOUN
cana-5238	131	3	)	)	PUNCT
cana-5238	131	4	=	=	SYM
cana-5238	131	5	ibh(ψ	ibh(ψ	PROPN
cana-5238	131	6	)	)	PUNCT
cana-5238	131	7	.	.	PUNCT
cana-5238	132	1	hence	hence	ADV
cana-5238	132	2	ψ	ψ	NOUN
cana-5238	132	3	is	be	AUX
cana-5238	132	4	db(c	db(c	NOUN
cana-5238	132	5	,	,	PUNCT
cana-5238	132	6	hg)-s	hg)-s	PUNCT
cana-5238	132	7	(	(	PUNCT
cana-5238	132	8	2	2	NUM
cana-5238	132	9	)	)	PUNCT
cana-5238	132	10	⇒	⇒	NOUN
cana-5238	132	11	(	(	PUNCT
cana-5238	132	12	1	1	X
cana-5238	132	13	)	)	PUNCT
cana-5238	132	14	let	let	VERB
cana-5238	132	15	ψ	ψ	NOUN
cana-5238	132	16	is	be	AUX
cana-5238	132	17	b	b	NUM
cana-5238	132	18	-	-	PUNCT
cana-5238	132	19	hg	hg	NOUN
cana-5238	132	20	-	-	NOUN
cana-5238	132	21	o	o	NOUN
cana-5238	132	22	and	and	CCONJ
cana-5238	132	23	db(c	db(c	NOUN
cana-5238	132	24	,	,	PUNCT
cana-5238	132	25	hg)-s	hg)-s	PROPN
cana-5238	132	26	then	then	ADV
cana-5238	132	27	ψ	ψ	X
cana-5238	132	28	=	=	X
cana-5238	132	29	ibh	ibh	X
cana-5238	132	30	(	(	PUNCT
cana-5238	132	31	ψ	ψ	NOUN
cana-5238	132	32	)	)	PUNCT
cana-5238	132	33	and	and	CCONJ
cana-5238	132	34	ig(ψ	ig(ψ	NOUN
cana-5238	132	35	)	)	PUNCT
cana-5238	132	36	=	=	SYM
cana-5238	132	37	ibh(ψ	ibh(ψ	PROPN
cana-5238	132	38	)	)	PUNCT
cana-5238	132	39	.	.	PUNCT
cana-5238	133	1	therefore	therefore	ADV
cana-5238	133	2	ig(ψ)=ψ	ig(ψ)=ψ	PROPN
cana-5238	133	3	.	.	PROPN
cana-5238	133	4	hence	hence	ADV
cana-5238	133	5	ψ	ψ	NOUN
cana-5238	133	6	is	be	AUX
cana-5238	133	7	gζ	gζ	NOUN
cana-5238	133	8	-	-	PUNCT
cana-5238	133	9	o.	o.	ADJ
cana-5238	133	10	remark	remark	NOUN
cana-5238	133	11	2.25	2.25	NUM
cana-5238	133	12	.	.	PUNCT
cana-5238	134	1	the	the	DET
cana-5238	134	2	notions	notion	NOUN
cana-5238	134	3	of	of	ADP
cana-5238	134	4	ψ	ψ	NOUN
cana-5238	134	5	is	be	AUX
cana-5238	134	6	b	b	NUM
cana-5238	134	7	-	-	PUNCT
cana-5238	134	8	hg	hg	NOUN
cana-5238	134	9	-	-	NOUN
cana-5238	134	10	o	o	NOUN
cana-5238	134	11	and	and	CCONJ
cana-5238	134	12	db(c	db(c	NOUN
cana-5238	134	13	,	,	PUNCT
cana-5238	134	14	hg)-s	hg)-	NOUN
cana-5238	134	15	are	be	AUX
cana-5238	134	16	independent	independent	ADJ
cana-5238	134	17	.	.	PUNCT
cana-5238	134	18	example	example	NOUN
cana-5238	135	1	2.26	2.26	NUM
cana-5238	135	2	.	.	PUNCT
cana-5238	136	1	assume	assume	VERB
cana-5238	136	2	z={1	z={1	PROPN
cana-5238	136	3	,	,	PUNCT
cana-5238	136	4	2	2	NUM
cana-5238	136	5	,	,	PUNCT
cana-5238	136	6	3	3	NUM
cana-5238	136	7	,	,	PUNCT
cana-5238	136	8	4	4	NUM
cana-5238	136	9	}	}	PUNCT
cana-5238	136	10	,	,	PUNCT
cana-5238	136	11	ζ={∅	ζ={∅	PROPN
cana-5238	136	12	,	,	PUNCT
cana-5238	136	13	{	{	PUNCT
cana-5238	136	14	1	1	NUM
cana-5238	136	15	,	,	PUNCT
cana-5238	136	16	3	3	NUM
cana-5238	136	17	}	}	PUNCT
cana-5238	136	18	,	,	PUNCT
cana-5238	136	19	{	{	PUNCT
cana-5238	136	20	2	2	NUM
cana-5238	136	21	,	,	PUNCT
cana-5238	136	22	3	3	NUM
cana-5238	136	23	}	}	PUNCT
cana-5238	136	24	,	,	PUNCT
cana-5238	136	25	{	{	PUNCT
cana-5238	136	26	1	1	NUM
cana-5238	136	27	,	,	PUNCT
cana-5238	136	28	2	2	NUM
cana-5238	136	29	,	,	PUNCT
cana-5238	136	30	3	3	NUM
cana-5238	136	31	}	}	PUNCT
cana-5238	136	32	,	,	PUNCT
cana-5238	136	33	{	{	PUNCT
cana-5238	136	34	1	1	NUM
cana-5238	136	35	,	,	PUNCT
cana-5238	136	36	4	4	NUM
cana-5238	136	37	}	}	PUNCT
cana-5238	136	38	,	,	PUNCT
cana-5238	136	39	{	{	PUNCT
cana-5238	136	40	1	1	NUM
cana-5238	136	41	,	,	PUNCT
cana-5238	136	42	3	3	NUM
cana-5238	136	43	,	,	PUNCT
cana-5238	136	44	4	4	NUM
cana-5238	136	45	}	}	PUNCT
cana-5238	136	46	,	,	PUNCT
cana-5238	136	47	z	z	NOUN
cana-5238	136	48	}	}	PUNCT
cana-5238	136	49	,	,	PUNCT
cana-5238	136	50	h={∅	h={∅	PROPN
cana-5238	136	51	,	,	PUNCT
cana-5238	136	52	{	{	PUNCT
cana-5238	136	53	1	1	NUM
cana-5238	136	54	}	}	PUNCT
cana-5238	136	55	,	,	PUNCT
cana-5238	136	56	{	{	PUNCT
cana-5238	136	57	2	2	NUM
cana-5238	136	58	}	}	PUNCT
cana-5238	136	59	}	}	PUNCT
cana-5238	136	60	.	.	PUNCT
cana-5238	137	1	then	then	ADV
cana-5238	137	2	ψ	ψ	X
cana-5238	137	3	=	=	PUNCT
cana-5238	137	4	{	{	PUNCT
cana-5238	137	5	2	2	NUM
cana-5238	137	6	}	}	PUNCT
cana-5238	137	7	is	be	AUX
cana-5238	137	8	b	b	NUM
cana-5238	137	9	-	-	PUNCT
cana-5238	137	10	hg	hg	NOUN
cana-5238	137	11	-	-	NOUN
cana-5238	137	12	o	o	NOUN
cana-5238	137	13	but	but	CCONJ
cana-5238	137	14	not	not	PART
cana-5238	137	15	db(c	db(c	PUNCT
cana-5238	137	16	,	,	PUNCT
cana-5238	137	17	hg)-s	hg)-s	PUNCT
cana-5238	137	18	and	and	CCONJ
cana-5238	137	19	m	m	PROPN
cana-5238	137	20	=	=	X
cana-5238	137	21	{	{	PUNCT
cana-5238	137	22	4	4	NUM
cana-5238	137	23	}	}	PUNCT
cana-5238	137	24	is	be	AUX
cana-5238	137	25	db(c	db(c	NOUN
cana-5238	137	26	,	,	PUNCT
cana-5238	137	27	hg)-s	hg)-s	PUNCT
cana-5238	137	28	but	but	CCONJ
cana-5238	137	29	not	not	PART
cana-5238	137	30	b	b	NOUN
cana-5238	137	31	-	-	PUNCT
cana-5238	137	32	hg	hg	NOUN
cana-5238	137	33	-	-	NOUN
cana-5238	137	34	o	o	NOUN
cana-5238	137	35	set	set	NOUN
cana-5238	137	36	.	.	PUNCT
cana-5238	138	1	3	3	X
cana-5238	138	2	.	.	X
cana-5238	138	3	new	new	ADJ
cana-5238	138	4	types	type	NOUN
cana-5238	138	5	of	of	ADP
cana-5238	138	6	sets	set	NOUN
cana-5238	138	7	definition	definition	NOUN
cana-5238	138	8	3.1	3.1	NUM
cana-5238	138	9	.	.	PUNCT
cana-5238	139	1	a	a	DET
cana-5238	139	2	subset	subset	NOUN
cana-5238	139	3	ψ⊂z	ψ⊂z	NOUN
cana-5238	139	4	is	be	AUX
cana-5238	139	5	called	call	VERB
cana-5238	139	6	1	1	NUM
cana-5238	139	7	.	.	PUNCT
cana-5238	140	1	α∗-hg	α∗-hg	NOUN
cana-5238	140	2	-	-	PUNCT
cana-5238	140	3	s	s	NOUN
cana-5238	140	4	,	,	PUNCT
cana-5238	140	5	if	if	SCONJ
cana-5238	140	6	igc∗ig(ψ	igc∗ig(ψ	PROPN
cana-5238	140	7	)	)	PUNCT
cana-5238	140	8	=	=	SYM
cana-5238	140	9	i(ψ	i(ψ	PROPN
cana-5238	140	10	)	)	PUNCT
cana-5238	140	11	.	.	PUNCT
cana-5238	141	1	2	2	X
cana-5238	141	2	.	.	X
cana-5238	141	3	σ∗-hg	σ∗-hg	NOUN
cana-5238	141	4	-	-	PUNCT
cana-5238	141	5	s	s	NOUN
cana-5238	141	6	,	,	PUNCT
cana-5238	141	7	if	if	SCONJ
cana-5238	141	8	c∗ig(ψ)=i(ψ	c∗ig(ψ)=i(ψ	PROPN
cana-5238	141	9	)	)	PUNCT
cana-5238	141	10	.	.	PUNCT
cana-5238	142	1	communications	communication	NOUN
cana-5238	142	2	on	on	ADP
cana-5238	142	3	applied	apply	VERB
cana-5238	142	4	nonlinear	nonlinear	ADJ
cana-5238	142	5	analysis	analysis	NOUN
cana-5238	142	6	issn	issn	NOUN
cana-5238	142	7	:	:	PUNCT
cana-5238	142	8	1074	1074	NUM
cana-5238	142	9	-	-	PUNCT
cana-5238	142	10	133x	133x	NUM
cana-5238	142	11	vol	vol	VERB
cana-5238	142	12	32	32	NUM
cana-5238	142	13	no	no	NOUN
cana-5238	142	14	.	.	PUNCT
cana-5238	143	1	10s	10	NOUN
cana-5238	143	2	(	(	PUNCT
cana-5238	143	3	2025	2025	NUM
cana-5238	143	4	)	)	PUNCT
cana-5238	143	5	1361	1361	NUM
cana-5238	143	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	143	7	3	3	X
cana-5238	143	8	.	.	X
cana-5238	143	9	π∗-hg	π∗-hg	NOUN
cana-5238	143	10	-	-	PUNCT
cana-5238	143	11	s	s	NOUN
cana-5238	143	12	,	,	PUNCT
cana-5238	143	13	if	if	SCONJ
cana-5238	143	14	igc∗(ψ)=i(ψ	igc∗(ψ)=i(ψ	NOUN
cana-5238	143	15	)	)	PUNCT
cana-5238	143	16	.	.	PUNCT
cana-5238	144	1	4	4	X
cana-5238	144	2	.	.	X
cana-5238	144	3	b∗-hg	b∗-hg	NOUN
cana-5238	144	4	-	-	PUNCT
cana-5238	144	5	s	s	NOUN
cana-5238	144	6	,	,	PUNCT
cana-5238	144	7	if	if	SCONJ
cana-5238	144	8	c∗ig(ψ)∪igc∗(ψ)=i(ψ	c∗ig(ψ)∪igc∗(ψ)=i(ψ	PROPN
cana-5238	144	9	)	)	PUNCT
cana-5238	144	10	.	.	PUNCT
cana-5238	145	1	5	5	X
cana-5238	145	2	.	.	X
cana-5238	145	3	β∗-hg	β∗-hg	NOUN
cana-5238	145	4	-	-	PUNCT
cana-5238	145	5	s	s	X
cana-5238	145	6	(	(	PUNCT
cana-5238	145	7	β∗-hg	β∗-hg	NOUN
cana-5238	145	8	-	-	PUNCT
cana-5238	145	9	s	s	NOUN
cana-5238	145	10	)	)	PUNCT
cana-5238	145	11	,	,	PUNCT
cana-5238	145	12	if	if	SCONJ
cana-5238	145	13	cigc∗(ψ	cigc∗(ψ	NUM
cana-5238	145	14	)	)	PUNCT
cana-5238	146	1	=	=	SYM
cana-5238	146	2	i(ψ	i(ψ	PROPN
cana-5238	146	3	)	)	PUNCT
cana-5238	146	4	.	.	PUNCT
cana-5238	147	1	remark	remark	PROPN
cana-5238	147	2	3.2	3.2	NUM
cana-5238	147	3	.	.	PUNCT
cana-5238	148	1	the	the	DET
cana-5238	148	2	notions	notion	NOUN
cana-5238	148	3	of	of	ADP
cana-5238	148	4	α∗-hg	α∗-hg	NOUN
cana-5238	148	5	-	-	PUNCT
cana-5238	148	6	s	s	X
cana-5238	148	7	(	(	PUNCT
cana-5238	148	8	resp	resp	NOUN
cana-5238	148	9	.	.	PUNCT
cana-5238	149	1	σ∗-hg	σ∗-hg	NOUN
cana-5238	149	2	-	-	PUNCT
cana-5238	149	3	s	s	NOUN
cana-5238	149	4	,	,	PUNCT
cana-5238	149	5	π∗-hg	π∗-hg	NOUN
cana-5238	149	6	-	-	PUNCT
cana-5238	149	7	s	s	NOUN
cana-5238	149	8	,	,	PUNCT
cana-5238	149	9	b∗-hg	b∗-hg	NOUN
cana-5238	149	10	-	-	PUNCT
cana-5238	149	11	s	s	NOUN
cana-5238	149	12	,	,	PUNCT
cana-5238	149	13	β∗-hg	β∗-hg	NOUN
cana-5238	149	14	-	-	PUNCT
cana-5238	149	15	s	s	NOUN
cana-5238	149	16	)	)	PUNCT
cana-5238	149	17	and	and	CCONJ
cana-5238	149	18	α	α	X
cana-5238	149	19	-	-	PUNCT
cana-5238	149	20	g	g	NOUN
cana-5238	149	21	-	-	PUNCT
cana-5238	149	22	o	o	NOUN
cana-5238	149	23	(	(	PUNCT
cana-5238	149	24	resp	resp	NOUN
cana-5238	149	25	.	.	PUNCT
cana-5238	150	1	σhg	σhg	NOUN
cana-5238	150	2	-	-	PUNCT
cana-5238	150	3	o	o	NOUN
cana-5238	150	4	,	,	PUNCT
cana-5238	150	5	π	π	PROPN
cana-5238	150	6	-	-	PUNCT
cana-5238	150	7	hg	hg	NOUN
cana-5238	150	8	-	-	NOUN
cana-5238	150	9	o	o	NOUN
cana-5238	150	10	,	,	PUNCT
cana-5238	150	11	b	b	X
cana-5238	150	12	-	-	PUNCT
cana-5238	150	13	hg	hg	NOUN
cana-5238	150	14	-	-	NOUN
cana-5238	150	15	o	o	NOUN
cana-5238	150	16	,	,	PUNCT
cana-5238	150	17	β	β	X
cana-5238	150	18	-	-	ADJ
cana-5238	150	19	hg	hg	NOUN
cana-5238	150	20	-	-	NOUN
cana-5238	150	21	o	o	NOUN
cana-5238	150	22	)	)	PUNCT
cana-5238	150	23	are	be	AUX
cana-5238	150	24	independent	independent	ADJ
cana-5238	150	25	.	.	PUNCT
cana-5238	150	26	example	example	NOUN
cana-5238	150	27	3.3	3.3	NUM
cana-5238	150	28	.	.	PUNCT
cana-5238	151	1	assume	assume	VERB
cana-5238	151	2	z={1	z={1	PROPN
cana-5238	151	3	,	,	PUNCT
cana-5238	151	4	2	2	NUM
cana-5238	151	5	,	,	PUNCT
cana-5238	151	6	3	3	NUM
cana-5238	151	7	,	,	PUNCT
cana-5238	151	8	4	4	NUM
cana-5238	151	9	}	}	PUNCT
cana-5238	151	10	,	,	PUNCT
cana-5238	151	11	ζ={∅	ζ={∅	PROPN
cana-5238	151	12	,	,	PUNCT
cana-5238	151	13	{	{	PUNCT
cana-5238	151	14	1	1	NUM
cana-5238	151	15	,	,	PUNCT
cana-5238	151	16	3	3	NUM
cana-5238	151	17	}	}	PUNCT
cana-5238	151	18	,	,	PUNCT
cana-5238	151	19	{	{	PUNCT
cana-5238	151	20	2	2	NUM
cana-5238	151	21	,	,	PUNCT
cana-5238	151	22	3	3	NUM
cana-5238	151	23	}	}	PUNCT
cana-5238	151	24	,	,	PUNCT
cana-5238	151	25	{	{	PUNCT
cana-5238	151	26	1	1	NUM
cana-5238	151	27	,	,	PUNCT
cana-5238	151	28	2	2	NUM
cana-5238	151	29	,	,	PUNCT
cana-5238	151	30	3},{1	3},{1	NUM
cana-5238	151	31	,	,	PUNCT
cana-5238	151	32	4	4	NUM
cana-5238	151	33	}	}	PUNCT
cana-5238	151	34	,	,	PUNCT
cana-5238	151	35	{	{	PUNCT
cana-5238	151	36	1	1	NUM
cana-5238	151	37	,	,	PUNCT
cana-5238	151	38	3	3	NUM
cana-5238	151	39	,	,	PUNCT
cana-5238	151	40	4	4	NUM
cana-5238	151	41	}	}	PUNCT
cana-5238	151	42	,	,	PUNCT
cana-5238	151	43	z	z	NOUN
cana-5238	151	44	}	}	PUNCT
cana-5238	151	45	,	,	PUNCT
cana-5238	151	46	h	h	NOUN
cana-5238	151	47	=	=	PRON
cana-5238	151	48	{	{	PUNCT
cana-5238	151	49	∅	∅	NOUN
cana-5238	151	50	,	,	PUNCT
cana-5238	151	51	{	{	PUNCT
cana-5238	151	52	1	1	NUM
cana-5238	151	53	}	}	PUNCT
cana-5238	151	54	,	,	PUNCT
cana-5238	151	55	{	{	PUNCT
cana-5238	151	56	2	2	NUM
cana-5238	151	57	}	}	PUNCT
cana-5238	151	58	}	}	PUNCT
cana-5238	151	59	.	.	PUNCT
cana-5238	152	1	then	then	ADV
cana-5238	152	2	ψ={1	ψ={1	PROPN
cana-5238	152	3	}	}	PUNCT
cana-5238	152	4	is	be	AUX
cana-5238	152	5	α	α	PROPN
cana-5238	152	6	-	-	PUNCT
cana-5238	152	7	hg	hg	NOUN
cana-5238	152	8	-	-	NOUN
cana-5238	152	9	o	o	X
cana-5238	152	10	(	(	PUNCT
cana-5238	152	11	resp	resp	NOUN
cana-5238	152	12	.	.	PUNCT
cana-5238	153	1	σ	σ	PROPN
cana-5238	153	2	-	-	PUNCT
cana-5238	153	3	hg	hg	NOUN
cana-5238	153	4	-	-	PROPN
cana-5238	153	5	o	o	NOUN
cana-5238	153	6	,	,	PUNCT
cana-5238	153	7	π	π	PROPN
cana-5238	153	8	-	-	PUNCT
cana-5238	153	9	hg	hg	NOUN
cana-5238	153	10	-	-	NOUN
cana-5238	153	11	o	o	NOUN
cana-5238	153	12	,	,	PUNCT
cana-5238	153	13	b	b	X
cana-5238	153	14	-	-	PUNCT
cana-5238	153	15	hg	hg	NOUN
cana-5238	153	16	-	-	NOUN
cana-5238	153	17	o	o	NOUN
cana-5238	153	18	,	,	PUNCT
cana-5238	153	19	β	β	X
cana-5238	153	20	-	-	ADJ
cana-5238	153	21	hg	hg	NOUN
cana-5238	153	22	-	-	NOUN
cana-5238	153	23	o	o	NOUN
cana-5238	153	24	)	)	PUNCT
cana-5238	153	25	but	but	CCONJ
cana-5238	153	26	not	not	PART
cana-5238	153	27	α∗-hg	α∗-hg	NOUN
cana-5238	153	28	-	-	PUNCT
cana-5238	153	29	s	s	X
cana-5238	153	30	(	(	PUNCT
cana-5238	153	31	resp	resp	NOUN
cana-5238	153	32	.	.	PUNCT
cana-5238	154	1	σ∗hg	σ∗hg	PROPN
cana-5238	154	2	-	-	PUNCT
cana-5238	154	3	s	s	X
cana-5238	154	4	,	,	PUNCT
cana-5238	154	5	π∗-hg	π∗-hg	NOUN
cana-5238	154	6	-	-	PUNCT
cana-5238	154	7	s	s	NOUN
cana-5238	154	8	,	,	PUNCT
cana-5238	154	9	b∗-hg	b∗-hg	NOUN
cana-5238	154	10	-	-	PUNCT
cana-5238	154	11	s	s	NOUN
cana-5238	154	12	,	,	PUNCT
cana-5238	154	13	β∗-hg	β∗-hg	NOUN
cana-5238	154	14	-	-	PUNCT
cana-5238	154	15	s	s	NOUN
cana-5238	154	16	)	)	PUNCT
cana-5238	154	17	and	and	CCONJ
cana-5238	154	18	m={2	m={2	PROPN
cana-5238	154	19	}	}	PUNCT
cana-5238	154	20	is	be	AUX
cana-5238	154	21	α∗-hg	α∗-hg	NOUN
cana-5238	154	22	-	-	PUNCT
cana-5238	154	23	s	s	X
cana-5238	154	24	(	(	PUNCT
cana-5238	154	25	resp	resp	NOUN
cana-5238	154	26	.	.	PUNCT
cana-5238	155	1	σ∗hg	σ∗hg	PROPN
cana-5238	155	2	-	-	PUNCT
cana-5238	155	3	s	s	X
cana-5238	155	4	,	,	PUNCT
cana-5238	155	5	π∗-hg	π∗-hg	NOUN
cana-5238	155	6	-	-	PUNCT
cana-5238	155	7	s	s	NOUN
cana-5238	155	8	,	,	PUNCT
cana-5238	155	9	b∗-hg	b∗-hg	NOUN
cana-5238	155	10	-	-	PUNCT
cana-5238	155	11	s	s	NOUN
cana-5238	155	12	,	,	PUNCT
cana-5238	155	13	β∗-hgs	β∗-hgs	ADJ
cana-5238	155	14	)	)	PUNCT
cana-5238	155	15	but	but	CCONJ
cana-5238	155	16	not	not	PART
cana-5238	155	17	α	α	PROPN
cana-5238	155	18	-	-	PUNCT
cana-5238	155	19	hg	hg	NOUN
cana-5238	155	20	-	-	NOUN
cana-5238	155	21	o	o	X
cana-5238	155	22	(	(	PUNCT
cana-5238	155	23	resp	resp	NOUN
cana-5238	155	24	.	.	PUNCT
cana-5238	156	1	σ	σ	PROPN
cana-5238	156	2	-	-	PUNCT
cana-5238	156	3	hg	hg	PROPN
cana-5238	156	4	o	o	PROPN
cana-5238	156	5	,	,	PUNCT
cana-5238	156	6	π	π	PROPN
cana-5238	156	7	-	-	PUNCT
cana-5238	156	8	hg	hg	NOUN
cana-5238	156	9	-	-	NOUN
cana-5238	156	10	o	o	NOUN
cana-5238	156	11	,	,	PUNCT
cana-5238	156	12	b	b	X
cana-5238	156	13	-	-	PUNCT
cana-5238	156	14	hg	hg	NOUN
cana-5238	156	15	-	-	NOUN
cana-5238	156	16	o	o	NOUN
cana-5238	156	17	,	,	PUNCT
cana-5238	156	18	β	β	X
cana-5238	156	19	-	-	PUNCT
cana-5238	156	20	hg	hg	NOUN
cana-5238	156	21	-	-	NOUN
cana-5238	156	22	o	o	NOUN
cana-5238	156	23	)	)	PUNCT
cana-5238	156	24	.	.	PUNCT
cana-5238	157	1	definition	definition	NOUN
cana-5238	157	2	3.4	3.4	NUM
cana-5238	157	3	.	.	PUNCT
cana-5238	158	1	the	the	DET
cana-5238	158	2	subset	subset	NOUN
cana-5238	158	3	ψ	ψ	X
cana-5238	158	4	⊂	⊂	PROPN
cana-5238	158	5	z	z	X
cana-5238	158	6	of	of	ADP
cana-5238	158	7	is	be	AUX
cana-5238	158	8	called	call	VERB
cana-5238	158	9	1	1	NUM
cana-5238	158	10	.	.	PUNCT
cana-5238	159	1	α∗-b	α∗-b	PROPN
cana-5238	159	2	-	-	PUNCT
cana-5238	159	3	hg	hg	NOUN
cana-5238	159	4	-	-	PUNCT
cana-5238	159	5	s	s	X
cana-5238	159	6	(	(	PUNCT
cana-5238	159	7	α∗-b	α∗-b	PROPN
cana-5238	159	8	-	-	PUNCT
cana-5238	159	9	hg	hg	NOUN
cana-5238	159	10	-	-	PUNCT
cana-5238	159	11	s	s	NOUN
cana-5238	159	12	)	)	PUNCT
cana-5238	159	13	,	,	PUNCT
cana-5238	159	14	if	if	SCONJ
cana-5238	159	15	ψ	ψ	VERB
cana-5238	159	16	=	=	NOUN
cana-5238	159	17	u∩v	u∩v	ADJ
cana-5238	159	18	,	,	PUNCT
cana-5238	159	19	where	where	SCONJ
cana-5238	159	20	u	u	NOUN
cana-5238	159	21	is	be	AUX
cana-5238	159	22	ζ	ζ	NOUN
cana-5238	159	23	-	-	PUNCT
cana-5238	159	24	o	o	NOUN
cana-5238	159	25	and	and	CCONJ
cana-5238	159	26	v	v	NOUN
cana-5238	159	27	is	be	AUX
cana-5238	159	28	α∗-hg	α∗-hg	ADV
cana-5238	159	29	-	-	PUNCT
cana-5238	159	30	s.	s.	PROPN
cana-5238	159	31	2	2	NUM
cana-5238	159	32	.	.	PUNCT
cana-5238	160	1	σ∗-b	σ∗-b	PROPN
cana-5238	160	2	-	-	PUNCT
cana-5238	160	3	hg	hg	NOUN
cana-5238	160	4	-	-	PUNCT
cana-5238	160	5	s	s	X
cana-5238	160	6	(	(	PUNCT
cana-5238	160	7	σ∗-b	σ∗-b	PROPN
cana-5238	160	8	-	-	PUNCT
cana-5238	160	9	hg	hg	NOUN
cana-5238	160	10	-	-	PUNCT
cana-5238	160	11	s	s	NOUN
cana-5238	160	12	)	)	PUNCT
cana-5238	160	13	,	,	PUNCT
cana-5238	160	14	if	if	SCONJ
cana-5238	160	15	ψ	ψ	VERB
cana-5238	160	16	=	=	NOUN
cana-5238	160	17	u∩v	u∩v	ADJ
cana-5238	160	18	,	,	PUNCT
cana-5238	160	19	where	where	SCONJ
cana-5238	160	20	u	u	NOUN
cana-5238	160	21	is	be	AUX
cana-5238	160	22	ζ	ζ	NOUN
cana-5238	160	23	-	-	PUNCT
cana-5238	160	24	o	o	NOUN
cana-5238	160	25	and	and	CCONJ
cana-5238	160	26	v	v	NOUN
cana-5238	160	27	is	be	AUX
cana-5238	160	28	σ∗-hg	σ∗-hg	NOUN
cana-5238	160	29	-	-	PUNCT
cana-5238	160	30	s.	s.	PROPN
cana-5238	160	31	3	3	NUM
cana-5238	160	32	.	.	PUNCT
cana-5238	161	1	π∗-b	π∗-b	PROPN
cana-5238	161	2	-	-	PUNCT
cana-5238	161	3	hg	hg	NOUN
cana-5238	161	4	-	-	PUNCT
cana-5238	161	5	s	s	X
cana-5238	161	6	(	(	PUNCT
cana-5238	161	7	π∗-b	π∗-b	PROPN
cana-5238	161	8	-	-	PUNCT
cana-5238	161	9	hg	hg	NOUN
cana-5238	161	10	-	-	PUNCT
cana-5238	161	11	s	s	NOUN
cana-5238	161	12	)	)	PUNCT
cana-5238	161	13	,	,	PUNCT
cana-5238	161	14	if	if	SCONJ
cana-5238	161	15	ψ	ψ	VERB
cana-5238	161	16	=	=	NOUN
cana-5238	161	17	u∩v	u∩v	ADJ
cana-5238	161	18	,	,	PUNCT
cana-5238	161	19	where	where	SCONJ
cana-5238	161	20	u	u	NOUN
cana-5238	161	21	is	be	AUX
cana-5238	161	22	ζ	ζ	NOUN
cana-5238	161	23	-	-	PUNCT
cana-5238	161	24	o	o	NOUN
cana-5238	161	25	and	and	CCONJ
cana-5238	161	26	v	v	NOUN
cana-5238	161	27	is	be	AUX
cana-5238	161	28	π∗-hg	π∗-hg	NOUN
cana-5238	161	29	-	-	PUNCT
cana-5238	161	30	s.	s.	PROPN
cana-5238	161	31	4	4	NUM
cana-5238	161	32	.	.	PUNCT
cana-5238	162	1	b∗-b	b∗-b	NOUN
cana-5238	162	2	-	-	PUNCT
cana-5238	162	3	hg	hg	NOUN
cana-5238	162	4	-	-	PUNCT
cana-5238	162	5	s	s	X
cana-5238	162	6	(	(	PUNCT
cana-5238	162	7	b∗-b	b∗-b	NOUN
cana-5238	162	8	-	-	PUNCT
cana-5238	162	9	hg	hg	NOUN
cana-5238	162	10	-	-	PUNCT
cana-5238	162	11	s	s	NOUN
cana-5238	162	12	)	)	PUNCT
cana-5238	162	13	,	,	PUNCT
cana-5238	162	14	if	if	SCONJ
cana-5238	162	15	ψ	ψ	VERB
cana-5238	162	16	=	=	NOUN
cana-5238	162	17	u∩v	u∩v	ADJ
cana-5238	162	18	,	,	PUNCT
cana-5238	162	19	where	where	SCONJ
cana-5238	162	20	u	u	NOUN
cana-5238	162	21	is	be	AUX
cana-5238	162	22	ζ	ζ	NOUN
cana-5238	162	23	-	-	PUNCT
cana-5238	162	24	o	o	NOUN
cana-5238	162	25	and	and	CCONJ
cana-5238	162	26	v	v	NOUN
cana-5238	162	27	is	be	AUX
cana-5238	162	28	b∗-hg	b∗-hg	NOUN
cana-5238	162	29	-	-	PUNCT
cana-5238	162	30	s.	s.	PROPN
cana-5238	162	31	5	5	NUM
cana-5238	162	32	.	.	PUNCT
cana-5238	163	1	β∗-b	β∗-b	PROPN
cana-5238	163	2	-	-	PUNCT
cana-5238	163	3	hg	hg	NOUN
cana-5238	163	4	-	-	PUNCT
cana-5238	163	5	s	s	X
cana-5238	163	6	(	(	PUNCT
cana-5238	163	7	β∗-b	β∗-b	NOUN
cana-5238	163	8	-	-	PUNCT
cana-5238	163	9	hg	hg	NOUN
cana-5238	163	10	-	-	PUNCT
cana-5238	163	11	s	s	NOUN
cana-5238	163	12	)	)	PUNCT
cana-5238	163	13	,	,	PUNCT
cana-5238	163	14	if	if	SCONJ
cana-5238	163	15	ψ	ψ	VERB
cana-5238	163	16	=	=	NOUN
cana-5238	163	17	u∩v	u∩v	ADJ
cana-5238	163	18	,	,	PUNCT
cana-5238	163	19	where	where	SCONJ
cana-5238	163	20	u	u	NOUN
cana-5238	163	21	is	be	AUX
cana-5238	163	22	ζ	ζ	NOUN
cana-5238	163	23	-	-	PUNCT
cana-5238	163	24	o	o	NOUN
cana-5238	163	25	and	and	CCONJ
cana-5238	163	26	v	v	NOUN
cana-5238	163	27	is	be	AUX
cana-5238	163	28	β∗-hg	β∗-hg	NOUN
cana-5238	163	29	-	-	PUNCT
cana-5238	163	30	s.	s.	PROPN
cana-5238	163	31	theorem	theorem	VERB
cana-5238	163	32	3.5	3.5	NUM
cana-5238	163	33	.	.	PUNCT
cana-5238	164	1	if	if	SCONJ
cana-5238	164	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	164	3	is	be	AUX
cana-5238	164	4	b	b	PROPN
cana-5238	164	5	-	-	PUNCT
cana-5238	164	6	hg	hg	NOUN
cana-5238	164	7	-	-	NOUN
cana-5238	164	8	o	o	NOUN
cana-5238	164	9	and	and	CCONJ
cana-5238	164	10	π∗-hg	π∗-hg	NOUN
cana-5238	164	11	-	-	PUNCT
cana-5238	164	12	s	s	NOUN
cana-5238	164	13	,	,	PUNCT
cana-5238	164	14	then	then	ADV
cana-5238	164	15	it	it	PRON
cana-5238	164	16	is	be	AUX
cana-5238	164	17	σ	σ	PROPN
cana-5238	164	18	-	-	PUNCT
cana-5238	164	19	hg	hg	NOUN
cana-5238	164	20	-	-	ADJ
cana-5238	164	21	o.	o.	ADJ
cana-5238	164	22	proof	proof	NOUN
cana-5238	164	23	.	.	PUNCT
cana-5238	165	1	let	let	VERB
cana-5238	165	2	ψ	ψ	NOUN
cana-5238	165	3	is	be	AUX
cana-5238	165	4	b	b	NUM
cana-5238	165	5	-	-	PUNCT
cana-5238	165	6	hg	hg	NOUN
cana-5238	165	7	-	-	NOUN
cana-5238	165	8	o	o	NOUN
cana-5238	165	9	and	and	CCONJ
cana-5238	165	10	π∗-hg	π∗-hg	PROPN
cana-5238	165	11	-	-	PUNCT
cana-5238	165	12	s.	s.	PROPN
cana-5238	165	13	then	then	ADV
cana-5238	165	14	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	165	15	)	)	PUNCT
cana-5238	165	16	and	and	CCONJ
cana-5238	165	17	igc∗(ψ)=ig(ψ	igc∗(ψ)=ig(ψ	NOUN
cana-5238	165	18	)	)	PUNCT
cana-5238	165	19	.	.	PUNCT
cana-5238	166	1	now	now	ADV
cana-5238	166	2	ψ	ψ	ADP
cana-5238	166	3	⊆	⊆	NUM
cana-5238	166	4	igc∗(ψ)∪c∗ig(ψ)⊆c∗ig(ψ)∪ig(ψ	igc∗(ψ)∪c∗ig(ψ)⊆c∗ig(ψ)∪ig(ψ	NOUN
cana-5238	166	5	)	)	PUNCT
cana-5238	166	6	=	=	SYM
cana-5238	166	7	c∗ig(ψ	c∗ig(ψ	NOUN
cana-5238	166	8	)	)	PUNCT
cana-5238	166	9	.	.	PUNCT
cana-5238	167	1	hence	hence	ADV
cana-5238	167	2	ψ	ψ	X
cana-5238	167	3	is	be	AUX
cana-5238	167	4	σ	σ	PROPN
cana-5238	167	5	-	-	PUNCT
cana-5238	167	6	hg	hg	NOUN
cana-5238	167	7	-	-	ADJ
cana-5238	167	8	o.	o.	NOUN
cana-5238	167	9	theorem	theorem	VERB
cana-5238	167	10	3.6	3.6	NUM
cana-5238	167	11	.	.	PUNCT
cana-5238	168	1	if	if	SCONJ
cana-5238	168	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	168	3	is	be	AUX
cana-5238	168	4	b	b	PROPN
cana-5238	168	5	-	-	PUNCT
cana-5238	168	6	hg	hg	NOUN
cana-5238	168	7	-	-	NOUN
cana-5238	168	8	o	o	NOUN
cana-5238	168	9	and	and	CCONJ
cana-5238	168	10	σ∗-hg	σ∗-hg	NOUN
cana-5238	168	11	-	-	PUNCT
cana-5238	168	12	s	s	NOUN
cana-5238	168	13	,	,	PUNCT
cana-5238	168	14	then	then	ADV
cana-5238	168	15	it	it	PRON
cana-5238	168	16	is	be	AUX
cana-5238	168	17	π	π	PROPN
cana-5238	168	18	-	-	PUNCT
cana-5238	168	19	hg	hg	ADJ
cana-5238	168	20	-	-	ADJ
cana-5238	168	21	o.	o.	ADJ
cana-5238	168	22	proof	proof	NOUN
cana-5238	168	23	.	.	PUNCT
cana-5238	169	1	let	let	VERB
cana-5238	169	2	ψ	ψ	NOUN
cana-5238	169	3	is	be	AUX
cana-5238	169	4	b	b	NUM
cana-5238	169	5	-	-	PUNCT
cana-5238	169	6	hg	hg	NOUN
cana-5238	169	7	-	-	NOUN
cana-5238	169	8	o	o	NOUN
cana-5238	169	9	and	and	CCONJ
cana-5238	169	10	σ∗-hg	σ∗-hg	PROPN
cana-5238	169	11	-	-	PUNCT
cana-5238	169	12	s.	s.	PROPN
cana-5238	169	13	then	then	ADV
cana-5238	169	14	ψ⊆igc∗(ψ	ψ⊆igc∗(ψ	NUM
cana-5238	169	15	)	)	PUNCT
cana-5238	169	16	∪	∪	ADP
cana-5238	169	17	c∗ig(ψ	c∗ig(ψ	PROPN
cana-5238	169	18	)	)	PUNCT
cana-5238	169	19	and	and	CCONJ
cana-5238	169	20	c∗ig(ψ	c∗ig(ψ	ADJ
cana-5238	169	21	)	)	PUNCT
cana-5238	169	22	=	=	SYM
cana-5238	169	23	ig(ψ	ig(ψ	NOUN
cana-5238	169	24	)	)	PUNCT
cana-5238	169	25	.	.	PUNCT
cana-5238	170	1	now	now	ADV
cana-5238	170	2	ψ⊆igc∗(ψ)∪c∗ig(ψ)⊆igc∗(ψ)∪ig(ψ)=igc∗(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ)⊆igc∗(ψ)∪ig(ψ)=igc∗(ψ	NUM
cana-5238	170	3	)	)	PUNCT
cana-5238	170	4	.	.	PUNCT
cana-5238	171	1	hence	hence	ADV
cana-5238	171	2	ψ	ψ	X
cana-5238	171	3	is	be	AUX
cana-5238	171	4	π	π	PROPN
cana-5238	171	5	-	-	PUNCT
cana-5238	171	6	hg	hg	ADJ
cana-5238	171	7	-	-	ADJ
cana-5238	171	8	o.	o.	ADJ
cana-5238	171	9	proposition	proposition	NOUN
cana-5238	171	10	3.7	3.7	NUM
cana-5238	171	11	.	.	PUNCT
cana-5238	172	1	let	let	AUX
cana-5238	172	2	(	(	PUNCT
cana-5238	172	3	z	z	NOUN
cana-5238	172	4	,	,	PUNCT
cana-5238	172	5	ζ	ζ	NOUN
cana-5238	172	6	,	,	PUNCT
cana-5238	172	7	h	h	NOUN
cana-5238	172	8	)	)	PUNCT
cana-5238	172	9	be	be	VERB
cana-5238	172	10	a	a	DET
cana-5238	172	11	strong	strong	ADJ
cana-5238	172	12	hgts	hgts	NOUN
cana-5238	172	13	and	and	CCONJ
cana-5238	172	14	ψ⊂z	ψ⊂z	NOUN
cana-5238	172	15	.	.	PUNCT
cana-5238	173	1	then	then	ADV
cana-5238	173	2	the	the	DET
cana-5238	173	3	following	follow	VERB
cana-5238	173	4	holds	hold	VERB
cana-5238	173	5	:	:	PUNCT
cana-5238	174	1	1	1	X
cana-5238	174	2	.	.	X
cana-5238	174	3	if	if	SCONJ
cana-5238	174	4	ψ	ψ	NOUN
cana-5238	174	5	is	be	AUX
cana-5238	174	6	α∗-hg	α∗-hg	ADV
cana-5238	174	7	-	-	PUNCT
cana-5238	174	8	s	s	NOUN
cana-5238	174	9	,	,	PUNCT
cana-5238	174	10	then	then	ADV
cana-5238	174	11	ψ	ψ	NOUN
cana-5238	174	12	is	be	AUX
cana-5238	174	13	α∗-b	α∗-b	PROPN
cana-5238	174	14	-	-	PUNCT
cana-5238	174	15	hg	hg	NOUN
cana-5238	174	16	-	-	PUNCT
cana-5238	174	17	s	s	X
cana-5238	174	18	,	,	PUNCT
cana-5238	174	19	2	2	NUM
cana-5238	174	20	.	.	PUNCT
cana-5238	175	1	if	if	SCONJ
cana-5238	175	2	ψ	ψ	NOUN
cana-5238	175	3	is	be	AUX
cana-5238	175	4	σ∗-hg	σ∗-hg	NOUN
cana-5238	175	5	-	-	PUNCT
cana-5238	175	6	s	s	NOUN
cana-5238	175	7	,	,	PUNCT
cana-5238	175	8	then	then	ADV
cana-5238	175	9	ψ	ψ	NOUN
cana-5238	175	10	is	be	AUX
cana-5238	175	11	σ∗-b	σ∗-b	PROPN
cana-5238	175	12	-	-	PUNCT
cana-5238	175	13	hg	hg	NOUN
cana-5238	175	14	-	-	PUNCT
cana-5238	175	15	s.	s.	PROPN
cana-5238	175	16	3	3	NUM
cana-5238	175	17	.	.	PUNCT
cana-5238	176	1	if	if	SCONJ
cana-5238	176	2	ψ	ψ	NOUN
cana-5238	176	3	is	be	AUX
cana-5238	176	4	π∗-hg	π∗-hg	NOUN
cana-5238	176	5	-	-	PUNCT
cana-5238	176	6	s	s	NOUN
cana-5238	176	7	,	,	PUNCT
cana-5238	176	8	then	then	ADV
cana-5238	176	9	ψ	ψ	NOUN
cana-5238	176	10	is	be	AUX
cana-5238	176	11	π∗-b	π∗-b	PROPN
cana-5238	176	12	-	-	PUNCT
cana-5238	176	13	hg	hg	NOUN
cana-5238	176	14	-	-	PUNCT
cana-5238	176	15	s	s	X
cana-5238	176	16	,	,	PUNCT
cana-5238	176	17	4	4	NUM
cana-5238	176	18	.	.	PUNCT
cana-5238	177	1	if	if	SCONJ
cana-5238	177	2	ψ	ψ	NOUN
cana-5238	177	3	is	be	AUX
cana-5238	177	4	b∗-hg	b∗-hg	NOUN
cana-5238	177	5	-	-	PUNCT
cana-5238	177	6	s	s	NOUN
cana-5238	177	7	,	,	PUNCT
cana-5238	177	8	then	then	ADV
cana-5238	177	9	ψ	ψ	NOUN
cana-5238	177	10	is	be	AUX
cana-5238	177	11	b∗-b	b∗-b	NOUN
cana-5238	177	12	-	-	PUNCT
cana-5238	177	13	hg	hg	NOUN
cana-5238	177	14	-	-	PUNCT
cana-5238	177	15	s	s	X
cana-5238	177	16	,	,	PUNCT
cana-5238	177	17	5	5	NUM
cana-5238	177	18	.	.	PUNCT
cana-5238	178	1	if	if	SCONJ
cana-5238	178	2	ψ	ψ	NOUN
cana-5238	178	3	is	be	AUX
cana-5238	178	4	β∗-hg	β∗-hg	NOUN
cana-5238	178	5	-	-	PUNCT
cana-5238	178	6	s	s	NOUN
cana-5238	178	7	,	,	PUNCT
cana-5238	178	8	then	then	ADV
cana-5238	178	9	ψ	ψ	NOUN
cana-5238	178	10	is	be	AUX
cana-5238	178	11	β∗-b	β∗-b	PROPN
cana-5238	178	12	-	-	PUNCT
cana-5238	178	13	hg	hg	NOUN
cana-5238	178	14	-	-	PUNCT
cana-5238	178	15	s.	s.	PROPN
cana-5238	178	16	proof	proof	NOUN
cana-5238	178	17	.	.	PUNCT
cana-5238	179	1	let	let	VERB
cana-5238	179	2	ψ	ψ	PART
cana-5238	179	3	be	be	AUX
cana-5238	179	4	a	a	DET
cana-5238	179	5	π∗-hg	π∗-hg	NOUN
cana-5238	179	6	-	-	PUNCT
cana-5238	179	7	s.	s.	PROPN
cana-5238	179	8	if	if	SCONJ
cana-5238	179	9	we	we	PRON
cana-5238	179	10	take	take	VERB
cana-5238	179	11	m	m	ADV
cana-5238	179	12	=	=	NOUN
cana-5238	179	13	z∈ζ	z∈ζ	NUM
cana-5238	179	14	,	,	PUNCT
cana-5238	179	15	then	then	ADV
cana-5238	179	16	ψ	ψ	VERB
cana-5238	179	17	=	=	VERB
cana-5238	179	18	m∩ψ	m∩ψ	NOUN
cana-5238	179	19	and	and	CCONJ
cana-5238	179	20	hence	hence	ADV
cana-5238	179	21	ψ	ψ	NOUN
cana-5238	179	22	is	be	AUX
cana-5238	179	23	a	a	DET
cana-5238	179	24	π∗-b	π∗-b	PROPN
cana-5238	179	25	-	-	PUNCT
cana-5238	179	26	hg	hg	NOUN
cana-5238	179	27	-	-	PUNCT
cana-5238	179	28	s.	s.	ADJ
cana-5238	179	29	proof	proof	NOUN
cana-5238	179	30	of	of	ADP
cana-5238	179	31	(	(	PUNCT
cana-5238	179	32	2	2	NUM
cana-5238	179	33	)	)	PUNCT
cana-5238	179	34	,	,	PUNCT
cana-5238	179	35	(	(	PUNCT
cana-5238	179	36	3	3	NUM
cana-5238	179	37	)	)	PUNCT
cana-5238	179	38	,	,	PUNCT
cana-5238	179	39	(	(	PUNCT
cana-5238	179	40	4	4	NUM
cana-5238	179	41	)	)	PUNCT
cana-5238	179	42	,	,	PUNCT
cana-5238	179	43	(	(	PUNCT
cana-5238	179	44	5	5	X
cana-5238	179	45	)	)	PUNCT
cana-5238	179	46	are	be	AUX
cana-5238	179	47	similar	similar	ADJ
cana-5238	179	48	of	of	ADP
cana-5238	179	49	proof	proof	NOUN
cana-5238	179	50	of	of	ADP
cana-5238	179	51	(	(	PUNCT
cana-5238	179	52	1	1	NUM
cana-5238	179	53	)	)	PUNCT
cana-5238	179	54	.	.	PUNCT
cana-5238	180	1	proposition	proposition	NOUN
cana-5238	180	2	3.8	3.8	NUM
cana-5238	180	3	.	.	PUNCT
cana-5238	181	1	for	for	ADP
cana-5238	181	2	a	a	DET
cana-5238	181	3	subset	subset	NOUN
cana-5238	181	4	ψ	ψ	ADP
cana-5238	181	5	a	a	DET
cana-5238	181	6	hgts	hgts	NOUN
cana-5238	181	7	(	(	PUNCT
cana-5238	181	8	z	z	NOUN
cana-5238	181	9	,	,	PUNCT
cana-5238	181	10	ζ	ζ	NOUN
cana-5238	181	11	,	,	PUNCT
cana-5238	181	12	h	h	NOUN
cana-5238	181	13	)	)	PUNCT
cana-5238	181	14	,	,	PUNCT
cana-5238	181	15	the	the	DET
cana-5238	181	16	following	follow	VERB
cana-5238	181	17	properties	property	NOUN
cana-5238	181	18	are	be	AUX
cana-5238	181	19	hold	hold	ADJ
cana-5238	181	20	:	:	PUNCT
cana-5238	182	1	1	1	X
cana-5238	182	2	.	.	X
cana-5238	182	3	if	if	SCONJ
cana-5238	182	4	ψ	ψ	NOUN
cana-5238	182	5	is	be	AUX
cana-5238	182	6	an	an	DET
cana-5238	182	7	σ∗-hg	σ∗-hg	NOUN
cana-5238	182	8	-	-	PUNCT
cana-5238	182	9	s	s	NOUN
cana-5238	182	10	and	and	CCONJ
cana-5238	182	11	gζ	gζ	PROPN
cana-5238	182	12	-	-	PUNCT
cana-5238	182	13	o	o	NOUN
cana-5238	182	14	,	,	PUNCT
cana-5238	182	15	then	then	ADV
cana-5238	182	16	ψ	ψ	NOUN
cana-5238	182	17	is	be	AUX
cana-5238	182	18	α∗-hg	α∗-hg	ADV
cana-5238	182	19	-	-	PUNCT
cana-5238	182	20	s.	s.	PROPN
cana-5238	182	21	2	2	NUM
cana-5238	182	22	.	.	PUNCT
cana-5238	183	1	if	if	SCONJ
cana-5238	183	2	ψ	ψ	NOUN
cana-5238	183	3	is	be	AUX
cana-5238	183	4	an	an	DET
cana-5238	183	5	π∗-hg	π∗-hg	NOUN
cana-5238	183	6	-	-	PUNCT
cana-5238	183	7	s	s	NOUN
cana-5238	183	8	and	and	CCONJ
cana-5238	183	9	gζ	gζ	PROPN
cana-5238	183	10	-	-	PUNCT
cana-5238	183	11	o	o	NOUN
cana-5238	183	12	,	,	PUNCT
cana-5238	183	13	then	then	ADV
cana-5238	183	14	ψ	ψ	NOUN
cana-5238	183	15	is	be	AUX
cana-5238	183	16	α∗-hg	α∗-hg	ADV
cana-5238	183	17	-	-	PUNCT
cana-5238	183	18	s.	s.	PROPN
cana-5238	183	19	3	3	NUM
cana-5238	183	20	.	.	PUNCT
cana-5238	184	1	if	if	SCONJ
cana-5238	184	2	ψ	ψ	NOUN
cana-5238	184	3	is	be	AUX
cana-5238	184	4	an	an	DET
cana-5238	184	5	b∗-hg	b∗-hg	NOUN
cana-5238	184	6	-	-	PUNCT
cana-5238	184	7	s	s	NOUN
cana-5238	184	8	,	,	PUNCT
cana-5238	184	9	then	then	ADV
cana-5238	184	10	ψ	ψ	NOUN
cana-5238	184	11	is	be	AUX
cana-5238	184	12	π∗-hg	π∗-hg	NOUN
cana-5238	184	13	-	-	PUNCT
cana-5238	184	14	s.	s.	PROPN
cana-5238	184	15	4	4	NUM
cana-5238	184	16	.	.	PUNCT
cana-5238	185	1	if	if	SCONJ
cana-5238	185	2	ψ	ψ	NOUN
cana-5238	185	3	is	be	AUX
cana-5238	185	4	an	an	DET
cana-5238	185	5	b∗-hg	b∗-hg	NOUN
cana-5238	185	6	-	-	PUNCT
cana-5238	185	7	s	s	NOUN
cana-5238	185	8	,	,	PUNCT
cana-5238	185	9	then	then	ADV
cana-5238	185	10	ψ	ψ	NOUN
cana-5238	185	11	is	be	AUX
cana-5238	185	12	σ∗-hg	σ∗-hg	NOUN
cana-5238	185	13	-	-	PUNCT
cana-5238	185	14	s.	s.	PROPN
cana-5238	185	15	proof	proof	NOUN
cana-5238	185	16	.	.	PUNCT
cana-5238	186	1	(	(	PUNCT
cana-5238	186	2	1	1	NUM
cana-5238	186	3	)	)	PUNCT
cana-5238	186	4	.	.	PUNCT
cana-5238	187	1	let	let	VERB
cana-5238	187	2	ψ	ψ	NOUN
cana-5238	187	3	is	be	AUX
cana-5238	187	4	σ∗-hg	σ∗-hg	NOUN
cana-5238	187	5	-	-	PUNCT
cana-5238	187	6	s	s	NOUN
cana-5238	187	7	and	and	CCONJ
cana-5238	187	8	gζ	gζ	PROPN
cana-5238	187	9	-	-	PUNCT
cana-5238	187	10	o.	o.	PROPN
cana-5238	187	11	then	then	ADV
cana-5238	187	12	igc∗ig(ψ)⊂c∗ig(ψ	igc∗ig(ψ)⊂c∗ig(ψ	PROPN
cana-5238	187	13	)	)	PUNCT
cana-5238	187	14	=	=	SYM
cana-5238	187	15	i(a	i(a	PROPN
cana-5238	187	16	)	)	PUNCT
cana-5238	187	17	.	.	PUNCT
cana-5238	188	1	communications	communication	NOUN
cana-5238	188	2	on	on	ADP
cana-5238	188	3	applied	apply	VERB
cana-5238	188	4	nonlinear	nonlinear	ADJ
cana-5238	188	5	analysis	analysis	NOUN
cana-5238	188	6	issn	issn	NOUN
cana-5238	188	7	:	:	PUNCT
cana-5238	188	8	1074	1074	NUM
cana-5238	188	9	-	-	PUNCT
cana-5238	188	10	133x	133x	NUM
cana-5238	188	11	vol	vol	VERB
cana-5238	188	12	32	32	NUM
cana-5238	188	13	no	no	NOUN
cana-5238	188	14	.	.	PUNCT
cana-5238	189	1	10s	10	NOUN
cana-5238	189	2	(	(	PUNCT
cana-5238	189	3	2025	2025	NUM
cana-5238	189	4	)	)	PUNCT
cana-5238	189	5	1362	1362	NUM
cana-5238	189	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	189	7	ζ	ζ	NOUN
cana-5238	189	8	ζ	ζ	NOUN
cana-5238	189	9	ζ	ζ	NOUN
cana-5238	189	10	therefore	therefore	ADV
cana-5238	189	11	igc∗ig(ψ)=i(a	igc∗ig(ψ)=i(a	PROPN
cana-5238	189	12	)	)	PUNCT
cana-5238	189	13	.	.	PUNCT
cana-5238	190	1	hence	hence	ADV
cana-5238	190	2	ψ	ψ	NOUN
cana-5238	190	3	is	be	AUX
cana-5238	190	4	α∗-hg	α∗-hg	ADV
cana-5238	190	5	-	-	PUNCT
cana-5238	190	6	s.	s.	PROPN
cana-5238	190	7	(	(	PUNCT
cana-5238	190	8	2	2	NUM
cana-5238	190	9	)	)	PUNCT
cana-5238	190	10	.	.	PUNCT
cana-5238	191	1	let	let	VERB
cana-5238	191	2	ψ	ψ	NOUN
cana-5238	191	3	is	be	AUX
cana-5238	191	4	π∗-hg	π∗-hg	NOUN
cana-5238	191	5	-	-	PUNCT
cana-5238	191	6	s	s	NOUN
cana-5238	191	7	and	and	CCONJ
cana-5238	191	8	gζ	gζ	PROPN
cana-5238	191	9	-	-	PUNCT
cana-5238	191	10	o.	o.	PROPN
cana-5238	191	11	then	then	ADV
cana-5238	191	12	igc∗ig(ψ)⊂igc∗(ψ)=i	igc∗ig(ψ)⊂igc∗(ψ)=i	PROPN
cana-5238	191	13	(	(	PUNCT
cana-5238	191	14	a	a	NOUN
cana-5238	191	15	)	)	PUNCT
cana-5238	191	16	.	.	PUNCT
cana-5238	192	1	therefore	therefore	ADV
cana-5238	192	2	igc∗ig(ψ)=i(a	igc∗ig(ψ)=i(a	PROPN
cana-5238	192	3	)	)	PUNCT
cana-5238	192	4	.	.	PUNCT
cana-5238	193	1	hence	hence	ADV
cana-5238	193	2	ψ	ψ	NOUN
cana-5238	193	3	is	be	AUX
cana-5238	193	4	α∗-hg	α∗-hg	ADV
cana-5238	193	5	-	-	PUNCT
cana-5238	193	6	s.	s.	PROPN
cana-5238	193	7	(	(	PUNCT
cana-5238	193	8	3	3	NUM
cana-5238	193	9	)	)	PUNCT
cana-5238	193	10	.	.	PUNCT
cana-5238	194	1	let	let	VERB
cana-5238	194	2	ψ	ψ	PRON
cana-5238	194	3	b∗-hg	b∗-hg	VERB
cana-5238	194	4	-	-	PUNCT
cana-5238	194	5	s.	s.	PROPN
cana-5238	194	6	then	then	ADV
cana-5238	194	7	igc∗(ψ)⊂igc∗(ψ)∪c∗ig(ψ)=i(ψ	igc∗(ψ)⊂igc∗(ψ)∪c∗ig(ψ)=i(ψ	PROPN
cana-5238	194	8	)	)	PUNCT
cana-5238	194	9	.	.	PUNCT
cana-5238	195	1	therefore	therefore	ADV
cana-5238	195	2	igc∗(ψ)=	igc∗(ψ)=	ADJ
cana-5238	195	3	i(ψ	i(ψ	PROPN
cana-5238	195	4	)	)	PUNCT
cana-5238	195	5	.	.	PUNCT
cana-5238	196	1	hence	hence	ADV
cana-5238	196	2	ψ	ψ	NOUN
cana-5238	196	3	is	be	AUX
cana-5238	196	4	π∗-hgs	π∗-hgs	ADJ
cana-5238	196	5	.	.	PUNCT
cana-5238	197	1	(	(	PUNCT
cana-5238	197	2	4	4	NUM
cana-5238	197	3	)	)	PUNCT
cana-5238	197	4	.	.	PUNCT
cana-5238	198	1	let	let	VERB
cana-5238	198	2	ψ	ψ	PRON
cana-5238	198	3	b∗-hg	b∗-hg	VERB
cana-5238	198	4	-	-	PUNCT
cana-5238	198	5	s.	s.	PROPN
cana-5238	198	6	then	then	ADV
cana-5238	198	7	c∗ig(ψ)⊂igc∗(ψ)∪c∗ig(ψ)=i(ψ	c∗ig(ψ)⊂igc∗(ψ)∪c∗ig(ψ)=i(ψ	PROPN
cana-5238	198	8	)	)	PUNCT
cana-5238	198	9	.	.	PUNCT
cana-5238	199	1	therefore	therefore	ADV
cana-5238	199	2	c∗ig(ψ)=i(ψ	c∗ig(ψ)=i(ψ	PROPN
cana-5238	199	3	)	)	PUNCT
cana-5238	199	4	.	.	PUNCT
cana-5238	200	1	hence	hence	ADV
cana-5238	200	2	ψ	ψ	X
cana-5238	200	3	is	be	AUX
cana-5238	200	4	σ∗-h	σ∗-h	PROPN
cana-5238	200	5	-	-	PUNCT
cana-5238	200	6	s.	s.	PROPN
cana-5238	200	7	theorem	theorem	VERB
cana-5238	200	8	3.9	3.9	NUM
cana-5238	200	9	.	.	PUNCT
cana-5238	201	1	let	let	AUX
cana-5238	201	2	(	(	PUNCT
cana-5238	201	3	z	z	NOUN
cana-5238	201	4	,	,	PUNCT
cana-5238	201	5	ζ	ζ	NOUN
cana-5238	201	6	,	,	PUNCT
cana-5238	201	7	h	h	NOUN
cana-5238	201	8	)	)	PUNCT
cana-5238	201	9	be	be	VERB
cana-5238	201	10	a	a	DET
cana-5238	201	11	strong	strong	ADJ
cana-5238	201	12	hgt	hgt	NOUN
cana-5238	201	13	s	s	PART
cana-5238	201	14	where	where	SCONJ
cana-5238	201	15	z	z	NOUN
cana-5238	201	16	is	be	AUX
cana-5238	201	17	c0	c0	PROPN
cana-5238	201	18	-space	-space	PROPN
cana-5238	201	19	and	and	CCONJ
cana-5238	201	20	ψ	ψ	X
cana-5238	201	21	⊂	⊂	PROPN
cana-5238	201	22	z.	z.	PROPN
cana-5238	202	1	then	then	ADV
cana-5238	202	2	the	the	DET
cana-5238	202	3	following	follow	VERB
cana-5238	202	4	conditions	condition	NOUN
cana-5238	202	5	are	be	AUX
cana-5238	202	6	equivalent	equivalent	ADJ
cana-5238	202	7	.	.	PUNCT
cana-5238	203	1	1	1	X
cana-5238	203	2	.	.	X
cana-5238	203	3	ψ	ψ	NOUN
cana-5238	203	4	is	be	AUX
cana-5238	203	5	ζ	ζ	NOUN
cana-5238	203	6	-	-	PUNCT
cana-5238	203	7	o	o	NOUN
cana-5238	203	8	,	,	PUNCT
cana-5238	203	9	2	2	NUM
cana-5238	203	10	.	.	PUNCT
cana-5238	203	11	ψ	ψ	NOUN
cana-5238	203	12	is	be	AUX
cana-5238	203	13	α	α	X
cana-5238	203	14	-	-	PUNCT
cana-5238	203	15	hg	hg	NOUN
cana-5238	203	16	-	-	NOUN
cana-5238	203	17	o	o	NOUN
cana-5238	203	18	and	and	CCONJ
cana-5238	203	19	α∗-b	α∗-b	PROPN
cana-5238	203	20	-	-	PUNCT
cana-5238	203	21	hg	hg	NOUN
cana-5238	203	22	-	-	PUNCT
cana-5238	203	23	s	s	NOUN
cana-5238	203	24	,	,	PUNCT
cana-5238	203	25	3	3	NUM
cana-5238	203	26	.	.	PUNCT
cana-5238	203	27	ψ	ψ	NOUN
cana-5238	203	28	is	be	AUX
cana-5238	203	29	σ	σ	PROPN
cana-5238	203	30	-	-	PUNCT
cana-5238	203	31	hg	hg	NOUN
cana-5238	203	32	-	-	NOUN
cana-5238	203	33	o	o	NOUN
cana-5238	203	34	and	and	CCONJ
cana-5238	203	35	σ∗-b	σ∗-b	PROPN
cana-5238	203	36	-	-	PUNCT
cana-5238	203	37	hg	hg	NOUN
cana-5238	203	38	-	-	PUNCT
cana-5238	203	39	s.	s.	PROPN
cana-5238	203	40	4	4	NUM
cana-5238	203	41	.	.	PUNCT
cana-5238	204	1	ψ	ψ	NOUN
cana-5238	204	2	is	be	AUX
cana-5238	204	3	π	π	PROPN
cana-5238	204	4	-	-	PUNCT
cana-5238	204	5	hg	hg	NOUN
cana-5238	204	6	-	-	NOUN
cana-5238	204	7	o	o	NOUN
cana-5238	204	8	and	and	CCONJ
cana-5238	204	9	π∗-b	π∗-b	PROPN
cana-5238	204	10	-	-	PUNCT
cana-5238	204	11	hg	hg	NOUN
cana-5238	204	12	-	-	PUNCT
cana-5238	204	13	s	s	X
cana-5238	204	14	,	,	PUNCT
cana-5238	204	15	5	5	NUM
cana-5238	204	16	.	.	PUNCT
cana-5238	204	17	ψ	ψ	NOUN
cana-5238	204	18	is	be	AUX
cana-5238	204	19	β	β	X
cana-5238	204	20	-	-	ADJ
cana-5238	204	21	hg	hg	NOUN
cana-5238	204	22	-	-	NOUN
cana-5238	204	23	o	o	NOUN
cana-5238	204	24	and	and	CCONJ
cana-5238	204	25	β∗-b	β∗-b	PROPN
cana-5238	204	26	-	-	PUNCT
cana-5238	204	27	hg	hg	NOUN
cana-5238	204	28	-	-	PUNCT
cana-5238	204	29	s.	s.	PROPN
cana-5238	204	30	proof	proof	NOUN
cana-5238	204	31	.	.	PUNCT
cana-5238	205	1	(	(	PUNCT
cana-5238	205	2	1	1	X
cana-5238	205	3	)	)	PUNCT
cana-5238	205	4	⇒	⇒	NOUN
cana-5238	205	5	(	(	PUNCT
cana-5238	205	6	2	2	NUM
cana-5238	205	7	)	)	PUNCT
cana-5238	205	8	,	,	PUNCT
cana-5238	205	9	(	(	PUNCT
cana-5238	205	10	1	1	X
cana-5238	205	11	)	)	PUNCT
cana-5238	205	12	⇒	⇒	NOUN
cana-5238	205	13	(	(	PUNCT
cana-5238	205	14	3	3	NUM
cana-5238	205	15	)	)	PUNCT
cana-5238	205	16	,	,	PUNCT
cana-5238	205	17	(	(	PUNCT
cana-5238	205	18	1	1	X
cana-5238	205	19	)	)	PUNCT
cana-5238	205	20	⇒	⇒	NOUN
cana-5238	205	21	(	(	PUNCT
cana-5238	205	22	4	4	NUM
cana-5238	205	23	)	)	PUNCT
cana-5238	205	24	,	,	PUNCT
cana-5238	205	25	are	be	AUX
cana-5238	205	26	obvious	obvious	ADJ
cana-5238	205	27	.	.	PUNCT
cana-5238	206	1	(	(	PUNCT
cana-5238	206	2	2)⇒(1	2)⇒(1	NOUN
cana-5238	206	3	)	)	PUNCT
cana-5238	206	4	.	.	PUNCT
cana-5238	207	1	let	let	VERB
cana-5238	207	2	ψ	ψ	NOUN
cana-5238	207	3	is	be	AUX
cana-5238	207	4	both	both	DET
cana-5238	207	5	α	α	PROPN
cana-5238	207	6	-	-	PUNCT
cana-5238	207	7	hg	hg	NOUN
cana-5238	207	8	-	-	NOUN
cana-5238	207	9	o	o	NOUN
cana-5238	207	10	and	and	CCONJ
cana-5238	207	11	α∗-b	α∗-b	PROPN
cana-5238	207	12	-	-	PUNCT
cana-5238	207	13	hg	hg	NOUN
cana-5238	207	14	-	-	PUNCT
cana-5238	207	15	s.	s.	PROPN
cana-5238	207	16	then	then	ADV
cana-5238	207	17	ψ⊆igc∗ig(ψ)=igc∗ig(m	ψ⊆igc∗ig(ψ)=igc∗ig(m	VERB
cana-5238	207	18	∩n	∩n	NOUN
cana-5238	207	19	)	)	PUNCT
cana-5238	207	20	,	,	PUNCT
cana-5238	207	21	where	where	SCONJ
cana-5238	207	22	m∈ζ	m∈ζ	NOUN
cana-5238	207	23	and	and	CCONJ
cana-5238	207	24	n	n	PRON
cana-5238	207	25	is	be	AUX
cana-5238	207	26	α∗-hg	α∗-hg	ADV
cana-5238	207	27	-	-	PUNCT
cana-5238	207	28	s.	s.	PROPN
cana-5238	207	29	hence	hence	ADV
cana-5238	207	30	ψ⊆igc∗ig(m)∩igc∗ig(n	ψ⊆igc∗ig(m)∩igc∗ig(n	ADV
cana-5238	207	31	)	)	PUNCT
cana-5238	207	32	.	.	PUNCT
cana-5238	208	1	now	now	ADV
cana-5238	208	2	ψ⊆m∩ψ⊆m∩[igc∗ig(m)∩i(n	ψ⊆m∩ψ⊆m∩[igc∗ig(m)∩i(n	X
cana-5238	208	3	)	)	PUNCT
cana-5238	208	4	]=	]=	NOUN
cana-5238	208	5	m∩i(n)=i(ψ	m∩i(n)=i(ψ	NOUN
cana-5238	208	6	)	)	PUNCT
cana-5238	208	7	.	.	PUNCT
cana-5238	209	1	hence	hence	ADV
cana-5238	209	2	ψ	ψ	NOUN
cana-5238	209	3	is	be	AUX
cana-5238	209	4	ζ	ζ	NOUN
cana-5238	209	5	-	-	NOUN
cana-5238	209	6	o.	o.	NOUN
cana-5238	209	7	(	(	PUNCT
cana-5238	209	8	3)⇒(1	3)⇒(1	PROPN
cana-5238	209	9	)	)	PUNCT
cana-5238	209	10	.	.	PUNCT
cana-5238	210	1	let	let	VERB
cana-5238	210	2	ψ	ψ	NOUN
cana-5238	210	3	is	be	AUX
cana-5238	210	4	both	both	DET
cana-5238	210	5	σ	σ	PROPN
cana-5238	210	6	-	-	PUNCT
cana-5238	210	7	hg	hg	NOUN
cana-5238	210	8	-	-	NOUN
cana-5238	210	9	o	o	NOUN
cana-5238	210	10	and	and	CCONJ
cana-5238	210	11	σ∗-b	σ∗-b	PROPN
cana-5238	210	12	-	-	PUNCT
cana-5238	210	13	hg	hg	NOUN
cana-5238	210	14	-	-	PUNCT
cana-5238	210	15	s.	s.	PROPN
cana-5238	210	16	then	then	ADV
cana-5238	210	17	ψ⊆c∗ig(ψ	ψ⊆c∗ig(ψ	PROPN
cana-5238	210	18	)	)	PUNCT
cana-5238	210	19	=	=	SYM
cana-5238	210	20	c∗ig(m∩n	c∗ig(m∩n	NOUN
cana-5238	210	21	)	)	PUNCT
cana-5238	210	22	,	,	PUNCT
cana-5238	210	23	where	where	SCONJ
cana-5238	210	24	m∈ζ	m∈ζ	NOUN
cana-5238	210	25	and	and	CCONJ
cana-5238	210	26	n	n	PRON
cana-5238	210	27	is	be	AUX
cana-5238	210	28	σ∗-hg	σ∗-hg	NOUN
cana-5238	210	29	-	-	PUNCT
cana-5238	210	30	s.	s.	PROPN
cana-5238	210	31	hence	hence	ADV
cana-5238	210	32	ψ⊆c∗ig(m)∩c∗ig(n	ψ⊆c∗ig(m)∩c∗ig(n	ADV
cana-5238	210	33	)	)	PUNCT
cana-5238	210	34	.	.	PUNCT
cana-5238	211	1	now	now	ADV
cana-5238	211	2	ψ⊆m∩ψ⊆m∩[c∗ig(m	ψ⊆m∩ψ⊆m∩[c∗ig(m	VERB
cana-5238	211	3	)	)	PUNCT
cana-5238	211	4	∩i(n)]=m∩i(n	∩i(n)]=m∩i(n	X
cana-5238	211	5	)	)	PUNCT
cana-5238	212	1	=	=	SYM
cana-5238	212	2	i(ψ	i(ψ	PROPN
cana-5238	212	3	)	)	PUNCT
cana-5238	212	4	.	.	PUNCT
cana-5238	213	1	hence	hence	ADV
cana-5238	213	2	ψ	ψ	NOUN
cana-5238	213	3	is	be	AUX
cana-5238	213	4	ζ	ζ	NOUN
cana-5238	213	5	-	-	NOUN
cana-5238	213	6	o.	o.	NOUN
cana-5238	213	7	(	(	PUNCT
cana-5238	213	8	4)⇒	4)⇒	X
cana-5238	213	9	(	(	PUNCT
cana-5238	213	10	1	1	NUM
cana-5238	213	11	)	)	PUNCT
cana-5238	213	12	.	.	PUNCT
cana-5238	214	1	let	let	VERB
cana-5238	214	2	ψ	ψ	NOUN
cana-5238	214	3	is	be	AUX
cana-5238	214	4	both	both	PRON
cana-5238	214	5	π	π	PROPN
cana-5238	214	6	-	-	PUNCT
cana-5238	214	7	hg	hg	NOUN
cana-5238	214	8	-	-	NOUN
cana-5238	214	9	o	o	NOUN
cana-5238	214	10	and	and	CCONJ
cana-5238	214	11	π∗-b	π∗-b	PROPN
cana-5238	214	12	-	-	PUNCT
cana-5238	214	13	hg	hg	NOUN
cana-5238	214	14	-	-	PUNCT
cana-5238	214	15	s.	s.	PROPN
cana-5238	214	16	then	then	ADV
cana-5238	214	17	ψ⊆igc∗(ψ)=igc∗(m∩n	ψ⊆igc∗(ψ)=igc∗(m∩n	PROPN
cana-5238	214	18	)	)	PUNCT
cana-5238	214	19	,	,	PUNCT
cana-5238	214	20	where	where	SCONJ
cana-5238	214	21	m∈ζ	m∈ζ	NOUN
cana-5238	214	22	and	and	CCONJ
cana-5238	214	23	n	n	PRON
cana-5238	214	24	is	be	AUX
cana-5238	214	25	π∗-hg	π∗-hg	NOUN
cana-5238	214	26	-	-	PUNCT
cana-5238	214	27	s.	s.	PROPN
cana-5238	214	28	hence	hence	ADV
cana-5238	214	29	ψ⊆igc∗(m)∩igc∗(n	ψ⊆igc∗(m)∩igc∗(n	ADJ
cana-5238	214	30	)	)	PUNCT
cana-5238	214	31	.	.	PUNCT
cana-5238	215	1	now	now	ADV
cana-5238	215	2	ψ⊆m∩ψ⊆m∩[igc∗(m)∩i	ψ⊆m∩ψ⊆m∩[igc∗(m)∩i	PUNCT
cana-5238	215	3	(	(	PUNCT
cana-5238	215	4	n	n	X
cana-5238	215	5	)	)	PUNCT
cana-5238	215	6	]=	]=	NOUN
cana-5238	215	7	m∩i	m∩i	NUM
cana-5238	215	8	(	(	PUNCT
cana-5238	215	9	n	n	NOUN
cana-5238	215	10	)	)	PUNCT
cana-5238	215	11	=	=	SYM
cana-5238	215	12	i(ψ	i(ψ	PROPN
cana-5238	215	13	)	)	PUNCT
cana-5238	215	14	.	.	PUNCT
cana-5238	216	1	hence	hence	ADV
cana-5238	216	2	ψ	ψ	NOUN
cana-5238	216	3	is	be	AUX
cana-5238	216	4	ζ	ζ	NOUN
cana-5238	216	5	-	-	NOUN
cana-5238	216	6	o.	o.	NOUN
cana-5238	216	7	(	(	PUNCT
cana-5238	216	8	5)⇒	5)⇒	NUM
cana-5238	216	9	(	(	PUNCT
cana-5238	216	10	1	1	NUM
cana-5238	216	11	)	)	PUNCT
cana-5238	216	12	.	.	PUNCT
cana-5238	217	1	let	let	VERB
cana-5238	217	2	ψ	ψ	NOUN
cana-5238	217	3	is	be	AUX
cana-5238	217	4	both	both	PRON
cana-5238	217	5	β	β	NOUN
cana-5238	217	6	-	-	PUNCT
cana-5238	217	7	hg	hg	NOUN
cana-5238	217	8	-	-	NOUN
cana-5238	217	9	o	o	NOUN
cana-5238	217	10	and	and	CCONJ
cana-5238	217	11	β∗-b	β∗-b	PROPN
cana-5238	217	12	-	-	PUNCT
cana-5238	217	13	hg	hg	NOUN
cana-5238	217	14	-	-	PUNCT
cana-5238	217	15	s.	s.	PROPN
cana-5238	217	16	then	then	ADV
cana-5238	217	17	ψ⊆cigc∗(ψ)=cigc∗(m∩n	ψ⊆cigc∗(ψ)=cigc∗(m∩n	NUM
cana-5238	217	18	)	)	PUNCT
cana-5238	217	19	,	,	PUNCT
cana-5238	217	20	where	where	SCONJ
cana-5238	217	21	m∈ζ	m∈ζ	NOUN
cana-5238	217	22	and	and	CCONJ
cana-5238	217	23	n	n	PRON
cana-5238	217	24	is	be	AUX
cana-5238	217	25	β∗-hg	β∗-hg	NOUN
cana-5238	217	26	-	-	PUNCT
cana-5238	217	27	s.	s.	PROPN
cana-5238	217	28	hence	hence	ADV
cana-5238	217	29	ψ⊆cigc∗(m)∩cigc∗(n	ψ⊆cigc∗(m)∩cigc∗(n	NUM
cana-5238	217	30	)	)	PUNCT
cana-5238	217	31	.	.	PUNCT
cana-5238	218	1	now	now	ADV
cana-5238	218	2	ψ⊆m∩ψ⊆m∩[cigc∗(m)∩i	ψ⊆m∩ψ⊆m∩[cigc∗(m)∩i	X
cana-5238	218	3	(	(	PUNCT
cana-5238	218	4	n)]=m∩i(n	n)]=m∩i(n	NOUN
cana-5238	218	5	)	)	PUNCT
cana-5238	218	6	=	=	SYM
cana-5238	218	7	i(ψ	i(ψ	PROPN
cana-5238	218	8	)	)	PUNCT
cana-5238	218	9	.	.	PUNCT
cana-5238	219	1	hence	hence	ADV
cana-5238	219	2	ψ	ψ	NOUN
cana-5238	219	3	is	be	AUX
cana-5238	219	4	ζ	ζ	NOUN
cana-5238	219	5	-	-	PUNCT
cana-5238	219	6	o.	o.	ADJ
cana-5238	219	7	remark	remark	NOUN
cana-5238	219	8	3.10	3.10	NUM
cana-5238	219	9	.	.	PUNCT
cana-5238	220	1	the	the	DET
cana-5238	220	2	notions	notion	NOUN
cana-5238	220	3	of	of	ADP
cana-5238	220	4	α	α	PROPN
cana-5238	220	5	-	-	PUNCT
cana-5238	220	6	hg	hg	NOUN
cana-5238	220	7	-	-	NOUN
cana-5238	220	8	o	o	X
cana-5238	220	9	(	(	PUNCT
cana-5238	220	10	resp	resp	NOUN
cana-5238	220	11	.	.	PUNCT
cana-5238	221	1	σ	σ	PROPN
cana-5238	221	2	-	-	PUNCT
cana-5238	221	3	hg	hg	NOUN
cana-5238	221	4	-	-	PROPN
cana-5238	221	5	o	o	NOUN
cana-5238	221	6	,	,	PUNCT
cana-5238	221	7	π	π	PROPN
cana-5238	221	8	-	-	PUNCT
cana-5238	221	9	hg	hg	NOUN
cana-5238	221	10	-	-	NOUN
cana-5238	221	11	o	o	NOUN
cana-5238	221	12	,	,	PUNCT
cana-5238	221	13	β	β	X
cana-5238	221	14	-	-	ADJ
cana-5238	221	15	hg	hg	NOUN
cana-5238	221	16	-	-	NOUN
cana-5238	221	17	o	o	NOUN
cana-5238	221	18	)	)	PUNCT
cana-5238	221	19	and	and	CCONJ
cana-5238	221	20	α∗-b	α∗-b	PROPN
cana-5238	221	21	-	-	PUNCT
cana-5238	221	22	hg	hg	NOUN
cana-5238	221	23	-	-	PUNCT
cana-5238	221	24	s	s	X
cana-5238	221	25	(	(	PUNCT
cana-5238	221	26	resp	resp	NOUN
cana-5238	221	27	.	.	PUNCT
cana-5238	222	1	σ∗-b	σ∗-b	PROPN
cana-5238	223	1	hg	hg	PROPN
cana-5238	223	2	-	-	PUNCT
cana-5238	223	3	s	s	PROPN
cana-5238	223	4	,	,	PUNCT
cana-5238	223	5	π∗-b	π∗-b	PROPN
cana-5238	223	6	-	-	PUNCT
cana-5238	223	7	hg	hg	NOUN
cana-5238	223	8	-	-	PUNCT
cana-5238	223	9	s	s	NOUN
cana-5238	223	10	,	,	PUNCT
cana-5238	223	11	β∗-b	β∗-b	PUNCT
cana-5238	223	12	-	-	PUNCT
cana-5238	223	13	hg	hg	NOUN
cana-5238	223	14	-s	-s	NOUN
cana-5238	223	15	)	)	PUNCT
cana-5238	223	16	are	be	AUX
cana-5238	223	17	independent	independent	ADJ
cana-5238	223	18	.	.	PUNCT
cana-5238	223	19	example	example	NOUN
cana-5238	224	1	3.11	3.11	NUM
cana-5238	224	2	.	.	PUNCT
cana-5238	225	1	assume	assume	VERB
cana-5238	225	2	z	z	NOUN
cana-5238	225	3	=	=	PUNCT
cana-5238	225	4	{	{	PUNCT
cana-5238	225	5	1	1	NUM
cana-5238	225	6	,	,	PUNCT
cana-5238	225	7	2	2	NUM
cana-5238	225	8	,	,	PUNCT
cana-5238	225	9	3	3	NUM
cana-5238	225	10	,	,	PUNCT
cana-5238	225	11	4	4	NUM
cana-5238	225	12	}	}	PUNCT
cana-5238	225	13	,	,	PUNCT
cana-5238	225	14	ζ={∅	ζ={∅	PROPN
cana-5238	225	15	,	,	PUNCT
cana-5238	225	16	{	{	PUNCT
cana-5238	225	17	1	1	NUM
cana-5238	225	18	,	,	PUNCT
cana-5238	225	19	3	3	NUM
cana-5238	225	20	}	}	PUNCT
cana-5238	225	21	,	,	PUNCT
cana-5238	225	22	{	{	PUNCT
cana-5238	225	23	2	2	NUM
cana-5238	225	24	,	,	PUNCT
cana-5238	225	25	3	3	NUM
cana-5238	225	26	}	}	PUNCT
cana-5238	225	27	,	,	PUNCT
cana-5238	225	28	{	{	PUNCT
cana-5238	225	29	1	1	NUM
cana-5238	225	30	,	,	PUNCT
cana-5238	225	31	2	2	NUM
cana-5238	225	32	,	,	PUNCT
cana-5238	225	33	3},{1	3},{1	NUM
cana-5238	225	34	,	,	PUNCT
cana-5238	225	35	4	4	NUM
cana-5238	225	36	}	}	PUNCT
cana-5238	225	37	,	,	PUNCT
cana-5238	225	38	{	{	PUNCT
cana-5238	225	39	1	1	NUM
cana-5238	225	40	,	,	PUNCT
cana-5238	225	41	3	3	NUM
cana-5238	225	42	,	,	PUNCT
cana-5238	225	43	4	4	NUM
cana-5238	225	44	}	}	PUNCT
cana-5238	225	45	,	,	PUNCT
cana-5238	225	46	z	z	NOUN
cana-5238	225	47	}	}	PUNCT
cana-5238	225	48	,	,	PUNCT
cana-5238	225	49	h={∅	h={∅	PROPN
cana-5238	225	50	,	,	PUNCT
cana-5238	225	51	{	{	PUNCT
cana-5238	225	52	1	1	NUM
cana-5238	225	53	}	}	PUNCT
cana-5238	225	54	,	,	PUNCT
cana-5238	225	55	{	{	PUNCT
cana-5238	225	56	2	2	NUM
cana-5238	225	57	}	}	PUNCT
cana-5238	225	58	}	}	PUNCT
cana-5238	225	59	.	.	PUNCT
cana-5238	226	1	then	then	ADV
cana-5238	226	2	ψ={1	ψ={1	PROPN
cana-5238	226	3	}	}	PUNCT
cana-5238	226	4	is	be	AUX
cana-5238	226	5	α	α	PROPN
cana-5238	226	6	-	-	PUNCT
cana-5238	226	7	hg	hg	NOUN
cana-5238	226	8	-	-	NOUN
cana-5238	226	9	o	o	X
cana-5238	226	10	(	(	PUNCT
cana-5238	226	11	resp	resp	NOUN
cana-5238	226	12	.	.	PUNCT
cana-5238	227	1	σ	σ	PROPN
cana-5238	227	2	-	-	PUNCT
cana-5238	227	3	hg	hg	NOUN
cana-5238	227	4	-	-	PROPN
cana-5238	227	5	o	o	NOUN
cana-5238	227	6	,	,	PUNCT
cana-5238	227	7	π	π	PROPN
cana-5238	227	8	-	-	PUNCT
cana-5238	227	9	hg	hg	NOUN
cana-5238	227	10	-	-	NOUN
cana-5238	227	11	o	o	NOUN
cana-5238	227	12	,	,	PUNCT
cana-5238	227	13	β	β	X
cana-5238	227	14	-	-	ADJ
cana-5238	227	15	hg	hg	NOUN
cana-5238	227	16	-	-	NOUN
cana-5238	227	17	o	o	NOUN
cana-5238	227	18	)	)	PUNCT
cana-5238	227	19	but	but	CCONJ
cana-5238	227	20	not	not	PART
cana-5238	227	21	α∗b	α∗b	NUM
cana-5238	227	22	-	-	PUNCT
cana-5238	227	23	hg	hg	NOUN
cana-5238	227	24	-	-	PUNCT
cana-5238	227	25	s	s	X
cana-5238	227	26	(	(	PUNCT
cana-5238	227	27	rep	rep	PROPN
cana-5238	227	28	.	.	PROPN
cana-5238	227	29	σ∗-bhg	σ∗-bhg	PROPN
cana-5238	227	30	-	-	PUNCT
cana-5238	227	31	s	s	PROPN
cana-5238	227	32	,	,	PUNCT
cana-5238	227	33	π∗-b	π∗-b	PROPN
cana-5238	227	34	-	-	PUNCT
cana-5238	227	35	hg	hg	NOUN
cana-5238	227	36	-	-	PUNCT
cana-5238	227	37	s	s	NOUN
cana-5238	227	38	,	,	PUNCT
cana-5238	227	39	β∗-b	β∗-b	PUNCT
cana-5238	227	40	-	-	PUNCT
cana-5238	227	41	hg	hg	NOUN
cana-5238	227	42	-	-	PUNCT
cana-5238	227	43	s	s	NOUN
cana-5238	227	44	)	)	PUNCT
cana-5238	227	45	and	and	CCONJ
cana-5238	227	46	m={2	m={2	PROPN
cana-5238	227	47	}	}	PUNCT
cana-5238	227	48	is	be	AUX
cana-5238	227	49	α∗-b	α∗-b	PROPN
cana-5238	227	50	-	-	PUNCT
cana-5238	227	51	hg	hg	NOUN
cana-5238	227	52	-	-	PUNCT
cana-5238	227	53	s	s	X
cana-5238	227	54	(	(	PUNCT
cana-5238	227	55	rep	rep	PROPN
cana-5238	227	56	.	.	PROPN
cana-5238	227	57	σ∗-b	σ∗-b	PROPN
cana-5238	227	58	-	-	PUNCT
cana-5238	227	59	hg	hg	PROPN
cana-5238	227	60	-s	-s	PROPN
cana-5238	227	61	,	,	PUNCT
cana-5238	227	62	π∗-b	π∗-b	PROPN
cana-5238	227	63	-	-	PUNCT
cana-5238	227	64	hg	hg	NOUN
cana-5238	227	65	-	-	PUNCT
cana-5238	227	66	s	s	NOUN
cana-5238	227	67	,	,	PUNCT
cana-5238	227	68	β∗-b	β∗-b	PUNCT
cana-5238	227	69	-	-	PUNCT
cana-5238	227	70	hg	hg	NOUN
cana-5238	227	71	-	-	PUNCT
cana-5238	227	72	s	s	NOUN
cana-5238	227	73	)	)	PUNCT
cana-5238	227	74	but	but	CCONJ
cana-5238	227	75	not	not	PART
cana-5238	227	76	α	α	PROPN
cana-5238	227	77	-	-	PUNCT
cana-5238	227	78	hg	hg	NOUN
cana-5238	227	79	-	-	NOUN
cana-5238	227	80	o	o	X
cana-5238	227	81	(	(	PUNCT
cana-5238	227	82	resp	resp	NOUN
cana-5238	227	83	.	.	PUNCT
cana-5238	228	1	σ	σ	PROPN
cana-5238	228	2	-	-	PUNCT
cana-5238	228	3	hg	hg	NOUN
cana-5238	228	4	-	-	PROPN
cana-5238	228	5	o	o	NOUN
cana-5238	228	6	,	,	PUNCT
cana-5238	228	7	π	π	PROPN
cana-5238	228	8	-	-	PUNCT
cana-5238	228	9	hg	hg	NOUN
cana-5238	228	10	-	-	NOUN
cana-5238	228	11	o	o	NOUN
cana-5238	228	12	,	,	PUNCT
cana-5238	228	13	β	β	X
cana-5238	228	14	-	-	PUNCT
cana-5238	228	15	hg	hg	NOUN
cana-5238	228	16	-	-	NOUN
cana-5238	228	17	o	o	NOUN
cana-5238	228	18	)	)	PUNCT
cana-5238	228	19	.	.	PUNCT
cana-5238	229	1	theorem	theorem	VERB
cana-5238	229	2	3.12	3.12	NUM
cana-5238	229	3	.	.	PUNCT
cana-5238	230	1	let	let	AUX
cana-5238	230	2	(	(	PUNCT
cana-5238	230	3	z	z	NOUN
cana-5238	230	4	,	,	PUNCT
cana-5238	230	5	ζ	ζ	NOUN
cana-5238	230	6	,	,	PUNCT
cana-5238	230	7	h	h	NOUN
cana-5238	230	8	)	)	PUNCT
cana-5238	230	9	be	be	VERB
cana-5238	230	10	a	a	DET
cana-5238	230	11	strong	strong	ADJ
cana-5238	230	12	hgt	hgt	NOUN
cana-5238	230	13	s	s	PART
cana-5238	230	14	where	where	SCONJ
cana-5238	230	15	z	z	NOUN
cana-5238	230	16	is	be	AUX
cana-5238	230	17	c0	c0	PROPN
cana-5238	230	18	-space	-space	PROPN
cana-5238	230	19	and	and	CCONJ
cana-5238	230	20	ψ⊂z	ψ⊂z	NOUN
cana-5238	230	21	.	.	PUNCT
cana-5238	231	1	then	then	ADV
cana-5238	231	2	the	the	DET
cana-5238	231	3	following	follow	VERB
cana-5238	231	4	conditions	condition	NOUN
cana-5238	231	5	are	be	AUX
cana-5238	231	6	equivalent	equivalent	ADJ
cana-5238	231	7	.	.	PUNCT
cana-5238	232	1	1	1	X
cana-5238	232	2	.	.	X
cana-5238	232	3	ψ	ψ	NOUN
cana-5238	232	4	is	be	AUX
cana-5238	232	5	ζ	ζ	NOUN
cana-5238	232	6	-	-	PUNCT
cana-5238	232	7	o	o	NOUN
cana-5238	232	8	,	,	PUNCT
cana-5238	232	9	2	2	NUM
cana-5238	232	10	.	.	PUNCT
cana-5238	232	11	ψ	ψ	NOUN
cana-5238	232	12	is	be	AUX
cana-5238	232	13	σ	σ	PROPN
cana-5238	232	14	-	-	PUNCT
cana-5238	232	15	hg	hg	NOUN
cana-5238	232	16	-	-	NOUN
cana-5238	232	17	o	o	NOUN
cana-5238	232	18	and	and	CCONJ
cana-5238	232	19	b∗-b	b∗-b	PROPN
cana-5238	232	20	-	-	PUNCT
cana-5238	232	21	hg	hg	NOUN
cana-5238	232	22	-	-	PUNCT
cana-5238	232	23	s	s	NOUN
cana-5238	232	24	communications	communication	NOUN
cana-5238	232	25	on	on	ADP
cana-5238	232	26	applied	apply	VERB
cana-5238	232	27	nonlinear	nonlinear	ADJ
cana-5238	232	28	analysis	analysis	NOUN
cana-5238	232	29	issn	issn	NOUN
cana-5238	232	30	:	:	PUNCT
cana-5238	232	31	1074	1074	NUM
cana-5238	232	32	-	-	PUNCT
cana-5238	232	33	133x	133x	NUM
cana-5238	232	34	vol	vol	VERB
cana-5238	232	35	32	32	NUM
cana-5238	232	36	no	no	NOUN
cana-5238	232	37	.	.	PUNCT
cana-5238	233	1	10s	10	NOUN
cana-5238	233	2	(	(	PUNCT
cana-5238	233	3	2025	2025	NUM
cana-5238	233	4	)	)	PUNCT
cana-5238	233	5	1363	1363	NUM
cana-5238	233	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	233	7	3	3	X
cana-5238	233	8	.	.	PUNCT
cana-5238	233	9	ψ	ψ	NOUN
cana-5238	233	10	is	be	AUX
cana-5238	233	11	π	π	PROPN
cana-5238	233	12	-	-	PUNCT
cana-5238	233	13	hg	hg	NOUN
cana-5238	233	14	-	-	NOUN
cana-5238	233	15	o	o	NOUN
cana-5238	233	16	and	and	CCONJ
cana-5238	233	17	b∗-b	b∗-b	PROPN
cana-5238	233	18	-	-	PUNCT
cana-5238	233	19	hg	hg	NOUN
cana-5238	233	20	-	-	PUNCT
cana-5238	233	21	s	s	PART
cana-5238	233	22	4	4	NUM
cana-5238	233	23	.	.	PUNCT
cana-5238	233	24	ψ	ψ	NOUN
cana-5238	233	25	is	be	AUX
cana-5238	233	26	b	b	AUX
cana-5238	233	27	-	-	PUNCT
cana-5238	233	28	hg	hg	NOUN
cana-5238	233	29	-	-	NOUN
cana-5238	233	30	o	o	NOUN
cana-5238	233	31	and	and	CCONJ
cana-5238	233	32	b∗-b	b∗-b	PROPN
cana-5238	233	33	-	-	PUNCT
cana-5238	233	34	hg	hg	NOUN
cana-5238	233	35	-	-	PUNCT
cana-5238	233	36	s	s	PART
cana-5238	233	37	proof	proof	NOUN
cana-5238	233	38	.	.	PUNCT
cana-5238	234	1	(	(	PUNCT
cana-5238	234	2	1	1	X
cana-5238	234	3	)	)	PUNCT
cana-5238	234	4	⇒	⇒	NOUN
cana-5238	234	5	(	(	PUNCT
cana-5238	234	6	2	2	NUM
cana-5238	234	7	)	)	PUNCT
cana-5238	234	8	⇒	⇒	NOUN
cana-5238	234	9	(	(	PUNCT
cana-5238	234	10	4	4	NUM
cana-5238	234	11	)	)	PUNCT
cana-5238	234	12	and	and	CCONJ
cana-5238	234	13	(	(	PUNCT
cana-5238	234	14	1	1	X
cana-5238	234	15	)	)	PUNCT
cana-5238	234	16	⇒	⇒	NOUN
cana-5238	234	17	(	(	PUNCT
cana-5238	234	18	3	3	NUM
cana-5238	234	19	)	)	PUNCT
cana-5238	234	20	⇒	⇒	NOUN
cana-5238	234	21	(	(	PUNCT
cana-5238	234	22	4	4	X
cana-5238	234	23	)	)	PUNCT
cana-5238	234	24	are	be	AUX
cana-5238	234	25	obvious	obvious	ADJ
cana-5238	234	26	,	,	PUNCT
cana-5238	234	27	since	since	SCONJ
cana-5238	234	28	z	z	NOUN
cana-5238	234	29	is	be	AUX
cana-5238	234	30	b∗-b	b∗-b	PROPN
cana-5238	234	31	-	-	PUNCT
cana-5238	234	32	hg	hg	NOUN
cana-5238	234	33	-	-	PUNCT
cana-5238	234	34	s	s	X
cana-5238	234	35	(	(	PUNCT
cana-5238	234	36	4	4	NUM
cana-5238	234	37	)	)	PUNCT
cana-5238	234	38	⇒	⇒	NOUN
cana-5238	234	39	(	(	PUNCT
cana-5238	234	40	1	1	NUM
cana-5238	234	41	)	)	PUNCT
cana-5238	234	42	.	.	PUNCT
cana-5238	235	1	let	let	VERB
cana-5238	235	2	ψ	ψ	NOUN
cana-5238	235	3	is	be	AUX
cana-5238	235	4	b	b	NUM
cana-5238	235	5	-	-	PUNCT
cana-5238	235	6	hg	hg	NOUN
cana-5238	235	7	-	-	NOUN
cana-5238	235	8	o	o	NOUN
cana-5238	235	9	and	and	CCONJ
cana-5238	235	10	b∗-b	b∗-b	PROPN
cana-5238	235	11	-	-	PUNCT
cana-5238	235	12	hg	hg	NOUN
cana-5238	235	13	-	-	PUNCT
cana-5238	235	14	s.	s.	PROPN
cana-5238	235	15	then	then	ADV
cana-5238	235	16	ψ⊆igc∗(ψ)∪c∗ig(ψ	ψ⊆igc∗(ψ)∪c∗ig(ψ	PROPN
cana-5238	235	17	)	)	PUNCT
cana-5238	235	18	=	=	SYM
cana-5238	235	19	igc∗(m∩n)∪c∗ig(m∩n	igc∗(m∩n)∪c∗ig(m∩n	NOUN
cana-5238	235	20	)	)	PUNCT
cana-5238	235	21	,	,	PUNCT
cana-5238	235	22	where	where	SCONJ
cana-5238	235	23	ψ	ψ	VERB
cana-5238	235	24	=	=	NOUN
cana-5238	235	25	m∩n	m∩n	ADJ
cana-5238	235	26	,	,	PUNCT
cana-5238	235	27	m∈ζ	m∈ζ	NOUN
cana-5238	235	28	and	and	CCONJ
cana-5238	235	29	v	v	NOUN
cana-5238	235	30	is	be	AUX
cana-5238	235	31	b∗-hg	b∗-hg	NOUN
cana-5238	235	32	-	-	PUNCT
cana-5238	235	33	s	s	PART
cana-5238	235	34	hence	hence	ADV
cana-5238	235	35	ψ⊆m∩ψ⊆m∩	ψ⊆m∩ψ⊆m∩	NOUN
cana-5238	236	1	[	[	X
cana-5238	236	2	igc∗(m∩n)∪c∗ig(m∩n)]⊆[m∩igc∗(m)∩igc∗(n)]∪[m∩c∗ig(m)∩c∗ig(n)]⊆[m∩igc∗(n)]∪[m∩c∗ig	igc∗(m∩n)∪c∗ig(m∩n)]⊆[m∩igc∗(m)∩igc∗(n)]∪[m∩c∗ig(m)∩c∗ig(n)]⊆[m∩igc∗(n)]∪[m∩c∗ig	ADJ
cana-5238	236	3	(	(	PUNCT
cana-5238	236	4	n)]=m∩[igc∗(n)∪c∗ig(n)]=m∩i(v)=i(ψ	n)]=m∩[igc∗(n)∪c∗ig(n)]=m∩i(v)=i(ψ	NOUN
cana-5238	236	5	)	)	PUNCT
cana-5238	236	6	.	.	PUNCT
cana-5238	237	1	remark	remark	PROPN
cana-5238	237	2	3.13	3.13	NUM
cana-5238	237	3	.	.	PUNCT
cana-5238	238	1	the	the	DET
cana-5238	238	2	notions	notion	NOUN
cana-5238	238	3	of	of	ADP
cana-5238	238	4	σ	σ	PROPN
cana-5238	238	5	-	-	PUNCT
cana-5238	238	6	hg	hg	NOUN
cana-5238	238	7	-	-	NOUN
cana-5238	238	8	o	o	X
cana-5238	238	9	(	(	PUNCT
cana-5238	238	10	resp	resp	NOUN
cana-5238	238	11	.	.	PUNCT
cana-5238	239	1	π	π	PROPN
cana-5238	239	2	-	-	PUNCT
cana-5238	239	3	hg	hg	NOUN
cana-5238	239	4	-	-	NOUN
cana-5238	239	5	o	o	NOUN
cana-5238	239	6	,	,	PUNCT
cana-5238	239	7	b	b	X
cana-5238	239	8	-	-	PUNCT
cana-5238	239	9	hg	hg	NOUN
cana-5238	239	10	-	-	NOUN
cana-5238	239	11	o	o	NOUN
cana-5238	239	12	)	)	PUNCT
cana-5238	239	13	and	and	CCONJ
cana-5238	239	14	b∗-b	b∗-b	PROPN
cana-5238	239	15	-	-	PUNCT
cana-5238	239	16	hg	hg	NOUN
cana-5238	239	17	-	-	PUNCT
cana-5238	239	18	s	s	PART
cana-5238	239	19	are	be	AUX
cana-5238	239	20	independent	independent	ADJ
cana-5238	239	21	.	.	PUNCT
cana-5238	240	1	example	example	NOUN
cana-5238	241	1	3.14	3.14	NUM
cana-5238	241	2	.	.	PUNCT
cana-5238	242	1	assume	assume	VERB
cana-5238	242	2	z	z	NOUN
cana-5238	242	3	=	=	PUNCT
cana-5238	242	4	{	{	PUNCT
cana-5238	242	5	1	1	NUM
cana-5238	242	6	,	,	PUNCT
cana-5238	242	7	2	2	NUM
cana-5238	242	8	,	,	PUNCT
cana-5238	242	9	3	3	NUM
cana-5238	242	10	,	,	PUNCT
cana-5238	242	11	4	4	NUM
cana-5238	242	12	}	}	PUNCT
cana-5238	242	13	,	,	PUNCT
cana-5238	242	14	ζ={∅	ζ={∅	PROPN
cana-5238	242	15	,	,	PUNCT
cana-5238	242	16	{	{	PUNCT
cana-5238	242	17	1	1	NUM
cana-5238	242	18	,	,	PUNCT
cana-5238	242	19	3	3	NUM
cana-5238	242	20	}	}	PUNCT
cana-5238	242	21	,	,	PUNCT
cana-5238	242	22	{	{	PUNCT
cana-5238	242	23	2	2	NUM
cana-5238	242	24	,	,	PUNCT
cana-5238	242	25	3	3	NUM
cana-5238	242	26	}	}	PUNCT
cana-5238	242	27	,	,	PUNCT
cana-5238	242	28	{	{	PUNCT
cana-5238	242	29	1	1	NUM
cana-5238	242	30	,	,	PUNCT
cana-5238	242	31	2	2	NUM
cana-5238	242	32	,	,	PUNCT
cana-5238	242	33	3},{1	3},{1	NUM
cana-5238	242	34	,	,	PUNCT
cana-5238	242	35	4	4	NUM
cana-5238	242	36	}	}	PUNCT
cana-5238	242	37	,	,	PUNCT
cana-5238	242	38	{	{	PUNCT
cana-5238	242	39	1	1	NUM
cana-5238	242	40	,	,	PUNCT
cana-5238	242	41	3	3	NUM
cana-5238	242	42	,	,	PUNCT
cana-5238	242	43	4	4	NUM
cana-5238	242	44	}	}	PUNCT
cana-5238	242	45	,	,	PUNCT
cana-5238	242	46	z	z	NOUN
cana-5238	242	47	}	}	PUNCT
cana-5238	242	48	,	,	PUNCT
cana-5238	242	49	h={∅	h={∅	PROPN
cana-5238	242	50	,	,	PUNCT
cana-5238	242	51	{	{	PUNCT
cana-5238	242	52	1	1	NUM
cana-5238	242	53	}	}	PUNCT
cana-5238	242	54	,	,	PUNCT
cana-5238	242	55	{	{	PUNCT
cana-5238	242	56	2	2	NUM
cana-5238	242	57	}	}	PUNCT
cana-5238	242	58	}	}	PUNCT
cana-5238	242	59	.	.	PUNCT
cana-5238	243	1	then	then	ADV
cana-5238	243	2	ψ={1	ψ={1	PROPN
cana-5238	243	3	}	}	PUNCT
cana-5238	243	4	is	be	AUX
cana-5238	243	5	σ	σ	PROPN
cana-5238	243	6	-	-	PUNCT
cana-5238	243	7	hg	hg	NOUN
cana-5238	243	8	-	-	NOUN
cana-5238	243	9	o	o	X
cana-5238	243	10	(	(	PUNCT
cana-5238	243	11	resp	resp	NOUN
cana-5238	243	12	.	.	PUNCT
cana-5238	244	1	π	π	PROPN
cana-5238	244	2	-	-	PUNCT
cana-5238	244	3	hg	hg	NOUN
cana-5238	244	4	-	-	NOUN
cana-5238	244	5	o	o	NOUN
cana-5238	244	6	,	,	PUNCT
cana-5238	244	7	b	b	X
cana-5238	244	8	-	-	PUNCT
cana-5238	244	9	hg	hg	NOUN
cana-5238	244	10	-	-	NOUN
cana-5238	244	11	o	o	NOUN
cana-5238	244	12	)	)	PUNCT
cana-5238	244	13	but	but	CCONJ
cana-5238	244	14	not	not	PART
cana-5238	244	15	b∗-b	b∗-b	PROPN
cana-5238	244	16	-	-	PUNCT
cana-5238	244	17	hg	hg	NOUN
cana-5238	244	18	-	-	PUNCT
cana-5238	244	19	s	s	NOUN
cana-5238	244	20	and	and	CCONJ
cana-5238	244	21	m={2	m={2	PROPN
cana-5238	244	22	}	}	PUNCT
cana-5238	244	23	is	be	AUX
cana-5238	244	24	b∗-bhg	b∗-bhg	NOUN
cana-5238	244	25	-	-	PUNCT
cana-5238	244	26	s	s	NOUN
cana-5238	244	27	but	but	CCONJ
cana-5238	244	28	not	not	PART
cana-5238	244	29	σ	σ	PROPN
cana-5238	244	30	-	-	PUNCT
cana-5238	244	31	hg	hg	NOUN
cana-5238	244	32	-	-	NOUN
cana-5238	244	33	o	o	X
cana-5238	244	34	(	(	PUNCT
cana-5238	244	35	resp	resp	NOUN
cana-5238	244	36	.	.	PUNCT
cana-5238	245	1	π	π	PROPN
cana-5238	245	2	-	-	PUNCT
cana-5238	245	3	hg	hg	NOUN
cana-5238	245	4	-	-	NOUN
cana-5238	245	5	o	o	NOUN
cana-5238	245	6	,	,	PUNCT
cana-5238	245	7	b	b	X
cana-5238	245	8	-	-	PUNCT
cana-5238	245	9	hg	hg	NOUN
cana-5238	245	10	-	-	NOUN
cana-5238	245	11	o	o	NOUN
cana-5238	245	12	)	)	PUNCT
cana-5238	245	13	.	.	PUNCT
cana-5238	246	1	theorem	theorem	VERB
cana-5238	246	2	3.15	3.15	NUM
cana-5238	246	3	.	.	PUNCT
cana-5238	247	1	let	let	AUX
cana-5238	247	2	(	(	PUNCT
cana-5238	247	3	z	z	NOUN
cana-5238	247	4	,	,	PUNCT
cana-5238	247	5	ζ	ζ	NOUN
cana-5238	247	6	,	,	PUNCT
cana-5238	247	7	h	h	NOUN
cana-5238	247	8	)	)	PUNCT
cana-5238	247	9	be	be	VERB
cana-5238	247	10	a	a	DET
cana-5238	247	11	strong	strong	ADJ
cana-5238	247	12	hgts	hgts	NOUN
cana-5238	247	13	,	,	PUNCT
cana-5238	247	14	where	where	SCONJ
cana-5238	247	15	z	z	NOUN
cana-5238	247	16	is	be	AUX
cana-5238	247	17	c0	c0	PROPN
cana-5238	247	18	-space	-space	PROPN
cana-5238	247	19	and	and	CCONJ
cana-5238	247	20	ψ⊂z	ψ⊂z	NOUN
cana-5238	247	21	.	.	PUNCT
cana-5238	248	1	then	then	ADV
cana-5238	248	2	the	the	DET
cana-5238	248	3	following	follow	VERB
cana-5238	248	4	conditions	condition	NOUN
cana-5238	248	5	are	be	AUX
cana-5238	248	6	equivalent	equivalent	ADJ
cana-5238	248	7	.	.	PUNCT
cana-5238	249	1	1	1	X
cana-5238	249	2	.	.	X
cana-5238	249	3	ψ	ψ	NOUN
cana-5238	249	4	is	be	AUX
cana-5238	249	5	ζ	ζ	NOUN
cana-5238	249	6	-	-	PUNCT
cana-5238	249	7	o	o	NOUN
cana-5238	249	8	,	,	PUNCT
cana-5238	249	9	2	2	NUM
cana-5238	249	10	.	.	PUNCT
cana-5238	249	11	ψ	ψ	NOUN
cana-5238	249	12	is	be	AUX
cana-5238	249	13	α	α	X
cana-5238	249	14	-	-	PUNCT
cana-5238	249	15	hg	hg	NOUN
cana-5238	249	16	-	-	NOUN
cana-5238	249	17	o	o	NOUN
cana-5238	249	18	and	and	CCONJ
cana-5238	249	19	σ∗-b	σ∗-b	PROPN
cana-5238	249	20	-	-	PUNCT
cana-5238	249	21	hgs	hgs	PROPN
cana-5238	249	22	,	,	PUNCT
cana-5238	249	23	3	3	NUM
cana-5238	249	24	.	.	PUNCT
cana-5238	249	25	ψ	ψ	NOUN
cana-5238	249	26	is	be	AUX
cana-5238	249	27	σ	σ	PROPN
cana-5238	249	28	-	-	PUNCT
cana-5238	249	29	hg	hg	NOUN
cana-5238	249	30	-	-	NOUN
cana-5238	249	31	o	o	NOUN
cana-5238	249	32	and	and	CCONJ
cana-5238	249	33	σ∗-b	σ∗-b	PROPN
cana-5238	249	34	-	-	PUNCT
cana-5238	249	35	hg	hg	NOUN
cana-5238	249	36	-	-	PUNCT
cana-5238	249	37	s	s	PART
cana-5238	249	38	.	.	PUNCT
cana-5238	250	1	proof	proof	NOUN
cana-5238	250	2	.	.	PUNCT
cana-5238	251	1	(	(	PUNCT
cana-5238	251	2	1	1	X
cana-5238	251	3	)	)	PUNCT
cana-5238	251	4	⇒	⇒	NOUN
cana-5238	251	5	(	(	PUNCT
cana-5238	251	6	2	2	NUM
cana-5238	251	7	)	)	PUNCT
cana-5238	251	8	.	.	PUNCT
cana-5238	252	1	let	let	VERB
cana-5238	252	2	a	a	DET
cana-5238	252	3	subset	subset	NOUN
cana-5238	252	4	ψ	ψ	X
cana-5238	252	5	of	of	ADP
cana-5238	252	6	z	z	PROPN
cana-5238	252	7	is	be	AUX
cana-5238	252	8	ζ	ζ	NOUN
cana-5238	252	9	-	-	NOUN
cana-5238	252	10	o.	o.	NOUN
cana-5238	253	1	then	then	ADV
cana-5238	253	2	it	it	PRON
cana-5238	253	3	is	be	AUX
cana-5238	253	4	α	α	NOUN
cana-5238	253	5	-	-	PUNCT
cana-5238	253	6	hg	hg	NOUN
cana-5238	253	7	-	-	NOUN
cana-5238	253	8	o	o	NOUN
cana-5238	253	9	and	and	CCONJ
cana-5238	253	10	σ∗-b	σ∗-b	PROPN
cana-5238	253	11	-	-	PUNCT
cana-5238	253	12	hg	hg	NOUN
cana-5238	253	13	-	-	PUNCT
cana-5238	253	14	s.	s.	PROPN
cana-5238	253	15	(	(	PUNCT
cana-5238	253	16	2)⇒(3	2)⇒(3	NUM
cana-5238	253	17	)	)	PUNCT
cana-5238	253	18	.	.	PUNCT
cana-5238	254	1	let	let	VERB
cana-5238	254	2	a	a	DET
cana-5238	254	3	subset	subset	NOUN
cana-5238	254	4	ψ	ψ	X
cana-5238	254	5	of	of	ADP
cana-5238	254	6	z	z	PROPN
cana-5238	254	7	is	be	AUX
cana-5238	254	8	both	both	DET
cana-5238	254	9	α	α	PROPN
cana-5238	254	10	-	-	PUNCT
cana-5238	254	11	hg	hg	NOUN
cana-5238	254	12	-	-	NOUN
cana-5238	254	13	o	o	NOUN
cana-5238	254	14	and	and	CCONJ
cana-5238	254	15	σ∗-b	σ∗-b	PROPN
cana-5238	254	16	-	-	PUNCT
cana-5238	254	17	hg	hg	NOUN
cana-5238	254	18	-	-	PUNCT
cana-5238	254	19	s.	s.	PROPN
cana-5238	254	20	then	then	ADV
cana-5238	254	21	it	it	PRON
cana-5238	254	22	is	be	AUX
cana-5238	254	23	both	both	DET
cana-5238	254	24	σ	σ	PROPN
cana-5238	254	25	-	-	PUNCT
cana-5238	254	26	hg	hg	NOUN
cana-5238	254	27	-	-	NOUN
cana-5238	254	28	o	o	NOUN
cana-5238	254	29	and	and	CCONJ
cana-5238	254	30	σ∗-b	σ∗-b	PROPN
cana-5238	254	31	-	-	PUNCT
cana-5238	254	32	hg	hg	NOUN
cana-5238	254	33	-	-	PUNCT
cana-5238	254	34	s.	s.	PROPN
cana-5238	254	35	(	(	PUNCT
cana-5238	254	36	3)⇒(1	3)⇒(1	PROPN
cana-5238	254	37	)	)	PUNCT
cana-5238	254	38	.	.	PUNCT
cana-5238	255	1	this	this	PRON
cana-5238	255	2	is	be	AUX
cana-5238	255	3	from	from	ADP
cana-5238	255	4	theorem	theorem	ADJ
cana-5238	255	5	3.9	3.9	NUM
cana-5238	255	6	.	.	PUNCT
cana-5238	255	7	remark	remark	PROPN
cana-5238	255	8	3.16	3.16	NUM
cana-5238	255	9	.	.	PUNCT
cana-5238	256	1	the	the	DET
cana-5238	256	2	notions	notion	NOUN
cana-5238	256	3	of	of	ADP
cana-5238	256	4	α	α	PROPN
cana-5238	256	5	-	-	PUNCT
cana-5238	256	6	hg	hg	NOUN
cana-5238	256	7	-	-	NOUN
cana-5238	256	8	o	o	NOUN
cana-5238	256	9	and	and	CCONJ
cana-5238	256	10	σ∗-b	σ∗-b	PROPN
cana-5238	256	11	-	-	PUNCT
cana-5238	256	12	hg	hg	NOUN
cana-5238	256	13	-	-	PUNCT
cana-5238	256	14	s	s	PART
cana-5238	256	15	are	be	AUX
cana-5238	256	16	independent	independent	ADJ
cana-5238	256	17	.	.	PUNCT
cana-5238	256	18	example	example	NOUN
cana-5238	257	1	3.17	3.17	NUM
cana-5238	257	2	.	.	PUNCT
cana-5238	257	3	assume	assume	VERB
cana-5238	257	4	z={1	z={1	PROPN
cana-5238	257	5	,	,	PUNCT
cana-5238	257	6	2	2	NUM
cana-5238	257	7	,	,	PUNCT
cana-5238	257	8	3	3	NUM
cana-5238	257	9	,	,	PUNCT
cana-5238	257	10	4	4	NUM
cana-5238	257	11	}	}	PUNCT
cana-5238	257	12	,	,	PUNCT
cana-5238	257	13	ζ={∅	ζ={∅	PROPN
cana-5238	257	14	,	,	PUNCT
cana-5238	257	15	{	{	PUNCT
cana-5238	257	16	1	1	NUM
cana-5238	257	17	,	,	PUNCT
cana-5238	257	18	3	3	NUM
cana-5238	257	19	}	}	PUNCT
cana-5238	257	20	,	,	PUNCT
cana-5238	257	21	{	{	PUNCT
cana-5238	257	22	2	2	NUM
cana-5238	257	23	,	,	PUNCT
cana-5238	257	24	3	3	NUM
cana-5238	257	25	}	}	PUNCT
cana-5238	257	26	,	,	PUNCT
cana-5238	257	27	{	{	PUNCT
cana-5238	257	28	1	1	NUM
cana-5238	257	29	,	,	PUNCT
cana-5238	257	30	2	2	NUM
cana-5238	257	31	,	,	PUNCT
cana-5238	257	32	3},{1	3},{1	NUM
cana-5238	257	33	,	,	PUNCT
cana-5238	257	34	4	4	NUM
cana-5238	257	35	}	}	PUNCT
cana-5238	257	36	,	,	PUNCT
cana-5238	257	37	{	{	PUNCT
cana-5238	257	38	1	1	NUM
cana-5238	257	39	,	,	PUNCT
cana-5238	257	40	3	3	NUM
cana-5238	257	41	,	,	PUNCT
cana-5238	257	42	4	4	NUM
cana-5238	257	43	}	}	PUNCT
cana-5238	257	44	,	,	PUNCT
cana-5238	257	45	z	z	NOUN
cana-5238	257	46	}	}	PUNCT
cana-5238	257	47	,	,	PUNCT
cana-5238	257	48	h={∅	h={∅	PROPN
cana-5238	257	49	,	,	PUNCT
cana-5238	257	50	{	{	PUNCT
cana-5238	257	51	1	1	NUM
cana-5238	257	52	}	}	PUNCT
cana-5238	257	53	,	,	PUNCT
cana-5238	257	54	{	{	PUNCT
cana-5238	257	55	2	2	NUM
cana-5238	257	56	}	}	PUNCT
cana-5238	257	57	}	}	PUNCT
cana-5238	257	58	.	.	PUNCT
cana-5238	258	1	then	then	ADV
cana-5238	258	2	ψ={1	ψ={1	PROPN
cana-5238	258	3	}	}	PUNCT
cana-5238	258	4	is	be	AUX
cana-5238	258	5	α	α	PROPN
cana-5238	258	6	-	-	PUNCT
cana-5238	258	7	hg	hg	NOUN
cana-5238	258	8	-	-	NOUN
cana-5238	258	9	o	o	NOUN
cana-5238	258	10	but	but	CCONJ
cana-5238	258	11	not	not	PART
cana-5238	258	12	σ∗-b	σ∗-b	PROPN
cana-5238	258	13	-	-	PUNCT
cana-5238	258	14	hg	hg	NOUN
cana-5238	258	15	-	-	PUNCT
cana-5238	258	16	s	s	NOUN
cana-5238	258	17	and	and	CCONJ
cana-5238	258	18	m={2	m={2	PROPN
cana-5238	258	19	}	}	PUNCT
cana-5238	258	20	is	be	AUX
cana-5238	258	21	σ∗-b	σ∗-b	PROPN
cana-5238	258	22	-	-	PUNCT
cana-5238	258	23	hg	hg	NOUN
cana-5238	258	24	-	-	PUNCT
cana-5238	258	25	s	s	X
cana-5238	258	26	but	but	CCONJ
cana-5238	258	27	not	not	PART
cana-5238	258	28	α	α	PROPN
cana-5238	258	29	-	-	PUNCT
cana-5238	258	30	hg	hg	NOUN
cana-5238	258	31	-	-	NOUN
cana-5238	258	32	o	o	NOUN
cana-5238	258	33	.	.	PUNCT
cana-5238	259	1	theorem	theorem	VERB
cana-5238	259	2	3.18	3.18	NUM
cana-5238	259	3	.	.	PUNCT
cana-5238	260	1	let	let	AUX
cana-5238	260	2	(	(	PUNCT
cana-5238	260	3	z	z	NOUN
cana-5238	260	4	,	,	PUNCT
cana-5238	260	5	ζ	ζ	NOUN
cana-5238	260	6	,	,	PUNCT
cana-5238	260	7	h	h	NOUN
cana-5238	260	8	)	)	PUNCT
cana-5238	260	9	be	be	VERB
cana-5238	260	10	a	a	DET
cana-5238	260	11	strong	strong	ADJ
cana-5238	260	12	hgts	hgts	NOUN
cana-5238	260	13	where	where	SCONJ
cana-5238	260	14	z	z	NOUN
cana-5238	260	15	is	be	AUX
cana-5238	260	16	c0	c0	PROPN
cana-5238	260	17	-space	-space	PROPN
cana-5238	260	18	and	and	CCONJ
cana-5238	260	19	ψ⊂z	ψ⊂z	NOUN
cana-5238	260	20	.	.	PUNCT
cana-5238	261	1	then	then	ADV
cana-5238	261	2	the	the	DET
cana-5238	261	3	following	follow	VERB
cana-5238	261	4	conditions	condition	NOUN
cana-5238	261	5	are	be	AUX
cana-5238	261	6	equivalent	equivalent	ADJ
cana-5238	261	7	.	.	PUNCT
cana-5238	262	1	1	1	X
cana-5238	262	2	.	.	X
cana-5238	262	3	ψ	ψ	NOUN
cana-5238	262	4	is	be	AUX
cana-5238	262	5	ζ	ζ	NOUN
cana-5238	262	6	-	-	PUNCT
cana-5238	262	7	o	o	NOUN
cana-5238	262	8	,	,	PUNCT
cana-5238	262	9	2	2	NUM
cana-5238	262	10	.	.	PUNCT
cana-5238	262	11	ψ	ψ	NOUN
cana-5238	262	12	is	be	AUX
cana-5238	262	13	α	α	X
cana-5238	262	14	-	-	PUNCT
cana-5238	262	15	hg	hg	NOUN
cana-5238	262	16	-	-	NOUN
cana-5238	262	17	o	o	NOUN
cana-5238	262	18	and	and	CCONJ
cana-5238	262	19	π∗-b	π∗-b	PROPN
cana-5238	262	20	-	-	PUNCT
cana-5238	262	21	hg	hg	NOUN
cana-5238	262	22	-	-	PUNCT
cana-5238	262	23	s	s	NOUN
cana-5238	262	24	,	,	PUNCT
cana-5238	262	25	3	3	NUM
cana-5238	262	26	.	.	PUNCT
cana-5238	262	27	ψ	ψ	NOUN
cana-5238	262	28	is	be	AUX
cana-5238	262	29	π	π	PROPN
cana-5238	262	30	-	-	PUNCT
cana-5238	262	31	hg	hg	NOUN
cana-5238	262	32	-	-	NOUN
cana-5238	262	33	o	o	NOUN
cana-5238	262	34	and	and	CCONJ
cana-5238	262	35	π∗-b	π∗-b	PROPN
cana-5238	262	36	-	-	PUNCT
cana-5238	262	37	hg	hg	NOUN
cana-5238	262	38	-	-	PUNCT
cana-5238	262	39	s.	s.	PROPN
cana-5238	262	40	proof	proof	NOUN
cana-5238	262	41	.	.	PUNCT
cana-5238	263	1	(	(	PUNCT
cana-5238	263	2	1)⇒(2	1)⇒(2	NUM
cana-5238	263	3	)	)	PUNCT
cana-5238	263	4	.	.	PUNCT
cana-5238	264	1	let	let	VERB
cana-5238	264	2	a	a	DET
cana-5238	264	3	subset	subset	NOUN
cana-5238	264	4	ψ	ψ	X
cana-5238	264	5	of	of	ADP
cana-5238	264	6	z	z	PROPN
cana-5238	264	7	is	be	AUX
cana-5238	264	8	ζ	ζ	NOUN
cana-5238	264	9	-	-	NOUN
cana-5238	264	10	o.	o.	NOUN
cana-5238	265	1	then	then	ADV
cana-5238	265	2	it	it	PRON
cana-5238	265	3	is	be	AUX
cana-5238	265	4	α	α	NOUN
cana-5238	265	5	-	-	PUNCT
cana-5238	265	6	hg	hg	NOUN
cana-5238	265	7	-	-	NOUN
cana-5238	265	8	o	o	NOUN
cana-5238	265	9	and	and	CCONJ
cana-5238	265	10	π∗-b	π∗-b	PROPN
cana-5238	265	11	-	-	PUNCT
cana-5238	265	12	hg	hg	NOUN
cana-5238	265	13	-	-	PUNCT
cana-5238	265	14	s.	s.	PROPN
cana-5238	265	15	(	(	PUNCT
cana-5238	265	16	2)⇒(3	2)⇒(3	NUM
cana-5238	265	17	)	)	PUNCT
cana-5238	265	18	.	.	PUNCT
cana-5238	266	1	let	let	VERB
cana-5238	266	2	a	a	DET
cana-5238	266	3	subset	subset	NOUN
cana-5238	266	4	ψ	ψ	X
cana-5238	266	5	of	of	ADP
cana-5238	266	6	z	z	PROPN
cana-5238	266	7	is	be	AUX
cana-5238	266	8	both	both	DET
cana-5238	266	9	α	α	PROPN
cana-5238	266	10	-	-	PUNCT
cana-5238	266	11	hg	hg	NOUN
cana-5238	266	12	-	-	NOUN
cana-5238	266	13	o	o	NOUN
cana-5238	266	14	and	and	CCONJ
cana-5238	266	15	π∗-b	π∗-b	PROPN
cana-5238	266	16	-	-	PUNCT
cana-5238	266	17	hg	hg	NOUN
cana-5238	266	18	-	-	PUNCT
cana-5238	266	19	s.	s.	PROPN
cana-5238	266	20	then	then	ADV
cana-5238	266	21	it	it	PRON
cana-5238	266	22	is	be	AUX
cana-5238	266	23	both	both	PRON
cana-5238	266	24	π	π	PROPN
cana-5238	266	25	-	-	PUNCT
cana-5238	266	26	hg	hg	NOUN
cana-5238	266	27	-	-	NOUN
cana-5238	266	28	o	o	NOUN
cana-5238	266	29	and	and	CCONJ
cana-5238	266	30	π∗-b	π∗-b	PROPN
cana-5238	266	31	-	-	PUNCT
cana-5238	266	32	hg	hg	NOUN
cana-5238	266	33	-	-	PUNCT
cana-5238	266	34	s.	s.	PROPN
cana-5238	266	35	(	(	PUNCT
cana-5238	266	36	3)⇒(1	3)⇒(1	PROPN
cana-5238	266	37	)	)	PUNCT
cana-5238	266	38	.	.	PUNCT
cana-5238	267	1	this	this	PRON
cana-5238	267	2	is	be	AUX
cana-5238	267	3	from	from	ADP
cana-5238	267	4	theorem	theorem	ADJ
cana-5238	267	5	3.9	3.9	NUM
cana-5238	267	6	.	.	PUNCT
cana-5238	268	1	theorem	theorem	VERB
cana-5238	268	2	3.19	3.19	NUM
cana-5238	268	3	.	.	PUNCT
cana-5238	269	1	let	let	AUX
cana-5238	269	2	(	(	PUNCT
cana-5238	269	3	z	z	NOUN
cana-5238	269	4	,	,	PUNCT
cana-5238	269	5	ζ	ζ	NOUN
cana-5238	269	6	,	,	PUNCT
cana-5238	269	7	h	h	NOUN
cana-5238	269	8	)	)	PUNCT
cana-5238	269	9	be	be	VERB
cana-5238	269	10	a	a	DET
cana-5238	269	11	strong	strong	ADJ
cana-5238	269	12	hgts	hgts	NOUN
cana-5238	269	13	where	where	SCONJ
cana-5238	269	14	z	z	NOUN
cana-5238	269	15	is	be	AUX
cana-5238	269	16	c0	c0	PROPN
cana-5238	269	17	-space	-space	PROPN
cana-5238	269	18	and	and	CCONJ
cana-5238	269	19	ψ⊂z	ψ⊂z	NOUN
cana-5238	269	20	.	.	PUNCT
cana-5238	270	1	then	then	ADV
cana-5238	270	2	the	the	DET
cana-5238	270	3	following	follow	VERB
cana-5238	270	4	conditions	condition	NOUN
cana-5238	270	5	are	be	AUX
cana-5238	270	6	equivalent	equivalent	ADJ
cana-5238	270	7	.	.	PUNCT
cana-5238	271	1	1	1	X
cana-5238	271	2	.	.	X
cana-5238	271	3	ψ	ψ	NOUN
cana-5238	271	4	is	be	AUX
cana-5238	271	5	ζ	ζ	NOUN
cana-5238	271	6	-	-	PUNCT
cana-5238	271	7	o	o	NOUN
cana-5238	271	8	,	,	PUNCT
cana-5238	271	9	2	2	NUM
cana-5238	271	10	.	.	PUNCT
cana-5238	271	11	ψ	ψ	NOUN
cana-5238	271	12	is	be	AUX
cana-5238	271	13	α	α	X
cana-5238	271	14	-	-	PUNCT
cana-5238	271	15	hg	hg	NOUN
cana-5238	271	16	-	-	NOUN
cana-5238	271	17	o	o	NOUN
cana-5238	271	18	and	and	CCONJ
cana-5238	271	19	β∗-b	β∗-b	PROPN
cana-5238	271	20	-	-	PUNCT
cana-5238	271	21	hg	hg	NOUN
cana-5238	271	22	-	-	PUNCT
cana-5238	271	23	s	s	NOUN
cana-5238	271	24	,	,	PUNCT
cana-5238	271	25	3	3	NUM
cana-5238	271	26	.	.	PUNCT
cana-5238	271	27	ψ	ψ	NOUN
cana-5238	271	28	is	be	AUX
cana-5238	271	29	β	β	X
cana-5238	271	30	-	-	ADJ
cana-5238	271	31	hg	hg	NOUN
cana-5238	271	32	-	-	NOUN
cana-5238	271	33	o	o	NOUN
cana-5238	271	34	and	and	CCONJ
cana-5238	271	35	β∗-b	β∗-b	PROPN
cana-5238	271	36	-	-	PUNCT
cana-5238	271	37	hg	hg	NOUN
cana-5238	271	38	-	-	PUNCT
cana-5238	271	39	s.	s.	PROPN
cana-5238	271	40	communications	communication	NOUN
cana-5238	271	41	on	on	ADP
cana-5238	271	42	applied	apply	VERB
cana-5238	271	43	nonlinear	nonlinear	ADJ
cana-5238	271	44	analysis	analysis	NOUN
cana-5238	271	45	issn	issn	NOUN
cana-5238	271	46	:	:	PUNCT
cana-5238	271	47	1074	1074	NUM
cana-5238	271	48	-	-	PUNCT
cana-5238	271	49	133x	133x	NUM
cana-5238	271	50	vol	vol	VERB
cana-5238	271	51	32	32	NUM
cana-5238	271	52	no	no	NOUN
cana-5238	271	53	.	.	PUNCT
cana-5238	272	1	10s	10	NOUN
cana-5238	272	2	(	(	PUNCT
cana-5238	272	3	2025	2025	NUM
cana-5238	272	4	)	)	PUNCT
cana-5238	272	5	1364	1364	NUM
cana-5238	272	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	272	7	ζ	ζ	NOUN
cana-5238	272	8	proof	proof	NOUN
cana-5238	272	9	.	.	PUNCT
cana-5238	273	1	(	(	PUNCT
cana-5238	273	2	1)⇒(2	1)⇒(2	NUM
cana-5238	273	3	)	)	PUNCT
cana-5238	273	4	.	.	PUNCT
cana-5238	274	1	let	let	VERB
cana-5238	274	2	a	a	DET
cana-5238	274	3	subset	subset	NOUN
cana-5238	274	4	ψ	ψ	X
cana-5238	274	5	of	of	ADP
cana-5238	274	6	z	z	PROPN
cana-5238	274	7	is	be	AUX
cana-5238	274	8	ζ	ζ	NOUN
cana-5238	274	9	-	-	NOUN
cana-5238	274	10	o.	o.	NOUN
cana-5238	275	1	then	then	ADV
cana-5238	275	2	it	it	PRON
cana-5238	275	3	is	be	AUX
cana-5238	275	4	α	α	NOUN
cana-5238	275	5	-	-	PUNCT
cana-5238	275	6	hg	hg	NOUN
cana-5238	275	7	-	-	NOUN
cana-5238	275	8	o	o	NOUN
cana-5238	275	9	and	and	CCONJ
cana-5238	275	10	β∗-b	β∗-b	PROPN
cana-5238	275	11	-	-	PUNCT
cana-5238	275	12	hg	hg	NOUN
cana-5238	275	13	-	-	PUNCT
cana-5238	275	14	s.	s.	PROPN
cana-5238	275	15	(	(	PUNCT
cana-5238	275	16	2)⇒(3	2)⇒(3	NUM
cana-5238	275	17	)	)	PUNCT
cana-5238	275	18	.	.	PUNCT
cana-5238	276	1	let	let	VERB
cana-5238	276	2	a	a	DET
cana-5238	276	3	subset	subset	NOUN
cana-5238	276	4	ψ	ψ	X
cana-5238	276	5	of	of	ADP
cana-5238	276	6	z	z	PROPN
cana-5238	276	7	is	be	AUX
cana-5238	276	8	both	both	DET
cana-5238	276	9	α	α	PROPN
cana-5238	276	10	-	-	PUNCT
cana-5238	276	11	hg	hg	NOUN
cana-5238	276	12	-	-	NOUN
cana-5238	276	13	o	o	NOUN
cana-5238	276	14	and	and	CCONJ
cana-5238	276	15	β∗-b	β∗-b	PROPN
cana-5238	276	16	-	-	PUNCT
cana-5238	276	17	hg	hg	NOUN
cana-5238	276	18	-	-	PUNCT
cana-5238	276	19	s.	s.	PROPN
cana-5238	276	20	then	then	ADV
cana-5238	276	21	it	it	PRON
cana-5238	276	22	is	be	AUX
cana-5238	276	23	both	both	PRON
cana-5238	276	24	β	β	NOUN
cana-5238	276	25	-	-	PUNCT
cana-5238	276	26	hg	hg	NOUN
cana-5238	276	27	-	-	NOUN
cana-5238	276	28	o	o	NOUN
cana-5238	276	29	and	and	CCONJ
cana-5238	276	30	β∗-b	β∗-b	PROPN
cana-5238	276	31	-	-	PUNCT
cana-5238	276	32	hg	hg	NOUN
cana-5238	276	33	-	-	PUNCT
cana-5238	276	34	s.	s.	PROPN
cana-5238	276	35	(	(	PUNCT
cana-5238	276	36	3)⇒(1	3)⇒(1	PROPN
cana-5238	276	37	)	)	PUNCT
cana-5238	276	38	.	.	PUNCT
cana-5238	277	1	this	this	PRON
cana-5238	277	2	is	be	AUX
cana-5238	277	3	from	from	ADP
cana-5238	277	4	theorem	theorem	ADJ
cana-5238	277	5	3.9	3.9	NUM
cana-5238	277	6	.	.	PUNCT
cana-5238	277	7	remark	remark	NOUN
cana-5238	277	8	3.20	3.20	NUM
cana-5238	277	9	.	.	PUNCT
cana-5238	278	1	the	the	DET
cana-5238	278	2	notions	notion	NOUN
cana-5238	278	3	of	of	ADP
cana-5238	278	4	α	α	PROPN
cana-5238	278	5	-	-	PUNCT
cana-5238	278	6	hg	hg	NOUN
cana-5238	278	7	-	-	NOUN
cana-5238	278	8	o	o	NOUN
cana-5238	278	9	and	and	CCONJ
cana-5238	278	10	π∗-b	π∗-b	PROPN
cana-5238	278	11	-	-	PUNCT
cana-5238	278	12	hg	hg	NOUN
cana-5238	278	13	-	-	PUNCT
cana-5238	278	14	s	s	PART
cana-5238	278	15	are	be	AUX
cana-5238	278	16	independent	independent	ADJ
cana-5238	278	17	.	.	PUNCT
cana-5238	278	18	example	example	NOUN
cana-5238	279	1	3.21	3.21	NUM
cana-5238	279	2	.	.	PUNCT
cana-5238	279	3	assume	assume	VERB
cana-5238	279	4	z={1	z={1	PROPN
cana-5238	279	5	,	,	PUNCT
cana-5238	279	6	2	2	NUM
cana-5238	279	7	,	,	PUNCT
cana-5238	279	8	3	3	NUM
cana-5238	279	9	,	,	PUNCT
cana-5238	279	10	4	4	NUM
cana-5238	279	11	}	}	PUNCT
cana-5238	279	12	,	,	PUNCT
cana-5238	279	13	ζ={∅	ζ={∅	PROPN
cana-5238	279	14	,	,	PUNCT
cana-5238	279	15	{	{	PUNCT
cana-5238	279	16	1	1	NUM
cana-5238	279	17	,	,	PUNCT
cana-5238	279	18	3	3	NUM
cana-5238	279	19	}	}	PUNCT
cana-5238	279	20	,	,	PUNCT
cana-5238	279	21	{	{	PUNCT
cana-5238	279	22	2	2	NUM
cana-5238	279	23	,	,	PUNCT
cana-5238	279	24	3	3	NUM
cana-5238	279	25	}	}	PUNCT
cana-5238	279	26	,	,	PUNCT
cana-5238	279	27	{	{	PUNCT
cana-5238	279	28	1	1	NUM
cana-5238	279	29	,	,	PUNCT
cana-5238	279	30	2	2	NUM
cana-5238	279	31	,	,	PUNCT
cana-5238	279	32	3},{1	3},{1	NUM
cana-5238	279	33	,	,	PUNCT
cana-5238	279	34	4	4	NUM
cana-5238	279	35	}	}	PUNCT
cana-5238	279	36	,	,	PUNCT
cana-5238	279	37	{	{	PUNCT
cana-5238	279	38	1	1	NUM
cana-5238	279	39	,	,	PUNCT
cana-5238	279	40	3	3	NUM
cana-5238	279	41	,	,	PUNCT
cana-5238	279	42	4	4	NUM
cana-5238	279	43	}	}	PUNCT
cana-5238	279	44	,	,	PUNCT
cana-5238	279	45	z	z	NOUN
cana-5238	279	46	}	}	PUNCT
cana-5238	279	47	,	,	PUNCT
cana-5238	279	48	h={∅	h={∅	PROPN
cana-5238	279	49	,	,	PUNCT
cana-5238	279	50	{	{	PUNCT
cana-5238	279	51	1	1	NUM
cana-5238	279	52	}	}	PUNCT
cana-5238	279	53	,	,	PUNCT
cana-5238	279	54	{	{	PUNCT
cana-5238	279	55	2	2	NUM
cana-5238	279	56	}	}	PUNCT
cana-5238	279	57	}	}	PUNCT
cana-5238	279	58	.	.	PUNCT
cana-5238	280	1	then	then	ADV
cana-5238	280	2	ψ={1	ψ={1	PROPN
cana-5238	280	3	}	}	PUNCT
cana-5238	280	4	is	be	AUX
cana-5238	280	5	α	α	PROPN
cana-5238	280	6	-	-	PUNCT
cana-5238	280	7	hg	hg	NOUN
cana-5238	280	8	-	-	NOUN
cana-5238	280	9	o	o	NOUN
cana-5238	280	10	but	but	CCONJ
cana-5238	280	11	not	not	PART
cana-5238	280	12	π∗-b	π∗-b	PROPN
cana-5238	280	13	-	-	PUNCT
cana-5238	280	14	hg	hg	NOUN
cana-5238	280	15	-	-	PUNCT
cana-5238	280	16	s	s	X
cana-5238	280	17	(	(	PUNCT
cana-5238	280	18	resp	resp	NOUN
cana-5238	280	19	.	.	PUNCT
cana-5238	281	1	β∗-b	β∗-b	PROPN
cana-5238	281	2	-	-	PUNCT
cana-5238	281	3	hg	hg	NOUN
cana-5238	281	4	-	-	NOUN
cana-5238	281	5	s	s	X
cana-5238	281	6	)	)	PUNCT
cana-5238	281	7	and	and	CCONJ
cana-5238	281	8	m={2	m={2	PROPN
cana-5238	281	9	}	}	PUNCT
cana-5238	281	10	is	be	AUX
cana-5238	281	11	π∗-b	π∗-b	PROPN
cana-5238	281	12	-	-	PUNCT
cana-5238	281	13	hg	hg	NOUN
cana-5238	281	14	-	-	PUNCT
cana-5238	281	15	s	s	X
cana-5238	281	16	(	(	PUNCT
cana-5238	281	17	resp	resp	NOUN
cana-5238	281	18	.	.	PUNCT
cana-5238	282	1	β∗-b	β∗-b	PROPN
cana-5238	282	2	-	-	PUNCT
cana-5238	282	3	hg	hg	NOUN
cana-5238	282	4	-	-	NOUN
cana-5238	282	5	s	s	X
cana-5238	282	6	)	)	PUNCT
cana-5238	282	7	but	but	CCONJ
cana-5238	282	8	not	not	PART
cana-5238	282	9	α	α	PROPN
cana-5238	282	10	-	-	PUNCT
cana-5238	282	11	hg	hg	NOUN
cana-5238	282	12	-	-	ADJ
cana-5238	282	13	o.	o.	NOUN
cana-5238	282	14	theorem	theorem	NOUN
cana-5238	282	15	3.22	3.22	NUM
cana-5238	282	16	.	.	PUNCT
cana-5238	283	1	let	let	AUX
cana-5238	283	2	(	(	PUNCT
cana-5238	283	3	z	z	NOUN
cana-5238	283	4	,	,	PUNCT
cana-5238	283	5	ζ	ζ	NOUN
cana-5238	283	6	,	,	PUNCT
cana-5238	283	7	h	h	NOUN
cana-5238	283	8	)	)	PUNCT
cana-5238	283	9	be	be	VERB
cana-5238	283	10	a	a	DET
cana-5238	283	11	strong	strong	ADJ
cana-5238	283	12	hgts	hgts	NOUN
cana-5238	283	13	where	where	SCONJ
cana-5238	283	14	z	z	NOUN
cana-5238	283	15	is	be	AUX
cana-5238	283	16	c0	c0	PROPN
cana-5238	283	17	-space	-space	PROPN
cana-5238	283	18	and	and	CCONJ
cana-5238	283	19	ψ⊂z	ψ⊂z	NOUN
cana-5238	283	20	.	.	PUNCT
cana-5238	284	1	then	then	ADV
cana-5238	284	2	the	the	DET
cana-5238	284	3	following	follow	VERB
cana-5238	284	4	conditions	condition	NOUN
cana-5238	284	5	are	be	AUX
cana-5238	284	6	equivalent	equivalent	ADJ
cana-5238	284	7	.	.	PUNCT
cana-5238	285	1	1	1	X
cana-5238	285	2	.	.	X
cana-5238	285	3	ψ	ψ	NOUN
cana-5238	285	4	is	be	AUX
cana-5238	285	5	ζ	ζ	NOUN
cana-5238	285	6	-	-	PUNCT
cana-5238	285	7	o	o	NOUN
cana-5238	285	8	,	,	PUNCT
cana-5238	285	9	2	2	NUM
cana-5238	285	10	.	.	PUNCT
cana-5238	285	11	ψ	ψ	NOUN
cana-5238	285	12	is	be	AUX
cana-5238	285	13	σ	σ	PROPN
cana-5238	285	14	-	-	PUNCT
cana-5238	285	15	hg	hg	NOUN
cana-5238	285	16	-	-	NOUN
cana-5238	285	17	o	o	NOUN
cana-5238	285	18	and	and	CCONJ
cana-5238	285	19	β∗-b	β∗-b	PROPN
cana-5238	285	20	-	-	PUNCT
cana-5238	285	21	hg	hg	NOUN
cana-5238	285	22	-	-	PUNCT
cana-5238	285	23	s	s	NOUN
cana-5238	285	24	,	,	PUNCT
cana-5238	285	25	3	3	NUM
cana-5238	285	26	.	.	PUNCT
cana-5238	285	27	ψ	ψ	NOUN
cana-5238	285	28	is	be	AUX
cana-5238	285	29	β	β	X
cana-5238	285	30	-	-	ADJ
cana-5238	285	31	hg	hg	NOUN
cana-5238	285	32	-	-	NOUN
cana-5238	285	33	o	o	NOUN
cana-5238	285	34	and	and	CCONJ
cana-5238	285	35	β∗-b	β∗-b	PROPN
cana-5238	285	36	-	-	PUNCT
cana-5238	285	37	hg	hg	NOUN
cana-5238	285	38	-	-	PUNCT
cana-5238	285	39	s	s	NOUN
cana-5238	285	40	.	.	PUNCT
cana-5238	286	1	proof	proof	NOUN
cana-5238	286	2	.	.	PUNCT
cana-5238	287	1	(	(	PUNCT
cana-5238	287	2	1	1	X
cana-5238	287	3	)	)	PUNCT
cana-5238	287	4	⇒	⇒	NOUN
cana-5238	287	5	(	(	PUNCT
cana-5238	287	6	2	2	NUM
cana-5238	287	7	)	)	PUNCT
cana-5238	287	8	.	.	PUNCT
cana-5238	288	1	let	let	VERB
cana-5238	288	2	a	a	DET
cana-5238	288	3	subset	subset	NOUN
cana-5238	288	4	ψ	ψ	X
cana-5238	288	5	of	of	ADP
cana-5238	288	6	z	z	PROPN
cana-5238	288	7	is	be	AUX
cana-5238	288	8	ζ	ζ	NOUN
cana-5238	288	9	-	-	NOUN
cana-5238	288	10	o.	o.	NOUN
cana-5238	289	1	then	then	ADV
cana-5238	289	2	it	it	PRON
cana-5238	289	3	is	be	AUX
cana-5238	289	4	σ	σ	PROPN
cana-5238	289	5	-	-	PUNCT
cana-5238	289	6	hg	hg	NOUN
cana-5238	289	7	-	-	NOUN
cana-5238	289	8	o	o	NOUN
cana-5238	289	9	and	and	CCONJ
cana-5238	289	10	β∗-b	β∗-b	PROPN
cana-5238	289	11	-	-	PUNCT
cana-5238	289	12	hg	hg	NOUN
cana-5238	289	13	-	-	PUNCT
cana-5238	289	14	s.	s.	PROPN
cana-5238	289	15	(	(	PUNCT
cana-5238	289	16	2)⇒	2)⇒	NUM
cana-5238	289	17	(	(	PUNCT
cana-5238	289	18	3	3	NUM
cana-5238	289	19	)	)	PUNCT
cana-5238	289	20	.	.	PUNCT
cana-5238	290	1	let	let	VERB
cana-5238	290	2	a	a	DET
cana-5238	290	3	subset	subset	NOUN
cana-5238	290	4	ψ	ψ	X
cana-5238	290	5	of	of	ADP
cana-5238	290	6	z	z	PROPN
cana-5238	290	7	is	be	AUX
cana-5238	290	8	both	both	DET
cana-5238	290	9	σ	σ	PROPN
cana-5238	290	10	-	-	PUNCT
cana-5238	290	11	hg	hg	NOUN
cana-5238	290	12	-	-	NOUN
cana-5238	290	13	o	o	NOUN
cana-5238	290	14	and	and	CCONJ
cana-5238	290	15	β∗-b	β∗-b	PROPN
cana-5238	290	16	-	-	PUNCT
cana-5238	290	17	hg	hg	NOUN
cana-5238	290	18	-	-	PUNCT
cana-5238	290	19	s.	s.	PROPN
cana-5238	290	20	then	then	ADV
cana-5238	290	21	it	it	PRON
cana-5238	290	22	is	be	AUX
cana-5238	290	23	both	both	PRON
cana-5238	290	24	β	β	NOUN
cana-5238	290	25	-	-	PUNCT
cana-5238	290	26	hg	hg	NOUN
cana-5238	290	27	-	-	NOUN
cana-5238	290	28	o	o	NOUN
cana-5238	290	29	and	and	CCONJ
cana-5238	290	30	β∗-b	β∗-b	PROPN
cana-5238	290	31	-	-	PUNCT
cana-5238	290	32	hg	hg	NOUN
cana-5238	290	33	-	-	PUNCT
cana-5238	290	34	s.	s.	PROPN
cana-5238	290	35	(	(	PUNCT
cana-5238	290	36	3)⇒	3)⇒	NUM
cana-5238	290	37	(	(	PUNCT
cana-5238	290	38	1	1	NUM
cana-5238	290	39	)	)	PUNCT
cana-5238	290	40	.	.	PUNCT
cana-5238	291	1	this	this	PRON
cana-5238	291	2	is	be	AUX
cana-5238	291	3	from	from	ADP
cana-5238	291	4	theorem	theorem	ADJ
cana-5238	291	5	3.9	3.9	NUM
cana-5238	291	6	.	.	PUNCT
cana-5238	291	7	remark	remark	NOUN
cana-5238	291	8	3.23	3.23	NUM
cana-5238	291	9	.	.	PUNCT
cana-5238	292	1	the	the	DET
cana-5238	292	2	notions	notion	NOUN
cana-5238	292	3	of	of	ADP
cana-5238	292	4	σ	σ	PROPN
cana-5238	292	5	-	-	PUNCT
cana-5238	292	6	hg	hg	NOUN
cana-5238	292	7	-	-	NOUN
cana-5238	292	8	o	o	NOUN
cana-5238	292	9	and	and	CCONJ
cana-5238	292	10	β∗-b	β∗-b	PROPN
cana-5238	292	11	-	-	PUNCT
cana-5238	292	12	hg	hg	NOUN
cana-5238	292	13	-	-	PUNCT
cana-5238	292	14	s	s	PART
cana-5238	292	15	are	be	AUX
cana-5238	292	16	independent	independent	ADJ
cana-5238	292	17	.	.	PUNCT
cana-5238	292	18	example	example	NOUN
cana-5238	293	1	3.24	3.24	NUM
cana-5238	293	2	.	.	PUNCT
cana-5238	294	1	assume	assume	VERB
cana-5238	294	2	z={1	z={1	PROPN
cana-5238	294	3	,	,	PUNCT
cana-5238	294	4	2	2	NUM
cana-5238	294	5	,	,	PUNCT
cana-5238	294	6	3	3	NUM
cana-5238	294	7	,	,	PUNCT
cana-5238	294	8	4	4	NUM
cana-5238	294	9	}	}	PUNCT
cana-5238	294	10	,	,	PUNCT
cana-5238	294	11	ζ={∅	ζ={∅	PROPN
cana-5238	294	12	,	,	PUNCT
cana-5238	294	13	{	{	PUNCT
cana-5238	294	14	1	1	NUM
cana-5238	294	15	,	,	PUNCT
cana-5238	294	16	3	3	NUM
cana-5238	294	17	}	}	PUNCT
cana-5238	294	18	,	,	PUNCT
cana-5238	294	19	{	{	PUNCT
cana-5238	294	20	2	2	NUM
cana-5238	294	21	,	,	PUNCT
cana-5238	294	22	3	3	NUM
cana-5238	294	23	}	}	PUNCT
cana-5238	294	24	,	,	PUNCT
cana-5238	294	25	{	{	PUNCT
cana-5238	294	26	1	1	NUM
cana-5238	294	27	,	,	PUNCT
cana-5238	294	28	2	2	NUM
cana-5238	294	29	,	,	PUNCT
cana-5238	294	30	3	3	NUM
cana-5238	294	31	}	}	PUNCT
cana-5238	294	32	,	,	PUNCT
cana-5238	294	33	{	{	PUNCT
cana-5238	294	34	1	1	NUM
cana-5238	294	35	,	,	PUNCT
cana-5238	294	36	4	4	NUM
cana-5238	294	37	}	}	PUNCT
cana-5238	294	38	,	,	PUNCT
cana-5238	294	39	{	{	PUNCT
cana-5238	294	40	1	1	NUM
cana-5238	294	41	,	,	PUNCT
cana-5238	294	42	3	3	NUM
cana-5238	294	43	,	,	PUNCT
cana-5238	294	44	4	4	NUM
cana-5238	294	45	}	}	PUNCT
cana-5238	294	46	,	,	PUNCT
cana-5238	294	47	z	z	NOUN
cana-5238	294	48	}	}	PUNCT
cana-5238	294	49	,	,	PUNCT
cana-5238	294	50	h={∅	h={∅	PROPN
cana-5238	294	51	,	,	PUNCT
cana-5238	294	52	{	{	PUNCT
cana-5238	294	53	1	1	NUM
cana-5238	294	54	}	}	PUNCT
cana-5238	294	55	,	,	PUNCT
cana-5238	294	56	{	{	PUNCT
cana-5238	294	57	2	2	NUM
cana-5238	294	58	}	}	PUNCT
cana-5238	294	59	}	}	PUNCT
cana-5238	294	60	.	.	PUNCT
cana-5238	295	1	then	then	ADV
cana-5238	295	2	ψ={1	ψ={1	PROPN
cana-5238	295	3	}	}	PUNCT
cana-5238	295	4	is	be	AUX
cana-5238	295	5	σ	σ	PROPN
cana-5238	295	6	-	-	PUNCT
cana-5238	295	7	hg	hg	NOUN
cana-5238	295	8	-	-	NOUN
cana-5238	295	9	o	o	NOUN
cana-5238	295	10	but	but	CCONJ
cana-5238	295	11	not	not	PART
cana-5238	295	12	β∗-b	β∗-b	PROPN
cana-5238	295	13	-	-	PUNCT
cana-5238	295	14	hg	hg	NOUN
cana-5238	295	15	-	-	PUNCT
cana-5238	295	16	s	s	NOUN
cana-5238	295	17	and	and	CCONJ
cana-5238	295	18	m={2	m={2	PROPN
cana-5238	295	19	}	}	PUNCT
cana-5238	295	20	is	be	AUX
cana-5238	295	21	β∗-b	β∗-b	PROPN
cana-5238	295	22	-	-	PUNCT
cana-5238	295	23	hg	hg	NOUN
cana-5238	295	24	-	-	PUNCT
cana-5238	295	25	s	s	X
cana-5238	295	26	but	but	CCONJ
cana-5238	295	27	not	not	PART
cana-5238	295	28	σ	σ	PROPN
cana-5238	295	29	-	-	PUNCT
cana-5238	295	30	hg	hg	NOUN
cana-5238	295	31	-	-	ADJ
cana-5238	295	32	o.	o.	NOUN
cana-5238	295	33	theorem	theorem	NOUN
cana-5238	295	34	3.25	3.25	NUM
cana-5238	295	35	.	.	PUNCT
cana-5238	296	1	let	let	VERB
cana-5238	296	2	(	(	PUNCT
cana-5238	296	3	z	z	NOUN
cana-5238	296	4	,	,	PUNCT
cana-5238	296	5	ζ	ζ	NOUN
cana-5238	296	6	,	,	PUNCT
cana-5238	296	7	h	h	NOUN
cana-5238	296	8	)	)	PUNCT
cana-5238	296	9	be	be	VERB
cana-5238	296	10	a	a	DET
cana-5238	296	11	strong	strong	ADJ
cana-5238	296	12	hgts	hgts	NOUN
cana-5238	296	13	,	,	PUNCT
cana-5238	296	14	where	where	SCONJ
cana-5238	296	15	z	z	NOUN
cana-5238	296	16	is	be	AUX
cana-5238	296	17	c0	c0	PROPN
cana-5238	296	18	-space	-space	PROPN
cana-5238	296	19	and	and	CCONJ
cana-5238	296	20	ψ⊂z	ψ⊂z	NOUN
cana-5238	296	21	.	.	PUNCT
cana-5238	297	1	then	then	ADV
cana-5238	297	2	the	the	DET
cana-5238	297	3	following	follow	VERB
cana-5238	297	4	conditions	condition	NOUN
cana-5238	297	5	are	be	AUX
cana-5238	297	6	equivalent	equivalent	ADJ
cana-5238	297	7	.	.	PUNCT
cana-5238	298	1	1	1	X
cana-5238	298	2	.	.	X
cana-5238	298	3	ψ	ψ	NOUN
cana-5238	298	4	is	be	AUX
cana-5238	298	5	ζ	ζ	NOUN
cana-5238	298	6	-	-	PUNCT
cana-5238	298	7	o	o	NOUN
cana-5238	298	8	,	,	PUNCT
cana-5238	298	9	2	2	NUM
cana-5238	298	10	.	.	PUNCT
cana-5238	298	11	ψ	ψ	NOUN
cana-5238	298	12	is	be	AUX
cana-5238	298	13	π	π	PROPN
cana-5238	298	14	-	-	PUNCT
cana-5238	298	15	hg	hg	NOUN
cana-5238	298	16	-	-	NOUN
cana-5238	298	17	o	o	NOUN
cana-5238	298	18	and	and	CCONJ
cana-5238	298	19	β∗-b	β∗-b	PROPN
cana-5238	298	20	-	-	PUNCT
cana-5238	298	21	hg	hg	NOUN
cana-5238	298	22	-	-	PUNCT
cana-5238	298	23	s	s	NOUN
cana-5238	298	24	,	,	PUNCT
cana-5238	298	25	3	3	NUM
cana-5238	298	26	.	.	PUNCT
cana-5238	298	27	ψ	ψ	NOUN
cana-5238	298	28	is	be	AUX
cana-5238	298	29	β	β	X
cana-5238	298	30	-	-	ADJ
cana-5238	298	31	hg	hg	NOUN
cana-5238	298	32	-	-	NOUN
cana-5238	298	33	o	o	NOUN
cana-5238	298	34	and	and	CCONJ
cana-5238	298	35	β∗-b	β∗-b	PROPN
cana-5238	298	36	-	-	PUNCT
cana-5238	298	37	hg	hg	NOUN
cana-5238	298	38	-	-	PUNCT
cana-5238	298	39	s.	s.	PROPN
cana-5238	298	40	proof	proof	NOUN
cana-5238	298	41	.	.	PUNCT
cana-5238	299	1	(	(	PUNCT
cana-5238	299	2	1	1	X
cana-5238	299	3	)	)	PUNCT
cana-5238	299	4	⇒	⇒	NOUN
cana-5238	299	5	(	(	PUNCT
cana-5238	299	6	2	2	NUM
cana-5238	299	7	)	)	PUNCT
cana-5238	299	8	.	.	PUNCT
cana-5238	300	1	let	let	VERB
cana-5238	300	2	a	a	DET
cana-5238	300	3	subset	subset	NOUN
cana-5238	300	4	ψ	ψ	X
cana-5238	300	5	of	of	ADP
cana-5238	300	6	z	z	PROPN
cana-5238	300	7	is	be	AUX
cana-5238	300	8	ζ	ζ	NOUN
cana-5238	300	9	-	-	NOUN
cana-5238	300	10	o.	o.	NOUN
cana-5238	301	1	then	then	ADV
cana-5238	301	2	it	it	PRON
cana-5238	301	3	is	be	AUX
cana-5238	301	4	π	π	PROPN
cana-5238	301	5	-	-	PUNCT
cana-5238	301	6	hg	hg	NOUN
cana-5238	301	7	-	-	NOUN
cana-5238	301	8	o	o	NOUN
cana-5238	301	9	and	and	CCONJ
cana-5238	301	10	β∗-b	β∗-b	PROPN
cana-5238	301	11	-	-	PUNCT
cana-5238	301	12	hg	hg	NOUN
cana-5238	301	13	-	-	PUNCT
cana-5238	301	14	s.	s.	PROPN
cana-5238	301	15	(	(	PUNCT
cana-5238	301	16	2)⇒(3	2)⇒(3	NUM
cana-5238	301	17	)	)	PUNCT
cana-5238	301	18	.	.	PUNCT
cana-5238	302	1	let	let	VERB
cana-5238	302	2	a	a	DET
cana-5238	302	3	subset	subset	NOUN
cana-5238	302	4	ψ	ψ	X
cana-5238	302	5	of	of	ADP
cana-5238	302	6	z	z	PROPN
cana-5238	302	7	is	be	AUX
cana-5238	302	8	both	both	PRON
cana-5238	302	9	π	π	PROPN
cana-5238	302	10	-	-	PUNCT
cana-5238	302	11	hg	hg	NOUN
cana-5238	302	12	-	-	NOUN
cana-5238	302	13	o	o	NOUN
cana-5238	302	14	and	and	CCONJ
cana-5238	302	15	β∗-b	β∗-b	PROPN
cana-5238	302	16	-	-	PUNCT
cana-5238	302	17	hg	hg	NOUN
cana-5238	302	18	-	-	PUNCT
cana-5238	302	19	s.	s.	PROPN
cana-5238	302	20	then	then	ADV
cana-5238	302	21	it	it	PRON
cana-5238	302	22	is	be	AUX
cana-5238	302	23	both	both	PRON
cana-5238	302	24	β	β	NOUN
cana-5238	302	25	-	-	PUNCT
cana-5238	302	26	hg	hg	NOUN
cana-5238	302	27	-	-	NOUN
cana-5238	302	28	o	o	NOUN
cana-5238	302	29	and	and	CCONJ
cana-5238	302	30	β∗-b	β∗-b	PROPN
cana-5238	302	31	-	-	PUNCT
cana-5238	302	32	hg	hg	NOUN
cana-5238	302	33	-	-	PUNCT
cana-5238	302	34	s.	s.	PROPN
cana-5238	302	35	(	(	PUNCT
cana-5238	302	36	3)⇒(1	3)⇒(1	PROPN
cana-5238	302	37	)	)	PUNCT
cana-5238	302	38	.	.	PUNCT
cana-5238	303	1	this	this	PRON
cana-5238	303	2	is	be	AUX
cana-5238	303	3	from	from	ADP
cana-5238	303	4	theorem	theorem	ADJ
cana-5238	303	5	3.9	3.9	NUM
cana-5238	303	6	.	.	PUNCT
cana-5238	303	7	remark	remark	NOUN
cana-5238	303	8	3.26	3.26	NUM
cana-5238	303	9	.	.	PUNCT
cana-5238	304	1	the	the	DET
cana-5238	304	2	notions	notion	NOUN
cana-5238	304	3	of	of	ADP
cana-5238	304	4	π	π	PROPN
cana-5238	304	5	-	-	PUNCT
cana-5238	304	6	hg	hg	NOUN
cana-5238	304	7	-	-	NOUN
cana-5238	304	8	o	o	NOUN
cana-5238	304	9	and	and	CCONJ
cana-5238	304	10	β∗-b	β∗-b	PROPN
cana-5238	304	11	-	-	PUNCT
cana-5238	304	12	hg	hg	NOUN
cana-5238	304	13	-	-	PUNCT
cana-5238	304	14	s	s	PART
cana-5238	304	15	are	be	AUX
cana-5238	304	16	independent	independent	ADJ
cana-5238	304	17	.	.	PUNCT
cana-5238	304	18	example	example	NOUN
cana-5238	305	1	3.27	3.27	NUM
cana-5238	305	2	.	.	PUNCT
cana-5238	306	1	assume	assume	VERB
cana-5238	306	2	z	z	NOUN
cana-5238	306	3	=	=	PUNCT
cana-5238	306	4	{	{	PUNCT
cana-5238	306	5	1	1	NUM
cana-5238	306	6	,	,	PUNCT
cana-5238	306	7	2	2	NUM
cana-5238	306	8	,	,	PUNCT
cana-5238	306	9	3	3	NUM
cana-5238	306	10	,	,	PUNCT
cana-5238	306	11	4	4	NUM
cana-5238	306	12	}	}	PUNCT
cana-5238	306	13	,	,	PUNCT
cana-5238	306	14	ζ={∅	ζ={∅	PROPN
cana-5238	306	15	,	,	PUNCT
cana-5238	306	16	{	{	PUNCT
cana-5238	306	17	1	1	NUM
cana-5238	306	18	,	,	PUNCT
cana-5238	306	19	3	3	NUM
cana-5238	306	20	}	}	PUNCT
cana-5238	306	21	,	,	PUNCT
cana-5238	306	22	{	{	PUNCT
cana-5238	306	23	2	2	NUM
cana-5238	306	24	,	,	PUNCT
cana-5238	306	25	3	3	NUM
cana-5238	306	26	}	}	PUNCT
cana-5238	306	27	,	,	PUNCT
cana-5238	306	28	{	{	PUNCT
cana-5238	306	29	1	1	NUM
cana-5238	306	30	,	,	PUNCT
cana-5238	306	31	2	2	NUM
cana-5238	306	32	,	,	PUNCT
cana-5238	306	33	3},{1	3},{1	NUM
cana-5238	306	34	,	,	PUNCT
cana-5238	306	35	4	4	NUM
cana-5238	306	36	}	}	PUNCT
cana-5238	306	37	,	,	PUNCT
cana-5238	306	38	{	{	PUNCT
cana-5238	306	39	1	1	NUM
cana-5238	306	40	,	,	PUNCT
cana-5238	306	41	3	3	NUM
cana-5238	306	42	,	,	PUNCT
cana-5238	306	43	4	4	NUM
cana-5238	306	44	}	}	PUNCT
cana-5238	306	45	,	,	PUNCT
cana-5238	306	46	z	z	NOUN
cana-5238	306	47	}	}	PUNCT
cana-5238	306	48	,	,	PUNCT
cana-5238	306	49	h={∅	h={∅	PROPN
cana-5238	306	50	,	,	PUNCT
cana-5238	306	51	{	{	PUNCT
cana-5238	306	52	1	1	NUM
cana-5238	306	53	}	}	PUNCT
cana-5238	306	54	,	,	PUNCT
cana-5238	306	55	{	{	PUNCT
cana-5238	306	56	2	2	NUM
cana-5238	306	57	}	}	PUNCT
cana-5238	306	58	}	}	PUNCT
cana-5238	306	59	.	.	PUNCT
cana-5238	307	1	then	then	ADV
cana-5238	307	2	ψ={1	ψ={1	PROPN
cana-5238	307	3	}	}	PUNCT
cana-5238	307	4	is	be	AUX
cana-5238	307	5	π	π	PROPN
cana-5238	307	6	-	-	PUNCT
cana-5238	307	7	hg	hg	NOUN
cana-5238	307	8	-	-	NOUN
cana-5238	307	9	o	o	NOUN
cana-5238	307	10	but	but	CCONJ
cana-5238	307	11	not	not	PART
cana-5238	307	12	β∗-b	β∗-b	PROPN
cana-5238	307	13	-	-	PUNCT
cana-5238	307	14	hg	hg	NOUN
cana-5238	307	15	-	-	PUNCT
cana-5238	307	16	s	s	NOUN
cana-5238	307	17	and	and	CCONJ
cana-5238	307	18	m={2	m={2	PROPN
cana-5238	307	19	}	}	PUNCT
cana-5238	307	20	is	be	AUX
cana-5238	307	21	β∗-b	β∗-b	PROPN
cana-5238	307	22	-	-	PUNCT
cana-5238	307	23	hg	hg	NOUN
cana-5238	307	24	-	-	PUNCT
cana-5238	307	25	s	s	X
cana-5238	307	26	but	but	CCONJ
cana-5238	307	27	not	not	PART
cana-5238	307	28	π	π	PROPN
cana-5238	307	29	-	-	PUNCT
cana-5238	307	30	hg	hg	ADJ
cana-5238	307	31	-	-	ADJ
cana-5238	307	32	o.	o.	ADJ
cana-5238	307	33	definition	definition	NOUN
cana-5238	307	34	3.28	3.28	NUM
cana-5238	307	35	.	.	PUNCT
cana-5238	308	1	a	a	DET
cana-5238	308	2	subset	subset	NOUN
cana-5238	308	3	ψ	ψ	NOUN
cana-5238	308	4	of	of	ADP
cana-5238	308	5	a	a	DET
cana-5238	308	6	hgts	hgts	NOUN
cana-5238	308	7	(	(	PUNCT
cana-5238	308	8	z	z	NOUN
cana-5238	308	9	,	,	PUNCT
cana-5238	308	10	ζ	ζ	NOUN
cana-5238	308	11	,	,	PUNCT
cana-5238	308	12	h	h	NOUN
cana-5238	308	13	)	)	PUNCT
cana-5238	308	14	is	be	AUX
cana-5238	308	15	called	call	VERB
cana-5238	308	16	1	1	NUM
cana-5238	308	17	.	.	PUNCT
cana-5238	309	1	ξ∗-hg	ξ∗-hg	NOUN
cana-5238	309	2	-	-	PUNCT
cana-5238	309	3	s	s	NOUN
cana-5238	309	4	,	,	PUNCT
cana-5238	309	5	igc∗ig(ψ)=ig(ψ	igc∗ig(ψ)=ig(ψ	NOUN
cana-5238	309	6	)	)	PUNCT
cana-5238	309	7	.	.	PUNCT
cana-5238	310	1	2	2	X
cana-5238	310	2	.	.	X
cana-5238	310	3	σ∗-hg	σ∗-hg	NOUN
cana-5238	310	4	-	-	PUNCT
cana-5238	310	5	s	s	NOUN
cana-5238	310	6	,	,	PUNCT
cana-5238	310	7	if	if	SCONJ
cana-5238	310	8	c∗ig(ψ)=ig(ψ	c∗ig(ψ)=ig(ψ	PROPN
cana-5238	310	9	)	)	PUNCT
cana-5238	310	10	.	.	PUNCT
cana-5238	311	1	3	3	X
cana-5238	311	2	.	.	X
cana-5238	311	3	π∗-hg	π∗-hg	NOUN
cana-5238	311	4	-	-	PUNCT
cana-5238	311	5	s	s	NOUN
cana-5238	311	6	,	,	PUNCT
cana-5238	311	7	if	if	SCONJ
cana-5238	311	8	igc∗(ψ)=ig(ψ	igc∗(ψ)=ig(ψ	NOUN
cana-5238	311	9	)	)	PUNCT
cana-5238	311	10	.	.	PUNCT
cana-5238	312	1	4	4	X
cana-5238	312	2	.	.	X
cana-5238	312	3	φ∗-hg	φ∗-hg	NOUN
cana-5238	312	4	-	-	PUNCT
cana-5238	312	5	s	s	NOUN
cana-5238	312	6	,	,	PUNCT
cana-5238	312	7	if	if	SCONJ
cana-5238	312	8	c∗ig(ψ)∪igc∗(ψ)=ig(ψ	c∗ig(ψ)∪igc∗(ψ)=ig(ψ	NOUN
cana-5238	312	9	)	)	PUNCT
cana-5238	312	10	.	.	PUNCT
cana-5238	313	1	communications	communication	NOUN
cana-5238	313	2	on	on	ADP
cana-5238	313	3	applied	apply	VERB
cana-5238	313	4	nonlinear	nonlinear	ADJ
cana-5238	313	5	analysis	analysis	NOUN
cana-5238	313	6	issn	issn	NOUN
cana-5238	313	7	:	:	PUNCT
cana-5238	313	8	1074	1074	NUM
cana-5238	313	9	-	-	PUNCT
cana-5238	313	10	133x	133x	NUM
cana-5238	313	11	vol	vol	VERB
cana-5238	313	12	32	32	NUM
cana-5238	313	13	no	no	NOUN
cana-5238	313	14	.	.	PUNCT
cana-5238	314	1	10s	10	NOUN
cana-5238	314	2	(	(	PUNCT
cana-5238	314	3	2025	2025	NUM
cana-5238	314	4	)	)	PUNCT
cana-5238	314	5	1365	1365	NUM
cana-5238	314	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	314	7	ζ	ζ	NOUN
cana-5238	314	8	ζ	ζ	NOUN
cana-5238	314	9	5	5	NUM
cana-5238	314	10	.	.	PUNCT
cana-5238	315	1	∆∗-hg	∆∗-hg	NOUN
cana-5238	315	2	-	-	PUNCT
cana-5238	315	3	s	s	NOUN
cana-5238	315	4	,	,	PUNCT
cana-5238	315	5	if	if	SCONJ
cana-5238	315	6	cigc∗(ψ)=ig(ψ	cigc∗(ψ)=ig(ψ	ADJ
cana-5238	315	7	)	)	PUNCT
cana-5238	315	8	.	.	PUNCT
cana-5238	316	1	definition	definition	NOUN
cana-5238	316	2	3.29	3.29	NUM
cana-5238	316	3	.	.	PUNCT
cana-5238	317	1	a	a	DET
cana-5238	317	2	subset	subset	NOUN
cana-5238	317	3	ψ	ψ	X
cana-5238	317	4	of	of	ADP
cana-5238	317	5	hgt	hgt	PROPN
cana-5238	317	6	s	s	PROPN
cana-5238	317	7	(	(	PUNCT
cana-5238	317	8	z	z	NOUN
cana-5238	317	9	,	,	PUNCT
cana-5238	317	10	ζ	ζ	NOUN
cana-5238	317	11	,	,	PUNCT
cana-5238	317	12	h	h	NOUN
cana-5238	317	13	)	)	PUNCT
cana-5238	317	14	is	be	AUX
cana-5238	317	15	called	call	VERB
cana-5238	317	16	1	1	NUM
cana-5238	317	17	.	.	PUNCT
cana-5238	318	1	ξ∗-b	ξ∗-b	NOUN
cana-5238	318	2	-	-	PUNCT
cana-5238	318	3	hg	hg	NOUN
cana-5238	318	4	-	-	PUNCT
cana-5238	318	5	s	s	X
cana-5238	318	6	,	,	PUNCT
cana-5238	318	7	if	if	SCONJ
cana-5238	318	8	ψ	ψ	VERB
cana-5238	318	9	=	=	NOUN
cana-5238	318	10	u∩v	u∩v	ADJ
cana-5238	318	11	,	,	PUNCT
cana-5238	318	12	where	where	SCONJ
cana-5238	318	13	u	u	NOUN
cana-5238	318	14	is	be	AUX
cana-5238	318	15	gζ	gζ	NOUN
cana-5238	318	16	-	-	PUNCT
cana-5238	318	17	o	o	NOUN
cana-5238	318	18	and	and	CCONJ
cana-5238	318	19	v	v	NOUN
cana-5238	318	20	is	be	AUX
cana-5238	318	21	ξ∗-hg	ξ∗-hg	NOUN
cana-5238	318	22	-	-	SYM
cana-5238	318	23	s	s	NOUN
cana-5238	318	24	2	2	NUM
cana-5238	318	25	.	.	PUNCT
cana-5238	318	26	σ∗-b	σ∗-b	PROPN
cana-5238	318	27	-	-	PUNCT
cana-5238	318	28	hg	hg	NOUN
cana-5238	318	29	-	-	PUNCT
cana-5238	318	30	s	s	X
cana-5238	318	31	,	,	PUNCT
cana-5238	318	32	if	if	SCONJ
cana-5238	318	33	ψ	ψ	VERB
cana-5238	318	34	=	=	NOUN
cana-5238	318	35	u∩v	u∩v	ADJ
cana-5238	318	36	,	,	PUNCT
cana-5238	318	37	where	where	SCONJ
cana-5238	318	38	u	u	NOUN
cana-5238	318	39	is	be	AUX
cana-5238	318	40	gζ	gζ	NOUN
cana-5238	318	41	-	-	PUNCT
cana-5238	318	42	o	o	NOUN
cana-5238	318	43	and	and	CCONJ
cana-5238	318	44	v	v	NOUN
cana-5238	318	45	is	be	AUX
cana-5238	318	46	σ∗-hg	σ∗-hg	NOUN
cana-5238	318	47	-	-	PUNCT
cana-5238	318	48	s.	s.	PROPN
cana-5238	318	49	3	3	NUM
cana-5238	318	50	.	.	PUNCT
cana-5238	319	1	π∗-b	π∗-b	PROPN
cana-5238	319	2	-	-	PUNCT
cana-5238	319	3	hg	hg	NOUN
cana-5238	319	4	-	-	PUNCT
cana-5238	319	5	s	s	X
cana-5238	319	6	,	,	PUNCT
cana-5238	319	7	if	if	SCONJ
cana-5238	319	8	ψ	ψ	VERB
cana-5238	319	9	=	=	NOUN
cana-5238	319	10	u∩v	u∩v	ADJ
cana-5238	319	11	,	,	PUNCT
cana-5238	319	12	where	where	SCONJ
cana-5238	319	13	u	u	NOUN
cana-5238	319	14	is	be	AUX
cana-5238	319	15	gζ	gζ	NOUN
cana-5238	319	16	-	-	PUNCT
cana-5238	319	17	o	o	NOUN
cana-5238	319	18	and	and	CCONJ
cana-5238	319	19	v	v	NOUN
cana-5238	319	20	is	be	AUX
cana-5238	319	21	π∗-hg	π∗-hg	NOUN
cana-5238	319	22	-	-	PUNCT
cana-5238	319	23	s.	s.	PROPN
cana-5238	319	24	4	4	NUM
cana-5238	319	25	.	.	PUNCT
cana-5238	320	1	φ∗-b	φ∗-b	VERB
cana-5238	320	2	-	-	PUNCT
cana-5238	320	3	hg	hg	NOUN
cana-5238	320	4	-	-	PUNCT
cana-5238	320	5	s	s	X
cana-5238	320	6	,	,	PUNCT
cana-5238	320	7	if	if	SCONJ
cana-5238	320	8	ψ	ψ	VERB
cana-5238	320	9	=	=	NOUN
cana-5238	320	10	u∩v	u∩v	ADJ
cana-5238	320	11	,	,	PUNCT
cana-5238	320	12	where	where	SCONJ
cana-5238	320	13	u	u	NOUN
cana-5238	320	14	is	be	AUX
cana-5238	320	15	gζ	gζ	NOUN
cana-5238	320	16	-	-	PUNCT
cana-5238	320	17	o	o	NOUN
cana-5238	320	18	and	and	CCONJ
cana-5238	320	19	v	v	NOUN
cana-5238	320	20	is	be	AUX
cana-5238	320	21	φ∗-hg	φ∗-hg	NOUN
cana-5238	320	22	-	-	PUNCT
cana-5238	320	23	s	s	NOUN
cana-5238	320	24	5	5	NUM
cana-5238	320	25	.	.	PUNCT
cana-5238	320	26	∆∗-b	∆∗-b	PROPN
cana-5238	320	27	-	-	PUNCT
cana-5238	320	28	hg	hg	NOUN
cana-5238	320	29	-	-	PUNCT
cana-5238	320	30	s	s	X
cana-5238	320	31	,	,	PUNCT
cana-5238	320	32	if	if	SCONJ
cana-5238	320	33	ψ	ψ	VERB
cana-5238	320	34	=	=	NOUN
cana-5238	320	35	u∩v	u∩v	ADJ
cana-5238	320	36	,	,	PUNCT
cana-5238	320	37	where	where	SCONJ
cana-5238	320	38	u	u	NOUN
cana-5238	320	39	is	be	AUX
cana-5238	320	40	gζ	gζ	NOUN
cana-5238	320	41	-	-	PUNCT
cana-5238	320	42	o	o	NOUN
cana-5238	320	43	and	and	CCONJ
cana-5238	320	44	v	v	NOUN
cana-5238	320	45	is	be	AUX
cana-5238	320	46	∆∗-hg	∆∗-hg	NOUN
cana-5238	320	47	-	-	PUNCT
cana-5238	320	48	s	s	NOUN
cana-5238	320	49	theorem	theorem	NOUN
cana-5238	320	50	3.30	3.30	NUM
cana-5238	320	51	.	.	PUNCT
cana-5238	321	1	if	if	SCONJ
cana-5238	321	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	321	3	is	be	AUX
cana-5238	321	4	both	both	PRON
cana-5238	321	5	π	π	PROPN
cana-5238	321	6	-	-	PUNCT
cana-5238	321	7	hg	hg	NOUN
cana-5238	321	8	-	-	NOUN
cana-5238	321	9	o	o	NOUN
cana-5238	321	10	and	and	CCONJ
cana-5238	321	11	ζ∗-closed	ζ∗-close	VERB
cana-5238	321	12	,	,	PUNCT
cana-5238	321	13	then	then	ADV
cana-5238	321	14	it	it	PRON
cana-5238	321	15	is	be	AUX
cana-5238	321	16	π∗-hg	π∗-hg	NOUN
cana-5238	321	17	-	-	PUNCT
cana-5238	321	18	s.	s.	PROPN
cana-5238	321	19	proof	proof	NOUN
cana-5238	321	20	.	.	PUNCT
cana-5238	322	1	let	let	VERB
cana-5238	322	2	ψ	ψ	NOUN
cana-5238	322	3	is	be	AUX
cana-5238	322	4	both	both	PRON
cana-5238	322	5	π	π	PROPN
cana-5238	322	6	-	-	PUNCT
cana-5238	322	7	hg	hg	NOUN
cana-5238	322	8	-	-	NOUN
cana-5238	322	9	o	o	NOUN
cana-5238	322	10	and	and	CCONJ
cana-5238	322	11	ζ∗-closed	ζ∗-close	VERB
cana-5238	322	12	.	.	PUNCT
cana-5238	323	1	then	then	ADV
cana-5238	323	2	ψ⊆igc∗(ψ	ψ⊆igc∗(ψ	NUM
cana-5238	323	3	)	)	PUNCT
cana-5238	323	4	and	and	CCONJ
cana-5238	323	5	c∗(ψ)⊂ψ	c∗(ψ)⊂ψ	NOUN
cana-5238	323	6	.	.	PUNCT
cana-5238	324	1	now	now	ADV
cana-5238	324	2	igc∗(ψ)⊂	igc∗(ψ)⊂	PROPN
cana-5238	324	3	c∗(ψ)⊂ψ	c∗(ψ)⊂ψ	NOUN
cana-5238	324	4	.	.	PUNCT
cana-5238	325	1	so	so	ADV
cana-5238	325	2	,	,	PUNCT
cana-5238	325	3	ψ	ψ	NOUN
cana-5238	325	4	=	=	NOUN
cana-5238	325	5	igc∗(ψ	igc∗(ψ	NUM
cana-5238	325	6	)	)	PUNCT
cana-5238	325	7	.	.	PUNCT
cana-5238	326	1	thus	thus	ADV
cana-5238	326	2	ig(ψ	ig(ψ	NOUN
cana-5238	326	3	)	)	PUNCT
cana-5238	326	4	=	=	PUNCT
cana-5238	326	5	igc∗(ψ	igc∗(ψ	NUM
cana-5238	326	6	)	)	PUNCT
cana-5238	326	7	.	.	PUNCT
cana-5238	327	1	hence	hence	ADV
cana-5238	327	2	ψ	ψ	NOUN
cana-5238	327	3	is	be	AUX
cana-5238	327	4	π∗-hg	π∗-hg	NOUN
cana-5238	327	5	-	-	PUNCT
cana-5238	327	6	s.	s.	PROPN
cana-5238	327	7	theorem	theorem	VERB
cana-5238	327	8	3.31	3.31	NUM
cana-5238	327	9	.	.	PUNCT
cana-5238	328	1	if	if	SCONJ
cana-5238	328	2	ψ⊂z	ψ⊂z	NOUN
cana-5238	328	3	is	be	AUX
cana-5238	328	4	both	both	DET
cana-5238	328	5	σ	σ	PROPN
cana-5238	328	6	-	-	PUNCT
cana-5238	328	7	hg	hg	NOUN
cana-5238	328	8	-	-	NOUN
cana-5238	328	9	o	o	NOUN
cana-5238	328	10	and	and	CCONJ
cana-5238	328	11	ζ∗-closed	ζ∗-close	VERB
cana-5238	328	12	,	,	PUNCT
cana-5238	328	13	then	then	ADV
cana-5238	328	14	it	it	PRON
cana-5238	328	15	is	be	AUX
cana-5238	328	16	ξ∗-hg	ξ∗-hg	NOUN
cana-5238	328	17	-	-	PUNCT
cana-5238	328	18	s	s	PART
cana-5238	328	19	proof	proof	NOUN
cana-5238	328	20	.	.	PUNCT
cana-5238	329	1	let	let	VERB
cana-5238	329	2	ψ	ψ	NOUN
cana-5238	329	3	is	be	AUX
cana-5238	329	4	both	both	DET
cana-5238	329	5	σ	σ	PROPN
cana-5238	329	6	-	-	PUNCT
cana-5238	329	7	hg	hg	NOUN
cana-5238	329	8	-	-	NOUN
cana-5238	329	9	o	o	NOUN
cana-5238	329	10	and	and	CCONJ
cana-5238	329	11	ζ∗-closed	ζ∗-close	VERB
cana-5238	329	12	.	.	PUNCT
cana-5238	330	1	then	then	ADV
cana-5238	330	2	ψ⊆c∗ig(ψ	ψ⊆c∗ig(ψ	PROPN
cana-5238	330	3	)	)	PUNCT
cana-5238	330	4	and	and	CCONJ
cana-5238	330	5	c∗(ψ)⊂ψ	c∗(ψ)⊂ψ	NOUN
cana-5238	330	6	.	.	PUNCT
cana-5238	331	1	now	now	ADV
cana-5238	331	2	c∗ig(ψ)⊂c∗(ψ)⊂ψ	c∗ig(ψ)⊂c∗(ψ)⊂ψ	PROPN
cana-5238	331	3	.	.	PUNCT
cana-5238	332	1	so	so	ADV
cana-5238	332	2	,	,	PUNCT
cana-5238	332	3	ψ	ψ	NOUN
cana-5238	332	4	=	=	NOUN
cana-5238	332	5	c∗ig(ψ	c∗ig(ψ	ADJ
cana-5238	332	6	)	)	PUNCT
cana-5238	332	7	.	.	PUNCT
cana-5238	333	1	thus	thus	ADV
cana-5238	333	2	ig(ψ	ig(ψ	NOUN
cana-5238	333	3	)	)	PUNCT
cana-5238	333	4	=	=	SYM
cana-5238	333	5	igc∗ig(ψ	igc∗ig(ψ	PROPN
cana-5238	333	6	)	)	PUNCT
cana-5238	333	7	.	.	PUNCT
cana-5238	334	1	hence	hence	ADV
cana-5238	334	2	ψ	ψ	NOUN
cana-5238	334	3	is	be	AUX
cana-5238	334	4	ξ∗-hg	ξ∗-hg	NOUN
cana-5238	334	5	-	-	PUNCT
cana-5238	334	6	s	s	NOUN
cana-5238	334	7	theorem	theorem	NOUN
cana-5238	334	8	3.32	3.32	NUM
cana-5238	334	9	.	.	PUNCT
cana-5238	335	1	let	let	AUX
cana-5238	335	2	(	(	PUNCT
cana-5238	335	3	z	z	NOUN
cana-5238	335	4	,	,	PUNCT
cana-5238	335	5	ζ	ζ	NOUN
cana-5238	335	6	,	,	PUNCT
cana-5238	335	7	h	h	NOUN
cana-5238	335	8	)	)	PUNCT
cana-5238	335	9	be	be	VERB
cana-5238	335	10	a	a	DET
cana-5238	335	11	strong	strong	ADJ
cana-5238	335	12	hgts	hgts	NOUN
cana-5238	335	13	where	where	SCONJ
cana-5238	335	14	z	z	NOUN
cana-5238	335	15	is	be	AUX
cana-5238	335	16	c0	c0	PROPN
cana-5238	335	17	-space	-space	PROPN
cana-5238	335	18	and	and	CCONJ
cana-5238	335	19	l⊂z	l⊂z	PROPN
cana-5238	335	20	.	.	PUNCT
cana-5238	336	1	then	then	ADV
cana-5238	336	2	the	the	DET
cana-5238	336	3	following	follow	VERB
cana-5238	336	4	conditions	condition	NOUN
cana-5238	336	5	are	be	AUX
cana-5238	336	6	equivalent	equivalent	ADJ
cana-5238	336	7	.	.	PUNCT
cana-5238	337	1	1	1	X
cana-5238	337	2	.	.	X
cana-5238	337	3	ψ	ψ	NOUN
cana-5238	337	4	is	be	AUX
cana-5238	337	5	gζ	gζ	NOUN
cana-5238	337	6	-	-	PUNCT
cana-5238	337	7	o	o	NOUN
cana-5238	337	8	,	,	PUNCT
cana-5238	337	9	2	2	NUM
cana-5238	337	10	.	.	PUNCT
cana-5238	337	11	ψ	ψ	NOUN
cana-5238	337	12	is	be	AUX
cana-5238	337	13	α	α	X
cana-5238	337	14	-	-	PUNCT
cana-5238	337	15	hg	hg	NOUN
cana-5238	337	16	-	-	NOUN
cana-5238	337	17	o	o	NOUN
cana-5238	337	18	and	and	CCONJ
cana-5238	337	19	ξ∗-b	ξ∗-b	PROPN
cana-5238	337	20	-	-	PUNCT
cana-5238	337	21	hg	hg	NOUN
cana-5238	337	22	-	-	PUNCT
cana-5238	337	23	s	s	NOUN
cana-5238	337	24	,	,	PUNCT
cana-5238	337	25	3	3	NUM
cana-5238	337	26	.	.	PUNCT
cana-5238	337	27	ψ	ψ	NOUN
cana-5238	337	28	is	be	AUX
cana-5238	337	29	σ	σ	PROPN
cana-5238	337	30	-	-	PUNCT
cana-5238	337	31	hg	hg	NOUN
cana-5238	337	32	-	-	NOUN
cana-5238	337	33	o	o	NOUN
cana-5238	337	34	and	and	CCONJ
cana-5238	337	35	σ∗-b	σ∗-b	PROPN
cana-5238	337	36	-	-	PUNCT
cana-5238	337	37	hg	hg	NOUN
cana-5238	337	38	-	-	PUNCT
cana-5238	337	39	s	s	NOUN
cana-5238	337	40	,	,	PUNCT
cana-5238	337	41	4	4	NUM
cana-5238	337	42	.	.	PUNCT
cana-5238	337	43	ψ	ψ	NOUN
cana-5238	337	44	is	be	AUX
cana-5238	337	45	π	π	PROPN
cana-5238	337	46	-	-	PUNCT
cana-5238	337	47	hg	hg	NOUN
cana-5238	337	48	-	-	NOUN
cana-5238	337	49	o	o	NOUN
cana-5238	337	50	and	and	CCONJ
cana-5238	337	51	π∗-b	π∗-b	PROPN
cana-5238	337	52	-	-	PUNCT
cana-5238	337	53	hg	hg	NOUN
cana-5238	337	54	-	-	PUNCT
cana-5238	337	55	s.	s.	PROPN
cana-5238	337	56	5	5	NUM
cana-5238	337	57	.	.	PUNCT
cana-5238	338	1	ψ	ψ	NOUN
cana-5238	338	2	is	be	AUX
cana-5238	338	3	β	β	X
cana-5238	338	4	-	-	ADJ
cana-5238	338	5	hg	hg	NOUN
cana-5238	338	6	-	-	NOUN
cana-5238	338	7	o	o	NOUN
cana-5238	338	8	and	and	CCONJ
cana-5238	338	9	∆∗-b	∆∗-b	PROPN
cana-5238	338	10	-	-	PUNCT
cana-5238	338	11	hg	hg	NOUN
cana-5238	338	12	-	-	PUNCT
cana-5238	338	13	s	s	PART
cana-5238	338	14	proof	proof	NOUN
cana-5238	338	15	.	.	PUNCT
cana-5238	339	1	(	(	PUNCT
cana-5238	339	2	1	1	X
cana-5238	339	3	)	)	PUNCT
cana-5238	339	4	⇒	⇒	NOUN
cana-5238	339	5	(	(	PUNCT
cana-5238	339	6	2	2	NUM
cana-5238	339	7	)	)	PUNCT
cana-5238	339	8	,	,	PUNCT
cana-5238	339	9	(	(	PUNCT
cana-5238	339	10	1	1	X
cana-5238	339	11	)	)	PUNCT
cana-5238	339	12	⇒	⇒	NOUN
cana-5238	339	13	(	(	PUNCT
cana-5238	339	14	3	3	NUM
cana-5238	339	15	)	)	PUNCT
cana-5238	339	16	,	,	PUNCT
cana-5238	339	17	(	(	PUNCT
cana-5238	339	18	1	1	X
cana-5238	339	19	)	)	PUNCT
cana-5238	339	20	⇒	⇒	NOUN
cana-5238	339	21	(	(	PUNCT
cana-5238	339	22	4	4	NUM
cana-5238	339	23	)	)	PUNCT
cana-5238	339	24	,	,	PUNCT
cana-5238	339	25	are	be	AUX
cana-5238	339	26	obvious	obvious	ADJ
cana-5238	339	27	.	.	PUNCT
cana-5238	340	1	(	(	PUNCT
cana-5238	340	2	2)⇒	2)⇒	NUM
cana-5238	340	3	(	(	PUNCT
cana-5238	340	4	1	1	NUM
cana-5238	340	5	)	)	PUNCT
cana-5238	340	6	.	.	PUNCT
cana-5238	341	1	let	let	VERB
cana-5238	341	2	ψ	ψ	NOUN
cana-5238	341	3	is	be	AUX
cana-5238	341	4	both	both	DET
cana-5238	341	5	α	α	PROPN
cana-5238	341	6	-	-	PUNCT
cana-5238	341	7	hg	hg	NOUN
cana-5238	341	8	-	-	NOUN
cana-5238	341	9	o	o	NOUN
cana-5238	341	10	and	and	CCONJ
cana-5238	341	11	ξ∗-b	ξ∗-b	PROPN
cana-5238	341	12	-	-	PUNCT
cana-5238	341	13	hg	hg	NOUN
cana-5238	341	14	-	-	PUNCT
cana-5238	341	15	s	s	X
cana-5238	341	16	then	then	ADV
cana-5238	341	17	ψ⊆igc∗ig(ψ)=igc∗ig(u∩v	ψ⊆igc∗ig(ψ)=igc∗ig(u∩v	NOUN
cana-5238	341	18	)	)	PUNCT
cana-5238	341	19	,	,	PUNCT
cana-5238	341	20	where	where	SCONJ
cana-5238	341	21	u	u	NOUN
cana-5238	341	22	is	be	AUX
cana-5238	341	23	gζ	gζ	NOUN
cana-5238	341	24	-	-	PUNCT
cana-5238	341	25	o	o	NOUN
cana-5238	341	26	and	and	CCONJ
cana-5238	341	27	v	v	NOUN
cana-5238	341	28	is	be	AUX
cana-5238	341	29	ξ∗-hg	ξ∗-hg	NOUN
cana-5238	341	30	-	-	NOUN
cana-5238	341	31	s	s	PART
cana-5238	341	32	hence	hence	ADV
cana-5238	341	33	ψ⊆igc∗ig(u)∩igc∗ig(v).now	ψ⊆igc∗ig(u)∩igc∗ig(v).now	PROPN
cana-5238	341	34	ψ⊆u∩ψ⊆u∩[igc∗ig(u)∩ig(v	ψ⊆u∩ψ⊆u∩[igc∗ig(u)∩ig(v	PRON
cana-5238	341	35	)	)	PUNCT
cana-5238	341	36	]	]	PUNCT
cana-5238	342	1	=	=	PUNCT
cana-5238	342	2	ig(u)∩ig(v)=ig(ψ	ig(u)∩ig(v)=ig(ψ	PROPN
cana-5238	342	3	)	)	PUNCT
cana-5238	342	4	.	.	PUNCT
cana-5238	343	1	hence	hence	ADV
cana-5238	343	2	ψ	ψ	X
cana-5238	343	3	is	be	AUX
cana-5238	343	4	gζ	gζ	NOUN
cana-5238	343	5	-	-	PUNCT
cana-5238	343	6	o.	o.	NOUN
cana-5238	343	7	(	(	PUNCT
cana-5238	343	8	3	3	NUM
cana-5238	343	9	)	)	PUNCT
cana-5238	343	10	⇒	⇒	NOUN
cana-5238	343	11	(	(	PUNCT
cana-5238	343	12	1	1	NUM
cana-5238	343	13	)	)	PUNCT
cana-5238	343	14	.	.	PUNCT
cana-5238	344	1	let	let	VERB
cana-5238	344	2	ψ	ψ	NOUN
cana-5238	344	3	is	be	AUX
cana-5238	344	4	both	both	DET
cana-5238	344	5	σ	σ	PROPN
cana-5238	344	6	-	-	PUNCT
cana-5238	344	7	hg	hg	NOUN
cana-5238	344	8	-	-	NOUN
cana-5238	344	9	o	o	NOUN
cana-5238	344	10	and	and	CCONJ
cana-5238	344	11	σ∗-b	σ∗-b	PROPN
cana-5238	344	12	-	-	PUNCT
cana-5238	344	13	hg	hg	NOUN
cana-5238	344	14	-	-	PUNCT
cana-5238	344	15	s.	s.	PROPN
cana-5238	344	16	then	then	ADV
cana-5238	344	17	ψ⊆c∗ig(ψ)=c∗ig(u∩v	ψ⊆c∗ig(ψ)=c∗ig(u∩v	NUM
cana-5238	344	18	)	)	PUNCT
cana-5238	344	19	,	,	PUNCT
cana-5238	344	20	where	where	SCONJ
cana-5238	344	21	u	u	NOUN
cana-5238	344	22	is	be	AUX
cana-5238	344	23	gζ	gζ	NOUN
cana-5238	344	24	-	-	PUNCT
cana-5238	344	25	o	o	NOUN
cana-5238	344	26	and	and	CCONJ
cana-5238	344	27	v	v	NOUN
cana-5238	344	28	is	be	AUX
cana-5238	344	29	σ∗-hg	σ∗-hg	NOUN
cana-5238	344	30	-	-	PUNCT
cana-5238	344	31	s.	s.	PROPN
cana-5238	344	32	hence	hence	ADV
cana-5238	344	33	ψ⊆c∗ig(u)∩c∗ig(v	ψ⊆c∗ig(u)∩c∗ig(v	NOUN
cana-5238	344	34	)	)	PUNCT
cana-5238	344	35	.	.	PUNCT
cana-5238	345	1	now	now	ADV
cana-5238	345	2	ψ⊆u∩ψ⊆u∩[c∗ig(u)∩ig(v	ψ⊆u∩ψ⊆u∩[c∗ig(u)∩ig(v	ADV
cana-5238	345	3	)	)	PUNCT
cana-5238	345	4	]	]	PUNCT
cana-5238	346	1	=	=	PUNCT
cana-5238	346	2	ig(u)∩ig(v)=ig(ψ	ig(u)∩ig(v)=ig(ψ	PROPN
cana-5238	346	3	)	)	PUNCT
cana-5238	346	4	.	.	PUNCT
cana-5238	347	1	hence	hence	ADV
cana-5238	347	2	ψ	ψ	X
cana-5238	347	3	is	be	AUX
cana-5238	347	4	gζ	gζ	NOUN
cana-5238	347	5	-	-	PUNCT
cana-5238	347	6	o.	o.	NOUN
cana-5238	347	7	(	(	PUNCT
cana-5238	347	8	4)⇒	4)⇒	X
cana-5238	347	9	(	(	PUNCT
cana-5238	347	10	1	1	NUM
cana-5238	347	11	)	)	PUNCT
cana-5238	347	12	.	.	PUNCT
cana-5238	348	1	let	let	VERB
cana-5238	348	2	ψ	ψ	NOUN
cana-5238	348	3	is	be	AUX
cana-5238	348	4	both	both	PRON
cana-5238	348	5	π	π	PROPN
cana-5238	348	6	-	-	PUNCT
cana-5238	348	7	hg	hg	NOUN
cana-5238	348	8	-	-	NOUN
cana-5238	348	9	o	o	NOUN
cana-5238	348	10	and	and	CCONJ
cana-5238	348	11	π∗-b	π∗-b	PROPN
cana-5238	348	12	-	-	PUNCT
cana-5238	348	13	hg	hg	NOUN
cana-5238	348	14	-	-	PUNCT
cana-5238	348	15	s.	s.	PROPN
cana-5238	348	16	then	then	ADV
cana-5238	348	17	ψ⊆igc∗(ψ	ψ⊆igc∗(ψ	PRON
cana-5238	348	18	)	)	PUNCT
cana-5238	348	19	=	=	SYM
cana-5238	348	20	igc∗(u∩v	igc∗(u∩v	NOUN
cana-5238	348	21	)	)	PUNCT
cana-5238	348	22	,	,	PUNCT
cana-5238	348	23	where	where	SCONJ
cana-5238	348	24	u	u	NOUN
cana-5238	348	25	is	be	AUX
cana-5238	348	26	gζ	gζ	NOUN
cana-5238	348	27	-	-	PUNCT
cana-5238	348	28	o	o	NOUN
cana-5238	348	29	and	and	CCONJ
cana-5238	348	30	v	v	NOUN
cana-5238	348	31	is	be	AUX
cana-5238	348	32	π∗-hg	π∗-hg	NOUN
cana-5238	348	33	-	-	PUNCT
cana-5238	348	34	s.	s.	PROPN
cana-5238	348	35	hence	hence	ADV
cana-5238	348	36	ψ⊆igc∗(u)∩igc∗(v	ψ⊆igc∗(u)∩igc∗(v	PRON
cana-5238	348	37	)	)	PUNCT
cana-5238	348	38	.	.	PUNCT
cana-5238	349	1	now	now	ADV
cana-5238	349	2	ψ⊆u∩ψ	ψ⊆u∩ψ	NOUN
cana-5238	349	3	⊆u∩[igc∗(u)∩ig(v	⊆u∩[igc∗(u)∩ig(v	NOUN
cana-5238	349	4	)	)	PUNCT
cana-5238	349	5	]	]	PUNCT
cana-5238	349	6	=	=	PUNCT
cana-5238	349	7	ig(u)∩ig(v)=ig(ψ	ig(u)∩ig(v)=ig(ψ	PROPN
cana-5238	349	8	)	)	PUNCT
cana-5238	349	9	.	.	PUNCT
cana-5238	350	1	hence	hence	ADV
cana-5238	350	2	ψ	ψ	X
cana-5238	350	3	is	be	AUX
cana-5238	350	4	gζ	gζ	NOUN
cana-5238	350	5	-	-	PUNCT
cana-5238	350	6	o.	o.	NOUN
cana-5238	350	7	(	(	PUNCT
cana-5238	350	8	5)⇒	5)⇒	NUM
cana-5238	350	9	(	(	PUNCT
cana-5238	350	10	1	1	NUM
cana-5238	350	11	)	)	PUNCT
cana-5238	350	12	.	.	PUNCT
cana-5238	351	1	let	let	VERB
cana-5238	351	2	ψ	ψ	NOUN
cana-5238	351	3	is	be	AUX
cana-5238	351	4	both	both	PRON
cana-5238	351	5	β	β	NOUN
cana-5238	351	6	-	-	PUNCT
cana-5238	351	7	hg	hg	NOUN
cana-5238	351	8	-	-	NOUN
cana-5238	351	9	o	o	NOUN
cana-5238	351	10	and	and	CCONJ
cana-5238	351	11	∆∗-b	∆∗-b	PROPN
cana-5238	351	12	-	-	PUNCT
cana-5238	351	13	hg	hg	NOUN
cana-5238	351	14	-	-	PUNCT
cana-5238	351	15	s	s	X
cana-5238	351	16	then	then	ADV
cana-5238	351	17	ψ⊆cigc∗(ψ	ψ⊆cigc∗(ψ	NOUN
cana-5238	351	18	)	)	PUNCT
cana-5238	352	1	=	=	NOUN
cana-5238	352	2	cigc∗(u∩v	cigc∗(u∩v	NOUN
cana-5238	352	3	)	)	PUNCT
cana-5238	352	4	,	,	PUNCT
cana-5238	352	5	where	where	SCONJ
cana-5238	352	6	u	u	NOUN
cana-5238	352	7	is	be	AUX
cana-5238	352	8	gζ	gζ	NOUN
cana-5238	352	9	-	-	PUNCT
cana-5238	352	10	o	o	NOUN
cana-5238	352	11	and	and	CCONJ
cana-5238	352	12	v	v	NOUN
cana-5238	352	13	is	be	AUX
cana-5238	352	14	∆∗-hg	∆∗-hg	NOUN
cana-5238	352	15	-	-	PUNCT
cana-5238	352	16	s	s	NOUN
cana-5238	352	17	hence	hence	NOUN
cana-5238	352	18	ψ	ψ	ADP
cana-5238	352	19	⊆	⊆	NUM
cana-5238	352	20	cigc∗(u)∩cigc∗(v	cigc∗(u)∩cigc∗(v	NOUN
cana-5238	352	21	)	)	PUNCT
cana-5238	352	22	.	.	PUNCT
cana-5238	353	1	now	now	ADV
cana-5238	353	2	ψ⊆u∩ψ⊆u∩[cigc∗(u)∩ig(v)]=ig(u)∩ig(v)=ig(ψ	ψ⊆u∩ψ⊆u∩[cigc∗(u)∩ig(v)]=ig(u)∩ig(v)=ig(ψ	PROPN
cana-5238	353	3	)	)	PUNCT
cana-5238	353	4	.	.	PUNCT
cana-5238	354	1	hence	hence	ADV
cana-5238	354	2	ψ	ψ	X
cana-5238	354	3	is	be	AUX
cana-5238	354	4	gζ	gζ	NOUN
cana-5238	354	5	-	-	PUNCT
cana-5238	354	6	o.	o.	ADJ
cana-5238	354	7	remark	remark	NOUN
cana-5238	354	8	3.33	3.33	NUM
cana-5238	354	9	.	.	PUNCT
cana-5238	355	1	the	the	DET
cana-5238	355	2	notions	notion	NOUN
cana-5238	355	3	of	of	ADP
cana-5238	355	4	α	α	PROPN
cana-5238	355	5	-	-	PUNCT
cana-5238	355	6	hg	hg	NOUN
cana-5238	355	7	-	-	NOUN
cana-5238	355	8	o	o	X
cana-5238	355	9	(	(	PUNCT
cana-5238	355	10	resp	resp	NOUN
cana-5238	355	11	.	.	PUNCT
cana-5238	356	1	σ	σ	PROPN
cana-5238	356	2	-	-	PUNCT
cana-5238	356	3	hg	hg	NOUN
cana-5238	356	4	-	-	PROPN
cana-5238	356	5	o	o	NOUN
cana-5238	356	6	,	,	PUNCT
cana-5238	356	7	π	π	PROPN
cana-5238	356	8	-	-	PUNCT
cana-5238	356	9	hg	hg	NOUN
cana-5238	356	10	-	-	NOUN
cana-5238	356	11	o	o	NOUN
cana-5238	356	12	,	,	PUNCT
cana-5238	356	13	β	β	X
cana-5238	356	14	-	-	ADJ
cana-5238	356	15	hg	hg	NOUN
cana-5238	356	16	-	-	NOUN
cana-5238	356	17	o	o	NOUN
cana-5238	356	18	)	)	PUNCT
cana-5238	356	19	and	and	CCONJ
cana-5238	356	20	ξ∗-b	ξ∗-b	PROPN
cana-5238	356	21	-	-	PUNCT
cana-5238	356	22	hg	hg	NOUN
cana-5238	356	23	-	-	PUNCT
cana-5238	356	24	s	s	X
cana-5238	356	25	(	(	PUNCT
cana-5238	356	26	resp	resp	NOUN
cana-5238	356	27	.	.	PUNCT
cana-5238	356	28	σ∗-bhg	σ∗-bhg	PROPN
cana-5238	356	29	-	-	PUNCT
cana-5238	356	30	s	s	PROPN
cana-5238	356	31	,	,	PUNCT
cana-5238	356	32	π∗-b	π∗-b	PROPN
cana-5238	356	33	-	-	PUNCT
cana-5238	356	34	hg	hg	NOUN
cana-5238	356	35	-	-	PUNCT
cana-5238	356	36	s	s	NOUN
cana-5238	356	37	,	,	PUNCT
cana-5238	356	38	∆∗-b	∆∗-b	PROPN
cana-5238	356	39	-	-	PUNCT
cana-5238	356	40	hg	hg	NOUN
cana-5238	356	41	-	-	PUNCT
cana-5238	356	42	s	s	NOUN
cana-5238	356	43	)	)	PUNCT
cana-5238	356	44	are	be	AUX
cana-5238	356	45	independent	independent	ADJ
cana-5238	356	46	.	.	PUNCT
cana-5238	357	1	communications	communication	NOUN
cana-5238	357	2	on	on	ADP
cana-5238	357	3	applied	apply	VERB
cana-5238	357	4	nonlinear	nonlinear	ADJ
cana-5238	357	5	analysis	analysis	NOUN
cana-5238	357	6	issn	issn	NOUN
cana-5238	357	7	:	:	PUNCT
cana-5238	357	8	1074	1074	NUM
cana-5238	357	9	-	-	PUNCT
cana-5238	357	10	133x	133x	NUM
cana-5238	357	11	vol	vol	VERB
cana-5238	357	12	32	32	NUM
cana-5238	357	13	no	no	NOUN
cana-5238	357	14	.	.	PUNCT
cana-5238	358	1	10s	10	NOUN
cana-5238	358	2	(	(	PUNCT
cana-5238	358	3	2025	2025	NUM
cana-5238	358	4	)	)	PUNCT
cana-5238	358	5	1366	1366	NUM
cana-5238	358	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	358	7	example	example	NOUN
cana-5238	358	8	3.34	3.34	NUM
cana-5238	358	9	.	.	PUNCT
cana-5238	358	10	assume	assume	VERB
cana-5238	358	11	z={1	z={1	PROPN
cana-5238	358	12	,	,	PUNCT
cana-5238	358	13	2	2	NUM
cana-5238	358	14	,	,	PUNCT
cana-5238	358	15	3	3	NUM
cana-5238	358	16	,	,	PUNCT
cana-5238	358	17	4	4	NUM
cana-5238	358	18	}	}	PUNCT
cana-5238	358	19	,	,	PUNCT
cana-5238	358	20	ζ={∅	ζ={∅	PROPN
cana-5238	358	21	,	,	PUNCT
cana-5238	358	22	{	{	PUNCT
cana-5238	358	23	1	1	NUM
cana-5238	358	24	,	,	PUNCT
cana-5238	358	25	3	3	NUM
cana-5238	358	26	}	}	PUNCT
cana-5238	358	27	,	,	PUNCT
cana-5238	358	28	{	{	PUNCT
cana-5238	358	29	2	2	NUM
cana-5238	358	30	,	,	PUNCT
cana-5238	358	31	3	3	NUM
cana-5238	358	32	}	}	PUNCT
cana-5238	358	33	,	,	PUNCT
cana-5238	358	34	{	{	PUNCT
cana-5238	358	35	1	1	NUM
cana-5238	358	36	,	,	PUNCT
cana-5238	358	37	2	2	NUM
cana-5238	358	38	,	,	PUNCT
cana-5238	358	39	3},{1	3},{1	NUM
cana-5238	358	40	,	,	PUNCT
cana-5238	358	41	4	4	NUM
cana-5238	358	42	}	}	PUNCT
cana-5238	358	43	,	,	PUNCT
cana-5238	358	44	{	{	PUNCT
cana-5238	358	45	1	1	NUM
cana-5238	358	46	,	,	PUNCT
cana-5238	358	47	3	3	NUM
cana-5238	358	48	,	,	PUNCT
cana-5238	358	49	4	4	NUM
cana-5238	358	50	}	}	PUNCT
cana-5238	358	51	,	,	PUNCT
cana-5238	358	52	z	z	NOUN
cana-5238	358	53	}	}	PUNCT
cana-5238	358	54	,	,	PUNCT
cana-5238	358	55	h={∅	h={∅	PROPN
cana-5238	358	56	,	,	PUNCT
cana-5238	358	57	{	{	PUNCT
cana-5238	358	58	1	1	NUM
cana-5238	358	59	}	}	PUNCT
cana-5238	358	60	,	,	PUNCT
cana-5238	358	61	{	{	PUNCT
cana-5238	358	62	2	2	NUM
cana-5238	358	63	}	}	PUNCT
cana-5238	358	64	}	}	PUNCT
cana-5238	358	65	.	.	PUNCT
cana-5238	359	1	then	then	ADV
cana-5238	359	2	ψ={3	ψ={3	PROPN
cana-5238	359	3	,	,	PUNCT
cana-5238	359	4	4	4	NUM
cana-5238	359	5	}	}	PUNCT
cana-5238	359	6	is	be	AUX
cana-5238	359	7	σ	σ	PROPN
cana-5238	359	8	-	-	PUNCT
cana-5238	359	9	hg	hg	NOUN
cana-5238	359	10	-	-	NOUN
cana-5238	359	11	o	o	X
cana-5238	359	12	(	(	PUNCT
cana-5238	359	13	resp	resp	NOUN
cana-5238	359	14	.	.	PUNCT
cana-5238	360	1	π	π	PROPN
cana-5238	360	2	-	-	PUNCT
cana-5238	360	3	hg	hg	NOUN
cana-5238	360	4	-	-	PROPN
cana-5238	360	5	o	o	NOUN
cana-5238	360	6	,	,	PUNCT
cana-5238	360	7	bπ	bπ	PROPN
cana-5238	360	8	-	-	PUNCT
cana-5238	360	9	hg	hg	NOUN
cana-5238	360	10	-	-	NOUN
cana-5238	360	11	o	o	NOUN
cana-5238	360	12	)	)	PUNCT
cana-5238	360	13	but	but	CCONJ
cana-5238	360	14	not	not	PART
cana-5238	360	15	φ∗-b	φ∗-b	VERB
cana-5238	360	16	-	-	PUNCT
cana-5238	360	17	hg	hg	NOUN
cana-5238	360	18	-	-	PUNCT
cana-5238	360	19	s	s	NOUN
cana-5238	360	20	and	and	CCONJ
cana-5238	360	21	m={2	m={2	PROPN
cana-5238	360	22	}	}	PUNCT
cana-5238	360	23	is	be	AUX
cana-5238	360	24	φ∗-bhg	φ∗-bhg	ADJ
cana-5238	360	25	-	-	PUNCT
cana-5238	360	26	s	s	NOUN
cana-5238	360	27	but	but	CCONJ
cana-5238	360	28	not	not	PART
cana-5238	360	29	σ	σ	PROPN
cana-5238	360	30	-	-	PUNCT
cana-5238	360	31	hg	hg	NOUN
cana-5238	360	32	-	-	NOUN
cana-5238	360	33	o	o	X
cana-5238	360	34	(	(	PUNCT
cana-5238	360	35	resp	resp	NOUN
cana-5238	360	36	.	.	PUNCT
cana-5238	361	1	π	π	PROPN
cana-5238	361	2	-	-	PUNCT
cana-5238	361	3	hg	hg	NOUN
cana-5238	361	4	-	-	NOUN
cana-5238	361	5	o	o	NOUN
cana-5238	361	6	,	,	PUNCT
cana-5238	361	7	b	b	X
cana-5238	361	8	-	-	PUNCT
cana-5238	361	9	hg	hg	NOUN
cana-5238	361	10	-	-	NOUN
cana-5238	361	11	o	o	NOUN
cana-5238	361	12	)	)	PUNCT
cana-5238	361	13	.	.	PUNCT
cana-5238	362	1	theorem	theorem	VERB
cana-5238	362	2	3.35	3.35	NUM
cana-5238	362	3	.	.	PUNCT
cana-5238	363	1	let	let	AUX
cana-5238	363	2	(	(	PUNCT
cana-5238	363	3	z	z	NOUN
cana-5238	363	4	,	,	PUNCT
cana-5238	363	5	ζ	ζ	NOUN
cana-5238	363	6	,	,	PUNCT
cana-5238	363	7	h	h	NOUN
cana-5238	363	8	)	)	PUNCT
cana-5238	363	9	be	be	VERB
cana-5238	363	10	a	a	DET
cana-5238	363	11	strong	strong	ADJ
cana-5238	363	12	hgts	hgts	NOUN
cana-5238	363	13	where	where	SCONJ
cana-5238	363	14	z	z	NOUN
cana-5238	363	15	is	be	AUX
cana-5238	363	16	c0	c0	PROPN
cana-5238	363	17	-space	-space	PROPN
cana-5238	363	18	and	and	CCONJ
cana-5238	363	19	ψ⊂z	ψ⊂z	NOUN
cana-5238	363	20	.	.	PUNCT
cana-5238	364	1	then	then	ADV
cana-5238	364	2	the	the	DET
cana-5238	364	3	following	follow	VERB
cana-5238	364	4	conditions	condition	NOUN
cana-5238	364	5	are	be	AUX
cana-5238	364	6	equivalent	equivalent	ADJ
cana-5238	364	7	.	.	PUNCT
cana-5238	365	1	1	1	X
cana-5238	365	2	.	.	X
cana-5238	365	3	ψ	ψ	NOUN
cana-5238	365	4	is	be	AUX
cana-5238	365	5	gζ	gζ	NOUN
cana-5238	365	6	-	-	PUNCT
cana-5238	365	7	o	o	NOUN
cana-5238	365	8	,	,	PUNCT
cana-5238	365	9	2	2	NUM
cana-5238	365	10	.	.	PUNCT
cana-5238	365	11	ψ	ψ	NOUN
cana-5238	365	12	is	be	AUX
cana-5238	365	13	σ	σ	PROPN
cana-5238	365	14	-	-	PUNCT
cana-5238	365	15	hg	hg	NOUN
cana-5238	365	16	-	-	NOUN
cana-5238	365	17	o	o	NOUN
cana-5238	365	18	and	and	CCONJ
cana-5238	365	19	φ∗-b	φ∗-b	PROPN
cana-5238	365	20	-	-	PUNCT
cana-5238	365	21	hg	hg	NOUN
cana-5238	365	22	-	-	PUNCT
cana-5238	365	23	s	s	NOUN
cana-5238	365	24	3	3	NUM
cana-5238	365	25	.	.	PUNCT
cana-5238	365	26	ψ	ψ	NOUN
cana-5238	365	27	is	be	AUX
cana-5238	365	28	π	π	PROPN
cana-5238	365	29	-	-	PUNCT
cana-5238	365	30	hg	hg	NOUN
cana-5238	365	31	-	-	NOUN
cana-5238	365	32	o	o	NOUN
cana-5238	365	33	and	and	CCONJ
cana-5238	365	34	φ∗-b	φ∗-b	PROPN
cana-5238	365	35	-	-	PUNCT
cana-5238	365	36	hg	hg	NOUN
cana-5238	365	37	-	-	PUNCT
cana-5238	365	38	s	s	PART
cana-5238	365	39	4	4	NUM
cana-5238	365	40	.	.	PUNCT
cana-5238	365	41	ψ	ψ	NOUN
cana-5238	365	42	is	be	AUX
cana-5238	365	43	b	b	PROPN
cana-5238	365	44	-	-	PUNCT
cana-5238	365	45	hg	hg	NOUN
cana-5238	365	46	-	-	NOUN
cana-5238	365	47	o	o	NOUN
cana-5238	365	48	and	and	CCONJ
cana-5238	365	49	φ∗-b	φ∗-b	PROPN
cana-5238	365	50	-	-	PUNCT
cana-5238	365	51	hg	hg	NOUN
cana-5238	365	52	-	-	PUNCT
cana-5238	365	53	s.	s.	PROPN
cana-5238	365	54	proof	proof	NOUN
cana-5238	365	55	.	.	PUNCT
cana-5238	366	1	(	(	PUNCT
cana-5238	366	2	1)⇒(2)⇒(4	1)⇒(2)⇒(4	NUM
cana-5238	366	3	)	)	PUNCT
cana-5238	366	4	and	and	CCONJ
cana-5238	366	5	(	(	PUNCT
cana-5238	366	6	1)⇒(3)⇒(4	1)⇒(3)⇒(4	NOUN
cana-5238	366	7	)	)	PUNCT
cana-5238	366	8	are	be	AUX
cana-5238	366	9	obvious	obvious	ADJ
cana-5238	366	10	,	,	PUNCT
cana-5238	366	11	since	since	SCONJ
cana-5238	366	12	z	z	PROPN
cana-5238	366	13	is	be	AUX
cana-5238	366	14	φ∗-b	φ∗-b	PROPN
cana-5238	366	15	-	-	PUNCT
cana-5238	366	16	hg	hg	NOUN
cana-5238	366	17	-	-	PUNCT
cana-5238	366	18	s	s	X
cana-5238	366	19	(	(	PUNCT
cana-5238	366	20	4)⇒(1	4)⇒(1	NOUN
cana-5238	366	21	)	)	PUNCT
cana-5238	366	22	.	.	PUNCT
cana-5238	367	1	let	let	VERB
cana-5238	367	2	ψ	ψ	NOUN
cana-5238	367	3	is	be	AUX
cana-5238	367	4	b	b	NUM
cana-5238	367	5	-	-	PUNCT
cana-5238	367	6	hg	hg	NOUN
cana-5238	367	7	-	-	NOUN
cana-5238	367	8	o	o	NOUN
cana-5238	367	9	and	and	CCONJ
cana-5238	367	10	φ∗-b	φ∗-b	PROPN
cana-5238	367	11	-	-	PUNCT
cana-5238	367	12	hg	hg	NOUN
cana-5238	367	13	-	-	PUNCT
cana-5238	367	14	s	s	X
cana-5238	367	15	then	then	ADV
cana-5238	367	16	ψ⊆igc∗(ψ)∪c∗ig(ψ)=igc∗(m∩n)∪c∗ig(m∩	ψ⊆igc∗(ψ)∪c∗ig(ψ)=igc∗(m∩n)∪c∗ig(m∩	PROPN
cana-5238	367	17	n	n	CCONJ
cana-5238	367	18	)	)	PUNCT
cana-5238	367	19	,	,	PUNCT
cana-5238	367	20	where	where	SCONJ
cana-5238	367	21	ψ	ψ	VERB
cana-5238	367	22	=	=	NOUN
cana-5238	367	23	m∩n	m∩n	X
cana-5238	367	24	,	,	PUNCT
cana-5238	367	25	m	m	VERB
cana-5238	367	26	is	be	AUX
cana-5238	367	27	gζ	gζ	NOUN
cana-5238	367	28	-	-	PUNCT
cana-5238	367	29	o	o	NOUN
cana-5238	367	30	and	and	CCONJ
cana-5238	367	31	v	v	NOUN
cana-5238	367	32	is	be	AUX
cana-5238	367	33	φ∗-hg	φ∗-hg	NOUN
cana-5238	367	34	-	-	PUNCT
cana-5238	367	35	s.	s.	PROPN
cana-5238	367	36	hence	hence	ADV
cana-5238	367	37	ψ⊆m∩ψ⊆m∩[igc∗(m∩n)∪c∗ig(m∩n)]⊆[m∩igc∗(m)∩igc∗(n)]∪[m∩c∗ig(m)∩c∗ig(n	ψ⊆m∩ψ⊆m∩[igc∗(m∩n)∪c∗ig(m∩n)]⊆[m∩igc∗(m)∩igc∗(n)]∪[m∩c∗ig(m)∩c∗ig(n	NOUN
cana-5238	367	38	)	)	PUNCT
cana-5238	367	39	]	]	PUNCT
cana-5238	368	1	⊆[m∩igc∗(n)]∪[m∩c∗ig(n)]=m∩[igc∗(n)∪c∗ig(n	⊆[m∩igc∗(n)]∪[m∩c∗ig(n)]=m∩[igc∗(n)∪c∗ig(n	NOUN
cana-5238	368	2	)	)	PUNCT
cana-5238	368	3	]=	]=	NOUN
cana-5238	368	4	m∩ig(v)=ig(ψ	m∩ig(v)=ig(ψ	NUM
cana-5238	368	5	)	)	PUNCT
cana-5238	368	6	.	.	PUNCT
cana-5238	369	1	remark	remark	VERB
cana-5238	369	2	3.36	3.36	NUM
cana-5238	369	3	.	.	PUNCT
cana-5238	370	1	the	the	DET
cana-5238	370	2	notions	notion	NOUN
cana-5238	370	3	of	of	ADP
cana-5238	370	4	σ	σ	PROPN
cana-5238	370	5	-	-	PUNCT
cana-5238	370	6	hg	hg	NOUN
cana-5238	370	7	-	-	NOUN
cana-5238	370	8	o	o	X
cana-5238	370	9	(	(	PUNCT
cana-5238	370	10	resp	resp	NOUN
cana-5238	370	11	.	.	PUNCT
cana-5238	371	1	π	π	PROPN
cana-5238	371	2	-	-	PUNCT
cana-5238	371	3	hg	hg	NOUN
cana-5238	371	4	-	-	NOUN
cana-5238	371	5	o	o	NOUN
cana-5238	371	6	,	,	PUNCT
cana-5238	371	7	b	b	X
cana-5238	371	8	-	-	PUNCT
cana-5238	371	9	hg	hg	NOUN
cana-5238	371	10	-	-	NOUN
cana-5238	371	11	o	o	NOUN
cana-5238	371	12	)	)	PUNCT
cana-5238	371	13	and	and	CCONJ
cana-5238	371	14	φ∗-b	φ∗-b	PROPN
cana-5238	371	15	-	-	PUNCT
cana-5238	371	16	hg	hg	NOUN
cana-5238	371	17	-	-	PUNCT
cana-5238	371	18	s	s	PART
cana-5238	371	19	are	be	AUX
cana-5238	371	20	independent	independent	ADJ
cana-5238	371	21	.	.	PUNCT
cana-5238	372	1	theorem	theorem	VERB
cana-5238	372	2	3.37	3.37	NUM
cana-5238	372	3	.	.	PUNCT
cana-5238	373	1	let	let	AUX
cana-5238	373	2	(	(	PUNCT
cana-5238	373	3	z	z	NOUN
cana-5238	373	4	,	,	PUNCT
cana-5238	373	5	ζ	ζ	NOUN
cana-5238	373	6	,	,	PUNCT
cana-5238	373	7	h	h	NOUN
cana-5238	373	8	)	)	PUNCT
cana-5238	373	9	be	be	VERB
cana-5238	373	10	a	a	DET
cana-5238	373	11	strong	strong	ADJ
cana-5238	373	12	hgts	hgts	NOUN
cana-5238	373	13	,	,	PUNCT
cana-5238	373	14	where	where	SCONJ
cana-5238	373	15	z	z	NOUN
cana-5238	373	16	is	be	AUX
cana-5238	373	17	c0	c0	PROPN
cana-5238	373	18	-space	-space	PROPN
cana-5238	373	19	and	and	CCONJ
cana-5238	373	20	ψ⊂z	ψ⊂z	NOUN
cana-5238	373	21	.	.	PUNCT
cana-5238	374	1	then	then	ADV
cana-5238	374	2	the	the	DET
cana-5238	374	3	following	follow	VERB
cana-5238	374	4	conditions	condition	NOUN
cana-5238	374	5	are	be	AUX
cana-5238	374	6	equivalent	equivalent	ADJ
cana-5238	374	7	.	.	PUNCT
cana-5238	375	1	1	1	X
cana-5238	375	2	.	.	X
cana-5238	375	3	ψ	ψ	NOUN
cana-5238	375	4	is	be	AUX
cana-5238	375	5	gζ	gζ	NOUN
cana-5238	375	6	-	-	PUNCT
cana-5238	375	7	o.	o.	ADJ
cana-5238	375	8	2	2	NUM
cana-5238	375	9	.	.	PUNCT
cana-5238	375	10	ψ	ψ	NOUN
cana-5238	375	11	is	be	AUX
cana-5238	375	12	α	α	X
cana-5238	375	13	-	-	PUNCT
cana-5238	375	14	hg	hg	NOUN
cana-5238	375	15	-	-	NOUN
cana-5238	375	16	o	o	NOUN
cana-5238	375	17	and	and	CCONJ
cana-5238	375	18	π∗-b	π∗-b	PROPN
cana-5238	375	19	-	-	PUNCT
cana-5238	375	20	hg	hg	NOUN
cana-5238	375	21	-	-	PUNCT
cana-5238	375	22	s.	s.	PROPN
cana-5238	375	23	3	3	NUM
cana-5238	375	24	.	.	PUNCT
cana-5238	376	1	ψ	ψ	NOUN
cana-5238	376	2	is	be	AUX
cana-5238	376	3	π	π	PROPN
cana-5238	376	4	-	-	PUNCT
cana-5238	376	5	hg	hg	NOUN
cana-5238	376	6	-	-	NOUN
cana-5238	376	7	o	o	NOUN
cana-5238	376	8	and	and	CCONJ
cana-5238	376	9	π∗-b	π∗-b	PROPN
cana-5238	376	10	-	-	PUNCT
cana-5238	376	11	hg	hg	NOUN
cana-5238	376	12	-	-	PUNCT
cana-5238	376	13	s.	s.	PROPN
cana-5238	376	14	proof	proof	NOUN
cana-5238	376	15	.	.	PUNCT
cana-5238	377	1	(	(	PUNCT
cana-5238	377	2	1)⇒(2	1)⇒(2	NUM
cana-5238	377	3	)	)	PUNCT
cana-5238	377	4	.	.	PUNCT
cana-5238	378	1	let	let	VERB
cana-5238	378	2	a	a	DET
cana-5238	378	3	subset	subset	NOUN
cana-5238	378	4	ψ	ψ	X
cana-5238	378	5	of	of	ADP
cana-5238	378	6	z	z	PROPN
cana-5238	378	7	is	be	AUX
cana-5238	378	8	gζ	gζ	PROPN
cana-5238	378	9	-	-	PUNCT
cana-5238	378	10	o.	o.	NOUN
cana-5238	379	1	then	then	ADV
cana-5238	379	2	it	it	PRON
cana-5238	379	3	is	be	AUX
cana-5238	379	4	α	α	NOUN
cana-5238	379	5	-	-	PUNCT
cana-5238	379	6	hg	hg	NOUN
cana-5238	379	7	-	-	NOUN
cana-5238	379	8	o	o	NOUN
cana-5238	379	9	and	and	CCONJ
cana-5238	379	10	π∗-b	π∗-b	PROPN
cana-5238	379	11	-	-	PUNCT
cana-5238	379	12	hg	hg	NOUN
cana-5238	379	13	-	-	PUNCT
cana-5238	379	14	s.	s.	PROPN
cana-5238	379	15	(	(	PUNCT
cana-5238	379	16	2)⇒(3	2)⇒(3	NUM
cana-5238	379	17	)	)	PUNCT
cana-5238	379	18	.	.	PUNCT
cana-5238	380	1	let	let	VERB
cana-5238	380	2	a	a	DET
cana-5238	380	3	subset	subset	NOUN
cana-5238	380	4	ψ	ψ	X
cana-5238	380	5	of	of	ADP
cana-5238	380	6	z	z	PROPN
cana-5238	380	7	is	be	AUX
cana-5238	380	8	both	both	DET
cana-5238	380	9	α	α	PROPN
cana-5238	380	10	-	-	PUNCT
cana-5238	380	11	hg	hg	NOUN
cana-5238	380	12	-	-	NOUN
cana-5238	380	13	o	o	NOUN
cana-5238	380	14	and	and	CCONJ
cana-5238	380	15	π∗-b	π∗-b	PROPN
cana-5238	380	16	-	-	PUNCT
cana-5238	380	17	hg	hg	NOUN
cana-5238	380	18	-	-	PUNCT
cana-5238	380	19	s.	s.	PROPN
cana-5238	380	20	then	then	ADV
cana-5238	380	21	it	it	PRON
cana-5238	380	22	is	be	AUX
cana-5238	380	23	both	both	PRON
cana-5238	380	24	ψ	ψ	NOUN
cana-5238	380	25	is	be	AUX
cana-5238	380	26	π	π	PROPN
cana-5238	380	27	-	-	PUNCT
cana-5238	380	28	hg	hg	NOUN
cana-5238	380	29	-	-	NOUN
cana-5238	380	30	o	o	NOUN
cana-5238	380	31	and	and	CCONJ
cana-5238	380	32	π∗-bhg	π∗-bhg	PROPN
cana-5238	380	33	-	-	PUNCT
cana-5238	380	34	s.	s.	PROPN
cana-5238	380	35	(	(	PUNCT
cana-5238	380	36	3)⇒(1	3)⇒(1	PROPN
cana-5238	380	37	)	)	PUNCT
cana-5238	380	38	.	.	PUNCT
cana-5238	381	1	this	this	PRON
cana-5238	381	2	is	be	AUX
cana-5238	381	3	from	from	ADP
cana-5238	381	4	theorem	theorem	ADJ
cana-5238	381	5	3.32	3.32	NUM
cana-5238	381	6	.	.	PUNCT
cana-5238	382	1	theorem	theorem	NOUN
cana-5238	382	2	3.38	3.38	NUM
cana-5238	382	3	.	.	PUNCT
cana-5238	383	1	let	let	AUX
cana-5238	383	2	(	(	PUNCT
cana-5238	383	3	z	z	NOUN
cana-5238	383	4	,	,	PUNCT
cana-5238	383	5	ζ	ζ	NOUN
cana-5238	383	6	,	,	PUNCT
cana-5238	383	7	h	h	NOUN
cana-5238	383	8	)	)	PUNCT
cana-5238	383	9	be	be	VERB
cana-5238	383	10	a	a	DET
cana-5238	383	11	strong	strong	ADJ
cana-5238	383	12	hgts	hgts	NOUN
cana-5238	383	13	where	where	SCONJ
cana-5238	383	14	z	z	NOUN
cana-5238	383	15	is	be	AUX
cana-5238	383	16	c0	c0	PROPN
cana-5238	383	17	-space	-space	PROPN
cana-5238	383	18	and	and	CCONJ
cana-5238	383	19	l⊂z	l⊂z	PROPN
cana-5238	383	20	.	.	PUNCT
cana-5238	384	1	then	then	ADV
cana-5238	384	2	the	the	DET
cana-5238	384	3	following	follow	VERB
cana-5238	384	4	conditions	condition	NOUN
cana-5238	384	5	are	be	AUX
cana-5238	384	6	equivalent	equivalent	ADJ
cana-5238	384	7	.	.	PUNCT
cana-5238	385	1	1	1	X
cana-5238	385	2	.	.	X
cana-5238	385	3	ψ	ψ	NOUN
cana-5238	385	4	is	be	AUX
cana-5238	385	5	gζ	gζ	NOUN
cana-5238	385	6	-	-	PUNCT
cana-5238	385	7	o	o	NOUN
cana-5238	385	8	,	,	PUNCT
cana-5238	385	9	2	2	NUM
cana-5238	385	10	.	.	PUNCT
cana-5238	385	11	ψ	ψ	NOUN
cana-5238	385	12	is	be	AUX
cana-5238	385	13	α	α	X
cana-5238	385	14	-	-	PUNCT
cana-5238	385	15	hg	hg	NOUN
cana-5238	385	16	-	-	NOUN
cana-5238	385	17	o	o	NOUN
cana-5238	385	18	and	and	CCONJ
cana-5238	385	19	σ∗-b	σ∗-b	PROPN
cana-5238	385	20	-	-	PUNCT
cana-5238	385	21	hg	hg	NOUN
cana-5238	385	22	-	-	PUNCT
cana-5238	385	23	s	s	NOUN
cana-5238	385	24	,	,	PUNCT
cana-5238	385	25	3	3	NUM
cana-5238	385	26	.	.	PUNCT
cana-5238	385	27	ψ	ψ	NOUN
cana-5238	385	28	is	be	AUX
cana-5238	385	29	σ	σ	PROPN
cana-5238	385	30	-	-	PUNCT
cana-5238	385	31	hg	hg	NOUN
cana-5238	385	32	-	-	NOUN
cana-5238	385	33	o	o	NOUN
cana-5238	385	34	and	and	CCONJ
cana-5238	385	35	σ∗-b	σ∗-b	PROPN
cana-5238	385	36	-	-	PUNCT
cana-5238	385	37	hg	hg	NOUN
cana-5238	385	38	-	-	PUNCT
cana-5238	385	39	s.	s.	PROPN
cana-5238	385	40	proof	proof	NOUN
cana-5238	385	41	.	.	PUNCT
cana-5238	386	1	(	(	PUNCT
cana-5238	386	2	1)⇒(2	1)⇒(2	NUM
cana-5238	386	3	)	)	PUNCT
cana-5238	386	4	.	.	PUNCT
cana-5238	387	1	let	let	VERB
cana-5238	387	2	a	a	DET
cana-5238	387	3	subset	subset	NOUN
cana-5238	387	4	ψ	ψ	X
cana-5238	387	5	of	of	ADP
cana-5238	387	6	z	z	PROPN
cana-5238	387	7	is	be	AUX
cana-5238	387	8	gζ	gζ	PROPN
cana-5238	387	9	-	-	PUNCT
cana-5238	387	10	o.	o.	NOUN
cana-5238	388	1	then	then	ADV
cana-5238	388	2	it	it	PRON
cana-5238	388	3	is	be	AUX
cana-5238	388	4	α	α	NOUN
cana-5238	388	5	-	-	PUNCT
cana-5238	388	6	hg	hg	NOUN
cana-5238	388	7	-	-	NOUN
cana-5238	388	8	o	o	NOUN
cana-5238	388	9	and	and	CCONJ
cana-5238	388	10	σ∗-b	σ∗-b	PROPN
cana-5238	388	11	-	-	PUNCT
cana-5238	388	12	hg	hg	NOUN
cana-5238	388	13	-	-	PUNCT
cana-5238	388	14	s.	s.	PROPN
cana-5238	388	15	(	(	PUNCT
cana-5238	388	16	2	2	NUM
cana-5238	388	17	)	)	PUNCT
cana-5238	388	18	⇒	⇒	NOUN
cana-5238	388	19	(	(	PUNCT
cana-5238	388	20	3	3	NUM
cana-5238	388	21	)	)	PUNCT
cana-5238	388	22	.	.	PUNCT
cana-5238	389	1	let	let	VERB
cana-5238	389	2	a	a	DET
cana-5238	389	3	subset	subset	NOUN
cana-5238	389	4	ψ	ψ	X
cana-5238	389	5	of	of	ADP
cana-5238	389	6	z	z	PROPN
cana-5238	389	7	is	be	AUX
cana-5238	389	8	both	both	DET
cana-5238	389	9	α	α	PROPN
cana-5238	389	10	-	-	PUNCT
cana-5238	389	11	hg	hg	NOUN
cana-5238	389	12	-	-	NOUN
cana-5238	389	13	o	o	NOUN
cana-5238	389	14	and	and	CCONJ
cana-5238	389	15	σ∗-b	σ∗-b	PROPN
cana-5238	389	16	-	-	PUNCT
cana-5238	389	17	hg	hg	NOUN
cana-5238	389	18	-	-	PUNCT
cana-5238	389	19	s.	s.	PROPN
cana-5238	389	20	then	then	ADV
cana-5238	389	21	it	it	PRON
cana-5238	389	22	is	be	AUX
cana-5238	389	23	both	both	DET
cana-5238	389	24	σ	σ	PROPN
cana-5238	389	25	-	-	PUNCT
cana-5238	389	26	hg	hg	NOUN
cana-5238	389	27	-	-	NOUN
cana-5238	389	28	o	o	NOUN
cana-5238	389	29	and	and	CCONJ
cana-5238	389	30	σ∗-b	σ∗-b	PROPN
cana-5238	389	31	-	-	PUNCT
cana-5238	389	32	hg	hg	NOUN
cana-5238	389	33	-	-	PUNCT
cana-5238	389	34	s.	s.	PROPN
cana-5238	389	35	(	(	PUNCT
cana-5238	389	36	3	3	X
cana-5238	389	37	)	)	PUNCT
cana-5238	389	38	⇒	⇒	NOUN
cana-5238	389	39	(	(	PUNCT
cana-5238	389	40	1	1	NUM
cana-5238	389	41	)	)	PUNCT
cana-5238	389	42	.	.	PUNCT
cana-5238	390	1	this	this	PRON
cana-5238	390	2	is	be	AUX
cana-5238	390	3	from	from	ADP
cana-5238	390	4	theorem	theorem	ADJ
cana-5238	390	5	3.32	3.32	NUM
cana-5238	390	6	.	.	PUNCT
cana-5238	391	1	theorem	theorem	NOUN
cana-5238	391	2	3.39	3.39	NUM
cana-5238	391	3	.	.	PUNCT
cana-5238	392	1	let	let	AUX
cana-5238	392	2	(	(	PUNCT
cana-5238	392	3	z	z	NOUN
cana-5238	392	4	,	,	PUNCT
cana-5238	392	5	ζ	ζ	NOUN
cana-5238	392	6	,	,	PUNCT
cana-5238	392	7	h	h	NOUN
cana-5238	392	8	)	)	PUNCT
cana-5238	392	9	be	be	VERB
cana-5238	392	10	a	a	DET
cana-5238	392	11	strong	strong	ADJ
cana-5238	392	12	hgts	hgts	NOUN
cana-5238	392	13	,	,	PUNCT
cana-5238	392	14	where	where	SCONJ
cana-5238	392	15	z	z	NOUN
cana-5238	392	16	is	be	AUX
cana-5238	392	17	c0	c0	PROPN
cana-5238	392	18	-space	-space	PROPN
cana-5238	392	19	and	and	CCONJ
cana-5238	392	20	l⊂z	l⊂z	PROPN
cana-5238	392	21	.	.	PUNCT
cana-5238	393	1	then	then	ADV
cana-5238	393	2	the	the	DET
cana-5238	393	3	following	follow	VERB
cana-5238	393	4	conditions	condition	NOUN
cana-5238	393	5	are	be	AUX
cana-5238	393	6	equivalent	equivalent	ADJ
cana-5238	393	7	.	.	PUNCT
cana-5238	394	1	communications	communication	NOUN
cana-5238	394	2	on	on	ADP
cana-5238	394	3	applied	apply	VERB
cana-5238	394	4	nonlinear	nonlinear	ADJ
cana-5238	394	5	analysis	analysis	NOUN
cana-5238	394	6	issn	issn	NOUN
cana-5238	394	7	:	:	PUNCT
cana-5238	394	8	1074	1074	NUM
cana-5238	394	9	-	-	PUNCT
cana-5238	394	10	133x	133x	NUM
cana-5238	394	11	vol	vol	VERB
cana-5238	394	12	32	32	NUM
cana-5238	394	13	no	no	NOUN
cana-5238	394	14	.	.	PUNCT
cana-5238	395	1	10s	10	NOUN
cana-5238	395	2	(	(	PUNCT
cana-5238	395	3	2025	2025	NUM
cana-5238	395	4	)	)	PUNCT
cana-5238	395	5	1367	1367	NUM
cana-5238	395	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	395	7	1	1	NUM
cana-5238	395	8	.	.	PUNCT
cana-5238	395	9	ψ	ψ	NOUN
cana-5238	395	10	is	be	AUX
cana-5238	395	11	gζ	gζ	NOUN
cana-5238	395	12	-	-	PUNCT
cana-5238	395	13	o	o	NOUN
cana-5238	395	14	,	,	PUNCT
cana-5238	395	15	2	2	NUM
cana-5238	395	16	.	.	PUNCT
cana-5238	395	17	ψ	ψ	NOUN
cana-5238	395	18	is	be	AUX
cana-5238	395	19	α	α	X
cana-5238	395	20	-	-	PUNCT
cana-5238	395	21	hg	hg	NOUN
cana-5238	395	22	-	-	NOUN
cana-5238	395	23	o	o	NOUN
cana-5238	395	24	and	and	CCONJ
cana-5238	395	25	∆∗-b	∆∗-b	PROPN
cana-5238	395	26	-	-	PUNCT
cana-5238	395	27	hg	hg	NOUN
cana-5238	395	28	-	-	PUNCT
cana-5238	395	29	s	s	NOUN
cana-5238	395	30	,	,	PUNCT
cana-5238	395	31	3	3	NUM
cana-5238	395	32	.	.	PUNCT
cana-5238	395	33	ψ	ψ	NOUN
cana-5238	395	34	is	be	AUX
cana-5238	395	35	β	β	X
cana-5238	395	36	-	-	ADJ
cana-5238	395	37	hg	hg	NOUN
cana-5238	395	38	-	-	NOUN
cana-5238	395	39	o	o	NOUN
cana-5238	395	40	and	and	CCONJ
cana-5238	395	41	∆∗-b	∆∗-b	PROPN
cana-5238	395	42	-	-	PUNCT
cana-5238	395	43	hg	hg	NOUN
cana-5238	395	44	-	-	PUNCT
cana-5238	395	45	s.	s.	PROPN
cana-5238	395	46	proof	proof	NOUN
cana-5238	395	47	.	.	PUNCT
cana-5238	396	1	(	(	PUNCT
cana-5238	396	2	1	1	X
cana-5238	396	3	)	)	PUNCT
cana-5238	396	4	⇒	⇒	NOUN
cana-5238	396	5	(	(	PUNCT
cana-5238	396	6	2	2	NUM
cana-5238	396	7	)	)	PUNCT
cana-5238	396	8	.	.	PUNCT
cana-5238	397	1	let	let	VERB
cana-5238	397	2	a	a	DET
cana-5238	397	3	subset	subset	NOUN
cana-5238	397	4	ψ	ψ	X
cana-5238	397	5	of	of	ADP
cana-5238	397	6	z	z	PROPN
cana-5238	397	7	is	be	AUX
cana-5238	397	8	gζ	gζ	PROPN
cana-5238	397	9	-	-	PUNCT
cana-5238	397	10	o.	o.	NOUN
cana-5238	398	1	then	then	ADV
cana-5238	398	2	it	it	PRON
cana-5238	398	3	is	be	AUX
cana-5238	398	4	α	α	NOUN
cana-5238	398	5	-	-	PUNCT
cana-5238	398	6	hg	hg	NOUN
cana-5238	398	7	-	-	NOUN
cana-5238	398	8	o	o	NOUN
cana-5238	398	9	and	and	CCONJ
cana-5238	398	10	∆∗-b	∆∗-b	PROPN
cana-5238	398	11	-	-	PUNCT
cana-5238	398	12	hg	hg	NOUN
cana-5238	398	13	-	-	PUNCT
cana-5238	398	14	s.	s.	PROPN
cana-5238	398	15	(	(	PUNCT
cana-5238	398	16	2	2	NUM
cana-5238	398	17	)	)	PUNCT
cana-5238	398	18	⇒	⇒	NOUN
cana-5238	398	19	(	(	PUNCT
cana-5238	398	20	3	3	NUM
cana-5238	398	21	)	)	PUNCT
cana-5238	398	22	.	.	PUNCT
cana-5238	399	1	let	let	VERB
cana-5238	399	2	a	a	DET
cana-5238	399	3	subset	subset	NOUN
cana-5238	399	4	ψ	ψ	X
cana-5238	399	5	of	of	ADP
cana-5238	399	6	z	z	PROPN
cana-5238	399	7	is	be	AUX
cana-5238	399	8	both	both	DET
cana-5238	399	9	α	α	PROPN
cana-5238	399	10	-	-	PUNCT
cana-5238	399	11	hg	hg	NOUN
cana-5238	399	12	-	-	NOUN
cana-5238	399	13	o	o	NOUN
cana-5238	399	14	and	and	CCONJ
cana-5238	399	15	∆∗-b	∆∗-b	PROPN
cana-5238	399	16	-	-	PUNCT
cana-5238	399	17	hg	hg	NOUN
cana-5238	399	18	-	-	PUNCT
cana-5238	399	19	s	s	VERB
cana-5238	399	20	then	then	ADV
cana-5238	399	21	it	it	PRON
cana-5238	399	22	is	be	AUX
cana-5238	399	23	both	both	PRON
cana-5238	399	24	β	β	NOUN
cana-5238	399	25	-	-	PUNCT
cana-5238	399	26	hg	hg	NOUN
cana-5238	399	27	-	-	NOUN
cana-5238	399	28	o	o	NOUN
cana-5238	399	29	and	and	CCONJ
cana-5238	399	30	∆∗-b	∆∗-b	PROPN
cana-5238	399	31	-	-	PUNCT
cana-5238	399	32	hg	hg	NOUN
cana-5238	399	33	-	-	PUNCT
cana-5238	399	34	s.	s.	PROPN
cana-5238	399	35	(	(	PUNCT
cana-5238	399	36	3	3	X
cana-5238	399	37	)	)	PUNCT
cana-5238	399	38	⇒	⇒	NOUN
cana-5238	399	39	(	(	PUNCT
cana-5238	399	40	1	1	NUM
cana-5238	399	41	)	)	PUNCT
cana-5238	399	42	.	.	PUNCT
cana-5238	400	1	this	this	PRON
cana-5238	400	2	is	be	AUX
cana-5238	400	3	from	from	ADP
cana-5238	400	4	theorem	theorem	ADJ
cana-5238	400	5	3.32	3.32	NUM
cana-5238	400	6	.	.	PUNCT
cana-5238	401	1	theorem	theorem	NOUN
cana-5238	401	2	3.40	3.40	NUM
cana-5238	401	3	.	.	PUNCT
cana-5238	402	1	let	let	AUX
cana-5238	402	2	(	(	PUNCT
cana-5238	402	3	z	z	NOUN
cana-5238	402	4	,	,	PUNCT
cana-5238	402	5	ζ	ζ	NOUN
cana-5238	402	6	,	,	PUNCT
cana-5238	402	7	h	h	NOUN
cana-5238	402	8	)	)	PUNCT
cana-5238	402	9	be	be	VERB
cana-5238	402	10	a	a	DET
cana-5238	402	11	strong	strong	ADJ
cana-5238	402	12	hgts	hgts	NOUN
cana-5238	402	13	,	,	PUNCT
cana-5238	402	14	where	where	SCONJ
cana-5238	402	15	z	z	NOUN
cana-5238	402	16	is	be	AUX
cana-5238	402	17	c0	c0	PROPN
cana-5238	402	18	-space	-space	PROPN
cana-5238	402	19	and	and	CCONJ
cana-5238	402	20	l⊂z	l⊂z	PROPN
cana-5238	402	21	.	.	PUNCT
cana-5238	403	1	then	then	ADV
cana-5238	403	2	the	the	DET
cana-5238	403	3	following	follow	VERB
cana-5238	403	4	conditions	condition	NOUN
cana-5238	403	5	are	be	AUX
cana-5238	403	6	equivalent	equivalent	ADJ
cana-5238	403	7	.	.	PUNCT
cana-5238	404	1	1	1	X
cana-5238	404	2	.	.	X
cana-5238	404	3	ψ	ψ	NOUN
cana-5238	404	4	is	be	AUX
cana-5238	404	5	gζ	gζ	NOUN
cana-5238	404	6	-	-	PUNCT
cana-5238	404	7	o	o	NOUN
cana-5238	404	8	,	,	PUNCT
cana-5238	404	9	2	2	NUM
cana-5238	404	10	.	.	PUNCT
cana-5238	404	11	ψ	ψ	NOUN
cana-5238	404	12	is	be	AUX
cana-5238	404	13	σ	σ	PROPN
cana-5238	404	14	-	-	PUNCT
cana-5238	404	15	hg	hg	NOUN
cana-5238	404	16	-	-	NOUN
cana-5238	404	17	o	o	NOUN
cana-5238	404	18	and	and	CCONJ
cana-5238	404	19	∆∗-b	∆∗-b	PROPN
cana-5238	404	20	-	-	PUNCT
cana-5238	404	21	hg	hg	NOUN
cana-5238	404	22	-	-	PUNCT
cana-5238	404	23	s	s	NOUN
cana-5238	404	24	,	,	PUNCT
cana-5238	404	25	3	3	NUM
cana-5238	404	26	.	.	PUNCT
cana-5238	404	27	ψ	ψ	NOUN
cana-5238	404	28	is	be	AUX
cana-5238	404	29	β	β	X
cana-5238	404	30	-	-	ADJ
cana-5238	404	31	hg	hg	NOUN
cana-5238	404	32	-	-	NOUN
cana-5238	404	33	o	o	NOUN
cana-5238	404	34	and	and	CCONJ
cana-5238	404	35	∆∗-b	∆∗-b	PROPN
cana-5238	404	36	-	-	PUNCT
cana-5238	404	37	hgs	hgs	PROPN
cana-5238	404	38	.	.	PUNCT
cana-5238	405	1	proof	proof	NOUN
cana-5238	405	2	.	.	PUNCT
cana-5238	406	1	(	(	PUNCT
cana-5238	406	2	1	1	X
cana-5238	406	3	)	)	PUNCT
cana-5238	406	4	⇒	⇒	NOUN
cana-5238	406	5	(	(	PUNCT
cana-5238	406	6	2	2	NUM
cana-5238	406	7	)	)	PUNCT
cana-5238	406	8	.	.	PUNCT
cana-5238	407	1	let	let	VERB
cana-5238	407	2	a	a	DET
cana-5238	407	3	subset	subset	NOUN
cana-5238	407	4	ψ	ψ	X
cana-5238	407	5	of	of	ADP
cana-5238	407	6	z	z	PROPN
cana-5238	407	7	is	be	AUX
cana-5238	407	8	gζ	gζ	PROPN
cana-5238	407	9	-	-	PUNCT
cana-5238	407	10	o.	o.	NOUN
cana-5238	408	1	then	then	ADV
cana-5238	408	2	it	it	PRON
cana-5238	408	3	is	be	AUX
cana-5238	408	4	σ	σ	PROPN
cana-5238	408	5	-	-	PUNCT
cana-5238	408	6	hg	hg	NOUN
cana-5238	408	7	-	-	NOUN
cana-5238	408	8	o	o	NOUN
cana-5238	408	9	and	and	CCONJ
cana-5238	408	10	∆∗-b	∆∗-b	PROPN
cana-5238	408	11	-	-	PUNCT
cana-5238	408	12	hg	hg	NOUN
cana-5238	408	13	-	-	PUNCT
cana-5238	408	14	s	s	X
cana-5238	408	15	(	(	PUNCT
cana-5238	408	16	2	2	NUM
cana-5238	408	17	)	)	PUNCT
cana-5238	408	18	⇒	⇒	NOUN
cana-5238	408	19	(	(	PUNCT
cana-5238	408	20	3	3	NUM
cana-5238	408	21	)	)	PUNCT
cana-5238	408	22	.	.	PUNCT
cana-5238	409	1	let	let	VERB
cana-5238	409	2	a	a	DET
cana-5238	409	3	subset	subset	NOUN
cana-5238	409	4	ψ	ψ	X
cana-5238	409	5	of	of	ADP
cana-5238	409	6	z	z	PROPN
cana-5238	409	7	is	be	AUX
cana-5238	409	8	both	both	DET
cana-5238	409	9	σ	σ	PROPN
cana-5238	409	10	-	-	PUNCT
cana-5238	409	11	hg	hg	NOUN
cana-5238	409	12	-	-	NOUN
cana-5238	409	13	o	o	NOUN
cana-5238	409	14	and	and	CCONJ
cana-5238	409	15	∆∗-b	∆∗-b	PROPN
cana-5238	409	16	-	-	PUNCT
cana-5238	409	17	hg	hg	NOUN
cana-5238	409	18	-	-	PUNCT
cana-5238	409	19	s.	s.	PROPN
cana-5238	409	20	then	then	ADV
cana-5238	409	21	it	it	PRON
cana-5238	409	22	is	be	AUX
cana-5238	409	23	both	both	PRON
cana-5238	409	24	β	β	NOUN
cana-5238	409	25	-	-	PUNCT
cana-5238	409	26	hg	hg	NOUN
cana-5238	409	27	-	-	NOUN
cana-5238	409	28	o	o	NOUN
cana-5238	409	29	and	and	CCONJ
cana-5238	409	30	∆∗-b	∆∗-b	PROPN
cana-5238	409	31	-	-	PUNCT
cana-5238	409	32	hgs	hgs	PROPN
cana-5238	409	33	.	.	PUNCT
cana-5238	410	1	(	(	PUNCT
cana-5238	410	2	3	3	X
cana-5238	410	3	)	)	PUNCT
cana-5238	410	4	⇒	⇒	NOUN
cana-5238	410	5	(	(	PUNCT
cana-5238	410	6	1	1	NUM
cana-5238	410	7	)	)	PUNCT
cana-5238	410	8	.	.	PUNCT
cana-5238	411	1	this	this	PRON
cana-5238	411	2	is	be	AUX
cana-5238	411	3	from	from	ADP
cana-5238	411	4	theorem	theorem	ADJ
cana-5238	411	5	3.32	3.32	NUM
cana-5238	411	6	.	.	PUNCT
cana-5238	412	1	theorem	theorem	VERB
cana-5238	412	2	3.41	3.41	NUM
cana-5238	412	3	.	.	PUNCT
cana-5238	413	1	let	let	AUX
cana-5238	413	2	(	(	PUNCT
cana-5238	413	3	z	z	NOUN
cana-5238	413	4	,	,	PUNCT
cana-5238	413	5	ζ	ζ	NOUN
cana-5238	413	6	,	,	PUNCT
cana-5238	413	7	h	h	NOUN
cana-5238	413	8	)	)	PUNCT
cana-5238	413	9	be	be	VERB
cana-5238	413	10	a	a	DET
cana-5238	413	11	strong	strong	ADJ
cana-5238	413	12	hgts	hgts	NOUN
cana-5238	413	13	,	,	PUNCT
cana-5238	413	14	where	where	SCONJ
cana-5238	413	15	z	z	NOUN
cana-5238	413	16	is	be	AUX
cana-5238	413	17	c0	c0	PROPN
cana-5238	413	18	-space	-space	PROPN
cana-5238	413	19	and	and	CCONJ
cana-5238	413	20	ψ⊂z	ψ⊂z	NOUN
cana-5238	413	21	.	.	PUNCT
cana-5238	414	1	then	then	ADV
cana-5238	414	2	the	the	DET
cana-5238	414	3	following	follow	VERB
cana-5238	414	4	conditions	condition	NOUN
cana-5238	414	5	are	be	AUX
cana-5238	414	6	equivalent	equivalent	ADJ
cana-5238	414	7	.	.	PUNCT
cana-5238	415	1	1	1	X
cana-5238	415	2	.	.	X
cana-5238	415	3	ψ	ψ	NOUN
cana-5238	415	4	is	be	AUX
cana-5238	415	5	gζ	gζ	NOUN
cana-5238	415	6	-	-	PUNCT
cana-5238	415	7	o	o	NOUN
cana-5238	415	8	,	,	PUNCT
cana-5238	415	9	2	2	NUM
cana-5238	415	10	.	.	PUNCT
cana-5238	415	11	ψ	ψ	NOUN
cana-5238	415	12	is	be	AUX
cana-5238	415	13	π	π	PROPN
cana-5238	415	14	-	-	PUNCT
cana-5238	415	15	hg	hg	NOUN
cana-5238	415	16	-	-	NOUN
cana-5238	415	17	o	o	NOUN
cana-5238	415	18	and	and	CCONJ
cana-5238	415	19	∆∗-b	∆∗-b	PROPN
cana-5238	415	20	-	-	PUNCT
cana-5238	415	21	hg	hg	NOUN
cana-5238	415	22	-	-	PUNCT
cana-5238	415	23	s	s	NOUN
cana-5238	415	24	,	,	PUNCT
cana-5238	415	25	3	3	NUM
cana-5238	415	26	.	.	PUNCT
cana-5238	415	27	ψ	ψ	NOUN
cana-5238	415	28	is	be	AUX
cana-5238	415	29	β	β	X
cana-5238	415	30	-	-	ADJ
cana-5238	415	31	hg	hg	NOUN
cana-5238	415	32	-	-	NOUN
cana-5238	415	33	o	o	NOUN
cana-5238	415	34	and	and	CCONJ
cana-5238	415	35	∆∗-b	∆∗-b	PROPN
cana-5238	415	36	-	-	PUNCT
cana-5238	415	37	hg	hg	NOUN
cana-5238	415	38	-	-	PUNCT
cana-5238	415	39	s.	s.	PROPN
cana-5238	415	40	proof	proof	NOUN
cana-5238	415	41	.	.	PUNCT
cana-5238	416	1	(	(	PUNCT
cana-5238	416	2	1	1	X
cana-5238	416	3	)	)	PUNCT
cana-5238	416	4	⇒	⇒	NOUN
cana-5238	416	5	(	(	PUNCT
cana-5238	416	6	2	2	NUM
cana-5238	416	7	)	)	PUNCT
cana-5238	416	8	.	.	PUNCT
cana-5238	417	1	let	let	VERB
cana-5238	417	2	a	a	DET
cana-5238	417	3	subset	subset	NOUN
cana-5238	417	4	ψ	ψ	X
cana-5238	417	5	of	of	ADP
cana-5238	417	6	z	z	PROPN
cana-5238	417	7	is	be	AUX
cana-5238	417	8	gζ	gζ	PROPN
cana-5238	417	9	-	-	PUNCT
cana-5238	417	10	o.	o.	NOUN
cana-5238	418	1	then	then	ADV
cana-5238	418	2	it	it	PRON
cana-5238	418	3	is	be	AUX
cana-5238	418	4	π	π	PROPN
cana-5238	418	5	-	-	PUNCT
cana-5238	418	6	hg	hg	NOUN
cana-5238	418	7	-	-	NOUN
cana-5238	418	8	o	o	NOUN
cana-5238	418	9	and	and	CCONJ
cana-5238	418	10	∆∗-b	∆∗-b	PROPN
cana-5238	418	11	-	-	PUNCT
cana-5238	418	12	hg	hg	NOUN
cana-5238	418	13	-	-	PUNCT
cana-5238	418	14	s	s	X
cana-5238	418	15	(	(	PUNCT
cana-5238	418	16	2)⇒(3	2)⇒(3	NUM
cana-5238	418	17	)	)	PUNCT
cana-5238	418	18	.	.	PUNCT
cana-5238	419	1	let	let	VERB
cana-5238	419	2	a	a	DET
cana-5238	419	3	subset	subset	NOUN
cana-5238	419	4	ψ	ψ	X
cana-5238	419	5	of	of	ADP
cana-5238	419	6	z	z	PROPN
cana-5238	419	7	is	be	AUX
cana-5238	419	8	both	both	PRON
cana-5238	419	9	π	π	PROPN
cana-5238	419	10	-	-	PUNCT
cana-5238	419	11	hg	hg	NOUN
cana-5238	419	12	-	-	NOUN
cana-5238	419	13	o	o	NOUN
cana-5238	419	14	and	and	CCONJ
cana-5238	419	15	∆∗-b	∆∗-b	PROPN
cana-5238	419	16	-	-	PUNCT
cana-5238	419	17	hgs	hgs	PROPN
cana-5238	419	18	.	.	PUNCT
cana-5238	420	1	then	then	ADV
cana-5238	420	2	it	it	PRON
cana-5238	420	3	is	be	AUX
cana-5238	420	4	both	both	PRON
cana-5238	420	5	β	β	NOUN
cana-5238	420	6	-	-	PUNCT
cana-5238	420	7	hg	hg	NOUN
cana-5238	420	8	-	-	NOUN
cana-5238	420	9	o	o	NOUN
cana-5238	420	10	and	and	CCONJ
cana-5238	420	11	∆∗-b	∆∗-b	PROPN
cana-5238	420	12	-	-	PUNCT
cana-5238	420	13	hg	hg	NOUN
cana-5238	420	14	-	-	PUNCT
cana-5238	420	15	s.	s.	PROPN
cana-5238	420	16	(	(	PUNCT
cana-5238	420	17	3	3	X
cana-5238	420	18	)	)	PUNCT
cana-5238	420	19	⇒	⇒	NOUN
cana-5238	420	20	(	(	PUNCT
cana-5238	420	21	1	1	NUM
cana-5238	420	22	)	)	PUNCT
cana-5238	420	23	.	.	PUNCT
cana-5238	421	1	this	this	PRON
cana-5238	421	2	is	be	AUX
cana-5238	421	3	from	from	ADP
cana-5238	421	4	theorem	theorem	ADJ
cana-5238	421	5	3.32	3.32	NUM
cana-5238	421	6	.	.	NOUN
cana-5238	421	7	4	4	NUM
cana-5238	421	8	.	.	PUNCT
cana-5238	421	9	decomposition	decomposition	NOUN
cana-5238	421	10	of	of	ADP
cana-5238	421	11	(	(	PUNCT
cana-5238	421	12	gζ	gζ	PROPN
cana-5238	421	13	,	,	PUNCT
cana-5238	421	14	ξ	ξ	NOUN
cana-5238	421	15	)	)	PUNCT
cana-5238	421	16	-continuity	-continuity	ADJ
cana-5238	421	17	definition	definition	NOUN
cana-5238	421	18	4.1	4.1	NUM
cana-5238	421	19	.	.	PUNCT
cana-5238	422	1	a	a	DET
cana-5238	422	2	map	map	NOUN
cana-5238	422	3	ν:(z	ν:(z	NOUN
cana-5238	422	4	,	,	PUNCT
cana-5238	422	5	ζ	ζ	NOUN
cana-5238	422	6	,	,	PUNCT
cana-5238	422	7	h)→	h)→	NUM
cana-5238	422	8	(	(	PUNCT
cana-5238	422	9	w	w	PROPN
cana-5238	422	10	,	,	PUNCT
cana-5238	422	11	ξ	ξ	NOUN
cana-5238	422	12	)	)	PUNCT
cana-5238	422	13	is	be	AUX
cana-5238	422	14	(	(	PUNCT
cana-5238	422	15	bhg	bhg	PROPN
cana-5238	422	16	,	,	PUNCT
cana-5238	422	17	ξ)-c	ξ)-c	VERB
cana-5238	422	18	,	,	PUNCT
cana-5238	422	19	if	if	SCONJ
cana-5238	422	20	j−1(v	j−1(v	PROPN
cana-5238	422	21	)	)	PUNCT
cana-5238	422	22	is	be	AUX
cana-5238	422	23	b	b	NUM
cana-5238	422	24	-	-	PUNCT
cana-5238	422	25	hg	hg	NOUN
cana-5238	422	26	-	-	NOUN
cana-5238	422	27	o	o	NOUN
cana-5238	422	28	for	for	ADP
cana-5238	422	29	each	each	DET
cana-5238	422	30	ξ	ξ	PROPN
cana-5238	422	31	-	-	NOUN
cana-5238	422	32	o	o	NOUN
cana-5238	422	33	set	set	NOUN
cana-5238	422	34	v	v	NOUN
cana-5238	422	35	in	in	ADP
cana-5238	422	36	(	(	PUNCT
cana-5238	422	37	w	w	PROPN
cana-5238	422	38	,	,	PUNCT
cana-5238	422	39	ξ	ξ	NOUN
cana-5238	422	40	)	)	PUNCT
cana-5238	422	41	.	.	PUNCT
cana-5238	423	1	definition	definition	NOUN
cana-5238	423	2	4.2	4.2	NUM
cana-5238	423	3	.	.	PUNCT
cana-5238	424	1	a	a	DET
cana-5238	424	2	map	map	NOUN
cana-5238	424	3	ν:(z	ν:(z	NOUN
cana-5238	424	4	,	,	PUNCT
cana-5238	424	5	ζ	ζ	NOUN
cana-5238	424	6	,	,	PUNCT
cana-5238	424	7	h)→(w	h)→(w	ADV
cana-5238	424	8	,	,	PUNCT
cana-5238	424	9	ξ	ξ	X
cana-5238	424	10	)	)	PUNCT
cana-5238	424	11	is	be	AUX
cana-5238	424	12	(	(	PUNCT
cana-5238	424	13	r∗g	r∗g	ADJ
cana-5238	424	14	,	,	PUNCT
cana-5238	424	15	ξ)-c	ξ)-c	VERB
cana-5238	424	16	(	(	PUNCT
cana-5238	424	17	(	(	PUNCT
cana-5238	424	18	r∗g	r∗g	ADJ
cana-5238	424	19	,	,	PUNCT
cana-5238	424	20	ξ)-c	ξ)-c	NOUN
cana-5238	424	21	)	)	PUNCT
cana-5238	424	22	,	,	PUNCT
cana-5238	424	23	(	(	PUNCT
cana-5238	424	24	resp	resp	NOUN
cana-5238	424	25	.	.	PUNCT
cana-5238	425	1	(	(	PUNCT
cana-5238	425	2	(	(	PUNCT
cana-5238	425	3	db(c	db(c	X
cana-5238	425	4	,	,	PUNCT
cana-5238	425	5	hg	hg	NOUN
cana-5238	425	6	)	)	PUNCT
cana-5238	425	7	,	,	PUNCT
cana-5238	425	8	ξ)c	ξ)c	NOUN
cana-5238	425	9	)	)	PUNCT
cana-5238	425	10	,	,	PUNCT
cana-5238	425	11	if	if	SCONJ
cana-5238	425	12	ν−1(v	ν−1(v	PROPN
cana-5238	425	13	)	)	PUNCT
cana-5238	425	14	is	be	AUX
cana-5238	425	15	r∗g	r∗g	NUM
cana-5238	425	16	set	set	VERB
cana-5238	425	17	(	(	PUNCT
cana-5238	425	18	resp	resp	NOUN
cana-5238	425	19	.	.	PUNCT
cana-5238	426	1	(	(	PUNCT
cana-5238	426	2	db(c	db(c	X
cana-5238	426	3	,	,	PUNCT
cana-5238	426	4	hg)-s	hg)-s	PROPN
cana-5238	426	5	for	for	ADP
cana-5238	426	6	each	each	DET
cana-5238	426	7	ξ	ξ	PROPN
cana-5238	426	8	-	-	NOUN
cana-5238	426	9	o	o	NOUN
cana-5238	426	10	set	set	NOUN
cana-5238	426	11	v	v	NOUN
cana-5238	426	12	in	in	ADP
cana-5238	426	13	(	(	PUNCT
cana-5238	426	14	w	w	PROPN
cana-5238	426	15	,	,	PUNCT
cana-5238	426	16	ξ	ξ	NOUN
cana-5238	426	17	)	)	PUNCT
cana-5238	426	18	.	.	PUNCT
cana-5238	427	1	definition	definition	NOUN
cana-5238	427	2	4.3	4.3	NUM
cana-5238	427	3	.	.	PUNCT
cana-5238	428	1	a	a	DET
cana-5238	428	2	function	function	NOUN
cana-5238	428	3	ν:(z	ν:(z	NOUN
cana-5238	428	4	,	,	PUNCT
cana-5238	428	5	ζ	ζ	NOUN
cana-5238	428	6	,	,	PUNCT
cana-5238	428	7	h)→(w	h)→(w	ADV
cana-5238	428	8	,	,	PUNCT
cana-5238	428	9	ξ	ξ	X
cana-5238	428	10	)	)	PUNCT
cana-5238	428	11	is	be	AUX
cana-5238	428	12	said	say	VERB
cana-5238	428	13	to	to	PART
cana-5238	428	14	be	be	AUX
cana-5238	428	15	(	(	PUNCT
cana-5238	428	16	α∗-b	α∗-b	PROPN
cana-5238	428	17	-	-	PUNCT
cana-5238	428	18	hg	hg	NOUN
cana-5238	428	19	,	,	PUNCT
cana-5238	428	20	ξ)-c	ξ)-c	NOUN
cana-5238	428	21	(	(	PUNCT
cana-5238	428	22	resp	resp	NOUN
cana-5238	428	23	.	.	PUNCT
cana-5238	429	1	(	(	PUNCT
cana-5238	429	2	π∗-b	π∗-b	PROPN
cana-5238	429	3	-	-	PUNCT
cana-5238	429	4	hg	hg	NOUN
cana-5238	429	5	,	,	PUNCT
cana-5238	429	6	ξ)-c	ξ)-c	NOUN
cana-5238	429	7	,	,	PUNCT
cana-5238	429	8	(	(	PUNCT
cana-5238	429	9	σ∗b	σ∗b	NOUN
cana-5238	429	10	-	-	PUNCT
cana-5238	429	11	hg	hg	NOUN
cana-5238	429	12	,	,	PUNCT
cana-5238	429	13	ξ)-c	ξ)-c	NOUN
cana-5238	429	14	,	,	PUNCT
cana-5238	429	15	(	(	PUNCT
cana-5238	429	16	b∗-b	b∗-b	NOUN
cana-5238	429	17	-	-	PUNCT
cana-5238	429	18	hg	hg	NOUN
cana-5238	429	19	,	,	PUNCT
cana-5238	429	20	ξ)-c	ξ)-c	NOUN
cana-5238	429	21	,	,	PUNCT
cana-5238	429	22	(	(	PUNCT
cana-5238	429	23	β∗-b	β∗-b	NOUN
cana-5238	429	24	-	-	PUNCT
cana-5238	429	25	hg	hg	NOUN
cana-5238	429	26	,	,	PUNCT
cana-5238	429	27	ξ)-c	ξ)-c	NOUN
cana-5238	429	28	)	)	PUNCT
cana-5238	429	29	,	,	PUNCT
cana-5238	429	30	if	if	SCONJ
cana-5238	429	31	ν−1(v	ν−1(v	PROPN
cana-5238	429	32	)	)	PUNCT
cana-5238	429	33	is	be	AUX
cana-5238	429	34	α∗-b	α∗-b	PROPN
cana-5238	429	35	-	-	PUNCT
cana-5238	429	36	hg	hg	NOUN
cana-5238	429	37	-	-	PUNCT
cana-5238	429	38	s	s	X
cana-5238	429	39	(	(	PUNCT
cana-5238	429	40	resp	resp	NOUN
cana-5238	429	41	.	.	PUNCT
cana-5238	430	1	π∗-b	π∗-b	PROPN
cana-5238	430	2	-	-	PUNCT
cana-5238	430	3	hg	hg	NOUN
cana-5238	430	4	-	-	PUNCT
cana-5238	430	5	s	s	NOUN
cana-5238	430	6	,	,	PUNCT
cana-5238	430	7	σ∗-b	σ∗-b	PROPN
cana-5238	430	8	-	-	PUNCT
cana-5238	430	9	hg	hg	NOUN
cana-5238	430	10	-	-	PUNCT
cana-5238	430	11	s	s	NOUN
cana-5238	430	12	,	,	PUNCT
cana-5238	430	13	b∗b	b∗b	NUM
cana-5238	430	14	-	-	PUNCT
cana-5238	430	15	hg	hg	NOUN
cana-5238	430	16	-	-	PUNCT
cana-5238	430	17	s	s	NOUN
cana-5238	430	18	,	,	PUNCT
cana-5238	430	19	β∗-b	β∗-b	PUNCT
cana-5238	430	20	-	-	PUNCT
cana-5238	430	21	hg	hg	NOUN
cana-5238	430	22	-	-	PUNCT
cana-5238	430	23	s	s	NOUN
cana-5238	430	24	)	)	PUNCT
cana-5238	430	25	for	for	ADP
cana-5238	430	26	each	each	DET
cana-5238	430	27	ξ	ξ	PROPN
cana-5238	430	28	-	-	NOUN
cana-5238	430	29	o	o	NOUN
cana-5238	430	30	set	set	NOUN
cana-5238	430	31	v	v	NOUN
cana-5238	430	32	in	in	ADP
cana-5238	430	33	(	(	PUNCT
cana-5238	430	34	w	w	PROPN
cana-5238	430	35	,	,	PUNCT
cana-5238	430	36	ξ	ξ	NOUN
cana-5238	430	37	)	)	PUNCT
cana-5238	430	38	.	.	PUNCT
cana-5238	431	1	definition	definition	NOUN
cana-5238	431	2	4.4	4.4	NUM
cana-5238	431	3	.	.	PUNCT
cana-5238	432	1	a	a	DET
cana-5238	432	2	function	function	NOUN
cana-5238	432	3	ν:(z	ν:(z	NOUN
cana-5238	432	4	,	,	PUNCT
cana-5238	432	5	ζ	ζ	NOUN
cana-5238	432	6	,	,	PUNCT
cana-5238	432	7	h)→(w	h)→(w	ADV
cana-5238	432	8	,	,	PUNCT
cana-5238	432	9	ξ	ξ	X
cana-5238	432	10	)	)	PUNCT
cana-5238	432	11	is	be	AUX
cana-5238	432	12	said	say	VERB
cana-5238	432	13	to	to	PART
cana-5238	432	14	be	be	AUX
cana-5238	432	15	(	(	PUNCT
cana-5238	432	16	ξ∗-b	ξ∗-b	NOUN
cana-5238	432	17	-	-	PUNCT
cana-5238	432	18	hg	hg	NOUN
cana-5238	432	19	,	,	PUNCT
cana-5238	432	20	ξ)-c	ξ)-c	NOUN
cana-5238	432	21	(	(	PUNCT
cana-5238	432	22	resp	resp	NOUN
cana-5238	432	23	.	.	PUNCT
cana-5238	433	1	(	(	PUNCT
cana-5238	433	2	π∗-b	π∗-b	PROPN
cana-5238	433	3	-	-	PUNCT
cana-5238	433	4	hg	hg	NOUN
cana-5238	433	5	,	,	PUNCT
cana-5238	433	6	ξ)-c	ξ)-c	NOUN
cana-5238	433	7	,	,	PUNCT
cana-5238	433	8	(	(	PUNCT
cana-5238	433	9	σ∗b	σ∗b	NOUN
cana-5238	433	10	-	-	PUNCT
cana-5238	433	11	hg	hg	NOUN
cana-5238	433	12	,	,	PUNCT
cana-5238	433	13	ξ)-c	ξ)-c	NOUN
cana-5238	433	14	,	,	PUNCT
cana-5238	433	15	(	(	PUNCT
cana-5238	433	16	φ∗-b	φ∗-b	NOUN
cana-5238	433	17	-	-	PUNCT
cana-5238	433	18	hg	hg	NOUN
cana-5238	433	19	,	,	PUNCT
cana-5238	433	20	ξ)-c	ξ)-c	NOUN
cana-5238	433	21	,	,	PUNCT
cana-5238	433	22	(	(	PUNCT
cana-5238	433	23	∆∗-b	∆∗-b	PROPN
cana-5238	433	24	-	-	PUNCT
cana-5238	433	25	hg	hg	NOUN
cana-5238	433	26	,	,	PUNCT
cana-5238	433	27	ξ)-c	ξ)-c	NOUN
cana-5238	433	28	)	)	PUNCT
cana-5238	433	29	,	,	PUNCT
cana-5238	433	30	if	if	SCONJ
cana-5238	433	31	ν−1(v	ν−1(v	PROPN
cana-5238	433	32	)	)	PUNCT
cana-5238	433	33	is	be	AUX
cana-5238	433	34	ξ∗-b	ξ∗-b	PROPN
cana-5238	433	35	-	-	PUNCT
cana-5238	433	36	hg	hg	NOUN
cana-5238	433	37	-	-	PUNCT
cana-5238	433	38	s	s	X
cana-5238	433	39	(	(	PUNCT
cana-5238	433	40	resp	resp	NOUN
cana-5238	433	41	.	.	PUNCT
cana-5238	434	1	π∗-b	π∗-b	PROPN
cana-5238	434	2	-	-	PUNCT
cana-5238	434	3	hg	hg	NOUN
cana-5238	434	4	-	-	PUNCT
cana-5238	434	5	s	s	NOUN
cana-5238	434	6	,	,	PUNCT
cana-5238	434	7	∆∗-b	∆∗-b	PROPN
cana-5238	434	8	-	-	PUNCT
cana-5238	434	9	hg	hg	NOUN
cana-5238	434	10	-	-	NOUN
cana-5238	434	11	s	s	PART
cana-5238	434	12	)	)	PUNCT
cana-5238	434	13	for	for	ADP
cana-5238	434	14	each	each	DET
cana-5238	434	15	ξ	ξ	X
cana-5238	434	16	o	o	NOUN
cana-5238	434	17	set	set	VERB
cana-5238	434	18	v	v	NOUN
cana-5238	434	19	in	in	ADP
cana-5238	434	20	(	(	PUNCT
cana-5238	434	21	w	w	PROPN
cana-5238	434	22	,	,	PUNCT
cana-5238	434	23	ξ	ξ	NOUN
cana-5238	434	24	)	)	PUNCT
cana-5238	434	25	.	.	PUNCT
cana-5238	435	1	communications	communication	NOUN
cana-5238	435	2	on	on	ADP
cana-5238	435	3	applied	apply	VERB
cana-5238	435	4	nonlinear	nonlinear	ADJ
cana-5238	435	5	analysis	analysis	NOUN
cana-5238	435	6	issn	issn	NOUN
cana-5238	435	7	:	:	PUNCT
cana-5238	435	8	1074	1074	NUM
cana-5238	435	9	-	-	PUNCT
cana-5238	435	10	133x	133x	NUM
cana-5238	435	11	vol	vol	VERB
cana-5238	435	12	32	32	NUM
cana-5238	435	13	no	no	NOUN
cana-5238	435	14	.	.	PUNCT
cana-5238	436	1	10s	10	NOUN
cana-5238	436	2	(	(	PUNCT
cana-5238	436	3	2025	2025	NUM
cana-5238	436	4	)	)	PUNCT
cana-5238	436	5	1368	1368	NUM
cana-5238	436	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	436	7	theorem	theorem	VERB
cana-5238	436	8	4.5	4.5	NUM
cana-5238	436	9	.	.	PUNCT
cana-5238	437	1	for	for	ADP
cana-5238	437	2	a	a	DET
cana-5238	437	3	map	map	NOUN
cana-5238	437	4	ν:(z	ν:(z	NOUN
cana-5238	437	5	,	,	PUNCT
cana-5238	437	6	ζ	ζ	NOUN
cana-5238	437	7	,	,	PUNCT
cana-5238	437	8	h)→(w	h)→(w	ADV
cana-5238	437	9	,	,	PUNCT
cana-5238	437	10	ξ	ξ	NOUN
cana-5238	437	11	)	)	PUNCT
cana-5238	437	12	where	where	SCONJ
cana-5238	437	13	z	z	NOUN
cana-5238	437	14	is	be	AUX
cana-5238	437	15	c0	c0	PROPN
cana-5238	437	16	-space	-space	PROPN
cana-5238	437	17	,	,	PUNCT
cana-5238	437	18	the	the	DET
cana-5238	437	19	following	follow	VERB
cana-5238	437	20	results	result	NOUN
cana-5238	437	21	are	be	AUX
cana-5238	437	22	equivalent	equivalent	ADJ
cana-5238	437	23	.	.	PUNCT
cana-5238	438	1	1	1	X
cana-5238	438	2	.	.	X
cana-5238	438	3	ν	ν	NOUN
cana-5238	438	4	is	be	AUX
cana-5238	438	5	(	(	PUNCT
cana-5238	438	6	gζ	gζ	NOUN
cana-5238	438	7	,	,	PUNCT
cana-5238	438	8	ξ)-c	ξ)-c	NOUN
cana-5238	438	9	,	,	PUNCT
cana-5238	438	10	2	2	NUM
cana-5238	438	11	.	.	PUNCT
cana-5238	439	1	ν	ν	NOUN
cana-5238	439	2	is	be	AUX
cana-5238	439	3	(	(	PUNCT
cana-5238	439	4	b	b	X
cana-5238	439	5	-	-	PUNCT
cana-5238	439	6	hg	hg	NOUN
cana-5238	439	7	,	,	PUNCT
cana-5238	439	8	ξ)c	ξ)c	NOUN
cana-5238	439	9	and	and	CCONJ
cana-5238	439	10	(	(	PUNCT
cana-5238	439	11	db(c	db(c	X
cana-5238	439	12	,	,	PUNCT
cana-5238	439	13	hg	hg	NOUN
cana-5238	439	14	)	)	PUNCT
cana-5238	439	15	,	,	PUNCT
cana-5238	439	16	ξ)-c	ξ)-c	NOUN
cana-5238	439	17	.	.	PUNCT
cana-5238	440	1	proof	proof	NOUN
cana-5238	440	2	.	.	PUNCT
cana-5238	441	1	the	the	DET
cana-5238	441	2	proof	proof	NOUN
cana-5238	441	3	is	be	AUX
cana-5238	441	4	clear	clear	ADJ
cana-5238	441	5	by	by	ADP
cana-5238	441	6	theorem	theorem	ADJ
cana-5238	441	7	2.53	2.53	NUM
cana-5238	441	8	.	.	PUNCT
cana-5238	442	1	theorem	theorem	VERB
cana-5238	442	2	4.6	4.6	NUM
cana-5238	442	3	.	.	PUNCT
cana-5238	443	1	for	for	ADP
cana-5238	443	2	a	a	DET
cana-5238	443	3	map	map	NOUN
cana-5238	443	4	ν:(z	ν:(z	NOUN
cana-5238	443	5	,	,	PUNCT
cana-5238	443	6	ζ	ζ	NOUN
cana-5238	443	7	,	,	PUNCT
cana-5238	443	8	h)→(w	h)→(w	ADV
cana-5238	443	9	,	,	PUNCT
cana-5238	443	10	ξ	ξ	NOUN
cana-5238	443	11	)	)	PUNCT
cana-5238	443	12	where	where	SCONJ
cana-5238	443	13	z	z	NOUN
cana-5238	443	14	is	be	AUX
cana-5238	443	15	c0	c0	PROPN
cana-5238	443	16	-space	-space	PROPN
cana-5238	443	17	,	,	PUNCT
cana-5238	443	18	the	the	DET
cana-5238	443	19	following	follow	VERB
cana-5238	443	20	results	result	NOUN
cana-5238	443	21	are	be	AUX
cana-5238	443	22	equivalent	equivalent	ADJ
cana-5238	443	23	.	.	PUNCT
cana-5238	444	1	1	1	X
cana-5238	444	2	.	.	X
cana-5238	444	3	ν	ν	NOUN
cana-5238	444	4	is	be	AUX
cana-5238	444	5	(	(	PUNCT
cana-5238	444	6	ζ	ζ	NOUN
cana-5238	444	7	,	,	PUNCT
cana-5238	444	8	ξ)-c	ξ)-c	NOUN
cana-5238	444	9	,	,	PUNCT
cana-5238	444	10	2	2	NUM
cana-5238	444	11	.	.	PUNCT
cana-5238	445	1	ν	ν	NOUN
cana-5238	445	2	is	be	AUX
cana-5238	445	3	(	(	PUNCT
cana-5238	445	4	α	α	NOUN
cana-5238	445	5	-	-	PUNCT
cana-5238	445	6	hg	hg	NOUN
cana-5238	445	7	,	,	PUNCT
cana-5238	445	8	ξ)-c	ξ)-c	VERB
cana-5238	445	9	and	and	CCONJ
cana-5238	445	10	(	(	PUNCT
cana-5238	445	11	α∗-b	α∗-b	PROPN
cana-5238	445	12	-	-	PUNCT
cana-5238	445	13	hg	hg	NOUN
cana-5238	445	14	,	,	PUNCT
cana-5238	445	15	ξ)-c	ξ)-c	NOUN
cana-5238	445	16	,	,	PUNCT
cana-5238	445	17	3	3	X
cana-5238	445	18	.	.	PUNCT
cana-5238	446	1	ν	ν	NOUN
cana-5238	446	2	is	be	AUX
cana-5238	446	3	(	(	PUNCT
cana-5238	446	4	σ	σ	PROPN
cana-5238	446	5	-	-	PUNCT
cana-5238	446	6	hg	hg	NOUN
cana-5238	446	7	,	,	PUNCT
cana-5238	446	8	ξ)-c	ξ)-c	VERB
cana-5238	446	9	and	and	CCONJ
cana-5238	446	10	(	(	PUNCT
cana-5238	446	11	σ∗-b	σ∗-b	PROPN
cana-5238	446	12	-	-	PUNCT
cana-5238	446	13	hg	hg	NOUN
cana-5238	446	14	,	,	PUNCT
cana-5238	446	15	ξ)-c	ξ)-c	NOUN
cana-5238	446	16	,	,	PUNCT
cana-5238	446	17	4	4	NUM
cana-5238	446	18	.	.	PUNCT
cana-5238	447	1	ν	ν	NOUN
cana-5238	447	2	is	be	AUX
cana-5238	447	3	(	(	PUNCT
cana-5238	447	4	π	π	PROPN
cana-5238	447	5	-	-	PUNCT
cana-5238	447	6	hg	hg	NOUN
cana-5238	447	7	,	,	PUNCT
cana-5238	447	8	ξ)-c	ξ)-c	VERB
cana-5238	447	9	and	and	CCONJ
cana-5238	447	10	(	(	PUNCT
cana-5238	447	11	π∗-b	π∗-b	PROPN
cana-5238	447	12	-	-	PUNCT
cana-5238	447	13	hg	hg	NOUN
cana-5238	447	14	,	,	PUNCT
cana-5238	447	15	ξ)-c	ξ)-c	NOUN
cana-5238	447	16	,	,	PUNCT
cana-5238	447	17	5	5	NUM
cana-5238	447	18	.	.	PUNCT
cana-5238	448	1	ν	ν	NOUN
cana-5238	448	2	is	be	AUX
cana-5238	448	3	(	(	PUNCT
cana-5238	448	4	β	β	NOUN
cana-5238	448	5	-	-	PUNCT
cana-5238	448	6	hg	hg	NOUN
cana-5238	448	7	,	,	PUNCT
cana-5238	448	8	ξ)-c	ξ)-c	VERB
cana-5238	448	9	and	and	CCONJ
cana-5238	448	10	(	(	PUNCT
cana-5238	448	11	β∗-b	β∗-b	NOUN
cana-5238	448	12	-	-	PUNCT
cana-5238	448	13	hg	hg	NOUN
cana-5238	448	14	,	,	PUNCT
cana-5238	448	15	ξ)c	ξ)c	NOUN
cana-5238	448	16	.	.	PUNCT
cana-5238	449	1	proof	proof	NOUN
cana-5238	449	2	.	.	PUNCT
cana-5238	450	1	the	the	DET
cana-5238	450	2	proof	proof	NOUN
cana-5238	450	3	is	be	AUX
cana-5238	450	4	clear	clear	ADJ
cana-5238	450	5	by	by	ADP
cana-5238	450	6	theorem	theorem	ADJ
cana-5238	450	7	3.9	3.9	NUM
cana-5238	450	8	.	.	PUNCT
cana-5238	451	1	theorem	theorem	VERB
cana-5238	451	2	4.7	4.7	NUM
cana-5238	451	3	.	.	PUNCT
cana-5238	452	1	for	for	ADP
cana-5238	452	2	a	a	DET
cana-5238	452	3	map	map	NOUN
cana-5238	452	4	ν:(z	ν:(z	NOUN
cana-5238	452	5	,	,	PUNCT
cana-5238	452	6	ζ	ζ	NOUN
cana-5238	452	7	,	,	PUNCT
cana-5238	452	8	h)→(w	h)→(w	ADV
cana-5238	452	9	,	,	PUNCT
cana-5238	452	10	ξ	ξ	NOUN
cana-5238	452	11	)	)	PUNCT
cana-5238	452	12	where	where	SCONJ
cana-5238	452	13	z	z	NOUN
cana-5238	452	14	is	be	AUX
cana-5238	452	15	c0	c0	PROPN
cana-5238	452	16	-space	-space	PROPN
cana-5238	452	17	,	,	PUNCT
cana-5238	452	18	the	the	DET
cana-5238	452	19	following	follow	VERB
cana-5238	452	20	results	result	NOUN
cana-5238	452	21	are	be	AUX
cana-5238	452	22	equivalent	equivalent	ADJ
cana-5238	452	23	.	.	PUNCT
cana-5238	453	1	1	1	X
cana-5238	453	2	.	.	X
cana-5238	453	3	ν	ν	NOUN
cana-5238	453	4	is	be	AUX
cana-5238	453	5	(	(	PUNCT
cana-5238	453	6	ζ	ζ	NOUN
cana-5238	453	7	,	,	PUNCT
cana-5238	453	8	ξ)-c	ξ)-c	NOUN
cana-5238	453	9	,	,	PUNCT
cana-5238	453	10	2	2	NUM
cana-5238	453	11	.	.	PUNCT
cana-5238	454	1	ν	ν	NOUN
cana-5238	454	2	is	be	AUX
cana-5238	454	3	(	(	PUNCT
cana-5238	454	4	σ	σ	PROPN
cana-5238	454	5	-	-	PUNCT
cana-5238	454	6	hg	hg	NOUN
cana-5238	454	7	,	,	PUNCT
cana-5238	454	8	ξ)-c	ξ)-c	VERB
cana-5238	454	9	and	and	CCONJ
cana-5238	454	10	(	(	PUNCT
cana-5238	454	11	b∗	b∗	ADJ
cana-5238	454	12	b	b	X
cana-5238	454	13	hg	hg	PROPN
cana-5238	454	14	,	,	PUNCT
cana-5238	454	15	ξ	ξ	PROPN
cana-5238	454	16	)	)	PUNCT
cana-5238	454	17	c	c	NOUN
cana-5238	454	18	,	,	PUNCT
cana-5238	454	19	3	3	NUM
cana-5238	454	20	.	.	PUNCT
cana-5238	455	1	ν	ν	NOUN
cana-5238	455	2	is	be	AUX
cana-5238	455	3	(	(	PUNCT
cana-5238	455	4	π	π	PROPN
cana-5238	455	5	-	-	PUNCT
cana-5238	455	6	hg	hg	NOUN
cana-5238	455	7	,	,	PUNCT
cana-5238	455	8	ξ)-c	ξ)-c	VERB
cana-5238	455	9	and	and	CCONJ
cana-5238	455	10	(	(	PUNCT
cana-5238	455	11	b∗	b∗	ADJ
cana-5238	455	12	b	b	X
cana-5238	455	13	hg	hg	PROPN
cana-5238	455	14	,	,	PUNCT
cana-5238	455	15	ξ	ξ	PROPN
cana-5238	455	16	)	)	PUNCT
cana-5238	455	17	c	c	NOUN
cana-5238	455	18	,	,	PUNCT
cana-5238	455	19	4	4	NUM
cana-5238	455	20	.	.	PUNCT
cana-5238	456	1	ν	ν	NOUN
cana-5238	456	2	is	be	AUX
cana-5238	456	3	(	(	PUNCT
cana-5238	456	4	b	b	PROPN
cana-5238	456	5	hg	hg	X
cana-5238	456	6	,	,	PUNCT
cana-5238	456	7	ξ	ξ	PROPN
cana-5238	456	8	)	)	PUNCT
cana-5238	456	9	c	c	NOUN
cana-5238	456	10	and	and	CCONJ
cana-5238	456	11	(	(	PUNCT
cana-5238	456	12	b∗	b∗	PROPN
cana-5238	456	13	b	b	X
cana-5238	456	14	hg	hg	PROPN
cana-5238	456	15	,	,	PUNCT
cana-5238	456	16	ξ	ξ	NOUN
cana-5238	456	17	)	)	PUNCT
cana-5238	456	18	c.	c.	NOUN
cana-5238	456	19	proof	proof	NOUN
cana-5238	456	20	.	.	PUNCT
cana-5238	457	1	the	the	DET
cana-5238	457	2	proof	proof	NOUN
cana-5238	457	3	is	be	AUX
cana-5238	457	4	clear	clear	ADJ
cana-5238	457	5	by	by	ADP
cana-5238	457	6	theorem	theorem	ADJ
cana-5238	457	7	3.12	3.12	NUM
cana-5238	457	8	.	.	PUNCT
cana-5238	458	1	theorem	theorem	NOUN
cana-5238	458	2	4.8	4.8	NUM
cana-5238	458	3	.	.	PUNCT
cana-5238	459	1	let	let	AUX
cana-5238	459	2	(	(	PUNCT
cana-5238	459	3	z	z	NOUN
cana-5238	459	4	,	,	PUNCT
cana-5238	459	5	ζ	ζ	NOUN
cana-5238	459	6	,	,	PUNCT
cana-5238	459	7	h	h	NOUN
cana-5238	459	8	)	)	PUNCT
cana-5238	459	9	be	be	VERB
cana-5238	459	10	a	a	DET
cana-5238	459	11	strong	strong	ADJ
cana-5238	459	12	hgts	hgts	NOUN
cana-5238	459	13	for	for	ADP
cana-5238	459	14	a	a	DET
cana-5238	459	15	function	function	NOUN
cana-5238	459	16	ν:(z	ν:(z	NOUN
cana-5238	459	17	,	,	PUNCT
cana-5238	459	18	ζ	ζ	NOUN
cana-5238	459	19	,	,	PUNCT
cana-5238	459	20	h	h	NOUN
cana-5238	459	21	)	)	PUNCT
cana-5238	459	22	→(w	→(w	NOUN
cana-5238	459	23	,	,	PUNCT
cana-5238	459	24	ξ	ξ	X
cana-5238	459	25	)	)	PUNCT
cana-5238	459	26	,	,	PUNCT
cana-5238	459	27	z	z	PROPN
cana-5238	459	28	is	be	AUX
cana-5238	459	29	c0	c0	PROPN
cana-5238	459	30	-space	-space	PROPN
cana-5238	459	31	.	.	PUNCT
cana-5238	460	1	then	then	ADV
cana-5238	460	2	the	the	DET
cana-5238	460	3	following	follow	VERB
cana-5238	460	4	conditions	condition	NOUN
cana-5238	460	5	are	be	AUX
cana-5238	460	6	equivalent	equivalent	ADJ
cana-5238	460	7	.	.	PUNCT
cana-5238	461	1	1	1	X
cana-5238	461	2	.	.	X
cana-5238	461	3	ν	ν	NOUN
cana-5238	461	4	is	be	AUX
cana-5238	461	5	(	(	PUNCT
cana-5238	461	6	ζ	ζ	NOUN
cana-5238	461	7	,	,	PUNCT
cana-5238	461	8	ξ)-c	ξ)-c	NOUN
cana-5238	461	9	,	,	PUNCT
cana-5238	461	10	2	2	NUM
cana-5238	461	11	.	.	PUNCT
cana-5238	462	1	ν	ν	NOUN
cana-5238	462	2	is	be	AUX
cana-5238	462	3	(	(	PUNCT
cana-5238	462	4	α	α	NOUN
cana-5238	462	5	-	-	PUNCT
cana-5238	462	6	hg	hg	NOUN
cana-5238	462	7	,	,	PUNCT
cana-5238	462	8	ξ)-c	ξ)-c	VERB
cana-5238	462	9	and	and	CCONJ
cana-5238	462	10	(	(	PUNCT
cana-5238	462	11	σ∗-b	σ∗-b	PROPN
cana-5238	462	12	-	-	PUNCT
cana-5238	462	13	hg	hg	NOUN
cana-5238	462	14	,	,	PUNCT
cana-5238	462	15	ξ)c	ξ)c	NOUN
cana-5238	462	16	,	,	PUNCT
cana-5238	462	17	3	3	NUM
cana-5238	462	18	.	.	PUNCT
cana-5238	463	1	ν	ν	NOUN
cana-5238	463	2	is	be	AUX
cana-5238	463	3	(	(	PUNCT
cana-5238	463	4	σhg	σhg	NOUN
cana-5238	463	5	,	,	PUNCT
cana-5238	463	6	ξ)-c	ξ)-c	NOUN
cana-5238	463	7	and	and	CCONJ
cana-5238	463	8	(	(	PUNCT
cana-5238	463	9	σ∗-bhg	σ∗-bhg	ADJ
cana-5238	463	10	,	,	PUNCT
cana-5238	463	11	ξ)-c	ξ)-c	NOUN
cana-5238	463	12	.	.	PUNCT
cana-5238	464	1	proof	proof	NOUN
cana-5238	464	2	.	.	PUNCT
cana-5238	465	1	the	the	DET
cana-5238	465	2	proof	proof	NOUN
cana-5238	465	3	is	be	AUX
cana-5238	465	4	clear	clear	ADJ
cana-5238	465	5	by	by	ADP
cana-5238	465	6	theorem	theorem	NOUN
cana-5238	465	7	3.15	3.15	NUM
cana-5238	465	8	.	.	PUNCT
cana-5238	466	1	theorem	theorem	NOUN
cana-5238	466	2	4.9	4.9	NUM
cana-5238	466	3	.	.	PUNCT
cana-5238	467	1	let	let	AUX
cana-5238	467	2	(	(	PUNCT
cana-5238	467	3	z	z	NOUN
cana-5238	467	4	,	,	PUNCT
cana-5238	467	5	ζ	ζ	NOUN
cana-5238	467	6	,	,	PUNCT
cana-5238	467	7	h	h	NOUN
cana-5238	467	8	)	)	PUNCT
cana-5238	467	9	be	be	VERB
cana-5238	467	10	a	a	DET
cana-5238	467	11	strong	strong	ADJ
cana-5238	467	12	hgts	hgts	NOUN
cana-5238	467	13	for	for	ADP
cana-5238	467	14	a	a	DET
cana-5238	467	15	function	function	NOUN
cana-5238	467	16	ν:(z	ν:(z	NOUN
cana-5238	467	17	,	,	PUNCT
cana-5238	467	18	ζ	ζ	NOUN
cana-5238	467	19	,	,	PUNCT
cana-5238	467	20	h)→(w	h)→(w	ADV
cana-5238	467	21	,	,	PUNCT
cana-5238	467	22	ξ	ξ	NOUN
cana-5238	467	23	)	)	PUNCT
cana-5238	467	24	,	,	PUNCT
cana-5238	467	25	z	z	PROPN
cana-5238	467	26	is	be	AUX
cana-5238	467	27	c0	c0	PROPN
cana-5238	467	28	-space	-space	PROPN
cana-5238	467	29	.	.	PUNCT
cana-5238	468	1	then	then	ADV
cana-5238	468	2	the	the	DET
cana-5238	468	3	following	follow	VERB
cana-5238	468	4	conditions	condition	NOUN
cana-5238	468	5	are	be	AUX
cana-5238	468	6	equivalent	equivalent	ADJ
cana-5238	468	7	.	.	PUNCT
cana-5238	469	1	1	1	X
cana-5238	469	2	.	.	X
cana-5238	469	3	ν	ν	NOUN
cana-5238	469	4	is	be	AUX
cana-5238	469	5	(	(	PUNCT
cana-5238	469	6	ζ	ζ	NOUN
cana-5238	469	7	,	,	PUNCT
cana-5238	469	8	ξ)-c	ξ)-c	NOUN
cana-5238	469	9	,	,	PUNCT
cana-5238	469	10	2	2	X
cana-5238	469	11	.	.	PUNCT
cana-5238	470	1	ν	ν	NOUN
cana-5238	470	2	is	be	AUX
cana-5238	470	3	(	(	PUNCT
cana-5238	470	4	α	α	NOUN
cana-5238	470	5	hg	hg	NOUN
cana-5238	470	6	,	,	PUNCT
cana-5238	470	7	ξ)-c	ξ)-c	VERB
cana-5238	470	8	and	and	CCONJ
cana-5238	470	9	(	(	PUNCT
cana-5238	470	10	π∗-b	π∗-b	PROPN
cana-5238	470	11	-	-	PUNCT
cana-5238	470	12	hg	hg	NOUN
cana-5238	470	13	,	,	PUNCT
cana-5238	470	14	ξ)-c	ξ)-c	NOUN
cana-5238	470	15	,	,	PUNCT
cana-5238	470	16	3	3	X
cana-5238	470	17	.	.	PUNCT
cana-5238	471	1	ν	ν	NOUN
cana-5238	471	2	is	be	AUX
cana-5238	471	3	(	(	PUNCT
cana-5238	471	4	π	π	PROPN
cana-5238	471	5	-	-	PUNCT
cana-5238	471	6	hg	hg	NOUN
cana-5238	471	7	,	,	PUNCT
cana-5238	471	8	ξ)-c	ξ)-c	VERB
cana-5238	471	9	and	and	CCONJ
cana-5238	471	10	(	(	PUNCT
cana-5238	471	11	π∗-b	π∗-b	PROPN
cana-5238	471	12	-	-	PUNCT
cana-5238	471	13	hg	hg	NOUN
cana-5238	471	14	,	,	PUNCT
cana-5238	471	15	ξ)-c	ξ)-c	NOUN
cana-5238	471	16	.	.	PUNCT
cana-5238	472	1	proof	proof	NOUN
cana-5238	472	2	.	.	PUNCT
cana-5238	473	1	the	the	DET
cana-5238	473	2	proof	proof	NOUN
cana-5238	473	3	is	be	AUX
cana-5238	473	4	clear	clear	ADJ
cana-5238	473	5	by	by	ADP
cana-5238	473	6	theorem	theorem	ADJ
cana-5238	473	7	3.18	3.18	NUM
cana-5238	473	8	.	.	PUNCT
cana-5238	474	1	theorem	theorem	VERB
cana-5238	474	2	4.10	4.10	NUM
cana-5238	474	3	.	.	PUNCT
cana-5238	475	1	let	let	AUX
cana-5238	475	2	(	(	PUNCT
cana-5238	475	3	z	z	NOUN
cana-5238	475	4	,	,	PUNCT
cana-5238	475	5	ζ	ζ	NOUN
cana-5238	475	6	,	,	PUNCT
cana-5238	475	7	h	h	NOUN
cana-5238	475	8	)	)	PUNCT
cana-5238	475	9	be	be	VERB
cana-5238	475	10	a	a	DET
cana-5238	475	11	strong	strong	ADJ
cana-5238	475	12	hgts	hgts	NOUN
cana-5238	475	13	for	for	ADP
cana-5238	475	14	a	a	DET
cana-5238	475	15	function	function	NOUN
cana-5238	475	16	ν:(z	ν:(z	NOUN
cana-5238	475	17	,	,	PUNCT
cana-5238	475	18	ζ	ζ	NOUN
cana-5238	475	19	,	,	PUNCT
cana-5238	475	20	h)→(w	h)→(w	ADV
cana-5238	475	21	,	,	PUNCT
cana-5238	475	22	ξ	ξ	NOUN
cana-5238	475	23	)	)	PUNCT
cana-5238	475	24	,	,	PUNCT
cana-5238	475	25	where	where	SCONJ
cana-5238	475	26	z	z	NOUN
cana-5238	475	27	is	be	AUX
cana-5238	475	28	c0	c0	NOUN
cana-5238	475	29	space	space	NOUN
cana-5238	475	30	.	.	PUNCT
cana-5238	476	1	then	then	ADV
cana-5238	476	2	the	the	DET
cana-5238	476	3	following	follow	VERB
cana-5238	476	4	conditions	condition	NOUN
cana-5238	476	5	are	be	AUX
cana-5238	476	6	equivalent	equivalent	ADJ
cana-5238	476	7	.	.	PUNCT
cana-5238	477	1	1	1	X
cana-5238	477	2	.	.	X
cana-5238	477	3	ν	ν	NOUN
cana-5238	477	4	is	be	AUX
cana-5238	477	5	(	(	PUNCT
cana-5238	477	6	ζ	ζ	NOUN
cana-5238	477	7	,	,	PUNCT
cana-5238	477	8	ξ)-c	ξ)-c	NOUN
cana-5238	477	9	,	,	PUNCT
cana-5238	477	10	2	2	NUM
cana-5238	477	11	.	.	PUNCT
cana-5238	478	1	ν	ν	NOUN
cana-5238	478	2	is	be	AUX
cana-5238	478	3	(	(	PUNCT
cana-5238	478	4	α	α	NOUN
cana-5238	478	5	-	-	PUNCT
cana-5238	478	6	hg	hg	NOUN
cana-5238	478	7	,	,	PUNCT
cana-5238	478	8	ξ)-c	ξ)-c	VERB
cana-5238	478	9	and	and	CCONJ
cana-5238	478	10	(	(	PUNCT
cana-5238	478	11	β∗-b	β∗-b	NOUN
cana-5238	478	12	-	-	PUNCT
cana-5238	478	13	hg	hg	NOUN
cana-5238	478	14	,	,	PUNCT
cana-5238	478	15	ξ)-c	ξ)-c	NOUN
cana-5238	478	16	,	,	PUNCT
cana-5238	478	17	3	3	X
cana-5238	478	18	.	.	PUNCT
cana-5238	479	1	ν	ν	NOUN
cana-5238	479	2	is	be	AUX
cana-5238	479	3	(	(	PUNCT
cana-5238	479	4	β	β	NOUN
cana-5238	479	5	-	-	PUNCT
cana-5238	479	6	hg	hg	NOUN
cana-5238	479	7	,	,	PUNCT
cana-5238	479	8	ξ)-c	ξ)-c	VERB
cana-5238	479	9	and	and	CCONJ
cana-5238	479	10	(	(	PUNCT
cana-5238	479	11	β∗-b	β∗-b	NOUN
cana-5238	479	12	-	-	PUNCT
cana-5238	479	13	hg	hg	NOUN
cana-5238	479	14	,	,	PUNCT
cana-5238	479	15	ξ)-c	ξ)-c	NOUN
cana-5238	479	16	.	.	PUNCT
cana-5238	480	1	proof	proof	NOUN
cana-5238	480	2	.	.	PUNCT
cana-5238	481	1	the	the	DET
cana-5238	481	2	proof	proof	NOUN
cana-5238	481	3	is	be	AUX
cana-5238	481	4	clear	clear	ADJ
cana-5238	481	5	by	by	ADP
cana-5238	481	6	theorem	theorem	NOUN
cana-5238	481	7	3.19	3.19	NUM
cana-5238	481	8	.	.	PUNCT
cana-5238	482	1	communications	communication	NOUN
cana-5238	482	2	on	on	ADP
cana-5238	482	3	applied	apply	VERB
cana-5238	482	4	nonlinear	nonlinear	ADJ
cana-5238	482	5	analysis	analysis	NOUN
cana-5238	482	6	issn	issn	NOUN
cana-5238	482	7	:	:	PUNCT
cana-5238	482	8	1074	1074	NUM
cana-5238	482	9	-	-	PUNCT
cana-5238	482	10	133x	133x	NUM
cana-5238	482	11	vol	vol	VERB
cana-5238	482	12	32	32	NUM
cana-5238	482	13	no	no	NOUN
cana-5238	482	14	.	.	PUNCT
cana-5238	483	1	10s	10	NOUN
cana-5238	483	2	(	(	PUNCT
cana-5238	483	3	2025	2025	NUM
cana-5238	483	4	)	)	PUNCT
cana-5238	483	5	1369	1369	NUM
cana-5238	483	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	483	7	theorem	theorem	VERB
cana-5238	483	8	4.11	4.11	NUM
cana-5238	483	9	.	.	PUNCT
cana-5238	484	1	let	let	AUX
cana-5238	484	2	(	(	PUNCT
cana-5238	484	3	z	z	NOUN
cana-5238	484	4	,	,	PUNCT
cana-5238	484	5	ζ	ζ	NOUN
cana-5238	484	6	,	,	PUNCT
cana-5238	484	7	h	h	NOUN
cana-5238	484	8	)	)	PUNCT
cana-5238	484	9	be	be	VERB
cana-5238	484	10	a	a	DET
cana-5238	484	11	strong	strong	ADJ
cana-5238	484	12	hgts	hgts	NOUN
cana-5238	484	13	for	for	ADP
cana-5238	484	14	a	a	DET
cana-5238	484	15	function	function	NOUN
cana-5238	484	16	ν:(z	ν:(z	NOUN
cana-5238	484	17	,	,	PUNCT
cana-5238	484	18	ζ	ζ	NOUN
cana-5238	484	19	,	,	PUNCT
cana-5238	484	20	h)→(w	h)→(w	ADV
cana-5238	484	21	,	,	PUNCT
cana-5238	484	22	ξ	ξ	NOUN
cana-5238	484	23	)	)	PUNCT
cana-5238	484	24	,	,	PUNCT
cana-5238	484	25	where	where	SCONJ
cana-5238	484	26	z	z	NOUN
cana-5238	484	27	is	be	AUX
cana-5238	484	28	c0	c0	NOUN
cana-5238	484	29	space	space	NOUN
cana-5238	484	30	.	.	PUNCT
cana-5238	485	1	then	then	ADV
cana-5238	485	2	the	the	DET
cana-5238	485	3	following	follow	VERB
cana-5238	485	4	conditions	condition	NOUN
cana-5238	485	5	are	be	AUX
cana-5238	485	6	equivalent	equivalent	ADJ
cana-5238	485	7	.	.	PUNCT
cana-5238	486	1	1	1	X
cana-5238	486	2	.	.	X
cana-5238	486	3	ν	ν	NOUN
cana-5238	486	4	is	be	AUX
cana-5238	486	5	(	(	PUNCT
cana-5238	486	6	ζ	ζ	NOUN
cana-5238	486	7	,	,	PUNCT
cana-5238	486	8	ξ)-c	ξ)-c	NOUN
cana-5238	486	9	,	,	PUNCT
cana-5238	486	10	2	2	NUM
cana-5238	486	11	.	.	PUNCT
cana-5238	487	1	ν	ν	NOUN
cana-5238	487	2	is	be	AUX
cana-5238	487	3	(	(	PUNCT
cana-5238	487	4	σ	σ	PROPN
cana-5238	487	5	-	-	PUNCT
cana-5238	487	6	hg	hg	NOUN
cana-5238	487	7	,	,	PUNCT
cana-5238	487	8	ξ)-c	ξ)-c	VERB
cana-5238	487	9	and	and	CCONJ
cana-5238	487	10	(	(	PUNCT
cana-5238	487	11	β∗-b	β∗-b	NOUN
cana-5238	487	12	-	-	PUNCT
cana-5238	487	13	hg	hg	NOUN
cana-5238	487	14	,	,	PUNCT
cana-5238	487	15	ξ)-c	ξ)-c	NOUN
cana-5238	487	16	,	,	PUNCT
cana-5238	487	17	3	3	X
cana-5238	487	18	.	.	PUNCT
cana-5238	488	1	ν	ν	NOUN
cana-5238	488	2	is	be	AUX
cana-5238	488	3	(	(	PUNCT
cana-5238	488	4	β	β	NOUN
cana-5238	488	5	-	-	PUNCT
cana-5238	488	6	hg	hg	NOUN
cana-5238	488	7	,	,	PUNCT
cana-5238	488	8	ξ)-c	ξ)-c	VERB
cana-5238	488	9	and	and	CCONJ
cana-5238	488	10	(	(	PUNCT
cana-5238	488	11	β∗-bhg	β∗-bhg	ADJ
cana-5238	488	12	,	,	PUNCT
cana-5238	488	13	ξ)-c	ξ)-c	NOUN
cana-5238	488	14	.	.	PUNCT
cana-5238	489	1	proof	proof	NOUN
cana-5238	489	2	.	.	PUNCT
cana-5238	490	1	the	the	DET
cana-5238	490	2	proof	proof	NOUN
cana-5238	490	3	is	be	AUX
cana-5238	490	4	clear	clear	ADJ
cana-5238	490	5	by	by	ADP
cana-5238	490	6	theorem	theorem	NOUN
cana-5238	490	7	3.22	3.22	NUM
cana-5238	490	8	.	.	PUNCT
cana-5238	491	1	theorem	theorem	NOUN
cana-5238	491	2	4.12	4.12	NUM
cana-5238	491	3	.	.	PUNCT
cana-5238	492	1	let	let	AUX
cana-5238	492	2	(	(	PUNCT
cana-5238	492	3	z	z	NOUN
cana-5238	492	4	,	,	PUNCT
cana-5238	492	5	ζ	ζ	NOUN
cana-5238	492	6	,	,	PUNCT
cana-5238	492	7	h	h	NOUN
cana-5238	492	8	)	)	PUNCT
cana-5238	492	9	be	be	VERB
cana-5238	492	10	a	a	DET
cana-5238	492	11	strong	strong	ADJ
cana-5238	492	12	hgts	hgts	NOUN
cana-5238	492	13	for	for	ADP
cana-5238	492	14	a	a	DET
cana-5238	492	15	function	function	NOUN
cana-5238	492	16	ν:(z	ν:(z	NOUN
cana-5238	492	17	,	,	PUNCT
cana-5238	492	18	ζ	ζ	NOUN
cana-5238	492	19	,	,	PUNCT
cana-5238	492	20	h)→(w	h)→(w	ADV
cana-5238	492	21	,	,	PUNCT
cana-5238	492	22	ξ	ξ	NOUN
cana-5238	492	23	)	)	PUNCT
cana-5238	492	24	,	,	PUNCT
cana-5238	492	25	where	where	SCONJ
cana-5238	492	26	z	z	NOUN
cana-5238	492	27	is	be	AUX
cana-5238	492	28	c0	c0	NOUN
cana-5238	492	29	space	space	NOUN
cana-5238	492	30	.	.	PUNCT
cana-5238	493	1	then	then	ADV
cana-5238	493	2	the	the	DET
cana-5238	493	3	following	follow	VERB
cana-5238	493	4	conditions	condition	NOUN
cana-5238	493	5	are	be	AUX
cana-5238	493	6	equivalent	equivalent	ADJ
cana-5238	493	7	.	.	PUNCT
cana-5238	494	1	1	1	X
cana-5238	494	2	.	.	X
cana-5238	494	3	ν	ν	NOUN
cana-5238	494	4	is	be	AUX
cana-5238	494	5	(	(	PUNCT
cana-5238	494	6	ζ	ζ	NOUN
cana-5238	494	7	,	,	PUNCT
cana-5238	494	8	ξ)-c	ξ)-c	NOUN
cana-5238	494	9	,	,	PUNCT
cana-5238	494	10	2	2	NUM
cana-5238	494	11	.	.	PUNCT
cana-5238	495	1	ν	ν	NOUN
cana-5238	495	2	is	be	AUX
cana-5238	495	3	(	(	PUNCT
cana-5238	495	4	π	π	PROPN
cana-5238	495	5	-	-	PUNCT
cana-5238	495	6	hg	hg	NOUN
cana-5238	495	7	,	,	PUNCT
cana-5238	495	8	ξ)-c	ξ)-c	VERB
cana-5238	495	9	and	and	CCONJ
cana-5238	495	10	(	(	PUNCT
cana-5238	495	11	β∗-bhg	β∗-bhg	ADJ
cana-5238	495	12	,	,	PUNCT
cana-5238	495	13	ξ)c	ξ)c	NOUN
cana-5238	495	14	,	,	PUNCT
cana-5238	495	15	3	3	X
cana-5238	495	16	.	.	PUNCT
cana-5238	496	1	ν	ν	NOUN
cana-5238	496	2	is	be	AUX
cana-5238	496	3	(	(	PUNCT
cana-5238	496	4	β	β	NOUN
cana-5238	496	5	-	-	PUNCT
cana-5238	496	6	hg	hg	NOUN
cana-5238	496	7	,	,	PUNCT
cana-5238	496	8	ξ)-c	ξ)-c	VERB
cana-5238	496	9	and	and	CCONJ
cana-5238	496	10	(	(	PUNCT
cana-5238	496	11	β∗-bhg	β∗-bhg	ADJ
cana-5238	496	12	,	,	PUNCT
cana-5238	496	13	ξ)-c	ξ)-c	NOUN
cana-5238	496	14	.	.	PUNCT
cana-5238	497	1	proof	proof	NOUN
cana-5238	497	2	.	.	PUNCT
cana-5238	498	1	the	the	DET
cana-5238	498	2	proof	proof	NOUN
cana-5238	498	3	is	be	AUX
cana-5238	498	4	clear	clear	ADJ
cana-5238	498	5	by	by	ADP
cana-5238	498	6	theorem	theorem	ADJ
cana-5238	498	7	3.25	3.25	NUM
cana-5238	498	8	.	.	PUNCT
cana-5238	499	1	theorem	theorem	NOUN
cana-5238	499	2	4.13	4.13	NUM
cana-5238	499	3	.	.	PUNCT
cana-5238	500	1	let	let	AUX
cana-5238	500	2	(	(	PUNCT
cana-5238	500	3	z	z	NOUN
cana-5238	500	4	,	,	PUNCT
cana-5238	500	5	ζ	ζ	NOUN
cana-5238	500	6	,	,	PUNCT
cana-5238	500	7	h	h	NOUN
cana-5238	500	8	)	)	PUNCT
cana-5238	500	9	be	be	VERB
cana-5238	500	10	a	a	DET
cana-5238	500	11	strong	strong	ADJ
cana-5238	500	12	hgts	hgts	NOUN
cana-5238	500	13	,	,	PUNCT
cana-5238	500	14	where	where	SCONJ
cana-5238	500	15	z	z	NOUN
cana-5238	500	16	is	be	AUX
cana-5238	500	17	c0	c0	NOUN
cana-5238	500	18	space	space	NOUN
cana-5238	500	19	for	for	ADP
cana-5238	500	20	a	a	DET
cana-5238	500	21	function	function	NOUN
cana-5238	500	22	ν:(z	ν:(z	NOUN
cana-5238	500	23	,	,	PUNCT
cana-5238	500	24	ζ	ζ	NOUN
cana-5238	500	25	,	,	PUNCT
cana-5238	500	26	h)→(w	h)→(w	ADV
cana-5238	500	27	,	,	PUNCT
cana-5238	500	28	ξ	ξ	NOUN
cana-5238	500	29	)	)	PUNCT
cana-5238	500	30	,	,	PUNCT
cana-5238	500	31	where	where	SCONJ
cana-5238	500	32	z	z	NOUN
cana-5238	500	33	is	be	AUX
cana-5238	500	34	c0	c0	PROPN
cana-5238	500	35	-space	-space	PROPN
cana-5238	500	36	.	.	PUNCT
cana-5238	501	1	then	then	ADV
cana-5238	501	2	the	the	DET
cana-5238	501	3	following	follow	VERB
cana-5238	501	4	conditions	condition	NOUN
cana-5238	501	5	are	be	AUX
cana-5238	501	6	equivalent	equivalent	ADJ
cana-5238	501	7	.	.	PUNCT
cana-5238	502	1	1	1	X
cana-5238	502	2	.	.	X
cana-5238	502	3	ν	ν	NOUN
cana-5238	502	4	is	be	AUX
cana-5238	502	5	(	(	PUNCT
cana-5238	502	6	gζ	gζ	NOUN
cana-5238	502	7	,	,	PUNCT
cana-5238	502	8	ξ)-c	ξ)-c	NOUN
cana-5238	502	9	,	,	PUNCT
cana-5238	502	10	2	2	NUM
cana-5238	502	11	.	.	PUNCT
cana-5238	503	1	ν	ν	NOUN
cana-5238	503	2	is	be	AUX
cana-5238	503	3	(	(	PUNCT
cana-5238	503	4	α	α	NOUN
cana-5238	503	5	-	-	PUNCT
cana-5238	503	6	hg	hg	NOUN
cana-5238	503	7	,	,	PUNCT
cana-5238	503	8	ξ)-c	ξ)-c	VERB
cana-5238	503	9	and	and	CCONJ
cana-5238	503	10	(	(	PUNCT
cana-5238	503	11	ξ∗-b	ξ∗-b	NOUN
cana-5238	503	12	-	-	PUNCT
cana-5238	503	13	hg	hg	NOUN
cana-5238	503	14	,	,	PUNCT
cana-5238	503	15	ξ)-c	ξ)-c	NOUN
cana-5238	503	16	,	,	PUNCT
cana-5238	503	17	3	3	X
cana-5238	503	18	.	.	PUNCT
cana-5238	504	1	ν	ν	NOUN
cana-5238	504	2	is	be	AUX
cana-5238	504	3	(	(	PUNCT
cana-5238	504	4	σ	σ	PROPN
cana-5238	504	5	-	-	PUNCT
cana-5238	504	6	hg	hg	NOUN
cana-5238	504	7	,	,	PUNCT
cana-5238	504	8	ξ)-c	ξ)-c	VERB
cana-5238	504	9	and	and	CCONJ
cana-5238	504	10	(	(	PUNCT
cana-5238	504	11	σ∗-b	σ∗-b	PROPN
cana-5238	504	12	-	-	PUNCT
cana-5238	504	13	hg	hg	NOUN
cana-5238	504	14	,	,	PUNCT
cana-5238	504	15	ξ)-c	ξ)-c	NOUN
cana-5238	504	16	,	,	PUNCT
cana-5238	504	17	4	4	NUM
cana-5238	504	18	.	.	PUNCT
cana-5238	505	1	ν	ν	NOUN
cana-5238	505	2	is	be	AUX
cana-5238	505	3	(	(	PUNCT
cana-5238	505	4	πhg	πhg	ADJ
cana-5238	505	5	,	,	PUNCT
cana-5238	505	6	ξ)-c	ξ)-c	VERB
cana-5238	505	7	and	and	CCONJ
cana-5238	505	8	(	(	PUNCT
cana-5238	505	9	π∗-b	π∗-b	PROPN
cana-5238	505	10	-	-	PUNCT
cana-5238	505	11	hg	hg	NOUN
cana-5238	505	12	,	,	PUNCT
cana-5238	505	13	ξ)-c	ξ)-c	NOUN
cana-5238	505	14	,	,	PUNCT
cana-5238	505	15	5	5	NUM
cana-5238	505	16	.	.	PUNCT
cana-5238	506	1	ν	ν	NOUN
cana-5238	506	2	is	be	AUX
cana-5238	506	3	(	(	PUNCT
cana-5238	506	4	β	β	NOUN
cana-5238	506	5	-	-	PUNCT
cana-5238	506	6	hg	hg	NOUN
cana-5238	506	7	,	,	PUNCT
cana-5238	506	8	ξ)-c	ξ)-c	VERB
cana-5238	506	9	and	and	CCONJ
cana-5238	506	10	(	(	PUNCT
cana-5238	506	11	∆∗-b	∆∗-b	PROPN
cana-5238	506	12	-	-	PUNCT
cana-5238	506	13	hg	hg	NOUN
cana-5238	506	14	,	,	PUNCT
cana-5238	506	15	ξ)-c	ξ)-c	NOUN
cana-5238	506	16	.	.	PUNCT
cana-5238	507	1	proof	proof	NOUN
cana-5238	507	2	.	.	PUNCT
cana-5238	508	1	the	the	DET
cana-5238	508	2	proof	proof	NOUN
cana-5238	508	3	is	be	AUX
cana-5238	508	4	clear	clear	ADJ
cana-5238	508	5	by	by	ADP
cana-5238	508	6	theorem	theorem	ADJ
cana-5238	508	7	3.32	3.32	NUM
cana-5238	508	8	.	.	PUNCT
cana-5238	509	1	theorem	theorem	VERB
cana-5238	509	2	4.14	4.14	NUM
cana-5238	509	3	.	.	PUNCT
cana-5238	510	1	let	let	AUX
cana-5238	510	2	(	(	PUNCT
cana-5238	510	3	z	z	NOUN
cana-5238	510	4	,	,	PUNCT
cana-5238	510	5	ζ	ζ	NOUN
cana-5238	510	6	,	,	PUNCT
cana-5238	510	7	h	h	NOUN
cana-5238	510	8	)	)	PUNCT
cana-5238	510	9	be	be	VERB
cana-5238	510	10	a	a	DET
cana-5238	510	11	strong	strong	ADJ
cana-5238	510	12	hgts	hgts	NOUN
cana-5238	510	13	,	,	PUNCT
cana-5238	510	14	where	where	SCONJ
cana-5238	510	15	z	z	NOUN
cana-5238	510	16	is	be	AUX
cana-5238	510	17	c0	c0	NOUN
cana-5238	510	18	space	space	NOUN
cana-5238	510	19	for	for	ADP
cana-5238	510	20	a	a	DET
cana-5238	510	21	function	function	NOUN
cana-5238	510	22	ν:(z	ν:(z	NOUN
cana-5238	510	23	,	,	PUNCT
cana-5238	510	24	ζ	ζ	NOUN
cana-5238	510	25	,	,	PUNCT
cana-5238	510	26	h)→(w	h)→(w	ADV
cana-5238	510	27	,	,	PUNCT
cana-5238	510	28	ξ	ξ	NOUN
cana-5238	510	29	)	)	PUNCT
cana-5238	510	30	,	,	PUNCT
cana-5238	510	31	where	where	SCONJ
cana-5238	510	32	z	z	NOUN
cana-5238	510	33	is	be	AUX
cana-5238	510	34	c0	c0	PROPN
cana-5238	510	35	-space	-space	PROPN
cana-5238	510	36	.	.	PUNCT
cana-5238	511	1	then	then	ADV
cana-5238	511	2	the	the	DET
cana-5238	511	3	following	follow	VERB
cana-5238	511	4	conditions	condition	NOUN
cana-5238	511	5	are	be	AUX
cana-5238	511	6	equivalent	equivalent	ADJ
cana-5238	511	7	.	.	PUNCT
cana-5238	512	1	1	1	X
cana-5238	512	2	.	.	X
cana-5238	512	3	ν	ν	NOUN
cana-5238	512	4	is	be	AUX
cana-5238	512	5	(	(	PUNCT
cana-5238	512	6	gζ	gζ	NOUN
cana-5238	512	7	,	,	PUNCT
cana-5238	512	8	ξ)-c	ξ)-c	NOUN
cana-5238	512	9	,	,	PUNCT
cana-5238	512	10	2	2	NUM
cana-5238	512	11	.	.	PUNCT
cana-5238	513	1	ν	ν	NOUN
cana-5238	513	2	is	be	AUX
cana-5238	513	3	(	(	PUNCT
cana-5238	513	4	σ	σ	PROPN
cana-5238	513	5	-	-	PUNCT
cana-5238	513	6	hg	hg	NOUN
cana-5238	513	7	,	,	PUNCT
cana-5238	513	8	ξ)-c	ξ)-c	VERB
cana-5238	513	9	and	and	CCONJ
cana-5238	513	10	(	(	PUNCT
cana-5238	513	11	φ∗-b	φ∗-b	NOUN
cana-5238	513	12	-	-	PUNCT
cana-5238	513	13	hg	hg	NOUN
cana-5238	513	14	,	,	PUNCT
cana-5238	513	15	ξ)-c	ξ)-c	NOUN
cana-5238	513	16	,	,	PUNCT
cana-5238	513	17	3	3	X
cana-5238	513	18	.	.	PUNCT
cana-5238	514	1	ν	ν	NOUN
cana-5238	514	2	is	be	AUX
cana-5238	514	3	(	(	PUNCT
cana-5238	514	4	π	π	PROPN
cana-5238	514	5	-	-	PUNCT
cana-5238	514	6	hg	hg	NOUN
cana-5238	514	7	,	,	PUNCT
cana-5238	514	8	ξ)-c	ξ)-c	VERB
cana-5238	514	9	and	and	CCONJ
cana-5238	514	10	(	(	PUNCT
cana-5238	514	11	φ∗-bhg	φ∗-bhg	ADJ
cana-5238	514	12	,	,	PUNCT
cana-5238	514	13	ξ)-c	ξ)-c	VERB
cana-5238	514	14	,	,	PUNCT
cana-5238	514	15	4	4	NUM
cana-5238	514	16	.	.	PUNCT
cana-5238	515	1	ν	ν	NOUN
cana-5238	515	2	is	be	AUX
cana-5238	515	3	(	(	PUNCT
cana-5238	515	4	b	b	X
cana-5238	515	5	-	-	PUNCT
cana-5238	515	6	hg	hg	NOUN
cana-5238	515	7	,	,	PUNCT
cana-5238	515	8	ξ)-c	ξ)-c	VERB
cana-5238	515	9	and	and	CCONJ
cana-5238	515	10	(	(	PUNCT
cana-5238	515	11	φ∗-bhg	φ∗-bhg	ADJ
cana-5238	515	12	,	,	PUNCT
cana-5238	515	13	ξ)-c	ξ)-c	NOUN
cana-5238	515	14	.	.	PUNCT
cana-5238	516	1	proof	proof	NOUN
cana-5238	516	2	.	.	PUNCT
cana-5238	517	1	the	the	DET
cana-5238	517	2	proof	proof	NOUN
cana-5238	517	3	is	be	AUX
cana-5238	517	4	clear	clear	ADJ
cana-5238	517	5	by	by	ADP
cana-5238	517	6	theorem	theorem	ADJ
cana-5238	517	7	3.35	3.35	NUM
cana-5238	517	8	.	.	PUNCT
cana-5238	518	1	theorem	theorem	VERB
cana-5238	518	2	4.15	4.15	NUM
cana-5238	518	3	.	.	PUNCT
cana-5238	519	1	let	let	AUX
cana-5238	519	2	(	(	PUNCT
cana-5238	519	3	z	z	NOUN
cana-5238	519	4	,	,	PUNCT
cana-5238	519	5	ζ	ζ	NOUN
cana-5238	519	6	,	,	PUNCT
cana-5238	519	7	h	h	NOUN
cana-5238	519	8	)	)	PUNCT
cana-5238	519	9	be	be	VERB
cana-5238	519	10	a	a	DET
cana-5238	519	11	strong	strong	ADJ
cana-5238	519	12	hgts	hgts	NOUN
cana-5238	519	13	,	,	PUNCT
cana-5238	519	14	where	where	SCONJ
cana-5238	519	15	z	z	NOUN
cana-5238	519	16	is	be	AUX
cana-5238	519	17	c0	c0	NOUN
cana-5238	519	18	space	space	NOUN
cana-5238	519	19	for	for	ADP
cana-5238	519	20	a	a	DET
cana-5238	519	21	function	function	NOUN
cana-5238	519	22	ν:(z	ν:(z	NOUN
cana-5238	519	23	,	,	PUNCT
cana-5238	519	24	ζ	ζ	NOUN
cana-5238	519	25	,	,	PUNCT
cana-5238	519	26	h)→(w	h)→(w	ADV
cana-5238	519	27	,	,	PUNCT
cana-5238	519	28	ξ	ξ	NOUN
cana-5238	519	29	)	)	PUNCT
cana-5238	519	30	,	,	PUNCT
cana-5238	519	31	where	where	SCONJ
cana-5238	519	32	z	z	NOUN
cana-5238	519	33	is	be	AUX
cana-5238	519	34	c0	c0	PROPN
cana-5238	519	35	-space	-space	PROPN
cana-5238	519	36	.	.	PUNCT
cana-5238	520	1	then	then	ADV
cana-5238	520	2	the	the	DET
cana-5238	520	3	following	follow	VERB
cana-5238	520	4	conditions	condition	NOUN
cana-5238	520	5	are	be	AUX
cana-5238	520	6	equivalent	equivalent	ADJ
cana-5238	520	7	.	.	PUNCT
cana-5238	521	1	1	1	X
cana-5238	521	2	.	.	X
cana-5238	521	3	ν	ν	NOUN
cana-5238	521	4	is	be	AUX
cana-5238	521	5	(	(	PUNCT
cana-5238	521	6	gζ	gζ	NOUN
cana-5238	521	7	,	,	PUNCT
cana-5238	521	8	ξ)-c	ξ)-c	NOUN
cana-5238	521	9	,	,	PUNCT
cana-5238	521	10	2	2	NUM
cana-5238	521	11	.	.	PUNCT
cana-5238	522	1	ν	ν	NOUN
cana-5238	522	2	is	be	AUX
cana-5238	522	3	(	(	PUNCT
cana-5238	522	4	α	α	NOUN
cana-5238	522	5	-	-	PUNCT
cana-5238	522	6	hg	hg	NOUN
cana-5238	522	7	,	,	PUNCT
cana-5238	522	8	ξ)-c	ξ)-c	VERB
cana-5238	522	9	and	and	CCONJ
cana-5238	522	10	(	(	PUNCT
cana-5238	522	11	π∗-b	π∗-b	PROPN
cana-5238	522	12	-	-	PUNCT
cana-5238	522	13	hg	hg	NOUN
cana-5238	522	14	,	,	PUNCT
cana-5238	522	15	ξ)-c	ξ)-c	NOUN
cana-5238	522	16	,	,	PUNCT
cana-5238	522	17	3	3	X
cana-5238	522	18	.	.	PUNCT
cana-5238	523	1	ν	ν	NOUN
cana-5238	523	2	is	be	AUX
cana-5238	523	3	(	(	PUNCT
cana-5238	523	4	πhg	πhg	ADJ
cana-5238	523	5	,	,	PUNCT
cana-5238	523	6	ξ)-c	ξ)-c	VERB
cana-5238	523	7	and	and	CCONJ
cana-5238	523	8	(	(	PUNCT
cana-5238	523	9	π∗-b	π∗-b	PROPN
cana-5238	523	10	-	-	PUNCT
cana-5238	523	11	hg	hg	NOUN
cana-5238	523	12	,	,	PUNCT
cana-5238	523	13	ξ)-c	ξ)-c	NOUN
cana-5238	523	14	.	.	PUNCT
cana-5238	524	1	proof	proof	NOUN
cana-5238	524	2	.	.	PUNCT
cana-5238	525	1	the	the	DET
cana-5238	525	2	proof	proof	NOUN
cana-5238	525	3	is	be	AUX
cana-5238	525	4	clear	clear	ADJ
cana-5238	525	5	by	by	ADP
cana-5238	525	6	theorem	theorem	ADJ
cana-5238	525	7	3.37	3.37	NUM
cana-5238	525	8	.	.	PUNCT
cana-5238	526	1	theorem	theorem	VERB
cana-5238	526	2	4.16	4.16	NUM
cana-5238	526	3	.	.	PUNCT
cana-5238	527	1	let	let	AUX
cana-5238	527	2	(	(	PUNCT
cana-5238	527	3	z	z	NOUN
cana-5238	527	4	,	,	PUNCT
cana-5238	527	5	ζ	ζ	NOUN
cana-5238	527	6	,	,	PUNCT
cana-5238	527	7	h	h	NOUN
cana-5238	527	8	)	)	PUNCT
cana-5238	527	9	be	be	VERB
cana-5238	527	10	a	a	DET
cana-5238	527	11	strong	strong	ADJ
cana-5238	527	12	hgts	hgts	NOUN
cana-5238	527	13	,	,	PUNCT
cana-5238	527	14	where	where	SCONJ
cana-5238	527	15	z	z	NOUN
cana-5238	527	16	is	be	AUX
cana-5238	527	17	c0	c0	NOUN
cana-5238	527	18	space	space	NOUN
cana-5238	527	19	for	for	ADP
cana-5238	527	20	a	a	DET
cana-5238	527	21	function	function	NOUN
cana-5238	527	22	ν:(z	ν:(z	NOUN
cana-5238	527	23	,	,	PUNCT
cana-5238	527	24	ζ	ζ	NOUN
cana-5238	527	25	,	,	PUNCT
cana-5238	527	26	h)→(w	h)→(w	ADV
cana-5238	527	27	,	,	PUNCT
cana-5238	527	28	ξ	ξ	NOUN
cana-5238	527	29	)	)	PUNCT
cana-5238	527	30	.	.	PUNCT
cana-5238	528	1	then	then	ADV
cana-5238	528	2	the	the	DET
cana-5238	528	3	following	follow	VERB
cana-5238	528	4	conditions	condition	NOUN
cana-5238	528	5	are	be	AUX
cana-5238	528	6	equivalent	equivalent	ADJ
cana-5238	528	7	.	.	PUNCT
cana-5238	529	1	1	1	X
cana-5238	529	2	.	.	X
cana-5238	529	3	ν	ν	NOUN
cana-5238	529	4	is	be	AUX
cana-5238	529	5	(	(	PUNCT
cana-5238	529	6	gζ	gζ	NOUN
cana-5238	529	7	,	,	PUNCT
cana-5238	529	8	ξ)-c	ξ)-c	NOUN
cana-5238	529	9	,	,	PUNCT
cana-5238	529	10	communications	communication	NOUN
cana-5238	529	11	on	on	ADP
cana-5238	529	12	applied	apply	VERB
cana-5238	529	13	nonlinear	nonlinear	ADJ
cana-5238	529	14	analysis	analysis	NOUN
cana-5238	529	15	issn	issn	NOUN
cana-5238	529	16	:	:	PUNCT
cana-5238	529	17	1074	1074	NUM
cana-5238	529	18	-	-	PUNCT
cana-5238	529	19	133x	133x	NUM
cana-5238	529	20	vol	vol	VERB
cana-5238	529	21	32	32	NUM
cana-5238	529	22	no	no	NOUN
cana-5238	529	23	.	.	PUNCT
cana-5238	530	1	10s	10	NOUN
cana-5238	530	2	(	(	PUNCT
cana-5238	530	3	2025	2025	NUM
cana-5238	530	4	)	)	PUNCT
cana-5238	530	5	1370	1370	NUM
cana-5238	530	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	530	7	2	2	X
cana-5238	530	8	.	.	PUNCT
cana-5238	531	1	ν	ν	NOUN
cana-5238	531	2	is	be	AUX
cana-5238	531	3	(	(	PUNCT
cana-5238	531	4	αhg	αhg	NOUN
cana-5238	531	5	,	,	PUNCT
cana-5238	531	6	ξ)-c	ξ)-c	NOUN
cana-5238	531	7	and	and	CCONJ
cana-5238	531	8	(	(	PUNCT
cana-5238	531	9	σ∗-b	σ∗-b	PROPN
cana-5238	531	10	-	-	PUNCT
cana-5238	531	11	hg	hg	NOUN
cana-5238	531	12	,	,	PUNCT
cana-5238	531	13	ξ)-c	ξ)-c	NOUN
cana-5238	531	14	,	,	PUNCT
cana-5238	531	15	3	3	X
cana-5238	531	16	.	.	PUNCT
cana-5238	532	1	ν	ν	NOUN
cana-5238	532	2	is	be	AUX
cana-5238	532	3	(	(	PUNCT
cana-5238	532	4	σ	σ	PROPN
cana-5238	532	5	-	-	PUNCT
cana-5238	532	6	hg	hg	NOUN
cana-5238	532	7	,	,	PUNCT
cana-5238	532	8	ξ)-c	ξ)-c	VERB
cana-5238	532	9	and	and	CCONJ
cana-5238	532	10	(	(	PUNCT
cana-5238	532	11	σ∗-b	σ∗-b	PROPN
cana-5238	532	12	-	-	PUNCT
cana-5238	532	13	hg	hg	NOUN
cana-5238	532	14	,	,	PUNCT
cana-5238	532	15	ξ)-c	ξ)-c	NOUN
cana-5238	532	16	.	.	PUNCT
cana-5238	533	1	proof	proof	NOUN
cana-5238	533	2	.	.	PUNCT
cana-5238	534	1	the	the	DET
cana-5238	534	2	proof	proof	NOUN
cana-5238	534	3	is	be	AUX
cana-5238	534	4	clear	clear	ADJ
cana-5238	534	5	by	by	ADP
cana-5238	534	6	theorem	theorem	ADJ
cana-5238	534	7	3.38	3.38	NUM
cana-5238	534	8	.	.	PUNCT
cana-5238	535	1	theorem	theorem	VERB
cana-5238	535	2	4.17	4.17	NUM
cana-5238	535	3	.	.	PUNCT
cana-5238	536	1	let	let	AUX
cana-5238	536	2	(	(	PUNCT
cana-5238	536	3	z	z	NOUN
cana-5238	536	4	,	,	PUNCT
cana-5238	536	5	ζ	ζ	NOUN
cana-5238	536	6	,	,	PUNCT
cana-5238	536	7	h	h	NOUN
cana-5238	536	8	)	)	PUNCT
cana-5238	536	9	be	be	VERB
cana-5238	536	10	a	a	DET
cana-5238	536	11	strong	strong	ADJ
cana-5238	536	12	hgts	hgts	NOUN
cana-5238	536	13	,	,	PUNCT
cana-5238	536	14	where	where	SCONJ
cana-5238	536	15	z	z	NOUN
cana-5238	536	16	is	be	AUX
cana-5238	536	17	c0	c0	NOUN
cana-5238	536	18	space	space	NOUN
cana-5238	536	19	for	for	ADP
cana-5238	536	20	a	a	DET
cana-5238	536	21	function	function	NOUN
cana-5238	536	22	ν:(z	ν:(z	NOUN
cana-5238	536	23	,	,	PUNCT
cana-5238	536	24	ζ	ζ	NOUN
cana-5238	536	25	,	,	PUNCT
cana-5238	536	26	h)→(w	h)→(w	ADV
cana-5238	536	27	,	,	PUNCT
cana-5238	536	28	ξ	ξ	NOUN
cana-5238	536	29	)	)	PUNCT
cana-5238	536	30	.	.	PUNCT
cana-5238	537	1	then	then	ADV
cana-5238	537	2	the	the	DET
cana-5238	537	3	following	follow	VERB
cana-5238	537	4	conditions	condition	NOUN
cana-5238	537	5	are	be	AUX
cana-5238	537	6	equivalent	equivalent	ADJ
cana-5238	537	7	.	.	PUNCT
cana-5238	538	1	1	1	X
cana-5238	538	2	.	.	X
cana-5238	538	3	ν	ν	NOUN
cana-5238	538	4	is	be	AUX
cana-5238	538	5	(	(	PUNCT
cana-5238	538	6	gζ	gζ	NOUN
cana-5238	538	7	,	,	PUNCT
cana-5238	538	8	ξ)-c	ξ)-c	NOUN
cana-5238	538	9	,	,	PUNCT
cana-5238	538	10	2	2	NUM
cana-5238	538	11	.	.	PUNCT
cana-5238	539	1	ν	ν	NOUN
cana-5238	539	2	is	be	AUX
cana-5238	539	3	(	(	PUNCT
cana-5238	539	4	α	α	NOUN
cana-5238	539	5	-	-	PUNCT
cana-5238	539	6	hg	hg	NOUN
cana-5238	539	7	,	,	PUNCT
cana-5238	539	8	ξ)-c	ξ)-c	VERB
cana-5238	539	9	and	and	CCONJ
cana-5238	539	10	(	(	PUNCT
cana-5238	539	11	∆∗-b	∆∗-b	PROPN
cana-5238	539	12	-	-	PUNCT
cana-5238	539	13	hg	hg	NOUN
cana-5238	539	14	,	,	PUNCT
cana-5238	539	15	ξ)-c	ξ)-c	NOUN
cana-5238	539	16	,	,	PUNCT
cana-5238	539	17	3	3	X
cana-5238	539	18	.	.	PUNCT
cana-5238	540	1	ν	ν	NOUN
cana-5238	540	2	is	be	AUX
cana-5238	540	3	(	(	PUNCT
cana-5238	540	4	β	β	NOUN
cana-5238	540	5	-	-	PUNCT
cana-5238	540	6	hg	hg	NOUN
cana-5238	540	7	,	,	PUNCT
cana-5238	540	8	ξ)-c	ξ)-c	VERB
cana-5238	540	9	and	and	CCONJ
cana-5238	540	10	(	(	PUNCT
cana-5238	540	11	∆∗-bhg	∆∗-bhg	X
cana-5238	540	12	,	,	PUNCT
cana-5238	540	13	ξ)-c	ξ)-c	NOUN
cana-5238	540	14	.	.	PUNCT
cana-5238	541	1	proof	proof	NOUN
cana-5238	541	2	.	.	PUNCT
cana-5238	542	1	the	the	DET
cana-5238	542	2	proof	proof	NOUN
cana-5238	542	3	is	be	AUX
cana-5238	542	4	clear	clear	ADJ
cana-5238	542	5	by	by	ADP
cana-5238	542	6	theorem	theorem	ADJ
cana-5238	542	7	3.39	3.39	NUM
cana-5238	542	8	.	.	PUNCT
cana-5238	543	1	theorem	theorem	NOUN
cana-5238	543	2	4.18	4.18	NUM
cana-5238	543	3	.	.	PUNCT
cana-5238	544	1	let	let	AUX
cana-5238	544	2	(	(	PUNCT
cana-5238	544	3	z	z	NOUN
cana-5238	544	4	,	,	PUNCT
cana-5238	544	5	ζ	ζ	NOUN
cana-5238	544	6	,	,	PUNCT
cana-5238	544	7	h	h	NOUN
cana-5238	544	8	)	)	PUNCT
cana-5238	544	9	be	be	VERB
cana-5238	544	10	a	a	DET
cana-5238	544	11	strong	strong	ADJ
cana-5238	544	12	hgts	hgts	NOUN
cana-5238	544	13	,	,	PUNCT
cana-5238	544	14	where	where	SCONJ
cana-5238	544	15	z	z	NOUN
cana-5238	544	16	is	be	AUX
cana-5238	544	17	c0	c0	NOUN
cana-5238	544	18	space	space	NOUN
cana-5238	544	19	for	for	ADP
cana-5238	544	20	a	a	DET
cana-5238	544	21	function	function	NOUN
cana-5238	544	22	ν:(z	ν:(z	NOUN
cana-5238	544	23	,	,	PUNCT
cana-5238	544	24	ζ	ζ	NOUN
cana-5238	544	25	,	,	PUNCT
cana-5238	544	26	h)→(w	h)→(w	ADV
cana-5238	544	27	,	,	PUNCT
cana-5238	544	28	ξ	ξ	NOUN
cana-5238	544	29	)	)	PUNCT
cana-5238	544	30	.	.	PUNCT
cana-5238	545	1	then	then	ADV
cana-5238	545	2	the	the	DET
cana-5238	545	3	following	follow	VERB
cana-5238	545	4	conditions	condition	NOUN
cana-5238	545	5	are	be	AUX
cana-5238	545	6	equivalent	equivalent	ADJ
cana-5238	545	7	.	.	PUNCT
cana-5238	546	1	1	1	X
cana-5238	546	2	.	.	X
cana-5238	546	3	ν	ν	NOUN
cana-5238	546	4	is	be	AUX
cana-5238	546	5	(	(	PUNCT
cana-5238	546	6	gζ	gζ	NOUN
cana-5238	546	7	,	,	PUNCT
cana-5238	546	8	ξ)-c	ξ)-c	NOUN
cana-5238	546	9	,	,	PUNCT
cana-5238	546	10	2	2	NUM
cana-5238	546	11	.	.	PUNCT
cana-5238	547	1	ν	ν	NOUN
cana-5238	547	2	is	be	AUX
cana-5238	547	3	(	(	PUNCT
cana-5238	547	4	σ	σ	PROPN
cana-5238	547	5	-	-	PUNCT
cana-5238	547	6	hg	hg	NOUN
cana-5238	547	7	,	,	PUNCT
cana-5238	547	8	ξ)-c	ξ)-c	VERB
cana-5238	547	9	and	and	CCONJ
cana-5238	547	10	(	(	PUNCT
cana-5238	547	11	∆∗-b	∆∗-b	PROPN
cana-5238	547	12	-	-	PUNCT
cana-5238	547	13	hg	hg	NOUN
cana-5238	547	14	,	,	PUNCT
cana-5238	547	15	ξ)-c	ξ)-c	NOUN
cana-5238	547	16	,	,	PUNCT
cana-5238	547	17	3	3	X
cana-5238	547	18	.	.	PUNCT
cana-5238	548	1	ν	ν	NOUN
cana-5238	548	2	is	be	AUX
cana-5238	548	3	(	(	PUNCT
cana-5238	548	4	β	β	NOUN
cana-5238	548	5	-	-	PUNCT
cana-5238	548	6	hg	hg	NOUN
cana-5238	548	7	,	,	PUNCT
cana-5238	548	8	ξ)-c	ξ)-c	VERB
cana-5238	548	9	and	and	CCONJ
cana-5238	548	10	(	(	PUNCT
cana-5238	548	11	∆∗-b	∆∗-b	PROPN
cana-5238	548	12	-	-	PUNCT
cana-5238	548	13	hg	hg	NOUN
cana-5238	548	14	,	,	PUNCT
cana-5238	548	15	ξ)-c	ξ)-c	NOUN
cana-5238	548	16	.	.	PUNCT
cana-5238	549	1	proof	proof	NOUN
cana-5238	549	2	.	.	PUNCT
cana-5238	550	1	the	the	DET
cana-5238	550	2	proof	proof	NOUN
cana-5238	550	3	is	be	AUX
cana-5238	550	4	clear	clear	ADJ
cana-5238	550	5	by	by	ADP
cana-5238	550	6	theorem	theorem	ADJ
cana-5238	550	7	3.40	3.40	NUM
cana-5238	550	8	.	.	PUNCT
cana-5238	551	1	theorem	theorem	NOUN
cana-5238	551	2	4.19	4.19	NUM
cana-5238	551	3	.	.	PUNCT
cana-5238	552	1	let	let	AUX
cana-5238	552	2	(	(	PUNCT
cana-5238	552	3	z	z	NOUN
cana-5238	552	4	,	,	PUNCT
cana-5238	552	5	ζ	ζ	NOUN
cana-5238	552	6	,	,	PUNCT
cana-5238	552	7	h	h	NOUN
cana-5238	552	8	)	)	PUNCT
cana-5238	552	9	be	be	VERB
cana-5238	552	10	a	a	DET
cana-5238	552	11	strong	strong	ADJ
cana-5238	552	12	hgts	hgts	NOUN
cana-5238	552	13	,	,	PUNCT
cana-5238	552	14	where	where	SCONJ
cana-5238	552	15	z	z	NOUN
cana-5238	552	16	is	be	AUX
cana-5238	552	17	c0	c0	NOUN
cana-5238	552	18	space	space	NOUN
cana-5238	552	19	for	for	ADP
cana-5238	552	20	a	a	DET
cana-5238	552	21	function	function	NOUN
cana-5238	552	22	ν:(z	ν:(z	NOUN
cana-5238	552	23	,	,	PUNCT
cana-5238	552	24	ζ	ζ	NOUN
cana-5238	552	25	,	,	PUNCT
cana-5238	552	26	h)→(w	h)→(w	ADV
cana-5238	552	27	,	,	PUNCT
cana-5238	552	28	ξ	ξ	NOUN
cana-5238	552	29	)	)	PUNCT
cana-5238	552	30	.	.	PUNCT
cana-5238	553	1	then	then	ADV
cana-5238	553	2	the	the	DET
cana-5238	553	3	following	follow	VERB
cana-5238	553	4	conditions	condition	NOUN
cana-5238	553	5	are	be	AUX
cana-5238	553	6	equivalent	equivalent	ADJ
cana-5238	553	7	.	.	PUNCT
cana-5238	554	1	1	1	X
cana-5238	554	2	.	.	X
cana-5238	554	3	ν	ν	NOUN
cana-5238	554	4	is	be	AUX
cana-5238	554	5	(	(	PUNCT
cana-5238	554	6	gζ	gζ	NOUN
cana-5238	554	7	,	,	PUNCT
cana-5238	554	8	ξ)-c	ξ)-c	NOUN
cana-5238	554	9	,	,	PUNCT
cana-5238	554	10	2	2	NUM
cana-5238	554	11	.	.	PUNCT
cana-5238	555	1	ν	ν	NOUN
cana-5238	555	2	is	be	AUX
cana-5238	555	3	(	(	PUNCT
cana-5238	555	4	π	π	PROPN
cana-5238	555	5	-	-	PUNCT
cana-5238	555	6	hg	hg	NOUN
cana-5238	555	7	,	,	PUNCT
cana-5238	555	8	ξ)-c	ξ)-c	VERB
cana-5238	555	9	and	and	CCONJ
cana-5238	555	10	(	(	PUNCT
cana-5238	555	11	∆∗-b	∆∗-b	PROPN
cana-5238	555	12	-	-	PUNCT
cana-5238	555	13	hg	hg	NOUN
cana-5238	555	14	,	,	PUNCT
cana-5238	555	15	ξ)-c	ξ)-c	NOUN
cana-5238	555	16	,	,	PUNCT
cana-5238	555	17	3	3	X
cana-5238	555	18	.	.	PUNCT
cana-5238	556	1	ν	ν	NOUN
cana-5238	556	2	is	be	AUX
cana-5238	556	3	(	(	PUNCT
cana-5238	556	4	β	β	NOUN
cana-5238	556	5	-	-	PUNCT
cana-5238	556	6	hg	hg	NOUN
cana-5238	556	7	,	,	PUNCT
cana-5238	556	8	ξ)-c	ξ)-c	VERB
cana-5238	556	9	and	and	CCONJ
cana-5238	556	10	(	(	PUNCT
cana-5238	556	11	∆∗-bhg	∆∗-bhg	X
cana-5238	556	12	,	,	PUNCT
cana-5238	556	13	ξ)-c	ξ)-c	NOUN
cana-5238	556	14	.	.	PUNCT
cana-5238	557	1	proof	proof	NOUN
cana-5238	557	2	.	.	PUNCT
cana-5238	558	1	the	the	DET
cana-5238	558	2	proof	proof	NOUN
cana-5238	558	3	is	be	AUX
cana-5238	558	4	clear	clear	ADJ
cana-5238	558	5	by	by	ADP
cana-5238	558	6	theorem	theorem	NOUN
cana-5238	558	7	3.41	3.41	NUM
cana-5238	558	8	.	.	PUNCT
cana-5238	559	1	reference	reference	NOUN
cana-5238	559	2	[	[	X
cana-5238	559	3	1	1	NUM
cana-5238	559	4	]	]	PUNCT
cana-5238	559	5	a.csaszar	a.csaszar	NOUN
cana-5238	559	6	,	,	PUNCT
cana-5238	559	7	generalized	generalize	VERB
cana-5238	559	8	topology	topology	NOUN
cana-5238	559	9	generalized	generalize	VERB
cana-5238	559	10	continuity	continuity	NOUN
cana-5238	559	11	acta	acta	PROPN
cana-5238	559	12	mathematica	mathematica	PROPN
cana-5238	559	13	hungarica	hungarica	PROPN
cana-5238	559	14	96	96	NUM
cana-5238	559	15	(	(	PUNCT
cana-5238	559	16	2002	2002	NUM
cana-5238	559	17	)	)	PUNCT
cana-5238	559	18	,	,	PUNCT
cana-5238	559	19	351	351	NUM
cana-5238	559	20	-	-	SYM
cana-5238	559	21	357	357	NUM
cana-5238	559	22	.	.	PUNCT
cana-5238	560	1	[	[	X
cana-5238	560	2	2	2	NUM
cana-5238	560	3	]	]	PUNCT
cana-5238	560	4	a.csaszar	a.csaszar	NOUN
cana-5238	560	5	,	,	PUNCT
cana-5238	560	6	generalized	generalize	VERB
cana-5238	560	7	open	open	ADJ
cana-5238	560	8	sets	set	NOUN
cana-5238	560	9	in	in	ADP
cana-5238	560	10	generalized	generalized	ADJ
cana-5238	560	11	topologies	topology	NOUN
cana-5238	560	12	acta	acta	PROPN
cana-5238	560	13	mathematica	mathematica	PROPN
cana-5238	560	14	hungarica	hungarica	PROPN
cana-5238	560	15	106	106	NUM
cana-5238	560	16	(	(	PUNCT
cana-5238	560	17	2005	2005	NUM
cana-5238	560	18	)	)	PUNCT
cana-5238	560	19	,	,	PUNCT
cana-5238	560	20	53	53	NUM
cana-5238	560	21	-	-	SYM
cana-5238	560	22	56	56	NUM
cana-5238	560	23	.	.	PUNCT
cana-5238	561	1	[	[	X
cana-5238	561	2	3	3	NUM
cana-5238	561	3	]	]	PUNCT
cana-5238	561	4	a.csaszar	a.csaszar	NOUN
cana-5238	561	5	,	,	PUNCT
cana-5238	561	6	modification	modification	NOUN
cana-5238	561	7	of	of	ADP
cana-5238	561	8	generalized	generalized	ADJ
cana-5238	561	9	topologies	topology	NOUN
cana-5238	561	10	via	via	ADP
cana-5238	561	11	hereditary	hereditary	ADJ
cana-5238	561	12	classes	class	NOUN
cana-5238	561	13	acta	acta	PROPN
cana-5238	561	14	mathematica	mathematica	PROPN
cana-5238	561	15	hungarica	hungarica	PROPN
cana-5238	561	16	115(2007	115(2007	NUM
cana-5238	561	17	)	)	PUNCT
cana-5238	561	18	,	,	PUNCT
cana-5238	561	19	29	29	NUM
cana-5238	561	20	-	-	SYM
cana-5238	561	21	36	36	NUM
cana-5238	561	22	.	.	PUNCT
cana-5238	562	1	[	[	X
cana-5238	562	2	4	4	X
cana-5238	562	3	]	]	PUNCT
cana-5238	562	4	s.maragathavalli	s.maragathavalli	PROPN
cana-5238	562	5	,	,	PUNCT
cana-5238	562	6	m.	m.	PROPN
cana-5238	562	7	sheik	sheik	PROPN
cana-5238	562	8	john	john	PROPN
cana-5238	562	9	and	and	CCONJ
cana-5238	562	10	d.	d.	PROPN
cana-5238	562	11	sivaraj	sivaraj	VERB
cana-5238	562	12	on	on	ADP
cana-5238	562	13	g	g	NOUN
cana-5238	562	14	-	-	PUNCT
cana-5238	562	15	closed	close	VERB
cana-5238	562	16	sets	set	NOUN
cana-5238	562	17	in	in	ADP
cana-5238	562	18	generalized	generalized	ADJ
cana-5238	562	19	topological	topological	ADJ
cana-5238	562	20	spaces	space	NOUN
cana-5238	562	21	,	,	PUNCT
cana-5238	562	22	journal	journal	NOUN
cana-5238	562	23	of	of	ADP
cana-5238	562	24	advanced	advanced	ADJ
cana-5238	562	25	research	research	NOUN
cana-5238	562	26	in	in	ADP
cana-5238	562	27	pure	pure	ADJ
cana-5238	562	28	mathematics	mathematic	NOUN
cana-5238	562	29	(	(	PUNCT
cana-5238	562	30	2)(2010	2)(2010	NOUN
cana-5238	562	31	)	)	PUNCT
cana-5238	562	32	,	,	PUNCT
cana-5238	562	33	3:24	3:24	NUM
cana-5238	562	34	-	-	SYM
cana-5238	562	35	33	33	NUM
cana-5238	562	36	.	.	PUNCT
cana-5238	563	1	[	[	X
cana-5238	563	2	5	5	X
cana-5238	563	3	]	]	PUNCT
cana-5238	563	4	w.	w.	PROPN
cana-5238	563	5	k.	k.	PROPN
cana-5238	563	6	min	min	PROPN
cana-5238	563	7	,	,	PUNCT
cana-5238	563	8	generalized	generalize	VERB
cana-5238	563	9	continuous	continuous	ADJ
cana-5238	563	10	maps	map	NOUN
cana-5238	563	11	defined	define	VERB
cana-5238	563	12	by	by	ADP
cana-5238	563	13	generalized	generalized	ADJ
cana-5238	563	14	open	open	ADJ
cana-5238	563	15	sets	set	NOUN
cana-5238	563	16	on	on	ADP
cana-5238	563	17	generalized	generalized	ADJ
cana-5238	563	18	topological	topological	ADJ
cana-5238	563	19	spaces	space	NOUN
cana-5238	563	20	acta	acta	PROPN
cana-5238	563	21	mathematica	mathematica	PROPN
cana-5238	563	22	hungarica	hungarica	PROPN
cana-5238	563	23	128(4	128(4	NUM
cana-5238	563	24	)	)	PUNCT
cana-5238	563	25	(	(	PUNCT
cana-5238	563	26	2010)pp	2010)pp	NUM
cana-5238	563	27	299	299	NUM
cana-5238	563	28	-	-	SYM
cana-5238	563	29	306	306	NUM
cana-5238	563	30	.	.	PUNCT
cana-5238	564	1	[	[	X
cana-5238	564	2	6	6	NUM
cana-5238	564	3	]	]	PUNCT
cana-5238	564	4	m.	m.	NOUN
cana-5238	564	5	rajamani	rajamani	NOUN
cana-5238	564	6	,	,	PUNCT
cana-5238	564	7	v.	v.	ADP
cana-5238	564	8	inthumathi	inthumathi	ADV
cana-5238	564	9	and	and	CCONJ
cana-5238	564	10	r.	r.	PROPN
cana-5238	564	11	ramesh	ramesh	PROPN
cana-5238	564	12	,	,	PUNCT
cana-5238	564	13	some	some	DET
cana-5238	564	14	new	new	ADJ
cana-5238	564	15	generalized	generalized	ADJ
cana-5238	564	16	topologies	topology	NOUN
cana-5238	564	17	via	via	ADP
cana-5238	564	18	hereditary	hereditary	ADJ
cana-5238	564	19	classes	class	NOUN
cana-5238	564	20	bol	bol	NOUN
cana-5238	564	21	.	.	PUNCT
cana-5238	565	1	soc	soc	PROPN
cana-5238	565	2	.	.	PUNCT
cana-5238	566	1	paran	paran	PROPN
cana-5238	566	2	.	.	PUNCT
cana-5238	567	1	mat	mat	PROPN
cana-5238	567	2	.	.	NOUN
cana-5238	568	1	30(2)(2012	30(2)(2012	NUM
cana-5238	568	2	)	)	PUNCT
cana-5238	568	3	,	,	PUNCT
cana-5238	568	4	71	71	NUM
cana-5238	568	5	-	-	SYM
cana-5238	568	6	77	77	NUM
cana-5238	568	7	.	.	PUNCT
cana-5238	569	1	[	[	X
cana-5238	569	2	7	7	X
cana-5238	569	3	]	]	X
cana-5238	569	4	m.	m.	NOUN
cana-5238	569	5	rajamani	rajamani	NOUN
cana-5238	569	6	,	,	PUNCT
cana-5238	569	7	v.	v.	ADP
cana-5238	569	8	inthumathi	inthumathi	ADV
cana-5238	569	9	and	and	CCONJ
cana-5238	569	10	r.	r.	PROPN
cana-5238	569	11	ramesh	ramesh	PROPN
cana-5238	569	12	,	,	PUNCT
cana-5238	569	13	(	(	PUNCT
cana-5238	569	14	ωµ	ωµ	ADP
cana-5238	569	15	,	,	PUNCT
cana-5238	569	16	λ	λ	NOUN
cana-5238	569	17	)	)	PUNCT
cana-5238	569	18	-continuity	-continuity	NOUN
cana-5238	569	19	in	in	ADP
cana-5238	569	20	generalized	generalized	ADJ
cana-5238	569	21	topological	topological	ADJ
cana-5238	569	22	spaces	space	NOUN
cana-5238	569	23	,	,	PUNCT
cana-5238	569	24	international	international	ADJ
cana-5238	569	25	journal	journal	NOUN
cana-5238	569	26	of	of	ADP
cana-5238	569	27	mathematical	mathematical	ADJ
cana-5238	569	28	archive	archive	NOUN
cana-5238	569	29	,	,	PUNCT
cana-5238	569	30	3(10)(2012	3(10)(2012	NUM
cana-5238	569	31	)	)	PUNCT
cana-5238	569	32	,	,	PUNCT
cana-5238	569	33	3696	3696	NUM
cana-5238	569	34	-	-	SYM
cana-5238	569	35	3703	3703	NUM
cana-5238	569	36	.	.	PUNCT
cana-5238	570	1	communications	communication	NOUN
cana-5238	570	2	on	on	ADP
cana-5238	570	3	applied	apply	VERB
cana-5238	570	4	nonlinear	nonlinear	ADJ
cana-5238	570	5	analysis	analysis	NOUN
cana-5238	570	6	issn	issn	NOUN
cana-5238	570	7	:	:	PUNCT
cana-5238	570	8	1074	1074	NUM
cana-5238	570	9	-	-	PUNCT
cana-5238	570	10	133x	133x	NUM
cana-5238	570	11	vol	vol	VERB
cana-5238	570	12	32	32	NUM
cana-5238	570	13	no	no	NOUN
cana-5238	570	14	.	.	PUNCT
cana-5238	571	1	10s	10	NOUN
cana-5238	571	2	(	(	PUNCT
cana-5238	571	3	2025	2025	NUM
cana-5238	571	4	)	)	PUNCT
cana-5238	571	5	1371	1371	NUM
cana-5238	571	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5238	572	1	[	[	X
cana-5238	572	2	8	8	NUM
cana-5238	572	3	]	]	X
cana-5238	572	4	r.	r.	PROPN
cana-5238	572	5	ramesh	ramesh	PROPN
cana-5238	572	6	and	and	CCONJ
cana-5238	572	7	r.	r.	PROPN
cana-5238	572	8	mariappan	mariappan	PROPN
cana-5238	572	9	generalized	generalize	VERB
cana-5238	572	10	open	open	ADJ
cana-5238	572	11	sets	set	NOUN
cana-5238	572	12	in	in	ADP
cana-5238	572	13	hereditary	hereditary	ADJ
cana-5238	572	14	generalized	generalized	ADJ
cana-5238	572	15	topological	topological	ADJ
cana-5238	572	16	spaces	space	NOUN
cana-5238	572	17	,	,	PUNCT
cana-5238	572	18	j.	j.	PROPN
cana-5238	572	19	math	math	PROPN
cana-5238	572	20	.	.	PUNCT
cana-5238	573	1	comput	comput	NOUN
cana-5238	573	2	.	.	PUNCT
cana-5238	574	1	sci	sci	PROPN
cana-5238	574	2	.	.	PROPN
cana-5238	574	3	,	,	PUNCT
cana-5238	574	4	5(2	5(2	NUM
cana-5238	574	5	)	)	PUNCT
cana-5238	574	6	(	(	PUNCT
cana-5238	574	7	2015	2015	NUM
cana-5238	574	8	)	)	PUNCT
cana-5238	574	9	,	,	PUNCT
cana-5238	574	10	149	149	NUM
cana-5238	574	11	-	-	SYM
cana-5238	574	12	159	159	NUM
cana-5238	574	13	.	.	PUNCT
cana-5238	575	1	[	[	X
cana-5238	575	2	9	9	NUM
cana-5238	575	3	]	]	X
cana-5238	575	4	r.	r.	PROPN
cana-5238	575	5	ramesh	ramesh	PROPN
cana-5238	575	6	and	and	CCONJ
cana-5238	575	7	ahmad	ahmad	PROPN
cana-5238	575	8	al	al	PROPN
cana-5238	575	9	-	-	PUNCT
cana-5238	575	10	omari	omari	PROPN
cana-5238	575	11	,	,	PUNCT
cana-5238	575	12	b	b	PROPN
cana-5238	575	13	hσ	hσ	NOUN
cana-5238	575	14	-open	-open	NOUN
cana-5238	575	15	sets	set	NOUN
cana-5238	575	16	in	in	ADP
cana-5238	575	17	hgts	hgts	NOUN
cana-5238	575	18	,	,	PUNCT
cana-5238	575	19	poincare	poincare	PROPN
cana-5238	575	20	journal	journal	PROPN
cana-5238	575	21	of	of	ADP
cana-5238	575	22	analysis	analysis	NOUN
cana-5238	575	23	and	and	CCONJ
cana-5238	575	24	applications	application	NOUN
cana-5238	575	25	9(1	9(1	NUM
cana-5238	575	26	)	)	PUNCT
cana-5238	575	27	(	(	PUNCT
cana-5238	575	28	2022	2022	NUM
cana-5238	575	29	)	)	PUNCT
cana-5238	575	30	,	,	PUNCT
cana-5238	575	31	31	31	NUM
cana-5238	575	32	-	-	SYM
cana-5238	575	33	40	40	NUM
cana-5238	575	34	.	.	PUNCT
cana-5238	576	1	[	[	X
cana-5238	576	2	10	10	NUM
cana-5238	576	3	]	]	X
cana-5238	576	4	r.	r.	PROPN
cana-5238	576	5	ramesh	ramesh	PROPN
cana-5238	576	6	and	and	CCONJ
cana-5238	576	7	ahmad	ahmad	PROPN
cana-5238	576	8	al	al	PROPN
cana-5238	576	9	-	-	PUNCT
cana-5238	576	10	omari	omari	PROPN
cana-5238	576	11	,	,	PUNCT
cana-5238	576	12	decomposition	decomposition	NOUN
cana-5238	576	13	of	of	ADP
cana-5238	576	14	(	(	PUNCT
cana-5238	576	15	α	α	NOUN
cana-5238	576	16	-	-	PUNCT
cana-5238	576	17	hg	hg	NOUN
cana-5238	576	18	,	,	PUNCT
cana-5238	576	19	λ	λ	NOUN
cana-5238	576	20	)	)	PUNCT
cana-5238	576	21	-continuity	-continuity	PROPN
cana-5238	576	22	,	,	PUNCT
cana-5238	576	23	poincare	poincare	PROPN
cana-5238	576	24	journal	journal	PROPN
cana-5238	576	25	of	of	ADP
cana-5238	576	26	analysis	analysis	NOUN
cana-5238	576	27	and	and	CCONJ
cana-5238	576	28	applications	application	NOUN
cana-5238	576	29	,	,	PUNCT
cana-5238	576	30	10(1	10(1	NUM
cana-5238	576	31	)	)	PUNCT
cana-5238	576	32	(	(	PUNCT
cana-5238	576	33	2023	2023	NUM
cana-5238	576	34	)	)	PUNCT
cana-5238	576	35	,	,	PUNCT
cana-5238	576	36	155	155	NUM
cana-5238	576	37	-	-	SYM
cana-5238	576	38	163	163	NUM
cana-5238	576	39	.	.	PUNCT
cana-5238	577	1	[	[	X
cana-5238	577	2	11	11	NUM
cana-5238	577	3	]	]	X
cana-5238	577	4	m.s	m.s	PROPN
cana-5238	577	5	.	.	PROPN
cana-5238	577	6	sarsak	sarsak	PROPN
cana-5238	577	7	,	,	PUNCT
cana-5238	577	8	on	on	ADP
cana-5238	577	9	some	some	DET
cana-5238	577	10	properties	property	NOUN
cana-5238	577	11	of	of	ADP
cana-5238	577	12	generalized	generalized	ADJ
cana-5238	577	13	open	open	ADJ
cana-5238	577	14	sets	set	NOUN
cana-5238	577	15	in	in	ADP
cana-5238	577	16	generalized	generalized	ADJ
cana-5238	577	17	topological	topological	ADJ
cana-5238	577	18	spaces	space	NOUN
cana-5238	577	19	,	,	PUNCT
cana-5238	577	20	demonstratio	demonstratio	PROPN
cana-5238	577	21	math	math	PROPN
cana-5238	577	22	.	.	PUNCT
cana-5238	578	1	(	(	PUNCT
cana-5238	578	2	2013	2013	NUM
cana-5238	578	3	)	)	PUNCT
cana-5238	578	4	.	.	PUNCT
cana-5238	579	1	[	[	X
cana-5238	579	2	12	12	NUM
cana-5238	579	3	]	]	X
cana-5238	579	4	ge	ge	PROPN
cana-5238	579	5	xun	xun	PROPN
cana-5238	579	6	and	and	CCONJ
cana-5238	579	7	ge	ge	PROPN
cana-5238	579	8	ying	ying	PROPN
cana-5238	579	9	,	,	PUNCT
cana-5238	579	10	µ	µ	DET
cana-5238	579	11	-separations	-separation	NOUN
cana-5238	579	12	in	in	ADP
cana-5238	579	13	generalized	generalized	ADJ
cana-5238	579	14	topological	topological	ADJ
cana-5238	579	15	spaces	space	NOUN
cana-5238	579	16	,	,	PUNCT
cana-5238	579	17	appl	appl	PROPN
cana-5238	579	18	.	.	PROPN
cana-5238	579	19	math	math	PROPN
cana-5238	579	20	.	.	PUNCT
cana-5238	580	1	j.	j.	PROPN
cana-5238	580	2	chinese	chinese	PROPN
cana-5238	580	3	univ	univ	PROPN
cana-5238	580	4	.	.	PROPN
cana-5238	580	5	,	,	PUNCT
cana-5238	580	6	25(2)(2010	25(2)(2010	NUM
cana-5238	580	7	)	)	PUNCT
cana-5238	580	8	,	,	PUNCT
cana-5238	580	9	243	243	NUM
cana-5238	580	10	-	-	SYM
cana-5238	580	11	252	252	NUM
cana-5238	580	12	.	.	PUNCT